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The phase structure of two color QCD and charged pion condensation phenomenon

T. G. Khunjua 1, K. G. Klimenko2, Roman N. Zhokhov3¶

1The University of Georgia, GE-0171 Tbilisi, Georgia

2Logunov Institute for High Energy Physics, NRC ”Kurchatov Institute”, 142281,
Protvino, Moscow Region, Russia

3Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave
Propagation (IZMIRAN), 108840 Troitsk, Moscow, Russia

<EMAIL_ADDRESS>
Abstract

The phase structure of dense quark matter in the two color case has been
investigated with nonzero baryon $\mu_{B}$, isospin $\mu_{I}$ and chiral
isospin $\mu_{I5}$ chemical potentials. It has been shown in the mean-field
approximation that there exist three dualities, one of them between phases
with spontaneous chiral symmetry breaking and condensation of charged pions,
found in the three color case. The other two dual symmetries between the phase
with condensation of charged pions and the phase with diquark condensation and
between chiral symmetry breaking and diquark condensation phenomena. It has
been demonstrated that due to the duality properties the phase diagram is
extremely symmetric and the whole phase diagram in two color case can be
obtained just by dualities from the phase structure of three color case. This
shows that the dualities are rather usefull tool. It is shown that chiral
imbalance generates charged pion condensation phenomenon in conditions of
matter in neutron stars, i. e. electrically neutral and $\beta$-equilibrated
dense quark matter. And that diquark condensation does not prohibit the
generation of the charged PC phase by chiral imbalance at least in the two
color case.

##  1 Introduction

It is contemplated that QCD is the theory of strong interaction and it should
be used for the studies of strongly interacting matter at large temperatures
and baryon densities. However, the perturbation theory cannot be used in the
conditions realized in heavy ion collisions or compact star matter. Lattice
simulations of QCD, which are nonperturbative ab-initio method and are quite
successful at finite temperatures, due to a notorious sign problem cannot be
applied to the case of non-zero baryon density, especially at rather large
values of baryon . Thus, there arise the interest to QCD-like models, for
example, to Nambu–Jona-Lasinio (NJL)-like models [1, 2].

In addition to finite temperature and non-zero baryon density, there can be
important various other quantities in quark matter. Tor example, isospin
imbalance, the difference between the number of $u$ and $d$ quarks, is an
obvious property of neutron stars. Rather recently it was demonstrated that
chiral imbalance, i. e. difference between densities of left and right-handed
quarks, is a quite interesting feature of quark matter [3, 4, 5]. An this led
to the whole new notion of anomalous transport phenomena, for example, the so-
called chiral magnetic effect [6]. Chiral imbalance can be accounted for by
chiral chemical potential $\mu_{5}$. There is another possibility of different
chiral chemical potentials for $u$ and $d$ quarks, $\mu_{u5}\neq\mu_{d5}$, and
chiral isospin chemical potential $\mu_{I5}=\mu_{u5}-\mu_{5d}$.

Recently, attention is paid to two-color QCD and QCD-like models [7, 8, 9, 10,
11, 12, 13, 14, 15, 16]. Although a two color and tree color cases have
substantial differences, the investigations of SU(2) QCD can provide us with
valuable insight about the properties of QCD at non-zero baryon density. As an
additional motivation let us recall that in the QCD with two colors there is
no sign problem and ab initio lattice simulations are possible. Also one can
stress that two-color QCD phase diagram is rather rich and can be quite
interesting by its own.

The paper is organized in the following manner. In Sec. 2 we introduce a two
color NJL model with two quark flavors ($u$ and $d$ quarks). It contains both
quark-antiquark and diquark channels of interaction and is a low energy
effective model of the two-color QCD (see [9]). Sec. III contains the
discussion on three duality properties (dual symmetries) of the model. In Sec.
4 the phase structure of the models is investigated. We discuss the story of
the prediction of charged pion condensation in dense quark matter in Sec. 5.

##  2 Two color NJL model

The Lagrangian of effective tow-color four-fermion NJL model with baryon
$\mu_{B}$, isospin $\mu_{I}$ and chiral isospin $\mu_{I5}$ chemical potentials
is

$\displaystyle L=\bar{q}\Big{[}i\hat{\partial}-m_{0}+{\cal
M}\gamma^{0}\Big{]}q+H\Big{[}(\bar{q}q)^{2}+(\bar{q}i\gamma^{5}\vec{\tau}q)^{2}+\big{(}\bar{q}i\gamma^{5}\sigma_{2}\tau_{2}q^{c}\big{)}\big{(}\overline{q^{c}}i\gamma^{5}\sigma_{2}\tau_{2}q\big{)}\Big{]},$
(1)

where $q\equiv q_{i\alpha}$ is a flavor ($i=u,d$) and color ($\alpha=1,2$)
doublet and a four-component Dirac spinor as well.
$\hat{\partial}\equiv\gamma^{\rho}\partial_{\rho}$ and charge-conjugated
spinors are $q^{c}=C\bar{q}^{T}$, $\overline{q^{c}}=q^{T}C$, where
$C=i\gamma^{2}\gamma^{0}$. Moreover, one uses the notations
$\vec{\tau}\equiv(\tau_{1},\tau_{2},\tau_{3})$ and $\sigma_{2}$ for usual
Pauli matrices acting in the two-dimensional flavor and color spaces,
respectively.

Chemical potentials are included in the Lagrangian (1) in the following term

$\bar{q}{\cal
M}\gamma^{0}q=\bar{q}\Big{[}\frac{\mu_{B}}{2}+\frac{\mu_{I}}{2}\tau_{3}+\frac{\mu_{I5}}{2}\gamma^{5}\tau_{3}\Big{]}q.$

It contains baryon $\mu_{B}$-, isospin $\mu_{I}$\- and chiral isospin
$\mu_{I5}$ chemical potentials. Videlicet, this model (1) is able to describe
the properties of quark matter with nonzero baryon
$n_{B}=(n_{u}+n_{d})/2\equiv n/2$, isospin $n_{I}=(n_{u}-n_{d})/2$ and chiral
isospin $n_{I5}=(n_{u5}-n_{d5})/2$ densities. These quantities are
thermodynamically conjugated to the following chemical potentials $\mu_{B}$,
$\mu_{I}$ and $\mu_{I5}$, respectively. The densities $n_{B}$, $n_{I}$ and
$n_{I5}$ can be obtained if one differentiate the thermodynamic potential
(TDP) of the model (1) with respect to the corresponding chemical potentials.

The Lagrangian (1) without term with chemical potentials is invariant under
color $SU(2)_{c}$ and baryon $U(1)_{B}$ symmetries. Moreover, in the chiral
limit ($m_{0}=0$), it has the same Pauli-Gursey flavor $SU(4)$ symmetry as
two-color QCD.

Now let us note that in the chiral limit ($m_{0}=0$) the Lagrangian (1) with
the term containing chemical potentials is no longer invariant under Pauli-
Gursey $SU(4)$ symmetry. So, apart from the color $SU(2)_{c}$ group, it is
invariant under $U(1)_{B}\times U(1)_{I_{3}}\times U(1)_{AI_{3}}$ group, where

$\displaystyle
U(1)_{B}:~{}q\to\exp(\mathrm{i}\alpha/2)q;~{}U(1)_{I_{3}}:~{}q\to\exp(\mathrm{i}\alpha\tau_{3}/2)q;~{}U(1)_{AI_{3}}:~{}q\to\exp(\mathrm{i}\alpha\gamma^{5}\tau_{3}/2)q.$
(2)

Note also that the quantities $n_{B}$, $n_{I}$ and $n_{I5}$ are the ground
state expectation values of the densities of conserved charges corresponding
to $U(1)_{B}$, $U(1)_{I_{3}}$ and $U(1)_{AI_{3}}$ symmetry groups and one has
$n_{B}=\langle\bar{q}\gamma^{0}q\rangle/2$,
$n_{I}=\langle\bar{q}\gamma^{0}\tau^{3}q\rangle/2$ and
$n_{I5}=\langle\bar{q}\gamma^{0}\gamma^{5}\tau^{3}q\rangle/2$. In addition,
the Lagrangian (1) is invariant under the electromagnetic $U(1)_{Q}$ group,

$\displaystyle U(1)_{Q}:~{}q\to\exp(\mathrm{i}Q\alpha)q,$ (3)

where $Q$ is a matrix of electric charges.

