Parton Distribution Functions from Ioffe time pseudo-distributions from lattice caclulations; approaching the physical point

We present results for the unpolarized parton distribution function of the nucleon computed in lattice QCD at the physical pion mass. This is the first study of its kind employing the method of Ioffe time pseudo-distributions. Beyond the reconstruction of the Bjorken-$x$ dependence we also extract the lowest moments of the distribution function using the small Ioffe time expansion of the Ioffe time pseudo-distribution. We compare our findings with the pertinent phenomenological determinations.

Introduction.-The determination and understanding of the internal quark and gluon structure of the proton is a crucial aspect of the precision phenomenology program of the current and future hadron collider experiments, especially the Large Hadron Collider (LHC) and the upcoming Electron-Ion Collider (EIC).The framework of collinear factorization quantifies the hadronic structure in terms of Parton Distribution Functions (PDFs) which encapsulate the pertinent information regarding the momentum distributions of quarks and gluons within the nucleon.Till very recently, the intrinsic non-perturbative nature of the PDFs was prohibiting an ab-initio computation and the conventional approach is to employ a variety of experimental data together with advanced fitting methodologies in order to extract the PDFs via global fits.The studies of PDFs are of paramount importance precisely due to the fact that their uncertainties play a crucial role in many LHC applications.They affect the measurement of precision SM parameters, such as the W mass, the strong coupling constant and the determination of the couplings of the Higgs boson where discrepancies from the stringently fixed SM predictions would serve as indisputable evidence of BSM physics [1].
The possibility to determine the PDFs with first principle lattice calculations is the object of a long endeavor which recently lead to a culmination of results.The primary difficulty impeding a first principle implementation is associated with the fact that the matrix elements defining the PDFs involve light-cone separated fields.In his seminal article that stimulated the recent efforts, X. Ji [2] proposed to compute matrix elements of fields separated by a purely space-like distance z = z 3 that define the so-called quasi-PDF, the distribution in the longitudinal momentum p 3 .In the large p 3 limit, they can be factorized into the light-cone PDF, f (x, µ 2 ).Subsequently, many articles studying quasi-PDFs, as well as the pion quasi-distribution amplitude (DA) appeared in the literature .
Ioffe time pseudo-distributions. -Another positionspace formulation was proposed in [39].In this approach, the basic object is the Ioffe time pseudo-distribution function (pseudo-ITD) M(ν, z 2 ).The Lorentz invariant ν = p • z is known as the Ioffe time [40,41].The pseudo-ITD is the invariant amplitude for a matrix element with space-like separated quark fields.
In renormalizable theories, the pseudo-ITD exhibits a logarithmic singularity at small values of z 2 .These shortdistance singularities can be factorized into the PDF and a perturbatively calculable coefficient function.The pseudo-ITD can also be considered as a LCS.A series of works implemented this formalism and studied its efficiency [42][43][44][45][46][47].For the sake of completeness, the main points of our formalism are summarized below, but we refer the reader to [46,48] for a detailed discussion.
