# Helicity-dependent distribution of strange quarks in the proton
from nonlocal chiral effective theory

Fangcheng He  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, CAS, Beijing 100190, China  Chueng-Ryong Ji  Department
of Physics, North Carolina State University, Raleigh, North Carolina 27695,
USA W. Melnitchouk  Jefferson Lab, Newport News, Virginia 23606, USA  Y.
Salamu  School of Physics and Electrical Engineering, Kashi University, Kashi,
844000, Xinjiang, China A. W. Thomas  CSSM and ARC Centre for Dark Matter
Particle Physics, Department of Physics, University of Adelaide, Adelaide SA
5005, Australia  P. Wang  Institute of High Energy Physics, CAS, Beijing
100049, China  College of Physics Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China X. G. Wang  CSSM and ARC Centre for Dark
Matter Particle Physics, Department of Physics, University of Adelaide,
Adelaide SA 5005, Australia

###### Abstract

The helicity-dependent strange quark distribution in the proton, $\Delta s$,
is calculated in a nonlocal chiral SU(3) effective field theory. The hadronic
proton to meson plus octet or decuplet baryon splitting functions are derived
at the one-loop level, with loop integrals rendered finite by correlation
functions introduced in the nonlocal Lagrangian. Within the convolution
framework, the proton strange helicity distribution is obtained using spin-
flavor symmetry to constrain the input valence quark distributions in the
hadronic intermediate states. The polarized strange quark distribution is
found to be quite small, with the lowest moment of $\Delta s$ negative, but
consistent with recent global QCD analyses.

††preprint: ADP-21-18/T1165, JLAB-THY-22-3576

##  I Introduction

It has been over 30 years since the European Muon Collaboration (EMC)
published their polarized deep-inelastic scattering (DIS) measurement of the
proton’s spin-dependent structure function, $g_{1}$ Ashman  _et al._ (1988).
The result suggested that only a very small fraction of the proton’s spin was
carried by quarks — an initially shocking discovery which contradicted the
prevailing quark model view in which constituent quarks accounted for the
proton’s global quantum numbers, including its spin. Subsequent experiments
with increasing precision and kinematic reach were performed at SLAC Anthony
_et al._ (1996); Abe  _et al._ (1998, 1997); Anthony  _et al._ (2000, 1999,
2003), HERMES Ackerstaff  _et al._ (1997); Airapetian  _et al._ (2007, 2012),
SMC Adeva  _et al._ (1998, 1999), COMPASS Alexakhin  _et al._ (2007); Alekseev
_et al._ (2010), Jefferson Lab Zheng  _et al._ (2004); Dharmawardane  _et al._
(2006); Prok  _et al._ (2009, 2014); Guler  _et al._ (2015); Fersch  _et al._
(2017); Parno  _et al._ (2015); Posik  _et al._ (2014); Solvignon  _et al._
(2015); Armstrong  _et al._ (2019), and RHIC Adamczyk  _et al._ (2014); Adare
_et al._ (2016); Adam  _et al._ (2019), and various global QCD analyses of
these data in terms of spin-dependent parton distribution functions (PDFs)
have been carried out de Florian  _et al._ (2009, 2014); Hirai and Kumano
(2009); Blümlein and Böttcher (2010); Leader  _et al._ (2011, 2015); Khanpour
_et al._ (2017); Nocera  _et al._ (2014); Sato  _et al._ (2016a, b); Ethier
_et al._ (2017); Zhou  _et al._ . A recent analysis from the JAM
Collaboration, for instance, gives a total fraction $\Delta\Sigma=0.36\pm
0.09$ of the proton’s spin carried by quarks at a scale of $Q^{2}=1$ GeV2
Ethier  _et al._ (2017).

One early explanation proposed for the small value of $\Delta\Sigma$ was that
the contribution to the proton spin from strange quarks, which is much less
well determined than that from up and down quarks, was large and negative.
Within the assumptions traditionally made in phenomenological analyses, such
as SU(3) flavor symmetry and the equivalence of the strange and antistrange
polarizations, $\Delta s=\Delta\bar{s}$, the integrated strange quark
polarization has typically come in at around $\Delta s^{+}\equiv\Delta
s+\Delta\bar{s}\approx-0.1$. In this scenario, the nonsinglet axial charge,
$a_{8}$, is extracted from hyperon beta decays to be $a_{8}=\Delta
u^{+}+\Delta d^{+}-2\Delta s^{+}=0.58\pm 0.03$ Close and Roberts (1993), in
which case a strange quark polarization of $\approx-0.1$ would give a total
quark spin contribution $\Delta\Sigma\equiv\Delta u^{+}+\Delta d^{+}+\Delta
s^{+}=a_{8}+3\Delta s^{+}$ that would be close to the phenomenological result.

The accuracy of the flavor SU(3) symmetry assumption has been questioned, on
the other hand, in several analyses Jaffe and Manohar (1990); Ratcliffe
(2004); Bass and Thomas (2010) that have suggested that the uncertainty could
be as large as $\approx 20\%$. A re-evaluation of the nucleon’s axial-charges
in the cloudy bag model Thomas  _et al._ (1981); Thomas (1984), for example,
taking into account the effect of the one gluon exchange hyperfine interaction
and the meson cloud, led to the value $a_{8}=0.46\pm 0.05$ Bass and Thomas
(2010). Recent lattice simulations directly including the effects of
disconnected quark loops have yielded smaller magnitudes for the strange quark
polarization, $\Delta s^{+}_{\rm latt}=-0.046\pm 0.008$ Alexandrou  _et al._
(2020), while an analysis of the proton spin taking into account the angular
momentum carried by the pion cloud Schreiber and Thomas (1988); Thomas (2008);
Myhrer and Thomas (2008) favors a value $\approx-0.01$ Bass and Thomas (2010);
Yamaguchi  _et al._ (1989). The recent JAM global QCD analysis Ethier  _et
al._ (2017), which used data from inclusive and semi-inclusive DIS in order to
relax the SU(3) symmetry constraint, also supports a smaller magnitude for the
strange quark polarization, $\Delta s^{+}_{\rm JAM}=-0.03\pm 0.10$, at a scale
$Q^{2}=1$ GeV2, but with a somewhat larger uncertainty. For a review of the
status of global QCD analyses and lattice QCD simulations, see Ref. Lin  _et
al._ (2018), while for an explanation of the importance of the mismatch
between the scale appropriate to quark models and that of lattice QCD and DIS,
see Ref. Thomas (2008).

On the theoretical front, considerable progress has been made in the last few
years in developing the formalism and feasibility of extracting the momentum
dependence of quark distributions from lattice QCD calculations of quasi-PDFs
and pseudo-PDFs Alexandrou  _et al._ (2019); Joó  _et al._ (2019), including
contributions from the $q\bar{q}$ sea, and exploring ways in which lattice
data could constrain the phenomenological distributions Bringewatt  _et al._
(2021). For continuum-based approaches, early model-dependent work focused on
the effects of proton fluctuations to kaon-hyperon intermediate states on
strange nucleon observables Signal and Thomas (1987); Melnitchouk and Malheiro
(1997, 1999); Holtmann  _et al._ (1996); Zamani and Saranchak (2001); Cao and
Signal (2003), although reliably quantifying such effects has proven
challenging. A more systematic methodology for quantifying the effects of
virtual meson loops on PDFs Thomas (1983) was formulated subsequently in the
framework of chiral effective field theory (EFT), and used to study the
unpolarized light quark asymmetry $\bar{d}-\bar{u}$, the strange–antistrange
asymmetry $s-\bar{s}$, as well as the strange quark helicity in the proton
Wang  _et al._ (2020); Salamu  _et al._ (2015); Wang  _et al._ (2016a, b).