In order to find the TDP and study the phase structure one starts from an
equivalent semi-bosonized Lagrangian $\widetilde{L}$ with auxiliary bosonic
fields $\sigma(x)$, $\vec{\pi}=(\pi_{1}(x),\pi_{2}(x),\pi_{3}(x))$,
$\Delta(x)$ and $\Delta^{*}(x)$

$\displaystyle\widetilde{L}=\bar{q}\Big{[}i\hat{\partial}-m_{0}+{\cal
M}\gamma^{0}-\sigma-i\gamma^{5}\vec{\tau}\vec{\pi}\Big{]}q-\frac{\sigma^{2}+\vec{\pi}^{2}+\Delta^{*}\Delta}{4H}-\frac{\Delta}{2}\Big{[}\bar{q}i\gamma^{5}\sigma_{2}\tau_{2}q^{c}\Big{]}-\frac{\Delta^{*}}{2}\Big{[}\overline{q^{c}}i\gamma^{5}\sigma_{2}\tau_{2}q\Big{]},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2$
(4)

It is clear that the Lagrangians (1) and (4) are equivalent, as can be seen by
using the equations of motion for auxiliary bosonic fields that have the form

$\displaystyle\sigma(x)=-2H(\bar{q}q),$
$\displaystyle\Delta(x)=-2H\Big{[}\overline{q^{c}}i\gamma^{5}\sigma_{2}\tau_{2}q\Big{]}=-2H\Big{[}q^{T}Ci\gamma^{5}\sigma_{2}\tau_{2}q\Big{]},$
$\displaystyle~{}\vec{\pi}(x)=-2H(\bar{q}i\gamma^{5}\vec{\tau}q),$
$\displaystyle\Delta^{*}(x)=-2H\Big{[}\bar{q}i\gamma^{5}\sigma_{2}\tau_{2}q^{c}\Big{]}=-2H\Big{[}\bar{q}i\gamma^{5}\sigma_{2}\tau_{2}C\bar{q}^{T}\Big{]}.$
(5)

One can easily see from (5) that $\sigma(x)$ and $\pi_{a}(x)$ ($a=1,2,3$) are
Hermitian, i.e. real, bosonic fields, and the fields $\Delta^{*}(x)$ and
$\Delta(x)$ are Hermitian conjugated to each other. Indeed,
$(\sigma(x))^{\dagger}=\sigma(x)$, $(\pi_{a}(x))^{\dagger}=\pi_{a}(x)$,
$(\Delta(x))^{\dagger}=\Delta^{*}(x)$ and
$(\Delta^{*}(x))^{\dagger}=\Delta(x)$, where $\dagger$ denotes the Hermitian
conjugation. Bosonic fields $\pi_{3}(x)$, $\pi^{\pm}(x)=(\pi_{1}(x)\mp
i\pi_{2}(x))/\sqrt{2}$ could be identified with the pion fields. If
$\langle\sigma(x)\rangle\neq 0$ or $\langle\pi_{0}(x)\rangle\neq 0$, then one
can see that chiral symmetry is broken down and we will call this phase chiral
symmetry breaking (CSB) one. If $\langle\pi_{1,2}(x)\rangle\neq 0$, then both
the isospin $U(1)_{I_{3}}$ and the electromagnetic $U(1)_{Q}$ symmetries are
broken down, this phase is called charged pion condensation phase (PC). If
$\langle\Delta(x)\rangle\neq 0$, then the baryon symmetry gets broken down
spontaneously and this phase will be called baryon superfluid phase (BSF). One
can see that the ground state expectation values of all bosonic fields (5) are
$SU(2)_{c}$ invariant, hence in this model the color symmetry can not be
broken down dynamically.

Upon introduction of Nambu-Gorkov field $\Psi$

$\Psi=\left({q\atop
q^{c}}\right),~{}~{}\Psi^{T}=(q^{T},\bar{q}C^{-1});~{}~{}\quad\overline{\Psi}=(\bar{q},\overline{q^{c}})=(\bar{q},q^{T}C)=\Psi^{T}\left(\begin{array}[]{cc}0~{}~{},&C\\\
C~{}~{},&0\end{array}\right)\equiv\Psi^{T}Y,$ (6)

the semi-bosonised Lagrangian would take a form

$\displaystyle\widetilde{L}=-\frac{\sigma^{2}+\vec{\pi}^{2}+\Delta^{*}\Delta}{4H}+\frac{1}{2}\Psi^{T}(YZ)\Psi,$
(7)

where matrix $Y$ is given in Eq. (6) and

$Z=\left(\begin{array}[]{cc}D^{+},&K\\\
K^{*}~{}~{},&D^{-}\end{array}\right)\equiv\left(\begin{array}[]{cc}i\hat{\partial}-m_{0}+{\cal
M}\gamma^{0}-\sigma-i\gamma^{5}\vec{\tau}\vec{\pi},&-i\gamma^{5}\sigma_{2}\tau_{2}\Delta\\\
~{}~{}~{}~{}~{}~{}~{}~{}-i\gamma^{5}\sigma_{2}\tau_{2}\Delta^{*}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},&i\hat{\partial}-m_{0}-\gamma^{0}{\cal
M}-\sigma-i\gamma^{5}(\vec{\tau})^{T}\vec{\pi}\end{array}\right).$ (8)

In the mean-field (or one fermion loop) approximation, the effective action
${\cal S}_{\rm{eff}}(\sigma,\vec{\pi},\Delta,\Delta^{*})$ can be expressed in
terms of the path integral over quark fields

$\displaystyle\exp(i{\cal
S}_{\rm{eff}}(\sigma,\vec{\pi},\Delta,\Delta^{*}))=N^{\prime}\int[d\bar{q}][dq]\exp\Bigl{(}i\int\widetilde{L}\,d^{4}x\Bigr{)},$
(9)

and if one express it in a more useful form

$\displaystyle{\cal S}_{\rm{eff}}(\sigma,\vec{\pi},\Delta,\Delta^{*})=-\int
d^{4}x\left[\frac{\sigma^{2}(x)+\vec{\pi}^{2}(x)+|\Delta(x)|^{2}}{4H}\right]+\widetilde{\cal
S}_{\rm{eff}},$ (10)

where the quark contribution is

$\exp(i\tilde{\cal
S}_{\rm{eff}})=N^{\prime}\int[d\bar{q}][dq]\exp\Bigl{(}\frac{i}{2}\int\Big{[}\Psi^{T}(YZ)\Psi\Big{]}d^{4}x\Bigr{)}.$
(11)

One can express the path integration to $\Psi$ field

$\exp(i\tilde{\cal
S}_{\rm{eff}})=\int[d\Psi]\exp\left\\{\frac{i}{2}\int\Psi^{T}(YZ)\Psi
d^{4}x\right\\}=\mbox{det}^{1/2}(YZ)=\mbox{det}^{1/2}(Z),$ (12)

One can define from ${\cal S}_{\rm{eff}}$ the thermodynamic potential (TDP)
$\Omega(\sigma,\vec{\pi},\Delta,\Delta^{*})$ in the mean-field approximation

${\cal
S}_{\rm{eff}}~{}\bigg{|}_{~{}\sigma,\vec{\pi},\Delta,\Delta^{*}=\rm{const}}=-\Omega(\sigma,\vec{\pi},\Delta,\Delta^{*})\int
d^{4}x.$

The ground state expectation values of
$\langle\sigma(x)\rangle\equiv\sigma,~{}\langle\vec{\pi}(x)\rangle\equiv\vec{\pi},~{}\langle\Delta(x)\rangle\equiv\Delta,~{}\langle\Delta^{*}(x)\rangle\equiv\Delta$,
can be found as solutions of the following equations (the so-called gap
equations)

$\displaystyle\frac{\partial\Omega}{\partial\pi_{a}}=0,~{}~{}~{}~{}~{}\frac{\partial\Omega}{\partial\sigma}=0,~{}~{}~{}~{}~{}\frac{\partial\Omega}{\partial\Delta}=0,~{}~{}~{}~{}~{}\frac{\partial\Omega}{\partial\Delta^{*}}=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}20$
(13)

It is assumed that $\sigma,\vec{\pi},\Delta,\Delta^{*}$ do not depend on the
space coordinates $x$.

