# Quark-model relations among TMDs in the parton model

F. Aslan1, S. Bastami2, A. Mahabir1, A. Tandogan1,3, P. Schweitzer1 1
Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.
2 Department of Physics, Kennesaw State University, 30144 Kennesaw, GA, U.S.A.
3 Department of Physics, University of Connecticut, Hartford, CT 06103, U.S.A.

###### Abstract

The covariant parton model (CPM) is a consequent application of the parton
model concept to the nucleon structure. In this model, there is a choice to
put quarks either in a pure-spin state or in a mixed-spin state. We show that
the mixed-spin version of the CPM does not support the quark-model relations
among transverse momentum dependent parton distributions (TMDs) which were
shown to hold in a large class of quark models. One can enforce the quark-
model relations to be valid in the CPM by imposing a condition which is
equivalent to putting the quarks in a pure-spin state. This gives a
complementary perspective on the connection of the pure- and mixed-spin state
CPM versions, and provides a fresh view on the question whether the quark-
model relations could be realized in QCD as “approximate relations” with some
useful numerical accuracy.

##  I Introduction

TMDs entail the description of the nucleon structure in deep-inelastic
scattering (DIS) processes when in the final state one detects an adequate
transverse momentum which is small compared to the hard scale $Q$ of the
process Collins-book . For the understanding of the nonperturbative properties
of TMDs, quark models play an important role in two ways. First, undistracted
by technical complexities inherent in a full QCD treatment, in models one may
investigate in a simpler theoretical framework the significance of a specific
physical aspect and gain in this way valuable insights. Second, in situations
where some of the TMDs are still not yet well known, results from models may
be helpful to interpret first data or give useful estimates for counting rates
in future experiments such as the Electron-Ion Collider Accardi:2012qut . In
this way, models complement phenomenology and lattice QCD studies.

In this work, we will study the covariant parton model (CPM) which is based on
Feynman’s parton model concept Feynman:1969ej ; Feynman:1973xc . The latter
played a historically important role for the interpretation of DIS processes
and establishing QCD and can, in a certain sense, be viewed as a “zeroth order
approximation” to QCD Ellis:1978ty ; Collins-book . The parton model provides
often an effective first step towards an understanding of QCD processes. For
instance, the “generalized parton model” of Refs. Anselmino:1994tv ;
Anselmino:1999pw ; Anselmino:2005nn ; Anselmino:2005sh ; Anselmino:2011ch
helped to pave the way to modern TMD phenomenology. The exploration of the
parton model concept for the sake of studying TMDs and their nonperturbative
properties was carried out in Refs. Zavada:1996kp ; Zavada:2001bq ;
Zavada:2002uz ; Efremov:2004tz ; Zavada:2007ww ; Efremov:2009ze ;
Zavada:2009ska ; Efremov:2010mt ; Zavada:2011cv ; Zavada:2013ola ;
Zavada:2015gaa ; Zavada:2019yom ; Bastami:2020rxn ; DAlesio:2009cps ;
Aslan:2022wqc . Further applications of the parton model concept can be found
in Refs. Blumlein:1996tp ; Blumlein:1996vs ; Blumlein:1998nv ; Roberts:1996ub
; Jackson:1989ph .

Due to the absence of interactions, the description of the nucleon structure
in the parton model is particularly lucid and the TMDs are described in terms
of covariant functions depending on the variable $P\cdot k$ where $P^{\mu}$
denotes the nucleon momentum and $k^{\mu}$ quark momentum. Despite the
simplicity of the model, there was an interesting puzzle. One group claimed
that the description of TMDs requires two independent covariant functions
Efremov:2009ze ; Zavada:2009ska ; Efremov:2010mt ; Zavada:2011cv ;
Zavada:2013ola ; Zavada:2015gaa ; Zavada:2019yom ; Bastami:2020rxn , while the
other group claimed that one needs three independent covariant functions
DAlesio:2009cps . This puzzle was resolved recently by showing that the
results of the two groups are equivalent except for the treatment of the quark
polarization state Aslan:2022wqc . In Zavada:2001bq ; Zavada:2002uz ;
Efremov:2004tz ; Zavada:2007ww ; Efremov:2009ze ; Zavada:2009ska ;
Efremov:2010mt ; Zavada:2011cv ; Zavada:2013ola ; Zavada:2015gaa ;
Zavada:2019yom ; Bastami:2020rxn  the quarks were chosen to be in a pure-spin
state, while in DAlesio:2009cps  they were (implicitly) assumed to be in a
mixed-spin state. Other than that, the results of the two groups are
equivalent Aslan:2022wqc .

Here we will take a different point of view as compared to Ref. Aslan:2022wqc
where the focus was on technical aspects of the quark correlator. In this work
the starting point is the quark model aspect of the approach: the CPM is after
all a quark model, i.e. a model without gauge field degrees of freedom. In
several models of such type it was observed that certain relations exist
between different TMDs to which we shall refer as quark model relations
(QMRs). Not all quark models support the QMRs, but it is worth stressing that
a wide class of very different models does.

The goal of this work is to investigate whether the TMDs in the mixed-spin
state version of the CPM DAlesio:2009cps ; Aslan:2022wqc  obey the QMRs. We
shall see that this is not the case and show that imposing the validity of the
QMRs in this model yields the same condition as when one chooses the quarks to
be in pure-spin state. In other words, if one starts with the mixed-spin state
version of the CPM and demands the model to comply with the QMRs supported in
other quark models, then one must introduce the pure-spin state model.

Our study is insightful in two ways. First, it gives insights on the CPM and
its relation to other quark models. Second, it opens a new perspective on QMRs
and may shed light on the question whether they could hold in QCD as
approximate relations with a potentially useful numerical accuracy in some
range of $x$ and $k_{T}$.

The structure of this work is as follows. In Sec. II we present the QMRs and
briefly discuss their understanding within quark models. In Sec. III we review
the CPM and present the results for TMDs in the mixed-spin state version of
the CPM. In Sec. IV we investigate the linear and non-linear QMRs in the CPM.
In Sec. V we discuss the physical implications of our findings, and in Sec. VI
we draw conclusions and give an outlook for future studies.

##  II Quark model relations among TMDs

In contrast to QCD, in quark models relations among different TMDs can exist
due to the simpler model dynamics or due to model symmetries. Some of these
relations such as, e.g, the quark-model Lorentz-invariance relations (qLIRs),
are generic in the sense that they hold in quark models which respect Lorentz
symmetry and contain no gauge field degrees of freedom Tangerman:1994eh ;
Mulders:1995dh ; Boer:1997nt , but are not valid in QCD Kundu:2001pk ;
Goeke:2003az ; Goeke:2005hb . We quote here only one qLIR, namely

$h_{T}^{q}(x,k_{T})-h_{T}^{\perp q}(x,k_{T})=h_{1L}^{\perp q}(x,k_{T})\,,$ (1)

on which it will be instructive to follow up below. A discussion of other
qLIRs can be found for instance in Ref. Metz:2008ib . We remark that the
notation in (1) and throughout this work is $k_{T}=|\vec{k}_{T}|$ and
$k_{T}^{2}$ will always denote $|\vec{k}_{T}|^{2}$.

