Regularization dependence on phase diagram in Nambu-Jona-Lasinio model

0.58 MB·English

by  T. Inagaki

Regularization dependence on phase diagram in Nambu–Jona-Lasinio model
T. Inagaki
Information Media Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8521, Japan
D. Kimura
General Education, Ube National College of Technology, Ube, Yamaguchi 755-8555, Japan
H. Kohyama
Department of Physics, National Taiwan University, Taipei, Taiwan 10617
(Dated: January 5, 2015)
We study the regularization dependence on meson properties and the phase diagram of quark
5 matterbyusingthetwoflavorNambu–Jona-Lasiniomodel. Wefindthatthemesonpropertiesand
1 thephasestructuredonotshowdrastically differencedependingtheregularization procedures. We
0 also find that the location or the existence of the critical end point highly depends on the regular-
2 ization methods and the model parameters. Then we think that regularization and parameters are
carefullyconsideredwhenoneinvestigatestheQCDcriticalendpointintheeffectivemodelstudies.
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I. INTRODUCTION
The phase structure of quark matter on finite temperature and density has actively been studied for decades [1].
Under usual condition, meaning low temperature and density, quarks are confined inside hadrons and they never be
able to observed as a single particle. On the other hand, due to the nature of the asymptotic freedom [2], quarks
andgluonscanbe freefromthe confinementathightemperatureanddensity,becausethe couplingstrengthbecomes
weak at high energy. It is, therefore,expected that quark matter undergoes the confined/deconfinedphase transition
at some temperature and density. This is important subject both in theoretical and experimental studies since it
crucially relates to the quark matter properties at relativistically high energy collisions and extremely dense stellar
objects such as neutron stars.
The first principle for quarks and gluons is Quantum Chromodynamics (QCD) which is a non-Abelian gauge field
theory for fermions. Our goals is to evaluate the phase structure based on this first principle QCD, however, it is
difficult to extract theoretical predictions due to the nature of complicated strongly interacting system. One of the
mostreliableapproachesistousethediscretisedversionofQCDcalledtheLatticeQCD(LQCD)inwhichtheoretical
calculation is performed on the discrete spacetime [3]. Although the LQCD works well at finite temperature T for
smallchemicalpotentialµ 0,thereis thetechnicaldifficulty calledthe“signproblem”whenonetriesto investigate
≃
the system at intermediate chemical potential. There effective models maybe nicely adopted because some models
can consistently treat the system at finite temperature and chemical potential.
Forthesakeofevaluatingthe phasestructureofquarkmatteratfinite temperatureandchemicalpotential,wewill
employ the Nambu–Jona-Lasinio (NJL) model [4] which is the most frequently used one in this context (there are a
lotofnicereviewpapersonthemodel,see,e.g.[5–9].) Themodelisconstructedbyincorporatingthefourpointquark
interactionintothe modelLagrangian,soitisnotrenormalizabledue tothis higherdimensionaloperator. Therefore,
the physical predictions of the model inevitably depend on the regularization procedure and the model parameters
chosen. The resulting phase diagram on the T µ plane is as well affected by the parameters and regularization
−
prescriptions. Soitis animportantissueto studywhether the phasestructure obtainedinoneregularizationmethod
is consistent with the ones from different regularizationmethods.
Inthispaper,wearegoingtostudythephasestructureofquarkmatterintheNJLmodelwithvariousregularization
ways,whicharethreedimensional(3D)momentumcutoff,fourdimensional(4D)momentumcutoff,Pauli-Villars(PV)
regularization,proper-time(PT)regularization,andthedimensionalregularization(DR).The3Dcutoffschemeisthe
most popular method in this model and a lot of works have been done in this way. The 4D cutoff method preserves
the Lorentzsymmetryinwhichspaceandtime aretreatedonequalfooting. The Pauli-Villarsregularizationis based
on the subtraction of the amplitude considering the virtually heavy particle to suppress the unphysical high energy
contribution coming from loop integrals [10–12]. The proper-time regularization makes integrals finite through the
exponentiallydumping factor[11,13]. Thedimensionalregularizationanalyticallycontinuesthe spacetimedimension
in the loop integralsto a non-integer value, then try to obtain finite contribution fromthe integrals [14]. Beside from
the frequently used3Dcutoff way,there havebeena lotofworksby using the 4D[6,7], PV[6,15–17], PT [6,18–26],
and DR [27–32]. The physical consequences depend on the regularization[33].
