Equivalence of Many-Gluon Green Functions in Duffin-Kemmer-Petieu and Klein-Gordon-Fock Statistical Quantum Field Theories

0.1 MB·English

by  B. M. Pimentel

Equivalence of Many-Gluon Green Functions
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0 in Duffin-Kemmer-Petieu and
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Klein-Gordon-Fock Statistical Quantum Field

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a Theories
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B.M.Pimentel∗ and V.Ya.Fainberg†
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l Abstract
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Weprovetheequivalenceofmany-gluonGreenfunctionsinDuffin-

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Kemmer-Petieu(DKP)andKlein-Gordon-Fock(KGF)statisticalquan-
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0 tumfieldtheories. Theproofisbasedonthefunctionalintegralformu-
2 lation for the statistical generating functional in a finite-temperature
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quantum field theory. As an illustration, we calculate one-loop po-
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0 larization operators in both theories and show that their expressions
5 indeed coincide.
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n Keywords: statistical quantum field theory, gluon Green functions, path in-
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l tegral, renormalization, equivalence.
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X 1 Introduction
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This work is a straightforward generalization of the articles [1]–[3] which
established the equivalence of many-photon Green functions in DKP and
KGF statistical quantum field theories.
In Section 2 we present a general proof of equivalence using the func-
tional integral method in statistical quantum field theory. From the physical
∗Instituto de Fisica, Universidade Estadual Paulista, Sa˜o Paulo, Brazil, E-mail:
[email protected].
†P.N.Lebedev Institute of Physics, Moscow, Russia, E-mail: [email protected].
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viewpoint, our result is understandable qualitatively: in non-zero tempera-
ture conditions gluons do not become massive (and thus do not acquire any
chemical potential), so that their intrinsic properties remain the same. To
illustrate this, in Section 3 we calculate one-loop polarization operators in
both theories and show that these operators actually coincide. Section 4
contains conclusive remarks.
2 Coincidence of Many-Gluon Green Func-
tions in DKP and KGF Theories for Finite
Temperatures
To construct agenerating functionalZ(J,J¯,J )for theGreenfunctions (GF)
µ
in statistical theory, we should perform transition to Euclidean space and
then limit the integration area along x : 0 ≤ x ≤ β, where β = 1/T, T is
4 4
the temperature, and J,J¯,J are the external currents. For simplicity, from
µ
now on we restrict ourselves to the case of fundamental representation of the
SU(N) group (see [4], [5]).
In DKP theory, the functional integral describing interaction between the
gluon field Aa and charged particles of spin-0 and mass m has the following
µ
form (in the α-gauge):
1 1
Z (Ji,J¯j,J ) = Z DAaDψiDψ¯iexp − Fa Fa − (∂ Aa)2
DKP µ 0 µ 4 µν µν 2α µ µ
Zβ (cid:26)(cid:20)
−c¯i(∂ Dijcj)+ψ¯i iβ Dij −mδij ψj +J¯iψi +Jiψ¯i +AaJa d4x . (1)
µ µ µ µ µ µ
(cid:21) (cid:27)
(cid:16) (cid:17)
Here a = 1,2,...N2 −1 is the group index; i,j = 1,2,...N ,
j
ij
Dij = δij∂ −ig Aata
µ µ µ
[t ,t ] = if t ,(cid:16) (cid:17) (2)
a b − abc c
where f are the SU(N) group structure constants.
abc
The Euclidean β -matrices in Eq.(1) are chosen as in [3] (see Eq.(4)); the
µ
fields ci,c¯j are the Faddeev-Popov ghosts [6]. As for the functional integra-
tions in this formula, they are understood as
N2−1
DAa = dAa. (3)
µ µ
Zβ Z 0≤Yx4≤β −∞≤Yxi≤+∞ aY=1 Yµ
To prove the coincidence of many-gluon Green functions, let us integrate
out the ψi and ψ¯j fields in Eq.(1). We get:
2
1 1
Z (Ji,J¯j,J ) = Z DAa exp − d4x Fa Fa + (∂ Aa)2
DKP µ 0 µ 4 µν µν 2α µ µ
Zβ (cid:26) Zβ (cid:20)
−c¯i(∂ Dijcj)+JaAa +Tr lnSii(x,x,Aa)
µ µ µ µ µ
(cid:21)
− d4xd4yJ¯i(x)Sij(x,y,Aa)Jj(y) , (4)
Zβ (cid:27)
where
−1
Sij(x,y,A) = β Dij −mδij δ(x−y) (5)
µ µ
istheGreenfunctionfortheDKP(cid:16)-particleinth(cid:17)eexternalYang-MillsfieldAa.
