In this paper we study the Kotzinian-Mulders effect of a single hadron production in semi-inclusive deep inelastic scattering (SIDIS) within the framework of transverse momentum dependent (TMD) factorization. The asymmetry is contributed by the convolution of the Kotzinian-Mulders function $g_{1T}$ and the unpolarized fragmentation function $D_1$. As a TMD distribution, the Kotzinian-Mulders function in the coordinate space in the perturbative region can be represented as the convolution of the $C$-coefficients and the corresponding collinear correlation function. The Wandzura-Wilczek approximation is used to obtain this correlation function. We perform a detailed phenomenological numerical analysis of the Kotzinian-Mulders effect in the SIDIS process within TMD factorization at the kinematics of the HERMES and COMPASS measurements. It is found that the obtained $x_B$-, $z_h$- and $P_{h\perp}$-dependent Kotzinian-Mulders effect are basically consistent with the HERMES and COMPASS measurements.
We find the light-cone wavefunction representations of the Sivers and Boer-Mulders distribution functions. A necessary condition for the existence of these representations is that the light-cone wavefunctions have complex phases. We induce the complex phases by incorporating the final-state interactions into the light-cone wavefunctions. For the scalar and axial-vector diquark models for nucleon, we calculate explicitly the Sivers and Boer-Mulders distribution functions from the light-cone wavefunction representations. We obtain the results that the Sivers distribution function has the opposit…
We present a phenomenological analysis of the $\cos 2 ϕ$ asymmetry recently measured by the COMPASS and HERMES collaborations in unpolarized semi-inclusive deep inelastic scattering. In the kinematical regimes explored by these experiments the asymmetry arises from transverse-spin and intrinsic transverse-momentum effects. We consider the leading-twist contribution, related to the so-called Boer-Mulders transverse-polarization distribution $h_1^{\perp}(x, k_T^2)$, and the twist-4 Cahn contribution, involving unpolarized transverse-momentum distribution functions. We show that a reasonably good…
Chirally odd generalized parton distributions (GPDs) and the Boer-Mulders function provide valuable information about spin-orbit correlations for quarks in nucleons and other hadrons. We compare results for the relevant GPD $\bar{E}^q_T$ from a variety of phenomenological models as well as recent lattice results. We find that $\bar{E}^q_T>0$ for nucleons as well as the pion and for both $u$ and $d$ quark. As a result, the corresponding Boer Mulders functions are all expected to be negative. The sign of $\bar{E}^q_T$ arises from the relative sign between the upper and lower Dirac components for…
The fractured Boer-Mulders functions describe an inriguing class of polarization effects for the production of baryons in the target fragmentation region of deep-inelastic processes. These functions characterize transverse momentum asymmetries related to the spin orientation for different flavors of axial vector diquarks in an unpolarized ensemble of protons just as the familiar Boer-Mulders functions characterize transverse momentum asymmetries connected to the spin orientation of quarks in unpolarized targets. The asymmetries of the fractured Boer-Mulders effect originating in the proton dis…
At present, the Boer-Mulders function for a given quark flavour has been extracted from data on semi-inclusive deep inelastic scattering using the simplifying, but theoretically inconsistent, assumption that it is proportional to the Sivers function for each quark flavour. In this paper, using the latest semi-inclusive deep inelastic COMPASS deuteron data on the $\langle\cosϕ_h\rangle$ and $\langle\cos 2 ϕ_h\rangle$ asymmetries we extract the collinear $x_B$-dependence of the Boer-Mulders function for the sum of the valence quarks $Q_V=u_V + d_V$ in an essentially model independent way, and fi…
We present a phenomenological analysis of the $q_{\mathrm{T}}$-weighted transverse spin dependent azimuthal asymmetry recently measured by the COMPASS Collaboration in polarized pion-proton Drell-Yan process. In the kinematical regimes explored by experiments, we consider the leading-twist contributions from the Boer-Mulders distribution functions $h_{1}^{\perp}(x,k_{\mathrm{T}}^{2})$ of both the pion and the proton, the transversity distribution $h_{1}(x,k_{\mathrm{T}}^{2})$ and the pretzelosity distribution $h_{1\mathrm{T}}^{\perp}(x,k_{\mathrm{T}}^{2})$ of the proton, with the unpolarized t…
