Non-commutative U(1) Super-Yang-Mills Theory: Perturbative Self-Energy Corrections

18 Pages·0.24 MB·English

by  A. A. Bichl

hep-th/0203141
TUW 02-05
UWThPh-2002-09
CERN-TH/2002-051
Non-commutative U(1) Super-Yang–Mills Theory:
Perturbative Self-Energy Corrections
A. A. Bichl1#, M. Ertl2∗, A. Gerhold3∗, J. M. Grimstrup4∗, H. Grosse5&,
L. Popp6∗, V. Putz7$, M. Schweda8∗, R. Wulkenhaar9$
3
0 # Theory Division, CERN
0
CH-1211 Geneva 23, Switzerland
2
∗ Institut fu¨r Theoretische Physik, Technische Universita¨t Wien
n
a Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria
J
& Institut fu¨r Theoretische Physik, Universita¨t Wien
7
1 Boltzmanngasse 5, A-1090 Vienna, Austria

2 $ Max-Planck-Institute for Mathematics in the Sciences
v Inselstraße 22–26, D-04103 Leipzig, Germany
1
4
1
3 Abstract. The quantization of the non-commutative N = 1, U(1)
0
super-Yang–Mills action is performed in the superfield formalism.
2
0 We calculate the one-loop corrections to the self-energy of the
/ vector superfield. Although the power-counting theorem predicts
h
t quadratic ultraviolet and infrared divergences, there are actually
-
p
only logarithmic UV and IR divergences, which is a crucial feature
e
h of non-commutative supersymmetric field theories.
:
v
i
X
r
a [email protected], work supported in part by “Fonds zur F¨orderung der Wissenschaftlichen
Forschung” (FWF) under contract P14639-TPH.
[email protected], work supported by “Fonds zur F¨orderung der Wissenschaftlichen Forschung”
(FWF) under contract P13125-TPH.
[email protected], work supported in part by “Fonds zur F¨orderung der Wissenschaft-
lichen Forschung” (FWF) under contract P13126-TPH.
[email protected], work supported by the Danish Research Agency.
5
[email protected].
[email protected], worksupportedinpartby “Fondszur F¨orderungder Wissenschaftlichen
Forschung” (FWF) under contract P13125-PHY.
[email protected],work supported in part by “Fonds zur F¨orderungder Wissenschaftlichen
Forschung” (FWF) under contract P13126-TPH.
[email protected].
[email protected],Schloeßmann Fellow.
Dedicated to Olivier Piguet on the occasion of his 60th birthday
1 Introduction
We know that the concept of space-time as a differentiable manifold cannot be reason-
ably applied to extremely short distances [1]. Simple heuristic arguments show that it
is impossible to locate a particle with arbitrarily small uncertainty [2]. An interesting
concept in order to replace standard differential geometry is non-commutative geometry
pioneered by Connes [3, 4]. Non-commutative geometry can be regarded as an extension
of the principles of quantum mechanics to geometry itself: space-time coordinates become
non-commutative operators.
The general strategy in non-commutative geometry is to generalize the mathematical
structures encountered in ordinary physics. Standard quantum field theories deal with
problems of interactions at short distances. Quantum field theory (QFT) on spaces with
different short-distance structure may therefore show interesting features. Since singular-
ities in standard QFT are a consequence of point-like interactions, there has been hope
that ‘smearing out the points’ [5] avoids these UV divergences. However, it was first
noticed by Filk [6] that divergences are not avoided on non-commutative R4. This raised
the question of whether the QFT is renormalizable, or not.
Scalar field theories were investigated in [7, 8], where a crucial feature of non-
commutative field theories appears—the UV/IR mixing. On the one-loop level the ques-
tion of renormalizability was affirmed for Yang–Mills theory on non-commutative R4
[9, 10, 11] and the non-commutative 4-torus [12] as well as for supersymmetric Yang–
Mills theory in (2+1) dimensions, with space being the non-commutative 2-torus [13].
QED on non-commutative R4 was treated in [14, 15] and BF–Yang–Mills theory in [16].
The Chern–Simons model on non-commutative space was treated in [17], see also [18].
