The BRS invariance of noncommutative U(N) Yang-Mills theory at the one-loop level

22 Pages·0.25 MB·English

by  C. P. Martin

FT/UCM–60–2000
The BRS invariance of noncommutative U(N) Yang-Mills theory
at the one-loop level
C. P. Mart´ın* and D. S´anchez-Ruiz†
Departamento de F´ısica Te´orica I, Universidad Complutense, 28040 Madrid, Spain
1
0
0 We show that U(N) Yang-Mills theory on noncommutativeMinkowski space-time
2
canbe renormalized,ina BRS invariantway, at the one-loop level,by multiplicative

n dimensional renormalization of its coupling constant, its gauge parameter and its
a fields. It is shown that the Slavnov-Taylor equation, the gauge-fixing equation and
J
the ghost equation hold, up to order ¯h, for the MS renormalized noncommutative
5
U(N) Yang-Mills theory. We give the value of the pole part of every 1PI diagram
2
which is UV divergent.

3
v
4
2
0
2
1
0
0 1.- Introduction
/
h
t
- Noncommutative field theories occur both in the ordinary (commutative space-time)
p
e field theory setting and in the realm of string theory. The study of the large N limit of
h
ordinary field theories naturally leads to field theories over noncommutative space [1, 2].
:
v General relativity and Heisenberg’s uncertainty principles give rise, when strong gravita-
i
X tional fields are on, to space-times defined by noncommuting operators [3], whereupon it
r arises the need to define quantum field theories over noncommutative space-times. Su-
a
per Yang-Mills theories on noncommutative tori occur in compactifications of M(atrix)-
theory [4], M(atrix)-theory on noncommutative tori being the subject of a good many
papers [5]. Theories of strings ending on D-branes in the presence of a NS-NS B-field lead
to noncommutative space-times; their infinite tension limit being –if unitarity allows it–
certain noncommutative field theories [6]. It is therefore no wonder that a sizeable amount
of work has been put in understanding, either in the continuum [7, 8] or on the lattice [9],
whether quantum field theories make sense on noncommutative space-times. Applications
to collider physics and Cosmology have just begun to come up [10].
Quantum field theories on noncommutative space-time present a characteristic con-
nection between UV and IR scales: the virtual high-momenta modes contributing to a
given Green function yield, when moving around a planar loop, an UV divergence, but
* email: [email protected]
† email: [email protected]
1
give rise to an IR divergence -even if the classical Lagrangian has only massive fields- as
they propagate along a nonplanar loop. This is the UV/IR mixing unveiled in ref. [11],
which has been further investigated in refs. [12]. The new –as regards to quantum field
theoryoncommutativespace-time– IRdivergences thatoccurinnoncommutativefieldthe-
ories makes it impossible [13], beyond a few loops, that these theories be renormalizable
a´ la Bologiubov-Parasiuk-Hepp-Zimmerman [14], if supersymmetry is not called in [15].
Besides, they lack a Wilsonian action [11], which puts in jeopardy the implementation in
noncommutative field theories of Wilson’s renormalization group program [16]; and, hence,
the existence of a continuum limit for these theories. The existence of a continuum limit
for noncommutative field theories has been studied in ref. [17].
It is well known [7] that, at the one-loop level, only if a diagram is planar it can be
UV divergent and that the momentum structure of this divergence, should it exist, is the
product of a polynomial of the appropriate dimension of the external momenta (the UV
degree of divergence of the Feynman loop integral) by suitable Moyal phases. Besides, if
the noncommutative diagram has an ordinary counterpart (the diagram when space-time
is commutative), the polynomial of the external momenta which carries the UV divergence
is the same for both diagrams. This might lead us to think that noncommutative field
theories are always one-loop renormalizable, if their ordinary counterpart is; which would
in turn render almost trivial the issue of the one-loop renormalizability of noncommutative
field theories. One cannot be more mistaken. There are certain ∗-deformations of λφ4 that
are shown not to be renormalizable at the one-loop level: see ref. [18].
