Chiral and trace anomalies in Deeply Virtual Compton Scattering Shohini Bhattacharya1Yoshitaka Hatta1 2yand Werner Vogelsang3z 1Physics Department Brookhaven National Laboratory Upton NY 11973 USA

2025-05-01 0 0 496.74KB 14 页 10玖币

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Chiral and trace anomalies in Deeply Virtual Compton Scattering

Shohini Bhattacharya,1, ∗Yoshitaka Hatta,1, 2, †and Werner Vogelsang3, ‡

1Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

2RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA

3Institute for Theoretical Physics, T¨ubingen University,

Auf der Morgenstelle 14, 72076 T¨ubingen, Germany

Inspired by recent work by Tarasov and Venugopalan, we calculate the one-loop quark box diagrams

relevant to polarized and unpolarized Deep Inelastic Scattering (DIS) by introducing oﬀ-forward

momentum lµas an infrared regulator. In the polarized case, we rederive the pole 1/l2related to

the axial (chiral) anomaly. In addition, we obtain the usual logarithmic term and the DIS coeﬃcient

function. We interpret the result in terms of the generalized parton distributions (GPDs) ˜

Hand

Eand discuss the possible violation of QCD factorization for the Compton scattering amplitude.

Remarkably, we also ﬁnd poles in the unpolarized case which are remnants of the trace anomaly. We

argue that these poles are canceled by the would-be massless glueball poles in the GPDs Hand Eas

well as in their moments, the nucleon gravitational form factors A, B and D. This mechanism sheds

light on the connection between the gravitational form factors and the gluon condensate operator

Fµν Fµν .

I. INTRODUCTION

The role of the UA(1) axial (chiral) anomaly in polarized Deep Inelastic Scattering (DIS) has a long and winding

history. Originally in the late 1970s it was used to constrain the one-loop gluonic correction to the ﬁrst moment of

the singlet g1(x) structure function, as well as the two-loop anomalous dimension of the quark helicity contribution

∆Σ to the proton spin [1]. Soon after the discovery of the ‘spin crisis’ by the European Muon Collaboration (EMC)

in 1988 [2], it was suggested that an anomaly-induced gluon helicity ∆Gcontribution to the ‘intrinsic’ quark helicity

∆˜

∆Σ = ∆˜

Σ−nfαs

2π∆G , (1)

could be the key to explaining the unexpectedly small value of ∆Σ [3, 4]. While such a scenario became popular at

the time, subsequent decades-long experiments and global analyses did not ﬁnd evidence for a suﬃciently large ∆G

to make it phenomenologically viable [5–7]. More importantly, the identiﬁcation of the axial anomaly contribution as

gluon helicity also met with much theoretical objection from the very start [8–11]. The gluonic contribution in (1)

comes from the infrared region of the triangle diagram which contains the Adler-Bell-Jackiw anomaly. As pointed

out by Jaﬀe and Manohar [8], a proper way to regulate the infrared singularity of this diagram is to calculate it in

oﬀ-forward kinematics. The anomaly then manifests itself as a pole in momentum transfer l=p2−p1in the matrix

element of the singlet axial current Jµ

5=Pf¯

ψfγµγ5ψf,

hp2|Jµ

5|p1i=nfαs

4π

ilµ

l2hp2|Fαβ

a˜

αβ |p1i,(2)

where the incoming and outgoing gluons are on shell and the nfquarks in the loop are massless. The appearance of the

pole 1/l2and the twist-four pseudoscalar operator F˜

Fseem alarming, as they signal some underlying nonperturbative

physics that does not ﬁt into the standard perturbative QCD framework. Yet, in ordinary perturbative calculations

done in forward kinematics, this problem is superﬁcially avoided by certain choices of infrared regularization such as

dimensional regularization.

∗Electronic address: sbhattach@bnl.gov

†Electronic address: yhatta@bnl.gov

‡Electronic address: werner.vogelsang@uni-tuebingen.de

arXiv:2210.13419v2 [hep-ph] 16 Jan 2023

FIG. 1: Box diagrams for the Compton amplitude in oﬀ-forward kinematics.

Decades after the initial controversy, infrared sensitivity and the subtleties of taking the forward limit seemed to

have been largely forgotten. Nowadays, forward kinematics is routinely used in the higher-order computations of

polarized cross sections and asymmetries. However, recently the issue of the anomaly pole has been rekindled by

Tarasov and Venugopalan [12, 13] who pursued and crystallized the original suggestion by Jaﬀe and Manohar. They

have demonstrated, within the worldline formalism, that the box diagram (see the left diagram in Fig. 1) contains a

pole 1/l2if it is calculated in oﬀ-forward kinematics. This may be viewed as a nonlocal generalization of the local

relation (2) unintegrated in the Bjorken variable x. As envisaged in [8] and elaborated in [13], at least after the x-

integration, the pole should be canceled by another massless pole due to the exchange of the η0meson, the would-be

Nambu-Goldstone boson of UA(1) symmetry breaking. This requirement leads to an independent derivation of the

UA(1) Goldberger-Treiman relation [14–16] between the pseudoscalar and pseudovector form factors.

Motivated by these developments, in this paper we further explore the physics of anomaly poles in two diﬀerent

directions. First, we calculate the box diagram in oﬀ-forward kinematics in the standard perturbation theory. This is

a useful cross-check of the result obtained in the worldline formalism [12]. In addition to reproducing the pole term,

we obtain the ‘usual’ perturbative corrections to the g1(x) structure function which features the DGLAP splitting

function and a coeﬃcient function. We then interpret the result in terms of the generalized parton distributions

(GPDs) ˜

Hand ˜

E. The emergence of the pole is potentially problematic for the QCD factorization of the Compton

amplitude. We discuss how factorization may still be justiﬁed following the possibility of cancellation of poles.

