diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbcvb" "b/data_all_eng_slimpj/shuffled/split2/finalzzbcvb" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbcvb" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nHybrid codes simultaneously encode classical and quantum information into quantum digits such that the information is protected against errors when transmitted through a quantum channel. The simultaneous transmission of classical and quantum information was first investigated by Devetak and Shor \\cite{Devetak2005}, who characterized the set of admissible rate pairs. Notably, they showed that, at least for certain small error rates, time-sharing a quantum channel is inferior to simultaneous transmission. Constructions of hybrid codes were first studied by Kremsky, Hsieh, and Brun \\cite{Kremsky2008} in the context of entanglement-assisted stabilizer codes and by B{\\'e}ny, Kempf, and Kribs \\cite{Beny2007a, Beny2007b} who outlined an operator-theoretic construction.\n\nMore recently, Grassl, Lu, and Zeng \\cite{Grassl2017} gave linear programming bounds for a class of hybrid codes and constructed a number of hybrid stabilizer codes with parameters better than those of hybrid codes constructed from quantum stabilizer codes. In particular, these genuine hybrid codes outperform ``trivial'' hybrid codes regardless of the error rate of the channel. Additional work on hybrid codes has been done from both a coding theory approach \\cite{Nemec2018} and from an operator-theoretic approach \\cite{Majidy2018}, as well as over a fully correlated quantum channel where the space of errors is spanned by $I^{\\otimes n}$, $X^{\\otimes n}$, $Y^{\\otimes n}$, and $Z^{\\otimes n}$ \\cite{Li2019}. While they are still relatively unstudied, multiple uses for hybrid codes have already become apparent, including protecting hybrid quantum memory \\cite{Kuperberg2003} and constructing hybrid secret sharing schemes \\cite{Zhang2011}.\n\nIn this paper we give some general results regarding hybrid codes, most notably that at least one of the quantum codes comprising a genuine hybrid code must be impure, as well as show that a hybrid code can always detect more errors than a comparable quantum code. We also generalize the weight enumerators given by Grassl et al. \\cite{Grassl2017} for hybrid stabilizer codes to more general nonadditive hybrid codes and use them to derive linear programming bounds. Finally, we give multiple constructions for infinite families of hybrid codes with good parameters. The first of these families are single error-detecting hybrid stabilizer codes with parameters $\\left[\\!\\left[n,n-3\\!:\\!1,2\\right]\\!\\right]_{2}$ where the length $n$ is odd, where an $\\left[\\!\\left[n,k\\!:\\!m,d\\right]\\!\\right]_{2}$ hybrid code encodes $k$ logical qubits and $m$ logical bits into $n$ physical qubits with minimum distance $d$. The second is a collection of families of single error-correcting hybrid codes constructed using stabilizer pasting, where we paste together stabilizers from Gottesman's $\\left[\\!\\!\\!\\:\\left[2^{j},2^{j}-j-2,3\\right]\\!\\!\\!\\:\\right]_{\\!\\!\\:2}$ stabilizers codes \\cite{Gottesman1996b} and the small distance 3 hybrid codes with $n=7,9,10,11$ from \\cite{Grassl2017}. Each of these families of hybrid codes were inspired by families of nonadditive quantum codes, especially those constructed by Rains \\cite{Rains1999a} and Yu, Chen, and Oh \\cite{Yu2015}.\n\n\\section{Hybrid Codes}\nA quantum code is a subspace of a Hilbert space that allows for encoded quantum information to be recovered in the presence of errors on the physical qudits. Here our encoded message is a unit vector in the Hilbert space $$H = \\bigotimes_{\\ell=1}^{n} \\mathbb{C}^{q} \\cong \\mathbb{C}^{q^{n}}.$$ We say a qantum code has parameters $\\left(\\!\\left(n,K\\right)\\!\\right)_{q}$ if and only if it can encode a superposition of $K$ orthogonal quantum states into $n$ quantum digits with $q$ levels.\n\nNow suppose that we want to simultaneously transmit classical and quantum messages. Our goal will be to encode them into the state of $n$ quantum digits that have $q$-levels each, so that the encoded message can be transmitted over a quantum channel. A hybrid code has the parameters $\\left(\\!\\left(n,K\\!:\\!M\\right)\\!\\right)_{q}$ if and only if it can simultaneously encode one of $M$ different classical messages and a superposition of $K$ orthogonal quantum states into $n$ quantum digits with $q$ levels.\n\nWe can understand the hybrid code as a collection of $M$ orthogonal $K$-dimensional quantum codes $\\mathcal{C}_{m}$ that are indexed by the classical messages $m\\in\\left[M\\right] \\coloneqq \\left\\{1, 2, \\ldots, M\\right\\}$. If we want to transmit a classical message $m\\in\\left[M\\right]$ and a quantum state $\\ket{\\varphi}$, then we need to encode $\\ket{\\varphi}$ into the quantum code $\\mathcal{C}_{m}$. We will refer to the each of the quantum codes $\\mathcal{C}_{m}$ as \\emph{inner codes} and the collection $\\mathcal{C}=\\left\\{\\mathcal{C}_{m} \\mid m\\in\\left[M\\right]\\right\\}$ as the \\emph{outer code}.\n\n\\subsection{Error Detection}\\label{seced}\n\nThe encoded states will be subject to errors when transmitted through a quantum channel. Our first task will be to characterize the errors that can be detected by the hybrid code. We will set up a projective measurement that either upon receipt of a state $\\ket{\\psi}$ in $H$ either (a) returns $\\epsilon$ to indicate that an error happened or (b) claims that there is no error and returns a classical message $m$ and a projection of $\\ket{\\psi}$ onto $\\mathcal{C}_{m}$.\n\nLet $P_{m}$ denote the orthogonal projector onto the quantum code $\\mathcal{C}_{m}$ for all integers $m$ in the range $1\\leq m\\leq M$. For distinct integers $a$ and $b$ in the range $1\\leq a,b\\leq M$, the quantum codes $\\mathcal{C}_{a}$ and $\\mathcal{C}_{b}$ are orthogonal, so $P_bP_a=0$. It follows that the orthogonal projector onto $\\mathcal{C} =\\bigoplus_{m=1}^{M}\\mathcal{C}_{m}$ is given by $$P=P_{1}+P_{2}+\\cdots+P_{M}.$$ We define the orthogonal projection onto $\\mathcal{C}^\\perp$ by $P_\\epsilon = 1-P$. For the hybrid code $\\left\\{\\mathcal{C}_{m} \\mid m\\in\\left[M\\right]\\right\\}$, we can define a projective measurement $\\mathcal{P}$ that corresponds to the set $$\\left\\{ P_{1}, P_{2}, \\dots, P_{M}, P_{\\epsilon}\\right\\}$$ of projection operators that partition unity.\n\nWe can now define the concept of a detectable error. An error $E$ is called \\textit{detectable}\\\/ by the hybrid code $\\left\\{\\mathcal{C}_{m} \\mid m\\in \\left[M\\right]\\right\\}$ if and only if for each index $a, b$ in the range $1\\leq a, b \\leq M$, we have \n\\begin{equation}\n\\label{hkl}\nP_{b} E P_{a} = \\begin{cases}\n\\lambda_{E,a} P_a & \\text{if $a=b$}, \\\\\n0 & \\text{if $a\\neq b$}\n\\end{cases}\n\\end{equation}\nfor some scalar $\\lambda_{E,a}$. \n\nThe motivation for calling an error $E$ detectable is the following simple protocol. Suppose that we encode a classical message $m$ and a quantum state into a state $\\ket{v_{m}}$ of $\\mathcal{C}_{m}$, and transmit it through a quantum channel that imparts the error $E$. If the error is detectable, then measurement of the state $E\\ket{v_{m}} = EP_{m}\\ket{v_{m}}$ with the projective measurement $\\mathcal{P}$ either \n\\begin{compactenum}[(E1)]\n\\item returns $\\epsilon$, which signals that an error happened, or \n\\item returns $m$ and corrects the error by projecting the state back\n onto a scalar multiple $\\lambda_{E,m}\\ket{v_{m}} = P_{m}EP_{m}\\ket{v_{m}}$ of the state\n $\\ket{v_m}$.\n\\end{compactenum}\nThe definition of a detectable error ensures that the measurement $\\mathcal{P}$ will never return an incorrect classical message $d$, since $P_{d} E P_{m} \\ket{v_{m}}= 0$ for all $d\\neq m$, so the probability of detecting an incorrect message is zero. An error that is not detectable by the hybrid code can change the encoded classical, the encoded quantum information, or both.\n\nThe condition in Equation (\\ref{hkl}) is equivalent to the hybrid Knill-Laflamme condition \\cite[Theorem 4]{Grassl2017} for detectable errors: an error $E$ is detectable by a hybrid code $\\mathcal{C}$ with orthonormal basis states $\\left\\{\\ket{c_{i}^{\\left(a\\right)}}\\mid i\\in\\left[K\\right],a\\in\\left[M\\right]\\right\\}$ if and only if \\begin{equation}\\label{klvec}\\bra{c_{i}^{\\left(b\\right)}}E\\ket{c_{j}^{\\left(a\\right)}}=\\lambda_{E,a}\\delta_{ij}\\delta_{ab}.\\end{equation} Compared to the original Knill-Laflamme conditions for fully quantum codes \\cite{Knill1997} where the scalar only depended on the detectable error, these hybrid conditions allow for scalars $\\lambda_{E,a}$ that may depend on both the detectable error $E$ and the classical message $a$, allowing more flexibility in the design of codes. However, this flexibility comes at the price of no longer being able to send a superposition of all of the codewords.\n\nThe next proposition shows that hybrid codes can always detect more errors than a comparable quantum code that encodes both classical and quantum information. This is remarkable given that the advantages are much less apparent when one considers minimum distance, see \\cite{Grassl2017}. \n\n\\begin{proposition}[{\\cite{Nemec2018}}]\n\\label{detectset}\nThe subset $\\mathcal{D}$ of detectable errors in $B\\!\\left(H\\right)$ of an $\\left(\\!\\left(n,K\\!:\\!M\\right)\\!\\right)_{q}$ hybrid code form a vector space of dimension $$\\dim \\mathcal{D} = q^{2n} - (MK)^2 + M.$$ In particular, a $\\left(\\!\\left(n, K\\!:\\!M\\right)\\!\\right)_{q}$ hybrid code with $M>1$ can detect more errors than an $\\left(\\!\\left(n, KM\\right)\\!\\right)_{q}$ quantum code.\n\\end{proposition}\n\\begin{proof}\nIt is clear that any linear combination of detectable errors is detectable. If we choose a basis adapted to the orthogonal decomposition $H=\\mathcal{C} \\oplus \\mathcal{C}^{\\perp}$ with $$\\mathcal{C}=\\mathcal{C}_{1}\\oplus\\mathcal{C}_{2}\\oplus \\cdots \\oplus \\mathcal{C}_{M},$$ then an error $E$ is represented by a matrix of the form \n$$ \n\\left(\\begin{array}{cc}\nA & R \\\\ \nS & T\n\\end{array}\\right),\n$$\nwhere the blocks $A$ and $T$ correspond to the subspaces $\\mathcal{C}$ and $\\mathcal{C}^{\\perp}$ respectively. Since $E$ is detectable, the $MK\\times MK$ matrix $A$ must satisfy $$A = \\lambda_{E,1} 1_K \\oplus \\lambda_{E,2} 1_{K} \\oplus \\cdots \\oplus\n\\lambda_{E,M} 1_{K},$$ where $1_{K}$ denote a $K\\times K$ identity matrix, but $R$, $S$, and $T$ can be arbitrary. Therefore, the dimension of the vector space of detectable errors is given by $q^{2n} - \\left(MK\\right)^{2} + M$.\n\nIn the case of an $\\left(\\!\\left(n, KM\\right)\\!\\right)_{q}$ quantum code, $A$ must satisfy $A=\\lambda_{E} 1_{KM}$, so the vector space of detectable errors has dimension $q^{2n}-\\left(KM\\right)^{2}+1$, which is strictly less than $q^{2n} - \\left(MK\\right)^{2} + M$ when $M>1$.\n\\end{proof}\n\nWe briefly recall the concept of a nice error basis (see \\cite{Klappenecker2002, Klappenecker2003, Knill1996} for further details), so that we can define a suitable notion of weight for the errors. Let ${G}$ be a group of order $q^{2}$ with identity element~1 and $\\mathcal{U}\\!\\left(q\\right)$ be the group of $q\\times q$ unitary matrices. A \\textit{nice error basis}\\\/ on $\\mathbb{C}^{q}$ is a set ${\\cal E}=\\{\\rho(g)\\in {\\cal U}(q) \\,|\\, g\\in {G}\\}$ of unitary matrices such that\n\\begin{tabbing}\ni)\\= (iiiii) \\= \\kill\n\\>(i) \\> $\\rho(1)$ is the identity matrix,\\\\[1ex]\n\\>(ii) \\> $\\trace\\rho(g)=0$ for all $g\\in G\\setminus \\{1\\}$,\\\\[1ex]\n\\>(iii) \\> $\\rho(g)\\rho(h)=\\omega(g,h)\\,\\rho(gh)$ for all $g,h\\in{G}$,\n\\end{tabbing}\nwhere $\\omega(g,h)$ is a nonzero complex number depending on $(g,h)\\in G\\times G$; the function $\\omega\\colon G\\times G\\rightarrow\\mathbb{C}^\\times$ is called the factor system of $\\rho$. We call $G$ the \\textit{index group}\\\/ of the error basis ${\\cal E}$. The nice error basis that we have introduced so far generalizes the Pauli basis to systems with $q\\ge 2$ levels. \n\nWe can obtain a nice error basis $\\mathcal{E}_n$ on $H\\cong \\mathbb{C}^{q^n}$ by tensoring $n$ elements of $\\mathcal{E}$, so $$ \\mathcal{E}_n = \\mathcal{E}^{\\otimes n} = \\{ E_1 \\otimes E_2\\otimes\\cdots \\otimes E_n \\mid E_k \\in \\mathcal{E}, 1\\le k\\le n\\}.$$ The weight of an element in $\\mathcal{E}_n$ are the number of non-identity tensor components. We write $\\wt(E)=d$ to denote that the element $E$ in $\\mathcal{E}_n$ has weight $d$. A hybrid code with parameters $\\left(\\!\\left(n,K\\!:\\!M,d\\right)\\!\\right)_{q}$ has \\emph{minimum distance} $d$ if it can detect all errors of weight less than $d$.\n\n\\begin{example}\n\\label{nonaddex}\nTo construct our nonadditive hybrid code $\\mathcal{C}$ we will combine two known degenerate stabilizer codes. The first code $\\mathcal{C}_{a}$ is the $\\left[\\!\\left[6,1,3\\right]\\!\\right]_{2}$ code constructed by extending the $\\left[\\!\\left[5,1,3\\right]\\!\\right]_{2}$ Hamming code, see \\cite{Calderbank1998}, where the stabilizer is given by $$\\left\\langle XXZIZI, ZXXZII, IZXXZI, ZIZXXI, IIIIIX\\right\\rangle.$$ The second code $\\mathcal{C}_{b}$ is a $\\left[\\!\\left[6,1,3\\right]\\!\\right]_{2}$ code not equivalent to $\\mathcal{C}_{a}$, see \\cite{Shaw2008}. Its stabilizer is given by $$\\left\\langle YIZXXY, ZXIIXZ, IZXXXX, IIIZIZ, ZZZIZI\\right\\rangle.$$\n\nWe can check that the resulting two codes are indeed orthogonal to each other. The resulting code $\\mathcal{C}$ is a $\\left(\\!\\left(6,2\\!:\\!2,1\\right)\\!\\right)_{2}$ nonadditive hybrid code, since there are several errors of weight one such that $P_{b}EP_{a}\\neq0$, for example $E=IIIIXI$. This shows that even though $\\mathcal{C}_{a}$ and $\\mathcal{C}_{b}$ are optimal quantum codes on their own, together they make a hybrid code with an extremely poor minimum distance. Later we will see how to construct hybrid codes with better minimum distances.\n\\end{example}\n\n\\subsection{Genuine Hybrid Codes}\n\nIn general, it is not difficult to construct hybrid codes using quantum stabilizer codes. As Grassl et al. \\cite{Grassl2017} pointed out, there are three simple constructions of hybrid codes that do not offer any real advantage over quantum error-correcting codes:\n\n\\begin{proposition}[{\\cite{Grassl2017}}]\\label{trivcon}\nHybrid codes can be constructed using the following ``trivial\" constructions:\n\\begin{enumerate}\n\\item Given an $\\left(\\!\\left(n,KM,d\\right)\\!\\right)_{q}$ quantum code of composite dimension $KM$, there exisits a hybrid code with parameters $\\left(\\!\\left(n,K\\!:\\!M,d\\right)\\!\\right)_{q}$.\n\\item Given an $\\left[\\!\\left[n,k\\!:\\!m,d\\right]\\!\\right]_{q}$ hybrid code with $k>0$, there exists a hybrid code with parameters $\\left[\\!\\left[n,k-1\\!:\\!m+1,d\\right]\\!\\right]_{q}$.\n\\item Given an $\\left[\\!\\left[n_{1},k_{1},d\\right]\\!\\right]_{q}$ quantum code and an $\\left[n_{2},m_{2},d\\right]_{q}$ classical code, there exists a hybrid code with parameters $\\left[\\!\\left[n_{1}+n_{2},k_{1}\\!:\\!m_{2},d\\right]\\!\\right]_{q}$.\n\\end{enumerate}\n\\end{proposition}\n\nWe say that a hybrid code is \\emph{genuine} if it cannot be constructed using one of the above constructions, following the work of Yu et al. on genuine nonadditive codes \\cite{Yu2015}. We also refer to a hybrid stabilizer code that provides an advantage over quantum stabilizer codes as a genuine hybrid stabilizer code. While all known genuine hybrid codes are in fact hybrid stabilizer codes, the linear programming bounds in Section \\ref{lpb} do not prohibit genuine nonadditive hybrid codes, and may give us some hints as to their parameters.\n\nMultiple genuine hybrid stabilizer codes with small parameters were constructed by Grassl et al. in \\cite{Grassl2017}, all of which have degenerate inner codes. Having degenerate inner codes can allow for a more efficient packing of the inner codes inside the outer code than is possible when using nondegenerate codes, giving a hybrid code with parameters superior to those using the first construction of Proposition \\ref{trivcon}. However, they do not exclude the possibility that there is a genuine hybrid code where all of the inner codes are nondegenerate. Here, we show that for a genuine hybrid code, at least one of its inner codes must be impure. Recall that a quantum code is \\emph{pure} if trace-orthogonal errors map the code to orthogonal subspaces. A code that is not pure is called \\emph{impure}.\n\n\\begin{proposition}\nSuppose $\\mathcal{C}$ is a genuine $\\left(\\!\\left(n,K\\!:\\!M,d\\right)\\!\\right)_{q}$ hybrid code. Then at least one inner code $\\mathcal{C}_{m}$ of the hybrid code $\\mathcal{C}$ is impure.\n\\end{proposition}\n\\begin{proof}\nSeeking a contradiction, suppose that every inner code of the hybrid code $\\mathcal{C}$ is pure. For $m\\in\\left[M\\right]$, let $P_{m}$ denote the orthogonal projector onto the $m$-th inner code of the hybrid code $\\mathcal{C}$. For every nonscalar error operator $E$ of weight less than $d$, we have \n$$ P_{a} E P_{b} = 0,$$\nwhere $a, b\\in\\left[M\\right]$. Let $P=P_{1} + P_{2} + \\cdots + P_{M}$ denote the projector onto the $KM$-dimensional vector space spanned by the inner codes. Then \n$$ PEP=0,$$\nso the image of $P$ is an $\\left(\\!\\left(n,KM,d\\right)\\!\\right)_{q}$ quantum code, contradicting that the hybrid code $\\mathcal{C}$ is genuine. \n\\end{proof}\n\nSince for stabilizer codes the definitions of impure and degenerate codes coincide, genuine hybrid stabilizer codes necessarily require that one of the inner codes is degenerate. Therefore, one of the difficulties in constructing families of genuine codes is finding nontrivial degenerate codes. Unfortunately, there are few known families of impure or degenerate codes, see for example \\cite{Aly2006, Aly2007}, and they typically have minimum distances much lower than optimal quantum codes, suggesting they are not particularly suitable to use in constructing genuine hybrid codes.\n\n\\subsection{Hybrid Stabilizer Codes}\n\nAll of the hybrid codes constructed by Grassl et al. \\cite{Grassl2017} were given using the codeword stabilizer (CWS)\/union stabilizer framework, see \\cite{Cross2009, Grassl2008}, which we will briefly describe here. Starting with a quantum code $\\mathcal{C}_{0}$, we choose a set of $M$ coset representatives $t_{i}$ from the normalizer of $\\mathcal{C}_{0}$ (we will always take $t_{1}$ to be $I$), and then construct the code $$\\mathcal{C}=\\bigcup\\limits_{i\\in\\left[M\\right]}t_{i}\\mathcal{C}_{0}.$$ In the case of hybrid codes, $t_{i}\\mathcal{C}_{0}$ are our inner codes and $\\mathcal{C}$ is our outer code. If both $\\mathcal{C}_{0}$ and $\\mathcal{C}$ are stabilizer codes, we say that $\\mathcal{C}$ is a hybrid stabilizer code.\n\nThe generators that define a hybrid code can be divided into those that generate the quantum stabilizer $\\mathcal{S}_{\\mathcal{Q}}$ which stabilizes the outer code $\\mathcal{C}$ and those that generate the classical stabilizer $\\mathcal{S}_{\\mathcal{C}}$ which together with $\\mathcal{S}_{\\mathcal{Q}}$ stabilizes the inner code $\\mathcal{C}_{0}$ \\cite{Kremsky2008}. The generators that define the $\\left[\\!\\left[7,1\\!:\\!1,3\\right]\\!\\right]_{2}$ hybrid stabilizer code given in \\cite{Grassl2017} are given in (\\ref{gen7}), where the generators of $\\mathcal{S}_{\\mathcal{Q}}$ are given above the dotted line, the generators of $\\mathcal{S}_{\\mathcal{C}}$ are between the dotted and solid line, the normalizer of the inner code $\\mathcal{C}_{0}$ is generated by all elements above the double line, and the normalizer of the outer code is generated by all of the elements. \n\n\\begin{equation}\n\\label{gen7}\n\\left(\\mkern-5mu\n\\begin{tikzpicture}[baseline=-.5ex]\n\\matrix[\n matrix of math nodes,\n column sep=.25ex, row sep=-.25ex\n] (m)\n{\nX & I & I & Z & Y & Y & Z \\\\\nZ & X & I & X & Z & I & X \\\\\nZ & I & X & X & I & Z & X \\\\\nZ & I & Z & Z & X & I & I \\\\\nI & Z & I & Z & I & X & X \\\\\nZ & I & I & I & I & I & X \\\\\nI & I & I & X & Z & Z & X \\\\\nI & I & I & Z & X & X & I \\\\\nI & I & I & I & X & Y & Y \\\\\n};\n\\draw[line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth]\n ([yshift=.2ex] m-5-1.south west) -- ([yshift=.2ex] m-5-7.south east);\n\\draw[line width=.5pt]\n ([yshift=.2ex] m-6-1.south west) -- ([yshift=.2ex] m-6-7.south east);\n\\draw[line width=.5pt]\n ([yshift=.22ex] m-8-1.south west) -- ([yshift=.2ex] m-8-7.south east);\n\\draw[line width=.5pt]\n ( m-8-1.south west) -- ( m-8-7.south east);\n\\end{tikzpicture}\\mkern-5mu\n\\right)\n\\end{equation}\n\nFollowing Kremsky et al. \\cite{Kremsky2008}, we will often only include the stabilizer generators, as they are sufficient to fully define the hybrid code, as shown in the following proposition:\n\n\\begin{proposition}\n\\label{hybgenconstr}\nLet $\\mathcal{C}$ be an $\\left[\\!\\left[n,k\\!:\\!m,d\\right]\\!\\right]_{p}$ hybrid stabilizer code over a finite field of prime order $p$ with quantum stabilizer $\\mathcal{S}_{\\mathcal{Q}}$ and classical stabilizer $\\mathcal{S}_{\\mathcal{C}}=\\left\\langle g_{1}^{\\mathcal{C}}, \\dots, g_{m}^{\\mathcal{C}}\\right\\rangle$. Then the stabilizer code $\\mathcal{C}_{c}$ associated with classical message $c\\in\\mathbb{F}_{p}^{m}$ is given by the stabilizer $$\\left\\langle \\mathcal{S}_{\\mathcal{Q}}, \\omega^{c_{1}}g_{1}^{\\mathcal{C}}, \\dots, \\omega^{c_{m}}g_{m}^{\\mathcal{C}}\\right\\rangle,$$ where $c_{i}$ is the $i$-th entry of $c$ and $\\omega$ is a primitive complex $p$-th root of unity.\n\\end{proposition}\n\\begin{proof}\nThere are $p^{k+m}$ codewords stabilized by $\\mathcal{S}_{\\mathcal{Q}}$. Each of these codewords is an eigenvector of $g_{i}^{\\mathcal{C}}$, which naturally partitions the code into $p$ cosets based on eigenvalues. Repeating this with all of the classical generators, we get $p^{m}$ cosets of codewords each of size $p^{k}$. Since $v$ being an eigenvector of $g_{i}^{\\mathcal{C}}$ with eigenvalue $\\omega^{-1}$ means that it is a $+1$ eigenvector of $\\omega g_{i}^{\\mathcal{C}}$, therefore each coset is the $+1$ eigenspace of a stabilizer of the form $\\left\\langle \\mathcal{S}_{\\mathcal{Q}}, \\omega^{c_{1}}g_{1}^{\\mathcal{C}}, \\dots, \\omega^{c_{m}}g_{m}^{\\mathcal{C}}\\right\\rangle$, where the string $c\\in\\mathbb{F}_{p}^{m}$ can be used to index the stabilizer codes.\n\\end{proof}\n\n\\section{Weight Enumerators and\\\\ Linear Programming Bounds}\n\nWeight enumerators for quantum codes were introduced by Shor and Laflamme \\cite{Shor1997}, and as with their classical counterparts they can be used to give good bounds on code parameters using linear programming, see \\cite{Ashikhmin1999, Ketkar2006}. Grassl et al. \\cite{Grassl2017} gave weight enumerators and linear programming bounds for hybrid stabilizer codes, but these weight enumerators will not work for nonadditive hybrid codes such as the one given in Example \\ref{nonaddex}. In this section, we define weight enumerators for general hybrid codes following the approach of Shor and Laflamme \\cite{Shor1997} and Rains \\cite{Rains1998} and use them to derive linear programming bounds for general hybrid codes.\n\n\\subsection{Weight Enumerators}\n\nFor an $\\left(\\!\\left(n,K\\!:\\!M,d\\right)\\!\\right)_{q}$ hybrid code $\\mathcal{C}$ defined by the projector $P=P_{1}+\\cdots+P_{M}$ and a nice error base $\\mathcal{E}_{n}$ as defined in Section \\ref{seced}, we define the two weight enumerators of the code following Shor and Laflamme \\cite{Shor1997}: $$A\\!\\left(z\\right)=\\sum\\limits_{d=0}^{n}A_{d}z^{d}\\text{ and }B\\!\\left(z\\right)=\\sum\\limits_{d=0}^{n}B_{d}z^{d},$$ where the coefficients are given by $$A_{d}=\\frac{1}{K^{2}M^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(EP\\right)\\Tr\\!\\left(E^{*}P\\right)$$ and $$B_{d}=\\frac{1}{KM}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(EPE^{*}P\\right).$$\n\nWe can also define weight enumerators using the inner code projectors $P_{a}$. Let $$A^{\\left(a,b\\right)}\\!\\left(z\\right)=\\sum\\limits_{d=0}^{n}A_{d}^{\\left(a,b\\right)}z^{d}\\text{ and }B^{\\left(a,b\\right)}\\!\\left(z\\right)=\\sum\\limits_{d=0}^{n}B_{d}^{\\left(a,b\\right)}z^{d},$$ where $$A_{d}^{\\left(a,b\\right)}=\\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(EP_{a}\\right)\\Tr\\!\\left(E^{*}P_{b}\\right)$$ and $$B_{d}^{\\left(a,b\\right)}=\\frac{1}{K}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(EP_{a}E^{*}P_{b}\\right).$$ Note that $A^{\\left(a,a\\right)}\\!\\left(z\\right)$ and $B^{\\left(a,a\\right)}\\!\\left(z\\right)$ are the weight enumerators of the quantum code associated with projector $P_{a}$. We can then write the weight enumerators for the outer code in terms of the weight enumerators for the inner codes:\n\n\\begin{lemma}\nThe weight enumerators of $\\mathcal{C}$ can be written as $$A\\!\\left(z\\right)=\\frac{1}{M^{2}}\\sum\\limits_{a,b=1}^{M}A^{\\left(a,b\\right)}\\!\\left(z\\right)\\text{ and }B\\!\\left(z\\right)=\\frac{1}{M}\\sum\\limits_{a,b=1}^{M}B^{\\left(a,b\\right)}\\!\\left(z\\right).$$\n\\end{lemma}\n\\begin{proof}\nBy linearity of the projector $P$ we have \\begin{align*} A_{d} & = \\frac{1}{K^{2}M^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(EP\\right)\\Tr\\!\\left(E^{*}P\\right) \\\\ & = \\frac{1}{K^{2}M^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\sum\\limits_{a,b=1}^{M}\\Tr\\!\\left(EP_{a}\\right)\\Tr\\!\\left(E^{*}P_{b}\\right) \\\\ & = \\frac{1}{M^{2}}\\sum\\limits_{a,b=1}^{M}A_{d}^{\\left(a,b\\right)}. \\end{align*} We can then rewrite the weight enumerator as \\begin{align*} A\\!\\left(z\\right) & = \\sum\\limits_{d=0}^{n}A_{d}z^{d} \\\\ & = \\frac{1}{M^{2}}\\sum\\limits_{d=0}^{n}\\sum\\limits_{a,b=1}^{M}A_{d}^{\\left(a,b\\right)}z^{d} \\\\ & =\\frac{1}{M^{2}}\\sum\\limits_{a,b=1}^{M}A^{\\left(a,b\\right)}\\!\\left(z\\right). \\end{align*} The result for $B\\!\\left(z\\right)$ follows from the same argument.\n\\end{proof}\n\nWhile the weight enumerator $B\\!\\left(z\\right)$ is the same as the one introduced by the authors in \\cite{Nemec2018}, the weight enumerator $A\\!\\left(z\\right)$ is different. There the $A^{\\left(a,b\\right)}\\!\\left(z\\right)$ weight enumerators with $a\\neq b$ were ignored, causing $A\\!\\left(z\\right)$ and $B\\!\\left(z\\right)$ to not satisfy the MacWilliams identity. The approach presented in this paper is more natural, as it treats both the inner and outer codes as quantum codes. The following result may be found in \\cite{Rains1998, Shor1997}, which we include for completeness:\n\n\\begin{lemma}[{\\cite{Rains1998, Shor1997}}]\n\\label{cauchyschwarz}\nLet $\\mathcal{C}$ be a $\\left(\\!\\left(n,K\\!:\\!M\\right)\\!\\right)_{q}$ hybrid code with weight distributions $A_{d}$ and $B_{d}$. Then for all integers $d$ in the range $0\\leq d\\leq n$ and all $a\\in\\left[M\\right]$ we have\n\\begin{enumerate}\n\\item $0\\leq A_{d}\\leq B_{d}$\n\\item $0\\leq A_{d}^{\\left(a,a\\right)}\\leq B_{d}^{\\left(a,a\\right)}$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFor every orthogonal projector $\\Pi:\\mathbb{C}^{q^{n}}\\rightarrow\\mathbb{C}^{q^{n}}$ of rank $K$, we have\n\\begin{equation*}\n0\\leq\\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(E\\Pi\\right)\\Tr\\!\\left(E^{*}\\Pi\\right)\n\\end{equation*}\nby the non-negativity of the trace inner product. Furthermore, we can write this inequality in the form\n\\begin{align*}\n0 & \\leq \\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(E\\Pi\\right)\\Tr\\!\\left(E^{*}\\Pi\\right) \\\\\n& = \\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\left\\vert\\Tr\\!\\left(E\\Pi\\right)\\right\\vert^{2} \\\\\n& = \\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\left\\vert\\Tr\\!\\left(\\left(\\Pi E\\Pi\\right)\\Pi\\right)\\right\\vert^{2}.\n\\end{align*}\nUsing the Cauchy-Schwarz inequality, we obtain\n\\begin{align*}\n0 & \\leq \\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(\\left(\\Pi E\\Pi\\right)\\left(\\Pi E\\Pi\\right)^{*}\\right)\\Tr\\!\\left(\\Pi^{*}\\Pi\\right) \\\\\n& = \\frac{1}{K}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\Tr\\!\\left(E\\Pi E^{*}\\Pi\\right).\n\\end{align*}\nSubstituting $\\Pi=P$ implies (1) and substituting $\\Pi=P_{a}$ implies (2).\n\\end{proof}\n\nThe main utility of weight enumerators for quantum codes is that they allow for a complete characterization of the error-correction capability of the code in terms of the minimum distance of the code. In the following proposition, we prove a similar result for the weight enumerators of hybrid codes.\n\n\\begin{proposition}\n\\label{wtenumnecsuf}\nLet $\\mathcal{C}$ be a $\\left(\\!\\left(n,K\\!:\\!M\\right)\\!\\right)_{q}$ hybrid code with weight distributions $A_{d}$ and $B_{d}$. Then $\\mathcal{C}$ can detect all errors in $\\mathcal{E}_{n}$ of weight $d$ if and only if $A_{d}^{\\left(a,a\\right)}=B_{d}^{\\left(a,a\\right)}$ for all $a\\in\\left[M\\right]$ and $B_{d}^{\\left(a,b\\right)}=0$ for all $a,b\\in\\left[M\\right],a\\neq b$.\n\\end{proposition}\n\\begin{proof}\nRecall that an error is detectable by a code if and only if it satisfies the hybrid Knill-Laflamme conditions in Equation (\\ref{klvec}), and that a projector onto one of the inner codes $\\mathcal{C}_{a}$ may be written as $P_{a}=\\sum_{i=1}^{K}\\ket{c_{i}^{\\left(a\\right)}}\\bra{c_{i}^{\\left(a\\right)}}$, where $\\left\\{\\ket{c_{i}^{\\left(a\\right)}}\\mid i\\in\\left[K\\right]\\right\\}$ is an orthonormal basis for $\\mathcal{C}_{a}$. Suppose that all errors of weight $d$ are detectable by $\\mathcal{C}$. Then \\begin{align*}\nA_{d}^{\\left(a,a\\right)} & = \\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}} \\Tr\\!\\left(EP_{a}\\right)\\Tr\\!\\left(E^{*}P_{a}\\right) \\\\\n& = \\frac{1}{K^{2}}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\left\\vert\\sum_{i=1}^{K}\\bra{c_{i}^{\\left(a\\right)}}E\\ket{c_{i}^{\\left(a\\right)}}\\right\\vert^{2} \\\\\n& = \\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\left\\vert\\alpha_{E}^{\\left(a\\right)}\\right\\vert^{2}.\n\\end{align*}\nSimilarly, we have \\begin{align*}\nB_{d}^{\\left(a,a\\right)} & = \\frac{1}{K}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}} \\Tr\\!\\left(EP_{a}E^{*}P_{a}\\right) \\\\\n& = \\frac{1}{K}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\sum\\limits_{i,j=1}^{K}\\left\\vert\\bra{c_{i}^{\\left(a\\right)}}E\\ket{c_{j}^{\\left(a\\right)}}\\right\\vert^{2} \\\\\n& = \\frac{1}{K}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\sum\\limits_{i=1}^{K}\\left\\vert\\bra{c_{i}^{\\left(a\\right)}}E\\ket{c_{i}^{\\left(a\\right)}}\\right\\vert^{2} \\\\\n& = \\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\left\\vert\\alpha_{E}^{\\left(a\\right)}\\right\\vert^{2}.\n\\end{align*}\nTherefore, we have that $A_{d}^{\\left(a,a\\right)}=B_{d}^{\\left(a,a\\right)}$. Additionally, if $a\\neq b$, then by Equation (\\ref{klvec}) we have $\\bra{c_{i}^{\\left(a\\right)}}E\\ket{c_{j}^{\\left(b\\right)}}=0$. Therefore, \\begin{align*}\nB_{d}^{\\left(a,b\\right)} & = \\frac{1}{K}\\sum\\limits_{\\substack{E\\in \\mathcal{E}_n\\\\ \\wt(E)=d}}\\sum\\limits_{i,j=1}^{K}\\left\\vert\\bra{c_{i}^{\\left(a\\right)}}E\\ket{c_{j}^{\\left(b\\right)}}\\right\\vert^{2} \\\\\n& = 0.\n\\end{align*}\n\nConversely, suppose that (a) $A_{d}^{\\left(a,a\\right)}=B_{d}^{\\left(a,a\\right)}$ for all $a\\in\\left[M\\right]$ and (b) $B_{d}^{\\left(a,b\\right)}=0$ for all $a,b\\in\\left[M\\right],a\\neq b$. Condition (a) implies that equality holds for each $E$ in the Cauchy-Schwarz inequality. Therefore, we have that $P_{a}EP_{a}$ and $P_{a}$ must be linearly dependent, so there must be a constant $\\alpha_{E}^{\\left(a\\right)}\\in\\mathbb{C}$ such that $P_{a}EP_{a}=\\alpha_{E}^{\\left(a\\right)}$, or equivalently, $\\bra{c_{i}^{\\left(a\\right)}}E\\ket{c_{j}^{\\left(a\\right)}}=\\alpha_{E}^{\\left(a\\right)}\\delta_{i,j}$, for all errors of weight $d$. Condition (b) implies that $\\bra{c_{i}^{\\left(a\\right)}}E\\ket{c_{j}^{\\left(b\\right)}}=0$ if $a\\neq b$, for all errors of weight $d$. Putting these together, we get the hybrid Knill-Laflamme conditions, so all errors of weight $d$ are detectable.\n\\end{proof}\n\n\\subsection{Linear Programming Bounds}\\label{lpb}\n\nOne of the more useful properties of weight enumerators is that they satisfy the Macwilliams identity \\cite{Shor1997}:\n\\begin{equation}\nB^{\\left(a,b\\right)}\\!\\left(z\\right)=\\frac{K}{q^{n}}\\left(1+\\left(q^{2}-1\\right)z\\right)^{n}A^{\\left(a,b\\right)}\\!\\left(\\frac{1-z}{1+\\left(q^{2}-1\\right)z}\\right).\n\\end{equation}\nThe MacWilliams identities, along with the results from Lemma \\ref{cauchyschwarz} and Proposition \\ref{wtenumnecsuf} and the shadow inequalities for qubit codes \\cite{Rains1999b} allow us to define linear programming bounds on the parameters of general hybrid codes (see \\cite{Ashikhmin1999, Calderbank1998, Rains1998} for linear programming bounds on quantum codes). Let \\begin{equation}K_{j}\\!\\left(r\\right)=\\sum\\limits_{k=0}^{j}\\left(-1\\right)^{k}\\left(q^{2}-1\\right)^{j-k}\\binom{r}{k}\\binom{n-r}{j-k}\\end{equation} denote the $q^{2}$-ary Krawtchouk polynomials.\n\n\\begin{proposition}\nThe parameters of an $\\left(\\!\\left(n,K\\!:\\!M,d\\right)\\!\\right)_{q}$ hybrid code must satisfy the following conditions:\n\\begin{enumerate}\n\\item $A_{j}=\\frac{1}{M^{2}}\\sum\\limits_{a,b=1}^{M}A_{j}^{\\left(a,b\\right)}$\n\\item $B_{j}=\\frac{1}{M}\\sum\\limits_{a,b=1}^{M}B_{j}^{\\left(a,b\\right)}$\n\\item $A_{0}^{\\left(a,b\\right)}=1$\n\\item $B_{0}^{\\left(a,b\\right)}=\\begin{cases} 1 & \\text{ if } a=b \\\\ 0 & \\text{ if } a\\neq b \\end{cases}$\n\\item $A_{j}^{\\left(a,a\\right)}=B_{j}^{\\left(a,a\\right)}$, for all $0\\leq j2$? For small lengths ($n\\leq19$) this family achieves the linear programming bounds for general hybrid codes given in Section \\ref{lpb}, and we suspect that $M=2$ is optimal for all odd $n$.\n\n\\section{Families of Hybrid Codes from\\\\ Stabilizer Pasting}\n\nIn this section, we construct two families of single-error correcting hybrid codes that can encode one or two classical bits. An infinite family of nonadditive quantum codes was constructed by Yu et al. \\cite{Yu2015} by pasting together (see \\cite{Gottesman1996a}) the stabilizers of Gottesman's $\\left[\\!\\!\\!\\:\\left[2^{j},2^{j}-j-2,3\\right]\\!\\!\\!\\:\\right]_{\\!\\!\\:2}$ codes \\cite{Gottesman1996b} with the non-Pauli observables of the $\\left(\\!\\left(9,12,3\\right)\\!\\right)_{2}$ and $\\left(\\!\\left(10,24,3\\right)\\!\\right)_{2}$ nonadditive CWS codes \\cite{Yu2007, Yu2008} which function in the same role as the Pauli stabilizers in stabilizer codes.\n\nBelow we give the generators of the hybrid codes originally given by Grassl et al. \\cite{Grassl2017} that we will use in the construction of our families. The generators for the $\\left[\\!\\left[7,1\\!:\\!1,3\\right]\\!\\right]_{2}$ code was previously given in (\\ref{gen7}), while those for the $\\left[\\!\\left[9,2\\!:\\!2,3\\right]\\!\\right]_{2}$, $\\left[\\!\\left[10,3\\!:\\!2,3\\right]\\!\\right]_{2}$, and $\\left[\\!\\left[11,4\\!:\\!2,3\\right]\\!\\right]_{2}$ hybrid stabilizer codes are (\\ref{gen9}), (\\ref{gen10}), and (\\ref{gen11}) respectively:\n\n\\begin{equation}\n\\label{gen9}\n\\left(\\mkern-5mu\n\\begin{tikzpicture}[baseline=-.65ex]\n\\matrix[\n matrix of math nodes,\n column sep=.25ex, row sep=-.25ex\n] (m)\n{\nX & I & I & Z & Y & Z & X & X & Y \\\\\nZ & X & I & Z & Y & X & Y & I & Z \\\\\nI & Z & X & Z & Z & I & X & I & X \\\\\nI & Z & Z & I & Y & X & X & Y & I \\\\\nZ & Z & I & X & X & I & X & Z & I \\\\\nZ & I & I & I & I & X & I & I & I \\\\\nI & Z & I & I & I & I & X & I & I \\\\\n};\n\\draw[line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth]\n ([yshift=.2ex] m-5-1.south west) -- ([yshift=.2ex] m-5-9.south east);\n\\end{tikzpicture}\\mkern-5mu\n\\right)\n\\end{equation}\n\n\\begin{equation}\n\\label{gen10}\n\\left(\\mkern-5mu\n\\begin{tikzpicture}[baseline=-.65ex]\n\\matrix[\n matrix of math nodes,\n column sep=.25ex, row sep=-.25ex\n] (m)\n{\nX & X & I & Z & I & Z & Y & Z & Y & Z \\\\\nX & I & Y & X & I & X & Z & X & X & Y \\\\\nX & Z & X & Y & Z & Y & Y & I & I & Y \\\\\nI & I & Z & Z & X & X & Y & Y & I & I \\\\\nZ & I & I & I & Z & Z & X & X & I & X \\\\\nZ & I & I & I & I & I & I & I & I & X \\\\\nI & I & Z & Z & I & I & I & I & I & I \\\\\n};\n\\draw[line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth]\n ([yshift=.2ex] m-5-1.south west) -- ([yshift=.2ex] m-5-10.south east);\n\\end{tikzpicture}\\mkern-5mu\n\\right)\n\\end{equation}\n\n\\begin{equation}\n\\label{gen11}\n\\left(\\mkern-5mu\n\\begin{tikzpicture}[baseline=-.65ex]\n\\matrix[\n matrix of math nodes,\n column sep=.25ex, row sep=-.25ex\n] (m)\n{\nI & Z & X & I & X & Z & I & Z & X & X & X \\\\\nI & Z & Z & X & I & I & Z & X & X & Y & Y \\\\\nZ & I & I & Z & X & X & Z & X & X & X & I \\\\\nX & X & I & X & Y & X & I & Y & Y & Y & X \\\\\nY & Y & I & X & X & Y & Y & Z & Y & I & Y \\\\\nZ & I & I & I & I & I & I & I & X & I & I \\\\\nI & Z & I & I & I & I & I & I & X & I & I \\\\\n};\n\\draw[line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth]\n ([yshift=.2ex] m-5-1.south west) -- ([yshift=.2ex] m-5-11.south east);\n\\end{tikzpicture}\\mkern-5mu\n\\right)\n\\end{equation}\n\nNote that in each case, the generators above the dotted line define a pure $\\left[\\!\\left[n,n-5,2\\right]\\!\\right]_{2}$ quantum code.\n\nThe next theorem describes families of hybrid quantum codes. Notice that $2^{2m+5} \\equiv 2^5 \\pmod{3}$, so the length $n$ given in the theorem is well-defined.\n\n\\begin{theorem} \nLet $m$ be a nonnegative integer and $n$ a positive integer given by\n$$n=\\frac{2^{2m+5}-32}{3}+a,$$\nwhere the parameter $a$ is a small positive integer that is specified below. Then there exists \n\\begin{compactenum}[(a)]\n\\item an $\\left[\\!\\left[n,n-2m-6\\!:\\!1,3\\right]\\!\\right]_{2}$ hybrid code for $a=7$ and \n\\item an $\\left[\\!\\left[n,n-2m-7\\!:\\!2,3\\right]\\!\\right]_{2}$ hybrid code for $a=9,10,11$.\n\\end{compactenum}\n\\end{theorem}\n\\begin{proof}\nRoughly speaking, we construct our code by partitioning the first $\\left(2^{2m+5}-32\\right)\\!