The Higgs Mechanism in Non-commutative Gauge Theories

34 Pages·0.31 MB·English

by  Frank J. Petriello

SLAC-PUB-8754
January 2001
The Higgs Mechanism in Non-commutative
Gauge Theories ∗
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2 Frank J. Petriello

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J Stanford Linear Accelerator Center

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3 Stanford University

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Stanford CA 94309, USA
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1 Abstract
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This paper investigates the non-commutative version of the Abelian Higgs model
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h at the one loop level. We find that the BRST invariance of the theory is maintained
t
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p at this order in perturbation theory, rendering the theory one-loop renormalizable.
e
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Upon removing the gauge field from the theory we also obtain a consistent continuum
:
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i renormalization of the broken O(2) linear sigma model, contradicting results found in
X
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a the literature. The beta functions for the various couplings of the gauged U(1) theory
are presented, as are the divergent contributions to every one particle irreducible (1PI)
function. We find that all physical couplings and masses are gauge independent. A
brief discussion concerning the symmetries P, C, and T in this theory is also given.
∗
Work supported by the Department of Energy, Contract DE-AC03-76SF00515
1 Introduction
This paper studies the perturbative aspects of spontaneous symmetry breaking in noncom-
mutative gauge theories. Non-commutative field theories have been the subject of great
activity recently (see, for example, references [1-34]), since being found to arise naturally in
limits of string theory formulated in the presence of background gauge fields [1, 2, 3]. Later
works have treated non-commutative field theories as objects worthy of study independent
of their string theory origins, a point of view adhered to in this paper.
A hallmark of these theories is the mixing of UV and IR divergences [4]; UV di-
vergences in the commutative theory can become IR divergences in the noncommutative
theory. This calls into question the renormalizability of non-commutative field theories.
Several papers have explicitly shown that such theories as φ4 and U(N) gauge theories, when
formulated on a non-commutative space, are one loop renormalizable [5, 6, 7, 8, 9, 10]. How-
ever, results in the literature [11] have shown that non-commutativity renders impossible the
continuum renormalization of the spontaneously broken linear sigma model. They find that
Goldstone’s theorem is violated at the one loop level, and the Goldstone mode obtains a
mass dependent upon the theory’s UV cutoff. The situtation in spontaneously broken gauge
theories seems to also merit investigation, as both an interesting question and as a preface
to any attempts to embed the Standard Model within a non-commutative framework. In
particular, the gauge dependence of the spontaneously broken theory should be checked, as
the analog of the problem seen in [11] would be a gauge dependent shift of one of the masses
in the theory. In this paper we examine the non-commutative Abelian Higgs model at the
one-loop level. We work in an arbitrary R gauge, and show that the resulting BRST invari-
ξ
ance of the action holds when one loop corrections are calculated by finding a counterterm
set capable of removing the divergences from the 1PI functions. We find that the physical
1
couplings and masses are gauge-independent. Upon taking the gauge coupling to zero, we
obtain a continuum renormalization of the broken O(2) linear sigma model, contradicting
the results in [11]. We show that the proper ordering of the NC generalization of φ 4 term in
| |
the globally symmetric theory is that consistent with the local realization of the symmetry.
We then summarize some of the properties of the theory, such as the beta functions for the
various couplings and violations of the discrete symmetries P, C, and T for certain types of
non-commutativity.
The paper is organized as follows. In section 2 we briefly review the essential ideas of
non-commutative (NC) field theory; for a more detailed introduction the reader is referred
to the references in the bibliography. Section 3 reviews the commutative Abelian Higgs
