One-loop Energy Corrections in the Nonabelian Gauge Field Theory in 2+1 Dimensions

7 Pages·0.1 MB·English

by  V. Ch. Zhukovsky

One-loop Energy Corrections in the Nonabelian Gauge Field
Theory in 2+1 Dimensions
V. Ch. Zhukovsky, N. A. Peskov
∗
Physical Faculty, Moscow State University,
119899, Moscow, Russia
Abstract
Exact constant solutions of field equations in the classical nonabelian SU(2) field
0
0 theory with the Chern-Simons topological mass in 2 + 1-dimensional space-time have
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been obtained. One-loop contributions of boson and fermion fluctuations to the gauge
2
field energy have been calculated.

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E-mail addresses: [email protected] peskov [email protected].
It is well known that a topological term, the so-called Chern-Simons term, which plays the
role of the gauge field mass, can be introduced in odd-dimensional gauge theories [1, 2].
As an example of such theories, we consider the three-dimensional topologically massive
SU(2)-gluodynamics described by the Lagrangian [1]
1 θ g
= Fa Faµν εµνα Fa Aa εabcAaAbAc . (1)
L −4 µν − 4 µν α − 3 µ ν α
(cid:18) (cid:19)
Here µ,ν = 0,1,2; a = 1,2,3, potentials A τaAa/2, where τa are Pauli matrices in color
µ ≡ µ
space, the color field tensor is given by Fa = ∂ Aa ∂ Aa +gεabcAbAc. The coefficient θ in
µν µ ν − ν µ µ ν
front of the last term in (1) (Chern-Simons term) is the Chern-Simons (CS) mass of the gauge
field.
In this case the field equations take the form
θ
DacFaµβ εβµνFc = 0, (2)
µ − 2 µν
where Dac = δac∂ gεabcAb. It is easily verified that these equations are satisfied by the
µ µ − µ
following constant nonabelian potentials
θ
Aaµ = δaµχ(a), (3)
2g λω
(a)
with normalized constant vector χ = (λi, λωi, ω), where λ = 1 and ω = 1. Note that
λω ± ±
λ and ω take its values independently. The Kronecker delta δaµ in (3) implies that directions
1, 2, 3 in the color space correspond to directions 1, 2, 0 in the Minkowski 2 + 1 space-time,
respectively. The following relations are easily verified:
θ θ2 θ2
Fa = ε Aaα, AaµAb = δab, AaµAaν = gµν.
µν −2 µνα µ 4g2 4g2
The gauge field energy density E corresponds to the 00 component of the symmetric
T
energy-momentum tensor µν obtained by the standard method [3]
T
1
µν = Tµν +∂ Tαµν = Faµ Faν gµν Fa Faα.
T α ∗ ∗ − 2 ∗ α∗
Here Tµν is the canonical energy-momentum tensor, T = Aa Fa + 1θε Aaβ , and
αµν − µ αν 2 ανβ
Fa = 1ε Faαβ, the dual field tensor, is in fact a vector in 2 + 1(cid:16)-dimensional space(cid:17)-time.
∗ µ 2 µαβ
Taking the solution (3) into account we can now represent µν in the form
T
θ4
µν = gµν.
T −32g2
The negative sign of the energy density E = 00 means that, for the given value of θ = 0, the
T 6
energy of this solution is lower than that of the trivial vacuum solution Aµ = 0, obtained in
the perturbation theory.
Next, we consider the one-loop effective potential
V = E + (1) + (1)
eff EG EQ
1
withconsiderationforbothgluon (1) andquark (1) loops. Westartwiththegluoncontribution
EG EQ
(1). The total gauge field aµ is represented as the sum of the constant classical background
EG A
field (3) and quantum fluctuations aaµ, i.e.
aµ = Aaµ +aaµ.
A
Introducing the gauge-fixing term
1 2
= ababµ ,
Lgf −2ξ ∇µ
(cid:16) (cid:17)
where ab = δab∂ +gεabcAc is the background covariant derivative, we obtain the fluctuation
∇µ µ µ
Lagrangian for the gauge ξ = 1 in the form
1 θ2
(ab Ab) = (ab) (∂ abµ)2 (abµ)2 +gεabcAafbµνac gεabcAcabµ∂ aaν.