The main focus of this work is another set of relations which have been
observed in several very different quark models. These relations, to which we
will refer to in the following as quark model relations (QMRs), are given by

$\displaystyle g_{1T}^{\perp q}(x,k_{T})$ $\displaystyle=$ $\displaystyle-
h_{1L}^{\perp q}(x,k_{T}),$ (2a) $\displaystyle g_{T}^{\perp q}(x,k_{T})$
$\displaystyle=$ $\displaystyle-h_{1T}^{\perp q}(x,k_{T}),$ (2b)
$\displaystyle g_{L}^{\perp q}(x,k_{T})$ $\displaystyle=$ $\displaystyle-
h_{T}^{q}(x,k_{T}),$ (2c) $\displaystyle
g_{1}^{q}(x,k_{T})-h_{1}^{q}(x,k_{T})$ $\displaystyle=$ $\displaystyle
h_{1T}^{\perp(1)q}(x,k_{T}),$ (2d) $\displaystyle
g_{T}^{q}(x,k_{T})-h_{L}^{q}(x,k_{T})$ $\displaystyle=$ $\displaystyle
h_{1T}^{\perp(1)q}(x,k_{T}),$ (2e)

In addition to the linear QMRs (2) also two nonlinear QMRs have been found
which are given by

$\displaystyle\frac{1}{2}\biggl{[}h_{1L}^{\perp q}(x,k_{T})\biggr{]}^{2}$
$\displaystyle=$ $\displaystyle-\,h_{1}^{q}(x,k_{T})\,h_{1T}^{\perp
q}(x,k_{T})\,,$ (3a) $\displaystyle\frac{1}{2}\biggl{[}g_{1T}^{\perp
q}(x,k_{T})\biggr{]}^{2}$ $\displaystyle=$ $\displaystyle g_{1T}^{\perp
q}(x,k_{T})\,g_{L}^{\perp q}(x,k_{T})+g_{T}^{q}(x,k_{T})\,g_{T}^{\perp
q}(x,k_{T})\,.$ (3b)

The transverse moment of a TMD is defined as

$h_{1T}^{\perp(1)q}(x,k_{T})=\frac{k_{T}^{2}}{2M^{2}}\,h_{1T}^{\perp
q}(x,k_{T})\,.$ (4)

The relations (2, 3) hold in a wide class of quark models which are based on
very different model concepts including the spectator model, bag model, or
light-front constituent quark model Jakob:1997wg ; Avakian:2007mv ;
Avakian:2008dz ; Avakian:2009jt ; She:2009jq ; Avakian:2010br ;
Pasquini:2008ax ; Lorce:2011zta . The QMRs (2a, 2d, 3a) involving twist-2 TMDs
were shown to arise from a certain rotational symmetry of the model lightcone
wave functions Lorce:2011zta . This symmetry is effectively present in many
models including, e.g., the pure-spin version of the CPM Efremov:2009ze ;
Bastami:2020rxn .

It is important to remark that not all models support the QMRs. For instance,
certain spectator model variants, where (to allow more flexible modeling) a
larger number of free model parameters was introduced Bacchetta:2008af , do
not support QMRs. Another example is quark-target model Meissner:2007rx  where
the presence of gluon degrees of freedom spoils QMRs.

##  III Quark correlator and TMDs in CPM

In this section, we first review the general structure of the quark correlator
in quark models and then discuss the specific results for the correlator and
TMDs in the CPM briefly commenting on the two versions of this model.

###  III.1 Quark correlator in a generic quark model

In a theory without explicit gauge degrees of freedom the quark correlator for
the nucleon is defined as follows

$\displaystyle\Phi_{ij}^{q}(k,P,S)$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{4}z}{(2\pi)^{4}}\;\mathrm{e}^{ikz}\,\langle
N|\,\overline{\Psi}_{j}^{\,q}(0)\;\Psi_{i}^{q}(z)\,|N\rangle\,,$ (5)

where $k^{\mu}$ is the quark 4-momentum, and $P^{\mu}$ and $S^{\mu}$ are the
nucleon 4-momentum and polarization vectors satisfying $P^{2}=M^{2}$,
$S^{2}=-1$, $P\cdot S=0$. In quark models, the Lorentz-structure of the
correlator (5) is described in terms of $k^{\mu}$, $P^{\mu}$, $S^{\mu}$ as
follows (we use the convention $\varepsilon^{0123}=1$ and assume a covariant
normalization of nucleon states) Boer:1997nt

$\displaystyle\Phi^{q}(k,P,S)$
$\displaystyle=MA_{1}^{q}+\not{P}A_{2}^{q}+\not{k}A_{3}^{q}+\frac{i}{2M}\;[\not{P},\not{k}]\,A_{4}^{q}+i(k\cdot
S)\gamma_{5}A_{5}^{q}+M\not{S}\gamma_{5}A_{6}^{q}+\frac{(k\cdot
S)}{M}\not{P}\gamma_{5}A_{7}^{q}+\frac{(k\cdot
S)}{M}\not{k}\gamma_{5}A_{8}^{q}$
$\displaystyle+\frac{[\not{P},\not{S}]}{2}\gamma_{5}A_{9}^{q}+\frac{[\not{k},\not{S}]}{2}\gamma_{5}A_{10}^{q}+\frac{(k\cdot
S)}{2M^{2}}[\not{P},\not{k}]\gamma_{5}A_{11}^{q}+\frac{1}{M}\varepsilon^{\mu\nu\rho\sigma}\gamma_{\mu}P_{\nu}k_{\rho}S_{\sigma}A_{12}^{q}\,,$
(6)

The amplitudes $A_{i}^{q}=A_{i}^{q}(P\cdot k,k^{2})$ in (III.1) are real
functions of the Lorentz scalars $P\cdot k$ and $k^{2}$ Mulders:1995dh ;
Boer:1997nt . The amplitudes $A_{i}^{q}$ are chiral-even for
$i=2,\,3,\,6,\,7,\,8,\,12$ and chiral-odd for $i=1,\,4,\,5,\,9,\,10,\,11$. In
QCD and in models with gauge field degrees of freedom, in the definition of
the quark correlator (5) Wilson lines must be included which run along a
nearly light-like 4-vector $n^{\mu}$ dictated by hard-momentum flow in the
considered process Collins-book . The presence of the additional vector
$n^{\mu}$ allows for 20 further Lorentz structures which are often denoted as
$B_{i}$-amplitudes Kundu:2001pk ; Goeke:2003az ; Goeke:2005hb . The T-odd
amplitudes $A_{i}^{q}$ for $i=4,\,5,\,12$ vanish in quark models as do the
pertinent T-odd TMDs because their modeling requires explicit gauge field
degrees of freedom Pobylitsa:2002fr . The T-odd amplitudes are included in
(III.1) merely for completeness. In this work, we will focus on T-even TMDs,
see Appendix A for the explicit expressions.

