Nambu-Goldstone Mechanism in Real-Time Thermal Field Theory

11 Pages·0.16 MB·English

by  Bang-Rong Zhou

Nambu-Goldstone mechanism in real-time thermal field theory
Bang-Rong Zhou
Department of Physics, Graduate School at Beijing, University of Science and Technology of China
Academia Sinica, Beijing 100039, China∗
and
The Abdus Salam International Centre for Theoretical Physics, P.O.Box 586, 34100 Trieste, Italy
()
In a one-generation fermion condensate scheme of electroweak symmetry breaking, it is proved
based on the Schwinger-Dyson equation in real-time thermal field theory in the fermion bubble
diagram approximation that, at finite temperature T below the symmetry restoration temperature
Tc, a massive Higgs boson and three massless Nambu-Goldstone bosons could emerge from spon-
taneous breaking of the electroweak group SUL(2)×UY(1) → UQ(1) if the two fermion flavors
in the one generation are mass degenerate, thus the Goldstone theorem is rigorously valid in this
9
case. However, if thetwo fermion flavorshaveunequalmasses, owing to ”thermal fluctuation”, the
9
Goldstone theoremwill betrueonlyapproximatelyfor averylarge momentumcutoffΛinthezero
9
1 temperature fermion loop or for low energy scales. All possible pinch singularities are proved to
cancel each other, as expected in a real-time thermal field theory.
n
a 14.80.Mz, 11.10.Wx, 11.30.Qc, 12.15.-y
J

1
1

I. INTRODUCTION
1
v
5 Spontaneous symmetry breaking at finite temperature has been investigated extensively [1-6]. However, most re-
3 searchby now has concentratedon the discussions of phase transition and criticaltemperature based on the effective
0
potential approachat finite temperature, and relatively less work is reportedabout the theoretical explorationof the
1
Nambu-Goldstonetheorem[7]atfinitetemperature,especiallyinmodelsofdynamicalsymmetrybreakingsuchasthe
0
9 Nambu-Jona-Lasinio(NJL)modelwithfour-fermioninteractions[7]. ResearchintotheNambu-Goldstonemechanism
9 atfinite temperature couldprovideus adeeper understandingofthe spontaneousbreakingofa continuoussymmetry
/ at finite temperature and is certainly quite interesting. The key point of such research lies in verifying existence of
h
t Nambu-Goldstonebosons,i.e.,determiningthe physicalmassesofthe fermionicscalarandpseudoscalarboundstates
-
p which appear as the products of spontaneous symmetry breaking. In a model of NJL form, the mass determination
e can be simply made by using Schwinger-Dyson equations, i.e., calculating directly the gap equation of fermion mass
h and the bound state propagators induced by the four-fermion interactions. On the other hand, it is possible that a
:
v scalar or pseudoscalar bound state is composed of fermions with different masses or is a combination of the scalar or
i pseudoscalar bilinears of these fermions [8-10]. Although a conventional effective potential approach is applicable to
X
such models if one introduces axialiary scalar fields and replaces the four-fermion interactions by Yukawa couplings
r
a between axialiary scalar fields and fermion fields [6], it is unknown, by the introduction of axialiary scalar fields,
whether one could include the effect of the mass diffrence of the constituent fermions of a bound state. However,
we believe that the Schwinger-Dyson equation can surely do that. Therefore, to determine directly the masses of
the bound states and to examine fully the effect of the fermion mass difference at finite temperature, we prefer the
Schwinger-Dyson equation at finite temperature to a temperature effective potential. All the calculations will be
conducted in the fermion bubble diagram approximation which amounts to the leading order of the 1/N expansion.
Since the determinationof physicalmasses of the bound states is the key point of our research;it is certainly more
convenientto take the real-timeformalismofthermalfield theory[4]than the imaginary-timeformalism. In this way
we will be able to avoid the cumbersome analytic continuation which is necessary in the latter formalism. However,
again, because of the possible mass differences of the constituent fermions inside the bound states, the use of the
real-time formalism will present a new question, i.e., whether this formalism consistent is with Nambu-Goldstone
mechanism. In a simple model with U (N) U (N)-invariant four-fermion interactions, it has been proved that in
L R
×
the real-time formalism of thermal field theory the Nambu-Goldstone mechanism works indeed, but it is under the
∗Permanent address.
1
assumption that the fermions are of equal masses and equal chemical potentials [11]. In this paper, among other
things, to examine further the stated consistency, we will consider a more realistic model – a one-generationfermion
condensate scheme of electroweak symmetry breaking [9]. The extension to the many-generationcase [10] is direct.
