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In the framework of large-momentum effective theory at one-loop matching accuracy, we perform a lattice calculation of the Collins-Soper kernel, which governs the rapidity evolution of transverse-momentum-dependent (TMD) distributions. We first obtain the quasi-TMD wave functions at three different meson momenta, on a lattice with valence clover quarks on a dynamical highly improved staggered quark sea and lattice spacing

Understanding the internal three-dimensional structure of hadrons, such as the proton, is an important goal in nuclear and particle physics. In this regard, the transverse momentum-dependent (TMD) parton distribution functions (TMDPDFs)

In contrast to the TMDPDFs that encode the probability density of parton momenta in hadrons, the transverse momentum-dependent wave functions (TMDWFs) offer a probability amplitude description of the partonic structure of hadrons, from which one can potentially calculate various quark/gluon distributions. In the QCD factorization involving transverse momentum, they are the most important ingredients to predict physical observables in exclusive processes, for instance, weak decays of heavy

A common feature of TMDPDFs and TMDWFs is that they depend both on the longitudinal momentum fraction

This has been realized in the framework of large-momentum effective theory

A very important result of LaMET development is that the TMDPDFs and TMDWFs can be calculated through the Euclidean quasi-TMDPDFs and quasi-TMDWFs, as well as a universal soft function (factor)

In this work, we present a state-of-the-art calculation of the CS kernel, based on a lattice QCD analysis of quasi-TMDWFs with

The remainder of this paper is organized as follows. In Sec.

In this section, we review the necessary theoretical background for the present calculation. We present the definitions of the CS kernel and rapidity evolution and introduce the quasi-TMD wave functions. We then discuss the factorization of the quasi-TMDWFs and its connection with the CS kernel.

Unlike the collinear light-cone PDFs and distribution amplitudes, the TMDPDFs and TMDWFs depend on both the renormalization scale

The CS kernel

In the past decades, the CS kernel has been widely studied in global fits of TMD parton distributions

As stated above, one can define the quasi-TMDWFs for a highly boosted pseudoscalar meson along the

Illustration of the staple-shaped gauge link included in unsubtracted quasi-TMDWFs and the related Wilson loop. The blue and red double lines in the upper panel represent the

Since the linear divergence is associated with the gauge link, it can be removed by a similar gauge link with the same total length. An optional choice is to make use of the Wilson loop, denoted as

With the help of the soft function, the infrared contributions in the subtracted quasi-TMDWFs can be properly accounted for such that the infrared structures for the quasi-TMDWFs and light-cone ones are matched. This implies a multiplicative factorization theorem in the framework of LaMET

A characteristic behavior of Eq.

Leading power reduced graph for pseudoscalar meson quasi-TMDWFs. Here

From Eq.

In order to extract the CS kernel

In this section, we present our lattice QCD results. We start with the lattice setup, followed by results for quasi-TMDWFs with two-point correlations. The Wilson loop results are discussed in Sec.

Our numerical simulations use

To further improve the statistical signals, we adopt hypercubic (HYP) smeared fat links

In order to calculate the quasi-TMDWFs defined in Eq.

By generating the wall-source propagators with quark momenta

In the above parametrization, the excited-state contributions are collected into the

Comparison of two-state fit and one-state fit to extract

The unsubtracted quasi-TMDWF matrix elements

The linear divergence in

The heavy quark effective potential term

The logarithmic divergence

In this work, the Wilson loop renormalization method

Results for the

For a pseudoscalar meson on a Euclidean lattice, both

Figure

According to our numerical simulations, the subtracted quasi-TMDWFs in coordinate space

Examples for subtracted quasi-TMDWFs in coordinate space. Here we take the cases of

Examples for subtracted quasi-TMDWFs in momentum space with hadron momentum

The CS kernel governs the rapidity evolution and thus is independent of the momentum fraction of the involved parton. But as indicated in Eq.

