aainstitutetext: ELTE Eötvös Loránd University, Institute for Theoretical
Physics,
Pázmány Péter sétány 1/A, H-1117, Budapest, Hungary

# Localisation of Dirac modes in gauge theories and Goldstone’s theorem at
finite temperature

Matteo Giordano<EMAIL_ADDRESS>
###### Abstract

I discuss the possible effects of a finite density of localised near-zero
Dirac modes in the chiral limit of gauge theories with $N_{f}$ degenerate
fermions. I focus in particular on the fate of the massless quasi-particle
excitations predicted by the finite-temperature version of Goldstone’s
theorem, for which I provide an alternative and generalised proof based on a
Euclidean ${\rm SU}(N_{f})_{A}$ Ward-Takahashi identity. I show that localised
near-zero modes can lead to a divergent pseudoscalar-pseudoscalar correlator
that modifies this identity in the chiral limit. As a consequence, massless
quasi-particle excitations can disappear from the spectrum of the theory in
spite of a non-zero chiral condensate. Three different scenarios are possible,
depending on the detailed behaviour in the chiral limit of the ratio of the
mobility edge and the fermion mass, which I prove to be a renormalisation-
group invariant quantity.

###### Keywords:

Nonperturbative Effects, Spontaneous Symmetry Breaking, Random Systems

##  1 Introduction

The low-energy physics of QCD at zero temperature is largely determined by two
non-perturbative phenomena, namely spontaneous chiral symmetry breaking and
confinement. The spontaneous breaking of chiral symmetry in the limit of
massless quarks and the associated appearance of massless Goldstone bosons
explain the lightness of pions for physical quark masses and their low-energy
dynamics. The origin of confinement can be seen instead in the opposite
infinite-mass, “quenched” (pure gauge) limit, where a linearly rising quark-
antiquark potential forces quarks to be confined inside hadrons.

At finite temperature, confinement in SU(3) pure gauge theory is signalled by
a divergent quark free energy, inferred from the vanishing of the Polyakov
loop expectation value, which is in turn a consequence of the centre symmetry
of the theory being unbroken. For finite quark masses this symmetry is
explicitly broken but only mildly, and confinement persists at low
temperatures, signalled by a disordered Polyakov loop in typical gauge
configurations, resulting in a small expectation value of the Polyakov loop.
At low temperatures also the effects of the spontaneous breaking of chiral
symmetry in the massless limit are still clearly present.

At higher temperatures QCD undergoes a rapid but analytic crossover to a phase
where chiral symmetry is approximately restored, and centre symmetry is
spontaneously broken (on top of the explicit breaking mentioned above) with
the Polyakov loop getting ordered, and quarks and gluons are liberated in the
quark-gluon plasma Borsanyi:2010bp ; Bazavov:2016uvm . Despite the very
different origin of chiral symmetry breaking and confinement, both chiral and
confining properties change dramatically at the transition. A similar
situation is found quite generally in gauge theories at finite temperature
Boyd:1996bx ; Damgaard:1998yv ; Karsch:1998qj ; Engels:2005te ; Karsch:2001nf
; deForcrand:2003vyj ; deForcrand:2008vr ; Bergner:2019dim , with
deconfinement always leading to a “more chirally symmetric” system, but the
reasons for this behaviour are still not fully understood.

The QCD crossover is characterised also by another drastic change, that of the
localisation properties of the low-lying modes of the Euclidean Dirac
operator. In fact, while delocalised in the low-temperature phase, these modes
become spatially localised on the scale of the inverse temperature in the
high-temperature phase Gockeler:2001hr ; Gattringer:2001ia ;
GarciaGarcia:2005vj ; GarciaGarcia:2006gr ; Gavai:2008xe ; Kovacs:2009zj ;
Bruckmann:2011cc ; Kovacs:2012zq ; Cossu:2016scb ; Holicki:2018sms . More
precisely, above the transition temperature the low modes are localised up to
a critical point, $\lambda_{c}$, in the spectrum, known as “mobility edge”,
beyond which they are again delocalised.111Localised modes are most likely
found also in the deep ultraviolet region of the spectrum (see footnote 22),
which, however, has no physical relevance. For a recent comprehensive review
of localisation in gauge theories, the interested reader can consult Ref.
Giordano:2021qav .

The relation between localisation and deconfinement has been extensively
studied in QCD and QCD-like gauge theories in recent years. These studies have
shown that these two phenomena are indeed intimately connected, with localised
modes appearing exactly at the critical point when the phase transition is a
genuine (i.e., non-analytic) transition Kovacs:2017uiz ; Vig:2020pgq ;
Giordano:2019pvc ; Giordano:2016nuu ; Giordano:2016vhx ; Kovacs:2010wx ;
Bruckmann:2017ywh ; Bonati:2020lal ; Baranka:2021san ; Cardinali:2021fpu . A
qualitative understanding of this connection is provided by the “sea/islands”
picture of localisation Bruckmann:2011cc ; Giordano:2015vla ; Giordano:2016cjs
; Giordano:2016vhx ; Giordano:2021qav , according to which the “islands” of
fluctuations in the “sea” of ordered Polyakov loops in the deconfined phase
provide “energetically” favourable regions where the low Dirac eigenmodes can
localise. This is possible since the ordering of the Polyakov loop around 1
opens a pseudogap in the spectrum, which is then populated by a low but finite
density of localised modes. Numerical evidence for the sea/islands picture has
been provided in Refs. Bruckmann:2011cc ; Cossu:2016scb ; Holicki:2018sms ;
Baranka:2021san . Remarkably, the existence of an ordered phase is all that is
required for this mechanism to be at work, so that one expects localisation of
low Dirac modes to appear in the deconfined phase of a generic gauge theory.
This has been so far confirmed in a variety of different gauge theories
Kovacs:2017uiz ; Vig:2020pgq ; Giordano:2019pvc ; Giordano:2016nuu ;
Giordano:2016vhx ; Kovacs:2010wx ; Bruckmann:2017ywh ; Bonati:2020lal ;
Baranka:2021san ; Cardinali:2021fpu , supporting the existence of a very close
relation between deconfinement and the localisation properties of low Dirac
modes.

The connection with chiral symmetry breaking has received instead less
attention in recent years, despite being the original motivation for the study
of localisation in gauge theories at finite temperature GarciaGarcia:2005vj ;
GarciaGarcia:2006gr . As is well known, chiral symmetry breaking is related to
the accumulation of near-zero Dirac modes Banks:1979yr . In the “disordered
medium scenario” Diakonov:1995ea  this is explained in terms of the mixing of
topological zero modes of the Dirac operator caused by the overlapping of
instantons and anti-instantons. While the “unperturbed” zero modes are
localised at finite temperature, they become delocalised after mixing, with
their eigenvalues broadening into a finite band around zero. It has been
observed, however, that in the high-temperature phase the low modes of
topological origin are not sufficient to quantitatively explain the amount of
localised modes Bruckmann:2011cc ; Vig:2021oyt . The disordered medium
scenario cannot therefore fully account for localised low modes, as is made
clear by the fact that these are found also in theories without instantons
Giordano:2019pvc ; Baranka:2021san . Nonetheless, topological modes are very
likely to play an important role, complementing the sea/islands picture
discussed above (see Ref. Vig:2021oyt ).

It is clear, in any case, that the low-lying eigenvalues of the Dirac operator
and the corresponding eigenvectors are sensitive both to confinement and
chiral symmetry breaking. The study of localisation could then lead not only
to a better understanding of these two phenomena individually, but also to
clarify their connection, with localisation possibly providing the mechanism
through which deconfinement improves the chiral symmetry properties of gauge
systems with fermions.

The appearance of localised modes in the spectrum of the Dirac operator
becomes somewhat less surprising if one notices its formal analogy with the
Hamiltonian of a disordered system. Such systems are known to display
eigenmode localisation since the seminal work of Anderson Anderson:1958vr ,
and have been intensely studied in the condensed-matter community for more
than sixty years (see Refs. thouless1974electrons ; lee1985disordered ;
kramer1993localization ; Evers:2008zz ; anderson50 ; manybody  for a review).
In fact, $-i{\not{D}}$ can be interpreted exactly as the Hamiltonian of a
quantum system in the background of random gauge fields, fluctuating according
to the distribution determined by the path-integral integration measure. With
this insight, the critical features of localisation observed at the mobility
edge Giordano:2013taa ; Nishigaki:2013uya ; Giordano:2014qna ;
Ujfalusi:2015nha  are understood as a consequence of universality, and of
suitable symmetry considerations. On the other hand, the appearance of
localised modes at the band centre (i.e., near zero) only in the high-
temperature phase is surprising, calling for an explanation that is
qualitatively provided by the sea/islands picture mentioned above.

The formal analogy with disordered systems, however, does not translate
directly into an analogy between physical phenomena observed on its gauge
theory side and on its condensed matter side. The reason for this is that,
while energy levels and eigenvectors of a Hamiltonian have a direct physical
meaning in condensed matter systems, it is not so for the Dirac eigenmodes in
gauge theories, where observables are obtained only after summing over all the
modes. The properties of individual points, and perhaps even of regions of the
Dirac spectrum, are then difficult to connect to phenomenology. A notable
exception to this state of affairs is the chiral limit of massless quarks,
where only near-zero modes have physical relevance. This is famously
exemplified by the Banks-Casher relation Banks:1979yr , stating that the
chiral condensate is proportional to the density of near-zero Dirac modes in
the chiral limit. It is then possible that in this limit the localisation
properties of near-zero modes have clear observable consequences.

The investigation of this issue is made more pressing by the known
consequences of localisation for the Goldstone excitations associated with the
spontaneous breaking of a continuous symmetry Goldstone:1962es  also at finite
temperature Lange:1965zz ; Kastler:1966wdu ; Swieca:1966wna ; Morchio:1987wd ;
Strocchi:2008gsa . It has been known for a long time that localisation can
lead to the disappearance of Goldstone modes in non-relativistic disordered
systems McKane:1980fs . This phenomenon has been rediscovered more recently in
the context of relativistic lattice gauge theories at zero temperature
Golterman:2003qe , although for an unphysical system, namely quenched SU(3)
gauge theory with Wilson fermions, where it explains the disappearance of
Goldstone bosons in the supercritical region outside the Aoki phase
Aoki:1983qi . Localisation and the fate of quasi-particle Goldstone
excitations in finite-temperature gauge theories have been studied only
recently in Refs. giordano_GT_lett ; Giordano:2021nat , reaching similar
conclusions: the presence of a finite density of localised near-zero modes
possibly leads to the disappearance of Goldstone excitations from the spectrum
of the theory.

In the absence of concrete models where localised near-zero modes are
explicitly shown to be sufficiently dense, the results of Refs.
giordano_GT_lett ; Giordano:2021nat  would probably be only of academic
interest, closing a loophole in the proofs of Goldstone’s theorem at zero and
finite temperature. There are, however, interesting results that could make
them more than simply a curiosity. The most intriguing one is the peak of
localised near-zero modes in QCD right above the pseudocritical temperature,
observed on the lattice studying overlap spectra in the background of HISQ
configurations for near-physical quark masses Dick:2015twa . While
localisation properties have not been studied further, this peak has been
observed to persist also at lower-than-physical light-quark masses
Ding:2020xlj ; Kaczmarek:2021ser ; Ding:2021gdy ; Ding:2021jtn . This peak is
usually ascribed to the topological modes mentioned above, and it is argued
HotQCD:2012vvd ; Dick:2015twa ; Ding:2020xlj ; Kaczmarek:2021ser  that it
should shrink in the chiral limit as topological excitations become a non-
interacting gas, so that its effects disappear except for what concerns the
anomalous ${\rm U}(1)_{A}$ symmetry. However, it is not clear what mechanism
should lead to instantons and anti-instantons forming a free gas in the chiral
limit, and the evidence for the suggested behaviour of the peak is not
conclusive. While there is no conclusive evidence for the peak surviving as a
finite peak either, the possibility should not be excluded at this stage that
a finite density of localised near-zero modes is found in the chiral limit.

Another interesting finding is the presence of two separate phase transitions
in SU(3) gauge theory with $N_{f}=2$ flavours of massless adjoint quarks
Karsch:1998qj ; Engels:2005te . In this theory a deconfining transition at
temperature $T_{\rm dec}$ separates a low-temperature, confining phase where
chiral symmetry is broken by a non-zero chiral condensate, and an
intermediate, deconfined phase where chiral symmetry is still broken but the
chiral condensate is reduced. A second transition at a higher temperature
$T_{\chi}>T_{\rm dec}$ separates this phase from the high-temperature,
deconfined and chirally-restored phase. In the intermediate phase one surely
finds a finite density of near-zero modes via the Banks-Casher relation; since
this phase is deconfined, the sea/islands picture suggests that these modes
will be localised, but no more or less direct evidence of this exists.
Moreover, the results of Refs. Karsch:1998qj ; Engels:2005te  are consistent
with the presence of massless Goldstone excitations. It is possible that, as a
consequence of the relatively small volumes employed in those studies,
localisation and its consequences could not manifest yet, and so larger
volumes would be needed to make conclusive statements. However, if confirmed
in larger volumes, their results do not contradict the analysis of Refs.
giordano_GT_lett ; Giordano:2021nat , and a detailed study of correlation
functions would be needed to test quantitatively the scenarios proposed there.

Although the basic approach is very similar to that of Refs. McKane:1980fs ;
Golterman:2003qe , the argument of Refs. giordano_GT_lett ; Giordano:2021nat
requires to deal with a certain number of technical complications, mostly
originating in the loss of full O(4) invariance at finite temperature. While
these complications can be overcome, they were discussed only briefly in Refs.
McKane:1980fs ; Golterman:2003qe , not to obscure the main points. In this
paper I present the argument in full depth, providing details on the various
aspects of the calculation. Before being able to discuss the fate of Goldstone
quasi-particles, a few intermediate results need to be derived, that I believe
are of interest in their own right. These include:

  * •

a “Euclidean” derivation of Goldstone’s theorem at finite temperature, and a
generalisation thereof, in the case of broken non-singlet axial flavour
symmetry, based on the corresponding Ward-Takahashi identities;

  * •

the possible appearance in the chiral limit of a $1/m$ divergence in the
pseudoscalar-pseudoscalar correlator, where $m$ is the quark mass, in the
presence of a finite density of localised near-zero modes;

  * •

the renormalisation of the spectral correlators appearing in the mode
decomposition of the pseudoscalar-pseudoscalar correlator, and the related
proof that the ratio $\lambda_{c}/m$ between the mobility edge and the quark
mass is a renormalisation-group invariant quantity.

After a brief review of finite-temperature quantum field theory in Section 2,
and of gauge theories in Section 3, the alternative proof of Goldstone’s
theorem at finite temperature is discussed in Section 4. The study of the
pseudoscalar-pseudoscalar correlator is reported in Section 5, including the
renormalisation of the spectral correlators and of the mobility edge, and the
fate of Goldstone excitations in the presence of localised modes. Conclusions
and discussion of future studies are found in Section 6. To improve
readability, most of the technical details are relegated to Appendices A to H,
which include: details on analyticity and reality properties of Euclidean
correlation functions (Appendix A); derivation of non-singlet axial Ward-
Takahashi identities (Appendix B) and their renormalisation (Appendix C);
another “Euclidean” proof of Goldstone’s theorem at finite temperature in
coordinate space (Appendix D); details about the pseudoscalar-pseudoscalar
correlator calculation, in particular estimates of the contribution of
exponentially localised modes (Appendix E), the study of the chiral limit
(Appendix F) and the renormalisation of the corresponding spectral correlators
(Appendix G), and the discussion of a bound on their large-distance behaviour
(Appendix H).

##  2 Finite-temperature quantum field theory

In this Section I review a few relevant aspects of quantum field theory at
finite temperature, mostly to set the notation. See, e.g., Refs.
Kapusta:2006pm ; Laine:2016hma ; fulling1987temperature ; Bros:1996mw ;
Cuniberti:2001hm ; Meyer:2010ii ; Meyer:2011gj ; Strocchi:2008gsa  for further
details. The expectation value of an observable $O$ for a system in thermal
equilibrium at temperature $T=1/\beta$ is obtained as follows from the density
matrix of the canonical ensemble,

$\langle\\!\langle\hat{O}\rangle\\!\rangle_{\beta}=\lim_{V\to\infty}\langle\\!\langle\hat{O}\rangle\\!\rangle_{\beta,V}\equiv\lim_{V\to\infty}\frac{{\rm
Tr}\,e^{-\beta\hat{H}_{V}}\hat{O}}{{\rm Tr}\,e^{-\beta\hat{H}_{V}}}\,,$ (1)

where $\hat{H}_{V}$ is the finite-volume Hamiltonian, and the volume $V$ of
the system is eventually sent to infinity in the thermodynamic limit. Here a
caret denotes operators acting on the (zero-temperature) Hilbert space of the
system, and ${\rm Tr}$ the trace over this space.

For (possibly composite) local bosonic field operators $\hat{\phi}_{1,2}(x)$,
$x=(t,\vec{x})$, one defines the thermal (real-time) two-point correlation
functions

$\displaystyle G^{(+)}_{\phi_{1}\phi_{2}}(t,\vec{x})$
$\displaystyle\equiv\langle\\!\langle\hat{\phi}_{1}(t,\vec{x})\hat{\phi}_{2}(0)\rangle\\!\rangle_{\beta}\equiv\lim_{V\to\infty}\langle\\!\langle
e^{it\hat{H}_{V}}\hat{\phi}_{1}(0,\vec{x})e^{-it\hat{H}_{V}}\hat{\phi}_{2}(0)\rangle\\!\rangle_{\beta,V}\,,$
(2) $\displaystyle G^{(-)}_{\phi_{1}\phi_{2}}(t,\vec{x})$
$\displaystyle\equiv\langle\\!\langle\hat{\phi}_{2}(0)\hat{\phi}_{1}(t,\vec{x})\rangle\\!\rangle_{\beta}\equiv\lim_{V\to\infty}\langle\\!\langle\hat{\phi}_{2}(0)e^{it\hat{H}_{V}}\hat{\phi}_{1}(0,\vec{x})e^{-it\hat{H}_{V}}\rangle\\!\rangle_{\beta,V}\,.$

Throughout this paper, an arrow denotes collectively the three spatial
components of a four-vector. Here and everywhere else in this paper, the
temporal evolution of $\hat{\phi}_{1,2}(t,\vec{x})$ in the infinite-volume
limit (also for complex time argument) is understood in the sense of the weak
limit of $H_{V}$ implied by Eq. (2). Under suitable convergence conditions on
the infinite-volume limit of the temporal evolution, the correlation functions
Eq. (2) satisfy Haag:1967sg  the KMS condition Kubo:1957mj ; Martin:1959jp .
Together with the relativistic locality condition,
$[\hat{\phi}_{i}(x),\hat{\phi}_{j}(y)]=0$ for spacelike Minkowski separation
$(x-y)^{2}<0$, this implies that $G^{(\pm)}_{\phi_{1}\phi_{2}}(t,\vec{x})$ are
the boundary values of an analytic function $G_{\phi_{1}\phi_{2}}(z,\vec{x})$,
analytic in the cut complex plane ${\cal
D}\equiv\mathbb{C}\backslash\\{z~{}|~{}|{\rm Re}\,z|\geq|\vec{x}|,\,{\rm
Im}\,z=n\beta,\,n\in\mathbb{Z}\\}$, and furthermore periodic in the imaginary
$z$ direction with period $\beta$,
$G_{\phi_{1}\phi_{2}}(z+in\beta,\vec{x})=G_{\phi_{1}\phi_{2}}(z,\vec{x})$ (see
Refs. fulling1987temperature ; Bros:1996mw ; Cuniberti:2001hm ;
Strocchi:2008gsa ). The correlation functions
$G^{(\pm)}_{\phi_{1}\phi_{2}}(t,\vec{x})$ for $t\in\mathbb{R}$ are then
recovered as

$\displaystyle G^{(+)}_{\phi_{1}\phi_{2}}(t,\vec{x})$
$\displaystyle=G_{\phi_{1}\phi_{2}}(t-i\epsilon,\vec{x})\,,$ (3)
$\displaystyle G^{(-)}_{\phi_{1}\phi_{2}}(t,\vec{x})$
$\displaystyle=G_{\phi_{1}\phi_{2}}(t+i\epsilon,\vec{x})=G_{\phi_{1}\phi_{2}}(t-i(\beta-\epsilon),\vec{x})\,,$

where the limit $\epsilon\to 0^{+}$ at the end of the calculation is
understood.

