Relativistic Bose gases at finite density

0.18 MB·

by  Jens O. Andersen

Relativistic Bose gases at finite density
Jens O. Andersen
Nordita,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
(Dated: February 2, 2008)
We consider a massive relativistic Bose gas with N complex scalars at finite density. At zero
temperature, we calculate the pressure, charge density and the speed of sound in the one-loop
approximation. In the nonrelativistic limit, we obtain the classic results for the dilute Bose gas.
We also discuss finite-temperature effects. In particular, we consider the problem of calculating
the critical temperature for Bose-Einstein condensation. Dimensional reduction and effective-field-
theory methods are used to perturbatively calculate the effects of the nonstatic Matsubara modes.
5
CalculationsofT intheeffective3dtheoryrequirenonperturbativemethods. UsingtheMonteCarlo
c
0
simulationsofX.Sun[Phys. Rev. E67,066702 (2003)]andtheseven-loopvariationalperturbation
0
theory (VPT) calculations of B. Kastening [Phys. Rev. A70, 043621 (2004)], we obtain T for
2 c
N =2 to second order in theinteraction.

n
a PACSnumbers:
J

3
I. INTRODUCTION critical density n calculated from Eq. (1) is
1 c

mT 3/2
2 The realization of Bose-Einstein condensation (BEC) of n =ζ 3 . (2)
v trappedalkaliatomsalmosttenyearsagohascreatedanenor- c 2 2π
4 mousinterestinthepropertiesoftheweaklyinteractingBose Inverting this equation, on(cid:0)e (cid:1)o(cid:16)btains(cid:17)the well-known result
9 gas [1, 2, 3]. The temperature at which these systems Bose T =2π/m n/ζ 3 2/3 . Similarly, for m 0, one finds
0 condense is of the order 10−5 Kelvin, which is many orders c 2 →
01 othfemoangsneittuodfesuhpiegrhfleuridthitayninth4eHceon(2d.e1n7sKa)t.ioBnottehmtpraeprapteudrealaknaldi (cid:2) (cid:0) (cid:1)(cid:3)T = 3n 1/2 . (3)
5 gases and 4Heare examples of nonrelativistic systems. c m
0 (cid:16) (cid:17)
BEC in relativistic Bose systems typically takes place in Thus, in the ultrarelativistic limit m = 0, T is infinite, or
/ c
h matter under extreme conditions. For example, kaons may equivalently, the critical charge density is zero. All charge
p condense in the color-flavor locked phase of high-density resides in theground state irrespective of thetemperature.
- QCD [4, 5, 6]. This phase is a superconducting phase of In the context of nonrelativistic field theory, the problem
p
QCDwhicharisesfromaninstabilityoftheFermisurface;In of calculating the transition temperature for Bose-Einstein
e
h analogy with ordinary BCS-theory, a weak attraction among condensation with aweak interaction has averylong history
: the quarks in one or more channels results in the formation and conflicting results have appeared in the literature [3]. A
v of Cooper pairs and the spontaneous breakdown of the color first-orderperturbativecalculation gives nocorrection tothe
Xi symmetry of QCD [7, 8, 9]. Such a phase may be found ideal-gas result, while higher-order calculations are plagued
in the interior of compact stars if the density is sufficiently with infrared divergences. This is a typical example of in-
ar high. The linear SU(2)L×SU(2)R-symmetric sigma model frared divergences that arise in thevicinity of asecond-order
at finite chemical potential µ for the hypercharge is used as phasetransition. Thelong-distancephysicsinthecritical re-
a toy model for the description of kaon condensation in the gionisnonperturbativeandonehastosumupaninfiniteset
color-flavor locked phase of QCD [10, 11]. ofdiagramstoobtainafiniteresult. Theproblemwassolved
Calculation of the critical temperature or the critical den- only recently by Baym et al. [13] who realized that it can
sity for Bose-Einstein condensation of an ideal Bose gas be reduced to nonperturbative calculations using a classical
has been a standard text-book calculation for a number of three-dimensional field theory. Once it was understood how
years [12]. The charge density n of excited bosons as a func- to organize the problem, the calculation of T has later been
c
tion of temperature is given by carried out using several methods. These include 1/N tech-
niques[14,15],latticesimulations[16,17],thelineardeltaex-
d3p 1 1
n = , (1) pansion [18, 19], variational perturbation theory [20, 21, 22],
(2π)3 eβ(ω−µ) 1 − eβ(ω+µ) 1 and renormalization group methods [23, 24]. In particular,
Z (cid:20) − − (cid:21)
the use of effective field theory methods to obtain an effec-
where ω= p2+m2, β=1/T and µ is thechemical poten- tivethree-dimensionalfieldtheorycombinedwithhighpreci-
tial. We have set ¯h = k = 1. Bose-Einstein condensation sionlatticecalculationshassettledtheissueinaveryelegant
B
takes placepwhen the chemical potential is equal to the mass way [16, 25].
