# Nucleon parton distributions from hadronic quantum fluctuations

Andreas Ekstedt<EMAIL_ADDRESS>Department of Physics and
Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden  Hazhar
Ghaderi<EMAIL_ADDRESS>Department of Physics and Astronomy,
Uppsala University, Box 516, SE-751 20 Uppsala, Sweden  Gunnar Ingelman
<EMAIL_ADDRESS>Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 20 Uppsala, Sweden  Swedish Collegium for Advanced
Study, Thunbergsvägen 2, SE-752 38 Uppsala, Sweden  Stefan Leupold
<EMAIL_ADDRESS>Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 20 Uppsala, Sweden

(June 19, 2019 )

###### Abstract

A physical model is presented for the non-perturbative parton distributions in
the nucleon. This is based on quantum fluctuations of the nucleon into baryon-
meson pairs convoluted with Gaussian momentum distributions of partons in
hadrons. The hadronic fluctuations, here developed in terms of hadronic chiral
perturbation theory, occur with high probability and generate sea quarks as
well as dynamical effects also for valence quarks and gluons. The resulting
parton momentum distributions $f(x,Q_{0}^{2})$ at low momentum transfers are
evolved with conventional DGLAP equations from perturbative QCD to larger
scales. This provides parton density functions $f(x,Q^{2})$ for the gluon and
all quark flavors with only five physics-motivated parameters. By tuning these
parameters, experimental data on deep inelastic structure functions can be
reproduced and interpreted. The contribution to sea quarks from hadronic
fluctuations explains the observed asymmetry between $\bar{u}$ and $\bar{d}$
in the proton. The strange-quark sea is strongly suppressed at low $Q^{2}$, as
observed.

##  I Introduction

The parton distribution functions (PDFs) of the nucleon are of great
importance. One reason is that they provide insights into the structure of the
proton and neutron as bound states of quarks and gluons, which is still a
largely unsolved problem due to our limited understanding of strongly coupled
QCD. Another reason is their use for calculations of cross-sections for high-
energy collision processes. These factorize in a hard parton level scattering
process, calculated in perturbation theory, and the flux of incoming partons
given by the PDFs.

This involves the factorization of processes that occur at momentum-transfer
scales of significantly different magnitudes. Of particular importance here is
that the PDFs $f(x,Q^{2})$ have the property that for $Q^{2}>Q_{0}^{2}\sim
1\,$GeV2 the dependence on $Q^{2}$ can be calculated by the Dokshitzer-Gribov-
Lipatov-Altarelli-Parisi equations (DGLAP) Altarelli and Parisi (1977);
Dokshitzer (1977); Gribov and Lipatov (1972) derived from perturbative QCD
(pQCD), which is well-established theoretically and experimentally confirmed.
However, the $x$-dependence needed at the starting scale $Q_{0}^{2}$ is not
known from fundamental principles and instead parametrized to reproduce proton
structure function data. This typically requires $x$-shapes given in terms of
five parameters for each parton flavor, resulting in $\sim 30$ free parameters
to account for valence quarks, gluons and sea quarks ($u$, $d$, $g$,
$\bar{u}$, $\bar{d}$, $s$, $\bar{s}$). There are different collaborations
Dulat  _et al._ (2016); Alekhin  _et al._ (2017); Ball  _et al._ (2017);
Harland-Lang  _et al._ (2015) performing such PDF parametrizations with DGLAP-
based $Q^{2}$-evolution that give good fits of proton structure data and are
excellent tools for cross-section calculations. However, the basic
$x$-dependence at $Q_{0}^{2}$ originating from the bound-state proton is here
only parametrized, but not understood.

To understand the basic shape of the parton momentum distributions in physical
terms, we here develop the theoretical basis for our earlier proposed model
Edin and Ingelman (1998); Alwall and Ingelman (2004, 2005) and elaborate on
its phenomenologically successful results. The first basic idea is to use the
uncertainty relation in position and momentum, $\Delta x\Delta p\sim\hbar/2$,
to give the basic momentum scale of partons confined in the length scale
$\Delta x$ given by the hadron diameter $D$. In the hadron rest frame it is
natural to assume a spherically symmetric Gaussian momentum distribution with
a typical width $\sigma\sim\hbar/(2D)$. The Gaussian is a convenient
mathematical form which cuts off large momenta that correspond to rare
fluctuations. It may be motivated as resulting from many soft interactions
within the hadron that add up to a Gaussian. The strength of this approach
lies in its simplicity and its small number of parameters.

The second basic idea is that whereas the valence quark and gluon
distributions are essentially given by the basic description of the bound
state bare nucleon and its quantum numbers, the sea quark distributions are
given by the hadronic fluctuations of the nucleon. For example, the proton
quantum state
$\ket{P}=\alpha_{\text{bare}}\ket{P}_{\text{bare}}+\alpha_{P\pi^{0}}\ket{P\pi^{0}}+\alpha_{n\pi^{+}}\ket{n\pi^{+}}+\cdots$
contains not only the bare proton but also nucleon-pion fluctuations with
probability amplitudes $\alpha_{N\pi}$. The point is that one should consider
the dominant quantum fluctuations in terms of least energy fluctuation and
thereby most long-lived Sullivan (1972); Ericson and Thomas (1983). It is
expected that pionic fluctuations dominate due to the small mass of the pion.
In turn, its smallness compared to a typical hadronic scale $\sim 1\,$GeV is a
consequence of spontaneous chiral symmetry breaking, which leads to the
identification of pions as Goldstone bosons Peskin and Schroeder (1995);
Scherer (2003). From these dominant fluctuations of the proton state, with the
presence of $\pi^{+}$ but lack of $\pi^{-}$, one expects an asymmetry in the
proton sea such that $\bar{d}>\bar{u}$ Thomas (1983); Melnitchouk  _et al._
(1991); Wakamatsu (1991); Henley and Miller (1990); Carvalho  _et al._ (2000);
Holtmann  _et al._ (1996); Kumano (1998); Speth and Thomas (1997); Kofler and
Pasquini (2017); Pobylitsa  _et al._ (1999); Dressler  _et al._ (2000), as is
also observed in data Towell  _et al._ (2001). In addition, an asymmetry in
the $\bar{s}-s$ distributions is expected because the dominant $\Lambda K^{+}$
fluctuation gives a harder momentum distribution of the heavier $\Lambda$, and
thereby of its $s$-quark, compared to the lighter $K^{+}$ and its $\bar{s}$
Alwall and Ingelman (2004, 2005); Signal and Thomas (1987); Melnitchouk and
Malheiro (1997, 1999).

This kind of hadronic fluctuations can in a simple phenomenological model Edin
and Ingelman (1998); Alwall and Ingelman (2005) be handled by having the
different baryon-meson ($BM$) fluctuation probabilities $|\alpha_{BM}|^{2}$ as
free parameters fitted to data. Here, we instead follow the theoretically
well-founded approach using the leading-order Lagrangian of three-flavor
chiral perturbation theory Jenkins and Manohar (1991); Pascalutsa and
Timmermans (1999); Pascalutsa  _et al._ (2007); Ledwig  _et al._ (2014) to
describe the proton state as the Fock expansion

$\displaystyle\ket{P}=$
$\displaystyle\alpha_{\text{bare}}\ket{P}_{\text{bare}}+\alpha_{P\pi^{0}}\ket{P\pi^{0}}+\alpha_{n\pi^{+}}\ket{n\pi^{+}}$
(1) $\displaystyle+$
$\displaystyle\alpha_{\Delta^{++}\pi^{-}}\ket{\Delta^{++}\pi^{-}}+\alpha_{\Delta^{+}\pi^{0}}\ket{\Delta^{+}\pi^{0}}$
$\displaystyle+$
$\displaystyle\alpha_{\Delta^{0}\pi^{+}}\ket{\Delta^{0}\pi^{+}}+\alpha_{\Lambda^{0}K^{+}}\ket{\Lambda^{0}K^{+}}+\cdots.$

For a detailed account of the chiral symmetry basis for the baryon-meson Fock
components, and subtle renormalization issues, we refer to Arndt and Savage
(2002); Chen and Ji (2001, 2002); Thomas  _et al._ (2000); Ji  _et al._
(2013); Wang  _et al._ (2016). The different terms in Eq. (1) are
theoretically well defined and related to each other with only three coupling
constants that are known from hadronic processes and weak decays of baryons.
In addition to the probability for the different hadronic fluctuations, the
theoretical formalism gives the hadron momentum distribution of the
fluctuations. Incorporating the hadronic momentum distributions with the above
parton momentum distributions in a hadron provides an improved model for the
parton momenta of the proton quantum state.

The PDFs are closely related to the proton structure functions that are
measured in deep inelastic scattering (DIS) of leptons on protons. The most
precise data are from electron and muon scattering, where the exchanged
virtual photon has high resolution power and couples to quarks in the proton.
The photon may therefore couple to a quark in the bare proton or in either the
baryon or the meson in a baryon-meson fluctuation.

Before giving the detailed account of model and results, we first elaborate on
the model’s general concepts and contrast it with other models. QCD covers
various energy regimes where different methods apply. For the sector of light
quarks, one has two energy scales at the microscopic level: the quark masses
of a couple of MeV, and $\Lambda_{\rm QCD}$ of a couple of 100 MeV—above which
the running coupling changes from strong to weak. Related to these microscopic
scales one can on the observable hadronic side identify the low-energy regime
on the order of the Goldstone boson (pion) masses (about 100 MeV), the medium-
energy regime on the order of the typical hadron masses (about 1 GeV) and the
high-energy regime much larger than the hadron masses. For the first and third
of these regimes there exists systematic tools used to quantify the
uncertainty of theoretical calculations and to improve on this uncertainty.
The tools at our disposal are chiral perturbation theory for the low-energy
regime and perturbative QCD for infrared safe quantities at high energies.
Unfortunately, there are no systematic tools available at medium energies
(except for lattice QCD Ma and Qiu (2018); Orginos  _et al._ (2017); Lin  _et
al._ (2015); Ji (2013)) to bridge the gap between these low- and high-energy
regimes. Thus, for the medium-energy regime and for the non-perturbative
quantities at high energies, one has to develop phenomenological models that
preferably link as much as possible to the systematic approaches.

In our model, we make sure that the hadronic part matches to chiral
perturbation theory at low energies. To extrapolate the PDFs from the medium-
energy regime to higher energies we use the DGLAP-equations of perturbative
QCD. The essential new part is the explicit construction of the
phenomenological model at a starting scale in the medium-energy regime. We
assume, first, hadronic fluctuations play an important role because we operate
in the medium-energy regime. Second, the quarks and gluons confined in a
hadron experience so many soft interactions that their momentum distributions
can be described by Gaussians. Clearly, these are assumptions as every
phenomenological model is based on some assumptions.

