数理解析研究所講究録
第 1904 巻 2014 年 1-12

1

Weak Solution of Renormalization Group Equation
Ken-Ichi Aoki, Shin-Ichiro Kumamoto and Daisuke Sato*titute for Theoretical, Kanazawa University$In\mathcal{S}$$Phy_{\mathcal{S}}ics$

Abstract
In the approach of non-perturbative renormalization group (NPRG), the spontaneous chiral symmetry breaking induces a singularity in its solution, e.g., theflow of the 4-fermi coupling constant blows up at a critical renormalization group(RG) scale. Thus, as long as directly solving the NPRG equation as a partial
diﬀerential equation, the RG flow cannot go beyond the critical scale to obtain infrared quantities such as the chiral condensates. In order to treat this singularity
in a mathematically rigorous way, we introduce the notion of weak solution of the
NPRG equation. The weak solution is found to give a unique global solution toward
the infrared limit, and we can calculate infrared quantities without any ambiguities.

1

Introduction

The almost 100 percent of mass is originated from spontaneous chiral symmetry breaking$(S\chi SB)$ , while a few percent of mass
is given by the so-called Higgs mechanism. The$S\chi SB$ is included by the strong
interaction between quarks at the low energy scale which
is described by quantum chromodynamics (QCD). Because of the strong interaction, theperturbation theory does not work, and so we need non-perturbative methods such as thelattice simulation and the Schwinger-Dyson (SD) approach.
In this article, we use the approach of non-perturbative renormalization group (NPRG)
that is originated from the Wilsonian idea. This approach does not have the sign problem
at finite chemical potential just like the lattice simulation, and can improve the gauge
dependence of physical quantities, which the SD approach suﬀers, in systematic approximations [1, 2].
For simplicity, we are limited to the analysis using the Nambu-Jona Lasinio (NJL)
model with a simplified discrete chiral symmetry, which is a low energy model of QCD
explaining the $S\chi SB$ . Its Lagrangian is given by$\mathcal{L}=\overline{\psi}\emptyset\psi-\frac{G_{0}}{2}(\overline{\psi}\psi)^{2}$,(1)

where and is a quark field and an antiquark field, respectively Here the discretechiral symmetry is that the Lagrangian is invariant under the following discrete chiraltransformation:$\psi$$\overline{\psi}$$\psiarrow\gamma_{5}\psi, \overline{\psi}arrow-\overline{\psi}\gamma_{5}$$*e$-mail address: satodai@hep.s.kanazawa-u.ac.jp

.

(2)

2
and the chiral condensates
The chiral symmetry thus forbids the mass term
However, when the 4-fermi coupling constantis larger than a critical coupling constant,its strong coupling induces the $S\chi SB$ . If using the mean field approximation, the criticalis the ultraviolet cutoﬀ scale. Note that the, whereiscoupling constantmean field approximation is equivalent to the self-consistency equation limited to thelarge-N leading, where $N$ is the number of quark flavors.
Many NPRG analyses of $S\chi SB$ have been performed by introducing the bosonization of the multi-fermi interactions [3-6], which is the so-called auxiliary field method or
Hubbard-Stratonovich transformation. However the analysis without the bosonization isdiﬃcult because in the RG procedure the 4-fermi coupling constant blows up at a criticalscale as a signal of $S\chi SB[7$ , 8 . Consequently we cannot go beyond the critical scale toobtain infrared physical quantities such as the chiral condensates.
The goal of this article is to analyze $S\chi SB$ in the NPRG approach without introducing
the bosonization. For this goal we adopt the method of weak solution [9], which has firstly
been introduced in the NPRG approach by the authors [10]. As the NPRG equation is
given by a partial diﬀerential equation (PDE), the weak solution satisfies the integral-form(weak) equation of the PDE. Since the weak solution is globally defined, it can include
singularities such as the explosive behavior of the 4-fermi coupling constant.
This article is organized as follows. In Sect. 2, we briefly explain the Wegner-Houghton
equation that is a formulation of the NPRG and the diﬃculty of the NPRG analysis of$S\chi SB$ without the bosonization.
In Sect3, the method of weak solution is adopted toovercome this diﬃculty. In Sect4, the bare mass of quark is introduced to define thechiral order parameters. In Sect5, the method of weak solution is applied to the firstorder phase transition at finite chemical potential, and the convexity of the eﬀective
potential given by the weak solution is discussed. Finally we summarize this article in$m\overline{\psi}\psi$$\langle\overline{\psi}\psi\rangle.$$G_{0}$$G_{c}$$4\pi^{2}/\Lambda_{0}^{2}$$\Lambda_{0}$$]$Sect 6.

