Lattice-QCD Calculations of TMD Soft Function Through Large-Momentum Effective Theory
Qi-An Zhang, Jun Hua, Yikai Huo, Xiangdong Ji, Yizhuang Liu, Yu-Sheng Liu, Maximilian Schlemmer, Andreas Schäfer, Peng Sun, Wei Wang, Yi-Bo Yang
aa r X i v : . [ h e p - l a t ] M a y Lattice-QCD Calculations of TMD Soft Function ThroughLarge-Momentum Effective Theory ( Lattice Parton Collaboration (LPC))
Qi-An Zhang, Jun Hua, Yikai Huo,
2, 3
Xiangdong Ji,
1, 4
Yizhuang Liu, Yu-Sheng Liu, Maximilian Schlemmer, Andreas Sch¨afer, Peng Sun, Wei Wang, ∗ and Yi-Bo Yang † Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China INPAC, SKLPPC, MOE KLPPC, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China Zhiyuan College, Shanghai Jiao Tong University, Shanghai 200240, China Department of Physics, University of Maryland, College Park, MD 20742, USA Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany Nanjing Normal University, Nanjing, Jiangsu, 210023, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China (Dated: June 1, 2020)The transverse-momentum-dependent (TMD) soft function is a key ingredient in QCD factor-ization of Drell-Yan and other processes with relatively small transverse momentum. We presenta lattice QCD study of this function at moderately large rapidity on a 2+1 flavor CLS dynamicensemble with a = 0 .
098 fm. We extract the rapidity-independent (or intrinsic) part of the softfunction through a large-momentum-transfer pseudo-scalar meson form factor and its quasi-TMDwave function using leading-order factorization in large-momentum effective theory. We also inves-tigate the rapidity-dependent part of the soft function—the Collins-Soper evolution kernel—basedon the large-momentum evolution of the quasi-TMD wave function.
Introduction.
For high-energy processes such as Higgsproduction at the Large-Hadron Collider, quantum chro-modynamics (QCD) factorization and parton distribu-tion functions (PDFs) have been essential for making the-oretical predictions [1, 2]. But for processes involving ob-servation of a relatively small transverse momentum, Q ⊥ such as in Drell-Yan (DY) production and semi-inclusivedeep inelastic scattering, a new non-perturbative quan-tity called soft function is required to capture the physicsof non-cancelling soft gluon-radiation at fixed Q ⊥ [3–6].Physically, the soft function in DY is a cross section fora pair of high-energy quark and anti-quark (or gluon)traveling in the opposite light-cone directions to radi-ate soft gluons of total transverse momentum Q ⊥ beforethey annihilate. Although much progress has been madein calculating the soft function in perturbation theory at Q ⊥ ≫ Λ QCD [7, 8], it is intrinsically non-perturbativewhen Q ⊥ is O (Λ QCD ). Calculating the non-perturbativetransverse-momentum-dependent (TMD) soft functionfrom first principles became feasible only recently [9].The main difficulty in calculating the TMD soft func-tion in lattice QCD is that it involves two light-likeWilson lines along directions n ± = √ (1 ,~ ⊥ , ±
1) in( t, ⊥ , z ) coordinates, making direct simulations in Eu-clidean space impractical. However, much progress hasbeen made in recent years in calculating physical quan-tities such as light-cone PDFs using the framework oflarge-momentum effective theory (LaMET) [10, 11]. The ∗ Corresponding author: [email protected] † Corresponding author: [email protected] key observation of LaMET is that the collinear quarkand gluon modes, usually represented by light-like fieldcorrelators [12–15], can be accessed for large-momentumhadron states. A detailed review of LaMET and its ap-plications to collinear PDFs and other light-cone distri-butions can be found in Refs.[16, 17]. More recently,some of the present authors have proposed that theTMD soft function can be extracted from a special large-momentum-transfer form factor of either a light meson ora pair of quark-antiquark color sources [9]. Once calcu-lated, the TMD factorization of the Drell-Yan and sim-ilar processes can be made with entirely lattice-QCD-computable non-perturbative quantities [18–23].The TMD soft function is often defined and applied notin momentum space but in transverse coordinate space interms of the Fourier transformation variable b ⊥ . In addi-tion, it also depends on the ultraviolet (UV) renormaliza-tion scale µ (often defined in dimensional regularizationand minimal subtraction or MS) and rapidity regulators Y + Y ′ [9, 12], S ( b ⊥ , µ, Y + Y ′ ) = e ( Y + Y ′ ) K ( b ⊥ ,µ ) S − I ( b ⊥ , µ ) (1)where the first factor is related to rapidity evolution[described by the Collin-Soper (CS) kernel K ], and thesecond factor S I is the intrinsic, rapidity independent,part of the soft contribution. The rapidity-regulator-independent CS-kernel K is found calculable by tak-ing ratio of the quasi-TMDPDF at two different mo-menta [20–25]. On the other hand, calculating the intrin-sic soft function on the lattice has never been attemptedbefore.In this paper we present the first lattice QCD cal-culation of the intrinsic soft function S I with severalmomenta on a 2+1 flavor CLS ensemble with a =0 .
