diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzemgj" "b/data_all_eng_slimpj/shuffled/split2/finalzzemgj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzemgj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\nThe study of isometric group actions on Riemannian manifolds has seen a number of important applications in Riemannian geometry.\n\nMany of them fall under the umbrella of the so-called \\emph{Grove's program}, whose goal is to study the properties of Riemannian manifolds with non-negative (or even almost non-negative) sectional curvature in the presence of symmetry. This program has been extremely fruitful both in producing new examples of manifolds with non-negative sectional curvature, and in proving important conjectures in the area when some symmetry is added (cf. \\cite{KWW21}, \\cite{GKS20}, \\cite{FGT17}, \\cite{GW14}, \\cite{GZ00}, \\cite{GVZ11}, \\cite{Dea11}, etc.)\n\nThe concept of an isometric group action can be generalized by a \\emph{singular Riemannian foliation}, \nwhich roughly speaking is the partition of a Riemannian manifold into smooth and equidistant submanifolds of possibly varying dimensions, called leaves (and the leaves can be thought as a generalization of the orbits of an isometric group action). \nIt turns out that, while being more flexible than group actions (cf. for example \\cite{Rad14}), singular Riemannian foliations\nstill retain a lot of the same structure of isometric group actions (cf. \\cite{MR19}, \\cite{GGR15}, \\cite{GR15}, \n\\cite{CM20}, \\cite{Mor19}, etc.).\n\nGiven the action of a compact Lie group, the orbits are homogeneous spaces and thus have a very restricted topology, \nwhich can be employed to extrapolate topological properties of the ambient manifold (e.g. \\cite{GZ12} and \\cite{GYW19}). \nIn \\cite{GYW19}, the authors ask to what extent the leaves of a singular Riemannian foliation on a non-negatively curved space \nare also topologically restricted. In \\cite{GGR15}, Galaz-Garcia and the first author proved that if $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$, then the fundamental group of a generic leaf is a product $A\\times K_2$ \nof an abelian group $A$ and a 2-step nilpotent 2-group $K_2$ - in particular, it is nilpotent. In the present paper, \nwe continue exploring the topology of the leaves of singular Riemannian foliations $(M,\\mathcal{F})$.\n\nThe first result states that if $M$ is simply connected, then a generic leaf $L_0$ of $\\mathcal{F}$ is a \\emph{nilpotent space}, i.e. $\\pi_1(L_0)$ acts nilpotently on $\\pi_n(L_0)$ for all $n>1$:\n\n\\begin{maintheorem}\\label{main-thm:leaves-nilpotent}\nIf $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$, \nthen the principal leaves of $\\mathcal{F}$ are nilpotent spaces. Furthermore, all leaves are finitely covered by a nilpotent space.\n\\end{maintheorem}\n\nThis answers the first part of Problem 4.8 in \\cite{GYW19}:\n\\begin{nonumberquestion}\nLet $\\mathcal{F}$ be a closed singular Riemannian foliation on a closed (simply connected) Riemannian manifold M of almost non-negative curvature. Are the leaves of $\\mathcal{F}$ finitely covered by a nilpotent space, which moreover is rationally elliptic?\n\\end{nonumberquestion}\n\nOur result does not in fact use the curvature assumption. On the rationally elliptic part of the question, we make the following remarks:\n\\begin{enumerate}\n\\item The very question of whether the leaves are rationally elliptic, only makes sense the moment we know that the leaves are (virtually) nilpotent spaces: these are in fact the spaces on which rational homotopy theory applies, and the rational dichotomy of rationally elliptic vs. rationally hyperbolic spaces holds.\n\\item Assuming the question above to be true, and applying it to the product foliation $(M\\times \\mathbb{S}^n,M\\times \\{pts.\\})$ with $M$ simply connected and almost non-negatively curved, would imply that every simply connected, almost non-negatively curved Riemannian manifold is rationally elliptic, which is the statement of the celebrated (and out of reach) Bott-Halperin-Grove Conjecture. \nIn particular, the rationally elliptic part of the question is so far out of reach.\n\\end{enumerate}\n\nThe second result analyzes more in detail the structure of the fundamental group of a generic leaf $L_0$ \nof a singular Riemannian foliation $(M,\\mathcal{F})$ with $M$ simply connected:\n \n\\begin{maintheorem}\\label{main-thm:non-abelian part}\nLet $(M,\\mathcal{F})$ be a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$. \nIf $L_0$ is a principal leaf of $\\mathcal{F}$, then the non-abelian part $K_2$ of the fundamental group of $L_0$ is of the form \n$$K_2\\cong (\\prod_{j=1}^s \\Z_{2^{a_j}}\\times \\Z_2^b\\times \\prod_{i=1}^k G_i)\/({\\Z_2^{c}\\times\\Z_4^{d}}),$$\nwhere each $G_{i}$ is isomorphic to a central product of copies of $Q_8$, with possibly one copy of $D_8$ or $\\Z_4$.\n\\end{maintheorem}\n\nThe groups $G_i$ in the theorem are called \\emph{generalized extraspecial}. These 2-groups already occur as fundamental groups of orbits of orthogonal representations and hence are impossible to avoid (e.g. $\\mathrm{SO}(3)$ acting on $\\mathbb{S}^4$), \nsee also a family of examples from Section \\ref{SS:examples}.\n\nFinally, we extend Theorem A from \\cite{GGR15} by showing that when $M$ has virtually nilpotent fundamental group, the leaves of any closed singular Riemannian foliation $(M,\\mathcal{F})$ have virtually nilpotent fundamental group as well:\n\n\\begin{maintheorem}\\label{main-thm:virtually nilpotent}\nSuppose $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on compact Riemannian manifold $M$ with virtually nilpotent fundamental group. Then the leaves of $\\mathcal{F}$ have virtually nilpotent fundamental group as well.\n\\end{maintheorem}\n\nIn the fundamental paper \\cite{KPT10}, the authors show that every Riemannian manifold with almost non-negative \nsectional curvature is finitely covered by a nilpotent space. With this in mind, Theorem \\ref{main-thm:virtually nilpotent} gives the following steaightforward corollary:\n\n\\begin{maincorollary}\nGiven a closed singular Riemannian foliation $(M, \\mathcal{F})$ on an almost non-negatively curved manifold $M$, the leaves have virtually nilpotent fundamental group.\n\\end{maincorollary}\n\n\nThis paper is organized as follows. In Section \\ref{S:preliminaries}, we collect some preliminaries about topological results \nfor singular Riemannian foliations, and the main notation for bilinear and quadratic forms we need in the proof of Theorem \\ref{main-thm:non-abelian part}. In Section \\ref{S:topology of leaves}, we prove Theorem \\ref{main-thm:leaves-nilpotent}. \nIn Section \\ref{S:fundamental group}, we prove Theorem \\ref{main-thm:non-abelian part} and provide a family of examples showing that the generalized extraspecial groups can indeed appear in the fundamental group of principal orbits of orthogonal representations. Finally, in Section \\ref{S:nilpotent fundamental group}, we prove Theorem \\ref{main-thm:virtually nilpotent}.\n \n\\section{Preliminaries}\\label{S:preliminaries}\n\n\\subsection{Singular Riemannian foliations}\n\nLet $M$ be a Riemannian manifold. A singular Riemannian foliation on $M$ is a partition $\\mathcal{F}$\nof $M$ into connected, injectively immersed submanifolds called leaves such that every geodesic that starts perpendicular \nto a leaf remains perpendicular to all the leaves it meets, and moreover, M admits a family of smooth vector fields \nthat spans the leaves at all points.\n\nA singular Riemannian foliation is called closed if all of its leaves are closed in $M$. \nGiven a singular Riemannian foliation $(M,\\mathcal{F})$ on a complete manifold $M$ we define the \\emph{dimension \nof $\\mathcal{F}$}, denoted $\\dim\\mathcal{F}$, as the maximal dimension of its leaves. \nThe codimension of $\\mathcal{F}$ is defined by $\\dim M-\\dim\\mathcal{F}$.\n\nA leaf $L$ of the foliation $\\mathcal{F}$ is called regular if its dimension is maximal, or equivalently, $\\dim L=\\dim\\mathcal{F}$.\nThe union of all regular leaves is an open, dense and connected submanifold, which is called the principal stratum of $M$ \nand is denoted by $M_0$. The union of all other leaves is called the singular stratum of $(M,\\mathcal{F})$ \nand the connected components of the singular stratum are called singular strata. \n\nFor a closed singular Riemannian foliation $(M,\\mathcal{F})$, the canonical projection $\\pi:M\\to M\/{\\mathcal{F}}$\ninduces a metric space structure on the leaf space $M\/{\\mathcal{F}}$, where the metric \nis given by $d_{M\/{\\mathcal{F}}}(\\pi(p),\\pi(q))=d_M(L_p,L_q)$. If in addition all the leaves of $\\mathcal{F}$ are regular, \nthen the leaf space is a Riemannian orbifold. In particular, given a closed singular Riemannian foliation $(M,\\mathcal{F})$, \nthe quotient space ${M_0}\/{\\mathcal{F}}$ is an orbifold.\n\nWe then call a leaf $L\\subset M_0$ \\emph{principal} if it projects to a manifold point of $M_0\/\\mathcal{F}$. Clearly, the set of principal leaves is open and dense in $M_0$.\n\n\n\\subsection{Slice Theorem}\\label{SS:Slice Theorem}\nIn this section we describe the structure of a singular Riemannian foliation around a leaf. For more details, we refer the interested reader to \\cite{MR19}.\n\nLet $(M, \\mathcal{F})$ be a closed singular Riemannian foliation, let $p\\in M$, and let $L_p$ denote the leaf through $p$. Define the \\emph{horizontal space to $\\mathcal{F}$ at $p$}, $\\nu_pL_p\\subseteq T_pM$, as the subspace perpendicular to $T_pL_p$. Then there exists a singular Riemannian foliation $(\\nu_pL_p,\\mathcal{F}_p)$ called the \\emph{infinitesimal foliation of $\\mathcal{F}$ at $p$}, with two important properties:\n\\begin{enumerate}\n\\item $\\mathcal{F}_p$ is invariant under rescalings,\n\\item In an $\\epsilon$-neighbourhood $\\nu_p^{\\epsilon}L_p$ of the origin in $\\nu_pL_p$, the exponential map $\\exp_p:\\nu_p^\\epsilon L_p\\to M$ takes the leaves of $\\mathcal{F}_p$ onto the connected components of the intersections $L\\cap \\exp \\nu_p^\\epsilon L_p$, with $L\\in \\mathcal{F}$. \n\\end{enumerate}\nFurthermore, there is a group of isometries $K\\subseteq O(\\nu_pL_p)$, sending leaves of $L_p$ to (possibly different) leaves of $\\mathcal{F}_p$, with the property that for any $v\\in \\nu_p^\\epsilon L_p$, the leaf $L_v\\in \\mathcal{F}_p$ satisfies the following:\n\\[\n\\exp_p(K\\cdot L_v)=L_{\\exp_p(v)}\\cap \\exp_p\\nu_p^\\epsilon L_p\n\\]\nIn other words, two leaves of $\\mathcal{F}_p$ are in the same $K$-orbit if and only if they exponentiate to different connected components of an intersection $L \\cap \\exp_p\\nu_p^\\epsilon L_p$, for some $L\\in \\mathcal{F}$.\n\nIn \\cite{MR19}, the following Slice Theorem establishes a model for a singular Riemannian foliation around a leaf:\n\\begin{nonumbertheorem}[Foliated Slice Theorem]\nGiven a closed singular Riemannian foliation $(M, \\mathcal{F})$ and a point $p\\in M$, let $(\\nu_pL_p, \\mathcal{F}_p)$ be the infinitesimal foliation of $\\mathcal{F}$ at $p$. Then there exists a compact Lie group $K\\subset O(\\nu_pL_p)$ and a principal $K$-bundle $P\\to L_p$ such that the foliation $\\mathcal{F}$ in an $\\epsilon$-neighbourhood of $L_p$ is foliated diffeomorphic to\n\\[\n(P\\times_K\\nu_pL,P\\times_K\\mathcal{F}_p)\n\\]\n\\end{nonumbertheorem}\n\nIt follows directly from the Slice Theorem that all principal leaves are diffeomorphic to each other, and for any leaf $L_p$, there is a locally trivial fiber bundle $L_0\\to L_p$ from a principal leaf $L_0$, whose fiber is an orbit $K\\cdot L_v$ for some principal point $v\\in (\\nu_pL_p, \\mathcal{F}_p)$, and it consists of a finite disjoint union of principal leaves of $\\mathcal{F}_p$.\n\n\n\n\\subsection {The Molino bundle}\\label{SS:molino}\n\nLet $(M,\\mathcal{F})$ be a closed singular Riemannian foliation of codimension $q$ on a compact Riemannian manifold $M$. \nThe principal $\\mathrm{O}(q)$-bundle $\\hat{M}\\to M_0$, where $\\hat{M}$ is the collection of orthonormal frames \nof ${TM_0}\/{T\\mathcal{F}}$, is called the Molino bundle. The foliation $\\mathcal{F}$ lifts to a foliation \n$\\hat{\\mathcal{F}}$ on $\\hat{M}$ whose leaves are diffeomorphic to the leaves of $\\mathcal{F}$ \non an open dense set. Moreover, the leaves of $\\hat{\\mathcal{F}}$ are given by fibers of a submersion \n$\\theta:\\hat{M}\\to W$, where $W$ is the frame bundle of the orbifold ${M_0}\/{\\mathcal{F}}$. \n\nConsider the fibration $\\hat{\\theta}:{\\hat{M}}_{\\mathrm{O}(q)}\\to W_{\\mathrm{O}(q)}$ induced by $\\theta$,\nwhere ${\\hat{M}}_{\\mathrm{O}(q)}={\\hat{M}}\\times_{\\mathrm{O}(q)}\\mathrm{EO}(q)$ and $W_{\\mathrm{O}(q)}=W\\times_{\\mathrm{O}(q)}\\mathrm{EO}(q)$ denote the Borel constructions of $\\hat{M}$ and $W$, respectively. \nNote that $\\hat{\\theta}:{\\hat{M}}_{\\mathrm{O}(q)}\\to W_{\\mathrm{O}(q)}$ and $\\theta:\\hat{M}\\to W$ \nhave the same fibers and hence the fiber of $\\hat{\\theta}$ is diffeomorphic to $L_0$, where $L_0$ is a principal leaf \nof $\\mathcal{F}$. In addition, ${\\hat{M}}_{\\mathrm{O}(q)}$ is homotopy equivalent to ${\\hat{M}}\/{\\mathrm{O}(q)}=M_0$ \nand $W_{\\mathrm{O}(q)}$ coincides with the Haefliger's classifying space $B$ of ${M_0}\/{\\mathcal{F}}$. \nTherefore, we get the following fibration (up to homotopy):\n$$L_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B.$$\n\n\\subsection{Bilinear and quadratic forms over $\\Z_2$}\\label{SS:quadratic}\n\nLet $V$ be a finite dimensional vector space over a field $F$. A quadratic form on $V$ is a map $Q:V\\to F$ \nsuch that $Q(\\lambda v)=\\lambda^2 Q(v)$ for all $\\lambda\\in F$ and $v\\in V$, and moreover, the map \n$B_Q:V\\times V\\to F$ defined by $B_Q(u,v)=Q(u+v)-Q(u)-Q(v)$ is a bilinear form. \nGiven a basis $\\{v_1,\\ldots,v_{\\ell}\\}$ of $V$, it follows that\n\\begin{equation}\\label{eq:quadratic form}\nQ(x_1v_1+\\ldots+x_{\\ell}v_{\\ell})=\\sum_{i=1}^{\\ell} Q(v_i)x_i^2+\\sum_{1\\leq i2$,\nwe can assume that we only have singular strata of codimension $\\leq 2$. Furthermore, it is known that there are no strata \nof codimension one, which reduces $\\mathcal{F}$ to only having strata of codimension two. \n\nLet $\\Sigma_1,\\ldots,\\Sigma_m$ denote the singular strata of $\\mathcal{F}$ of codimension two.\nFor $i=1,\\ldots, m$, choose a singular leaf $L'_i$ in $\\Sigma_i$, and let $L_i$ be a principal leaf at some distance $\\epsilon_i$ \nfrom $L'_i$. For $\\epsilon_i$ small enough, the foot-point projection $\\pi_i:L_i\\to L'_i$ is a circle bundle.\nFix a point $p_i\\in L_i$, and let $[c_i]\\in \\pi_1(L_i,p_i)$ be the element represented by the fiber $c_i$ \nof $\\pi_i$ through $p_i$.\n\nFixing a principal leaf $L_0$ and $p_0\\in L_0$, we can choose, for each $i=1,\\ldots, m$, a diffeomorphism $h_i:L_i\\to L_0$, \nand define $k_i=(h_i)_*([c_i])\\in\\pi_1(L_0,p_0)$. The group $K$ generated by the elements $k_i$ is then a normal subgroup \nof $\\pi_1(L_0,p_0)$. Furthermore, there exists a homotopy fibration\n\\[\nL_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B,\n\\]\nas described in Section \\ref{SS:molino}. One has the following (see the proof of Theorem A in \\cite{GGR15}):\n\\begin{enumerate}\n\\item $\\pi_1(L_0,p_0)$ is generated by the subgroup $K$ and the image of the boundary map \n$\\partial:\\pi_2(B,b_0)\\to\\pi_1(L_0,p_0)$.\n\\item $H:=\\im(\\partial)$ is central in $\\pi_1(L_0,p_0)$\n\\item Any two non-commuting generators $k_i$ and $k_j$ of $K$ satisfy $k_ik_j=k_j^{-1}k_i$.\n\\item Let $N\\subseteq K$ be the subgroup generated by the non-central $k_i$'s, and let $Z_{(2)}$ \ndenote the Sylow $2$-subgroup of $Z(K)$. Then $\\pi_1(L_0,p_0)$ is nilpotent, and equal to $A\\times K_2$, \nwhere $A$ is abelian and $K_2=N\\cdot Z_{(2)}$.\n\\end{enumerate}\n\n\\subsection{Proof of Theorem \\ref{main-thm:leaves-nilpotent}}\n\nAs discussed in Section \\ref{SS:known-results}, the principal leaves of $\\mathcal{F}$ have nilpotent fundamental groups. \nAs a first step towards the proof of Theorem \\ref{main-thm:leaves-nilpotent}, we prove that the principal leaves \nare nilpotent spaces:\n\n\\begin{proposition}\\label{P:princ-leaves-nilp}\nSuppose $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$. \nLet $L_0$ denote a principal leaf of $\\mathcal{F}$ and let $p_0\\in L_0$. Then $\\pi_1(L_0,p_0)$ acts trivially on $\\pi_n(L_0,p_0)$ \nfor $n\\geq 2$.\n\\end{proposition}\n\n\\begin{proof}\nLet $[\\gamma]\\in\\pi_1(L_0,p_0)$ and $[\\omega]\\in\\pi_n(L_0,p_0)$. The goal is to prove that $[\\gamma]$\nacts trivially on $[\\omega]$. By the discussion in Section \\ref{SS:known-results}, we may assume that either $[\\gamma]\\in H$ \nor $[\\gamma]=k_i$ for some $i$. \n\\par \nFirst, consider the case in which $[\\gamma]=k_i$ for some $i$. Note that ${\\bf p}_i:=\\pi_i\\circ h_i^{-1}:L_0\\to L'_i$ \nis a circle bundle whose fiber is represented by $k_i$. This means that $k_i\\in\\ker(({\\bf p}_i)_*)$, where $({\\bf p}_i)_*$\nis the induced map on $\\pi_n$. Hence we have:\n$$({\\bf p}_i)_*([\\gamma]\\cdot[\\omega])=({\\bf p}_i)_*(k_i\\cdot[\\omega])=(({\\bf p}_i)_*(k_i))\\cdot(({\\bf p}_i)_*([\\omega]))=({\\bf p}_i)_*([\\omega]).$$\nBy the long exact sequence of homotopy groups associated to the fibration \n$\\mathbb{S}}\\newcommand{\\Ca}{\\mathrm{Ca}}\\newcommand{\\pp}{\\mathbb{P}^1\\to L_0\\overset{{\\bf p}_i}{\\rightarrow} L'_i$, it follows that the homomorphism \n$({\\bf p}_i)_*$ is injective in $\\pi_n$ for $n\\geq 2$. \nThis, together with $({\\bf p}_i)_*([\\gamma]\\cdot[\\omega])=({\\bf p}_i)_*([\\omega])$, implies that $[\\gamma]$ \nacts trivially on $[\\omega]$.\n\\par\nSuppose now that $[\\gamma]\\in H=\\im(\\partial)$ and choose $[\\beta]\\in\\pi_2(B,b_0)$ such that $[\\gamma]=\\partial([\\beta])$.\nConsider the fibration \n$$L_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B.$$\nNote that the action of $\\pi_1(L_0,p_0)$ on $\\pi_n(L_0,p_0)$ satisfies $[\\gamma]\\cdot[\\omega]=(\\iota_0)_*([\\gamma])\\cdot[\\omega]$ (see \\cite[Exercise 4.3.10]{Hat02}). Therefore, \n$$[\\gamma]\\cdot[\\omega]=(\\iota_0)_*([\\gamma])\\cdot[\\omega]=(\\iota_0)_*(\\partial([\\beta]))\\cdot[\\omega]=e\\cdot[\\omega]=[\\omega].$$\nThis completes the proof.\n\\end{proof}\n\nMoving to the non-principal leaves, we first prove that every leaf has a virtually nilpotent fundamental group.\n\n\\begin{lemma}\\label{L:other-leaves}\nSuppose $(M,\\mathcal{F})$ is a closed singular Riemannian foliation with principal leaf $L_0$. If $\\pi_1(L_0)$ is virtually nilpotent, \nthen so is the fundamental group $\\pi_1(L)$ of every leaf $L$ of $\\mathcal{F}$.\n\\end{lemma}\n\n\\begin{proof}\nFor any leaf $L$ of $\\mathcal{F}$, the foliated Slice Theorem (cf. Section \\ref{SS:Slice Theorem}) implies that there is a fibration $L_0\\to L$ whose fiber $F$ \nhas finitely many connected components. From the long exact sequence in homotopy one then has\n\\[\n\\pi_1(L_0)\\to \\pi_1(L)\\to \\pi_0(F)\n\\]\nfrom which it follows that $\\pi_1(L)$ is a finite extension of a quotient of $\\pi_1(L_0)$, therefore it is virtually nilpotent as well.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm:leaves-nilpotent}]\nThe statement about principal leaves has been proved in Proposition \\ref{P:princ-leaves-nilp}, so we now have to only consider non-principal leaves.\n\nGiven a leaf $L$, choose $p\\in L$. Recall that, by the Foliated Slice Theorem (cf. \\ref{SS:Slice Theorem}), there is a locally trivial fibration $\\phi:L_0\\to L$ whose fiber $F$ has finitely many connected components, all diffeomorphic to a principal leaf of the infinitesimal foliation $(\\nu_p L_p,\\mathcal{F}_p)$. Furthermore, the action $\\pi_1(L)\\to \\operatorname{Diff}(F)$ induces an action $\\pi_1(L)\\to \\operatorname{Aut}(\\pi_*(F))$, which factors as $\\pi_1(L)\\stackrel{\\psi}{\\to} \\pi_0(K)\\to \\operatorname{Aut}(\\pi_*(F))$. In particular:\n\\begin{enumerate}\n\\item The subgroup $G_1:=\\ker\\psi \\subseteq \\pi_1(L)$ has finite index in $\\pi_1(L)$ and it acts trivially on $\\pi_*(F)$.\n\\item The fibration induces a map $\\pi_1(L_0)\\stackrel{\\phi_*}{\\to} \\pi_1(L)\\to \\pi_0(F)$. Thus $G_2:=\\phi_*(\\pi_1(L_0))$ is a nilpotent subgroup of $\\pi_1(L)$ with finite index.\n\\end{enumerate}\nConsider $G:=G_1\\cap G_2\\subseteq \\pi_1(L)$, which is by the points above a nilpotent subgroup with finite index. We will now show that $G$ acts nilpotently on each $\\pi_n(L)$, i.e. the \\emph{lower central series} $\\Gamma^m_G(\\pi_n(L))\\subseteq\\pi_n(L)$ defined iteratively by\n\\[\n\\Gamma_G^1(\\pi_n(L))=\\pi_n(L), \\qquad \\Gamma_G^{m+1}(\\pi_n(L))=\\{\\gamma\\cdot \\alpha-\\alpha\\mid \\gamma\\in G,\\,\\alpha\\in \\Gamma_G^{m}(\\pi_n(L))\\}\n\\]\neventually becomes trivial.\n\nConsider the long exact sequence\n\\[\n\\cdots\\to\\pi_n(F)\\to\\pi_n(L_0)\\stackrel{\\phi_*}{\\to}\\pi_n(L)\\stackrel{\\partial}{\\to}\\pi_{n-1}(F)\\to\\cdots\n\\]\nLet $\\alpha\\in \\pi_n(L)$, and $\\gamma=\\phi_*(\\gamma_0)\\in G$, where $\\gamma_0\\in \\pi_1(L_0)$. Recall that $\\partial(\\gamma\\cdot \\alpha)=\\gamma\\cdot \\partial(\\alpha)$, where the action on the left is $\\pi_1(L)$ acting on $\\pi_*(L)$, while on the right we have the $\\pi_1(L)$-action on $\\pi_*(F)$. Since $G\\subseteq G_1$, we have\n\\[\n\\partial(\\gamma\\cdot\\alpha)=\\partial (\\alpha)\\quad\\Rightarrow \\quad \\partial(\\gamma\\cdot \\alpha-\\alpha)=0\n\\]\nand therefore\n\\[\n\\Gamma_G^2(\\pi_n(L))\\subseteq \\ker (\\partial)=\\phi_*(\\pi_n(L_0))=\\phi_*(\\Gamma_{\\pi_1(L_0)}^1(\\pi_n(L_0))).\n\\]\nFinally, we notice that if $\\alpha=\\phi_*(\\alpha_0)$ with $\\alpha_0\\in \\pi_n(L_0)$ then\n\\[\n\\gamma\\cdot \\alpha=(\\phi_*(\\gamma_0))\\cdot(\\phi_*(\\alpha_0))=\\phi_*(\\alpha_0)\\Rightarrow \\gamma\\cdot \\alpha-\\alpha=\\phi_*(\\gamma_0\\cdot \\alpha_0-\\alpha_0).\n\\]\n\nBy induction on $m$, one then has\n\\[\n\\Gamma_G^{m+1}(\\pi_n(L))\\subseteq \\phi_*\\big(\\Gamma_{\\pi_1(L_0)}^{m}(\\pi_n(L_0))\\big).\n\\]\nSince by Proposition \\ref{P:princ-leaves-nilp}, $\\Gamma_{\\pi_1(L_0)}^{2}(\\pi_n(L_0))=0$, we have $\\Gamma_G^{3}(\\pi_n(L))=0$ which proves that $G$ acts nilpotently on $\\pi_n(L)$, hence finishing the proof.\n\\end{proof}\n\n\\section{Fundamental groups of the principal leaves}\\label{S:fundamental group}\n\nThis section consists of two parts. The first part is devoted to the proof of Theorem \\ref{main-thm:non-abelian part}.\nIn the second part, we provide examples of singular Riemannian foliations whose principal leaves have fundamental groups \nof the form discussed in Theorem \\ref{main-thm:non-abelian part}.\n\nSuppose that $(M,\\mathcal{F})$ is a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold $M$. \nFix a principal leaf $L_0$ of $\\mathcal{F}$ and $p_0\\in L_0$. Let $N$ and $K_2$ be the subgroups of $\\pi_1(L_0,p_0)$ \ndiscussed in Section \\ref{SS:known-results}.\n\nConsider the graph $\\Gamma$ with vertices the generators of $N$ and an edge between $k_i$ and $k_j$ \nif and only if $k_ik_jk_i^{-1}=k_j^{-1}$. Note that for every generator $k_i$ of $N$, there exists another generator \nwhich does not commute with $k_i$. Therefore, $\\Gamma$ does not contain any isolated vertices. \nNote moreover that for every connected component $\\Gamma_i$ of $\\Gamma$, all vertices of $\\Gamma_i$ \nsquare to the same element $c_i$. In addition, by proof of Theorem A in \\cite{GGR15}, for any generator $k_i$ of $N$,\nwe have $k_i^4=1$ and $k_i^2$ is central in $K$. Therefore, $c_i$ is a central element of $N$ of order two.\nAltogether, we get that there is a map $C:\\pi_0(\\Gamma)\\to Z(N)$ defined by $C(\\Gamma_i)=c_i$.\n\n\\begin{notation}\\label{N:N_c}\nFrom now on, we fix an element $c$ of $Z(N)$ which is of the form $k_i^2$ for some generator $k_i$ of $N$.\nMoreover, $N_c$ denotes the subgroup of $N$ that is generated by all the vertices in $\\Gamma_c:=C^{-1}(c)$.\n\\end{notation}\n\nRecall that given a group $G$, the \\emph{Frattini subgroup} $\\Phi(G)$ is the intersection of all the maximal subroups of $G$. Furthermore, we recall the following:\n\n\\begin{definition}\\label{def:Generalized ES}\nA $2$-group $G$ is called \\emph{generalized extraspecial} if $\\Phi(G)$ is central, and $\\Phi(G)=[G,G]=\\Z_2$.\n\\end{definition}\n\nWe prove two important properties of the groups $N_c$. \n\n\\begin{lemma}\\label{L:abt-N_c}\nLet $\\{N_c\\}_{c\\in \\textrm{Im}(C)}$ be the collection of groups defined above. Then:\n\\begin{enumerate}\n\\item For $c\\neq c'$, the groups $N_{c}$ and $N_{c'}$ commute.\n\\item Each $N_c$ is a generalized extraspecial $2$-group.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFirst, we prove Statement (1). Let $k_1,\\ldots, k_{\\ell}$ be the generators of $N_c$, and $k'_1,\\ldots, k'_r$ be the generators\nof $N_{c'}$. As vertices of $\\Gamma$, there is no edge between any $k_i$ and any $k'_j$, which means that each $k_i$ commutes with any $k'_j$ in $K$. Hence the result follows.\n\\par\nAs for Statement (2), if $k_1,\\ldots, k_{\\ell}$ denote the generators of $N_c$, then $V={N_c}\/{\\langle c\\rangle}$ \nis isomorphic to $\\Z_2^{\\ell}$ and is generated by $[k_1],\\ldots,[k_{\\ell}]$. \nIt follows that $N_c$ fits into a short exact sequence\n\\[\n1\n\\to \\langle c\\rangle \\to N_c\\to V\\to 1\n\\]\nand in particular one has that both $N_c^2:=\\langle g^2\\mid g\\in N_c\\rangle$ and the commutator subgroup $[N_c,N_c]$ \ncoincide with $\\langle c \\rangle\\simeq \\Z_2$. Therefore, the same is true for the Frattini subgroup $\\Phi(N_c)$\nsince for a $2$-group $G$, one has $\\Phi(G)=G^2\\cdot [G,G]$.\n\\end{proof}\n\nGiven generalized extraspecial groups $G_1$ and $G_2$, with Frattini subgroups generated by $c_1$ and $c_2$, respectively, define the \\emph{central product} $G_1*G_2$ by $G_1*G_2:={(G_1\\times G_2)}\/\\langle (c_1,c_2)\\rangle$. \nThis is again a generalized extraspecial group, since\n\\[\\Phi(G_1*G_2)=\\Phi(G_1)\\times_{\\Z_2}\\Phi(G_2) \\cong \\Z_2.\\]\n\nThe $*$ operation is furthermore associative, and thus it makes sense to define, for a generalized extraspecial group $G$, \nthe central product powers\n\\[\n(G)^{*m}:=\\underbrace{G*G*\\ldots*G}_\\text{m\\textrm{ times}}\n\\]\n\nGeneralized extraspecial 2-groups are, as the name suggests, a generalization of \\emph{extraspecial $2$-groups}, \nthat is 2-groups such that $\\Phi(G)=Z(G)=[G,G]\\cong \\Z_2$. These groups have been thoroughly studied at least since the 60's \\cite{Hal56}. They are extremely simple: an extraspecial group has the form $(Q_8)^{*m}$ or $(Q_8)^{*(m-1)}*D_8$ \nfor some $m\\geq 1$, where $Q_8$ is the quaternion group and $D_8$ is the dihedral group of order $8$\n(cf. Theorem 2.2.11 of \\cite{LGM05}). It then follows from Lemma 3.2 in \\cite{Sta02} that\n\n\\begin{theorem}\nA generalized extraspecial $2$-group is of the form $G\\times \\Z_2^n$, where $G$ is one of\n\\[\nQ_8^{*m},\\qquad Q_8^{*(m-1)}*D_8,\\qquad Q_8^{*(m-1)}*\\Z_4.\n\\]\n\\end{theorem}\n\n\\subsection{The associated quadratic form}\\label{SS:associated quadratic form}\n\nLet $G$ be a generalized extraspecial $2$-group with $\\Phi(G)=G^2=\\langle c\\rangle$,\nand let $V:=G\/{\\langle c\\rangle}$. It is easy to check that $V$ is a vector space over $\\Z_2$. \n\nDefine the function $Q_G:V\\to\\Z_2$ by $Q_G([g])=k$, where $g^2=c^k$. Since $c$ is central in $G$ and has order two, \nfor any $g\\in G$, we have $(cg)^2=cgcg=c^2g^2=g^2$ and thus $Q_G([cg])=Q_G([g])$. \nTherefore, $Q:=Q_G$ is well-defined and in fact a quadratic form as defined in Section \\ref{SS:quadratic}. \nFurthermore, the bilinear form $B_Q$ associated to $Q$ (cf. Section \\ref{SS:quadratic}) satisfies \n$$ghg^{-1}h^{-1}=c^{B_Q([g],[h])},~{\\text{for}}~g,h\\in G.$$ \nIn order to see this, note that both $g^2$ and $h^2$ are central elements of $G$. Therefore,\n$$c^{B_Q([g],[h])}=c^{Q([g]+[h])}c^{-Q([g])}c^{-Q([h])}=(gh)^2g^{-2}h^{-2}=ghg^{-1}h^{-1}.$$\n\n\nThe quadratic form of each generalized extraspecial group can be explicitly computed. For this, consider the quadratic forms:\n\\begin{alignat*}{3}\n & H_+: \\Z_2^2\\to \\Z_2 && \\qquad H_-:\\Z_2^2\\to\\Z_2 && \\qquad Q_1:\\Z_2\\to \\Z_2\\\\\n & H_+(x,y)=xy && \\qquad H_-(x,y)=x^2+y^2+xy && \\qquad Q_1(x)=x^2.\n\\end{alignat*}\nWe have the following:\n\n\\begin{proposition}\\label{prop:quad-forms}\nSuppose that $G$ is a generalized extraspecial $2$-group and let $V:= G\/\\Phi(G)$. \n\\begin{enumerate}\n\\item If $G=(Q_8)^{*m}$, then $V\\simeq\\Z_2^{2m}$ and\n\\[\nQ_G=H_-^{\\oplus m}=\\begin{cases}\nH_+^{\\oplus m}& m~{\\rm{even}}\\\\\nH_-\\oplus H_+^{\\oplus (m-1)} & m~{\\rm{odd}}\n\\end{cases}\n\\]\n\\item If $G=(Q_8)^{*(m-1)}*D_8$, then $V\\simeq\\Z_2^{2m}$ and\n\\[\nQ_G=H_-^{\\oplus (m-1)}\\oplus H_+=\\begin{cases}\nH_+^{\\oplus m} & m~{\\rm{odd}}\\\\\nH_-\\oplus H_+^{\\oplus (m-1)}& m~{\\rm{even}}\n\\end{cases}\n\\]\n\\item If $G=(Q_8)^{m}*\\Z_4$, then $V\\simeq\\Z_2^{2m+1}$ and\n\\[\nQ_G=H_+^{\\oplus m}\\oplus Q_1=H_-^{\\oplus m}\\oplus Q_1\\]\n\\item If $G=G'\\times \\Z_2^n$ with $G'$ as in the previous points, then $V\\simeq V'\\oplus \\Z_2^n$ and $Q_G=Q_{G'}\\oplus0^{\\oplus n}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThis proposition follows easily from the following straightforward facts:\n\\begin{enumerate}\n\\item For $G=Q_8$, $G\/\\Phi(G)\\simeq \\Z_2^2$ and $Q_G=H_-$.\n\\item For $G=D_8$, $G\/\\Phi(G)\\simeq \\Z_2^2$ and $Q_G=H_+$.\n\\item For $G=\\Z_4$, $G\/\\Phi(G)\\simeq \\Z_2$ and $Q_G=Q_1$.\n\\item Given $G_1$ and $G_2$ with quotients $V_i=G_i\/\\Phi(G_i)$, one has\n\\[\n(G_1*G_2)\/\\Phi(G_1*G_2)=V_1\\oplus V_2\\quad\\textrm{and}\\quad Q_{G_1*G_2}=Q_{G_1}\\oplus Q_{G_2}.\n\\]\n\\item Given $G$ with quotient $V=G\/\\Phi(G)$, one has\n\\[\n(G\\times \\Z_2^n)\/\\Phi(G\\times \\Z_2^n)\\simeq V\\oplus \\Z_2^n\\quad\\textrm{and}\\quad Q_{G\\times \\Z_2^n}=Q_G\\oplus 0^{\\oplus n}.\n\\qedhere\\]\n\\end{enumerate}\n\\end{proof}\n\n\\begin{remark}\\label{remark:N_c}\nThe group $N_c$ discussed above is generated by elements of order four, that is the $k_i$'s. \nMoreover, for each $k_i$, there exists $k_j$ such that $k_ik_jk_i^{-1}k_j^{-1}=c$.\nThis is reflected in the corresponding quadratic form $Q:V\\to\\Z_2=\\{0,1\\}$ as follows.\nThere exists a basis $\\{v_1,\\ldots,v_{\\ell}\\}$ of $V\\cong\\Z_2^\\ell$ with the property that $Q(v_i)=1$ for all $i$, \nand for each $v_i$, there exists $v_j$ such that $B_Q(v_i,v_j)=1$. We call such quadratic forms \\emph{admissible}. \n\\end{remark}\n\nThe next step consists of understanding which of the quadratic forms in Proposition \\ref{prop:quad-forms} is admissible. \nWe start by reducing the problem to quadratic forms without trivial summands:\n\\begin{lemma}\\label{lem:splitting}\nLet $Q:V\\to\\Z_2$ be a quadratic form. If there exists a splitting $V=V_1\\oplus V_2$ such that $Q$ splits as $Q=q\\oplus 0^{\\oplus n}$ with $Q|_{V_1}=q$ and $Q|_{V_2}=0^{\\oplus n}$, then $Q$ is admissible if and only if $q$ is admissible.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $Q$ is admissible and choose a basis\n$$\\{(v_1,w_1),\\ldots, (v_{m+n},w_{m+n})\\}$$\nof $V_1\\oplus V_2$ with the property that $Q(v_i,w_i)=1$, and for every $(v_i,w_i)$ there exists $(v_j,w_j)$\nwith $B_Q((v_i,w_i),(v_j,w_j))=1$. After possibly rearranging basis elements of $V_1\\oplus V_2$, \nwe may assume that $\\{v_1,\\ldots, v_m\\}$ forms a basis for $V_1$. \nSince $Q(v_i,w_i)=q(v_i)$ and $B_Q((v_i,w_i),(v_j,w_j))=B_q(v_i,v_j)$, the basis $\\{v_1,\\ldots, v_m\\}$ of $V_1$ \nis admissible for $q$. On the other hand, if $\\{v_1,\\ldots, v_m\\}$ is admissible for $q$ and $\\{w_1,\\ldots, w_n\\}$ \nis any basis of $V_2$, then \n$$\\{(v_i,{\\bf{0}})\\mid i=1,\\ldots, m\\}\\cup\\{(v_1,w_j)\\mid j=1,\\ldots, n\\},$$\nforms an admissible basis for $Q$.\\end{proof}\n\nWe now apply Lemma \\ref{lem:splitting} to classify the admissible quadratic forms.\n\n\\begin{theorem}\\label{thm:admissible}\nAny admissible quadratic form $Q:\\Z_2^{\\ell}\\to\\Z_2$ is isometric to one of the following, up to orthogonal sum \nwith $0^{\\oplus n}$:\n\\begin{equation}\\label{eq:admissible}\nH_+^{\\oplus m}~(m\\geq 2),\\qquad H_-\\oplus H_+^{\\oplus m-1},\\qquad H_+^{\\oplus m}\\oplus Q_1~(m\\geq 2).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nSince the quadratic forms over $\\Z_2$ are classified (see Proposition \\ref{prop:quadratic form}), \nwe only need to check the admissibility condition. By Lemma \\ref{lem:splitting}, we may assume that $Q$ does not split \nas $q\\oplus 0^{\\oplus m}$. We break the proof into cases.\n\n\\smallskip\n\n{\\textbf{Case 1:}} $Q=H_-\\oplus H_+^{\\oplus m-1}$, where $2m=\\ell$. The quadratic form $Q$ is given by\n$$Q(x,y,z_1,z_2,\\ldots,z_{2m-2})=x^2+xy+y^2+z_1z_2+\\ldots+z_{2m-3}z_{2m-2}.$$\nLet $e_1,\\ldots,e_{\\ell}$ denote the standard basis elements of $\\Z_2^{\\ell}$ and consider the following basis:\n\\begin{align*}\n & v_1=e_1+e_2, \\quad v_2=e_3+e_4,\\quad\\ldots\\quad v_{m}=e_{2m-1}+e_{2m},\\\\\n & v_{m+1}=e_1,\\\\\n & v_{m+2}=e_1+e_3, \\quad v_{m+3}=e_1+e_5,\\quad \\ldots\\quad v_{2m}=e_1+e_{2m-1}.\n\\end{align*}\nThen $Q(v_i)=1$ for all $i$, and for every $v_i$, there exists $v_j$ such that $B_Q(v_i,v_j)=1$. Hence $Q$ is admissible. \n\n\\smallskip\n\n{\\textbf{Case 2:}} $Q=H_+^{\\oplus m}$, where $2m=\\ell$. Note that the only element of $\\Z_2^2$ \nthat is mapped to $1$ by $H_+$ is $(1,1)$. Therefore, $H_+$ is not admissible. However, if $m\\geq 2$, \nthen the following basis of $\\Z_2^{\\ell}$ is admissible for $Q$:\n\\begin{align*}\n &v_1=e_1+e_2, \\quad v_2=e_3+e_4\\quad \\ldots\\quad v_{m}=e_{2m-1}+e_{2m},\\\\\n &v_{m+1}=e_1+e_{2m-1}+e_{2m},\\\\\n & v_{m+2}=e_1+e_2+e_4, \\quad v_{m+3}=e_3+e_4+e_6,\\ldots \\\\\n &v_{2m}=e_{2m-3}+e_{2m-2}+e_{2m}.\n\\end{align*}\n\n\\smallskip\n\n{\\textbf{Case 3:}} $Q=H_+^{\\oplus m}\\oplus Q_1$, where $m\\geq 2$ and $2m+1=\\ell$. \nLet $\\{v_1,\\ldots, v_{2m}\\}$ denote the basis constructed for $H_+^{\\oplus m}$ in Case 2, \nand let $v_{2m+1}=e_1+e_{2m+1}$. Then $\\{v_1,\\ldots, v_{2m+1}\\}$ forms an admissible basis for $Q$.\n\nFor $Q=H_+\\oplus Q_1$, the elements with non-zero quadratic form are $(1,1,0)$, $(1,0,1)$, $(0,1,1)$, $(0,0,1)$. \nAmong these, the only vectors with non-zero bilinear form are the first three, which are linearly dependent \nand thus do not form a basis. Hence $H_+\\oplus Q_1$ is not admissible.\n\\end{proof}\n\nRecall that the group $N_c$ (cf. Notation \\ref{N:N_c}) is a generalized extraspecial group with an admissible basis. \nFrom the previous theorem, we then get:\n\n\\begin{corollary}\\label{C:N_c}\nIf $N_c$ is a generalized extraspecial group whose corresponding quadratic form is admissible,\nthen, up to a direct product with copies of $\\Z_2$, the group $N_c$ is isomorphic to one of the following:\n\\begin{equation}\\label{eq:N_c}\n(Q_{8})^{*m_1},\\qquad (Q_{8})^{*(m_1-1)}*D_8~(m_1\\geq 2),\\qquad (Q_{8})^{*m_1}*\\Z_4~(m_1\\geq 2).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nThis follows trivially by comparing the quadratic forms in Proposition \\ref{prop:quad-forms} with the classification of admissible quadratic forms in Theorem \\ref{thm:admissible}.\n\\end{proof}\n\nFinally, we prove Theorem \\ref{main-thm:non-abelian part}. \n\n\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm:non-abelian part}]\nFix $p_0\\in L_0$. As discussed in Section \\ref{SS:known-results}, the non-abelian part $K_2$ of $\\pi_1(L_0,p_0)$ \nis a $2$-group of the form $K_2=N\\cdot Z_{(2)}$, where $N$ is generated by the non-central generators of $K$\nand $Z_{(2)}$ denotes the Sylow $2$-subgroup of $Z(K)$. Furthermore, by the discussion in Section \\ref{S:fundamental group}, $N=N_{c_1}\\cdot \\ldots\\cdot N_{c_k}$, where the elements $c_i\\in Z(K)$ have order two.\nBy Corollary \\ref{C:N_c}, each $N_{c_i}$ is of the form $G_i\\times \\Z_2^{a_i}$, where $G_i$ is one of the groups \nlisted in Equation \\eqref{eq:N_c}. Let $a=\\sum_i a_i$. Finally, since all the groups $N_{c_i}$ commute \nwith one another by Lemma \\ref{L:abt-N_c}, one has $N_{c_i}\\cap N_{c_j}\\subseteq Z(N_{c_i})\\cap Z(N_{c_j})$, \nand $Z(N_{c_i})\\subseteq Z(K_2)$. Therefore\n\\[\nK_2\\cong (Z_{(2)}\\times \\prod_{i=1}^kN_{c_i})\/Z'=(Z_{(2)}\\times \\Z_2^a\\times \\prod G_i)\/Z',\n\\]\nwhere $Z'\\subseteq Z_{(2)}\\times \\prod_i Z(N_{c_i})$ is the subgroup of $K_2$ generated by the intersections $H_{ij}=N_{c_i}\\cap N_{c_j}$ and $H_{0j}=Z_{(2)}\\cap N_{c_j}$. Since the groups $H_{ij}$, $H_{0j}$ are all abelian \nand central, commute with one another, and have elements of order $2$ or $4$ (because $Z(N_{c_i})=\\Z_2^{a_i}\\times \\Z_2$ \nor $\\Z_2^{a_i}\\times \\Z_4$) it follows that $Z'=\\Z_2^\\alpha\\times Z_4^\\beta$ for some $\\alpha$ and $\\beta$.\n\\end{proof}\n\n\n\\subsection{Examples of fundamental groups of principal leaves}\\label{SS:examples}\n\nThe family of examples below shows that the non-abelian groups $G_i$ discussed in Theorem \\ref{main-thm:non-abelian part} \nactually arise as fundamental groups of principal leaves of homogeneous singular Riemannain foliations.\n\nLet $\\{e_1,\\ldots, e_n\\}$ be the standard basis of $\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H}^n$. The Clifford algebra $Cl(0,n)$ on $\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H}^n$\nis defined as the associative algebra generated by $e_1,\\ldots, e_n$, where multiplication of the elements $e_i$ \nis given by:\n$$e_i^2=-1,\\hspace{0.3cm}e_ie_j=-e_je_i.$$\nConsider the subset $E(n)=\\{\\pm e_{i_1}\\ldots e_{i_{2k}}\\}\\subseteq Cl(0,n)$ containing products of even numbers of the $e_i$'s. This is easily seen to be a group under the product of $Cl(0,n)$. In \\cite{CHM09}, Czarnecki, Howe, and McTavish prove that for the action of $G=\\mathrm{SO}(n)\\times\\mathrm{SO}(n)$ \non $M_{n\\times n}(\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H})$ defined by $(g,h)\\cdot A=g^TAh$, the fundamental group of a principal orbit is of the form $E(n)\\times\\Z_2$. In this section, we investigate the structure of $E(n)$.\n\n\\begin{lemma}\nLet $G_{0,n-1}$ be the group defined by generators $-1,e_1,\\ldots,e_{n-1}$ and relations\n$$(-1)^2=1,\\qquad (e_i)^2=-1,\\hspace{0.5cm}[e_i,e_j]=-1~(i\\neq j),\\qquad [e_i,-1]=1.$$\nThen the groups $E(n)$ and $G_{0,n-1}$ are isomorphic.\n\\end{lemma}\n\n\\begin{proof}\nWe have: \n$$G_{0,n-1}=\\{\\pm e_{i_1}\\ldots e_{i_{\\ell}}\\mid 1\\leq i_j\\leq n-1, e_i^2=-1, e_ie_j=-e_je_i\\}.$$\nGiven an ordered set $I=(i_1,\\ldots, i_m)$ with indices $i_j$ in $\\{1, \\ldots, n-1\\}$, let $e_I=e_{i_1}\\ldots e_{i_m}$.\nNotice that if $I=(i_1,\\ldots, i_m)$ and $J=(j_1,\\ldots, j_p)$, then $e_Ie_J=e_{I\\cup J}$,\nwhere $I\\cup J=(i_1,\\ldots, i_m,j_1,\\ldots, j_p)$. Now, define the map $\\psi:G_{0,n-1}\\to E(n)$ by\n\\begin{equation*}\n\\psi(e_I)=\n\\begin{cases}\ne_I & |I|~{\\text{even}}\\\\\ne_{I\\cup\\{n\\}} & |I|~{\\text{odd}}\n\\end{cases}\n\\end{equation*}\nFirst, we claim that $\\psi(e_Ie_J)=\\psi(e_I)\\psi(e_J)$ for multi-indices $I$ and $J$.\n\n\\smallskip \n\n{\\textbf{Case 1.}} $|I|$ and $|J|$ are both even. In this case, we have:\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J}=e_Ie_J=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip \n\n{\\textbf{Case 2.}} $|I|$ and $|J|$ are both odd. In this case, we have:\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J}=e_Ie_J=e_Ie_J(-e_ne_n)=e_{I\\cup\\{n\\}}e_{J\\cup\\{n\\}}=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip \n\n{\\textbf{Case 3.}} If $|I|$ is even and $|J|$ is odd, then\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J\\cup\\{n\\}}=e_Ie_{J\\cup\\{n\\}}=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip \n\n{\\textbf{Case 4.}} If $|I|$ is odd and $|J|$ is even, then\n$$\\psi(e_Ie_J)=\\psi(e_{I\\cup J})=e_{I\\cup J\\cup\\{n\\}}=e_{I\\cup\\{n\\}}e_J=\\psi(e_I)\\psi(e_J).$$\n\n\\smallskip\n\nTherefore, $\\psi$ is a homomorphism. It is easy to see that $\\psi$ is injective, and hence an isomorphism\nsince the groups $G_{0,n-1}$ and $E(n)$ have the same order.\n\\end{proof}\n\nThe groups $G_{0,n-1}$ have been classified by Salingaros \\cite{Sal81,Sal82,Sal84} (cf. \\cite{AVW18}).\nWe use this classification to write the group $E(n)\\cong G_{0,n-1}$ as a central product.\nThis gives rise to the following list for fundamental groups of the principal orbits of the $G$-action on $M_{n\\times n}(\\mathbb{R}}\\newcommand{\\C}{\\mathbb{C}}\\newcommand{\\HH}{\\mathbb{H})$:\\\\\n\\begin{equation*}\nE(n)\\times\\Z_2\\cong\n\\begin{cases}\n((Q_8)^{*\\frac{n-4}{2}}*D_8)\\times \\Z_2^2 & n\\equiv 0~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n(Q_8)^{*\\frac{n-1}{2}}\\times\\Z_2 & n\\equiv 1, 3~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n((Q_8)^{*\\frac{n-2}{2}}*\\Z_4)\\times\\Z_2 & n\\equiv 2, 6~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n(Q_8)^{*\\frac{n-2}{2}}\\times\\Z_2^2 & n\\equiv 4~({\\text{mod}}~8)\\vspace{0.1cm}\\\\\n((Q_8)^{*\\frac{n-3}{2}}*D_8)\\times\\Z_2 & n\\equiv 5, 7~({\\text{mod}}~8)\n\\end{cases}\n\\end{equation*}\n\nWe do not know, however, whether \\emph{all} groups in Theorem \\ref{main-thm:non-abelian part} do in fact arise \nas fundamental groups of principal leaves in a simply connected manifold.\n\\smallskip\n\n\\section{Virtually nilpotent fundamental group}\\label{S:nilpotent fundamental group}\n\nIn this section, we consider singular Riemannian foliations $(M,\\mathcal{F})$, where the fundamental group of $M$ is virtually nilpotent.\nAs the following example shows, the fundamental group of a principal leaf is not necessarily nilpotent in this case.\n\n\\begin{example}\nLet $\\hat{M}={\\mathbb{C}}^2\\times{\\mathbb{S}}^1$ and consider the homogeneous foliation $\\hat{\\mathcal{F}}$ \non $\\hat{M}$ induced by the linear action of $T^3=T^2\\times S^1$. Let $M={\\hat{M}}\/{\\Z_2}$, where \nthe non-trivial element $g$ of $\\Z_2$ acts by $g\\cdot(z_1,z_2,t)=({\\bar{z}}_1,{\\bar{z}}_2,t+\\frac{1}{2})$.\nNote that $M$ inherits a singular Riemannian foliation $\\mathcal{F}={\\hat{\\mathcal{F}}}\/{\\Z_2}$.\n\\par\nThe manifold $M$ is orientable, and is homotopy equivalent to ${\\mathbb{S}}^1$. In particular, $M$ is nilpotent.\nHowever, the principal leaf of $\\mathcal{F}$ is $T^3\/{\\Z_2}$ which has fundamental group\n$$G=\\Z^2\\rtimes\\Z=\\langle a, b, c: cac^{-1}=a^{-1}, cbc^{-1}=b^{-1}, ab=ba\\rangle.$$\nSince $G_{\\ell}=\\langle a^{2^{\\ell}}, b^{2^{\\ell}}\\rangle$ for any $\\ell$, $G$ is not nilpotent.\n\\end{example}\n\nNevertheless, in what follows, we prove that the fundamental groups of the leaves contain a nilpotent subgroup of finite index. \n\n\\begin{notation}\nThroughout the rest of this section, $L_0$ denotes a principal leaf of $\\mathcal{F}$. Furthermore, we fix $p_0\\in L_0$, \nand $K=\\langle k_1,\\ldots, k_m\\rangle$ denotes the normal subgroup of $\\pi_1(L_0,p_0)$ discussed at the beginning \nof Section \\ref{S:fundamental group}. Recall that there is a homotopy fibration\n$$L_0\\overset{\\iota_0}{\\rightarrow}M_0\\overset{\\hat{\\theta}}{\\rightarrow}B.$$\nwhich induces a long exact sequence\n\\[\n0\\to H\\to \\pi_1(L_0,p_0)\\stackrel{(\\iota_0)_*}{\\to} \\pi_1(M_0,p_0)\\stackrel{\\hat{\\theta}_*}{\\to} \\pi_1(B,b)\\to 1,\n\\]\nwhere $H=\\partial(\\pi_2(B))$, as well as an action of $\\pi_1(B,b)$ on $L_0$. Denote by $\\hat{K}$ the group generated by $H$ \nand $c\\cdot K$, for $c\\in \\pi_1(B,b)$. Notice that for every $\\gamma\\in \\pi_1(M_0,p_0)$ with $c=\\hat{\\theta}_*(\\gamma)$, \nand every $g\\in\\pi_1(L_0,p_0)$, $(\\iota_0)_*(c\\cdot g)=\\gamma (\\iota_0)_*(g)\\gamma^{-1}$.\n\\end{notation}\n\\begin{lemma}\\label{lemma:central}\nLet $(M,\\mathcal{F})$ be a closed singular Riemannian foliation on a compact Riemannian manifold $M$.\nIf $\\pi_1(M)$ is $n$-step nilpotent, then $(\\pi_1(L_0,p_0))_{n+1}\\subseteq \\hat{K}$, where $(\\pi_1(L_0,p_0))_{n+1}$ \ndenotes the $(n+1)$-th group in the lower central series of $\\pi_1(L_0,p_0)$.\n\\end{lemma}\n\n\\begin{proof}\nSince removing strata of codimension $> 2$ does not change the fundamental group of $M$, we can assume that $M$ \nonly contains singular strata of codimension $\\leq 2$. In particular, we use the notation and results in Section \n\\ref{SS:known-results}.\n\nLetting $\\iota:L_0\\to M$ denote the inclusion, one then has\n\\[\n\\iota_*((\\pi_1(L_0,p_0))_{n+1})\\subseteq (\\pi_1(M,p_0))_{n+1}=1.\n\\]\nTherefore, given any curve $\\alpha$ representing an element of $(\\pi_1(L_0,p_0))_{n+1}$, there exists a disk \n$\\bar{\\iota}:\\mathbb{D}^2\\to M$ extending $\\iota(\\alpha)$. By transversality, this can be deformed to only intersect, \ntransversely, the singular strata $\\Sigma_1,\\ldots,\\Sigma_m$ of codimension 2, and the intersection consists of finitely many points $\\{q_1,\\ldots q_r\\}$ with $q_j\\in \\Sigma_{i_j}$. For each $j=1,\\ldots, r$, let $q'_j$ be a point in $\\bar\\iota(\\mathbb{D}^2)$ \nclose to $q_j$, let $u_j$ be a curve in $\\bar\\iota(\\mathbb{D}^2)$ connecting $p_0$ to $q'_j$, and let $\\psi_j$ a small loop \nin $\\bar\\iota(\\mathbb{D}^2)$ based at $q'_j$, turning once around $q_j$. Finally, let $\\gamma_j=u_j\\star \\psi_j\\star u_j^{-1}$. Then:\n\\begin{enumerate}\n\\item For each $i=1,\\ldots, r$, $[\\gamma_j]\\in \\pi_1(M_0,p_0)$ is conjugate to $(\\iota_0)_*(k_{i_j})$ \nwith $k_{i_j}\\in K\\subseteq \\pi_1(L_0,p_0)$. By the discussion before the proposition, it follows that $[\\gamma_j]=(\\iota_0)_*(c_j\\cdot k_{i_j})$ for some $c_j\\in \\pi_1(B,b)$.\n\\item $(\\iota_0)_*[\\alpha]=[\\gamma_1]\\star\\cdots\\star[\\gamma_r]=(\\iota_0)_*((c_1\\cdot k_{i_1})\\star\\cdots\\star (c_r\\cdot k_{i_r}))$ in $\\pi_1(M_0,p_0)$.\n\\end{enumerate}\n\nSince $H=\\ker((\\iota_0)_*)$, it follows that $[\\alpha]=h((c_1\\cdot k_{i_1})\\star\\cdots\\star (c_r\\cdot k_{i_r}))$ \nfor some $h\\in H$. In particular, $[\\alpha]\\in \\hat{K}$, and therefore $(\\pi_1(L_0,p_0))_{n+1}\\subseteq \\hat{K}$.\n\\end{proof}\n\nWe are finally ready to prove that if $(M, \\mathcal{F})$ is a closed singular Riemannian foliation with $\\pi_1(M)$ virtually nilpotent, then the fundamental group of every leaf is virtually nilpotent as well.\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm:virtually nilpotent}]\nNotice that if $\\pi:\\hat{M}\\to M$ is a finite cover, and $(\\hat{M},\\hat{\\mathcal{F}})$ is the lifted singular Riemannian foliation, one has that a leaf $\\hat{L}\\in \\hat{\\mathcal{F}}$ has virtually nilpotent fundamental group if and only the corresponding leaf $\\pi(\\hat{L})\\in \\mathcal{F}$ does. Therefore, up to replacing $M$ with a finite cover $\\hat{M}$, we can assume that $\\pi_1(M)$ is nilpotent.\n\nLet $L_0$ be a principal leaf, and consider the Hurewicz homomorphism $h:\\pi_1(L_0,p_0)\\to H_1(L_0;\\Z)$ and let $G=h^{-1}(2\\cdot H_1(L_0;\\Z))$. \nClearly, $G$ has finite index in $\\pi_1(L_0,p_0)$, Since $\\pi_1(L_0,p_0)\/G\\cong H_1(L_0;\\Z)\/2\\cdot H_1(L_0;\\Z)$ \nis finite. We claim that if $\\pi_1(M)$ is $n$-step nilpotent, then $G$ is $(n+1)$-step nilpotent.\n\nBy Lemma \\ref{lemma:central}, $G_{n+1}\\subseteq G\\cap \\hat{K}$. The proof is complete once we prove that $G$ \ncommutes with $\\hat{K}$. Notice that $\\hat{K}$ is generated by $H$, and elements of the form $c\\cdot k_i$ for $c\\in \\pi_1(B,b)$ \nand $k_i$ one of the generators of $K$. Recall that $H$ is central in $\\pi_1(L_0,p_0)$ (in particular, $G$ commutes with $H$),\nand for each $g\\in \\pi_1(L_0,p_0)$, $gk_ig^{-1}=k_i^{\\pm 1}$. Since $\\pi_1(B,b)$ acts on $\\pi_1(L_0,p_0)$ \nby group automorphisms, it also follows that for every $g\\in \\pi_1(L_0,p_0)$, $g(c\\cdot k_i)g^{-1}=(c\\cdot k_i)^{\\pm 1}$.\n\nNotice that if $g(c\\cdot k_i)g^{-1}=(c\\cdot k_i)^\\epsilon$ (for $\\epsilon=\\pm1$), then $g^{-1}(c\\cdot k_i)g=(c\\cdot k_i)^{\\epsilon}$ as well. In particular, for every $g_1,g_2\\in \\pi_1(L_0,p_0)$, and every $(c\\cdot k_i)\\in \\hat{K}$, one has:\n\\[\n[g_1,g_2]\\cdot (c\\cdot k_i)[g_1,g_2]^{-1}=(c\\cdot k_i).\n\\]\nThe main observation is that, by definition, any element $g\\in G$ can be written as $g=g_3^2[g_1,g_2]\\cdots [g_{2k-1},g_{2k}]$ \nfor some $g_1,\\ldots g_{2k}\\in \\pi_1(L_0,p_0)$ and therefore, for any generator $(c\\cdot k_i)$ of $\\hat{K}$, one has:\n\\begin{align*}\ng(c\\cdot k_i)g^{-1}=&g_3^2[g_1,g_2]\\cdots [g_{2k-1},g_{2k}](c\\cdot k_i)[g_{2k-1},g_{2k}]^{-1}\\cdots [g_1,g_2]^{-1}g_3^{-2}\\\\\n=&g_3^2(c\\cdot k_i)g_3^{-2}=g_3(c\\cdot k_i)^\\epsilon g_3^{-1}\\\\\n=&(c\\cdot k_i)^{\\epsilon^2}=(c\\cdot k_i).\n\\end{align*}\nTherefore, $G$ commutes with $\\hat{K}$ and hence $G_{n+2}=[G,G_{n+1}]\\subseteq [G,\\hat{K}]=\\{1\\}$.\n\nThis proves that the principal leaves of $\\mathcal{F}$ have virtually nilpotent fundamental group. The corresponding statement for the non-principal leaves then follows from Lemma \\ref{L:other-leaves}.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nDirect photon production is of special importance in relativistic heavy-ion collisions (for reviews, see \\cite{Gale:2009gc,Chatterjee:2009rs}). Since photons couple with the quark-gluon plasma (QGP) only through the electromagnetic interaction, their mean free path is much longer than the dimension of the reaction zone and they can escape from it carrying the information of QGP at the instance of their emission. These photons are called thermal photons, whose observation is regarded as an indirect evidence for the formation of hot thermalized states of quarks and gluons in relativistic heavy-ion collisions \\cite{Shuryak:1978ij,Kapusta:1991qp}. There are still other various photon sources over the time evolution in a heavy-ion collision event in addition to thermal photons from the QGP phase. For example, at the very initial stage, partonic hard collisions (such as the gluon Compton scattering $gq\\to \\gamma q$ and quark annihilation $q\\bar q \\to \\gamma g$) produce the photons, which are called prompt photons \\cite{Owens:1986mp,Turbide:2007mi,Klasen:2013mga}. At the late stage after hadronization, scattering processes between hadrons also produce photons \\cite{Kapusta:1991qp,Holt:2015cda}. Thus, identifying and quantifying photon sources become a vital issue for extracting direct information on QGP from the photon observation.\n\n\nMeasurements of the photons in relativistic heavy-ion collisions have been performed both at RHIC and LHC, and the results of transverse momentum ($p_T$) distributions $dN^\\gamma\/dp_\\perp^2 dy$ and elliptic flow coefficients ($v_2^\\gamma$) at mid-rapidity $y\\sim 0$ are reported \\cite{Adare:2014fwh,Adare:2015lcd,Adam:2015lda,Lohner:2012ct} (see also \\cite{David:2019wpt} for recent experimental review). From the exponential slopes of the $p_T$ distributions, ``{\\it effective temperatures}'' of produced photons at RHIC ($\\sqrt{s_{NN}}=200~$GeV) and LHC ($\\sqrt{s_{NN}}=2.76~$TeV) are found to be $T_{\\rm eff}\\sim 230~$MeV and $T_{\\rm eff}\\sim 300$~MeV, respectively. Very interestingly, the elliptic flow $v_2^\\gamma$ of the direct photon distribution is found be as large as that of hadrons. Notice that although the ``decay photons'' produced in hadron decays (mainly due to $\\pi^0\\to 2 \\gamma$) have been subtracted from the total yield to obtain the direct photon yield, it is still a mixture of photons from various sources. Despite many theoretical attempts to reproduce these experimental results, any theoretical model so far seems to be incapable of explaining the photon data adequately. For example, hydrodynamic models which explain the hadronic spectra and anisotropic flow very well, tend to underpredict the amplitude of photon elliptic flow (e.g., see \\cite{Chatterjee:2021gwa}). In particular, even an up-to-date hydrodynamic model calculation with various possible effects included still underestimates the photon yield \\cite{Paquet:2017wji,Gale:2021emg}. Other model calculations such as blast-wave type fireball model and ideal hydrodynamic model also give rise to smaller thermal photon yield and elliptic flow than the experimental data by PHENIX an ALICE \\cite{vanHees:2014ida}. Furthermore, the parton-hadron-string dynamics (PHSD) model shares the same tendency in direct photon yield and elliptic flow \\cite{Linnyk:2015tha}. Meanwhile, the pre-equilibrium Glasma stage is discussed in Refs.~\\cite{Berges:2017eom,Monnai:2019vup} as a possible photon production source, but its contribution is also likely to suppress the elliptic flow.\n\nThis situation is referred to as the \"direct photon puzzle\".\nThe difficulty comes from two seemingly contradictive aspects of the photon data, large yields and strong collective flow. The large yield could be attributed to early stage of the evolution with higher temperatures, while the strong collective flow prefers large photon emission at later stage when momentum anisotropy of QGP is well developed. Therefore, it is not easy to explain these two points within a single theoretical model. However we should note that there is a reservation about the experimental results because the large yield of photons measured by PHENIX has not been confirmed by STAR.\n\nGiven this situation, it should be very worthwhile to examine another source of photons which has been overlooked so far and is inherent to late stages of the QGP time evolution. This brought us to think of a possibility of photon radiations at hadronization of QGP, which is, in fact, natural from the viewpoint of ordinary electromagnetic plasmas. It is well-known that an electromagnetic plasma radiates numerous photons when it goes back to an atomic gas. This process is called the ``radiative recombination\" and is seen in various astrophysical situations (for an overview see \\cite{graham2012recombination, hahn1997electron}. This is natural because photon emission is advantageous for satisfying energy conservation in the formation of bound states (electrically neutral atoms). We can expect similar phenomena occur in the hadronization processes, where valence quarks and antiquarks form bound states. In the present paper, we formulate photon production at the hadronization stage in analogy with the radiative recombination in electromagnetic plasmas and investigate properties of the produced photons. Photon production during QCD phase transition and hadronization has been attracting much attention indeed, and been discussed in different frameworks, for example, in \\cite{vanHees:2014ida,Campbell:2015jga,Young:2015adw}. \n\n\nThe present paper is organized as follows. In the next section, we explain the radiative recombination in ordinary electromagnetic plasma and comment on similarities to and differences from the QGP. Then in Sec.~III, we provide the basic theoretical framework for radiative recombination applied to hadronization.\nNumerical results are presented in Sec.~IV, where we include the thermal photon contributions obtained by a hydrodynamic simulation to compare the results to the observed data.\nDiscussions and summary are given in Sec.~V. \n\nIn Appendix A, we present photon production formulas in radiative recombination model.\nIn Appendix B, we give a brief description of thermal photon calculation in a hydrodynamic model \\cite{Miyachi-Nonaka}.\n\n\n\n\\section{Radiative recombination in electromagnetic plasmas}\n\nIn this section, we explain what is known about radiative recombination in ordinary plasmas and in other relevant processes, which are helpful to understand how the radiative recombination in QGP could be formulated. The primary and simplest example reads an electromagnetic plasma made of electrons and protons. When the temperature is decreased, the plasma decays and neutralizes into a hydrogen gas through the microscopic mechanism of the radiative recombination $e^-+p^+\\to {\\rm H}^0 +\\gamma$. Emission of a photon compensates the energy difference between the initial continuum state and the final bound state. The ordinary plasma flashes when it decays. This process is well-known in plasma physics and is also called ``free-bound transition\" \\cite{fujimoto2004plasma}. The secondary example is the glow discharge which is seen as a typical picture of a plasma. The third example is radiative recombination in the early universe: There was a ``recombination era\" and without treating the radiative recombination we would not be able to accurately evaluate the ``recombination temperature'' which is essential for understanding the cosmic microwave background (CMB) data \\cite{weinberg2008cosmology}. The fourth example is gas nebulae, such as Orion Nebula, which have beautiful radiations containing continuum spectra due to the radiative recombination \\cite{osterbrock2006astrophysics}. Lastly, in addition to these examples induced by the electromagnetic interaction, we also have examples in nuclear reactions. We know that in the sun there are two important processes called the ``pp chain\" and the ``CNO cycle,\" both of which include formation of bound states accompanied by a photon emission (such as $D + p \\to {}^{3}{\\rm He}+\\gamma$ in the pp chain and $p+{}^{12}{\\rm C}\\to {}^{13}{\\rm N}+\\gamma$ in the CNO cycle) \\cite{clayton1968principles}. All these examples indicate that radiation is naturally inherent to the formation of bound states, which suggests the possibility of photon emissions at hadronization, provided that hadronization can be modeled as a coalescence process of valence partons.