This paper is organized as follows; Section II introduces the model Lagrangian,and show the model treatment on
the meson properties and the explicit formalismat finite temperature and chemicalpotential. In Sec. III, we present
various regularization procedures, 3D, 4D, PV, PT and DR prescriptions with explicit equations. We then perform
the parameter fitting in Sec. IV. In Sec. V, the numerical results of the meson properties are shown. We then draw
the phase diagrams with several parameter sets using various regularization methods in Sec. VI. We also study the
phase diagram with the parameters fixed under the condition with the same constituent quark mass in Sec. VII.
In Sec. VIII, we give the discussions on the obtained results. Finally, we write the concluding remarks in Sec. IX.
Several detailed calculations are shown in Appendix.
II. TWO FLAVOR NJL MODEL
In this paper we consider two light quarks with equal mass. The model has SU (2) SU (2) flavor symmetry at
L R
⊗
the massless limit, m 0.
→
A. The Lagrangian and gap equation
The Lagrangianof the two flavor NJL model is given by
=ψ¯(i∂ mˆ)ψ+G (ψ¯ψ)2+(ψ¯iγ τaψ)2 , (1)
5
L 6 −
(cid:2) (cid:3)
3
where mˆ is the diagonal mass matrix mˆ = diag(m ,m ) and G is the effective coupling strength of the four point
u d
interaction. We set m =m in this paper. The application of the mean-field approximation
d u
σ
ψψ (2)
h i≃−2G
leads the following mean-field Lagrangian
σ2
˜=ψ¯(i∂ m∗)ψ , (3)
L 6 − − 4G
with the constituent mass m∗ = m +σ. The flavor symmetry is broken down, SU (2) SU (2) SU (2), by
u L R L+R
⊗ →
non-vanishing current quark mass, m , and dynamically generated σ. Thanks to the simple form of the Lagrangian,
u
one can easily evaluate the effective potential, = lnZ/V where Z is the partition function
eff
V −
Z = [ψ]exp i d4x ˜ (4)
D L
Z (cid:20) Z (cid:21)
and V is the volume of the system. After some algebra,we see
σ2 d4k
∗
(σ)= lndet(k m ). (5)
Veff 4G − i(2π)4 6 −
Z
The detailed derivation of the effective potential is presented in [9].
The gap equation is obtained through the extreme condition of the potential with respect to σ, namely,
∂
eff
V =0. (6)
∂σ
This condition leads the following gap equation
∗
σ =2N G itrS(m ), (7)
f
·
with the number of flavors N and
f
d4k 1
∗
itrS(m )= tr , (8)
− i(2π)4 k m∗+iε
Z 6 −
wheretracetakesthe spinorandcolorindices. This is the keyequationinthe modelbecauseit determinesthe values
of the chiral condensate ψψ and the constituent quark mass m∗.
