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我们提出了一种新的拟标量介子环校正的新公式,用于在非局部协变手性有效场理论内的核子parton分布,包括SU(3)八位位组和十重子重子的贡献。 受局部量规不变性和Lorentz不变紫外线正则化要求的约束,非局部拉格朗日数产生与量规链接相关的其他相互作用。 我们使用它们来计算完整的质子→介子+重子分裂函数集,除了零动量的δ函数项外,它还包含壳内和壳外贡献以及与有限大小相关的非局部贡献 质子 我们使用偶极子调节器的简单示例以数字方式说明各种局部和非局部函数的形状。
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Parton distributions from nonlocal chiral SU(3) effective theory:
Splitting functions
Y. Salamu,
1,2
Chueng-Ryong Ji,
3
W. Melnitchouk,
4
A. W. Thomas,
5
and P. Wang
1,6
1
Institute of High Energy Physics, CAS, Beijing 100049, China
2
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
3
North Carolina State University, Raleigh, North Carolina 27695, USA
4
Jefferson Lab, Newport News, Virginia 23606, USA
5
CoEPP and CSSM, Department of Physics, University of Adelaide, Adelaide SA 5005, Australia
6
Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China
(Received 20 June 2018; published 29 January 2019; corrected 24 July 2019)
We present a new formulation of pseudoscalar meson loop corrections to nucleon parton distributions
within a nonlocal covariant chiral effective field theory, including contributions from SU(3) octet and
decuplet baryons. The nonlocal Lagrangian, constrained by requirements of local gauge invariance and
Lorentz-invariant ultraviolet regularization, generates additional interactions associated with gauge links.
We use these to compute the full set of proton→ meson þ baryon splitting functions, which in general
contain on-shell and off-shell contributions, in addition to δ-function terms at zero momentum, along with
nonlocal contributions associated with the finite size of the proton. We illustrate the shapes of the various
local and nonlocal functions numerically using a simple example of a dipole regulator.
DOI: 10.1103/PhysRevD.99.014041
I. INTRODUCTION
The important role played by chiral symmetry in
hadron physics has been documented for man y d ecades.
Traditionally the purview of low-energy hadron and nuclear
physics, more recently the relevance of chiral symmetry in
QCD has become more prominent also in high-energy
reactions, in which the quark and gluon (or parton) sub-
structure of hadrons is manifest. One of the most striking
expressions of the chiral symmetry and its approximate
breaking is in the nonperturbative structure of the sea quark
distributions of the nucleon [1,2]. In particular, the breaking
of chiral SU(3) symmetry was anticipated [3] to generate
unequal strange and (light) nonstrange sea quark distribu-
tions, and, ev en more dramatically, an excess of
d antiquarks
over
u. The latter was confirmed in proton-proton and
proton-deuteron Drell-Yan experiments at CERN [4] and
Fermilab [5], following earlier indirect indications from
inclusive [6] and semi-inclusiv e [7] deep-inelastic scattering
(DIS) data on proton and deuteron targets.
The observation of a large
d − u asymmetry has also
served to motivate more challenging searches for other
nonperturbative asymmetries, such as those between strange
and antistrange quarks in the proton, s −
s [8–10],or
between the helicity dependent light antiquark distributions,
Δ
d − Δu [11]. The phenomenological success in describing
the
d − u asymmetry, in particular, in terms of nonpertur-
bative models of the nucleon in which its peripheral structure
is modeled by a pseudoscalar meson cloud suggested that
signatures of chiral symmetry breaking may also be found
in other types of parton distribution functions (PDFs)
[8,12–18].
While considerable experience has been accumulated
with nonperturbative models, a challenge has been to
compute the chiral symmetry breaking effects on the
PDFs in a model-independent way from QCD. An impor-
tant step in establishing a direct connection with QCD was
made with the observation [19] that the leading nonanalytic
(LNA) behavior of moments of the nonsinglet PDFs,
expanded in powers of the pion mass, m
π
, could be
obtained from chiral effective field theory, which encodes
the same chiral symmetry properties as present in QCD
[20–22]. In addition to demonstrating how lattice QCD
data on PDF moments and other observables simulated at
unphysically large pion masses could be extrapolated to the
physical point [23], the result [19] demonstrated unambig-
uously that a nonzero component of
d − u arises as a direct
consequence of the infrared structure of QCD.
