Lattice QCD study of the Boer-Mulders effect in a pion

Lattice QCD study of the Boer-Mulders effect in a pion

M. Engelhardt Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA    P. Hägler Institut für Theoretische Physik, Universität Regensburg, Regensburg, Germany    B. Musch Institut für Theoretische Physik, Universität Regensburg, Regensburg, Germany    J. Negele Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA    A. Schäfer Institut für Theoretische Physik, Universität Regensburg, Regensburg, Germany

The three-dimensional momenta of quarks inside a hadron are encoded in transverse momentum-dependent parton distribution functions (TMDs). This work presents an exploratory lattice QCD study of a TMD observable in the pion describing the Boer-Mulders effect, which is related to polarized quark transverse momentum in an unpolarized hadron. Particular emphasis is placed on the behavior as a function of a Collins-Soper evolution parameter quantifying the relative rapidity of the struck quark and the initial hadron, e.g., in a semi-inclusive deep inelastic scattering (SIDIS) process. The lattice calculation, performed at the pion mass , utilizes a definition of TMDs via hadronic matrix elements of a quark bilocal operator with a staple-shaped gauge connection; in this context, the evolution parameter is related to the staple direction. By parametrizing the aforementioned matrix elements in terms of invariant amplitudes, the problem can be cast in a Lorentz frame suited for the lattice calculation. In contrast to an earlier nucleon study, due to the lower mass of the pion, the calculated data enable quantitative statements about the physically interesting limit of large relative rapidity. In passing, the similarity between the Boer-Mulders effects extracted in the pion and the nucleon is noted.

I Introduction

Transverse momentum-dependent parton distributions (TMDs) Boer:2011fh () constitute one of the pillars on which the three-dimensional tomography of hadrons rests. Together with the three-dimensional spatial information derived from generalized parton distributions (GPDs), they permit a comprehensive reconstruction of hadron substructure and thus have a bearing on seminal topics in hadron physics, such as orbital angular momentum contributions to nucleon spin, or spin-orbit correlations in hadrons. Through the selection of particular parton spin and transverse momentum components, a variety of effects can be probed, including naively time-reversal odd (T-odd) quantities such as the Sivers and Boer-Mulders functions; these only exist by virtue of initial or final state interactions in corresponding physical processes, introducing a preferred chronology in the description of the process. For example, in semi-inclusive deep inelastic scattering (SIDIS), the operative element is final-state interactions between the struck quark and the hadron remnant; on the other hand, in the Drell-Yan (DY) process, initial state interactions before the lepton pair production enable T-odd effects. TMDs thus in general have to be considered in the context of specific physical processes, within a factorization framework appropriate for the process in question, separating the hard reaction from the TMD and other elements such as fragmentation functions. In the case of T-odd effects, the process-dependence manifests itself in the prediction of a sign change of the Sivers and Boer-Mulders functions between the SIDIS and DY processes signchange ().

In view of the fundamental importance of TMDs and the rich spectrum of effects that can be probed, TMDs have been, and continue to be the target of a variety of experimental efforts. Deep-inelastic scattering experiments performed by COMPASS COMPASS (), HERMES HERMES () and Jefferson Lab jlab () have yielded TMD data including evidence for the T-odd Sivers effect. Complementary Drell-Yan experiments at COMPASS compassdy () and Fermilab fnaldy () are envisaged, which could, in particular, test the aforementioned sign change between the SIDIS and DY processes. Related transverse single-spin asymmetries have been measured at RHIC in polarized proton-proton collisions BNL (). Further experimental efforts at RHIC are projected to provide insight into strong QCD evolution effects expected for the Sivers TMD rhic (). TMDs furthermore constitute a central focus of the proposed Electron-Ion Collider facility eicwhite ().