\/3$ qubits into disjoints sets, forming a perfect code on each partition, and use one of the four small hybrid codes on the remaining last $a$ qubits. These codes are then ``glued\" to one another by using stabilizer pasting. Other than a small number of degenerate errors introduced by the small hybrid code that must be handled individually, each single-qubit Pauli error has a unique syndrome, allowing for the correction of any single-qubit error.\n\nWe will now describe the code construction in more detail. We \ntake the $n=\\left(2^{2m+5}-32\\right)\\!\/3+a$ qubits and partition them into disjoint sets \n$$U_{m}\\cup U_{m-1}\\cup\\cdots\\cup U_{1}\\cup V_{a},$$ \nwhere $\\left\\lvert U_{k}\\right\\rvert=2^{2k+3}$ and $\\left\\lvert V_{a}\\right\\rvert=a$.\nThe set $U_{m}$ contains the first $2^{2m+3}$ qubits, $U_{m-1}$ the next $2^{2m+1}$ qubits, and so forth. The final $a$ qubits are contained in $V_a$. \n\nLet $k$ be an integer in the range $1\\le k\\le m$. On the qubits in the set $U_{k}$, we can construct a stabilizer code of length $2^{2k+3}$ with $2k+5$ stabilizer generators, following Gottesmann~\\cite{Gottesman1996b}. The $2k+5$ stabilizer generators are given as follows. Two of these generators are the tensor product of only Pauli-$X$ and $Z$ operators, which we call $X_{U_{k}}$ and $Z_{U_{k}}$ respectively. We define the other $2k+3$ stabilizers by\n\\begin{equation*}\n\\mathcal{S}_{j}^{k}=X^{h_{j}}Z^{h_{j-1}+h_{1}+h_{2k+3}},\n\\end{equation*}\nfor $j\\in\\left[2k+3\\right]$. Here we let $h_{j}$ be the $j$-th row of the $\\left(2k+3\\right)\\times2^{2k+3}$ matrix $H_{k}$, whose $i$-th column is the binary representation of $i$, $h_{0}$ is defined to be the all-zero vector, and $X^{h_{j}}=X^{h_{j,0}}X^{h_{j,1}}\\dots X^{h_{j,2^{2k+3}-1}}$, with $Z^{h_{j}}$ defined similarly.\n\nFor the set $V_{a}$, let $H_{j}^{\\mathcal{Q}}$ be the generators of the quantum stabilizer $\\mathcal{S}_{\\mathcal{Q}}$ of the length $a$ hybrid code defined by the generators in (\\ref{gen7}), (\\ref{gen9}), (\\ref{gen10}), or (\\ref{gen11}), and $H_{j}^{\\mathcal{C}}$ be the generators of the classical stabilizer $\\mathcal{S}_{\\mathcal{C}}$ (since the length 7 hybrid code only has one generator in $\\mathcal{S}_{\\mathcal{C}}$, we can remove $H_{2}^{\\mathcal{C}}$). The stabilizer can be pasted together as shown in (\\ref{stabpastgen}), where suitable identity operators should be inserted in the blank spaces:\n\n\\begin{equation}\n\\label{stabpastgen}\n\\left(\\mkern-5mu\n\\begin{tikzpicture}[baseline=-.65ex]\n\\matrix[\n matrix of math nodes,\n column sep=.25ex, row sep=-.25ex\n] (m)\n{\nX_{U_{m}} & & & & & \\\\\nZ_{U_{m}} & & & & & \\\\\nS_{1}^{m} & X_{U_{m-1}} & & & & \\\\\nS_{2}^{m} & Z_{U_{m-1}} & & & & \\\\\n\\vdots & \\vdots & \\ddots & & & \\\\\nS_{2m-6}^{m} & S_{2m-8}^{m-1} & \\cdots & & & \\\\\nS_{2m-5}^{m} & S_{2m-7}^{m-1} & \\cdots & X_{U_{2}} & & \\\\\nS_{2m-4}^{m} & S_{2m-6}^{m-1} & \\cdots & Z_{U_{2}} & & \\\\\nS_{2m-3}^{m} & S_{2m-5}^{m-1} & \\cdots & S_{1}^{2} & X_{U_{1}} & \\\\\nS_{2m-2}^{m} & S_{2m-4}^{m-1} & \\cdots & S_{2}^{2} & Z_{U_{1}} & \\\\\nS_{2m-1}^{m} & S_{2m-3}^{m-1} & \\cdots & S_{3}^{2} & S_{1}^{1} & H_{1}^{\\mathcal{Q}} \\\\\nS_{2m}^{m} & S_{2m-2}^{m-1} & \\cdots & S_{4}^{2} & S_{2}^{1} & H_{2}^{\\mathcal{Q}} \\\\\nS_{2m+1}^{m} & S_{2m-1}^{m-1} & \\cdots & S_{5}^{2} & S_{3}^{1} & H_{3}^{\\mathcal{Q}} \\\\\nS_{2m+2}^{m} & S_{2m}^{m-1} & \\cdots & S_{6}^{2} & S_{4}^{1} & H_{4}^{\\mathcal{Q}} \\\\\nS_{2m+3}^{m} & S_{2m+1}^{m-1} & \\cdots & S_{7}^{2} & S_{5}^{1} & H_{5}^{\\mathcal{Q}} \\\\\n & & & & & H_{1}^{\\mathcal{C}} \\\\\n & & & & & H_{2}^{\\mathcal{C}} \\\\\n};\n\\draw[line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth]\n ([yshift=.2ex] m-15-1.south west) -- ([yshift=.2ex] m-15-6.south east);\n\\end{tikzpicture}\\mkern-5mu\n\\right)\n\\end{equation}\n\nSuppose that we have an single-qubit Pauli error on the block $U_{m}$. Since the code is pure, the syndrome of each error will be distinct and such that the Pauli-$X$, $Y$, and $Z$ sydromes will start with $01$, $11$, and $10$ respectively. However, this leaves all of the syndromes starting with $00$ unused, so Pauli-$X$, $Y$, and $Z$ errors on the block $U_{m-1}$ will have distinct syndromes starting with $0001$, $0011$, and $0010$ respectively. Continuing on, any single-qubit Pauli error occurring on the block $U_{k}$ will have a distinct syndrome starting with $2\\left(m-k\\right)$ $0$s.\n\nAll of the syndromes of errors occurring on the block $V_{a}$ start with $2m$ $0$s. Here our code is not pure, but it is almost pure, with the only degenerate errors being the weight 2 errors in $\\mathcal{S}_{\\mathcal{C}}$. For example, when $V_{a}$ has 11 qubits, it will have three weight 1 degenerate errors: $Z_{1}$ (a Pauli-$Z$ on the first qubit of the block), $Z_{2}$, and $X_{9}$, each with the syndrome $00011$ (preceeded by $2m$ zeros). If we measure this syndrome, we apply the operator $ZZIIIIIIXII$ to the state, which maps the original codeword to itself up to a global phase. Note, however, that while this global phase is the same for codewords of the same inner code for a given error, it may differ for codewords from different inner codes. In fact, this is exactly what prevents the outer code from being a distance 3 quantum code rather than a distance 3 hybrid code. The argument for when $V_{a}$ has 7, 9, and 10 qubits is similar.\n\nSince we know how to correct any single-qubit Pauli error based on its syndrome, each of the codes must have minimum distance 3.\n\\end{proof}\n\nHere we show that these hybrid codes are better than optimal quantum stabilizer codes using a result of Yu et al. \\cite{Yu2013}.\n\n\\begin{proposition}\nLet $m$ be a nonnegative integer and $n$ a positive integer given by\n$$n=\\frac{2^{2m+5}-32}{3}+a,$$\nwhere $a\\in\\left\\{7,9,10,11\\right\\}$. Then there does not exist an $\\left[\\!\\left[n,n-2m-5,3\\right]\\!\\right]_{2}$ stabilizer code.\n\\end{proposition}\n\\begin{proof}\nWhen $a=7,9,10$, we have \n\\begin{align*}\nn & = \\frac{2^{2m+5}-32}{3}+a \\\\\n& = \\frac{2^{2m+5}-8}{3}+\\left(a-8\\right) \\\\\n& = \\frac{8}{3}\\left(4^{m+1}-1\\right)+\\left(a-8\\right).\n\\end{align*}\nBy a result of Yu et al. \\cite[Theorem 1]{Yu2013}, distance 3 stabilizer codes with lengths of the form $$\\frac{8}{3}\\left(4^{k}-1\\right)+b,$$ where $b\\in\\left\\{-1,1,2\\right\\}$, can exist if and only if $$2m+5\\geq \\left\\lceil\\log_{2}\\!\\left(3n+1\\right)\\right\\rceil+1.$$ But in this case we have\n\\begin{align*}\n\\left\\lceil\\log_{2}\\!\\left(3n+1\\right)\\right\\rceil+1 & = \\left\\lceil\\log_{2}\\!\\left(2^{2m+5}+3a-31\\right)\\right\\rceil+1\\\\\n & > \\left\\lceil\\log_{2}\\!\\left(2^{2m+5}-2^{2m+4}\\right)\\right\\rceil+1 \\\\\n & = 2m+5,\n\\end{align*}\nso when $a=7,9,10$, there is no distance 3 stabilizer code of length $n$.\n\nWhen $a=11$, a different case of \\cite[Theorem 1]{Yu2013} applies, so distance 3 stabilizer codes with lengths of this form can exist if and only if $$2m+5\\geq \\left\\lceil\\log_{2}\\!\\left(3n+1\\right)\\right\\rceil.$$ However, this gives us\n\\begin{align*}\n\\left\\lceil\\log_{2}\\!\\left(3n+1\\right)\\right\\rceil & = \\left\\lceil\\log_{2}\\!\\left(2^{2m+5}+2\\right)\\right\\rceil\\\\\n & > \\left\\lceil\\log_{2}\\!\\left(2^{2m+5}\\right)\\right\\rceil \\\\\n & = 2m+5,\n\\end{align*}\nso when $a=11$, there is likewise no distance 3 stabilizer code of length $n$.\n\\end{proof}\n\nAs with our family of error-detecting hybrid codes, it would be interesting to know whether any of these codes meet the linear programming bounds from Section \\ref{lpb}. Since none of the hybrid codes we started with meet these bounds, it is doubtful that any of the hybrid codes constructed from stabilizer pasting would also meet this bound, leaving it unclear whether or not these codes are optimal among all hybrid codes.\n\n\\section{Conclusion and Discussion}\nIn this paper we have proven some general results about hybrid codes, showing that they can always detect more errors than comparable quantum codes. Furthermore we proved the necessity of impurity in the construction of genuine hybrid codes. Additionally, we generalized weight enumerators for hybrid stabilizer codes to nonadditive hybrid codes, allowing us to develop linear programming bounds for nonadditive hybrid codes. Finally, we have constructed several infinite families of hybrid stabilizer codes that provide an advantage over optimal stabilizer codes.\n\nBoth of our families of hybrid codes were inspired by the construction of nonadditive quantum codes. In hindsight this is not very surprising, as the examples of hybrid codes with small parameters given by Grassl et al. \\cite{Grassl2017} were constructed using a CWS\/union stabilizer construction. Most interesting is that all known good nonadditive codes with small parameters have a hybrid code with similar parameters. This would suggest that looking at larger nonadditive codes such as the quantum Goethals-Preparata code \\cite{Grassl2008} or generalized concatenated quantum codes \\cite{Grassl2009} might be helpful in constructing larger hybrid codes. Alternatively, it may be possible to use the existence of hybrid codes to point to where nonadditive codes may be found. For instance the existence of an $\\left[\\!\\left[11,4\\!:\\!2,3\\right]\\!\\right]_{2}$ hybrid code suggests a nonadditive code with similar parameters might exist.\n\nAs previously suggested by Grassl et al. \\cite{Grassl2017}, one possible way to construct new hybrid codes with good parameters is to start with degenerate quantum codes with good parameters. Another possible approach to constructing new hybrid stabilizer codes is to find codes such that there are few small weight errors that are in the normalizer but not in the stabilizer, and then add those small weight errors to the generating set of the stabilizer to get a degenerate code. Here, the original code becomes the outer code of the hybrid code and the degenerate code the inner code.\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nThe top partner holds a special place in many extensions of the\nStandard Model~\\cite{bsm_review}. As the fermion with the\nlargest coupling to the Higgs field, the top gives the largest\nquadratic correction to the Higgs mass term. To have a natural and\nuntuned cancellation of this term, we would expect the supersymmetric\ntop squark --- the stop (${\\tilde t}$) --- to be close in mass to the top itself.\nAdditionally, in generic supersymmetric flavor models the large top Yukawa\ndrives the mixing of left-- and right--handed stops and pushes the\nlightest stop mass eigenstate to be the lightest squark. Experimentally, however,\nno evidence of a relatively light stop has been obtained in collider searches.\nA combination of \nATLAS~\\cite{atlas_stops}\nand\nCMS~\\cite{cms_stops}\nresults at 7 and 8 TeV excludes stop pair production decaying to final\nstates containing an invisible, stable supersymmetric particle ({\\em e.g.}, the lightest neutralino, $\\tilde{\\chi}^0$) for stop\nmasses in the range of $100- 750$~GeV, assuming a massless invisible\ndecay product.\\bigskip\n\nNevertheless, in the two-dimensional plane of ${\\tilde t}$ and $\\nz{}$ masses, there remains a\nnotable window in the experimental exclusion regions: neither experiment has ruled out the\npossibility that stop pair production events may be buried top in production when the mass difference\n$\\mst -(\\mne{} + m_t)$ becomes small.\nThere is a simple explanation for this lack of sensitivity to stop \nproduction near the ``degeneracy line:'' when the mass\nsplitting is small, the invisible particles ($\\nz{}$) carry little momentum,\nso the final state from stop pair production closely mimics \nthat of top pairs in the Standard Model. In principle, measurable differences in the\nmissing transverse energy ($\\slashed{E}_T$) distributions would for fully hadronic top decays would appear if\nstop events are also present, a feature that might allow discovery or exclusion of degenerate stops~\\cite{degenerate_stops_had}.\nIn practice, however, such searches face challenging jet combinatorics and require\nprecise understanding of the background $\\slashed{E}_T$. In the di- or semi-leptonic channels, kinematic variables built from the decay\nproducts of the top are nearly identical for $t\\bar{t}$\nand $\\st{} \\st{}^*$ events, assuming the stop decays to either (a) an on-shell\nor off-shell top and an invisible $\\nz{}$, or (b) a bottom quark and a\nchargino, where the latter decays into a $\\nz{}$ and a $W^{(\\ast)}$ boson. Analyzing differences in the top production angles or top\ndecay products have been suggested~\\cite{more_light_stops} to\nsearch for stop pairs contaminating the top sample, but the possible\nimprovement is small and can be washed out by necessary trigger and\nselection criteria.\\bigskip\n\n\n\nIn this study we\nexplore an alternative approach for distinguishing top and stop pair production that avoids these difficulties. Specifically, \nwe show how correlations between tagging jets can be used to search\nfor stop pairs in the top pair sample at the\nLHC~\\cite{kaoru} independent of the stop decays. In particular, we consider the difference in the\nazimuthal angles $\\Delta\\phi$ of forward jets produced in association\nwith the top or stop pair in vector boson fusion (VBF)\nevents.\\footnote{Here, the fusing vector bosons are primarily gluons,\n justifying the term ``VBF.''} These jets arise from initial state\nradiation. The information in their $\\Delta \\phi$ distribution can be\nused regardless of decay channels, as long as we can manage to extract a\nsignal-rich sample. As was originally demonstrated in the context of\nHiggs\nphysics~\\cite{delta_phi,higgs_spin},\nthe difference in azimuthal angle between the two forward jets $\\Delta\n\\phi$ from weak--boson--fusion events inherit information about the\nhelicities of the weak bosons involved in the production. From the\nunderlying argument it is obvious that this technique can be generalized to gluon\nfusion~\\cite{higgs_spin,delta_phi_gg}. The helicities that can participate in a\ngiven process are set by the Lorentz structure of the production\nmatrix element, and so for pair production the distribution of\n$\\Delta\\phi$ is sensitive to properties of pair-produced particles such as\nspin and CP assignment.\\bigskip\n\nFor the pair production process of interest here, the resulting differential cross section has the form\n\\begin{equation}\n\\label{eq:delphidist}\n\\frac{d\\sigma}{d\\Delta\\phi} = \nA_0 + A_1 \\cos \\Delta\\phi+A_2 \\cos (2\\Delta\\phi)\\ \\ \\ ,\n\\end{equation}\nwhere the expansion coefficients $A_k$ encode the interplay of the underlying pair production\namplitude and the helicity of the fusing gluons. As shown in our earlier work~\\cite{matt_michael}, \nthe sign of $A_2$ is set by the spins of the produced particles: $A_2>0$ for scalars and $A_2<0$ \nfor fermions. In general, this sensitivity could provide a powerful technique for\ndiagnosing the spin of any new particles that may be discovered at the LHC~\\cite{matt_michael}. \nThis is also the case for top pair production close to threshold, while in the\nrelativistic limit the sum of the two azimuthal angles is the more\nsensitive observable~\\cite{kaoru}. \nIn the present context, we show how one may exploit the same effect to identify or exclude the presence\nof stop pairs in the region of parameter space near the degeneracy line. Moreover,\nwe describe how the $\\cos (2\\Delta\\phi)$ correlation between initial state radiation jets\ncan be reliably described in event simulations that take into account parton showering and realistic\ndetector jet identification and show that the correlation is not washed \nout through azimuthal decorrelation \\cite{daCosta:2011ni,Khachatryan:2011zj} To our knowledge, this study represents\nthe first such demonstration, indicating that study of azimuthal tagging jet correlations may be a realistic\ntool in other contexts as well~\\cite{matt_michael,kaoru}. \\bigskip\n\nBefore determining if the degenerate stop production could be hiding in top pair production at the LHC, one should ask whether the measured cross section for top\npair production allows for such a scenario. This rate has been measured\nnumerous\ntimes~\\cite{atlas_top_0,cms_top_0,atlas_top_1,cms_top_1,tevatron_top}\nand agrees with theoretical predictions~\\cite{Czakon:2013goa} within\nuncertainties. In Table~\\ref{tab:xsection}, we show the measured top\npair cross sections at the Tevatron and the LHC, along with the\ntheoretical predictions and the supersymmetric stop pair production\ncross sections for light stop masses of 175 and 200~GeV. At first glance,\nthe measured cross section would appear to rule out the addition\nof a stop with mass near that of the top. However, it is unclear how\nthe top cross section measurements would respond to an admixture of\nstop events, and there may be a degeneracy between the cross section\nand top mass measurements. Short of a detailed analysis of this question that goes beyond the scope of the present study, we cannot rule out the possibility -- however unlikely -- that a 175~GeV stop could be hiding inside the top sample. Moreover, a stop with mass around 200~GeV, still within the degeneracy window, is not in significant tension with the\nexperimental results, given the uncertainties. Consequently, we will consider two benchmark cases, corresponding to \n$(\\mst,\\mne{})=(175,1)$ GeV and $(200, 25)$ GeV, respectively.\n\n\\begin{table}[t]\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n$\\sqrt{s}$~[TeV] & $\\sigma_{t\\bar{t}}~$[pb] & $\\sigma_{t\\bar{t}}~$[pb] & $\\sigma_{\\st{} \\st{}^*}$~[pb] & $\\sigma_{\\st{} \\st{}^*}$~[pb] \\\\ & experiment & theory & $\\mst = 175~{\\ensuremath\\rm GeV}$ & $\\mst = 200~{\\ensuremath\\rm GeV}$ \\\\ \\hline\n1.96 & $7.68\\pm0.20_\\text{stat}\\pm0.36_\\text{sys}$~(CDF+D\\O\\ \\cite{tevatron_top}) \n & $7.164{^{+0.110}_{-0.200}}_\\text{scale}{^{+0.169}_{-0.122}}_\\text{pdf}$ & 0.587 & 0.252 \\\\ \\hline\n7 & $\\begin{array}{c} 177\\pm3_\\text{stat} \\pm {^8_7}_\\text{sys}\\pm7_\\text{lumi}~\\text{(ATLAS)} \\\\ \\pm3_\\text{stat} \\pm {^8_77}_\\text{sys}\\pm7_\\text{lumi}~\\text{(CMS)} \\end{array}$\n & $172.0{^{+4.4}_{-5.8}}_\\text{scale}{^{+4.7}_{-4.8}}_\\text{pdf}$ & 24.0 & 11.9 \\\\ \\hline\n8 & $\\begin{array}{c} 238\\pm2_\\text{stat} \\pm 7_\\text{sys}\\pm7_\\text{lumi}\\pm 4_{\\text{beam~$E$}}~\\text{(ATLAS~\\cite{atlas_top_1})} \\\\ 227 \\pm 3_\\text{stat} \\pm 11_\\text{sys}\\pm 10_\\text{lumi}~\\text{(CMS \\cite{cms_top_1})} \\end{array}$\n & $245.8{^{+6.2}_{-8.4}}_\\text{scale}{^{+6.2}_{-6.4}}_\\text{pdf}$ & 34.5 & 17.3 \\\\ \\hline\n14 & -- & $953.6{^{+22.7}_{-33.9}}_\\text{scale}{^{+16.2}_{-17.8}}_\\text{pdf}$ & 135 & 72.1 \\\\ \\hline\n\\end{tabular}\n\\caption{Cross sections for top and stop pair production at the 1.96\n TeV Tevatron and 7, 8, and 14 TeV LHC. The theoretical predictions\n for the $t\\bar{t}$ cross sections are calculated at NNLO+NNLL, for\n $m_t = 173.3$~GeV~\\cite{Czakon:2013goa}. Cross sections for stop\n pair production are calculated at NLO in {\\tt\n Prospino2}~\\cite{prospino}\n with a light $\\st{1}$ and all other supersymmetric particles\n decoupled.}\n\\label{tab:xsection}\n\\end{table}\n\nOur discussion is organized as follows. In Section~\\ref{sec:spin} we explain the physics behind the\n$\\Delta\\phi$ correlations of VBF tagging jets in the specific cases of\ntop and stop pair production. In Section~\\ref{sec:simulation} we then\ndiscuss the simulation of these events including multi-jet merging in\n{\\tt MadGraph5}. While in the default setup the correlations between\nthe tagging jets are not guaranteed to be included we show how they can be accounted\nfor. In the same section we study the tagging jet correlations at\nparton level and show how a dedicated analysis can separate top and\nstop contributions to a mixed event sample. In\nSection~\\ref{sec:searches} we confirm that using realistic cuts and a\nfast detector simulation these results can be reproduced.\n\n\\section{Tagging jet correlations}\n\\label{sec:spin}\n\nWe are interested in top and stop pairs with two associated tagging\njets, produced primarily via initial state radiation, or equivalently,\nthrough VBF\ndiagrams~\\cite{tagging}. Eventually, to separate\nVBF production from all other sources of jets we will\nemploy strict selection cuts, primarily requiring the jets to be\nforward. A representative Feynman diagram is shown in\nFigure~\\ref{fig:feynman}, defining our notation for the different\nmomenta. The full gauge-invariant matrix element will be the sum of\nmany diagrams, but the cuts will emphasize this topology's contribution\nto the amplitude. In our simulations, we will\ninclude all initial parton states, though in practice gluons dominate for\nthe parameter range of interest.\nIt is most convenient to write the relevant kinematics\nin the three frames shown in\nFigure~\\ref{fig:kinematics}~\\cite{delta_phi}. The emission of\nthe fusing vector bosons (gluons in our case) from the incoming\npartons are described in the Breit frames (frames~I and II), defined\nby the gluon momenta being purely space--like and in the\n$z$-direction:\n\\begin{alignat}{5}\nq_1^\\mu & = k_1^\\mu - k_3^\\mu = (0,0,0,Q_1), \\notag \\\\\nq_2^\\mu & = k_2^\\mu - k_4^\\mu = (0,0,0,-Q_2) \\; .\n\\end{alignat}\nThe top\/stop pair production frame shown as frame~X in\nFigure~\\ref{fig:kinematics} is defined as the frame in which\n$q_1^\\mu+q_2^\\mu = (\\sqrt{\\hat{s}},\\vec{0})$, where $\\hat{s} \\equiv\n(p_1+p_2)^2$ is the invariant mass of the top or stop\npair.\\bigskip\n\n\\begin{figure}[b!]\n\\includegraphics[width=0.23\\textwidth]{.\/feyn_VBFtop.pdf}\n\\caption{A representative Feynman diagram for the VBF process $pp\\to\n t\\bar{t}+j j$ with two tagging jets. Similar diagrams exist for stop\n pair production. The initial and final state partons can be quarks,\n anti-quarks, or gluons. The different channels contributing to the\n hard $gg \\to t\\bar{t}$ scattering are denoted by a solid dot.}\n\\label{fig:feynman}\n\\end{figure}\n\n\\begin{figure}[t]\n \\includegraphics[width=0.6\\textwidth]{.\/kinematics.pdf}\n\\caption{Kinematics for VBF events, showing the two Breit frames~I and\n II and the production frame~X~\\cite{delta_phi}.}\n\\label{fig:kinematics}\n\\end{figure}\n\nWe now focus on the dependence of the differential cross section on\nthe azimuthal angles $\\phi_1$ and $\\phi_2$. As long as the tagging\njets with the momenta $k_3$ and $k_4$ are forward, the $z$-axis shared by frames~I, II,\nand X is nearly collinear with the experimental beam axis. As a first\nstep we can approximate the observed azimuthal angles in the\nlaboratory frame by the angles in the plane orthogonal to the top or\nstop momenta~\\cite{higgs_spin}. The matrix element for the full\nVBF event takes the form\n\\begin{equation}\n{\\cal M} = \\sum_{h_1,h_2} \n{\\cal M}_\\text{I}^\\mu(h_1,\\phi_1,\\theta_1)\n{\\cal M}_\\text{II}^\\nu(h_2,\\phi_2,\\theta_2)\n{\\cal M}_\\text{X}^{\\mu\\nu}(h_1,h_2,\\Theta) \\; ,\n\\end{equation}\nwhere $h_1,h_2=-1,0,+1$ are the helicities of the gluons $q_1$ and\n$q_2$, measured relative to the $z$-axis, so $h_1 = +1$ is positive\nangular momentum for $q_1$, but $h_2 = -1$ is positive angular\nmomentum for $q_2$. We suppress the dependence on the color factors. A\nboost is required to take each matrix element from its individual\nframe to a common center--of--mass frame. All these boosts will be in\n$z$-direction and will not induce additional dependence on the\nazimuthal angles $\\phi_i$. Therefore, $\\phi_1$ and $\\phi_2$ enter\nonly as phases of the Breit matrix elements,\n\\begin{alignat}{5}\n{\\cal M}_\\text{I}(h_1,\\phi_1,\\theta_1) &= \n{\\cal M}_\\text{I}(h_1,0,\\theta_1) \\; e^{+ih_1\\phi_1}, \\notag \\\\\n{\\cal M}_\\text{II}(h_2,\\phi_2,\\theta_2) &= \n{\\cal M}_\\text{II}(h_2,0,\\theta_2) \\; e^{-ih_2\\phi_2}.\n\\end{alignat}\nWe can rewrite $\\phi_1$ and $\\phi_2$ in terms of their difference\n$\\Delta \\phi \\equiv \\phi_1-\\phi_2$ and their sum $\\phi_+ \\equiv\n\\phi_1+\\phi_2$~\\cite{delta_phi,higgs_spin}. The angle $\\phi_+$ is\nphysically unobservable without reference to the top or stop\nproduction plane, which we will not attempt to reconstruct, and so it\ncan be integrated over. Abbreviating the six-body phase space factors as\n$({\\cal PS})$ and the integration over all other angles as $d\\Omega$,\nthe differential cross section with respect to $\\Delta\\phi$ can be\nwritten as\n\\begin{equation}\n\\frac{d\\sigma}{d\\Delta \\phi} = ({\\cal PS}) \\int d\\Omega\\sum_{h_1^{(')},h_2^{(')}} \ne^{i\\Delta h \\; \\Delta\\phi\/2}\n\\left[{\\cal M}_\\text{I}^\\mu(h_1){\\cal M}_\\text{I}^{\\mu'*}(h_1')\\right]\n\\left[{\\cal M}_\\text{II}^\\nu(h_2){\\cal M}_\\text{II}^{\\nu'*}(h_2')\\right]\n\\left[{\\cal M}_\\text{X}^{\\mu\\nu}(h_1,h_2){\\cal M}_\\text{X}^{\\mu'\\nu'*}(h_1',h_2') \\right] \\; ,\n\\end{equation}\nwith $\\Delta h = h_1-h_1'+h_2-h_2'$. This distribution has to be\ninvariant under the shift $\\Delta \\phi \\to \\Delta \\phi+2\\pi$, which\ntranslates into the condition $\\Delta h = 0, \\pm 2, \\pm 4$. Terms\nwith odd $\\Delta h$ must vanish, and larger values of $\\Delta h$\ncannot be generated for $|h_j| \\le 1$ (allowing for\noff-shell gluons). We then expand the exponential\nwith the helicities in sines and cosines and, assuming CP conservation,\nignore the complex sine contributions. The three allowed helicity\nchanges $\\Delta h$ give rise to the three coefficients of\nEq.~(\\ref{eq:delphidist}),\n\\begin{alignat}{5}\nA_n & = ({\\cal PS}) \\int d\\Omega \\sum_{\\Delta h = \\pm n} \n\\left[{\\cal M}_\\text{I}^\\mu(h_1){\\cal M}_\\text{I}^{\\mu'*}(h_1')\\right]\n\\left[{\\cal M}_\\text{II}^\\nu(h_2){\\cal M}_\\text{II}^{\\nu'*}(h_2')\\right]\n\\left[{\\cal M}_\\text{X}^{\\mu\\nu}(h_1,h_2){\\cal M}_\\text{X}^{\\mu'\\nu'*}(h_1',h_2') \\right] \\; .\n\\label{eq:diffsigma}\n\\end{alignat}\nWe will be most interested in $A_2$, where $\\Delta h= \\pm 4$. This can only\nbe satisfied by the unique configuration $h_1 = h_2 = \\pm 1$ and $h_i'\n= -h_i$.\\bigskip\n\nFrom explicit calculation, the contribution from the matrix\nelements for gluon emission, {\\sl i.e.} \\,${\\cal M}_\\text{I}(h_1)^\\mu{\\cal\n M}_\\text{I}(-h_1)^{\\mu'*}$ and ${\\cal M}_\\text{II}(h_2)^\\nu{\\cal\n M}_\\text{II}(-h_2)^{\\nu'*}$ for $h_i = \\pm 1$, are all\npositive~\\cite{delta_phi}. As a result, the sign of $A_2$\ndepends only on the sign of the pair production interference terms\n${\\cal M}_\\text{X}^{\\mu\\nu}(h,h){\\cal M}_\\text{X}^{\\mu'\\nu'*}(-h,-h)$,\nwith $h = \\pm 1$. That is, the sign of $A_2$ depends on the relative\nsign between the matrix element for pair production where the total\nincoming $z$-component of angular momentum is $+2$, and the matrix\nelement where the incoming $J_z = -2$ .\n\nAn explicit calculation of these interference terms in the case of the\nfusion of abelian gauge bosons shows that, for the production of\nscalars, these interference terms are overall positive, while for\nfermion production, the terms are overall\nnegative~\\cite{matt_michael}. We can now repeat this calculation in\nthe case of QCD-coupled heavy quarks~\\cite{kaoru} or squarks. The\nresults are made more clear by multiplying the matrix elements in\nframe~X by polarization vectors for the virtual gluons $q_1$ and\n$q_2$, treating them as approximately on-shell. Recalling that\npositive helicity for both gluons is defined relative to the $z$-axis,\nrather than relative to the gluon momentum, both sets of polarization\nvectors can be written as $\\epsilon_{1\/2}^\\pm = \n(0,1,\\pm i,0)\/\\sqrt{2}$.\\bigskip\n\nWe begin with the fermionic case. For top pairs, the relevant\nproduction matrix elements times polarization vectors in frame~X are\n\\begin{alignat}{5}\n\\left[{\\cal M}^{\\mu\\nu} _\\text{X}(h,h)\\right]^{s,s} \\epsilon_\\mu(h)\\epsilon_\\nu(h) = & \n- \\; \\; \\; ig_s^2 \\; 2s \\; \n\\left( \\{T^a,T^b\\}+\\beta\\cos\\Theta [T^a,T^b] \\right) \\; \n\\beta \\sqrt{1-\\beta^2} \\; \\frac{\\sin^2\\Theta}{1-\\beta^2\\cos^2\\Theta} \\notag \\\\\n\\left[{\\cal M}^{\\mu\\nu} _\\text{X}(h,h)\\right]^{s,-s} \\epsilon_\\mu(h)\\epsilon_\\nu(h) = & \n- h \\; ig_s^2 \\; 2s \\; \n\\left(\\{T^a,T^b\\}+\\beta\\cos\\Theta [T^a,T^b] \\right) \\; \n\\beta \\qquad \\sin\\Theta \\frac{1- 2s h\\cos\\Theta}{1-\\beta^2\\cos^2\\Theta} \\; . \n\\label{eq:fermionspin}\n\\end{alignat}\nThe angle $\\Theta$ is defined in Figure~\\ref{fig:kinematics}. The\nsuperscripts $s,s$ or $s,-s$ for $s= \\pm 1\/2$ denote the helicities of\nthe top and anti-top, measured relative to each of their momenta. In terms of the total\nproduction energy $\\hat{s}$ the\nvelocity of the top and anti-top $\\beta$ is $\\beta= \\sqrt{1-4m^2\/\\hat{s}}$.\n\nNotably, the matrix elements for production of a $t\\bar{t}$ pair with\nthe same helicity assignments Eq.~\\eqref{eq:fermionspin} do not have\nthe property that ${\\cal M}_X(+1,+1) \\times {\\cal M}_\\text{X}(-1,-1)^*\n< 0$, contrary to our expectations. However, the signs of the $s,-s$\nmatrix elements with opposite helicity are manifestly asymmetric, as ${\\cal\n M}_\\text{X}^{s,-s}(h,h) \\propto h$, so this product is indeed\nnegative. The fact that one term is not clearly negative could be\nconcerning for our argument, but by inspection it is clear that the negative\nterms are strictly larger in magnitude than the positive\ncontributions. It is possible that the $\\beta$ dependence of the\n$A_2$ term could be useful in an experimental analysis. Cuts placed on\nthe top decay products could be used to enhance particular ranges of $\\beta$~\\cite{kaoru},\nenhancing or suppressing the interference effect and providing useful \nside-bands. We will not further investigate this possibility in this paper. \n\\bigskip\n\nTurning to the stop pair production, the relevant matrix elements are\n\\begin{equation}\n{\\cal M}_\\text{X}^{\\mu\\nu}(h,h)\\epsilon_\\mu(h)\\epsilon_\\nu(h) = \nig_s^2 \\; \n\\left(\\{T^a,T^b\\}+\\beta\\cos\\Theta [T^a,T^b] \\right) \\; \n\\frac{\\beta^2\\sin^2\\Theta}{1-\\beta^2\\cos^2\\Theta} \\; .\n\\label{eq:scalarM}\n\\end{equation}\nClearly this does not depend on the gluon helicities $h$, and so the\ninterference terms are positive. This results in a positive $A_2$ term\nfor stop pair production, and thus, the sign of $A_2$ can be used to\ndistinguish the production of scalar stops and fermionic tops. Note\nthat these two calculations only demonstrate that the top and stop\ndistributions will have opposite signs of their $A_2$ components,\nwithout addressing the relative magnitudes. To answer that question,\nwe must turn to Monte Carlo simulation.\n\n\\section{Simulating VBF (S)Tops}\n\\label{sec:simulation}\n\n\\begin{figure}[b!]\n\\includegraphics[width=0.245\\textwidth]{.\/matching_pt_0.pdf}\n\\includegraphics[width=0.245\\textwidth]{.\/matching_pt_1.pdf}\n\\includegraphics[width=0.245\\textwidth]{.\/matching_pt_2.pdf}\n\\includegraphics[width=0.245\\textwidth]{.\/matching_pt_3.pdf}\n\\caption{Normalized $p_T$ distributions of the four leading jets in\n the merged $t\\bar{t}$ samples, with {\\tt xqcut}=20 (black), 40 (red),\n and 60~GeV (blue). We use anti-$k_T$ jets with $\\Delta R = 0.5$,\n and require $p_T> 20$~GeV and $|\\eta_j|<5$.}\n\\label{fig:qcut}\n\\end{figure} \n\nIn order to extract information on the spin of the heavy top or stop\nparticles from tagging jets we need to ensure that our simulation\nkeeps all relevant spin correlations. Naively, this can be guaranteed by\ngenerating events for the hard processes $\\st{}\\st{}^* jj$ and\n$t\\bar{t}jj$~\\cite{skands,matt_michael,kaoru}.\nHowever, the transverse momentum of the tagging jets will often be\nsignificantly below the energy scale of this hard\nprocess. In that region of phase space, for example the transverse momentum \nspectrum of jet radiation is only properly\ndescribed once we include the parton shower or other implementation of Sudakov factors. In standard\nshowering algorithms the\nprobabilistic parton shower is (usually) averaged over the helicities\nof the participating partons. In such simulations, any apparent spin\ncorrelation between the hard process and the tagging jets --or between\nthe tagging jets themselves-- comes only from kinematic\nconstraints~\\cite{skands}, rather than from a combination of kinematics and underlying \ninterference effects. What we need is a merged description\nof the parton shower and the hard matrix element, where the tagging jets are\ngenerated through the matrix element.\\bigskip\n\nTo that end, we consider two benchmark parameter points for stop signals for stop pair\nproduction followed by a decay into a top and a missing energy particle,\n\\begin{alignat}{5}\npp \\to \\st{} \\st{}^* \\to (t \\nz{}) \\; (\\bar{t} \\nz{}) \n\\qquad \\qquad \n(\\mst,\\mne{}) =\n\\begin{cases} (175, 1)~{\\ensuremath\\rm GeV} \\\\ (200,25)~{\\ensuremath\\rm GeV} \\; .\\end{cases}\n\\end{alignat}\nThe invisible particles coming from a\nprompt decay can be a neutralino or a gravitino. As we are not\nclosely investigating the stop and top decay patterns we will refer to the\ngeneric missing energy particle as $\\nz{}$.\n\nFor the background and each signal benchmark we generate events for\nthe pair production of stops and tops at the 14 TeV LHC with up to\nthree extra jets in {\\tt MadGraph5}~\\cite{mg5,mlm}, matching the\njets to {\\tt Pythia6}~\\cite{pythia} and using anti-$k_T$\njets with $R=0.5$~\\cite{fastjet} down to a matching scale {\\tt\n xqcut}=20~GeV. This choice (endorsed by the {\\tt MadGraph} authors \\cite{madgraphonline})\nensures that the spin correlations in\nthe tagging jets are kept, provided the two tagging jets are chosen\nfrom the three leading jets that do not originate from top decay. We will compare these results to\nunmatched hard $t\\bar{t}jj$ and $\\st{} \\st{}^*jj$\nevents~\\cite{kaoru}. In this section we do not keep\ntrack of the top and stop decays. The two tagging jets are the two\nhardest jets which fulfill all $p_T$ and $\\Delta \\eta$\nrequirements.\\bigskip\n\nIn order to ensure that all final state jets in {\\tt MadGraph5} are\ngenerated by the matrix element and hence include all spin and angular\ncorrelations, we can move the matching scale to values below the\ntransverse momenta for all potential tagging jets, {\\tt xqcut}$<\np_{T,j}$.\\footnote{We have confirmed that for events with {\\tt\n xqcut}$> p_{T,j}$ the correlations between the tagging jets in\n {\\tt MadGraph} are indeed lost.} While this choice will hugely\ndecrease the efficiency of the event generation, because a very large\nfraction of events will be vetoed to generate the Sudakov suppression,\nit will ensure that our events include all the necessary\ninformation. Because the matching scale is not a physical parameter,\nit can be varied within a reasonable range, where we will see that the\ndefinition of `reasonable' is different for kinematic distributions\nand the total rate.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.245\\textwidth]{.\/pythia_dphi1.pdf}\n\\includegraphics[width=0.245\\textwidth]{.\/pythia_dphi2.pdf}\n\\includegraphics[width=0.245\\textwidth]{.\/pythia_dphi3.pdf}\n\\includegraphics[width=0.245\\textwidth]{.\/pythia_dphi4.pdf}\n\\caption{Normalized $\\Delta\\phi$ distributions for the two\n highest-$p_T$ forward jets at parton level, requiring $\\Delta \\eta_{jj} >\n 1,2,3,4$. We show top pairs (blue) and stop pairs (red) matched to three jets,\n as well as the unmatched two-jet samples for tops (cyan) and stops (purple). \n We also show the best fits to the functional form $A_0+A_1\\cos\\Delta\\phi+A_2\\cos\n (2\\Delta\\phi)$. For the stop samples, the $(\\mst,\\mne{}) = (175,1)$~GeV scenario is shown\n with a solid line, while $(\\mst,\\mne{}) = (200,25)$~GeV is shown\n with a dotted line.}\n\\label{fig:partondphi}\n\\end{figure}\n\nBefore we study the spin correlation between the tagging jets we test\nif our choice of the matching scale, {\\tt xqcut}=20~GeV,\nleads to stable and consistent results. To this end we show the\n$p_T$ distributions for the first four jets for top pair production in\nFigure~\\ref{fig:qcut}. This distribution directly probes the Sudakov\nsuppression and should therefore be most sensitive to artifacts from\nthe choice of the matching scale. We vary the matching scale from\n20~GeV to 40~GeV and the default value of 60~GeV. We see that the\ndistributions are essentially indistinguishable between the three\nsamples over the entire range of $p_T$, so our choice of scales does \nnot present any problems for the tagging jet distributions.\n\nOn the other hand, the combined cross sections from {\\tt MadGraph}\nshow a wider variation, with $\\sigma_{t\\bar{t}} = 2.9,~ 1.3,$ $0.94$,\nand $0.71$~nb for {\\tt xqcut}=20, 40, 60, and 100~GeV. Given that\nmulti-jet merging is based on a combination of leading order matrix\nelements and a leading logarithmic parton shower, this variation\nreflects the uncertainty of a leading order cross section with four\npowers of $\\alpha_s$. For smaller values of {\\tt xqcut} we include\nmore and more real emission as described by the full matrix element,\nbut only compensated for by approximate virtual corrections in the\nSudakov factor. If we apply an external normalization of the total\nproduction rate, for example to the precision predictions shown in\nTable~\\ref{tab:xsection} we can use a {\\tt MadGraph} event samples\nwith the matching scale of 20~GeV to accurately simulate the\nproduction of top or stop pairs plus jets.\\bigskip\n\nWe can now consider the distribution of forward jets in top or stop events. \nIn this Section, we will focus on confirming\nthe existence and the sign of the $A_2$ terms, as derived from the interference pattern described in \nSection~\\ref{sec:spin}. Moreover, we need to test if our event generation \nindeed captures all relevant physics.