# The Quark Pauli Principle and the Transmutation of Nuclear Matter

Larry McLerran  Institute for Nuclear Theory, University of Washington, Box
351550, Seattle, WA 98195, USA<EMAIL_ADDRESS>Gerald A. Miller  Department
of Physics, University of Washington, Seattle, WA 98195, USA<EMAIL_ADDRESS>
###### Abstract

The phase space density, $\rho^{Q}$, of quarks in nuclei is studied using
realistic models of unintegrated quark distributions, known as transverse
momentum densities (TMDs). If this density exceeds unity for matter at normal
nuclear densities, the effects of the quark Pauli principle must play a role
in nuclei, and models in which the nucleon density at low momentum is small
(Quarkyonic matter) may become a starting point for an entirely new
description of nuclei. We denote the nuclear density for which $\rho^{Q}=1$ to
be a transmutation density, $n_{T}$, because quark degrees of freedom must be
relevant at that density. Including the TMDs of [G. de Teramond et. al, Phys.
Rev. Lett. 120, 182002, (2018)] for the valence quarks and phenomenological
TMDs for the sea quarks we find that $n_{T}=0.17\pm 0.04\,\rm fm^{-3}$, the
density of normal nuclear matter. Some of fhe implications of this finding are
discussed.

††preprint: NT@UW-24-07

Introduction

In Ref.McLerran and Pisarski (2007), a class of solvable models of Quarkyonic
matter, corresponding to a filled quark Fermi sea with a baryonic Fermi
surface were proposed. These Ideal Dual Quarkyonic (IdylliQ) models correspond
to a free gas of nucleons composed of quarks with a constraint that the phase
space density of quarks cannot exceed one Fujimoto  _et al._ (2024). The phase
space density of quarks was determined by the probability to find a quark
inside a nucleon convoluted with the nucleon density. For a particular choice
of probability density, this model was explicitly solved. The IdylliQ models
are constructed to be explicitly dual between the nucleon and the quark
descriptions. There are two phases. The low density phase is an ideal gas of
nucleons and nowhere does the quark phase space density equal or exceed one.
On the other hand, the high density phase corresponds to a filled Fermi sphere
of quarks with a diffuse Fermi surface. The nucleon sits on a shell in
momentum space at a momentum of order $k_{F}^{N}\sim N_{c}k_{F}^{Q}$ where
$k_{F}^{Q}$ is the Fermi momentum of the filled quark Fermi sea and $N_{c}$ is
the number of quark colors. In addition to the shell of nucleons at the Fermi
surface, there is an under-occupied density of nucleons which corresponds to
the Fermi sea of quarks. The onset density where this occurs is determined
from the probability density to find a quark inside a hadron at zero quark
momentum becoming equal to one. In Quarkyonic matter, a hard equation of
state, $P\sim{1\over 3}\epsilon$, is rapidly approached McLerran and Reddy
(2019).

In Ref. Koch  _et al._ (2024), the Quarkyonic model of noninteracting nucleons
was generalized to include pion and sigma meson interactions. A choice of the
probability distribution to find a quark inside the nucleon was made that
allowed an exact solution of the theory, with scalar mesons ($\sigma$) and
$\pi$ exchange interactions included in the mean field approximation. When the
momentum scale in this probability distribution, and the scalar meson coupling
constants were tuned to produce the binding energy and density of isospin
symmetric nuclear matter, it was found that the Quarkyonic transition occurred
slightly below the density of nuclear matter, and an acceptable phenomenology
of the bulk properties of nuclear matter was extracted.