Concerning supersymmetry, also a deformationof theanticommutator ofthe fermionic
superspace coordinates was considered [19], but this deformation is not compatible with
supertranslations andchiralfields. At thecomponent level, renormalizability oftheWess–
Zumino model to all orders of perturbation theory was shown in [20]. A superspace
formulation (at the classical level) of the Wess–Zumino model and of super-Yang–Mills
theory was given in [21]. Eventually, renormalizability of the Wess–Zumino model in
the non-commutative superspace formalism was established in [22]. Non-commutative
N = 1,2 super-Yang–Mills theories were studied by Zanon in [23], using the background
field method, with the result that at oneloopthere areonly logarithmic divergences in the
self-energy. This is remarkable because the power-counting theorem predicts quadratic
divergences for N = 1 super-Yang–Mills theory, which would lead, according to the
power-counting analysis of non-commutative field theories by Chepelev and Roiban [24],
to non-renormalizability on non-commutative space-time. The lowering of the degree of
divergence from quadratic to logarithmic seems to be governed by non-renormalization
theorems, see [25].
In this paper we reinvestigate the question of UV/IR mixing in non-commutative N =
1 super-Yang–Mills theory, where we work in the non-commutative superfield formalism
[22]. It turns out that the one-loop self-energy of the superfield suffer indeed only from
logarithmic IR divergences. UV divergences are multiplicatively renormalizable as usual.
1
Assuming that this behaviour continues to all orders, non-commutative N = 1 super-
Yang–Mills theory would be renormalizable, according to [24], provided that commutants-
type divergences are absent.
Therefore, non-commutativity does not spoil the cancellation of quadratic and linear
divergences in supersymmetric theories, as stated already in the literature [26, 20, 27, 23,
28, 29, 30].
On the other hand, non-commutative non-supersymmetric theories suffer from
quadratic (linear) IR divergences which would prevent renormalizability at higher loop
order. Possible ways out could be hard non-commutative loops resummation [31] or the
use of field redefinitions [32].
The paper is organized as follows: Section 2 presents the Moyal product applied to
superfields, while section 3 treats the action of our model. In section 4 the Legendre
transformation and the perturbative expansion are performed and, after a short power-
counting argument given in section 5, the self-energy of the vector superfield is calculated
at the one-loop level (section 6). Appendices contain some calculations and conventions.
2 Moyal Product for Superfields
We consider a non-commutative (N = 1) superspace characterized by the algebra
[xˆµ,xˆν] = iΘµν, (1)
where Θµν is an antisymmetric, constant and real matrix. We do not deform the anti-
commuting coordinates θ and θ¯α˙, i.e. we assume
α
{θ ,θ } = {θ¯α˙,θ¯β˙} = {θ ,θ¯α˙} = [xˆµ,θ ] = [xˆµ,θ¯α˙] = 0. (2)
α β α α
The non-commutative algebra is represented on an ordinary manifold by the Moyal prod-
uct [6]. The Moyal product of two vector superfields can be written as [22]
(φ⋆φ′)(x,θ ,θ¯ ) = dP dP δ˜ (1,2)δ˜ (1,3)
1 1 V2 V3 V V
Z
× φ˜(p ,θ ,θ¯ )φ˜′(p ,θ ,θ¯ )e−i(p2+p3)xe−ip2∧p3. (3)
2 2 2 3 3 3
The Moyal product has the important property
dV (φ⋆φ′)(1) = dV (φ′ ⋆φ)(1) = dV φ(1)φ′(1). (4)
1 1 1
Z Z Z
This implies in particular that one can perform cyclic rotations of the fields under the
integral.
For definiteness we have used vector superfields in (3) and (4). Of course, one can
easily write down analogous formulae for (anti-)chiral superfields.