\n\n\n\nThere are two key equations for the description of radiative recombination in electromagnetic plasmas \\cite{rybicki2008radiative,weinberg2008cosmology}. One is the Kramers-Milne relation which relates recombination cross section to that of its inverse process (photo-ionization $\\gamma + A \\to e^- + A^+$) and corresponds to the detailed balance relation in thermal equilibrium states. The other is the Saha equation which gives ionization ratio $X=n_{\\rm ion}\/(n_{\\rm ion}+n_{\\rm atom})$ between the atom and ion numbers, $n_{\\rm atom}, n_{\\rm ion}$, as a function of a temperature when the electrons, ions, and atoms are in thermal equilibrium. The Kramers-Milne relation is useful because the photo-ionization rate is easily measured by experiments, and the Saha equation applies to an ordinary plasma because one can control the lifetime of a plasma much longer than the microscopic time scale of radiation reaction. These relations are, unfortunately, not suitable to the hadronization from QGP in heavy-ion collisions since it should be treated as a nonequilibrium process. Hadronization occurs in a small lump of QGP at the time scale of the strong interaction, and therefore the photons are just emitted without re-absorption. To this situation we cannot use the relations which assume a system in bulk equilibrium. \n\n\nWhen the density of an electromagnetic plasma is relatively high, another type of recombination will be possible. It is the ``three-body recombination\" $2 e^- + A^+\\to e^- + A$, where the energy conservation is satisfied by the spectator electron. \nIf one defines the effective photon emission rate at some density, it will be reduced by the presence of the three-body, or in general, multi-body recombination. We may expect similar phenomena in the hadronization process. Formation of bound states without photon emission will be possible if the valence partons interact with other particles in the medium. We will be able to absorb this kind of effects into an effective recombination rate including density dependence. In fact, as we will discuss later, since we treat the recombination rate as a parameter, we expect that such kind of effects are included in the overall normalization parameter which is determined to fit the experimental data. Alternatively, effects of multi-body recombination without a photon emission will be described by ``off-shell'' valence partons. However, within our framework in the present analysis, it is not easy to define off-shell partons in QGP. \n\nLastly, we comment on a very similar process in $e^+e^-$ collisions. Recall that the $e^+e^-$ collisions have been used to discover new particles by changing the invariant mass. Pronounced resonances such as $\\rho, \\omega, \\phi$ and $J\/\\psi$ mesons are measured with energies {\\it below} the invariant mass of the $e^+e^-$ system. This is called the {\\it radiative return}, and is quite important for the analysis of $R$-ratio around threshold and $(g-2)$ of leptons (for reviews, see \\cite{Actis:2010gg, Druzhinin:2011qd}). Notice that the formation of a resonance below the $e^+ e^-$ invariant mass is accompanied by photon emissions and the typical radiative return is represented as \n$\ne^+ + e^- \\to \\text{hadrons} + n\\gamma\\, ,\n$\nwhere the number $n$ of emitted photons is not necessarily one, $n\\ge 1$. In particular, a clean process with a single meson and a single photon \n$\ne^++e^-\\to \\text{meson} + \\gamma\n$ \nis experimentally measured. For example, $J\/\\psi$ production was observed at BaBar \\cite{Aubert:2003sv}, and more recently $\\chi_c$ and $\\eta_c$ at BESIII \\cite{Ablikim:2014hwn, Ablikim:2017ove}. The counterpart in purely QED case such as $e^+e^-\\to \\mu^+\\mu^-\\gamma$ can be perturbatively calculated (though quite tedious), and one can study interplay between the initial state radiation and the final state radiation. On the other hand, hadron production suffers from ambiguity related to the coupling between the virtual photon and a composite hadron. For example, $e^++e^- \\to \\gamma^* \\to \\text{meson}+\\gamma$ includes the transition of a virtual photon into a meson which may be phenomenologically described by the vector meson dominance. Heavy quarkonium production will have less ambiguity, but the NRQCD formalism developed for the calculation of heavy-quarkonium production involves nonperturbative matrix elements (see for example, \\cite{Chung:2008km, Sang:2009jc}). \n\n\nThere are Monte-Carlo generators for the radiative return called PHOKHARA (for real photon emission) and EKHARA (for virtual photon emission) \\cite{Czyz:2017veo}. These generators treat radiation by the vertices like $meson^*\\to meson + \\gamma\\, ({\\rm or}\\ \\gamma^*)$ which are given by effective lagrangian \\cite{Czyz:2012nq}. Here, $meson^*$ could be a virtual (or off-shell) meson. This kind of picture will be useful in our problem. Another lesson from the radiative return is that the final state could involve several hadrons, typically light mesons such as pions and kaons. For example, final states with four mesons like $\\pi^+ \\pi^- K^+ K^-$ were extensively studied \\cite{Actis:2010gg}. We expect similar multiple hadron production in the radiative hadronization. As we will discuss in the next section, we will adopt the ``Recombination model\" and modify it so that it allows photon emission. This recombination model provides the number of produced mesons by the overlapping between a $q\\bar q$ state and a meson state, and implicitly assumes a single meson production from a $q\\bar q$ state. However, the multiple hadron production seen in the $e^+e^-$ collisions suggests that the one-to-one correspondence between a $q\\bar q$ state and a meson should not exactly hold. Still, we may be able to effectively absorb such effects into the recombination rate. We should be aware that the overall recombination rate could contain many different physical effects. Having said all these suggestions and caveats, we are now ready for the problem how to formulate the radiative recombination at hadronization. \n\n\n\n\n\\section{Radiative hadronization: formulation}\n\nHadronization is a nonperturbative process because it takes place around the critical temperature $T\\sim T_c$ and the typical strong coupling $\\alpha_s(T\\sim T_c)$ is not small. It is also a nonequilibrium phenomenon in the sense mentioned in the previous section. One of the possible frameworks to describe the hadronization will be to work in an effective theory that includes both hadronic and constituent-quark degrees of freedom (see \\cite{Young:2015adw} for an analysis in the quark-meson coupling theory). Within this framework, one can compute the hadron production cross sections similarly to the radiation return. We will, however, take an alternative approach. In fact, as already commented before, we know a simple and phenomenologically successful model for hadronization which is based on the coalescence of constituent quark degrees of freedom. It is the recombination model and we utilize it to describe the radiative hadronization. Below we first explain briefly the basic strategy of the recombination model, and then discuss how to modify it to include the photon emission. \n\n\n\n\n\\subsection{Recombination model}\n\nThe recombination\/coalescence models provide a phenomenological description of hadronization for hadron production in the intermediate $p_T$ region ($2 \\simle p_T \\simle 5$~GeV\/c) and give a natural explanation for intriguing phenomena such as the anomalous baryon\/meson ratio and constituent quark number (CQN) scaling of the elliptic flow \\cite{Fries:2003vb,Greco:2003xt,Molnar:2003ff,Hwa:2003bn} (for a review, see \\cite{Fries:2008hs}). There are several recombination\/coalescence models with some differences \\cite{Fries:2003kq, Greco:2003mm,Hwa:2004ng}, and we will adopt the ReCo model that was developed by Duke group \\cite{Fries:2003kq}. \n\n\nThe ReCo model starts by defining the number of hadrons that one can find in the quark\/antiquark distributions. For example, the total number of mesons is defined as an overlap between the meson state $|M;\\bm{P}\\rangle$ and the reduced two-body density matrix $\\hat \\rho_{ab}$ which represents the partonic system with partons, $a$ and $b$, undergoing hadronization:\n\\begin{eqnarray}\nN_M=\\sum_{ab}\\int \\frac{d^3\\bm{P}}{(2\\pi)^3}\\, \\langle M; \\bm{P}|\\, \n\\hat \\rho_{ab} \\, |M;\\bm{P}\\rangle \\, ,\n\\end{eqnarray}\nwhere summation is taken over all the possible combinations of partons $a, b$ that have nonzero overlap with a mesonic state.\nIn this sense, the ReCo model simply projects the partonic picture of the QGP onto the hadron picture, and does not describe dynamical processes of hadron formation. However, since the formula includes matrix elements like $\\langle M;\\bm{P}|\\bm{r}_1,\\bm{r}_2\\rangle$ with $|\\bm{r}_1,\\bm{r}_2\\rangle$ being a state having a quark at $\\bm{r}_1$ and an antiquark at $\\bm{r}_2$, it allows for an intuitive understanding of coalescence processes\\footnote{Note however that this matrix element will be interpreted as a quark-antiquark component of a meson wavefunction.} like $q\\bar q\\to M(\\text{meson})$ and $qqq\\to B(\\text{baryon})$. \n\n\nAfter some manipulations (see \\cite{Fries:2003kq} for details), one obtains the momentum distribution of mesons made of partons $a$ and $b$ as \n\\begin{equation}\nE\\frac{dN_M}{d^3\\bm{P}}=C_M \\int_\\Sigma \\frac{P\\cdot u(R)}{(2\\pi)^3}\\int_0^1 dx\\, w_a(R;x\\bm{P})\\, \\left|\\phi^{}_M(x)\\right|^2\\, w_b(R;(1-x)\\bm{P}) \\, , \n\\label{dN\/dP}\n\\end{equation}\nwhere $P^\\mu=(E,\\bm{P})$ is the four-momentum of the meson $M$, and $R$ is a four-vector specifying a point on the hypersurface $\\Sigma$ where the hadronization takes place, and $u^\\mu (R)$ is a unit vector orthogonal to the hypersurface $\\Sigma$ at $R$. The function $\\phi_M(x)$ is the light-cone wavefunction of a meson with $x$ being the momentum fraction of one of the two quarks, and $w_a(R;x\\bm{P})$ is a one-particle phase space distribution of parton $a$,\n\n\nWe assume the longitudinal boost-invariant expansion (Bjorken expansion) of the QGP so that the hadronization hypersurface $\\Sigma$ has a constant longitudinal proper time $\\tau=\\sqrt{t^2-z^2}=\\, $const and a point $R^\\mu$ on it is specified as\n$$\nR^\\mu = (t,x,y,z)=(\\tau \\cosh \\eta,\\, \\rho \\cos \\phi,\\, \\rho \\sin \\phi ,\\, \\tau \\sinh \\eta)\\, \n$$\nwith the space-time rapidity $\\eta$, the transverse radial coordinate $\\rho$, and the azimuthal angle $\\phi$. Then the forward normal vector $u^\\mu(R)$ orthogonal to the hypersurface $u\\cdot dR|_{\\tau= \\rm const.} =0$, is given by $u^\\mu(R)=(\\cosh \\eta,0,0,\\sinh \\eta)$.\n\n\nRegarding $\\phi_M(x)$, we expect that our results are insensitive to its details and take $|\\phi_M(x)|^2=\\delta(x-1\/2)$ for analytic evaluation and $\\phi_M(x) =\\sqrt{30}\\, x(1-x)$ for numerical evaluation. Notice that both examples of the wavefunction have a peak at $x=1\/2$, and therefore $w_a(R,\\tfrac{1}{2}\\bm{P})w_b(R,\\tfrac{1}{2}\\bm{P})\\sim {\\rm e}^{-P\/T}$ gives rise to the dominant configuration in the $x$ integration. \nThe overall factor $C_M$ counts state degeneracy. The corresponding formula for baryon production has the factor $C_B$ and the wavefunctions of the three-quark state should have a peak around $x=1\/3$, and therefore $w_a(R,\\tfrac{1}{3}\\bm{P})w_b(R,\\tfrac{1}{3}\\bm{P})w_c(R,\\tfrac{1}{3}\\bm{P})\\sim {\\rm e}^{-P\/T}$. Thus, this model predicts that the ratio of the proton to the pion yield at the common $P_T$ at mid-rapidity, $R_{p\/\\pi}\\equiv \\frac{dN_p}{d^2P_Tdy}\\big{\/}\\frac{dN_{\\pi^0}}{d^2P_Tdy}$, is essentially given by a ratio $C_B\/C_M$, which amounts to $\\sim 2$. This is indeed consistent with the experimental result known as the anomalous baryon\/meson ratio, which is in contrast to the expectation from parton fragmentation processes in perturbative QCD, $R_{p\/\\pi}\\sim 0.2$.\n\n\nFor $w_a(R;x\\bm{P})$, \nwe assume that at the onset of hadronization the quarks\/antiquarks are in local thermal equilibrium with a fluid flow velocity $\\bar v^\\mu(R)$, which we parameterize as \n\\begin{equation}\n\\bar v^\\mu(R)=(\\cosh \\eta^{}_L \\cosh \\eta^{}_T,\\, \\sinh \\eta^{}_T \\cos \\phi, \\, \\sinh \\eta^{}_T \\sin \\phi,\\, \\sinh \\eta^{}_L \\cosh \\eta^{}_T)\\, , \\label{normalized_flow_vector}\n\\end{equation}\nwhere $\\eta^{}_L$ and $\\eta^{}_T$ are longitudinal and transverse flow rapidities, respectively. This four-velocity is normalized as $\\bar v_\\mu \\bar v^\\mu=1$.\nIn the Bjorken expansion, which we assume in this work, the longitudinal flow rapidity $\\eta_L$ is identified with the space-time rapidity $\\eta$, i.e., $\\eta^{}_L=\\eta$, and the longitudinal flow velocity is $v_L=\\tanh \\eta = z\/t$. On the other hand, the transverse rapidity $\\eta_T$ is related to the transverse flow velocity $v^{}_T=\\sqrt{v_x^2+v_y^2}$ by\n\\begin{equation}\n v^{}_T=\\tanh \\eta^{}_T \\, ,\n \\label{transverse_velocity}\n\\end{equation}\nat mid-rapidity ($\\cosh \\eta_L = 1$).\nWe assume that $v_T$ is independent of the transverse radius $\\rho$ for computational simplicity.\n\n\nBy using this flow velocity $\\bar v^\\mu(R)$, we take the one-body phase space distribution of parton $a$ as \n\\begin{equation}\nw_a(R;p)=\\gamma_a \\, {\\rm e}^{-p\\cdot \\bar v(R)\/T} {\\rm e}^{-\\eta^2 \/2\\Delta^2}\nf(\\rho, \\phi)\\, ,\n\\label{phase_space_dist}\n\\end{equation}\nwhere $\\gamma_a$ is the fugacity factor of parton $a$.\nThe factor ${\\rm e}^{-\\eta^2 \/2\\Delta^2} f(\\rho, \\phi)$ describes the spatial profile of the hot medium.\nFor central collisions, one may simply assume a constant profile $f(\\rho,\\phi)=\\theta(\\rho_0-\\rho)$\nwithin the transverse radius $\\rho_0$ of the fireball at the recombination time $\\tau$.\nIn more general cases, the transverse profile will be adjusted to reproduce the collision-centrality dependence of observed meson yields.\nMeanwhile, the $\\eta$-dependence may be ignored as far as the mid rapidity region is concerned. These approximations will be used in analytic evaluations of the ReCo model below.\n\n\n\nWe have assumed that the quark momentum distribution in $w_a$ has the thermal profile\n${\\rm e}^{-p \\cdot \\bar v(R)\/T}$\nof temperature $T$, boosted by the collective flow $\\bar v(R)$.\nWe introduce elliptic anisotropy in this momentum distribution by a weak modulation of the transverse flow rapidity $\\eta_T$\naround the mean $\\overline{\\eta}_T$:\n\\begin{align}\n \\eta_T(\\phi;p_T)=\\overline{\\eta}_T (1 - h(p_T)\\cos 2\\phi) \\, .\n\\label{etaT_phi}\n\\end{align}\nThe modulation amplitude $h(p_T)$ is assumed to be $p_T$-dependent:\n\\begin{align}\n h(p_T)=\\frac{\\alpha}{1+(p_T\/p_0)^a}\n\\label{hpt-profile}\n\\end{align}\nwith $\\alpha=(1-r)\/(1+r)$ fixed by the transverse aspect ratio $r$ of the almond-shape collision zone,\nand with $a$ and $p_0$ the constant parameters controlling the momentum dependence.\nWe then calculate the parton distribution eq.~(\\ref{phase_space_dist}),\nto find the elliptic flow coefficient $v_2^a(p^{}_T)$ of parton $a$ defined by \n\\begin{eqnarray}\nw_a(R;p)=\\overline w_a(R;p)\\Big(1+2v_2^a(p^{}_T)\\cos 2\\phi \\Big)\\, ,\n\\end{eqnarray}\nwhere $\\overline w_a(R;p)$ is the part independent of the azimuthal angle $\\phi$.\nInserting this distribution into the formula (\\ref{dN\/dP}), we can compute the elliptic flow coefficient for\na meson transverse momentum ($P_T$) distribution at mid-rapidity:\n\\begin{equation}\n v_2^M(P_T)\\equiv \\langle \\cos 2\\Phi \\rangle^{}_{P_L=0}\n =\\frac{{\\displaystyle \\int d\\Phi \\cos 2\\Phi \\left(\\frac{dN_M}{d^2P_TdP_L}\\Big|_{P_L=0}\\right)} }\n {{\\displaystyle \\int d\\Phi \\left(\\frac{dN_M}{d^2P_TdP_L}\\Big|_{P_L=0}\\right)}}\\, ,\n\\end{equation}\nwhere $\\Phi$ is the azimuthal angle of the produced meson momentum.\nIf we adopt the $\\delta$-function approximation for the light-cone wavefunction $|\\phi_M(x)|^2=\\delta(x-1\/2)$\nand assume a universal elliptic flow coefficient for all quark flavors,\n$v_2^q(p_T)\\equiv v^a_2(p_T)$,\nthen the elliptic flow of the meson momentum distribution is analytically evaluated as \n\\begin{eqnarray}\nv_2^M(P_T)=\\frac{2\\, v^q_2(\\frac12 P_T)}{1+2\\, v^q_2(\\frac12 P_T)^2}\n\\, ,\n\\end{eqnarray}\nwhich simplifies further for $v^q_2(p_T)\\ll 1$ to\n\\begin{eqnarray}\nv_2^M(P_T)\\simeq 2v^q_2(P_T\/2)\\, .\n\\end{eqnarray}\nSimilarly, for baryons one finds \n\\begin{eqnarray}\nv_2^B(P_T)\\simeq 3v^q_2(P_T\/3)\\, .\n\\end{eqnarray}\nTherefore, the elliptic flow coefficient of a hadron with $n$ constituents satisfies\nthe following scaling:\n\\begin{eqnarray}\nv_2^h(P_T)\\simeq nv^q_2(P_T\/n)\\, .\n\\label{eq:CQNscaling}\n\\end{eqnarray}\nIf we plot $v_2^h(P_T)\/n$ as a function of $P_T\/n$ for various hadrons, the results will collapse into a single curve which is determined by the quark elliptic flow coefficient $v_2^q$. This ``CQN scaling\" is indeed observed in experimental data, and is regarded as one of the evidences for the quark recombination in hadronization and also for formation of a thermalized QGP in relativistic heavy-ion collision experiments. Nevertheless, as is obvious from the above derivation of the scaling, it appears only in an idealized situation and a certain deviation is expected from the scaling limit even when the recombination mechanism dominates in hadronization. Such deviations will have different sources, from which we can extract physical information on hadronization.\n\n\n\nAs we emphasized before, the recombination model describes the meson (baryon) formation as a 2-to-1 (3-to-1) process. If we take this literally, it implies that the model violates the energy conservation law because the invariant mass of the initial state with two (three) partons cannot be the same as the energy of a bound state. In Ref.~\\cite{Fries:2003kq}, it is argued that the energy conservation would be preserved by the effects of interactions with the medium (which generates a width in parton dispersion) and that omission of explicit treatment of such effects would not significantly affect the bulk features of hadron production.\nThe medium effect on the recombination reminds us of the three-body recombination in electromagnetic plasmas, where one of the three is just a spectator to keep the energy conservation.\n\n\nBut, at the same time, we notice another process which satisfies the energy conservation law in electromagnetic plasmas. It is the radiative recombination. The counterpart process in hadronization should be also possible.\\footnote{While it is possible to make energy conserved within a dynamical model including the effects of resonances as developed in \\cite{Ravagli:2007xx}, we will take a different picture. } In the next subsection we will discuss how to modify the ReCo model so that it describes the radiative hadronization. \n\n\n\\subsection{Radiative hadronization model}\n\nWe modify the ReCo model so as to allow for a photon emission, which we call radiative ReCo model.\nThen, the meson formation process becomes a 2-to-2 process,\n\\begin{equation}\n q+\\bar q \\to M+\\gamma\n \\, ,\n \\label{radiative_hadronization}\n\\end{equation}\nwhich fulfills the energy conservation law. A similar modification for baryon formation with a photon emission: $qqq\\to B + \\gamma$ should be also possible.\n\n\n\\begin{figure}[t] \n\\begin{center}\n\\vspace*{-1mm}\n \\includegraphics[width=0.7\\hsize]{NewReCo.pdf}\n\\end{center}\n\\caption{Radiative ReCo model with a photon emission.}\n\\label{fig:ReCo}\n\\end{figure}\n\n\nIn our radiative ReCo model, we re-interpret the original ReCo model \\cite{Fries:2003kq} as a tool for picking up a ``preformed state'' consisting of two partons and assume the preformed state emits a photon to form a bound state (see Fig.~\\ref{fig:ReCo}). Notice that we do not consider this preformed state as any physical resonance but just as an intermediate state in radiative meson production.\n\nThe preformed state consisting of two partons with momenta $p_1^\\mu=(E_1, \\bm{p}_1)$ and $p_2^\\mu=(E_2, \\bm{p}_2)$,\nhas the invariant mass $M_*$ and momentum $\\bm{P}$ determined by\n\\begin{eqnarray}\nE \\equiv \\sqrt{M_*^2+\\bm{P}_{}^2}&=&\\sqrt{m_1^2+\\bm{p}_1^2}\n+\\sqrt{m_2^2+\\bm{p}_2^2}\\, ,\\\\\n\\bm{P}&=&\\bm{p}_1 + \\bm{p}_2\\, ,\n\\end{eqnarray}\nwhere $m_1$ and $m_2$ are the constituent quark masses.\nThe invariant mass $M_*$ is a function of $\\bm{p}_1$ and $\\bm{p}_2$ and is evidently larger than $m_1+m_2$:\n$$\nM_*\\ge m_1+m_2\\, .\n$$\nOn the other hand, in the constituent quark picture, the mass of the bound state $M$ should be smaller than $m_1+m_2$ by the binding energy: $M < m_1+m_2$. The surplus energy of $M_* - M>0$ should be carried away by a photon emission here.\nThen, the number of the photons emitted in the formation of a meson $M$ reads\n\\begin{equation}\nE_\\gamma \\frac{d N_\\gamma}{d^3k}=\\kappa \\int dM_*\\, \\varrho(M_*)\\int d^3P \n\\left(\\frac{dN_{M_*}}{d^3P}\\right)\\left(E_\\gamma \\frac{dn_\\gamma(k; M_*,P)}{d^3k }\\right) \\, ,\n\\label{photon_distribution}\n\\end{equation}\nwhich means that it is given by the product of the number of preformed states and the photon distribution emitted from a preformed state. We explain each ingredient below.\n\nFirst of all, ${dN_{M_*}}\/{d^3P}$ is the number of the preformed states which is characterized by momentum ${\\bm P}$ and invariant mass $M_*$. We evaluate this part by the original ReCo model. Although we do not know the light-cone wavefunction of the preformed state, we expect that the results are insensitive to its details as we already commented concerning the original ReCo model. We use the same light-cone wavefunctions as in the original ReCo model. \n\nNext, $E_\\gamma {dn_\\gamma(k; M_*,P)}\/{d^3k}$ corresponds to the photon distribution emitted from the preformed state moving with the momentum $\\bm{P}$. We adopt a tree picture $M_*\\to M+\\gamma$ which is the leading order with respect to QED coupling $\\alpha$. We assume no specific polarization of the preformed states, and the photon distribution is treated as isotropic in the rest frame of the preformed state. More explicitly, we use the following photon distribution:\n\\begin{equation}\n E_\\gamma \\frac{dn_\\gamma}{d^3 k}\n = \\frac{1}{4\\pi k_0}\\, \\delta (E_{\\gamma _{\\rm CM}} - k_0)\n = \\frac{M_*}{4\\pi k_0}\\, \\delta (k \\cdot P - k_0 M_*), \n\\label{photon_CM_dist}\n\\end{equation}\nwhere $E_{\\gamma _{\\rm CM}}$ is the photon energy\nand $k_0\\equiv (M^2_*-M^2)\/(2M_*)$ is the momentum magnitude of the photon and the meson in the center-of-mass (CM) frame of the preformed state ($M_*$-rest frame) with $M$ being the mass of the accompanying meson. The photon distribution is normalized as $\\int dn_\\gamma =1$.\n\n\nWe also introduced $\\varrho(M_*)$ to represent an invariant mass distribution of the preformed states of two partons. It should have a support for $M_*\\ge m_1+m_2$. Considering the thermal distributions of quarks and antiquarks, we expect that the number of preformed states will rapidly decrease with increasing $M_*$. Therefore, we will replace it with the threshold contribution, {\\it i.e.}, $\\varrho(M_*)= \\delta(M_*-(m_1+m_2))$ in this paper.\n\n\nFinally and importantly, we comment on the overall factor $\\kappa$ which is introduced to reflect other possible effects on radiative hadronization. Consider the recombination process mediated by a preformed state $q+\\bar q \\to M_* \\to X$ where $X$ stands for any physical final states, including the radiative hadronization $X=M+\\gamma$.\nIn general, however, $X$ can be multiple hadron states (with photons), as we discussed previously in relation to the radiation return in $e^+e^-$ collisions.\nOnce we compute all possible diagrams for the decay of $M_*$, we are able to define the ``branching ratio\" for the radiative hadronization which corresponds to the factor $\\kappa$. Since the transition probability for $M_*\\to M+\\gamma$ would be proportional to the QED coupling $\\alpha=1\/137$, one may naively expect that $\\kappa$ would be of the order of $\\alpha$. On the other hand, we also know empirically that the CQN scaling works very well up to several GeV of meson momentum $p_T$. This fact suggests that a single meson formation would be the dominant process over multiple meson production. If so, the single photon emission attached to this dominant process may have different value of $\\kappa$ from the naive expectation.\nTherefore, we leave the overall factor $\\kappa$ as a parameter to be determined\nby the experimental data.\\footnote{One can consider gluon radiation,\n $M+g$, but the gluons are strongly interacting and will be re-absorbed into medium.}\n\n\nWe remark here that the number of mesons is given by the same formula as eq.~\\eqref{photon_distribution},\nwith the last factor replaced by the meson distribution emitted from the preformed state:\n\\begin{equation}\nE_M \\frac{d N_M^{\\rm radReCo}}{d^3K}= \\kappa \\int dM_*\\, \\varrho(M_*)\\int d^3P \n\\left(\\frac{dN_{M_*}}{d^3P}\\right)\n\\left(E_M \\frac{dn_M(K; M_*,P)}{d^3K}\\right) \\, ,\n\\label{meson_distribution}\n\\end{equation}\nwhere $K^\\mu$ is a momentum of the produced meson, satisfying $P_\\mu=K_\\mu + k_\\mu^\\gamma$. The distribution $dn_M\/d^3K$ can be defined in a similar way to the photon case. \n\n\nRecall that the original ReCo model naturally explains the CQN scaling.\nIn our modified ReCo model, on the other hand, the scaling should appear at the level of the distribution of preformed states $M_*$ in the integrand of eq.~(\\ref{meson_distribution}) and\nthe meson production accompanying a photon emission may violate the CQN scaling to some extent.\nWe check this point both analytically and numerically in this paper.\nBut we emphasize here that the main contribution to hadron yield at around 2 GeV is given by the original ReCo model\nand the radiative hadronization \\eqref{meson_distribution} is a subdominant process of order $\\kappa$ at most.\nMoreover, since the photon carries away a fraction of the preformed-state momentum, the meson spectrum of the radiative hadronization is shifted to the lower momentum region and therefore at a given momentum $p_T$ its yield is more suppressed than the value simply expected by the factor $\\kappa$.\nHowever, we stress here that to photon production the radiative hadronization can give a significant contribution.\n\n\n\n\\subsection{Characteristics of the radiative hadronization}\n\n\n\nIn order to understand characteristics of the radiative hadronization, let us evaluate the momentum distributions of photons (\\ref{photon_distribution}) and mesons (\\ref{meson_distribution}) under certain approximations.\n\n\n\\subsubsection{Distribution of the preformed states}\n\n\nWe compute the number of preformed states by using the formula (\\ref{dN\/dP}) of the original ReCo model \\cite{Fries:2003kq} and we recap the formulas of\nReCo model here. The transverse momentum spectrum of the preformed state at mid rapidity $\\eta= 0$ is given by \\cite{Fries:2003kq}\n\\begin{equation}\nE_{M_*}\\left.\\frac{dN_{M_*}}{d^3P}\\right|_{\\eta=0}\\sim C_{M_*} M_{*T} \\frac{\\tau A_T}{(2\\pi)^3} \\; 2\\gamma_a\\gamma_b I_0\\left(\\frac{P_T\\sinh \\eta_T}{T_{\\rm reco}}\\right)\n\\int_0^1 dx|\\phi_{M_*}(x)|^2 k_{M_*}(x,P_T)\n\\end{equation}\nwith \n\\begin{equation}\nk_{M_*}(x,P_T) \\equiv K_1\\left(\\frac{\\cosh \\eta_T}{T_{\\rm reco}}\\left\\{\\sqrt{m_a^2+x^2P_T^2}+\\sqrt{m_b^2+(1-x)^2P_T^2}\\right\\}\\right)\\, ,\n\\end{equation}\nwhere $M_{*T}=\\sqrt{P_T^2+M_*^2}$ is the transverse mass,\n$A_T$ is the transverse area of the parton system at hadronization, representing the collision geometry, and $I_0(x)$ and $K_1(x)$ are the modified Bessel functions.\nThe parameter $T_{\\rm reco}$ is the recombination temperature at which quark recombination $q+\\bar q\\to M_*$ occurs.\nFor analytic evaluation, we simply take $|\\phi_{M_*}(x)|^2=\\delta(x-1\/2)$\nand perform the integration over $x$ for pion production:\n$$\n\\int_0^1 dx|\\phi_{M_*}(x)|^2 k_{M_*}(x,P_T)\\simeq K_1\\left(\\frac{\\cosh \\eta_T}{T_{\\rm reco}}M_{*T}\\right)\\, ,\n$$\nwhere $M_{*T}$ appears in the argument because we have taken $m_a=m_b=m$\nand $M_*= 2m$ (recall that we adopt $\\varrho(M_*)=\\delta (M_*-(m_1+m_2))$). By using the asymptotic forms of the modified Bessel functions $I_0(z)\\sim {\\rm e}^z\/\\sqrt{2\\pi z}$ and $K_1(z)\\sim \\sqrt{\\pi\/2z}\\ {\\rm e}^{-z}$ for large $z\\gg 1$ and large $P_T$ approximation $M_{*T}\\simeq P_T$,\nwe find the following approximate form for the $P_T$ distribution:\n\\begin{equation}\nE_{M_*}\\left.\\frac{dN_{M_*}}{d^3P}\\right|_{\\eta=0}\\sim \n{\\rm e}^{-P_T \/ T_{\\rm eff}^*}\\, ,\n\\label{M*pt_distribution}\n\\end{equation}\nwhere $T_{\\rm eff}^*$ is the effective temperature for the preformed state defined by\n\\begin{equation}\nT_{\\rm eff}^*=T_{\\rm reco}\\ {\\rm e}^{\\eta^{}_T}=T_{\\rm reco} \\, \\sqrt{\\frac{1+v^{}_T}{1-v^{}_T}}\\label{eff_temp}\n\\; .\n\\end{equation} \nThe recombination temperature $T_{\\rm reco}$ is identified with\nthe hadronization temperature in the original ReCo model.\nThe multiplicative factor $e^{\\eta_T}>1$ reflects the effect of the nonzero transverse flow of the quarks,\nwhich blue-shifts the inverse slope parameter of the preformed state distribution\nfrom $T_{\\rm reco}$ to $T_{\\rm eff}^*$.\nFor example, this factor $T_{\\rm eff}^*\/T_{\\rm reco}$ amounts to $\\sim $1.7 for $v_T=0.5$ and $2$ for $v_T=0.6$. \n\n\n\\subsubsection{Transverse momentum distributions of photons and mesons}\n\nGiven the distribution of the preformed states, let us discuss the\nphoton distribution eq.~(\\ref{photon_distribution}) at mid-rapidity $k_L=0$.\nWe can perform the integration over $\\Phi$ in eq.~(\\ref{photon_distribution}) with the $\\delta$-function\nin the photon distribution (\\ref{photon_CM_dist}) in the laboratory frame,\n\\begin{align}\n E_\\gamma \\frac{dn^\\gamma}{d^2k_T dk_L}\n &= \\frac{M_*}{4\\pi k_0}\\delta(\n k E_* - k_T P_T \\cos(\\Phi-\\phi) - k_0 M_*)\n ,\n\\end{align}\nwhere $\\Phi$ ($\\phi$) is the azimuthal angle of the preformed state (photon) from the reaction plane,\nas shown in Fig.~{\\ref{fig:decay}.\nThen we obtain\n\\begin{align}\n \\left .\n E_\\gamma \\frac{dN_\\gamma}{d^2 k_{T} dk_L}\n \\right |_{k_L=0} &=\n\\int_{-P_{L\\rm max}}^{P_{L\\rm max}} dP_L \\int_{P_{T\\rm min}}^{P_{T\\rm max}} d P_T \\, \n\\frac{dN_*}{d^2P_T dP_L}\\,\n\\frac{1}{4\\pi k_0}\n \\frac{2M_*}{k_T |\\sin \\theta|}\n \\, ,\n\\label{PT_integral}\n\\end{align}\nwhere \n\\begin{align}\n \\cos \\theta =\\cos(\\Phi-\\phi)=\n \\frac{k E_{M*} -k_0 M_*}{k_T P_T} \n \\, .\n \\label{eq:theta}\n\\end{align}\nThe integration ranges of the longitudinal and transverse momenta,\n$\\pm P_{L\\rm max}$ and $P_{T\\rm min,max}$, of the preformed state are determined by decay kinematics (See Appendix A).\n\n\n\\begin{figure}[t] \n\\begin{center}\n \\includegraphics[width=0.4\\textwidth,bb=0 0 600 600]{Decay-v3.png}\n\\hspace{0.05\\textwidth}\n\\includegraphics[width=0.35\\textwidth,bb=0 0 450 300]{Decay-align-v4.png}\n\\end{center}\n\\caption{Left: Kinematics of the photon emission from the preformed state, ${\\boldsymbol P} \\to {\\boldsymbol k} + {\\boldsymbol K}$.\n Right: Momentum boost from the CM to the laboratory frame\n for the photon momentum parallel (top) and anti-parallel (bottom) to ${\\boldsymbol P}$.\n}\\vspace{-2mm}\n \\label{fig:decay}\n\\end{figure}\n\n\n\n\nWe derive here approximate formulas valid for large photon momentum $k_T \\gg M_*$.\nThe distribution of the preformed states (\\ref{M*pt_distribution}) is a steeply-falling\nfunction of $P_T$ for large $P_T \\gg T_{\\rm eff}^*$,\nand therefore dominant contribution to the integral comes from the lower end of the $P_T$ integration\nwith $P_L \\simeq 0$.\nIn other words,\nit comes from the configurations in which ${\\boldsymbol P}$ is almost parallel to ${\\bm k}$, \n{\\it i.e.}, $\\cos \\theta \\sim 1$ and $P_L \\simeq 0$.\nIn this configuration,\nthe momentum $k_T$ is simply related with the CM-frame momentum $k_0$ by the transverse boost along $P_T = M_* \\sinh y_T^*$ via\n\\begin{align}\n k_T=k_0 \\cosh y_T^* + k_0 \\sinh y_T^* \\sim 2k_0\\frac{P_T}{M_*}\n =(1-\\frac{M^2}{M_*^2}) P_T\n\\label{mom-shift}\n\\end{align}\nas $\\cosh y_T^* \\sim \\sinh y_T^*=P_T\/M_*$.\nThe momentum deficit is carried by the accompanying meson emitted in the opposite direction in the\n$M_*$-rest frame: $K_T= -k_0 \\cosh y_T^* + \\sqrt{M^2+k_0^2} \\sinh y_T^*\n\\sim\n(M^2\/M_*^2)P_T ,\n$\nand $k_T + K_T = P_T$ holds as it should (Fig.~\\ref{fig:decay} top-right). \n\nSince the number of the radiated photons is proportional to that\nof preformed states, the photon distribution should behave as\n(See Appendix A for more details)\n\\begin{align}\n\\left . E_\\gamma \\frac{dN_\\gamma}{d^2 k_T dk_L}\n\\right |_{k_L=0} \\sim\n\\exp \\left ( -\\frac{P_{T}}{T^*_{\\rm eff}(M_*)} \\right ) \n\\sim\n \\exp \\left ( -\\frac{k_{T}}{T^\\gamma_{\\rm eff}(M_*)} \\right )\n\\end{align}\nwith\n\\begin{align}\n T^\\gamma_{\\rm eff}(M_*) = \\left ( 1-\\frac{M^2}{M_*^2} \\right )\n T^*_{\\rm eff} \\, .\n \\label{gamma_T}\n\\end{align}\nThus we find that the effective temperature of the photon $T^\\gamma_{\\rm eff}$ becomes lower than $T_{\\rm eff}^*$.\n\n\n\nOn the other hand, the main contributions for the meson production by the radiative ReCo model come\nfrom the configuration in which the meson momentum is parallel to $P_T$ in the $M_*$-rest frame\n(See the bottom-right panel in Fig.~\\ref{fig:decay}), \nthe same calculation yields\n\\begin{align}\n K_T = k_0 \\cosh y_T^* + \\sqrt{M^2+k_0^2} \\sinh y_T^* \\sim P_T,\n\\end{align}\nand the accompanied photon has nearly zero energy and momentum $k_T \\sim 0$ due to its massless nature.\nAccordingly, we find\n\\begin{align}\n \\left . E_M \\frac{dN_M}{d^2K_TdK_L}\\right\\vert_{K_L=0}\n \\sim\n \\exp \\left(-\\frac{K_T}{T_{\\rm eff}^{\\rm meson}(M_*)}\\right)\n\\end{align}\nwith\n\\begin{align}\n T_{\\rm eff}^{\\rm meson}(M_*)=T_{\\rm eff}^*\n \\, .\n\\label{eq:meson_T}\n\\end{align}\nThe meson has the same effective temperature as the preformed state, up to some corrections.\n\n\n\nThe radiative hadronization predicts that there is an ordering in the effective temperatures of\nphotons, mesons, and preformed states: \n\\begin{equation}\n T_{\\rm eff}^\\gamma(M_*) < T_{\\rm eff}^{\\rm meson}(M_*)\n \\sim T_{\\rm eff}^* \\, .\n \\label{eq:ordering}\n\\end{equation}\nNote that the thermal exponential form of the photon and meson distributions reflects the shape of the parton distributions\nand thus the origin of higher effective temperatures of photons and mesons than the hadronization temperature $T_{\\rm reco}$\ncan be attributed to the partonic collective flow built up during the QGP evolution.\n\n\n\n\n\\subsubsection{$v_2$ of photons and mesons}\n\nWithin the same approximations, we can evaluate the elliptic flow\ncoefficient for the photons, $v_2^\\gamma(k_T)$, defined by \n\\begin{equation}\n v_2^\\gamma(k_T)\\equiv \\frac\n {{\\displaystyle\n \\int d\\phi \\cos 2\\phi\n \\left(\\left. k \\frac{dN_\\gamma}{d^2k_Tdk_L }\\right|_{k_L=0} \\right)}}\n {{\\displaystyle\n \\int d\\phi\n \\left(\\left. k \\frac{dN_\\gamma}{d^2k_T dk_L }\\right|_{k_L=0}\\right)}}\\, .\n \\label{v2_gamma}\n\\end{equation}\nThe distribution of the preformed state is computed with the original ReCo model,\n\\begin{equation}\n\\frac{dN_*}{d^2P_T dP_L}=\n \\overline{ \\frac{dN_*}{d^2P_T dP_L}}\\,\n \\Big(1+2v_2^*(P_T)\\cos 2\\Phi \\Big)\\, ,\n\\label{v2_preformed}\n\\end{equation}\nwhere \n$\\overline{dN_*\/d^2 P_T dP_L}$ is $\\Phi$-independent part of the spectrum and the nonzero elliptic flow $v_2^*(P_T)$ is inherited from the quark\/anti-quark elliptic flow.\nThen, in place of eq.~(\\ref{PT_integral}), the integration over $\\Phi$ with the $\\delta$-function\nyields\n\\begin{align}\n \\left .\n k \\frac{dN_\\gamma}{d^2 k_{T} dk_L}\n \\right |_{k_L=0} &=\n\\int_{-P_{L\\rm max}}^{P_{L\\rm max}} dP_L \\int_{P_{T\\rm min}}^{P_{T\\rm max}} d P_T \\,\n \\overline{\\frac{dN_*}{d^2P_T dP_L}}\\,\n (1+2v_2^*(P_T) \\cos 2 \\phi \\cos 2 \\theta)\\frac{1}{4\\pi k_0}\n \\frac{2M_*}{k_T |\\sin \\theta|}\n,\n\\label{gamma_dist_2}\n\\end{align}\nwhere we have used $\\sum_{i=\\pm}\\cos 2\\Phi_i = 2 \\cos 2\\phi \\cos 2\\theta$\nwith $\\Phi_\\pm = \\phi \\pm \\theta$.\nWe insert eq.~(\\ref{gamma_dist_2}) into the definition (\\ref{v2_gamma}),\nand evaluate the momentum integral approximately with its threshold value near $P_T \\sim P_{T\\rm min}$ and $P_L \\sim 0$,\nwhere we can put $\\cos 2\\theta \\sim 1$.\nThus we find that, after the $\\phi$ integration, the elliptic flow coefficient is unchanged\nbut its momentum argument is replaced with $k_T = (1-M^2\/M_*^2)P_T$~:\n\\begin{equation}\n v_2^\\gamma(k_T) \\sim v_2^*\\left(\\frac{k_T}{1-M^2\/M_*^2}\\right)\n \\, .\n\\label{photon_v2_scaling}\n\\end{equation}\nIn the same manner, we find the coefficient for the meson distribution in radiative ReCo model as \n\\begin{equation}\nv_2^{\\rm meson}(K_T)\\sim v_2^*\\left(K_T \\right)\\, .\n\\label{meson_v2_scaling}\n\\end{equation} \nWe emphasize here that the elliptic flow coefficients $v_2$ of the photons and mesons are both given by $v_2^*$ of the preformed states, \nwhich approximately satisfies the CQN scaling, $v_2^*(P_T) \\sim 2 v_2^q(P_T\/2)$~(see eq.~\\eqref{eq:CQNscaling}). \n\n\nWe have shown that the effective temperature and the elliptic flow coefficient of \nthe meson distribution are the same as those of the preformed-state distribution, \nwhile for the photon distribution these parameters are estimated in the similar manner but with a simple momentum shift \\eqref{mom-shift}.\nThis means that even if the contributions of the radiative hadronization are added,\nnot only the meson elliptic flow still follows the CQN scaling,\nbut also the photon elliptic flow does so approximately. \n\n\n\n\\section{Numerical results}\n\nFor our numerical study we employ a 2-dimensional (2D) model, neglecting the longitudinal momentum of the preformed state $M_*$ ($P_L=0$),\nsince we have seen that the $P_L$-integration plays only a minor role in the modification of the photon and meson distributions from that of the preformed state.\n\n\nThe photon distribution of 2D radiative ReCo model reads \n\\begin{align}\n \\left .\n k \\frac{d N_\\gamma}{d^2k_T dk_L} \\right |_{k_L=0}\n &=\\kappa \\int_{P_{T\\rm min}}^{P_{T\\rm max}} dP_T \\,\n \\sum_{i=\\pm} \\frac{dN_{M_*}}{d^2P_TdP_L}(P_T, \\Phi_i,0)\n \\frac{1}{2\\pi} \\frac{M_*}{k_T |\\sin \\theta |}\n \\, ,\n \\label{2d_photon_distribution}\n\\end{align}\nwhere \n$\\Phi_\\pm=\\phi\\pm \\theta$ with $\\theta$ defined in eq.~\\eqref{eq:theta}.\nWe generate the distribution of the preformed states of mass $M_*$,\nusing the original ReCo model \\cite{Fries:2003kq,Fries:2003vb}.\nBut we change the recombination temperature $T_{\\rm reco}$ to 155 MeV, which is within the range of the pseudo-critical temperature obtained\nin lattice QCD calculations \\cite{Borsanyi:2010cj,Bazavov:2011nk}.\nAccordingly the transverse flow at hadronization is set to $v_T=0.6$ to reproduce the $p_T$ distribution of the mesons,\nand the freeze-out time $\\tau$ and the fireball radius $\\rho_0$ are also adjusted adequately (see Table \\ref{tab:ParamSet}).\nWe set $M_* = 2M_{ud}$ for the preformed state based on the constituent quark model picture. \n\n\n\n\\begin{table}\n \\begin{tabular}{cccccccccc}\n \\hline\n& $T_{\\rm reco}$~\/MeV & ~ $v_T$ ~ & ~$\\tau$~\/fm ~ & ~ $\\rho_0$~\/fm ~ &~ $\\gamma_{u,d}$~ & ~$\\gamma_{\\bar u,\\bar d}$~ & $M_{ud}$~\/MeV & $p_0$~\/GeV & $a$ \\\\\n \\hline\n RHIC & 155 & 0.6 & 8.0 & 12.5 & 1 & 0.9 & 260 & 1.0 & 2.5\\\\\n LHC & 155 & 0.65 & 15.0 & 20.0 & 1 & 1 & 260 & 1.1 & 2.5\\\\ \n\\hline\n\\end{tabular}\n \\caption{\n Model parameters.\n }\n\\label{tab:ParamSet}\n\\end{table} \n\n\n\\subsection{Model characteristics}\nFirst let us numerically test the characteristics of particle distributions of our radiative ReCo model,\nwhich was discussed in the previous section.\nNote that in this subsection we set the normalization factor $\\kappa=1$ of the radiative ReCo model to study the model characteristics,\nwhile in the next subsections we will adjust the parameter $\\kappa$ so that the model reproduces the observed photon distributions.\n\n\n\\subsubsection{Transverse momentum spectrum}\nWe show $p_T$ distributions of the pions and photons produced by the radiative ReCo model at $\\sqrt{s_{NN}}=200$ GeV\nin Fig.~\\ref{fig:model_char_pion} (left), along with the distribution of the preformed states.\nAt a given momentum, yields of the pions and photons are much lower than that of the preformed state because\neach of them shares a fraction of the momentum of the preformed state,\nand therefore their distributions are shifted to the lower $p_T$ region.\nBut their $p_T$ slopes are similar to each other in the relevant momentum region.\n\n\nWe define the effective temperature $T_{\\rm eff}$ here\nby fitting each of the $p_T$ distributions with an exponential function $\\exp(-p_T\/T_{\\rm eff})$\nin the momentum range $2 < p_T < 5$ GeV.\\footnote{%\n %\n The slope parameter $T_{\\rm eff}$ is centrality-independent in this model\n since the quark distribution \\eqref{phase_space_dist}\n depends on collision centrality only by the weak modulation of the transverse flow \\eqref{etaT_phi}\n and by the overall factor of the hot-zone size $f(r,\\phi)$.\n}~\nIn Fig.~\\ref{fig:model_char_pion} (right) we check the $M_*$ dependence of these effective temperatures,\n$T_{\\rm eff}^{\\pi}$ of pions and $T_{\\rm eff}^\\gamma$ of photons produced in the radiative ReCo model.\nThey follow the ordering eq.~\\eqref{eq:ordering} derived in the previous section.\n$T_{\\rm eff}^\\pi = T_{\\rm eff}^*$ is predicted by eq.~\\eqref{eq:meson_T}, and \nthe difference between $T_{\\rm eff}^{\\pi}$ and $T^*_{\\rm eff}$ may be understood\nas a correction due to subleading $p_T$-dependence of the distributions.\nOn the other hand, $T_{\\rm eff}^\\gamma$ of photons is lower than $T^*_{\\rm eff}$ and\napproaching it with increasing $M_*$, as predicted by eq.~\\eqref{gamma_T}. \n\nOur model restricts the invariant mass of the preformed state to $M^*= 2 M_{ud}=520$~MeV for pion production,\nbut in more general treatments, the preformed states with $M^* \\ge 2M_{ud}$ may well contribute to the pion production.\nHowever, from Fig.~\\ref{fig:model_char_pion} (right), the slope parameter $T_{\\rm eff}$ of the photons and pions from the radiative ReCo model\nseem rather insensitive to the $M_*$ value, as far as $M_*$ is much larger than the meson mass $M$.\n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=7.5cm]{radReCo_Npt_MstMeGam.pdf}\n \\hfil\n\\includegraphics[width=7.5cm]{radReCo_Teff_vs_Mst.pdf}\n \\caption{%\n Left: Comparison of transverse momentum distributions $d^2N\/(2\\pi p_T dp_T dy)$\n of the photons (green bold solid), pions (blue thin solid), and preformed state (black dashed) in radiative ReCo model\n ($M_* = 2M_{ud}$, $\\kappa=1$ with $b=5.5$ fm at $\\sqrt{s_{NN}}=200$ GeV).\n Right: \n$M^*$-dependence of the slope parameters of the photons $T_{\\rm eff}^\\gamma$ (green circle),\n of pions $T_{\\rm eff}^\\pi$ (blue triangle), and of preformed states $T_{\\rm eff}^*$ (black square).\n The parameter $T$ is determined by fitting the function $\\propto e^{-p_T\/T}$ \n to the $p_T$ distributions in the momentum range $2 < p_T < 5$ GeV.\n \\label{fig:model_char_pion} } \n\\end{figure}\n\n\n\nWe show the results for kaon production with $M_* = M_s + M_{ud}$ and $M_s = 460$ MeV in Fig.~\\ref{fig:model_char_kaon},\nwhere the particle yields become smaller at $p_T \\lesssim 2$ GeV than the pion case due to the mass effect. \nMoreover, the $p_T$ slope of the photon distribution is much steeper than those of the kaon and preformed state distributions, \nbecause, unlike the pion mass, the kaon mass $M_K=495$~MeV is comparable to $M_*=720$~MeV here (see eq.~\\eqref{gamma_T}).\nWe see that the photon yield associated with radiative hadronization of the kaons is small compared with that of the pions,\nand we neglect it in this work.\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=7.5cm]{radReCo_Npt_MstMeGam_kaon.pdf}\n \\caption{%\n The same plot as in Fig.~\\ref{fig:model_char_pion} (right),\n but associated with kaon production ($M_s =460$ MeV).\n \\label{fig:model_char_kaon}\n }\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=7.5cm]{radReCo_v2_scaling.pdf}\n \\caption{%\n Rescaled elliptic flow coefficients $v_2(p_T\/n_q)\/n_q$\n of the pions (blue thin solid), photons (green thick solid),\n and preformed state (black dashed) at $b=5.5$ fm with $n_q=2$.\n The flow coefficient $v_2(p_T)$ of the quarks (purple dotted)\n is shown for comparison.\n Parameters are the same as in Fig.~\\ref{fig:model_char_pion}. \n \\label{fig:check_QNS_v2_rhic}\n }\n\\end{figure}\n\n\n\\subsubsection{Elliptic flow $v_2$}\n\nWe introduce an azimuthal angle dependence of the quark\/antiquark flow $v_T(p_T)$\nby the modulation amplitude $h(p_T)$ of the transverse flow rapidity $\\eta_T(\\phi; p_T)$ as in eq.~(\\ref{etaT_phi}).\nThe magnitude of $h(p_T)$ is determined by the aspect ratio of the initial collision zone.\nIn Fig.~\\ref{fig:check_QNS_v2_rhic} shown is the result of the elliptic flow coefficients $v_2$\nof the pions (cyan thin) and photons (green bold) as well as that of the preformed states (black dashed),\nafter divided by the constituent quark number $n_q=2$.\nThe quark\/antiquark elliptic flow coefficient $v_2^q$ is also shown (purple dotted) for comparison.\n\n\nWe find that all these $v_2$ have the same magnitude,\nwhich is taken over from the quark\/antiquark flow through the radiative recombination.\nWe also note that the flow coefficient $v_2^\\pi$ of the pions exactly follows\nthat of the preformed state $v_2^*(p_T)$ at $p_T \\gtrsim 1$ GeV,\nwhile the photon $v_2^\\gamma(p_T)$ lies slightly below them.\nIndeed, we have confirmed that,\nwhen plotted in the shifted momentum $\\bar p_T=p_T\/(1-M^2\/M_*^2)$,\nthe photon $v_2^\\gamma(p_T)$ curve overlaps with that of the preformed states at larger $p_T$,\nas discussed in eq.~\\eqref{photon_v2_scaling}.\n\n\n\n\n\n\n\n\\subsection{RHIC}\n\n\nNext we study the contributions of radiative hadronization to the photon production in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.\nIn order to make a comparison of the photon yield to the direct photon data at RHIC \\cite{Adler:2003qi}, \nwe include the thermal photon contributions evaluated with a 3-dimensional viscous hydrodynamic model with the kinetic freezeout temperature $T_{\\rm fo}=116$ MeV \\cite{Miyachi-Nonaka}\n(see Appendix B).\nWe assign the impact parameter $b=3.0, 5.5, 7.5$ and $9.0$ fm in our model for the centrality classes, 0--10, 10--20, 20--30 and 30--40 $\\%$ of the collision events, respectively, based on the Glauber model estimate \\cite{PHENIX:2003iij}. \n\n\n\nIn Fig.~\\ref{fig:centrality_rhic} we compare the $p_T$ distributions of $\\pi^0$ obtained\nby the ReCo (black solid) and radiative ReCo (cyan dashed) models to PHENIX data at different centralities \\cite{Adler:2003qi}.\nThe parameter $\\kappa=0.2$ of the radiative ReCo model is determined so that the sum of the photons from radiative hadronization and\nthe thermal photons reproduces the observed photon yield (see Fig.~\\ref{fig:photon_centrality_rhic} below).\nWe are reassured here that the original ReCo model reproduces the pion $p_T$ distributions for different centrality classes, in the $p_T$ range from 2 to 4 GeV,\nwhere the quark recombination is regarded as the dominant hadronization mechanism.\nOutside this region, other production mechanisms, hydrodynamic process at the lower $p_T$ and parton fragmentation at the higher $p_T$, are important.\nIn contrast, the contribution from the radiative ReCo model takes only a small fraction of the pion yield (less than 10 \\% of the original ReCo model) at a given momentum $p_T$,\nand it brings no obstruction on the success of the original ReCo model in describing meson production in this momentum region \\cite{Fries:2003kq,Fries:2003vb}. \n\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{radReCo_Npt_wData-NoGam.pdf}\n \\caption{Transverse momentum distributions of $\\pi^0$\n computed with ReCo model (black solid) and radiative ReCo model (cyan dashed)\n for impact parameter $b=3$, 5.5, 7.5 and 9 fm in Au+Au collisions\n at $\\sqrt{s_{NN}}=200$ GeV.\n The $\\pi^0$ data set of 0--10 \\%, 10--20 \\%, 20--30 \\% and 30--40 \\% centrality classes\n is adopted from \\cite{Adler:2003qi}.\n \\label{fig:centrality_rhic} } \n\\end{figure}\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{Prompt+Thermal+radReCo_photons_Npt.pdf}\n \\caption{\n Transverse momentum distributions of direct photons \n computed with radiative ReCo model (green dashed) for impact parameter $b=5.5$ (left) and $9$ fm (right)\n in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.\n Thermal photon distribution obtained by a viscous hydrodynamic model (purple dotted),\n rescaled prompt photons (black dot-dashed), and their sum (red solid) are also shown.\n The data is adopted from \\cite{Adare:2014fwh}, and the parameter $\\kappa=0.2$ is determined to fit the data. \n\\label{fig:photon_centrality_rhic}\n}\n\\end{figure}\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{radReCo_v2pt_pi.pdf}\n \\caption{\n Pion elliptic flow coefficient $v_2$ from ReCo model (black solid)\n and radiative ReCo model (blue dashed) as a function of $P_T$\n for $b=5.5$ fm (left) and $b=9.0$ fm (right) in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV. \n Data $\\pi^0$ (blue circles) are taken from PHENIX \\cite{Adare:2013wop}.\n \\label{fig:v2_rhic}\n}\n\\end{figure}\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=10cm]{Prompt+Thermal+radReCo_photons_v2pt.pdf}\n \\caption{\n Elliptic flow coefficient $v_2^\\gamma$ of the direct photons (red solid) for impact parameter\n $b=5.5$ (left) and 9 fm (right) in Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV.\n The $v_2^\\gamma$ of the photons from a viscous hydrodynamic model \n and $v_2^\\gamma$ of the photons from radiative ReCo model are shown in purple dotted and green dashed curves, respectively.\nThe normalization $\\kappa=0.2$ for the radiative ReCo model is adopted.\n Data for direct photons (blue solid stars) is adopted from \\cite{Adare:2015lcd}.\n \\label{fig:v2_photon_rhic}\n}\n\\end{figure}\n\n\n\nNext in Fig.~\\ref{fig:photon_centrality_rhic}\nwe show the $p_T$ distributions of the photons emitted in radiative pion production (green dashed)\nfor $b=5.5$ fm (left panel) and $b=9.0$ fm (right panel),\ntogether with those of the thermal photons (purple dotted) and the prompt photons (black dot-dashed),\nand the total (red solid).\nFor thermal photon production, we adopt the thermal photon rates of QGP in \\cite{Arnold:2001ms} and\nthat of the hadronic phase in \\cite{Holt:2015cda,Turbide:2003si,Heffernan:2014mla},\nand integrate these rates over the evolution profile obtained by a 3D viscous hydrodynamic simulation (See Appendix B for details).\nOur estimate of the thermal photon contribution is consistent with other model studies \\cite{Paquet:2015lta}.\nRegarding prompt photon production in AA collisions, which dominates the total photon distribution at higher $p_T$,\nwe use the empirical fit of the photon distribution in pp collisions, $a_1 (1+p_T^2\/a_2)^{a_3}$ ($a_{1,2,3}$ are constants),\nscaled with the number of nucleon collisions for AA collisions, \nas is done by PHENIX \\cite{Adare:2014fwh}.\nThe experimental data of the direct photons in the 0--20 \\% (for $b=5.5$ fm) and 20--40 \\% (for $b=9.0$ fm) centrality classes\nare taken from PHENIX \\cite{Adare:2014fwh}\\footnote{%\n A new data analysis is published in Ref.~\\cite{PHENIX:2022rsx} and\n consistent with that in Ref.~\\cite{Adare:2014fwh}.\n}. \n\n\nWe set the normalization of the radiative ReCo model to $\\kappa=0.2$ so that the sum of the two photon contributions, thermal radiation and radiative hadronization, reproduces the observed photon yield for $p_T < 3$ GeV.\nIndeed, the photon $p_T$ distributions for two centrality classes $0-20$ \\% and $20-40$ \\% are reproduced fairly well with the same normalization $\\kappa=0.2$.\nWe notice that the photon yield from the radiative ReCo model is estimated to be several times larger than that from the thermal radiation and that the $p_T$ slope of the resultant photon distribution is mostly determined by the contribution from the radiative ReCo model for $2 0.4 \\, \\textrm{arcmin}$. This makes sure that the galaxies are large enough in order for the morphology to be determined. The g-band images for each of these 84\\,723 objects were downloaded from the SDSS Science Archive \\citep[DR12;][]{sdss-dr12}. We used the SDSS mosaicking service\\footnote{https:\/\/dr12.sdss.org\/mosaics\/} which combines the maximum number of scans possible for the final image. The mosaicking service employs Swarp \\citep{swarp}, which aligns and combines the background offsets in the separate images. The result is a deep, background-subtracted image of each galaxy in the g-band. We limited ourselves to a field of view of $1.125\\cdot D_{25}$ since even the deep SDSS mosaics rarely reach the $25\\, \\textrm{mag}\\,\\textrm{arcsec}^{-2}$ surface brightness level. \n \n\\subsection{Data preprocessing}\n \nThe images were star subtracted using PTS\\footnote{http:\/\/www.skirt.ugent.be\/pts}, the python toolkit for SKIRT (\\citealt{skirt}; Verstocken et al., in prep.). This makes use of the SDSS point source catalogue as a prior for star positions, and then tries to find a peak around the positions which resembles a true point source. These point sources are then replaced by the local background using bicubic interpolation.\n\nThe star-subtracted images were used to calculate a few features (such as the total g-band luminosity), further discussed in Sect. \\ref{sec-ml}. After the extraction of these features, we log-scaled the images in order to emphasize lower brightnesses, especially at the outskirts of galaxies. First, the image flux was linearly rescaled to the interval [0, 1]. We then log-scaled the pixels in the following way:\n \n\\begin{equation}\n F' = \\frac{\\log \\left(1 + a F \\right)}{\\log \\left( 1 + a \\right)}.\n\\label{eq-scauto}\n\\end{equation}\n \nHere, $F$ is the original pixel value (between 0 and 1). The log-scaled pixel value $F'$ also ranges from 0 to 1, and these then serve as input for the machine learning (Sect.~\\ref{sec-ml}). The 1 inside the log prevents very small values from dominating the output scale. The \\textit{scaling value a} determines how much lower brightnesses are emphasized. Small values of $a$ ($a < 1$) result in a nearly linear scaling. Large values of $a$ result in a pure log scaling, which boosts fainter regions. The scaling value was determined independently for each of the objects. First, the noise level of the input $F$ was determined by sigma clipping the image, and then defining the noise level as two standard deviations above the mean. The scaling value was then fitted so the output (i.e. log-scaled) noise level equals 0.2. The result is that images with a high signal to noise ($S\/N$) have a larger scaling value, which allows for their fainter features to stand out. Images where the noise is more prevalent get a smaller scaling value, which in turn prevents the noise from being mistaken with a feature of the galaxy. This is demonstrated for a low, median, and high scaling value galaxy in Fig.~\\ref{fig-scauto}, where the bottom row shows the automatic scaling value procedure. The result is a more consistent background, boosting features without blowing up the noise. The background noise value of 0.2 was picked visually to distinguish faint features from noise (see bottom panel of Fig.~\\ref{fig-scauto}). We argue that if humans can distinguish the two, deep neural networks should also be able to learn the difference.\n \n\\begin{figure}\n \\centering\n \\includegraphics[width=\\hsize]{figures\/scauto_examples.pdf}\n \\caption{Demonstration of the automatic scaling value. The rows present different scalings, with the top row being a linear scale (equivalent to a scaling value $a \\ll 1$), the middle row using a constant scaling value of 78 (the median of the automatic scaling values), and the bottom row using the automatic scaling value (which fixes the output noise level to 0.2). For the bottom row, the determined scaling values are given as an inset for the different galaxies. The columns show three different galaxies, which from left to right have an increasing $S\/N$ (and thus an increasing scaling value).}\n \\label{fig-scauto}\n\\end{figure}\n \nSo far, we discussed how the input of the machine learning (the g-band image) was processed. Using GSWLC 2, we combined the Bayesian estimate of the stellar mass with a Bayesian estimate of $L_g$ to produce our target $M\/L$. The Bayesian luminosities were taken directly from the GSWLC SED models. A flat prior over the parameter range of the model grid is used, so the Bayesian values are likelihood-weighted averages. Contrary to a least $\\chi_r^2$ method, this allows us to get an uncertainty on $M\/L$ for each galaxy. It should be noted that there is little difference between best-model (i.e. least $\\chi_r^2$) and Bayesian estimates of the $M\/L$, since stellar mass is one of the parameters that can be derived most accurately from SED fitting \\citep{conroy-sedfit-review}. To further improve the $M\/L$ estimate, GSWLC uses a two-component star-formation history (SFH), which allows for a larger old stellar component. The current SFR then only fixes the young component, without constraining the older stellar population (which happens for a single component SFH). This greatly reduces the outshining bias \\citep{gswlc}.\n \n \\begin{figure}\n \t\\centering\n \t\\includegraphics[width=\\hsize]{figures\/ml_hist.pdf}\n \t\\caption{Histogram of the $M\/L$ values from GSWLC 2, after applying the threshold $D_{25} > 0.4$ arcmin. The galaxies with $\\chi_r^2 >= 5$ were not used for the machine learning. From the remaining galaxies, $59\\,637$ were used for training, $6\\,627$ for validation and $7\\,363$ galaxies for testing, as described in Sect.~\\ref{ssec-optimization-setup}. These samples were randomly drawn, and hence their distribution is similar.}\n \t\\label{fig-ml-hist}\n\\end{figure}\n \nOur sample so far is only limited by the minimum angular size ($D_{25} > 0.4$ arcmin), which results in 84\\,723 galaxies. We found that the distribution on $M\/L$ was quite broad, with some galaxies having $M\/L < 0.1$ and others having $M\/L > 10$ (all $M\/L$ are given in solar units). Most of these outliers can however be removed by setting an upper limit on the fitting $\\chi_r^2$ (i.e. the goodness of the CIGALE fit). A large $\\chi_r^2$ means that even the best model did not fit the observed fluxes well, and hence the resulting properties can be inaccurate. These high $\\chi_r^2$ objects are possible mismatches between optical and UV sources, or sources where the UV was compromised by a lower resolution. We decided to use only galaxies for which the $\\chi_r^2$ was below 5. This significantly reduced the number of outliers: the number of galaxies with a $M\/L$ below 0.1 is now 2 (from 36), while 20 galaxies have a $M\/L$ above 10 (from 50). These two criteria ($\\chi_r^2$ and angular resolution) result in our final sample, which contains 73\\,627 galaxies. This sample has a minimum, median, and maximum $M\/L$ of 0.09, 2.7 and 16.6, while without the $\\chi_r^2$ cut-off we had a minimum and maximum $M\/L$ of 0.04 and 30.8 respectively. The distribution of $M\/L$ can be seen in Fig.~\\ref{fig-ml-hist}, for the different subsamples (see Sect.