diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzenmg" "b/data_all_eng_slimpj/shuffled/split2/finalzzenmg" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzenmg" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\nDelfs \\cite{Delfs85} considered a sheaf cohomology theory for\n(abstract) semialgebraic sets over arbitrary closed fields and proved\na semialgebraic version of homotopy invariance. In \\cite{EdmundoJP05}\nthis was generalized to definable sets and maps over an o-minimal\nexpansion of a group. If one further assumes that the o-minimal\nstructure expands a field, then one can use the triangulation theorem\nto show that the cohomology groups of a definable set are finitely\ngenerated and invariant under both elementary extensions and\nexpansions of the language. We prove that this continues to hold for\narbitrary o-minimal expansions of groups, provided we restrict\nourselves to definably compact sets. \n\nWorking without the field assumption entails various difficulties. \nTo begin with one cannot make use of the apparatus of singular cohomology.\nSo we work, following the above authors, with sheaf cohomology. More\nprecisely, given a definable set $X\\subset M^n$, the set of types $\\widetilde\nX$ of $X$ with the ``spectral topology'' is quasi-compactification of\n$X$, and we define $\\Hom^i(X;\\ca F ) := \\Hom^i(\\widetilde X ; \\ca F)$, where $\\ca\nF$ is a sheaf of Abelian groups on $\\widetilde X$.\n\nNow consider the case when $G$ is an Abelian group and $\\ca F$ is the\nconstant sheaf $G$ (i.e.\\xspace\\ the sheaf generated by the presheaf with\nconstant value $G$). Assuming that $X$ is {\\bf definably compact}\n(i.e.\\xspace\\ closed and bounded) we prove that $\\Hom^i(X;G)$ is finitely\ngenerated and invariant under both elementary extensions $N\\succ M$\n(i.e.\\xspace\\ $\\Hom^i(X;G) = \\Hom^i(X(N);G)$) and expansions of the language of $M$\n(note that expanding the language leaves $X$ invariant but alters $\\widetilde\nX$).\n\nThis would be easy to prove if $M$ expands a field. In fact in this\ncase by the triangulation theorem $X$ is definably homeomorphic to the\ngeometrical realization $|K|$ (in $M$) of a finite simplicial complex\n$K$, and a routine Mayer-Vietoris argument (together with the\nacyclicity of simplexes) shows that $\\Hom^i(X;G) \\cong \\Hom^i(K;G)$, where\nthe latter is the $i$-th simplicial cohomology group of $K$.\n\nIf $M$ does not expand a field we do not have the triangulation\ntheorem but we still have the cell decomposition theorem (see\n\\cite{Dries98}). One could then be tempted to invoke the uniqueness\ntheorem for cohomology functors satisfying the (appropriate form) of\nthe Eilenberg-Steenrod axioms. However, despite the cell\ndecomposition theorem, we have no guarantee that a definable set in an\no-minimal expansion of a group is a sort of definable CW-complex, so\nthe uniqueness theorem does not apply. The problem is that in the\ndefinition of a CW-complex one requires that the cells come equipped\nwith an attaching map that extends continuously to the boundary, while\nfor the o-minimal cells we do not have any such control of the\nboundary. \n\nAs standard references on sheaf cohomology and \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology we\nuse \\cite{Godement73} and \\cite{Bredon97}. For the reader's\nconvenience we give in the appendixes the relevant definitions and\nresults. Sheaf cohomology is defined for arbitrary topological\nspaces, but many results are proved in the quoted texts under the\nadditional assumption that the space (or the family of supports) is\nHausdorff and paracompact. This is potentially a source of problems\nsince, given a definable set $X$, the spectral space $\\widetilde X$ associated\nto it is in general not Hausdorff. In some cases we can reduce to the\ncompact Hausdorff situation using the fact that $\\widetilde X$ has a continuous\nretraction onto a compact Hausdorff subspace ${\\widetilde X}^{\\max}$ with the\nsame cohomology groups (see \\cite{CarralC83}), but in some other cases\nit is more convenient to show that the proofs of the relevant results\nin \\cite{Godement73} or \\cite{Bredon97} work with the Hausdorff\nhypothesis being replaced by normality (according to our convention\nnormality does not imply Hausdorff). The latter approach has the\nadvantage that one can do without the spectrality hypothesis. In\nparticular it can be shown that for normal paracompact spaces (not\nassumed to be Hausdorff or spectral) sheaf cohomology coincides with\n\\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology with coefficients in the given sheaf. This is\nreported in appendix~\\ref{AP:D},\nbut we shall not really need it.\nInstead we do need some results connecting the cohomology of\na subspace with the cohomology of its neighbourhoods. Such results\nhave been proved in \\cite{Delfs85,Jones06,EdmundoJP05} in the spectral\nsituation (see Corollary \\ref{taut}), but they can also been\nestablished under more general hypothesis (Theorem\n\\ref{LEM:TAUT-PCN}). As a corollary we obtain that\n$\\Hom^*(\\bigcap_{t>0} Y_t) \\cong \\varinjlim_t \\Hom^*(Y_t)$, whenever\n$(Y_t \\mid t>0)$ is a definable decreasing\nfamily of definably compact sets~$Y_t$\n(Corollary~\\ref{LEM:TAUT-DEFINABLE}). We will also show\n(Theorem~\\ref{THM:DEFINABLE}) that if the o-minimal structure $M$\nexpands a field, then $\\varinjlim_t \\Hom^*(Y_t) \\cong \\Hom^*(Y_{t_0})$ for\nall sufficiently small $t_0$, but we are not able to prove this fact\nwithout the field assumption.\n\n\\section{Topological preliminaries}\\label{preliminaries} \n\nLet $X$ be a topological space. $X$ is {\\bf normal} if every pair of\ndisjoint closed subsets of $X$ can be separated by open\nneighbourhoods. $X$ is {\\bf paracompact} if every open covering of $X$\nhas a locally finite refinement. Unlike other authors, in both\ndefinitions we {\\bf do not require that $X$ be Hausdorff}. We shall\ncall a space {\\bf PcN\\xspace} if it is paracompact and normal. Note that a\nparacompact Hausdorff space is PcN\\xspace.\n\nLet us recall that a {\\bf quasi-compact} space is a topological space $X$ in\nwhich every open covering has a finite refinement, or equivalently every\nfamily of closed sets with the finite intersection property has a non-empty\nintersection. So a compact space is an Hausdorff quasi-compact space.\n\nNote that a quasi-compact space is \\emph{a fortiori} paracompact.\nMoreover, it is a well-known fact that a in a PcN\\xspace space $X$\nevery open covering $\\Cov{U}$ admits a shrinking $\\Cov{V}$:\ni.e.\\xspace, $\\Cov{V}$ is an open covering of~$X$,\nand for every $V \\in \\Cov{V}$ there exists $U \\in \\Cov{U}$ containing~$\\overline V$, the closure of~$V$.\n\nA {\\bf spectral space} is a\nquasi-compact space having a basis of quasi-compact open sets stable\nunder finite intersections and such that every irreducible closed set\nis the closure of a unique point. The prime spectrum of a commutative\nring with its Zariski topology is an example of a spectral\nspace. Another example is the set of prime filters of a lattice (see\n\\cite{CarralC83}). The set of $n$-types (ultrafilters of definable\nsets) of a first order topological structure $M$ in the sense of\n\\cite{Pillay87} can also be endowed with a spectral topology (see\n\\cite{Pillay88}). In particular if $M$ is a real closed field one\nobtains in this way the real spectrum of the polynomial ring $M[x_1,\n\\ldots, x_n]$ (see \\cite{CosteR82}).\n\n\\section{Compactifications}\n\nIn this section we discuss a variant of the Wallman (or\nStone-\\texorpdfstring{\\v{C}ech}{Cech}\\xspace) compactification of a normal topological space. The\nvariant depends on the particular choice of a basis of open sets. \n\n\\begin{definition} \\label{comp} \nGiven a topological space $X$ with a fixed basis of open sets $\\basis$\n(that we always assume closed under finite intersections) we can\ndefine a spectral space $\\widetilde X = \\widetilde {(X,\\basis)}$ (depending on $\\basis$) as follows. A\n{\\bf constructible set} is a boolean combination of basic open sets\n$U\\in \\basis$. Let $\\widetilde X$ be the set of {\\bf\nultrafilters} of constructible sets (i.e.\\xspace\\ maximal families of\nconstructible sets closed under finite intersections and not\ncontaining the empty set). For $b\\subset X$ constructible, let\n$\\tilde b = \\{p\\in \\widetilde X \\mid b\\in p\\}$. So $p\\in \\tilde b \\Longleftrightarrow b\\in\np$. The {\\bf spectral topology} on $\\widetilde X$ is defined as follows. As a\nbasis of open sets of $\\widetilde X$ we take the sets of the form $\\tilde b$\nwith $b$ an open constructible subset of $X$.\n\\end{definition}\n\n\\begin{lemma} $\\widetilde X$ is a spectral space. \\end{lemma} \n\n\\begin{proof} To prove that $\\widetilde X$ is quasi-compact consider a family $\\{C_i\n\\mid i\\in I\\}$ of closed sets $C_i \\subset \\widetilde X$ with the finite\nintersection property. We must prove that $\\bigcap_{i\\in I}C_i$ is\nnon-empty. Without loss of generality we can assume that $\\{C_i\\mid\ni\\in I\\}$ is closed under finite intersections. Let $x\\in \\widetilde X$ be an\nultrafilter containing all the closed constructible sets $b$ with $\\widetilde\nb \\supset C_i$ for some $i$. Then $x\\in \\bigcap_i C_i$, so $\\widetilde X$ is\nquasi-compact. The same argument shows that the sets $\\widetilde b$ with $b$\nconstructible are quasi-compact. So the sets $\\widetilde b$, with $b$ open\nand constructible, form a basis of quasi-compact open sets of $\\widetilde X$\nstable under finite intersections. To finish the proof we must show\nthat given an irreducible closed set $C$ of $\\widetilde X$, there is a unique\npoint $x\\in C$ with $C = Cl(x)$. To this aim, let $x$ be an\nultrafilter of constructible sets containing all the closed\nconstructible sets $b$ with $\\widetilde b \\supset C$ and the complements of\nthe closed constructible sets $c$ such that $\\widetilde c \\cap C$ is a proper\nsubset of $C$ (this family has the finite intersection property by the\nirreducibility of $C$). Then clearly $Cl(x) = C$. To prove that $x$\nis unique, suppose $Cl(x) = Cl(y)$. Then $x$ and $y$ contain the same\nclosed constructible sets. But the closed constructible sets generate\nthe boolean algebra of all the constructible sets. So $x$ and $y$ must\ncontain the same constructible sets, and are therefore equal. \\end{proof}\n\n\\begin{remark}\nNote that a point $x \\in \\widetilde X$ is closed if and only if $x$\ncontains a maximal family of closed constructible sets with the finite intersection property. \nSo by Zorn's lemma every closed subset $C$ of $\\widetilde X$ contains a closed point. \n\\end{remark}\n\n\\noindent For $x\\in X$ let \\[\\type x := \\{b \\mid x \\in b\\} \\in \\widetilde X.\\]\nSince a constructible set $b$ is empty if and only if $\\widetilde b$ is empty,\nthe map \\[\\type {~} \\colon X\\to \\widetilde X\\] has dense image. Moreover this\nmap is injective whenever $X$ is a $T_1$-space. So in this case,\nidentifying $x$ with $\\type x$, we have $X\\subset \\widetilde X$, and it is\neasy to see that the original topology on $X$ coincides with the\ntopology induced by $\\widetilde X$ (use the fact that for $A$ constructible,\n$\\widetilde A \\cap X = A$). Therefore:\n\n\\begin{lemma} \\label{T1} If $X$ is $T_1$, then for every open basis $\\basis$ of~$X$,\n$\\widetilde{(X, \\basis)}$ is a quasi-compactification of~$X$. \\end{lemma}\n\nWe say that $(X,\\basis)$ is {\\bf constructibly\nnormal} if any pair of disjoint constructible open sets can be\nseparated by closed constructible sets. \n\n\\begin{lemma} If $(X,\\basis)$ is constructibly normal, then $\\widetilde X$ is\nnormal (not necessarily Hausdorff). \\end{lemma}\n\n\\begin{proof} Indeed given two disjoint closed subsets $A$ and $B$ of $\\widetilde X$, by\nquasi-compactness there are disjoint closed constructible sets $A',B'$\nin $X$ with $\\widetilde {A'} \\supset A$ and $\\widetilde {B'} \\supset B$. By the\nassumption $A',B'$ can be separated by disjoint open constructible\nsets $U\\supset A'$ and $V \\supset B'$. So $\\widetilde U$ and $\\widetilde V$ are open\nsets separating $A,B$. \n\\end{proof} \n\nIn a normal spectral space $Y$, the subset $Y^{\\max}$ of the closed\npoints of $Y$ is compact Hausdorff (see \\cite{CarralC83}). Also note\nthat, if $x$ is a closed point of $X$ and the singleton $\\{x\\}$ is\nconstructible, then $\\type x$ is a closed point of $\\widetilde X$. So we have:\n\n\\begin{lemma} If $(X,\\basis)$ is constructibly normal and the points of $X$ are\nclosed and constructible, then ${\\widetilde X}^{\\max}$ is a compactification\nof $X$, namely it is a compact Hausdorff space containing $X$ as a\ndense subspace.\\end{lemma}\n\nIf $X$ is a normal Hausdorff space, and we take as a basis of $X$ the family\n$\\basis$ of {\\em all} its open subsets, then ${\\widetilde X}^{\\max}$ is the\n{\\bf Wallman compactification} of $X$ \\cite{Wallman38}. For\nnormal Hausdorff spaces it coincides with the {\\bf Stone-\\texorpdfstring{\\v{C}ech}{Cech}\\xspace\ncompactification} (see \\cite[Thm. 3.6.22]{Engelking89}).\n\n\\begin{example} \nConsider the space $\\Q$ with a basis $\\basis$ of open sets given by the open intervals \n$(a,b)\\subset \\Q$, where we allow $a=-\\infty$ or $b = +\\infty$. Then \n\\[\\widetilde {(\\Q,\\basis)} = \\{a^-, a^+\\}_{a\\in \\Q} \\cup \\R \\cup \\{\\pm \\infty\\}\\]\nwhere: $a^+$ is the unique ultrafilter containing all sets of the form\n$(a,b)$ with $b>a$; similarly $a^-$ contains all sets $(b,a)$ with\n$b0$\nand every Abelian group $G$. \\end{lemma}\n\nIndeed it is easy to see the lemma holds for an arbitrary interval $I$, not necessarily closed and bounded\n(note that in any case $\\widetilde I$ is quasi-compact). \n\n\\section{Contractibility of cells}\n\nLet $M$ be an o-minimal expansion of a group. \n\n\\begin{lemma} \\label{contraction1} Let $I$ be a bounded interval in $M$ \\rom(closed,\nhalf-closed, or open\\rom). Then $I$ is definably contractible to a\npoint. \\end{lemma}\n\n\\begin{proof} Let us consider the case $I = (a,b)$. Given $0< t \\leq \\frac{b-a}\n2$, there is a (unique) definable continuous function $f_t \\colon\n(a,b) \\to (a,b)$ such that $f_t$ is the identity on $[a+t,b-t]$ and it is\nconstant on both $[a,a+t]$ and $[b-t,b]$, with values $a+t$ and $b-t$\nrespectively (so the image of $f_t$ is $[a+t,b-t]$). Define $f_0 =\nf$. Then $(f_t)_{0\\leq t \\leq \\frac{b-a} 2}$ is a deformation retract\nof $(a,b)$ to the point $\\frac {a+b} 2$. The other cases are\nsimilar. \\end{proof}\n\nWe will employ the following notation: given $B \\subseteq M^{n-1}$ and \\mbox{$f,g : B \\to M$},\n\\[\\begin{aligned}{}\n(f,g)_B &:= \\set{(x,y)\\in M^{n-1} \\times M: x \\in B \\ \\& \\ f(x) < y < g(x)},\\\\\n[f,g]_B &:= \\set{(x,y)\\in M^{n-1} \\times M: x \\in B \\ \\& \\ f(x) \\leq y \\leq g(x)},\\\\\n\\Gamma(f) &:= \\set{(x,y)\\in M^{n-1} \\times M: x \\in B \\ \\& \\ f(x) = y},\n\\text{ the graph of } f.\n\\end{aligned}\\]\n\n\\begin{lemma} \\label{contraction2} If $C$ is a bounded cell of dimension $m>0$\nin $M^n$ then there is a deformation retract of $C$ onto a cell of\nstrictly lower dimension. So by induction every bounded cell is\ndefinably contractible to a point. \\end{lemma}\n\n\\begin{proof} If $C$ is the graph of a function we can reason by induction on\nthe dimension of the ambient space. So the only interesting case is\nwhen $m>1$ and\n$C = (f,g)_B$.\nLet $h = \\frac {f+g} 2$.\nWe will define\na deformation retract from $C$ to $\\Gamma(h)$. We can assume that\n$h$ is a constant function, since we can reduce to this case by a\ndefinable homeomorphism which fixes all but the last coordinate (just\ntake any constant function $h_1\\colon B \\to M$, and define $f_1,g_1$\nso that they differ from $h_1$ by the same amount in which $f,g$\ndiffer from $h$). Since $C$ is bounded, there are constants $a,b\\in M$\nsuch that $h$ is the constant function $\\frac {a+b} 2$ and $(f,g)_B\n\\subset B\\times (a,b)$. By (the proof of) Lemma \\ref{contraction1}\nthere is deformation retract of $(a,b)$ onto $\\{\\frac {a+b} 2\\}$,\nwhich induces a deformation retract of $B\\times (a,b)$ onto $B \\times\n\\{\\frac {a+b} 2\\}$, namely onto the graph of $h$. \\end{proof}\n\nBy Fact \\ref{homotopy} we obtain:\n\n\\begin{corollary} \\label{acyclic-cell} If $C$ is a bounded cell of dimension $m$ in\n$M^n$ then $\\Hom^p(C;G) = 0$ for all $p>0$ and every Abelian group~$G$.\n\\end{corollary}\n\nIf we generalize slightly the definition of definable homotopy and allow the parameter of a homotopy to vary in the interval $[-\\infty, + \\infty]$, we get that Fact~\\ref{homotopy} is still true, and therefore in Lemmata~\\ref{contraction1} and~\\ref{contraction2} we can drop the ``boundedness'' hypothesis.\nThus, Corollary~\\ref{acyclic-cell} is true also for unbounded cells.\n\n\\section{Cells with non-acyclic closure} \n\nLet $M$ be an o-minimal expansion of a group and let $X\\subset M^n$ be\na definable set. We will prove (Theorem \\ref{fin-gen}) that the\ncohomology groups $\\Hom^p(X;G)$ of $X$ are finitely generated. An\nimportant special case is when $X$ is the closure $\\overline C$ of a bounded\ncell $C$. One may be tempted to conjecture that $\\Hom^i(\\overline C ; G)=0$ in\ndimension $i>0$, but Example \\ref{strange-cell} shows that in general\nthis is false. Indeed similar examples show that $\\Hom^1(\\overline C ; G)$ can\nhave arbitrarily large finite rank.\n\n\\begin{example} \\label{strange-cell} Let $M=(\\R,<,+,\\cdot)$. There is a bounded\ncell $C$ of dimension $2$ in $\\R^4$ whose closure $\\overline C$ has an ``hole'',\nnamely it is definably homotopic to a circle. \\end{example}\n\n\n\\begin{floatingfigure}[r]{15ex}\n\\input{annulus.latex} \n\\caption{\n\\label{annulus}\n\\end{floatingfigure}\nBefore giving the example, let us observe that\nin Figure~\\ref{annulus} (an annulus with a ray removed) we have\na space homeomorphic to a disk whose closure is homotopic to a circle.\nHowever the space of Figure~\\ref{annulus} is not a cell in\nthe sense of o-minimal cell decompositions. So we have to proceed\ndifferently. \n\nAs a preliminary step we show that there is a\ntwo-dimensional cell $D$ in $\\R^3$ which is homeomorphic to an open\ndisk minus a ray via an homeomorphism which extends to the\nclosures. An example is the cell $D$ depicted on the top-right part of\nFigure~\\ref{cell}.\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{cell5.latex} \n\\caption{A two-dimensional cell in $\\R^3$}\n\\label{cell}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{cell} must be interpreted as follows. The projection of\n$D$ on the first two coordinates is the open square on the\nbottom-right part of Figure~\\ref{cell}, with the five triangles\npartitioning the square corresponding to the five regions in $D$. So $D$\nis the graph of a function $f$ from the square to $\\R$. $D$ is\nhomeomorphic to an open disk with a ray removed via an homeomorphism\nthat extends to the closures and sends the ray of the disk to the\nvertical dashed line $I \\subset \\partial D$ depicted in the Figure.\nThe three triangular regions of the cell $D$ are level sets of $f$,\nwith the central triangle being in a lower level, and the two top\ntriangles being on a common higher level.\n\nWe are now ready to describe the cell $C$ of Example\n\\ref{strange-cell}. The idea is the following. Given a disk without a\nray as in Figure~\\ref{cell}, we can ``bend'' it going in a higher\ndimension, so as to create a ``hole'' (as in Figure~\\ref{annulus}) by\nseparating the two sides of the middle part of the ray. More\nprecisely, consider a definable continuous function $g\\colon D \\to \\R$\nwith the following properties: 1) $g$ takes non-zero values only for\npoints $x\\in D$ sufficiently close to the middle point $x_0$ of the\ndashed segment $I \\subset \\partial D$; 2) if $g(x) > 0$, then $x$ is\non the ``left hand side'' of $I$ and $g(x) \\to 1$ for $x\\to x_0$ from\nthe left; 3) if $g(x) < 0$, then $x$ is on the ``right-hand-side'' of\n$I$ and $g(x) \\to -1$ for $x\\to x_0$ from the right. Let $C$ be the\ngraph of $g$. Then $H^1(\\overline C; G) \\neq 0$.\n\n\\begin{remark} It is possible to show that $\\Hom^1(X;G)$ can have arbitrarily high\nrank even for a definable set $X$ that can be decomposed as a disjoint\nunion of only two bounded cells. \nTherefore, unlike the case of triangulations,\nthe abstract structure of a cell decomposition\n(namely the dimensions and the adjacency relation between cells) by no\nmeans determines the cohomology of a definable set. It is not however\nexcluded that by a refined version of the cell decomposition theorem one \ncould obtain decompositions that do determine the cohomology groups. \\end{remark}\n\n\\section{Acyclic coverings}\nIn this section we give a sufficient condition, not based on\ndeformation retracts, to prove that an inclusion $X\\subset Y$ of\ntopological spaces induces an isomorphism in cohomology (see Lemma\n\\ref{iso}).\n\nGiven an open cover $\\ca U = \\{U_i \\mid i \\in I\\}$ of a topological\nspace $X$, and a subset $J\\subset I$, we write $U_J$ for the\nintersection $\\bigcap_{i\\in J} U_i$. The following theorem of Leray\nsays that, given an ``acyclic covering'', the cohomology of a sheaf\ncan be computed as the \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology of the covering\n(cf.\\xspace~Appendix~\\ref{AP:D}). \n\n\\begin{fact}\\label{Leray}\n{\\rm(\\cite[Thm. 4.13, p. 193]{Bredon97}, \\cite[\\S 5.4, p. 213]{Godement73})}\nLet $\\ca F$ be\na sheaf on a topological space $X$ and let $\\ca U = \\{U_i \\mid i \\in\nI\\}$ be an open covering of $X$ having the property that $\\Hom^p(U_J; \\ca\nF) = 0$ for every $p>0$ and every finite $J\\subset I$. \nThen the canonical homomorphism $\\check{\\Hom}\\vphantom{\\Hom}^*(\\ca U ; \\ca F)\\to \\Hom^*(X;\\ca F)$\nis an isomorphism.\n\\end{fact}\n\nIf moreover we assume that $\\ca F$ is the constant sheaf $G$ and $\\Hom^0(U_J; \\ca F ) = G$ for every $J$ with $U_J$ non-empty, then from the\ndefinition of \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology it follows that $\\check{\\Hom}\\vphantom{\\Hom}^*(\\ca U;G)$\ncoincides with the $i$-th simplicial cohomology group with\ncoefficients in $G$ of the nerve $N({\\ca U})$ of $\\ca U$. So we have:\n\n\\begin{corollary} \\label{good-cover} \nIf there exists a finite open cover $\\ca U= \\{U_i \\mid i \\in I\\}$ of $X$ with \n $\\Hom^p(U_J; G ) = 0$ for every $p > 0$ and $\\Hom^0(U_J; G) = G$ for every finite\n$J\\subset I$ with $U_J$ non-empty, then $\\Hom^i(X;G)$ is isomorphic to the $i$-th \nsimplicial cohomology group of the nerve $N({\\ca U})$ of the covering, so in particular it is \nfinitely generated. \n\\end{corollary} \n\nUnfortunately we do not know the answer to the following:\n\n\\begin{question} Let $X$ be a definably compact set in an o-minimal expansion $M$ of a\ngroup. Does $\\widetilde X$ have a cover as in Corollary \\ref{good-cover}? \\end{question} \n\nThe answer is positive if $M$ expands a field, as a simple application\nof the triangulation theorem shows.\n\n\\begin{remark} \\label{inducedh}\nLet $f\\colon X\\to Y$ be a continuous function, let ${\\cal V} =\n\\{V_i \\mid i\\in I\\}$ be an (indexed) open cover of $Y$ and consider\nthe (indexed) open cover $f^{-1}({\\cal V}):= \\{ f^{-1}(V_i)\\mid i \\in I\\}$ of\n$X$. It follows easily from the definition of the induced\nhomomorphism $\\check{f}$ in \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology (see \\cite[IX, \\S 4]{EilenbergS52} or \\cite[III.4.1.4]{Bredon97}) that there is a commutative\ndiagram:\n\n\\DIAGV{90} {\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal V};G)} \\n{\\Ear{a}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p(f^{-1}({\\cal V});G)} \\nn \n {\\Sar{b}} \\n{} \\n{\\Sar{c}} \\nn \n {\\check{\\Hom}\\vphantom{\\Hom}^p(Y; G)} \\n{\\Ear{\\check{f}}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p(X; G)} \n\\diag \n\n\n\\noindent where $b,c$ are the natural morphisms (see \\cite{Godement73})\nand $a$ is induced by the simplicial map on the nerves of\nthe indexed coverings sending $f^{-1}(V_i)$ to $V_i$. \\end{remark}\n\n\n\\begin{lemma} \\label{iso} Let $X\\subset Y$ be definable sets. Suppose that there\nare coverings ${\\cal U} = \\{ U_i \\mid i\\in I\\}$ of $X$ and ${\\cal V} =\n\\{ V_i \\mid i \\in I\\}$ of $Y$ indexed by the same finite set $I$ such\nthat:\n\\begin{enumerate}\n\\item $U_i \\subset V_i$ for all $i,j\\in I$.\n\\item For all finite $F\\subset I$, $U_F$ is non-empty iff $V_F$ is\nnon-empty (i.e.\\xspace\\ the natural map among the nerves of the coverings is\nan isomorphism).\n\\item For each finite $F \\subset I$ the sets $U_F$ and $V_F$ are either empty or connected, \nand for all $q>0$, $\\Hom^q(U_F; G) = \\Hom^q(V_F; G) = 0$. \n\\end{enumerate}\nThen the inclusion map $X\\subset Y$ induces an isomorphism $\\Hom^*(Y;\nG) \\to \\Hom^*(X; G)$ for any Abelian group $G$. \\end{lemma}\n\nNote that we do not require that $V_i \\cap X = U_i$. \n\n\\begin{proof} We are going to apply Remark \\ref{inducedh} to the case when\n$X\\subset Y$ and $f$ is the inclusion map. In this case $f^{-1}({\\cal\nV}) = {\\ca V} \\cap X$ (by definition). Consider the following\ncommutative diagram, where $a,b,c$ are as in Remark \\ref{inducedh},\n$d$ is induced by the natural simplicial isomorphism on the nerves of\nthe coverings, $e$ is the natural morphism form the\n\\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology of a covering to the \\texorpdfstring{\\v{C}ech}{Cech}\\xspace cohomology of the\nspace, and $p,q$ are the natural maps from \\texorpdfstring{\\v{C}ech}{Cech}\\xspace to sheaf\ncohomology.