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Prepared for submission to JHEP
Nucleon relativistic weak-neutral axial-vector
four-current distributions
Yi Chena,∗
aInterdisciplinary Center for Theoretical Study and Department of Modern Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China
E-mail: physchen@mail.ustc.edu.cn
Abstract: Relativistic full weak-neutral axial-vector four-current distributions inside a
general spin-1
2hadron are systematically studied for the first time, where the second-class
current contribution associated with the induced pseudotensor form factor (FF) is included.
We clearly demonstrate that the 3D axial charge distribution, being parity-odd in the Breit
frame, is in fact related to the induced pseudotensor FF GZ
T(Q2)rather than the axial FF
GZ
A(Q2). We study the frame-dependence of full axial-vector four-current distributions for
a moving hadron, and compare them with their light-front counterparts. We revisit the
role played by the Melosh rotation, and understand more easily and intuitively the origins
of distortions appearing in light-front distributions (relative to the Breit frame ones) using
the conjecture that we propose in this work. In particular, we show that the second-class
current contribution, although explicitly included, does not contribute in fact to the mean-
square axial and spin radii. We finally illustrate our results in the case of a proton using
the weak-neutral axial-vector FFs extracted from experimental data.
*Corresponding author
arXiv:2411.12521v3 [hep-ph] 9 Mar 2025
Contents
1 Introduction 1
2 Weak-neutral axial-vector form factors 3
3 Quantum phase-space formalism 5
4 Breit frame distributions 6
4.1 BF weak-neutral axial-vector four-current distributions 6
4.2 BF mean-square radii 9
5 Elastic frame distributions 10
5.1 EF weak-neutral axial-vector four-current distributions 10
5.2 EF mean-square transverse radii 14
6 Light-front distributions 15
6.1 LF weak-neutral axial-vector four-current distributions 16
6.2 LF amplitudes via proper IMF limit of EF amplitudes 18
6.3 LF mean-square transverse radii 21
7 Summary 21
A Parametrization of nucleon weak-neutral axial-vector FFs 22
B Breakdown of Abel transformation for axial charge distributions 26
1 Introduction
Nucleons (i.e. protons and neutrons) are key hadrons to study for understanding quan-
tum chromodynamics (QCD), since they are responsible for more than 99% of the visible-
matter mass in the universe [1]. Protons, in particular, also hold another unique role of
being the only stable composite building blocks in nature [2]. Due to the complicated
nonperturbative dynamics of their quark and gluon degrees of freedom, nucleons inherit
particularly rich and intricate internal structures. When it comes to the study of prop-
erties and internal structures of the nucleon in the weak sector via elastic or quasielastic
(anti)neutrino-(anti)nucleon scatterings [3–22], nucleon axial-vector form factors (FFs) be-
come particularly important.
Axial-vector FFs are Lorentz-invariant functions that describe how the hadron re-
acts with the incoming (anti)neutrino in a scattering reaction, encoding therefore very
clean internal axial charge and spin information of the hadron in the weak sector since
– 1 –
(anti)neutrinos participate only in weak interactions. In the Standard Model, there are in
general two types of axial-vector FFs of a hadron in the weak sector: the weak-charged
ones via the weak-charged current interactions mediated by the W±bosons, and the weak-
neutral ones via the weak-neutral current interactions mediated by the Z0bosons. These
axial-vector FFs also serve as important quantities for constraining the systematic uncer-
tainties of high-precision measurements in (anti)neutrino oscillation experiments [23–31].
On the theory side, tremendous progress has been reported in the last few years from the
first-principle lattice QCD side [32–57]; theoretical evaluations of the nucleon axial-vector
FFs and their contributions to the associated cross sections based on chiral perturbation
theory and various models/approaches are still rapidly developing [58–91]. For (recent)
reviews on nucleon axial-vector FFs and associated physics of (anti)neutrino interactions,
see e.g. Refs. [92–108].
According to textbooks, charge distributions can be defined in the Breit frame (BF)
in terms of three-dimensional (3D) Fourier transform of the Sachs electric FF [1,109,110].
However, relativistic recoil corrections spoil their interpretation as probabilistic distribu-
tions [111–116]. In position space, a probabilistic density interpretation is tied to Galilean
symmetry that implies the invariance of inertia under the change of frames. In a relativistic
theory, however, inertia becomes a frame-dependent concept because of Lorentz symmetry.
The only way out is to switch to the light-front (LF) formalism [117] where a Galilean sub-
group of the Lorentz group is singled out [118,119], allowing therefore a nice probabilistic
interpretation [120–130]. The price to pay is that besides losing the longitudinal spatial
dimension1, these LF distributions also exhibit various distortions owing to the particular
LF perspective, Lorentz effects and complicated Wigner-Melosh rotations [132–137], which
are sometimes hard to reconcile with an intuitive picture of the system in 3D at rest.
The quantum phase-space formalism distinguishes itself by the fact that the require-
ment of a strict probabilistic interpretation is relaxed and replaced by a milder quasiprob-
abilistic picture [138–140]. This approach is quite appealing since it allows one to define in
a consistent way relativistic spatial distributions inside a target with arbitrary spin and ar-
bitrary average momentum [141–155], providing a smooth and natural connection between
the BF and essentially the LF pictures2, and allowing one to explicitly trace the spatial
distortions caused by Lorentz boosts and Wigner rotations for any spin-jhadrons under
the protection of Poincaré symmetry.
As an extension of our recent work [150], we study in this work in detail the relativistic
full weak-neutral axial-vector four-current distributions inside a general spin-1
2hadron (e.g.,
the proton), where the second-class current contribution associated with the induced pseu-
dotensor FF is newly taken into account. We explicitly demonstrate that the relativistic
3D weak-neutral axial charge distribution, being parity-odd in the BF, is in fact related to
the weak-neutral induced pseudotensor FF GZ
T(Q2)rather than the axial FF GZ
A(Q2). This
1We note that Miller and Brodsky [131] recently demonstrated at the wavefunction level that frame-
independent and three-dimensional LF coordinate-space wavefunctions can be obtained by using the di-
mensionless, frame-independent longitudinal coordinate ˜z.
2Strictly speaking, the smooth connection is between the BF and infinite-momentum frame (IMF) dis-
tributions, see e.g., Refs. [143,144,146,148–155]. However, IMF distributions coincide most of the time
with the corresponding LF distributions [149].
– 2 –
clarifies and reconfirms that the quantity R2
A, see the Eq. (1) of Ref. [150], is evidently not
the 3D mean-square axial radius of a spin-1
2target.
We also study the full weak-neutral axial-vector four-current distributions for a moving
hadron focusing on their frame-dependence, and compare them with their LF counterparts.
In particular, we explicitly reproduce the LF axial-vector four-current amplitudes via the
proper IMF limit of the corresponding elastic frame (EF) amplitudes, which together with
our recent works [148–152] inspires us to propose the following conjecture: Any light-front
amplitudes for well-defined light-front distributions in principle can be explicitly reproduced
from the corresponding elastic frame amplitudes in the proper infinite-momentum frame
limit. As a reward, we can understand more easily and intuitively the origins of distortions
appearing in LF distributions (relative to the BF ones). For completeness, both 3D and
2D transverse mean-square axial and spin radii in different frames are rederived. We show
in particular that the second-class current contribution, although explicitly included in our
calculations, does not contribute in fact to the mean-square axial and spin radii.
The paper is organized as follows. In Sec. 2, we present the full matrix elements
of the weak-neutral axial-vector four-current operator and associated weak-neutral axial-
vector FFs for a general spin- 1
2hadron. The key ingredients of the quantum phase-space
formalism are briefly given in Sec. 3. We start our analysis in Sec. 4with the 3D Breit frame
distributions of the weak-neutral axial-vector four-current densities inside a general spin-1
2
hadron. We then present in Sec. 5the generic elastic frame distributions for a moving
spin-1
2hadron at arbitrary average momentum. In Sec. 6, we present the corresponding
light-front distributions, and the explicit demonstration of light-front amplitudes via the
proper infinite-momentum frame limit of elastic frame amplitudes. In Sec. 4to Sec. 6, we
also rederive 3D and 2D transverse mean-square axial and spin radii. Finally, we summarize
our findings in Sec. 7. For completeness, we also provide details for the parametrization of
nucleon weak-neutral axial-vector form factors in Appendix A, and further discussions on
the breakdown of Abel transformation of axial charge distributions in Appendix B.
2 Weak-neutral axial-vector form factors
Based on solely Lorentz covariance and the hermiticity property of the weak-neutral
axial-vector four-current operator ˆ
jµ
5(x)≡ˆ
¯
ψ(x)γµγ5ˆ
ψ(x), there are in general three weak-
neutral axial-vector FFs of a spin- 1
2hadron in the weak sector [61,93]. These three axial-
vector FFs together describe the internal weak-neutral axial-vector content of the system
in response to external weak-neutral current interactions; see e.g., Fig. 1for the tree-level
Feynman diagram of the neutrino-nucleon elastic scattering.
In the case of a spin-1
2hadron, matrix elements of the weak-neutral axial-vector four-
current operator ˆ
jµ
5in general can be parametrized as [61,65,93,95]3
⟨p′, s′|ˆ
jµ
5(0)|p, s⟩= ¯u(p′, s′)Γµ(P, ∆)u(p, s),(2.1)
3We note that matrix elements of the divergence of ˆ
jµ
5have been investigated recently in Refs. [156,157].
