# The role of the chiral anomaly in polarized deeply inelastic scattering II:
Topological screening and transitions from emergent axion-like dynamics

Andrey Tarasov1,2 Raju Venugopalan3 1Department of Physics, The Ohio State
University, Columbus, OH 43210, USA
2Joint BNL-SBU Center for Frontiers in Nuclear Science (CFNS) at Stony Brook
University, Stony Brook, NY 11794, USA
3Physics Department, Brookhaven National Laboratory, Bldg. 510A, Upton, NY
11973, U.S.A.

###### Abstract

In Tarasov and Venugopalan (2020), we demonstrated that the structure function
$g_{1}(x_{B},Q^{2})$ measured in polarized deeply inelastic scattering (DIS)
is dominated by the triangle anomaly in both the Bjorken limit of large
$Q^{2}$ and the Regge limit of small $x_{B}$. In the worldline formulation of
quantum field theory, the triangle anomaly arises from the imaginary part of
the worldline effective action. We show explicitly how a Wess-Zumino-Witten
term coupling the topological charge density to a primordial isosinglet
${\bar{\eta}}$ arises in this framework. We demonstrate the fundamental role
played by this contribution both in topological mass generation of the
$\eta^{\prime}$ and in the cancellation of the off-forward pole arising from
the triangle anomaly in the proton’s helicity $\Sigma(Q^{2})$. We recover the
striking result by Shore and Veneziano that
$\Sigma\propto\sqrt{\chi^{\prime}(0)}$, where $\chi^{\prime}$ is the slope of
the QCD topological susceptibility in the forward limit. We construct an
axion-like effective action for $g_{1}$ at small $x_{B}$ that describes the
interplay between gluon saturation and the topology of the QCD vacuum. In
particular, we outline the role of “over-the-barrier” sphaleron-like
transitions in spin diffusion at small $x_{B}$. Such topological transitions
can be measured in polarized DIS at a future Electron-Ion Collider.

##  I Introduction

In our previous paper Tarasov and Venugopalan (2020) (henceforth Paper I), we
discussed the role of the chiral anomaly in the inclusive polarized deeply
inelastic scattering (DIS) process

$\displaystyle e(k)+N(P,S)\to e(k^{\prime})+X\,,$ (1)

where $k$ denotes the four-momentum of the lepton ($e$) which scatters off a
polarized target hadron ($N$) with four-momentum
$P=(P^{+},M^{2}/2P^{+},0_{\perp})$ and four-spin $S$ (with $S^{2}=-1$) via the
exchange of a virtual photon $\gamma^{\ast}$ with four-momentum
$q=k-k^{\prime}$. We showed, within a powerful worldline formalism, that the
anomaly provides the dominant contribution to the spin-dependent structure
function $g_{1}(x_{B},Q^{2})$ in the Bjorken and Regge limits of QCD. The
former, for center-of-mass energies $\sqrt{s}\rightarrow\infty$, corresponds
to the DIS kinematics $Q^{2}=-q^{2}\rightarrow\infty$ and the Bjorken variable
$x_{B}=Q^{2}/(2P\cdot q)$ kept fixed; the latter refers to the limit
$x_{B}\rightarrow 0$ and fixed $Q^{2}$.

We further demonstrated in Paper I that the leading perturbative contributions
to $g_{1}$ have a power law divergence in the Mandelstam variable $t$ in the
forward scattering limit $t\rightarrow 0$ in both Bjorken and Regge
asymptotics. We noted that the nonperturbative dynamics which regulates this
divergence is also what resolves the $U_{A}(1)$ problem in QCD. Indeed the
fundamental role of this nonperturbative dynamics was previously argued111Note
that $\frac{1}{9}\Sigma(Q^{2})\approx\int_{0}^{1}dx_{B}\,g_{1}(x_{B},Q^{2})$,
with the $\approx$ sign denoting that a term needs to be added on the l.h.s
corresponding to a linear combination of the isotriplet and isooctet axial
vector charges of the proton. These contributions are weakly dependent on
$Q^{2}$ and we will ignore them henceforth to focus on isosinglet
contributions to $g_{1}(x_{B},Q^{2})$. to be true Jaffe and Manohar (1990);
Veneziano (1989); Shore and Veneziano (1990) for the first moment of
$g_{1}(x_{B},Q^{2})$, the isosinglet quark helicity $\Sigma(Q^{2})$.

In particular, Shore and Veneziano Shore and Veneziano (1990, 1992) showed
that $\Sigma(Q^{2})\propto\sqrt{d\chi/dt}|_{t\rightarrow 0}$, where $\chi$ is
the topological susceptibility of the QCD vacuum. The scale controlling this
quantity is the $\eta^{\prime}$ mass, which is finite even in the chiral limit
Witten (1979); Veneziano (1979). The derivation in Shore and Veneziano (1990,
1992) extensively employed functional chiral Ward identities that follow from
the Wess-Zumino action for QCD coupled to external sources Zumino (1970); Wess
and Zumino (1971).

Further, invoking QCD sum rule arguments to compute $\chi(t)$, Narison, Shore
and Veneziano Narison  _et al._ (1995, 1999) showed that their results for
$\Sigma(Q^{2})$ are in good agreement with HERMES Airapetian  _et al._ (2007a)
and COMPASS Alekseev  _et al._ (2010a) data. A comprehensive review of this
“topological screening” picture of why the quark helicity is anomalously small
(when compared to simple quark model expectations based on the OZI rule Ellis
and Jaffe (1974)) can be found in Shore (2008).

Another nonperturbative approach to computing the proton’s helicity follows ’t
Hooft’s seminal work ’t Hooft (1976, 1986) relating classical instanton
configurations in the QCD vacuum to $U_{A}(1)$ breaking and the origin of the
mass of the $\eta^{\prime}$. This description of the anomaly in the language
on instantons, while by no means unique, is consistent with Veneziano’s
approach Veneziano (1979). For the instanton picture of the quark helicity in
polarized DIS, we refer the reader to Forte (1990); Forte and Shuryak (1991);
Dorokhov  _et al._ (1993); Qian and Zahed (2016); Schäfer and Shuryak (1998).

In this paper, we will develop an alternative formulation of the problem in a
generalization of the worldline framework we discussed in Paper I to include
the coupling of the Dirac fields to scalar, pseudoscalar and axial vector
fields representing low energy degrees of freedom in the QCD effective action.
As we noted in Paper I, the coupling of the isosinglet axial vector current to
low energy dynamics of gauge fields (represented by the topological charge
density) arises from the imaginary part of the worldline effective action. The
generalization of the worldline action to include all possible low energy
degrees of freedom will therefore include additional such imaginary terms that
must be taken into account to fully describe the dynamics of the anomaly.

In a certain sense, as will become apparent, this worldline approach to the
role of the chiral anomaly in proton spin threads a line between the two
aforementioned approaches, the Shore-Veneziano approach employing chiral Ward
identities and that of instanton based approaches. This third way will prove
especially beneficial when we turn our attention to $g_{1}(x_{B},Q^{2})$ at
small $x_{B}$.

Our first objective is to understand in detail the cancellation of the anomaly
pole in $\Sigma(Q^{2})$; we will show how one recovers the Shore-Veneziano
results. Specifically, in the Bjorken limit, the anomaly requires we replace
the isosinglet current $J_{5}^{\mu}$ (whose expectation value in the polarized
proton ground state is $\Sigma(Q^{2})$) with

$J_{5}^{\mu}\rightarrow\frac{l^{\mu}}{l^{2}}\Omega\,,$ (2)

where $\Omega=\frac{\alpha_{s}}{4\pi}{\rm Tr}\left(F{\tilde{F}}\right)$ is the
topological charge density expressed in terms of the field strength tensor and
its dual $F_{\mu\nu}$ defined as
$\tilde{F}_{\mu\nu}=\frac{1}{2}\,\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$.
Here $l=P^{\prime}-P$ is the four vector corresponding to the four-momentum
transfer from the proton to the DIS probe, with $l^{2}=t$, the Mandelstam
variable. The r.h.s of Eq. (2) corresponds to a massless exchange from the
topological charge density $\Omega$ which couples nonperturbatively to the
proton. However this massless exchange can also be mediated (in the chiral
limit) by a massless flavor singlet pseudoscalar field ${\bar{\eta}}$, with a
similar nonperturbative coupling to the proton. One may in the chiral limit,
and at large $N_{c}$, interpret ${\bar{\eta}}$ as the “primordial” ninth
Goldstone boson. The absence of this field in the hadron spectrum is of course
the $U_{A}(1)$ problem. It is resolved by the nontrivial susceptibility of the
QCD vacuum which generates the large mass of the $\eta^{\prime}$ Witten
(1979); Veneziano (1979).

We will derive explicitly in our approach the Wess-Zumino-Witten (WZW) Wess
and Zumino (1971); Witten (1983) term in the imaginary part of the effective
action that couples the isosinglet pseudoscalar field ${\bar{\eta}}$ to the
topological charge density $\Omega$. This term plays a fundamental role in the
cancellation of the anomaly pole because an identical pole exists in the
${\bar{\eta}}$ exchange with the proton. We will show in detail how this
cancellation arises in our approach both at leading order in the $\Omega$ and
${\bar{\eta}}$ exchange, and to all orders. We will further show how this
interplay results in an anomalous Goldberger-Treiman relation Veneziano (1989)
and in topological mass generation of the $\eta^{\prime}$. Thus as noted, the
role played by the chiral anomaly in the proton’s spin is deeply tied to the
resolution of the $U_{A}(1)$ problem.

While these conclusions, if not the approach, are familiar from the work of
Veneziano and collaborators, our framework can be extended to the computation
of $g_{1}(x_{B},Q^{2})$ in the Bjorken and Regge asymptotics of QCD. This is
because, as demonstrated in Paper I, the triangle anomaly in Eq. (2) is the
dominant contribution to $g_{1}(x_{B},Q^{2})$ in both limits. In the Bjorken
limit, as we will discuss briefly, the formalism of Shore and Veneziano goes
through for $g_{1}(x_{B},Q^{2})$, precisely as for its first moment
$\Sigma(Q^{2})$. The corresponding matrix element can be computed on the
lattice similarly to prior lattice computations Liang  _et al._ (2018);
Alexandrou  _et al._ (2020); Mejía-Díaz  _et al._ (2018); Lin  _et al._
(2018). For discussions of how to extract $\Sigma(Q^{2})$ directly from the
slope (with $t$) of the topological susceptibility, see Giusti  _et al._
(2002); a recent review of lattice extractions of the topological
susceptibility can be found in Bali  _et al._ (2021).

The situation is quite different at small $x_{B}$ because of the phenomenon of
gluon saturation and the emergence of a corresponding saturation scale $Q_{S}$
Gribov  _et al._ (1983); Mueller and Qiu (1986). In Regge asymptotics, this
scale is larger than the scales governing intrinsically nonperturbative
dynamics in QCD. In the unpolarized proton, the gauge configurations
representing the saturated state are static classical configurations McLerran
and Venugopalan (1994a, b, c) and their dynamics is described by the Color
Glass Condensate (CGC) Effective Field Theory (EFT) Gelis  _et al._ (2010);
Kovchegov and Levin (2012). In the polarized proton, such configurations can
be dynamical on the time scales of spin diffusion. Thus the classical
configurations responsible for spin diffusion in this high energy asymptotics
are not the energy degenerate instanton solutions describing tunneling between
different $\theta$-vacua (each corresponding to distinct integer valued Chern-
Simons number) but “over the barrier” topological transitions that are
enhanced by the large dynamical saturation scale. A well-known example of such
transitions are the sphaleron solutions Klinkhamer and Manton (1984)
conjectured to play a major role in electroweak baryogenesis Kuzmin  _et al._
(1985). Similar sphaleron-like topological transitions have also been
discussed in the QCD context both in-and out-of equilibrium McLerran  _et al._
(1991); Moore and Tassler (2011); Shuryak and Zahed (2003); Mace  _et al._
(2016).

With these considerations in mind, we will write down an “axion-like”
effective action for $g_{1}(x_{B},Q^{2})$ at small $x_{B}$ that captures both
the physics of gluon saturation and spin diffusion, which are respectively
controlled by $Q_{S}$ and the Yang-Mills topological susceptibility $\chi_{\rm
YM}$. Depending on their relative magnitude (specifically of $Q_{S}$ and
$m_{\eta^{\prime}}$, the $\eta^{\prime}$ mass), the gauge field configurations
are either, as noted, “conventionally” sphaleron-like (for
$m_{\eta^{\prime}}>Q_{S}$) or novel topological shock wave configurations (for
$Q_{S}>m_{\eta^{\prime}}$). We will discuss the consequences of this interplay
and other qualitative features of the dynamics captured by the effective
action. A quantitative study of QCD evolution in this framework, and
phenomenological consequences thereof for polarized DIS measurements at the
Electron-Ion Collider (EIC) Accardi  _et al._ (2016); Aschenauer  _et al._
(2019), will be discussed in follow-up work Tarasov and Venugopalan .

The paper is organized as follows. In the next section, we will briefly
recapitulate the worldline derivation of $g_{1}(x_{B},Q^{2})$ in Paper I which
demonstrated the dominance of the triangle graph of the anomaly in both
Bjorken and Regge asymptotics. We will emphasize that the triangle graph, and
indeed all dynamical effects of the anomaly, can be computed directly from the
imaginary part of the worldline effective action. In Section III, we will
discuss the extension of the imaginary part of the worldline effective action
to include, in addition to gauge fields, the coupling of the fermions to
scalar, pseudoscalar, and axial vector fields, which capture the dynamics of
low energy modes in the QCD effective action. We will then show in this
formalism how the WZW term coupling ${\bar{\eta}}$ to $\Omega$ arises. The
profound consequences of this result for the cancellation of the pole of the
anomaly is discussed in detail in Section IV. In Section V, we will write down
the effective action for $g_{1}(x_{B},Q^{2})$ and sketch its key features. We
end in Section VI with a summary, outlook on future work, and a discussion of
some of the larger implications of our work.

In Appendix A, we provide details of the derivation of the WZW term from the
imaginary part of the worldline effective action. In Appendix B, we will
outline the derivation of the CGC effective action in the worldline formalism
for the case where the hadron is not polarized. In Appendix C, we will extend
this discussion to the polarized proton case. In particular, we will present
an argument for the failure of high energy expansions in perturbative QCD for
operators in the polarized proton that are sensitive to the anomaly.

##  II Anomaly dominance of $g_{1}(x_{B},Q^{2})$

We will first briefly recapitulate222We refer interested readers to Paper I
for more details Tarasov and Venugopalan (2020). here the worldline derivation
in Paper I where we demonstrated that the triangle graph of the anomaly
dominates $g_{1}(x_{B},Q^{2})$ in both the Bjorken and Regge limits of DIS. We
will also discuss the result in Paper I showing that the triangle graph can be
recovered directly from the imaginary part of the worldline effective action.
This will serve to motivate our focus in the rest of this paper on the role of
the imaginary part of the effective action. As is well-known Schubert (2001),
its contributions can be fundamentally understood as arising from the
noninvariance of the measure of the QCD path integral under a global chiral
rotation Fujikawa (1979).

The $g_{1}(x_{B},Q^{2})$ structure function can be extracted most generally
from the antisymmetric piece of the hadron tensor Anselmino  _et al._ (1995),
which can be expressed as

$\displaystyle\tilde{W}_{\mu\nu}(q,P,S)=\frac{2M_{N}}{P\cdot
q}\epsilon_{\mu\nu\alpha\beta}\,q^{\alpha}\Big{\\{}S^{\beta}g_{1}(x_{B},Q^{2})+\Big{[}S^{\beta}-\frac{(S\cdot
q)P^{\beta}}{P\cdot q}\Big{]}g_{2}(x_{B},Q^{2})\Big{\\}}\,,$ (3)

where $M_{N}$ denotes the proton mass and the totally antisymmetric Levi-
Civita tensor $\epsilon_{\mu\nu\alpha\beta}$ is defined with
$\epsilon_{0123}=-1$. For a longitudinally polarized target,
$S^{\mu}(\lambda)\simeq\frac{2{\tilde{\lambda}}_{P}}{M_{N}}P^{\mu}$, with
${\tilde{\lambda}}_{P}=\pm\frac{1}{2}$ representing the proton’s helicity; in
this case, the $g_{2}$ structure function does not contribute.

The full hadron tensor itself can be expressed as the imaginary part of the
expectation value of the polarization tensor:

$\displaystyle W^{\mu\nu}(q,P,S)=\frac{1}{\pi e^{2}}{\rm Im}\ \int
d^{4}x~{}e^{iqx}\langle P,S|\frac{\delta^{2}\Gamma[a,A]}{\delta
a_{\mu}(\frac{x}{2})\delta a_{\nu}(-\frac{x}{2})}|P,S\rangle\,,$ (4)

where $\Gamma[a,A]$ is the QED+QCD worldline effective action, $a_{\mu}(x)$
denotes the QED electromagnetic field and $A$ is the four-vector denoting the
QCD gauge field. Its antisymmetric piece, which appears on the l.h.s of Eq.
(3), can be written as

$\displaystyle i\tilde{W}^{\mu\nu}(q,P,S)=\frac{1}{2\pi e^{2}}\,{\rm Im}\ \int
d^{4}x\,e^{-iqx}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{3}}{(2\pi)^{4}}e^{-ik_{1}\frac{x}{2}}e^{ik_{3}\frac{x}{2}}\langle
P,S|{\tilde{\Gamma}}^{\mu\nu}_{A}[k_{1},k_{3}]|P,S\rangle\,.$ (5)

Here
${\tilde{\Gamma}}^{\mu\nu}_{A}[k_{1},k_{3}]\equiv{\tilde{\Gamma}}^{\mu\nu}[k_{1},k_{3}]-(\mu\leftrightarrow\nu)$,
with

$\displaystyle{\tilde{\Gamma}}^{\mu\nu}[k_{1},k_{3}]\equiv\int
d^{4}z_{1}d^{4}z_{3}\frac{\delta^{2}\Gamma[a,A]}{\delta a_{\mu}(z_{1})\delta
a_{\nu}(z_{3})}|_{a=0}\,e^{ik_{1}z_{1}}e^{ik_{3}z_{3}}\,,$ (6)

where $k_{1}$ and $k_{3}$ denote the incoming photon four-momenta. Because the
r.h.s of Eq. (5) corresponds to the same in-out ground state of the proton,
$k_{1}=-k_{3}=-q$ in the forward limit. However to extract the infrared pole
of the anomaly, as discussed at length in Paper I, one needs to keep the
incoming photon momenta distinct in computing the off-forward matrix element
$\langle P^{\prime}|\dots|P\rangle$ in Eq. (4), with $P^{\prime}-P\equiv l$
and $t=l^{2}$, and then subsequently take $t\to 0$ in the final expression.

