Non-perturbative approach to the effective potential of the $λφ^{4} theory at finite temperature

14 Pages·0.19 MB·English

by  Tomohiro Inagaki

Non-perturbative approach to the effective potential of the λφ4
theory at finite temperature
Tomohiro Inagakia∗, Kenzo Ogureb†and Joe Satoc‡
a Department of Physics, Kobe University,
8
9 Rokkoudai, Nada, Kobe 657, Japan
9 bInstitute for Cosmic Ray Research, University of Tokyo,
1
Midori-cho, Tanashi, Tokyo 188-8502, Japan

n c Department of Physics, University of Tokyo,
a
J Bunkyo-ku, Hongo, Tokyo 133-0033, Japan

9
2 February 1, 2008

3
v
3
Abstract
3
1 We construct a non-perturbative method to calculate the effective potential of the λφ4 theory
5 at finite temperature. We express the derivative of the effective potential with respect to the mass
0
square in terms of the full propagator. We reduce it to the partial differential equation for the
7
effectivepotentialusinganapproximation. Wenumericallysolveitandobtaintheeffectivepotential
9
non-perturbatively. Wefind that thephase transition is second order as it should be. Wedetermine
/
h several critical exponents.
t
-
p PACS numbers: 05.70.Fh; 11.10.Wx; 11.15.Tk; 11.30.Qc
e Keywords: λφ4 theory; finitetemperature field theory; effectivepotential
h
:
v 1 Introduction
i
X
r It is often expected that broken symmetries are restored at high temperature [1]. The temperature-
a
induced phase transition will be observed in relativistic heavy ion collisions, interior of neutron stars,
and the early stage of the universe. We may probe new physics through the phase transition at high
temperature.
It is, however, very difficult to examine the phase transition. For example, the perturbation theory
oftenbreaksdownathightemperature. Asiswell-knowninfinitetemperaturefieldtheorieshigherorder
contributions of the loop expansion are enhanced for Bose fields by many interactions in the thermal
bath [2, 3]. In the λφ4 theory physical quantities are expanded in terms of λT2/m2 and λT/m at
finite temperature. The ordinary loop expansion is improved by resumming the daisy diagram which
includes all the higher order contributions of O λT2/m2 n [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] The loop
expansionparameterisλT/maftertheresummat(cid:16)i(cid:0)on. Itme(cid:1)an(cid:17)sthattheperturbationtheorybreaksdown
atT > m/λ[9]. Aroundthe criticaltemperature the ratiom/T is alwaysofO(λ) soa non-perturbative
∼
analysis is necessary to study the phase transition in λφ4 theory[5].
A variety of methods is used to investigate the phase transition, for example, lattice simulation
[14, 15, 16, 17, 18, 19], C.J.T. method [20, 21], ε-expansion [22], effective three dimensional theory
∗e-mailaddress: [email protected]
†e-mailaddress: [email protected]
‡e-mailaddress: [email protected]
1
[23, 24, 25, 26, 27], gap equation method [28], non-purturbative renormalization group method[29, 30,
31,32,33]andso on. All the same westill needanothermethodto study the phasetransitionsince they
are applicable to limited situations.
In Ref.[34] a new non-perturbative approach was suggested to avoid the infrared divergence which
appearsinthepressure[35]. Theydifferentiatedthegeneratingfunctionalwithrespecttothemasssquare
and found the infrared finite expression for the pressure in thermal equilibrium.
InthepresentpaperweemploytheideaofRef.[34]anddevelopanewmethodtocalculatetheeffective
potential. Differentiatingtheeffectivepotentialwithrespecttothemasssquare,weexpressthederivative
intermsofthefullpropagator. Weconstructthepartialdifferentialequationfortheeffectivepotentialby
approximating the full propagator. We calculate the effective potential beyond the perturbation theory
by solving this equation.
In section 2 we consider the λφ4 theory at finite temperature and show the exact expression of the
∂V
derivative of the effective potential . We approximate it and obtain the partial differential equation
∂m2
for the effective potential. We give the reasonable initial condition to solve this equation. In Sec.3 we
solve it and get the effective potential numerically. We obtain the susceptibility, field expectation value,
and specific heat from it. We determine the several critical exponents by observing their behaviours as
T varies. The Sec.4 is devoted to the concluding remarks.