Subsequent work [24–29] computed the full set of lowest
order corrections to PDFs arising from pseudoscalar meson
loops, both for the PDF moments and the Bjorken-x
dependence. The LNA behavior of the various contribu-
tions can be established model-independently by consid-
ering the infrared limit; however, the computation of the
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published articl e’s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D 99, 014041 (2019)
2470-0010=2019=99(1)=014041(25) 014041-1 Published by the American Physical Society
full amplitude requires specific choices for regularizing the
divergences in the loop integrals. In the literature, regu-
larization prescriptions such as transverse momentum cut-
offs, Pauli-Villars, dimensional regularization or infrared
regularization have been used, as well as form factors or
finite-range regulators. The latter take into account the
finite size of hadrons [30,31], while the others are generally
more suitable for theories that treat hadrons as pointlike.
In practice, the extended structure of the nucleon and
other baryons does become important in many traditional
hadronic physics applications. In nonrelativistic calcula-
tions, if the regulators are in three-dimensional momentum
space, such as for finite-range regularization, charge con-
servation, which is related to the time component of the
current, is respected in the presence of form factors. In
relativistic calculations, on the other hand, simply replacing
the nonrelativistic regulator by a covariant one can lead to
violation of local gauge symmetry and charge conservation.
The problem of preserving gauge invariance in theories
with hadronic form factors can be formally alleviated by
introducing nonlocal interactions into the gauge invariant
local Lagrangian, which allows one to consistently generate
a covariant regulator. A method for constructing nonlocal
Lagrangians with gauge fields was described by Terning
[32], based on the path-ordered exponential introduced by
Wilson [33] and earlier by Bloch [34].Variantsofthe
method were subsequently used in phenomenological appli-
cations to strange vector form factors and other nucleon
matrix elements by a number of authors [35–37]. The pion
and σ meson properties have been studied by gauging
nonlocal meson–quark interactions in relativistic quark
models [38,39]. The nonlocal Lagrangian at the hadron
lev el was also recently constructed and applied to electro-
magnetic form factors of the nucleon [40–42].
The presence of gauge links in the nonlocal Lagrangian
connecting different spacetime coordinates generates addi-
tional diagrams which are needed to ensure the local gauge
invariance of the theory. This guarantees that the proton
and neutron charges, for example, are unaffected by meson
loops, or that contributions to the strangeness in the
nucleon from diagrams with intermediate state kaons
and hyperons sum to zero. These basic features of the
theory are not guaranteed for a local Lagrangian with a
covariant regulator, but arise automatically in the nonlocal
theory in which the Ward identities and charge conserva-
tion are necessarily satisfied. In fact, a nonlocal formulation
may be preferable on physical grounds, as this more
naturally represents the extended structure of hadrons.
In this paper we describe how the nonlocal formulation
of the chiral SU(3) effective theory can be used to derive the
contributions from pseudoscalar meson loops to PDFs in
the nucleon. We include both the SU(3) octet and decuplet
baryons, using a covariant regulator generated through the
nonlocal Lagrangian that respects Lorentz and gauge
symmetry. In the present paper we focus on the formalism
and the derivation of the proton → baryon þ meson split-
ting functions from the nonlocal chiral Lagrangian; a
follow-up paper [43] will report on the results for the
nucleon PDFs, computed through convolutions of the
splitting functions and PDFs in the virtual mesons and
baryons in the loops.
We begin by reviewing in Sec. II the familiar local
effective Lagrangian in the standard chiral SU(3) effective
field theory. The generalization of the effective Lagrangian
to the nonlocal case is described in Sec. III, a procedure
which allows the preservation of gauge invariance in the
presence of covariant vertex functions for the nucleon–
baryon–meson interaction. The main results for the
proton → meson þ baryon splitting functions are derived
in Sec. IV for the full set of lowest order diagrams,
including rainbow, bubble, tadpole and Kroll-Ruderman
contributions, as well as additional terms that arise from the
gauge links generated from the nonlocal interactions. Here
we present the model independent results for the nonana-
lytic behavior of the moments of the splitting functions,
and illustrate the relative shapes and magnitudes of the
various functions using a simple example of a covariant
dipole vertex form factor. Finally, in Sec. V we summarize
our results and outline future applications of the new
formalism.