\nTo be independent of the details of the top decay,\nwe use Monte Carlo truth to distinguish between associated\njets and those from top decay. For specific top decays it should be\nstraightforward to distinguish between ISR jets and decay jets, as has been shown\nfor direct production of supersymmetric particles~\\cite{susy_isr}, for \nweak--boson--fusion pair production of supersymmetric\nparticles~\\cite{wbf_isr}, and for sgluon pair\nproduction~\\cite{sgluon_isr}, as we will demonstrate shortly. We then place selection criteria on\nour 3-jet matched or 2-jet unmatched samples in order to isolate VBF-type production from all other\ndiagrams that generate two or more jets in association with stops or\ntops. Adapting the criteria used for WBF Higgs\nselection~\\cite{delta_phi,wbf_isr}, we begin by requiring at\nleast two parton--level jets in the merged sample with\n\\begin{equation}\np_{T,j} > 20~{\\ensuremath\\rm GeV}, \\qquad \\qquad \\qquad\n|\\eta_j|<5, \\qquad \\qquad \\qquad \n\\Delta \\eta_{jj} > 1, 2, 3, 4 \\; .\n\\label{eq:vbf_cuts}\n\\end{equation}\nThe increasing rapidity separation should emphasize the VBF-induced \nangular correlations between the tagging jets~\\cite{higgs_spin}.\nMore realistic selection criteria will be put in place once we include\na fast detector simulation in Section~\\ref{sec:searches}.\\bigskip\n\n\\begin{table}[t]\n\\begin{footnotesize}\n\\begin{tabular}{ll|c|c|c|c|c|c|c|c} \\hline\n& & \\multicolumn{2}{c|}{$|\\Delta \\eta_{jj}|>1$} & \\multicolumn{2}{c|}{$|\\Delta \\eta_{jj}|>2$} \n & \\multicolumn{2}{c|}{$|\\Delta \\eta_{jj}|>3$} & \\multicolumn{2}{c}{$|\\Delta \\eta_{jj}|>4$} \\\\ \n& & $A_1\/A_0$ & $A_2\/A_0$ & $A_1\/A_0$ & $A_2\/A_0$ & $A_1\/A_0$ & $A_2\/A_0$ & $A_1\/A_0$ & $A_2\/A_0$ \\\\ \\hline \n\\multirow{2}{*}{$t\\bar{t}$} \n& 2-jet & $-0.016\\pm0.03$ & $+0.005\\pm 0.001$ & $-0.07\\pm0.01$ & $-0.021\\pm0.004$ & $-0.08\\pm0.01$ & $-0.035\\pm0.006$ & $-0.07\\pm0.01$ & $-0.05\\pm0.01$ \\\\\n& 3-jet & $-0.08\\pm0.01$ & $+0.009\\pm0.002$ & $-0.13\\pm0.02$ & $-0.018\\pm0.003$ & $-0.13\\pm0.02$ & $-0.048\\pm0.008$ & $-0.12\\pm0.02$ & $-0.07\\pm0.01$ \\\\ \\hline\n$\\st{}\\st{}^*$\n& 2-jet & $-0.0023\\pm0.0003$ & $+0.07\\pm0.01$ & $-0.06\\pm 0.01$ & $+0.08\\pm 0.01$ & $-0.07\\pm0.01$ & $+0.12\\pm0.02$ & $-0.06\\pm0.02$& $+0.15\\pm0.02 $ \\\\ \n(175,1)\n& 3-jet & $-0.07\\pm0.01$ & $+0.10\\pm0.02$ & $-0.12\\pm0.02$ & $+0.12\\pm0.02$ & $-0.12\\pm0.02$ & $+0.18\\pm0.03$ & $-0.11\\pm0.02$ & $+0.25\\pm0.04$ \\\\ \\hline\n$\\st{}\\st{}^*$\n& 2-jet & $+0.007\\pm0.001$ & $+0.07\\pm0.01$ & $-0.05\\pm0.01$ & $+0.07\\pm0.01$ & $-0.06\\pm 0.01$ & $+0.11\\pm 0.02$ & $-0.05\\pm0.01$ & $+0.15\\pm0.02$ \\\\ \n(200,25) \n& 3-jet & $-0.06\\pm0.01$ & $+0.10\\pm0.02$ & $-0.10\\pm0.02$ & $+0.12\\pm0.02$ & $-0.11\\pm0.02$ & $+0.17\\pm0.03$ & $-0.09\\pm0.02$& $+0.24\\pm0.04$ \\\\ \\hline\n\\end{tabular}\n\\end{footnotesize}\n\\caption{Best-fit values for the $\\cos\\Delta\\phi$ and $\\cos\n (2\\Delta\\phi)$ coefficients\n defined in Eq.~\\eqref{eq:diffsigma}. The fits are\n performed at parton level, corresponding to \n Figure~\\ref{fig:partondphi}. The 3-jet matched (2-jet unmatched) top background sample before any\n cuts consists of $1.95 \\times 10^6$ ($3.09 \\times 10^6$) events, the $\\mst =175$~GeV\n stop sample is $5.65 \\times 10^5$ ($6.18 \\times 10^6$) events, and the $\\mst =200$~GeV\n sample is $1.08\\times 10^6$ ($9.24\\times 10^5$) events.}\n\\label{tab:partonvbf}\n\\end{table}\n\nIn Figure~\\ref{fig:partondphi} we plot the normalized $\\Delta\\phi$\ndistributions between the two highest-$p_T$ parton--level tagging jets\ndefined in the laboratory frame, requiring $\\Delta \\eta_{jj} > 1$,\n2, 3, and 4 in the successive panels. As can be seen, there is a clear difference between the\ntagging jet correlations from stop and top events, corresponding to\nthe sign of the $\\cos (2\\Delta\\phi)$ term. It induces a clearly\nvisible minimum in the stop sample around $\\Delta \\phi = \\pi\/2$, especially\nnoticeable when compared to the slight excess here in the top sample.\nTop pairs are dominated by a slight preference\nfor back-to-back tagging jets.\n\nWithout a $\\Delta \\eta_{jj}$ cut, the non-trivial azimuthal dependence\nwould be highly suppressed. This is expected, since central jets do\nnot predominantly come from the ISR diagrams and do not reflect\ninformation about the helicity of fusing gluons through interference\npatterns in our reference frame. As we enforce increasingly large $\\Delta\\eta_{jj}$ cuts we\nsee a finite $\\cos (2\\Delta\\phi)$ component develop in both the top\nand stop samples; with the appropriate signs for fermionic and scalar\npairs.\n\\bigskip\n\nIn Table~\\ref{tab:partonvbf}, we show the relative size of the\n$\\cos\\Delta\\phi$ ($A_1$) and $\\cos (2\\Delta\\phi)$ ($A_2$) modes for\nthe top background and stop benchmark points, normalized to the\nconstant term $A_0$. The coefficients are obtained from the normalized\nten--bin histograms at parton level, using the standard {\\tt ROOT} fitting\nalgorithm. It is apparent that the non-trivial $A_2$ term is present\nin the unmatched two-jet sample, and survives after the addition of a\nthird jet in the matching scheme. The magnitude of the $A_1$ term\nsignificantly increases for the matched samples. \n\nComparing the events with three merged jets and the events with\nonly two hard jets we see that the merged sample shows an\nadditional shift towards larger azimuthal tagging jet separation. The\nreason is that with a third jet recoiling against the hard top or stop\npair system we now have a choice to pick the two tagging jets. We\nsystematically bias the selection towards an effectively larger\n$\\Delta \\eta_{jj}$ separation translating into more back-to-back\ntagging jets. However, this shift mostly affects the $\\cos \\Delta\n\\phi$ distribution, while the critical $\\cos (2\\Delta\\phi)$ mode is\nsymmetric around $\\Delta \\phi = \\pi\/2$ and therefore just slightly\ntilted. The fact that for top pair production the kinematic effect\nfrom additional jet radiation looks similar to the $\\cos \\Delta \\phi$\nmode from spin correlations explains the surprising finding of \nRef.~\\cite{skands} that the parton shower simulation seems to capture\nsome of the expected spin correlations while it should not.\n\nThe size of $A_2$ is only slightly affected by the\ndifferent simulational approaches shown in Table~\\ref{tab:partonvbf}, {\\sl i.e.} \\, \nthe theory-driven unmerged 2-jet setup and the more realistic merged\n3-jet case. If anything, the effect in $\\cos (2\\Delta\\phi)$ is more \npronounced in the multi-jet case, contrary to what is observed as \nazimuthal decorrelation in 2-jet production. The two stop mass benchmarks are\nconsistent with each other. Already for $\\Delta \\eta_{jj} >2$ we\nobserve the expected sign difference between the fermionic and scalar\nprocesses. It will become an experimental issue how wide a\nrapidity separation of the two tagging jets is needed to extract the\nmost information with a limited sample size.\n\n\\section{Stop Searches}\n\\label{sec:searches} \n\n\\begin{table}[b!]\n\\begin{tabular}{ll|c|c|c|c}\n\\hline\n & & \\multicolumn{2}{c|}{$|\\eta_j|<2.5,~|\\Delta \\eta_{jj}|>2$} & \\multicolumn{2}{c}{$|\\eta_j|<4.5,~|\\Delta \\eta_{jj}|>3$} \\\\\n & & di-leptonic & semi-leptonic & di-leptonic & semi-leptonic \\\\ \\hline \n\\multirow{5}{*}{$t\\bar{t}$} & leptons & 3.2\\% & 29\\% & 3.2\\% & 29\\%\\\\\n & +$b$-tag \\& jets & 0.17\\% & 0.98\\% & 0.23\\% & 1.5\\%\\\\\n & +$W$-mass & -- & 0.19\\% & -- & 0.25\\%\\\\\n & +$|\\Delta \\eta|$ & 0.053\\% & 0.066\\% & 0.061\\% & 0.064\\%\\\\\n & Final $\\sigma$ & 505~fb & 629~fb & 582~fb & 610~fb\\% \\\\ \\hline\n\\multirow{5}{*}{$\\st{} \\st{}^*$ (175,1)} & leptons & 3.3\\% & 29\\% & 3.3\\% & 29\\%\\\\\n & +$b$-tag \\& jets & 0.14\\% & 0.87\\% & 0.19\\% & 1.3\\%\\\\\n & +$W$-mass & -- & 0.17 \\% & -- & 0.23\\%\\\\\n & +$|\\Delta \\eta|$& 0.041\\% & 0.060\\% & 0.048\\% & 0.058\\% \\\\\n & Final $\\sigma$ & 55~fb & 81~fb & 65~fb & 78~fb \\\\ \\hline\n\\multirow{5}{*}{$\\st{} \\st{}^*$ (200,25)} & leptons & 3.3\\% & 29\\% & 3.3\\% & 29\\%\\\\\n& +$b$-tag \\& jets & 0.17\\% & 1.1\\% & 0.23\\% & 1.6\\%\\\\\n & +$W$-mass & -- & 0.22\\% & -- & 0.28\\%\\\\\n & +$|\\Delta \\eta|$ & 0.050\\% & 0.076\\% & 0.057\\% & 0.069\\% \\\\ \n & Final $\\sigma$ & 36~fb & 55~fb & 41~fb & 50~fb \\\\ \\hline\n\\end{tabular}\n\\caption{Cumulative efficiencies, including branching\n ratios, after detection selection criteria, in both di- and\n semi-leptonic channels. Also shown is the cross section\n after all cuts are applied. The ``leptons'' cut requires two (one) $e$ or $\\mu$\n for the di-lepton (semi-leptonic) channel. Two $b$-tagged and two (four)\n or more non-$b$-tagged jets are required to pass ``$b$-tag \\& jets,'' and the semi-leptonic\n $W$-mass reconstruction is defined in the text. The final $|\\Delta \\eta|$ criteria\n is applied for both jet selection criteria as defined\n in Eq.~\\eqref{eq:jet_def}.}\n\\label{tab:efficiencies}\n\\end{table}\n\n\nThe results obtained in the last section at parton level and using Monte--Carlo truth\nclearly demonstrate the analytic argument of Section~\\ref{sec:spin}.\nOnce all helicity information is taken into account and kinematic cuts\nrestrict events to the VBF phase space, the stop events have a\npositive coefficient $A_2$, while the top background has a negative\n$A_2$. However, these results do not yet demonstrate that this\ndifference between scalars and fermions can be used to enhance the\nstop sample among tops in a real experiment. One might worry that the identification of the \ntagging jets, combinatorics, or detector effects could wash out\nthese correlations and make them experimentally invisible.\\bigskip\n\nTo confirm the experimental accessibility of the azimuthal correlation\nas a way to separate top pairs from stop pairs we now hadronize the\nparton level event samples with {\\tt Pythia} and apply the fast\ndetector simulation {\\tt Delphes3}~\\cite{delphes} with\nconfiguration files provided by the Snowmass Energy Frontier\nsimulations~\\cite{snowmass}. Jets\nare clustered using the anti-$k_T$~\\cite{fastjet} algorithm\nwith $R = 0.5$. All decays are included via {\\tt Pythia}, so we do\nnot systematically account for spin correlations and interference\npatterns in the production and decay processes. From the last section\nit is clear that the details of the top and stop decays play no role\nin our analysis, beyond triggering and combinatorial challenges. In\nour analysis we include both semi-leptonic and di-leptonic top pair\ndecays. Fully hadronic decays of tops could be added once we resolve\nQCD and combinatorical issues, discussed for example in\nRefs.~\\cite{combinatorics}.\n\nWe generate the equivalent of 4.8~fb$^{-1}$ of 14 TeV LHC data for the\ntop background and both stop signal points. Although this is much less than the\nplanned integrated luminosity of the next stage of LHC running, generating the corresponding full data set\nwould be extremely resource intensive and not essential for purposes of demonstrating\nthe feasibility of the $\\Delta\\phi$ technique. Indeed, as we will show below, even\nwith only $\\sim 5$ fb$^{-1}$, the interference effect can already make stops known\nin the top sample, though additional luminosity would be required to improve\nthe statistical significance.\\bigskip\n\nDepending on the assumed decay channel we require one or two electrons\nand muons, required to have \n\\begin{alignat}{5}\np_{T,\\ell} > 20~{\\ensuremath\\rm GeV} \\quad \\text{and} \\quad |\\eta_\\ell|<2.5 \\; . \n\\end{alignat}\nRegardless of the selection criteria of forward\njets, we require exactly two $b$-tagged jets \nwith \n\\begin{alignat}{5}\np_{T,b}> 50~{\\ensuremath\\rm GeV} \\quad \\text{and} \\quad |\\eta_b|<2.5 \\; ,\n\\end{alignat}\nusing the {\\tt Delphes3} efficiency of approximately 70\\% per $b$-tag.\nFor the upcoming 14~TeV runs of the LHC, where pile-up and jet energy\ncalibration might be an issue, we follow two potential choices for the\njet requirements,\n\\begin{alignat}{5}\n(1) \\qquad p_{T,j} &> 20~{\\ensuremath\\rm GeV} \\quad \\mbox{and} \\quad |\\eta_j| <2.5 \\notag \\\\\n(2) \\qquad p_{T,j} &> 20~{\\ensuremath\\rm GeV} \\quad \\mbox{and} \\quad |\\eta_j| <4.5 \\; .\n\\label{eq:jet_def}\n\\end{alignat} \nWhile the conservative assumption will prove to be sufficient to\nreveal the presence of degenerate stops, including tagging jets to\n$|\\eta|<4.5$ will improve the physics reach in\nthis type of search.\n\nFor the di-leptonic channel, we require two or more light-flavor\njets. In the semi-leptonic channel we require four or more jets. Due\nto limited statistics, in the di-leptonic channel we do not subdivide the events \ninto different lepton flavor\ncombinations, though this could be useful for a full experimental\nanalysis. Similarly, a full experimental analysis might find it\nuseful to include a systematic multi-jet analysis for tagging jets as well\nas decay jets~\\cite{moments_wbf},\nbut in this paper we limit ourselves to the cleanest possible\nsignature.\n\nTo differentiate the $W$-decay jets from the VBF tagging jets in the\nsemi-leptonic channel, we suggest the following reconstruction\nalgorithm: of all pairs of central ($|\\eta_j|<1$) jets passing\na staggered cut $p_{T,j} > 60,30$~GeV we take the pair with an invariant mass\nclosest to $m_W$. If an event has such a pair of jets and their\ninvariant mass is within 30~GeV of the $m_W$, it is retained for the\nVBF selection criteria. The two highest-$p_T$ QCD jets remaining must then have an\ninvariant mass of either less than 50~GeV or greater than 100~GeV, to\navoid possible misidentification with the $W$-boson decay\nproducts. This strict set of requirements provides a very clean sample\nof events where the two VBF jets are well separated from all other\nhadronic activity in the detector, though the efficiency is\ncorrespondingly low, and improvements on this algorithm are obviously possible.\n\nThe highest-$p_T$ non-$W$-tagged jets in the semi-leptonic sample and the\nhighest-$p_T$ jets in di-leptonic events are likely be the two tagging jets, so we apply\nthe $\\Delta\\eta_{jj}$ cut. In the conservative jet selection\nscenario~(1) with $|\\eta_j|<2.5$ we only require $|\\Delta \\eta_{jj}| >\n2$, in order not to cut too deeply into the efficiency. For the more\noptimistic situation~(2) with $|\\eta_j|<4.5$ we can also require a\nlarger jet separation: $|\\Delta \\eta_{jj}| > 3$. From all events\npassing this final cut we construct the $\\Delta\\phi$ distribution. The\nfinal efficiencies and effective cross sections for both the di- and\nsemi-leptonic channels are shown in\nTable~\\ref{tab:efficiencies}, including the efficiencies of each cut\nleading up to the final $\\Delta \\eta$ selection.\\bigskip\n\n\\begin{table}[t]\n\n\\begin{tabular}{cc|c|c|c|c} \\hline\n & & \\multicolumn{2}{c|}{$|\\eta_j|<2.5,~|\\Delta \\eta_{jj}|>2$} & \\multicolumn{2}{c}{$|\\eta_j|<4.5,~|\\Delta \\eta_{jj}|>3$} \\\\ \n & & di-leptonic $A_2\/A_0$ & semi-leptonic $A_2\/A_0$ & di-leptonic $A_2\/A_0$ & semi-leptonic $A_2\/A_0$ \\\\ \\hline \n\\multicolumn{2}{c|}{$t\\bar{t}$} & $-0.10 \\pm 0.03$ & $-0.05\\pm 0.03$ & $-0.12\\pm 0.03$ & $-0.08 \\pm 0.03$ \\\\ \\hline\n\\multirow{2}{*}{$\\st{} \\st{}^*$ (175,1)} & $\\st{} \\st{}^*$ only & $+0.20\\pm0.09$ & $+0.10\\pm0.07$ & $+0.16\\pm0.09$ & $+0.18 \\pm 0.07$ \\\\ \n & $\\st{} \\st{}^* + t\\bar{t}$ & $-0.07\\pm0.03$ & $-0.03\\pm0.02$& $-0.09 \\pm 0.03$& $-0.05\\pm 0.02$ \\\\ \\hline\n\\multirow{2}{*}{$\\st{} \\st{}^*$ (200,25)} & $\\st{} \\st{}^*$ only & $+0.22\\pm0.11$ & $+0.03\\pm0.08$ & $+0.18 \\pm 0.11$ & $+0.16\\pm 0.10$ \\\\\n & $\\st{} \\st{}^* + t\\bar{t}$ & $-0.08\\pm0.03$ & $-0.04\\pm0.01$ & $-0.10 \\pm 0.03$ & $-0.06\\pm 0.03$ \\\\ \\hline\n\\end{tabular}\n\n\\caption{Best-fit values for the $\\cos\n (2\\Delta\\phi)$ coefficients $A_2$, normalized to the constant term $A_0$,\n defined in Eq.~\\eqref{eq:diffsigma}, for di-leptonic and semi-leptonic events\n corresponding to 4.8~fb$^{-1}$ of luminosity, after fast detector simulation. \n Fits to the two stop signal points are performed for signal only as well as\n signal plus top background.}\n\\label{tab:delphesvbf}\n\\end{table}\n\nBased on the 4.8~fb$^{-1}$ of simulated signal and background data\n given in Table~\\ref{tab:delphesvbf}, we can extrapolate\nwhat integrated luminosities would be required to observe a significant number\nof stop pair events inside the top sample. Clearly, the statistical errors from 5~fb$^{-1}$\nof integrated luminosity would be too large to make any statement,\nas the difference between the background distribution and the background\nplus signal is equivalent to the fit uncertainties. \n\nHowever, by taking the central fit values of the $d\\sigma\/\\Delta\\phi$ differential\ndistribution as the `true' parameter values, we can determine the statistical \npower for a given amount of data. \nThe luminosity from the first year of LHC14 running is expected to be around 25~fb$^{-1}$. \nThis data set would reduce the statistical errors on the $A_2\/A_0$ parameter to \napproximately $1\\%$. This would allow a $\\sim 1.5\\sigma$ statistical\ndifferentiation between background and background plus signal for 175~GeV\ntops in the di-leptonic channel ($\\sim 1\\sigma$ for 200~GeV stops) in the current detector \nconfiguration, and somewhat less in the semi-leptonic channel. With improved jet tracking in the forward region, this might be\nimproved to $1.7\\sigma$ with a year's luminosity. With a data set of 100~fb$^{-1}$, \n$3.2\\sigma$ observation would be possible in both channels for 175~GeV\nstops, and $2\\sigma$ discovery for 200~GeV stops, assuming the\nconservative $|\\eta|$ requirements. This would be improved to\n$3.7\\sigma$ for 175 GeV ($2.4\\sigma$ for 200 GeV) stops assuming the\ndetector performance allows for $|\\eta| <4.5$ in the tagging jets.\n\nSuch statements do not include systematic errors, which are clearly of concern for an observable \nso dependent on jet reconstruction and identification. However, analysis of tagging jets\nhas already been proven to work in Higgs studies with the 8~TeV run.\nMoreover, as noted in this paper, several handles are available to allow experimental \ncontrol of these issues. The signal will be visible in \nboth semi-leptonic and di-leptonic decays, and with sufficient luminosity the\ndi-leptonic channel could be further broken down into the different flavor\ncombinations. The turn-on of the non-trivial $A_2$ signal as the $\\Delta \\eta$\ncut is instituted provides an important cross check, and it is possible\nthat selection cuts intended to isolate the $\\beta$ dependence~\\cite{kaoru} of the\ntop and stop signals will also define useful side-bands.\n\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nDuring the first LHC run tagging jets have been shown to be powerful tools\nin observing Higgs decays to photons, $W$-boson, and tau-leptons. In the \ncoming LHC runs with almost twice the collider energy their role will become\neven more pronounced, also reaching beyond Higgs analyses.\nSimilar to the spin and CP studies based on weak--boson--fusion Higgs\nevents~\\cite{delta_phi,higgs_spin}, we can test top quark properties\nin top pair production with two forward\njets~\\cite{kaoru,matt_michael}. This tagging jet analysis has the general \nadvantage that it does not rely on the reconstruction of the hard process, in our\ncase the top pair. Instead, we can use the dependence on the azimuthal angle\n$\\Delta\\phi$ between the tagging jets to search for non-standard events in the top\nsample at the LHC. Specficially, the coefficient $A_2$ of the $\\cos (2 \\Delta \\phi)$ \nterm in the distribution is negative for top pair production, whereas \nlight scalar top pairs will give a significant positive\ncontribution to this observable.\\bigskip\n\nWe first showed how the different signs can be understood in terms of\nthe gluon helicity combinations contributing to the total rate. We\nthen established and tested a non-standard {\\tt MadGraph5} setup which\nallows us to simulate events with all angular correlations between the\nISR tagging jets intact. Using this modified generation tool we showed\nthat the precision on the extraction of $A_2$ increases with the\nrapidity separation of the tagging jets. We also saw that the $A_2$\nmode is not sensitive to the details of the ISR tagging jet simulation\nand the model parameters in the stop decays. Finally, we estimated\nthat such an analysis should give $>3\\sigma$ results in multiple channels\nwith around 100~inverse femtobarns of data at a 14~TeV LHC. Because the analysis is purely\nbased on the tagging jets is can be generalized to any hard process in\nand beyond the Standard Model.\\bigskip\n\n\\begin{center}\n{\\bf Acknowledgments}\n\\end{center}\n\nWe would like to thank Stefan Prestel for checking that our {\\tt\n MadGraph5} simulation makes sense. MB would like to\nthank Maria Spiropulu, Joe Lykken, Yuri Gershtein, and John-Paul Chou\nfor helpful discussion and resources. MB and MJRM thank the Aspen\nCenter for Physics, where this project was originally conceived,\nwhile TP foolishly skipped the workshop. Finally, TP would like to \nthank Frank Krauss for deep insights into azimuthal decorrelation. This work was supported in part by U.S. Department of Energy contract DE-SC0011095 (MJRM).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\nThe very first stage in a high-energy heavy-ion collision is dominated by extremely strong {\\it chromo-electromagnetic} (chromo-EM) fields reflecting colliding nuclei filled with high-density gluons (color glass condensate). Such a state with strong fields is called a ``glasma'' which is named since it is a transitional state between a color {\\it glass} condensate (before the collision) and a quark-gluon {\\it plasma} (QGP)~\\cite{Lappi:2006fp}. The glasma is characterized by a field strength ${\\cal F}$ of the order of the saturation scale: $g{\\cal F}\\sim Q_{s}^2$ (with $g$ being the QCD coupling). Notice that the saturation scale $Q_s$ is a semihard scale representing a typical transverse momentum of gluons in a colliding nucleus and can become large enough, at high energies, compared to light quark masses $Q_s\\gg m_q$. Besides, it has long been known that heavy-ion collisions, with electrically charged nuclei, are accompanied by {\\it electromagnetic} (EM) fields, but only recently was it seriously recognized that the strong EM fields could affect time evolution of heavy-ion collision events since the strength $F$ of the EM fields could be as large as or even greater than the nonperturbative QCD scale $\\Lambda_{\\rm QCD}$, namely $eF\\simge \\Lambda_{\\rm QCD}^2$ and thus $eF\\gg m_q^2$ \\cite{Kharzeev:2007jp, Skokov:2009qp, Bzdak:2011yy, Deng:2012pc}. Since both the chromo-EM and EM fields created in heavy-ion collisions can be strong enough compared with the light quark masses, the effects of strong fields cannot be treated as perturbation (even though the coupling constants are small), but must be treated in a nonperturbative way. Then we expect nonlinear and nonperturbative phenomena associated with the strong fields to occur. Typical examples of such phenomena include particle productions (quarks, antiquarks and gluons) from these strong fields (the Schwinger mechanism), which must be a key towards understanding the formation of QGP.\n\n\nWhile the (coherent) chromo-EM fields will disappear as the QGP is formed, the EM fields could survive longer due to Faraday's law, which works in the presence of a conducting medium~\\cite{Tuchin:2013ie, Gursoy:2014aka}. If the EM fields survive at a strong enough level until the formation of QGP, and even until the end of the QGP's lifetime, we need to describe the QCD phase transition \nwith the effects of strong EM fields taken into account. Notice that the effects of strong {\\it magnetic} fields on thermodynamical or fundamental quantities of QGP can be investigated in lattice QCD simulations, and are indeed found to be large. For example, at zero temperature, lattice QCD simulations confirmed the ``magnetic catalysis'' as predicted in several effective models \\cite{Gusynin:1994re, Gusynin:1994xp, Kashiwa:2011js, Gatto:2010pt, Kamikado:2013pya, Cohen:2007bt, Andersen:2012dz, Andersen:2012zc} in which the value of chiral condensate increases with increasing magnetic field strength. On the other hand, at finite temperature, lattice QCD simulations almost at the physical point concluded \\cite{Bali:2012zg, Bali:2011qj} that the magnetic catalysis does not necessarily occur at all the temperature regions, but rather gets weakened and even shows opposite behavior with increasing temperature. Such behavior of the chiral condensate around the critical temperature is called ``magnetic inhibition'' \\cite{Fukushima:2012kc} or ``inverse magnetic catalysis'', which eventually gives rise to decreasing critical temperature.\nFor recent reviews on the phase diagram of chiral phase transitions in strong magnetic fields, see, e.g., Refs.~\\cite{Andersen:2014xxa, Miransky:2015ava}.\nFurthermore, it is reported~\\cite{Bruckmann:2013oba} that the (pseudo)critical temperature of the confinement-deconfinement phase transition (for the Polyakov loop) also decreases with increasing magnetic field. This is achieved by increasing Polyakov loop expectation values. Probably, these two phenomena are related to each other. However, so far, there is no clear explanation about the physical mechanism behind this (for recent attempts, see Refs.~\\cite{Kojo:2012js, Kojo:2013uua} and \\cite{Braun:2014fua, Mueller:2015fka}).\n\n\nWe can investigate these two aspects, namely the nonlinear and nonperturbative dynamics of strong fields (including particle production) and the phase transition under strong external fields, within a single framework of an effective action. So far, effective actions for QED and QCD in various external conditions have been extensively explored. First of all, Euler and Heisenberg derived a nonlinear effective action for constant EM fields at the electron's one-loop level, known as the Euler-Heisenberg (EH) action~\\cite{Heisenberg:1935qt}. Later, Schwinger reproduced the same action in a field-theoretical manner, which is the so-called Schwinger proper time method~\\cite{Schwinger:1951nm}. The EH action at finite temperature is computed in imaginary time formalism~\\cite{Dittrich:1979ux, Gies:1998vt} as well as in real time formalism~\\cite{Cox:1984vf, Loewe:1991mn}. Furthermore, an analog of the EH action in QCD (for chromo-EM fields) has been evaluated too within a similar method at zero and finite temperatures \\cite{Savvidy:1977as, Matinyan:1976mp, Nielsen:1978rm, Leutwyler:1980ma, Schanbacher:1980vq, Dittrich:1983ej, Cea:1987ku, Cho:2002iv, Dittrich:1980nh, Gies:2000dw}. Lastly, the most recent progress was to compute the EH action at zero temperature when both the EM and chromo-EM fields are present, which was done by one of the authors and B.~V.~Galilo and S.~N.~Nedelko independently~\\cite{Ozaki:2013sfa, Galilo:2011nh}. The author of Ref.~\\cite{Ozaki:2013sfa} used this effective action to investigate the QCD vacuum (gluon condensate) in the presence of strong magnetic fields. Though all of these are about the effective action for strong fields and choromo-EM condensates, it should be possible to include the Polyakov loop at finite temperature. \nIndeed, an effective action (or potential) for the Polyakov loop at the one-loop level was computed independently by D.~J.~Gross, R.~D.~Pisarski, and L.~G.~Yaffe~\\cite{Gross:1980br}, and by N.~Weiss~\\cite{Weiss:1980rj, Weiss:1981ev}, and the result is called the Weiss potential.\nIn the present paper, we are going to derive an analog of the EH effective action in QCD+QED at finite temperature with the Polyakov loops included. Thus, the result may be collectively called the ``Euler-Heisenberg-Weiss action.\" Our result is also a generalization of the one obtained by H.~Gies~\\cite{Gies:2000dw}, who computed an effective action for the Polyakov loop and the chromo-electric field.\n\n\nThe paper is organized as follows: In the next section, we will derive the effective action for QCD+QED at finite temperature by using the Schwinger proper time method. \nVariables of the effective action are the EM and chromo-EM fields as well as the Polyakov loop, and one can reproduce the previous results (the EH action with QCD+QED fields, the Weiss potential, etc.) in various limits. Then, we discuss some applications of our effective action in Sec. III. First, we investigate quark-antiquark pair production in QCD+QED fields at zero temperature. We obtain the quark production rate in the presence of QCD+QED fields, which allows us to study the quark pair production with arbitrary angle between the EM and chromo-EM fields. Next, we study an effective potential for the Polyakov loop with electromagnetic fields. We find that the magnetic field enhances the explicit center symmetry breaking, while the electric field reduces it. This indicates that the (pseudo)critical temperature of the confinement-deconfinement phase transition decreases (increases) with increasing magnetic (electric) field. Finally, we conclude our study in Sec. IV.\n\n\\begin{comment}\n\\com{[below has been totally rewritten from here]}\n\nIn relativistic heavy ion collisions, extremely strong chromo-electromagnetic fields, so-called glasma \\cite{Lappi:2006fp}, are generated.\nThe strength of the chromo-electromagnetic field might be of order of the saturation scale $\\sim Q_{s}$.\nFurthermore, it has been recognized that strong electromagnetic fields are also created in relativistic heavy ion collisions, whose strengths would reach QCD scale $\\Lambda_{QCD}$ or even exceed it \\cite{Kharzeev:2007jp, Skokov:2009qp, Bzdak:2011yy, Deng:2012pc}. \nTherefore, at the initial stage of relativistic heavy ion collisions, both strong chromoelctromagnetic fields and electromagnetic fields can coexist.\nAfter the impact of the heavy ion collision, matters namely quarks and gluons are produced from the strong chromoelectromagnetic fields (glasma) and thermalized into quark and gluon plasma (QGP).\nProduction mechanisms of matters are thus key to understand the initial condition of relativistic heavy ion collisions and the formation of QGP.\nElectromagnetic fields created there could also give large contributions to quark productions.\n\n\nAs the temperature decreases, QCD phase transitions such as confinement-deconfinement phase transition and chiral phase transition will occur.\nIf the strong fields are still remaining during phase transitions, the fields must strongly affect QCD phase transitions.\nIn association with phase transitions in strong fields, lattice QCD calculations can simulate strongly interacting quark and gluon systems in the presence of strong magnetic fields.\nThey are able to explore $T$-$B$ phase diagram of QCD without notorious sign problem as appeared in finite density lattice QCD.\n\nSeveral effective models including NJL type models \\cite{Gusynin:1994re, Gusynin:1994xp, Kashiwa:2011js, Gatto:2010pt} and chiral perturbation theory \\cite{Cohen:2007bt, Andersen:2012dz, Andersen:2012zc} predict an increasing chiral condensate in the presence of magnetic fields at zero temperature.\nThis phenomenon is known as magnetic catalysis.\nLattice QCD indeed observe magnetic catalysis in both quenched approximation and full QCD simulations.\nHowever at finite temperature, there is a discrepancy between effective models and lattice QCD.\nNamely, lattice QCD at physical point predicts a decreasing critical temperature of chiral phase transition in strong magnetic fields \\cite{Bali:2012zg, Bali:2011qj},\ncalled magnetic inhibition or inverse magnetic catalysis,\nwhile effective models fail to reproduce that.\nFurthermore, F. Bruckmann et. al. \\cite{Bruckmann:2013oba} report that the (pseudo)-critical temperature of confinement-deconfinement phase transition also decreases with increasing magnetic field.\nProbably, those two inverse magnetic catalysises are related to each other.\n\nRecently, authors in \\cite{Braun:2014fua} and \\cite{Mueller:2015fka} successfully reproduce the inverse magnetic catalysis of chiral sector from functional approaches such as Dyson-Schwinger equations and functional renormalization group. \nYet, the physical mechanism behind the inverse magnetic catalysis is not so clear and thus still under discussion.\n\n\nIn order to investigate systems where strong QED and QCD fields coexist, an effective action of the theories can be an important tool.\nMoreover, an effective action containing a order parameter is also useful to study a phase transition.\nSo far, effective actions for QED and QCD in various external conditions have been extensively explored.\nEuler and Heisenberg firstly derive the non-linear effective action for QED at one-loop level, known as the Euler-Heisenberg action \\cite{Heisenberg:1935qt}.\nSchwinger reproduces the same action in a field theoretical manner, which is so-called the Schwinger's proper time method \\cite{Schwinger:1951nm}.\nThe Euler-Heisenberg action at finite temperature has been obtained from imaginary time formalism \\cite{Dittrich:1979ux, Gies:1998vt} as well as real time formalism \\cite{Cox:1984vf, Loewe:1991mn}.\nQCD effective action is also evaluated at zero and finite temperatures in literatures \\cite{Savvidy:1977as, Matinyan:1976mp, Nielsen:1978rm, Leutwyler:1980ma, Schanbacher:1980vq, Dittrich:1983ej, Cea:1987ku, Cho:2002iv, Dittrich:1980nh, Gies:2000dw}.\nRecently, one of the authors, and B. V. Galilo and S. N. Nedelko derive the Euler-Heisenberg action for QCD+QED at zero temperature \\cite{Ozaki:2013sfa, Galilo:2011nh}.\nBy using the effective action, the author of \\cite{Ozaki:2013sfa} investigates QCD vacuum in the presence of the strong magnetic fields.\nIn this paper, we derive effective action for QCD+QED at finite temperatures and discuss some applications of the action.\n\nThis paper is organized as follows.\nIn section II, we derive effective action for QCD+QED by using the Schwinger's proper time method.\nThen, we discuss some applications of our effective action in section III.\nFirst, we investigate quark-antiquark pair productions in QCD+QED fields at zero temperature.\nWe obtain the quark production rate in the presence of QCD+QED fields which allows us to study the quark pair production with arbitrary field configurations.\nNext we study an effective potential for the Polyakov loop with electromagnetic fields.\nWe find that the magnetic field enhances the explicit center symmetry breaking while the electric field reduces it.\nThis indicates that the (pseudo)-critical temperature of confinement-deconfinement phase transition decreases (increases) with increasing magnetic (electric) field.\nFinally, we conclude our study in Section IV.\n\n\\com{[up to here]}\n\\end{comment}\n\n\n\\section{one-loop effective action for QCD+QED at finite temperature}\n\n\nIn this section, we derive the one-loop effective action for QCD+QED at finite temperature. \nThe effective action will be a function of chromo-EM and EM fields, as well as the Polyakov loop. Notice that both the strong fields and the Polyakov loop can be treated as {\\it background fields} so that the background field method is applicable. We will take quantum fluctuations around the background fields up to the second order in the action, and integrate them in the path integral. This corresponds to computing the action at the one-loop level.\n\n\nWe shall begin with the four-dimensional QCD action of the SU$(N_{c})$ gauge group with $N_{f}$ flavor quarks interacting with EM fields:\n\\begin{eqnarray}\nS_{\\rm QCD+QED}\n&=& \\int d^{4}x \\left\\{-\\frac{1}{4} F_{\\mu \\nu}^{a} F^{a \\mu \\nu} - \\frac{1}{4} f_{\\mu \\nu} f^{\\mu \\nu} + \\bar{q} \\left( i \\gamma_{\\mu} D^{\\mu} - M_{q} \\right) q\\right\\} \\, ,\n\\label{QCD+QEDaction}\n\\end{eqnarray}\nwhere the covariant derivative contains gluon fields\\footnote{Throughout the paper, we use $a,b,c$ (and $h$) for adjoint color indices ($a,b,c=1, \\ldots,N_c^2-1$), $i$ for fundamental color indices $(i=1,\\ldots,N_c)$, $\\mu,\\nu,\\alpha,\\beta$ for Lorentz indices, and $f$ for flavor indices $(f=1,\\ldots,N_f)$.} $A^{a}_{\\mu}$ $(a=1,\\ldots,N_c^2-1)$ and U(1) gauge fields $a_{\\mu}$ as\n\\begin{eqnarray}\nD_{\\mu} = \\partial_{\\mu} - igA_{\\mu}^{a} T^{a} - ieQ_{q}a_{\\mu}\\, ,\n\\label{covderivative-all}\n\\end{eqnarray}\nand the gluon and EM field-strength tensors are given by\n$\nF_{\\mu \\nu}^{a}\n= \\partial_{\\mu} A_{\\nu}^{a} - \\partial_{\\nu}A_{\\mu}^{a} + gf^{abc} A_{\\mu}^{b} A_{\\nu}^{c} $ and $\nf_{\\mu \\nu}\n= \\partial_{\\mu} a_{\\nu} - \\partial_{\\nu} a_{\\mu}\\, ,\n$\nrespectively. In this paper, we treat the EM fields just as background fields, and assume that the field strengths are constant so that $\\partial f = 0$. We abbreviate color, flavor, and spinor indices of the quark field in Eq.~(\\ref{QCD+QEDaction}).\nMass and charge matrices of quarks are given by $M_{q} = {\\rm{diag}}(m_{q_{1}}, m_{q_{2}}, \\ldots, m_{q_{N_{f}}} )$ and $Q_{q} = {\\rm{diag}}( Q_{q_{1}}, Q_{q_{2}}, \\ldots, Q_{q_{N_{f}}} )$. As for the gluon field, we apply the background field method and decompose the gluon field into a slowly varying background field ${\\cal A}_{\\mu}^{a}$ and a quantum fluctuation $\\tilde{A}_{\\mu}^{a}$ as\n\\begin{eqnarray}\nA_{\\mu}^{a} = {\\cal A}_{\\mu}^{a} + \\tilde{A}_{\\mu}^{a}\\, .\n\\end{eqnarray}\nHere we employ the covariantly constant field as a background field, which obeys the following condition \\cite{Batalin:1976uv, Gyulassy:1986jq, Tanji:2011di}:\n\\begin{eqnarray}\n{\\cal D}_{\\rho}^{ac} {\\cal F}^{c}_{\\mu \\nu} = 0\\, ,\n\\label{CondtionCovariantConstant}\n\\end{eqnarray}\nwhere the covariant derivative ${\\cal D}_\\mu$ is defined only with respect to the gluon background field: \n\\begin{eqnarray}\n{\\cal D}_{\\mu}^{ac} = \\partial_{\\mu} \\delta^{ac} + g f^{abc} {\\cal A}^{b}_{\\mu}\n\\, , \\label{covderivative-gluon}\n\\end{eqnarray} \nand ${\\cal F}_{\\mu \\nu}^{a} = \\partial_{\\mu} {\\cal A}_{\\nu}^{a} - \\partial_{\\nu} {\\cal A}^{a}_{\\mu} + g f^{abc} {\\cal A}_{\\mu}^{b} {\\cal A}_{\\nu}^{c}$. \nFrom the condition (\\ref{CondtionCovariantConstant}), the field-strength tensor ${\\cal F}^{a}_{\\mu \\nu}$ can be factorized as\n${\\cal F}^{a}_{\\mu \\nu} = {\\cal F}_{\\mu \\nu} n^{a}$, where ${n}^a$ is a unit vector in color space, normalized as ${n}^{a}{n}^{a} = 1$,\nwhereas ${\\cal F}_{\\mu \\nu}$ expresses the magnitude of the chromo-EM field.