2 Evolution equation for the effective potential
As mentioned in Sec.1, the loop expansion loses its validity at high temperature. We need a non-
perturbative method to calculate the effective potential. The effective potential, in general, satisfies the
following relation,
m2 ∂V
V(m2)= dm2+V(M2). (1)
ZM2 (cid:18)∂m2(cid:19)
∂V
Once we know and V(M2), we cancalculate the effective potential for arbitrarym2. Following the
∂m2
idea,weconstructanevolutionequationfortheeffectivepotentialoftheλφ4 theoryatfinitetemperature.
∂V
In the following we give and an appropriate initial condition V(M2).
∂m2
We consider the λφ4 theory which is defined by the Lagrangiandensity
1 ∂φ 2 1 1 λ
L =− − (∇φ)2− m2φ2− φ4+L +Jφ, (2)
E ct
2 ∂τ 2 2 4!
(cid:18) (cid:19)
where L represents the counter term and J is an external source function. If m2 is negative, the scalar
ct
fieldφdevelopsthenon-vanishingfieldexpectationvalueatT =0. Itisexpectedthatthefieldexpectation
value decreases as T increases and the phase transition takes place at the critical temperature T . We
c
can explore properties of this phase transition by studying the effective potential at finite temperature.
Following the standard procedure of dealing with the Matsubara Green function [36], we introduce
the temperature to the theory. The generating functional at finite temperature is given by
1/T
Z = D[φ]exp dτ d3xL . (3)
T E
Z Z0 Z !
∂V
In the λφ4 theory the derivative of the effective potential is expressedby the full propagatorof the
∂m2
scalar field (See Appendix A),
∂V ∂V ∂V ∂V ∂V
tree 1 2 ct
= + + + . (4)
∂m2 ∂m2 ∂m2 ∂m2 ∂m2
2
V is the tree part,
tree
∂V 1
tree ≡ φ¯2. (5)
∂m2 2
The non-perturbative effects are contained in V and V ,
1 2
∂V 1 +i∞+ǫ d3p 1 1
1
≡ dp (6)
∂m2 2πiZ−i∞+ǫ 0Z (2π)3−p20+p2+m2+ λ2φ¯2+Πep0/T −1
∂V 1 +i∞ d3p 1
2
≡ dp . (7)
∂m2 4πiZ−i∞ 0Z (2π)3−p20+p2+m2+ λ2φ¯2+Π
V is the counter term part,
ct
∂V 1
ct ≡ (Z Z −1) φ¯2
∂m2 m φ 2
(cid:20)
1 +i∞+ǫ d3p 1 1
+ dp
2πiZ−i∞+ǫ 0Z (2π)3−p20+p2+m2+ λ2φ¯2+Πep0/T −1
1 +i∞ d3p 1
+ dp . (8)
4πiZ−i∞ 0Z (2π)3−p20+p2+m2+ λ2φ¯2+Π(cid:21)
∂V
Here Π=Π(p2,−p2,φ¯,m2,T) describes the full self-energy. The third term 2 on the right hand side
0 ∂m2
ofEq.(4)isdivergent. Thisdivergenceisremovedbythecounterterm(8)aftertheusualrenormalization
procedure is adopted at T =0. The counter term which is determined at T =0 removes the ultra-violet
divergence even in finite temperature [7, 12, 37].
We give the initial condition at M2 ∼ O(T2) where the loop expansion is valid, (λT/M ∼ λ ≪ 1).
WecalculateV(M2)bytheperturbationtheoryuptotheonelooporder. Aftertherenormalizationwith
MS scheme at the renormalization scale µ¯, the one-loop effective potential is
V(M2)=V (M2)+V (M2)+V (M2)+V (M2), (9)
tree 1 2 ct
where V ,V and V +V are given by
tree 1 2 ct
1 λ
V (M2)= M2φ¯2+ φ¯4, (10)
tree
2 4!
∞
T 1 λ
V (M2)= drr2log 1−exp − r2+M2+ φ¯2 , (11)
1 2π2 T 2
Z0 " r !#
M2+ λφ¯2 2 M2+ λφ¯2 3
V (M2)+V (M2)= 2 log 2 − . (12)
2 ct 64π2 µ¯2 2
(cid:0) (cid:1) " ! #
Note that we need not resum the daisy diagram which has only a negligible contribution of O(λ) for
M2 ∼O(T2).
