The description of the internal structure of hadrons is a fundamental problem of QCD. The present description of high-energy processes is founded on factorization theorems, which express the cross sections of reactions in terms of calculable perturbative parts and universal nonperturbative functions. In the modern era, the emphasis of studies is shifting toward multidimensional observables [1,2] and multidimensional parton distributions, such as transverse momentum dependent parton distribution functions (TMDPDFs) [3]. TMDPDFs encode information about the 3D parton momenta inside a hadron. TMDPDFs are assumed to be universal in the perturbative domain [small b in Eq. (1)], i.e., to depend on the types of parton and hadron but not on the process. This universality is the cornerstone of the factorization approach. In the case of TMDPDFs, it has been only indirectly confirmed by many phenomenological extractions, which utilize multiple processes [4–11].

The evolution of TMDPDFs with the rapidity scale ζ is described by 2ζddζlnΦf/h(x,b,μ,ζ)=K(b,μ),(1)which also provides the simplest way to access the CS kernel K(b,μ)–the topic of this study. For large b, where there exists only little experimental data, the status of the universality of the CS kernel is questionable. It can at present only be tested by lattice simulations like ours. Φf/h is a TMDPDF of flavor f in hadron h with x being the longitudinal momentum fraction, and b being the transverse distance, at scale μ. The evolution equation (1) is predicted to hold for TMDPDFs of any kind, including also twist-3 TMDPDFs, as was derived recently in [12–14] and is verified for the first time by lattice calculation in this work. In fact, the CS kernel is one of the most fundamental nonperturbative functions in QCD since it describes the interaction of a parton with the QCD vacuum [15] and appears in the description of many types of processes, including inclusive ones [16], exclusive ones [17], and jet-production [18]. Being a vacuum-determined function, the CS kernel obeys a stronger universality—it is independent of any quantum numbers except the color representation of the probe (quark or gluon). The confirmation of this universality for the CS-kernel is of fundamental importance for QCD.

Traditionally, the CS kernel is determined from fits of scattering data, along with TMDPDFs; see Refs. [7,8,19] for examples. However, this approach requires assumption of a functional form and, thus, is biased. Recently a number of more direct ways were proposed. All these methods suggest to determine the CS kernel from the ratio of properly constructed observables—cross sections [20] or quasi-TMDPDFs [21–23]. The latter can be achieved by lattice QCD simulations, which have been done in Refs. [21,24–27]. So far, all simulations were done for unpolarized quasi-TMDPDFs of the proton. The only exception is [25], where also polarized quasi-TMDPDFs are used.

In this paper, we present a new set of lattice computations of the CS kernel. We compare results for four basically independent calculations, namely for proton and pion targets, and twist-two and twist-three quasi-TMDPDF operators. The agreement between the results confirms universality of the CS kernel and further constrains its form. This is a proof of principle. The precision of such tests will improve continuously in the future.

We consider the following matrix element (quasi-TMDPDF) Wf/h[Γ](b;ℓ,L;v,P,S;μ)=⟨h(P,S)|q¯f(b+ℓv)ΓU[C(ℓ,v,b,L)]qf(0)|h(P,S)⟩,(2)where |h(P,S)⟩ is a single-hadron state with momentum P (Pμ=(EP,Px,0,0) in this study) and spin S (which is suppressed in the following for brevity). Γ is a Dirac matrix and f indicates the flavor of the quark field. The staple-shaped Wilson link U is of length L stretching in the direction vμ=(0,±1,0,0) and of width b→ pointing in a transverse spatial direction. The quark-antiquark pair connected by the Wilson link is positioned in the same imaginary time slice. The offset of the quark-antiquark pair along v is denoted by ℓ. The structure of the matrix element is sketched in Fig. 1.