– 3 –
Z0*
νℓνℓ
NN
Figure 1. Illustration of the tree-level Feynman diagram of the t-channel weak-neutral current
elastic scattering reaction νℓ(k) + N(p)→νℓ(k′) + N(p′)in the first Born approximation (i.e. one
virtual Z0boson exchange), with ℓ=e, µ, τ . The four-momentum transfer is ∆ = k−k′=p′−p.
with
Γµ(P, ∆) ≡γµγ5GZ
A(Q2) + ∆µγ5
2MGZ
P(Q2)−σµν ∆νγ5
2MGZ
T(Q2),(2.2)
where ˆ
ψ(x)denotes the quark field operator (which can either be a flavor-singlet or flavor-
multiplet), P= (p′+p)/2,∆ = p′−p, and p2=p′2=M2are the on-mass-shell relations
for the same hadron of mass Min the initial and final states. Here, GZ
A(Q2),GZ
P(Q2)and
GZ
T(Q2)are called axial, induced pseudoscalar and induced pseudotensor FFs, respectively.
The hermiticity condition implies that all these three FFs are real in the spacelike region as
functions of the four-momentum transfer squared Q2≡ −∆2≥0. Since the explicit (diago-
nal) matrix forms of generators Taassociated with the operator ˆ
jµ
5,a(x)≡ˆ
¯
ψ(x)γµγ5Taˆ
ψ(x),
see e.g. Ref. [95], for a given flavor-space fundamental representation group SU (n)f(n≥2
and n∈Z) of a given (anti)neutrino-hadron elastic scattering reaction are known, it is
convenient to work in the U(1)frepresentation without loss of generality.
Now, let us take a close look why these weak-neutral axial-vector FFs GZ
A(Q2),GZ
P(Q2)
and GZ
T(Q2)are real in Eq (2.2). Since ˆ
jµ
5= (ˆ
jµ
5)†, the hermiticity condition implies that
⟨p′, s′|ˆ
jµ
5(0)|p, s⟩=⟨p, s|ˆ
jµ
5(0)|p′, s′⟩†,(2.3)
which amounts to Γµ(P, ∆) = γ0Γµ,†(P, −∆)γ0, or equivalently Γµ,†(P, ∆) = γ0Γµ(P, −∆)γ0.
Explicit evaluation of this condition using the full vertex function (2.2) shows that
GZ
X(Q2) = GZ
X(Q2)∗, X =A, P, T , (2.4)
where we have used the following identities for the Dirac γmatrices:
γ0γµγ5γ0=γ5(γµ)†, γ0γ5γ0=−γ5, γ0σµν γ5γ0=−γ5(σµν )†.(2.5)
According to Weinberg [158], one can use the G-parity to classify all possible currents
formed by Dirac bilinears in the literature in term of the first- and second-class currents.
– 4 –
The G-parity is defined as the combination of the charge-conjugation C-parity after an
rotation of a 180◦angle around the y-axis in isospin space, namely G≡Cexp (iπIy),
where Iy=σy/2is the y-component isospin matrix. As the ne plus ultra, one can easily
demonstrate that the vector and axial-vector four-currents jµ=¯
ψγµψand jµ
5=¯
ψγµγ5ψ
which transform in the following manner
GjµG−1= +jµ,Gjµ
5G−1=−jµ
5(2.6)
are classified as the first-class currents, whereas those transform in the opposite manner
GjµG−1=−jµ,Gjµ
5G−1= +jµ
5(2.7)
are classified as the second-class currents.
Using Weinberg’s classification [158], we can identify GZ
Aand GZ
Pin Eq. (2.2) as the
FFs associated with the first-class currents, while identify GZ
Tas the FF associated with
the second-class current [61,93,159]. In the presence of exact isospin symmetry or G-
parity invariance, the second-class current contribution vanishes identically [61,65,95], i.e.
GZ
T(Q2)=0. It is thus common in the literature that the second-class current contribution
of a hadron is usually not much discussed or calculated, see e.g. Refs. [58–60,63–86,88],
since the G-parity invariance is well respected by the strong interaction or QCD4.
However, the G-parity invariance is in general not preserved in electromagnetic and
weak interactions, owing to either the quark electric charge or quark mass differences [160].
This is the partial motivation that we wish to take into account the induced pseudotensor
FF GZ
T(Q2). Another motivation is that following our recent work [150] we wish to figure
out whether the 3D nucleon axial charge distribution in the BF is finite or not, and to check
whether the nucleon 3D axial (charge) radius in a more general case exists or not, when the
full vertex function (2.2) is taken into account without assuming the G-parity invariance.
3 Quantum phase-space formalism
Although FFs are objects defined in momentum space and extracted from experimental
data involving particles with well-defined momenta, their physical interpretation actually
resides in position space [149]. Because of Lorentz symmetry, the notion of relativistic
spatial distributions in general depends on the target average momentum, hindering there-
fore in general a probabilistic interpretation in position space. We are therefore natu-
rally led to switch our perspective to a phase-space picture, which is quasiprobabilistic at
the quantum level owing to Heisenberg’s uncertainty principle. In this section, we only
present the key ingredients of the quantum phase-space formalism, and refer readers to
Refs. [143,144,148,149] for more details.
In quantum field theory, it has been known for a long time that the expectation value
4In strong interactions or QCD, e.g. the strong decays of mesons, the G-parity invariance is exact.
– 5 –
of any an operator ˆ
Oin a physical state |Ψ⟩can be expressed as [138–140]
⟨Ψ|ˆ
O(x)|Ψ⟩=X
s′,s Zd3P
(2π)3d3R ρs′s
Ψ(R,P)⟨ˆ
O⟩s′s
R,P(x),(3.1)
where ρs′s
Ψ(R,P)defines the Wigner distribution interpreted as the quantum weight for
finding the system at average position R=1
2(r′+r)and average momentum P=1
2(p′+p).
Probabilistic densities are then recovered upon integration over either average position
or momentum variables [144,149]. A compelling feature of the quantum phase-space for-
malism is that wave-packet details have been cleanly factored out in Eq. (3.1). We can then
interpret the phase-space amplitude
⟨ˆ
O⟩s′s
R,P(x) = Zd3∆
(2π)3ei∆·R⟨P+∆
2, s′|ˆ
O(x)|P−∆
2, s⟩
2pp′0p0(3.2)
as the internal distribution associated with a state localized in the Wigner sense around
average position Rand average momentum P[142–149,153–155].
4 Breit frame distributions
The BF is specified by the condition P=0. From a phase-space perspective, the BF
can be regarded as the average rest frame of the system, where spin structure assumes its
simplest form [83,144,146,149–152,161]. In this frame, one can obtain fully relativistic 3D
spatial distributions of a generic composite system for its static internal structures in the
Wigner sense. This is also the reason why so many 3D mean-square radii of a hadron, e.g.
(electric) charge ⟨r2
ch⟩, mass ⟨r2
mass⟩, mechanical ⟨r2
mech⟩and spin ⟨r2
spin⟩radii, are usually
defined in this frame [1,149,150,162–165].
Since the energy transfer ∆0=P·∆/P 0vanishes automatically in this frame, internal
distributions in the BF do not depend on time x0arising due to the translation invariance
of the matrix elements (3.2). Following Refs. [144,148–150], relativistic 3D axial-vector
four-current distributions in the BF are defined as
Jµ
5,B(r)≡Zd3∆
(2π)3e−i∆·r⟨p′
B, s′
B|ˆ
jµ
5(0)|pB, sB⟩
2P0
BP=0
,(4.1)
where r=x−Ris the distance relative to the center R=0of the system, p′
B=−pB=
∆/2,Q2=∆2,τ≡Q2/(4M2), and P0
B=p′0
B=p0
B=M√1 + τ.
4.1 BF weak-neutral axial-vector four-current distributions
Evaluating the matrix elements (2.1) in the BF leads to [59,61,62,64,150]
A0
B=√1 + τ(i∆·σ)s′
BsBGZ
T(∆2),
AB= 2P0
Bσ−∆(∆·σ)
4P0
B(P0
B+M)s′
BsB
GZ
A(∆2)−
∆(∆·σ)s′
BsB
2MGZ
P(∆2),(4.2)
– 6 –
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
J5,B
0(r) [fm-3]
-2-1 0 1 2
-0.06
-0.04
-0.02
0.00
0.02
0.04
rz[fm]
4πr2·J5,B
0(r) [fm-1]
Figure 2. The 3D weak-neutral axial charge distributions J0
5,B (r)(upper panel) and 4πr2·J0
5,B (r)
(lower panel) in the BF along the z-axis inside a longitudinally polarized (i.e. ˆ
s=ez) proton, using
proton’s weak-neutral induced pseudotensor FF GZ
T(Q2)given in Appendix A.
where Aµ
B≡ ⟨p′
B, s′
B|ˆ
jµ
5(0)|pB, sB⟩, and explicit canonical polarization indices will be omit-
ted for better legibility hereafter unless necessary.
Inserting the BF amplitudes (4.2) into Eq. (4.1) leads to the following relativistic 3D
BF weak-neutral axial charge and axial-vector current distributions [150]
J0
5,B(r) = Zd3∆
(2π)3e−i∆·ri∆·σ
2MGZ
T(∆2) = −∇r·σ
2MZd3∆
(2π)3e−i∆·rGZ
T(∆2),
J5,B(r) = Zd3∆
(2π)3e−i∆·rσ−∆(∆·σ)
4P0
B(P0
B+M)GZ
A(∆2)−∆(∆·σ)
4M P 0
B
GZ
P(∆2).
(4.3)
We find that the 3D axial charge distribution J0
5,B(r)is in fact related to the weak-neutral
induced pseudotensor FF GZ
T(Q2)rather than the axial FF GZ
A(Q2). We stress that above
results (4.3) are also well consistent with our previous work [150], where the G-parity in-
variance of QCD is further applied to the matrix elements (2.1), eliminating therefore the
second-class current contribution associated with GZ
T(Q2)[61,65,95]. Besides, we notice
that J0
5,B(r)is parity-odd owing to the associated parity-odd factor (r·σ)coming from
the Fourier transform of (∆·σ). In Fig. 2, we illustrate the parity-odd 3D axial charge
distributions J0
5,B(r)and 4πr2·J0
5,B(r)with r=|r|along the z-axis inside a longitudinally
polarized (i.e. the unit polarization vector5ˆ
s=ez) proton, using proton’s weak-neutral
induced pseudotensor FF given in Appendix A.