~\\ref{ssec-optimization-setup}). We clearly see a bimodality, which (after inspecting the individual images) roughly correspond to elliptical galaxies for high $M\/L$ and disk galaxies for low $M\/L$. This distribution is specific for our sample: the low $M\/L$ spirals tend to be less luminous and hence they more often fall outside our selection criteria (both $D_{25}$ and the brightness cut from GSWLC). Our sample has a minimum, median, and maximum pixelsize of 0.08 kpc, 0.44 kpc, and 2.44 kpc respectively. The $D_{25}$ criterion selects mostly the more nearby galaxies, so the median redshift is now 0.05. The median seeing for the SDSS g-band is 1.4 arcsec.\n \n\\subsection{Optimization setup}\n\\label{ssec-optimization-setup}\n\nIn order to learn from the data, the machine learning algorithm minimizes an optimization objective (also called a \\textit{loss function}). Since we have access to a Bayesian $M\/L$ as well as its uncertainty, we decided to use a L1 loss that takes into account the uncertainty on $M\/L$. We denote it with $\\mathcal{L}_1$ to distinguish it from the standard L1 without uncertainty. It is defined as follows:\n\n\\begin{equation}\n\t\\mathcal{L}_1 = \\frac{1}{N}\\sum_{i = 1}^N \\left|\\frac{\\Upsilon_{\\textrm{pred}, i} - \\Upsilon_{\\textrm{true}, i}}{\\Delta \\Upsilon_{\\textrm{true}, i}}\\right|.\n\\label{eq-l1loss}\n\\end{equation}\n\n$\\Upsilon_{\\textrm{true}, i}$ denotes the `true' $M\/L$ for the $i^{\\textrm{th}}$ galaxy, which is the Bayesian estimate from GSWLC. $\\Delta\\Upsilon_{\\textrm{true}, i}$ is the corresponding Bayesian error, and $\\Upsilon_{\\textrm{pred}, i}$ is the value predicted by our machine learning method. $N$ is the number of galaxies in the considered set. We usually define a separate $\\mathcal{L}_1$ for the training, validation, and test set (see below). Optimizing $\\mathcal{L}_1$ is equivalent to optimizing a weighted mean absolute error (MAE), with the weights defined as $w_i = 1 \/ \\left( \\Delta \\Upsilon_{\\textrm{true}, i} \\right)$. Since $\\Delta \\Upsilon_{\\textrm{true}, i}$ is derived from the likelihood over the model grid in CIGALE, and does not take into account systematic uncertainties, some galaxies can have a very low error. In order to prevent a few galaxies from dominating the weights, we used a minimum relative error on the Bayesian $M\/L$ of 5\\% (this affects about a quarter of our sample). Galaxies with a high $M\/L$ typically have a larger $\\Delta \\Upsilon_{\\textrm{true}, i}$. We first experimented with a squared loss ($\\mathcal{L}_2$), but found that this was dominated by a few outliers (mostly samples with a low $M\/L$ and hence a low error on $M\/L$). The $\\mathcal{L}_1$ loss ensures that we focus more on the general trend \\citep{mae}. In Appendix~\\ref{app-loss-functions}, we experiment with different loss functions, and show that our results still apply for other common loss functions (such as the L2 variant $\\mathcal{L}_2$). The $\\mathcal{L}_1$ loss performs well on a range of metrics. We will use the term `loss' to describe the optimization criterion on the training set, while `metric' is used for how well the predictor performs on the test set.\n\nWe split our sample randomly in three parts, thus creating subsets that are representative of the whole $M\/L$ distribution (see Fig.~2). First, 10\\% is kept apart as a test set (7\\,363 galaxies). From the remaining set, another 10\\% is split off as a validation set (6\\,627 galaxies). The other 59\\,637 make up the training set. The goal of the algorithm is to minimize $\\mathcal{L}_1$ (the loss) on unseen examples. The main way this is accomplished is by minimizing $\\mathcal{L}_1$ on the training set. However, care has to be taken that the algorithm does not \\textit{overfit}. Overfitting happens when the model is too complex, and the behaviour of individual training samples is learned (instead of the general trend). The result is a low training set loss but a high test and validation set loss. The purpose of the validation set is to prevent overfitting. We optimize the algorithm's hyperparameters (which determine the model complexity), including the duration of training, by using the ones that produce the lowest validation set loss. This means that the validation set is no longer an unbiased estimate of how our algorithm performs on unseen examples, which is why we set apart a test set. This is a typical setup for machine learning \\citep{ml-goodfellow, ml-python}.\n \n \n\\section{Machine learning}\n\\label{sec-ml}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=17cm]{figures\/pipeline.pdf}\n\t\\caption{Schematic overview of the machine learning pipeline. The black arrows denote the order in which properties are derived from each other. The pink boxes show the optimization objective: three CNNs are optimizing the Galaxy Zoo 2 probabilities, using a mean squared error (L2) loss. The LightGBM and the fourth CNN optimize $M\/L$ according to a $\\mathcal{L}_1$ loss. The initial weights of this last CNN were set to the final values of the first CNN (pretraining). The boxes with the bright green outline were used as features for the LightGBM, after disregarding the ones that always were zero (dead neurons). For each of the neural layers or blocks, the output dimension is provided on the left.}\n\t\\label{fig-pipeline}\n\\end{figure*}\n\n\nOur algorithm can be subdivided in two parts. The first part consists of four CNNs \\citep{lecuncnn} which are trained to detect morphology information. The second part is a gradient boosting machine \\citep[GBM;][]{gradient-boosting}, more specifically Microsoft's LightGBM \\citep{lightgbm}. The GBM combines the morphology information from the CNNs with other information, such as redshift and total image luminosity, in order to predict the $M\/L$. A schematic overview of the complete pipeline can be found in Fig.~\\ref{fig-pipeline}. For more information on the machine learning terminology used here, we refer to Appendix~\\ref{app-terminology}.\n\nThe benefit of using this two-part algorithm is that the task of predicting $M\/L$ from the images is split in two easier tasks. The first part detects what features are present in the image (spiral structure, a bar, a possible merger, etc.). The second part then determines how this morphological information correlates with $M\/L$. We have tried using a single CNN trained on $M\/L$, but this often got stuck in local minima, predicting an average $M\/L$ for all samples. Using this two part algorithm also allows us to better interpret the results, since we can directly correlate the $M\/L$ with the morphology features.\n\n\n\\subsection{CNN - detecting morphology features}\n\\label{ssec-cnn}\n\nCNNs are a type of neural network that make use of the 2D image structure. It consists primarily of convolutional layers, each having multiple convolutional kernels (also called filters). These kernels are trained through gradient based optimization, in order to minimize the training loss. In our networks, most kernels are of size $3\\times 3$; the number of trainable parameters is drastically reduced compared to fully connected layers. The kernels in early layers detect simple features such as edges. The implicit assumption in CNNs is translational invariance: a kernel that detects a feature in one part will detect the same feature in other parts of the image. Throughout the architecture, the image typically gets downscaled, giving rise to higher level features (which in our case can learn to detect spiral structure, bulges, bars, etc.). The final layers are often fully connected (also referred to as dense), combining all features into the final prediction. \n\nWhile the final goal is to learn $M\/L$, we started with training our networks on the morphology information from Galaxy Zoo 2 \\citep[GZ2;][]{gz2,gz2-hart}. Since this made use of the SDSS DR7, we crossmatched GZ2 with our catalogue and used the 58\\,966 galaxies ($\\sim 80$\\% of our sample) for which the sky separation was less than 3.6 arcsec. This sky separation was chosen to cleanly separate our matches (>~99\\% of which are closer than 1 arcsec) from possible mismatches (>~99\\% separated by more than 10 arcsec). While we could probably find a one-to-one relation between each of their DR7 and our DR12 galaxies, it is beneficial to train the morphology on a subsample, in order for the GBM to also learn on training samples for which the morphology is known less precisely (the GBM trains on the full training set, but the CNNs are only trained on the subset that has a GZ2 match). Unlike past endeavors to predict morphology information from GZ2 \\citep{dieleman,sanchez}, our CNN only uses the g-band image. \n\nGZ2 contains 11 questions, with 37 answers in total. We decided to use the weighted vote fractions as probabilities for each answer. We did not use the distance debiased vote fractions, since the GBM has access to redshift and can apply any necessary corrections. Since some questions are only answered after particular answers of previous questions, we converted the weighted vote fractions to unconditional probabilities (i.e. multiplying by the probability of the question being asked). For example, answer four gives the probability of being an edge-on disk, which has to be smaller than or equal to the probability of the galaxy having a disk or feature (answer two, the parent question). For a list of all GZ2 questions and answers, see Fig.~2 of \\citet{gz2-hart}. The last dense layer of all networks have a ReLU activation, making sure the output for an answer is larger than or equal to zero (this is a regression approach also taken in \\citealt{dieleman}). A post-processing layer then takes care of the normalization. First, all answers for a particular question have to sum to one. Then, all answers for that question are multiplied by the estimated probability of that question being asked, determined by the features from higher up answers. This way, the network automatically produces valid unconditional probabilities. All steps of the post-processing layer are differentiable.\n\nInstead of using a single CNN, we used an ensemble of CNNs. Different network architectures will make different errors, and combining the extracted features leads to more robust results \\citep{ensembling}. Since the purpose of the CNNs is to detect the morphology, we used these different CNNs as input to our GBM. The GBM can learn in which scenarios a particular CNN is more accurate than another, and can make use of the combined information. Our final model is based on five extracted feature layers from four different networks. Different setups can lead to similar results, and it might be possible to significantly simplify the setup without too much degradation of the test $\\mathcal{L}_1$. The first architecture is the ResNet50, part of the residual learning framework which won the ILSVRC2015 competition \\citep{resnet}. The residual blocks ensure that only residuals from the previous layer have to be learned, making it possible to build much deeper networks. The second architecture is Xception \\citep{xception}, which was based on an inception architecture \\citep{inception}. The idea is to separate spatial features from depth (channel) features by doing multiple convolutions in parallel, starting from a pointwise convolution. These two networks produce state of the art results on many imaging datasets. We applied the transfer learning technique, starting the network weights from their Imagenet values \\citep{imagenet}. We used the keras python library\\footnote{https:\/\/keras.io\/}, in which these models are already implemented. We only kept the convolutional part, after which we applied global average pooling, a dense-256 layer (i.e. a fully connected layer with 256 neurons) with ReLU activation, followed by a dense-37 layer (matching the 37 answers in GZ2). This was then followed by the probability normalization layer, described above. The optimization objective was to minimize the L2 loss (regular RMSE) of the predicted and ground truth probabilities. The ResNet50 architecture used $197\\times 197$ images as input (the minimum required), while the Xception architecture used $128\\times 128$ images. Prior to the training of these networks, all training samples are scaled to the corresponding resolution (pixel area interpolation for shrinking, bicubic interpolation for zooming), with all networks keeping the same field of view per galaxy. Since these networks are pretrained on ImageNet, which has three input colour channels, we duplicated each image across the three channels to avoid changing the architecture.\n\nA third CNN architecture is a more traditional, shallow network (further called the \\textit{custom} network). It consists of 4 convolutional layers followed by two fully connected layers. The number of channels (depth) in the consecutive layers is: 32, 64, 128, 128, 512, 37. The first three convolutional layers are followed by $2\\times2 $ max pooling, after which dropout is applied \\citep{dropout}. The last convolutional layer is followed by a global max pooling but no further dropout. The first and second convolutional kernels are $5\\times 5$ and $4\\times 4$, respectively, and the last two convolutional layers are $3\\times 3$. No zero padding is applied. This architecture is inspired by \\citet{dieleman}, the main differences being that we only have one input channel and that we do not use their view preprocessing pipeline. The input dimensions are $69\\times 69$. Since there is no pretrained variant of this network, we used Glorot uniform random initialization \\citep{glorot-initialization}.\n\nFor these first three networks, we used the 37 estimated GZ2 answer probabilities as features for the GBM. We then used the ResNet50 architecture to extract more features. First, we extract the 2048 features that followed the convolutional part (before the fully connected layers). Furthermore, we took the whole architecture and replaced the probability normalization layer by a dense-1 layer. This network was then further trained to predict $M\/L$ (minimizing $\\mathcal{L}_1$). This again is a form of transfer learning: we pretrain the network on morphology, and then train on $M\/L$. This makes training easier, and we experienced fewer problems with local minima. From this retrained network, we extract the 37 features from the next to last layer, which no longer directly correspond to the 37 answer probabilities (although they are primed on them). Just like the other networks, these features only make use of the log-scaled (Equation \\ref{eq-scauto}) image, without any extra input such as luminosity or redshift. This means that they are still purely morphological features (i.e. depending only on galaxy structure), even though they do not have a clear interpretation like the GZ2 probabilities do. We will refer to the four CNNs as CNN 1, 2, 3 and 4, where we use the order in which they appear in Fig.~\\ref{fig-pipeline} (from left to right).\n\nIn order to make the networks generalize better, we applied data augmentation at training time. The images were randomly rotated (between 0 and 360 degrees), zoomed (between 0.7 and 1.3), and flipped (horizontal and vertical). This means that for every pass through the training set (epoch), the networks see slightly different images. We used the Adam optimizer \\citep{adam} with Nesterov's momentum. We applied a factor of 0.3 learning rate decay when the validation loss did not improve for 4 epochs, and stopped training after 30 epochs (since the validation loss did not seem to improve further). \n\n\\subsection{GBM - combining all information}\n\\label{ssec-gbm}\n\nSo far, the different CNN architectures produced the following morphological features: the custom CNN, ResNet50 and Xception each produce 37 GZ2 features, the ResNet50 provides 2048 features from the last convolutional layer, and a retrained (on $M\/L$) ResNet50 gives 37 features which are primed on GZ2. In total, these account for 2196 features. In addition, the GBM uses features extracted from the images. The following luminosity features are used, where we used the corresponding flux and multiplied by $4\\pi D^2$ (where $D$ is the distance in Mpc): sum (over all pixels), mean, maximum, minimum, standard deviation, central pixel, mean around central pixel ($5\\times 5$ pixels around the centre), the original image size, and the scaling value (as described in Sect.~\\ref{sec-methods}). After the log-scaling (which also scales the images between 0 and 1), we also extract some image statistics: the sum, mean, standard deviation, central pixel value, and mean around the central pixel. We also added two features which required extra metadata: the redshift and pixel size (in kpc). These allow the network to distinguish between a faint but nearby galaxy and a bright, distant galaxy. After removing 17 features that were always zero (dead neurons, 10 from CNN 3 and 7 from the inner part of CNN 1), we are left with 2195 features. The units of all the features are of no importance at this point: decision trees -- on which the GBM is based -- are scale invariant.\n\nGradient boosting traditionally makes use of decision trees as a base classifier. The trees are built sequentially, where each tree learns from the mistakes made from previous ones (as with ResNets, we learn residuals). LightGBM makes use of several optimizations compared to traditional gradient boosting. Since the morphology features were already extracted via the CNN, training was fast (around five minutes on a dual-core CPU). This allowed us to do 5-fold cross-validation in order to optimize the hyperparameters. We ended up using trees with 40 leaves, with a minimum of 150 samples per leaf. Each tree only had access to a (random) subset of 40\\% of the features, and 80\\% of the data (bagging). The validation set was used for early stopping, in order to prevent overfitting. \n \n \n\\section{Results and discussion}\n\\label{sec-results}\n\n\\subsection{Single band predictions}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\hsize]{figures\/g_predictors.pdf}\n\t\\caption{ \\textit{Left:} Comparing the predicted $M\/L$ to the true $M\/L$ for a gradient boosting machine without morphology (top) and with morphology (bottom). Both predictors only make use of the g-band. The colour of the points is a 2D gaussian kernel density estimate. \\textit{Right:} histogram of dex error (top) and $\\mathcal{L}_1$ (bottom) for each galaxy. Purple is used for the predictor without morphology, while green is used for the predictor that includes morphology. For both quantities, closer to zero is better, positive numbers denote overpredictions, and negative values are underpredictions. The figure only includes galaxies from the test set. The dashed lines show 0.15 dex errors.}\n\t\\label{fig-g-predictors}\n\\end{figure}\n\nOur first goal is to investigate how good a single g-band image can constrain $M\/L$. As described in Sect.~\\ref{ssec-gbm}, our GBM combines the morphology information from the CNNs with other information (luminosity statistics, distance, and pixelsize). To evaluate the benefit of morphology, we compare this pipeline to a similar predictor that does not make use of morphology, nor any other resolved data (such as the luminosity features and the pixel size). Instead, we use two features: the g-band luminosity $L_g$ (calculated from the SDSS modelmag flux and distance), and the redshift. This reference method is shown in the top left panel of Fig.~\\ref{fig-g-predictors}, while our method (including morphology) is shown in the bottom left panel. These panels plot the predicted $M\/L$ against the ground truth (the Bayesian $M\/L$ from GSWLC). Even though the reference method only uses $L_g$ and redshift, it does not perform all that bad. We note that this works differently than using a single 3.4~$\\mu$m band, constant $M\/L$ assumption. This reference estimator can use the trend that low $M\/L$ spirals tend to have fewer stars (and hence they are less luminous) than a typical high $M\/L$ elliptical \\citep{review-hubble-sequence}. As discussed in Sect.~\\ref{ssec-interpretation}, it can also use the redshift feature to make use of Malmquist bias.\n\nIt is however clear that including morphology features clearly improves the results. The test set $\\mathcal{L}_1$ (Eq.~\\ref{eq-l1loss}) improves from 4.52 to 2.29. If we disregard the Bayesian uncertainty on the ground truth, we can use the root mean square logarithmic error (RMSLE), which improves from 0.227 dex to 0.124 dex. Including morphology also leads to less biased estimates, as seen on the right panels of Fig.~\\ref{fig-g-predictors}. For our estimator, 85.4\\% of the test set falls within 0.15 dex, while this is only 55.1\\% for the reference method without morphology. The reference method seems to be biased towards underpredictions (although there is a long tail towards overpredictions extending outside the histogram). This is mainly caused by the lack of predictions above 3.3: it seems like these high $M\/L$ cases are not easily found by $L_g$ and redshift alone. We will see in Sect.~\\ref{ssec-interpretation} that these galaxies mainly correspond to edge-on disks.\n\nInterestingly, the outliers in $\\Upsilon_{\\textrm{pred}, i} \/ \\Upsilon_{\\textrm{true}, i}$ do not necessarily match the outliers regarding $\\mathcal{L}_1$. For example, most underpredictions have a large error on $\\Upsilon_{\\textrm{true}, i}$. The largest outliers in $\\mathcal{L}_1$ (when including morphology) are overestimations. These have a low actual $M\/L$, which often results in a lower error on $M\/L$, and hence these datapoints are punished harder by our loss. A different loss function will weight galaxies differently, but we found that this does not change our conclusions (see Appendix~\\ref{app-loss-functions}). \n \nIncluding morphology allows us to detect galaxies with $M\/L > 3.3$. However, the highest $M\/L$ that is predicted is 5.6 (while the ground truth values run up to 16.6). For one, there are only a limited number of these extreme cases, and it is safer to predict a lower $M\/L$. Moreover, this suggests that there is no easy way to detect these samples from the g-band images alone. These samples also have a larger error on $M\/L$, and hence a more conservative estimate is not punished as hard for these samples.\n \n\\subsection{Interpretation}\n\\label{ssec-interpretation}\n\n\\begin{figure}\n \t\\centering\n \t\\includegraphics[width=\\hsize]{figures\/permutation_importances.pdf}\n \t\\caption{Feature importance ranking, by the amount the validation $\\mathcal{L}_1$ increases after permuting that feature. The CNN 4 features come from the ResNet50 that was retrained on $M\/L$ (the rightmost network in Fig.~\\ref{fig-pipeline}). All luminosity features are derived from the raw (star subtracted but not scaled) images, as described in Sect.~\\ref{ssec-gbm}. \\textit{Top:} standard predictor as described in Sect.~\\ref{sec-ml}, using all features. \\textit{Bottom:} a (freshly trained) predictor which does not make use of CNN 4.}\n \t\\label{fig-feat-importances}\n\\end{figure}\n\nOne of the useful properties of our pipeline is that it decouples morphology extraction and $M\/L$ prediction. The morphology detection by a CNN can be understood by inspecting the different layers. The first layers learn simple features such as edges, while deeper layers can learn to detect spiral arms, bars, or other features \\citep{dieleman}. The LightGBM can be interpreted by looking at the feature importances. These are presented in Fig.~\\ref{fig-feat-importances}. We have used \\textit{permutation importances}, which proved to be a robust feature importance measure in the study of \\citet{permutation-importance}. A certain feature's importance is calculated by permuting all observations of that feature, calculating the validation set $\\mathcal{L}_1$, and subtracting the non-permuted $\\mathcal{L}_1$ from this. The permuting leads to randomizing that feature without losing the distribution's properties. If the GBM heavily relies on a particular important feature, the permutation should increase $\\mathcal{L}_1$ considerably, leading to a larger importance.\n\nFrom the top panel of Fig.~\\ref{fig-feat-importances}, we can see that the ten most important features contain a mix of luminosity features, distance related features, and morphology features. The morphology features which are used the most are the ones from CNN 4, which was retrained to optimize $M\/L$. As discussed further below, these tend to correlate directly with $M\/L$. They no longer directly correspond to the GZ2 probabilities, but since CNN 4 only uses the log-normalized image, its features only depend on the galaxy's morphology. CNN 4 essentially eases the work for the GBM by moving part of the $M\/L$ prediction to that CNN. Since this hinders the interpretability of the model, we also show the feature importance for a GBM that does not make use of CNN 4 (bottom panel of Fig.~\\ref{fig-feat-importances}). This predictor is slightly worse, with a test $\\mathcal{L}_1$ of 2.41 (instead of 2.29). Although simple GZ2 questions such as the presence of galaxy features (e.g. spiral arms) correlate well with $M\/L$ (see below), they are not part of the most important features. Instead, the luminosity statistics seem to be more robust features. Since ellipticals and spirals have different brightness profiles, the luminosity statistics (such as the ratio of the mean centre luminosity and total luminosity) do provide morphological information. The GZ2 features that are most important (if CNN 4 is not present) look for a lack of bulge, and for irregular galaxies. Apparently, these two features can not be easily substituted by luminosity statistics.\n\nDue to the large number of features, the model can be resistant against the removal of some features. For example, if we remove the total luminosity feature (sum over all pixels), there is still the mean around the central pixel luminosity which can serve as a proxy. So if we train the LightGBM after leaving out the total luminosity feature, the test set $\\mathcal{L}_1$ only increases by 0.02. This shows an important difference between the permutation importance and so-called drop-out importance (increase in $\\mathcal{L}_1$ after retraining the model without that feature). If we have highly correlated features, retraining the model without one of those features will allow the similar features to make up for its lack. Our permutation importance measures something different: how important is that feature in the current model. We found that by only using the top 50 features (and redoing the cross-validation), the results stay the same. The computational time, however, decreases dramatically, with training only taking about 40 CPU seconds (from 16.7 CPU minutes) and evaluating on the training set taking only 4.2 CPU seconds (instead of 20 CPU seconds), using two threads on a Intel i5 processor.\n\n\\newcommand\\Tstrut{\\rule{0pt}{2.6ex}} \n\\newcommand\\Bstrut{\\rule[-0.9ex]{0pt}{0pt}}\n\\newcommand{\\TBstrut}{\\Tstrut\\Bstrut}\n\\begin{table}\n\t\\caption{Test set $\\mathcal{L}_1$ for a LightGBM model that uses only the features which are checked. CNN 1-3 refers to the first three CNNs (from the left) in Fig~\\ref{fig-pipeline}, all of which are only trained on the Galaxy Zoo probabilities. CNN 4 is the ResNet50 which was retrained on $M\/L$. When leaving out the distance feature (redshift and pixelsize), we also replace all luminosity features by the corresponding flux features. The baseline $\\mathcal{L}_1$ (i.e. minimizing $\\mathcal{L}_1$ when predicting a single value) is 6.52.} \n\t\\label{tab-feature-removing} \n\t\\centering \n\t\\begin{tabular}{c c c c c} \n\t\t\\hline\\hline \n\t\tCNN 1-3 & CNN 4 & Luminosity & Distance & Test $\\mathcal{L}_1$ \\TBstrut \\\\ \n\t\t\\hline \n\t\t\\checkmark & \\checkmark & \\checkmark & \\checkmark & 2.29 \\Tstrut \\\\\n\t\t& \\checkmark & \\checkmark & \\checkmark& 2.37 \\\\\n\t\t\\checkmark & & \\checkmark & \\checkmark & 2.41 \\\\\n\t\t\\checkmark & \\checkmark & & \\checkmark& 2.32 \\\\\n\t\t\\checkmark & \\checkmark & \\checkmark & & 2.67 \\\\\n\t\t&& \\checkmark& \\checkmark & 3.38 \\\\\n\t\t\\hline \n\t\\end{tabular}\n\\end{table}\n\n\\begin{figure*} \n\t\\centering\n\t\\includegraphics[width=17cm]{figures\/scaling_relations.pdf}\n\t\\caption{Influence of a few features on $M\/L$, for the training set. Due to the large number of datapoints, we use (hexagonal) bins. The opacity of each bin corresponds to the number of galaxies in the bin, in a non-linear way (ensuring that lower densities are still visible). The total luminosity denotes the total g-band luminosity in $L_{\\odot, g}$, and is shown in log space. For the top left panel, the $P_{\\rm{feature}}$ feature from CNN 1 estimates the GZ2 probability of the galaxy having a feature or disk. For the bottom left panel, the morphology is determined from the last layer of CNN 1. A galaxy is defined as irregular if the predicted probability of being irregular is larger than 20\\%, it is considered edge-on if it is not irregular but has a probability of being edge-on larger than 40\\%, and it is a feature or disk if the corresponding probability is larger than 40\\% (but it is not in the previous two categories). The ellipticals are the remaining datapoints. }\n\t\\label{fig-scaling-relations}\n\\end{figure*}\n\nMaybe even more important is what happens when we leave out a group of features. The results of such an ablation study can be seen in Table~\\ref{tab-feature-removing}. We see that a reliable distance estimate (in our case the redshift from SDSS) is quite important. It should be noted that we need a distance estimate to go from $M\/L$ to stellar mass anyway. The luminosity is less important in order to estimate $M\/L$. This does not contradict with the total luminosity being an important feature: as discussed further below, the luminosity features allow the machine to roughly distinguish between high and low $M\/L$. However, in the absence of luminosity features the morphology features can take their place. We also see that CNN 4 and CNN 1-3 complement each other well: we see a clear improvement when all CNNs are combined. The results are clearly worse when no CNN is present (bottom row, $\\mathcal{L}_1 = 3.38$), although this predictor still has access to the resolved luminosity features, allowing it to outperform the reference method (top panel of Fig.~\\ref{fig-g-predictors}; $\\mathcal{L}_1 = 4.52$).\n \nOf course, our model does not use the actual GZ2 probabilities as input: this ensures that no human interaction is required when making new predictions. CNN 1 to 3 exist to estimate the GZ2 probabilities. These estimations are not perfect, and we might wonder what the effect of these errors might be on the final prediction. To determine this, we replaced the custom network (CNN 3) by the actual GZ2 cumulative probabilities. The resulting $\\mathcal{L}_1$ of this cheating model is 2.25 (compared to 2.29 for the standard estimator). This shows that only minor improvements can be made by further improving the GZ2 predictions. It also shows that we can trust our CNNs to make good morphology detections, and hence that decisions made regarding the CNN pipeline are not negatively impacting our further analysis.\n\nWe can see the effect of the different features by looking at their influence on $M\/L$. In Fig.~\\ref{fig-scaling-relations}, we show how the target $M\/L$ correlates with some of the features. These correlations are the driving force behind the machine learning. In the top left panel, we can see that the luminosity feature can distinguish roughly between high $M\/L$ and low $M\/L$ galaxies: galaxies with $L_g < 10^{6.5} L_{\\odot, g}$ tend to have a low $M\/L$, while galaxies with $L_g > 10^{7.5} L_{\\odot, g}$ tend to have a high $M\/L$. With the help of the Galaxy Zoo probabilities estimated by the ResNet50 architecture (CNN 1), we can further distinguish between the two groups even in the case of intermediate luminosities. In this case, the GZ2 probability $P_{\\textrm{feature}}$ is used as a colour scale, where $P_{\\textrm{feature}}$ gives the probability of a morphological feature or disk being present (in contrast to being smooth, or a ``star or artifact''). For constant luminosity, galaxies with features tend to have a lower $M\/L$. One exception is the cloud of feature galaxies with $M\/L > 4$, which is explained in the next paragraph. Looking at lower $M\/L$ (< 2), we see that the probability of the galaxy having a feature increases with luminosity. This can be attributed to a distance-dependent classification bias \\citep{gz2-hart}. Essentially, the spiral structure is hard (or impossible) to see for fainter, more distant galaxies (with a lower $S\/N$). The algorithm can detect these low $S\/N$ galaxies (through luminosity features, redshift and scaling value), and react by predicting a lower $M\/L$ than is typical when $P_{\\rm{feature}}$ is low. \n\nThe bottom left panel is similar to the top left, but has combined three Galaxy Zoo features ($p_{\\textrm{feature}}$, $p_{\\textrm{edge-on}}$ and $p_{\\textrm{irregular}}$) to create four categories. We notice that irregulars have a very low $M\/L$, probably because a merger-triggered star formation burst leads to a young stellar population \\citep{merger-starburst}. As expected, there is the bimodality between disk galaxies and ellipticals, where ellipticals are believed to be more evolved objects with an older stellar population (and hence a higher $M\/L$). The big exception here is edge-on disks, which seem to have a very high $M\/L$. This is the result of our definition of the stellar luminosity $L$, where we directly multiplied the flux by $4\\pi D^2$. Edge-on disks are more attenuated and hence we receive less light, resulting in a higher $M\/L$. While our definition ignores anisotropy (the luminosity now depends on the viewing angle), the $M\/L$ only serves as a bridge to estimate the total stellar mass. The CIGALE models assume no particular geometry. We have inspected the influence of $p_{\\textrm{edge-on}}$ on the total stellar mass $M$, and found no clear trend: these two variables have a Spearman correlation coefficient of only 0.03. This suggests that we can still estimate the stellar mass, even when attenuation in edge-on disks causes the observed luminosity to be lower than the intrinsic luminosity (averaged over all directions).\n\nThe top right panel of Fig.~\\ref{fig-scaling-relations} clearly shows the effects of Malmquist bias. Our main selection criterion is $D_{25} > 0.4$ arcmin. At higher redshift, we only include very large (and thus often luminous) objects. These tend to have a high $M\/L$. The GBM makes use of this bias by predicting a high $M\/L$ for higher redshift galaxies. The result is that for our sample, the predictions are actually more accurate for further away galaxies. This stresses the importance that the test set (or any set on which the machine learning is evaluated) should have the same selection criteria as the training set. We learn by example, and so the assumption is that new samples are similar to the training set. \n\nThe CNN 1 features from the left two panels are actually not often present in the trees of the GBM, due to the presence of CNN 4. CNN 4 was trained to correlate more directly with $M\/L$, as seen in the bottom right panel. The result is that the CNN 4 features are no longer directly interpretable. Leaving out CNN 4 increases the test $\\mathcal{L}_1$ by only 0.12, as seen from Table~\\ref{tab-feature-removing}, so the relations from the left two panels do give us some insight in the behaviour of the machine learning. \n\n\\subsection{Using colour and morphology}\n\\label{ssec-gi-morph}\n\n\\begin{figure*} \n\t\\centering\n\t\\includegraphics[width=17cm]{figures\/gi_predictors.pdf}\n\t\\caption{\\textit{Top left, top right and bottom left}: predicted $M\/L$ vs ground truth for $g-i$ power law, global luminosity GBM and the morphology GBM respectively. Only the galaxies in the test set are shown. \\textit{Bottom left}: histogram of dex errors. }\n\t\\label{fig-gi-predictors}\n\\end{figure*}\n \nSo far, we have shown that it is possible to make reasonable $M\/L$ (and hence stellar mass) predictions with observations in only one band (and ideally a distance estimate). This of course does not replace traditional stellar mass methods, but shows that the morphology of a galaxy does provide valuable information. Now we can wonder: does morphology give the same information as colour, or is there a benefit in using morphology in addition to global g and i luminosities? To investigate this, we added the g-band luminosity $L_g$, i-band luminosity $L_i$ and g - i colour $L_g\/L_i$ as features to the LightGBM. These are derived from the SDSS modelmags, which were also used for the SED fitting \\citep{gswlc, gswlc2}. After training has completed, we selected the 50 features that were used the most in the GBM, and retrained using only those. The result is shown in the bottom left panel of Fig.~\\ref{fig-gi-predictors}. The resulting test set $\\mathcal{L}_1$ is 1.12. We compare this against a standard method to estimate the $M\/L$ from a single colour: a power law between $M\/L$ and $g-i$ colour \\citep{zibetti2009}. The two power law parameters were fit on the training set, minimizing $\\mathcal{L}_1$ (just like the machine learning). The result is shown in the top left panel of Fig.~\\ref{fig-gi-predictors}, although the test metrics exclude two datapoints with extreme mispredictions. This already shows one of the drawbacks of this method: it is not applicable if the two fluxes are `incompatible' (e.g. due to large uncertainties or observational artifacts). To make a fairer comparison, we also compare against a more sophisticated single colour method. Instead of assuming a power law, we used a LightGBM regressor (which was also used for the morphology method). This method made use of four features: redshift, $L_i$, $L_g$ and $L_g \/ L_i$. The last feature is beneficial since the individual decision trees only split based on one feature. This method, which does not make use of morphology, is shown in the top right panel of Fig.~\\ref{fig-gi-predictors} and achieves a $\\mathcal{L}_1$ of 1.26. \n \nThere is a clear improvement when going from a power law to a GBM. The power law is unable to make a good fit for both low and high $M\/L$ (low and high $M\/L$ refer to the bimodality also seen in Fig.~\\ref{fig-ml-hist}). There's also a large number of outliers, and these influence the fit to keep the $\\mathcal{L}_1$ under control. A GBM can easily improve on this: every point in the feature space is assigned a $M\/L$ which minimizes the corresponding $\\mathcal{L}_1$, which hence avoids the bias that can be seen in the power law. This can also be verified by looking at the distribution of dex errors, in the bottom right panel of Fig.~\\ref{fig-gi-predictors}. The $g-i$ power law has a strong tail towards overpredictions: 5.1\\% of galaxies have a logarithmic error larger than 0.15 dex (overpredictions), while only 1.8\\% have a logarithmic error smaller than -0.15 dex (underpredictions). For the GBM method (without morphology), only 2.3\\% have a logarithmic error outside 0.15 dex (over- and underpredictions).\n \nAdding morphology to the GBM (bottom left panel of Fig.~\\ref{fig-gi-predictors}) further reduces the test $\\mathcal{L}_1$ to 1.12, resulting in a better estimator. An important question to investigate is whether a test $\\mathcal{L}_1$ of 1.12 (with morphology) is \\textrm{significantly} better than a test $\\mathcal{L}_1$ of 1.26 (without morphology). The reported $\\mathcal{L}_1$ are the mean $\\mathcal{L}_1$ over the $7\\,363$ test samples. Hence, we can look at the distribution of the individual $\\mathcal{L}_1$. First, we did a Kolmogorov-Smirnov two-sample test, for a one-sided comparison. The null hypothesis, which states that the model with morphology does not have a significantly lower $\\mathcal{L}_1$ than the model without morphology, can be rejected with very high confidence (p-value $7.6\\times 10^{-9}$). Next, we took 100\\,000 bootstrap samples of the $\\mathcal{L}_1$ distribution of both models. The mean $\\mathcal{L}_1$ of each of these bootstrap samples has no overlap for the two models. The bootstrap estimate for the mean $\\mathcal{L}_1$ of the method without morphology is $1.26 \\pm 0.01$, while the estimate for the method with morphology is $1.12 \\pm 0.01$. Hence, we can conclude that adding the morphology features gives a significant improvement.\n \nWe notice most improvement at $M\/L > 4$, which are dominated by edge-on galaxies, as can be seen from the bottom left panel of Fig.~\\ref{fig-scaling-relations}. We can quantify the improvement for each of the morphology classes from that panel. By including morphology features, $\\mathcal{L}_1$ improves from 1.15 to 1.08 for irregulars, from 1.31 to 1.18 for feature (disk) galaxies, from 1.14 to 1.07 for ellipticals, and from 1.52 to 1.14 for edge-on galaxies (the biggest improvement). A single colour often underestimates the stellar mass for these edge-on galaxies, and morphology information can be used to prevent this. \n \nWhen comparing these single colour predictors to the g-band image predictor (bottom left panel of Fig.~\\ref{fig-g-predictors}, $\\mathcal{L}_1 = 2.29$), we see that morphology can not replace colour. It is, however, impressive that a g-band $M\/L$ predictor gets close to a $g-i$ power law, even though the power law is also fitted to this dataset. This refitting was necessary, since the original \\citet{zibetti2009} power law has a test set $\\mathcal{L}_1$ of 7.47 (mainly due to the different SPS model grid than GSWLC). Due to its easy application, a single colour power law is commonly used \\citep[e.g.][]{scpl-2, scpl-1}, but as shown here it should be used with caution. Machine learning techniques can help improve these estimates without needing to do SED fitting at evaluation time. The morphology can help assist predictions of $M\/L$. Since it is available for more nearby galaxies anyway, it can be a valuable improvement over techniques that only use global flux information (and hence dismiss all information from the resolved image).\n\n\\subsection{Morphology assisted $M\/L$ with multiple colours: limitations of the ground truth}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\hsize]{figures\/ugriz_buildup_morphology.pdf}\n\t\\caption{Comparison of the performance (test set $\\mathcal{L}_1$) of different GBM predictors. The predictors only make use of the denoted broadbands. The yellow bars only make use of global information (luminosity, distance and colours), while the green bars also make use of morphology.}\n\t\\label{fig-ugriz-buildup}\n\\end{figure}\n\nAfter looking at single band and single colour predictors, we might wonder what happens when multiple colours are available. Similar to Sect.~\\ref{ssec-gi-morph}, we train gradient boosting machines with and without morphology. For a sequence of available SDSS broadbands, the results are shown in Fig.~\\ref{fig-ugriz-buildup}. We see that the performance stagnates at an $\\mathcal{L}_1$ of 1, when the predictions are as accurate as the uncertainty on the ground truth. Including morphology improves the results for all cases, although the stagnation at $\\mathcal{L}_1 = 1$ limits the benefit when multiple broadbands are available.\n\nThe problem is that we are limited by our ground truth (i.e. the prediction target): the GSWLC $M\/L$ come from Bayesian SED fitting to \\textrm{global} fluxes. This means that the morphology can only make up for missing broadbands, but not for the uncertainties that come from using only global fluxes. It is possible to apply SED fitting pixel-by-pixel, and then sum the stellar masses of the individual pixels. The assumption that a spectrum is the sum of SSPs with a simple attenuation law applies better to individual pixels than to complete galaxies. So while pixel-by-pixel SED fitting is believed to be more accurate \\citep{unresolved-sed1, unresolved-sed2}, it is also more expensive. Pixel-matched panchromatic datasets are required, where the band with the worst resolution effectively sets the working resolution. A high $S\/N$ is required for all relevant pixels. This method is also much more computationally intensive, and hence the number of models that can be fit is limited. The result is that at the time of writing, no large pixel-by-pixel SED fitted catalogues exist. Should they become available in the future, our method can be retrained which can make it possible to beat global flux methods. \n\nAnother way to improve the ground truth is by using more information. Currently, GSWLC uses the WISE observations to estimate the total infrared luminosity $L_{IR}$ \\citep{gswlc2}. Assuming energy balance, this then constrains the total energy absorbed by the dust, allowing us to make better estimates of the unattenuated stellar spectrum. Although the uncertainty on this $L_{IR}$ estimation is only 0.08 dex, it uses just a single WISE band. This can make it troublesome for galaxies with large uncertainties on that WISE band, or for galaxies where the correction for AGN contribution leads to additional uncertainties. The best way to constrain $L_{IR}$ is still to measure it with FIR observations, and hence galaxies with FIR data will have a slightly better ground truth $M\/L$. In addition to using UV-FIR broadbands, spectroscopy can be used as an additional constraint for the SED fitting \\citep{beagle, beagle-spectroscopy, bagpipes, prospector}. Limiting the training to samples were this additional information (more broadbands and\/or spectroscopy) is available unfortunately implies that the size of the training set will be smaller.\n\nOf course, the best case scenario would be that our ground truth were the actual stellar $M\/L$. Unfortunately, there is no way to directly measure stellar mass, instead of estimating it through SPS. There is however a situation in which we know the stellar mass: cosmological simulations. With radiative transfer, it is possible to create mock observations of these simulated galaxies \\citep[e.g.][]{camps2018}. These then could serve as a good training target, since we no longer have to deal with the limitations of SED fitting. The radiative transfer treats the effects of dust rigorously (in contrast to an empirical attenuation law), and star forming regions can be treated with subgrid prescriptions. Recently, some successes have been achieved with training CNNs on cosmological simulations, while testing them on real galaxies \\citep[e.g.][]{simulation-cnn-1,simulation-cnn-2}. The main limitation of this approach is that there are still discrepancies between the observed and the simulated universes.\n\n \n\\subsection{Applications and discussion}\n\nThe success of using morphology information to predict stellar mass depends on the quality of the images. In this work, we limited ourselves to $D_{25} > 0.4$ arcmin to make sure that we have enough pixels for each galaxy. Upcoming surveys will allow for deeper and higher resolution observations, drastically increasing the number of galaxies that are well resolved. In particular, Euclid will have a very broad optical band ($r+i+z$) which is useful to get deeper images. These will be combined with ground-based photometry ($griz$) and Euclid photometry ($YJH$) \\citep{euclid}. The goal is to have 1.5 billion galaxies with very accurate morphometric information. These will be an excellent target for training and testing morphological stellar mass estimates. Besides Euclid, LSST \\citep{lsst} and WFIRST \\citep{wfirst} will also provide wide-field optical\/NIR imaging which could benefit from our method. As discussed in the previous section, the hardest but most rewarding challenge to solve will be to acquire more accurate ground truth $M\/L$, such as from pixel-by-pixel SED fitting. The morphology can then use the resolved information to improve on a global colour estimate. \n\nGSWLC 1 contains about $700~000$ galaxies and is one of the largest SED fitted catalogues to date. This already shows that even global SED fitting will be computationally challenging for Euclid's 1.5 billion galaxies, without significantly reducing the number of fitted models. A machine learning approach (with or without morphology) can be a good alternative. With the use of a single GPU, CNN evaluation is more than an order of magnitude faster than (global) SED fitting on a 100 core CPU cluster. If we train on pixel-by-pixel SED fits, we further avoid the outshining bias. So training on a small but accurate $M\/L$ subset of Euclid, and evaluating on the remaining $> 1$ billion galaxies seems promising. We found that if we train the GBM with only half of the data, the test $\\mathcal{L}_1$ degrades only slightly to 2.31 (from 2.29), confirming that the quality of the training data is more important than the quantity.\n \nOur pipeline can also be used to predict other galaxy properties, such as SFR or metallicity. For both of these properties, spectroscopy can be a valuable constraint on the ground truth. We hope that this two-step process (CNN + GBM) can further improve our understanding of which morphological properties best correlate with the physical properties of a galaxy. This can then further constrain galaxy evolution models. \n \n\n\\section{Summary and conclusions}\n\\label{sec-conclusions}\n\nWe made use of a machine learning framework to make morphology assisted $M\/L$ predictions. First, we predicted $M\/L$ from a single g-band image. The pipeline can be split in two parts: a first part estimates morphology features such as the probability of the galaxy being featureless, edge-on, merging, etc. This information is then combined with redshift, pixel size, and a few g-band luminosity features in order to predict $M\/L$. We optimized a $\\mathcal{L}_1$ loss that weights down samples with a large uncertainty on $M\/L$. Our best model has a test set $\\mathcal{L}_1$ of 2.29, and a RMSLE of 0.124 dex. The morphology from the g-band can partially make up for a lack of observed colour. These predictions are made possible because featureless ellipticals tend to have a higher $M\/L$ than galaxies with features such as spirals (left two panels of Fig.~\\ref{fig-scaling-relations}). Irregular galaxies tend to have a low small $M\/L$, while highly inclined disk galaxies tend to have a very high $M\/L$. Even though the spiral features can not be detected for more distant, dimmer galaxies, the algorithm is trained to produce unbiased results. \n\nObserving multiple bands does lead to a better constrained $M\/L$. A $g-i$ power law, recalibrated on our dataset achieves a $\\mathcal{L}_1$ of 1.90 (compared to 2.29 for our g-band only method). The $g-i$ power law has trouble fitting both small and large $M\/L$. This can be avoided by using a GBM (or other machine learning method). We find that a GBM that makes use of global $g$ and $i$ fluxes and a distance estimate achieves a $\\mathcal{L}_1$ of 1.26. Including the g-band morphology features further improves the $\\mathcal{L}_1$ to 1.12, showing that morphology information does have an added benefit over only global colours. Even though this improvement is small, we have shown that it is significant. Most of the improvement happens for edge-on disk galaxies. With global fluxes only, it is hard to distinguish a more inclined and hence attenuated galaxy from an older one (both effects lead to redder colours), and we find that the $M\/L$ tends to be underpredicted in those cases.\n\nIn future work, we hope that this machine learning framework can be trained on better target estimates for $M\/L$. Currently, our target values are derived from unresolved fluxes, limiting the benefit of our method over global colour methods. Our method can be fit to reproduce pixel-by-pixel SED fitted $M\/L$, but has less strict requirements on pixel $S\/N$, and is faster at evaluation time. \n\n\n \\begin{acknowledgements}\n We thank the anonymous referee for helpful comments which improved this paper. W.D., S.V. and M.B. gratefully acknowledge support from the Flemish Fund for Scientific Research (FWO-Vlaanderen). W.D. is a pre-doctoral researcher of the FWO-Vlaanderen. Special thanks to the Flemish Supercomputer Centre (VSC) for providing computational resources and support. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http:\/\/www.sdss3.org\/. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.\n \\end{acknowledgements}\n\n \n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Bethe--Sommerfeld conjecture is the following statement: for any $d \\geq 2$ and any periodic function $V:{\\mathbb R}^d \\to {\\mathbb R}$, the spectrum of the Schr\\\"odinger operator \n\\[\nL_V := -\\nabla^2 + V\\]\nhas only finitely many gaps. This was studied by many people with important advances in \\cite{HelMoh98,Karp97,PopSkr81,Skr79,Skr84,Skr85,Vel88}, and culminating in the paper of Parnovskii \\cite{Parn2008AHP}. One way to think about the Bethe--Sommerfeld conjecture is that any energy $E$ that is very large relative to the potential $V$ lies in the spectrum of $L_V$. Since discrete Schr\\\"odinger operators are bounded, the high-energy region is absent, so the appropriate discrete version of the Bethe--Sommerfeld conjecture lies in the region of small $V$. Discrete versions of the conjecture were proved on square lattices by Embree--Fillman in dimension $d=2$ \\cite{EmbFil2017} and by Han--Jitomirskaya in arbitrary dimensions $d \\geq 2$ \\cite{HanJit2017}. In those works, the spectrum of a discrete periodic Schr\\\"odinger operator on the square lattice $\\ell^2({\\mathbb Z}^d)$ with a small potential was shown to consist of at most two intervals. Moreover, they showed that as soon as at least one period of the potential is odd, then the spectrum is an interval, and, in the event that a gap opens perturbatively, it must happen at the exceptional energy $E=0$.\n\nMany interesting physical models occur with different underlying lattice geometries beyond the standard square lattice. One of the most prominent such models is supplied by graphene, a two-dimensional material that consists of carbon atoms at the vertices of a hexagonal lattice. The fascinating properties of graphene have led to a substantial amount of attention in mathematics and physics, see e.g.\\ \\cite{BZ2018,BHJ2018,CGPNG,DelMon2010,FW12,HKR2016,KP07,N11} and references therein. \nIn view of this, we are motivated to study the Bethe--Sommerfeld conjecture for the hexagonal lattice and for the corresponding dual lattice (the triangular lattce).\n\nIn addition to the hexagonal and triangular lattices, we also study the square lattice with next-nearest neighbor interactions, which is motivated by the extended Harper model (EHM). The EHM was proposed by Thouless \\cite{Thouless83} and has also led to a lot of study in mathematics and physics \\cite{AJM17,H17,H18,HJ17,Thouless94,JM15}; it corresponds to an electron in a square lattice that interacts not only with its nearest neighbors but also its next-nearest neighbors.\nIn the following, we will refer to square lattice with next-nearest neighbor interactions as the {\\it EHM lattice}, in order to distinguish it from the standard square lattice.\n\n\n\n\n\n\n{Let us mention in particular the closely related work \\cite{HKR2016}.} In \\cite{HKR2016}, Helffer, Kerdelhu\\'e and Royo-Letelier developed a Chambers analysis for magnetic Laplacians on the hexagonal lattice (and its dual lattice: triangular lattice) with rational flux.\nThey showed that for a non-trivial rational flux $p\/q\\notin{\\mathbb Z}$, the magnetic Laplacians on hexagonal and triangular lattices have non-overlapping (possibly touching) bands.\nThis recovers a similar feature of the square lattice \\cite{BelSim82}.\nHowever, unlike the square lattice that has no touching bands except at the center for $q$ even \\cite{VMou89}, \nthey were able to give an explicit example of non-trivial touching bands for hexagonal and triangular lattices. \nIndeed they showed that the triangular Laplacian has touching bands at energy $E=-\\sqrt{3}$ for $p\/q=1\/6$, and the hexagonal Laplacian has touching bands at energies $E=\\pm \\sqrt{3}$ and $0$ for $p\/q=1\/2$.\nTherefore, the underlying geometry is greatly responsible for the formation of touching bands.\nBut it {has remained} unclear that whether there will be other touching bands for different fluxes (and if any, what are the locations).\nIn our work we are able to give a sharp criterion of the formation of touching bands for the free Laplacians on these lattices and the EHM lattice, see Theorems \\ref{t:bsc:tri}, \\ref{t:bsc:hex} and \\ref{t:bsc:nnn}.\n\nMotivated by these models, we prove the Bethe--Sommerfeld conjecture for the triangular, hexagonal, and EHM lattices. \nSimilar to the square lattice case, we show that small perturbations of the free Laplacian may only open gaps at certain {\\it exceptional energies}.\nOur proof uses the perturb-and-count technique developed in \\cite{HanJit2017}.\nThe overall strategy is to argue by contradiction. \nNamely, we assume two adjacent spectral bands of the free Laplacian have a trivial overlap containing a single energy $E$.\nThen, we carefully choose a Floquet parameter and perturb all the Floquet eigenvalues along two different directions. \nIt is then argued that different directions lead to different counting of eigenvalues that move above\/below $E$, hence a contradiction.\nAt the exceptional energies, we are able to develop a {\\it sharp} criterion, in terms of the periods, of whether the gaps could possibly open {under an infinitesimal perturbation}.\nWe also construct potentials that do open (the theoretically existing) gaps at these exceptional energies.\n\nAlthough the general strategy follows that of \\cite{HanJit2017}, {there are several challenges to overcome in the present work.}:\n\\begin{itemize}\n\\item\nThe Floquet parameters and perturbation directions that we choose in the perturb-and-count technique are strongly model-dependent in a subtle fashion. For example, at non-exceptional energies, we locate Floquet parameters and a perturbation direction in a way such that the Floquet eigenvalues with vanishing linear terms have quadratic terms of the same sign along this direction. \nAt the exceptional energy of the triangular lattice, we choose two directions such that the eigenvalues with vanishing gradients have quadratic terms of different signs along the two directions; for a more detailed discussion, see Remark~\\ref{rem:tri}.