Poincare' covariant light-front spectral function and transverse momentum distribution

TL;DR

This paper derives the light-front spectral function and transverse momentum distributions for fermions in the valence approximation, linking them to light-front wave functions and clarifying their twist structure.

Contribution

It provides explicit expressions for transverse momentum distributions from light-front wave functions and relates twist-three distributions to leading twist transverse distributions.

Findings

Explicit formulas for transverse momentum distributions derived

Twist-three distributions expressed as linear combinations of leading twist distributions

Clarification of the relation between fermion correlator and spectral function in valence approximation

Abstract

In valence approximation the fermion correlator is simply related to the light-front spectral function. Then the leading twist time-reversal even transverse momentum distributions can be explicitly obtained from the light-front wave function of the system and the twist-three distributions are linear combinations of the transverse distributions at leading twist.

Equations

$$\displaystyle n^{\tau}_{\sigma^{\prime}\sigma}(x,{\bf k}_{\perp};{\cal M},{\bf S})~{}=~{}\int\!\!\ \!\!\!\!\!\ \!\!\!\!\!\sum d\epsilon~{}{1\over 2~{}{(2\pi)^{3}}}~{}{1\over 1-x}~{}{E_{S}\over\kappa^{+}}~{}{\cal P}^{\tau}_{{\cal M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)\quad\quad\quad$$

$$\displaystyle{{n^{\tau}_{\sigma^{\prime}\sigma}(x,{\bf k}_{\perp};{\cal M},{\bf S})={1\over 2}\left\{b_{0,{\cal M}}+{\mbox{\boldmath{$\sigma$}}}\cdot{\mbox{\boldmath{$f$}}_{{\cal M}}(x,{\bf k}_{\perp};\bf S)}\right\}_{\sigma^{\prime}\sigma}}}$$

$$\displaystyle{\mbox{\boldmath{$f$}}}_{{\cal M}}(x,{\bf k}_{\perp};{\bf S})$$

$$n_{\sigma {\sigma }^{\prime }}^{\tau }(x,{\mathbf{k}}_{\perp };\mathcal{M},\mathbf{S})=\frac{2(-1{)}^{\mathcal{M}+1/2}}{(1-x)}\int d{k}_{23}\sum _L{\mathcal{Z}}_{\sigma {\sigma }^{\prime }}^{\tau }(x,{\mathbf{p}}_{\perp },k_{23},L,\mathbf{S})\text{vspace}-3mm$$

$$\displaystyle{\Phi^{\tau}_{\alpha,\beta}(p,P,S)}={1\over 2}\int{d\xi^{-}d\xi^{+}d{\mbox{\boldmath{$\xi$}}}_{T}}~{}e^{ip\xi}\left\langle P,S,A|\bar{\psi}^{\tau}_{\beta}(0)\psi^{\tau}_{\alpha}(\xi)|A,S,P\right\rangle$$

$$\displaystyle\Phi^{\tau p}_{\alpha,\beta}(p,P,S)~{}=~{}{D\over 2p^{+}}~{}\sum_{\sigma\sigma^{\prime}}\left\{u_{\alpha}({\tilde{\bf p}},\sigma^{\prime})~{}{\cal P}^{\tau}_{{\cal M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)~{}{\bar{u}}_{\beta}({\tilde{\bf p}},\sigma)\right\},~{}$$

$$\mathrm{Tr}({\gamma }^{+}{\Phi }^p)$$

$$\mathrm{Tr}({\gamma }^{+}{\gamma }_5{\Phi }^p)$$

$$\mathrm{Tr}(/{\mathbf{p}}_{\perp }{\gamma }^{+}{\gamma }_5{\Phi }^p)$$

$$f(x,|{\mathbf{p}}_{\perp }{|}^2)=b_0$$

$$g_{1T}(x,|{\mathbf{p}}_{\perp }{|}^2)=\frac{M}{|{\mathbf{p}}_{\perp }|}{b}_{4,\mathcal{M}}$$

$$h_{1L}^{\perp }(x,|{\mathbf{p}}_{\perp }{|}^2)=\frac{M}{|{\mathbf{p}}_{\perp }|}{b}_{3,\mathcal{M}}$$

$$\Delta f={\Delta }_T^{\prime }f+\frac{|{\mathbf{p}}_{\perp }{|}^2}{2M^2}{h}_{1T}^{\perp }{g}_{1T}=-h_{1L}^{\perp }$$

Full text

1 Introduction

Transverse momentum distributions (TMDs) are a powerful tool to study hadron structure [1]. Light-cone models have been used to study the three-dimensional hadron structure, to disentangle contributions from different angular momentum components and to investigate possible relations among the TMDs, with the aim to offer a guide for the extraction of TMDs from experimental data [2, 3].

In this paper a Poincaré covariant, light-front (LF) spin-dependent spectral function is considered to investigate hadrons within the LF Hamiltonian dynamics in valence approximation. We present both the most general expression for the spin-dependent momentum distribution in terms of six scalar functions and a linear relation between the LF spectral function and the fermion correlator. This link implies approximate relations between the six time-reversal even (T-even) TMDs, as well relations between the leading twist and the twist-three TMDs [4].