To complement these efforts, providing nonperturbative QCD input from first principles to the analysis of TMD effects, a project to calculate TMD observables within lattice QCD was initiated and developed in straightlett (); straightlinks (); tmdlat (). The present work constitutes a continuation of this project. As described in detail below, the formal definition of TMDs is based on nonlocal operators, specifically quark bilocal operators with a gauge connection that takes the shape of a staple. The path followed by the gauge connection is in principle infinite in length, and thus it cannot be straightforwardly treated in terms of an operator product expansion, as is commonly done, e.g., for ordinary parton distributions or generalized parton distributions. In view of this genuinely nonlocal character of the operators, lattice QCD explorations of corresponding hadronic matrix elements directly at the nonlocal operator level were undertaken in straightlett (); straightlinks (), concentrating initially on the simpler case of straight gauge links connecting the quark operators. The nonlocal nature of the operators in particular raises novel questions regarding regularization and renormalization, which were addressed in considerable detail in straightlinks (). Whereas these questions deserve further study, the aforementioned explorations suggest that it is a viable working assumption to treat nonlocal lattice operators in analogy to the fashion in which they are treated in continuum QCD collbook (), namely, by absorbing divergences into multiplicative soft factors. These soft factors can then be canceled in appropriate ratios; this scheme was used to construct TMD observables in the subsequent investigation tmdlat (), and will be used in the present work. Formally related studies of nonlocal lattice operators, in which a gauge link in the (Euclidean) time direction originates from the propagation of a heavy auxiliary quark, have been carried out in detmold (); also, a direct approach to light-cone distribution amplitudes based on nonlocal lattice operators was laid out in vbraun (). Moreover, the comprehensive framework for investigating parton physics within lattice QCD put forward in Ji_1 () and developed and explored in Ji_2 (); Ji_3 (); steffens () relies on a direct treatment of such nonlocal lattice operators.

The present work focuses on a TMD observable related to the Boer-Mulders effect in a pion. Lattice QCD studies of pion structure, predominantly focusing on form factors, have been previously reported in bonnet (); ff1 (); ff2 (); pion_GPD (); ff3 (); ff4 (); ff5 (); ff6 (); ff7 (). Choosing the pion as the hadron state is motivated by the principal goal of the investigation presented here, namely, understanding the behavior of TMDs as a function of an evolution parameter quantifying the rapidity difference between the hadron momentum and a vector describing the trajectory of the struck quark. Details are furnished further below. In the previous nucleon study tmdlat (), no definite conclusions regarding the limit of large rapidity difference proved possible. By virtue of its lower mass, the pion provides a larger rapidity difference at given momentum, and this choice of hadron state thus aids in approaching the limit of physical interest. In addition, the spinless nature of the pion permits additional spatial averaging to suppress statistical uncertainties. Indeed, the chief advance of the present work lies in providing quantitative insight into the limit of large evolution parameter. Preliminary accounts of this work were given in latt13 (); qcdevol14 (); lc2014 ().

Ii Definition of TMD observables

ii.1 Correlation functions

Quark transverse momentum-dependent parton distributions (TMDs) can be defined in terms of the fundamental correlator


where denotes the momentum of the hadron state; the present work focuses on pions, and thus no spin is attached to the state. represents an arbitrary Dirac matrix structure. The quark operators at positions and are connected by the gauge link , which connects the points listed in its argument by straight-line segments; thus, the gauge link has the shape of a staple, cf. Fig. 1, with the unit vector specifying the staple direction and its length. One is ultimately interested in the limit , which in a concrete lattice calculation is of course reached by extrapolation. This gauge link form incorporates final state interactions between the struck quark and the hadron remnant in semi-inclusive deep inelastic scattering (SIDIS) pijlman (), and analogously initial state interactions in the Drell-Yan process (DY).

Figure 1: Gauge link structure in the correlator (1). In a concrete lattice calculation, the limit has to be taken numerically.

The ellipsis in the argument of indicates that the correlator will depend on various further parameters, related, e.g., to regularization, specified below as needed.

Fourier transformation of (1),


leads to the momentum space correlator , which ultimately will be parametrized in terms of TMDs, cf. below. The position space correlator (1), as written, requires regularization not only of the quark operator self-energies, but also of the self-energy of the Wilson line (this is indicated by the subscript “unsubtr.”). The regularization of the latter is effected by dividing by the soft factor . The detailed structure of the soft factor depends on the concrete factorization approach employed. For example, in the scheme developed in collbook (); spl1 (), which, for reasons discussed further below, provides the phenomenological framework for the present study, the soft factor takes the form


where is a vacuum expectation value of Wilson line structures extending, initially, into space-like directions; some of them remain at finite rapidity , whereas others are taken to the light-cone limit, i.e., infinite rapidity. This particular form of the soft factor, containing more than one rapidity, cannot be cast in a Lorentz frame in which it exists at a single time, and for this reason, it is not suited for evaluation within lattice QCD. However, the observables that will be defined further below are ratios in which the soft factors cancel tmdlat (). There is, therefore, no obstacle to evaluating those observables within lattice QCD, and the detailed form of the soft factor is immaterial. An alternative construction of TMD soft factors which, in principle, is amenable to lattice QCD evaluation has been put forward in Ji_3 () within the framework laid out in Ji_1 ().

