Transverse momentum structure of proton within the basis light-front quantization framework

We obtain the leading-twist valence quark transverse-momentum-dependent parton distribution functions (TMD PDFs) for the proton within the basis light-front quantization (BLFQ) framework. Our results are consistent with lattice QCD calculations and our previous results for the collinear limit. We also obtain consistency with the Soffer-type bounds. Within our approach, we find that six T-even TMDs in the leading twist are all independent of each other, and previously found model-dependent relations do not hold. This is a promising sign that our results are representative of future, more extensive treatments of QCD. Furthermore, we obtain a non-trivial $ x $-dependence of the $ \left<(p^\perp)^2\right>$ and some consistency with the Gaussian ansatz but only in the small $ (p^\perp)^2 $ region. Those features suggest our results may be a useful alternative in future experimental extractions.

In this work, we investigate the quark TMDs of the proton within basis light-front quantization (BLFQ), which provides an alternative non-perturbative framework for solving relativistic many-body bound state problems in quantum field theories [44][45][46]. Previously, this approach has been successfully applied to explore the TMDs of the electron in QED [47]. Here we consider the light-front effective Hamiltonian for the nucleon in the constituent valence quark Fock space and solve for its mass eigenstates and light-front wavefunctions (LFWFs). Parameters in our Hamiltonian have been fixed to reproduce the nucleon mass and the flavor Dirac form factors [48,49]. The LFWFs in this calculation have been successfully applied to compute nucleon properties such as the electromagnetic and axial form factors, radii, PDFs, GPDs, angular momentum distributions etc. [48][49][50] Here, we extend those investigations to study the proton TMDs at the leading twist.

BLFQ framework
Basis light-front quantization (BLFQ) [44] is a nonperturbative framework for calculating the internal structures of a hadron's bound state. BLFQ starts with the light-front eigenvalue equation [51,52] and adopts basis states to express it as a hermitian matrix eigenvalue problem.
The current work truncates the Fock sector expansion [51,53] of the proton system to the leading threequark sector [48,49] |P, Λ = λ1,λ2,λ3 Here, P = (P + , M 2 P + , 0 ⊥ ), M and Λ are the momentum, mass and light-front helicity [54] of the proton, respectively. p ⊥ i is the transverse momentum of the ith quark, x i = p + i P + is its longitudinal momentum fraction, λ i is its light-front helicity, and roman alphabet subscripts run through the three quarks. ψ Λ λ1,λ2,λ3 is the light-front threequark helicity amplitude.
With quarks being the only explicit degrees of freedom, the following effective Hamiltonian is diagonalized to obtain the light-front wavefunction (LFWF) of the proton state [48] The confinement potential 1 2 i,j V conf.
i,j includes both the transverse and the longitudinal confinements. The trans-verse confining potential is adopted from light-front holographic QCD [55]. We also employ a complementary longitudinal confining potential [56]. The total confinement potential reduces to the 3-dimensional harmonic oscillator potential in the nonrelativistic limit [48,49,57]. The one-gluon exchange (OGE) term, a QCD version of the corresponding term in QED [58], encodes the interactions among the three active quarks arising from the exchange of a gluon.
With the help of 2-dimensional harmonic oscillator (2D HO) basis states in the transverse direction 1 plane-wave state in the longitudinal direction confined in a box with length L with an anti-periodic boundary condition, and also light-cone helicity state [54] in the spin space, Eq. (1) is transformed to a hermitian matrix eigenvalue problem. The above basis choice introduces four quantum numbers for every quark single-particle state: n, m for the transverse degree of freedom (d.o.f.), k for the longitudinal d.o.f. (longitudinal momentum is 2πk L with k taking half odd-integer values) and λ for the spin d.o.f. Two basis space truncations, N max and K, are added to render the resulting matrix finite [48]. N max introduces truncation in the transverse direction for the total energy of the 2D HO basis states i (2n i + |m i | + 1) ≤ N max , and K represents the resolution in the longitudinal direction In this paper, all the calculations are performed with N max = 10, K = 16.5. The physical parameters in the effective Hamiltonian (Eq. (3)), which include the quark mass (m q/k ) in the kinetic energy ( ), the quark mass (m q/g ) and coupling constant (α s ) entering the OGE term (V OGE ), and the strength (κ) of the confinement potential (V conf. i,j ), are listed in Table 1. Along with the value of 2D HO basis scale b = 0.6 GeV, those parameters are determined by fitting the nucleon mass and the flavor form factors as in Refs. [48,49]. By fitting the same observables, the values of those parameters display decreasing changes with increasing basis truncations, N max and K [48]. We surmise that our TMD distribution would change slightly when varying N max and K.
After diagonalizing the Hamiltonian matrix, we obtain the proton mass M = 1.018 GeV, and the corresponding LFWF in momentum space expressed via Eq. (2) in terms of three-quark helicity amplitudes Table 1: Model parameters for the basis truncations Nmax = 10 and K = 16.5 [48,49].
Here α is the set of all four quantum numbers k, n, m, λ and ψ(α 1 , α 2 , α 3 ) are the amplitudes of the LFWF expressed in the BLFQ basis.
In the current study, we only retain the zeroth-order expansion of the gauge link This choice is very common in practice [28,30,35,36], under which all T-odd TMDs reduce to zero.
In the leading twist, in general one would find eight TMDs, which are parameterized as follows [59, 60] where the convention 12 ⊥ = 1 is used. However, two of them, f ⊥ 1T and h ⊥ 1 , are T-odd and thus vanish under the current gauge link approximation, Eq. (11).
With the help of the rotation matrix in the S = 1 2 representation of the SU (2) group [61,62], all six T-even leading-twist TMDs are expressed in terms of the lightfront helicity amplitudes ψ Λ λ1,λ2,λ3 (p 1 , p 2 , p 3 ) as Here p R = p 1 + ip 2 , and we omit the arguments (x, (p ⊥ ) 2 ) for quantities on the left and (p 1 , p 2 , p 3 ) for quantities on the right.