In particular, $G^{(\pm)}_{\phi_{1}\phi_{2}}$ can be obtained by analytic
continuation from the Euclidean correlation function
${\mathcal{G}}_{\phi_{1}\phi_{2}}(t,\vec{x})$,

$\displaystyle{\mathcal{G}}_{\phi_{1}\phi_{2}}(t,\vec{x})$
$\displaystyle\equiv\langle\\!\langle\theta(t)\hat{\phi}_{1}(-it,\vec{x})\hat{\phi}_{2}(0)+\theta(-t)\hat{\phi}_{2}(0)\hat{\phi}_{1}(-it,\vec{x})\rangle\\!\rangle_{\beta}$
(4) $\displaystyle\equiv\langle\\!\langle
T\\{\hat{\phi}_{E1}(t,\vec{x})\hat{\phi}_{E2}(0)\\}\rangle\\!\rangle_{\beta}\,,$

where $\hat{\phi}_{E1,2}(t,\vec{x})\equiv\hat{\phi}_{1,2}(-it,\vec{x})$, and
$T$ stands for time-ordering in Euclidean (imaginary) time $t$. This function
is the restriction to real $z$ of
${\mathcal{G}}_{\phi_{1}\phi_{2}}(z,\vec{x})=G_{\phi_{1}\phi_{2}}(-iz,\vec{x})$,
analytic in the cut complex plane $i{\cal D}$, and periodic in the real
direction,
${\mathcal{G}}_{\phi_{1}\phi_{2}}(z+n\beta,\vec{x})={\mathcal{G}}_{\phi_{1}\phi_{2}}(z,\vec{x})$.
One then has

$\displaystyle G^{(+)}_{\phi_{1}\phi_{2}}(t,\vec{x})$
$\displaystyle={\mathcal{G}}_{\phi_{1}\phi_{2}}(\epsilon+it,\vec{x})\,,$
$\displaystyle G^{(-)}_{\phi_{1}\phi_{2}}(t,\vec{x})$
$\displaystyle={\mathcal{G}}_{\phi_{1}\phi_{2}}(-\epsilon+it,\vec{x})\,.$ (5)

In particular, the thermal expectation value of the commutator
$[\hat{\phi}_{1}(x),\hat{\phi}_{2}(0)]$ equals the discontinuity of
${\mathcal{G}}_{\phi_{1}\phi_{2}}$ along the imaginary axis (corresponding to
real time),

$\displaystyle\langle\\!\langle[\hat{\phi}_{1}(t,\vec{x}),\hat{\phi}_{2}(0)]\rangle\\!\rangle_{\beta}$
$\displaystyle=G^{(+)}_{\phi_{1}\phi_{2}}(t,\vec{x})-G^{(-)}_{\phi_{1}\phi_{2}}(t,\vec{x})={\mathcal{G}}_{\phi_{1}\phi_{2}}(\epsilon+it,\vec{x})-{\mathcal{G}}_{\phi_{1}\phi_{2}}(-\epsilon+it,\vec{x})$
(6)
$\displaystyle={\mathcal{G}}_{\phi_{1}\phi_{2}}(\epsilon+it,\vec{x})-{\mathcal{G}}_{\phi_{1}\phi_{2}}(\beta-\epsilon+it,\vec{x})\,,$

where $t\in\mathbb{R}$ and the second line follows from periodicity. As a
further consequence of periodicity in Euclidean time,
${\mathcal{G}}_{\phi_{1}\phi_{2}}(t,\vec{x})$ with $t\in\mathbb{R}$ can be
written as a mixed Fourier sum/Fourier transform as follows,

${\mathcal{G}}_{\phi_{1}\phi_{2}}(t,\vec{x})=\frac{1}{\beta}\sum_{n}\int\frac{d^{3}k}{(2\pi)^{3}}\,e^{-i(\omega_{n}t+\vec{k}\cdot\vec{x})}\tilde{{\mathcal{G}}}_{\phi_{1}\phi_{2}}(\omega_{n},\vec{k})\,,$
(7)

where $\omega_{n}$ are the bosonic Matsubara frequencies,
$\omega_{n}=\frac{2\pi n}{\beta}$, $n\in\mathbb{Z}$, and

$\tilde{{\mathcal{G}}}_{\phi_{1}\phi_{2}}(\omega_{n},\vec{k})\equiv\int_{0}^{\beta}dt\int
d^{3}x\,e^{i(\omega_{n}t+\vec{k}\cdot\vec{x})}{\mathcal{G}}_{\phi_{1}\phi_{2}}(t,\vec{x})\equiv\int_{\beta}d^{4}x\,e^{i(\omega_{n}t+\vec{k}\cdot\vec{x})}{\mathcal{G}}_{\phi_{1}\phi_{2}}(t,\vec{x})\,,$
(8)

with the subscript $\beta$ denoting compactification of the temporal
direction.

A central role in this paper is played by the spectral function,

$\displaystyle\tilde{\rho}_{\phi_{1}\phi_{2}}(\omega,\vec{k})$
$\displaystyle\equiv\int d^{4}x\,e^{i(\omega
t-\vec{k}\cdot\vec{x})}\langle\\!\langle[\hat{\phi}_{1}(t,\vec{x}),\hat{\phi}_{2}(0)]\rangle\\!\rangle_{\beta}$
(9) $\displaystyle=\int d^{4}x\,e^{i(\omega
t-\vec{k}\cdot\vec{x})}\left({\mathcal{G}}_{\phi_{1}\phi_{2}}(\epsilon+it,\vec{x})-{\mathcal{G}}_{\phi_{1}\phi_{2}}(-\epsilon+it,\vec{x})\right)\,,$

and by the closely related retarded and advanced propagators,

$\displaystyle\tilde{r}_{\phi_{1}\phi_{2}}(\omega,\vec{k})$
$\displaystyle\equiv i\int d^{4}x\,e^{i(\omega
t-\vec{k}\cdot\vec{x})}\theta(t)\langle\\!\langle[\hat{\phi}_{1}(t,\vec{x}),\hat{\phi}_{2}(0)]\rangle\\!\rangle_{\beta}\,,$
(10) $\displaystyle\tilde{a}_{\phi_{1}\phi_{2}}(\omega,\vec{k})$
$\displaystyle\equiv-i\int d^{4}x\,e^{i(\omega
t-\vec{k}\cdot\vec{x})}\theta(-t)\langle\\!\langle[\hat{\phi}_{1}(t,\vec{x}),\hat{\phi}_{2}(0)]\rangle\\!\rangle_{\beta}\,.$

These are the boundary values of analytic functions, analytic respectively for
${\rm Im}\,\omega>0$ and ${\rm Im}\,\omega<0$, from which one obtains the
spectral function as follows,

$\displaystyle\tilde{\rho}_{\phi_{1}\phi_{2}}(\omega,\vec{k})=-i\left(\tilde{r}_{\phi_{1}\phi_{2}}(\omega+i\epsilon,\vec{k})-\tilde{a}_{\phi_{1}\phi_{2}}(\omega-i\epsilon,\vec{k})\right)\,.$
(11)

More directly, one can use the analytic continuation relations between the
retarded and advanced propagators and the Fourier coefficients of the
Euclidean correlator,
$\tilde{{\mathcal{G}}}_{\phi_{1}\phi_{2}}(\omega_{n},\vec{k})$, to reconstruct
$\tilde{r}_{\phi_{1}\phi_{2}}$ and $\tilde{a}_{\phi_{1}\phi_{2}}$, and so
$\tilde{\rho}_{\phi_{1}\phi_{2}}$, by analytic interpolation in the sense of
Carlson’s theorem, under reasonable hypotheses of moderate asymptotic growth
in the complex time variable Bros:1996mw ; Cuniberti:2001hm . For $n\neq 0$
one has Bros:1996mw ; Cuniberti:2001hm ; Meyer:2010ii ; Meyer:2011gj

$\tilde{{\mathcal{G}}}_{\phi_{1}\phi_{2}}(\omega_{n},\vec{k})=\left\\{\begin{aligned}
&\tilde{r}_{\phi_{1}\phi_{2}}(i\omega_{n},-\vec{k})\,,&&&n&>0\,,\\\
&\tilde{a}_{\phi_{1}\phi_{2}}(i\omega_{n},-\vec{k})\,,&&&n&<0\,.\end{aligned}\right.$
(12)

These results apply also to the case $n=0$ if the spectral function is regular
at the origin. If instead there is a transport peak, i.e., if

$\tilde{\rho}_{\phi_{1}\phi_{2}}(\omega,\vec{k})=2\pi
A_{\phi_{1}\phi_{2}}(\vec{k})\omega\delta(\omega)+B_{\phi_{1}\phi_{2}}(\omega,\vec{k})\,,$
(13)

with $B_{\phi_{1}\phi_{2}}$ regular at $\omega=0$, one has Meyer:2010ii ;
Meyer:2011gj

$\tilde{{\mathcal{G}}}_{\phi_{1}\phi_{2}}(0,\vec{k})-\tilde{r}_{\phi_{1}\phi_{2}}(i\epsilon,-\vec{k})=\tilde{{\mathcal{G}}}_{\phi_{1}\phi_{2}}(0,\vec{k})-\tilde{a}_{\phi_{1}\phi_{2}}(-i\epsilon,-\vec{k})=A_{\phi_{1}\phi_{2}}(-\vec{k})\,.$
(14)

Further details are given in Appendix A.

Time-ordered Euclidean field correlators at finite temperature can be
expressed in terms of a path integral (see, e.g., Refs. Kapusta:2006pm ;
Laine:2016hma ),

$\langle\\!\langle
T\\{\hat{\phi}_{E1}(x_{1})\ldots\hat{\phi}_{En}(x_{n})\\}\rangle\\!\rangle_{\beta}=\langle\phi_{1}(x_{1})\ldots\phi_{n}(x_{n})\rangle_{\beta}\equiv\frac{\int_{\beta}[{\mathscr{D}}\chi]\,e^{-S_{E}[\chi]}\phi_{1}[\chi(x_{1})]\ldots\phi_{n}[\chi(x_{n})]}{\int_{\beta}[{\mathscr{D}}\chi]\,e^{-S_{E}[\chi]}}\,,$
(15)

where $S_{E}$ is a suitable Euclidean action, and the path integration
$\int_{\beta}[{\mathscr{D}}\chi]$ is over sets of bosonic ($c$-number) and
fermionic (Grassmann) field variables that are respectively periodic or
antiperiodic in the time direction. Thermal correlation functions are then
reconstructed by means of analytic continuation in the time coordinate
according to Eq. (5).

In the axiomatic setting, the analyticity properties of the real-time
correlation functions follow from the general properties expected of quantum
fields, and so the properties of the imaginary-time correlators are a
consequence of the basic assumptions (see, e.g., Ref. Bros:1996mw ). If one
instead takes the Euclidean theory defined by Eq. (15) as the starting point,
these properties become necessary conditions that the theory must satisfy in
order to be able to ultimately obtain a local relativistic quantum field
theory. An example is discussed below in Section 4.2.

##  3 Gauge theories with Dirac fermions

In this Section I briefly describe gauge theories with Dirac fermions
quantised in the path-integral approach. The discussion is quite general, and
applies to four-dimensional gauge theories of compact Lie groups, minimally
coupled to $N_{f}$ degenerate “flavours” of Dirac fermions of mass $m$,
transforming in some representation of the group, at finite temperature and in
the imaginary-time formalism. The fermionic part of the Euclidean action is
$S_{\rm F}=\int_{\beta}d^{4}x\,{\mathscr{L}}_{\rm F}$, with
${\mathscr{L}}_{\rm F}$ the following Euclidean Lagrangean density,

${\mathscr{L}}_{\rm
F}=\bar{\psi}({\not{D}}+m)\psi\,,\qquad{\not{D}}=\gamma_{\mu}D_{\mu}\,,\qquad
D_{\mu}=\partial_{\mu}+igB_{\mu}\,.$ (16)

Here $\psi$ and $\bar{\psi}$ denote collectively two independent sets of
Grassmann variables $\psi_{f\eta c}(x)$ and $\bar{\psi}_{f\eta c}(x)$, with
flavour index $f=1,\ldots,N_{f}$, “Dirac” index $\eta=1,\ldots,4$, and
“colour” index $c=1,\ldots,N_{c}$ corresponding to the gauge group
representation. In Eq. (16) and in the following, summation over the
suppressed discrete indices, as well as over repeated explicit indices, is
understood, unless explicitly stated otherwise. The (generally non-Abelian)
Hermitean gauge fields $B_{\mu}(x)=B_{\mu}^{\dagger}(x)$ read
$B_{\mu}=B_{\mu}^{a}T^{a}$, with $B_{\mu}^{a}$ real fields and
$T^{a}=T^{a{\dagger}}$ a set of $N_{c}\times N_{c}$ matrices providing a
representation of the gauge group generators. Both $B_{\mu}$ and the Dirac
operator ${\not{D}}$ in the background of the gauge field act trivially on
flavour space. The Euclidean Hermitean gamma matrices
$\gamma_{\mu}=\gamma_{\mu}^{\dagger}$, $\mu=1,\ldots,4$, satisfy the
anticommutation relations $\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}$;
the fifth gamma matrix
$\gamma_{5}\equiv-\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}$ satisfies
$\gamma_{5}=\gamma_{5}^{\dagger}$, $\gamma_{5}^{2}=\mathbf{1}$, and
$\\{\gamma_{5},\gamma_{\mu}\\}=0$.

Expectation values are formally defined in terms of a path integral as

$\displaystyle\langle{\mathcal{O}}\rangle_{\beta}$ $\displaystyle\equiv
Z_{\beta}^{-1}\int_{\beta}[{\mathscr{D}}B]\,e^{-S_{\rm
G}[B]}\int_{\beta}[{\mathscr{D}}\psi][{\mathscr{D}}\bar{\psi}]\,e^{-S_{\rm
F}[\psi,\bar{\psi},B]}\,{\mathcal{O}}[\psi,\bar{\psi},B]\,,$ (17)
$\displaystyle Z_{\beta}$
$\displaystyle\equiv\int_{\beta}[{\mathscr{D}}B]\,e^{-S_{\rm
G}[B]}\int_{\beta}[{\mathscr{D}}\psi][{\mathscr{D}}\bar{\psi}]\,e^{-S_{\rm
F}[\psi,\bar{\psi},B]}\,.$

Following for definiteness the Faddeev-Popov-De Witt gauge-invariant approach
as in Ref. Bernard:1974bq , $S_{\rm G}$ is the gauge part of the action,
including the usual Yang-Mills action and a gauge-fixing term, while the
Faddeev-Popov determinant necessary to restore gauge invariance is included in
the integration measure $\int_{\beta}[{\mathscr{D}}B]$.222Strictly speaking,
the gauge-invariant continuum integration measure is not well defined beyond
perturbation theory due to the existence of Gribov copies Gribov:1977wm ;
Singer:1978dk . This issue is dealt with by first formulating the theory in a
gauge-invariant way on a lattice and taking eventually the continuum limit.
The symbol $\int_{\beta}[{\mathscr{D}}B]$ should then be understood as a
shorthand for this procedure. Here I prefer to avoid the technical
complications of the lattice formulation for the sake of clarity, as they pose
no obstacle to the development of the main argument (see Section 3.5 for
further comments). Here the subscript $\beta$ denotes the periodicity
condition $B_{\mu}(\beta,\vec{x})=B_{\mu}(0,\vec{x})$ to be imposed on the
gauge fields. Integration over the fermion fields is done in the sense of
Berezin integration of Grassmann variables, over field configurations
satisfying the antiperiodicity conditions
$\psi(\beta,\vec{x})=-\psi(0,\vec{x})$ and
$\bar{\psi}(\beta,\vec{x})=-\bar{\psi}(0,\vec{x})$, denoted again by the
subscript $\beta$. Since the fermionic action is quadratic, one formally has

$\displaystyle\int_{\beta}[{\mathscr{D}}\psi][{\mathscr{D}}\bar{\psi}]\,e^{-S_{\rm
F}[\psi,\bar{\psi},B]}={\rm Det}({\not{D}}[B]+m)\,,$ (18)

where ${\rm Det}$ denotes the functional determinant, and the dependence on
the gauge fields has been made explicit. In general, after integrating out the
fermion fields one is left with

$\displaystyle\langle{\mathcal{O}}\rangle_{\beta}$
$\displaystyle=Z_{\beta}^{-1}\int_{\beta}[{\mathscr{D}}B]\,e^{-S_{\rm
G}[B]}{\rm Det}({\not{D}}[B]+m){\mathcal{O}}_{\rm
G}[B]\equiv\langle{\mathcal{O}}_{\rm G}\rangle_{\beta}\,,$ (19)
$\displaystyle{\mathcal{O}}_{\rm G}[B]$ $\displaystyle\equiv\left({\rm
Det}({\not{D}}[B]+m)\right)^{-1}\int_{\beta}[{\mathscr{D}}\psi][{\mathscr{D}}\bar{\psi}]\,e^{-S_{\rm
F}[\psi,\bar{\psi},B]}\,{\mathcal{O}}[\psi,\bar{\psi},B]\,,$

with ${\mathcal{O}}_{G}$ built out of gauge fields and of fermionic
propagators in a fixed gauge-field background, $({\not{D}}[B]+m)^{-1}$.

###  3.1 Dirac eigenmodes and localisation

The eigenmodes of the Euclidean Dirac operator play an essential role in this
paper. The Dirac operator is anti-Hermitean, with purely imaginary
eigenvalues. At finite temperature and in a finite spatial box of volume $V$,
imposing periodic spatial boundary conditions on the gauge fields to preserve
translation invariance, these eigenvalues form a discrete set
$\\{i\lambda_{n}\\}$, $\lambda_{n}\in\mathbb{R}$, with corresponding
eigenvectors $\psi_{n}$, ${\not{D}}\psi_{n}=i\lambda_{n}\psi_{n}$, obeying
antiperiodic boundary conditions in the temporal direction and periodic
boundary conditions in the spatial directions. Since the Dirac operator is
trivial in flavour space, one treats the eigenmodes as carrying only Dirac and
colour indices on top of the spacetime coordinate, $\psi_{n}=\psi_{n\,\eta
c}(x)$. For future utility I introduce the following notation,

$(\psi_{n^{\prime}}(x),\Gamma\psi_{n}(x))\equiv\sum_{\eta^{\prime},c^{\prime},\eta,c}\psi_{n^{\prime}\,\eta^{\prime}c^{\prime}}(x)^{*}\Gamma_{\eta^{\prime}c^{\prime}\,\eta
c}\psi_{n\,\eta
c}(x)\,,\qquad\|\psi_{n}(x)\|^{2}\equiv(\psi_{n}(x),\psi_{n}(x))\,,$ (20)

i.e., the scalar product $(\cdot,\cdot)$ is restricted to Dirac and colour
space, while the coordinate $x$ is kept fixed. Throughout this paper I will
always assume that Dirac modes have been orthonormalised, i.e.,

$\int_{\beta}d^{4}x\,(\psi_{n^{\prime}}(x),\psi_{n}(x))=\delta_{n^{\prime}n}\,.$
(21)

Due to the chiral property $\\{\gamma_{5},{\not{D}}\\}=0$, nonzero eigenvalues
appear in pairs $\pm i\lambda_{n}$, with corresponding eigenvectors $\psi_{n}$
and $\gamma_{5}\psi_{n}$.333In case of degenerate eigenvalues one can always
choose suitable eigenvectors that satisfy this relation. Moreover, one can
choose the zero modes $\psi_{n_{0}}$, ${\not{D}}\psi_{n_{0}}=0$, to have
definite chirality, i.e., to obey
$\gamma_{5}\psi_{n_{0}}=\xi_{n_{0}}\psi_{n_{0}}$ with $\xi_{n_{0}}=\pm 1$. For
the fermionic determinant, Eq. (18), one formally has in terms of the Dirac
eigenvalues ${\rm
Det}({\not{D}}+m)=\prod_{n}(i\lambda_{n}+m)^{N_{f}}=m^{N_{0}N_{f}}\prod_{n,\lambda_{n}>0}(\lambda_{n}^{2}+m^{2})^{N_{f}}$,
with $N_{0}$ the number of exact zero modes.