At large L and hadron momentum, it is assumed that the matrix element (2) can be factorized [21–23,28,29] after having been transformed to momentum space. The structure of the factorization theorem crucially depends on Γ. In particular, for Γ={γ0,v,…} (the complete set can be found in Ref. [30]) one has the so-called leading power (LP) expression Wf/h[Γ](x;b;P;μ)=(2|x|(P+)2ζ)K(b,μ)/2×CH(xP+,μ)Φf/h[Γ](x,b;μ,ζ)+O(λ2),(3)where P+=(EP+Px)/2 and Φ is the physical TMDPDF and CH is the coefficient function. The coefficient functions CH are known at next-to-leading order (NLO) in the QCD coupling constant [21–23]. The variable x is the momentum fraction, Fourier-conjugate to ℓPx, and M is the mass of the hadron. O(λ2) contains various power-suppressed terms, O(λ2)=O(M2(xP+)2,1(bP+)2,bL,1ML).(4)

The left-hand side of Eq. (3) is independent of ζ. Thus, the ratio of quasi-TMDPDFs which differ only in their momenta is simply R[Γ](x,b,μ;P1,P2)=Wf/h[Γ](x,b;P1,S;μ)Wf/h[Γ](x,b;P2,S;μ)=(P1+P2+)K(b,μ)CH(xP1+,μ)CH(xP2+,μ)+O(λ2).(5)Inverting this relation one determines the CS-kernel.

One should note that for this procedure, a Fourier transformation of the quasi-TMDPDF from coordinate space (ℓPx) to momentum-fraction space (x) is required. Such a transformation requires model assumptions concerning the tail of the quasi-TMDPDF [27], which introduces additional systematic uncertainty. Furthermore, in [31] it is shown that the CS kernel extracted in this way can be sensitive to x-dependent higher-twist effects, which are much stronger than those in the TMD wave function case. This complication is avoided when the ratio of the first Mellin moments (ℓ=0, accordingly x is suppressed in the following) of the two quasi-TMDPDFs is considered [23]. It reads R[Γ](P1,P2;b)=(P1+P2+)K(b,μ)r[Γ](b,μ;P1,P2),(6)where r is [23] r[Γ](b,μ;P1,P2)=1+4CFαs(μ)4πln(P1+P2+)×[1-ln(2P1+P2+μ2)-2M[Γ](b,μ)].(7)The function M contains the residual terms of the perturbative expansion, and depends on the quantum numbers of the quasi-TMDPDF. A key argument underlying this method is that the function M[Γ](b,μ) is almost independent of b. This assumption is based on the weak correlation of b and x dependencies of TMDPDFs, which has been verified by fitting experimental data with the unpolarized TMDPDFs of proton and pion [6,7]. The value of M can be found by comparing Eq. (7) and its value in perturbation theory at b∼1  GeV-1 and μ0=2  GeV (where both perturbation theory and the factorization theorem should be valid). For details see Ref. [25]. This method is much simpler than evaluating Eq. (5) but cannot be improved beyond NLO.

The description of the cases Γ={1,γ5,…} requires the next-to-leading power (NLP) factorization theorem [30]. NLP factorization has a much more involved form and expresses a single quasi-TMDPDF by a sum of various physical TMDPDFs and new lattice-related nonperturbative functions Ψ21(b) and Ψ12(b) [30]. For particular combinations of Γ and polarization, the NLP factorization simplifies to the form of Eq. (3) (with a different coefficient function). In these cases, one can use Eq. (6) to determine the CS-kernel (note that Eq. (5) is not helpful due to the x-dependence of CH at NLP). These simple cases include Γ=1 for the TMDPDF e(x,b).