\n\n\\DIAGV{90} {\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal V};G)} \\n{\\Ear{a}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal V}\\cap X;G)} \\n{\\Ear{d}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p({\\cal U}; G)}\\nn \n {\\Sar{b}} \\n{} \\n{\\Sar{c}} \\n{\\raisebox{1em}{\\SWAR{\\raisebox{1em}{e}}}} \\n{} \\n{} \\nn\n {\\check{\\Hom}\\vphantom{\\Hom}^p(Y; G)} \\n{\\Ear{\\check{f}}} \\n{\\check{\\Hom}\\vphantom{\\Hom}^p(X; G)} \\n{} \\n{} \\nn\n {\\Sar{p}} \\n{} \\n{\\Sar{q}} \\n{} \\n{} \\nn\n {\\Hom^p(Y; G)} \\n{\\Ear{f^* }} \\n{\\Hom^p(X; G)} \\n{} \\n{} \n\\diag\n\n\\noindent By our assumptions on the coverings, $a$ and $d$ are isomorphisms. \nBy Fact \\ref{Leray} $p\\circ b$ and $q\\circ e$ are isomorphisms. So \n $f^*: \\Hom^*(Y;G) \\to \\Hom^*(X; G)$ is\nan isomorphism. \\end{proof}\n\n\\section{Finiteness results for cohomology} \n\nAs usual let $M$ be an o-minimal expansion of group. Given a definably\ncompact set $X$ we want to prove that $\\Hom^i(X;G)$ is finitely generated\nfor every $i$ and every Abelian group $G$. An important special case\nis when $X$ is the closure $\\overline C$ of a bounded cell $C$. We will show\n(as a consequence of Corollary \\ref{X-C}) that there is a point $a\\in\nC$ such that $\\Hom^p(\\overline C \\setminus \\{a\\};G) \\cong \\Hom^p(\\partial C;G)$,\nwhere $\\partial C := \\overline C \\setminus C$ is the boundary of\n$C$. Granted this, since $\\partial C$ has smaller dimension than $\\overline\nC$, by induction on the dimension we can assume that the cohomology\ngroups of $\\partial C$ are finitely generated and carry on with the\ninductive proof. At first sight the fact that $\\Hom^p(\\overline C \\setminus\n\\{a\\};G) \\cong \\Hom^p(\\partial C;G)$ looks rather intuitive: one may even\nbe tempted to conjecture that $\\partial C$ is a definable deformation\nretract of $\\overline C \\setminus \\{a\\}$ (as it would be the case for the\ncells of a CW-complex), or at least that these two spaces are\ndefinably homotopy equivalent. However we are not able to prove this.\nWe proceed instead in a different manner, with the role of definable\ndeformation retracts being taken by Lemma \\ref{iso}. There is however\na further complication. We are not able to apply Lemma~\\ref{iso}\ndirectly to the pair of sets $(X,Y) = (\\overline C \\setminus \\{a\\}, \\partial\nC)$, but only to pairs of sets of the form $(C \\setminus \\{a\\},\nC\\setminus C_t)$, where $(C_t)_{t>0}$ is a suitable definable\ncollection of sets with $\\bigcup_{t>0} C_t = C$, and consequently $\\bigcap_{t>0}(\\overline C \\setminus C_t) = \\partial C$ (the singleton $\\{a\\}$ is one of the\n$C_t$). So we first prove $\\Hom^p(C \\setminus \\{a\\}) \\cong \\Hom^p(C\\setminus\nC_t)$. Then we deduce $\\Hom^p(\\overline C \\setminus \\{a\\}) \\cong \\Hom^p(\\overline\nC\\setminus C_t)$ by the excision theorem. Finally, with the help of\nCorollary \\ref{LEM:TAUT-DEFINABLE}, we let $t\\to 0$ to obtain $\\Hom^p(\\overline\nC \\setminus \\{a\\}) \\cong \\Hom^p(\\partial C)$. This is the idea. Let us\nnow come to the details.\n\n\\begin{lemma} \\label{cell-covers}\nLet $C\\subset M^n$ be a bounded cell of dimension $m$.\nThere is a definable family $\\{ C_t \\mid t>0 \\}$ of definably compact sets\n$C_t\\subset C$ such that:\n\\begin{enumerate}\n\\item $C = \\bigcup_{t>0}C_t$.\n\\item If $01$ and $C = \\Gamma(f)$, where $f\\colon B\\to M$. By\ninduction $B_t$ is defined and we set $C_t = \\Gamma(f \\rest{B_t})$. \n\\item \nLet $n>1$ and $C = (f,g)_B$. By induction $B_t$ is defined.\nWe put\n$C_t = [f+\\gamma_t, g-\\gamma_t]_{B_t}$,\nwhere $\\gamma_t := \\min(\\frac{f - g}{2}, t)$.\n\\end{enumerate} \n\nWith this definition we have:\n\n\\begin{claim} \\label{cell-cover} For each $t>0$ \nthere is a covering $\\ca U = \\{U_i\\mid i\\in I\\}$ of\n$C\\setminus C_t$ such that:\n\\begin{enumerate}\n\\item \nThe index set $I$ is the family of the closed faces of an\n$m$-dimensional cube, where $m= \\dim(C)$. \\rom(So $|I|=2m$\\rom).\n\\item If $F\\subset I$, then $U_F := \\bigcap_{i\\in F}U_i$ is either\nempty or a cell. \\rom(So in particular $\\Hom^p(U_F; G) = 0$ for all $p>0$\nand, if $U_F \\neq \\emptyset$, $\\Hom^0(U_F;G) = G$.\\rom)\n\\item For $F\\subset I$, $U_F \\neq \\emptyset$ iff the faces of the\ncubes belonging to $F$ have a non-empty intersections. \\rom(So the nerve of\n$\\ca U$ is isomorphic to the nerve of a covering of an $m$-cube by its\nclosed faces.\\rom)\n\\end{enumerate} \n\\end{claim} \n\\begin{minipage}{\\textwidth}\n\\begin{floatingfigure}[r]{25ex}\n\\input{cover2.latex} \n\\caption{}\n\\label{cover}\n\\end{floatingfigure}\nFor example for $m=2$ we have four open sets $U_i$ which intersect each other\nas in Figure~\\ref{cover}.\nNote that the claim implies, by Corollary~\\ref{good-cover}, that $C\n\\setminus C_t$ has the same cohomology groups of an $m-1$ dimensional\nsphere. To prove the claim we define $\\ca U$ by induction on the\ndimension $n$ of the ambient space. We distinguish four cases\naccording to the definition of~$C_t$.\n\\end{minipage}\n\\begin{enumerate}\n\\item{If $n=1$ and $C$ is a singleton, then $\\ca U$ is the covering\nconsisting of one open set (given by the whole space $C$).}\n\\item{If $n=1$ and $C = (a,b)$, then $C\\setminus C_t$ is the union of\nthe two open subsets $(a, a + \\gamma_t)$ and $(b- \\gamma_t, b)$,\nand we define $\\ca U$ as the covering consisting of these two sets.}\n\\item Let $n>1$ and $C = \\Gamma(f)$, where $f\\colon B\\to M$. By\ninduction we have a covering $\\ca V$ of $B\\setminus B_t$ with the\nstated properties, and we define $\\ca U$ to be the covering of\n$C\\setminus C_t$ induced by the natural homeomorphism between the\ngraph of $f$ and its domain.\n\\item\nLet $n > 1$ and $C = (f,g)_B$. By definition $C_t = (f + \\gamma_t, g -\n\\gamma_t)_{B_t}$. By induction $B\\setminus B_t$ has a covering $\\ca V\n= \\{V_j \\mid j\\in J\\}$ with the stated properties, where $J$ is the\nset of closed faces of the cube $[0,1]^{m-1}$. Define a covering $\\ca\nU = \\{U_i \\mid i\\in I\\}$ of $C\\setminus C_t$ as follows. As index set\n$I$ we take the closed faces of the cube $[0,1]^m$. Thus $|I| = |J| +\n2$, the two extra faces corresponding to the ``top'' and ``bottom''\nface of $[0,1]^m$. \nWe associate to the top face \nthe open set $(g- \\gamma_t,g)_B \\subset C\\setminus C_t$ and to the\nbottom face the open set $(f, f + \\gamma_t)_B \\subset C \\setminus\nC_t$. The other open sets of the covering are the preimages of the sets $V_j$ \nunder the projection $M^n\\to M^{n-1}$. This\ndefines a covering of $C\\setminus C_t$ with the stated properties.\n\\end{enumerate}\n\nIt remains to show that the inclusion map $C\\setminus C_{t'} \\subset\nC\\setminus C_{t}$ induces an isomorphism $\\Hom^p(C \\setminus C_t; G) \\to\n\\Hom^p(C \\setminus C_{t'}; G)$. To this aim it suffices to observe that\nby (the proof of) Claim 1 there are coverings $\\ca U$ of $C \\setminus\nC_{t'}$ and $\\ca V$ of $C \\setminus C_t$ satisfying the assumptions of\nLemma \\ref{iso}. \n\\hfill\\textsquare\n\n\\begin{lemma} \\label{x-c} Let $X$ be a definably compact set, $C$~be a cell of\nmaximal dimension in~$X$, and $G$ be an Abelian group.\nThen for each $00$ the inclusion map\n$X\\setminus C \\subset X\\setminus C_t$ induces an isomorphism \\[\\Hom^p(X\n\\setminus C_t; G) \\cong \\Hom^p(X \\setminus C; G).\\] \\end{corollary}\n\n\\begin{proof} For $t>0$ we have\n\\[ \\begin{array}{ccc}\n\\Hom^*(X\\setminus C_t;G) & \\cong & \\varinjlim_s \\Hom^*(X \\setminus C_s;G) \\\\\n & \\cong & \\Hom^*(X \\setminus C;G)\n\\end{array} \\]\nwhere the first isomorphism follows from Lemma~\\ref{x-c} and\nRemark~\\ref{lim}, and the second one follows from the equality $X\n\\setminus C = \\bigcap_s (X \\setminus C_s)$ and\nCorollary~\\ref{LEM:TAUT-DEFINABLE}. \\end{proof}\n\nNote that for $t$ big enough, $C_t$ is a singleton. So in particular,\napplying the theorem to $X = \\overline{C}$, we have proved that there is a\npoint $a\\in C$ such that:\n \\begin{equation} \\label{partial} \\Hom^p(\\overline{C}\n\\setminus \\{a\\}; G) \\cong \\Hom^p(\\partial C; G). \\end{equation} We do not\nknow however whether $\\partial C$ is a definable deformation retract of $\\overline C\n\\setminus \\{a\\}$.\n\n\\begin{theorem} \\label{fin-gen} Let $X \\subset M^n$ be a definably compact\nset and $G$ be an Abelian group.\nThen, for each $p$, $\\Hom^p(X;G)$ is finitely generated. Moreover\n$\\Hom^p(X;G)=0$ for $p>\\dim(X)$. \\end{theorem}\n\n\\begin{proof}\nBy a Mayer-Vietoris argument. Decompose $X$ into cells. Let $C$ be\na cell of $X$ of maximal dimension. Let $t>0$ be such that $C_t \\neq\n\\emptyset$ and write $X$ as the union of $X\\setminus C_t$ and\n$C$. Consider the Mayer-Vietoris sequence (see \\cite[eqn. 32,\np. 98]{Bredon97}) associated to this union:\n\\begin{equation} \\label{EQ:MAYER}\n\\ldots \\to \\Hom^{p-1}(C \\setminus C_t) \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C_t)\n\\oplus \\Hom^p(C) \\to \\Hom^p(C\\setminus C_t) \\to \\ldots\n\\end{equation}\nwhere we have omitted the coefficients $G$ in the notation. By Corollary \\ref{X-C} the\ninclusion $X\\setminus C_t \\subset X\\setminus C$ induces an isomorphism\nin cohomology, so composing with this isomorphism we obtain \n\\begin{equation} \\label{mv} \n\\ldots \\to \\Hom^{p-1}(C \\setminus C_t) \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to \\Hom^p(C\\setminus C_t) \\to \\ldots\n\\end{equation}\nNow $C$ has the cohomology of a point and $C\\setminus C_t$ has the\ncohomology of an $(m-1)$-dimensional sphere. So the displayed part of\nthe sequence above has the form:\n\\begin{align}\\label{mv1} \n& G \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to 0,\\\\\n\\text{ or}\\quad &\n\\label{mv2} \n0 \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to G,\\\\\n\\text{ or} \\quad &\n\\label{mv3} \n0 \\to \\Hom^p(X) \\to \\Hom^p(X\\setminus C) \\oplus\n\\Hom^p(C) \\to 0;\n\\end{align}\nwhere (\\ref{mv3}) applies for $p \\in \\{m, m-1\\}$. \n By induction on the number of cells \n$\\Hom^p(X\\setminus C)$ is finitely generated, and vanishes for $p\\geq m$.\n From the above sequences it then follows that the same holds for $\\Hom^p(X)$.\n\\end{proof}\n \n\n\\section{Elementary extensions and change of language}\n\nLet $M$ be an o-minimal expansion of a group and let $X\\subset M^n$ be\na definable set. We have seen that we can associate to $X$ the\nspectral space $\\widetilde X$ of all its types over $M$ (such a type can\nbe identified with an ultrafilter of $M$-definable sets that contains \n$X$). The cohomology of $X$ has been defined as the cohomology of $\\widetilde X$. \nIf $N\\succ M$ is an elementary extension we may also associate to $X$ the spectral\nspace $\\widetilde{X(N)}$ (ultrafilters of $N$-definable sets that contains $X(N)$). \n\n\\begin{theorem} \\label{theta} Let $\\theta \\colon \\widetilde{X(N)} \\to \\widetilde X$ be\nthe map that sends a type over $N$ to its restriction over $M$. If $X$\nis definably compact then $\\theta$ induces an isomorphism\n$\\Hom^*(\\widetilde X; G) \\to \\Hom^*(\\widetilde{X(N)}; G)$ for any Abelian group $G$. \\end{theorem}\n\n\\begin{proof}\nFor both $X$ and $X(N)$ we have an exact sequence as in\nequation~\\eqref{EQ:MAYER}\nabove. The terms of the two exact sequences so obtained are connected by the\nhomomorphisms induced by $\\theta$ in such a way that the resulting \ndiagram commutes (it is important that in equation~\\eqref{EQ:MAYER}\nwe take the parameter $t$ in the small model~$M$).\nThe desired result then follows\narguing as in Theorem \\ref{fin-gen} by induction on the number of cells. \\end{proof} \n\n \nWe now extend the above result to certain type-definable sets. As above let $N\\succ M$. \n\n\\begin{theorem} Let $X\\subset M^n$ be a definably compact set. Let $A\\subset\n\\widetilde X$ be a type-definable closed subset and let $A(N) := \\theta^{-1}(A)\n\\subset \\widetilde{X(N)}$. Then $\\theta$ induces an isomorphism\n$\\Hom^*(A; G) \\to \\Hom^*(A(N) ; G)$. In particular if\n$p$ is a closed type in $\\widetilde X$, then the set $\\theta^{-1}(p)$ of all\ntypes of $\\widetilde{X(N)}$ which restrict to $p$ has the same\ncohomology of a point \\rom(so in particular it is connected\\rom). \\end{theorem}\n\\begin{proof} Each closed type-definable set $A\\subset \\widetilde X$ can be written as an\nintersection $\\bigcap_{i\\in I}X_i$ of definably compact sets $X_i\n\\subset M^n$. Now observe that $A(N) =\n\\bigcap_{i\\in I}\\widetilde{X_i(N)}$. By Corollary \\ref{LEM:TAUT-DEFINABLE} and Theorem\n\\ref{theta} $\\Hom^*(A(N);G) = \\varinjlim_{i\\in I}\\Hom^*(\\widetilde{X_i(N)};G) =\n\\varinjlim_{i\\in I}\\Hom^*(X_i;G) = \\Hom^*(A;G)$.\n\\end{proof}\nIn similar way we can prove: \n\n\\begin{theorem} If $M_1$ is an o-minimal expansion of $M$ to a bigger language and\n$X\\subset {M}^n$ is a definably compact set in $M$, then the map\n$\\widetilde{X(M_1)} \\to \\widetilde X$ sending each type in the language\n$L_1$ to its restriction to $L$ induces an isomorphism $\\Hom^*(\\widetilde\nX; G) \\to \\Hom^*(\\widetilde{X(M_1)}; G)$ for any Abelian group $G$.\\end{theorem}\n\n\\section{Definable families in expansions of fields}\n\\begin{definition}\nGiven definable maps $f : X \\to Y$ and $f' : X' \\to Y'$, we say that $f$ and $f'$ have the same definable topological type iff there exist definable homeomorphisms $\\lambda: X \\to X'$ and $\\mu: Y \\to Y'$ making the following diagram commute:\n\\DIAGV{50}\n{}\\n{X} \\n{\\Ear{f}} \\n{X'} \\nn\n{}\\n{\\Sar{\\lambda}} \\n{} \\n{\\saR{\\mu}} \\nn\n{}\\n{Y} \\n{\\Ear{g}} \\n{Y'}\n\\diag\n\\end{definition}\n\n\\begin{definition}\nLet $(Y_t)_{t > 0}$ be a family of subsets of~$M^n$.\nWe will say that $(Y_t)_{t>0}$ is decreasing if $Y_t \\subseteq Y_{t'}$ for every $t \\leq t'$.\n\\end{definition}\n\nIf $M$ is an o-minimal expansion of a\nfield, Corollary \\ref{LEM:TAUT-DEFINABLE} can be strengthened as follows. \n\n\\begin{theorem} \\label{THM:DEFINABLE}\nAssume that $M$ is an o-minimal expansion of a field.\nLet $\\Pa{Y_t}_{t > 0}$ be a definable decreasing\nfamily of definably compact subsets of some definable set~$Y$.\nLet $A := \\bigcap_{t > 0} Y_t$. \nThen for every sufficiently small $t$ we have a natural isomorphism\ninduced by the inclusion\n\\[\n\\Hom^*\\Pa{ A; {\\ca F}} \\cong \\Hom^*\\Pa{{Y_t}; {\\ca F}},\n\\]\nfor every sheaf ${\\ca F}$ on~$\\widetilde Y$.\n\\end{theorem}\nIn the above theorem we cannot weaken the hypothesis to $Y_t$ closed\n(instead of definably compact). For instance, let $Y_t :=\n[\\frac{1}{t}, \\infty) \\subseteq M$. \n\nNote that Corollary \\ref{X-C} is a special case of Theorem \\ref{THM:DEFINABLE}.\nTo prove the theorem, we need the following lemmata.\nNote that for the lemmata we do not need that $M$ expands a field,\nbut only that it expands a group.\n\\begin{lemma} \\label{REM:MINIMA} Let $M$ be an o-minimal expansion of a group. \nLet $Y \\subseteq M^n$ be definable, and $f: Y \\to M$ be a definable function \\rom(not necessarily continuous\\rom).\nLet $A \\subseteq Y$ be the set of local minima of~$f$.\nThen, $f(A)$ is finite.\n\\end{lemma} \n\\begin{proof}\nIf not, let $a < b \\in M$ such that, for every $t \\in (a,b)$ there\nexists $\\gamma(t) \\in Y$ such that $f(\\gamma(t)) = t$ and $\\gamma(t)$\nis a local minimum for~$f$. By definable choice, we can assume that\n$\\gamma$ is a definable continuous function. It follows that in\n any neighbourhood of $\\gamma(t)$ there are points of the\nform $\\gamma(t')$ with $t' 0}$ be a decreasing definable family of closed subsets of~$M^n$.\nThen, there exists $t_0 >0$ such that, for every $t \\in M$ with $0 < t < t_0$ we have\n\\[\nY_t = \\bigcap_{t < u < t_0} Y_u\n\\]\n\\end{lemma}\n\\begin{proof}\nLet $Y := \\bigcup_{t > 0} Y_t$.\nDefine\n\\begin{align*}\nf &: Y \\to M^{\\geq 0}\\\\\nf(x) &:= \\inf \\set{t: x \\in Y_t}.\n\\end{align*}\nDefine also $Z_t := f^{-1}([0,t])$.\nNote that $Z_t = \\bigcap_{t < u < t_0} Y_u$, and\n$Y_t \\subseteq Z_t$.\nTherefore,\nthe conclusion is equivalent to saying that, for every $0 < t < t_0$, we have that $Y_t = Z_t$.\n\n\\begin{claim}\nFor every $t \\geq 0$, \nif $x \\in Z_t \\setminus Y_t$, then $x$ is a local minimum for~$f$.\n\\end{claim}\nIf not, let $x \\in Z_t \\setminus Y_t$ such that $x$ is not a local minimum for~$f$. Note that $f(x) = t$.\nLet $\\gamma: (0, \\varepsilon) \\to Y$ be a definable function such that $\\lim_{s \\to 0}\\gamma_s = x$ and $f(\\gamma_s) < f(x)$.\nThen, $\\gamma_s \\in Y_t$.\nSince $Y_t$ is closed, we have that $x \\in Y_t$, absurd.\n\nBy Lemma~\\ref{REM:MINIMA}, there exists $t_0 >0$ such that, for every \n$x \\in Y$, if $0 < f(x) < t_0$, then $x$ is not a local minimum for~$f$.\nThe claim implies the conclusion.\n\\end{proof}\n\n\\begin{remark}\nA few remarks about the above Lemma and its proof.\n\\begin{enumerate}\n\\item The function $f$ is lower semi-continuous, because $Z_t$ is closed for every~$t$.\n\\item The hypothesis that the $Y_t$ are closed is necessary.\n\\item Even if the function $f$ in the proof is continuous, we cannot conclude that $Y_t = Z_t$ always.\n\\item It is not true that there exists $t_0 > 0$ such that $f$ is continuous on~$Y_{t_0}$.\n\\end{enumerate}\n\\end{remark} \n\n\\begin{proof}[Proof of Theorem~\\ref{THM:DEFINABLE}]\nBy Corollary~\\ref{LEM:TAUT-DEFINABLE},\n\\[\n\\Hom^*(\\widetilde A; {\\ca F}) = \\varinjlim_{t \\to 0} \\Hom^*(\\widetilde{Y_t}; {\\ca F}).\n\\]\nSince $\\Pa{Y_t}_{t > 0}$ is a definable family, and $M$ expands a\nfield, the (definable) topological type of the $Y_t$ is eventually\nconstant (this is a consequence of the trivialization theorem holding\nin o-minimal expansions of fields, see \\cite{Dries98}). Hence we can\nassume w.l.o.g.\\xspace\\ that the $Y_t$ are all definably homeomorphic, and\ntherefore that their cohomology groups over $G$ are isomorphic to each\nother; let us denote by $P$ such group. Thus, for every $t \\leq t'$,\nthe map induced by the inclusion\n\\[\n\\phi^{t'}_{t}: \\Hom^*(\\widetilde{Y}_{t'}; {\\ca F}) \\to \\Hom^*(\\widetilde{Y}_{t}; {\\ca F})\n\\]\nis an endomorphism of~$P$.\nIt suffices to prove that $\\phi^{t'}_{t}$ is an isomorphism for all sufficiently small $t'$ to prove the theorem.\nBy Lemma~\\ref{LEM:FAMILY}, w.l.o.g.\\xspace we can assume that, for every $t >0$,\n\\[\nY_t = \\bigcap_{t' >t} Y_{t'}.\n\\]\n\nFor convenience, define $\\phi^{t'}_t := \\phi^{t}_{t'}$ if $t' \\leq t$, and $I := M^{>0}$.\nLet $E$ be the equivalence relation on $I$ given by\n\\[\nt E t' \\Leftrightarrow \\phi^{t'}_{t} \\text{is an isomorphism}.\n\\]\n\\begin{claim}\n$E$~is a definable subset of $I^2$.\n\\end{claim}\nIn fact, by the trivialization theorem, there exists a finite\npartition $\\{ C_1, \\dotsc, C_n \\}$ of $\\{(t,t') \\in M^2: 0 < t \\leq\nt' \\}$ into definable sets, such that the definable topological type of\nthe inclusion map $Y_t \\to Y_{t'}$ is constant on each~$C_k$.\nHence, the isomorphism type of the map $\\phi^{t'}_t$ is constant on each\n$C_k$, and thus $E$ is a finite union of some of the~$C_k$.\n\n\\begin{claim}\\label{CL:OPEN}\nFor every $t \\in I$ there exists $t_0 >t$ such that, for every $t' \\in I$,\n\\[\nt \\leq t' < t_0 \\Rightarrow t E t'.\n\\]\n\\end{claim}\\noindent\nSince $Y_t = \\bigcap_{t' >t} Y_{t'}$, by Corollary\n\\ref{LEM:TAUT-DEFINABLE} we have $\\Hom^*(\\widetilde{Y_t}; {\\ca F}) =$%\n\\linebreak[2]%\n\\raisebox{0pt}[2ex][0pt]{$\\displaystyle{\\varinjlim_{t' \\to t^+}} \\Hom^*(\\widetilde{Y}_{t'}; {\\ca F})$};\nnamely,\n\\[\nP = \\varinjlim_{t' \\to t^+} \\set{P, \\phi^{t'}_{t}}.\n\\]\nBy definition of inductive limit, for every $p \\in P$ there exists $t'>t$ and $p' \\in P$ such that $p = \\phi^{t'}_{t}(p')$.\nHowever, $P$ is finitely generated, and therefore there exists $t_0 >t$ such that $\\phi^{t'}_t$ is surjective for every $t_0 > t' >t$.\nSince $P$ is finitely generated, using Nakayama's lemma, we can conclude that $\\phi^{t'}_t$ is an isomorphism.\n\n\\begin{claim}\\label{CL:E-FINITE}\n$I\/E$ is finite\n\\end{claim}\nClaim~\\ref{CL:OPEN} implies that each equivalence class of $E$ has\nnon-empty interior. Since $E$ is definable, $I\/E$ must be finite.\n\nIt follows from Claim~\\ref{CL:E-FINITE} that there exists a left\nneighbourhood $J$ of $0$ such that, for every $t,t' \\in J$, $t E t'$.\nTherefore, for every $t \\leq t'$ in that neighbourhood $\\phi^{t'}_t$\nis an isomorphism, and we conclude by Remark~\\ref{lim}.\n\\end{proof}\n\n\\begin{remark}\nIn the situation of Theorem~\\ref{THM:DEFINABLE}, let\n\\begin{align*}\nf &: Y \\to M^{\\geq 0}\\\\\nf(x) &:= \\inf \\set{t: x \\in Y_t}.\n\\end{align*}\nWe have remarked that $f$ might not be continuous.\nAssume that there exists $t_0 >0$ such that $f$ is continuous on $Y_{t_0}$.\nThen, the proof of Theorem~\\ref{THM:DEFINABLE} can be simplified.\nIn fact, by the trivialization theorem, we can assume that there exists a continuous definable map $\\lambda: Y_{t_0} \\setminus A \\to F$ (where $F := f^{-1}(t_0)$), such that the map\n\\begin{align*}\n\\mu:= (f,\\lambda) &: Y_{t_0}\\setminus A \\to (0,t_0) \\times F\n\\end{align*}\nis a homeomorphism.\nLet $0 < t \\leq t' \\leq t_0$.\nLet $\\theta: [0,1] \\times [0,t'] \\to [0,t']$ be a definable \nstrong deformation retraction between $[0,t']$ and $[0,t]$.\nDefine\n\\[\\begin{array}{r@{\\,}ll}\n\\Lambda: [0,1] \\times Y_{t'} &\\to Y_{t'}\\\\\n\\Lambda\\Pa{s, \\mu^{-1}(u,x)} &= \\mu^{-1}\\Pa{\\theta(s,u),x} &\\text{if } 0 0$ such that $A$ is a deformation retract of $Y_t$ for every $0\n< t < t_0$? \\end{question} \n\n\nNote that in the proof of Theorem~\\ref{THM:DEFINABLE}\nwe have used heavily the trivialization theorem, which holds only for\nexpansions of fields. \n\n\\begin{question} Can we weaken the hypothesis of\nTheorem~\\ref{THM:DEFINABLE} dropping the condition that $M$ expands a\nfield \\rom(but saying instead that it expands a group\\rom)? \\end{question}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n Resolved stellar populations and star formation histories (SFHs) provide a powerful tool to understand galaxy formation and evolution in all scales (e.g., Cignoni \\& Tosi \\cite{sl_cignoni10}). Detailed SFHs of Local Group (LG) dwarf galaxies have shown that these are complex systems (e.g., Mateo \\cite{sl_mateo98}; Tolstoy, Hill \\& Tosi \\cite{sl_tolstoy09}). The complexity of SFHs observed in LG dwarfs has been explored by means of numerical simulations and the results show that their diversity can be understood by invoking the action of ram pressure stripping, tidal stirring, and\/or cosmic ultraviolet (UV) background radiation upon gas-rich dwarf galaxies (e.g., Mayer \\cite{sl_mayer10}; Kazantzidis et al.~\\cite{sl_kazantzidis11}; Nichols \\& Bland-Hawthorn \\cite{sl_nichols11}; and references therein). From an observational point of view, the comparison of SFHs between dwarf galaxies in the LG and outside the LG shows that the former are representative of their kind (Weisz et al.~\\cite{sl_weisz11a}), so that a deeper study of individual SFHs is valuable to uncover and understand the intrinsic and environmental differences affecting their evolution (e.g., Weisz et al.~\\cite{sl_weisz11b}). \n\n One such interesting galaxy to focus on is the Sculptor Dwarf Irregular Galaxy (SDIG). First discovered by Laustsen et al.~(\\cite{sl_laustsen77}), SDIG is a low-luminosity (Heisler et al.~\\cite{sl_heisler97}), gas-rich dwarf irregular (dIrr) galaxy (Cesarsky, Falgarone \\& Lequeux \\cite{sl_cesarsky77}; Lequeux \\& West \\cite{sl_lequeux81}; C\\^ot\\'e et al.~\\cite{sl_cote97}). Atomic hydrogen observations show that SDIG has a smooth distribution without the presence of structures such as shells, holes or bubbles (C\\^ot\\'e et al.~\\cite{sl_cote00}). SDIG contains no $\\ion {H}{ii}$ regions and this is suggestive of a quiescent phase with very low current star formation rate (SFR; Miller \\cite{sl_miller96}; Heisler et al.~\\cite{sl_heisler97}; Skillman, C\\^ot\\'e \\& Miller \\cite{sl_skillman03}). Bouchard et al.~(\\cite{sl_bouchard09}) detect two H$\\alpha$ point sources and estimate a very low current SFR (3.8$\\times$10$^{-5}$~M$_{\\sun}$ yr$^{-1}$); however, Lee et al.~(\\cite{sl_lee11}) detect far-ultraviolet (FUV) and near-ultraviolet (NUV) emission that appears to have a clumpy morphology and suggest main-sequence B stars as the source of this emission. \n\n With a distance of 3.2~Mpc and based on its radial velocity, SDIG is a member of a loose galaxy triplet (Karachentsev et al.~\\cite{sl_karachentsev06}), of which the other two galaxy members are NGC\\,7793 at a distance of 3.61$\\pm$0.53~Mpc (Vlajic, Bland-Hawthorn \\& Freeman \\cite{sl_vlajic11}) and UGCA442 at a distance of 3.8$^{+0.8} _{-0.3}$~Mpc (Mould \\cite{sl_mould05}). This triplet, the NGC\\,7793 subgroup (Karachentsev et al.~\\cite{sl_karachentsev03}) lies at the far side of the Sculptor group. SDIG therefore lies near the limit of the ACS Nearby Galaxy Survey Treasury galaxy sample (ANGST; Dalcanton et al.~\\cite{sl_dalcanton09}), so a study of the SFH of this subgroup represents an opportunity to learn more about the Sculptor group, a low-density and extended filamentary structure with several embedded subgroups (e.g., Jerjen, Freeman \\& Binggeli \\cite{sl_jerjen98}). The filamentary structure of the Sculptor group extends from 1.5~Mpc to 4~Mpc, and points to an unvirialised state, with galaxies still falling in (Jerjen, Freeman \\& Binggeli \\cite{sl_jerjen98,sl_jerjen00}; Karachentsev \\cite{sl_karachentsev05}). The early evolutionary state and the low density of the Sculptor group provide an ideal environment in which its galaxy members may be considered as still evolving in isolation.\n\n In the present work, we examine the resolved stellar populations and model the color-magnitude diagram (CMD) of SDIG using archival Hubble Space Telescope (HST) observations. The structure of this work is as follows. In \\S2 we present the observations and photometry, including photometric error analysis. In \\S3 we show the CMD and discuss the stellar content of SDIG, including an examination of the spatial distribution of stellar populations selected in several stellar evolutionary phases. In \\S4 we briefly review the CMD modelling technique. In \\S5 we show the results for the SFH of SDIG. We summarise and discuss our findings in \\S6.