###  III.2 Quark correlator in the CPM

In the CPM, one can explore the equation of motion for the quark fields
$(i\not{\partial}-m_{q})\,\Psi^{q}(z)=0$ in order to derive the following
results for the amplitudes DAlesio:2009cps ; Aslan:2022wqc

$\displaystyle A_{1}^{q}=\frac{m_{q}}{M}\,A_{3}^{q},$ $\displaystyle
A_{2}^{q}=0,\quad A_{4}^{q}=0,\quad A_{5}^{q}=0,\quad
A_{6}^{q}=\frac{m_{q}}{M}\,A_{10}^{q},$ $\displaystyle
A_{7}^{q}=-\frac{m_{q}}{M}\,A_{11}^{q},$ $\displaystyle A_{9}^{q}=0,\quad
A_{10}^{q}=\frac{(P\cdot
k)}{M^{2}}\,A_{11}^{q}-\frac{m_{q}}{M}\,A_{8}^{q}\,,\quad A_{12}^{q}=0.$ (7)

The T-odd amplitudes $A_{4}^{q}$, $A_{5}^{q}$, $A_{12}^{q}$ vanish in the CPM
which is a general quark model prediction due to the absence of gauge field
degrees of freedom Pobylitsa:2002fr . Interestingly, also the T-even
amplitudes $A_{2}^{q}$ and $A_{9}^{q}$ vanish which is a specific feature of
the CPM, and is in general not the case in other quark models. The amplitudes
$A_{1}^{q}$, $A_{6}^{q}$, $A_{7}^{q}$ are proportional to current quark masses
and hence negligibly small for the light quark flavors. At this stage the
relations (7) imply that in the CPM 3 independent amplitudes exist which can
be chosen to be the unpolarized amplitude $A_{3}^{q}$, the chiral-even
polarized amplitude $A_{8}^{q}$, and the chiral-odd polarized amplitude
$A_{11}^{q}$.

As shown in Aslan:2022wqc  one has two choices to treat quark polarization
effects. First, one can work with a CPM with 3 independent amplitudes
DAlesio:2009cps  which corresponds to quarks in a mixed-spin state (as long as
the inequality $|A_{8}^{q}|<|A_{11}^{q}|$ is valid) Aslan:2022wqc .
Alternatively, one can put the quarks in a pure-spin state in which case
$|A_{8}^{q}|=|A_{11}^{q}|$ with the physical-sign solution corresponding to
$A_{8}^{q}=-A_{11}^{q}$ determined from a comparison to other models and
lattice QCD studies Aslan:2022wqc . This corresponds to a CPM with 2
independent amplitudes which can be chosen to be the unpolarized amplitude
$A_{3}^{q}$ and the polarized amplitude $A_{8}^{q}$. The two choices can be
summarized as follows

mixed-spin state CPM $\displaystyle\Leftrightarrow$ $\displaystyle\mbox{3
linearly independent amplitudes:}\quad A_{3}^{q},\quad A_{8}^{q},\quad
A_{11}^{q},\phantom{\frac{1}{1}}$ pure-spin state CPM
$\displaystyle\Leftrightarrow$ $\displaystyle\mbox{2 linearly independent
amplitudes:}\quad A_{3}^{q},\quad A_{8}^{q}=-A_{11}^{q}.$ (8)

###  III.3 TMDs in the mixed-spin state version of the CPM

The starting point for our investigation is the CPM with quarks in the mixed-
spin state. In this section, we quote the results for T-even TMDs starting
with the model expressions for unpolarized TMDs (we define
$k^{\pm}=\frac{1}{\sqrt{2}}(k^{0}\pm k^{3})$)

$\displaystyle f_{1}^{q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\biggl{[}xA_{3}^{q}(P\cdot k,k^{2})\biggr{]}_{k^{+}=xP^{+}},$ (9a)
$\displaystyle f^{\perp q}(x,k_{T})$ $\displaystyle=$ $\displaystyle
2P^{+}\int dk^{-}\biggl{[}A_{3}^{q}(P\cdot k,k^{2})\biggr{]}_{k^{+}=xP^{+}},$
(9b) $\displaystyle e^{q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\biggl{[}\frac{m_{q}}{M}\,A_{3}^{q}(P\cdot
k,k^{2})\biggr{]}_{k^{+}=xP^{+}}.$ (9c)

The expressions for chiral-even polarized TMDs in the mixed-spin state parton
model are given by

$\displaystyle g_{1}^{q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}\frac{x^{2}M^{2}-x\,P\cdot
k+m_{q}^{2}}{M^{2}}\,A_{8}^{q}(P\cdot
k,k^{2})-\frac{m_{q}}{M}\,x\,A_{11}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (10a) $\displaystyle g_{1T}^{\perp
q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}xA_{8}^{q}(P\cdot k,k^{2})-\dfrac{m_{q}}{M}A_{11}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (10b) $\displaystyle g_{T}^{q}(x,k_{T})$
$\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}\dfrac{\vec{k}_{T}^{2}+2m_{q}^{2}}{2M^{2}}\,A_{8}^{q}(P\cdot
k,k^{2})-\dfrac{m_{q}}{M}\;\dfrac{P\cdot k}{M^{2}}\,A_{11}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (10c) $\displaystyle g_{L}^{\perp
q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}\frac{x\,M^{2}-P\cdot k}{M^{2}}\,A_{8}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (10d) $\displaystyle g_{T}^{\perp
q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}A_{8}^{q}(P\cdot k,k^{2})\Biggr{]}_{k^{+}=xP^{+}}.$ (10e)

Finally, the model expressions for chiral-odd polarized TMDs are given by

$\displaystyle h_{1}^{q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}\dfrac{\vec{k}_{T}^{2}-2\,x\,P\cdot
k}{2M^{2}}\,A_{11}^{q}(P\cdot k,k^{2})+x\,\dfrac{m_{q}}{M}\,A_{8}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (11a) $\displaystyle h_{1L}^{\perp
q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}xA_{11}^{q}(P\cdot k,k^{2})-\dfrac{m_{q}}{M}A_{8}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (11b) $\displaystyle h_{1T}^{\perp
q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}A_{11}^{q}(P\cdot k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (11c)
$\displaystyle h_{L}^{q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}\frac{x^{2}M^{2}-2\,x\,P\cdot k}{M^{2}}\,A_{11}^{q}(P\cdot
k,k^{2})+\dfrac{m_{q}}{M}\;\dfrac{P\cdot k}{M^{2}}\,A_{8}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (11d) $\displaystyle h_{T}^{q}(x,k_{T})$
$\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}\frac{x\,M^{2}-P\cdot k}{M^{2}}\,A_{11}^{q}(P\cdot
k,k^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (11e) $\displaystyle h_{T}^{\perp
q}(x,k_{T})$ $\displaystyle=$ $\displaystyle 2P^{+}\int
dk^{-}\Biggl{[}-\dfrac{P\cdot k}{M^{2}}A_{11}^{q}(P\cdot
k,k^{2})+\dfrac{m_{q}}{M}A_{8}^{q}(P\cdot k,k^{2})\Biggr{]}_{k^{+}=xP^{+}}.$
(11f)

For massless quarks in the mixed-spin state version of the CPM, the chiral-
even (chiral-odd) polarized TMDs are given entirely in terms of the chiral-
even (chiral-odd) amplitude $A_{8}^{q}$ ($A_{11}^{q}$).