It is common lore that U(N) Yang-Mills theories on noncommutative Minowski are
one-loop renormalizable. Indeed, if one assumes that gauge, better, BRS invariance is
preserved attheone-looplevel, itisdifficulttothinkotherwise. However, statementsabout
the BRS invariance of a field theory are rigorous only if they are based either on explicit
computations or on the Quantum Action Principle [19] plus BRS cohomology techniques
[20]. Since we lack a Quantum Action Principle for noncommutative field theories, we
should better carry out explicit computations, lest our statements will be erected on shaky
ground. Even if we had a Quantum Action Principle at our disposal, it would always be
advisable to check general results by performing explicit computations up to a few loops.
In this paper we shall compute the complete UV divergent contribution to the 1PI
functional of 4-dimensional noncommutative U(N) Yang-Mills field theory for an arbitrary
Lorentz gauge-fixing condition. We shall use dimensional regularization to carry out the
computations. We shall thus show by explicit computation that this 1PI functional is the
sum of two integrated polynomials (with respect to the Moyal product) of the field and
its derivatives. The first term is the noncommutative Yang-Mills action. This term is
nontrivial in the cohomology of the noncommutative Slavnov-Taylor operator and gives
rise to the renormalization of the coupling constant. The second term is exact in the
cohomology of the noncommutative Slavnov-Taylor operator and gives rise to the wave
function and gauge-fixing parameter renormalizations. This result constitutes a highly
nontrivial check of the one-loop BRS invariance of the the theory; the high nontriviality
of the check stemming from the fact that the UV divergence of each planar diagram
contributing to the 4-point function of the gauge field has a structure which differs very
much from that of the 4-point tree-level contribution. The BRS invariance of the MS
2
(minimal subtraction) UV divergent part of the one-loop 1PI functional leads, as we shall
see, to a renormalized BRS invariant 1PI functional up to order ¯h.
The layout of this paper is as follows. In Section 2, we set the notation and display the
Feynman rules for noncommutative U(N) Yang-Mills field theory in an arbitrary Lorentz
gauge. In Section 3, we give the UV divergent divergent contribution to the one-loop 1PI
Green functions in the MS scheme of dimensional regularization and, from these data we
obtain the MS UV divergent part of 1PI functional written in an explicitly BRS invariant
form. Section 4 is devoted to comments and conclusions. In the Appendices we give for
the record the UV divergent contribution to each one-loop 1PI diagram in the MS scheme.
2.- Notation and Feynman rules
The classical U(N) field theory on noncommutative Minkowski space-time is given by
the Yang-Mills functional
1
S = − Tr(F ⋆Fµν)(x), (1)
YM µν
4g2T
R
Z
where F (x) is given by
µν
F (x) = ∂ A −∂ A −i{A ,A }(x),
µν µ µ ν µ µ ν
with {A ,A }(x) = (A ⋆A )(x)−(A ⋆A )(x).
µ ν µ ν ν µ
The gauge field, A , is an N × N hermitian matrix, (A )i = N2−1Aa (T )i . We
µ µ j a=0 µ a j
shall take the hermitian U(N) generators, Ta, normalized so that TrT T = T δ , if
a b R ab
a,b ≥ 1, T0 = tR1N×N with (tR)2N = TR. Aaµ is a real vector fPunction on IR4. The
symbol ∗ denotes the Moyal, or star, product defined as follows
d4q d4p
(f ⋆g)(x) = ei(q+p)ei 21θµνqµpν f(q)g(p).
(2π)4 (2π)4
Z Z
Here f(q) and q(p) are, respectively, the Fourier transforms of f(x) and g(x), the latter
being two rapidly decreasing functions at infinity. θµν will be taken either magnetic or
light-like, thus unitarity holds [21]. The reader is referred to ref. [22] for introductions to
Noncommutative geometry. We shall denote 1θµν p q by ω(p,q).
2 µ ν
To quantize the classical noncommutative U(N) Yang-Mills theory we shall use the
path integral defined by means of the BRS quantization method. We introduce then the
noncommutative ghost field, c, the noncommutative antighost field, c¯, and the auxiliary
field B. The fields c, c¯and B are functions on IR4 with values on the Lie algebra of U(N).