\nWe further assume that ${\\cal F}_{\\mu \\nu}$ is very slowly varying, satisfying $\\partial_{\\sigma} {\\cal F}_{\\mu \\nu} = 0$, which allows us to obtain the analytic expression of the EH action for QCD, just as in QED. Both ${\\cal F}_{\\mu \\nu}$ and ${n}^a$ are space-time independent. The background field ${\\cal A}^a_\\mu$ is proportional to the color unit vector ${n}^a$ as\n\\begin{eqnarray}\n{\\cal A}_{\\mu}^{a}\n&=& {\\cal A}_{\\mu} {n}^{a}\\, ,\n\\label{BackgroundField}\n\\end{eqnarray}\nand the field-strength tensor ${\\cal F}_{\\mu \\nu}$ has an Abelian form, ${\\cal F}_{\\mu \\nu} = \\partial_{\\mu} {\\cal A}_{\\nu} - \\partial_{\\nu} {\\cal A}_{\\mu}.\n$\nThis background field (\\ref{BackgroundField}) indeed satisfies the condition (\\ref{CondtionCovariantConstant}).\nBy using the background field and the quantum fluctuation, the full gluon field-strength tensor can be decomposed as\n\\begin{eqnarray}\nF^{a}_{\\mu \\nu}\n&=& {\\cal F}_{\\mu \\nu} {n}^{a} + ( {\\cal D}_{\\mu}^{ac} \\tilde{A}_{\\nu}^{c} - {\\cal D}_{\\nu}^{ac} \\tilde{A}_{\\mu}^{c}) + gf^{abc} \\tilde{A}_{\\mu}^{b} \\tilde{A}_{\\nu}^{c}\\, .\n\\end{eqnarray} \nApplying the background gauge for the quantum fluctuation,\n\\begin{eqnarray}\n{\\cal D}^{ac}_{\\mu} \\tilde{A}^{c}_{\\mu} \n&=& 0\\, ,\n\\end{eqnarray}\nwe get the gauge fixed action in the presence of EM fields,\n\\begin{eqnarray}\nS_{\\rm QCD+QED}\n&=& \\int d^{4}x \n\\left[ - \\frac{1}{4} \\left\\{ \n {\\cal F}_{\\mu \\nu} {n}^{a} \n + \\left( {\\cal D}_{\\mu}^{ac} \\tilde{A}_{\\nu}^{c} \n - {\\cal D}_{\\nu}^{ac} \\tilde{A}_{\\mu}^{c} \\right) \n + g f^{abc} \\tilde{A}_{\\mu}^b \\tilde{A}_{\\nu}^{c} \n \\right\\}^{2} \n - \\frac{1}{2 \\xi} ( {\\cal D}^{ac}_{\\mu} \\tilde{A}^{c \\mu} )^{2} \\right. \n \\nonumber \\\\\n&& \\qquad \\quad \\left. \n - \\bar{c}^{a} \\left( {\\cal D}_{\\mu} D^{\\mu} \\right)^{ac} c^{c} \n + \\bar{q}\\left(i \\gamma_{\\mu} D^{\\mu} - M_{q} \\right) q \n - \\frac{1}{4} f_{\\mu \\nu} f^{\\mu \\nu}\n\\right]\\, ,\n\\end{eqnarray}\nwhere $c$ is the ghost field and $\\xi$ is the gauge parameter. \nNotice that one of the covariant derivatives in the ghost kinetic term $D_{\\mu}^{ac}$ and the one in the quark kinetic term $D_{\\mu}$ defined in Eq.~(\\ref{covderivative-all}) contain all the gauge fields.\nThe effective action for the background fields ${\\cal A}_\\mu$ \nand $a_\\mu$\ncan be obtained through the functional integral as \n\\begin{eqnarray}\n{\\rm{exp}} \\Big( i S_{\\rm eff}[{\\cal A}_\\mu, a_\\mu] \\Big)\n&\\equiv& \\int {\\mathscr D} \\tilde{A} {\\mathscr D}c {\\mathscr D} \\bar{c} {\\mathscr D}q {\\mathscr D} \\bar{q} \\ \\ {\\rm{exp}} \\left( i \\int d^{4}x S_{\\rm QCD+QED} \\right)\\, .\n\\end{eqnarray}\nWe perform the functional integral with fluctuations taken up to the second order. This corresponds to evaluating the one-loop diagrams as shown in Fig.~1. \n\\begin{figure}\n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=1.0 \\textwidth]{diagrams.pdf}\n\\vskip -0.1in\n\\end{center}\n\\end{minipage}\n\\caption{\nTypical loop diagrams contributing to the effective action. \nThe field $\\mathcal{A}$ contains both the chromo-EM fields and the Polyakov loop.\n}\n\\end{figure}\nThe gluon, ghost, and quark loop integrations can be separately done, and one finds, respectively,\n\\begin{eqnarray}\n&&\\!\\!\\int\\!\\! {\\mathscr D} \\tilde{A}\\, {\\rm{exp}}\n\\left\\{ \\int \\! d^{4}x \\frac{-i}{2} \\tilde{A}^{a \\mu} \\left[\n- ( {\\cal D}^{2})^{ac} g_{\\mu \\nu} - 2 g f^{abc} {\\cal F}^{b}_{\\mu \\nu} \n\\right] \\tilde{A}^{c \\nu} \\right\\\n\\! ={\\rm{det}} \\! \\left[ - ( {\\cal D}^{2})^{ac} g_{\\mu \\nu} \n - 2 g f^{abc} {\\cal F}_{\\mu \\nu}^{b} \n \\right]^{-\\frac12}, \\nonumber \\\\\n&&\\!\\!\\int\\! {\\mathscr D} c {\\mathscr D} \\bar{c} \\ {\\rm{exp}}\n\\left\\{ i \\int d^{4}x \\ \\bar{c}^{a} \\left[ - ( {\\cal D}^{2} )^{ac} \\right] c^{c} \\right\\}\n= {\\rm{det}} \\left[ - ( {\\cal D}^{2} )^{ac} \\right]^{+1}, \\label{full_actions} \n\\\\\n&&\\!\\!\\int\\! {\\mathscr D} q {\\mathscr D} \\bar{q} \\ {\\rm{exp}} \n\\left\\{ i \\int d^{4}x \\ \\bar{q} \\left(\ni \\gamma_{\\mu} \\hat{\\cal D}^{\\mu} - M_{q} \\right) q \\right\\}\n= {\\rm{det}} \\left[ i \\gamma_{\\mu} \\hat{\\cal D}^{\\mu} - M_{q} \\right]^{+1}.\n\\nonumber\n\\end{eqnarray}\nHere we have taken the Feynman gauge, $\\xi = 1$.\nIn the quark one-loop contribution, the covariant derivative $\\hat{\\cal D}_{\\mu}$ contains both of the background fields ${\\cal A}_\\mu$ and $a_\\mu$:\n\\begin{eqnarray}\n\\hat{\\cal D}_{\\mu}\n&=& {\\cal D}_{\\mu} -ieQ_{q} a_{\\mu}\\nonumber\\\\\n&=& \\partial_{\\mu} - ig {\\cal A}_{\\mu}^{a} T^{a} -ieQ_{q} a_{\\mu}\\, .\n\\label{CovariantDerivativeQuark}\n\\end{eqnarray}\nOn the other hand, the gluon and ghost one-loop contributions contain \n${\\cal D}_\\mu^{ac}$ and ${\\cal F}_{\\mu\\nu}^a$, which only depend on the gluon background field $\\mathcal{A}_\\mu$. This is, of course, because the gluon and ghost fields do not have electric charge and thus cannot interact with EM fields. Since these contributions are the same as in the pure Yang-Mills (YM) theory, we may call these the YM part.\n\n\nSo far, we have not specified the background field ${\\cal A}_\\mu$, but it can contain both the chromo-EM fields and the Polyakov loop. Let us briefly explain how the Polyakov loop is described within our framework. In the pure Yang-Mills theory at finite temperature, there is a confinement-deconfinement transition whose order parameter is given by the Polyakov loop. It is defined by the (closed) Wilson line along the imaginary time ($\\tau$) direction:\n\\begin{eqnarray}\n\\Phi (\\vec{x})\n&=& \\frac{1}{N_{c}} {\\rm{Tr}} \\ \\mathcal{P} \\ {\\rm{exp}} \\left\\{ ig \\int^{\\beta}_{0} d\\tau {A}_{4}^{a} (\\tau, \\vec{x}) T^{a} \\right\\}\\, ,\n\\label{defPolyakovLoop}\n\\end{eqnarray}\nwhere $\\beta = 1\/T$ is the inverse temperature and ${\\cal P}$ stands for a path-ordered product along the imaginary time direction. Indeed, $\\langle \\Phi \\rangle \\to 0$ ($\\langle \\Phi \\rangle \\neq 0$) corresponds to a confining (deconfined) phase, since the negative logarithm of the expectation value of the Polyakov loop can be identified with the free energy of a static quark (a vanishing value of the Polyakov loop implies that the energy of a single quark state is infinity). These two phases are distinguished by the center symmetry. The gauge fields at finite temperature are not necessarily periodic in the direction of imaginary time and can have ambiguity related to the center subgroup $Z_{N_c}$ of the gauge symmetry SU$(N_c)$. This residual symmetry is called the center symmetry and the theory is invariant under gauge transformations which differ at $\\tau = 0$ and $\\tau = \\beta$ by a center element of the gauge group. The Polyakov loop $\\Phi$ transforms as $\\Phi \\to {\\rm e}^{2\\pi i n\/N_{c}}\\Phi$ $(n=0,1,2, \\ldots, N_{c}-1)$. Thus, the values of $\\Phi$ distinguish the center symmetric (confining) phase and the center broken (deconfined) phase. Dynamical quarks, however, explicitly break the center symmetry. Therefore, in QCD, the Polyakov loop should be understood as an approximated order parameter. Still, we can compute an effective action for the Polyakov loop and discuss how a phase transition occurs when external parameters such as temperature are varied. \n\nAn effective action for the Polyakov loop in the pure Yang-Mills theory was obtained in Refs.~\\cite{Gross:1980br, Weiss:1980rj} in the following way:\nWorking in what we now call the ``Polyakov gauge\" for a time-independent field $A_4^a(\\vec x)=\\phi(\\vec x)\\delta^{a3}$ in the SU(2) case, \nthe authors of Refs.~\\cite{Gross:1980br, Weiss:1980rj} performed a functional integral with respect to fluctuations around the field $\\phi(\\vec x)$. \nThis procedure is nothing but the one we explained above where we treated the gluon field $A_\\mu^a$ as a background ${\\cal A}_\\mu^a$ with a fluctuation around it. Besides, as long as we consider a spatially homogeneous and time-independent order parameter $\\bar{\\mathcal{A}}_4^a$, we can have both the Polyakov loop and the chromo-EM fields at the same time. We divide the background field into the constant part and the coordinate-dependent part as \n$\\mathcal{A}_{\\mu}^{a}(x) = (\\bar{\\mathcal{A}}_{\\mu} + \\hat{\\mathcal{A}}_{\\mu}(x)) n^{a}$.\nThe second term gives the real (physical) chromo-EM fields so that $\\mathcal{F}_{\\mu \\nu}^{a} = \\partial_{\\mu} \\mathcal{A}_{\\nu}^{a}(x) - \\partial_{\\nu} \\mathcal{A}_{\\mu}^{a}(x) = (\\partial_{\\mu} \\hat{\\mathcal{A}}_{\\nu}(x) - \\partial_{\\nu} \\hat{\\mathcal{A}}_{\\mu}(x)) n^{a}$, while the first constant term $\\bar{\\mathcal{A}}_{\\mu}$ does not. \nWe want to treat both the chromo-EM fields and the Polyakov loop, and the latter is described at finite temperature. In order to have the both, we specify the transformation of the temporal component of the background field $\\mathcal{A}_0^a(x)$ under the Wick rotation of the coordinate, $x_{0} \\to -ix_{4} = -i\\tau$ and $x_{i} \\to x_{i} \\ (i=1,2,3)$, as follows:\n$\\mathcal{A}_{0}^{a}(x) = (\\bar{\\mathcal{A}}_{0} + \\hat{\\mathcal{A}}_{0}(x)) n^{a} \\to (i\\bar{\\mathcal{A}}_{4} + \\hat{\\mathcal{A}}_{0}(x)) n^{a}$.\nIn this way, the first term gives the Polyakov loop defined in Eq.~(\\ref{defPolyakovLoop}), while the second term remains unchanged to give the real chromo-EM fields.\nWe work in the Polyakov gauge for $\\bar{\\mathcal{A}}_{4}^{a}$ \\cite{Weiss:1980rj}\\footnote{In the literature, the fourth component of the gauge field $\\bar{\\mathcal{A}}^{a}_{4}$ in the Polyakov gauge is often expressed in terms of $N_{c}-1$ real scalar fields. In our formalism, these fields are properly encoded in the color eigenvalues $\\omega_{i} \\ (i=1, \\ldots, N_{c})$ and $v_{h} \\ (h=1, \\ldots, N_{c}^{2} -1)$, which will be defined later. Here, choosing the third \ndirection of the color unit vector---$n^{a} = \\delta^{a 3}$ at finite temperature---we pick up the one particular field $\\bar{\\mathcal{A}}_{4}$ which provides a simple expression for the Poyakov loop as shown in Eq.~(\\ref{simple_Polyakov_loop}). However, in the finial expression of our effective action, it is quite straightforward to keep all the $N_{c}-1$ scalar fields in the color eigenvalues $\\omega_{i}$ and $v_{h}$.}:\n\\begin{eqnarray}\n\\bar{\\mathcal{A}}_{4}^{a} = \\bar{\\mathcal{A}}_{4}\\, \\delta^{3 a}, \\ \\ \\ \\partial_{4} \\bar{\\mathcal{A}}_{4} = 0\\, , \n\\end{eqnarray}\nwhich does not conflict with the covariantly constant condition in Eq.~(\\ref{CondtionCovariantConstant}). Notice that we use this gauge with $\\delta^{a3}$ even for the SU($N_c$) case, and the color unit vector $n^a$ introduced in Eq.~(\\ref{BackgroundField}) should be understood as $n^a=\\delta^{3a}$ at finite temperature.\\footnote{Still, we keep the expression $n^a$ because we will discuss the case at zero temperature.} Following Ref.~\\cite{Weiss:1980rj}, we also introduce a dimensionless field $C$ as \n\\begin{eqnarray}\nC = \\frac{g {\\bar{\\mathcal{A}}_{4} } }{ 2 \\pi T }, \n\\end{eqnarray}\nso that the Polyakov loop is simply given as \n\\begin{eqnarray}\n\\Phi\n&=& {\\rm{cos}} (\\pi C) \\qquad \\qquad \\quad \\ \\ {\\rm{for \\ SU(2) }}\\, , \\nonumber \\\\\n\\Phi\n&=& \\frac{1}{3} \\Big\\{ 1 + 2{\\rm{cos}}( \\pi C) \\Big\\} \\quad \\ {\\rm{for\\ SU(3)}}\\, . \\label{simple_Polyakov_loop}\n\\end{eqnarray} \n\n\n\\begin{comment}\n\\com{[I have rewritten the text from here]}\n\nAt high temperature, we expect that color degrees of freedom are released and a perturbative approach becomes valid. \nAs the temperature decreases, confinement-deconfinement phase transition occurs at a certain critical temperature, which has been observed in lattice QCD simulations.\nAn approximated (exact in pure YM) order parameter for confinement-deconfinment phase transition in QCD is the Polyakov loop, which is a Wilson line closing around the imaginary time direction \\cite{Weiss:1980rj, Gies:2000dw},\n\\begin{eqnarray}\n\\Phi (\\vec{x})\n&=& \\frac{1}{N_{c}} {\\rm{Tr}} \\ \\mathcal{T} \\ {\\rm{exp}} \\left\\{ ig \\int^{\\beta}_{0} d\\tau {A}_{0}^{a} (\\tau, \\vec{x}) T^{a} \\right\\}\n\\end{eqnarray}\nwhere $\\beta = 1\/T$ is the inverse temperature. $\\mathcal{T}$ stands for time ordering. $\\hat{A}_{0}^{A}$ is the zeroth component of the background gauge field. The negative logarithm of the expectation value of the Polyakov loop can be identified as a free energy of a static quark. $\\langle \\Phi \\rangle \\to 0$ being an infinite free energy corresponds to a confining phase, while $\\langle \\Phi \\rangle \\neq 0$ indicates a deconfining phase. Under gauge transformations which differ at $x_{0} = 0$ and $x_{0} = \\beta$ by a center element of the gauge group, $\\Phi$ transforms as $\\Phi \\to {\\rm e}^{2\\pi i n\/N_{c}}$, $(n=0,1,2, \\ldots, N_{c}-1)$. This implies that $\\langle \\Phi \\rangle = 0$ indicates a center symmetric phase, while $\\langle \\Phi \\rangle \\neq 0$ a broken phase of the center symmetry. Dynamical quarks, however, explicitly break the center symmetry. Therefore, in QCD the Polyakov loop is an approximated order parameter of confinement(center symmetric)-deconfinement(broken) phase transition. \nIn the Polyakov gauge,\n\\begin{eqnarray}\n\\hat{A}_{0}^{A}(x_{0}, \\vec{x}) = \\bar{\\mathcal{A}}_{0}(x_{0}, \\vec{x}) \\delta^{3 A}, \\ \\ \\ \\partial_{0} \\bar{\\mathcal{A}}_{0} (x_{0}, \\vec{x}) = 0,\n\\end{eqnarray}\nthe Polyakov loop is given as\n\\begin{eqnarray}\n\\Phi(\\vec{x})\n&=& {\\rm{cos}} (\\pi c) \\ \\ {\\rm{for}} \\ SU(2) \\nonumber \\\\\n\\Phi(\\vec{x})\n&=& \\frac{1}{3} \\left( 1 + 2{\\rm{cos}}( \\pi c) \\right) \\ \\ {\\rm{for}} SU(3).\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nc(\\vec{x}) = \\frac{g \\bar{\\mathcal{A}}_{0}(\\vec{x}) }{ 2 \\pi T }\n\\end{eqnarray}\nIn this study, we consider a homogeneous order parameter $\\bar{\\mathcal{A}}_{0}(\\vec{x}) = \\bar{\\mathcal{A}}_{0}$, where the Polyakov loop is independent of the space coordinate $\\vec{x}$.\n\n\\com{[up to here]}\n\\end{comment}\n\n\\subsection{Yang-Mills part of effective action}\n\nNow, we consider the Yang-Mills part (gluon and ghost contributions) of the one-loop effective action. In the one-loop level, the effect of EM fields is not included in gluon and ghost loops, since these do not directly interact with EM fields. From Eq.~(\\ref{full_actions}), the effective actions of gluon and ghost parts are given, respectively, as\n\\begin{eqnarray}\niS_{\\rm gluon}\n&\\equiv & {\\rm{ln}} \\ {\\rm{det}} \\left[ - ({\\cal D}^{2})^{ac} g_{\\mu \\nu} - 2 g f^{abc} {\\cal F}^{b}_{\\mu \\nu} \\right]^{-\\frac12}, \\label{gluon-part}\\\\\niS_{\\rm ghost}\n&\\equiv & {\\rm{ln}} \\ {\\rm{det}} \\left[ - ({\\cal D}^{2})^{ac} \\right]^{+1}.\n\\label{ghost-part}\n\\end{eqnarray}\nLet us first explore the gluon part (\\ref{gluon-part}). By using the proper time integral,\\footnote{We use the following identity:\n$$\n\\ln (\\hat M -i\\delta)=\\frac{1}{\\epsilon}- \\frac{i^\\epsilon}{\\epsilon \\Gamma(\\epsilon)}\\int_0^\\infty \\frac{ds}{s^{1-\\epsilon}}\\, {\\rm e}^{-is (\\hat M -i\\delta)} $$ \nin the limit $\\epsilon\\to 0$ and $\\delta\\to 0$. We ignore the first divergent term, since it does not depend on the fields.} the gluon part of the effective action can be rewritten in the following form (the limit $\\epsilon,\\delta\\to 0$ is always implicit and should be taken after the calculation): \n\\begin{eqnarray}\niS_{\\rm gluon}\n&=& -\\frac{1}{2} {\\rm{Tr}} \\ {\\rm{ln}} \n\\left[ - ({\\cal D}^{2})^{ac} g_{\\mu \\nu} \n - 2 g f^{abc} {\\cal F}^{b}_{\\mu \\nu} \n\\right] \\nonumber \\\\\n&=& \\int d^{4}x \\frac{i^{\\epsilon}}{2} \n \\sum_{h=1}^{N_{c}^{2}-1} \\int^{\\infty}_{0} \\frac{ds}{s^{1-\\epsilon}} \n {\\rm{tr}} \\langle x | \n {\\rm e}^{- i \\left( - {\\cal D}_{v_{h}}^{2} g_{\\mu \\nu} \n + 2i g v_{h} {\\cal F}_{\\mu \\nu} -i \\delta \\right) s}\n | x \\rangle \\nonumber \\\\\n&=& \\int d^{4}x \\frac{i^{\\epsilon}}{2} \\sum_{h=1}^{N_{c}^{2}-1} \n \\int^{\\infty}_{0} \\frac{ds}{s^{1-\\epsilon}} \n {\\rm e}^{-\\delta s} \n \\left\\{ {\\rm e}^{-i(2gv_{h}\\mathfrak{a} )s} + {\\rm e}^{ -i ( - 2gv_{h}\\mathfrak{a} )s} \n + {\\rm e}^{-i ( igv_{h}\\mathfrak{b} )s } + {\\rm e}^{ -i ( -2igv_{h}\\mathfrak{b} )s } \n \\right\\} \\nonumber \\\\\n&& \\times \\langle x | {\\rm e}^{ - i ( - {\\cal D}_{v_{h}}^{2} ) s} |x \\rangle \\, .\n\\end{eqnarray}\nWhile the capital trace ``Tr\" in the first line is taken with respect to colors, Lorentz indices, and coordinates, ``tr\" in the second line is only for Lorentz indices. Also, in the second line, we have introduced real quantities $v_{h}$ $(h=1,\\ldots, N_c^2-1)$ that are eigenvalues of a Hermitian matrix $V^{ac}\\equiv if^{abc} {n}^{b}$ (i.e., $V^{ac}\\varphi^c=v_h \\varphi^a$), and Lorentz-invariant quantities\n$\\mathfrak{a}$, $\\mathfrak{b}$ defined by\n\\begin{eqnarray}\n\\mathfrak{a}\n\\equiv \\frac{1}{2} \\sqrt{ \\sqrt{ \\mathcal{F}^{4} + (\\mathcal{F}\\cdot \\tilde{\\mathcal{F}})^{2} } + \\mathcal{F}^{2} }\\, , \\ \\ \\ \\ \n\\mathfrak{b}\n\\equiv \\frac{1}{2} \\sqrt{ \\sqrt{ \\mathcal{F}^{4} + (\\mathcal{F}\\cdot \\tilde{\\mathcal{F}})^{2} } - \\mathcal{F}^{2} }\\, ,\n\\end{eqnarray}\nwith the dual field-strength tensor $\\tilde{\\mathcal{F}}^{\\mu \\nu} = \\frac{1}{2} \\epsilon^{\\mu \\nu \\alpha \\beta} \\mathcal{F}_{\\alpha \\beta}$ (or equivalently, by $\\mathfrak{a}^2-\\mathfrak{b}^2=\\frac12 \\mathcal{F}^2$ and $\\mathfrak{a} \\mathfrak{b} = \\frac14 \\mathcal{F}\\cdot \\tilde \\mathcal{F}$).\nThe covariant derivative is defined as ${\\cal D}_{v_{h} \\mu} = \\partial_{\\mu} - ig v_{h} \\mathcal{A}_{\\mu}$. \nThe calculation up to now is in fact the same as in the case at zero temperature which was done in Ref.~\\cite{Ozaki:2013sfa}. \nAt finite temperature, however, one needs to be careful in evaluating the matrix element $\\langle x | {\\rm e}^{ - i ( - {\\cal D}_{v_{h}}^{2} ) s} |x \\rangle$.\n\\begin{comment}\nTaking the Fock-Schwinger gauge in the presence of the order parameter $\\bar{\\mathcal{A}}_{0}$,\n\\begin{eqnarray}\nA_{\\mu} \n&=& i\\bar{\\mathcal{A}}_{0} \\delta_{0\\mu} - \\frac{1}{2} F_{\\mu \\nu} (x - x^{\\prime})^{\\nu},\n\\end{eqnarray}\n\\end{comment}\nNamely, it can be now written as the Matsubara summation:\n\\begin{eqnarray}\n\\langle x | {\\rm e}^{ - i ( - {\\cal D}_{v_{h}}^{2} ) s} |x \\rangle\n&=& \\left. i T \\sum_{n=-\\infty}^{\\infty} \\int \\frac{ d^{3}p }{ (2\\pi)^{3} } \n\\, {\\rm e}^{- p_\\alpha X_{h}^{\\alpha\\beta}(is) p_\\beta }\\, {\\rm e}^{-Y_{h}(is) } \\right|_{p_{0} = igv_{h}\\bar{\\mathcal{A}}_{4} - i 2\\pi n T}\\, ,\n\\label{Matrix-element-T} \n\\end{eqnarray}\nwhere the functions $X_{h}^{\\alpha \\beta}(\\bar{s})$ and $Y_{h}(\\bar{s})$ have been defined as \\cite{Dittrich:2000zu}\n\\begin{eqnarray}\nX_{h}^{\\alpha \\beta}(\\bar{s})\n&=& \\left[(gv_{h}\\mathcal{F} )^{-1} {\\rm{tan}}(gv_{h}\\mathcal{F} \\bar{s})\\right]^{\\alpha\\beta}, \\nonumber \\\\\nY_{h}(\\bar{s})\n&=& \\frac{1}{2} {\\rm{tr}} \\ {\\rm{ln}} \\ {\\rm{cos}}(gv_{h}\\mathcal{F} \\bar{s}).\n\\end{eqnarray}\nIn the presence of the Polyakov loop $\\bar{\\mathcal{A}}_{4}$, the periodic boundary condition of the gluon in the imaginary time direction is modified.\nThen, the Matsubara frequency is shifted by the Polyakov loop as in Eq.~(\\ref{Matrix-element-T}).\nPerforming the three-dimensional momentum integral and applying the Poisson resummation~\\cite{Dittrich:2000zu}, one can obtain the matrix element in terms of \n$\\mathfrak{a}$ and $\\mathfrak{b}$ as\n\\begin{eqnarray}\n\\!\\!\\langle x | {\\rm e}^{ - i ( - {\\cal D}_{v_{h}}^{2} ) s} |x \\rangle\n&=& -\\frac{i}{16 \\pi^{2}} \\frac{ gv_{h}\\mathfrak{a} s}{{\\rm{sin}}(gv_{h}\\mathfrak{a} s)} \\frac{gv_{h}\\mathfrak{b} s}{ {\\rm{sinh}}(gv_{h}\\mathfrak{b} s) } \\left[\n1 + 2\\sum_{n=1}^{\\infty} {\\rm e}^{ i \\frac{\\mathfrak{h}(s)}{4T^{2}}n^{2}} {\\rm{cos}}\\left( \\frac{ gv_{h} \\bar{\\mathcal{A}}_{4}}{T} n \\right) \\right],\n\\label{KarnelG}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\mathfrak{h}(s)\n&=& \\frac{\\mathfrak{b}^{2} - {\\mathfrak{e}}^{2} }{\\mathfrak{a}^{2}+\\mathfrak{b}^{2}}\\, g v_{h} \\mathfrak{a}\\, {\\rm{cot}}(gv_{h} \\mathfrak{a} s) + \\frac{ \\mathfrak{a}^{2} + {\\mathfrak{e}}^{2} }{ \\mathfrak{a}^{2} + \\mathfrak{b}^{2} }\\, g v_{h} \\mathfrak{b}\\, {\\rm{coth}} (gv_{h}\\mathfrak{b} s)\\, ,\n\\end{eqnarray}\nwith\n\\begin{eqnarray} \n{\\mathfrak{e}}^{2} = (u_{\\alpha} \\mathcal{F}^{\\alpha \\mu})( u_{\\beta} \\mathcal{F}^{\\beta}_{\\mu} ).\n\\end{eqnarray}\nThe vector $u^{\\mu}$ is the heat-bath four-vector, which is $(1,0,0,0)$ in the rest frame of the heat bath. The first (second) term in Eq.~(\\ref{KarnelG}) corresponds to the zero-(finite-)temperature contribution. \nThe gluon part of the effective action is then given as\n\\begin{eqnarray}\niS_{\\rm gluon} \n&=& - \\frac{i^{1+\\epsilon}}{ 32 \\pi^{2} } \\int d^{4}x \\sum_{h=1}^{N_{c}^{2}-1} \n\\int^{\\infty}_{0} \\frac{ds}{s^{3-\\epsilon}} {\\rm e}^{-\\delta s} \n\\left\\{ {\\rm e}^{-i(2gv_{h}\\mathfrak{a} )s} + {\\rm e}^{ -i ( - 2gv_{h}\\mathfrak{a} )s} \n + {\\rm e}^{-i ( igv_{h}\\mathfrak{b} )s } + {\\rm e}^{ -i ( -2igv_{h}\\mathfrak{b} )s } \n\\right\\} \\nonumber \\\\\n&&\\qquad\\qquad\\ \\ \\times \\frac{ gv_{h}\\mathfrak{a} s}{{\\rm{sin}}(gv_{h}\\mathfrak{a} s)} \n\\frac{gv_{h}\\mathfrak{b} s}{ {\\rm{sinh}}(gv_{h}\\mathfrak{b} s) } \\left[\n1 + 2\\sum_{n=1}^{\\infty} {\\rm e}^{ i \\frac{\\mathfrak{h}(s)}{4T^{2}}n^{2}} \n{\\rm{cos}}\\left( \\frac{ gv_{h} \\bar{\\mathcal{A}}_{4}}{T} n \\right) \\right].\\label{action_gluon}\n\\end{eqnarray}\nSimilarly, we obtain the ghost part as\n\\begin{eqnarray}\niS_{\\rm ghost}\n&=& \\frac{i^{1+\\epsilon}}{ 32 \\pi^{2} } \\int d^{4}x \\sum_{h=1}^{N_{c}^{2}-1} \n\\int^{\\infty}_{0} \\frac{ds}{s^{3-\\epsilon}} {\\rm e}^{-\\delta s} \n\\left\\{ 2 \\right\\} \\nonumber \\\\\n&& \\times \\frac{ gv_{h}\\mathfrak{a} s}{{\\rm{sin}}(gv_{h}\\mathfrak{a} s)} \n\\frac{gv_{h}\\mathfrak{b} s}{ {\\rm{sinh}}(gv_{h}\\mathfrak{b} s) } \\left[\n1 + 2\\sum_{n=1}^{\\infty} {\\rm e}^{ i \\frac{\\mathfrak{h}(s)}{4T^{2}}n^{2}} \n{\\rm{cos}}\\left( \\frac{ gv_{h} \\bar{\\mathcal{A}}_{4}}{T} n \\right) \\right].\n\\label{action_ghost}\n\\end{eqnarray}\nIn both parts, the first terms in the square brackets are the results at zero temperature and agree with the known results \\cite{Ozaki:2013sfa}. As discussed in detail in Ref.~\\cite{Ozaki:2013sfa}, each term has an ultraviolet (UV) divergence, which, however, can be absorbed by renormalizing the coupling $g$ and fields $\\mathcal{A}_\\mu$ \\cite{Savvidy:1977as, Matinyan:1976mp}. On the other hand, the finite-temperature contributions do not have UV divergence, and thus we do not need an additional renormalization procedure for the finite-temperature contributions. \nWe regard the coupling and fields as renormalized ones and focus on UV-finite pieces in Eqs.~(\\ref{action_gluon}) and (\\ref{action_ghost}).\n\n\nOur results (\\ref{action_gluon}) and (\\ref{action_ghost}) are effective actions for chromo-EM fields as well as the Polyakov loop at finite temperature. These are generalizations of the previous results in two cases. Indeed, if we consider the pure chromo-{\\it electric} background with a Polyakov loop (${\\cal B}=0,\\ {\\cal E}\\neq 0, \\ \\mathcal{A}_0\\neq 0$), we find $\\mathfrak{a} \\to i{\\cal E}$, $\\mathfrak{b} \\to 0$ and reproduce Gies's effective action at finite temperature \\cite{Gies:2000dw}. Moreover, in the case of the pure chromo-{\\it magnetic} background (${\\cal E}=0,\\ {\\cal B}\\neq 0$, $\\bar{\\mathcal{A}}_{4}=0$), we find $\\mathfrak{a} \\to {\\cal B}$, $\\mathfrak{b} \\to 0$ and reproduce the results obtained in Refs.~\\cite{Dittrich:1980nh, Kapusta:1981nf}. \n\n\n\\begin{comment}\n\\com{[Below has been rewritten]}\n\nIn the vacuum part, there is a UV divergence.\nHowever, the UV divergence can be renormalized in the coupling and fields \\cite{Savvidy:1977as, Matinyan:1976mp}.\nOn the other hand, in the finite temperature part, there is no UV divergence and thus we do not need an additional renormalization procedure for the finite temperature part. If we consider the pure chromo-electric background: $\\mathfrak{a} \\to iE_{c}$ and $\\mathfrak{b} \\to 0$, we reproduce Gies' effective action at finite temperature \\cite{Gies:2000dw}.\nOn the other hand, in the case of the pure chromo-magnetic background: $\\mathfrak{a} \\to H_{c}$ and $\\mathfrak{b} \\to 0$ with $\\bar{\\mathcal{A}}_{4}=0$, we reproduce the results obtained in \\cite{Dittrich:1980nh, Kapusta:1981nf}.\n\n\\com{[up to here]}\n\\end{comment}\n\n\\subsection{Quark part of effective action}\n\n\nFor the quark part of the effective action, we follow basically the same procedures as in the Yang-Mills part.\nFrom the functional integral (\\ref{full_actions}), the quark part of the one-loop effective action reads\n\\begin{eqnarray}\niS_{\\rm quark}\n&=& {\\rm{ln}} \\ {\\rm{det}} \\left[ i \\gamma_{\\mu} \\hat{\\cal D}^{\\mu} - M_{q} \\right].\n\\end{eqnarray}\nUtilizing the proper time integral, we evaluate the effective action as\n\\begin{eqnarray}\niS_{\\rm quark}\n&=& {\\rm{Tr}} \\ {\\rm{ln}} \\left[ i \\gamma_{\\mu} \\hat{\\cal D}^{\\mu} - M_{q} \\right] \\nonumber \\\\\n&=& -\\int dx^{4} \\frac{i^{\\epsilon}}{2} \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{ds}{s^{1-\\epsilon}} \n{\\rm e}^{-i (m_{q_{f}}^{2} - i \\delta ) s} {\\rm{tr}} \\langle x | \n{\\rm e}^{-is \\left( -\\mathbb{D}_{i,f}^{2} - \\frac{1}{2} \\sigma \\cdot \\mathbb{F}_{i,f} \\right) } \n|x \\rangle,\n\\end{eqnarray}\nwhere $\\mathbb{D}_{i,f}^{\\mu} = \\partial^{\\mu} - i \\mathbb{A}_{i,f}^{\\mu}$ with the field $\\mathbb{A}_{i,f}^{\\mu}$ being a linear combination of the gluon field $\\mathcal{A}_{\\mu}$ and the photon field $a^{\\mu}$ as \n\\begin{eqnarray}\n\\mathbb{A}_{i,f}^{\\mu}\n&=& g\\omega_{i} \\mathcal{A}^{\\mu} + eQ_{q_{f}} a^{\\mu}. \\label{linear_combination}\n\\end{eqnarray}\nThis covariant derivative $\\mathbb{D}_{i,f}^{\\mu}$ can be obtained from $\\hat{\\cal D}^{\\mu}$ defined in Eq. (\\ref{CovariantDerivativeQuark}) with the covariantly constant field employed as the background field.\nHere $\\omega_{i}\\ (i=1,\\ldots,N_c)$ are eigenvalues of an $N_c\\times N_c$ matrix ${n}^{a} T^{a}$ and satisfy\\footnote{Let $\\Omega$ be a diagonal matrix with eigenvalues $\\omega_i$, i.e., $\\Omega={\\rm diag}(\\omega_1,\\ldots,\\omega_{N_c})=Un^aT^aU^\\dagger$. Then, $\\sum_{i=1}^{N_c}\\omega_i=\\, {\\rm tr}\\, \\Omega= n^a\\, {\\rm tr}\\, T^a=0$ and $\\sum_{i=1}^{N_c}\\omega_i^2=\\, {\\rm tr}\\, \\Omega^2=\\, {\\rm tr}\\, (T^aT^b)n^a n^b=1\/2.$ } $\\sum_{i=1}^{N_c}\\omega_i=0$ and $\\sum_{i=1}^{N_c}\\omega_i^2=1\/2$.\nThe field-strength tensor $\\mathbb{F}_{i,f}^{\\mu \\nu}$ can be expressed in terms of constant chromo-EM fields $\\vec{\\mathcal{E}}$, $\\vec{\\mathcal B}$, and EM fields $\\vec{E}$, $\\vec{B}$ as [with the notation $\\vec{V}=(V_x,V_y,V_z)$]\n\\begin{eqnarray}\n\\mathbb{F}_{i,f}^{\\mu \\nu}\n&=& g \\omega_{i} {\\cal F}^{\\mu \\nu} + eQ_{q_{f}} f^{\\mu \\nu} \\nonumber \\\\\n&=&\ng \\omega_{i} \\left(\n\\begin{array}{cccc}\n0 & \\mathcal{E}_{x} & \\mathcal{E}_{y} & \\mathcal{E}_{ z } \\\\\n-\\mathcal{E}_{x} & 0 & \\mathcal{B}_{z} & - \\mathcal{B}_{y} \\\\\n-\\mathcal{E}_{y} & - \\mathcal{B}_{z} & 0 & \\mathcal{B}_{x} \\\\\n-\\mathcal{E}_{z} & \\mathcal{B}_{y} & - \\mathcal{B}_{x} & 0 \\\\\n\\end{array}\n\\right)\n+\ne Q_{q_{f}} \\left(\n\\begin{array}{cccc}\n0 & E_{x} & E_{y} & E_{z } \\\\\n-E_{x} & 0 & B_{z} & - B_{y} \\\\\n-E_{y} & - B_{z} & 0 & B_{x} \\\\\n-E_{z} & B_{y} & - B_{x} & 0 \\\\\n\\end{array}\n\\right).\n\\label{matrix_field}\n\\end{eqnarray}\nThe eigenvalues of the field-strength tensor $\\mathbb{F}^{\\mu \\nu}_{i,f}$ are given by $\\pm i\\mathfrak{a}_{i,f}$ and $\\pm \\mathfrak{b}_{i,f}$\nwith\n\\begin{eqnarray}\n\\mathfrak{a}_{i,f}\n = \\frac{1}{2} \\sqrt{ \\sqrt{ \\mathbb{F}_{i,f}^{4} + ( \\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f} )^{2} } + \\mathbb{F}_{i,f}^{2} } \\ , \\ \\ \\ \\ \\ \n\\mathfrak{b}_{i,f}\n = \\frac{1}{2} \\sqrt{ \\sqrt{ \\mathbb{F}_{i,f}^{4} + ( \\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f} )^{2} } - \\mathbb{F}_{i,f}^{2} } \\ .\n\\end{eqnarray}\nThe dual field-strength tensor $\\tilde{\\mathbb{F}}^{\\mu \\nu}_{i,f}$ is defined as $\\tilde{\\mathbb{F}}^{\\mu \\nu}_{i,f} = \\frac{1}{2} \\epsilon^{\\mu \\nu \\alpha \\beta} \\mathbb{F}_{i,f \\alpha \\beta}$. By using Eq.~(\\ref{matrix_field}), $\\mathbb{F}_{i,f}^{2}=2(\\mathfrak{a}_{i,f}^2-\\mathfrak{b}_{i,f}^2)$ and $\\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f}=4\\mathfrak{a}_{i,f}\\mathfrak{b}_{i,f}$ can be expressed in terms of chromo-EM fields and EM fields as\n\\begin{comment}\n\\begin{eqnarray}\n\\mathbb{F}_{i,f}^{2}\n&=& 2 \\left[ (g \\omega_{i})^{2}( \\vec{\\mathcal{B}}^{2} - \\vec{\\mathcal{E}}^{2} ) + (eQ_{q_{f}})^{2} ( \\vec{B}^{2} - \\vec{E}^{2} ) + 2 g \\omega_{i}eQ_{q_{f}}( \\vec{\\mathcal{B}} \\cdot \\vec{B} - \\vec{\\mathcal{E}} \\cdot \\vec{E}) \\right], \\nonumber \\\\\n\\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f}\n&=& -4 \\left[ (g \\omega_{i})^{2} \\vec{\\mathcal{E}}\\cdot \\vec{\\mathcal{B}} + (eQ_{q_{f}})^{2} \\vec{E}\\cdot \\vec{B} + g \\omega_{i} eQ_{q_{f}} ( \\vec{\\mathcal{E}} \\cdot \\vec{B} + \\vec{E} \\cdot \\vec{\\mathcal{B}} ) \\right]. \\label{FFtilde}\n\\end{eqnarray}\n\\end{comment}\n\\begin{eqnarray}\n\\mathbb{F}_{i,f}^{2}\n&=& 2( \\vec{ \\mathcal{B} }_{i,f}^{2} - \\vec{ \\mathcal{E} }_{i,f}^{2} ), \\nonumber \\\\\n\\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f}\n&=& -4 \\vec{ \\mathcal{E} }_{i,f} \\cdot \\vec{ \\mathcal{B} }_{i,f},\n\\label{FFtilde}\n\\end{eqnarray}\nwhere we have defined the combined electromagnetic fields as $\\vec{ \\mathcal{E} }_{i,f} = g \\omega_{i} \\vec{ \\mathcal{E} } + eQ_{q_{f}} \\vec{E}$ and $\\vec{ \\mathcal{B} }_{i,f} = g \\omega_{i} \\vec{ \\mathcal{B} } + eQ_{q_{f}} \\vec{ B}$. \nTaking the trace of the matrix $\\langle x | {\\rm e}^{-is \\left( -\\mathbb{D}_{i,f}^{2} - \\frac{1}{2} \\sigma \\cdot \\mathbb{F}_{i,f} \\right) } |x \\rangle$ at finite temperature, we get\n\\begin{eqnarray}\n&&{\\rm{tr}} \\langle x | {\\rm e}^{-is \\left( -\\mathbb{D}_{i,f}^{2} - \\frac{1}{2} \\sigma \\cdot \\mathbb{F}_{i,f} \\right) } |x \\rangle \\nonumber \\\\\n&&\\qquad = \\left. i T \\sum_{n=-\\infty}^{\\infty} \\int \\frac{ d^{3} p }{ (2\\pi)^{3} }\\, {\\rm e}^{ - p_\\alpha \\mathbb{X}^{\\alpha\\beta}_{i,f}(is) p_\\beta } {\\rm e}^{- \\mathbb{Y}_{i,f}(is) }\\, {\\rm{tr}} \\, {\\rm e}^{ \\frac{i}{2} \\sigma \\cdot \\mathbb{F}_{i,f}s } \\right|_{p_{0} = ig \\omega_{i} \\bar{\\mathcal{A}}_{4} - i \\pi (2n+1) T}\\, .\n\\label{matrix_quark}\n\\end{eqnarray}\nHere, the functions $\\mathbb{X}_{i,f}^{\\alpha \\beta} (\\bar{s})$ and $\\mathbb{Y}_{i,f} (\\bar{s})$ have been defined as~\\cite{Dittrich:2000zu}\n\\begin{eqnarray}\n\\mathbb{X}_{i,f}^{\\alpha \\beta} (\\bar{s})\n&=& \\left[ \\mathbb{F}_{i,f}^{-1}\\, {\\rm{tan}} ( \\mathbb{F}_{i,f} \\bar{s} ) \\right]^{\\alpha \\beta}, \\nonumber \\\\\n\\mathbb{Y}_{i,f} (\\bar{s})\n&=& \\frac{1}{2} {\\rm{tr}} \\ {\\rm{ln}} \\ {\\rm{cos}} ( \\mathbb{F}_{i,f} \\bar{s} )\\, .\n\\end{eqnarray} \nIn the presence of the Polyakov loop $\\bar{\\mathcal{A}}_{4}$, the antiperiodic boundary condition for the quark is also modified. Then, the temporal component of the four-momentum vector has been replaced by the Polyakov loop and the Matsubara frequency for a fermion in Eq.~(\\ref{matrix_quark}). \nThe third part, ${\\rm{tr}}\\, {\\rm e}^{ \\frac{i}{2} \\sigma \\cdot \\mathbb{F}_{i,f} s}$, is common with the case at zero temperature and was computed in Ref.~\\cite{Ozaki:2013sfa}. The result is\n\\begin{eqnarray}\n{\\rm{tr}} \\, {\\rm{exp}} \\left( \\frac{i}{2} \\sigma \\cdot \\mathbb{F}_{i,f} s \\right)\n&=& 4 {\\rm{cos}}( \\mathfrak{a}_{i,f}s ) {\\rm{cosh}} (\\mathfrak{b}_{i,f} s).\n\\end{eqnarray}\nNow, performing the three-dimensional momentum integral and using the Poisson resummation, we find from Eq.~(\\ref{matrix_quark})\n\\begin{eqnarray}\n{\\rm{tr}} \\langle x | {\\rm e}^{-is \\left( -\\mathbb{D}_{i,f}^{2} - \\frac{1}{2} \\sigma \\cdot \\mathbb{F}_{i,f} \\right) } |x \\rangle \n&=& - \\frac{ i }{ 4 \\pi^{2} s^{2} } \\frac{ ( \\mathfrak{a}_{i,f}s )( \\mathfrak{b}_{i,f}s ) }{ {\\rm{sin}}( \\mathfrak{a}_{i,f} s ) {\\rm{sinh}} ( \\mathfrak{b}_{i,f} s ) } {\\rm{cos}}( \\mathfrak{a}_{i,f}s ) {\\rm{cosh}}( \\mathfrak{b}_{i,f}s ) \\nonumber \\\\\n&& \\times \\left\\{ 1 + 2 \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{ \\frac{i}{4T^{2}} \\mathfrak{h}_{i,f}(s) n^{2} } {\\rm{cos}} \\left( \\frac{ g \\omega_{i}\\bar{\\mathcal{A}}_{4} n }{ T} \\right) \\right\\} \\, ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\mathfrak{h}_{i,f}(s)\n&=& \\frac{ \\mathfrak{b}_{i,f}^{2} - {\\mathfrak{e}}_{i,f}^{2} }{\\mathfrak{a}_{i,f}^{2} + \\mathfrak{b}_{i,f}^{2}}\\mathfrak{a}_{i,f} {\\rm{cot}}(\\mathfrak{a}_{i,f}s) + \\frac{ \\mathfrak{a}_{i,f}^{2} + {\\mathfrak{e}}_{i,f}^{2} }{ \\mathfrak{a}_{i,f}^{2} + \\mathfrak{b}_{i,f}^{2} } \\mathfrak{b}_{i,f} {\\rm{coth}}(\\mathfrak{b}_{i,f}s)\\, ,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n{\\mathfrak{e}}_{i,f}^{2}\n&=& (u_{\\alpha} \\mathbb{F}_{i,f}^{\\alpha \\mu}) ( u_{\\beta} \\mathbb{F}^{\\ \\beta}_{i,f \\mu})\\, .\n\\end{eqnarray}\nIn the heat-bath rest frame, we have $u^{\\mu} = (1,0,0,0)$ and then\n$\n{\\mathfrak{e}}_{i,f}^{2} = \\vec{ \\mathcal{E} }_{i,f}^{2} = ( g \\omega_{i} \\vec{\\mathcal{E}} + eQ_{q_{f}} \\vec{E} )^{2}\n$.\nTherefore, the quark part of the one-loop effective action reads\n\\begin{eqnarray} \niS_{\\rm quark}\n&=& \\frac{i^{1+\\epsilon}}{8\\pi^{2}} \\int d^{4}x \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\int_{0}^{\\infty} \\frac{ds}{s^{3-\\epsilon}} \n{\\rm e}^{-i(m_{q_f}^{2}-i\\delta)s} (\\mathfrak{a}_{i,f}s)(\\mathfrak{b}_{i,f}s) {\\rm{cot}}(\\mathfrak{a}_{i,f}s) {\\rm{coth}}(\\mathfrak{b}_{i,f}s) \\nonumber \\\\\n&&\\qquad \\times \\left[ 1 + 2 \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{\\frac{i}{4T^{2}} \\mathfrak{h}_{i,f}(s) n^{2} } {\\rm{cos}}\\left( \\frac{ g\\omega_{i} \\bar{\\mathcal{A}}_{4}n}{T} \\right) \\right]\\, .\n\\label{action_quark}\n\\end{eqnarray}\nAs in the YM part, the first (second) term corresponds to the zero-(finite-)temperature contribution. The zero-temperature contribution agrees with the previous result obtained in Ref.~\\cite{Ozaki:2013sfa}. \n\nAgain, the first term contains UV divergences. \nThese divergences have two origins: QCD and QED \\cite{Ozaki:2013sfa}. \nThis is because the resummed quark one-loop diagrams contain contributions from the diagrams with only two EM field insertions (QED) and only two chromo-EM field insertions (QCD).\nThe UV divergence coming from purely QCD dynamics is additive to the one which we encounter in the YM part. Then, we can absorb all the UV divergences by renormalizing the coupling $g,e$ and fields $\\mathcal{A}_{\\mu}, a_{\\mu}$. \nFrom the renormalization procedure at zero temperature, we have obtained the correct beta functions of both QCD and QED in Ref.~\\cite{Ozaki:2013sfa}. The sum of the three parts (\\ref{action_gluon}), (\\ref{action_ghost}), and (\\ref{action_quark}) may be called the Euler-Heisenberg-Weiss action in QCD+QED at finite temperature. This result can be applied to several systems where strong EM fields and chromo-EM fields coexist at zero and finite temperatures. In the next section, we will show some applications of our effective actions.\\\\\n\n\n\n\\section{Applications of Euler-Heisenberg-Weiss action in QCD+QED}\n\nIn this section we will discuss two applications of our results. \nThe first one is the quark pair production in the presence of both EM and chromo-EM fields. We treat the effective action at zero temperature. The second application is to investigate the effects of EM fields on the effective potential for the Polyakov loop at finite temperature. We will discuss the possible implication for the inverse magnetic catalysis.\n\n\n\n\n\n\\subsection{Quark pair production in QCD+QED fields}\n\nLet us first discuss quark-antiquark pair production in constant QCD+QED fields as an application of our effective action. For this problem, only the quark part (\\ref{action_quark}) is relevant.\n\nIn the early stage of relativistic heavy-ion collisions, extremely strong chromo-EM fields and EM fields could coexist. \nNotice that the strong {\\it{electric}} field in addition to the strong magnetic field could be created on an event-by-event basis~\\cite{Deng:2012pc}.\nThe strength of the chromo-EM fields is approximately of the order of the saturation scale: $|g\\vec{\\mathcal{B}}|, |g\\vec{\\mathcal{E}}| \\sim Q_{s}^2 $, whereas strengths of EM fields would reach the QCD nonperturbative scale $|e\\vec E|, |e\\vec B|\\sim \\Lambda_{QCD}^2$, or even exceed it. Under such strong QCD+QED fields, a number of quark-antiquark pairs must be created through the Schwinger mechanism. The pair-production rate per unit space-time volume can be obtained from the imaginary part of the quark effective Lagrangian at zero-temperature. \nTaking the zero temperature contribution in Eq.~(\\ref{action_quark}), one finds\n\\begin{eqnarray}\n\\mathcal{L}_{\\rm quark}\n= \\frac{ S_{\\rm quark} }{ \\int d^{4}x } \n= \\frac{1}{8\\pi^{2}} \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} {\\rm e}^{-is(m_{q_{f}}^{2} - i \\delta) } (\\mathfrak{a}_{i,f}s)(\\mathfrak{b}_{i,f}s) {\\rm{cot}}(\\mathfrak{a}_{i,f}s) {\\rm{coth}}(\\mathfrak{b}_{i,f}s)\\, .\n\\end{eqnarray}\nThis is the same as the result obtained in Ref.~\\cite{Ozaki:2013sfa}. \nThe imaginary part of the effective Lagrangian thus reads\n\\begin{eqnarray}\n{\\Im }m\\, \\mathcal{L}_{\\rm quark}\n&=& - \\frac{1}{8\\pi^{2}} \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} {\\rm e}^{- \\delta s } {\\rm{sin}}(m_{q_{f}}^{2} s) \\times (\\mathfrak{a}_{i,f}s)(\\mathfrak{b}_{i,f}s) {\\rm{cot}}(\\mathfrak{a}_{i,f}s) {\\rm{coth}}(\\mathfrak{b}_{i,f}s) \\nonumber \\\\\n&=& \\frac{1}{2 i } \\frac{1}{8 \\pi^{2} } \\sum_{i=1}^{N_{c}^{2}} \\sum_{f=1}^{N_{f}} \\left\\{\n\\int^{0}_{-\\infty} \\frac{ds}{s^{3}}\\, {\\rm e}^{-is(m_{q_{f}}^{2} + i\\delta ) } + \\int^{\\infty}_{0} \\frac{ds}{s^{3}}\\, {\\rm e}^{-is(m_{q_{f}}^{2} - i \\delta ) } \\right\\} \\nonumber \\\\\n&& \\qquad \\times (\\mathfrak{a}_{i,f}s)(\\mathfrak{b}_{i,f}s) {\\rm{cot}}(\\mathfrak{a}_{i,f}s) {\\rm{coth}}(\\mathfrak{b}_{i,f}s)\\, .