µ
The term Tr lnS(x,x,A) gives rise to all vacuum diagrams in perturbation
theory.
On the other hand,
expTr lnSij(x,x,Aa) = detSij(x,y,Aa)
= DψiDψ¯jexp − d4xψ¯i(x) β Dij +δijm ψj(x) , (6)
µ µ
Zβ (cid:26) Zβ (cid:16) (cid:17) (cid:27)
where ψi(x) is the column vector
φi(x)
∂ φi(x)
 4 
ψi(x) = ∂ φi(x) , (7)
 1 
 ∂ φi(x) 
 2 
 ∂3φi(x) 
 
 
and thus we can rewrite Eq.(6) in the component form as:
expTr lnSij(x,x,Aa) = detSij(x,y,Aa)
4
= D φ∗iD φjDφ∗iDφjexp − d4x φ∗iDijφj
µ µ µ µ
Zβµ=1 (cid:26) Zβ (cid:20)
Y
+φ∗iDijφj +m φiφ∗i +φ∗iφi . (8)
µ µ µ µ
(cid:16) (cid:17)(cid:21)(cid:27)
After integrating over φ∗i and φi we obtain
µ µ
detSij(x,y,Aa) ≡ detGij(x,y,A) = expTr lnGii(x,x,A)
1
= DφiDφ∗jexp + d4xφ∗i (D2)ij −m2δij φj , (9)
m µ
Zβ (cid:26) Zβ (cid:16) (cid:17) (cid:27)
where
Gij(x,y,Aa) = −(D2)ij +m2δij δ4(x−y) (10)
µ
is the Green function of the KG(cid:16)F equation in the(cid:17)external field Aa(x).
µ
It follows from Eqs.(8)–(10) that many-gluon Green functions coincide in
DKP and KGF theories. This completes the proof of equivalence of many-
gluon Green functions in these theories.
3
3 Polarization Operator in One-Loop Approx-
imation
In order to prove the equality of one-loop polarization operators in DKP and
KGF statistical theories it is sufficient to consider the loops formed by scalar
massive particles, since allother one-loopdiagramscoincide inthese theories.
The one-looppolarizationoperatorinDKPtheoryhasthefollowing form:
g2
ΠDKP(k) = Tr dpβ (ta)ijSjk(p+k)β (ta)klSli(p), (11)
µν (2π)2β µ ν
Z
where
Sjk(p) = δjk(ipˆ−m)−1. (12)
One easily checks that
δik ipˆ(ipˆ+m)
Sjk = δjk(ipˆ−m)−1 = − +1 , (13)
m p2 +m2 !
2πn
pˆ= β p , p2 = p2 +p2, p = , −∞ < n < +∞,
µ µ 4 4 β
(ipˆ−m)ijSjk(pˆ) = δik. (14)
Using Eq.(11)–(14) we obtain the polarization operator in DKP theory
(in g2-approximation):
g2 (2p+k) (2p+k)
ΠDKP(k) = (t )ij(t )ji dp µ ν
µν (2π)2β a a (p2 +m2)((p+k)2 +m2)
p4 Z (cid:18)
X
δ δ
µν µν
− − . (15)
p2 +m2 (p+k)2 +m2
(cid:19)
In KGF theory, the one-loop polarization operator equals to:1
g2(ta)ij(ta)ji (2p+k) (2p+k) 2δµν
ΠKGF(k) = dp µ ν − .
µν (2π)3β p4 Z (p2 +m2)((p+k)2 +m2) p2 +m2!