Some other models do not have proper matching to the low-energy regime, in
particular using a pseudoscalar instead of a pseudovector pion-nucleon
coupling. The Goldstone theorem demands that all interactions of the pions
vanish with the pion momenta, which is satisfied with the pseudovector
interaction but not with the pseudoscalar interaction. Other approaches use
sophisticated quark models to describe the quark-gluon aspects and/or
interlink the hadronic and quark degrees of freedom, as e.g. in pion-quark
models. In contrast to that, we clearly separate the hadronic fluctuations
from the quark-gluon distributions inside of the hadrons, because these are
described by different sets of quantum basis states with different degrees of
freedom. Wherever proper QCD theory (governed by chiral perturbation theory or
perturbative QCD) is not available, we use simplicity as a useful guiding
principle to get insights into the unknown dynamics of strongly interacting
systems. The strength of our model lies in the clear links to the better known
QCD regimes at higher and lower energies and otherwise in the simplicity of
our model. We want to explore how far we can come with such a model and which
insights can be obtained from it.

In this paper we present the complete model we have constructed based on these
basic ideas. Section II presents the formalism for DIS on the proton with its
hadronic fluctuations, where some more technical details are provided in
appendices at the end of the paper. In Section III we present our model for
the parton distributions in a probed hadron, i.e. the $x$-shape at the
starting scale $Q_{0}^{2}$ for pQCD evolution. Results are then presented in
Section IV in terms of obtained parton momentum distributions and their
ability to reproduce data on proton structure functions and quark sea
asymmetries. We give our conclusions in Section V.

##  II DIS on a nucleon with hadron fluctuations

The cross-section for deep inelastic lepton-nucleon scattering is
theoretically well known as a product of the leptonic and hadronic tensors,
$\mathrm{d}\sigma\propto l^{\mu\nu}W_{\mu\nu}$. The leptonic tensor is
straightforward to calculate and well known for photon exchange,
$l^{\mu\nu}=\operatorname{tr}[\not{p}_{l}^{\prime}\gamma^{\mu}\not{p}_{l}\gamma^{\nu}]/2$,
as well as for $W$ or $Z$ exchange. We consider both electromagnetic and weak
interactions.

(a)

(b)

(c)

Figure 1: Deep inelastic scattering on (a) the bare proton and on (b) the
meson or (c) the baryon in a baryon-meson quantum fluctuation of the proton.

The hadronic tensor $W_{\mu\nu}$ is a much more complex object and is of prime
interest here. In order to take into account the proton target with its
hadronic fluctuations, as illustrated in Fig. 1, we decompose the hadronic
tensor to include the possibilities to probe either the bare proton or the
meson or baryon in a fluctuation as follows

$\displaystyle
W_{\mu\nu}=W_{\mu\nu}^{\text{bare}}+W_{\mu\nu}^{H}=W_{\mu\nu}^{\text{bare}}+\sum_{BM}\left(W_{\mu\nu}^{MB}+W_{\mu\nu}^{BM}\right)$
(2)

where the notation $MB$ and $BM$ denotes probing the meson and baryon,
respectively. The general form of the hadronic tensor is Jaffe (1985)

$\displaystyle
W_{\mu\nu}=\frac{1}{4\pi}\int\\!\mathrm{d}^{4}\xi\,\mathrm{e}^{\mathrm{i}q\xi}\Braket{P}{J_{\mu}(\xi)J_{\nu}(0)}{P}$
(3)

in terms of the hadronic current $J_{\mu}(\xi)$ as a function of the spacetime
coordinate $\xi$. Using light-cone time-ordered perturbation theory Brodsky
_et al._ (1998) we calculate the here introduced part corresponding to the
hadronic fluctuations giving

$\displaystyle
W_{\mu\nu}^{H}=\frac{1}{4\pi}\int\\!\mathrm{d}^{4}\xi\,\mathrm{e}^{\mathrm{i}q\xi}\sum_{BM,\lambda}\int_{0}^{1}\\!\frac{\mathrm{d}y}{y}$
$\displaystyle\Big{\\{}f_{MB}^{\lambda}(y)\Braket{M(p_{M}^{+}=yp^{+})}{\left[J^{M}_{\mu}(\xi),J^{M}_{\nu}(0)\right]}{M(p_{M}^{+}=yp^{+})}$
(4)
$\displaystyle\phantom{\sum_{BM,\lambda}\int_{0}^{1}\\!\frac{\mathrm{d}y}{y}\,\Big{\\{}}+$
$\displaystyle
f_{BM}^{\lambda}(y)\Braket{B^{\lambda}(p_{B}^{+}=yp^{+})}{\left[J^{B}_{\mu}(\xi),J^{B}_{\nu}(0)\right]}{B^{\lambda}(p_{B}^{+}=yp^{+})}\Big{\\}}$

where the first term is for DIS probing the meson ($M$) and the second term
for probing the baryon ($B$). Expressions equivalent to (4) can be found in
the literature Kofler and Pasquini (2017).

The integration variable is the fraction $y$ of the proton’s energy-momentum
carried by the meson or baryon. Following common practice in DIS theory we use
light-cone momenta $p^{+}=p^{0}+p^{3}$ and $p^{-}=p^{0}-p^{3}$, and thereby
$y_{i}=\frac{p_{i}^{+}}{p^{+}}~{}(i=B,M)$. This has the advantage of being
independent of longitudinal boosts, e.g. from the proton rest frame to the
commonly used infinite-momentum frame. The light-cone momenta $p_{i}^{-}$ are
given by the on-shell condition which in the $\boldsymbol{p}_{\perp}=0$ frame
become $p_{i}^{-}=\frac{m_{i}^{2}+k_{\perp}^{2}}{p_{i}^{+}},~{}(i=B,M)$.

In (4) the sum runs over all baryon-meson pairs, with helicity $\lambda$ of
the baryon. We have included all baryons in both the octet and decuplet of
flavor SU(3), and all the Goldstone bosons represented by the mesons in the
spin-zero octet. Naturally, the fluctuations with a pion will dominate due to
its exceptionally low mass. Kaons are needed to get the leading contribution
for the strange-quark sea. Table 1 shows the relative strengths of different
fluctuations due to the couplings to be discussed further below.

The dynamical behavior depends on the hadronic distribution functions

$\displaystyle f_{BM}(y)=\sum_{\lambda}f^{\lambda}_{BM}(y),$ (5)

which are probability distributions for the physical proton to fluctuate to a
baryon-meson pair. The baryon carries a light-cone fraction $y$ and the meson
the remaining momentum fraction, i.e. satisfying the relation

$\displaystyle f_{BM}(y)=f_{MB}(1-y),$ (6)

giving flavor and momentum conservation for each particular hadronic
contribution. This ensures that all parton momentum sum rules come out
correctly Szczurek and Speth (1993). The hadronic distribution functions are
explicitly given in Eqs. (LABEL:E:_HadDist,LABEL:E:_HadDist2). Their explicit
form depends on the Lagrangian used for the hadronic fluctuations, to which we
now turn.

The relevant part of the leading-order chiral Lagrangian describing the
interaction of spin 1/2 and spin 3/2 baryons with spin 0 mesons (as Goldstone
bosons) is given by Jenkins and Manohar (1991); Pascalutsa and Timmermans
(1999); Pascalutsa  _et al._ (2007); Ledwig  _et al._ (2014)

$\displaystyle\mathcal{L}_{\text{int}}=\frac{D}{2}\operatorname{tr}(\bar{B}\gamma^{\mu}\gamma_{5}\\{u_{\mu},B\\})+\frac{F}{2}\operatorname{tr}(\bar{B}\gamma^{\mu}\gamma_{5}[u_{\mu},B])-\frac{h_{A}}{m_{R}}\frac{\epsilon_{ade}g_{\mu\nu}\epsilon^{\rho\mu\alpha\beta}}{2\sqrt{2}}\left[\left(\partial_{\alpha}\bar{T}_{\beta}^{abc}\right)\gamma_{5}\gamma_{\rho}u^{\nu}_{bd}B_{ce}+\bar{B}_{ec}u^{\nu}_{db}\gamma_{5}\gamma_{\rho}\partial_{\alpha}T^{abc}_{\beta}\right],$
(7)

where ‘tr’ refers to flavor trace. Here, the $B_{ab}$ are the matrix elements
of the matrix $B$ representing the octet baryons. The decuplet baryons are
represented by the totally symmetric flavor tensor $T_{abc}^{\mu}$. Similarly,
the spin 0 octet mesons are represented by a matrix $\Phi$ appearing in the
Lagrangian through $u_{\mu}$ given by
$u_{\mu}\equiv\mathrm{i}u^{\dagger}(\nabla_{\mu}U)u^{\dagger}=u_{\mu}^{\dagger}$
where $u^{2}\equiv U=\exp(\mathrm{i}\Phi/F_{\pi})$. For further details see
Appendix A.