II. LOCAL CHIRAL EFFECTIVE LAGRANGIAN
In this section we review the standard local chiral
effective theory for mesons and baryons. The lowest-
order Lagrangian, consistent with chiral SUð3Þ
L
×
SUð3Þ
R
symmetry, describing the interaction of pseudo-
scalar mesons (ϕ) with octet (B) and decuplet (T
μ
) baryons,
is given by [44,45]
L ¼ Tr½
BðiD − M
B
ÞB −
D
2
Tr½
Bγ
μ
γ
5
fu
μ
;Bg
−
F
2
Tr ½
Bγ
μ
γ
5
½u
μ
;B þ T
ijk
μ
ðiγ
μνα
D
α
− M
T
γ
μν
ÞT
ijk
ν
−
C
2
½ϵ
ijk
T
ilm
μ
Θ
μν
ðu
ν
Þ
lj
B
mk
þ H:c:
−
H
2
T
ijk
μ
γ
α
γ
5
ðu
α
Þ
kl
T
ijl
μ
þ
f
2
4
Tr ½D
μ
UðD
μ
UÞ
†
; ð1Þ
where M
B
and M
T
are the octet and decuplet masses, D and
F are the meson–octet baryon coupling constants, C and H
are the meson–octet–decuplet and meson–decuplet–
decuplet baryon couplings, respectively, f ¼ 93 MeV is
the pseudoscalar decay constant, and “H.c.” denotes the
Hermitian conjugate. The tensor ϵ
ijk
is the antisymmetric
tensor in flavor space, and we define the tensors γ
μν
¼
1
2
½γ
μ
; γ
ν
and γ
μνα
¼
1
2
fγ
μν
; γ
α
g in terms of the Dirac γ-
matrices. The octet–decuplet transition tensor operator Θ
μν
is defined as
SALAMU, JI, MELNITCHOUK, THOMAS, and WANG PHYS. REV. D 99, 014041 (2019)
014041-2
Θ
μν
¼ g
μν
−
Z þ
1
2
γ
μ
γ
ν
; ð2Þ
where Z is the decuplet off-shell parameter. The SU(3)
baryon octet fields B
ij
include the nucleon N (¼ p, n), Λ,
Σ
;0
and Ξ
−;0
fields, and are given by the matrix
B ¼
0
B
B
@
1
ffiffi
2
p
Σ
0
þ
1
ffiffi
6
p
ΛΣ
þ
p
Σ
−
−
1
ffiffi
2
p
Σ
0
þ
1
ffiffi
6
p
Λ n
Ξ
−
Ξ
0
−
2
ffiffi
6
p
Λ
1
C
C
A
: ð3Þ
The baryon decuplet fields T
ijk
μ
, which include the Δ, Σ
,
Ξ
and Ω
−
fields, are represented by symmetric tensors
with components
T
111
¼Δ
þþ
;T
112
¼
1
ffiffiffi
3
p
Δ
þ
;T
122
¼
1
ffiffiffi
3
p
Δ
0
;T
222
¼Δ
−
;
T
113
¼
1
ffiffiffi
3
p
Σ
þ
;T
123
¼
1
ffiffiffi
6
p
Σ
0
;T
223
¼
1
ffiffiffi
3
p
Σ
−
;
T
133
¼
1
ffiffiffi
3
p
Ξ
0
;T
233
¼
1
ffiffiffi
3
p
Ξ
−
;
T
333
¼Ω
−
: ð4Þ
In the meson sector, the operator U in Eq. (1) is defined in
terms of the matrix of pseudoscalar fields ϕ,
U ¼ u
2
; with u ¼ exp
i
ϕ
ffiffiffi
2
p
f
; ð5Þ
