#### Anomalous dimensions of higher spin currents in large N CFTs

Received: October
Anomalous dimensions of higher spin currents in large
Yasuaki Hikida 0 1
Taiki Wada 0
Open Access
c The Authors.
0 Department of Physical Sciences, College of Science and Engineering, Ritsumeikan University
1 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University
We examine anomalous dimensions of higher spin currents in the critical O(N ) scalar model and the Gross-Neveu model in arbitrary d dimensions. These two models are proposed to be dual to the type A and type B Vasiliev theories, respectively. reproduce the known results on the anomalous dimensions to the leading order in 1=N by using conformal perturbation theory. This work can be regarded as an extension of previous work on the critical O(N ) scalars in 3 dimensions, where it was shown that the bulk computation for the masses of higher spin elds on AdS4 can be mapped to the boundary one in conformal perturbation theory. The anomalous dimensions of the both theories agree with each other up to an overall factor depending only on d, and we discuss the coincidence for d = 3 by utilizing N = 2 supersymmetry.
AdS-CFT Correspondence; Conformal Field Theory; Higher Spin Symmetry
Contents
1 Introduction 2 3 Methods
O(N ) scalar model
3.2 Integral I1
3.3 Integral I2
Integral I1(1) Integral I1(2)
Gross-Neveu model
Integral I1(1)
Integral I1(2)
4.3 Integral I2
Generalizations
Anomalous dimensions of higher spin currents
4.2 Integral I1
Non-singlet currents
U(N~ ) scalar model
Conclusion and discussions
A Alternative method for Bk;l
B Technical details for the theory of fermions
B.1 Two point function
B.2 Three point function
B.3 Elements for integral I1(2)
Anomalous dimensions of higher spin currents
Introduction
be read o from the scaling dimension
s of dual current as
Ms2 =
d) (d + s
interpretations on the symmetry breaking.
model is for the preparation of the application.
; N ) has a relevant deformation
of the double-trace type as
S =
ddxO (x)O (x) ;
set Od 2 =
an IR
xed point of the RG
(i = 1; 2;
in conformal perturbation theory.
the masses of dual higher spin
elds on AdS4.1 We do not repeat in this paper but it is
Gross-Neveu model in d dimensions.
collect technical materials.
Methods
polarization vector
s with
2 + s,
hJs (x1)Js (x2)i = Ns
(x^12)2s
x2) =
already proposed in [27, 28].
hO (x1)O (x2)i =
Y Ai(xi)
= hQin=1 Ai(xi)e
tation as
which leads to
O (p) =
(2 )d=2
ddxO (x)e ip x ;
point function is given by
with a normalization factor C
O . We compute correlation functions in the presence of
After the deformation, the two point function becomes
We will be interested in the regime with f
d=222
(p2)d=2
a( ) =
hO (p)O ( p)if = F (p)
f F (p)2 + f 2F (p)3
F (p)
1 + f F (p)
F (p)
hO (p)O ( p)i0 =
d=22d 2 a( )
(p2)d=2
S =
ddpO (p)O ( p) :
hO (p)O ( p)if
or in the coordinate representation
1, where we nd
which is de ned by
hO (x1)O (x2)if
G (x12)
G (x12) =
da( ) (x212)d
2 + s)
(x^12)2s
(x^12)2s
s log(x122)) +
dimensions from the term proportional to log(x212).
puted as
ddx3hJs (x1)Js (x2)O (x3)O (x3)i0
ddx3ddx4hJs (x1)Js (x2)O (x3)O (x3)O (x4)O (x4)i0 +
functions involving J
I1 =
I2 =
1 Z
1 Z
ddx3ddx4G (x34)hJs (x1)Js (x2)O (x3)O (x4)i0 ;
Stratonovich auxiliary eld
S = R ddx O .
scheme, we change the propagator (2.10) as
G (x12) =
da( ) (x212)d
S =
to log(x212) as (see [33])
I1 + I2 = (x122 2
1 Ires + I1 n + O( )
1 Ires + I2 n + O( )
(I1res + I2res) + log(x122 2) (I1res + 2I2res) + I1 n + I2 n + O( ) :
Gross-Neveu model in d dimensions.
mulas in [38]. For the chain integral, we use
ddx3 (x213) 1 (x223) 2
v( 1; 2; 3) =
d=2 Y a( i) :
Here 3 = d
ddx4 (x214) 1 (x224) 2 (x234) 3
; (2.20)
O(N ) scalar model
; N ) in d dimensions. The
obtained in [39] as2
with @^ =
the Wick contraction
Here we set
is obtained as
where the coe cient can be computed as [40]
@^l @^s k
(x^12)2s
Js =
X ak@^s k i ^k i ;
@
ak =
h i(x1) j (x2)i = C
( 1)k ks k+d=2 2
s+d 4
s+d 4
d=2 2
1 + s)s
denote the two point function as
hJs (x1)Js (x2)i
for later use.