Along these lines, a nonlocal chiral effective theory was recently proposed,
which allows the study of hadron properties at relatively large momentum
transfer Wang (2014); He and Wang (2018a, b), while consistently taking into
account the finite size of hadrons from the underlying chiral Lagrangian. This
framework was used to compute electromagnetic form factors of the nucleon He
and Wang (2018a, b), as well as collinear parton distributions Salamu  _et
al._ (2019a, b) and transverse momentum dependent distributions He and Wang
(2019), such as the Sivers function, for the sea quarks in the nucleon.
Furthermore, if the nonlocal behavior is assumed to be a general property of
all the interactions, it produces interesting results when applied to the
lepton anomalous magnetic moments He and Wang (2020).

In this paper, we extend our previous analysis Salamu  _et al._ (2019b); He
and Wang (2019) of the chiral loop contributions to the nonperturbative
strange quark PDF in the polarized sector within the framework of nonlocal
chiral effective theory. In Sec. II, we review the derivation of the nonlocal
chiral Lagrangian starting from the local effective Lagrangian. From this the
spin-dependent hadronic splitting functions are computed in Sec. III, for the
case of a covariant dipole form factor, and numerical results are presented in
Sec. IV. Finally, Sec. V contains a summary of our findings and suggestions
for future research.

##  II Effective Lagrangian

Our analysis is based on the chiral SU(3)${}_{L}\times$SU(3)R effective
Lagrangian, describing interactions of octet ($B$) and decuplet ($T_{\mu}$)
baryons with pseudoscalar mesons ($\phi$) Jenkins and Manohar (1991); Bernard
(2008); Hacker  _et al._ (2005),

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\rm
Tr}\big{[}\bar{B}(i\not{D}-M_{B})B\big{]}-\frac{D}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}\\{u_{\mu},B\\}\big{]}-\frac{F}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}[u_{\mu},B]\big{]}$ (1)
$\displaystyle-$ $\displaystyle\frac{F-D}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}B\big{]}{\rm
Tr}\big{[}u_{\mu}\big{]}-\frac{{\cal
C}}{2}\left(\epsilon^{ijk}\,\overline{T}_{\mu}^{ilm}\Theta^{\mu\nu}(u_{\nu})^{lj}B^{mk}+{\rm
H.c.}\right)$ $\displaystyle+$
$\displaystyle\overline{T}_{\mu}^{ijk}\big{(}i\gamma^{\mu\nu\alpha}D_{\alpha}-M_{T}\gamma^{\mu\nu}\big{)}T_{\nu}^{ijk}-\frac{{\cal
H}}{2}\,\overline{T}_{\mu}^{ijk}\gamma^{\mu\nu\alpha}\gamma_{5}(u_{\alpha})^{kl}\,T_{\nu}^{ijl},$

where $M_{B}$ and $M_{T}$ are the masses of the octet and decuplet baryons,
and $D$, $F$, $\cal C$ and $\cal H$ denote the baryon-meson coupling
constants. The octet-decuplet baryon transition operator $\Theta^{\mu\nu}$ is
given by

$\Theta^{\mu\nu}=g^{\mu\nu}-\big{(}Z+\tfrac{1}{2}\big{)}\gamma^{\mu}\gamma^{\nu},$
(2)

where $Z$ is the decuplet off-shell parameter (usually chosen to be $Z=-1/2)$.
The pseudoscalar mesons couple to baryons through the vector and axial vector
combinations involving the field $u=\exp\big(i\phi/\sqrt{2}\,f\big{missing})$,

$\displaystyle\Gamma_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(u^{\dagger}\partial_{\mu}u+u\partial_{\mu}u^{\dagger}\right)+\frac{i}{2}\left(u^{\dagger}\lambda^{a}u-u\lambda^{a}u^{\dagger}\right)a_{\mu}^{a},$
(3) $\displaystyle u_{\mu}$ $\displaystyle=$ $\displaystyle
i\left(u^{\dagger}\partial_{\mu}u-u\partial_{\mu}u^{\dagger}\right)-\left(u^{\dagger}\lambda^{a}u+u\lambda^{a}u^{\dagger}\right)a_{\mu}^{a},$
(4)

where $f$ is the pseudoscalar decay constant, $a_{\mu}^{a}$ corresponds to the
external axial-vector fields, and $\lambda^{a}$ ($a=0,\ldots,8$) represent the
unit matrix and the eight Gell-Mann matrices. For further details about the
SU(3) chiral Lagrangian see the discussions in Ref. Jenkins and Manohar
(1991); Bernard (2008); Hacker  _et al._ (2005); Salamu  _et al._ (2019a).

From the chiral Lagrangian (1) one can derive the electromagnetic currents
that couple to the external field $a_{\mu}^{a}$,

$\displaystyle J_{A}^{\mu,a}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}[u\lambda^{a}u^{\dagger}-u^{\dagger}\lambda^{a}u,B]\big{]}+\frac{D}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}\left\\{u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u,B\right\\}\big{]}$
(5) $\displaystyle+$ $\displaystyle\frac{F}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}[u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u,B]\big{]}+\frac{F-D}{2}\,{\rm
Tr}\big{[}\bar{B}\gamma^{\mu}\gamma_{5}B\big{]}{\rm
Tr}\big{[}u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u\big{]}$
$\displaystyle+$ $\displaystyle\frac{\cal
C}{2}\,\big{(}\overline{T}_{\nu}\Theta^{\nu\mu}(u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u)B+{\rm
H.c.}\big{)}+\frac{\cal
H}{2}\,\overline{T}_{\nu}\gamma^{\nu\alpha\mu}\left(u\lambda^{a}u^{\dagger}+u^{\dagger}\lambda^{a}u,T_{\alpha}\right),$

where the notations are as defined in Ref. Salamu  _et al._ (2019a). Following
the methodology discussed in Refs. Terning (1991); Holdom (1992); Abe  _et
al._ (1997); Faessler  _et al._ (2003); Wang and Banerjee (1996); He and Wang
(2018a, b), we write the nonlocal baryon–meson interaction Lagrangian for the
meson coupling to a proton as Salamu  _et al._ (2019a)

$\displaystyle{\cal L}^{\rm(nonloc)}_{\rm had}(x)$ $\displaystyle=$
$\displaystyle\bar{p}(x)\bigg{(}\frac{C_{B\phi}}{f}\gamma^{\mu}\gamma^{5}B(x)\,+\frac{C_{T\phi}}{f}\Theta^{\mu\nu}T_{\nu}(x)\bigg{)}\int\differential^{4}{a}F(a)\,\partial_{\mu}\phi(x+a)+{\rm
H.c.}$ (6) $\displaystyle+$
$\displaystyle\frac{iC_{\phi\phi^{\dagger}}}{2f^{2}}\,\bar{p}(x)\gamma^{\mu}p(x)\int\differential^{4}{a}\,F(a)\,\phi(x+a)\int\differential^{4}{b}\,F(b)\,\big{(}\partial_{\mu}\phi^{\dagger}(x+b)+{\rm
H.c.}\big{)},~{}~{}$

where $C_{B\phi}$, $C_{T\phi}$ and $C_{\phi\phi^{\dagger}}$ are the coupling
constants for the $pB\phi$, $pT\phi$ and $pp\phi\phi^{\dagger}$ interactions,
respectively (see Table I of Ref. Salamu  _et al._ (2019a)).