The BRS transformations read thus
sA (x) = D c(x) = ∂ c(x)−i{A ,c}(x), sc¯(x) = b, sb(x) = 0, sc(x) = i(c⋆c)(x).
µ µ µ µ
The renormalizationofthe compositetransformationssA (x)and sc(x)demands that
µ
theexternalfieldsJ (x)andH(x),whichcoupletothem,beusheredin. TheBRSinvariant
µ
4-dimensional classical action for an arbitrary Lorentz gauge-fixing condition reads
S = S +S +S , (2)
cl YM gf ext
3
where S has been given in eq. (1) and
YM
1 λ
S = d4x Tr s[c¯⋆( B +∂ Aµ)](x),
gf µ
T 2
R
Z
S = d4x Tr Jµ ⋆sA +H ⋆sc (x),
ext µ
Z (cid:16) (cid:17)
where λ is the gauge-fixing parameter.
Taking into account that
N2−1
(Ta)j1 (Ta)j2 = T δj1 δj2,
i1 i2 R i2 i1
a=0
X
one readily shows that the gauge field propagator reads
d4p (−i) pµ1pµ2
e−ip(x1−x2) T g2δj2 δj1 gµ1µ2 +(λ′ −1) ,
(2π)4 p2 R i1 i2 p2
Z (cid:20) (cid:21)
where λ′ ≡ λ/g2. The remaining propagators and tree-level vertices are obtained from
eq. (2) by using standard path integral techniques. We have gathered the Feynman rules
in Figures 1 and 2. Note that the Feynman rules are rendered in ‘t Hooft’s double index
notation [23]. This notation is very appropriate to work out the colour contribution to a
given U(N) Yang-Mills diagram.
For the Noncommutative U(N) gauge theory to be BRS invariant at the quantum
level the Slavnov-Taylor identity for the 1PI functional Γ[A ,B,c,¯c;J ,H] must hold.
µ µ
This identity reads
δΓ δΓ δΓ δΓ δΓ
S(Γ) ≡ d4x Tr + + B = 0. (3)
δJµ δA δH δc δc¯
µ
Z h i
For the one-loop, Γ , contribution to Γ[A ,B,c,¯c;J ,H], the Slavnov-Taylor equation
1 µ µ
boils down to
B Γ = 0, (4)
1
where
δS δ δS δ δS δ δS δ δ
B = d4x Tr cl + cl + cl + cl + B . (5)
δJµ δA δAµ δJ δH δc δc δH δc¯
µ µ
Z h i
B is the linearized noncommutative Slavnov-Taylor operator.
Since the formal Feynman diagrams contributing to Γ[A ,B,c,¯c;J ,H] present UV
µ µ
divergences, it is not straightforward that the renormalized, would it exist, 1PI functional
defining the quantum theory satisfies the Slavnov-Taylor identity: anomalies may occur.
We shall see in this paper that at the one-loop level the theory at hand can be renormalized
in such a way that eq. (4) holds for the MS renormalized action.
4
m 1ji11 k m 2ji22 GA(0A)µ1µ2ji11ij22(k) = TRg2 δij12δij21 (−k2i) gµ1µ2 +(λ′−1)kµ1kk2µ2
(cid:2) (cid:3)
m j m j
1i11 3i33 iS µ1µ2µ3j1j2j3(k ,k ,k =−k −k )=
AAA i1i2i3 1 2 3 1 2
k k
1 k2 3 g2iTR δij13δij21δij32 eiω(k1,k2)−δij12δij23δij31 e−iω(k1,k2)
m j [gµ1µ2h(k1−k2)µ3 +gµ2µ3(k2−k3)µ1 +gµ3µ1(k3−ik1)µ2]
2
2i
2
m j m j
1i11 4i44 iSAµA1µA2Aµ3µ4ij11ij22ij33ij44(k1,k2,k3,k4=−k1−k2−k3)=
k k
1 4 i δj4δj1δj2δj3 ei[ω(k1,k2)+ω(k3,k4)]
g2TR i1 i2 i3 i4
k2 k3 (2gµ1µ3gµ2µ4 −gµ1µ4gµ2µ3 −gµ1µ2gµ3µ4)
m j m j
2 3 +(1243)+(1324)+(1342)+(1423)+(1432)
2i 3i
2 3
j j
i11 k i22 G(c0c¯)ji11ij22(k)= TR δij12δij21 ki2
m j
i iScc¯Aij22ij11ijµ(k2,k1,k)=
k
j1 j2 ik1µ e−iω(k1,k2)δj2δj δj1 −eiω(k1,k2)δj δj1δj2
i1 i2 TR i1 i2 i i1 i2 i
k1 k2 h i
Figure 1: Feynman rules for noncommutative U(N) Yang-Mills theory: Propagators and
vertices with no external fields. (ijkl) denotes a permutation of (1234).