We obtain the leading twist T-odd quark transverse momentum dependent parton distribution functions (TMDs) of the proton, namely the Sivers function, $f_{1T}^{\perp q}(x,p_\perp^2)$, and the Boer-Mulders function, $h_1^{\perp q}(x,p_\perp^2)$, in a light-front quark-diquark model constructed with the wave functions predicted by the soft-wall AdS/QCD. The gluon rescattering is crucial to predict a nonzero T-odd TMDs. We study the utility of a nonperturbative $SU(3)$ gluon rescattering kernel going beyond the usual approximation of perturbative $U(1)$ gluons. The spin asymmetries in semi-inclusi…
We show that measuring the $\cos 2ϕ$ angular dependence in unpolarized Drell-Yan processes with $π^-$ beams colliding on proton and deuteron targets can determine the ratio of the Boer-Mulders functions for $d$ and $u$ quarks inside the proton $h_1^{\perp,d}/h_1^{\perp,u}$, which is still lack of theoretical constraint. The comparison of the $\cos 2 ϕ$ asymmetries measured in unpolarized $π^- p$ and $π^- D$ Drell-Yan processes, which are accessible at CERN by the COMPASS collaboration, can help to discriminate whether $h_1^\perp$ effects or QCD vacuum effects are preferred by data.
A major experimental program is presently underway to determine the Sivers, Boer-Mulders and transversity distributions, vital for understanding the internal structure of the nucleon. To this end we consider the Sivers, Boer-Mulders and transversity azimuthal asymmetries of the difference cross sections of hadrons with opposite charges in SIDIS reactions with unpolarized and transversely polarized target l+N\to l'+h+X, h=π^\pm, K^\pm, h^\pm. We show that on deuteron target these asymmetries are particularly simple and determine the sum of the valence-quark Q_V=u_V+d_V transverse momentum depen…
The naive time reversal odd (``T-odd'') parton distribution $h_{1}^{\perp}$, the so-called Boer-Mulders function, for both up ($u$) and down ($d$) quarks is considered in the diquark spectator model. While the results of different articles in the literature suggest that the signs of the Boer-Mulders function in semi-inclusive DIS for both flavors $u$ and $d$ are the same and negative, a previous calculation in the diquark-spectator model found that $h_{1}^{\perp(u)}$ and $h_{1}^{\perp(d)}$ have different signs. The flavor dependence is of significance for the analysis of the azimuthal $\cos(2ϕ…
We investigate the theoretical framework of the $\cos 2ϕ$ azimuthal asymmetry contributed by the coupling of two Boer-Mulders functions in the dilepton production unpolarized $πp$ Drell-Yan process by applying the transverse momentum dependent factorization at leading order. We adopt the model calculation results of the unpolarized distribution function $f_1$ and Boer-Mulders function $h_1^\perp$ of pion meson from the light-cone wave functions. We take into account the transverse momentum evolution effects for both the distribution functions of pion and proton by adopting the existed extracti…
A major experimental program is presently underway worldwide to determine the fundamental non-perturbative functions, the Sivers, Boer-Mulders and transversity distributions, which are vital for an understanding of the internal structure of the nucleon. However, at present, many simplifying assumptions are used in extracting these functions from the data. We demonstrate that if the binning of the data in Q^2 is small enough so that Q^2-evolution can be neglected inside a bin, then one can obtain stringent tests of these assumptions. Failure to satisfy these tests implies that the presently ext…
We apply the MIT bag model to study the Boer-Mulders function of the pion, a $T$-odd function that describes the transverse polarization distribution of the quark inside the pion. We simulate the effect of the gauge link through the "one-gluon-exchange" approximation. We consider both the quark helicity nonflip and double-flip contributions. The result in the MIT bag model is compared with those in the spectator models.