The nonlocal interaction between a quark $q$ in a hadron and the external
axial-vector field $a_{\mu}$ is given by

$\displaystyle{\cal L}^{\rm(nonloc)}_{q\,\rm ext}(x)$ $\displaystyle=$
$\displaystyle\int\differential^{4}{a}\,F(a)\Big{(}C_{B}^{q}\,\bar{B}(x)\gamma^{\mu}\gamma^{5}B(x)\,+C_{T}^{q}\,\overline{T}_{\alpha}\gamma^{\mu}\gamma^{5}T^{\alpha}(x)\Big{)}\,a_{\mu}(x+a)$
(7) $\displaystyle+$ $\displaystyle
C_{BT}^{q}\,\int\differential^{4}{a}\,F(a)\Big{(}\bar{B}(x)\,\Theta^{\mu\nu}T_{\nu}(x)+{\rm
H.c.}\Big{)}\,a_{\mu}(x+a)$ $\displaystyle+$
$\displaystyle\frac{iC_{B\phi}^{q}}{f}\int\differential^{4}{a}\,F(a)\int\differential^{4}{b}\,F(b)\,\Big{(}\bar{p}(x)\gamma^{\mu}B(x)\,\phi(x+a)+{\rm
H.c.}\Big{)}\,a_{\mu}(x+b)$ $\displaystyle+$
$\displaystyle\frac{C_{\phi\phi^{\dagger}}^{q}}{f^{2}}\int\differential^{4}{a}\,F(a)\int\differential^{4}{b}\,F(b)\int\differential^{4}{c}\,F(c)\,\bar{p}(x)\gamma^{\mu}\gamma^{5}p(x)\,$
$\displaystyle\hskip 156.49014pt\times\
\phi(x+a)\,\phi^{\dagger}(x+b)\,a_{\mu}(x+c),$

where $C_{j}^{q}$ ($j=B$, $T$, $BT$, $B\phi$, or $\phi\phi^{\dagger}$) are the
coupling constants (axial charges) for the interaction between the quark $q$
in the hadronic configuration $j$ and the axial-vector field. In the present
work we focus on the strange quark contribution, for which the corresponding
couplings $C_{j}^{s}$ are listed in Table 1.

Table 1: Coupling constants $C_{B}^{s}$, $C_{T}^{s}$, $C_{BT}^{s}$,
$C_{B\phi}^{s}$ and $C_{\phi\phi^{\dagger}}^{s}$ for the interaction between a
strange quark $s$ and an external axial-vector field in the corresponding
hadronic configurations.
$B$ |  $\Lambda$ |  $\Sigma^{0}$ |  $\Sigma^{+}$
---|---|---|---
$C_{B}^{s}$ |  $F+\frac{1}{3}D$ |  $F-D$ |  $F-D$
$T$ |  $\Sigma^{*0}$ |  $\Sigma^{*+}$ |
$C_{T}^{s}$ |  $\frac{1}{3}{\cal H}$ |  $\frac{1}{3}{\cal H}$ |
$BT$ | $\Sigma^{0}\Sigma^{*0}$ | $\Sigma^{+}\Sigma^{*+}$ |
$C_{BT}^{s}$ | $\frac{1}{3}{\cal C}$ | $-\frac{1}{3}{\cal C}$ |
$B\phi$ | $\Lambda K^{+}$ | $\Sigma^{0}K^{+}$ | $\Sigma^{+}K^{0}$
$C_{B\phi}^{s}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$
$\phi\phi^{\dagger}$ | $K^{0}\overline{K}^{0}$ | $K^{+}K^{-}$ |
$C_{\phi\phi^{\dagger}}^{s}$ | $\frac{1}{2}(F-D)$ | $F$ |

For the electromagnetic form factors and unpolarized PDFs discussed in Refs.
He and Wang (2018a, b); Salamu  _et al._ (2019a) the path integral of the
vector field in the gauge link was necessary to guarantee local gauge
invariance. In the current application to interactions with the axial-vector
field and spin-dependent PDFs, there is no associated conserved charge and
hence no gauge link term in the nonlocal Lagrangian. The corresponding
nonlocal axial-vector current can be obtained from Eq. (7) and is given by

$\displaystyle J_{A}^{\mu,q}(x)$ $\displaystyle=$
$\displaystyle\int\differential^{4}{a}\,F(a)\,\Big{(}C_{B}^{q}\,\bar{B}(x-a)\gamma^{\mu}\gamma^{5}B(x-a)\,+C_{T}^{q}\,\overline{T}_{\alpha}(x-a)\gamma^{\mu}\gamma^{5}\,T^{\alpha}(x-a)\Big{)}$
(8) $\displaystyle+$ $\displaystyle
C_{BT}^{q}\int\differential^{4}{a}\,F(a)\,\Big{(}\bar{B}(x-a)\,\Theta^{\mu\nu}\,T_{\nu}(x-a)+{\rm
H.c.}\Big{)}$ $\displaystyle+$
$\displaystyle\frac{iC_{B\phi}^{q}}{f}\int\differential^{4}{a}\,F(a)\int\differential^{4}{b}\,F(b)\Big{(}\bar{p}(x-b)\gamma^{\mu}B(x-b)\,\phi(x+a-b)+{\rm
H.c.}\Big{)}\,$ $\displaystyle+$
$\displaystyle\frac{C_{\phi\phi^{\dagger}}^{q}}{f^{2}}\int\differential^{4}{a}\,F(a)\int\differential^{4}{b}\,F(b)\int\differential^{4}{c}\,F(c)\,\bar{p}(x-c)\gamma^{\mu}\gamma^{5}\,p(x-c)$
$\displaystyle\hskip 156.49014pt\times\ \phi(x+a-c)\,\phi^{\dagger}(x+b-c).$

From the nonlocal axial current and Lagrangian, in the next section we will
compute the proton to baryons + meson splitting functions necessary for the
helicity-dependent strange quark distribution.

##  III Hadronic splitting functions

The spin-dependent hadronic splitting functions, $\Delta f_{j}$, can be
evaluated from the matrix elements of the hadronic operators of the axial
current, which correspond to the one meson loop diagrams in Fig. 1 Salamu  _et
al._ (2019a); Wang  _et al._ (2020). The matrix elements of the hadronic
operators give rise to the octet rainbow, tadpole, Kroll-Ruderman (KR),
decuplet rainbow, and octet-decuplet transition splitting functions, as
illustrated by the diagrams in Fig. 1. A detailed derivation of the splitting
functions corresponding to the processes illustrated in Fig. 1 was presented
in Ref. Wang  _et al._ (2020). In this section, we summarise those results,
giving the splitting functions as a function of the light-cone variable
$y=k^{+}/p^{+}$, where $k^{\mu}$ is the four-momentum of the kaon and
$p^{\mu}$ is the four-momentum of the external proton. Following Refs. Salamu
_et al._ (2019a, b), in our numerical calculation we choose for simplicity the
Fourier transformation of the correlation function $F(a)$ in the nonlocal
Lagrangian to take a dipole form,

$\widetilde{F}(k)=\bigg{(}\frac{\Lambda^{2}-m_{\phi}^{2}}{\Lambda^{2}-k^{2}}\bigg{)}^{2},$
(9)

where $\Lambda=\Lambda_{B,T}$ are the cutoff parameters for the octet baryon-
meson and decuplet baryon-meson vertices, and $m_{\phi}$ is the meson mass.

Figure 1: One-loop contributions to the spin-dependent PDFs of the nucleon
from (a) octet rainbow, (b) tadpole, (c) Kroll-Ruderman, (d) decuplet rainbow,
and (e) octet-decuplet transition diagrams. The octet baryons, decuplet
baryons and pseudoscalar mesons are represented by the solid, double-solid and
dashed lines, respectively, while the symbol $\otimes$ denotes insertion of
the hadronic axial current operator in Eq. (8).