From this Lagrangian we derive the non-zero terms when applied to our cases of
a proton fluctuating into a meson together with an octet or decuplet baryon

$\displaystyle\mathcal{L}^{P\to B_{\text{oct}}M}=$
$\displaystyle\Bigg{[}\frac{-D-F}{\sqrt{2}F_{\pi}}\bar{n}\gamma^{\mu}\gamma_{5}(\partial_{\mu}\pi^{-})-\frac{D+F}{2F_{\pi}}\bar{P}\gamma^{\mu}\gamma_{5}(\partial_{\mu}\pi^{0})+\frac{D-3F}{2\sqrt{3}F_{\pi}}\bar{P}\gamma^{\mu}\gamma_{5}(\partial_{\mu}\eta)$
(8)
$\displaystyle-\frac{D-F}{2F_{\pi}}\bar{\Sigma}^{0}\gamma^{\mu}\gamma_{5}(\partial_{\mu}K^{-})-\frac{D-F}{\sqrt{2}F_{\pi}}\bar{\Sigma}^{+}\gamma^{\mu}\gamma_{5}(\partial_{\mu}\bar{K}^{0})+\frac{D+3F}{2\sqrt{3}F_{\pi}}\bar{\Lambda}\gamma^{\mu}\gamma_{5}(\partial_{\mu}K^{-})\Bigg{]}P+\text{h.c.}$

and

$\displaystyle\mathcal{L}^{P\to B_{\text{dec}}M}=$
$\displaystyle\frac{h_{A}\varepsilon^{\rho\mu\alpha\beta}}{2m_{R}F_{\pi}}\Bigg{[}\sqrt{\frac{1}{3}}(\partial_{\alpha}\bar{\Sigma}^{*+}_{\beta})\gamma_{5}\gamma_{\rho}(\partial_{\mu}\bar{K}^{0})-\sqrt{\frac{1}{6}}(\partial_{\alpha}\bar{\Sigma}^{*0}_{\beta})\gamma_{5}\gamma_{\rho}(\partial_{\mu}K^{-})$
(9) $\displaystyle+$
$\displaystyle(\partial_{\alpha}\bar{\Delta}^{++}_{\beta})\gamma_{5}\gamma_{\rho}(\partial_{\mu}\pi^{+})-\sqrt{\frac{2}{3}}(\partial_{\alpha}\bar{\Delta}^{+}_{\beta})\gamma_{5}\gamma_{\rho}(\partial_{\mu}\pi^{0})-\sqrt{\frac{1}{3}}(\partial_{\alpha}\bar{\Delta}^{0}_{\beta})\gamma_{5}\gamma_{\rho}(\partial_{\mu}\pi^{-})\Bigg{]}P+\\!\text{h.c.}$

respectively. The effective nature of the hadronic theory —manifested by the
appearance of the derivative couplings of the form
$\sim\gamma_{5}\gamma^{\mu}\partial_{\mu}M(z)$ in the Lagrangians— introduces
a slight ambiguity for the meson momentum $p_{M}$ appearing in the numerators
in the application of the light-cone time-ordered framework. In the
literature, there are two common choices for the meson momentum appearing in
the numerators Holtmann  _et al._ (1996); Kofler and Pasquini (2017),

$\displaystyle
p_{M}^{(A)}=\left(p^{+}_{P}-p^{+}_{B},p^{-}_{P}-p^{-}_{B},\boldsymbol{p}_{P\perp}-\boldsymbol{p}_{B\perp}\right),$
(10) $\displaystyle
p_{M}^{(B)}=\left(p^{+}_{P}-p^{+}_{B},\frac{m_{M}^{2}+p_{M\perp}^{2}}{p_{M}^{+}},\boldsymbol{p}_{P\perp}-\boldsymbol{p}_{B\perp}\right).$
(11)

We find that these two choices give nearly identical results concerning the
extracted values of our model’s parameters and hence both choices yield
similar conclusions. But even though choice (10) gives a slightly better shape
for the flavor asymmetry, to be discussed in Section IV.3, we will use choice
(11) because this choice is in line with the Goldstone theorem Goldstone  _et
al._ (1962) whereas choice (10) is not, as explicitly shown in Appendix B.

As discussed in Appendix A, the parameter values are as follows Granados  _et
al._ (2017). The pion decay constant $F_{\pi}=92.4\text{ MeV}$ and the
couplings $D=0.80,F=0.46$ Cabibbo  _et al._ (2003) and $h_{A}=2.7\pm 0.3$ with
an uncertainty range to include partial decay width data on $\Delta\rightarrow
N\pi$ and $\Sigma^{*}\rightarrow\Lambda\pi$ as well as the large-$N_{C}$ limit
Dashen and Manohar (1993); Pascalutsa and Vanderhaeghen (2006)
$h_{A}^{\text{large-$N_{C}$}}=\frac{3}{\sqrt{2}}g_{A}=2.67$ where
$g_{A}=F+D=1.26$ is well constrained by the beta decay of the neutron
Patrignani  _et al._ (2016).

Using light-cone time-ordered perturbation theory, the Lagrangians (8,9) lead
to the hadronic distribution functions

$\displaystyle
f^{\lambda}_{BM}(y)=\frac{1}{2y(1-y)}\int\\!\frac{\mathrm{d}^{2}k_{\perp}}{(2\pi)^{3}}\,\left|g_{BM}~{}G(y,k_{\perp}^{2},\Lambda^{2})\frac{S^{\lambda}\left(y,\boldsymbol{k}_{\perp}\right)}{m_{P}^{2}-m^{2}(y,k_{\perp}^{2})}\right|^{2},$
(12) $\displaystyle
f^{\lambda}_{MB}(y)=\frac{1}{2y(1-y)}\int\\!\frac{\mathrm{d}^{2}k_{\perp}}{(2\pi)^{3}}\,\left|g_{BM}~{}G(1-y,k_{\perp}^{2},\Lambda^{2})\frac{S^{\lambda}\left(1-y,\boldsymbol{k}_{\perp}\right)}{m_{P}^{2}-m^{2}(1-y,k_{\perp}^{2})}\right|^{2}$
(13)

for the baryon and meson, respectively, probed in the fluctuation. As
required, they satisfy $f_{BM}(y)=f_{MB}(1-y)$. The various hadronic couplings
$g_{BM}$ are provided in Table 1 and the vertex functions
$S^{\lambda}(y,\boldsymbol{k}_{\perp})$ are given in Appendix B. The
suppression of the energy fluctuation is seen as the propagator with the
difference of the squared masses of the proton and the baryon-meson system
given by

$\displaystyle
m^{2}(y,k_{\perp}^{2})\equiv\frac{m_{B}^{2}+k_{\perp}^{2}}{y}+\frac{m_{M}^{2}+k_{\perp}^{2}}{1-y}.$
(14)

Table 1: The proton to baryon-meson fluctuations, with couplings and their strength relative to the respective largest coupling $g_{BM}^{\text{max}}$, where $g_{DM}^{\text{max}}=g_{\Delta^{++}\pi^{-}}$ and $g_{OM}^{\text{max}}=g_{n\pi^{+}}$.  $BM$ | $\Delta^{++}\pi^{-}$ | $\Delta^{+}\pi^{0}$ | $\Delta^{0}\pi^{+}$ | $\Sigma^{*+}K^{0}$ | $\Sigma^{*0}K^{+}$ | $n\pi^{+}$ | $P\pi^{0}$ | $\Lambda K^{+}$ | $\Sigma^{+}K^{0}$ | $\Sigma^{0}K^{+}$ | $P\eta$
---|---|---|---|---|---|---|---|---|---|---|---
$g_{BM}$ | $\frac{h_{A}}{2m_{R}F_{\pi}}$ | $\frac{-h_{A}}{\sqrt{6}m_{R}F_{\pi}}$ | $\frac{-h_{A}}{2\sqrt{3}m_{R}F_{\pi}}$ | $\frac{h_{A}}{2\sqrt{3}m_{R}F_{\pi}}$ | $\frac{-h_{A}}{2\sqrt{6}m_{R}F_{\pi}}$ | $\frac{-D-F}{\sqrt{2}F_{\pi}}$ | $\frac{-D-F}{2F_{\pi}}$ | $\frac{D+3F}{2\sqrt{3}F_{\pi}}$ | $\frac{-D+F}{\sqrt{2}F_{\pi}}$ | $\frac{-D+F}{2F_{\pi}}$ | $\frac{D-3F}{2\sqrt{3}F_{\pi}}$
$\left|\frac{g_{BM}}{g_{BM}^{\text{max}}}\right|^{2}$ | 1 | 0.67 | 0.33 | 0.33 | 0.17 | 1 | 0.5 | 0.5 | 0.08 | 0.04 | 0.03

The function $G(y,k_{\perp}^{2},\Lambda_{H}^{2})$ is a cut-off form factor,
which is used to avoid the integral getting an unphysical divergence. The
physics issue to account for is the fact that the description in terms of
hadronic degrees of freedom is only valid at hadronic scales, whereas for
higher momentum-transfer scales parton degrees of freedom should be used. To
phase out the hadron formalism it is convenient to introduce a suitably
constructed form factor.

In practice, it is conceivable to cut on the virtuality of the fluctuation
Holtmann  _et al._ (1996) or on the modulus of the three-momentum (in a proper
reference frame). While the first option sounds plausible from a point of view
of Heisenberg’s uncertainty relation (or Fermi’s Golden Rule), this quantum-
mechanical aspect is already accounted for by the just mentioned propagator in
Eqs. (LABEL:E:_HadDist,LABEL:E:_HadDist2). An additional such cut is therefore
artificial. Instead we choose to cut off the three-momentum of the hadrons in
the fluctuation as seen in the rest frame of the proton. If relevant at all,
high-momenta fluctuations should be of partonic not hadronic nature.

To conserve the condition $f_{BM}(y)=f_{MB}(1-y)$ it is necessary to use a
symmetric combination of the meson/baryon three-momentum and a natural choice
is to use the average of the squares of the three-momenta of the meson and
baryon. To make this manifestly frame-independent we write its value in the
proton rest frame expressed in a Lorentz-invariant form and take the form
factor to be

$\displaystyle
G\left(y,k_{\perp}^{2},\Lambda_{H}^{2}\right)=\exp\left[-\frac{\mathcal{A}^{2}\left(y,k_{\perp}\right)}{2\Lambda_{H}^{2}}\right]$
(15)

where $\Lambda_{H}$ is the parameter that regulates the suppression of larger
scales. Because this is related to the switch to partonic degrees of freedom,
one would expect it to be of the same order as the starting scale $Q_{0}$ of
the pQCD formalism. The function $\mathcal{A}^{2}$ in the form factor is given
by

$\displaystyle\mathcal{A}^{2}\left(y,k_{\perp}\right)\equiv\left(\boldsymbol{p}^{2}_{B}+\boldsymbol{p}^{2}_{M}\right)|_{p^{+}=m_{P}}=\frac{(p\cdot
p_{B})^{2}+(p\cdot p_{M})^{2}}{m_{P}^{2}}-m_{B}^{2}-m_{M}^{2}$ (16)
$\displaystyle=\left(\frac{m_{B}^{2}+k_{\perp}^{2}}{2m_{P}y}\right)^{2}+\left(\frac{m_{M}^{2}+k_{\perp}^{2}}{2m_{P}(1-y)}\right)^{2}+k_{\perp}^{2}-\frac{m_{B}^{2}+m_{M}^{2}}{2}+\frac{m_{P}^{2}}{4}\left[(1-y)^{2}+y^{2}\right],$

where light-cone momenta $p_{B}^{+}=yp^{+}$ and $p_{M}^{+}=(1-y)p^{+}$ have
been used to obtain the last expression. This form factor regularizes any
potential end-point ($y=0,1$) singularities. Furthermore, high values of
$k_{\perp}$ are largely suppressed which renders the integrals in Eqs.
(LABEL:E:_HadDist,LABEL:E:_HadDist2) finite and restricts the hadronic
fluctuations to the low-momentum scales where the hadronic language is
applicable.