2
3 The Action
For simplicity we choose the gauge group U(1). We introduce a vector superfield φ whose
gauge transformation is given by [21]:
(eφ′) = (e−iΛ¯) ⋆(eφ) ⋆(eiΛ) , (5)
⋆ ⋆ ⋆ ⋆
with a chiral superfield Λ (gauge parameter). With the help of the Baker–Campbell–
Hausdorff formula we obtain the infinitesimal gauge transformation of φ itself:
i i
φ′ = φ+i(Λ−Λ¯)+ [φ,Λ+Λ¯] + [φ,[φ,Λ−Λ¯] ] +..., (6)
⋆ ⋆ ⋆
2 12
where the dots denote terms that contain three or more powers of φ. The gauge-invariant
NCSYM action is given by
1
S = − dSWαW , (7)
inv α
128g2
Z
with
W := D¯2 (e−φ) ⋆D (eφ) . (8)
α ⋆ α ⋆
We perform a Taylor expansion of the integrand,
(cid:0) (cid:1)
1
S = − dV −φDαD¯2D φ−(D¯2Dαφ)[φ,D φ]
inv α α ⋆
128g2
Z (cid:20)
1 1
− [φ,D¯2Dαφ] [φ,D φ] + [φ,Dαφ] D¯2[φ,D φ] +O(φ5) .
⋆ α ⋆ ⋆ α ⋆
3 4
(cid:21)
(9)
In order to prepare the quantization, we introduce a chiral superfield B (multiplier field)
and two chiral anticommuting superfields c (ghost) and c (antighost). The BRS trans-
+ −
formations are given by:
1 1
sφ = c −c¯ + [φ,c +c¯ ] + [φ,[φ,c −c¯ ] ] +...
+ + + + ⋆ + + ⋆ ⋆
2 12
=: Q (φ,c ),
s +
sc = −c ⋆c , sc¯ = −c¯ ⋆c¯ ,
+ + + + + +
sc = B, sc¯ = B¯,
− −
sB = 0, sB¯ = 0. (10)
Now we can write down the BRS-invariant total action:
S = S +S +S , (11)
tot inv gf φπ
where S is given by (9) and the gauge fixing and the Faddeev–Popov terms are given
inv
by [33]:
1
S = − dV(B +B¯)φ, (12)
gf
128
Z
1
S = dV(c +c¯ )Q (φ,c ). (13)
φπ − − s +
128
Z
3
Using (10), the Faddeev–Popov term can be rewritten as
1
S = dV c¯ c −c c¯
φπ − + − +
128

Z
1 1
+ c +c¯ [φ,c +c¯ ] + [φ,[φ,c −c¯ ] ] +... . (14)
− − + + ⋆ + + ⋆ ⋆
2 12
!
(cid:18) (cid:19)
(cid:16) (cid:17)
Again the dots denote terms with three or more powers of φ.
In the following we will also include a mass term in the total action,
1
S = dVM2φ2, (15)
mass
16g2
Z
in order to avoid an IR divergence in the propagator of the vector superfield.
4 Generating Functionals
The generating functional of connected Green’s functions for the free theory can be ob-