For the octet baryon rainbow diagram of Fig. 1(a), the splitting function
$\Delta f_{B\phi}^{(\rm rbw)}$ can be expressed as a sum of the on-shell, off-
shell and $\delta$-function contributions,

$\Delta f_{B\phi}^{(\rm
rbw)}(y)=\frac{C_{B\phi}^{2}\overline{M}_{\\!B}^{2}}{(4\pi f)^{2}}\left[\Delta
f_{B}^{\rm(on)}(y)+\Delta f_{B}^{(\rm off)}(y)+\Delta
f_{B}^{(\delta)}(y)\right],$ (10)

where $M$ is the nucleon mass and $\overline{M}_{\\!B}=M_{B}+M$. The coupling
constants $C_{B\phi}$ are given in terms of the usual SU(3) coefficients $D$
and $F$. The on-shell function is written as

$\displaystyle\Delta f^{(\mathrm{on})}_{B}(y)$ $\displaystyle=$
$\displaystyle\overline{\Lambda}_{B}^{8}\int\\!\differential{k}^{2}_{\perp}\,\frac{y\,\big{[}\\!-\\!k^{2}_{\perp}+(\Delta_{B}+yM)^{2}\big{]}\big{(}4D_{B\phi}+D_{B\Lambda_{B}}\big{)}}{\bar{y}^{2}D^{2}_{B\phi}D^{5}_{B\Lambda_{B}}}\,,$
(11)

where $\bar{y}=1-y$ is the light-cone fraction carried by the baryon, and we
define for shorthand $\Delta_{B}=M_{B}-M$ and
$\overline{\Lambda}_{B}^{2}=\Lambda_{B}^{2}-m_{\phi}^{2}$. The functions
$D_{B\phi}$ and $D_{B\Lambda_{B}}$ are defined as

$D_{B\phi}=-\frac{1}{\bar{y}}\big{(}k^{2}_{\bot}+yM_{B}^{2}+\bar{y}\,m_{\phi}^{2}-y\bar{y}\,M^{2}\big{)},$
(12a) and
$D_{B\Lambda_{B}}=-\frac{1}{\bar{y}}\big{(}k^{2}_{\bot}+yM_{B}^{2}+\bar{y}\,\Lambda_{B}^{2}-y\bar{y}\,M^{2}\big{)}.$
(12b)

In the limit where $\Lambda_{B}\to\infty$, the local on-shell splitting
function reduces to

$\displaystyle\Delta f^{(\mathrm{on})}_{B}(y)$ $\displaystyle=$
$\displaystyle\int\\!\differential{k}^{2}_{\perp}\,\frac{y\,\big{[}\\!-\\!k^{2}_{\perp}+(\Delta_{B}+yM)^{2}\big{]}}{\bar{y}^{2}D^{2}_{B\phi}},$
(13)

which coincides with the results obtained in Refs. Holtmann  _et al._ (1996);
Melnitchouk and Malheiro (1997, 1999) in this limit. The off-shell splitting
function in Eq. (10) is given by

$\Delta
f_{B}^{(\mathrm{off})}(y)=\frac{2\overline{\Lambda}_{B}^{8}}{\overline{M}_{\\!B}\\!}\int\\!\differential{k}_{\perp}^{2}\,\frac{\big{(}\Delta_{B}+yM\big{)}}{\bar{y}\,D_{B\phi}D_{B\Lambda}^{4}}.$
(14)

For the $\delta$-function term, $\Delta f_{\phi}^{(\delta)}$, which arises
from meson loops with zero light-cone momentum ($k^{+}=0$), one has

$\Delta
f_{B}^{(\delta)}(y)=\frac{1}{6\overline{M}^{2}_{\\!B}}\bigg{(}2\Lambda_{B}^{2}+3m_{\phi}^{2}\Big{(}1+2\log\frac{m_{\phi}^{2}}{\Lambda_{B}^{2}}\Big{)}-\frac{6m_{\phi}^{4}}{\Lambda_{B}^{2}}+\frac{m_{\phi}^{6}}{\Lambda_{B}^{4}}\bigg{)}\,\delta(y).$
(15)

The splitting functions of the tadpole diagram in Fig. 1(b) for the charged
and neutral kaon loop contributions are given by

$\Delta f_{K^{+}}^{\rm(tad)}(y)=\Delta
f_{K^{0}}^{\rm(tad)}(y)\equiv-\frac{\overline{M}_{\\!B}^{2}}{(4\pi
f)^{2}}\,\Delta f_{\phi}^{(\delta)}(y)\,,$ (16)

where the generic tadpole function $\Delta f_{\phi}^{(\delta)}$ is related to
the $\delta$-function term in the rainbow diagram in Eq. (15),

$\Delta f_{\phi}^{(\delta)}(y)=-\Delta f_{B}^{(\delta)}(y)\,.$ (17)

The splitting function of the Kroll-Ruderman diagrams in Fig. 1(c) can be
written in terms of the off-shell and $\delta$-function contributions as

$\displaystyle\Delta f^{\rm(KR)}_{B\phi}(y)$ $\displaystyle=$
$\displaystyle\frac{C_{B\phi}\,\overline{M}_{\\!B}^{2}}{(4\pi
f)^{2}}\,\Big{[}\Delta f_{B}^{(\rm off)}(y)+2\Delta
f_{B}^{(\delta)}(y)\Big{]},$ (18)

with the off-shell function $\Delta f_{B}^{(\rm off)}$ given in Eq. (14) and
the $\delta$-function component, $\Delta f_{B}^{(\delta)}$, in Eq. (15). Note
that Eq. (18) has the opposite sign relative to Eq. (49) of Ref. Wang  _et
al._ (2020) because the coefficient $C_{B\phi}$ used here has the opposite
sign compared to that in Wang  _et al._ (2020).

For the decuplet intermediate states, because of the higher spin of the baryon
the polarized splitting functions are somewhat more complicated. As in the
octet case, the splitting function associated with the decuplet rainbow
diagram in Fig. 1(d) can be decomposed as a sum of on-shell, off-shell and
$\delta$-function contributions,

$\Delta f_{T\phi}^{(\rm
rbw)}(y)=\frac{C_{T\phi}^{2}\overline{M}_{\\!T}^{2}}{(4\pi
f_{\phi})^{2}}\,\Big{[}\Delta f^{(\rm on)}_{T}(y)+\Delta f^{(\rm
off)}_{T}(y)+\Delta f_{T}^{(\delta)}(y)\Big{]},$ (19)

where $\overline{M}_{T}=M_{T}+M$. The couplings $C_{T\phi}$ are given in terms
of the coefficient $\cal C$. In our analysis, we will take ${\cal C}=-2D$ from
SU(6) symmetry. The on-shell part of the splitting function is given by

$\displaystyle\Delta f^{(\rm on)}_{T}(y)$ $\displaystyle=$
$\displaystyle-\frac{\overline{\Lambda}_{T}^{8}}{18M_{T}^{2}\overline{M}_{\\!T}^{2}}\int\\!\differential{k}^{2}_{\bot}\,\Big{[}k^{2}_{\bot}+\big{(}\overline{M}_{\\!T}-yM\big{)}^{2}\Big{]}\Big{[}k^{4}_{\bot}-8\bar{y}MM_{T}\,k^{2}_{\bot}-\big{(}M_{T}^{2}-\bar{y}^{2}M^{2}\big{)}^{2}\Big{]}$
(20) $\displaystyle\hskip
99.58464pt\times\,\frac{y\,\big{(}4D_{T\phi}+D_{T\Lambda_{T}}\big{)}}{\bar{y}^{4}D_{T\phi}^{2}D_{T\Lambda_{T}}^{5}},$

where $\overline{\Lambda}_{T}^{2}=\Lambda_{T}^{2}-m_{\phi}^{2}$, and
$D_{T\phi}$, $D_{T\Lambda}$ are defined in analogy with Eq. (12a),

$\displaystyle D_{T\phi}$ $\displaystyle=$
$\displaystyle-\frac{k^{2}_{\bot}+yM_{T}^{2}+\bar{y}\,m_{\phi}^{2}-y\bar{y}\,M^{2}}{\bar{y}},$
(21a) $\displaystyle D_{T\Lambda_{T}}$ $\displaystyle=$
$\displaystyle-\frac{k^{2}_{\bot}+yM_{T}^{2}+\bar{y}\,\Lambda_{T}^{2}-y\bar{y}\,M^{2}}{\bar{y}}\,.$
(21b)