To compute $i\tilde{W}^{\mu\nu}(q,P,S)$, and hence $g_{1}(x_{B},Q^{2})$, we
employed a powerful worldline formalism in Paper I; to one loop accuracy, the
QED+QCD effective action in this formalism333As we will discuss shortly, and
in greater detail in Section III, we will replace $\Gamma\rightarrow{\cal W}$,
a more general expression which includes additional couplings with scalar,
pseudoscalar and axial vector fields; we will split the latter into real and
imaginary pieces. can be expressed as Schubert (2001)

$\displaystyle\Gamma[a,A]=-\frac{1}{2}\int^{T}_{0}\frac{dT}{T}{\rm
Tr_{c}}\int\mathcal{D}x\int\mathcal{D}\psi\exp\Big{\\{}-\int^{T}_{0}d\tau\Big{(}\frac{1}{4}\dot{x}^{2}+\frac{1}{2}\psi_{\mu}\dot{\psi}^{\mu}+ig\dot{x}^{\mu}(A_{\mu}+a_{\mu})-ig\psi^{\mu}\psi^{\nu}F_{\mu\nu}(A+a)\Big{)}\Big{\\}}\,,$
(7)

where $x^{\mu}(\tau)$ and $\psi^{\mu}(\tau)$ are respectively 0+1-dimensional
scalar coordinate and Grassmann variables coupled to the background
electromagnetic ($a_{\mu}$) and gluon ($A_{\mu}$) fields. Note that the scalar
functional integral has periodic (P) boundary conditions while the
Grassmannian functional integral has anti-periodic (AP) boundary conditions.

In this formalism, ${\tilde{\Gamma}}^{\mu\nu}_{A}[k_{1},k_{3}]$ on the r.h.s
of Eq. (5) can be written, to one loop accuracy, as

$\displaystyle{\tilde{\Gamma}}^{\mu\nu}_{A}[k_{1},k_{3}]=\int\frac{d^{4}k_{2}}{(2\pi)^{4}}\int\frac{d^{4}k_{4}}{(2\pi)^{4}}~{}\Gamma^{\mu\nu\alpha\beta}_{A}[k_{1},k_{3},k_{2},k_{4}]~{}{\rm
Tr_{c}}({\tilde{A}}_{\alpha}(k_{2}){\tilde{A}}_{\beta}(k_{4}))\,,$ (8)

where the ${\tilde{A}}$ denote the Fourier transforms of the background gauge
fields, the trace is over their color degrees of freedom and the box diagram
in the r.h.s takes the form,

$\displaystyle\Gamma^{\mu\nu\alpha\beta}_{A}[k_{1},k_{3},k_{2},k_{4}]=-\frac{g^{2}e^{2}e_{f}^{2}}{2}\int^{\infty}_{0}\frac{dT}{T}~{}\int\mathcal{D}x\int\mathcal{D}\psi~{}\exp\Big{\\{}-\int^{T}_{0}d\tau\Big{(}\frac{1}{4}\dot{x}^{2}+\frac{1}{2}\psi\cdot\dot{\psi}\Big{)}\Big{\\}}$
$\displaystyle\times\prod^{4}_{k=1}\int^{T}_{0}d\tau_{k}~{}\Big{[}\sum^{9}_{n=1}\mathcal{C}^{\mu\nu\alpha\beta}_{n;(\tau_{1},\tau_{2},\tau_{3},\tau_{4})}[k_{1},k_{3},k_{2},k_{4}]-(\mu\leftrightarrow\nu)\Big{]}e^{i\sum^{4}_{i=1}k_{i}x_{i}}\,.$
(9)

The coordinate ($x_{i}\equiv x(\tau_{i})$) and Grassmann variables
($\psi_{i}\equiv\psi(\tau_{i})$) in the coefficients
$\mathcal{C}^{\mu\nu\alpha\beta}_{n;(\tau_{1},\tau_{2},\tau_{3},\tau_{4})}[k_{1},k_{3},k_{2},k_{4}]$
on the r.h.s depend on the proper time coordinates of the interaction of the
worldlines with the external electromagnetic and gauge fields. Fully general
expressions for these coefficients were provided in Paper I.

The worldline representation of the box diagram provides a useful intuition by
mapping the ordering of the momentum labels of the four vertices to that of
the corresponding proper times. This allows one to understand the usual
triangle limit of the box diagram in Bjorken asymptotics as “pinching”
$\tau_{1}\rightarrow\tau_{3}$ as $k_{1}\rightarrow k_{3}$ (with corrections of
order $1/Q$). More unexpectedly, it allows one to interpret Regge asymptotics
as $\tau_{2}\rightarrow\tau_{4}$ in the shockwave limit $k_{2}\rightarrow
k_{4}$ of the gauge fields with corrections of order $O(\frac{x_{B}}{x})$, as
$x_{B}\rightarrow 0$ for gluon momenta carrying a finite but small fraction
$x$ of the hadron’s large “+” momentum. This gives rise to an “inverted
triangle” which too is sensitive to the anomaly.

Indeed, we showed in Paper I that the computation of Eq. (8) in either limit
gives identically,

$\displaystyle
S^{\mu}g_{1}(x_{B},Q^{2})=\sum_{f}e_{f}^{2}\frac{\alpha_{s}}{i\pi
M_{N}}\int^{1}_{x_{B}}\frac{dx}{x}~{}\Big{(}1-\frac{x_{B}}{x}\Big{)}\int\frac{d\xi}{2\pi}e^{-i\xi
x}\lim_{l_{\mu}\to 0}\frac{l^{\mu}}{l^{2}}\langle P^{\prime},S|{\rm
Tr_{c}}F_{\alpha\beta}(\xi n)\tilde{F}^{\alpha\beta}(0)|P,S\rangle\,.$ (10)

Here $\alpha_{s}$ is the QCD coupling, the sum is over $n_{f}$ quark
flavors444We will assume these to be massless and $n_{f}=3$ for our
discussion. with electric charge $e_{f}$. The structure of the r.h.s is
dominated by the triangle graph in either limit; therefore the operator that
governs the r.h.s is the topological charge density
$\Omega=\frac{\alpha_{s}}{4\pi}{\rm Tr}\left(F{\tilde{F}}\right)$, with
$\tilde{F}_{\mu\nu}=\frac{1}{2}\,\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$.
While the structure of the operator is identical in both limits, we will see
that one obtains qualitatively different results in the two limits. The other
noteworthy feature of Eq. (10) is the pole $l^{\mu}/l^{2}$, which is a
consequence of the anomaly equation for $J_{\mu}^{5}$; the cancellation of
this pole will be the topic of Section IV.

We also showed in Paper I that the anomaly can be extracted directly and with
relative ease Schubert (2001); Mueller and Venugopalan (2017) from the
imaginary part of the effective action. Contributions to the imaginary part
${\cal W}_{I}$ (as we will discuss shortly) can be extracted by adding
auxiliary terms to the worldline effective action that contain odd powers of
$\gamma_{5}$. To extract the triangle graph, it is sufficient to include the
interaction term with the axial vector field $\not{B}\gamma_{5}$ McKeon and
Schubert (1998); Mondragon  _et al._ (1996); D’Hoker and Gagne (1996a);
Schubert (2001); Mueller and Venugopalan (2017, 2018):

$\displaystyle{\cal W}_{I}[A,B]=-\frac{1}{2}{\rm
Tr_{c}}\int^{\infty}_{0}\frac{dT}{T}\,\int\mathcal{D}x\int_{AP}\mathcal{D}\psi$
$\displaystyle\times\exp\Big{\\{}-\int^{T}_{0}d\tau\Big{(}\frac{1}{4}\dot{x}^{2}+\frac{1}{2}\psi_{\mu}\dot{\psi}^{\mu}+ig\dot{x}^{\mu}A_{\mu}-ig\psi^{\mu}\psi^{\nu}F_{\mu\nu}-2i\psi_{5}\dot{x}^{\mu}\psi_{\mu}\psi_{\nu}B^{\nu}+i\psi_{5}\partial_{\mu}B^{\mu}+(D-2)B^{2}\Big{)}\Big{\\}}\,,$
(11)

where $D$ is the number of spacetime dimensions and $\psi_{5}$ is the
Grassmann counterpart of the $\gamma_{5}$ matrix in the worldline framework.
The structure of the $B$-dependent terms, as we will briefly review in Section
III, comes from exponentiating the phase of the Dirac determinant in the QCD
effective action McKeon and Schubert (1998); Mondragon  _et al._ (1996);
D’Hoker and Gagne (1996a).

Since $J_{5}^{\mu}$ couples to $B^{\mu}$, its expectation value is obtained by
taking the functional derivative of ${\cal W}_{I}[A,B]$ with respect to $B$
and then setting the latter equal to zero:

$\displaystyle\langle P^{\prime},S|J^{\kappa}_{5}|P,S\rangle=\int
d^{4}y\,\frac{\partial{\cal W}_{I}[A,B]}{\partial
B_{\kappa}(y)}\Big{|}_{B_{\kappa}=0}e^{ily}\equiv\Gamma^{\kappa}_{5}[l]\,,$
(12)

which gives,

$\displaystyle\Gamma^{\kappa}_{5}[l]$ $\displaystyle=$
$\displaystyle\frac{i}{2}{\rm
Tr_{c}}\int^{\infty}_{0}\frac{dT}{T}\,\int\mathcal{D}x\int_{AP}\mathcal{D}\psi~{}\int^{T}_{0}d\tau_{l}~{}\psi_{5}\Big{(}il^{\kappa}+2\psi^{\kappa}_{l}\dot{x}_{l}\cdot\psi_{l}\Big{)}e^{ilx_{l}}$
(13) $\displaystyle\times$
$\displaystyle\exp\Big{\\{}-\int^{T}_{0}d\tau\Big{(}\frac{1}{4}\dot{x}^{2}+\frac{1}{2}\psi_{\mu}\dot{\psi}^{\mu}+ig\dot{x}^{\mu}A_{\mu}-ig\psi^{\mu}\psi^{\nu}F_{\mu\nu}\Big{)}\Big{\\}}\,,$

where $\tau_{l}$ is the proper time coordinate of the $B$-field insertion into
the worldline, and $l$ is the incoming momentum. We use the shorthand notation
$x_{l}\equiv x(\tau_{l})$, $\psi_{l}\equiv\psi(\tau_{l})$.

Figure 1: The triangle graph representing the vector-vector-axial vector (VVA)
coupling of the chiral anomaly.

Expanding the phase in Eq. (13) to second order in the coupling constant,

$\displaystyle\Gamma^{\kappa}_{5}[l]=-\frac{ig^{2}}{2}{\rm
Tr_{c}}\int^{\infty}_{0}\frac{dT}{T}\,\int\mathcal{D}x\int_{AP}\mathcal{D}\psi~{}\int^{T}_{0}d\tau_{l}~{}\psi_{5}\Big{(}il^{\kappa}+2\psi^{\kappa}_{l}\dot{x}_{l}\cdot\psi_{l}\Big{)}e^{ilx_{l}}\int^{T}_{0}d\tau_{2}\int^{T}_{0}d\tau_{4}$
$\displaystyle\times\Big{(}\dot{x}^{\alpha}_{2}A_{\alpha}(x_{2})-2\psi^{\lambda}_{2}\psi^{\alpha}_{2}\partial_{\lambda}A_{\alpha}(x_{2})\Big{)}\Big{(}\dot{x}^{\beta}_{4}A_{\beta}(x_{4})-2\psi^{\eta}_{4}\psi^{\beta}_{4}\partial_{\eta}A_{\beta}(x_{4})\Big{)}\exp\Big{\\{}-\int^{T}_{0}d\tau\Big{(}\frac{1}{4}\dot{x}^{2}+\frac{1}{2}\psi_{\mu}\dot{\psi}^{\mu}\Big{)}\Big{\\}}\,,$

we can rewrite this equation as

$\displaystyle\Gamma^{\kappa}_{5}[l]=\int\frac{d^{4}k_{2}}{(2\pi)^{4}}\int\frac{d^{4}k_{4}}{(2\pi)^{4}}~{}\Gamma^{\kappa\alpha\beta}_{5}[l,k_{2},k_{4}]~{}{\rm
Tr_{c}}A_{\alpha}(k_{2})A_{\beta}(k_{4})\,,$ (15)

where the VVA vertex function shown in Fig. 1 can be expressed as

$\displaystyle\Gamma^{\kappa\alpha\beta}_{5}[l,k_{2},k_{4}]\equiv-\frac{ig^{2}}{2}\int^{\infty}_{0}\frac{dT}{T}\,\int\mathcal{D}x\int_{AP}\mathcal{D}\psi~{}\int^{T}_{0}d\tau_{l}~{}\psi_{5}\Big{(}il^{\kappa}+2\psi^{\kappa}_{l}\dot{x}_{l}\cdot\psi_{l}\Big{)}e^{ilx_{l}}$
(16)
$\displaystyle\times\int^{T}_{0}d\tau_{2}\int^{T}_{0}d\tau_{4}\Big{(}\dot{x}^{\alpha}_{2}+2i\psi^{\alpha}_{2}\psi^{\lambda}_{2}k_{2\lambda}\Big{)}e^{ik_{2}x_{2}}\Big{(}\dot{x}^{\beta}_{4}+2i\psi^{\beta}_{4}\psi^{\eta}_{4}k_{4\eta}\Big{)}e^{ik_{4}x_{4}}\exp\Big{\\{}-\int^{T}_{0}d\tau\Big{(}\frac{1}{4}\dot{x}^{2}+\frac{1}{2}\psi_{\mu}\dot{\psi}^{\mu}\Big{)}\Big{\\}}\,.$

This three-point function $\Gamma^{\kappa\alpha\beta}_{5}[l,k_{2},k_{4}]$ has
a $\psi_{5}$ in the argument of the Grassmannian functional integral which
changes the boundary condition from being antiperiodic (AP) to being periodic
(P). As a result, the Grassmann variables in the functional integral acquire a
zero mode, which can be separated out from the nonzero modes in the action,
and in the measure, as

$\displaystyle\psi^{\mu}(\tau)=\psi^{\mu}_{0}+\xi^{\mu}(\tau)\,;\ \ \
\int_{P}\mathcal{D}\psi=\int d^{4}\psi_{0}\int_{P}\mathcal{D}\xi\,;\ \ \
\int^{T}_{0}d\tau\,\xi(\tau)=0\,.$ (17)

Separating out the zero mode thus, we obtain

$\displaystyle\Gamma^{\kappa\alpha\beta}_{5}[l,k_{2},k_{4}]\equiv-\frac{ig^{2}}{2}\int^{\infty}_{0}\frac{dT}{T}\,\int\mathcal{D}x\int
d^{4}\psi_{0}\int_{P}\mathcal{D}\xi~{}\int^{T}_{0}d\tau_{l}~{}\Big{(}il^{\kappa}+2\psi^{\kappa}_{l}\dot{x}_{l}\cdot\psi_{l}\Big{)}e^{ilx_{l}}$
$\displaystyle\times\int^{T}_{0}d\tau_{2}\int^{T}_{0}d\tau_{4}\Big{(}\dot{x}^{\alpha}_{2}+2i\psi^{\alpha}_{2}\psi^{\lambda}_{2}k_{2\lambda}\Big{)}e^{ik_{2}x_{2}}\Big{(}\dot{x}^{\beta}_{4}+2i\psi^{\beta}_{4}\psi^{\eta}_{4}k_{4\eta}\Big{)}e^{ik_{4}x_{4}}\exp\Big{\\{}-\int^{T}_{0}d\tau\Big{(}\frac{1}{4}\dot{x}^{2}+\frac{1}{2}\psi_{\mu}\dot{\psi}^{\mu}\Big{)}\Big{\\}}\Big{|}_{\psi=\psi_{0}+\xi}\,.$

The evaluation of the functional integrals over $x$ and $\xi$, as well as the
integral over zero mode $\psi_{0}$, is straightforward and discussed at length
in Tarasov and Venugopalan (2019, 2020). We obtain,

$\displaystyle\Gamma^{\kappa\alpha\beta}_{5}[l,k_{2},k_{4}]=\frac{1}{2\pi^{2}}\frac{k^{\kappa}_{2}+k^{\kappa}_{4}}{(k_{2}+k_{4})^{2}}~{}\epsilon^{\alpha\sigma\beta\lambda}k_{2\sigma}k_{4\lambda}(2\pi)^{4}\delta^{4}(l+k_{2}+k_{4})\,,$
(19)

which agrees with the result given in Ref. Schubert (2001).

Substituting the VVA vertex function back into Eq. (15), we obtain,

$\displaystyle\Gamma^{\kappa}_{5}[l]=\frac{1}{4\pi^{2}}\frac{l^{\kappa}}{l^{2}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}\int\frac{d^{4}k_{4}}{(2\pi)^{4}}~{}{\rm
Tr_{c}}F_{\alpha\beta}(k_{2})\tilde{F}^{\alpha\beta}(k_{4})~{}(2\pi)^{4}\delta^{4}(l+k_{2}+k_{4})\,,$
(20)

the operator structure of which, up to kinematic factors, is identical to the
expressions that lead the principal result of Paper I (given here in Eq.
(10)).

A common interpretation of the first moment $\Sigma$ is that of a local
operator because the corresponding integral over $x_{B}$ can be written as the
local operator $J_{\mu}^{5}$, which is often interpreted as being
qualitatively distinct from the operator in Eq. (10). However, as also
emphasized previously in Jaffe and Manohar (1990), the presence of the
infrared pole ensures that $J_{\mu}^{5}$ receives an intrinsically nonlocal
contribution; indeed, as we shall see, $J_{\mu}^{5}$ can be expressed in terms
of the QCD topological susceptibility, which is manifestly nonlocal. Further
corroboration follows from our result that the anomaly dominates both in
Bjorken and Regge asymptotics. While the operator product expansion may be
employed in the former limit, it cannot be presumed to hold in the latter. One
must therefore interpret the r.h.s of Eq. (10) as a smearing of the
topological charge density $\Omega$. The treatment of $\Omega$ as an intrinsic
low energy degree of freedom, on par with the Goldstone modes of chiral
symmetry breaking, is discussed extensively in Veneziano (1989); Shore and
Veneziano (1990, 1992); Shore (2008) and will be be addressed in the following
sections.

##  III  WZW ${\bar{\eta}}$ term from the imaginary part of the worldline
effective action

From Eq. (10), we see that the triangle anomaly generates a contribution,
proportional to the topological charge density, that diverges in the forward
limit. This is of course untenable and there must be other nonperturbative
contributions that cancel this power law divergence. As observed previously in
the literature Jaffe and Manohar (1990); Veneziano (1989); Shore and Veneziano
(1990, 1992), this cancellation can be understood as arising (in the chiral
limit and at large $N_{c}$) from the exchange of a “primordial” ninth
Goldstone boson ${\bar{\eta}}$ arising from the spontaneous symmetry breaking
of the flavor group $U_{L}(3)\times U_{R}(3)$ to the vector group $U_{V}(3)$.

There is of course no ${\bar{\eta}}$ Goldstone pole just as there is no
anomaly pole in the QCD spectrum; the appearance of both, and their
cancellation, are features of a particular limit of the theory that do not
survive when one fully accounts for its rich nonperturbative dynamics. The
important point to note however is that the same physics that ensures the
former by generating a massive $\eta^{\prime}$ meson (the famous “$U_{A}(1)$
problem”) is also what ensures the latter555The $U_{A}(1)$ problem is resolved
by the nontrivial susceptibility of the QCD vacuum; as noted, an attractive
mechanism that generates this susceptibility is provided by instanton mediated
interactions ’t Hooft (1976, 1986).. In other words, the dynamical interplay
between the physics of the anomaly, and that of the isosinglet pseudoscalar
$U_{A}(1)$ sector of QCD resolves both problems simultaneously: the lifting of
the ${\bar{\eta}}$ pole by topological mass generation of the $\eta^{\prime}$
and the cancellation of the anomaly pole. This will be shown explicitly in
Section IV.