The non-local matrix element, with U being a straight Wilson line, p = (p + , m 2 2p + , 0 T ), z = (0, z − , 0 T ) and γ a = γ + in light-cone coordinates, defines the MS ITD (introduced in [41]), given a regularization is made for the z 2 = 0 singularity.For z 2 = 0, this arXiv:2004.01687v1[hep-lat] 3 Apr 2020 matrix element has the following Lorentz decomposition The pseudo-ITD M(ν, z 2 ) contains the leading twist contribution, while N is a higher-twist term.In the kinematics p = (E, 0, 0, p 3 ), z = (0, 0, 0, z 3 ), the choice α = 0 isolates M. Nonetheless, it still contains higher twist contaminations O(z 2 Λ 2 QCD ).In the limit of small z 2 , where higher twist terms are suppressed, M is factorizable into the ITD (or equivalently, the PDF) and a perturbative coefficient function, provided that one removes Wilson line-related UV divergences that appear at finite z 2 .These UV divergences are eliminated if one considers the reduced pseudo-ITD [39,42] given by the ratio It contains the same singularities in the z 2 = 0 limit as M, and can be related to the MS light-cone ITD, Q(ν, µ 2 ), by the NLO matching relation [49][50][51] where is the Altarelli-Parisi kernel [52], and Extracting the matrix element.-The numerical computation of our matrix elements relies on Gaussian smearing [55] and momentum-smearing [56] for constructing the nucleon interpolating field, as well as the summation method for better control of the excited state contamination.The latter is intimately related to the Feynman-Hellmann (FH) theorem [57] and has been widely used in Lattice calculations of PDFs [18,20,21,42,43,46,47].The matrix element is determined from a ratio of correlation functions where C 2,3 are standard two and three point correlation functions, t is the Euclidean separation between the source and sink interpolating fields, and the operator insertion time τ is summed over the entire temporal range.The effective matrix element M eff is then constructed as The leading excited-state effects can be parameterized by with ∆ being the energy gap between the ground state and the lowest excited state.
The summation method has a clear advantage over the typical ratio method.The excited state contamination scales as exp(−∆t) instead of exp(−∆t/2), which allows for smaller t to be used to control excited state effects.Since correlation functions' errors grow exponentially, the summation method requires significantly fewer measurements to obtain a desirable statistical precision for data with controlled excited states.This feature is important for calculations at large momenta, where energy gaps can be small and the error decays much faster than for low momenta.Lattice QCD calculation.-In this study, three ensembles of configurations with decreasing value of the pion mass have been employed.In Tab.I, we list all the parameters of our analysis.The pion masses of this study are 172 MeV, 278 MeV, and 358 MeV.These ensembles allow for a controlled extrapolation to the precise physical pion mass which constitutes an important limit to be taken in order to safely compare with the PDF determinations of global fits but also for the first time we can study the pion mass effects on the ITD.As was done in [46], correlation functions with several different smearings were simultaneously fit to determine the matrix element from Eq. ( 8).The matrix elements extracted from fitting correlation functions to Eq. ( 8) are shown in Fig. 1.Moments of the PDF.-Following our suggestion in [44], we can use the reduced pseudo-ITD to compute the moments of the PDF.Valuable information for the PDF can be extracted from the data without dealing with the pitfalls of the inverse problem.The moments of the MS PDF, a n (µ 2 ), are related multiplicatively to those of the Fourier transform of the reduced pseudo-ITD, where C n are the Mellin moments of the matching kernel C(u, µ 2 z 2 ) with respect to u.To NLO accuracy, where are the moments of the Altarelli-Parisi kernel, and ) where is the Altarelli-Parisi ker and is the non-logarithmic part.
Extracting the matrix element -The numerical tation of our matrix elements relies on Gaussian ing [51] and momentum-smearing [52] for cons the nucleon interpolating field, as well as the sum method for better control of the excited state c nation.The latter is intimately related to the Fe Hellmann (FH) theorem [53] and has been widely Lattice calculations of PDFs [17,18,38,39,42, The matrix element is determined from a rati relation functions .
where C 2,3 are standard two and three point tion functions, t is the Euclidean separation betw source and sink interpolating fields and the ope sertion time ⌧ is summed over the entire tempora The e↵ective matrix element M e↵ is then constr The leading excited-state e↵ects can be paramete removes Wilson line-related UV divergences that appear at finite z 2 .These UV divergences are eliminated if one considers the reduced pseudo-ITD [35,38] given by the ratio It contains the same singularities in the z 2 = 0 limit as M, and can be related to the MS light-cone ITD by the NLO matching relation [45][46][47] M(⌫, z 2 ) =Q(⌫, µ 2 ) where and is the non-logarithmic part.