###  III.4 Onshellness and a useful identity

In the CPM the quarks are on-shell, i.e. the amplitudes $A_{i}^{q}(P\cdot
k,k^{2})$ are actually functions of the type

$A_{i}^{q}(P\cdot k,k^{2})=F^{q}_{i}(P\cdot k)\,\delta(k^{2}-m_{q}^{2})\,.$
(12)

The explicit expressions for the functions $F_{i}^{q}(P\cdot k)$ can be found
in Refs. Bastami:2020rxn ; DAlesio:2009cps ; Aslan:2022wqc  and will not be
needed in this work. However, we will need an identity among the kinematic
variables which holds under the $k^{-}$ integration, and can be derived as
follows. Obviously, due to (12) we have

$\displaystyle 0=\int dk^{-}\biggl{(}k^{2}-m_{q}^{2}\biggr{)}A_{i}^{q}(P\cdot
k,k^{2})\Biggr{|}_{k^{+}=xP^{+}}=\int
dk^{-}\biggl{(}2k^{+}k^{-}-\vec{k}_{T}^{2}-m_{q}^{2}\biggr{)}A_{i}^{q}(P\cdot
k,k^{2})\Biggr{|}_{k^{+}=xP^{+}}\,.$

Next, we notice that $2k^{+}k^{-}=2xP^{+}k^{-}=2xP\cdot
k-2xP^{-}k^{+}=2xP\cdot k-2x^{2}P^{+}P^{-}=2xP\cdot k-x^{2}M^{2}$ holds under
the integral where $k^{+}=xP^{+}$. Inserting this in the above intermediate
step, dividing by $2x$ and rearranging, we obtain

$\displaystyle\int dk^{-}\biggl{(}P\cdot k\biggr{)}A_{i}^{q}(P\cdot
k,k^{2})\Biggr{|}_{k^{+}=xP^{+}}=\int
dk^{-}\biggl{(}\frac{x^{2}M^{2}+\vec{k}_{T}^{2}+m_{q}^{2}}{2x}\biggr{)}A_{i}^{q}(P\cdot
k,k^{2})\Biggr{|}_{k^{+}=xP^{+}}\,.$ (13)

Thus we see that under the $k^{-}$ integral due to the mass-shell condition
implicit in the amplitudes, cf. Eq. (12), we can replace the variable $P\cdot
k$ by an expression determined in terms of $x$, $k_{T}$ and the nucleon and
quark masses. This identity will be helpful in the following.

##  IV Consequences of imposing QMRs in mixed-spin state CPM

Before we investigate the QMRs in the CPM, it is instructive to discuss first
the example of the qLIR in Eq. (1). Here and in the following it is convenient
to reformulate the relations such that all TMDs appear on one side of the
equation. Inserting the model expressions (11b, 11e, 11f) for $h_{T}^{q}$,
$h_{T}^{\perp q}$, $h_{1L}^{\perp q}$ we obtain

$\displaystyle h_{T}^{q}(x,k_{T})$ $\displaystyle-h_{T}^{\perp
q}(x,k_{T})-h_{1L}^{\perp q}(x,k_{T})=2P^{+}\int
dk^{-}\Biggl{[}\frac{x\,M^{2}-P\cdot k}{M^{2}}\,A_{11}^{q}(P\cdot k,k^{2})$
$\displaystyle-\Bigl{(}-\dfrac{P\cdot k}{M^{2}}A_{11}^{q}(P\cdot
k,k^{2})+\dfrac{m_{q}}{M}A_{8}^{q}(P\cdot
k,k^{2})\Bigr{)}-\Bigl{(}xA_{11}^{q}(P\cdot
k,k^{2})-\dfrac{m_{q}}{M}A_{8}^{q}(P\cdot
k,k^{2})\Bigr{)}\Biggr{]}_{k^{+}=xP^{+}}=0.$

We see that the qLIR (1) is valid for any $A_{8}^{q}(P\cdot k,k^{2})$ and
$A_{11}^{q}(P\cdot k,k^{2})$. This was to be expected. The qLIRs only require
the absence of gauge field degrees of freedom, and thus must be valid in every
quark model respecting Lorentz invariance. The investigation of this and other
qLIRs is a useful cross check for the theoretical consistency of a model, but
does not yield new insights. In this respect, the QMRs are more insightful as
we shall discuss next.

###  IV.1 QMR between gear-worm functions, Eq. (2a)

The TMDs $g_{1T}^{\perp q}(x,k_{T})$ and $h_{1L}^{\perp q}(x,k_{T})$ are
sometimes called gear-worm functions. In the spectator model study of Ref.
Jakob:1997wg  the QMR (2a) between these TMDs was derived which was later
confirmed in several other quark models. Inserting the CPM expressions (10b)
for $g_{1T}^{\perp q}(x,k_{T})$ and (11b) for $h_{1L}^{\perp q}(x,k_{T})$, the
relation (2a) can be expressed as

$g_{1T}^{\perp q}(x,k_{T})+h_{1L}^{\perp q}(x,k_{T})=2P^{+}\int
dk^{-}\Bigl{(}x-\dfrac{m_{q}}{M}\Bigr{)}\Biggl{[}A_{8}^{q}(P\cdot
k,k^{2})+A_{11}^{q}(P\cdot k,k^{2})\Biggr{]}_{k^{+}=xP^{+}}.$ (14)

Clearly, in the mixed-spin state version of the CPM, where the amplitudes
$A_{8}^{q}$ and $A_{11}^{q}$ are unrelated, the relation (2a) is not valid. If
we would like the CPM to comply with this QMR, then this is possible if and
only if we impose the condition $A_{8}^{q}=-A_{11}^{q}$ which corresponds to
the pure-spin state version of the CPM, cf. Eq. (8).

###  IV.2 QMR between $g_{T}^{\perp q}$ and $h_{1T}^{\perp q}$, Eq. (2b)

The QMR (2b) connecting the TMDs $g_{T}^{\perp q}$ and $h_{1T}^{\perp q}$ was,
to the best of our knowledge, first discussed in Ref. Avakian:2009jt .
Inserting CPM expressions (10e) and (11c) for $g_{T}^{\perp q}(x,k_{T})$ and
$h_{1T}^{\perp q}(x,k_{T})$ respectively into Eq. (2b) yields

$\displaystyle g_{T}^{\perp q}(x,k_{T})+h_{1T}^{\perp q}(x,k_{T})$
$\displaystyle=$ $\displaystyle 2P^{+}\int dk^{-}\Biggl{[}A_{8}^{q}(P\cdot
k,k^{2})+A_{11}^{q}(P\cdot k,k^{2})\Biggr{]}_{k^{+}=xP^{+}}\,.$ (15)

Again we see that if the amplitudes $A_{8}^{q}$ and $A_{11}^{q}$ are
unrelated, then the QMR (2b) is not valid which is the case in the mixed-spin
state version of the model. For the CPM to comply with this QMR, it is
necessary to introduce the condition $A_{8}^{q}=-A_{11}^{q}$ which brings us
to the pure-spin state version of the CPM, cf. Eq. (8).

