Comments on the role of field redefinition on renormalisation of $N=\frac{1}{2}$ supersymmetric pure gauge theory

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by  A. F. Kord

Comments on the role of field redefinition on
renormalisation of N = 1 supersymmetric pure
2
gauge theory
A. F. Kord 1,2, M. Haddadi Moghaddam 1
4 1: Department of Physics, Hakim Sabzevari University (HSU), P.O.Box 397,
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Sabzevar, Iran
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2: Institute for Research in Fundamental Sciences(IPM),P.O. Box 19395-5531,
J

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Tehran, Iran

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p E-mail: [email protected]
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1 Abstract
v
1
4
WestudyoneloopcorrectionstoN = 1 supersymmetricSU(N)×
6 2
1
. U(1) pure gauge theory. We calculate divergent contributions of the
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4 1PI graphs contain the non-anti-commutative parameter C up to one
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:
v loop corrections. We find the disagreement between component for-
i
X
malismandsuperspaceformalismisbecauseofthefieldredefinitionin
r
a
component case. We modify gaugino field redefinition and lagrangian.
We show extra terms of lagrangian have been generated by λ redefini-
tion and are necessary for the renormalisation of the theory. Finally
we prove N = 1 supersymmetric gauge theory is renormalisable up
2
one loop corrections using standard method of renormalisation
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1 Introduction
Theories defined on non-anti-commutative superspace have been studied exten-
sively during last ten years [1, 2]. Superspace in such non-anti-commutative
theories is a superspace whose fermionic supercoordinates are not anticommuta-
tive. One could construct a field theory in non-anti-commutative superspace in
terms of superfields with the star-product where lagrangian is deformed from the
original theory by the non-anti-commutative parameters.
Recentlysomerenormalisabilityaspectsofthenon-anti-commutativefieldtheories
have been studies. It has been shown non-anti-commutative field theories are not
power-counting renormalisable; however it has been discussed that they could be
renormalisableifsomeadditionaltermshavebeenaddedtothelagrangianinorder
to divergences to all orders [3]-[8]. The renormalisability of non-anti-commutative
versionsoftheWess-Zuminomodelhasbeendiscussed [3,4], withexplicitcompu-
tations up to two loops [5]. The renormalisability of non-anti-commutative gauge
theory with N = 1 supersymmetry has been studied in [6, 8]. The authors in [6]
2
show that the theory is renormalisable to all order of perturbation theory. The
conditions of the renormalisability of non-anticommutative (NAC) field theories
have been studied with explicit computations up one and two loops [9]-[15].
The renormalisability of supersymmetric gauge field theories has been discussed in
WZ gauge [6, 7]. The explicit one loop corrections in component formalism have
been done in [9]-[11]. The authors in [10, 11] have claimed the precise form of
the lagrangian is not preserved by renormalisation. They have shown by explicit
calculation that there are problems with assumption of gauge invariance which is
required to rule out some classes of divergent structure in non-anti-commutative
theory.From their calculation, one can see even at one loop divergent non-gauge-
invariant terms are generated. In order to remove the non-gauge-invariant terms
2
and restore gauge invariance at one loops they introduce a one loop divergent field
redefinition in the case of pure N = 1 supersymmetry (i.e. no chiral matter).
2
On the other hand, the authors in [12, 13] have started from superspace formalism
and discussed renormalisability and supergauge invariance. They proved that the
field redefinition is not necessary and the original effective action is not only gauge
but also supergauge invariant up one loop corrections. The disagreement between
two approaches put a big question mark which approaches we should relay on in
N = 1 supersymmetric gauge theory.
2
In this paper we investigate the renormalisability of N = 1 supersymmetric pure
2
gauge theory at one-loop perturbative corrections in component formalism. We
shallshowN = 1 supersymmetricgaugetheoryisrenormalisableinausualmanner
2
without any needs for field redefinition (there is not theoretical justification or
interpretation for the field redefinition as mentioned by authors [10]) which leads
to the lagrangian change. Therefore we shall prove two approaches lead to the
same conclusion.