X
(16)
The term proportional to δ in Eq.(16) plays an important role in the proof
µν
of transversality of Π (i.e., k Π = 0). This term appears because of the
µν µ µν
term ∼ (Aa(x))2 entering (D2)ij in Eq.(10).
µ µ
After the substitution (p + k) → p in the term δ ((p + k)2 + m2)−1 of
µν
Eq.(15) it changes into δ (p2+m2)−1, so that the right-hand sides of Eq.(15)
µν
andEq.(16) coincide, andbecome formallygauge-invariant. Thiscoincidence
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The expressions(15) and (16) coincide with Eqs.(14)and (10) in [3] up to the substi-
tution e2 →g2(t )ij(t )ji.
a a
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of ΠDKP and ΠKGF in one-loop approximation confirms the general proof of
µν µν
equivalence presented in Section 2.
In relativistic quantum field theory, the Π (k) tensor has the form:
µν
Π = k k −k2δ Π(k2). (17)
µν µ ν µν
(cid:16) (cid:17)
In quantum statistics, Π depends on the two vectors: k and u , the
µν µ µ
latter being the unit vector of the media velocity. Thus, the most general
expression reads (see [7], page 75)
k k k u (ku)
µ ν µ ν
Π = δ − A + u u −
µν µν k2 1 µ ν k2
(cid:18) (cid:19) (cid:18)
k u (ku) k k (ku)2
− ν µ + µ ν A ≡ Φ1 A +Φ2 A . (18)
k2 k2 2 µν 1 µν 2
(cid:19)
Introducing the notation (in any approximation)
a ≡ Π = 3A +λA
1 µµ 1 2
(ku)2
a = u Π u = λ(A +λA ), λ = 1− , (19)
2 µ µν ν 1 2 k2
we have
1 1 1 3
A = a − a , A = −a + a . (20)
1 1 2 2 1 2
2 λ 2λ λ
(cid:18) (cid:19) (cid:18) (cid:19)
If the system is at rest2
k2
λ = 1− 4 (21)
k2
and
k2 k2 2
a = 1− 4 A + 1− 4 A . (22)
2 k2! 1 k2! 2
To proceed further, it is convenient to represent a and a in the form
1 2
a = aR +aβ, i = 1,2.
i i i
Here the terms aR do not depend on β and must be renormalized; aβ
i i
depend on β and must vanish when β → ∞:
lim aβ = 0. (23)
i
β→∞
Now Eq.(18) may be rewritten in the following form:
1 1 1 1 3 3
Π = Φ1 (aR − aR +aβ − aβ)+ Φ2 (−aR + aR −aβ + aβ).(24)
µν 2 µν 1 λ 2 1 λ 2 2λ µν 1 λ 2 1 λ 2
2
Onecanshowthattherepresentation(18)is,strictlyspeaking,validonlyifthesystem
is at rest: u=0,u4 =1 (see [7], Chapter 11, Section 7).
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The terms ∼ Φ2 should vanish when β → ∞. Correspondingly, in this
µν
limit we get the Π tensor of Euclidean quantum field theory. Since aR and
µν 1
aR do not depend on β, we obtain after renormalization:
2
λ
aR = aR. (25)
2 3 1
Thus
1
lim Π = Φ1 aR, or Π = aR. (26)
β→∞ µν 3 µν 1 µν 1
Let us calculate a and a using the general formula for summation over
1 2
p in the r.h.s. of Eq.(15). We may drop the terms proportional to δ in
4 µν
r.h.s. of Eqs.(10) and (14), because such terms vanish after regularization
and renormalization. Then
g2(N2 −1) (2p+k)2
a = − dp , (27)
1 (2π)32β (p2 +m2)((p+k)2 +m2)
p4 Z
X
g2(N2 −1) (2p+k)2
a = − dp 4 , (28)
2 (2π)32β (p2 +m2)((p+k)2 +m2)
p4 Z
X
on account of (see [5], p.114, Eq.(7.32))
N2 −1
(ta)ij(ta)ji = . (29)
2
The general formula for summation over p reads (see [7], p.123, Appendix
4
3, and [8], p.299):
+∞ +∞+iǫ
1 2πn 1 1 f(ω,K)+f(−ω,K)
f( ,K) = dωf(ω,K)+ dω . (30)
β β 2π 2π e−iβω −1
Xn −Z∞ −∞Z+iǫ
After juxtapositionofEqs.(27),(28) with Eqs.(27),(28) forstatisticalQED [3]
and substitution of g2(N2−1)/2 for e2 we obtain corresponding expressions
for statistical quantum gluodynamics.