Non-perturbative renormalization group2

definedIn the NPRG approach, a central object is the Wilsonian eﬀective action
with momentums higher than theby integrating the microscopic degrees of freedom
scale :
$S_{eff}[\phi_{)}\Lambda]$$\phi_{H}$$\Lambda$$\int \mathcal{D}\phi_{H}e^{-S_{0}[\phi_{L_{\rangle}}\phi_{H},\Lambda_{0}]}=e^{-S_{eff}[\phi_{L},\Lambda]}$(3),

is a bare action with the ultraviolet cutoﬀ scale
where
the cutoﬀ scale by a dimensionless scale such that
$S_{0}[\phi;\Lambda_{0}]$$\Lambda$$\Lambda(t)=\Lambda_{0}e^{-t}$The -dependence of the eﬀective action
following functional partial diﬀerential equation:$t$$\Lambda_{0}$. Now we parametrize$t$$S_{eff}[\phi;\Lambda]$.

(4)

is given by a NPRG equation as the$\partial_{t}S_{eff}[\phi;t]=\beta_{WH}[\frac{\delta S_{eff}}{\delta\phi}, \frac{\delta^{2}S_{eff}}{\delta\phi^{2}};t]$,(5)which is called the Wegner-Houghton (WH) equation [11] (see Ref. [12] for the detail form
. The WH equation is the exact equation that provides macroscopic informations,of$\beta_{WH})$

3such as the chiral condensates and the eﬀective quark mass, by setting the bare action to
the initial condition at $t=0$ and solving it as a diﬀerential equation toward the infrared
scale $(tarrow\infty)$ . Of course, it cannot be solved exactly, but various non-perturbative
approximation to solve it are available.
In this article, the WH equation is applied to the NJL model (3). As an approximation,
we restrict the full interaction space of the eﬀective action
to be the subspace
relevant to $S\chi SB$ as follows:$S_{0}$$S_{eff}[\psi, \overline{\psi};t]$$S_{eff}[ \psi, \overline{\psi};t]=\int d^{4}x\{\overline{\psi}\beta\psi-V_{W}(x;t)\}$(6),

where a scalar fermion-bilinear field,, is introduced. The potential term $V_{W}(x;t)$ iscalled the fermion potential here, whose initial condition is set to $V_{W}(x;t=0)=(G_{0}/2)x^{2}$according to the NJL Lagrangian (1).
In addition to the restriction of the interaction space, the large-N non-leading partsof the WH equation (3) are ignored. Then, the NPRG equation for the fermion potential
in the large-N approximation is given by the following partial diﬀerential equation:$x=\overline{\psi}\psi$$\partial_{t}V_{W}(x;t)=\frac{\Lambda^{4}}{4\pi^{2}}\log(1+\frac{1}{\Lambda^{2}}(\partial_{x}V_{W})^{2})\equiv-F(\partial_{x}V_{W};t)$.

(7)Here the momentum cutoﬀ A have been performed with respect to the length of four
Euclidean momentum :. Note that the approximation used here isequivalent to the mean field one,
Now we introduce the mass function, $M(x;t)=\partial_{x}V_{W}(x;t)$ , to interpret the $S\chi SB$ inthis framework. The value of the mass function at the origin is the coeﬃcient of massterm
in the eﬀective action as its name suggests. The chiral symmetry is realized by
the invariance of the fermion potential under the chiral transformation, $xarrow-x$ , given by
Eq. (2): $V(-x;t)=V(x;t)$ , and then $M(-x;t)=-M(x;t)$ . The NPRG equation (7) withthe chiral-invariant fermion potential is also invariant under the chiral transformation, andthus its solution with the chiral-invariant initial condition maintain its chiral-invariant
structure at all scales. If the mass function is analytic, its value at the origin vanishes
since the mass function with the chiral invariance is odd with respect toWhile the NPRG equation does not spontaneously break the chiral-invariant structureof the fermion potential, its second derivative at the origin that is the 4-fermi coupling
constant
blows up at a critical scale
if its initial coupling
constantis larger than the critical coupling constant $G_{c}[7,8]$ . This explosive behavior isnothing but a signal of the $S\chi SB$ , and suggests that the $S\chi SB$ solution of the mass function
after has a finitejump at the origin with the chiral-invariant structure. Mathematically,such a singular solution of the PDE cannot be authorized. However the singular solution
can be defined as a weak solution and predict the physical quantities as shown in the next
section.
$p_{\mu}$$\sum_{\mu=1}^{4}p_{\mu}^{2}\leq\Lambda$$\overline{\psi}\psi$$x.$$G(t)\equiv\partial^{2}V(x;t)/\partial x^{2}|_{x=0}$$t_{c}$$G_{0}$$t_{c}$