In series of papers the Boer-Mulders function for a given quark flavour has been extracted: (i) from data on semi-inclusive deep inelastic scattering, using the simplifying, but theoretically inconsistent, assumption that it is proportional to the Sivers function for each quark flavour and (ii) from data on Drell-Yan reactions. In earlier papers, using the semi-inclusive deep inelastic COMPASS deuteron data on the $\langle\cosϕ_h\rangle$ and $\langle\cos 2 ϕ_h\rangle$ asymmetries, we extracted the collinear $x_{_{\!B}}$ dependence of the Boer-Mulders function for the sum of the valence quarks …
We calculate the two leading-twist transverse momentum dependent distribution functions of the pion meson, the unpolarized distribution $f_{1π}(x,\bm{k}^2_T)$ and the Boer-Mulders function $h_{1π}^\perp (x,\bm{k}^2_T)$, using the pion wave functions derived from a light-cone approach. The evolution effect of the first $\bm k_T$-moment of the pion Boer-Mulders function is studied by employing an approximate evolution kernel. Using the model resulting distributions, we predict the transverse momentum weighted $\cos2ϕ$ azimuthal asymmetry in the unpolarized $π^- p$ Drell-Yan process which can be …
Sign Reversal of Boer-Mulders Functions from Semi-inclusive Deep-Inelastic Scattering to the Drell-Yan Process
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；National Central University（中央大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；South Texas College（南德克萨斯学院）
We compute the leading-twist T-odd quasi-distributions of the proton in a spectator model with scalar and axial-vector diquarks: the quasi Sivers function $\tilde{f}_{1T}^\perp(x, k_T^2; P_z)$ and the quasi Boer-Mulders function $\tilde{h}_1^\perp(x, k_T^2; P_z)$. We obtain the quark-quark correlators in the four-dimensional Euclidian space by replacing $γ^+$ and $σ^{i+}$ in the light-cone frame with $γ_z$ and $σ_{iz}$. We show by analytical calculation that the results of $\tilde{f}_{1T}^\perp$ and $\tilde{h}_1^\perp$ derived from the correlators can reduce to the expressions of the correspon…
A general formalism for the evaluation of time reversal odd parton distributions is applied here to calculate the Boer-Mulders function. The same formalism when applied to evaluate the Sivers function led to results which fulfill the Burkardt sum rule quite well. The calculation here has been performed for two different models of proton structure: a constituent quark model and the MIT bag model. In the latter case, important differences are found with respect to a previous evaluation in the same framework, a feature already encountered in the calculation of the Sivers function. The results obt…
We predict the twist-2 Transverse Momentum Dependent parton distribution functions (TMDs) of the pion, namely the unpolarized quark TMD, $f_{1}(x, k_\perp)$, and the transversely polarized quark TMD, also known as the Boer-Mulders function, $h^\perp_{1}(x, k_\perp)$, using a holographic light-front pion wavefunction with dynamical spin effects. These spin effects, in conjunction with gluon rescattering, are crucial to predict a non-zero holographic Boer-Mulders function. We investigate the use of a non-perturbative SU(3) gluon rescattering kernel, thus going beyond the usual approximation of p…
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 c…
At leading twist the transverse-momentum dependent parton distributions of the pion consist of two functions, the unpolarized $f_{1,π}(x,\boldsymbol k^{\, 2}_\perp)$ and the Boer-Mulders function $h_{1,π}^{\perp}(x,\boldsymbol k^{\, 2}_\perp)$. We study both functions within a light-front constituent model of the pion, comparing the results with different pion models and the corresponding nucleon distributions from a light-front constituent model. After evolution from the model scale to the relevant experimental scales, the results for the collinear pion valence parton distribution function $f…