Eq.(36) in [3] is replaced by
k k g2(N2 −1) ∞ (1−4m2/z2)3/2
lim Π = − µ ν +δ k4 dz . (31)
β→∞ µν k2 µν! 96π2 ! z2(z2 +k2)
4mZ2
The same expression for Π also follows from Eq.(11) found in [9], where
µν
the photon Green function was calculated using the dispersion approach.
From Eq.(25) and (35) from [??] we find the expression for aR:
2
λ g2(N2 −1)k2 ∞ (1−4m2/z2)3/2
aR = aR = (k4 −k2) dz . (32)
2 3 1 96π2 z2(z2 +k2)
4mZ2
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Finally, we get for aβ and aβ (µ 6= 0):
1 2
∞
g2(N2 −1) pdp −1
aβ = (4m2 +k2) eβ(E−µ) −1
1 32π2 E|k|
Z0 (cid:16) (cid:17)
(k2 +2p|k|)2+4E2k2
×ln 4 (33)
(k2 −2p|k|)2+4E2k2
4
∞
g2(N2 −1) p2dp −1
aβ = eβ(E−µ) −1 (E2 −k2)
2 32π2 Ep|k| 4
Z0 (cid:16) (cid:17) (cid:26)
(k2 +2p|k|)2+4E2k2 (k2 +2iEk )2 −4p2k2
×ln 4 +2iEk ln 4 ,(34)
(k2 −2p|k|)2+4E2k2 4 (k2 −2iEk )2 −4p2k2
4 4 (cid:27)
where E = (p2 +m2)1/2.
4 Conclusions
We have proven the equivalence of many-gluon Green functions in DKP and
KGFstatisticalfieldtheories(Section2)andcalculatedone-looppolarization
operators to illustrate this equivalence (Section 3).
Thus, the series of our works [1–3,9] prove that both theories lead to
identical results for observable physical quantities. In this respect, it would
be interesting to prove the equivalence of the results related to the processes
which involve unstable particles and, in particular, to apply the methods of
DKP theory to the Standard Model.
Acknowledgments. V.Ya.Fainberg is grateful to I.V.Tyutin for useful dis-
cussions, RFFI Fund (Grant No. 02-01-0056) and Scientific Schools Fund
(Grant No. NS-1578.2003.2).
References
[1] B.M.Pimentel, V.Ya.Fainberg, TMP, 124 (2000) 445
[2] R.Casana, V.Ya.Fainberg, B.M.Pimentel, J.S.Valverde, Phys. Lett.,
A316 (2003) 33–43
[3] J.S.Valverde, B.M.Pimentel, V.Ya.Fainberg, TMP, 140 (2004) 44
[4] A.A.Slavnov, L.D.Faddeev, Introduction to Quantum Theory of Gauge
Fields, 2nd ed. corr., Moscow., Nauka, 1988, p. 272.
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[5] M.B.Voloshin, K.A.Ter-Martirosyan, The Theory of Gauge Interaction of
Elementary Particles, Energoatomizdat, 1994
[6] V.N.Popov, L.D.Faddeev, Sov. Phys. Uskekhi, 1979, Vol.III, p.437
[7] E.S.Fradkin, Proc. of the Lebedev Phys. Inst., Vol.29, 1965
[8] H.G.Rothe, Lattice Gauge Theories: An Introduction, Singapore, World
Scientific, 1996
[9] V.Ya.Fainberg, B.M.Pimentel, J.S.Valverde, Dispersion Methods in DKP
Theory: in Proc. of the Int. Meeting on Quantization, Gauge Theories
and Strings dedicated to the memory of E.S.Fradkin (Moscow, June 5–
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Scientific World, 2001, p.79
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