A simple relationship that determines the onset density of Quarkyonic matter
was found in Ref. McLerran and Pisarski (2007) and was exploited in Ref. Koch
_et al._ (2024). This relation is best expressed in terms of phase space
densities. The phase space density for a particle is determined from its
number density as

$\rho(k)={1\over N_{dof}}(2\pi)^{3}{{dn}\over{d^{3}k}}$ (1)

The factor of $N_{dof}$ accounts for the number of states due to degeneracy in
color, spin and isospin. We assume degeneracy in these degrees of freedom and
will consider isosinglet nuclear matter. The phase space density corresponds
to the quantum occupation number of states at a momentum $k$, and with the
factor of $N_{dof}$ extracted, will have a maximum value of 1 for fermions.
The phase space density of quarks at finite temperature and density system of
quarks would be $\rho^{Q}(E)={1\over{1+e^{(E-\mu_{Q})/T}}}$ where $T$ is the
temperature and $\mu_{Q}=\mu_{N}/N_{c}$ is the chemical potential for the
quarks, and there is a similar relation for nucleons,
$\rho^{N}(E)={1\over{1+e^{(E-\mu_{N})/T}}}$. If we ignore the anti-quarks, the
baryon number density $n_{B}$ satisfies,

$n_{B}=4\int~{}{{d^{3}p}\over{(2\pi)^{3}}}~{}\rho^{Q}(p)=4\int~{}{{d^{3}p}\over{(2\pi)^{3}}}~{}\rho^{N}(p)$
(2)

because quarks carry baryon number $1/N_{c}$ and the contribution from the
quark phase space density has an overall factor of $N_{c}$. When anti-quarks
are included in the nucleon wave function, the above sum rule represents the
difference between quark and antiquark phase space densities. Including anti-
quarks does not affect the value of $\rho^{Q}-\rho^{\overline{Q}}$, but when
we relate phase space densities of nucleons to quarks, $q\overline{q}$ pairs
will play a role in determining the phase space density of quarks, $\rho^{Q}$.
The phase space density represents the occupation number of quantum mechanical
states, and the Pauli principle imposes a restriction $\rho^{Q}\leq 1$. The
present work is concerned with the zero temperature limit.

Ref. Koch  _et al._ (2024) assumed there are only valence quarks in the
nucleon. In that work, the quark and nucleon phase space densities are related
as

$\rho^{Q}(\vec{k})=\int{{d^{3}p}\over(2\pi)^{3}}K(\vec{k}-\vec{p}/N_{c})\rho^{N}(p)$
(3)

where $K$ is the normalized probability distribution to find a quark inside a
nucleon, $\int~{}{{d^{3}k}\over{(2\pi)^{3}}}K(\vec{k})=1$. This quark
distribution saturates when $\rho^{Q}(0)=1$, and an equation that determines
the onset density was obtained. The content of Ref.McLerran and Pisarski
(2007) was to analyze this IdylliQ model for a system of free nucleons
composed of quarks satisfying the constraint that $\rho^{Q}\leq 1$. There is a
transition from an ideal gas of nucleons to Quarkyonic matter when this
condition is first satisfied, $\rho^{Q}(k=0)=1$. In Ref. Koch  _et al._
(2024), it was argued that this transition might occur at densities below
nuclear matter and then assuming this is the case, with the inclusion of
interactions with a pion and sigma field, a reasonable phenomenology of
nuclear matter was constructed. An unusual nucleon occupation density in which
$\rho^{N}$ is $1/N_{c}^{3}$ for small nucleon momenta was obtained. A viable
description of quasi-elastic electron scattering from nuclear matter was
obtained despite this exceptional feature.

The purpose of this paper is to derive a condition for the onset of Quarkyonic
matter using experimentally constrained probability dsitributions for quarks
and anti-quarks in a nucleon. In particular, we will use the Light Front QCD
distribution functions of the HLFHS collaborationde Teramond  _et al._
(2018),Sufian  _et al._ (2017), as this formalism provide a very successful
hadron phenomenology with specific forms of distribution functions. We also
will employ the sea quark distributions of CTEQ10 Lai  _et al._ (2010).

Quark phase space density

We use light-front, Fock-Space wave functions to obtain $\rho^{Q}(\vec{k})$
Brodsky  _et al._ (1998). In such a description this quantity is the square of
a light-front wave function. Such a wave function depends on a longitudinal
momentum fraction $x$ and the transverse component of momentum, $\vec{k}_{T}$,
e.g. $|\psi(x,\vec{k}_{T})|^{2}$. These quantities are commonly called
unintegrated distributions or transverse momentum distributions Angeles-
Martinez  _et al._ (2015).