Using this theoretical formalism we illustrate the total fluctuation
probability for a proton to a $BM$ pair by calculating

$\displaystyle\lvert\alpha_{BM}(\Lambda_{H})\rvert^{2}=\int_{0}^{1}\\!\mathrm{d}y\,f_{BM}(y)$
(17)

for both momentum choices, Eqs. (10,11), giving the result shown in Fig. 2.
One observes that the probability for a proton to fluctuate into a baryon-
meson state is quite sizable. Notably, for a cut-off $\Lambda_{H}$ around
$1\,$GeV, the contribution from the baryon-decuplet members (mainly from the
$\Delta$’s) is comparable in size to the nucleon-pion fluctuations.

Figure 2: The baryon-meson fluctuation probability as a function of the cut-
off parameter $\Lambda_{H}$ for two different choices of meson momentum. The
solid [dashed] curves refer to choice (11) [(10)]. The two upper curves (blue)
are the sum of all the octet-baryon–meson probabilities. The two lower curves
(red) are the sum of all the decuplet-baryon–meson probabilities.

Due to the hadronic fluctuations, the PDFs for the proton are given by a
convolution of the hadronic distributions, Eqs.
(LABEL:E:_HadDist,LABEL:E:_HadDist2), and the PDFs for the hadron being
probed. Thus, the PDF for a parton $i$ in the proton can be written in the
form Holtmann  _et al._ (1996); Pasquini and Boffi (2006); Kofler and Pasquini
(2017)

$\displaystyle f_{i/P}$ $\displaystyle(x)=~{}f_{i/P}^{\text{bare}}(x)$ (18)
$\displaystyle+$ $\displaystyle\sum_{H\in
B,M}\int\\!\mathrm{d}y\,\mathrm{d}z\,\delta(x-yz)f^{\text{bare}}_{i/H}(z)f_{H/P}(y),$

taking into account the contributions from the bare proton and the $BM$
fluctuations. In our approach, the PDF for the ‘bare’ part in any of these
contributions (bare proton, baryon or meson in a fluctuation) is obtained from
a Gaussian as mentioned in the Introduction and to be discussed in Section
III. These bare distributions contain constituent quarks and gluons, but no
sea quarks.

In this work we include all the admissible octet-baryon–meson and decuplet-
baryon–meson pairs in the fluctuations, i.e. the $\ket{N\pi}$,
$\ket{\Delta\pi}$, $\ket{\Lambda K}$, $\ket{\Sigma K}$, and
$\ket{\Sigma^{*}K}$ fluctuations. The $\ket{P\eta}$ contribution can be
neglected due to mass suppression and its very small coupling to the proton
state, see Table 1. The nucleon-pion and the Delta-pion fluctuations give the
largest contributions, while the $\ket{\Lambda K}$, $\ket{\Sigma K}$ and
$\ket{\Sigma^{*}K}$ fluctuations act as small corrections. However, because
neither $\ket{N\pi}$ nor $\ket{\Delta\pi}$ contribute to the strange sea,
other fluctuations like $\ket{\Lambda K}$ while being small are the leading
hadronic contributions to the strange sea. It is found that the $\ket{\Lambda
K}$ fluctuation is most important, while the $\ket{\Sigma K}$ and
$\ket{\Sigma^{*}K}$ fluctuations are suppressed due to a small coupling and
the larger masses involved respectively, see Table 1.

Once the starting distributions have been obtained from the convolution model
at a particular starting scale $Q^{2}_{0}$, the PDFs are obtained for higher
$Q^{2}$ by DGLAP evolution. The DGLAP evolution is performed at next-to-
leading order (NLO) using the QCDNUM package Botje (2011).

##  III Generic model for parton distributions in a hadron

For any bare hadron we are considering the parton momentum distributions for
its valence quarks/antiquarks and a gluon component. This applies for both the
above considered bare proton as well as for the baryons and mesons in a
hadronic fluctuation.  We therefore now consider deep inelastic scattering
(DIS) on such a generic hadron. The DIS formalism was developed and is
conventionally interpreted in the infinite momentum frame (IMF) where the
hadron has a large momentum such that parton masses and transverse momenta are
kinematically negligible. The essential momentum axis is defined through the
measurement given by the probe, e.g. a virtual photon. The formalism is Zavada
(1997) also applicable in the target hadron rest frame and does not prefer any
special reference system, with IMF as a limiting case. Our model for PDFs of
the bound hadron state has a basic physics motivation originating in the
hadron rest frame, but because we define the parton’s energy-momentum fraction
$x={k^{+}}/{p_{H}^{+}}$ using light-cone momenta ($k^{+}$ for the parton and
$p_{H}^{+}$ for the hadron) this key variable is independent of longitudinal
boosts. Thereby our model should be applicable in any frame of interest for
DIS.

In the rest frame of such a generic bare hadron there is no preferred
direction. Therefore the spherical symmetry motivates the assumption that the
parton’s momentum distributions in $k_{x},k_{y}$ and $k_{z}$ are the same.
Assuming a Gaussian momentum distribution for these components provides a
convenient mathematical form which suppresses large momenta that should
correspond to rare momentum fluctuations.  Because a Gaussian results
mathematically from adding many small contributions, it can be argued to be
applicable here to represent the added effect of many soft momentum exchanges
within the bound-state hadron in the absence of a proper description derived
from QCD. The four-momentum distribution for a parton of type $i$ and mass
$m_{i}$ is therefore assumed to be given by Edin and Ingelman (1998); Alwall
and Ingelman (2005)

$\displaystyle F_{i/H}(k)$ $\displaystyle=N_{i/H}(\sigma_{i},m_{i})$ (19)
$\displaystyle\times$
$\displaystyle\exp\left(-\frac{(k_{0}-m_{i})^{2}+k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}{2\sigma_{i}^{2}}\right)$

where $N$ is a normalization factor.

The width $\sigma$ of this Gaussian is expected to be physically given by the
uncertainty relation $\Delta x\Delta p\sim\hbar/2$ that enforces increasing
momentum fluctuations for a particle confined in a smaller spatial range.
Thus, for a hadron of size $D$ (diameter) one expects $\sigma\sim\hbar/(2D)$
and therefore being typically of order $0.1\,$GeV.

The PDF for a parton $i=q$, $\bar{q}$, $g$ of mass $m_{i}$ in the hadron $H$
is then given by

$\displaystyle f^{\text{bare}}_{i/H}(x)={}$
$\displaystyle\int^{\prime}\\!\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\,\delta\left(\frac{k{{}^{+}}}{p_{H}^{+}}-x\right)F_{i/H}(k)$
(20)

where $x={k^{+}}/{p_{H}^{+}}$ is the discussed light-cone energy-momentum
fraction of a parton in the hadron. The prime on the integral sign in (20)
indicates the kinematical constraints that the quark four-momentum $k$ must
obey. We demand that the scattered parton must be on-shell or have a timelike
virtuality (causing final-state QCD radiation) limited by the mass of the
hadronic system, i.e. $m_{i}^{2}<j^{2}=(k+q)^{2}<(p_{H}+q)^{2}$. Likewise, the
hadron remnant must have a four-vector $r^{2}=(p_{H}-k)^{2}>0$, cf. Fig. 1a.

From a conceptual point of view it is interesting to note that the constraints
involving the photon momentum $q$ bring in the influence of the quantum
mechanical measurement process on the distribution. Thus the originally
spherically symmetric function (19) is reshaped into a distribution that
contains the directional information originating from the virtual photon that
probes the hadron.

As one consequence of these kinematical constraints, the light-cone energy-
momentum fraction $x$ is automatically restricted to its physical range
$0<x<1$. The constraint that the remnant should have timelike momentum,
$(p_{H}-k)^{2}>0$, has a related interesting consequence. In the hadron rest
frame this constraint translates to

$\displaystyle k_{\perp}^{2}<x(1-x)m_{H}^{2}-(1-x)k^{2}\,.$ (21)

Obviously this inequality is then true for any frame that leaves $k_{\perp}$
untouched, i.e. any frame between the hadron rest frame and the infinite-
momentum frame. On account of (21), the integral measure
$k_{\perp}\,\mathrm{d}k_{\perp}$, which is a part of $\mathrm{d}^{4}k$ in
(20), vanishes for $x\to 1$. Thus, in this limit the PDF smoothly vanishes. To
summarize, the kinematical constraints ensure that $f^{\text{bare}}_{i/H}(x)$
is only non-vanishing for $0<x<1$ and that $\lim\limits_{x\to
1}f^{\text{bare}}_{i/H}(x)=0$.

The normalizations $N_{q/H}(\sigma_{q},m_{q})$ are fixed by the flavor sum
rules, i.e. the integrals giving the correct numbers of different valence
quark flavors. $N_{g/H}(\sigma_{g},0)$ is fixed by the momentum sum rule, i.e.
to get the sum of $x$-weighted integrals to be unity.

Thus, the only free parameters are the Gaussian widths $\sigma_{g}$,
$\sigma_{1}$, $\sigma_{2}$, where the indices refer to the widths of the
distributions for the gluon and for the quark flavors represented by one quark
($\int_{0}^{1}\\!\mathrm{d}x\,f^{\text{bare}}_{q/H}(x)=1$) or two quarks
($\int_{0}^{1}\\!\mathrm{d}x\,f^{\text{bare}}_{q/H}(x)=2$) in the probed
hadron. For instance, $\sigma_{2}$ ($\sigma_{1}$) applies for $u$ ($d$) in the
proton.  Distributions of quarks appearing triply in a baryon, such as the $u$
distribution in the $\Delta^{++}$ baryon, could be given a different Gaussian
width $\sigma_{3}$. However, the final distributions are not very sensitive to
$\sigma_{3}$. For simplicity we choose to determine such distributions by
making use of isospin symmetry relations such as
$f^{\text{bare}}_{u/\Delta^{++}}(x)=2f^{\text{bare}}_{u/\Delta^{+}}(x)-f^{\text{bare}}_{d/\Delta^{+}}(x)$.

With this model we have chosen a minimalistic approach with the same Gaussian
distributions for all partons, having a width that only depends on the number
of same-flavor quarks, but not on the particular hadron considered. Of course,
one could introduce more complexity requiring more parameters, but we find it
more interesting to see what insights this minimal physics-motivated model can
give.

The above parametrization automatically conserves isospin (e.g.
$f^{\text{bare}}_{u/P}(x)=f^{\text{bare}}_{d/n}(x)$ and similarly for the
other hadrons). With the above widths for all possible hadrons, the
distributions only depend on mass effects via the mentioned kinematical
constraints.