As written in (2), the transverse components of the quark separation are Fourier conjugate to the quark transverse momentum , while the longitudinal component is Fourier conjugate to the longitudinal momentum fraction . The present work will be confined to the case , corresponding to evaluating the integral with respect to over the correlator and TMDs derived from it. It should be stressed, however, that there is no obstacle to extending calculations of the type presented here to a scan of the -dependence111In a practical calculation, the range of accessible is limited by the available and , (where denotes the spatial momentum), leading to an increasing systematic uncertainty at small ., yielding upon Fourier transformation the -dependence of and the TMDs under consideration. Studies of the -dependence in the straight gauge link () case have already been carried out in straightlinks (), and further investigations in this direction are planned for future work. A related proposal to obtain the -dependence of parton distributions has been put forward and explored in Ji_1 (); Ji_2 (); Ji_3 (); steffens ().

Finally, (2) is evaluated at , in accordance with the standard phenomenological framework, which employs a Lorentz frame in which the hadron of mass propagates with a large momentum in 3-direction, ; then, the quark momentum components scale such that the correlator (2) and TMDs derived from it are principally functions of the quark longitudinal momentum fraction and the quark transverse momentum vector , whereas the dependence on the component becomes irrelevant in this limit. Correspondingly, (2) is regarded as having been integrated over , implying that the conjugate variable is to be set to zero, as written.

Before continuing with the parametrization of in terms of TMDs, it is important to note a further specification with respect to the staple direction . The Wilson lines along the legs of the staple represent an effective, resummed description of gluon exchanges between, in the case of SIDIS, the struck quark and the hadron remnant pijlman (). Accordingly, their direction should be taken to follow the path of the ejected quark, close to the light cone from the point of view of the hadron. Whereas, at tree level, there is no obstacle to the most straightforward choice, namely, a light-like , beyond tree level, this choice is associated with rapidity divergences rapidrev (). Various schemes have been advanced to treat these divergences collbook (); spl1 (); Ji:2004wu (); idilbi (), and the equivalence between some of them discussed in tmdequiv (). In particular, the scheme advanced in collbook (); spl1 () effects regularization by tilting the staple direction slightly off the light cone into the spacelike region. This feature is crucial for a concrete implementation of a lattice QCD evaluation of matrix elements of the type (1), as will be detailed further below. Thus, the phenomenological framework providing the backdrop for the present treatment is, specifically, the one advanced in collbook (); spl1 (). Choosing to be spacelike for the purposes of regularization implies a dependence of the calculation on an additional parameter, characterizing proximity of the staple to the light cone. Here, this parameter will be chosen as


where the absolute length of a four-vector is denoted by . In terms of , the light-cone limit corresponds to . The generated lattice data thus have to be extrapolated to the double limit , , i.e., the limit of infinite staple length, and staple direction converging toward the light cone.

Alternative to this purely kinematic characterization, also the parameter is frequently employed cs (), and viewed as a dynamical scale, to be compared to ; perturbative evolution equations in can then be derived at sufficiently large , cf., e.g., spl2 (). The question whether one has reached the asymptotic regime appropriate for the definition of TMDs presumably has both kinematic and dynamical aspects. The applicability of perturbation theory, e.g., for determining evolution equations is a dynamical issue most adequately characterized by considering the dimensionful parameter in relation to the characteristic QCD scale, i.e., requiring . On the other hand, this by itself does not necessarily guarantee kinematics close to the light cone, . Presumably, both conditions need to be taken into account in general. In the present work, no quantitative connection to perturbative evolution is attempted; the numerical data do not reach values of which lie clearly within the perturbative regime. The maximum spatial momentum employed is , and the maximum value of is , corresponding to . Instead, the dependence of the lattice data on the kinematical variable will be studied empirically, including exploration of ad hoc ansätze for the large- behavior, allowing for corresponding extrapolations. Characterizing the large- limit is in fact the primary goal of the present investigation. This limit was seen to present a considerable challenge in the previous study tmdlat (), which considered nucleon TMDs; no definite statements concerning the large- behavior proved possible. The present work, focusing on pions, permits accessing higher both by virtue of the lighter hadron mass (note that the hadron mass enters the denominator of (4)), and by employing additional spatial averaging facilitated by the spinless nature of the pion, enhancing statistics. As will be seen further below, the data extracted in the present investigation are of sufficient quality to yield a signal for the limit of the generalized Boer-Mulders shift defined in eq. (23) below. This constitutes the main advance of this work.

