Abstract
We calculate the soft function using lattice QCD in the framework of large momentum effective theory incorporating the one-loop perturbative contributions. The soft function is a crucial ingredient in the lattice determination of light cone objects using transverse-momentum-dependent (TMD) factorization. It consists of a rapidity-independent part called intrinsic soft function and a rapidity-dependent part called Collins-Soper kernel. We have adopted appropriate normalization when constructing the pseudoscalar meson form factor that is needed in the determination of the intrinsic part and applied Fierz rearrangement to suppress the higher-twist effects. In the calculation of CS kernel we consider a CLS ensemble other than the MILC ensemble used in a previous study. We have also compared the applicability of determining the CS kernel using quasi TMDWFs and quasi TMDPDFs. As an example, the determined soft function is used to obtain the physical TMD wave functions (WFs) of pion and unpolarized iso-vector TMD parton distribution functions (PDFs) of proton.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.C. Collins and D.E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. 213 (1983) 545] [INSPIRE].
J.C. Collins and D.E. Soper, Back-To-Back Jets: Fourier Transform from B to K-Transverse, Nucl. Phys. B 197 (1982) 446 [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].
S. Amoroso et al., Snowmass 2021 Whitepaper: Proton Structure at the Precision Frontier, Acta Phys. Polon. B 53 (2022) 12 [arXiv:2203.13923] [INSPIRE].
R. Abdul Khalek et al., Snowmass 2021 White Paper: Electron Ion Collider for High Energy Physics, arXiv:2203.13199 [INSPIRE].
I. Scimemi and A. Vladimirov, Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum, JHEP 06 (2020) 137 [arXiv:1912.06532] [INSPIRE].
R. Angeles-Martinez et al., Transverse Momentum Dependent (TMD) parton distribution functions: status and prospects, Acta Phys. Polon. B 46 (2015) 2501 [arXiv:1507.05267] [INSPIRE].
Z.-B. Kang, K. Samanta, D.Y. Shao and Y.-L. Zeng, Transverse momentum dependent distribution functions in the threshold limit, arXiv:2211.08341 [INSPIRE].
M. Bury, A. Prokudin and A. Vladimirov, Extraction of the Sivers Function from SIDIS, Drell-Yan, and W±/Z Data at Next-to-Next-to-Next-to Leading Order, Phys. Rev. Lett. 126 (2021) 112002 [arXiv:2012.05135] [INSPIRE].
A. Bacchetta et al., Transverse-momentum-dependent parton distributions up to N3LL from Drell-Yan data, JHEP 07 (2020) 117 [arXiv:1912.07550] [INSPIRE].
M.G. Echevarria, Z.-B. Kang and J. Terry, Global analysis of the Sivers functions at NLO+NNLL in QCD, JHEP 01 (2021) 126 [arXiv:2009.10710] [INSPIRE].
MAP (Multi-dimensional Analyses of Partonic distributions) collaboration, Unpolarized transverse momentum distributions from a global fit of Drell-Yan and semi-inclusive deep-inelastic scattering data, JHEP 10 (2022) 127 [arXiv:2206.07598] [INSPIRE].
F. Landry, R. Brock, G. Ladinsky and C.P. Yuan, New fits for the nonperturbative parameters in the CSS resummation formalism, Phys. Rev. D 63 (2001) 013004 [hep-ph/9905391] [INSPIRE].
F. Landry, R. Brock, P.M. Nadolsky and C.P. Yuan, Tevatron Run-1 Z boson data and Collins-Soper-Sterman resummation formalism, Phys. Rev. D 67 (2003) 073016 [hep-ph/0212159] [INSPIRE].
A.V. Konychev and P.M. Nadolsky, Universality of the Collins-Soper-Sterman nonperturbative function in gauge boson production, Phys. Lett. B 633 (2006) 710 [hep-ph/0506225] [INSPIRE].
P. Sun, J. Isaacson, C.-P. Yuan and F. Yuan, Nonperturbative functions for SIDIS and Drell–Yan processes, Int. J. Mod. Phys. A 33 (2018) 1841006 [arXiv:1406.3073] [INSPIRE].
U. D’Alesio, M.G. Echevarria, S. Melis and I. Scimemi, Non-perturbative QCD effects in qT spectra of Drell-Yan and Z-boson production, JHEP 11 (2014) 098 [arXiv:1407.3311] [INSPIRE].
M.G. Echevarria, A. Idilbi, Z.-B. Kang and I. Vitev, QCD Evolution of the Sivers Asymmetry, Phys. Rev. D 89 (2014) 074013 [arXiv:1401.5078] [INSPIRE].
