We present the results that are necessary in the ongoing lattice calculations
of the gluon parton distribution functions (PDFs) within the pseudo-PDF
approach. We give a classification of possible two-gluon correlator functions
and identify those that contain the invariant amplitude determining the gluon
PDF in the light-cone $z^2 \to 0$ limit. One-loop calculations have been
performed in the coordinate representation and in an explicitly gauge-invariant
form. We made an effort to separate ultraviolet (UV) and infrared (IR) sources
of the $\ln (-z^2)$-dependence at short distances $z^2$. The UV terms cancel in
the reduced Ioffe-time distribution (ITD), and we obtain the matching relation
between the reduced ITD and the light-cone ITD. Using a kernel form, we get a
direct connection between lattice data for the reduced ITD and the normalized
gluon PDF. We also show that our results may be used for a rather
straightforward calculation of the one-loop matching relations for quasi-PDFs.Comment: 8 pages, 5 figures. Gluon-quark term adde

Balitsky, Ian

Morris, Wayne

Radyushkin, Anatoly

English

arXiv

We present the results that are necessary in the ongoing lattice calculations of the gluon parton distribution functions (PDFs) within the pseudo-PDF approach. We give a classification of possible two-gluon correlator functions and identify those that contain the invariant amplitude determining the gluon PDF in the light-cone z2 → 0 limit. One-loop calculations have been performed in the coordinate representation and in an explicitly gauge-invariant form. We made an effort to separate ultraviolet (UV) and infrared (IR) sources of the ln⁡(−z2)-dependence at short distances z2. The UV terms cancel in the reduced Ioffe-time distribution (ITD), and we obtain the matching relation between the reduced ITD and the light-cone ITD. Using a kernel form, we get a direct connection between lattice data for the reduced ITD and the normalized gluon PDF. We also show that our results may be used for a rather straightforward calculation of the one-loop matching relations for quasi-PDFs

Old Dominion University

Gluon Pseudo-Distributions at Short Distances: Forward Case

We present the results that are necessary in the ongoing lattice calculations of the unpolarized and polarized gluon parton distribution functions (PDFs) within the pseudo-PDF approach. We give a classification of possible two-gluon correlator functions and identify those that contain the invariant amplitude determining the gluon PDF in the light-cone z2 → 0 limit. One-loop calculations have been performed in the coordinate representation and in an explicitly gauge-invariant form. We made an effort to separate ultraviolet (UV) and infrared (IR) sources of the ln(−z2)-dependence at short distances z2. The UV terms cancel in the reduced Ioffe-time distribution (ITD), and we obtain the matching relation between the reduced ITD and the light-cone ITD. Using a kernel form, we get a direct connection between lattice data for the reduced ITD and the normalized gluon PDF