098 fm [26], see Table I. In particular we perform sim-ulations of the large-momentum light-meson form factorand quasi-TMD wave functions (TMDWFs), whose ratiogives the intrinsic soft function [9]. The Wilson loop ma-trix element will be used to remove the linear divergencein the quasi-TMD wave function. The CS kernel, K , canalso be calculated from the external momentum depen-dence of the quasi-TMD wave function [16], and we willcalculate it as a by-product. Our result is consistent withthat of a quenched lattice study using TMDPDFs [25]. FIG. 1. Illustration of the pseudo-scalar meson form factor F calculated in this work. The initial and final momenta ofthe pion are large and opposite. The transition “current” ismade of two local operators at a fixed spatial separation b ⊥ . t sep is the time separation between the source and sink of thepion. Theoretical Framework.
The intrinsic soft function( S I ) can be obtained from the QCD factorization ofa large-momentum form factor of a non-singlet lightpseudo-scalar meson with constituents π = q γ q , withthe transition current made of two quark-bilinears witha fixed transverse separation ~b = ( ~n ⊥ b ⊥ , F ( b ⊥ , P z ) = h π ( − ~P ) | ( q Γ q )( ~b )( q Γ q )(0) | π ( ~P ) i c . (2)Here q , are light quark fields of different flavors, and ~P = ( ~ ⊥ , P z ). The initial and final mesons approachtwo opposite lightcone directions in the P z → ∞ limit.Only the connected diagram is important in the largemomentum limit, as illustrated in Fig. 1.It can be shown that the form factor defined in Eq. (2)is factorizable into the quasi-TMDWF Φ and the intrinsicsoft function S I [9, 16] F ( b ⊥ , P z ) = S I ( b ⊥ ) (3) × Z dx dx ′ H ( x, x ′ , P z )Φ † ( x ′ , b ⊥ , − P z ) Φ( x, b ⊥ , P z )where H is perturbative hard kernel. The quasi-TMDWFΦ is the Fourier transformation of the coordinate-space correlation function φ ( z, b ⊥ , P z ) = lim ℓ →∞ φ ℓ ( z, b ⊥ , P z , ℓ ) p Z E (2 ℓ, b ⊥ ) , (4) φ ℓ ( z, b ⊥ , P z , ℓ )= D (cid:12)(cid:12)(cid:12) q (cid:16) z n z + ~b (cid:17) Γ Φ W ( ~b, ℓ ) q (cid:16) − z n z (cid:17) (cid:12)(cid:12)(cid:12) π ( ~P ) E . In the above W ( ~b, ℓ ) is the spacelike staple-shaped gaugelink, W ( ~b, ℓ ) = P exp " ig s Z z/ − ℓ d s n z · A ( n z s + b ⊥ ) × P exp " ig s Z b ⊥ d s n ⊥ · A ( − ℓn z + sn ⊥ ) × P exp " ig s Z − ℓ − z/ d s n z · A ( n z s ) , (5) n z and n ⊥ are the unit vectors in z and transverse di-rections respectively. Z E (2 ℓ, b ⊥ ) is the vacuum expec-tation value of a rectangular spacelike Wilson loop withsize 2 ℓ × b ⊥ which removes the pinch-pole singularity andWilson-line self-energy in quasi-TMDWF [9].Since the UV divergence of the intrinsic soft functionis multiplicative [16], the ratio S I ( b ⊥ , /a ) /S I ( b ⊥ , , /a )calculable on lattice is UV renormalization-scheme inde-pendent, where b ⊥ , is a reference distance which is takensmall enough to be calculated perturbatively. Thus wecan obtain the result in the MS scheme through S I, MS ( b ⊥ , µ ) = (cid:18) S I ( b ⊥ , /a ) S I ( b ⊥ , , /a ) (cid:19) S I, MS ( b ⊥ , , µ ) (6)where S I, MS ( b ⊥ , , µ ) is perturbatively calculable, e.g., S I, MS ( b ⊥ , µ ) = 1 − α s C F π ln µ b ⊥ e − γ E + O ( α s ) . (7)In the present exploratory study, we will consider onlythe leading order matching in Eq. (3), for which the per-turbative kernel is H ( x, x ′ , P z ) = 1 / (2 N c ) + O ( α s ), inde-pendent of x and x ′ . Using φ (0 , b ⊥ , − P z ) = φ (0 , b ⊥ , P z )under parity transformation, we obtain S I ( b ⊥ ) = 2 N c F ( b ⊥ , P z ) | φ (0 , b ⊥ , P z ) | + O ( α s , (1 /P z ) ) , (8)where power corrections from finite P z are ignored. Since P z is related to the rapidity of the meson, we henceforthreplace it by the boost factor γ ≡ E π /m π . Eq. (6) canbe written as S I, MS ( b ⊥ , µ ) = F ( b ⊥ , P z ) F ( b ⊥ , , P z ) | φ (0 , b ⊥ , , P z ) | | φ (0 , b ⊥ , P z ) | + O ( α s , γ − ) . (9)The ratio on the right-hand side of the above expressionis independent of the renormalization scale µ since onlythe leading-order contribution is kept.On the other hand, the quasi-TMDWF can be usedto extract the Collins-Soper kernel K using a methodsimilar to [20] K ( b ⊥ , µ ) = 1ln( P z /P z ) ln (cid:12)(cid:12)(cid:12)(cid:12) C ( xP z , µ )Φ MS ( x, b ⊥ , P z , µ ) C ( xP z , µ )Φ MS ( x, b ⊥ , P z , µ ) (cid:12)(cid:12)(cid:12)(cid:12) (10)= 1ln( P z /P z ) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R d x Φ( x, b ⊥ , P z ) R d x Φ( x, b ⊥ , P z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( α s , γ − )= 1ln( P z /P z ) ln (cid:12)(cid:12)(cid:12)(cid:12) φ (0 , b ⊥ , P z ) φ (0 , b ⊥ , P z ) (cid:12)(cid:12)(cid:12)(cid:12) + O ( α s , γ − ) . (11)In the second line, again only use the leading ordermatching kernel C ( xP z , µ ) = 1 + O ( α s ) is used. Therenormalization factors for Φ are cancelled. The rapidity-scheme-independent CS kernel K is independent of µ inthis approximation because only the leading term hasbeen kept.While Eqs. (6) and (10) are exact and can be usedfor precision studied in the future, Eqs. (9) and (11) arethe leading-order approximation used in this pioneeringwork. TABLE I. Parameters used in the numerical simulation. Thefirst row shows the parameters of the 2+1 flavor clover fermionCLS ensemble (named A654) and the second one shows thenumber of the A654 configurations and valence pion massused for this calculation. β L × T a (fm) c sw κ sea l m sea π (MeV)3.34 24 ×
48 0.098 2.06686 0.13675 333 N cfg κ vl m vπ (MeV)868 0.13622 547 Simulation setup.