Extracting the matrix element -The numerical computation of our matrix elements relies on Gaussian smearing [51] and momentum-smearing [52] for constructing the nucleon interpolating field, as well as the summation method for better control of the excited state contamination.The latter is intimately related to the Feynman-Hellmann (FH) theorem [53] and has been widely used in Lattice calculations of PDFs [17,18,38,39,42,43,54].The matrix element is determined from a ratio of correlation functions where C 2,3 are standard two and three point correlation functions, t is the Euclidean separation between the source and sink interpolating fields and the operator insertion time ⌧ is summed over the entire temporal range.The e↵ective matrix element M e↵ is then constructed as in t.This means that if N measureme reach a specific statistical precision fo t rat with the ratio method, only p N required to reach the same precision a for the summation method.In both ca tion from excited states is the same.portant for calculations at high hadron excited state energy gaps can be smal noise ratio decays much faster than states.
Lattice QCD calculation.-In this stu of configurations with lattice spacing decreasing value of the pion mass ha In Tab.I, we list all the parameters o pion masses of this study are 172 M 390 MeV.These ensembles allow for a olation to the precise physical pion tutes an important limit to be taken compare with the PDF determination also for the first time we can study th on the ITD.As was done in [42], co with several di↵erent smearings were to determine the matrix element from trix elements extracted from fitting co to Eq. ( 8) are shown in Fig. 1. ) with respect to u.To NLO FIG. 1.The reduced pseudo-ITD calculated on ensembles with 358 MeV, 278 MeV, and 172 MeV pion masses.The upper and lower plots are the real and imaginary component respectively.There appears to be very small mass effects within this range of ν and z 2 .
The even and odd moments can be determined from the coefficients of polynomials which are fit to the real and imaginary components respectively.The order of the polynomial is chosen to minimize the χ 2 /d.o.f. for each z 2 separately.As an example, the first and second moments calculated on the ensemble a091m170 are shown in Fig. 2. The z 2 dependence of the resulting PDF moments can be used to check for the size of higher twist effects, which do not seem significant.
Matching to MS. -Similarly to Ref. [46], the reduced The first two moments of the pseudo and the MS lightcone PDF computed from the ensemble a091m170, compared to phenomenologically determined PDF moments from the NLO global fit CJ15nlo [58], the NNLO global fits MSTW2008nnlo68cl nf4 [59] and NNPDF31 nnlo pch as 0118 mc 164 [60] all evolved to 2 GeV.
pseudo-ITD from each ensemble is matched to the lightcone MS ITD at a given scale µ by inverting Eq. ( 4).As a result, we obtain a set of z 2 -independent curves for Q(ν, µ 2 ) at µ = 2 GeV, shown in Fig. 3a.
As seen in the moments, the matching procedure has a small O(α s /π) ∼ 0.1 effect on the distribution.The contributions from the convolution of B and L with the reduced pseudo-ITD appear with opposite signs.The convolution with L is slightly larger in magnitude, but by a factor which is approximately the same as the logarithmic coefficient of B. This feature may just be a coincidence at NLO, but it hints that higher order corrections may also be small.An NNLO or non-perturbative matching is required to check the effects of the perturbative truncation on the matching.(Upper) The MS ITD matched to 2 GeV from the reduced pseudo-ITD results calculated at 358 MeV, 278 MeV, and 172 MeV.(Lower) The nucleon valence distribution obtained from fitting the ITD to the form in Eq. ( 13) from each of those ensembles.
Determination of the PDF.-The inversion of the Fourier transform defining the ITD, given a finite amount of data, constitutes an ill-posed problem which can only be resolved by including additional information.As was shown in [45], the direct inverse Fourier transform can lead to numerical artifacts, such as artificial oscillations in the resulting PDF.Many techniques have been proposed to accurately calculate PDFs from lattice data [21,28,45,61].This issue also occurs in the determination of the PDF from experimental data.
