Pion and nucleon relativistic electromagnetic four-current distributions Yi Chen

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Pion and nucleon relativistic electromagnetic four-current

distributions

Yi Chen

Interdisciplinary Center for Theoretical Study and Department of Modern Physics,

University of Science and Technology of China, Hefei, Anhui 230026, China

Shanghai Institute of Applied Physics,

Chinese Academy of Sciences, Shanghai 201800, China and

University of Chinese Academy of Sciences, Beijing 100049, China

C´edric Lorc´e∗

CPHT, CNRS, ´

Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France

The quantum phase-space approach allows one to deﬁne relativistic spatial distri-

butions inside a target with arbitrary spin and arbitrary average momentum. We

apply this quasiprobabilistic formalism to the whole electromagnetic four-current op-

erator in the case of spin-0 and spin-1

2targets, study in detail the frame dependence

of the corresponding spatial distributions, and compare our results with those from

the light-front formalism. While former works focused on the charge distributions,

we extend here the investigations to the current distributions. We clarify the role

played by the Wigner rotation and argue that electromagnetic properties are most

naturally understood in terms of Sachs form factors, contrary to what the light-front

formalism previously suggested. Finally, we illustrate our results using the pion and

nucleon electromagnetic form factors extracted from experimental data.

∗Corresponding author: cedric.lorce@polytechnique.edu

arXiv:2210.02908v2 [hep-ph] 5 Dec 2022

I. INTRODUCTION

Pions and nucleons are key systems to study for understanding quantum chromodynam-

ics (QCD). Pions, the lightest bound states in QCD, play a special role since they are the

(pseudo) Nambu-Goldstone bosons associated with the dynamical breakdown of chiral sym-

metry [1]. Nucleons are by far the most abundant (known) hadrons in nature, responsible

for more than 99% of the visible matter in the universe [2]. Pions and nucleons have very

diﬀerent masses originating from their diﬀerent, rich and complicated internal structures,

which constitutes a fundamental puzzle for modern physics.

The electromagnetic structure of hadrons is encoded in Lorentz-invariant functions known

as form factors (FFs). They have been measured with extreme precision in various scattering

experiments over the past decades [3–23]. On the theory side, lattice QCD calculations of

these FFs have witnessed tremendous progress in the last few years [24–36]. Recent reviews

on the extraction and the physics associated with electromagnetic FFs can be found in

Refs. [2,15,37–39].

According to textbooks, electromagnetic FFs can be interpreted as 3D Fourier transforms

of charge and magnetization densities in the Breit frame (BF) [40,41]. However, relativistic

recoil corrections spoil their interpretation as probabilistic distributions [42–47]. Switching

to the light-front formalism allows one to deﬁne alternative 2D charge densities free of these

issues [48–56], but there is a price to pay. Besides losing one spatial dimension these light-

front distributions also display various distortions, which are sometimes hard to reconcile

with an intuitive picture of the system at rest.

The concept of relativistic spatial distribution has recently been revisited in several works

with the goal of clarifying the relation between 3D Breit frame and 2D light-front deﬁni-

tions; see e.g. [57–65]. In this paper, we adopt the quantum phase-space approach where

the physical interpretation is relaxed to a quasiprobabilistic1one, allowing one to deﬁne

relativistic spatial distributions inside a target with arbitrary spin and arbitrary average

momentum [68–73]. This formalism is particularly appealing since it provides a natural and

smooth connection between the Breit frame and the inﬁnite-momentum frame pictures, and

allows one to understand the distortions in the light-front distributions in terms of relativis-

tic kinematical eﬀects. While the discussions in the literature have essentially focused on

the charge distributions, we extend here the study to the whole electromagnetic four-current

and demonstrate the consistency of the phase-space picture.