The paper is arrangedas follows. In Sec. II we present the model, give its Lagrangianand derive the gap equation
atfinitetemperature. InSecs. III,IVandVwewillrespectivelycalculatepropagatorsofthescalar,thepseudoscalar,
andthecharged-scalarboundstatesatfinitetemperaturebymeansoftheSchwinger-Dysonequationssatisfiedbythe
fermionic four-point Green functions, determine the bound states’ masses, and discuss the pinch singularity problem
which is especially related to the real-time formalism of thermal field theory. Finally, in Sec. VI we come to our
conclusions.
II. ONE-GENERATION FERMION CONDENSATE MODEL AND GAP EQUATION
In this model of electroweak symmetry breaking, we have one generationof Q fermions forming a SU (2) U (1)
L Y
×
doublet (U,D) and are assigned in the representation R of the color group SU (3) with the dimension d (R). The
c Q
symmetry breaking is induced by the effective four-fermion interactions among the Q fermions below some high
momentum scale Λ which, in real-time thermal field theory, are described by the Lagrangian[4,9,10]
= S + P + C . (2.1)
L4F L4F L4F L4F
The neutral scalar couplings
2
1
LS4F = 4 (−1)a+1gQ′Q(Q¯′Q′)(a)(Q¯Q)(a),
a=1Q,Q′
X X
gQ′Q =gQ1/′2Q′gQ1/Q2, Q,Q′ =U,D, (2.2)
where a = 1 denotes physical fields and a = 2 ghost fields. The physical fields and the ghost fields interact only
through the propagators. The neutral pseudoscalar couplings
2
1
LP4F = 4 (−1)a+1gQ′ ′Q(Q¯′iγ5Q′)(a)Q¯iγ5Q)(a),
a=1Q,Q′
X X
gQ′ ′Q =(−1)IQ3′−IQ3gQ′Q, Q,Q′ =U,D, (2.3)
and I3 denotes the third component of the weak isospin of the Q fermions. The charged scalar couplings
Q
2
C = G ( 1)a+1(D¯Γ+U)(a)(U¯Γ−D)(a),
L4F 2 −
a=1
X
1
Γ± = [cosϕ sinϕ (cosϕ+sinϕ)γ ],
5
√2 − ±
G=g +g , cos2ϕ=g /G, sin2ϕ=g /G. (2.4)
UU DD UU DD
We indicate that Lagrangian (2.1) is the real-time thermal field theory version of the following zero-temperature
four-fermion Lagrangianfor n=1 [10]:
0 = G (φ0)2+(φ0)2+2φ+φ− , (2.5)
L4F 4 S P
h i
where
2
φ0 = cosϕ(U¯U)(1)+sinϕ(D¯D)(1),
S
φ0 = cosϕ(U¯iγ U)(1) sinϕ(D¯iγ D)(1),
P 5 − 5
φ− =(U¯Γ−D)(1), φ+ =(D¯Γ+U)(1) (2.6)
are, respectively, the configurations of the physical neutral scalar, neutral pseudoscalar, and charged scalar bound
states which are expressed by physical fermion fields with a =1. In zero-temperature field theory, one assumes that
S will lead to the vacuum expectation value g (Q¯Q)(1) =0 and this will induce spontaneous breaking
L4F Q=U,D QQh i6
of electroweak group. At finite temperature T and in the real-time formalism of thermal field theory, we will assume
the thermalexpectationvalue g (Q¯QP)(1) =0,whereonlythephysicalfields(a=1)areconsidered[12].