model, concentrating upon setting up the counterterm structure and upon defining gauge
independent physical parameters. We discuss this in detail as similar definitions will be used
when discussing the NC model. In section 4 we construct the NC Abelian Higgs model, and
show by explicit calculation that the theory is renormalizable. We present our conclusions
in section 5.
2 Field Theories on NC Spaces
The essential idea of NCQFT is a generalization of the usual d-dimensional space, Rd, asso-
ciated with commuting space-time coordinates to one which is non-commuting, Rd. In such
θ
a space the conventional coordinates are represented by operators which no longer commute:
[Xˆ ,Xˆ ] = iθ , (1)
µ ν µν
2
where θ represents a real anti-symmetric matrix. NCQFT can be phrased in terms of
µν
conventional commuting QFT through the application of the Weyl-Moyal correspondence
Aˆ(Xˆ) A(x), (2)
←→
where A represents a quantum field with Xˆ being the set of non-commuting coordinates
and x corresponding to the commuting set. However, in formulating NCQFT, one must be
ˆ ˆ ˆ ˆ
careful to preserve orderings in expressions such as A(X)B(X). This is accomplished with
the introduction of the Moyal product, Aˆ(Xˆ)Bˆ(Xˆ) = A(x) B(x), where the effect of the
×
commutation relation is absorbed into the cross. Introducing the Fourier transform pair
1
Aˆ(Xˆ) = dαeiαXˆ a(α)
(2π)d/2
Z
1
a(α) = dxe−iαx A(x), (3)
(2π)d/2
Z
with x and α being real n-dimensional variables, allows us to write the product of two fields
as
1
Aˆ(Xˆ)Bˆ(Xˆ) = dαdβeiαXˆ a(α)eiβXˆ b(β)
(2π)d
Z
= 1 dαdβ ei(α+β)Xˆ−21αµβν[Xˆµ,Xˆν]a(α)b(β). (4)
(2π)d
Z
We thus have the correspondence
Aˆ(Xˆ)Bˆ(Xˆ) A(x) B(x), (5)
←→ ×
provided we identify
A(x) B(x) e2iθµν∂ζµ∂ηνA(x+ζ)B(z +η) . (6)
× ≡
(cid:20) (cid:21)ζ=η=0
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Note that propagators are identical on commutative and NC spaces because quadratic forms
remain unchanged:
i ← →
d4xφ(x) φ(x) = φ(x)exp θ ∂ ∂ φ(x)
µν µ ν
× 2
Z Z (cid:26) (cid:27)
i → →
= φ(x)exp − θ ∂ ∂ φ(x) = d4xφ(x)φ(x), (7)
µν µ ν
2
Z (cid:26) (cid:27) Z
since θ is antisymmetric.
µν
3 Commutative Abelian Higgs Model
Here we review the commutative Abelian Higgs model in some detail, as much of our con-
struction will carry over into the non-commutative case. The commutative Abelian Higgs
model begins with the Lagrangian
1 λ
= − (F )2 + ∂ +igA 2 +µ2 φ 2 φ 4, (8)
AH µν µ µ
L 4 | | | | − 6 | |
where F = ∂ A ∂ A . This Lagrangian is invariant under the gauge transformation
µν µ ν ν µ
−
φ eieα(x)φ,
→
A A ∂ α. (9)
µ µ µ
→ −
The potential
λ
V[φ] = µ2 φ 2 φ 4
| | − 6 | |
has minima at
3µ2
φ 2 = ν2 = . (10)
min
| | λ
4
Expanding
h iσ
φ = ν + + , (11)
√2 √2
we arrive at the Lagrangian
1 1 1 1 1
= − (F )2 + M2A2 + (∂ σ)2 + (∂ h)2 m2h2
AH µν µ µ
L 4 2 2 2 − 2
√2λν √2λν λ λ
+MAµ∂ σ h3 hσ2 h2σ2 h4
µ
− 6 − 6 − 12 − 24
λ g2 g2
σ4 + σ2A2 + h2A2 +ghAµ∂ σ gσAµ∂ h
µ µ
−24 2 2 −
+√2g2νhA2. (12)
The Higgs field has acquired a mass m2 = 2λν2/3, while the gauge boson has acquired a
mass M2 = 2g2ν2. We will work in an R gauge, so to this we add the gauge fixing and
ξ
ghost Lagrangians
1
+ = − (∂ Aµ ξMσ)2 c¯ ∂2 +ξM2 +ξgMh c, (13)
gf gh µ
L L 2ξ − −
(cid:16) (cid:17)
which cancels the A σ cross term. The Feynman rules can be found in [35]. The total
−
Lagrangian + + is invariant under the BRST transformation
AH gf gh
L L L
δh = gσcΘ
−
δσ = McΘ+ghcΘ
δA = (∂ c)Θ
µ µ
−
1
δc¯ = (∂ Aµ ξMσ)Θ
µ
−ξ −
δc = 0 . (14)
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To study the renormalizability of the theory we define the following counterterms
relating the bare and physical quantities:
Aµ = Z1/2Aµ, φ = Z1/2φ, µ2 = Z−1Z µ2,
B A B φ B φ µ
λ = Z−2Z λm4−d, g = Z−1/2Z gm2−d/2, (15)
B φ λ D B A g D
where m is a constant with dimensions of mass used to account for units in dimensional
D
regularization. Note that if we expand
h iσ 3µ2
φ = ν + + , ν2 =
√2 √2 λ
as before, we will no longer be expanding around the minimum of the potential; the higgs
tadpole will acquire a nonzero value. It is convenient, though unnecessary, to introduce a
new counterterm Z , expand
ν
h iσ
φ = Z ν + + , (16)
ν √2 √2
and fix Z by requiring the higgs tadpole to vanish. We could, if desired, refrain from
ν
introducing Z , and include the Higgs tadpole in the calculation of other Green’s functions.
ν
Depending upon the gauge in which we work, Z will be UV divergent. Although it may
ν
seem strange to be expanding the scalar field around an infinite gauge-dependent vev, the
expansion point is not a physical obervable, so no contradiction arises. This procedure is
discussed in [36]. We now take the Lagrangian in eq. (8), written in terms of bare quantities,