5In general, the unit polarization vector is given by ˆ
s=χ†
h′σχh=σh′h.
– 7 –
s
=ex
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
rx[fm]
ry[fm]
J5,B(r) [fm-3]
Proton
0.0
0.5
1.0
1.5
2.0
2.5
Figure 3. The 3D weak-neutral axial-vector current distribution J5,B (r)in the BF in the transverse
plane inside a transversely polarized (i.e. ˆ
s=ex) proton, using proton’s weak-neutral axial-vector
FFs GZ
A(Q2)and GZ
P(Q2)given in Appendix A.
Moreover, we do notice that the 3D axial-vector current distribution J5,B (4.3) is
in fact independent of the induced pseudotensor FF GZ
T(Q2). In other words, J5,B is free
from the second-class current contribution, and therefore assumes the same expression as in
Ref. [150]. Using the QCD equation of motion [166], one can show that SB(r) = J5,B (r)/2
is the physically meaningful 3D (intrinsic) spin distribution in the BF [141,143,147,150],
characterizing how the spin is distributed in the weak-neutral sector. In Fig. 3, we illustrate
the 3D weak-neutral axial-vector current distribution J5,B(r)in the transverse plane inside
a transversely polarized proton, using proton’s weak-neutral axial-vector FFs given in Ap-
pendix A. We observe that the distribution is perfectly mirror symmetric (antisymmetric)
with respect to the x-axis (y-axis) in the transverse plane, exhibiting the so-called toroidal
mode [167,168] in the weak sector. This can be more easily understood from the multipole
decomposition [149] of J5,B (r). In doing so, we find that J5,B (r)consists of mirror sym-
metric (antisymmetric) monopole and quadrupole contributions solely with respect to the
x-axis (y-axis).
Recalling in the BF that p′
B=−pB=∆/2and p′0
B=p0
B=P0
B, we do recognize in
Eq. (4.2) the characteristic spin structure σ−pB(pB·σ)
p0
B(p0
B+M)defined relative to the center
of mass RM, which is the only relativistic center transforming as the spatial part of a
Lorentz four-vector, and corresponds therefore to a physically more natural and transparent
relativistic center of the system [142,145,149]. There are other choices of the relativistic
center [142,145,149], e.g. the center of energy RE=RM+P׈
s
2M(EP+M)and the center of
– 8 –
(canonical) spin Rc=RM+P׈
s
2MEPwith EP≡pP2+M2. Different from RM,REand
Rchowever will cause respectively sideways shifts P׈
s
2M(EP+M)and P׈
s
2MEPof pure relativistic
origin when (P׈
s)=0, e.g, when the spinning system is longitudinally moving while
it is transversely polarized. This in turn justifies that the parametrization (2.1) is indeed
physically clear and transparent, since the spin is defined relative to the center of mass.
For more details of the relativistic centers and sideways shifts, see Refs. [142,145] and
also the similar discussions for the parametrization of polarization-magnetization tensor in
Ref. [149].
4.2 BF mean-square radii
Although the induced pseudotensor FF GZ
T(Q2)is explicitly taken into account (2.2),
we note that the total axial charge of a general spin-1
2composite system in the BF vanishes
identically [150]:
Zd3r J0
5,B(r) = lim
∆→0
1
2M(i∆·σ)GZ
T(∆2)= 0,(4.4)
because of the parity-odd nature of the 3D axial charge distribution J0
5,B(r)itself. This
means that the definition of the standard mean-square axial (charge) radius
⟨r2
A⟩ ≡ Rd3r r2J0
5,B(r)
Rd3r J0
5,B(r)(4.5)
for a general spin-1
2hadron is in fact not well-defined. This is different from case for the
definition of the mean-square charge radius of the neutron where one can replace Gn
E(0) = 0
with Gp
E(0) = 1 [1,149] so as to make the definition well-defined6, since the induced
pseudotensor charge GZ
T(0) is in general not zero [169], see e.g., Eq. (A.10), while the
distribution itself is parity-odd.
In the case GZ
T(Q2) = 0 by using the G-parity invariance of QCD, J0
5,B(r)vanishes
identically and thus the axial (charge) radius does not exist [150], contrary to what is
usually stated in the literature [19,20,58,59,106,170–173] via
R2
A≡ − 6
GZ
A(0)
dGZ
A(Q2)
dQ2Q2=0
=1
GZ
A(0) −∇2
∆GZ
A(∆2)∆=0.(4.6)
In conclusion, in either GZ
T(Q2) = 0 or GZ
T(Q2)= 0 cases, the genuine 3D mean-square
axial (charge) radius ⟨r2
A⟩is not defined via Eq. (4.6), since the 3D axial charge distribution
itself is related to GZ
T(Q2)rather than GZ
A(Q2).
On the other hand, the 3D mean-square spin radius ⟨r2
spin⟩is a well-defined and physi-
cally meaningful quantity [150]. Using proton’s weak-neutral axial-vector FFs given in Ap-
pendix Aand the same formula for the mean-square spin radius given in Refs. [150,174],
6Note also the fact that the BF charge distribution of the neutron is spherically symmetric [144,149].
– 9 –
we find for the proton that ⟨r2
spin⟩ ≈ (2.1054 fm)2and R2
A≈(0.6510 fm)2. We reconfirm
that the relativistic contribution 1
4M21 + 2GZ
P(0)
GZ
A(0) indeed plays a dominant role [150].
Before we move forward, there are two key points deserving to be emphasized. The
first point is that the relation between a genuine 3D mean-square radius and the slope of
the corresponding FF is in general not so obvious and simple, see the examples given in
Refs. [150,163]. One needs to carefully define first the corresponding distribution, and then
derive the genuine mean-square radius based on that distribution. The second point is that
the concept of identifying a mean-square radius simply via the slope of the FF with same
naming is in general misleading and incorrect. A typical example is the 3D mean-square
axial (charge) radius for which the 3D axial charge distribution J0
5,B(r)is not even related
to the axial FF GZ
A(Q2), see Eq. (4.3) and Ref. [150]. Furthermore, our result of the axial
charge distribution (4.3) also explicitly reveals the breakdown of Abel transformation, see
the further discussions given in Appendix B.
5 Elastic frame distributions
BF distributions provide us the best proxy for picturing a system in 3D sitting in
average at rest around the origin in the Wigner sense, where the spin in the parametrization
(2.1) is defined relative to the relativistic center of mass. If one is however interested in
the internal structure of a moving system, one can employ the so-called elastic frame (EF)
distributions introduced in Ref. [141].
Following Refs. [144,149,150], the relativistic axial-vector four-current distributions
for a moving target in the generic EF are defined as
Jµ
5,EF(b⊥;Pz)≡Zd2∆⊥
(2π)2e−i∆⊥·b⊥⟨p′, s′|ˆ
jµ
5(0)|p, s⟩
2P0∆z=|P⊥|=0
,(5.1)
where the z-axis has been chosen along P= (0⊥, Pz)without loss of generality. Since the
energy transfer ∆0=P·∆/P 0vanishes automatically, EF distributions (5.1) are indeed
independent of time.
5.1 EF weak-neutral axial-vector four-current distributions
Evaluating directly the matrix elements (2.1) in the generic EF leads to [150]
A0
EF = 2P0(i∆⊥·σ⊥)
2MGZ
T(∆2
⊥) + Pz
P0σzGZ
A(∆2
⊥),
Az
EF = 2P0Pz
P0
(i∆⊥·σ⊥)
2MGZ
T(∆2
⊥) + σzGZ
A(∆2
⊥),
A⊥
EF = 2√P2P0+M(1 + τ)
(P0+M)√1 + τσ⊥+(ez×i∆⊥)⊥
2M
Pz
(P0+M)√1 + τGZ
A(∆2
⊥)
−∆⊥(∆⊥·σ⊥)
2GZ
A(∆2
⊥)
P0+M+GZ
P(∆2
⊥)
M,
(5.2)
– 10 –
where Aµ
EF ≡ ⟨p′, s′|ˆ
jµ
5(0)|p, s⟩,p′= (P0,∆⊥/2, Pz),p= (P0,−∆⊥/2, Pz),Q2=∆2
⊥, and
P0=pM2(1 + τ) + P2
z.
Poincaré symmetry can also be employed to determine how the matrix elements of the
axial-vector four-current operator in different Lorentz frames are related to each other. One
can write in general [132,133]
⟨p′, s′|ˆ
jµ
5(0)|p, s⟩=X
s′
B,sB
D†(j)
s′s′
B(p′
B,Λ)D(j)
sBs(pB,Λ) Λµν⟨p′
B, s′
B|ˆ
jν
5(0)|pB, sB⟩,(5.3)
where ⟨p′
B, s′
B|ˆ
jν
5(0)|pB, sB⟩represent the BF matrix elements, Λis the rotationless Lorentz
boost matrix from the BF to a generic Lorentz frame, and D(j)is the Wigner rotation matrix
for spin-jsystems. Alternatively, one can analytically reproduce above EF amplitudes
(5.2) by applying the covariant Lorentz transformation (5.3) on the BF amplitudes (4.2)
at ∆z= 0, with the help of Wigner rotation matrix D(1/2) given in Ref. [149]. In doing
so, we do explicitly reproduce our previous results [146,148–150] for the Wigner angular
Pz[GeV]
0
0.2
0.4
0.6
1.0
1.5
3.0
∞
0.0
0.5
1.0
1.5
J5,EF
0(b;Pz) [fm-2]
bA
2EF
0.0 0.5 1.0 1.5
0.0
0.5
1.0
b[fm]
2πb·J5,EF
0(b;Pz) [fm-1]
Figure 4. Frame-dependence of EF weak-neutral axial charge distributions J0
5,EF(b;Pz)(upper
panel) and 2πb ·J0
5,EF(b;Pz)(lower panel) as a function b=|b⊥|inside a longitudinally polarized
(i.e. ˆ
s=ez) proton, using proton’s weak-neutral axial FF GZ
A(Q2)given in Appendix A. Maxima
of 2πb ·J0
5,EF are indicated by the gray dot-dashed curve, and the corresponding root-mean-square
transverse axial (charge) radii p⟨b2
A⟩EF of J0
5,EF from Eq. (5.7) are also marked in the lower panel.