Soffer-type bounds
Since the current calculations subsume the gluon dynamics into effective interactions among the 3 valence quarks and we ignore the gluon contributions from the gauge link, we cannot directly access the dynamical role of the gluons. These choices result in many interesting relations connecting twist-2 and twist-3 TMDs [6,63,64]. But, due to the focus of this paper on twist-2 TMDs, we will defer the study of the validity of those relations to a future work.
Still, in Ref. [65], the authors investigated the bounds of the leading-twist TMDs from the point of view of the positivity of the matrix representing the quark helicity structure. In our current calculations, we obtain zero f ⊥ 1T and 3 h ⊥ 1 under the approximation Eq. (11). Thus, the bounds of Ref. [65] reduce to We identify the points where the left-hand side (LHS) quantities and the right-right side (RHS) quantities of those four bounds above are nearest to each other and list them in Table 2. We find that the BLFQ results fulfill all the above bounds, which serves as an important consistency check of the current BLFQ calculations of T-even TMDs.

Reduction to the collinear distributions
TMDs are the extension of collinear parton distributions (PDFs) that incorporate information in the transverse momentum direction. After integrating over the transverse momenta one should regain PDFs from TMDs. Out of the 8 leading-twist TMDs, only three of them survive after this integration. In Fig. (2), we plot the integration of TMDs and the corresponding PDFs calculated directly within the BLFQ framework. One observes that they compare well with each other to within small residual differences that provide metrics for our numerical uncertainties.
Further, Refs. [48,49] evolve the same leading-twist PDFs calculated within the BLFQ framework via DGLAP equations, and find good consistency between the evolved BLFQ results and experimental results. This suggests that our current TMD calculations may have the potential to explain experimental data, a research area for a future investigation.

Flavor-ratio results compared with the lattice QCD calculations
Reference [37] calculated TMDs using lattice QCD with the assumption of a straight-line gauge link. This nontrivial gauge link also leads to vanishing T-odd TMDs like our approximation Eq. (11). In addition to the bare results, Ref.
[37] also shows their results in the form of flavor-ratios dxf u dxf d . We adopt this quantity for crosscomparison, since it cancels, at least some of, the possible model-dependent overall factors and even scale evolution effects 2 . We show these comparisons in Fig. (3). Surprisingly, even though those results are obtained within two totally different frameworks, we still find qualitative agreement while interesting differences are visible. For example, we generally find that the magnitudes of our flavor-ratios decrease faster in the high (p ⊥ ) 2 region. We attribute this difference to the fact that, the BLFQ results for d quark are generally wider than those of the u quark in the transverse momentum (see also Fig. (4) and the surrounding discussions), while in the lattice QCD simulations, they tend towards similar widths (see Figs. (12,13) of Ref. [37]).

x − p ⊥ factorization and Gaussian ansatz
Many preliminary extractions of TMDs from experimental data, like Ref. [67] for f 1 , Ref. [26] for g 1T , Refs. [22,24,25] for h 1 and Ref. [27] for h ⊥ 1T , follow a simple functional form (the so-called Gaussian ansatz): Here, f q (x, (p ⊥ ) 2 ) is a generic notation of all TMDs for flavor q and f q (x) is its collinear part, i.e., f q (x) = d 2 p ⊥ f q (x, (p ⊥ ) 2 ). Apart from the specific Gaussian distribution, the most important implication of the above ansatz is that the averaged transverse momentum squared (p ⊥ ) 2 is distribution-dependent, but flavor and x independent.
But, more realistic extractions, like Refs. [66,[68][69][70] for f 1 and Ref. [23] for h 1 , do not support those simplifications. Thus, it would be very interesting to investigate   whether those assumptions hold for the BLFQ results, which may facilitate future comparisons with the experimental extractions and may even guide future extractions.