L µ| µ L µ − 2 µ − 4 µ ν − µ ν
Here, the first term in r.h.s. is the same Lagrangian as in (1), though defined in terms of the
field fluctuations ab and the corresponding tensor fb .
µ µν
The energy spectrum of gluon fluctuation modes ε , corresponding to all degrees of freedom
i
(physical and unphysical) that remain after gauge fixing, is determined by the equation
det δacgµν p2 + 1θ2 iθδacεαµνp 2igεabc Abνpµ +Abµpν gµνAbpα = 0.
2 − α − − α
h (cid:16) (cid:17) (cid:16) (cid:17)i
Nine solutions of the above equation are as follows:
θ2
ε2 = p2 +λ ,
i i 2
where λ = 3 2√2, λ = 5 2√6, and the other five degenerate spectrum branches with
1,2 3,4
± ±
λ = 1, corresponding to tachyonic modes, are unstable.
5,...,9
−
The one-loop energy correction due to gauge field fluctuations around the background is
given by the expression
(1) = 1 ε (p) = 1 [p2 +λ θ2/2]1/2. (4)
EG 2 i 2 i
p,i p,i
X X
Consideration for tachyonic modes contribution to expression (4) results in the appearance of
the imaginary part of the energy correction (1), defined as follows (see, e.g., [3]):
EG
Im (1) = 1 ε2(p) 1/2.
EG −2 | i |
εX2i<0
The imaginary part is finite and its calculation results in the following expression (with due
regard for degeneracy)
5 θ/√2 θ2 5
Im (1) = p2dp = θ 3.
EG −2(2π)2 Z|p|=0 s 2 − −24√2π| |
2
The real part of the energy correction corresponding to ε2 > 0 contribution to (4) is diver-
i
gent. The appropriate application of the dimensional regularization procedure yields:
Re (1)reg = 1µ2ǫ [p2 +λ θ2/2]1/2 ǫ, (5)
EG 2 i −
εX2i>0
where µ is an arbitrary scale mass parameter. We can now introduce the proper time repre-
sentation for this expression, and using the regulator ǫ > 1, we find
2
µ2ǫ 1 ds 4
Re (1)reg = ∞ dp e s(p2+λiθ2/2) +5 ∞ dpe s(p2 θ2/2) ,
EG 2(2π)2Γ(ǫ−1/2) Z0 s3/2−ǫ(cid:26)Z Xi=1 − Z|p|=θ/√2 − − (cid:27)
and this, upon integration over the two-dimensional momentum, yields
µ2ǫ ds 4
Re (1)reg = ∞ 5+ e sλiθ2/2 .
EG 8πΓ(ǫ 1/2) s5/2 ǫ −
− Z0 − (cid:26) Xi=1 (cid:27)
The (ultraviolet) divergent at s 0 term in Re (1)reg is determined by the expression
→ EG
µ2ǫ ds 4 θ2
Re (1)div = ∞ 5+ 1 sλ . (6)
EG 8πΓ(ǫ 1/2) s5/2 ǫ − i 2
− Z0 − (cid:26) Xi=1(cid:16) (cid:17)(cid:27)
The difference Re (1)ren Re (1)reg Re (1)div is finite in the limit ǫ 0 and its calculation
EG ≡ EG − EG →
can be reduced to Γ-function integrals. However, the two results, for imaginary and real parts,
stillincludecontributionsoftheunphysical modesintroducedbythegauge-fixingterm. Inorder
to get rid of these unphysical contributions, the ghost fields ηa and ηa with the Lagrangian
= ( abηb)(Dacµηc)
Lgh ∇µ
should be introduced. Here Dab = δab∂ +gεabc c is the total covariant derivative. The ghost
µ µ Aµ
energy spectrum is determined by the equation:
det( 2)ab = 0,
∇
where ( 2)ab = ac cbµ. Thus, three branches of the ghost spectrum are obtained ε2 =
∇ ∇µ ∇ ghi
p2 + 1ζ θ2, where ζ = 1, ζ = i, and the ghost contribution to the effective potential is
2 i 1 − 2,3 ±
given by the general formula:
(1) = ε (p).