m of the bosons. At that temperature, all the charge can no TheO(2)-symmetric relativistic Bose gas at finitetemper-
longerbeaccomodated intheexcitedstatesanditcondenses ature and chemical potential has been studied in detail by
into the ground state. Generally, T can only be calculated Benson, Bernstein, and Dodelson [26, 27], while the prob-
c
numerically, but in various limits analytic results can be ob- lemofcalculatingT wasaddressedinapaperbyBedingham
c
tained. For example, in the nonrelativistic (NR) limit, the and Evans employing the linear delta expansion [28]. In the
presentpaper,weexaminetherelativisticBosegasinamore stituting Eq.(5) into Eq. (4), theaction can be written as
general setting where we consider N coupled scalars. For
N = 1, it reduces to the standard case, while for N = 2,
the system is isomorphic to the linear SU(2)L × SU(2)R- S = S0+Sfree+Sint . (6)
symmetric sigma model and hence relevant for kaon conden-
sation in stars.
where
II. PERTURBATION THEORY
S = dt d3x µ2 m2 φ2 λφ4 , (7)
0 − 0− 0
In this section, we briefly discuss the perturbative frame- Z Z
(cid:2)(cid:0) (cid:1) (cid:3)
work for a massive Bose gas with N charged scalars at finite 1 ∂2
chemical potential µ. Theaction is Sfree = dt d3x 2φi − ∂t2 +∇2+µ2−m2
Z Z ( (cid:20)
S = Z dtZ d3x(cid:20)(∂0+iµ)Φ†(∂0−iµ)Φ −2λφ20−4δj1δj1λφ20 φi+iµ φ2∂∂φt1 −φ1∂∂φt2
(cid:21) h
− ∂iΦ† (∂iΦ)−m2Φ†Φ−λ Φ†Φ 2 , (4) +...+φ ∂φ2N−1 φ ∂φ2N , (8)
(cid:21) 2N ∂t − 2N−1 ∂t
(cid:0) (cid:1) (cid:0) (cid:1) )
i
where Φ = (Φ ,Φ ...,Φ ) and Φ is a complex scalar field.
1 2 N i
iWndeefipresntdpeanrtavmaecturuizmetehxepqecutaanttiounmvfiaeluldeΦφ1ainndtetrwmosroefaalqtuimane-- Sint = − dt d3x √2 m2−µ2+2λφ20 φ1φ0
0 Z Z (cid:20)
tum fluctuatingfields: (cid:0) (cid:1)
1
+√2λ φ2+φ2+...+φ2 φ φ + λ(φ φ )2 .(9)
1 1 2 2N 1 0 4 i i
Φ = φ + (φ +iφ ) . (5) (cid:21)
1 0 √2 1 2 (cid:0) (cid:1)
Similarly, the remaining complex fields Φ ,...,Φ are Thepropagators that correspond to thefree part S of the
2 N free
parametrized in terms of 2N 2 real fields φ ,...,φ . Sub- action are given by
3 2N
−
D (ω,p) = i ω2−p2−m21 2iµω , (10)
1 (ω2−ω12+)(ω2−ω12−) −2iµω ω2−p2−m22 !
D (ω,p) = i ω2−p2−m22 2iµω , (11)
2 (ω2−ω22+)(ω2−ω22−) −2iµω ω2−p2−m22 !