Before we get there, we will first show how such contributions arise in the
worldline formalism. In particular, we will derive the Wess-Zumino-Witten
(WZW) term that couples the pseudoscalar isosinglet ${\bar{\eta}}$ field to
the topological charge density. The presence of this term is crucial for our
discussion in Section IV.

The interplay of perturbative and nonperturbative dynamics is captured in the
worldline formalism by parametrizing the low frequency modes of the Dirac
operator in terms of scalar, pseudoscalar, and axial vector fields666For a
nice discussion of the underpinnings of this approach, see Leutwyler and
Smilga (1992). Note further that we have implicitly in mind the separation of
the gauge field configurations in $A_{\mu}$ into high energy gluon modes and
low energy nonperturbative modes which could be glueball or instanton
configurations. In this light, the topological charge density $\Omega$ must be
viewed as an intrinsically nonperturbative degree of freedom.. Restricting
ourselves to the isosinglet pseudoscalar sector of interest, the QCD fermion
action can be written as777We will follow here, for convenience, the
conventions and notations of D’Hoker and Gagne (1996a, b) since some of the
key results in these papers are central to this work. For another discussion,
with similar features, we refer the reader to Mondragon  _et al._ (1996,
1995); McKeon and Schubert (1998); both approaches are reviewed in Schubert
(2001).

$\displaystyle S_{\rm fermion}[{\bar{\Psi}},\Phi,\Pi,A,B,\Psi]=\int
d^{4}x\,{\bar{\Psi}}^{I}\left[i\not{\partial}-\Phi+i\gamma^{5}\Pi+\not{A}+\gamma^{5}\not{B}\right]^{IJ}\Psi^{J}\,.$
(21)

Here $\Phi$, $\Pi$, $A$ and $B$ denote respectively scalar, pseudoscalar,
vector and axial vector fields, whose couplings to the higher frequency
fermion fields ${\bar{\Psi}}$, $\Psi$, are absorbed into the field
definitions. The superscripts $I$ and $J$ denote the internal quantum numbers
of the fermion multiplet as well as those of the matrix valued “source”
fields.

Since the worldline effective action corresponds to computing a quark loop in
an arbitrary number of background fields, the corresponding perturbative
expression is simply

$\displaystyle{\cal W}[\Phi,\Pi,A,B]$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\cdots\frac{d^{4}k_{n}}{(2\pi)^{4}}\,\delta^{(4)}(k_{1}+\cdots+k_{n})\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{\not{q}+im}{q^{2}+m^{2}}$
(22) $\displaystyle\times$
$\displaystyle\left(i{\tilde{\Phi}}_{1}+\gamma_{5}\,{\tilde{\Pi}}_{1}+{\tilde{\not{A}}}_{1}+\gamma_{5}{\tilde{\not{B}}}_{1}\right)\frac{\not{q}-\not{k}_{1}-\cdots\not{k}_{n}+im}{\left(q-k_{1}-\cdots
k_{n}\right)^{2}+m^{2}}\cdots\left(i{\tilde{\Phi}}_{n}+\gamma_{5}\,{\tilde{\Pi}}_{n}+{\tilde{\not{A}}}_{n}+\gamma_{5}{\tilde{\not{B}}}_{n}\right)\,.$

Even numbers of insertions of scalar, pseudoscalar and axial vector fields
contribute to the real part of ${\cal W}$ while odd numbers contribute to the
imaginary part. The map between the worldline and Feynman diagram computations
of these was discussed previously in D’Hoker and Gagne (1996b); Mondragon  _et
al._ (1996); for DIS specifically, it was discussed in Tarasov and Venugopalan
(2019).

The (Euclidean) fermion effective action in the presence of these sources can
in general be written as

$-{\cal W}[A,B,\Phi,\Pi]={\rm Ln}\,{\rm Det}\,\left[{\cal D}\right]\,,$ (23)

with the Dirac operator,

${\cal D}=\not{p}-i\Phi(x)-\gamma_{5}\,\Pi\,-\not{A}-\gamma_{5}\not{B}\,.$
(24)

This effective action can split into real and imaginary parts, with D’Hoker
and Gagne (1996a),

$\displaystyle{\cal W}_{R}=-\frac{1}{2}{\rm Ln}\left({\cal D}^{\dagger}{\cal
D}\right)\,\,\,;\,\,\,{\cal W}_{I}=\frac{1}{2}{\rm Arg}\,{\rm Det}\left({\cal
D}^{2}\right)\,.$ (25)

Since the anomaly is sensitive to the imaginary part of the effective action,
we will focus on the latter alone in the rest of this paper888The derivation
of the real part of the effective action in the presence of sources is
discussed at length in D’Hoker and Gagne (1996a, b)..

An important observation in D’Hoker and Gagne (1996a) is that substituting Eq.
(24) into Eq. (25) leads to terms linear in the Grassmann variables which are
physically unappealing; the solution (which does not alter the effective
action), is to double the degrees of freedom in both the real and imaginary
parts of the effective action as

$\displaystyle{\cal W}_{R}=-\frac{1}{2}\,{\rm Ln}\,{\rm
Det}\left({\cal\theta}\right)\,\,\,\,{\rm
with}\,\,\,\,{\cal\theta}=\begin{pmatrix}0&{\cal D}\\\ {\cal
D}^{\dagger}&0\end{pmatrix}\,,$ (26)

and

$\displaystyle{\cal W}_{I}=-\frac{1}{2}\,{\rm Arg}\,{\rm
Det}\left({\tilde{\cal\theta}}\right)\,\,\,\,{\rm
with}\,\,\,\,{\tilde{\cal\theta}}=\begin{pmatrix}0&{\cal D}\\\ {\cal
D}&0\end{pmatrix}\,,$ (27)

In this “doubling’ framework, one also likewise replaces the $4\times 4$ gamma
matrices with the $8\times 8$ matrices,

$\displaystyle\Gamma_{\mu}=\begin{pmatrix}0&\gamma_{\mu}\\\
\gamma_{\mu}&0\end{pmatrix}\,\,\,;\,\,\,\Gamma_{5}=\begin{pmatrix}0&\gamma_{5}\\\
\gamma_{5}&0\end{pmatrix}\,\,\,;\,\,\,\Gamma_{6}=\begin{pmatrix}0&iI\\\
-iI&0\end{pmatrix}\,,$ (28)

where $I$ is the $4\times 4$ unit matrix and the six Hermitean
$\Gamma$-matrices satisfy
$\left\\{\Gamma_{A},\Gamma_{B}\right\\}=2\delta_{AB}\,I_{8\times 8}$.

In this representation, d’Hoker and Gagné, derived a remarkable expression for
the argument of the Dirac determinant D’Hoker and Gagne (1996b):

$\displaystyle{\cal
W}_{\mathcal{I}}=-\frac{i}{32}\int^{1}_{-1}d\alpha\int^{\infty}_{0}dT\,\mathcal{N}\int_{{\rm
PBC}}\mathcal{D}x\,\mathcal{D}\psi~{}{\rm
tr}~{}\chi\,\bar{\omega}(0)\exp\Big{[}-\int^{T}_{0}d\tau\mathcal{L}_{(\alpha)}(\tau)\Big{]}\,,$
(29)

which has a structure very similar to that of the real part, albeit with some
key differences we shall enumerate999We have set here, and elsewhere, the
value of the einbein $\mathcal{E}=2$. Further,  ${\cal
N}(T)=\int[Dp]\,\exp\left(-\int_{0}^{T}d\tau p^{2}(\tau)\right)\,,$ (30) is a
field-independent normalization factor. . Firstly,
$\mathcal{D}\psi=\mathcal{D}\psi_{\mu}\mathcal{D}\psi_{5}$ and ${\rm PBC}$
denotes periodic boundary conditions for both cooordinate and Grassmann
variables. This qualitatively differs from ${\cal W}_{R}$, where the Grassmann
integrals have anti-periodic boundary conditions. Other differences to the
real part of the effective action are,
i) the integration over $\alpha$, which explicitly breaks global chiral
invariance for $|\alpha|<1$,
ii) the proper time measure $dT\rightarrow dT/T$ in the real part,
iii) and not least, the factor $\chi\,{\bar{\omega}}(0)$, which is a direct
consequence of the anomaly.

This last term is given by

$\displaystyle\chi\bar{\omega}(0)=4\times$
$\displaystyle\begin{pmatrix}-i\psi^{5}_{0}(\partial^{\mu}B_{\mu}(x_{0})-2\Phi(x_{0})\Pi(x_{0}))+\frac{4i}{\mathcal{E}}\psi^{\mu}_{0}\psi^{\nu}_{0}\psi^{5}_{0}\dot{x}_{0\mu}B_{\nu}(x_{0})&\frac{2}{\mathcal{E}}\psi^{\mu}_{0}\dot{x}_{0\mu}\Phi(x_{0})\\\
\frac{2}{\mathcal{E}}\psi^{\mu}_{0}\dot{x}_{0\mu}\Phi(x_{0})&-i\psi^{5}_{0}(\partial^{\mu}B_{\mu}(x_{0})-2\Phi(x_{0})\Pi(x_{0}))+\frac{4i}{\mathcal{E}}\psi^{\mu}_{0}\psi^{\nu}_{0}\psi^{5}_{0}\dot{x}_{0\mu}B_{\nu}(x_{0})\end{pmatrix}\,.$

Strikingly, the worldline Lagrangian for the imaginary part of the effective
action is nearly identical to that for the real part except for the chiral
symmetry breaking “regulator” $\alpha$ multiplying $\Phi$ and $B_{\mu}$:

$\displaystyle\mathcal{L}_{(\alpha)}(\tau)=\mathcal{L}(\tau)\Big{|}_{\Phi\to\alpha\Phi,B\to\alpha
B}\,,$ (32)

where the worldline Lagrangian for the real part of the effective action is

$\displaystyle\mathcal{L}(\tau)=\frac{\dot{x}^{2}}{2\mathcal{E}}+\frac{1}{2}\psi\dot{\psi}-i\dot{x}^{\mu}\mathcal{A}_{\mu}+\frac{\mathcal{E}}{2}\mathcal{H}^{2}+i\mathcal{E}\psi^{\mu}\psi_{5}\mathcal{D}_{\mu}\mathcal{H}+\frac{i\mathcal{E}}{2}\psi^{\mu}\psi^{\nu}\mathcal{F}_{\mu\nu}\,.$
(33)

We have adopted here a two component notation combining respectively scalar
and pseudoscalar source fields, and likewise for the vector and axial vector
fields, These fields are defined as

$\displaystyle\mathcal{A}_{\mu}\equiv\begin{pmatrix}A^{L}_{\mu}&0\\\
0&A^{R}_{\mu}\end{pmatrix}=\begin{pmatrix}A_{\mu}+B_{\mu}&0\\\
0&A_{\mu}-B_{\mu}\end{pmatrix}\,.$ (34)
$\displaystyle\mathcal{H}\equiv\begin{pmatrix}0&iH\\\
-iH^{\dagger}&0\end{pmatrix}=\begin{pmatrix}0&i\Phi+\Pi\\\
-i\Phi+\Pi&0\end{pmatrix}\,.$ (35)

Note that if we turn off the scalar and pseudoscalar sources (set ${\cal
H}=0$), we will recover precisely101010One must first separate out the zero
and nonzero modes, as in Eq. (17), in the latter. the ${\cal W}_{I}$ in Eq.
(11) that was employed to compute the triangle graph and to relate it to the
topological charge density $\Omega$.

As noted earlier, in addition to the triangle graph, the imaginary part of the
worldline action contains all other anomalous contributions allowed by the
symmetries of the theory. This can be appreciated immediately by observing the
similarities between the worldline form of the effective action in the
presence of sources to the Wess-Zumino action Wess and Zumino (1971). As is
well-known, variations of the latter with respect to these sources generates
functional Ward identities (including anomalous ones) Shore (2008).

When we add the scalar and pseudoscalar sources to the mix, d’Hoker and Gagné
D’Hoker and Gagne (1996a) showed explicitly that ${\cal W}_{I}$ (when expanded
out to order $O(\Pi^{5})$) reproduces precisely the Wess-Zumino-Witten term
(WZW) Wess and Zumino (1971); Witten (1983) governing $\pi^{0}\rightarrow
2\,\gamma$. One sees in this derivation that it is essential that the scalar
$\Phi$ have a nonzero vacuum expectation value (vev). This is apparent from
Eq. (III) which contributes the odd power of $\Pi$ in the WZW action. The
presence of this vev can be interpreted as that acquired by the $\sigma$-field
in the linear sigma model after the spontaneous breaking of chiral
symmetry111111As shown by Weinberg Weinberg (1967), in the limit of large vev
masses, one recovers the non-linear sigma model which provides the scaffolding
for chiral perturbation theory. The computation by d’Hoker and Gagné of the
WZW-$\pi^{0}$ term is performed in this large mass limit..

We can likewise show that, expanding $W_{\mathcal{I}}$ up to order
$\Pi\,A^{2}$, gives the relation

$\displaystyle{\cal W}_{\mathcal{I}}[\Pi
A^{2}]=\frac{ig^{2}\,2n_{f}}{16\pi^{2}}\frac{1}{\Phi}\,{\rm tr}_{c}\int
d^{4}x\,\Pi(x)\,F_{\mu\nu}(x)\tilde{F}^{\mu\nu}(x)\,.$ (36)

The explicit derivation121212Here we have analytically continued the result in
Eq. (140) to Minkowski space, restored the gauge coupling, and taken into
account the sum over quark flavors. is given in Appendix A and is in
agreement131313To see this, note that $\Pi=\psi/3$ in Kaiser and Leutwyler
(2000). with the corresponding ${\cal L}_{\rm WZW}$ term in Kaiser and
Leutwyler (2000) which was derived from chiral perturbation theory for the
$U_{V}(3)$ nonet.

We define the primordial ${\bar{\eta}}$ isosinglet field as

$\displaystyle{\bar{\eta}}=-\sqrt{2n_{f}}\,\frac{\Pi}{\Phi}\,F_{{\bar{\eta}}}\,,$
(37)

where the relation

$\langle
0|J^{\mu}_{5}|{\bar{\eta}}\rangle=i\sqrt{2n_{f}}\,l^{\mu}\,F_{{\bar{\eta}}}(l^{2})\,,$
(38)

defines the ${\bar{\eta}}$ decay constant $F_{{\bar{\eta}}}$ in the forward
limit $l^{2}\rightarrow 0$.

We digress here briefly to note that in the description of the isosinglet
sector of the QCD chiral Lagrangian141414See Kaiser and Leutwyler (2000) for
the relevant discussion., the massless ${\bar{\eta}}$ field is understood as
the prodigal ninth Goldstone boson of $U_{V}(3)$ that survives the spontaneous
symmetry breaking of the global $U_{R}(3)\times U_{L}(3)$ symmetry that
generates the chiral condensate. The ground state of the broken symmetry phase
is invariant only under $U_{V}(3)$ and the dynamical fields of the low energy
effective theory can be represented by the matrix $U(x)$ that transforms under
this group. In the large $N_{c}$ limit, the ${\bar{\eta}}$ field corresponds
to the phase of the determinant of $U(x)$. This description is therefore
completely consistent with our worldline construction where we showed that Eq.
(36) follows from the phase of the determinant, as seen in Eq. (25).

In the large $N_{c}$ limit, this Goldstone description in terms of the
${\bar{\eta}}$ is exact and the decay constants of all the nonet pseudo-
Goldstone bosons are identical: $F_{{\bar{\eta}}}=F_{\pi}\approx 93$ MeV. For
finite quark masses and finite $N_{c}$, the decay constants mix amongst each
other; their values can be obtained from Dashen–Gell-Mann-Oakes-Renner
relations and the rates for radiative decays of $\eta$, $\eta^{\prime}$ to
photons. Remarkably, the diagonal components corresponding to the isosinglet
decay to $\eta^{\prime}$ constant and the isooctet decay to $\eta$ are
extracted to be very close to $F_{\pi}$, about $15\%$ larger. For a detailed
discussion, we refer the reader to Shore (2008); Gan  _et al._ (2020).

The WZW effective action for the ${\bar{\eta}}$ field is then given by

$\displaystyle S_{\rm
WZW}^{{\bar{\eta}}}=-i\frac{\sqrt{2\,n_{f}}}{F_{{\bar{\eta}}}}\int
d^{4}x\,{\bar{\eta}}\,\Omega\,,$ (39)

where the “i” indicates its origin in the imaginary part of the effective
action in Eq. (29).

This WZW term that couples the ${\bar{\eta}}$ to the topological charge
density $\Omega$ plays a fundamental role in QCD. Firstly, the mixing of
${\bar{\eta}}$ and $\Omega$ ensures that the ${\bar{\eta}}$ is “eaten up” by
the latter, leaving a physical massive $\eta^{\prime}$ meson in the spectrum.
Secondly, this term plays a key role the cancellation of the anomaly pole, and
in generating $\Sigma(Q^{2})$, the proton’s helicity. We will now discuss the
relation of these two fundamental issues, namely, topological mass
generation151515For an elegant discussion of topological mass generation, see
Dvali (2005); Dvali  _et al._ (2006). and the proton’s helicity.

##  IV Topological screening of the proton helicity

We noted previously two early bodies of work relevant to our discussion here
that addressed the role of anomaly cancellation in the proton’s helicity
$\Sigma(Q^{2})$. Jaffe and Manohar Jaffe and Manohar (1990) argued that the
infrared pole obtained in the perturbative computation of the triangle anomaly
must be cancelled by a like contribution in the pseudoscalar sector but did
not discuss this mechanism in detail, specifically its relation to topological
mass generation. In contrast, the approach of Shore and Veneziano Shore and
Veneziano (1990, 1992); Shore (2008), developing previous work161616See also
related work in Hatsuda (1990); Ji (1990); Liu (1992). by Veneziano Veneziano
(1989), was fully nonperturbative, extensively employing chiral Ward
identities derived from the Wess-Zumino Wess and Zumino (1971) effective
action. They did not however discuss the cancellation of the anomaly pole,
which is only implicit in their approach.

We will discuss here a diagrammatic treatment that reconciles these two
approaches and in particular, recovers the key results of Shore and
Veneziano’s “topological screening” description of the proton’s helicity.
Since the triangle graph gives the dominant contribution to
$g_{1}(x_{B},Q^{2})$ in both Bjorken and Regge asymptotics, we will for
simplicity focus here on $\Sigma(Q^{2})$; we will take up the question of the
$x_{B}$ dependence of the triangle graph in Section V.

###  IV.1 $\Sigma(Q^{2})$ and the anomalous Goldberger-Treiman relation

We begin by examining closely how the cancellation of the infrared pole in the
matrix element of the axial vector current occurs. Our starting point is the
general decomposition of the off-forward matrix element of the flavor singlet
axial vector current171717We thank Elliot Leader for a discussion of off-
forward matrix elements and for bringing Bakker  _et al._ (2004) to our
attention where the properties of such matrix elements are discussed.; we will
eventually take the forward limit. Introducing the spinor $u(P,S)$ for the
nucleon target of mass $M_{N}$ with momentum $P$ and spin $S$, we can write
the matrix element of $J^{\mu}_{5}$ in terms of the axial and pseudoscalar
form factors $G_{A}$ and $G_{P}$ defining the coupling of the current to the
target at finite momentum transfer as

$\displaystyle\langle
P^{\prime},S|J^{\mu}_{5}|P,S\rangle=\bar{u}(P^{\prime},S)\Big{[}\gamma^{\mu}\gamma_{5}G_{A}(l^{2})+l^{\mu}\gamma_{5}G_{P}(l^{2})\Big{]}u(P,S)\,.$
(40)

Here $l^{\mu}={P^{\prime}}^{\mu}-P^{\mu}$ is the momentum transfer between the
outgoing and incoming nucleon.