The matrix element Eq. (2) can be calculated as the ratio of a three-point and a two-point function on the lattice, W[Γ]=2EPlim0≪τ≪tC3ptΓ(P→,C,t,τ,Γ)C2pt(P→,t),(8)where EP is the energy of the hadron extracted from the two-point function. In the continuum limit, the lattice definition Eq. (8) reproduces the continuum definition, see, e.g., Ref. [32]. The parameters t and τ are the source-sink separation and the temporal distance between the source and the inserted nonlocal quark current. The three-point function is defined as C3ptΓ(P→,C,t,τ,Γ)≡⟨tr{ΓSO(P→,t)JΓ(C,τ)O¯(P→,0)}⟩(9)and calculated using the sequential source method [33] with hadron interpolator O(P→,t). The nonlocal quark current reads JΓ(C,τ)≡q¯(b,τ)ΓU[C(v,b,L)]q(0,τ),(10)where q can be either up or down quark, and U[C(v,b,L)] is the staple-shaped Wilson link in Fig. 1. The two-point function is C2pt(P→,t)≡⟨tr{ΓSO(P→,t)O¯(P→,0)}⟩,(11)where ΓS=(1+γ4)(1-iγ2γ1)/2 is needed for the proton to project out the desired parity and spin. For the pion ΓS is not necessary and is set to unity. We adopt HYP smearing for the gauge links [34] and use momentum smearing [35] to improve the signal. We analyse the CLS ensemble H101 generated using Nf=2+1 flavors of clover-improved Wilson fermions [36]. The lattice setup is summarized in Table I.

The analysis for the proton reuses the data generated in [25], where the source-sink separation is tsnk-tsrc=11a and the valence quark is the same as the sea quark. In the pion case, the simulation with the same setup is much noisier. To reduce the noise, we use a heavier valence quark corresponding to mπval=686  MeV. We do not expect a substantial mass dependence of the resulting CS kernel. In fact, in the physical limit, at large boost factors, the CS kernel depends only weakly on the quark mass, see, e.g., Ref. [26]. Besides, we use a smaller source-sink separation (9a) to further increase the signal for the three- and two-point functions.

For the pion, we fit the ratio to a constant in the interval τ∈[4a,6a], where the excited states are suppressed. The simulation has been done for six momenta P1∈{0,1,2,3,4,5}2πaNσ, but only the first three nonzero momenta have good enough signal/noise ratio to be processed further. We have confirmed that the extracted energies respect the dispersion relation EP=M2+P12 within statistical errors. We have also confirmed that our results respect the charge conservation condition C3pt[γ0](t,τ,γ0)/C2pt(t)=1/ZV, where ZV is the renormalization constant for the quark current in the vector channel [37] and C3pt[γ0](t,τ,γ0) is the local three-point function. For nonlocal correlations, we consider transverse separations in the y- or z-direction (or a combination of both), with lengths {1,2,2,…,8,62,9}a. The size of the staple-link L is taken as large as possible under the conditions that Eq. (4) is small and the signal for the three-point function is acceptable.

As detailed in Refs. [32,38–40], the hadron matrix elements Eq. (8) in different channels (for different Γ) can be parameterized using invariant amplitudes A˜i and B˜i. For the pion this parametrization is [39] W[γ0]=EPA˜2,W[γ1]=PxA˜2+M2v[1](v·P)B˜1,W[1]=MA˜1.(12)Thus the relevant amplitudes can be obtained from a combination of matrix elements in the three channels γ0, γ1 and 1. Combining the available amplitudes which satisfy the LP (f1) and NLP (e) factorization theorem [30], we obtain f1(b2,P+)=P+(A˜2(b2)+M2v+(v·P)P+B˜1(b2)),e(b2,P+)=P+A˜1(b2).(13)

The momentum pair Px={2,3}×2πaNσ, used for both pion and proton, corresponds to P-/P+={0.13,0.07} for the pion and P-/P+={0.25,0.14} for the proton, such that M2/(xP+)2 is not large in Eq. (4). We also analyzed the momentum pair Px={1,2}×2πaNσ for the pion. This case has larger systematic uncertainties due to the power corrections in Eq. (4) which we cannot reliably quantify. On the other hand, it has much smaller statistical uncertainties and we do not know for which combination the total uncertainty is larger. We present, therefore, our final results in Fig. 3 for the momenta P1/P2=3/2, i.e., with the larger statistical errors, and give the analog for P1/P2=2/1 in Appendix A. 1/(bP+)2≪1 for Eq. (4) is fulfilled if b≫{1.8a,1.1a}, implying that our extracted CS kernel is valid at b>0.15  fm. Finally, to make b/L and 1/(ML) small, L is chosen as large as possible. We observed plateaus in the interval [4a,7a] for both f1 and e at all values of b for the pion, see Fig. 2. The analogous figures for the proton are presented in Appendix B. Still larger values of L can also be included into the fit, but have negligible impact on the fit’s quality. To increase the statistics and reduce systematic uncertainties, we have combined the data with L pointing into the positive and negative v directions.