3

Method of weak solution

In this $section_{\}}$ we define a weak solution [9] of the mass function and show how toconstruct it. Diﬀerentiating the PDE (7) with respect to , we then obtain that for the$x$

4mass function,$\partial_{t}M(x;t)=-\frac{\partial}{\partial x}F(M(x;t);t)$$=- \frac{\partial}{\partial M}F(\Lambda I;t)\cdot\frac{\partial JI}{\partial x}$(8).

This equation belongs to a class of conservation low type which includes the famous
Bergers’ equation without viscosity. To define the singular solution with finite jumps, we
extend the PDE into the following weak equation:$\int_{0}^{\infty}dt\int_{-\infty}^{\infty}dx(M\frac{\partial\varphi}{\partial t}+F\frac{\partial\varphi}{\partial x})+\int_{-\infty}^{\infty}dxM\varphi|_{t=0}=0$,(9)

is any smooth and bounded test function. Compared to the strong equation (8), the derivative of the mass function is gotten rid oﬀ in the weak equation (9),which can then have a singular solution with finite jumps. In general, weak solutions are
not uniquely determined depending on the initial conditions. However the physical initialcondition is expected to give the unique weak solution.
An important fact derived from the weak equation,(9) is the Rankine-Hugoniot (RH)condition,

where$\varphi(x;t)$$(M_{L}-M_{R})dS(t)=[F(M_{L})-F(M_{R})]dt$ ,(10)

are values of right and left limits of theandwhere $S(t)$ is a jump position, andmass function at $x=S(t)$ . This RH condition will be used to construct the weak solution.
In the rest of this section, the method of characteristics to construct the weak solution, and the mass function on
is shown. We now consider a characteristic curve,it,, which satisfy the following coupled ordinary diﬀerential equations(ODEs):$M_{R}$$M_{L}$$x=\overline{x}(t)$$\overline{M}(t)=M(\overline{x};t)$$\frac{d\overline{x}(t)}{dt}=\frac{\partial}{\partial M^{-}}F(\overline{M};t)$

(11)

,

$\frac{d\overline{M}(t)}{dt}=\frac{\partial}{\partial\overline{x}}F(\overline{M};t)=0$

(12)

.

Here the initial condition is given by$\overline{x}(t=0)=x_{0}$

(13)
(14)

,

$\overline{M}(t=0)=\partial_{x}V_{W}(x;t)|_{x=x0,t=0}$.

is the value of the “local” strong solution of Eq. (8) on the
We now emphasize that
. Thus the initial value problem of the PDE (8) is transformed tocharacteristic curve
the partially equivalent one of the coupled ODEs (11), (12), although the set of (local”
strong solutions is not necessarily the global solution of the original PDE as will be seen
later. We can now easily construct the set of local strong solutions by varying the valueof the fermion potential onof and solving the coupled ODEs. Moreover the value$x(t)$ is obtained by solving the following ODE with Eqs. (11), (12):$\overline{M}(t)$$\overline{x}(t)$$\overline{V}_{W}$$x_{0}$$\frac{d\overline{V}_{W}(t)}{dt}=M^{-}\frac{\partial F(\overline{M};t)}{\partial M^{-}}-F(J^{-}I;t)$

.

(15)

5
Note that the PDE (7) is a Hamiltonian-Jacobi type equation well-known in the analytical$V_{W}(x;t)$ , $M(x;t)$ and $F(M;t)$ correspond to the time, the coordimechanics, where
nate, the action, the momentum and the time-dependent Hamiltonian, respectively. Thenthe coupled ODEs (11), (12) are nothing but the canonical equations of Hamiltonian.
The numerical solution of the characteristic curves and the set of local strong solutions$M(x;t)$ constructed by them are shown in Fig. 2 (a), (b). The characteristic curves
can be regarded as the contour lines of $lII(x;t)$ because the right-hand side of Eq. (12)vanishes. After , the contour lines cross each other, and then the set of the local strong
solutionshas a folding structure. Thus the set, which has the multi values, canno longer be the global solution of the PDE (8). On the other hand, the weak solution
can be constructed by the patchwork of the local strong solutions, which is determinedusing the RH condition (10).
$t,$

