The non-anticommutative supersymmetric U(1) gauge theory

0.24 MB·English

by  I. Jack

LTH818
The non-anticommutative supersymmetric
U gauge theory
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9 I. Jack, D.R.T. Jones and R. Purdy
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2 Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK

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9 Abstract
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Wediscussthenon-anticommutative ( = 1)supersymmetricU gaugetheory
] N 2 1
in fourdimensions, includingasuperpotential. We performtheone-loop renormal-
h
t isation of the model, including the complete set of terms necessary for renormalis-
-
p ability, showing in detail how the eliminated and uneliminated forms of the theory
e
lead to equivalent results.
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v
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1 Introduction
Deformed quantum field theories have been subject to renewed attention in recent years
due to their natural appearance in string theory. Initial investigations focussed on
theories on non-commutative spacetime in which the commutator of the spacetime co-
ordinates becomes non-zero. More recently [1–9], non-anticommutative supersymmetric
theories have been constructed by deforming the anticommutators of the Grassmann co-
ordinates θα (while leaving the anticommutators of the θα˙ unaltered). Consequently, the
anticommutators of the supersymmetry generators Q are deformed while those of the
α˙
Q areunchanged. It isstraightforwardtoconstruct non-anticommutative versions ofor-
α
dinary supersymmetric theories by taking the superspace action and replacing ordinary
products by the Moyal -product [10] which implements the non-anticommutativity.
∗
Non-anticommutative versions of the Wess-Zumino model and supersymmetric gauge
theories have been formulated in four dimensions [10,11] and their renormalisability
discussed [12–16], with explicit computations up to two loops [17] for the Wess-Zumino
modelandone loopfor gaugetheories [18–22]. Even morerecently, non-anticommutative
theories in two dimensions have been constructed [23,25–28], and their one-loop diver-
gences computed [24,29]. In Ref. [30] we returned to a closer examination of the non-
anticommutative Wess-Zumino model (with a superpotential) in four dimensions, and
showed that to correctly obtain results for the theory where the auxiliary fields have
been eliminated, from the corresponding results for the uneliminated theory, it is neces-
sary to include in the classical action separate couplings for all the terms which may be
generated by the renormalisation process.
Itseemsnaturaltoextendtheabovecalculationstothegaugedcase, forwhichweseek
thesimplestpossiblegaugedextensionoftheWess-Zuminomodelwitha(trilinear)super-
potential. Generalgauged non-commutative theories were considered earlier [18–22], and
in particular gauged interacting theories in Ref. [22]; however there we only considered a
trilinear superpotential in the adjoint SU case, and a mass term in the fundamental U
N N
case. The simplest model with a trilinear superpotential is the three-field Wess-Zumino
model with a U gauge invariance, and it is this model we shall consider here. We shall
1
consider the one-loop renormalisation of this model in its entirety; the divergent contri-
butions in the absence of a superpotential can be extracted from Refs. [18], [19], while
even some of the contributions with a superpotential may be extracted from Ref. [22] by
judicious adaptation of the results there presented for the case of the fundamental U
N
case with mass terms; while a number of the divergent contributions will require a fresh
diagrammatic computation. We start by considering the uneliminated theory and then
proceed to compare with the results from the corresponding theory with the auxiliary
fields eliminated.
2 Action
In this section we shall give the action for an = 1 supersymmetric U gauge theory
N 2 1
coupled to chiral matter with a superpotential [10] [11] [22]. This is obtained by the re-
1
duction to components of the deformed, i.e. non-anticommutative, action in superspace.