\nThis is similar to what was done in \\cite{HanJit2017} for the square lattice case.\nHowever, for the EHM lattice, any direction will lead to the same number of positive and negative quadratic terms; see Remark \\ref{rem:sqn}.\nThis issue is resolved by a new construction: we find a direction that moves approximately $2\/3$ of the degenerate eigenvalues up while the other $1\/3$ move down.\nAll these constructions depend heavily on the Floquet representation of the eigenvalues, and thus get more difficult as the underlying geometry gets more complicated.\n\\item Applying the perturb-and-count ideas directly to the hexagonal lattice is quite difficult,\ndue to the fact that the Floquet eigenvalues do not have simple expressions; compare \\eqref{eq:hexTriBandRel}.\nHowever, one can relate Laplacians and Floquet matrices for the triangular and hexagonal lattices in a fairly elegant fashion ({see \\cite{HKR2016} and our \\eqref{eq:hexSquareTri}}). Thus, we prove the Bethe--Sommerfeld conjecture directly for the triangular lattice and then derive the corresponding statement for the hexagonal lattice via a somewhat soft argument.\n\\item Because of the more complicated structure of the lattices involved, constructing potentials that open gaps at the exceptional energies is substantially more difficult than in the square lattice. \nIn particular, we need to construct (2,2)-periodic potentials that live on eight vertices for the hexagonal lattice, and (3,3)-periodic potential for the EHM lattice.\nIn this paper we develop an {robust} technique to study these finite volume problems in a sharp way.\nIndeed, we can not only prove that a gap exists, but also estimate its size up to a constant factor (see Theorems~\\ref{thm:triExampleGapLength}, \\ref{thm:hexQ}, and \\ref{thm:nnnExGapLength}). \nIn the case of the triangular lattice, we are even able to use our technique \\emph{exactly} compute the gap, not only estimate its size (Theorem~\\ref{thm:triExampleGapLength}).\n\\end{itemize}\n\n\\bigskip\n\n\\subsection{Main Results}\n\nLet us now describe more precisely the setting in which we work and the results that we prove. By a \\textit{graph}, we shall mean a pair $\\Gamma = ({\\mathcal V}, {\\mathcal E})$ where ${\\mathcal V}$ is a nonempty set and ${\\mathcal E}$ is a nonempty subset of ${\\mathcal V} \\times {\\mathcal V}$ with the following properties:\n\\begin{itemize}\n\\item For no $v \\in {\\mathcal V}$ does one have $(v,v) \\in {\\mathcal E}$;\n\\item If $(u,v) \\in {\\mathcal E}$, then $(v,u) \\in {\\mathcal E}$.\n\\end{itemize}\nIf $(u,v) \\in \\mathcal{E}$, we write $u\\sim v$ and we say that $u$ and $v$ are neighbors or neighboring vertices. We think of ${\\mathcal E}$ as the set of \\emph{directed edges}; $(u,v)$ represents the edge that originates at $u$ and terminates at $v$.\n\nGiven such a graph, we consider $\\mathcal{H}_\\Gamma = \\ell^2(\\mathcal{V})$ and the associated \\emph{graph Laplacian} $\\Delta_\\Gamma: \\mathcal{H}_\\Gamma \\to \\mathcal{H}_\\Gamma$, which acts via\n\\[\n[\\Delta_\\Gamma \\psi]_u\n=\n\\sum_{v \\sim u} \\psi_v,\n\\quad\nu \\in \\mathcal{V}, \\; \\psi \\in \\mathcal{H}_\\Gamma.\n\\]\nTechnically, this is the adjacency operator of the graph. Other authors use $\\psi_v - \\psi_u$ where we have only $\\psi_v$. Our convention is slightly more natural for the setting in which we wish to work. Concretely, all of the graphs that we consider in the present work have uniform degree (all vertices in a given graph have the same number of incident edges), and hence leaving off the $-\\psi_u$ term merely costs us a multiple of the identity operator, and it simplifies the appearance of a few calculations.\n\nBy a \\emph{Schr\\\"odinger operator} on $\\Gamma$, we mean an operator of the form $H_Q = H_{\\Gamma,Q} = \\Delta_\\Gamma + Q$, where $Q:{\\mathcal V} \\to {\\mathbb R}$ is a bounded function that acts on ${\\mathcal H}_\\Gamma$ by multiplication:\n\\[\n[Q\\psi]_u\n=\nQ(u) \\psi_u,\n\\quad\nu \\in \\mathcal{V}, \\; \\psi \\in \\mathcal{H}_\\Gamma.\n\\]\n\\begin{figure*}[t]\n\\begin{tikzpicture}[yscale=.6,xscale=.6]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,6);\n\\draw [-,line width=.06cm] (0,2) -- (6,2);\n\\draw [-,line width = .06cm] (2,0) -- (2,6);\n\\draw [-,line width=.06cm] (0,4) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (4,6);\n\\draw [-,line width=.06cm] (0,6) -- (6,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](0,2) circle (.2);\n\\filldraw[color=black, fill=black](2,2) circle (.2);\n\\filldraw[color=black, fill=black](4,2) circle (.2);\n\\filldraw[color=black, fill=black](6,2) circle (.2);\n\\filldraw[color=black, fill=black](0,4) circle (.2);\n\\filldraw[color=black, fill=black](2,4) circle (.2);\n\\filldraw[color=black, fill=black](4,4) circle (.2);\n\\filldraw[color=black, fill=black](6,4) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](2,6) circle (.2);\n\\filldraw[color=black, fill=black](4,6) circle (.2);\n\\filldraw[color=black, fill=black](6,6) circle (.2);\n\\end{tikzpicture}\n\\caption{The square lattice.}\n\\end{figure*}\nIn the present work, we study ${\\mathbb Z}^2$-periodic graphs. That is, we consider graphs whose vertices $\\mathcal{V}$ comprise a subset of ${\\mathbb R}^2$ and for which there exist linearly independent translations $\\bm{a}_1, \\bm{a}_2 \\in {\\mathbb R}^2$ which leave $\\Gamma$ invariant. That is to say:\n\\begin{itemize}\n\\item For any vertex $v \\in {\\mathcal V}$, $v + \\bm{a}_j \\in {\\mathcal V}$ for $j=1,2$;\n\\item For any edge $(u,v) \\in {\\mathcal E}$, $(u + \\bm{a}_j,v + \\bm{a}_j) \\in {\\mathcal E}$ for $j=1,2$.\n\\end{itemize}\nWe will then be most interested in studying the case when the potential $Q$ is itself periodic. In general, we will say that $Q:{\\mathcal V} \\to {\\mathbb R}$ is $\\bm{p} = (p_1,p_2)$-periodic for some $p_1,p_2 \\in {\\mathbb Z}_+$ if and only if \n\\[\nQ(u+p_1 \\bm{a}_1) = Q(u + p_2 \\bm{a}_2) = Q(u),\\quad\n\\text{for all } u \\in {\\mathcal V}.\n\\]\n\nThe square lattice is the graph with vertices ${\\mathcal V}_{\\mathrm{sq}} = {\\mathbb Z}^2$ and where \n\\[\n\\bm{n} \\sim \\bm{n}'\n\\iff\n\\| \\bm{n} - \\bm{n'} \\|\n=\n1.\n\\]\nHere and throughout the paper, $\\|\\cdot\\|$ denotes the Euclidean norm on ${\\mathbb R}^2$. It is easy to see that the associated Laplacian acts on $\\ell^2({\\mathbb Z}^2)$ via\n\\[\n[\\Delta_{\\mathrm{sq}} \\psi]_{n,m}\n=\n\\psi_{n-1,m} + \\psi_{n+1,m} + \\psi_{n,m-1} + \\psi_{n,m+1}.\n\\]\n\n\n\n\nPart of the motivation for the present work comes from \\cite{EmbFil2017, HanJit2017, KrugPreprint}. In \\cite{EmbFil2017}, Embree and Fillman showed that if $Q: {\\mathbb Z}^2 \\to {\\mathbb R}$ is $(p_1,p_2)$-periodic and sufficiently small, then $\\sigma(\\Delta_{\\mathrm{sq}}+Q)$ consists of one or two intervals and that the spectrum consists of exactly one interval whenever at least one of $p_1$ or $p_2$ is odd, which generalized the work of Kr\\\"uger, who proved a similar result under the stricter condition that the periods were coprime \\cite{KrugPreprint}. In \\cite{HanJit2017}, Han and Jitomirskaya showed that if $Q:{\\mathbb Z}^d \\to {\\mathbb R}$ is $(p_1,\\ldots,p_d)$-periodic and small, then the same results hold true: the spectrum has no more than one gap and has no gaps as long as at least one period is odd.\n\n\n\n\\subsection{The Triangular Lattice} \n\\begin{figure*}[b]\n \\begin{minipage}{0.45\\textwidth}\n \\centering\n\\begin{tikzpicture}[yscale=.85,xscale=.85]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,{sqrt(3)}) -- (6,{sqrt(3)});\n\\draw [-,line width = .06cm] (0,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .06cm] (0,{3*sqrt(3)}) -- (6,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (0,{2*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (0,0) -- (3,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (2,0) -- (5,{3*sqrt(3)});\n\\draw [-,line width=.06cm] (4,0) -- (6,{2*sqrt(3)});\n\\draw [-,line width=.06cm] (0,{2*sqrt(3)}) -- (2,0);\n\\draw [-,line width=.06cm] (1,{3*sqrt(3)}) -- (4,0);\n\\draw [-,line width=.06cm] (3,{3*sqrt(3)}) -- (6,0);\n\\draw [-,line width=.06cm] (5,{3*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [->,line width=.06cm,color=blue] (1,{sqrt(3)}) -- (2,{2*sqrt(3)});\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](1,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](3,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](5,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](2,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](1,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](3,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](5,{3*sqrt(3)}) circle (.2);\n\\draw [->,line width=.06cm,color=blue] (1,{sqrt(3)}) -- (2,{2*sqrt(3)});\n\\draw [->,line width=.06cm,color=blue] (1,{sqrt(3)}) -- (3,{sqrt(3)});\n\\node [above] at (1,{sqrt(3)+.3}) {\\cold{$\\bm{a}_2$}};\n\\node [below] at (1.7,{sqrt(3)-.1}) {\\cold{$\\bm{a}_1$}};\n\\end{tikzpicture}\n\\caption{A portion of the triangular lattice}\\label{fig:trilat}\n\\end{minipage}\n\\hfill\n \\begin{minipage}{0.45\\textwidth}\n \\centering\n\\begin{tikzpicture}[yscale=.75,xscale=.75]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,6);\n\\draw [-,line width=.06cm] (0,2) -- (6,2);\n\\draw [-,line width = .06cm] (2,0) -- (2,6);\n\\draw [-,line width=.06cm] (0,4) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (4,6);\n\\draw [-,line width=.06cm] (0,6) -- (6,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\draw [-,line width = .06cm] (0,6) -- (6,0);\n\\draw [-,line width = .06cm] (2,0) -- (0,2);\n\\draw [-,line width = .06cm] (4,0) -- (0,4);\n\\draw [-,line width = .06cm] (2,6) -- (6,2);\n\\draw [-,line width = .06cm] (6,4) -- (4,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](0,2) circle (.2);\n\\filldraw[color=black, fill=black](2,2) circle (.2);\n\\filldraw[color=black, fill=black](4,2) circle (.2);\n\\filldraw[color=black, fill=black](6,2) circle (.2);\n\\filldraw[color=black, fill=black](0,4) circle (.2);\n\\filldraw[color=black, fill=black](2,4) circle (.2);\n\\filldraw[color=black, fill=black](4,4) circle (.2);\n\\filldraw[color=black, fill=black](6,4) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](2,6) circle (.2);\n\\filldraw[color=black, fill=black](4,6) circle (.2);\n\\filldraw[color=black, fill=black](6,6) circle (.2);\n\\end{tikzpicture}\n\\caption{The triangular lattice after shearing.}\\label{fig:trishear}\n\\end{minipage}\n\\end{figure*}\n\nThe first graph that we consider is the \\emph{triangular lattice}. The graph has vertices\n\\[\n{\\mathcal V}_{\\mathrm{tri}}\n=\n\\set{n \\bm{a}_1 + m \\bm{a}_2 : n,m \\in {\\mathbb Z}},\n\\]\nwhere the generating vectors are\n\\[\n\\bm{a}_1\n=\n\\begin{bmatrix}\n1 \\\\ 0\n\\end{bmatrix},\n\\quad\n\\bm{a}_2\n=\n\\frac{1}{2} \\begin{bmatrix}\n1 \\\\ \\sqrt{3}\n\\end{bmatrix}.\n\\]\nOne then declares $v \\sim w$ for $v,w \\in \\mathcal{V}$ if $\\|v - w\\| = 1$. Thus, every $v \\in \\mathcal{V}$ has 6 neighbors; more specifically, if $v = n\\bm{a}_1 + m \\bm{a}_2$, then $v$ has neighbors\n\\[\n(n \\pm1) \\bm{a}_1 + m \\bm{a}_2, \\quad\nn\\bm{a}_1 + (m\\pm 1) \\bm{a}_2,\\quad\n(n \\pm 1) \\bm{a}_1 + (m\\mp 1) \\bm{a}_2.\n\\]\nConsequently, after identifying $n \\bm{a}_1 + m \\bm{a}_2$ with the point $(n,m) \\in {\\mathbb Z}^2$, we may view the Laplacian on the triangular lattice as an operator on $\\ell^2({\\mathbb Z}^2)$ via\n\\begin{equation} \\label{eq:triLaplacianSqVersion}\n[\\Delta_{\\rm tri} \\psi]_{n,m}\n=\n[\\Delta_{\\rm sq} \\psi]_{n,m}\n+\n\\psi_{n-1,m+1} + \\psi_{n+1,m-1}.\n\\end{equation}\nThis correspondence amounts to shearing and stretching the the triangular lattice, and essentially maps the triangular lattice to the square lattice with skewed next-nearest-neighbor interactions added. See Figures~\\ref{fig:trilat} and \\ref{fig:trishear}.\n\n\n\\begin{theorem}[Bethe--Sommerfeld for the triangular lattice] \\label{t:bsc:tri}\nFor all $\\bm{p} = (p_1,p_2 )\\in {\\mathbb Z}_+^2$, there is a constant $c = c_{\\bm{p}} > 0$ such that, if $Q:{\\mathcal V}_{\\mathrm{tri}} \\to {\\mathbb R}$ is $\\bm{p}$-periodic and $\\|Q\\|_\\infty \\leq c$, the following hold true for $H_Q = \\Delta_{\\mathrm{tri}} + Q$:\n\\begin{enumerate}\n\\item[{\\rm(1)}] $\\sigma(H_Q)$ consists of no more than two intervals.\n\\item[{\\rm(2)}] If at least one of $p_1$ or $p_2$ is odd, then $\\sigma(H_Q)$ consists of a single interval.\n\\end{enumerate}\nMoreover, the gap in the first setting may only open at the energy $E = -2$.\n\\end{theorem}\n\nThis theorem is sharp vis-\\`a-vis the number of intervals in the spectrum and the arithmetic restrictions on the periods. Concretely, we exhibit a $(2,2)$-periodic potential that perturbatively opens a gap at $-2$.\n\n\\begin{theorem} \\label{t:triExamples}\nThere exists $Q:{\\mathcal V}_{\\mathrm{tri}} \\to {\\mathbb R}$ which is $(2,2)$-periodic, such that $\\sigma(H_{\\lambda Q})$ has exactly two connected components for any sufficiently small $\\lambda > 0$.\n\\end{theorem}\n\n\\subsection{The Hexagonal Lattice} The set of vertices of the hexagonal lattice is closely related to that of the triangular lattice. Concretely, define $\\bm{b}_\\pm$ by\n\\[\n\\bm{b}_\\pm\n=\n\\frac{1}{2}\n\\begin{bmatrix}\n 3 \\\\ \\pm \\sqrt{3}\n\\end{bmatrix}.\n\\]\nThen, we obtain the hexagonal lattice by deleting the centers of some of the hexagons formed by the triangular lattice; more precisely,\n\\[\n{\\mathcal V}_{\\mathrm{hex}}\n=\n\\set{n \\bm{a}_1 + m \\bm{a}_2 \\in {\\mathcal V}_{\\mathrm{tri}}:n,m \\in {\\mathbb Z}} \\setminus \\{- \\bm{a}_1 + k \\bm{b}_+ + \\ell \\bm{b}_- : k,\\ell \\in {\\mathbb Z}\\}.\n\\]\nEquivalently, it is not hard to check that $\\{0, \\bm{a}_1\\}$ is a fundamental set of vertices and hence every $v \\in {\\mathcal V}_{\\mathrm{hex}}$ may be written uniquely as either $n \\bm{b}_+ + m \\bm{b}_-$ or $\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-$ for integers $n,m$, so we have\n\\begin{align*}\n{\\mathcal V}_{\\mathrm{hex}}\n& =\n\\set{n \\bm{b}_+ + m \\bm{b}_- : n,m \\in {\\mathbb Z}} \\cup \\set{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_- : n,m \\in {\\mathbb Z}}.\n\\end{align*}\nWe define ${\\mathcal E}_{\\mathrm{hex}}$ by declaring $u \\sim v$ for $u, v \\in {\\mathcal V}_{\\mathrm{hex}}$ if $\\|u - v\\|_2 = 1$. After some calculations, we see that\n\\begin{align*}\n[\\Delta_{\\mathrm{hex}} \\psi]_{n \\bm{b}_+ + m \\bm{b}_-}\n& =\n\\psi_{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-} + \\psi_{\\bm{a}_1 + n \\bm{b}_+ + (m-1) \\bm{b}_-} + \\psi_{\\bm{a}_1 + (n-1) \\bm{b}_+ + m \\bm{b}_-} \\\\\n[\\Delta_{\\mathrm{hex}} \\psi]_{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-}\n& =\n\\psi_{n \\bm{b}_+ + m \\bm{b}_-} + \\psi_{n \\bm{b}_+ + (m+1) \\bm{b}_-} + \\psi_{(n+1) \\bm{b}_+ + m \\bm{b}_-}\n\\end{align*}\n\n\n\\begin{figure*}[t]\n\\begin{tikzpicture}[yscale=.75,xscale=.75]\n\\draw [-,line width = .06cm] (0,0) -- (1,{sqrt(3)});\n\\draw [-,line width = .06cm] (0,{2*sqrt(3)}) -- (1,{sqrt(3)});\n\\draw [-,line width = .06cm] (0,{2*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (0,{4*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .06cm,color=red] (1,{sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .06cm] (1,{3*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .06cm] (4,0) -- (3,{sqrt(3)});\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (4,{4*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (4,0) -- (6,0);\n\\draw [-,line width = .06cm] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .06cm] (4,{4*sqrt(3)}) -- (6,{4*sqrt(3)});\n\\draw [-,line width = .06cm] (6,0) -- (7,{sqrt(3)});\n\\draw [-,line width = .06cm] (6,{2*sqrt(3)}) -- (7,{sqrt(3)});\n\\draw [-,line width = .06cm] (6,{2*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\draw [-,line width = .06cm] (6,{4*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=red, fill=red](1,{sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](3,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](1,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](3,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{4*sqrt(3)}) circle (.2);\n\\draw [->,line width = .06cm,color=blue] (1.2,{sqrt(3)-.1}) -- (3.8,{.1});\n\\draw [->,line width = .06cm,color=blue] (1.2,{sqrt(3)+.1}) -- (3.8,{2*sqrt(3)-.1});\n\\node at (2,{1.8*sqrt(3)}) {\\cold{$\\bm{b}_+$}};\n\\node at (2,{.2*sqrt(3)}) {\\cold{$\\bm{b}_-$}};\n\\end{tikzpicture}\n\\caption{A portion of the hexagonal lattice. A fundamental domain is highlighted in red.}\\label{fig:hexlat}\n\\end{figure*}\nSee Figure~\\ref{fig:hexlat}. The formula for $\\Delta_{\\mathrm{hex}}$ can be made more compact if we view the associated Hilbert space as \n\\[\n\\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)\n=\n\\set{\\Psi:{\\mathbb Z}^2 \\to {\\mathbb C}^2 : \\sum_{n,m} \\|\\Psi_{n,m}\\|^2 < \\infty},\n\\]\nwhere the standard basis of ${\\mathbb C}^2$ corresponds to the left and right vertices of the fundamental domain, respectively. More precisely, given $\\psi \\in \\ell^2({\\mathcal V}_{\\mathrm{hex}})$, define $\\Psi \\in \\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)$ by \n\\[\n\\Psi_{n,m}\n=\n\\begin{bmatrix}\n\\psi_{n \\bm{b}_+ + m \\bm{b}_-} \\\\ \\psi_{\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-}\n\\end{bmatrix}.\n\\]\nIdentifying $\\ell^2({\\mathcal V}_{\\mathrm{hex}})$ and $\\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)$ in this fashion, the Laplacian for the hexagonal lattice is given by\n\\begin{align*}\n[\\Delta_{\\mathrm{hex}} \\Psi]_{n,m}\n& =\nU(\\Psi_{n,m-1}+ \\Psi_{n-1,m}) + L(\\Psi_{n,m+1}+ \\Psi_{n+1,m}) + J \\Psi_{n,m},\n\\end{align*}\nwhere\n\\[\nU\n=\n\\begin{bmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{bmatrix},\n\\quad\nL \n=\nU^\\top\n=\n\\begin{bmatrix}\n0 & 0 \\\\ 1 & 0\n\\end{bmatrix},\n\\quad\nJ\n=\nU+L\n=\n\\begin{bmatrix}\n0 & 1 \\\\ 1 & 0\n\\end{bmatrix}.\n\\]\n{Equivalently, if we denote by $S_1, S_2 : \\ell^2({\\mathbb Z}^2) \\to \\ell^2({\\mathbb Z}^2)$ the shift operators\n\\[\n[S_1 \\psi]_{n,m}\n=\n\\psi_{n+1,m},\n\\quad\n[S_2\\psi]_{n,m}\n=\n\\psi_{n,m+1},\n\\]\nwe have\n\\[\n\\Delta_{\\mathrm{hex}} \\Psi\n=\n\\begin{bmatrix}\n(S_1^* + S_2^* + {\\mathbb I})\\psi^- \\\\ (S_1 + S_2 + {\\mathbb I})\\psi^+\n\\end{bmatrix}\n\\quad\n\\text{for any} \\quad \n\\Psi \n=\n\\begin{bmatrix} \\psi^+ \\\\ \\psi^- \\end{bmatrix}\n\\in \\ell^2({\\mathbb Z}^2,{\\mathbb C}^2).\n\\]\nAbbreviating somewhat, we write:\n\\begin{equation} \\label{eq:hexDecomp}\n\\Delta_{\\mathrm{hex}}\n=\\begin{bmatrix}\n0 & S_1^* + S_2^* + {\\mathbb I} \\\\\nS_1 + S_2 + {\\mathbb I} & 0\n\\end{bmatrix}.\n\\end{equation}}\n\n\\begin{theorem}[Bethe--Sommerfeld for the hexagonal lattice] \\label{t:bsc:hex}\nFor all $\\bm{p} = (p_1,p_2 )\\in {\\mathbb Z}_+^2$, there is a constant $c = c_{\\bm{p}} > 0$ such that, if $Q:{\\mathcal V}_{\\mathrm{hex}} \\to {\\mathbb R}$ is $\\bm{p}$-periodic and $\\|Q\\|_\\infty \\leq c$, the following statements hold true for $H_Q = \\Delta_{\\mathrm{hex}} + Q$:\n\\begin{enumerate}\n\\item[{\\rm(1)}] $\\sigma(H_Q)$ consists of no more than four intervals.\n\\item[{\\rm(2)}] If at least one of $p_1$ or $p_2$ is odd, then $\\sigma(H_Q)$ consists of no more than two intervals.\n\\end{enumerate}\nMoreover, gaps may only open at $0$ and $\\pm1$ in the first case, and only at zero in the second case.\n\\end{theorem}\n\nMoreover, this theorem is sharp in the following sense: there exists a $(1,1)$-periodic potential $Q_1$ which infinitesimally opens a gap at zero, and there is a $(2,2)$-periodic potential $Q_2$ which infinitesimally opens gaps at $-1$, $0$, and $1$ in the following sense:\n\n\\begin{theorem} \\label{t:hexExamples}\n\\begin{enumerate}\n\\item[{\\rm(1)}] There exists $Q_1: {\\mathcal V}_{\\mathrm{hex}} \\to {\\mathbb R}^2$ which is $(1,1)$-periodic such that $\\sigma(H_{\\lambda Q_1})$ has exactly two connected components for all $\\lambda>0$.\n\\item[{\\rm(2)}] There exists $Q_2: {\\mathcal V}_{\\mathrm{hex}} \\to {\\mathbb R}^2$ which is $(2,2)$ periodic such that $\\sigma(H_{\\lambda Q_2})$ has exactly four connected components for any sufficiently small $\\lambda > 0$.\n\\end{enumerate}\n\\end{theorem}\n\nLet us remark that Theorem~\\ref{t:hexExamples}.(1) is well-known; we merely list it for completeness. The example in Theorem~\\ref{t:hexExamples}.(2) is novel.\n\n\\subsection{The EHM Lattice}\n\nThe EHM lattice also has vertex set ${\\mathcal V}_{\\mathrm{sqn}} = {\\mathcal V}_{\\mathrm{sq}} = {\\mathbb Z}^2$. However, now, one connects $\\bm{n}$ and $\\bm{n}'$ if and only if they are nearest neighbors or next-nearest-neighbors in the square lattice. Equivalently, one declares\n\\[\n\\bm{n} \\sim \\bm{n}'\n\\iff\n\\|\\bm{n} - \\bm{n}' \\|_\\infty\n=\n1.\n\\] \nThe associated Laplacian acts on $\\ell^2({\\mathbb Z}^2)$ via\n\\[\n[\\Delta_{\\mathrm{sqn}} \\psi]_{n,m}\n=\n[\\Delta_{\\mathrm{sq}}]_{n,m}+\\psi_{n-1,m-1}+\\psi_{n-1,m+1}+\\psi_{n+1,m-1}+\\psi_{n+1,m+1}.\n\\]\nSee Figure~\\ref{fig:ehmlat}.\n\\begin{figure*}[t]\n\n\\begin{tikzpicture}[yscale=.6,xscale=.6]\n\\draw [-,line width = .06cm] (0,0) -- (6,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,6);\n\\draw [-,line width=.06cm] (0,2) -- (6,2);\n\\draw [-,line width = .06cm] (2,0) -- (2,6);\n\\draw [-,line width=.06cm] (0,4) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (4,6);\n\\draw [-,line width=.06cm] (0,6) -- (6,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\draw [-,line width = .06cm] (0,0) -- (6,6);\n\\draw [-,line width = .06cm] (2,0) -- (6,4);\n\\draw [-,line width = .06cm] (4,0) -- (6,2);\n\\draw [-,line width = .06cm] (0,2) -- (4,6);\n\\draw [-,line width = .06cm] (0,4) -- (2,6);\n\\draw [-,line width = .06cm] (6,0) -- (6,6);\n\\draw [-,line width = .06cm] (0,6) -- (6,0);\n\\draw [-,line width = .06cm] (2,0) -- (0,2);\n\\draw [-,line width = .06cm] (4,0) -- (0,4);\n\\draw [-,line width = .06cm] (2,6) -- (6,2);\n\\draw [-,line width = .06cm] (6,4) -- (4,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](2,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](0,2) circle (.2);\n\\filldraw[color=black, fill=black](2,2) circle (.2);\n\\filldraw[color=black, fill=black](4,2) circle (.2);\n\\filldraw[color=black, fill=black](6,2) circle (.2);\n\\filldraw[color=black, fill=black](0,4) circle (.2);\n\\filldraw[color=black, fill=black](2,4) circle (.2);\n\\filldraw[color=black, fill=black](4,4) circle (.2);\n\\filldraw[color=black, fill=black](6,4) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](2,6) circle (.2);\n\\filldraw[color=black, fill=black](4,6) circle (.2);\n\\filldraw[color=black, fill=black](6,6) circle (.2);\n\\end{tikzpicture}\n\\caption{A portion of the EHM lattice.}\\label{fig:ehmlat}\n\\end{figure*}\n\n\\begin{theorem}[Bethe--Sommerfeld for the EHM lattice] \\label{t:bsc:nnn}\nFor all $\\bm{p} = (p_1,p_2 )\\in {\\mathbb Z}_+^2$, there is a constant $c = c_{\\bm{p}} > 0$ such that, if $Q:{\\mathcal V}_{\\mathrm{sqn}} \\to {\\mathbb R}$ is $\\bm{p}$-periodic and $\\|Q\\|_\\infty \\leq c$, the following hold true for $H_Q = \\Delta_{\\mathrm{sqn}} + Q$:\n\\begin{enumerate}\n\\item[{\\rm(1)}] $\\sigma(H_Q)$ consists of no more than two intervals.\n\\item[{\\rm(2)}] If at least one of $p_1$ or $p_2$ is not divisible by three, then $\\sigma(H_Q)$ consists of a single interval.\n\\end{enumerate}\nMoreover, the gap in the first setting may only open at the energy $E = -1$.\n\\end{theorem}\nThis theorem is also sharp:\n\\begin{theorem} \\label{t:nnnExamples}\nThere exists $Q:{\\mathbb Z}^2 \\to {\\mathbb R}$ which is $(3,3)$-periodic such that $\\sigma(H_{\\lambda Q})$ has exactly two connected components for any sufficiently small $\\lambda>0$.\n\\end{theorem}\n\n\\bigskip\n\nThe remainder of the paper is organized as follows. Section~\\ref{sec:floquet} recalls Floquet theory for ${\\mathbb Z}^2$-periodic graphs. We work with the triangular lattice in Section~\\ref{sec:tri}, proving Theorems~\\ref{t:bsc:tri} and \\ref{t:triExamples}. We then work with the hexagonal lattice in Section~\\ref{sec:hex}, proving Theorems~\\ref{t:bsc:hex} and \\ref{t:hexExamples}. Finally, we conclude with the EHM lattice in Section~\\ref{sec:nnn}, proving Theorems~\\ref{t:bsc:nnn} and \\ref{t:nnnExamples}.\n\n\\section{Floquet Theory for Periodic Schr\\\"odinger Operators on Periodic Graphs}\n\\label{sec:floquet}\n\nLet $\\Gamma= ({\\mathcal V},{\\mathcal E})$ be a ${\\mathbb Z}^2$-periodic graph with translation symmetries $\\bm{a}_1, \\bm{a}_2 \\in {\\mathbb R}^2$, and suppose $Q:{\\mathcal V} \\to {\\mathbb R}$ is $\\bm{p} = (p_1,p_2)$-periodic, that is,\n\\[\nQ(u + p_j \\bm{a}_j)\n=\nQ(u),\\quad\nu \\in {\\mathcal V},\\; j = 1,2.\n\\]\nWe will briefly describe Floquet theory for $H_Q = \\Delta_\\Gamma + Q$, following \\cite{KorSab2014}. The main purpose of this section is to establish notation, so we do not give any proofs. One may write $H_Q$ as a constant-fiber direct integral over the fundamental domain. Concretely, let\n\\[\n{\\mathcal V}_{\\rm f}\n=\n{\\mathcal V} \\cap \\set{s \\bm{a}_1 + t \\bm{a}_2 : 0 \\le s < p_1, \\; 0 \\le t < p_2}.\n\\]\nBy periodicity, $|{\\mathcal V}_{\\rm f}| = P := p_0 p_1 p_2$, where\n\\[\np_0\n=\n|{\\mathcal V} \\cap \\set{s\\bm{a}_1 + t \\bm{a}_2 : 0 \\le s,t < 1}|.\n\\]\nHere, and throughout the paper, we use $|S|$ to denote the cardinality of the set $S$. For each edge $(u,v) \\in {\\mathcal E}$ there exist unique vertices $u_{\\rm f}, v_{\\rm f} \\in {\\mathcal V}_{\\rm f}$ and unique integers $n,m,n',m' \\in {\\mathbb Z}$ with\n\\[\nu = u_{\\rm f} + np_1 \\bm{a}_1 + mp_2 \\bm{a}_2,\n\\quad\nv = v_{\\rm f} + n'p_1\\bm{a}_1 + m'p_2 \\bm{a}_2,\n\\]\nWe then define the \\emph{index} of $(u,v)$ by $\\tau(u,v) = (n'-n,m'-m)$. Finally, for $u,v \\in {\\mathcal V}_{\\rm f}$, we define $B(u,v)$ to be the set of all translates of $v$ that connect to $u$ via an edge of $\\Gamma$:\n\\[\nB(u,v)\n=\n\\set{w \\in {\\mathcal V} : w \\sim u \\text{ and } w = v + np_1 \\bm{a}_1 + mp_2 \\bm{a}_2 \\text{ for some }n,m \\in {\\mathbb Z}}.\n\\]\n\nThen, for each $\\bm{\\theta} =(\\theta_1,\\theta_2) \\in {\\mathbb R}^2$, the corresponding Floquet matrix is a self-adjoint operator on ${\\mathcal H}_{\\rm f}:= \\ell^2({\\mathcal V}_{\\rm f}) = {\\mathbb C}^{{\\mathcal V}_{\\rm f}}$ defined by\n\\begin{equation} \\label{eq:floqMatDef}\n\\langle \\delta_u, H_Q(\\bm{\\theta}) \\delta_v \\rangle\n=\n\\sum_{w \\in B(u,v)} \\exp\\Big(i \\big\\langle \\tau(u,w), \\bm{\\theta} \\big\\rangle\\Big).\n\\end{equation}\nIn the event that the sum in \\eqref{eq:floqMatDef} is empty, $\\langle \\delta_u, H_Q(\\bm{\\theta}) \\delta_v \\rangle = 0$. Clearly, if $\\theta_j' - \\theta_j \\in 2\\pi{\\mathbb Z}$ for $j=1,2$, then $H_Q(\\bm{\\theta}) = H_Q(\\bm{\\theta}')$, so $H_Q(\\bm{\\theta})$ descends to a well-defined function of $\\bm{\\theta} \\in {\\mathbb T}^2 := {\\mathbb R}^2 \/ (2\\pi{\\mathbb Z})^2 \\cong [0,2\\pi)^2$. We will freely use $\\bm{\\theta} \\in {\\mathbb R}^2$ or $\\bm{\\theta} \\in {\\mathbb T}^2$ depending on which is more convenient in a given setting.\n\nInformally, \\eqref{eq:floqMatDef} represents the restriction of $H_Q$ to the discrete torus \n\\[\n({\\mathbb Z}\\bm{a}_1 \\oplus {\\mathbb Z}\\bm{a}_2) \/ (p_1 {\\mathbb Z} \\bm{a}_1 \\oplus p_2 {\\mathbb Z} \\bm{a}_2)\n\\cong\n{\\mathbb Z}_{p_1} \\oplus {\\mathbb Z}_{p_2}.\n\\]\nwith the following boundary conditions: wrapping once around the torus in the positive $\\bm{a}_1$ direction accrues a phase $e^{i\\theta_1}$ and wrapping around once in the positive $\\bm{a}_2$ direction accrues a phase $e^{i\\theta_2}$. More precisely, we may view $H_Q(\\bm{\\theta})$ in the following manner. The operator $H_Q$ acts on the space ${\\mathbb C}^{\\mathcal V}$ of arbitrary (not necessarily square-summable) functions ${\\mathcal V} \\to {\\mathbb C}$. When $Q$ is $(p_1,p_2)$-periodic, then for each $\\bm{\\theta} \\in {\\mathbb T}^2$, $H_Q$ preserves the subspace\n\\[\n\\mathcal{H}(\\bm{\\theta})\n=\n\\set{\\psi \\in {\\mathbb C}^{\\mathcal V} : \\psi(u+p_j \\bm{a}_j) = e^{i\\theta_j} \\psi(u)}.\n\\]\nThen, $H_Q(\\bm{\\theta})$ is equivalent to the restriction of $H_Q$ to ${\\mathcal H}(\\bm{\\theta})$.\n\nFor each $\\bm{\\theta}$, order the eigenvalues of $H_Q(\\bm{\\theta})$ as\n\\[\nE_1(\\bm{\\theta})\n\\leq\n\\cdots\n\\leq\nE_P(\\bm{\\theta})\n\\]\nwith each eigenvalue listed according to its multiplicity. Then, for $1 \\le j \\le P$, the $j$th spectral \\emph{band} of $H_Q$ is defined by\n\\[\nF_j\n=\nF_j(Q)\n:=\n\\mathrm{ran}(E_j)\n=\n\\set{E_j(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb T}^2}\n=\n\\set{E_j(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb R}^2}.\n\\]\n\n\\begin{theorem} \\label{t:floquet}\nWith notation as above,\n\\[\n\\sigma(H_Q)\n=\n\\bigcup_{\\bm{\\theta} \\in {\\mathbb T}^2} H_Q(\\bm{\\theta})\n=\n\\bigcup_{j=1}^P F_j.\n\\]\n\\end{theorem}\n\nWe will use Theorem~\\ref{t:floquet} in the following way. Making the dependence on the potential $Q$ explicit, one may write\n\\[\nF_j = F_j(Q)\n=\n[E_j^-(Q),E_j^+(Q)].\n\\]\nThe key fact is the following: by standard perturbation theory for self-adjoint operators, $E_j^\\pm(Q)$ are 1-Lipschitz functions of $Q$. Here, one views $Q$ as an element of ${\\mathbb R}^P$ and the perturbation is with respect to the uniform metric thereupon. In particular, if an energy $E$ satisfies $E \\in \\mathrm{int}(F_j(Q))$, then $(E-\\delta,E+\\delta) \\subseteq F_j(Q)$ for some positive $\\delta$, and it follows that $E \\in F_j(Q') \\subseteq \\sigma(H_{Q'})$ for any $(p_1,p_2)$-periodic $Q'$ with $\\|Q-Q'\\|_\\infty < \\delta$. Note that here it is very important that one views the periods as fixed: one may only perturb within ${\\mathbb R}^P$ for a fixed $P$. Thus, our analysis revolves around determining for a given energy $E$, whether $E$ belongs to the interior of some band of the Laplacian, where the Laplacian is viewed as a degenerate $(p_1,p_2)$-periodic operator. \n\n\n\n\n\\section{Triangular Laplacian}\\label{sec:tri}\n\nWe view the triangular Laplacian as acting on the square lattice $\\ell^2({\\mathbb Z}^2)$, but with extra connections as in \\eqref{eq:triLaplacianSqVersion}:\n\\begin{align*}\n[\\Delta_{\\rm tri} u]_{n,m}\n& =\nu_{n-1,m} + u_{n+1,m} + u_{n,m-1} + u_{n,m+1} + u_{n-1,m+1} + u_{n+1,m-1} \\\\\n& =\n[\\Delta_{\\rm sq}u]_{n,m} + u_{n-1,m+1} + u_{n+1,m-1}.\n\\end{align*}\nNow, given $p_1,p_2 \\in {\\mathbb Z}_+$, we view $\\Delta_{\\mathrm{tri}}$ as a $\\bm{p}$-periodic operator and perform the Floquet decomposition. Define $P:=p_1p_2$ as in Section~\\ref{sec:floquet}, and put\n\\[\n\\Lambda :=\n{\\mathbb Z}^2 \\cap \\Big( [0,p_1) \\times [0,p_2) \\Big).\n\\]\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb R}^2$, it is straightforward to check that\n\\[\n\\sigma(H(\\bm{\\theta}))\n=\n\\set{e_{\\bm{\\ell}}^{\\Lambda}(\\bm{\\theta}) : \\bm{\\ell}\\in \\Lambda },\n\\]\nwhere $\\bm{\\ell}=(\\ell_1,\\ell_2)$ and\n\\[\ne^{\\Lambda}_{\\bm{\\ell}}(\\bm{\\theta})\n=\n2\\cos\\left( \\frac{\\theta_1+2\\pi \\ell_1}{p_1}\\right) \n+ 2\\cos\\left(\\frac{\\theta_2+2\\pi \\ell_2}{p_2}\\right) \n+ 2\\cos\\left(\\frac{\\theta_1 + 2\\pi \\ell_1}{p_1} - \\frac{\\theta_2 + 2\\pi \\ell_2}{p_2}\\right).\n\\]\nLet us point out that one needs to be somewhat careful at this point; namely, $e^\\Lambda_{\\bm{\\ell}}(\\bm{\\theta})$ is not a well-defined function of $\\bm{\\theta} \\in {\\mathbb T}^2$. However, the error incurred in using a different coset representative of $\\bm{\\theta} \\in {\\mathbb T}^2$ is simply a change in the index $\\bm{\\ell}$, and one can check that the \\emph{family} $\\set{e^\\Lambda_{\\bm{\\ell}}(\\bm{\\theta}) : \\bm{\\ell} \\in \\Lambda}$ is a well-defined function on ${\\mathbb T}^2$ (as well it should, since the \\emph{operator} $H(\\bm{\\theta})$ is itself a well-defined function of $\\bm{\\theta} \\in {\\mathbb T}^2$). In any case, the ambiguity disappears when one considers the covering space ${\\mathbb R}^2$, which we do for most of the paper. One could also use the minimal covering space ${\\mathbb R}^2 \/ (p_1{\\mathbb Z} \\oplus p_2 {\\mathbb Z})$ on which the $e_{\\bm{\\ell}}^\\Lambda$ are well-defined, but this does not accrue any benefits vis-\\`a-vis the present work, so we simply use ${\\mathbb R}^2$.\n\nAs in Section~\\ref{sec:floquet}, we label these eigenvalues in increasing order according to multiplicity by\n\\[\nE_1^{\\Lambda}(\\bm{\\theta})\n\\le \nE_2^{\\Lambda}(\\bm{\\theta})\\le \\cdots E_P^{\\Lambda}(\\bm{\\theta})\n\\]\nand denote the $P$ spectral bands by\n\\[\nF_k^{\\Lambda}\n=\n\\set{E_k^{\\Lambda}(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb R}^2},\n\\quad\n1 \\le k \\le P.\n\\]\nStraightforward computations shows that $\\sigma(\\Delta_{\\rm tri})=[-3,6]$, and thus\n\\[\n\\bigcup_{k=1}^P F_k^{\\Lambda}\n=\n[-3,6].\n\\]\nHenceforth, we view $p_1$ and $p_2$ as fixed and so we drop $\\Lambda$ from the superscripts.\nOur main theorem of this section is the following.\n\\begin{theorem}\\label{thm:trimain}\nLet $p_1,p_2 \\in {\\mathbb Z}_+$ be given.\n\\begin{enumerate}\n\\item[{\\rm 1.}]\nEach $E\\in (-3, 6)\\setminus \\{-2\\}$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\item[{\\rm 2.}] If one of the periods $p_1, p_2$ is odd, then $E=-2$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}[Proof of Theorem~\\ref{t:bsc:tri}]\nAs already discussed, this follows immediately from Theorem~\\ref{thm:trimain}.\n\\end{proof}\n\\subsection{Proof of Theorem \\ref{thm:trimain}} \nWe will divide the proof into two different cases: $E\\neq -2$ and $E=-2$.\nOur general strategy is to argue by contradiction.\nMore specifically, we assume $E=\\min F_{k+1}=\\max F_k$ for some $1\\leq k\\leq P-1$, and show that this leads to a contradiction. We will use the following two lemmas, whose proofs we provide at the end of the present section.\n\\begin{lemma}\\label{lem:constructiontri}\nFor any $E \\in {[-3,6]}$, there exist $x, y \\in [0,2\\pi)$ such that\n\\begin{align}\n\\label{eq:xyCondA} \\cos(x) + \\cos(y) + \\cos(x-y) & = \\frac{E}{2} \\\\\n\\label{eq:xyCondB} \\sin(x) + \\sin(y) & = 0.\n\\end{align}\nFurthermore, {if $E \\neq -2$}, we have\n\\begin{align}\n\\label{eq:xyCondC}\\cos(x) + \\cos(y)=-1+\\sqrt{E+3} \\neq 0\n\\end{align}\nfor any $x,y$ that satisfy conditions \\eqref{eq:xyCondA} and \\eqref{eq:xyCondB}.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lem:triJ0empty}\nConsider the following system:\n\\begin{equation} \\label{eq:triJ0syst}\n\\begin{cases}\n\\cos(x) + \\cos(y) + \\cos(x-y) = \\frac{E}{2},\\\\\n\\sin(x)+\\sin(x-y)=0,\\\\\n\\sin(y)-\\sin(x-y)=0.\n\\end{cases}\n\\end{equation}\nFor any $E \\in (-3,6) \\setminus \\{-2\\}$, the solution set of \\eqref{eq:triJ0syst} is empty. For $E = -2$, the solutions of \\eqref{eq:triJ0syst} in $[0,2\\pi)^2$ are $(0,\\pi)$, $(\\pi,0)$ and $(\\pi, \\pi)$.\n\\end{lemma}\n\nWe will use Lemma \\ref{lem:constructiontri} in the $E\\neq -2$ case, and Lemma \\ref{lem:triJ0empty} in the $E=-2$ case.\n\n\\subsubsection{$E\\neq -2$}\\\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:trimain}.1]\nLet $E \\in (-3,6) \\setminus\\{-2\\}$ be given and suppose for the purpose of establishing a contradiction that $E = \\max F_k = \\min F_{k+1}$ for some $1 \\le k < P$. Let $(x,y)$ denote a solution to \\eqref{eq:xyCondA} and \\eqref{eq:xyCondB} from Lemma \\ref{lem:constructiontri}, and take $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1,\\widetilde{\\theta}_2)\\in [0,2\\pi)^2$ and $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)},\\ell_2^{(1)})\\in \\Lambda$ such that \n\\[p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)})=x,\\, \\ \\text{and }\\ \\ p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)})=y.\n\\]\nIt is clear that $\\widetilde{\\bm{\\theta}}$ and $\\bm{\\ell}^{(1)}$ are uniquely determined by $x$ and $y$.\nLet us also note that \\eqref{eq:xyCondA} is equivalent to \n\\[e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=E.\\]\nDefine $\\Lambda_E(\\widetilde{\\bm{\\theta}}) \\subseteq \\Lambda$ to be the set of all $\\bm{\\ell} \\in \\Lambda$ such that $e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = E$. Then $r := |\\Lambda_E(\\widetilde{\\bm{\\theta}})|$ is the multiplicity of $E \\in \\sigma(H(\\widetilde{\\bm{\\theta}}))$ and clearly $\\bm{\\ell}^{(1)} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}})$.\n\n\nSince $E \\in F_k$ by assumption, let $s\\in {\\mathbb Z}\\cap [1,r]$ be chosen so that\n\\[E_{k-s}(\\widetilde{\\bm{\\theta}})0$ small enough such that \n\\[E_{k-s}(\\bm{\\theta})0\\}.\n\\end{aligned}\n\\end{equation}\nConsequently, we always have \n\\begin{align}\\label{eq:sumJbetatri}\n|\\mathcal{J}_{\\bm{\\beta}}^0|+|\\mathcal{J}_{\\bm{\\beta}}^+|+|\\mathcal{J}_{\\bm{\\beta}}^-|=r.\n\\end{align}\nWe also define $\\mathcal{J}_0$ as follows\n\\begin{align}\\label{def:J0tri}\n\\mathcal{J}_0\n=\n{\\mathcal J}_0(\\widetilde{\\bm{\\theta}})\n:=\\{ \\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) :\\ \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = \\bm{0}\\}.\n\\end{align}\nSince $E \\neq -2$, Lemma~\\ref{lem:triJ0empty} clearly implies $\\mathcal{J}_0=\\emptyset$.\n\nWe choose $\\bm{\\beta}_1=(\\beta_{1,1},\\beta_{1,2})=(p_1,p_2)\/\\sqrt{p_1^2+p_2^2}$. Then \\eqref{eq:xyCondB} is equivalent to \n\\[\\bm{\\beta}_1 \\cdot \\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=0,\\]\nhence $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$. \n\n\nNext we are going to perturb the point $\\widetilde{\\bm{\\theta}}$ and count the eigenvalues.\nSince $\\mathcal{J}_0 = \\emptyset$, we can choose a unit vector $\\bm{\\beta}_2$ such that \n\\begin{align}\\label{eq:beta2trinon-empty}\n\\bm{\\beta}_2\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq 0,\n\\end{align}\nholds for any $\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}})$.\nThus, $\\mathcal{J}_{\\bm{\\beta}_2}^0=\\emptyset$, so one concludes\n\\begin{align}\\label{eq:beta2trinon-empty'}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_{\\bm{\\beta}_2}^-|=r.\n\\end{align}\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_2$}\nWe first perturb the eigenvalues along the $\\bm{\\beta}_2$ direction.\nSince $\\mathcal{J}_{\\bm{\\beta}_2}^0 = \\emptyset$, we will always employ \\eqref{eq:pertgeneralbetatri1order}.\n\nFor $t > 0$ small enough, we have the following.\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq r-s.\n\\end{align}\n\n\\item If ${\\bm{\\ell} } \\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k-s}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:beta2trinon-empty'}, Equations~\\eqref{eq1:Jbeta2+tri} and \\eqref{eq2:Jbeta2+tri} imply\n\\begin{equation} \\label{eq:Jbeta2MinusCard+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\n=\ns.\n\\end{equation}\nUpon realizing that $\\mathcal{J}_{-\\bm{\\beta}_2}^0 = \\emptyset$ and $\\mathcal{J}_{-\\bm{\\beta}_2}^\\pm = \\mathcal{J}_{\\bm{\\beta}_2}^\\mp$, we may apply the analysis above with $\\bm{\\beta}_2$ replaced by $-\\bm{\\beta}_2$ and conclude that\n\\begin{equation} \\label{eq:JMinusBeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+| = |\\mathcal{J}_{-\\bm{\\beta}_2}^-| = s.\n\\end{equation}\nIn particular, \\eqref{eq:Jbeta2MinusCard+tri} and \\eqref{eq:JMinusBeta2+tri} imply\n\\begin{align}\\label{eq4:Jbeta2tri}\nr=2s.\n\\end{align}} \n\n\\begin{comment}\n\\begin{align}\\label{eq3:Jbeta2+tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|=r-s.\n\\end{align}\n\n\n\nWhen $t$ is small enough and $t < 0$, we have the following.\n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1+t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2-tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2-tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:beta2trinon-empty'}, \\eqref{eq1:Jbeta2-tri} and \\eqref{eq2:Jbeta2-tri} into account, we have\n\\begin{align}\\label{eq3:Jbeta2-tri}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|=s.\n\\end{align}\nCombining \\eqref{eq3:Jbeta2+tri} with \\eqref{eq3:Jbeta2-tri}, we arrive at\n\\begin{align}\\label{eq4:Jbeta2tri}\nr=2s.\n\\end{align}\n\\end{comment}\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_1$}\nNow we perturb the eigenvalues along $\\bm{\\beta}_1$.\nThe case when ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^{\\pm}$ is similar to that of $\\bm{\\beta}_2$.\nThe difference here is $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$.\n\nBy Lemma~\\ref{lem:constructiontri}, we have\n\\begin{align}\n\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big) + \\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)\n=\n-1+\\sqrt{E+3}\n\\neq\n0\n\\end{align}\nfor ${\\bm{\\ell} = (\\ell_1,\\ell_2)} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0$.\nThus, by employing \\eqref{eq:pertgeneralbetatri2order}, we obtain\n\\begin{align}\n&e_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\\notag\\\\\n&=\nE - {\\frac{ t^2}{2(p_1^2+p_2^2)}} \\Big(\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big) + \\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big) \\Big) + O(t^3)\\notag\\\\\n& = E - {\\frac{ t^2}{2(p_1^2+p_2^2)}} \\Big(-1+\\sqrt{E+3} \\Big)+O(t^3).\\label{eq:Jbeta10tri}\n\\end{align}\nNotice that the choice of $\\bm{\\beta}_1$ causes the {third} $t^2$ term of \\eqref{eq:pertgeneralbetatri2order} to drop out.\n\nWithout loss of generality, we assume $E\\in (-2, 6)$. The other case can be handled similarly.\nFor $E\\in (-2,6)$, \\eqref{eq:Jbeta10tri} implies that \n\\begin{align}\\label{eq:Jbeta10tri'}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1) < E = \\min F_{k+1},\n\\end{align}\nholds for $|t|>0$ small enough and for any $\\bm{\\ell} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0$. \n\nCombining \\eqref{eq:Jbeta10tri'} with \\eqref{eq:pertgeneralbetatri1order}, we have the following.\n\nFor $t>0$ small enough,\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq r-s\n=\ns,\n\\end{align}\n{where the equality follows from \\eqref{eq4:Jbeta2tri}.}\n\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0 \\bigcup \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k-s-1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:sumJbetatri} and \\eqref{eq4:Jbeta2tri}, Equations~\\eqref{eq1:Jbeta1+tri} and \\eqref{eq2:Jbeta1+tri} yield\n\\begin{equation} \\label{eq3:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\ns.\n\\end{equation}\nAs before, we may observe that ${\\mathcal J}_{-\\bm{\\beta}_1}^0 = {\\mathcal J}_{\\bm{\\beta}_1}^0$ and ${\\mathcal J}_{-\\bm{\\beta}_1}^\\pm = {\\mathcal J}_{\\bm{\\beta}_1}^\\mp$. Then, the analysis above applied with $\\bm{\\beta}_1$ replaced by $-\\bm{\\beta}_1$ forces\n\\begin{equation} \\label{eq3:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\ns.\n\\end{equation}\nTaken together, \\eqref{eq3:Jbeta1+tri} and \\eqref{eq3:Jbeta1-tri} imply $|{\\mathcal J}_{\\bm{\\beta}_1}^0| = 0$, which contradicts ${\\mathcal J}_{\\bm{\\beta}_1}^0 \\neq \\emptyset$.\n}\n\n\\begin{comment}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta1+tri} and \\eqref{eq2:Jbeta1+tri} into account, we have\n\\begin{align}\\label{eq3:Jbeta1+tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|=r-s.\n\\end{align}\n\n\n\nFor $t<0$ small enough, \n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_1}^0 \\bigcup \\mathcal{J}_{\\bm{\\beta}_1}^+ $, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta1-tri} and \\eqref{eq2:Jbeta1-tri} into account, we have\n\\begin{align}\\label{eq3:Jbeta1-tri}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|=s.\n\\end{align}\nCombining \\eqref{eq3:Jbeta1+tri} with \\eqref{eq3:Jbeta1-tri}, we arrive at\n\\begin{align}\\label{eq4:Jbeta1tri}\nr=2s-|\\mathcal{J}_{\\bm{\\beta}_1}^0|.\n\\end{align}\nSince $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$, \\eqref{eq4:Jbeta1tri} is in contradiction with \\eqref{eq4:Jbeta2tri}.\n\\end{comment}\n\n\\end{proof}\n\n\\subsubsection{$E=-2$}\\\n\nFirst, we would like to make a remark on our strategy of the proof of the $E=-2$ case, and on the importance of one of the periods being odd.\n\\begin{remark}\\label{rem:tri}\nWe will choose $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1, \\widetilde{\\theta}_2)$ and $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)}, \\ell_2^{(1)})$ such that $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=-2$ and \n$\\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=\\bm{0}$. \nLemma \\ref{lem:triJ0empty} yields three possibilities $(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)}), p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)}))=(0,\\pi)$, $(\\pi, 0)$ or $(\\pi, \\pi)$.\nDepending on which one of $p_1, p_2$ is odd, we will choose $(0,\\pi)$ (if $p_1$ is odd), or $(\\pi, 0)$ (if $p_2$ is odd).\nThis choice guarantees that the only eigenvalue located at $-2$ with vanishing gradient is $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})$.\nConsequently, it suffices to control the second order perturbation of (a single eigenvalue) $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})$ along a given direction $(\\beta_1, \\beta_2)$.\nWhen $p_1$ is odd, this is equivalent to controlling the sign of the following expression (compare \\eqref{eq:remtri}):\n$$-\\beta_{2}\\Big(\\frac{\\beta_{1}}{p_1}-\\frac{\\beta_{2}}{p_2}\\Big).$$\nWe can easily choose two directions such that the expression above has different signs, which leads to un-even {eigenvalue counts and hence to the desired contradiction}.\n\n{{\\it A posteriori}, the existence of a $(2,2)$-periodic potential satisfying the conclusion of Theorem~\\ref{t:triExamples} implies that this argument must fail if both $p_1$ and $p_2$ are even; let us briefly describe why this must be the case.} If both $p_1, p_2$ are even, there will be three eigenvalues at $-2$ with vanishing gradients, corresponding to all three solutions $(0,\\pi)$, $(\\pi, 0)$, $(\\pi, \\pi)$.\nTrying to control the second order perturbations of all these three eigenvalues along $(\\beta_1, \\beta_2)$ is equivalent to controlling the signs of the following three expressions simultaneously\n\\begin{align*}\n-\\beta_{2}\\Big(\\frac{\\beta_{1}}{p_1}-\\frac{\\beta_{2}}{p_2}\\Big),\\ \\ \n\\beta_{1}\\Big(\\frac{\\beta_{1}}{p_1}-\\frac{\\beta_{2}}{p_2}\\Big),\\ \\ \\text{and}\\ \\ \n\\beta_1\\beta_2.\n\\end{align*}\nA simple inspection of these three expressions yields that two of them are always non-negative with the other one being non-positive. \nTherefore we can never choose two different directions that lead to un-even {eigenvalue counts}.\nThis explains why at least one of the periods must be odd for our argument to work. \n\\end{remark}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:trimain}.2]\nNow let us give a detailed proof. Without loss of generality, assume $p_1$ is odd, let $E = -2$, and assume for the sake of contradiction that $E = \\max F_k = \\min F_{k+1}$ for some $k$.\nWe choose $\\widetilde\\bm{\\theta}$ and $\\bm{\\ell}^{(1)}$ via\n\\begin{align*}\n\\widetilde{\\theta}_1=0,\\ {\\ell_1^{(1)}} = 0,\\ \\ \\ \n(\\widetilde{\\theta}_2, {\\ell_2^{(1)}})=\n\\begin{cases}\n\\Big(0, \\frac{p_2}{2}\\Big),\\ \\ \\text{if } p_2 \\text{ is even},\\\\\n\\Big(\\pi, \\frac{p_2-1}{2}\\Big),\\ \\ \\text{if } p_2 \\text{ is odd}.\n\\end{cases}\n\\end{align*}\n\nWith these choices of $\\bm{\\ell}^{(1)}$ and $\\widetilde{\\bm{\\theta}}$, one can check that $e_{{\\bm{\\ell}^{(1)}}}(\\widetilde\\bm{\\theta}) = -2 = E$. As before, let $r$ denote the multiplicity of $E$ and let $\\Lambda_E(\\widetilde{\\bm{\\theta}})$ denote the set of $\\bm{\\ell} \\in \\Lambda$ with $e_{\\bm{\\ell}}(\\widetilde\\bm{\\theta}) = -2$.\nNote that we also have $\\nabla e_{{\\bm{\\ell}^{(1)}}}(\\widetilde\\bm{\\theta}) = \\bm{0}$, and thus $\\mathcal{J}_0\\neq \\emptyset$. Moreover, we claim that ${\\mathcal J}_0 = \\{\\bm{\\ell}^{(1)}\\}$. To see this, suppose there exists $\\bm{\\ell} \\neq \\bm{\\ell}^{(1)}$ in ${\\mathcal J}_0$. In view of Lemma~\\ref{lem:triJ0empty}, we must have\n\\[\\frac{\\widetilde{\\theta}_1+2\\pi\\ell_1}{p_1}=\\pi,\\]\nwhich implies $p_1 = 2\\ell_1$, which is impossible, since $p_1$ is odd.\nConsequently,\n\\[\\mathcal{J}_0=\n{\\{ \\bm{\\ell}^{(1)} \\}}.\\]\n\nLet us choose $\\bm{\\beta}_1=(\\beta_{1,1},\\beta_{1,2})=(0,1)$ and a unit vector\n\\[\\bm{\\beta}_2=(\\beta_{2,1},\\beta_{2,2}) \\sim (2p_1, p_2)\/\\sqrt{4p_1^2+p_2^2}\\] \nsuch that \n\\begin{align}\\label{eq:beta2triodd1}\n\\beta_{2,2}\\Big(\\frac{\\beta_{2,1}}{p_1}-\\frac{\\beta_{2,2}}{p_2}\\Big)>0,\n\\end{align}\nand\n\\begin{align}\\label{eq:beta2triodd2}\n\\bm{\\beta}_2\\cdot \\nabla e_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}})\\neq 0\\ \\ \\text{holds for any }\\ \\ell \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) \\setminus \\{\\bm{\\ell}^{(1)}\\}.\n\\end{align}\n\nWe will use \\eqref{eq:beta2triodd1} to control the perturbation of $e_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}})$ along the $\\bm{\\beta}_2$ direction. \nWe also note that \\eqref{eq:beta2triodd2} simply says \n\\begin{align}\\label{eq:beta20triodd}\n\\mathcal{J}_{\\bm{\\beta}_2}^0=\\mathcal{J}_{0}\n=\n\\{{\\bm{\\ell}^{(1)}}\\}.\n\\end{align}\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_2$}\nWe first perturb the eigenvalues along the $\\bm{\\beta}_2$ direction.\\\n\nBy \\eqref{eq:beta20triodd}, we need only consider first-order perturbation theory as in \\eqref{eq:pertgeneralbetatri1order} for $\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}) \\setminus \\{\\bm{\\ell}^{(1)}\\}$.\nSince ${\\bm{\\ell}^{(1)}} \\in \\mathcal{J}_{0}$, we need to employ \\eqref{eq:pertgeneralbetatri2order} for $e_{{\\bm{\\ell}^{(1)}}}$.\nIndeed, by \\eqref{eq:pertgeneralbetatri2order}, we have for $|t|>0$ small enough,\n\\begin{align}\\label{eq:remtri}\ne_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n& =\ne_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}})\n-\\frac{t^2}{2}\n\\Bigg[\\frac{\\beta_{2,1}^2}{p_1^2}-\\frac{\\beta_{2,2}^2}{p_2^2}-\\Big(\\frac{\\beta_{2,1}}{p_1}-\\frac{\\beta_{2,2}}{p_2}\\Big)^2\\Bigg]+O(t^3) \\notag\\\\\n& =-2-\\frac{\\beta_{2,2}}{p_2}\\Big(\\frac{\\beta_{2,1}}{p_1}-\\frac{\\beta_{2,2}}{p_2}\\Big)t^2+O(t^3)\\\\\n& <-2 \\notag\\\\\n& = \\min F_{k+1}, \\notag\n\\end{align}\nwhere we used \\eqref{eq:beta2triodd1} in the last inequality.\n\nFor $t > 0$ small enough, we then have the following.\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq r-s.\n\\end{align}\n\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_2}^-\\bigcup \\mathcal{J}_0$, we have \n\\[\nE_{k-s}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_2)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|+|\\mathcal{J}_0|\\leq s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, {\\eqref{eq:beta20triodd},} \\eqref{eq1:Jbeta2+triodd}, and \\eqref{eq2:Jbeta2+triodd} into account, we have\n\\begin{align}\\label{eq3:Jbeta2+triodd}\n{|\\mathcal{J}_{\\bm{\\beta}_2}^-|=s-1}.\n\\end{align}\n{Replacing $\\bm{\\beta}_2$ by $-\\bm{\\beta}_2$ as in previous phases of the argument, we arrive at\n\\begin{equation} \\label{eq3:Jbeta2-triodd}\n|{\\mathcal J}_{\\bm{\\beta}_2}^+|\n=\ns-1.\n\\end{equation}}\n\n\\begin{comment}\nWhen $t$ is small enough and $t < 0$, we have the following.\n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^+\\bigcup \\mathcal{J}_0$, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1+t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta2-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_0|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta2-triodd} and \\eqref{eq2:Jbeta2-triodd} into account, we have\n\\begin{align}\\label{eq3:Jbeta2-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_0|=s.\n\\end{align}\n\\end{comment}\nCombining \\eqref{eq3:Jbeta2+triodd} with \\eqref{eq3:Jbeta2-triodd}, we arrive at\n\\begin{align}\\label{eq4:Jbeta2triodd}\n{r\n= |{\\mathcal J}_{\\bm{\\beta}_2}^+| + |{\\mathcal J}_{\\bm{\\beta}_2}^-| + |{\\mathcal J}_0|\n=\n2s-1.}\n\\end{align}\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_1$}\nNow we perturb the eigenvalues along $\\bm{\\beta}_1 = (0,1)$.\nThe case when ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^{\\pm}$ is similar to that of $\\bm{\\beta}_2$.\nThe difference here is the {behavior of} perturbations of $e_{{\\bm{\\ell}^{(1)}}}$ {in the direction $\\bm{\\beta}_1$.}\nIndeed, by \\eqref{eq:pertgeneralbetatri2order}, we have\n\\begin{align*}\ne_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n&=e_{{\\bm{\\ell}^{(1)}}}(\\widetilde{\\bm{\\theta}})\n-\\frac{t^2}{2}\n\\Bigg[\\frac{\\beta_{1,1}^2}{p_1^2}-\\frac{\\beta_{1,2}^2}{p_2^2}-\\Big(\\frac{\\beta_{1,1}}{p_1}-\\frac{\\beta_{1,2}}{p_2}\\Big)^2\\Bigg]+O(t^3)\\\\\n&=-2+\\frac{t^2}{p_2^2}+O(t^3)\\\\\n&>-2=\\max F_{k}.\n\\end{align*}\nThus, the perturbations of $e_{{\\bm{\\ell}^{(1)}}}$ {in the direction $\\bm{\\beta}_1$} always move up.\n\nFor $t>0$ small enough,\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^+\\bigcup \\mathcal{J}_0$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|+|\\mathcal{J}_0| \\leq r-s.\n\\end{align}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k-s-1}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t\\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1+triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-| \\leq s.\n\\end{align}\n\\end{itemize}\nIn view of \\eqref{eq:sumJbetatri}, Equations~\\eqref{eq1:Jbeta1+triodd} and \\eqref{eq2:Jbeta1+triodd} yield\n\\begin{align}\\label{eq3:Jbeta1+triodd}\n{|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\ns.}\n\\end{align}\n{Applying the usual symmetry argument, we also arrive at $|{\\mathcal J}_{\\bm{\\beta}_1}^+| = s$, which leads to\n\\[\nr\n=\n|{\\mathcal J}_{\\bm{\\beta}_1}^+| + |{\\mathcal J}_{\\bm{\\beta}_1}^-| + |{\\mathcal J}_0|\n=\n2s+1,\n\\]\nwhich in turn contradicts \\eqref{eq4:Jbeta2triodd}.}\n\\end{proof}\n\n\\begin{comment}\nFor $t<0$ small enough, \n\\begin{itemize}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_1}^+ $, we have\n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq s.\n\\end{align}\n\\item If $(\\ell_m,k_m)\\in \\mathcal{J}_{\\bm{\\beta}_1}^-\\bigcup \\mathcal{J}_0$, we have \n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\ne_{\\ell_m,k_m}(\\widetilde{\\theta}_1 + t\\beta_{1,1}, \\widetilde{\\theta}_2 + t\\beta_{1,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|+|\\mathcal{J}_0|\\leq r-s.\n\\end{align}\n\\end{itemize}\nTaking \\eqref{eq:sumJbetatri}, \\eqref{eq1:Jbeta1-triodd} and \\eqref{eq2:Jbeta1-triodd} into account, we have\n\\begin{align}\\label{eq3:Jbeta1-triodd}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|=s.\n\\end{align}\nCombining \\eqref{eq3:Jbeta1+triodd} with \\eqref{eq3:Jbeta1-triodd}, we arrive at\n\\begin{align}\\label{eq4:Jbeta1triodd}\nr=2s+|\\mathcal{J}_0|=2s+1.\n\\end{align}\nThis contradicts \\eqref{eq4:Jbeta2triodd}.\n\\end{comment}\n\n\n\n\\subsection{Proof of Lemmas \\ref{lem:constructiontri} and \\ref{lem:triJ0empty}}\\label{sec:constructionlemmaproof}\n\\begin{proof}[Proof of Lemma~\\ref{lem:constructiontri}]\nLet {$E \\in [-3,6]$ be given}, let $x$ be as-yet-unspecified, set $y = 2\\pi - x$, and note that \\eqref{eq:xyCondB} holds. Then, using $y = 2\\pi - x$, we note that\n\\begin{align*}\n\\cos(x) + \\cos(y) + \\cos(x-y)\n& =\n2\\cos(x) + \\cos(2x) \\\\\n& =\n2\\cos(x) + 2\\cos^2(x) - 1.\n\\end{align*}\nSetting $z = \\cos(x)$, we seek to solve $2z+2z^2-1 = E\/2$, which gives\n\\[\nz^2 + z - \\frac{1}{2} - \\frac{E}{4} = 0\n\\implies\nz = \n\\frac{-1 \\pm \\sqrt{3+E}}{2}.\n\\]\nThus, we may take $x$ so that\n\\[\n\\cos(x)\n=\n\\frac{-1+\\sqrt{3+E}}{2}.\n\\]\nIn fact, since $-3 \\le E \\le 6$, we may take $0 \\le x \\le 2\\pi\/3$. Thus, with this choice of $x$ (and $y = 2\\pi - x$), we get \\eqref{eq:xyCondA}. \n\nFinally, suppose $x$ and $y$ solve \\eqref{eq:xyCondA} and \\eqref{eq:xyCondB} for $E \\neq -2$. From \\eqref{eq:xyCondB}, we deduce that either $x+y = 2\\pi$ or $|x-y|=\\pi$. The second option leads to $E = -2$, so we must have $y=2\\pi - x$. Then, $E \\neq -2$ guarantees\n\\[\n\\cos(x) + \\cos(2\\pi - x)\n=\n2\\cos(x)\n=\n-1 + \\sqrt{3+E}\n\\neq 0,\n\\]\nwhich proves \\eqref{eq:xyCondC}.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:triJ0empty}]\nSuppose that $x$ and $y$ solve\n\\begin{align}\n\\label{eq:triJ0empty:cossum}\n\\cos(x) + \\cos(y) + \\cos(x-y)\n& =\n\\lambda \\\\\n\\label{eq:triJ0empty:sin1}\n\\sin(x) + \\sin(x-y) & = 0 \\\\\n\\label{eq:triJ0empty:sin2}\n\\sin(y) - \\sin(x-y) & = 0\n\\end{align}\nfor some $\\lambda \\in (-3\/2,3)$. Adding \\eqref{eq:triJ0empty:sin1} and \\eqref{eq:triJ0empty:sin2}, we arrive at \n\\[\n\\sin(x) = - \\sin(y).\n\\]\nFor $(x,y) \\in [0,2\\pi)^2$, this forces either $|x-y| = \\pi$ or $x+y = 2\\pi$. In the case $|x-y| = \\pi$, substituting in to \\eqref{eq:triJ0empty:sin1} and \\eqref{eq:triJ0empty:sin2} gives $\\sin(x) = \\sin(y) = 0$, forcing $x,y \\in \\{0,\\pi\\}$. Plugging the various possibilities into \\eqref{eq:triJ0empty:cossum}, one either gets $\\lambda = 3 \\notin (-3\/2,3)$ (when $x=y=0$) or $\\lambda = -1$ (when at least one of $x$ or $y$ is $\\pi$).\n\nAlternatively, if $x = 2\\pi - y$, \\eqref{eq:triJ0empty:sin1} yields $\\sin(x) + \\sin(2x) = 0$, which leads to\n\\[\n\\sin(x)(1 + 2\\cos(x))\n=\n0.\n\\]\nSetting $\\sin(x) = 0$ yields $x \\in \\{0,\\pi\\}$ which leads to the same solutions as before. Setting $1 +2\\cos(x) = 0$ yields $(x,y) = (2\\pi\/3,4\\pi\/3)$ or $(x,y) = (4\\pi\/3, 2\\pi\/3)$. Plugging in either possibility into \\eqref{eq:triJ0empty:cossum} yields\n\\[\n\\cos(x) + \\cos(y) + \\cos(x-y)\n=\n-\\frac{3}{2} \\notin (-3\/2,3),\n\\]\nas claimed.\n\\end{proof}\n\n\n\n\n\n\\subsection{\\boldmath Opening a Gap at $-2$}\nLet us exhibit a $(2,2)$-periodic potential that perturbatively opens a gap at energy $E = -2$ for the triangular lattice.