The off-shell decuplet function part is given by

$\displaystyle\Delta f^{(\rm off)}_{T}(y)$ $\displaystyle=$
$\displaystyle\frac{\overline{\Lambda}_{T}^{8}}{\big{(}3M_{T}^{2}\overline{M}_{\\!T}\big{)}^{2}}\int\differential{k}^{2}_{\bot}\,\frac{1}{\bar{y}^{3}D_{T\phi}D^{4}_{T\Lambda_{T}}}\Big{[}k^{6}_{\bot}+\big{(}\bar{y}^{2}M^{2}-M_{T}^{2}-3\bar{y}M_{T}M\big{)}\,k^{4}_{\bot}$
(22) $\displaystyle\hskip
56.9055pt-\,\big{(}3M_{T}^{4}+2\bar{y}M_{T}^{3}M+4\bar{y}^{2}M_{T}^{2}M^{2}+6\bar{y}^{3}M_{T}M^{3}+\bar{y}^{4}M^{4}\big{)}\,k^{2}_{\bot}$
$\displaystyle\hskip
56.9055pt-\,\big{(}M_{T}^{3}-2\bar{y}M_{T}^{2}M+\bar{y}^{3}M^{3}\big{)}\big{(}M_{T}+\bar{y}M\big{)}^{3}\Big{]}.$

For the $\delta$-function contribution, we perform the $k_{\perp}$ integration
analytically to obtain

$\displaystyle\Delta f^{(\delta)}_{T}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\big{(}3M_{T}\overline{M}_{\\!T}\big{)}^{2}}\bigg{\\{}\frac{\overline{\Lambda}_{T}^{2}}{12\Lambda_{T}^{4}M_{T}^{2}}\Big{[}4\Lambda_{T}^{8}+\Lambda_{T}^{6}\,\Big{(}29M_{T}\overline{M}_{\\!T}+M_{T}M+14M^{2}-14m_{\phi}^{2}\Big{)}$
(23)
$\displaystyle+\,\Lambda_{T}^{4}\,\Big{(}22m_{\phi}^{4}-5m_{\phi}^{2}\big{(}14M^{2}+30MM_{T}+29M_{T}^{2}\big{)}+\,4\overline{M}_{\\!T}^{2}\big{(}2M^{2}+2MM_{T}-3M_{T}^{2}\big{)}\Big{)}$
$\displaystyle-\,2\Lambda_{T}^{2}\,m_{\phi}^{2}\Big{(}m_{\phi}^{2}\big{(}14M^{2}+30MM_{T}+29M_{T}^{2}\big{)}-\,5\big{(}2M^{2}+2MM_{T}-3M_{T}^{2}\big{)}\overline{M}_{\\!T}^{2}\Big{)}$
$\displaystyle-\,2m_{\phi}^{4}\overline{M}_{\\!T}^{2}\Big{(}2M^{2}+2MM_{T}-3M_{T}^{2}\Big{)}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{1}{2M_{T}^{2}}\Big{[}2\overline{M}_{T}^{2}\big{(}2M^{2}+2MM_{T}-3M_{T}^{2}\big{)}-m_{\phi}^{2}\big{(}14M^{2}+30MM_{T}+29M_{T}^{2}\big{)}+2m_{\phi}^{4}\Big{]}$
$\displaystyle\hskip
28.45274pt\times\,m_{\phi}^{2}\log\frac{m_{\phi}^{2}}{\Lambda_{T}^{2}}\bigg{\\}}\
\delta(y)\,.$

For the octet-decuplet rainbow transition diagrams in Fig. 1(e), the splitting
function can also be written as a sum of three terms,

$\Delta
f_{TB\phi}^{\rm(rbw)}(y)=-\frac{C_{T\phi}C_{B\phi}\overline{M}_{\\!T}\overline{M}_{\\!TB}}{(4\pi
f)^{2}}\Big{[}\Delta f^{(\rm on)}_{TB}(y)+\Delta f^{(\rm off)}_{TB}(y)+\Delta
f_{TB}^{(\delta)}(y)\Big{]},$ (24)

where $\overline{M}_{TB}=M_{T}+M_{B}$. Explicit calculation of the on-shell
octet-decuplet transition function in Eq. (24) gives

$\displaystyle\Delta f^{(\rm on)}_{TB}(y)$ $\displaystyle=$
$\displaystyle\frac{\overline{\Lambda}_{B}^{4}\overline{\Lambda}_{T}^{4}}{3M_{T}^{2}\overline{M}_{\\!TB}\Delta_{TB}}\int\frac{\differential{k}_{\bot}^{2}}{\bar{y}^{2}}\,\left(\frac{1}{D_{T\phi}D^{2}_{T\Lambda_{B}}D^{2}_{T\Lambda_{T}}}-\frac{1}{D_{B\phi}D^{2}_{B\Lambda_{B}}D^{2}_{B\Lambda_{T}}}\right)$
(25)
$\displaystyle\times\Big{[}k_{\bot}^{4}-\bar{y}M(3M_{T}-M_{B})\,k_{\bot}^{2}-2M_{T}\Delta_{TB}\,k_{\bot}^{2}$
$\displaystyle\quad-\,\big{(}\Delta_{B}+yM\big{)}\big{(}\Delta_{T}+yM\big{)}\big{(}\overline{M}_{\\!T}-yM\big{)}^{2}\Big{]},$

where $\Delta_{TB}=M_{T}-M_{B}$, while for the off-shell transition function
we have

$\displaystyle\Delta f^{(\rm off)}_{TB}(y)$ $\displaystyle=$
$\displaystyle\frac{\overline{\Lambda}_{B}^{4}\overline{\Lambda}_{T}^{4}}{3M_{T}^{2}\overline{M}_{\\!T}\overline{M}_{\\!TB}}\int\frac{\differential{k}_{\bot}^{2}}{\bar{y}^{2}}\Bigg{\\{}\frac{\overline{M}_{\\!T}\big{(}k_{\bot}^{2}(\bar{y}M+2M_{T})-(M_{T}-\bar{y}M)(\overline{M}_{\\!T}-yM)^{2}\big{)}}{D_{T\phi}D^{2}_{T\Lambda_{B}}D^{2}_{T\Lambda_{T}}}$
(26) $\displaystyle+$
$\displaystyle\frac{1}{D_{B\phi}D^{2}_{B\Lambda_{B}}D^{2}_{B\Lambda_{T}}}\Big{[}k_{\bot}^{4}+\big{(}\bar{y}M(3M+4M_{B})+3(1\\!+\\!\bar{y})M_{B}M_{T}+MM_{T}\\!-\\!2M_{T}^{2}\big{)}k_{\bot}^{2}$
$\displaystyle\quad-\,(M_{B}-\bar{y}M)\Big{(}y^{3}M^{3}-y^{2}M\big{(}2M_{B}M_{T}+(4M+M_{B})\overline{M}_{\\!T}\big{)}$
$\displaystyle\qquad\qquad\qquad\quad-\,y\,\big{(}M_{B}^{2}(M+3M_{T})+MM_{B}\Delta_{T}-M(5M-M_{T})\overline{M}_{\\!T}\big{)}$
$\displaystyle\qquad\qquad\qquad\quad+\,M_{B}^{3}+M_{T}^{3}+2M\Delta_{B}\overline{M}_{\\!B}+M_{T}(2M+M_{B})\overline{M}_{\\!TB}\Big{)}\Big{]}\Bigg{\\}}.$

Finally, for the $\delta$-function contribution to the octet-decuplet
transition, we have