Fig. 2 shows the contributions to the form factors $G_{A}$ and $G_{P}$ and we
will discuss each of these at length. We first note that the diagrams in Figs.
2a and 2b representing the coupling of the isosinglet axial vector current to
the nucleon target are fully analogous to similar diagrams representing the
couplings of the non-anomalous axial currents to the nucleon. In contrast, the
diagrams in Figs. 2c and 2d are generated by the triangle anomaly.

Figure 2: Diagrams representing the coupling of the axial vector current
$J^{\mu}_{5}$ to the nucleon target. See text for details.

The diagram in Fig. 2a represents the direct coupling of the isosinglet axial
vector current to the nucleon target. It is fundamentally different from other
diagrams in Fig. 2 since it is the only diagram which contributes to the axial
form factor $G_{A}(l^{2})$:

$\langle P^{\prime},S|J^{\mu}_{5}|P,S\rangle|_{\rm
Fig.\ref{fig:2nn}a}=G_{A}(l^{2})\,\bar{u}(P^{\prime},S)\,\gamma^{\mu}\gamma_{5}\,u(P,S)\,,$
(41)

while the diagrams in Figs. 2b-d contribute to the pseudoscalar form factor
$G_{P}(l^{2})$. With regard to the latter diagrams, we first observe that as a
consequence of the dynamical breaking of $U_{A}(1)$, no massless isosinglet
pseudoscalar particle can exist in the physical spectrum. The requirement that
the form factor $G_{P}(l^{2})$ cannot have a pole at $l^{2}=0$ can be
expressed as

$\displaystyle\lim_{l\to 0}\Big{[}\langle
P^{\prime},S|J^{\mu}_{5}|P,S\rangle|_{\rm Fig.\ref{fig:2nn}b}+\langle
P^{\prime},S|J^{\mu}_{5}|P,S\rangle|_{\rm
Figs.\ref{fig:2nn}c+\ref{fig:2nn}d}\Big{]}=0\,.$ (42)

This of course implies that in the forward limit the matrix element in Eq.
(40) is solely determined by the contribution in Fig. 2a:

$\langle P,S|J^{\mu}_{5}|P,S\rangle=\langle P,S|J^{\mu}_{5}|P,S\rangle|_{\rm
Fig.\ref{fig:2nn}a}=2M_{N}\,G_{A}(0)\,S^{\mu}\,.$ (43)

As this expression indicates, $\Sigma(Q^{2})=2\,G_{A}(0)$: the proton’s
helicity is equal to twice its isosinglet axial vector form factor, which can
be extracted from the first moment of $g_{1}$ in combination with results for
the isotriplet ($G_{A}^{(3)}$) and isooctet ($G_{A}^{(8)}$) axial vector
charges extracted respectively from nucleon and hyperon beta decay. From these
extractions, the COMPASS collaboration determines that
$G_{A}=2\,\Sigma(Q^{2})=0.32\pm 0.03\,(stat.)\pm 0.03\,(syst.)$ at $Q^{2}=3$
GeV2 Alekseev  _et al._ (2010b) which is compatible with the HERMES
collaboration’s extraction Airapetian  _et al._ (2007b) at $Q^{2}=5$ GeV2 of
$G_{A}=2\,\Sigma(Q^{2})=0.330\pm 0.011\,(th.)\pm 0.025\,(exp.)\pm
0.028\,(evol.)$.

One could view Eq. (43) as our final result. However it by itself provides
little insight into the numbers quoted, and in particular, the “spin puzzle”
Kuhn  _et al._ (2009) of why it is much smaller from the so-called OZI
expectation Ellis and Jaffe (1974) that $G_{A}|_{\rm
OZI}=2\,\sqrt{3}\,G_{A}^{(8)}\equiv 2\,\sqrt{3}\times(0.167\pm 0.006)=0.579\pm
0.021$. To understand how it can be computed, and its connection to the
$U_{A}(1)$ problem, we need to delve more deeply into the dynamics underlying
the individual contributions in Fig. 2 and further, into relations that can be
deduced181818An alternate decomposition of the pseudoscalar contributions into
a sum of terms with one proportional to the gluon helicity and the other
proportional to a dimension six operator has been proposed in the literature
Hatta (2020). However as also discussed in some detail in Jaffe and Manohar
(1990), we believe such a decomposition must be interpreted with care. amongst
these.

Indeed one of the relations, as we will discuss later, explicitly ensures that
Eq. (42) is satisfied. Another relation that must be satisfied is of course
the anomaly equation for the divergence of the singlet axial current in the
chiral limit,

$\displaystyle\langle
P^{\prime},S|\partial_{\mu}J^{\mu}_{5}|P,S\rangle=\langle
P^{\prime},S|2\,n_{f}\,\Omega|P,S\rangle\,.$ (44)

Recall that $\Omega\equiv\frac{\alpha_{s}}{4\pi}{\rm Tr}(F\tilde{F})$ is the
topological charge density. Since the l.h.s of Eq. (44) is given by the sum of
the diagrams in Fig. 2, it imposes an important constraint on the individual
dynamical contributions.

Turning now to these, the diagram in Fig. 2b represents the exchange, between
the axial current and the nucleon, of an ($\langle
0|u{\bar{u}}+d{\bar{d}}|0\rangle$) $\eta_{0}$ projection of the ($\langle
0|u{\bar{u}}+d{\bar{d}}+s{\bar{s}}|0\rangle$) ${\bar{\eta}}$ field we
discussed in the previous section. This is necessary because ${\bar{\eta}}$
contains an $s{\bar{s}}$ component that cannot couple directly to the nucleon
and requires “OZI violating” gluon exchanges to propagate Liu (1992).
Nevertheless, the projection of $J_{\mu}^{5}$ on the $\eta_{0}$ state is
nonzero and one can therefore define, in analogy to Eq. (38),

$\displaystyle\langle
0|J^{\mu}_{5}|\eta_{0}\rangle=i\sqrt{2{\tilde{n}}_{f}}\,l^{\mu}\,F_{\bar{\eta}}(l^{2})\,,$
(45)

where $l$ denotes the four-momentum of the intermediate $\eta_{0}$ field. Here
${\tilde{n}}_{f}=2$, is used to represent the two up and down flavors191919We
do so to avoid carrying factors of 2 and 3 (denoting $n_{f}=3$ in the
${\bar{\eta}}$ case) around in intermediate steps.. In other words, since
$J_{\mu}^{5}$ is flavor blind, the only change is the normalization with
respect to the number of flavors.

The contribution of Fig. 2b can thus be expressed as

$\langle P^{\prime},S|J^{\mu}_{5}|P,S\rangle|_{\rm
Fig.\ref{fig:2nn}b}=g_{\eta_{0}NN}\,\bar{u}(P^{\prime},S)\,\gamma_{5}\,u(P,S)\cdot\frac{i}{l^{2}}\cdot
i\sqrt{2{\tilde{n}}_{f}}\,l^{\mu}F_{\bar{\eta}}(l^{2})\,,$ (46)

where we have further parametrized the $\eta_{0}$-nucleon interaction by the
$\eta_{0}NN$ coupling $g_{\eta_{0}NN}$ in the effective Lagrangian,

$\displaystyle\Delta\mathcal{L}=ig_{\eta_{0}NN}\,\eta_{0}\,\bar{N}\,\gamma_{5}\,N\,.$
(47)

As their structure suggests, the diagrams in Figs. 2c and 2d are generated by
the triangle anomaly and following our discussion in Sec. II, their
contribution can be formally written in terms of the matrix element of
$\Omega$ as

$\langle P^{\prime},S|J^{\mu}_{5}|P,S\rangle|_{\rm
Figs.\ref{fig:2nn}c+\ref{fig:2nn}d}=-i\frac{l^{\mu}}{l^{2}}\langle
P^{\prime},S|2\,n_{f}\,\Omega|P,S\rangle\,.$ (48)

Substituting the divergence of each of Eqs. (41), (46) and (48) into the l.h.s
of Eq. (44) gives,

$\displaystyle
iG_{A}(l^{2})\bar{u}(P^{\prime},S)\not{l}\gamma_{5}u(P,S)-ig_{\eta_{0}NN}\bar{u}(P^{\prime},S)\gamma_{5}u(P,S)\sqrt{2{\tilde{n}}_{f}}F_{\bar{\eta}}(l^{2})+\langle
P^{\prime},S|2n_{f}\Omega|P,S\rangle=\langle
P^{\prime},S|2n_{f}\Omega|P,S\rangle\,.$ (49)

Then using equation of motion for the spinor $u(P,S)$, we can rewrite this
equation as

$\displaystyle\bar{u}(P^{\prime},S)\Big{[}2M_{N}\,G_{A}(l^{2})-g_{\eta_{0}NN}\sqrt{2{\tilde{n}}_{f}}F_{\bar{\eta}}(l^{2})\Big{]}\gamma_{5}\,u(P,S)=0\,,$
(50)

This equation relates $G_{A}$ to $g_{\eta_{0}NN}\,F_{\bar{\eta}}$ which
yields, in the forward limit $l\to 0$,

$\displaystyle
G_{A}(0)=\frac{\sqrt{2{\tilde{n}}_{f}}}{2M_{N}}\,F_{\bar{\eta}}\,g_{\eta_{0}NN}\,.$
(51)

This expression is the generalization of the well-known Goldberger-Treiman
relation to the isosinglet axial vector current, as first suggested by
Veneziano Veneziano (1989) and developed further by Shore and Veneziano Shore
and Veneziano (1990, 1992).

If we substitute Eq. (51) into Eq. (43), we can alternatively formulate our
result in Eq. (43) as

$\langle
P,S|J^{\mu}_{5}|P,S\rangle=\sqrt{2{\tilde{n}}_{f}}\,F_{\bar{\eta}}\,g_{\eta_{0}NN}S^{\mu}\,.$
(52)

Thus the anomalous Goldberger-Treiman relation allows us to express the matrix
element of $J^{\mu}_{5}$ equivalently in terms of the vacuum decay constant of
the primordial pseudoscalar $\eta_{0}$ field and its coupling thereof to the
polarized nucleon.

We obtained Eq. (52) from Eq. (43) due to the relation between the diagrams in
Figs. 2a and 2b and making further use of the anomaly relation in Eq. (44). In
the forward limit, we can also employ the constraint in Eq. (42) relating the
diagrams in Fig. 2b-d, to rewrite Eq. (52) in yet another form. Substituting
Eqs. (46), and (48) into Eq. (42) we obtain

$\displaystyle\lim_{l\to
0}\Big{[}g_{\eta_{0}NN}\bar{u}(P^{\prime},S)\gamma_{5}u(P,S)\cdot\frac{i}{l^{2}}\cdot
i\sqrt{2{\tilde{n}}_{f}}l^{\mu}F_{\bar{\eta}}(l^{2})-i\frac{l^{\mu}}{l^{2}}\langle
P^{\prime},S|2n_{f}\Omega|P,S\rangle\Big{]}=0\,.$ (53)

Thus the pole of the triangle anomaly that we computed perturbatively in Paper
I can be understood as being cancelled, as conjectured in Jaffe and Manohar
(1990), by the t-channel exchange of a massless primordial $\eta_{0}$ meson,
with the further proviso that there be no such pole in the physical spectrum.

We will now take into account the fact that the matrix element of $\Omega$ is
given by the two diagrams in Figs. 2c-d that couple the anomaly to the
nucleon. The diagrams in Figs. 2c-d can be written formally as202020Here, and
later in the text, the correlators $\langle 0|T\,\Omega\eta_{0}|0\rangle$ and
$\langle 0|T\,\Omega\Omega|0\rangle$ are to be understood as the Fourier
transforms of the corresponding correlators in coordinate space.

$\langle P^{\prime},S|J^{\mu}_{5}|P,S\rangle|_{\rm
Fig.\ref{fig:2nn}c}=-i\frac{l^{\mu}}{l^{2}}\langle
P^{\prime},S|2n_{f}\Omega|P,S\rangle|_{\rm
Fig.\ref{fig:2nn}c}=i\frac{l^{\mu}}{l^{2}}2n_{f}\cdot\langle
0|T\,\Omega\eta_{0}|0\rangle\cdot
g_{\eta_{0}NN}\,\bar{u}(P^{\prime},S)\,\gamma_{5}\,u(P,S)\,,$ (54)

and

$\langle P^{\prime},S|J^{\mu}_{5}|P,S\rangle|_{\rm
Fig.\ref{fig:2nn}d}=-i\frac{l^{\mu}}{l^{2}}\langle
P^{\prime},S|2n_{f}\Omega|P,S\rangle|_{\rm
Fig.\ref{fig:2nn}d}=i\frac{l^{\mu}}{l^{2}}2n_{f}\cdot\langle
0|T\,\Omega\Omega|0\rangle\cdot g_{\Omega
NN}\,\bar{u}(P^{\prime},S)\,\gamma_{5}\,u(P,S)\,.$ (55)

Here “T” denotes that the vacuum correlators $\langle
0|T\,\Omega\eta_{0}|0\rangle$ and $\langle 0|T\,\Omega\Omega|0\rangle$ are
time-ordered.

Substituting these equations into Eq. (53), we obtain,

$\displaystyle\sqrt{2{\tilde{n}}_{f}}\,F_{\bar{\eta}}\,g_{\eta_{0}NN}=2n_{f}\,\lim_{l\to
0}\Big{[}i\,\langle 0|T\,\Omega\eta_{0}|0\rangle\,g_{\eta_{0}NN}+i\,\langle
0|T\,\Omega\Omega|0\rangle\,g_{\Omega NN}\Big{]}\,.$ (56)

In Section IV.2, we will employ the WZW term to derive general expressions for
the correlators in the r.h.s. of Eq. (56). We will show in particular212121It
is useful to compare this result with that in Liu (1992). Firstly, in that
work, our $g_{\Omega NN}$ is denoted as $g_{{\bar{\eta}}NN}$. Secondly, in the
relevant discussion of this contribution, a “subtraction term” is introduced
to impose by hand that this contribution be nonzero in order to recover the
anomalous Goldberger-Treiman relation. This is because Liu (1992) did not
explicitly consider the contribution shown in Fig. 2b, which as we saw,
naturally gives the anomalous Goldberger-Treiman relation. that the correlator
$\langle 0|T\,\Omega\Omega|0\rangle$ (whose Fourier transform is the
topological susceptibility) doesn’t survive in the forward limit due to a
shift of the infrared pole of the ${\bar{\eta}}$ to $m^{2}_{\eta^{\prime}}$.
Anticipating this result, we obtain

$\displaystyle\sqrt{2{\tilde{n}}_{f}}\,F_{\bar{\eta}}=2n_{f}\,\lim_{l\to
0}i\,\langle 0|T\,\Omega\eta_{0}|0\rangle\,.$ (57)

Eq. (57) relates the ${\bar{\eta}}$ decay constant to the vacuum correlator
$\langle 0|T\,\Omega\eta_{0}|0\rangle$ in the forward limit. Substituting Eq.
(57) into Eq. (52), we obtain

$\langle P,S|J^{\mu}_{5}|P,S\rangle=2n_{f}\,\lim_{l\to 0}i\,\langle
0|T\,\Omega\eta_{0}|0\rangle\,g_{\eta_{0}NN}\,S^{\mu}\,.$ (58)

Eqs. (58), (43) and (52) represent different equivalent expressions for the
matrix element of the axial vector current in the forward limit. Each of these
expressions provides unique insight into the nonperturbative dynamics that
generates the proton’s helicity.

Eq. (58) makes explicit the connection of the proton’s helicity to the
topological charge density of the QCD vacuum. As we will further show, the
r.h.s of Eq. (58) is proportional to the slope of the QCD topological
susceptibility at $l^{2}\rightarrow 0$. In the chiral limit, the topological
susceptibility of the QCD vacuum is strictly zero; its slope is therefore
small. Thus the screening of the topological charge explains why the proton’s
helicity is small providing a natural resolution to the proton spin puzzle. We
turn now to a more detailed discussion of topological screening.

###  IV.2 Anomaly cancellation and topological screening

We will compute here respectively $\langle 0|T\Omega\Omega|0\rangle$ and
$\langle 0|T\Omega\eta_{0}|0\rangle$ employing the WZW term in Eq. (39).
Specifically, we’ll study the effect of the WZW coupling in the two-point
Green functions and demonstrate how it generates a nonzero
$m^{2}_{\eta^{\prime}}$. We’ll then show how the pole cancellation in Eq. (53)
gives us the result noted earlier.

####  IV.2.1 The WZW-${\bar{\eta}}$ term and the topological susceptiblity

Consider the topological susceptibility corresponding to the Fourier transform
of $i\langle 0|T\Omega\Omega|0\rangle$ in Eq. (56):

$\displaystyle\chi(l^{2})=i\int d^{4}x\,e^{ilx}\langle
0|T\,\Omega(x)\Omega(0)|0\rangle\,.$ (59)

To leading order (Fig. 3a), this correlator is nothing but the Fourier
transform $\chi_{\rm YM}(l^{2})$ of the Yang-Mills topological susceptibility;
as is well-known, it is a smooth function of momentum and doesn’t have a pole
Kogut and Susskind (1975).

Figure 3: a: The Yang-Mills topological susceptibility
$\langle\Omega\Omega\rangle_{\rm YM}$; b: First correction to
$\langle\Omega\Omega\rangle_{\rm YM}$.

The first correction to Fig. 3a, shown in Fig. 3b, is given by222222A classic
discussion of this $1/N_{c}$ expansion in the topological susceptibility can
be found in Witten (1979).

$\displaystyle\langle 0|T\,\Omega\Omega|0\rangle|_{\rm
Fig.\ref{figsup1}b}=\Big{[}-i\chi_{\rm
YM}(l^{2})\Big{]}\cdot\Big{[}-i\frac{\sqrt{2n_{f}}}{F_{\bar{\eta}}}\Big{]}\cdot\frac{i}{l^{2}}\cdot\Big{[}-i\frac{\sqrt{2n_{f}}}{F_{\bar{\eta}}}\Big{]}\cdot\Big{[}-i\chi_{\rm
YM}(l^{2})\Big{]}\,,$ (60)

where we take into account the ${\bar{\eta}}$ vacuum propagator $\langle
0|T{\bar{\eta}}{\bar{\eta}}|0\rangle=\frac{i}{l^{2}}$ and the coupling between
${\bar{\eta}}$ and $\Omega$ as specified by the WZW action in Eq. (39). Adding
the two contributions, we obtain,

$\displaystyle\langle 0|T\Omega\Omega|0\rangle|_{\rm
Fig.\ref{figsup1}a+Fig.\ref{figsup1}b}=-i\chi_{\rm
YM}(l^{2})\Big{\\{}1-\frac{1}{l^{2}}\frac{2n_{f}\,\chi_{\rm
YM}(l^{2})}{F^{2}_{{\bar{\eta}}}}\Big{\\}}\,.$ (61)

Further iterating Fig.3(b) to all orders, we can express the resummed result
as

$\displaystyle\chi(l^{2})=\chi_{\rm
YM}(l^{2})\frac{1}{1+\frac{1}{l^{2}}\frac{2n_{f}\chi_{YM}(l^{2})}{F^{2}_{\bar{\eta}}}}\,.$
(62)

Note that since we are interested in the correlator in the forward limit $l\to
0$, we can rewrite it as

$\displaystyle\chi(l^{2})=l^{2}\frac{1}{l^{2}-m^{2}_{\eta^{\prime}}}\chi_{\rm
YM}(l^{2})\,,$ (63)

where we introduced

$\displaystyle
m^{2}_{\eta^{\prime}}\equiv-\frac{2\,n_{f}}{F^{2}_{{\bar{\eta}}}}\chi_{\rm
YM}(0)\,.$ (64)

This last expression is the well-known Witten-Veneziano formula Witten (1979);
Veneziano (1979) for the mass of the $\eta^{\prime}$ meson.