One may relate $\rho^{q}(\vec{k})$ to $\rho^{q}(x,\vec{k}_{T})$ by using the
transformation

$x={k_{z}+E(\vec{k})\over p^{+}}$ (4)

where the $p^{+}$ is the sum of the nucleon’s energy and $z$-component of
momentum, and ${\partial x\over\partial k_{z}}=x/p^{+}.$ Preserving the
integral over all values of $\vec{k}$ leads to the equivalent expressions:
${d^{3}n_{q}\over d^{3}k}={x\over E}{d^{3}n_{q}\over dxd^{2}k_{\perp}}$ so
that

$\displaystyle n_{q}(\vec{k})={x\over E}{d^{3}n\over dxd^{2}k_{\perp}}={x\over
E}|\psi(x,\vec{k}_{T})|^{2}$ (5)

We shall be guided by Ref. de Teramond  _et al._ (2018) that presents models
of wave functions for specific Fock-space components of valence quark wave
functions. These do not carry the entire momentum of the nucleon. It is well-
known that gluons and the sea carry the remainder. Therefore, we use that
paper’s valence quark distributions and include also the effects of the quark
sea using the CTEQ10 distribution functions.

Light-front wave functions

The light-front valence wave functions of de Teramond  _et al._ (2018) are
given for parton numbers $\tau=3,4$ by

$\psi_{\rm
eff}^{\tau}(x,\vec{k}_{T})=4\sqrt{2}\pi\frac{\sqrt{q_{\tau}(x)f(x)}}{1-x}\,\exp\left[-\frac{2f(x)}{(1-x)^{2}}\,\vec{k}_{T}^{2}\right],$
(6)

with normalization
$\int_{0}^{1}dx\int\frac{d^{2}\vec{k}_{T}}{(2\pi)^{3}}\left|\psi_{\it
eff}(x,\vec{k}_{T})\right|^{2}=1.$ The quark distribution functions are given
by

$\displaystyle
q_{\tau}(x)=\frac{\Gamma(\tau-1/2)}{\Gamma(\tau-1)\sqrt{\pi}}\big{(}1-w(x)\big{)}^{\tau-2}\,w(x)^{-{\frac{1}{2}}}\,w^{\prime}(x)$
(7) $\displaystyle w(x)=x^{1-x}e^{-a(1-x)^{2}},$ (8)

with. $f(x)={1\over 4\lambda}\log{1/w(x)}$, and $a$ a flavor independent
parameter $a=0.531\pm 0.037$, and $\lambda=(0.548\,{\rm GeV})^{2}$. The
valence $u$ and $d$ quark distributions are given by:

$\displaystyle u_{\rm v}(x)$ $\displaystyle=$ $\displaystyle{3\over
2}q_{\tau=3}(x)+\frac{1}{2}\,q_{\tau=4}(x),$ (9) $\displaystyle d_{\rm v}(x)$
$\displaystyle=$ $\displaystyle\,q_{\tau=4}(x),$ (10)

The normalization is $\int_{0}^{1}dx\,q_{\tau}(x)=1$. These distributions are
in good agreement with experimental values and the nucleon electromagnetic
form factors derived from these wave functions are also in good accord with
measured data.

These considerations enable us to develop the necessary more detailed versions
of Eq. (5), to be able to use parton distribution functions and TMDs that are
functions of light cone variables. We rewrite Eq. (5) using
$E(\vec{k})=\sqrt{\vec{k}^{2}+\omega_{q}^{2}}$, where $\omega_{q}$ is a
suitable quark energy. In the non-relativistic quark model $\omega_{q}$ is a
quark mass and in the bag model it is a mode energy.