This paper is organized as follows. In Sec. II, we quickly review the concept of generic

elastic frame distributions within the quantum phase-space approach. We start our analysis

in Sec. III with a spin-0 target, introducing relativistic electromagnetic four-current distri-

1In the probabilistic picture, the state is perfectly localized in position space and the expectation value of

an operator Ois written as hOi=Rd3R|Ψ(R)|2O(R) with Ψ(R) the position space wave packet. In the

quasiprobabilistic picture, the same expectation value is expressed as hOi=Rd3P

(2π)3d3R ρΨ(R,P)O(R,P)

with ρΨ(R,P) the Wigner distribution, a real-valued function constructed from Ψ(R) and describing the

localization of the system in phase space [66,67]. Due to Heisenberg’s uncertainty principle, Wigner dis-

tributions are negative in some regions and cannot therefore be interpreted as strict probability densities.

butions and studying their frame dependence. Then we compare with the corresponding

light-front distributions and illustrate our ﬁndings using the pion (π+) electromagnetic form

factor extracted from experimental data. We proceed in Sec. IV with a spin-1

2target. We

discuss in detail the non-trivial role played by the Wigner spin rotation and show that elec-

tromagnetic properties are most naturally understood in terms of the Sachs form factors.

Here we also compare with the light-front formalism and illustrate our results using the nu-

cleon electromagnetic form factors extracted from experimental data. Finally, we summarize

our ﬁndings in Sec. V, and provide further discussions and details in three Appendices.

II. ELASTIC FRAME DISTRIBUTIONS

Two-dimensional spatial distributions in a generic elastic frame (EF) have been intro-

duced in [68] to study the angular momentum inside the nucleon. They are fully relativistic

objects that can be interpreted as the internal distributions associated with a target lo-

calized in phase-space (i.e. with deﬁnite average momentum and position) in the Wigner

sense [69,70,74]. Although they have in general only a quasiprobabilistic interpretation,

they provide a natural connection between spatial distributions deﬁned in the BF and in

the inﬁnite-momentum frame (IMF) [75].

For convenience, we choose the z-axis along the average total momentum of the target

P=1

2(p0+p) = (0⊥, Pz). The spatial distributions of the electromagnetic four-current are

then deﬁned within the phase-space formalism as [71]

Jµ

EF(b⊥;Pz)≡Zd2∆⊥

(2π)2e−i∆⊥·b⊥hp0, s0|ˆ

jµ(0)|p, si

2P0∆z=0

,(1)

where ˆ

jµ(x) is the electromagnetic four-current operator, ∆ = p0−pis the four-momentum

transfer, and b⊥is the transverse position relative to the average center of the target. The

four-momentum eigenstates are normalized as hp0, s0|p, si= 2p0(2π)3δ(3)(p0−p)δs0swith

sand s0the usual canonical spin labels. The elastic condition ∆0=P·∆/P 0= 0 is

automatically ensured by the restriction2∆z= 0 and implies that the resulting distribution

does not depend on time [68]. Note that the factor 2P0in the denominator of Eq. (1)

appears naturally in the phase-space formalism and ensures that the total electric charge

q=Zd2b⊥J0

EF(b⊥;Pz)s0=s=hp, s|ˆ

j0(0)|p, si

2p0(2)

2In the BF there is no need to set ∆z= 0 since by deﬁnition P=0, and so one can deﬁne in that case a

static three-dimensional spatial distribution.

transforms as a Lorentz scalar [71,76]. One can also formally write

hp, s|ˆ

j0(0)|p, si

2p0=hp, s|Rd3rˆ

j0(r)|p, si

hp, s|p, si,(3)

indicating that the total charge does not depend on the particular choice made for the

normalization of the four-momentum eigenstates.

Using Poincar´e and discrete spacetime symmetries, the oﬀ-forward matrix elements

hp0, s0|ˆ

jµ(0)|p, sifor a spin-jtarget can be parameterized in terms of (2j+ 1) Lorentz-

invariant electromagnetic FFs [77–80]. Diﬀerent sets of FFs for a given spin value jhave

been considered in the literature. These sets are all physically equivalent since they simply

correspond to diﬀerent choices for the basis of Lorentz tensors, and hence are linearly related

to each other. A particularly convenient and physically transparent basis is provided by

the multipole expansion in the BF, i.e. for P=0. In that frame, the multipole structure

appears to be the same as in the non-relativistic theory: the charge distribution consists of

a tower of electric multipoles of even order, while the electric current can be expressed in

terms of a tower of magnetic multipoles of odd order [79,81,82].