\n\n\\section{Observations and photometry}\n\n We use archival observations taken with the Advanced Camera for Surveys (SNAP\\,9771; Karachentsev et al.~\\cite{sl_karachentsev06}) with a total exposure time of 900~sec in the F814W-band, and 1200~sec in the F606W-band. We retrieve the pre-reduced images through the ST-ECF Hubble Science Archive and perform stellar point source photometry using the ACS module of DOLPHOT, a modified version of the photometry package HSTphot (Dolphin \\cite{sl_dolphin00}). We apply point source photometry simultaneously on all the individual, calibrated and flat-fielded \"FLT\" images, and we use as a reference image the drizzled image in the F814W filter. We proceed with the photometry reduction steps as described in the ACS module of DOLPHOT. DOLPHOT provides magnitudes in both the ACS\\,\/\\,WFC and the Landolt UBVRI photometric systems and we choose to use the ACS\\,\/\\,WFC filter system for our study. The final photometric catalogue includes 11363~stars selected to have ${\\rm S\/N} >$~5, sharpness $|$sharp$_{\\rm F606W}+$sharp$_{\\rm F814W}|<$1, and crowding $($crowd$_{\\rm F606W}+$crowd$_{\\rm F814W})<$1 (e.g., Williams et al.~\\cite{sl_williams09}). \n\n We conduct artificial star tests in order to quantify the photometric errors and completeness of our data, using the utilities provided within DOLPHOT. To that end, we run artificial star tests with 10$^5$~stars per ACS\\,\/\\,WFC chip, with range in F606W from 21~mag to 30~mag, and in (F606W-F814W) from $-$1.5~mag to 3.5~mag. In order to better quantify the photometric errors in the faintest magnitudes, which we need for the CMD modelling, we run complementary artificial star tests with an additional number of 2$\\times$10$^5$ stars per ACS\\,\/\\,WFC chip and covering magnitudes fainter than 25~mag. The high number of stars inserted during the artificial star tests is not an issue for self-induced crowding, since each star is inserted and photometered one at a time (Dolphin \\cite{sl_dolphin00}; see also Perina et al.~\\cite{sl_perina09}). \n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm,clip]{20193fg1a.eps}\n \\includegraphics[width=8cm,clip]{20193fg1b.eps}\n \\caption{Photometric errors derived from artificial star tests and estimated as the difference between output and input magnitudes, as a function of the output magnitude in the F814W filter (upper panel) and in the F606W filter (lower panel).}\n \\label{sl_figure1}%\n \\end{figure}\n The photometric errors derived using the artificial star tests are shown in Fig.~\\ref{sl_figure1}. The mean photometric error and standard deviation is 0.03$\\pm$0.1~mag in F814W, and 0.02$\\pm$0.09~mag in F606W. \n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm,clip]{20193fg2a.eps}\n \\includegraphics[width=8cm,clip]{20193fg2b.eps}\n \\caption{Completeness factors as a function of the F606W-band magnitude, upper panel, and F814W-band magnitude, lower panel. The red arrow in each panel indicates the 50\\% completeness factor.}\n \\label{sl_figure2}%\n \\end{figure}\nIn Fig.~\\ref{sl_figure2} we show the completeness factors as a function of magnitude. The 50\\% completeness factor occurs at the magnitude F606W~$=\\sim$26.9~mag, and F814W~$=\\sim$25.9~mag. \n\n\\begin{table}[t]\n \\caption[]{Properties of SDIG.}\n \\label{table1} \n \\begin{tabular}{lc}\n\\hline\\hline\n Quantity & Value \\\\\n\\hline\n Type & dIrr \\\\\n RA~(J2000.0) & $00^h08^m13.36^s$ \\\\\n Dec~(J2000.0) & $-34^\\circ 34' 42.0''$ \\\\\n $\\rm E(B-V)$~(mag) & 0.012 \\\\\n $A_{\\rm F606W}, A_{\\rm F814W}$~(mag) & 0.034, 0.022 \\\\\n $F814W_{\\rm TRGB}$~(mag) & 23.44$\\pm$0.05 \\\\\n $(m-M)_0$~(mag) & 27.51$\\pm$0.06 \\\\\n Distance~(Mpc) &3.2$\\pm$0.1 \\\\\n M$_{\\rm V}$~(mag) &$-11.87$ \\\\\n M$_{\\rm {\\ion {H}{i}}}$~(M$_{\\sun}$) &2.5$\\times$10$^7$ \\\\\n x$_{\\rm C}$~(arcsec) & 105.1$\\pm$1.7 \\\\\n y$_{\\rm C}$~(arcsec) & 114.5$\\pm$1.3 \\\\\n $e$ & $0.05$ \\\\\n PA~(degrees) & 0 \\\\\n $r_{\\rm eff}$~(arcsec) & $29.4\\pm1.6$ \\\\\n\\hline\n\\end{tabular} \n\\end{table}\n Table~\\ref{table1} lists the basic properties for SDIG. Each row shows: (1) the galaxy type; (2) and (3) equatorial coordinates of the field center (J2000.0); (4) the reddening adopted from Schlegel et al.~(\\cite{sl_schlegel98}); (5) the galactic foreground extinction in the F606W and F814W filters, computed using the extinction ratios for a G2 star listed in Sirianni et al.~(\\cite{sl_sirianni05}; their Table~14); (6) the magnitude of the tip of the red giant branch (TRGB) we derive from the data; (7) the distance modulus we derive from the data; (8) the derived distance in Mpc; (9) the absolute V-band magnitude adopted from Georgiev et al.~(\\cite{sl_georgiev09}), who use the distance of Karachentsev et al.~\\cite{sl_karachentsev06}; (10) the {\\ion {H}{i}} mass adopted from Koribalski et al.~(\\cite{sl_koribalski04}); (11) and (12) the magnitude-weighted coordinates of the center, x$_{\\rm C}$ and y$_{\\rm C}$, respectively, which we compute from the data (see $\\S$3); (13) and (14) the ellipticity and position angle adopted from Kirby et al.~(\\cite{sl_kirby08}); (15) the effective radius adopted from Kirby et al.~(\\cite{sl_kirby08}).\n\n\\section{Color-magnitude diagram and stellar content}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm,clip]{20193fg3.eps}\n \\caption{Color-magnitude diagram of SDIG. The gray dashed line shows the 50\\% completeness factors. The error-bars correspond to the photometric errors derived using artificial star tests.}\n \\label{sl_figure3}%\n \\end{figure}\n The CMD of SDIG shown in Fig.~\\ref{sl_figure3} reveals the presence of several stellar evolutionary features. The most prominent feature is the red giant branch (RGB). The RGB indicates that stars with ages from $\\sim$1.5~Gyr and older are present (e.g, Salaris, Cassisi \\& Weiss \\cite{sl_salaris02}). Intermediate-age stars, with ages between 1~Gyr and 10~Gyr, are revealed in the presence of luminous asymptotic giant branch (AGB) stars, while the young main-sequence (MS) and blue-loop (BL) stars indicate the presence of stars with ages younger than 1~Gyr. Therefore, SDIG contains stars spanning a wide range in age, from $\\sim$Myr to older ages. \n\n In the following analyses and unless otherwise noted, we use the TRGB magnitude and distance modulus we measure in the ACS\\,\/\\,WFC filter system. We derive the distance modulus for SDIG using the absolute F814W-band magnitude of the TRGB computed through the calibration of Rizzi et al.~(\\cite{sl_rizzi07}): $M_{\\rm F814W} = -4.06 + 0.20\\times [({\\rm F606W-F814W}) - 1.23]$, and the F814W-band magnitude of the TRGB computed through the Sobel-filtering technique on the F814W-band luminosity function (e.g., Lee, Freedman \\& Madore \\cite{sl_lee93}; Sakai et al.~\\cite{sl_sakai96}). We find a TRGB F814W-band magnitude of 23.44$\\pm$0.05~mag, while measuring a dereddened color at the TRGB magnitude level of 1.05$\\pm$0.03~mag, we derive ${\\rm M}_{\\rm F814W}=-$4.09$\\pm$0.02~mag. The quoted uncertainties take into account the photometric errors from the artificial star tests and the width of the Sobel filter response. These values along with the extinction listed in Table~\\ref{table1} give a distance modulus of 27.51$\\pm$0.06~mag, which translates to a distance of 3.2$\\pm$0.1~Mpc. The quoted uncertainties take into account the width of the Sobel filter response for the TRGB magnitude detection, uncertainties related to the TRGB calibration, and the photometric errors. Our derived distance modulus and distance are consistent with the values of 27.53~mag and 3.21~Mpc, respectively, found in Karachentsev et al.~(\\cite{sl_karachentsev06}), who use the same method on the transformed V and I filters and assign an uncertainty to their distance of 8\\%. We list the derived F814W-band magnitude of the TRGB, distance modulus, and distance in Table~\\ref{table1}.\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=6cm,clip]{20193fg4a.eps}\n \\includegraphics[width=6cm,clip]{20193fg4b.eps}\n \\includegraphics[width=6cm,clip]{20193fg4c.eps}\n \\caption{Color-magnitude diagrams within three elliptical radius annuli. The first annulus corresponds to an elliptical radius equal to 1~$r_{\\rm eff}$, the second between 1 and 2~$r_{\\rm eff}$, and the third beyond 2~$r_{\\rm eff}$, shown from left to right, respectively. In the left panel, we overplot Padova isochrones with ages of 20~Myr, 40~Myr, 60~Myr, 80~Myr, and 150~Myr, and for all isochrones a Z metallicity of 0.001. Also shown in the left panel with the error bars the photometric errors. In the right panel, we show with open circles the galactic foreground contamination.}\n \\label{sl_figure4}%\n \\end{figure*}\n Fig.~\\ref{sl_figure4} shows the CMD of stars distributed within three elliptical radius annuli: within 1 effective radius, $r_{\\rm eff}$; between 1~$r_{\\rm eff}$ and 2~$r_{\\rm eff}$; and beyond 2~$r_{\\rm eff}$. The elliptical radius is defined as: $r = [{\\rm x}^{2}+{\\rm y}^{2}\/(1 - e)^{2}]^{1\/2}$, where x and y are the distances along the major and minor axis, respectively, and $e$ is the ellipticity. We adopt the ellipticity, position angle, and effective radius from Kirby et al.~(\\cite{sl_kirby08}), listed in Table~\\ref{table1}. Also listed in Table~\\ref{table1} are the x$_{\\rm C}$ and y$_{\\rm C}$ coordinates of the magnitude-weighted center of SDIG, which we compute from our data using the equation: ${\\rm x_{C}}=\\Sigma({\\rm mag}_{i}\\times {\\rm x}_{i})\/\\Sigma {\\rm mag}_{i}$, where ${\\rm mag}_{i}$ corresponds to the F814W-band magnitude of each star, and ${\\rm x}_{i}$ to their x (or y for the y$_{\\rm C}$) coordinates, in pixels. We use stars with magnitudes brighter than the magnitude with 50\\% completeness factors. In the right panel of Fig.~\\ref{sl_figure4} we show with the open circles the Galactic foreground contamination, which was estimated using the TRILEGAL code (Girardi et al.~\\cite{sl_girardi05}; Vanhollebeke, Groenewegen \\& Girardi \\cite{sl_vanhollebeke09}). We expect a total of 58 foreground stars in the same color and magnitude location of the SDIG stars, which translates to less than 0.5\\% of the total number of SDIG stars. In the left panel of Fig.~\\ref{sl_figure4} we overplot Padova isochrones (Marigo et al.~\\cite{sl_marigo08}; Girardi et al.~\\cite{sl_girardi10}), with ages between 20~Myr and 150~Myr. Fig.~\\ref{sl_figure4} indicates that the bulk of the young MS stars are confined within the central parts of SDIG. \n\n We examine the cumulative distribution functions of the MS, BL, luminous AGB, and RGB stars, versus the elliptical radius, r, defined in the preceding paragraph. We select those MS stars with their color ranging from $-$0.5~mag to $-$0.1~mag and their F814W-band magnitude ranging from 23.8~mag to 25.25~mag; the BL stars are selected with their color ranging from $-$0.05~mag to 0.35~mag and with their F814W-band magnitude ranging from 21~mag to 25~mag. The RGB stars are selected between the magnitude of the TRGB and 1~mag below, while the color selection ranges such as to follow the slope of the RGB. The luminous AGB stars are selected to be brighter by 0.15~mag than the TRGB (Armandroff et al.~\\cite{sl_armandroff93}) and to lie within 1~mag above (${\\rm I_{TRGB}}-0.15$)~mag, and within the color range of 1~$<({\\rm V-I})_0<$~3.5~(mag). We count a number of 80 luminous AGB stars, which translates into a fraction, i.e. number of luminous AGB stars versus the number of RGB stars within 1~mag below the TRGB magnitude, N(AGB)~\/~N(RGB)=13\\%, comparable to that of Sculptor group dwarf galaxies (Lianou et al.~\\cite{sl_lianou12}).\n \\begin{figure}\n \\centering\n \\includegraphics[width=6cm,clip]{20193fg5.eps}\n \\caption{Cumulative distribution functions for the selected stellar populations. The vertical line shows the effective radius, r$_{eff}$.} \n \\label{sl_figure5}%\n \\end{figure}\nThe cumulative distribution functions in Fig.~\\ref{sl_figure5} show that the MS and BL stars are confined to the central parts of SDIG, while the luminous AGB and the RGB stars are more spatially extended. Performing a two-sided Kolmogorov-Smirnov (K-S) test between each of the cumulative distributions for the MS, BL, and luminous AGB stars with the cumulative distribution of the RGB stars, we derive results consistent with spatially separated populations at the 99\\% confidence level in all cases.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=4.4cm,clip]{20193fg6a.eps}\n \\includegraphics[width=4.4cm,clip]{20193fg6b.eps}\n \\includegraphics[width=4.4cm,clip]{20193fg6c.eps}\n \\includegraphics[width=4.4cm,clip]{20193fg6d.eps}\n \\caption{Gaussian-smoothed stellar density maps for stars selected in the MS, BL, luminous AGB, and RGB phases, expressed in terms of angular offsets from the dwarf's center. The density maps are color coded such that black corresponds to a density of 0.16 stars per sq.~arcsec and the density declines toward lighter grey. The ellipse has a major axis equal to 1~$r_{\\rm eff}$, while the crosses emphasise the location of the center of the dwarf. The red arrows in the panel of the RGB population indicates the directions of North and East, and is the same for all panels.}\n \\label{sl_figure6}%\n \\end{figure}\n Comparing the spatial distribution of the luminous AGB stars with that of the RGB stars, in the stellar density maps shown in Fig.~\\ref{sl_figure6}, the former are more confined in the inner parts of SDIG. The selected BL stars are confined to the central parts of the dwarf when we compare their spatial distribution with the one of the luminous AGB and RGB stars. The stellar density map of the young MS stars shows a population concentrated towards the central parts of SDIG, while in the same time its spatial location is not coincident with the very center of SDIG, having an offset of 8.2~arcsecs, or 130~pc at a distance of 3.2~Mpc. We note that a similar characteristic of off-center local sites of young star formation characterise the majority of the transition-type dwarfs in the Sculptor group (Jerjen \\& Rejkuba \\cite{sl_jerjen01}; Lianou et al.~\\cite{sl_lianou12}), while the same is also observed for the young stellar populations of other LG dwarf galaxies (e.g, Dohm-Palmer et al.~\\cite{sl_dohm-palmer97}; Gallagher et al.~\\cite{sl_gallagher98}; Mateo \\cite{sl_mateo98}; Cole et al.~\\cite{sl_cole99}; McConnachie et al.~\\cite{sl_mcconnachie06}).\n\n\\section{Color-magnitude diagram modelling}\n\n The basic idea using the modelling of the CMD to uncover the evolutionary history of a galaxy is that its composite stellar populations can be regarded as a linear combination of simple stellar populations of different ages and metallicities. Thus, the SFH can be obtained by comparing the observed features in the CMD with those in a model CMD. There are several features in a CMD that can be used as age and metallicity indicators (e.g., Gallart, Zoccali \\& Aparicio \\cite{sl_gallart05}). Synthetic populations are constructed using a set of stellar evolution models, adopting an initial mass function (IMF) and applying a star formation and a chemical enrichment law (e.g., Gallart, Zoccali \\& Aparicio \\cite{sl_gallart05}; Cignoni \\& Tosi \\cite{sl_cignoni10}). In practice, the density of stars within a defined color and magnitude grid along the observed CMD is compared with the density of the synthetic stars drawn from the model CMD (for a review see Tolstoy, Hill \\& Tosi \\cite{sl_tolstoy09}; Aparicio \\cite{sl_aparicio02}), and the best model is adopted as the SFH. \n\n The approach taken here was developed by Cole to study the stellar populations of dIrr galaxies in the LG (see, e.g., Skillman et al.~\\cite{sl_skillman03b}; Cole et al.~\\cite{sl_cole07}; Cignoni et al.~\\cite{sl_cignoni12}). As with any such approach, the power of the method is greatly reduced by lack of photometric depth (e.g., Weisz et al.~\\cite{sl_weisz11b}). Detailed inferences about the SFH will be restricted to age ranges for which the age-sensitive MS, BL, and to a lesser extent AGB stars are visible. It is still possible to place constraints on long-term average SFRs and mean metallicities based on inferences from the RGB and the regions of the CMD that are badly affected by incompleteness. However, these inferences are by necessity more model-dependent and of lower precision, and the resulting uncertainties are dominated by systematic effects.\n\n The SFH-fitting starts with theoretical isochrones interpolated to a fine grid of age and metallicity in order to create a synthetic CMD with no gaps. The most recent models from the Padova models are used. The synthetic CMDs are binned in age and metallicity (Z) to avoid ``overfitting'' noise in the CMDs. The bins are initially spaced by a default value of $\\Delta$log(age) = 0.10, but adjacent bins are merged when lack of information in the CMDs leads to instability in the solutions. In this case, ages greater than 1~Gyr are only represented on the RGB and AGB, and the age resolution is decreased to $\\Delta$log(age) = 0.30 (factor of two). There are not many stars younger than 30~Myr in the CMD, so the youngest age bin spans the entire range from 4--30~Myr as a way to populate the CMD bins sufficiently to draw SFH inferences. \n\n The CMD is divided into a regular grid of color-magnitude cells, and the expectation value of the number of stars in each cell for a SFR of 1 M$_{\\odot}$ yr$^{-1}$ is calculated from the isochrones. Several parameters are assumed to be fixed during the solution: the distance modulus and reddening, IMF, fraction of binaries, and the binary mass ratio distribution function. The adopted IMF is from Chabrier (\\cite{sl_chabrier03}), and the binary fraction and mass ratios are parameterized based on Duquennoy \\& Mayor (\\cite{sl_duquennoy91}) and Mazeh et al.~(\\cite{sl_mazeh92}). We take 35\\% of stars to be single and the rest to be binary. The binaries are divided into ``wide'' and ``close'' binaries in a 3:1 ratio; the secondaries in the wide systems are drawn from the same IMF as the primaries, but in the close systems the secondary masses are drawn from a flat IMF. The distance and reddening are initially constrained to the values given in Table~1, but are varied if the resulting synthetic CMDs are mismatched to the data.\n\n No age-metallicity relation is explicitly assumed, but a range of metallicities at each age is allowed, constrained by the color range of the data. Because of the shallow photometric depth of the data, there are few strong constraints on the metallicity, and those that exist are model-dependent. The RGB is affected by age-metallicity degeneracy, but indicates a range of metallicities between 0.0004 $\\leq$ Z $\\leq$ 0.002. These numbers are consistent with the Padova AGB star tracks, and the color separation between the BL stars and MS. There is no metallicity constraint from $\\ion {H}{ii}$ regions, but the inferred values are consistent with the metallicity-luminosity relationship for dIrr galaxies in Lee, Zucker \\& Grebel (\\cite{sl_lee07}) (see discussion below).\n\n The IMF-weighted, color-magnitude binned isochrones are convolved with color and magnitude error distributions derived from the artificial star tests in order to create synthetic CMDs with the same properties as the data. Linear combinations of the synthetic CMDs are created by the fitting routine, which uses a maximum likelihood test based on the Poisson distribution of counts in each bin (Cash \\cite{sl_cash79}) to find the combination that is most likely to produce the observed CMD. We use a simulated annealing technique to find the best fit while avoiding false, local maxima in likelihood space. The SFH errorbars are calculated by successive perturbation of each age-metallicity bin; because the total number of stars is fixed by the observations, a decrease in SFR in one bin results in an increase in adjacent bins; this gives anticorrelations between adjacent bins. In general, the code is driven by the most populous cells of the CMD to its preferred solution; because most stars are observed on the MS and low-mass stars vastly outnumber those of higher mass, the most significant cells in determining the SFH are frequently below the 50\\% completeness threshold, and the solution depends sensitively on the artificial star test results.\n\n Our data are not deep enough to reach the oldest MS turn-offs, and due to the age-metallicity degeneracy on the RGB, the solution of the CMD modelling for ages higher than $\\sim$1.5~Gyr bears larger uncertainties. \n \\begin{figure*}\n \\centering\n \\includegraphics[width=17cm, clip]{20193fg7.eps}\n \\caption{Left panel: logarithmically-scaled, binned CMD used in the maximum-likelihood fit for SFH; right panel: best-fit model. Without differential reddening, there is clear separation between the MS and BL stars, which are mingled together in the data. The AGB stands out as being poorly-fit; the mismatch occurs because of the large number of stars fainter than m814 $\\approx$26, which are part of the same stellar population as the AGB stars.}\n \\label{sl_figure7}%\n \\end{figure*}\nThe observed and modelled CMDs for SDIG are shown in Fig.~\\ref{sl_figure7}. From the CMD modelling, we derive a distance modulus of 27.54~mag, while the reddening required is E(B-V)=0.04. The higher reddening, as compared to the value from Schlegel et al.~(\\cite{sl_schlegel98}), is consistent with the one that Heisler et al.~(\\cite{sl_heisler97}) derive for the very central parts of SDIG.\n\n The best-fit solution under the restrictions of the fit procedure matches the mean stellar density and colors of the major stellar sequences well, but there are some notable differences between the simulated CMD and the data. The MS and BL in the data are blurred, with no clear color gap between them; however, in the models these features are well-separated. Presumably this is due to differential reddening, with a tendency for young stars to be associated with dust clouds and circumstellar material. \n\n \\section{Star formation history}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm,clip]{20193fg8a.eps}\n \\includegraphics[width=8cm,clip]{20193fg8b.eps}\n \\caption{Star formation rate as a function of age for SDIG based on the maximum-likelihood fit, shown in the upper panel with a linear scale on age, and in the lower panel with a logarithmic scale with age. Note in the upper panel the break in age scale at 750~Myr that reflects the decreasing time resolution of the CMD with increasing age.}\n \\label{sl_figure8}%\n \\end{figure}\n We show the best-fit SFH for SDIG in Fig.~\\ref{sl_figure8}. Overall, the best-fit SFH is characterised by the following features:\n\n\\begin{itemize}\n\n\\item An average age for formation of stars of $\\langle \\tau \\rangle$ = 6.4$^{+1.6}_{-1.4}$~Gyr and a total astrated mass (time-integrated SFR) of 17.7$\\times$10$^6$M$_{\\sun}$.\n\n\\item The average metallicity is inferred to be Z $\\approx$0.0006 ([M\/H] $\\approx$ $-$1.5~dex), with some tendency to increase with time, as the BL stars are more consistent with a metallicity closer to Z = 0.001.\n\n\\item A long-term SFH that is consistent with a constant SFR over several Gyr. The average SFR is 1.3$^{+0.4}_{-0.3}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$. Any bursts or fallow periods are of short enough duration and\/or amplitude that they do not strongly perturb the mean SFR averaged over factor of two periods in age. This is broadly consistent with the typical dIrr galaxy found in Weisz et al.~(\\cite{sl_weisz11b}); however the HST SDIG observations analysed here were not deep enough to reveal factor of 2--3 variation in SFR over sub-Gyr intervals at intermediate or old ages, and these are not excluded by the CMD analysis.\n\n\\item The age resolution improves dramatically for ages less than $\\approx$800--1000~Myr, but strong variations over intervals of a few hundred Myr are not observed, with two exceptions:\n\n\\begin{enumerate}\n\\item Over the past 100~Myr the mean SFR increases by a factor of $\\approx$2 over its long-term average value, but then declines slightly. Variations of this duration and amplitude would be completely hidden by photometric incompleteness and age-metallicity degeneracy for ages older than $\\approx$1~Gyr; this behaviour may be typical of the history of low-luminosity dIrrs (e.g., Cole \\cite{sl_cole10}). Because the fraction of evolved stars is low, the derived SFR is susceptible to uncertainties in the modelling of incompleteness and possible variations in the IMF. Stars in this age range would contribute to the SFR as measured from FUV data, but not to the H$\\alpha$ luminosity owing to the low absolute SFR with a value of $\\sim$2.5$^{+0.7}_{-0.6}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$ from 0--10~Myr.