where thedispersion relations are
1 1
ω (p) = p2+2µ2+ (m2+m2) (m2+m2)2+2µ(2µ2+m2+m2)+4µ2p2, (12)
1± 2 1 2 ± 2 1 2 1 2
r
p
ω (p) = p2+µ2+m2 µ. (13)
2± 2±
p
Here thetree-level masses m2 and m2 are From these equations, we see that there are two massless
1 2
modes that in thelong-wavelength behaveas
m2 = µ2+m2+6λφ2 , (14)
1 − 0
m2 = µ2+m2+2λφ2 . (15) µ2 m2
2 − 0 ω1−(p) = 3µ2− m2 p, (18)
r −
mIn22t=he0m,iannidmusomthofetdhiespcelarssisoicnalrealcattiioonn,smre21du=ce2(tµo2−m2)and ω2−(p) = 2pµ2 . (19)
Theotherexcitationsω+(p)andω+(p)aregappedwithgaps
ω1±(p) = p2+3µ2 m2 (3µ2 m2)2+4µ2p2 , 1 2
− ± − ∆ = 2(3µ2 m2) and ∆ = 2µ. In the case N = 2, the
q p (16) ga1pless particle−s ω− and ω−2carry the quantum numbers of
ω (p) = p2+µ2 µ. (17) K+ anpdK0,while1themass2ivemodesω+ andω+ carrythose
2± ± 1 2
p
2
of K− and K¯0 [10, 11]. Note that there are only N massless where ∆ m2, ∆ λ,and ∆ are theone-loop mass countert-
1 1 1
E
modes despite the fact that the potential has 2N 1 flat di- erm, coupling constant counterterm, and vacuum countert-
−
rections which also is the numberof broken generators. This erm, respectively. After integrating over the energy ω, we
isinagreementwiththecountingrulederivedbyNielsenand obtain
Chadba [29], which states that the modes with a quadratic
1
dispersion relation must be counted twice. Secondly, due to Ω (µ,φ ) = ω (p)+(N 1)ω (p) +∆ m2φ2
1 0 2 1± − 2± 1 0
thequadraticdispersion relation for small p,theLandau cri- Zp
terion [30] for superfluidity can never be satisfied, except for +∆ λ(cid:2)φ4+∆ . (cid:3) (27)
N =1. Thus despite the presence of a Bose condensate, the 1 0 1E
system is not a superfluid. The integral involving ω can be calculated analytically in
2±
dimensional regularization, but the integral of ω cannot.
1±
Inordertoextractthedivergencesanalytically,wemakesub-
III. ZERO TEMPERATURE tractions in the integrand that render the integral finite in
d = 3 dimensions and then extract the poles in d 3 from
−
the subtracted integrals. The substraction term Ω should
In this section, we apply perturbation theory at zero tem- sub
not introduce any infrared divergences. Our choice for the
perature to calculate the pressure, charge density, and the
subtracted integral is
speed of sound in theone-loop approximation.
The partition function Z is given bythe path integral Ω = p+ m2+4λφ20 m4+8m2λφ20+20λ2φ40 .
sub 2p − 8(p2+µ2)3/2
= Φ† ΦeiS (20) Zp(cid:20) (cid:21)
Z D D (28)
Z
where theaction S is given by (4). Thepressure is
P The first two terms in Ωsub vanish identically in dimensional
ln regularization since there is no mass scale in the integrand.
P(µ) = −i VTZ , (21) The last term is given in Eq. (A.3). The one-loop thermody-
namics potential can then be written as
whereVT isthespace-timevolumeofthesystem. Thecharge
densitycanbefoundbydifferentingthepressurewithrespect
1 1 3
to µ: Ω1 = −2(4π)2 m4 N ǫ +2L + 2(N −1)
∂ (µ) (cid:26) h (cid:16) 1 (cid:17) 3 i
n(µ) = P . (22) +4m2λφ2 (N +1) +2L + (N 1)
∂µ 0 ǫ 2 −
h (cid:16) (cid:17) i
The speed of sound c is given by the coefficient of ω (p) 1 3
1− +4λ2φ4 (N+4) +2L + (N 1)
as p 0. Corrections to the tree-level result can be found 0 ǫ 2 −
by ca→lculating the dispersion relation in the long-wavelength h (cid:16) (cid:17) i(cid:27)
+∆ m2φ2+∆ λφ4+∆ , (29)
limitincludingtheself-energyfunctionΠ1−(ω,p). Itcanalso 1 0 1 0 1E
be derived once we knowthecharge density and is given by
where L = ln Λ and g is a function of the ratio m/µ that
µ
n∂µ must be evaluated numerically:
c2 = . (23) (cid:0) (cid:1)
µ∂n
1
Thechemicalpotentialmeasurestheamountofenergyneeded g(m/µ) = 2 ω1±−Ωsub . (30)
to add a particle to the system, and in the nonrelativistic Zp