\n\n\\begin{figure*}[t]\n\n\\begin{tikzpicture}[yscale=.8,xscale=.8]\n\\draw [-,line width = .06cm] (0,0) -- (9,0);\n\\draw [-,line width = .06cm] (0,0) -- (0,9);\n\\draw [-,line width=.06cm] (0,9) -- (9,9);\n\\draw [-,line width = .06cm] (9,0) -- (9,9);\n\\draw [-,line width = .06cm] (0,9) -- (9,0);\n\\draw [-,line width = .06cm] (3,0) -- (3,9);\n\\draw [-,line width = .06cm] (6,0) -- (6,9);\n\\draw [-,line width = .06cm] (0,3) -- (9,3);\n\\draw [-,line width = .06cm] (0,6) -- (9,6);\n\\draw [-,line width = .06cm] (0,3) -- (3,0);\n\\draw [-,line width = .06cm] (0,6) -- (6,0);\n\\draw [-,line width = .06cm] (3,9) -- (9,3);\n\\draw [-,line width = .06cm] (6,9) -- (9,6);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](0,3) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=black, fill=black](0,9) circle (.2);\n\\filldraw[color=black, fill=black](3,0) circle (.2);\n\\filldraw[color=red, fill=red](3,3) circle (.2);\n\\filldraw[color=red, fill=red](3,6) circle (.2);\n\\filldraw[color=black, fill=black](3,9) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=red, fill=red](6,3) circle (.2);\n\\filldraw[color=red, fill=red](6,6) circle (.2);\n\\filldraw[color=black, fill=black](6,9) circle (.2);\n\\filldraw[color=black, fill=black](9,0) circle (.2);\n\\filldraw[color=black, fill=black](9,3) circle (.2);\n\\filldraw[color=black, fill=black](9,6) circle (.2);\n\\filldraw[color=black, fill=black](9,9) circle (.2);\n\\draw [-,line width = .06cm,color=red] (3,3) -- (3,6);\n\\draw [-,line width = .06cm,color=red] (3,3) -- (6,3);\n\\draw [-,line width = .06cm,color=red] (6,3) -- (6,6);\n\\draw [-,line width = .06cm,color=red] (3,6) -- (6,6);\n\\draw [-,line width = .06cm,color=red] (6,3) -- (3,6);\n\\node at (3.8,3.5) {$\\hot{q_1=1}$};\n\\node at (6.8,3.5) {$\\hot{q_2=1}$};\n\\node at (3.8,6.5) {$\\hot{q_3=1}$};\n\\node at (7,6.5) {$\\hot{q_4=-1}$};\n\\end{tikzpicture}\n\\caption{A $(2,2)$ periodic potential on the triangular lattice with a gap at $E = -2$ for all positive coupling constants.}\\label{fig:tri2x2fundDomain}\n\\end{figure*}\n\n\\begin{theorem} \\label{thm:triExampleGapLength}\nDefine\n\\[\nQ_{n,m} \n= \n(-1)^{mn}\n=\n\\begin{cases}\n1 & \\text{ if } m \\text{ or } n \\text{ is even}, \\\\\n-1 & \\text{ if both } m \\text{ and } n \\text{ are odd,}\n\\end{cases}\n\\]\nand denote $H_\\lambda = \\Delta_{\\mathrm{tri}} + \\lambda Q$. For all $\\lambda > 0$, $\\sigma(H_\\lambda)$ has two connected components. Moreover, for all $\\lambda > 0$ sufficiently small, the gap that opens about $E=-2$ is precisely equal to\n\\[\n\\mathfrak{g}_\\lambda\n=\n\\left(-\\sqrt{4+\\lambda^2},-2+\\lambda \\right).\n\\]\nIn particular,\n\\[\n|\\mathfrak{g}_\\lambda|\n=\n\\lambda + \\left( \\sqrt{4+\\lambda^2}-2 \\right)\n\\sim\n\\lambda + \\frac{\\lambda^2}{2},\n\\]\nso the gap opens linearly as $\\lambda \\downarrow 0$.\n\\end{theorem}\n\nThe following lemma will be used:\n\n\\begin{lemma} \\label{lem:triGapLength:trigPolyEst}\nFor all $\\bm{\\theta} \\in {\\mathbb T}^2$ and all $0 \\leq a \\leq 54$,\n\\[\n4(\\sin\\theta_1 + \\sin\\theta_2 - \\sin(\\theta_1+\\theta_2))^2\n+ a(1+\\cos\\theta_1+\\cos\\theta_2+\\cos(\\theta_1+\\theta_2))\n\\geq 0.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nDefine\n\\[\ng(\\theta_1,\\theta_2,a)\n=\n4(\\sin\\theta_1 + \\sin\\theta_2 - \\sin(\\theta_1+\\theta_2))^2\n+ a(1+\\cos\\theta_1+\\cos\\theta_2+\\cos(\\theta_1+\\theta_2)).\n\\]\nWe begin by checking the boundary of ${\\mathbb T}^2 \\times [0,54]$. It is easy to see that $g \\geq 0$ if $a = 0$. For $a = 54$, define $h(\\bm{\\theta}) = g(\\bm{\\theta},54)$. Using the identities \n\\begin{align*}\n\\sin x+\\sin y-\\sin(x+y)&=4\\sin\\left(\\frac{x}{2}\\right) \\sin\\left(\\frac{y}{2}\\right) \\sin\\left(\\frac{x+y}{2}\\right),\\\\\n\\cos x-\\cos (x+y)&=2\\sin\\left(\\frac{y}{2}\\right) \\sin\\left(x+\\frac{y}{2}\\right)\\\\\n\\sin x+\\sin (x+y)&=2\\cos\\left(\\frac{y}{2}\\right) \\sin\\left(x+\\frac{y}{2}\\right),\n\\end{align*}\nwe may simplify $\\nabla h$ to get\n\\begin{align}\\label{eq:A3}\n\\frac{\\partial h}{\\partial \\theta_1}\n& =\n4\\sin\\left(\\theta_1+\\frac{\\theta_2}{2}\\right)\n\\Bigg[\n16\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_2}{2}\\right)\n\\Bigg],\\\\\n\\label{eq:A4}\n\\frac{\\partial h}{\\partial \\theta_2}\n& =\n4\\sin\\left(\\theta_2+\\frac{\\theta_1}{2}\\right)\n\\Bigg[\n16\\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_1}{2}\\right)\n\\Bigg].\n\\end{align}\nConsequently, setting $\\nabla h = 0$ leads to four cases. For notational convenience, define\n\\[\n\\alpha\n=\n\\arcsin\\sqrt[4]{\\frac{27}{32}} .\n\\]\n\\noindent \\textbf{Case 1.}\n\\[\\sin\\left(\\theta_1+\\frac{\\theta_2}{2}\\right)=\\sin\\left(\\theta_2+\\frac{\\theta_1}{2}\\right)=0.\\]\nThis implies $\\theta_1 + \\frac{1}{2}\\theta_2 \\in \\pi {\\mathbb Z}$ and $\\theta_2 + \\frac{1}{2}\\theta_1 \\in \\pi {\\mathbb Z}$. Solving the resulting systems for solutions in $[0,2\\pi)$ yields three points:\n\\[\\bm{\\theta}=(0,0),\\ \\left( \\frac{2\\pi}{3},\\frac{2\\pi}{3}\\right),\\ \\left( \\frac{4\\pi}{3},\\frac{4\\pi}{3}\\right).\\]\n\\noindent \\textbf{Case 2.}\n\\[\\sin\\left(\\theta_1+\\frac{\\theta_2}{2}\\right)=16\\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_1}{2}\\right)=0.\\]\nAs before, the first condition forces $\\theta_1 + \\frac{1}{2} \\theta_2 \\in \\pi{\\mathbb Z}$. Plugging the various possibilities that this yields into the second condition gives three solutions:\n\\begin{align*}\n\\bm{\\theta}=(\\pi, 0),\\ \\ (2\\alpha,2\\pi - 4\\alpha),\\ \\ (2\\pi-2\\alpha,4\\alpha).\n\\end{align*}\n\\noindent \\textbf{Case 3.}\n\\[\\sin\\left(\\theta_2+\\frac{\\theta_1}{2}\\right)=16\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_2}{2}\\right)=0.\\]\nArguing as in Case~2, there are three solutions:\n\\begin{align*}\n\\bm{\\theta}=(0,\\pi),\\ \\ (2\\pi - 4\\alpha,2\\alpha),\\ \\ (4\\alpha,2\\pi-2\\alpha).\n\\end{align*}\n\\noindent \\textbf{Case 4.}\n\\begin{align}\n\\label{eq:triGapLength:trigPolyEst:Case4a}\n16\\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_1}{2}\\right)&=0\\\\\n\\label{eq:triGapLength:trigPolyEst:Case4b}\n16\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right) \n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)- 27 \\cos\\left(\\frac{\\theta_2}{2}\\right)&=0.\n\\end{align}\nMultiply \\eqref{eq:triGapLength:trigPolyEst:Case4a} by $\\sin(\\theta_2\/2)$, multiply \\eqref{eq:triGapLength:trigPolyEst:Case4b} by $\\sin(\\theta_1\/2)$, and subtract the results to obtain\n\\[\n\\sin\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)\n=\n0.\n\\]\nUsing this, we see that the solutions are \n\\begin{align*}\n\\bm{\\theta}=(\\pi, \\pi),\\ \\ (2\\alpha,2\\alpha),\\ \\ (2\\pi - 2\\alpha,2\\pi - 2\\alpha)\n\\end{align*}\nEvaluating $g$ at these points, we find out $\\max h(\\bm{\\theta})=216$ attained at $(0,0)$, \n$\\min h(\\bm{\\theta})= 0$, attained at\n\\[\n\\bm{\\theta}\n=\n\\left( \\frac{2\\pi}{3},\\frac{2\\pi}{3}\\right),\n\\;\n\\left( \\frac{4\\pi}{3},\\frac{4\\pi}{3}\\right),\n\\; (\\pi,0), \\; (0,\\pi), \\; (\\pi,\\pi).\n\\]\n\n\n\nFinally, we need to look at critical points of $g$ in the interior of ${\\mathbb T}^2 \\times [0,54]$. However, this is easy. Any zero of $\\nabla g$ must in particular satisfy $\\frac{\\partial g}{\\partial a} = 0$, which forces\n\\[\n1+\\cos\\theta_1 + \\cos\\theta_2 + \\cos(\\theta_1+\\theta_2)=0,\n\\]\nwhich clearly implies $g \\geq 0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:triExampleGapLength}]\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb T}^2$, denote by $H_\\lambda(\\bm{\\theta})$ the Floquet matrix corresponding to $H_\\lambda$. Ordering the vertices of the $2\\times 2$ fundamental domain as shown in Figure~\\ref{fig:tri2x2fundDomain}, we obtain\n\\[\nH_\\lambda(\\bm{\\theta}) -(-2+\\varepsilon) {\\mathbb I}\n=\n\\begin{bmatrix}\n2+\\lambda-\\varepsilon & 1+ e^{-i\\theta_1} & 1 + e^{-i\\theta_2}& 1 + e^{-i(\\theta_1+\\theta_2)} \\\\\n1 + e^{i\\theta_1}& 2+\\lambda -\\varepsilon & e^{i\\theta_1} + e^{-i\\theta_2} & 1 + e^{-i\\theta_2} \\\\\n1 + e^{i\\theta_2} & e^{-i\\theta_1} + e^{i\\theta_2} & 2+\\lambda -\\varepsilon & 1 + e^{-i\\theta_1} \\\\\n1 + e^{i(\\theta_1+\\theta_2)}& 1 + e^{i\\theta_2} & 1 + e^{i\\theta_1} & 2 - \\lambda -\\varepsilon\n\\end{bmatrix}\n\\]\n\n\n\n\nFor $\\bm{\\theta} \\in {\\mathbb T}^2$, $\\lambda>0$, and $\\varepsilon \\in {\\mathbb R}$, define\n\\[\np(\\bm{\\theta},\\lambda,\\varepsilon)\n=\n\\det(H_\\lambda(\\bm{\\theta}) - (-2+\\varepsilon) {\\mathbb I}).\n\\]\nAfter some calculations, one observes that\n\\begin{align*}\np(\\bm{\\theta},\\lambda,\\varepsilon)\n=\n-\\lambda^4 & - 4\\lambda^3 + X(\\bm{\\theta}) - 4\\varepsilon \\lambda\\left(3 - \\frac{\\lambda^2}{2}- \\cos\\theta_1 - \\cos\\theta_2 - \\cos(\\theta_1+\\theta_2) \\right) \\\\\n & + 4\\varepsilon^2 (3+3\\lambda - \\cos\\theta_1 - \\cos\\theta_2 - \\cos(\\theta_1+\\theta_2)) \\\\\n & - 2\\varepsilon^3(4 + \\lambda) + \\varepsilon^4,\n\\end{align*}\nwhere \n\\[\nX(\\bm{\\theta}) = \n-4\\big(\\sin\\theta_1+\\sin\\theta_2 - \\sin(\\theta_1+\\theta_2)\\big)^2\n\\]\nClearly $X(\\bm{\\theta}) \\leq 0$ for all $\\bm{\\theta}$, so we have\n\\[\n\\det(H_\\lambda(\\bm{\\theta}) + 2{\\mathbb I})\n=\np(\\bm{\\theta},\\lambda,0)\n\\leq\n-\\lambda^4 - 4\\lambda^3 <0\n\\]\nfor all $\\lambda > 0$; consequently $-2 \\notin \\sigma(H_\\lambda)$ for all $\\lambda > 0$, which proves the first claim of the theorem. Introducing $W_1(\\lambda,\\varepsilon) := -\\lambda^4 -4\\lambda^3 + 2 \\varepsilon\\lambda^3 + 12\\varepsilon^2\\lambda - 2\\varepsilon^3(4+\\lambda) + \\varepsilon^4$, we may rewrite $p$ as\n\\begin{align} \\label{eq:triPrewrittenW1}\np(\\bm{\\theta},\\lambda,\\varepsilon)\n=\n X(\\bm{\\theta}) - 4\\varepsilon(\\lambda-\\varepsilon)(3-\\cos\\theta_1-\\cos\\theta_2-\\cos(\\theta_1+\\theta_2)) +W_1(\\lambda,\\varepsilon).\n \\end{align}\nBy standard eigenvalue perturbation theory, we know that $|g_\\lambda^\\pm+2| \\leq \\lambda$, so we need only concern ourselves with $|\\varepsilon| \\leq \\lambda$. Since $X(\\bm{\\theta}) \\leq 0$ for all $\\bm{\\theta}$ and the second term of \\eqref{eq:triPrewrittenW1} is nonpositive whenever $0 \\leq \\varepsilon \\leq \\lambda$, we arrive at\n\\[\np(\\bm{\\theta},\\lambda,\\varepsilon)\n\\leq\n-\\lambda^4 -4\\lambda^3 + 2 \\varepsilon\\lambda^3 + 12\\varepsilon^2\\lambda - 2\\varepsilon^3(4+\\lambda) + \\varepsilon^4 \n= \n W_1(\\lambda,\\varepsilon)\n\\]\nfor all $\\bm{\\theta} \\in {\\mathbb T}^2$, all $\\lambda > 0$, and all $0 \\leq \\varepsilon \\leq \\lambda$. Moreover, we observe that $p(\\bm{0},\\lambda,\\varepsilon) = W_1(\\lambda,\\varepsilon)$, so this bound is sharp. Factoring $W_1$, we arrive at\n\\[\nW_1(\\lambda,\\varepsilon)\n=\n(\\lambda - \\varepsilon)^2(\\varepsilon^2 - 8\\varepsilon - \\lambda^2 - 4\\lambda).\n\\]\nConsequently, we see that $W_1(\\lambda,\\varepsilon) < 0$ for $\\varepsilon \\in [0,\\lambda)$, which implies that $p(\\bm{\\theta},\\lambda,\\varepsilon) < 0$ for all $\\bm{\\theta} \\in {\\mathbb T}^2$, all $\\lambda > 0$, and all $0 \\le \\varepsilon < \\lambda$; consequently, $[-2,-2+\\lambda) \\cap \\sigma(H_\\lambda) = \\emptyset$, which is to say:\n\\begin{equation} \\label{eq:triGapRightSide}\n[-2,-2+\\lambda) \\subseteq \\mathfrak{g}_\\lambda.\n\\end{equation}\nOn the other hand, $p(\\bm{0},\\lambda,\\lambda) = 0$, so \n\\begin{equation} \\label{eq:triGapRightSide-2+lambda}\n-2+\\lambda \\in \\sigma(H_\\lambda(\\bm{0})) \\subseteq \\sigma(H_\\lambda)\n\\end{equation} \nAlternatively, $-2+\\lambda \\in \\sigma(H_\\lambda)$ is clear from eigenvalue perturbation theory as soon as one has $[-2,2+\\lambda)\\cap \\sigma(H_\\lambda) = \\emptyset$.\n\n\nNow, for $- \\lambda \\leq \\varepsilon \\leq 0$, we have to be more careful with the term\n\\[\nq(\\bm{\\theta},\\lambda,\\varepsilon)\n:=\n-4\\varepsilon(\\lambda - \\varepsilon)(3- \\cos\\theta_1 - \\cos\\theta_2 - \\cos(\\theta_1+\\theta_2)),\n\\]\nas $q$ can be positive when $-\\lambda < \\varepsilon < 0$. Naively, one can bound\n\\[\n3-\\cos\\theta_1-\\cos\\theta_2 - \\cos(\\theta_1+\\theta_2)\n\\leq\n\\frac{9}{2},\n\\]\nwhich leads to the upper bound of $X(\\bm{\\theta}) + q(\\bm{\\theta},\\lambda,\\varepsilon) \\leq -18\\varepsilon(\\lambda - \\varepsilon)$. However, the maximum of $q$ occurs at the global \\emph{minimum} of $X$, so we can do better. Indeed, for $\\lambda > 0$ small and $-\\lambda \\leq \\varepsilon \\leq 0$, we have\n\\begin{equation} \\label{eq:triGapRefinedXQbound}\nX(\\bm{\\theta}) + q(\\bm{\\theta},\\lambda,\\varepsilon) \\leq - 16\\varepsilon(\\lambda-\\varepsilon).\n\\end{equation}\nIn particular, by Lemma~\\ref{lem:triGapLength:trigPolyEst}, the bound in \\eqref{eq:triGapRefinedXQbound} holds for all $\\varepsilon$ such that $-\\lambda \\leq \\varepsilon\\leq 0$ as long as $8 \\lambda^2 < 54$, i.e.\\ $0 < \\lambda < \\frac{3\\sqrt{3}}{2}$. This then leads us to\n\\begin{align*}\np(\\bm{\\theta},\\lambda,\\varepsilon)\n\\leq\nW_2(\\lambda,\\varepsilon)\n:&=\n-\\lambda^4 -4\\lambda^3 + 2 \\varepsilon\\lambda^3 + 12\\varepsilon^2\\lambda - 2\\varepsilon^3(4+\\lambda) + \\varepsilon^4 - 16\\varepsilon(\\lambda-\\varepsilon) \\\\\n& =\nW_1(\\lambda,\\varepsilon) - 16\\varepsilon(\\lambda-\\varepsilon)\n\\end{align*}\nfor $\\lambda > 0$ small and $-\\lambda \\leq \\varepsilon \\leq 0$. Factoring $W_2$ yields\n\\[\np(\\bm{\\theta},\\lambda,\\varepsilon)\n\\leq\nW_2(\\lambda,\\varepsilon)\n=\n(\\varepsilon - \\lambda)(\\varepsilon - \\lambda- 4)(\\varepsilon^2 - 4\\varepsilon - \\lambda^2)\n\\]\nfor $\\lambda > 0$ small and $-\\lambda \\leq \\varepsilon \\leq 0$. It is straightforward to find the roots of $W_2$ and to observe that $W_2(\\lambda,\\varepsilon) < 0$ when\n\\[\n2 - \\sqrt{4+\\lambda^2}\n<\n\\varepsilon\n\\leq \n0.\\]\nAs a result, this implies $p(\\bm{\\theta},\\lambda,\\varepsilon) < 0$ for all $\\bm{\\theta}$, all $\\lambda > 0$ small, and all $\\varepsilon \\in (2-\\sqrt{4+\\lambda^2},0]$, which in turn yields\n\\begin{equation} \\label{eq:triGapLeftSide}\n(-\\sqrt{4+\\lambda^2},-2] \\subseteq \\mathfrak{g}_\\lambda.\n\\end{equation}\nOn the other hand, \n\\[\np\\left((\\pi,\\pi),\\lambda,2 - \\sqrt{4+\\lambda^2}\\right) \n=\n W_2\\left(\\lambda,2 - \\sqrt{4+\\lambda^2}\\right)\n=\n0,\n\\] \nwhich leads us to conclude \n\\begin{equation} \\label{eq:triGapLeftSide-sqrt4+lambda2}\n-\\sqrt{4+\\lambda^2} \\in \\sigma(H_\\lambda(\\pi,\\pi)) \\subseteq \\sigma(H_\\lambda).\n\\end{equation}\nPutting together \\eqref{eq:triGapRightSide}, \\eqref{eq:triGapRightSide-2+lambda}, \\eqref{eq:triGapLeftSide}, and \\eqref{eq:triGapLeftSide-sqrt4+lambda2}, we obtain\n\\[\n\\mathfrak{g}_\\lambda\n=\n\\left(-\\sqrt{4+\\lambda^2},-2+\\lambda \\right)\n\\]\nfor small $\\lambda$, as promised.\n\n\\end{proof}\n\n\n\n\nThe effort involved in proving Lemma~\\ref{lem:triGapLength:trigPolyEst} in order to improve the constant ``18'' to ``16'' is nontrivial, but worthwhile. In particular, this is exactly what enables the exact factorization of $W_2$ and hence the ability to exactly compute the gap edges.\n\n\n\\section{Hexagonal Laplacian} \\label{sec:hex}\n\n\nWe now continue with the Laplacian on the hexagonal lattice. Let $\\Gamma_{\\mathrm{hex}} = ({\\mathcal V}_{\\mathrm{hex}},{\\mathcal E}_{\\mathrm{hex}})$ and\n\\[\n\\bm{b}_\\pm\n=\n\\frac{1}{2} \\begin{bmatrix} 3 \\\\ \\pm \\sqrt{3} \\end{bmatrix}\n\\]\nbe as in the introduction, let periods $p_1,p_2 \\in {\\mathbb Z}_+$ be given, and view $H = \\Delta_{\\mathrm{hex}}$ as a $(p_1,p_2)$-periodic operator.\nFor this setting, there are two vertices of ${\\mathcal V}_{\\mathrm{hex}}$ in $\\set{s\\bm{b}_+ + t \\bm{b}_- : 0 \\le s, t < 1}$, so our Floquet operator $H(\\bm{\\theta})$ will be $P \\times P$ with $P = 2p_1p_2$. As usual, define $\\Lambda = \\big( [0,p_1)\\times[0,p_2)\\big) \\cap {\\mathbb Z}^2$, denote the eigenvalues of $H(\\bm{\\theta})$ by\n\\[\nE_1^\\Lambda(\\bm{\\theta})\n\\leq\n\\cdots\n\\leq E_P^\\Lambda(\\bm{\\theta}),\n\\]\nand let $F_k^\\Lambda$ for $1 \\le k \\le P$ denote the bands of the spectrum. Our main theorem in this section is the following result.\n\n\n\\begin{theorem} \\label{thm:hexmain}\nLet $p_1,p_2 \\in {\\mathbb Z}_+$ be given.\n\\begin{enumerate}\n\\item Every $E \\in (-3,3) \\setminus \\set{-1,0,1}$ belongs to $\\mathrm{int}(F_j)$ for some $1 \\le j \\le P$.\n\\item If at least one of $p_1$ or $p_2$ is odd, then $-1 \\in \\mathrm{int}(F_k)$ and $+1 \\in \\mathrm{int}(F_\\ell)$ for some $1 \\le k \\le \\ell \\le P$\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:bsc:hex}]\nThis follows immediately from Theorem~\\ref{thm:hexmain}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:hexmain}] \nFrom \\eqref{eq:hexDecomp}, we have\n\\[\n\\Delta_{\\mathrm{hex}}\n=\n\\begin{bmatrix}\n0 & S_1^* + S_2^* + {\\mathbb I} \\\\ S_1 + S_2 + {\\mathbb I} & 0 \\end{bmatrix},\n\\]\nwhere $S_j: \\ell^2({\\mathbb Z}^2) \\to \\ell^2({\\mathbb Z}^2)$ denote the shifts\n\\[\n[S_1 \\psi]_{n,m} = \\psi_{n+1,m},\n\\quad\n[S_2 \\psi]_{n,m} = \\psi_{n,m+1}.\n\\]\nIt is easy to see that\n\\[\nS_1 + S_1^* + S_2 + S_2^* + S_1S_2^* + S_1^*S_2 = \\Delta_{\\mathrm{tri}},\n\\]\nthe triangular Laplacian. Thus, a simple calculation shows that\n\\begin{equation} \\label{eq:hexSquareTri}\n[\\Delta_{\\mathrm{hex}}^2 \\Psi]_{\\bm{n}}\n\\begin{bmatrix} [\\Delta_{\\mathrm{tri}} \\psi^+]_{\\bm{n}} + 3\\psi_{\\bm{n}}^+ \\\\ [\\Delta_{\\mathrm{tri}} \\psi^-]_{\\bm{n}} + 3\\psi_{\\bm{n}}^- \\end{bmatrix}\n\\quad\n\\text{for } \\Psi = \\begin{bmatrix} \\psi^+ \\\\ \\psi^- \\end{bmatrix} \\in \\ell^2({\\mathbb Z}^2,{\\mathbb C}^2).\n\\end{equation}\nThis calculation extends to the Floquet matrices, so we see that for each $1 \\le k \\le P$, the bands of $H = \\Delta_{\\mathrm{hex}}$ obey\n\\[\nF^\\Lambda_{k,{\\mathrm{hex}}}\n=\n-F^\\Lambda_{P+1-k,{\\mathrm{hex}}}\n\\]\nand \n\\begin{equation} \\label{eq:hexTriBandRel}\nF_{k,{\\mathrm{hex}}}^\\Lambda\n=\n\\begin{cases}\n\\sqrt{F^\\Lambda_{k - \\frac{P}{2},{\\mathrm{tri}}}+3} & \\frac{P}{2} < k \\leq P \\\\[3mm]\n-\\sqrt{F^\\Lambda_{\\frac{P}{2}+1-k,{\\mathrm{tri}}}+3} & 1 \\leq k \\leq \\frac{P}{2}\n\\end{cases}\n\\end{equation}\nFrom this, we deduce that $E \\in (-3,3)$ lies in the interior of some $F_{k,{\\mathrm{hex}}}$ if and only if $E^2-3$ lies in the interior of some $F_{\\ell,{\\mathrm{tri}}}$. For $E \\in (-3,3) \\setminus \\set{-1,0,1}$, $E^2-3 \\in (-3,6) \\setminus \\{-2\\}$, while $(\\pm1)^2-3 = -2$. Thus, the conclusions of the theorem follow from Theorem~\\ref{thm:trimain}.\n\n\\end{proof}\n\n\\subsection{\\boldmath Opening gaps at $0$ and $\\pm 1$}\n\nDefine the $(1,1)$-periodic potential $Q_1$ on ${\\mathcal V}_{\\mathrm{hex}}$ by $Q_1(\\bm{0}) = 1$ and $Q_1(\\bm{a}_1) = -1$, that is,\n\\[\nQ_1(n \\bm{b}_+ + m \\bm{b}_-) = 1,\n\\quad\nQ_1(\\bm{a}_1 + n \\bm{b}_+ + m \\bm{b}_-) = -1,\n\\quad n,m \\in {\\mathbb Z}.\n\\]\nAfter identifying $\\ell^2({\\mathcal V}_{\\mathrm{hex}})$ with $\\ell^2({\\mathbb Z}^2,{\\mathbb C}^2)$ in the usual way, we get (as an operator) $[Q_1 \\Psi]_{\\bm{n}}= Z\\Psi_{\\bm{n}}$, where\n\\[\nZ=\\begin{bmatrix}\n1 & 0 \\\\ 0 & -1\n\\end{bmatrix}.\n\\]\n\nFrom the calculations $Z U = U = -UZ$ and $Z L= -L = -L Z$, we deduce that $Q_1 \\Delta_{\\mathrm{hex}} + \\Delta_{\\mathrm{hex}} Q_1 = 0$, and hence\n\\[\n(\\Delta_{\\mathrm{hex}} + \\lambda Q_1)^2\n=\n\\Delta_{\\mathrm{hex}}^2 + \\lambda^2 \n\\geq\n\\lambda^2.\n\\]\nConsequently, $(-\\lambda,\\lambda) \\cap \\sigma(\\Delta_{\\mathrm{hex}} + \\lambda Q_1) = \\emptyset$ and there is a gap at zero. In particular, the gap is precisely $(-\\lambda,\\lambda)$, and so opens linearly at the maximal possible rate.\n\\medskip\n\nLet us consider the $(2,2)$-periodic case. We parameterize our potential as $(q_1,\\ldots,q_8) \\in {\\mathbb R}^8$ as shown in Figure~\\ref{fig:hex22period}.\n\\begin{figure*}[t]\n\n\\begin{tikzpicture}[yscale=1]\n\\draw [-,line width = .1cm,color=black] (0,0) -- (1,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (0,{2*sqrt(3)}) -- (1,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (0,{2*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (0,{4*sqrt(3)}) -- (1,{3*sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (1,{sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (1,{3*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .1cm] (4,0) -- (3,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (4,{2*sqrt(3)}) -- (3,{sqrt(3)});\n\\draw [-,line width = .1cm,color=red] (4,{2*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (4,{4*sqrt(3)}) -- (3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (4,0) -- (6,0);\n\\draw [-,line width = .1cm,color=red] (4,{2*sqrt(3)}) -- (6,{2*sqrt(3)});\n\\draw [-,line width = .1cm] (4,{4*sqrt(3)}) -- (6,{4*sqrt(3)});\n\\draw [-,line width = .1cm] (6,0) -- (7,{sqrt(3)});\n\\draw [-,line width = .1cm] (6,{2*sqrt(3)}) -- (7,{sqrt(3)});\n\\draw [-,line width = .1cm] (6,{2*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (6,{4*sqrt(3)}) -- (7,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,0) -- (0,0);\n\\draw [-,line width = .1cm,color=red] (-2,{2*sqrt(3)}) -- (0,{2*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{4*sqrt(3)}) -- (0,{4*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,0) -- (-3,{sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{2*sqrt(3)}) -- (-3,{sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{2*sqrt(3)}) -- (-3,{3*sqrt(3)});\n\\draw [-,line width = .1cm] (-2,{4*sqrt(3)}) -- (-3,{3*sqrt(3)});\n\n\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](4,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=red, fill=red](1,{sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](3,{sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](0,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](4,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](6,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](1,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](3,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](7,{3*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](0,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](4,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](6,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-2,{0*sqrt(3)}) circle (.2);\n\\filldraw[color=red, fill=red](-2,{2*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-2,{4*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-3,{1*sqrt(3)}) circle (.2);\n\\filldraw[color=black, fill=black](-3,{3*sqrt(3)}) circle (.2);\n\\node at (-2.5,{2*sqrt(3)}){\\hot{$q_1$}};\n\\node at (.5,{2*sqrt(3)}){\\hot{$q_2$}};\n\\node at (.5,{3*sqrt(3)}){\\hot{$q_3$}};\n\\node at (3.5,{3*sqrt(3)}){\\hot{$q_4$}};\n\\node at (.5,{sqrt(3)}){\\hot{$q_5$}};\n\\node at (3.5,{sqrt(3)}){\\hot{$q_6$}};\n\\node at (3.5,{2*sqrt(3)}){\\hot{$q_7$}};\n\\node at (6.5,{2*sqrt(3)}){\\hot{$q_8$}};\n\\end{tikzpicture}\n\\caption{A portion of the hexagonal lattice. A fundamental domain for a $(2,2)$-periodic potential is highlighted in red.}\\label{fig:hex22period}\n\\end{figure*}\n\nWe now turn to the construction of a potential that opens gaps at $0$, $1$, and $-1$ simultaneously. We show that it opens gaps linearly at zero, quadratically at $\\pm1$. Later on, we will show that one cannot open gaps linearly at $\\pm 1$ on both sides.\n\n\\begin{theorem}\\label{thm:hexQ}\nOrder the vertices of a $2\\times 2$ fundamental cell of the hexagonal lattice as shown in Fig.~\\ref{fig:hex22period}, define a $(2,2)$-periodic potential $Q$ by\n\\[\n(q_1,\\ldots,q_8)\n=\n(1,-1,1,2,-2,-1,1,-1),\n\\]\nand denote $H_\\lambda = \\Delta_{\\mathrm{hex}} + \\lambda Q$. Then, for $ |\\lambda| > 0$ sufficiently small, $\\sigma(H_\\lambda)$ consists of four connected components. Moreover, if $\\mathfrak{g}_{E,\\lambda} = (g_{E,\\lambda}^{-}, g_{E,\\lambda}^{+})$ denote the gaps of $\\sigma(H_{\\lambda})$ that open at $E=0,\\pm 1$, one has\n\\[\n \\Big(\\pm 1-\\frac{\\lambda^2}{20}, \\pm 1+\\frac{\\lambda^2}{20}\\Big)\\subset \\mathfrak{g}_{\\pm 1,\\lambda} {\\subseteq \\Big(\\pm 1 - \\frac{1}{2}\\lambda^2,\\pm1 + \\frac{1}{2}\\lambda^2 \\Big)},\\]\nand\n\\[ \\ \\ \\ \\ \\Big(-\\frac{\\lambda}{{5}}, \\frac{\\lambda}{{5}}\\Big)\\subset \\mathfrak{g}_{0,\\lambda} \\subset \\Big(-\\frac{\\lambda}{4}, \\frac{\\lambda}{4}\\Big)\\]\nfor all $|\\lambda| > 0$ sufficiently small.\n\\end{theorem}\n\nWe point that we do not carefully optimize the constants; it is possible to get better constants than $1\/20$, ${1\/2}$, ${1\/5}$, and $1\/4$.\n\n\\begin{proof}\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb T}^2$, let $H_\\lambda(\\bm{\\theta})$ denote the Floquet matrix corresponding to $H_\\lambda$. Ordering the vertices of the fundamental domain as in Figure~\\ref{fig:hex22period}, we obtain:\n\\begin{align}\\label{eq:hexH}\nH_\\lambda(\\bm{\\theta})\n=\n\\begin{bmatrix}\n\\lambda & 1 & 0 & e^{-i\\theta_1} & 0 & e^{-i\\theta_2} & 0 & 0 \\\\\n1 & -\\lambda & 1 & 0 & 1 & 0 &0&0 \\\\\n0 & 1 & \\lambda & 1 & 0 & 0 & 0 & e^{-i\\theta_2} \\\\\ne^{i\\theta_1} & 0 & 1 & 2\\lambda & 0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 & - 2\\lambda & 1 & 0 & e^{-i\\theta_1} \\\\\ne^{i\\theta_2} & 0 & 0 & 0 & 1 & - \\lambda & 1 & 0 \\\\\n0 & 0 &0 & 1 & 0 & 1 & \\lambda & 1 \\\\\n0 & 0 & e^{i\\theta_2} & 0 & e^{i\\theta_1} & 0 & 1 & -\\lambda\n\\end{bmatrix}\n\\end{align}\n\n\nFirst, let us consider the gaps at $E=\\pm 1$.\nCalculations yield\n\\begin{align}\\label{eq:hexsumXZ}\n\\det\\big(H_{\\lambda}(\\bm{\\theta})- (\\pm 1+ s\\lambda^2) {\\mathbb I} \\big)\n= \nX_0^\\pm(\\bm{\\theta}) + X_4^\\pm(\\bm{\\theta},s) \\lambda^4 + X_6^\\pm(\\bm{\\theta},s) \\lambda^6 + O(\\lambda^8)\n\\end{align}\nin which\n\\begin{align*}\nX_0^\\pm(\\bm{\\theta})&=\n-4 (-\\sin(\\theta_1) + \\sin(\\theta_1-\\theta_2) + \\sin(\\theta_2))^2\\\\\nX_4^\\pm(\\bm{\\theta},s)\n&=\n8(s\\pm1)(2s\\mp 1) (3 -\\cos(\\theta_1)-\\cos(\\theta_1-\\theta_2)-\\cos(\\theta_2))\\\\\nX_6^\\pm(\\bm{\\theta},s)\n&=\n-1 \\mp 12 s + 72 s^2 \\mp 16 s^3-4 s^2 (\\pm 4 s+1)(\\cos(\\theta_1) + \\cos(\\theta_1-\\theta_2) + \\cos(\\theta_2)\n\\end{align*}\nIt is clear that \n\\begin{align}\\label{eq:hexX0Z0}\nX_0^\\pm(\\bm{\\theta})\\leq 0 \\quad \\text{for all }\\bm{\\theta} \\in {\\mathbb T}^2.\n\\end{align}\nSince $\\cos(\\theta_1) + \\cos(\\theta_1-\\theta_2) + \\cos(\\theta_2)\\leq 3$, we also have \n\\begin{align}\\label{eq:hexX4Z4}\nX_4^\\pm(\\bm{\\theta},s)\\leq 0 \\quad \\text{for all } \\bm{\\theta} \\in {\\mathbb T}^2, \\; |s|\\leq 1\/2.\n\\end{align}\nWe also have for $|s|\\leq 1\/4$,\n\\[\nX_6^+(\\bm{\\theta},s)\\leq -1-12 s+72 s^2-16 s^3+12 s^2 (4s+1)=:T(s),\n\\]\nand \n\\[\nX_6^-(\\bm{\\theta},s)= X_6^+(\\bm{\\theta},-s) \\leq T(-s).\n\\]\nOne easily checks that $T(s)$ is decreasing on $[-0.05, 0.05]$, and\n\\[T(-0.05)=-0.194.\\]\nHence for $|s|\\leq 0.05$,\n\\begin{align}\\label{eq:hexX6Z6}\nX_6^\\pm(\\bm{\\theta},s)\\leq -0.194.\n\\end{align}\nCombining \\eqref{eq:hexX0Z0}, \\eqref{eq:hexX4Z4}, and \\eqref{eq:hexX6Z6}, we obtain that for $|\\lambda|>0$ sufficiently small, and $|s| \\leq 1\/20$,\n\\[\\det(H_{\\lambda}(\\bm{\\theta})- (\\pm 1+ s\\lambda^2) {\\mathbb I})\\leq -0.1\\lambda^6<0.\\]\nThis proves the claimed lower bound on the gaps at $\\pm 1$.\n\nOn the other hand, let us note that $X_0^\\pm(0,0) =X_0^\\pm(\\pi,\\pi) = 0$, while\n\\begin{align*}\n\\begin{cases}\nX_4^+(\\bm{\\theta}, 0.5)= 0 \\quad \\text{and}\\quad X_6^+((\\pi,\\pi),0.5)=12,\\\\\nX_4^+((0,0),s)=0 \\quad \\text{and}\\quad X_6^+((0,0),-0.5)=28,\\\\\nX_4^-((0,0),s)=0\\quad \\text{and} \\quad X_6^-((0,0),0.5)=28,\\\\\nX_4^-(\\bm{\\theta}, -0.5)=0 \\quad \\text{and}\\quad X_6^-((\\pi, \\pi), -0.5)=12.\n\\end{cases}\n\\end{align*}\nThus for small $\\lambda>0$, we have\n\\begin{align*}\n\\begin{cases}\n\\det(H_\\lambda(\\pi,\\pi)-(1+0.5\\lambda^2){\\mathbb I}) > 0,\\\\\n\\det(H_\\lambda(0,0)-(1-0.5\\lambda^2){\\mathbb I})>0,\\\\\n\\det(H_\\lambda(0,0)-(-1+0.5\\lambda^2){\\mathbb I})>0,\\\\\n\\det(H_\\lambda(\\pi,\\pi)-(-1-0.5\\lambda^2){\\mathbb I})>0.\n\\end{cases}\n\\end{align*}\nWe also easily check that \n\\[X_0^\\pm\\left(\\frac{\\pi}{2},\\pi\\right)=-16,\\]\nwhich implies that for small $\\lambda>0$, we have\n\\[\\det\\left(H_\\lambda\\left(\\frac{\\pi}{2},\\pi \\right)-(\\pm 1\\pm 0.5 \\lambda^2 {\\mathbb I}) \\right)<0.\\]\nWe therefore conclude that \n\\[\\pm 1 + 0.5 \\lambda^2 \\in \\sigma(H_\\lambda) \\text{ and } \\pm 1 - 0.5 \\lambda^2 \\in \\sigma(H_\\lambda),\\]\nwhich proves the upper bounds on the gaps at $\\pm 1$.\n\nNow let us consider the gap at $E=0$. After calculations, we have\n\\begin{align}\\label{eq:hexsumY}\n\\det(H_{\\lambda}(\\bm{\\theta})- s\\lambda {\\mathbb I} )=Y_0(\\bm{\\theta})+Y_2(\\bm{\\theta},s)\\lambda^2+Y_4(\\bm{\\theta},s)+O(\\lambda^6),\n\\end{align}\nwhere \n\\begin{align*}\nY_0(\\bm{\\theta})\n& =\n15 + 2\\cos(2\\theta_1) - 4 \\cos(\\theta_1 - 2\\theta_2) + 2 \\cos(2\\theta_1-2\\theta_2) \n- 4 \\cos(2\\theta_1 - \\theta_2) \\\\ & \\qquad + 2\\cos(2\\theta_2) - 4 \\cos(\\theta_1 + \\theta_2),\n\\end{align*}\n\\[\nY_2(\\bm{\\theta}, s)=2[5-26 s^2+(2+4 s^2)(\\cos(\\theta_1)+\\cos(\\theta_1-\\theta_2)+\\cos(\\theta_2))],\n\\]\nand \n\\[\nY_4(\\bm{\\theta},s)=(1-s^2)[-3 - 42 s^2 + 4 (2 + s^2) (\\cos(\\theta_1)+\\cos(\\theta_1-\\theta_2)+\\cos(\\theta_2))]\n\\]\nWe claim that\n\\begin{equation}\n\\label{eq:hexY0geq0}\nY_0(\\bm{\\theta})\\geq 0\n\\quad\n\\text{for all } \\bm{\\theta} \\in {\\mathbb T}^2.\n\\end{equation} \nLet us see how to use \\eqref{eq:hexY0geq0} to prove the claimed gap at zero and defer the proof of \\eqref{eq:hexY0geq0} for a moment. Using\n\\[\\cos(\\theta_1)+\\cos(\\theta_1-\\theta_2)+\\cos(\\theta_2)\\in \\left[-\\frac{3}{2}, 3\\right],\\]\nwe obtain that for $|s|<1\/5$\n\\begin{align}\\label{eq:hexY2}\nY_2(\\bm{\\theta}, s)\\geq 2(5-26s^2-3(1+2s^2))=4(1-16 s^2)>\\frac{36}{25}.\n\\end{align}\nCombining \\eqref{eq:hexsumY} with \\eqref{eq:hexY2}, we obtain that for $|\\lambda|>0$ sufficiently small\n\\[\\det(H_{\\lambda}(\\bm{\\theta})- s\\lambda\\, {\\mathbb I} )>\\lambda^2.\\]\nThis proves the claimed lower bound of the gap at $0$, modulo the claim that $Y_0(\\bm{\\theta}) \\geq 0$ for all $\\bm{\\theta} \\in {\\mathbb T}^2$. \n\nTo prove the upper bound, we compute\n\\begin{align*}\n\\begin{cases}\nY_0\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right)=0,\\\\\nY_2\\left(\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right), s\\right)=4(1-16 s^2),\\\\\nY_4\\left(\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right), s\\right)=3(s^2-1)(16s^2+5),\n\\end{cases}\n\\end{align*}\nwhich implies that for small $\\lambda>0$,\n\\[\\det\\left(H_{\\lambda}\\left(\\frac{2\\pi}{3}, \\frac{4\\pi}{3}\\right)\\pm 0.25 \\lambda\\, {\\mathbb I} \\right)<0.\\]\nWe also compute that $Y_0(0,0)=9$, which shows for small $\\lambda>0$, \n\\[\\det(H_{\\lambda}(0,0)\\pm 0.25 \\lambda\\, {\\mathbb I})>0.\\]\nThus we conclude that\n\\[\\pm 0.25 \\lambda \\in \\sigma(H_\\lambda),\\]\nwhich proves the claimed upper bound of the gap at $0$.\n\nTo complete the argument, all that remains is to show $Y_0(\\bm{\\theta})\\geq 0$ for all $\\bm{\\theta}\\in {\\mathbb T}^2$.\nTo that end, introduce two auxiliary variables\n\\[\nz := \\cos\\left(\\frac{\\theta_1 - \\theta_2}{2} \\right),\n\\quad\nw := \\cos\\left(\\frac{\\theta_1 + \\theta_2}{2} \\right),\n\\]\nand write $g(z,w)$ to mean $Y_0(\\bm{\\theta})$ in the variables $z$ and $w$. Thus, to optimize $Y_0(\\bm{\\theta})$ on ${\\mathbb T}^2$, it suffices to optimize $g(z,w)$ on the square $[-1,1]^2$. To execute this change of variables, first note the following simple consequences of standard identities:\n\\begin{align*}\n\\cos(2\\theta_1) + \\cos(2\\theta_2)\n& =\n2(2z^2-1)(2w^2-1),\\\\\n\\cos(2\\theta_1 - 2\\theta_2)\n& =\n2(2z^2-1)^2-1,\\\\\n\\cos(\\theta_1 + \\theta_2)\n& =\n2w^2 -1,\\\\\n\\cos(\\theta_1 - 2\\theta_2) + \\cos(2\\theta_1 - \\theta_2)\n& =\n2 zw(4z^2 - 3).\n\\end{align*}\nPutting all this together,\n\\begin{align*}\ng(z,w)\n& =\n15+ 4(2z^2-1)(2w^2-1) - 8zw(4z^2-3)+ 2(2(2z^2-1)^2-1) - 4(2w^2-1).\n\\end{align*}\nIt is easy to check that $g \\geq 0$ holds on the boundary; concretely,\n\\begin{align*}\ng(\\pm 1,w)\n& =\n15 + 4(2w^2-1) \\mp 8w + 2 - 4(2w^2-1) \\\\\n& =\n17 \\mp 8w \\\\\n& \\geq\n17 - 9 \\\\\n& > 0.\n\\end{align*}\nand\n\\begin{align*}\ng(z,\\pm 1)\n& =\n15 + 4(2z^2-1) \\mp 8z(4z^2-3) + (16z^4-16z^2+2) - 4 \\\\\n& =\n16 z^4 \\mp 32 z^3 - 8z^2 \\pm 24z + 9 \\\\\n& =\n(3 \\pm 4z - 4z^2)^2 \\\\\n& \\geq 0.\n\\end{align*}\nSo, we now seek zeros of $\\nabla g$ for $|z| < 1$ and $|w| < 1$.\nOne easily computes $\\partial_z g$ and $\\partial_w g$:\n\\begin{align*}\n\\partial_z g\n& = \n8(w-2z)(3+4z(w-z)) \\\\\n\\partial_w g\n& =\n8(3z - 4z^3 + 4w(z^2-1)).\n\\end{align*}\nSetting $\\partial_w g = 0$ yields\n\\begin{equation} \\label{eq:Y0critical:wval}\nw\n=\n\\frac{4z^3 - 3z}{4(z^2-1)}.\n\\end{equation}\nSince we are working on the interior of $[-1,1]^2$, $z \\neq \\pm 1$ and the denominator does not vanish. Substituting this expression for $w$ into $\\partial_z g$ and simplifying, we get\n\\[\n\\partial_zg \\left(z,\\frac{4z^3 - 3z}{z^2-1} \\right)\n=\n2z \\left( \\frac{1}{(z^2-1)^2} - 16 \\right).\n\\]\nSetting this equal to zero, we obtain three values of $z$ with $|z| <1$: $0$ and $\\pm \\sqrt{3}\/2$. Inserting these $z$ values into \\eqref{eq:Y0critical:wval}, the corresponding $w$ values are all readily seen to be zero. Plugging in the three critical points $(0,0)$ and $(\\pm \\sqrt{3}\/2,0)$ into $g$ yields 25 and 16, respectively, which concludes the proof that $g \\geq 0$ and hence \n\\[\nY_0(\\bm{\\theta}) \\geq 0\n\\]\nfor all $\\bm{\\theta} \\in {\\mathbb T}^2$, proving \\eqref{eq:hexY0geq0}.\n\\end{proof}\n\n\nNext, we show that for any $(2,2)$-periodic potential, it is impossible that it opens linear order gaps on both sides of $E=\\pm 1$ simultaneously.\n\\begin{theorem}\\label{thm:E=pm1linear}\nFor any $(2,2)$-periodic potential $Q$ and any constant $c>0$, the following holds for all sufficiently small $\\lambda>0$:\n\\[ \\left( (-1-c\\lambda, -1+c\\lambda) \\cup (1-c\\lambda, 1+c\\lambda) \\right) \\cap \\sigma(H_{\\lambda})\\neq \\emptyset\\]\n\\end{theorem}\n\\begin{proof}\nLet $(q_1,q_2,\\ldots,q_8)$ be the potential on a $2\\times 2$ fundamental cell, as shown in Fig.~\\ref{fig:hex22period}. \nThe corresponding Floquet matrix $H_{\\lambda}(\\bm{\\theta})$ is \n\\[\nH_\\lambda(\\bm{\\theta})\n=\n\\begin{bmatrix}\n\\lambda q_1 & 1 & 0 & e^{-i\\theta_1} & 0 & e^{-i\\theta_2} & 0 & 0 \\\\\n1 & \\lambda q_2& 1 & 0 & 1 & 0 &0&0 \\\\\n0 & 1 & \\lambda q_3& 1 & 0 & 0 & 0 & e^{-i\\theta_2} \\\\\ne^{i\\theta_1} & 0 & 1 & \\lambda q_4& 0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 & \\lambda q_5& 1 & 0 & e^{-i\\theta_1} \\\\\ne^{i\\theta_2} & 0 & 0 & 0 & 1 & \\lambda q_6& 1 & 0 \\\\\n0 & 0 &0 & 1 & 0 & 1 & \\lambda q_7& 1 \\\\\n0 & 0 & e^{i\\theta_2} & 0 & e^{i\\theta_1} & 0 & 1 & \\lambda q_8\n\\end{bmatrix}\n\\]\nFor $0<|s|0>X_3^-(\\bm{0},s_0).\\]\nCombining this with \\eqref{eq:X0Y00}, we obtain\n\\begin{equation} \\label{eq:det00>0}\n\\det(H_\\lambda(\\bm{0})-(1+s_0\\lambda) {\\mathbb I})>0\n\\end{equation}\nfor small $\\lambda > 0$. We also have\n\\begin{align}\\label{eq:X0Y0Pi\/4}\nX_0^\\pm((\\pi\/4, 3\\pi\/4), s_0)=-4.\n\\end{align}\nIn particular, \\eqref{eq:X0Y0Pi\/4} implies that \n\\begin{align}\\label{eq:detPi\/4<0}\n\\det(H_\\lambda(\\pi\/4,3\\pi\/4))-(1+s_0\\lambda) {\\mathbb I})<0\n\\end{align}\nfor all $\\lambda \\geq 0$ small.\n\nCombining \\eqref{eq:detPi\/4<0} with \\eqref{eq:det00>0}, for any sufficiently small $\\lambda > 0$, there exists $\\bm{\\theta}$ such that\n\\[\\det(H_{\\lambda}(\\bm{\\theta})-(1+s_0\\lambda) {\\mathbb I})=0.\\]\nHence \n\\[(1-c\\lambda, 1+c\\lambda)\\cap \\sigma(H_{\\lambda})\\neq \\emptyset\\]\nas claimed.\n\\end{proof}\n\n\\section{Square Laplacian with Next-Nearest Neighbor Interactions}\\label{sec:nnn}\nWe now turn our attention to the EHM lattice, whose Laplacian is given by\n\\begin{align*}\n[\\Delta_{\\mathrm{sqn}} u]_{n,m}\n& =\nu_{n-1,m} + u_{n+1,m} + u_{n,m-1} + u_{n,m+1} + u_{n-1,m+1} + u_{n-1,m+1} + u_{n+1,m-1} + u_{n+1,m+1}\\\\\n& =\n[\\Delta_{\\rm sq}u]_{n,m} + u_{n-1,m-1} + u_{n-1,m+1} + u_{n+1,m-1} + u_{n+1,m+1} \\\\\n& =\n[\\Delta_{{\\mathrm{tri}}}u]_{n,m} + u_{n-1,m-1} + u_{n+1,m+1}.\n\\end{align*}\nNow, given $p_1,p_2 \\in {\\mathbb Z}_+$, we define $P = p_1p_2$ and $\\Lambda = {\\mathbb Z}^2 \\cap \\big([0,p_1) \\times [0,p_2)\\big)$ as before and view $\\Delta_{\\mathrm{sqn}}$ as a $(p_1,p_2)$-periodic operator and perform the Floquet decomposition. For $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb R}^2$, it is straightforward to check that\n\\[\n\\sigma(H(\\bm{\\theta}))\n=\n\\set{e_{\\bm{\\ell}}(\\bm{\\theta}) : \\bm{\\ell}\\in \\Lambda },\n\\]\nwhere $\\bm{\\ell}=(\\ell_1,\\ell_2)$ and\n\\begin{align*}\ne_{\\bm{\\ell}}(\\bm{\\theta})\n=\n2\\cos\\left( \\frac{\\theta_1+2\\pi \\ell_1}{p_1}\\right) \n+ 2\\cos\\left(\\frac{\\theta_2+2\\pi \\ell_2}{p_2}\\right) \n+ &2\\cos\\left(\\frac{\\theta_1 + 2\\pi \\ell_1}{p_1} - \\frac{\\theta_2 + 2\\pi \\ell_2}{p_2}\\right)\\\\\n+ &2\\cos\\left(\\frac{\\theta_1 + 2\\pi \\ell_1}{p_1} + \\frac{\\theta_2 + 2\\pi \\ell_2}{p_2}\\right).\n\\end{align*}\n\\begin{comment}\nWe now consider this as a $(p_1,p_2)$-periodic operator, and hence we seek (generalized) eigenfunctions $u$ with $u_{n+p_1,m} \\equiv e^{2\\pi i \\theta_1} u_{n,m}$ and $u_{n,m+p_2} \\equiv e^{2\\pi i y} u_{n,m}$ with $x,y \\in [0,1]$. Take\n\\[\nu_{n,m}^{(\\ell_1,\\ell_2)}\n=\n\\exp\\left(2\\pi i\\left(n \\frac{x+\\ell}{p_1} + m\\frac{y+k}{p_2} \\right) \\right)\n\\]\nwith $\\ell,k \\in {\\mathbb Z}$ and $0 \\le \\ell < p_1$, $0 \\le k < p_2$. It is straightforward to check that this leads to eigenvalues\n\nFor later use, we compute the derivative of $e_{\\ell,k}$:\n\\[-\\frac{1}{4 \\pi}\n\\nabla e_{\\ell,k}(x,y)\n=\n\\left( \\frac{1}{p_1} \\sin\\left(\\hat{x}\\right) + \\frac{1}{p_1}\\sin\\left(\\hat{x} - \\hat{y}\\right), \\frac{1}{p_2}\\sin(\\hat{y}) - \\frac{1}{p_2}\\sin(\\hat{x} - \\hat{y}) \\right)\n\\]\nwith $\\hat{x} = 2\\pi p_1^{-1} (x+\\ell)$ and $\\hat{y} = 2\\pi p_2^{-1} (y+k)$.\n\\end{comment}\nAs in Section~\\ref{sec:floquet}, we label these eigenvalues in increasing order according to multiplicity by\n\\[\nE_1(\\bm{\\theta})\n\\le \nE_2(\\bm{\\theta})\\le \\cdots E_P(\\bm{\\theta})\n\\]\nand denote the $P$ spectral bands by\n\\[\nF_k\n=\n\\set{E_k(\\bm{\\theta}) : \\bm{\\theta} \\in {\\mathbb R}^2},\n\\quad\n1 \\le k \\le P.\n\\]\nStraightforward computations shows that $\\sigma(\\Delta_{\\mathrm{sqn}})=[-4,8]$, hence \n\\[\n\\bigcup_{k=1}^P F_k\n=\n[-4,8].\n\\]\nOur main theorem of this section is\n\\begin{theorem}\\label{thm:sqnmain}\nLet $p_1,p_2 \\in {\\mathbb Z}_+$ be given.\n\\begin{enumerate}\n\\item[{\\rm 1.}]\nEach $E\\in (-4, 8)\\setminus \\{-1\\}$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\item[{\\rm 2.}] If one of the periods $p_1, p_2$ is not divisible by three, then $E=-1$ belongs to $\\mathrm{int}(F_k)$ for some $1\\leq k\\leq P$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:bsc:nnn}]\nThis follows immediately from Theorem~\\ref{thm:sqnmain}.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:sqnmain}} \n\nAs with the proof of Theorem~\\ref{thm:trimain}, we will divide the proof into two different cases: $E\\neq -1$ and $E=-1$ and argue by contradiction.\nTo that end, assume for the sake of establishing a contradiction that $E=\\min F_{k+1}=\\max F_k$ for some $1\\leq k\\leq P-1$.\n\nWe will use the following lemmas, whose proofs we provide at the end of the present section.\n\\begin{lemma}\\label{lem:constructionsqn}\nLet us consider the following system:\n\\begin{align}\\label{eq:xyCondABsqn}\n\\cos(x) + \\cos(y) + \\cos(x-y)+\\cos(x+y) & = \\frac{E}{2} \\\\\n\\sin(x) + \\sin(x-y)+\\sin(x+y) & = 0. \\notag\n\\end{align}\nFor any $E \\in (-4,8) \\setminus \\{-1\\}$, the solution set of \\eqref{eq:xyCondABsqn} in $[0,2\\pi)^2$ satisfies\n\\begin{align}\\label{eq:solutionx=0sqn}\nx=0,\\ \\ 1+2\\cos(y)=\\frac{E+1}{3},\n\\end{align}\nor \n\\begin{align}\\label{eq:solutionx=pisqn}\nx=\\pi,\\ \\ 1+2\\cos(y)=-(E+1).\n\\end{align}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:sqnJ0empty}\nConsider the following system:\n\\begin{equation} \\label{eq:sqnJ0syst}\n\\begin{cases}\n\\cos(x) + \\cos(y) + \\cos(x+y) +\\cos(x-y) = \\frac{E}{2},\\\\\n\\sin(x)+\\sin(x-y)+\\sin(x+y)=0,\\\\\n\\sin(y)-\\sin(x-y)+\\sin(x+y)=0.\n\\end{cases}\n\\end{equation}\nFor any $E \\in (-4,8) \\setminus \\{0, -1\\}$, the solution set of \\eqref{eq:sqnJ0syst} is empty. \nFor $E=0$, the unique solution of \\eqref{eq:sqnJ0syst} in $[0,2\\pi)^2$ is $(\\pi,\\pi)$.\nFor $E =-1$, the solutions of \\eqref{eq:sqnJ0syst} in $[0,2\\pi)^2$ are $({2\\pi}\/{3},{2\\pi}\/{3})$, $({2\\pi}\/{3},{4\\pi}\/{3})$, $({4\\pi}\/{3},{2\\pi}\/{3})$ and $({4\\pi}\/{3},{4\\pi}\/{3})$.\n\\end{lemma}\n\nWe will use Lemma \\ref{lem:constructionsqn} in the $E\\neq -1$ case, and Lemma \\ref{lem:sqnJ0empty} in the $E=-1$ case.\n\n\\subsubsection{$E\\neq -1$}\\\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:sqnmain}.1]\nLet $E \\in (-4,8)\\setminus\\set{-1}$ be given, and suppose towards a contradiction that $E = \\max F_k = \\min F_{k+1}$ for some $k$. Define $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1,\\widetilde{\\theta}_2)\\in [0,2\\pi)^2$ and $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)},\\ell_2^{(1)})\\in \\Lambda$ via\n\\begin{align}\\label{def:ttheellsqnE}\n\\widetilde{\\theta}_1=0,\\ \\ell_1^{(1)}=0,\\ \\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)}}{p_2}=\\arccos\\Big(\\frac{E-2}{6}\\Big)\\in (0,\\pi).\n\\end{align}\nNote that since $E\\in (-4, 8)$, we have $\\frac{E-2}{6}\\in (-1,1)$, hence $\\arccos\\Big(\\frac{E-2}{6}\\Big)$ is always well-defined.\nNote also that $\\widetilde{\\theta}_2$ and $\\ell_2^{(1)}$ are uniquely determined.\nUsing \\eqref{def:ttheellsqnE}, one easily checks that \n\\[e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=E,\\]\nand\n\\begin{align}\\label{eq:sqnJbeta10}\n(1,0)\\cdot \\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=0.\n\\end{align}\nAs in the proof of Theorem~\\ref{thm:trimain}, denote $\\Lambda_E(\\widetilde{\\bm{\\theta}}) = \\{\\bm{\\ell} \\in \\Lambda : e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = E\\}$, let $r := |\\Lambda_E(\\widetilde{\\bm{\\theta}})|$ be the multiplicity of $E$ as an eigenvalue of $H(\\widetilde{\\bm{\\theta}})$, and choose $s\\in {\\mathbb Z}\\cap [1,r]$ such that \n\\[E_{k-s}(\\widetilde{\\bm{\\theta}})0$ small enough such that \n\\[E_{k-s}(\\bm{\\theta})0\\}.\n\\end{aligned}\n\\end{equation}\nBy definition, we must have\n\\begin{align}\\label{eq:sumJbetasqn}\n|\\mathcal{J}_{\\bm{\\beta}}^0|+|\\mathcal{J}_{\\bm{\\beta}}^+|+|\\mathcal{J}_{\\bm{\\beta}}^-|=r\n\\end{align}\nfor any $\\bm{\\beta}$. We also define $\\mathcal{J}_0$ as follows\n\\begin{align}\\label{def:J0sqn}\n\\mathcal{J}_0\n={\\mathcal J}_0(\\widetilde{\\bm{\\theta}})\n:=\n\\{\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}}):\\ \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = \\bm{0}\\}.\n\\end{align}\nIf $E\\neq 0$, Lemma~\\ref{lem:sqnJ0empty} directly implies $\\mathcal{J}_0=\\emptyset$.\nIf $E=0$, $\\mathcal{J}_0$ is also empty. \nTo see this, suppose on the contrary that $\\bm{\\ell} = (\\ell_1,\\ell_2) \\in {\\mathcal J}_0$.\nLemma~\\ref{lem:sqnJ0empty} implies that\n\\begin{equation} \\label{eq:sqn:J0emptyE=0eq1}\n\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}=\\pi,\n\\end{equation}\nand \\eqref{def:ttheellsqnE} forces\n\\begin{equation} \\label{eq:sqn:J0emptyE=0eq2}\n\\frac{\\widetilde{\\theta}_2+2\\pi \\ell^{(1)}_2}{p_2}=\\arccos\\left(-\\frac{1}{3}\\right).\n\\end{equation}\nSubtracting \\eqref{eq:sqn:J0emptyE=0eq1} from \\eqref{eq:sqn:J0emptyE=0eq2} yields\n\\[\\frac{\\ell^{(1)}_2-\\ell_2}{p_2}=\\frac{1}{2\\pi}\\arccos\\left(-\\frac{1}{3}\\right)-\\frac{1}{2}.\\]\nHowever, this implies that $(2\\pi)^{-1}\\arccos\\left(-1\/3\\right)$ is a rational number, which contradicts the following well-known fact, whose proof we supply at the end of the present section.\n\\begin{lemma}\\label{lem:arccos1\/3}\n\\[\\frac{1}{2\\pi}\\arccos\\left(-\\frac{1}{3}\\right)\\in {\\mathbb R}\\setminus {\\mathbb Q}.\\]\n\\end{lemma}\nTherefore $\\mathcal{J}_0=\\emptyset$ for any $E\\neq -1$.\n\nWe choose $\\bm{\\beta}_1=(1,0)$. Then \\eqref{eq:sqnJbeta10} implies $\\bm{\\ell}^{(1)} \\in {\\mathcal J}_{\\bm{\\beta}_1}^0$, and hence\n\\begin{align}\\label{eq:beta1sqnnon-empty}\n\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset.\n\\end{align} \n\n\nNext we are going to perturb the point $\\widetilde{\\bm{\\theta}}$ and count the eigenvalues.\nSince $\\mathcal{J}_0=\\emptyset$, we can choose a unit vector $\\bm{\\beta}_2$ such that \n\\begin{align}\\label{eq:beta2sqnnon-empty}\n\\bm{\\beta}_2\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq 0,\n\\end{align}\nholds for any $\\bm{\\ell} \\in \\Lambda_E(\\widetilde{\\bm{\\theta}})$.\nThus $\\mathcal{J}_{\\bm{\\beta}_2}^0=\\emptyset$ and\n\\begin{align}\\label{eq:beta2sqnnon-empty'}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|+|\\mathcal{J}_{\\bm{\\beta}_2}^-|=r.\n\\end{align}\nArguing as in the proof of Theorem~\\ref{thm:trimain}.1, we deduce \\begin{comment}\nWe first perturb the eigenvalues along the $\\bm{\\beta}_2$ direction.\nSince $\\mathcal{J}_{\\bm{\\beta}_2}^0 = \\emptyset$, we will always employ \\eqref{eq:pertgeneralbetasqn1order}.\n\nFor $t > 0$ small enough, we have the following.\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_2}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies\n\\begin{align}\\label{eq1:Jbeta2+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+|\\leq r-s.\n\\end{align}\n\n\\item If ${\\bm{\\ell} } \\in \\mathcal{J}_{\\bm{\\beta}_2}^-$, we have \n\\[\nE_{k-s}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\theta}_1 + t\\beta_{2,1}, \\widetilde{\\theta}_2 + t\\beta_{2,2})\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta2+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:beta2sqnnon-empty'}, Equations~\\eqref{eq1:Jbeta2+sqn} and \\eqref{eq2:Jbeta2+sqn} imply\n\\begin{equation} \\label{eq:Jbeta2MinusCard+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^-|\n=\ns.\n\\end{equation}\nUpon realizing that $\\mathcal{J}_{-\\bm{\\beta}_2}^0 = \\emptyset$ and $\\mathcal{J}_{-\\bm{\\beta}_2}^\\pm = \\mathcal{J}_{\\bm{\\beta}_2}^\\mp$, we may apply the analysis above with $\\bm{\\beta}_2$ replaced by $-\\bm{\\beta}_2$ and conclude that\n\\begin{equation} \\label{eq:JMinusBeta2+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_2}^+| = |\\mathcal{J}_{-\\bm{\\beta}_2}^-| = s.\n\\end{equation}\nIn particular, \\eqref{eq:Jbeta2MinusCard+sqn} and \\eqref{eq:JMinusBeta2+sqn} imply \\end{comment}\n\\begin{align}\\label{eq4:Jbeta2sqn}\nr=2s.\n\\end{align}\n\n\n\n\\subsubsection*{Perturbation along $\\bm{\\beta}_1$}\nNow we perturb the eigenvalues along $\\bm{\\beta}_1=(1,0)$.\nThe case when ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^{\\pm}$ is similar to that of $\\bm{\\beta}_2$.\nThe difference here is that, according to \\eqref{eq:beta1sqnnon-empty}, $\\mathcal{J}_{\\bm{\\beta}_1}^0\\neq \\emptyset$.\n\nBy Lemma~\\ref{lem:constructionsqn}, we have that for $(\\ell_1,\\ell_2)\\in \\mathcal{J}_{\\bm{\\beta}_1}^0$,\n\\begin{align}\\label{eq:beta1sqnt2sign}\n(E+1)\\Bigg[\n\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big)\n+&\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}-\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)\\\\\n&\\qquad +\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}+\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big) \n\\Bigg]\n>0. \\notag\n\\end{align}\nIndeed, if $(\\ell_1,\\ell_2)\\in \\mathcal{J}_{\\bm{\\beta}_1}^0$, \n$(x,y) = (p_1^{-1} (\\widetilde{\\theta}_1 + 2\\pi \\ell_1), p_2^{-1}(\\widetilde{\\theta}_2 + 2\\pi \\ell_2))$ is a solution to \\eqref{eq:xyCondABsqn}.\nHence Lemma~\\ref{lem:constructionsqn} implies that \nwe have \neither \n\\[\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}=0,\\ \\ 1+2\\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)=\\frac{E+1}{3},\\]\nor \n\\[\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}=\\pi,\\ \\ 1+2\\cos\\Big(\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)=-(E+1).\\]\nClearly, both cases lead to \\eqref{eq:beta1sqnt2sign}.\n\nBy employing \\eqref{eq:pertgeneralbetasqn2order}, we obtain\n\\begin{align}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n=\nE & - {\\frac{ t^2}{2 p_1^2}} \\Bigg[\n\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}\\Big)\n+\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}-\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big)\\\\\n& \\qquad\\qquad\\qquad\n+\\cos\\Big(\\frac{\\widetilde{\\theta}_1 + 2\\pi {\\ell_1}}{p_1}+\\frac{\\widetilde{\\theta}_2 + 2\\pi {\\ell_2}}{p_2}\\Big) \n\\Bigg] + O(t^3)\\notag\n\\end{align}\nfor $\\bm{\\ell} \\in {\\mathcal J}_{\\bm{\\beta}_1}^0$. Combining this with \\eqref{eq:beta1sqnt2sign}, we obtain that for $|t|>0$ small enough\n\\begin{align}\\label{eq:Jbeta10sqn}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n\\begin{cases}\n0,\\\\\n>E,\\ \\ \\text{if } E+1<0.\n\\end{cases}\n\\end{align}\nNotice that the choice of $\\bm{\\beta}_1$ causes the second $t^2$ term of \\eqref{eq:pertgeneralbetasqn2order} to drop out.\n\nWithout loss of generality, we assume $E\\in (-1, 8)$. The complementary case when $E\\in (-4, -1)$ can be handled similarly.\nFor $E\\in (-1, 8)$, \\eqref{eq:Jbeta10sqn} implies that \n\\begin{align}\\label{eq:Jbeta10sqn'}\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\end{align}\nholds for $|t|>0$ small enough and for any $\\bm{\\ell} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0$. \n\nCombining \\eqref{eq:Jbeta10sqn'} with \\eqref{eq:pertgeneralbetasqn1order}, we have the following.\n\nFor $t>0$ small enough,\n\\begin{itemize}\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^+$, we have\n\\[\nE_{k+r-s+1}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n>\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n>\nE\n=\n\\max F_k,\n\\]\nwhich implies \n\\begin{align}\\label{eq1:Jbeta1+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\\leq r-s\n=\ns,\n\\end{align}\n{where the equality follows from \\eqref{eq4:Jbeta2sqn}.}\n\n\\item If ${\\bm{\\ell}} \\in \\mathcal{J}_{\\bm{\\beta}_1}^0 \\bigcup \\mathcal{J}_{\\bm{\\beta}_1}^-$, we have \n\\[\nE_{k-s-1}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n<\ne_{{\\bm{\\ell}}}(\\widetilde{\\bm{\\theta}} + t \\bm{\\beta}_1)\n<\nE\n=\n\\min F_{k+1},\n\\]\nwhich implies\n\\begin{align}\\label{eq2:Jbeta1+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\\leq s.\n\\end{align}\n\\end{itemize}\n{In view of \\eqref{eq:sumJbetasqn} and \\eqref{eq4:Jbeta2sqn}, Equations~\\eqref{eq1:Jbeta1+sqn} and \\eqref{eq2:Jbeta1+sqn} yield\n\\begin{equation} \\label{eq3:Jbeta1+sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\ns.\n\\end{equation}\nAs before, we may observe that ${\\mathcal J}_{-\\bm{\\beta}_1}^0 = {\\mathcal J}_{\\bm{\\beta}_1}^0$ and ${\\mathcal J}_{-\\bm{\\beta}_1}^\\pm = {\\mathcal J}_{\\bm{\\beta}_1}^\\mp$. Then, the analysis above applied with $\\bm{\\beta}_1$ replaced by $-\\bm{\\beta}_1$ forces\n\\begin{equation} \\label{eq3:Jbeta1-sqn}\n|\\mathcal{J}_{\\bm{\\beta}_1}^-|\n=\n|\\mathcal{J}_{\\bm{\\beta}_1}^0|+|\\mathcal{J}_{\\bm{\\beta}_1}^+|\n=\ns.\n\\end{equation}\nTaken together, \\eqref{eq3:Jbeta1+sqn} and \\eqref{eq3:Jbeta1-sqn} imply $|{\\mathcal J}_{\\bm{\\beta}_1}^0| = 0$, which contradicts \\eqref{eq:beta1sqnnon-empty}.\n}\n\\end{proof}\n\n\\subsubsection{$E=-1$}\\\n\nFirst, we would like to make a remark on our strategy of the proof of the $E=-1$ case, and on the importance of one of the period being not divisible by $3$.\n\\begin{remark}\\label{rem:sqn}\nFor the exceptional energy {$E = -1$} of the EHM lattice, we can not use eigenvalues with vanishing gradients to create un-even numbers of counting unless neither $p_1$ nor $p_2$ is divisible by $3$.\nThe reason is the following: suppose {\\it only} $p_1$ is not divisible by $3$ and we choose $\\widetilde{\\bm{\\theta}}=(\\widetilde{\\theta}_1, \\widetilde{\\theta}_2)$ and $\\bm{\\ell}^{(1)}=(\\ell_1^{(1)}, \\ell_2^{(1)})$ such that $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=-1$ and $\\nabla e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})=\\bm{0}$. \nLemma \\ref{lem:sqnJ0empty} yields four possibilities $(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)}), p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(1)}))=(2\\pi\/3,2\\pi\/3)$, $(2\\pi\/3,4\\pi\/3)$, $(4\\pi\/3,2\\pi\/3)$ or $(4\\pi\/3,4\\pi\/3)$.\nWithout loss of generality, we choose $(2\\pi\/3, 2\\pi\/3)$, the other three choices are essentially the same.\nSince $p_2$ is divisible by $3$, there exists $\\bm{\\ell}^{(2)}$, such that \n$(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(2)}), p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2^{(2)}))=(2\\pi\/3,4\\pi\/3)$.\nHence $e_{\\bm{\\ell}^{(2)}}(\\widetilde{\\bm{\\theta}})$ is also located at $-1$ with vanishing gradient.\nPerturbing $e_{\\bm{\\ell}^{(1)}}(\\widetilde{\\bm{\\theta}})$ and $e_{\\bm{\\ell}^{(2)}}(\\widetilde{\\bm{\\theta}})$ along a given direction $(\\beta_1, \\beta_2)$ is equivalent to controlling the signs of the following two expressions:\n$$\\beta_1\\beta_2\\ \\ \\text{and}\\ \\ -\\beta_1\\beta_2.$$\nThis means we can never choose two different directions that lead to un-even {counts}.\nTherefore we need to develop a new argument for this case.\n\nIndeed, when $p_1$ is not divisible by $3$, we choose $p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1^{(1)})=2\\pi\/3$ and $\\widetilde{\\theta}_2$ such that \n$p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2)\\notin \\{2\\pi\/3, 4\\pi\/3\\}$ {{\\it regardless} of the choice of $\\ell_2$}. Such choices guarantee that there are in total $p_2$ eigenvalues located at $-1$, which are \n$\\{e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}),\\ \\ \\ell_1=\\ell_1^{(1)}\\}$. It then suffices to control the movements of these eigenvalues along any given direction.\nA key observation is that along any direction, approximately $2p_2\/3$ eigenvalues will move up (down) while the other $p_2\/3$ eigenvalues move down (up), see \\eqref{eq:E=-1Jbbe}.\nThis leads to un-even counting that we need.\nLet us point out that if both $p_1, p_2$ are divisible by $3$, this argument does not work {(as it must, given the example constructed in Theorem~\\ref{t:nnnExamples})}: there will be $2p_2$ eigenvalues located at $-1$, and $p_2$ of them move up while the other $p_2$ of them move down along any given direction.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:sqnmain}.2]\nWithout loss of generality, we assume $p_1$ is not divisible by $3$.\nLet $p_j=3p_j'+k_j$, where $p_j',k_j \\in {\\mathbb Z}$ with $0 \\le k_j < 3$ and then define $\\widetilde\\bm{\\theta}$ by\n\\begin{align*}\n\\widetilde{\\theta}_1= \\frac{2\\pi k_1}{3},\\ \\ \n\\widetilde{\\theta}_2={\\frac{k_2+1}{4}\\pi}.\n\\end{align*}\n{As usual, denote $\\Lambda_E(\\widetilde{\\bm{\\theta}}) = \\Lambda_{-1}(\\widetilde{\\bm{\\theta}}) = \\{\\bm{\\ell} \\in \\Lambda : e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}}) = -1\\}$. We first claim that\n\\begin{equation} \\label{eq:sqnE=-1:multiplicity}\n\\Lambda_{-1}(\\widetilde{\\bm{\\theta}})\n=\n\\set{(p_1', \\ell_2) : 0 \\leq \\ell_2 < p_2 \\text{ and } \\ell_2 \\in {\\mathbb Z}}.\n\\end{equation}}\nLet us consider the trigonometric equation\n\\begin{align}\\label{eq:E=-1}\n\\cos(x)+\\cos(y)+\\cos(x-y)+\\cos(x+y)=-\\frac{1}{2}=\\frac{E}{2}.\n\\end{align}\nUsing the identity $\\cos(x-y)+\\cos(x+y)=2\\cos(x) \\cos(y)$, we see that \\eqref{eq:E=-1} is equivalent to \n\\[(2\\cos(x)+1) (2\\cos(y)+1)=0,\\]\nwhose solutions are $\\cos(x)=-1\/2$ or $\\cos(y)=-1\/2$. {With our choice of $\\widetilde{\\bm{\\theta}}$, it is clear that\n\\begin{equation} \\label{eq:E=-1the1ell1}\n\\frac{\\widetilde{\\theta}_1 + 2\\pi p_1'}{p_1}\n=\n\\frac{2\\pi}{3},\n\\quad\n\\cos\\left(\\frac{\\widetilde{\\theta}_1 + 2\\pi p_1'}{p_1}\\right)\n=\n-\\frac{1}{2}.\n\\end{equation}\nConsequently, \n\\begin{equation} \\label{eq:sqn:E=-1:LambdaEeq1}\ne_{(p_1',\\ell_2)}(\\widetilde{\\bm{\\theta}}) = -1 \\quad \\text{for every } 0 \\le \\ell_2 < p_2.\n\\end{equation}}\nDue to our choice of $\\widetilde{\\theta}_2$, we get\n\\begin{align}\\label{eq:E=-1J0}\n\\cos(p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2))\\neq -\\frac{1}{2}\\quad\n\\text{for any } \\ell_2 \\in [0,p_2) \\cap {\\mathbb Z}.\n\\end{align}\nIndeed, since $p_2^{-1}(\\widetilde{\\theta}_2+2\\pi\\ell_2) \\in [0,2\\pi)$, $\\cos(p_2^{-1}(\\widetilde{\\theta}_2+2\\pi \\ell_2))=-1\/2$ would force \n\\[\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2} \\in \\set{\\frac{2\\pi}{3}, \\frac{4\\pi}{3}},\\]\nwhich, after doing some algebra, leads to {\n\\[\n3(8\\ell_2+k_2+1) \\in \\{8p_2,16p_2\\},\n\\]\nwhich is plainly impossible, since $\\ell_2,p_2 \\in {\\mathbb Z}$ and $k_2 \\in \\{0,1,2\\}$.}\nAdditionally, due to our choice of $\\widetilde{\\theta}_1$, we also have\n\\begin{align}\\label{eq:E=-1Jbeta10}\n{\\cos(p_1^{-1}(\\widetilde{\\theta}_1+2\\pi \\ell_1))\\neq -1\/2 \\quad\n\\text{for any } \\ell_1 \\in \\big([0,p_1)\\cap{\\mathbb Z}\\big)\\setminus\\{p_1'\\}}.\n\\end{align}\nTo see this, suppose on the contrary that \\eqref{eq:E=-1Jbeta10} fails. This forces\n\\[\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}=\\frac{4\\pi}{3}\\]\nfor some $0 \\leq \\ell_1 < p_1$ with $\\ell_1\\neq p_1'$. Since\n\\[\\frac{\\widetilde{\\theta}_1+2\\pi p_1'}{p_1}=\\frac{2\\pi}{3},\\]\nthis implies\n\\[\\frac{2\\pi (\\ell_1-p_1')}{p_1}=\\frac{2\\pi}{3},\\]\nwhich is impossible since $p_1$ is not divisible by $3$. Combining \\eqref{eq:E=-1J0} and \\eqref{eq:E=-1Jbeta10} yields\n\\begin{equation} \\label{eq:sqn:E=-1:LambdaEeq2} \n{e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\neq - 1\n\\quad\n\\text{for any } \\bm{\\ell} =(\\ell_1,\\ell_2)\\in\\Lambda \\text{ such that } \\ell_1 \\neq p_1'.}\n\\end{equation}\nTaken together, \\eqref{eq:sqn:E=-1:LambdaEeq1} and \\eqref{eq:sqn:E=-1:LambdaEeq2} imply \\eqref{eq:sqnE=-1:multiplicity}.\n\nLet us choose $\\bm{\\beta}=(\\beta_1,\\beta_2)=(1,0)$. \nWe have that for any $\\bm{\\ell}\\in \\Lambda$:\n\\begin{align*}\n&\\bm{\\beta}\\cdot \\nabla e_{\\bm{\\ell}}(\\widetilde{\\bm{\\theta}})\\\\\n=&-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)\n-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}-\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\n-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}+\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\\\\n=&-\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)\\Bigg[1+2\\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\Bigg].\n\\end{align*}\nBy \\eqref{eq:sqnE=-1:multiplicity}, \\eqref{eq:E=-1the1ell1}, and \\eqref{eq:E=-1J0}, we have the following {for any $\\bm{\\ell} = (\\ell_1, \\ell_2) \\in \\Lambda_{-1}(\\widetilde{\\bm{\\theta}})$}:\n\\[\\sin\\Big(\\frac{\\widetilde{\\theta}_1+2\\pi \\ell_1}{p_1}\\Big)=\\frac{\\sqrt{3}}{2},\\ \\ \\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)\\neq -\\frac{1}{2}.\\]\nThis implies\n\\begin{align}\\label{eq:E=-1Jbbe}\n\\mathcal{J}_{\\bm{\\beta}}^0=\\emptyset,\\ \\ \\text{and }\\ \\ \n\\mathcal{J}_{\\bm{\\beta}}^{\\pm}\n=\n\\Bigg\\lbrace \\bm{\\ell} \\in \\Lambda_{-1}(\\widetilde{\\bm{\\theta}}) :\\ \\ \\mp \\frac{1}{2} \\mp \\cos\\Big(\\frac{\\widetilde{\\theta}_2+2\\pi \\ell_2}{p_2}\\Big)>0 \\Bigg\\rbrace.