###  IV.3 QMR between $g_{L}^{\perp q}$ and $h_{T}^{q}$, Eq. (2c)

The QMR (2c) connecting the twist-3 TMDs $g_{L}^{\perp q}$ and $h_{T}^{q}$ was
derived for the first time in Ref. Jakob:1997wg  and later confirmed in other
models. Inserting respectively the model expressions (10d) and (11f) for the
TMDs $g_{L}^{\perp q}(x,k_{T})$ and $h_{T}^{q}(x,k_{T})$ into Eq. (2c) leads
immediately to

$\displaystyle g_{L}^{\perp q}(x,k_{T})+h_{T}^{q}(x,k_{T})=2P^{+}\int
dk^{-}\left(\frac{x\,M^{2}-P\cdot k}{M^{2}}\right)\,\Biggl{[}A_{8}^{q}(P\cdot
k,k^{2})+A_{11}^{q}(P\cdot k,k^{2})\Biggr{]}_{k^{+}=xP^{+}}\,.$ (16)

Also in this case we see that in the mixed-spin state version of the CPM the
relation (2c) is not valid, unless we demand that $A_{8}^{q}=-A_{11}^{q}$
which is equivalent to introducing the pure-spin state version of the CPM, cf.
Eq. (8).

###  IV.4 QMR of helicity, transversity and pretzelosity, Eq. (2d)

This QMR was, to the best of our knowledge, first discussed in Avakian:2008dz
. The difference of $g_{1}^{q}$ and $h_{1}^{q}$ was known to be related in
models to quark orbital angular momentum Ma:1997gy ; Ma:1998ar  implying that
pretzelosity is related to quark orbital angular momentum She:2009jq .
Although only a model relation, this is the only connection of quark orbital
angular momentum to TMDs known so far, and attracted a lot of interest. The
QMR (2d) and its connection to quark orbital angular momentum have been
confirmed in several other model studies. Inserting the model expressions
(10a, 11a, 11c) into Eq. (2d) we obtain the lengthy expression

$\displaystyle
g_{1}^{q}(x,k_{T})-h_{1}^{q}(x,k_{T})-h_{1T}^{\perp(1)q}(x,k_{T})=2P^{+}\int
dk^{-}\Bigg{[}\Big{(}\dfrac{x^{2}M^{2}-xP\cdot
k+m_{q}^{2}-xm_{q}M}{M^{2}}\Big{)}A_{8}^{q}(P\cdot k,k^{2})$
$\displaystyle+\Big{(}\dfrac{xP\cdot
k-xm_{q}M-k_{T}^{2}}{M^{2}}\Big{)}A_{11}^{q}(P\cdot k,k^{2})$
$\displaystyle\Bigg{]}_{k^{+}=xP^{+}}\,.$

In order to proceed, we eliminate $P\cdot k$ under the integral by means of
the identity (13). After rearranging, the result can be expressed as

$\displaystyle
g_{1}^{q}(x,k_{T})-h_{1}^{q}(x,k_{T})-h_{1T}^{\perp(1)q}(x,k_{T})$
$\displaystyle=$ $\displaystyle 2P^{+}\int dk^{-}\dfrac{(xM-
m_{q})^{2}-k_{T}^{2}}{2M^{2}}\Bigg{[}A_{8}^{q}(P\cdot
k,k^{2})+A_{11}^{q}(P\cdot k,k^{2})\Bigg{]}_{k^{+}=xP^{+}}.$ (17)

As in the previous cases, we see that in the mixed-spin state version of the
CPM the relation (2d) is not supported. For this QMR to be valid in the CPM we
must introduce the condition $A_{8}^{q}=-A_{11}^{q}$ which is equivalent to
introducing the pure-spin state version of the CPM, cf. Eq. (8).

###  IV.5 QMR of twist-3 TMDs $g_{T}^{q}$ and $h_{L}^{q}$ to pretzelosity, Eq.
(2e)

We now turn our attention to the last linear QMR which connects
$g_{T}^{q}(x,k_{T})$, $h_{L}^{q}(x,k_{T})$ and the transverse moment of
pretzelosity. Inserting in (2e) the model expressions (10c, 11c, 11d) for
$g_{T}^{q}(x,k_{T})$, $h_{L}^{q}(x,k_{T})$ and $h_{1T}^{\perp q}(x,k_{T})$
gives

$g_{T}^{q}(x,k_{T})-h_{L}^{q}(x,k_{T})-h_{1T}^{\perp(1)q}(x,k_{T})=2P^{+}\int
dk^{-}\left(\dfrac{2m_{q}^{2}+k_{T}^{2}}{2M^{2}}-\dfrac{m_{q}}{M}\dfrac{P\cdot
k}{M^{2}}\right)\Bigg{[}A_{8}^{q}(P\cdot k,k^{2})+A_{11}^{q}(P\cdot
k,k^{2})\Bigg{]}_{k^{+}=xP^{+}}.$ (18)

As in the previous cases, we observe that the QMR (2e) is not valid in the
mixed-spin state version of the CPM, and can be satisfied only when one
introduces the condition $A_{8}^{q}=-A_{11}^{q}$, i.e. the pure-spin state
version of the model.

###  IV.6 Nonlinear QMR between $h_{1}^{q}$, $h_{1L}^{\perp q}$,
$h_{1T}^{\perp q}$ in Eq. (3a)

The nonlinear QMR (3a) was derived in Ref. Avakian:2010br . Inserting the
model expressions (11a, 11b, 11c) into the nonlinear relation (3a) we obtain

$\displaystyle 2\,h_{1}^{q}(x,k_{T})\,h_{1T}^{\perp q}(x,k_{T})+h_{1L}^{\perp
q}$ $\displaystyle(x,k_{T})^{2}$ $\displaystyle=(2P^{+})^{2}\iint
dk^{-}d{k^{\prime}}^{-}$
$\displaystyle\Biggl{[}x\,\frac{m_{q}}{M}\Biggl{(}A_{8}^{q}(P\cdot
k,k^{2})A_{11}^{q}(P\cdot k^{\prime},k^{\prime}{}^{2})-A_{8}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})A_{11}^{q}(P\cdot k,k^{2})\Biggr{)}$
$\displaystyle+\frac{m_{q}^{2}}{M^{2}}\Biggl{(}A_{8}^{q}(P\cdot
k,k^{2})A_{8}^{q}(P\cdot k^{\prime},k^{\prime}{}^{2})-A_{11}^{q}(P\cdot
k,k^{2})A_{11}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})\Biggr{)}\Biggr{]}_{k^{+}=xP^{+}},$ (19)

where $k=(k^{+},k^{-},\vec{k}_{T})$ and
$k^{\prime}=(k^{+},k^{\prime}{}^{-},\vec{k}_{T})$ and we used the identity
(13) to eliminate the variable $P\cdot k$ under the integral. In order to
proceed, we repeat the calculation leading to (19) with the dummy integration
variables $k^{-}$ and ${k^{\prime}}^{-}$ interchanged, and take the average of
the two results. In this way, the ”mixed terms” in the first term on the
right-hand side of (19) with $A_{8}^{q}A_{11}^{q}$ cancel out, and we obtain