Old Dominion University ODU Digital Commons Physics Faculty Publications Physics 2022 Unpolarized and Polarized Gluon Pseudo-Distributions at Short Distances: Forward Case Ian Balitsky Old Dominion University, ibalitsk@odu.edu Wayne Morris Old Dominion University, wmorris@odu.edu Anatoly Radyushkin Old Dominion University, aradyush@odu.edu Follow this and additional works at: https://digitalcommons.odu.edu/physics_fac_pubs  Part of the Quantum Physics Commons Original Publication Citation Balitsky, I., Morrisa, W., & Radyushkina, A. (2022). Unpolarized and polarized gluon pseudo-distributions at short distances: Forward case. 38th International Symposium on Lattice Field Theory, Zoom/Gather@MIT. https://doi.org/10.22323/1.396.0589 This Conference Paper is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact digitalcommons@odu.edu. PoS(LATTICE2021)589Unpolarized and polarized gluon pseudo-distributions atshort distances: Forward caseI. Balitsky,a,b W. Morrisa,b,∗ and A. Radyushkina,baPhysics Department, Old Dominion University,4600 Elkhorn Ave, Norfolk, VA 23529, USAbThomas Jefferson National Accelerator Facility,12000 Jefferson Avenue, Newport News, VA 23606, USAE-mail: balitsky@jlab.org, wmorr001@odu.edu, radyush@jlab.orgWe present the results that are necessary in the ongoing lattice calculations of the unpolarizedand polarized gluon parton distribution functions (PDFs) within the pseudo-PDF approach. Wegive a classification of possible two-gluon correlator functions and identify those that containthe invariant amplitude determining the gluon PDF in the light-cone z2 → 0 limit. One-loopcalculations have been performed in the coordinate representation and in an explicitly gauge-invariant form. We made an effort to separate ultraviolet (UV) and infrared (IR) sources ofthe ln(−z2)-dependence at short distances z2. The UV terms cancel in the reduced Ioffe-timedistribution (ITD), and we obtain the matching relation between the reduced ITD and the light-cone ITD. Using a kernel form, we get a direct connection between lattice data for the reducedITD and the normalized gluon PDF.The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021Zoom/Gather@Massachusetts Institute of Technology∗Speaker© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/flY4iil PROCEEDINGS ~ OF SCIENCE PoS(LATTICE2021)589Gluon pseudo-distributions W. Morris1. IntroductionLattice calculations of parton distribution functions (PDFs) are now a subject of considerableinterest. Modern efforts aim at the extractions of PDFs f (x) themselves rather than their xNmoments. On the lattice, this may be achieved by switching from local operators to equal-timecorrelators [1–4]. We use the pseudo-PDF approach [4], which is coordinate-space oriented, andparton distributions are extracted there by taking the short-distance z3 → 0 limit.Since the z3 → 0 limit is singular, one needs matching relations to convert the Euclidean lattice datainto the usual light-cone PDFs. Our goal is to outline the pseudo-PDF approach to the extractionof unpolarized gluon PDFs, and also to find one-loop matching conditions.In the gluon case, the calculation is complicated by strict requirements of gauge invariance. In thissituation, a very effective method is provided by the coordinate-representation approach of Ref. [5].It is based on the background-field method and the heat-kernel expansion. It allows, starting withthe original gauge-invariant bilocal operator, to find its modification by one-loop corrections. Theresults are obtained in an explicitly gauge-invariant form.2. Matrix elements2.1 