In order to investigate the temperature-induced phase transition we consider the theory with non-
vanishing field expectation value at T = 0 (i.e. m2 takes a negative value, m2 = −µ2). We calculate
V(−µ2) with the effective potential (9) by
−µ2 ∂V ∂V ∂V ∂V
V(−µ2) = tree + 1 + 2 + ct dm2
ZM2 (cid:18) ∂m2 ∂m2 ∂m2 ∂m2(cid:19)
+V (M2)+V (M2)+V (M2)+V (M2). (13)
tree 1 2 ct
∂V
For m2 ≪T2 the contribution from 1 is enhanced by the Bose factor. The contribution from V can
∂m2 1
be the same order as that from the tree part around the critical temperature.
3
The quantity V +V will have a negligible contribution:
2 ct
−µ2 ∂V ∂V
2 + ct dm2+V (M2)+V (M2)
ZM2 (cid:18)∂m2 ∂m2(cid:19) 2 ct
=V (−µ2)+V (−µ2). (14)
2 ct
We can show that V (−µ2)+V (−µ2) is really small at the leading order of the loop expansion. At one
2 ct
loop level with daisy diagram resummation we find
V (−µ2) + V (−µ2)
2 ct
−µ2+ λφ¯2+Π 2 −µ2+ λφ¯2+Π 3
= 2 log 2 − . (15)
64π2 µ¯2 2
(cid:0) (cid:1) " ! #
The self-energy satisfies Π∼µ2 aroundthe critical temperature for the second-orderor the weakly first-
order phase transition [5]. Because we are interested in the effective potential at small φ¯ region only to
investigate the phase structure, we neglect (14) in the following calculations.
Furthermore, we ignore the momentum dependence of the self-energy Π and generate the two point
function from the effective potential V. We replace as follows in Eq.(4),
λ ∂2V
m2+ φ¯2+Π(0,0,φ¯,m2,T)→ . (16)
2 ∂φ¯2
We obtain the partial differential equation for the effective potential by integrating over p and angle
0
variables in Eq.(4),
∞
∂V 1 1 1 1
= φ¯2+ drr2 . (17)
∂m2 2 4π2
Z0 ∂2V 1 ∂2V
r2+ exp r2+ −1
s ∂φ¯2 Ts ∂φ¯2!
3 Numerical results
We calculate the effective potential by solving the partial differential equation (17) with the initial con-
dition V +V in Eq.(9). We solve the equation numerically and show the phase structure of λφ4
tree 1
theory.
3.1 analytic continuation
∂2V
The integral in (17) is well defined in the region where is real and positive. The effective potential
∂φ¯2
V(φ¯) is, however, complex for small φ¯ below the critical temperature, T < T . We have to find the
c
analytic continuation in order to calculate the effective potential there.
To make the analytic continuation, we change the variable of integration r to z through
r2
z = +Z2−Z, (18)
T2
r
and rewrite the differential equation (17),
∂V 1 T2 ∞ z(z+2Z)
= φ¯2+ dz . (19)
∂m2 2 4π2 ez+Z −1
Z0 p
1 ∂2V
Here Z is the double valued function which is given by Z = .
sT2 ∂φ¯2
4
The imaginary part of the effective potential is interpreted as a decay rate of the unstable state [38].
∂V
It is natural that we assume such an imaginary part is negative. The imaginary part of should
∂m2
be positive in order that the imaginary part of the effective potential may be negative. We have to
1 ∂2V ∂V
select the branch of Z = so that imaginary part of will be positive. We calculate the
sT2 ∂φ¯2 ∂m2
effectivepotentialinthisbranchandthe imaginarypartofitisalwaysnegativeaswewillseeinthenext
subsection.
3.2 numerical result
Putting the initial condition V +V in Eq.(9) at M2 = T2 we numerically solve Eq.(19) and obtain
tree 1
the effective potential at m2 =−µ2. We use the explicit differencing method [39]. In this subsection we
show the effective potential and calculate critical exponents.
We illustrate the behaviour of the effective potential at λ = 1 in Fig.1 (a). The field expectation
value φ is the minimum point of the effective potential. It seems to disappear smoothly at the critical
c
temperature. We find that the phase transition is second order as it should be.
For comparison, in Figs.1 (b), (b’) and (c) we show the effective potential calculated by the per-
turbation theory at one and two loop order with daisy diagram resummation.1At the one loop order
an extremely small gap appears at the critical temperature as is clearly seen in Fig.1 (b’). The phase
transition is first order at the one loop order.
