# On the impossibility of spontaneous vector flavor symmetry breaking on the
lattice

Matteo Giordano<EMAIL_ADDRESS>ELTE Eötvös Loránd University,
Institute for Theoretical Physics, Pázmány Péter sétány 1/A, H-1117, Budapest,
Hungary

###### Abstract

I show that spontaneous breaking of vector flavor symmetry on the lattice is
impossible in gauge theories with a positive functional-integral measure, for
discretized Dirac operators linear in the quark masses, if the corresponding
propagator and its commutator with the flavor symmetry generators can be
bounded in norm independently of the gauge configuration and uniformly in the
volume. Under these assumptions, any order parameter vanishes in the symmetric
limit of fermions of equal masses. I show that these assumptions are satisfied
by staggered, minimally doubled and Ginsparg-Wilson fermions for positive
fermion mass, for any value of the lattice spacing, and so in the continuum
limit if this exists. They are instead not satisfied by Wilson fermions, for
which spontaneous vector flavor symmetry breaking is known to take place in
the Aoki phase. The existence of regularizations unaffected by residual
fermion doubling for which the symmetry cannot break spontaneously on the
lattice establishes rigorously (at the physicist’s level) the impossibility of
its spontaneous breaking in the continuum for any number of flavors.

##  I Introduction

The importance of symmetries and of the way in which they are realized in
quantum field theories can hardly be overemphasized. In the context of strong
interactions and its microscopic theory, i.e., QCD, an important role is
played by the approximate vector flavor symmetry involving the lightest two or
three types (“flavors”) of quarks, which holds exactly in the limit of quarks
of equal masses; and by its enhancement to chiral flavor symmetry in the limit
of massless quarks. Vector flavor symmetry and the pattern of its explicit
breaking largely determine the structure of the hadronic spectrum; chiral
flavor symmetry and its spontaneous breaking down to vector flavor symmetry
explain the lightness of pions and their dynamics, as well as the absence of
parity partners of hadrons. The full symmetry group at the classical level
includes also the $\mathrm{U}(1)_{B}$ symmetry responsible for baryon number
conservation, and the axial $\mathrm{U}(1)_{A}$ symmetry, that does not
survive the quantization process and becomes anomalous in the quantum theory.

An interesting question is whether baryon number and vector flavor symmetry
can break down spontaneously in general vector gauge theories, where the
fermions’ left-handed and right-handed chiralities are coupled in the same way
to the gauge fields. This could in principle happen for exactly degenerate
massive fermions, leading to the appearance of massless Goldstone bosons; and
in the chiral limit of massless fermions it could lead to a different symmetry
breaking pattern than the usual one, and so to a different set of Goldstone
bosons. This question has been essentially answered in the negative by Vafa
and Witten in a famous paper [1]. There they actually prove a stronger result,
namely the impossibility of finding massless particles in the spectrum of a
gauge theory with positive functional-integral measure that couple to
operators with nonvanishing baryon number or transforming nontrivially under
vector flavor transformations. This is done by deriving a bound on the fermion
propagator that guarantees its exponential decay with the distance as long as
the fermion mass is nonzero. Since massless bosons coupling to the operators
mentioned above would appear in the spectrum as a consequence of Goldstone’s
theorem [2, 3, 4] if those symmetries were spontaneously broken, the
impossibility of spontaneous breaking follows.

The elegant and powerful argument of Vafa and Witten is developed using the
“mathematical fiction” of the functional integral formalism for interacting
quantum field theories in continuum (Euclidean) spacetime. The crucial issue
of the regularization of the functional integral, generally required to make
it a mathematically well defined object, is discussed only briefly. In
particular, the possibility of formulating the argument using a lattice
regularization is mentioned, but not discussed in detail. The general validity
of this statement is called into question by the existence of examples of
spontaneous breaking of vector flavor symmetry on the lattice, namely in the
Aoki phase [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25] of lattice gauge theories with Wilson fermions [26]. While this is
not in contradiction with the argument of Vafa and Witten in the continuum
[22], it also makes clear that this argument does not trivially extend to the
lattice in a general setting. It would then be desirable to identify
conditions that guarantee the impossibility of baryon number and vector flavor
symmetry breaking on the lattice, at least for small lattice spacing, which
could help in putting Vafa and Witten’s “theorem” on more solid ground.

The strategy of widest generality is to directly prove a lattice version of
Vafa and Witten’s bound on the propagator, which would allow one to recover
all the conclusions of Ref. [1] in a rigorous way (under the tacit assumption
of the existence of the continuum limit). This was done for staggered fermions
[27, 28, 29] in Ref. [30], so excluding completely the possibility of breaking
baryon number symmetry and the vector flavor symmetry of several staggered
fields on the lattice using this discretization. However, in four dimensions
one flavor of staggered fermions on the lattice describes four degenerate
“tastes” of fermions in the continuum limit, and while the spontaneous
breaking of the corresponding extended flavor symmetry is excluded by the
result of Ref. [30], this limits the impossibility proof to a number of
physical fermion species that is a multiple of four (and of
$2^{\left[{d}/{2}\right]}$ in $d$ dimensions). The extension to an arbitrary
number of fermion species requires the “rooting trick” [31, 32, 33], whose
theoretical status is still not fully clear (see Refs. [34, 35]). It would
then be nice to extend the proof of Ref. [30] or derive a similar bound also
for other discretizations that describe a single fermion species. However, the
proof makes essential use of the anti-Hermiticity and ultralocality of the
operator: while it can probably be extended quite straightforwardly to other
discretizations that share these properties, e.g., the minimally doubled
fermions of Karsten and Wilczek [36, 37] and of Creutz and Boriçi [38, 39]
(that are, however, still describing two fermion species in the continuum
limit), it is not clear how to do so with discretizations that do not, e.g.,
Ginsparg-Wilson fermions [40, 41, 42, 43, 44, 45, 46, 47, 48, 49].

A less general strategy, still sufficient to prove the impossibility of
spontaneous symmetry breaking on the lattice, is to show that the
corresponding order parameters must vanish. Partial results for vector flavor
symmetry following this strategy are present in the literature. Already in
Ref. [1] the authors show that vector flavor symmetry cannot be spontaneously
broken by the formation of the simplest symmetry-breaking bilinear fermion
condensate, when approaching the symmetric case of degenerate fermion masses
starting from the non-degenerate case. Their argument works only for
discretizations of the Dirac operator that are anti-Hermitean, so it applies
again only to staggered and minimally doubled (and obviously to naive)
fermions. In Ref. [50] the authors show that the simplest symmetry-breaking
condensate must vanish also for Ginsparg-Wilson fermions. They do not add any
symmetry-breaking term to the action, applying instead the formalism of
probability distribution functions [51, 52] to the relevant operator to show
the absence of degenerate vacua. More precisely, their result shows that if
degenerate non-symmetric vacua are present, they cannot be distinguished by
the (vanishing) expectation value of this operator.

In this paper I pursue this second strategy and present a simple argument that
spontaneous vector flavor symmetry breaking is impossible on the lattice for
gauge theories with a positive integration measure, as long as the
discretization of the Dirac operator satisfies certain reasonable assumptions.
More precisely, I show that any localized order parameter for vector flavor
symmetry breaking must vanish in the symmetric limit of fermions of equal
masses (taken of course after the thermodynamic limit), for massive lattice
Dirac operators $\mathrm{D}_{M}$ that

  1. 0.

are linear in the fermion masses, $\mathrm{D}_{M}=D^{(0)}+M\Delta D$, with
$D^{(0)}$ and $\Delta D$ trivial in flavor space, and $M$ a Hermitean mass
matrix;

  2. 1.

have inverse bounded in norm by a configuration- and volume-independent
constant, finite in the symmetric limit;

  3. 2.

have derivative with respect to the fermion masses, $\Delta D$, also bounded
in norm by a configuration- and volume-independent constant, finite in the
symmetric limit.

Assumption (0.) is rather natural, and assumption (2.) is not really
restrictive; both are satisfied by all common discretizations. Assumption (1.)
is instead crucial, and it means that the propagator corresponding to
$\mathrm{D}_{M}$ is bounded in norm for all configurations, uniformly in the
volume. This may in general not be the case, for example if a finite density
of near-zero modes of $\mathrm{D}_{M}$ develops in the thermodynamic limit, as
it happens with Wilson fermions in the Aoki phase. For staggered111I refer
here to the exact flavor symmetry of several staggered fields with the same
mass, that holds at any lattice spacing, not to the approximate taste symmetry
that becomes exact only in the continuum limit., minimally doubled, and
Ginsparg-Wilson fermions222The result applies to $\gamma_{5}$-Hermitean
Ginsparg-Wilson fermions with $2R=\mathbf{1}$, see Section V.2., assumption
(1.) holds as long as the fermion masses are nonzero, and the functional-
integration measure is positive for nonnegative fermion masses, so that for
these discretizations the spontaneous breaking of vector flavor symmetry is
impossible at finite positive fermion mass.

My argument is clearly of narrower scope than the one in Ref. [1] and its
counterpart for staggered fermions in Ref. [30], and limited to quadratic
fermion actions with the usual symmetry-breaking terms. On the other hand, it
is mathematically rigorous for a physicist’s standard, leaving little room for
loopholes, and applies to more general discretizations than staggered
fermions. The strategy of proof is standard: one starts from the explicitly
broken case with fermions of different masses, and shows that observables
related by a vector flavor transformation have the same expectation value in
the symmetric limit of equal masses, taken after the infinite-volume limit.
This is achieved by proving two rather elementary bounds on the fermion
propagator and on its commutator with the generators of the vector flavor
symmetry group, that hold independently of the lattice size under assumptions
(0.)–(2.). This results in the magnitude of the difference between the
expectation values of observables related by a vector flavor transformation
obeying a bound proportional to the spread in mass of the fermions, uniformly
in the volume. In the symmetric limit such expectation values are then equal,
and any order parameter for symmetry breaking must therefore vanish. A few
remarks are in order.

(i) The geometry of the lattice, the boundary conditions imposed on the
fields, the type of gauge action, the temperature of the system, and the value
of the lattice spacing and of the other parameters of the theory play no role
as long as positivity of the integration measure and the boundedness
assumptions (1.) and (2.) (or more generally the derived bounds on the
propagator and on its commutator with the symmetry generators) hold.

(ii) The restriction to localized observables is natural, as Goldstone’s
theorem involves observables that are localized in spacetime, and in space in
the finite temperature case [4]. Their counterparts on a finite lattice
involve lattice fields associated with a finite number of lattice sites or
edges (links), that remains unchanged as the system size grows. In particular,
this means that they are polynomial in the fermion fields, of degree
independent of the lattice size.

(iii) If assumptions (0.)–(2.) hold for any lattice spacing, or at least for
any sufficiently small spacing, then all the relevant order parameters vanish
in the symmetric infinite-volume theory also in the continuum limit, if this
exists (notice the order of limits: thermodynamic first, then symmetric,
continuum last). Vector flavor symmetry will then be realized in the
continuum. For staggered, minimally doubled, and Ginsparg-Wilson fermions this
is the case for any positive fermion mass.

(iv) The fate of vector flavor symmetry in the chiral limit, both on the
lattice and in the continuum, can be discussed following the argument
presented in Ref. [1]: barring accidental degeneracies of the ground states,
vector flavor symmetry must remain unbroken.

(v) The restriction to quadratic actions is not a limitation as far as the
eventual continuum limit is concerned. Renormalizable higher-order operators
with the right global and local symmetries are available only in dimension
lower than or equal to two, where spontaneous breaking of a continuous
symmetry is forbidden [55, 56, 57]. The inclusion of symmetry-breaking non-
renormalizable operators in the action may lead to spontaneously broken phases
on the lattice, but does not affect the long-distance physics in the continuum
limit. Since lattice discretizations exist that guarantee the realization of
vector flavor symmetry in the continuum limit, any hypothetical phase where it
is spontaneously broken on the lattice should shrink as this limit is
approached. This is the case also for the spontaneously broken phases possibly
appearing on the lattice for discretizations that do not satisfy the
assumptions of this paper, e.g., the Aoki phase found with Wilson fermions.

(vi) The existence of regularizations unaffected by residual fermion doubling
in the continuum limit for which the symmetry cannot break spontaneously on
the lattice at any spacing (e.g., Ginsparg-Wilson fermions) establishes
rigorously (at the physicist’s level of rigor) the impossibility of its
spontaneous breaking in continuum gauge theories for any number of physical
fermion species.

The plan of the paper is the following. After briefly reviewing gauge theories
on the lattice to set up the notation in Section II, and vector flavor
symmetry in Section III, I derive the relevant bounds and prove the main
statement in Section IV. The cases of staggered, Ginsparg-Wilson, Wilson, and
minimally doubled fermions are discussed in Section V. A brief summary is
given in Section VI. A few technical details are given in Appendix A.