In the mean field approximation the expression for the TDP can be obtained and
it has the following form

$\displaystyle\Omega(M,\vec{\pi},\Delta,\Delta^{*})$ $\displaystyle=$
$\displaystyle\frac{(M-m_{0})^{2}+\vec{\pi}^{2}+|\Delta|^{2}}{4H}+i\int\frac{d^{4}p}{(2\pi)^{4}}\ln\det
L(p).$ (14)

The expressions for $\det L(p)$ can be obtained analytically. The integration
over $\vec{p}$ is performed up to the $|\vec{p}|<\Lambda$, i. e. sharp
momentum cutoff with $\Lambda=657$ MeV. The values of $H$ is taken to be
$H=7.23$ GeV-2 [12, 13]. The chiral limit, $m_{0}=0$, will be used to simplify
numerical calculations.

##  3 Duality properties

###  3.1 The case with $\mu\neq 0$, $\nu\neq 0$ and $\nu_{5}\neq 0$

Above we considered the TDP as a function of all six condensates, i. e. $M$,
$\vec{\pi}$, $\Delta$ and $\Delta^{*}$, but if one consider the symmetries of
the model the number of condensates could be reduced. As it has been discussed
above, in the chiral limit and without chemical potentials, i. e. if all of
them are equal to zero, the Lagrangian is invariant with respect to
$SU(4)\times U(1)_{B}\times SU(2)_{c}$ group. If chemical potentials are non-
zero then the Lagrangian is invariant with respect to reduced symmetry group
$U(1)_{B}\times U(1)_{I_{3}}\times U(1)_{AI_{3}}\times SU(2)_{c}$. So in this
case the TDP depends only on the following combinations $|\Delta|^{2}$,
$\pi_{1}^{2}+\pi_{2}^{2}$ and $M^{2}+\pi_{0}^{2}$, and without loss of
generality one can put $\pi_{2}=0$. Hence, below the TDP is assumed to depend
on $M,\pi_{1}$ and Moreover, it can be shown that if $m_{0}\neq 0$ the global
minimum point (GMP) of the TDP always have $\pi_{0}=0$. Hence, below we assume
that the TDP depends only on $M,\pi_{1}$ and $|\Delta|$.

It can be shown with the use of any analytical calculation program that the
expression for the TDP is invariant under the so-called dual transformation
${\cal D}_{1}$,

$\displaystyle{\cal
D}_{1}:~{}~{}~{}~{}\mu\longleftrightarrow\nu,~{}~{}~{}\pi_{1}\longleftrightarrow|\Delta|.$
(15)

This dual property of the TDP was noticed for the first time in the framework
of the two-color NJL model in [8, 13] but for the case of $\nu_{5}=0$. In
particular, it was shown that the PC and BSF phases are arranged symmetrically
on the $(\mu,\nu)$-phase diagram of the model.

Furthermore, it has been obtained that the TDP is invariant also with respect
to the following dual transformations ${\cal D}_{2}$ and ${\cal D}_{3}$

$\displaystyle{\cal
D}_{2}:~{}~{}\mu\longleftrightarrow\nu_{5},~{}~{}M\longleftrightarrow|\Delta|;~{}~{}~{}~{}~{}~{}{\cal
D}_{3}:~{}~{}\nu\longleftrightarrow\nu_{5},~{}~{}M\longleftrightarrow\pi_{1}.$
(16)

##  4 Phase diagram

Now let us discuss the arguments why there is no mixed phase in the phase
diagram in our case and hence one can assume that the global minimum point
(GMP) $(M,\pi_{1},|\Delta|)$ of the TDP has only one nonzero coordinate. There
is a rather cogent argument for absence of mixed phases and it is based on
dual properties discussed above. The phase diagram concerning chiral symmetry
breaking and charged pion condensation phenomena has the same structure in two
color and three color cases in framework of NJL models. These phenomena in the
three color case has been investigated in [17, 18, 19] and it has been
displayed that (I) chiral and charged pion condensates do not take non-zero
values simultaneously (meaning no mixed phase). Hence, this holds also for two
color NJL model. Then, in order to show that (II) there is no such a region,
where chiral and diquark condensates are non-zero simultaneously, the duality
${\cal D}_{2}$, which turns pion condensate to the diquark one, should be
used. The absence of mixed phase with non-zero pion and diquark condensates
can be shown applying either ${\cal D}_{2}$ to the former case (I) or the
${\cal D}_{3}$ to the latter one (II). This does not show that there could not
be a phase with all three non-zero condensates (chiral, pion and diquark), but
it seems to be quite unlikely possibility. Without dualities showing the
absence of mixed phases would be rather hard and can be done only numerically.

Let us talk about possible phases in the system that can be realized. (i) the
chiral symmetry breaking (CSB) phase: GMP has the form $(M\neq
0,\pi_{1}=0,|\Delta|=0)$. (ii) charged pion condensation (PC) phase: GMP of
the form $(M=0,\pi_{1}\neq 0,|\Delta|=0)$. (iii) baryon superfluid (BSF)
phase: GMP is $(M=0,\pi_{1}=0,|\Delta|\neq 0)$. (iv) symmetrical phase: GMP
has the form $(M=0,\pi_{1}=0,|\Delta|=0)$.

###  4.1 The case of one chemical potential

The phase diagram of the two color NJL model with only nonzero baryon chemical
potential (although with nonzero temperature $T$) has been investigated in
[9]. It was shown that at non-zero (although not for too large) $\mu$ and zero
temperature $T=0$ there is the BSF phase. If one apply to this phase diagram
the duality transformation ${\cal D}_{1}$ one would get the phase diagram with
only nonzero isospin chemical potential $\nu$. And one can see that at $\nu>0$
there is the charged PC phase. Now acting by the other duality transformation
${\cal D}_{2}$ upon the phase diagram with nonzero $\mu$ (original one) one
would get the phase structure at only non-zero $\nu_{5}$ and when $\nu_{5}>0$
observe CSB phase is realized in the system. One can conclude that there exist
one-to-one correspondence between different phenomena, chiral symmetry
breaking, charged pion condensation and diquark condensation, and various
chemical potentials, chiral isospin chemical potential $\nu_{5}$, isospin
chemical potential $\nu$ and quark chemical potential $\mu$.

###  4.2 The case of two chemical potentials

The $(\mu,\nu)$-phase diagram at $\nu_{5}=0$ has been already considered in
[8, 13]. The obtained phase diagram is depicted in Fig. 1(a) and one can note
that it is self-dual, i. e. BSF and charged PC phases are mirror symmetrical
with respect to the $\mu=\nu$ line. This is due to the duality ${\cal D}_{1}$
of TDP

If one employs the dual transformation ${\cal D}_{2}$ and act on this diagram,
one obtain the $(\nu_{5},\nu)$-phase diagram of the model at $\mu=0$ (it is
presented in Fig. 1(b)). In a similar fashion, acting on the diagram of Fig.
1(a) with the dual transformation ${\cal D}_{3}$, one can get the
$(\mu,\nu_{5})$-phase diagram of the model at $\nu=0$ (see Fig. 1(c)). One can
note that CSB and BSF phases are located mirror symmetrically with respect to
the line $\nu_{5}=\mu$. Here one can also clearly see the discussed above one-
to-one correspondence between different phenomena and corresponding chemical
potentials.

Figure 1:  (a) $(\mu,\nu)$-phase diagram at zero $\nu_{5}$. (b)
$(\nu_{5},\nu)$-phase diagram at zero $\mu$. (c) $(\mu,\nu_{5})$-phase diagram
at zero $\nu$.  Figure 2:  (a) $(\mu,\nu)$-phase diagram at $\nu_{5}=0.1$ GeV.
(b) $(\nu_{5},\nu)$-phase diagram at $\mu=0.1$ GeV. (c) $(\mu,\nu_{5})$-phase
diagram at $\nu=0.1$ GeV. All the notations are the same as in previous Fig.
1.

###  4.3 Phase structure of the general case: $\mu\neq 0,\nu\neq 0$ and
$\nu_{5}\neq 0$

In this section let us consider the general case when all three chemical
potentials $\mu$, $\nu$, $\nu_{5}$ are non-zero, i. e.
$(\mu,\nu,\nu_{5})$-phase diagram of the model. One can use the assumption
that there is no mixed phase, which was proved (excluding three nonzero
condensates case) above, and perform numerical calculations only for
projections of the TDP on the axis of condensates ($M$, $\pi$) (one need only
to find minima of the projections and compare them).