h i
B. Meson properties
The propertiesofthe pionandsigmamesoncanbe studiedbasedonthe modelwith the determinedchiralconden-
sate. The interacting Lagrangianof the pion and quarks is written by
=ig ψ¯γ τ πψ, (9)
πqq πqq 5
L ·
where τ are 2 2 matrices in the flavor space and πi represent the pion fields. The explicit expression is τ π =
i
τ−π−+τ+π++×τ0π0, with τ± =(τ1 τ2)/√2 and τ0 =τ3 where τi are the Pauli matrices. ·
±
Byapplyingtherandomphaseapproximation,wecanwritethepionpropagatorasthesummationofthegeometrical
series of the one-loop diagram, which gives
g2 2G
∆ (p2)= πqq , (10)
π p2 m2 ≃ 1 2GΠπ(p2)
− π −
where Ππ is the following quark loop contribution
d4k
Ππ(p2)= 2 tr[γ S(k)γ S(k p)], (11)
− i(2π)4 5 5 −
Z
with the quark propagator
1
S(k)= . (12)
k m∗+iǫ
6 −
4
TheexplicitderivationofEq.(10)isdiscussedinthereviewpaper[6]. Thepionmassiscalculatedatthepoleposition
of the propagator,so the condition reads
1 2GΠπ(p2) =0. (13)
− p2=m2π
It shouldbe noted that the residue at the pol(cid:2)e p2 =m2 coin(cid:3)c(cid:12)ides with the squareof the coupling strengthg2 so we
π (cid:12) πqq
have the relation
∂Ππ −1
g2 = . (14)
πqq ∂p2
(cid:18) (cid:19) (cid:12)p2=m2π
(cid:12)
In the similar manner, the sigma meson mass is evaluated at(cid:12)the pole of the propagator,
(cid:12)
g2 2G
∆ (p2)= σqq , (15)
σ p2 m2 ≃ 1 2GΠσ(p2)
− σ −
with
d4k
Πσ(p2)= 2 tr[S(k)S(k p)]. (16)
− i(2π)4 −
Z
Therefore the condition which determines the sigma meson mass becomes
1 2GΠσ(p2) =0. (17)
− p2=m2σ
The pion decay constant is calculated from(cid:2)the following (cid:3)eq(cid:12)uation
(cid:12)
τ
iδ pµf = 0ψ¯ jγµγ ψ π . (18)
ij π 5 i
h | 2 | i
The explicit form becomes
1 d4k
pµf = tr[γµγ g S(k)γ S(k p)]. (19)
π 2 i(2π)4 5 πqq 5 −
Z
Thus the equations Eqs.(13), (17) and (19) are the ones whichdetermine the pionmass, sigmamass, andthe pion
decay constant.
C. Explicit formalism at finite temperature
Since our purpose is to study the phase structure on temperature T and chemical potential µ, we need to extend
the equations to finite temperature. According to the imaginary time formalism, the integral region of the temporal
component becomes finite due to the periodic or anti-periodic condition of fields as
β
Z = [ψ]exp dτ d3x ˜+µψ¯γ ψ . (20)
0
Z D "Z0 Z (cid:16)L (cid:17)#
where τ is imaginary time and β is the inverse temperature 1/T. Consequently, continuous integral in the temporal
direction is replaced by the following discrete summation,
∞
d4k d3k
F(k ,k) T F(iω +µ,k), (21)
i(2π)4 0 → (2π)3 n
Z n=−∞Z
X
where ω = 2nπT or (2n+1)πT depending on the statistical property of field, i.e., for bosons or fermions, and the
n
chemical potential seen in Eq. (20) appears in the way iω +µ. In this paper, we only treat fermionic quark loop
n
contributions then ω =(2n+1)πT is always the case.
n
With the help of the formalism Eq.(21), we see that the gap equation at finite temperature becomes
σ =2N G [trS0+trST], (22)
f
·
d3k 1
trS0 = N m∗ , (23)
− c (2π)32E
Z
d3k 1
trST =N m∗ f(E±) , (24)
c (2π)32E
Z " ± #
X
5
whereN is the number ofcolors,E =√k2+m∗2, E± =E µ andf(E)=1/(1+eβE). Itis importantto note that
c
the contributionscanbe expressedbythe summationofthe±T independentpart(trS0)andT dependentpart(trST).
This characteristic is general if one takes the infinite number of the frequency summation in finite temperature field
theory and crucial when we apply the regularization procedures to the appearing integrals.
Since the gap equation is derived by differentiating the effective potential with respect to σ, then the effective
potential can be obtained by integrating the gap equation (see, for example, [35]),
σ2 σ
′ ′
(σ)= N dσ itrS(m +σ ). (25)
f u
V 4G −
Z0
wherewehavedroppedthesuffixin andjustwritten fornotationalsimplicity. Thereaftertheeffectivepotential
eff
at finite temperature = σ+ 0+V T is evaluated as V
V V V V
σ2
σ = , (26)
V 4G
d3k
0 = 2N N E, (27)
V − f c (2π)3
Z
T = 2N N T d3k ln 1+e−βE± . (28)
V − f c (2π)3
Z X± h i
It is important to note that, if we apply some regularizations,the results between the direct calculationfrom Eq. (5)
and the one after integrating the gap equation may become different, because regularization essentially means the
subtraction and there are several ways of subtractions. Therefore, in this paper, we persistently use the latter way
shown in Eq. (25) so that the model treatment becomes consistent. It should be noticed that the finite temperature
correction, T, contains no divergent integral. A finite result can be obtained for the finite temperature correction
V
without applying any regularizations.