It should be noted that these PDFs can be analytically evaluated in terms of
error functions Alwall and Ingelman (2005), but in practice it is more
convenient to evaluate them numerically. As discussed, these bare
distributions will only contain valence quarks and gluons, whereas the sea
distributions will be entirely generated by hadronic fluctuations. All the
resulting PDFs are at the low hadronic scale to be used as starting
distributions at $Q_{0}^{2}$ for DGLAP evolution to large scales $Q^{2}$.

##  IV Model results based on data comparison

###  IV.1 The few adjustable parameters

The model introduced above has few parameters which are expected to lie in a
limited range in order for the model to make sense.

The description of hadronic fluctuations is controlled by three coupling
strengths with values already fixed by data from various hadronic processes.
As discussed in connection with Table 1 above, the coupling $g_{A}=F+D=1.26$
is constrained to the $1\%$ level from the beta decay of the neutron
Patrignani  _et al._ (2016) whereas $D=0.80$ and $F=0.46$ may vary
independently by $\sim\pm 5\%$ as long as their sum is fixed Cabibbo  _et al._
(2003). Because it is their sum that appears in the most probable
fluctuations, a variation in $D$ and $F$ has a negligible effect on the
results. For the decuplet coupling we take $h_{A}=2.7\pm 0.3$. Because
$h_{A}/m_{R}$, with $m_{R}$ the resonance mass (basically $m_{\Delta}$),
appears as the effective coupling in the decuplet Lagrangian (9), we vary the
ratio

$\displaystyle
1.737~{}\text{GeV}^{-1}<h_{A}/m_{\Delta}<2.435~{}\text{GeV}^{-1}$ (22)

to see the resulting sensitivity on this uncertainty (see Appendix A for
details).

The only newly introduced parameter in the hadron fluctuation model is the
regulator for the high-momentum suppression, $\Lambda_{H}$. This parameter is
constrained to have a value large enough to allow hadronic fluctuations of
some baryon-meson configurations, i.e. energy fluctuations of at least a few
hundred MeV. On the other hand, it must be small enough to ensure a separation
between the hadronic and partonic degrees of freedom. Thus, a reasonable
expectation is a value in the range
$0.5~{}\text{GeV}\lesssim\Lambda_{H}\lesssim 1$ GeV.

Based on the model it is expected that the $\sigma$ parameters have values on
the order of $0.1\,$GeV and the $Q_{0}$ a value on the order of $1\,$GeV. The
former is given, as discussed above, by the inverse size of hadrons and the
latter by the factorization scale of non-perturbative bound hadron state
dynamics from the pQCD description  at higher momentum scales. For the
evolution of parton density functions at $Q^{2}>Q_{0}^{2}$ we use the NLO
DGLAP equations with the running coupling $\alpha_{s}(Q^{2})$, with
$\alpha_{s}(m_{Z}^{2})=0.1190$ in agreement with the measured value Tanabashi
_et al._ (2018). In addition, we expect that $Q_{0}\sim\Lambda_{H}$. However,
because $Q_{0}$ and $\Lambda_{H}$ are defined in two different formalisms, the
partonic and hadronic respectively, and there is no theoretically well-defined
link between these two descriptions, one cannot a priori take them as being
the same parameter. Still, as will be seen below they do come out to have the
same value within their uncertainties.

###  IV.2 Comparison with proton structure function data

The values of the just discussed parameters are obtained from inclusive deep
inelastic lepton-proton scattering giving the proton structure functions
$F_{2}$ and $xF_{3}$. Figures 3-4 show $\mu P$ data from NMC Arneodo  _et al._
(1995) and BCDMS Benvenuti  _et al._ (1989), neutrino data from CDHSW, NuTeV
and CHORUS Berge  _et al._ (1991); Tzanov  _et al._ (2006); Önengüt  _et al._
(2006) and $eP$ data from H1 Adloff  _et al._ (2001) in comparison to our
model results.  Our objective is not to obtain the best possible fit in terms
of lowest $\chi^{2}$ in a global fit of all relevant data and thereby compete
with conventional PDF parametrizations of the different $xf_{i}(x,Q_{0}^{2})$
having some 30 free parameters. Instead, our aim is to gain understanding
through our physically motivated model with only 5 parameters of physical
significance and with expected values in order for the model to make sense. We
have therefore not made a global fit to all data, but rather investigated the
importance of our few parameters for different observables.  The parameters
$\sigma_{1}$, $\sigma_{2}$, $\Lambda_{H}$, and $Q_{0}$ can be nailed down
using $F_{2}$ and $xF_{3}$ data. $Q_{0}$ and $\sigma_{g}$ are given by the
small-$x$ $F_{2}$ data: With $Q_{0}$ given, it’s always possible to fit data
by varying $\sigma_{g}$. We find that the following parameter values give the
best overall result

$\displaystyle\sigma_{1}=0.11~{}\text{GeV},\sigma_{2}=0.22~{}\text{GeV},\sigma_{g}=0.028~{}\text{GeV},$
(23) $\displaystyle\Lambda_{H}=0.87~{}\text{GeV},Q_{0}=0.88~{}\text{GeV}.$

(a)

(b)

(c)

Figure 3:  Model predictions for proton structure functions
LABEL:sub@Fig:_F2Large $F_{2}$, LABEL:sub@Fig:_F3nutevCDHSW $xF_{3}$, and
LABEL:sub@Fig:_F2neu $F_{2}^{\nu}$ as function of $Q^{2}$ for different
$x$-bins compared to data on fixed target $\mu P$ scattering from the New Muon
Collaboration (NMC) Arneodo  _et al._ (1995) and BCDMS Benvenuti  _et al._
(1989), and from neutrino-scattering experiments CDHS Berge  _et al._ (1991),
NuTeV Tzanov  _et al._ (2006), and CHORUS Önengüt  _et al._ (2006).

Figure 4: The proton structure function $F_{2}$ as a function of $x$ for
various $Q^{2}$-bins. Our model curve compared to data from the H1 $eP$
collider experiment Adloff  _et al._ (2001).

Notice that the fit results in $\Lambda_{H}$ and $Q_{0}$ being practically the
same, confirming our expectation that this scale constitutes the transition
from hadron to parton degrees of freedom in the model. Moreover, the Gaussian
widths are found to be of the expected magnitude $\sim 0.1$ GeV. The gluon
distribution is particularly soft, which may seem surprising. However, the
above argument based on the uncertainty relation gives
$\sigma\sim\hbar/(2D)=56$ MeV for the proton charge radius 0.875 fm Patrignani
_et al._ (2016). In view of the symmetry properties of two-particle wave
functions of indistinguishable states it should not be surprising that the
momentum distribution for quark flavors that appear singly in the hadron
differ from the one for quark flavors that appear pairwise.

Considering the fact that the model has effectively only four parameters,
which are also constrained by the physics assumptions of the model, it is
remarkable that such a large amount of structure function data can be
reasonably well described. Admittedly, there are some kinematical regions of
some experimental data sets where deviations do occur, but the general
behavior is reproduced and substantial $(x,Q^{2})$ ranges are well fitted.
The large $Q^{2}$-range covered by the HERA data in Fig. 4 is quite well
described as given by the DGLAP equations and can be further improved by
switching to NNLO DGLAP evolution. Because we are also interested in polarized
parton distributions Ekstedt  _et al._ (2018) where splitting functions are
only known to NLO precision, we choose to use NLO evolution for both the
polarized and the unpolarized distributions.

It is therefore of interest to look into some details on the $x$-shapes of
individual parton densities as they emerge from the model including both the
hadronic fluctuations and the probed hadron’s generic parton density
description, but without any pQCD evolution. This is shown in the top panel of
Fig. 5, where the overall shape of the valence quark distributions is quite
similar to conventional PDF parametrizations.  The fact that the distributions
go smoothly to zero, $xf_{i}(x)\to 0$, for $x\to 1$ is not due to a choice of
a particular form of the PDFs, such as including a factor $(1-x)^{a}$ as in
most parametrizations of PDFs. Instead this behavior is due to the kinematical
constraints on the DIS process, as explained above in connection with Eq.
(20). This is particularly important for a proper model of the valence quark
distributions. Further characteristic features are that the gluon distribution
is quite large for smaller $x$ and the sea quarks are suppressed but not at
all negligible.  So there is a non-trivial contribution of non-perturbatively
generated sea quarks in the bound-state proton. Examining the sea quark
distributions one notes the different distributions for $\bar{u}$ and
$\bar{d}$, on the one hand, and for $s$ and $\bar{s}$, on the other. This is
the basis for asymmetries in the light sea and strange sea, as will be further
discussed below.

The effect on the PDFs from pQCD evolution using the DGLAP equations is shown
in the middle and lower panels of Fig. 5. Due to the $\log{Q^{2}}$-dependent
evolution there is a quick increase from $Q_{0}^{2}$ so that already at
$Q^{2}=1.3$ GeV2 the perturbatively generated sea quarks and gluons dominate
at small $x$ over the originally non-perturbative sea.

The PDFs obtained at the starting scale $Q^{2}_{0}$ are evaluated numerically.
However, for illustrative purposes the starting distributions for a parton $i$
can be parametrized in the convenient form $xf_{i}(x)=a~{}x^{b}(1-x)^{c}$. The
fitted coefficients for the various distributions are given in Table 2.

Table 2: Parametrization $xf_{i}(x)=ax^{b}(1-x)^{c}$ at $Q_{0}^{2}$.  Distribution |  $a$ |  $b$ |  $c$
---|---|---|---
$x\bar{s}$ |  0.81 |  1.4 |  14.
$xs$ |  6.7 |  2.2 |  16.
$x\bar{d}$ |  0.97 |  0.69 |  7.9
$xd$ |  11. |  1.1 |  5.8
$x\bar{u}$ |  0.54 |  0.71 |  8.6
$xu$ |  5.5 |  0.84 |  2.4
$xg$ |  5700. |  1.6 |  47.
Figure 5: The resulting PDFs $xf_{i}(x)$ of the proton at the starting scale
$Q_{0}^{2}=0.77~{}\text{GeV}^{2}$ and at $Q^{2}=1.3~{}\text{GeV}^{2}$ and
$Q^{2}=10~{}\text{GeV}^{2}$.