where ϕ includes the π, K and η mesons,
ϕ ¼
0
B
B
@
1
ffiffi
2
p
π
0
þ
1
ffiffi
6
p
ηπ
þ
K
þ
π
−
−
1
ffiffi
2
p
π
0
þ
1
ffiffi
6
p
η K
0
K
−
K
0
−
2
ffiffi
6
p
η
1
C
C
A
: ð6Þ
The pseudoscalar mesons couple to the baryon fields
through the vector and axial vector combinations
Γ
μ
¼
1
2
ðu
†
∂
μ
u þu∂
μ
u
†
Þ −
i
2
ðu
†
λ
a
u þuλ
a
u
†
Þυ
a
μ
; ð7Þ
u
μ
¼ iðu
†
∂
μ
u − u∂
μ
u
†
Þþðu
†
λ
a
u − uλ
a
u
†
Þυ
a
μ
; ð8Þ
where υ
a
μ
corresponds to an external vector field, and λ
a
(a ¼ 1; …; 8) are the Gell-Mann matrices. The covariant
derivatives of the octet and decuplet baryon fields in the
chiral Lagrangian (1) are defined as [46,47]
D
μ
B ¼ ∂
μ
B þ½Γ
μ
;B − ihλ
0
iυ
0
μ
B; ð9Þ
D
μ
T
ijk
ν
¼ ∂
μ
T
ijk
ν
þðΓ
μ
;T
ν
Þ
ijk
− ihλ
0
iυ
0
μ
T
ijk
ν
; ð10Þ
where υ
0
μ
denotes an external singlet vector field, λ
0
is
the unit matrix, and hi denotes a trace in flavor space.
For the covariant derivative of the decuplet field, we use the
notation
ðΓ
μ
;T
ν
Þ
ijk
¼ðΓ
μ
Þ
i
l
T
ljk
ν
þðΓ
μ
Þ
j
l
T
ilk
ν
þðΓ
μ
Þ
k
l
T
ijl
ν
: ð11Þ
For the pseudoscalar meson fields, the covariant derivarive
is written
D
μ
U ¼ ∂
μ
U þðiUλ
a
− iλ
a
UÞυ
a
μ
: ð12Þ
Expanding the Lagrangian (1) to leading order in the
baryon and meson fields, the relevant interaction part for a
meson and baryon coupling to a proton can be written
explicitly as
L
int
¼
ðD þFÞ
2f
ð
pγ
μ
γ
5
p∂
μ
π
0
þ
ffiffiffi
2
p
pγ
μ
γ
5
n∂
μ
π
þ
Þ −
ðD þ3FÞ
ffiffiffiffiffi
12
p
f
pγ
μ
γ
5
Λ∂
μ
K
þ
þ
ðD − FÞ
2f
ð
ffiffiffi
2
p
pγ
μ
γ
5
Σ
þ
∂
μ
K
0
þ pγ
μ
γ
5
Σ
0
∂
μ
K
þ
Þ −
D − 3F
ffiffiffiffiffi
12
p
f
pγ
μ
γ
5
p∂
μ
η
þ
C
ffiffiffiffiffi
12
p
f
ð−2
pΘ
νμ
Δ
þ
μ
∂
ν
π
0
−
ffiffiffi
2
p
pΘ
νμ
Δ
0
μ
∂
ν
π
þ
þ
ffiffiffi
6
p
pΘ
νμ
Δ
þþ
μ
∂
ν
π
−
− pΘ
νμ
Σ
0
μ
∂
ν
K
þ
þ
ffiffiffi
2
p
pΘ
νμ
Σ
þ
μ
∂
ν
K
0
þ H:c:Þ
þ
i
4f
2
pγ
μ
p½ðπ
þ
∂
μ
π
−
− π
−
∂
μ
π
þ
Þþ2ðK
þ
∂
μ
K
−
− K
−
∂
μ
K
þ
ÞþðK
0
∂
μ
K
0
− K
0
∂
μ
K
0
Þ: ð13Þ
The terms involving the coupling H are not present because of the restriction to proton initial states. The current calculations
below also do not involve the terms with the coupling H for the proton initial states.