O(x) = i i ;
hO(x1)O(x2)i = C
O = 2N C2 :
deformation, the propagator (2.16) becomes
G2 (x12)
G(x12) =
(x212)2
2) sin( d=2) (d
= d(d
2 sin( d=2) (d
2) (d=2 + 1)
e.g., [35, 36].
= 0.
The results are summarized as
terms. The expressions for s
3 are new. The anomalous dimension s is then read o as
s = 2
(s + d=2
2)(s + d=2
which agrees with the known result in [34].
Integral I1
+ (x3 $ x4) ;
Accordingly, we separate the integral I1 into two parts as
I1 = I1(1) + I1(2) ;
I(a) =
ddx3ddx4G(x34)Ka(xi)
4(s + d=2
2)(s + d=2
(d + 1) (s + 1)
1) (d + s
3)(s + d=2
2)(s + d=2
D0s =
1)(s + d
(s + d=2
2)(s + d=2
1)(s + d
(d + 1) (s + 1)
1) (d + s
; (3.10)
the other integral I1(2).
Integral I1(1)
We would like to compute the residue of
I(1) = 16N C4C
X akal @^1k@^2s l
ddx3ddx4 (x223) (x234) +2
1 Z
@^l @^s k
ddx3ddx4 (x223) (x234) +2
= v( ; + 2
; )v( + 1
( + 2) ( ) (x212)
propagators with dimension
, respectively. Derivatives may be acted on the solid lines.
Thus we have
where we have used (3.3) and (3.8).
Integral I1(2)
We then examine the integral
8C2C
( + 2) ( ) D0s =
I(2) = 8N C4C
Bk;l =
ddx3ddx4 @
(x234)2
The integral can be expressed graphically as in
gure 1. It is argued to be possible to
formula (2.6), the integral can be rewritten as
(ip^1)k(ip^2)s l(ip^3)s k(ip^4)l
p21p22p23p24(p25)d=2 2+
with p^n =
pn and
ei(p1 x13+p2 x23+p3 x14+p4 x24+p5 x34)
CB = (C ) 4C2
d=224 2
p1 = p
p3 = q ;
p4 =
This also means p2 = q0
p and p5 = q0
q. With these variables, the integral is now
Bk;l =
ddpddqddq0
p^))s l(iq^)s k
( iq^0)l
q)2(p
q0)2q2q02((q
q0)2)d=2 2+
: (3.18)
We rst evaluate the q-integral as
q^))k(iq^)s k
(2 )d q2(p
q)2((q
q0)2)d=2 2+
Introducing the Feynman parameters x; y we arrive at
(d=2 +
(d=2 +
(i( w^ + (1
q^))k(iq^)s kyd=2 3+
yq0)2 + x(1
x)p2 + y(1
2xyp q0)d=2+
(w2 + x(1
x)p2 + y(1
2xyp q0)d=2+
(4 )d=2 (d=2
x)p2 + y(1
2xyp q0)
even n, the integral with w 1
contracting with
i , we have (
(4 )d=2 (d=2
The integrals over x and y are
(i)s X
a=0 b=0
= (i)s X
= (i)s X
a=0 b=0
a=0 b=0
{ 10 {
: (3.23)
k ( 1)ap^s a bq^0a+b Z 1
a + b + d=2
k ( 1)ap^s a bq^0a+b
a + b + d=2
x)k+b+d=2 2xs k b
(k + b + d=2
1) (s
b + 1)
(s + d=2)
The q0-integral can be evaluated similarly as
ddq0 (p^
= ( i)s (2
= ( i)s (2
(4 )d=2
(4 )d=2
q^0)s l(q^0)l+a+b
d=2) Z 1
q0)2q02
z)p^)s l(zp^)l+a+b
z)p2)2 d=2
(s + a + b
2 + d)
d=2) (s
1 + d=2) (l + a + b
1 + d=2) p^s+a+b
(p2)2 d=2 :
The p-integral is
H1(2)(k; l) =
Thus we nd
Bk;l =
a=0 b=0
(k + b + d=2
1) (s
(s + d=2)
d=222d 4 (d
2) 22s (2s + 2 ) (x^12)2s
= ( 1)s
3d=222d+1(2s + 2
1) (2
d=2)( ( ))2N (s + 1)
1 (2s + d
3) (d
4N C4 (s + 1)( ( ))2
3 + s) D0sH1(2)(k; l) + O( 0
a + b + d=2
b + 1) (s
1 + d=2) (l + a + b
1 + d=2)
(s + a + b
2 + d)
I(2) =
1 2(2s + d
3) (d
3 + s)
(s + 1)( ( ))2
C D0s X
akalH1(2)(k; l) + O( 0) :
After summing over k; l; a; b,3 we nd
2(s + d=2
2)(s + d=2
Combining (3.16), we thus nd the residue of I1 at
= 0 as in (3.9).