Z.-B. Kang, A. Prokudin, P. Sun and F. Yuan, Extraction of Quark Transversity Distribution and Collins Fragmentation Functions with QCD Evolution, Phys. Rev. D 93 (2016) 014009 [arXiv:1505.05589] [INSPIRE].
A. Bacchetta et al., Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production, JHEP 06 (2017) 081 [Erratum ibid. 06 (2019) 051] [arXiv:1703.10157] [INSPIRE].
I. Scimemi and A. Vladimirov, Analysis of vector boson production within TMD factorization, Eur. Phys. J. C 78 (2018) 89 [arXiv:1706.01473] [INSPIRE].
V. Bertone, I. Scimemi and A. Vladimirov, Extraction of unpolarized quark transverse momentum dependent parton distributions from Drell-Yan/Z-boson production, JHEP 06 (2019) 028 [arXiv:1902.08474] [INSPIRE].
M. Bury et al., PDF bias and flavor dependence in TMD distributions, JHEP 10 (2022) 118 [arXiv:2201.07114] [INSPIRE].
M. Horstmann, A. Schafer and A. Vladimirov, Study of the worm-gear-T function g1T with semi-inclusive DIS data, Phys. Rev. D 107 (2023) 034016 [arXiv:2210.07268] [INSPIRE].
P. Hagler, B.U. Musch, J.W. Negele and A. Schafer, Intrinsic quark transverse momentum in the nucleon from lattice QCD, EPL 88 (2009) 61001 [arXiv:0908.1283] [INSPIRE].
B.U. Musch et al., Sivers and Boer-Mulders observables from lattice QCD, Phys. Rev. D 85 (2012) 094510 [arXiv:1111.4249] [INSPIRE].
B. Yoon et al., Lattice QCD calculations of nucleon transverse momentum-dependent parton distributions using clover and domain wall fermions, in the proceedings of the 33rd International Symposium on Lattice Field Theory, SISSA (2015) [arXiv:1601.05717] [INSPIRE].
B. Yoon et al., Nucleon Transverse Momentum-dependent Parton Distributions in Lattice QCD: Renormalization Patterns and Discretization Effects, Phys. Rev. D 96 (2017) 094508 [arXiv:1706.03406] [INSPIRE].
X. Ji, Parton Physics on a Euclidean Lattice, Phys. Rev. Lett. 110 (2013) 262002 [arXiv:1305.1539] [INSPIRE].
X. Ji, Parton Physics from Large-Momentum Effective Field Theory, Sci. China Phys. Mech. Astron. 57 (2014) 1407 [arXiv:1404.6680] [INSPIRE].
LPC collaboration, Nonperturbative determination of the Collins-Soper kernel from quasitransverse-momentum-dependent wave functions, Phys. Rev. D 106 (2022) 034509 [arXiv:2204.00200] [INSPIRE].
M.-H. Chu et al., Transverse-Momentum-Dependent Wave Functions of Pion from Lattice QCD, arXiv:2302.09961 [INSPIRE].
X. Ji, P. Sun, X. Xiong and F. Yuan, Soft factor subtraction and transverse momentum dependent parton distributions on the lattice, Phys. Rev. D 91 (2015) 074009 [arXiv:1405.7640] [INSPIRE].
X. Ji et al., Large-momentum effective theory, Rev. Mod. Phys. 93 (2021) 035005 [arXiv:2004.03543] [INSPIRE].
Lattice Parton collaboration, Lattice-QCD Calculations of TMD Soft Function Through Large-Momentum Effective Theory, Phys. Rev. Lett. 125 (2020) 192001 [arXiv:2005.14572] [INSPIRE].
X. Ji, Y. Liu and Y.-S. Liu, TMD soft function from large-momentum effective theory, Nucl. Phys. B 955 (2020) 115054 [arXiv:1910.11415] [INSPIRE].
Y. Li et al., Lattice QCD Study of Transverse-Momentum Dependent Soft Function, Phys. Rev. Lett. 128 (2022) 062002 [arXiv:2106.13027] [INSPIRE].
Z.-F. Deng, W. Wang and J. Zeng, Transverse-momentum-dependent wave functions and soft functions at one-loop in large momentum effective theory, JHEP 09 (2022) 046 [arXiv:2207.07280] [INSPIRE].
M.A. Ebert, I.W. Stewart and Y. Zhao, Determining the Nonperturbative Collins-Soper Kernel From Lattice QCD, Phys. Rev. D 99 (2019) 034505 [arXiv:1811.00026] [INSPIRE].