– 11 –
conditions:
cos θ=P0+M(1 + τ)
(P0+M)√1 + τ,sin θ=−√τPz
(P0+M)√1 + τ.(5.4)
As a result, EF amplitudes (5.2) can be equivalently rewritten as [150]
A0
EF = 2Mγ√1 + τ(i∆⊥·σ⊥)
2MGZ
T(∆2
⊥) + βσzGZ
A(∆2
⊥),
Az
EF = 2Mγ√1 + τβ(i∆⊥·σ⊥)
2MGZ
T(∆2
⊥) + σzGZ
A(∆2
⊥),
A⊥
EF = 2M√1 + τcos θσ⊥−(ez×i∆⊥)⊥
2M√τsin θGZ
A(∆2
⊥)
−∆⊥(∆⊥·σ⊥)
4M√1 + τGZ
A(∆2
⊥)
P0+M+GZ
P(∆2
⊥)
M,
(5.5)
with γ≡P0/√P2and β≡Pz/P 0. Inserting the EF amplitudes (5.2) or (5.5) into Eq. (5.1)
leads to the following relativistic EF weak-neutral axial-vector four-current distributions:
J0
5,EF(b⊥;Pz) = Zd2∆⊥
(2π)2e−i∆⊥·b⊥(i∆⊥·σ⊥)s′s
2MGZ
T(∆2
⊥) + Pz
P0(σz)s′sGZ
A(∆2
⊥),
Jz
5,EF(b⊥;Pz) = Zd2∆⊥
(2π)2e−i∆⊥·b⊥Pz
P0(i∆⊥·σ⊥)s′s
2MGZ
T(∆2
⊥)+(σz)s′sGZ
A(∆2
⊥),
J⊥
5,EF(b⊥;Pz) = Zd2∆⊥
(2π)2e−i∆⊥·b⊥−∆⊥(∆⊥·σ⊥)s′s
4P0GZ
A(∆2
⊥)
P0+M+GZ
P(∆2
⊥)
M
+√P2
P0cos θ(σ⊥)s′s−(ez×i∆⊥)⊥
2M√τsin θ δs′sGZ
A(∆2
⊥).
(5.6)
We remind that SEF(b⊥;Pz) = J5,EF (b⊥;Pz)/2is the EF spin distribution [141,143,
147,150]. In Fig. 4, we illustrate the EF axial charge distributions J0
5,EF(b;Pz)and 2πb ·
J0
5,EF(b;Pz)as a function of b=|b⊥|inside a longitudinally polarized (i.e. ˆ
s=ez) proton at
different Pz, using proton’s weak-neutral axial FF given in Appendix A. We see that J0
5,EF
in Fig. 4strongly depends on Pzand vanishes identically when Pz= 0, reflecting the fact
that it is an induced effect for a longitudinally polarized spin- 1
2target. This observation is
also consistent with the fact that J0
5,B is entirely contributed by GZ
T(Q2)in the BF (4.3).
According to Eq. (5.6), it is clear that J0
5,EF(b⊥;∞) = Jz
5,EF(b⊥;∞), which is quite
reminiscent of the result in the electromagnetic case [148,149]. In Fig. 5, we illustrate the
longitudinal axial-vector current distributions Jz
5,EF(b⊥;Pz)and 2πb ·Jz
5,EF(b⊥;Pz)along
the x-axis at different Pzinside a transversely polarized proton, using proton’s weak-neutral
induced pseudotensor FF given in Appendix A. We see that the distributions in Fig. 5are
indeed parity-odd similar as Fig. 2, and have strong Pz-dependence. We note that if the
proton is longitudinally polarized, Jz
5,EF will be axially symmetric and frame-independent,
and it coincides exactly with the Pz=∞case of J0
5,EF(b;Pz)in Fig. 4.
We do observe that the transverse EF axial-vector current distribution J⊥
5,EF is in-
– 12 –
Pz[GeV]
0
0.2
0.4
0.6
1.0
1.5
3.0
∞
-0.04
-0.02
0.00
0.02
0.04
J5,EF
z(b⟂;Pz) [fm-2]
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-0.04
-0.02
0.00
0.02
0.04
bx[fm]
2πb·J5,EF
z(b⟂;Pz) [fm-1]
Figure 5. Frame-dependence of EF weak-neutral longitudinal axial-vector current distributions
Jz
5,EF(b⊥;Pz)(upper panel) and 2πb ·Jz
5,EF(b⊥;Pz)(lower panel) along the x-axis inside a trans-
versely polarized (i.e. ˆ
s=ex) proton, using proton’s weak-neutral induced pseudotensor FF
GZ
T(Q2)given in Appendix A. The maxima (minima) are indicated by the gray dot-dashed curves.
dependent of GZ
T(Q2), since the transverse part of the BF axial-vector four-current am-
plitudes (4.2) does not get mixed under longitudinal Lorentz boosts (5.3). In Fig. 6, we
illustrate the transverse EF weak-neutral axial-vector current distribution J⊥
5,EF(b⊥;Pz)in
the transverse plane inside a transversely polarized proton at Pz= 2 GeV, using proton’s
weak-neutral axial-vector FFs given in Appendix A. We observe that the EF distribution
J⊥
5,EF at Pz= 2 GeV, compared with the BF one at Pz= 0 in Fig. 3, is no longer mirror
symmetric with respect to the x-axis but is still mirror antisymmetric with respect to the
y-axis in the transverse plane, and it does exhibit the toroidal mode [167,168] similar as
Fig. 3. The key reason is that on top of the monopole and quadrupole contributions, J⊥
5,EF
at finite Pzalso contains a dipole contribution, which explicitly breaks the up-down mirror
symmetry (with respect to the x-axis) but still preserves the left-right mirror antisymme-
try (with respect to the y-axis) in the transverse plane for a transversely polarized spin-1
2
target.
Furthermore, owing to the Lorentz mixing of temporal and longitudinal components of
the axial-vector four-current amplitudes under a longitudinal Lorentz boost, we also notice
that as long as Pzis nonvanishing the generic EF distributions J0
5,EF and Jz
5,EF will depend
on GZ
A(Q2)and GZ
T(Q2)respectively, in comparison with those BF ones (4.3). Moreover,
– 13 –
s
=ex
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
bx[fm]
by[fm]
J5,EF
⟂(b⟂;Pz=2 GeV) [fm-2]
Proton
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Figure 6. The transverse EF weak-neutral axial-vector current distribution J⊥
5,EF(b⊥;Pz)in the
transverse plane inside a transversely polarized (i.e. ˆ
s=ex) proton at Pz= 2 GeV, using proton’s
weak-neutral axial-vector FFs GZ
A(Q2)and GZ
P(Q2)given in Appendix A.
we note that both J0
5,EF and Jz
5,EF in the generic EF are free from the Wigner rotation (5.3)
while J⊥
5,EF suffers from it (see, e.g., the cos θand sin θfactors in J⊥
5,EF), since the Wigner
rotation (5.3) mixes (σ⊥)s′swhile leaves (σz)s′sand (∆⊥·σ⊥)s′sunchanged [149].
5.2 EF mean-square transverse radii
Following Refs. [149,150], we then rederive the mean-square transverse axial and spin
radii using the EF distributions (5.6). Although GZ
Tenters both J0
5,EF and Jz
5,EF (5.6), we
obtain exactly the same mean-square transverse radii as Ref. [150]
⟨b2
A⟩EF(Pz) = 1
2E2
P
+2
3R2
A,
⟨b2
spin,L⟩EF (Pz) = 2
3R2
A,
(5.7)
with EP=pM2+P2
z. Likewise, since GZ
T(Q2)does not enter J⊥
5,EF, see Eq. (5.6), we ex-
pect the same result for the mean-square transverse spin radius ⟨b2
spin,T ⟩EF(Pz)as Ref. [150].
From these results, we conclude that the second-class current contribution associated with
GZ
T(Q2), although explicitly included in our calculations, does not contribute to the mean-
square transverse axial and spin radii in the generic EF.
Using the same expression of ⟨b2
spin,T ⟩EF(Pz)as Refs. [150,174] and the weak-neutral
axial-vector FFs given in Appendix A, we show in Fig. 7the mean-square transverse spin
– 14 –
0←Pz
Pz→∞
Pz≈3.0149 GeV
10-210-1100101102103104
4.275
4.280
4.285
4.290
4.295
Pz[GeV]
〈bspin,T
2〉EF(Pz) [fm2]
Figure 7. The mean-square transverse spin radius ⟨b2
spin,T ⟩EF(Pz)of the proton as a function of Pz,
using proton’s weak-neutral axial-vector FFs GZ
A(0) and GZ
P(0) given in Appendix A. The minimum
of ⟨b2
spin,T ⟩EF(Pz)is located at Pz≃3.2132 Mp≈3.0149 GeV.
radius ⟨b2
spin,T ⟩EF(Pz)for the proton as a function of Pz. In particular, we find that the
minimum of ⟨b2
spin,T ⟩EF(Pz)is located at Pz≃3.2132 Mp≈3.0149 GeV, while the maximum
of ⟨b2
spin,T ⟩EF(Pz)is located at Pz= 0. In the IMF limit Pz→ ∞, the value of ⟨b2
spin,T ⟩EF(∞)
is entirely contributed by the Pz-independent terms, see the Eq. (52) of Ref. [174], lying in
between ⟨b2
spin,T ⟩EF(0) and ⟨b2
spin,T ⟩EF(Pz≃3.2132 Mp).