For the present study, we use con-figurations generated with 2+1 flavor clover fermionsand tree-level Symanzik gauge action configuration bythe CLS collaboration using periodic boundary condi-tions [26]. The detailed parameters are listed in Table I.Note that m π = 547 MeV instead of 333 MeV is used forvalence quarks in order to have a better signal. Physi-cally, the soft function becomes independent of the mesonmass for large boost factors γ .To calculate the form factor in Eq.(2), we generate thewall source propagator, S w ( x, t, t ′ ; ~p ) = X ~y S ( t, ~x ; t ′ , ~y ) e i~p · ( ~y − ~x ) , (12)on the Coulomb gauge fixed configurations at t ′ = 0 and t sep for both the initial and final meson states. S is thequark propagator from ( t ′ , ~y ) to ( t, ~x ). Then we can con-struct the three point function (3pt) corresponding to the form factor in Eq. (2), C ( b ⊥ , P z ; p z , t sep , t ) (13)= 1 L X x Tr h S † w ( ~x + ~b, t, − ~p ) γ Γ S w ( ~x + ~b, t, t sep ; ~p ) × S † w ( ~x, t, t sep ; − ~P + ~p ) γ Γ S w ( ~x, t, ~P − ~p ) i . The quark momentum ~p = ( ~ ⊥ , p z ), and the relation γ S † ( x, y ) γ = S ( y, x ) have been applied for the anti-quark propagator. We have tested several choices of Γ,and will use the unity Dirac matrix Γ = I as it has thebest signal and describes the leading twist light-cone con-tribution in the large P z limit. Notice that the Γ = γ case is subleading in large P z limit, although the excitedstate contamination might be smaller.By generating the wall source propagators at all the 48time slices with quark momentum p z = ( − , − , , , × π/ ( La ), we can maximize the statistics of the 3pt func-tion with all the meson momenta P z from 0 to 8 π/ ( La )( ∼ . t and t sep . C ( b ⊥ , P z , t sep , t )is related to the bare F ( b ⊥ , P z ) using standard parame-terization of 3pt with one excited sate, C ( b ⊥ , P z ; p z , t sep , t ) = A w ( p z ) (2 E ) e − Et sep (cid:2) F ( b ⊥ , P z )+ c ( e − ∆ Et + e − ∆ E ( t sep − t ) ) + c e − ∆ Et sep (cid:3) . (14) A w is the matrix element of the Coulomb gauge fixed wall(CFW) source pion interpolation field, E = p m π + P z is the pion energy, ∆ E is the mass gap between pion andits first excited state, c , are parameters for the excitedstate contamination. Note that the p z dependence factor A w will cancelThe same wall source propagators can be used to cal-culate the two-point function related to the bare quasi-TMDWF, C ( b ⊥ , P z ; p z , ℓ, t ) = 1 L p Z E (2 ℓ, b ⊥ ) X x Tr e i ~P · ~x × h S † w ( ~x + ~b, t, − ~p ) W ( ~b, ℓ ) γ Γ Φ S w ( ~x, t, P z − ~p ) i = A w ( p z ) A p E e − Et φ ℓ (0 , b ⊥ , P z , ℓ )(1 + c e − ∆ Et ) , (15)where again we parameterize the mixing with one excitedstate. A p is the matrix element of the point sink pion in-terpolation field. It will be removed when we normalize φ ℓ (0 , b ⊥ , P z , ℓ ) with φ ℓ (0 , , P z , Φ = γ t γ to define the wave function amplitude in Eq. (4). Basedon the quasi-TMDPDF study in Ref. [25] with the sim-ilar staple-shaped gauge link operator, the mixing effectfrom most of the Dirac matrices are at 1% level exceptin the cases with certain Dirac matrix which has verylarge statistical uncertainties. Thus the mixing effect isignored in this work.The dispersion relation of the pion state, statisticalchecks for the measurement histogram, and informa-tion on the autocorrection between configurations canbe found in the supplemental materials [27]. FIG. 2. Results for the ℓ dependence of the quasi-TMDWFwith z = 0, and also the square root of the Wilson loopwhich is used for the subtraction, taking the { P z , b ⊥ , t } = { π/L, a, a } case as a example. All the results are normal-ized with their values at ℓ = 0. Numerical Results.