Results for the naive-time-reversal-odd quark distributions in a light-cone quark model are presented. The final-state interaction effects are generated via single-gluon exchange mechanism. The formalism of light-cone wave functions is used to derive general expressions in terms of overlap of wave-function amplitudes describing the different orbital angular momentum components of the nucleon. In particular, the model predictions show a dominant contribution from S- and P-wave interference in the Sivers function and a significant contribution also from the interference of P and D waves in the B…
We present Monte Carlo simulations of both the Sivers and the Boer-Mulders effects in the polarized Drell-Yan $π^\pm p^\uparrow \to μ^+ μ^- X$ process at the center-of-mass energy $\sqrt{s} \sim 14$ GeV reachable at COMPASS with pion beams of energy 100 GeV. For the Sivers effect, we adopt two different parametrizations for the Sivers function to explore the statistical accuracy required to extract unambiguous information on this parton density. In particular, we verify the possibility of checking its predicted sign change between Semi-Inclusive Deep-Inelastic Scattering (SIDIS) and Drell-Yan …
We study the $\cos 2 ϕ$ azimuthal asymmetry in unpolarized semi-inclusive DIS, taking into account both the perturbative contribution (gluon emission and splitting) and the non perturbative effects arising from intrinsic transverse motion and transverse spin of quarks. In particular we explore the possibility to extract from $<\cos 2 ϕ>$ some information about the Boer--Mulders function $h_1^{\perp}$, which represents a transverse--polarization asymmetry of quarks inside an unpolarized hadron. Predictions are presented for the HERMES, COMPASS and JLab kinematics, where $<\cos 2 ϕ>$ is dominate…
We use the Boer-Mulders functions parameterized from unpolarized $p+D$ Drell-Yan data by the FNAL E866/NuSea Collaboration combined with recently extracted Collins functions to calculate the $\cos 2 ϕ$ asymmetries in unpolarized semi-inclusive deeply inelastic scattering (SIDIS) processes both for ZEUS at Hadron Electron Ring Accelerator (HERA) and Jefferson Lab experiments (JLab), and to compare our results with their data. We also give predictions for the $\cos 2 ϕ$ asymmetries of SIDIS in the kinematical regime of HERMES Collaboration, and the forthcoming JLab experiments. We predict that t…
Fixed-target unpolarized Drell-Yan experiments often feature an acceptance depending on the polar angle of the lepton tracks in the laboratory frame. Typically leptons are detected in a defined angular range, with a dead zone in the forward region. If the cutoffs imposed by the angular acceptance are independent of the azimuth, at first sight they do not appear dangerous for a measurement of the cos(2ϕ)-asymmetry, relevant because of its association with the violation of the Lam-Tung rule and with the Boer-Mulders function. On the contrary, direct simulations show that up to 10 percent asymmet…
We correct our analysis of the HERMES experiment for the azimuthal sin(phi)-spin asymmetry in semi-inclusive hadroproduction in DIS on longitudinally (with respect to the lepton momentum) polarized target because of discovered misprint in sign in the paper P.J.Mulders and R.D.Tangerman, Nucl. Phys. B 461 (1996) 197.
I discuss the relation between the Qiu-Sterman effects on one hand and the Collins, Sivers and Boer-Mulders effects on the other hand. It was suggested before that some of these effects are in fact the same, thus providing interesting connections between transverse-momentum dependent twist-2 functions and collinear twist-3 functions. Here I propose an alternative way to reach similar conclusions.
We study the process dependent nuclear $k_\perp$ broadening effect by employing the transverse momentum dependent(TMD) factorization approach in combination with the Mclerran-Venugopalan(MV) model. More specifically, we investigate how the parton transverse momentum distributions are affected by the process dependent gauge links in cold nuclear matter. In particular, our analysis also applies to the polarized cases including the nuclear quark Boer-Mulders function and the linearly polarized gluon distribution. Our main focus is on the nuclear TMDs at intermediate or large x.