###  IV.3 The $\bar{d}$-$\bar{u}$ asymmetry

From a pQCD point of view, the momentum distribution of the $\bar{d}$ and
$\bar{u}$ sea in the proton should be similar because
$m_{u},m_{d}\ll\Lambda_{\text{QCD}}$, $Q_{0}$. This is, however, not the case
as seen in data from e.g. Towell  _et al._ (2001) where a clear asymmetry is
seen (cf. Fig. 6). Such an asymmetry arises naturally from hadronic
fluctuations of the proton where the non-perturbative sea distributions are
dominantly generated by the pions Thomas (1983); Melnitchouk  _et al._ (1991);
Wakamatsu (1991); Henley and Miller (1990); Carvalho  _et al._ (2000);
Holtmann  _et al._ (1996); Kumano (1998); Speth and Thomas (1997); Kofler and
Pasquini (2017). The energy-wise lowest fluctuations are $P\pi^{0}$ and
$n\pi^{+}$, where the former does not contribute to the $\bar{d}$-$\bar{u}$
asymmetry Because the $\pi^{0}$ is symmetric in $\bar{d}d$ and $\bar{u}u$.
Taking only these nucleonic fluctuations into account gives already decent
agreement with data on the difference $x\bar{d}-x\bar{u}$ as shown by the
dotted curve in Fig. 6 (upper panel). However, these nucleonic fluctuations
are not sufficient to explain the ratio $\bar{d}(x)/\bar{u}(x)$ as shown by
the dotted curve in the lower panel of Fig. 6.

Figure 6: The light-quark sea asymmetry in terms of the difference
$x\bar{d}(x)-x\bar{u}(x)$ (upper panel) and the ratio $\bar{d}(x)/\bar{u}(x)$
(lower panel). Data from Fermilab E866/NuSea collaboration Towell  _et al._
(2001) compared to our model result. The dotted curve takes into account only
nucleon-pion ($N\pi$) fluctuations. The shaded band accounts for all
fluctuations, with $\Delta\pi$ being most important among the decuplet
contributions. The shaded band is obtained by a variation in the decuplet
coupling $h_{A}/m_{\Delta}$ where the solid (dashed) curve refers to the
largest (smallest) value of the decuplet coupling in Eq. (22).

The results become better when also including fluctuations with other baryons.
In particular the $\ket{\Delta^{++}\pi^{-}}$ state, having the largest
decuplet coupling (see Table 1) and having $\bar{u}$ in the $\pi^{-}$,
contributes significantly to bring the curves down to the data points. The
full octet and decuplet contribution is shown in Fig. 6 where the band
represents a variation in the decuplet coupling $h_{A}/m_{\Delta}$, with the
largest (smallest) value in Eq. (22) corresponding to the solid (dashed)
curve. As seen in the figure an $n\%$ variation in the coupling results in an
$n\%$ variation in the difference $x\bar{d}-x\bar{u}$ for small $x\lesssim
0.15$. The variation has a slightly smaller impact on the ratio
$\bar{d}/\bar{u}$, but the variation is essentially of the same order of
magnitude as that of $x\bar{d}-x\bar{u}$.

###  IV.4 The strange sea of the proton

Due to the possibility of the proton to fluctuate into $\ket{\Lambda K^{+}}$,
$\ket{\Sigma K}$ and $\ket{\Sigma^{*}K}$ states a non-perturbative strange sea
will arise, as shown in Fig. 5. It is suppressed relative to the light-quark
sea partly due to the kinematical suppression of these fluctuations with
higher-mass hadrons, but also due to the smaller hadronic couplings shown in
Table 1. Moreover, the $x$-distributions of $s$ and $\bar{s}$ are not the
same, but $s$ has a harder momentum distribution than $\bar{s}$ Signal and
Thomas (1987); Melnitchouk and Malheiro (1997, 1999); Alwall and Ingelman
(2004). This is a kinematic effect arising from that the $s$ quark is in the
baryon, due to its higher mass, gets a harder $y$-spectrum in the hadronic
fluctuation than the lighter meson containing the $\bar{s}$. The dominance of
kaons in the low-$x$ region and similarly the dominance of strange baryons in
the higher-$x$ region is clearly seen in the ratio $(s-\bar{s})/(s+\bar{s})$
in Fig. 7. Here one can also see how the additional symmetric $s\bar{s}$ from
$g\to s\bar{s}$ in pQCD reduces this ratio with increasing $Q^{2}$. Because
pQCD fills up the low-$x$ region to a higher degree, the kaon effect is more
depleted than the ‘baryon peak’, which is, however, shifted to lower $x$.

Figure 7: The ratio $(s-\bar{s})/(s+\bar{s})$ as a function of $x$ evaluated
for different values of $Q^{2}$.

The symmetric $s\bar{s}$ sea from the $\log{Q^{2}}$ DGLAP evolution builds up
quickly and dominates at small-$x$ already for $Q^{2}=1.3~{}\text{GeV}^{2}$,
as shown in the middle panel of Fig. 7. Thus, the asymmetry is only expected
to be visible at quite low $Q^{2}$ and therefore hard to observe
experimentally.

The extraction of the strange sea from data is not at all trivial because it
requires some additional observable to signal that an $s$ or $\bar{s}$ has
been probed. In Fig. 8 our model is compared to data on the total strange sea
$\left(xs(x)+x\bar{s}(x)\right)/2$. The CCFR data Bazarko  _et al._ (1995) are
obtained from neutrino-nucleon scattering producing a charm quark decaying
semileptonically giving an opposite sign dimuon signature, i.e.
$\nu_{\mu}+N\to\mu^{-}+c+X$ where $c\to s+\mu^{+}+\nu_{\mu}$ or
$\bar{\nu}_{\mu}+N\to\mu^{+}+\bar{c}+X$ where
$\bar{c}\to\bar{s}+\mu^{-}+\bar{\nu}_{\mu}$. The charged-current subprocess
$W^{+}s\to c$ or $W^{-}\bar{s}\to\bar{c}$ is here the essential point. Other
sources of charm production, such as $W^{+}g\to c\bar{s}$ or
$W^{-}g\to\bar{c}s$, or other sources of dimuon production from other decays
must be taken into account to extract a proper measure of the strange sea, as
discussed in Bazarko  _et al._ (1995). The result shows that although the
shape difference between the $xs(x)$ and $x\bar{s}(x)$ distributions is
consistent with zero, it has large uncertainties. CCFR assumed
$xs(x)=x\bar{s}(x)$ for extracting the data points shown in Fig. 8.

The more recent result of HERMES Airapetian  _et al._ (2014) is obtained from
data on the multiplicities of charged kaons in semi-inclusive deep-inelastic
electron-proton scattering. This requires a detailed and non-trivial analysis
of the fragmentation function into kaons to extract the contribution from
initial-state strange quarks in the basic DIS process $\gamma s\to s$ or
$\gamma\bar{s}\to\bar{s}$. As seen in Fig. 8 the CCFR and HERMES results
differ substantially and do not provide a clear result on the strange sea. Our
model result agrees reasonably well with the HERMES result, but compared to
CCFR it has a too small strange sea at low $Q^{2}$. Because the strange-quark
sea is not yet well determined, we contribute with some further
investigations.

Figure 8: The strange-quark sea, $(xs+x\bar{s})/2$, as a function of $x$ for
different values of $Q^{2}$, with our model results compared to data from
Bazarko  _et al._ (1995); Airapetian  _et al._ (2014). The CCFR analysis
assumes $xs(x)=x\bar{s}(x)$.

The strange-quark content of the proton can be characterized by the momentum
fraction carried by the strange sea relative to the light-quark sea or the
non-strange quark content Bazarko  _et al._ (1995)

$\displaystyle\kappa=\frac{\int_{0}^{1}\\!\mathrm{d}x\,[xs\left(x,Q^{2}\right)+x\bar{s}\left(x,Q^{2}\right)]}{\int_{0}^{1}\\!\mathrm{d}x\,[x\bar{u}\left(x,Q^{2}\right)+x\bar{d}\left(x,Q^{2}\right)]},$
(24)
$\displaystyle\eta=\frac{\int_{0}^{1}\\!\mathrm{d}x\,[xs\left(x,Q^{2}\right)+x\bar{s}\left(x,Q^{2}\right)]}{\int_{0}^{1}\\!\mathrm{d}x\,[x{u}\left(x,Q^{2}\right)+x{d}\left(x,Q^{2}\right)]},$

where $\kappa=1$ would mean a flavor SU(3) symmetric sea. These ratios are
shown in Fig. 9 versus $Q^{2}$, where the qualitative behavior is
understandable within our model. At $Q_{0}^{2}$ there is, as discussed, only a
small non-perturbative strange-quark sea from hadron fluctuations. With
increasing $Q^{2}$ the perturbative $\log{Q^{2}}$ evolution first builds up
the $s\bar{s}$ sea quickly and then flattens off at larger scales (note the
logarithmic $Q^{2}$ scale in the figure).

Figure 9:  The ratios $\kappa$ (upper panel) and $\eta$ (lower panel) in Eq.
(LABEL:E:_Kappa-Eta) for the strange-quark content of the proton as calculated
in our model. The data points at $Q^{2}=22.2~{}\text{GeV}^{2}$ are from the
CCFR Collaboration Bazarko  _et al._ (1995).

The proton sea is, however, not flavor SU(3) symmetric as indicated by the
value of $\kappa$ and quantified by the strange-sea suppression factor

$\displaystyle
r_{s}(x,Q^{2})=\frac{s(x,Q^{2})+\bar{s}(x,Q^{2})}{2\bar{d}(x,Q^{2})}.$ (25)

Our results for this quantity are shown in Fig. 10 together with ATLAS data
Aad  _et al._ (2012, 2014). As seen for $Q^{2}$ slightly larger than the
starting value for the QCD evolution $Q_{0}^{2}$, the suppression factor is
constant and near unity for $x\lesssim 0.01$. For low $x$ this is in agreement
with the $ePWZ$-fit of Aad  _et al._ (2012). For larger $x$ our model gives
$r_{s}\left(0.023,1.9~{}\text{GeV}^{2}\right)\approx 0.62$, which is
consistent within uncertainties of experimental observations:
$r_{s}(0.023,1.9~{}\text{GeV}^{2})=0.56\pm 0.04$ Alekhin  _et al._ (2015),
$r_{s}(0.023,1.9~{}\text{GeV}^{2})=1.00^{+0.25}_{-0.28}$ Aad  _et al._ (2012)
and $r_{s}(0.023,1.9~{}\text{GeV}^{2})=0.96^{+0.26}_{-0.30}$ Aad  _et al._
(2014). As seen in Fig. 10, $r_{s}\rightarrow 1$ as $x\rightarrow 0$, this
supports the hypothesis that the quark sea at low $x$ is flavor symmetric.