ii.2 Parametrizations

Returning to the momentum space correlator , its parametrization in terms of the relevant Lorentz structures yields, at leading twist,


For spinless particles such as the pion, there are only two leading twist TMDs, in contrast to the eight which arise for spin- particles tmd1 (); tmd2 (); tmd3 (). The TMD is simply the unpolarized quark distribution, whereas the Boer-Mulders function bm () encodes the distribution of transversely polarized quarks in the pion. The Boer-Mulders function is odd under time reversal (T-odd). Physically, it only exists by virtue of the final and initial state interactions, in the SIDIS and DY processes, respectively, which break the symmetry of the processes under time reversal. Formally, it is the introduction of the additional vector describing the staple direction in the staple-shaped gauge link which breaks the symmetry; the Boer-Mulders function vanishes for a straight gauge link, . Correspondingly, the generalized Boer-Mulders shift defined in eq. (23) below will be an odd function of .

On the other hand, one can also decompose the position space correlator into invariant amplitudes tmdlat (). The general decomposition is (the combinations corresponding specifically to the leading twist TMDs (5) and (6) will be considered further below):


The present treatment focuses on the special case , which in the context of TMDs, defined in a frame in which and , also implies tmdlat (). Under these constraints, the above relations are readily inverted. The amplitudes needed below are, explicitly,


Note that and are regular for owing to the prefactor. Of particular interest are the leading twist objects


(where denote transverse spatial indices), given in terms of the amplitude combinations


Also these combinations are regular for by virtue of the prefactors in (12),(14). Note that, in the case of vanishing spatial momentum, one cannot identify “forward” and “backward” directions for ; there is then only a single branch in , the sign of which is a matter of definition. Although only the combinations and appear in (15),(16), for the numerical analysis to follow, it will be valuable to be able to consider , , and individually, not just those combinations.

Given that and are related via a Fourier transformation, the quantities arising in the respective decompositions (5)-(6) and (15)-(16) must be similarly related, i.e., the amplitudes must be related to Fourier-transformed TMDs. Indeed, denoting -moments of generic Fourier-transformed TMDs by


where denotes the Bessel functions of the first kind, one finds tmdlat ()


Note the appearance of the soft factors on the right hand sides.

ii.3 Boer-Mulders shift

As already indicated further above, one obtains an observable in which the soft factors cancel by forming a suitable ratio, namely, the “generalized Boer-Mulders shift”


Note that ratios of this type also cancel -independent multiplicative field renormalization constants attached to the quark operators in (1) at finite physical separation . It should be emphasized that the construction thus far is a continuum perturbative QCD construction. That this construction carries across into lattice QCD, i.e., that the lattice operators are similarly regularized and renormalized by multiplicative soft factors which cancel in ratios, is a working assumption which was already explored in considerable detail in straightlinks (), and which will be investigated further in future work. Physically, this assumption appears plausible at least at separations substantially larger than the lattice spacing, where the lattice operators are expected to approximate the corresponding continuum operators.

To interpret the generalized Boer-Mulders shift, note that the limit of the quantities defined in (20) formally corresponds to -moments of TMDs,


Thus, in the formal limit, the generalized Boer-Mulders shift reduces to the “Boer-Mulders shift”