and insert the physical quantities and counterterms, while expanding the field φ as in eq.
(16). Our new Lagrangian contains two pieces: the Lagrangian of eq. (12) written in terms
of the physical parameters, and the counterterm Lagrangian, which is used to subtract the
divergences in the physical Green’s functions. The counterterm Lagrangian generated from
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the original Lagrangian plus is
gf
L
1 1 3 1 1
cnt = νm2 Z Z Z3Z h m2 Z2Z Z 1 h2 m2 Z2Z Z σ2
LAH+gf √2 ν µ − ν λ − 2 2 ν λ − 2 µ − − 4 ν λ − µ
(cid:16) (cid:17) (cid:18) (cid:19) (cid:16) (cid:17)
1 1 1 1
+ (Z 1)(∂ h)2 + (Z 1)(∂ σ)2 (Z 1)λh4 (Z 1)λσ4
φ µ φ µ λ λ
2 − 2 − − 24 − − 24 −
1 √2 √2
(Z 1)λh2σ2 (Z Z 1)λνh3 (Z Z 1)λνhσ2
λ ν λ ν λ
−12 − − 6 − − 6 −
Z
A (F )2 +√2(Z Z Z 1)gνAµ∂ σ + Z Z2Z2 1 g2ν2A2
− 4 µν g φ ν − µ φ g ν −
(cid:16) (cid:17)
1 1
+ Z Z Z2 1 g2νhA2 + Z Z2 1 g2σ2A2 + Z Z2 1 g2h2A2
φ ν g − 2 φ g − 2 φ g −
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+(Z Z 1)gAµ[h∂ σ σ∂ h] . (17)
φ g µ µ
− −
This expression uses the fact that is already written in terms of the physical parameters
gf
L
and fields. To determine the counterterms for , we first note that the Higgs-ghost inter-
gh
L
action is super-renormalizable, and therefore doesn’t need a counterterm. Returning to the
gauge transformation of eq. (9) written in terms of the unbroken fields, and expanding as in
eq. (16), we arrive at the counterterm Lagrangian
cnt = (Z 1)ξM2c¯c. (18)
Lgh − ν −
The super-renormalizability of the Higgs-ghost interaction means that we do not need to
introduce a wave-function renormalization constant for the ghost field. The new Lagrangian
is invariant under a “renormalized” BRST symmetry, which is identical to eq. (14) with
M Z M in the δσ transformation.
ν
→
The terms listed above illustrate the subtlety involved with the renormalization of
spontaneously broken theories; a limited number of counterterms are needed to subtract a
large number of divergences. The above theory is renormalizable in spite of these difficulties.
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An explicit one loop calculation reveals the counterterms
g2 3g2 ξg2 ξg2
Z = 1, Z = 1 , Z = 1+ , Z = 1+ ,
g A − 24π2ǫ φ 8π2ǫ − 8π2ǫ ν 8π2ǫ
5λ 9g4 ξg2 λ ξg2
Z = 1+ + , Z = 1+ , (19)
λ 24π2ǫ 4π2λǫ − 4π2ǫ µ 12π2ǫ − 8π2ǫ
where ǫ = 4 d can account for the one loop divergences in this theory, and we have used
−
the minimal subtraction prescription.
In preparation for our discussion of the NC case, let us discuss how to obtain gauge-
independent couplings and masses. Eq. (15) gives the relations between the bare couplings
and physical couplings; solving the equations for the physical couplings in terms of the bare
couplings and renormalization constants, and inserting the expressions of eq. (19) for the
renormalization constants, gives ξ independent expressions for the physical couplings. The
calculationofthephysical couplingsatvariousrenormalizationpointsisfacilitatedbyfinding
their beta functions. We find the following values:
∂λ 5λ2 3λg2 9g4
β(λ) = m = +
D
∂m 24π2 − 4π2 4π2
D
∂g2 g4
β(g2) = m = . (20)
D∂m 24π2
D
These quantities are in agreement with those found in [37]. We can solve these differential
equations to find the relations between physical couplings at various renormalization points;
for example, we find
g2
g2 = 0 , (21)
1 g02 ln mD
− 24π2 mD0
(cid:16) (cid:17)
where g is the coupling at the renormalization point m . Similarly for the masses, we have
0 D0
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the following relations between bare and physical masses:
2
m2 = λ ν2 = Z Z−1m2
B 3 B B µ φ
M2 = 2g2ν2 = Z2Z−1Z Z Z−1M2 (22)
B B B g A µ φ λ
Note that ν = 3µ2/λ ; Z just defines the shift of the expansion point, and does not
B B B ν
enter this expression. We can check that these lead to gauge independent definitions of the
physical masses; calculation yields
λ 3g2
m2 = m2 1 +
B" − 12π2ǫ 8π2ǫ#
λ 5g2 9g4
M2 = M2 1+ + , (23)
B" 8π2ǫ − 12π2ǫ 4π2λǫ#
where the bare masses are infinite in order to cancel the 1/ǫ poles. The important point is
the gauge independence of these results; we will find that the same definitions of the physical
parameters give gauge independent results in the NC theory.
4 Non-commutative Abelian Higgs Model
4.1 Setup of NC Symmetry Breaking in U(1)
We now examine the non-commutative extension of the Abelian Higgs model, following the
procedure introduced in the previous section. Non-commutative U(1) gauge theory coupled
to a complex scalar field is defined by the Lagrangian
1 λ
= − F Fµν +D φ (Dµφ)∗ +µ2 φ 2 φ∗ φ φ∗ φ, (24)
AH µν µ
L 4 × × | | − 6 × × ×
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