Figure

The fit results of

Theoretically, the physical CS kernel is purely real; however, there still exists a residual imaginary part at one-loop matching. Based on the perturbative results given in Ref.

On the lattice side, we have found that the quasi-TMDWFs are indeed complex. However, when taking the ratio of quasi-TMDWFs with different momentum, the imaginary parts mostly cancel. As a result, this leaves a mismatch between the imaginary parts of the perturbative matching kernel and quasi-TMDWFs, though both are complex. Shown as Fig.

Comparison of ratios of one-loop matching kernels and quasi-TMDWFs with

For now, we can only consider this imaginary part as systematic uncertainty of our final results, and it can be expressed as

It should be noticed that the perturbative matching kernel

One should notice that the Wilson loop renormalized quasi-TMDWF on the lattice [Eq.

The extracted CS kernel from the combined fit of the ratios of quasi-TMDWFs with different momenta are shown as the red data points in Fig.

Upper: comparison of our results

As mentioned above, the imaginary parts might come from higher-order/power corrections in the matching kernel, and here we consider them as one possible source of systematical uncertainties. In practice, we use the average of

Comparison of the Collins-Soper kernel

As a comparison, we also give the tree-level matching result for the CS kernel. With the leading order matching kernel

We compare our results with the ones from perturbative calculations and phenomenological extractions, as well as the lattice results determined by other collaborations.

The black solid and dashed lines in the upper panel of Fig.

Similar to this work, the results of LPC

In another way, the SWZ

The lower panel of Fig.

In this work, we have calculated the CS kernel on a MILC lattice configuration in the large-momentum effective theory framework. Comparing with our previous study

For our future studies, we need to use lattice configurations with multiple lattice spacing to understand the finite lattice spacing effects. We would use a valence quark mass consistent with the sea-quark one to reduce the nonunitarity effects. One such effect might be the imaginary part of the meson wave function, which seems inconsistent with perturbative calculation at the present time. Clearly, all these explorations will take more computational resources.

We thank Xu Feng, Yizhuang Liu, and Feng Yuan for useful discussions. This work is supported in part by Natural Science Foundation of China under Grants No. 11735010, No. 11911530088, No. U2032102, No. 11653003, No. 11975127, No. 11975051, No. 12005130, and No. 12147140. M. C., J. H., and W. W. are also supported by Natural Science Foundation of Shanghai under Grant No. 15DZ2272100. P. S. is also supported by the Jiangsu Specially Appointed Professor Program. Y. B. Y. is also supported by the Strategic Priority Research Program of Chinese Academy of Sciences, Grants No. XDB34030303 and No. XDPB15. A. S., P. S., W. W., Y. B. Y., and J. H. Z. are also supported by the NSFC-DFG joint grant under Grants No. 12061131006 and No. SCHA 458/22. X. J. is partially supported by the U.S. Department of Energy under Award No. DE-SC0020682. The calculation was supported by Advanced Computing East China Subcenter and the

In Sec.

Four examples for comparing two-state fit and one-state fit to extract the

In Sec.

Results showing the dependence on the gauge line length

In Sec.

Examples of comparisons for

Comparison of weight factors with

In our analysis, we adopt the brute-force Fourier transformation to obtain the quasi-TMDWFs in momentum space. In order to figure out whether the finite separation and truncation will introduce some bias, we do the following analysis.

First, in order to explicitly demonstrate the impact from the tail at large

Quasi-TMDWFs after Fourier transformation with different

Comparison of the CS kernel extracted from the quasi-TMDWFs with different truncation at

Second, we made a further exploration on the discreteness of data. Using the available data, we first made an extrapolation and then performed the Fourier transformation. The corresponding results are shown in Fig.

Comparison of results with and without interpolation.

It has been indicated in the main text that different fit ranges as well as parametrization formulas might introduce some arbitrariness into the joint fit. The end-point range can be roughly estimated by the largest attainable

Comparison of the jointly fitted results for the constant fit and the parametrization in Eq.