Unpolarized caseThe spin-averagedmatrix element for operators comprised of twogluonfieldswith uncontractedindices is Mµα;νβ (z, p) ≡ 〈p|Gµα (z) [z, 0]Gνβ (0) |p〉 ,where [z, 0] is the standard straight-linegauge link in the adjoint representation. Accounting for the antisymmetry of Gρσ with respectto indices, and the available four-vectors p, and z, the decomposition of the matrix element intoinvariant amplitudes is:Mµα;νβ(z, p) =(gµνgαβ − gµβgαν)Mgg(ν, z2)+(gµνpαpβ − gµβpαpν − gανpµpβ + gαβpµpν)Mpp(ν, z2)+(gµνzαzβ − gµβzαzν − gανzµzβ + gαβzµzν)Mzz(ν, z2)+(gµνzαpβ − gµβzαpν − gανzµpβ + gαβzµpν)Mzp(ν, z2)+(gµνpαzβ − gµβpαzν − gανpµzβ + gαβpµzν)Mpz(ν, z2)+(pµzαpνzβ − pαzµpνzβ − pµzαpβzν + pαzµpβzν)Mppzz(ν, z2) , (1)where ν = −p · z is the Ioffe time[6]. The light-cone distribution is obtained fromgαβM+α,+β (z, p), where z is taken to be in the “minus” direction z = z−, so:gαβM+α,β+ (z−, p) = −2p2+Mpp(ν, z2), (2)thus the PDF is determined byMpp:−Mpp (ν, 0) =12∫ 1−1dxe−ixνx fg (x) . (3)The procedure is then to take projections of the matrix element that contain theMpp structure, andlittle of anything else. The projection that best meets this condition is M0i;i0, whose decomposition2PoS(LATTICE2021)589Gluon pseudo-distributions W. Morrisis M0i;i0 = 2Mgg + 2p20Mpp . We can remove the contaminating Mgg structure by adding theprojection Mi j;ji, whose decomposition is Mi j;ji = −2Mgg .While all projections of the matrix element are individually multiplicatively renormalizable [7],they won’t necessarily carry the same anomalous dimension; however, this addition works becauseboth projections, M0i;i0 and Mi j;ji, have the same anomalous dimension at one loop.2.2 Polarized caseWith polarized gluonswe have thematrix element comprised of a gluon field strength tensor anda dual field strength tensor: mµα;νβ (z, p) ≡ 〈p, s |Gµα (z) [z, 0]G̃µβ (0) |p, s〉, where the dual tensoris defined by G̃µβ ≡ 12εµβρλGρλ. Taking the z-odd combination Mµα;νβ (z, p) ≡ mµα;νβ (z, p) −mµα;νβ (−z, p), we perform a similar decomposition into invariant amplitudes, and find that the LChelicity distribution is defined by:gαβ M̃+α;β+ (z−, p) = −2p+s+[M(+)ps (ν, 0) − νMpp (ν, 0)](4)where s+ is related by sµ = mSµ to the usual gluon polarization vector Sµ, andM(+)ps ≡ Mps +Msp.In the case of pseudodistributions, a promising combination of projections on this matrixelement is M̃0i;i0+ M̃ji;i j , whose decomposition is 2p3p0[M(+)ps(ν, z2)−(1 + m2/p23)νMpp(ν, z2)].For p3 →∞, this result approaches the decomposition structure in the LC case.3. One-loop corrections3.1 Link self energy and ultraviolet divergencesThe link self energy correction, given byg2Nc8π2Γ (1 − εUV )(−z2µ2UV + iε)εUV(2εUV − 1) εUVGµα (z)Gνβ (0) , (5)should, in principle, be zero on the light-cone. However, this result contains linear (εUV = 1/2) andlogarithmic (εUV = 0) UV poles and a singularity on the light-cone z2 = 0 after expansion in εUV .In order to account for this, we explicitly separate the z2 dependence generated by the UV singularterms and those in the QCD (or DGLAP ) evolution logarithms ln(−z2µ2IR).Figure 1: Self-energy-type correction for the gauge link and vertex diagrams with gluons coming out of thegauge link.3z tz O z tz 0 Pl PoS(LATTICE2021)589Gluon pseudo-distributions W. Morris3.2 Vertex contributionsWhen one uses the background-field technique, with gluon propagator in the background-Feynman (bF) gauge [8], the three-gluon vertex differs from the usual Yang-Mills vertex. Therefore,the Feynman diagrams we use do not correspond one-to-one with the usual Feynman diagrams. Aconsequence of this