tained from the bilinear part S of S +S via a Legendre transformation:
bil tot mass
Zc = S + dVJφ+ dS(J B +η c +η c )
bil bil B − + + −
Z Z
+ dS¯ J B¯ +η¯ c¯ +η¯ c¯
B¯ − + + −
Z
(cid:0) (cid:1)
= S + dP J˜ φ˜ + dP J˜ B˜ +η˜ c˜ +η˜ c˜
bil V −p p S B,−p p −,−p +,p +,−p −,p
Z Z (cid:16) (cid:17)
+ dP J˜ B˜¯ +η˜¯ c˜¯ +η˜¯ c˜¯ , (16)
S¯ B¯,−p p −,−p +,p +,−p −,p
Z
(cid:16) (cid:17)
where φ, B, B¯, c and c¯ are replaced by the inverse solutions of
± ±
δ S
V bil = −J˜,
δ φ˜ p
V −p
δ S δ S
S bil = −J˜ , S¯ bil = −J˜ ,
δ B˜ B,p δ B˜¯ B¯,p
S −p S¯ −p
δ S δ S
S bil S bil
= η˜ , = η˜ ,
δ c˜ +,p δ c˜ −,p
S −,−p S +,−p
δ S δ S
S¯ bil = η˜¯ , S¯ bil = η˜¯ . (17)
δ c˜¯ +,p δ c˜¯ −,p
S¯ −,−p S¯ +,−p
This leads to
1
Zc = dP dP J˜ ∆ (1,2)J˜ + dP dP J˜ ∆ (1,2)J˜
bil V1 V2 2 −p1 φφ −p2 V1 S2 −p1 φB B,−p2
Z Z
+ dP dP J˜ ∆ (1,2)J˜ + dP dP J˜ ∆ (1,2)J˜
V1 S¯2 −p1 φB¯ B¯,−p2 S1 S¯2 B,−p1 BB¯ B¯,−p2
Z Z
− dP dP η˜¯ ∆ (1,2)η˜ − dP dP η˜¯ ∆ (1,2)η˜ , (18)
S¯1 S2 −,−p1 c¯+c− +,−p2 S¯1 S2 +,−p1 c¯−c+ −,−p2
Z Z
4
where the propagators are given by
1− 1θ2 θ¯2 p2
∆ (1,2) = −g2(2π)4δ4(p +p )ep2,µ(θ2σµθ¯1−θ1σµθ¯2) 4 21 21 2,
φφ 1 2 p2(p2 −M2)
2 2
1−(θ σρθ¯ )p + 1θ2 θ¯2 p2
∆ (1,2) = 8(2π)4δ4(p +p )ep2,µ(θ2σµθ¯1−θ1σµθ¯2) 21 21 2,ρ 4 21 21 2,
φB 1 2 p2
2
1+(θ σρθ¯ )p + 1θ2 θ¯2 p2
∆φB¯(1,2) = 8(2π)4δ4(p1+p2)ep2,µ(θ2σµθ¯1−θ1σµθ¯2) 21 21 p22,ρ 4 21 21 2,
2
∆ (1,2) = ∆ (1,2) = ∆ (1,2),
c¯+c− c¯−c+ φB
16M2
∆ (1,2) = ∆ . (19)
BB¯ g2 φB¯
The generating functional of general Green’s functions is given by
Z = N−1e~iΓinte~iZbcil, (20)
where N is a normalization factor such that Z[0] = 1 and
1 ~ 4 δ4
Γ = dP dP dP dP Γ (1,2,3,4) V
int 4! i V1 V2 V3 V4 φ4 δ J˜ δ J˜ δ J˜ δ J˜
(cid:18) (cid:19) Z V −p1 V −p2 V −p3 V −p4
1 ~ 3 δ3
+ dP dP dP Γ (1,2,3) V
3! i V1 V2 V3 φ3 δ J˜ δ J˜ δ J˜
(cid:18) (cid:19) Z V −p1 V −p2 V −p3
~ 3 δ2δ
+ dP dP dP Γ (1,2,3) S V
i V1 V2 V3 c−c+φ δ η˜ δ η˜ δ J˜
(cid:18) (cid:19) Z S −,−p2 S +,−p1 V −p3
~ 3 δ δ δ
+ dP dP dP Γ (1,2,3) S S¯ V
i V1 V2 V3 c¯−c+φ δ η˜ δ η˜¯ δ J˜
(cid:18) (cid:19) Z S −,−p2 S¯ +,−p1 V −p3
~ 3 δ δ δ
+ dP dP dP Γ (1,2,3) S¯ S V