PARTON DISTRIBUTIONS FROM NONLOCAL CHIRAL … PHYS. REV. D 99, 014041 (2019)
014041-3
From the Lagrangian (1) one can also obtain the form of the electromagnetic current that couples to the external field υ
a
μ
,
J
μ
a
¼
1
2
Tr½
Bγ
μ
½uλ
a
u
†
þ u
†
λ
a
u; Bþ
D
2
Tr½
Bγ
μ
γ
5
fuλ
a
u
†
− u
†
λ
a
u; Bg þ
F
2
Tr½
Bγ
μ
γ
5
½uλ
a
u
†
− u
†
λ
a
u; B
þ
1
2
T
ν
γ
ναμ
ðuλ
a
u
†
þ u
†
λ
a
u; T
α
Þþ
C
2
ð
T
ν
Θ
νμ
ðuλ
a
u
†
− u
†
λ
a
uÞB þ H:c:Þ
þ
f
2
4
Tr½∂
μ
UðU
†
iλ
a
− iλ
a
U
†
ÞþðUiλ
a
− iλ
a
UÞ∂
μ
U
†
: ð14Þ
For the SU(3) flavor singlet current coupling to the external field υ
0
μ
, one has
J
μ
0
¼hλ
0
iTr½Bγ
μ
Bþhλ
0
iT
ν
γ
ναμ
T
α
; ð15Þ
where again λ
0
is the unit matrix and hi denotes a trace in flavor space.
The currents for a given quark flavor are then expressed as combinations of the SU(3) singlet and octet currents,
J
μ
u
¼
1
3
J
μ
0
þ
1
2
J
μ
3
þ
1
2
ffiffiffi
3
p
J
μ
8
; ð16aÞ
J
μ
d
¼
1
3
J
μ
0
−
1
2
J
μ
3
þ
1
2
ffiffiffi
3
p
J
μ
8
; ð16bÞ
J
μ
s
¼
1
3
J
μ
0
−
1
ffiffiffi
3
p
J
μ
8
; ð16cÞ
where J
μ
3
and J
μ
8
are the a ¼ 3 and 8 components of the octet current, respectively. Using Eqs. (14), (15) and (16), the
currents J
μ
u
, J
μ
d
and J
μ
s
can be written explicitly as
J
μ
u
¼ 2pγ
μ
p þnγ
μ
n þ Λγ
μ
Λ þ 2Σ
þ
γ
μ
Σ
þ
þ Σ
0
γ
μ
Σ
0
−
1
2f
2
ðpγ
μ
pπ
þ
π
−
þ 2pγ
μ
pK
þ
K
−
Þþ3Δ
þþ
α
γ
αβμ
Δ
þþ
β
þ 2Δ
þ
α
γ
αβμ
Δ
þ
β
þ Δ
0
α
γ
αβμ
Δ
0
β
þ 2Σ
þ
α
γ
αβμ
Σ
þ
β
þ Σ
0
α
γ
αβμ
Σ
0
β
þ iðπ
−
∂
μ
π
þ
− π
þ
∂
μ
π
−
ÞþiðK
−
∂
μ
K
þ
− K
þ
∂
μ
K
−
Þ
−
iðD þFÞ
ffiffiffi
2
p
f
pγ
μ
γ
5
nπ
þ
þ
iðD þ3FÞ
ffiffiffiffiffi
12
p
f
pγ
μ
γ
5
ΛK
þ
−
iðD − FÞ
2f
pγ
μ
γ
5
Σ
0
K
þ
þ
C
ffiffiffiffiffi
12
p
f
ði
ffiffiffi
6
p
pΘ
μν
Δ
þþ
ν
π
−
þ i
ffiffiffi
2
p
pΘ
μν
Δ
0
ν
π
þ
þ ipΘ
μν
Σ
0
ν
K
þ
þ H:c:Þ; ð17aÞ
J
μ
d
¼ pγ
μ
p þ2nγ
μ
n þ 2 Σ
−
γ
μ
Σ
−
þ Σ
0
γ
μ
Σ
0
þ Λγ
μ
Λ þ
1
2f
2
ðpγ
μ
pπ
þ
π
−
− pγ
μ
pK
0
K
0
ÞþΔ
þ
α
γ
αβμ
Δ
þ
β
þ 2Δ
0
α
γ
αβμ
Δ
0
β
þ 3Δ
−
α
γ
αβμ
Δ
−
β
þ Σ
0
α
γ
αβμ
Σ
0
β
þ 2Σ
0−
α