##  II Gauge theories on the lattice

I will consider $d$-dimensional vector gauge theories with $N_{f}$ flavors of
fermions, all transforming in the same $N_{c}$-dimensional representation of a
compact gauge group, discretized on a finite lattice containing
${\mathcal{V}}$ sites. Suitable boundary conditions are assumed on the gauge
and fermion fields. The shape of the lattice and the boundary conditions play
no distinctive role in the following; in particular, the discussion applies to
systems both at zero and finite temperature. The partition function and the
expectation values of the theory are given by

$\displaystyle Z$
$\displaystyle\equiv\int[\mathrm{D}U]\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{G}}[U]-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}\,,$
(1) $\displaystyle\langle{\mathcal{O}}\rangle$
$\displaystyle\equiv\frac{1}{Z}\int[\mathrm{D}U]\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{G}}[U]-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}{\mathcal{O}}[\psi,\bar{\psi},U]\,,$

where $[\mathrm{D}U]=\prod_{\ell}dU_{\ell}$ is the product of the Haar
measures associated with the gauge variables $U_{\ell}$ attached to the
lattice links $\ell$, and
$[\mathrm{D}\psi\mathrm{D}\bar{\psi}]=\prod_{xfa\alpha}d\psi_{fa\alpha}(x)d\bar{\psi}_{fa\alpha}(x)$
is the Berezin integration measure associated with the Grassmann variables
$\psi_{fa\alpha}(x)$ and $\bar{\psi}_{fa\alpha}(x)$ attached to the lattice
sites $x$. Here $f$ and $a$ are the discrete indices associated with the
flavor and color (i.e., gauge group) degrees of freedom, $f=1,\ldots,N_{f}$,
$a=1,\ldots,N_{c}$, and $\alpha$ is the Dirac index, typically
$\alpha=1,\ldots,2^{\left[{d}/{2}\right]}$, but possibly absent altogether
(e.g., for staggered fermions). The full set of discrete indices will be
collectively denoted as $A=fa\alpha$; when needed, the color and Dirac indices
will be denoted together as ${A_{\star}}=a\alpha$. Finally,
$\mathcal{S}_{\mathrm{G}}$ and $\mathcal{S}_{\mathrm{F}}$ denote the gauge and
fermionic parts of the action. The fermionic action is taken to be of the form

$\displaystyle\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]$
$\displaystyle=\sum_{x,y,A,B}\bar{\psi}_{A}(x)\left(\mathrm{D}_{M}[U]\right)_{AB}(x,y)\psi_{B}(y)$
(2) $\displaystyle=\bar{\psi}\mathrm{D}_{M}[U]\psi\,,$

where in the last passage I introduced the matrix notation that will be used
repeatedly. Here $\mathrm{D}_{M}$ is the massive Dirac operator, whose
dependence on the gauge links $U_{\ell}$ will be often omitted for simplicity.
Expectation values are computed in two steps. For a generic observable
${\mathcal{O}}[\psi,\bar{\psi},U]$, integration over Grassmann variables
yields

$\displaystyle\left\langle{\mathcal{O}}\right\rangle_{\mathrm{F}}$
$\displaystyle\equiv\frac{\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}{\mathcal{O}}[\psi,\bar{\psi},U]}{\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}}$
(3)
$\displaystyle=\frac{\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}{\mathcal{O}}[\psi,\bar{\psi},U]}{\det\mathrm{D}_{M}[U]}\,.$

The expectation value of ${\mathcal{O}}$ is then obtained from Eq. (3) by
averaging over gauge fields,
$\langle{\mathcal{O}}\rangle=\langle\langle{\mathcal{O}}\rangle_{\mathrm{F}}\rangle_{\mathrm{G}}$,
where for a purely gluonic observable $\tilde{{\mathcal{O}}}[U]$

$\displaystyle\langle\tilde{{\mathcal{O}}}\rangle_{\mathrm{G}}$
$\displaystyle\equiv\frac{\int[\mathrm{D}U]\,e^{-\mathcal{S}_{\mathrm{G}}[U]}\det\mathrm{D}_{M}[U]\tilde{{\mathcal{O}}}[U]}{\int[\mathrm{D}U]\,e^{-\mathcal{S}_{\mathrm{G}}[U]}\det\mathrm{D}_{M}[U]}$
(4)
$\displaystyle=\frac{1}{Z}\int[\mathrm{D}U]\,e^{-\mathcal{S}_{\mathrm{G}}[U]}\det\mathrm{D}_{M}[U]\tilde{{\mathcal{O}}}[U]\,.$

I assume that the full gluonic integration measure
$d\mu_{\mathrm{G}}=[\mathrm{D}U]\,e^{-\mathcal{S}_{\mathrm{G}}[U]}\det\mathrm{D}_{M}[U]$
is positive, i.e., $e^{-\mathcal{S}_{\mathrm{G}}[U]}\det\mathrm{D}_{M}[U]\geq
0$ (and does not vanish everywhere). The gluonic action is otherwise
unspecified, besides its being gauge-invariant. I consider massive Dirac
operators of the form

$\mathrm{D}_{M}=\mathbf{1}_{F}D^{(0)}+M\Delta D\,,$ (5)

with $M$ a constant Hermitean matrix carrying only flavor indices and
independent of coordinates and gauge links. The symbol $\mathbf{1}_{F}$, and
similarly $\mathbf{1}_{C}$ and $\mathbf{1}_{D}$, denote the identity in flavor
($F$), color ($C$), and Dirac ($D$) space; $\mathbf{1}$ will denote the
identity in the full flavor, color, Dirac and coordinate space. The operators
$D^{(0)}$ and $\Delta D$ carry only color, Dirac, and coordinate indices,
i.e., $(D^{(0)})_{{A_{\star}}{B_{\star}}}(x,y)$ and $(\Delta
D)_{{A_{\star}}{B_{\star}}}(x,y)$. Since one can diagonalize $M$ with a
unitary transformation, and reabsorb this into a redefinition of the fermion
fields that does not affect the Berezin integration measure, one can consider
a diagonal mass matrix $M=\mathrm{diag}(m_{1},\ldots,m_{N_{f}})$ without loss
of generality, and write

$\displaystyle\mathrm{D}_{M}$
$\displaystyle=\mathrm{diag}\left(D^{(m_{1})},\ldots,D^{(m_{N_{f}})}\right)\,,$
(6) $\displaystyle D^{(m)}$ $\displaystyle\equiv D^{(0)}+m\Delta D\,,$
$\displaystyle(\mathrm{D}_{M})_{f{A_{\star}}\,g{B_{\star}}}(x,y)$
$\displaystyle=\delta_{fg}(D^{(m_{f})})_{{A_{\star}}{B_{\star}}}(x,y)\,.$

The fermion propagator is then

$\displaystyle\mathrm{S}_{M}$
$\displaystyle=\mathrm{D}_{M}^{-1}=\mathrm{diag}\left(S^{(m_{1})},\ldots,S^{(m_{N_{f}})}\right)\,,$
(7) $\displaystyle S^{(m)}$ $\displaystyle\equiv(D^{(m)})^{-1}\,,$
$\displaystyle(\mathrm{S}_{M})_{f{A_{\star}}\,g{B_{\star}}}(x,y)$
$\displaystyle=\delta_{fg}S^{(m_{f})}_{{A_{\star}}{B_{\star}}}(x,y)\,,$

and the fermion determinant is

$\det\mathrm{D}_{M}={\prod_{f=1}^{N_{f}}}\det D^{(m_{f})}\,.$ (8)

The trace over all indices, i.e., flavor, color, Dirac, and coordinates, will
be denoted by ${\rm Tr}$. The trace over one or more of the discrete indices
will be denoted by ${\rm tr}$ with one or more of the subscripts $F$, $C$,
$D$, indicating which indices are being traced. Matrix multiplication is
understood not to involve the indices displayed explicitly: for matrices
$\mathrm{P},\mathrm{Q}$ carrying all indices, and matrices $P,Q$ carrying all
but flavor indices,

$\displaystyle\left(\mathrm{P}\mathrm{Q}\right)_{AB}(x,y)$
$\displaystyle=\sum_{z,C}\mathrm{P}_{AC}(x,z)\mathrm{Q}_{CB}(z,y)\,,$ (9)
$\displaystyle\left(\mathrm{P}(x,z)\mathrm{Q}(z,y)\right)_{AB}$
$\displaystyle=\sum_{C}\mathrm{P}_{AC}(x,z)\mathrm{Q}_{CB}(z,y)\,,$
$\displaystyle\left(PQ\right)_{{A_{\star}}{B_{\star}}}(x,y)$
$\displaystyle=\sum_{{C_{\star}},z}P_{{A_{\star}}{C_{\star}}}(x,z)Q_{{C_{\star}}{B_{\star}}}(z,y)\,,$
$\displaystyle\left(P(x,z)Q(z,y)\right)_{{A_{\star}}{B_{\star}}}$
$\displaystyle=\sum_{{C_{\star}}}P_{{A_{\star}}{C_{\star}}}(x,z)Q_{{C_{\star}}{B_{\star}}}(z,y)\,.$

A similar convention applies to Hermitean conjugation,

$\displaystyle(\mathrm{P}^{\dagger})_{AB}(x,y)$
$\displaystyle=\mathrm{P}_{BA}(y,x)^{*}\,,$ (10)
$\displaystyle(\mathrm{P}(x,y)^{\dagger})_{AB}$
$\displaystyle=\mathrm{P}_{BA}(x,y)^{*}\,,$
$\displaystyle(P^{\dagger})_{{A_{\star}}{B_{\star}}}(x,y)$
$\displaystyle=P_{{B_{\star}}{A_{\star}}}(y,x)^{*}\,,$
$\displaystyle(P(x,y)^{\dagger})_{{A_{\star}}{B_{\star}}}$
$\displaystyle=P_{{B_{\star}}{A_{\star}}}(x,y)^{*}\,.$

At a certain point I will assume that the propagator and the operator $\Delta
D$ are suitably bounded in norm. In the finite-dimensional case, the operator
norm $\|\mathrm{A}\|$ of an operator $\mathrm{A}$ equals the largest of the
eigenvalues $a_{n}^{2}$ of the positive Hermitean operator
$\mathrm{A}^{\dagger}\mathrm{A}$,

$\displaystyle\|\mathrm{A}\|^{2}$ $\displaystyle=\sup_{\psi\neq
0}\frac{(\mathrm{A}\psi,\mathrm{A}\psi)}{(\psi,\psi)}=\sup_{\psi\neq
0}\frac{(\psi,\mathrm{A}^{\dagger}\mathrm{A}\psi)}{(\psi,\psi)}=\max_{n}a_{n}^{2}\,,$
(11)

where $(\psi,\phi)$ denotes the standard Hermitean inner product. I will
assume that

  1. 1.

$\|\mathrm{S}_{M}\|\leq m_{0}^{-1}<\infty$, with $m_{0}$ independent of the
gauge configuration and of the lattice volume ${\mathcal{V}}$, and finite in
the limit of equal fermion masses;

  2. 2.

$\|\Delta D\|\leq\Delta D_{\mathrm{max}}<\infty$, with $\Delta
D_{\mathrm{max}}$ independent of the gauge configuration and of the lattice
volume, and finite in the limit of equal fermion masses.