To regularize the Feynman integrals of our theory will shall use dimensional regu-
larization. To define θµν in dimensional regularization we shall follow the philosophy in
ref. [24] and say that θµν is an algebraic object which satisfies the following equations
θµν = −θνµ, ηˆ θρν = 0, p θµρη θσνp ≥ 0,∀p .
µρ µ ρσ ν µ
Here η and ηˆ are, respectively, the “D-dimensional” and “D-4-dimensional” metrics as
ρσ µρ
defined in ref. [24]. It is not difficult to convince oneself that, with the previous definition
of θ , the one-loop Feynman integrals do have a mathematically well-defined expressions
µν
and that the techniques used in ref. [24] to prove the Quantum Action Principle for dimen-
sionally regularized ordinary field theories can also be employed here to conclude that at
5
j m j
i11 k 2i22 Sc(0;s)Aij11;ji22µ2(k1)= ik1µ2 δij12δij21
1
m j
3i33 S(0) j1j2 j3 (k ,k ,k )=
cA;sAi1i2µ2;i3µ3 1 2 3
m 2ji2 k2 k1 ji1 −i hδij12δij23δij31 e−iω(k1,k2)−δij13δij21δij32 eiω(k1,k2)i gµ2µ3
2 1
j
3
i
3 Sc(0c);scji11ij22;ij33(k1,k2,k3)=
i δj2δj3δj1 e−iω(k1,k2)−δj3δj1δj2 eiω(k1,k2)
i1 i2 i3 i1 i2 i3
j j
i2 k2 k1 i1 h i
2 1
Figure 2: Feynman rules for noncommutative U(N) Yang-Mills theory: Vertices with
insertions of BRS variations.
the one-loop level our dimensionally regularized noncommutative U(N) is BRS invariant.
The so regularized Feynman diagrams are meromorphic functions of D, with simple poles
at D = 4, if they are planar, and no poles, if they are nonplanar and P θµρη θσνP > 0
µ ρσ ν
for any linear combination, P, of the external momenta.
3.- The MS UV divergent part of the 1PI functional
The fact that at the one-loop level only planar diagrams can be UV divergent and
the fact that planar diagrams have the same UV degree as their ordinary field theory
counterparts readily leads to the conclusion that the one-loop UV divergent 1PI functions
are following: Γ , Γ , Γ , Γ , Γ , Γ , Γ and Γ , with obvious notation.