Up to an unimportant factor of $i$, ${\not{D}}[B]$ for a fixed background
field can be seen as the Hamiltonian of a four-dimensional quantum-mechanical
system evolving in an additional, fictitious time. This system is effectively
three-dimensional due to the compactification of the temporal direction.
Moreover, for purely fermionic observables ${\mathcal{O}}$ the corresponding
${\mathcal{O}}_{\rm G}$ in Eq. (19) can be expressed in terms of the
eigenvalues and eigenvectors of this Hamiltonian, with the remaining
integration being in practice a (gauge-invariant) average over the background
gauge fields. This is formally identical to the ensemble average of a
disordered system with (Hermitean) Hamiltonian $-i{\not{D}}[B]$ with energy
levels $\lambda_{n}[B]$ and eigenvectors $\psi_{n}[B]$, with gauge field
configurations providing different realisations of disorder, distributed
according to the probability distribution determined by the path-integral
measure after fermions have been integrated out, Eq. (19).

It is well known that for disordered systems the eigenmodes can become
localised, as was first realised by Anderson in his seminal paper
Anderson:1958vr . Typically, localised and delocalised modes are found in
disjoint spectral regions, separated by so-called mobility edges where a
second-order phase transition takes place, known as Anderson transition. The
subject of Anderson localisation and Anderson transitions has been and still
is a very active area of research in condensed matter physics (see Refs.
thouless1974electrons ; lee1985disordered ; kramer1993localization ;
Evers:2008zz ; anderson50 ; manybody  for a review), and has recently become
of interest also in high-energy physics after the observation of localised
modes in the high-temperature phase of gauge theories on the lattice
Gockeler:2001hr ; Gattringer:2001ia ; GarciaGarcia:2005vj ;
GarciaGarcia:2006gr ; Gavai:2008xe ; Kovacs:2009zj ; Bruckmann:2011cc ;
Kovacs:2012zq ; Cossu:2016scb ; Holicki:2018sms ; Giordano:2021qav ;
Kovacs:2010wx ; Kovacs:2017uiz ; Giordano:2019pvc ; Giordano:2016nuu ;
Giordano:2016vhx ; Vig:2020pgq ; Bruckmann:2017ywh ; Bonati:2020lal ;
Baranka:2021san ; Cardinali:2021fpu ; Giordano:2015vla ; Giordano:2016cjs ;
Vig:2021oyt ; Giordano:2013taa ; Nishigaki:2013uya ; Giordano:2014qna ;
Ujfalusi:2015nha .

For the purposes of this paper, very little information is needed about
localisation, besides the generic characterisation of localised and
delocalised modes. Qualitatively, localised modes are supported essentially
only in a finite spatial region whose size remains basically unchanged as the
system size is increased. Delocalised modes, instead, extend over the whole
system and keep spreading out as the system size grows, although not
necessarily at the same rate. More precisely, the localisation properties of
modes in a given spectral region are determined quantitatively by the scaling
with the volume of the inverse participation ratio,

${\rm IPR}_{n}=\int_{\beta}d^{4}x\,\|\psi_{n}(x)\|^{4}\,,$ (22)

averaged over gauge configurations (i.e., realisations of disorder). Notice
that since $\gamma_{5}^{2}=\mathbf{1}$, one has that
$\|\psi_{n}(x)\|^{2}=\|\gamma_{5}\psi_{n}(x)\|^{2}$, and so $\psi_{n}$ and
$\gamma_{5}\psi_{n}$ have the same IPR. Working in a box of fixed temporal
size $\beta$ and varying spatial volume $V$, if modes in the given spectral
region are typically non-negligible only in a spatial region of size
$O(V^{\alpha})$ one finds

${\rm IPR}\sim V^{\alpha}\left(V^{-\alpha}\right)^{2}=V^{-\alpha}$ (23)

where the exponent $\alpha$ is the fractal dimension of the modes.444While
other definitions of fractal dimension can be adopted (e.g., the “infrared
dimension” of Refs. Alexandru:2021pap ; Alexandru:2021xoi ; Horvath:2021zjk ),
it is the one obtained from the IPR that turns out to be important for our
purposes (see Section 5.2). For localised modes $\alpha=0$, while delocalised
modes have $0<\alpha\leq 1$.555Since
$1=\left|\int_{\beta}d^{4}x\,\|\psi_{n}(x)\|^{2}\right|^{2}\leq\int_{\beta}d^{4}x\,\|\psi_{n}(x)\|^{4}\int_{\beta}d^{4}x\,1=\beta
V\cdot{\rm IPR}_{n}\,,$ one necessarily has $1-\alpha\geq 0$. In the condensed
matter literature the term “delocalised” is usually reserved to the case
$\alpha=1$, while modes with $0<\alpha<1$ are called “critical”. For our
purposes there is no need to distinguish these two cases (see Section 5.2).

###  3.2 Flavour symmetries and Ward-Takahashi identities

Besides local gauge symmetry, the fermionic action is manifestly invariant
under a set of spacetime symmetries, namely translations, spatial rotations,
and reflections through the hyperplanes orthogonal to the temporal and spatial
directions. In the context of the Faddeev-Popov-De Witt approach, these are
manifest also in the gauge action if one uses a covariant gauge, such as
Lorenz gauge. The fermionic action also enjoys flavour symmetries related to
transformations in the internal flavour space. In particular, the fermionic
Lagrangean is invariant under a group of ${\rm U}(N_{f})={\rm
U}(1)_{B}\times{\rm SU}(N_{f})_{V}$ transformations, where ${\rm U}(1)_{B}$
corresponds to a common change of phase of all the different flavours, and
${\rm SU}(N_{f})_{V}$ corresponds to special unitary rotations of the flavour
components. For massless fermions, $m=0$, the two chiral components of the
fermionic fields are decoupled, and the symmetry is further enhanced to the
chiral symmetry ${\rm U}(N_{f})_{L}\times{\rm U}(N_{f})_{R}={\rm
U}(1)_{B}\times{\rm U}(1)_{A}\times{\rm SU}(N_{f})_{L}\times{\rm
SU}(N_{f})_{R}$. The ${\rm SU}(N_{f})_{L}\times{\rm SU}(N_{f})_{R}$ factor
contains the subgroups of ${\rm SU}(N_{f})_{V}$ (vector) and ${\rm
SU}(N_{f})_{A}$ (axial) transformations, given respectively by

$\displaystyle\psi$ $\displaystyle\to e^{i\alpha_{a}t^{a}}\psi\,,$
$\displaystyle\bar{\psi}$ $\displaystyle\to\bar{\psi}e^{-i\alpha_{a}t^{a}}\,,$
$\displaystyle\alpha_{a}$ $\displaystyle\in\mathbb{R}\,,$ (24)
$\displaystyle\psi$ $\displaystyle\to e^{i\beta_{a}t^{a}\gamma_{5}}\psi\,,$
$\displaystyle\bar{\psi}$
$\displaystyle\to\bar{\psi}e^{i\beta_{a}t^{a}\gamma_{5}}\,,$
$\displaystyle\beta_{a}$ $\displaystyle\in\mathbb{R}\,,$ (25)

where $t^{a}$, $a=1,\ldots N_{f}^{2}-1$ are the Hermitean and traceless
generators of ${\rm SU}(N_{f})$, obeying $[t^{a},t^{b}]=if^{abc}t^{c}$ with
totally antisymmetric and real structure constants $f^{abc}$. Normalisation is
chosen so that $f^{abc}f^{abd}=N_{f}\delta^{cd}$ and ${\rm
tr}\,t^{a}t^{b}=\frac{1}{2}\delta^{ab}$. The ${\rm U}(1)_{B}$ and ${\rm
SU}(N_{f})_{V}$ symmetries are expected not to break spontaneously, also in
the massless limit Vafa:1983tf . Spontaneous breaking of the ${\rm
SU}(N_{f})_{A}$ symmetry is instead possible. The ${\rm U}(1)_{A}$ symmetry is
known to be anomalous Adler:1969gk ; Bell:1969ts  due to non-invariance of the
functional integration measure Fujikawa:1979ay ; Fujikawa:1980eg  and will not
be considered in this paper.

The symmetry under the transformations Eq. (25) implies an infinite set of
Ward-Takahashi identities Ward:1950xp ; Takahashi:1957xn . Their derivation is
rather standard, and is briefly reviewed for completeness in Appendix B for
the case at hand; here I only report the results, which are also not new (see,
e.g., Ref. Brandt:2014qqa  for $N_{f}=2$). Defining the infinitesimal,
$x$-dependent transformation

$\delta_{A}\psi(x)=i\epsilon_{a}(x)t^{a}\gamma_{5}\psi(x)\,,\qquad\delta_{A}\bar{\psi}(x)=i\epsilon_{a}(x)\bar{\psi}(x)t^{a}\gamma_{5}\,,$
(26)

one obtains for any observable ${\mathcal{O}}$ the identity

$\left\langle\left(-\partial_{\mu}A^{a}_{\mu}(x)+2mP^{a}(x)\right){\mathcal{O}}\right\rangle_{\beta}=\left\langle-i\frac{\delta_{A}{\mathcal{O}}}{\delta\epsilon_{a}(x)}\right\rangle_{\beta}\,,$
(27)

where $A^{a}_{\mu}$ are the flavour non-singlet axial-vector currents, and
$P^{a}$ are the flavour non-singlet pseudoscalar densities,

$A^{a}_{\mu}(x)\equiv\bar{\psi}(x)\gamma_{\mu}\gamma_{5}t^{a}\psi(x)\,,\qquad
P^{a}(x)\equiv\bar{\psi}(x)\gamma_{5}t^{a}\psi(x)\,.$ (28)

Of particular interest here is the case ${\mathcal{O}}=P^{b}(y)$. A
straightforward calculation leads to

$-\partial_{\mu}\langle A^{a}_{\mu}(x)P^{b}(0)\rangle_{\beta}+2m\langle
P^{a}(x)P^{b}(0)\rangle_{\beta}=\delta^{(4)}(x)\delta^{ab}\Sigma\,,$ (29)

where $\Sigma$ is the chiral condensate, defined by
$\langle\bar{\psi}_{f}\psi_{g}\rangle_{\beta}\equiv\delta_{fg}\Sigma$, which
follows from vector flavour symmetry. The four-dimensional Dirac delta in Eq.
(29) is understood to be periodic in time,

$\delta^{(4)}(x)=\delta_{P}(t)\delta^{(3)}(\vec{x})\,,\qquad\delta_{P}(t)=\sum_{n=-\infty}^{\infty}\delta(t-n\beta)\,.$
(30)

Exploiting vector flavour symmetry further, one finds

$\displaystyle\langle A^{a}_{\mu}(x)P^{b}(0)\rangle_{\beta}$
$\displaystyle\equiv\delta^{ab}{\mathcal{G}}_{AP\,\mu}(x)\,,$
$\displaystyle\langle P^{a}(x)P^{b}(0)\rangle_{\beta}$
$\displaystyle\equiv\delta^{ab}{\mathcal{G}}_{PP}(x)\,,$ (31)

and Eq. (29) can be recast as

$-\partial_{\mu}{\mathcal{G}}_{AP\,\mu}(x)+2m{\mathcal{G}}_{PP}(x)=\delta^{(4)}(x)\Sigma\,.$
(32)

The momentum-space version of this identity, obtained through a Fourier
transform [see Eq. (8)], reads

$i\omega_{n}\tilde{{\mathcal{G}}}_{AP\,4}(\omega_{n},\vec{k})+i\vec{k}\cdot\vec{\widetilde{\mkern
0.0mu{\mathcal{G}}}}_{AP}(\omega_{n},\vec{k})+2m\tilde{{\mathcal{G}}}_{PP}(\omega_{n},\vec{k})=\Sigma\,.$
(33)

###  3.3 Time-reflection symmetry

The fermionic action is invariant under the following “time reflection”
transformation,

$\displaystyle\psi(t,\vec{x})$
$\displaystyle\to\gamma_{4}\gamma_{5}\psi(\beta-t,\vec{x})\,,$
$\displaystyle\bar{\psi}(t,\vec{x})$
$\displaystyle\to\bar{\psi}(\beta-t,\vec{x})\gamma_{5}\gamma_{4}\,,$ (34)
$\displaystyle B_{\mu}(t,\vec{x})$
$\displaystyle\to\zeta_{\mu}B_{\mu}(\beta-t,\vec{x})\,,$

with $\zeta_{4}=-1$ and $\zeta_{1,2,3}=1$. No summation over $\mu$ is implied
here and in the following equations. Using the Faddeev-Popov-De Witt procedure
in a covariant gauge, the gauge action is also invariant under the
transformation Eq. (34), so this leaves the full action invariant. Under time
reflection one has for the pseudoscalar densities and axial-vector currents

$\displaystyle P^{a}(t,\vec{x})$ $\displaystyle\to-P^{a}(\beta-t,\vec{x})\,,$
$\displaystyle A^{a}_{\mu}(t,\vec{x})$
$\displaystyle\to-\zeta_{\mu}A^{a}_{\mu}(\beta-t,\vec{x})\,.$ (35)

For the correlators ${\mathcal{G}}_{AP\,\mu}$ and ${\mathcal{G}}_{PP}$ [see
Eq. (31)] one then finds

$\displaystyle\begin{split}{\mathcal{G}}_{AP\,\mu}(t,\vec{x})&=\zeta_{\mu}\langle
A^{a}_{\mu}(\beta-t,\vec{x})P^{a}(\beta,\vec{0})\rangle_{\beta}=\zeta_{\mu}\langle
A^{a}_{\mu}(\beta-t,\vec{x})P^{a}(0,\vec{0})\rangle_{\beta}=\zeta_{\mu}{\mathcal{G}}_{AP\,\mu}(\beta-t,\vec{x})\,,\end{split}$
(36) $\displaystyle{\mathcal{G}}_{PP}(t,\vec{x})$ $\displaystyle=\langle
P^{a}(\beta-t,\vec{x})P^{a}(\beta,\vec{0})\rangle_{\beta}=\langle
P^{a}(\beta-t,\vec{x})P^{a}(0,\vec{0})\rangle_{\beta}={\mathcal{G}}_{PP}(\beta-t,\vec{x})\,,$
(37)

where antiperiodicity of $\psi$ and $\bar{\psi}$ has been used to replace
$P^{a}(\beta,\vec{0})\to P^{a}(0,\vec{0})$, and no summation over $a$ is
implied. The symmetry properties of the coordinate-space correlators translate
into the following relations for the momentum-space correlators,

$\tilde{{\mathcal{G}}}_{AP\,\mu}(\omega_{n},\vec{k})=\zeta_{\mu}\tilde{{\mathcal{G}}}_{AP\,\mu}(-\omega_{n},\vec{k})\,,\qquad\tilde{{\mathcal{G}}}_{PP}(\omega_{n},\vec{k})=\tilde{{\mathcal{G}}}_{PP}(-\omega_{n},\vec{k})\,.$
(38)

###  3.4 Analytic continuation

Since they will be used repeatedly, it is convenient to summarise the relevant
analytic continuation relations needed to reconstruct the real-time,
Minkowskian thermal expectation values from the imaginary-time, Euclidean
correlation functions. The relevant Minkowskian operators are the axial-vector
current and pseudoscalar density operators,

$\displaystyle\hat{A}^{a}_{\mu}(x)\equiv\bar{\hat{\psi}}(x)\tilde{\gamma}_{\mu}\tilde{\gamma}_{5}t^{a}\hat{\psi}(x)\,,\qquad\hat{P}^{a}(x)\equiv\bar{\hat{\psi}}(x)\tilde{\gamma}_{5}t^{a}\hat{\psi}(x)\,.$
(39)

Here $\tilde{\gamma}^{\mu}$, $\mu=0,\ldots,3$ are the Minkowskian gamma
matrices, obeying
$\\{\tilde{\gamma}^{\mu},\tilde{\gamma}^{\nu}\\}=2\eta^{\mu\nu}$ with
$\eta^{\mu\nu}={\rm diag}(1,-1,-1,-1)$, and
$\tilde{\gamma}^{5}=i\tilde{\gamma}^{0}\tilde{\gamma}^{1}\tilde{\gamma}^{2}\tilde{\gamma}^{3}$.
These are related with the Euclidean gamma matrices $\gamma_{\mu}$ and
$\gamma_{5}$ as $\gamma_{4}=\tilde{\gamma}^{0}$,
$\gamma_{j}=-i\tilde{\gamma}^{j}$, and $\gamma_{5}=\tilde{\gamma}^{5}$. In Eq.
(39), $\bar{\hat{\psi}}=\hat{\psi}^{\dagger}\tilde{\gamma}^{0}$, as usual.
Using the general analytic continuation relation Eq. (5), one finds that
Euclidean and Minkowskian two-point correlation functions are related as
follows,

$\displaystyle\langle\\!\langle\hat{A}^{a0}(t,\vec{x})\hat{P}^{a}(0)\rangle\\!\rangle_{\beta}$
$\displaystyle={\mathcal{G}}_{AP\,4}(\epsilon+it,\vec{x})\,,$
$\displaystyle\langle\\!\langle\hat{P}^{a}(0)\hat{A}^{a0}(t,\vec{x})\rangle\\!\rangle_{\beta}$
$\displaystyle={\mathcal{G}}_{AP\,4}(-\epsilon+it,\vec{x})\,,$ (40)
$\displaystyle\langle\\!\langle\hat{A}^{aj}(t,\vec{x})\hat{P}^{a}(0)\rangle\\!\rangle_{\beta}$
$\displaystyle=i{\mathcal{G}}_{AP\,j}(\epsilon+it,\vec{x})\,,$
$\displaystyle\langle\\!\langle\hat{P}^{a}(0)\hat{A}^{aj}(t,\vec{x})\rangle\\!\rangle_{\beta}$
$\displaystyle=i{\mathcal{G}}_{AP\,j}(-\epsilon+it,\vec{x})\,,$
$\displaystyle\langle\\!\langle\hat{P}^{a}(t,\vec{x})\hat{P}^{a}(0)\rangle\\!\rangle_{\beta}$
$\displaystyle={\mathcal{G}}_{PP}(\epsilon+it,\vec{x})\,,$
$\displaystyle\langle\\!\langle\hat{P}^{a}(0)\hat{P}^{a}(t,\vec{x})\rangle\\!\rangle_{\beta}$
$\displaystyle={\mathcal{G}}_{PP}(-\epsilon+it,\vec{x})\,.$