6 Light-front distributions
For completeness, we finally study the relativistic full weak-neutral axial-vector four-
current distributions using the LF formalism [117], where LF distributions in some cases
can be interpreted as strict probabilistic densities [120–130] since the symmetry group
associated with the transverse LF plane is the Galilean subgroup singled out from the
Lorentz group [118,119].
The x+-independent symmetric LF frame [120,127,141] is specified by the conditions7:
P⊥=0⊥and ∆+= 0, which ensure that the LF energy transfer ∆−= (P⊥·∆⊥−
P−∆+)/P +vanishes automatically. Following Refs. [144,149,150], the weak-neutral axial-
vector four-current distributions in the symmetric LF frame are defined as
Jµ
5,LF(b⊥;P+)≡Zd2∆⊥
(2π)2e−i∆⊥·b⊥LF⟨p′, λ′|ˆ
jµ
5(0)|p, λ⟩LF
2P+∆+=|P⊥|=0
,(6.1)
where P+is treated as an independent variable in the LF formalism, and LF helicity states
|p, λ⟩LF are covariantly normalized as LF⟨p′, λ′|p, λ⟩LF = 2p+(2π)3δ(p′+−p+)δ(2)(p′
⊥−
7One can relax the condition P⊥=0⊥, provided that LF distributions are restricted to x+= 0 [175,176].
– 15 –
p⊥)δλ′λ. Besides, |p, λ⟩LF can be related to the canonical spin states |p, s⟩via the Melosh
rotation [134]
|p, λ⟩LF =X
s
|p, s⟩⟨p, s|
⟨p, s|p, s⟩|p, λ⟩LF =X
s|p, s⟩M(j)
sλ (p),M(j)
sλ (p)≡⟨p, s|p, λ⟩LF
⟨p, s|p, s⟩,(6.2)
where M(j)(p)denotes the unitary Melosh rotation matrix for spin-jsystems.
In the spin-1
2case, the generic 2×2unitary Melosh rotation matrix M(1/2) is explicitly
given by [135,148,149]
M(1/2)(p) = cos θM
2−e−iϕpsin θM
2
eiϕpsin θM
2cos θM
2!,(6.3)
with
cos θM=(p0+pz+M)2− |p⊥|2
(p0+pz+m)2+p2
⊥
,sin θM=−2(p0+pz+M)|p⊥|
(p0+pz+m)2+p2
⊥
,(6.4)
where p⊥=|p⊥|(cos ϕp,sin ϕp), and θMis the Melosh rotation angle. It is easy to verify
that the condition cos2θM+ sin2θM= 1 is indeed automatically guaranteed. In the limit
pz→ ∞, the Melosh rotation matrix M(1/2) becomes a 2×2identity matrix, and therefore
the canonical spin polarization states |s⟩coincide with the LF helicity states |λ⟩LF in the
IMF limit, namely limpz→∞ |s⟩=|λ⟩LF.
We note that although the Melosh rotation matrix M(j)becomes an trivial identity
matrix in the IMF limit, it does not necessarily mean that LF distributions are completely
free from relativistic artifacts caused by Melosh rotations [135,144,146,148,149]. For ex-
ample, the change of polarization basis (6.2) modifies the normalization of four-momentum
eigenstates (i.e. P0→P+), which usually brings the explicit P+-dependence for transverse
LF distributions, e.g. J⊥
5,LF(b⊥;P+)in Eq. (6.7) and J⊥
LF(b⊥;P+)in Ref. [148]. Typically
when a LF distribution is completely free from P+, it should also free from relativistic
artifacts caused by Melosh rotations and therefore assumes physically clear probabilistic in-
terpretation [120–122,125,127,128,177], e.g. the LF electric charge distribution J+
LF [148]
and the LF axial charge distribution J+
5,LF (6.7). From the Melosh rotation perspective, this
also explains (to some extent) why the LF components O+and O−are usually regarded as
the “good” and “bad” components in the literature, respectively. In this sense, we believe
that the pictures provided by those LF densities that depend explicitly on P+can not be
considered as realistic representations of the system on the average at rest [149].
6.1 LF weak-neutral axial-vector four-current distributions
By analogy with Eq. (2.1), matrix elements of the axial-vector four-current operator in
terms of LF helicity states |p, λ⟩LF for a general spin-1
2hadron are parametrized as
LF⟨p′, λ′|ˆ
jµ
5(0)|p, λ⟩LF = ¯uLF(p′, λ′)γµGZ
A+∆µ
2MGZ
P−σµν ∆ν
2MGZ
Tγ5uLF(p, λ),(6.5)
– 16 –
where uLF(p, λ)denotes the LF helicity Dirac spinor, and the explicit Q2-dependence of
these axial-vector FFs GZ
A(Q2),GZ
P(Q2), and GZ
T(Q2)for clarity is omitted.
Evaluating directly the matrix elements (6.5) in the symmetric LF frame in terms of
LF helicity Dirac spinors leads to [150]
A+
LF = 2P+(i∆⊥·σ⊥)λ′λ
2MGZ
T(∆2
⊥)+(σz)λ′λGZ
A(∆2
⊥),
A−
LF = 2P−(i∆⊥·σ⊥)λ′λ
2MGZ
T(∆2
⊥)−(σz)λ′λGZ
A(∆2
⊥),
A⊥
EF = 2M(σ⊥)λ′λ+(ez×i∆⊥)⊥
2Mδλ′λGZ
A(∆2
⊥)−∆⊥(∆⊥·σ⊥)λ′λ
4M2GZ
P(∆2
⊥),
(6.6)
with Aµ
LF ≡LF⟨p′, λ′|ˆ
jµ
5(0)|p, λ⟩LF. We observe that A⊥
EF is free from the second-class
current contribution associated with GZ
T(Q2), while A+
LF and A−
LF do receive contributions
from that, similar as the case of the EF amplitudes (5.2).
Inserting the LF amplitudes (6.6) into Eq. (6.1) leads to the following LF weak-neutral
axial-vector four-current distributions [150,178]:
J+
5,LF(b⊥;P+) = Zd2∆⊥
(2π)2e−i∆⊥·b⊥(i∆⊥·σ⊥)λ′λ
2MGZ
T(∆2
⊥)+(σz)λ′λGZ
A(∆2
⊥),
J−
5,LF(b⊥;P+) = Zd2∆⊥
(2π)2e−i∆⊥·b⊥P−
P+(i∆⊥·σ⊥)λ′λ
2MGZ
T(∆2
⊥)−(σz)λ′λGZ
A(∆2
⊥),
J⊥
5,LF(b⊥;P+) = Zd2∆⊥
(2π)2e−i∆⊥·b⊥M
P+(σ⊥)λ′λ+(ez×i∆⊥)⊥
2Mδλ′λGZ
A(∆2
⊥)
−∆⊥(∆⊥·σ⊥)λ′λ
4M2GZ
P(∆2
⊥).
(6.7)
We remind that J+
5,LF is also called the LF helicity distribution [120,178], and S⊥
LF(b⊥;P+) =
J⊥
5,LF(b⊥;P+)/2is the transverse LF spin distribution [141,143,147,150].
According to Eqs. (6.7) and (5.6), it is clear that
J+
5,LF(b⊥;P+) = J0
5,EF(b⊥;∞) = Jz
5,EF(b⊥;∞),(6.8)
where we used the fact that canonical spin polarizations coincide with LF helicities in the
IMF limit. Above result (6.8) is quite reminiscent of the similar results in the electromag-
netic case [148,149]. It is therefore clear that for a longitudinally polarized (i.e. ˆ
s=ez)
proton, J+
5,LF(b;P+)and 2πb ·J+
5,LF(b;P+)will be exactly the same as the Pz=∞case
of J0
5,EF(b;Pz)and 2πb ·J0
5,EF(b;Pz)in Fig. 4, respectively. Similarly for a transversely
polarized (i.e. ˆ
s=ex) proton, J+
5,LF(b⊥;P+)and 2πb ·J+
5,LF(b⊥;P+)will be exactly the
same as the Pz=∞case of Jz
5,EF(b⊥;Pz)and 2πb ·Jz
5,EF(b⊥;Pz)in Fig. 5, respectively.
In Fig. 8, we illustrate the (scaled) transverse LF axial-vector current distribution
P+
MJ⊥
5,LF(b⊥;P+)in the transverse plane inside a transversely polarized proton, using pro-
ton’s weak-neutral axial-vector FFs given in Appendix A. Similarly as Fig. 6, the distribu-
tion P+
MJ⊥
5,LF in the transverse plane is not mirror symmetric with respect to the x-axis
– 17 –
s
=ex
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
bx[fm]
by[fm]
P+
MJ5,LF
⟂(b⟂;P+) [fm-2]
Proton
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Figure 8. The (scaled) transverse LF weak-neutral axial-vector current distribution
P+
MJ⊥
5,LF(b⊥;P+)in the transverse plane inside a transversely polarized (i.e. ˆ
s=ex) proton,
using proton’s weak-neutral axial-vector FFs GZ
A(Q2)and GZ
P(Q2)given in Appendix A.
but is still mirror antisymmetric with respect to the y-axis, and it also exhibits the toroidal
mode [167,168] in the weak sector similar as Fig. 6. This is consistent with our expectation,
since J⊥
5,LF also contains monopole, dipole and quadrupole contributions similar as J⊥
5,EF.