Taking the forward limit, we find

$\displaystyle\chi(0)=0\,,$ (65)

which follows from topological mass generation: the WZW mixing of the massless
${\bar{\eta}}$ field with the topological charge density induces a massive
$\eta^{\prime}$ field. As we saw in the previous section, combining this
result with Eq. (56) yielded Eq. (57) relating the ${\bar{\eta}}$ decay
constant $F_{\bar{\eta}}$ to the vacuum correlator $\langle
0|T\,\Omega\eta_{0}|0\rangle$ in the forward limit. We will now show that the
latter is given by the slope of the topological susceptibility
$\chi^{\prime}(l^{2})$.

Figure 4: a: Leading order contribution to $\langle\Omega\eta_{0}\rangle$; b:
The full correlator for $\langle\Omega\eta_{0}\rangle$ replacing $\chi_{\rm
YM}\rightarrow\chi$, as represented by the grey blob.

####  IV.2.2 The WZW-${\bar{\eta}}$ term and the correlator $\langle
0|T\Omega\eta_{0}|0\rangle$

The leading order diagram contributing to $\langle 0|T\Omega\eta_{0}|0\rangle$
is shown in Fig. 4a. Its computation, following our diagrammatic rules, is
straightforward, and gives232323Our derivation of Eq. (39) was flavor blind so
the only difference in the coupling of $\Omega$ to $\eta_{0}$ relative to its
coupling to ${\bar{\eta}}$ is to replace $n_{f}\rightarrow{\tilde{n}}_{f}$.,

$\displaystyle\langle 0|T\Omega\eta_{0}|0\rangle|_{\rm
Fig.\ref{figsup2}a}=\Big{[}-i\chi_{\rm
YM}(l^{2})\Big{]}\cdot\Big{[}-i\frac{\sqrt{2\tilde{n}_{f}}}{F_{\bar{\eta}}}\Big{]}\cdot\frac{i}{l^{2}}\,.$
(66)

To obtain the full correlator from Fig. 4b, one can simply replace $\chi_{\rm
YM}(l^{2})\to\chi(l^{2})$:

$\displaystyle\langle 0|T\Omega\eta_{0}|0\rangle|_{\rm
Fig.\ref{figsup2}b}=-i\frac{1}{l^{2}}\frac{\sqrt{2\tilde{n}_{f}}}{F_{\bar{\eta}}}\chi(l^{2})\,.$
(67)

We note that this result agrees with the parametrization of this two-point
Green function in Shore and Veneziano (1992); Shore (2008).

Substituting this result into the r.h.s of Eq. (57), we find

$F^{2}_{\bar{\eta}}=\lim_{l\to 0}\frac{2n_{f}}{l^{2}}\chi(l^{2})\,.$ (68)

Expanding the topological susceptibility $\chi(l^{2})$ in a Taylor series
around $l^{2}=0$ as

$\displaystyle\chi(l^{2})=\chi(0)+l^{2}\chi^{\prime}(0)+\dots\,,$ (69)

where $\chi^{\prime}(l^{2})\equiv\frac{d}{dl^{2}}\chi(l^{2})$, and taking into
account that $\chi(0)=0$ in the chiral limit, we can substitute the second
term of the expansion into Eq. (68), which yields

$F^{2}_{\bar{\eta}}=2n_{f}\chi^{\prime}(0)\,.$ (70)

As a result, using Eq.(52), we obtain,

$\langle
P^{\prime},S|J^{\mu}_{5}|P,S\rangle=\sqrt{\frac{2}{3}}\,2n_{f}\,g_{\eta_{0}NN}\,\sqrt{\chi^{\prime}(0)}\,S^{\mu}\rightarrow\Sigma(Q^{2})=\sqrt{\frac{2}{3}}\,\frac{2n_{f}}{M_{N}}\,g_{\eta_{0}NN}\sqrt{\chi^{\prime}(0)}\,.$
(71)

This expression for the proton helicity in terms of square root of the slope
of the QCD topological susceptibility was first obtained by Shore and
Veneziano Shore and Veneziano (1992); Shore (2008) from manipulations of the
anomalous chiral Ward identities; we have provided here a complementary and
intuitive derivation.

As mentioned earlier, this nontrivial result provides a simple explanation for
why $\Sigma(Q^{2})$ is anomalously small; the forward topological
susceptibility $\chi(0)$ in the chiral limit is strictly zero, and a non-zero
contribution to $\Sigma(Q^{2})$ can only arise from small deviations from it,
as represented by its slope at $l^{2}=0$, the scale for which is set by the
large value of $m_{\eta^{\prime}}$. Narison, Shore and Veneziano Narison  _et
al._ (1995, 1999) employed QCD sum rules to evaluate this expression obtaining
results in agreement with the HERMES Airapetian  _et al._ (2007a) and COMPASS
Alekseev  _et al._ (2010a) data we quoted after Eq. (43). A very recent update
to these sum rule determinations is given in Narison (2021). One can in
principle compute $G_{A}$ in Eq. (43) directly on the lattice by computing the
off-forward matrix element of $J_{\mu}^{5}$; however one has to ensure that
its anomalous Ward identity is satisfied Liu (1992). Alternatively, one can
instead determine $\Sigma$ from Eq. (71) by computing $\chi^{\prime}(0)$ on
the lattice Giusti  _et al._ (2002); for a discussion of the current status of
computations of the topological susceptibility and relevant references, we
refer the reader to Bali  _et al._ (2021).

####  IV.2.3 ${\bar{\eta}}$ effective action

In Eq. (39), we obtained the form of the ${\bar{\eta}}\,\Omega$ WZW coupling
from the imaginary part of the worldline effective action. Further, from Eqs.
(29), (33) and (32), we can deduce a kinetic term for the ${\bar{\eta}}$ field
Mondragon  _et al._ (1995); Schubert (2001),

$\displaystyle S_{\rm kinetic}=\int
d^{4}x\,\frac{1}{2}\,(\partial_{\mu}{\bar{\eta}})\,(\partial^{\mu}{\bar{\eta}})\,.$
(72)

Indeed, we employed this kinetic term in our diagrammatic analysis.

There is an additional “$\theta$-term” contribution from the imaginary part of
the worldline effective action which has the same structure as Eq. (39), and
is given by Kaiser and Leutwyler (2000)

$\displaystyle S_{\theta}=\int d^{4}x\,\theta\,\Omega\,.$ (73)

Finally, there is also a term (see Witten (1980) and references therein)
representing the free energy of the $\theta$-vacuum given by

$\displaystyle S_{\rm vac}=\frac{1}{2}\int d^{4}x\,\chi_{\rm
YM}\,\theta^{2}\,,$ (74)

whose second derivative with respect to theta, at $\theta=0$, defines the
Yang-Mills topological susceptibility.

Putting everything together, we can write the low energy ${\bar{\eta}}$
effective action as

$\displaystyle S_{{\bar{\eta}}}=\int
d^{4}x\left[\frac{1}{2}\,(\partial_{\mu}{\bar{\eta}})\,(\partial^{\mu}{\bar{\eta}})+\left(\theta-\frac{\sqrt{2n_{f}}}{F_{{\bar{\eta}}}}{\bar{\eta}}\right)\,\Omega+\frac{\chi_{\rm
YM}}{2}\,\theta^{2}\,\right]\,.$ (75)

Since there is no kinetic term for $\theta$, it acts as a constraint and can
be eliminated using the equations of motion:

$\displaystyle\frac{\delta
S_{{\bar{\eta}}}}{\delta\theta}=0\rightarrow\theta=-\frac{\Omega}{\chi_{\rm
YM}}\,.$ (76)

Plugging this back into Eq.(75), we get

$\displaystyle S_{{\bar{\eta}}}=\int
d^{4}x\left[\frac{1}{2}\,(\partial_{\mu}{\bar{\eta}})\,(\partial^{\mu}{\bar{\eta}})-\frac{\sqrt{2n_{f}}}{F_{{\bar{\eta}}}}{\bar{\eta}}\,\Omega-\frac{\Omega^{2}}{2\,\chi_{\rm
YM}}\right]\,.$ (77)

This form of the action is identical to that of Shore and Veneziano Shore
(2008) (see also Hatsuda (1990)) which they argue to be the simplest effective
action consistent with the anomalous Ward identities of QCD. We will employ
its equivalent representation in Eq. (75) in our discussion in Sec. V.

Defining the $\eta^{\prime}$ field as

$\displaystyle\eta^{\prime}=\frac{F_{\eta^{\prime}}}{F_{{\bar{\eta}}}}{\bar{\eta}}\,,$
(78)

where $F_{\eta^{\prime}}$ is the $\eta^{\prime}$ decay constant (see Section 3
of Shore (2008)) and introducing a glueball field defined as

$\displaystyle G=\Omega+\frac{\sqrt{2n_{f}}}{F_{\eta^{\prime}}}\,\chi_{\rm
YM}\,\eta^{\prime}\,,$ (79)

one can reexpress the effective action in Eq. (77) as

$\displaystyle S_{{\eta^{\prime}}}=\int
d^{4}x\left[-\frac{1}{2}\,\eta^{\prime}\,\left(\partial^{2}+m_{\eta^{\prime}}^{2}\right)\eta^{\prime}-\frac{G^{2}}{2\chi_{\rm
YM}}\right]\,,$ (80)

where $m_{\eta^{\prime}}$ is the $\eta^{\prime}$ mass given by the Witten-
Veneziano formula in Eq. (64). Thus the mixing between ${\bar{\eta}}$ and
$\Omega$ in Eq. (75) generates an effective action describing the physical
massive $\eta^{\prime}$ and a non-propagating glueball field $G$, which
decouples from the hadron spectrum. Note further that as
$N_{c}\rightarrow\infty$, one has $\eta^{\prime}\rightarrow{\bar{\eta}}$,
since $m_{\eta^{\prime}}\rightarrow 0$. In this “OZI limit” of QCD, the
anomaly vanishes restoring $U_{A}(1)$ and the ${\bar{\eta}}$ is the prodigal
ninth Goldstone boson.

Recall from Section III that Eqs. (39), (72) and (73) can be understood as
arising from the phase of the Dirac determinant in the QCD effective action,
where the relevant low energy degrees of freedom are parametrized with scalar,
pseudoscalar, vector and axial vector degrees of freedom. There is of course
the path integral over the gauge field configurations to consider. In ’t
Hooft’s ’t Hooft (1976, 1986) explanation of the $U_{A}(1)$ problem, classical
(Euclidean) instanton gauge field configurations are the dominant
configurations responsible for the coupling of the topological charge density
to fermion zero modes Leutwyler and Smilga (1992); hence $\chi_{\rm YM}$ in
Eq. (74) in this picture is saturated by the dynamics of such configurations
Schäfer and Shuryak (1998). However as pointed out by Veneziano Veneziano
(1979), that while sufficient, instanton configurations are not required for
the solution of the $U_{A}(1)$ problem. Indeed the discussion above did not
invoke the instanton picture at all though it is consistent with it. We will
argue in the next section that while instanton-anti-instanton configurations
may dominate at large $x_{B}$, the physics of gluon saturation suggests that
other classical configurations increasingly begin to play a role on the short
time scales probed by a DIS probe with decreasing $x_{B}$.

##  V Axion-like action at small $x_{B}$: Gluon saturation and sphaleron
transitions

The triangle graph, as noted previously here, and in detail in Paper I,
dominates the box diagram contributing to $g_{1}(x_{B},Q^{2})$ in both Bjorken
and Regge asymptotics. The dynamics underlying the $x_{B}$ dependence is
therefore contained in the diagrams shown in Fig 2, whose interplay we
discussed on general grounds in the previous section. We will now consider
these in greater detail and point to novel features that emerge at small
$x_{B}$.

Figure 5: Diagram illustrating isosinglet quark exchange in the t-channel.
Such reggeon contributions are allowed at both large and small $x_{B}$, though
they are suppressed at small $x_{B}$.

A subtle point which will govern our analysis must be noted at the outset. As
we observed in Eq. (43), it is sufficient to compute the $x_{B}$-dependent
generalization of the coupling of the axial charge to the nucleon, as shown in
Fig. 2(a). Alternately, one could employ the anomalous Goldberger-Treiman we
derived in Eq. (51) and compute instead the $x_{B}$-dependent generalization
of Fig. 2(b), specifically that of the product $F_{\eta_{0}}\,g_{\eta_{0}NN}$.

The dynamics underlying Figs. 2 (a) and (b) is illustrated in Fig. 5. Fig. 2
(a) corresponds to a direct axial coupling at a given $x_{B}$ to a valence
quark in Fig. 5. Though Fig. 2 (b) is formally represented as a Feynman
diagram in Fig. 5, its physics (due to its sensitivity to the off-forward
pole) is governed primarily by low frequency modes of the fermion determinant
and is fundamentally nonperturbative. Spin diffusion along the
t-channel242424One may ask whether spin diffusion can occur instead due to
spin precession in a background field. In Eq. (7), these contributions to
$g_{1}$ would correspond to terms linear in $F_{12}$. However as discussed at
length in Appendix C, for operators sensitive to the anomaly, terms that are
naively sub-leading (relative to this linear term) in an eikonal expansion
cannot be ignored due to the off-forward pole in the t-channel. This leads to
Fig. 2 (b) being sensitive to the anomaly, as seen clearly in Eq. (53). can be
viewed as being mediated by reggeon exchange (corresponding to the $\eta_{0}$)
in the isosinglet sector. At large $x_{B}$, $g_{1}$ can be computed using
lattice QCD methods Gockeler  _et al._ (1996); Liang  _et al._ (2018);
Alexandrou  _et al._ (2020); Mejía-Díaz  _et al._ (2018); Lin  _et al._
(2018); Giusti  _et al._ (2002); Bali  _et al._ (2021). However at small
$x_{B}$, particularly in Regge asymptotics, lattice computations are
challenging due to the difficulties posed by i) computing higher moments of
local operators, ii) boosting the proton to high energies on the lattice
Shanahan (2018).

While one might consider this situation challenging for small $x_{B}$
computations, the “anomaly diagram” in Fig. 2 (c) provides a crucial assist as
we will describe shortly. Firstly, note that we showed explicitly (in the
discussion culminating in Eq. (65)) that the other possible anomaly diagram
Fig. 2(d) is zero in the chiral limit. Because of the anomaly contribution in
Fig. 2(c), we showed in Eq. (58) that the matrix element for $J_{\mu}^{5}$ can
be expressed in terms of the matrix element of the topological charge density,
or equivalently, in terms of $\chi^{\prime}(0)$, as shown in Eq. (71). The
latter expression is manifestly finite in the limit $l\rightarrow 0$.

Figure 6: Diagram illustrating the t-channel propagation of an isosinglet
$\eta_{0}$ pseudoscalar field Bjorken and Regge asymptotics. This $\eta_{0}$
field has a nonperturbative distribution that couples to polarized valence
partons with large momentum fractions in the proton. As it propagates along
the t-channel, the $\eta_{0}$ couples to the vacuum topological charge density
represented by the correlator $\langle\Omega\eta_{0}\rangle$ (illustrated in
Fig. 4 (b)), acquiring a mass in the process before its coupling to the
triangle graph via the anomaly.

We can further explore the structure of Eq. (58) by rewriting the vacuum
correlator as a functional integral over $A$ and $\bar{\eta}$ fields as

$\displaystyle\langle P,S|J^{\mu}_{5}|P,S\rangle=2\,n_{f}\,i\,\int
d^{4}y\,\int{D{\bar{\eta}}}\,\,{\tilde{W}}_{P,S}[{\bar{\eta}}]\,\int[DA]$
$\displaystyle\times\Omega(0)\,\eta_{0}(y)\,\exp\left(iS_{\rm YM}+i\int
d^{4}x\left[\frac{1}{2}\,(\partial_{\mu}{\bar{\eta}})\,(\partial^{\mu}{\bar{\eta}})-\frac{\sqrt{2n_{f}}}{F_{{\bar{\eta}}}}{\bar{\eta}}\,\Omega\right]\right)\,S^{\mu}\,,$
(81)

where we generalized $g_{\eta_{0}NN}$ in Eq. (58) by introducing the weight
functional ${\tilde{W}}_{P,S}[{\bar{\eta}}]$; it represents the
nonperturbative distribution of ${\bar{\eta}}$ determined (with the
$\sqrt{\frac{2}{3}}$ normalization) from the pseudoscalar coupling of the
$\eta_{0}$ field to the polarized proton with the normalization

$\int[D{\bar{\eta}}]\,{\tilde{W}}_{P,S}[{\bar{\eta}}]=g_{\eta_{0}NN}\,.$ (82)

Comparing Eq. (V) with the contribution of the triangle anomaly to the matrix
element of the axial vector current given in Eq. (48), we observe that the
regularization of the infrared pole is equivalent to replacing
$\frac{l^{\mu}}{l^{2}}$ in the matrix element by the above functional
integral.

The ${\bar{\eta}}\,\Omega$ term in Eq. (V), after expanding out to linear
order in ${\bar{\eta}}$, gives

$\langle{\bar{\eta}}{\bar{\eta}}\rangle\,\langle\Omega\Omega\rangle\,,$ (83)

where the first expectation value gives the ${\bar{\eta}}$-propagator
$\frac{i}{l^{2}}$ and the second, the Yang-Mills topological susceptibility
$\chi_{{\rm YM}}(l^{2})$, as seen previously in Eq. (66). Subsequent expansion
to $O({\bar{\eta}}^{3})$, and higher odd powers in ${\bar{\eta}}$ then give
Eq. (60), illustrated by the correction to Fig. 3a shown in Fig. 3b,

$\langle{\bar{\eta}}{\bar{\eta}}\rangle\,\langle\Omega\Omega\rangle\langle{\bar{\eta}}{\bar{\eta}}\rangle\,\langle\Omega\Omega\rangle\cdots\,,$
(84)

with the resummation of such contributions to all orders generating the
$\chi(l^{2})$ in Eq. (62).