Q=U,D QQh iT 6
We thus obtain the mass of the Q fermion,
P
1
m (T,µ) m = g1/2 g1/2 (Q¯′Q′)(1) , (2.7)
Q ≡ Q −2 QQ Q′Q′h iT
Q′=U,D
X
which will lead to the relation
mQ/gQ1/Q2 =mQ′/gQ1/′2Q′ (2.8)
and the gap equation at finite temperature T,
1= g I , (2.9)
QQ Q
Q=U,D
X
with
1 d (R) d4l
I = (Q¯Q)(1) = Q tr[iS11(l,m )]
Q −2mQh iT 2mQ Z (2π)4 Q
d4l i
=2d (R) 2πδ(l2 m2)sin2θ(l0,µ ) , (2.10)
Q Z (2π)4 "l2−m2Q+iε − − Q Q #
where we have used the thermal propagatorof fermion in the matrix form
iS11(l,m ), iS12(l,m ) i/(l m +iε), 0
Q Q = 6 − Q
iS21(l,m ), iS22(l,m ) 0, i/(l m iε)
Q Q Q
(cid:18) (cid:19) (cid:18) − 6 − − (cid:19)
2π(l+m )δ(l2 m2) sin2θ(l0,µQ), 12eβµQ/2sin2θ(l0,µQ) , (2.11)
− 6 Q − Q (cid:18)−12e−βµQ/2sin2θ(l0,µQ), sin2θ(l0,µQ) (cid:19)
with the chemical potential µ of the Q fermion and the denotations
Q
θ(l0) θ( l0)
sin2θ(l0,µ )= + − (2.12)
Q exp[β(l0 µ )]+1 exp[β( l0+µ )]+1
Q Q
− −
and β = 1/T. The gap equation (2.9) could be satisfied merely at lower temperature T than T , where T is the
c c
critical temperature above which Eq. (2.9) is no longer valid and thus electroweak symmetry restoration is implied
[12,13]. In view of this, in the following discussion we will assume T < T so that the gap equation (2.9) can always
c
be used.
III. SCALAR BOUND STATE MODE
The propagators for fermionic bound states correspond to the four-point Green functions of the fermions. Since
there exist two types of four-fermion interaction vertices (a = 1,2) in real-time thermal field theory, the four-point
Green functions will also be a matrix with the row and the column denoted by the index a. The four-point functions
3
for the transition from (Q¯Q)(a) to (Q¯′Q′)(b) can be denoted by ΓQ′bQa(p); then, from Eq. (2.2), they will obey the
S
following linear algebraic equations [10]:
ΓQS′bQ′′c(p) δQ′′Qδca−NQca′′(p)gQ′′Q(−1)a+1 = 2igQ′Qδba(−1)a+1, Q′,Q=U,D, b,a=1,2, (3.1)
Xc XQ′′ h i
where p is the four-momentum of the bound state, and 2iNca represents the contribution of the Q-fermion loop
− Q
with an a-type and a c-type scalar coupling vertex [Eq.(2.2)], i.e.,
i d4l
Nca(p)= d (R) tr[iSca(l,m )iSac(l+p,m )]. (3.2)
Q −2 Q (2π)4 Q Q
Z
Equations (3.1) have the solutions
ΓQS′bQa(p)= 2∆i(p)gQ′Qδb11+ gQQNQ22(p)−δb2 gQQNQ21(p)δa1
S
 XQ XQ
   

δb2 1 g N11(p) +δb1 g N12(p) δa2 , Q′,Q=U,D, b,a=1,2, (3.3)
−  − QQ Q  QQ Q  
XQ XQ 
   
where the coefficient determinant of Eqs. (3.1), 
∆ (p)= 1 g N11(p) 1+ g N22(p) + g N12(p) g N21(p) . (3.4)
S  − QQ Q  QQ Q   QQ Q  QQ Q 
Q Q Q Q
X X X X
     
The propagatorfor the physical scalar bound state φ0 expressed in Eq. (2.6) is
S
Γφ0S(p)=cos2ϕΓU1U1(p)+sin2ϕΓD1D1(p)+sinϕcosϕ ΓD1U1(p)+ΓU1D1(p)
S S S S
(cid:2) (cid:3)
=iG 1+ g N22(p) /2∆ (p). (3.5)
 QQ Q  S
Q
X
 