Thissituationismodifiedatthetwolooporder. Weobservenogapandfindthatthephasetransition
is second order as shown in Fig.1 (c). Though Fig.1 (a) and Fig.1 (c) show the similar behaviour, it will
be accident. The effective potential calculated up to the two loop order includes the contribution from
thegraphs,Fig.2(a)andFig.2(b),withdaisyresummation. Ontheotherhandwecantakeintoaccount
thecontributionfromalltheothergraphsinadditiontoFig.2(a)andFig.2(b)withintheapproximation
(16) by solving Eq.(19) automatically. The Fig.1 (a) accidentally coincides with Fig.1 (c).
ForT <T theeffectivepotentialdevelopsanon-vanishingimaginarypartatsmallφ¯range. Weshow
c
it in Fig.3. It should be noted that the sign of the imaginary part is always negative. It is consistent
with the discussion in the previous subsection.
Evaluating the effective potential with varying the temperature, T, and the coupling constant, λ, we
obtainthe criticaltemperature as a function ofλ where the field expectationvalue disappears. We show
the phase boundary on T-λ plain in Fig.4.
The critical exponents are defined for the second-orderphase transition. Around the critical temper-
ature we expect that the susceptibility χ, the expectation value φ , and the specific heat C behave as
c
[40]
χ∝|t|−γ,φ ∝|t|+β,C ∝|t|−α, (20)
c
where t = (T −T )/T. Analysing the effective potential more precisely we can calculate the critical
c
exponents γ,β, and α. The susceptibility χ satisfies the following relation,
ξ ∝ρ−1, (21)
where ρ is the curvature of the effective potential at φ .
c
Since thespecificheatC isgivenbythe secondderivativeofthe effectivepotentialaroundthecritical
temperature, the effective potential V(φ ) behaves as
c
V(φ )∝|t|2−α. (22)
c
We examine the behaviour of ρ,φ ,and V(φ ) around the critical temperature and find the critical
c c
exponents γ,β and α. In Fig.5 the critical behaviour of ρ,φ and V(φ ) are shown as a function of
c c
the temperature. We numerically calculate the critical exponents from them. Our numerical results
5
0.004 0.004
T =5.1µ
0.002 T =5.1µ 0.002
T =5.05µ
V(φ) V(φ)
ℜ 0 ℜ 0
µ4 T =5.05µ µ4 T =5.0µ
T =5.0µ
-0.002 -0.002
T =4.95µ
T =4.95µ
-0.004 -0.004
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
φ/µ φ/µ
(a) non-perturbative method (b) perturbation at 1-loop level
1.5×10−5
T =4.9957µ
0.004
1×10−5
0.002 T =5.1µ
5×10−6
V(φ)
ℜ 0
V(φ) µ4 T =5.05µ
ℜ 0 T =4.9952µ
µ4
-0.002
T =5.0µ
−5×10−6 T =4.95µ
T =4.9947µ
-0.004
−1×10−5
0 0.1 0.2 0.3 0.4
0 0.2 0.4 0.6 0.8 1
φ/µ
φ/µ
(b’) perturbation at 1-loop level around the
(c) perturbation at 2-loop level
critical temperature
Figure 1: The behaviour of the effective potential V is shown for fixed λ(= 1) as the function of the
temperature. We find no qualitative change for other values λ(= 0.5,0.1,0.05). We normalise that
V(0)=0.
6
(a) (b)
Figure 2: Two loop diagrams that contribute to the effective potential.
ℑV(φ)/µ4
0
-0.0001
-0.0002 T =4.95µ
-0.0003
T =5.0µ
-0.0004
-0.0005
0 0.2 0.4 0.6 0.8 1
φ/µ
Figure 3: Imaginary part of the effective potential near the critical temperature
7
70
60
50 φc =0
40
T /µ
c
30
20
10
φ 6=0
c
0
0 0.2 0.4 0.6 0.8 1
λ
Figure 4: Phase boundary
Table 1: Critical exponents
our results Landau theory experimental results [41]
β ∼0.5 0.5 0.33
γ ∼1 1 1.24
α ∼0 0 0.11
are presented in the table 1.2 The critical exponents within our approximation are independent of the
couplingconstantλ. Wenotethatourresultsdescribedinthepresentsubsectionremainunchangedeven
when we put the initial mass scale M2 =T2/4 or M2 =4T2.