Here and in the rest of this subsection no summation over the flavour index
$a$ is implied. Of particular interest is the reconstruction of the spectral
function, Eq. (9), through that of the retarded and advanced propagators, Eq.
(10). In this work I need the spectral functions $\tilde{\rho}_{A^{0a}P^{a}}$
and $\tilde{\rho}_{P^{a}P^{a}}$, that are independent of $a$ thanks to vector
flavour invariance. For brevity I will denote them as follows,

$\displaystyle\tilde{c}(\omega,\vec{k})$
$\displaystyle\equiv\tilde{\rho}_{A^{0a}P^{a}}(\omega,\vec{k})\,,$
$\displaystyle\tilde{c}^{P}(\omega,\vec{k})$
$\displaystyle\equiv\tilde{\rho}_{P^{a}P^{a}}(\omega,\vec{k})\,.$ (41)

These spectral functions can be obtained by analytic continuation using
directly Eq. (40) [see Eq. (9)], or from the retarded and advanced
propagators, which in turn can be reconstructed by analytic interpolation from
the Fourier coefficients of the Euclidean correlator Bros:1996mw ;
Cuniberti:2001hm . Setting for brevity ($\mu=0,\ldots,3$)

$\displaystyle\tilde{r}^{\mu}(\omega,\vec{k})$
$\displaystyle\equiv\tilde{r}_{A^{\mu a}P^{a}}(\omega,\vec{k})\,,$
$\displaystyle\tilde{a}^{\mu}(\omega,\vec{k})$
$\displaystyle\equiv\tilde{a}_{A^{\mu a}P^{a}}(\omega,\vec{k})\,,$ (42)
$\displaystyle\tilde{r}^{P}(\omega,\vec{k})$
$\displaystyle\equiv\tilde{r}_{P^{a}P^{a}}(\omega,\vec{k})\,,$
$\displaystyle\tilde{a}^{P}(\omega,\vec{k})$
$\displaystyle\equiv\tilde{a}_{P^{a}P^{a}}(\omega,\vec{k})\,,$

one has from Eq. (12) that for $n\neq 0$

$\displaystyle\tilde{{\mathcal{G}}}_{AP\,4}(\omega_{n},\vec{k})$
$\displaystyle=\tilde{r}^{0}(i\omega_{n},-\vec{k})\,,$ $\displaystyle n$
$\displaystyle>0\,,$
$\displaystyle\tilde{{\mathcal{G}}}_{AP\,4}(\omega_{n},\vec{k})$
$\displaystyle=\tilde{a}^{0}(i\omega_{n},-\vec{k})\,,$ $\displaystyle n$
$\displaystyle<0\,,$ (43) $\displaystyle
i\tilde{{\mathcal{G}}}_{AP\,j}(\omega_{n},\vec{k})$
$\displaystyle=\tilde{r}^{j}(i\omega_{n},-\vec{k})\,,$ $\displaystyle n$
$\displaystyle>0\,,$ $\displaystyle
i\tilde{{\mathcal{G}}}_{AP\,j}(\omega_{n},\vec{k})$
$\displaystyle=\tilde{a}^{j}(i\omega_{n},-\vec{k})\,,$ $\displaystyle n$
$\displaystyle<0\,,$
$\displaystyle\tilde{{\mathcal{G}}}_{PP}(\omega_{n},\vec{k})$
$\displaystyle=\tilde{r}^{P}(i\omega_{n},-\vec{k})\,,$ $\displaystyle n$
$\displaystyle>0\,,$
$\displaystyle\tilde{{\mathcal{G}}}_{PP}(\omega_{n},\vec{k})$
$\displaystyle=\tilde{a}^{P}(i\omega_{n},-\vec{k})\,,$ $\displaystyle n$
$\displaystyle<0\,.$

These relations hold also for $n=0$ if a transport peak is absent.

###  3.5 Regularisation and renormalisation

The discussion so far has been entirely formal, ignoring the ill-defined
nature of path integrals. As is well known, these require a suitable
regularisation to become mathematically well defined, and an appropriate
renormalisation procedure to remove the divergences appearing when the
regularisation is removed. Since regularisation usually breaks some of the
symmetries, their recovery after renormalisation is carried out is not
guaranteed in the general case, and this can spoil the formal results
discussed above. In particular, Ward-Takahashi identities will be violated in
the regulated theory if the regularisation breaks the corresponding symmetry,
and it is not obvious that they can be recovered in the same form after
renormalisation.

Before drawing any conclusion from the symmetry properties discussed in the
previous Subsections, it is important to make sure that they can be enforced
in the finite, renormalised theory. The best way to enforce a symmetry is
obviously to choose a regularisation that does not break it, in which case
renormalisation will not spoil it. The best known non-perturbative
regularisation of path integrals is the lattice regularisation (see, e.g.,
Refs. Rothe:1992nt ; Montvay:1994cy ; Gattringer:2010zz ), which is especially
convenient when dealing with gauge theories, since it allows one to mantain
manifest gauge invariance. In the lattice approach, the formal functional
integral is replaced with a well-defined finite-dimensional integral over
fields defined only on the discrete elements of a finite lattice, eventually
taking the limits of infinite volume and zero lattice spacing. This approach
clearly breaks most of the spacetime symmetry; using a hypercubic lattice with
periodic boundary conditions one can still retain symmetry under translations
by multiples of the lattice spacing, a discrete subgroup of the SO(4) group,
and reflections [so that Eqs. (36) and (37) hold also in the regulated
theory]. Nonetheless, for asymptotically free theories it is widely believed,
and supported by a vast amount of numerical evidence, that a continuum limit
exists (after a suitable renormalisation procedure) where spacetime symmetries
are fully restored.

While vector flavour symmetry can be implemented exactly on the lattice, the
axial flavour symmetry is problematic due to the known difficulties of
implementing exact chiral symmetry for Dirac operators discretised on the
lattice Nielsen:1981hk ; Nielsen:1980rz ; Nielsen:1981xu . Nonetheless, for
lattice Dirac operators satisfying the Ginsparg-Wilson relation
Ginsparg:1981bj , such as the fixed-point action Hasenfratz:1993sp ;
DeGrand:1995ji , domain-wall fermions Kaplan:1992bt ; Shamir:1993zy , and
overlap fermions Narayanan:1993sk ; Narayanan:1993ss ; Neuberger:1997fp ;
Neuberger:1998wv , one has an exact chiral-type symmetry that holds on any
finite lattice Luscher:1998pqa , and that reduces to the usual chiral symmetry
in the formal continuum limit. This implies exact Ward-Takahashi identities
for suitably defined lattice currents and densities, that hold for any lattice
spacing and tend to the continuum identities as the spacing goes to zero
Luscher:1998pqa ; Hasenfratz:1998jp ; Kikukawa:1998py ; Hasenfratz:2002rp ,
and guarantee that the desired symmetry can be enforced in the renormalised
theory. In particular, renormalised continuum correlation functions will
satisfy the continuum Ward-Takahashi identities – of course, assuming that
such a limit exists.

Taking the continuum limit and the associated restoration of spacetime
symmetries for granted, one can use the continuum Ward-Takahashi identities as
fully meaningful relations between renormalised quantities, and ignore where
they came from. This will suffice for the discussion of the finite-temperature
version of Goldstone’s theorem in Section 4. On the other hand, when studying
the pseudoscalar-pseudoscalar correlator in detail in Section 5 one has to
keep track of the effects of renormalisation. Instead of dealing with the
technical complications of the lattice approach, it is simpler to discuss the
issue of renormalisation directly in the continuum limit. In fact, if this
limit exists, then a renormalised theory with the desired symmetries can also
be obtained directly in the continuum, starting from a regularised theory
where the representation of fermionic observables in terms of sums over the
eigenmodes of the continuum Dirac operator is cut off symmetrically at some
ultraviolet scale $\Lambda$.666A suitable regularisation of the integration
over gauge fields is also required. However, this does not affect the
argument. Ward-Takahashi identities are not exact anymore in this case, but
their violations should disappear after appropriate renormalisation and
removal of the cutoff.

A detailed discussion of the renormalisation issues related to the Ward-
Takahashi identity Eq. (29), in continuum language, is provided in Appendix C.
As mentioned above, using Ginsparg–Wilson fermions Ginsparg:1981bj  in the
lattice regularisation of the theory one can use the lattice Ward-Takahashi
identities implied by the exact lattice chiral symmetry Luscher:1998pqa  to
show nonperturbatively Hasenfratz:1998jp  that $m$ renormalises only
multiplicatively; that the composite operators $A^{a}_{\mu}$ require no
renormalisation after the usual mass and coupling renormalisations have been
carried out; and that the multiplicative renormalisation constants $Z_{P}$ and
$Z_{S}$ of the non-singlet pseudoscalar and singlet scalar densities satisfy
$Z_{P}=Z_{S}=Z_{m}^{-1}$ with $Z_{m}$ the mass renormalisation constant.
Combined with the properties of bilinear correlators under the “${\cal
R}_{5}$-parity” transformation Frezzotti:2003ni , the lattice chiral symmetry
implies that all additive divergent contact terms drop from the lattice
analogue of Eq. (29) in the chiral limit. Based on the argument given above,
it is then safe to use the continuum identity Eq. (29) in its regularised
version to discuss renormalisation issues in a simpler continuum language.
After renormalisation, Eq. (29) in its renormalised version can be used as the
starting point for an alternative derivation of the finite temperature version
of Goldstone’s theorem Lange:1965zz ; Kastler:1966wdu ; Swieca:1966wna ;
Morchio:1987wd ; Strocchi:2008gsa . While renormalisation is an essential part
of the construction of the theory itself, and deserves a careful discussion as
such, it will become clear that it plays a limited role in the arguments of
this paper concerning the chiral limit, which is in fact dominated by the low-
end, infrared part of the Dirac spectrum.

##  4 Goldstone’s theorem at finite temperature

In this Section I discuss an alternative derivation of the finite temperature
analogue of Goldstone’s theorem Lange:1965zz ; Kastler:1966wdu ;
Swieca:1966wna ; Morchio:1987wd ; Strocchi:2008gsa , including a slight but
useful generalisation, based on the Ward-Takahashi identity Eq. (29). In this
Section all Euclidean quantities are understood to be renormalised.

###  4.1 Review of the standard derivation

For quantum field theories at finite temperature, the analogue of Goldstone’s
theorem Goldstone:1962es  proved in Refs. Lange:1965zz ; Kastler:1966wdu ;
Swieca:1966wna ; Morchio:1987wd  states that the spontaneous breaking of a
continuous symmetry in a theory invariant under spatial translations leads to
a gapless spectrum of “quasi-particle” excitations (see Ref. Strocchi:2008gsa
for a detailed discussion and a full list of references). A heuristic proof of
this “Goldstone’s theorem at finite temperature” is based on the following
observations. Let $\hat{J}^{\mu}$ be the conserved Noether current associated
with the symmetry, $\partial_{\mu}\hat{J}^{\mu}=0$, and let

$\hat{Q}_{V}(t)\equiv\int_{V}d^{3}x\,\hat{J}^{0}(t,\vec{x})\,,$ (44)

be the corresponding charge, regularised by restricting spatial integration to
a finite volume $V$. If a nonzero expectation value is found for the
commutator

$\lim_{V\to\infty}\langle\\!\langle[i\hat{Q}_{V}(0),\hat{{\mathcal{O}}}]\rangle\\!\rangle_{\beta}=b\neq
0\,,$ (45)

for some local observable ${\mathcal{O}}$, then current conservation and
relativistic locality imply

$\lim_{V\to\infty}\langle\\!\langle[i\hat{Q}_{V}(t),\hat{{\mathcal{O}}}]\rangle\\!\rangle_{\beta}=b\,,~{}~{}\forall
t\,.$ (46)

Taking the Fourier transform (in the sense of distributions) of Eq. (46) one
then finds

$\lim_{\vec{k}\to
0}i\tilde{\rho}_{J^{0}{\mathcal{O}}}(\omega,\vec{k})=\lim_{\vec{k}\to 0}\int
d^{4}x\,e^{i(\omega
t-\vec{k}\cdot\vec{x})}\langle\\!\langle[i\hat{J}^{0}(t,\vec{x}),\hat{{\mathcal{O}}}]\rangle\\!\rangle_{\beta}=2\pi
b\delta(\omega)\,,$ (47)

from which one infers the existence of massless quasi-particle excitations,
i.e., such that their energy vanishes and their lifetime becomes infinite in
the zero-momentum limit.777The same conclusions hold more generally for non-
relativistic systems, replacing the requirement of relativistic locality with
Swieca’s condition Swieca:1966wna  for the commutator of the spatial part of
the current with the relevant observable (see Ref. Strocchi:2008gsa ). I will
refer to these as Goldstone excitations or quasi-particles throughout this
paper.

In the case at hand, the relevant symmetry is the non-singlet axial part of
chiral symmetry. After analytic continuation to Minkowski spacetime, Eq. (27)
expresses conservation of the non-singlet axial currents $\hat{A}_{\mu}^{a}$
in the chiral limit $m\to 0$, under the usual assumption that the second term
on the left-hand side can be dropped. Equations (29) and (32) further show
that axial flavour symmetry is spontaneously broken, in the sense of Eq. (45),
if $\Sigma\neq 0$. In fact, integrating Eq. (32) in the $m\to 0$ limit over
space and over the infinitesimal time interval $[-\epsilon,\epsilon]$ one gets
as $\epsilon\to 0$,

$\Sigma_{*}=-\int
d^{3}x\,\left[{\mathcal{G}}_{AP\,4}(\epsilon,\vec{x})-{\mathcal{G}}_{AP\,4}(-\epsilon,\vec{x})\right]=\int
d^{3}x\,\langle\\!\langle[i\hat{A}^{a0}(0,\vec{x}),i\hat{P}^{a}(0)]\rangle\\!\rangle_{\beta}\,,$
(48)

where no summation over $a$ is implied and $\Sigma_{*}$ is the chiral
condensate in the chiral limit. Here I used continuity of $\int
d^{3}x\,\vec{\nabla}\cdot\vec{{\mathcal{G}}}_{AP}(t,\vec{x})$ at $t=0$,888This
follows from $\int
d^{3}x\,\vec{\nabla}\cdot\vec{{\mathcal{G}}}_{AP}(t,\vec{x})=\lim_{R\to\infty}\int_{\partial
B_{R}}d^{2}\vec{\Sigma}\cdot\vec{{\mathcal{G}}}_{AP}(t,\vec{x})$, with $B_{R}$
the ball of radius $R$ and $d^{2}\vec{\Sigma}$ the corresponding infinitesimal
surface element oriented outwards, and from continuity in $t$ of
$\vec{{\mathcal{G}}}_{AP}(t,\vec{x})$ for $t<|\vec{x}|$ (see Section 2). It is
assumed that $\vec{{\mathcal{G}}}_{AP}(t,\vec{x})$ vanishes sufficiently fast
at spatial infinity (at least like $|\vec{x}|^{-2}$) so that the surface
integral is convergent (but not necessarily zero). and the analytic
continuation relations Eq. (40). Clearly, Eq. (48) is nothing but Eq. (45) at
$t=0$ with $\hat{Q}_{V}(0)=\int_{V}d^{3}x\,\hat{A}^{a0}(0,\vec{x})$ the
finite-volume axial charge and $\hat{{\mathcal{O}}}=i\hat{P}^{a}(0)$. One can
now use current conservation and relativistic locality to complete the
argument, obtaining Eq. (47) with $-\tilde{c}$ [see Eq. (41)] on the left-hand
side, and $b=\Sigma_{*}$ on the right-hand side, and infer the existence of
pseudoscalar Goldstone quasi-particles, i.e., “quasi-pions” [see Eq. (63)
below].

The standard argument outlined above makes essential use of current
conservation as an operator equation to infer Eq. (46) from Eq. (45). In the
following I discuss a more direct argument that requires only knowledge of the
Ward-Takahashi identity Eq. (29) in its energy-momentum-space form, Eq. (33),
works directly with Euclidean quantities, and allows for a simple but useful
generalisation. This argument has been presented briefly in Ref.
giordano_GT_lett ; here I provide a more detailed discussion. In Appendix D I
discuss the coordinate-space version of the argument, which is new. Of course,
appropriate analyticity conditions must be satisfied in order to be able to
reconstruct the physical, Minkowskian correlation functions. Moreover, a
suitable regularity condition must also be satisfied to guarantee relativistic
locality of the reconstructed theory. This regularity condition plays an
important role and will be discussed next.

###  4.2 Regularity condition

In this Subsection I “reverse-engineer” a condition that has to be imposed on
$\vec{\mkern 0.0mu\widetilde{{\mathcal{G}}}}_{AP}$ in order to obtain the
desired locality properties of quantum field theory in Minkowski space.
Starting from the relativistic locality condition, $[\vec{\mkern
0.0mu\hat{A}}^{a}(x),\hat{P}^{b}(0)]=0$ for $x^{2}<0$, one finds (in the sense
of distributions) that

$\displaystyle\lim_{k\to 0}\vec{k}\cdot\int
d^{3}x\,e^{-i\vec{k}\cdot\vec{x}}\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{b}(0)]\rangle\\!\rangle_{\beta}=\lim_{k\to
0}i\int
d^{3}x\,\left\\{\vec{\nabla}e^{-i\vec{k}\cdot\vec{x}}\right\\}\cdot\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{b}(0)]\rangle\\!\rangle_{\beta}$ (49)
$\displaystyle=$ $\displaystyle\lim_{k\to 0}i\int
d^{3}x\,\vec{\nabla}\cdot\left\\{e^{-i\vec{k}\cdot\vec{x}}\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{b}(0)]\rangle\\!\rangle_{\beta}\right\\}-i\int
d^{3}x\,e^{-i\vec{k}\cdot\vec{x}}\,\vec{\nabla}\cdot\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{b}(0)]\rangle\\!\rangle_{\beta}$
$\displaystyle=$ $\displaystyle-i\int
d^{3}x\,\vec{\nabla}\cdot\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{b}(0)]\rangle\\!\rangle_{\beta}=0\,,$

since the first term on the second line and the term on the third line vanish
due to finiteness of the support of the commutator at fixed $t$.999More
formally, after smearing over $t$ with a function of compact support $h(t)$,
the quantity $\vec{C}(\vec{x}\,)=\int dt\,h(t)\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{a}(0)]\rangle\\!\rangle_{\beta}$ is a
distribution in $\vec{x}$ of compact support, and so by the Paley-Wiener
theorem for distributions its Fourier transform $\vec{\mkern
0.0mu\widetilde{C}}(\vec{k})=\int
d^{3}x\,e^{-i\vec{k}\cdot\vec{x}}\vec{C}(\vec{x}\,)$ is an entire function, so
that $\vec{k}\cdot\vec{\mkern 0.0mu\widetilde{C}}(\vec{k})\to 0$ as
$\vec{k}\to 0$ follows. This holds independently of the quark mass $m$, and
should be true in particular in the chiral limit if one is to obtain a decent
quantum field theory. In this limit, the final equality implies the time-
independence of the regularised charge commutator if the Noether current is
conserved, so that Eq. (46) follows from Eq. (45) (see the discussion in Ref.
Strocchi:2008gsa , Section 15.2.II).