The paper is organized as follows: First we briefly review NAC supersymmetric
gauge theories and their lagrangian. Then an explicit one-loop calculation of the
three and four-point functions of the theory in the C-deformed sector is carried
out to calculate the corrections. We show some anomaly terms appears in the
1PI functions which spoil the renormalisability of theory. Next we introduce extra
termstotheoriginallagrangianinordertorenormaliseNACpuregaugesupersym-
metric theory and calculate corrections which come from these new terms. Finally
we discus the source of the extra lagrangian, and show that these new terms have
hidden because of the component λ redefinition [1, 20]), so in order to reproduce
them one should reverse gaugino field redefinition.
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2 The pure gauge supersummetric action of
NAC gauge theory
The original non-anticommutative theory defined in superfields appears to require
a U(N) gauge group. Here, at first we would like to consider U(N) gauge the-
ory for non-(anti)commutative (NAC) superspace. The action for an N = 1/2
supersymmetric U(N) pure gauge theory is given by:
(cid:90) (cid:104) 1
S = d4x Tr{− FµνF −2iλ¯σ¯µ(D λ)+D2}
µν µ
2
(cid:105)
−2igCµνTr{F λ¯λ¯}+g2 | C |2 Tr{(λ¯λ¯)2} , (1)
µν
where
F = ∂ A −∂ A +ig[A ,A ],
µν µ ν ν µ µ ν
D λ = ∂ λ+ig[A ,λ], (2)
µ µ µ
and
A = AARA, λ = λARA, D = DARA, (3)
µ µ
Corresponding to any index a for SU(N) we introduce the index A = (0,a), so
that A runs from 0 to N2−1. with RA being the group matrices for U(N) in the
fundamental representation. These satisfy
[RA,RB] = ifABCRC, {RA,RB} = dABCRC, (4)
wherefABC iscompletelyantisymmetric, fabc isthesameasforSU(N)andf0bc =
0, while dABC is completely symmetric; dabc is the same as for SU(N), d0bc =
(cid:113)
(cid:112)2/Nδbc,d00c = 0 and d000 = (cid:112)2/N. In particular, R0 = 1 1 . We have also
2N
1
Tr{RARB} = δAB (5)
2
4
The following identities hold in U(N) group and will be extensively used below
fABLfLCD +fACLfLDB +fADLfLBC = 0, (6)
fABLdLCD +fACLdLDB +fADLdLBC = 0, (7)
fADLfLBC = dABLdLCD −dACLdLDB, (8)
N
fIAJfJBKfKCI = − fABC, (9)
2
N
dIAJfJBKfKCI = − dABCdAcBcC. (10)
2
Where dA = 1+δ0A , cA = 1−δ0A.
Upon substituting the above relations in eq. (1), we obtain the action in the U(N)
case in the form:
(cid:90) (cid:104) 1 1
S = d4x − FµνAFA −iλ¯Aσ¯µ(D λ)A+ DADA
4 µν µ 2
1 1 (cid:105)
− igdABCCµνFAλ¯Bλ¯C + g2dABEdCDE | C |2 (λ¯Aλ¯B)(λ¯Cλ¯D)
2 µν 8
, (11)
With gauge coupling g, gauge field A and gaugino λ.
µ
Beside, definition for F and D λa are given by:
µν µ
FA = ∂ AA−∂ AA−gfABCABAC,
µν µ ν ν µ µ ν
D λA = ∂ λA−gfABCABλC, (12)
µ µ µ
Cµν is related to the non-anti-commutativity parameter Cαβ by:
Cµν = Cαβ(cid:15) σµν γ (13)
βγ α
also, we have:
1
Cαβ = (cid:15)αγσµν βC , (14)
2 γ µν
where
1
σµν β = (σµ σ¯νρ˙β −σν σ¯µρ˙β), (15)
α 4 αρ˙ αρ˙
1
σ¯µνα˙ = (σ¯µα˙ρσν −σ¯να˙ρσµ ). (16)
β˙ 4 ρβ˙ ρβ˙
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The useful identity is:
| C |2 = CµνC . (17)
µν
There are some properties of C in App A. In above Eqs., Cαβ is the non-anti-
commutativity parameter, and our conventions are consistent with ref [1]. The
action for pure N = 1/2 supersymmetric gauge theory (Eq. 11) is invariant under
the standard U(N) gauge transformations and the N = 1/2 supersymmetry trans-
formations. The standard U(N) version of the NAC gauge theory is not renormal-
isable [1]. Therefore; we would like to present a SU(N)×U(1) lagrangian which
has N = 1 supersymmetric properties, so we introduce the following action:
2
(cid:90) (cid:104) 1 1
S = d4x − FµνAFA −iλ¯Aσ¯µ(D λ)A+ DADA
4 µν µ 2
1 1 (cid:105)
− iγABCdABCCµνFAλ¯Bλ¯C + γABCDEdABEdCDE | C |2 (λ¯Aλ¯B)(λ¯Cλ¯D)
2 µν 8
, (18)
One beauty of the above equation is one could easily switch between the original
U(N)theoryandSU(N)×U(1)theory. Inourworkwedefineγabcde = γ , γabcd0 =
0
γ , γ0b0de = γa0c0e = γ0bc0e = γa00de = γ . Indeed we give them in terms of g and
1 2
g . They are given by:
0
g2
γabc = g, γab0 = γa0b = g , γ0ab = (19)
0
g
0
g2
γ = g2, γ = ( )2, γ = g2h (20)
0 1 g 2 0
0
Where h = 1. The above action is similar to the SU(N)×U(1) action in ref [11].