which, in view of the structure of the right-hand side, formally takes the form of the average transverse momentum in -direction of quarks polarized in the transverse (“”) -direction, in an unpolarized (“”) pion, normalized to the corresponding number of valence quarks. This is in accord with the structure of the correlator (16), where represents the quark spin. The numerator in (25) sums over the contributions from quarks and antiquarks, whereas the denominator contains the difference between quark and antiquark contributions, thus giving the number of valence quarks straightlinks (); muldtan (). It should be noted, however, that the -moments of TMDs (24) appearing in (25) are in general divergent bacchetta () at large and thus not well-defined absent further regularization. The generalized quantity (23) is a natural regularization, with finite effectively acting as a regulator through the associated Bessel weighting, cf. (20). This Bessel weighting also is advantageous in the analysis of experimental asymmetries bweight1 (); bweight2 (). In the present work, lattice QCD data for the generalized Boer-Mulders shift (23) will be obtained and presented at finite . The path by which these data can be obtained proceeds via lattice QCD evaluation of the fundamental correlator (1) for a range of Dirac and staple link structures, extraction of the relevant invariant amplitudes (11)-(14), and construction of the ratio (23). As already mentioned further above, for this lattice QCD calculational scheme to be viable, it is necessary to employ a phenomenological framework such as the one advanced in collbook (); spl1 (), in which all separations in the correlator (1) are spacelike, including the staple direction . Such a scheme implies dependence on the Collins-Soper-type evolution parameter , cf. (4), quantifying proximity of the staple to the light cone. The principal focus of the present investigation is, indeed, the dependence of the generalized Boer-Mulders shift on , including its asymptotic behavior.

Iii Lattice QCD calculations

Lattice QCD employs a Euclidean time coordinate, serving to project out hadronic ground states via the associated exponentially decaying time evolution. As a consequence, when evaluating matrix elements of operators in hadronic states, no Minkowski time separations in those operators can be accomodated; one is restricted to operators which are defined at one single time. This is the reason why it is imperative to employ a framework in which all separations in the fundamental correlator (1) are spacelike. Only in this case is there no obstacle to boosting the problem to a Lorentz frame in which the operator in (1) exists at a single time, and performing the lattice calculation in that particular frame.

The decomposition of the resulting correlators into invariant amplitudes, cf. (11)-(14), is a further crucial element of the present treatment. Expressed in this fashion, the results of the lattice calculation are immediately applicable also in the original Lorentz frame in which (1) was initially defined. Finally, as already discussed above, the construction of ratios of amplitudes in which soft factors cancel serves to connect the results to phenomenological observables such as the generalized Boer-Mulders shift (23).

The lattice QCD data for the present exploration were obtained within a mixed action scheme employing domain wall valence quarks on an dynamical asqtad quark gauge ensemble provided by the MILC collaboration milc (). Since the principal focus lies on understanding the systematics of the large limit, which proved inaccessible in previous investigations, a fairly high pion mass, , was chosen for this study to alleviate statistical fluctuations. Further details of the ensemble are given in Table 1.

(fm) (MeV) (GeV) #conf. #meas.
0.11849(14)(99) 0.02 0.05 518.4(07)(49) 1.348(09)(13) 486 3888
Table 1: Parameters of the lattice ensemble. Note that the lattice spacing was determined in a different scheme than in LHPC_1 (); LHPC_2 (), and consequently it, as well as the pion and nucleon masses quoted, differ slightly from the ones given in the aforementioned references tmdlat (). The first error quoted for the pion and nucleon masses is statistical, the second stems from the conversion to physical units using . The bare asqtad quark masses are denoted . Eight measurements were made on each gauge configuration.

This mixed action scheme, including the specific ensemble employed here, has been used extensively by the LHP Collaboration for studies of hadron structure, cf., e.g., LHPC_1 (); LHPC_2 (). It also provided the basis for the previous nucleon TMD investigation reported in tmdlat ().

To extract the correlator (1), one evaluates both three-point functions and two-point functions with pion sources and sinks of definite spatial momentum222In practice, momentum conservation eliminates the need for projection at the source, provided one projects onto zero momentum transfer at the operator insertion in instead. ,


where , and are source time, operator insertion time and sink time, respectively, and denotes an interpolating field with the quantum numbers of the pion. Wuppertal-smeared quark fields were employed to construct these pion sources and sinks, which were separated by . Only connected contractions contributing to were evaluated; disconnected contributions, which are expected to be small, are omitted in all results presented below. In the case of the meson, -quark and -quark distributions coincide; contrary to the nucleon case, there is therefore no nontrivial quark combination in which disconnected contributions exactly cancel.

The fundamental correlator (1) is then obtained from plateaus in for in the three-point to two-point function ratio bonnet (),


where is the energy of the pion state and is taken to be the operator in (1). It should be noted that, at the employed source-sink separation of , significant excited state contaminations in the plateaus extracted from (28) cannot be excluded a priori. This issue was not investigated in the present exploratory study at the fairly high pion mass . However, in future work at lower pion masses, where excited state contaminations are exacerbated, it will present an additional challenge.