limit, weintroducethenonrelativsticchemical potential µNR The countertermsnecessary to cancel thepoles in ǫ are [20]:
by µ = m+ µ . In the NR limit, Eq. (23) is therefore
NR
replaced by Nm4
∆ = , (31)
n ∂µ 1E 2(4π)2ǫ
c2 = NR (24)
m ∂n ∆ m2 = 2(N +1)m2λ , (32)
1 (4π)2ǫ
2(N +4)λ2
A. Pressure ∆ λ = . (33)
1 (4π)2ǫ
The mean-field pressure is found by evaluating minus Theone-loopcontributiontothepressure isgivenby Ω
0 1 1
P P −
the classical thermodynamic potential Ω (µ,φ ) at themini- evaluated at the classical minimum. After renormalization,
0 0
mum of theclassical action S : thepressure through one loop reducesto
0
P0(µ) = 41λ µ2−m2 2 . (25) (µ) = 1 µ2 m2 2+ 1 2m4L 6m2µ2L
P0+1 4λ − (4π)2 −
The one-loop contribution to th(cid:0)eeffectiv(cid:1)epotential is (
(cid:0) (cid:1)
1 dω 3
Ω1(µ,φ0) = 2i 2π lndetD1(ω,p) +µ4 (N+4)L+ 4(N −1) +µ4g(m/µ). (34)
Z Zp )
+(N 1)lnd(cid:2)etD (ω,p) +∆ m2φ2 h i
− 2 1 0 WenextconsidertheNRlimitofthepressure. Inthislimit,
+∆1λφ40+∆1E , (cid:3) (26) µNR ≪m. Moreover,thekineticenergyismuchsmallerthan
3
m and so we can expandphysical quantitiesin powers of the where we have eliminated µ in favor of the density n.
|rmNR
dimensionless quantities µ /m and k2/2m2. This yields Notethat theexpansion parameterin theNRlimit isthedi-
NR
mensionlessquantity√na3 whichisreferredtoasthethegas
ω2 (p) = p2 p2+4mµ , (35) parameter. This result (42) was first derived by Beliaev [32],
1− 4m2 NR whocalculated theleading corrections tothedispersion rela-
ω2 (p) = p4 (cid:0). (cid:1) (36) tion (36) in the low-momentum limit 2
2− 4m2
Theotherquasiparticleexcitationshaveω =ω =2mand
1+ 2+
sotheircontributioncanbeneglected. InNRfieldtheory,itis IV. FINITE TEMPERATURE
customarytoset2m=1andwewilldosointheremainderof
this section. Introducingthe scattering length a=λ/8πm= We next discuss the behavior of the system defined by
λ/4π, thepressure becomes Eq.(4) at finitetemperature.
µ2 1
= NR ω +(N 1)ω . (37)
P0+1 16πa − 2 1− − 2−
Zp A. Low-temperature effects
(cid:2) (cid:3)
Notethatω doesnotcontributetothepressuresincethere
2−
is there is no scale in the integral and so it set to zero in Wefirstconsiderthethermalcorrectionstothepressureat
dimensional regularization. Using a simple ultraviolet cutoff temperaturesT muchlowerthanthechemicalpotentialµ. In
Λ to regulate the integral, thedivergence would becancelled thisregime,thethermodynamicsisdominatedbythemassless
by a vacuumcounterterm 1. Using Eq.(A.4),we obtain modes. Wecan thenapproximate thedispersion relations by
ω (p)andω (p)bytheirlow-momentumlimits(18)and(19).
1 2
= µ2NR 1 32 2µNRa2 . (38) The pressure in theone-loop approximation is
P0+1 16πa − 15π
" p #
1
= ln P2+ω2 (p)
P0+1 −2 0 1±
ZP
B. Charge density and the speed of sound 1P(N 1(cid:2)) ln P2+(cid:3)ω2 (p) . (43)
−2 − 0 2±
ZP
We next consider the speed of sound, which is given by P (cid:2) (cid:3)
Omitting the contribution from the massive modes, neglect-
Eq.(23). Forsimplicity we consideronly theultrarelativistic
ingthezero-temperaturepieces,andusing(A.14)–(A.16),we
andnonrelativisticlimits. Intheselimits,thetree-levelresults
obtain
are c=1/√3 and c=√2µ, respectively.
andTh(3e4)c:harge density can be calculated using Eqs. (22) P0T+1 = √33π02T4 +(N −1)T µ2Tπ 3/2ζ 52 . (44)
n= µ3 1+ λ 4(N +4)L+2N 7+64π2g(0) . Similarly, in thenonrelativistic limit, o(cid:0)ne fi(cid:1)nds (cid:0) (cid:1)
λ (4π2)2 −
(cid:26) (cid:2) (cid:3)(cid:27)(39) T = π2T4 +(N 1)T T 3/2ζ 5 , (45)
P0+1 90(2µ )3/2 − 4π 2
NR
Inverting (39) and using (23), we obtain the speed of sound (cid:0) (cid:1) (cid:0) (cid:1)
where we again have set 2m=1. For N =1, this reduces to
dueto interactions in themedium:
theold result of Lee and Yang[33].