6.2 LF amplitudes via proper IMF limit of EF amplitudes
Following Refs. [148,151,152], we can explicitly reproduce the genuine LF amplitudes
(6.6) by making use of the EF amplitudes (5.2) in the proper IMF limit [i.e. expand the
amplitudes in 1/P +at large P+, and keep only the leading term]8. Starting from Eq. (5.2),
we can first construct the following amplitudes A±
EF in the generic EF:
⟨p′, s′|ˆ
j+
5(0)|p, s⟩=A0
EF +Az
EF
√2= 2P+(i∆⊥·σ⊥)s′s
2MGZ
T(∆2
⊥)+(σz)s′sGZ
A(∆2
⊥),
⟨p′, s′|ˆ
j−
5(0)|p, s⟩=A0
EF − Az
EF
√2= 2P−(i∆⊥·σ⊥)s′s
2MGZ
T(∆2
⊥)−(σz)s′sGZ
A(∆2
⊥),
(6.9)
with ˆ
j±
5≡(ˆ
j0
5±ˆ
jz
5)/√2. Clearly, above amplitudes ⟨p′, s′|ˆ
j±
5(0)|p, s⟩are not proper LF am-
plitudes since they are still defined in terms of the canonical (or instant-form) polarization
states |s⟩rather than the LF helicity states |λ⟩LF [148,151,152].
8More specifically, we first switch variables from {P0, Pz}to {P+, P −}in the constructed EF amplitudes,
and then expand the amplitudes in 1/P +at large P+, and finally keep only the leading term.
– 18 –
We are now ready to take the proper IMF limit of ⟨p′, s′|ˆ
j±
5(0)|p, s⟩and ⟨p′, s′|ˆ
j⊥
5(0)|p, s⟩.
It then follows that the constructed amplitudes are given by
⟨p′, s′|ˆ
j+
5(0)|p, s⟩proper IMF = 2P+(i∆⊥·σ⊥)s′s
2MGZ
T(∆2
⊥)+(σz)s′sGZ
A(∆2
⊥),
⟨p′, s′|ˆ
j−
5(0)|p, s⟩proper IMF = 2P−(i∆⊥·σ⊥)s′s
2MGZ
T(∆2
⊥)−(σz)s′sGZ
A(∆2
⊥),
⟨p′, s′|ˆ
j⊥
5(0)|p, s⟩proper IMF = 2M(σ⊥)s′s+(ez×i∆⊥)⊥
2Mδs′sGZ
A(∆2
⊥)
−∆⊥(∆⊥·σ⊥)s′s
2MGZ
P(∆2
⊥),
(6.10)
We note that in the IMF limit (i.e., Pz→ ∞), similarly as in the electromagnetic case [148],
the amplitude ⟨p′, s′|ˆ
j+
5(0)|p, s⟩will be enhanced while the amplitude ⟨p′, s′|ˆ
j−
5(0)|p, s⟩will
be suppressed, owing to the associated global factors P+and P−, respectively. Using the
fact that limPz→∞ |s⟩=|λ⟩LF since M(1/2)(p) = M†(1/2) (p′) = 12×2at Pz→ ∞, we can
simply apply the replacements s→λand s′→λ′to the constructed amplitudes (6.10).
In doing so, we indeed explicitly reproduce the genuine LF amplitudes (6.6). Following
this procedure, one can also explicitly reproduce the LF amplitudes (6.6) by using the EF
amplitudes (5.5) in terms of Wigner rotation angle θ, with the help that [148]9
lim
Pz→∞ cos θ=1
√1 + τ,lim
Pz→∞ sin θ=−√τ
√1 + τ.(6.11)
We remind that similar procedure has been used for the cross check of the LF polarization
and magnetization amplitudes in Ref. [149].
Explicit demonstrations of LF amplitudes via the proper IMF limit of corresponding
EF amplitudes in electromagnetic four-current [148,151,152], polarization-magnetization
tensor [149], and axial-vector four-current (6.9-6.11) cases inspire us to propose the following
conjecture:
Conjecture. Any light-front (LF) amplitudes for well-defined LF distributions in prin-
ciple can be explicitly reproduced from the corresponding elastic frame (EF) amplitudes
in the proper infinite-momentum frame (IMF) limit.
As a reward, the origins of distortions appearing in LF distributions (relative to the
BF ones) can be understood more easily and intuitively using the covariant Lorentz trans-
formation and the above conjecture, with the help of the Melosh rotation (6.2). We should
note that to obtain LF amplitudes one does not necessarily need to first perform the covari-
ant Lorentz transformation and then reproduce the LF amplitudes using above conjecture,
9We note that there is a typo in the Eq. (41) of Ref. [148] that “limPz→∞ tan θ=−1/√τ” should be
corrected as “limPz→∞ tan θ=−√τ”, see Eq. (6.11).
– 19 –
since one can always straightforwardly obtain the LF amplitudes by directly evaluating LF
helicity Dirac bilinears. However, it is usually not so easy and intuitive to understand the
distortions appearing in LF distributions via the direct evaluation of LF helicity Dirac bi-
linears. Our procedure provides a legitimate but more easy and intuitive way to understand
the origins of distortions appearing in LF distribution (relative to the BF ones).
Hence, we can classify more easily the origins of distortions appearing in LF distribu-
tions (relative to the BF ones) into three key sources. The first key source of distortions
in LF distributions is due to the peculiar LF perspective for defining the O+and O−
components [148,149]. In the cases of electromagnetic and axial-vector four-current am-
plitudes, the expressions of LF amplitudes e
O±
LF involve γ(1 ±β)e
O±
Bunder a longitudinal
Lorentz boost, where e
O±
B≡(e
O0
B±e
O3
B)/√2stand for the constructed BF amplitudes. The
γ(1 ±β)factors will result in the overall P±factors in e
O±
LF, respectively. According to the
generic definition of LF distributions Oµ
LF(b⊥;P+)[148–150], the P+factor in e
O+
LF will be
canceled out by the same P+factor in the definition of Oµ
LF(b⊥;P+)arising due to the
normalization of LF four-momentum eigenstates. Therefore, LF distributions O+
LF(b⊥;P+)
are usually P+-independent, allowing therefore physically clear probabilistic interpreta-
tion [120,127,177]. However, the P−factor in e
O−
LF can not be canceled out and results in
an Q-dependent factor P−/P +=M2(1 + τ)/[2(P+)2]within the Fourier transform, which
together with e
O−
Busually causes strange distortions and distributions. As a result, LF dis-
tributions O−
LF(b⊥;P+)are usually regarded as the “bad” components and are considered
as complicated objects without clear physical interpretation [148]. For example, physically
clear F1(Q2)and F2(Q2)in J+
LF become physically unclear G1(Q2)and G2(Q2)in J−
LF in
the electromagnetic case [148], and the sum of the two terms associated respectively with
GZ
T(Q2)and GZ
A(Q2)in J+
5,LF (6.7) becomes the difference between these two terms in J−
5,LF
(6.7).
The second key source of distortions originates from complicated Wigner rotations and
the Lorentz mixing of temporal and longitudinal components under longitudinal Lorentz
boosts according to the covariant Lorentz transformation for amplitudes from BF to the
IMF [i.e. e
Oµ
B→e
Oµ
IMF], see e.g. Eq. (5.3). This is the only source of physical distortions
for EF and IMF distributions. Based on the conjecture above, one needs to take the proper
IMF limit of EF amplitudes, which in general causes kinematical suppression (or Lorentz
contraction) for terms in the EF amplitudes involving P0in the denominator while there
is no Pz-dependent factor in the numerator, e.g. the term involving GZ
A(Q2)/(P0+M)in
A⊥
EF (5.2) is completely suppressed at Pz→ ∞.
The third key source of distortions originates from relativistic artefacts caused by
Melosh rotations (6.2). The change of polarization basis modifies the normalization of
four-moment eigenstates, which usually brings the explicit P+-dependence for the trans-
verse components of LF distributions, see e.g. J⊥
5,LF(b⊥;P+)in Eq. (6.7) and J⊥
LF(b⊥;P+)
in Ref. [148].
– 20 –
6.3 LF mean-square transverse radii
Following Refs. [150,178], we also rederive the mean-square transverse axial (or helicity)
and spin radii using the LF distributions (6.7). We find that
⟨b2
A⟩LF(P+) = ⟨b2
spin,L⟩LF (P+) = 2
3R2
A,
⟨b2
spin,T ⟩LF(P+) = 2
3R2
A+1
2M2
GZ
P(0)
GZ
A(0),
(6.12)
which do coincide with Ref. [150] although we have taken into account GZ
T(Q2), see Eq. (6.7).
From above results (6.12), we conclude that the second-class current contribution associated
with the induced pseudotensor FF GZ
T(Q2), although explicitly included in our calculations,
does not contribute in fact to the mean-square transverse axial and spin radii on the LF.
7 Summary
In this paper, we extended our recent work [150] of the relativistic weak-neutral axial-
vector four-current distributions inside a general spin-1
2composite system, where the second-
class current contribution associated with the weak-neutral induced pseudotensor form
factor is newly included. To the best of our knowledge, this is the first time that the
full weak-neutral axial-vector four-current distributions inside a general spin-1
2hadron are
systematically studied in terms of relativistic Breit frame, elastic frame and light-front
distributions.
When the system is on the average at rest, we clearly demonstrated that the 3D weak-
neutral axial charge distribution J0
5,B in the Breit frame is in fact related to the induced
pseudotensor form factor GZ
T(Q2)rather than the axial form factor GZ
A(Q2). Besides, we
found that J0
5,B is of parity-odd nature (4.3), due to which the definition of the standard
mean-square axial (charge) radius (4.5) is in fact not well-defined. We also tested for the
first time the Abel and its inverse transforms of axial charge distributions in the spin-1
2
case, through which we explicitly revealed the breakdown of Abel transformation for the
connection in physics between 2D light-front and 3D Breit frame axial charge distributions,
see Appendix B. On the other hand, we found that the 3D axial-vector current distribu-
tion J5,B is free from the second-class current contribution, and is closely related to the
physically meaningful 3D (intrinsic) spin distribution [141,143,147,150].