For the purposes of this paper it suffices to consider the most general
localized gauge-invariant observable, so polynomial in the fermion fields and
dependent on finitely many link variables, with fermion number zero333On a
finite lattice observables are necessarily polynomial in the fermion fields
and dependent on finitely many link variables: the restriction is relevant in
the thermodynamic limit. Nonzero fermion number immediately entails a
vanishing expectation value. Any non-gauge-invariant observable can be
replaced with its average over gauge orbits without changing its expectation
value.. For notational purposes it is convenient to write it with its discrete
indices contracted with the most general matrix carrying flavor, color and
Dirac indices, dependent on the link variables (and possibly also explicitly
on the lattice coordinates, and on the parameters of the theory), and having
the right transformation properties under gauge transformations to make the
observable gauge invariant. I will then consider

$\displaystyle{\mathcal{O}}_{{\mathcal{M}}}[\psi,\bar{\psi},U]$ (12)
$\displaystyle\equiv{\sum_{\begin{subarray}{c}A_{1},\ldots,A_{n},\\\
B_{1},\ldots,B_{n}\end{subarray}}}({\mathcal{M}}[U])_{A_{1}\ldots
A_{n}B_{1}\ldots B_{n}}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})$
$\displaystyle\phantom{\equiv\sum_{A_{1},\ldots,A_{n},}}\times{\prod_{i=1}^{n}}\psi_{B_{i}}(y_{i})\bar{\psi}_{A_{i}}(x_{i})$
$\displaystyle=\sum_{\mathbf{A},\mathbf{B}}({\mathcal{M}}[U])_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y}){\prod_{i=1}^{n}}\psi_{B_{i}}(y_{i})\bar{\psi}_{A_{i}}(x_{i})\,.$

For brevity I will write
${\mathcal{M}}[U]_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})$, using bold
typeface to denote collectively a set of indices or variables. I will
generally omit the dependence on $U$ when unimportant. The transformation
properties of ${\mathcal{M}}$ under gauge transformations are easily obtained
from those of the fermionic fields, and do not play any role in the following.
Notably, the quantity

${\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})\equiv{\rm
tr}_{FCD}\\{{\mathcal{M}}(\mathbf{x},\mathbf{y}){\mathcal{M}}(\mathbf{x},\mathbf{y})^{\dagger}\\}$
(13)

is gauge invariant. The expectation value
$\langle{\mathcal{O}}_{\mathcal{M}}\rangle=\langle\langle{\mathcal{O}}_{\mathcal{M}}\rangle_{\mathrm{F}}\rangle_{\mathrm{G}}$
is obtained averaging $\langle{\mathcal{O}}_{\mathcal{M}}\rangle_{\mathrm{F}}$
over gauge fields using Eq. (4). From Eq. (3) one finds using Wick’s theorem

$\displaystyle\left\langle{\mathcal{O}}_{{\mathcal{M}}}\right\rangle_{\mathrm{F}}$
$\displaystyle=\sum_{\mathbf{A},\mathbf{B}}{\mathcal{M}}[U]_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})\left\langle\prod_{i=1}^{n}\psi_{B_{i}}(y_{i})\bar{\psi}_{A_{i}}(x_{i})\right\rangle_{\mathrm{F}}$
(14)
$\displaystyle=\sum_{\mathbf{A},\mathbf{B}}{\mathcal{M}}[U]_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})$
$\displaystyle\phantom{=}\times\sum_{\mathrm{P}\in\mathrm{S}_{n}}\sigma_{\mathrm{P}}\prod_{i=1}^{n}\mathrm{S}_{M}[U]_{B_{i}A_{\mathrm{P}(i)}}(y_{i},x_{\mathrm{P}(i)})\,,$

with $\mathrm{P}$ a permutation of $n$ elements and $\sigma_{\mathrm{P}}=\pm
1$ its signature.

Restrictions on ${\mathcal{M}}$ are required in order for the integration over
link variables to yield finite results. In the physically relevant cases
${\mathcal{M}}$ is a product of Wilson lines, suitably connecting the fermion
fields to achieve gauge invariance, and so polynomial in the link variables.
Imposing this restriction on ${\mathcal{M}}$, or asking more generally for
${\mathcal{M}}$ to be continuous in the link variables, guarantees finiteness
of the Haar integrals thanks to the compactness of the gauge group. In this
case ${\mathcal{K}}_{\mathcal{M}}$ is bounded from above by its maximum on the
compact integration manifold. The thermodynamic limit is taken while keeping
${\mathcal{M}}$ fixed as a function of the link variables, so it should affect
the convergence properties of the Haar integrals only mildly (in particular,
the possible dependence on the lattice coordinates is unchanged and cannot
cause convergence problems). Under the assumption of continuity of
${\mathcal{M}}$ in the link variables, this limit does not affect at all the
upper bound on ${\mathcal{K}}_{\mathcal{M}}$: assumption (1.) on the
propagator then suffices to show convergence of
$\langle{\mathcal{O}}_{\mathcal{M}}\rangle$, see below in Section IV. Finally,
any mass dependence of ${\mathcal{M}}$ is assumed to be continuous, at least
in the symmetric limit of equal fermion masses. The results of this paper can
be proved also if one relaxes the requirement of continuity to absolute
integrability of the entries ${\mathcal{M}}_{\mathbf{A}\mathbf{B}}$: this is
discussed in Appendix A.

##  III Vector flavor transformations

Vector flavor transformations are defined by

$\displaystyle\psi_{f{A_{\star}}}(x)$
$\displaystyle\to\textstyle{\sum_{g}}V_{fg}\psi_{g{A_{\star}}}(x),$ (15)
$\displaystyle\bar{\psi}_{f{A_{\star}}}(x)$
$\displaystyle\to\textstyle{\sum_{g}}\bar{\psi}_{g{A_{\star}}}(x)V^{\dagger}_{gf}\,,$

where $V_{fg}$ are the entries of a unitary unimodular $N_{f}\times N_{f}$
matrix $V\in\mathrm{SU}(N_{f})$. This can be written as
$V=e^{i\theta_{a}t^{a}}\equiv e^{i\bm{\theta}\cdot\bm{t}}\equiv
V(\bm{\theta})$, with $\theta_{a}\in\mathbb{R}$ and with $t^{a}$ the Hermitean
and traceless generators of $\mathrm{SU}(N_{f})$, taken with the standard
normalization $2\,{\rm tr}_{F}\,t^{a}t^{b}=\delta^{ab}$. The task is to show
that any localized observable ${\mathcal{O}}={\mathcal{O}}[\psi,\bar{\psi},U]$
and its transformed ${\mathcal{O}}^{\bm{\theta}}$,

${\mathcal{O}}^{\bm{\theta}}[\psi,\bar{\psi},U]\equiv{\mathcal{O}}[V(\bm{\theta})\psi,\bar{\psi}V(\bm{\theta})^{\dagger},U]\,,$
(16)

have the same expectation value in the infinite-volume theory in the symmetric
limit $M\to m\mathbf{1}_{F}$, i.e.,

$\lim_{M\to
m\mathbf{1}_{F}}\lim_{{\mathcal{V}}\to\infty}\left\langle{\mathcal{O}}^{\bm{\theta}}-{\mathcal{O}}\right\rangle=0\,.$
(17)

On a finite lattice all observables are obviously localized. In the
thermodynamic limit ${\mathcal{V}}\to\infty$, every localized observable is a
linear combination of finitely many of the ${\mathcal{O}}_{\mathcal{M}}$
discussed above, Eq. (12), where ${\mathcal{M}}$ is understood to be a fixed
function of the link variables, independent of ${\mathcal{V}}$. Moreover,
since $\mathrm{SU}(N_{f})$ is a Lie group, any finite transformation can be
obtained by composition of infinitesimal ones. It suffices then to consider
observables ${\mathcal{O}}_{\mathcal{M}}$ and transformations with
$\theta_{a}\ll 1$ in Eq. (17), i.e., one has to show

$\lim_{M\to
m\mathbf{1}_{F}}\lim_{{\mathcal{V}}\to\infty}\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\bigg{|}_{\bm{\theta}=0}=0\,.$
(18)

An explicit proof that Eq. (18) implies Eq. (17) is given in Appendix A.1. In
Appendix A.2 I show that Eq. (17) implies that order parameters for vector
flavor symmetry, i.e., expectation values of observables that transform
nontrivially under $\mathrm{SU}(N_{f})$, must vanish in the symmetric limit.

To efficiently study the effect of a vector flavor transformation on the
expectation value of an arbitrary observable, it is convenient to make use of
the corresponding well known integrated Ward-Takahashi identity, derived for
completeness in Appendix A.1,

$\displaystyle
i\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}^{\bm{\theta}}\right\rangle\bigg{|}_{\bm{\theta}=0}$
$\displaystyle=\left\langle{\mathcal{C}}^{a}{\mathcal{O}}\right\rangle\,,$
(19) $\displaystyle{\mathcal{C}}^{a}[\psi,\bar{\psi},U]$
$\displaystyle\equiv\bar{\psi}[t^{a},\mathrm{D}_{M}[U]]\psi\,.$

Integrating fermions out one finds for a generic observable ${\mathcal{O}}$

$\displaystyle
i\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}^{\bm{\theta}}\right\rangle_{\mathrm{F}}\bigg{|}_{\bm{\theta}=0}$
$\displaystyle=\left\langle{\mathcal{C}}^{a}{\mathcal{O}}\right\rangle_{\mathrm{F}}=\left\langle{\mathcal{C}}^{a}\rangle_{\mathrm{F}}\langle{\mathcal{O}}\right\rangle_{\mathrm{F}}-{\rm
Tr}\left\\{F^{a}O\right\\}\,,$ (20)

with

$O_{AB}(x,y)\equiv\left\langle\frac{\partial}{\partial_{L}\psi_{B}(y)}\frac{\partial}{\partial_{L}\bar{\psi}_{A}(x)}{\mathcal{O}}\right\rangle_{\mathrm{F}}\,,$
(21)

where $\partial_{L}$ denotes the usual left derivative with respect to
Grassmann variables, and

$\displaystyle
F^{a}\equiv\mathrm{S}_{M}[t^{a},\mathrm{D}_{M}]\mathrm{S}_{M}=[\mathrm{S}_{M},t^{a}]\,,$
(22)

i.e., the commutator of the propagator with the generators of
$\mathrm{SU}(N_{f})$. The first term in Eq. (20) vanishes since

$\displaystyle\left\langle{\mathcal{C}}^{a}\right\rangle_{\mathrm{F}}$
$\displaystyle=-{\rm
Tr}\left\\{[t^{a},\mathrm{D}_{M}]\mathrm{S}_{M}\right\\}=-{\rm
Tr}\left\\{t^{a}\left[\mathrm{D}_{M},\mathrm{S}_{M}\right]\right\\}=0\,.$ (23)

Specializing now to observables of the form Eq. (12), one finds, for
$i,j=1,\ldots,n$,

$\displaystyle\frac{\partial}{\partial_{L}\psi_{B_{j}}(y_{j})}\frac{\partial}{\partial_{L}\bar{\psi}_{A_{i}}(x_{i})}{\mathcal{O}}_{\mathcal{M}}[\psi,\bar{\psi},U]$
(24)
$\displaystyle=-s_{ij}({\mathcal{M}}[U])_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})\prod_{k=1}^{n-1}\psi_{B_{k}^{(j)}}(y_{k}^{(j)})\bar{\psi}_{A_{k}^{(i)}}(x_{k}^{(i)})\,.$

Here the superscript $(i)$ means that the $i$th element is omitted from the
set of indices while keeping their ordering unchanged, i.e.,
$\mathbf{A}^{(i)}=\\{{A}^{(i)}_{1},\ldots,{A}^{(i)}_{n-1}\\}=\\{A_{1},\ldots,A_{i-1}\\}\cup\\{A_{i+1},\ldots,A_{n}\\}$,
and similarly for the other sets. The sign factor $-s_{ij}=(-1)^{i-j-1}$
appears when reordering the Grassmann variables to be in the same form as in
Eq. (12). Using now Eq. (14) one finds explicitly

$\displaystyle O_{{\mathcal{M}}\,A_{i}B_{j}}(\mathbf{x},\mathbf{y})$
$\displaystyle\equiv\left\langle\frac{\partial}{\partial_{L}\psi_{B_{j}}(y_{j})}\frac{\partial}{\partial_{L}\bar{\psi}_{A_{i}}(x_{i})}{\mathcal{O}}_{\mathcal{M}}\right\rangle_{\mathrm{F}}=-s_{ij}\sum_{\mathrm{P}\in\mathrm{S}_{n-1}}\sigma_{\mathrm{P}}\sum_{{\mathbf{A}}^{(i)},{\mathbf{B}}^{(j)}}{\mathcal{M}}_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})\prod_{k=1}^{n-1}(\mathrm{S}_{M})_{{B}^{(j)}_{k}{A}^{(i)}_{\mathrm{P}(k)}}({y}^{(j)}_{k},{x}^{(i)}_{\mathrm{P}(k)})\,,$
(25)

where $\mathrm{P}$ is now a permutation of $n-1$ elements and
$\sigma_{\mathrm{P}}$ its signature. Notice that $A_{i}$ and $B_{j}$ are not
contracted in Eq. (25), and that the dependence of
$O_{{\mathcal{M}}\,A_{i}B_{j}}$ on $x_{i},y_{j}$ is only through that of
${\mathcal{M}}$. One can finally write

$\displaystyle
i\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\Big{|}_{\bm{\theta}=0}=\left\langle{\mathcal{C}}^{a}{\mathcal{O}}_{\mathcal{M}}\right\rangle=\sum_{i,j=1}^{n}s_{ij}\sum_{\mathrm{P}\in\mathrm{S}_{n-1}}\sigma_{\mathrm{P}}\left\langle\sum_{\mathbf{A},\mathbf{B}}{\mathcal{M}}_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})F^{a}_{B_{j}A_{i}}(y_{j},x_{i})\prod_{k=1}^{n-1}(\mathrm{S}_{M})_{{B}^{(j)}_{k}{A}^{(i)}_{\mathrm{P}(k)}}({y}^{(j)}_{k},{x}^{(i)}_{\mathrm{P}(k)})\right\rangle_{\mathrm{G}}\,.$
(26)

##  IV Proof of the main result

Equation (26) is the starting point for establishing a bound on the magnitude
of the variation of the expectation value of ${\mathcal{O}}_{\mathcal{M}}$
under an infinitesimal vector flavor transformation. The argument is quite
elementary, and should be understandable in spite of my best attempts at
obscuring it with cumbersome notation. As shown above, after performing the
Wick contractions the variation of $\langle{\mathcal{O}}_{\mathcal{M}}\rangle$
is written as a sum of terms of the form
$\langle{\mathcal{M}}F\prod\mathrm{S}\rangle_{\mathrm{G}}$, differing by
permutations of the indices. By standard techniques, if the integration
measure is positive this is bounded in magnitude by a sum of terms of the form
$\langle({\rm tr}\\{{\mathcal{M}}{\mathcal{M}}^{\dagger}\\}{\rm
tr}\\{FF^{\dagger}\\}\prod{\rm
tr}\\{\mathrm{S}\mathrm{S}^{\dagger}\\})^{\frac{1}{2}}\rangle_{\mathrm{G}}$,
which under the assumptions of Section II on the Dirac operator and on the
continuity of ${\mathcal{M}}$ can be bounded uniformly in the volume by a
constant times the spread in mass of the fermions. In the symmetric limit one
then finds that the variation of $\langle{\mathcal{O}}_{\mathcal{M}}\rangle$
under any infinitesimal vector flavor transformation vanishes. By a similar
argument one can show that $\langle{\mathcal{O}}_{\mathcal{M}}\rangle$ is
finite, independently of the fermion masses, also in the thermodynamic limit.