The LF spectral function, {\cal P}^{\tau}_{{\cal M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{\kappa}}}},\epsilon,S), was defined in Ref. [5], starting from the LF wave function for a three-body system with spin $1/2$, third component $\cal M$ and polarization vector $\bf S$. The energy $\epsilon$ is the energy of a fully interacting two-particle [23] subsystem and the variable \tilde{\mbox{\boldmath{\kappa}}}=(\kappa^{+},\mbox{\boldmath{\kappa}}_{\perp}) is the LF momentum for particle 1 in the intrinsic reference frame of the cluster [1,(23)]. The spectral function is defined through the overlaps between the LF wave function of the system and the tensor product of a plane wave of momentum \tilde{\mbox{\boldmath{\kappa}}} and the state which describes the intrinsic motion of the two-particle spectator subsystem. The mentioned tensor product allows one to take care of macrocausality and to introduce a new effect of binding in the spectral function. The LF spectral function, through the Bakamjian-Thomas construction of the Poincaré generators [6], allows one to embed the successful phenomenology for few-nucleon systems in a Poincaré covariant framework and to satisfy at the same time normalization and momentum sum rule. As a first test of our approach the EMC effect for ${}^{3}He$ is being evaluated. Preliminary results show encouraging improvements with respect to a convolution approach with a momentum distribution [7].

2 Light-Front spin-dependent spectral function and transverse momentum distributions

2.1 Spin-dependent momentum distribution

Integration of the LF spectral function on the intrinsic energy $\epsilon$ of the $(A-1)$ system, gives the LF spin-dependent momentum distribution [5]

[TABLE]

where $\kappa^{+}=x~{}{\cal M}_{0}[1,(23)]$, with ${\cal M}_{0}[1,(23)]$ the free mass of the cluster [1,(23)], and E_{S}=\sqrt{4m^{2}+4m\epsilon+|{\mbox{\boldmath{\kappa}}}|^{2}}. Within the LF approach, the momentum distribution can be expressed through the three available independent vectors : i) the polarization vector $\bf S$; ii) the unit vector $\hat{n}$ (identified with $\hat{z}$) which defines the $\pm$ LF components, $v^{\pm}=v^{0}\pm{\hat{n}}\cdot{\bf v}$, and iii) the transverse (with respect to the $z$ axis) momentum component {\bf k}_{\perp}={\bf p}_{\perp}=\mbox{\boldmath{\kappa}}_{\perp} of the momentum $\bf p$

[TABLE]

where {{\mbox{\boldmath{f}}}}_{{\cal M}}(x,{\bf k}_{\perp};\bf S) is a pseudovector

[TABLE]

The functions $b_{i,{\cal M}}(x,{\bf k}_{\perp})$ $(i=0,1,...,5)$ and then $n^{\tau}_{\sigma^{\prime}\sigma}(x,{\bf k}_{\perp};{\cal M},{\bf S})$ can be obtained from the LF wave function of the system in momentum space. It results that the spin-dependent momentum distribution is an integral on the relative intrinsic momentum $k_{23}$ of the interacting spectator pair [4]

[TABLE]

where $L$ is the orbital angular momentum of the contributions to the one-body off-diagonal density matrix (only the values $L=0$ or $L=2$ are allowed).

2.2 Fermion correlator and LF spectral function

The fermion correlator in terms of the LF coordinates is [1]

[TABLE]

where $|A,S,P\rangle$ is the A-particle state and $\psi^{\tau}_{\alpha}(\xi)$ the particle field (e.g. a nucleon of isospin ${\tau}$ in a nucleus, or a quark in a nucleon). A linear relation exists between the correlator in valence approximation, $\Phi^{\tau p}$, and the spectral function [4]

[TABLE]

where $D=[(P^{+})^{2}~{}\pi~{}E_{S}]/\{p^{+}~{}m~{}{\cal{M}}_{0}[1,(23)]\}$. Then, traces of $\Phi^{p}$ can be expressed by traces of the spectral function :

[TABLE]

The proper integration on $p^{-}$ of Eqs. (7,8,9) and taking $p^{+}=xP^{+}$ gives relations between the TMDs at leading twist and the functions  $b_{i,{\cal M}}$ in Eqs. (2,3), viz

[TABLE]

Linear equations between transverse parton distributions were discussed in [2]

[TABLE]

From the explicit expressions of the functions ${b}_{i,{\cal M}}$ in terms of the wave function of the system, one finds that these equalities hold exactly in valence approximation when the contribution to the transverse distributions from the angular momentum $L=2$ is absent. This implies a vanishing value of the orbital angular momentum of the particle in the system wave function [4].

On the contrary the quadratic relation presented in [2] does not hold in our approach, even if the contribution from the angular momentum $L=2$ is absent, because of the presence of $\int dk_{23}$ in the expressions (4) of the transverse momentum distributions.

By evaluating proper traces of both the correlator and the spectral function, one can also obtain the twist-three TMDs in terms of the functions ${{b}_{i,{\cal M}}}$ and relate twist-three and twist-two TMDs. In our approximation the same linear relations found in [8] hold, once the gluon contributions are disregarded. Obviously the T-odd TMDs vanish in valence approximation [4].

3 Conclusions and perspectives

The LF spin-dependent momentum distribution for a spin $1/2$ system composed of three fermions (as ${}^{3}He$ or a nucleon in valence approximation) can be expressed through six functions  ${b}_{i,{\cal M}}$, that can be written in terms of the LF wave function of the system in momentum space. A simple relation exists between the fermion correlator in valence approximation and the LF spectral function. Then it follows that the TMDs are combinations of the functions ${b}_{i,{\cal M}}$.

As a result we found that the linear relations proposed between the T-even twist-two TMDs hold in valence approximation whenever the contribution from the L=2 orbital angular momentum term in the one-body off-diagonal density matrix is absent, while the quadratic relation does not hold even in this case. Furthermore, in valence approximation the proper relations between the twist-three and the twist-two TMDs hold, once the gluon contributions are disregarded.

In the close future we will evaluate the transverse momentum distributions for a nucleon in ${}^{3}He$, that could be extracted from measurements of appropriate spin asymmetries in ${}^{3}{\overrightarrow{He}}({\overrightarrow{e}},e^{\prime}p)$ experiments at high momentum transfer.