When the system is boosted, the situation gets more complicated since both the Wigner
rotation and the Lorentz mixing effect will play the roles. We did observe clear frame-
dependence of the axial charge distribution J0
5,EF (longitudinal axial-vector current dis-
tribution Jz
5,EF) in the generic elastic frame for a longitudinally (transversely) polarized
target. The frame-dependence of J0
5,EF and Jz
5,EF is solely due to the Lorentz mixing of
temporal and longitudinal components of the axial-vector four-current amplitudes under
longitudinal Lorentz boosts (5.3), since they are both free from the Wigner rotation. On
the contrary, the transverse axial-vector current distribution J⊥
5,EF (5.6) does not get mixed
under longitudinal Lorentz boosts, but it suffers from the Wigner rotation (5.5).
– 21 –
We also studied the full weak-neutral axial-vector four-current distributions using the
light-front formalism. We revisited the role played by Melosh rotations, and further pro-
posed the conjecture based on our recent works [148–152] that any light-front amplitudes
for well-defined light-front distributions in principle can be explicitly reproduced from the
corresponding elastic frame amplitudes in the proper infinite-momentum frame limit. As a
reward, we can understand more easily and intuitively the origins of distortions appearing
in light-front distributions (relative to the Breit frame ones). For completeness, we also
rederived the mean-square transverse axial and spin radii in different frames. We showed
in particular that the second-class current contribution, although explicitly included in our
calculations, does not contribute in fact to the mean-square axial and spin radii.
To get a more intuitive picture of the weak-neutral axial-vector structure of a spin-1
2
hadron, we also numerically illustrated our results in the case of a proton, using proton’s
weak-neutral axial-vector form factors extracted from experimental data, see Appendix A.
It should be emphasize that our analytic formulas apply to any spin-1
2hadrons (e.g. Λ0,Σ0,
Ξ0, etc.) and can be easily generalized to higher spin systems, as long as the corresponding
weak-neutral axial-vector form factors are available.
A Parametrization of nucleon weak-neutral axial-vector FFs
In the literature [9,10,15,18,95], the nucleon weak-neutral GZ
A(Q2)and weak-charged
GW
A(Q2)axial FFs are usually parametrized in terms of a standard dipole model ansätz :
GL
A(Q2) = GL
A(0)
1 + Q2
(ML
A)22,(A.1)
where L=Z, W and ML
Ais the corresponding (axial) dipole mass. We note that the weak-
charged nucleon axial FF GW
A(Q2)has been extracted from quasielastic (anti)neutrino-
nucleon and (anti)neutrino-nuclei scattering data with MW
A≈(1.026 ±0.021) GeV [95].
The weak-charged axial charge (or coupling constant) GW
A(0) = (1.2754 ±0.0013) [179] is
very precisely determined in neutron beta decay reaction n→pe−¯νe.
According to Refs. [9,10,180–184], the weak-neutral axial-vector FFs GZ
X(Q2)for
X=A, P, T can be related to the corresponding weak-charged ones GW
X(Q2)via
GZ
X(Q2) = X
f
gf
AGf
X(Q2)
≃1
2hGW
X(Q2)−Gs
X(Q2) + Gc
X(Q2)−Gb
X(Q2) + Gt
X(Q2)i,
(A.2)
where GW
X≃Gu
X−Gd
X≡G(u−d)
X, and Gf
Xdenotes the FF contribution from the f-flavor
quark with f=u, d, s, c, b, t. We note that the axial-vector couplings of quarks to the Z0
boson in the Standard Model are given by gu,c,t
A=1
2and gd,s,b
A=−1
2, which explain the
overall factor 1
2in Eq. (A.2). It should be noted that GZ
A(t=−Q2)and GZ
P(t=−Q2)can
also be accessed via measurements of generalized parton distributions (GPDs) ˜
Hf(x, ξ, t)
– 22 –
and ˜
Ef(x, ξ, t)at JLab (Hall A/B/C), LHC, HERA (HERMES, H1, ZEUS), NICA, EIC,
EicC, etc. through the first-moment sum rules [185–187]
"GZ
A(t)
GZ
P(t)#=X
f
gf
AZ1
−1
dx"˜
Hf(x, ξ, t)
˜
Ef(x, ξ, t)#.(A.3)
In general, the contributions from the heavy-flavor quarks (i.e. c,b,t) are very small for
the nucleon and are therefore usually neglected in practical calculations. In this analysis,
we also neglect the contributions from heavy-flavor quarks for the nucleon.
Using directly the extracted experimental data of the proton weak-neutral axial FF
GZ
A(Q2)from Ref. [184] based on recent MiniBooNE measurements [15,18] and performing
the standard dipole model (A.1) fit to the data (denoted as “Dipole fit”), we find
MZ
A≈(1.0500 ±0.0107) GeV,(A.4)
where GZ
A(0) = (0.65520 ±0.00465) is fixed using both world average experimental data of
GW
A(0) = (1.2754 ±0.0013) from Particle Data Table [179] and the strange quark contri-
bution to the nucleon spin ∆s≡Gs
A(0) ≈(−0.0350 ±0.0092) in the continuum limit and
physical pion point from lattice QCD [188]; see also Refs. [21,22,183] for recent analyses
of Gs
A(0).
Alternative to Eq. (A.1) with MZ
Agiven in Eq. (A.4), we can also obtain GZ
A(Q2)in
terms of GW
A(Q2)and Gs
A(Q2)via Eq. (A.2)10. For the strange quark contributions Gs
A(Q2)
and Gs
P(Q2),Gs
X(Q2)for X=A, P are also usually parametrized in the literature in terms
of the standard dipole model [46]
Gs
X(Q2) = Gs
X(0)
1 + Q2
(Ms
X)22,(A.5)
where the corresponding dipole parameters Gs
A(0) = −0.044(8),Ms
A= 0.992(164) GeV,
Gs
P(0) = −1.325(406) and Ms
P= 0.609(89) GeV can be found in Ref. [46].
In the upper panel of Fig. 9, we show the direct “Dipole fit” (red dot-dashed curve) result
of GZ
A(Q2)using the extracted MiniBooNE data [15,18,184] (circle markers) via Eq. (A.1)
and the reconstructed GZ
A(Q2) = [GW
A(Q2)−Gs
A(Q2)]/2labeled by “Quasi-νN ” (blue solid
curve) using Eq. (A.2) in terms of GW
A(A.1) extracted from quasielastic (anti)neutrino
scattering data and Gs
A(A.5), in comparison with GZ
A(Q2)from the BNL E734 measure-
ments [9,10] (orange dotted curve) and the recent ETM lattice QCD calculations [44,46]
(green dot-dot-dashed curve), where confidence bands at 95% confidence level are also
shown. Within error bands, all these experimental-data-based results (MiniBooNE, Dipole
fit, BNL E734, and Quasi-νN ) are well consistent with each other, which in turn validates
the correctness of Eq. (A.2). In contrast, the ETM lattice result of GZ
A(Q2)[44,46] shows
sizeable deviation from the other (i.e. MiniBooNE, Dipole fit, BNL E734 and Quasi-νN)
10We note that the direct (anti)neutrino-nucleon elastic scattering data [15,18,184] of GZ
A(Q2)in turn
can help us to test whether the relation (A.2) between GZ
A(Q2)and GW
A(Q2)is valid or not.
– 23 –
MiniBooNE
Dipole fit
BNL E734
Quasi-νN
ETM lattice
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
Q2[GeV2]
GA
Z(Q2)
μ-capture (1981)
μ-capture (2013)
Saclay exp.
PPD +Dipole fit
PPD +Quasi-νN
NLO χPT
ETM lattice
0.0 0.1 0.2 0.3 0.4
0
30
60
90
120
150
Q2[GeV2]
GP
Z(Q2)
Figure 9. Comparison of the nucleon weak-neutral axial GZ
A(Q2)(upper panel) and induced pseu-
doscalar GZ
P(Q2)(lower panel) FFs as a function of Q2using different methods, where confidence
bands of GZ
A(Q2)at 95% confidence level are also shown. See texts for more details.
results.
To the best of our knowledge, there is currently no direct experimental data of the nu-
cleon weak-neutral induced pseudoscalar FF GZ
P(Q2)in the literature. To obtain GZ
P(Q2),
we need the knowledge of GW
P(Q2)and Gs
P(Q2)according to Eq. (A.2). The result of
Gs
P(Q2)is given in Eq. (A.5). The remaining task is to obtain GW
P(Q2). In chiral perturba-
tion theory (χPT), the nucleon weak-charged induced pseudoscalar FF GW
P(Q2)from full
chiral structure up to next-to-leading-order (NLO) is given by [60,95]
GW
P(Q2) = gπ±pn
2(Mp+Mn)Fπ
Q2+M2
π−2GW
A(0)(Mp+Mn)2
(MW
A)2+O(Q2;M2
π),(A.6)
where gπ±pn is the pseudoscalar pion-nucleon coupling constant, Mp(Mn) is the proton
(neutron) mass, Mπis the charged pion mass, Fπ=fπ/√2≈(92.0653 ±0.8485) MeV [179]
is the pion decay constant for the π+→µ+νµreaction, and MW
A≈(1.026±0.021) GeV [95].
Based on the recent combined analysis of experimental data using chiral effective field theory
– 24 –
for f2
π±pn = 0.0769(5)a(0.9)b[189], we find [179]
gπ±pn =√4π(Mp+Mn)
Mπ
fπ±pn ≈(13.22613 ±0.04369),(A.7)
where uncertainties of f2
π±pn from the first (a) and second (b) errors are added in quadrature.
Alternatively, we can also obtain GW
P(Q2)by assuming the pion-pole dominance (PPD)
hypothesis [41,83], which is based on the low-energy QCD relations—the partially conserved
axial-vector current (PCAC) relation and the Goldberger-Treiman relation [190], namely
GW
P(Q2) = (Mp+Mn)2
M2
π+Q2GW
A(Q2),(A.8)
where GW
A(Q2)is extracted from quasielastic (anti)neutrino scattering data [95] via Eq. (A.1).