P. Shanahan, M. Wagman and Y. Zhao, Collins-Soper kernel for TMD evolution from lattice QCD, Phys. Rev. D 102 (2020) 014511 [arXiv:2003.06063] [INSPIRE].
P. Shanahan, M. Wagman and Y. Zhao, Lattice QCD calculation of the Collins-Soper kernel from quasi-TMDPDFs, Phys. Rev. D 104 (2021) 114502 [arXiv:2107.11930] [INSPIRE].
M. Schlemmer et al., Determination of the Collins-Soper Kernel from Lattice QCD, JHEP 08 (2021) 004 [arXiv:2103.16991] [INSPIRE].
H.-T. Shu et al., Universality of the Collins-Soper kernel in lattice calculations, arXiv:2302.06502 [INSPIRE].
Y. Li and H.X. Zhu, Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum Resummation, Phys. Rev. Lett. 118 (2017) 022004 [arXiv:1604.01404] [INSPIRE].
A.A. Vladimirov, Correspondence between Soft and Rapidity Anomalous Dimensions, Phys. Rev. Lett. 118 (2017) 062001 [arXiv:1610.05791] [INSPIRE].
LPC collaboration, Unpolarized Transverse-Momentum-Dependent Parton Distributions of the Nucleon from Lattice QCD, arXiv:2211.02340 [INSPIRE].
X. Xiong, X. Ji, J.-H. Zhang and Y. Zhao, One-loop matching for parton distributions: Nonsinglet case, Phys. Rev. D 90 (2014) 014051 [arXiv:1310.7471] [INSPIRE].
X. Ji, Y. Liu and Y.-S. Liu, Transverse-momentum-dependent parton distribution functions from large-momentum effective theory, Phys. Lett. B 811 (2020) 135946 [arXiv:1911.03840] [INSPIRE].
M.A. Ebert, S.T. Schindler, I.W. Stewart and Y. Zhao, Factorization connecting continuum & lattice TMDs, JHEP 04 (2022) 178 [arXiv:2201.08401] [INSPIRE].
X. Ji and Y. Liu, Computing light-front wave functions without light-front quantization: A large-momentum effective theory approach, Phys. Rev. D 105 (2022) 076014 [arXiv:2106.05310] [INSPIRE].
[Lattice Parton Collaboration (LPC)] collaboration, Renormalization of Transverse-Momentum-Dependent Parton Distribution on the Lattice, Phys. Rev. Lett. 129 (2022) 082002 [arXiv:2205.13402] [INSPIRE].
A.A. Vladimirov, Self-contained definition of the Collins-Soper kernel, Phys. Rev. Lett. 125 (2020) 192002 [arXiv:2003.02288] [INSPIRE].
HPQCD and UKQCD collaborations, Highly improved staggered quarks on the lattice, with applications to charm physics, Phys. Rev. D 75 (2007) 054502 [hep-lat/0610092] [INSPIRE].
A. Hasenfratz and F. Knechtli, Flavor symmetry and the static potential with hypercubic blocking, Phys. Rev. D 64 (2001) 034504 [hep-lat/0103029] [INSPIRE].
G.S. Bali, B. Lang, B.U. Musch and A. Schäfer, Novel quark smearing for hadrons with high momenta in lattice QCD, Phys. Rev. D 93 (2016) 094515 [arXiv:1602.05525] [INSPIRE].
MILC collaboration, Lattice QCD Ensembles with Four Flavors of Highly Improved Staggered Quarks, Phys. Rev. D 87 (2013) 054505 [arXiv:1212.4768] [INSPIRE].
X. Ji, J.-H. Zhang and Y. Zhao, Renormalization in Large Momentum Effective Theory of Parton Physics, Phys. Rev. Lett. 120 (2018) 112001 [arXiv:1706.08962] [INSPIRE].
T. Ishikawa, Y.-Q. Ma, J.-W. Qiu and S. Yoshida, Renormalizability of quasiparton distribution functions, Phys. Rev. D 96 (2017) 094019 [arXiv:1707.03107] [INSPIRE].
J. Green, K. Jansen and F. Steffens, Nonperturbative Renormalization of Nonlocal Quark Bilinears for Parton Quasidistribution Functions on the Lattice Using an Auxiliary Field, Phys. Rev. Lett. 121 (2018) 022004 [arXiv:1707.07152] [INSPIRE].