γ
αβμ
Σ
−
β
− iðπ
−
∂
μ
π
þ
− π
þ
∂
μ
π
−
ÞþiðK
0
∂
μ
K
0
− K
0
∂
μ
K
0
Þ
þ
iðD þFÞ
ffiffiffi
2
p
f
pγ
μ
γ
5
nπ
þ
−
iðD − FÞ
ffiffiffi
2
p
f
pγ
μ
γ
5
Σ
þ
K
0
−
C
ffiffiffi
6
p
f
ði
ffiffiffi
3
p
pΘ
μν
Δ
þþ
ν
π
−
þ ipΘ
μν
Δ
0
ν
π
þ
þ ipΘ
μν
Σ
þ
ν
K
0
þ H:c:Þ;
ð17bÞ
J
μ
s
¼ Σ
þ
γ
μ
Σ
þ
þ Σ
0
γ
μ
Σ
0
þ Λγ
μ
Λ þ
1
2f
2
ð2pγ
μ
pK
þ
K
−
þ pγ
μ
pK
0
K
0
ÞþΣ
þ
α
γ
αβμ
Σ
þ
β
þ Σ
0
α
γ
αβμ
Σ
0
β
− iðK
−
∂
μ
K
þ
− K
þ
∂
μ
K
−
Þ − iðK
0
∂
μ
K
0
− K
0
∂
μ
K
0
Þþ
iðD − FÞ
ffiffiffi
2
p
f
pγ
μ
γ
5
Σ
þ
K
0
þ
iðD − FÞ
2f
pγ
μ
γ
5
Σ
0
K
þ
−
iðD þ3FÞ
ffiffiffiffiffi
12
p
f
pγ
μ
γ
5
ΛK
þ
þ
C
ffiffiffiffiffi
12
p
f
ð−i
pΘ
μν
Σ
0
ν
K
þ
þ i
ffiffiffi
2
p
pΘ
μν
Σ
þ
ν
K
0
þ H:c:Þ; ð17cÞ
SALAMU, JI, MELNITCHOUK, THOMAS, and WANG PHYS. REV. D 99, 014041 (2019)
014041-4
where the terms involving the doubly-strange baryons Ξ
0;−
and Ξ
0;−
and the triply-strange Ω
−
are not present because they
cannot couple to the proton initial states.
III. NONLOCAL CHIRAL LAGRANGIAN
In this section we describe the generation of the nonlocal Lagrangian from the local meson–baryon Lagrangian in Sec. II.
Evaluating the traces in Eq. (1) and introducing the minimal substitution for the electromagnetic field A
μ
, the local
Lagrangian density can be rewritten more explicitly in the form
L
ðlocalÞ
ðxÞ¼BðxÞðiγ
μ
D
μ;x
− M
B
ÞBðxÞþ
C
Bϕ
f
½
pðxÞγ
μ
γ
5
BðxÞD
μ;x
ϕðxÞþH:c:þT
μ
ðxÞðiγ
μνα
D
α;x
− M
T
γ
μν
ÞT
ν
ðxÞ
þ
C
Tϕ
f
½
pðxÞΘ
μν
T
ν
ðxÞD
μ;x
ϕðxÞþH:c:þ
iC
ϕϕ
†
2f
2
pðxÞγ
μ
pðxÞ½ϕðxÞðD
μ;x
ϕÞ
†
ðxÞ − D
μ;x
ϕðxÞϕ
†
ðxÞ
þ D
μ;x
ϕðxÞðD
μ;x
ϕÞ
†
ðxÞþ; ð18Þ
where for the interaction part we show only those terms that contribute to a meson–baryon coupling to a proton, and we
keep the dependence on the space-time coordinate x explicitly. The covariant derivatives here are written so as to indicate
the coordinate with respect to which the derivative is taken,
D
μ;x
BðxÞ¼½∂
μ
− ie
q
B
A
μ
ðxÞBðxÞ; ð19aÞ
D
μ;x
T
ν
ðxÞ¼½∂
μ
− ie
q
T
A
μ