A U gauge-invariant superpotential requires at least three chiral fields; we shall take
1
exactly three, with scalar, fermion, auxiliary components denoted φ , ψ , F , i = 1,2,3.
i i i
The corresponding U charges are denoted q , i = 1,2,3. For simplicity we shall consider
1 i
a massless superpotential. For convenience we split the action into kinetic and potential
terms, namely
S = S +S (1)
0 kin pot
where
S = d4x 1FµνF iλ¯σµ(D λ)+ 1D2
kin −4 µν − µ 2
Z (cid:20)
igCµνF λ¯λ¯ +F F iψ σµ(D ψ) (Dµφ) (D φ)
µν i i i µ i i µ i
− − −
+√2gCµν(D φ) λ¯σ ψ +igCµνφ F F + 1 C 2g2F φ λ¯λ¯
µ i ν i i µν i 4| | i i
+ gq φ Dφ +i√2gq (φ λψ ψ λ¯φ )
i i i i i i − i i
Xi n
γ Cµνg √2(D φ) λ¯σ ψ +√2φ λ¯σ (D ψ) +iφ F F , (2)
i µ i ν i i ν µ i i µν i
−
(cid:21)
h io
and
S = d4x (F G yφ ψ ψ yφ ψ ψ yφ ψ ψ )+h.c.
pot i i 1 2 3 2 3 1 3 1 2
− { − − − }
Z h
+2igyCµνF φ φ φ 1y C 2F F F , (3)
µν 1 2 3 − 4 | | 1 2 3
i
where
G = yφ φ , (4)
1 2 3
and similarly for G , G (corresponding to a superpotential W(Φ) = yΦ Φ Φ ). The
2 3 1 2 3
covariant derivative is defined by
(D φ) = (∂ +igq A )φ . (5)
µ i µ i µ i
In Eq. (2), Cµν is related to the non-anti-commutativity parameter Cαβ by
Cµν = Cαβǫ σµνγ, (6)
βγ α
where
σµν = 1(σµσν σνσµ),
4 −
σµν = 1(σµσν σνσµ), (7)
4 −
and
C 2 = CµνC . (8)
µν
| |
Our conventions are in accord with Ref. [10]; in particular,
σµσν = ηµν +2σµν. (9)
−
2
The definition of C 2 is similarly well-established although C2 might be a preferable
| |
notation for this quantity.
For gauge invariance of S we require
pot
q +q +q = 0, (10)
1 2 3
while anomaly cancellation leads to
q q q = 0 (11)
1 2 3
so that the allowed set of charges is in fact (q, q,0). This means that in fact the most
−
general trilinear superpotential is in fact W = yΦ Φ Φ +y′Φ3 (assuming Φ to be the
1 2 3 3 3
neutral field). We choose, however, to retain W = yΦ Φ Φ and to present formulae in
1 2 3
a manner explicitly symmetric under q permutations; for example for later convenience
i
we denote
Q = q2 +q2 +q2. (12)
1 2 3
Note also that it follows from Eqs. (10),(11) that superpotential mass terms are allowed
in general; however as remarked earlier we will restrict ourselves to the massless case.
It is interesting to note that the constraints Eqs. (10),(11) mean that if we set q =
1
q = q and y = √2gq then the undeformed theory has = 2 supersymmetry.
2
− N
It is easy to show that S is invariant under
0
δA = iλ¯σ ǫ,
µ µ
−
δλ = iǫ D +(σµνǫ) F + 1iC λ¯λ¯ , δλ¯ = 0,
α α α µν 2 µν α˙
δD = ǫσµD λ¯,
µ (cid:2) (cid:3)
−
δφ = √2ǫψ , δφ = 0,
i i i
δψα = √2ǫαF , δψ = i√2(D φ )(ǫσµ) ,
i i iα˙ − µ i α˙
δF = 0, δF = i√2D ψ σµǫ 2igq φ ǫλ+2CµνgD (φ ǫσ λ¯). (13)
i i − µ i − i i µ i ν
The set of terms multiplied by γ are separately = 1 invariant under the transfor-
i N 2
mations of Eq. (13); they are not in fact produced by the reduction to components of the
superspace action, but we have anticipated the need for them later when we renormalise
the theory. It will be sufficient to take γ to consist purely of divergent contributions.