Next, we carry on the integral in the meson properties; the one loop contribution can be written as
2trS
Ππ(p2)= +p2I(p2), (29)
− m∗
with
d4k 1
I(p2)=4N . (30)
c i(2π)4(k2 m∗2)[(k p)2 m∗2]
Z − − −
Since trS is already evaluated above, the remaining task is to calculate I(=I0+IT), and it becomes
d3k 1
I0(p2)=4N , (31)
c (2π)3E(4E2 p2)
Z −
IT(p2)= 4N d3k ±f(E±) . (32)
− c (2π)3E(4E2 p2)
Z P −
Similarly, the one-loop diagram of the scalar channel can be written as
2trS
Πσ(p2)= +(p2 4m∗2)I(p2). (33)
− m∗ −
We now have already evaluated all the ingredients of Πσ above in Eqs. (23), (24), (31) and (32), so we do not need
further calculations.
Finally, let us derive the equation for the pion decay constant. After a bit of algebra we obtain the relation,
∗
f =g m I(0). (34)
π πqq
Here we evaluate f at p2 =0 following [6].
π
III. REGULARIZATION PROCEDURES
Since the integrals obtained in the previous section include infinities, we need to apply some regularizationso that
the model leads finite quantities. As mentioned in the introduction, the model is not renormalizable,then the model
predictions inevitably depend on regularization procedures chosen. Here, we shall present possible regularization
methods in this section.
6
A. Three dimensional cutoff scheme
The idea of the three dimensional (3D) cutoff is simple; one drops high frequency mode by introducing the cutoff
scale Λ into the integrals. We work in the 3-dimensional polar coordinate system and cut the radial coordinate as
3D
d4k dk Λ3Dk2dk
0
dΩ . (35)
(2π)4 → 2π (2π)3 3
Z Z Z0 Z
By performing the integrals, we have for the gap equation σ =2N GtrS,
f
N m∗ Λ + Λ2 +m∗2
trS0 = c Λ Λ2 +m∗2 m∗2ln 3D 3D , (36)
3D 2π2 3D 3D − pm∗ !
q
N m∗ Λ3D k2
trST = c dk f(E±) . (37)
3D − π2 Z0 E " ± #
X
The effective potential can also be calculated as
N N Λ Λ2 +m∗2
0 (σ)= c f Λ Λ2 +m∗2(2Λ2 +m∗2) m∗4ln 3D 3D , (38)
V3D − 8π2 " 3D 3D 3D − p m∗ #
q
T (σ)= NcNfT Λ3Ddk k2 ln(1+e−βE±) . (39)
V3D − π2 Z0 " ± #
X
The quark loop integral in the meson properties I(p2) reads
2N Λ3D k2
I0 = c dk , (40)
3D π2 E(4E2 p2)
Z0 −
2N Λ3D k2
IT = c dk f(E±) . (41)
3D − π2 Z0 E(4E2−p2)" ± #
X
Note that the integraldiverges around4E2 p2, and we apply the principal integralto avoidthis divergence [34]. It
may be worth mentioning that the integral≃can be performed analytically when p2 = 0 for T = 0, then one has for
the pion decay constant,
N m∗2 Λ Λ + Λ2 +m∗2
f2 = c 3D +ln 3D 3D . (42)
π3D 2π2 − Λ23D+m∗2 pm∗ !
We thus obtain the required quantities in evalupating the phase diagram and meson properties.
B. Four dimensional cutoff scheme
Inthe four dimensional(4D)cutoff regularizationscheme,we introducethe cutoff scaleΛ inthe Euclideanspace
4D
after performing the Wick rotation,
d4k Λ4Dk3dk
E E E dΩ . (43)
(2π)4 → (2π)4 4
Z Z0 Z
This is wellknownfour dimensionalcutoff method forT =0 case. As the naturalextensionto finite temperature, we
introduce the cutoff scale by
d4k L4 √Λ24D−ωn2 k2dk
E
T dΩ , (44)
(2π)4 → (2π)3 3
Z n=−XL4−1Z0 Z
where L is the maximum integer which does not exceed Λ /(2πT) 1/2.