$x,$

$\overline{x}(t)$$t_{c}$$\Lambda\ell(x;t)$Figure 1: Equal (vanishing) area rule.
For the practical purpose, we here convert the RH condition to a geometric one equivalent to it as follows. The total derivatives of left and right limits of the fermion potential$V_{W}(x;t)$ on the jump position $S(t)$ is given by$dV_{W}^{L,R}= \frac{\partial V}{\partial x}|_{L,R}dS+\frac{\partial V}{\partialt}|_{L,R}dt$$=M|_{L,R}dS-F|_{L,R}dt$ ,(16)

The diﬀerence between the left and right limits of Eq. (16) vanishes because of the RH
condition: $d(V_{W}^{L}-V_{W}^{R})=0$ . Since no singular point doesn’t exist at the initial condition,
the fermion potential is entirely continuous even at the jump position $S(t)$ :$V_{W}^{L}=V_{W}^{R}$

.

(17)

Next, we integrate the set of the local strong solutions of $lII(x;t)$ as follows:$\int_{L}^{R}Mdx=\int_{L}^{R}\frac{\partial V_{W}}{\partial x}dx$$=V_{W}^{R}-V_{W}^{L}$$=0$ .

(18)

6
Thus the jump position can geometrically be determined by the equal (vanishing) arearule (Fig. 1). In Fig. 2, we now show the weak solution constructed from the local strong
solutions at $G_{0}=1.005G_{c}$ using the equal area rule. The jump appears at the origin
after the critical scale . The uniqueness of our weak solution is proved because of theentropy condition, which guarantees the uniqueness of weak solution when the selectedcharacteristic curves fill the x-t plane [9] as shown in Fig. 2 (a’).
$t_{c}$

(a)

(b)

(a’)

(b’)

Figure 2: (a) Characteristics. (b) Set of local strong solutions of mass function given by
the characteristics. (a’) Characteristics selected by the RH condition and jump
(discontinuity). (b’) Weak solution of mass function. The jump position obey
the equal (vanishing) area rule.

4

Bare mass

In the previous section, we have obtained the $S\chi SB$ weak solution with a finite jump at
the origin. However the physical mass of quark as an order parameter of chiral symmetry
cannot be determined since the mass function at the origin is not defined. To define the, which explicitly breaks theorder parameter, we introduce the bare mass termchiral symmetry, to the Lagrangian (1). The bare mass term modifies the initial conditionof the PDE (8): $M(x;t=0)=G_{0}x+m_{0}$ . Then, because of the translation invariance of
the PDE with respect to , the mass function at $m_{0}\neq 0$ is given by the one at $m_{0}=0$ as$m_{0}\overline{\psi}\psi$$x$7follows:(19)$M(x;t, m_{0})=M(x+m_{0}/G;t, m_{0}=0)$ .

Then the mass function at the origin is well defined because the jump appears not at theorigin but at $x=-m_{0}/G$ . Thus, taking the limit $m_{0}arrow+0$ (called the chiral limit), we
can define the -dependent eﬀective mass $M_{phs}(t)$ as the order parameter:$t$$M_{phys}(t)= \lim_{m0arrow+0}M(x;t, m_{0})|_{x=0}$

(20)

.

Fig. 3 shows the RG evolutions of the physical masses in the chiral limit and at thenon-zero bare mass.1 The physical mass in the chiral limit shows the second order phasetransition due to the singular behavior of the mass function at the origin, while thephysical mass at $m_{0}\neq 0$ shows the cross over. The reader may think that the weaksolution method is not necessary if $m_{0}\neq 0$ . However global methods, such as the weaksolution, is needed at the small bare mass compared to the physical mass since the massfunction has the jump near the origin. Actually, Ref. [2] shows that the Taylor expansion
to solve the PDE7 does not work at the small bare mass.