An analogous regularization of the infrared pole in the expression for the
structure function $g_{1}$ in Eq. (10) yields

$\displaystyle
g_{1}(x_{B},Q^{2})=\left(\sum_{f}e_{f}^{2}\right)\frac{n_{f}\alpha_{s}}{\pi
M_{N}}i\int
d^{4}y\,\int^{1}_{x_{B}}\frac{dx}{x}~{}\Big{(}1-\frac{x_{B}}{x}\Big{)}\int\frac{d\xi}{2\pi}e^{-i\xi
x}\,\int{D{\bar{\eta}}}\,{\tilde{W}}_{P,S}[{\bar{\eta}}]\,\int[DA]$
$\displaystyle\times\,{\rm Tr_{c}}F_{\alpha\beta}(\xi
n)\tilde{F}^{\alpha\beta}(0)\,\eta_{0}(y)\,\exp\left(iS_{\rm YM}+i\int
d^{4}x\left[\frac{1}{2}\,(\partial_{\mu}{\bar{\eta}})\,(\partial^{\mu}{\bar{\eta}})-\frac{\sqrt{2n_{f}}}{F_{{\bar{\eta}}}}{\bar{\eta}}\,\Omega\right]\right)\,.$
(85)

The last two terms in the exponential, corresponding to the dynamics of the
${\bar{\eta}}$, are identical to that describing the coupling of a putative
axion particle Wilczek (1978); Weinberg (1978); Kim (1987) to QCD matter. The
underlying dynamics of the functional integral representation of Eq. (V) is
illustrated in Fig. 6. This expression is consistent with Eq. (77) if we
assume the latter to be saturated by nonperturbative classical configurations.

We will now argue that novel dynamics emerges in Regge asymptotics that allows
one to compute the dynamics inside the blob in Fig. 6 in a weak coupling
framework. This dynamics is due to the phenomenon of gluon saturation Gribov
_et al._ (1983); Mueller and Qiu (1986) at small $x_{B}$ corresponding to the
close packing of gluons in the hadron. At maximal occupancies of
$O(1/\alpha_{s})$, the dynamics is controlled by a saturation scale
$Q_{S}(x_{B})\gg\Lambda_{\rm QCD}$, which screens color charge beyond this
close packing scale. Since $\alpha_{s}(Q_{S})\ll 1$ in the Regge limit, the
high occupancy of closely packed glue within a radius $1/Q_{S}(x_{B})$ inside
the proton forms a classical lump. Further, in this limit, its dynamics can be
studied systematically in weak coupling McLerran and Venugopalan (1994a, b,
c).

Gluon saturation has been studied extensively within the framework of the
Color Glass Condensate (CGC) effective field theory Iancu and Venugopalan
(2003); Gelis  _et al._ (2010); Kovchegov and Levin (2012); Blaizot (2017). In
short, large $x$ color charges in a hadron or nucleus are treated as static
classical color charges with color charge density $\rho$ coupled to classical
gauge field configurations $A_{\rm cl}[\rho]\sim 1/g$, where $g$ is the gauge
coupling. Sources and fields are separated at the scale
$x_{0}=\Lambda^{+}/P^{+}$; at small $x$, logarithmically enhanced gluon
emissions (LLx) $\alpha_{s}\ln(x_{0}/x_{0}^{\prime})~{}O(1)$ from the fields
can be absorbed into a new source distribution $\rho^{\prime}$ at the scale
$x_{0}^{\prime}={\Lambda^{\prime}}^{+}$, and iterated, satisfying a Wilsonian
renormalization group evolution equation described by a JIMWLK Hamiltonian
Jalilian-Marian  _et al._ (1998a, b, 2000); Iancu  _et al._ (2001); Ferreiro
_et al._ (2002). The JIMWLK Hamilitonian describes the energy evolution of the
saturation scale which specifies the nonperturbative distribution
$W_{Y_{0}}[\rho]$ of color sources at the initial rapidity scale
$Y_{0}=\ln(1/x_{0})$.

A detailed derivation in the worldline formalism of the expectation value of
operators in an unpolarized proton or nucleus is given in Appendix B. One
obtains (see Eqs. (165) and (166)),

$\displaystyle\langle\mathcal{O}\rangle_{\rm
unpol.}=\int\mathcal{D}\rho~{}W_{Y}[\rho]~{}\int\mathcal{D}A~{}\mathcal{O}[A]~{}e^{iS_{\rm
CGC}[A,\rho]}\,,$ (86)

where $Y$ is the rapidity of interest that $W_{Y}[\rho]$ has been evolved to,
and

$\displaystyle S_{\rm CGC}[A,\rho]=-\frac{1}{4}\int
d^{4}xF^{\mu\nu}_{a}F^{a}_{\mu\nu}+\frac{i}{N_{c}}\int d^{2}x_{\perp}~{}{\rm
tr}_{c}\big{[}\rho(x_{\perp})\ln\big{(}U_{[\infty,-\infty]}(x_{\perp})\big{)}\big{]}\,,$
(87)

with
$U_{[\infty,-\infty]}(x_{\perp})=\exp\Big{[}-ig\int^{\infty}_{-\infty}dx^{+}A^{-}(x^{+},z_{\perp})\Big{]}$.
The shock wave classical field $A_{\rm cl.}^{\mu}[\rho]$ corresponding to the
saddle point of this effective action is well-known; we will discuss it
further in Sec. V.2.

Our interest here is in deriving the spin-dependent effective action in the
Regge limit of $x_{B}\rightarrow 0$. From our general discussion in Appendix
B, in addition to the evolution of the initial density matrix of the polarized
proton in coordinate/momentum phase space, and in color, we must consider its
evolution in spin and flavor. As we have discussed at length, the evolution of
the density matrix in the flavor isosinglet sector is governed by the
imaginary part of the worldline effective action, specifically the WZW term in
Eq. (39) and the corresponding kinetic term for the ${\bar{\eta}}$ field in
Eq. (72). Combining these with the CGC effective action (87) we obtain

$\displaystyle\langle\mathcal{O}\rangle_{\rm pol.}^{\rm
Regge}=\int\mathcal{D}\rho\,W_{Y}[\rho]\int{D{\bar{\eta}}}\,{\tilde{W}}_{P,S}[{\bar{\eta}}]\int\mathcal{D}A~{}\mathcal{O}[A]~{}e^{iS_{\rm
pCGC}[A,\rho,{\bar{\eta}}]}\,,$ (88)

where the spin-polarized CGC effective action is

$\displaystyle S_{\rm pCGC}[A,\rho,{\bar{\eta}}]=S_{\rm CGC}[A,\rho]+\int
d^{4}x\left[\frac{1}{2}\,(\partial_{\mu}{\bar{\eta}})\,(\partial^{\mu}{\bar{\eta}})-\frac{\sqrt{2n_{f}}}{F_{{\bar{\eta}}}}{\bar{\eta}}\,\Omega\right]\,.$
(89)

In particular, Eq. (V) for $g_{1}$ in the Regge limit is252525As discussed in
Paper I, a consistent treatment in the worldline formalism would set $x_{B}=0$
in the argument of Eq. (V).

$\displaystyle g_{1}^{\rm
Regge}(x_{B},Q^{2})=\left(\sum_{f}e_{f}^{2}\right)\frac{n_{f}\alpha_{s}}{\pi
M_{N}}i\int
d^{4}y\,\int^{1}_{x_{B}}\frac{dx}{x}~{}\int\frac{d\xi}{2\pi}e^{-i\xi
x}\,\int\mathcal{D}\rho~{}W_{Y}[\rho]\,\int{D{\bar{\eta}}}\,{\tilde{W}}_{P,S}[{\bar{\eta}}]\,\int[DA]$
$\displaystyle\times\,{\rm Tr_{c}}F_{\alpha\beta}(\xi
n)\tilde{F}^{\alpha\beta}(0)\,\eta_{0}(y)\,\exp\left(iS_{\rm CGC}+i\int
d^{4}x\left[\frac{1}{2}\,(\partial_{\mu}{\bar{\eta}})\,(\partial^{\mu}{\bar{\eta}})-\frac{\sqrt{2n_{f}}}{F_{{\bar{\eta}}}}{\bar{\eta}}\,\Omega\right]\right)\,.$
(90)

This functional integral, describing spin diffusion of $S^{\mu}$ at small
$x_{B}$, is illustrated in Fig. 7.

Figure 7: Left: Diagram similar to Fig. 2 (c) but now including the coupling
to color sources with each $\rho$ insertion denoted by the “operator” symbol.
Right: The correlator $\langle\Omega{\bar{\eta}}\rangle$ in the diagram on the
left in the background field generated by static classical color sources
$\rho$ at small $x_{B}$.

The path integral over $\rho$ causes $g_{1}^{\rm Regge}$ to differ
qualitatively from the corresponding expression in the Bjorken limit. In Fig.
7 (left), the coupling of the color sources to $\chi(l^{2})$ is illustrated
with the “operator” symbol here representing the $\rho$ insertions; the figure
on the right represents the correlator $\langle\Omega{\bar{\eta}}\rangle$
inside Fig. 7 (left).

In the absence of the coupling to the ${\bar{\eta}}$ field, the topological
charge density $\Omega\propto{\bf E}\cdot{\bf B}$ in the CGC is zero Kharzeev
_et al._ (2002); we note though that a finite $\Omega$ is generated in a
nuclear collision Kharzeev  _et al._ (2002); Lappi and McLerran (2006);
Kharzeev  _et al._ (2008); Fillion-Gourdeau and Jeon (2009); Mace  _et al._
(2016); Jokela  _et al._ (2020). The interplay of the axion-like coupling with
the CGC was also considered recently for the case of a physical axion
interacting with the CGC, which bears strong similarity with our problem of
the ${\bar{\eta}}$ interacting with the CGC gauge fields Jokela  _et al._
(2020). It is also analogous to the problem of an axion or axion-like field
propagating in a hot non-Abelian plasma, which is relevant in a number of
cosmological contexts McLerran  _et al._ (1991); Berghaus  _et al._ (2020).
The axion dynamics considered in McLerran  _et al._ (1991) and Jokela  _et
al._ (2020), respectively, are especially relevant because, as we will now
discuss, they bookend two model approaches to computing the effective action
in Eq. (V) that apply in different kinematic regimes of interest.

###  V.1 Spin diffusion via over-the-barrier topological transitions

Figure 8: The vacuum energy of $\theta$-vacua as function of the Chern-Simons
number $N_{\rm CS}$ labeled by positive and negative integers. The arrows
denote instanton/anti-instanton tunneling configurations and over-the-barrier
sphaleron configurations; both of these effects result in a change of the
Chern-Simons number.

In the first approach, the saddle point in Eq. (V) of the path integral in
over gauge fields, after an analytic continuation to Euclidean space262626The
color sources $\rho$ are static on the relevant time scales., is given by
instanton classical fields $A_{\mu}^{\rm inst.}$ satisfying
$F_{\mu\nu}=\pm{\tilde{F}}_{\mu\nu}$. As we noted previously, without the term
in Eq. (87) coupling the gauge fields to $\rho$, the effective action in Eq.
(V) should be identical to the effective action in Sec. IV.2.3 with the
instanton fields saturating the topological charge density Schäfer and Shuryak
(1998). Recall that this action reproduces the results of Sec. IV.

In this approach, the $J\cdot A$ coupling of the color sources to the gauge
fields, along with the average over the color density matrix $W_{Y}[\rho]$ can
be considered analogous to a thermal average272727The evolution equation for
$W_{Y}[\rho]$ satisfies the Kossakowski-Lindblad form of the density matrix
for open quantum systems Armesto  _et al._ (2019)., with the saturation scale
$Q_{S}$ playing an analogous role to the temperature.

The computation then follows along the lines282828One must understand ${\bf
E}\equiv{\bf E(\rho)}$ and ${\bf B}\equiv{\bf B(\rho)}$ in the Hamiltonian
employed in that derivation. of McLerran  _et al._ (1991). From the equations
of motion in Eq. (V), one has

$\frac{\sqrt{2n_{f}}}{F_{\eta^{\prime}}}\Omega=-\partial_{\mu}\partial^{\mu}{\eta^{\prime}}\,,$
(91)

The explicit derivation in McLerran  _et al._ (1991), performed in the real-
time Schwinger-Keldysh formalism, gives after thermal averaging (here
corresponding to the averaging of sources),

$\frac{\partial^{2}{\eta^{\prime}}}{\partial
t^{2}}=-\gamma\frac{\partial{\eta^{\prime}}}{\partial
t}-m_{\eta^{\prime}}^{2}{\eta^{\prime}}\,.$ (92)

A subtle point, discussed at length in McLerran  _et al._ (1991), is that the
coupling of the gauge fields to the color charges does not alter topological
mass generation whereby ${\bar{\eta}}\rightarrow\eta^{\prime}$. Both
${\bar{\eta}}$ and $\eta^{\prime}$ couple identically to $\Omega$, with the
only difference being the strength of the coupling given by the difference in
their respective decay constants.

In particular, note that the only difference in the equations of motion
relative to that derived from Eq. (80) is the term with the friction
coefficient $\gamma$. This term reflects the drag on $\eta^{\prime}$
propagation due to the coupling of the color sources to the gauge field. In
our picture, this is fundamentally what causes the quenching of the
coefficient ($g_{1}$) of the spin four-vector $S^{\mu}$ reflecting the
efficiency of spin diffusion292929In high energy DIS, it is more convenient to
represent Eq. (91) in lightcone coordinates. Clearly, this choice of
coordinates should not alter our discussion of the physics of spin diffusion..

The underlying dynamics is illustrated in Fig. 8. In ‘t Hooft’s picture ’t
Hooft (1976, 1986), tunneling instanton-anti-instanton configurations generate
the nontrivial Yang-Mills topological susceptibility which, we have seen, are
responsible for the large $\eta^{\prime}$ mass. The effect of the coupling to
large $x$ sources, and the averaging over $W_{Y}[\rho]$ is to introduce the
saturation momentum $Q_{S}>\Lambda_{\rm QCD}$, which can lead to over-the-
barrier sphaleron transitions as shown in Fig. 8. For the finite temperature
case, the friction coefficient $\gamma$ is proportional to the sphaleron
transition rate McLerran  _et al._ (1991): $\gamma=2n_{f}\Gamma_{\rm
sphaleron}/F_{\bar{\eta}}^{2}T$, where303030The Chern-Simons current
$K^{\mu}=\frac{g^{2}}{8\pi^{2}}\varepsilon^{\mu\nu\rho\sigma}{\rm
Tr}\left(A_{\nu}\partial_{\rho}A_{\sigma}-\frac{2ig}{3}A_{\nu}A_{\rho}A_{\sigma}\right)$,
which satisfies $\partial_{\mu}K^{\mu}=\Omega$.

$\Gamma_{\rm sphaleron}=\lim_{\delta t\rightarrow\infty}\frac{1}{V\delta
t}\langle\left(N_{\rm CS}(t+\delta t)-N_{\rm CS}(t)\right)^{2}\rangle\,,$ (93)

with $N_{\rm CS}(t)=\int d^{3}x\,K^{0}$. Here $V$ denotes the three
dimensional volume of the system. At finite temperature, $\Gamma_{\rm
sphaleron}=\kappa\alpha_{s}^{5}\,T^{4}$, where $\kappa$ is a nonperturbative
constant Moore and Tassler (2011). In the CGC, from parametric arguments
alone313131Interestingly, in numerical simulations of the hot and dense Glasma
Kharzeev  _et al._ (2002); Lappi and McLerran (2006); Mace  _et al._ (2016)
produced in a nuclear collision, one finds that the sphaleron transition rate
scales with the string tension of a spatial Wilson loop in the Glasma Mace
_et al._ (2016). , one can deduce that $\Gamma_{\rm sphaleron}\propto
Q_{S}^{4}$ and $\gamma\propto 2n_{f}\frac{Q_{S}^{3}}{F_{\bar{\eta}}^{2}}$.
Parametrically, for $t\sim 1/Q_{S}$, the interaction time of the probe with
the shock wave, the first term on the r.h.s will dominate over the second for
when $\gamma^{2}>m_{\eta^{\prime}}^{2}$, or equivalently, when
$\frac{2n_{f}}{F^{2}_{\bar{\eta}}}Q^{6}_{S}>\chi_{\rm YM}(0)$. When the
friction term dominates, ${\bar{\eta}}\propto m_{\eta^{\prime}}\,\exp(-\gamma
t)$ . From Eq. (91), we then have

$\langle\Omega\eta_{0}\rangle\propto F_{\bar{\eta}}\,\chi_{\rm
YM}(0)\,\left(\frac{Q_{S}^{2}}{F_{\bar{\eta}}^{2}}\right)^{3}\exp\left(-4n_{f}\,C\,\frac{Q_{S}^{2}}{F_{\bar{\eta}}^{2}}\right)\,,$
(94)

for $t\sim\frac{1}{Q_{S}}$, with $\langle\cdots\rangle$ denoting the average
over the path integrals in Eq. (V). Here $C$ is a nonperturbative constant and
we have employed Eq. (64), the Witten-Veneziano formula. Substituting this
expression in Eq (V), we obtain323232In writing this expression we have
assumed that the four-volume corresponding to the $\eta^{\prime}$ field is
sensitive only to scales $1/Q_{S}$ over which a sphaleron transition takes
place inducing the drag, but is homogeneous over longer spacetime scales, as
suggested by Eq. (92). Since the sphaleron transition rate is defined per unit
four volume, the two factors effectively cancel. We have also assumed that the
phase does not contribute at small $x_{B}$.

$\displaystyle g_{1}^{\rm
Regge}(x_{B},Q^{2})\propto\frac{Q_{S}^{2}m_{\eta^{\prime}}^{2}}{F_{\bar{\eta}}^{3}M_{N}}\,\exp\left(-4\,n_{f}C\,\frac{Q_{S}^{2}}{F_{\bar{\eta}}^{2}}\right)\,,$
(95)

We have not specified the prefactors of the expression in this model
computation (along the lines of McLerran  _et al._ (1991)) of the effective
action in Eq. (V) because, unless $C$ is much smaller than $O(1)$, they do not
affect the takeaway message that $g_{1}$ is exponentially quenched with
increasing $Q_{S}$, already for $Q_{S}$ of a few hundred MeV.

A detailed derivation of the arguments outlined above and predictions for
polarized DIS measurements at the EIC Accardi  _et al._ (2016); Aschenauer
_et al._ (2019) will be the subject of Paper III Tarasov and Venugopalan . The
kinematic regime where they are valid is where the coupling of color sources
to the gauge fields can be treated as a perturbation to the instanton-anti-
instanton configurations populating the QCD vacuum. More specifically, one
requires small $x_{B}$ values where the color sources can be approximated as
classical color charge configurations but one still has
$Q_{S}<m_{\eta^{\prime}}$. We will now turn our attention to the strict Regge
regime where $x_{B}\rightarrow 0$, giving $Q_{S}\gg m_{\eta^{\prime}}$.

###  V.2 Spin diffusion through topological shock wave configurations

Figure 9:  Left: The shock wave gauge field configuration $A_{\mu}$ of the MV
model McLerran and Venugopalan (1994a, b) generated by static classical color
sources $\rho$. Right: The modified shock wave configuration
$A_{\mu}+a_{\mu}$, where $a_{\mu}$ is generated by the axion-like current
resulting from the WZW ${\bar{\eta}}\,\Omega$ coupling.

In Regge asymptotics, the coupling term in Eq. (87) is as large as the Yang-
Mills action, with both being $O(1/g^{2})$. It can no longer be thought of as
a perturbation to the instanton-anti-instanton configurations. The saddle
point solution that minimizes the action in Eq. (87) corresponds to the CGC
shock wave solutions McLerran and Venugopalan (1994a, b); Jalilian-Marian  _et
al._ (1997); Kovchegov (1996). These configurations by themselves do not carry
any topological charge. However the presence of the ${\bar{\eta}}\,\Omega$
term in Eq. (V) changes this result qualitatively.