\n\\end{eqnarray}\nThe integrand has infinitely many poles along the real axis [from cot$( \\mathfrak{a}_{i,f}s)$] and along the imaginary axis [from coth$(\\mathfrak{b}_{i,f}s)$]. With a small positive number $\\delta>0$, the integral contour along the real axis is inclined. Closing the contour in the lower half of the $s$ plane as depicted in Fig. 2 and picking up the poles lying on the imaginary axis $s_{\\rm poles} = - i n \\pi \/ \\mathfrak{b}_{i,f}$, we find\n\\begin{figure}[t]\n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=0.8 \\textwidth]{lower_contour.pdf}\n\\vskip -0.1in\n\\end{center}\n\\end{minipage}\n\\caption{\nContour on the complex $s$ plane. The contour along the real axis is inclined by an infinitesimal number $\\delta>0$.}\n\\end{figure}\n\\begin{eqnarray}\n{\\Im}m\\, \\mathcal{L}_{\\rm quark}\n&=& \\frac{1}{8 \\pi^{2}} \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\mathfrak{a}_{i,f} \\mathfrak{b}_{i,f} \\sum_{n=1}^{\\infty} \\frac{1}{n}\\, {\\rm e}^{ - \\frac{ m_{q_{f}}^{2} }{\\mathfrak{b}_{i,f}} n\\pi }\n {\\rm{coth}} \\left( \\frac{\\mathfrak{a}_{i,f}}{\\mathfrak{b}_{i,f}} n \\pi \\right).\n\\label{ImLq_full}\n\\end{eqnarray}\nBy using this expression, we can investigate quark-antiquark pair productions under arbitrary configurations of constant chromo-EM and EM fields. \nThe production rate per unit space-time volume is given by \n$\nw_{q\\bar{q}} = 2{\\Im}m\\, \\mathcal{L}_{\\rm quark}.\n$\nWhen we take $N_{c} = N_{f}=1$, $Q=1$, $g\\to 0$, $B\\to 0$ and replace $m_{q} \\to m_{e}$ in Eq.~(\\ref{ImLq_full}), we reproduce the well-known Schwinger formula for the production rate of $e^+e^-$ pairs in an electric field \\cite{Schwinger:1951nm}:\n\\begin{eqnarray}\nw_{e^{+}e^{-}} = 2{\\Im m}\\, \\mathcal{L}_{\\rm EH} \n= \\frac{(eE)^{2}}{4 \\pi^{3}} \\sum_{n=1}^{\\infty} \\frac{1}{n^{2}} {\\rm e}^{ - \\frac{ m_{e}^{2} }{eE} n\\pi },\n\\label{rate_e}\n\\end{eqnarray}\nas we expected.\nOn the other hand, in the pure chromo-electric field case, we obtain the same formula for quark productions derived by G.C.~Nayak \\cite{Nayak:2005pf}.\n\n\n\\begin{comment}\n\\com{[I am not sure if we should discuss the following from here]}\n\nFor small quark masses limit, the imaginary part of the effective Lagrangian has a logarithmic dependence on the masses as\n\\begin{eqnarray}\n{\\rm{Im}}\\, \\mathcal{L}_{\\rm quark}\n&\\sim& \\frac{1}{8 \\pi^{2}} \\sum_{a=1}^{N_{c}} \\sum_{i=1}^{N_{f}} a_{A,i} b_{A,i} {\\rm{ln}} \\left( \\frac{ b_{A,i} }{ m_{q_{i}}^{2} \\pi } \\right).\n\\end{eqnarray}\nSimilar logarithmic dependences are also discussed in \\cite{Hidaka:2011dp, Hidaka:2011fa, Hashimoto:2014dza} for QED case.\n\\com{[up to here]}\n\\end{comment}\n\n\n\\subsubsection{Quark pair production in purely electric background}\n\nFirst, we shall consider quark pair production in a purely electric background with vanishing magnetic fields: $\\vec{B}, \\vec{\\mathcal{B}} \\to 0$.\nIn this case, the production rate for $q\\bar q$ pairs of flavor $f$ becomes\n\\begin{eqnarray}\nw_{q_f \\bar q_f}\n&=& \\frac{1}{4 \\pi^{3}} \\sum_{i=1}^{N_{c}} \\mathfrak{b}_{i,f}^{2} \\sum_{n=1}^{\\infty} \\frac{1}{n^{2}}\\, {\\rm e}^{- \\frac{ m_{q_{f}}^{2}}{\\mathfrak{b}_{i,f}} n \\pi}, \n\\label{EcEformula}\n\\end{eqnarray}\nwhere \n$\\mathfrak{b}_{i,f}= \\sqrt{ \\vec{ \\mathcal{E} }_{i,f}^{2} } = \\sqrt{ (g \\omega_{i})^{2} \\mathcal{E}^{2} \n + (eQ_{q_{f}})^{2} E^{2} \n + 2g \\omega_{i}eQ_{q_{f}} \\mathcal{E} E \n {\\rm{cos}}\\theta_{\\mathcal{E}E} \n }$,\nwith \n$E = \\sqrt{ \\vec{E}^{2} }$, $\\mathcal{E} = \\sqrt{ \\vec{ \\mathcal{E} }^{2} }$,\nand $\\theta_{\\mathcal{E}E}$ being the angle between $\\vec{E}$ and $\\vec{\\mathcal{E}}$. \nFor $N_{c} = 3$, the eigenvalues $\\omega_{i}$ are given by $\\omega_{1} = 1\/2$, $\\omega_{2} = -1\/2$, and $\\omega_{3}=0$. \nRecall that a factor $g\\omega_i$ plays the role of an effective coupling between the chromo-EM field and quarks [see Eq.~(\\ref{linear_combination})]. Thus, a quark (or an antiquark) with $\\omega_3=0$ does not interact with the chromo-EM field in this representation. Still, since there is always a coupling with the EM fields, $q\\bar q$ production with $\\omega_3=0$ is possible due to electric fields, i.e., $\\mathfrak{b}_{i=3,f}=|eQ_{q_f} E|\\neq 0$. \n\nLet us see the dependences of production rates on the quark mass $m_q$ and the angle $\\theta_{\\mathcal{E}E}$. We first consider the case with light quark masses $m_{q_f}^2\\ll \\mathfrak{b}_{i,f}$. \nThe left panel of Fig. 3 shows the light (up) quark production rate with $m_{q}=5$ MeV and $Q_{q}=+2\/3$. \nThe chromo-electric field is fixed to $g\\mathcal{E}=1$~GeV$^{2}$, which is a typical value realized in heavy-ion collisions at RHIC and LHC, while we take several values of strength for the $E$ field. The production rate increases with increasing $E$ field, which is an expected behavior of the usual Schwinger mechanism, but it does not show dependence on the angle $\\theta_{{\\cal E}E}$, while $\\mathfrak{b}_{i,f}$ certainly depends on $\\theta_{{\\cal E}E}$.\nThis unexpected behavior can be understood as follows:\nWhen the quark mass is small enough, $m_{q}^2 \\ll \\mathfrak{b}_{i,f}$, we can approximate the production rate as\n\\begin{eqnarray}\nw_{q_f\\bar q_f\n\\sim \\frac{1}{4 \\pi^{3}} \\sum_{i=1}^{N_{c}} \\mathfrak{b}_{i}^{2} \\sum_{n=1}^{\\infty} \\frac{1}{n^{2}} = \\frac{1}{4\\pi^{3}} \\left\\{ \\frac{ (g\\mathcal{E})^{2} }{2} + N_{c}(eQ_{q}E)^{2} \\right\\} \\zeta(2)\\, ,\n\\end{eqnarray}\nwhere $\\zeta(2) = \\pi^{2}\/6$ and $\\mathfrak{b}_{i} = \\sqrt{ (g \\omega_{i})^{2} \\mathcal{E}^{2} + (eQ_{q})^{2} E^{2} + 2g \\omega_{i}eQ_{q} \\mathcal{E} E {\\rm{cos}}\\theta_{\\mathcal{E}E} }$.\nNotice that the angle dependence in $\\mathfrak{b}_i$ drops out thanks to the relations $\\sum_{i=1}^{N_{c}} \\omega_{i}^{2} = 1\/2$ and $\\sum_{i=1}^{N_{c}} \\omega_{i} = 0$.\nTherefore, the production rate is independent of the angle $\\theta_{\\mathcal{E}E}$. \n\n\n\nWe next discuss the production of heavy quark-antiquark pairs. Since the heavy quark limit just implies that the pair creation does not occur, we consider the case where quark masses are comparable to the background field $m_q^2 \\sim \\mathfrak{b}_{i,f}$. This is realized for charm quarks if we again take the typical value of the chromo-electric field $g{\\cal E}=1~$GeV$^2$. For $m_{c}=1.25$~GeV and $Q_{q}= Q_{\\rm charm} = +2\/3$, the production rate of a charm quark pair is shown in the right panel of Fig.~3. This time, while the production rate becomes small, one can see a clear dependence on the angle $\\theta_{{\\cal E}E}$. Both effects (small production rate and angle dependence) come from the exponential factor in Eq.~(\\ref{EcEformula}).\nIn particular when the electric field is parallel (or antiparallel) to the chromo-electric field, the production rate has a maximum.\nSince the exponential factor is very sensitive to the change of $\\mathfrak{b}_{i,f}$, the rate is largely enhanced at $\\theta_{{\\cal E}E}= 0, \\pi$. \nSymmetric shape of the angle dependence with respect to $\\theta_{{\\cal E}E}=\\pi\/2$ is not so trivial. Notice that the effective field strengths of the combined field at $\\theta_{{\\cal E}E}= 0$ and $\\pi$ are not equivalent for a fixed value of $i$; namely, it is the strongest for the parallel configuration (for $\\omega_i>0$) $\\mathfrak{b}_{i,{\\rm charm}}(\\theta_{{\\cal E}E}= 0)=\\sqrt{ (g \\omega_{i})^{2} \\mathcal{E}^{2} \n + (eQ_{\\rm charm})^{2} E^{2} \n + 2g \\omega_{i}eQ_{\\rm charm} \\mathcal{E} E \n }$ and the weakest for the antiparallel configuration\n$\\mathfrak{b}_{i,{\\rm charm}}(\\theta_{{\\cal E}E}= \\pi)=\\sqrt{ (g \\omega_{i})^{2} \\mathcal{E}^{2} \n + (eQ_{\\rm charm})^{2} E^{2} \n - 2g \\omega_{i}eQ_{\\rm charm} \\mathcal{E} E \n }$, implying that pair production is most enhanced for the parallel configuration. This is true for any index of $i$ giving a positive eigenvalue $\\omega_{i} > 0$. However, this eigenvalue appears with a partner $\\omega_{j}$ having an opposite sign $\\omega_j=-\\omega_i$ [for SU(3) we have $\\omega_1=-\\omega_2=1\/2$], and the antiparallel configuration gives the strongest effective field for the index $j$, $\\mathfrak{b}_{j,{\\rm charm}}(\\theta_{{\\cal E}E}=\\pi)=\\mathfrak{b}_{i,{\\rm charm}}(\\theta_{{\\cal E}E}=0)$. Therefore, after summing over all the pairwise modes $i$, we obtain the angle dependence symmetric with respect to $\\theta_{{\\cal E}E}=\\pi\/2$.\n \n \n\n\n\\begin{figure*}[t]\n\\begin{tabular}{cc}\n\\begin{minipage}{0.55\\hsize}\n\\includegraphics[width=0.8 \\textwidth, bb = 160 50 750 600]{eE-and-angle-dep_2ImLq_only_EcE_ver2.pdf}\n\\end{minipage}\n\\begin{minipage}{0.55\\hsize}\n\\includegraphics[width=0.8 \\textwidth, bb = 160 50 750 600]{eE-and-angle-dep_charm-prod_only_EcE_ver2.pdf}\n\\end{minipage}\n\\end{tabular}\n\\caption{ Quark production rate as a function of the angle $\\theta_{E_{\\rm{chro}}E}$, which stands for $\\theta_{\\mathcal{E}E}$.\nThe left panel is the light (up) quark production rate, while the right panel is the heavy (charm) quark production rate.\nThe chromo-electric field is fixed as $g\\mathcal{E} = 1$ GeV$^{2}$. \n}\n\\end{figure*}\n\n\n\n\n\\subsubsection{Quark pair production in purely chromo-EM background}\n\nNext, we investigate quark pair production under chromo-EM fields in the absence of EM fields. \nLorentz-invariant quantities $\\mathbb{F}_{i,f}^{2}$ and $\\mathbb{F}_{i,f}\\cdot \\tilde{\\mathbb{F}}_{i,f}$ are now explicitly given as [see Eq.~(\\ref{FFtilde})]\n\\begin{eqnarray}\n\\mathbb{F}_{i,f}^{2} = 2 (g \\omega_{i})^{2} (\\mathcal{B}^{2} - \\mathcal{E}^{2} )\\, , \\ \\ \\ \\ \\ \\ \n\\mathbb{F}_{i,f}\\cdot \\tilde{\\mathbb{F}}_{i,f} = -4 (g \\omega_{i})^{2} \\mathcal{E} \\mathcal{B} {\\rm{cos}}\\, \\theta_{\\mathcal{E} \\mathcal{B} }\\, ,\n\\end{eqnarray}\nwhere $\\mathcal{B} = \\sqrt{ \\vec{ \\mathcal{B} }^{2} }$, and $\\theta_{\\mathcal{E} \\mathcal{B}}$ stands for the angle between $\\vec{\\mathcal{E}}$ and $\\vec{\\mathcal{B}}$. When $\\theta_{\\mathcal{E} \\mathcal{B} } = \\pm \\pi\/2$ and $\\mathcal{E} > \\mathcal{B}$, we can move into a system with pure chromo-electric fields with\n$\n\\mathfrak{a}_{i,f} = \\mathfrak{a}_{i} = 0 $ and\n$\\mathfrak{b}_{i,f} = \\mathfrak{b}_{i} = |g \\omega_{i}| \\sqrt{\\mathcal{E}^{2} - \\mathcal{B}^{2} }$ by the Lorentz transformation. Then, the production rate for a certain flavor of quark becomes\n\\begin{eqnarray}\n2 {\\Im m}\\, \\mathcal{L}_{\\rm quark}\n&=& \\frac{1}{4\\pi^{3}} \\sum_{i=1}^{N_{c}} \\mathfrak{b}_{i}^{2} \\sum_{n=1}^{\\infty} \\frac{1}{n^{2}} \\, {\\rm e}^{- \\frac{ m_{q}^{2} }{\\mathfrak{b}_{i}} n \\pi },\n\\end{eqnarray}\nwhich decreases as $\\mathcal{B}$ increases. \nFurthermore, for $\\mathcal{B} \\ge \\mathcal{E}$ the production rate vanishes since in this case the system is equivalent to the pure chromo-magnetic field system.\nWhen $\\theta_{\\mathcal{E} \\mathcal{B}} = 0, \\pi$, which would be relevant configurations for relativistic heavy-ion collisions, $\\mathfrak{a}_{i}$ and $\\mathfrak{b}_{i}$ become\n$\n\\mathfrak{a}_{i} = | g \\omega_{i} \\mathcal{B}|$, $\\mathfrak{b}_{i} = |g \\omega_{i} \\mathcal{E}|$.\nThen, the production rate reads\n\\begin{eqnarray}\n2 {\\Im m}\\, \\mathcal{L}_{\\rm quark}\n&=& \\frac{1}{4\\pi^{2}} \\sum_{i=1}^{N_{c}} | g \\omega_{i} \\mathcal{B} | | g \\omega_{i} \\mathcal{E}| \\sum_{n=1}^{\\infty} \\frac{1}{n}\\, {\\rm e}^{ - \\frac{ m_{q}^{2} }{ | g \\omega_{i} \\mathcal{E} | } n \\pi }\n{\\rm{coth}} \\left( \\frac{ \\mathcal{B} }{ \\mathcal{E} } n \\pi \\right).\n\\label{chromoEB}\n\\end{eqnarray}\nThis production rate is the same result as obtained in Refs.~\\cite{Suganuma:1991ha, Tanji:2008ku}.\nIt increases as either the chromo-electric field or the chromo-magnetic field increases.\nFigure~4 shows $\\theta_{\\mathcal{E} \\mathcal{B}}$ dependence of the light quark production rate with a fixed value of the chromo-electric field, $g\\mathcal{E}=1$ GeV$^{2}$.\nThe maxima appear when the chromo-magnetic field is parallel (or antiparallel) to the chromo-electric field.\n\n\n\\begin{figure}\n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=0.8 \\textwidth]{angle-dep_2ImLq_EcHc_ver2.pdf}\n\\vskip -0.1in\n\\end{center}\n\\end{minipage}\n\\caption{\nLight (up) quark production rate as a function of $\\theta_{E_{\\rm{chro}}B_{\\rm{chro}}}$, which stands for $\\theta_{\\mathcal{E}\\mathcal{B}}$ with vanishing electromagnetic fields.\nWe take the strength of the chromo-electric field as $g\\mathcal{E} = 1$ GeV$^{2}$. \n}\n\\end{figure}\n\n\n\\subsubsection{Quark pair production in a glasma with EM fields}\n\nNow we shall consider a specific configuration of chromo-EM fields that are relevant for relativistic heavy-ion collisions accompanied by EM fields. Suppose that the chromo-electric field and the chromo-magnetic field are parallel to each other, $\\vec{\\mathcal{B}} \\parallel \\vec{\\mathcal{E}}$, and that these strengths are approximately equal to the saturation scale: $|g\\vec{\\mathcal{B}}| = |g\\vec{\\mathcal{E}}| = 1$~GeV$^{2} \\sim Q_{s}^2$. \nThis configuration of chromo-EM fields is indeed realized at the very early stage of the glasma evolution.\nUnder this condition, we investigate light (up) quark productions with $m_{q}=0.5$ MeV and $Q_{q} = +2\/3$.\n\\begin{comment}\nWith vanishing electromagnetic fields, the quark production rate is\n\\begin{eqnarray}\n2 {\\rm{Im}} \\mathcal{L}_{q}\n&=& 0.111 \\ [{\\rm{GeV}}^{4}].\n\\end{eqnarray}\nComparing to electron production rate (\\ref{rate_e}) with the critical electric field $eE = eE_{c} = m_{e}^{2}$, we get\n\\begin{eqnarray}\nw_{q\\bar{q}} \/ w_{e^{+}e^{-}} \\sim 4.20 \\times 10^{15}.\n\\end{eqnarray}\nThis indicates a huge production rate can be obtained in chromoelectromagnetic fields created in relativistic heavy ion collisions.\n\\end{comment}\nLet us turn on the EM fields. \nIn the heavy-ion collisions, the dominant EM field is the magnetic field perpendicular to the beam direction (equivalent to the direction of the glasma fields). But here we consider the case $|e\\vec{B}| \\neq 0$ and $|e\\vec{E}|=0$, with arbitrary orientation. Then, the quantities $\\mathbb{F}_{i,f}^{2}$, and $\\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f}$ read [see Eq.~(\\ref{FFtilde})]\n\\begin{eqnarray}\n\\mathbb{F}_{i,f}^{2}\n&=& 2 \\left[ (eQ_{q})^{2} B^{2} + 2 g \\omega_{i} eQ_{q} \\mathcal{B}B {\\rm{cos}} \\theta_{\\mathcal{B}B} \\right], \\nonumber \\\\\n\\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f}\n&=& -4 \\left[ (g \\omega_{i})^{2} \\mathcal{E} \\mathcal{B} + g \\omega_{i} eQ_{q} \\mathcal{E} B {\\rm{cos}} \\theta_{\\mathcal{B}B} \\right],\n\\end{eqnarray} \nwith $B = \\sqrt{ \\vec{B}^{2} }$. Here we have used the fact that ${\\rm{cos}} \\theta_{\\mathcal{E} B} = {\\rm{cos}} \\theta_{\\mathcal{B}B}$. \nNote that in the case of antiparallel configuration of $\\vec{\\mathcal{B}}$ and $\\vec{\\mathcal{E}}$, results are the same as those of the parallel case, since this changes $\\mathbb{F}_{i,f}\\cdot \\tilde{\\mathbb{F}}_{i,f} \\to - \\mathbb{F}_{i,f}\\cdot \\tilde{\\mathbb{F}}_{i,f}$, but it is squared in $\\mathfrak{a}_{i,f}$ and $\\mathfrak{b}_{i,f}$. \n\nFigure~5 shows the quark production rate as a function of the angle $\\theta_{\\mathcal{B}B}$ with several strengths of the magnetic field.\nAt the angle relevant for relativistic heavy-ion collisions, $\\theta_{\\mathcal{B} B} = \\pi\/2,$ the production rate slightly decreases with increasing $B$ field. This can be understood from Eq.~(\\ref{ImLq_full}) as follows:\nIn this case, the quantity $\\mathfrak{a}_{i,f} = \\frac{1}{2} \\sqrt{ \\sqrt{ 4(eQ_{q})^{4} B^{4} + 16 (g \\omega_{i})^{4} \\mathcal{E} \\mathcal{B} } + 2(eQ_{q})^{2} B^{2} }$ (or $\\mathfrak{b}_{i,f} = \\frac{1}{2} \\sqrt{ \\sqrt{ 4(eQ_{q})^{4} B^{4} + 16 (g \\omega_{i})^{4} \\mathcal{E} \\mathcal{B} } - 2(eQ_{q})^{2} B^{2} }$ ) increases (decreases) with increasing $B$ field, while the product $\\mathfrak{a}_{i,f} \\mathfrak{b}_{i,f} = | \\vec{ \\mathcal{E} }_{i,f} \\cdot \\vec{ \\mathcal{B} }_{i,f} |= (g \\omega_{i})^{2} \\mathcal{E} \\mathcal{B}$ is independent of $B$ field.\nTherefore, at $\\theta_{\\mathcal{B} B} = \\pi\/2$, the quark production rate monotonically decreases due to the exponential factor ${\\rm exp}\\{- (m_{q}^{2}\/\\mathfrak{b}_{i,f}) n \\pi\\} $. \nThis result is independent of the sign of $\\omega_{i}$.\n\nOn the other hand, Fig.~5 shows that the quark production rate increases with increasing $B$ field at $\\theta_{\\mathcal{B}B} = 0$ and $\\pi$. This can be understood as follows:\nAt $\\theta_{\\mathcal{B}B} = 0, \\pi$, the quark production rate reads from Eq.~(\\ref{ImLq_full})\n\\begin{eqnarray}\n2 {\\Im m}\\, \\mathcal{L}_{\\rm{quark}}\n&=& \\frac{1}{4\\pi^{2}} \\sum_{i=1}^{N_{c}} |g\\omega_{i}| \\mathcal{E} \\mathcal{B}_{i,f} \\sum_{n=1}^{\\infty}\n\\frac{1}{n} {\\rm e}^{- \\frac{m_{q}^{2}}{|g\\omega_{i}|\\mathcal{E}} n \\pi } \\coth \\left( \\frac{ \\mathcal{B}_{i,f} }{ |g\\omega_{i}| \\mathcal{E} } n \\pi \\right),\n\\label{choromoEBandB}\n\\end{eqnarray}\nwhere the strength of the combined magnetic field has been defined as $\\mathcal{B}_{i,f} = |g\\omega_{i} \\mathcal{B} + eQ_{q}B | $ for $\\theta_{\\mathcal{B}B} = 0$, whereas $\\mathcal{B}_{i,f} = |g\\omega_{i} \\mathcal{B} - eQ_{q}B | $ for $\\theta_{\\mathcal{B}B} = \\pi$. \nThis production rate has a similar form with Eq.~(\\ref{chromoEB}).\nFirst, we consider the case $|g\\omega_{i} \\mathcal{B}| > |eQ_{q}B|$. When the chromo-magnetic field and the magnetic field are (anti)parallel to each other, $\\theta_{\\mathcal{B}B} = 0$ ($\\theta_{\\mathcal{B}B} = \\pi$), with $\\omega_{i} > 0$ ($\\omega_{i} < 0$), the strength of the combined magnetic field $\\mathcal{B}_{i,f}$ linearly increases with increasing $B$ field, and thus $\\coth \\left( \\frac{ \\mathcal{B}_{i,f} }{ |g\\omega_{i}| \\mathcal{E} } n \\pi \\right)$ slightly decreases and approaches unity.\nWhen $\\theta_{\\mathcal{B}B} = 0$ ($\\theta_{\\mathcal{B}B} = \\pi$) with $\\omega_{i} < 0$ ($\\omega_{i} > 0$), the field strength $\\mathcal{B}_{i,f}$ linearly decreases with increasing $B$ field, but $\\coth \\left( \\frac{ \\mathcal{B}_{i,f} }{ |g\\omega_{i}| \\mathcal{E} } n \\pi \\right)$ increases. Then, after summing over all the modes $i$, the production rate (\\ref{choromoEBandB}) at $\\theta_{\\mathcal{B}B} = 0$ ($\\theta_{\\mathcal{B}B} = \\pi$) monotonically increases with increasing $B$ field.\nIn the case of $|g\\omega_{i} \\mathcal{B}| \\le |eQ_{q}B|$, the production rate of both modes $i=1,2$ increases with increasing $B$ field regardless of the sign of $\\omega_{i}$, and thus the total production rate also monotonically increases.\n\\begin{comment}\n{\\color{red}[I replaced the following sentence by above.}\nWhen the chromo-magnetic field and the magnetic field are (anti-)parallel to each other $\\theta_{\\mathcal{B}B} = 0$ ($\\theta_{\\mathcal{B}B} = \\pi$) with $\\omega_{i} > 0$, the production rate increases (decreases) with increasing $B$-field.\nThe opposite behavior happens when $\\omega_{i} < 0$.\nAfter summing over all the modes $i$, the production rate at $\\theta_{\\mathcal{B}B} = 0, \\pi$ monotonically increases with increasing $B$-field, which can be seen in Fig. 5. \n{\\color{red}up to here.]}\n\\end{comment}\nFurthermore, we again obtain the angle dependence symmetric with respect to $\\theta_{\\mathcal{B}B} = \\pi\/2$ in the production rate.\n\n\n\\begin{figure}\n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=0.8 \\textwidth]{light_quark_prod_eB_angle-dep_ver2.pdf}\n\\vskip -0.1in\n\\end{center}\n\\end{minipage}\n\\caption{\nLight (up) quark production rate in a $B$ field as a function of $\\theta_{B_{\\rm{chro}B}}$, which stands for $\\theta_{\\mathcal{B} B}$ with a parallel configuration of $\\vec{\\mathcal{E}}$ and $\\vec{\\mathcal{B}}$. \nWe take strengths of chromo-electromagnetic fields as $g\\mathcal{B} = g \\mathcal{E} = 1$~GeV$^{2}$.\n}\n\\end{figure}\n\n\nNext we consider the case with $|e\\vec{E}| \\neq 0$ and $|e\\vec{B}|=0$.\nIn this case, $\\mathbb{F}_{i,f}^{2}$ and $\\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f}$ become [see Eq.~(\\ref{FFtilde})]\n\\begin{eqnarray}\n\\mathbb{F}_{i,f}^{2}\n&=& 2 \\left[- (eQ_{q_{f}})^{2} E^{2} - 2g \\omega_{i}eQ_{q_{i}} \\mathcal{E} E {\\rm{cos}} \\theta_{\\mathcal{E} E} \\right], \\nonumber \\\\\n\\mathbb{F}_{i,f} \\cdot \\tilde{\\mathbb{F}}_{i,f}\n&=& -4 \\left[ (g \\omega_{i})^{2} \\mathcal{E} \\mathcal{B} + g \\omega_{i}eQ_{q_{f}} \\mathcal{B} E {\\rm{cos}} \\theta_{\\mathcal{E}E} \\right]\\, .\n\\end{eqnarray}\nIn this expression, we have used ${\\rm{cos}} \\theta_{\\mathcal{B}E} = {\\rm{cos}} \\theta_{\\mathcal{E}E}$.\nAgain, the results are the same as those of the case where $\\vec{\\mathcal{B}}$ is antiparallel to $\\vec{\\mathcal{E}}$.\nFigure~6 shows the quark production rate as a function of the angle $\\theta_{\\mathcal{E}E}$ with several values of strength of the electric field.\nAs the electric field increases, the production rate increases for whole angle regions. \nThis can be understood in a similar way to the previous case as follows:\nAt $\\theta_{\\mathcal{E} E} = \\pi\/2$, the factor $\\mathfrak{a}_{i,f} \\mathfrak{b}_{i,f} = | \\vec{ \\mathcal{E} }_{i,f} \\cdot \\vec{ \\mathcal{B} }_{i,f} |= (g \\omega_{i})^{2} \\mathcal{E} \\mathcal{B}$ is independent of the electric field.\nAs for each factor, $\\mathfrak{a}_{i,f} = \\frac{1}{2} \\sqrt{ \\sqrt{ 4(eQ_{q})^{4} E^{4} + 16 (g \\omega_{i})^{4} \\mathcal{E} \\mathcal{B} } - 2(eQ_{q})^{2} E^{2} }$ decreases with increasing electric field, while $\\mathfrak{b}_{i,f} = \\frac{1}{2} \\sqrt{ \\sqrt{ 4(eQ_{q})^{4} E^{4} + 16 (g \\omega_{i})^{4} \\mathcal{E} \\mathcal{B} } + 2(eQ_{q})^{2} E^{2} }$ increases. \nThese behaviors are opposite to those of the previous case with $|e\\vec{E}| = 0$ and $|e\\vec{B}| \\neq 0$, and thus the production rate at $\\theta = \\pi\/2$ monotonically increases. At $\\theta_{\\mathcal{E} E} = 0, \\pi$, the quark production rate (\\ref{ImLq_full}) can be rewritten as\n\\begin{eqnarray}\n2 {\\Im m}\\, \\mathcal{L}_{\\rm{quark}}\n&=& \\frac{1}{4\\pi^{2}} \\sum_{i=1}^{N_{c}} \\mathcal{E}_{i,f} |g\\omega_{i}| \\mathcal{B} \\sum_{n=1}^{\\infty}\n\\frac{1}{n} {\\rm e}^{- \\frac{m_{q}^{2}}{\\mathcal{E}_{i,f}} n \\pi } \\coth \\left( \\frac{ |g\\omega_{i} |\\mathcal{B} }{ \\mathcal{E}_{i,f} } n \\pi \\right),\n\\label{chromoEBandE}\n\\end{eqnarray}\nwhere the strength of the combined electric field has been defined as $\\mathcal{E}_{i,f} = | g\\omega_{i} \\mathcal{E} + e Q_{q}E| $ for $\\theta_{\\mathcal{E} E} = 0$ and $\\mathcal{E}_{i,f} = | g\\omega_{i} \\mathcal{E} - e Q_{q}E| $ for $\\theta_{\\mathcal{E} E} = \\pi$. \nIn the case of $|g\\omega_{i} \\mathcal{E}| > |e Q_{q}E|$, when the chromo-electric field and the electric field are (anti)parallel to each other, $\\theta_{\\mathcal{E} E} = 0$ ($\\theta_{\\mathcal{E} E} = \\pi$), with $\\omega_{i} > 0$ ($\\omega_{i} < 0$), the strength of the combined electric field $\\mathcal{E}_{i,f}$ linearly increases with increasing $E$ field, and thus $\\coth \\left( \\frac{ |g\\omega_{i} |\\mathcal{B} }{ \\mathcal{E}_{i,f} } n \\pi \\right)$ monotonically increases.\nWhen $\\theta_{\\mathcal{E} E} = 0$ ($\\theta_{\\mathcal{E} E} = \\pi$) with $\\omega_{i} < 0$ ($\\omega_{i} > 0$), the field strength $\\mathcal{E}_{i,f}$ linearly decreases with increasing $E$ field, and $\\coth \\left( \\frac{ |g\\omega_{i} |\\mathcal{B} }{ \\mathcal{E}_{i,f} } n \\pi \\right)$ slightly decreases and approaches unity. \nThen, after summing over all the modes $i$, the production rate (\\ref{chromoEBandE}) at $\\theta_{\\mathcal{E} E} = 0$ ($\\theta_{\\mathcal{E} E} = \\pi$) monotonically increases with increasing $E$ field.\nOn the other hand, in the case of $|g\\omega_{i} \\mathcal{E}| \\le |e Q_{q}E|$, the production rate of both modes $i=1,2$ increases with increasing $E$ field regardless of the sign of $\\omega_{i}$, and thus the total production rate also monotonically increases.\n\\begin{comment}\n{\\color{red}[I replaced the following sentence by above}\nThis production rate is the same form as Eq. (\\ref{choromoEBandB}) and thus enhanced as the strength of electric field $E$ increases.\n{\\color{red}up to here.]}\n\\end{comment}\nFrom these results, we expect that strong EM fields created in the early stage of relativistic heavy-ion collisions would largely affect quark productions from a glasma (chromo-EM fields) depending on the field configurations, and would thus possibly influence the formation of QGP.\n\n\n\\begin{figure}\n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=0.8 \\textwidth]{light_quark_prod_eE_ver2.pdf}\n\\vskip -0.1in\n\\end{center}\n\\end{minipage}\n\\caption{\nLight (up) quark production rate in an $E$ field as a function of $\\theta_{E_{\\rm{chro}}E}$, which stands for $\\theta_{\\mathcal{E} E}$ with a parallel configuration of $\\vec{\\mathcal{E}}$ and $\\vec{\\mathcal{B}}$. \nWe take strengths of chromo-electromagnetic fields as $g\\mathcal{B} = g \\mathcal{E} = 1$ GeV$^{2}$.\n}\n\\end{figure}\n\n\n\\subsection{Weiss potential with electromagnetic fields}\n\nIn this subsection, we will investigate the effects of EM fields on the confinement-deconfinement phase transition by using the effective potential of the Polyakov loop in the presence of EM fields. \n\nPrior to going into the details, let us briefly explain the effective potential without external fields being imposed. The one-loop calculation at finite temperature in SU(2) gauge theory and in the massless fermion limit yields the effective potential for the temporal component of the gauge field $(C = \\frac{ g \\bar{\\mathcal{A}}_{4} }{ 2 \\pi T })$ as~\\cite{Weiss:1980rj,Weiss:1981ev, Gross:1980br}\n\\begin{eqnarray}\nV^{\\rm Weiss}[C]=V_{\\rm YM}^{\\rm Weiss}[C]+V^{\\rm Weiss}_{\\rm quark}[C]\\, ,\n\\end{eqnarray}\nwhere the YM and quark parts are given, respectively, by \n\\begin{eqnarray}\nV_{\\rm YM}^{\\rm Weiss}[C]&=&- \\frac{ 3 }{ 45 } \\pi^{2} T^{4} + \\frac{3}{4} \\pi^{2} T^{4} C^{2} (1-C)^{2}\\, ,\\label{Weiss_YM}\\\\\nV_{\\rm quark}^{\\rm Weiss}[C]&=&- \\frac{7}{90} \\pi^{2} T^{4} + \\frac{1}{6} \\pi^{2} T^{4} C^{2} ( 2 - C^{2} )\\, .\\label{Weiss_quark}\n\\end{eqnarray}\nThis result is called the Weiss potential. In Fig.~7, we show the Weiss potential $V^{\\rm Weiss}[C]$ and its breakdown. \nWe see that in the YM part, the minima appear at $C=0$ and $C = 1$, reflecting the center symmetry $C\\to C+1$ in SU(2). Thus, selecting one of the two minima spontaneously breaks the center symmetry.\nSince the system should be in the deconfined phase in the high-temperature region where a perturbative approach becomes valid,\nthis result seems to be natural. \nThe quark part of the effective potential explicitly breaks the center symmetry, and $C=0$ and $C=1$ are no longer degenerated. \nIn the presence of the quark part, $C=0$ is favored, which corresponds to the deconfined phase. We are now going to investigate how this picture is modified by the presence of external EM fields.\n\\begin{figure}\n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=0.8 \\textwidth]{weiss_potentials.pdf}\n\\vskip -0.1in\n\\end{center}\n\\end{minipage}\n\\caption{Weiss potential as a function of $C$. \nConstant terms which are independent of $C$ are subtracted.\n}\n\\end{figure}\n\n\nNow we come back to our most general results (\\ref{action_gluon}), (\\ref{action_ghost}), and (\\ref{action_quark}). \nTaking the vanishing limit of the chromo-EM fields, $\\vec{\\mathcal{E}}, \\vec{\\mathcal{B}} \\to 0$, but keeping the Polyakov loop $\\bar{\\mathcal{A}}_{4}$ and EM fields nonzero in the results , we obtain the effective potential\n\\begin{eqnarray}\nV_{\\rm eff} [\\bar{\\mathcal{A}}_{4}, E, B]\n&=& - \\frac{S_{\\rm eff}}{\\int dx^{4} } \\nonumber \\\\\n&=& \\frac{1}{32 \\pi^{2}} \\sum_{h=1}^{N_{c}^{2}-1} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} \\left\\{ 4-2 \\right\\} 2 \\sum_{n=1}^{\\infty} {\\rm e}^{ i\\frac{n^{2}}{4T^{2}s} }\n{\\rm{cos}} \\left( \\frac{ g v_{h} \\bar{\\mathcal{A}}_{4} }{T} n \\right) \\nonumber \\\\\n&& - \\frac{1}{8\\pi^{2}} \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} {\\rm e}^{-im_{q_{f}}^{2}s} (\\mathfrak{a}_{f}s)( \\mathfrak{b}_{f}s ){\\rm{cot}}(\\mathfrak{a}_{f}s) {\\rm{coth}}(\\mathfrak{b}_{f}s) \\nonumber \\\\\n&& \\times 2 \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{i \\frac{1}{4T^{2}} \\mathfrak{h}_{f}(s) n^{2}} {\\rm{cos}} \\left( \\frac{ g \\omega_{i} \\bar{\\mathcal{A}}_{4} }{T} n \\right).\n\\end{eqnarray}\nwhere $\\mathfrak{a}_f$ and $\\mathfrak{b}_f$ are just given by the EM fields as\n\\begin{eqnarray}\n\\mathfrak{a}_{f}\n= \\frac{1}{2} \\sqrt{ \\sqrt{ F_{f}^{4} + (F_{f}\\cdot \\tilde{F}_{f})^{2} } + F_{f}^{2} }\\, , \\qquad \n\\mathfrak{b}_{f}\n= \\frac{1}{2} \\sqrt{ \\sqrt{ F_{f}^{4} + (F_{f} \\cdot \\tilde{F}_{f} )^{2} } - F_{f}^{2} }\\, ,\n\\end{eqnarray}\nwith $F_{f}^{2} = 2 (eQ_{q_{f}})^{2} ( \\vec{B}^{2} - \\vec{E}^{2} ) $ and $F_{f} \\cdot \\tilde{F}_{f} = -4 (eQ_{q_{f}})^{2} \\vec{E} \\cdot \\vec{B}$. \nThe factor $\\mathfrak{h}_{f}(s)$ is given by\n\\begin{eqnarray}\n\\mathfrak{h}_{f}(s)\n&=& \\frac{ \\mathfrak{b}_{f}^{2} - {\\mathfrak{e}}_{f}^{2} }{ \\mathfrak{a}_{f}^{2} + \\mathfrak{b}_{f}^{2} } \\mathfrak{a}_{f} {\\rm{cot}}( \\mathfrak{a}_{f}s ) + \\frac{ \\mathfrak{a}_{f}^{2} + {\\mathfrak{e}}_{f}^{2} }{ \\mathfrak{a}_{f}^{2} + \\mathfrak{b}_{f}^{2} } \\mathfrak{b}_{f} {\\rm{coth}}(\\mathfrak{b}_{f} s)\\, ,\n\\end{eqnarray}\nwhere ${\\mathfrak{e}}_{f}^{2} = (u_{\\alpha} F_{f}^{\\alpha \\mu})( u_{\\beta} F_{f \\mu}^{\\beta}) = (eQ_{q_{f}})^{2} E^{2} $ with $u_{\\mu} = (1,0,0,0)$.\nHere we have subtracted divergences appearing in the zero-temperature contribution, which are independent of\n$\\bar{\\mathcal{A}}_{4}$.\n\n\\subsubsection{Weiss potential in magnetic fields}\n\nConsider a pure magnetic field case, $\\vec{E} \\to 0$, $\\vec{B} \\neq 0$.\nThen, the effective potential reads,\n\\begin{eqnarray}\nV_{\\rm eff} [\\bar{\\mathcal{A}}_{4}, B]\n&=&\\frac{1}{32 \\pi^{2}} \\sum_{h=1}^{N_{c}^{2}-1} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} \\left\\{ 4-2 \\right\\} 2 \\sum_{n=1}^{\\infty} {\\rm e}^{ i\\frac{n^{2}}{4T^{2}s} }\n\\, {\\rm{cos}} \\left( \\frac{ g v_{h} \\bar{\\mathcal{A}}_{4} }{T} n \\right) \\nonumber \\\\\n&& - \\frac{1}{8\\pi^{2}} \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} {\\rm e}^{-im_{q_f}^{2}s} ( e|Q_{q_{f}}|B s) {\\rm{cot}}( e|Q_{q_{f}}|Bs) \\nonumber \\\\\n&&\\qquad \\times 2 \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{i \\frac{ n^{2} }{ 4T^{2}s } }\\, {\\rm{cos}} \\left( \\frac{ g \\omega_{i}\\bar{\\mathcal{A}}_{4} }{T} n \\right).\n\\label{VB}\n\\end{eqnarray} \nWe rewrite the proper time integrals in two steps. Recall that the integral should be defined with an infinitesimally small number $\\delta$ which makes the contour slightly inclined to avoid the poles along the real axis (in the second term). Then we can easily change the contour from $[0,\\infty]$ along the real axis to $[-i\\infty,0]$ along the imaginary axis (the Wick rotation), since there is no pole along the imaginary axis. Finally, by renaming the variable $s$ as $-i\\sigma$, we obtain the following representation with integrals defined by real functions\\footnote{The second line of Eq.~(\\ref{PolyakovLoopwithB}) coincides with Eq.~(B.6) in the appendix of Ref.~\\cite{Bruckmann:2013oba}.}:\n\\begin{eqnarray}\nV_{\\rm eff} [\\bar{\\mathcal{A}}_{4}, B]\n&=& - \\frac{ 1 }{ 8\\pi^{2}} \\sum_{h=1}^{N_{c}^{2}-1} \\int^{\\infty}_{0} \\frac{d\\sigma }{\\sigma^{3}} \\sum_{n=1}^{\\infty} {\\rm e}^{- \\frac{n^{2}}{4T^{2}\\sigma} }\\, {\\rm{cos}}\\left( \\frac{ g v_{h} \\bar{\\mathcal{A}}_{4} }{ T } n \\right) \\nonumber \\\\\n&& +\\frac{1}{4\\pi^{2}} \\sum_{i=1}^{N_{c}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{d\\sigma}{\\sigma^{2}} {\\rm e}^{-m_{q_{f}}^{2}\\sigma} (e|Q_{q_{f}}|B) {\\rm{coth}}(e|Q_{q_{f}}|B\\sigma ) \\nonumber \\\\\n&&\\qquad \\times \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{ - \\frac{n^{2}}{4T^{2}\\sigma} }\\, {\\rm{cos}} \\left( \\frac{ g \\omega_{i}\\bar{\\mathcal{A}}_{4} }{ T} n \\right).\n\\label{PolyakovLoopwithB}\n\\end{eqnarray}\nFor simplicity, we shall restrict ourselves to $N_{c}=2$, which provides us with all the essential features of the perturbative effective potential in the presence of EM fields. In this case, the eigenvalues $\\omega_{i}$ and $v_{h}$ are simply given by $\\omega_{i} = \\pm 1\/2$ and $v_{h} = 0, \\pm 1$.\nThe effective potential reads,\n\\begin{eqnarray}\nV_{\\rm eff}[ C, B ]\n&=& - \\frac{ 3 }{ 45 } \\pi^{2} T^{4} + \\frac{3}{4} \\pi^{2} T^{4} C^{2} (1-C)^{2} \\label{effectiveVTB} \\\\\n&& + \\frac{1}{2\\pi^{2}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{d\\sigma }{\\sigma^{2}} {\\rm e}^{-m_{q_{f}}^{2}\\sigma} (e|Q_{q_{f}}|B) {\\rm{coth}}(e|Q_{q_{f}}|B\\sigma ) \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{ - \\frac{n^{2}}{4T^{2}\\sigma} } {\\rm{cos}} \\left( C\\pi n \\right)\\, . \\nonumber \n\\end{eqnarray}\nThe first line does not depend on the magnetic field and corresponds to the YM part $V_{\\rm YM}$. This is nothing but the Weiss potential~(\\ref{Weiss_YM}) \\cite{Weiss:1980rj}. The second line corresponds to the quark part $V_{\\rm quark}$, and the integral and summation over $n$ can be easily performed numerically. \nFrom now on, we further restrict ourselves to the one flavor $f=1$ with the electric charge $Q_{q_{f}}=1$ for simplicity.\nNow, analytic expressions are available in two limiting cases: One is the $B\\to 0$ and $m_q\\to 0$ limit, where the quark part of the effective potential is reduced to that of the Weiss potential (\\ref{Weiss_YM}):\n\\begin{eqnarray}\nV_{\\rm quark} [C]\n&= & - \\frac{7}{90} \\pi^{2} T^{4} + \\frac{1}{6} \\pi^{2} T^{4} C^{2} ( 2 - C^{2} )=V^{\\rm Weiss}_{\\rm quark}[C]\\, .\n\\end{eqnarray}\nThe other is the strong magnetic field limit: $eB \\gg m_{q}^{2}$, where the quark part can be written as\n\\begin{eqnarray}\nV_{\\rm quark} [C, B]\n&=& - 2 \\frac{ (eB) }{ \\pi^{2} } T^{2} \\left\\{ \\frac{ \\pi^{2} }{ 12 } - \\frac{ (C\\pi)^{2} }{ 4 } \\right\\}. \\label{quarkVstrongB}\n\\end{eqnarray}\nFigure~8 shows the magnetic field dependence of the quark part of the effective potential which is given by the second line of Eq.~(\\ref{effectiveVTB}). \nHere, we show only one flavor contribution with $x=m_q^2\/T^2=0.5$. An important observation is that as the magnetic field increases, the explicit breaking of the center symmetry is enhanced, and $C=0$ (deconfined phase) becomes more stable. This is qualitatively consistent with the analytic representation at strong magnetic fields [see Eq.~(\\ref{quarkVstrongB})] in that the potential value at $C=0$ becomes more negative and the rising behavior becomes steeper with increasing magnetic field. The enhancement of the center symmetry-breaking effects due to increasing magnetic field indicates that the quark loop interacting with magnetic fields can be one of the important sources for reducing the (pseudo)critical temperature $T_{c}$ of confinement-deconfinement phase transition, as observed in recent lattice QCD simulations \\cite{Bruckmann:2013oba}. In the last part of this subsection, we will see within a phenomenological model that this is indeed the case.\n\n\n\n\\begin{figure}[t]\n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=0.8 \\textwidth]{potential_eB_full_paper.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{Quark part of the effective potential as a function of $C$ for several values of magnetic fields. $x$ and $y$ are given as $x = m_{q}^{2}\/T^{2}$ and $y = eB\/T^{2}$, respectively. \n}\n\\end{figure}\n\n\n\n\n\\subsubsection{Weiss potential in electric fields}\n\n\nIn the case of a pure electric field, $\\vec{B} \\to 0$ and $\\vec{E} \\neq 0$, the situation is a bit subtle. \nThe effective potential of the quark part can be written as\n\\begin{eqnarray}\nV_{\\rm quark}[\\bar{\\mathcal{A}}_{4}, E]\n&=& - \\frac{ 1}{2\\pi^{2}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} {\\rm e}^{-im_{q_{f}}^{2}s} \\left( e|Q_{q_{f}}|Es \\right) {\\rm{coth}} \\left( e| Q_{q_{f}}| Es \\right) \\nonumber \\\\\n&& \\quad \\times \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{i \\frac{n^2}{4T^{2}s} \\left( e |Q_{q_{f}}| Es \\right) {\\rm{coth}} \\left( e |Q_{q_{f}}|Es \\right) } {\\rm{cos}} \\left( \\frac{ g\\bar{\\mathcal{A}}_{4} }{2T } n \\right).\n\\end{eqnarray}\nNote that we cannot reach this result from Eq.~(\\ref{VB}) by replacing $B$ with $iE$, unlike the zero-temperature contribution. This is due to the form of the factor $\\mathfrak{h}_{f}(s)=(e|Q_{q_f}|Es){\\rm coth}(e|Q_{q_f}|Es)$ in the exponential. Because of this factor, the full calculation (even numerical evaluation) is rather difficult. Furthermore, since there are singularities (poles) on the imaginary axis, we cannot perform the Wick rotation of the proper time $s$, unlike the Weiss potential in magnetic fields. \nTo avoid these difficulties, we expand the effective potential with respect to the electric field. Using $x{\\rm{coth}}x \\sim 1 + x^{2}\/3 \\cdots$, we get\n\\begin{eqnarray}\nV_{\\rm quark}[\\bar{\\mathcal{A}}_{4}, E]\n&=& - \\frac{1}{2\\pi^{2}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{ds}{s^{3}} {\\rm e}^{-im_{q_{f}}^{2}s } \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{i \\frac{ n^{2} }{4T^{2}s} } {\\rm{cos}} \\left( \\frac{ g \\bar{\\mathcal{A}}_{4} }{2T } n \\right) \\nonumber \\\\\n&& - \\frac{ 1}{6 \\pi^{2}} \\sum_{f=1}^{N_{f}} (e|Q_{q_{f}}| E)^{2} \\int^{\\infty}_{0} \\frac{ds}{s} {\\rm e}^{-im_{q_{f}}^{2}s} \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{i \\frac{ n^{2}}{4T^{2}s} } \\left( 1 + \\frac{ n^{2} }{ 4 T^{2} s } \\right) {\\rm{cos}} \\left( \\frac{ g \\bar{\\mathcal{A}}_{4} }{ 2T } n \\right) \\nonumber \\\\\n&& + {\\cal O}(E^{4})\\, .\n\\end{eqnarray}\nAt this stage, we can perform the Wick rotation for the proper time $s$. Then, the effective potential reads\n\\begin{eqnarray}\nV_{\\rm quark}[C, E]\n&=& \n\\frac{1}{2\\pi^{2}} \\sum_{f=1}^{N_{f}} \\int^{\\infty}_{0} \\frac{d\\sigma}{\\sigma^{3}} {\\rm e}^{-m_{q_{f}}^{2}\\sigma} \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{- \\frac{n^{2}}{4T^{2}\\sigma} } {\\rm{cos}} \\left( \n C \\pi n \\right) \\nonumber \\\\\n && - \\frac{1}{6\\pi^{2}} \\sum_{f=1}^{N_{f}} (e|Q_{q_{f}}|E )^{2} \\int^{\\infty}_{0} \\frac{d\\sigma}{\\sigma} {\\rm e}^{-m_{q_{f}}^{2}\\sigma} \n \\sum_{n=1}^{\\infty} (-1)^{n} {\\rm e}^{- \\frac{ n^{2} }{ 4T^{2} \\sigma } } \\left( 1 - \\frac{ n^{2} }{ 4 T^{2} \\sigma } \\right) {\\rm{cos}} \\left( C \\pi n \\right) \\nonumber \\\\\n && + {\\cal O}(E^{4})\\, .