AA AAA AAAA c¯c c¯Ac Jc JAc Hcc
We have computed the one-loop UV divergent contribution to all the divergent 1PI
Green functions. Taking into account the results presented in the Appendices, one obtains
the following values for these UV divergent part in the MS scheme:
1 10
Γ(AA),(pole)j1j2(p) = −(λ′ −1) N δj2 δj1 p2g −p p
µ1µ2 i1i2 (4π)2ε 3 i1 i2 µ1µ2 µ1 µ2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
1 4 3
Γ(AAA),(pole)j1j2j3(p ,p ,p ) = − (λ′ −1)
µ1µ2µ3 i1i2i3 1 2 3 (4π)2ε 3 2
(cid:16) (cid:17) (cid:16) (cid:17)
N e−iω(p1,p2)δj2 δj3δj1 −eiω(p1,p2)δj3 δj1δj2
i1 i2 i3 i1 i2 i3
h i
(p −p ) g +(p −p ) g +(p −p ) g
1 2 µ3 µ1µ2 2 3 µ1 µ2µ3 3 1 µ2 µ1µ3
(cid:16) (cid:17)
6
1 2
Γ(AAAA),(pole)j1j2j3j4(p ,p ,p ,p ) = +2(λ′ −1)
µ1µ2µ3µ4 i1i2i3i4 1 2 3 4 (4π)2ε 3
(cid:16) (cid:17) (cid:16) (cid:17)
N δj4 δj1δj2δj3 ei[ω(p1,p2)+ω(p3,p4)]
i1 i2 i3 i4
(2g g −g g −g g ) +
µ1µ3 µ2µ4 µ1µ4 µ2µ3 µ1µ2 µ3µ4
(1243)+(1324)+(1342)+(1423)+(1432)
′
1 λ −1
Γ(cc¯),(pole)j2j1(p ) =− g2 1− N δj2δj1 p2
i2i1 1 (4π)2ε 2 i1 i2 1
(cid:16) (cid:17) (cid:16) (cid:17)
1
Γ(cc¯A),(pole)j2j1 j3 (p ,p ,p ) = g2 1+(λ′ −1)
i2i1 i3µ 2 1 3 (4π)2ε
(cid:16) (cid:17)
N e−iω(p1,p2(cid:0))δj2 δj3 δj1 −(cid:1)eiω(p1,p2)δj3 δj1 δj2 p
i1 i2 i3 i1 i2 i3 1µ
h 1 λ′ −1 i
Γ(cJ),(pole)j1j2 (p ) =−i g2T 1− δj2 δj1 p
i1i2µ2 1 (4π)2ε R 2 i1 i2 1µ2
(cid:16) (cid:17) (cid:16) (cid:17)
1
Γ(cAJ),(pole)j1j2 j3 (p ,p ,p ) =−i g2T 1+(λ′ −1)
i1i2µ2i3µ3 1 2 3 (4π)2ε R
(cid:16) (cid:17)
N e−iω(p1,p2)δj2 δj3(cid:0)δj1 −eiω(p1,p(cid:1)2)δj3 δj1 δj2 g
i1 i2 i3 i1 i2 i3 µ2µ3
h 1 i
Γ(ccH),(pole)j1j2j3(p ,p ,p ) =i g2T 1+(λ′ −1)
i1i2i3 1 2 3 (4π)2ε R
(cid:16) (cid:17)
N e−iω(p1,p2)δj2(cid:0)δj3 δj1 −eiω((cid:1)p1,p2)δj3 δj1 δj2 , (6)
i1 i2 i3 i1 i2 i3
h i
where D = 4−2ε, (ijkl) stands for the corresponding permutation of the indices and the
following convention has been taken, in order to take properly account of signs:
δΓ δsφ ·Γ
Γ(Φ1Φ2...Φn) = , Γ(φ1φ2...φnKφ) = i ,
i1i2...in δΦi1δΦi2 ...δΦin i1i2...ini δφi1δφi2 ...δφin
1 2 n (cid:12)0 1 2 n (cid:12)0
(cid:12) (cid:12)
with Φ standing for any, internal or ex(cid:12)ternal, field and K for the external fiel(cid:12)d coupled
(cid:12) φ (cid:12)
to sφ. The already known results for the U(1) theory are retrieved by setting N = T = 1
R
in the previous expressions.