$\displaystyle 2\,h_{1}^{q}(x,k_{T})\,h_{1T}^{\perp q}(x,k_{T})+h_{1L}^{\perp
q}$ $\displaystyle(x,k_{T})^{2}$ $\displaystyle=(2P^{+})^{2}\iint
dk^{-}d{k^{\prime}}^{-}$
$\displaystyle\frac{m_{q}^{2}}{M^{2}}\Biggl{[}A_{8}^{q}(P\cdot
k,k^{2})A_{8}^{q}(P\cdot k^{\prime},k^{\prime}{}^{2})-A_{11}^{q}(P\cdot
k,k^{2})A_{11}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})\Biggr{]}_{k^{+}=xP^{+}},$ (20)

It is convenient to rewrite this result in the following equivalent way

$\displaystyle 2\,h_{1}^{q}(x,k_{T})\,h_{1T}^{\perp q}(x,k_{T})+h_{1L}^{\perp
q}$ $\displaystyle(x,k_{T})^{2}$ $\displaystyle=(2P^{+})^{2}\iint
dk^{-}d{k^{\prime}}^{-}$
$\displaystyle\frac{m_{q}^{2}}{M^{2}}\Biggl{[}\Biggl{(}A_{8}^{q}(P\cdot
k,k^{2})+A_{11}^{q}(P\cdot k,k^{2})\Biggr{)}\Biggl{(}A_{8}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})-A_{11}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})\Biggr{)}\Biggr{]}_{k^{+}=xP^{+}}.$ (21)

In order to show that (21) is equivalent to (20) one can apply the trick with
repeating the calculation with the dummy integration variables $k^{-}$ and
$k^{\prime}{}^{-}$ interchanged, and taking the average.

As in the case of linear QMRs, the nonlinear relation (3a) is in general not
valid in the CPM version with quarks in a mixed-spin state. Interestingly and
in contrast to the linear case, the violation of the nonlinear QMR (3a) is,
however, a small effect proportional to $m_{q}^{2}/M^{2}$ which is numerically
of the order ${\cal O}(10^{-6})$ for the light $u$\- and $d$-flavors. This
observation may have interesting consequences on which we shall comment in
Sec. V.

If we insist on the nonlinear QMR (3a) to be exactly valid for $m_{q}\neq 0$,
then we see from the final expression (21) that there are two solutions:
$A_{8}^{q}=\pm A_{11}^{q}$. It is not surprising to find two solutions, as we
deal with a quadratic equation. Both solutions were encountered in
Aslan:2022wqc , and $A_{8}^{q}=+A_{11}^{q}$ was recognized to be an unphysical
solution as it would imply opposite signs for quark helicity and transversity
TMDs in contradiction to results from other models and lattice QCD. The
solution $A_{8}^{q}=-A_{11}^{q}$ leads to like signs for quark helicity and
transversity TMDs in agreement with other models and lattice QCD and
constitutes therefore the physical solution Aslan:2022wqc . Thus, the CPM with
massive quarks complies exactly with the nonlinear QMR (3a) if and only if we
use the pure-spin version of the model.

###  IV.7 Nonlinear QMR between $g_{1T}^{\perp q}$, $g_{L}^{\perp q}$,
$g_{T}^{q}$, $g_{T}^{\perp q}$ in Eq. (3b)

The nonlinear QMR (3b) was also derived in Ref. Avakian:2010br . Inserting the
model expressions (10b, 10c, 10d, 10e) into the nonlinear relation (3b) we
obtain

$\displaystyle 2\,g_{1T}^{\perp q}(x,k_{T})\,g_{L}^{\perp q}(x,k_{T})+$
$\displaystyle 2\,g_{T}^{q}(x,k_{T})\,g_{T}^{\perp q}(x,k_{T})-g_{1T}^{\perp
q}(x,k_{T})^{2}$ $\displaystyle=(2P^{+})^{2}\iint dk^{-}d{k^{\prime}}^{-}$
$\displaystyle\Biggl{[}x\,\frac{m_{q}}{M}\Biggl{(}A_{8}^{q}(P\cdot
k,k^{2})A_{11}^{q}(P\cdot k^{\prime},k^{\prime}{}^{2})-A_{8}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})A_{11}^{q}(P\cdot k,k^{2})\Biggr{)}$
$\displaystyle+\frac{m_{q}^{2}}{M^{2}}\Biggl{(}A_{8}^{q}(P\cdot
k,k^{2})A_{8}^{q}(P\cdot k^{\prime},k^{\prime}{}^{2})-A_{11}^{q}(P\cdot
k,k^{2})A_{11}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})\Biggr{)}\Biggr{]}_{k^{+}=xP^{+}},$ (22)

where we used the identity (13) to eliminate the variable $P\cdot k$ in the
coefficient of the $A_{8}^{q}(k\cdot P,k^{2})A_{8}^{q}(P\cdot
k^{\prime},k^{\prime}{}^{2})$-term (in other cases $P\cdot k$ cancels out).
The expression under the integral of (22) coincides with the expression in
(19) and the further steps continue from here in the same way as in Sec. IV.6
including all considerations and conclusions.

##  V Discussion of the results

In order to better understand the physical implications of our results it is
instructive to briefly review the relation between the two versions of the CPM
Aslan:2022wqc . In Ref. Aslan:2022wqc  it was recognized that for massive
quarks, $m_{q}\neq 0$, the quark correlator can be expressed compactly be
introducing an axial vector $w_{q}^{\mu}$ which has the properties of a quark
polarization vector and satisfies $k\cdot w_{q}=0$. One then has111 The
polarization of massless quarks cannot be described in terms of a polarization
vector. But ultimately in the massless case, one has the same choice of
putting a quark in a pure-spin vs mixed-spin state. In this work, it is more
insightful to work with the more general case $m_{q}\neq 0$. But if desired,
the current quark mass effects can be neglected at any stage, cf.
Aslan:2022wqc . a choice: a quark can be in a pure-spin state with
$w_{q}^{2}=-1$, or mixed-spin state with $-1<w_{q}^{2}<0$. These two choices
lead to the two versions of the model, cf. Eq. (8).

It is an interesting question which of the two CPM versions might be more
realistic from phenomenological point of view. At first glance, one could
suspect the mixed-spin state version of the CPM to be phenomenologically more
realistic due to a larger flexibility with three independent covariant
functions which can be uniquely determined from parametrizations of
unpolarized, helicity, and transversity parton distribution functions Harland-
Lang:2014zoa ; deFlorian:2014yva ; Kang:2015msa ; Sato:2016tuz ;
Radici:2018iag ; Bailey:2020ooq ; Cammarota:2020qcw . This question can be
answered by future studies, when more of the TMDs become better known and
constrained by data.