i
The C 2F F F and C 2F φ λ¯λ¯ terms in Eqs. (2), (3) are also each separately = 1
| | 1 2 3 | | i i N 2
invariant, and therefore could be omitted from our action without spoiling the = 1
N 2
invariance. However, once we do include the C 2F F F and C 2F φ λ¯λ¯ terms, it is
1 2 3 i i
| | | |
necessary for the renormalisation of the model to include all possible terms which may
be generated, as was explained in the ungauged case in Ref. [30]. It is easy to list these
terms [16] [22]. The action has a “pseudo R-symmetry” under
φ e−iαφ , F eiαF , λ e−iαλ, Cαβ e−2iαCαβ, y eiαy, (14)
i i i i
→ → → → →
F , φ , λ¯ and y transforming with opposite charges to F , φ , λ and y respectively, and
i i i i
all other fields being neutral; and also a “pseudo chiral symmetry” under
φ eiγφ , y e−3iγy, (15)
i i
→ →
3
F and ψ transforming in a similar fashion to φ and barred quantities transforming
i i i
with opposite charges; the gauge fields being unaffected. The divergent terms which can
arise subject to these invariances, for the massless U case and suppressing the 1,2,3
1
subscripts, consist of (in addition to those already present in the action)
C 2F2φ2, y C 2Fφ4, y2 C 2φ6, y C 2λ¯λ¯φ3. (16)
| | | | | | | |
The combination
y−1[F ψ (Cψ )+F ψ (Cψ )+F ψ (Cψ )] (17)
1 2 3 2 3 1 3 1 2
(where (Cψ) = C ψβ) is allowed by the above symmetries and = 1 invariant, but
α αβ N 2
we shall see later that it is not in fact generated as a divergence in the U theory (at least
1
at one loop) if it is not already present in the classical Lagrangian, and so we choose to
2
omit it. Terms of the generic form φ ψ(Cψ) are allowed by the above symmetries but
it is impossible to construct an = 1 invariant combination which includes them. We
N 2
have included in (16) the appropriate factors of y for invariance under the pseudo-chiral
symmetry. These factors are not uniquely determined since yy is invariant under this
symmetry; thechoicewehavemadeisbothconciseandmotivatedbylaterconsiderations.
We must include all the terms in (16) with their own coefficient in the action and
therefore we are led to our complete action
S = S +S (18)
0 gen
where S is given in Eq. (1) and
0
S = d4x y−1 C 2 (k 1yy)F F F +k (F F G +F F G +F F G )
gen | | { 1 − 4 1 2 3 2 1 2 3 2 3 1 3 1 2
Z h
+k (F G G +F G G +F G G )+k G G G
3 1 2 3 2 3 1 3 1 2 4 1 2 3
}
+ C 2 K 1g2 F φ +K yφ φ φ λ¯λ¯ . (19)
| | 1 − 4 i i 2 1 2 3
i
(It is natural to imp(cid:8)o(cid:0)se the sam(cid:1)e cyclic symmetry(cid:9)on S as already present in the
gen
superpotential). The F F F and F φ λ¯λ¯ terms are now effectively assigned an arbitrary
1 2 3 i i
coefficient since the fact that they are separately = 1 invariant (as are all the terms
N 2
in S ) means there is no reason for their renormalisation to be accounted for purely
gen
by replacing quantities in S by the corresponding bare ones; = 1 invariance will not
0 N 2
preserve the values of their coefficients derived from the deformed superfield action.
We use the standard gauge-fixing term
1
S = d4x(∂.A)2 (20)
gf
2α
Z
with its associated ghost terms. The gauge propagator is given by
1 p p
µ ν
∆ = η +(α 1) (21)
µν µν
−p2 − p2
(cid:18) (cid:19)
and the fermion propagator is
p σµ
∆ = µ αα˙, (22)
αα˙ p2
where the momentum enters at the end of the propagator with the undotted index.
4
a 2W
1
−
b W
1
c W
1
−
d 0
Table 1: Divergent contributions from Fig. 2
3 Renormalisation
In this section we discuss the renormalisation of the gauged non-anticommutative Wess-
Zumino model at one loop.