is, where one would have two linear UV divergences that cancel after additionof two diagrams, in our case that same divergence cancels implicitly. This can be seen from theevolution part of the vertex correction:g2Nc8π2Γ (d/2 − 2)(d − 3)(−z2)d/2−2 ∫ 10 du [u3−d − 1]+Gµα (ūz)Gνβ (0) , (6)where the linear divergences present in the “u3−d” part and the “−1” part cancel.The full uncontracted vertex calculation contains also a UV divergent and constant part, howeverthis term is zero for the M0i;i0 and the Mi j;ji projection.3.3 Box and self-energy contributionsThe “box” correction is free of UV divergences, but gives a more complicated structure, gen-erating a mixture of different operators corresponding to different projections of Gµα (ūz)Gνβ (0).The full uncontracted result is too long for this paper, but it should be noted that the DGLAP partdoes not have the necessary plus-prescription form. To get it, one should add the contribution of thegluon self-energy diagrams. The relevant part of the result is the coefficient to theMpp structure,2(−u3 + u2 − 2u + 1), which integrates to 1/6.Figure 2: Box diagram and gluon self-energy-type insertions into the right leg.The self-energy diagrams contain both UV and collinear divergences that generate logarithmic termln(µ2IR/µ2UV). In order to obtain the necessary plus prescription with the “box” diagram, one canseparate this term into the difference ln(z23µ2IR)− ln(z23µ2UV).The self-energy result isg2Nc8π212 − d/2[2 −β02Nc]Gµα (z)Gνβ (0) , (7)where β0 = 11Nc/3 in gluodynamics, and 12−d/2 is to be replaced by ln(z23µ2IR)− ln(z23µ2UV).Substituting in the value of β0, we get 1/6, giving us the needed plus-prescription.3.4 Polarized caseBecause the one-loop calculationwas performed at the operator level with uncontracted indices,the results for Gµα (z) G̃ρσ (0) can be easily obtained by contracting the previous results from thissection with 12ενβρσ . The DGLAP evolution structure and matching relation for the polarized4z O z O z O R  PoS(LATTICE2021)589Gluon pseudo-distributions W. Morrisgluon pseudodistribution will be given in an upcoming paper. Here, we just list the results of theM̃0i;i0 + M̃ji;i j combination:M̃0i;i0(z, p) + M̃ji;i j(z, p)→g2Nc8π2166(1εUV+ log(z23eγ)) (M̃0i;i0(z, p) + M̃ji;i j(z, p))+g2Nc8π2∫ 10du{4δ(ū) − 2ūu + 2(1ū− ū)+−[4uū+4 log(1 − u)ū]++(1εIR− log(z23eγ)) [{4uū + 2[u2/ū]+}−12(β0Nc+ 6)δ(ū)]}×(M̃0i;i0(uz, p) + M̃ji;i j(uz, p))(8)4. DGLAP evolution structure, unpolarized case4.1 Reduced Ioffe-time distributionIn order to handle the link and self-energy UV divergences, we use the reduced ITD:M(ν, z23)≡Mpp(ν, z23)Mpp(0, z23) . (9)Thismethodworks because our operator ismultiplicatively renormalizable, and because theDGLAPevolution logarithm drops out ofMpp(0, z23), so we’re only and entirely removing the non-DGLAPrelated z3 dependence.4.2 Matching relationsCombining the one-loop gluon corrections, and the gluon-quark mixing term (that containsthe gq evolution kernel Bgq (u) = 1 + (1 − u)2 ) , we obtain the matching relation (excluding higherFigure 3: Gluon-quark mixing diagram.twist terms, or rather keeping onlyMpp):M(ν, z23)Ig(0, µ2) = Ig(ν, µ2) −αsNc2π∫ 10duIg(uν, µ2){ln(z23µ2e2γE /4)Bgg(u)+4[u + log(ū)ū]++23[1 − u3]+}−αsCF2πln(z23µ2e2γE /4) ∫ 10dw[IS(wν, µ2) − IS(0, µ2)]Bgq(w) , (10)5- --z 0 H PoS(LATTICE2021)589Gluon pseudo-distributions W. Morriswhere M(ν, z23), the reduced ITD, is our “lattice function”, and Ig(ν, µ2) and IS(uν, µ2) are thelight-cone ITDs. The gluon Altarelli-Parisi kernel is given byBgg(u) = 2[(1 − ūu)2ū]+. (11)The gluon light-cone ITD can be directly related