Meanwhile, one could also try to address this question based on what is known
about TMDs from other quark models. A striking observation is that a large
class of quark models supports the QMRs. Thus, one could wonder whether, based
on a comparison to other models, for instance the linear QMRs (2) should also
hold in the CPM. If one would like the CPM to comply with the QMRs, then one
must introduce a condition between the polarized amplitudes, namely
$A_{8}^{q}=-A_{11}^{q}$ as shown in Sec. IV, which leads at once to the pure-
spin state version of the CPM.

To be more precise, when one approaches the issue from the point of view of a
quark polarization vector $w_{q}^{\mu}$, the pure-spin condition
$w_{q}^{2}=-1$ only tells us that $|A_{8}^{q}|=|A_{11}^{q}|$, and the CPM per
se is not able to predict the sign of the chiral-odd TMDs. It is necessary to
resort to results from other models and lattice QCD to determine the physical
solution Aslan:2022wqc . Here the situation is different. The linear QMRs
already “encode” the information from other models about the relative signs of
the polarized chiral-even and chiral-odd TMDs. By imposing the linear QMRs in
the mixed-spin version of the CPM, one is unambiguously lead to the condition
$A_{8}^{q}=-A_{11}^{q}$ without encountering any spurious unphysical solution.

Thus, there are two ways to introduce the CPM with quarks in a pure-spin
state: (a) by demanding that $w_{q}^{2}=-1$ and determining the physical
solution, or (b) by demanding that the model be compliant with the QMRs
observed in other quark models. The two procedures are conceptually quite
different, but nevertheless equivalent. This is an interesting observation in
itself, and gives new insights on the CPM. Notice that this observation is
independent of whether one considers massive quarks or neglects quark mass
effects, cf. footnote 1.

These considerations are of interest beyond the CPM and give rise to a
question regarding the spin state of quarks in other models which, to the best
of our knowledge, has not been addressed in literature. Considering that the
TMDs in the CPM comply with QMRs if and only if the quarks are in pure-spin
state, one may wonder whether the reverse is true: if a quark model supports
the QMRs, are the quarks in this model necessarily in a pure-spin state? We do
not know the answer. It will be interesting to address this question in other
models.

The above remarks about the QMRs leading to the condition
$A_{8}^{q}=-A_{11}^{q}$ refer to the linear case (2). For the non-linear QMRs
(3) the situation is different. These relations are quadratic in TMDs, and
hence it is not surprising to encounter two solutions $A_{8}^{q}=\pm
A_{11}^{q}$ one of which is physical and the other unphysical. As a
consequence, with non-linear QMRs (3) alone, we would need to use additional
constraints to determine the physical and eliminate the unphysical solution —
analogously as it was done with the two solutions of the condition
$w_{q}^{2}=-1$ in Aslan:2022wqc .

However, there is an interesting difference between the ways the CPM can
comply with linear and non-linear QMRs which bears an unexpected observation.
In the more general mixed-spin state version of the CPM, the violation of the
non-linear QMRs (3) is proportional to the square of the current quark masses.
In other words, already in the mixed-spin version of the CPM the non-linear
QMRs (3) are supported modulo current quark mass effects proportional to
$m_{q}^{2}/M^{2}$ which is numerically an effect of order $10^{-6}$ for the
up- and down-flavors.

This is an interesting observation for the following reason. The description
of TMDs in QCD becomes equivalent to that in the parton model in the Wandzura-
Wilczek-type (WW-type) approximation Avakian:2007mv . This approximation
consists in exploring the QCD equations of motion for twist-3 TMDs to relate
them to the better known twist-2 TMDs and the so-called tilde terms which are
contributions due to quark-gluon-quark matrix elements and current quark mass
terms. Neglecting the tilde- and current quark mass terms constitutes the WW-
type approximation. (The attribute “type” is added to distinguish the more
complex TMD case from the original WW-approximation for the colinear function
$g_{T}^{q}(x)$ Wandzura:1977qf .) The exploration of the free equation of
motion in the parton model generates exactly the same mass terms as in QCD but
of course no tilde-terms. In this sense, the predictions of the parton model
are equivalent to the description of TMDs in QCD in the WW-type approximation.

The linear QMRs (2) hold in the CPM only if one introduces an additional
constraint which is equivalent to putting the quarks in a pure-spin state. It
remains to be seen whether this leads to a realistic modelling of the nucleon
structure from phenomenological point of view. However, the non-linear QMRs
(3) do not require such an additional condition, and are valid also for
(light) quarks in a mixed-spin state. This could imply that the non-linear
QMRs (3) are more likely to be supported in QCD because no additional (pure-
spin state) condition is required for their validity.

The observation that the non-linear QMRs (3) could be valid in the WW-type
approximation is interesting. The WW-type approximation has been explored for
phenomenological applications for instance in Bastami:2018xqd . The quality of
this approximation cannot be determined a priori, and it needs to be
investigated on a case by case basis because different quark-gluon-quark
matrix elements are neglected in each case. In some cases the WW-type
approximation was shown to work with a phenomenologically useful approximation
Bastami:2018xqd ; Bhattacharya:2021twu  and in one case there is support from
lattice QCD Bhattacharya:2021moj . It will be very interesting to investigate
whether the non-linear QMRs (3) could be valid in QCD with a similarly useful
approximation

##  VI Conclusions and outlook

In this work, we have investigated the quark-model relations (QMRs) in the
mixed-spin version of the covariant-parton model (CPM). The equations of
motion in the CPM imply some conditions among the amplitudes in the quark
correlator, but leave the amplitudes $A_{8}^{q}$ and $A_{11}^{q}$ unrelated.
We have shown that the linear QMRs are not valid, unless one introduces the
condition $A_{8}^{q}=-A_{11}^{q}$. This condition is equivalent to putting the
quarks in a pure-spin state (more precisely: the pure-spin state condition
only implies $|A_{8}^{q}|=|A_{11}^{q}|$ and does not determine the relative
sign).

Our results are of interest because they give insights on the CPM and raise
interesting questions about quark models and QMRs. The observation that
imposing linear QMRs is equivalent to putting the quarks in a pure-spin state
is primarily an insight about the CPM but may be of interest also beyond this
model for the following reason. In the CPM the two statements, (i) quarks are
in the pure-spin state and (ii) model complies with QMRs, are equivalent. It
will be interesting to investigate whether this is the case also in other
models: if a quark model obeys the QMRs, are then the quarks in this model in
a pure-spin state? This aspect deserves further investigations.

We also learn an interesting lesson about QMRs. In QCD, each TMD is an
independent function describing a different aspect of the nucleon structure,
and no relations among TMDs exist. In quark models, the situation can be
simpler and relations among TMDs may exist. Such relations become particularly
interesting if they are supported by a wide class of different quark models as
is the case with the QMRs which arise from a certain symmetry of the nucleon
wave function which is present in many (though not all) quark models.