1 (N +4)λ
c = 1+ . (40)
√3 24π2
(cid:20) (cid:21)
B. Dimensional reduction
Thesignofthecorrectionisdeterminedbythebeta-function.
In the NR limit, the charge density follows from Eqs. (22)
and (38): Effectivefield theory methodscan convenientlybeused to
organize the calcuation of physical quantities whenever two
n = µNR 1 8 2µNRa2 . (41) or more momentum scales are well separated. The conden-
sation temperature for BEC is an ideal problem for applying
8πa − 3π
" p # effective field theory. At finite temperature there are two
The speed of sound then becomes characteristic scales in the system. The first is the correla-
tion length which is associated with the effective chemeical
na3 potential. Sincethephasetransitionissecondorder,thecor-
c = 4√πan 1+8 , (42) relation length becomes infinite at T . The second scale is
π c
" r #
2 Intheorginalderivation,Beliaevexpressedhisresultintermsof
1 In the NR limit, all the divergences at the one-loop level are thecondensatedensityn0,whichisdifferentfromthetotalden-
power divergences and hence the parameters require no renor- sity due to the depletion of the condensate caused by quantum
malization if one uses dimensional regularization. See Ref. [3]
forathoroughdiscussion. fluctuations. Wehaven=n0 1+ 83 n0πa3 .
(cid:20) q (cid:21)
4
associatedwiththenonzeroMatsubaramodesandisoforder will be taken care of by theeffective theory. Loop correction
T. For distances much larger than the inverse temperature to the free energy in the 3d theory vanish since there is no
and fortemperatures sufficientlyclose tothecritical temper- momentum scale in the loop integrals 3. Thus f is directly
ature,sothattheeffectivechemicalpotentialismuchsmaller given by (47).
thanthetemperature,thenonstaticMatsubaramodesdecou-
Nπ2 5(N +1)λ
ple and the long-distance physics can be described in terms f = T4 1
of an effective three-dimensional field theory for the zeroth − 45 − 16
(cid:18) (cid:19)
Matsubara mode. Theaction for this effective theory is 1 µ2 (N +1)λ
Nµ2T2 1 +
−6 − 4π2T2 8π2
S = d3x (∂ Φ†)(∂ Φ)+m2Φ†Φ (cid:18) (cid:19)
3d − i i 3 1 3µ2
Z h + Nm2T2 1 (48)
+λ3 Φ†Φ 2+f +... , (46) 6 (cid:18) − 4π2T2(cid:19)
wherethedotsindicate(cid:0)highe(cid:1)r-orderopeirators. f isareferred
to as the coefficient of the unit operator and represents the
contributiontothefreeenergydensityfromthenonzeroMat-
subara modes. The parameters in Eq. (46) are functions of
T and the coefficients of the underlying theory (4), and are
renormalizedduetotheircouplingtothenonstaticMatsubara
frequencies. Theseparameterscanbedeterminedbyintegrat-
ingoutnonstaticmodesexplicitlyasdonebyBedinghamand
Evans in Ref. [28], but perhaps a more streamlined way of FIG. 2: One -and two-loop Feynman diagrams for the self-
calculatingthemisbymatchingstaticGreen’sfunctions[31]. energy Π(p ,p). A dot indicates an insertion of theoperator
0
For the purpose of matching, the chemical potential can for- µ2 m2 Φ†Φ+iµ(Φ†∂ Φ ∂ Φ†Φ).