I now present the detailed proof. Using standard inequalities for the absolute
value, the assumed positivity of the integration measure, and the Cauchy-
Schwarz inequality for inner products, one has

$\displaystyle\left|\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\Big{|}_{\bm{\theta}=0}\right|$
$\displaystyle\leq\sum_{i,j=1}^{n}\sum_{\mathrm{P}\in\mathrm{S}_{n-1}}\left|\left\langle\sum_{\mathbf{A},\mathbf{B}}{\mathcal{M}}_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})F^{a}_{B_{j}A_{i}}(y_{j},x_{i})\prod_{k=1}^{n-1}(\mathrm{S}_{M})_{{B}^{(j)}_{k}{A}^{(i)}_{\mathrm{P}(k)}}({y}^{(j)}_{k},{x}^{(i)}_{\mathrm{P}(k)})\right\rangle_{\mathrm{G}}\right|$
(27)
$\displaystyle\leq\sum_{i,j=1}^{n}\sum_{\mathrm{P}\in\mathrm{S}_{n-1}}\left\langle\left|\sum_{\mathbf{A},\mathbf{B}}{\mathcal{M}}_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})F^{a}_{B_{j}A_{i}}(y_{j},x_{i})\prod_{k=1}^{n-1}(\mathrm{S}_{M})_{{B}^{(j)}_{k}{A}^{(i)}_{\mathrm{P}(k)}}({y}^{(j)}_{k},{x}^{(i)}_{\mathrm{P}(k)})\right|\right\rangle_{\mathrm{G}}$
$\displaystyle\leq\sum_{i,j=1}^{n}\sum_{\mathrm{P}\in
S_{n-1}}\left\langle\left({\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y}){\mathcal{F}}^{a}(y_{j},x_{i})\prod_{k=1}^{n-1}{\mathcal{S}}({y}^{(j)}_{k},{x}^{(i)}_{\mathrm{P}(k)})\right)^{\frac{1}{2}}\right\rangle_{\mathrm{G}}\,,$

where ${\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})$ is defined in Eq.
(13) and

$\displaystyle{\mathcal{F}}^{a}(y,x)$ $\displaystyle\equiv{\rm
tr}_{FCD}\\{F^{a}(y,x)F^{a}(y,x)^{\dagger}\\}\,,$ (28)
$\displaystyle{\mathcal{S}}(y,x)$ $\displaystyle\equiv{\rm
tr}_{FCD}\\{\mathrm{S}_{M}(y,x)\mathrm{S}_{M}(y,x)^{\dagger}\\}\,.$

The quantity ${\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})$ is a
positive gauge-invariant function of the link variables and their Hermitean
conjugates, polynomial (or more generally continuous) if ${\mathcal{M}}$ is
polynomial (continuous), defined on the compact domain given by the direct
product of finitely many compact gauge-group manifolds. It is therefore
bounded from above in magnitude by its maximum, which depends on the details
of ${\mathcal{M}}$ but is otherwise a configuration- and volume-independent
quantity,

${\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})={\rm
tr}_{FCD}\\{{\mathcal{M}}(\mathbf{x},\mathbf{y}){\mathcal{M}}(\mathbf{x},\mathbf{y})^{\dagger}\\}\leq{C}_{\mathcal{M}}\,.$
(29)

Moreover, since $D^{(0)}$ and $\Delta D$ are trivial and $M$ is diagonal in
flavor space, one has $[t^{a},\mathrm{D}_{M}]=[t^{a},M]\Delta D$,

$\displaystyle
F^{a}_{g{B_{\star}}\,f{A_{\star}}}(y,x)=(\mathrm{S}_{M}[t^{a},M]\Delta
D\mathrm{S}_{M})_{g{B_{\star}}\,f{A_{\star}}}(y,x)$ (30)
$\displaystyle=t^{a}_{gf}(m_{f}-m_{g})(S^{(m_{g})}\Delta
DS^{(m_{f})})_{{B_{\star}}{A_{\star}}}(y,x)\,,$

and so

$\displaystyle{\mathcal{F}}^{a}(y,x)$
$\displaystyle=\textstyle{\sum_{f,g}}|t^{a}_{gf}|^{2}(m_{f}-m_{g})^{2}$ (31)
$\displaystyle\phantom{=\textstyle{\sum_{f,g}}}\times{\rm
tr}_{CD}\\{(S^{(m_{g})}\Delta DS^{(m_{f})})(y,x)$
$\displaystyle\phantom{=\textstyle{\sum_{f,g}}{\rm
tr}_{CD}}\times(S^{(m_{g})}\Delta DS^{(m_{f})})(y,x)^{\dagger}\\}\,.$

The partial traces appearing in Eqs. (28) and (31) can be bounded using an
elementary lemma, proved in Appendix A.3: given a multi-indexed matrix
$\mathrm{A}$, the partial trace of $\mathrm{A}^{\dagger}\mathrm{A}$ over a
subset of its indices is bounded by the dimension of the corresponding space
times the square of the operator norm of $\mathrm{A}$. One has then the
following bound on the propagator [see Eq. (74)], valid for arbitrary lattice
coordinates $x$ and $y$,

${\mathcal{S}}(y,x)\leq N_{f}N_{c}N_{D}\|\mathrm{S}_{M}\|^{2}\,,$ (32)

from which one obtains the bound

$\displaystyle\prod_{k=1}^{n-1}{\mathcal{S}}({y}^{(j)}_{k},{x}^{(i)}_{\mathrm{P}(k)})\leq\left(N_{f}N_{c}N_{D}\|\mathrm{S}_{M}\|^{2}\right)^{n-1}\,.$
(33)

Since the bound Eq. (32) is independent of the coordinates, the bound Eq. (33)
is independent of the particular choice of $i$ and $j$ and of the permutation
$\mathrm{P}$. Using again Eq. (74) one finds

$\displaystyle{\rm tr}_{CD}\\{(S^{(m_{g})}\Delta
DS^{(m_{f})})(y,x)(S^{(m_{g})}\Delta DS^{(m_{f})})(y,x)^{\dagger}\\}$ (34)
$\displaystyle\leq N_{c}N_{D}\|S^{(m_{g})}\Delta DS^{(m_{f})}\|^{2}$
$\displaystyle\leq N_{c}N_{D}\|S^{(m_{g})}\|^{2}\|\Delta
D\|^{2}\|S^{(m_{f})}\|^{2}$ $\displaystyle\leq
N_{c}N_{D}\|\mathrm{S}_{M}\|^{4}\|\Delta D\|^{2}\,,$

where I used the well known inequality $\|AB\|\leq\|A\|\|B\|$, and the obvious
fact that
$\|S^{(m_{f})}\|\leq\max_{f^{\prime}}\|S^{(m_{f^{\prime}})}\|=\|\mathrm{S}_{M}\|$.
From Eqs. (31) and (34) one obtains the following bound on the commutator of
the propagator with the $\mathrm{SU}(N_{f})$ generators,

$\displaystyle{\mathcal{F}}^{a}(y,x)$
$\displaystyle\leq\sum_{f,g}|t^{a}_{gf}|^{2}(m_{f}-m_{g})^{2}N_{c}N_{D}\|\mathrm{S}_{M}\|^{4}\|\Delta
D\|^{2}$ (35) $\displaystyle\leq{\textstyle\frac{1}{2}}(\delta
m)^{2}N_{c}N_{D}\|\mathrm{S}_{M}\|^{4}\|\Delta D\|^{2}\,,$

where I denoted with $\delta m\equiv\max_{f,g}|m_{f}-m_{g}|$ the spread in
mass of the fermions, and I used ${\rm tr}_{F}(t^{a})^{2}=\frac{1}{2}$. Also
this bound is independent of the coordinates, and so each of the $n^{2}(n-1)!$
terms appearing in the sums over $i$, $j$ and $\mathrm{P}$ in Eq. (27) obeys
the same bound. Collecting now Eqs. (29), (33) and (35), one finds

$\displaystyle\left|\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\Big{|}_{\bm{\theta}=0}\right|$
$\displaystyle\leq\delta
m\,\tilde{{C}}_{\mathcal{M}}\left\langle\|\mathrm{S}_{M}\|^{n+1}\|\Delta
D\|\right\rangle_{\mathrm{G}}\,,$ (36)

having set $\tilde{{C}}_{\mathcal{M}}\equiv
n\,n!\left(N_{f}N_{c}N_{D}\right)^{\frac{n}{2}}\left(\frac{{C}_{\mathcal{M}}}{2N_{f}}\right)^{\frac{1}{2}}$.
I now make the assumptions that $\|\mathrm{S}_{M}\|\leq m_{0}^{-1}<\infty$ and
$\|\Delta D\|\leq\Delta D_{\mathrm{max}}<\infty$, with $m_{0}$ and $\Delta
D_{\mathrm{max}}$ independent of the gauge configuration. With these
assumptions one concludes

$\displaystyle\left|\left\langle{\mathcal{C}}^{a}{\mathcal{O}}_{\mathcal{M}}\right\rangle\right|$
$\displaystyle=\left|\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\Big{|}_{\bm{\theta}=0}\right|\leq\delta
m\frac{\tilde{{C}}_{\mathcal{M}}\Delta D_{\mathrm{max}}}{m_{0}^{n+1}}\,.$ (37)

Using also the assumption that $m_{0}$ and $\Delta D_{\mathrm{max}}$ are
independent of the lattice size ${\mathcal{V}}$, the bound Eq. (37) is volume-
independent and therefore holds also in the thermodynamic limit; using the
continuity in mass of ${C}_{\mathcal{M}}$ and the assumed finiteness of
$m_{0}$ and $\Delta D_{\mathrm{max}}$ in the symmetric limit one concludes
that

$\displaystyle\lim_{\delta m\to
0}\lim_{{\mathcal{V}}\to\infty}\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\Big{|}_{\bm{\theta}=0}=0\,,$
(38)

which is what had to be proved [see Eq. (18)].

Notice that under the same assumptions used above one can show that the
expectation values $\left\langle{\mathcal{O}}_{\mathcal{M}}\right\rangle$ are
indeed finite in the thermodynamic limit, independently of the choice of
masses. In fact, a bound similar to Eq. (27) is obtained for
$\langle{\mathcal{O}}_{\mathcal{M}}\rangle$ starting from Eq. (14),

$\displaystyle\left|\left\langle{\mathcal{O}}_{\mathcal{M}}\right\rangle\right|$
$\displaystyle\leq\sum_{\mathrm{P}\in\mathrm{S}_{n}}\left\langle\left({\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})\prod_{k=1}^{n}{\mathcal{S}}({y}_{k},{x}_{\mathrm{P}(k)})\right)^{\frac{1}{2}}\right\rangle_{\mathrm{G}}\,.$
(39)

Under the boundedness assumptions on $\|\mathrm{S}_{M}\|$ and the continuity
assumption on ${\mathcal{M}}$

$\displaystyle\left|\left\langle{\mathcal{O}}_{\mathcal{M}}\right\rangle\right|$
$\displaystyle\leq
n!\,\left(N_{f}N_{c}N_{D}\right)^{\frac{n}{2}}\left\langle{\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})^{\frac{1}{2}}\|\mathrm{S}_{M}\|^{n}\right\rangle_{\mathrm{G}}$
(40)
$\displaystyle\leq\frac{n!\,\left(N_{f}N_{c}N_{D}\right)^{\frac{n}{2}}{C}_{\mathcal{M}}^{\frac{1}{2}}}{m_{0}^{n}}\,,$

which is a finite bound, independent of ${\mathcal{V}}$. The extension of this
result and of Eq. (38) to the case of absolutely integrable
${\mathcal{M}}_{\mathbf{A}\mathbf{B}}$ is discussed in Appendix A.4.