i V1 V2 V3 c−c¯+φ δ η˜¯ δ η˜ δ J˜
(cid:18) (cid:19) Z S¯ −,−p2 S +,−p1 V −p3
~ 3 δ2δ
+ dP dP dP Γ (1,2,3) S¯ V
i V1 V2 V3 c¯−c¯+φ δ η˜¯ δ η˜¯ δ J˜
(cid:18) (cid:19) Z S¯ −,−p2 S¯ +,−p1 V −p3
1 ~ 4 δ2δ2
+ dP dP dP dP Γ (1,2,3,4) S V
2! i V1 V2 V3 V4 c−c+φ2 δ η˜ δ η˜ δ J˜ δ J˜
(cid:18) (cid:19) Z S −,−p2 S +,−p1 V −p3 V −p4
1 ~ 4 δ δ δ2
+ dP dP dP dP Γ (1,2,3,4) S S¯ V
2! i V1 V2 V3 V4 c¯−c+φ2 δ η˜ δ η˜¯ δ J˜ δ J˜
(cid:18) (cid:19) Z S −,−p2 S¯ +,−p1 V −p3 V −p4
1 ~ 4 δ δ δ2
+ dP dP dP dP Γ (1,2,3,4) S¯ S V
2! i V1 V2 V3 V4 c−c¯+φ2 δ η˜¯ δ η˜ δ J˜ δ J˜
(cid:18) (cid:19) Z S¯ −,−p2 S +,−p1 V −p3 V −p4
1 ~ 4 δ2δ2
+ dP dP dP dP Γ (1,2,3,4) S¯ V ,
2! i V1 V2 V3 V4 c¯−c¯+φ2 δ η˜¯ δ η˜¯ δ J˜ δ J˜
(cid:18) (cid:19) Z S¯ −,−p2 S¯ +,−p1 V −p3 V −p4
(21)
5
where we have defined
δ3S
Γ (1,2,3) = V int ,
φ3 δ φ˜ δ φ˜ δ φ˜
V p1 V p2 V p3(cid:12)0
(cid:12)
δ4S
Γφ4(1,2,3,4) = δ φ˜ δ φ˜V δintφ(cid:12)(cid:12)˜ δ φ˜ ,
V p1 V p2 V p3 V p4(cid:12)0
(cid:12)
δ3S
Γc−c+φ(1,2,3) = δ c˜ δV c˜int δ φ˜ ,(cid:12)(cid:12)
V −,p1 V +,p2 V p3(cid:12)0
(cid:12)
δ4S
Γc−c+φ2(1,2,3,4) = δ c˜ δ c˜V inδt φ˜ δ(cid:12)(cid:12) φ˜ , (22)
V −,p1 V +,p2 V p3 V p4(cid:12)0
(cid:12)
and similarly for the terms Γ (1,2,3), Γ (1,2,3),Γ (1,(cid:12)2,3), Γ (1,2,3,4),
c¯−c+φ c−c¯+φ c¯−c¯+φ (cid:12) c¯−c+φ2
Γ (1,2,3,4) and Γ (1,2,3,4). Here, S is the interaction part of S , and the
c−c¯+φ2 c¯−c¯+φ2 int tot
subscript 0 means that all the fields have to be set to zero after the functional deriva-
tives have been performed. The mixture of (anti-)chiral and vectorial field derivatives
in the ghost sector is due to our convention that source terms for the ghosts involve the
(anti-)chiral measure (16), whereas interactions between ghosts and vector superfields are
defined in terms of the vectorial measure (14). One should notice here that these are all
necessary vertices for the calculation of the one-loop self-energy part of the vector super-
field. The final results for these vertex functions (22) are rather complicated and can be
looked up in appendix A.