There is another way, which is even much simpler, one can use the dual
properties of the model. Let us first note that chiral symmetry breaking and
pion condensation phenomena in the framework of NJL model are very similar in
the cases of three and two colors, what is different is diquark condensation.
The $(\nu,\nu_{5})$ phase diagram at different $\mu$ as well as $(\mu,\nu)$
phase diagram at different $\nu_{5}$, where only chiral symmetry breaking and
pion condensation phenomena were accounted for, have been investigated in the
case of three colors in [18]. Therefore, one knows the behavior of PC and CSB
phases in $(\nu,\nu_{5})$ and $(\mu,\nu)$ phase diagrams in the case of two
colors. Then, the duality transformations ${\cal D}_{1}$, ${\cal D}_{2}$ and
${\cal D}_{3}$ can be used to get the information about BSF phase and hence
obtain the whole phase diagram. It is quite remarkable that the whole phase
structure of two color NJL model can be obtained from some results of three
color case. Duality constrain the phase portrait and make it so highly
symmetric that this gets possible.

Now we can discuss the phase structure itself. Let us start with several
$(\mu,\nu)$ phase diagrams at various values of $\nu_{5}$ that are not that
large (see Figs. 1(a), 2(a) and 3(a,b)). One can note that the diquark
condensation (BSF phase) is realized in the domain, where $\mu>\nu,\nu_{5}$,
whereas if $\nu>\mu,\nu_{5}$ the charged PC phase is present. This behavior
qualitatively can be explained in a similar way as in [13], where the case
$\nu_{5}=0$ was considered. $\nu_{5}$ in our case (at phase diagrams
discussed) is not very large and, for simplicity, one can put it
$\nu_{5}\approx 0$. If $\nu>\mu$, $u$ and $\bar{d}$ quarks form Fermi seas and
the condensation of Cooper pairs $u\bar{d}$ is possible, and one can see that
it is charged PC phase. Whereas, if $\mu>\nu$, then $u$ and $d$ quarks form
Fermi seas and in this case the formation of Cooper pairs $ud$ and their
condensation can happen and this leads to BSF phase.

Now if the values of $\nu_{5}$ is rather large (Fig 3(c)) the BSF phase is
realized mainly in the domain, where $\mu<\nu\approx\nu_{5}$, and the charged
PC phase in the domain, where $\nu<\mu\approx\nu_{5}$). This is a different
behavior in comparison with the one discussed above for rather small values of
$\nu_{5}$ Figs. 1(a), 2(a) and 3(a,b). This difference can be qualitatively
explained by the following arguments. The Fermi surfaces of the left and
right-handed $u$ and $d$ quarks has the form

$\displaystyle\mu_{uR}=\mu+\nu+\nu_{5},~{}~{}\mu_{uL}=\mu+\nu-\nu_{5},~{}~{}\mu_{dR}=\mu-\nu-\nu_{5},~{}~{}\mu_{dL}=\mu-\nu+\nu_{5}.$
(17)

At $\mu<\nu\approx\nu_{5}$ for the quarks $d_{R}$ the chemical potential
$\mu_{dR}$ is negative, hence the Fermi sea of charge conjugated $d_{R}^{c}$
quarks can be formed, so as the Fermi sea of right-handed $u_{R}$ quarks,
which $\mu_{uR}$ is also greater than zero. The formation of particle-hole
Cooper pair $\bar{u}_{R}d_{R}^{c}$, which has quantum numbers of
$\Delta^{*}(x)$, is possible and its condensation leads to appearance of BSF
phase.

But to get the idea of the phase structure at other rather large values of
$\nu_{5}$, one can imagine that the charged PC phase (the BSF phase) of this
diagram (see Fig. 3(c)) has the form of a boot sole pointing by its tip to the
value $\mu=\nu_{5}=0.4$ GeV of the $\mu$-axis (to the value $\nu=\nu_{5}=0.4$
GeV of the $\nu$-axis). (And in the figure, these “boot soles” intersect with
each other, as well as with the strip of the CSB phase along the line
$\mu=\nu$.) Then, to find (approximately) the $(\mu,\nu)$-phase diagram at
another rather high value of $\nu_{5}$, one should, starting from the diagram
of Fig. 3(c), simply shift the “boot sole” of the charged PC phase parallel to
the $\mu$-axis in a position, in which its tip is directed to the point of
this axis, where $\mu=\nu_{5}$. In a similar way the BSF phase should be
shifted along the $\nu$-axis.

Note, that already in Figs. 3(a,b) one can see the beginning of the process of
forming the charged PC and BSF phases in the form of the “boot soles”. At
rather large value of $\nu_{5}$, say at $\nu_{5}=0.24$ GeV, a small part of
the BSF phase, a tip of the “boot sole”, penetrates into the region where
$\mu<\nu$ (see in Fig. 3(b)). And at even higher values of $\nu_{5}$ in all
the $(\mu,\nu)$-phase diagrams obtained in this way, the BSF phase (the
charged PC phase) will be located mainly in the region $\mu<\nu\approx\nu_{5}$
(in the region $\nu<\mu\approx\nu_{5}$).

Figure 3:  (a) $(\mu,\nu)$-phase diagram at $\nu_{5}=0.2$ GeV. (b)
$(\mu,\nu)$-phase diagram at $\nu_{5}=0.24$ GeV. (c) $(\mu,\nu)$-phase diagram
at $\nu_{5}=0.4$ GeV. All the notations are the same as in Fig. 1.

##  5 Charged pion condensation in dense quark matter

Let us briefly discuss the story with charged pion condensation phenomenon in
dense quark matter without chiral imbalance, i.e. at $\mu_{I5}=0$ and
$\mu_{5}=0$. First, there has been predicted the possibility of the existence
of the charged PC in dense quark matter with isospin imbalance but without the
requirements of its $\beta$ equilibrium and electrical neutrality. This
investigation ($\mu_{B}\neq 0$, $\mu_{I}\neq 0$, but at $\mu_{Q}=0$, when the
contribution of electrons are not taken into account) in the chiral limit was
performed in the framework of the NJL model in [20]. It has been shown that at
nonzero baryon density (and at comparatively small isospin chemical potential
$\mu_{I}$) there is the charged PC phase in the system. Then in [21] it was
shown that this prediction stays valid also under the conditions of neutron
star matter, namely if the contribution of electrons as well as electric
neutrality and $\beta$-equilibrium requirements are taken into account. The
investigation was performed in terms of the NJL model and, for simplicity, the
consideration, as very often is done in similar situations, has been performed
in the chiral limit.

In the case of the $\beta$-equilibrated matter there is no isospin chemical
potential $\mu_{I}$ in the Lagrangian but there is electric charge chemical
potential $\mu_{Q}$. It is possible to draw the $(\mu,\mu_{Q})$-phase diagram
of NJL model. After that, One can then draw a curve of zero electric charge
density $\langle n_{Q}\rangle$ on the phase diagram.

After that, in [22] it was shown that if one accurately include into the NJL
model considerations the electric neutrality and $\beta$-equilibrium
conditions, combined with the taking physical value of the current quark mass
$m_{0}$ (though it is rather small but non-zero) it was shown that the charged
PC phase is completely washed away from the phase diagram and there is no any
indication of charged pion condensation in dense isospin asymmetric matter at
least in the NJL model approach (see Fig. 4(a)). Similar conclusions were made
also in terms of hadronic models.

Then several years after these considerations there have been found several
external physically realistic parameters that create charged PC phase in dense
quark/baryonic matter or promote its the generation (see the recent review
[23]). In [24] it was shown, in particular, that if one allows the existence
of spatially inhomogeneous condensates in the system then this phase is
possible. Also it was revealed in [25] that if one take into account the
finite size of the system this phase can also be realized. Additionally, it
was shown rather recently that chiral imbalance in the system can lead to the
appearance of charged PC phase in dense quark (baryonic) matter as well [18,
17].

###  5.1 Charged pion condensation in electric neutral matter

It was interesting to check if the generation of the charged PC phase can also
be realized in the conditions of neutron star matter (additionally assuming
chiral imbalance), namely in electrically neutral and $\beta$-equilibrated
dense quark matter (at $\mu_{I5}=2\nu_{5}\neq 0$ and $\mu_{5}\neq 0$ in
addition), or the electric charge neutrality condition (together with
$\beta$-equilibrium condition) would completely or partially destroy this
effect.