Notice that, upon formal generalization of S in eq. (2) to the “D-dimensional” space
cl
of dimensional regularization, the momentum structure of the singular contributions dis-
played above is the same as the corresponding term in S . So, one would expect that these
cl
1PI contributions can be subtracted by MS (minimal subtraction) multiplicative renormal-
ization of the fields and parameters in the BRS invariant action. And, indeed, this is so,
if we perform the following infinite renormalizations
g = µ2εZ g, λ = Z λ, A = Z A , B = Z B,
0 g 0 λ 0µ A µ 0 B
J = Z J , H = Z H, c = Z c and c¯ = Z c¯,
0µ J µ 0 H 0 c 0 c¯
where the subscript 0 labels the bare quantities. Now,
S0 = S −Z(1)(2S +2S +2S )+
YM YM g AA AAA AAAA
(1)
Z (2S +3S +4S ),
A AA AAA AAAA
7
so that
1 5 1
−2Z(1) +2Z(1) = 2N − (λ′ −1) g2T ,
g A (4π)2ε 3 2 R
(cid:20) (cid:21)
1 2 3
−2Z(1) +3Z(1) = 2N − (λ′ −1) g2T ,
g A (4π)2ε 3 4 R
(cid:20) (cid:21)
1 1
−2Z(1) +4Z(1) = 2N − −(λ′ −1) g2T .
g A (4π)2ε 3 R
(cid:20) (cid:21)
Hence,
1 11 1 1
Z = 1− 2N g2T , Z = 1− 2N [1+ (λ′ −1)]g2T . (7)
g R A R
(4π)2ε 6 (4π)2ε 4
Analogously,
′
1 3−λ 1
Z Z = 1+ g2T N, Z Z Z = 1− λ′g2T N,
c¯ c (4π)2ε 2 R c¯ A c (4π)2ε R
1
Z Z2 = 1− λ′g2T N , Z = Z−1, Z = Z2 and Z = Z . (8)
H c (4π)2ε R B A λ A J c¯
That the Zs in eqs. (7) and (8) render UV finite the one-loop 1PI functional is a
consequence of BRS invariance. Indeed, in view of eq. (6), it is not difficult to show that
the singular contribution, Γ(pole), to the dimensionally regularized 1PI functional can be
cast into the form
a
Γ(pole) = dDx Tr (F ⋆Fµν)(x)+B X, (9)
µν D
4g2T
R
Z
where
X = dDx Tr a (J +∂ c¯)⋆A −a H ⋆c (x),
1 µ µ µ 2
Z (cid:16) ′ (cid:17)
1 22 1 3+λ 1
a = N T g2, a = + N T g2, a = + λ′N T g2,
(4π)2ε 3 R 1 (4π)2ε 2 R 2 (4π)2ε R
and B is the linearized Slavnov-Taylor operator acting upon the space of formal ∗-
D
polynomials constructed with “D-dimensional” monomials of the fields and their deriva-
tives. B is defined as follows
D
δS δ δS δ δS δ δS δ δ
B = dDx Tr cl + cl + cl + cl + B .
D δJµ δA δAµ δJ δH δc δc δH δc¯
µ µ
Z h i
The operator B is the “D-dimensional” counterpart of the operator B defined in eq. (5).
D
Eq. (9) gives, in an explicitly BRS invariant form, the UV divergent contribution to
the 1PI functional in the MS scheme of Dimensional Regularization. Note that Γ(pole) is
8
the sum of two terms: the first term, the Yang-Mills term, is B -closed and the second
D
term is B -exact (recall that B2 = 0). The analogy with ordinary SU(N) Yang-Mills
D D
theory is apparent. And, indeed, as as in ordinary four-dimensional Yang-Mills theory we
have
a
Z = 1− , Z = 1+a , Z Z = 1−a +a , Z Z Z = 1+a ,
g A 1 c¯ c 1 2 c¯ A c 2
2
Z Z2 = 1+a , Z = Z−1, Z = Z2 and Z = Z .
H c 2 B A λ A J c¯
Eqs. (7) and (8) are thus retrieved. Let us remark that the values we have obtained for the
Zs agree with the corresponding values of the Zs of SU(N) Yang-Mills theory on ordinary
Minkowski space-time.