4.1.1.
Flavor and x dependence of (p ⊥ ) 2 within the BLFQ framework We compute the averaged transverse momentum squared (p ⊥ ) 2 for the BLFQ results as where f q BLFQ are TMDs obtained within the BLFQ framework. The averaged transverse momentum squared for f 1 , g 1T , h 1 and h ⊥ 1T are shown in Fig. (4). One can see that within the BLFQ framework, (p ⊥ ) 2 do exhibit a strong flavor and x dependence. It is also observed that (p ⊥ ) 2 for d quarks is generally larger than that of u quark.
We further fit the x dependence of (p ⊥ ) 2 q f for different TMDs and flavors using the following function We find that, excluding h ⊥d 1T , the x dependences of (p ⊥ ) 2 q f from all other TMDs are generally very uniform, with parameters very close to the following average values (unit of all the dimension-2 values are GeV 2 ): Qualitatively, the above results are consistent with those experimental extractions which do take into account the x-dependence of (p ⊥ ) 2 , like Refs. [68,70]. The preliminary results, Eqs. (28,29), are useful as an alternative for the functional form of the averaged transverse momentum squared for future experimental extractions.

Gaussian ansatz and the BLFQ results
We then investigate the compatibility of the BLFQ results with the Gaussian ansatz. For this purpose, we use two methods to 'fit' the Gaussian width. As the first method, we calculate the x-independent (p ⊥ ) 2 commonly used in the literature as: As the second method, we determine the x-dependent (p ⊥ ) 2 by demanding that the Gaussian distributions coincide with the BLFQ results at (p ⊥ ) 2 = 0 3 . This is a commonly used strategy, like in Ref. [28], to investigate the compatibility between Gaussian ansatz and TMD calculations. We have: Using those two Gaussian widths we construct two different Gaussian-type distributions as and compare them with the BLFQ results in the small ( 0.4 GeV 2 ) (p ⊥ ) 2 region in Fig. (5). From these comparisons, it is evident that in the small (p ⊥ ) 2 region, if, and only if we include proper x-dependence of the Gaussian width, then the Gaussian distribution would be a good approximation for the BLFQ results.
In Fig. (6), we show the comparisons between the BLFQ results and the Gaussian-type distributions in the linearlog plot to investigate their large momentum behaviors.  Since the large momentum behaviors of the Gaussian-type distributions are very similar to each other, here we only compare with the Gaussian type results obtained from Eqs. (31,32). From these comparisons, it is evident that the BLFQ results decrease more slowly than the Gaussiantype distributions. This is expected, since in the large (p ⊥ ) 2 region, TMDs would decrease as an inverse power of p ⊥ [71], which we believe is reasonably approximated within BLFQ's dynamics.
Due to the space limits, we only show the plots of f 1 for the above two comparisons, but the observations are similar for other five leading-twist T-even TMDs.

Model-dependent relations
In full QCD, all TMDs should be independent of each other. But, some non-trivial relations between TMDs are also observed in many quark models. 4 We find that none of those previously found relations are satisfied by the BLFQ results, suggesting that the current BLFQ results of the leading-twist T-even TMDs may indeed be independent of each other.
We surmise that having independent T-even TMDs provides support for our underlying non-perturbative framework. The absence of model-dependent relations, along with the fact that our results follow the universal Soffertype bounds, implies that the BLFQ framework are heading in a valuable direction for simulating full QCD.

Conclusions
Basis Light-front Quantization (BLFQ) has been proposed as a non-perturbative framework for solving quantum field theory. In this work, we have calculated the quark TMD PDFs for the proton from its light-front wave functions within the framework of BLFQ. These wave functions have been obtained from the eigenvectors of an effective light-front Hamiltonian in the leading Fock sector incorporating a three-dimensional confining potential and a one-gluon exchange interaction with fixed coupling.
In this study, the gauge link has been set to unity, which leaves us six nonzero TMDs (T-even) out of the eight leading-twist TMDs. We compare our results with the previous PDF calculations within the same framework and with the lattice QCD simulation, and find good consistency in both cases. The validity of the universal Soffer-type inequalities and the absence of all the previously found model-dependent relations together imply that the BLFQ framework captures key elements of the non-perturbative physics from QCD. Increasing the number of Fock sectors would generate more independent helicity amplitudes, and thus more independent TMDs from higher twist or from the T-odd domain. One would then expect that extensions to higher Fock sectors would bring us closer to our ultimate goal, the description of full QCD.
Our calculations do not support the x − p ⊥ factorization commonly used in the preliminary phenomenological studies [22,[24][25][26][27]67]. More specifically, the non-trivial x dependence of (p ⊥ ) 2 q f precludes the x − p ⊥ factorization of the form where N = d 2 p ⊥f q ((p ⊥ ) 2 ). We also compare the BLFQ results with Gaussian-type distributions and find that Gaussian distribution is only useful for describing the BLFQ results in the small (p ⊥ ) 2 region. Future developments will focus on the inclusion of a non-trivial gauge link that will provide a prediction of the Boer-Mulders and the Sivers functions and their application to spin-asymmetries. Another major development will focus on the extension to higher Fock sectors, especially the |qqqqq and |qqqg Fock sectors, to evaluate gluon and sea-quark TMDs. Our approach can also be utilized to calculate higher-twist TMDs.