To work out the implications of Eq. (49) in the Euclidean setting, one uses
the relations Eq. (43) between the retarded and advanced propagators
$\tilde{r}^{j}$ and $\tilde{a}^{j}$ and the Euclidean correlator
$\tilde{{\mathcal{G}}}_{AP\,j}$. Plugging them into Eq. (49), one finds
($n>0$, no summation over $a$)

$\displaystyle\lim_{\vec{k}\to 0}\vec{k}\cdot\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP}(\omega_{\pm n},\vec{k})$
$\displaystyle=\pm\lim_{\vec{k}\to
0}\vec{k}\cdot\int_{-\infty}^{\infty}dt\,\theta(\pm t)e^{-\omega_{\pm n}t}\int
d^{3}x\,e^{i\vec{k}\cdot\vec{x}}\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{a}(0)]\rangle\\!\rangle_{\beta}$ (50)
$\displaystyle=\pm\int_{-\infty}^{\infty}dt\,\theta(\pm
t)e^{-\omega_{n}|t|}\lim_{\vec{k}\to 0}\vec{k}\cdot\int
d^{3}x\,e^{i\vec{k}\cdot\vec{x}}\langle\\!\langle[\vec{\mkern
0.0mu\hat{A}}^{a}(t,\vec{x}),\hat{P}^{a}(0)]\rangle\\!\rangle_{\beta}=0\,,$

where the exchange of the limit $\vec{k}\to 0$ and integration over $t$ is
justified by the exponential damping factor. From the Euclidean perspective,
this is a necessary condition that the Euclidean correlators must obey in
order to reconstruct a decent Minkowskian theory. I then assume that
$\vec{\mkern 0.0mu\widetilde{{\mathcal{G}}}}_{AP}$ obeys the regularity
condition $\vec{k}\cdot\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP}(\omega_{n},\vec{k})\to 0$ as $\vec{k}\to
0$ for $n\neq 0$.

###  4.3 Euclidean proof of Goldstone’s theorem in momentum space

To obtain a Euclidean proof of Goldstone’s theorem at finite temperature, one
starts from the Ward-Takahashi identity in energy-momentum-space Eq. (33).
Setting

${\mathcal{R}}(x)\equiv
2m{{\mathcal{G}}}_{PP}(x)\,,\qquad\widetilde{{\mathcal{R}}}(\omega_{n},\vec{k})\equiv
2m\widetilde{{\mathcal{G}}}_{PP}(\omega_{n},\vec{k})\,,$ (51)

this identity becomes

$i\omega_{n}\widetilde{{\mathcal{G}}}_{AP\,4}(\omega_{n},\vec{k})+i\vec{k}\cdot\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP}(\omega_{n},\vec{k})+\widetilde{{\mathcal{R}}}(\omega_{n},\vec{k})=\Sigma\,.$
(52)

The symmetry properties Eq. (38) imply that
$\widetilde{{\mathcal{G}}}_{AP\,4}(\omega_{n},\vec{k})=-\widetilde{{\mathcal{G}}}_{AP\,4}(-\omega_{n},\vec{k})$,
so in particular $\widetilde{{\mathcal{G}}}_{AP\,4}(0,\vec{k})=0$, while the
regularity condition Eq. (50) requires $\vec{k}\cdot\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP}(\omega_{n},\vec{k})\to 0$ as $\vec{k}\to
0$ for $n\neq 0$ (also in the chiral limit). Notice also
$\widetilde{{\mathcal{R}}}(-\omega_{n},\vec{k})=\widetilde{{\mathcal{R}}}(\omega_{n},\vec{k})$.
One then obtains from Eq. (52)

$\displaystyle i\vec{k}\cdot\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP}(0,\vec{k})+\widetilde{{\mathcal{R}}}(0,\vec{k})$
$\displaystyle=\Sigma\,,$ (53) $\displaystyle\lim_{\vec{k}\to
0}\left\\{i\omega_{n}\widetilde{{\mathcal{G}}}_{AP\,4}(\omega_{n},\vec{k})+\widetilde{{\mathcal{R}}}(\omega_{n},\vec{k})\right\\}$
$\displaystyle=\Sigma\,,\quad n\neq 0\,.$ (54)

It is usually (although perhaps implicitly) assumed that the pseudoscalar-
pseudoscalar correlator is sufficiently regular as a function of $m$ in the
chiral limit, so that $\widetilde{{\mathcal{R}}}(\omega_{n},\vec{k})\to 0$ as
$m\to 0$. I will refer to this as the standard scenario. As I show below in
Section 5, this may not be the case if a finite density of localised near-zero
Dirac modes is present. More precisely, in the presence of localised near-zero
modes the pseudoscalar-pseudoscalar correlator can develop a $1/m$ divergence,
that exactly cancels out the factor of $m$ and leaves behind a finite
$\widetilde{{\mathcal{R}}}$ in the chiral limit. I will refer to this as the
non-standard scenario.

Denoting by a subscript $*$ the quantities obtained in the chiral limit,
taking now $m\to 0$ in Eq. (52) followed by $\vec{k}\to 0$ one obtains in full
generality

$\displaystyle i\vec{k}\cdot\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP\,*}(0,\vec{k})+\widetilde{{\mathcal{R}}}_{*}(0,\vec{k})$
$\displaystyle=\Sigma_{*}\,,$ (55) $\displaystyle\lim_{\vec{k}\to
0}\left\\{i\omega_{n}\widetilde{{\mathcal{G}}}_{AP\,4\,*}(\omega_{n},\vec{k})+\widetilde{{\mathcal{R}}}_{*}(\omega_{n},\vec{k})\right\\}$
$\displaystyle=\Sigma_{*}\,,\quad n\neq 0\,.$ (56)

Exploiting rotation invariance, one finds from the equation for zero Matsubara
frequency, Eq. (55), that

$\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP\,*}(0,\vec{k})=-i\vec{k}\left(\frac{\Sigma_{*}-\widetilde{{\mathcal{R}}}_{*}(0,\vec{k})}{\vec{k}^{2}}\right)\mathop{\to}_{\vec{k}\to
0}-i\left(\Sigma_{*}-\widetilde{{\mathcal{R}}}_{*}(0,\vec{0})\right)\frac{\vec{k}}{\vec{k}^{2}}\,,$
(57)

and so it follows that $\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP\,*}(0,\vec{k})$ has a pole at $\vec{k}=0$
if $\Sigma_{*}-\widetilde{{\mathcal{R}}}_{*}(0,\vec{0})\neq 0$, so in
particular in the standard scenario, where
$\widetilde{{\mathcal{R}}}_{*}(0,\vec{0})=0$, if $\Sigma_{*}\neq 0$. However,
the existence of this pole does not imply a priori that massless Goldstone
quasi-particles are present. This is different from the zero-temperature case,
where one would find $\widetilde{{\mathcal{G}}}_{AP\,\mu\,*}\propto
p_{\mu}/p^{2}$ due to O(4) invariance. In that case, after analytic
continuation $(p_{4},\vec{p}\,)\to(-ip^{0},\vec{p}\,)$ to Minkowski space one
finds a pole at $(p^{0})^{2}-\vec{p}^{\,2}=0$ in the axial-vector-pseudoscalar
correlator, which directly implies the presence of massless particles in the
spectrum. At finite temperature full O(4) invariance is lost, and the
connection with the spectrum is encoded in the axial-vector-pseudoscalar
spectral function, whose reconstruction requires the analytic interpolation of
the discrete Fourier components of the Euclidean correlator. The presence of a
pole in $\vec{\mkern 0.0mu\widetilde{{\mathcal{G}}}}_{AP\,*}$ at zero
frequency is therefore by itself not sufficient to infer the relevant
properties of the spectral function at zero frequency.

To make progress one needs to exploit Eq. (56). To this end, one sets

${\rm G}_{*}(\omega_{n})\equiv\lim_{\vec{k}\to
0}\widetilde{{\mathcal{G}}}_{AP\,4\,*}(\omega_{n},\vec{k})\,,\qquad{\rm
R}_{*}(\omega_{n})\equiv\lim_{\vec{k}\to
0}\widetilde{{\mathcal{R}}}_{*}(\omega_{n},\vec{k})\,,$ (58)

and using Eq. (56) one finds

$\displaystyle{\rm G}_{*}(\omega_{n})$ $\displaystyle=\frac{\Sigma_{*}-{\rm
R}_{*}(\omega_{n})}{i\omega_{n}}\,,~{}~{}~{}~{}n\neq 0\,,$ (59)

while ${\rm G}_{*}(0)=0$. It is instructive to discuss first the standard
scenario in which $\widetilde{{\mathcal{R}}}_{*}$ vanishes and so ${\rm
R}_{*}=0$. In this case Eq. (59) entirely determines ${\rm
G}_{*}(\omega_{n})$, up to the value of $\Sigma_{*}$. It is then easy to
obtain its unique analytic interpolation (in the sense of Carlson’s theorem)
to a function $\bar{\rm G}_{*}(\Omega)$ of a complex variable
$\Omega\in\mathbb{C}$, and reconstruct the relevant Minkowskian quantities by
analytic continuation Bros:1996mw ; Cuniberti:2001hm . One
finds101010Formally, one defines $F(\Omega)=\frac{\Sigma}{i\Omega}-\bar{\rm
G}_{*}(\Omega)$, $\Omega\neq 0$, and $F(0)=0$, and looks for analytic
interpolations $\bar{\rm G}_{*}(\Omega)$ obeying Eq. (59) $\forall n\neq 0$.
This function obeys $F(\omega_{n})=0~{}\forall n$, and Carlson’s theorem
implies that its unique interpolation analytic in the upper half of the
complex plane is $F(\Omega)\equiv 0$, which further extends uniquely to
$F(\Omega)\equiv 0$ on the whole complex plane. This leads to Eq. (60). Notice
that $\lim_{\Omega\to 0}\bar{\rm G}_{*}(\Omega)\neq{\rm G}_{*}(0)=0$.

$\bar{\rm G}_{*}(\Omega)=\frac{\Sigma_{*}}{i\Omega}\,,$ (60)

which is analytic in the whole complex plane except for a pole at $\Omega=0$.
This is enough to reconstruct the spectral function in the zero-momentum
limit. In fact, using Eq. (43), one sees that the interpolation Eq. (60) for
${\rm Im}\,\Omega>0$ and ${\rm Im}\,\Omega<0$ corresponds respectively to the
retarded and advance propagators $\tilde{r}^{0}$ and $\tilde{a}^{0}$ in the
chiral and zero-momentum limit,

$\displaystyle\lim_{\vec{k}\to 0}\tilde{r}_{*}^{0}(\Omega,\vec{k})$
$\displaystyle=\frac{\Sigma_{*}}{\Omega}~{}\text{for}~{}{\rm Im}\,\Omega>0\,,$
$\displaystyle\lim_{\vec{k}\to 0}\tilde{a}_{*}^{0}(\Omega,\vec{k})$
$\displaystyle=\frac{\Sigma_{*}}{\Omega}~{}\text{for}~{}{\rm Im}\,\Omega<0\,.$
(61)

Since the spectral density is given by

$i\tilde{c}(\omega,\vec{k})=\tilde{r}^{0}(\omega+i\epsilon,\vec{k})-\tilde{a}^{0}(\omega-i\epsilon,\vec{k})\,,\qquad\omega\in\mathbb{R}\,,$
(62)

one finds in the chiral limit

$\lim_{\vec{k}\to
0}i\tilde{c}_{*}(\omega,\vec{k})=\frac{\Sigma_{*}}{\omega+i\epsilon}-\frac{\Sigma_{*}}{\omega-i\epsilon}=-2\pi
i\Sigma_{*}\delta(\omega)\,,$ (63)

and so if $\Sigma_{*}\neq 0$ one finds massless Goldstone excitations in the
spectrum. This completes this alternative proof of Goldstone’s theorem at
finite temperature under the usual assumptions on the symmetry breaking term.

In the non-standard scenario where $\widetilde{{\mathcal{R}}}_{*}\neq 0$,
$\bar{\rm G}_{*}(\Omega)$ is not fully determined, but one can still use Eq.
(59) to relate it with the analytic interpolation of ${\rm
R}_{*}(\omega_{n})$, which will be denoted with $\bar{\rm R}_{*}(\Omega)$. One
finds

$\displaystyle\bar{\rm G}_{*}(\Omega)=\frac{\Sigma_{*}-\bar{\rm
R}_{*}(\Omega)}{i\Omega}\,.$ (64)

Using Eq. (62) and the well known result $\frac{1}{\omega\pm i\epsilon}={\rm
PV}\frac{1}{\omega}\mp i\pi\delta(\omega)$, together with the symmetry
property $\bar{\rm R}_{*}(-\Omega)=\bar{\rm R}_{*}(\Omega)$ that follows from
Eq. (38) by analytic continuation, one finds for the spectral function in the
chiral limit

$\displaystyle\lim_{\vec{k}\to 0}i\tilde{c}_{*}(\omega,\vec{k})$
$\displaystyle=\bar{\rm G}_{*}(\epsilon-i\omega)-\bar{\rm
G}_{*}(-\epsilon-i\omega)$ (65)
$\displaystyle=-i\pi\delta(\omega)\left[2\Sigma_{*}-\bar{\rm
R}_{*}(\epsilon)-\bar{\rm R}_{*}(-\epsilon)\right]-{\rm
PV}\frac{1}{\omega}\left[\bar{\rm R}_{*}(\epsilon-i\omega)-\bar{\rm
R}_{*}(-\epsilon-i\omega)\right]$
$\displaystyle=-i2\pi\delta(\omega)\left[\Sigma_{*}-\bar{\rm
R}_{*}(\epsilon)\right]-{\rm PV}\frac{1}{\omega}\left[\bar{\rm
R}_{*}(\epsilon-i\omega)-\bar{\rm R}_{*}(\epsilon+i\omega)\right]\,.$

The quantity in the second square bracket on the right-hand side of Eq. (65)
is manifestly antisymmetric, so that one can drop the principal-value
prescription. To discuss its regularity properties it is convenient to express
it in terms of the pseudoscalar spectral function. Recalling Eq. (43), one
sees that

$\displaystyle\bar{\rm R}_{*}(\epsilon-i\omega)$
$\displaystyle=\lim_{\vec{k}\to 0}\lim_{m\to
0}2m\tilde{r}^{P}(i\epsilon+\omega,-\vec{k})\,,$ (66) $\displaystyle\bar{\rm
R}_{*}(-\epsilon-i\omega)$ $\displaystyle=\lim_{\vec{k}\to 0}\lim_{m\to
0}2m\tilde{a}^{P}(-i\epsilon+\omega,-\vec{k})\,,$

and so

$\displaystyle\bar{\rm R}_{*}(\epsilon-i\omega)-\bar{\rm
R}_{*}(-\epsilon-i\omega)$ $\displaystyle=\lim_{\vec{k}\to 0}i\int
d^{4}x\,e^{i(\omega t-\vec{k}\cdot\vec{x})}\lim_{m\to
0}2m\langle\\!\langle[\hat{P}^{a}(t,\vec{x}),\hat{P}^{a}(0)]\rangle\\!\rangle_{\beta}$
(67) $\displaystyle=i\lim_{\vec{k}\to 0}\lim_{m\to
0}2m\tilde{c}^{P}(\omega,\vec{k})\,.$

In the last passage it is assumed that the chiral limit can be exchanged with
the Fourier transform at finite $\vec{k}$.

The pseudoscalar spectral function at $\vec{k}=0$ is an antisymmetric function
of $\omega$ [see Eq. (135) in Appendix A], expected to be regular at
$\omega=0$ and so vanishing at least like $\omega$ (see Ref. Meyer:2011gj ).
Moreover, no transport peak $\propto\omega\delta(\omega)$ is expected in the
pseudoscalar channel. These expectations are supported both by analytic
perturbative results (also in the chiral limit) and by numerical lattice
calculations Karsch:2003wy ; Aarts:2005hg ; Burnier:2017bod . It is then
reasonable to assume that the transport peak is absent, and that
$\tilde{c}^{P}$ is regular at $\omega=0$, and remains so as $m\to
0$.111111These assumptions can be weakened. The effects of a transport peak
can be taken into account, see footnote 12. Concerning the behaviour at
$\omega=0$, it is sufficient to assume that $\tilde{c}^{P}$ has an integrable
singularity, which can further be demanded only in the relevant chiral and
zero-momentum limits. Under these assumptions one finds from Eq. (14)

$\bar{\rm R}_{*}(\epsilon)=\lim_{\vec{k}\to 0}\lim_{m\to
0}2m\tilde{r}^{P}(i\epsilon,-\vec{k})=\lim_{\vec{k}\to 0}\lim_{m\to
0}2m\widetilde{{\mathcal{G}}}_{PP}(0,\vec{k})=\lim_{\vec{k}\to
0}\widetilde{{\mathcal{R}}}_{*}(0,\vec{k})={\rm R}_{*}(0)\,,$ (68)

so that Eq. (65) reduces to

$\lim_{\vec{k}\to 0}i\tilde{c}_{*}(\omega,\vec{k})=-2\pi
i\delta(\omega)[\Sigma_{*}-{\rm R}_{*}(0)]-\frac{1}{\omega}[\bar{\rm
R}_{*}(\epsilon-i\omega)-\bar{\rm R}_{*}(\epsilon+i\omega)]\,,$ (69)

with the second term regular as $\omega\to 0$.121212In the presence of a
transport peak $\tilde{c}^{P}(\omega,\vec{k})|_{\rm tp}=2\pi
A_{PP}(\vec{k})\omega\delta(\omega)$, the principal value prescription must be
kept in Eq. (69), and since ${\rm
PV}\frac{1}{\omega}\omega\delta(\omega)=\frac{\omega^{2}}{\omega^{2}+\epsilon^{2}}\delta(\omega)=0$
one finds no contribution to $\tilde{c}_{*}(\omega,\vec{k}\to 0)$ from the
second term. Setting $A_{*}=\lim_{\vec{k}\to 0}\lim_{m\to
0}2mA_{PP}(\vec{k})$, from Eq. (14) it follows $\lim_{\epsilon\to 0}\bar{\rm
R}_{*}(\epsilon)={\rm R}_{*}(0)-A_{*}$, and so the delta term is changed to
$-2\pi i\delta(\omega)[\Sigma_{*}-{\rm R}_{*}(0)+A_{*}]$. Only the first term
affects the presence of massless Goldstone excitations in the spectrum, which
exist if $\Sigma_{*}-{\rm R}_{*}(0)\neq 0$. This completes the proof of the
generalised Goldstone’s theorem at finite temperature in the presence of a
nonzero ${\rm R}_{*}(0)$.

It is worth commenting on the result above, especially in relation with the
usual Goldstone’s theorem at finite temperature. As already discussed above, a
nonzero ${\rm R}_{*}(0)$ can only appear if the pseudoscalar-pseudoscalar
correlator develops a $1/m$ divergence as $m\to 0$, cancelling out the factor
of $m$ appearing in the Ward-Takahashi identity and leaving a finite
contribution in the chiral limit. This mechanism is reminiscent of the
formation of anomalies, although here one is sensitive to the infrared rather
than the ultraviolet regime of the theory. For this reason, I will refer to a
nonzero ${\rm R}_{*}(0)$ as an anomalous remnant.

It is clear from Eq. (69) that in principle ${\rm R}_{*}(0)$ could cancel
$\Sigma_{*}$, so that in spite of the apparent spontaneous breaking of a
continuous symmetry by a nonzero expectation value of an order parameter one
would find no Goldstone excitations, as one would expect from the usual
Goldstone’s theorem at finite temperature. However, an anomalous remnant
radically modifies the usual Ward-Takahashi identity in the chiral limit,
signalling that the non-singlet axial currents are not conserved in this
limit. More precisely, the anomalous remnant makes the non-singlet axial
flavour symmetry explicitly broken even in the chiral limit. As a consequence,
one evades Goldstone’s theorem at finite temperature, since this has current
conservation as one of its main hypotheses. The presence of massless quasi-
particle excitations is not guaranteed by a nonzero condensate alone, and it
is rather the difference between the amount of spontaneous breaking, measured
by $\Sigma_{*}$, and of explicit breaking, measured by ${\rm R}_{*}(0)$, that
determines the fate of Goldstone excitations.131313I show below in Section 5.7
that $|\Sigma_{*}|-|{\rm R}_{*}(0)|=-(\Sigma_{*}-{\rm R}_{*}(0))\geq 0$.