It has been shown that charged pion condensation phenomenon is triggered by
chiral imbalance in dense (i.e. at $\mu_{B}\neq 0$ when baryon density in
nonzero) with conditions of electric neutrality (with account of electrons)
and $\beta$-equilibrium both in the chiral limit (at $m_{0}=0$) and in the
physical point ($m_{0}\neq 0$). The discussed generation of charged PC can be
induced even by only chiral isospin chemical potential $\mu_{I5}$ (at zero
chiral chemical potential, $\mu_{5}=0$). But if both forms of chiral imbalance
are non-zero in the system ($\mu_{5}$ and $\nu_{5}$ chemical potentials), then
the generation of charged PC in dense baryon matter is a quite inevitable
phenomenon. In Fig. 4(b) the phase diagram of electrically neutral and
$\beta$-equilibrated dense quark matter is depicted and one can see that the
neutrality line $n_{Q}=0$ intersects the huge chunk of the PCd phase.

Figure 4: (a) $(\mu,\nu\equiv\mu_{Q}/2)$-phase diagram of quark matter in the
$\beta$-equilibrium with electrons at $\nu_{5}=0$ and $\mu_{5}=0$. (b)
$(\nu_{5},\nu)$-phase diagram at $\mu=400$ MeV and $\mu_{5}=300$ MeV. PC
denotes the phase with charged pion condensation (in PCd, d means non-zero
baryon density). CSB denotes the phase where there is no pion condensation and
dynamical quark mass is rather large. SYM and ApprSYM are phases where quark
mass is zero or around the current quark mass value, respectively. The dashed
line is ”neutrality line” where the density of electric charge $\langle
n_{Q}\rangle$ is equal to zero.

###  5.2 Charged pion condensation in dense quark matter and diquark
condensation

It was shown chiral imbalance generate charged PC in dense baryon matter. But
it is well known that in three color case in matter with rather large baryon
density the color superconductivity phenomenon (diquark condensation) can take
place. It starts from some value of baryon density but there is probably
rather large region where both phenomena, namely generation of charged PC and
diquark condensation, can exist. And there is natural question if diquark
condensation phenomenon can destroy the generation of charged PC in dense
baryonic matter (at least in a wide range of baryon densities).

One can note that diquark condensation appear in the two color case if there
is no other chemical potentials at even infinitesimally small values of baryon
density. And probably this phenomenon is even more widespread in two color
case than it is in three color case.

In the two color case we have discussed in the phase diagram of two color
dense quark matter in Sec. 4 and one can see e. g. in Fig. 3(c) that diquark
condensation (BSF phase) almost does not intersects with the PCd phase at all.
And this means that diquark condensation does not prohibit the generation of
the charged PC phase by chiral imbalance at least in the two color case.

##  6 Summary

Let us now sum up the central results of our studies of dense quark matter
with isospin and chiral isospin imbalances in the framework of the two-color
NJL model.

  * •

It has been found that the phase structure of (two color) dense quark matter
with isospin and chiral isospin imbalances is quite rich.

It has been shown in the framework of two-color NJL model that there phase
diagram of dense quark matter with isospin and chiral isospin imbalances has a
very rich structure.

  * •

The phenomena of chiral symmetry breaking, charged pion condensation and
diquark condensation have been shown to be in a way connected with the
corresponding chemical potentials, $\nu_{5}$,$\mu_{I}$ and $\mu_{B}$,
respectively.

  * •

It has been shown that $(\mu,\nu,\nu_{5})$ phase structure of quark matter in
the two color case is highly symmetric. Behind this symmetry lies the dual
properties (dualities) between CSB, charged PC and BSF phenomena. One of these
dualities is similar to the one found in the case of three colors.

  * •

The dualities are very powerful tools and can greatly simplify the studies of
the phase structure. Employing only duality properties the whole phase diagram
two color NJL model can be obtained from the results of previous studies of
NJL model in the three color case.

  * •

The previously discussed generation of charged PC by chiral imbalance was
shown to be a quite inevitable phenomenon and to take place also in the
conditions pertinent to the neutron stars, i. e. in electrically neutral and
$\beta$-equilibrated dense quark matter.

  * •

It has been revealed that in dense (nonzero baryon density) quark matter the
generation by chiral imbalance $\nu_{5}$ of the charged PC phase is not
impeded in any way by diquark condensation at least in two color case.

## Acknowledgment

R.N.Z. is grateful for support of Russian Science Foundation under the grant
№19-72-00077. The work is also supported by the Foundation for the Advancement
of Theoretical Physics and Mathematics BASIS grant.

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# Average Evolution and Size-Topology Relations for Coarsening 2d Dry Foams

Anthony T. Chieco1, James P. Sethna2 and Douglas J. Durian1 1Department of
Physics and Astronomy, University of Pennsylvania, Philadelphia PA, USA
2Department of Physics, Cornell University, Ithaca NY, USA

###### Abstract

Two-dimensional dry foams coarsen according to the von Neumann law as
$dA/dt\propto(n-6)$ where $n$ is the number of sides of a bubble with area
$A$. Such foams reach a self-similar scaling state where area and side-number
distributions are stationary. Combining self-similarity with the von Neumann
law, we derive time derivatives of moments of the bubble area distribution and
a relation connecting area moments with averages of the side-number
distribution that are weighted by powers of bubble area. To test these
predictions, we collect and analyze high precision image data for a large
number of bubbles squashed between parallel acrylic plates and allowed to
coarsen into the self-similar scaling state. We find good agreement for
moments ranging from two to twenty.

##  I Introduction

Dry two-dimensional foams can be made by squashing bubbles between parallel
plates and letting most of the liquid drain out. Viewed from normal to the
plates, neighboring bubbles are separated by thin soap films that are circular
arcs and that – according to Plateau’s Laws – meet at $120^{\circ}$ at 3-fold
vertices Weaire and Rivier (1984); Weaire and Hutzler (1999); Cantat  _et al._
(2013); Langevin (2020). As a consequence, Euler’s rule implies that the
average number of sides is $\langle n\rangle=6$ if the sample is sufficiently
large. It is commonly observed that bubbles with more sides $n$ tend to be
bigger. Various size-topology relations have been proposed to quantify this
behavior Rivier (1985); Chiu (1995). One of the oldest is Lewis’s Law Lewis
(1930), which states that the area of $n$ sided-bubbles is a linear function
of $n$. Desch’s Law analogously holds that the perimeter of $n$-sided bubbles
is linear in $n$. These “laws” are actually empirical approximations, found to
hold to varying degrees and with different linear relations for different
kinds of cellular structures, _e.g._ plant and animal tissues or foams or
grains in an alloy. For foams, systematic deviations from the Lewis and Desch
laws were recently observed and accounted for by a simplified “granocentric”
model Newhall  _et al._ (2012) in which a central particle is uniformly
surrounded by $n$ equidistant bubbles of the same size Roth  _et al._ (2013).

Even in the absence of soap film rupture and drainage of the liquid between
bubbles, foams coarsen with time. Locally, gas diffuses between neighboring
bubbles according to their pressure difference. This tends to make small
bubbles shrink and large bubbles grow. Based on little more than geometry, von
Neumann’s law von Neumann (1952); Mullins (1956); Stavans (1993a) states that
the area $A_{i}$ of a bubble $i$ with $n_{i}$ sides changes at rate

$\frac{dA_{i}}{dt}=K_{o}(n_{i}-6)$ (1)

where $K_{o}$ is a materials constant proportional to surface tension, the
solubility and diffusivity of the gas, and inversely proportional to film
thickness. This law is exact, no matter what the sizes and shapes of the
neighboring bubbles. If an initial foam sample is not too pathological or
small, it will evolve into a statistically self-similar scaling state where
the side-number distribution and dimensionless ratios of bubble area moments
are all independent of time Mullins (1986) (see Refs. Glazier  _et al._
(1987); Marder (1987); Beenakker (1988); Weaire and Lei (1990); Glazier  _et
al._ (1990); Herdtle and Aref (1992) on the approach to the scaling state, and
Refs. Stavans (1990); Roth  _et al._ (2013) on the demonstration of scaling,
for dry 2d foams). In such a state, the von Neumann law can be used to show
that the average coarsening rate is constant and equal to

$\frac{d\langle A\rangle}{dt}=2K_{o}\frac{\langle A\rangle^{2}}{\langle
A^{2}\rangle}\big{[}\langle\langle n\rangle\rangle-6\big{]}$ (2)

where $\langle A\rangle=(\sum_{i=1}^{N}A_{i})/N=A_{tot}/N$ is the average
bubble size, $N$ is the total number of bubbles, $A_{tot}$ is the sample area,
$\langle A^{2}\rangle=(\sum_{i=1}^{N}{A_{i}}^{2})/N$, $\langle\langle
n\rangle\rangle=\sum nF(n)$ is the area-weighted average side number, and
$F(n)$ is the area-weighted side number distribution – _i.e._ the fraction of
sample area inside $n$-sided bubbles Roth  _et al._ (2013). While the
probability $P(n)$ that a randomly-chosen _bubble_ is $n$-sided has been
widely studied, the probability $F(n)$ that a randomly chosen _point_ is
inside an $n$-sided bubble is more directly important for the average
coarsening behavior. According to Eq. (2), the average coarsening rate depends
on both bubble sizes, through the moment ratio $\langle A^{2}\rangle/\langle
A\rangle^{2}$, as well as on topology, through $\langle\langle
n\rangle\rangle$.