\n\n\\item A similar-amplitude SFR enhancement is inferred to have taken place from 600--1100~Myr ago. The younger stars in this interval are directly observable in the CMD as BL stars at F606W-band magnitude $\\approx$ 26.5--27~mag on the blue side of the RGB. The older stars are confused with the RGB and highly subject to incompleteness, resulting in the large uncertainties in the SFR at this age range. Because of the high incompleteness, the best-fit SFR has high uncertainty (reflected in the model overproduction of AGB stars associated with this age range). Based on this data alone the balance of evidence favours the existence of a $\\lesssim$500-Myr long episode of increased SFR $\\approx$1~Gyr ago, but is consistent with a flat SFH given the small number of stars and the statistical nature of the problem. As above, this type of event would be undetectable at older ages based on this data set alone. \n\\end{enumerate}\n\\end{itemize}\n\n Integrating the SFR over the entire history gives a total astrated stellar mass M$_{\\star}$=~1.77$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$. For ages less than 1~Gyr, we estimate a stellar mass M$_{\\star, <1 {\\rm Gyr}} =$~1.68$^{+0.69}_{-0.75}$~$\\times$10$^{6}$~M$_{\\sun}$, which consists a fraction of $\\sim$10\\% of the total stellar mass. Because the maximum mass of surviving stars decreases with age, only $\\approx$85\\% of the total astrated mass is still present in the form of stars, i.e., the current stellar mass M$_{\\star, {\\rm current}} =$~1.50$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$. Kirby et al.~(\\cite{sl_kirby08}) derive a total stellar mass equal to 1.6$\\times$10$^{7}$~M$_{\\sun}$, using deep near-infrared observations that trace the old stellar populations and assuming an H-band stellar mass-to-light ratio of 1. The agreement between our current stellar mass and the total stellar mass of Kirby et al.~(\\cite{sl_kirby08}) is very good. \n\n The current stellar mass is comparable to the {\\ion {H}{i}} mass in SDIG (see Table~\\ref{table1}; Koribalski et al.~\\cite{sl_koribalski04}) with a ratio of M$_{\\rm {\\ion {H}{i}}}$~\/~M$_{\\star} =$~0.6. We compute the baryonic gas fraction, M$_{\\rm gas}$~\/~M$_{\\rm baryonic}$, and compare it with the baryonic gas fraction of the ANGST sample galaxies (Weisz et al.~\\cite{sl_weisz11b}; their Fig.~9). The baryonic gas fraction of SDIG is estimated equal to M$_{\\rm gas}$~\/~M$_{\\rm baryonic} =$0.7, assuming a gas mass M$_{\\rm gas}=$1.4$\\times$M$_{\\rm {\\ion {H}{i}}}$ and a baryonic mass M$_{\\rm baryonic}=$M$_{\\rm gas} +$M$_{\\star}$. These values are listed in Table~\\ref{table2}. Placing SDIG in Fig.~9 of Weisz et al.~(\\cite{sl_weisz11b}) locates it in the upper left region occupied by dIrrs, with SDIG slightly above ESO321-G014. \n\n Based on our current stellar mass estimate and the V-band absolute magnitude listed in Table~\\ref{table1}, we compute a V-band stellar mass-to-light ratio equal to M$_{\\star}$~\/~L$_{\\rm V} =$ 3.2~M$_{\\sun}$~\/~L$_{\\sun}$. This is higher than transition-type LG galaxies such as LGS3 and DDO\\,216 (Hunter, Elmegreen \\& Ludka \\cite{sl_hunter10}). Using the B-band magnitude listed in Karachentsev et al.~(\\cite{sl_karachentsev04}; column (6) in their Table~1), we compute a B-band stellar mass-to-light ratio equal to M$_{\\star}$~\/~L$_{\\rm B} =$1.4~M$_{\\sun}$~\/~L$_{\\sun}$. The B-band stellar mass-to-light ratio of SDIG is lower than the average B-band stellar mass-to-light ratio (equal to 2.4~M$_{\\sun}$~\/~L$_{\\sun}$) of the sample of star-burst dwarf galaxies listed in McQuinn et al.~(\\cite{sl_mcquinn10}), but is higher than the average value of $\\sim$1.1~M$_{\\sun}$~\/~L$_{\\sun}$ for dIrrs with $-$11~mag $\\leq$ M$_B$ $\\leq$ $-$13~mag (Weisz et al.~\\cite{sl_weisz11b}). \n\n SDIG is a late-type dwarf galaxy and the dominant population has intermediate ages, with an average age of 6.4$^{+1.6}_{-1.4}$~Gyr. Thus, the age-metallicity degeneracy on the RGB makes it difficult to derive individual stellar metallicities. On the other hand, the {\\em mean} stellar metallicity of mixed stellar populations remains rather robust against the age-metallicity degeneracy on the RGB (Lianou et al.~\\cite{sl_lianou11}). Therefore, we use the average age of the stellar populations in SDIG \n \\begin{figure}\n \\centering\n \\includegraphics[width=7.5cm,clip]{20193fg9.eps}\n \\caption{CMD focused on the RGB stars. The grey circles indicate the RGB stars used to compute the fiducial line, with the latter shown with the open black circles. The solid red line shows the Dartmouth 6.4~Gyr, solar-scaled isochrone with metallicity $-$1.6~dex in [M\/H] that best matches the RGB fiducial, while the black dotted line shows the Padova 6.4~Gyr isochrone with metallicity $-$1.5~dex in [M\/H]. Towards fainter magnitudes, the two isochrones overlap. }\n \\label{sl_figure9}%\n \\end{figure}\nin order to constrain its {\\em mean} stellar metallicity. To that end, we use the fiducial of RGB stars, with magnitudes ranging from the TRGB to 1~mag fainter and shown in Fig.~\\ref{sl_figure9} with the open circles, to which we overplot the best-matched isochrone from the Dartmouth (Dotter et al.~\\cite{sl_dotter07,sl_dotter08}) and Padova Stellar Evolution Databases, with an age equal to the average age of the stellar population in SDIG. Here, we use the colour excess and distance modulus we derive from the SFH modelling (see $\\S$5). The best-matched isochrones are shown in Fig.~\\ref{sl_figure9}, with the red solid line and black dotted line, respectively, for the Dartmouth and Padova isochrones. The metallicity of the best-matched Dartmouth isochrone is $-$1.6~dex, while the metallicity of the best-matched Padova isochrone is $-$1.5~dex. The metallicity of the Padova isochrone that best matches the RGB fiducial is consistent with the average metallicity we derive from the SFH-modelling. The inferred stellar metallicity translates to an [O\/H] value of $-$1.13~dex (Mateo \\cite{sl_mateo98}), or to a 12+${\\rm log}$[O\/H] value of 7.78, therefore is consistent with the metallicity-luminosity relation in Lee et al.~(\\cite{sl_lee07}). Moreover, the stellar metallicity of SDIG is consistent with the stellar metallicity-luminosity relation in Cloet-Osselaer et al.~(\\cite{sl_cloet-osselaer12}). \n\n\\begin{table}[t]\n \\caption[]{Derived properties of SDIG.}\n \\label{table2} \n \\begin{tabular}{lc}\n\\hline\\hline\n Quantity & Value \\\\\n\\hline\n$\\langle {\\rm SFR} \\rangle$ & 1.3$^{+0.4}_{-0.3}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$ \\\\\nSFR$_{\\rm current}$ & 2.5$^{+0.7}_{-0.6}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$ \\\\\nSFR$_{\\rm recent}$ & 2.7$\\pm$0.5$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$ \\\\\n$\\langle {\\rm \\tau} \\rangle$ & 6.4$^{+1.6}_{-1.4}$~Gyr \\\\ \nM$_{\\star}$ & 1.77$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$ \\\\\nM$_{\\star, <1 {\\rm Gyr}}$ & 1.68$^{+0.69}_{-0.75}$~$\\times$10$^{6}$~M$_{\\sun}$ \\\\\nM$_{\\star, {\\rm current}}$ & 1.50$^{+0.71}_{-0.72}$~$\\times$10$^{7}$~M$_{\\sun}$ \\\\\nM$_{\\rm gas}$ & 3.5$\\times$10$^{7}$~M$_{\\sun}$ \\\\\nM$_{\\rm baryonic}$ & 5$\\times$10$^{7}$~M$_{\\sun}$ \\\\\nM$_{\\rm gas}$~\/~M$_{\\rm baryonic}$ & 0.7 \\\\\nM$_{\\star}$~\/~L$_{\\rm V}$ & 3.2~M$_{\\sun}$~\/~L$_{\\sun}$ \\\\\nM$_{\\star}$~\/~L$_{\\rm B}$ & 1.4~M$_{\\sun}$~\/~L$_{\\sun}$ \\\\\n${\\rm [M\/H]}_{\\rm SFH}$ & $-$1.5~dex \\\\\n${\\rm [M\/H]}_{\\rm Padova}$ & $-$1.5~dex \\\\\n${\\rm [M\/H]}_{\\rm Dartmouth}$ & $-$1.6~dex \\\\\n\\hline\n\\end{tabular} \n\\end{table}\nThe derived properties of SDIG are listed in Table~\\ref{table2}. Each row lists the following information: (1) the average SFR; (2) and (3) the current and recent SFRs, respectively (see below); (4) the average age of the stellar populations; (5) the astrated stellar mass; (6) the stellar mass for stars younger than 1~Gyr; (7) the current stellar mass, M$_{\\star, {\\rm current}}$; (8) the gas mass, computed as 1.4$\\times$M$_{\\rm {\\ion {H}{i}}}$; (9) the baryonic mass, computed as M$_{\\rm gas}+$M$_{\\star, {\\rm current}}$; (10) the baryonic gas fraction, computed as M$_{\\rm gas}\/$M$_{\\rm baryonic}$; (11) the current V-band stellar mass-to-light ratio; (12) the current B-band stellar mass-to-light ratio; (13) the average metallicity based on the SFH results; (14) the stellar metallicity based on the RGB fiducial and Padova isochrones; (15) the stellar metallicity based on the RGB fiducial and Dartmouth isochrones. \n\nWe can compare the SFR in our youngest age bin to current SFR indicators like H$\\alpha$ and FUV emission.\nThe current SFR is estimated from our data assuming it is equal to the average SFR in our first age bin (4--30~Myr), 2.5$^{+0.7}_{-0.6}$~$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$. H$\\alpha$ emission is sensitive to SFR over the past 10~Myr but can be biased to lower values than the true current SFR when the SFR is less than $\\approx$3$\\times$10$^{-3}$~M$_{\\odot}$~yr$^{-1}$ (e.g., Lee et al.~\\cite{sl_lee09}) because it relies on ionising photons from O stars, which may not be fully sampled by the IMF in cases of low SFR. We detect of order a dozen stars bright enough and blue enough to be candidate OV stars, so stochastic effects are expected to be important. Despite this caveat, the current SFR we derive from the CMD modelling is consistent with the upper limit of 2$\\times$10$^{-3}$~M$_{\\sun}$~$yr^{-1}$ that Heisler et al.~(\\cite{sl_heisler97}) derive based on their 3~$\\sigma$ detection of the H$\\alpha$ surface brightness across the continuum-subtracted image of SDIG. However, Bouchard et al.~(\\cite{sl_bouchard09}) derive a lower current SFR of 3.8$\\times$10$^{-5}$~M$_{\\sun}$~yr$^{-1}$ based on the H$\\alpha$ flux in two detected knots. Their SFR is just 3\\% of the value we find. This could be taken as evidence for a quiescent period of low SFR, or for random variance due to the low numbers of massive stars formed. \n\nWe can estimate the number of O-type stars expected to be produced by the recent SFR using the stellar population synthesis code Starburst99 (Leitherer et al.~\\cite{sl_leitherer99}). As a rough estimate, we assume continuous star formation at a constant rate equal to the recent SFR (see Table~\\ref{table2}), a Salpeter IMF, and solar metallicity. The results provide us with the total number of ionising photons emitted during the most recent time step of 2~Myr, and over a longer period of 100~Myr. Using the ionising flux Q$_0$ for an O5V star (Q$_0$ = 10$^{49.22}$ s$^{-1}$, M $\\approx$37~M$_{\\sun}$; Martins, Schaerer \\& Hillier~\\cite{sl_martins05}), we estimate that of order 18 such stars would have been produced over 100~Myr given our CMD-derived SFR. Of these, only a small fraction would be expected to be observed today owing to their short lifetimes. As an additional way to estimate the number of massive stars present, we employ a Monte Carlo approach to estimate the variance in massive star numbers given an overall low SFR. To that end, we assume a Salpeter IMF (Salpeter~\\cite{sl_salpeter}) with lower and upper mass limits of 0.5 to 150~M$_{\\sun}$, and draw from this distribution until 2.7$\\times$10$^{5}$~M$_{\\sun}$ total mass of stars are formed; this number corresponds to our recent SFR integrated over 10$^8$ yr. The process is repeated 100 times, allowing us to calculate that on average 1216$\\pm$40 stars with masses appropriate to O stars (here assumed to be 18--100~M$_{\\sun}$) are formed over this time period. At any given moment we would expect to see $\\approx$120 O stars if the IMF was being fully sampled, assuming an average lifetime of 10~Myr. The fact that we observe about an order of magnitude less confirms that SDIG is in a ``down'' state of star formation lasting at least the duration of a typical O star lifetime. However, given the SFR determined over our youngest age bin (4--30~Myr) is higher than the lifetime average for the galaxy and the high gas content of SDIG, it seems more likely that this represents a combination of short-term SFR fluctuation and stochastic effects due to IMF sampling than to any physically meaningful suppression of star formation on longer timescales.\n\nBecause H$\\alpha$ is known to be problematic in cases of very low SFR, the FUV emission provides a better estimator of the recent SFR, sampling stars down to a few solar masses, and ages of order 100~Myr (Lee et al.~\\cite{sl_lee11}). We use the FUV asymptotic magnitude from Table~2 of Lee et al.~\\cite{sl_lee11} in conjunction with the FUV-SFR relation from Lee et al.~(\\cite{sl_lee09}; their $\\S$3) to estimate a recent SFR of 9.6$\\times$10$^{-4}$~M$_{\\sun}$~yr$^{-1}$. For comparison, the SFR we derive from the CMD is 2.7$\\pm$0.5~$\\times$10$^{-3}$~M$_{\\sun}$~yr$^{-1}$. We therefore find improved agreement between the recent SFR indicator and the CMD modelling, but a factor of 2--3 discrepancy remains, presumably due to the fact that a galaxy that is deficient in O stars will also be deficient in FUV flux.\n\n\\section{Summary and discussion}\n\n SDIG is a gas-rich, low-luminosity dIrr (C\\^ot\\'e et al.~\\cite{sl_cote97}; Heisler et al.~{\\cite{sl_heisler97}}). It is dominated by intermediate-age stars with an average age ${\\rm \\tau}=$6.4$^{+1.6}_{-1.4}$~Gyr, similar to the mean mass-weighted age of LG dIrrs (e.g., Orban et al.~\\cite{sl_orban08}). The average SFR is similar to other Sculptor group dwarfs, listed in Weisz et al.~(\\cite{sl_weisz11b}; their Table~2), with ESO540-G030 and ESO540-G032 having the lowest and highest average SFRs, respectively. The non-detection of $\\ion {H}{ii}$ regions and low H$\\alpha$-based SFR have been regarded as indicative of SDIG experiencing a currently quiescent phase (Miller \\cite{sl_miller96}; Heisler et al.~\\cite{sl_heisler97}; Skillman et al.~\\cite{sl_skillman03}; Bouchard et al.~\\cite{sl_bouchard09}). However, our SFH modelling shows elevated star formation for ages younger than 100~Myr, consistent with the star formation levels detected in the FUV emission. Our current data do not allow us to resolve star formation events lasting $\\sim$10$^8$~Myr for ages older than $\\sim$1~Gyr. There are intriguing hints of a strong previous enhancement of SFR at ages of 600--1100~Myr, although the amplitude and timing of this event are model-dependent. Similar patterns of globally decreasing SFR during the past 1--2~Gyr have been observed in a number of LG galaxies with similar luminosity to SDIG, e.g. Leo~A (Cole et al.~\\cite{sl_cole07}).\n\n SDIG is a member of the NGC\\,7793 subgroup located at the far-side of the Sculptor group (Jerjen et al.~\\cite{sl_jerjen98}; Karachentsev et al.~\\cite{sl_karachentsev06}), and NGC\\,7793 is identified as its main disturber, i.e. the neighbouring galaxy producing the maximum tidal action (Karachentsev et al.~\\cite{sl_karachentsev04}). NGC\\,7793 is a spiral galaxy and stellar population analyses of several halo and disk fields detect a metallicity gradient (Vlajic et al.~\\cite{sl_vlajic11}), as well as indications for substantial stellar radial migration (Radburn-Smith et al.~\\cite{sl_radburn12}). The SFH of a halo field in NGC\\,7793 indicates that a burst of star formation has occurred within the last 20~Myr and with an amplitude three times that of the average SFR, most likely associated with the spiral arm feature contaminating the halo field (Dalcanton et al.~\\cite{sl_dalcanton12}; Figs.~1, 2, and 9 for NGC\\,7793). The deprojected distance between SDIG and NGC\\,7793 is 450$^{+340}_{-270}$~kpc, estimated using the method in Karachentsev et al.~(\\cite{sl_karachentsev04}), i.e. ${\\rm R}^{2}={\\rm D}^{2} + {\\rm D}^{2}_{{\\rm MD}} - 2 {\\rm D} {\\rm D}_{{\\rm MD}}cos\\Theta$, where ${\\rm D}_{{\\rm MD}}$ is the distance in Mpc of NGC\\,7793 adopted from Vlajic et al.~(\\cite{sl_vlajic11}), ${\\rm D}$ is the distance in Mpc of SDIG, and $\\Theta$ is their angular separation, calculated using NASA\/IPAC Extragalactic Database (NED). The uncertainty in the deprojected distance takes into account the uncertainties in the distance of NGC\\,7793 and SDIG, as well as their angular projected separation as the minimum physical distance between them. We note that many gas-rich irregular and transition dwarfs are similarly distant from the Milky Way, for example NGC\\,6822 or Leo\\,T (McConnachie 2012). Given the mass of NGC\\,7793 from Carignan \\& Puche (\\cite{sl_car90}) the orbital timescale for SDIG around NGC\\,7793 would appear to be too long to be responsible for star formation activity on timescales shorter than a Hubble time, but other signs of disturbance are present.\n\n C\\^ot\\'e et al.~(\\cite{sl_cote00}) discuss how the low value of SDIG's maximum rotational velocity implies that random motions may significantly contribute to the dynamical support and may thus lead to the puffing-up of an initially thin disk. To that end, the twisted {\\ion {H}{i}} disk of SDIG (C\\^ot\\'e et al.~\\cite{sl_cote00}) may indicate an interaction having occurred in this dwarf's past. NGC\\,7793 has been mapped with Chandra (Pannuti et al.~\\cite{sl_pannuti11}), and the results indicate an asymmetry in the detected X-ray sources. Pannuti et al.~(\\cite{sl_pannuti11}) discuss an interaction between SDIG and NGC\\,7793 as a plausible way to explain this asymmetry in the X-ray sources, although they do not regard it as strong a possibility. Radburn-Smith et al.~(\\cite{sl_radburn12}) discuss that the detected stellar radial migration in NGC\\,7793 may also be due to a past interaction event having occurred in NGC\\,7793. While the deprojected distance of SDIG shows its current location and its actual orbit remains unknown (e.g. Bellazzini et al.~\\cite{sl_bellazzini96}), the timescale of the star formation enhancements and the crossing timescale of SDIG around NGC\\,7793 suggest that the elevated SFRs in SDIG are not due to gravitational interactions between them. The SF enhancements and disturbed {\\ion {H}{i}} kinematics seen in SDIG may be due to the effect of gravitational interactions with a lower luminosity, gas-poor dwarf galaxy companion that has thus far remained elusive, as such tidal interactions between dwarf galaxies now start to become revealed (e.g., Martinez-Delgado et al.~\\cite{sl_delgado12}; Rich et al.~\\cite{sl_rich12}; Hunter et al.~\\cite{sl_hunter98}). Tidal interactions between dwarf galaxies may also provide an important link to understanding the evolution and characteristics of dwarf galaxies in the isolated far-outskirts of the LG, as the case of the transition-type dwarf galaxy VV124 shows (e.g., Kirby et al. 2012).\n\n It is interesting in this respect that based on photometric arguments alone, Heisler et al.~(\\cite{sl_heisler97}) conclude that it is not possible to rule out that SDIG-like systems may evolve to dSphs. It is thus intriguing that the off-center local sites of star formation observed in the young MS stars of SDIG, as well as other LG dwarfs (Dohm-Palmer et al.~\\cite{sl_dohm-palmer97}; Gallagher et al.~\\cite{sl_gallagher98}; Mateo \\cite{sl_mateo98}; Cole et al.~\\cite{sl_cole99}; McConnachie et al.~\\cite{sl_mcconnachie06}), are observed in the majority of the transition-type dwarfs in the Sculptor group (Lianou et al.~\\cite{sl_lianou12}; Jerjen \\& Rejkuba \\cite{sl_jerjen01}). Environmental effects on gas-rich dwarfs provide a possible avenue to their evolution into gas-poor systems (e.g., Mayer \\cite{sl_mayer10}; and references therein), and such environmental effects seem to affect their recent star formation activity (e.g., Bouchard et al.~\\cite{sl_bouchard09}; Thomas et al.~\\cite{sl_thomas10}; Weisz et al.~\\cite{sl_weisz11b}). However, the detection of a recent modest upturn in SFR, and hints of a similar episode $\\approx$0.6-1~Gyr ago is further indication that small galaxies are able to sustain complex star formation histories without strong tidal interactions from larger neighbours.\n\n\\begin{acknowledgements} The authors would like to thank an anonymous referee for the thoughtful comments. We are grateful to Margrethe Wold and Claus Leitherer for useful discussions on Starburst 99. \\\\\n This research has made use of the following facilities: NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration; NASA's Astrophysics Data System Bibliographic Services; SAOImage DS9 developed by Smithsonian Astrophysical Observatory; Aladin.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThin metal films of nanometric size can have physical properties different from their bulk counterparts due to quantum size effects (QSE) caused by \nthe valence electron confinement in the direction normal to the film surfaces \\cite{chiang,tringides,jia}. Since thin films are among the basic \ncomponents of modern nanodevices an increasing attention has been addressed to the quantum vacuum fluctuations forces (van der Waals and Casimir \nforces) between them. Theoretical determinations of these forces have been based mainly on a continuum description of the material dielectric \nproperties, neglecting modifications of the electronic structures due to the boundaries, i.e. under the assumption that the film dielectric \nproperties be the same of the bulk material \\cite{esquivel,benassi,lambrecht2,pirozhenko2,lenac}. This assumption is not valid for small systems with low electron \ndensity and\/or large surface to volume ratio. In such cases a significant size dependence of the basic properties, like the Fermi energy and the \ndensity of electron states, arises as a consequence of the electron confinement \\cite{rogers1}.\\\\\nIn a previous paper \\cite{benassi3} we have shown that the inclusion of QSE in the plasma model of a free electron metal leads to modifications of the \nforce intensity between nanometric size films with respect to the bulk plasma model which range from several to few percent depending upon the film \nelectron density and the separation distance. The calculated forces show quantum size oscillations and are less intense compared \nto those determined with the bulk plasma model. The calculations were performed using the \\emph{particle in a box} model \\cite{wood} (hereafter indicated as PBM) in which independent electrons \nare confined along the direction normal to the film surfaces by hard walls and behave as a two dimensional gas parallel to the \nsurfaces. Such model represents a simplified picture of the one electron potential in the film.\\\\\nIn this paper we improve our description of the film dielectric properties along two lines of development: \nfirst we introduce a finite well model for the one-electron potential along the surface normal, second we include intraband absorption by introducing the relaxation time in a manner that allows to keep number conservation.\nAs to the first issue, we notice that the use of a soft confining potential like a finite well, besides being closer to the real shape of the one-electron potential as determined by first \nprinciples calculations \\cite{schulter,feibelman,ciraci,ogando}, allows for a better treatment of QSE for films of small thickness, since it does not \nintroduce a priori a distance \nwithin which electrons are to be confined and leaves the Fermi energy free to oscillate with the film size \\cite{rogers1,rogers2}, a feature that is \nnot present in the PBM model, where the Fermi energy is kept equal to the bulk value. \nThe inclusion of Drude absorption in vacuum force calculations has been a subject of considerable debate, mainly in relation with finite temperature \ncorrections \\cite{boestrom3,bezerra,brevik,bezerra2,hoye2,brevik2}. Very accurate measurements have been reported that shows that the simple plasma model of bulk dielectric can give better agreement with the experiments than the model with finite relaxation times \\cite{decca2}.\nOur results refer to the $T=0^{\\circ}$K case where difficulties do not seem to be present \\cite{lamoreaux}. Our purpose is to understand how QSE may affect the force between thin films when intraband absorption is included.\nIn this paper we introduce such corrections and we show their importance in the \ncalculation of the intensity of the force between metallic films.\\\\\n\\section{Theoretical framework}\nWe consider two identical metal films of thickness $D$ with plane boundaries separated by a distance $\\ell$ (see figure \\ref{fig1} (a)). $D$ represents \nthe extension, along the $z$-direction normal to the surface, of the positive ion distribution, which is supposed to be uniform with the same density of the bulk system. By extending \nprevious results for isotropic slabs to the case of films with anisotropic dielectric tensor we can write the expression of the force per unit area \nas \\cite{lifshitz,dzyaloshinskii,zhou,bordag}:\n\\begin{equation}\nF=-\\frac{\\hbar }{2 \\pi^2}\\int_{0}^{\\infty} k\\: dk\\int_{0}^{\\infty}d\\omega\\: \\gamma(\\omega)\n\\bigg[\\frac{Q_{TM}(i\\omega)^2}{1-Q_{TM}(i\\omega)^2}+\n+\\frac{Q_{TE}(i\\omega)^2}{1-Q_{TE}(i\\omega)^2}\\bigg] \n\\end{equation}\n\\begin{equation}\nQ_{TM}=\\frac{\\rho_{TM}(1-e^{-2 \\gamma_{TM} D})}{1-\\rho_{TM}^2e^{-2 \\gamma_{TM} D}}e^{-\\gamma \\ell}\\qquad\nQ_{TE}=\\frac{\\rho_{TE}(1-e^{-2 \\gamma_{TE} D})}{1-\\rho_{TE}^2e^{-2 \\gamma_{TE} D}}e^{-\\gamma \\ell}\n\\end{equation}\n\\begin{equation}\n\\rho_{TM}=\\frac{\\gamma_{TM}(\\omega)-\\gamma\\epsilon_{xx}(\\omega)}{\\gamma_{TM}(\\omega)+\\gamma\\epsilon_{xx}(\\omega)} \n\\qquad\\rho_{TE}=\\frac{\\gamma_{TE}(\\omega)-\\gamma(\\omega)}{\\gamma_{TE}(\\omega)+\\gamma(\\omega)}\n\\end{equation}\n\\begin{equation}\n\\gamma_{TE}(\\omega)=\\sqrt{k^2-\\frac{\\omega^2}{c^2}\\epsilon_{xx}(\\omega)}\\qquad\n\\gamma_{TM}(\\omega)=\\sqrt{\\bigg(\\frac{k^2}{\\epsilon_{zz}(\\omega)}-\\frac{\\omega^2}{c^2}\\bigg)\\epsilon_{xx}(\\omega)}\n\\end{equation}\n\\begin{equation}\n\\gamma(\\omega)=\\sqrt{k^2-\\frac{\\omega^2}{c^2}}\n\\end{equation}\nHere $\\epsilon_{xx}$ and $\\epsilon_{zz}$ are the diagonal components of the dielectric tensor along the planar directions and along the surface normal respectively. We assume $\\epsilon_{xx}=\\epsilon_{yy}$ and the off diagonal components to be\nzero. \n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm,angle=0]{fig1.eps}\n\\caption{\\label{fig1} (a) Sketch of the two identical interacting films. (b) Infinitely deep quantum well with artificial spill-out. (c) Finite quantum well with natural spill-out.