It is interesting to notice that the presence of a pole at zero spatial
momentum in the correlator $\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP}(0,\vec{k})$ is after all sufficient to
infer the existence of massless Goldstone excitations. In fact, the residue at
this pole equals
$\Sigma_{*}-\widetilde{{\mathcal{R}}}_{*}(0,\vec{0})=\Sigma_{*}-{\rm
R}_{*}(0)$ up to a constant factor [see Eq. (57)], so that the pole is present
if and only if a Dirac-delta term is present in the axial-vector-pseudoscalar
spectral function.

As a final remark, I mention that one can extend the calculation done above to
the case of finite fermion mass without any difficulty. For the singular part
of the spectral function $\tilde{c}(\omega,\vec{k})$ one finds

$\lim_{\vec{k}\to 0}i\tilde{c}(\omega,\vec{k})|_{\rm sing}=-2\pi
i\delta(\omega)[\Sigma-{\rm R}(0)]\,,$ (70)

analogously to Eq. (69). At finite quark mass one generally finds a nonzero
${\rm R}(0)=\lim_{\vec{k}\to 0}\widetilde{{\mathcal{R}}}(0,\vec{k})=\int
d^{4}x\,2m{{\mathcal{G}}}_{PP}(x)=2m\chi_{\pi}$, where $\chi_{\pi}$ is the
pseudoscalar susceptibility, and a nonvanishing condensate $\Sigma$. However,
one can show that at finite $m$

$\Sigma={\rm R}(0)=2m\chi_{\pi}\,,$ (71)

and so the singular part of $\tilde{c}$ vanishes and no massless quasi-
particle excitation appears, as one expects when $m\neq 0$. This requires one
to show that there is no pole in $\vec{\mkern
0.0mu\widetilde{{\mathcal{G}}}}_{AP}(0,\vec{k})$ at $\vec{k}=0$ [see Eq.
(53)]. Equivalently, one notices that Eq. (71) is just the usual integrated
Ward-Takahashi identity relating the chiral condensate and the pseudoscalar
susceptibility, which holds if one can drop the boundary term when integrating
Eq. (32) over spacetime. Both the absence of a pole and the vanishing of the
boundary term follow if $\vec{{\mathcal{G}}}_{AP}(x)$ falls off sufficiently
fast at large distances. This can be shown to be the case. An argument by Vafa
and Witten Vafa:1983tf ; Nussinov:1999sx  establishes an exponential bound on
the two-point correlation function of any flavour non-singlet gauge-invariant
local operator, at any nonzero $m$ in theories with a positive path-integral
measure. This applies in particular to the axial-vector currents
$A^{a}_{\mu}$, and to the finite-temperature gauge theories considered here.
Using the Ward-Takahashi identity

$\partial_{\mu}\langle A^{a}_{\mu}(x)A^{b}_{\nu}(0)\rangle_{\beta}=2m\langle
P^{a}(x)A^{b}_{\nu}(0)\rangle_{\beta}=2m\delta^{ab}{{\mathcal{G}}}_{AP\,\nu}(x)\,,$
(72)

obtained by setting ${\mathcal{O}}=A^{b}_{\nu}(0)$ in Eq. (27), this implies
an exponential bound on $\vec{{\mathcal{G}}}_{AP}$, and so the desired result
follows.141414In fact, the bound of Ref. Vafa:1983tf  is easily extended to
the two-point function of any pair of flavour non-singlet local bilinear
operators like $A^{a}_{\mu}$ and $P^{a}$, so it applies directly to
$\vec{{\mathcal{G}}}_{AP}(x)$. Notice that if $\Sigma_{*}\neq 0$, Eq. (71)
implies a divergent pseudoscalar susceptibility in the chiral limit, and a
finite limit for ${\rm R}(0)\to\Sigma_{*}$ as $m\to 0$. This limit in general
does not coincide with ${\rm R}_{*}(0)$, which is obtained by taking first the
chiral limit followed by the zero-momentum limit [see comment before Eq.
(55)], since the two limits generally do not commute. While the localisation
properties of Dirac modes play no role in establishing the integrated Ward-
Takahashi identity Eq. (71), I will show below in Section 5 that they are
crucial in the determination of ${\rm R}_{*}(0)$.

##  5 Pseudoscalar correlator in the chiral limit

As discussed in the previous Section, it is usually assumed that the
contribution ${{\mathcal{R}}}$ [see Eq. (51)] of the pseudoscalar-pseudoscalar
correlator to the non-singlet axial Ward-Takahashi identity Eq. (29) vanishes
in the chiral limit. As anticipated, I argue now that this is not the case if
a finite density of localised modes is found near the origin of the Dirac
spectrum: Under certain technical conditions, such modes lead to the
development of a $1/m$ infrared divergence, that compensates the factor of $m$
and thus gives a finite contribution to the Ward-Takahashi identity also in
the chiral limit. As discussed above at the end of Section 4.3, this leads to
an anomalous remnant ${\rm R}_{*}(0)$ [see Eq. (58)] that competes with the
chiral condensate to determine the fate of Goldstone excitations.

In this Section I initially work with “bare”, unrenormalised quantities
(denoted by a subscript $B$), appearing in a suitably regularised version of
the path integral, Eq. (17). This could be, e.g., a lattice regularisation
with Ginsparg-Wilson fermions Ginsparg:1981bj ; Hasenfratz:1993sp ;
DeGrand:1995ji ; Kaplan:1992bt ; Shamir:1993zy ; Narayanan:1993sk ;
Narayanan:1993ss ; Neuberger:1997fp ; Neuberger:1998wv , that guarantees
control over the chiral properties of the theory Luscher:1998pqa ;
Hasenfratz:1998jp ; Kikukawa:1998py ; Hasenfratz:2002rp . However, as
explained in Section 3.5, it is justified to work directly in the continuum,
which allows for a simpler and clearer treatment of the main issues, without
having to deal with the technicalities of the lattice approach.

The starting point is the decomposition of the bare pseudoscalar correlator
$\langle P_{B}^{a}(x)P_{B}^{b}(0)\rangle_{\beta}$ in terms of the eigenmodes
of the Dirac operator. For infrared (IR) regularisation purposes one works in
a finite spatial volume $V$, imposing periodic boundary conditions in the
spatial directions. Antiperiodic boundary conditions are imposed instead in
the (compact) time direction due to the antiperiodicity condition on fermion
fields, see Section 3. In this setting the spectrum of ${\not{D}}$ becomes
discrete, and so the eigenvalues $i\lambda_{n}$ and the corresponding
eigenvectors $\psi_{n}$ will be labeled by an index $n$ taking integer values.
Moreover, as discussed above in Section 3.5, for ultraviolet (UV)
regularisation purposes one cuts off the spectrum at some scale $\Lambda$,
including in the mode decompositions only modes with
$|\lambda_{n}|\leq\Lambda$. Cutting off modes in this way makes the chiral
condensate and the relevant correlation functions finite, introducing
violations in the Ward-Takahashi identity that, however, disappear as
$\Lambda\to\infty$. These violations, as well as UV divergences, are of no
concern here, since they originate in the UV part of the Dirac spectrum, while
in the chiral limit only the low-end, IR part of the spectrum plays a role, as
will become clear below. Nonetheless, in spite of the fact that they have no
physical effect, UV modes should be carefully handled to obtain physically
meaningful, renormalised quantities.

###  5.1 Mode decomposition

A straightforward calculation gives the following result for the pseudoscalar-
pseudoscalar correlator expressed in terms of Dirac modes,

$-\langle
P_{B}^{a}(x)P_{B}^{b}(0)\rangle_{\beta}=\lim_{V\to\infty}\frac{\delta^{ab}}{2}\left\langle\sum_{n,n^{\prime}}\frac{{\mathcal{O}}^{\gamma_{5}}_{n^{\prime}n}(x){\mathcal{O}}^{\gamma_{5}}_{nn^{\prime}}(0)}{(i\lambda_{n}+m_{B})(i\lambda_{n^{\prime}}+m_{B})}\right\rangle_{\beta}\equiv-\delta^{ab}{\mathcal{G}}_{PP\,B}(x)\,.$
(73)

Here and in the following equations, only expectation values of functionals
${\mathcal{O}}_{\rm G}[B_{B}]$ of the (bare) gauge fields appear. In Eq. (73)
$m_{B}$ denotes the bare fermion mass, and ${\rm tr}\,$ denotes the trace over
Dirac and colour indices. The dependence of ${\mathcal{G}}_{PP\,B}$ on $m_{B}$
and on the temperature $T=1/\beta$ is left implicit. Moreover [see Eq. (20)],

${\mathcal{O}}_{n^{\prime}n}^{\Gamma}(x)\equiv(\psi_{n^{\prime}}(x),\Gamma\psi_{n}(x))\,.$
(74)

In this paper I will be concerned only with $\Gamma=\mathbf{1},\gamma_{5}$, in
which case the following properties hold,

$\displaystyle{\mathcal{O}}_{-n^{\prime}-n}^{\Gamma}(x)$
$\displaystyle={\mathcal{O}}_{n^{\prime}n}^{\Gamma}(x)\,,$
$\displaystyle{\mathcal{O}}_{n^{\prime}n}^{\Gamma}(x)^{*}$
$\displaystyle={\mathcal{O}}_{nn^{\prime}}^{\Gamma}(x)\,,$
$\displaystyle{\mathcal{O}}_{-n^{\prime}n}^{\Gamma}(x)$
$\displaystyle={\mathcal{O}}_{n^{\prime}n}^{\gamma_{5}\Gamma}(x)\,,$ (75)

where the notation $-n$ indicates that the mode
$\psi_{-n}\equiv\gamma_{5}\psi_{n}$ is involved. As mentioned above, the sums
over modes in Eq. (73), as well as the product in the determinant ${\rm
Det}({\not{D}}+m)=\prod_{n}(i\lambda_{n}+m)^{N_{f}}$ appearing in Eq. (19),
are restricted to $|\lambda_{n}|\leq\Lambda$.

Taking into account the symmetry of the spectrum, the relation between
eigenvectors implied by the chiral property discussed in Section 3.1, and the
properties Eq. (75), one can recast Eq. (73) as

$-{\mathcal{G}}_{PP\,B}(x)=\lim_{V\to\infty}\frac{1}{2}\int_{-\Lambda}^{\Lambda}d\lambda\int_{-\Lambda}^{\Lambda}d\lambda^{\prime}\,\frac{(m_{B}^{2}-\lambda\lambda^{\prime})C^{\gamma_{5}}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})}{(\lambda^{2}+m_{B}^{2})(\lambda^{\prime\,2}+m_{B}^{2})}\,,$
(76)

where I introduced the spectral correlators

$C^{\Gamma}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})\equiv\bigg{\langle}\sum_{n,n^{\prime}}\delta(\lambda-\lambda_{n})\delta(\lambda^{\prime}-\lambda_{n^{\prime}}){\rm
Re}\,\\{{\mathcal{O}}_{n^{\prime}n}^{\Gamma}(x){\mathcal{O}}_{nn^{\prime}}^{\Gamma}(0)\\}\bigg{\rangle}_{\beta}\,,$
(77)

and made explicit their dependence on the bare fermion mass, the volume, and
the UV regulator. It is convenient to separate the cases
$\lambda_{n}=\pm\lambda_{n^{\prime}}$ from the rest, and set151515Strictly
speaking, in the definition of $\bar{C}^{\,\Gamma}_{V,\,\Lambda}$ the
condition $n\neq\pm n^{\prime}$ should read
$\lambda_{n}\neq\pm\lambda_{n^{\prime}}$, while contributions from distinct
but exactly degenerate modes, $\lambda_{n}=\lambda_{n^{\prime}}$ with $n\neq
n^{\prime}$, should be included in $C_{s\,V,\,\Lambda}^{\Gamma}$. However,
zero modes do not contribute in the thermodynamic limit (see below), and
degenerate nonzero modes are expected to appear only on a set of
configurations of zero measure, and can be ignored. See footnote 20 for
further comments.

$\displaystyle C^{\Gamma}_{s\,V,\,\Lambda}(\lambda;x;m_{B})$
$\displaystyle\equiv\bigg{\langle}\sum_{n}\delta(\lambda-\lambda_{n}){\mathcal{O}}_{nn}^{\Gamma}(x){\mathcal{O}}_{nn}^{\Gamma}(0)\bigg{\rangle}_{\beta}\,,$
(78)
$\displaystyle\bar{C}^{\,\Gamma}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})$
$\displaystyle\equiv\bigg{\langle}\sum_{\begin{subarray}{c}n,n^{\prime}\\\
n\neq\pm
n^{\prime}\end{subarray}}\delta(\lambda-\lambda_{n})\delta(\lambda^{\prime}-\lambda_{n^{\prime}}){\rm
Re}\,\\{{\mathcal{O}}_{n^{\prime}n}^{\Gamma}(x){\mathcal{O}}_{nn^{\prime}}^{\Gamma}(0)\\}\bigg{\rangle}_{\beta}\,,$

in terms of which one has

$\displaystyle C^{\gamma_{5}}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})$
$\displaystyle=\delta(\lambda+\lambda^{\prime})C^{\mathbf{1}}_{s\,V,\,\Lambda}(\lambda;x;m_{B})+\delta(\lambda-\lambda^{\prime})C^{\gamma_{5}}_{s\,V,\,\Lambda}(\lambda;x;m_{B})$
(79)
$\displaystyle\phantom{=}+\bar{C}^{\,\gamma_{5}}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})\,.$

In the language of random Hamiltonians (see Section 3.1), the quantities
defined in Eqs. (77) and (78) are a type of Green’s functions measuring the
correlation between eigenmodes. Using Eq. (75) one finds the following
symmetry relations,

$\displaystyle\bar{C}^{\,\Gamma}_{V,\,\Lambda}(-\lambda,-\lambda^{\prime};x;m_{B})$
$\displaystyle=\bar{C}^{\,\Gamma}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})\,,$
$\displaystyle\bar{C}^{\,\Gamma}_{V,\,\Lambda}(\lambda^{\prime},\lambda;x;m_{B})$
$\displaystyle=\bar{C}^{\,\Gamma}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})\,,$
(80)
$\displaystyle\bar{C}^{\,\Gamma}_{V,\,\Lambda}(-\lambda,\lambda^{\prime};x;m_{B})$
$\displaystyle=\bar{C}^{\,\gamma_{5}\Gamma}_{V,\,\Lambda}(\lambda,\lambda^{\prime};x;m_{B})\,,$
$\displaystyle C^{\Gamma}_{s\,V,\,\Lambda}(-\lambda;x;m_{B})$
$\displaystyle=C^{\Gamma}_{s\,V,\,\Lambda}(\lambda;x;m_{B})\,.$

As they hold in any volume, these relations will hold also in the
thermodynamic limit $V\to\infty$.

###  5.2 Large-volume limit

In this Subsection I discuss the large-volume limit of the spectral
correlators, and show that here the localisation properties of the Dirac
eigenmodes play a crucial role: while they do not affect the large-volume
behaviour of $\bar{C}^{\,\Gamma}_{V,\,\Lambda}$, they strongly affect whether
$C^{\Gamma}_{s\,V,\,\Lambda}$ survives the infinite-volume limit or not.

In order to see how the localisation properties of the modes affect their
contribution to the spectral correlators, Eq. (78), notice first the exact
bounds

$|{\mathcal{O}}^{\Gamma}_{n^{\prime}n}(x)|^{2}=|(\psi_{n^{\prime}}(x),\Gamma\psi_{n}(x))|^{2}\leq\|\Gamma\|^{2}\|\psi_{n^{\prime}}(x)\|^{2}\,\|\psi_{n}(x)\|^{2}\,,$
(81)

where the matrix norm $\|\Gamma\|=1$ for $\Gamma=\mathbf{1},\gamma_{5}$, and

$\displaystyle\left|\langle{\mathcal{O}}^{\Gamma}_{n^{\prime}n}(x){\mathcal{O}}^{\Gamma}_{nn^{\prime}}(0)\rangle_{\beta}\right|$
$\displaystyle\leq\langle\left|{\mathcal{O}}^{\Gamma}_{n^{\prime}n}(x)\right|\left|{\mathcal{O}}^{\Gamma}_{nn^{\prime}}(0)\right|\rangle_{\beta}$
(82)
$\displaystyle\leq\|\Gamma\|^{2}\langle\|\psi_{n^{\prime}}(x)\|\,\|\psi_{n}(x)\|\,\|\psi_{n^{\prime}}(0)\|\,\|\psi_{n}(0)\|\rangle_{\beta}$
$\displaystyle\leq\|\Gamma\|^{2}\sqrt{\langle\|\psi_{n^{\prime}}(x)\|^{2}\,\|\psi_{n}(x)\|^{2}\rangle_{\beta}}\sqrt{\langle\|\psi_{n^{\prime}}(0)\|^{2}\,\|\psi_{n}(0)\|^{2}\rangle_{\beta}}$
$\displaystyle=\|\Gamma\|^{2}\langle\|\psi_{n^{\prime}}(x)\|^{2}\,\|\psi_{n}(x)\|^{2}\rangle_{\beta}\,.$

These follow from the Cauchy-Schwarz inequality, on the third line applied to
the expectation value $\langle\ldots\rangle_{\beta}$ which is defined by a
path-integral with positive-definite integration measure. Translation
invariance is used in the last passage. In a first approximation, one can
treat these bounds more loosely as estimates of the magnitude of
${\mathcal{O}}^{\Gamma}_{n^{\prime}n}$ and their correlation functions, in
order to obtain the volume dependence of the individual contributions to Eq.
(78). While these estimates should be supplemented by suitable factors in
order to take into account the dependence on $\Gamma$ and, more importantly,
on the correlation between modes $n$ and $n^{\prime}$ and on the distance
between $0$ and $x$, they should suffice if all one is interested in is their
volume dependence.