The problem is reduced to the calculation of Nab(p). By using the formula
Q
1 X
= iπδ(X) (3.6)
X +iε X2+ε2 −
and through direct but rather lengthy derivation we obtain
1
N11(p)=I + (p2 4m2 +iε)[K (p)+H (p) iS (p)]
Q Q 2 − Q Q Q − Q
= N22(p) ∗,
− Q
i
N12(p)=N(cid:2)21(p)=(cid:3) (p2 4m2)R (p), (3.7)
Q Q −2 − Q Q
where
id4l 1
K (p)= 2d (R)
Q − Q (2π)4(l2 m2 +iε)[(l+p)2 m2 +iε]
Z − Q − Q
1
d (R) Λ2+M2 Λ2
= Q dx ln Q , M2 =m2 p2x(1 x), (3.8)
8π2 Z MQ2 − Λ2+MQ2! Q Q− −
0
with the four-dimension Euclidean momentum cutoff Λ,
4
d4l (l+p)2 m2
H (p)=4πd (R) − Q +(p p) δ(l2 m2)sin2θ(l0,µ )
Q Q (2π)4 ([(l+p)2 m2]2+ε2 →− ) − Q Q
Z − Q
∞ ⇀ ⇀ ⇀ ⇀ 2
= 1 d| l || l | ln(p2−2ωQlp0+2| l || p |) +ε2 +(p0 p0)
16π2|⇀p |Z0 ωQl  (p2−2ωQlp0+2|⇀l ||⇀p |)2+ε2 →− 
 
1 1 ⇀2
(cid:20)exp[β(ωQl−µQ)]+1 + exp[β(ωQl+µQ)]+1(cid:21), ωQl =rl +m2Q, (3.9)
d4l
S (p)=4π2d (R) δ(l2 m2)δ[(l+p)2 m2][sin2θ(l0+p0,µ )cos2θ(l0,µ )+cos2θ(l0+p0,µ )sin2θ(l0,µ )],
Q Q (2π)4 − Q − Q Q Q Q Q
Z
(3.10)
and
d4l
R (p)=2π2d (R) δ(l2 m2)δ[(l+p)2 m2]sin2θ(l0,µ )sin2θ(l0+p0,µ ). (3.11)
Q Q (2π)4 − Q − Q Q Q
Z
In Eq.(3.9) the limit ε 0 will be taken only after the total calculations are completed. Substituting Eq. (3.7) into
→
Eq. (3.5) and using the gap equation (2.9) and the relation
g /G=m2/ m2 (3.12)
QQ Q Q
Q
X
derived from Eq. (2.8), we obtain
Γφ0S(p)=−i Qm2Q/ (p2−4m2Q+iε)m2Q[KQ(p)+HQ(p)−iSQ(p)]
P XQ
[ (p2 4m2)m2R (p)]2
 Q − Q Q . (3.13)
− Q(p2−4mP2−iε)m2Q[KQ(p)+HQ(p)+iSQ(p)])
The mass squared m2φ0 is determined byPthe pole’s position of Γφ0S(p), i.e., by the equation
S
2 2 2
(p2 4m2)m2[K (p)+H (p)] + (p2 4m2)m2S (p) = (p2 4m2)m2R (p) . (3.14)
 − Q Q Q Q   − Q Q Q   − Q Q Q 
XQ  XQ  XQ 
Wenotice that when Λ2 >>M2,     
Q
d (R) Λ2
Q
K (p)= ln +1
Q 8π2 m2
Q !
λ 1arctan 1 if λ >1,
dQ(R) Q− √λQ−1 Q ,λ =4m2/p2. (3.15)
− 8π2 p 1 λ ln1+√1−λQ, if λ <1, Q Q
 − Q 1−√1−λQ Q
p
Hence K (p) is realand positive definite. The same conclusionis true with H (p) and S (p) since the integrands in
Q Q Q
Eqs. (3.9) and (3.10) are real and positive, but not applicable to R (p). In addition, we have the inequalities
Q
d4l
S (p) R (p)=4π2d (R) δ(l2 m2)δ[(l+p)2 m2]sin2[θ(l0+p0,µ ) θ(l0,µ )] 0. (3.16)
Q ± Q Q (2π)4 − Q − Q Q ± Q ≥
Z
Let
5
m2K (p)=k, m2H (p) =h, m2S (p)= s, m2R (p)=r,
Q Q Q Q Q Q Q Q
Q Q Q Q
X X X X
m4K (p)=k˜, m4H (p)=h˜, m4S (p)=s˜, m4R (p)=r˜. (3.17)
Q Q Q Q Q Q Q Q
Q Q Q Q
X X X X
then Eq. (3.14) can be rewritten as
[(k+h)p2 4(k˜+h˜)]2+(sp2 4s˜)2 =(rp2 4r˜)2. (3.18)
− − −
It has the solutions
[(k+h)(k˜+h˜)+ss˜ rr˜]
p2 =4 − 1 A ,
(k+h)2+s2 r2 { ± }
−
1/2
[(k+h)2+s2 r2][(k˜+h˜)2+s˜2 r˜2]
A= 1 − − . (3.19)
( − [(k+h)(k˜+h˜)+ss˜ rr˜]2 )
−
Equation (3.19) shows that we could obtain two different m2 . However, we indicate that, first of all, in the special
φ0
S