}\n\\end{figure}\nThis assumption is consistent with the two dimensional gas behavior of the electrons parallel to the surface. The anisotropy of the dielectric tensor is a consequence of the finite extension of the film along the $z$-direction and it is the main feature introduced by the size quantization. To calculate the force one needs the expression of the dielectric tensor, which has to be derived from the film electronic structure. \nWe assume an independent particle model with the one electron potential $V(z)$. The electron energies are given by\n\\begin{equation}\nE_{\\textbf{k}_{\\parallel},n}=\\frac{\\hbar^2}{2 m}\\textbf{k}_{\\parallel}^2+E_{n}\n\\end{equation} \ni.e. they are described by the continuous quantum number $\\textbf{k}_{\\parallel}$ and by the discrete sub-band index $n$ coming from the quantization of the perpendicular wavevector, $m$ being the electron mass. The corresponding wavefunctions are given by\n\\begin{equation}\n\\psi_{\\textbf{k}_{\\parallel},n}(\\textbf{r}_{\\parallel},z)=\\frac{1}{\\sqrt{A}}e^{i\\textbf{k}_{\\parallel}\\cdot\\textbf{r}_{\\parallel}}\\phi_{n}(z)\n\\label{wfc}\n\\end{equation} \nhere $A$ is the surface area, $\\textbf{k}_{\\parallel}$ and $\\textbf{r}_{\\parallel}$ are two dimensional wavevectors parallel to the surface, and $\\phi_{n}(z)$ is supposed to be independently normalized. The functions \n$\\phi_{n}(z)$ are the solutions of the equation\n\\begin{equation}\n\\bigg\\{\\frac{\\hbar^2}{2 m}\\frac{\\partial^2}{\\partial z^2}+V(z)\\bigg\\}\\phi_{n}(z)=E_n\\phi_{n}(z)\n\\label{schrod}\n\\end{equation}\nwith the proper boundary conditions. The Fermi energy $E_{F}$ is obtained through the \\emph{aufbau} procedure i.e. by arranging the eigenvalues in \nascending numerical order and counting until the number of states needed to accommodate all the electrons in the film is reached. This procedure \nleads to a Fermi energy that depends upon the film size and is generally different from the bulk value \\cite{rogers2}. This can be understood \nby noting that, to ensure charge neutrality, the number of electrons and the number of ions per unit area have to be equal. The electron density \n$n(z)$ can be simply obtained from the wavefunctions:\n\\begin{equation}\nn(z)=\\frac{1}{2 \\pi}\\sum_{m=1}^{m_0}\\bigg(\\frac{2 m E_F}{\\hbar^2}-E_n\\bigg)\\vert\\phi_{m}(z)\\vert^2\n\\end{equation} \nwhere $m_0$ is the label of the last occupied state, while the ion density is simply given by $n_0=k_{F_{B}}^3\/3 \\pi^2$, where $k_{F_{B}}$ is the bulk Fermi wavevector. By integrating the densities along the $z$ axis and imposing that both give the same number of charges per unit area, one gets the relation\n\\begin{equation}\n\\frac{1}{2 \\pi}\\sum_{m=1}^{m_0}\\bigg(\\frac{2 m E_F}{\\hbar^2}-E_n\\bigg)=n_0 D\n\\label{neucond}\n\\end{equation}\nwhich, for finite $D$ values, is generally not satisfied if one replaces $E_F$ with its bulk counterpart $E_{F_{B}}=\\hbar^2 k_{F_{B}}^2\/2 m$. \nIn the case of the PBM model this equation is not satisfied, since one assumes that the Fermi level is the same as in the bulk. To obtain charge neutrality one has to \nimpose the additional condition that the electron density be confined on a length $d$ larger than $D$. This artificially introduces the electronic charges spill-out but has the consequence that the average \nelectronic density is lower than the ion density \\cite{rogers1,benassi3,czoschke,czoschke2}.\\\\ \nFor the purpose of the present study we assume the potential to be a finite well $V(z) = -V_0$ inside the film and zero outside. For such finite well\nmodel (FWM) the energies of the bound states can be written as \n\\begin{equation}\nE_{\\textbf{k}_{\\parallel},n}=\\frac{\\hbar^2}{2 m}\\textbf{k}_{\\parallel}^2+\\frac{\\hbar^2}{2 m}k_{zn}^2-V_0\n\\end{equation}\nwhere $k_{zn}$ are the quantized transverse wavevectors. They are obtained from the equation giving the condition for the existence of bound states \nin a quantum well \\cite{davydov}:\n\\begin{equation}\nk_{zn}=\\frac{n \\pi}{D}-\\frac{2}{D}sin^{-1}\\bigg(\\frac{k_{zn}}{k_0}\\bigg)\n\\label{russi}\n\\end{equation}\nwith $k_0=\\sqrt{2 m V_0}\/\\hbar$. Notice that the first term at the second member is the value of the transverse wavevector for an infinite well \nmodel of size $D$ and the second term goes to zero as $V_0$ goes to infinity. This implies that, for given film size and number of electrons, \nthe Fermi energy referred to the well bottom is higher in the infinite well model.\nNotice that when $V_0$ goes to infinity one does not recover the PBM, since the Fermi energy is varied with respect to the bulk value in order to \nsatisfy the charge neutrality condition (\\ref{neucond}). Figure \\ref{fig1} (b) and (c) illustrate the difference: in the PBM the electronic charge density is \nconfined within a distance $d=D+\\Delta$, larger than the size of the ionic charge distribution, to allow for the electronic charge spill-out and to \nensure global neutrality for $E_F=E_{F_{B}}$. In the FWM the charge spill-out results naturally from the behaviour of the single particle \nstates while the charge neutrality is achieved by varying the Fermi energy with respect to the bulk value. Obviously in the limit of infinitely deep well \n(hereafter indicated as IWM) the electronic charge turns out to be entirely localized within the length $D$ and the Fermi energy is strongly \nincreased compared to its bulk value.\\\\\nOnce the electron energies and wavefunctions have been obtained, one can calculate the dielectric tensor from the expression \\cite{wood}:\n\\begin{eqnarray}\n\\nonumber\n\\epsilon_{\\alpha\\alpha}(\\omega)&=1-\\frac{\\omega_{P}^{2}}{\\omega^{2}}-\\frac{8\\pi e^{2}}{A d m^{2}\n\\omega^{2}}\\sum_{{\\bf k}_{\\parallel},n}\\sum_{{\\bf k}_{\\parallel}',n'}\nf(E_{\\bf{k}_{\\parallel},n})(E_{\\bf{k}_{\\parallel},n}-E_{\\bf{k}_{\\parallel}',n'})\\times\\\\\n&\\times\\frac{\\vert\\langle\\psi_{{\\bf k}_{\\parallel},n}\\vert\\hat{p_{\\alpha}}\\vert\\psi_{{\\bf k}_{\\parallel}',n'}\n\\rangle\\vert^{2}}{(E_{\\bf{k}_{\\parallel},n}-E_{\\bf{k}_{\\parallel}',n'})^2-\\hbar^2\\omega^2}\n\\label{dieltens}\n\\end{eqnarray}\nhere $\\alpha=x,y,z$ labels the cartesian component of the tensor, $\\hat{p}_{\\alpha}$ indicates the component of the electron linear momentum,\n$\\omega_{P}=\\Omega_{P} n(D)\/n_0$ is the plasma frequency of the quantized electron gas ($\\Omega_{P}=\\sqrt{4\\pi e^2 N_0\/m}$ is the free electron plasma frequency and $n(D)$ is the average electron density of the film) and $f(E_{\\bf{k}_{\\parallel},n})$ is the occupation factor of the $(\\bf{k}_{\\parallel},n)$ state. \nIn the IWM and the FWM $\\omega_P = \\Omega_P$, while in PBM $n(D)$ is smaller than $n_0$, since the electronic charge is distributed over a larger distance than the ionic charge (see Fig.\\ref{fig1} (b)).\nThe off-diagonal component are equal to zero. This expression differs from the \nplasma model dielectric function adopted in previous studies in that: \n(i) it has a tensor character with $\\epsilon_{xx}=\\epsilon_{yy}\\neq\\epsilon_{zz}$, (ii) through the double sum in the second member it accounts for transitions between lateral sub-bands, whose probability amplitude is expressed by the momentum matrix element between the one electron wavefunctions (\\ref{wfc}). It can be easily shown that these \ntransitions do not affect the lateral components of the dielectric tensor, which are given by the simple expression of the plasma dielectric function\n\\begin{equation}\n\\epsilon_{xx}(\\omega)=\\epsilon_{yy}(\\omega)=1-\\frac{\\omega_{P}^{2}}{\\omega^2}\n\\label{drudeded}\n\\end{equation}\nbecause the momentum matrix element for $x$ and $y$ component vanishes.\n\\section{Results for finite well potential}\nIn this section we present the results of calculations for finite well potentials. We take $Al$, $Ag$ and $Cs$ corresponding to a radius per electron \nin Bohr units $r_s\/a_0$ equal to $2.07$, $3.02$ and $5.62$ respectively, in order to illustrate QSE at different densities and potential depths. The \nvalue of the potential depth is obtained by summing the metal work function $W$ with the calculated Fermi energy. \n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig2.eps}\n\\caption{\\label{fig2} Fermi energy normalized to its bulk value for $Al$, $Ag$ and $Cs$ using the IWM (dashed line) and the FWM (continuous line). The bulk Fermi energies and the work functions have been taken from ref. \\cite{ashcroft}.}\n\\end{figure}\nFigure \\ref{fig2} reports the calculated Fermi energies as a function of the product between the Fermi wavevector and the film thickness. This allows \nto better point out the oscillations and the cusps arising from the crossing of the Fermi energy by new subbands upon varying the film size. \nThe figure shows the $V_0$ value appropriate to the bulk and to the large $D$ limit. \nWe give in the same figure the results obtained by assuming an infinitely deep potential (IWM). The comparison allows to illustrate the effects of the potential softening. In agreement with previously published results \\cite{schulter,rogers2,sernelius3} we find that: \n\\begin{itemize}\n\\item $E_F$ is systematically larger than $E_{F_{B}}$ and goes to the bulk value as $D$ goes to infinity. The difference is more pronounced and the bulk \nlimit is achieved at larger size in the low density systems, as it is clearly shown by the comparison between $Cs$ and $Al$ curves;\n\\item As expected from the discussion of the previous section, the softening of the potential leads to less pronounced deviations from the bulk \nvalues. Because of the stronger electron confinement, the IWM has a larger Fermi energy that the FWM;\n\\item The cusps correspond to integer values of half of the Fermi wavelength in the IWM case. This feature is only approximately verified for the FWM.\n\\end{itemize} \nAs a consequence of size quantization, the $z$-component of the dielectric tensor is expected to go to a finite value $\\epsilon_{zz}(0)$ as the \nfrequency goes to zero. This value increases proportionally to $D^2$ in the large size limit \\cite{wood}. In Figure \\ref{fig3} we plot the quantity \n$\\epsilon_{zz}(0)\/D^2$ for the three cases under study. \n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig3.eps}\n\\caption{\\label{fig3} Static value of the $zz$ component of the dielectric tensor for $Al$, $Ag$ and $Cs$ using the IWM (dashed lines), the PBM (dotted lines) and the FWM (continuous lines).}\n\\end{figure}\nAgain the results show the cusps due to the filling of new subbands as $D$ increases \n\\cite{benassi3}. The asymptotic limit is obtained for $k_{F}D\/\\pi$ of the order of $5\\div 6$ in the three cases. Significant differences appear in the \nlarge $D$ behaviour when the FWM results are compared with those from the IWM model: $\\epsilon_{zz}(0)$ is larger for finite wells of small \nsize, while it is smaller at high $D$ values. The convergence to the asymptotic limit is considerably slower for the infinite well, specially in the \nlow density metals. This behaviour reflects the differences in the distribution of the eigenvalues of equation (\\ref{schrod}). For the infinite well \nthere are infinite bound states whose energy scales like $n^2$, see equation (\\ref{russi}), and the separation between two successive levels \nincreases linearly with $n$. Such behaviour is not present in the FWM, for which equation (\\ref{schrod}) has a finite number of eigenvalues \ncorresponding to bound states and a continuum spectrum at positive energies. The PBM curve takes values closer to the FWM than to the IWM. This is primarily a consequence of the fact that PBM allows for electron charge spill-out, while in IWM the electron distribution is confined within $D$ i.e. it has the same size of the positive charge.\\\\\nIn the following we show the changes in the force caused by the size quantization with respect to the results obtained by using the isotropic continuum \nplasma model, where the $z$-component of the film dielectric tensor is equal to the planar components i.e. is given by equation (\\ref{drudeded}). We \nuse the symbol $F_Q$ to indicate the force per unit area calculated for the quantized film, while $F_P$ is the force per unit area calculated in the \nisotropic plasma model. To better illustrate the results, in figures \\ref{fig4} and \\ref{fig5} we plot the quantity\n\\begin{equation}\n\\delta_{P}=\\frac{F_P-F_Q}{F_P}\n\\end{equation}\nas a function of the separation distance $\\ell$ for films of $1$ and $5$ nm thickness respectively. In each figure we display the results for the \nthree cases under study and we compare the finite well with the IWM at the same density. This allows us to point out the modifications caused by the \npotential softening. We also show the curves appropriate to the PBM.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig4.eps}\n\\caption{\\label{fig4} Relative percentual difference for the force between two identical films of thickness $D=1$ nm as a function of the films separation $\\ell$, for $Al$, $Ag$ and $Cs$. Dashed lines have been obtained using the PBM, dotted lines have been obtained using the IWM and the continuous lines represents the finite well results.}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig5.eps}\n\\caption{\\label{fig5} Relative percentual difference for the force between two identical films of thickness $D=5$ nm as a function of the films separation $\\ell$, for $Al$, $Ag$ and $Cs$. Dashed lines have been obtained using the PBM, dotted lines have been obtained using the IWM and the continuous lines represents the finite well results.}\n\\end{figure}\nIn agreement with our previous findings we observe that \n\\begin{itemize}\n\\item QSE tend to reduce the intensity of the force;\n\\item the reduction is more significant at low density ($Cs$) than at high density ($Al$); \n\\item it may be considerably higher than $10\\%$ for $1$ nm thickness and reduce to few per cent at $5$ nm;\n\\item it can be appreciable over a distance interval up to $10\\div 50$ nm.\n\\end{itemize}\nThe most important conclusion that can be drawn from the figures is that the potential shape is important and can lead to a substantial \nmodifications of the quantum size effects both at small and at large distances. The models which confine the electronic charge tend to overestimate \nthe force reduction induced by size quantization. The curves for the PBM show large force reduction (greater $\\delta_P$ values) over a wide interval of distances. \nOn passing to the IWM case one notice that the \nremoval of the constriction that the Fermi energy be equal to the bulk value, still keeping an infinitely deep potential, leads to smaller $\\delta_P$ \nvalues and to a more rapid decay of the curves at large distances. Reducing the well depth to finite values has a similar effect: it causes a general \ndecrease of $\\delta_P$ and a narrowing of the distance interval over which QSE are appreciable. \nThis also implies that any increase of the confining potential depth at fixed ion density leads to higher $\\delta_P$ values and to more significant \nQSE.\nThe large values taken by $\\delta_{P}$ in the PBM case do not arise from the charge confinement only, since, as pointed out before, the constraint on \nthe charge distribution is weaker that in the IWM. To a large extent they are a consequence of the plasma frequency normalization caused by the \ndecrease in the average electron charge density that it is necessary in order to achieve global charge neutrality \\cite{benassi3,czoschke,czoschke2}. \nIn the isotropic plasma model one takes $\\omega_{P}=\\Omega_{P}$. In the PBM this value is obtained only at large film thickness. Neglecting this \nnormalization i.e. taking the free electron plasma frequency in the parallel components of the dielectric tensor (but not in $\\epsilon_{zz}$) would \nlead to $\\delta_{P}$ values closer to the well potential models.\n\\section{Intraband absorption effects}\nTo introduce intraband absorption we have to modify the dielectric tensor in a way that allows to include relaxation time effects in the parallel \ncomponents and to recover the Drude behaviour in the large $D$ limit. This cannot be done by simply introducing an imaginary part of the frequency \n$\\omega$, since this violates the continuity equation locally \\cite{mermin,garik}. The appropriate recipe is to replace into equation \n(\\ref{dieltens}) $\\omega^2$ with $\\omega(\\omega+i 2 \\pi\/\\tau)$, where $\\tau$ is the relaxation time.\nFor the parallel components this leads to the Drude dielectric function\n\\begin{equation}\n\\epsilon(\\omega)=1+\\frac{\\omega_{P}^2}{\\omega(\\omega+i \\gamma)}\n\\end{equation}\nwhere $\\gamma=2 \\pi \/\\tau$\nThe results of the calculation of the force per unit area with the modified dielectric tensor are displayed in Figures \\ref{fig6} and \\ref{fig7} for \n$1$ nm and $5$ nm films. We report the quantity\n\\begin{equation}\n\\delta_{D}(\\gamma)=\\frac{F_D(\\gamma)-F_{QD}(\\gamma)}{F_D(\\gamma)}\n\\end{equation}\nwhere $F_D$ is the force calculated using the bulk Drude model with a given relaxation time and $F_{QD}$ in the one obtained by the calculation with \nthe same relaxation time and with size quantization included. For $\\tau\\rightarrow\\infty$ we recover the plasma model so that \n$\\delta_P=\\delta_D(0)$.\nThe calculations have been performed assuming a finite potential well and for two values of the relaxation frequency $\\gamma$. We have taken values \nthat approximately correspond to those reported for the metals under consideration \\cite{ashcroft}. The figures show a comparison with the curves \nobtained by the plasma model with QSE. \n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig6.eps}\n\\caption{\\label{fig6} Relative percentual difference for the force between two identical films of thickness $D=1$ nm as a function of the films \nseparation $\\ell$, for $Al$, $Ag$ and $Cs$. Continuous lines represent the results for $\\gamma=0$. Dotted lines have been obtained using \n$\\gamma=5\\times 10^{13}$ rad\/s for $Ag$ and $Cs$ and $\\gamma=10^{14}$ rad\/s for $Al$. Dashed lines have been obtained using $\\gamma=10^{14}$ rad\/s \nfor $Ag$ and $Cs$ and $\\gamma=10^{15}$ rad\/s for $Al$.}\n\\end{figure}\nIt is clear from these results that the main effect of the inclusion of intraband absorption is to increase $\\delta$ i.e. to increase the difference \nwith respect to the calculations with the bulk dielectric function. The smaller is the relaxation time the larger is the reduction of the force. The \neffect is qualitatively the same in the three metals under study, but it depends upon the well depth and the film size.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig7.eps}\n\\caption{\\label{fig7} Relative percentual difference for the force between two identical films of thickness $D=5$ nm as a function of the films \nseparation $\\ell$, for $Al$, $Ag$ and $Cs$. Continuous lines represent the results for $\\gamma=0$. Dotted lines have been obtained using \n$\\gamma=5\\times 10^{13}$ rad\/s for $Ag$ and $Cs$ and $\\gamma=10^{14}$ rad\/s for $Al$. Dashed lines have been obtained using $\\gamma=10^{14}$ rad\/s \nfor $Ag$ and $Cs$ and $\\gamma=10^{15}$ rad\/s for $Al$.}\n\\end{figure}\nThe influence of the shape of the potential is illustrated in Fig.\\ref{fig8} showing the curves of $Ag$ films at a given relaxation frequency for the \nvarious models. Again it should be noticed that the FWM gives the lower $\\delta_D$ values. The PBM results show large $\\delta_D$ values over a very \nwide interval of distances. To a large extent this behaviour has to be imputed to the renormalization of the plasma frequency.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig8.eps}\n\\caption{\\label{fig8} Relative percentual difference for the force between two identical $Ag$ films of thickness $D=5$ nm as a function of the films \nseparation $\\ell$, $\\gamma=10^{14}$ rad\/s. Continuous lines represent the FWM, dotted line the IWM and dashed line the PBM.}\n\\end{figure}\n \\begin{figure}\n\\centering\n\\includegraphics[width=8cm,angle=0]{fig9.eps}\n\\caption{\\label{fig9} Relative percentual difference for the force between two identical $Ag$ films separated by a distance $\\ell=5$ nm as a function of the films thickness $D$. Continuous line represents the result for $\\gamma=0$ whereas the dashed line has been obtained with $\\gamma=10^{14}$ rad\/s. }\n\\end{figure}\nFigure \\ref{fig9} shows typical curves of $\\delta_D$ as a function of the film thickness for different values of the relaxation frequencies at a \ngiven separation distance of $5$ nm. As expected $\\delta_D$ decreases with $D$, but the slope at large thicknesses ($D$ of the order of $10\\div50$ \nnm) depends significantly upon the relaxation frequency. \n\\section{Conclusions and lines of development}\nWe have presented a rather complete set of theoretical results illustrating the possible role of size quantization effects in the electromagnetic \nvacuum force between very thin films and showing how the determination of these effects depends upon the description of the film electronic structure \nand upon the inclusion of intraband effects. However we want to point out that our analysis is still a mere indication of the corrections to the \nsimple picture that assumes the same dielectric function for films and bulk solids. Although the basic features of size quantization (confinement of \nthe electronic charge, anisotropy of the dielectric tensor, presence of inter-subbands transitions in the dielectric function) are already present in \nthe models we have studied, there is room for substantial improvements before an accurate comparison with experimental data, like those obtained by \nLisanti et al. for $Pd$ films \\cite{lisanti}, can be done. A more detailed treatment should include (i) band structure effects, (ii) non-locality of the dielectric response and (iii) non-local treatment of the reflectivity.\nA self consistent first principles calculation of the inverse dielectric matrix for a slab of the appropriate size and symmetry, from which a \nmacroscopic dielectric function can be derived with band structure and non-local effects included, could provide the appropriate treatment of the \nfirst two issues \\cite{li,sernelius2}. \nCorrections to the Fresnel optics, along the lines indicated by several authors \\cite{appel,feibelman2,kempa}, can lead to important modifications \nof the reflectivity even in the case of a free electron gas film. We are currently investigating these matters and the results will be presented \nelsewhere.\nStill when comparing theory with experiments for thin films one should consider the fact that measured relaxation frequency turn out to depend upon \nthe film size and morphology \\cite{jalochowski2,fahsold,sotelo,pribil,hoffmann}. This is largely due to the so called classical size effects arising from the scattering of the electrons at the \nfilm boundaries. In view of the sensitivity of QSE to the value of the relaxation time, comparison with experimental data may be possible only if \na realistic estimate of the modifications in the relaxation times caused by surface scattering is available.\n\\ack AB thanks \\emph{CINECA Consorzio Interuniversitario} ({\\tt\nwww.cineca.it}) for funding his Ph.D. fellowship.\n\\section*{Reference}\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\@startsection {section}{1}{\\z@}{-3.5ex plus-1ex minus\n -.2ex}{2.3ex plus.2ex}{\\reset@font\\small\\bf\\uppercase}}\n\\makeatother\n\n\\title{\\large \\bf Electroweak symmetry breaking in Higgs mechanism with\ncomposite operators and solution of naturalness }\n\\author{V.V. Kiselev,\\\\\n{\\small\\it State Research Center of Russia \"Institute for High\nEnergy Physics\"},\\\\ {\\small\\sl Protvino, Moscow region, 142281 Russia}\\\\\n{\\small\\sl E-mail: kiselev@th1.ihep.su, Fax: +7-(0965)-744739 }}\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\n{Introducing a source for a bi-local composite operator motivated by the\nperturbative expansion in gauge couplings, we calculate its effective potential\nin the renormalization group of Standard Model with no involvement of\ntechnicolor. The potential indicates the breaking of electroweak symmetry below\na scale $M$ due to a nonzero vacuum expectation value of neutral component for\nthe ${\\sf SU(2)}$-doublet operator. At virtualities below a cut off $\\Lambda$\nwe introduce the local higgs approximation for the effective fields of sources\ncoupled to the composite operators. The value of $\\Lambda\\approx 600$ GeV is\nfixed by the measured masses of gauge vector bosons. The exploration of\nequations for infrared fixed points of calculated Yukawa constants allows us to\nevaluate the masses of heaviest fermion generation with a good accuracy, so\nthat $m_t(m_t) = 165\\pm 4$ GeV, $m_b(m_b) = 4.18\\pm 0.38$ GeV and\n$m_\\tau(m_\\tau) = 1.78\\pm 0.27$ GeV. After a finite renormalization of\neffective fields for the sources of composite operators, the parameters of\neffective Higgs field potential are calculated at the scale of matching with\nthe local theory $\\Lambda$. The fixed point for the Yukawa constant of $t$\nquark and the matching condition for the null effective potential at $M$ drive\nthe $M$ value to the GUT scale. The equation for the infrared fixed point of\nquartic self-action allows us to get estimates for two almost degenerate scalar\nparticles with $m_H= 306\\pm 5$ GeV, while third scalar coupled with the $\\tau$\nlepton is more heavy: $m_{H_\\tau} = 552\\pm 9$ GeV. Some phenomenological\nimplications of the offered approach describing the effective scalar field, and\na problem on three fermion generations are discussed.