\n\\end{eqnarray}\nThe systematic expansion with respect to the $E$ field is possible, and the integral and sum can be performed numerically at each order.\n\n\nIn Fig. 9 we show the electric field dependence of the quark part of the effective potential. From this figure, we see that the electric field decreases the explicit breaking of the center symmetry. This is completely opposite to the $B$ dependence of the effective potential. Thus, we expect that $T_{c}$ increases with increasing $E$ field and approaches the $T_{c}$ of the pure YM theory.\n\n\n\\begin{figure}[t] \n\\begin{minipage}{0.8\\hsize}\n\\begin{center}\n\\includegraphics[width=0.8 \\textwidth]{potential_eE_OrderE2_paper.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{ Quark part of the effective potential as a function of $C$ for several values of electric fields. $x$ and $y$ are given as $x = m_{q}^{2}\/T^{2}$ and $y = eE\/T^{2}$, respectively. \n}\n\\end{figure}\n\n\n\\subsubsection{Phenomenological analysis on $T_c(B)$}\n\nWe have seen that imposing magnetic fields enhances the explicit breaking of the center symmetry. What we have evaluated is a perturbative contribution (in the sense that we assume that the coupling is small enough), and thus we discussed how the Weiss potential (that is also evaluated in a perturbative framework) is modified in the presence of the EM fields. Within this perturbative calculation, we are not able to approach the region where phase transition will take place. Indeed, even if the quark part of the effective potential depends on the magnetic fields $V_{\\rm quark}[C,B]$, the total effective potential $V_{\\rm eff}[C,B]=V_{\\rm YM}[C]+V_{\\rm quark}[C,B]$ selects the center broken state $C=0$, and thus confinement-deconfinement phase transition never occurs within this perturbative framework. However, recall that the magnetic field can affect the effective potential of the Polyakov loop only through the quark loop at leading order. Therefore, we expect that even the perturbative evaluation of the quark part $V_{\\rm quark}[C,B]$ can make sense if combined with some nonperturbative effective potential $V_{\\rm YM}^{\\rm nonpert}[C]$ for study of the effects of magnetic fields on the phase transition. Here we discuss whether this is indeed the case. \n\n\n\nLet us introduce a simple model of a gluonic potential reproducing confinement-deconfinement phase transition,\n\\begin{eqnarray}\n\\mathcal{U}[C]\n&=& -\\frac{1}{2}a(T) \\Phi^{2} + b(T)\\, {\\rm{ln}} \\left[ 1 - 6 \\Phi^{2} + 8 \\Phi^{3} - 3 \\Phi^{4} \\right]\n\\label{phenomenological_potential}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\na(T) = a_{0} + a_{1}(T_{0}\/T) + a_{2}(T_{0} \/ T )^{2}, \\ \\ \\ \\ b(T) = b_{3}(T_{0}\/T)^{3}.\n\\end{eqnarray}\nNow, we consider the $N_{c}=3$ case.\nHere the parameters are\n$a_{0} = 3.51,\\, a_{1} = -2.47,\\, a_{2} = 15.2,\\, b_{3} = -1.75$, and $T_{0} = 270$ MeV, which are fixed to reproduce the quenched lattice QCD results \\cite{Roessner:2006xn}.\nInstead of $V_{YM}$, we employ this phenomenological potential (\\ref{phenomenological_potential}) and combine it with $V_{\\rm quark} [C,B]$. In this way, we can study how the temperature dependence of the Polyakov loop changes with magnetic fields. \nNotice that the quark part of the perturbative effective potential $V_{\\rm quark}[C,B]$ with $N_{c}=3$ is the same as that of the one with $N_{c}=2$, since the quark with $\\omega_{3}=0$ does not contribute to the potential.\nTherefore, we can use the same potential evaluated in the second line of Eq.~(\\ref{effectiveVTB}).\nThe result is shown in Fig. 10. In this analysis, we have used $\\omega_{i} = \\pm 1\/2, 0$ and a constituent quark mass $m_{q} = 350$ MeV. Thanks to the explicit center symmetry breaking, the Polyakov loop increases with increasing $B$ field, in particular below the phase transition temperature, \nwhich eventually brings about decreasing pseudocritical temperature $T_c(B)2$. Let $\\eta$ be an umbilical normal vector of $M^n$. If the normal connection is flat, then $\\eta$ is in the direction of the distinguished normal vector $\\xi$.\n\\end{thm}\n\nIt is proved in \\cite{YTST} that there exists neither totally geodesic real hypersurfaces nor totally umbilical real hypersurfaces of a complex projective space. From Theorem 1.1, we can generalize this result to the following\n\n\\begin{cor}\n In $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$ ($n>2$) there exists neither totally geodesic CR submanifolds of maximal CR dimension nor totally umbilical CR submanifolds of maximal CR dimension, whose normal connections are flat.\n\\end{cor}\n\nNext we consider the converse of Theorem 1.1. For 3-dimensional submanifolds, we prove the following theorem.\n\n\\begin{thm}\n Let $M^3$ be a 3-dimensional CR submanifold of maximal CR dimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$, $p>1$. If the normal connection is flat, then $p=3$ and the distinguished normal vector $\\xi$ is umbilical.\n\\end{thm}\n\nAs the application of Theorem 1.1 and Theorem 1.3, we prove the non-existence of a class of CR submanifolds of maximal CR dimension of a complex projective space.\n\n\\begin{thm}\n In $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$ $(p>1)$ there exist no 3-dimensional pseudo-umbilical CR submanifolds of maximal CR dimension with flat normal connection.\n\\end{thm}\n\n\\begin{cor}\n In $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$ $(p>1)$ there exist no 3-dimensional minimal CR submanifolds of maximal CR dimension with flat normal connection.\n\\end{cor}\n\n\\begin{remark}\nWe should note that for some other ambient spaces there may exist pseudo-umbilical submanifolds with flat normal connection. For instance, from results of \\cite{BYC1} we know that minimal surfaces of a hypersurface of a Euclidean space $\\mathbf{E}^m$ and the product of two plane circles in $\\mathbf{E}^4$ are both pseudo-umbilical with flat normal connection.\n\\end{remark}\n\n\\section{\\bf Preliminaries}\n\\vskip 0.4 true cm\n\nLet $M^n$ be a CR submanifold of maximal CR dimension of $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$. For each point $x\\in M^n$, the real dimension of the holomorphic tangent space $H_x(M^n)$ is $n-1$. Therefore $M^n$ is necessarily odd-dimensional and there exists a unit normal vector $\\xi_x$ such that\n$$JT_x(M^n)\\subset T_x(M^n)\\oplus span\\{\\xi_x\\}.$$\nWrite\n\\begin{equation}\\label{defU}\n U_x=-J\\xi_x.\n\\end{equation}\nIt is easy to see that $U_x$ is a unit tangent vector of $M^n$ which spans the totally real tangent space $R_x(M^n)$. So a tangent vector $Z_x$ of $M^n$ is a holomorphic tangent vector, i.e., $Z_x\\in H_x(M^n)$ if and only if $Z_x$ is orthogonal to $U_x$. For any $X\\in TM^n$, we may write\n\\begin{equation}\\label{defFu}\n JX=FX+u(X)\\xi,\n\\end{equation}\nwhere $F$ is a skew-symmetric endomorphism acting on $TM^n$, $u$ is the one form dual to $U$. It is proved in \\cite{MDMO1} that\n\\begin{equation}\\label{F2}\n F^2X=-X+u(X)U,\n\\end{equation}\n\\begin{equation}\\label{FU}\n u(FX)=0,\\ FU=0,\n\\end{equation}\nwhich imply $M^n$ has an almost contact structure.\n\nLet $T^{\\bot}_1(M^n)$ be the subbundle of the normal bundle $T^{\\bot}(M^n)$ defined by\n$$\n T^{\\bot}_1(M^n)=\\{\\eta\\in T^{\\bot}(M^n)|\\langle \\eta,\\xi\\rangle=0\\},\n$$\nwhere $\\langle,\\rangle$ is the inner product of the tangent space of $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$. Since $T^{\\bot}_1(M^n)$ is $J$-invariant, we can choose a local orthonormal basis of $T^{\\bot}(M^n)$ in the following way:\n\\begin{equation}\\label{nframe}\n \\xi,\\ \\xi_1,\\cdots,\\xi_q,\\ \\xi_{1^*},\\cdots,\\xi_{q^*},\n\\end{equation}\nwhere $\\xi_{a^*}=J\\xi_a,\\ a=1,\\cdots,q$ and $q=\\frac{p-1}{2}$.\n\nLet $A, A_a, A_{a^*}$ denote the shape operators for the normals $\\xi, \\xi_a, \\xi_{a^*}$, respectively. Write\n$$\n D\\xi=\\sum_a(s_a\\xi_a+s_{a^*}\\xi_{a^*}),\n$$\n$$\n D\\xi_a=-s_a\\xi+\\sum_b(s_{ab}\\xi_b+s_{ab^*}\\xi_{b^*}),\n$$\n$$\n D\\xi_{a^*}=-s_{a^*}\\xi+\\sum_b(s_{a^*b}\\xi_b+s_{a^*b^*}\\xi_{b^*}),\n$$\nwhere $s$'s are the coefficients of the normal connection $D$. Let $\\overline\\nabla$ be the connection of $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$. By using the classical Weingarten formula and noting that $\\overline\\nabla J=0$, one can obtain the following relations (\\cite{MDMO1}):\n\\begin{equation}\\label{Aa*}\n A_{a^*}X=FA_aX-s_a(X)U,\n\\end{equation}\n\\begin{equation}\\label{Aa}\n A_{a}X=-FA_{a^*}X+s_{a^*}(X)U,\n\\end{equation}\n\\begin{equation}\\label{trAa*}\n {\\rm trace} A_{a^*}=-s_a(U),\\ {\\rm trace} A_a=s_{a^*}(U),\n\\end{equation}\n\\begin{equation}\\label{sa*}\n s_{a^*}(X)=\\langle A_aU,X\\rangle,\n\\end{equation}\n\\begin{equation}\\label{sa}\n s_{a}(X)=-\\langle A_{a^*}U,X\\rangle,\n\\end{equation}\n\\begin{equation}\\label{sab}\n s_{a^*b^*}=s_{ab},\\ s_{a^*b}=-s_{ab^*},\n\\end{equation}\n\\begin{equation}\\label{gbU}\n \\nabla_XU=FAX,\n\\end{equation}\nwhere $X,Y$ are tangent to $M^n$, $\\nabla$ is the connection induced from $\\overline\\nabla$, and $a,b=1,\\cdots,q$.\n\nTo prove our theorems, we need to write the classical equations of Codazzi and Ricci for submanifolds. For the sake of convenience, set $\\xi_0=\\xi$ and $\\alpha,\\beta=0,1,\\cdots,q,1^*,\\cdots,q^*$. Recall the equation of Codazzi for the normal vector $\\xi$ is given by \\cite{MDMO1}\n\\begin{align}\\label{CodazziA}\n (\\nabla_XA)Y-(\\nabla_YA)X =&-(\\overline{R}(X,Y)\\xi)^{\\top}+\\sum_b\\{s_b(X)A_bY-s_b(Y)A_bX\\}\\notag\\\\\n & +\\sum_b\\{s_{b^*}(X)A_{b^*}Y-s_{b^*}(Y)A_{b^*}X\\},\n\\end{align}\nwhere $\\overline{R}$ is the Riemannian curvature tensor of $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$, $X,Y$ are tangent to $M^n$, $(\\overline{R}(X,Y)\\xi)^{\\top}$ is the tangent part of $\\overline{R}(X,Y)\\xi$, and $(\\nabla_XA)Y$ is defined as\n\\begin{equation}\\label{deflA}\n (\\nabla_XA)Y=\\nabla_XAY-A(\\nabla_XY).\n\\end{equation}\nRecall that the equation of Ricci is given by \\cite{MDMO1}\n\\begin{equation}\\label{eqRicci}\n \\langle R^{\\bot}(X,Y)\\xi_{\\alpha},\\xi_{\\beta}\\rangle=\\langle \\overline{R}(X,Y)\\xi_{\\alpha},\\xi_{\\beta}\\rangle+\\langle [A_{\\alpha},A_{\\beta}]X,Y\\rangle,\n\\end{equation}\nwhere $R^{\\bot}$ is the curvature tensor of the normal connection, and\n\\begin{equation*}\n [A_{\\alpha},A_{\\beta}]=A_{\\alpha}\\circ A_{\\beta}-A_{\\beta}\\circ A_{\\alpha}.\n\\end{equation*}\n\n\nNote that the Riemannian curvature tensor $\\overline{R}$ of $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$ is given by\n\\begin{equation}\\label{curvatureP}\n\\overline{R}(\\overline{X},\\overline{Y})\\overline{Z}= \\langle\\overline{Y},\\overline{Z}\\rangle\\overline{X}\n-\\langle\\overline{X},\\overline{Z}\\rangle\\overline{Y}+\\langle J\\overline{Y},\\overline{Z}\\rangle J\\overline{X}\n-\\langle J\\overline{X},\\overline{Z}\\rangle J\\overline{Y}+2\\langle \\overline{X},J\\overline{Y}\\rangle J\\overline{Z},\n\\end{equation}\nwhere $\\overline{X},\\overline{Y},\\overline{Z}$ are tangent to $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$. From (\\ref{curvatureP}),(\\ref{defU}),(\\ref{defFu}), we calculate\n\\begin{equation*}\n \\overline{R}(X,Y)\\xi=u(Y)FX-u(X)FY+2\\langle FX,Y\\rangle U,\n\\end{equation*}\n\\begin{equation*}\n \\overline{R}(X,Y)\\xi_a=-2\\langle FX,Y\\rangle\\xi_{a^*},\n\\end{equation*}\n\\begin{equation*}\n \\overline{R}(X,Y)\\xi_{a^*}=2\\langle FX,Y\\rangle\\xi_{a}.\n\\end{equation*}\nTherefore the equation of Codazzi (\\ref{CodazziA}) becomes \\cite{MDMO1}\n\\begin{align}\\label{equationCodazziA}\n (\\nabla_XA)Y-(\\nabla_YA)X = & u(X)FY-u(Y)FX-2\\langle FX,Y\\rangle U \\notag\\\\\n & +\\sum_b\\{s_b(X)A_bY-s_b(Y)A_bX\\}\\notag\\\\\n &+\\sum_b\\{s_{b^*}(X)A_{b^*}Y-s_{b^*}(Y)A_{b^*}X\\}.\n\\end{align}\nThe equation of Ricci (\\ref{eqRicci}) becomes\n\\begin{equation}\\label{eqRicciAAa}\n \\langle R^{\\bot}(X,Y)\\xi,\\xi_{a}\\rangle=\\langle [A,A_{a}]X,Y\\rangle,\n\\end{equation}\n\\begin{equation}\\label{eqRicciAAa*}\n \\langle R^{\\bot}(X,Y)\\xi,\\xi_{a^*}\\rangle=\\langle [A,A_{a^*}]X,Y\\rangle,\n\\end{equation}\n\\begin{equation}\\label{eqRicciAaAb}\n \\langle R^{\\bot}(X,Y)\\xi_a,\\xi_b\\rangle=\\langle [A_a,A_b]X,Y\\rangle,\n\\end{equation}\n\\begin{equation}\\label{eqRicciAaAb*}\n \\langle R^{\\bot}(X,Y)\\xi_a,\\xi_{b^*}\\rangle=-2\\langle FX,Y\\rangle\\delta_{ab}+\\langle [A_a,A_{b^*}]X,Y\\rangle,\n\\end{equation}\n\\begin{equation}\\label{eqRicciAa*Ab*}\n \\langle R^{\\bot}(X,Y)\\xi_{a^*},\\xi_{b^*}\\rangle=\\langle [A_{a^*},A_{b^*}]X,Y\\rangle.\n\\end{equation}\n\n\n\\section{\\bf The Position of the Umbilical Normal Vector in the Normal Bundle}\n\\vskip 0.4 true cm\n\nLet $M^n$ be a CR submanifold of maximal CR dimension of $\\mathbf{P}^{\\frac{n+p}{2}}(\\mathbf{C})$. The normal connection $D$ is said to be {\\it flat}, if the curvature tensor $R^{\\bot}$ of $D$ vanishes. In this section we discuss the position of the umbilical normal vector in the normal bundle for this kind of submanifolds. Recall that a normal vector $\\eta$ is said to be {\\it umbilical}, if the shape operator with respect to $\\eta$ is given by\n\\begin{equation}\\label{3AelxMn}\n A_{\\eta}=\\lambda id: T_x(M^n)\\to T_x(M^n),\n\\end{equation}\nwhere $\\lambda=\\langle\\eta,\\zeta\\rangle$, $\\zeta$ is the mean curvature vector, and $id:T_x(M^n)\\to T_x(M^n)$ is the identity map. Specially, if $\\zeta$ is umbilical, then the submanifold $M^n$ is called {\\it pseudo-umbilical}. It is obvious that minimal submanifolds must be pseudo-umbilical (see \\cite{BYC2}).\n\nFrom equations of Ricci (\\ref{eqRicciAAa})-(\\ref{eqRicciAa*Ab*}), we see that flat normal connection implies that\n\\begin{equation}\\label{3AAaAa0}\n [A,A_a]=0,\\ [A,A_{a^*}]=0,\n\\end{equation}\n\\begin{equation}\\label{3AaAAb0}\n [A_a,A_b]=0,\\ [A_{a^*},A_{b^*}]=0,\n\\end{equation}\n\\begin{equation}\\label{3AaAdab}\n [A_{a},A_{b^*}]=2\\delta_{ab}F,\n\\end{equation}\nwhere $a,b=1\\cdots q$.\n\n\\begin{proof}[Proof of Theorem 1.1]\n The result trivially holds when $p=1$. In the following, we assume $p>1$. For the umbilical normal vector $\\eta$, we decompose it as $\\eta=\\eta_1+\\eta_2$, where $\\eta_1\\in span\\{\\xi\\}, \\eta_2\\bot\\xi$. Choose the unit normal vector $\\xi_1$ such that $\\eta_2=|\\eta_2|\\xi_1$, then\n \\begin{equation*}\n \\eta=|\\eta_1|\\xi+|\\eta_2|\\xi_1.\n \\end{equation*}\n From the definition of the umbilicity of $\\eta$ (see (\\ref{3AelxMn})), we deduce that\n \\begin{align*}\n 0 & =[A_{\\eta},A_{1^*}]=[|\\eta_1|A+|\\eta_2|A_1,A_{1^*}]\\notag\\\\\n & =|\\eta_1|[A,A_{1^*}]+|\\eta_2|[A_1,A_{1^*}].\n \\end{align*}\n Substituting (\\ref{3AAaAa0}) and (\\ref{3AaAdab}) into the above formula, we get\n \\begin{equation*}\n 2|\\eta_2|F=0.\n \\end{equation*}\n Since $n>2$ and $rank F=n-1$, we conclude that $|\\eta_2|=0$. Therefore $\\eta=|\\eta|\\xi$.\n\\end{proof}\n\nTo prove Theorem 1.3, we need the following lemmas. The first one is an easy linear algebra result which can be obtained by direct calculations.\n\n\\begin{lem}\n Let $(V,\\langle,\\rangle)$ be an $n$-dimensional inner product space and $f:V\\to V$ be a linear transformation. Suppose there exist $\\lambda\\in \\mathbf{R}$ and $X\\in V$ such that $f(X)=\\lambda X$. If the linear transformations $f_1,f_2:V\\to V$ are both commutative with $f$, then we have\n \\begin{equation*}\n f(f_1X)=\\lambda f_1X,\\ f(f_2X)=\\lambda f_2X,\\ f([f_1,f_2]X)=\\lambda [f_1,f_2]X.\n \\end{equation*}\n\\end{lem}\n\n\n\\begin{lem}\n Let $M^3$ be a 3-dimensional CR submanifold of maximal CR dimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$, $p>1$. If the normal connection is flat, then $p=3$.\n\\end{lem}\n\n\\begin{proof}\n Otherwise\\ $p>3$, then we may choose orthonormal frame\n \\begin{equation*}\n \\xi,\\ \\xi_1,\\ \\xi_2,\\cdots,\\xi_q,\\ \\xi_{1^*},\\ \\xi_{2^*},\\cdots,\\xi_{q^*}\n \\end{equation*}\n of $T^{\\bot}(M^3)$. In the following we consider the eigenvalues and eigenvectors of the shape operator $A_1$. We prove first that if there exists an eigenvalue of $A_1$, say $\\alpha$, such that $U$ is not the eigenvector corresponding to $\\alpha$, then the multiplicity of $\\alpha$ is 2. In fact, since the normal connection is flat, from (\\ref{3AaAAb0}) and (\\ref{3AaAdab}), we have\n \\begin{equation*}\n [A_1,A_2]=0,\\ [A_1,A_{2^*}]=0.\n \\end{equation*}\n According to Lemma 3.1, if $X$ is an eigenvector corresponding to $\\alpha$, then\n \\begin{equation*}\n A_1([A_2,A_{2^*}]X)=\\alpha [A_2,A_{2^*}]X.\n \\end{equation*}\n Noting that $[A_2,A_{2^*}]=2F$, the above formula becomes\n \\begin{equation*}\n A_1(FX)=\\alpha FX.\n \\end{equation*}\n It is easy to see that if $X\\not\\in span\\{U\\}$, then $X$ and $FX$ are linearly independent. Hence the above formula implies the multiplicity of $\\alpha$ is at least 2. This combined with Theorem 1.1 shows that the multiplicity of $\\alpha$ is 2.\n\n Next we prove $A_1$ has two distinct eigenvalues, and $U$ is the eigenvector corresponding to the simple one, while all the holomorphic tangent vectors are eigenvectors corresponding to the other one whose multiplicity is 2. In fact, Theorem 1.1 guarantees that $A_1$ has at least two distinct eigenvalues, say $\\alpha$ and $\\beta$. From the declaration above, we know that $U$ is an eigenvector corresponding to $\\alpha$ or $\\beta$, say $\\beta$ (otherwise dim$M^3\\geqq 4$). Then the eigenvectors of $\\alpha$ are orthonormal to $U$. Also from the declaration above, we see that $\\alpha$ has multiplicity 2.\n\n In entirely the same way we can prove that $A_{1^*}$ also has two distinct eigenvalues, and $U$ is an eigenvector corresponding to the simple one, while all the holomorphic tangent vectors are eigenvectors corresponding to the other one whose multiplicity is 2.\n\n Take a holomorphic tangent vector $X\\not=0$. Assume that\n \\begin{equation*}\n A_1X=\\alpha X,\\ A_{1^*}X=\\alpha^*X.\n \\end{equation*}\n By a direct calculation, we have\n \\begin{equation*}\n [A_1,A_{1^*}]X=0.\n \\end{equation*}\n On the other hand, (\\ref{3AaAdab}) implies that $[A_1,A_{1^*}]X=2FX\\not=0$. This contradiction shows that $p=3$.\n\\end{proof}\n\n\\begin{lem}\n Let $M^3$ be a 3-dimensional CR submanifold of maximal CR dimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$, $p>1$. If the normal connection is flat, then either the distinguished normal vector $\\xi$ is umbilical, or the shape operator $A$ has two distinct eigenvalues. In the latter case, $U$ is an eigenvector corresponding to the simple eigenvalue, while all the holomorphic tangent vectors are eigenvectors corresponding to the eigenvalue with multiplicity 2. In this case, $U$ is also the eigenvector of $A_1$ and $A_{1^*}$.\n\\end{lem}\n\n\\begin{proof}\n From Lemma 3.2, we know that $p=3$. Choose orthonormal frame $\\xi,\\ \\xi_1,\\ \\xi_{1^*}$ of $T^{\\bot}(M^3)$. Since the normal connection is flat, from (\\ref{3AAaAa0}) and (\\ref{3AaAdab}), we have\n \\begin{equation}\\label{flatcom}\n [A,A_1]=0,\\ [A,A_{1^*}]=0,\\ [A_1,A_{1^*}]=2F.\n \\end{equation}\n By the same discussion as in the proof of Lemma 3.2, we know that if $A$ has at least two distinct eigenvalues, then $A$ has two distinct eigenvalues and $U$ is an eigenvector corresponding to the simple eigenvalue, while all the holomorphic tangent vectors are eigenvectors corresponding to the eigenvalue with multiplicity 2. Assume that $AU=\\mu U$. From Lemma 3.1 and (\\ref{flatcom}), we have\n \\begin{equation*}\n A(A_1U)=\\mu A_1U,\\ A(A_{1^*}U)=\\mu A_{1^*}U.\n \\end{equation*}\n Noting that $\\mu$ is the simple eigenvalue of $A$ and $U$ is the corresponding eigenvector, it follows that there exist $\\mu_1,\\mu_{1^*}\\in\\mathbf{R}$, such that $A_1U=\\mu_1U,\\ A_{1^*}U=\\mu_{1^*}U$, which imply that $U$ is also the eigenvector of $A_1$ and $A_{1^*}$.\n\\end{proof}\n\nWith the above three lemmas, we can prove Theorem 1.3.\n\n\\begin{proof}[Proof of Theorem 1.3]\n Lemma 3.2 shows that $p=3$. Now we prove $\\xi$ is umbilical. Otherwise, from Lemma 3.3, we know that $A$ has two distinct eigenvalues, say $\\lambda,\\mu$. Assume $\\mu$ is the simple one, then\n \\begin{equation}\\label{3AUmZlZ}\n AU=\\mu U,\\ AZ=\\lambda Z,\n \\end{equation}\n where $Z$ is any holomorphic tangent vector of $M^3$.\n\n Let $\\zeta$ be the mean curvature vector, we decompose it as $\\zeta=\\zeta_1+\\zeta_2$, where $\\zeta_1\\in span\\{\\xi\\}, \\zeta_2\\bot\\xi$. Choose the unit normal vector $\\xi_1$ such that $\\zeta_2=|\\zeta_2|\\xi_1$, then\n \\begin{align*}\n \\zeta = & |\\zeta_1|\\xi+|\\zeta_2|\\xi_1\\\\\n = & \\frac{1}{3}({\\rm trace} A)\\xi+\\frac{1}{3}({\\rm trace} A_1)\\xi_1+\\frac{1}{3}({\\rm trace} A_{1^*})\\xi_{1^*}.\n \\end{align*}\n This implies that\n \\begin{equation}\\label{3traA10}\n {\\rm trace} A=3|\\zeta_1|,\\ {\\rm trace} A_1=3|\\zeta_2|,\\ {\\rm trace} A_{1^*}=0.\n \\end{equation}\n Combining (\\ref{trAa*}) and (\\ref{3traA10}), we see that\n \\begin{equation}\\label{3s1U3z2}\n s_1(U)=0,\\ s_{1^*}(U)=3|\\zeta_2|.\n \\end{equation}\n Further, it follows from Lemma 3.3, (\\ref{sa*}) and (\\ref{sa}) that\n \\begin{equation}\\label{3A1UUU0}\n A_{1^*}U=\\langle A_{1^*}U,U\\rangle U=-s_1(U)U=0,\n \\end{equation}\n \\begin{equation}\\label{3A1UZ2U}\n A_{1}U=\\langle A_{1}U,U\\rangle U=s_{1^*}(U)U=3|\\zeta_2|U.\n \\end{equation}\n Then for any $X\\in T(M^3),X\\bot U$, we have\n \\begin{equation}\\label{3s1XUX0}\n s_1(X)=-\\langle A_{1^*}U,X\\rangle=0,\\ s_{1^*}(X)=\\langle A_1U,X\\rangle=0.\n \\end{equation}\n\n Note that (\\ref{3A1UUU0}) implies $0$ is an eigenvalue of $A_{1^*}$. According to Theorem 1.1, there must exist a non-zero eigenvalue of $A_{1^*}$, say $\\alpha$. Then (\\ref{3traA10}) shows that $-\\alpha$ is also an eigenvalue of $A_{1^*}$. Assume that $X\\in T(M^3), X\\bot U, |X|=1$, and\n \\begin{equation}\\label{A1*X}\n A_{1^*}X=\\alpha X.\n \\end{equation}\n Write $Y=FX$, then\n \\begin{equation}\\label{A1*Y}\n A_{1^*}Y=-\\alpha Y.\n \\end{equation}\n From (\\ref{Aa}),(\\ref{3s1XUX0}),(\\ref{A1*X}),and (\\ref{A1*Y}), we have\n \\begin{equation}\\label{3A1XYAX}\n A_1X=-\\alpha Y,\\ A_1Y=-\\alpha X.\n \\end{equation}\n By a direct calculation, one can easily get\n \\begin{equation}\\label{A1A1*com}\n [A_1,A_{1^*}]X=-2\\alpha^2 Y.\n \\end{equation}\n On the other hand, it follows from (\\ref{3AaAdab}) that\n \\begin{equation}\\label{A1A1*comX}\n [A_1,A_{1^*}]X=2FX=2Y.\n \\end{equation}\n Comparing (\\ref{A1A1*com}) and (\\ref{A1A1*comX}), we get $\\alpha^2=-1$. This is impossible, since the shape operator $A_{1*}$ is symmetric and its eigenvalues are all real numbers. This contradiction shows that $\\xi$ is umbilical.\n\n\n\\end{proof}\n\n\n\n\\section{\\bf None Existence of a Class of CR Submanifolds of Maximal CR Dimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$ with Flat Normal Connection}\n\\vskip 0.4 true cm\n\nIn this section we prove the non-existence of 3-dimensional pseudo-umbilical CR submanifolds of maximal CR dimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$ with flat normal connection. Otherwise, let $M^3$ be such a submanifold. We first study the position of the mean curvature vector $\\zeta$ in the normal bundle.\n\n\\begin{lem}\n Let $M^3$ be a 3-dimensional pseudo-umbilical CR submanifolds of maximal CR dimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$, $p>1$. If the normal connection is flat, then the mean curvature vector $\\zeta$ is in the direction of $\\xi$.\n\\end{lem}\n\n\\begin{proof}\n We decompose $\\zeta$ as $\\zeta=\\zeta_1+\\zeta_2$, where $\\zeta_1\\in span\\{\\xi\\},\\ \\zeta_2\\bot\\xi$. We need to prove $\\zeta_2=0$. Otherwise, $\\zeta_2\\not=0$. From Theorem 1.1 we see that $\\zeta_2$ is not umbilical. From Theorem 1.3 we know that $\\zeta_1$ is umbilical. Hence $\\zeta$ is not umbilical. This contradicts our assumption that $M^3$ is pseudo-umbilical. Therefore, $\\zeta_2=0$, i.e.,$\\zeta\\in span\\{\\xi\\}$.\n\\end{proof}\n\n\\begin{remark}\n The method we used in the proof of Lemma 4.1 is due to B.Y.Chen who, in \\cite{BYCGL}, studied the umbilical normal vectors of submanifolds of a submanifold.\n\\end{remark}\n\nFrom Theorem 1.3, $p=3$. So\n\\begin{equation*}\n \\zeta = \\frac{1}{3}({\\rm trace} A)\\xi+\\frac{1}{3}({\\rm trace} A_1)\\xi_1+\\frac{1}{3}({\\rm trace} A_{1^*})\\xi_{1^*}.\n\\end{equation*}\nCombined this with Lemma 4.1, we have\n\\begin{equation}\\label{4traA10}\n {\\rm trace} A=3|\\zeta|,\\ {\\rm trace} A_1=0,\\ {\\rm trace} A_{1^*}=0.\n\\end{equation}\nFurther, it follows from (\\ref{trAa*}) that\n\\begin{equation}\\label{4s1U1U0}\n s_1(U)=0,\\ s_{1^*}(U)=0.\n\\end{equation}\n\n\\begin{lem}\n Let $M^3$ be a 3-dimensional pseudo-umbilical CR submanifolds of maximal CR dimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$, $p>1$. If the normal connection is flat, then $A_1U$ is a non-zero holomorphic tangent vector of $M^3$.\n\\end{lem}\n\n\\begin{proof}\n From (\\ref{Aa}) and (\\ref{4s1U1U0}),\n \\begin{equation*}\n A_1U=-FA_{1^*}U.\n \\end{equation*}\n Then the first formula of (\\ref{FU}) implies that $A_1U$ is orthogonal to $U$, which shows that $A_1U$ is a holomorphic tangent vector. In the following, we prove $A_1U\\not=0$. Otherwise, $A_1U=0$. Combining (\\ref{Aa*}) and (\\ref{4s1U1U0}), we also have\n \\begin{equation*}\n A_{1^*}U=0.\n \\end{equation*}\n Then (\\ref{sa*}) and (\\ref{sa}) give that\n \\begin{equation}\\label{4s10s10}\n s_1=0,\\ s_{1^*}=0.\n \\end{equation}\n From (\\ref{4traA10}) and Theorem 1.1, we see that $A_{1^*}$ has non-zero eigenvalues $\\alpha$ and $-\\alpha$. By the same discussion as in the latter part of the proof of Theorem 1.3, one can deduce that $\\alpha^2=-1$ which contradicts the fact that $\\alpha$ is a real number. So $A_1U\\not=0$. This completes the proof.\n\\end{proof}\n\nNow write\n\\begin{equation}\\label{4XA1YFX}\n X=A_1U,\\ Y=FX.\n\\end{equation}\nFrom (\\ref{4s1U1U0}) and (\\ref{Aa*}), it is easy to see that\n\\begin{equation}\\label{Y}\n Y=FX=FA_1U=A_{1^*}U.\n\\end{equation}\nNote that $\\{X,Y,U\\}$ are orthogonal to each other.\n\n\\begin{lem}\n With respect to the frame $\\{X,Y,U\\}$ chosen above, we have\n \\begin{equation*}\n |X|^2=|Y|^2=1,\n \\end{equation*}\n \\begin{equation*}\n s_1(X)=0,\\ s_1(Y)=-1,\\ s_{1^*}(X)=1, s_{1^*}(Y)=0,\n \\end{equation*}\n and the mean curvature $|\\zeta|=constant$.\n\\end{lem}\n\n\\begin{proof}\n From (\\ref{sa*}), (\\ref{sa}), (\\ref{4XA1YFX}) and (\\ref{Y}), we have\n \\begin{equation}\\label{4s1XYX0}\n s_1(X)=-\\langle A_{1^*}U,X\\rangle=-\\langle Y,X\\rangle=0,\n \\end{equation}\n \\begin{equation}\\label{4s1YXY0}\n s_{1^*}(Y)=\\langle A_{1}U,Y\\rangle=\\langle X,Y\\rangle=0,\n \\end{equation}\n \\begin{equation}\\label{4s1YYY2}\n s_1(Y)=-\\langle A_{1^*}U,Y\\rangle=-|Y|^2,\n \\end{equation}\n \\begin{equation}\\label{4s1XXX2}\n s_{1^*}(X)=\\langle A_{1}U,X\\rangle=|X|^2.\n \\end{equation}\nApplying (\\ref{4s1U1U0}),(\\ref{Y}) and (\\ref{4s1XYX0}) to the equation of Codazzi (\\ref{equationCodazziA}), we get\n\\begin{equation}\\label{4XAU1XY}\n (\\nabla_XA)U-(\\nabla_UA)X=-Y+s_{1^*}(X)Y.\n\\end{equation}\nOn the other hand, we calculate\n\\begin{align}\\label{4XAUUzX}\n & (\\nabla_XA)U-(\\nabla_UA)X \\notag\\\\\n = & \\nabla_X(|\\zeta|U)-A(\\nabla_XU)-\\nabla_U(|\\zeta|X)+A(\\nabla_UX)\\notag\\\\\n = & (X|\\zeta|)U-(U|\\zeta|)X.\n\\end{align}\nIn the above calculation we use the fact that $AZ=|\\zeta|Z$ for any tangent vector $Z$ of $M^3$, which can be deduced from the pseudo-umbilicity of $M^3$ and Lemma 4.1. Combining (\\ref{4XAU1XY}) and (\\ref{4XAUUzX}), we get\n\\begin{equation*}\n (X|\\zeta|)U-(U|\\zeta|)X+(1-s_{1^*}(X))Y=0.\n\\end{equation*}\nSo\n\\begin{equation}\\label{4Xz01X1}\n X|\\zeta|=0,\\ U|\\zeta|=0,\\ s_{1^*}(X)=1.\n\\end{equation}\nSimilarly, applying the equation of Codazzi (\\ref{equationCodazziA}) to $(\\nabla_YA)U-(\\nabla_UA)Y$, we get\n\\begin{equation}\\label{4Yz01Y1}\n Y|\\zeta|=0,\\ U|\\zeta|=0,\\ s_{1}(Y)=-1.\n\\end{equation}\nFrom (\\ref{4s1YYY2}), (\\ref{4s1XXX2}), (\\ref{4Xz01X1}), and (\\ref{4Yz01Y1}), we know that\n\\begin{equation*}\n |X|^2=1,\\ |Y|^2=1,\\ |\\zeta|=constant.\n\\end{equation*}\n\\end{proof}\n\n\n\\begin{lem}\n For the holomorphic tangent vector $X,Y$ defined by (\\ref{4XA1YFX}) and (\\ref{Y}), we have\n \\begin{equation*}\n A_1X=U,\\ A_1Y=0,\\ A_{1^*}X=0,\\ A_{1^*}Y=U.\n \\end{equation*}\n \\end{lem}\n\n\\begin{proof}\n From (\\ref{Aa*}), (\\ref{Aa}) and Lemma 4.3, we have\n \\begin{equation}\\label{4A1X1XU}\n A_1X=-FA_{1^*}X+U,\n \\end{equation}\n \\begin{equation}\\label{4A1YA1Y}\n A_1Y=-FA_{1^*}Y,\n \\end{equation}\n \\begin{equation}\\label{4A1XA1X}\n A_{1^*}X=FA_{1}X,\n \\end{equation}\n \\begin{equation}\\label{4A1Y1YU}\n A_{1^*}Y=FA_{1}Y+U.\n \\end{equation}\n From (\\ref{3AaAdab}), we get\n \\begin{equation*}\n [A_1,A_{1^*}]U=0.\n \\end{equation*}\n Substituting (\\ref{4XA1YFX}) and (\\ref{Y}) into the above formula, we have\n \\begin{equation}\\label{4A1YA1X}\n A_1Y=A_{1^*}X.\n \\end{equation}\n From (\\ref{4A1X1XU}), (\\ref{4A1YA1X}) and the skew-symmetry of $F$, we calculate\n \\begin{equation}\\label{4A1X1YY}\n \\langle A_1X,X\\rangle=-\\langle FA_{1^*}X,X\\rangle=\\langle A_{1^*}X,Y\\rangle=\\langle A_1Y,Y\\rangle.\n \\end{equation}\n From (\\ref{4traA10}), (\\ref{4s1U1U0}) and (\\ref{sa*}), we calculate\n \\begin{align}\\label{40tr1YY1}\n 0 = & {\\rm trace} A_1=\\langle A_1X,X\\rangle+\\langle A_1Y,Y\\rangle+\\langle A_1U,U\\rangle\\notag\\\\\n = & \\langle A_1X,X\\rangle+\\langle A_1Y,Y\\rangle+s_{1^*}(U)\\notag\\\\\n = & \\langle A_1X,X\\rangle+\\langle A_1Y,Y\\rangle.\n \\end{align}\n Combining (\\ref{4A1X1YY}) and (\\ref{40tr1YY1}), we know that\n \\begin{equation}\\label{4A1XYY0}\n \\langle A_1X,X\\rangle=0,\\ \\langle A_1Y,Y\\rangle=0.\n \\end{equation}\n From (\\ref{4A1XA1X}) and the skew-symmetry of $F$, we calculate\n \\begin{equation}\\label{4A1X1XY}\n \\langle A_{1^*}X,X\\rangle=\\langle FA_1X,X\\rangle=-\\langle A_1X,Y\\rangle.\n \\end{equation}\n On the other hand, from (\\ref{4A1YA1X}),\n \\begin{equation}\\label{4A1X1XY2}\n \\langle A_{1^*}X,X\\rangle=\\langle A_1Y,X\\rangle=\\langle A_1X,Y\\rangle.\n \\end{equation}\n Combining (\\ref{4A1X1XY}) and (\\ref{4A1X1XY2}), we have\n \\begin{equation}\\label{4A1XXY0}\n \\langle A_{1^*}X,X\\rangle=0,\\ \\langle A_{1}X,Y\\rangle=0.\n \\end{equation}\n From (\\ref{4traA10}), (\\ref{4s1U1U0}), (\\ref{4A1XXY0}) and (\\ref{sa}), we calculate\n \\begin{align}\\label{40tr1YY}\n 0 = & {\\rm trace} A_{1^*}=\\langle A_{1^*}X,X\\rangle+\\langle A_{1^*}Y,Y\\rangle+\\langle A_{1^*}U,U\\rangle\\notag\\\\\n = & \\langle A_{1^*}X,X\\rangle+\\langle A_{1^*}Y,Y\\rangle-s_{1}(U)\\notag\\\\\n = & \\langle A_{1^*}Y,Y\\rangle.\n \\end{align}\n From (\\ref{4A1YA1X}) and (\\ref{4A1XYY0}), we have\n \\begin{equation}\\label{4A1YYY0}\n \\langle A_{1^*}Y,X\\rangle=\\langle A_{1^*}X,Y\\rangle=\\langle A_1Y,Y\\rangle=0.\n \\end{equation}\n Noting that $\\{X,Y,U\\}$ are orthonormal, by using (\\ref{4A1XYY0}), (\\ref{4A1XXY0}) and Lemma 4.3, we get\n \\begin{align*}\n A_1X = & \\langle A_{1}X,X\\rangle X+\\langle A_{1}X,Y\\rangle Y+\\langle A_{1}X,U\\rangle U\\notag\\\\\n = & s_{1^*}(X)U=U.\n \\end{align*}\n Similarly, by using (\\ref{sa*}), (\\ref{sa}), (\\ref{4A1XYY0}), (\\ref{4A1XXY0}), (\\ref{40tr1YY}), (\\ref{4A1YYY0}) and Lemma 4.3, we also have\n \\begin{equation*}\n A_1Y=0,\\ A_{1^*}X=0,\\ A_{1^*}Y=U.\n \\end{equation*}\n\\end{proof}\n\nNow we can prove Theorem 1.4.\n\n\\begin{proof}[Proof of Theorem 1.4]\n Since the co-dimension $p>1$, it follows from Theorem 1.3 that $p=3$. Let $M^3$ be a pseudo-umbilical CR submanifold of maximal CRdimension of $\\mathbf{P}^{\\frac{3+p}{2}}(\\mathbf{C})$ whose normal connection is flat. Let $X,Y$ be the holomorphic tangent vectors defined by (\\ref{4XA1YFX}) and (\\ref{Y}). From (\\ref{3AaAdab}), we have\n \\begin{equation}\\label{4A1AX2Y}\n [A_1,A_{1^*}]X=2Y.\n \\end{equation}\n On the other hand, it follows from Lemma 4.4 and (\\ref{Y}) that\n \\begin{equation}\\label{4A1A1UY}\n [A_1,A_{1^*}]X=A_1A_{1^*}X-A_{1^*}A_1X=-A_{1^*}U=-Y.\n \\end{equation}\n This is a contradiction which proves the non-existence of such submanifolds. This completes the proof.\n\\end{proof}\n\n\n\n\n\n\\vskip 0.5 true cm\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1} \\setcounter{equation}{0}}\n\\def$\\bullet${$\\bullet$}\n\\def{\\bar Q}_1{{\\bar Q}_1}\n\\def{\\bar Q}_p{{\\bar Q}_p}\n\n\\def\\quad{\\quad}\n\n\\defB_\\circ{B_\\circ}\n\n\n\\let\\a=\\alpha \\let\\bigskip=\\beta \\let\\g=\\gamma \\let\\partial=\\delta \\let\\e=\\epsilon\n\\let\\c=\\chi \\let\\th=\\theta \\let\\k=\\kappa\n\\let\\l=\\lambda \\let\\m=\\mu \\let\\n=\\nu \\let\\x=\\xi \\let\\rightarrow=\\rho\n\\let\\s=\\sigma \\let\\tilde=\\tau\n\\let\\vp=\\varphi \\let\\vep=\\varepsilon\n\\let\\w=\\omega \\let\\G=\\Gamma \\let\\D=\\Delta \\let\\Th=\\Theta\n \\let\\P=\\Pi \\let\\S=\\Sigma\n\n\n\n\\def{1\\over 2}{{1\\over 2}}\n\\def\\tilde{\\tilde}\n\\def\\rightarrow{\\rightarrow}\n\\def\\nonumber\\\\{\\nonumber\\\\}\n\\let\\bm=\\bibitem\n\\def{\\tilde K}{{\\tilde K}}\n\\def\\bigskip{\\bigskip}\n\n\\let\\partial=\\partial\n\n\n\\begin{flushright}\n\\end{flushright}\n\\vspace{20mm}\n\\begin{center}\n{\\LARGE Comments on black holes I:}\n\\\\\n\\vspace{5mm}\n{\\LARGE The possibility of complementarity}\n\\\\\n\\vspace{18mm}\n{\\bf Samir D. Mathur ~and~ David Turton }\\\\\n\n\\vspace{8mm}\nDepartment of Physics,\\\\ The Ohio State University,\\\\ Columbus,\nOH 43210, USA\\\\ \\vskip .2 in mathur.16@osu.edu\\\\turton.7@osu.edu\n\\vspace{4mm}\n\\end{center}\n\\vspace{10mm}\n\\thispagestyle{empty}\n\\begin{abstract}\n\\bigskip\nWe comment on a recent paper of Almheiri, Marolf, Polchinski and Sully who argue against black hole complementarity based on the claim that an infalling observer `burns' as he attempts to cross the horizon. We show that measurements made by an infalling observer outside the horizon are statistically identical for the cases of vacuum at the horizon and radiation emerging from a stretched horizon. This forces us to follow the dynamics all the way to the horizon, where we need to know the details of Planck-scale physics. We note that in string theory the fuzzball structure of microstates does not give any place to `continue through' this Planck regime. AMPS argue that interactions near the horizon preclude traditional complementarity. But the conjecture of `fuzzball complementarity' works in the opposite way: the infalling quantum is absorbed by the fuzzball surface, and it is the resulting dynamics that is conjectured to admit a complementary description. \n\n\n\n\\bigskip\n\n\n\n\n\n\n\\end{abstract}\n\\vskip 1.0 true in\n\n\\newpage\n\n\\numberwithin{equation}{section}\n\\setcounter{tocdepth}{1}\n\\tableofcontents\n\n\\baselineskip=16pt\n\\parskip=3pt\n\n\\section{Introduction}\n \n The quantum theory of black holes has proven to be rich territory for the exploration of the most fundamental laws of physics. The discoveries of black hole entropy \\cite{bek}, and Hawking radiation \\cite{hawking} provided deep links between gravity and thermodynamics, while raising a serious problem in the form of the information paradox. One suggestion that arose in this context was the notion of black hole complementarity \\cite{complementarity}. String theory provides a microscopic explanation for the entropy of black holes \\cite{sv}, and the fuzzball structure of microstates provides a solution to the information paradox \\cite{lm4,fuzzballs,fuzzball3,fuzzball4,cern,plumpre,plumberg,otherfcrefs}.\n \n Recently there have appeared several papers discussing the relations between the information paradox, entanglement theorems, complementarity and other issues involving the quantum theories of black holes \\cite{amps,ampsfollowups}\\footnote{See also the earlier work of \\cite{Braunstein:2009my}.}. Since there are several interrelated issues in the area of black holes, we have split our discussion into a set of papers, each addressing a different question. In this article we comment on some of the arguments used in the paper of Almheiri, Marolf, Polchinski and Sully (AMPS) \\cite{amps} and argue that they do not address the conjecture of `fuzzball complementarity' developed in \\cite{plumpre,plumberg,otherfcrefs}.\n \n We note that the fuzzball program provides a consistent picture of all issues in the quantum dynamics of black holes (see \\cite{reviews} for reviews). We will keep this fact at the back of our mind, since in many cases the fuzzball description provides us an explicit model to judge the validity of abstract arguments. \n \n We begin with some definitions and basic facts about black holes and the information paradox. We then make two observations:\n \n\n \n (a) It is often assumed that if an infalling observer `hits something' at the horizon, then there cannot be a `complementary' description where he goes through. While traditional complementarity may have this feature, the kind of complementarity suggested by fuzzballs is different. We use a toy example provided by AdS\/CFT duality to observe that in one description an infalling quantum `breaks up', while in another description it continues its trajectory unscathed. We note that the case of the black hole is somewhat different from the AdS\/CFT case, and explain how complementarity can arise for hard-impact processes involving quanta with energy $E\\gg kT$ falling freely into the black hole\\footnote{Here $E$ refers to the conserved Killing energy of the infalling quantum, and $T$ is the temperature of the black hole as measured from infinity.}. \n \n\n \n (b) One might think that an observer falling into the traditional black hole sees nothing as he falls up to the horizon, but an observer falling towards a body radiating `real quanta' from a stretched horizon would get `burnt' by the highly energetic photons encountered close to this horizon. We show that observations of Hawking quanta made outside the horizon actually yield similar results in both cases. Switching off a detector before crossing the horizon of a traditional black hole creates excitations from vacuum fluctuations, and these excitations have the same spectrum as excitations created by `real quanta' from a stretched horizon.