$\displaystyle\Delta f^{(\delta)}_{TB}(y)$ $\displaystyle=$
$\displaystyle\frac{\delta(y)}{6M_{T}^{2}\overline{M}_{T}\overline{M}_{TB}\big{(}\Lambda_{B}^{2}-\Lambda_{T}^{2}\big{)}^{3}}\Bigg{\\{}2\big{(}\Lambda_{B}^{2}+\Lambda_{T}^{2}\big{)}\overline{\Lambda}_{B}^{4}\overline{\Lambda}_{T}^{4}\,\log\frac{\Lambda_{B}^{2}}{\Lambda_{T}^{2}}$
(27) $\displaystyle+$
$\displaystyle\overline{\Lambda}_{T}^{4}\log\frac{m_{\phi}^{2}}{\Lambda_{B}^{2}}\,\Big{[}2m_{\phi}^{2}\,\big{(}(\Lambda_{B}^{2}+\Lambda_{T}^{2})(\overline{M}_{T}\overline{M}_{TB}+\Delta_{B}\overline{M}_{B})-\Lambda_{B}^{2}\Lambda_{T}^{2}-2\Lambda_{B}^{4}\big{)}$
$\displaystyle\qquad\qquad\,+\,2m_{\phi}^{4}\,(\Lambda_{B}^{2}+\Lambda_{T}^{2})-\Lambda_{B}^{4}\big{(}4\overline{M}_{TB}\overline{M}_{T}+4\Delta_{B}\overline{M}_{B}-\Lambda_{B}^{2}-\Lambda_{T}^{2}\big{)}\Big{]}$
$\displaystyle-$
$\displaystyle\overline{\Lambda}_{B}^{4}\log\frac{m_{\phi}^{2}}{\Lambda_{T}^{2}}\,\Big{[}2m_{\phi}^{2}\big{(}(\Lambda_{B}^{2}+\Lambda_{T}^{2})(\overline{M}_{T}\overline{M}_{TB}+\Delta_{B}\overline{M}_{B})-\Lambda_{B}^{2}\Lambda_{T}^{2}-2\Lambda_{T}^{4}\big{)}$
$\displaystyle\qquad\qquad\,+\,2m_{\phi}^{4}(\Lambda_{B}^{2}+\Lambda_{T}^{2})-\Lambda_{T}^{4}\big{(}4\overline{M}_{TB}\overline{M}_{T}+4\overline{M}_{B}\Delta_{B}-\Lambda_{B}^{2}-\Lambda_{T}^{2}\big{)}\Big{]}$
$\displaystyle+$
$\displaystyle(\Lambda_{B}^{2}-\Lambda_{T}^{2})\overline{\Lambda}_{B}^{2}\overline{\Lambda}_{T}^{2}\,\Big{[}m_{\phi}^{2}\big{(}4\overline{M}_{TB}\overline{M}_{T}+4\Delta_{B}\overline{M}_{B}+\Lambda_{B}^{2}+\Lambda_{T}^{2}\big{)}$
$\displaystyle\qquad\qquad\qquad\quad-\,2(\Lambda_{T}^{2}+\Lambda_{B}^{2})(\overline{M}_{TB}\overline{M}_{T}+\Delta_{B}\overline{M}_{B})-2\Lambda_{B}^{2}\Lambda_{T}^{2}\Big{]}\Bigg{\\}}\,.$

In the local limit, where the regulator parameters $\Lambda_{B,T}\to\infty$,
all of the splitting functions presented here are consistent with those
obtained using Pauli-Villars (PV) regularization in Ref. Wang  _et al._
(2020).

Using the above set of splitting functions, the strange quark PDF can be
computed in the form of convolutions with the strange quark PDFs in the
hadronic configurations in terms of the explicit hadronic configurations as
Wang  _et al._ (2020); Salamu  _et al._ (2019b)

$\displaystyle\Delta s(x)$ $\displaystyle=$
$\displaystyle\sum_{B\phi}\Big{(}\Delta\bar{f}_{B\phi}^{(\rm
rbw)}\otimes\Delta s_{B}+\Delta\bar{f}_{B\phi}^{(\rm KR)}\otimes\Delta
s_{B}^{(\rm KR)}\Big{)}+\ \sum_{\phi}\Delta\bar{f}_{\phi}^{(\rm
tad)}\otimes\Delta s_{\phi}^{(\rm tad)}$ (28) $\displaystyle+$
$\displaystyle\sum_{T\phi}\Delta\bar{f}_{T\phi}^{(\rm rbw)}\otimes\Delta
s_{T}+\sum_{TB\phi}\Delta\bar{f}_{TB\phi}\otimes\Delta s_{TB}\ ,$

where the symbol “$\otimes$” represents a convolution integral, and for
notational convenience we define the splitting functions
$\Delta\bar{f}_{j}(y)\equiv\Delta f_{j}(\bar{y})$. The input PDFs for the
strange quark in the hadronic configurations on the right-hand side of Eq.
(28) can be related to the valence quark unpolarized $u(x)$ and $d(x)$ PDFs
and polarized $\Delta u(x)$ and $\Delta d(x)$ PDFs in the proton by comparing
the coefficients of the axial-vector operators using SU(6) symmetry Wang  _et
al._ (2020). Specifically, the input PDFs $\Delta s_{B}$ in the rainbow
diagram with intermediate octet states are related to the polarized PDFs in
the proton as

$\displaystyle\Delta s_{\Lambda}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\big{(}2\Delta u(x)-\Delta
d(x)\big{)},~{}~{}~{}~{}~{}\Delta s_{\Sigma^{+}}(x)=\Delta
s_{\Sigma^{0}}(x)=\Delta d(x).$ (29)

The input PDFs in the Kroll-Ruderman diagrams, $\Delta s_{B}^{(\rm KR)}$, are
related to the unpolarized distributions in the proton by

$\displaystyle\Delta s_{\Lambda}^{(\rm KR)}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}\big{(}2u(x)-d(x)\big{)},~{}~{}~{}~{}~{}\Delta
s_{\Sigma^{+}}^{(\rm KR)}(x)=\sqrt{2}\Delta s_{\Sigma^{0}}^{(\rm
KR)}(x)=\frac{1}{\sqrt{2}}d(x).$ (30)

For $\Delta s_{\phi}^{(\rm tad)}$ in the tadpole diagram, we have the
relationship

$\displaystyle\Delta s_{K^{+}}^{(\rm tad)}(x)=\frac{1}{2}\Delta
u(x),~{}~{}~{}~{}~{}\Delta s_{K^{0}}^{(\rm tad)}(x)=\frac{1}{2}\Delta d(x).$
(31)

Finally, for the decuplet $\Delta s_{T}$ and octet-decuplet transition $\Delta
s_{TB}$, these can be expressed as

$\displaystyle\Delta s_{\Sigma^{*+}}(x)$ $\displaystyle=$ $\displaystyle\Delta
s_{\Sigma^{*0}}(x)=\frac{1}{2}\big{(}\Delta u(x)-2\Delta d(x)\big{)},$ (32)
$\displaystyle\Delta s_{\Sigma^{*+}\Sigma^{+}}(x)$ $\displaystyle=$
$\displaystyle-\Delta
s_{\Sigma^{*0}\Sigma^{0}}(x)=\frac{1}{\sqrt{3}}\big{(}\Delta u(x)-2\Delta
d(x)\big{)}.$ (33)

With these inputs, we can proceed to compute the strange helicity PDF in the
proton in Eq. (28) numerically, as we discuss next in the following section.

##  IV Numerical results

In this section we discuss the numerical results for the spin-dependent
strange quark distributions in the proton from the present analysis. In the
previous analysis of meson loop contributions to the spin-averaged strange
quark PDFs in the proton Salamu  _et al._ (2019a, b), the regulator parameter
$\Lambda$ was determined by fitting the cross section data for inclusive
baryon production in high-energy $pp$ scattering, $pp\to BX$, for different
species of baryon $B$. The best fit yielded the values $\Lambda_{B}=1.1(1)$
GeV and $\Lambda_{T}=0.8(1)$ GeV for the octet and decuplet intermediate
cases, respectively. In the present analysis of spin-dependent PDFs we use the
same parameters $\Lambda_{B}$ and $\Lambda_{T}$ and the coupling constants
$C_{B\phi}$ and $C_{T\phi}$ to compute the splitting functions numerically.

###  IV.1 Splitting functions

Figure 2: On-shell “(on)” and off-shell “(off)” contributions to the spin-
dependent splitting functions in the nonlocal EFT for (a) the octet baryon
$\Delta f_{B}$ and (b) the decuplet baryon $\Delta f_{T}$ and octet-decuplet
interference $\Delta f_{TB}$ intermediate states. The bands correspond to
regulator parameter values $\Lambda_{B}=1.0-1.2$ GeV for octet and
$\Lambda_{T}=0.7-0.9$ GeV for decuplet baryons.