(1),MS
We shall define as usual the renormalized one-loop 1PI functional, Γ , in the MS
ren
scheme:
Γr(e1n),MS = LIMε→0 Γ(D1R)eg −Γ(pole) .
h i
(1)
Here Γ denotes the dimensionally regularized 1PI functional at order h¯ and the func-
DReg
tional Γ(pole) is given in eq. (9). The limit ε → 0 is taken after performing the subtraction
of the pole and replacing every “D-dimensional” algebraic object with its 4-dimensional
counterpart [24]; this is why we have denoted it by LIM. We shall not discuss in this
paper how to make sense out of the noncommutative IR divergences [11] that occur in
(1),MS
Γ .
ren
(1)
Since Γ is BRS invariant, i.e., it satisfies the Slavnov-Taylor identity at order h¯
DReg
(1)
B Γ = 0,
D DReg
(1),MS
the MS renormalized 1PI functional Γ is also BRS invariant:
ren
BΓ(1),MS = 0.
ren
The operator B has been defined in eq. (5). We thus conclude that the Slavnov-Taylor
identity (eq. (3)) holds for the renormalized theory at order h¯.
By using standard textbook techniques, one can work out the renormalization group
equation-expressing theinvariance ofthe observables under changes ofthe renormalization
scale µ- for ΓMS:
ren
∂ ∂ ∂ δ
µ +β −δ − γ d4xφ(x) ΓMS[φ;g,θ,λ]= 0.
∂µ ∂g2 λ∂λ φ δφ(x) ren
h Xφ Z i
The fields are denoted by φ. It should be noticed that θ is a dimensionful parameter
µν
which is not renormalized in the MS renormalization scheme.
The one-loop beta function of the theory is easily computed to be
dg2 1 22
β(g2) ≡ µ = − N T g4.
R
dµ 8π2 3
9
The other renormalization group coefficients read at the one-loop level:
′
1 3+λ 1
γ = + N T g2, γ = + λ′N T g2,
A R c R
8π2 2 8π2
γ = γ = γ(cid:0) = −γ(cid:1) , δ = −2γ λ, γ = −γ .
J c¯ B A λ A H c
In the proof of the renormalizabilityof ordinarySU(N) Yang-Millstheory, besides the
Slavnov-Taylor equation, two equations play an important role, namely, the gauge-fixing
and ghost equation [20]. Do these equations also hold for noncommutative U(N) Yang-
Mills theory? It is not difficult to show that Γ(pole) in eq. (9) verifies both the gauge-fixing
equation and the ghost equation:
δΓ(pole) δΓ(pole) δΓ(pole)
= 0, +∂ = 0.
µ
δB δc¯ δJ
µ
Hence, uptoorderh¯, theMSrenormalized1PIfunctionaldoessatisfyboththegauge-fixing
equation and the ghost equation:
δΓMS
T ren = λB +∂A + O(h¯2),
R
δB
δΓMS δΓMS
ren +∂ ren = 0 + O(h¯2).
µ
δc¯ δJ
µ
4.- Conclusions and comments
In this paper we have computed, within the MS scheme of dimensional regularization,
all the one-loop UV divergent contributions to the 1PI functional for noncommutative
U(N) Yang-Mills theory in an arbitrary Lorentz gauge. We have shown that these con-
tributions satisfy the Slavnov-Taylor equation, the gauge-fixing equation and the ghost
equation and, hence, that the MS renormalized 1PI functional satisfies those equations
as well. That the one-loop UV divergent contributions to the 1PI functional satisfy the
Slavnov-Taylor equation has been shown by casting the pole part of the 1PI functional
in an explicitly BRS invariant form. As with ordinary SU(N) Yang-Mills theory, this
explicitly BRS invariant form is the sum of a B-closed term -which is proportional to the
Yang-Mills action- and B-exact term; being B the linearized noncommutative BRS oper-
ator. Our computations lead us to believe that the ∗-deformation of the ordinary BRS
cohomology techniques [20] are to play an important role in the proof of the perturbative
renormalizability of noncommutative gauge theories.
That the 2- and 3-point functions of noncommutative U(N) theory can be renormal-
ized in a BRS invariant way does not come as a surprise once one shows, as we have done
in this paper, that the sum of a given diagram with its permutations, the diagrams being
planar, is proportional to the tree-level 1PI Green function. This generalizes the results
in refs. [1, 7]. The reader is referred to the Appendices for further details; where he can
realize, in particular, that every contribution listed there grows with N as corresponds to
the fact that they come only from planar diagrams.
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