\n}\n\\end{abstract}\n\n\\vspace*{1cm}\nPACS numbers: 12.60.Rc, 11.15.Ex, 12.60.Fr, 14.80.Cp\n\n\\section{Introduction}\nAt present, the Standard Model exhibits almost a total success in experimental\nmeasurements \\cite{SM}. The only question being a white spot on its body, is\nthe empirical verification of mechanism for the spontaneous breaking of\nelectroweak symmetry. In this respect, the minimal model involving a single\nlocal Higgs field brings a disadvantage: the stability of potential under the\nquantum loop corrections requires a restriction of quadratic divergency in the\nself-action by the introduction of ``low'' energy cut-off $\\Lambda\\sim 10^3$\nGeV, which is not a natural physical scale standing far away from what can be\ndesirable \\cite{tHoft}: the GUT scale, $M_{\\rm GUT}\\sim 10^{16}$ GeV\n\\cite{GUT}, or even the Planck mass, $M_{\\rm Pl}\\sim 10^{19}$ GeV. The reason\nfor putting the $\\Lambda$ so small, has to originate beyond the Standard\nModel. Two highways to a ``new physics'' merit the most popularity. The first\none is a technicolor \\cite{tc} postulating an extra-strong interaction for new\ntechnifermions, which form some ``QCD-like'' condensates, breaking down the\nelectroweak symmetry and giving the masses to the ordinary gauge bosons.\nDespite some problems with the generation of realistic mass values for the\nquarks and leptons and suppression of flavor changing neutral currents, the\nextended technicolor \\cite{etc} provides quite a clear picture for what happens\nin the region deeper than $10^3$ GeV. However, the most strict objection\nagainst such the way is the comparison with the current measurements, which\ndisfavor the technicolor models possessing the calculability \\cite{PEP}.\nA general consideration of models with the condensation of heavy fermions is\nreviewed in ref.\\cite{cvet}, while a brilliant presentation of both the\nideas on the electroweak symmetry breaking with composite operators and\ntechniques as well as results is given in a comprehensive survey by C.T.Hill\nand E.H.Simmons \\cite{HiSi}. However, the condensation, in general, does not\nprovide us with the solution of naturalness. In fact, this approach\nreformulates the problem as a fine-tuning phenomenon, since the separation of\ndynamics responsible for the composite operators at a high scale from the\nlow-energy electroweak physics takes place at effective couplings tuned to some\ncritical values. Therefore, we need an additional argumentation in order to\naddress the naturalness in the framework of condensation mechanism with\ncomposite operators. We present an idea toward this direction below.\n\nThe second way is a supersymmetry \\cite{SUSY} reforming the\nquadratic divergency in the self-action of Higgs field into the\nlogarithmic one, so that it prescribes the scale $\\Lambda$ to be a\nsplitting between the particles of Standard Model and their\nsuper-partners. Therefore, the supersymmetry has to be broken in a\nmanner conserving the logarithmic behavior of renormalization,\nwhich is an additional challenge to study and a degree of\nambiguity. However, the advantage is the stability of Higgs\npotential, so that $\\Lambda$ certainly is a reasonable scale\nreflecting the physics in the supersymmetric theory. What remains\nis the question: why the basic SUSY scale is so ``low'' in\ncomparison with the GUT scale? Hence, the naturalness is again the\nproblem standing in the higher-quality context.\n\nIf the ultraviolet cut off energy in the loop calculations is placed close to\nthe Planck scale (see Fig.\\ref{SM}\\footnote{The figure originally appeared in\nref. \\cite{HR}, and it is taken from ref. \\cite{Bj}, while the two-loop\nconsideration recently was done in ref. \\cite{PZ}.}), the\nStandard Model suffers from the inherent inconsistency except a narrow window\nin the range of possible values of Higgs particle mass: $m_H = 160\\pm 20$ GeV,\nwhich does not contradict the value following from the precise measurements of\nelectroweak parameters in the electron-positron annihilation at the $Z$ boson\npeak.\n\n\\begin{figure}[th]\n\\setlength{\\unitlength}{1mm}\n\\begin{center}\n\\begin{picture}(80,80)\n\\put(0,0){\\epsfxsize=9cm \\epsfbox{7.eps}}\n\\end{picture}\n\\end{center}\n\\caption{The region of higgs mass constrained by requirements of the SM\nconsistency.}\n\\label{SM}\n\\end{figure}\n\n\nThe reasons for such the inconsistency are the following: At lower masses the\nvacuum stability is broken, i.e., the quartic coupling constant of scalar field\nchanges its sign \\cite{AI}. At higher masses the theory enters the strong\nself-interaction regime, which indicates that the quartic coupling constant\nbecomes infinite (alike the Landau pole) at a scale less than the offered cut\noff \\cite{Elias}. If the scale of ultraviolet cut off in the SM is much lower\nthan the Planck scale, then the region of higgs masses providing the SM\nconsistency, is more wide. However, such the scales are not natural. A low cut\noff scale should indicate a new dynamics. While the vacuum instability is an\nunavoidable physical constraint, the phase of strong higgs self-interaction\ncould be treated in the framework of the following representation: The scalar\nhiggs can be described in terms of the local field below an ultraviolet cut off\n$\\Lambda$ placed close to the region of strong regime. At virtualities higher\nthan $\\Lambda$, the strongly self-coupled higgs is not fundamental and local\nquantum. The dynamics should be described by means of weakly interacting\nparticles, so that a composite operator with appropriate quantum numbers has to\ncorrespond to the higgs in the `dual' limit implying that the effective\npotential of the composite operator yields a development of vacuum expectation\nvalue for the global (independent of space-time point) source of operator. This\nstrong self-interaction regime could be realized with no involvement of an\nextended underlying theory alike the technicolor dynamics, since some composite\noperators can develop an appropriate effective potential in the framework of\nstandard electroweak symmetry.\n\nOur assumptions are the followings:\n\n1. We choose a form of composite operators describing the nonlocal phase of\nhiggs in the strong self-interaction regime (SSIR) and suppose the connection\nof such the operators to the higgses. The suggestion on the nonlocality of\nhiggses allows us to replace the strong self-interaction regime in the theory\nwith the local Higgs fields by the weak self-interaction regime (WSIR) of\nsources for the composite operators.\n\n2. The interactions of fermions and gauge bosons in the SSIR are given by the\ndynamics of SM with no local scalar higgses as well as no extensions like a\ntechnicolor or so.\n\n3. Concerning the position of scale $\\Lambda$ denoting the infrared cut off in\nthe calculations with the composite operators as well as the ultraviolet cut\noff in the local theory with the scalar higgs, we put it into the (infrared)\nfixed point for the Yukawa coupling constants of heaviest fermion in the local\ntheory. the numerical value of $\\Lambda$ is given by the masses of\nweak-interaction gauge bosons.\n\n4. We consider Yukawa couplings of the only heaviest fermion generation in the\nSM.\n\n5. In the SSIR we introduce the ultraviolet cut off $M\\gg \\Lambda$. At $M$ the\nelectroweak symmetry is exactly restored.\n\n6. We match the effective potential of sources for the composite operators with\nthe potential of corresponding local scalar fields at the scale $\\Lambda$.\n\nThe corresponding divisions of virtualities are presented in Fig.\n\\ref{division}.\n\n\\begin{figure}[th]\n\\setlength{\\unitlength}{1mm}\n\\begin{center}\n\\begin{picture}(100,45)\n\\put(0,0){\\epsfxsize=11cm \\epsfbox{8.eps}}\n\\put(15,-4){$\\Lambda$}\n\\put(86,-4){$M$}\n\\put(100,0){log$_{\\rm 10}\\mu [{\\rm GeV}]$}\n\\put(-10,30){Local higgs}\n\\put(-4,25){phase}\n\\put(27,27){`Strong self-interaction regime'}\n\\put(34,33){Composite operators:}\n\\put(92,33){GUT or}\n\\put(92,27){Quantum Gravity}\n\\end{picture}\n\\end{center}\n\\caption{The division of virtualities as accepted in the calculations\nthroughout of this paper.}\n\\label{division}\n\\end{figure}\n\nIt is important to stress that the global sources of composite operators\ndevelop the Higgs-like potential in the region of $[\\Lambda;M]$, so that the\ncorresponding couplings of self-interaction as well as Yukawa constants fall\noff to zero under the increase of virtuality from $\\Lambda$ to $M$. Therefore,\nthe dynamics of local interactions is perturbative in the region of $[\\Lambda;\nM]$, while the notion on the ``strong self-interaction regime'', strictly\nspeaking, concerns for a theory with the local Higgs field, i.e. we replace\n$\\left.{\\rm SSIR}\\right|_{\\rm local}\\to \\left.{\\rm WSIR}\\right|_{\\rm\ncomposite}$.\n\nPostponing a supersymmetric extension in a time, in this paper we develop a\nnew insight into the breaking of electroweak symmetry by means of exploring the\ndynamics of SM to calculate an effective potential for a source of bi-local\noperator with no technicolor interactions. The physical reasoning for the\nchoice of operator under study was hinted in ref.\\cite{nz}. So, in the second\norder of perturbation theory we write down the following contribution to the\naction:\n\\begin{equation}\n i S_{2m} = - \\int dx dy\\;\\; {\\bf T}[\\bar L_L(x) \\slashchar{B}(x) L_L(x) \\cdot\n\\bar L_R(y) \\slashchar{B}(y) L_R(y)]\\cdot 4\\pi \\alpha_Y \\cdot\n\\frac{Y_L}{2}\\cdot \\frac{Y_R}{2},\n \\label{eq:2}\n\\end{equation}\nwhere we have introduced the notations $L_L$ for the left-handed doublets and\n$L_R$ for the right-handed singlets, $B$ is the gauge field of weak hypercharge\n$Y$, $\\alpha_Y$ is its coupling constant. Note, that the gauge field of local\n${\\sf U(1)}$-group is the only one interacting with both the left-handed and\nright-handed fermions. If we suggest a nontrivial vacuum correlators with the\ncharacteristic distance $r\\sim 1\/v$\n\\begin{eqnarray}\n \\langle 0|{\\bf T}[ \\slashchar{B}(x) L_L(x)\\cdot\n\\bar L_R(y) \\slashchar{B}(y)] |0\\rangle &\\Rightarrow&\n\\frac{\\delta(x-y)}{v^4} \\langle 0| {\\bf T}[\\slashchar{B}(x) L_L(x)\\cdot\n\\bar L_R(x) \\slashchar{B}(x)] |0\\rangle \\nonumber \\\\ & \\sim &\n\\delta(x-y)\\; v ,\n \\label{eq:3}\n\\end{eqnarray}\nsupposing that the scales of expectations for $BB$ and $L_L \\bar L_R$ are\ndriven by $v^2$ and $v^3$, respectively, then the Dirac masses of fermions are\ndetermined by the action\\footnote{By the way, Eq.(\\ref{eq:3a}) implies that\nin the SM the neutrino is massless since its right-handed component is\ndecoupled, $Y_R=0$.}\n\\begin{equation}\n \\label{eq:3a}\n S_{fm} \\sim \\int dx\\; \\bar L_L(x) L_R(x)\\cdot v\\cdot\n4\\pi \\alpha_Y \\cdot\n\\frac{Y_L}{2}\\cdot \\frac{Y_R}{2} +{\\rm h.c.}\n\\end{equation}\nIn this way we extend the SM action by the initial bi-local bare $J$-term\n$$\nS_{ib}=\\int dx dy\\; N_J\\cdot J(x,y)\\; [\\bar L_R(x)\\;\n\\underbrace{\\slashchar{B}^{\\perp}(x) \\slashchar{B}^{\\perp}(y)} \\;\nL_L(y)] - \\int dx \\phi(x) J(x,x)+{\\rm h.c.},\n$$\nwhere $N_J=\\pi\\alpha_Y\\cdot Y_l\\cdot Y_R$, and\n$\\underbrace{\\cdot\\;\\cdot} $ denotes the propagation of\ntransversal U(1)-gauge field $B_{\\mu}^{\\perp}\n=(g_{\\mu\\nu}-\\partial_\\mu\\partial_\\nu\/\\partial^2)B^{\\nu}$, which\nis independent of the longitudinal mode, so that\n$$\n\\underbrace{B_\\mu^{\\perp}(x) B_\\nu^{\\perp}(0)} = -i g_{\\mu\\nu}\\int\n\\frac{d^4 p}{(2\\pi)^4} e^{ipx} \\frac{1}{p^2}\n$$\nto the leading order of perturbative theory. To the bare order the\nequation of motion for the bi-local field results in the\nstraightforward substitution of local field $\\phi$, as it stands\nin the above consideration for the correlators, developing the\nvacuum expectation values. After the analysis of divergences in\nthe $J$-dependent Green functions, the corresponding contra-terms\nmust be added to the action. Then the $J$-source can be integrated\nout or renormalized, that results in a Higgs-like action,\ncontaining some couplings to fermions as well as a suitable\npotential to develop the spontaneous breaking of electroweak\nsymmetry.\n\nWe stress that there are no other suitable composite operators appearing in the\nsecond order of SM gauge symmetry with the quantum numbers relevant to the\nHiggs interactions providing the generation of fermion masses through the\nYukawa-like couplings except the operators described above.\n\nIn this paper we calculate the effective potential up to the quartic term for\nthe sources corresponding to the bi-local composite operators of quarks and\nleptons to the one-loop accuracy of renormalization in the SM. The\nnormalization condition of potential parameters: $\\mu^2$ and $\\lambda$ standing\nin\n$$\nV(J^\\dagger,J) = -\\mu^2\\cdot J^\\dagger J+\\lambda \\cdot (J^\\dagger J)^2,\n$$\nis strictly defined in the SM, since we do not involve some additional\ninteractions. Therefore, both $\\mu^2$ and $\\lambda$ for a nonfundamental source\nmust be equal to zero, exactly, i.e. $V=0$, which, however, can be satisfied at\na single scale $M$ because of logarithmic renormalization for couplings, so\nthat\n\\begin{equation}\n\\mu^2(M) =0,\\;\\;\\; \\lambda(M)=0.\n\\label{matchnull}\n\\end{equation}\nIt is essential that the choice of composite operators is conformed to the\neffective action of SM in the second order over the gauge couplings. Otherwise,\nthe introduction of arbitrary composite operators with the given properties\nwith respect to the gauge symmetry generally does not imply the imposition of\nmatching condition in (\\ref{matchnull}), which is extremely important, since it\nremoves an uncertainty of the potential due to a finite renormalization of\nparameters.\n\nBelow $M$, the mass parameter $\\mu^2(\\Lambda)$ depending on the ``infrared\ncut-off $\\Lambda$, is positive, and the electroweak symmetry is broken down.\nSo, we suppose that the bi-local representation is valid in the range of\nvirtualities: $[\\Lambda;M]$, and below $\\Lambda$ we can explore the local\nHiggs fields.\n\nAs was shown in ref. \\cite{HaHa}, a variety of composite operators appropriate\nfor the Higgs quantum numbers can be rearranged so that practically arbitrary\nvalues of higgs mass or $t$ quark mass could be derived. In other words, in\norder to get a definite description of higgs sector, one should to suppress\ncontributions by a lot of composite operators except the special ones. this\ndominance of several composite operators is usually motivated by an extended\ndynamics beyond the SM, for instance, by the thechnicolor providing the\ndominance of some bound channels.\n\nIn the present paper, the choice of dominant composite operators is dictated by\nthe SM gauge symmetry, since we isolate the only composite structure in the\nsecond order over the gauge couplings, while the appearance of other operators\ntakes place at higher orders, and, hence, their contributions must be\nsuppressed\\footnote{This suppression becomes even better at higher virtualities\nbecause of the asymptotic freedom of nonabelian interactions, while the abelian\ncharge remains small up to the GUT scale.}. Moreover, this motivation on the\nform of composite operators makes us to add the matching condition of\n(\\ref{matchnull}), which is a new idea for the composite models, and it is\ncertainly due to the electroweak nature of composite operators.\n\nThus, for a walker travelling from low scales to higher ones, the\nwhole picture of electroweak symmetry breaking looks as the\nfollowing:\n\\begin{itemize}\n\\item[1.] The SM extension with several Higgs fields is the local theory with\nthe ultraviolet cut-off $\\Lambda$.\n\\item[2.] The parameters of Higgs potential at the scale $\\Lambda$ is matched\nwith the effective potential of bi-local source, calculated in the range\n$[\\Lambda; M]$, so that $M$ denotes the scale, where the potential is exactly\nzero.\n\\end{itemize}\nThe value of $\\Lambda$, hence, can be related with the masses of\ngauge bosons, or the vacuum expectation value ({\\sc vev}) $v_{\\rm\nSM}$ for the Higgs field in the SM. The value of $M$ with respect\nto $\\Lambda$ is fixed by two simple requirements: at the matching\npoint $\\Lambda$ the Yukawa constant of $t$-quark calculated with\nthe composite operators is determined by the condition of infrared\nfixed point in the local theory \\cite{Hill}, while the Yukawa\nconstant is expressed in terms of abelian gauge coupling at the\nscales of $\\Lambda$ and $M$ due to the consistent matching\ncondition of potential (\\ref{matchnull}). At fixed $\\Lambda$ this\nsupposition makes $M$ to grow to the GUT scale, that implies the\nsolution of naturalness. So, we can read off the third point:\n\\begin{itemize}\n\\item[3.] The Yukawa constants of heaviest fermions in the local theory have\nthe matching conditions at $\\Lambda$ to the couplings given by the bi-local\nrepresentation, so that the infrared fixed point for the $t$-quarks is exactly\nreached.\n\\end{itemize}\nThe masses of $b$-quark and $\\tau$-lepton can be also calculated after the use\nof both the definite matching at $\\Lambda$ and infrared fixed points in the RG\nequations below $\\Lambda$.\n\nFinally, the potential of Higgs fields at $\\Lambda$ can serve to estimate the\nmasses of neutral scalar particles by means of RG evolution and the infrared\nfixed point for the quartic vertex $\\lambda$.\nThe important property of fixed points under consideration is that the Yukawa\nconstants and quartic coupling are given by appropriate combinations of gauge\ncoupling constants.\n\nThus, the local theory with the local Higgs fields and the electroweak symmetry\nbreaking can be certainly matched to the effective potential of sources for the\nbi-local composite operators of quarks and leptons at the scale $\\Lambda$ and\nto the corresponding Yukawa constants, which are calculable in the region of\nvirtualities $[\\Lambda; M]$, so that the fixed point matching of $t$-quark\ncoupling and the symmetry matching-condition of null effective potential\n(\\ref{matchnull}) result in $M$ living in the GUT area.\n\nThen we find the following general results:\n\n1. Three bi-local composite operators formed by the fermions of heaviest\ngeneration, develop the effective potential of their sources, so that nonzero\n{\\sc vev}'s break the electroweak symmetry. We treat these dynamics above the\nscale $\\Lambda$ as the strong self-interaction regime for three independent\nscalar higgses as equivalent to the weak self-interaction regime for three\nindependent sources of composite operators.\n\n2. The position of matching point $\\Lambda=633$ GeV is fixed by the measured\nmasses of gauge bosons, after the higgs sector is given by three independent\nscalar fields.\n\n3. At $\\Lambda$ the infrared fixed point condition is satisfied for the\nYukawa coupling of $t$ quark, only, while the couplings of $b$ quark and $\\tau$\nlepton evolve to the fixed points at lower scales in agreement with the current\ndata available. The masses of higgses evolve too.\n\n4. Under the item 3, the position of ultraviolet cut off $M\\sim\n10^{12}-10^{19}$ GeV with respect to $\\Lambda$ is given by the condition of\nzero effective potential for the sources of composite operators, that is\ngoverned by the renormalization group for the ${\\sf U}(1)$ hypercharge, so that\nwe find the natural hierarchy for $M$ and $\\Lambda$.\n\nThe paper is organized in the following way: Section II is devoted to the\ndefinition of sources for the bi-local composite operators and calculation of\neffective potential to the one-loop accuracy. The masses of gauge bosons and\nYukawa constants of fermions are evaluated in Section III at the scale\n$\\Lambda$. The exploration of infrared fixed point conditions for the Yukawa\nconstants and quartic Higgs coupling is considered in Section IV. Numerical\nestimates of masses for the heaviest fermions as well as the Higgs fields are\ngiven in Section V. In Section VI we shortly discuss the problem of generations\nand the vacuum structure. The obtained results and the points of discussion are\nsummarized in Conclusion.\n\n\\section{Sources of composite operators and effective potential}\n\nLet us define the following bare actions for the sources of bi-local operators\n\\begin{eqnarray}\nS_\\tau &=& \\int dx dy\\; N_\\tau\\cdot J_{\\tau}^\\dagger (x,y)\\; [\\bar\n\\tau_R(x)\\; \\underbrace{\\slashchar{B}^{\\perp}(x)\n\\slashchar{B}^{\\perp}(y)}\n\\; \\tau_L(y)] +{\\rm h.c.},\\nonumber \\\\\nS_t &=& \\int dx dy\\; N_t\\cdot J_t^\\dagger (x,y)\\; [\\bar\nt_R(x)\\cdot n\\; \\underbrace{\\slashchar{B}^{\\perp}(x)\n\\slashchar{B}^{\\perp}(y)}\n\\; \\bar n\\cdot t_L(y)] +{\\rm h.c.}, \\label{act}\\\\\nS_b &=& \\int dx dy\\; N_b\\cdot J_b^\\dagger (x,y)\\; [\\bar\nb_R(x)\\cdot n\\; \\underbrace{\\slashchar{B}^{\\perp}(x)\n\\slashchar{B}^{\\perp}(y)} \\; \\bar n\\cdot b_L(y)] +{\\rm\nh.c.},\\nonumber\n\\end{eqnarray}\nwhere we have introduced the ${\\sf SU(3)}$-triplet unit-vector $n_i$, so that\n$\\bar n\\cdot n=1$, and the $n$-dependent terms in the effective action after\nthe account for the loop-corrections have to be averaged over $n_i$ to restore\nthe explicit invariance under the transformations of ${\\sf SU(3)}$. For\ninstance, since we generally have $n_i\\bar n_j= \\frac{1}{3} \\delta_{ij}+\n\\frac{1}{\\sqrt{3}}\\lambda^a_{ij} F_a$, we can straightforwardly check that\n$$\n\\langle n_i \\bar n_j\\rangle = \\frac{1}{3}\\delta_{ij},\\;\\;\\;\n\\langle F_a\\rangle=0,\\;\\;\\; \\langle F_a F_b\\rangle = \\frac{1}{8}\\delta_{ab},\n$$\nand so on.\n\nFor nonzero $Y_L$ and $Y_R$, which are under consideration, we can redefine\nthe factors $N_J$ to include the hypercharges into the definition of sources,\nso that $\\tilde N_p = \\alpha_Y$ and $\\tilde J_p =\\pi\\cdot Y_L \\cdot Y_R J_p$\nfor $p=t,\\; b,\\; \\tau$, which will not change the final results concerning for\nthe physical quantities: masses and couplings. In what follows we will omit the\ntildes for the sake of briefness.\n\n\\setlength{\\unitlength}{1mm}\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(70,40)\n\\put(7,0){\\epsfxsize=5.5cm \\epsfbox{1.ps}}\n\\put(33.5,30){$J^\\dagger$}\n\\put(10,10){$L$}\n\\put(60,10){$R$}\n\\end{picture}\n\\end{center}\n\\caption{The vertex of global source $J^\\dagger$ for the bi-local operator of\nleft-handed and right-handed fermions, where the huge dot denotes the\npropagation of hypercharge gauge boson.}\n\\label{vert}\n\\end{figure}\n\nIn the calculations of effective potential we consider the global values of\nsources independent of local coordinates: $\\partial_{x,y} J(x,y) \\equiv 0$. The\ncorresponding vertex derived from actions (\\ref{act}) is shown in Fig.\n\\ref{vert}.\nFor the $t$-quark it has the form\n\\begin{equation}\n\\Gamma_t = i\\alpha_Y\\; J^\\dagger\\; \\bar t_R(p)\\cdot n\\; \\frac{-4i}{p^2}\\;\n\\bar n\\cdot t_L(p)+{\\rm h.c.}\n\\end{equation}\n\nThe diagrams for the calculation of quadratic and quartic terms of effective\npotential are shown in Figs. \\ref{mu} and \\ref{lam}, respectively.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(70,40)\n\\put(7,0){\\epsfxsize=5.5cm \\epsfbox{2.ps}}\n\\put(33.5,35){$L$}\n\\put(33.5,0){$R$}\n\\put(33.5,26){$p$}\n\\put(33.5,9){$p$}\n\\put(8,17){$J$}\n\\put(59,17){$J^\\dagger$}\n\\end{picture}\n\\end{center}\n\\caption{The $J^\\dagger J$-term in the effective potential.}\n\\label{mu}\n\\end{figure}\nThe parameters of potential\n\\begin{equation}\nV(J^\\dagger,J) = -\\mu^2\\cdot J^\\dagger J+\\lambda \\cdot (J^\\dagger J)^2,\n\\end{equation}\ncan be written down in the euclidean space as\n\\begin{eqnarray}\ni\\mu^2_B & = & -i N_J^2 \\int_{\\Lambda^2}^{M^2} \\frac{d^4 p}{(2\\pi)^4}\\;\n\\frac{4^2{\\rm tr}[P_L \\slashchar{p} \\slashchar{p}]}{(p^2)^4},\\\\\n-i4\\lambda_B & = & -2 i N_J^4 \\int_{\\Lambda^2}^{M^2} \\frac{d^4 p}{(2\\pi)^4}\\;\n\\frac{4^4{\\rm tr}[P_L \\slashchar{p} \\slashchar{p} \\slashchar{p}\n\\slashchar{p}]}{(p^2)^8},\n\\end{eqnarray}\n\\noindent\nwhich are independent of the fermion flavor. Here $P_L=\n\\frac{1}{2}(1-\\gamma_5)$ is the projector on the left-handed fermions.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(110,40)\n\\put(7,0){\\epsfxsize=4cm \\epsfbox{3.ps}}\n\\put(43.5,36){$J^\\dagger$}\n\\put(43.5,0){$J$}\n\\put(2,0){$J^\\dagger$}\n\\put(2,36){$J$}\n\\put(2,17.5){$R$}\n\\put(43.5,17.5){$R$}\n\\put(67,0){\\epsfxsize=4cm \\epsfbox{4.ps}}\n\\put(103.5,36){$J^\\dagger$}\n\\put(103.5,0){$J$}\n\\put(62,0){$J^\\dagger$}\n\\put(62,36){$J$}\n\\put(62,17.5){$L$}\n\\put(103.5,17.5){$L$}\n\\end{picture}\n\\end{center}\n\\caption{The $(J^\\dagger J)^2$-term in the effective potential.}\n\\label{lam}\n\\end{figure}\n\nSupposing $M^2\\gg \\Lambda^2$, we find\n\\begin{eqnarray}\n-\\mu^2_B & = & N_J^2\\; \\frac{2}{\\pi^2}\\; \\frac{1}{\\Lambda^2},\\\\\n\\lambda_B & = & N_J^4\\; \\frac{4}{\\pi^2}\\; \\frac{1}{\\Lambda^8}.\n\\end{eqnarray}\n\nAs we have already mentioned in the Introduction, the effective potential has\nto be subtracted, so that at the scale $M$ it equals zero, exactly, since we\ndeal with the source of composite operators not involving some interactions\nbeyond the gauge interactions of SM. Then, we get\n\\begin{eqnarray}\n\\mu^2_R(\\Lambda) & = & \\frac{2}{\\pi^2}\\; \\frac{1}{\\Lambda^2}\\; \\alpha_Y^2(M)\n(1-\\varkappa^2(\\Lambda)),\\\\\n\\lambda_R(\\Lambda) & = & \\frac{4}{\\pi^2}\\; \\frac{1}{\\Lambda^8}\\; \\alpha_Y^4(M)\n(1-\\varkappa^2(\\Lambda))^2,\n\\end{eqnarray}\nwhere we have introduced the notation for\n$$\n\\varkappa(\\Lambda) = \\frac{\\alpha_Y(\\Lambda)}{\\alpha_Y(M)},\n$$\nwith the normalization $\\varkappa(M)=1$. The scale-independent factors\n$\\alpha_Y^{2,4}(M)$ can be removed by the redefinition of sources: $J^\\prime =\n\\alpha_Y J$, which we imply below. In addition we introduce $J(\\Lambda) =\n\\frac{1}{\\Lambda^2} J^\\prime$ to obtain more usual notations. Then,\n\\begin{eqnarray}\n\\mu^2(\\Lambda) & = & \\frac{2}{\\pi^2}\\;\n(1-\\varkappa^2(\\Lambda))\\;{\\Lambda^2},\\\\\n\\lambda(\\Lambda) & = & \\frac{4}{\\pi^2}\\; (1-\\varkappa^2(\\Lambda))^2.