Integral I2
function is computed as
and real dimension d by
Mathealso appendix B.1.
Thus the integral I2 is written as
Ck;l =
ddx3ddx4ddx5ddx6
(x234) (x256) (x235)2
(x246)2
is evaluated to the order
0 in [31] as
C0;0js=0 =
2d(a( ))3(a(2))3a(d
+ 4B(2)
3B( )
where B(x) =
the e ects of derivatives.
the residue at
the order
= 0 (see
diagrams have di erent structure from the one in
gure 2, and it turns out to give wrong
which has the same structure.
the exponent of propagator between x3 and x4 as
as a convenient choice. After
{ 12 {
Ck0;l =
ddx3ddx4ddx5ddx6
(x234) + (x256) (x235)2
Using (2.20) the integral over x3 can be evaluated as
ddx3 (x213) (x234) + (x235)2
d=2 (1
) (d=2
= v( ; +
The propagator between x1 and x4 can be rewritten as
(x214)d=2 2+
( )s k(d=2
Integration over x4 then gives
ddx4 (x214)d 3+ x245(x246)2
= v(d
d=2 (3
) ( ) (d=2
(x256)3 d=2
(x216) (x215)d=2 2+
(x256)3 d=2
(x215)d=2 2+
k + ) (k
a + d=2
2) (a + 1) (s
b + d=2
a + d
3) (s
{ 13 {
We re-express the propagator between x1 and x5 as
(x215)d=2 2+
)a(d=2
in (3.27) with (3.28).
Summing over all contributions, we nd
Ck0;l =
1 (2s + d
3) (d
4N C4 (s + 1)( ( ))3
H2(k; l) =
a=0 b=0
x245(x215)1
(x214)d=2 2+
(x214)d=2 2+
(x246)2
: (3.36)
: (3.37)
We can see that there is no extra zero or pole at
Ck;l =
2 Ck0;l + O( 0) :
The integral I2 thus becomes
1 2(2s + d
3) (d
3 + s) (d + 1)
(s + 1)( ( ))2( (
1))2(d
D0s X akalH2(k; l) + O( 0) :
The summation over k; l can be performed as in (3.9).
Gross-Neveu model
be resolved by introducing
to be nite in the limit
anomalous dimensions previously obtained in [37].
Anomalous dimensions of higher spin currents
; N~ , which transform
can be written in terms of bi-linears of free
fermions as4
We also use ^
= =.
Js =
X a~k@^k i^@^s k 1 i ;
s+d 3
d=2 1
s+d 3
k+d=2 1 ;
a~k = ( 1)k
a~s 1 k = ( 1)s 1
(x^12)2s
by applying the Wick contraction as
Using (4.1) the two point function can be rewritten as
D~ 0s =
C2N~ X a~ka~lg4
3 @^2s l 1@^1k@2 4 (x221)
Here we have set
gn = tr( 1
(2x^12)2s
Here we have set N
N~ tr1 . Thus we obtain C~s in (4.3) as
where we have used
as shown in appendix (B.1).
of (1.2). In the current case, the scalar operator is
O~(x) =
whose normalization is
hO~(x1)O~(x2)i =
N~ C2tr @=1 (x212)
= CO~ (x212)d 1
C ~ = 4N 2C2 : (4.12)
Here we have used tr(
) = tr1
. The operator is dual to the scalar eld in the type
{ 15 {
Gd 1(x12)
It might be convenient to express the coe cient C~ as5
G~(x12) = C~ (x212)1
C~ =
d=2) sin( d=2) (d
C~ = 2d
N (2
d=2) (d=2 + 1) (d=2)2
4(s + d=2
2)(s + d=2
Ps (d + 1) (s + 1)
1) (d + s
3)(s + d=2
2)(s + d=2
D~ 0s =
1)(s + d
(s + d=2
2)(s + d=2
Ps =
nd agreement. For spin s
(s + d=2
2)(s + d=2
1)(s + d
Ps (d + 1) (s + 1)
1) (d + s
; (4.17)
results are summarized as
read o as
s = 2
{ 16 {
if we replace
Integral I1
function may be divided into two parts as
+ (x3 $ x4) ;
(x234)1
K~2(xi)= 2N~ C4 X
a~ka~lg6@^1k@4 1
In terms of K~a, the integral I1 in (2.15) becomes
I1 = I
d x3d x4K~a(xi)G~(x34) :
d d
1 Z
and I
= 0.