Furthermore, we mention the generating functional of connected Green’s functions,
which is given by
~
Zc = lnZ. (23)
i
5 Power-Counting
We note that, apart from the exponentials and the θ-factors in the numerator, the vector
field propagators are of order 1 and the ghost propagators of order 1 . We consider the
(p2)2 p2
exponentials and the θ-factors. From the invariance of Green’s functions with respect to
translations and supersymmetry transformations one finds that a one-particle irreducible
Green’s function in momentum space can always be written as [33]:
δnΓ[φ ]
Γ˜(1,...,n) = i
δφ˜(p )...δφ˜(p )
1 n
n
= (2π)4δ4( pj)e− ni=2pi,µ(θiσµθ¯1−θ1σµθ¯i)f˜(−p2,...,−pn,θi1,θ¯i1). (24)
j=1 P
X
This general structure is true in particular for propagators and vertices. Thus, the gen-
eral structure of the integrand of the superspace integral corresponding to an arbitrary
Feynman graph is
I = exp − l (θ σµθ¯ −θ σµθ¯) f˜ (−l ,θ ,θ¯ ). (25)
ij,τ,µ i j j i ij,τ ij,τ ij ij
(cid:16) Xi≤j Xτ (cid:17)Yi≤j Yτ
E(p,k,θl,θ¯l)
| {z }
6
Here, l is the momentum running from point i to point j, and τ counts momenta
ij,τ
running between the same pair of points. We have chosen a basis for (l ) = (p,k),
ij,τ
where p and k are the external and internal momenta, respectively. With momentum
conservation,
l = p , (26)
ij,τ i
j,τ
X
we find
E(p,k,θ ,θ¯) = E(p,k,θ ,θ¯ )− (θ σµθ¯ −θ σµθ¯)p . (27)
l l l1 l1 i 1 1 i i,µ
i
X
Therefore, the exponentials appearing in the formulae for the propagators and vertices
can be rewritten as
ekµ(θiσµθ¯j−θjσµθ¯i) ⇒ ekµ(θi1σµθ¯j1−θj1σµθ¯i1), (28)
if and only if k is an internal momentum. A Taylor expansion of the exponentials shows
that from the θ-factors and the exponential we will at most get terms like θ2 θ¯2 k2. The
i1 i1
highest power of k2 that canappear is just thenumber of independent differences θ (with
ij
j = 1 in the calculation above) that can be constructed, which is exactly n−1, n being
the number of vertices. So we find for the superficial divergence degree of an 1PI-graph
d(Γ) = 4L−2G−4V +2(n +n −1)+2n . (29)
G V V
Here, L is the number of loop integrations, G and V are the numbers of ghost and vector
superfield propagators, respectively, n and n count the ghost-vector superfield and the
G V
pure-vector superfield vertices. The last term, 2n , has to be included in (29) because of
V
the four covariant derivatives that appear in the parts of the Lagrangian corresponding
to the three and four vertices of the vector superfield.
Using the topological relation L = G+V −n −n +1 and charge conservation for
G V
the ghost fields, 2n = 2G+N (N being the number of external ghost fields), we find
G G G
d(Γ) = 2−N . (30)
G
Forthevectorsuperfieldself-energy(N = 0),thismeansasuperficialdegreeofdivergence
G
of 2.
6 Self-Energy of the Vector Superfield
The following Feynman graphs contribute to the self-energy of the vector superfield at the
one-loop level (continuous lines denote vector superfield propagators, dotted lines ghost
7
propagators):
4 3 4 ¯3
+ −
1 2 1 2
I : I :
1 2
(cid:1) (cid:2)
¯4 3 4 5
+ −
1 2 1 2
I : I :
3 4
3 6
(cid:3) (cid:4)
4 ¯5 ¯4 5
+ − + −
1 2 1 2