We demonstrate the Wilson-linelength ℓ dependence of the norm of the quasi-TMDWFsin Fig. 2. As one can see from this figure, with { P z , b ⊥ , t } = { π/L, a, a } , both the quasi-TMDWF φ ℓ (0 , b ⊥ , P z , ℓ ) and the square root of the Wilson loop Z E decay exponentially with the length ℓ , but the subtractedquasi-TMDWF is length independent when ℓ ≥ . P z , b ⊥ , and t can be foundin the supplemental materials [27]. Based on this ob-servation, we will use ℓ = 7 a = 0 .
686 fm as asymptoticresults for all cases in the following calculation.
FIG. 3. The ratios C ( b ⊥ , P z , t sep , t ) /C (0 , P z , , t sep ) (datapoints) which converge to the ground state contribution at t, t sep → ∞ (gray band) as function of t sep and t , with { P z , b ⊥ } = { π/L, a } . As in this figure, our data in gen-eral agree with the predicted fit function (colored bands). We applied the joint fit of the form factor andquasi-TMDWF with the same P z and b ⊥ with theparameterization in Eqs. (14) and (15), and the ra-tios C ( b ⊥ , P z , t sep , t ) /C (0 , P z , , t sep ) with different t sep and t for the { P z , b ⊥ } = { π/L, a } case are shown inFig. 3, with ground state contribution (gray band) andthe fitted results at finite t and t (colored bands). In the calculation, the excited state contribution is properly de-scribed by the fit with χ / d . o . f . = 0 .
6. The details ofthe joint fit, and also more fit quality checks are shownin the supplemental materials [27], with similar fittingqualities.
FIG. 4. The intrinsic soft factor as a function of b ⊥ with b ⊥ , = a as in Eq. (9). With different pion momentum P z ,the results are consistent with each other. The dashed curveshows the result of the 1-loop calculation, see Eq. (7), withthe strong coupling α s (1 /b ⊥ ). The resulting soft factor as function of b ⊥ is plotted inFig. 4, at γ = 2.17, 3.06 and 3.98, which corresponds to P z = { , , } π/L = { . , . , . } GeV respectively.As in Fig. 4, the results at different large γ are consistentwith each other, demonstrating that the asymptotic limitis stable within errors. We also compare the intrinsic softfunction extracted from the lattice to the one-loop resultin Eq. (7), with α s ( µ = 1 /b ⊥ ) evolving from α s ( µ =2 GeV) ≈ .
3. Notice that the b ⊥ dependence of theformer comes purely from the lattice simulation, whilethat for the latter is from perturbation theory.We can see a clear P z dependence in thequasi-TMDWF | φ ℓ (0 , b ⊥ , P z , ℓ ) | normalized with φ ℓ (0 , , P z , /γ . Thus we use Eq. (11) to extract the kernel in thetree level approximation, and compare the result in thelower panel of Fig. 5 with the calculation in Ref. [25]and up to 3-loop perturbative one with α s ( µ = 1 /b ⊥ ).We estimate the systematic uncertainty by combiningin quadrature the statistical errors with contributionsfrom the imaginary part of the quasi-TMDWF, whichshould be identically zero. For details see the supple-mental materials [27], in particular Fig. 11. Our resultappears consistent with a quenched calculation basedon TMDPDFs from Ref. [25], but the results based on P z /P z = 3 / P z /P z = 4 / b ⊥ which might come from insufficient statisticsat P z =2.11 GeV, or potential systematic uncertainties. Summary and Outlook.