QMRs become even more interesting if they require only general model
assumptions. In the CPM, the linear QMRs require a strong model assumption,
namely the quarks must be in pure spin state. The situation is different for
the nonlinear QMRs. These relations become exact in the CPM for quarks in a
pure-spin state and/or for massless quarks. However, even in the most general
case in the CPM, i.e. for massive quarks in a mixed-spin state, the nonlinear
QMRs are still valid to a very good approximation, namely up to negligibly
small quadratic quark mass effects $\propto m_{q}^{2}/M^{2}$.

Thus, the nonlinear QMRs are practically supported in the CPM independently of
the quark spin state. In other words, the nonlinear QMRs require no strong
model assumption (like the pure-spin condition). From the point of view of the
CPM, all that is required for the nonlinear QMRs is the absence of
interactions. From the point of view of QCD, this in turn means that the non-
linear QMRs could be valid in the WW-type approximation.

It is important to remark, that even if the QMRs were valid at one scale, due
to the different evolution equations of the different TMDs, they would not be
valid at other scales. However, considering their crude nature, the ”accuracy”
of quark models can be expected to be around ${\cal O}$(30-40$\,\%$)
Boffi:2009sh ; Pasquini:2011tk , the TMD evolution effects are not a dominant
uncertainty. It will be interesting to see whether phenomenological
extractions or lattice QCD results will support, at some scale, predictions
from quark models like the CPM within such model accuracy.

The spin state of a quark in QCD is not easy to determine Dalitz:1988aq ;
Efremov:1992pe ; Collins:1992kk . The comparison of the CPM predictions to
phenomenological results for TMD extractions will constitute one way to infer
to which extent the quarks in QCD can be viewed as being in a pure- or mixed-
spin state. It will be interesting to shed more light on the polarization
state of the quarks in the nucleon based on dedicated phenomenological, model
and lattice QCD studies.

Acknowledgments. This work was partly supported by NSF under the Award No.
1812423 and Award No. 2111490, by the U.S. Department of Energy, under the
contract no. DE-AC05-06OR23177 under which Jefferson Science Associates, LLC
operates Jefferson Lab, and within the framework of the TMD Collaboration.

##  Appendix A Quark model expressions for T-even TMDs

In this Appendix, to make this work self-contained, we list the quark model
expressions for T-even TMDs in terms of the amplitudes defined in (III.1).
These expressions are valid in all models without gauge field degrees of
freedom. In QCD, the TMDs depend on the renormalization scale $\mu^{2}$ and
the scale $\zeta$ at which lightcone divergences are regulated. In this work,
we do not indicate the scales for brevity. The determination of these scales
in a model calculation is is an important part of the modelling. In previous
works in the CPM, the scales were assumed to be
$\mu^{2}=\zeta\simeq=(\mbox{3-4})\,{\rm GeV}^{2}$.

In the twist-2 case, the expressions for TMDs read

$\displaystyle f_{1}^{q}(x,k_{T})\;$ $\displaystyle=$
$\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}A_{2}^{q}+xA_{3}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$ (23)
$\displaystyle g_{1}^{q}(x,k_{T})\;$ $\displaystyle=$
$\displaystyle\;2P^{+}\int dk^{-}\biggl{[}-A_{6}^{q}-\frac{P\cdot
k-M^{2}x}{M^{2}}(A_{7}^{q}+xA_{8}^{q})\biggr{]}_{k^{+}=xP^{+}}\,,$ (24)
$\displaystyle g_{1T}^{\perp q}(x,k_{T})\;$ $\displaystyle=$
$\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}A_{7}^{q}+xA_{8}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$ (25)
$\displaystyle h_{1}^{q}(x,k_{T})\;$ $\displaystyle=$
$\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}-A_{9}^{q}-xA_{10}^{q}+\frac{\vec{k}_{T}^{\,2}}{2M^{2}}\;A_{11}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$
(26) $\displaystyle h^{\perp q}_{1L}(x,k_{T})\;$ $\displaystyle=$
$\displaystyle\;2P^{+}\int dk^{-}\biggl{[}A_{10}^{q}-\frac{P\cdot
k-M^{2}x}{M^{2}}\;A_{11}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$ (27) $\displaystyle
h^{\perp q}_{1T}(x,k_{T})\;$ $\displaystyle=$ $\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}A_{11}^{q}\biggr{]}_{k^{+}=xP^{+}}\,.$ (28)

In the twist-3 case, the expressions are given by

$\displaystyle e^{q}(x,k_{T})\;$ $\displaystyle=$ $\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}A_{1}^{q}\;\biggr{]}_{k^{+}=xP^{+}}\,,$ (29) $\displaystyle
f^{\perp q}(x,k_{T})\;$ $\displaystyle=$ $\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}A_{3}^{q}\;\biggr{]}_{k^{+}=xP^{+}}\,,$ (30) $\displaystyle
g_{T}^{q}(x,k_{T})\;$ $\displaystyle=$ $\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}-A_{6}^{q}+\frac{\vec{k}_{T}^{\,2}}{2M^{2}}\;A_{8}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$
(31) $\displaystyle g^{\perp q}_{L}(x,k_{T})\;$ $\displaystyle=$
$\displaystyle\;2P^{+}\int dk^{-}\biggl{[}-\frac{P\cdot
k-M^{2}x}{M^{2}}\;A_{8}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$ (32) $\displaystyle
g^{\perp q}_{T}(x,k_{T})\;$ $\displaystyle=$ $\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}A_{8}^{q}\;\biggr{]}_{k^{+}=xP^{+}}\,,$ (33) $\displaystyle
h_{L}^{q}(x,k_{T})\;$ $\displaystyle=$ $\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}-A_{9}^{q}-\frac{P\cdot k}{M^{2}}\;A_{10}^{q}+\frac{(P\cdot
k-M^{2}x)^{2}}{M^{4}}\,A_{11}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$ (34)
$\displaystyle h_{T}^{q}(x,k_{T})\;$ $\displaystyle=$
$\displaystyle\;2P^{+}\int dk^{-}\biggl{[}-\frac{P\cdot
k-M^{2}x}{M^{2}}\;A_{11}^{q}\biggr{]}_{k^{+}=xP^{+}}\,,$ (35) $\displaystyle
h^{\perp q}_{T}(x,k_{T})\;$ $\displaystyle=$ $\displaystyle\;2P^{+}\int
dk^{-}\biggl{[}-A_{10}^{q}\biggr{]}_{k^{+}=xP^{+}}\,.$ (36)

In QCD also $B_{i}^{q}$ amplitudes enter, see e.g. Metz:2008ib  for the full
expressions. But in quark models the 14 T-even TMDs are expressed in terms of
9 T-even $A_{i}^{q}$ amplitudes. This implies 5 relations, namely the qLIRs
mentioned in Sec. II.

We also remark that, in contrast to QCD, in the CPM no UV- or rapidity
divergences appear. This allows one to relate TMDs and colinear parton
distribution functions simply as $f_{1}^{q}(x)=\int
d^{2}k_{T}f_{1}^{q}(x,k_{T})$ with a finite $k_{T}$-integration, which in QCD
Collins-book  as well as in some models Schweitzer:2012hh ; Aslan:2021new2  is
spoiled by the appearance of divergences.

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