Egh − ghi
p,i
X
Here, the “minus” sign arises as a result of the Fermi statistics applied to ghosts. The branch
ε makes a nonzero contribution only to the imaginary part of effective potential. This is
gh1
easily demonstratedincalculationslike thoseforthegluoncontribution totheenergyimaginary
part. The total result is the following
7
Im (1) +Im (1) = θ 3.
EG Egh −24√2π| |
It is also easy to see that the other energy branches make contribution to the real part only
µ2ǫ ds sθ2
Re (1)reg = µ2ǫ [p2 + 1ζ θ2]1/2 ǫ = ∞ cos .
Egh − 2 i − −2πΓ(ǫ 1) s5/2 ǫ 2
p,Xi=2,3 − 2 Z0 −
3
Finally, we obtain the following result for the boson contribution to the renormalized effective
potential, where only the gluon physical degrees of freedom have been retained
θ4 θ 3 3√6
V E +Re (1) +Re (1) = | | 1+ .
eff ≡ EG Egh −32g2 − 2π 4
(cid:18) (cid:19)
We observe in this connection that the standard prescription for the dimensional regular-
ization integrals, similar to (6), implies that they are assumed to be equal to zero [4]. The
terms like 1/ǫ evidently do not appear in the divergent parts. Hence, one may say that the
divergent part of the effective potential is put equal to zero by definition, i.e., Re (1)div = 0
EG
and Re (1)div = 0, and moreover, masses and coupling constants are not renormalized in this
Egh
case, which is in agreement with the fact that gauge theories are super-renormalizable in three
dimensions [2, 5].
It should also be reminded that the tachyon modes of the gluon spectrum make no contri-
bution to the real part of the correction (4).
We now pass to calculation of the fermion contribution to the effective potential. The quark
energy spectrum in the external field specified by (3) can be found from the Dirac equation
[(i∂ γµ m)+ 1gτaAaγµ]ψ(x) = 0. (7)
µ − 2 µ
Here, the following representation for the γ-matrices in 2 + 1 dimensions is chosen: γ0 = σ3,
γ1,2 = iσ1,2, with σi as Pauli matrices. The γ-matrices obey the following relation: γµγν =
gµν iεµναγ .
α
−
Two quantum states of quark in the SU(2) fundamental representation correspond to two
different colors. Spinors in 2 + 1-dimensions are two-component objects, which corresponds
to two degrees of freedom of a fermion (particle–antiparticle) and, hence, spin coefficients are
closely related to the sign of the particle energy. The mass term mψψ in the Lagrangian is not
invariant under -inversion of coordinates: x x, y y, or x x, y y. Recall that
P → − → → → −
simultaneous reflection of two spatial coordinates is equivalent to rotation. The Lagrangian
is invariant only under the combined operation that involves -inversion and the substitution
P
m m [1, 6]. Therefore, the spin operator for two-dimensional fermions is naturally intro-
→ −
duced in the form Σ = (sgnm)γ0/2. Since the matrix γ0 simultaneously determines the sign of
the energy, the positive- and negative-energy states correspond to opposite signes of the spin
polarization.
Solving equation (7) in the gauge field (3) results in two different branches of the quark
energy spectrum
ε2 = p2 +m2 , ε2 = p2 +m2 ,
1 eff1 2 eff2
where m2 = (m θ˜)2, m2 = (m θ˜)(m+ 3θ˜) and θ˜ = θ/4. It is interesting to note that
eff1 − eff2 ˜ −
in the special case, when m = θ, the quark effective mass in this gauge field vanishes, and
the two branches of the energy spectrum coincide. For values of the mass lying in the interval
θ˜ 2 θ˜ < m < θ˜+2 θ˜, the energy squared, ε2, becomes negative for certain values of the
− − | | − | | 2
quark momentum p, and tachyonic modes appear in the quark spectrum.