Figure 3: RG evolution of the physical masses in $m_{0}arrow 0$ and $m_{0}=$ O. The NPRG
equation given by Eq. (21) $(\mu=0, G=1.7G_{c})$ is used for evaluating thephysical mass to compare the result at finite chemical potential$\mu\neq 0.$

5

First order phase transition at finite chemical potential

Let us consider the first order phase transition at finite chemical potential $(\mu\neq 0)$ using
the weak-solution method. The first order phase transition is more non-trivial than thesecond order phase transition because the RG evolution of the physical mass has a finite
jump even at $m_{0}\neq 0$ (as shown in Fig. 4). Moreover the non-uniqueness of weak solution
is associated with the fact that the eﬀective potential, which is non-convex, has the multilocal minima in the mean-field analysis. In this section, we show that the weak solution

8

Figure 4: RG evolutions of the physical mass at finite density.

at finite chemical potential is uniquely determined, and the eﬀective potential constructedby the weak solution is automatically “convexised” with only the one correct minimum.
The chemical potential is introduced by adding the term
to the Lagrangian (1).
For the simple NPRG equation, we use the spacial momentum cutoﬀ:. Thenthe right-hand side of Eq. (8) changes$\mu\overline{\psi}\gamma_{0}\psi$$\mu$$\sum_{i=1}^{3}p_{i}^{2}\leq\Lambda^{2}$$to^{2}$$-F(x;t)= \frac{\Lambda^{3}}{\pi^{2}}[\sqrt{\Lambda^{2}+M^{2}}+(\mu-\sqrt{\Lambda^{2}+M^{2}})\cdot\Theta(\mu-\sqrt{\Lambda^{2}+M^{2}})]$

,

(21)

where
is the Heaviside step function. The characteristic curve is consequently givenby the following ODE:$\Theta(x)$$\frac{d\overline{x}(t)}{dt_{ノ}}=-\frac{\Lambda^{3}M}{\pi^{2}\sqrt{\Lambda^{2}+M^{2}}}\Theta(\sqrt{\Lambda^{2}+M^{2}}-\mu)$

(22)

.

In Fig. 5, we show the characteristic curves and those selected by the RH condition which
are evaluated at $m_{0}=0,$ $\mu=0.7$ . and $G_{0}=1.7G_{c}$ . Fig. 5 (b) shows the uniqueness of
our weak solution because the entropy condition is satisfied. Fig. 6 (a), (b) show the weaksolutions of the mass function and the fermion potential at $m_{0}=0.01\Lambda_{0}$ . These figures
shows that in the RG procedure the two jumps simultaneously appear, move toward eachother, and finally merge into one. Thus the RG evolution of the physical mass shows thefirst order phase transition as shown in Fig. 4.
In the rest of this section, we discuss the convexity of the Legendre eﬀective potential
constructed by the weak solution. At first, we define the free energy $W(j;t)$ by intro, which
ducing the external source for the chiral condensates: its source term
is distinguished from the mass term, is added to the Lagrangian (1). Then the initialcondition of the fermion potential is$\langle\overline{\psi}\psi\rangle$$V_{W}(x;t=0,j)=m_{0}x+ \frac{G_{0}}{2}x^{2}+jx$ .

lIn the real world the bare mass has the non-zero value given by the Higgs mechanism.
2The critical coupling constant also changes:$G_{c}=2\pi^{2}/\Lambda_{0}.$$j\overline{\psi}\psi$

(23)

9

(a)

(b)

Figure 5: (a) Characteristics. (b) Characteristics selected by the RH condition and jump
(discontinuity).
Now the free energy and the chiral condensates are given by(24)$W(j;t)=V_{W}(x=0;t,j)$ ,$\phi(j;t)\equiv\langle\overline{\psi}\psi\rangle_{j}=\frac{\partial W(j;t)}{\partial j}$

,

(25)

respectively. We eventually define the Legendre eﬀective potential of the chiral condensate
as follows:$V_{L}(\phi;t)=j\phi(j;t)-W(j;t)$ ,

where

(26)

is satisfied.
As seen in the previous section, because of the translation invariance of the PDE withrespect to , the fermion potential at $j\neq 0$ is given by the one at $j=0$ :$\partial V_{L}/\partial\phi=j$$x$$V_{W}(x;t, j)=V_{W}(x+j/G_{0};t,j=0)- \frac{m_{0}j}{G_{0}}-\frac{j^{2}}{2G_{0}}$

(27)

,

Thus the free energy and the chiral condensates are given by the quantities at $j=0$ as
follows:$W(j;t)=V_{W}(x=0;t,j)=V_{W}(j/G_{0};t,j=0)- \frac{m_{0}j}{G_{0}}-\frac{j^{2}}{2G_{0}}$$\phi(j;t)=\frac{1}{G_{0}}[M(j/G_{0};t, j=0)-m_{0}-j]$ .