\n\\end{align}\nHence we expect that $|{\\mathcal J}_{\\bm{\\beta}}^+|\\sim p_2\/3$, and $|{\\mathcal J}_{\\bm{\\beta}}^-|\\sim 2p_2\/3$.\nMore precisely, we note that\n\\[\n{{\\mathcal J}_{\\bm{\\beta}}^+\n=\n\\Big\\lbrace (p_1', \\ell_2):\\ \\ \\frac{2\\pi}{3} < \\frac{(k_2+1)\\pi\/4 + 2\\pi \\ell_2}{p_2} < \\frac{4\\pi}{3} \\Big\\rbrace}.\\]\nUsing $p_2=3p_2'+k_2$, we obtain\n\\[\n{{\\mathcal J}_{\\bm{\\beta}}^+\n=\n\\Big\\lbrace (p_1',\\ell_2):\\ \\ p_2'+\\frac{5k_2-3}{24} < \\ell_2 < 2p_2'+\\frac{13k_2-3}{24} \\Big\\rbrace}.\n\\]\nConsequently,\n\\begin{align*}\n{\\mathcal J}_{\\bm{\\beta}}^+=\n\\begin{cases}\n\\{(p_1', \\ell_2):\\ \\ p_2'\\leq \\ell_2 \\leq 2p_2'-1\\},\\ \\ \\text{if } k=0,\\\\\n\\{(p_1', \\ell_2):\\ \\ p_2'+1\\leq \\ell_2 \\leq 2p_2'\\},\\ \\ \\text{if } k=1, 2.\n\\end{cases}.\n\\end{align*}\nTherefore \n\\begin{align}\\label{eq:Jbetapm}\n(|{\\mathcal J}_{\\bm{\\beta}}^+|, |{\\mathcal J}_{\\bm{\\beta}}^-|)=\n\\begin{cases}\n(p_2', 2p_2'),\\ \\ \\text{if } k=0,\\\\\n(p_2', 2p_2'+1),\\ \\ \\text{if } k=1,\\\\\n(p_2', 2p_2'+2),\\ \\ \\text{if } k=2.\n\\end{cases}\n\\end{align}\nNote that $p_2'\\geq 1$ whenever $k_2 = 0$.\nThus, a direct consequence of \\eqref{eq:Jbetapm} is\n\\begin{align}\\label{eq:Jbetapmneq}\n|{\\mathcal J}_{\\bm{\\beta}}^+|\\neq |{\\mathcal J}_{\\bm{\\beta}}^-|.\n\\end{align}\nOn the other hand, since ${\\mathcal J}_{\\bm{\\beta}}^0 = \\emptyset$, following the same argument as in the proof of Theorems~\\ref{thm:trimain}.1 yields $|{\\mathcal J}_{\\bm{\\beta}}^+|=|{\\mathcal J}_{\\bm{\\beta}}^-|$, which contradicts \\eqref{eq:Jbetapmneq}.\n\\end{proof}\n\n\\subsection{Proofs of Lemmas \\ref{lem:constructionsqn}, \\ref{lem:sqnJ0empty}, and \\ref{lem:arccos1\/3}}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:constructionsqn}]\nLet $x$ and $y$ solve \\eqref{eq:xyCondABsqn} with $E \\neq -1$. The second condition therein yields\n\\[\n\\sin(x) + 2\\sin(x)\\cos(y) = 0,\n\\]\nleading to two possibilities: $\\sin(x) = 0$ or $\\cos(y) = -1\/2$. If $\\sin(x) = 0$, we get $x = 0$ or $x = \\pi$, which yields \\eqref{eq:solutionx=0sqn} and \\eqref{eq:solutionx=pisqn} upon plugging in to the first condition in \\eqref{eq:xyCondABsqn}. In the event that $\\cos(y) = -1\/2$, we arrive at\n\\begin{align*}\n\\cos(x) + \\cos(y) + \\cos(x-y) + \\cos(x+y)\n& =\n\\cos(x) + \\cos(y) + 2\\cos(x)\\cos(y) \\\\\n& =\n\\cos(x) - \\frac{1}{2} - \\cos(x) \\\\\n& =\n-\\frac{1}{2},\n\\end{align*}\nin contradiction with $E \\neq -1$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:sqnJ0empty}]\nSuppose $x$ and $y$ satisfy \\eqref{eq:sqnJ0syst}. From the proof of Lemma~\\ref{lem:constructionsqn}, the second condition of \\eqref{eq:sqnJ0syst} implies $\\sin(x) = 0$ or $\\cos(y) = -1\/2$. Thus, $x = 0$, $x = \\pi$, $y=2\\pi\/3$, or $y = 4\\pi\/3$. \nWhen $\\sin(x) = 0$, the third condition of \\eqref{eq:sqnJ0syst} forces $\\sin(y) = 0$. The four points so obtained yield $E = 8$ when $(x,y) = (0,0)$, $E=-4$ when $(x,y) = (0,\\pi),(\\pi,0)$ and $E = 0$ when $(x,y) = (\\pi,\\pi)$. \nAlternatively, when $\\cos(y) = -1\/2$, the third condition of \\eqref{eq:sqnJ0syst} yields $\\cos(x) = -1\/2$, which impies $x = 2\\pi\/3$ or $x = 4\\pi\/3$. As in the Proof of Lemma~\\ref{lem:constructionsqn}, the four points corresponding to\n\\[\nx,y \\in \\set{\\frac{2\\pi}{3}, \\frac{4\\pi}{3}}\n\\]\nall yield $E = -1$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:arccos1\/3}]\nSuppose \n\\begin{align}\\label{eq1:cos1\/3}\n\\cos\\left(\\frac{2\\pi m}{n}\\right)=-\\frac{1}{3},\n\\end{align}\nfor $m\/n\\in {\\mathbb Q}$.\nLet $T_n(\\cdot)$ denote the $n$-th degree Cheybeshev polynomial so that\n\\begin{align}\\label{eq2:cos1\/3}\nT_n\\left(\\cos\\left(\\frac{2\\pi m}{n}\\right)\\right)=\\cos(2\\pi m)=1.\n\\end{align}\nIt is well-known that $T_n(x)=\\sum_{k=0}^n a_k x^k$, where $a_n=2^{n-1}$ and $a_k\\in {\\mathbb Z}$ for any $k$.\nHence \\eqref{eq1:cos1\/3} and \\eqref{eq2:cos1\/3} imply\n\\[2^{n-1}\\left(-\\frac{1}{3}\\right)^n+\\sum_{k=0}^{n-1} a_k \\left(-\\frac{1}{3}\\right)^{k} =1.\\]\nMultiplying by $(-3)^n$ on both sides of the equation, we obtain\n\\[2^{n-1}-3\\sum_{k=0}^{n-1} a_k (-3)^{n-k-1}=(-3)^n,\\]\nwhich implies $2^{n-1}$ is divisible by $3$.\nContradiction.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{\\boldmath Opening a gap at $-1$}\n\n\\begin{figure*}[b]\n\\begin{tikzpicture}[yscale=0.8, xscale=0.8]\n\\draw [-,line width = .1cm] (0,0) -- (12,0);\n\\draw [-,line width = .1cm] (0,0) -- (0,12);\n\\draw [-,line width= .1cm] (0,3) -- (12,3);\n\\draw [-,line width= .1cm] (3,0) -- (3,12);\n\\draw [-,line width= .1cm] (0,6) -- (12,6);\n\\draw [-,line width= .1cm] (6,0) -- (6,12);\n\\draw [-,line width= .1cm] (0,9) -- (12,9);\n\\draw [-,line width= .1cm] (9,0) -- (9,12);\n\\draw [-,line width= .1cm] (0,12) -- (12,12);\n\\draw [-,line width= .1cm] (12,0) -- (12,12);\n\\draw [-,line width= .1cm] (0,0) -- (12,12);\n\\draw [-,line width= .1cm] (3,0) -- (12,9);\n\\draw [-,line width= .1cm] (6,0) -- (12,6);\n\\draw [-,line width= .1cm] (9,0) -- (12,3);\n\\draw [-,line width= .1cm] (0,3) -- (9,12);\n\\draw [-,line width= .1cm] (0,6) -- (6,12);\n\\draw [-,line width= .1cm] (0,9) -- (3,12);\n\\draw [-,line width= .1cm] (3,0) -- (0,3);\n\\draw [-,line width= .1cm] (6,0) -- (0,6);\n\\draw [-,line width= .1cm] (9,0) -- (0,9);\n\\draw [-,line width= .1cm] (12,0) -- (0,12);\n\\draw [-,line width= .1cm] (12,3) -- (3,12);\n\\draw [-,line width= .1cm] (12,6) -- (6,12);\n\\draw [-,line width= .1cm] (12,9) -- (9,12);\n\\filldraw[color=black, fill=black](0,0) circle (.2);\n\\filldraw[color=black, fill=black](3,0) circle (.2);\n\\filldraw[color=black, fill=black](6,0) circle (.2);\n\\filldraw[color=black, fill=black](9,0) circle (.2);\n\\filldraw[color=black, fill=black](0,3) circle (.2);\n\\filldraw[color=red, fill=red](3,3) circle (.2);\n\\filldraw[color=red, fill=red](6,3) circle (.2);\n\\filldraw[color=red, fill=red](9,3) circle (.2);\n\\filldraw[color=black, fill=black](0,6) circle (.2);\n\\filldraw[color=red, fill=red](3,6) circle (.2);\n\\filldraw[color=red, fill=red](6,6) circle (.2);\n\\filldraw[color=red, fill=red](9,6) circle (.2);\n\\filldraw[color=black, fill=black](0,9) circle (.2);\n\\filldraw[color=red, fill=red](3,9) circle (.2);\n\\filldraw[color=red, fill=red](6,9) circle (.2);\n\\filldraw[color=red, fill=red](9,9) circle (.2);\n\\filldraw[color=black, fill=black](12,0) circle (.2);\n\\filldraw[color=black, fill=black](12,3) circle (.2);\n\\filldraw[color=black, fill=black](12,6) circle (.2);\n\\filldraw[color=black, fill=black](12,9) circle (.2);\n\\filldraw[color=black, fill=black](12,12) circle (.2);\n\\filldraw[color=black, fill=black](0,12) circle (.2);\n\\filldraw[color=black, fill=black](3,12) circle (.2);\n\\filldraw[color=black, fill=black](6,12) circle (.2);\n\\filldraw[color=black, fill=black](9,12) circle (.2);\n\\draw [-,line width= .1cm,color=red] (3,3) -- (3,9);\n\\draw [-,line width= .1cm,color=red] (6,3) -- (6,9);\n\\draw [-,line width= .1cm,color=red] (9,3) -- (9,9);\n\\draw [-,line width= .1cm,color=red] (3,3) -- (9,3);\n\\draw [-,line width= .1cm,color=red] (3,6) -- (9,6);\n\\draw [-,line width= .1cm,color=red] (3,9) -- (9,9);\n\\draw [-,line width= .1cm,color=red] (3,6) -- (6,9);\n\\draw [-,line width= .1cm,color=red] (3,3) -- (9,9);\n\\draw [-,line width= .1cm,color=red] (6,3) -- (9,6);\n\\draw [-,line width= .1cm,color=red] (3,6) -- (6,3);\n\\draw [-,line width= .1cm,color=red] (3,9) -- (9,3);\n\\draw [-,line width= .1cm,color=red] (6,9) -- (9,6);\n\\node at (3.9,3.3){\\hot{$q_1$}};\n\\node at (6.9,3.3){\\hot{$q_2$}};\n\\node at (9.9,3.3){\\hot{$q_3$}};\n\\node at (3.9,6.3){\\hot{$q_4$}};\n\\node at (6.9,6.3){\\hot{$q_5$}};\n\\node at (9.9,6.3){\\hot{$q_6$}};\n\\node at (3.9,9.3){\\hot{$q_7$}};\n\\node at (6.9,9.3){\\hot{$q_8$}};\n\\node at (9.9,9.3){\\hot{$q_9$}};\n\\end{tikzpicture}\n\\caption{A $3\\times 3$ potential on the square lattice that opens a gap at $E=-1$ with small positive positive coupling.}\\label{fig:sqn33period}\n\\end{figure*}\n\n\\begin{theorem} \\label{thm:nnnExGapLength}\nEnumerate the vertices of a $3\\times 3$ fundamental cell of the square lattice as in Figure~\\ref{fig:sqn33period}, denote $r=\\sqrt{4-\\sqrt{15}}$, define a $(3,3)$-periodic potential $Q$ on ${\\mathbb Z}^2$ via\n\\[\n(q_1,\\ldots,q_9)\n=\n\\Big(-r-\\frac{1}{r}+2,\\ -r,\\ -r+\\frac{1}{r}-2,\\ -\\frac{1}{r},\\ 0,\\ +\\frac{1}{r},\\ r-\\frac{1}{r}-2,\\ r,\\ r+\\frac{1}{r}+ 2\\Big),\n\\]\nand denote $H_\\lambda = \\Delta_{\\mathrm{sqn}} + \\lambda Q$. Then, for all $ \\lambda > 0$ sufficiently small, $\\sigma(H_\\lambda)$ consists of two connected components. Moreover, if $\\mathfrak{g}_\\lambda$ denotes the gap that opens at energy $-1$, one has\n\\[\n\\left(-1 - \\frac{\\lambda}{10}, -1 + \\frac{\\lambda}{10} \\right)\n\\subseteq\n\\mathfrak{g}_\\lambda\n\\subseteq\n\\left(-1 - \\frac{\\lambda}{4},-1+\\frac{\\lambda}{4}\\right).\n\\]\nIn particular, the gap opens linearly.\n\\end{theorem}\n\nLet us observe that the proof below can be refined a bit to yield sharper constants than $1\/10$ and $1\/4$.\n\n\\begin{proof}\nFor $\\bm{\\theta} = (\\theta_1,\\theta_2) \\in {\\mathbb T}^2$, let $H_\\lambda(\\bm{\\theta})$ denote the Floquet matrix corresponding to $H_\\lambda$. Ordering the vertices of the fundamental domain as in Figure~\\ref{fig:sqn33period}, we obtain:\n\\[\nH_\\lambda(\\bm{\\theta})\n=\n\\begin{bmatrix}\n\\lambda q_1 & 1 & e^{-i\\theta_1} & 1 & 1 & e^{-i\\theta_1} & e^{-i \\theta_2} & e^{-i\\theta_2} &e^{-i(\\theta_1+\\theta_2)} \\\\\n1 & \\lambda q_2 & 1 & 1 & 1 & 1 & e^{-i \\theta_2}& e^{-i\\theta_2} & e^{-i\\theta_2} \\\\\ne^{i \\theta_1} & 1 & \\lambda q_3& e^{i\\theta_1} & 1 & 1 & e^{i(\\theta_1-\\theta_2)} & e^{-i\\theta_2} & e^{-i\\theta_2}\\\\\n1 & 1 & e^{-i\\theta_1} & \\lambda q_4 & 1 & e^{-i\\theta_1} & 1 & 1 &e^{-i\\theta_1} \\\\\n1 & 1 & 1 & 1 & \\lambda q_5 & 1 & 1 & 1 &1 \\\\\ne^{i\\theta_1} & 1 & 1 & e^{i\\theta_1} & 1 & \\lambda q_6 & e^{i\\theta_1} & 1 & 1 \\\\\ne^{i\\theta_2} & e^{i\\theta_2} & e^{-i(\\theta_1-\\theta_2)} & 1 & 1 &e^{-i\\theta_1} & \\lambda q_7& 1 & e^{-i\\theta_1} \\\\\ne^{i\\theta_2} & e^{i\\theta_2} & e^{i\\theta_2} & 1 & 1 & 1 & 1 & \\lambda q_8 & 1 \\\\\ne^{i(\\theta_1+\\theta_2)} & e^{i\\theta_2} & e^{i\\theta_2} & e^{i\\theta_1} &1 &1 & e^{i\\theta_1} & 1 &\\lambda q_9 \n\\end{bmatrix}.\n\\]\n\\begin{comment}\nStraightforward (Mathematica!) calculations reveal\n\\begin{align*}\n\\det(H_{\\lambda}(\\bm{\\theta}) + {\\mathbb I})\n& =\nQ_0(\\bm{\\theta})+\\sum_{k=1}^9 Q_k(\\bm{\\theta}) \\lambda^k.\n\\end{align*}\nwhere\n\\begin{align*}\nQ_0(\\bm{\\theta})& =\n4096 \\sin^6\\left(\\frac{\\theta_1}{2}\\right) \\sin^6\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_1(\\bm{\\theta})& =0\\\\\nQ_2(\\bm{\\theta})&=10240 \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_3(\\bm{\\theta})&=1024 \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_4(\\bm{\\theta})&=5824 \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_5(\\bm{\\theta})&=1024 \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nQ_6(\\bm{\\theta})&=176 - 80 \\cos(\\theta_1)-80 \\cos(\\theta_2) - 16 \\cos(\\theta_1) \\cos(\\theta_2) - 8 \\sin(\\theta_1) \\sin(\\theta_2)\\\\\nQ_8(\\bm{\\theta})&=12\\\\\nQ_9(\\bm{\\theta})&=0.\n\\end{align*}\nNext we will show $Q_6\\geq 0$, which will lead to $\\det(H_{\\lambda}(\\bm{\\theta}) + {\\mathbb I})\\geq 12\\lambda^8>0$.\n\nTo this end, we compute\n\\begin{align*}\n\\nabla Q_6(\\bm{\\theta})=(&80\\sin(\\theta_1)-8\\cos(\\theta_1) \\sin(\\theta_2)+16\\cos(\\theta_2)\\sin(\\theta_1),\\\\\n &\\qquad 80 \\sin(\\theta_2) -8\\cos(\\theta_2) \\sin(\\theta_1)+16 \\cos(\\theta_1) \\sin(\\theta_2) ).\n\\end{align*}\nSetting $\\nabla Q_6(\\bm{\\theta})=(0,0)$, we arrive tat\n\\begin{align}\n10\\sin(\\theta_1)-\\cos(\\theta_1)\\sin(\\theta_2)+2\\sin(\\theta_1)\\cos(\\theta_2)=0,\\label{eq1}\\\\\n10\\sin(\\theta_2)-\\cos(\\theta_2)\\sin(\\theta_1)+2\\sin(\\theta_2)\\cos(\\theta_1)=0.\\label{eq2}\n\\end{align}\nAdding \\eqref{eq1} and \\eqref{eq2}, we obtain\n\\begin{align}\\label{eq3}\n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)\\left(10\\cos\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)+\\cos\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)\\right)=0.\n\\end{align}\nSubtracting \\eqref{eq2} from \\eqref{eq1}, we obtain\n\\begin{align}\\label{eq4}\n\\sin\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)\\left(10\\cos\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)+3\\cos\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)\\right)=0.\n\\end{align}\nSolving \\eqref{eq3} and \\eqref{eq4} yields\n\\[\n\\cos\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)=\\cos\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)=0,\n\\]\nor \n\\[\n\\sin\\left(\\frac{\\theta_1+\\theta_2}{2}\\right)=\\sin\\left(\\frac{\\theta_1-\\theta_2}{2}\\right)=0.\n\\]\nHence all the four solutions are $\\bm{\\theta}=(0,0), (0, \\pi), (\\pi, 0), (\\pi, \\pi)$.\n\nEvaluating $Q_6(\\bm{\\theta})$ at these points yields\n\\[Q_6(0,0)=0,\\ \\ Q_6(0,\\pi)=192,\\ \\ Q_6(\\pi,0)=192,\\ \\ Q_6(\\pi,\\pi)=320.\\]\nIn conclusion, we have shown $Q_6(\\bm{\\theta})\\geq 0$ for all $\\bm{\\theta}$.\nHence $\\det(H_{\\lambda,\\theta_1,\\theta_2} + {\\mathbb I})\\geq 12\\lambda^8>0$, thus $E=-1$ is not in the spectrum.\n\\end{comment}\nFor $s\\in (-1,1)$, let us consider \n\\[\n\\det(H_{\\lambda}(\\bm{\\theta}) + (1+ s\\lambda) {\\mathbb I})\n=\n\\sum_{k=0}^9 X_k(\\bm{\\theta}, s) \\lambda^k.\n\\]\nOur goal is to show $\\det(H_{\\lambda}(\\bm{\\theta}) + (1+ s\\lambda) {\\mathbb I})$ never vanishes for sufficiently small $\\lambda>0$ and for $|s| < 0.1$.\nDirect computations yield\n\\begin{align*}\nX_0(\\bm{\\theta}, s)&=4096 \\sin^6\\left(\\frac{\\theta_1}{2}\\right) \\sin^6\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_1(\\bm{\\theta}, s)&=0\\\\\nX_2(\\bm{\\theta}, s)&=Y_2(s) \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_3(\\bm{\\theta}, s)&=Y_3(s) \\sin^4\\left(\\frac{\\theta_1}{2}\\right) \\sin^4\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_4(\\bm{\\theta}, s)&=Y_4(s) \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_5(\\bm{\\theta}, s)&=Y_5(s) \\sin^2\\left(\\frac{\\theta_1}{2}\\right) \\sin^2\\left(\\frac{\\theta_2}{2}\\right)\\\\\nX_6(\\bm{\\theta}, s)&=Y_{6,1}(s)+Y_{6,2}(s)\\cos(\\theta_1)+Y_{6,3}(s)\\cos(\\theta_2)\\\\ & \\qquad +Y_{6,4}(s)\\cos(\\theta_1)\\cos(\\theta_2)+Y_{6,5}(s) \\sin(\\theta_1)\\sin(\\theta_2)\\\\\nX_7(\\bm{\\theta}, s)&=0\\\\\nX_8(\\bm{\\theta}, s)&=Y_8(s)\\\\\nX_9(\\bm{\\theta}, s)&=Y_9(s),\n\\end{align*}\nin which\n\\begin{align*}\nY_2(s)&=512 (20-9 s^2)\\\\\nY_3(s)&=256 (4 - 20 s + 3 s^3)\\\\\nY_4(s)&=16(364 + 144 s - 504 s^2 + 81 s^4)\\\\\nY_5(s)&=16 (64 - 196 s - 48 s^2 + 104 s^3 - 9 s^5)\\\\\nY_{6,1}(s)&=176 + 704 s - 3132 s^2 - 496 s^3 + 1376 s^4 - 96 s^6\\\\\nY_{6,2}(s)&=-80 + (96 \\sqrt{15}-320) s + (1380 +144 \\sqrt{15}) s^2 + 208 s^3 - (584+54 \\sqrt{15}) s^4 + 42 s^6\\\\\nY_{6,3}(s)&=-80 - (320 +96 \\sqrt{15}) s + (1380 - 144 \\sqrt{15}) s^2 + 208 s^3 - (584- 54 \\sqrt{15}) s^4 + 42 s^6\\\\\nY_{6,4}(s)&=-16 - 64 s + 372 s^2 + 80 s^3 - 208 s^4 + 12 s^6\\\\\nY_{6,5}(s)&=8(2s-1)^3\\\\\nY_8(s)&=12 + 32 s - 360 s^2 - 512 s^3 + 1025 s^4 + 96 s^5 - 224 s^6 + 9 s^8\\\\\nY_9(s)&=12 s + 16 s^2 - 120 s^3 - 128 s^4 + 205 s^5 + 16 s^6 - 32 s^7 + s^9.\\\\\n\\end{align*}\nOne simple observation is that \n\\begin{align}\\label{eq:sumY6}\nY_{6,1}(s)+Y_{6,2}(s)+Y_{6,3}(s)+Y_{6,4}(s)=0.\n\\end{align}\nIt is easy to see that for $|s|<0.1$, \n\\[Y_2(s), Y_3(s),Y_5(s)>0.\\]\nIt is easy to compute that\n\\[\nY'_9(s)=12 + 32 s - 360 s^2 - 512 s^3 + 1025 s^4 + 96 s^5 - 224 s^6 + 9 s^8\n=\nY_8(s).\n\\]\nThus, \n\\begin{equation} \\label{eq:Y9prime(s)}\nY_9'(s)>12 - 32 \\times 0.1 - 360 \\times (0.1)^2 - 512 \\times (0.1)^3 - 96 \\times (0.1)^5 - 224\\times (0.1)^6{>4.5}>0\\end{equation}\nfor $|s|<0.1$, which implies\n\\begin{align}\\label{eq:Y9s}\nY_9(s)\\geq Y_9(-0.1)>-1\n\\end{align}\nfor all $|s| < 0.1$. Carefully estimating $Y_4(s)$ and $Y_8(s)$ will help us bound the $\\lambda^6$ order term from below using the AM-GM inequality.\n\\begin{align}\\label{eq:Y4Y8s}\nY_4(s)&\\geq 16(364-144\\times 0.1-504\\times (0.1)^2 {-81 \\times (0.1)^4})>5500,\\\\\nY_8(s)&\\geq 12 - 32 \\times 0.1 - 360 \\times (0.1)^2 - 512 \\times (0.1)^3 - 96 \\times (0.1)^5 - 224 \\times (0.1)^6>4.5. \\notag\n\\end{align}\n{In fact, since $Y_8 = Y_9'$, the second inequality already follows from \\eqref{eq:Y9prime(s)}.}\nFor the $Y_{6,j}$ terms, we have\n\\begin{equation}\\label{eq:Y6s}\n\\begin{aligned}\nY_{6,1}(s)&\\geq 176 - 704 \\times 0.1 - 3132 \\times (0.1)^2 - 496 \\times (0.1)^3 - 96 \\times (0.1)^6>0,\\\\\n Y_{6,2}(s)&\\leq -80 + (96 \\sqrt{15}-320) \\times 0.1 + (1380 +144 \\sqrt{15}) \\times (0.1)^2 \\\\ & \\qquad\\qquad + 208 \\times (0.1)^3 + 42\\times (0.1)^6<0 \\\\\n Y_{6,3}(s)&\\leq -80 + (320 + 96 \\sqrt{15}) \\times 0.1 + (1380 - 144 \\sqrt{15}) \\times (0.1)^2 \\\\ & \\qquad\\qquad+ 208 \\times (0.1)^3 + 42 \\times (0.1)^6<0, \\\\\nY_{6,4}(s)&\\leq -16 + 64 \\times 0.1 + 372 \\times (0.1)^2 + 80 \\times (0.1)^3 + 12 \\times (0.1)^6<0, \\\\\n-14 & \\leq Y_{6,5}(s)< 0.\n\\end{aligned}\n\\end{equation}\nUsing \\eqref{eq:sumY6} and \\eqref{eq:Y6s}, we obtain\n\\begin{align} \\label{eq:X6LB}\n\\notag\nX_6(\\bm{\\theta}) & \\geq Y_{6,1}(s)+Y_{6,2}(s)+Y_{6,3}(s)+Y_{6,4}(s) + Y_{6,5}(s)\\sin(\\theta_1)\\sin(\\theta_2) \\\\\n\\notag\n& =\nY_{6,5}(s)\\sin(\\theta_1)\\sin(\\theta_2) \\\\\n& \\geq\n-14 |\\sin(\\theta_1)\\sin(\\theta_2)|.\n\\end{align}\nIn particular, the first line uses $Y_{6,2}, Y_{6,3}, Y_{6,4} < 0$, the second line uses \\eqref{eq:sumY6}, and the final line uses $-14 \\leq Y_{6,5} < 0$.\n\nNow we combine our estimates together. \nNote that \n\\begin{align}\\label{eq:sumX0235}\nX_0(\\bm{\\theta},s)+X_2(\\bm{\\theta},s)\\lambda^2+X_3(\\bm{\\theta},s)\\lambda^3+X_5(\\bm{\\theta},s)\\lambda^5\\geq 0.\n\\end{align}\nUsing $a^2+b^2\\geq 2|ab|$, we obtain the following from \\eqref{eq:Y4Y8s}\n\\[\nX_4(\\bm{\\theta},s)\\lambda^4+\\frac{1}{2}X_8(\\bm{\\theta},s)\\lambda^8 \\geq 2\\sqrt{2.25\\times 5500} \\left|\\sin\\left(\\frac{\\theta_1}{2}\\right) \\sin\\left(\\frac{\\theta_2}{2}\\right)\\right| \\lambda^6.\n\\]\nUsing $2|\\sin(x\/2)|\\geq 2|\\sin(x\/2)\\cos(x\/2)|=|\\sin(x)|$, we obtain from above that\n\\[X_4(\\bm{\\theta},s) {\\lambda^4} + \\frac{1}{2}X_8(\\bm{\\theta},s) {\\lambda^8} \n\\geq \n55 |\\sin(\\theta_1)\\sin(\\theta_2)|{\\lambda^6}.\\]\nCombining this with \\eqref{eq:X6LB}, we have\n\\begin{equation} \\label{eq:sumX486}\nX_4(\\bm{\\theta},s)\\lambda^4+\\frac{1}{2}X_8(\\bm{\\theta},s)\\lambda^8+X_6(\\bm{\\theta},s)\\lambda^6 \n\\geq\n41 |\\sin(\\theta_1)\\sin(\\theta_2)|\\lambda^6\n\\geq 0.\n\\end{equation}\n\nFinally using \\eqref{eq:Y9s} and \\eqref{eq:Y4Y8s}, we have\n\\begin{align}\\label{eq:sumY89}\n\\frac{1}{2}X_8(\\bm{\\theta},s)\\lambda^8+X_9(\\bm{\\theta},s)\\lambda^9=\\frac{1}{2}Y_8(s)\\lambda^8+Y_9(s)\\lambda^9\\geq 2.25\\lambda^8-\\lambda^9>0.25 \\lambda^8,\n\\end{align}\nprovided that $\\lambda<2$.\nCombining \\eqref{eq:sumX0235}-\\eqref{eq:sumY89}, we have\n\\[\\det(H_{\\lambda}(\\bm{\\theta}) + (1+ s\\lambda) {\\mathbb I}) \\geq 0.25\\lambda^8>0,\\]\nfor any $\\bm{\\theta}\\in {\\mathbb T}^2$ and $|s|<0.1$. \nThis proves the lower bound on the gap.\n\n{For the upper bound, observe that $X_j\\big((\\pi,0),s\\big) = 0$ for all $s$ and for every $0 \\le j \\le 5$ and\n\\[\nX_6\\big((\\pi,0),\\pm 1\/4\\big)\n<-85.\n\\]\nThus, for small $\\lambda > 0$, \n\\[\n\\det\\big(H_\\lambda(\\pi, 0) + (1\\pm\\lambda\/4){\\mathbb I} \\big) < -85\\lambda^6 + O(\\lambda^8)<0.\n\\]\nIt is also clear that $X_0((0,0),s)=4096$,\nwhich implies \n\\[\\det \\big(H_\\lambda(0, 0) +(1\\pm\\lambda\/4){\\mathbb I} \\big) =4096+O(\\lambda)>0.\\]\nThus we conclude that \n\\[1\\pm \\frac{\\lambda}{4}\\in \\sigma(H_\\lambda),\\]\nwhich concludes the proof of the upper bound on the length of the gap.}\n\\end{proof}\n\n\\section*{Acknowledgement}\nWe would like to thank Svetlana Jitomirskaya for comments on an earlier version of the manuscript, and Tom Spencer for useful discussions.\nR.H. would like to thank IAS, Princeton, for its hospitality during the 2017-18 academic year,\nand Virginia Tech for its hospitality during which part of the work was done.\nR.H. is supported in part by the National Science Foundation under Grant No. DMS-1638352. \nJ.F.\\ was supported in part by an AMS-Simons Travel Grant 2016--2018.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nNowadays, the axion mechanism represents our best solution to the long standing strong CP puzzle, that is, the non-observation of CP violation in the strong interactions that should have manifested itself as an electric dipole moment for the neutron~\\cite{Abel:2020pzs}.\n\nThe axion mechanism relies on the spontaneous breaking of a new symmetry, the PQ symmetry~\\cite{PQ}, and on the subsequent realignment of the associated Goldstone boson, the axion~\\cite{Weinberg:1977ma,Wilczek:1977pj}, by strong interaction effects that kills off any CP violation in the QCD Lagrangian. This solution is thus tailored to the problem it is intended to solve and, as such, may appear a bit ad-hoc. In addition, unsuccessful experimental searches for the axion have ruled out its simplest incarnation, leaving us with essentially two classes of scenarios in which the axion is extremely light (well below the eV scale) and very weakly coupled to normal matter: the KSVZ~\\cite{KSVZ} framework in which new very heavy colored fermions are introduced, and the DFSZ~\\cite{DFSZ} scenario in which at least two Higgs doublets are required. Though the strong CP puzzle is extremely serious, additional motivations appear desirable to motivate such departures from the Standard Model (SM) matter content. To that avail, knowing that the axion could also make up for the observed dark matter (DM) offers a strong incentive to pursue this route~\\cite{DMaxion}. \n\nYet, current axion models cannot explain why the DM relic density is so close to that of baryonic matter. Though this may be totally coincidental, it nevertheless suggests a link between DM and baryogenesis~\\cite{Kaplan:1991ah}, another prominent cosmological enigma. Actually, it suggests DM is not foreign to baryon $\\mathcal{B}$ or lepton $\\mathcal{L}$ number (see Ref.~\\cite{Alonso-Alvarez:2021oaj} and references therein for a recent analysis), or that DM is somehow related to $\\mathcal{B}$ being spontaneously broken~\\cite{Dulaney:2010dj}. In parallel, there have been many attempts at involving axions in the baryogenesis mechanism, see e.g. Refs.~\\cite{Craig:2010au,Servant:2014bla,Jeong:2018ucz,Co:2019wyp,Krauss:2022usd,Domcke:2020kcp}, though in general still relying on the SM anomalous $\\mathcal{B}+\\mathcal{L}$ effects. \n\nOur goal here is to go one step further and entangle the PQ symmetry with $\\mathcal{B}$ and $\\mathcal{L}$ from the start. As a matter of principle, accidental symmetries are not particularly attractive, but while we can live with the PQ symmetry, assuming some dynamics hide behind it, $\\mathcal{B}$ and $\\mathcal{L}$ cannot be viable since, as said before, the electroweak non-perturbative dynamics break them, and baryogenesis asks for their violation. By unifying the PQ symmetry with $\\mathcal{B}$ and $\\mathcal{L}$, all three are broken spontaneously, but a single Goldstone field remains, the axion (for some recent works along this line, see Refs.~\\cite{Reig:2018yfd,Ohata:2021rkh}). In this way, the complex scalar field whose pseudoscalar component is the axion becomes charged under $\\mathcal{B}$ and $\\mathcal{L}$ and, at the high scale, protects the model from additional $\\mathcal{B}$ and\/or $\\mathcal{L}$ violation. At the same time, though the axion has no charge, it inherits a $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating phenomenology. Whether this is sufficient to relate the DM and baryonic relic densities remains to be seen, and is beyond the scope of the present paper, but we think these constructions may direct us in the right direction.\n\nIn this paper, we will use scalar and vector leptoquarks and diquarks to entangle the PQ, $\\mathcal{B}$, and $\\mathcal{L}$ symmetries. Such states are well motivated in various theoretical settings (see Ref.~\\cite{Dorsner:2016wpm} for a review) and, furthermore, supported by a number of anomalies like the $W$ boson mass, B decays~\\cite{LQreview} or $(g-2)_{\\mu}$~\\cite{Dorsner:2019itg}, or even combinations of them~\\cite{Athron:2022qpo,Bhaskar:2022vgk}. Our goal is to systematically analyze the $\\mathcal{B}$ and\/or $\\mathcal{L}$ symmetry breaking patterns that can arise combining the DFSZ and KSVZ scenarios with leptoquarks and diquarks and, in each case, to analyze the impact on the axion phenomenology.\n\nThe paper is organized as follows. In section~\\ref{Sec2a}, we briefly introduce the KSVZ and DFSZ axion models and, in Section~\\ref{Sec2b}, discuss in some details the ambiguities arising from the $\\mathcal{B}$ and $\\mathcal{L}$ fermionic currents~\\cite{Quevillon:2020hmx,Quevillon:2020aij}. Then in section~\\ref{Sec2c}, we set up the leptoquark and diquark sector, describing all the possible $\\mathcal{B}$ and $\\mathcal{L}$ explicit breaking patterns achievable with these states. This forms the basis for combining the axion and leptoquark\/diquark sectors in Sec.~\\ref{SecAxLQDQ}. We analyze first the KSVZ setting in Sec.~\\ref{SecKSVZ} and describe the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm1), (2,0), (1,\\pm3)$ spontaneous breaking patterns, further adding to them a spontaneously generated $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ seesaw mechanism for neutrino masses. These scenarios are then trivially adapted to the DFSZ setting in Sec.~\\ref{SecDFSZ}. In the final Sec.~\\ref{SecSpont}, we show how to force $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ effects to involve one or more axion fields. The phenomenology is then quite different, and we briefly describe some possible consequences for the neutron lifetime anomaly or neutron-antineutron oscillation experiments. Finally, our results are summarized in Sec.~\\ref{Ccl}.\n\n\\section{Axion and leptoquark models}\n\nIn this section, the KSVZ~\\cite{KSVZ} and DFSZ~\\cite{DFSZ} axion models are introduced, and their connection to baryon and lepton numbers, $\\mathcal{B}$ and $\\mathcal{L}$, are detailed. Then, we introduce separately the leptoquarks and diquarks that can be coupled to SM fermions, and discuss how their couplings drive specific $\\mathcal{B}$ and $\\mathcal{L}$ violating patterns. This sets the stage for the next section, where both axion models and leptoquarks\/diquarks will be put together.\n\n\\subsection{Introducing the KSVZ and DFSZ models\\label{Sec2a}}\n\nIn both the KSVZ and DFSZ constructions, the axion emerges as the pseudoscalar component of a complex scalar field. This state is neutral under all the SM gauge interactions, $\\phi=(\\mathbf{1},\\mathbf{1},0)$ under $SU(3)_{C}\\otimes SU(2)_{L}\\otimes U(1)_{Y}$, but its kinetic term is invariant under the rephasing $\\phi\\rightarrow e^{i\\alpha}\\phi$. This invariance is promoted to a\nspontaneously broken symmetry $U(1)_{\\phi}$ by postulating a rephasing invariant scalar potential with the usual Mexican hat shape, $V(\\phi^{\\dagger}\\phi)=\\mu^{2}\\phi^{\\dagger}\\phi+\\lambda(\\phi^{\\dagger}\\phi)^{4}$, $\\mu^{2}<0$ and $\\lambda>0$. In that case, the components of $\\phi$ can be written\n\\begin{equation}\n\\phi=\\frac{1}{\\sqrt{2}}(v_{\\phi}+\\rho)\\exp(i\\eta_{\\phi}\/v_{\\phi})\\ ,\n\\label{PhiPolar}\n\\end{equation}\nwith $\\eta_{\\phi}$ the associated Goldstone boson and $v_{\\phi}^{2}=-\\mu^{2}\/\\lambda$ the vacuum expectation value (VEV). As the breaking scale $v_{\\phi}$ naturally tunes all the $\\eta_{\\phi}$ couplings, it is assumed much higher than the electroweak scale to avoid exclusion bounds.\n\nTo solve the strong CP puzzle, $\\eta_{\\phi}$ must interact with SM particles~\\cite{Weinberg:1977ma,Wilczek:1977pj}, in particular with gluons via a $\\eta_{\\phi}G^{\\alpha,\\mu\\nu}\\tilde{G}_{\\mu\\nu}^{a}$ coupling~\\cite{PQ}. What differentiates the KSVZ and DFSZ models is how these couplings are introduced. The former~\\cite{KSVZ} adds a vector-like colored fermion $\\Psi_{L,R}\\sim(\\mathbf{R},\\mathbf{T},Y)$ for some complex representation $\\mathbf{R}$ of $SU(3)_{C}$, but otherwise arbitrary weak representation $\\mathbf{T}$ and hypercharge $Y$, and postulates the Lagrangian (the rest of the SM couplings are understood)\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ}} & =\\partial_{\\mu}\\phi^{\\dagger}\\partial^{\\mu\n}\\phi-V(\\phi)+\\bar{\\Psi}_{L,R}(i\\slashed D)\\Psi_{L,R}+(y\\phi\\bar{\\Psi}_{L}\\Psi_{R}+h.c.)\\nonumber\\\\\n& -\\bar{u}_{R}\\mathbf{Y}_{u}q_{L}H-\\bar{d}_{R}\\mathbf{Y}_{d}q_{L}H^{\\dagger\n}-\\bar{e}_{R}\\mathbf{Y}_{e}\\ell_{L}H^{\\dagger}-\\bar{\\nu}_{R}\\mathbf{Y}_{\\nu\n}\\ell_{L}H+h.c.\\ . \\label{KSVZ0}%\n\\end{align}\nThe covariant derivative acting on $\\Psi_{L,R}$ is as appropriate to its chosen gauge quantum numbers. What characterizes this model is first that the Goldstone boson of the PQ symmetry does not mix with that of the $SU(2)_{L}\\otimes U(1)_{Y}$ breaking (the phase of the Higgs doublet $H$). Thus, the axion is simply $a^{0}=\\eta_{\\phi}$, and it has no direct coupling to any of the SM particles. It only couples to $\\Psi_{L}$ and $\\Psi_{R}$, which necessarily have different charges under $U(1)_{\\phi}$. Then, axion to SM gauge boson couplings first arise at one-loop, via anomalous $\\Psi_{L,R}$ triangle loops, while those to SM fermions require a further gauge boson loop. Since $\\Psi_{L,R}$ can be massive in the electroweak unbroken phase, its loops do not break $SU(2)_{L}\\otimes U(1)_{Y}$ and the couplings to gauge bosons have the $SU(2)_{L}\\otimes U(1)_{Y}$ invariant form~\\cite{Georgi:1986df}\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ}}^{eff}=-\\frac{1}{16\\pi^{2}v_{\\phi}}a^{0}(g_{s}%\n^{2}d_{L}C_{C}G_{\\mu\\nu}^{a}\\tilde{G}^{a,\\mu\\nu}+g^{2}d_{C}C_{L}W_{\\mu\\nu}%\n^{i}\\tilde{W}^{i,\\mu\\nu}+g^{\\prime2}d_{L}d_{C}C_{Y}B_{\\mu\\nu}\\tilde{B}^{\\mu\n\\nu})\\ , \\label{GaugeCouplAno}\n\\end{equation}\nwith the quadratic invariants and dimensions of the $\\mathbf{R}$ and $\\mathbf{T}$ representations denoted $C_{C,L}$ and $d_{C,L}$, and $C_{Y}=Y^{2}\/4$.\n\nThe DFSZ model~\\cite{DFSZ} does not introduce new fermions, but requires two Higgs doublets. The important couplings are%\n\\begin{align}\n\\mathcal{L}_{\\mathrm{DFSZ}} & =\\partial_{\\mu}\\phi^{\\dagger}\\partial^{\\mu\n}\\phi-V(\\phi^{\\dagger}\\phi)+\\phi^{2}H_{u}^{\\dagger}H_{d}+V(H_{u}^{\\dagger\n}H_{u},H_{d}^{\\dagger}H_{d})\\nonumber\\\\\n& -\\bar{u}_{R}\\mathbf{Y}_{u}q_{L}H_{u}-\\bar{d}_{R}\\mathbf{Y}_{d}q_{L}%\nH_{d}^{\\dagger}-\\bar{e}_{R}\\mathbf{Y}_{e}\\ell_{L}H_{d}^{\\dagger}-\\bar{\\nu}%\n_{R}\\mathbf{Y}_{\\nu}\\ell_{L}H_{u}+h.c.\\ . \\label{DFSZ0}%\n\\end{align}\nThe potentials and Yukawa couplings are invariant under three independent $U(1)$s, corresponding to the rephasing of $\\phi$, $H_{u}$, and $H_{d}$. A combination of these is explicitly removed by the mixing term $\\phi^{2}H_{u}^{\\dagger}H_{d}$ (we could equally take $\\phi H_{u}^{\\dagger}H_{d}$, but at the cost of introducing a new mass scale), so that only two Goldstone bosons arise. Explicitly, if we adopt for $H_{u,d}$ a polar representation similar as in Eq.~(\\ref{PhiPolar}), with their pseudoscalar components denoted as $\\eta_{u,d}$ and their VEVs as $v_{u,d}$, the $\\phi^{2}H_{u}^{\\dagger}H_{d}$ coupling translates as a mass term for the combination $\\pi^{0}\\sim2\\eta_{\\phi}\/v_{\\phi}-\\eta_{u}\/v_{u}+\\eta_{d}\/v_{d}$. One of the two remaining Goldstone bosons is eaten by the $Z$ boson. Since $H_{u,d}$ have the same hypercharge, the would-be Goldstone state $G^{0}$ must be $G^{0}\\sim v_{u}\\eta_{u}+v_{d}\\eta_{d}$. The last remaining Goldstone mode, orthogonal to both $\\pi^{0}$ and $G^{0}$, stays massless and is the axion:\n\\begin{equation}\na^{0}\\sim\\eta_{\\phi}+\\frac{v_{EW}}{v_{\\phi}}\\sin2\\beta(\\cos\\beta\\eta_{u}\n-\\sin\\beta\\eta_{d})+\\mathcal{O}(v_{EW}^{2}\/v_{\\phi s}^{2})\\ , \\label{DFSZA0}%\n\\end{equation}\nwith $\\tan\\beta=v_{u}\/v_{d}$ and $v_{EW}^{2}=v_{u}^{2}+v_{d}^{2}\\approx (246\\,$GeV)$^{2}$. The net result of all this is that the axion components in $H_{u,d}$ are suppressed by $v_{u,d}\/v_{\\phi}$. The leading couplings of the axion to SM particles come from the Yukawa couplings, with\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ}}^{eff}=-i\\frac{v_{EW}}{v_{\\phi}}\\sin2\\beta\n\\sum_{f=u,d,e}\\frac{m_{f}}{v_{EW}}\\chi_{P}^{f}\\,a^{0}\\bar{\\psi}_{f}\\gamma\n_{5}\\psi_{f}\\ ,\\ \\ \\chi_{P}^{u}=\\frac{1}{\\tan\\beta}\\ ,\\ \\chi_{P}^{d}=\\chi\n_{P}^{e}=\\tan\\beta\\ . \\label{PseudoCoupl}%\n\\end{equation}\nTo reach this form, the mass terms are identified as $\\sin\\beta v_{EW}\\mathbf{Y}_{u}\\equiv\\sqrt{2}\\mathbf{m}_{u}$ and $\\cos\\beta v_{EW}\\mathbf{Y}_{d,e}\\equiv\\sqrt{2}\\mathbf{m}_{d,e}$ and the fermions are rotated to their mass basis. In the DFSZ setting, the axion couplings to gauge bosons only arise through SM fermion loops. As shown in Ref.~\\cite{Quevillon:2019zrd} (see also Refs.~\\cite{Bonnefoy:2020gyh,Quevillon:2021sfz}), starting from the pseudoscalar couplings in Eq.~(\\ref{PseudoCoupl}), the final couplings to gauge boson do not have the form shown in Eq.~(\\ref{GaugeCouplAno}), but instead explicitly break $SU(2)_{L}\\otimes U(1)_{Y}$ invariance. Naively, this is easily understood since SM fermions only acquire masses after the $SU(2)_{L}\\otimes U(1)_{Y}$ breaking.\n\n\\subsection{Introducing baryon and lepton numbers\\label{Sec2b}}\n\nIn the following, when introducing leptoquark states, baryon and lepton numbers $\\mathcal{B}$ and $\\mathcal{L}$ will play a central role. The purpose in this section is to gather a few important facts about the interplay of these accidental symmetries with the PQ symmetry. Additional information on this topic can be found in Ref.~\\cite{Quevillon:2020hmx}.\n\nBy definition, the $U(1)$ symmetry associated to the axion state is called the PQ symmetry. Given the scalar couplings described in the previous section, the PQ charges of all the scalar states is well-defined in the KSVZ and DFSZ models. Explicitly, we have in the KSVZ setting\n\\begin{equation}\n\\begin{tabular}[c]{ccc}\\hline\nKSVZ & $\\phi$ & $H$\\\\\\hline\n$U(1)_{\\phi}$ & $1$ & $0$\\\\\n$U(1)_{H}$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\ \\ \\ \\ \\ \\Longrightarrow%\n\\begin{tabular}[c]{ccc}\\hline\nKSVZ & $\\phi$ & $H$\\\\\\hline\n$U(1)_{PQ}$ & $1$ & $0$\\\\\n$U(1)_{Y}$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nand in the DFSZ, choosing the two independent $U(1)$ symmetries as those associated to Higgs doublet rephasings\\footnote{The PQ charges of $\\phi$, $H_{u}$ and $H_{d}$ are simply the coefficients of $\\eta_{\\phi,u,d}$ in Eq.~(\\ref{DFSZA0}), up to a choice of normalization.},\n\\begin{equation}\n\\begin{tabular}[c]{cccc}\\hline\nDFSZ & $\\phi$ & $H_{u}$ & $H_{d}$\\\\\\hline\n$U(1)_{Hu}$ & $1\/2$ & $1$ & $0$\\\\\n$U(1)_{Hd}$ & $-1\/2$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\ \\ \\ \\ \\ \\Longrightarrow\n\\begin{tabular}[c]{cccc}\\hline\nDFSZ & $\\phi$ & $H_{u}$ & $H_{d}$\\\\\\hline\n$U(1)_{PQ}$ & $(x+1\/x)\/2$ & $x$ & $-1\/x$\\\\\n$U(1)_{Y}$ & $0$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\ \\ \\label{DFSZScalars}\n\\end{equation}\nwith the conventional notation $\\tan\\beta\\equiv1\/x$. Note that the $U(1)_{Y}$ and $U(1)_{PQ}$ charges of the two Higgs doublets are not `orthogonal', reflecting the fact that the original $U(1)_{Hu}$ and $U(1)_{Hd}$ charges for the three states $(\\phi,H_{u},H_{d})$ were not. Also, it is important to keep in mind that though well-defined, these PQ charges are only defined in the electroweak broken phase, since they are function of $x\\equiv v_{d}\/v_{u}$.\n\nFor fermions, identifying the PQ charge is less trivial because the Yukawa couplings allow for two accidental symmetries, $\\mathcal{B}$ and $\\mathcal{L}$ (no particular structure is assumed for $\\mathbf{Y}_{u,d,e,\\nu}$, so individual flavors are not conserved a priori). Looking at the Lagrangian, the KSVZ model prescribes\n\\begin{equation}\n\\begin{tabular}[c]{ccccccccc}\\hline\nKSVZ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ &\n$e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\alpha$ & $\\alpha-1$ & $\\beta$ & $\\beta$ & $\\beta$ & $\\gamma$ &\n$\\gamma$ & $\\gamma$\\\\\n$U(1)_{Y}$ & $Y$ & $Y$ & $1\/3$ & $4\/3$ & $-2\/3$ & $-1$ & $-2$ & $0$\\\\\\hline\n\\end{tabular}\n\\label{KSVZfermions}\n\\end{equation}\nwhere $\\alpha$, $\\beta$, and $\\gamma$ are arbitrary, and correspond to conserved $\\Psi$ number, baryon number, and lepton number, respectively. Similarly, for the DFSZ model,\n\\begin{equation}\n\\begin{tabular}[c]{ccccccc}\\hline\nDFSZ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\beta$ & $\\beta+x$ & $\\beta-1\/x$ & $\\gamma$ & $\\gamma-1\/x$ &\n$\\gamma+x$\\\\\n$U(1)_{Y}$ & $1\/3$ & $4\/3$ & $-2\/3$ & $-1$ & $-2$ & $0$\\\\\\hline\n\\end{tabular}\n\\label{DFSZfermions}\n\\end{equation}\nSince $\\beta$ and $\\gamma$ are aligned with baryon and lepton numbers, it is tempting to set $\\beta=\\gamma=0$. This is not acceptable. For the DFSZ scenario, all the SM fermions do couple to the axion, but these couplings are not $SU(2)_{L}\\otimes U(1)_{Y}$ invariant. Looking at Eq.~(\\ref{PseudoCoupl}), no value of $\\beta$ or $\\gamma$ make perfect sense since the PQ charge of the Dirac $u$ and $d$ states are different, so that of $q_{L}$ cannot be defined. The situation appears simpler in the KSVZ case, where it seems rather natural to set $\\beta=\\gamma=0$ since the SM fermions are not directly coupled to the scalar field $\\phi$. Yet, even that is not tenable.\n\nTo see this, let us set off a seesaw mechanism~\\cite{TypeI}. Given the quantum numbers of the $\\nu_{R}$ field, we can either allow for a Majorana mass term $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, a coupling $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, or a coupling $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$. These three cases are mutually exclusive since they impose different PQ charges to $\\nu_{R}$. Let us consider the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ case, which in effect identifies the PQ symmetry with lepton number symmetry, and the axion with the Majoron~\\cite{Langacker:1986rj,Shin:1987xc,Clarke:2015bea} (see also \\cite{Heeck:2019guh}). It imposes non-zero values for $\\gamma$~\\cite{Quevillon:2020hmx}\n\\begin{align}\n\\text{KSVZ} & :\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}\\rightarrow\\gamma=\\frac{1}{2}\\ ,\\\\\n\\text{DFSZ} & :\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}\\rightarrow\\gamma=\\frac{1-3x^{2}}{4x}\\ .\n\\end{align}\nIn both cases, the PQ current acquires a component aligned with the lepton number current, $J_{\\mathcal{L}}^{\\mu}=\\bar{\\ell}_{L}\\gamma^{\\mu}\\ell_{L}+\\bar{e}_{R}\\gamma^{\\mu}e_{R}+\\bar{\\nu}_{R}\\gamma^{\\mu}\\nu_{R}$. In other words, $\\ell_{L}$ and\/or $e_{R}$ do end up PQ charged also. Yet, in the KSVZ case, a look at the Lagrangian shows that neither are directly coupled to $\\phi$. Because of $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, the axion does end up coupled to right-handed neutrinos, with a $a^{0}\\rightarrow\\nu_{R}\\nu_{R}$ vertex, but no such $\\Delta\\mathcal{L}=2$ coupling exists with the other leptons since it is forbidden by hypercharge. Only at the cost of extra Higgs doublet insertions could a $a^{0}\\rightarrow\\nu_{L}\\nu_{L}$ exist, as arising from an effective PQ- and hypercharge-neutral operator $\\phi^{\\dagger}H\\ell_{L}H\\ell_{L}$ (or $\\phi^{\\dagger}H_{u}\\ell_{L}H_{u}\\ell_{L}$ in the DFSZ model), while obviously, any $\\Delta\\mathcal{L}=2$ coupling to charged lepton would require either extra gauge fields, or charged Higgs bosons.\n\nThe ambiguous nature of the PQ charges of fermions is not purely academic. In most phenomenological studies of the axion, the starting point is the effective Lagrangian that is obtained by reparametrizing fermion fields to make them PQ neutral (even if that is usually not explicitly stated):\n\\begin{equation}\n\\psi\\rightarrow\\exp(-iPQ(\\psi)a^{0}\/v_{\\phi})\\psi\\ , \\label{ReparamG}%\n\\end{equation}\nwhere $\\psi$ denotes generically the PQ-charged fermions. Since the underlying physics is PQ neutral, this looks innocuous. Yet, it modifies the Lagrangian of the model in two important ways. First, it removes the axion field from Yukawa interactions (both for the SM and heavy fermions, if present), and replace them by shift-symmetric derivative couplings of the axion to the\nfermionic PQ current, as adequate for a Goldstone boson\n\\begin{equation}\n\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}=\\frac{\\partial_{\\mu}a^{0}}{v_{\\phi}%\n}J_{PQ}^{\\mu}\\ ,\\ J_{PQ}^{\\mu}=\\sum_{\\psi}PQ(\\psi)\\bar{\\psi}\\gamma^{\\mu}\\psi\\ .\n\\end{equation}\nSecond, the PQ symmetry being anomalous, the fermion reparametrizations in Eq.~(\\ref{ReparamG}) change the fermionic measure. To account for this, one must introduce anomalous couplings to the gauge bosons,\n\\begin{equation}\n\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}=\\frac{a^{0}}{16\\pi^{2}v_{\\phi}}\\left(\ng_{s}^{2}\\mathcal{N}_{C}G_{\\mu\\nu}^{a}\\tilde{G}^{a,\\mu\\nu}+g^{2}%\n\\mathcal{N}_{L}W_{\\mu\\nu}^{i}\\tilde{W}^{i,\\mu\\nu}+g^{\\prime2}\\mathcal{N}%\n_{Y}B_{\\mu\\nu}\\tilde{B}^{\\mu\\nu}\\frac{{}}{{}}\\right) \\;,\n\\end{equation}\nwhere the coefficients $\\mathcal{N}_{C,L,Y}$ are functions of the PQ charges of all the fermions, and generically given by\n\\begin{equation}\n\\mathcal{N}_{X}=\\sum_{\\psi}PQ(\\psi)C_{X}(\\psi)\\ ,\n\\end{equation}\nwith $C_{C,L,Y}(\\psi)$ the quadratic invariant of the field $\\psi$ under $SU(3)_{C}$, $SU(2)_{L}$ or $U(1)_{Y}$. The effective Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\text{\\textrm{Eff}}}=\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}%\n}+\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}\\ , \\label{AxionEL}%\n\\end{equation}\nis in general the basis in which the axion phenomenology is studied, with the common further assumption that $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ is model-dependent and subleading compared to the model independent $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$. Yet, since the PQ charge of the fermions are ambiguous, both $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ and $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$ are also ambiguous. This is most striking in the DFSZ case, where $\\mathcal{N}_{L}\\sim3\\beta+\\gamma$. This conundrum was analyzed in Ref.~~\\cite{Quevillon:2019zrd}, where in particular it was shown that $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ and $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$ do in fact contribute at the same order to physical observables, and that this ensures all the ambiguities in $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ and $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$ cancel each other systematically. This means that the couplings to (chiral) gauge bosons cannot be read off $\\delta\\mathcal{L}_{\\text{\\textrm{Jac}}}$, and that $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ cannot be neglected.\n\nFor our purpose, it is important to emphasize how this translates for the baryon and lepton numbers. Thus, consider the KSVZ scenario with the fermion charges in Eq.~(\\ref{KSVZfermions}), keeping $\\alpha$, $\\beta$, and $\\gamma$ arbitrary, and let us perform the reparametrization of Eq.~(\\ref{ReparamG}) for all the fermions. The PQ current is then identified as\n\\begin{equation}\nJ_{PQ}^{\\mu}=\\bar{\\Psi}_{R}\\gamma^{\\mu}\\Psi_{R}+\\alpha J_{\\Psi}^{\\mu}+3\\beta\nJ_{\\mathcal{B}}^{\\mu}+\\gamma J_{\\mathcal{L}}^{\\mu}\\ ,\n\\end{equation}\nwhere%\n\\begin{align}\nJ_{\\Psi}^{\\mu} & =\\bar{\\Psi}_{L}\\gamma^{\\mu}\\Psi_{L}+\\bar{\\Psi}_{R}%\n\\gamma^{\\mu}\\Psi_{R}=\\bar{\\Psi}\\gamma^{\\mu}\\Psi\\ ,\\\\\nJ_{\\mathcal{B}}^{\\mu} & =\\frac{1}{3}\\bar{q}_{L}\\gamma^{\\mu}q_{L}+\\frac{1}%\n{3}\\bar{u}_{R}\\gamma^{\\mu}u_{R}+\\frac{1}{3}\\bar{d}_{R}\\gamma^{\\mu}d_{R}%\n=\\frac{1}{3}\\bar{u}\\gamma^{\\mu}u+\\frac{1}{3}\\bar{d}\\gamma^{\\mu}d\\ ,\\ \\\\\nJ_{\\mathcal{L}}^{\\mu} & =\\bar{\\ell}_{L}\\gamma^{\\mu}\\ell_{L}+\\bar{e}%\n_{R}\\gamma^{\\mu}e_{R}+\\bar{\\nu}_{R}\\gamma^{\\mu}\\nu_{R}=\\bar{e}\\gamma^{\\mu\n}e+\\bar{\\nu}\\gamma^{\\mu}\\nu\\ .\n\\end{align}\nAt first sight, one may think to discard the vector currents $J_{\\Psi}^{\\mu}$, $J_{\\mathcal{B}}^{\\mu}$, and $J_{\\mathcal{L}}^{\\mu}$ from the derivative interactions since upon integration by part, $\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi=-a^{0}\\partial_{\\mu}\\bar{\\psi}\\gamma^{\\mu}\\psi=-a^{0}\\bar{\\psi}(m-m)\\psi=0$. This is incorrect though. The vector Ward identity does not survive to the presence of chiral gauge interactions. While $J_{\\Psi}^{\\mu}$ can indeed safely be discarded since $\\Psi$ is vector-like, the baryon and lepton currents are anomalous in the presence of chiral gauge fields:\n\\begin{equation}\n\\partial_{\\mu}J_{\\mathcal{B}}^{\\mu}=\\partial_{\\mu}J_{\\mathcal{L}}^{\\mu}%\n=-\\frac{N_{f}}{16\\pi^{2}}\\left( \\frac{1}{2}g^{2}W_{\\mu\\nu}^{i}\\tilde\n{W}^{i,\\mu\\nu}-\\frac{1}{2}g^{\\prime2}B_{\\mu\\nu}\\tilde{B}^{\\mu\\nu}\\right) \\ .\n\\end{equation}\nObviously, these contributions trivially cancel the $\\beta$ and $\\gamma$-dependent Jacobian terms generated by the fermion reparametrization, which have precisely the same form and origin. Thus, in the KSVZ setting, it seems\nthat the sole role of the SM fermions derivative interactions aligned with the $\\mathcal{B}$ and $\\mathcal{L}$ current is to kill the correspondingly spurious anomalous gauge interactions.\n\nThere is a problem in this reasoning though. This cancellation occurs whether a $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling is assumed initially present or not, since the value of $\\gamma$ is irrelevant. This is puzzling since in the presence of $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, the axion should retain some couplings to $\\nu_{R}$. In the above argument, the step at which we lost the $a\\nu_{R}\\nu_{R}$ coupling is in the Ward identity. After the spontaneous symmetry breaking (SSB), $\\mathcal{L}$, as part of the PQ symmetry, is no longer conserved and the equation of motion (EoM) of $\\nu_{R}$ breaks explicitly the anomalous vector Ward identity. In practice, $(\\partial_{\\mu}a^{0}\/v)\\bar{\\nu}_{R}\\gamma^{\\mu}\\nu_{R}$ does generate the $(M_{R}\/v)a^{0}\\nu_{R}\\nu_{R}$ coupling. This means that whether the axion is coupled to $\\nu_{R}$ or not is not apparent at the level of the effective axion Lagrangian, but hides in the EoM of $\\nu_{R}$.\\ Further, these EoM spoil the $1\/v_{\\phi}$ scaling of the effective Lagrangian operators, since they contain terms of $\\mathcal{O}(v_{\\phi})$. Phenomenologically, this failure of the effective interactions to manifestly exhibit all the possible axion interactions is clearly an important point to keep in mind.\n\nTo conclude, let us stress again:\n\n\\begin{itemize}\n\\item The PQ symmetry has some room for $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating effects. In the presence of such violation, the PQ symmetry eats part of the $\\mathcal{B}$ and $\\mathcal{L}$ accidental $U(1)$s, and the PQ current inherits some $J_{\\mathcal{B}}^{\\mu}$ and\/or $J_{\\mathcal{L}}^{\\mu}$ components.\n\n\\item Incorporating a $\\mathcal{B}$ and\/or $\\mathcal{L}$ component in the PQ current does not modify the leading order axion to gauge boson couplings.\n\n\\item The $\\mathcal{B}$ and\/or $\\mathcal{L}$ components of PQ current do not tell us much about the couplings of the axion to SM fermions. Most of the $\\partial_{\\mu}a^{0}J_{\\mathcal{B}}^{\\mu}$ and $\\partial_{\\mu}a^{0}J_{\\mathcal{L}}^{\\mu}$ couplings are just there to cancel spurious local anomalous terms.\n\n\\item Any $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings must break explicitly the (already anomalous) $\\mathcal{B}$ and\/or $\\mathcal{L}$ vector Ward identities. In their presence, the EoM of the SM fermions will ensure the derivative interactions $\\partial_{\\mu}a^{0}J_{\\mathcal{B}}^{\\mu}$ and $\\partial_{\\mu}a^{0}J_{\\mathcal{L}}^{\\mu}$ do include the expected $\\Delta\\mathcal{B}$ and\/or $\\Delta\\mathcal{L}$ couplings of the axion.\n\\end{itemize}\n\nAs we will see in the following, introducing leptoquark states often forces us to entangle $\\mathcal{B}$ and\/or $\\mathcal{L}$ with the PQ symmetry. These points are thus crucial to understand the phenomenological consequences.\n\n\\subsection{Introducing leptoquarks and diquarks\\label{Sec2c}}\n\nLeptoquarks (LQ) are scalars or vectors that couple simultaneously to a quark-lepton pair, while diquarks (DQ) couple to quark pairs (for a review, see e.g. Ref.~\\cite{Dorsner:2016wpm}). Given the quantum numbers of the SM fermions, only a finite number of LQ and DQ can couple to normal matter, and only a few of them can have both LQ and DQ couplings. Though the full list of possible LQ and DQ states is well-known, let us nevertheless go through this construction as it will play an important role in the following, and permits to conveniently introduce our notations.\n\nAll the LQ are color triplets, while DQ are triplets (using $1\\supset 3\\otimes3\\otimes3$) or sexplets (using $1\\supset3\\otimes3\\otimes\\bar{6}$). From the point of view of $SU(2)_{L}$, these states can be either triplet, doublets, or singlets, depending on the involved SM fermions. Once $SU(2)_{L}\\otimes SU(3)_{C}$ contractions are set, the hypercharge is then fixed to accommodate specific couplings to SM fermions. In this regard, one should remember that scalars couple to $\\bar{\\psi}_{L}\\psi_{R}$ or $\\bar{\\psi}_{R}\\psi_{L}$, vectors to $\\bar{\\psi}_{R}\\gamma_\\mu\\psi_{R}$ or $\\bar{\\psi}_{L}\\gamma_\\mu\\psi_{L}$, and that charge conjugation $\\mathrm{C}$ flips the chirality. This means that a scalar can couple to $\\bar{\\psi}_{R}^{\\mathrm{C}}\\psi_{R}$ for example. By constructing all possible pairs of SM leptons, including conjugate fields, the standard list of possible states are recovered, with the scalars LQ states\n\\begin{align}\n(\\mathbf{3},\\mathbf{2},+1\/3) & :S_{2}^{1\/3}\\times(\\bar{d}_{R}\\ell_{L}\\ ,\\ \\bar{q}_{L}\\nu_{R})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{2},+7\/3) & :S_{2}^{7\/3}\\times(\\bar{u}_{R}\\ell_{L}\\ ,\\ \\bar{q}_{L}e_{R})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-2\/3) & :S_{1}^{2\/3}\\times(\\bar{d}_{R}\\nu_{R}^{\\mathrm{C}}\\ \\ ,\\ \\bar{u}_{R}e_{R}^{\\mathrm{C}}\\ ,\\ \\bar{q}_{L}\\ell_{L}^{\\mathrm{C}})\\ ,\\ \\ (\\mathbf{3},\\mathbf{3},-2\/3):S_{3}^{2\/3}\\times\\bar\n{q}_{L}\\ell_{L}^{\\mathrm{C}}\\ ,\\label{LQ1}\\\\\n(\\mathbf{3},\\mathbf{1},+4\/3) & :S_{1}^{4\/3}\\times\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-8\/3) & :S_{1}^{8\/3}\\times\\bar{d}_{R}e_{R}^{\\mathrm{C}}\\ ,\\nonumber\n\\end{align}\nand the vector LQ states$\\ $%\n\\begin{align}\n(\\mathbf{3},\\mathbf{2},+1\/3) & :V_{2,\\mu}^{1\/3}\\times(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}\\ ,\\ \\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{2},-5\/3) & :V_{2,\\mu}^{5\/3}\\times(\\bar{d}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}\\ ,\\ \\bar{q}_{L}\\gamma^{\\mu}e_{R}^{\\mathrm{C}})\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},+4\/3) & :V_{1,\\mu}^{4\/3}\\times(\\bar{u}_{R}\\gamma^{\\mu}\\nu_{R}\\ ,\\ \\bar{d}_{R}\\gamma^{\\mu}e_{R}\\ ,\\ \\bar{q}_{L}\\gamma^{\\mu}\\ell_{L})\\ ,\\ \\ (\\mathbf{3},\\mathbf{3},+4\/3):V_{3,\\mu}^{4\/3}\\times\\bar{q}%\n_{L}\\gamma^{\\mu}\\ell_{L}\\ ,\\label{LQ2}\\\\\n(\\mathbf{3},\\mathbf{1},10\/3) & :V_{1,\\mu}^{10\/3}\\times\\bar{u}_{R}\\gamma^{\\mu}e_{R}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-2\/3) & :V_{1,\\mu}^{2\/3}\\times\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\ .\\nonumber\n\\end{align}\nMany notations exist for these states, in particular $S_{i}$, $\\tilde{S}_{i}$, $\\bar{S}_{i}$ when several states occur with the same $SU(3)_{C}\\otimes SU(2)_{L}$ quantum numbers~\\cite{Dorsner:2016wpm}. Here, we denote all states as color triplets $S_{t}^{y}$ or $V_{t}^{y}$, with $t$ the $SU(2)_{L}$ dimensionality and $y$ the absolute value of the $U(1)_{Y}$ hypercharge. Note also that $V_{1,\\mu}^{2\/3}$ and $S_{1}^{4\/3}$ exist only in the presence of $\\nu_{R}$, and are thus often discarded. Concerning diquarks, there are only six possible combinations of quark fields, leading to%\n\\begin{align}\n(\\mathbf{3},\\mathbf{2},+1\/3) & :V_{2,\\mu}^{1\/3}\\times\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{2},-5\/3) & :V_{2,\\mu}^{5\/3}\\times\\bar{u}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-2\/3) & :S_{1}^{2\/3}\\times(\\bar{q}_{L}^{\\mathrm{C}}q_{L}\\ ,\\ \\bar{d}_{R}^{\\mathrm{C}}u_{R})\\ ,\\ (\\mathbf{3},\\mathbf{3},-2\/3):S_{3}^{2\/3}\\times\\bar{q}_{L}^{\\mathrm{C}}q_{L}\\ ,\\label{LQ3}\\\\\n(\\mathbf{3},\\mathbf{1},+4\/3) & :S_{1}^{4\/3}\\times\\bar{d}_{R}^{\\mathrm{C}}d_{R}\\ ,\\nonumber\\\\\n(\\mathbf{3},\\mathbf{1},-8\/3) & :S_{1}^{8\/3}\\times\\bar{u}_{R}^{\\mathrm{C}}u_{R}\\ .\\nonumber\n\\end{align}\nAll these states are already present in the LQ list. Note that each of the above quark state can also couple to a DQ transforming like $\\mathbf{\\bar{6}}$ under $SU(3)_{C}$, with the same $SU(2)_{L}\\otimes U(1)_{Y}$ quantum numbers. In that case, they do not have LQ couplings. We will adopt the same notation for these states, relying on the context to make clear whether they transform as $\\mathbf{3}$ or $\\mathbf{\\bar{6}}$.\n\n\n\\begin{table}[t] \\centering\n\\begin{tabular}[c]{l}\n\\begin{tabular}[c]{lllllll}\\hline\n$\\Delta\\mathcal{B}$ & $\\Delta\\mathcal{L}$ & Dim. &\n\\multicolumn{2}{l}{Operators (no $\\nu_{R}$)} & & \\\\\\hline\n$+0$ & $+2$ & $5$ & $H^{\\dagger2}\\ell_{L}^{2}$ & & & \\\\\n$+1$ & $+1$ & $6$ & $q_{L}^{3}\\ell_{L}$ & $u_{R}^{2}d_{R}e_{R}$ & $q_{L}%\nu_{R}d_{R}\\ell_{L}$ & $q_{L}^{2}u_{R}e_{R}$\\\\\n$+1$ & $-1$ & $7$ & $H^{\\dagger}d_{R}^{3}\\ell_{L}^{\\mathrm{C}}$ & $Hd_{R}%\n^{2}q_{L}e_{R}^{\\mathrm{C}}$ & $Hd_{R}^{2}u_{R}\\ell_{L}^{\\mathrm{C}}$ &\n$Hq_{L}^{2}d_{R}\\ell_{L}^{\\mathrm{C}}$\\\\\n$+2$ & $+0$ & $9$ & $d_{R}^{4}u_{R}$ & $d_{R}^{3}u_{R}q_{L}^{2}$ & $d_{R}%\n^{2}q_{L}^{4}$ & \\\\\n$+1$ & $+3$ & $9$ & $u_{R}^{2}q_{L}\\ell_{L}^{3}$ & $u_{R}^{3}\\ell_{L}^{2}%\ne_{R}$ & & \\\\\n$+1$ & $-3$ & $10$ & $Hd_{R}^{3}\\ell_{L}^{\\mathrm{C},3}$ & & &\n\\end{tabular}\n\\\\\n\\begin{tabular}[c]{llllllll}\\hline\n$\\Delta\\mathcal{B}$ & $\\Delta\\mathcal{L}$ & Dim. &\n\\multicolumn{2}{l}{Operators (one $\\nu_{R}$)} & & & \\\\\\hline\n$+0$ & $+2$ & $5$ & $H^{\\dagger2}e_{R}\\nu_{R}$ & & & & \\\\\n$+1$ & $+1$ & $6$ & $q_{L}^{2}d_{R}\\nu_{R}$ & $d_{R}^{2}u_{R}\\nu_{R}$ & & &\n\\\\\n$+1$ & $-1$ & $7$ & $H^{\\dagger}d_{R}^{2}q_{L}\\nu_{R}^{\\mathrm{C}}$ &\n$Hd_{R}q_{L}u_{R}\\nu_{R}^{\\mathrm{C}}$ & $Hq_{L}^{3}\\nu_{R}^{\\mathrm{C}}$ & &\n\\\\\n$+2$ & $+0$ & $9$ & -- & & & & \\\\\n$+1$ & $+3$ & $9$ & $d_{R}u_{R}^{2}\\ell_{L}^{2}\\nu_{R}$ & $d_{R}q_{L}u_{R}%\n\\ell_{L}^{2}\\nu_{R}$ & $u_{R}^{3}e_{R}^{2}\\nu_{R}$ & $u_{R}^{2}q_{L}\\ell\n_{L}e_{R}\\nu_{R}$ & $q_{L}^{2}u_{R}\\ell_{L}^{2}\\nu_{R}$\\\\\n$+1$ & $-3$ & $10$ & $Hd_{R}^{3}\\ell_{L}^{\\mathrm{C}}e_{R}^{\\mathrm{C}}\\nu\n_{R}^{\\mathrm{C}}$ & $Hd_{R}^{2}q_{L}\\ell_{L}^{\\mathrm{C},2}\\nu_{R}%\n^{\\mathrm{C}}$ & & & \\\\\\hline\n\\end{tabular}\n$\\ $\n\\end{tabular}\n$\\ $\n\\caption{Leading effective operators with non-trivial $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ charges in the SM, involving no or one $\\nu_R$ field.\nWe do not include redundant patterns, e.g. all the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=n\\times(0,2),n\\times\n(1,1),...$ with $n=2,3,...$ operators, or operators of higher dimensions within each $(\\Delta\\mathcal{B},\\Delta\\mathcal\n{L})$ class. With even more fields, the next unique patterns involve eight fermions, and induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,5)$ transitions at dimension 12, and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-5)$ transitions at dimension 13 (with an extra Higgs field). All these processes involve at least one $\\nu\n_{R}$ field at these orders. Still higher in dimensionality, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(3,1)$ and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,7)$ come at the ten-fermion level, via dimension-15 operators. Only the SM Higgs doublet $H$ is used in the Table together with SM fermions, but the extension to the THDM is trivial.}%\n\\label{TableLQBL}\n\\end{table}\n\nIntroducing scalar or vector states that couple to quarks and leptons can impact the accidental $\\mathcal{B}$ and $\\mathcal{L}$ symmetries (for a recent review, see e.g. Ref.~\\cite{Assad:2017iib}). Depending on which states are introduced and, if several of them are present, depending also on how they are coupled, the symmetry pattern can be quite different. Actually, these symmetry patterns are reminiscent of those of the possible effective operators involving SM fields but carrying non-trivial $\\mathcal{B}$ and\/or $\\mathcal{L}$ charges~\\cite{Weinberg,WeinbergPRD22,Weldon:1980gi}. Those are listed in Table~\\ref{TableLQBL}. This connection is easily understood from tree diagrams with the external fermions linked together by virtual LQ\/DQ exchanges\\footnote{The notation LQ\/DQ generically refers to any of the pure LQ, pure DQ, or mixed LQ\/DQ state introduced in Eqs.~(\\ref{LQ1}),~(\\ref{LQ2}), and~(\\ref{LQ3}).}. Obviously, these external fermion states must be $SU(3)_{C}\\otimes SU(2)_{L}\\otimes U(1)_{Y}$ invariant since the LQ\/DQ are. Further, operators with six or less fermions are the most relevant when only renormalizable interactions among the LQ\/DQ are present. Being colored, these states can at most have quadratic or cubic interactions, hence induce four or six fermion interactions. More complicated fermion interactions can arise, but they would require multiple cubic interactions, and would not open additional phenomenologically interesting channels. Indeed, the above set contains already the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ operators for neutrino masses, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators for neutron-antineutron oscillations, and all the others for proton decay. Note, finally, that one can understand why some states have both LQ and DQ couplings while others do not from the fact that dimension-six operators are necessarily $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$, see Table~\\ref{TableLQBL}. As tree-level exchanges of states with both LQ and DQ couplings (Fig.