P. Shanahan, M.L. Wagman and Y. Zhao, Nonperturbative renormalization of staple-shaped Wilson line operators in lattice QCD, Phys. Rev. D 101 (2020) 074505 [arXiv:1911.00800] [INSPIRE].
X. Ji et al., A Hybrid Renormalization Scheme for Quasi Light-Front Correlations in Large-Momentum Effective Theory, Nucl. Phys. B 964 (2021) 115311 [arXiv:2008.03886] [INSPIRE].
Lattice Parton Collaboration (LPC) collaboration, Self-renormalization of quasi-light-front correlators on the lattice, Nucl. Phys. B 969 (2021) 115443 [arXiv:2103.02965] [INSPIRE].
Y. Ji, J.-H. Zhang, S. Zhao and R. Zhu, Renormalization and mixing of staple-shaped Wilson line operators on the lattice revisited, Phys. Rev. D 104 (2021) 094510 [arXiv:2104.13345] [INSPIRE].
RQCD collaboration, Nonperturbative Renormalization in Lattice QCD with three Flavors of Clover Fermions: Using Periodic and Open Boundary Conditions, Phys. Rev. D 103 (2021) 094511 [Erratum ibid. 107 (2023) 039901] [arXiv:2012.06284] [INSPIRE].
HotQCD collaboration, Heavy Quark Diffusion from 2+1 Flavor Lattice QCD with 320 MeV Pion Mass, Phys. Rev. Lett. 130 (2023) 231902 [arXiv:2302.08501] [INSPIRE].
R. Babich et al., Adaptive multigrid algorithm for the lattice Wilson-Dirac operator, Phys. Rev. Lett. 105 (2010) 201602 [arXiv:1005.3043] [INSPIRE].
J.C. Osborn et al., Multigrid solver for clover fermions, PoS LATTICE2010 (2010) 037 [arXiv:1011.2775] [INSPIRE].
SciDAC et al. collaborations, The Chroma software system for lattice QCD, Nucl. Phys. B Proc. Suppl. 140 (2005) 832 [hep-lat/0409003] [INSPIRE].
QUDA collaboration, Solving Lattice QCD systems of equations using mixed precision solvers on GPUs, Comput. Phys. Commun. 181 (2010) 1517 [arXiv:0911.3191] [INSPIRE].
QUDA collaboration, Scaling Lattice QCD beyond 100 GPUs, in the proceedings of the SC11 International Conference for High Performance Computing, Networking, Storage and Analysis, (2011) [https://doi.org/10.1145/2063384.2063478] [arXiv:1109.2935] [INSPIRE].
QUDA collaboration, Accelerating Lattice QCD Multigrid on GPUs Using Fine-Grained Parallelization, arXiv:1612.07873 [INSPIRE].
Y.-J. Bi et al., Lattice QCD package GWU-code and QUDA with HIP, PoS LATTICE2019 (2020) 286 [arXiv:2001.05706] [INSPIRE].
Acknowledgments
We acknowledge the Rechenzentrum of Regensburg for providing computer time on the Athene Cluster. We thank the CLS Collaboration for sharing the ensembles used to perform this study. We thank Wolfgang Söldner for valuable discussions on the X650 ensemble. The LQCD calculations were performed using the multigrid algorithm [66, 67], Chroma software suite and QUDA [69,70,71] through HIP programming model . This work is supported in part by Natural Science Foundation of China under grant No. U2032102, 12125503, 12205106, 12175073, 12222503, 12293062, 12147140, 12205180. The computations in this paper were run on the Siyuan-1 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University, and Advanced Computing East China Subcenter. J.H and J.L are also supported by Guangdong Major Project of Basic and Applied Basic Research No. 2020B0301030008, the Science and Technology Program of Guangzhou No. 2019050001. Y.B.Y is also supported by the Strategic Priority Research Program of Chinese Academy of Sciences, Grant No. XDB34030303 and XDPB15. J.H.Z. is supported in part by National Natural Science Foundation of China under grant No. 11975051. J.Z. is also supported by Project funded by China Postdoctoral Science Foundation under Grant No. 2022M712088. A.S., H.T.S, W.W, Y.B.Y and J.H.Z are also supported by a NSFC-DFG joint grant under grant No. 12061131006 and SCHA 458/22.
Author information
Authors and Affiliations
Consortia
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2306.06488
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
The Lattice Parton Collaboration (LPC)., Chu, MH., He, JC. et al. Lattice calculation of the intrinsic soft function and the Collins-Soper kernel. J. High Energ. Phys. 2023, 172 (2023). https://doi.org/10.1007/JHEP08(2023)172
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2023)172