The value of the constant M is determined following the procedure described in Ref. [25]. We use the reference transverse separation b0=3a=0.26  fm=1.3  GeV-1, for which the value of the CS kernel is safely known from perturbative computations [41,42,45,46] and from phenomenological extractions [7,9] (all agree with each other up to small corrections). At the same time, the terms in Eq. (4) are small. We normalize the value of the CS kernel at this point (explicitly, we use the values of the phenomenological extraction SV19 [7] at N3LO). Our estimate for M is -0.83(0.73) for f1(pion), -4.98(0.61) for e(pion), -0.57(0.34) for f1(proton), and -1.04(1.32) for e(proton). A cross check on the determination of M using other lattice CS kernel data can be found in Appendix C. The uncertainty in the estimation of M results in a fully correlated uncertainty for the CS kernel δK. The values of δK are {0.15,0.10,0.08,0.27} for f1(pion), e(pion), f1(proton), and e(proton), correspondingly. Note that δK is not shown in Fig. 3 and the figures in the appendices as it is dominated by the statistical uncertainty at the normalization point and, thus, cannot be simply added as an independent uncertainty.

The resulting values of the CS kernel are plotted in Fig. 3. In the left plot, we compare our high momentum results for the CS kernel with two phenomenological extractions, SV19 [7] and MAP22 [9], one 3-loop perturbative calculation [41,42] and a “Literature combined” result shown as a yellow band that summarizes previous lattice calculations [24,26,27,43,44] in a way described in Appendix D. In the right plot we show the difference of the other three extractions performed in this study to the most accurate one, obtained at twist-3 in pion states.

All lattice results display qualitatively similar behavior. The differences between them are probably mainly due to systematic effects, since most calculations differ in important aspects. For example, the computation [27] is based on 1-loop matching while the computation in [24] is based on tree-level matching. This observation underlines the relevance of our results: Various systematic effects should differ markedly between pion and proton as well as between twist-2 and twist-3. Therefore, the close agreement of our four sets of data for the CS kernel does not only confirm its universality and the results of [12–14], but it suggests also that the uncertainties are still dominated by statistics.

Note also that, since pion states possess higher symmetry, fewer amplitudes are involved in the parametrization for the pion than for the proton, leading to reduced uncertainties when solving for the amplitudes. Therefore, calculating the CS-kernel for a pion as we pioneered with this paper should be especially reliable.

In Fig. 3, the data points with b>7a are plotted with light colors and gray error bars to indicate that the extracted values suffer from uncontrolled systematic uncertainties. In this region, plateaus of R are not reached even for the largest values of L, as can be seen in Fig. 2. Additionally, the points at small b<0.8  GeV-1 are contaminated by power corrections ∼b-2. Therefore, our main results are the points in the intermediate region.

We have extracted the CS kernel from the first Mellin moment of twist-2 and twist-3 pion and proton quasi-TMDPDFs on the CLS ensemble H101. At present the CS kernel for nonperturbative transverse distances can only be obtained from lattice simulations. Therefore, this is a prime example for combined analyses of experimental and lattice data being needed to obtain relatively complex observables such as TMDPDFs. The fact that we compare for the first time four qualitatively different cases, namely the TMDPDFs f1 and e of proton and pion, is the primary merit of our investigation. The fact that all four sets of results agree confirms the universality of the CS kernel and suggests that for intermediate transverse distances 0.8  GeV-1≲b≲2.6  GeV-1 systematic errors are not important. Our results are consistent with previous work within uncertainties. The main challenge for future work is to quantify all sources of systematic uncertainties. We have demonstrated that combining twist-2 and twist-3 TMDPDFs for different hadrons is conducive to this end.