I : I :
5 6
3 ¯6 ¯3 6+
− + −
(cid:5) (cid:6)
¯4 5 4 ¯5
+ − + −
1 2 1 2
I : I : (31)
7 8
3 ¯6 ¯3 6
− + − +
(cid:7) (cid:8)
From the generating functional (23) we obtain the following integrals corresponding to
these graphs (after amputation of the external lines):
~
I = dP dP Γ (3,4,1,2)∆ (3,4),
1 2i V3 V4 φ4 φφ
Z
~
I = − dP dP Γ (3,4,1,2)∆ (3,4),
2 i V3 V4 c¯−c+φ2 c¯−c+
Z
~
I = − dP dP Γ (3,4,1,2)∆ (3,4),
3 i V3 V4 c−c¯+φ2 c−c¯+
Z
~
I = dP dP dP dP Γ (3,4,1)∆ (3,6)Γ (5,6,2)∆ (5,4),
4 2i V3 V4 V5 V6 φ3 φφ φ3 φφ
Z
~
I = − dP dP dP dP Γ (3,4,1)∆ (3,6)Γ (5,6,2)∆ (5,4),
5 i V3 V4 V5 V6 c−c+φ c−c¯+ c¯−c¯+φ c¯−c+
Z
~
I = − dP dP dP dP Γ (3,4,1)∆ (3,6)Γ (5,6,2)∆ (5,4),
6 i V3 V4 V5 V6 c¯−c¯+φ c¯−c+ c−c+φ c−c¯+
Z
~
I = − dP dP dP dP Γ (3,4,1)∆ (3,6)Γ (5,6,2)∆ (5,4),
7 i V3 V4 V5 V6 c−c¯+φ c−c¯+ c−c¯+φ c−c¯+
Z
~
I = − dP dP dP dP Γ (3,4,1)∆ (3,6)Γ (5,6,2)∆ (5,4). (32)
8 i V3 V4 V5 V6 c¯−c+φ c¯−c+ c¯−c+φ c¯−c+
Z
(Note that ∆ (3,6) = −∆ (6,3).) We now insert the explicit expressions for the
c−c¯+ c¯+c−
propagatorsandverticesintotheeightintegralsin(32). Aftersomelengthysimplifications
of the integrands (see appendix A) we arrive at
~ 1
I = δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) d4k sin2(p ∧k)
1 128i 1 2 1 k2(k2 −M2)
Z
8
2 1
× − + θ2 θ¯2 (2k2 +p2) , (33)
3 6 12 12 1
(cid:18) (cid:19)
~ 1
I = δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) d4k sin2(p ∧k) θ2 θ¯2 , (34)
2 384i 1 2 1 k2 12 12
Z
~ 1
I = δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) d4k sin2(p ∧k) θ2 θ¯2 , (35)
3 384i 1 2 1 k2 12 12
Z
~
I = δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) d4k sin2(p ∧k)
4 1 2 1
128i
Z
1
×
k2(k2 −M2)(k +p )2((k +p )2 −M2)
1 1
1
× −4(k2)2 −8k2(kp )−5k2p2 −p2(kp )− θ2 θ¯2 −4(k2)2p2
1 1 1 1 4 12 12 1
(cid:18)
(cid:16)
+8k2(kp )2 +8(kp )3 +8(kp )2p2 +(kp )(p2)2 −3(p2)2k2 , (36)
1 1 1 1 1 1 1
(cid:19)
~ 1 (cid:17)
I = δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) 1−(θ σρθ¯ )p + θ2 θ¯2 p2
5 256i 1 2 12 12 1,ρ 4 12 12 1
(cid:18) (cid:19)
1
× d4k sin2(p ∧k) , (37)
1 k2(p +k)2
1
Z
~ 1
I = δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) 1+(θ σρθ¯ )p + θ2 θ¯2 p2
6 256i 1 2 12 12 1,ρ 4 12 12 1
(cid:18) (cid:19)
1
× d4k sin2(p ∧k) , (38)
1
k2(p +k)2
1
Z
~ 1
I = − δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) d4k sin2(p ∧k)
7 256i 1 2 1 k2(p +k)2
1
Z
1
× 1−(θ σρθ¯ )(p +2k) +θ2 θ¯2 p2 +p k +k2 , (39)
12 12 1 ρ 12 12 4 1 1
(cid:18) (cid:18) (cid:19)(cid:19)
~ 1
I = − δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) d4k sin2(p ∧k)
8 256i 1 2 1 k2(p +k)2
1
Z
1
× 1+(θ σρθ¯ )(p +2k) +θ2 θ¯2 p2 +p k +k2 . (40)
12 12 1 ρ 12 12 4 1 1
(cid:18) (cid:18) (cid:19)(cid:19)
This gives up to terms in the integrand, which evaluate for M 6= 0 to finite quantities
8 ~ sin2(p ∧k)
I = δ4(p +p )e−p1,µ(θ1σµθ¯2−θ2σµθ¯1) d4k 1
i 128i 1 2 k2(k +p )2
1
i=1 Z
X
14 1 26 8(kp )2
× − − θ2 θ¯2 −4kp − p2 + 1 . (41)
3 4 12 12 1 3 1 k2 −M2
(cid:16) (cid:16) (cid:17)(cid:17)
As usual we write sin2(p ∧k) = 1 − 1 cos(2p ∧ k) and refer to the part corresponding
1 2 2 1
to 1 as ‘planar’ and the part corresponding to 1 cos(2p ∧k) as ‘non-planar’. The planar
2 2 1
9