,

(28)

(29)

Since the set of local strong solutions of the mass function $M(j/G_{0};t,j=0)$ is multivalued, that of $\phi(j;t)$ is so. Obeying the equal area rule, the weak solution of $\phi(j;t)$ isthen constructed from its local strong solutions as well as the mass function.
The set of strong solutions of
is not convex and has multi local minima as
shown Fig. 6 (c). On the other hand, we can prove that its weak solution is the “convexised”’ potential whose minimum agrees with the global minimum of the set of the local$V_{L}(\phi;t)$10strong solution as follows. Because of the continuity of the fermion potential (17), the freeenergy is also continuous at the jump position of the mass function $M(j/G_{0};t,j=0)$ :$j_{s}$$W^{L}-W^{R}=V_{W}^{L}-V_{W}^{R}=0$

.

(30)

Using this continuity of the free energy and Eq. (26), we obtain$\frac{V_{L}^{L}-V_{L}^{R}}{\phi^{L}-\phi^{R}}=j_{s}$

(31)

,

at the two
which means that the line connecting the Legendre eﬀective potential. Thus the weak solutionagrees with the envelope since
positionsof the eﬀective potential is automatically convexised and has the correct minimum which
agrees with the global minimum of the local strong solutions as shown in Fig. 6.
$V_{L}(\phi;t)$$\phi^{L},$6$\phi^{R}$$\partial V_{L}/\partial\phi|_{L,R}=j_{S}$Summary

In this article, we have introduced the weak solution to define the singular $S\chi SB$ solutionof NPRG equation that can predict physical quantities such as the physical quark massand the chiral condensates. The weak solution satisfies the integral-form (weak) of the
PDE. Specifically we have evaluated the weak solution of the large-N NPRG equation forthe mass function which is the first derivative of the fermion potential with respect to thescalar bilinear-fermion field
We have constructed the weak solution by the method of characteristics. The set of
local strong solutions given by the characteristics is multi-valued and thus no longer is theglobal solution of the PDE. The weak solution can geometrically be constructed by thepatchwork of the local strong solutions using the equal area rule, which is derived by the
Rankine-Hugoniot condition. The uniqueness of the weak solution has been guaranteedby the entropy condition. Then we have obtained the $S\chi SB$ weak solution of the massfunction with a finite jump at the origin.
The method of weak solution has also been applied to the first order phase transitionat finite chemical potential. We have shown that in the RG procedure two jump appear
simultaneously, move toward each other, and finally merge into one. This RG evolution
is nothing but the first order phase transition. Finally we have discussed the convexityof the Legendre eﬀective potential of the chiral condensates which is constructed by the
weak solution of the fermion potential. The weak solution of the eﬀective potential, whichshows the first order phase transition in the RG procedure, is automatically convexised”
with only one correct minimum, while the eﬀective potential obtained by the mean-field
analysis has the multi local minima.
$\overline{\psi}\psi.$“

Acknowledgements
The authors greatly appreciate helpful lectures by Prof. Akitaka Matsumura who told us
how to construct the weak solution.

11

(1a)

(1b)

(3a)

(1c)

(3b)

(3c)

$00291$$000365$$\hat{tf)}$$\overline{\eta.}$$>^{\lrcorner}\vee\hat{\Leftrightarrow}0$$>^{\geq}\vee o_{-}x$$\backslash$:$C$$002*001$$000 \int_{02}$$g$$0001$$\theta$

(1a)

(1b)

(1c)

(3a)

(3b)

(3c)

002

Figure 6: RG evolution of physical quantities by weak solution with non-zero bare mass$(G_{0}=1.7G_{c}, m_{0}=0.01\Lambda_{0}, \mu=0.7, t=0.3,0.4,0.5,0.5615, \infty)$ . (a) Mass
function. (b) Fermion potential. (c) Legendre eﬀective potential. The thicksolid lines denote the weak solution, and the dashed lines denote the local strong
solutions dropped by the equal area rule. The thin solid line in (c) denotes theenvelope. Origins of -axis in (b) and (c) are arbitrary.
$y$

12

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[2] K.-I. Aoki and D. Sato, PTEP 2013, 043B04 (2013), 1212.0063.
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[8] J. Braun, J.Phys. G39, 033001 (2012), 1108.4449.
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[10] K.-I. Aoki, S.-I. Kumamoto, and D. Sato, in preparation (2014).

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