##  V Application to specific discretizations

In this Section I discuss explicitly several lattice discretizations of the
single-flavor Dirac operator. A superscript is used to distinguish them and
the corresponding propagators, i.e.,
$D^{\mathrm{X}(m)}=D^{\mathrm{X}(0)}+m\Delta D^{\mathrm{X}}$,
$\mathrm{S}_{M}^{\mathrm{X}}=\mathrm{diag}(S^{\mathrm{X}(m_{1})},\ldots,S^{\mathrm{X}(m_{N_{f}})})$,
$S^{\mathrm{X}(m)}=(D^{\mathrm{X}(m)})^{-1}$. I discuss in particular
staggered fermions (S), Ginsparg-Wilson fermions (GW), Wilson fermions (W),
and minimally doubled fermions (KW, BC), on hypercubic lattices. Lattice sites
are labelled by coordinates $x_{\mu}=0,\ldots,L_{\mu}-1$, where $L_{\mu}$ is
the linear size in direction $\mu$, with $\mu=1,\ldots,d$. The lattice
oriented edges connect $x$ and $x+\hat{\mu}$, with $\hat{\mu}$ the unit vector
in direction $\mu$; the associated link variables are denoted with
$U(x,x+\hat{\mu})$, and $U(x,x-\hat{\mu})\equiv U(x-\hat{\mu},x)^{\dagger}$
denotes the link variable associated with the oppositely oriented edge.
Finally, $\delta(x,y)=\prod_{\mu=1}^{d}\delta_{x_{\mu}\,y_{\mu}}$. Standard
boundary conditions (periodic for link variables and periodic/antiperiodic in
space/time for Grassmann variables) are understood, although they do not play
any particular role.

###  V.1 Staggered fermions

The case of staggered fermions [27, 28, 29] is the most straightforward. The
corresponding discretization of the lattice Dirac operator carries no Dirac
index, and reads

$\displaystyle D^{\mathrm{S}(0)}(x,y)$
$\displaystyle={\textstyle\frac{1}{2}}\textstyle{\sum_{\mu=1}^{d}}U(x,y)\left(\eta_{\mu}(x)\delta(x+\hat{\mu},y)\right.$
(41)
$\displaystyle\phantom{=\frac{1}{2}\textstyle{\sum_{\mu=1}^{d}}U(x,y)}\left.-\eta_{\mu}(y)\delta(x-\hat{\mu},y)\right)\,,$
$\displaystyle\Delta D^{\mathrm{S}}(x,y)$
$\displaystyle=\mathbf{1}_{C}\delta(x,y)\,,$

where $\eta_{\mu}(x)=(-1)^{\sum_{\alpha<\mu}x_{\alpha}}$. Notice that
$L_{\mu}$ must be an even number for every $\mu$. The staggered operator is
anti-Hermitean, and obviously commutes with $\Delta D^{\mathrm{S}}$. Let
$i\lambda_{n}$, $\lambda_{n}\in\mathbb{R}$, be its purely imaginary
eigenvalues. Since $D^{\mathrm{S}(0)}$ has the chiral property
$\\{\varepsilon,D^{\mathrm{S}(0)}\\}=0$, where
$\varepsilon_{ab}(x,y)=(-1)^{\sum_{\alpha}x_{\alpha}}\delta_{ab}\delta(x,y)$,
these come in complex conjugate pairs $\pm i\lambda_{n}$ or vanish, and so

$\det
D^{\mathrm{S}(m)}=m^{{\mathcal{N}}_{0}^{\mathrm{S}}}\prod_{n,\,\lambda_{n}>0}(\lambda_{n}^{2}+m^{2})\,,$
(42)

where ${\mathcal{N}}_{0}^{\mathrm{S}}$ is the number of exact zero modes,
which must be an even number. The integration measure $d\mu_{\mathrm{G}}$ is
therefore positive for any choice of $m_{f}$. For the propagator one has

$\displaystyle S^{\mathrm{S}(m)\,{\dagger}}S^{\mathrm{S}(m)}$
$\displaystyle=\left(m^{2}-D^{\mathrm{S}(0)}{}^{\,2}\right)^{-1}\,,$ (43)
$\displaystyle\|S^{\mathrm{S}(m)}\|^{2}$
$\displaystyle=\frac{1}{m^{2}+\min_{n}\lambda_{n}^{2}}\leq\frac{1}{m^{2}}\,,$

and so

$\displaystyle\|\mathrm{S}_{M}^{\mathrm{S}}\|^{2}$
$\displaystyle=\max_{f}\|S^{\mathrm{S}(m_{f})}\|^{2}=\frac{1}{\min_{f}m_{f}^{2}+\min_{n}\lambda_{n}^{2}}$
(44)
$\displaystyle\leq\frac{1}{\min_{f}m_{f}^{2}}\equiv\frac{1}{m_{0}^{2}}\,.$

Obviously $\|\Delta D^{\mathrm{S}}\|=1$. All the assumptions used in Section
IV hold, and so vector flavor symmetry cannot be spontaneously broken, as long
as the common fermion mass in the symmetric limit is nonzero. This was already
known [30]. The result holds at any lattice spacing, and remains true for any
choice of boundary conditions or inclusion of external fields (e.g., an
imaginary chemical potential or a magnetic field) that preserves the anti-
Hermiticity and the chiral property of $D^{\mathrm{S}(0)}$.

###  V.2 Ginsparg-Wilson fermions

Massless Ginsparg-Wilson fermions are characterized by the relation [40]

$\\{D^{\mathrm{GW}(0)},\gamma_{5}\\}=2D^{\mathrm{GW}(0)}R\gamma^{5}D^{\mathrm{GW}(0)}\,,$
(45)

with $R$ a local operator, satisfied by the corresponding lattice
discretization $D^{\mathrm{GW}(0)}$ of the Dirac operator. Most of the known
examples [41, 42, 43, 44, 45, 46, 47, 48, 49] satisfy this relation with
$2R=\mathbf{1}$, and moreover are $\gamma_{5}$-Hermitean,

$\gamma_{5}D^{\mathrm{GW}(0)}\gamma_{5}=D^{\mathrm{GW}(0)}{}^{\,{\dagger}}\,.$
(46)

If these extra assumptions hold it is easy to show that

$\displaystyle(D^{\mathrm{GW}(0)}-\mathbf{1})(D^{\mathrm{GW}(0)}-\mathbf{1})^{\dagger}$
$\displaystyle=\mathbf{1}\,,$ (47)

i.e., $D^{\mathrm{GW}(0)}=\mathbf{1}+{\mathcal{U}}$ with ${\mathcal{U}}$
unitary. For massive Ginsparg-Wilson fermions one uses

$\Delta
D^{\mathrm{GW}}=\mathbf{1}-{\textstyle\frac{1}{2}}D^{\mathrm{GW}(0)}\,,$ (48)

and so

$D^{\mathrm{GW}(m)}=\left(1+{\textstyle\frac{m}{2}}\right)\mathbf{1}+\left(1-{\textstyle\frac{m}{2}}\right){\mathcal{U}}\,.$
(49)

This is a normal operator with spectrum lying on a circle of radius
$|1-\frac{m}{2}|$ centered at $1+\frac{m}{2}$, so its eigenvalues are bounded
in magnitude from below by the square root of

$\displaystyle\min_{\varphi}\left|\left(1+{\textstyle\frac{m}{2}}\right)+\left(1-{\textstyle\frac{m}{2}}\right)e^{i\varphi}\right|^{2}=\min(m^{2},4)\,.$
(50)

It follows that the propagator obeys
$\|\mathrm{S}^{\mathrm{GW}}_{M}\|\leq\max\left(\frac{1}{\min_{f}|m_{f}|},\frac{1}{2}\right)$,
which is a finite bound if $m_{f}\neq 0$ $\forall f$. For $\Delta
D^{\mathrm{GW}}$ one has

$\displaystyle\|\Delta
D^{\mathrm{GW}}\|^{2}\leq{\textstyle\frac{1}{2}}\max_{\varphi}\left|1-e^{i\varphi}\right|^{2}=2\equiv\Delta
D_{\mathrm{max}}^{2}<\infty\,.$ (51)

As a consequence of Eq. (46),
$\gamma_{5}{\mathcal{U}}\gamma_{5}={\mathcal{U}}^{\dagger}$, and so if
$\psi_{n}$ is a common eigenvector of $D^{\mathrm{GW}(0)}$ and
$D^{\mathrm{GW}(0)}{}^{\,{\dagger}}$ with eigenvalues
$\mu_{n}=1+e^{i\varphi_{n}}$ and $\mu_{n}^{*}=1+e^{-i\varphi_{n}}$,
respectively, then
$D^{\mathrm{GW}(0)}\gamma_{5}\psi_{n}=\gamma_{5}D^{\mathrm{GW}(0)}{}^{\,{\dagger}}\psi_{n}=\mu_{n}^{*}\gamma_{5}\psi_{n}$.
It follows that complex eigenvalues come in complex-conjugate pairs; for the
real eigenvalues $\mu_{n}=\mu_{n}^{*}=0,2$, one can instead choose chiral
eigenvectors $\psi_{\pm}$, satisfying $\gamma_{5}\psi_{\pm}=\pm\psi_{\pm}$.
For the determinant of $D^{\mathrm{GW}(m)}$ one finds

$\displaystyle\det D^{\mathrm{GW}(m)}$
$\displaystyle=m^{{\mathcal{N}}_{0}^{\mathrm{GW}}}2^{{\mathcal{N}}_{2}^{\mathrm{GW}}}$
(52)
$\displaystyle\phantom{=}\times\prod_{n,\,\sin\varphi_{n}>0}\left[\left(2\cos{\textstyle\frac{\varphi_{n}}{2}}\right)^{2}+\left(m\sin{\textstyle\frac{\varphi_{n}}{2}}\right)^{2}\right]\,,$

with ${\mathcal{N}}_{0,2}^{\mathrm{GW}}$ the degeneracies of the two real
eigenvalues. It follows that the integration measure $d\mu_{\mathrm{G}}$ is
positive if $m_{f}\geq 0$ $\forall f$, and more generally for an even number
of negative masses. Vector flavor symmetry cannot be spontaneously broken in
the symmetric limit as long as the common fermion mass is positive, or just
nonzero if $N_{f}$ is even, at any value of the lattice spacing. The use of
different boundary conditions or the inclusion of external fields in
$D^{\mathrm{GW}(0)}$ does not change this result, as long as the operator
remains of the form $D^{\mathrm{GW}(0)}=\mathbf{1}+{\mathcal{U}}$ and the
$\gamma_{5}$-Hermiticity property Eq. (46) holds.

###  V.3 Wilson fermions

For Wilson fermions [26] the massless operator
$D^{\mathrm{W}(0)}=D^{\mathrm{n}(0)}+R^{\mathrm{W}}$ is obtained adding the
naive discretization $D^{\mathrm{n}(0)}$ of the massless Dirac operator and
the Wilson term $R^{\mathrm{W}}$, while $\Delta D^{\mathrm{W}}$ is the
identity in color, Dirac, and coordinate space,

$\displaystyle D^{\mathrm{n}(0)}(x,y)$
$\displaystyle={\textstyle\frac{1}{2}}\textstyle{\sum_{\mu=1}^{d}}U(x,y)\gamma_{\mu}\left(\delta(x+\hat{\mu},y)\right.$
(53)
$\displaystyle\phantom{={\textstyle\frac{1}{2}}\textstyle{\sum_{\mu=1}^{d}}U(x,y)\gamma_{\mu}()}\left.-\delta(x-\hat{\mu},y)\right)\,,$
$\displaystyle R^{\mathrm{W}}(x,y)$
$\displaystyle=-{\textstyle\frac{r}{2}}\mathbf{1}_{D}\textstyle{\sum_{\mu=1}^{d}}\left(U(x,y)\left(\delta(x+\hat{\mu},y)\right.\right.$
$\displaystyle\phantom{=-{\textstyle\frac{r}{2}}\mathbf{1}_{D}\textstyle{\sum_{\mu=1}^{d}}U(x,y)()}\left.\left.+\delta(x-\hat{\mu},y)\right)\right.$
$\displaystyle\phantom{=-{\textstyle\frac{r}{2}}\mathbf{1}_{D}\textstyle{\sum_{\mu=1}^{d}}()}\left.-2\mathbf{1}_{C}\delta(x,y)\right)\,,$
$\displaystyle\Delta D^{\mathrm{W}}(x,y)$
$\displaystyle=\mathbf{1}_{C}\mathbf{1}_{D}\delta(x,y)\,,$

with $r$ a nonzero real parameter. This operator is not anti-Hermitean and not
even normal, satisfying only the $\gamma_{5}$-Hermiticity condition
$\gamma_{5}D^{\mathrm{W}(m)}\gamma_{5}=D^{\mathrm{W}(m)\,{\dagger}}$. The
spectrum of $D^{\mathrm{W}(m)}$ is generally complex, and while
$\gamma_{5}$-Hermiticity guarantees that $\det D^{\mathrm{W}(m)}$ is real, one
is not guaranteed to find a positive integration measure $d\mu_{\mathrm{G}}$,
unless an even number of fermions with the same mass is present. Moreover,
while $\Delta D^{\mathrm{W}}$ is obviously bounded, no general lower bound
applies to the spectrum of $D^{\mathrm{W}(m)\,{\dagger}}D^{\mathrm{W}(m)}$,
even in the massive case, and so no uniform upper bound on the norm of the
propagator is available. The result of the previous Section therefore does not
apply to Wilson fermions. This is not surprising since it is known that vector
flavor symmetry is spontaneously broken in the Aoki phase [5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. One can, however,
refine the discussion and see more precisely how things fail for Wilson
fermions.