In this paper, we generalize three ways upon Eq. (2), and we use one of the
results to predict a generalized size-topology relation. Then we make
experimental tests.

##  II Predictions

To assist with derivations, we use single angle brackets to denote numeric
averages and double angle brackets to denote weighted averages. Specifically,
we write the average $p$-th power of bubble area as

$\langle A^{p}\rangle=\frac{1}{N}\sum_{i=1}^{N}{A_{i}}^{p}$ (3)

And we write the $A^{p}$-weighted average side numbers as

$\langle\langle
n\rangle\rangle_{p}=\frac{\sum_{i=1}^{N}n_{i}{A_{i}}^{p}}{\sum_{i=1}^{N}{A_{i}}^{p}}$
(4)

Note that $\langle\langle n\rangle\rangle_{1}$ is the area-weighted average
side number $\langle\langle n\rangle\rangle$ used in Eq. (2), and
$\langle\langle n\rangle\rangle_{0}=\langle n\rangle=6$ is the familiar
numeric average size number.

For our first generalization of Eq. (2), we begin by writing and rearranging
the following identity:

$\displaystyle\langle A\rangle^{p}$ $\displaystyle=$ $\displaystyle\langle
A\rangle^{p}\times\frac{\langle A\rangle}{\langle
A\rangle}\times\frac{\sum{A_{i}}^{p}}{\sum{A_{i}}^{p}}\times\frac{\sum{A_{i}}^{p+1}}{N\langle
A^{p+1}\rangle}$ (5) $\displaystyle=$ $\displaystyle\frac{1}{N\langle
A\rangle}\times\frac{\langle A\rangle^{p+1}}{\langle
A^{p+1}\rangle}\times\frac{\sum{A_{i}}^{p}}{\sum{A_{i}}^{p}}\times\sum{A_{i}}^{p+1}$
(6)

where the sums run over all bubbles, from $i=1$ to $i=N$. Note that the first
three terms in Eq. (6) are all independent of time: The first is the
reciprocal of sample area, the second is constant when the sample is in a
self-similar scaling state, and the third equals one. Therefore, it is
straightforward to differentiate both sides with respect to time. Using von
Neumann’s law on the fourth term on the right hand side, then recognizing
$\sum{A_{i}}^{p}/N=\langle A^{p}\rangle$ and rearranging, gives

$p\langle A\rangle^{p-1}\frac{d\langle A\rangle}{dt}=\frac{\langle
A\rangle^{p}\langle A^{p}\rangle}{\langle
A^{p+1}\rangle}\frac{\sum(p+1){A_{i}}^{p}K_{o}(n_{i}-6)}{\sum{A_{i}}^{p}}$ (7)

which simplifies to the following generalization of Eq. (2):

$\frac{d\langle A\rangle}{dt}=K_{o}\left(1+\frac{1}{p}\right)\frac{\langle
A\rangle\langle A^{p}\rangle}{\langle A^{p+1}\rangle}\big{[}\langle\langle
n\rangle\rangle_{p}-6\big{]}$ (8)

This holds for any nonzero value of $p$, not just integers. Near $p=0$ it
implies $d\langle A\rangle/dt=K_{o}\lim_{p\to 0}[\langle\langle
n\rangle\rangle_{p}-6]/p$, which proves $\langle n\rangle=6$ for steady state
without use of Euler’s rule. Since the left hand side of Eq. (8) is the same
for any value of $p$, we may equate the right hand sides evaluated at $p$ and
at $p\rightarrow q$. This gives the final result

$\frac{p(q+1)\langle A^{p+1}\rangle\langle A^{q}\rangle}{q(p+1)\langle
A^{p}\rangle\langle A^{q+1}\rangle}=\frac{\langle\langle
n\rangle\rangle_{p}-6}{\langle\langle n\rangle\rangle_{q}-6}$ (9)

relating bubbles sizes, on the left, and network topology, on the right, which
holds in the self-similar scaling state. It will be tested experimentally in
later sections for the special case $q=p-1$, where it can be rewritten as

$\frac{\langle A^{p}\rangle^{2}}{\langle A^{p-1}\rangle\langle
A^{p+1}\rangle}\frac{\langle\langle n\rangle\rangle_{p}-6}{\langle\langle
n\rangle\rangle_{p-1}-6}=\frac{p^{2}}{p^{2}-1}$ (10)

Here, the product of size and topology ratios decrease towards one as $p$
increases towards infinity. As per Eq. (8), it holds for general values of
$p$. Note that both sides vanish in the limit $p\rightarrow 0$ and both sides
diverge in the limit $p\rightarrow 1$ owing to $\langle\langle
n\rangle\rangle_{0}\equiv\langle n\rangle=6$. We note, too, that Eq. (9) can
be derived from Eq. (10).

For a second generalization of Eq. (2), we consider the time derivative of
$\langle A^{p}\rangle=\frac{1}{N}\sum{A_{i}}^{p}$. Substituting $1/N=\langle
A\rangle/A_{tot}$ and using the product differentiation rule, then von
Neumann’s law, gives

$\displaystyle\frac{d\langle A^{p}\rangle}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{A_{tot}}\frac{d\langle
A\rangle}{dt}\sum{A_{i}}^{p}+\frac{\langle
A\rangle}{A_{tot}}\frac{d}{dt}\sum{A_{i}}^{p}$ (11) $\displaystyle=$
$\displaystyle\frac{\langle A^{p}\rangle}{\langle A\rangle}\frac{d\langle
A\rangle}{dt}+\frac{1}{N}\sum p{A_{i}}^{p-1}K_{o}(n_{i}-6)$ (12)
$\displaystyle=$ $\displaystyle\frac{\langle A^{p}\rangle}{\langle
A\rangle}\frac{d\langle A\rangle}{dt}+pK_{o}\langle
A^{p-1}\rangle\big{[}\langle\langle n\rangle\rangle_{p-1}-6\big{]}$ (13)

For $p=1$ these expressions reduce to $d\langle A\rangle/dt=d\langle
A\rangle/dt$ and hence give nothing new. For $p>1$ we may simplify further by
assuming that the foam is in a self-similar scaling state, which permits
$d\langle A\rangle/dt$ to be evaluated with Eq. (8) at $p\rightarrow p-1$.
This gives the final result:

$\frac{d\langle A^{p}\rangle}{dt}=K_{o}\frac{p^{2}}{p-1}\langle
A^{p-1}\rangle\big{[}\langle\langle n\rangle\rangle_{p-1}-6\big{]}$ (14)

which grows in proportion to $t^{p-1}$ and holds for $p>1$.

For the third and perhaps prettiest generalization of Eq. (2), we compute the
time derivative of $\langle A^{p+1}\rangle/\langle
A^{p}\rangle=\sum{A_{i}}^{p+1}/\sum{A_{i}}^{p}$. This quantity has units of
area, so in the scaling state its time derivative should be constant. Indeed,
using the quotient rule with von Neumann’s law, then simplifying with Eq.
(10), gives

$\frac{d}{dt}\frac{\langle A^{p+1}\rangle}{\langle
A^{p}\rangle}=K_{o}\left(1+\frac{1}{p}\right)\big{[}\langle\langle
n\rangle\rangle_{p}-6\big{]}$ (15)

This result can be alternatively derived using Eq. (14). To test if the foam
is in a scaling state, experimental data for average bubble growth should
collapse to a line of slope $K_{o}$ if plotted as $\langle
A^{p+1}\rangle/\langle A^{p}\rangle$ versus $(1+1/p)[\langle\langle
n\rangle\rangle_{p}-6]t$ for several values of $p$.