The on-shell and off-shell contributions to the spin-dependent splitting
functions in the nonlocal EFT calculation for the strange octet, decuplet and
octet-decuplet baryon interference intermediate states are shown in Fig. 2,
with the bands corresponding to regulator cutoff values $\Lambda_{B}=1.0-1.2$
GeV and $\Lambda_{T}=0.7-0.9$ GeV for the octet and decuplet baryons,
respectively. The results are qualitatively similar to those found for the
splitting functions in the local EFT calculation with Pauli-Villars
regularization Wang  _et al._ (2020), although the magnitude there was
somewhat smaller. For the octet baryon case, the on-shell splitting function
$\Delta f_{B}^{(\rm on)}$ is negative at small meson momentum fractions $y$,
but changes sign to become positive for $y\gtrsim 0.3$. The off-shell
contribution $\Delta f_{B}^{(\rm off)}$ remains negative for all $y$ values,
and is $\approx 4$ times larger in magnitude at the peak $y\approx 0.1$ than
the on-shell contribution. Note that because of the mass difference between
the hyperon and nucleon, the off-shell function is nonzero at $y=0$.
Interestingly, compared with the corresponding spin-averaged splitting
functions from Ref. Salamu  _et al._ (2019a), the spin-dependent off-shell
function is identical, while the spin-averaged on-shell function is positive.

For intermediate state involving decuplet baryons, the splitting functions are
generally smaller in magnitude than for the octet baryons. Both the decuplet
on-shell and off-shell splitting functions are positive, and as in the octet
case the on-shell contributions vanish at $y=0$, while the off-shell
contributions remain nonzero. The octet-decuplet transition splitting
functions are significantly larger in magnitude than the decuplet functions,
indicating a greater role played by the $BT$ transition in spin-dependent
observables than for the diagonal $T$ contributions. The on-shell transition
function is negative, peaking at $y\approx 0.2$, while the off-shell
transition function is positive and decreases with $y$. Compared with the
corresponding splitting functions for the local EFT calculation with Pauli-
Villars regularization Wang  _et al._ (2020), the shapes are quite similar;
however, the magnitude of the transition contributions in the nonlocal case
are again larger.

###  IV.2 Strange quark polarization

Using the convolution formula (28), the polarized strange quark PDF is
evaluated in terms of the derived hadronic splitting functions and the PDFs in
the the hadronic intermediate state configurations. The $\delta(y)$ terms in
the splitting functions, although not shown in Fig. 2, make significant
contributions to the strange quark PDFs via the convolutions. With the SU(3)
relations, the strange quark PDFs for the intermediate states can be given in
terms of the spin-dependent and spin-averaged $u$ and $d$ quark PDFs in the
proton Wang  _et al._ (2020), which are determined from global QCD analyses of
high-energy polarized Ethier  _et al._ (2017); Nocera  _et al._ (2014);
Hartland and Nocera (2013) and unpolarized Ball  _et al._ (2017); Sato  _et
al._ (2020); Abrams  _et al._ ; Cocuzza  _et al._ (2021) scattering data. In
this analysis, for the spin-averaged $u$ and $d$ quark distributions in the
proton we use the CJ15 parametrization from Ref. Accardi  _et al._ (2016a),
while the polarized $\Delta u$ and $\Delta d$ PDFs are taken from the JAM
analysis in Ref. Ethier  _et al._ (2017).

Figure 3: Contributions to the strange quark polarization in the proton,
$x\Delta s(x)$, at $Q^{2}=1$ GeV2 from the octet rainbow, Kroll-Ruderman, and
tadpole diagrams [(a), (b)]; and the decuplet rainbow and decuplet-octet
transition rainbow diagrams [(c), (d)]. The uncertainty bands correspond to
the range of $\Lambda_{B}=1.0$–1.2 GeV for octet and $\Lambda_{T}=0.7$–0.9 GeV
for the decuplet and decuplet-octet transition. The left column [(a), (c)]
shows the decomposition according to the type of diagram, while the right
column [(b), (d)] shows the decomposition according to the type of splitting
function.

The contributions to the polarized strange PDF $x\Delta s$ from the various
terms in Eq. (28) are shown in Fig. 3, illustrating both decompositions in
terms of diagram types and in terms of splitting function types. In Fig. 3(a)
we observe large cancellations between contributions from the positive tadpole
and negative KR diagrams. In contrast, the magnitude of the octet rainbow
diagram contribution is relatively small, changing sign from negative at small
$x$ to positive at larger $x$. The sum of these contribution is positive for
$x\gtrsim 0.2$ with magnitude $\lesssim 0.004$. Compared with the calculation
in Ref. Wang  _et al._ (2020) which used PV regularization, the tadpole
contribution in the nonlocal case is significantly larger, and the
contribution from the KR diagram has the opposite sign. This is mainly because
the $\delta$-function contribution in the nonlocal calculation is much larger,
as can be seen in the decomposition in Fig. 3(b). Here, the (positive)
$\delta$-function term gives the largest contribution, while the (mostly
negative) on-shell and off-shell terms are somewhat smaller. The on-shell
contribution changes smoothly from positive at small $x$ to negative at
$x\gtrsim 0.2$, whereas the off-shell term has largest magnitude at low $x$.
The overall behavior of the octet contribution is driven by the
$\delta$-function contribution.

The contributions from diagrams involving decuplet baryons in the intermediate
states are shown in Fig. 3(c) and (d). There are strong cancellations seen
between the positive decuplet rainbow and the negative octet-decuplet
transition contributions, resulting in a total result that is negative with a
small magnitude ($\lesssim 0.002$), comparable to the total octet
contribution. Furthermore, in contrast to the octet case, the off-shell
decuplet contributions are positive, but cancelled somewhat by the negative
on-shell and $\delta$-function terms. Compared with the earlier calculation
Wang  _et al._ (2020) with PV regularization, the signs of each of the
decuplet rainbow and octet-decuplet transition terms is the same, but somewhat
different magnitudes result in a net contribution which is negative.

In Fig. 4 we plot the full results for the strange quark PDF, $x\Delta
s^{+}(x)$, including all octet, decuplet and transition contributions (the
contribution from meson loops to polarized antistrangeness $\Delta\bar{s}$ is
zero). For comparison, the results for $x\Delta s^{+}(x)$ from the NNPDF
Nocera  _et al._ (2014) and JAM Ethier  _et al._ (2017) global QCD analyses at
$Q^{2}=1$ GeV2 are also shown. The overall magnitude of the calculated strange
polarization in the nonlocal chiral effective theory is relatively small, with
$x\Delta s$ starting out negative at $x\lesssim 0.25$ and becoming positive at
larger $x$ values. The large uncertainty on the PDF parametrizations reflect
the weak constraints that currently exist on $\Delta s$ from data. Other
analyses in which the strange quark helicity-dependent distributions have been
extracted have been under the assumption of $\Delta s=\Delta\bar{s}$ de
Florian  _et al._ (2008); Leader  _et al._ (2010); Borsa  _et al._ (2020);
Blümlein and Böttcher (2010); Khorramian  _et al._ (2011). Similar to our
result, the sign change of $x\Delta s$ from negative to positive with
increasing $x$ was found in Refs. de Florian  _et al._ (2008); Leader  _et
al._ (2010); Borsa  _et al._ (2020), although the zero appeared in the
smaller-$x$ region. In some other extractions Blümlein and Böttcher (2010);
Khorramian  _et al._ (2011), $\Delta s$ was negative in the entire $x$ region
without any sign change, driven by the assumption of SU(3) flavor symmetry for
the axial vector charges. Recent lattice simulations of $x\Delta(s)$
Alexandrou  _et al._ (2021) also suggest a sign change at $x\sim 0.3$,
consistent with our result.