Then with $\vec{k}_{T}=0$ and Eq. (4), one finds

$\displaystyle k_{z}={x^{2}M^{2}-\omega_{q}^{2}\over 2xM},$ (11)

so that $k_{z}(\tau)=0$ corresponds to a transmutation value of $x_{\rm
trans}=\omega_{q}/M$. The function $\omega_{q}$ depends on the given Fock-
Space component, so that $x_{\rm trans}$ varies with the component even though
$\vec{k}$ vanishes for all components.

This result together with Eqn. 5, gives the quark probability density per unit
momentum at $k=0$ where $x_{0}$ refers to the average $x$ value for a
particular Fock space component as

$n_{q}(\vec{k}=0)={1\over M}|\psi(x_{\rm trans},\vec{k}_{T})|^{2}$ (12)

It is interesting to note that this result would be obtained immediately by
considering the phase space density to be ${d^{3}n\over dk^{+}d^{2}k_{\perp}}$
with $k^{+}=xM$.

The next step is to estimate the values of $x_{\rm trans}$. We assume that the
glue accounts for one-half of the nucleon mass. Then for $\tau=3$ and taking
all Fermions to have an equal contribution to the nucleon mass, we find
$\omega_{q}(\tau=3)={1\over 6}M,$ so that $x_{\rm trans}(3)={1\over 6}$ for
that component. The component with $\tau=4$ is a valence contribution in the
theory of Ref. de Teramond  _et al._ (2018), so that if we consider the system
to be three quarks along with a valence gluon, we find
$\omega_{q}(\tau=4)={1\over 8}M,$ so that $x_{\rm trans}(4)={1\over 8}$. It is
useful to compare these values with the average values of $x$ computed using
$q_{3,4}$ of Eq. (7). Defining $x_{\tau}\equiv\int_{0}^{1}dx\,q_{\tau}(x)$ we
find $x_{3}=0.184$ and $x_{4}=0.136$ in good agreement with the estimates
$1/6$ and $1/8$. This means that on average the values of $x$ are close to
$x_{\rm trans}$. Using Eq. (9) and Eq. (10) we find the average value of $x$
for the up quark is 0.172 and that for the down quark is 0.126. These are in
good accord with the values from CTEQ10 of 0.153 and 0.126.

The formalism of de Teramond  _et al._ (2018) is not designed to give
expressions for sea quark distributions. Instead, we modify Eq. (6) by
replacing $q_{\tau}$ of Eq. (7) by sea quark distributions $u_{s}(x)$ and
$d_{s}$ taken from CTEQ10. This procedure follows the logic of de Teramond
_et al._ (2018) because the dependence on $\vec{k}_{T}$ is universal in that
logic. Doing this maintains the spatial extent of the configuration to of the
nucleon size. We next need to apply Eq. (11), again assuming that sea gluons
account for half of the nucleon mass. Three quarks plus one $q\bar{q}$ pair
would lead to a five-parton system and $x=1/10.$ This value is an upper limit,
since this is the minimal state which has quark-antiquark pairs in it. Other
possible values of examined below.

Nuclear densities

The next step is to place the nucleon into the nucleus. The standard
convolution formula, when applied to the quark phase space density reads

$\displaystyle n_{q}^{\tau}(x,\vec{k}_{T})={1\over M}\int_{x}^{\infty}{dy\over
y}|\psi_{\rm eff}^{\tau}(x/y,\vec{k}_{T})|^{2}f_{N}(y),$ (13)

where $\psi_{\rm eff}^{\tau}(x/y,\vec{k}_{T})$ is the light-front wave
function for the given Fock-space component and $f_{N}(y)$ is the probability
that a nucleon carries a momentum fraction $y=(E(p)+p_{z}/M$, where $\vec{p}$
is the nucleon momentum and $M$ is the nucleon mass. The derivation of this
formula for the integrated (over $\vec{k}_{T}$) is discussed in many reviews
Geesaman  _et al._ (1995); Norton (2003); Hen  _et al._ (2013, 2017). One
simple derivation, based on Feynman diagrams is in Miller and Smith (2002).