Presumably, for an even number of flavors and sufficiently small $\delta m$
the sign problem of the integration measure affects only a set of gauge
configurations of zero measure. If so, in this case the integration measure
would effectively be positive, and so one could follow the derivation of the
previous Section up to Eq. (36), obtaining

$\displaystyle\left|\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\Big{|}_{\theta=0}\right|$
$\displaystyle\leq\delta
m\,\tilde{{C}}_{\mathcal{M}}\left\langle\|\mathrm{S}_{M}^{\mathrm{W}}\|^{n+1}\right\rangle_{\mathrm{G}}$
(54) $\displaystyle=\delta
m\,\tilde{{C}}_{\mathcal{M}}\int_{0}^{\infty}ds\,p(s)s^{n+1}\,,$

where

$p(s)\equiv\left\langle\delta\left(s-\|\mathrm{S}_{M}^{\mathrm{W}}\|\right)\right\rangle_{\mathrm{G}}\,.$
(55)

One could then still exclude the spontaneous breaking of vector flavor
symmetry if $p(s)$ vanished faster than any polynomial as $s\to\infty$, for
example if $p(s)=0$ for $s>s_{0}$ for some $s_{0}$, or if it vanished
exponentially. If $p(s)$ vanished only as a power law $p(s)\sim s^{-n_{0}}$,
or not at all, the argument above would not provide a viable bound for
$n>n_{0}-2$, and spontaneous breaking could not be excluded. In fact, in the
Aoki phase one finds a finite spectral density of near-zero modes of the
Hermitean operator $H=\gamma_{5}D^{\mathrm{W}(m)}$ [7, 20, 21]. Since

$\displaystyle\|\mathrm{S}^{\mathrm{W}(m)}\|^{2}$
$\displaystyle=\sup_{\psi\neq
0}\frac{(\psi,\left(D^{\mathrm{W}(m)}D^{\mathrm{W}(m)\,{\dagger}}\right)^{-1}\psi)}{(\psi,\psi)}$
(56) $\displaystyle=\sup_{\psi\neq
0}\frac{(\gamma_{5}\psi,\left(H^{2}\right)^{-1}\gamma_{5}\psi)}{(\gamma_{5}\psi,\gamma_{5}\psi)}=\frac{1}{\min_{n}h_{n}^{2}}\,,$

where $h_{n}\in\mathbb{R}$ are the eigenvalues of $H$, one finds that as the
lattice volume grows and the lowest mode of $H$ on typical configurations
tends to zero $p(s)$ becomes more and more peaked at a larger and larger value
of $s$, eventually tending to infinity in the thermodynamic limit, and the
right-hand side of Eq. (54) blows up, making the bound useless. Outside the
Aoki phase the spectrum of $H$ is gapped around the origin, the right-hand
side of Eq. (54) is finite, and the bound prevents spontaneous flavor symmetry
breaking.

###  V.4 Minimally doubled fermions

For the minimally doubled fermions of Karsten and Wilczek (KW) [36, 37], and
of Creutz and Boriçi (BC) [38, 39], the massless Dirac operator is of the form
$D^{\mathrm{X}(0)}=D^{\mathrm{n}(0)}+R^{\mathrm{X}}$,
$\mathrm{X}=\mathrm{KW},\mathrm{BC}$, where the naive operator
$D^{\mathrm{n}(0)}$ is defined in Eq. (53), and the inclusion of the terms

$\displaystyle R^{\mathrm{KW}}(x,y)$
$\displaystyle=-{\textstyle\frac{ir}{2}}\gamma_{d}\textstyle{\sum_{\mu=1}^{d-1}}\left(U(x,y)\left(\delta(x+\hat{\mu},y)\right.\right.$
(57)
$\displaystyle\phantom{=-{\textstyle\frac{ir}{2}}\gamma_{d}\textstyle{\sum_{\mu=1}^{d-1}}()U(x,y)}\left.\left.+\delta(x-\hat{\mu},y)\right)\right.$
$\displaystyle\phantom{=-{\textstyle\frac{ir}{2}}\gamma_{d}\textstyle{\sum_{\mu=1}^{d-1}}()}\left.-2\mathbf{1}_{C}\delta(x,y)\right)\,,$
$\displaystyle R^{\mathrm{BC}}(x,y)$
$\displaystyle=-{\textstyle\frac{ir}{2}}\textstyle{\sum_{\mu=1}^{d}}\gamma_{\mu}^{\prime}\left(U(x,y)\left(\delta(x+\hat{\mu},y)\right.\right.$
$\displaystyle\phantom{=-{\textstyle\frac{ir}{2}}\textstyle{\sum_{\mu=1}^{d}}()U(x,y)\gamma_{\mu}^{\prime}}\left.\left.+\delta(x-\hat{\mu},y)\right)\right.$
$\displaystyle\phantom{=-{\textstyle\frac{ir}{2}}\textstyle{\sum_{\mu=1}^{d}}()\gamma_{\mu}^{\prime}}\left.-2\mathbf{1}_{C}\delta(x,y)\right)\,,$

where $\gamma_{\mu}^{\prime}\equiv\Gamma\gamma_{\mu}\Gamma$ and
$\Gamma\equiv\frac{1}{\sqrt{d}}\sum_{\nu=1}^{d}\gamma_{\nu}$, reduces the
number of doublers to two when $r=1$. The massive operator is obtained in both
cases using the trivial mass term $\Delta D^{\mathrm{KW}}=\Delta
D^{\mathrm{BC}}=\Delta D^{\mathrm{W}}$, see again Eq. (53). For both types of
fermions the massless operator is anti-Hermitean and chiral,
$\\{D^{\mathrm{X}(0)},\gamma_{5}\\}=0$, and obviously commutes with the mass
term. One can then diagonalize $D^{\mathrm{X}(0)}$ obtaining purely imaginary
eigenvalues $i\lambda_{n}^{\mathrm{X}}$ and a symmetric spectrum, and so the
single-flavor propagators $S^{\mathrm{X}(m)}$ obey

$\|S^{\mathrm{X}(m)}\|^{2}=\frac{1}{m^{2}+\min_{n}\lambda_{n}^{\mathrm{X}\,2}}\leq\frac{1}{m^{2}}\,,\qquad\mathrm{X}=\mathrm{KW},\mathrm{BC}\,.$
(58)

The fermionic determinant reads

$\det
D^{\mathrm{X}(m)}=m^{{\mathcal{N}}_{0}^{\mathrm{X}}}\prod_{n,\,\lambda_{n}>0}(\lambda_{n}^{\mathrm{X}\,2}+m^{2})\,,\qquad\mathrm{X}=\mathrm{KW},\mathrm{BC}\,,$
(59)

with ${\mathcal{N}}_{0}^{\mathrm{X}}$ the number of exact zero modes, so it is
positive for nonnegative fermion masses, and for an even number of negative
masses. The same argument therefore applies as with staggered fermions, and
vector flavor symmetry cannot break spontaneously as long as the common
fermion mass is positive (or just nonzero if $N_{f}$ is even) in the symmetric
limit, independently of the lattice spacing.

##  VI Conclusions

In this paper I have shown that under quite general assumptions on the
discretization of the Dirac operator $\mathrm{D}_{M}$, one can rigorously
exclude the possibility of spontaneous breaking of vector flavor symmetry on
the lattice, in gauge theories with a positive functional-integral measure.
These assumptions are:

  1. 0.

$\mathrm{D}_{M}$ is linear in the fermion masses,
$\mathrm{D}_{M}=D^{(0)}+M\Delta D$, with $D^{(0)}$ and $\Delta D$ trivial in
flavor space, and $M$ a Hermitean mass matrix;

  2. 1.

the norm of the propagator $\mathrm{D}_{M}^{-1}$ can be bounded by a
configuration- and volume-independent quantity, that remains finite in the
symmetric limit of fermions of equal masses, $M\to m\mathbf{1}_{F}$;

  3. 2.

the norm of the derivative of $\mathrm{D}_{M}$ with respect to the fermion
masses, $\Delta D$, can be bounded by a configuration- and volume-independent
quantity, that remains finite as $M\to m\mathbf{1}_{F}$.

The impossibility of spontaneous flavor symmetry breaking on the lattice is
proved by showing that any localized order parameter must vanish in the
symmetric limit, taken after the thermodynamic limit. If the assumptions above
hold for any (or at least for sufficiently small) lattice spacing, this result
remains true also in the continuum limit, if this exists. My argument applies
in particular to staggered fermions [27, 28, 29]; to the minimally doubled
fermions of Karsten and Wilczek [36, 37] and of Creutz and Boriçi [38, 39];
and to Ginsparg-Wilson fermions [40, 41, 42, 43, 44, 45, 46, 47, 48, 49] that
are $\gamma_{5}$-Hermitean and satisfy the Ginsparg-Wilson relation with
$2R=\mathbf{1}$ [see Eq. (45)]. For these discretizations one can exclude
spontaneous breaking of vector flavor symmetry on the lattice for any spacing,
and so in the continuum limit as well, for any positive common fermion mass
$m$ (and for any nonzero $m$ for an even number of flavors). Quite
unsurprisingly, the argument fails in the case of Wilson fermions [26], where
such a spontaneous breaking is known to happen in the Aoki phase [5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. While for
staggered fermions spontaneous breaking of vector flavor symmetry (as well as
of baryon number symmetry) was already completely excluded by the results of
Ref. [30], for Ginsparg-Wilson fermions only partial results were previously
available [50].

My result is clearly not as powerful as that obtained by Vafa and Witten
working with the continuum functional integral in Ref. [1], and by Aloisio et
al. in Ref. [30] working with staggered fermions on the lattice. In
particular, although it excludes the possibility of Goldstone bosons appearing
in the spectrum due to spontaneous flavor symmetry breaking, it cannot exclude
completely the presence of massless bosons, as Refs. [1, 30] do. On the other
hand, the use of a properly regularized functional integral rather than the
continuum one used in Ref. [1] makes the present argument mathematically fully
rigorous. Since the ultralocality and anti-Hermiticity of the staggered
operator are not used, as they are in Ref. [30], my argument works also for
more general discretizations, in particular allowing one to treat the case of
an arbitrary number of physical fermion flavors in the continuum limit without
resorting to the “rooting trick”.

The bound on the variation of expectation values under a vector flavor
transformation [see Eq. (37)] proved here to derive the main result is
probably far less than optimal, as it does not take into account the
cancellations present in fermionic observables due to the oscillating sign of
the contributions of the various field contractions. The bound on the
propagator [see Eq. (32)] is also likely to be suboptimal, and one suspects
that a lattice analogue of the Vafa-Witten bound could be obtained also for
more general discretizations than staggered fermions, for which it was proved
in Ref. [30]. A direct extension of the proof of Ref. [30] to minimally
doubled fermions seems feasible, while a different approach is probably needed
for Ginsparg-Wilson fermions. It is worth noting that a global bound on the
propagator, instead of the local one of Refs. [1, 30], suffices to prove the
impossibility of vector flavor symmetry breaking.

The present argument does not rule out the appearance of phases with
spontaneously broken vector flavor symmetry on the lattice if terms of order
higher than quadratic are included in the fermionic action, even if the
quadratic terms satisfy assumptions (0.)–(2.). Non-symmetric vacua may in fact
exist, degenerate with the symmetric one in the symmetric limit, but with
ground state energy increased by the standard symmetry-breaking term used
here. These vacua could not be reached with the procedure used here, and would
require the addition of different symmetry-breaking terms to the symmetric
action in order to select them. This possibility is of limited interest in the
physical case of QCD, since in this theory vector flavor symmetry is broken
explicitly precisely by the differences in the quark masses, and the symmetric
limit of interest where one should investigate the possibility of its
spontaneous breaking is the one considered in this paper. More generally,
while such spontaneously broken phases on the lattice could be problematic for
numerical simulations, they should be unphysical and not survive the continuum
limit.