In this work, we have pre-sented an exploratory lattice calculation of the intrinsic
FIG. 5. Quasi-TMDWF (upper panel) and extracted Collins-Soper kernel (lower panel), as functions of b ⊥ . The visible P z dependence of the quasi-TMDWF can be primarily under-stood by that from the Collins-Soper kernel, as the kernel weobtained with tree level matching is consistent with up to 3-loop perturbative calculations (at small b ⊥ ) with the strongcoupling α s at the scale 1 /b ⊥ , and also the non-perturbativeresult from the pion quasi-TMDPDF. Results from quenchedlattice calculations [25] are also shown for comparison. soft function by simulating the light-meson form factorof four-quark non-local operators and quasi-TMD wave functions. Our result shows a mild hadron momentumdependence, which allows a future precision study toeliminate the large momentum dependence using pertur-bative matching [16]. As a reliability check, the agree-ment between the CS kernel obtained from our quasi-TMDWF result and the previous calculations showsthat the systematic uncertainties including the partiallyquenching effect, the leading perturbative matching andmissing power corrections 1 /γ in LaMET expansionmight be sub-leading. Still our calculation paves the waytowards the first principle predictions of physical crosssections for, e.g., Drell-Yan and Higgs productions atsmall transverse momentum. Acknowledgment. — We thank Xu Feng, Jianhui Zhangand Yong Zhao for valuable discussions. We thank theCLS Collaboration for sharing the lattice ensembles usedto perform this study. The LQCD calculations were per-formed using the Chroma software suite [28]. The nu-merical calculation is supported by Chinese Academy ofScience CAS Strategic Priority Research Program of Chi-nese Academy of Sciences, Grant No. XDC01040100,Center for HPC of Shanghai Jiao Tong University,HPC Cluster of ITP-CAS, and Jiangsu Key Lab forNSLSCS. J. Hua is supported by NSFC under grant No.11735010 and 11947215. Y.-S. Liu is supported by Na-tional Natural Science Foundation of China under grantNo.11905126. M. Schlemer and A. Sch¨afer were sup-ported by the cooperative research center CRC/TRR-55of DFG. P. Sun is supported by Natural Science Foun-dation of China under grant No. 11975127 as well asJiangsu Specially Appointed Professor Program. W.Wang is supported in part by Natural Science Founda-tion of China under grant No. 11735010, 11911530088,by Natural Science Foundation of Shanghai under grantNo. 15DZ2272100. Q.-A. Zhang is supported by theChina Postdoctoral Science Foundation and the NationalPostdoctoral Program for Innovative Talents (Grant No.BX20190207). [1] R. K. Ellis, W. J. Stirling and B. R. Webber, Camb.Monogr. Part. Phys. Nucl. Phys. Cosmol. , 1 (1996).[2] H. W. Lin et al. , Prog. Part. Nucl. Phys. , 107 (2018)doi:10.1016/j.ppnp.2018.01.007 [arXiv:1711.07916 [hep-ph]].[3] J. C. Collins and D. E. Soper, Nucl. Phys. B ,381 (1981) Erratum: [Nucl. Phys. B , 545 (1983)].doi:10.1016/0550-3213(81)90339-4[4] J. C. Collins, D. E. Soper and G. F. Sterman, Nucl. Phys.B , 199 (1985). doi:10.1016/0550-3213(85)90479-1[5] X. d. Ji, J. p. Ma and F. Yuan, Phys. Rev.D , 034005 (2005) doi:10.1103/PhysRevD.71.034005[hep-ph/0404183].[6] X. d. Ji, J. P. Ma and F. Yuan, Phys. Lett.B , 299 (2004) doi:10.1016/j.physletb.2004.07.026[hep-ph/0405085].[7] M. G. Echevarria, I. Scimemi and A. Vladimirov, Phys. Rev. D , no. 5, 054004 (2016)doi:10.1103/PhysRevD.93.054004 [arXiv:1511.05590[hep-ph]].