The function $f_{N}(y)$ is given by

$\displaystyle
f_{N}(y)=\int{d^{2}p_{\perp}dp^{+}\over(2\pi)^{3}}\delta(y-{p^{+}\over
M})n_{N}(\vec{p})$ (14)

with $p_{z}={1\over 2}(p^{+}-{{M}^{2}+p_{\perp}^{2}\over p^{+}})$. In the
present work we take the nucleon momentum density
$n(p)=\Theta(k_{F}^{2}-p_{\perp}^{2}-p_{z}^{2})$, a standard Fermi gas
function. In that case $\int dyf_{N}(y)=1/(6\pi^{2})k_{F}^{3}$, where $k_{F}$
is the Fermi momentum and the integral is one-fourth of the nucleon density of
nuclear matter, $\rho_{NM}/4$. For such a density the integral above is
approximated by using $f_{N}(y)=\delta(1-y)$, accurate to better than 1 % for
the values of $x$ that are of interest here. Thus for a given value of
$\omega_{q}$:

$\displaystyle\rho_{q}(\vec{k}=0)={1\over M}|\psi_{\rm
eff}^{\tau}({\omega_{q}\over M},\vec{k}_{T}=0)|^{2}{\rho_{NM}\over 4}$ (15)

Pauli principle condition

We are concerned with isospin-symmetric nuclear matter ($N=Z)$, so that the
isoscalar wave function at $\vec{k}=0$ is given by

$\displaystyle|\psi_{v}|^{2}\equiv{1\over 2}(|\psi_{\rm eff}^{3}({1\over
6},0)|^{2}+|\psi_{\rm eff}^{4}({1\over 8},0)|^{2}),$ (16)

where the arguments are obtained from Eq. (11). The isoscalar wave function
for the sea is given, in a first estimate, by

$\displaystyle|\psi_{s}|^{2}(x)\equiv{1\over 3}(|\psi_{\rm
u\,sea}({x},0)|^{2}+|\psi_{\rm d\,sea}({x},0)|^{2}),$ (17)

with the factor 1/3 accounting for the fact that there are three colors of
quarks. As noted above, the sea could contain many anti-quarks. A 7 fermion
system would have an average value of $x$ of 1/14=0.07, and a 9 quark system
would have an average value of $x$ of 1/18=0.055. We taking the probability of
having between 5 and 7 fermions to be evenly distributed so that, in a first
estimate, we use

$\displaystyle\overline{|\psi_{\rm s\,}|^{2}}\equiv{1\over 3}(|\psi_{\rm
s}({.1},0)|^{2}+|\psi_{\rm s}({1/14},0)|^{2}+|\psi_{\rm s}({1/18},0)|^{2}).$
(18)

Eq. (15) shows that the quark number density increases as the value of
$\rho_{NM}$ increases. Since $\rho^{q}\leq 1$ there will be a maximum value of
$\rho_{NM}$ that causes $\rho^{q}(\vec{k}=0)$ to be unity. Let us define that
critical value of nuclear density, for which the quark Pauli principle is
unity to be the transmutation density $\rho_{T}$. Then Eq. (15) tells us that

$\displaystyle\rho_{T}^{\rm
in}=\frac{4M}{|\psi_{v}|^{2}+\overline{|\psi_{s}|^{2}}},$ (19)

where the superscript refers to an initial estimate The standard value of the
density of nuclear matter is $\rho_{0}=0.17\,\rm fm^{-3}$ Bohr and Motttelson
(1998). If $\rho_{T}\geq\rho_{0}$ then the quark Pauli principle must enter
into the physics of nuclei.

Using Eq. (19) leads to the result that $\rho_{T}^{\rm in}=0.168=\rho_{0}$, so
that normal nuclear matter may be quarkyonic! In this calculation the valence
quark contribution to the denominator of Eq. (19) is about 55% of the total
and the sea quark contribution is 45%. The valence quark contribution to
$\rho_{T}$ is consistent with the estimate obtained in Ref. Koch  _et al._
(2024) using a Gaussian model of the nucleon momentum density constrained by
the measured charge radius of the proton.