Consider first the case $n\neq\pm n^{\prime}$. If modes $n$ near $\lambda$
have fractal dimension $\alpha(\lambda)$, i.e., are mostly supported in
regions whose size scales like $V^{\alpha(\lambda)}$, then
$\|\psi_{n}(x)\|^{2}\sim V^{-\alpha(\lambda)}$ inside the supporting region
while being negligible outside. Translation invariance implies that the
probability of finding a given spacetime point inside the support of the mode
is $V^{\alpha(\lambda)}/V$. Finally, the correlation between modes is expected
to decrease as $|\lambda_{n}-\lambda_{n^{\prime}}|$ increases, and so it will
be small for most pairs of modes. One can then estimate

$\langle\|\psi_{n^{\prime}}(x)\|^{2}\,\|\psi_{n}(x)\|^{2}\rangle_{\beta}\sim\left(\frac{1}{V^{\alpha(\lambda^{\prime})}}\frac{V^{\alpha(\lambda^{\prime})}}{V}\right)\left(\frac{1}{V^{\alpha(\lambda)}}\frac{V^{\alpha(\lambda)}}{V}\right)=\frac{1}{V^{2}}\,,$
(83)

irrespectively of the localisation properties of the modes. From the last line
in Eq. (82) one then estimates

$\left|\langle{\mathcal{O}}^{\Gamma}_{n^{\prime}n}(x){\mathcal{O}}^{\Gamma}_{nn^{\prime}}(0)\rangle_{\beta}\right|\sim\frac{1}{V^{2}}\,.$
(84)

Since $\bar{C}_{V,\,\Lambda}^{\,\Gamma}$ involves a double sum over modes, and
since the number of modes per unit spectral interval typically scales like the
volume,161616The case in which the growth is faster than $O(V)$, leading to
points in the spectrum where the spectral density has an integrable
singularity in the infinite-volume limit (van Hove singularity), is not
considered here. one has $O(V^{2})$ contributions of order $O(1/V^{2})$. One
then expects
$\bar{C}_{\Lambda}^{\,\Gamma}(\lambda,\lambda^{\prime};x;m_{B})\equiv\lim_{V\to\infty}\bar{C}_{V,\,\Lambda}^{\,\Gamma}(\lambda,\lambda^{\prime};x;m_{B})$
to be nonzero as long as both spectral densities $\rho_{B}(\lambda)$ and
$\rho_{B}(\lambda^{\prime})$ are nonzero, independently of the localisation
properties of modes near $\lambda$ and $\lambda^{\prime}$. The bare spectral
density is defined as

$\rho_{B,V}(\lambda)\equiv\frac{1}{\beta
V}\left\langle\sideset{}{{}^{\prime}}{\sum}_{n}\delta(\lambda-\lambda_{n})\right\rangle_{\beta}\,,\qquad\rho_{B}(\lambda)\equiv\lim_{V\to\infty}\rho_{B,V}(\lambda)\,,$
(85)

with $\rho_{B,V}$ the bare spectral density in a finite volume. Here
$\sum_{n}^{\prime}$ denotes the sum over nonzero modes only. Exact zero modes
are explicitly excluded even though they would not contribute in the
thermodynamic limit, since their number scales only like $N_{0}\sim\sqrt{V}$
at large volume. As a further consequence, since the estimate Eq. (84) applies
in particular to pairs of distinct zero modes, and pairs of a zero and a
nonzero mode, one finds that the corresponding total contributions scale
respectively like $V/V^{2}$ and $V\sqrt{V}/V^{2}$, and so contributions
involving zero modes can be dropped from $\bar{C}_{V,\,\Lambda}^{\,\Gamma}$ in
Eq. (78).

The situation is different in the case $n=\pm n^{\prime}$, where correlations
cannot be neglected. Using the last line in Eq. (82), one can estimate for
modes near $\lambda$ that

$\left|\langle{\mathcal{O}}^{\Gamma}_{nn}(x){\mathcal{O}}^{\Gamma}_{nn}(0)\rangle_{\beta}\right|\sim\langle\|\psi_{n}(x)\|^{4}\rangle_{\beta}=\frac{1}{\beta
V}\langle{\rm IPR}_{n}\rangle_{\beta}\sim\frac{1}{VV^{\alpha(\lambda)}}\,,$
(86)

where translation invariance was used. Since a single sum over modes appears
in $C_{s\,V,\,\Lambda}^{\Gamma}$, there are $O(V)$ such contributions in any
spectral region with a finite density of modes, and so one expects
$C_{s\,V,\,\Lambda}^{\Gamma}(\lambda;x)\to 0$ in the large-volume limit except
if modes near $\lambda$ are localised, in which case one expects to find a
finite value. Equation (86) can be turned into a rigorous bound showing that
$C_{s\,V,\,\Lambda}^{\Gamma}(\lambda;x)$ vanishes in the large-volume limit if
the fractal dimension of modes near $\lambda$ is nonzero, i.e., if they are
delocalised. From the last line in Eq. (82), in fact, one has171717The Dirac
deltas in Eq. (87) can be handled rigorously, without changing the result, by
integrating first over small intervals around $\lambda,\lambda^{\prime}$,
using then Eq. (82), and finally taking the limit of infinitesimal intervals.

$\displaystyle|C_{s\,V,\,\Lambda}^{\Gamma}(\lambda;x;m_{B})|$
$\displaystyle\leq\bigg{\langle}\sum_{n}\delta(\lambda-\lambda_{n})|{\mathcal{O}}_{nn}^{\Gamma}(x)||{\mathcal{O}}_{nn}^{\Gamma}(0)|\bigg{\rangle}_{\beta}\leq\bigg{\langle}\sum_{n}\delta(\lambda-\lambda_{n})\|\psi_{n}(x)\|^{4}\bigg{\rangle}_{\beta}$
(87) $\displaystyle=\frac{1}{\beta
V}\bigg{\langle}\sum_{n}\delta(\lambda-\lambda_{n}){\rm
IPR}_{n}\bigg{\rangle}_{\beta}=\rho_{B,V}(\lambda)\overline{\rm
IPR}_{V}(\lambda)+\delta(\lambda)\frac{N_{0}}{\beta V}\overline{\rm
IPR}_{V}^{0}\,,$

where I used translation invariance, and where

$\overline{\rm IPR}_{V}(\lambda)\equiv\frac{1}{\beta
V\rho_{B,V}(\lambda)}\bigg{\langle}\displaystyle\sideset{}{{}^{\prime}}{\sum}_{n}\delta(\lambda-\lambda_{n}){\rm
IPR}_{n}\bigg{\rangle}_{\beta}\,,\qquad\overline{\rm
IPR}_{V}^{0}\equiv\frac{1}{N_{0}}\bigg{\langle}\sum_{\begin{subarray}{c}n_{0}\\\
\lambda_{n_{0}}=0\end{subarray}}{\rm IPR}_{n_{0}}\bigg{\rangle}_{\beta}\,,$
(88)

is the average ${\rm IPR}$ computed locally in the spectrum.181818The
dependence of $\overline{\rm IPR}_{V}$ on $\Lambda$ and $m_{B}$ is irrelevant
here and has been suppressed. In computing $\overline{\rm IPR}_{V}^{0}$ one
should average over the degenerate zero-mode subspace using the procedure of
Ref. Baranka:2021san , treating separately the two chiralities, but this would
not change the fact that zero modes do not contribute here (see below).
Assuming that modes have fractal dimension $\alpha(\lambda)$ near $\lambda$,
one has $\overline{\rm IPR}_{V}(\lambda)\sim V^{-\alpha(\lambda)}$. If modes
are delocalised near $\lambda\neq 0$, i.e., $\alpha(\lambda)>0$, one finds
that $C_{s\,V,\,\Lambda}^{\Gamma}$ vanishes in the thermodynamic limit. On the
other hand, if modes are localised then $\alpha(\lambda)=0$ and
$C_{s\,V,\,\Lambda}^{\Gamma}$ need not vanish, and as shown above one expects
that it does not.191919While capturing correctly the overall volume
dependence, the simple estimate Eq. (86) entirely misses the $x$-dependence of
the spectral correlators. Taking localisation literally, one would find
$\left|\langle{\mathcal{O}}^{\Gamma}_{nn}(x){\mathcal{O}}^{\Gamma}_{nn}(0)\rangle_{\beta}\right|\sim\frac{1}{\ell^{3}}$
for spatial separation smaller than the typical localisation length $\ell$,
and zero otherwise. A more precise analysis is carried out in Appendix E using
a more realistic exponential envelope of localised modes, that leads to expect
an exponential suppression in $|\vec{x}|$ rather than strictly finite support.
One then expects $C_{{\rm
loc}\,\Lambda}^{\Gamma}\equiv\lim_{V\to\infty}C_{s\,V,\,\Lambda}^{\Gamma}$ to
be finite where a finite density of localised modes is present and only there,
as signalled by the subscript “${\rm loc}$”. An explicit calculation shows
that this is the case (see Section 5.6 below). Notice that since the last term
in Eq. (87) scales like $\sqrt{V}/V$, exact zero modes give no contribution to
$C_{s\,V,\,\Lambda}^{\Gamma}$ in the thermodynamic limit independently of
their localisation properties.202020Exactly degenerate but distinct nonzero
modes could also contribute to $C_{s\,V,\,\Lambda}^{\Gamma}$. For these modes
the estimate Eq. (84) applies so that their contribution is suppressed like
$N_{\rm deg}V/V^{2}=N_{\rm deg}V^{-1}$ in the thermodynamic limit, with
$N_{\rm deg}$ their typical (possibly $\lambda$-dependent) degeneracy, unless
they are strongly spatially correlated. In this case, the estimate Eq. (86)
applies instead, leading to a total contribution of order $N_{\rm
deg}V^{-\alpha}$. These modes can then be relevant only if localised, strongly
spatially correlated, and appearing on a set of configurations of finite
measure. This seems very unlikely to happen.

Summarising, in the thermodynamic limit one finds that
$\bar{C}_{\Lambda}^{\,\Gamma}$ is nonzero where the spectral density is
nonzero, while $C_{{\rm loc}\,\Lambda}^{\Gamma}$ is nonzero only in spectral
regions where modes are localised.212121As pointed out in Section 3, localised
and delocalised modes usually do not coexist in the same spectral region.
However, the qualitative estimate given above under Eq. (88) does not depend
on this non-coexistence assumption. If for some system one could separate
localised and delocalised modes in a given spectral region and define the
density $\rho_{\rm loc}(\lambda)$ of localised modes, then $C_{{\rm
loc}\,\Lambda}^{\Gamma}$ would be nonzero where $\rho_{\rm loc}\neq 0$. In the
infinite-volume limit Eq. (76) then reduces to

$\displaystyle-{\mathcal{G}}_{PP\,B}(x)$
$\displaystyle=\int_{0}^{\Lambda}d\lambda\,\left(\frac{C_{{\rm
loc}\,\Lambda}^{\mathbf{1}}(\lambda;x;m_{B})}{\lambda^{2}+m_{B}^{2}}+\frac{(m_{B}^{2}-\lambda^{2})C_{{\rm
loc}\,\Lambda}^{\gamma_{5}}(\lambda;x;m_{B})}{(\lambda^{2}+m_{B}^{2})^{2}}\right)$
(89)
$\displaystyle\phantom{=}+\frac{1}{2}\int_{-\Lambda}^{\Lambda}d\lambda\int_{-\Lambda}^{\Lambda}d\lambda^{\prime}\,\frac{(m_{B}^{2}-\lambda\lambda^{\prime})\bar{C}_{\Lambda}^{\gamma_{5}}(\lambda,\lambda^{\prime};x;m_{B})}{(\lambda^{2}+m_{B}^{2})(\lambda^{\prime\,2}+m_{B}^{2})}\,,$

having used the symmetry relations Eq. (80).

###  5.3 Renormalisation

The bare pseudoscalar-pseudoscalar correlator needs to be suitably
renormalised before the UV cutoff $\Lambda$ is removed. As discussed in detail
in Appendix C, both additive and multiplicative renormalisation are required,
and the renormalised pseudoscalar-pseudoscalar correlator reads

${\mathcal{G}}_{PP}(x)=\lim_{\Lambda\to\infty}Z_{m}^{2}\left[{\mathcal{G}}_{PP\,B}(x)-{\rm
CT}_{PP}(x)\right]\,,$ (90)

where ${\rm CT}_{PP}(x)$ are divergent contact terms and $Z_{m}$ is the mass
renormalisation constant, with the renormalised mass defined as
$m_{B}=Z_{m}m$. Divergent contact terms originate from large Dirac eigenvalues
in Eq. (89). However, they are polynomial in the fermion mass and so drop from
${\mathcal{R}}$ in the chiral limit, and therefore can essentially be ignored
as far as the Ward-Takahashi identity, Eqs. (29) and (32), is concerned. More
generally, all the contributions to ${\mathcal{R}}$ coming from large
eigenvalues vanish in the chiral limit, including any finite term remaining
after the subtraction procedure, since they yield at most a constant term in
the pseudoscalar-pseudoscalar correlator as $m\to 0$. Multiplicative
renormalisation, on the other hand, is required also in the chiral limit. To
identify the divergent contact terms, one splits the integrals in Eq. (89) at
a suitably chosen $m$-independent subtraction scale. To disentangle additive
and multiplicative divergences, it is convenient to work with renormalised
spectral correlators. This also allows one to see more clearly how the
remaining finite terms behave in the chiral limit.

The procedure is most easily illustrated in the case of the chiral condensate.
The bare chiral condensate $\Sigma_{B}$ is obtained from the spectral density,
Eq. (85), via the Banks-Casher relation Banks:1979yr ,

$-\Sigma_{B}=\int_{0}^{\Lambda}d\lambda\,\rho_{B}(\lambda,m_{B})\frac{2m_{B}}{\lambda^{2}+m_{B}^{2}}\,.$
(91)

Here the dependence of $\rho_{B}$ on the bare mass has been made explicit. The
spectral density grows like $\lambda^{3}$ at large $\lambda$, thus leading to
quadratic and logarithmic additive divergences, which originate from the
mixing of the scalar density with the identity operator (see Appendix C). A
possible renormalisation scheme to take care of them is the following (see,
e.g, Ref. Leutwyler:1992yt ). One splits the integral at some (mass-
independent) subtraction scale $\mu_{B}$, and for $\lambda>\mu_{B}$ one
expands the denominator in powers of $m_{B}^{2}/\lambda^{2}$, obtaining

$\displaystyle-\Sigma_{B}$
$\displaystyle=\int_{0}^{\mu_{B}}d\lambda\,\rho_{B}(\lambda,m_{B})\frac{2m_{B}}{\lambda^{2}+m_{B}^{2}}+\int_{\mu_{B}}^{\Lambda}d\lambda\,\rho_{B}(\lambda,m_{B})\frac{2m_{B}^{5}}{\lambda^{4}(\lambda^{2}+m_{B}^{2})}$
(92)
$\displaystyle\phantom{=}+\int_{\mu_{B}}^{\Lambda}d\lambda\,\rho_{B}(\lambda,m_{B})\left(\frac{2m_{B}}{\lambda^{2}}-\frac{2m_{B}^{3}}{\lambda^{4}}\right)=\Sigma_{B}^{(1)}+\Sigma_{B}^{(2)}\,.$

The integrals on the first line, that define $\Sigma_{B}^{(1)}$, are formally
convergent as $\Lambda\to\infty$, i.e., if one ignores that the bare spectral
density is also $\Lambda$-dependent, and furthermore are made finite by
multiplicative renormalisation. In fact, multiplying by $Z_{m}$ one finds
after a change of variables

$\displaystyle Z_{m}\Sigma_{B}^{(1)}$
$\displaystyle=\int_{0}^{\frac{\mu_{B}}{Z_{m}}}d\lambda\,Z_{m}\rho_{B}(Z_{m}\lambda,Z_{m}m)\frac{2m}{\lambda^{2}+m^{2}}+\int_{\frac{\mu_{B}}{Z_{m}}}^{\frac{\Lambda}{Z_{m}}}d\lambda\,Z_{m}\rho_{B}(Z_{m}\lambda,Z_{m}m)\frac{2m^{5}}{\lambda^{4}(\lambda^{2}+m^{2})}\,.$
(93)

As it was shown in Refs. DelDebbio:2005qa ; Giusti:2008vb , the quantity

$\rho(\lambda,m)\equiv\lim_{\Lambda\to\infty}Z_{m}\rho_{B}(Z_{m}\lambda,Z_{m}m)$
(94)

is finite, and so Eq. (93) has a finite limit as $\Lambda\to\infty$ if
$\mu\equiv\mu_{B}/Z_{m}$ is kept fixed. Notice that since $Z_{m}$ depends on
$\Lambda$ only logarithmically, the (bare) subtraction scale
$\mu_{B}=Z_{m}\mu$ has to depend logarithmically on the cutoff, while
$\Lambda/Z_{m}$ still diverges as $\Lambda\to\infty$. The terms on the second
line of Eq. (92), that define $\Sigma_{B}^{(2)}$, remain divergent also after
multiplication by $Z_{m}$, and need to be subtracted. One then defines the
renormalised condensate as

$\displaystyle-\Sigma\equiv\lim_{\Lambda\to\infty}-Z_{m}\Sigma_{B}^{(1)}=\int_{0}^{\mu}d\lambda\,\rho(\lambda,m)\frac{2m}{\lambda^{2}+m^{2}}+\int_{\mu}^{\infty}d\lambda\,\rho(\lambda,m)\frac{2m^{5}}{\lambda^{4}(\lambda^{2}+m^{2})}\,.$
(95)

The second term is at least of order $O(m^{5})$ and vanishes in the chiral
limit. For the first term one finds instead the well-known result Banks:1979yr

$\displaystyle-\Sigma_{*}$ $\displaystyle=\lim_{m\to
0}\rho(0,m)\int_{0}^{\frac{\mu}{m}}dz\,\frac{2}{1+z^{2}}+\lim_{m\to
0}\int_{0}^{\mu}d\lambda\,\frac{2mf(\lambda)}{\lambda^{2}+m^{2}}=\pi\rho(0,0)\,,$
(96)

where $\rho(0,m)\equiv\lim_{\lambda\to 0}\rho(\lambda,m)$ and
$\rho(0,0)\equiv\lim_{m\to 0}\rho(0,m)$, and where
$f(\lambda)=\rho(\lambda,m)-\rho(0,m)$ is assumed to vanish at least as fast
as some power law as $\lambda\to 0$, i.e., $\lambda^{-\gamma}f(\lambda)\to 0$
as $\lambda\to 0$ for some $\gamma>0$. In fact, in this case, the second term
in Eq. (96) is of order at most $m^{\frac{1}{2^{N}}}$ for some sufficiently
large $N$ such that $2^{N}\gamma>1$, as shown in Appendix F [see Eq. (193)].