Similarly to the discussion in Jokela  _et al._ (2020), the equations of
motion are

$\displaystyle D_{\mu}F^{\mu\nu}=J^{\nu}+j_{\rm
ax}^{\nu}\,\,\,;\,\,\,D_{\mu}{\tilde{F}}^{\mu\nu}=0\,,$ (96)

with

$\displaystyle j_{\rm
ax}^{\nu}=-\partial_{\mu}{\bar{\eta}}\,{\tilde{F}}^{\mu\nu}\,\,\,{\rm
with}\,\,\,D_{\mu}j_{\rm ax}^{\mu}=0\,.$ (97)

In contrast to Sec. V.1, we will not consider the backreaction on the
${\bar{\eta}}$ fields from $\Omega$, since the shock wave fields are much
shorter-lived on the relevant time scales in Regge asymptotics.

In the absence of the axion current $j_{\rm ax}^{\mu}$, the equations of
motion can be solved exactly. For a nucleus moving with large
$P^{+}\rightarrow\infty$, for the corresponding lightcone current given by
$J^{\mu}=\delta^{\mu+}\rho(x^{-},x_{\perp})$, one obtains (in Lorenz gauge
$\partial_{\mu}A_{\rm cl}^{\mu}=0$)

$A_{\rm cl}^{+}=-\frac{\rho}{\nabla_{\perp}^{2}}\,\,\,;\,\,\,\,A_{\rm
cl}^{-}=0\,\,\,\,;\,\,\,\,A_{\rm cl}^{i}=0\,.$ (98)

These gauge configurations are static in $x^{+}$ since $D_{\mu}J^{\mu}=0$,
where $D_{\mu}$ represents the covariant derivative. They are also pure gauge
configurations satisfying $F^{ij}=0$, with $i,j=1,2$.

Turning on the current $j_{\rm ax}^{\mu}$ induces a gauge field $a_{\rm
cl}^{\mu}$; solving the equations of motion for $A_{\rm cl}^{\mu}+a_{\rm
cl}^{\mu}$, is equivalent to solving the small fluctuation equations,

$\left(D^{2}g^{\nu\mu}-D^{\nu}D^{\mu}-2igF_{\rm cl}^{\nu\mu}\right)a_{{\rm
cl},\mu}=j_{{\rm ax},{\rm cl}}^{\nu}\,,$ (99)

where the covariant derivatives are understood to be those of the classical
fields, and $j_{{\rm ax},{\rm
cl}}^{\nu}=-\partial_{\mu}{\bar{\eta}}\,{\tilde{F}}_{\rm cl}^{\mu\nu}$. Here
${\tilde{F}}_{\rm cl}^{\mu\nu}$ is dual field strength in $A^{-}=0$ gauge,
whose only nonzero component is

${\tilde{F}}_{\rm cl}^{+i,a}=\epsilon^{ij}\,U^{ab}\,\partial_{j}A_{\rm
cl}^{+,b}\,,$ (100)

expressed in terms of the field strength in Lorenz gauge, with $A_{\rm
cl}^{+,a}$ given in Eq. (98) and $U^{ab}$ is an adjoint matrix which will be
defined below shortly.

Since this equation is linear in $a_{{\rm cl},\mu}$, it is straightforward to
solve, with

$a_{{\rm cl},\mu}(x)=\int d^{4}y\,G_{\mu\nu}(x-y)\,j_{{\rm ax},{\rm
cl}}^{\nu}(y)\,,$ (101)

with

$\left(D^{2}g^{\mu\rho}-D^{\mu}D^{\rho}-2igF_{\rm
cl}^{\mu\rho}\right)G_{\rho\nu}(x-y)=i\delta^{\mu}_{\nu}\delta^{(4)}(x-y)\,,$
(102)

where $G_{\mu\nu}(x-y)$ is the shock wave gluon propagator in the axial gauge.
Its Fourier transform333333We define
$G^{0}(x-y)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{ip\cdot(x-y)}G^{0}(p)\,\,\,{\rm
and}\,\,\,G(x,y)=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}q}{(2\pi)^{4}}e^{ip\cdot
x+iq\cdot y}{\tilde{G}}(p,q)\,.$ can be written as McLerran and Venugopalan
(1994c); Ayala  _et al._ (1995); Balitsky and Belitsky (2002)

${\tilde{G}}_{\mu\nu;ab}(p,q)={\tilde{G}}_{\mu\rho;ac}^{0}(p){\cal
T}_{g}^{\rho\sigma;cd}(p,q){\tilde{G}}_{\sigma\nu;db}^{0}(q)\,,$ (103)

with the free propagator in $a^{-}=0$ gauge ($n^{\mu}=\delta^{\mu+}$,
satisfying $n\cdot a=a^{-}$),

${\tilde{G}}_{\mu\nu;ab}^{0}(p)=\frac{i}{p^{2}+i\epsilon}\left(-g_{\mu\nu}+\frac{p_{\mu}n_{\nu}+p_{\nu}n_{\mu}}{p\cdot
n}\right)\delta_{ab}\,,$ (104)

and the effective vertex

${\cal T}_{g}^{\mu\nu;ab}(p,q)=-2\pi\delta(p^{-}-q^{-})(2p^{-})\,{\rm
sign}(p^{-})\,g^{\mu\nu}\int d^{2}{\bf z}_{\perp}e^{-i(p-q)_{\perp}\cdot{\bf
z}_{\perp}}\left(U^{ab}\right)^{{\rm sign}(p^{-})}({\bf z}_{\perp})\,.$ (105)

The dependence of the shock wave propagator on the color sources $\rho$ is
contained in the adjoint Wilson line

$U(x_{\perp})={\cal P}_{-}\exp\left(-ig\int
dz^{-}\frac{1}{\nabla_{\perp}^{2}}\rho^{a}(z^{-},x_{\perp})T^{a}\right)\,,$
(106)

where ${\cal P}_{-}$ denotes path ordering in $x^{-}$ variable and $T^{a}$ are
matrices representing the adjoint generators of the $SU(3)$ color algebra.

Following Jokela  _et al._ (2020), we can write

$\Omega=\frac{\alpha_{s}}{2\pi}\,\left(\partial_{+}a_{i}^{a}\right)\,{\tilde{F}}_{\rm
cl}^{+i,a}\,,$ (107)

where the parenthesis denotes the electric field induced by the ${\bar{\eta}}$
field and the other term is the CGC background magnetic field given by Eq.
(100). This result for $\Omega$, illustrated in Fig. 9, is a nontrivial
functional of $\rho$, and linear in ${\bar{\eta}}$.

Since the only nonzero component the classical dual field strength tensor is
${\tilde{F}}^{+i}=-\epsilon^{ij}\,F_{-j}$, and since our choice of gauge gives
$a^{-}=0$, we only obtain the contribution from $a_{i}$ given by

$\partial_{+}a_{i}=-\int
d^{4}y\,(\partial_{+}G_{ij})\,(\partial_{+}{\bar{\eta}})\,{\tilde{F}}_{\rm
cl}^{+j}(y)\,.$ (108)

Substituting Eq. (108) into Eq. (107), we now have all the ingredients to
compute $\langle\Omega\rangle$. Thus similarly to Sec. V.1, we can write

$\displaystyle\langle\Omega\eta_{0}\rangle\propto\langle(\partial_{+}{\bar{\eta}}){\bar{\eta}}\rangle_{{\bar{\eta}}}\,\langle{\tilde{F}}_{\rm
cl}^{+i}(x)\,{\tilde{F}}_{\rm cl}^{+j}(y)\rangle_{\rho}\,,$ (109)

where the second term on the r.h.s is the correlator of transverse magnetic
fields. The detailed computation of the r.h.s is quite subtle and will be a
subject addressed further in Paper III Tarasov and Venugopalan . However on
general grounds, one expects the correlator of magnetic fields in the CGC to
obey

$\displaystyle\langle{\tilde{F}}_{\rm cl}^{+i}(x)\,{\tilde{F}}_{\rm
cl}^{+j}(y)\rangle_{\rho}\propto\exp\left(-Q_{S}^{2}(x-y)^{2}\right)\approx\exp\left(-{\bar{C}}\,\frac{Q_{S}^{2}}{F_{\bar{\eta}}^{2}}\right)\,,$
(110)

where ${\bar{C}}$ is a constant factor that must be computed and the typical
scale for the propagation of the ${\bar{\eta}}$ field can reasonably be set to
be its decay constant. Hence as in the case of the model computation in Sec.
V.1, from Eqs. (109) and (110), one expects $g_{1}(x_{B},Q_{S}^{2})$ in the
Regge limit to be similarly exponentially suppressed with increasing $Q_{S}$.
As noted, a more detailed computation of the path integral in Eq. (V), and
phenomenological consequences thereof, beyond these simple model estimates,
will be discussed separately in Paper III Tarasov and Venugopalan .

##  VI Summary and outlook

In our previous paper Tarasov and Venugopalan (2020), we computed the
contribution of the box diagram to the polarized structure function
$g_{1}(x_{B},Q^{2})$ employing the worldline representation of the fermion
determinant in QCD. We demonstrated that the isosinglet triangle anomaly
dominates the structure of the box diagram in both the Bjorken and Regge
asymptotics of QCD. Specifically, $S^{\mu}\,g_{1}$ is proportional to forward
limit $l^{\mu}\rightarrow 0$ of the off-forward matrix element of the nonlocal
operator $\frac{l^{\mu}}{l^{2}}\,\Omega$ , where $\Omega$ is the topological
charge density in a polarized proton with the spin four-vector $S^{\mu}$. That
this result holds when $x_{B}\rightarrow 0$ is remarkable and strongly
suggestive of the fundamental role of the topology of the QCD vacuum in the
proton’s spin.

In this paper, we significantly developed the framework introduced in Tarasov
and Venugopalan (2020). A major focus was to demonstrate how the off-forward
pole of the anomaly cancels in the forward limit and the consequences thereof.
In order to do so, we reaffirmed that the anomaly arises from the imaginary
part of the worldline effective action, which corresponds to the phase of the
fermion determinant. We then discussed a generalization of the worldline
effective action that takes into account the coupling of fermion modes to low
energy scalar, pseudoscalar, vector and axial vector degrees of freedom.
Limiting ourselves to isosinglet contributions to the imaginary part of the
worldline effective action, we showed explicitly (in Appendix A) the existence
of a Wess-Zumino-Witten term that couples the topological charge density
$\Omega$ to a massless isosinglet pseudoscalar field ${\bar{\eta}}$. While
this particular WZW contribution is well-known in the chiral perturbation
theory literature, our derivation of this term in the worldline framework is
new.

We then demonstrated the fundamental role played by the WZW term in the
cancellation of the anomaly pole in the off-forward matrix element of the
isosinglet axial vector current $J_{\mu}^{5}$ in the polarized proton; in the
forward limit, this matrix element determines the proton’s helicity
$\Sigma(Q^{2})$. We first identified the axial vector and pseudoscalar
contributions to this matrix element and derived the anomalous Goldberger-
Treiman relation that connects the two. Specifically, as first suggested by
Veneziano, the axial vector charge $G_{A}$ representing the direct coupling of
$J_{\mu}^{5}$ to the polarized proton can be equated to the product of the
isosinglet coupling to the proton times its decay constant.

Due to the WZW term, this pseudoscalar exchange can also be mediated through
the anomaly, specifically the QCD topological susceptibility $\chi(l^{2})$.
The leading contribution to this quantity is the Yang-Mills topological
susceptibility $\chi_{\rm YM}(l^{2})$, which is of the order of typical
nonperturbative QCD scales. However, as shown by Witten and Veneziano, higher
order $1/N_{c}$ contributions to $\chi_{\rm YM}(l^{2})$ from ${\bar{\eta}}$
exchange mediated via the WZW term results in the topological generation of
the $\eta^{\prime}$ mass, resolving the $U_{A}(1)$ problem. The resulting
topological screening of the ${\bar{\eta}}$ pole then ensures that, in the
chiral limit, $\chi(l^{2})\rightarrow 0$ when $l^{2}\rightarrow 0$. We showed
explicitly how this topological screening results in the cancellation of the
pole of the anomaly and recovered the striking result of Shore and Veneziano
(1990, 1992) that $\Sigma(Q^{2})\propto\sqrt{\chi^{\prime}(0)}$, where
$\chi^{\prime}(0)$ is the slope of the QCD topological susceptibility in the
forward limit. In the topological screening picture, the fact that $\chi(0)=0$
in the chiral limit provides a natural explanation of the “spin puzzle” of why
the measured isosinglet axial charge is much smaller than its octet
counterpart Veneziano (1989); Shore (2008); Kuhn  _et al._ (2009). In other
words, the underlying physics that resolves the $U_{A}(1)$ problem also
resolves the proton’s spin puzzle.

In Tarasov and Venugopalan (2020), we showed that $g_{1}$ in the Regge limit
is represented by the same matrix element that contributes in Bjorken
asymptotics. However the computation of the matrix element in the two limits
is quite different because the former is strongly influenced by the physics of
gluon saturation which introduces a large emergent scale $Q_{S}(x)$. Using the
insights provided by the worldline effective action outlined in Appendices
A-C, we constructed an axion-like effective action for $g_{1}$ that captures
the physics of gluon saturation and is consistent with anomalous chiral Ward
identities. The underlying dynamics of this action is controlled by the
saturation scale and the Yang-Mills topological susceptibility $\chi_{\rm
YM}$, or equivalently, the $\eta^{\prime}$ mass. In the absence of the
coupling to a large number of color sources (represented by $Q_{S}$), our
formulation is compatible with a picture of spin diffusion mediated by
instanton-anti-instanton configurations, as suggested by ’t Hooft’s
explanation of the $U_{A}(1)$ problem.

However when classical sources begin to play a role at small $x_{B}$, the
additional presence of a dynamical momentum scale $Q_{S}$ can induce over-the-
barrier sphaleron-like topological transitions (as opposed to the instanton-
anti-instanton tunneling transitions) between different $\theta$-vacua labeled
by integer-valued Chern-Simons numbers. This is very similar to the
temperature induced sphaleron transitions that have been studied previously.
In particular, one can map our axion-like effective action in this case to
earlier work McLerran  _et al._ (1991) describing the propagation of an axion
in a hot QCD plasma. Similarly to that study, the presence of classical color
sources does not prevent topological mass generation of the $\eta^{\prime}$;
it however experiences a drag force that strongly impacts spin diffusion.

This picture of spin diffusion is plausible for $x_{B}$ values in the
kinematic window corresponding to $\Lambda_{\rm QCD}<Q_{S}<m_{\eta^{\prime}}$.
However at very small $x_{B}$ when $Q_{S}\gg m_{\eta^{\prime}}$, our effective
action suggests that it is more likely that static shock wave CGC
configurations dominate over instanton/sphaleron-like configurations. In this
case, the WZW ${\bar{\eta}}\,\Omega$ coupling generates a current that induces
a nontrivial topological charge density in the gluon shock wave. The problem
here, with essential modifications, is similar to recent work Jokela  _et al._
(2020) on the interaction of a putative axion with QCD matter in the presence
of classical color sources. The influence of such a perturbation on the shock
wave diminishes rapidly with increasing $Q_{S}$ leading to a rapid quenching
of spin diffusion in the Regge limit.

This first study can be quantified further to provide concrete predictions for
$g_{1}$ at small $x_{B}$. It is equally important is to understand how
renormalization works in our framework analogously to previous work in
perturbative QCD Kodaira (1980); Vogelsang (1991); Zijlstra and van Neerven
(1994); Vogelsang (1996); de Florian and Vogelsang (2019). While the anomaly
equation holds both for bare and renormalized quantities Espriu and Tarrach
(1982), the correspondence at small $x_{B}$ between the smearing of the
topological charge density and that of the gluon shock wave needs to be better
understood. Work in this direction is in progress. Not least, for quantitative
precision, we will need to extend our computation beyond the chiral limit and
take into account the influence of light quark masses. Following the
pioneering work in Leutwyler (1996); Herrera-Siklody  _et al._ (1997); Kaiser
and Leutwyler (2000), there has been considerable work in the chiral
perturbation theory of the $U(3)$ nonet both on the phenomenology Gan  _et
al._ (2020) of $\eta^{\prime}$ mixings and decays and on high order precision
computations in this framework Vonk  _et al._ (2019). With regard to the
latter, finite temperature computations are especially relevant Gómez Nicola
_et al._ (2019).

It will also be important to identify other signatures of the topological
screening picture, given the exciting possibility that $g_{1}$ at small
$x_{B}$ could be sensitive to sphaleron-like transitions. One possibility
discussed previously in the literature is to measure semi-inclusive hadron
production in DIS, off polarized proton and deuteron targets, in the target
fragmentation region Shore and Veneziano (1998); de Florian  _et al._ (1997).
Specifically, it was argued that first $x_{B}$ moments of so-called “fracture”
functions Grazzini  _et al._ (1998) (of the momentum fraction of the nucleon
carried by the hadron) satisfies the following. i) It is sensitive to the
ratio of the isosinglet and isotriplet axial charges; this ratio, a
quantitative measure of OZI violation, is proportional to $\chi^{\prime}(0)$.
ii) It is independent of the target. Our work suggests that such an OZI
suppression may be strongly sensitive to $x_{B}$ and in this kinematic regime,
may also have a target dependence due to the differing color charge densities
probed. For a discussion of effects of the anomaly in the context of quark
fragmentation, see Kang and Kharzeev (2011). One can also pursue in parallel
similar signatures in polarized proton-proton collisions343434We thank Werner
Vogelsang for a discussion on this point.. These and other such possible
phenomenological consequences with be pursued separately.

We now turn to our outlook on the possible implications of this work for QCD
spin studies. The presence of the off-forward infrared pole of the anomaly in
$\Sigma$ (and, as we have argued, in $g_{1}$) suggests i) that $\Sigma$ can be
expressed as an intrinsically nonlocal operator, and ii) $\Sigma$ has a
nontrivial dependence on infrared physics, in particular that governing the
physics of the $\eta^{\prime}$. Indeed, our final result for $\Sigma$ is
proportional to the square root of the slope of the topological susceptibility
in the forward limit. Our work (and preceding work in Veneziano (1989); Shore
and Veneziano (1990)) therefore brings into question the applicability353535A
corollary to this question would be to better understand in this topological
screening framework the role of entanglement Beane and Ehlers (2019) in the
proton’s spin. of QCD factorization for quantities such as $g_{1}$ that are
sensitive to the anomaly Collins (1993). Likewise, similarly to our concerns
regarding the applicability of collinear kinematics, we discuss at length in
Appendix C why the eikonal expansion, frequently employed in studies of spin
at small $x_{B}$, is not applicable to observables sensitive to the anomaly.
This is because the latter couples to zero modes of the Dirac operator
Leutwyler and Smilga (1992), the physics of which is missed when higher order
terms in this expansion are omitted. Not least in importance are the possible
implications of this work for spin sum rules and their interpretation Aidala
_et al._ (2013); Ji  _et al._ (2021).

Looking beyond QCD spin, the possibility of laboratory measurements of
sphaleron transitions is of great interest in phenomena spanning a wide range
in energy scales. As is well-known, sphaleron transitions are conjectured to
play a key role in electroweak baryogenesis Kuzmin  _et al._ (1985); Cohen
_et al._ (1993); Riotto and Trodden (1999). Sphaleron-like topological
transitions can also produce a Chiral Magnetic Effect (CME) Kharzeev  _et al._
(2008); Fukushima  _et al._ (2008) in heavy-ion collisions. In this case,
topological transitions in drive charge separation in an external magnetic
field, signatures of which can be extracted in the heavy-ion experiments.
While the CME has been ruled out Abdallah  _et al._ (2021) at the highest
energies studied at the Relativistic Heavy Ion Collider (RHIC), prospects for
its measurement exist at lower energies An  _et al._ (2021). Complicating the
extraction of a definitive signal of this effect is the large experimental and
theoretical background Koch  _et al._ (2017) in the complex environment of the
RHIC collisions. Our study suggests that polarized DIS measurements at small
$x_{B}$ may provide an alternate route to empirically establish the existence
of such topological transitions in nature.