[8] Y. Li and H. X. Zhu, Phys. Rev. Lett. , no.2, 022004 (2017) doi:10.1103/PhysRevLett.118.022004[arXiv:1604.01404 [hep-ph]].[9] X. Ji, Y. Liu and Y. S. Liu, arXiv:1910.11415 [hep-ph].[10] X. Ji, Phys. Rev. Lett. , 262002 (2013)doi:10.1103/PhysRevLett.110.262002 [arXiv:1305.1539[hep-ph]].[11] X. Ji, Sci. China Phys. Mech. Astron. , 1407(2014) doi:10.1007/s11433-014-5492-3 [arXiv:1404.6680[hep-ph]].[12] J. Collins, Camb. Monogr. Part. Phys. Nucl. Phys. Cos-mol. , 1 (2011).[13] C. W. Bauer, S. Fleming, D. Pirjol andI. W. Stewart, Phys. Rev. D , 114020 (2001) doi:10.1103/PhysRevD.63.114020 [hep-ph/0011336].[14] C. W. Bauer and I. W. Stewart, Phys. Lett. B , 134 (2001) doi:10.1016/S0370-2693(01)00902-9[hep-ph/0107001].[15] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev.D , 054022 (2002) doi:10.1103/PhysRevD.65.054022[hep-ph/0109045].[16] X. Ji, Y. S. Liu, Y. Liu, J. H. Zhang and Y. Zhao,arXiv:2004.03543 [hep-ph].[17] K. Cichy and M. Constantinou, Adv. High EnergyPhys. , 3036904 (2019) doi:10.1155/2019/3036904[arXiv:1811.07248 [hep-lat]].[18] X. Ji, P. Sun, X. Xiong and F. Yuan, Phys. Rev.D , 074009 (2015) doi:10.1103/PhysRevD.91.074009[arXiv:1405.7640 [hep-ph]].[19] X. Ji, L. C. Jin, F. Yuan, J. H. Zhang andY. Zhao, Phys. Rev. D , no. 11, 114006 (2019)doi:10.1103/PhysRevD.99.114006 [arXiv:1801.05930[hep-ph]].[20] M. A. Ebert, I. W. Stewart and Y. Zhao,Phys. Rev. D , no. 3, 034505 (2019)doi:10.1103/PhysRevD.99.034505 [arXiv:1811.00026 [hep-ph]].[21] M. A. Ebert, I. W. Stewart and Y. Zhao, JHEP , 037(2019) doi:10.1007/JHEP09(2019)037 [arXiv:1901.03685[hep-ph]].[22] X. Ji, Y. Liu and Y. S. Liu, arXiv:1911.03840 [hep-ph].[23] A. A. Vladimirov and A. Schfer, Phys. Rev. D ,no. 7, 074517 (2020) doi:10.1103/PhysRevD.101.074517[arXiv:2002.07527 [hep-ph]].[24] M. A. Ebert, I. W. Stewart and Y. Zhao, JHEP , 099(2020) doi:10.1007/JHEP03(2020)099 [arXiv:1910.08569[hep-ph]].[25] P. Shanahan, M. Wagman and Y. Zhao,arXiv:2003.06063 [hep-lat].[26] M. Bruno et al. , JHEP , 043 (2015)doi:10.1007/JHEP02(2015)043 [arXiv:1411.3982 [hep-lat]].[27] Supplemental materials.[28] R. G. Edwards et al. [SciDAC and LHPC andUKQCD Collaborations], Nucl. Phys. Proc. Suppl. , 832 (2005) doi:10.1016/j.nuclphysbps.2004.11.254[hep-lat/0409003]. SUPPLEMENTAL MATERIALSA. Simulation checks
FIG. 6. The dispersion relation of the pion state with the pion mass from the 2pt function. The data up to 8 π/L ( ∼ E π = √ m π + c P + c P a with c = 0 . c = − . π/L from the continuum limit is around 2%. Fig. 6 shows the dispersion relation with the pion mass we used. The curve shows the fit based on the formula E π = p m π + c P + c P a , where the last term in the square root parameterizes discretization errors. We usedmomenta up to 8 π/L ( ∼ c = 0 . c = − . E π = p m π + P in the continuum limit.Taking the form factor with P z = 6 π/L , b ⊥ =3, t =8 and t = t / ×
48 (time slides) = 41664(measurements) with 54 exceptional measurements being dropped in the analysis since they give results deviatingfrom the central ones by much more than 5 σ . After we average the measurements over the same configuration, wefind that the autocorrection effect is negligible, since no obvious bin size dependence of the result is observed, as FIG. 7. Statistical check on the simulation, taking the form factor with P z = 6 π/L , b ⊥ =3, t =8 and t = t / ×
48 time slides - 54 exceptional measurements),and the right panel shows the bin size dependence after we averaged all the measurements on the same configuration. shown in the right panel of Fig. 7.