4 4D
−
In the 4D cutoff way, it is difficult to divide the contribution into the temperature independent and dependent
parts, since there is also cutoff in the frequency summation.
7
The explicit form of trS and the effective potential become
N m∗T L4 √Λ24D−ωn2 1
trS = c dkk2 , (45)
4D 2π2 (ω−)2+E2
n=−XL4−1Z0 n
(σ)= NcNfT L4 √Λ24D−ωn2dkk2ln((ω−)2+E2), (46)
V4D − 4π2 n
n=−XL4−1Z0
where ω− =ω iµ.
n n−
For T =0, the integral can be performed analytically by using the Feynman parameter method,
N m∗ Λ2 +m∗2
trS0 = c Λ2 m∗2ln 4D , (47)
4D π2 4D− m∗2
(cid:20) (cid:18) (cid:19)(cid:21)
N N Λ2 +m∗2
0 (σ)= c f Λ2 m∗2 m∗4ln 4D +Λ∗4 ln(Λ2 +m∗2) . (48)
V4D − 8π2 4D − m∗2 4D 4D
(cid:20) (cid:21)
One should give the special attention in calculating I (p2), because the integral includes divergence to be cured as
4D
seen in the 3D cutoff case. The analytic expression of I0 (p2) will be given in appendix A
4D
Again we show the explicit form for the pion decay constant at T =0,
N m∗2 Λ2 Λ2 +m∗2
f2 = c 4D +ln 4D . (49)
π4D 4π2 −Λ2 +m∗2 m∗2
(cid:20) 4D (cid:18) (cid:19)(cid:21)
C. Pauli-Villars regularization
Inthis regularization,thedivergencesfromloopintegralsaresubtractedbyintroducingvirtuallyheavyparticlesas
1 1 a
i
. (50)
k2 m2 −→ k2 m2 − k2 Λ2
− − i − i
X
This manipulation induces virtual frictional force so that the contribution from unphysical high frequency mode is
suppressed.
In evaluation the gap equation, we apply the following subtraction
1 a a
1 2
, (51)
p2 m∗2 − p2 Λ2 − p2 Λ2
− − 1 − 2
where
m∗2 Λ2 Λ2 m∗2
a = − 2, a = 1− . (52)
1 Λ2 Λ2 2 Λ2 Λ2
1− 2 1− 2
By setting the cutoff scales Λ =Λ =Λ after the subtraction, we have
1 2 PV
N m∗ m∗2
trS0 = c Λ2 m∗2+m∗2ln , (53)
PV 4π2 PV− Λ∗2
(cid:18) PV(cid:19)
d3p f(E±) Λ2 m∗2 f(E±)
trST = 2N m∗ m 1+ PV− Λ . (54)
PV − c (2π)3 E − 2p2 E
Z (cid:20) m (cid:18) (cid:19) Λ (cid:21)
where E =√k2+m∗2 and E = k2+Λ2 . By integrating the above equation, we obtain the effective potential
m Λ PV
N N p 3 1 m∗2
0 = c f Λ2 m∗2 m∗4+ m∗4ln , (55)
VPV − 8π2 PV − 4 2 Λ2
(cid:20) PV(cid:21)
T = NcNfT dk k2ln(1+e−βEm±) m2(4k2+2Λ2 m∗2)f(EΛ±) . (56)
VPV − π2 − 8 − E
X± Z h Λ i
8
Since the divergence coming from the integral I(p2) is order of log, one subtractionis enough to make it finite, so we
get
2N 1 1
I0 = c dkk2 , (57)
PV π2 E (4E2 p2) − E (4E2 p2)
Z (cid:20) m m− Λ Λ− (cid:21)
IT = 2Nc dkk2 ±f(Em±) ±f(EΛ±) . (58)
PV − π2 Z "EmP(4Em2 −p2) − EPΛ(4EΛ2 −p2)#
The pion decay constant at T =0 becomes
N m∗2 Λ2 Λ2
f2 = c 1+ PV ln PV . (59)
πPV 4π2 − Λ2 m∗2 m∗2
(cid:18) PV− (cid:19)