The N=1/2 supersymmetry transformation is:
δAA = −iλ¯Aσ¯ (cid:15) , (21)
µ µ
1
δλA = i(cid:15) DA+(σµν(cid:15)) [FA + iC γABCdABCλ¯Bλ¯C], δλ¯A = 0 , (22)
α α α µν 2 µν α˙
δDA = −(cid:15)σµD λ¯A. (23)
µ
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3 One-loop corrections
In our calculation, we use standard gauge fixing term
(cid:90)
S = 1 d4x(∂.A)2 (24)
gf 2α
and consider the Feynman rules in the super-Fermi-Feynman gauge(α = 1).
In this section we first review the one-loop perturbative corrections to the unde-
formed N = 1 part of the theory. It has been shown that the quantum corrections
of N = 1 part of the theory are not affected by C-deformation [9, 10]. Therefore;
gaugefieldandgauginoanomalousdimensionsandgaugeβ-functionsarethesame
as those in the ordinary N = 1 case. The C-independent part of the bare action
can be written as:
(cid:90) (cid:104) 1 1 (cid:105)
S = d4x − Fµν AFA −iλ¯Aσ¯µ∂ λA+igfABCλ¯Aσ¯µABλC + DADA (25)
C=0 4 µν µ µ 2
The C-independent part of the action is renormalisable if one introduce bare fields
and couplings according to:
1 1
A = Z2A , λ = Z2λ, g = Z g, (26)
Bµ A µ B λ B g
thatZ ,Z andZ areknownasarenormalisationconstants. Alsoonecandefine:
A λ g
1
δ = Z −1,δ = Z −1, δ = Z Z2Z −1, (27)
1 A 2 λ 3 g A λ
finally, one should add the following counter terms to the lagrangian of theory in
order to renormalise theory:
1
L = − δ Fµν AFA −iδ λ¯Aσ¯µ∂ λA+δ igfABCλ¯Aσ¯µABλC, (28)
counter−terms 4 1 µν 2 µ 3 µ
where,
Z = 1+2NL, Z = 1−2NL, Z = 1−3NL, (29)
A λ g
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and L is given by:
g2
L = . (30)
16π2ε
Here ε = 4−D is the regulator.
(We have given here the renormalisation constants corresponding to the SU(N)
sector of the U(N) theory; those for the U(1) sector, namely Z ,Z ,Z are
λ0 A0 g0
given by ommiting the terms in N and replacing g by g . )
0
3.1 One-loop C deformed Corrections
In this part we will present on-loop graphs contributing to the new terms arising
from C-deformed part of the action. The one-loop one-particle-irreducible (1PI)
graphs of the C-deform Aλ¯λ¯ three point functions are depicted in Figs(2). Using
Feynman rules one could compute the divergent contributions from the graphs.
As an example we calculate the one loop corrections to fig(2-a). It is given by:
Figure 1: The diagram (2-a).