case where only single-flavor Q fermions exist (e.g., in the top-quark condensate scheme [8]) or all the m are equal
Q
(mass degenerate), the A will be identical to zero and we still have only a single mφ0. Next, in general case, if the
S
momentum cut-off Λ is large enough, then, considering Eq. (3.16), we always have
(k+h)2 >>s2 r2 >0, (k˜+h˜)2 >>s˜2 r˜2 >0, (k+h)(k˜+h˜)>>ss˜ rr˜,
− − −
and thus A 0. In fact, A = 0 can be explained as thermal fluctuation of the squared mass of φ0. In any way, we
≈ 6 S
may consider that physically the mass of φ0 is determined by the equation
S
4
m2 =p2 [(k+h)(k˜+˜h)+ss˜ rr˜]
φ0S ≃ (k+h)2+s2−r2 − (cid:12)p2=m2φ0S
(cid:12)
4(k+h)(k˜+h˜)+2[(s+r)(s˜ r˜)+(s˜+r˜)(s(cid:12) r)]
= − − . (3.20)
(k+h)2+s2 r2 (cid:12)
− (cid:12)(cid:12)p2=m2φ0S
(cid:12)
(cid:12)
Based on Eq. (3.16) we have s r 0 and s˜ r˜ 0; thus it is deduced from Eq. (3.20) that
− ≥ − ≥
2(mQ)min ≤mφ0S ≤2(mQ)max, (3.21)
where (m ) and (m ) are, respectively, the minimal and the maximal mass among the Q fermions. The
Q min Q max
limitation (3.21) is formally the same as the one at zero temperature [10], but it should be understood that m
Q
≡
m (T,µ ) is now the Q-fermion mass at T =0.
Q Q
6
When p 0, by Eq. (3.9), H (p) = 0 and by Eqs. (3.10) and (3.11), S (p) and R (p) contain the pinch
Q Q Q
→
singularities. However,we obtain from Eq. (3.16) that
[S (p) R (p)] =0 or (s˜ r˜) =0 (3.22)
Q Q p→0 p→0
− | − |
and this will make the pinch singularities contained in the denominator of Γφ0S(p) cancel each other. This result is
just expected in real-time thermal field theory.
Inaddition, wealsoindicate that itis easyto verifyby Eq. (3.3)that, similarto the zero-temperaturecase,for the
orthogonalcombination to φ0,
S
φ˜0 = sinϕ(U¯U)(1)+cosϕ(D¯D)(1), (3.23)
S −
its propagator Γφ˜0S(p) = 0, i.e., the configuration φ˜0, does not exist. We only have the single neutral scalar bound
S
state φ0 left.
S
6
IV. PSEUDOSCALAR BOUND STATE MODE
The calculation of the propagator for pseudoscalar bound state is similar to the one for scalar bound state. The
four-pointfunction fortransitionfrom(Q¯iγ Q)(a) to (Q¯′iγ Q′)(b) is denotedbyΓQ′bQa; then fromEq. (2.3)they will
5 5 P
obey the linear algebraic equations [10]
ΓPQ′bQ”c(p) δQ”Qδca−NQca”5(p)gQ′ ”Q(−1)a+1 = 2igQ′ ′Qδba(−1)a+1, Q′,Q=U,D, b,a=1,2. (4.1)
Xc XQ” h i
We have used the denotation
i d4l
Nca(p)= d (R) tr[iγ iSca(l,m )iγ iSac(l+p,m )], (4.2)
Q5 −2 Q (2π)4 5 Q 5 Q
Z
and 2iNca(p) represents the contribution of the Q-fermion loop with an a-type and a c-type pseudoscalar coupling
− Q5
vertex [Eq. (2.3)]. By comparing Eqs. (3.1) with Eqs. (4.1), it is easy to find that the solutions of Eqs. (4.1) can be
obtained from the solutions (3.3) by the substitutions
NQca(p) → NQca5(p), gQ′Q → gQ′ ′Q.