ðxÞT
ν
ðxÞ; ð19bÞ
D
μ;x
ϕðxÞ¼½∂
μ
− ie
q
ϕ
A
μ
ðxÞϕðxÞ; ð19cÞ
where e
q
B
, e
q
T
and e
q
ϕ
are the quark flavor charges of the octet baryon B, decuplet baryon T and meson ϕ, respectively. For
example, for the proton one has the charges e
u
p
¼ 2e
d
p
¼ 2, e
s
p
¼ 0, while for the Σ
þ
hyperon e
u
Σ
þ
¼ 2e
s
Σ
þ
¼ 2, e
d
Σ
þ
¼ 0, and
so forth. For the mesons, the flavor charges for the π
þ
are e
u
π
þ
¼ −e
d
π
þ
¼ 1 but e
q
π
0
¼ 0 for all q, and for the K
þ
these are
e
u
K
þ
¼ −e
s
K
þ
¼ 1, e
d
K
þ
¼ 0, and similarly for the charge conjugate states. These flavor charges may be read off from the
currents given in Eqs. (17a)–(17c). The coefficients C
Bϕ
and C
Tϕ
in Eq. (18) depend on the coupling constants D, F and C,
and are given explicitly in Table I for the processes discussed in this work.
Using the methods described in Refs. [32,37–42], the nonlocal version of the local Lagrangian (18) can be written as
L
ðnonlocÞ
ðxÞ¼BðxÞðiγ
μ
D
μ;x
− M
B
ÞBðxÞþT
μ
ðxÞðiγ
μνα
D
α;x
− M
T
γ
μν
ÞT
ν
ðxÞ
þ
pðxÞ
C
Bϕ
f
γ
μ
γ
5
BðxÞþ
C
Tϕ
f
Θ
μν
T
ν
ðxÞ
Z
d
4
aG
q
ϕ
ðx; x þ aÞFðaÞD
μ;xþa
ϕðx þa ÞþH:c:
þ
iC
ϕϕ
†
2f
2
pðxÞγ
μ
pðxÞ
Z
d
4
a
Z
d
4
bG
q
ϕ
ðx þb; x þ aÞF ðaÞFðbÞ
× ½ϕðx þ aÞðD
μ;xþb
ϕÞ
†
ðx þbÞ − D
μ;xþa
ϕðx þaÞϕ
†
ðx þ bÞþ D
μ;x
ϕðxÞðD
μ;x
ϕÞ
†
ðxÞþ; ð20Þ
TABLE I. Coupling constants C
Bϕ
, C
Tϕ
and C
ϕϕ
†
for the pBϕ, pTϕ and ppϕϕ
†
interactions, respectively, for the
various allowed flavor channels.
ðBϕÞðpπ
0
Þðnπ
þ
ÞðΣ
þ
K
0
ÞðΣ
0
K
þ
ÞðΛK
þ
Þ
C
Bϕ
1
2
ðD þ FÞ
1
ffiffi
2
p
ðD þ FÞ
1
ffiffi
2
p
ðD − FÞ
1
2
ðD − FÞ −
1
ffiffiffiffi
12
p
ðD þ 3FÞ
(Tϕ) ðΔ
0
π
þ
ÞðΔ
þ
π
0
ÞðΔ
þþ
π
−
ÞðΣ
þ
K
0
ÞðΣ
0
K
þ
Þ
C
Tϕ
−
1
ffiffi
6
p
C −
1
ffiffi
3
p
C
1
ffiffi
2
p
C
1
ffiffi
6
p
C −
1
ffiffiffiffi
12
p
C
ðϕϕ
†
Þðπ
þ
π
−
ÞðK
0
¯
K
0
ÞðK
þ
K
−
Þ
C
ϕϕ
†
1
2
1
2
1
PARTON DISTRIBUTIONS FROM NONLOCAL CHIRAL … PHYS. REV. D 99, 014041 (2019)
014041-5
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