Poincar´e symmetry can also be used to determine how matrix elements of the electro-

magnetic four-current operator in diﬀerent Lorentz frames are related to each other. One

can write in general [77,83]

hp0, s0|ˆ

jµ(0)|p, si=X

B,sB

D∗(j)

Bs0(p0

B,Λ)D(j)

sBs(pB,Λ) Λµνhp0

B, s0

B|ˆ

jν(0)|pB, sBi,(4)

where hp0

B, s0

B|ˆ

jν(0)|pB, sBiis the BF matrix element, Λµνis the Lorentz boost from the BF

to a generic Lorentz frame, and D(j)is the Wigner rotation matrix for spin-jtargets, see

Appendix C. Since temporal and spatial components of the electromagnetic four-current get

mixed under a Lorentz boost, the odd magnetic multipoles in the BF will induce odd electric

multipoles in a generic Lorentz frame. Similarly, even electric multipoles in the BF will

induce even magnetic multipoles in a generic Lorentz frame. These odd electric and even

magnetic multipoles do not break parity (P) nor time-reversal (T) symmetries, and should

therefore not be confused with the P- and T-breaking ones which are not considered in this

work. Wigner rotations complicate further the relation (4) by reorganizing the multipole

weights. Namely, any particular multipole in the BF will usually generate a contribution to

all 3multipoles in a generic Lorentz frame [71].

In the following we will focus on the spin-0 and spin-1

2targets, and apply our formalism

to map the electromagnetic four-current distributions inside a pion and a nucleon using the

electromagnetic FFs extracted from experimental data. While the EF charge distributions

3A similar mechanism explains why relations between transverse-momentum dependent parton distribu-

tions and orbital angular momentum appear in various models of the nucleon [84,85].

EF have already been discussed in Refs. [59,71], the EF currents JEF will be studied here

for the ﬁrst time.

III. SPIN-0TARGET

Let us start with the simplest case, namely a spin-0 target. The matrix elements of the

electromagnetic four-current operator are parametrized in terms of a single FF

hp0|ˆ

jµ(0)|pi=e2PµF(Q2) (5)

with Q2=−∆2and ethe electric charge of a proton.

A. Breit frame distributions

The BF electromagnetic four-current distributions are deﬁned as

Jµ

B(r)≡Zd3∆

(2π)3e−i∆·rhp0

B|ˆ

jµ(0)|pBi

2P0

(6)

with p0

B=−pB=∆/2 and p00

B=p0

B=P0

B=M√1 + τ. We introduced for convenience

the Lorentz invariant quantity τ=Q2/4M2=∆2/4M2which measures the magnitude of

relativistic eﬀects.

The BF charge distribution is obtained by considering the µ= 0 component in Eq. (6).

Like in the non-relativistic theory, it corresponds simply to the 3D Fourier transform of the

electromagnetic FF [86]

B(r) = eZd3∆

(2π)3e−i∆·rF(∆2).(7)

It is spherically symmetric since there is no preferred spatial direction when P=0.

The BF current distribution for a spin-0 target is directly proportional to Pand hence

vanishes in the BF

JB(r) = 0,(8)

which is consistent with the interpretation of the BF as the average rest frame of the system

within the quantum phase-space approach [69–71].

The BF picture agrees with our naive expectation for a spin-0 system at rest. In order

to see what happens when the system has non-zero average momentum, we need to switch

to the concept of EF distributions.

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Pionandnucleonrelativisticelectromagneticfour-currentdistributionsYiChenInterdisciplinaryCenterforTheoreticalStudyandDepartmentofModernPhysics,UniversityofScienceandTechnologyofChina,Hefei,Anhui230026,ChinaShanghaiInstituteofAppliedPhysics,ChineseAcademyofSciences,Shanghai201800,ChinaandUniversityof...

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