\n \n\n \n We then address the argument made in AMPS \\cite{amps}. In brief outline, the AMPS argument goes as follows: \n\\begin{enumerate}[(i)]\n\t\\item If Hawking evaporation is unitary, then the state near the horizon is not the vacuum in an infalling observer's frame, but involves high-energy excitations.\n\t\\vspace{-3mm}\n\t\\item If there are high-energy excitations near the horizon, then an infalling observer will measure physical high energy quanta emerging from the black hole, and get burnt.\n\t\\vspace{-3mm}\n\t\\item If the observer gets burnt, then we cannot have any complementary description where he falls through without noticing anything at the horizon.\n\\end{enumerate}\n \n \n From points (a) and (b) above, we find that the AMPS gedanken experiment does not lead to the conclusions they suggest. If one wishes to avoid Planck-scale physics, then one should restrict to measurements made outside the stretched horizon. For such measurements\n point (b) shows that an infalling observer will see the traditional black hole and a radiating stretched horizon as statistically similar systems.\n The underlying reason for this equivalence is that there is too little time for him to detect the Hawking quanta before he reaches the horizon. More importantly, point (a) shows that even if the infalling observer were to hit the stretched horizon violently, this fact would not by itself invalidate the possibility of complementarity; in fact it is this very interaction that is expected to admit a complementary description.\n \n In the Discussion (Section \\ref{secsix}) we summarize the essential physics involved in the conjecture of fuzzball complementarity to show precisely why it is not addressed by the AMPS argument. \n \n The reader who is already familiar with fuzzballs and the conjecture of fuzzball complementarity may skip directly to Section \\ref{secfour}.\n\n \n \n\\section{The information paradox and the fuzzball proposal}\\label{basics}\n \n In this section we review the resolution of the information paradox through the fuzzball construction in string theory. Though the later arguments will be more abstract, the steps below will help us decide the validity of these arguments.\n \n \n \\bigskip\n \n\\noindent{ {\\bf (a) The traditional black hole}}\n\n\n The information paradox arises from the way Hawking radiation is emitted from the {\\it traditional black hole}. We define the traditional black hole as follows. There is a horizon, and a neighbourhood of the horizon with the following property. One can choose good slices in this neighbourhood, and in these good coordinates physics is `normal'. Here `normal' physics means exactly what we mean by normal physics in the lab: evolution of long wavelength modes ($\\lambda\\gg l_p$) is given by local quantum field theory on curved space, with corrections controlled by a small parameter $\\epsilon$. These corrections can come from any quantum gravity effect, local or nonlocal, and all we require is that $\\epsilon\\rightarrow 0$ as $M\\rightarrow \\infty$, where $M$ is the mass of the black hole. \n\n\\bigskip\n\n\n\\noindent{ {\\bf (b) The information paradox}}\n\n The traditional black hole arose from a study of gravitational collapse that leads to the Schwarzschild metric\n\\begin{equation}\nds^2=-(1-{2M\\over r})dt^{2}+{dr^2\\over 1-{2M\\over r}}+r^{2}{d\\Omega_2^{2}}\n\\label{one}\n\\end{equation}\nIf we use semiclassical gravity to follow the evolution of quantum modes during the collapse, we get the traditional black hole. We have the a vacuum region around the horizon which indeed gives `lab' physics in a good slicing (i.e., in Kruskal coordinates). Evolution of vacuum modes at this horizon leads to entangled pairs being created, with one member of the pair staying in the black hole and the other escaping to infinity as Hawking radiation. The entangled pair can be modeled for simplicity by \\cite{cern}\\footnote{Further analysis of such `bit models' can be found in \\cite{plumpre,plumberg,bits,giddings}.}\n\\begin{equation}\n|\\psi\\rangle_{pair}={1\\over \\sqrt{2}}\\left ( |0\\rangle_{in}|0\\rangle_{out}+|1\\rangle_{in}|1\\rangle_{out}\\right )\n\\label{two}\n\\end{equation}\nThe entanglement between the inside and outside grows by $\\ln 2$ with each emission. Near the endpoint of evaporation this would leave just two possibilities: information loss or a remnant \\cite{hawking,cern}. Both of these look unsatisfactory; we would like a pure state of Hawking radiation carrying all the information of the black hole. \n\n\\bigskip\n\n\\noindent{ {\\bf (c) The theorem controlling small corrections}}\n\n The problem would be resolved if gravitational collapse led to a state other than the traditional black hole. But the traditional black hole solution appeared to admit no deformations, leading to the phrase `black holes have no hair'. Exactly the same problem holds for black holes in AdS. Thus AdS\/CFT duality cannot by itself help to resolve the problem (for a detailed discussion of this issue, see \\cite{cern,conflicts}).\n\nThis situation led many string theorists to the following belief. Hawking computed the pair creation at leading order, but there can always be small quantum gravity corrections to the wavefunction (\\ref{two})\n\\begin{equation}\n|\\psi\\rangle_{pair}={1\\over \\sqrt{2}}\\left ( |0\\rangle_{in}|0\\rangle_{out}+|1\\rangle_{in}|1\\rangle_{out}\\right )+\\epsilon {1\\over \\sqrt{2}}\\left ( |0\\rangle_{in}|0\\rangle_{out}-|1\\rangle_{in}|1\\rangle_{out}\\right )\n\\label{twenty}\n\\end{equation}\nwhere we have added a small amount of an orthogonal state for the pair. The correction $\\epsilon$ for each pair must be small since the horizon geometry is smooth, but the number of emitted quanta is large ($\\sim (M\/m_p)^2$), and the net effect of the small corrections may accumulate in such a way that the overall state of the radiation would not be entangled with the black hole. \n\n\n\n\nBut in \\cite{cern} it was shown that this hope is false; the change in entanglement $\\delta S_{ent}$, compared to the entanglement $S_{ent}$ of the leading-order Hawking process, is bounded by\n\\begin{equation}\n{\\delta S_{ent}\\over S_{ent}}<2\\epsilon \\,.\n\\label{three}\n\\end{equation}\nThis inequality is the essential reason why the Hawking argument has proved so robust over the years -- no small corrections can save the situation. We will make use of (\\ref{three}) many times; many arguments in the other papers we discuss are also based on this inequality. \n\n\\bigskip\n\n\n\n\\noindent{ {\\bf (d) The fuzzball structure of microstates}}\n\n\n\n In \\cite{emission} it was found that a bound state in string theory {\\it grows} in size with the number of branes in the bound state and with the coupling, so that its wavefunctional is always spread over a radius which is order the Schwarzschild radius. This growth in size is a very stringy effect; it arises from the phenomenon of `fractionation' \\cite{dasmathur} which uses the extended nature of fundamental objects in the theory. Such horizon sized wavefunctionals are termed `fuzzballs'. \n\n\nThe size of fuzzball states is estimated by using the entropy of brane bound states, together with the physics of fractionation. Thus this size estimate involves {\\it all} the states of the black hole. To study the properties of fuzzballs further, it is useful to look at states where we place `many quanta in the same mode'. This is analogous to black body radiation, where placing a large number $N$ of quanta in the same harmonic gives a laser beam, with quantum fluctuations suppressed as $\\sim {1\\over N}$.\\footnote{This study of low fluctuation states has led some to be confused about the nature of fuzzballs. They ask: are fuzzballs just solutions to supergravity or do they involve stringy degrees of freedom? As can be seen from the above discussion, there is no fundamental classical\/quantum divide between states; all we can do is look at states with small or large fluctuations. In particular the non-BPS states studied in \\cite{ppwave} using the pp-wave technique were given in terms of strings placed in a fuzzball geometry. The correct question is not; `how messy is the fuzzball'; the only relevant question is `do we get a traditional black hole (with `lab physics' around a horizon) or do we not'. The only feature common to all fuzzballs is that we never form a traditional horizon.} One find that the fuzzballs generate a spacetime that resembles the traditional black hole far away from the horizon, but which ends\\footnote{The word `end' should be understood as follows. In all known examples, individual black hole microstates are described by solutions of string theory involving smooth geometry far from the black hole, no horizon, and thus no interior (where `interior' refers to the space-time inside the horizon of the corresponding classical black hole solution). For generic states, the structure at the scale of the would-be horizon may be expected to have Planck-scale degrees of freedom (see also Footnote 4). In general, since there is no interior, we say that space-time ends outside the would-be horizon.} in a set of string theory sources before reaching the horizon \\cite{lm4,fuzzballs,fuzzball3}. This is pictured in Fig.\\;\\ref{fdiss}.\\footnote{For the two-charge BPS black hole, all states have been shown to be fuzzballs. For other black holes, some fraction of the states have been constructed, and in each case have been found to be fuzzballs.}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.4]{fdiss.eps}\n\\caption{(a) The traditional black hole; small corrections at the horizon {\\it cannot} get information out in the Hawking radiation. (b) The fuzzball picture of black hole microstates; spacetime ends in stringy theory sources just before the horizon is reached. }\n\\label{fdiss}\n\\end{center}\n\\vspace{-2mm}\n\\end{figure}\n\n\\bigskip\n\n\n\n\\noindent{\\bf (e) Resolution of the paradox}\n\nGiven the existence of fuzzballs, the information paradox is resolved as follows. Fuzzballs do not radiate by pair creation from an `information-free horizon'; instead the radiation emerges from the surface of the fuzzball and carries information just like any normal body. This radiation has been explicitly worked out for simple fuzzballs; the rate of radiation agrees exactly with the Hawking emission rate expected for those fuzzballs but the details of the fuzzball state are seen to be imprinted in the spectrum of emitted quanta \\cite{radiation}. \n\nIf we start with a collapsing shell, then its wavefunction spreads over the enormous phase space of fuzzball states \\cite{tunnel}, and then these fuzzball states radiate like any other warm body. The time for this spread can be estimated to be much smaller than the Hawking evaporation time \\cite{rate}\n\\begin{equation}\n t_{fuzzball}\\ll t_{hawking}\n \\label{four}\n \\end{equation}\nThis solves the information paradox. \n\n\n\n\\section{Traditional complementarity vs Fuzzball complementarity}\\label{seccomp}\n\nIn this section we will explain what we mean by having a `complementary description'. We start by giving a toy example: the case of AdS\/CFT duality \\cite{maldacena}. This toy model is new. We briefly recall the traditional notion of complementarity, and then turn to how complementarity is conjectured to arise in the fuzzball description of microstates. This `fuzzball complementarity' has things in common to the toy example of AdS\/CFT duality, but also differs from it in a crucial way.\n\n\\subsection{Toy example of complementarity: AdS\/CFT duality}\\label{ads}\n\nWe start with an example that illustrates what we mean by having a complementary description. In this example an infalling quantum will encounter some degrees of freedom and appear to `go splat'; i.e. get `destroyed'. Yet there will be an alternative description where it continues unscathed. When a description of the latter kind exists, we will say that we have a `complementary' description of the degrees of freedom in the former description.\n\n\n\n\\begin{figure}[!]\n\\includegraphics[scale=.85]{fz2p.eps}\n\\caption{AdS\/CFT duality, traditional complementarity and fuzzball complementarity.}\n\\label{fz2p} \n\\end{figure}\n\n\nConsider IIB string theory compactified on $S^1\\times T^4$. Let $y$ be the coordinate along $S^1$ and $z_1, \\dots z_4$ be the coordinates on $T^4$. \nWe consider a bound state of $n_1$ D1 branes wrapped on $S^1$ and $n_5$ D5 branes wrapped on $S^1\\times T^4$. This bound state is depicted in Fig.\\;\\ref{fz2p}(a), where the direction along the branes is the $S^1$. \n\n\nWe are working in the context of a D-brane bound state in flat space, where in one description we have a CFT coupled to flat space, and in the other description we have a geometry with flat asymptotics and an AdS throat. The degrees of freedom deep inside the AdS throat (on the gravity side) will not play a role in the following.\n\nTo be more specific, we take the AdS radius $R_{AdS}$ to be macroscopically large. On the gravity side, we consider a throat which is very long in units of $R_{AdS}$ (measured by proper distance along a radial geodesic). We fix a CFT location $r=R_{CFT}$ in the usual way. We then consider the trajectory of an infalling quantum along a radial geodesic from say one AdS radius of proper distance above $r=R_{CFT}$ to one AdS radius of proper distance below this location (on the gravity side).\n\nIn the description involving a CFT coupled to flat space, the transition from an infalling graviton in flat space to CFT degrees of freedom is described by the corresponding CFT operator which describes the absorption (see e.g.~\\cite{Avery:2009tu}).\n\n\nA graviton with both indices on the $T^4$ is a scalar in the remaining dimensions. Consider in particular the graviton $h_{12}$, arriving at the brane bound state as shown in Fig.\\;\\ref{fz2p}(a).\n\n\nIn the CFT description, on hitting the brane bound state, the energy of the graviton gets converted to vibrations of the branes (open strings);\\footnote{The actual evolution on the branes is more complicated when we consider interactions in the CFT, but this simple picture illustrates the point we wish to make. Note that we are considering the gravity description at weak coupling, and so the CFT description is at strong coupling. But the important fact is that there are {\\it two} descriptions at the same coupling; one using strongly interacting CFT of freedom, and one using the spin 2 graviton and higher closed string modes. In the former description the incoming $h_{12}$ appears to break up into pieces, while in the latter it remains intact.} a vibration polarized in the direction $X^1$ moves up along the $S^1$ and a vibration polarized in the direction $X^2$ moves down the $S^1$ \\cite{comparing,interactions,malstrom}.\n\nOne may say that the graviton has `gone splat' on hitting the branes, to such an extent that it has split into two parts, $X^1$ and $X^2$. These two products obtained after impact certainly do not look like the single graviton $h_{12}$ that was arriving towards the brane bound state\\footnote{At strong coupling, the graviton is absorbed into degrees of freedom of the strongly coupled CFT, where we cannot make precise statements. Nevertheless, the graviton may still be described as having `gone splat' in the sense that, in the CFT description, it is no longer a graviton and has been converted into strongly coupled CFT degrees of freedom.}. But as we well know, there is an alternative description of this physics where we replace the brane bound state by an AdS region. In the latter description, the graviton $h_{12}$ falls smoothly into the AdS region, remaining as a single entity $h_{12}$ (Fig.\\;\\ref{fz2p}(b)). We can call this latter description a `complementary' description of the interaction with the D1D5 branes. Now we can ask our question: when the incoming graviton broke up on the D1D5 brane bound state, did it go `splat' or not?\n\nTo better understand how to interpret this situation, we look at a more detailed example where we start with {\\it two} gravitons, $h_{12}$ and $h_{34}$, separated by a distance $D$. We can think of this pair of gravitons as being an `object'; if the separation of the gravitons is increased or decreased, we can say that the object has `been damaged' and `feels pain'. \n\nAt zero coupling in the CFT, the evolution of these gravitons proceeds as follows \\cite{lm4}. First $h_{12}$ hits the D1D5 bound state, and changes to excitations $X^1, X^2$ which travel at the speed of light in opposite directions along $y$. At a later time $h_{34}$ hits the bound state and changes to vibrations $X^3, X^4$, again separating at the speed of light. But the separation $D$ between the initial gravitons can be recovered from the open string excitations. Let $y_i$ be the location along the $S^1$ of the excitation $X^i$, for $i=1,2,3,4$. We have\n\\begin{equation}\ny_1=t, ~~y_2=-t, ~~y_3=t-D, ~~y_4=-t+D\n\\end{equation}\nso the value of $D$ is encoded in the vibrations $X^i$ as\n\\begin{equation}\nD={1\\over 2}[(y_1-y_2)-(y_3-y_4)]\n\\end{equation}\nIn the dual gravity description, the two gravitons fall smoothly into the AdS, maintaining their separation $D$ and thus showing no indication of `damage' or `pain'. But given that the CFT description is a faithful copy of the gravity description, and that we can recover the same value $D$ from the CFT, it looks correct to say that there is no damage or pain felt in the brane description either.\\footnote{Again, to make the toy example more accurate, one should consider the CFT at strong coupling. The basic result is unchanged: in the CFT description the incoming graviton is absorbed into degrees of freedom of the strongly coupled CFT, which in no way resemble the incoming gravitons, yet which somehow encode the value of $D$.}\n\nBy contrast, when we throw an object onto a normal concrete wall, we do not expect to find a complementary description. Let us analyze what was special about the D1D5 brane case which did allow for complementarity.\n\n\nIn the D1D5 example, the Hilbert space of the incoming gravitons mapped faithfully into the Hilbert space of vibrations of the branes. That is, if we write the eigenstates of the incoming graviton as $|\\psi_E\\rangle$ and the eigenstates of the D1D5 system as $|\\tilde\\psi_E\\rangle$, then we find\n\\begin{equation}\n\\int\\, dE \\, C(E)\\, |\\psi_E\\rangle ~\\rightarrow~ \\int \\, dE \\, C(E)\\, |\\tilde \\psi_E\\rangle\n\\label{eone}\n\\end{equation}\n The {\\it nature} of the excitations changed completely - they changed from being gravitons to being vibrations of branes - but this is {\\it not} important. What is important is that the amplitude for a given energy remained the same (or approximately the same). A important input for getting a relation like (\\ref{eone}) is that the D1D5 bound state had a very closely spaced set of energy levels. This high density of levels leads to a `fermi-golden-rule' absorption of the graviton, and in such an absorption each incoming energy level $E_k$ transfers its amplitude to energy levels $\\tilde E_k$ that are very close to $E_k$. (In \\cite{interactions} the absorption of the graviton onto the brane bound state was computed by such a fermi-golden rule process.) \n \n\nWhat {\\it does} cause `damage' or `pain' is the situation where the levels available in the absorbing system are not sufficiently continuous. In this situation we will find in general \n\\begin{equation}\n\\int \\, dE \\, C(E)\\, |\\psi_E\\rangle ~\\rightarrow~ \\int \\, dE \\, C'(E) \\, |\\tilde \\psi_E\\rangle, ~~~C(E)\\ne C'(E)\n\\label{eoneq}\n\\end{equation}\nIn particular, a concrete wall will not have the same energy levels as the object hitting it, and so the incoming object will not be mapped faithfully into excitations of the concrete wall. In this situation we do not expect a complementary description of the impact. \n\nTo summarize, we cannot just say: `If we go `splat' on hitting some degrees of freedom, then we cannot have complementarity'. The impact transfers excitation energy to the degrees of freedom that are encountered. To know if we can have a complementary description we have to ask if the Hilbert space of the infalling object maps faithfully into a subspace of the Hilbert space of the encountered degrees of freedom. \n\n\\subsection{Traditional complementarity}\\label{trad}\n\n\nIn the early works on black hole complementarity \\cite{complementarity}, the physics that was proposed is depicted in Fig.\\;\\ref{fz2p}(c),(d). It was assumed that we can place a `stretched horizon' just outside $r=2M$, and that incoming quanta could be taken to interact with degrees of freedom on this stretched horizon. In the complementary description, we have just the smooth infall through the horizon.\n\nThe problem with this proposal is discovered when we ask for the physical origin of the degrees of freedom on the stretched horizon. It was argued that since the Schwarzschild coordinates break down at $r=2M$, there will be violent fluctuations of the gravitational degrees of freedom as we approach $r=2M$. It was further argued that these violent fluctuations are indicative of the fact that physics outside the horizon is self-consistent, and the stretched horizon provides the natural boundary beyond which we need not look.\n\nSuch an argument is, however, unsatisfactory. The breakdown of Schwarzschild coordinates means that we should use better coordinates, not that we are entitled to assume new physics. But there is an even more serious difficulty with this proposal, which we can see by returning to our basic question: how does the information paradox get resolved? There is a `smooth slicing' of the geometry where we see the creation of entangled pairs (\\ref{two}). The defenders of traditional complementarity argued that the inner and outer parts of the horizon should not be considered in the same Hilbert space, since an observer who falls in has strong limitations on how he can communicate with the outside; thus the state (\\ref{two}) makes no sense. But no mechanism was proposed to implement such a drastic change to normal physics. The skeptics of complementarity simply noted that there {\\it is} a good slicing of the geometry which we should use to do physics at the horizon, and with this slicing there appears to be no reason to not have a single Hilbert space that includes both the inner and outer parts of the horizon. \n\nFor these reasons, the traditional picture of complementarity remained an unresolved issue. It is important to note the difference between the traditional black hole case and the example of AdS\/CFT that we presented in Section \\ref{ads}. In the AdS\/CFT example of Fig.\\;\\ref{fz2p}(a),(b), the boundary where we get a complementary description is not a horizon, and there is no particle creation there. Thus we do not have the information problem. But in the case of a black hole there is no way to stop the creation of entangled pairs in any picture where a smooth horizon is assumed, and then we cannot scape the information paradox. As we will see now, the way complementarity can arise with the fuzzball picture in string theory is somewhat different, and needs us to recognize that real degrees of freedom appear at the location of the horizon. \n\n\\subsection{The proposal of fuzzball complementarity}\n\nWith the explicit construction of black hole microstates in string theory (fuzzballs) we find that things work out differently from the traditional picture of complementarity. The general idea of `fuzzball complementarity' is developed in \\cite{plumpre,plumberg,otherfcrefs}. The notion of making spacetime by entanglement \\cite{raamsdonk,israel,eternal} is very useful in this approach. Here we just give an outline of how things work:\n\n\\parskip=10pt\n\n(a) Complementarity does {\\it not} arise because of a choice of coordinates (Schwarzschild vs Kruskal). Instead, the construction of microstates is fully covariant.\n\n(b) In the traditional black hole we have {\\it vacuum} around the horizon. But in string theory, spacetime has a `boundary' where it ends with in a set of string theory sources just outside $r=2M$, before the horizon is reached. The details of these sources encode the choice of microstate. \n\n(c) Hawking radiation arises as quanta radiated from the details of microstate structure near the boundary. For simple microstates this radiation has been explicitly computed, and it arises from `ergoregion emission' \\cite{radiation} near the boundary. The details of the ergoregion structure depend on the choice of microstate. \n\n\n(d) Since we have `real' degrees of freedom at the horizon, the $E\\sim kT$ quanta radiated from the microstate are able to carry out the information of the microstate. We {\\it cannot} have a complementary picture where we replace the physics of such quanta by the vacuum physics seen at the horizon of the traditional black hole. In this way our complementarity differs from traditional complementarity.\nWhat we have to do is make a distinction between $E\\sim kT$ quanta (relevant for the information problem) and $E\\gg kT$ quanta (relevant for the `infall problem' of heavy observers). It was conjectured in~\\cite{plumberg} that the complementary description should describe measurements in the frame of a lab (composed of $E\\gg kT$ quanta) falling freely from infinity to the surface of the fuzzball. We can describe such a process as a `hard-impact' process.\n\n(e) Let us restate the previous point another way. In the fuzzball scenario, the exact state near the horizon is not the vacuum state of an infalling observer, or anything close to it; it is expected to have Planck-scale degrees of freedom. Thus we cannot say that we have low energy effective field theory at the horizon, and then use this low energy field theory for the purpose of describing all possible low energy observations of an infalling observer. Instead, we conjecture a complementary description for hard-impact processes involving $E\\gg kT$ quanta.\n\n(f) The complementarity conjecture is now the following (Fig.\\;\\ref{fz2p}(e),(f)). \nGiven a hard-impact process involving $E\\gg kT$ quanta, the resulting dynamics can be reproduced to a first approximation by the geometry of the black hole interior, for times of order crossing time (i.e. before the quanta reach the singularity). This description emerges from the fuzzball dynamics as follows. The $E\\gg kT$ quanta excite collective modes of the fuzzball. To a first approximation, the evolution of these modes is insensitive to the precise choice of fuzzball microstate (assuming we have taken a generic microstate). The evolution of these collective modes in this leading approximation is to be encoded in the complementary description. Thus, let the initial state of the hole have mass $M$ and be the linear combination of fuzzball states $\\sum_i C_i |F_i\\rangle$. When a quantum of energy $E\\gg kT$ impacts hard onto the fuzzball surface, the wavefunction of the fuzzball shifts to a combination $\\sum_j C'_j F'_j$ over the fuzzball states with mass $M+E$:\n\\begin{equation}\n\\sum_i C_i |F_i\\rangle~\\rightarrow ~\\sum_j C'_j F'_j\n\\label{evolve}\n\\end{equation}\nIf $E\\gg kT$, then the number of coefficients $C'_j$ is much larger than the number of $C_i$. The leading order evolution of the coefficients $C'_j$ is to be captured by the complementary description. \n\n(g) We can now see the similarities and differences with the toy example of AdS\/CFT duality discussed in Section \\ref{ads}: \\vspace{-5mm}\n\\begin{enumerate}[(i)]\n\t\\item The D1D5 brane degrees of freedom are analogous to degrees of freedom at the `boundary' of the fuzzball microstate. \\vspace{-2mm}\n \\item The D1D5 branes were taken to be in their ground state,\\footnote{We can take excited states of the D1D5 branes, but in AdS\/CFT duality we take these to be low energy excitations, and their effect in the dual gravitational description will occur near $r=0$, not near the place where the CFT is placed.} while the fuzzball structure differs microscopically from state to state. Thus we get only approximate complementarity in the black hole case, by looking at hard-impact, $E\\gg kT$ processes where the details of the fuzzball microstate become irrelevant. \\vspace{-2mm}\n \\item In the AdS\/CFT case the complementary description was possible because of the closely spaced levels of the D1D5 brane system. In the black hole case we again have a close spacing of levels, which is guaranteed by the large number $Exp[S_{bek}]$ of fuzzball microstates. \n\\end{enumerate}\n\n\\parskip=3pt\n\n\\subsection{Summary}\n\nTo summarize, we have observed the following:\n\n(a) In our toy example of AdS\/CFT duality, we have a brane description, where an incoming quantum appears to hit some degrees of freedom violently and `break up'. In a `complementary description', the incoming quantum smoothly through into an AdS region. There is no radiation from the AdS boundary itself, so there is no creation of entangled pairs at that location.\n\n(b) In traditional complementarity, one argues that there are two equivalent descriptions, a fact allowed by the limitations on communication between observers inside and outside the hole. In one description (that of the outside observer)\nincoming quanta are reflected back as Hawking radiation from a stretched horizon, while in another description (that for an infalling observer) the horizon is a smooth place. Since there is a horizon, there is a creation of entangled pairs (\\ref{two}) in a smooth slicing at that location, and there is no clear mechanism to remove this entanglement.\n\n(c) In fuzzball complementarity, there are real degrees of freedom at the horizon which arise from the fact for each black hole microstate, the compact directions pinch off in a mess of string sources and spacetime ends before we reach $r=2M$. The details of this `fuzzball' differs from microstate to microstate; there is no Hawking type creation of entangled pairs and the radiation from the fuzzball surface can be explicitly seen carry information of the microstate. Since the fuzzball surface differs from microstate to microstate, complementarity can only be obtained in an approximation where the effect of these differences is small. The conjecture is that when $E\\gg kT$ quanta impact the fuzzball, they excite collective modes that are relatively insensitive to the precise choice of microstate; the evolution of these modes (\\ref{evolve})\n can be approximated by evolution in a spacetime that mimics the black hole interior. \n \n\\section{Limits on measurements made outside the horizon}\\label{secfour} \n\n\n\nIn this section we address the following question. If we measure the radiation outside a black hole, then can we tell the difference between a traditional black hole and an object that radiates unitarily at the same temperature $T$ from a surface just outside $2M$?\n\nThe measurements we are interested in are close to the horizon ($|r-2M|\\ll 2M$), so we can consider the near horizon geometry depicted in Fig.\\;\\ref{fz6}. In Fig.\\;\\ref{fz6}(a) we have the traditional black hole, which has vacuum around the horizon, so the near horizon region looks like Minkowski space when seen in Kruskal coordinates. In Fig.\\;\\ref{fz6}(b) we have a warm surface placed just outside $r=2M$ (indicated by the jagged line), and this surface is assumed to radiate quanta at the temperature $T$ of the black hole.\n\n\n \\begin{figure}[htbt]\n\\begin{center}\n\\includegraphics[scale=.75]{fz6.eps}\n\\end{center}\n\\caption{(a) An inertial detector in Minkowski space, making a measurement using only the indicated part of its trajectory. Vacuum fluctuations excite the detector. (b) A similar detection, but for case of a warm body radiating into the right Rindler wedge. The wavelength of quanta is of the same order as the distance from the horizon. (c) Radiation from a `hot' body, where the wavelength is much shorter than the distance from the horizon.}\n\\label{fz6} \n\\end{figure} \n\nAt first it may appear that the case of Fig.\\;\\ref{fz6}(b) has real radiation that can `burn', while there is no real radiation in Fig.\\;\\ref{fz6}(a). We {\\it can} see quanta in Minkowski spacetime by taking a detector that accelerates. But our interest in in freely falling observers, which are indicated by the straight line trajectory in Fig.\\;\\ref{fz6}(a). One may expect that a detector moving in straight line in Minkowski space should not detect any quanta. But the situation we have is a little special. We are asking if we can distinguish the physical situations of Fig.\\;\\ref{fz6}(a) and Fig.\\;\\ref{fz6}(b) by observations {\\it outside the horizon}. Thus a detector trying to make a measurement would have to do this task by using only a section of its trajectory like that indicated in Fig.\\;\\ref{fz6}(a).\n\nBut if we place conditions on how long a detector has to make a measurement, then we run into the problem that we pick up vacuum fluctuations. We discuss the scales involved in the problem in Section \\ref{sectime}. Suppose we are considering radiation at the Hawking temperature $T$. The wavelength of these quanta at a distance $d$ from the horizon is $d\\sim \\lambda$. The infalling detector trying to measure such quanta has a limited time to make this measurement, and we argue that this available time is less than the time required to make the desired measurement.\n\nIn Section \\ref{rindler} we note that the above estimates reflect a general fact: for generic states of the radiating body in Fig.\\;\\ref{fz6}(b), observations of radiation do not appear statistically different from the vacuum fluctuations picked up by the detector of Fig.\\;\\ref{fz6}(a). The arguments we give are very basic to the theory of particle detection, and are implicit in many treatments of Rindler space (see e.g. the review \\cite{Crispino:2007eb} and references within).\\footnote{We also thank Bill Unruh for an earlier conversation about detectors in Minkowski spacetime.}\n\n\n\n\\subsection{Time needed for detector response}\\label{sectime}\n\nLet us examine what kind of quanta a detector can actually pick up in a measurement process. In Appendix \\ref{detect} we show that if we wish to measure a quantum of wavelength $\\lambda$, the we need a proper time $\\gtrsim\\lambda$ to elapse along the detector trajectory:\n\\begin{equation}\n\\Delta \\tau_{needed}\\gtrsim \\lambda\n\\label{ten}\n\\end{equation}\n In Fig.\\;\\ref{fz6} we note two different possibilities for the location of the quantum of wavelength $\\lambda$. In Fig.\\;\\ref{fz6}(a),(b) the quantum is at a distance $d\\sim \\lambda$ from the black hole surface. In Fig.\\;\\ref{fz6}(c) the quantum is at a distance $d\\gg \\lambda$ from the black hole surface. In Appendix \\ref{wavelength} we show that the Hawking quanta radiated from the black hole surface are of the former type; the typical wavelength found at a distance $\\lambda$ from the horizon is $\\sim\\lambda$ itself:\n \\begin{equation}\n \\lambda \\sim d\n \\label{el}\n \\end{equation}\n We now begin the see the source of difficulty in catching high energy Hawking quanta: we are already very close to the horizon when we encounter them, and then we may have too little time left to interact with them. Before proceeding, there is one effect that we must take into account. Because the detector is infalling, it sees the outgoing quantum as being Lorentz contracted; thus the wavelength of the quantum appears shorter than the distance $d$ measured along a $t=const$ slice. We take a local Lorentz frame oriented along the Schwarzschild $t, r$ directions, and let the proper velocity of the detector in this frame be\n \\begin{equation}\nU^{\\hat t}=\\cosh\\alpha, ~~~U^{\\hat r}=-\\sinh\\alpha\n\\end{equation}\nThen, as shown in Appendix \\ref{wavelength}, the effective wavelength of the Hawking quanta encountered by the infalling detector is\n\\begin{equation}\n\\lambda_{eff}\\sim d e^{-\\alpha}\n\\end{equation}\nNow we consider the proper time available to an infalling detector to measure the Hawking quantum; this detection must be made between the time the detector is at a distance $\\sim d$ from the horizon and the time it falls through the horizon. In Appendix \\ref{time} we show that for a detector falling in from far outside the horizon, this proper time is\n \\begin{equation}\n \\Delta \\tau_{available}< de^{-\\alpha}\n \\label{tw}\n \\end{equation}\n Putting together (\\ref{ten}), (\\ref{el}) and (\\ref{tw}) we get\n \\begin{equation}\n \\Delta \\tau_{available}< \\Delta \\tau_{needed}\n \\end{equation}\n so we conclude that an infalling detector cannot reliably pick up Hawking quanta being radiated from a black hole surface. We now turn to comparing the behavior of detectors in the situations of Fig.\\;\\ref{fz6}(a) and Fig.\\;\\ref{fz6}(b).\n\n\n\\subsection{Detectors in Rindler space and detectors near warm bodies} \\label{rindler}\n\n\nLet us consider the following question. We look at the situation of Fig.\\;\\ref{fz6}(a), where we have an inertial detector in empty Minkowski space, but the detection is required to be made before the detector crosses the Rindler horizon. We can therefore capture our physics by using Rindler coordinates covering the right Rindler wedge \n\\begin{equation}\nt_M=r_R \\sinh t_R, ~~~x_M=r_R \\cosh t_R\n\\end{equation}\nwhere $t_M, x_M$ are the Minkowski coordinates and $r_R, t_R$ are the Rindler coordinates. Now consider the behavior of the detector as seen in these Rindler coordinates. The space near the horizon looks very hot; it is full of Rindler quanta. Would these quanta `burn' the infalling detector?\n\n\nAt first one may think that an inertial detector in Minkowski space should see nothing. But we have already noted above that the limits placed on the measuring time causes the detector to be excited by vacuum fluctuations. We will now see that such an excitation is of the same kind as that expected in Fig.