The restriction to a quadratic lattice action is in fact not really a
limitation as far as the usual continuum limit is concerned. For continuum
gauge theories in dimension $d>2$ (in $d\leq 2$ the spontaneous breaking of a
continuous symmetry is forbidden [55, 56, 57]) there are no perturbatively
renormalizable fermionic operators with the right global and local symmetries
other than the quadratic ones, approximated on the lattice by the action used
here. The inclusion of higher order terms in the lattice action only adds
perturbatively non-renormalizable interactions that do not affect the long-
distance physics in the usual continuum limit. Hypothetical spontaneously
broken phases on the lattice should then shrink as the continuum limit is
approached, with vector flavor symmetry being realized in the continuum
theory. Phases with spontaneously broken vector flavor symmetry may still be
found in the continuum if unconventional continuum limits exist, but this
would concern a different type of continuum theories. Universality of the
continuum limit also implies that the spontaneously broken phases potentially
found on the lattice for quadratic actions not satisfying assumptions
(0.)–(2.) should shrink in the usual continuum limit, as is the case for the
Aoki phase of Wilson fermions.

In conclusion, the existence of lattice discretizations of the Dirac operator,
free of doublers, for which spontaneous vector flavor symmetry breaking for
finite positive fermion mass is impossible at any lattice spacing (i.e., the
Ginsparg-Wilson fermions discussed above) implies the same impossibility in
the continuum limit, if this exists, for an arbitrary number of fermion
species. This settles the issue of spontaneous vector flavor symmetry breaking
in a rigorous manner (for a physicist’s standard of rigor).

###### Acknowledgements.

I thank C. Bonati and M. D’Elia for discussions, and V. Azcoiti for
discussions and a careful reading of the manuscript. This work was partially
supported by the NKFIH grant KKP-126769.

##  Appendix A Technical details

###  A.1 Integrated Ward-Takahashi identity and finite flavor transformations

Since the Berezin integration measure is invariant under vector flavor
transformations, Eq. (15), one has after changing variables [see Eq. (16) for
the notation]

$\displaystyle\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}{\mathcal{O}}^{\bm{\theta}}[\psi,\bar{\psi},U]$
$\displaystyle=\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\bar{\psi}V(\bm{\theta})\mathrm{D}_{M}[U]V(\bm{\theta})^{\dagger}\psi}{\mathcal{O}}[\psi,\bar{\psi},U]\,.$
(60)

For infinitesimal $\theta_{a}$, expanding both sides of the equation to
leading order in $\theta_{a}$, one finds

$\displaystyle\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}\left({\mathcal{O}}[\psi,\bar{\psi},U]+\sum_{a}\theta_{a}\frac{\partial}{\partial\theta_{a}}{\mathcal{O}}^{\bm{\theta}}[\psi,\bar{\psi},U]\Big{|}_{\bm{\theta}=0}+O(\theta^{2})\right)$
(61)
$\displaystyle=\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}\left({\mathcal{O}}[\psi,\bar{\psi},U]-i\left(\bar{\psi}[\bm{\theta}\cdot\bm{t},\mathrm{D}_{M}[U]]\psi\right)\,{\mathcal{O}}[\psi,\bar{\psi},U]+O(\theta^{2})\right)\,,$

from which Eq. (19) follows,

$\displaystyle
i\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}^{\bm{\theta}}\right\rangle\bigg{|}_{\bm{\theta}=0}$
$\displaystyle=\left\langle\left(\bar{\psi}[t^{a},\mathrm{D}_{M}]\psi\right){\mathcal{O}}\right\rangle=\left\langle{\mathcal{C}}^{a}{\mathcal{O}}\right\rangle\,,$
$\displaystyle{\mathcal{C}}^{a}[\psi,\bar{\psi},U]$
$\displaystyle\equiv\bar{\psi}[t^{a},\mathrm{D}_{M}[U]]\psi\,.$ (62)

For finite transformations, since $V(\bm{\theta})$ is analytic in $\theta_{a}$
one can write the trivial identity

$\displaystyle\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}\left({\mathcal{O}}^{\bm{\theta}}[\psi,\bar{\psi},U]-{\mathcal{O}}[\psi,\bar{\psi},U]\right)=\int_{0}^{1}d\alpha\,\frac{d}{d\alpha}\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}{\mathcal{O}}^{\alpha\bm{\theta}}[\psi,\bar{\psi},U]\,.$
(63)

Using now Eq. (60) one finds the following result for the change of the
expectation value of an observable under a finite vector flavor
transformation,

$\displaystyle\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}\left({\mathcal{O}}^{\bm{\theta}}[\psi,\bar{\psi},U]-{\mathcal{O}}[\psi,\bar{\psi},U]\right)$
(64)
$\displaystyle=-\int_{0}^{1}d\alpha\,\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\bar{\psi}V(\alpha\bm{\theta})\mathrm{D}_{M}[U]V(\alpha\bm{\theta})^{\dagger}\psi}\left(\bar{\psi}V(\alpha\bm{\theta})\left[V(\alpha\bm{\theta})^{\dagger}\frac{dV(\alpha\bm{\theta})}{d\alpha},\mathrm{D}_{M}[U]\right]V(\alpha\bm{\theta})^{\dagger}\psi\right)\,{\mathcal{O}}[\psi,\bar{\psi},U]$
$\displaystyle=-i\int_{0}^{1}d\alpha\,\int[\mathrm{D}\psi\mathrm{D}\bar{\psi}]\,e^{-\mathcal{S}_{\mathrm{F}}[\psi,\bar{\psi},U]}\,\left(\bar{\psi}\left[\bm{\theta}\cdot\bm{t},\mathrm{D}_{M}[U]\right]\psi\right)\,{\mathcal{O}}^{\alpha\bm{\theta}}[\psi,\bar{\psi},U]\,.$

This implies

$\displaystyle\left\langle{\mathcal{O}}^{\bm{\theta}}-{\mathcal{O}}\right\rangle$
$\displaystyle=-i\sum_{a}\theta_{a}\int_{0}^{1}d\alpha\,\left\langle{\mathcal{C}}^{a}\,{\mathcal{O}}^{\alpha\bm{\theta}}\right\rangle\,.$
(65)

For the observables ${\mathcal{O}}_{\mathcal{M}}$ of interest, the effect of a
vector flavor transformation $V(\bm{\theta})$ can be fully accounted for by
replacing ${\mathcal{M}}\to{\mathcal{M}}^{\bm{\theta}}$, i.e.,
${\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}={\mathcal{O}}_{{\mathcal{M}}^{\bm{\theta}}}$,
with

${\mathcal{M}}^{\bm{\theta}}_{\mathbf{A}\mathbf{B}}\equiv{\mathcal{M}}_{A_{1}^{\prime}\ldots
A_{n}^{\prime}B_{1}^{\prime}\ldots
B_{n}^{\prime}}V(\bm{\theta})^{\dagger}_{A_{1}A_{1}^{\prime}}\ldots
V(\bm{\theta})^{\dagger}_{A_{n}A_{n}^{\prime}}V(\bm{\theta})_{B_{1}^{\prime}B_{1}}\ldots
V(\bm{\theta})_{B_{n}^{\prime}B_{n}}\,,$ (66)

where
$V_{A_{i}A_{i}^{\prime}}=V_{f_{i}f_{i}^{\prime}}\,\delta_{a_{i}a_{i}^{\prime}}\delta_{\alpha_{i}\alpha_{i}^{\prime}}$.
Notice that

$\displaystyle{\mathcal{K}}_{{\mathcal{M}}^{\bm{\theta}}}(\mathbf{x},\mathbf{y})$
$\displaystyle={\rm
tr}_{FCD}\\{{\mathcal{M}}^{\bm{\theta}}(\mathbf{x},\mathbf{y}){\mathcal{M}}^{\bm{\theta}}(\mathbf{x},\mathbf{y})^{\dagger}\\}={\rm
tr}_{FCD}\\{{\mathcal{M}}(\mathbf{x},\mathbf{y}){\mathcal{M}}(\mathbf{x},\mathbf{y})^{\dagger}\\}={\mathcal{K}}_{{\mathcal{M}}}(\mathbf{x},\mathbf{y})\,,$
(67)

which implies that the same upper bound applies to ${\mathcal{M}}$ and
${\mathcal{M}}^{\bm{\theta}}$ in Eq. (29), i.e.,
${C}_{{\mathcal{M}}^{\bm{\theta}}}={C}_{\mathcal{M}}$, and the same constant
$\tilde{{C}}_{\mathcal{M}}=\tilde{{C}}_{{\mathcal{M}}^{\bm{\theta}}}$ appears
in the bound Eq. (37). Using now Eq. (65) one finds

$\displaystyle\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}-{\mathcal{O}}_{\mathcal{M}}\right\rangle$
$\displaystyle=\left\langle{\mathcal{O}}_{{\mathcal{M}}^{\bm{\theta}}}-{\mathcal{O}}_{\mathcal{M}}\right\rangle=-i\sum_{a}\theta_{a}\int_{0}^{1}d\alpha\,\left\langle{\mathcal{C}}^{a}\,{\mathcal{O}}_{{\mathcal{M}}^{\alpha\bm{\theta}}}\right\rangle\,.$
(68)

The integrand on the right-hand side obeys the bound Eq. (37), and since
$\tilde{{C}}_{{\mathcal{M}}^{\alpha\bm{\theta}}}=\tilde{{C}}_{\mathcal{M}}$
are independent of $\alpha$ one finds

$\displaystyle\left|\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}-{\mathcal{O}}_{\mathcal{M}}\right\rangle\right|$
$\displaystyle\leq\sum_{a}|\theta_{a}|\int_{0}^{1}d\alpha\,\left|\left\langle{\mathcal{C}}^{a}\,{\mathcal{O}}_{{\mathcal{M}}^{\alpha\bm{\theta}}}\right\rangle\right|\leq\delta
m\frac{\tilde{{C}}_{\mathcal{M}}\Delta
D_{\mathrm{max}}}{m_{0}^{n+1}}\sum_{a}|\theta_{a}|\,.$ (69)

Since this bound is independent of ${\mathcal{V}}$ and remains finite in the
symmetric limit, one has

$\lim_{\delta m\to
0}\lim_{{\mathcal{V}}\to\infty}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}-{\mathcal{O}}_{\mathcal{M}}\right\rangle=0\,.$
(70)

This implies
$\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle=\left\langle{\mathcal{O}}_{\mathcal{M}}\right\rangle$
in the thermodynamic and symmetric limit, for any $\bm{\theta}$, and so vector
flavor symmetry cannot be spontaneously broken if the assumptions of Section
II are satisfied.

###  A.2 Vanishing of order parameters

Under $\mathrm{SU}(N_{f})$ vector flavor transformations, the matrix
${\mathcal{M}}$ transforms in the product representation with $n$ fundamental
and $n$ antifundamental factors. This is a unitary representation that can be
decomposed in the direct sum of unitary irreducible representations. The most
general ${\mathcal{M}}$ is then the linear combination of matrices
${\mathcal{M}}^{(R)}_{r}(\mathbf{x},\mathbf{y})$,

$\big{(}{\mathcal{M}}^{(R)}_{r}\big{)}_{f_{1}{A_{\star}}_{1}\ldots
f_{n}{A_{\star}}_{n}g_{1}{B_{\star}}_{1}\ldots
g_{n}{B_{\star}}_{n}}(\mathbf{x},\mathbf{y})=({\mathcal{T}}^{(R)}_{r})_{f_{1}\ldots
f_{n}g_{1}\ldots
g_{n}}\tilde{\mathcal{M}}_{{A_{\star}}_{1}\ldots{A_{\star}}_{n}{B_{\star}}_{1}\ldots{B_{\star}}_{n}}(\mathbf{x},\mathbf{y})\,,$
(71)

where $\tilde{\mathcal{M}}$ is a matrix acting only on color and Dirac space
that can depend on the gauge variables in a suitably gauge-covariant way, and
where ${\mathcal{T}}^{(R)}_{r}$, $r=1,\ldots,d_{R}$, are constant tensors in
flavor space, independent of the gauge variables and of any feature of the
theory other than the number of flavors, that transform irreducibly under
vector flavor transformations [see Eq. (66) for the notation],

${\mathcal{T}}^{(R)\,\bm{\theta}}_{r}=\sum_{r^{\prime}}{\mathcal{T}}^{(R)}_{r^{\prime}}D^{(R)}\left(V(\bm{\theta})\right)_{r^{\prime}r}\,,$
(72)

with $D^{(R)}(V)$ the representative of $V\in\mathrm{SU}(N_{f})$ in the
irreducible unitary representation $R$ of dimension $d_{R}$. For these
quantities one has in the thermodynamic and symmetric limit [see Eq. (70)]