It might be interesting to similarly compute the time derivative of quantities
such as $\langle A^{p+q+1}\rangle/[\langle A^{p}\rangle\langle A^{q}\rangle]$
or $\langle A^{pq+1}\rangle/\langle A^{p}\rangle^{q}$, which have units of
area and hence are also constant in the scaling state. Note that Eq. 10 gives
us one equation for every two unknowns. If we could get another size-topology
relation for every $p$, it might be possible to combine with the above
identities to solve for $\langle A^{p}\rangle/\langle A\rangle^{p}$ and
$\langle\langle n\rangle\rangle_{p}$, and conceivably derive (for example) an
analytic form for the bubble size distribution.

##  III Materials and Methods

To test the predictions by experiment we generate aqueous foams of Nitrogen
bubbles that have no film rupture and that, as we will show, coarsen in a self
similar scaling state. These foams are made from a solution that is 92%
deionized water and 8% Dawn Ultra Concentrated dish detergent. The foam is
generated inside a sample cell constructed from two 1.91 cm-thick acrylic
plates separated by a spacing $H=0.21$ cm and sealed with two concentric
o-rings, the inner of which has a 23 cm diameter; this is the same apparatus
used in Roth  _et al._ (2013); Chieco and Durian (2021), where additional
details may be found.

Foams are produced as follows. First the cell is filled completely with
foaming solution. It is then flushed with Nitrogen and sealed when a desired
amount of liquid remains. The entire sample cell is vigorously shaken for
several minutes until the gas is uniformly dispersed as bubbles that are
smaller than the gap between the plates. This ensures a large number of small
bubbles and aids in repeatability, tested later. The foam is thus initially
very wet, opaque, and three-dimensional. We stand the cell so that the plane
of the foam is vertical and place it between a Vista Point A light box and a
Nikon D850 camera with a Nikkor AF-S 300mm 1:2.8D lens. Images are acquired
every 5 minutes for up to 24 hours. Several minutes after production most of
the liquid drains out of the foam; after an hour the bubbles become large
compared to the gap. The resulting foam is dry and quasi- two-dimensional, and
only subsequent data are kept. In this regime, the radius of the Plateau
borders is about 0.03 mm, so the thin soap films span more than 97% of the gap
$H$ between the plates and gas transport across the borders is negligible.

The areas of individual bubbles are found directly from the images. The images
undergo some some slight post processing to enhance contrast and then we
binarize, skeletonize, and watershed them. Bubbles are the watershedding
basins of the skeletonized images and the number of pixels within each basin
is converted into the bubble area. Thus the choice of camera is important and
the D850 has an $8256\times 5504~{}{p_{o}}^{2}$ pixel array where $p_{o}$ is
the pixel size; combining it with a telephoto lens placed $1.5$ m away allows
us to have both a large number of bubbles and an accurate measurement of their
area.

The vertices of each bubble are also found from the watershed image as the
pixels where three basins are in contact. Each of the three bubbles are
assigned that vertex and the total number of vertices of each bubble equals
its number of sides. We track the number of sides of each bubble and find they
only change when there is a topological rearrangement; thus we verify there
are no film ruptures throughout the experiment. Only bubbles that do not
overlap the boundary of the $19\times 6.5$ cm2 region of interest are kept for
analysis. With the resulting collection of relevant size and topology
information for bubbles from each image we can now test predictions from the
previous section.

##  IV Experimental Results and Discussion

We are interested in examining the predictions made by two generalizations of
Eq. 2. The first is Eq. 15 which uses distributions of the bubble areas and
sides to predict a coarsening rate for the entire foam. And the second is Eq.
10 which is a generalized size-topology relationship. These equations were
derived for 2-dimensional foams in a scaling state so we first establish the
foam is self-similar. Once this is determined, we test the expectations
against the results garnered from three separate experiments. To distinguish
the samples and gauge repeatability we identify them as Foam A, B, and C. Each
foam is produced using the method stated in the previous section; the
experiments differ by the specific initial distributions of 3-dimensional
bubble sizes and the dates they were performed; the data for Foam C was
collected several months before the data for Foam A and B. We keep images only
when the foams have become quasi-2d. Fig. 1 show representative zoomed-in
images, where the bubbles appear as polygons with curved edges. The three data
sets consist of $\\{81,101,69\\}$ images for the three samples, respectively,
as they coarsen: the corresponding initial numbers of bubbles are
$\\{1745,3025,2588\\}$, and the final numbers of bubble are
$\\{728,688,846\\}$; the corresponding initial and final average areas of the
bubbles are $\\{6.7,3.5,4.4\\}$ mm2, and $\\{15.6,14.4,12.7\\}$ mm2,
respectively; In supplementary material we provide all bubble area and side
number data sorted by time for the three experiments.

###  IV.1 Self-Similar Scaling State

Self-similarity is well documented and has been observed in experiment and
simulation. Here, a qualitative demonstration is illustrated by Fig. 1. There,
we show two images separated by 340 minutes and zoom in to a portion from each
that is $\left[L_{x},L_{y}\right]=[15,7.5]\sqrt{\langle A\rangle}$, so that
the average bubble occupies the same visual space in each. Bubbles in the
younger foam are smaller and thus appear more pixelated. Otherwise the two
images appear very similar, as though coming from different regions of the
same sample and hence as having the same apparent bubble size distributions.

Figure 1: Images of coarsening foam at different times, zoomed in so that the
distributions of sizes appear the same by eye. Here, the image side lengths
are both $L_{x}=15\sqrt{\langle A\rangle}$ and $L_{y}=7.5\sqrt{\langle
A\rangle}$. This is a qualitative demonstration of self-similarity in the
scaling state.

For a quantitative demonstration that the foams are in a scaling state, we
consider several metrics. Fig. 2(a) shows that the average area increases
linearly with time for all three of our foams. Such linearity holds only in
the self-similar regime (too-monodisperse and too-polydisperse sample have
initial coarsening rates that are respectively slower or faster Glazier  _et
al._ (1987); Beenakker (1988)). Furthermore, this linear behavior is predicted
by Eq. 2 for a self similar foam because both the ratio $\langle
A^{2}\rangle/\langle A\rangle^{2}$ and the area-weighted average side number
$\langle\langle n\rangle\rangle$ are independent of time. These quantities are
plotted in parts (b) and (c), respectively; both are constant in time and are
nearly the same for the three foam samples. For each foam sample, we calculate
$K_{o}$ from Eq. 2, or Eq. 8 for $p=1$ (since they are the same equation)
using the slopes from part (a) as well as the individual values of $\langle
A^{2}\rangle/\langle A\rangle^{2}$ and $\langle\langle n\rangle\rangle$ for
each foam; we find that $K_{o}=\left[0.022\pm 0.001,0.025\pm 0.001\right]$
mm2/min where the first value is for Foam A and B, and the second value is for
Foam C. These experimental values depend on the choice of gas and solution,
which are the same for all three foams; however the larger value for $K_{o}$
for Foam C is likely due to the chemical aging of the foaming solution. But
the data for Foam A and B, which were acquired one day apart and collapse,
demonstrate repeatability. Taking all metrics together we establish the three
foam samples are in the self-similar regime.

Finally, as an additional check on the data, Fig. 2(c) shows average side
number $\langle n\rangle\approx 6$ for each of the three samples. By Euler’s
rule this should equal six for large enough samples, whether or not they are
in the scaling state. We note that for each quantity in Fig. 2(b) and (c) the
values are nearly the same between foam samples. This along with the collapse
of all of the area distributions between the three samples in Fig. 4, are
further proof of good repeatability regardless of the physical chemistry of
the foaming solution.

Figure 2: (a) Average area, (b) second moment divided by the average area
squared, and (c) area-weighted average side number $\langle\langle
n\rangle\rangle$ as well as the average side number $\langle n\rangle$ versus
time. The different symbols represent data from the three different foam
samples. In part (a) statistical error bars are included for Foam C as
$\sigma_{A}=\langle A\rangle/\sqrt{N}$ where $N$ is the number of bubbles. The
statistical uncertainties for the other samples are comparable, and correspond
well to the bumps and wiggles in the data and also to the differences between
samples A and B. Also in (a) the dot-dashed represent proportionality fits,
giving $K_{o}$ values as labeled.

Having established the foam is self-similar it follows that the distributions
for side-number $P(n)$ and area-weighted side number $F(n)$ must also collapse
throughout time. Therefore all of the data are averaged together for all times
and for all three foam samples, giving one final distribution for $P(n)$ and
one for $F(n)$; these are plotted in Fig. 3 parts (a) and (b), respectively.
For both, the distributions are almost the same when comparing between
experiments and the error bar span the calculated values for each $n$. Fig.
3(a) shows that 5 and 6-sided bubbles are most prevalent. From $P(n)$, the
average number of sides is computed to be $\langle n\rangle=5.97\pm 0.02$.
This is slightly smaller than 6, as expected, because the samples are finite.