0 0
mallybetreatedasaperturbationonthesamefootingasthe − − −
quarticcoupling. Intheeffectivetheorym2 isalso treatedas (cid:0) (cid:1)
3 The mass parameter is found by matching the two-point
a perturbation. The matching is carried out by calculating
functions in the two theories at zero external momenta. The
Green’s functions perturbatively in the two theories and de-
diagrams that contribute to the self-energy function in the
mandtheybethesamefordistancesRmuchlargerthan1/T.
full theory through two loops are shown in Fig. 2. The self-
This way of calculating static correlators introduces infrared
energy in the effective theory vanishes for the same reason
divergences at an intermediate stage and must be regular-
as did the loop corrections to the free energy. This implies
ized. Note thate these infrared divergences that appear are
that the mass parameter m2 is given directly by evaluating
the same in the two theories and hence they cancel in the 3
theFeynman diagrams in Fig. 1. Weobtain
matching procedure. We use dimensional regularization as
discussed in theappendix. 1
m2 µ2+m2+2(N +1)Z λ
3 ≈ − λ P2
ZP
P 1 P2
+2(N+1) µ2 m2 λ 4 0
− P4 − P6
Z (cid:20) (cid:21)
(cid:0) (cid:1)1P
4(N+1)2λ2
− P2Q4
ZPQ
P 1
4(N+1)λ2 , (49)
FIG. 1: One -and two-loop vacuum diagrams for f. A − P2Q2(P +Q)2
ZPQ
dotindicatesaninsertionoftheoperator µ2 m2 Φ†Φ+ P
− − where we again have neglected terms of higher-order terms.
iµ(Φ†∂0Φ−∂0Φ†Φ). (cid:0) (cid:1) Zλ is therenormalization constant for thecoupling λ:
(N+4)λ
InFig.1,weshowthevacuumdiagramsthroughtwoloops Zλ = 1+ 8π2ǫ . (50)
in thefull theory and theexpression is
Afterrenormalization, themass term reduces to
F ≈ NP1ZNPµln4P2−N1 (cid:0)µ28−P02m+2(cid:1)8PZPP04(cid:20)P12 −2PP024(cid:21) m23 = −µ32λ+m12+ (N+26(1N)λ+T12)(cid:20)1− 3ζ(cid:0)′µ(22π12−T)m2 2(cid:1)
−2 P4 − P6 P8 + +2 γ+2 −
ZP (cid:20) (cid:21) 8π2 ǫ − 3 ζ( 1)
P 1 P2 1 (cid:18) −
+Nµ2m2 4 0 +N(N +1)λ (4 2N) Λ
P4 − P6 P2Q2 + − ln . (51)
ZP (cid:20) (cid:21) ZPQ 3 4πT
P 1 P2 P (cid:19)(cid:21)
+2N(N +1)λµ2 4 0 , (47)
P4Q2 − P6Q2
ZPQ(cid:20) (cid:21)
P
where we have omitted terms of order m4, µ6 etc. The sign 3 Recall that m2 for the purpose of matching is treated as a per-
3
is reminder that we are neglecting infrared physics which turbationandthus thepropagators aremassless.
≈
5
Note that the mass parameter has a UV divergence after At the critical point, the renormalized mass is by dimen-
renormalization. Theremainingdivergenceisexactlytheone sionalanalysisproportionaltoλ2. ThecaseN =2isrelevant
3
arising at the two-loop level in the 3d effective theory (see to kaon condensates in stars and we therefore consider this
also Sec. IVC). case in the following. In the remainder of this section, we
Finally, we need λ at the tree level. By comparing the also restrict ourselves to the ultrarelativistic limit. Its value
3
coefficients of the operator Φ†Φ 2 in the two theories and was determined bySun [34] using lattice simulations:
taking the different normalization of the fields into account,
one finds (cid:0) (cid:1) m23(Λ=λ3/3) = 0.002558(16) , (58)
λ2
3
λ = λT . (52)
3
where the renormalization scale was chosen to be Λ= λ /3.
3
The critical chemical potential then reducesto
C. Critical temperature
1 3λ ζ′( 1)
µc = 2λT 1+ (4π)2 −2γ+2ζ(−1) −0.1346 .
After having determined the parameters in the effective r (cid:20) (cid:18) − (cid:19)(cid:21)
theory, the strategy for calculating T is as follows. First we (59)
c
determine the critical chemical potential µ as a function of
temperature and the critical value of the mc2. Then we cal- The expectation value Φ†Φ cannot be calculated in per-
3 h i
turbation theory due to infrared divergences. They depend
culate the density as a function of T and µ and obtain the
on nonperturbative physics and can e.g. be determined us-
critical density as a function of T by the substituing the ex-
ing lattice simulations or the 1/N-expansion. At the critical
pressionforµ . Finally,thecriticaltemperatureisdetermined
c
pointandforN =2,itwascomputedbySun[34]usingMonte
by invertingthecritical density as a function of T.