The set of combinations of pion momenta and staple-shaped gauge link paths used is listed in Table 2. It should be noted that, in the case of either or extending into a direction which does not coincide with a lattice axis, there is more than one optimal approximation of the corresponding continuum path by a lattice link path; e.g., if , where denotes the lattice link vector in -direction, both the sequence of links and the sequence of links equally well approximate the continuum path. In such a case, was always averaged over all equivalent lattice link paths, for both and vectors. This symmetry improvement of the lattice operators is important to preserve the manifest time-reversal transformation properties present for the continuum staple-shaped gauge link path operators.

Note also that, in the mixed action scheme used for these calculations, before evaluating domain wall propagators for valence quarks, the asqtad gauge configurations are HYP-smeared to reduce dislocations (or rough fields) that would otherwise allow right-handed states on one domain wall to mix with left-handed states on the other domain wall. The lattice gauge link paths in (1) were constructed using those same HYP-smeared gauge configurations. This has the advantageous consequence that renormalization constants are closer to their tree-level values, while it would have no effect in the continuum limit. On the other hand, as shown in ref. steffens (), there are significant differences between 0 and 2 steps of HYP-smearing in the direct calculation of parton distributions using straight gauge link paths instead of staples (see Figs. 3-5 therein), so the optimal use of HYP smearing requires further study.

, , ,
, ,
, , ,
, ,
, , ,
, ,
, , , ,
, , ,
, , ,
, , ,
, , ,
, ,
, , ,
, ,
, ,
, ,
, ,
, ,
Table 2: Sets of staple-shaped gauge link paths and spatial pion momenta used on the lattice. Gauge link paths are characterized by the quark separation vector and the staple vector , cf. Fig. 1. The surveyed range of , parameterized in the table by the integer , was always chosen to extend from zero to well beyond the point where a numerical signal ceases to be discernible. The maximal magnitude of the Collins-Soper parameter attained in these sets is , for paired with and rotations thereof.

Iv Numerical results

iv.1 SIDIS and DY limits

The first step in the analysis of the obtained data concerns the behavior as a function of staple length . For ease of notation, both positive and negative are considered for a fixed to distinguish staples oriented in the forward and backward directions with respect to the pion momentum. Of particular physical interest is the asymptotic behavior for , corresponding to the SIDIS and DY limits. Fig. 2 displays results for the -quark generalized Boer-Mulders shift as a function of at a fixed , with each of the four panels corresponding to a successively larger transverse quark separation . The T-odd behavior of the observable is evident. As the SIDIS and DY limits are approached, a clear plateau behavior in is observed up to moderate values of ; as rises, statistical uncertainties increase (as indicated by the jackknife error estimates in the plots), and the identification of the plateaus becomes more tenuous, cf., e.g., the lower right panel, for . Plateau values are extracted by averaging over the regions and , respectively, as indicated by the fit lines in the plots; finally, the SIDIS and DY limits are obtained imposing T-oddness, i.e., the two plateau values in each plot are averaged with a relative minus sign to yield the asymptotic SIDIS and DY estimates also displayed in the panels (open symbols). The asymptotic values slightly decrease in magnitude as rises.

Figure 2: Generalized Boer-Mulders shift for -quarks as a function of staple extent , for fixed ; the panels illustrate data obtained at a succession of quark separations . Plateau fits and extraction of asymptotic values (open symbols) are described in the main text.

Figure 3: Generalized Boer-Mulders shift for -quarks in the SIDIS limit as a function of the quark separation , for three different values of the Collins-Soper parameter . The shown uncertainties are statistical jackknife errors.