~\\ref{Fig1}$a$) must match onto these operators, only $V_{2}^{y}$ and $S_{1}^{y}$ can occur since they couple to a quark-lepton (or antiquark-antilepton) pair\\footnote{This condition is sometimes quantified using $\\mathcal{F}=3\\mathcal{B}+\\mathcal{L}$ as a quantum numbers~\\cite{Dorsner:2016wpm}, so that those states with both LQ and DQ couplings have $\\mathcal{F}=\\pm2$, and the others $\\mathcal{F}=0$. We prefer here to use $\\mathcal{B}\\pm\\mathcal{L}$.}.\n\nWith the above picture in mind, let us see in more details how the various\n$(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ patterns of Table~\\ref{TableLQBL} can arise:\n\n\\begin{enumerate}\n\\item[A.] Exact $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}:$ Whenever a given $S$ or $V$ state with only LQ or DQ coupling is present, $\\mathcal{B}$ and $\\mathcal{L}$ can still be unambiguously defined. The LQ or DQ state simply carries some specific $\\mathcal{B}$ and $\\mathcal{L}$ quantum numbers, but overall, $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ is still exact. This remains true even in the presence of several different states, so long as they do not couple together.\n\n\\item[B.] Exact $U(1)_{\\mathcal{B}-\\mathcal{L}}:$ When a state with both LQ and DQ couplings is present, the symmetry gets reduced to $U(1)_{\\mathcal{B}-\\mathcal{L}}$, with the $\\mathcal{B}-\\mathcal{L}$ quantum numbers $-2\/3$ for $S_{1}^{y}$ and $V_{2}^{y}$, $+1\/3$ and $-1$ for quarks and leptons, respectively. This remains true if more than one DQ\/LQ state is present provided any couplings among them is compatible with these charge assignments, which further requires the $\\mathcal{B}-\\mathcal{L}$ quantum numbers of $S_{2}^{y}$ and $V_{1}^{y}$ to be $+4\/3$. For example, a scenario with $S_{1}^{2\/3}$ and $S_{1}^{4\/3}$ but without an $S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}$ interaction, or with $S_{2}^{7\/3}$, $S_{1}^{2\/3}$ and a coupling $H^{\\dagger}S_{2}^{7\/3}S_{1}^{2\/3}S_{1}^{2\/3}$, or with $S_{2}^{1\/3}$, $S_{1}^{2\/3}$ and a coupling $HS_{2}^{1\/3}S_{1}^{2\/3}S_{1}^{2\/3}$ all preserve $U(1)_{\\mathcal{B}-\\mathcal{L}}$ (note that the antisymmetric color contraction requires at least two different $S_{1}^{2\/3}$). For all these scenarios, the $S$ and\/or $V$ mass has to be pushed at the GUT scale since $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators induce proton decay (Fig.~\\ref{Fig1}$a$).\n\n\\item[C.] Exact $U(1)_{3\\mathcal{B}+\\mathcal{L}}:$ A peculiar situation arises for the $S_{2}^{1\/3}$ state, because the $H^{\\dagger}S_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ is allowed by all the SM gauge symmetries. In its presence, the active symmetry gets reduced to $U(1)_{3\\mathcal{B}+\\mathcal{L}}$ since $S_{2}^{1\/3}$ is neutral for that specific combination. This situation is again problematic because $H^{\\dagger}S_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ collapses to $Hd_{R}^{\\dagger3}\\ell_{L}^{3}$ and can induce a $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ proton decay. No other combination of scalar leptoquarks contributes to this $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ channel.\n\n\\item[D.] No exact $U(1):$ In the presence of two states having different $\\mathcal{B}-\\mathcal{L}$ quantum numbers, there is no remaining symmetry whenever those states have all their gauge-allowed couplings to SM fermions turned on, and when they are coupled together. For example, introducing both $S_{2}^{1\/3}$ and $S_{1}^{2\/3}$ with a $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling, $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ are entirely broken. As seen earlier, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ proton decay is induced by $S_{1}^{2\/3}$, pushing its mass to the GUT range. But the total absence of accidental $U(1)$s means the other classes of $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ operators are also generated. The simplest is the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ operator, generating neutrino masses via the diagram of Fig.~\\ref{Fig1}$b$.\n\n\\item[E.] Exact $U(1)_{\\mathcal{B}}:$ Adding to the scenarios A a seesaw mechanism for neutrino masses, i.e., a $\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ term, then $U(1)_{\\mathcal{L}}$ is explicitly broken but $U(1)_{\\mathcal{B}}$ remains exact, preventing proton decay. The same pattern can be obtained using mixing terms among some carefully chosen LQ\/DQ states, such that an effective neutrino mass term is generated but proton decay cannot occur. For example, introducing $S_{2}^{1\/3}$, $S_{1}^{2\/3}$, the mixing term $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ but turning off the DQ couplings of $S_{1}^{2\/3}$ (or alternatively, with the mixing term $\\mu S_{2}^{1\/3}S_{2}^{1\/3}S_{1}^{2\/3}$ but turning off the LQ couplings of $S_{1}^{2\/3}$), the dimension-five $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ operator arises, see Fig.~\\ref{Fig1}$b$. In these scenarios, $S_{2}^{1\/3}$, $S_{1}^{2\/3}$ acquire well defined $\\mathcal{B}$ numbers, $U(1)_{\\mathcal{B}}$ is conserved, and proton decay is forbidden.\n\n\\item[F.] Exact $U(1)_{\\mathcal{B}+\\mathcal{L}}:$ Another possible symmetry pattern corresponds to taking again $S_{2}^{1\/3}$, $S_{1}^{2\/3}$, and the $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling but turning off the LQ couplings of $S_{1}^{2\/3}$ (or with $\\mu S_{2}^{1\/3}S_{2}^{1\/3}S_{1}^{2\/3}$ but turning off the DQ couplings of $S_{1}^{2\/3}$). In this case, no neutrino masses can be generated, but proton decay is back. Yet, the proton decay channels do not match those induced by the dimension-six Weinberg operators. With the $\\mu HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling, the simplest processes lead to the dimension-seven $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ effective operators, see Fig.~\\ref{Fig1}$c$, while the $\\mu S_{2}^{1\/3}S_{2}^{1\/3}S_{1}^{2\/3}$ coupling generates $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ transitions but with an extra lepton-antilepton pair.\n\n\\item[G.] Exact $U(1)_{\\mathcal{L}}:$ Another pattern is obtained by introducing several states but now allowing only for DQ couplings, and turning on some mixing terms (this kind of construction was considered recently e.g. in Refs.~\\cite{Arnold:2012sd,FileviezPerez:2015mlm}). These latter mixings are necessary since otherwise, $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ remains exact. The simplest scenarios are those with $S_{1}^{2\/3}$, $S_{1}^{4\/3}$, and the cubic coupling $\\mu S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}$, or $S_{1}^{4\/3}$, $S_{1}^{8\/3}$, and the cubic coupling $\\mu S_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}$. In both cases, only the DQ couplings are allowed, and $S_{1}^{4\/3}$ ($S_{1}^{8\/3}$) must transform as $\\mathbf{\\bar{6}}$ in the first (second) case, respectively. As a result, neither neutrino masses nor proton decay are induced, but the dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators do arise, and contribute to neutron-antineutron oscillations, see Fig.~\\ref{Fig1}$d$.\n\n\\item[H.] Exact $U(1)_{3\\mathcal{B}-\\mathcal{L}}:$ As for the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ case, dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ operators are attainable by taking $S_{1}^{2\/3}$, $S_{1}^{4\/3}$, and the cubic coupling $\\mu S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}$, or $S_{1}^{4\/3}$, $S_{1}^{8\/3}$, and the cubic coupling $\\mu S_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}$, but turning on only the LQ couplings (since all LQ transform as $\\mathbf{3}$, the color contraction requires three different LQ to be present). Yet, only interactions involving $\\nu_{R}$ can occur because of the LQ coupling of $S_{1}^{4\/3}$ to $\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}$, so proton decay is suppressed. The dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ operators not involving $\\nu_{R}$ require a combination of scalar and vector LQ, for example $S_{1}^{2\/3}V_{2}^{1\/3}V_{2}^{1\/3}$ can induce both $\\bar{q}_{L}\\ell_{L}^{\\mathrm{C}}\\bar{u}_{R}\\gamma_{\\mu}\\ell_{L}^{\\mathrm{C}}\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}$ and $\\bar{u}_{R}e_{R}^{\\mathrm{C}}\\bar{u}_{R}\\gamma_{\\mu}\\ell_{L}^{\\mathrm{C}}\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}$.\n\\end{enumerate}\n\n\\begin{figure}[ptb]\n\\begin{center}\n\\includegraphics[height=2.3134in,width=5.3195in]{Fig1.jpg}\n\\caption{LQ\/DQ processes inducing proton decay via $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators ($a.$), a neutrino Majorana mass term ($b.$), proton decay via $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators ($c.$), and neutron-antineutron oscillations via a $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operator ($d.$).}%\n\\label{Fig1}\n\\end{center}\n\\end{figure}\n\nThis concludes our list of symmetry patterns. It is quite remarkable that a relatively simple scenario exists for all the possible patterns of Table~\\ref{TableLQBL}, with in each case the `orthogonal' $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ pattern remaining as an exact accidental $U(1)$ symmetry. What this list does not show is that actually, not so many other scenarios do exist to generate most of these symmetry-breaking patterns. Indeed, in most cases, allowing for several LQ\/DQ states, both scalar and vector, and some couplings among them, one simply ends up with no accidental symmetries. The interesting situations in which some accidental symmetries do remain are quite constrained, and those can be classified once and for all.\n\nFirst, notice that at the renormalizable level, there are only two classes of couplings among the LQ\/DQ: those with bilinear color contractions, typically $\\mathbf{3}\\otimes\\mathbf{\\bar{3}}$ or $\\mathbf{6}\\otimes\\mathbf{\\bar{6}}$, and those with cubic contractions, typically $\\mathbf{3}\\otimes\\mathbf{3}\\otimes\\mathbf{3}$ or $\\mathbf{3}\\otimes\\mathbf{3}\\otimes\\mathbf{\\bar{6}}$. For the former, barring partial derivatives acting on the LQ\/DQ fields, the only non-trivial LQ\/DQ bilinear couplings compatible with the SM gauge symmetries are%\n\\begin{equation}\nHS_{2}^{1\/3\\dagger}S_{1}^{2\/3}\\ ,\\ HS_{1}^{4\/3\\dagger}S_{2}^{1\/3}%\n\\ ,\\ HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}\\ ,\\ HV_{2,\\mu}^{1\/3\\dagger}V_{1}^{2\/3,\\mu\n}\\ ,\\ HV_{1,\\mu}^{4\/3\\dagger}V_{2}^{1\/3,\\mu}\\ ,\\ HV_{1,\\mu}^{2\/3\\dagger}%\nV_{2}^{5\/3,\\mu}\\ . \\label{LQOpsBmL}\n\\end{equation}\nThe $HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ coupling was used to illustrate the symmetry patterns, but all the others are completely similar: $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ are entirely broken when all the LQ\/DQ couplings are present (case D), $U(1)_{\\mathcal{B}}$ stays exact with only LQ couplings (case E), or $U(1)_{\\mathcal{B}+\\mathcal{L}}$ remains if $S_{1}^{y}$ or $V_{2}^{y}$ have only DQ couplings (case F). This last situation is probably the most interesting phenomenologically since each coupling in Eq.~(\\ref{LQOpsBmL}) produces a specific subset of the dimension-seven $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators in Table~\\ref{TableLQBL}.\n\nFor cubic interactions, though there are a total of $37$ such couplings, most of them involve LQ\/DQ of different $\\mathcal{B}-\\mathcal{L}$ charges and conserve either $U(1)_{\\mathcal{B}-\\mathcal{L}}$ (case B) or $U(1)_{\\mathcal{B}+\\mathcal{L}}$ (case F). Yet, compared to the dimension 6 and 7 operators in Table~\\ref{TableLQBL}, they necessarily produce an extra lepton-antilepton pair. The symmetry patterns typical of six-fermion states, i.e., leading to the dimension 9 or 10 operators in Table~\\ref{TableLQBL}, are obtained with three LQ\/DQ with the same $\\mathcal{B}-\\mathcal{L}$ charge, and this leaves only eight possibilities:\n\\begin{align}\n& S_{1}^{2\/3}V_{2,\\mu}^{1\/3}V_{2}^{1\/3,\\mu}\\ ,\\ S_{1}^{4\/3}V_{2,\\mu}%\n^{1\/3}V_{2}^{5\/3,\\mu}\\ ,\\ S_{1}^{2\/3}S_{1}^{2\/3}S_{1}^{4\/3}\\ ,\\ S_{1}%\n^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}\\ ,\\label{LQOpsB2}\\\\\n& H^{\\dagger}S_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}\\ ,\\ H^{\\dagger}S_{2}%\n^{1\/3}V_{1,\\mu}^{2\/3}V_{1}^{4\/3,\\mu}\\ ,\\ H^{\\dagger}S_{2}^{7\/3}V_{1,\\mu\n}^{\\prime2\/3}V_{1}^{2\/3,\\mu}\\ ,\\ HS_{2}^{1\/3}V_{1,\\mu}^{\\prime2\/3}%\nV_{1}^{2\/3,\\mu}\\ . \\label{LQOpsBL3}%\n\\end{align}\nThe scenarios in the first line lead to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ or $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ operators (case G and H), and those in the second line to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ operators (case C). Note that for $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm3)$ transitions, the LQ must transform as $\\mathbf{3}$, and the color contraction is necessarily antisymmetric. When this is not compensated by an antisymmetric $SU(2)_L$ contraction, the three LQs must be different (hence one of the two $V_{1,\\mu}^{2\/3}$ fields is primed in the last two operators of Eq.~(\\ref{LQOpsBL3})). This does not apply to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators, for which it is always possible to take one of the DQ to transform as a symmetric $\\mathbf{\\bar{6}}$. As a final remark, it should be noted that scalar or vector color-singlet dileptons could also be introduced, opening the door to quartic couplings among the new states, and correspondingly, to eight-fermion $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,\\pm2)$ operators~\\cite{Helset:2021plg}. This will not be considered here.\n\nThroughout this paper, when estimating bounds on LQ\/DQ masses from proton decay or neutron-antineutron oscillations, the LQ\/DQ couplings to SM fermions is assumed flavor universal, or at the very least non-hierarchical in flavor space. As was shown in Ref.~\\cite{MFVBandL}, this is a strong assumption for $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating operators. The $SU(3)^{5}$ flavor\nsymmetry would ask instead for a strong hierarchy because of the systematic presence of the three quark generations in all the operators in Table~\\ref{TableLQBL}. In the present context, such hierarchies would first require LQ\/DQ to carry flavor quantum numbers, and then to extend the minimal flavor violating formalism to the LQ\/DQ sector~\\cite{Davidson:2010uu}. This will not be analyzed here, but such kind of flavor suppression should be kept in mind, especially given the context in B physics. There, a number of puzzles in leptonic and semileptonic decays can be explained by introducing new LQ states with particular flavor hierarchies (for a recent review, see e.g. Ref.~\\cite{LQreview}). Typically, the favored LQ is $V_{1,\\mu}^{4\/3}\\sim(\\mathbf{3},\\mathbf{1},+4\/3)$ thanks to its $q_{L}\\gamma^{\\mu}\\ell_{L}$ couplings, but other states could also occur in principle. The connection of some of these models with axions has been investigated e.g. in Ref.~\\cite{Fuentes-Martin:2019bue} (for some considerations of axions in the context of the B physics anomalies see e.g. \\cite{Baek:2020ovw}, whereas axions in a more broad flavor context have also been studied in Refs.~\\cite{Ema:2016ops,Calibbi:2016hwq,Arias-Aragon:2017eww,Bonnefoy:2020llz}, but to our knowledge, no systematic studies has been performed yet. In the present paper, our goal is mainly to analyze symmetry breaking patterns involving both LQ\/DQ and axions, so the LQ\/DQ couplings to SM fermions will simply be assumed $\\mathcal{O}(1)$ for all flavors whenever deriving bounds on their masses. Turning on non-trivial flavor structures is left for future studies.\n\n\\section{Coupling axions to leptoquarks and diquarks\\label{SecAxLQDQ}}\n\nIn the previous section, we have established the possible global accidental symmetries in the presence of LQ and DQ states. Here, we want to add to these scenarios a KSVZ or DFSZ sector. The consequences are rather different for both models, since the SM fermions can be PQ neutral in the former case, but not in the latter. Yet, so long as the $\\phi$ (and the heavy KSVZ fermions $\\Psi_{L,R}$) are not directly coupled to the LQ\/DQ states, the axion stays rather insensitive to the possible $\\mathcal{B}$ and\/or $\\mathcal{L}$ violation.\n\nTo illustrate this, consider the KSVZ scenario. Without direct couplings of $\\phi$ or $\\Psi_{L,R}$ to the LQ\/DQ states, the $U(1)_{\\phi}$ symmetry stays separate from the accidental symmetries $U(1)_{\\mathcal{B},\\mathcal{L}}$, so the PQ breaking proceeds trivially as\n\\begin{equation}\nU(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}\\overset\n{\\text{Explicit}}{\\rightarrow}U(1)_{\\phi}\\otimes U(1)_{X}\\simeq U(1)_{PQ}%\n\\otimes U(1)_{X}\\overset{\\text{Spontaneous}}{\\rightarrow}U(1)_{X}\\ ,\\ \\\n\\end{equation}\nThe specific LQ\/DQ scenario fixes which accidental symmetry\n\\begin{equation}\nU(1)_{X}=U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}},\\ U(1)_{\\mathcal{B}%\n\\pm\\mathcal{L}}\\ ,U(1)_{\\mathcal{B}}\\ ,\\ U(1)_{\\mathcal{L}}%\n,\\ U(1)_{3\\mathcal{B}\\pm\\mathcal{L}}\\ ,...\\ , \\label{SurvivingU1}%\n\\end{equation}\nsurvives, by introducing couplings that explicitly break $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}\\backslash U(1)_{X}$. Yet, the axion does not break $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ or $U(1)_{X}$, only the dynamics of the SM and LQ\/DQ fields does. Of course, the axion being coupled to SM gauge fields and SM fermions, it does end up coupled to leptoquarks and possibly acquires some $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating decay channels, but this is indirect. A good example for this situation is the KSVZ model with a Majorana mass $M_{R}\\nu_{R}\\nu_{R}$. The Majorana mass term explicitly breaks $U(1)_{\\mathcal{L}}$ at all scale, but such that $U(1)_{X}=U(1)_{\\mathcal{B}}$ stays exact at all scales. Clearly, the axion dynamics does not break $U(1)_{\\mathcal{L}}$, only neutrino masses do. Thus, any $\\Delta\\mathcal{L}=2$ effect would come indirectly, e.g. as in $a^{0}\\rightarrow\\bar{\\nu}_{R}\\nu_{L}\\rightarrow\\nu_{R}\\nu_{L}$. The situation in the DFSZ scenario is similar, though the $U(1)_{PQ}$ arises from a specific combination of $U(1)_{\\phi}$ and $U(1)_{Y}$, see Eq.~(\\ref{DFSZScalars}). This situation also corresponds to that often found in simple GUT models. For example, in $SU(5)$, gauge interactions break $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ down to $U(1)_{\\mathcal{B}-\\mathcal{L}}$ independently of the axion field (for a detailed account of how the PQ, $\\mathcal{B}$, and $\\mathcal{L}$ symmetries are entangled in the $SU(5)$ setting, see Ref.~\\cite{Quevillon:2020aij}).\n\nOur goal is to consider situations in which the symmetry above the PQ scale entangles $U(1)_{\\phi}$ within $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$. Breaking $U(1)_{\\phi}$ spontaneously then means breaking a linear combination of $\\mathcal{B}$ and $\\mathcal{L}$ (or both) spontaneously. Taking again the KSVZ scenario for illustration, this is accomplished by introducing some set of couplings that are only invariant under a subgroup of $U(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$. In most cases of interests, $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ stays active at the high scale, but $\\phi$ carries some definite $\\mathcal{B}$ and\/or $\\mathcal{L}$ quantum numbers, so that the breaking chain becomes\n\\begin{equation}\nU(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}\\overset\n{\\text{Explicit}}{\\rightarrow}U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}%\n}\\simeq U(1)_{PQ}\\otimes U(1)_{X}\\overset{\\text{Spontaneous}}{\\rightarrow\n}U(1)_{X}\\ .\n\\end{equation}\nThe simplest example illustrating this situation is the KSVZ model with the $\\phi^{\\dagger}\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ couplings, so that $\\phi$ becomes a $(\\mathcal{B},\\mathcal{L})=(0,2)$ state, $U(1)_{PQ}=U(1)_{\\mathcal{L}}$ is spontaneously broken, but $U(1)_{X}=U(1)_{\\mathcal{B}}$ stays exact. Compared to the previous case, the main difference is that the axion has a $\\Delta\\mathcal{L}=2$ coupling $a^{0}\\rightarrow\\nu_{R}\\nu_{R}$ of $\\mathcal{O}(1)$. Of course, phenomenologically, whether one adds $M_{R}\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ or $\\phi^{\\dagger}\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ is irrelevant, but this may not be the case for scenarios in which $U(1)_{\\mathcal{B}}$ is spontaneously broken. Our goal here is to systematically study these scenarios, taking advantage of the fact that LQ\/DQ open many routes to entangle $U(1)_{\\phi}$ within $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ at the renormalizable level (with only SM fields, the $\\phi^\\dagger\\bar{\\nu}^\\mathrm{C}_{R}\\nu_{R}$ coupling is the only possibility). Note, finally, that in the KSVZ context, there is actually an extra accidental symmetry corresponding to $\\Psi$ number, $U(1)_{\\Psi}$, that will either survive or be incorporated within $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ via explicit breaking terms independent of $\\phi$. In this way, the final surviving $U(1)_{X}$ accidental symmetry is independent of $U(1)_{\\Psi}$, and still given by Eq.~(\\ref{SurvivingU1}).\n\nIn practice, to entangle the $U(1)_{\\phi}$ symmetry with the accidental symmetries, the strategy is to turn on some direct couplings between $\\phi$ and the LQ\/DQ, and for these latter, to turn on some or all of their couplings to SM fields such that no direct $\\mathcal{B}$ and\/or $\\mathcal{L}$ violation occurs. It is important to stress that we do not assign $U(1)$ charges to the fields. Instead, we let the theory tell us what are the exact accidental $U(1)$ symmetries, and what are the charges of the fields. Indeed, it is well-known that symmetries and charges are entirely fixed given a set of couplings in the Lagrangian, but often one identifies them by inspection, or starts from the charges to infer the allowed couplings. In the present case, as we will see, the set of couplings can be quite large, and the surviving $U(1)$s assign quite intricate charges to the fields. Typically, a naive inspection of the Lagrangian couplings would most likely miss some of the surviving $U(1)$s, or outright fail to identify possible scenarios. In practice, starting from the Lagrangian also provides a very systematic procedure: to find the surviving $U(1)$ symmetries, it suffices to express the charge constraint corresponding to each coupling, and solve this system of equations. When this system is under-determined, each parametric under-determination corresponds to a surviving $U(1)$. The charges of $\\phi$ under these $U(1)$ then tell us which combination is spontaneously broken.\n\n\\subsection{KSVZ scenarios with leptoquarks and diquarks\\label{SecKSVZ}}\n\nOur requirements for the KSVZ scenarios are first that there should be only one Higgs doublet, neutral under the PQ and all accidental symmetries, and no direct mixing of the heavy fermions $\\Psi_{L,R}$ with SM quarks to avoid FCNC or CKM unitarity constraints. Also, our goal is to force proton decay, neutron-antineutron oscillations, or a Majorana mass terms for $\\nu_{R}$ (or more generally, neutrino-less double beta decays~\\cite{Deppisch:2012nb}) to only arise through the spontaneous symmetry breaking of $U(1)_{\\phi}$. Thus, none of these observables should be immediately allowed by LQ\/DQ transitions. Typically, the strategy to achieve this is, starting from some Lagrangian with a specific set of couplings among $\\phi$, some chosen LQ\/DQs, and the SM fermions, to identify the accidental symmetries, and then make sure these accidental symmetries forbid any other renormalizable Lagrangian couplings. This will be made clear going through specific examples. But, before that, let us describe some generic features of the scenarios and their consequence for the axion effective Lagrangian.\n\nIn all scenarios, there will be some $\\phi^{2}S_{i}^{\\dagger}S_{j}$, $\\phi HS_{i}^{\\dagger}S_{j}$, and\/or $\\phi S_{i}S_{j}S_{k}$ couplings. In this representation, the axion ends up coupled to the LQ\/DQ, as can be seen plugging in Eq.~(\\ref{PhiPolar}) in these couplings (remember $\\eta_{\\phi}=a^{0}$ in the KSVZ setting). Importantly, these couplings are never suppressed by the PQ breaking scale, since for example\n\\begin{equation}\n\\phi S_{i}S_{j}S_{k}\\rightarrow\\frac{1}{\\sqrt{2}}(v_{\\phi}+i\\eta_{\\phi}+...)S_{i}S_{j}S_{k}\\ . \n\\label{ExampleSSS}\n\\end{equation}\nThough as a matter of principle, the axion $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings are not suppressed by $v_{\\phi}$, this scale nevertheless indirectly limits them. Indeed, the leading $v_{\\phi}$ term produces a direct coupling among the LQ\/DQ such that one falls into any one of the situations described in Sec.~\\ref{Sec2c}, with some $U(1)_{X}$ smaller than $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ remaining exact. At low energy, these LQ\/DQ couplings can induce $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating processes, hence set rather strong bounds on the LQ\/DQ masses. Now, the largest $v_{\\phi}$ is, the tightest these bounds are, so indirectly, the $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating axion couplings to SM fermions decrease for increasing $v_{\\phi}$.\n\nComing back to the point of principle, one may wonder how is it that the axion couplings is not suppressed by $v_{\\phi}$ in the effective axion Lagrangian language of Eq.~(\\ref{AxionEL}). Indeed, as a result of the $\\phi^{2}S_{i}^{\\dagger}S_{j}$, $\\phi HS_{i}^{\\dagger}S_{j}$, and\/or $\\phi S_{i}S_{j}S_{k}$ couplings, some or all of the LQ\/DQ become charged under $U(1)_{PQ}$. This means that if, along with Eq.~(\\ref{ReparamG}) for the fermions, we reparametrize them as\n\\begin{equation}\nS_{i}\\rightarrow\\exp(-iPQ(S_{i})a^{0}\/v_{\\phi})S_{i}\\ , \n\\label{ReparamS}\n\\end{equation}\nthe axion field is entirely removed from all the Lagrangian couplings. Indeed, the Lagrangian is PQ-symmetric, so the $\\exp(ia^{0}\/v_{\\phi})$ factors always compensate exactly. Their kinetic terms $D_{\\mu}S_{i}^{\\dagger}D^{\\mu}S_{i}$ are not invariant under the reparametrization though, and as for fermions, this is embodied in dimension-five interactions\n\\begin{equation}\n\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}=\\frac{1}{v_{\\phi}}\\partial_{\\mu}%\na^{0}J_{PQ}^{\\mu}\\text{ ,\\ \\ \\ }J_{PQ}^{\\mu}=\\sum_{i}PQ(S_{i})(S_{i}^{\\dagger\n}(D^{\\mu}S_{i})-(D^{\\mu}S_{i}^{\\dagger})S_{i})+...\n\\end{equation}\nThis representation is deceptive because the axion couplings to LQ\/DQ appear suppressed by $v_{\\phi}$. Yet, they are not suppressed because the EoM of the $S_{i}$ have $\\mathcal{O}(v_{\\phi})$ terms, like that coming from a $v_{\\phi}S_{i}S_{j}S_{k}$ coupling in the example of Eq.~(\\ref{ExampleSSS}). The same happens if LQ\/DQ are integrated out before the reparametrization Eq.~(\\ref{ReparamS}). They then do not occur in $J_{PQ}^{\\mu}$, but SM fermions do, and their EoM now have inherited $\\mathcal{O}(v_{\\phi}\/M^{n}$)\nterms for some $n$, with $M$ the LQ\/DQ mass scale. In all cases, the axion keeps its $\\mathcal{O}(v_{\\phi}^{0})$ $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings, as it should.\n\nThis shows explicitly that the shift-symmetric $\\delta\\mathcal{L}_{\\text{\\textrm{Der}}}$ is not well-suited to these scenarios, at least for what concerns couplings to matter fields. For gauge boson, the situation is a bit different. The fermion reparametrization Eq.~(\\ref{ReparamG}) generates spurious anomalous interactions to chiral gauge fields that are cancelled by the anomalies in the $J_{PQ}^{\\mu}$ current, exactly as before, but the LQ\/DQ obviously do not. Thus, for them, the axion effective Lagrangian after the reparametrization of Eq.~(\\ref{ReparamS}) correctly captures the fact that triangle graphs with LQ\/DQ running in the loop are not anomalous, and vanish at the dimension-five level for a massless axion. Thus, none of the axion to gauge boson couplings is affected by the LQ\/DQ at leading order.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}+\\mathcal{L}$}\n\nWe have seen that $\\mathcal{B}+\\mathcal{L}$ is immediately broken whenever a given $S_{i}$ or $V_{i}$ has both LQ and DQ couplings. For example, $S_{1}^{8\/3}$ with its couplings to $\\bar{d}_{R}e_{R}^{\\mathrm{C}}$ and $\\bar{u}_{R}^{\\mathrm{C}}u_{R}$ can induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators and proton decay. A possible strategy to adapt this scenario and force these operators to appear only through the SSB of $\\phi$ is to consider two such states, one LQ and one DQ, with a $\\phi$-dependent mixing term:\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{8\/3}\\bar\n{d}_{R}e_{R}^{\\mathrm{C}}+\\tilde{S}_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}%\nu_{R}+\\phi^{2}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}+h.c.\\ , \\label{LagrKSVZ1a}%\n\\end{equation}\nwith $\\mathcal{L}_{\\mathrm{KSVZ}}$ given in Eq.~(\\ref{KSVZ0}), and LQ\/DQ kinetic terms are understood. We also do not write explicitly the LQ\/DQ scalar potential terms made of bilinears like $S_{1}^{8\/3\\dagger}S_{1}^{8\/3}$ or $\\tilde{S}_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}$ since those are neutral under any $U(1)$ symmetry. Solving for the $U(1)$ charges of all the fields under the requirement that the Higgs doublet is neutral (to avoid mixing with $U(1)_{Y}$), a triple under-determination remains, which we can identify as\\footnote{Evidently, the normalization of each line is free, but that for $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ is chosen to reproduce conventional quark and lepton $\\mathcal{B}$ and $\\mathcal{L}$ of $1\/3$ and $1$, respectively.}\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{8\/3}$ & $\\tilde{S}_{1}^{8\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\Psi}$ & $0$ & $0$ & $0$ & $1$ & $1$ & $0$ & $0$ & $0$ & $0$ & $0$ &\n$0$\\\\\n$U(1)_{\\mathcal{B}}$ & $1\/2$ & $1\/3$ & $-2\/3$ & $-1\/2$ & $0$ & $1\/3$ & $1\/3$ &\n$1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $1\/2$ & $1$ & $0$ & $-1\/2$ & $0$ & $0$ & $0$ & $0$ &\n$1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\label{ChargKSVZ1a}\n\\end{equation}\n\n\nWhat this table shows is that $\\phi$ carries a $U(1)_{\\mathcal{B}+\\mathcal{L}}$ charge, which thus gets spontaneously broken, while $U(1)_{\\mathcal{B}-\\mathcal{L}}$ stays exact. This model is essentially identical to that introduced long ago in Ref.~\\cite{WeinbergPRD22}, except that the Goldstone boson is here identified with the axion. This pattern of symmetry breaking is easily understood from the Lagrangian couplings and the diagram in Fig.~\\ref{Fig2}. Plugging in the polar representation of $\\phi$, Eq.~(\\ref{PhiPolar}), the effective operator at the low-scale is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff}=\\exp\n(2ia^{0}\/v_{\\phi})\\frac{v_{\\phi}^{2}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\bar{u}%\n_{R}^{\\mathrm{C}}u_{R}\\bar{d}_{R}^{\\mathrm{C}}e_{R}+h.c.\\ , \\label{EffHKSVZ1}%\n\\end{equation}\nwhere we have identified $\\eta_{\\phi}$ as the axion $a^{0}$, and denoted the $S_{1}^{8\/3}$ and $\\tilde{S}_{1}^{8\/3}$ masses as $m_{S}$ and $m_{\\tilde{S}}$, respectively. Note well that this operator arises entirely through the SSB: the charges in Eq.~(\\ref{ChargKSVZ1a}) explicitly prevent a DQ coupling for $S_{1}^{8\/3}$, and a LQ coupling for $\\tilde{S}_{1}^{8\/3}$. Expanding the exponential, the leading term involves only SM particles, and contributes to proton decay. Thus, $m_{S}$ and $m_{\\tilde{S}}$ have to be pushed quite high, though a bit lower that in the usual GUT scenarios. For instance, while the scale of the dim-6 operators is typically pushed above $10^{14}$~GeV, we only need $m_{S}\\approx m_{\\tilde{S}}>10^{11}$~GeV when $v_{\\phi}=10^{9}$~GeV. With these parameters, the proton decay modes involving the axion are thus totally negligible. Finally, notice that the axion totally disappears from $\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff}$ under the reparametrization Eq.~(\\ref{ReparamG}), with the PQ charges identified as $(\\mathcal{B}+\\mathcal{L})\/2$. As stated earlier, the $v_{\\phi}a^{0}\\bar{u}_{R}^{\\mathrm{C}}u_{R}\\bar{d}_{R}^{\\mathrm{C}}e_{R}$ effective coupling would then hide in the $\\partial_{\\mu}a^{0}J_{PQ}^{\\mu}\/v_{\\phi}$ terms since the quarks and leptons inherit from $v_{\\phi}^{2}\\bar{u}_{R}^{\\mathrm{C}}u_{R}\\bar{d}_{R}^{\\mathrm{C}}e_{R}$ some $\\mathcal{O}(v_{\\phi}^{2})$ terms in their EoM.%\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.3586in,width=1.5826in]{Fig2.jpg}\n\\caption{Proton decay operator generated by the spontaneous breaking of $U(1)_{\\mathcal{B}+\\mathcal{L}}$.}%\n\\label{Fig2}\n\\end{center}\n\\end{figure}\n\nAs shown in Eq.~(\\ref{ChargKSVZ1a}), $U(1)_{\\Psi}$ remains as an exact accidental symmetry, which means that the $\\mathcal{B}+\\mathcal{L}$ charges of $\\Psi_{L,R}$ are not unambiguously defined. To fix them requires $\\Psi_{L,R}$\nto couple to SM fermions, and this is possible only for some specific gauge quantum numbers. If we further ask that $S_{1}^{8\/3}$ ($\\tilde{S}_{1}^{8\/3}$) should always (never) couple to leptons, the only possibilities are%\n\\begin{equation}%\n\\begin{tabular}\n[c]{c|cccccc}\\hline\n$Y,\\mathcal{B},\\mathcal{L}$ & $S_{1}^{8\/3}\\bar{\\Psi}_{L}\\ell_{L}^{\\mathrm{C}}$\n& $S_{1}^{8\/3}\\bar{\\Psi}_{R}e_{R}^{\\mathrm{C}}$ & $S_{1}^{8\/3}\\bar{\\Psi}%\n_{R}\\nu_{R}^{\\mathrm{C}}$ & $\\tilde{S}_{1}^{8\/3}\\bar{\\Psi}_{L}^{\\mathrm{C}%\n}q_{L}$ & $\\tilde{S}_{1}^{8\/3}\\bar{\\Psi}_{R}^{\\mathrm{C}}u_{R}$ & $\\tilde\n{S}_{1}^{8\/3}\\bar{\\Psi}_{R}^{\\mathrm{C}}d_{R}$\\\\\\hline\n$\\Psi_{L}$ & $-\\dfrac{5}{3},\\dfrac{1}{3},0\\rule[-0.14in]{0in}{0.36in}$ &\n$-\\dfrac{2}{3},\\dfrac{5}{6},\\dfrac{1}{2}$ & $-\\dfrac{8}{3},\\dfrac{5}{6}%\n,\\dfrac{1}{2}$ & $\\dfrac{7}{3},\\dfrac{1}{3},0$ & $\\dfrac{4}{3},\\dfrac{5}%\n{6},\\dfrac{1}{2}$ & $\\dfrac{10}{3},\\dfrac{5}{6},\\dfrac{1}{2}$\\\\\n$\\Psi_{R}$ & $-\\dfrac{5}{3},-\\dfrac{1}{6},-\\dfrac{1}{2}\\rule[-0.14in]%\n{0in}{0.36in}$ & $-\\dfrac{2}{3},\\dfrac{1}{3},0$ & $-\\dfrac{8}{3},\\dfrac{1}%\n{3},0$ & $\\dfrac{7}{3},-\\dfrac{1}{6},-\\dfrac{1}{2}$ & $\\dfrac{4}{3},\\dfrac\n{1}{3},0$ & $\\dfrac{10}{3},\\dfrac{1}{3},0$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThese couplings are mutually exclusive since they impose different hypercharges for $\\Psi_{L,R}$. Also, $S_{1}^{8\/3}\\bar{\\Psi}_{R}e_{R}^{\\mathrm{C}}$ and $\\tilde{S}_{1}^{8\/3}\\bar{\\Psi}_{R}^{\\mathrm{C}}u_{R}$ would allow for direct $\\bar{\\Psi}_{R}d_{R}$ and $\\bar{\\Psi}_{R}u_{R}$ couplings, respectively, hence must be discarded. Note how the peculiar choice of couplings completely twists the $\\mathcal{B},\\mathcal{L}$ charges, in the sense that they do not correspond to the naive assignments of $\\mathcal{B}=1\/3$ and $\\mathcal{L}=0$ one may have expected for the \"heavy quarks\" of the KSVZ mechanism. As said before, the charges of the fields have to be deduced from the set of couplings of the Lagrangian, and not the other way around.\n\nSimilar scenarios can be constructed using $S_{1}^{2\/3}$, $S_{1}^{4\/3}$, $V_{2}^{1\/3}$ or $V_{2}^{5\/3}$. Actually, $S_{1}^{2\/3}$ was considered in Ref.~\\cite{Reig:2018yfd}, though the model built there is more complicated (here the PQ symmetry is directly identified with $\\mathcal{B}+\\mathcal{L}$ and only a single Higgs doublet is introduced instead of four). Each time, two such states are taken, with one having LQ couplings, and the other DQ couplings, and a $\\phi$-driven mixing term is introduced. The only difference in each case is the specific $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operator(s) that can be spontaneously generated, see Table~\\ref{TableLQBL}, and thereby, the induced pattern of proton decay modes. In this respect, it is worth to look at the $S_{1}^{4\/3}$ scenario, since it has only LQ couplings to $\\nu_{R}$:\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{4\/3}\\bar\n{u}_{R}\\nu_{R}^{\\mathrm{C}}+\\tilde{S}_{1}^{4\/3}\\bar{d}_{R}^{\\mathrm{C}}%\nd_{R}+\\phi^{2}S_{1}^{4\/3\\dagger}\\tilde{S}_{1}^{4\/3}+\\phi\\bar{\\nu}%\n_{R}^{\\mathrm{C}}\\nu_{R}+h.c.\\ . \\label{LagrKSVZ1b}%\n\\end{equation}\nLet us also turn on a coupling to $\\Psi_{L,R}$, to fix its charges, and for definiteness, let us take $S_{1}^{4\/3}\\bar{\\Psi}_{L}\\ell_{L}^{\\mathrm{C}}$. Because of the $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling, solving for the $U(1)$ charges of all the fields now leaves a single under-determination:\n\\begin{equation}%\n\\begin{tabular}\n[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{4\/3}$ & $\\tilde{S}_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $2$ & $2\/3$ & $-10\/3$ & $5\/3$ & $-1\/3$ & $5\/3$ & $5\/3$ & $5\/3$ &\n$-1$ & $-1$ & $-1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThis time, neither $\\mathcal{B}$ nor $\\mathcal{L}$ survives. Starting from $U(1)_{\\phi}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$, two $U(1)$s are explicitly broken by $\\mathcal{L}_{\\mathrm{KSVZ+LQ}}$, while the remaining exact $U(1)$ is identified with $U(1)_{PQ}$ and spontaneously broken by $\\phi$. The interest in this scenario is that $S_{1}^{4\/3}$ couples only to $\\nu_{R}$, whose mass is pushed at the PQ breaking scale by the $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling. At the low-energy scale, the leading proton decay operator will scale as%\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff}=\\exp\n(2ia^{0}\/v_{\\phi})\\frac{v_{\\phi}^{2}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\bar{d}%\n_{R}d_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}+h.c.\\rightarrow\n\\exp(2ia^{0}\/v_{\\phi})\\frac{v_{\\phi}v_{EW}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\bar{d}%\n_{R}d_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{L}^{\\mathrm{C}}+h.c.\\ .\n\\label{FinalScale11}%\n\\end{equation}\nThanks to this extra suppression, the PQ breaking scale, which is also the neutrino seesaw scale, and the LQ\/DQ mass scale, can all sit at around $10^{9}$ GeV. They could thus naturally have a common UV origin.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}-\\mathcal{L}$}\n\nWith only LQ\/DQ, scenarios in which $\\mathcal{B}-\\mathcal{L}$ is explicitly broken typically arise from any one of the $HS_{i}^{\\dagger}S_{j}$ or $HV_{i}^{\\dagger}V_{j}$ couplings in Eq.~(\\ref{LQOpsBmL}). Those couplings always involve a pure LQ state together with a mixed LQ\/DQ state. The $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ pattern arises when the latter has only DQ couplings. All these scenarios can be adapted to force $\\mathcal{B}-\\mathcal{L}$ to be broken spontaneously instead of explicitly. Let us take the $HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ case as an example, the others being totally similar. To entangle the KSVZ symmetry with $\\mathcal{B}-\\mathcal{L}$, we start from the Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{2}^{7\/3}(\\bar{u}_{R}\\ell_{L}+\\bar{q}_{L}%\ne_{R})+\\phi HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}+h.c.\\ , \\label{LagrKSVZ2}%\n\\end{equation}\nwhere again kinetic terms and LQ\/DQ potential terms are understood. For definiteness, we also include the $S_{1}^{4\/3}\\bar{\\Psi}_{L}\\ell_{L}^{\\mathrm{C}}$ coupling to get rid of $U(1)_{\\Psi}$ and fix the quantum numbers of $\\Psi_{L,R}$. Then, there remain only a $U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ symmetry with charges\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{2}^{7\/3}$ & $S_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1$ & $1\/3$ & $-2\/3$ & $-2\/3$ & $-5\/3$ & $1\/3$ & $1\/3$\n& $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-1$ & $-1$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0$ & $1$\n& $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nand $U(1)_{\\mathcal{B}-\\mathcal{L}}$ is spontaneously broken when $\\phi$ acquires its vacuum expectation value. Note that these charges prevent the LQ couplings of $S_{1}^{4\/3}$ (taking $\\phi HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ instead, they would further forbid the $HS_{2}^{1\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ coupling). The final operators are part of the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ dimension-seven ones in Table~\\ref{TableLQBL} because of the Higgs doublet appearing in the $\\phi HS_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ mixing term (see Fig.~\\ref{Fig3}):\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\exp\n(ia^{0}\/v_{\\phi})\\frac{v_{\\phi}}{m_{S}^{2}m_{\\tilde{S}}^{2}}H\\bar{d}_{R}%\nd_{R}^{\\mathrm{C}}(\\bar{u}_{R}\\ell_{L}+\\bar{q}_{L}e_{R})+h.c.\\ .\n\\end{equation}\nThe situation is thus similar to that in Eq.$~$(\\ref{FinalScale11}). Further lowering the LQ\/DQ scale by about an order of magnitude is possible starting from the $HV_{1,\\mu}^{2\/3\\dagger}V_{2}^{5\/3,\\mu}$ coupling, as $V_{1,\\mu}^{2\/3\\dagger}$ couples only to $\\nu_{R}$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.3984in,width=1.9009in]{Fig3.jpg}\n\\caption{Proton decay operator generated by the spontaneous breaking of $U(1)_{\\mathcal{B}-\\mathcal{L}}$.}\n\\label{Fig3}\n\\end{center}\n\\end{figure}\n\nIn this regard, note that all these scenarios are again compatible with a seesaw mechanism. Adding either a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, or $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling to $\\mathcal{L}_{\\mathrm{KSVZ+LQ}}$ in Eq.~(\\ref{LagrKSVZ2}), a single exact $U(1)$ remains at the PQ scale, with charges\n\\begin{equation}\n\\begin{tabular}[c]{ccccccccccccc}\\hline\n& & $\\phi$ & $S_{2}^{7\/3}$ & $\\tilde{S}_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$\n& $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ & $4\/3$ & $-2\/3$\n& $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $-1$ & $-1$ & $-1$\\\\\n$\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2\/3$ & $0$\n& $-2\/3$ & $-1$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$\\\\\n$M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $1$ & $1\/3$ & $-2\/3$\n& $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ & $0$ & $0$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nFor all these cases, the axion still emerges as a massless Goldstone boson, and is associated to both $U(1)_{\\mathcal{B}-\\mathcal{L}}$ and $U(1)_{\\mathcal{L}}$ spontaneous breakings.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}$}\n\nThe spontaneous breaking of $\\mathcal{B}$ first arose at the dimension-9 level in Table~\\ref{TableLQBL} since it necessarily involves six fermions. As seen in Sec.~\\ref{Sec2c}, typical scenarios thus require a cubic coupling among DQ states. Let us start with\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}u_{R}+\\phi\nS_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}+h.c.\\ , \\label{LagrKSVZ3}%\n\\end{equation}\nwhere $S_{1}^{4\/3}\\sim(\\mathbf{3},\\mathbf{1},+4\/3)$ and $S_{1}^{8\/3}\\sim(\\mathbf{\\bar{6}},\\mathbf{1},-8\/3)$. Though not compulsory, we add the coupling $S_{1}^{8\/3}\\bar{\\Psi}_{L}^{\\mathrm{C}}q_{L}$ to break $U(1)_{\\Psi}$ and fix the charges of $\\Psi_{L,R}$. With this Lagrangian, only two $U(1)$s are exact:\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{2}^{8\/3}$ & $S_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $2$ & $-2\/3$ & $-2\/3$ & $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$\n& $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThus, $U(1)_{PQ}=U(1)_{\\mathcal{B}}$ is broken spontaneously by two units, but $U(1)_{\\mathcal{L}}$ stays exact. This model is actually very similar to that of Ref.~\\cite{Barbieri:1981yr} (see also Ref.~\\cite{Berezhiani:2015afa}), except that the Goldstone boson associated to the $U(1)_{\\mathcal{B}}$ breaking is identified with the axion. In turn, the axion ends up coupled to neutron pairs, via the diagram shown in Fig.~\\ref{Fig4a}. The corresponding operator is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)}^{eff}=\\exp\n(ia^{0}\/v_{\\phi})\\frac{v_{\\phi}}{m_{S^{4\/3}}^{4}m_{S^{8\/3}}^{2}}\\bar{d}%\n_{R}^{\\mathrm{C}}d_{R}\\bar{d}_{R}^{\\mathrm{C}}d_{R}\\bar{u}_{R}^{\\mathrm{C}%\n}u_{R}+h.c.\\ . \\label{ScaleNN}%\n\\end{equation}\nTypical bounds on the scale of the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators are at around $100$~TeV~\\cite{Mohapatra:2009wp,Phillips:2014fgb} if the couplings implicit in Eq.~(\\ref{LagrKSVZ3}) are all $\\mathcal{O}(1)$. The PQ scale of $10^{9}$~GeV pushes the DQ scale slightly higher than those $100\\ \\text{TeV}$, but given that their masses appear to the sixth power, this is marginal (less than an order of magnitude). The presence of the axion also leads to an effective operator%\n\\begin{equation}\n\\frac{1}{m_{S^{4\/3}}^{4}m_{S^{8\/3}}^{2}}a^{0}\\bar{d}_{R}^{\\mathrm{C}}d_{R}%\n\\bar{d}_{R}^{\\mathrm{C}}d_{R}\\bar{u}_{R}^{\\mathrm{C}}u_{R}+h.c.\\ \\rightarrow\n\\frac{\\Lambda_{QCD}^{6}}{m_{S^{4\/3}}^{4}m_{S^{8\/3}}^{2}}a^{0}\\bar\n{n}^{\\mathrm{C}}\\gamma_{5}n+h.c.\\ ,\n\\end{equation}\nwith the QCD confinement scale $\\Lambda_{QCD}$ of the order of $300$~MeV. Because the DQ mass scale is pushed rather high by the dimension-nine operator in Eq.~(\\ref{ScaleNN}), this direct coupling is very suppressed. Note, though, that it could have consequences in a cosmological context~\\cite{Mohapatra:2009wp}.%\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.8844in,width=1.7167in]{Fig4a.jpg}\n\\caption{Neutron-antineutron oscillation operators generated by the spontaneous breaking of $U(1)_{\\mathcal{B}}$.}\n\\label{Fig4a}\n\\end{center}\n\\end{figure}\n\nAs for the previous two scenarios, a seesaw mechanism can be implemented by adding a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, or $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling to the Lagrangian in Eq.~(\\ref{LagrKSVZ3}). For the former two, this identifies the axion as the Majoron~\\cite{Barbieri:1981yr}. The only change is, in some sense, to assign a $\\mathcal{B}$ number to $\\nu_{R}$, hence by extension, to the leptons:\n\\begin{equation}\n\\begin{tabular}[c]{ccccccccccccc}\\hline\n& & $\\phi$ & $S_{2}^{8\/3}$ & $S_{1}^{4\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ & $-2\/3$ & $-2\/3$\n& $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $-1$ & $-1$ & $-1$\\\\\n$\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ &\n$-2\/3$ & $-2\/3$ & $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1$ & $1$ & $1$\\\\\n$M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}:$ & $U(1)_{PQ}$ & $2$ & $-2\/3$ &\n$-2\/3$ & $1\/3$ & $-5\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ & $0$ & $0$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nNote that the charges imposed by the presence of $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ open the door to the $S_{1}^{4\/3}\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}$ coupling also, and thus to direct proton decay via an $S_{1}^{4\/3}$ Fermi interaction. For the other two scenarios, proton decay remains forbidden since all its decay modes include an odd number of leptons, but only $\\Delta\\mathcal{L}=2n$ transitions are made possible by the Lagrangian couplings.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}\\pm3\\mathcal{L}$}\n\nFrom Eq.~(\\ref{LQOpsB2}), it is clear that the scenarios leading to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ can be adapted to generate $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ effects. All that is needed is to replace all DQ couplings by LQ couplings. The only difficulty is to account for the antisymmetric color contraction, since LQ necessarily transform as $\\mathbf{3}$ under $SU(3)_{C}$. If we insist on introducing at most two different LQ, the only available scenario is%\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{2\/3}(\\bar\n{d}_{R}\\nu_{R}^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}}+\\bar{q}_{L}\\ell\n_{L}^{\\mathrm{C}})+V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}%\n^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})+\\phi S_{1}%\n^{2\/3}V_{2,\\mu}^{1\/3}V_{2}^{1\/3,\\mu}+h.c.\\ . \\label{LagrKSVZ4}%\n\\end{equation}\nAs usual, the $U(1)_{\\Psi}$ is broken explicitly, this time by adding $V_{2,\\mu}^{1\/3}\\bar{\\Psi}_{L}\\gamma^{\\mu}e_{R}^{\\mathrm{C}}$ to force the hypercharge of $\\Psi_{L,R}$ to be different from that of SM quarks. If instead of the LQ couplings, DQ couplings were allowed, this scenario produces the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ symmetry pattern discussed in the previous section. Now, with these LQ couplings and no DQ couplings, the charges are\n\\begin{equation}%\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2}^{1\/3}$ & $\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1$ & $1\/3$ & $1\/3$ & $1\/3$ & $-2\/3$ & $1\/3$ & $1\/3$ &\n$1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $3$ & $1$ & $1$ & $0$ & $-3$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThe PQ symmetry is identified with $U(1)_{\\mathcal{B}+3\\mathcal{L}}$, and dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ proton decay operators appear at the low scale, see Fig.~\\ref{Fig4b}$a$ (a similar LQ model was proposed in Ref.~\\cite{WeinbergPRD22} to break $\\mathcal{B}+3\\mathcal{L}$ spontaneously). The fact that these operators are dimension-nine allows to lower the LQ scale, but qualitatively, this scenario is not very different from the $\\mathcal{B}\\pm\\mathcal{L}$ ones. Also, a seesaw mechanism can be added with either $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ or $M_{R}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, but not with $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ as this would allow back the DQ couplings of both $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$. It should be noted that these charges allow for the $D^{\\mu}HS_{1}^{2\/3}V_{2,\\mu}^{1\/3\\dagger}$ and $HD^{\\mu}S_{1}^{2\/3}V_{2,\\mu}^{1\/3\\dagger}$couplings. If not initially present, they are immediately generated via a fermion loop. Yet, these operators carry $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,0)$ and cannot help create simpler proton decay processes. They could turn on some new four-fermion semileptonic FCNC operators though, but these effects are beyond our scope.%\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=2.2329in,width=4.9536in]{Fig4b.jpg}\n\\caption{Proton decay operators generated by the spontaneous breaking of $U(1)_{\\mathcal{B}+3\\mathcal{L}}$ ($a.$) and $U(1)_{\\mathcal{B}-3\\mathcal{L}}$ ($b.$).}\n\\label{Fig4b}\n\\end{center}\n\\end{figure}\n\nThe final pattern is $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$, and this one is quite difficult to induce spontaneously. The operators in Eq.~(\\ref{LQOpsBL3}) being already of dimension four, we cannot proceed as for the other cases and simply multiply them by $\\phi$. One way to proceed is to start with an operator from the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$\nclass in Eq.~(\\ref{LQOpsB2}), and then switch $\\mathcal{L}$ by six units using $\\Delta\\mathcal{L}=2$ operators of Eq.~(\\ref{LQOpsBmL}). For instance, the Lagrangian\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} & =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{2}%\n^{1\/3}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})+V_{1,\\mu}^{2\/3}\\bar{d}%\n_{R}\\gamma^{\\mu}\\nu_{R}\\nonumber\\\\\n& \\ \\ \\ \\ +\\phi(HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}+H^{\\dagger}V_{1}%\n^{2\/3\\dagger,\\mu}V_{2,\\mu}^{1\/3}+S_{1}^{2\/3\\dagger}V_{2,\\mu}^{1\/3\\dagger}%\nV_{2}^{1\/3,\\mu\\dagger})+h.c.\\ , \\label{LagrKSVZ5}%\n\\end{align}\ndoes lead to the desired $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ pattern, as shown in Fig.~\\ref{Fig4b}$b$. With four LQ states, it is certainly more complex than the other scenarios, though it should be noted that there is a certain symmetric flavor to the presence of $S_{2}^{1\/3}$, $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$, $V_{1,\\mu}^{2\/3}$. Also, it is not compulsory for $\\phi$ to appear in all of the last three couplings, but when it does, only some combinations do give a $\\mathcal{B}-3\\mathcal{L}$ charge to $\\phi$. Further adding $V_{1,\\mu}^{2\/3}\\bar{\\Psi}_{R}\\gamma^{\\mu}e_{R}$ to fix the $\\Psi_{L,R}$ quantum numbers, we find%\n\\begin{equation}\n\\begin{tabular}[c]{cccccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2}^{1\/3}$ & $S_{2}^{1\/3}$ & $V_{1}^{2\/3}$ &\n$\\Psi_{L}$ & $\\Psi_{R}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ &\n$\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1\/4$ & $1\/12$ & $1\/12$ & $1\/3$ & $1\/3$ & $7\/12$ &\n$1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-3\/4$ & $-1\/4$ & $-1\/4$ & $-1$ & $-1$ & $-3\/4$ & $0$ &\n$0$ & $0$ & $0$ & $1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThe PQ symmetry is thus indeed $U(1)_{\\mathcal{B}-3\\mathcal{L}}$. Note that these charges forbid all the SM fermion couplings of $S_{1}^{2\/3}$ and $V_{2}^{1\/3}$, as well as all other possible cubic interactions among the LQ and DQ states\\footnote{Some derivative interactions are possible though, but those necessarily involve the LQs whose SM fermion couplings are forbidden, hence they do not alter the symmetry breaking pattern, and would lead to more suppressed proton decay operators.}. However, given the complicated structure shown in Fig.~\\ref{Fig4b}$b$, the final operators are of dimension 16 instead of dimension 10:\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)}^{eff} &\n=\\frac{\\phi^{4}(H^{\\dagger}H)}{m_{S1\/3}^{2}m_{V2\/3}^{4}m_{S2\/3}^{2}%\nm_{V1\/3}^{4}}H^{\\dagger}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})\\bar{d}%\n_{R}\\gamma_{\\mu}\\nu_{R}\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\nonumber\\\\\n& \\rightarrow\\frac{v_{\\phi}^{4}v_{EW}^{3}}{m_{S1\/3}^{2}m_{V2\/3}^{4}m_{S2\/3}%\n^{2}m_{V1\/3}^{4}}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})\\bar{d}_{R}%\n\\gamma_{\\mu}\\nu_{R}\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\ .\n\\end{align}\nBesides, turning on a seesaw mechanism with $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ (as $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ would allow back some $S_{1}^{2\/3}$ and $V_{2}^{1\/3}$ couplings to SM fermions), a further suppression of $(v_{EW}\/v_{\\phi})^{2}$ to connect two $\\nu_{R}$ to light fermions arises. Altogether, assuming a common scale for all the LQs, their mass can be as low as around 100 TeV when $v_{\\phi}\\approx10^{9}$ GeV. This is much lower than in GUT scenarios, and actually falls within the ballpark of the scale required by neutron-antineutron oscillation from Eq.~(\\ref{ScaleNN}).\n\n\\subsection{DFSZ scenarios with leptoquarks and diquarks\\label{SecDFSZ}}\n\nAll the scenarios discussed in the KSVZ case can readily be adapted to the DFSZ model. Basically, one removes the $\\Psi_{L,R}$ state but introduces a $\\phi^{2}H_{u}^{\\dagger}H_{d}$ coupling, while the $\\phi$ couples to various combinations of LQ\/DQ states exactly as in the KSVZ scenarios. A number of peculiarities are worth mentioning though:\n\n\\begin{enumerate}\n\\item The symmetry patterns are more difficult to analyze in the DFSZ case because the PQ and hypercharge symmetries are entangled, see Eq.~(\\ref{DFSZScalars}). Thus, further entangling $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ with the $U(1)$s associated to $H_{u}$ and $H_{d}$ rephasing blurs the picture completely. In practice, the PQ charges of $\\phi$, $H_{u}$, and $H_{d}$ are always fixed to $PQ(H_{u})=x$, $PQ(H_{d})=-1\/x$, $PQ(\\phi)=\\left( x+1\/x\\right) \/2$, see Eq.~(\\ref{DFSZScalars}), no matter the amount of $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ that is entangled within $U(1)_{PQ}$.\n\n\\item Because $H_{u}$, and $H_{d}$ carry PQ charges, so does the SM fermions, even without the presence of LQ\/DQ states. As shown in Eq.~(\\ref{DFSZfermions}), these charges have ambiguities reflecting the exact accidental symmetries. Thus, any entanglement of $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ with $U(1)_{PQ}$ will be reflected in that arbitrariness. Typically, only one free parameter will remain instead of the $\\beta$ and $\\gamma$ parameters of Eq.~(\\ref{DFSZfermions}). Thus, looking at this remaining arbitrariness permits to identify the combination of $\\beta$ and $\\gamma$, i.e., $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$, that has been spontaneously broken.\n\n\\item Because $U(1)_{PQ}$ has always a component within $U(1)_{Hu}\\otimes U(1)_{Hd}$, the PQ charge of the SM fermions are never fully aligned with some combinations of $\\mathcal{B}$ and $\\mathcal{L}$. As a result, LQ states are often restricted to couple to only a single SM fermion LQ or DQ pair. For example, the gauge symmetries allow both $S_{2}^{1\/3}\\bar{d}_{R}\\ell_{L}$ and $S_{2}^{1\/3}\\bar{q}_{L}\\nu_{R}$, but the PQ charge do not since $PQ(\\bar{d}_{R}\\ell_{L})=\\gamma-\\beta+1\/x$ and $PQ(\\bar{q}_{L}\\nu_{R})=\\gamma-\\beta+x$, and this is true independently of $\\beta$ and $\\gamma$. In some cases, this actually makes the choice of LQ\/DQ couplings more natural than in the KSVZ case, since once some of them are selected, the others are immediately forbidden.\n\n\\item With $H_{u,d}$ at hand, many new couplings to LQ\/DQ states are a priori possible already in the scalar potential. For instance, replacing $H$ by $H_{u}$ or $H_{d}$ in any of the couplings in Eqs.