D. Proper-time regularization
The basic idea of the proper-time regularization is based on the following manipulation of the Gamma function,
∞
1 1
dτ τn−1e−Aτ, (60)
An → Γ[n]Z1/Λ2PT
where the lower cut 1/Λ2 induces the dumping factor into the original propagator since, for example with n=1,
PT
∞
1 dτe−Aτ = 1 e−(kE2+m∗2)/Λ2PT. (61)
kE2 +m∗2 →Z1/Λ2PT kE2 +m∗2
Thereforeinthisregularizationhighfrequencycontributionisdumpedbythefactore−kE2/Λ2PT,sotheoriginaldivergent
integral turns out to be finite. For A contains a imaginary part, Eq. (60) is modified as,
1 in ∞
dτ τn−1e−iAτ, (Im(A)<0,Re(n)>0). (62)
An → Γ[n]
Z+0
Under this procedure, the integral of trS in the gap equation becomes
∗
trS0 = Ncm Λ2 e−m∗2/Λ2PT +m∗2Ei( m∗2/Λ2 ) , (63)
PT 4π2 PT − PT
trS = Ncm∗Th ∞ ∞ dτ e−iπ/4e−i (ωn−)2+m∗2 τi+c.c. . (64)
PT 2π3/2 τ3/2 { }
nX=0Z+0 h i
where Ei( x) is the exponential-integral function. For m∗2 Λ2 , Eq. (63) is expanded as
− ≪ PT
N m∗ m∗2 m∗2
trS0 c Λ2 m∗2+m∗2 ln +γ . (65)
PT ≃ 4π2 PT− Λ2 E − 2Λ2
(cid:20) (cid:18) PT PT(cid:19)(cid:21)
We rotate the contourofthe integrationin Eq. (64) to the imaginaryaxis of τ [23–25]. For ω2 µ2+m∗2 >0, the
0−
trace becomes
trS = Ncm∗T ∞ ∞ dτ cos(2ω µτ)e−(ωn2−µ2+m∗2)τ, (66)
PT π3/2 n=0Z1/Λ2PT τ3/2 n
X
and for ω2 µ2+m∗2 <0,
0−
trS = Ncm∗T ∞ ∞ dτ cos(2ω µτ)e−(ωn2−µ2+m∗2)τ
PT π3/2 nX>[N]Z1/Λ2PT τ3/2 n
[N] ∞
dτ sin(2ω µτ)e(ωn2−µ2+m∗2)τ
−n=0(Z1/Λ2PT τ3/2 n
X
π/2
Λ Re eiπ/4 dθe−iθ/2exp i (ω−)2+m∗2 eiθ/Λ2 , (67)
− PT Z−π/2 − { n } PT !)#
(cid:2) (cid:3)
9
where N = µ2 m∗2/(πT) 1 /2 . Similarly, the effective potential can be calculated through
{ − − }
p (σ)= NcNfT ∞ ∞ dτ e−3iπ/4e−i (ωn−)2+m∗2 τ +c.c. . (68)
VPT 4π3/2 τ5/2 { }
nX=0Z+0 h i
For ω2 µ2+m∗2 >0, one has
0−
∞ ∞
(σ)= NcNfT dτ cos(2ω µτ)e−(ωn2−µ2+m∗2)τ, (69)
VPT 2π3/2 n=0Z1/Λ2PT τ5/2 n
X
and for ω2 µ2+m∗2 <0,
0−
∞ ∞
(σ)= NcNfT dτ cos(2ω µτ)e−(ωn2−µ2+m∗2)τ
VPT 2π3/2 nX>[N]Z1/Λ2PT τ5/2 n
[N] ∞
+ dτ sin(2ω µτ)e(ωn2−µ2+m∗2)τ
n=0(Z1/Λ2PT τ5/2 n
X
π/2
+ Λ3Re e−iπ/4 dθe−3iθ/2exp i (ω−)2+m∗2 eiθ/Λ2 . (70)
Z−π/2 − { n } PT !)#
(cid:2) (cid:3)
I can also be calculated by
PT
I (p2)= NcT ∞ 1/2dα ∞ dτ e−3iπ/4e−i (ωn−)2+∆ τ +c.c. . (71)
PT −π3/2 τ1/2 { }
nX=0Z0 Z+0 h i
where α is the Feynman integration parameter and ∆=m∗2 p2/4+α2p2. Then the integral can be written
−
I (p2)= 2NcT 1/2dα ∞ θ(W (α)) ∞ dτ cos(2ω µτ)e−(ωn2−µ2+∆)τ
PT π3/2 Z0 n=0" n Z1/Λ2PT τ1/2 n
X
∞
+θ( W (α)) dτ sin(2ω µτ)e(ωn2−µ2+∆)τ
− n (Z1/Λ2PT τ1/2 n
1 π/2
Re e−iπ/4 dθeiθ/2exp i (ω−)2+∆ eiθ/Λ2 , (72)
− Λ Z−π/2 − { n } PT !)#
(cid:2) (cid:3)
where W (α)=ω2 +m∗2+(α2 1/4)p2 µ2.