Two branches of the energy spectrum are related to opposite projections of the particle
color spin corresponding to two eigenvalues of the operator
J = Jaτa/2 = θ˜ 1gAµp ωγ0τ3(p γµ +θ˜ m) (8)
− µ µ
− −
4
thatisdefined fortheplane-wave solutionsoftheDiracequation(7)ψ (x) = exp( iε t+ip~x)Ψ
s s s
−
with s = 1,2:
JΨ = ( 1)s(θ˜ m)Ψ ,
s s
− −
where Ψ are constant spinors, corresponding to the Dirac Hamiltonian eigenvalues ε . We
s s
˜
note, that the r.h.s. of the above equation vanishes in the case m = θ. Moreover, under this
condition, it appears that Ψ = Ψ .
1 2
The one-loop quark contribution to the vacuum energy is described by the general equation
(1) = Re (1) +iIm (1) = ε (p) i ε (p) . (9)
EQ EQ EQ − s − | s |
ε2s(Xp)>0 ε2s(Xp)<0
The real part of the correction is defined by the expression
1 1
Re (1) = dpε ∞ dpε .
EQ −(2π)2 Z 1 − (2π)2 Z|p|=p∗ 2
Here p = [ m2 ]1/2Θ( m2 ) is the lowest of the quark momentum values that provide a
∗ − eff2 − eff2
real contribution to the quark energy in the absence of the tachyonic modes. Function Θ(x) is
the Heaviside (unit step) function. To find the finite part of the quark energy correction, the
same procedure as in the case of gluons can be applied. The negative sign in front of the real
part of the energy in (9) is due to the fact that quarks are fermions. The quark contribution
to the effective potential finally reads
1 3 m m
Re (1) = m θ˜ (m θ˜)2 | | m+3θ˜ | |(m θ˜)(5m θ˜)+
EQ 6π | − | − − 4 | | − 4 − −
n (cid:16) (cid:17)
m θ˜2 +[(m θ˜)(m+3θ˜)]3/2Θ((m+θ˜)2 4θ˜2) .
| | − −
o
We note that the sign of this expression is opposite to that of the gluon contribution.
Like in the gluon case, the imaginary part of the energy does not diverge and takes nonzero
values only if tachyonic modes are present. The final expression for the imaginary part can be
written in the following simple form
1
Im (1) = (m θ˜)(m+3θ˜) 3/2Θ(4θ˜2 (m+θ˜)2).
EQ −6π| − | −
In particular, in the limit of the vanishing quark mass, m 0, we obtain
→
θ˜3 √3 θ˜3
Re (1) = | | , Im (1) = | | .
EQ 6π EQ − 2π
It should be emphasized that it is the tachyonic modes in the quark spectrum that lead to
the appearance of an imaginary part in the energy of vacuum fluctuations. This means that the
constant gauge field configuration in question is unstable. However, such factors, as boundaries
and finite volume of the spatial region occupied by the gauge field, finite temperature, as
well as consideration for inhomogeneous modes and higher loop contributions may lead to its
stabilization.
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References
[1] Deser S., Jackiw R., Templeton S. Ann. Phys. (N.Y.) 1982. Vol. 140. p. 372.
[2] Jackiw R., Templeton S. Phys. Rev. 1981. Vol. D23. p. 2291.
[3] Sokolov A.A. et al. Kalibrovochnye polya (Gauge Fields), Moscow: Moscow University
Press, 1986, [in Russian].
[4] Faddeev L.D., Slavnov A.A. Gauge Fields: Introduction to Quantum Theory, Benjamin-
Cummings Publishing Co., 1980. Trottier H.D. Phys. Rev. 1991. Vol. D44. p. 464.
[5] Templeton S. Phys. Rev. 1981. Vol. D24. p. 3134.
[6] Deser S., Jackiw R., Templeton S. Phys. Rev. Lett. 1982. Vol. 48. p. 975.
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