## Acknowledgements

We thank Constantia Alexandrou, Elke Aschenauer, Silas Beane, Daniel Boer, Gia
Dvali, Renee Fatemi, Stefano Forte, Yoshitaka Hatta, Bob Jaffe, Xiangdong Ji,
Yuri Kovchegov, Elliot Leader, Kei-Fei Liu, Rob Pisarski, Werner Vogelsang and
Feng Yuan for conversations that have influenced this work.

A.T.’s work is supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics under Award Number DE-SC0004286. R.V.’s work is
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, under Contracts No. de-sc0012704. A.T and R.V.’s work on this
topic is also supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, within the framework of the TMD Theory Topical
Collaboration.

##  Appendix A Detailed derivation of the WZW action for the ${\bar{\eta}}$

We begin with the expression in Eq.(29):

$\displaystyle
W_{\mathcal{I}}=-\frac{i}{32}\int^{1}_{-1}d\alpha\int^{\infty}_{0}dT\mathcal{N}\int_{P}\mathcal{D}x\mathcal{D}\psi~{}{\rm
tr}~{}\chi\bar{\omega}(0)\exp\Big{[}-\int^{T}_{0}d\tau\mathcal{L}_{(\alpha)}(\tau)\Big{]}$
(111)

To isolate the WZW term of interest, we will first expand $W_{\mathcal{I}}$ up
to order $\Pi A^{2}$ in a presence of the scalar field363636This scalar field
can be understood, via a Legendre transform, as the functional derivative of
the Wess-Zumino effective action Zumino (1970); Wess and Zumino (1971) with
respect to the chiral condensate. We will take this field to be constant for
the rest of the discussion. $\Phi$, assuming that all fields commute with each
other. We start by expanding $W_{\mathcal{I}}$ to the linear power in $\Pi$
and taking the trace of two component matrices:373737We will set the einbein
$\mathcal{E}=2$ at the end of the derivation.,

$\displaystyle
W_{\mathcal{I}}[\Pi]=\frac{\mathcal{E}\Phi}{4}\int^{1}_{-1}d\alpha\int^{\infty}_{0}dT\mathcal{N}\int_{P}\mathcal{D}x\mathcal{D}\psi~{}\exp\Big{[}-\int^{T}_{0}d\tau\Big{\\{}\frac{\dot{x}^{2}}{2\mathcal{E}}+\frac{1}{2}\psi\dot{\psi}-i\dot{x}^{\mu}A_{\mu}+\frac{i\mathcal{E}}{2}\psi^{\mu}\psi^{\nu}F_{\mu\nu}\Big{\\}}\Big{]}\exp\Big{[}-T\frac{\mathcal{E}\alpha^{2}\Phi^{2}}{2}\Big{]}$
$\displaystyle\times{\rm
tr}_{c}\Big{(}\psi_{5}(\tau_{0})\Pi(x_{0})-\psi^{\mu}(\tau_{0})\dot{x}_{\mu}(\tau_{0})\int^{T}_{0}d\tau_{1}\psi^{\nu}(\tau_{1})\psi_{5}(\tau_{1})\partial_{\nu}\Pi(x_{1})\Big{)}$
(112)

where we use the shorthand notation $x_{i}\equiv x(\tau_{i})$. The first term
in this equation corresponds to expansion of the insertion $\bar{\omega}(0)$
in Eq. (111) to the leading power in $\Pi$ while the second term is obtained
by expansion of the exponential factor.

Now expanding the exponential factor to the order $A^{2}$ we obtain,

$\displaystyle W_{\mathcal{I}}[\Pi
A^{2}]=-\frac{\mathcal{E}\Phi}{4}\int^{1}_{-1}d\alpha\int^{\infty}_{0}dT\mathcal{N}\int_{P}\mathcal{D}x\mathcal{D}\psi~{}\exp\Big{[}-\int^{T}_{0}d\tau\Big{\\{}\frac{\dot{x}^{2}}{2\mathcal{E}}+\frac{1}{2}\psi\dot{\psi}\Big{\\}}\Big{]}\exp\Big{[}-T\frac{\mathcal{E}\alpha^{2}\Phi^{2}}{2}\Big{]}$
$\displaystyle\times{\rm
tr}_{c}\Big{(}\psi_{5}(\tau_{0})\Pi(x_{0})-\psi^{\mu}(\tau_{0})\dot{x}_{\mu}(\tau_{0})\int^{T}_{0}d\tau_{1}\psi^{\nu}(\tau_{1})\psi_{5}(\tau_{1})\partial_{\nu}\Pi(x_{1})\Big{)}V_{2}V_{3}$
(113)

where the interaction with the background gluon field is defined by the
worldline vertex,

$\displaystyle
V_{i}\equiv\int^{T}_{0}d\tau_{i}\Big{(}\dot{x}^{\rho}(\tau_{i})+\mathcal{E}\psi^{\rho}(\tau_{i})\psi^{\alpha}(\tau_{i})\partial_{\alpha}\Big{)}A_{\rho}(x_{i})$
(114)

To calculate the functional integrals in Eq. (113), it is convenient to
rewrite the background fields in terms of their Fourier transforms, which
gives383838In the second term of the equation, we used rotational invariance
to replace the integration over $\tau_{1}$ by $\tau_{0}$, and subsequently
substituted $\tau_{0}\leftrightarrow\tau_{1}$.,

$\displaystyle W_{\mathcal{I}}[\Pi
A^{2}]=-\frac{\mathcal{E}\Phi}{4}\int^{1}_{-1}d\alpha\int^{\infty}_{0}dT\mathcal{N}\int_{P}\mathcal{D}x\mathcal{D}\psi\exp\Big{[}-\int^{T}_{0}d\tau\Big{\\{}\frac{\dot{x}^{2}}{2\mathcal{E}}+\frac{1}{2}\psi\dot{\psi}\Big{\\}}\Big{]}\exp\Big{[}-T\frac{\mathcal{E}\alpha^{2}\Phi^{2}}{2}\Big{]}$
$\displaystyle\times\prod_{k=0,2,3}\int\frac{d^{4}p_{k}}{(2\pi)^{4}}\,{\rm
tr}_{c}\psi_{5}(\tau_{0})\Big{(}1+ip_{0\nu}\psi^{\nu}(\tau_{0})\int^{T}_{0}d\tau_{1}\psi^{\mu}(\tau_{1})\dot{x}_{\mu}(\tau_{1})\Big{)}\,\Pi(p_{0})\,e^{ip_{0}x_{0}}\,\tilde{V}_{2}\,e^{ip_{2}x_{2}}\,\tilde{V}_{3}\,e^{ip_{3}x_{3}}\,,$
(115)

where

$\displaystyle\tilde{V}_{i}\equiv\int^{T}_{0}d\tau_{i}\Big{[}\dot{x}^{\rho}(\tau_{i})+i\mathcal{E}\psi^{\rho}(\tau_{i})\psi^{\alpha}(\tau_{i})p_{i\alpha}\Big{]}A_{\rho}(p_{i})$
(116)

With this form for Eq. (115), we can begin to perform the functional
integration over the coordinate and Grassmann worldlines. Considering first
the Grassmanian integrals, since they satisfy periodic boundary conditions, we
first separate their zero modes as

$\displaystyle\psi(\tau)=\psi+\xi(\tau)\,,$ (117)

and substitute

$\displaystyle\int_{P}\mathcal{D}\psi\exp\Big{[}-\int^{T}_{0}d\tau\frac{1}{2}\psi\dot{\psi}\Big{]}\to\int
d^{5}\psi\int_{P}\mathcal{D}\xi\exp\Big{[}-\int^{T}_{0}d\tau\frac{1}{2}\xi\dot{\xi}\Big{]}$
(118)

The integration over the zero modes in Eq.(115) can be easily done using the
Grassmann identity,

$\displaystyle\int
d^{5}\psi\,\psi^{\mu}\psi^{\nu}\psi^{\rho}\psi^{\sigma}\psi^{5}=\epsilon^{\mu\nu\rho\sigma}\,,$
(119)

where the convention we use for the Levi-Civita tensor in Euclidean space is
$\epsilon_{1234}=1$.

The remaining integral over $\xi$ can be straightforwardly calculated (see
Refs. Strassler (1992); D’Hoker and Gagne (1996a); Schubert (2001) for
details) using

$\displaystyle\int_{P}\mathcal{D}\xi\exp\Big{[}-\int^{T}_{0}d\tau\frac{1}{2}\xi\dot{\xi}\Big{]}\xi^{\mu}(\tau_{1})\xi^{\nu}(\tau_{2})=-\frac{1}{2}g^{\mu\nu}\dot{G}_{B}(\tau_{1},\tau_{2})\,,$
(120)

where the derivative of the bosonic worldline propagator is

$\displaystyle\dot{G}_{B}(\tau_{1},\tau_{2})={\rm
sign}(\tau_{1}-\tau_{2})-\frac{2(\tau_{1}-\tau_{2})}{T}\,.$ (121)

Note that before applying Eqs. (119) and (120) , one should properly arrange
the variables in Eq.(115), taking into account the anti-commuting property of
Grassmann variables.

After a rather laborious calculation, we obtain,

$\displaystyle W_{\mathcal{I}}[\Pi
A^{2}]=-\frac{\mathcal{E}^{2}\Phi}{4}\int^{1}_{-1}d\alpha\int^{\infty}_{0}dT\mathcal{N}\int_{P}\mathcal{D}x\exp\Big{[}-\int^{T}_{0}d\tau\frac{\dot{x}^{2}}{2\mathcal{E}}\Big{]}\exp\Big{[}-T\frac{\mathcal{E}\alpha^{2}\Phi^{2}}{2}\Big{]}\prod_{k=0,2,3}\int\frac{d^{4}p_{k}}{(2\pi)^{4}}\int^{T}_{0}d\tau_{2}\int^{T}_{0}d\tau_{3}$
$\displaystyle\times\,{\rm
tr}_{c}\,\Big{(}\mathcal{E}p_{2\alpha}p_{3\beta}\epsilon^{\rho\alpha\sigma\beta}-p_{0\nu}\int^{T}_{0}d\tau_{1}\dot{x}_{1\mu}\mathcal{S}^{\mu\nu;\sigma\rho}_{\tau_{1},\tau_{2},\tau_{3}}(p_{2},p_{3})\Big{)}\,\Pi(p_{0})\,e^{ip_{0}x_{0}}\,A_{\rho}(p_{2})\,e^{ip_{2}x_{2}}\,A_{\sigma}(p_{3})\,e^{ip_{3}x_{3}}\,,$
(122)

where

$\displaystyle\mathcal{S}^{\mu\nu;\sigma\rho}_{\tau_{1},\tau_{2},\tau_{3}}(p_{2},p_{3})\equiv
p_{2\alpha}\dot{x}^{\sigma}_{3}\epsilon^{\mu\rho\alpha\nu}+p_{3\beta}\dot{x}^{\rho}_{2}\epsilon^{\mu\sigma\beta\nu}+i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\rho\alpha\sigma}g^{\beta\nu}\dot{G}_{B}(\tau_{3},\tau_{0})$
$\displaystyle-i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\rho\alpha\beta}g^{\sigma\nu}\dot{G}_{B}(\tau_{3},\tau_{0})+i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\rho\sigma\beta}g^{\alpha\nu}\dot{G}_{B}(\tau_{2},\tau_{0})-i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\rho\sigma\nu}g^{\alpha\beta}\dot{G}_{B}(\tau_{2},\tau_{3})$
$\displaystyle+i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\rho\beta\nu}g^{\alpha\sigma}\dot{G}_{B}(\tau_{2},\tau_{3})-i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\alpha\sigma\beta}g^{\rho\nu}\dot{G}_{B}(\tau_{2},\tau_{0})+i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\alpha\sigma\nu}g^{\rho\beta}\dot{G}_{B}(\tau_{2},\tau_{3})$
$\displaystyle-i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\mu\alpha\beta\nu}g^{\rho\sigma}\dot{G}_{B}(\tau_{2},\tau_{3})+i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\rho\alpha\sigma\beta}g^{\mu\nu}\dot{G}_{B}(\tau_{1},\tau_{0})-i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\rho\alpha\sigma\nu}g^{\mu\beta}\dot{G}_{B}(\tau_{1},\tau_{3})$
$\displaystyle+i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\rho\alpha\beta\nu}g^{\mu\sigma}\dot{G}_{B}(\tau_{1},\tau_{3})-i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\rho\sigma\beta\nu}g^{\mu\alpha}\dot{G}_{B}(\tau_{1},\tau_{2})+i\frac{\mathcal{E}}{2}p_{2\alpha}p_{3\beta}\epsilon^{\alpha\sigma\beta\nu}g^{\mu\rho}\dot{G}_{B}(\tau_{1},\tau_{2})\,.$
(123)

The integration over the coordinate worldlines can be performed in a similar
way. We begin again by separating out the zero modes as,

$\displaystyle x(\tau_{i})=\bar{y}+y(\tau_{i})\,,$ (124)

and replacing

$\displaystyle\int_{P}\mathcal{D}x\to\int d^{4}\bar{y}\int_{P}\mathcal{D}y\,.$
(125)

Performing the change of variables (Eq.(124)) in Eq. (122), the integral over
the zero modes $\bar{y}$ yields the overall 4-momentum conserving delta-
function:

$\displaystyle\int
d^{4}\bar{y}\,e^{ip_{0}\bar{y}}e^{ip_{2}\bar{y}}e^{ip_{3}\bar{y}}=(2\pi)^{4}\delta^{4}(p_{0}+p_{2}+p_{3})\,.$
(126)

The remaining integral over $y(\tau)$ can be calculated using the identity

$\displaystyle\mathcal{N}\int_{P}\mathcal{D}y\exp\Big{[}-\int^{T}_{0}d\tau\frac{\dot{y}^{2}}{2\mathcal{E}}\Big{]}\,\mathcal{Y}\,e^{ip_{0}y_{0}}e^{ip_{2}y_{2}}e^{ip_{3}y_{3}}=\frac{1}{(2\pi\mathcal{E}T)^{2}}\,\langle\mathcal{Y}\,e^{ip_{0}y_{0}}e^{ip_{2}y_{2}}e^{ip_{3}y_{3}}\rangle\,,$
(127)

where $\mathcal{Y}$ is an arbitrary product of factors $\dot{y}^{\alpha}_{i}$
(such as those appearing in Eq. (122)) and the notation
$\langle\mathcal{Y}e^{ip_{0}y_{0}}e^{ip_{2}y_{2}}e^{ip_{3}y_{3}}\rangle$
denotes all possible Wick contractions between the trajectories $y$ in the
product $\mathcal{Y}e^{ip_{0}y_{0}}e^{ip_{2}y_{2}}e^{ip_{3}y_{3}}$. The
resulting Wick contractions can be calculated using

$\displaystyle\langle
y^{\mu}(\tau_{1})y^{\nu}(\tau_{2})\rangle=-g^{\mu\nu}G_{B}(\tau_{1},\tau_{2})\,\,\,{\rm
with}\,\,\,G_{B}(\tau_{1},\tau_{2})=\frac{\mathcal{E}}{2}|\tau_{1}-\tau_{2}|-\mathcal{E}\frac{(\tau_{1}-\tau_{2})^{2}}{2T}\,,$
(128)

and

$\displaystyle\langle y^{\mu}(\tau_{1})e^{iky(\tau_{2})}\rangle=i\langle
y^{\mu}(\tau_{1})y^{\nu}(\tau_{2})\rangle\,k_{\nu}e^{iky(\tau_{2})}\,.$ (129)

After the computations using these Wick contractions, the expectation value of
the remaining exponential factors should be performed, and give,

$\displaystyle\langle
e^{ip_{0}y_{0}}e^{ip_{2}y_{2}}e^{ip_{3}y_{3}}\rangle=\exp\Big{[}p_{0}\cdot
p_{2}G_{B}(\tau_{0},\tau_{2})+p_{0}\cdot
p_{3}G_{B}(\tau_{0},\tau_{3})+p_{2}\cdot
p_{3}G_{B}(\tau_{2},\tau_{3})\Big{]}\,.$

As a result of these manipulations, and assuming that the background gluon
fields are on the mass-shell, we obtain the formula:

$\displaystyle W_{\mathcal{I}}[\Pi
A^{2}]=-\frac{\mathcal{E}^{2}\Phi}{4}\int^{1}_{-1}d\alpha\int^{\infty}_{0}dT\frac{1}{(2\pi\mathcal{E}T)^{2}}\exp\Big{[}-T\frac{\mathcal{E}\alpha^{2}\Phi^{2}}{2}\Big{]}\prod_{k=0,2,3}\int\frac{d^{4}p_{k}}{(2\pi)^{4}}\int^{T}_{0}d\tau_{2}\int^{T}_{0}d\tau_{3}$
$\displaystyle\times{\rm
tr}_{c}\big{(}\mathcal{E}+\int^{T}_{0}d\tau_{1}\mathcal{T}_{\tau_{1},\tau_{2},\tau_{3}}(p_{2},p_{3})\big{)}\epsilon^{\rho\alpha\sigma\beta}p_{2\alpha}A_{\rho}(p_{2})p_{3\beta}A_{\sigma}(p_{3})\,\Pi(p_{0})$
$\displaystyle\times\exp\Big{(}p_{2}\cdot
p_{3}[-G_{B}(\tau_{0},\tau_{2})-G_{B}(\tau_{0},\tau_{3})+G_{B}(\tau_{2},\tau_{3})]\Big{)}(2\pi)^{4}\delta^{4}(p_{0}+p_{2}+p_{3})\,,$
(130)

where

$\displaystyle\mathcal{T}_{\tau_{1},\tau_{2},\tau_{3}}(p_{2},p_{3})\equiv-\frac{\partial^{2}}{\partial\tau_{1}\partial\tau_{3}}G_{B}(\tau_{1},\tau_{3})-\frac{\partial^{2}}{\partial\tau_{1}\partial\tau_{2}}G_{B}(\tau_{1},\tau_{2})-\frac{\mathcal{E}p_{2}\cdot
p_{3}}{2}\Big{\\{}\big{(}\dot{G}_{B}(\tau_{0},\tau_{1})+\dot{G}_{B}(\tau_{1},\tau_{3})+\dot{G}_{B}(\tau_{3},\tau_{0})\big{)}^{2}$
$\displaystyle+\big{(}\dot{G}_{B}(\tau_{0},\tau_{1})+\dot{G}_{B}(\tau_{1},\tau_{2})+\dot{G}_{B}(\tau_{2},\tau_{0})\big{)}^{2}-\big{(}\dot{G}_{B}(\tau_{2},\tau_{1})+\dot{G}_{B}(\tau_{1},\tau_{3})+\dot{G}_{B}(\tau_{3},\tau_{2})\big{)}^{2}-\dot{G}^{2}_{B}(\tau_{3},\tau_{0})-\dot{G}^{2}_{B}(\tau_{2},\tau_{0})$