Second, we point out that entirely analogous poles can arise in unpolarized DIS, or more precisely, in the symmetric

(in Lorentz indices µν) part of the Compton scattering amplitude Tµν in oﬀ-forward kinematics. Just as the pole

in the polarized sector is related to the axial (chiral) anomaly, that in the unpolarized sector is related to the trace

anomaly. Indeed, it is known in QED and other gauge theories [17–19] that the oﬀ-forward photon matrix element of

the energy momentum tensor Θµν has an anomaly pole:

hp2|Θµν |p1i ∼ 1

l2hp2|Fαβ Fαβ |p1i,(3)

again from the triangle diagram. The residue is proportional to the matrix element of the twist-four scalar operator

hFαβ Fαβ i(or the ‘gluon condensate’ in QCD) which characterizes the trace anomaly. We shall derive the unintegrated

(in x) version of (3) by evaluating the quark box diagrams and interpret the result in terms of the unpolarized GPDs

Hand E. We then make a connection to the gravitational form factors of the proton and discuss the possibility of

cancellation of poles.

II. PRELIMINARIES

In this section, we set up our notations for the kinematical variables that enter the calculation of the quark box

diagrams in DIS. More precisely, since we generalize the calculation to oﬀ-forward kinematics as explained in the

Introduction, we consider the Compton scattering amplitude

Tµν =iZd4y

2πeiq·yhP2|T{Jµ(y/2)Jν(−y/2)}|P1i=Tµν

sym +iT µν

asym ,(4)

where Jµ=Pfef¯

ψfγµψf, with f=u, d, s, .. being a ﬂavor index, is the electromagnetic current and the subscript

‘sym/asym’ refers to the symmetric/antisymmetric part in the photon polarization indices µ, ν.|P1,2iare the proton

single-particle states. q=q1+q2

2is the average of the incoming and outgoing virtual photon momenta, and the

momentum transfer is denoted by P2−P1=q1−q2=l. We assume −q2≡Q2>0 is large and neglect ‘higher-twist’

terms of order O(M2/Q2) where M2=P2

1=P2

2is the proton mass squared. Their inclusion is understood in the

literature [20]. We also assume that t≡l2<0 is much smaller than the hard scale |t|  Q2and neglect terms of

order l2/Q2. We deﬁne the following variables

P=P1+P2

2, xB=Q2

2P·q, ξ =q2

2−q2

4P·q≈−l+

2P+,(5)

where xBis the generalized Bjorken variable and ξis the skewness parameter. In forward scattering, xBcoincides

with the usual Bjorken variable in DIS. In Deeply Virtual Compton Scattering (DVCS) where q2

2= 0, xB≈ξ.

The quark box diagrams of interest are part of the perturbative expansion of (4) at one-loop. There are three

topologies, see Fig. 1. The diagrams consist of two insertions of photon ﬁelds with momenta (q1, q2) and two insertions

of gluon ﬁelds with partonic momenta (p1, p2) which we parametrize as

p1=p−l

2, p2=p+l

2, x ≡p·q

P·q.(6)

Note that p2−p1=P2−P1=l, so the momentum transfer t=l2<0 is the same in both the hadronic and partonic

processes. We assume the incoming partons to be massless, p2

1=p2

2= 0, which means p·l= 0 and p2=−l2/4. The

kinematical variables at the partonic level are deﬁned as

ˆx=Q2

2p·q=xB

x,ˆ

ξ=q2

2−q2

4p·q=−q·l

2p·q=ξ

x.(7)

The relation ˆxq ·l=−ˆ

ξQ2will be used in the calculation below. The photon virtualities can be written as

1=−Q2ˆx+ˆ

ˆx+l2

4, q2

2=Q2ˆ

ξ−ˆx

ˆx+l2

4.(8)

Finally, the polarization vectors of the incoming and outgoing gluons, (p1)≡1,∗(p2)≡∗

2, respectively, satisfy the

physical conditions

1·(p−l/2) = 0 →1·p=1·l

∗

2·(p+l/2) = 0 →∗

2·p=−∗

2·l

2.(9)

We note that throughout this paper we use the conventions γ5=iγ0γ1γ2γ3and 0123 = +1. This becomes relevant

for the antisymmetric part of the Compton amplitude (4), to which we turn ﬁrst.

III. ANTISYMMETRIC PART

In the antisymmetric case we deﬁne

Jα≡ −αβµν PβImTasym

µν .(10)

In the forward limit lµ→0, Jα=g1(xB)¯u(P)γαγ5u(P)=2g1(xB)Sαis proportional to the g1structure function

in polarized DIS. We have calculated the box diagrams in the near-forward region |lµ|  Q, using the Mathematica

package ‘Package-X’ [21]. The result is, for massless quarks in the loop,

Jα|box ≈1

αs

2π

X

f

¯u(P2)∆Pqg ln Q2

−l2+δCoﬀ

g⊗∆G(xB)γαγ5+lα

l2δCanom

g⊗˜

F(xB)γ5u(P1),(11)

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ChiralandtraceanomaliesinDeeplyVirtualComptonScatteringShohiniBhattacharya,1,YoshitakaHatta,1,2,yandWernerVogelsang3,z1PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA2RIKENBNLResearchCenter,BrookhavenNationalLaboratory,Upton,NY11973,USA3InstituteforTheoreticalPhysics,TubingenUnive...

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Chiral and trace anomalies in Deeply Virtual Compton Scattering Shohini Bhattacharya1Yoshitaka Hatta1 2yand Werner Vogelsang3z 1Physics Department Brookhaven National Laboratory Upton NY 11973 USA

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