Fig. 3 summarizes the results obtained in the SIDIS limit as a function of the quark separation , for three different values of the Collins-Soper parameter . Note that the data at small , up to , may be affected by discretization artefacts, but at larger , the data are expected to well approximate the continuum limit. For larger , cf. also further examples below, the statistical fluctuations rapidly increase, and no useful signal was obtained beyond in the case . The data appear to approach well-defined limits as either or becomes large. The behavior as a function of will be discussed in greater detail below; the behavior as becomes large seems plausible: Physically, once exceeds the size of the pion, the correlator (1) cannot anymore probe correlations inside the pion; it only contains vacuum-vacuum and vacuum-pion correlations. The -dependence of these correlations is then expected to be dominated by the typical exponential fall-off with observed in the vacuum333Heuristically, the expectation value of the gauge link staple, which, after integrating out the quark fields, may be thought of as being completed into a closed loop by the (fluctuating) world lines of dynamically propagating quarks, is expected to be determined by the chromodynamic flux piercing the loop. This is, e.g., the origin of the Wilson loop area law demonstrating confinement in Yang-Mills theory greensite (). Once exceeds the size of the pion, at most one of the legs of the staple can traverse the pion; the other runs entirely within the vacuum. Consider now varying by shifting the latter leg; the area being added or removed from the loop lies purely within the vacuum. Only vacuum chromodynamic flux is being added or subtracted, while the chromodynamic flux influenced by the pion remains fixed. Thus, the variation of the expectation value with is determined purely by vacuum properties (which is not to say that the expectation value becomes entirely independent of the properties of the pion; only its -dependence does). This argument is unchanged if one subsequently averages over different positions of the leg of the staple traversing the pion.; neglecting all other dependences in comparison, and canceling the dominant behavior in the ratio (23) leads to the expectation of a constant asymptotic behavior in .

As increases, one furthermore would expect the aforementioned constant to converge to an asymptotic limit. Remarkably, however, even the data appear to already approach the same large- constant as the data at higher . Possibly, this may be understood as a consequence of the Lorentz invariance of the vacuum; only vacuum-vacuum and vacuum-pion correlations are probed at large , and it seems plausible that these would be independent of the pion momentum. It would be desirable to gain a more definite understanding of this property. The most remarkable feature of the data, however, is the apparent tendency of the generalized Boer-Mulders shift to become constant in as is increased, not only for asymptotic values of , but for all . No obvious reason for this behavior at low to intermediate is apparent, and it would be very interesting to develop an understanding of it. The constant behavior implies that, in the relevant transverse momentum range corresponding to the probed range of , the transverse momentum spectrum of polarized quarks is the same as for unpolarized ones.

iv.2 Evolution in

A special focus of the present investigation is the behavior as a function of and the large limit, i.e., studying in detail the behavior of the sequence of data seen in Fig. 3 at a fixed value of . Fig. 4 shows two further panels analogous to the ones in Fig. 2, but with varying between the panels and fixed instead. Thus, in terms of a sequence, the lower left panel of Fig. 2 lies in between the two panels displayed in Fig. 4. As already mentioned further above, for pion spatial momentum , corresponding to the left panel in Fig. 4, there is only one branch as a function of , as shown. The right panel in Fig. 4, corresponding to , illustrates the rapid deterioration of signal as the pion momentum is increased. Nevertheless, at the moderate used here, a plateau can still be extracted.

Figure 4: Generalized Boer-Mulders shift for -quarks as a function of staple extent , analogous to Fig. 2, but at fixed and varying between the panels instead. In terms of a sequence, the lower left panel of Fig. 2 lies in between the two panels shown here. For pion spatial momentum (left panel), there is only one branch as a function of .
Figure 5: Generalized Boer-Mulders shift for -quarks in the SIDIS limit as a function of the Collins-Soper parameter , for fixed quark separation (left) and (right). Filled symbols correspond to the full generalized Boer-Mulders shift, whereas open symbols correspond to the partial contribution obtained by replacing in (23) by , for further discussion, cf. main text. Spatial pion momenta along various directions, as labeled, were employed; labels in general subsume more than one magnitude along the specified axis, cf. Table 2 for the complete set of momenta for which data were generated. Data labeled include averaging over 90 degree rotations thereof, i.e., momenta along any of the three spatial lattice axes are included. A given value of can be accessed using different pairs of and staple direction ; in such cases, data corresponding to different directions are slightly offset from one another horizontally for better visibility. The shown uncertainties are statistical jackknife errors.