\n\\end{eqnarray}\n\nThe vacuum expectation value, {\\sc vev}, is given by $\\langle J^\\dagger\nJ\\rangle = \\frac{\\mu^2}{2\\lambda}$, so that\n\\begin{equation}\n\\langle J^\\dagger(\\Lambda) J(\\Lambda)\\rangle = \\frac{1}{4}\\;\n\\frac{1}{1-\\varkappa^2(\\Lambda)}\\; \\Lambda^2.\n\\end{equation}\nRemember, that the potential parameters are the same for all charged heavy\nfermions: $t$-quark, $b$-quark and $\\tau$-lepton. The density of vacuum energy\nis independent of flavor, too,\n$$\nV({\\rm vac}) = -\\frac{\\mu^4}{4\\lambda} = -\\frac{1}{4 \\pi^2}\\;\\Lambda^4.\n$$\nThen the action represented as the sum of terms over the space-time intervals\nwith $d^4 x \\sim 1\/\\Lambda^4$, has the form\n$$\nS({\\rm vac}) = -\\int d^4 x\\; V({\\rm vac})\\sim \\sum \\frac{1}{4\\pi^2},\n$$\nand it is independent of $\\Lambda$.\n\n\\section{Masses of gauge bosons and Yukawa constants}\n\nThe diagrams, which result in the masses of gauge bosons, are shown in Fig.\n\\ref{mw}, where the permutations over the gauge bosons are implied.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(140,60)\n\\put(7,0){\\epsfxsize=5.5cm \\epsfbox{5.ps}}\n\\put(38,39){$A_j$}\n\\put(28,39){$A_i$}\n\\put(9,17){$J$}\n\\put(56,17){$J^\\dagger$}\n\\put(67,0){\\epsfxsize=5.5cm \\epsfbox{6.ps}}\n\\put(98,-5){$A_j$}\n\\put(88,39){$A_i$}\n\\put(69,17){$J$}\n\\put(116,17){$J^\\dagger$}\n\\end{picture}\n\\end{center}\n\\caption{The $(A_i A_j)$-terms in the effective potential of gauge bosons.}\n\\label{mw}\n\\end{figure}\n\n\nWe straightforwardly find that the couplings of gauge bosons are proportional\nto the differences of their charges, so that\n$$\nm_{12}^2 A_1^\\mu A_2^\\nu g_{\\mu\\nu} \\sim (Q_1^L-Q_1^R) (Q_2^L-Q_2^R) A_1^\\mu\nA_2^\\nu g_{\\mu\\nu},\n$$\nwhere $Q^{L,R}$ denote the charges of left-handed and right-handed fermions.\nThis implies that the vector-like gauge bosons, i.e. when $Q^L=Q^R$, remain\nmassless.\n\nFor the $W$- and $Z$-bosons after the subtraction procedure of $\\varkappa^2 \\to\n(1-\\varkappa^2)$, we find\n\\begin{eqnarray}\nm_W^2 & = & \\frac{4\\pi \\alpha_2}{2} \\sum_p \\frac{\\Lambda_p^2}{4\\pi^2},\\\\\nm_Z^2 & = & m_W^2 \\; \\frac{1}{\\cos^2 \\theta_W},\n\\end{eqnarray}\nwhere $\\theta_W$ is the Weinberg angle \\cite{Wein}, as usual, and the sum is\ntaken over the heavy flavors $p=t,\\; b,\\; \\tau$. As we have seen in the\nprevious section $\\Lambda_p=\\Lambda$ is independent of flavor, and, hence, we\ncan introduce the Higgs field $h_p$ with the {\\sc vev}, $\\langle h_p\\rangle=v$,\nso that\n$$\nv=\\frac{\\Lambda}{2\\pi},\\;\\;\\;\\; h_p(v) = \\frac{1}{\\pi}\n\\sqrt{1-\\varkappa^2(v)}\\;\nJ_p(v).\n$$\nThus, we get\n\\begin{equation}\nm_W^2 = \\frac{4\\pi \\alpha_2}{2}\\; 3 v^2,\n\\end{equation}\nso that $v_{\\rm SM}^2 = 3 v^2 \\approx (174\\; {\\rm GeV})^2$, when the potential\nat the scale $\\Lambda$ has the form\\footnote{There is a possibility to change\nthe convention on the prescription of scale by replacing $\\Lambda\\to v$.}\n\\begin{equation}\nV(h_p,h_p^\\dagger) = -2 \\Lambda^2\\; h_p^\\dagger\\cdot h_p + (2\\pi)^2\\;\n(h_p^\\dagger\\cdot h_p)^2.\n\\end{equation}\nWe check that the quadratic term $-2\\Lambda^2$ is exactly given by the one-loop\ncalculation in the local $\\phi^4$-theory with $\\lambda=(2\\pi)^2$ and cut-off\n$\\Lambda$.\n\nThe masses of fermions at the same scale can be derived from the diagram shown\nin Fig. \\ref{vert} by putting the fermion momenta to the given virtuality,\n$p^2=\\Lambda^2$. Then, after the appropriate subtraction [$\\varkappa\\to\n(1-\\varkappa)$] we\nget\n$$\nm_p = \\lambda_p \\cdot v,\n$$\nwith\n\\begin{eqnarray}\n\\lambda_t(v) = \\lambda_b(v) &=& \\frac{4\\pi}{3\\sqrt{2}}\\;\n\\sqrt{\\frac{1-\\varkappa(v)}{1+\\varkappa(v)}},\\\\\n\\lambda_\\tau(v) &=& 3 \\lambda_t(v).\n\\end{eqnarray}\nReplacing $v$ by $v_{\\rm SM}$, we find $\\lambda_p^{\\rm SM} =\n\\lambda_p\/\\sqrt{3}$.\n\nThus, we have calculated the masses of gauge bosons, Yukawa constants of\nheaviest fermions and the parameters of Higgs potential at the scale $\\Lambda$,\nwhich have to be matched with the quantities of local theory valid below\n$\\Lambda$.\n\n\\section{Infrared fixed points}\nTo the moment we have the local theory with three neutral Higgs fields, which\nare coupled with the appropriate heavy fermions in each sector, with the\ncut-off $\\Lambda$, where the Yukawa couplings have to be matched with the\nvalues calculated in the effective potential of sources for the composite\noperators.\n\nThe one-loop RG equations for the couplings\\footnote{The corresponding two-loop\nRG equations are given in ref. \\cite{Machacek:1983fi}. We shortly comment the\ninfluence of two-loop corrections below.} have the form \\cite{Pok}\n\\begin{eqnarray}\n\\frac{d\\ln \\lambda_t}{d\\ln \\mu} & = & \\frac{1}{(4\\pi)^2}\\; \\left[ \\frac{9}{2}\n\\lambda_t^2 -8 g_3^2 -\\frac{17}{12} g_Y^2 -\\frac{9}{4} g_2^2\\right],\\nonumber\\\\\n\\frac{d\\ln \\lambda_b}{d\\ln \\mu} & = & \\frac{1}{(4\\pi)^2}\\; \\left[ \\frac{9}{2}\n\\lambda_b^2 -8 g_3^2 -\\frac{5}{12} g_Y^2 -\\frac{9}{4} g_2^2\\right],\n\\label{RG1}\\\\\n\\frac{d\\ln \\lambda_\\tau}{d\\ln \\mu} & = & \\frac{1}{(4\\pi)^2}\\; \\left[\n\\frac{5}{2} \\lambda_\\tau^2 -\\frac{15}{4} g_Y^2 -\\frac{9}{4}\ng_2^2\\right],\\nonumber\n\\end{eqnarray}\nwhere $g_3^2 = 4\\pi \\alpha_s$ is the QCD coupling, $g_Y^2 = 4\\pi \\alpha_Y$ is\nthe hypercharge coupling, and $g_2^2 = 4\\pi \\alpha_2$ is the $\\sf SU(2)$-group\ncoupling. At ``low'' virtualities about $v\\sim 100$ GeV, the dominant\ncontribution to the $\\beta$-functions of quark couplings is given by QCD. We\nsuppose that the value of matching point $\\Lambda$ is dictated by the fixed\npoint condition for the $t$-quark: $\\frac{d\\ln \\lambda_t}{d\\ln \\mu}=0$\n\\cite{Hill}, i.e.\n\\begin{equation}\n\\lambda_t^2(v) = \\frac{64\\pi}{9}\\; \\alpha_s(v)+\\frac{34\\pi}{27}\\;\n\\alpha_Y(v)+2\\pi \\alpha_2(v)\\approx \\frac{64\\pi}{9}\\; \\alpha_s(v),\n\\label{app1}\n\\end{equation}\nwhen the matching gives\n\\begin{equation}\n\\lambda_t^2(v) = \\frac{8\\pi^2}{9}\\; \\frac{1-\\varkappa(v)}{1+\\varkappa(v)}.\n\\end{equation}\nTherefore, we find\n\\begin{equation}\n\\varkappa(v) = \\frac{1-\\frac{8\\alpha_s(v)}{\\pi}}{1+\\frac{8\\alpha_s(v)}{\\pi}}.\n\\end{equation}\nDue to the contribution by the hypercharge the difference between the RG\nequations for $\\lambda_b$ and $\\lambda_t$ causes the reach of infrared fixed\npoint for the $b$-quark at a lower scale than for the $t$-quark. Indeed, the\nfixed point condition for the $b$-quark reads off\n\\begin{equation}\n\\frac{9}{2}(\\lambda_b^2(\\mu)-\\lambda_t^2(\\mu)) = -g_Y^2(\\mu).\n\\label{fb}\n\\end{equation}\nMaking use of matching condition $\\lambda_b(v)=\\lambda_t(v)$, we can write down\n$$\n\\frac{d\\ln \\lambda_t\/\\lambda_b}{d\\ln \\mu}= - \\frac{1}{(4\\pi)^2}\\; g_Y^2,\n$$\nfor small changes, so that\n\\begin{equation}\n\\lambda_b(\\mu)-\\lambda_t(\\mu)\\approx -\\frac{\\lambda_t(\\mu)}{(4\\pi)^2}\\; g_Y^2\\;\n\\ln\\frac{\\Lambda}{\\mu}.\n\\label{rb}\n\\end{equation}\nThen, we can derive from (\\ref{fb}) and (\\ref{rb}) the following estimate of\ncurrent mass for the $b$-quark\n\\begin{equation}\n\\ln\\frac{m_t}{m_b(\\hat v_b)} = \\frac{\\pi}{4\\alpha_s(m_b(\\hat v_b))},\n\\end{equation}\nwhere the current mass of $t$-quark is given by\n$$\nm_t(m_t) = \\frac{8}{3} \\sqrt{\\pi \\alpha_s(v)}\\cdot v,\n$$\nsince the evolution of $t$-quark mass above the scale $v$ is determined by the\nrunning of effective constant, which is negligibly small in the interval\n$[v,m_t]$, and, hence, $m_t(m_t)\\approx m_t(v)$ with quite a high accuracy.\nThe scale of $b$-quark normalization is given by the following\n$$\nm_b(\\hat v_b) = \\frac{8}{3}\\sqrt{\\pi\\alpha_s(\\hat v_b)}\\cdot \\hat v_b,\n$$\nand we use the QCD evolution to extract the current mass of $b$-quark at the\nscale of its value\n$$\nm_b(m_b) = m_b(\\hat v_b)\\left(\\frac{\\alpha_s(m_b(m_b))}{\\alpha_s(\\hat\nv_b)}\\right)^{12\/25}.\n$$\n\nNext, we can evaluate the mass of $\\tau$-lepton in the same manner. At low\nenergies we modify the RG equation for the $\\tau$-coupling, neglecting the\nfour-fermion weak interactions and taking into account the photon contribution.\nSo, we have\n\\begin{equation}\n\\frac{d\\ln \\lambda_\\tau}{d\\ln \\mu} = \\frac{1}{(4\\pi)^2}\\; \\left[\n\\frac{5}{2} \\lambda_\\tau^2 -24\\pi \\alpha_{em}\\right],\n\\end{equation}\nand the infrared fixed point condition reads off\n\\begin{equation}\n\\lambda_\\tau^2 = \\frac{48\\pi}{5}\\; \\alpha_{em}.\n\\label{ftau}\n\\end{equation}\nThe change of $\\lambda_\\tau$ from the matching value\n$\\lambda_\\tau^2=9\\lambda_t^2 = 64\\pi\\alpha_s$ can be found in the solution of\n\\begin{equation}\n\\frac{d\\ln \\lambda_\\tau}{d\\ln \\mu} \\approx \\frac{1}{(4\\pi)^2}\\;\n\\frac{5}{2} \\cdot 9 \\lambda_t^2 \\approx 40\\; \\frac{\\alpha_s}{4\\pi},\n\\label{tauev}\n\\end{equation}\nso that\n\\begin{equation}\n\\lambda_\\tau(\\mu) =\n\\lambda_\\tau(v)\\cdot\\left(\\frac{\\alpha_s(\\mu)}\n{\\alpha_s(v)}\\right)^{-\\frac{40}{2 b_3}},\n\\label{rtau}\n\\end{equation}\nwhere $b_3 = 11-\\frac{2}{3}n_f =9$ at $n_f=3$. From (\\ref{ftau}), (\\ref{rtau})\nand $\\lambda_t^2= 64\\pi \\alpha_s\/9$ we deduce the relation\n\\begin{equation}\n\\alpha_s(m_\\tau) =\n\\alpha_s(v)\\cdot\\left(\\frac{3}{20}\\;\\frac{\\alpha_{em}(m_\\tau)}{\\alpha_s(v)}\n\\right)^{-\\frac{9}{40}}.\n\\end{equation}\nNote, that the one-loop evolution to such the large change of scales is quite a\nrough approximation. To improve the estimate of $\\tau$-lepton mass we integrate\n(\\ref{tauev}) numerically with the same boundary conditions and extract the\nvalue under consideration.\n\nLet us consider the way to estimate the masses of neutral Higgs bosons. The RG\nequations for the quartic couplings of scalar particles with the heaviest\nfermions are represented by the following:\n\\begin{equation}\n\\frac{d \\lambda}{d\\ln\\mu} = \\frac{3}{2\\pi^2}\\left[\\lambda^2-\\frac{a_p}{4}\\;\n\\lambda_p^4\\right],\n\\end{equation}\nwhere $a_t=a_b=1$, $a_\\tau=\\frac{1}{3}$, and we neglect the contribution given\nby the electroweak gauge couplings. This approximation is quite reasonable,\nsince at $\\Lambda(v)$ the quartic couplings $\\lambda(v)=(2\\pi)^2$ dominate. For\nthe Higgs fields coupled to the $t$- and $b$-quarks, the infrared fixed points\ncoincide with each other to the order under consideration, so that\n$$\n\\lambda(\\mu_H) \\approx \\frac{1}{2}\\; \\lambda^2_{t,b}(\\mu_H) = \\frac{32\\pi}{9}\\;\n\\alpha_s(\\mu_H),\n$$\nwhich implies that the corresponding masses of scalars are degenerated with a\nhigh accuracy. Let us evaluate the scale of reaching the infrared fixed point.\nThe evolution can be approximated at large $\\lambda$ by the equation\n$$\n\\frac{1}{\\lambda(\\mu_H)} =\n\\frac{1}{\\lambda(v)}+\\frac{3}{2\\pi^2}\\;\\ln\\frac{v}{\\mu_H},\n$$\nso that we derive\n\\begin{equation}\n\\ln \\frac{v}{\\mu_H} \\approx\n\\frac{3\\pi}{16\\alpha_s(\\mu_H)}-\\frac{1}{6},\n\\label{str-h}\n\\end{equation}\nIf we use the RG evolution for the QCD\ncoupling \\cite{PEP}\n$$\n\\frac{1}{\\alpha_s(\\mu_H)} = \\frac{1}{\\alpha_s(v)}-\n\\frac{b_3}{2\\pi}\\; \\ln\\frac{v}{\\mu_H},\n$$\nat $n_f=5$, we arrive to\n\\begin{equation}\n\\ln \\frac{v}{\\mu_H} \\approx \\frac{6}{55}\\;\\frac{\\pi}{\\alpha_s(v)},\n\\end{equation}\nalthough the straightforward equation for the scale in (\\ref{str-h}) can be\nmore accurate numerically.\n\nThus, following the general relation for the mass of Higgs field,\n$$\nm_H(\\mu) = 2 \\sqrt{\\lambda(\\mu)}\\cdot v,\n$$\nwe have the estimates\n\\begin{eqnarray}\nm_H(v) &=& 4\\pi\\cdot v,\\\\\nm_H(\\mu_H) &=& \\frac{8}{3}\\sqrt{2\\pi\\alpha_s(\\mu_H)} \\cdot v.\n\\end{eqnarray}\nAs for the Higgs field coupled to the $\\tau$-lepton, it is quite easily\nrecognize that the corresponding scale $\\mu$ is much greater than for the\nscalars coupled with the heaviest quarks, and, hence, its mass is greater than\nwe have considered above. Indeed, we can use the evolution of $\\lambda_\\tau$ at\nlarge scales, where it is driven as $\\lambda_\\tau = 3 \\lambda_t$, so that we\nderive the relation analogous to (\\ref{str-h})\n\\begin{equation}\n\\ln \\frac{v}{\\mu_{H_\\tau}} \\approx\n\\frac{\\pi}{16\\sqrt{3}\\alpha_s(\\mu_{H_\\tau})}-\\frac{1}{6},\n\\label{str-htau}\n\\end{equation}\nand\n$$\nm_{H_\\tau}(\\mu_{H_\\tau}) =\n{8}\\sqrt{\\frac{2}{\\sqrt{3}}\\pi\\alpha_s(\\mu_{H_\\tau})} \\cdot v.\n$$\nTo the moment we are ready to get numerical estimates.\n\n\\section{Numerical evaluation and the naturalness}\n\nFirst of all, the {\\sc vev}'s of Higgs fields are directly given by the masses\nof gauge bosons, so that\n$$\nv = 100.8\\pm 0.1\\; {\\rm GeV},\n$$\nand the cut-off\n$$\n\\Lambda = 2\\pi v= 633.0\\pm 0.6\\; {\\rm GeV,}\n$$\nwhere we use the experimental data shown in Table \\ref{tab}.\n\n\\begin{table}[th]\n\\begin{center}\n\\begin{tabular}{||p{2cm}|p{2.5cm}||}\n\\hline\n$m_W$, GeV & $80.41\\pm 0.09 $ \\\\\n\\hline\n$\\alpha_2^{-1}$ & $29.60\\pm 0.04 $ \\\\\n\\hline\n$m_t$, GeV & $174\\pm 5 $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The experimental data on the electroweak parameters\n[5,6].}\n\\label{tab}\n\\end{table}\n\nThe estimates for the masses of fermions depend on the values of QCD coupling\nconstant. We put the value\\footnote{The central value is slightly displaced\nfrom the ``world average'' $\\alpha_s(m_Z)=0.119\\pm 0.002$ \\cite{PDG}, though it\nis within the current uncertainty. However, this parameter corresponds to the\nLEP fit \\cite{SM} as well as to the recent global fit of structure functions\n\\cite{AK}.}\n$$\n\\alpha_s(m_Z)=0.122\\pm 0.003,\n$$\nwhich corresponds to the $\\Lambda^{(5)}_{\\overline{\\rm MS}}=255\\pm 45$ MeV in\nthe three-loop approximation for the $\\beta$-function. We suppose that the\nthreshold values for the changing the number of active quark flavors are equal\nto $\\hat m_b=4.3$ GeV and $\\hat m_c=1.3$ GeV. The variation of threshold values\nis not so important in the estimates in contrast to the uncertainty in\n$\\alpha_s$, which dominates in the error-bars.\n\nThen we can numerically solve the equations in the previous section to find the\ncurrent masses\n\\begin{eqnarray}\nm_t(m_t) & = & 165\\pm 1\\; {\\rm GeV,} \\nonumber\\\\\nm_b(m_b) & = & 4.18\\pm 0.38\\; {\\rm GeV,} \\\\\nm_\\tau(m_\\tau) & = & 1.78\\pm 0.27\\; {\\rm GeV.}\\nonumber\n\\end{eqnarray}\nThe one-loop relation of perturbative QCD for the pole mass of quark \\cite{PEP}\nis given by\n$$\nm^{(p)} = m(m) \\left(1+\\frac{4}{3\\pi}\\; \\alpha_s(m)\\right).\n$$\nThen we estimate\n\\begin{eqnarray}\nm_t^{(p)} &=& 173\\pm 2\\; {\\rm GeV},\\nonumber\\\\\nm_b^{(p)} &=& 4.62\\pm 0.40\\; {\\rm GeV}.\\nonumber\n\\end{eqnarray}\nThe QED correction to the $\\tau$-lepton mass is negligibly small.\n\nWe see that the $t$-quark mass is in a good agreement with the direct\nmeasurements. The $b$-quark mass is in the desirable region. It is close to\nthat of estimated in the QCD sum rules \\cite{SVZ}, where $m_b(m_b)=4.25\\pm\n0.15$ GeV \\cite{BM}, and in the potential approach \\cite{KKO}, where\n$m_b(m_b)=4.20\\pm 0.06$ GeV. It is worth to note that the pole mass is not the\nvalue, which has a good convergency in the OPE approach (see references in\n\\cite{BM,KKO}), so we present it to the first order for the sake of reference.\nHowever, we stress also that the deviations from the central values are caused\nby the uncertainties in the $\\alpha_s$ running.\n\nThe infrared fixed masses of neutral scalars, coupled with the $t$- and\n$b$-quarks and the $\\tau$-lepton, equal\n\\begin{equation}\nm_H = 306\\pm 5\\; {\\rm GeV,}\\;\\;\\;\\; m_{H_\\tau} = 552\\pm 9\\; {\\rm GeV,}\n\\label{hg}\n\\end{equation}\nwhich can be compared with the global fit of SM at LEP yielding $m_H=\n76^{+85}_{-47}$ GeV \\cite{SM}. The central value of this fit was recently\nexcluded by the direct searches at modern LEP energies, where the constraint\nwas obtained $m_H > 95$ GeV \\cite{SM,PEP}. We expect, however, that\nmany-doublet models of Higgs sector have a different connection to the LEP\ndata. Indeed, the fit of SM with the single Higgs particle yields the value for\nthe logarithm $l_H=\\log_{10} m_H^{\\rm SM}[{\\rm GeV}] = 1.88^{+0.33}_{-0.41}$,\nwhereas this correction basically contributes into the observed quantities due\nto the coupling to the massive gauge bosons. Then, we can write down the\nfollowing approximation for this value in the model under consideration:\n$$\nl_H = \\frac{1}{3}\\sum_p \\kappa_p \\log_{10}\nm_{H_p}[{\\rm GeV}],\n$$\nwhere the factor $\\frac{1}{3}$ represents the fraction of scalar coupling in\nthe squares of gauge boson masses, respectively for $p=t,\\; b,\\; \\tau$, and\n$\\kappa_p$ stands for the possible formfactors at high virtualities of the\norder of masses of Higgs fields. To test, we put the simple approximation\n$$\n\\kappa_p \\approx \\frac{1}{1+\\frac{m^2_{H_p}}{\\Lambda^2}},\n$$\nwhich results in $\\kappa_t=\\kappa_b$ close to unit, and $\\kappa_\\tau\\sim\n\\frac{1}{4}$, so that the value under consideration is equal to\n$$\nl_H \\approx 1.86,\n$$\nthat is optimistically close to what was observed at LEP. So, the values in\n(\\ref{hg}) are not in contradiction with the current data.\n\nNext, since we deal with the strongly coupled version of Higgs sector\n(remember, that $m_H(v)\\approx 1267$ GeV), we need more careful consideration\nof effective potential to take into account the higher dimensional operators,\nrepresenting the multi-higgs couplings. So, we keep (\\ref{hg}) as soft\nestimates of masses for the Higgs fields, which implies that the decays into\nthe massive gauge bosons are the dominant modes for these scalar particles .\n\nFinally, we evaluate the scale $M$, where the electroweak symmetry has to be\nexactly restored. The value of $\\varkappa(v)$ is equal to\n$$\n\\varkappa(v) = \\frac{\\alpha_Y(v)}{\\alpha_Y(M)} = 0.532\\pm 0.005,\n$$\nwhich implies $\\alpha_1^{-1}(M) \\approx 32$. The implication of $\\varkappa$ for\n$M$ depends on the running of $\\alpha_Y=\\frac{3}{5}\\alpha_1$ \\cite{PEP}:\n$$\n\\frac{1}{\\alpha_1(M)} = \\frac{1}{\\alpha_1(v)}+\n\\frac{b_1}{2\\pi}\\; \\ln\\frac{M}{v},\n$$\nwhere $b_1$ is model-dependent. So, in the SM $b_1=-\\frac{4}{3}\\;\nn_g-\\frac{1}{10}\\; n_h$ with $n_g=3$ being the number of fermion generations,\n$n_h$ is the number of Higgs doublets, we obtain\\footnote{Numerically, we put\n$\\alpha_1^{-1} = 58.6$ for the order-of-magnitude estimate.}\n$$\nM_{\\rm SM} \\approx 2.5\\cdot 10^{19}\\; {\\rm GeV,}\n$$\nwhen in the SUSY extension $b_1=-2\\; n_g-\\frac{3}{10}\\; n_h$, so that\n$$\nM_{\\rm SUSY} \\approx 7\\cdot 10^{12}\\; {\\rm GeV.}\n$$\nHence, we obtain the broad constraints\n$$\nM = 7\\cdot 10^{12}-2.5\\cdot 10^{19}\\; {\\rm GeV,}\n$$\nand the value strongly depends on the set of fields in the region above the\ncut-off $\\Lambda$. At present, we cannot strictly draw a conclusion on a\npreferable point. However, we can state that the offered mechanism for the\nbreakdown of the electroweak symmetry solves the problem of naturalness, since\nthe observed ``low'' scale of gauge boson masses is reasonably related to the\n``high'' scale of GUT or even Planck mass.\n\nFinally, we comment on possible uncertainties of numerical\nestimates and a role of two-loop corrections. First, we analyze\nthe subleading terms in eq.(\\ref{app1}). The gauge charges\nneglected in the fixed point condition of (\\ref{app1}) result in\nthe displacement of $t$ quark mass by a value about 4 GeV, if we\ndo not change the normalization of QCD coupling constant. In this\nway, we note that under the account of gauge charge corrections in\n(\\ref{app1}) the same central value of $t$ quark mass, i.e. 165\nGeV, is reproduced at $\\alpha_s(m_Z^2) =0.118$, which coincides\nwith the Particle Data Group ``world-average''. Next, the two-loop\ncorrections in the RG equations for the Yukawa couplings\n\\cite{Machacek:1983fi} as applied to the $t$ quark lead to an\nadditional displacement of fixed point value. However, in this\ncase we have to take into account the one-loop correction to the\nrelation between the current mass and the pole mass of $t$ quark\ndue to the Higgs sector, that results in the following additive\nrenormalization of $m_t(m_t)$ \\cite{HK}\n$$\n\\frac{\\delta m_t(m_t)}{m_t(m_t)} = - \\frac{1}{16\\pi^2}\\,\\frac{9}{2}\\,\n\\frac{m_t^2}{v_{\\rm SM}^2},\n$$\nat the higgs mass $m_H\\approx 2 m_t$. The above correction compensates the\ndisplacement due to the two-loop modification of infrared fixed-point condition\nfor the $t$ quark.\n\nSecond, we study the two-loop corrections to the fixed point condition for the\n$b$ quark. The corresponding modification of (\\ref{fb}) reads off\n\\begin{equation}\n\\frac{9}{2}(\\lambda_b^2(\\mu)-\\lambda_t^2(\\mu)) \\approx -g_Y^2(\\mu)\\left(1 -\n\\frac{1}{16\\pi^2}\\,\\frac{4}{3}\\,\\lambda_t^2(\\mu)\\right)+O(g^4),\n\\label{fb2}\n\\end{equation}\nthat results in the appropriate correction in (\\ref{rb}), {\\it\nviz.}, in the small change of slope in front of log about 2\\%. The\nsolution of equations for the running $b$ quark mass under the\nvariations caused by the introduction of two-loop corrections and\nthe uncertainty in the coupling constant of QCD is shown in Fig.\n\\ref{uncert}. We can straightforwardly see that the variation of\nslope in the RG equation for the Yukawa constant of $b$ quark due\nto the two-loop corrections results in uncertainties, which are\nmuch less than the variation of $b$ quark mass caused by the\nuncertainties in the running coupling constant of QCD at moderate\nvirtualities about the $b$ quark mass. Therefore, the dominant\norigin of uncertainty for the $b$ quark mass is the normalization\nof $\\alpha_s$. The same conclusion can be drawn for the mass of\n$\\tau$ lepton. Therefore, the uncertainty in the estimates of\nmasses for the $b$ quark and the $\\tau$ lepton is not essentially\nchanged by the introduction of two-loop corrections, while the\nvalue of $t$ quark mass depends on the normalization of $\\alpha_s$\nas well as the two-loop corrections combined, so that the\nuncertainty in the current mass can reach 4 GeV in $m_t$.\n\n\\begin{figure}[th]\n\\begin{center}\n\\begin{picture}(150,50)\n\\put(0,0){\\epsfxsize=7cm \\epsfbox{9.eps}}\n\\put(80,0){\\epsfxsize=7cm \\epsfbox{10.eps}}\n\\put(0,45){$\\lambda_b\/\\lambda_t$}\n\\put(80,45){$\\lambda_b\/\\lambda_t$}\n\\put(63,-2){$\\ln m_t\/\\mu$}\n\\put(140,-2){$\\ln m_t\/\\mu$}\n\\end{picture}\n\\end{center}\n\\caption{The variation of RG solution for the $b$ quark Yukawa constant under\nthe introduction of two-loop corrections with respect to the one-loop result\n(solid lines) and the uncertainty caused by the change in the normalization of\n$\\alpha_s$ with $\\Lambda_{\\overline{\\rm MS}}^{(5)} = 200$ MeV and\n$\\Lambda_{\\overline{\\rm MS}}^{(5)} = 280$ MeV (the band). At the right figure\nwe scale the square region marked in the left picture.}\n\\label{uncert}\n\\end{figure}\n\nAs for the estimates of masses for the scalar fields, we emphasize that they\ngive preliminary results, and further investigations are in progress, since we\nshould, first, sum up subleading terms with higher powers of the higgs field\nsquared in the effective potential and, second, consider complete RG equations\nfor the quartic self-coupling, including suppressed terms. Nevertheless, our\npreliminary estimates show that the scalar fields should be significantly\nheavy.\n\n\\section{Generations, the number of Higgs fields and vacuum}\n\nIn the previous sections we have introduced three independent global sources\nfor the bi-local operators composed by the fermions of the heaviest generation,\ni.e., $t$ quark, $b$ quark and $\\tau$ lepton. In the SSIR, these sources\nacquire the effective potentials providing the spontaneous breaking of\nelectroweak symmetry. Below the scale $\\Lambda$ we assume the connection of\nsuch the potentials with the potentials of local Higgs fields. Thus, we suppose\nthe introduction of three independent local Higgs doublets at the low energies.\nTherefore, we suggest the condensation of sources related with the heaviest\ngeneration only.\n\nWe have found the `democratic' form for the potentials of independent sources\nin the SSIR. All three potentials have the same values of quadratic and quartic\ncouplings, while we suggest evidently broken `democracy' for the fermion\ngenerations, since we do not introduce the condensation of sources for the\ncomposite operators built of junior fermions.\n\nIn this section we describe a possible development on the problem of fermion\ngenerations and the structure of vacuum in the Higgs sector.\n\nSo, let us introduce the notation of normalized {\\sc vev}'s for the global\nsources connected with the Higgs fields as follows:\n$$\n\\chi_p = h_p\/v, \\;\\;\\; p=\\tau,\\, t,\\, b,\n$$\nand the corresponding vacua $| 0_p\\rangle$, so that\n\\begin{equation}\n\\langle 0_p| \\chi_{p'} |0_{p''}\\rangle = \\delta_{pp'}\\delta_{p'p''}.\n\\label{norm}\n\\end{equation}\nThen we easily find that the mass terms\n\\begin{eqnarray}\n{\\cal L}^\\tau_{Y} & \\sim & \\bar \\tau_R \\tau_L \\cdot \\chi_\\tau + {\\rm h.c.},\n\\nonumber \\\\\n{\\cal L}^t_{Y} & \\sim & \\bar t_R t_L \\cdot \\chi_t\n+ {\\rm h.c.}, \\\\\n{\\cal L}^b_{Y} & \\sim & \\bar b_R b_L \\cdot \\chi_b\n+ {\\rm h.c.}, \\nonumber\n\\end{eqnarray}\ncould be represented by means of fields\n\\begin{equation}\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\\;\n\\left(\\begin{array}{ccc} 1 & 1 & 1 \\\\ 1 & \\omega & \\omega^2 \\\\ 1 & \\omega^2 &\n\\omega \\end{array}\\right)\n\\cdot\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n = U \\cdot\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right) ,\n\\label{trans}\n\\end{equation}\nas follows\n\\begin{eqnarray}\n{\\cal L}^\\tau_{Y} & \\sim & \\bar \\tau_R \\tau_L \\cdot\n(\\varphi_1+\\varphi_2+\\varphi_3) +\n{\\rm h.c.