Γµα˙β˙ = −g2γAJIdAJIfBDJfCDICµν(cid:15)γ˙δ˙(p +p ) σ¯r α˙γσ¯t β˙δσmσn g
a 1 2 ν γγ˙ δδ˙ rt
(cid:90) ddk (p −k) (p +k)
1 k 2 λ
× (31)
(2π)dk2(p −k)2(p +k)2
1 2
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Using Feynman parameter in App. B:
1 (cid:90) 1 (cid:90) 1 x
= 2 dx dy , (32)
abc [axy+bx(1−y)+c(1−x)]3
0 0
we can simplify denominator of Eq. (31)
1 (cid:90) 1 (cid:90) 1
= 2 dx dy
k2(p −k)2(p +k)2
1 2 0 0
x
× . (33)
[k2+2k.[p x(y−1)+p (1−x)]+p2x(1−y)+p2(1−x)]3
1 2 1 2
By changing variables to
k(cid:48) = k+p x(y−1)+p (1−x),
1 2
the denominator of integral in Eq. (33) is given by:
1 (cid:90) 1 (cid:90) 1 x
= 2 dx dy
k2(p −k)2(p +k)2 [k2−∆]3
1 2 0 0
where
∆ = [p x(2y−1)+2p (1−x)]2
1 2
so, the integral of Eq. (32) is given by:
(cid:90) 1 (cid:90) 1 (cid:90) ddk x(p −k) (p +k)
1 k 2 λ
2 dx dy (34)
(2π)d [k2−∆]3
0 0
then we arrive:
(cid:90) 1 (cid:90) 1 (cid:90) ddk −k k −ig
k λ kλ
2 dx dy = (35)
(2π)d[k2−∆]3 32π2ε
0 0
we finally have for Eq. (31):
Γµα˙β˙ = 4iNLγABCdABCdAcBcC(cid:15)α˙β˙Cµρ(p +p ) (36)
a 1 2 ρ
Moreover, as it be seen in Fig. (2-a), we have momentum - energy conserving in
the loop:
q = (p +p )
ν 1 2 ν
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The divergent contributions up to one loop correction to diagrams in Fig. 2 are
given by:
Γ(1)µα˙β˙ = 4iNLγABCdABCdAcBcC(cid:15)α˙β˙Cµνq
2−a ν
Γ(1)µα˙β˙ = iNLγABCdABCcAdBcC[1(cid:15)α˙β˙Cµνq + 1(Yµν)α˙β˙(p −p ) ]
2−b 2 ν 3 1 2 ν
+one permutation
Γ(1)µα˙β˙ = 1iNLγABCdABCcAdBcC[(cid:15)α˙β˙Cµν(4p +5p ) − 2(Yµν)α˙β˙(2p +7p ) ]
2−c 4 1 2 ν 3 1 2 ν
+one permutation
Γ(1)µα˙β˙ = −3iNLγABCdABCcA(cid:15)α˙β˙Cµνq
2−d ν
Γ(1)µα˙β˙ = −1iNLγABCdABCcAdBcC[(cid:15)α˙β˙Cµν −2(Yµν)α˙β˙]p
2−e 2 2ν
+one permutation (37)
Yµνis been defined,
(Yµν)α˙β˙ = (cid:15)α˙θ˙Cµρg (σ¯λν)β˙ (38)
ρλ θ˙
Where tensor Yµν is symmetric respect to both Lorentz and spinor indices and
tensor Cµν is anti-symmetric. In our computation permutations has taken into
account by changing the position of C as well as symmetry factors. Adding the
different divergent contributions from the diagrams of fig 2 corresponding to dif-
ferent U(1) and SU(N) parts, we have:
Σe Γ(1)µα˙β˙ = 15iNLγabcdabc(cid:15)α˙β˙Cµνq +8iNLγ0bcd0bc(cid:15)α˙β˙Cµνq
i=a 2−i 4 ν ν
1 1
− iNLγa0cda0c(cid:15)α˙β˙Cµνp − iNLγab0dab0(cid:15)α˙β˙Cµνp
2ν 1ν
2 2
1
+ iNLγabcdabc(Yµν)α˙β˙(p −p )
1 2 ν
2
−iNLγa0cda0c(Yµν)α˙β˙p +iNLγab0dab0(Yµν)α˙β˙p (39)
2ν 1ν
LetusnowcontinuewiththerelevantdiagramscontainingonlyC-deformedvertex
and contributing to the four point functions (Fig. 3 and Fig. 4 ). Using the
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