Fig. 5 summarizes all results obtained in the SIDIS limit at as a function of the Collins-Soper parameter (left panel), and analogous data for the nearby value (right panel). From the figure, the good rotational properties of the calculation are evident; a given value of can be accessed using different directions of the pion spatial momentum and the staple direction , including both on- and off-axis directions, as shown. The corresponding results coincide, indicating that potential lattice artefacts are under control, and thus buttressing the physical significance of the data obtained. To assess the asymptotic behavior at large , it is advantageous to consider not only the full generalized Boer-Mulders shift, but also the partial contribution obtained by replacing in (23) by , omitting the contribution from , cf. eq. (18). Both quantities are displayed Fig. 5; as is evident from the figure, the partial contribution vanishes at , but monotonically increases in magnitude as is increased. By contrast, the remaining contribution from to the full generalized Boer-Mulders shift, dominant at , decreases in magnitude as rises. This matches the behavior expected from eq. (18), according to which the contribution from becomes insignificant for under the assumption that the amplitude stays finite. Altogether, the full generalized Boer-Mulders shift also decreases in magnitude as increases. Thus, the behavior of the data suggests that, by considering both quantities, one has access to both lower and upper bounds for the generalized Boer-Mulders shift, considerably increasing the confidence in the extrapolations to large discussed below. The comparison between the partial and full quantities also permits an assessment of the extent to which evolution in has progressed towards the asymptotic limit. Evidently, according to Fig. 5, already about half of the magnitude of the full generalized Boer-Mulders shift is subsumed in the partial contribution at . A significant part of the evolution has thus already been achieved at that value of . In this respect, the present study yields a much clearer picture than was obtained in the previous nucleon investigation tmdlat ().

Figure 6: Least-squares fits to the generalized Boer-Mulders shift data shown in Fig. 5 (left). Separate fits to filled and open data points are shown. Left panel displays fits obtained with ansatz ; right panel displays fits obtained with ansatz . Error bands show statistical jackknife uncertainties of the fits. Data points that lie outside the horizontal range of the displayed fit curves were not included in fits; in the case of the open symbols, one less data point was taken into account because it already lies in the region of an apparent inflection point, at which the simple asymptotic fit ansatz certainly is not applicable anymore. The asymptotic values displayed include the separate fits to the filled and open data points (filled and open squares), as well as the results of combined fits (filled circles); for details, cf. main text. The shown uncertainties are statistical jackknife errors.
Figure 7: Least-squares fits to the generalized Boer-Mulders shift data shown in Fig. 5 (right). Separate fits to filled and open data points are shown. Left panel displays fits obtained with ansatz ; right panel displays fits obtained with ansatz . Error bands show statistical jackknife uncertainties of the fits. Data points that lie outside the horizontal range of the displayed fit curves were not included in fits. The asymptotic values displayed include the separate fits to the filled and open data points (filled and open squares), as well as the results of combined fits (filled circles); for details, cf. main text. The shown uncertainties are statistical jackknife errors.
Figure 8: Least-squares fits to generalized Boer-Mulders shift data at , analogous to Figs. 6 and 7. The data were all obtained using spatial pion momenta along the lattice axes ( in the nomenclature of Fig. 5, including multiples and 90 degree rotations thereof, cf. Table 2 for the complete set of momenta for which data were generated). Separate fits to filled and open data points are shown. Left panel displays fits obtained with ansatz ; right panel displays fits obtained with ansatz . Error bands show statistical jackknife uncertainties of the fits. Data points that lie outside the horizontal range of the displayed fit curves were not included in fits. The asymptotic values displayed include the separate fits to the filled and open data points (filled and open squares), as well as the results of combined fits (filled circles); for details, cf. main text. The shown uncertainties are statistical jackknife errors.

To obtain quantitative statements about the large limit, least-squares fits444Whereas the data at higher generally display the larger statistical uncertainties, the data at lower are likely to deviate systematically to a larger degree from any putative simple asymptotic behavior. It therefore does not seem appropriate to bias the fits towards the lower data by performing a fit taking into account the statistical uncertainties. to the -dependences of the data were performed, using power-correction ansätze of the form and . It should be noted that the values of for which data were obtained in the present study do not reach clearly into the perturbative regime within which a reliable connection to perturbative evolution can be guaranteed. The aforementioned fit functions should therefore be regarded as ad hoc ansätze. On the one hand, separate fits to the full generalized Boer-Mulders shift data and to the partial contribution were performed; on the other hand, a combined fit to both quantities with a common constant (but, of course, separate coefficients ) was carried out. The results are shown in Figs. 6-8, and the asymptotic values summarized in Table 3. Fig. 6 displays the fits to the data of Fig. 5 (left), at , whereas Fig. 7 displays the fits to the data of Fig. 5 (right), at . Fig. 8 additionally displays fits to data obtained at a lower , namely, .

Fit function Full B.-M. shift Contribution Combined fit RMS deviation of
(GeV) only (GeV) (GeV) combined fit (GeV)
-0.146(26) -0.141(36) -0.145(25) 0.00755
-0.166(16) -0.110(22) -0.148(15) 0.01695