As was done in Ref. [46], the approach which is used here (and is common amongst phenomenological determi-nations) is to include information in the form of a modeldependent PDF parametrization.The parameterization used here is where N normalizes the PDF.The fits to that form are shown in Fig. 3b.The results of these fits are largely consistent with each other.The heaviest pion mass PDF has notably larger statistical error than the others.This effect is due to a larger variance in the highly correlated c and d parameters.In the lighter two pion masses, the correlation between these parameters appears stronger, leading to a smaller statistical error in the resulting PDFs.Extrapolation to the physical pion mass.-In order to determine the valence PDF for physical pion mass, our results must be extrapolated to 135 MeV.To do this, the central values of these curves are extrapolated and the errors are propagated.We have performed the extrapolation including and excluding the statistically noisy result from the heaviest pion ensemble.When using all three ensembles, we extrapolate the results using the form where ∆m π = m π −m 0 and m 0 is the physical pion mass.When using only the two lighter pion mass ensembles, we fix either a or b to be zero.Though, these extrapolations are not guaranteed to satisfy the normalization of the PDF, we have found them to be close within statistical precision.The extrapolated PDFs are shown in Fig 4a .The linear extrapolation with the lightest two ensembles is compared to phenomenological determinations in Fig 4b.
The PDF obtained from this fit, for x > ∼ 0.2 is larger than the phenomenological fits.This feature is consistent with the larger value of the second moment compared to the global fits in Fig. 2. Other remaining systematic errors could explain this discrepancy.This calculation was performed on ensembles with a fairly coarse lattice spacing and uses data with ap ∼ O(1).Discretization errors have been shown [46] to be potentially significant.Future calculations at smaller lattice spacings are required to control these effects.There also exist potentially notable finite volume corrections.Conclusions.-We presented the first calculation of the nucleon PDF based on the method of Ioffe time pseudodistributions performed at the physical pion mass.This was an important step that had to be taken in order to have a more meaningful comparison with the pertinent phenomenological results.Also, by studying three ensembles with different pion masses, we were able to investigate the dependence of the ITD on the pion mass.We saw that it is relatively mild compared to expectations stemming from the studies of x [62] and calculations of quasi-PDFs [14].(Upper) The extrapolations of the nucleon valence PDF to physical pion mass.(Lower) The nucleon valence distribution compared to phenomenological determinations from the NLO global fit CJ15nlo [58] (green), and the NNLO global fits MSTW2008nnlo68cl nf4 [59] (red) and NNPDF31 nnlo pch as 0118 mc 164 [60] (blue) at a reference scale of 2 GeV.
Compared to similar studies, our analysis capitalizes on three key factors.First, the ratio of matrix elements that yields a clean way to avoid all pitfalls and systematics of fixed gauge non-perturbative renormalization.Second, the short distance factorization, that allows for matching to MS without relying on large momentum data with their large statistical noise and potential discretization errors.Third, the summation method, that allows for a better control of the excited state contamination.Having studied finite volume effects and discretization errors in [46], in our upcoming work we plan to study in a systematic way the continuum extrapolation as well as effects stemming from excited state contamination.
FIG. 2.The first two moments of the pseudo and the MS lightcone PDF computed from the ensemble a091m170, compared to phenomenologically determined PDF moments from the NLO global fit CJ15nlo[58], the NNLO global fits MSTW2008nnlo68cl nf4[59] and NNPDF31 nnlo pch as 0118 mc 164[60] all evolved to 2 GeV.
FIG. 3.(Upper) The MS ITD matched to 2 GeV from the reduced pseudo-ITD results calculated at 358 MeV, 278 MeV, and 172 MeV.(Lower) The nucleon valence distribution obtained from fitting the ITD to the form in Eq. (13) from each of those ensembles.
FIG. 4.(Upper) The extrapolations of the nucleon valence PDF to physical pion mass.(Lower) The nucleon valence distribution compared to phenomenological determinations from the NLO global fit CJ15nlo[58] (green), and the NNLO global fits MSTW2008nnlo68cl nf4[59] (red) and NNPDF31 nnlo pch as 0118 mc 164[60] (blue) at a reference scale of 2 GeV.

TABLE I .
[54]meters for the lattices generated by the JLab/W&M collaboration[53]using 2+1 flavors of stout-smeared clover Wilson fermions and a tree-level tadpole-improved Symanzik gauge action.More details about these ensembles can be found in[54].