FIG. 8. Figure for the ℓ dependence of | φ ℓ (0 , b ⊥ , P z , ℓ ) | with { P z , b ⊥ , t } = { π/L, a, a } (top left, the case shown in Fig. 2), { P z , b ⊥ , t } = { π/L, a, a } (top right), { P z , b ⊥ , t } = { π/L, a, a } (bottom left), and { P z , b ⊥ , t } = { π/L, a, a } (bottomright). B. ℓ dependence of TMDWF In Fig. 8, we give the ℓ dependence of | φ ℓ (0 , b ⊥ , P z , ℓ ) | for a few more cases, similar to the { P z , b ⊥ , t } = { π/L, a, a } case shown in Fig. 2 but with larger P z , b ⊥ and also t . C. Two-state fit of the form factors
In this work, we use the following joint fit to obtain the norm of the subtracted quasi-TMDWF | φ ℓ (0 , b ⊥ , P z , ℓ ) | and soft factor S I ( b ⊥ ) (with ℓ = 7 a ), C ( b ⊥ , P z , t sep , t ) C (0 , P z , , t sep ) = | ˜ φ ℓ (0 , b ⊥ , P z , ℓ ) | ˜ S I ( b ⊥ ) + C ( e − ∆ Et + e − ∆ E ( t sep − t ) ) + C e − ∆ Et sep C e − ∆ Et sep ,C ( b ⊥ , P z , , t ) C (0 , P z , , t ) = | ˜ φ ℓ (0 , b ⊥ , P z , ℓ ) | e θ ( b ⊥ ,P z ,ℓ ) (1 + C e − ∆ Et )1 + C e − ∆ Et , (16)where ˜ φ ℓ (0 , b ⊥ , P z , ℓ ) = φ ℓ (0 , b ⊥ , P z , ℓ ) φ ℓ (0 , , P z , , ˜ S I ( b ⊥ ) = φ L (0 , , P z , A w EA p S I ( b ⊥ ) , (17)and θ ( b ⊥ , P z , ℓ ) is the phase of the quasi-TMDWF. The additional factor in the definition of ˜ S I will be cancelled by˜ S I ( b ⊥ = a ) in the ratio of Eq. (6). FIG. 9. The ratios C ( b ⊥ , P z , t sep , t ) /C (0 , P z , , t sep ) as function of t sep and t , with { P z , b ⊥ } = { π/L, a } (left panel), { P z , b ⊥ } = { π/L, a } (right panel) and { P z , b ⊥ } = { π/L, a } (lower panel). In Fig. 9, we shows the ratios C ( b ⊥ , P z , t sep , t ) /C (0 , P z , , t sep ) with P z = 6 π/L , b ⊥ = { a, a, a } , comparedwith the two-state fit predictions (colored bands) and fitted ground state contribution (gray band). All of them showa good agreement between data and fits.As another check, we also consider the differential summed ratio R ( b ⊥ , P z , t sep ) ≡ SR ( b ⊥ , P z , t sep ) − SR ( b ⊥ , P z , t sep −
1) = | ˜ φ ℓ (0 , b ⊥ , P z , ℓ ) | ˜ S I ( b ⊥ ) + O ( e − ∆ Et sep ) ,SR ( b ⊥ , P z , t sep ) ≡ X