The same procedure can now be repeated for the pseudoscalar-pseudoscalar
correlator. In this case, after splitting the integrals and multiplying by the
renormalisation constant $Z_{m}^{2}$, one finds

${\mathcal{G}}_{PP\,B}(x)=\sum_{i,j=1}^{2}{\mathcal{G}}^{(ij)}_{PP\,B}(x)\,,$
(97)

with

$\displaystyle-Z_{m}^{2}{\mathcal{G}}_{PP\,B}^{(ij)}(x)$
$\displaystyle=\delta_{ij}\int_{I_{i}}d\lambda\,\frac{Z_{m}C_{{\rm
loc}\,\Lambda}^{\mathbf{1}}(Z_{m}\lambda;x;Z_{m}m)}{\lambda^{2}+m^{2}}$ (98)
$\displaystyle\phantom{=}+\delta_{ij}\int_{I_{i}}d\lambda\,\frac{(m^{2}-\lambda^{2})Z_{m}C_{{\rm
loc}\,\Lambda}^{\gamma_{5}}(Z_{m}\lambda;x;Z_{m}m)}{(\lambda^{2}+m^{2})^{2}}$
$\displaystyle\phantom{=}+\int_{I_{i}}d\lambda\int_{I_{j}}d\lambda^{\prime}\,\frac{(m^{2}+\lambda\lambda^{\prime})Z_{m}^{2}\bar{C}_{\Lambda}^{\mathbf{1}}(Z_{m}\lambda,Z_{m}\lambda^{\prime};x;Z_{m}m)}{(\lambda^{2}+m^{2})(\lambda^{\prime\,2}+m^{2})}$
$\displaystyle\phantom{=}+\int_{I_{i}}d\lambda\int_{I_{j}}d\lambda^{\prime}\,\frac{(m^{2}-\lambda\lambda^{\prime})Z_{m}^{2}\bar{C}_{\Lambda}^{\gamma_{5}}(Z_{m}\lambda,Z_{m}\lambda^{\prime};x;Z_{m}m)}{(\lambda^{2}+m^{2})(\lambda^{\prime\,2}+m^{2})}\,,$

where $I_{1}=[0,\mu]$ and $I_{2}=[\mu,\Lambda/Z_{m}]$. Here the properties Eq.
(80) have been used to restrict the double integral on the second line of Eq.
(89) to the positive part of the spectrum. It is clear that additive
divergences can originate only from ${\mathcal{G}}_{PP\,B}^{(12)}$,
${\mathcal{G}}_{PP\,B}^{(21)}$, and ${\mathcal{G}}_{PP\,B}^{(22)}$. The
symmetry of the integrand under $\lambda\leftrightarrow\lambda^{\prime}$
implies ${\mathcal{G}}^{(12)}_{PP\,B}={\mathcal{G}}^{(21)}_{PP\,B}$. Using an
argument similar to that of Refs. DelDebbio:2005qa ; Giusti:2008vb , I show in
Appendix G that the functions

$\displaystyle C_{{\rm loc}}^{\Gamma}(\lambda;x;m)$
$\displaystyle\equiv\lim_{\Lambda\to\infty}Z_{m}C_{{\rm
loc}\,\Lambda}^{\Gamma}(Z_{m}\lambda;x;Z_{m}m)\,,$ (99)
$\displaystyle\bar{C}^{\,\Gamma}(\lambda,\lambda^{\prime};x;m)$
$\displaystyle\equiv\lim_{\Lambda\to\infty}Z_{m}^{2}\bar{C}_{\Lambda}^{\,\Gamma}(Z_{m}\lambda,Z_{m}\lambda^{\prime};x;Z_{m}m)\,,$

are renormalised, finite quantities. This makes
${\mathcal{G}}_{PP}^{(11)}\equiv\lim_{\Lambda\to\infty}Z_{m}^{2}{\mathcal{G}}_{PP\,B}^{(11)}$
finite in the large-$\Lambda$ limit. It remains only to identify and subtract
the additively divergent contributions to ${\mathcal{G}}_{PP\,B}^{(12)}$ and
${\mathcal{G}}_{PP\,B}^{(22)}$. For the purposes of this paper this need not
be done explicitly: all that matters is that the remaining finite terms stay
finite also in the chiral limit, which is easy to show. Details are provided
in Appendix F. One then concludes that

$Z_{m}^{2}\left({\mathcal{G}}_{PP\,B}^{(12)}(x)+{\mathcal{G}}_{PP\,B}^{(21)}(x)+{\mathcal{G}}_{PP\,B}^{(22)}(x)\right)=Z_{m}^{2}{\rm
CT}_{PP}(x)+F(x;m)+\ldots\,,$ (100)

with omitted terms vanishing as $\Lambda\to\infty$, and $F$ a finite
$\Lambda$-independent quantity that remains finite also as $m\to 0$. One has
then ${\mathcal{G}}_{PP}={\mathcal{G}}_{PP}^{(11)}+F$, and for the quantity of
interest, namely ${\mathcal{R}}_{*}=\lim_{m\to 0}{\mathcal{R}}$ [see Eq. (51)]
one finds ${\mathcal{R}}_{*}=\lim_{m\to 0}2m{\mathcal{G}}_{PP}^{(11)}$, and
the following spectral representation,

$\displaystyle-{\mathcal{R}}_{*}(x)$ $\displaystyle=\lim_{m\to
0}2m\int_{0}^{\mu}d\lambda\,\left(\frac{C_{{\rm
loc}}^{\mathbf{1}}(\lambda;x;m)}{\lambda^{2}+m^{2}}+\frac{(m^{2}-\lambda^{2})C_{{\rm
loc}}^{\gamma_{5}}(\lambda;x;m)}{(\lambda^{2}+m^{2})^{2}}\right)$ (101)
$\displaystyle\phantom{=\lim_{m\to
0}}+2m\int_{0}^{\mu}d\lambda\int_{0}^{\mu}d\lambda^{\prime}\,\frac{(m^{2}+\lambda\lambda^{\prime})\bar{C}^{\mathbf{1}}(\lambda,\lambda^{\prime};x;m)+(m^{2}-\lambda\lambda^{\prime})\bar{C}^{\gamma_{5}}(\lambda,\lambda^{\prime};x;m)}{(\lambda^{2}+m^{2})(\lambda^{\prime\,2}+m^{2})}\,.$

###  5.4 Renormalisation of the mobility edge

The result Eq. (99) has an important consequence for the renormalisation
properties of the mobility edges. As discussed above in Section 5.2, the
unrenormalised spectral correlators $C_{{\rm
loc}\,\Lambda}^{\Gamma}(\lambda;x;m_{B})$ have support only in disjoint
regions where modes are localised, separated by mobility edges
$\lambda^{(i)}_{c\,B}$ from regions where modes are instead delocalised. Since
renormalisation consists only in a multiplicative factor and a rescaling of
$\lambda$ and $m_{B}$, the support of $C_{{\rm loc}}^{\Gamma}(\lambda;x;m)$ is
still made of disjoint regions delimited by the renormalised mobility edges
$\lambda^{(i)}_{c}=\lim_{\Lambda\to\infty}Z_{m}^{-1}\lambda^{(i)}_{c\,B}$. In
other words, the mobility edges renormalise like the fermion mass, so that the
ratios $\lambda_{c\,B}^{(i)}/m_{B}$ are renormalisation-group-invariant
quantities free from UV divergences, i.e., the equality

$\frac{\lambda_{c\,B}^{(i)}}{m_{B}}=\frac{\lambda_{c}^{(i)}}{m}+o(\Lambda^{0})$
(102)

holds up to corrections that vanish as the UV regulator is removed. This had
been suggested before Kovacs:2012zq , and was supported by numerical results
on the lattice Kovacs:2012zq ; Cardinali:2021fpu , but had not been proved
yet. Details are provided in Appendix G.

###  5.5 Chiral limit

The final step in order to obtain ${\mathcal{R}}_{*}$ is to determine the
behaviour of the various terms appearing in Eq. (101) as $m\to 0$. To proceed
it is necessary to make assumptions about the position of the mobility
edge(s), if present. It is assumed from now on that modes are localised in the
range $[0,Z_{m}\lambda_{c}]$ in the UV-regulated theory, and so in the range
$[0,\lambda_{c}]$ of the “renormalised” spectrum, with
$\lambda_{c}=\lambda_{c}(m)$ the renormalised mobility edge. It is also
assumed that, if other regions with localised modes are present, then these
are found above some renormalised lower mobility edge $\lambda_{c}^{\prime}$
and remain separated from $\lambda=0$ in the chiral limit.222222Localised
modes have been observed at the high end ($|\lambda|>\lambda_{c}^{\prime}$) of
the spectrum of staggered fermions in pure gauge $\mathbb{Z}_{2}$ gauge theory
in 2+1 dimensions on the lattice Baranka:2021san , and it is likely that this
feature is found also in other theories. However, these are ultraviolet modes
that should not affect the continuum physics. In particular, it is unlikely
that $\lambda_{c}^{\prime}$ reaches down to the origin in the chiral limit.
With these assumptions, Eq. (101) becomes

${\mathcal{R}}_{*}(x)={\mathcal{R}}_{*}^{(1)}(x)+{\mathcal{R}}_{*}^{(2)}(x)+{\mathcal{R}}_{*}^{(3)}(x)\,,$
(103)

where

$-{\mathcal{R}}_{*}^{(1)}(x)=\lim_{m\to
0}\phantom{+}2m\int_{0}^{\lambda_{c}}d\lambda\,\left(\frac{C_{{\rm
loc}}^{\mathbf{1}}(\lambda;x;m)}{\lambda^{2}+m^{2}}+\frac{(m^{2}-\lambda^{2})C_{{\rm
loc}}^{\gamma_{5}}(\lambda;x;m)}{(\lambda^{2}+m^{2})^{2}}\right)$ (104)

and232323The upper limit of integration is set to $\mu$ for generality, and
does not mean that modes are localised in the whole interval
$[\lambda_{c}^{\prime},\mu]$.

$-{\mathcal{R}}_{*}^{(2)}(x)=\lim_{m\to
0}2m\int_{\lambda_{c}^{\prime}}^{\mu}d\lambda\,\left(\frac{C_{{\rm
loc}}^{\mathbf{1}}(\lambda;x;m)}{\lambda^{2}+m^{2}}+\frac{(m^{2}-\lambda^{2})C_{{\rm
loc}}^{\gamma_{5}}(\lambda;x;m)}{(\lambda^{2}+m^{2})^{2}}\right)$ (105)

receive contributions only from localised modes, while

$-{\mathcal{R}}_{*}^{(3)}(x)=\lim_{m\to
0}2m\int_{0}^{\mu}d\lambda\int_{0}^{\mu}d\lambda^{\prime}\,\frac{(m^{2}+\lambda\lambda^{\prime})\bar{C}^{\mathbf{1}}(\lambda,\lambda^{\prime};x;m)+(m^{2}-\lambda\lambda^{\prime})\bar{C}^{\gamma_{5}}(\lambda,\lambda^{\prime};x;m)}{(\lambda^{2}+m^{2})(\lambda^{\prime\,2}+m^{2})}$
(106)

receives contributions from both localised and delocalised modes. It is shown
in Appendix F that in the chiral limit the integrals appearing in
${\mathcal{R}}_{*}^{(1)}$, Eq. (104), behave as follows as functions of $m$,

$\displaystyle\int_{0}^{\lambda_{c}}d\lambda\,\frac{C_{{\rm
loc}}^{\mathbf{1}}(\lambda;x;m)}{\lambda^{2}+m^{2}}$
$\displaystyle=\frac{1}{m}\left(\arctan\left(\frac{\lambda_{c}}{m}\right)\,C_{{\rm
loc}}^{\mathbf{1}}(0;x;0)+o(m^{0})\right)\,,$ (107)
$\displaystyle\int_{0}^{\lambda_{c}}d\lambda\,\frac{(m^{2}-\lambda^{2})C_{{\rm
loc}}^{\gamma_{5}}(\lambda;x;m)}{(\lambda^{2}+m^{2})^{2}}$
$\displaystyle=\frac{1}{m}\left(\frac{\frac{\lambda_{c}}{m}}{1+\frac{\lambda_{c}^{2}}{m^{2}}}C_{{\rm
loc}}^{\gamma_{5}}(0;x;0)+o(m^{0})\right)\,,$

so possibly diverging like $1/m$ and leading to a finite ${\mathcal{R}}_{*}$,
while the integrals appearing in ${\mathcal{R}}_{*}^{(2,3)}$ tend to constants
in the chiral limit, and so give no contribution to ${\mathcal{R}}_{*}$. These
results are valid under the rather mild technical assumptions that finite
limits exist for the spectral correlators as $\lambda\to 0$ and/or
$\lambda^{\prime}\to 0$, and that such limits are approached at least as fast
as some power law, and moreover that the resulting quantities have a finite
limit as $m\to 0$. One then obtains the main result of this Subsection,

$\displaystyle-{\mathcal{R}}_{*}(x)$ $\displaystyle=\pi\xi C_{{\rm
loc}}^{\mathbf{1}}(0;x;0)+\eta C_{{\rm loc}}^{\gamma_{5}}(0;x;0)\,,$ (108)

where

$\xi\equiv\frac{2}{\pi}\arctan\kappa\,,\qquad\eta\equiv\frac{2\kappa}{1+\kappa^{2}}\,,\qquad\kappa\equiv\lim_{m\to
0}\frac{\lambda_{c}(m)}{m}\,.$ (109)

Notice that $\kappa$ is renormalisation-group invariant. Notice also that
$C_{{\rm loc}}^{\Gamma}(0;x;0)$ are obtained using the following order of
limits, starting from the bare, finite-volume spectral correlators,

$C_{{\rm loc}}^{\Gamma}(0;x;0)=\lim_{m\to 0}\lim_{\lambda\to
0}\lim_{\Lambda\to\infty}\lim_{V\to\infty}Z_{m}C_{s\,V,\,\Lambda}^{\Gamma}(Z_{m}\lambda;x;Z_{m}m_{B})\,.$
(110)

Finally, notice that the subtraction scale $\mu$ does not affect the final
result, as it should be.

Assuming that $C_{{\rm loc}}^{\Gamma}(0;x;0)$ is nonzero in the chiral limit,
there are three possible scenarios depending on how the mobility edge scales
with $m$ in the chiral limit.

#### (i)

If $\lambda_{c}$ vanishes faster than $m$, then $\kappa\to 0$. In this case
one finds ${\mathcal{R}}_{*}=0$, so that the presence of localised near-zero
modes does not affect the Ward-Takahashi identity Eq. (29). If
$\lambda_{c}\sim m^{\delta+1}$, then $\kappa\sim m^{\delta}$, and (i-a) if
$\delta\geq 1$, the pseudoscalar-pseudoscalar correlator remains finite in the
chiral limit, while (i-b) if $0<\delta<1$ it develops an infrared divergence
$1/m^{1-\delta}$.242424If $\delta\geq 1$, no divergence can appear in the
terms omitted in Eq. (107), which are subleading as long as $\delta>0$, see
Eq. (204) and subsequent discussion in Appendix F.

#### (ii)

If $\lambda_{c}$ vanishes as fast as $m$, then $\kappa\to\text{constant}$. In
this case the pseudoscalar-pseudoscalar correlator develops an infrared
divergence $1/m$, and one finds ${\mathcal{R}}_{*}\neq 0$. One finds for the
coefficient of the first term $\pi\xi<\pi$, and the second term is finite.

#### (iii)

If $\lambda_{c}$ vanishes more slowly than $m$, including not vanishing at
all,252525The quite unlikely case of $\lambda_{c}$ diverging in the chiral
limit is contained in case (iii). In fact, below some value of $m$ one would
find $\lambda_{c}>\mu$, and so $\lambda_{c}$ would be replaced by the cutoff
$\mu$ in the calculations above. then $\kappa\to\infty$. Also in this case the
pseudoscalar-pseudoscalar correlator develops an infrared divergence $1/m$,
and one finds ${\mathcal{R}}_{*}\neq 0$. The coefficient of the first term is
maximal and equal to $\pi$ in this case, while the second term vanishes.

###  5.6 Anomalous remnant

The anomalous remnant ${\rm R}_{*}(0)$ is obtained by integrating
${\mathcal{R}}_{*}(x)$ over Euclidean spacetime. If one can exchange the order
of integration and of the sequence of limits appearing in Eq. (110), then the
calculation is straightforward. Set

$\displaystyle-{\rm R}_{*}(0)$
$\displaystyle=-\int_{\beta}d^{4}x\,{\mathcal{R}}_{*}(x)=\pi\xi
I_{*}^{\mathbf{1}}+\eta I_{*}^{\gamma_{5}}\,,$ (111)

where [see Eqs. (78) and (99)]

$\displaystyle I_{*}^{\Gamma}$ $\displaystyle=\int_{\beta}d^{4}x\,\lim_{m\to
0}\lim_{\lambda\to 0}C_{{\rm loc}}^{\Gamma}(\lambda;x;m)$ (112)
$\displaystyle=\int_{\beta}d^{4}x\,\lim_{m\to 0}\lim_{\lambda\to
0}\lim_{\Lambda\to\infty}\lim_{V\to\infty}Z_{m}\sum_{n}\left\langle\delta(Z_{m}\lambda-\lambda_{n}){\mathcal{O}}_{nn}^{\Gamma}(x){\mathcal{O}}_{nn}^{\Gamma}(0)\right\rangle_{\beta}\,,$

and denote $\textrm{Lim}\equiv\lim_{m\to 0}\lim_{\lambda\to
0}\lim_{\Lambda\to\infty}\lim_{V\to\infty}$. Under the interchangeability
assumption made above one finds

$\displaystyle I_{*}^{\mathbf{1}}$
$\displaystyle=\textrm{Lim}\,Z_{m}\sum_{n}\left\langle\delta(Z_{m}\lambda-\lambda_{n})\int_{\beta}d^{4}x\,{\mathcal{O}}_{nn}^{\mathbf{1}}(x){\mathcal{O}}_{nn}^{\mathbf{1}}(0)\right\rangle_{\beta}$
(113)
$\displaystyle=\textrm{Lim}\,Z_{m}\sum_{n}\left\langle\delta(Z_{m}\lambda-\lambda_{n}){\mathcal{O}}_{nn}^{\mathbf{1}}(0)\right\rangle_{\beta}=\textrm{Lim}\,Z_{m}\frac{1}{\beta
V}\sum_{n}\left\langle\delta(Z_{m}\lambda-\lambda_{n})\right\rangle_{\beta}$
$\displaystyle=\textrm{Lim}\,Z_{m}\rho_{B,V}(Z_{m}\lambda,Z_{m}m)=\lim_{m\to
0}\lim_{\lambda\to
0}\lim_{\Lambda\to\infty}Z_{m}\rho_{B}(Z_{m}\lambda,Z_{m}m)$
$\displaystyle=\lim_{m\to 0}\lim_{\lambda\to 0}\rho(\lambda,m)=\lim_{m\to
0}\rho(0,m)=\rho(0,0)\,,$

where
$\int_{\beta}d^{4}x\,{\mathcal{O}}_{n^{\prime}n}^{\mathbf{1}}(x)=\delta_{n^{\prime}n}$
was used [see Eq. (21)] and exact zero modes were dropped, while

$\displaystyle I_{*}^{\gamma_{5}}$
$\displaystyle=\textrm{Lim}\,Z_{m}\sum_{n}\left\langle\delta(Z_{m}\lambda-\lambda_{n})\int_{\beta}d^{4}x\,{\mathcal{O}}_{nn}^{\gamma_{5}}(x){\mathcal{O}}_{nn}^{\gamma_{5}}(0)\right\rangle_{\beta}=0\,,$
(114)

since
$\int_{\beta}d^{4}x\,{\mathcal{O}}_{n^{\prime}n}^{\gamma_{5}}(x)=\int_{\beta}d^{4}x\,{\mathcal{O}}_{n^{\prime}-n}^{\mathbf{1}}(x)=\delta_{n^{\prime}-n}$.
In conclusion,

$-{\rm R}_{*}(0)=\pi\xi\rho(0,0)\,.$ (115)

This is the main result of this Section: in the presence of a finite density
of localised near-zero modes, and if $\xi\neq 0$, one finds a nonvanishing
anomalous remnant ${\rm R}_{*}(0)$.

Of course, one can consider the spacetime integral in Eq. (112) also away from
the chiral and zero-eigenvalue limits. Under the assumption that the other
limits can be suitably interchanged, the calculation above shows that

$I^{\Gamma}(\lambda;m)=\int_{\beta}d^{4}x\,C_{{\rm
loc}}^{\Gamma}(\lambda;x;m)=\delta_{\Gamma,\mathbf{1}}\rho(\lambda,m)\,,$
(116)

if $\alpha(\lambda)=0$, and zero otherwise, and so $C_{{\rm
loc}}^{\mathbf{1}}$ must be nonzero in the presence of a finite density of
localised modes, as anticipated in Section 5.2.

The result Eq. (115) crucially depends on the possibility of integrating over
spacetime before taking the various limits. For localised modes this is
justified as follows.

  * •

In the finite-volume and UV-regularised theory (e.g., on a finite lattice),
the mode sum is over a finite number of modes, and can certainly be exchanged
with integration.

  * •

Averaging over gauge fields in the regularised setting should cause no
problem. For example, in the lattice regularisation the discretised gauge
field is represented in terms of link variables which are elements of the
gauge group. Averaging then consists in a multiple compact Haar integral over
the gauge group, that can certainly be exchanged with spacetime integration,
i.e., with summation over lattice sites.

  * •

The infinite-volume limit and the removal of the UV cutoff are difficult to
control analytically. Nonetheless, it is expected that at finite fermion mass
the pseudoscalar-pseudoscalar correlator is bounded exponentially as a
function of $|\vec{x}|$ (or at least by some integrable function),
independently of the localisation properties of the modes,262626For localised
modes this is shown below in Appendix E. For delocalised modes, averaging over
gauge fields leads to destructive interference effects among modes, and one
expects an exponential decay of the correlator. uniformly in $V$ and