Thus
ΓPQ′bQa(p)= 2∆i(p)gQ′ ′Qδb11+ gQQNQ225(p)−δb2 gQQNQ215(p)δa1
S
 XQ XQ
   

δb2 1 g N11(p) +δb1 g N12(p) δa2 , Q′,Q=U,D, b,a=1,2, (4.3)
−  − QQ Q5  QQ Q5  
XQ XQ 
   
where 
∆ (p)= 1 g N11(p) 1+ g N22(p) + g N12(p) g N21(p) . (4.4)
P  − QQ Q5  QQ Q5   QQ Q5  QQ Q5 
Q Q Q Q
X X X X
     
The propagatorof the physical neutral pseudoscalar bound state φ0 defined in Eq. (2.6) is
P
Γφ0P(p)=cos2ϕΓU1U1(p)+sin2ϕΓD1D1(p) sinϕcosϕ ΓD1U1(p)+ΓU1D1(p)
P P − P P
(cid:2) (cid:3)
=iG 1+ g N22(p) /2∆ (p). (4.5)
 QQ Q5  P
Q
X
 
For the orthogonalcombination φ˜0 to φ0, we have
P P
Γφ˜0P(p)=0, φ˜0P =sinϕ(U¯iγ5U)(1)+cosϕ(D¯iγ5D)(1); (4.6)
hence, only the single neutral pseudoscalar bound state φ0 exists. The calculations of Nab in Eq. (4.2) are similar
P Q5
to the ones of Nab in Eq. (3.2) and the results can be obtained by the substitutions
Q
Nab(p)= Nab(p) . (4.7)
Q5 Q p2−4m2→p2
Q
(cid:12)
By means of the gap equation (2.9) and the relation (3.12)(cid:12)we obtain from Eqs. (4.5), (4.4), (4.7), and (3.7) that
[ m2R (p)]2
Γφ0P(p)=−i m2Q/(p2+iε) m2Q[KQ(p)+HQ(p)−iSQ(p)]− m2 [K Q(p)+QHQ (p)+iS (p)]
XQ XQ Q Q PQ Q Q 
r2 P
= i m2/(p2+iε)k+h is . (4.8)
− Q − − k+h+is
Q (cid:18) (cid:19)
X
7
We notice that when p 0, h = 0 and s = r based on Eq. (3.21); so the terms containing the pinch singularities
in the denominator of E→q. (4.8) will cancel each other and Γφ0P(p) becomes finite. The expression (4.8) shows that
p2 =0 is a single pole of the propagator Γφ0P(p) and thus φ0 is a massless neutral pseudoscalar composite particle.
P
It is interesting to indicate that when p2 = 0, S (p) = R (p) = 0 in Eqs. (3.10) and (3.11) since the constraints
Q Q
l2 =m2, (l+p)2 =m2, and p2 =0 could not be submitted simultaneously and in addition, based onEq. (3.9), also
Q Q
H (p)=0; so we will have
Q
Γφ0P(p)= i m2/(p2+iε)k if p2 0, (4.9)
− Q →
Q
X
which has the identical form to the propagator for the neutral pseudoscalar bound state at T = 0 [10], except that
the Q-fermion mass m (T =0) is replaced by m (T,µ ). The result implies that the mass of φ0 is not affected by
Q Q Q P
a finite temperature completely.
V. CHARGED SCALAR BOUND STATE MODE
For the calculation of the propagator for charged scalar bound state, a new feature is that the fermion loop is
constituted by the propagators of the U and D fermions possibly with different masses. Denote the four-point
function for the transition from (U¯Γ−D)(a) to (D¯Γ+U)(b) by Γba (p); then, based on Eq. (2.4), they will obey the
φ−
linear algebraic equations
G
Γbc (p)[δca Lca(p)G( 1)a+1]=i δba( 1)(a+1), b,a=1,2, (5.1)
φ− − − 2 −
c=1,2
X
where
i d4l
Lca(p)= d (R) tr Γ−iSca(l,m )Γ+iSac(l+p,m ) (5.2)
−2 Q (2π)4 U D
Z
(cid:2) (cid:3)
and 2iLca(p) represents the contribution of the fermion loop composed of U-fermion and D-fermion propagators
−
with an a-type Γ+ coupling vertex and a c-type Γ− coupling vertex. The solutions of Eqs. (5.1) are
iG
Γba (p)= [1+GL22(p)]δb1 GL21(p)δb2 δa1 [1 GL11(p)]δb2+GL12(p)δb1 δa2 , (5.3)
φ− 2∆ (p) − − −