It is necessary to account for the uncertainty that arises from using
different estimates of $\omega_{q}$. A rough estimate of the uncertainty in
this estimate is obtained by using the spread of $x-$values between $1/10$ and
$1/18$ instead of the average in Eq. (18). This gives

$\rho_{T}\approx 0.17\pm 0.04\,\rm fm^{-3}.$ (20)

We also examine the variation caused by varying the value of 1/6 and 1/8
appearing by by plus or minus 10 % to account for differences between the
theory and CTEQ10 for the valence quarks. The effect of this variation along
with that of varying the value of $x_{s}$ at which the sea quark distributions
is shown in Fig. 1. The central result is that it is reasonable to expect that
the quark Pauli principle plays a role in nuclear physics at normal densities.
Normal nuclear matter might very well be quarkyonic.

Figure 1:  Variation of transition density, $\rho^{T}$ with $x_{s}$. The three
different curves are obtained from Eq. (19) and then varying the arguments of
the valence quark distributions by $\pm 10$%.

Implications and Discussion

In this paper, we determined the transmutation density to Quarkyonic matter
using phenomenologically justified transverse momentum distributions, parton
distribution functions unintegrated in $k_{T}$. This advances the treatment of
Ref. Koch  _et al._ (2024), which used a probability distribution, $K(k,p)$,
chosen to simplify the computations, might be criticized for lack of support
from direct measurements of such distributions. Our computations here strongly
suggest that the transmutation density is quite close to that of nuclear
matter at normal density, even though there are significant uncertainties in
evaluating the kinematics at which $\vec{k}=0$ holds.

Suppose the Pauli repulsion of quarks, that enters when $\rho^{T}$=1, is
indeed the origin of the mechanism which stabilizes nuclear matter against
collapse. This suggests that $\omega$ meson repulsion or repulsive contact
interactions, the stabilization mechanisms of many nuclear computations, has
its origin in the Fermi repulsion of quarks arising from the Pauli principle.
The treatment of such repulsive forces results Koch  _et al._ (2024) in much
different Fermi distributions of nucleons than is the case for mean field
treatments of nuclear matter. This is because the overlap of quarks in the
core leads to a filling of quark energy levels, which imply that the
corresponding nucleon distributions will be under-occupied. This leads to a
shell-like structure for the momentum distributions of nucleons. The resulting
physical picture of nuclear matter has fundamental differences from that of a
mean field description. Indeed, nuclear matter might not be a normal Fermi
liquid.

We recognize that the ideas of this paper must be tested against existing
measurements of nuclear properties. Ref. Koch  _et al._ (2024) already shows
that Quarkyonic matter is not inconsistent with data taken in the region of
the quasi-elastic peak of electron scattering, but further exploration is
necessary. Quantum fluctuations of nuclear density must be considered. Short-
ranged correlations occur when two nucleons overlap in space. If so, the
density can be higher than $\rho_{0}$, and the effects of quark degrees of
freedom must be relevant i.e. short-ranged nucleon-nucleon correlations must
be associated with quark degrees of freedom. The influences of quarks cause
the nuclear modification of parton distribution functions, the EMC effect
Geesaman  _et al._ (1995); Norton (2003); Hen  _et al._ (2013, 2017). A much
more detailed analysis of the relationship between Quarkyonic matter and the
EMC effect is warranted. Nevertheless, it is fair to infer that the
considerations of this paper strengthen the idea that there is a causal
relation between between the EMC effect and short-ranged correlations as in
the review Hen  _et al._ (2017).

## Acknowledgements

L.M. acknowledges the support by the US Department of Energy under contract
number DE-FG02-00ER41132. and G.A.M. acknowledges the DOE grant No. DE-
FG02-97ER41014. We thank Dmitriy N. Kim for providing the CTEQ10 quark
distributions.

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