n n − −
The pion decay constant at T =0 reads the following simple form,
N m∗2
f2 = c Ei( m∗2/Λ2 ). (73)
πPT − 4π2 − PT
For m∗2 Λ2 , we have
≪ PT
N m∗2 m∗2 Λ2
f2 c γ + +ln PT . (74)
πPT ≃ 4π2 − E Λ2 m∗2
(cid:26) PT (cid:27)
E. Dimensional regularization
Inthedimensionalregularizationmethod,weobtainfinitequantitiesthroughanalyticallycontinuingthedimension
in the loop integral to a non-integer value, D, as
d4k dDk
M4−D , (75)
(2π)4 → 0 (2π)D
Z Z
where the scale parameter M is inserted so as to adjust the mass dimension of physical quantities. The method is
0
well known since it preserves most of symmetries. Note that this result is the same as the result obtained from the
proper-time integral (0<τ < ) and expressed by the poles of the Gamma function.
∞
10
The trace trS in the gap equation reads
N M4−Dm∗ D
trS0 = − c 0 Γ 1 (m∗2)D/2−1, (76)
DR (2π)D/2 − 2
(cid:18) (cid:19)
kD−2
trST = A m∗ dk f(E±) , (77)
DR − D Z 2E " ± #
X
where
N 22−D/2M4−D
A = c 0 . (78)
D π(D−1)/2Γ((D 1)/2)
−
The effective potential becomes
N N M4−D D
0 = c f 0 Γ (m∗2)D/2, (79)
VDR 2(2π)D/2 −2
(cid:16) (cid:17)
T = A N T dk kD−2 ln 1+e−βE± . (80)
VDR − D f
Z X± h i
In the similar manner, the integral I(p2) appearing in the meson propagatoris calculated as
kD−2
I0 =A dk , (81)
DR D E(4E2 p2)
Z −
kD−2
IT =A dk − f(E±) . (82)
DR D E(4E2 p2)
Z − " ± #
X
Note we need to perform the principal integration for m∗2 <p2/4.
The pion decay constant at T =0 reads the following simple form,
N M4−D D
f2 = c 0 Γ 2 (m∗2)D/2−1. (83)
πDR (2π)D/2 − 2
(cid:18) (cid:19)
We show a concrete example of trS0 and f2 for D 2,3,4 in appendix B.
DR πDR ≃
IV. MODEL PARAMETERS
Havingobtainedtheequationwhichdeterminesthepionmassandpiondecayconstant,wearenowreadytoperform
the parameter fitting. In the previous section we suppose that all the cutoff scales are equal in each regularization
to reduce the parameters. Thus the model has three parameters: the cutoff scale Λ, the current quark mass m and
u
the coupling strength G. Whereas in the DR there appears one more parameter, so the total number becomes four:
the currentmassm , dimensionD, scaleparameterM andthe coupling G, asdiscussed inRef. [36]. In this section,
u 0
we shall set the model parameters by fitting the pion mass and decay constant. The actual values we use are shown
below
m =135MeV, f =94MeV. (84)
π π
Forthe casewiththeDR,weperformfittingwithonemorequantity,the neutralpiondecayconstanttotwophotons,
which will be discussed later.
A. Parameters in various regularizations
Here we align the model parameters in various regularizations in this subsection.
Table I, II, III and IV show how the parameters change according to the current quark mass m , where we first
u
set the value of m then search the parameters Λ and G which lead m = 135MeV and f = 94MeV. One sees the
u π π