C
(cid:0)(cid:8) (cid:9) (cid:8) (cid:9) (cid:1)
where
∆ (p)=[1 GL11(p)][1+GL22(p)]+G2L12(p)L21(p). (5.4)
C
−
The propagatorfor physical charged scalar bound state φ− is
Γφ−(p) Γ11 (p)=iG/2 1 GL11(p)+G2L12(p)L21(p) , (5.5)
≡ φ− − 1+GL22(p)
(cid:20) (cid:21)
and for the orthogonal combination φ˜− to φ− we have
Γφ˜−(p)=0, φ˜− = 1 U¯[cosϕ+sinϕ+(cosϕ sinϕ)γ ]D, (5.6)
5
√2 −
since C containsnoconfigurationφ˜−. Itisnoticedthatφ− anditshermitianconjugateφ+ havethesamepropagator
L4F
and they become the only two charged scalar bound states. Direct calculations give
1 (p2+iε) 1 (m2 m2 )2
L11(p)= g I + [K (p)+H (p) iS (p)]+ E (p)+i U − D S (p)
G Q QQ Q 2 UD UD − UD 2" UD m2U +m2D UD #
X
= L22(p) ∗,
−
(cid:2)i (cid:3)(m2 m2 )2
L12(p)= p2 U − D R (p)exp[β(µ µ )/2],
−2" − m2U +m2D # UD U − D
i (m2 m2 )2
L21(p)= p2 U − D R (p)exp[ β(µ µ )/2], (5.7)
−2" − m2U +m2D # UD − U − D
8
where
1
d (R) m2(1 x)+m2 x Λ2+M2 (p) Λ2
K (p)= Q dx U − D ln UD ,
UD 4π2 m2 +m2 M2 (p) − Λ2+M2 (p)
Z U D (cid:20) UD UD (cid:21)
0
M2 (p)=m2(1 x)+m2 x p2x(1 x), (5.8)
UD U − D − −
d4l (l+p)2 m2
H (p)=4πd (R) − D δ(l2 m2)sin2θ(l0,µ )+(p p,m m ,µ µ ) ,
UD Q (2π)4 [(l+p)2 m2 ]2+ε2 − U U →− U ↔ D U ↔ D
Z (cid:26) − D (cid:27)
(5.9)
m2 m2 d4l
E (p)=4πd (R) U − D
UD Q m2 +m2 (2π)4
U D Z
[(l+p)2 m2][(l+p)2 m2 ]
− U − D δ(l2 m2)sin2θ(l0,µ ) (p p,m m ,µ µ ) , (5.10)
[(l+p)2 m2 ]2+ε2 − U U − →− U ↔ D U ↔ D
(cid:26) − D (cid:27)
d4l
S (p)=4π2d (R) δ(l2 m2)δ[(l+p)2 m2 ]
UD Q (2π)4 − U − D
Z
sin2θ(l0,µ )cos2θ(l0+p0,µ )+cos2θ(l0,µ )sin2θ(l0+p0,µ ) , (5.11)
U D U D
(cid:8) (cid:9)
d4l
R (p)=2π2d (R) δ(l2 m2)δ[(l+p)2 m2 ]sin2θ(l0,µ )sin2θ(l0+p0,µ ). (5.12)
UD Q (2π)4 − U − D U D
Z
Considering the gap equation (2.6) we obtain from Eqs. (5.5) and (5.7) the propagator for φ−:
Γφ−(p)= i/ (p2+iε)[K (p)+H (p) iS (p)]+E (p)+iM¯2S (p)
UD UD UD UD UD
− −
(p2 M¯2)2R2 (p)
(cid:8) − UD (5.13)
−(p2 iε)[K (p)+H (p)+iS (p)]+E (p) iM¯2S (p)}
UD UD UD UD UD
− −
with
M¯2 =(m2 m2 )2/(m2 +m2 ). (5.14)
U − D U D
The mass of φ−(φ+) will be determined by the zero point of the denominator of Γφ−(p). An interesting question is
that under what conditions p2 0 is the pole of Γφ−(p) so that φ− and φ+ would become massless bound states.
→
Let us discuss this problem in two cases.
(1) m = m = m . That is, the two fermion flavors in one generation are mass degenerate. In this case, we have
U D Q
K (p)=K (p), H (p)=H (p), S (p)=S (p), R (p)=R (p), E (p)=0, and M¯2 =0. Thus
UD Q UD Q UD Q UD Q UD
R2(p)
Γφ−(p)= i/(p2+iε) K (p)+H (p) iS (p) Q , (5.15)
Q Q Q
− " − − KQ(p)+HQ(p)+iSQ(p)#
which has a form similar to the propagator (4.8) of pseudoscalar bound state except that now no sum of Q with the
weight m2 exists. Therefore, it follows from Eq. (5.14) that p2 = 0 is the single pole of Γφ−(p) and φ− and φ+ will
Q
be exactly massless charged bound states — charged Nambu-Goldstone bosons. In addition, similar to the case of
Γφ0P, when p 0 the pinch singularities appearing in SQ(p) and RQ(p) also cancel each other.