Figure 10: The strange-sea suppression factor $r_{s}=(s+\bar{s})/(2\bar{d})$
as a function of $x$ evaluated for different values of $Q^{2}$. Data from the
ATLAS collaboration Aad  _et al._ (2012, 2014).

Figure 11: Upper panel: The ratios $\kappa$ and $\eta$, evaluated at
$Q^{2}=22.2~{}\text{GeV}^{2}$, as function of the hadron fluctuation regulator
$\Lambda_{H}$. The middle curve is the ratio $\kappa/\eta$. Lower panel: The
probability for all the hadronic fluctuations containing strangeness (dashed
curve) and not containing any strangeness (solid curve), as a function of
$\Lambda_{H}$. The ratio of these probabilities is also shown (dot-dashed
curve). Notice the two different scales on the vertical axes.

For completeness we show in Fig. 11 (top panel) the dependence of the strange-
sea ratios $\kappa$ and $\eta$ on the hadron fluctuation regulator
$\Lambda_{H}$. Whereas $\kappa$ strongly depends on $\Lambda_{H}$, $\eta$ is
almost independent of $\Lambda_{H}$. This can be understood from the plot in
the lower panel of the same figure which compares the non-strange fluctuation
probability $\mathcal{P}_{ns}$ (e.g. for $\ket{N\pi}$ and $\ket{\Delta\pi}$)
and the probability $\mathcal{P}_{s}$ that the proton fluctuates into a hadron
pair that does contain strangeness. Not only is
$\mathcal{P}_{s}\ll\mathcal{P}_{ns}$, but also its slope is much smaller
implying that for increasing $\Lambda_{H}$ the rate of population growth is
much larger for those fluctuations that contain $\bar{u}$ and $\bar{d}$
quarks, than those containing $s$ and $\bar{s}$ quarks
($\Delta\mathcal{P}_{s}/\Delta\Lambda_{H}\approx 5\%~{}\text{GeV}^{-1}$ and
$\Delta\mathcal{P}_{ns}/\Delta\Lambda_{H}\approx 90\%~{}\text{GeV}^{-1}$
between $0.5~{}\text{GeV}\leq\Lambda_{H}\leq 1.0~{}\text{GeV}$). Hence
$\kappa$ depends much more strongly on $\Lambda_{H}$ than does $\eta$ due to
appearance of $\bar{u}$ and $\bar{d}$ distributions in its definition. As
shown in the lower panel of Fig. 11, at the regulator value of
$\Lambda_{H}=0.87$ GeV, roughly 1% of the fluctuations contain strangeness.
This can be compared to the result obtained in Ref. Alwall and Ingelman
(2005), where the strangeness fluctuations had to constitute 5% in order to
reproduce the then available CCFR data.

If it turns out to be a need for a larger non-perturbative strange-quark sea
than in our present model, this might be remedied by a minor modification of
the model. One option could be a flavor-dependent momentum cutoff
$\Lambda_{H}$, but to keep our model as simple as possible we refrained from
introducing more parameters. An alternative explanation might come from the
importance of additional degrees of freedom not considered so far. In the
strangeness $S=-1$ sector there are four baryonic states below the antikaon-
nucleon threshold: $\Lambda$, $\Sigma$, $\Sigma^{*}(1385)$ and
$\Lambda^{*}(1405)$. The first three have been taken into account in our
approach as the strangeness counterparts of the nucleon and the $\Delta(1232)$
considered in the pion-baryon fluctuations. But we have not included the
$\Lambda^{*}(1405)$ in our framework. On the one hand, we found that the
comparatively heavy $K\Sigma^{*}(1385)$ fluctuation is much less important
than the lighter $K\Lambda$. This suggests that also $K\Lambda^{*}(1405)$ is
negligible. On the other hand, the negative-parity $\Lambda^{*}(1405)$ couples
with an $s$-wave to nucleon-antikaon while all our interactions are of
$p$-wave nature. This can enhance the importance of the $\Lambda^{*}(1405)$.
The ultimate reason why we have not explored its influence in the present work
is the absence of unambiguous experimental information about the coupling
strength between a nucleon and $K\Lambda^{*}(1405)$. This is related to the
long-standing question about the nature of the $\Lambda^{*}(1405)$. Being
lighter than all non-strange baryons with negative parity, it has been
speculated because a long time Dalitz  _et al._ (1967) that the
$\Lambda^{*}(1405)$ is merely an antikaon-nucleon bound state instead of a
three-quark state; see, for instance Hall  _et al._ (2015) for further
discussion and references. This would point to a relatively large coupling
strength. Yet in view of these theoretical uncertainties we have not pursued a
detailed analysis of the $K\Lambda^{*}(1405)$ fluctuation as long as there is
no clear need for an enhancement of the strange sea.

###  IV.5 Pion PDF

With our model parameters fixed by DIS data, it is possible to find the
distributions for other hadrons. The pion PDFs can be obtained by the method
given in Section III. Here we do not take into account hadronic fluctuations
for the following reasons. From the point of view of many-body theory the
first contributions that one would consider are hadronic two-body
fluctuations. However, due to parity conservation the pion cannot fluctuate
into a pair of pions. Two-body states with a pion and a mesonic resonance are
already quite far away from the pion mass shell. This applies even more to
nucleon-antinucleon fluctuations. Next one might consider three-body
fluctuations, in particular three pions. But for the nucleon we have not taken
into account three-body states like two pions and a nucleon. There one might
even run into a double-counting problem because the $\Delta$ is essentially an
elastic resonance in the pion-nucleon channel, i.e. pion-$\Delta$ fluctuations
contain a significant part of the three-body fluctuations into two pions and a
nucleon. These reservations apply also to the pion. In addition, the three-
pion fluctuations are suppressed at low energies due to the chiral dynamics of
the pions. In technical terms, the derivative couplings of the Goldstone
bosons lead to a suppression of the two-loop diagrams Scherer (2003) that
correspond to the three-pion fluctuations. For all these reasons we will
neglect hadronic effects for the pion.

Figure 12: Model predictions for the valence quark distribution
$xv^{\pi}=xu^{\pi^{+}}-x\bar{u}^{\pi^{+}}$, compared with parametrizations
from Barry  _et al._ (2018) (left) and Aicher  _et al._ (2010) (right).

Because experimental data are limited to the pion valence region, we choose to
focus on this region. The comparison of our model with parametrizations from
other groups Barry  _et al._ (2018); Aicher  _et al._ (2010) is shown in Fig.
12. Obviously, we obtain a quite good agreement, which gives further credits
to our generic model for the parton distributions in a hadron. It would be
interesting to compare our predictions for other mesons with data. However,
because our primary interest lies in the nucleon we leave this comparison for
a dedicated study.

##  V Conclusions

This study has demonstrated that the momentum distribution of partons in the
proton, and thereby the observed proton structure functions, can be understood
in terms of basic physical processes. We thereby obtain new knowledge
regarding the poorly understood non-perturbative dynamics of the bound-state
proton. Using the well-established pQCD DGLAP equations for the
$Q^{2}$-dependence above the scale $Q_{0}^{2}$, our model developed here
addresses the basic shape in the distribution of the energy-momentum fraction
$x$ carried by different parton species at $Q_{0}^{2}$. Thus, the model treats
the physics at the transition from bound-state hadron degrees of freedom to
the internal parton degrees of freedom. It does so by convoluting hadronic
quantum fluctuations with partonic fluctuations. To describe the former we use
the leading-order Lagrangian of chiral perturbation theory, the low-energy
effective theory that respects the symmetries of QCD as the underlying theory.
The partonic fluctuations arise quantum mechanically due to confinement within
the small size of a hadron, as given by the uncertainty relation in position
and momentum.

Interestingly the fit that gives best agreement with the structure functions
$F_{2}$ and $xF_{3}$ data yields a value where the hadronic language ends and
QCD evolution begins to be $\Lambda_{H}=Q_{0}=0.87~{}\text{GeV}$.

Thus, having a model with effectively only four dimensionful parameters with
physically meaningful values, it is highly non-trivial that we obtain a very
satisfying reproduction of a large amount of data.  The model’s assumption on
the origin of non-perturbative sea quarks at the scale $Q_{0}^{2}$ being only
given by the hadronic fluctuations, works well to describe data when also
including the perturbatively generated sea quarks by the DGLAP evolution at
higher $Q^{2}$. Thus, there is no need to complicate the model with some
additional, unknown source of a non-perturbative quark sea. In particular we
find that the $n\pi^{+}$ and the $\Delta\pi$ fluctuations generate the flavor
asymmetry $x\bar{d}-x\bar{u}>0$ in the proton sea, to a large extent
consistent with experimental data. This shows that the model captures the
essential physics observed.

Regarding the strange-quark sea of the proton, arising from proton
fluctuations into strange hadrons, we find that it is substantially suppressed
due to the larger masses of strange hadrons. An asymmetry in terms of
different $x$-distribution for $s$ and $\bar{s}$, with $s(x)$ being harder, is
found. However, this effect is reduced at larger $Q^{2}$ due to the
development of the symmetric $s\bar{s}$ sea from $g\to s\bar{s}$ in pQCD. The
remaining asymmetry at observed $Q^{2}$ is too small to be seen in present
data. Further details of the non-perturbative strange sea of our model are
given to promote future studies, including the potentially interesting
inclusion of the $\Lambda^{*}(1405)$ in the hadronic fluctuations.

We have here considered PDFs of the proton and the pion where most
experimental information is available for testing our model. The model is,
however, quite general and can give the parton momentum distributions in any
hadron.

Based on the phenomenological success of the model and its theoretical basis,
spin degrees of freedom and the proton spin puzzle are studied in another
paper Ekstedt  _et al._ (2018).

## Acknowledgments

We acknowledge helpful discussions with C. G. Granados and U. Aydemir at an
early stage of this project. This work was supported by the Swedish Research
Council under contract 621-2011-5107.