Carlo calculations:
The relation between the bare mass and the renormalized
mass is Φ†Φ
= 0.00289(18) . (60)
m2 = m2 + (N +1)λ23 . (53) (cid:28) λ3 (cid:29) −
3 3,ren (4π)2ǫ
InsertingEqs.(59)and(60)intoEq.(57),thecriticaldensity
This relation is exact due to the fact that the effective 3d becomes
theory (46) is superrenormalizable. Since the bare mass is
independent of the renormalization scale, the renormalized 2 3λ ζ′( 1)
mass satisfies an evolution equation. This equation relates nc = 9λT3 1+ (4π)2 2−2γ+2ζ(−1) +0.3217 .
the value of m2 evaluated at two different normalization r (cid:20) (cid:18) − (cid:19)(cid:21)
3,ren (61)
points Λ and Λ :
0
(N+1)λ2 Λ Invertingthisequation,weobtainthecriticaltemperatureas
m23,ren(Λ0) = m23,ren(Λ)+ 4π2 3 ln Λ0 . (54) a function of thedensity
Using Eq. (53), Eq. (51) for the chemical potential becomes Tc = 29λ 1/6n1/3 1− (4λπ)2 2−2γ+2ζζ′((−11))
µ2 m2 1+ (N+1)λ = (N +1)λT2 1 (cid:16) (cid:17) (cid:20) (cid:18) −
− 4π2 6 +0.3217 . (62)
(cid:18) (cid:19) (cid:20)
(cid:0) (cid:1) 3λ 2(N +1) ζ′( 1) (cid:19)(cid:21)
+ 2 γ+2 −
8π2 − 3 ζ( 1) Theleading-orderresultistheusualperturbativelycalculable
(cid:18) − (cid:19)
high-temperatureresult,whilethesecond-orderterminvolves
(4 2N) Λ 16π2 m2 (Λ)
+ − ln 3,ren .(55) nonperturbative physics. In contrast, the first order correc-
3 4πT − (N+1) λ23 (cid:21) tiontoTcinthenonrelativisticBosegascannotbedetermined
inperturbationtheory[13]. Notealsothatinaccordancewith
The charge density is given by
Eq. (3), T becomes infinite in the absence of interactions.
c
Finally, the impressive seven-loop VPT calculations of Kas-
∂S
n = 3d tening [21, 22] give 0.002586(17) and 0.002796(192) for the
− ∂µ −
(cid:28) (cid:29) quantities in Eqs. (58) and (60). Thus the critical tempera-
= ∂f + Φ†Φ ∂m23 +λ Φ†Φ 2 . (56) ture is within errors in complete agreement with the lattice
−∂µ h i ∂µ 3 prediction.
(cid:10)(cid:0) (cid:1) (cid:11)
The quantity Φ†Φ 2 is by dimensional analysis propor-
tionaltoλ2. Itscontribution tothedensityisthereforethird V. SUMMARY
3 (cid:10)(cid:0) (cid:1) (cid:11)
orderintheinteractionandcanbeomittedinasecond-order
calculation. Using Eqs. (48), (51), and (56), the charge den-
In this paper, we have discussed the thermodynamics of
sity becomes
relativisticBosegasesatzeroandfinitetemperature. Atzero
1 3m2 µ2 (N+1)λ temperature, thermodynamic quantities can be expanded in
n = NµT2 1+ + aloopexpansion,andwecalculatedthepressure,chargeden-
3 4π2T2 − 2π2T2 8π2
(cid:20) (cid:21) sityandspeedofsoundintheone-loopapproximation. Inthe
2µ Φ†Φ . (57) nonrelativistic limit, one easily obtains the standard results
− h i
6
for the dilute Bose gas. In the critical region, perturbation In the imaginary-time formalism for thermal field theory,
theory breaks down due to infrared divergences. However, the4-momentumP =(ω ,p)isEuclideanwithP2=ω2+p2.