},\n\\nonumber \\\\\n{\\cal L}^t_{Y} & \\sim & \\bar t_R t_L \\cdot (\\varphi_1+\\omega^2 \\varphi_2+\\omega\n\\varphi_3)\n+ {\\rm h.c.}, \\label{generation}\\\\\n{\\cal L}^b_{Y} & \\sim & \\bar b_R b_L \\cdot (\\varphi_1+\\omega \\varphi_2+\\omega^2\n\\varphi_3)\n+ {\\rm h.c.}, \\nonumber\n\\end{eqnarray}\nwhere we omit the Yukawa couplings and use the matrix $U$ defined in terms of\n$\\omega = \\exp(i\\frac{2\\pi}{3})$.\n\nSuch the transformation in (\\ref{trans}) relates the `heavy' basis of $\\chi_p$\nwith the `democratic' basis of $\\varphi_i$. So, the definition (\\ref{trans})\ncan be equivalently changed by permutations of $\\chi_\\tau \\leftrightarrow\n\\chi_t \\leftrightarrow \\chi_b$ or permutations of columns in the matrix $U$.\nSuch the permutations correspond to the finite cyclic group ${\\mathbb Z}_3$\nwith the\nbasis $\\omega$, so that complex phases of $\\varphi_i$ are given by\n$e^{iq\\frac{2\\pi}{3}}$ with the charges $q=(0,-1,1)$ of\n$(\\varphi_1,\\varphi_2,\\varphi_3)$.\n\nFurther, we can note that the vacuum fields have simple connections as follows:\n\\begin{eqnarray}\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{l} 1^{~} \\\\ 1\\\\ 1\\end{array}\\right) & \\Rightarrow &\n\\langle 0_\\tau| \\boldsymbol \\chi_1 |0_\\tau \\rangle =\n\\langle 0_\\tau|\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n|0_\\tau \\rangle =\n\\left(\\begin{array}{c} 1 \\\\ 0\\\\ 0\\end{array}\\right), \\nonumber\\\\[4mm]\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{l} 1 \\\\ \\omega\\\\ \\omega^2\\end{array}\\right) & \\Rightarrow &\n\\langle 0_t| \\boldsymbol \\chi_2 |0_t \\rangle =\n\\langle 0_t|\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n|0_t \\rangle=\n\\left(\\begin{array}{c} 0 \\\\ 1\\\\ 0\\end{array}\\right), \\\\[4mm]\n\\left(\\begin{array}{c} \\varphi_1 \\\\ \\varphi_2\\\\ \\varphi_3\\end{array}\\right) =\n\\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{l} 1 \\\\ \\omega^2\\\\ \\omega\\end{array}\\right) & \\Rightarrow &\n\\langle 0_b| \\boldsymbol \\chi_3 |0_b \\rangle =\n\\langle 0_b|\n\\left(\\begin{array}{c} \\chi_\\tau \\\\ \\chi_t\\\\ \\chi_b\\end{array}\\right)\n|0_b \\rangle =\n\\left(\\begin{array}{c} 0 \\\\ 0\\\\ 1\\end{array}\\right), \\nonumber\n\\end{eqnarray}\nwhere the conditions of normalization (\\ref{norm}) are reproduced.\n\nLet us {\\it postulate the extended definition of the vacuum}\n\\begin{equation}\n|vac \\rangle = |0_\\tau \\rangle \\otimes |0_t \\rangle \\otimes |0_b \\rangle ,\n\\label{vacuum}\n\\end{equation}\nwhich implies the ${\\mathbb Z}_3$ symmetry of the vacuum. Then, the couplings\nintroduced\nin (\\ref{generation}) are extended to three generations of fermions, whereas\nthe only generation of $\\tau$, $t$ and $b$ is heavy, while two junior\ngenerations are massless.\n\nThe vacuum definition (\\ref{vacuum}) could be treated as the following\nassumption:\n\n\\noindent\n{\\sf The number of generations equals the number of charged flavors in the\ngeneration as well as the number of Higgs fields in the local phase.}\n\nThus, we postulate the ${\\mathbb Z}_3$ symmetry of the vacuum as the\nfundamental\ndynamical principal of the theory. Moreover, we suggest that this symmetry of\nthe vacuum is exact, so that it is conserved under radiative corrections to the\nYukawa constants of fermions.\n\nA realistic description of generations, i.e., a model with nonzero masses of\njunior fermions is not the problem under the current consideration, and it is\nbeyond the scope of this work. Nevertheless, we add two notes.\n\nFirst, a general structure of Yukawa interactions with the ${\\mathbb Z}_3$\nsymmetry of\nthe vacuum has the form\n\\begin{eqnarray}\n{\\cal L}^\\tau_{Y} & \\sim & \\bar \\tau_R \\tau_L \\cdot (g_{1}^\\tau\n\\varphi_1+g_{2}^\\tau\n\\varphi_2+ g_{3}^\\tau\n\\varphi_3) + {\\rm h.c.},\n\\nonumber \\\\\n{\\cal L}^t_{Y} & \\sim & \\bar t_R t_L \\cdot ( g_{1}^t \\varphi_1+ g_{2}^t\n\\omega^2 \\varphi_2+\ng_{3}^t \\omega \\varphi_3)\n+ {\\rm h.c.}, \\label{real}\\\\\n{\\cal L}^b_{Y} & \\sim & \\bar b_R b_L \\cdot ( g_{1}^b \\varphi_1+ g_{2}^b \\omega\n\\varphi_2+\ng_{3}^b \\omega^2 \\varphi_3)\n+ {\\rm h.c.}, \\nonumber\n\\end{eqnarray}\nwhere the constants $g_i$ can be restricted by the following conditions:\n$g_{2,3}$ are real, while $g_1$ can be complex. So, the symmetric point $$\ng_1=g_2=g_3=1,\n$$\nrestores the hierarchy of single heavy generation and two massless generations.\nThe development of {\\it ansatz} (\\ref{real}) with the realistic values of\nparameters consistent with the current data on the quark masses and mixing CKM\nmatrix of charged quark currents was given in Ref. \\cite{generations}.\n\nSecond, the same form of mass matrix following from (\\ref{real}) could result\nin the leading order symmetry in the sector of neutrinos. Indeed, if we put\n$$\ng_1=e^{i\\frac{2\\pi}{3}},\\;\\;\\; |g_1|=g_2=g_3=1,\n$$\nthen we get the completely degenerate neutrinos, while small deviations in the\n$g_1$ phase and absolute values of $g_i$ will result in small differences of\nneutrino masses squared as observed in the neutrino oscillations\n\\cite{neutrinos}.\n\n\\section{Conclusion}\n\nIn this work we have argued that there are three starting points for the\npresented consideration of Higgs sector of electroweak theory. These\nmotivations are the following:\n\nFirst, as well known, in the Standard Model the Yukawa coupling constant of $t$\nquark obtained from the measurement of $t$ quark mass, is close to its value in\nthe infrared fixed point derived from the renormalization group equation. If\nthe Higgs sector is extended to three scalar fields separately coupled to\nthree heaviest charged fermions, i.e., the $t$, $b$ quarks and $\\tau$ lepton,\nwith the same vacuum expectation values of Higgs fields, then the Yukawa\ncoupling constant of $t$ quark is {\\bf exactly} posed to the infrared fixed\npoint. The challenge is whether this coincidence is accidental or not. We treat\nthis fact as the fundamental feature of dynamics determining the development of\nmasses. Moreover, the problem acquires an additional insight because the only\nfermion generation is heavy, while two junior generations are approximately\nmassless. These features can be attributed by introducing the fundamental\n${\\mathbb Z}_3$\nsymmetry of the vacuum, so that this symmetry is conserved under the radiative\nloop corrections responsible for the development of nonzero masses of junior\ngenerations.\n\nSecond, the strong self-interaction regime in the Higgs sector of Standard\nModel at large virtualities can be treated as the indication of nonlocality,\ni.e., the compositeness of operators relevant to the electroweak symmetry\nbreaking. We have introduced a separation of virtuality regions: the local\nHiggs phase in the range of $[0;\\, \\Lambda]$, the nonlocal strong\nself-interaction regime in the range $[\\Lambda;\\, M]$, and the symmetric phase\nabove $M$. We have determined a form of composite operators and their\nconnection to the local phase by considering the second order of effective\naction in the SM.\n\nThird, the development of effective potential for the global sources of\ncomposite operators from the point of symmetric phase $M\\sim 10^{12}-10^{19}$\nGeV is stopped in the infrared fixed point $\\Lambda \\approx 633$ GeV for the\nYukawa coupling constant of $t$ quark. If the dynamics of evolution is given by\nthe same electroweak group, then the large logarithm of $\\ln M\/\\Lambda$ is\nclose to the value appearing in the calculation of GUT scale. So, since the\nbreaking of electroweak symmetry and the fermion mass generation involve the\ncomposite operators with both left and right handed fermions, the gauge\ninteraction of ${\\sf U}(1)$ group determines the evolution of parameters of the\neffective potential in the strong self-interaction regime. Then the logarithm\nof $\\ln M\/\\Lambda$ in the coupling $g_1$ has the value depending on the set of\nfundamental fields above the scale $\\Lambda$. Anyway, $M$ should be close to\nthe\nGUT scale, and this fact implies the solution of problem on the naturalness.\n\nThen, following the above motivations, we have evaluated the basic parameters\nof the model. We have calculated the effective potential for the sources of\ncomposite operators, responsible for the breaking down the electroweak symmetry\nand generation of masses for the gauge bosons and heaviest fermions. The\ncorresponding couplings serve as the matching values for the quadratic and\nquartic constants in the potential of local Higgs fields as well as the Yukawa\ninteractions at the scale $\\Lambda$, which is the ultraviolet cut-off for the\nlocal theory and the low boundary of $[\\Lambda;M]$-range for the effective\npotential of sources coupled with the bi-local composite operators of quarks\nand leptons. At $M$ the local gauge symmetry is restored, so that the effective\npotential is exactly equal to zero.\n\nPosing the matching of Yukawa constant for the $t$-quark to the infrared fixed\npoint at the scale $\\Lambda$, related to the gauge boson masses, we have found\nthe null-potential value $M$ in the range of GUT park, which indicates the\nsolution of naturalness. The exploration of fixed points has resulted in the\nfollowing current masses of heaviest fermions: $m_t(m_t) = 165\\pm 4$ GeV,\n$m_b(m_b) = 4.18\\pm 0.38$ GeV and $m_\\tau(m_\\tau) = 1.78\\pm 0.27$ GeV. Two\ndegenerated neutral Higgs fields have the infrared fixed mass $m_H= 306\\pm 5$\nGeV, and the third scalar has the mass $m_{H_\\tau}= 552\\pm 9$ GeV. So, the\nestimates do not contradict with the current constraints, coming from the\nexperimental data.\n\nSome questions need for an additional consideration. To the moment, discussing\nno possible ways to study, we focus on the directions requiring a progress.\n\n\\begin{itemize}\n\\item[1.]\nWhat is a picture for the generation of Yukawa constants, responsible for the\nmasses of ``junior'' fermions?\n\\end{itemize}\nAs we have supposed in the paper, three sectors of Higgs fields are coupled to\nthe appropriate heavy fermions, so that we need speculations based on a\nsymmetry causing the junior generations to be massless to the leading order.\n\\begin{itemize}\n\\item[2.]\nWhat are the constraints on the model parameters as follows from the current\ndata on the flavor changing neutral currents and precision measurements at LEP?\n\\end{itemize}\nSo, we expect that this point is not able to bring serious objections against\nthe model, since we do not involve any interactions distinct from the gauge\nones, composing the SM.\n\\begin{itemize}\n\\item[3.]\nThe most constructive question is a supersymmetric extension of mechanism under\nconsideration. Can SUSY provide new features or yield masses of super-partners?\n\\end{itemize}\nTo our opinion, the SUSY extension is more complicated, since there are many\ndifferent relations between the mixtures of various sparticles, which all are\nexpected to be essentially massive ($\\tilde m\\sim \\Lambda$) in contrast to the\nSM, wherein the junior generations are decoupled from the Higgs fields to the\nleading order.\n\\begin{itemize}\n\\item[4.]\nA simple application, we think, is an insertion of the model into the TeV-scale\nKaluza-Klein ideology, being under intensive progress now \\cite{KK}.\n\\end{itemize}\nSo, $\\varkappa(v)$ transforms its logarithmic behavior to the\npower dependence on the scale. Then, the $M$ returns to a value\nnot far away from the matching point $\\Lambda\\sim 1$ TeV, as it\nshould be in the KK approach.\n\nThus, we have offered the model of electroweak symmetry breaking, which\nprovides a positive connection to the naturalness as well as needs some deeper\nstudies under progress.\n\nThis work is in part supported by the Russian Foundation for Basic Research,\ngrants 99-02-16558, 01-02-99315, 01-02-16585 and 00-15-96645, the Russian\nMinistry on the education, grant E00-3.3-62.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec1}\n\n\nDynamics on networks is a topic on which extensive works have been\ndone in recent years. Phenomena which have been studied include evolution of spin systems, \nopinion dynamics, disease spreading dynamics, etc. \\cite{Barrat,psen-book}. \nThe dynamical picture is quite different from that on regular lattices due to \nthe topological features of network. For example, one can define the dynamics in different ways for the voter model on a network while these \nrules become identical on lattices \\cite{caste}. \n\n\nThe study of Ising model\non networks has revealed a number of interesting features when \nstatic properties are considered. \nEven in one dimension, when randomly new links are added (or existing links rewired) as in a Watts Strogatz (WS) network \\cite{WS},\n one gets a phase transition \\cite{Weigt,Gitterman,Niko} which occurs\nwith mean field criticality \\cite{Kim,Herrero,Hong}. On Euclidean networks, indications of both mean field type and finite dimensional-like phase transitions have been \nshown to exist by varying the relevant parameter \\cite{achat-psen}. \nOn scale free networks \\cite{Leone,Golt,Igloi}, the transition temperature shows a logarithmic\nincrease with the system size which is perhaps the most surprising result \\cite{Alek,Bian,Herrero1,Viana}.\n\nWhile considering ordering dynamics on regular lattices at zero temperature for the Ising model using Glauber dynamics, \nit is known that for any dimension greater than one, freezing occurs with a probability dependent on the dimension \\cite{redner}. This happens \nwhen one considers a completely random initial condition. On random graphs and \nnetworks, one encounters similar freezing phenomena which depend on the \ndensity of added links \\cite{svenson,hagg,boyer,pratap}. \n\n\nOn random networks or graphs, the\n evolution of the Ising model from a completely random state\nshows that the system does not order.\nA freezing effect was observed and although there could\nbe an emergent majority of nodes with either spin up or spin down state, domains of nodes with opposing spins\nsurvive \\cite{svenson,hagg}. \nCareful observations show that the disordered state is not an absorbing state \\cite{castellano}. \nIt is instead a stationary active state,\nwith some spins flipping, while keeping the energy constant. The number of\ndomains remaining in the system is just two. The qualitative picture is then the same as on\nregular lattices for $d > 2$ \\cite{redner}, the system wanders forever in an iso-energy set of states.\nThe distribution of the steady state residual energy (which is identical to the number of bonds between oppositely oriented spins apart from a constant) \nfor the Ising model on a random networks\nwas investigated \\cite{baek}.\nIt was found that the distribution typically shows two peaks, one very close to the actual ground state where the residual\nenergy is zero and one far away from it.\n\n\nDynamics of the Ising model on the WS network with restricted rewiring has also been considered.\n Here initially\na spin is connected to its four nearest neighbours\n and then only the second nearest neighbour links are rewired with probability $p$.\nThe system therefore always remains connected.\nUnder the zero temperature Glauber dynamics,\nfreezing effect was observed\nfor any $p\\neq0$ \\cite{biswas}\n\n\nIn this paper, we have considered in detail the variation of the relevant thermodynamic quantities as \nfunctions of time for the zero temperature dynamics of Ising model\non various networks. Our main emphasis is on the Ising model on scale free networks at temperature $T=0$ as such studies \nhave not been made so far to the best of our knowledge. Apart from the question whether\nthe equilibrium state is reached or not, we have also explored the nature of the state in case it does not.\nWe are interested to see whether \nany active-absorbing phase transition occurs as the system parameters are\nvaried. \nBoth the random scale free network (RSF) and the Barab\\'{a}si-Albert (BA) network have been considered for the study.\nAlthough many results are known for the WS network, we have explored specifically the \npossibility of an active-absorbing phase transition in this network.\nThe BA model is studied to make direct comparison with the random scale free network, where results can be quite different \\cite{albert1}.\nAlso, comparison with respect to\nissues like freezing and absorbing phase transition \nmay be made for the RSF and the WS networks.\n\nIn section II, we describe the network models and the dynamical evolution.\nThe quantities which have been calculated are defined in Section III. The results are presented in the next section and in the last section we summarise and \ndiscuss the studies made. \n\n\\section{THE NETWORK MODELS and DYNAMICS}\nWe have considered \n three different types of network: (a) Random scale free, (b) Barab\\'{a}si-Albert and (3)\n Watts Strogatz (addition type) \\cite{WS} network. We describe in brief how these networks are generated\nand the dynamical evolution process. In this section we also include a brief discussion of how the numerical data had been analysed in previous studies.\n\n\\subsection{Random scale free (RSF) network} In the random scale free network the degree distribution follows a power law\nbut otherwise the network is random.\nTo generate random scale free network \\cite{albert1,doro,boguna,newman,albert,psen}, we assigned the degree of each node using\nthe power law:\n\\begin{equation}\n{ \\mathcal P}(k) \\sim k^{-\\gamma}, \n\\end{equation}\nwhere $k$ is the degree of node and $\\gamma$ is the characteristic degree exponent. The minimum value of $k$ is $1$ and \nthe maximum cut-off value is $\\surd N$, where $N$ number of nodes.\nThis cut-off value ensures that there is no correlation \\cite{cutoff}. \nStarting from the node with the maximum degree, links have been established with randomly selected distinct nodes.\n\n\\subsection{ Barab\\'{a}si-Albert (BA) network} \nBarab\\'{a}si-Albert network is a growing network where new nodes are joined to existing nodes\nwith preferential attachment. We start with the three fully connected nodes.\nSubsequently, a single node is added at a time to the network which is linked to \none existing node. The probability that the new node is connected\nto the existing $i$th node with degree $k_i$ is given by \\cite{albert,albert1}, \n\\begin{equation}\n \\Pi (k_i) = k_i\/ {\\sum_{j} {k_j}}.\n\\end{equation}\nDegree distribution of the network is a power law with exponent $\\gamma=3$;\n$ P(k) \\sim k^{-3}$ \\cite{albert1}.\n\n\n\\subsection{Watts Strogatz (addition type) model} Addition type WS network is a one dimensional regular\nchain with two nearest neighbour links as well as with some extra randomly connected long range links.\nHere the long range links have been added with probability $q\/N$ (total long range links $\\sim O(N)$), where $N$ is the number of nodes and $q$ is a parameter, \nwhich denotes the number of extra long range links per node on an average. So the average degree per node of this network is $2+q$,\nwhich is a finite quantity as $q\/N \\to 0$ in the thermodynamic limit. It is known \\cite{Gitterman} that an order-disorder transition occurs at $q=1$.\n\n\\subsection{Dynamics of Ising model on networks}\n\\label{ising-net}\nThe Hamiltonian of the Ising system in these networks can be expressed as\n\\begin{equation}\nH= -{\\sum_{i 0$. We find that for $q < 1$ the freezing probability reaches unity in the thermodynamic limit \nwhile it remains fairly constant beyond this value (Fig. \\ref{sat_ws}b). This is consistent with the \nactive-absorbing phase transition stipulated to take place at $q=1$; in the active state, one can never reach the equilibrium ground state configuration. \n\n\n\n\\begin{figure}[!h]\n\\resizebox{43mm}{!}{\\includegraphics {flip_add_sat.eps}}\n\\resizebox{42mm}{!}{\\includegraphics {freeze_inset.eps}}\n\\caption{(Color online) WS: Variation of (a) saturation value of fraction of spin flips $P_{sat}$\nand (b) freezing probability $F(q)$ as a function of $q$ for different system sizes $N$ (right panel). Inset of (b) shows the variation of peak value of $F(q)$\nwith the system size.\n \\label{sat_ws}\n}\n\\end{figure}\n\n\n\n$F(q)$ shows a peak at $q=0.2$ and the position of the peak is independent of system size which\nshows that the system is maximally disordered here.\nThe peak values of freezing probability as a function of system size $N$ is fitted with\nthe form $F_{peak}(N)=1-e^{-aN^b}$ and the estimated values of the exponents\nare $a\\approx 0.134$ and $b\\approx 0.540$ (inset of Fig. \\ref{sat_ws}b). It may be noted that the same form is obeyed in the BA model.\n\n\n\n\n\\section{Discussions and conclusions}\nWe have studied zero temperature Glauber dynamics of the Ising model on three types of networks and compared the results.\nFrozen state is observed in all the three types of network models. For random scale free network, freezing probability is\n unity for $\\gamma \\gtrsim 2$, i.e., the system never reaches the global equilibrium\n but for lower value of the parameter $\\gamma$, it decreases with system size and shows a system size independent \nbehaviour for $\\gamma \\lesssim 1$. This behaviour of freezing probability suggests a freezing transition point at $\\gamma_c^f \\simeq 2.0$.\nWe also find an order disorder transition point taking place very close to this point; in fact we believe that they occur at the same point\nand the difference is only a finite size effect. Also close to this point, the first active-absorbing (A-A) phase transition takes place; \nthe disordered phase for $2\\lesssim \\gamma \\lesssim 3$ is active while for $\\gamma \\lesssim 2$, \none gets an absorbing phase. A second A-A transition takes place close to $\\gamma \\approx 3.0$\nand the system evolves to an absorbing disordered state beyond this value. In all probability these two A-A transitions take place at $\\gamma=2.0$ and\n$3.0$ in absence of finite size effects; these two points are significant as for $\\gamma\\leq 2.0$, the average degree diverges while\nfor $\\gamma\\leq 3.0$, the degree variance diverges in the thermodynamic limit.\n\nOne can compare the results of RSF network and BA network for the same characteristic degree exponent $\\gamma =3.0$.\nThe residual energy shows an increase with system size $N$ in both cases in a power law manner with an exponent which is fairly\n close. Also, the saturation value of magnetisation decreases with $N$ in both networks in a power law manner, corresponding exponents are\n however quite different. The saturation energy and magnetisation behaviour are consistent with the fact that the state is disordered for both RSF and BA networks.\n However freezing probability for BA model and RSF network\nshow different behaviour with system size. For RSF, the freezing probability shows a system size independent behaviour while for BA model, it has a nonlinear dependence.\nAs far as the spin flip probability is concerned RSF (at $\\gamma = 3.0$) again differs from the BA network.\nThe RSF network and BA model are intrinsically different, BA is a growing network generated with a particular strategy.\nThere is no loop in this network. In the case of RSF, the structure is completely different, there may be loops. \nAlthough in a numerical study, the value of $\\gamma$ may not be exactly 3 in either the RSF or BA networks\ndue to finite size effects, we believe that this cannot be the reason for the results being \nqualitatively different. In fact \nprevious studies on RSF and BA have shown that characteristic features may be quite different for the two networks \\cite{albert1,psen}.\n\nThe results obtained for the WS model can also be compared to those found for the RSF network and BA network.\nIn the WS network an A-A transition is observed at $q \\simeq 1.0$ which is also the order-disorder transition point. Hence this \nis similar to the occurrence of an A-A and an order-disorder transition occurring simultaneously in the RSF. However in the WS network,\nthe entire disordered phase is active. For small average degree the system is in an active and for large degree in an absorbing phase.\nFor the RSF network, two A-A transitions exist where absorbing phase is observed for {\\it both} large degree and small degree and an active \nstate exists in between these two absorbing phases. In both RSF and WS another interesting feature is present, the ordered \nstate shows a finite freezing probability with negligible system size dependence.\nIn the WS network maximum value of the freezing probability $F(q)$ occurs at $q\\approx 0.2$ and \nshows a behaviour similar to the freezing probability in BA network\nas a function of system size.\n\nTo summarise, a systematic study of ordering dynamics of the Ising model on scale free networks has been made for the first time to the best of our knowledge.\nIt is observed that the system freezes to a non equilibrium steady state for all values of the relevant parameters in the random scale free\nnetworks (RSF) and the Barab\\'{a}si-Albert model (BA).\nThe presence of two active-absorbing phase transitions in the RSF makes it different from the WS network where only one such transition \nis observed. It is also concluded that in RSF one of the active-absorbing phase transition takes place at the order-disorder transition\npoint which is similar to what is observed in the WS network.\n\\label{discussion}\n\nAcknowledgements: The authors thank Soham Biswas for his suggestions and encouragement.\nAK acknowledges financial support from UGC sanction no. F.7-48\/2007(BSR). PS acknowledges financial support from CSIR \nproject. Computations made on HP cluster financed by DST (FIST scheme), India.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}