##  Appendix A The relevant Lagrangians

For the metric we use $g=\text{diag}(+1,-1,-1,-1)$ and $\epsilon^{0123}=+1$.
The relevant part of the leading-order chiral Lagrangian describing the
interaction of Goldstone bosons with nucleons and spin 3/2 baryons is given by
Jenkins and Manohar (1991); Pascalutsa and Timmermans (1999); Pascalutsa  _et
al._ (2007); Ledwig  _et al._ (2014)

$\displaystyle\mathcal{L}_{\text{int}}={}$
$\displaystyle\frac{D}{2}\operatorname{tr}(\bar{B}\gamma^{\mu}\gamma_{5}\\{u_{\mu},B\\})+\frac{F}{2}\operatorname{tr}(\bar{B}\gamma^{\mu}\gamma_{5}[u_{\mu},B])$
(26) $\displaystyle+$
$\displaystyle\frac{1}{2\sqrt{2}}h_{A}\epsilon_{ade}g_{\mu\nu}\left(\bar{T}^{\mu}_{abc}u^{\nu}_{bd}B_{ce}+\bar{B}_{ec}u^{\nu}_{db}T_{abc}^{\mu}\right).$

Relativistic Rarita-Schwinger fields exhibit some problematic features related
to how to handle its spin-1/2 components. Apart from exchanging spin-3/2
resonances, the Lagrangian (26) induces an additional unphysical contact
interaction. This can be cured by subscribing to the Pascalutsa prescription,
which in our case means making the substitution Pascalutsa and Timmermans
(1999); Pascalutsa  _et al._ (2007)

$\displaystyle
T^{\mu}\rightarrow-\frac{1}{m_{R}}\epsilon^{\nu\mu\alpha\beta}\gamma_{5}\gamma_{\nu}\partial_{\alpha}T_{\beta},$
(27)

where $m_{R}$ refers to the resonance mass
($m_{R}=m_{\Delta},m_{\Sigma^{*}}$). Note that this substitution induces an
explicit flavor breaking but these effects are beyond leading order.

In (26) $B_{ab}$ is the entry in the $a$th row, $b$th column of the matrix
representing the octet baryons

$\displaystyle
B=\begin{pmatrix}\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&\Sigma^{+}&P\\\
\Sigma^{-}&-\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2}{\sqrt{6}}\Lambda\end{pmatrix}.$ (28)

The Goldstone bosons are contained in

$\displaystyle\Phi=\begin{pmatrix}\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-\frac{2}{\sqrt{3}}\eta\end{pmatrix}$ (29)

and $u_{\mu}$ is essentially given by

$\displaystyle
u_{\mu}\equiv\mathrm{i}u^{\dagger}(\partial_{\mu}U)u^{\dagger}=u_{\mu}^{\dagger}~{}\text{where}~{}u^{2}\equiv
U=\exp(\mathrm{i}\Phi/F_{\pi}).$ (30)

Finally, the decuplet is represented by a totally symmetric flavor tensor

$\displaystyle
T^{111}=\Delta^{++},T^{112}=\frac{1}{\sqrt{3}}\Delta^{+},T^{122}=\frac{1}{\sqrt{3}}\Delta^{0},T^{222}=\Delta^{-}\\!,$
(31) $\displaystyle
T^{113}=\frac{1}{\sqrt{3}}\Sigma^{*+},T^{123}=\frac{1}{\sqrt{6}}\Sigma^{*0},T^{223}=\frac{1}{\sqrt{3}}\Sigma^{*-},$
$\displaystyle
T^{133}=\frac{1}{\sqrt{3}}\Xi^{*0},T^{233}=\frac{1}{\sqrt{3}}\Xi^{*-},T^{333}=\Omega.$

The couplings we use Granados  _et al._ (2017) are $F_{\pi}=92.4\text{
MeV},D=0.80,F=0.46$ Cabibbo  _et al._ (2003) and $h_{A}$ can be determined
from the partial decay width $\Sigma^{*}\rightarrow\Lambda\pi$ or from
$\Delta\rightarrow N\pi$ to be

$\displaystyle
h_{A}^{\Sigma^{*}\rightarrow\Lambda\pi}=2.4~{}\text{and}~{}h_{A}^{\Delta\rightarrow
N\pi}=2.88.$ (32)

In the large-$N_{C}$ limit Dashen and Manohar (1993); Pascalutsa and
Vanderhaeghen (2006), one also gets ($N_{C}=$ number of colors)

$\displaystyle h_{A}^{\text{large-$N_{C}$}}=\frac{3}{\sqrt{2}}g_{A}=2.67,$
(33)

where $g_{A}=F+D=1.26$. We will explore the range

$\displaystyle h_{A}^{\pm}=2.7\pm 0.3.$ (34)

Notice that after the substitution (27) it is the ratio $h_{A}/m_{R}$ that
appears with each decuplet-baryon–meson term [cf. Eq. (7)]. That is, on the
probability level one has schematically

$\displaystyle\mathcal{P}(\Lambda_{H})\sim
g_{OM}^{2}T_{\text{oct}}(\Lambda_{H})$ (35)
$\displaystyle+\left(\frac{h_{A}}{m_{\Delta}}\right)^{2}T_{\Delta}(\Lambda_{H})+\left(\frac{h_{A}}{m_{\Sigma^{*}}}\right)^{2}T_{\Sigma^{*}}(\Lambda_{H})+\cdots$

so that one could vary $h_{A}$ by $\sim 10\%$ for each of the separate
decuplet terms keeping the masses as shown but numerically it makes not much
of a difference to instead use $m_{R}=m_{\Delta}$ for both terms and vary the
ratio between its smallest and largest values
$1.737~{}\text{GeV}^{-1}<h_{A}/m_{\Delta}<2.435~{}\text{GeV}^{-1}$ coming from
[cf. (34)]

$\displaystyle\frac{h_{A}^{-}}{m_{\Sigma^{*}}}$
$\displaystyle=1.737\,\mbox{GeV}^{-1}<\frac{h_{A}}{m_{\Delta}}<\frac{h_{A}^{+}}{m_{\Delta}}=2.435\,\mbox{GeV}^{-1}.$
(36)

The effect of this variation is shown in Fig. 6 and its effects on the
probabilities are studied in Ref. Ghaderi (2018).

##  Appendix B The vertex functions

The functions $S^{\lambda}\left(y,\boldsymbol{k}_{\perp}\right)$ are the
amplitudes for a particular hadronic fluctuation of a proton with positive
helicity and we calculate them using (on-shell) light-front spinors and the
Lagrangian of Eq. (7). These amplitudes were calculated for a wide variety of
hadronic fluctuations in Holtmann  _et al._ (1996). Our results are basically
the same with minor differences due to a different choice of Lagrangian. Apart
from different normalizations, our results for the functions
$S^{\lambda}(y,\boldsymbol{k}_{\perp})$ agree with those found in Pasquini and
Boffi (2006). We now present the vertex functions for both choices of the
meson’s ‘derivative momentum’.

For meson momentum choice (10) the amplitudes are

$\displaystyle
S^{+\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=-\frac{(m_{B}+m_{P})(ym_{P}-m_{B})}{\sqrt{y}},$
(37) $\displaystyle
S^{-\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=-\frac{(m_{B}+m_{P})k_{\perp}\mathrm{e}^{\mathrm{i}\phi}}{\sqrt{y}},$

where the transverse momentum
$k_{\perp}=|\boldsymbol{k}_{\perp}|=|(k_{1},k_{2})|$ and $\phi\in[0,2\pi[$ is
the angle in the polar form of the object
$k_{1}+\mathrm{i}k_{2}=k_{\perp}\mathrm{e}^{\mathrm{i}\phi}$. For meson
momentum choice (11) the amplitudes are

$\displaystyle
S^{+\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=-\frac{k_{\perp}^{2}+m_{M}^{2}y-(1-y)^{2}m_{B}m_{P}}{(1-y)\sqrt{y}},$
(38) $\displaystyle
S^{-\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=-\frac{(m_{B}+m_{P})k_{\perp}\mathrm{e}^{\mathrm{i}\phi}}{\sqrt{y}}.$

For the case of a fluctuation into a spin-$3/2$ baryon and a spinless meson we
find that the different helicity configurations yield the following
amplitudes. For meson momentum choice (10) the amplitudes are

$\displaystyle
S^{+\frac{3}{2}}(y,\boldsymbol{k}_{\perp})=\frac{\mathrm{i}\mathrm{e}^{-\mathrm{i}\phi}m_{B}\left(m_{B}+ym_{P}\right)}{\sqrt{2y}y}k_{\perp},$
(39) $\displaystyle
S^{+\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=\frac{-\mathrm{i}}{\sqrt{6y}y}\left[k_{\perp}^{2}(2m_{B}+ym_{P})+(ym_{P}+m_{B})^{2}(ym_{P}-m_{B})\right],$
$\displaystyle
S^{-\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=\frac{\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}}{\sqrt{6y}y}\left[k_{\perp}^{2}+(m_{B}+ym_{P})(ym_{P}-2m_{B})\right]k_{\perp},$
$\displaystyle
S^{-\frac{3}{2}}(y,\boldsymbol{k}_{\perp})=\mathrm{i}\frac{\mathrm{e}^{2\mathrm{i}\phi}m_{B}}{\sqrt{2y}y}k_{\perp}^{2}.$

For meson momentum choice (11) the amplitudes are

$\displaystyle
S^{+\frac{3}{2}}(y,\boldsymbol{k}_{\perp})=\frac{\mathrm{i}\mathrm{e}^{-\mathrm{i}\phi}m_{B}\left(m_{B}+ym_{P}\right)}{\sqrt{2y}y}k_{\perp},$
(40) $\displaystyle
S^{+\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=\mathrm{i}\frac{k_{\perp}^{2}(m_{P}y-m_{B}(y-2))-\left(m_{B}^{2}(y-1)^{2}-m_{M}^{2}y^{2}\right)(m_{B}+m_{P}y)}{\sqrt{6}(y-1)y^{3/2}},$
$\displaystyle
S^{-\frac{1}{2}}(y,\boldsymbol{k}_{\perp})=-\mathrm{i}\frac{k_{\perp}\mathrm{e}^{\mathrm{i}\phi}\left[k_{\perp}^{2}+(y-1)m_{B}\left(ym_{P}-(y-2)m_{B}\right)+m_{M}^{2}y^{2}\right]}{\sqrt{6}(y-1)y^{3/2}},$
$\displaystyle
S^{-\frac{3}{2}}(y,\boldsymbol{k}_{\perp})=\mathrm{i}\frac{\mathrm{e}^{2\mathrm{i}\phi}m_{B}}{\sqrt{2y}y}k_{\perp}^{2}.$

Notice that as $k_{\perp}$ goes to zero only the spin-conserving amplitudes
$S^{+\frac{1}{2}}(y,\boldsymbol{k}_{\perp})$ remain in (LABEL:E:_VonehalfA-
LABEL:E:_S32ChoiceB), as expected.

We also note that in the chiral limit $m_{M}\to 0$ with $y\to 1$ and
$k_{\perp}\to 0$ the amplitudes corresponding to momentum choice (11) vanish,
in line with the Goldstone theorem Goldstone  _et al._ (1962).

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