We can thus construct GZ
P(Q2)=[GW
P(Q2)−Gs
P(Q2)]/2with known Gs
P(Q2)(A.5) by us-
ing GW
P(Q2)either from Eq. (A.6) which is labeled by “NLO χPT”, or from Eq. (A.8) which
is labeled by “PPD + Quasi-νN ”. The PPD hypothesis also inspires us to reconstruct
GZ
P(Q2)by using directly the “Dipole fit” GZ
A(Q2)from Eqs. (A.1) and (A.4), namely
GZ
P(Q2) = 4M2
M2
π+Q2GZ
A(Q2),(A.9)
which is labeled by “PPD + Dipole fit”.
In the lower panel of Fig. 9, we present our results of GZ
P(Q2)by using different methods:
“PPD + Dipole fit” (red dot-dashed curve), “PPD + Quasi-ν N ” (blue solid curve), and
“NLO χPT”, in comparison with the reconstructed GZ
P(Q2) = [GW
P(Q2)−Gs
P(Q2)]/2by
using Gs
P(Q2)(A.5) and the experimental data of GW
P(Q2)from the ordinary µ-capture
measurements [191,192] at11 Q2≈0.88 M2
µin the µ−+p→n+νµreaction labeled by “µ-
capture (1981)” [191] (square marker) and “µ-capture (2013)” [192] (triangle marker), and
from the low-energy charged pion electroproduction measurements [193] labeled by “Saclay
exp.” (circle markers). Besides, we also show the results of GZ
P(Q2)=[G(u−d)
P−Gs
P]/2
from the recent ETM lattice QCD calculations [44,46] labeled by “ETM lattice” (green
dot-dot-dashed curve). We find that all these results of GZ
Pare well consistent with each
other, which also indicates the validity of Eq. (A.2). Owing to the intensive overlaps of
these results, the confidence bands for GZ
P(Q2)at 95% confidence level are not shown in
Fig. 9.
For the nucleon weak-neutral induced pseudotensor FF GZ
T, there is currently no di-
rect experimental data at all. To some extent, this also explains why the second-class
current contribution of the nucleon associated with the induced pseudotensor FF GZ
T(Q2)
are scarcely discussed and calculated in the literature [58–60,63–85,88]. According to
the Fig. 7 of Ref. [169], one can assume that GW
T(Q2)can by roughly approximated by
GW
T(Q2)≡κTGW
A(Q2), where the factor κT≈0.1is roughly the mean value of the ratio
11More rigorously, Q2=h(Mµ+Mp)2−M2
n
Mµ(Mµ+Mp)−1iM2
µ=hMp(Mµ+Mp)−M2
n
Mµ(Mµ+Mp)iM2
µ≈0.88 M2
µ, where Mµis the
muon mass.
– 25 –
GW
T(0)/GW
A(0) in the Fig. 7 of Ref. [169] with GW
A=FAand GW
T= 2F3
A, and GW
A(Q2)is
given in Eq. (A.1) with MW
A≈(1.026 ±0.021) GeV [95].
As a result, Ref. [169] thus inspires us to propose the following ansätz for the nucleon
weak-neutral induced pseudotensor FF GZ
T(Q2):
GZ
T(Q2) = κTGZ
A(Q2),(A.10)
where GZ
A(Q2)is the direct dipole model (A.1) fit to the elastic (anti)neutrino-nucleon scat-
tering data [15,18,184]. We should emphasize that the reason why we relate GZ
T(Q2)to
GZ
A(Q2)via the ansätz (A.10) is simply because the assumption proposed in Ref. [169],
which is the only reference that we have ever found with experimentally reasonable and
useful relation for GW
T(Q2)[and thus for GZ
T(Q2)]. In the real word, GZ
T(Q2)is most
probably to be quite different from GZ
A(Q2)instead of the simple but naive scaling ansätz
(A.10), since GZ
T(Q2)is strongly constrained by many symmetries and conservation laws
while GZ
A(Q2)is not. This also provides a key motivation for future experimental mea-
surements of GZ
T(Q2), e.g. using the formulas of full tree-level unpolarized differential cross
sections [174], for the proton in (anti)neutrino-proton elastic scattering.
For numerical calculations and illustrations of the proton weak-neutral axial-vector
four-current distributions in Sec. 4to Sec. 6, we declare that we employ only nucleon
weak-neutral axial-vector FFs GZ
A(Q2)from Eq. (A.1) using the “Dipole fit”, GZ
P(Q2)from
Eq. (A.9) using the “Dipole fit” GZ
A(Q2)and the PPD hypothesis, and GZ
T(Q2)from
Eq. (A.10) using the “Dipole fit” GZ
A(Q2)and κT≈0.1.
B Breakdown of Abel transformation for axial charge distributions
The Abel and its inverse transforms, named after Niels H. Abel for integral transforms
in mathematics, have been visited recently [153,154,194–198] in the case of charge and
energy-momentum tensor spatial distributions with the goal of connecting 2D LF spatial
distributions with the corresponding 3D ones, where 2D LF spatial distributions are re-
garded as the 2D Abel images of the corresponds 3D spatial distributions. We notice that
some discussions and debates have been triggered in Refs. [195,199]. In this appendix, we
will explicitly show the breakdown of Abel and its inverse transforms for the connection in
physics between 2D LF and 3D BF axial charge distributions in the spin-1
2case.
According to textbooks, the standard definitions of the Abel and its inverse transforms
are given by [200]
A[g] (b)≡ G(b)=2Z∞
b
drr
√r2−b2g(r),
g(r) = −1
πZ∞
r
db1
√b2−r2
dG(b)
db,
(B.1)
where A[g] (b)≡ G(b)is called the 2D Abel image of the 3D spatial function g(r)[199]. It
is thus not difficult to obtain the following generic relation for the nth order Mellin moment
– 26 –
of the Abel image G(b)in connecting with the corresponding 3D spatial function g(r):
Z∞
0
db bn−1G(b) = √πΓn
2
Γn+1
2Z∞
0
dr rng(r), n ∈N+={1,2,3,···},(B.2)
which is believed to be valid as long as g(r)decreases faster than any order of rn[153].
Using the dipole model ansätz (A.1) for the axial FF GZ
A(Q2), one can easily obtain
the following analytic expression of the LF axial charge distribution from Eq. (6.7) for a
longitudinally polarized spin- 1
2target (with b=|b⊥|):
J+
5,LF(b⊥;P+)=(σz)λ′λGZ
A(0)b(MZ
A)3
4πK1(bMZ
A)≡J+
5,LF(b),(B.3)
which is axially symmetric and can be regarded as the 2D Abel image of a 3D spatial
distribution J0
5,naive(r). Applying the inverse Abel transform (B.1) to J+
5,LF(b), we find
J0
5,naive(r) = (σz)λ′λGZ
A(0)(MZ
A)3
8πe−rM Z
A.(B.4)
One can first check that the axial charge normalization condition
(σz)λ′λGZ
A(0) = Zd3r J0
5,naive(r) = Zd2b⊥J+
5,LF(b)(B.5)
seems to be automatically guaranteed. Besides, one can also check that J+
5,LF(b)and
J0
5,naive(r)indeed satisfy the generic relation (B.2) for the corresponding Mellin moment
at any order. In particular, we find that the mean-square radius of J0
5,naive(r)is given by
⟨r2
A⟩Abel
naive =Rd3r r2J0
5,naive(r)
Rd3r J0
5,naive(r)=12
(MZ
A)2≈(0.6510 fm)2
=−6
GZ
A(0)
dGZ
A(Q2)
dQ2Q2=0
=R2
A,
(B.6)
which exactly coincides with R2
A(4.6) widely employed in the literature [19,20,58,59,106,
170–173]. In contrast to J0
5,naive (B.4), the genuine 3D axial charge distribution (4.3) in the
BF for a longitudinally polarized spin- 1
2target is in fact given by
J0
5,B(r)=(σz)s′sZd3∆
(2π)3e−i∆·ri∆z
2MGZ
T(∆2),(B.7)
which is actually related to the induced pseudotensor FF GZ
T(Q2)rather than the axial
FF GZ
A(Q2). This explicitly demonstrates that even though we neglect the polarization
difference, the naive 3D distribution J0
5,naive(r)does not assume clear physical meaning for
quantifying the genuine 3D spatial distribution of weak-neutral axial charges in the BF
for a longitudinally polarized spin- 1
2target. This explicitly reveals for the first time the
breakdown of Abel and its inverse transforms for the connection in physics between 2D
– 27 –
LF and 3D BF axial charge distributions in the spin-1
2case. Our demonstration of the
breakdown of Abel transformation is well consistent with the fact that the LF Galilean
subgroup (singled out from the Lorentz group) does not have an SO(3) subgroup, and thus
LF dynamics does not preserve the 3D spherical symmetry [117,199].
Acknowledgments
We warmly thank Dr. Raza Sabbir Sufian for very helpful communications, and Profs.
Dao-Neng Gao, Ren-You Zhang and Guang-Peng Zhang for valuable discussions at an early
stage of this work. We are very grateful to Profs. Cédric Lorcé, Qun Wang and Yang Li for so
many valuable encouragements, discussions and suggestions during our collaboration [150].
We thank Profs. Chueng-Ryong Ji, Carlos Muñoz Camacho, Weizhi Xiong, Qin-Tao Song,
Tobias Frederico, João P. de Melo, Stanislaw D. Głazek, Craig D. Roberts, Wayne N. Poly-
zou, Ismail Zahed, Sanjin Benic, Feng-Kun Guo, Shan Cheng, Yu Jia and Drs. Ho-Yeon
Won, Poonam Choudhary, Sudeep Saha, Jani Penttala for the helpful communications
during the “Light-Cone 2024: Hadron Physics in the EIC era” conference. This work is sup-
ported in part by the National Natural Science Foundation of China (NSFC) under Grant
Nos. 12135011, 11890713 (a sub-Grant of 11890710), and by the Strategic Priority Research
Program of the Chinese Academy of Sciences (CAS) under Grant No. XDB34030102.
Note added. Recently, we were informed of Ref. [201] which works on a similar topic.
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