The divergent contributions from one-loop diagrams to terms in S can mostly be
kin
extracted from the results for the SU U case presented in Refs. [18], [19], and so
N 1
×
we shall just give the results (suppressing the well-known C-independent contributions)
without tabulating the contributions from individual diagrams; an exception is the yy-
dependent divergences, since in Ref. [22], where we incorporated a superpotential, we
did not consider the resulting new divergent contributions to terms in S . The corre-
kin
sponding diagrams are depicted in Figs. 1, 2. The contribution from Fig. 1 is simply
2√2yygLCµνφ λ¯σ ∂ ψ , (23)
i ν µ i
−
where
1
L = . (24)
16π2ǫ
The contributions from Fig. 2 are tabulated in Table 1, where
W = i√2yyg2CµνA q φ λ¯σ ψ . (25)
1 µ i i ν i
i
X
(In this and allthe following tables the factorsof Lare suppressed.) Taking into account
the contributions from Table 1, Eq. (23) and those which can be extracted from Ref. [19],
we obtain
Γpole = L d4x 2ig3QCµνF λ¯λ¯ 2√2gyyCµνφ λ¯σ D ψ
kin − µν − i ν µ i
Z h
+ 2√2αg3q2CµνD φ λ¯σ ψ 2ig3Cµνq2φ F F . (26)
i µ i ν i − i i µν i
Xi (cid:16) (cid:17)i
The contributions to S , however, need to be reassessed due to the different form
pot
for the potential, and we therefore show the relevant diagrams in Fig. 3 and list the
corresponding contributions in Table 2. In Table 2, W and W are defined by
2 3
W = iQg3CµνF φ φ φ
2 µν 1 2 3
W = ig3Cµν[q2∂ φ φ φ +q2∂ φ φ φ +q2∂ φ φ φ ]A .
3 1 µ 1 2 3 2 µ 2 3 1 3 µ 3 1 2 ν
(27)
5
a 4W +8W
2 3
b 4W
3
c 2W 12W
2 3
− −
d 8W
2
e 2αW
2
f 2W
2
g 4W 8W
2 3
− −
h 8W
3
i 2αW
2
−
j 2W
2
−
k 4W +8W
2 3
l 8W
3
−
Table 2: Divergent contributions from Fig. 3
The contributions from Table 2 add to
10iQg3L d4xyCµνF φ φ φ . (28)
µν 1 2 3
Z
Note that the contributions from Figs. 3(e)-(h) cancel those from 3(i)-(l); we shall subse-
quently omit several other pairs of diagrams where a similar cancellation occurs (in fact
we have done so already, since a potential divergent yyCµνF Fφ contribution cancels
µν
for this reason).
The divergent contributions to the F F F and F φ λ¯λ¯ terms will be given in detail
1 2 3 i i
shortlysince these terms have nowbeenassigned separate couplingsinS andso thedi-
gen
vergences cannot be extracted from earlier work. The remaining divergent contributions
are denoted by
Γpole = d4x C 2 y−1[X F F F +X F F G +X F F G +X F F G
rem − | | 1 1 2 3 2a 1 2 3 2b 2 3 1 2c 3 1 2
Z (cid:20) (cid:26)
+X F G G +X F G G +X F G G +X G G G
3a 1 2 3 3b 2 3 1 3c 3 1 2 4 1 2 3
+X′(F2φ2 +F2φ2 +F2φ2)+X′′(q φ F +q φ F +q φ F )2]
2 1 1 2 2 3 3 2 1 1 1 2 2 2 3 3 3
+ X F φ +X′ q2F φ +X yφ φ φ λ¯λ¯
5 i i 5 i i i 6 1 2 3
(cid:27)
h X i
+X (q2φ ψ +q2φ ψ +q2φ ψ )(q φ Cψ +q φ Cψ +q φ Cψ ) . (29)
7 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3
(cid:21)
(Note the overall minus sign, introduced to avoid a proliferation of negative signs later
on.) In Figs. 4-9 are depicted the divergent one-loop diagrams contributing to X , etc.