n n
one can take advantage of the fact that there is a separation The Euclidean energy p has discrete values: ω =2nπT for
0 n
of scales to simplify the problem of calculating static quan- bosons, where n is an integer. Loop diagrams involve sums
tities such as T . The effects of the nonstatic modes can be over ω and integrals over p. We define the dimensionally
c n
caclulated perturbatively employing dimensional reduction, regularized sum-integral by
while the effective three-dimensional theory must be treated
nonperturbatively. It is interesting to note that the leading eγΛ2 ǫ ddp
correction to T in the ultrarelativistic limit, is calculable in T . (A.5)
c ≡ 4π (2π)d
perturbation theory while in theNR limit it is not. The rea- PZP (cid:18) (cid:19) ωnX=2nπT Z
sonissimplythatthechemicalpotentialcouplesquadratically
to Φ†Φ in thefirst case and linearly in the latter. The specific sum-integrals needed are
By gauging the linear SU(2) SU(2) -symmetric sigma
L R
×
model in various ways, one obtains a number of interesting π2T4
lnP2 = , (A.6)
gauge theories [10, 11]. The Higgs mechanism and the Gold- − 45
stone mechanism are both realized in the conventional man- ZP
P 1 T2 ζ′( 1)
ner. The interest in these models is partly due to the fact = 1+ 2+2 − ǫ , (A.7)
that the rotational symmetry is broken as well and leads to P2 12 ζ( 1)
ZP (cid:20) (cid:18) − (cid:19) (cid:21)
a directional dependence of the dispersion relation which is P P2 T2 ζ′( 1)
linear for small wave vectors and roton-like for larger wave 0 = 1+2 − ǫ , (A.8)
P4 −24 ζ( 1)
vectors. These models may therefore be of interest for con- ZP (cid:20) − (cid:21)
densed matter systems such as superfluid helium. P 1 1 µ 2ǫ 1
= +2γ , (A.9)
P4 (4π)2 4πT ǫ
ZP (cid:16) (cid:17) h i
P P2 1 µ 2ǫ 1 1
Acknowledgments 0 = +2+2γ , (A.10)
P6 (4π)2 4πT 4 ǫ
ZP (cid:16) (cid:17) h i
P P4 1 µ 2ǫ 1 1 8
TheauthorwouldliketothankP.Arnoldforusefuldiscus- 0 = + +2γ ,(A.11)
P8 (4π)2 4πT 8 ǫ 3
sions. ZP (cid:16) (cid:17) h i
P 1
=0. (A.12)
P2Q2(P +Q)2
APPENDIX A: FORMULAS ZPQ
P
We also need to expand some sum-integrals about zero tem-
Dimensional regularization can be used to regularize both perature. The phonon part of the spectrum then dominates
the ultraviolet divergences and infrared divergences in three- thetemperature-dependentpartof thesum-integral. Wecan
dimensionalintegralsovermomenta. Thespatialdimensionis therefore approximate the dispersion relations ω1−(p) and
generalized tod=3 2ǫ dimensions. Integrals areevaluated ω2−(p)bytheirlow-momentumlimits(18)-(19),andthisgives
−
at avalueof d for which they convergeand then analytically theleading temperaturecorrection. These are
continued to d=3. Weuse theintegration measure
eγΛ2 ǫ d3−2ǫp ln P02+ω12−(p) (A.13)
. (A.1)
Zp ≡ (cid:18) 4π (cid:19) Z (2π)3−2ǫ PZP h iT ∞
= ω (p)+ dpp2ln 1 e−βω1−(p)
whereΛisanarbitrarymomentumscale. Thefactor(eγ/4π)ǫ 1− π2 −
Zp Z0
is introduced so that, after minimal subtraction of the poles π2T4 3µ2 m(cid:2)2 3/2 (cid:3)
imnaǫlidzauteiotnouscltarlaevoioflethtediMveSrgreenncoersm,Λaliczoaitniocindesschweimthe.therenor- =Zpω1−(p)− 45 (cid:18) µ2−−m2 (cid:19) +..., (A.14)
Weneed theregularized integrals
ln[P2+ω2 (p) (A.15)
0 2−
p2+µ2 = µ4 Λ 2ǫ 1 + 3 , (A.2) PZP iT ∞
Zpp1 −24(4π)2Λ(cid:18)µ2ǫ(cid:19)1hǫ 2i =Zpω2−(p)+ π2 Z0 dpp2ln(cid:2)1−e−βω2−(p)(cid:3)
(p2+µ2)3/2 = (4π)2 µ ǫ , (A.3) = ω (p) 2T µT 3/2ζ 5 +.... (A.16)
Zp µ5/2 (cid:18) (cid:19) h i Zp 2− − (cid:0)2π(cid:1) (cid:0)2(cid:1)
p p2+µ = . (A.4)
15π2
Zp
p
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