Figure 4: Meson loop contribution to the polarized strange quark PDF (red band) $x\Delta s^{+}\equiv x\Delta s+x\Delta\bar{s}$ compared with the results from the NNPDF Nocera  _et al._ (2014); Hartland and Nocera (2013) (orange band) and JAM Ethier  _et al._ (2017) (yellow band) global QCD analyses, at a scale $Q^{2}=1$ GeV2. The meson loop band reflects the range of cutoff parameters $\Lambda_{B}=1.0$–1.2 GeV and $\Lambda_{T}=0.7$–0.9 GeV for the octet and decuplet sectors, respectively.  Table 2:  Contributions from various diagrams to the integral of $\Delta s(x)$ at $Q^{2}=1$ GeV2, in units of $10^{-2}$. The sum of the total octet, tadpole, and total decuplet and octet-decuplet transition terms is in the range $\langle\Delta s\rangle=[-0.51,-0.26]\times 10^{-2}$. $\Lambda_{B}$ (GeV) |  $\langle\Delta s\rangle_{B\,\rm rbw}^{(\rm on)}$ |  $\langle\Delta s\rangle_{B\,\rm rbw}^{(\rm off)}$ |  $\langle\Delta s\rangle_{B\,\rm rbw}^{(\delta)}$ |  $\langle\Delta s\rangle_{\rm KR}^{(\rm off)}$ |  $\langle\Delta s\rangle_{\rm KR}^{(\delta)}$ |  total octet |  $\langle\Delta s\rangle_{\rm tad}^{(\delta)}$
---|---|---|---|---|---|---|---
1.0 |  0.02 | $-0.90$ |  0.73 |  0.87 | $-1.55$ | $-0.83$ |  0.84
1.1 |  0.00 | $-1.38$ |  1.18 |  1.32 | $-2.48$ | $-1.36$ |  1.35
1.2 | $-0.04$ | $-1.94$ |  1.73 |  1.86 | $-3.65$ | $-2.04$ |  1.99
$\Lambda_{T}$ (GeV) |  $\langle\Delta s\rangle_{T\,\rm rbw}^{(\rm on)}$ |  $\langle\Delta s\rangle_{T\,\rm rbw}^{(\rm off)}$ |  $\langle\Delta s\rangle_{T\,\rm rbw}^{(\delta)}$ |  $\langle\Delta s\rangle_{TB\,\rm rbw}^{(\rm on)}$ |  $\langle\Delta s\rangle_{TB\,\rm rbw}^{(\rm off)}$ |  $\langle\Delta s\rangle_{TB\,\rm rbw}^{(\delta)}$ |  total decuplet
0.7 |  0.04 |  0.02 |  0.04 | $-0.45$ |  0.42 | $-0.41$ |  $-0.34$
0.8 |  0.14 |  0.08 |  0.18 | $-0.83$ |  0.74 | $-0.77$ |  $-0.46$
0.9 |  0.33 |  0.22 |  0.53 | $-1.24$ |  1.05 | $-1.16$ |  $-0.27$

Integrating the strange polarized PDFs over all $x$, the resulting
contributions to the total moment $\langle\Delta
s\rangle\equiv\int_{0}^{1}\differential{x}\Delta s(x)$ are listed individually
in Table 2. This moment is especially interesting in view of its role in
neutrino transport in neutron stars, which is a key cooling mechanism Bollig
_et al._ (2017); Hobbs  _et al._ (2016). With the exception of the on-shell
component, the contributions from the octet states are larger than the
corresponding ones from the decuplet baryons, with the total octet
contribution several times larger than the total decuplet. For the range of
$\Lambda_{B,T}$ values considered in this analysis, the octet intermediate
state contribution to $\langle\Delta s\rangle$ is between $-0.0204$ and
$-0.0083$, while the contribution from decuplet intermediate states is between
$-0.0046$ and $-0.0027$. The large positive contribution from the tadpole
diagram cancels much of the (negative) octet and decuplet contributions,
leaving a net strange quark polarization in the proton to be in the range
$\langle\Delta s\rangle\approx[-0.0051,-0.0026]$. Note that although the first
moment of $\Delta s(x)$ is negative, the sign of the distribution is $x$
dependent. The overall negative value of $\langle\Delta s\rangle$ results from
the relatively larger negative $\Delta s(x)$ at small $x$, including at $x=0$,
compared with the smaller positive $\Delta s(x)$ at large $x$.

Interestingly, our results are comparable with those from Ref. Wang  _et al._
(2020) using PV regularization, where the lowest moment $\langle\Delta
s\rangle$ was also negative and in the range $[-0.0050,-0.0025]$. However, we
should note that although the total moments $\langle\Delta s\rangle$ turn out
to be similar, the individual contributions from the various terms are quite
different. In particular, the $\delta$-function terms for the intermediate
octet states in the nonlocal case are significantly larger than those in the
PV case. The calculated moment can also be compared with determinations from
the JAM global QCD analysis Ethier  _et al._ (2017), which yielded
$\langle\Delta s^{+}\rangle_{\scriptsize{\rm JAM}}=-0.03(10)$. Earlier
phenomenological analyses generally found more negative values of the first
moment of the strange quark polarization Leader  _et al._ (2010); Nocera  _et
al._ (2014); Khorramian  _et al._ (2011), which were, however, largely driven
by the assumption of SU(3) flavor symmetry for the axial vector charges.
Future data on semi-inclusive DIS and parity-violating inclusive DIS at the
planned Electron-Ion Collider Accardi  _et al._ (2016b) should reduce the
uncertainty on the extracted $\langle\Delta s^{+}\rangle$, and allow a better
discrimination between the $\Delta s$ and $\Delta\bar{s}$ distributions.

##  V Conclusion

Our main aim in this paper was to examine the generation of polarized
strangeness in the proton from meson loops computed within a nonlocal chiral
effective field theory at the one loop level. In contrast to previous
calculations using local versions of the effective theory, the nonlocal
implementation allows one to study hadron structure while a priori taking into
account the finite size of the hadrons in a natural and consistent way.

We derived explicit expressions for the spin-dependent proton to pseudoscalar
meson plus octet or decuplet baryon splitting functions, using a simple dipole
shape for the Fourier transform of the correlation function that describes the
hadrons’ extended structure. With the regulator parameters determined
phenomenologically from spin-averaged measurements of semi-inclusive hyperon
production in $pp$ collisions, the strange helicity PDF was computed from
convolutions of the splitting functions and the parton distributions
associated with the hadronic intermediate states. The contributions involving
octet baryons in the intermediate states were found to be several times larger
in magnitude than those involving decuplet baryons, and significant
cancellation was found between these and the tadpole contributions. The result
was a relatively small net polarized strange quark helicity, which was
negative at small $x$ and positive at high $x$.

Integrated over $x$, the lowest moment was found to lie in the range
$\langle\Delta s\rangle=[-5.1,-2.6]\times 10^{-3}$. Interestingly, this is
very similar to the results found in the previous calculations Wang  _et al._
(2020) using a local effective chiral theory with Pauli-Villars
regularization, even though the shape of the $\Delta s(x)$ distribution there
was somewhat different. This suggests that while the details of the meson loop
calculation depend on the prescription chosen to regularize the short-distance
behavior, the overall effect on the generated strange quark polarization is
relatively robust.

Our results are also qualitatively similar to those found in the recent global
QCD analysis by the JAM collaboration, $\langle\Delta
s^{+}\rangle_{\scriptsize{\rm JAM}}=-0.03(10)$ Ethier  _et al._ (2017), as
well as with the latest lattice QCD simulations from the ETM Collaboration,
$\langle\Delta s^{+}\rangle_{\scriptsize{\rm latt}}=-0.046(8)$ Alexandrou  _et
al._ (2020). We expect uncertainties in the determination of $\langle\Delta
s^{+}\rangle$ on both of these fronts to decrease as more experimental and
lattice data become available over the next few years.

## Acknowledgments

This work is supported by the Australian Research Council through the ARC
Discovery Project DP180100497 (AWT), the DOE Contract No. DE-AC05-06OR23177,
under which Jefferson Science Associates, LLC operates Jefferson Lab, DOE
Contract No. DE-FG02-03ER41260, and by the NSFC under Grant No. 11975241.

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