\\;\\ref{fz6}(b), where we have `real' quanta being radiated at the Rindler temperature by a surface placed just outside the Rindler horizon. \n\nLet the quanta being detected correspond to a scalar field $\\phi$, which is taken to be in the Minkowski vacuum state $|0\\rangle_M$. Since our observations are confined to the right Rindler wedge, we can use the expansion of the field operator in Rindler modes\n\\begin{equation}\n\\hat \\phi=\\sum_\\omega [f_\\omega(r_R)e^{-i\\omega t_R}\\, \\hat a_\\omega+\nf^*_\\omega(r_R)e^{i\\omega t_R}\\, \\hat a_\\omega^\\dagger]\n\\end{equation}\nLet the detector be a 2-level system. We will take it to start in the unexcited state $|i\\rangle$, and interactions with $\\phi$ can move it to the state $|f\\rangle$. The interaction is described by $\\int d\\tau \\, \\hat H_{int}(\\tau)$ where (see e.g.~\\cite{Crispino:2007eb})\n\\begin{equation}\n\\hat H_{int}(\\tau)=q \\, h(\\tau)\\, \\hat O(\\tau)\\, \\hat \\phi \\bigl( t_R(\\tau), r_R(\\tau)\\bigr)\n\\label{qint} \n\\end{equation}\nHere $\\hat O$ is an operator made out of the detector variables, $q$ is a coupling constant and $0\\le h(\\tau) \\le 1$ is a `switching function' that allows us to switch on and switch off the interaction of the detector with the scalar field $\\phi$. \n\nThe Minkowski vacuum $|0\\rangle_M$ can be written in terms of Rindler states of the left (L) and right (R) wedges\n\\begin{equation}\n|0\\rangle_M=C\\sum_k e^{-{E_k\\over 2}}|E_k\\rangle_L|E_k\\rangle_R, ~~~~~~~C=\\Big (\\sum_k e^{-E_k}\\Big )^{-{1\\over 2}}\n\\label{split}\n\\end{equation}\nNow suppose the interaction is switched on for a brief period as indicated in Fig.\\;\\ref{fz6}(a). Before the interaction is switched on, the state of the overall system is\n\\begin{equation}\n|\\Psi\\rangle_i=|i\\rangle \\otimes C\\sum_k e^{-{E_k\\over 2}}|E_k\\rangle_L|E_k\\rangle_R\n\\label{qstate}\n\\end{equation}\nUsing first order perturbation theory in the strength of the interaction $q$, we ask for the amplitude for the transition\n\\begin{equation}\n|i\\rangle\\otimes |E_k\\rangle_L|E_k\\rangle_R~\\rightarrow~|f\\rangle\\otimes |E_k\\rangle_L|E_{k'}\\rangle_R\n\\end{equation}\nThis amplitude is\n\\begin{equation}\n{\\cal A}_{kk'}=-i \\int_{-\\infty}^\\infty d\\tau\\, h(\\tau)\\, \\langle f | \\hat O |i\\rangle {}_R\\langle E_{k'} |\\hat\\phi \\bigl( t_R(\\tau), r_R(\\tau)\\bigr )|E_k\\rangle_R\n\\end{equation}\nThe quantity ${}_R\\langle E_{k'} |\\hat\\phi \\bigl(t_R(\\tau), r_R(\\tau)\\bigr)|E_k\\rangle_R$ can be easily computed by writing $|E_k\\rangle_R$ in terms of the occupation numbers for different Rindler modes and using the field expansion (\\ref{split}). Note that $h(\\tau)$ in nonzero only over the part of the detector trajectory indicated in Fig.\\;\\ref{fz6}(a). \n\nThe probability for the detector to get excited $|i\\rangle\\rightarrow |f\\rangle$ is then\n\\begin{equation}\nP_{Minkowski}=|C|^2\\sum_k e^{-E_k}\\sum_{k'}|{\\cal A}_{kk'}|^2\n\\end{equation}\nwhere the subscript on $P$ indicates that this computation was performed for the Minkowski vacuum situation of Fig.\\;\\ref{fz6}(a). Here the factor $e^{-E_k}$ reflects the fact that the probability of finding the state $|E_k\\rangle_R$ in the state (\\ref{split}) is\n\\begin{equation}\np_{E_k}=|C|^2e^{-E_k}\n\\label{wone}\n\\end{equation}\n\nNow consider a state that describes a warm body at the same temperature as Rindler space, as shown in Fig.\\;\\ref{fz6}(b). In terms of Rindler eigenstates, this state has a form\n\\begin{equation}\n|\\Psi\\rangle=\\sum_k C_k |E_k\\rangle\n\\label{stateq}\n\\end{equation}\nDifferent microstates of the warm body have different coefficients $C_k$, but the ensemble average over possible microstates will have\n\\begin{equation}\n\\langle |C_k|^2\\rangle=|C|^2 e^{-E_k}\n\\label{approximation}\n\\end{equation}\nin agreement with (\\ref{wone}).\nWe again consider the infalling detector with the same interaction (\\ref{qint}). With the state (\\ref{stateq}) the probability for the detector to get excited is\n\\begin{equation}\nP_{microstate}=\\sum_k \\sum_{k'}|C_k|^2| {\\cal A}_{kk'}|^2\n\\end{equation}\nUsing (\\ref{approximation}) we find that the the ensemble average of the excitation probability for radiation from `warm bodies' is the same as the excitation probability in the Minkowski vacuum when the detection range is confined to be outside the horizon\n\\begin{equation}\n\\langle P_{microstate}\\rangle=P_{Minkowski}\n\\end{equation}\nIn particular, if the infalling body is macroscopic so that it `measures' a large number of quanta, then the effect of radiation in any one microstate will be approximately the same as the effect of vacuum fluctuations when \nwe restrict to the part of the observer worldline that is outside the horizon:\n\\begin{equation}\n P_{microstate}\\approx P_{Minkowski}\n \\label{eqburn}\n\\end{equation}\nA similar effect is also obtained when we consider a detector that has fallen in from near infinity. Quanta at infinity with wavelength $\\lambda$ are wavepackets that have a transverse size $\\Delta \\gtrsim\\lambda$; this is necessary since otherwise the uncertainty principle will give the quantum more transverse momentum $\\sim 1\/\\Delta$ than radial momentum, and the quantum will not really be headed towards the black hole. As the quantum comes closer to the horizon, the wavelength in the radial directions becomes small by blue-shifting, while the transverse size $\\Delta$ remains unaffected. Thus all quanta falling in from infinity are `flattened' near the horizon. The largeness of $\\Delta$ compared to the radial wavelengths of Hawking quanta near the horizon means that several Hawking quanta at different angular positions along the horizon can interact with the infalling quantum. Thus we are again led to compute statistical averages, getting a result like (\\ref{eqburn}).\n\n\n\\subsection{Summary}\n\n\nTo summarize, we have compared measurements made by an infalling detector in the case of Minkowski space (Fig.\\;\\ref{fz6}(a)) and in the case of a warm body at the same temperature (Fig.\\;\\ref{fz6}(b)). These two cases are equivalent to the traditional black hole and to a black object with a radiating surface just outside the horizon. While one might at first think that the detector would measure very different things in the two cases, we find that the detector excitation probabilities are actually {\\it similar}. The underlying reason for the similarity is the fact that we need the detection to be completed before the detector reaches the horizon, and this causes vacuum fluctuation excitations in the Minkowski space case that resemble the `real' quanta picked up in the warm body case.\n\nWhile fuzzballs radiate at exactly the rate expected for Hawking emission, one may envisage a theory other than string theory where the quanta are emitted with energy \n\\begin{equation}\nE\\gg kT\n\\end{equation}\nwith $T$ the Hawking temperature. In other words, we may give up the thermal spectrum of emission, and have the situation pictured in Fig.\\;\\ref{fz6}(c) where the emitted quantum has wavelength $\\lambda\\ll d$ at a distance $d$ from the horizon. In this case it {\\it is} possible to make a reliable measurement of the quantum, since ample time is available before the detector reaches the black hole surface. But in this case the emitted radiation will not carry away all the information of the black hole. This follows because the entropy of Hawking radiation (at temperature $T$) is just $\\sim 1.3$ times the Bekenstein entropy $S_{bek}$ \\cite{zurek}. Taking $E\\gg kT$ will give us $N\\ll S_{bek}$ quanta to carry out the $S_{bek}$ bits in the black hole, and this is not possible since each quantum carries $\\sim 1$ bit of information.\n\n\n\n\n\\section{The AMPS argument}\\label{secfive}\n\nIn this section we examine the main argument of Almheiri, Marolf, Polchinski and Sully (AMPS). We will note that the measurement they envisage cannot be performed reliably in the given situation, and further, that no conclusions about fuzzball complementarity can be drawn from such a situation. \n\n\nIn outline, the AMPS argument goes as follows: \n\\begin{enumerate}[(i)]\n\t\\item If Hawking evaporation is unitary, then the state near the horizon is not the vacuum in an infalling observer's frame, but involves high-energy excitations.\n\t\\vspace{-3mm}\n\t\\item If there are high-energy excitations near the horizon, then an infalling observer will measure physical high energy quanta emerging from the black hole, and get burnt\n\t\\vspace{-3mm}\n\t\\item If the observer gets burnt, then we cannot have any complementary description where he falls through without noticing anything at the horizon.\n\\end{enumerate}\n \n\n\n\nWe examine each of these steps in turn.\n\n\\bigskip\n\n\\noindent{ {\\bf (i) The need for large corrections at the horizon}}\n\n In \\cite{cern} it was shown, using strong subadditivity, that semiclassical physics at the horizon cannot lead to the behavior of entanglement entropy $S_{ent}$ that is expected for normal bodies \\cite{page}. The behavior for $S_{int}$ is depicted in Fig.\\;\\ref{fz9}. AMPS try to summarize a version of this argument, but miss a crucial step. We would like to clarify this point since it is important, before continuing with the AMPS argument.\n\n \\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.75]{fz9.eps}\n\\end{center}\n\\caption{(a) The growth of entanglement entropy for the traditional black hole in the leading order Hawking computation (solid line), and with small corrections allowed (dashed line). (b) The entanglement entropy expected for a normal body \\cite{page}; $S_{ent}$ must return to zero when the body radiates away completely.}\n\\label{fz9} \n\\end{figure}\n\nConsider the Hawking pair (\\ref{two}) produced in the leading order Hawking process; let the outer and inner members of this pair be called $B, C$ respectively. AMPS consider this leading order process, a fact which is implicit in their assumption that\n$S_{BC}=0$;\ni.e., the produced pair is not entangled with anything else (Fig.\\;\\ref{fz7}(a)). They then use strong subadditivity to argue that $S_{ent}$ cannot return to zero like it should for normal bodies. But this situation does not need the powerful relation of strong subadditivity. In the leading order Hawking process the relation (\\ref{two}) tells us that the state of the created pairs is a tensor product of individual pairs (eq. (17) of \\cite{cern}), and so $S_{ent}=N\\ln 2$ after $N$ pairs have been produced. This gives the linearly increasing graph of Fig.\\;\\ref{fz9}(a), and we do not need strong subadditivity to prove that $S_{ent}$ does not return to zero.\n\n\\begin{figure}[htbt]\n\\begin{center}\n\\includegraphics[scale=.75]{fz7.eps}\n\\end{center}\n\\caption{(a) Creation of entangled pairs in the leading order Hawking computation. (b) Small corrections; if these could reproduce the graph Fig.\\;\\ref{fz9}(b) then we would not need a firewall. (c) A firewall that one can pass through; now one can detect the quanta near the horizon. (d) In a fuzzball spacetime ends before the horizon. Hawking radiation is an integral part of the dynamics of the fuzzball.}\n\\label{fz7} \n\\end{figure}\n\n\n\n\nThe important issue, as discussed in Section \\ref{basics}, is whether {\\it subleading} corrections to the leading order Hawking process can make $S_{ent}$ reproduce the behavior of a normal body.\\footnote{The possibility that this might happen was raised in \\cite{eternal}. Hawking's reversal of his belief that information is lost was also implicitly based on the assumption that exponentially small corrections to the leading order process would produce an unentangled state \\cite{hawkingreverse}.} If small corrections could do the job, then we {\\it cannot} conclude that there would be a firewall; we depict this in Fig.\\;\\ref{fz7}(b). To analyze small corrections we have to start from \n$S_{BC}=\\epsilon$\nand then we do need to use strong subadditivity\n to establish the required inequality (\\ref{three}).\n\nTo summarize, a smooth vacuum at the horizon leads to the creation of Hawking pairs (\\ref{two}), and with (\\ref{three}) we see that we cannot get information out in Hawking radiation. Thus if we do wish to have the radiation be unitary, then we must alter the structure of the modes involved in the Hawking process. One may try to restrict the required change to just these modes; this requires us to invoke as yet undiscovered nonlocal effects \\cite{giddings}. If we choose to not do this, then we have an alteration of the physics for {\\it all modes} at the horizon. AMPS take the latter route\\footnote{They consider the possibility of nonlocal effects in a separate discussion later.}, and then consider an experiment: they let an infalling observer fall into such a hole and argue he will get `burnt' by the altered structure at the horizon. Further, they argue that getting burnt in this way precludes the possibility of complementarity. We now examine each of these issues in turn.\n\n\n\\bigskip\n\n\\noindent{ {\\bf (ii) Getting burnt by Hawking quanta}}\n\n\nHere AMPS wish to distinguish the traditional black hole from a body that radiates at the Hawking temperature from a surface just outside the horizon. They argue that in the case of the radiating body an infalling observer will observe high energy quanta, while there will be no such quanta observed for the traditional black hole. Let us see what questions we can ask:\n\n\\bigskip\n\n(a) The temperature of the radiation is $T\\sim m_{p}$ at a distance $l_{p}$ from \nthe horizon. If we wish to avoid Planck-scale physics, the we can try to focus on the radiation a distance $d$ from the horizon with\n\\begin{equation}\nl_p\\ll d\\ll 2M\n\\end{equation}\nFor concreteness, let us think of $d\\sim 10^6 l_p$, where we expect the temperature to be high enough to `burn' but the physics is still not the unknown physics at Planck scale. For instance, in Fig.\\;1 of \\cite{amps}, one can ask if the infalling detector gets burnt during the part of its trajectory where it crosses the shaded region representing the outgoing Hawking quantum. To restrict our question to the required region, we consider the effect of the radiation on a detector which is switched off when we get to a distance closer than $\\sim 10^6 l_p$ from the horizon. But in such a situation we have noted in Section \\ref{rindler} that the excitation probability of the detector in the case of the traditional horizon and in the case of the firewall are statistically the {\\it same} (eq.(\\ref{eqburn})). Thus we cannot say that we will get burnt in one case and not the other. \n\n\n(b) The above equality of excitation probabilities resulted from the fact that we were allowed a limited time to make the detection; thus vacuum fluctuations excited the detector even for the traditional hole. We could allow ourselves a longer time for detection if we assumed that we could pass through the firewall to the other side of the horizon, and again find ourselves in a region of low temperature. Such a possibility is pictured in Fig.\\;\\ref{fz7}(b), and in this case we would excite the detector for the firewall but not for the traditional hole. But to have this `other side' to the firewall we need to pass through a `wall' of of Planck-scale physics. Since the strength of gravitational interactions increases with energy, we can expect that the largest interactions would be when the detector is crossing the region of Planck temperature, and so we cannot focus on the issue of detection of quanta with wavelength $\\sim 10^6 l_p$ without asking if the theory allows us to pass through the Planck temperature region. (In particular, it is hard to imagine a physical model reproducing Fig.\\;\\ref{fz7}(c) which has smooth space on both sides of a Planck energy region.)\nAs we note below, the fuzzball microstates of string theory do {\\it not} allow us to pass through the Planck temperature region; spacetime ends there in a stringy mess. Further, as we will note in part (iii) below, interaction with the Planck-scale degrees of freedom is not what precludes the kind of complementarity that we find with fuzzballs; instead, it is this interaction which transfers information to the collective modes of the fuzzball and leads to a complementary description.\n\n(c) In Fig.\\;\\ref{fz7}(c) we depict the situation with fuzzballs. The incoming quanta cannot pass through the fuzzball surface, and so they transfer their energy to excitations of the fuzzball. The fuzzball details and the radiation it emits are parts of the same structure: the radiation is the small time dependent part of the gravitational solution away from $r\\approx 2M$. The response of the Planck-scale degrees of freedom in encoded in the response (\\ref{evolve}) of the fuzzball, and this effect is expected to dominate over interactions with the radiation tail.\n\n\nOne thing is important to note about this interaction. Let the infalling observer be made of degrees of freedom that evolve slower than the Planck scale. Then the observer does not evolve significantly between the time that its coupling to the radiation becomes significant and the time it reaches the fuzzball boundary. Thus it is not clear what `burning' means in this context. The correct question to focus on is not the evolution of the infalling observer, but rather the evolution (\\ref{evolve}) of the fuzzball degrees of freedom that the observer impacts.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bigskip\n\n\\noindent{ {\\bf (iii) The possibility of complementarity}}\n\nFinally, let us address the issue of complementarity. The AMPS paper claims that their argument applies to the proposal of fuzzball complementarity. We can paraphrase this as claiming that, even for the simplest of hard-impact processes involving high-energy quanta, if an infalling quantum `burns up' at the Planck-temperature surface of the fuzzball, then there cannot be any other approximate complementary description involving free infall. This desire to avoid any interaction is suggested by the traditional proposal of complementarity. But as we have seen in Section \\ref{trad}, there are difficulties with traditional complementarity, and this is not the kind of complementarity that we have proposed. \n\nConsider first our toy example of AdS\/CFT duality. A graviton falling onto a D1D5 brane bound state {\\it did } interact strongly on reaching these branes and broke up into a pair of excitations. Yet there was a complementary description where it passed smoothly into an AdS region. Similarly, $E\\gg kT$ gravitons falling onto the fuzzball surface {\\it do} interact strongly with the surface and excite the collective dynamics of the fuzzball degrees of freedom; it is this collective dynamics (\\ref{evolve}) which will have a dual representation where to a first approximation the graviton will appear to fall through a horizon.\n\nThe moral we draw is that it is incorrect to conclude that complementarity would be impossible if an object encountered strong interactions near the horizon. The situation is quite the opposite: we need {\\it strong} interactions near the horizon to absorb the energy of the infalling quantum into the black hole's degrees of freedom to get `fuzzball complementarity'. If the absorption leads to an approximately faithful map of the infalling quantum's Hilbert space into a subspace of the black hole degrees of freedom, then we have the possibility of a complementary description of the infall. \n\n\n \n\n\\section{Discussion}\\label{secsix}\n\nIn this paper we have done two things: we summarized how complementarity is conjectured to work with fuzzballs, and we noted how the AMPS argument fails to address the underlying physics in this conjecture. In the discussion below we will put these two parts together, to see more directly where the AMPS argument goes wrong. In short, we will see that complementarity is a story of {\\it two} descriptions of the physics, while AMPS try to have elements of both descriptions in the {\\it same} setting. \n\nFor the discussion below, it is helpful to summarize one version of the AMPS argument as follows:\n\n\\bigskip\n\n(1) Suppose the infalling observer sees nothing around $r=2M$ in some\ndescription.\n\n(2) Then in this description we have a smooth patch of spacetime around\nthe horizon.\n \n(3) Evolution of vacuum modes in this smooth patch will lead to an entangled Hawking pair, and this\nwill lead to the information problem.\n\n\\bigskip\n\nThe problem with this argument is that the description in which we have (1) (i.e. smooth spacetime) is valid only as an approximation for describing the physics of hard-impact $E\\gg kT$ infalling quanta for short times (order $\\sim M$). One cannot use the effective smooth spacetime used in this approximation to describe the entanglement of Hawking pairs over the much longer Hawking evaporation time (order $\\sim M^2$); in particular one cannot relate the effective description of (1) to the information problem which needs us to talk about the details of $\\sim (M\/m_p)^2$ Hawking pairs.\n\n \n Let us now see in more detail how things actually work:\n \n \\bigskip\n \n (a) The microstates of the black hole are fuzzballs, which means that the gravitational solution ends just outside $r=2M$ when the compact directions pinch off; the structure at this location is a quantum mess of KK monopoles, strings, fluxes, etc. (i.e. the set of allowed sources in string theory). \n \n (b) $E\\sim kT$ radiation is emitted from these sources, carrying the information of the microstate. A simple model to keep in mind is the computation of \\cite{radiation}, where ergoregions near the fuzzball surface emit quanta by ergoregion emission. From this computation we learn that there is no sharp separation between the radiation and the fuzzball: the gravitational field in the ergoregion is unstable and radiates gravitons. If we follow these emitted gravitons back to their source, then we find more and more nonlinear gravitational physics, culminating in the `cap' where the fuzzball solution ends in KK monopoles etc. Thus whenever we ask if we interact with emitted quanta, we might as well go all the way and ask if we interact with the fully nonlinear `cap'. \n \n (c) We recalled the toy example of AdS\/CFT, which has similarities and differences with the black hole case. For now we look at the similarities. Suppose we have a bound state of $N$ D1 and $N$ D5 branes. The infalling quanta of Fig.\\;\\ref{fz2p}(a) impacts this collection of branes and transfers its energy into excitations of the branes. Similarly, a quantum falling onto the fuzzball transfers its energy to the string theoretic sources (KK monopoles etc) on the fuzzball surface. In (b) we had noted that the radiation from the fuzzball was just the tail end of the full nonlinear KK monopole `cap', so interactions with radiation near the horizon are included in this description. \n \n \n \n \n\n(d) But in the AdS\/CFT case, there is a second description, that of Fig.\\;\\ref{fz2p}(b), where the infalling quantum sails smoothly through into an AdS region. In this description we do not see the D1 and D5 branes as something that can be `hit'. In the analogous case of fuzzballs, an infalling object does not see the nonlinear KK monopoles etc. near $r=2M$, but instead sails through smoothly. In particular, it does not see the `tail end' of the nonlinear structure -- the radiation from the fuzzball -- as high energy quanta that can be `hit'. \n\n(e) To understand how it is possible for the infalling observer to sail through smoothly, consider first the AdS\/CFT example. If the D1,D5 branes were `inert'; i.e., they did not shift their internal state when the infalling object approached, then there would not be any description where the object `sailed through'. But in fact the D1D5 brane bound state has a {\\it vast} space of internal excitations, and this changes the situation: the approach of the infalling object creates excitations in this vast space of possibilities, and the dynamics of these excitations is the dominant physics of the combined branes+object system. It is this dynamics that is described by the smooth infall into AdS space.\\footnote{One often thinks of AdS\/CFT duality as saying that the gravity variables in AdS can be re-expressed in terms of gauge theory variables on the boundary of AdS. But the origin of this duality is in the context of absorption by D-branes and black holes, and in that context the natural process to consider is the infall of quanta from infinity onto the branes. The largeness of $N$, the number of branes, leads to the excitation spectrum of the branes as being very dense, and the effective AdS description emerges.}\n\n\n\n(f) The fuzzball has a similarly large phase space of deformations, since the number of fuzzball solutions is $Exp[S_{bek}]$. Now we see the basic element missing from the analysis of AMPS. They ask for the dynamics of the infalling object (what it measures etc.) but they ignore the fact that the much more important dynamics is the change of the state of the {\\it fuzzball}: $\\sum_i C_i |F_i\\rangle\\rightarrow \\sum_j C'_j F'_j$. This latter dynamics is so dominant that one must consider the infalling object and fuzzball as one unified system and then analyze the dynamics. When infalling quanta with energy $E\\gg kT$ fall freely onto the fuzzball from far away, the conjecture is that the resulting dynamics has an approximate description valid for short times (order $\\sim M$) that mimics infall through a smooth horizon \\cite{plumpre,plumberg,otherfcrefs}; this is analogous to how in the AdS\/CFT case the object falls through smoothly into an AdS space. \n\n(g) The approximate nature of the `smooth infall' description is important. Since this description is valid only over a time of order $\\sim M$, we cannot use this patch of smooth space to argue that entangled Hawking pairs will be created and will escape to large distances from the black hole. There is hardly time to create one pair in such a region. We cannot join together many such patches to argue that we have created many entangled pairs, since the description is only valid for short times and does not accurately track $E\\sim kT$ physics. The existence of many entangled pairs would have led to the information problem as discussed in Section \\ref{basics}(b),(c); this problem does not arise here since we cannot study the creation of a large set of such pairs in our approximate `smooth infall' description.\n\n\\bigskip\n\n \nTo summarize, the error in the AMPS argument can be seen by considering the infall of an observer into a stack of branes. These branes are in a particular internal state, which can be probed by patient low energy scattering experiments from infinity. But the infalling observer reports none of this structure as he approaches the branes; he feels as if it is falling through empty AdS space. This `magical disappearance' of the branes can be traced to the fact that the branes have a vast set of internal states, and the dominant effect of the approach of the observer is to alter the internal state of the branes. Thus the dynamics of the \nbranes+observer system is governed by the evolution of these newly created excitations, and not by the observer scattering off a fixed state of the branes. \n\nWe can now see the fundamental role that the fuzzball construction plays in resolving the puzzles with black holes. If we have the traditional Penrose diagram of the hole, with vacuum at the horizon, then we get the creation of entangled pairs, and we cannot evade the Hawking information loss problem \\cite{hawking,cern}. But in string theory we find that there is very nontrivial structure at the horizon: the KK monopoles etc at the fuzzball surface carry `real' degrees of freedom that radiate unitarily\nlike a normal body. This resolves the information paradox. But we can ask a different question: what happens when we consider hard impacts of high energy ($E\\gg kT$) quanta on the fuzzball surface? In this case the physics is analogous to what we find in AdS\/CFT: the KK monopole and other string theoretic degrees of freedom on the fuzzball surface act like the branes in the D1D5 system. The infalling observer reports nothing special as it approaches these objects, since the dominant dynamics is that of {\\it exciting} the fuzzball degrees of freedom, not the response of the observer. AMPS implicitly assume that they are falling towards a radiating surface that is {\\it inert} to such excitations, and thus miss the physics of free infall which is common to the fuzzball and the AdS\/CFT cases.\n\nIn the context of the argument (1)-(3) listed at the start of this section, we see that the implication (1) $\\rightarrow$ (2) is misleading. It is not that we don't have structure at the location of the branes; rather, the infalling observer does not report such structure. The patch of smooth spacetime in (2) is an effective description of the $\\sum_i C_i |F_i\\rangle\\rightarrow \\sum_j C'_j F'_j$ dynamics which describes the excitations of the impacted fuzzballs; it is not the actual gravitational solution at the horizon. The implication (2) $\\rightarrow$ (3) does not work since this effective description cannot be applied to a region larger than $\\sim M$ which would be needed to create a large number of entangled pair. In general there are {\\it two} descriptions involved: (i) the actual microscopic fuzzball which carries all information of the state and radiates unitarily, and (ii) the approximate short time description of collective modes, that mimics free infall. AMPS do not differentiate carefully these two descriptions, and that leads them to claim an apparent contradiction with fuzzball complementarity. \n\nIn conclusion, the AMPS argument does not apply to the process by which complementarity is conjectured to arise in the fuzzball picture. But it is a very interesting argument to consider, since it brings out clearly the various important physical principles involved in the quantum dynamics of black holes. \n\n\\section*{Acknowledgements}\n\nThis work was supported in part by DOE grant DE-FG02-91ER-40690. We thank the authors of \\cite{amps} as well as Iosif Bena, Borun Chowdhury, Stefano Giusto, Oleg Lunin, Lenny Susskind and Nick Warner for discussions. In particular we are grateful to Don Marolf for patiently explaining to us the nature of the AMPS argument.\n\n\n\\begin{appendix}\n\n\\section[Appendices]{Timescale for detection} \\label{detect}\n\nHere we note that a detector needs a proper time $\\Delta \\tau\\gtrsim \\lambda$ to detect a quantum of wavelength $\\lambda$. Since this argument is well known, we will describe it for the simple case of a detector at rest in the Minkowski vacuum; the extension to other situations is straightforward.\n\nWe assume for simplicity that the metric is time independent in our choice of coordinates. The field operator can be expanded as\n\\begin{equation}\n\\hat\\Phi=\\sum_k [{1\\over \\sqrt{2\\omega_k}} e^{i(kx-\\omega_k t)}\\hat a_k\n+{1\\over \\sqrt{2\\omega_k}}e^{-i(kx-\\omega_k t)}\\hat a_k^\\dagger], ~~~~[\\hat a_k, \\hat a_{k'}^\\dagger ] =\\delta _{k, k'}\n\\end{equation}\n We take the detector to be a harmonic oscillator\n\\begin{equation}\n\\hat\\Psi={1\\over \\sqrt{2\\Omega}} e^{-i\\Omega \\tau}\\hat A + {1\\over \\sqrt{2\\Omega}} e^{i\\Omega \\tau}\\hat A^\\dagger, ~~~[\\hat A, \\hat A^\\dagger]=1\n\\end{equation}\nThe interaction along the worldline is given by $\\int d\\tau \\, \\hat H_{int} (\\tau)$ where\n\\begin{equation}\n\\hat H_{int}(\\tau)=q \\, h(\\tau) \\, \\hat\\Phi\\bigl (t(\\tau), x(\\tau)\\bigr )\\hat \\Psi(\\tau)\n\\end{equation}\nHere $q$ is a coupling constant and $0\\le h(\\tau)\\le 1$ is a function that allows us to switch on and switch off the detector.\n\nWe start at $\\tau\\rightarrow -\\infty$ with the detector in the ground state: $\\hat A|0\\rangle_A=0$. Let us also take the spacetime to be empty of quanta: $\\hat a_k |0\\rangle_a=0$. We take first order perturbation theory in $q$. The amplitude to reach the state $|1\\rangle_A|1\\rangle_{k} \\equiv \\hat A^\\dagger \\hat a_k^\\dagger |0\\rangle_A|0\\rangle_a$ is\n\\begin{equation}\n{\\cal A}=-iq\\int_{-\\infty}^\\infty d\\tau {1\\over \\sqrt{2\\omega_k}}{1\\over \\sqrt{2\\Omega}}h(\\tau)e^{i\\Omega \\tau}e^{-ikx(\\tau)+i\\omega_k t(\\tau)}\n\\end{equation}\nWe take\n\\begin{equation}\nh(\\tau)=e^{- ({\\tau\\over \\Delta \\tau})^2}\n\\end{equation}\nwhich corresponds to making a measurement over an interval $\\sim \\Delta\\tau$. \nWe also let the detector trajectory to describe a detector at rest at $x=0$, which gives $x(\\tau)=0, t(\\tau)=\\tau$ for all $\\tau$. This gives \n\\begin{equation}\n{\\cal A}=-i q\\int _{-\\infty}^\\infty d\\tau {1\\over \\sqrt{2\\omega_k}}{1\\over \\sqrt{2\\Omega}}e^{- ({\\tau\\over \\Delta \\tau})^2}e^{i(\\Omega+\\omega_k)\\tau}=-iq{1\\over \\sqrt{2\\omega_k}}{1\\over \\sqrt{2\\Omega}}\\Delta\\tau\\sqrt{\\pi}e^{-{1\\over 4}(\\Delta \\tau)^2(\\Omega+\\omega_k)^2}\n\\end{equation}\nKeeping the detector on for all time is equivalent to taking $\\Delta\\tau\\rightarrow \\infty$, in which case we get ${\\cal A}=0$. So the detector does not get excited, which is expected since we started with empty Minkowski space. \n\nBut now consider a situation where the detector is switched on and off in a comparatively short interval, as would need to be the case if one was trying to detect a Hawking quantum by an infalling detector before the detector hit the black hole surface. For detection times shorter than the wavelengths we want to measure\n\\begin{equation}\n\\Delta\\tau\\lesssim {1\\over (\\Omega+\\omega_k)}\n\\end{equation}\nwe get\n\\begin{equation}\n{\\cal A}\\sim -iq{1\\over \\sqrt{2\\omega_k}}{1\\over \\sqrt{2\\Omega}}\\Delta\\tau\\sqrt{\\pi}\\ne 0\n\\end{equation}\nso we pick up vacuum fluctuations in the detector. \n\nTo summarize, suppose we make a detector with frequency $\\Omega$ to pick up quanta of wavelength $\\lambda\\sim \\Omega^{-1}$. Then the effect of vacuum fluctuations will be comparable to the effect of `real quanta' if\n\\begin{equation}\n\\Delta\\tau \\lesssim {1\\over (\\Omega+\\omega_k)} < {1\\over \\Omega}\\sim \\lambda\n\\end{equation}\n\n\n\n\\refstepcounter{section}\n\\section*{\\thesection \\quad Wavelength of Hawking quanta} \\label{wavelength}\n\n\n\nConsider the Schwarzschild black hole\n\\begin{equation}\nds^2=-(1-{2M\\over r})dt^{2}+{dr^2\\over 1-{2M\\over r}}+r^{2}{d\\Omega_2^{2}}\n\\label{oneq}\n\\end{equation}\nThe temperature is ${1\\over 8\\pi M}$, so the wavelength of Hawking quanta at infinity is $\\lambda_\\infty\\sim M$. The wavelength of such a quantum at any position $r$ is\n\\begin{equation}\n\\lambda\\sim (-g_{tt})^{1\\over 2}\\lambda_\\infty\\sim M(1-{2M\\over r})^{1\\over 2}\n\\end{equation}\nNear the horizon $(r-2M)\\ll 2M$ we can use Rindler coordinates\n\\begin{equation}\nt_R={t\\over 4M}, ~~r_R=\\sqrt{8M(r-2M)}\n\\label{sixt}\n\\end{equation}\nThis gives the metric in the time and radial directions\n\\begin{equation}\nds^2\\approx -r_R^2 dt_R^2+dr_R^2\n\\label{fift}\n\\end{equation}\nFrom now on we restrict attention to just these directions. \nIn this near-horizon region we have for the wavelength of radiated quanta\n\\begin{equation}\n\\lambda\\sim M^{1\\over 2} (r-2M)^{1\\over 2}\\sim r_R\n\\end{equation}\nFrom (\\ref{fift}) we see that the distance from the horizon measured on a constant $t_R$ slice is $d=r_R$. Thus if a black hole emits radiation at the Hawking temperature, then the wavelength of these quanta at a distance $d$ from the horizon is \n\\begin{equation}\n\\lambda\\sim d\n\\end{equation}\nThis is the wavelength measured along a slice of constant Schwarzschild time $t$. If this quantum is encountered by an infalling detector, then the effective wavelength will be Lorentz contracted. Let the proper velocity of the detector \nin a local Lorentz frame oriented along the Schwarzschild $t, r$ directions, be \n\\begin{equation}\nU^{\\hat t}=\\cosh\\alpha, ~~~U^{\\hat r}=-\\sinh\\alpha\n\\label{eqalpha}\n\\end{equation}\nThe momentum vector of an outgoing massless quantum in the local Lorentz frame is\n\\begin{equation}\n(p^{\\hat t}, p^{\\hat r})\\sim ({1\\over \\lambda}, {1\\over \\lambda})\n\\end{equation}\nThe energy of the quantum as measured by the detector is then\n\\begin{equation}\nE=-p_\\mu U^\\mu\\sim {1\\over \\lambda}(\\cosh\\alpha+\\sinh\\alpha)={1\\over \\lambda} e^\\alpha\n\\end{equation}\nand the effective wavelength that is seen by the infalling detector is then\n\\begin{equation}\n\\lambda_{eff}\\sim \\lambda e^{-\\alpha}\\sim d e^{-\\alpha}\n\\label{sevent}\n\\end{equation}\nwhere as above, $d$ is the distance measured from the horizon in the Schwarzschild frame along a $t=const$ slice.\n\n\n\n\n\\refstepcounter{section}\n\\section*{\\thesection \\quad Proper time along infalling geodesic} \\label{time}\n\n\n\nWe wish to ask how much proper time $\\Delta \\tau$ elapses along a geodesic between the time it is at a distance $d$ from $r=2M$ and the time it hits the black hole surface at $r=2M$. Since we are working near the horizon, we use the Rindler coordinates (\\ref{sixt}). The Kruskal-type coordinates appropriates to a freely falling observer are given locally by taking the Minkowski coordinates related to $t_R, r_R$ by\n\\begin{equation}\nt_M=r_R \\sinh t_R, ~~~x_M=r_R \\cosh t_R\n\\end{equation}\n\n \\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.65]{fz3.eps}\n\\end{center}\n\\caption{The Rindler coordinates near the horizon, and the corresponding Minkowski coordinates. The infalling geodesic starts at $r_R=d, t_R=0$ and ends at the Rindler horizon.}\n\\label{fz3} \n\\end{figure}\n\nFig.\\;\\ref{fz3} shows the geodesic that we follow. This geodesic is a straight line in the local Minkowski coordinates\n\\begin{equation}\nt_M=\\cosh\\alpha ~\\tau, ~~~x_M=-\\sinh\\alpha~\\tau+d\n\\end{equation}\nWe have taken the geodesic to start with $\\tau=0$ at position $r_R=d$ and time $t_R=0$.\nHere $\\alpha$ is a constant the gives the velocity of infall; note that it is the same $\\alpha$ as the one that appears in (\\ref{eqalpha}). The geodesic crosses the horizon $t_M=r_M$ at proper time $\\tau_f$ with\n\\begin{equation}\n\\cosh\\alpha ~\\tau_f =-\\sinh\\alpha ~ \\tau_f + d, ~~~\\Rightarrow~~~\\tau_f=d e^{-\\alpha}\n\\end{equation}\n Thus if an observer on an infalling trajectory tries to detect a quantum at distance $d$ from the horizon, then the time he has available to make the detection is\n\\begin{equation}\n\\Delta\\tau_{available}< \\tau_f=d e^{-\\alpha}\n\\end{equation}\n\n\\end{appendix}\n\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}