to the light-cone PDF throughIg(ν, µ2)=12∫ 1−1dxeixνx fg(x, µ2). (12)Because x fg(x, µ2)is an even function of x, the real part of Ig(ν, µ2)is given by the cosinetransform of x fg(x, µ2), while the imaginary part vanishes.The factor Ig(0, µ2)= 〈x〉µ2 is the fraction of the hadron momentum carried by the gluons. Itshould be found from an independent lattice calculation (see, e.g. [9]).The matching relation can be cast into a new kernel form in terms of the light-cone PDFs:M(ν, z23) =∫ 10dxx fg(x, µ2)〈x〉µ2Rgg(xν, z23µ2)+∫ 10dxx fS(x, µ2)〈x〉µ2Rgq(xν, z23µ2), (13)whereRgg(y, z23µ2)= cos y −αsNc2π{ln(z23µ2e2γE /4)RB(y) + RL(y) + RC(y)}, (14)andRgq(y, z23µ2)= −αsNc2πln(z23µ2e2γE /4)RB(y) (15)The various R terms are given by cosine transformations of the gluon kernel, log term, constantterm, and mixing kernel, and are all perturbatively calculable expressions.Using lattice data and models for fg(x, µ2) and fS(x, µ2), one can fit their parameters and αs.5. ConclusionWepresented the results of the calculations necessary in the ongoingwork to extract unpolarizedgluon PDFs from the lattice using the method of pseudo-PDFs, and provide an update on the soonto be released results for the polarized case. Specifically, we demonstrated M0i;i0 + Mi j;ji to bethe most promising combination of matrix elements for obtaining the unpolarized gluon PDF, andM̃0i;i0 + M̃ji;i j to be the likely combination for obtaining the gluon helicity PDF. We gave thematching relations, in the unpolarized case, between the reduced pseudo-ITD and the light-coneITD or light-cone PDF, demonstrating that lattice data and light-cone PDFs can be directly related.AcknowledgementsWe thank K. Orginos, J.-W. Qiu, D. Richards and S. Zhao for their interest and discussions.Funding information This work is supported by Jefferson Science Associates, LLC under U.S.DOE Contract #DE- AC05-06OR23177 and by U.S. DOE Grant #DE-FG02-97ER41028.6PoS(LATTICE2021)589Gluon pseudo-distributions W. MorrisReferences[1] V. Braun and D. Müller, Exclusive processes in position space and the pion distributionamplitude, Eur. Phys. J. C 55 (2008) 349 [hep-ph/0709.1348][2] X. Ji, Parton Physics on a Euclidean Lattice, Phys. Rev. Lett. 110 (2013) 262002[hep-ph/1305.1539][3] Y. Q. Ma and J. W. Qiu, Extracting Parton Distribution Functions from Lattice QCD Calcu-lations, Phys. Rev. D 98 (2018) no.7, 074021 [hep-ph/1404.6860][4] A. V. Radyushkin, Quasi-parton distribution functions, momentum distributions, and pseudo-parton distribution functions, Phys. Rev. D 96 (2017) no.3, 034025 [hep-ph/1705.01488][5] I. I. Balitsky and V. M. Braun, Evolution Equations for QCD String Operators, Nucl. Phys. B311 (1989) 541.[6] B. L. Ioffe, Space-time picture of photon and neutrino scattering and electroproduction cross-section asymptotics, Phys. Lett. 30B, 123 (1969)[7] Z. Y. Li, Y. Q. Ma and J. W. Qiu, Multiplicative Renormalizability of Opera-tors defining Quasiparton Distributions, Phys. Rev. Lett. 122 (2019) no.6, 062002[hep-ph/arXiv:1809.01836][8] L. F. Abbott, The Background Field Method Beyond One Loop, Nucl. Phys. B 185 (1981) 189[9] Y. B. Yang, M. Gong, J. Liang, H. W. Lin, K. F. Liu, D. Pefkou and P. Shanahan, Nonpertur-batively renormalized glue momentum fraction at the physical pion mass from lattice QCD,Phys. Rev. D 98 (2018) no.7, 074506 [hep-lat/1805.00531]7

Unpolarized and Polarized Gluon Pseudo-Distributions at Short Distances: Forward Case

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Gluon Pseudo-Distributions at Short Distances: Forward Case

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