Using the light-front holographic model, we study the transverse momentum dependent parton distributions (TMDs) for the case of pion. At leading twist, the unpolarized parton distribution function $ f_{1π}(x,\bfk^{2}) $ and the Boer-Mulders function $ h_{1π}^{\bot}(x,\bfk^{2}) $ are obtained for pion. We calculate both the functions using the light-front holographic model with spin improved wave function and compare the predicted results with available results of other models. In order to provide inputs in predicting future experimental data, a LO evolution is performed from model scale to exp…
We review the current status and future prospect for probing the transverse momentum dependent (TMD) parton distributions using the Drell-Yan process. We focus on the Boer-Mulders and Sivers functions, which are expected to undergo a sign-change from semi-inclusive deep-inelastic scattering (SIDIS) to Drell-Yan process. The constraints of existing Drell-Yan and SIDIS experiments on the signs of these functions are discussed. Future Drell-Yan measurements for the TMDs are also presented.
The Drell-Yan hadronic tensor for electromagnetic (EM) current is calculated in the Sudakov region $s\gg Q^2\gg q_\perp^2$ with ${1\over Q^2}$ accuracy, first at the tree level and then with the double-log accuracy. It is demonstrated that in the leading order in $N_c$ the higher-twist quark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion. The resulting tensor for unpolarized hadrons is EM gauge-invariant and depends on two leading-twist TMDs: $f_1$ responsible for total DY cross section, and Boer-Mulders function $h_1^\perp$. The order-of-magnitude estimates of ang…
This article presents the review of the current understanding on the pion-nucleon Drell-Yan process from the point of view of the TMD factorization. Using the evolution formalism for the unpolarized and polarized TMD distributions developed recently, we provide the theoretical expression of the relevant physical observables, namely, the unpolarized cross section, the Sivers asymmetry, and the $\cos2ϕ$ asymmetry contributed by the double Boer-Mulders effects. The corresponding phenomenology, particularly at the kinematical configuration of the COMPASS $πN$ Drell-Yan facility, is displayed numer…
Our understanding of the transverse spin structure of hadrons might definitely get improved by the information we gather on transverse momentum dependent (TMD) distributions. These new functions could also be crucial for a description of the observed transverse single spin asymmetries (SSA). In a hard scattering model for inclusive hadronic reactions, based on a generalized QCD factorization scheme, many mechanisms - namely the Sivers, Collins, Boer-Mulders effects - might contribute to a SSA. We show how the k_T dependent phases arising from the partonic kinematics together with a suitable ch…
The observed results for the azimuthal single spin asymmetries (SSAs) of the proton, measured in the semi-inclusive deep inelastic scattering (SIDIS), can be explained by the final-state interaction (FSI) from the gluon exchange between the outgoing quark and the target spectator system. SSAs require a phase difference between two amplitudes coupling the target with opposite spins to the same final state. We have used the model of light front wave functions (LFWFs) consisting of a spin-$\frac{1}{2}$ system as a composite of a spin-$\frac{1}{2}$ fermion and a spin-1 vector boson to estimate the…
We derive the leading order matching of the quark generated polarized transverse momentum dependent (TMD) distributions onto the collinear functions at small values of the transverse distance. Starting from the very definition of the TMD operator and performing the light-cone operator product expansion up to twist-3 order, we evaluate each distribution directly in position space. We primarily consider the cases of Sivers, Boer-Mulders and worm-gear functions. The effects of the TMD process dependence on the matching are explicitly shown. We also discuss the moments of TMD distributions which c…
The Drell-Yan process is studied in the framework of TMD factorization in the Sudakov region $s\gg Q^2\gg q_\perp^2$ corresponding to recent LHC experiments with $Q^2$ of order of mass of Z-boson and transverse momentum of DY pair $\sim$ few tens GeV. The DY hadronic tensors are expressed in terms of quark and quark-gluon TMDs with ${1\over Q^2}$ and ${1\over N_c^2}$ accuracy. It is demonstrated that in the leading order in $N_c$ the higher-twist quark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion. The resulting hadronic tensors depend on two leading-twist TMDs: $…