$\displaystyle u^{(R)}_{r}$
$\displaystyle\equiv\left\langle{\mathcal{O}}_{{\mathcal{T}}^{(R)}_{r}\tilde{\mathcal{M}}}\right\rangle=\left\langle{\mathcal{O}}_{{\mathcal{T}}^{(R)}_{r}\tilde{\mathcal{M}}}^{\bm{\theta}}\right\rangle=\sum_{r^{\prime}}\left\langle{\mathcal{O}}_{{\mathcal{T}}^{(R)}_{r^{\prime}}\tilde{\mathcal{M}}}\right\rangle
D^{(R)}(V(\bm{\theta}))_{r^{\prime}r}=\sum_{r^{\prime}}u^{(R)}_{r^{\prime}}D^{(R)}(V(\bm{\theta}))_{r^{\prime}r}\,,$
(73)

which expresses the invariance of the vector $u^{(R)}$ with components
$u^{(R)}_{r}$, $r=1\ldots,d_{R}$, under an arbitrary $\mathrm{SU}(N_{f})$
transformation. No non-zero invariant vector exists if $R$ is a nontrivial
representation444Proof: In matrix notation, Eq. (73) reads
$u^{(R)T}=u^{(R)T}D^{(R)}(V)$, $\forall V\in\mathrm{SU}(N_{f})$, implying also
$u^{(R)*}=D^{(R)}(V)^{\dagger}u^{(R)*}$. The projector $P=u^{(R)*}u^{(R)T}$ is
then left invariant by an irreducible unitary representation of
$\mathrm{SU}(N_{f})$,
$P=D^{(R)}(V)^{\dagger}PD^{(R)}(V)=D^{(R)}(V)^{-1}PD^{(R)}(V)$. By Schur’s
lemma $P$ is then proportional to the $d_{R}$-dimensional identity matrix, but
since it has only rank 1 it must vanish if $d_{R}\neq 1$., so $u^{(R)}=0$
follows unless $R$ is the one-dimensional trivial representation, and all
order parameters vanish.

###  A.3 Bound on partial traces

Let $\mathrm{A}$ be a matrix with entries labelled by pairs of an arbitrary
number of indices. Denote the subset of indices corresponding to an “internal”
space $\mathcal{I}$ collectively by $I$, and denote the remaining ones
collectively by $x$, i.e., the matrix entries are labelled as
$\mathrm{A}_{I^{\prime}I}(x^{\prime},x)$. The partial trace over the internal
space of the matrix
$\mathrm{A}(x^{\prime},x)^{\dagger}\mathrm{A}(x^{\prime},x)$, where Hermitean
conjugation and matrix multiplication are understood to apply only to the
internal indices, satisfies the bound

${\rm
tr}_{\mathcal{I}}\mathrm{A}(x^{\prime},x)^{\dagger}\mathrm{A}(x^{\prime},x)={\rm
tr}_{\mathcal{I}}\mathrm{A}(x^{\prime},x)\mathrm{A}(x^{\prime},x)^{\dagger}\equiv\sum_{I,I^{\prime}}\mathrm{A}_{I^{\prime}I}(x^{\prime},x)\mathrm{A}_{I^{\prime}I}(x^{\prime},x)^{*}\leq\mathrm{dim}\,\mathcal{I}\,\|\mathrm{A}\|^{2}\,,$
(74)

where $\|\mathrm{A}\|$ is the usual operator norm of $\mathrm{A}$. This is
equal to the square root of the largest eigenvalue of the matrix
$\mathrm{A}^{\dagger}\mathrm{A}$, where Hermitean conjugation and matrix
multiplication are understood to apply to all the indices [see Eqs. (9) and
(10) for the notation].

#### Proof

By the spectral theorem, the Hermitean matrix $\mathrm{A}^{\dagger}\mathrm{A}$
can be written as follows,

$\displaystyle\mathrm{A}^{\dagger}\mathrm{A}$
$\displaystyle=\sum_{n}a_{n}^{2}\phi_{n}\phi_{n}^{\dagger}\,,$
$\displaystyle\left(\mathrm{A}^{\dagger}\mathrm{A}\right)_{I^{\prime}I}(x^{\prime},x)$
$\displaystyle=\sum_{n}a_{n}^{2}\phi_{n\,I^{\prime}}(x^{\prime})\,\phi_{n\,I}(x)^{*}\,,$
(75)

where $a_{n}^{2}$ are its real positive eigenvalues and $\phi_{n}$ are a
complete set of orthonormal eigenvectors,
$(\mathrm{A}^{\dagger}\mathrm{A})\phi_{n}=a_{n}^{2}\phi_{n}$, with

$\displaystyle(\phi_{n^{\prime}},\phi_{n})$
$\displaystyle\equiv\sum_{x,I}\phi_{n^{\prime}\,I}(x)^{*}\phi_{n\,I}(x)=\delta_{n^{\prime}n}\,,$
$\displaystyle\sum_{n}\phi_{n\,I^{\prime}}(x^{\prime})\phi_{n\,I}(x)^{*}$
$\displaystyle=\delta_{I^{\prime}I}\delta(x^{\prime},x)\,.$ (76)

Since ${\rm
tr}_{\mathcal{I}}\mathrm{A}(x^{\prime},x)^{\dagger}\mathrm{A}(x^{\prime},x)\geq
0$ for any $x^{\prime}$, one finds

$\displaystyle{\rm
tr}_{\mathcal{I}}\mathrm{A}(x^{\prime},x)^{\dagger}\mathrm{A}(x^{\prime},x)$
$\displaystyle={\rm
tr}_{\mathcal{I}}(\mathrm{A}^{\dagger})(x,x^{\prime})\mathrm{A}(x^{\prime},x)\leq\sum_{x^{\prime}}{\rm
tr}_{\mathcal{I}}(\mathrm{A}^{\dagger})(x,x^{\prime})\mathrm{A}(x^{\prime},x)={\rm
tr}_{\mathcal{I}}(\mathrm{A}^{\dagger}\mathrm{A})(x,x)$ (77)
$\displaystyle=\sum_{I,n}a_{n}^{2}\phi_{n\,I}(x)\,\phi_{n\,I}(x)^{*}\leq\left(\max_{n^{\prime}}a_{n^{\prime}}^{2}\right)\sum_{I,n}\phi_{n\,I}(x)\,\phi_{n\,I}(x)^{*}=\|\mathrm{A}\|^{2}\sum_{I}\delta_{II}\delta(x,x)$
$\displaystyle=\mathrm{dim}\,\mathcal{I}\,\|\mathrm{A}\|^{2}\,.$

###  A.4 Extension to absolutely integrable
${\mathcal{M}}_{\mathbf{A}\mathbf{B}}$

The requirement of continuity of ${\mathcal{M}}_{\mathbf{A}\mathbf{B}}$, used
in Section IV to bound ${\mathcal{K}}_{\mathcal{M}}$ independently of the
gauge configuration and uniformly in the volume, can be relaxed to the
requirement that the entries ${\mathcal{M}}_{\mathbf{A}\mathbf{B}}$ be
absolutely integrable, a condition conveniently expressed as

$\left\langle{\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})^{\frac{1}{2}}\right\rangle_{\textrm{G}}=\left\langle\left({\rm
tr}_{FCD}\\{{\mathcal{M}}(\mathbf{x},\mathbf{y}){\mathcal{M}}(\mathbf{x},\mathbf{y})^{\dagger}\\}\right)^{\frac{1}{2}}\right\rangle_{\textrm{G}}<\infty\,.$
(78)

Since $|X_{1}|\leq\sqrt{\sum_{i}|X_{i}|^{2}}\leq\sum_{i}|X_{i}|$, this
condition is equivalent to
$\left\langle\left|{\mathcal{M}}_{\mathbf{A}\mathbf{B}}(\mathbf{x},\mathbf{y})\right|\right\rangle_{\textrm{G}}<\infty$
$\forall\mathbf{A},\mathbf{B}$. It is further assumed that Eq. (78) remains
true in the thermodynamic limit.

Under the assumption of a positive integration measure, and under assumption
(1.) in Section II on the norm of the propagator, this suffices to prove
convergence of $\langle{\mathcal{O}}_{\mathcal{M}}\rangle$. Using this
assumption in the first inequality in Eq. (40) one finds

$\left|\left\langle{\mathcal{O}}_{\mathcal{M}}\right\rangle\right|\leq
n!\,\left(N_{f}N_{c}N_{D}\right)^{\frac{n}{2}}\left\langle{\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})^{\frac{1}{2}}\|\mathrm{S}_{M}\|^{n}\right\rangle_{\mathrm{G}}\leq\frac{n!\,\left(N_{f}N_{c}N_{D}\right)^{\frac{n}{2}}}{m_{0}^{n}}\left\langle{\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})^{\frac{1}{2}}\right\rangle_{\mathrm{G}}\,,$
(79)

which under the absolute-integrability condition discussed above is a finite
bound that remains so as ${\mathcal{V}}\to\infty$.

Using also assumption (2.) in Section II one can prove the impossibility of
vector flavor symmetry breaking. Starting from Eq. (27), and using Eqs. (33)
and (35) and the assumptions on $\|\mathrm{S}_{M}\|$ and $\|\Delta D\|$, one
finds

$\displaystyle\left|\frac{\partial}{\partial\theta_{a}}\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}\right\rangle\Big{|}_{\bm{\theta}=0}\right|$
$\displaystyle\leq\delta
m\,\bar{{C}}_{n}\left\langle{\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})^{\frac{1}{2}}\|\mathrm{S}_{M}\|^{n+1}\|\Delta
D\|\right\rangle_{\mathrm{G}}\leq\delta m\frac{\bar{{C}}_{n}\Delta
D_{\mathrm{max}}}{m_{0}^{n+1}}\left\langle{\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})^{\frac{1}{2}}\right\rangle_{\mathrm{G}}\,,$
(80)

where $\bar{{C}}_{n}\equiv
n\,n!\left(N_{f}N_{c}N_{D}\right)^{\frac{n}{2}}\left(\frac{1}{2N_{f}}\right)^{\frac{1}{2}}$.
Since the last factor is finite with a finite thermodynamic limit thanks to
the assumption of absolute integrability, the desired result, Eq. (38),
follows.

The case of finite transformations is obtained by a straightforward extension
of the argument of Appendix A.1: since the right-hand side of Eq. (80) is
unchanged when replacing ${\mathcal{M}}\to{\mathcal{M}}^{\alpha\bm{\theta}}$,
one has

$\displaystyle\left|\left\langle{\mathcal{O}}_{\mathcal{M}}^{\bm{\theta}}-{\mathcal{O}}_{\mathcal{M}}\right\rangle\right|$
$\displaystyle\leq\sum_{a}|\theta_{a}|\int_{0}^{1}d\alpha\,\left|\left\langle{\mathcal{C}}^{a}\,{\mathcal{O}}_{{\mathcal{M}}^{\alpha\bm{\theta}}}\right\rangle\right|\leq\delta
m\frac{\bar{{C}}_{n}\Delta
D_{\mathrm{max}}}{m_{0}^{n+1}}\left\langle{\mathcal{K}}_{\mathcal{M}}(\mathbf{x},\mathbf{y})^{\frac{1}{2}}\right\rangle_{\mathrm{G}}\sum_{a}|\theta_{a}|\,,$
(81)

from which Eq. (70) follows under the assumption of absolute integrability.

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  * Note [1] I refer here to the exact flavor symmetry of several staggered fields with the same mass, that holds at any lattice spacing, not to the approximate taste symmetry that becomes exact only in the continuum limit.
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  * Note [3] On a finite lattice observables are necessarily polynomial in the fermion fields and dependent on finitely many link variables: the restriction is relevant in the thermodynamic limit. Nonzero fermion number immediately entails a vanishing expectation value. Any non-gauge-invariant observable can be replaced with its average over gauge orbits without changing its expectation value.
  * Note [4] Proof: In matrix notation, Eq. (73) reads $u^{(R)T}=u^{(R)T}D^{(R)}(V)$, $\forall V\in\mathrm{SU}(N_{f})$, implying also $u^{(R)*}=D^{(R)}(V)^{\dagger}u^{(R)*}$. The projector $P=u^{(R)*}u^{(R)T}$ is then left invariant by an irreducible unitary representation of $\mathrm{SU}(N_{f})$, $P=D^{(R)}(V)^{\dagger}PD^{(R)}(V)=D^{(R)}(V)^{-1}PD^{(R)}(V)$. By Schur’s lemma $P$ is then proportional to the $d_{R}$-dimensional identity matrix, but since it has only rank 1 it must vanish if $d_{R}\not=1$.