→
(2) m = m . In this case, we notice that no pinch singularity could appear. This may also be seen from the
U D
6
expressions (5.11) and (5.12) of S (p) and R (p) in which δ(l2 m2)δ[(l+p)2 m2 ] in the integrands will be
UD UD − U − D
equal to be zero if p = 0 and m = m . Therefore, we could calculate the propagator for φ− on the condition that
U D
6
the ghost fields with a=2 are omitted completely and obtain
9
Γφ−(p)=iG/2[1 GL11(p)]
−
= i/ (p2+iε)[K (p)+H (p) iS (p)]+E (p)+iM¯2S (p) . (5.16)
UD UD UD UD UD
− −
Equation(5.16)maybeobta(cid:8)inedapproximatelyfromEq. (5.13)byassumingthatthemoment(cid:9)umcutoffΛinK (p)
UD
is large enough to neglect terms containing R2 (p). The pole of Γφ−(p) is determined by the equation
UD
E (p)+iM¯2S (p)
p2 = UD UD . (5.17)
−K (p)+H (p) iS (p)
UD UD UD
−
Equation (5.17) shows that at finite temperature it is possible that the single pole of Γφ−(p) is not at p2 = 0 and
thus the masses of φ− and φ+ are not equal to zeros. However, it is seen from the right-hand side of Eq. (5.17) that
as long as the momentum cutoff Λ is large enough, the single pole of Γφ−(p) could still be approximately at p2 = 0.
⇀
In particular, we notice that when p2 = 0 and p0 = p 0, both E (p) and S (p) in the numerator of the
UD UD
| | →
right-hand side of Eq. (5.17) will approach zeros. This means that at a low energy scale, φ− and φ+ could still be
considered as massless bound states and identified with charged Nambu-Goldstone bosons.
VI. CONCLUSIONS
We have expounded electroweak symmetry breaking at finite temperature in a one-generation fermion condensate
scheme in the real-time formalism of thermal field theory and in the fermion bubble approximation. It is proved by
means of direct calculations of the propagators for bound states that, at the temperature T below the symmetry
restorationtemperatureT ,itis alwayspossibletoobtainamassiveneutralscalarboundstateφ0,amasslessneutral
c S
pseudoscalar bound state φ0, and two massless charged scalar bound states φ− and φ+ if the two flavors of the
P
one generation of fermions are mass degenerate. In this case, we can precisely identify φ0 with the Higgs boson
S
and φ0, φ∓ with the three Nambu-Goldstone bosons which appear as the products of the spontaneous breaking of
P
electroweakgroupSU (2) U (1) U (1). Inotherwords,theGoldstonetheoremisvalidrigorouslyinthefermion
L Y Q
× →
mass-degenerate case. On the other hand, when the two fermion flavors have unequal masses, we have seen that the
Higgs boson will show double masses due to the effect of ”thermal flactuation” except one of the two flavors being
massless, and the two charged scalar bosons φ∓ will also not be exactly massless. However, we find that as long as
the momentum cutoff Λ of the zero-temperature sectors of the fermion loops is sufficiently large or one is dealing
with low energy scalesof the bound states, then it is still possible approximatelyto obtain a single-Higgs-bosonmass
and almost massless φ∓. In this case we can say that the Goldstone theorem is only valid approximately at a finite
temperature. The well-known top-quark condensate scheme [8] certainely belongs to the latter case. Whether the
appearence of such a situation originates from the real-time formalism itself of thermal field theory deserves to be
examinedfurther. Nevertheless,ourdiscussionshaveshownthatallpossiblepinchsingularitiescanceleachotherand
do not emerge from the final expressions and this is just the result expected in a real-time thermal field theory. It is
worth researching further if the above results based on the Schwinger-Dyson equation in the real-time formalism of
thermal field theory could also appear in the imaginary-time formalism or in an effective potential approach.
ACKNOWLEDGMENTS
This work was done (in part) with the support of the Abdus Salam International Centre for Theoretical Physics,
Trieste,Italy. Itwasalsopartiallysupportedbythe NationalNaturalScienceFoundationofChina andbyGrantNo.
LWTZ-1298 of the Chinese Academy of Sciences.
[1] D.A.Kirzhnitsand A.D.Linde,Phys.Lett. 42B,471 (1972); S.Weinberg,Phys.Rev.D7,2887 (1973); 9,3357 (1974);
L. Dolan and R.Jackiw, ibid.9, 3320 (1974).
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