We investigate the single transverse-spin asymmetry with a $\sin (2ϕ-ϕ_S)$ modulation in the pion-induced Drell-Yan process within the theoretical framework of the transverse momentum dependent factorization. The asymmetry is contributed by the convolution of the Boer-Mulders function and the transversity. We adopt the model results for the distributions of the pion meson from the light-cone wavefunction approach and the available parametrization for the distributions of the proton to numerically estimate the $\sin (2ϕ-ϕ_S)$ asymmetry in $π^- p$ Drell-Yan at the kinematics of the COMPASS at CE…
We assess the current phenomenological status of transverse momentum dependent (TMD) parton distribution functions (PDFs) and fragmentation functions (FFs) and study the effect of consistently including perturbative QCD (pQCD) evolution. Our goal is to initiate the process of establishing reliable, QCD-evolved parametrizations for the TMD PDFs and TMD FFs that can be used both to test TMD-factorization and to search for evidence of the breakdown of TMD-factorization that is expected for certain processes. In this article, we focus on spin-independent processes because they provide the simplest…
Originally it was thought that the proton is not divisible into smaller particles. However, from collision experiments it follows that it consists of a 'soup' of countless 'colored' elementary particles such as quarks and gluons that are kept together by the strong nuclear force. The particles constituting the proton are also referred to as partons and their distributions are described by parton distribution functions (PDFs). As these functions cannot be calculated, they must be extracted from experiment. More knowledge on PDFs teaches us more about the inner workings of protons and enables us…
We reinterpret jet clustering as an axis-finding procedure which, along with the proton beam, defines the virtual-photon transverse momentum $q_T$ in deep inelastic scattering (DIS). In this way, we are able to probe the nucleon intrinsic structure using jet axes in a fully inclusive manner, similar to the Drell-Yan process. We present the complete list of azimuthal asymmetries and the associated factorization formulae at leading power for deep-inelastic scattering of a nucleon. The factorization formulae involve both the conventional time-reversal-even (T-even) jet function and the T-odd one,…
COMPASS is a fixed-target experiment in operation at the CERN North Area (SPS, M2 beam-line) since 2002. An important part of the broad physics programme of the experiment is dedicated to the exploration of the transverse spin-structure of the nucleon studying target transverse spin dependent azimuthal asymmetries (TSAs) arising in the Semi-Inclusive DIS (SIDIS) and Drell-Yan (DY) cross-sections. In addition to those measurements, COMPASS has recently studied also the TSAs weighted by powers of the hadron transverse momentum (in SIDIS) and virtual photon transverse momentum, $q_T$ (in DY). In …
We apply the formalism developed earlier for studying transverse momentum dependent parton distribution functions (TMDs) at small Bjorken $x$ to construct the small-$x$ asymptotics of the quark Sivers function. First, we explicitly construct the complete fundamental "polarized Wilson line" operator to sub-sub-eikonal order: this object can be used to study a variety of quark TMDs at small-$x$. We then express the quark Sivers function in terms of dipole scattering amplitudes containing various components of the "polarized Wilson line" and show that the dominant (eikonal) term which contributes…
The first measurement of transverse-spin-dependent azimuthal asymmetries in the pion-induced Drell-Yan (DY) process is reported. We use the CERN SPS 190 GeV/$c$, $π^{-}$ beam and a transversely polarized ammonia target. Three azimuthal asymmetries giving access to different transverse-momentum-dependent (TMD) parton distribution functions (PDFs) are extracted using dimuon events with invariant mass between 4.3 GeV/$c^2$ and 8.5 GeV/$c^2$. The observed sign of the Sivers asymmetry is found to be consistent with the fundamental prediction of Quantum Chromodynamics (QCD) that the Sivers TMD PDFs …