4 Conclusion
We constructed the non-perturbative method to investigate the phase structure of λφ4 theory. The
derivativeofthe effective potentialwithrespecttomasssquarewasexactlyexpressedintermsofthe full
propagator at finite temperature. We found the partial differential equation for the effective potential
with the replacement (16). We gave the initial condition by the 1 loop effective potential in the range
where the perturbation theory is reliable. We numerically solved the partial differential equation and
obtained the effective potential.
Though we made the approximation (16), we could find that the phase transition of λφ4 theory is
secondorderas it shouldbe. Our method is veryinteresting becauseit canshowthe correctorderofthe
phase transition. The approximation (16) may be fairy good.
We determined several critical exponents which roughly agree with those of Landau approximation.
Theyare,however,roughvaluesbecauseitisverydifficulttosolvethenonlinearpartialdifferentialequa-
tion (17) numerically. We need elaborate a numerical study to obtain more accurate critical exponents.
1 Weusetheequations inRef[5]todrawthem.
2 Due to the instability in the explicit differencing method, we can not see the fine structure of the effective potential
andcangetonlytheroughvalues ofthecriticalexponents. Weneedfurthernumericalstudytogetmoreprecisevalues.
8
φ /µ ρ/µ
c
1 0.25
0.8 0.2
0.6 0.15
0.4 0.1
0.2 0.05
0 0
4.8 4.9 5 5.1 5.2 4.8 4.9 5 5.1 5.2
T/µ T/µ
(a) Field expectation value φ (b) curvature ρ at the minimum
c
V(φ )/µ4
c
0
-0.005
-0.01
-0.015
-0.02
-0.025
4.8 4.9 5 5.1 5.2
T/µ
(c) Minimum of the effective potential V(φ )
c
Figure 5: Critical behaviour of φ , ρ and V(φ )
c c
9
The main problem of our non-perturbative method is how to improve the approximation to the
full propagator. We cannot estimate the error from the approximation (16). We need improve the
approximation to the full propagator in order to know the correction to the current result.
Ourmethodis verypromisingsinceitcanprobethe regionwherethe traditionalperturbationtheory
breaks down.
Acknowledgements
The authors would like to thank Akira Niegawa and Jiro Arafune for useful discussions.
A The derivative of the effective potential in terms of the full
propagator
∂V
Thederivativeoftheeffectivepotential canbe representedbythe fullpropagator. Inthisappendix
∂m2
∂V
we present details of the calculation of given in Eq.(4).
∂m2
We consider the Lagrangiandensity which is defined by
2
1 ∂φ 1 1 λ
L =− 0 − (∇φ )2− m2φ2− 0φ4+J φ , (23)
E 2 ∂τ 2 0 2 0 0 4! 0 0 0
(cid:18) (cid:19)
where the suffix 0 denotes the bare quantities.
We adopt the mass-independent renormalization procedure and represent the effective potential as a
function of renormalized quantities. The renormalization constants Z and renormalized quantities are
introduced through transformations
φ = Z1/2φ,
0 φ
m = Z1/2m,
0 m (24)
λ = Z λ,
0 λ
J = Z−1/2J.
0 φ
Using these renormalization constants and renormalized quantities, we separate the Lagrangian density
(23) into the tree part L and the counter term part L as [40, 42]
1 ct
L =L [φ]+L [φ]+(J +J )φ, (25)
E 1 ct 1 ct
where J +J ≡J. The laglangian density L and L are given by
1 ct 1 ct
2
1 ∂φ 1 1 λ
L [φ] ≡ − − (∇φ)2− m2φ2− φ4,
1
2 ∂τ 2 2 4!
 (cid:18) (cid:19)
2
 Lct[φ] ≡ −12(Zφ−1)"(cid:18)∂∂φτ(cid:19) +(∇φ)2#− 21(ZmZφ−1)m2φ2 (26)
λ
− (Z Z2−1)φ4.
Here we separatethe external sour4c!e Jλinφto J and J , which satisfy the following equations:
1 ct
∂L
1
+J = 0,
1
∂φ (27)
 (cid:12)φ=φ¯
 (cid:12)(cid:12) hφiJ = φ¯.
(cid:12)
We expand the field φ(x) around the classical background φ¯,
φ(x)=φ¯+η(x), (28)
10