~(\\ref{LQOpsB2}) or~(\\ref{LQOpsBL3}) would couple the axion to $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating operators. However, these situations correspond to breaking $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ at the electroweak scale, by entangling them with $U(1)_{Y}$. Indeed, $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating operators would involve $\\eta_{u}$ or $\\eta_{d}$ (the pseudoscalar components of $H_{u}$ and $H_{d}$), and thus also the Would-be Goldstone associated to $U(1)_{Y}$ since $G^{0}\\sim v_{u}\\eta_{u}+v_{d}\\eta_{d}$. The axion has only tiny $\\eta_{u}$ and $\\eta_{d}$ components, see Eq.~(\\ref{DFSZA0}). Turning on some $H_{u}^{\\dagger}H_{d}S_{j}^{\\dagger}S_{i}$ couplings would prevent any $G^{0}$ coupling, but would similarly lead to tiny axion couplings via its $\\cos\\beta\\eta_{u}-\\sin\\beta\\eta_{d}$ component. For these reasons, all the scenarios discussed below start from coupling $\\phi$ to the LQ\/DQ states, so that $\\mathcal{B}$ and\/or $\\mathcal{L}$ are broken at the PQ scale and the axion inherits some large $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating couplings. These scenarios are thus constructed exactly as in the KSVZ case.\n\\end{enumerate}\n\nAfter these general comments, let us briefly go through each of the $\\mathcal{B}$ and\/or $\\mathcal{L}$ spontaneous breaking scenario.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}+\\mathcal{L}$}\n\nBy analogy with the KSVZ scenario, Eq.~(\\ref{LagrKSVZ1a}), let us take%\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{8\/3}\\bar\n{d}_{R}e_{R}^{\\mathrm{C}}+\\tilde{S}_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}%\nu_{R}+\\phi^{2}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}+h.c.\\ , \\label{LagrDFSZ1}%\n\\end{equation}\nwith $\\mathcal{L}_{\\mathrm{DFSZ}}$ given in Eq.~(\\ref{DFSZ0}). Solving for the $U(1)$ charges under the constraint that $PQ(H_{u})=x$, $PQ(H_{d})=-1\/x$, which fixes $PQ(\\phi)=\\left( x+1\/x\\right) \/2$, leaves a single under-determination. In this way, we identify the remaining symmetry as $U(1)_{\\mathcal{B}-\\mathcal{L}}$, with%\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{8\/3}$ & $\\tilde{S}_{1}^{8\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ &\n$\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{1}{x}-x\\rule[-0.14in]{0in}{0.36in}$ & $-2x$ & $0$ & $x$\n& $\\dfrac{1}{x}$ & $-\\dfrac{1}{x}-x$ & $-x$ & $-\\dfrac{1}{x}$\\\\\n$U(1)_{\\mathcal{B}-\\mathcal{L}}$ & $-2\/3$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$-1$ & $-1$ & $-1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nwith $\\phi$, $H_{u}$, and $H_{d}$ neutral under $U(1)_{\\mathcal{B}-\\mathcal{L}}$. This shows that $U(1)_{\\mathcal{B}+\\mathcal{L}}\\subset U(1)_{PQ}\\subset U(1)_{Hu}\\otimes U(1)_{Hd}\\otimes U(1)_{\\mathcal{B}}\\otimes U(1)_{\\mathcal{L}}$ is spontaneously broken. Note well that the quoted $U(1)_{PQ}$ charges are just one possible choice, since $U(1)_{\\mathcal{B}-\\mathcal{L}}$ remains as an ambiguity. We could also have written the charges as\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{8\/3}$ & $\\tilde{S}_{1}^{8\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ &\n$\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{1}{x}-x-2\\xi\\rule[-0.14in]{0in}{0.36in}$ & $-2x-2\\xi$ &\n$\\xi$ & $x+\\xi$ & $\\dfrac{1}{x}+\\xi$ & $-\\dfrac{1}{x}-x-3\\xi$ & $-x-3\\xi$ &\n$-\\dfrac{1}{x}-3\\xi$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nwith $\\xi$ the free parameter corresponding to $U(1)_{\\mathcal{B}-\\mathcal{L}}$. We can also see that this corresponds to Eq.~(\\ref{DFSZfermions}) with $\\beta=\\xi$ and $\\gamma=-\\dfrac{1}{x}-x-3\\xi$. This shows that the dimension-five axion to gauge boson couplings are unaffected by the LQ\/DQ, since they are independent of $\\beta$ and $\\gamma$. Also, one should not conclude that the axion does not couple to $q_{L}$, even though that coupling is absent from the axion effective Lagrangian since $PQ(q_{L})$ is set to zero.\n\nConcerning the axion $\\mathcal{B}+\\mathcal{L}$ violating operator, the same effective interactions arises as in the KSVZ scenario, see Eq.~(\\ref{EffHKSVZ1}). This is evident from Fig.~\\ref{Fig2}, which is independent of how the axion emerges. The only difference is that the pseudoscalar component of $\\phi$ is not purely the axion, but this is only a totally negligible $\\mathcal{O}(v_{EW}\/vs)$ effect, see Eq.~(\\ref{DFSZA0}).\n\nFinally, exactly as in the KSVZ scenario, the remaining $U(1)_{\\mathcal{B}-\\mathcal{L}}$ freedom permits to set up a PQ-induced seesaw mechanism by adding $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ or $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$. In both cases, this simply fixes the parameter $\\xi$ and removes the remaining $U(1)_{\\mathcal{B}-\\mathcal{L}}$ ambiguity. Yet, the final PQ charges do not reflect at all the peculiar symmetry breaking pattern, with $U(1)_{\\mathcal{B}+\\mathcal{L}}$ and $U(1)_{\\mathcal{L}}$ being separately, but concurrently, spontaneously broken at the PQ scale. By the way, exactly the same PQ charges arise if $\\phi^{2}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}$ is replaced by $H_{d}^{\\dagger}H_{u}S_{1}^{8\/3\\dagger}\\tilde{S}_{1}^{8\/3}$, though as discussed before, the symmetry breaking chain is very different, as are the axion couplings.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}-\\mathcal{L}$}\n\nPursuing our adaptation of the KSVZ scenario, let us consider now%\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{2}^{7\/3}\\bar{u}_{R}\\ell_{L}+\\phi H_{u}%\nS_{2}^{7\/3\\dagger}S_{1}^{4\/3}+h.c.\\ . \\label{LagrDFSZ2}%\n\\end{equation}\nBoth fermionic couplings of $S_{2}^{7\/3}$ cannot be present at the same time for the PQ symmetry to exist, so we choose to keep $S_{2}^{7\/3}\\bar{u}_{R}\\ell_{L}$ and discard $S_{2}^{7\/3}\\bar{q}_{L}e_{R}$. Also, we introduced $H_{u}$ in the quartic scalar coupling, but could equally well have used $H_{d}$. From this Lagrangian, the PQ charges are found to be\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{4\/3}$ & $S_{2}^{7\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ &\n$e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $-\\dfrac{2}{x}\\rule[-0.14in]{0in}{0.36in}$ & $\\dfrac{3x}%\n{2}-\\dfrac{3}{2x}$ & $0$ & $x$ & $\\dfrac{1}{x}$ & $\\dfrac{3}{2x}-\\dfrac{x}{2}$\n& $\\dfrac{5}{2x}-\\dfrac{x}{2}$ & $\\dfrac{3}{2x}+\\dfrac{x}{2}$\\\\\n$U(1)_{\\mathcal{B}+\\mathcal{L}}$ & $-2\/3$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nSo, $U(1)_{\\mathcal{B}-\\mathcal{L}}$ is spontaneously broken at the PQ scale, but $U(1)_{\\mathcal{B}+\\mathcal{L}}$ remains. As before, we could rewrite the PQ charge introducing a free parameter to reflect the exact $U(1)_{\\mathcal{B}+\\mathcal{L}}$ symmetry, hence one should not interpret $PQ(q_{L})=0$ as meaning it has no coupling to the axion. The $U(1)_{\\mathcal{B}+\\mathcal{L}}$ ambiguity can then be used to allow for a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ or $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling, and set up the seesaw mechanism.\n\nThe final $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operator is again one of the dimension-seven operators listed in Table~\\ref{TableLQBL}. Note, though, that because the PQ symmetry restricts the LQ couplings to SM fermions, only a single operator is induced. This is a generic characteristic of the DFSZ implementation: compared to the KSVZ case, it is more restrictive. Phenomenologically, this could show up as definite decay patterns for the proton (if ever observed). Starting from Eq.~(\\ref{LagrDFSZ2}), the operator arising at tree level is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac{1}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\phi H_{u}\\bar{d}_{R}d_{R}^{\\mathrm{C}}\\bar\n{u}_{R}\\ell_{L}+h.c.\\ ,\n\\end{equation}\nNote that some other gauge and PQ invariant operators may arise at higher loops via Yukawa insertions, but those are more suppressed. The leading proton decay operator is thus proportional to $v_{\\phi}v_{u}\/m_{S}^{4}$, and the constraints are similar as in the KSVZ scenario. Concerning the axion coupling, notice that\n\\begin{equation}\n\\phi H_{u}\\ell_{L}\\rightarrow\\frac{1}{2}v_{u}v_{\\phi}\\exp i\\left( \\frac{\\eta\n_{u}}{v_{u}}+\\frac{v_{\\phi}}{v_{\\phi}}\\right) \\nu_{L}\\ ,\n\\end{equation}\nso the combination that occurs in the effective operator is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{v_{\\phi}v_{u}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\left( 1+i\\frac{G^{0}}{v}%\n+i\\frac{a^{0}}{v_{\\phi}}\\frac{3x^{2}+1}{x^{2}+1}+i\\frac{\\pi^{0}}{v}x\\right)\n\\bar{d}_{R}d_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{L}\\ .\n\\end{equation}\nFor comparison, the $\\mu H_{u}S_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ coupling would lead to\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac{\\mu\nv_{u}}{m_{S}^{2}m_{\\tilde{S}}^{2}}\\left( 1+i\\frac{G^{0}}{v}+i\\frac{a^{0}%\n}{v_{\\phi}}\\frac{2x^{2}}{x^{2}+1}+i\\frac{\\pi^{0}}{v}x\\right) \\bar{d}_{R}%\nd_{R}^{\\mathrm{C}}\\bar{u}_{R}\\nu_{L}\\ ,\n\\end{equation}\nwith $\\mu$ some mass scale. The $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operator arises at the $v_{\\phi}$ scale from $\\phi H_{u}S_{2}^{7\/3\\dagger}S_{1}^{4\/3}$, but at a lower scale from $H_{u}S_{2}^{7\/3\\dagger}S_{1}^{4\/3}$ since we would expect $\\mu$ to be at the LQ\/DQ scale, $\\mu\\sim m_{S}$, or even at the electroweak scale, $\\mu\\sim v$. Note that in both cases, the $G^{0}$ enters as expected for a would-be Goldstone, and would disappear in the unitary gauge. The axion coupling is $\\mathcal{O}(v_{EW}\/v_{\\phi})$ compared to the four-fermion operator, exactly like in the KSVZ scenario.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}$}\n\nNeutron-antineutron oscillations can be induced in the same way in the DFSZ and KSVZ models, see Fig.~\\ref{Fig4a}. Starting with\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{4\/3}\\bar\n{d}_{R}^{\\mathrm{C}}d_{R}+S_{1}^{8\/3}\\bar{u}_{R}^{\\mathrm{C}}u_{R}+\\phi\nS_{1}^{4\/3}S_{1}^{4\/3}S_{1}^{8\/3}+h.c.\\ , \\label{LagrDFSZ3}%\n\\end{equation}\nwhere $S_{1}^{4\/3}\\sim(\\mathbf{3},\\mathbf{1},+4\/3)$ and $S_{1}^{8\/3}\\sim(\\mathbf{\\bar{6}},\\mathbf{1},-8\/3)$, we get the PQ charges%\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{4\/3}$ & $S_{1}^{8\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ &\n$e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{x}{2}-\\dfrac{5}{6x}\\rule[-0.14in]{0in}{0.36in}$ &\n$\\dfrac{7}{6x}-\\dfrac{3x}{2}$ & $-\\dfrac{x}{4}-\\dfrac{7}{12x}$ & $\\dfrac\n{3x}{4}-\\dfrac{7}{12x}$ & $\\dfrac{5}{12x}-\\dfrac{x}{4}$ & $0$ & $\\dfrac{1}{x}$\n& $x$\\\\\n$U(1)_{\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThus, $U(1)_{\\mathcal{B}}$ is broken spontaneously, but $U(1)_{\\mathcal{L}}$ stays exact. The phenomenology is the same as in the KSVZ model, see Fig.~\\ref{Fig4a} and Eq.~(\\ref{ScaleNN}). Majorana neutrino masses can be generated spontaneously with $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$, but not with $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ as the PQ charges of the leptons would then allow for the $S_{1}^{4\/3}\\bar{u}_{R}\\nu_{R}^{\\mathrm{C}}$ coupling, and thereby to tree-level proton decay.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}\\pm3\\mathcal{L}$}\n\nThe last two scenarios are those producing exotic $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm3)$ proton decay operators. Let us start with the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ case, and the Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}}=\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}^{2\/3}\\bar\n{q}_{L}\\ell_{L}^{\\mathrm{C}}+V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell\n_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})+\\phi^{\\dagger}\nS_{1}^{2\/3}V_{2,\\mu}^{1\/3}V_{2}^{1\/3,\\mu}+h.c.\\ . \\label{LagrDFSZ4}%\n\\end{equation}\nThe $S_{1}^{2\/3}(\\bar{d}_{R}\\nu_{R}^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}})$ and $S_{1}^{2\/3}\\bar{q}_{L}\\ell_{L}^{\\mathrm{C}}$ couplings cannot both be present, and we take only the latter, while the $V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu _{R}^{\\mathrm{C}})$ couplings are compatible with each other. The $U(1)$ charges are then\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccc}\\hline\n& $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$\n& $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}$ & $\\dfrac{1}{6x}-\\dfrac{x}{2}\\rule[-0.14in]{0in}{0.36in}$ &\n$\\dfrac{x}{2}+\\dfrac{1}{6x}$ & $0$ & $x$ & $\\dfrac{1}{x}$ & $\\dfrac{1}%\n{6x}-\\dfrac{x}{2}$ & $\\dfrac{7}{6x}-\\dfrac{x}{2}$ & $\\dfrac{1}{6x}+\\dfrac\n{x}{2}$\\\\\n$U(1)_{3\\mathcal{B}-\\mathcal{L}}$ & $0$ & $0$ & $1$ & $1$ & $1$ & $-1$ & $-1$\n& $-1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThe $U(1)_{3\\mathcal{B}-\\mathcal{L}}$ symmetry remains, and its orthogonal combination $U(1)_{\\mathcal{B}+3\\mathcal{L}}$ is spontaneously broken. Dimension-nine $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ proton decay operators thus appear at the low scale (as well as semileptonic $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,0)$ operators since these charges allow for the $D^{\\mu}HS_{1}^{2\/3}V_{2,\\mu}^{1\/3\\dagger}$ couplings). Once more, there is enough room for a seesaw mechanism with $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ and\/or $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$. Depending on the LQ couplings of $S_{1}^{2\/3}$, it is always possible to choose the seesaw operator that sets PQ charges forbidding the DQ couplings of both $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$, and thus proton decay via dimension-six operators.\n\nConcerning the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ operators, we start from\n\\begin{align}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}} & =\\mathcal{L}_{\\mathrm{DFSZ}}+S_{2}%\n^{1\/3}\\bar{d}_{R}\\ell_{L}+V_{1,\\mu}^{2\/3}\\bar{d}_{R}\\gamma^{\\mu}\\nu\n_{R}\\nonumber\\\\\n& +\\phi(H_{u}S_{2}^{1\/3\\dagger}S_{1}^{2\/3}+H_{d}^{\\dagger}V_{1}%\n^{2\/3\\dagger,\\mu}V_{2,\\mu}^{1\/3}+S_{1}^{2\/3\\dagger}V_{2,\\mu}^{1\/3\\dagger}\nV_{2}^{1\/3,\\mu\\dagger})+h.c.\\ , \\label{LagrDFSZ5}%\n\\end{align}\nwith the $S_{2}^{1\/3}\\bar{q}_{L}\\nu_{R}$ removed. Several choices are possible for introducing the doublets $H_{u}$ and $H_{d}$ in these couplings, and we opt for the one most symmetrical with the Yukawa couplings, see Eq.~(\\ref{DFSZ0}). Only one of the fermionic couplings of $S_{2}^{1\/3}$ can be turned on, and we choose $S_{2}^{1\/3}\\bar{d}_{R}\\ell_{L}$. Then, the $U(1)$\ncharges are found to be\n\\begin{equation}%\n\\begin{tabular}\n[c]{ccccccccccc}\\hline\n& $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $S_{2}^{1\/3}$ & $V_{1,\\mu}^{2\/3}$ &\n$q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{PQ}\\rule[-0.14in]{0in}{0.39in}$ & $\\dfrac{x^{2}+5}{6x}$ & $\\dfrac\n{x^{2}-1}{6x}$ & $\\dfrac{5x^{2}+4}{3x}$ & $\\dfrac{2x^{2}+4}{3x}$ & $0$ & $x$ &\n$\\dfrac{1}{x}$ & $\\dfrac{5x^{2}+1}{-3x}$ & $\\dfrac{5x^{2}-2}{-3x}$ &\n$\\dfrac{2x^{2}+1}{-3x}$\\\\\n$U(1)_{3\\mathcal{B}+\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $1$ & $1$ & $1$ &\n$1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThis time, $U(1)_{3\\mathcal{B}+\\mathcal{L}}$ remains and $U(1)_{\\mathcal{B}-3\\mathcal{L}}$ is spontaneously broken. The induced operator, from a process easily adapted from that of Fig.~\\ref{Fig4b}, is\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)}^{eff}=\\frac\n{\\phi^{4}(H_{d}^{\\dagger}H_{d}^{\\dagger}H_{u})}{m_{S1\/3}^{2}m_{V2\/3}^{4}m_{S2\/3}^{2}%\nm_{V1\/3}^{4}}\\bar{d}_{R}\\ell_{L}\\bar{d}_{R}\\gamma_{\\mu}\\nu\n_{R}\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}\\ .\n\\end{equation}\nAgain, phenomenologically, there is not much difference between the DFSZ and KSVZ implementation.\n\n\\subsection{Axion-induced proton decay and neutron-antineutron oscillations\\label{SecSpont}}\n\nIn both the KSVZ and DFSZ cases, we can induce spontaneously proton decay or neutron-antineutron oscillations. But, in all the scenarios discussed up to now, the processes involving the axion were $\\mathcal{O}(v_{EW}\/v_{\\phi})$ with respect to that without it. The reason is of course that in all cases, some coupling of $\\phi$ to the LQ\/DQ states was introduced, and $\\phi=(v_{\\phi}+\\rho)\\exp(i\\eta_{\\phi}\/v_{\\phi})\\approx v_{\\phi}+\\rho+ia^{0}$ (see Eq.~(\\ref{ExampleSSS})). The purpose here is to kill off the leading term, leaving only axion-induced $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating processes. The only way to achieve this is to consider derivative couplings of $\\phi$ to pairs of LQ\/DQs, and there are only three renormalizable options\n\\begin{equation}\n\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}\\ ,\\ \\partial_{\\mu}\\phi\nS_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}\\ ,\\ \\partial_{\\mu}\\phi S_{1}^{4\/3\\dagger\n}V_{1}^{4\/3,\\mu}.\n\\label{DerScenars}\n\\end{equation}\nIn these cases, the axion enters as $\\partial_{\\mu}\\phi\\approx\\partial_{\\mu}\\rho+i\\partial_{\\mu}a^{0}$, without a leading term tuned by $v_{\\phi}$. Though we have not attempted at constructing UV complete models generating such interactions, their structure is evocative of that which could arise if both $\\phi$ and scalar LQ\/DQ were somehow related to the fields giving masses to the vector LQ\/DQ. Such a situation can happen in simple GUT models: In Ref.~\\cite{Wise:1981ry} for example, $\\phi$ is identified with the phase of the complex $H_{24}$ field breaking $SU(5)$ down to the SM gauge group. Note, though, that the PQ breaking scale and the LQ\/DQ mass scale would be related in such models. In the present section, the two will be kept independent, with the latter usually much lower than the former.\n\nLet us see which symmetry breaking patterns can be achieved with these building blocks. We will use the KSVZ setting throughout as the alignments of the PQ with some combination of $\\mathcal{B}$ and $\\mathcal{L}$ are manifest, but the adaptation to the DFSZ scenario is immediate. Also, we will discard $\\Psi_{L,R}$ from the discussion. As in Sec.~\\ref{SecKSVZ}, their charge can always be set separately by introducing some couplings to the LQ\/DQ.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}-\\mathcal{L}$}\n\nThe scenarios with $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators are immediately obtained using any one of the three couplings in Eq.~(\\ref{DerScenars}). For example, we can take\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}^{2\/3}(\\bar\n{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{1,\\mu}^{2\/3}%\n\\bar{d}_{R}\\gamma^{\\mu}\\nu_{R}+\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}%\nV_{1}^{2\/3,\\mu}+h.c.\\ , \\label{LagrSSB1}%\n\\end{equation}\nand get two active $U(1)$s, with charges\n\\begin{equation}%\n\\begin{tabular}[c]{cccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{1,\\mu}^{2\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ &\n$\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $0$ &\n$0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $1$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $1$ & $1$ &\n$1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThus, $\\phi$ spontaneously breaks $U(1)_{\\mathcal{B}-\\mathcal{L}}$, leaving $\\mathcal{B}+\\mathcal{L}$ as an exact accidental symmetry. As before, we can add a $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling to generate neutrino masses and forbid the LQ couplings of $S_{1}^{2\/3}$.\n\nThe situation starting from the other derivative interaction is similar, hence we can generate:\n\\begin{align}\n\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu} & \\rightarrow\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{2}m_{V}^{2}}\\partial_{\\mu}\\phi(\\bar{q}_{L}^{\\mathrm{C}}q_{L}%\n+\\bar{d}_{R}^{\\mathrm{C}}u_{R})\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}\\nu\n_{R}^{\\mathrm{C}}\\ ,\\\\\n\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu} & \\rightarrow\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{2}m_{V}^{2}}\\partial_{\\mu}\\phi\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu\n}q_{L}(\\bar{d}_{R}^{\\mathrm{C}}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}^{\\mathrm{C}%\n}\\nu_{R}^{\\mathrm{C}})\\ ,\\\\\n\\partial_{\\mu}\\phi S_{1}^{4\/3\\dagger}V_{1}^{4\/3,\\mu} & \\rightarrow\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{2}m_{V}^{2}}\\partial_{\\mu}\\phi\\bar{d}_{R}^{\\mathrm{C}}d_{R}(\\bar\n{u}_{R}^{\\mathrm{C}}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}}+\\bar{d}_{R}^{\\mathrm{C}%\n}\\gamma^{\\mu}e_{R}^{\\mathrm{C}}+\\bar{q}_{L}^{\\mathrm{C}}\\gamma^{\\mu}\\ell\n_{L}^{\\mathrm{C}})\\ .\n\\end{align}\nAll these situations induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators, see Fig.~\\ref{Fig5}$a$, but necessarily involving the axion, with for example\n\\begin{align}\n\\frac{\\partial_{\\mu}\\phi}{m_{S}^{2}m_{V}^{2}}\\bar{d}_{R}\\gamma^{\\mu}%\nq_{L}^{\\mathrm{C}}\\bar{d}_{R}\\ell_{L}+h.c. & \\rightarrow\\frac{1}{m_{S}^{2}%\nm_{V}^{2}}\\partial_{\\mu}a^{0}\\bar{d}_{R}\\gamma^{\\mu}q_{L}^{\\mathrm{C}}\\bar\n{d}_{R}\\ell_{L}+h.c. \\nonumber \\\\ \n & \\rightarrow\\frac{\\Lambda_{QCD}^{3}}{m_{S}^{2}m_{V}^{2}%\n}(\\partial_{\\mu}a^{0}\\bar{p}\\gamma^{\\mu}\\left( 1-\\gamma^{5}\\right)\n\\ell + \\partial_{\\mu}a^{0}\\bar{n}\\gamma^{\\mu}\\left( 1-\\gamma^{5}\\right)\n\\nu + ... + h.c.)\\ ,\n\\label{OpsAPL}\n\\end{align}\nwhere (...) denotes operators with additional light mesons. Given the proton decay bounds, and taking $\\Lambda_{QCD}\\approx 300$~MeV, this imposes a quite high bound $m_{S}\\approx m_{V}>10^{4}$~TeV, close to the PQ breaking scale and quite lower than the GUT scale. With those values, such $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators cannot affect significantly the phenomenology of the axion, as its coupling to photons or gluons remain much larger.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.5766in,width=4.7547in]{Fig5.jpg}%\n\\caption{The $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators involving one ($a.$) and two axions ($b.$).}%\n\\label{Fig5}%\n\\end{center}\n\\end{figure}\n\nThe situation can be different for an axion-like particle (ALP) with a mass above that of the proton. If the mass is just slightly above but below that of the neutron, $m_{p}-m_{e} 900\\ \\text{GeV} \\left( \\frac{10^{16}\\ \\text{GeV}}{f_a} \\right)^{1\/4}\\;.\n\\end{equation}\nPlugging this in Eq.~(\\ref{neutdec}), the branching ratio for $n\\rightarrow a^{0}+\\nu$ is at around $1\\%$ provided $f_{a}$ is pushed at the GUT scale, $f_a \\approx 10^{16}\\ \\text{GeV}$, see Fig.~\\ref{FigNeut}. Note that for that value of $f_{a}$, the ALP still decays mostly into $\\gamma\\gamma$ as $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell})>\\Gamma(a^{0}\\rightarrow\\gamma\\gamma)$ requires $f_{a}$ about an order of magnitude larger. The $p\\rightarrow\\ell(a^{0\\ast}\\rightarrow\\gamma\\gamma)$ decay can happen only for $\\ell=e,\\mu$, but the underlying LQ couplings could actually exhibit non-trivial flavor hierarchies. If they couple preferentially to the $\\tau$, then proton decay would be forced to occur via more suppressed channels, e.g. via $p\\rightarrow\\pi\\nu_{\\tau}(a^{0\\ast}\\rightarrow\\gamma\\gamma)$, and $f_{a}$ could be brought down by a few orders of magnitude. Thus, an ALP could realize the scenario proposed in Ref.~\\cite{NeutronTau} to solve the neutron lifetime puzzle, though it does not alleviate its inherent fine tuning of the dark particle mass. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=11cm]{Neutron.jpg}%\n\\caption{Evolution of the $a^{0}\\rightarrow\\gamma\\gamma$, $a^{0}\\rightarrow p\\bar{\\ell}$, $n\\rightarrow a^{0}+\\nu$ and $p\\rightarrow\\ell(a^{0\\ast}\\rightarrow\\gamma\\gamma)$ widths as functions of $f_a$. The dashed line indicates the observed neutron lifetime discrepancy, $\\Gamma_{bottle}-\\Gamma_{beam} \\approx 7.1\\times 10^{-30}$~GeV~\\cite{NeutronTau}. The ALP mass is kept fixed at $m_a =0.9384$~GeV. The LQ\/DQ mass is adjusted so that $\\tau \\left( p\\rightarrow\\ell(a^{0\\ast}\\rightarrow\\gamma\\gamma) \\right) = 10^{32}$~yr. For the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operator, Eq.~(\\ref{OpsAPL}), $m_{S}\\approx m_{V}$ then follows the indicated line (Case I), and must be a bit below the TeV to reproduce the observed $\\Gamma_{bottle}-\\Gamma_{beam}$ discrepancy. Concerning the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ scenario, for which leptons and antileptons should be interchanged, the extra factor of $f_a$ in Eq.~(\\ref{OpsAPantiL}) sets $m_{S}\\approx m_{V}\\approx 90$~TeV independently of $f_a$ (Case II).}\n\\label{FigNeut}%\n\\end{center}\n\\end{figure}\n\n\nFor heavier ALPs, neutron decay is kinematically closed, and the $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu})>\\Gamma(a^{0}\\rightarrow\\gamma\\gamma)$ pattern can arise for lower $f_{a}$ values (though still very large from the axion point of view), with for example $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu})>\\Gamma(a^{0}\\rightarrow\\gamma\\gamma)$ for $f_{a}>10^{13}$ GeV if $m_{a}=100~$GeV. This, however, requires also to boost the $a\\rightarrow p\\bar{\\ell},n\\bar{\\nu}$ rate by allowing light LQ\/DQ at around the TeV scale. Even if $\\Gamma(a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu})$ does not dominate, such decay channels could have some cosmological implications. From a baryogenesis point of view, it is interesting to remark that the present scenario has all the necessary ingredients. Provided the LQ\/DQ couple to more than one SM fermion states, several operators will simultaneously contribute to the $a^{0}\\rightarrow p\\bar{\\ell},n\\bar{\\nu}$ and $a^{0}\\rightarrow\\bar{p}\\ell,\\bar{n}\\nu$ decay processes, and since the LQ\/DQ couplings to SM fermions are a priori complex, their rates would be different (slightly, as rescattering is required). In this picture, note that if there are several LQ\/DQ states with a non-trivial mass spectrum, their decay chains may first generate an asymmetry when $m_{S,V}>m_{a}$~\\cite{Arnold:2012sd}, but it would be washed out and regenerated at a lower scale by the ALP decays. Whether this mechanism is sufficient to generate the observed baryon asymmetry is left for a future study.\n\nTo close this section, let us mention another scenario in which $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ operators require two derivative couplings, and proton decay is associated to axion pair production. Specifically, if we start from\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} & =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R}%\n)+S_{2}^{1\/3}(\\bar{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})\\nonumber\\\\\n& \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}+\\partial\n_{\\mu}\\phi V_{2}^{1\/3,\\mu\\dagger}S_{2}^{1\/3}+\\phi H^{\\dagger}V_{1,\\mu\n}^{2\/3\\dagger}V_{2}^{1\/3,\\mu}+h.c.\\ ,\n\\end{align}\nthe mixing between $S_{1}^{2\/3}$ and $S_{2}^{1\/3}$ can only occur through that of $V_{1,\\mu}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$. Specifically, with this specific choice of couplings,\n\\begin{equation}%\n\\begin{tabular}[c]{cccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{1,\\mu}^{2\/3}$ & $S_{2}^{1\/3}$ & $V_{2,\\mu\n}^{1\/3}$ & $q_{L}$ & $u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}%\n$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1\/3$ & $-2\/3$ & $-1\/3$ & $1\/3$ & $0$ & $1\/3$ & $1\/3$\n& $1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $1\/3$ & $0$ & $-1\/3$ & $-1$ & $-2\/3$ & $0$ & $0$ & $0$\n& $1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nThanks to these charges, which crucially follow from whether $\\phi$ or $\\phi^{\\dagger}$ are introduced in the couplings, no other SM fermion couplings of the LQ\/DQ, nor any other renormalizable couplings among the LQ\/DQ, is allowed. Turning on $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ does not change this picture, except for the operator $\\phi^{\\dagger}S_{1}^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}$. This is quite natural looking at the Lagrangian, since $\\phi^{\\dagger}S_{1}^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ followed by $S_{1}^{2\/3}\\rightarrow\\bar{q}_{L}^{\\mathrm{C}}q_{L}$ and $S_{2}^{1\/3}\\rightarrow\\bar{q}_{L}\\nu_{R}$ permits to recover $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ by closing the $q_{L}$ loops. As the $\\phi^{\\dagger}S_{1}^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}$ and $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ have the same quantum numbers, both carrying $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$, they are both able to generate neutrino masses only, and do not affect the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ pattern.\n\nThe leading operator for proton decay is now (Fig.~\\ref{Fig5}$b$):%\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{1}{m_{S}^{4}m_{V}^{4}}\\phi\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi H^{\\dagger\n}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})(\\bar{d}%\n_{R}^{\\mathrm{C}}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}^{\\mathrm{C}}\\nu\n_{R}^{\\mathrm{C}})\\ ,\n\\end{equation}\nand it contains in particular\n\\begin{equation}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)}^{eff}=\\frac\n{v_{\\phi}v\\Lambda_{QCD}^{3}}{m_{S}^{4}m_{V}^{4}}\\partial_{\\mu}a^{0}%\n\\partial^{\\mu}a^{0}\\bar{p}\\left( 1-\\gamma^{5}\\right) \\ell+...+h.c.\\ .\n\\end{equation}\nPhenomenologically, proton decay is suppressed, even for relatively low $m_{S}\\approx m_{V}$ of $\\mathcal{O}(10~$TeV$)$. On the other hand, if $a^{0}$ is an ALP with twice its mass above the proton but below the neutron mass, this setting is less interesting because the LQ\/DQ masses need to be too low to reach $B(n\\rightarrow a^{0}+a^{0}+\\bar{\\nu})\\approx1\\%$. Whether ALP or axions, the cosmological implications of this scenario would be worth further study though, as the consequences of opening up (possibly CP-violating) $a^{0}+p\\leftrightarrow a^{0}+\\ell$ and $a^{0}+\\bar{p}\\leftrightarrow a^{0}+\\bar{\\ell}$ scattering processes could provide a new baryogenesis mechanism.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}+\\mathcal{L}$}\n\nTo attain $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators, the trick is to start from the previous scenario, but use some additional LQ couplings to switch $\\mathcal{L}$ by two units. Specifically, we can consider%\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} & =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu\n}\\nu_{R}^{\\mathrm{C}})\\nonumber\\\\\n& \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}+\\phi\nH^{\\dagger}V_{1,\\mu}^{2\/3\\dagger}V_{2}^{1\/3,\\mu}+h.c.\\ . \\label{LagrSSB2}%\n\\end{align}\nProvided $V_{1,\\mu}^{2\/3}$ has no couplings to SM fermions, and only those two interactions among $\\phi$ and the LQ\/DQ are present, two $U(1)$s are present in the Lagrangian, with charges\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{1,\\mu}^{2\/3}$ & $V_{2}^{1\/3,\\mu}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1\/2$ & $-2\/3$ & $-1\/6$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$1\/3$ & $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-1\/2$ & $0$ & $1\/2$ & $1$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nSo, the $U(1)_{\\mathcal{B}+\\mathcal{L}}$ symmetry is spontaneously broken, while $\\mathcal{B}-\\mathcal{L}$ remains. If neutrino masses are generated by adding the $\\phi\\bar{\\nu}_{R}\\nu_{R}^{\\mathrm{C}}$ coupling, the remaining exact $U(1)$ symmetry suffices to keep off all other interactions among $\\phi$ and the LQ\/DQ, as well as the LQ\/DQ couplings to SM fermions not already present in the Lagrangian, except for a $\\phi^{\\dagger}S_{1}^{2\/3}V_{2}^{1\/3,\\mu}V_{2,\\mu}^{1\/3}$ which carries the same quantum number as $\\phi^{\\dagger}{\\bar{\\nu}_{R}}^{\\mathrm{C}}\\nu_{R}$. Neither is able to open new $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ patterns for proton decay.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.3517in,width=2.0254in]{Fig5b.jpg}\n\\caption{Axion-induced $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ operators.}\n\\label{Fig5b}\n\\end{center}\n\\end{figure}\n\nThe leading $\\mathcal{B}+\\mathcal{L}$ violating operator is (see Fig.~\\ref{Fig5b})\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)}^{eff} & =\\frac\n{1}{m_{S}^{2}m_{V}^{4}}\\phi\\partial_{\\mu}\\phi H^{\\dagger}(\\bar{q}%\n_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})(\\bar{u}_{R}^{\\mathrm{C}%\n}\\gamma^{\\mu}\\ell_{L}+\\bar{q}_{L}^{\\mathrm{C}}\\gamma^{\\mu}\\nu_{R})\\nonumber\\\\\n& \\rightarrow\\frac{v_{EW}v_{\\phi}}{m_{S}^{2}m_{V}^{4}}\\partial_{\\mu}a^{0}(\\bar\n{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})(\\bar{u}%\n_{R}^{\\mathrm{C}}\\gamma^{\\mu}e_{L}+\\bar{d}_{L}^{\\mathrm{C}}\\gamma^{\\mu}\\nu\n_{R})\\ .\n\\label{OpsAPantiL}\n\\end{align}\nPhenomenologically, thanks to the $v_{EW}v_{\\phi}$ from the $\\phi HV_{2,\\mu}^{1\/3\\dagger}V_{1}^{2\/3,\\mu}$ coupling, the LQ\/DQ scale can be lower by about an order of magnitude without violating proton decay bounds. For ALPs, the main difference with the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ scenario is that $f_{a}=v_{\\phi}$ occurs in the $n\\rightarrow a^{0}\\bar{\\nu}$ and $a^0\\rightarrow p\\ell$ amplitudes, but cancels out from the $p\\rightarrow\\bar{\\ell}(a^0\\rightarrow\\gamma\\gamma)$ rate. This means that $m_{S}\\approx m_{V}$ cannot be as low as before, but must above $90$~TeV. Yet, this high scale is compensated in the $n\\rightarrow a^{0}\\bar{\\nu}$ rate by the $v_{\\phi}$ factor, so its branching ratio can still reach $\\mathcal{O}(1\\%)$. Actually, the dependencies of the various rates on $f_a$ stays exactly as depicted in Fig.~\\ref{FigNeut}, but for $m_{S}\\approx m_{V}\\approx 90$~TeV.\n\nNote, finally, that $\\mathcal{B}+\\mathcal{L}$ violating operators are not easily forced to involve pairs of axions. The pattern of LQ\/DQ couplings to SM fermions, and their hypercharge, does not leave many options if only renormalizable operators are allowed. The simplest we could find would require two different Higgs doublets, so would be suitable for the DFSZ model\n\\begin{align}\n\\mathcal{L}_{\\mathrm{DFSZ+LQ}} & =\\mathcal{L}_{\\mathrm{DFSZ}}+S_{1}%\n^{4\/3}\\bar{d}_{R}^{\\mathrm{C}}d_{R}+S_{1}^{2\/3}(\\bar{d}_{R}\\nu_{R}%\n^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}}+\\bar{q}_{L}\\ell_{L}^{\\mathrm{C}%\n})\\nonumber\\\\\n& \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{1}^{2\/3\\dagger}V_{1}^{2\/3,\\mu}+\\partial\n_{\\mu}\\phi S_{1}^{4\/3\\dagger}V_{1}^{4\/3,\\mu}+H_{u}^{\\dagger}H_{d}^{\\dagger\n}V_{1,\\mu}^{2\/3\\dagger}V_{1}^{4\/3,\\mu}+h.c.\\ .\n\\end{align}\nAs the phenomenology is similar as that for $\\mathcal{B}-\\mathcal{L}$ violating operators, we do not detail this situation further.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}$}\n\nGiven that we want to start from the derivative couplings, which are all quadratic in the LQ\/DQ, we will need to add at least some cubic interactions. This quickly increases the number of new state needed, and phenomenologically, the longest the chain, the smallest the predicted rate given that LQ\/DQ masses are at least of a few TeV.\n\nThe simplest processes correspond to the skeleton graph $\\partial_{\\mu}\\phi\\rightarrow X_{i}(X_{l}\\rightarrow X_{j}X_{k})$, with the final\n$X_{i}X_{j}X_{k}$ set allowing for $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ transitions, so $X_{i,j,k}=S_{1}^{y}$ or $V_{2}^{y}$ for some $y$. If $X_{l}$ is integrated out, the effective operator involves $\\partial_{\\mu}\\phi X_{i}X_{j}X_{k}$ plus some Higgs fields. The simplest such operators are of dimension six, and only seven of them are compatible with the gauge symmetries,\n\\begin{align}\n& \\partial_{\\mu}\\phi~H^{\\dagger}~(S_{1}^{2\/3}S_{1}^{4\/3}V_{2}^{1\/3,\\mu\n},\\ S_{1}^{4\/3}S_{1}^{4\/3}V_{2}^{5\/3,\\mu},\\ V_{2}^{1\/3,\\mu}V_{2,\\nu}%\n^{1\/3}V_{2}^{1\/3,\\nu})\\ ,\\\\\n& \\partial_{\\mu}\\phi~H~(V_{2}^{5\/3,\\mu}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}%\n,\\ S_{1}^{2\/3}S_{1}^{2\/3}V_{2}^{1\/3,\\mu},\\ S_{1}^{2\/3}S_{1}^{4\/3}%\nV_{2}^{5\/3,\\mu},\\ S_{1}^{4\/3}S_{1}^{8\/3}V_{2}^{1\/3,\\mu})\\ ,\n\\end{align}\nwhere $\\partial_{\\mu}\\phi$ could be replaced by $\\partial_{\\mu}\\phi^{\\dagger}$ wherever required. Starting from the three derivative interactions of Eq.~(\\ref{DerScenars}), there are several ways to reach these operators using a $HX_{l}X_{j}X_{k}$ or $H^{\\dagger}X_{l}X_{j}X_{k}$ coupling. Since $X_{l}=S_{2}^{y}$ or $V_{1}^{y}$, these operators alone cannot induce $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ processes. Further, if $X_{l}$ transforms as a $6$, it does not couple to SM fermions hence these operators cannot lead to proton decay either. If $X_{l}$ transforms as a $3$, one must make sure the PQ charges forbid $X_{l}\\rightarrow\\ell q$. All this nevertheless leaves many possible mechanisms, though many of them turn out to be essentially equivalent phenomenologically, so let us take an example.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=1.9787in,width=4.7158in]{Fig6.jpg}\n\\caption{One and two axion induced neutron-antineutron oscillation $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators.}\n\\label{Fig6}\n\\end{center}\n\\end{figure}\n\nConsider the Lagrangian\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} & =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\nonumber\\\\\n& +\\partial_{\\mu}\\phi V_{1}^{2\/3,\\mu\\dagger}S_{1}^{2\/3}+HV_{1,\\mu}^{2\/3}%\nS_{1}^{2\/3}V_{2}^{1\/3,\\mu}+h.c.\\ , \\label{LagrSSB3a}%\n\\end{align}\nwhere $S_{1}^{2\/3}$ and $V_{1}^{2\/3,\\mu}$ transform as $\\mathbf{3}$, but $V_{2}^{1\/3,\\mu}\\sim\\mathbf{\\bar{6}}$ since the final operator $\\partial_{\\mu}\\phi HS_{1}^{2\/3}S_{1}^{2\/3}V_{2}^{1\/3,\\mu}$ would cancel for $V_{2}^{1\/3,\\mu}\\sim\\mathbf{3}$. Dropping the $\\Psi_{L,R}$, as their charge can independently be fixed by turning on some couplings to the LQ, the active $U(1)$s are then\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $V_{1,\\mu}^{2\/3}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $2$ & $-2\/3$ & $-2\/3$ & $4\/3$ & $1\/3$ & $1\/3$ & $1\/3$\n& $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $1$ & $1$ &\n$1$\\\\\\hline\n\\end{tabular}\n\\ \\label{SponB}%\n\\end{equation}\nTurning on any of the LQ couplings of $S_{1}^{2\/3}$ or $V_{1,\\mu}^{2\/3}$ would break $U(1)_{\\mathcal{B}+\\mathcal{L}}$, and induce proton decay (compare Eq.~(\\ref{LagrSSB3a}) with Eq.~(\\ref{LagrSSB2})). At this level, their presence is thus forbidden by the still active $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$ symmetries. For an even stricter protection, the PQ symmetry can be extended to prevent these couplings. It suffices to add a seesaw mechanism with the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling, something we should do anyway (the $\\phi\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ would instead allow all the LQ couplings). Note that $S_{1}^{2\/3}$ and $V_{2,\\mu}^{1\/3}$ can mix via a $D^{\\mu}HS_{1}^{2\/3\\dagger}V_{2,\\mu}^{1\/3}$ term, but this is inessential since they have the same $\\mathcal{B}$ and $\\mathcal{L}$ quantum numbers. This scenario lead to neutron-antineutron oscillation operators, with the diagram of Fig.~\\ref{Fig6}$a$, corresponding to\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)}^{eff} & =\\frac\n{1}{m_{S}^{8}}\\partial_{\\mu}\\phi H(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}%\n_{R}^{\\mathrm{C}}u_{R})(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}%\n}u_{R})\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}+h.c.\\ \\nonumber\\\\\n& \\rightarrow\\frac{v\\Lambda_{QCD}^{6}}{m_{S}^{8}}\\partial_{\\mu}a^{0}\\bar\n{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n + ... + h.c.\\ , \\label{SponB2}%\n\\end{align}\nwhere we have set all the DQ masses to a common $m_{S}$ value. Since there is no associated $n-\\bar{n}$ operator, this scale can in principle be quite low. The best low energy limits come from nuclear transitions, as this operator also contributes to $nn\\rightarrow a$, but those do not push $m_{S}$ well above the TeV scale~\\cite{Heeck:2020nbq}. The main constraint thus come from LHC searches~\\cite{CMS:2020gru,ATLAS:2020dsk,ATLAS:2020xov,CMS:2020wzx,ATLAS:2021oiz}. Note, though, that the generic leptoquark searches may not apply to this case: all these states decay to diquark pairs and, furthermore, $V_{1}^{2\/3,\\mu}$ could end up quite long lived if it is lighter than $V_{2}^{1\/3,\\mu}$ and $S_{1}^{2\/3,\\mu}$, and would show up in channels with at least four jets.\n\nEven if $m_{S}$ can be quite low, at around the TeV say, the $a^{0}nn$ coupling is significantly smaller than the other couplings, including to $n\\bar{n}$, as can be estimated setting $f_{a}\\equiv v_{\\phi}$:\n\\begin{equation}\n\\frac{1}{f_{a}}\\approx\\frac{v\\Lambda_{QCD}^{6}}{m_{S}^{8}}\\Leftrightarrow\nm_{S}\\approx10~\\text{GeV}\\times\\left( \\frac{f_{a}}{10^{9}\\ \\text{GeV}%\n}\\right) ^{1\/8}\\ ,\n\\end{equation}\nfor $\\Lambda_{QCD}\\approx300$ MeV. Even with $f_{a}$ close to the Planck scale, the LQ mass would need to be well below the TeV scale, which would again be ruled out by direct searches. For $m_{S}$ around the TeV, the $a^{0}nn$ coupling is at best $10^{-16}$ smaller than that to $a^{0}n\\bar{n}$. Thus, $a^{0}nn$ does not represent a competitive signature for direct axion searches.\n\nIndirectly, the $a^{0}nn$ coupling may nevertheless open new routes by relying instead on neutron-antineutron oscillation phenomena. Indeed, while $a^{0}nn$ cannot generate $n\\rightarrow\\bar{n}$ in vacuum, oscillations could now be catalyzed by an axion dark matter background. While the typical high frequency of the coherent axion field precludes any observation using standard beam searches for $n-\\bar{n}$ oscillations (the induced $\\delta m_{n-\\bar{n}}$ would average to zero), transient variations of the axion field may be observable in this way. Another possibility would be to exploit the magnetic splitting between $n$ and $\\bar{n}$ states, which in a $1~$T magnetic field would be of about $10^{-7}$~eV~\\cite{Phillips:2014fgb}, larger than the axion mass if $f_{a}>10^{14}$~GeV. Note that the neutron beam go through a 4.6~T magnet in neutron lifetime experiments, Ref.~\\cite{Nico:2004ie,Yue:2013qrc}, and that axion-induced mixing effects, if they occur, would not have been excluded by the recent mirror neutron search of Ref.~\\cite{Broussard:2021eyr}, which relies on hypothesized mirror neutrons capabilities to pass through normal matter.\n\nTwo other features compared to the usual neutron oscillations are worth mentioning: the coupling is axial, $\\bar{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n$, instead of the usual scalar $\\bar{n}^{\\mathrm{C}}n$ oscillation operator, so the spin dependencies are different~\\cite{Gardner:2014cma}, and the $\\partial_{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n$ coupling can be CP violating~\\cite{Berezhiani:2015uya,McKeen:2015cuz,Berezhiani:2018xsx} since the DQ couplings are a priori complex, so $n$ and $\\bar{n}$ may react differently to an axionic background. Also, compared to neutron-mirror neutron oscillations, like those invoked to explain the neutron lifetime anomaly~\\cite{Berezhiani:2018eds}, the antineutron would not be invisible but would either decay to antiproton, or annihilate with the surrounding matter. A quantitative analysis of these signatures is clearly called for but would require a detailed study, which go beyond our scope. Also, other manifestations of the $a^{0}nn$ coupling in an astrophysical and cosmological context are left for a future study.\n\nWith only three LQ, another rather simple scenario can lead to the $\\partial_{\\mu}\\phi H^{\\dagger}V_{2}^{1\/3,\\mu}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}$ operator by virtual $S_{2}^{1\/3}$ exchanges, and involves only states transforming as $\\mathbf{3}$:\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} & =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\nonumber\\\\\n& +\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}+H^{\\dagger}%\nS_{2}^{1\/3}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}+h.c.\\ . \\label{LagrSSB3b}%\n\\end{align}\nThe same $U(1)_{\\mathcal{B}}$ charges are found as in Eq.~(\\ref{SponB}), with $V_{1,\\mu}^{2\/3}\\rightarrow S_{2}^{1\/3}$. Also, as before, adding the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling prevents all the LQ couplings of $V_{2,\\mu}^{1\/3}$, $S_{1}^{2\/3}$, and $S_{2}^{1\/3}$. Proton decay is now forbidden by the existence of the PQ symmetry at the high scale, and does not arise at the low scale thanks to the specific $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2)$ symmetry breaking pattern. The final operator is phenomenologically similar to that in Eq.~(\\ref{SponB2}).\n\nMany other choices of DQ states are possible, but they lead to similar patterns. We will not investigate more complicated processes, except for the following that leads to a different phenomenology:\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} & =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{q}_{L}^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})+V_{2,\\mu\n}^{1\/3}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\mu}q_{L}\\nonumber\\\\\n& \\ \\ \\ \\ +\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}+\\partial\n_{\\mu}\\phi V_{1}^{2\/3,\\mu\\dagger}S_{1}^{2\/3}+\\phi^{\\dagger}S_{2}^{1\/3}V_{2,\\mu}%\n^{1\/3}V_{1}^{2\/3,\\mu}+h.c.\\ .\\label{LagrSSB3c}%\n\\end{align}\nIn some senses, it combines the previous two scenarios, and gives the same charges as in Eq.~(\\ref{SponB}), with $V_{1,\\mu}^{2\/3}$ and $S_{2}^{1\/3}$ having $\\mathcal{B}=4\/3$. Also, the $\\phi^{\\dagger}\\bar{\\nu}_{R}^{\\mathrm{C}}\\nu_{R}$ coupling now suffices to prevent the LQ couplings of the four states, $V_{2,\\mu}^{1\/3}$, $V_{1,\\mu}^{2\/3}$, $S_{1}^{2\/3}$, $S_{2}^{1\/3}$. What differs however is how the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ effects are induced at the low-energy scale. The two derivative couplings are needed, and $\\phi$ further occurs in the cubic DQ coupling, so the leading operator is (see Fig.~\\ref{Fig6}$b$)\n\\begin{align}\n\\mathcal{H}_{(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)}^{eff} & =\\frac\n{1}{m_{S}^{10}}\\phi\\partial_{\\mu}\\phi^\\dagger\\partial_{\\nu}\\phi^\\dagger(\\bar{q}_{L}%\n^{\\mathrm{C}}q_{L}+\\bar{d}_{R}^{\\mathrm{C}}u_{R})\\bar{d}_{R}^{\\mathrm{C}%\n}\\gamma^{\\mu}q_{L}\\bar{d}_{R}^{\\mathrm{C}}\\gamma^{\\nu}q_{L}+h.c.\\ \\nonumber\\\\\n& \\rightarrow\\frac{v_{\\phi}\\Lambda_{QCD}^{6}}{m_{S}^{10}}\\partial_{\\mu}%\na^{0}\\partial^{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma_{5}n+...+h.c.\\ .\n\\end{align}\nThough this operator is now of dimension 14 instead of that of dimension 12 in Eq.~(\\ref{SponB2}), the extra suppression is compensated by the $v_{\\phi}$ factor since $v_{\\phi}\\Lambda_{QCD}\/m_{S}^{2}$ is of $\\mathcal{O}(1)$ for $m_{S}$ around the tens of TeV scale and $v_{\\phi}$ at around $10^{6}$~TeV. The nuclear transition bounds are thus similar as in the single axion case, and in any case not competitive with direct collider searches for new colored states. Phenomenologically, neutron-antineutron conversion now requires pairs of axions, and would occur through scattering processes like $a^{0}+n\\leftrightarrow a^{0}+\\bar{n}$ or $n+n\\leftrightarrow a^0+a^0$ and $\\bar{n}+\\bar{n}\\leftrightarrow a^0+a^0$. Though unlikely to be ever observed, these processes could play a cosmological role.\n\n\\subsubsection{Spontaneous breaking of $\\mathcal{B}\\pm3\\mathcal{L}$}\n\nThe $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,3)$ scenarios are trivially obtained from any of the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ Lagrangians of the previous section by switching all DQ couplings to LQ couplings. For example, starting from Eq.~(\\ref{LagrSSB3b}),\n\\begin{align}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}} & =\\mathcal{L}_{\\mathrm{KSVZ}}+S_{1}%\n^{2\/3}(\\bar{d}_{R}\\nu_{R}^{\\mathrm{C}}+\\bar{u}_{R}e_{R}^{\\mathrm{C}}+\\bar\n{q}_{L}\\ell_{L}^{\\mathrm{C}})+V_{2,\\mu}^{1\/3}(\\bar{u}_{R}\\gamma^{\\mu}\\ell\n_{L}^{\\mathrm{C}}+\\bar{q}_{L}\\gamma^{\\mu}\\nu_{R}^{\\mathrm{C}})\\\\\n& +\\partial_{\\mu}\\phi S_{2}^{1\/3\\dagger}V_{2}^{1\/3,\\mu}+H^{\\dagger}%\nS_{2}^{1\/3}V_{2,\\nu}^{1\/3}V_{2}^{1\/3,\\nu}+h.c.\\ ,\n\\end{align}\nleads to the charges\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $V_{2,\\mu}^{1\/3}$ & $S_{1}^{2\/3}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $-1$ & $-2\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ &\n$0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-3$ & $-2$ & $1$ & $1$ & $0$ & $0$ & $0$ & $1$ & $1$ &\n$1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nBy analogy, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ transitions can be induced by taking the Lagrangian\n\\begin{equation}\n\\mathcal{L}_{\\mathrm{KSVZ+LQ}}=\\mathcal{L}_{\\mathrm{KSVZ}}+S_{2}^{1\/3}(\\bar\n{d}_{R}\\ell_{L}+\\bar{q}_{L}\\nu_{R})+V_{1,\\mu}^{2\/3}\\bar{d}_{R}\\gamma^{\\mu}%\n\\nu_{R}+\\partial_{\\mu}\\phi V_{1}^{2\/3,\\mu\\dagger}S_{1}^{2\/3}+\\phi S_{1}%\n^{2\/3}S_{2}^{1\/3}S_{2}^{1\/3}+h.c.\\ ,\n\\end{equation}\nwith the charges are\n\\begin{equation}%\n\\begin{tabular}[c]{ccccccccccc}\\hline\n& $\\phi$ & $S_{1}^{2\/3}$ & $S_{2}^{1\/3}$ & $V_{1,\\mu}^{2\/3}$ & $q_{L}$ &\n$u_{R}$ & $d_{R}$ & $\\ell_{L}$ & $e_{R}$ & $\\nu_{R}$\\\\\\hline\n$U(1)_{\\mathcal{B}}$ & $1\/2$ & $-1\/6$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$ & $1\/3$\n& $0$ & $0$ & $0$\\\\\n$U(1)_{\\mathcal{L}}$ & $-3\/2$ & $1\/2$ & $-1$ & $-1$ & $0$ & $0$ & $0$ & $1$ &\n$1$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{equation}\nNote that for each case, additional $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,0)$ couplings involving pairs of LQs are possible, like $\\phi HS_{2}^{1\/3\\dagger}S_{1}^{2\/3}$ or $D^{\\mu}HS_{2}^{1\/3\\dagger}V_{1,\\mu}^{2\/3}$ for the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-3)$ scenario. Those can neither affect the symmetry pattern, nor open new routes for proton decay.\n\nPhenomenologically, these scenarios are very similar to the $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,\\pm1)$ ones described before, so we will not detail them further. The main difference is the extra suppression of proton and neutron decays due to the higher dimensionality of the operators, and of the many particles in the final states. These scenarios thus have essentially the same phenomenology whenever these suppressions can be compensated by lowering the LQ mass scale without violating LHC bounds.\n\n\\section{Conclusions\\label{Ccl}}\n\nIn this paper, the opportunities arising from combining leptoquarks and diquarks with axions have been systematically analyzed. From a phenomenological standpoint, our main results are:\n\n\\begin{enumerate}\n\\item The PQ symmetry of which the axion is the Goldstone boson can be identified with any combination of baryon $\\mathcal{B}$ and lepton $\\mathcal{L}$ numbers. In this way, $\\mathcal{B}$ and $\\mathcal{L}$ appear less accidental, in the sense that one combination of $\\mathcal{B}$ and\/or $\\mathcal{L}$ is spontaneously broken, while the orthogonal combination remains exact and is actually protected by the PQ symmetry. Reminiscent of the possible $\\Delta\\mathcal{B}$ and\/or $\\Delta\\mathcal{L}$ operators made of SM fields (see Table~\\ref{TableLQBL}), the simplest scenarios identify $U(1)_{PQ}$ with $U(1)_{\\mathcal{B}\\pm\\mathcal{L}}$, $U(1)_{\\mathcal{B}\\pm3\\mathcal{L}}$, $U(1)_{\\mathcal{B}}$, or $U(1)_{\\mathcal{L}}$, and induce spontaneously either proton decay, neutron-antineutron oscillations, or a Majorana mass terms for $\\nu_{R}$ (or more generally, neutrinoless double beta decays).\n\n\\item All scenarios can be supplemented with a seesaw mechanism. The axion is then not only the Goldstone boson associated to $U(1)_{\\mathcal{B}\\pm\\mathcal{L}}$, $U(1)_{\\mathcal{B}\\pm3\\mathcal{L}}$, or $U(1)_{\\mathcal{B}}$ breaking, but becomes also the Majoron associated to the $U(1)_{\\mathcal{L}}$ breaking. Though no accidental symmetry (besides of course $U(1)_{PQ}$ itself) remains, each scenario retains a specific phenomenology. For example, when $U(1)_{PQ}$ is identified both with $U(1)_{\\mathcal{B}}$ and $U(1)_{\\mathcal{L}}$, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2n,0)$ and $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(0,2n)$ transitions are possible, but proton decay cannot occur.\n\n\\item For each pattern of symmetry breaking, it is also possible to prevent axion-free proton decay, neutron-antineutron oscillations, or neutrinoless double beta decays. In other words, one can make sure $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})$ effects always involve at least one axion field. Phenomenologically, $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,1)$ scenarios open the door to $p\\rightarrow a^{0}+\\ell,$ $n\\rightarrow a^{0}+\\nu$, $p\\rightarrow2a^{0}+\\ell,$ $n\\rightarrow2a^{0}+\\nu$, and scattering processes like $a^{0}+(p,n)\\leftrightarrow a^{0}+(\\ell,\\nu)$. Scenarios with $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(1,-1)$ or $(1,\\pm3)$ are similar. Following the strategy proposed in Ref.~\\cite{NeutronTau}, if $a^{0}$ is an ALP of just the right mass, such that proton decay is forbidden but neutron decay is not, these scenarios are able to solve the neutron lifetime puzzle, see Fig.~\\ref{FigNeut}.\n\n\\item When applied to $(\\Delta\\mathcal{B},\\Delta\\mathcal{L})=(2,0)$ operators, being forced to include an axion field could lead to very peculiar effects. The phenomenology of the $\\partial_{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma^{\\mu}\\gamma_{5}n$ and $\\partial_{\\mu}a^{0}\\partial^{\\mu}a^{0}\\bar{n}^{\\mathrm{C}}\\gamma_{5}n$ interactions have, to our knowledge, not been investigated in detail yet. Though a dedicated analysis is called for, we do not expect these interactions to be phenomenologically relevant in vacuum, but they could open interesting channels in an axionic dark matter background, or transitions like $n\\rightarrow\\bar{n}+a^{0}$ or $n\\rightarrow\\bar{n}+a^{0}+a^0$ in an intense magnetic field.\n\\end{enumerate}\n\nBesides these phenomenological aspects, we have also analyzed the consequences on the foundations of axion effective Lagrangians. Whenever the axion is associated to some patterns of $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ breaking, the SM fermions become charged under the PQ symmetry. Typically, they thus occur in the usual $f_{a}$-suppressed derivative interactions, but through vector current interactions, $\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi$ (since $\\mathcal{B}$ and $\\mathcal{L}$ are vectorial). Often, these interactions are discarded owing to the naive vector Ward identity, but this is incorrect for two reasons:\n\n\\begin{enumerate}\n\\item[5.] Axion-gauge field interaction are usually expected to be $(g_{X}^{2}\/f_{a})\\mathcal{N}_{X}a^{0}X_{\\mu\\nu}\\tilde{X}^{\\mu\\nu}$, $X=G^{a}$, $W^{i}$, $B$, with $\\mathcal{N}_{X}$ summing up the contribution of all the fields charged under both the PQ symmetry and the $X$ gauge interactions of strength $g_{X}$.\\ Thus, $\\mathcal{N}_{X}$ depend on the SM fermion charges, with in particular $\\mathcal{N}_{W}$ and $\\mathcal{N}_{B}$ depending on how $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ are embedded in $U(1)_{PQ}$. Yet, as shown in Ref.~\\cite{Quevillon:2019zrd}, the SM fermion contributions to $\\mathcal{N}_{W}$ and $\\mathcal{N}_{B}$ arising from $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ systematically cancel with that coming from triangle graphs built on the corresponding $\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi$ interactions.\\ At the end of the day, the $U(1)_{\\mathcal{B}}$ and\/or $U(1)_{\\mathcal{L}}$ components of $U(1)_{PQ}$ do not alter the axion to gauge boson couplings, even though this is not apparent at the level of the effective Lagrangian.\n\n\\item[6.] The counting rule in powers of $1\/f_{a}$, central in constructing the axion effective Lagrangian (see e.g. Ref.~\\cite{Georgi:1986df}), is invalid when $\\mathcal{B}$ and\/or $\\mathcal{L}$ are broken spontaneously along with the PQ symmetry. Indeed, the equations of motion of the SM fermions (or that of the leptoquarks if they have not been integrated out) inherit $\\mathcal{O}((f^{\\alpha})^{n}),$ $n\\geq1$ terms, so that $\\mathcal{O}(f_{a}^{n-1})$ interactions are hidden inside $f_{a}^{-1}\\partial_{\\mu}a^{0}\\bar{\\psi}\\gamma^{\\mu}\\psi$. In practice, in the present paper, all these interactions were suppressed by some relatively high power of the leptoquark masses, which are pushed above the TeV by direct collider searches. Thus, in all the scenarios considered here, the $\\mathcal{B}$ and\/or $\\mathcal{L}$ violating interactions are not expected to be dominant compared to e.g. the two photon or two gluon modes for $f_{a}$ below the Planck scale. Still, as this relative suppression has nothing to do with $f_{a}$, there is no guarantee it always happens.\n\\end{enumerate}\n\nIn conclusion, even if entangling the PQ symmetry with the accidental symmetries of the SM requires new leptoquarks states, and often several of them, these scenarios end up being more economical from a $U(1)$ global symmetry point of view. The axion becomes a central piece, not only solving the strong CP puzzle, and maybe making up for the observed dark matter, but also setting off the seesaw mechanism and introducing potentially CP violating baryon number violation. With all its capabilities, the axion could hold the keys to many of the standing cosmological enigmas.\n\n\\subsubsection*{Acknowledgements} This work is supported by the labex \\textit{Enigmass}, and by the\nCNRS\/IN2P3 Master project \\textit{Axions from Particle Physics to Cosmology}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}