1
Their divergent contributions are shown diagram by diagram in Tables 3-9 and given in
6
X X X X′
1 2a,b,c 3a,b,c 2
a 6k yy
2
b 8k yy 4k yy
2 2
c 4k yy 2k yy
1 1
d 8k yy 2k yy
3 3
e 12k yy
3
f 6k yy
2
g 8k yy
4
Table 3: Divergent contributions from Fig. 4
total by
X(1) = (6k 6g2)yyL,
1 2 −
X(1) = 4(k +2k +2k )yy +2(1+α)k q q g2 L,
2a { 1 2 3 2 1 2 }
X(1) = 4(k +2k +2k )yy +2(1+α)k q q g2 L,
2b { 1 2 3 2 2 3 }
X(1) = 4(k +2k +2k )yy +2(1+α)k q q g2 L,
2c { 1 2 3 2 3 1 }
X(1) = 2(3k +6k +4k )yy+(1+α)[2(k +2k )q q Qk ]g2 L,
3a { 2 3 4 1 2 2 3 − 3 }
X(1) = 2(3k +6k +4k )yy+(1+α)[2(k +2k )q q Qk ]g2 L,
3b { 2 3 4 1 2 3 1 − 3 }
X(1) = 2(3k +6k +4k )yy+(1+α)[2(k +2k )q q Qk ]g2 L,
3c { 2 3 4 1 2 1 2 − 3 }
X(1) = (1+α)(k +2k +2k )Qg2L,
4 − 2 3 4
X′(1) = 2(k +2k +k )yyL,
2 1 2 3
X′′(1) = 1(1+α)g4y,
2 −4
X(1) = [(4K +2K )yy g2yy)L,
5 1 2 −
X(1) = g2(8K 10g2)L,
5′ 1 −
X(1) = [2(7 α)K +(7 α)K +14g2]Qg2L,
6 − 1 − 2
X(1) = 16g4L. (30)
7
The terms involving X′, X′′ and X′ are not contained in the original action; while the
2 2 5
term involving X is not = 1 invariant. However, we shall see later that all these
7 N 2
terms may be removed (at least at one loop) by field redefinitions. Other diagrams which
potentially contribute divergences turn out to be zero or to cancel. Fig. (10) is in fact
zero by symmetry. The divergences from the diagrams of Fig. (11) are of the form
y−1[(q q )F ψ (Cψ )+(q q )F ψ (Cψ )+(q q )F ψ (Cψ )] (31)
2 3 1 2 3 3 1 2 3 1 1 2 3 1 2
− − −
which (in contrast to the similar combination in (17)) is also not = 1 invariant;
N 2
moreover there is no field redefinition which could remove these terms and so they must
and indeed do cancel.
7
X X X X X X X
2a 2b 2c 3a 3b 3c 4
a 2αk q q g2 2αk q q g2 2αk q q g2
2 1 2 2 2 3 2 3 1
b 2k q q g2 2k q q g2 2k q q g2
2 1 2 2 2 3 2 3 1
c αk Qg2 αk Qg2 αk Qg2
3 3 3
− − −
d k Qg2 k Qg2 k Qg2
3 3 3
− − −
e 2αk q q g2 2αk q q g2 2αk q q g2
1 2 3 1 3 1 1 1 2
f 2k q q g2 2k q q g2 2k q q g2
1 2 3 1 3 1 1 1 2
g 4αk q q g2 4αk q q g2 4αk q q g2
2 2 3 2 3 1 2 1 2
h 4k q q g2 4k q q g2 4k q q g2
2 2 3 2 3 1 2 1 2
i 2αk Qg2
4
−
j 2k Qg2
4
−
k αk g2Q
2
−
l k Qg2
2
−
m 2αk Qg2
3
−
n 2k Qg2
3
−
Table 4: Divergent contributions from Fig. 5
X X′′
1 2
a 6yyg2
−
b 1αg4
−4
c 1g4
−4
d 0
Table 5: Divergent contributions from Fig. 6
X X X′
5 6 5
a 8g2K
1
b 4K yy
1
c 2K yy
2
d 2αQg2K
1
−
e 2g2QK
1
−
f 16Qg2K
1
g 8Qg2K
2
h αQg2K
2
−
i Qg2K
2
−
Table 6: Divergent contributions from Fig. 7
8
X X X′
5 6 5
a g2yy
−
b 8g4
−
c 2g4
−
d 0
e 8g4Q
f 1αQg4
2
g 1Qg4
2
h 1(3+α)Qg4
2
i 4Qg4
j αQg4
−
k 0
Table 7: Divergent contributions from Fig. 8
X
7
a 4αg4
−
b 4(3+α)g4
c 4αg4
−
d 4αg4
e 4g4
Table 8: Divergent contributions from Fig. 9
9