Figure 3: (a) Side-number distribution and (b) area-weighted side-number
distribution for the three coarsening foam samples, averaged together over
time. The error bars span the values coming from the three different foam
samples. The average number side number $\langle n\rangle$ and the area-
weighted average side number $\langle\langle n\rangle\rangle$ are indicated by
arrows as labeled.

Fig. 3(b) shows that the average $F(n)$ distribution for the three foams is
skewed more towards cells with large $n$ in comparison to $P(n)$, particularly
for bubbles with $n=\\{7,8\\}$ sides. This is understood because bubbles with
a larger number of sides tend have larger areas; the area-weighted average
side number, $\langle\langle n\rangle\rangle=6.83\pm 0.03$, is therefore
larger than $\langle n\rangle$.

Since the samples are self-similar, the cumulative area distributions (CDFs)
must be independent of time when plotted versus $A/\langle A\rangle$. This is
exemplified in Fig. 4 for each of the three coarsening foams, where the gray
curves are data from different times and where $y=1-\mathcal{N}_{CDF}$ is the
fraction of bubbles whose normalized areas are larger than $x=A/\langle
A\rangle$. The gray curves collapse to the CDF made from all the normalized
bubble areas collected into one distribution for each foam. The three total
distributions also collapse and the data are found to follow the same
slightly-compressed exponential as was found in Roth  _et al._ (2013), and is
consistent with other works Glazier and Weaire (1992); Stavans (1993b). This
behavior, while expected, is important because it demonstrates that all the
normalized moments of the area distribution are independent of time; this fact
is a main ingredient in both the derivation and expectations of the size-
topology relations to be tested next. Note, however, that the data have a
cutoff where the CDF plummets to zero above the largest observed bubbles,
whereas the compressed exponential fit has no such cutoff. Therefore large-$p$
moments, which emphasize the largest bubbles, will have significant systematic
differences that depend on sample size.

Figure 4: Cumulative distribution function for bubble areas for the three
coarsening samples, separately averaged over time and shown as dotted curves;
the largest five observed bubble areas for each sample are plotted as symbols.
Solid gray curves are distributions at different times, for three different
foam samples, which contribute to the averages. The black dot-dashed curve is
an exponential and the dashed curve is a compressed exponential.

###  IV.2 Testing the Size-Topology Relations

We are interested in testing two of our generalized forms of Eq. 2. The first
is Eq. 15, relating the rate of change of two of the area moments to the area-
weighted side number. Since the foam is self-similar the equation can be
solved to show that $\langle A^{p+1}\rangle/\langle A^{p}\rangle$ grows
linearly in time. By choosing any value of $p$ and evaluating the equation the
data should collapse to a line with slope $K_{o}$. This is done for several
values of $p$ and Fig. 5(a) shows data for Foam A and B, while Fig. 5(b) shows
data for Foam C since it has a different $K_{o}$ value per Fig. 2a. For Foams
A and B the data collapse to the same line with slope $K_{o}$ that is
consistent with the values calculated from Fig. 2 data. Foam C shows the best
collapse of the three data sets and also has $K_{o}$ within error of the prior
analysis.

Figure 5: Area moments normalized by preceding moment versus weighted time for
various powers of $p$ as labeled. Part (a) show Foam A and B because they have
the same coarsening rate as determined by Fig. 2 and part (b) shows Foam C.
Plotted as such, the data collapse and have proportionality constant $K_{o}$
taken from the Fig. 2a analysis, in accord with the prediction of Eq. 15.

Eq. 15 may be the cleanest looking of our generalizations but it is similar to
Eq. 2 in that it is a rate of change. However, Eq. 10 relates the sizes of the
bubbles directly to their topology. There are two main ingredients in the
equation and those are the dimensionless moments of the area distribution and
the $A^{p}$-weighted side number. We evaluate these quantities using data from
all times and they are individually plotted in Fig. 6(a) and (b),
respectively. We note that while Eq. 10 calls for values of $p\geq 2$ this is
not the case of its components which can be calculated for any power.
Therefore the $x$-axis in Fig. 6 extends to negative numbers and we also
evaluate the quantities for non integer values of $p$.

Figure 6: Dimensionless moments of the area distribution (a) and weighted side
number for areas raised to different powers (b) versus the power use in
calculation. These quantities are calculated using data from all times for
each of the three samples as labeled. Part (a) shows gamma functions as black
dash-dotted and red dash curves that respectively correspond to the
exponential and compressed exponential CDFs shown in Fig.4. The gray lines in
both parts point to special cases for each quantity: part (a) they show where
$\langle A^{p}\rangle/\langle A\rangle^{p}=1$ and part (b) shows where
$\langle\langle n\rangle\rangle_{p}=6$.

One striking feature of the figure is that the data for Foam A appears
separate from the data from Foam B and C; this is different than the
coarsening behavior where Foam A and B are similar and Foam C is different.
This separation does not happen until values of $p\gtrsim 7$ as evidenced by
the inset in Fig. 6(a). The inset demonstrates how the data are well described
by the moment generating function for the compressed exponential we fit to the
area distribution. However the expectation deviates from the data for large
$p$, likely due to finite size effects and/or because there is no cutoff for
the assumed size distribution. It matches the data for Foam A best because it
has the largest bubble(s) of all three data sets. This is similar to what we
see from Fig. 6(b) where the large-$p$ data for Foam A separates from the
other two data sets. This too is explained by the largest bubble being in Foam
A because large bubbles also have a large number of sides. Therefore for
large-$p$ the calculation of $\langle\langle n\rangle\rangle_{p}$ is dominated
by the bubbles with the most number of sides which also have very large areas.
What is important for the size-topology equation though is how these
quantities relate to the ones evaluated at a $p$ integer step above them and
that they are time independent. We have already shown the latter and now want
to evaluate the entirety of Eq. 10 to see if the data follow the expectation.

Using the various dimensionless moments of the area distribution and the
$A^{p}$-weighted average side number we now compute the size-topology
relationship of Eq. 10. It is evaluated for $2\leq p\leq 20$; the expectation
asymptotes to 1 so the resulting values have $1$ subtracted from them and are
plotted in Fig. 7 for each of our three foams. Performing this subtraction
along with plotting the $y$-axis on a log-scale is done to get a closer look
at the data. This reveals good agreement with the expectation. However, there
are deviations at large $p$ but they are explained by finite size effects and
the sensitivity of large-$p$ moments to the largest bubbles in the sample.

Figure 7: The generalized size-topology identity versus power. The different
symbols represent three different coarsening foams. The data collapse to the
expectation which is plotted as a black curve.

##  V Conclusion

We have derived several identities for 2d foams in a self-similar scaling
state. The ones we tested experimentally are: Eq. 15, which compares the rate
of change of successive moments of the area distribution to the
$A^{p}$-weighted average side number; Eq. 10, which is a generalized size-
topology relation that relates moments in the area distribution to the
$A^{p}$-weighted side number. Both equations are derived using only von
Neumann’s law and the fact that the foams are in a self-similar scaling state.
We tested these relationships for three different foam samples. After showing
the foam is self-similar we found the data agree well with both predictions. A
natural extension would be to study 3d foams; however, since the von Neumann
like expression for domain growth in 3d is not purely topological MacPherson
and Srolovitz (2007), any size-topology relationships would be approximate.
Instead, future research might focus on the dynamic size-topology equation
that was derived here in the form of Eq. 14 but was not tested. Another avenue
for future work would be to consider how the size-topology relations constrain
the form of the bubble size distribution and if they could permit it to be
derived. It would be particularly interesting to investigate if there is a
cutoff in the distribution, _i.e._ if there is a maximum possible bubble size
in comparison with the average. Further work could also explore how $d\langle
A^{p}\rangle/dt$ is affected by nonzero wetness, where von Neumann’s law must
be modified to account for transport across the Plateau borders as predicted
in Schimming and Durian (2017), and tested in Chieco and Durian (2021); this
is challenging because the corrections to von Neumann depend not only on the
size of the Plateau borders but also on the shapes of the bubbles.

## Author Contributions

All authors conceived and designed the study. DJD performed the calculations,
while ATC performed the experiments and analyses with guidance by DJD and JPS.
DJD and ATC wrote the manuscript with input from JPS.

## Funding

This work was supported by NASA grant 80NSSC19K0599.

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