Integral I
We start from the simplest integral I
. Rewriting
(x234)d=2
and integrating by part, we obtain
d=2)(x12) (x122)1 d=2
{ 17 {
2N~ C~C4
a~ka~lg6( 1)l@^s 1 k+l 2 4 @ 3
1 1
@^1k@1 1 @^s 1 l@ 5 @ 6
2 2 2
3 3
d x3d x4
(x241) (x223) (x234)d=2
4N C~C4
3 3
d x3d x4
a~ka~l( 1)l@^s k+l
1
(x241) (x223) (x234)d=2
In the second equality, we have used
and (4.7).
@ 5 @ 6 = g
d d
d x3d x4
(x241) (x223) (x234)d=2
= v( ; d=2
)v( ; d=2
( ( ))2 (d=2) (x212)d=2 2
(x212)d=2 2
we nd
I(1) =
Here we have also used (4.8).
Integral I1(2)
( ) (d=2 + 1) k;l=0
(2x^12)2s
transform formula in (2.6). Then the integral I1(2) becomes
I(2) =
N~ C~C1
ddx3ddx4
p21p22p23p24(p25)d=2 1+
d=222 2
Thus we may replace
p1 = p
p3 =
p5 = q
with p2 = p5
p3 = q and p4 =
p5 = q0
p. The integral is now given by
I(2) =
N C~C1
q)2]) :
where Jk;l are de ned by
Jk;l (p2)2 d=2+
ddqddq0
q^)k(q^)s 1 k(q^0)l(p^
q)2q2q02(p
q0)2((q
q0)2)d=2 1+
in more convenient form. Using (4.7) and
= 0, we have
p^)q0 (p
q^)q0 (q0
q^)q (q0
p^)q (p
p2 + q2]
q0)2] + q^0(p^
Since p
q, q, p
integrals with p2, (p
K1(2) and K1(3), respectively.
q)2 and (q0
q)2 in the numerator, which may be called as K1(1),
example of integral as
Kk;l;m;n (p2)2 d=2+
(p^)k+l+m+n
ddqddq0
q2q02(p
q^)k(q^)l(q^0)m(p^
q0)2((q
q0)2)d=2 1+
de ne
Bk;l;m;n (p2)2 d=2+
(p^)k+l+m+n
ddqddq0
q^)k(q^)l(q^0)m(p^
q)2q2q02(p
q0)2((q
q0)2)d=2 2+
Using (3.26) and
we have
Jk;l = Ks l;l;s k;k
Bk;s k;l;s l + Ks k;k;s l;l
Kl;s 1 l;k+1;s k
Ks 1 l;l;s k;k+1 (4.35)
+ Kk+1;s 1 k;l+1;s 1 l
Bk+1;s k 1;l+1;s l 1 + Kl+1;s 1 l;k+1;s 1 k
Ks 1 k;k;s l;l+1
Kk;s 1 k;l+1;s l :
D0s = 22s 1N C2 (s + 1)(2
(x^12)2s
3 + s)D~ 0s ;
(p2)2 d=2
= ( 1)s
3d=222d 1(d + 2s
3) (d
d=2)( ( ))2N (s)
Then the integral I1(2) in (4.30) becomes
Performing the summation over k; l we nd
I(2) = ( 1)s d(d + 2s
3) (d
2 + s)
d=2) ( ) (s)
D~ 0s X
(I1(2))res =
2(s + d=2
2)(s + d=2
Combining the result in (4.26), the reside of I1 at
Wick contraction (4.4) as
hJs (x1)O~(x2)O~(x3)i0 =
We can rewrite
as (see appendix B.2)
Ts(xi) =
(x223)d=2
N~ C3Xa~ktr @^1k@=3 (x231)
^@^1s 1 k =
+(x2 $ x3) :
with ak in (3.1) used for the scalar case.
The integral I2 in (2.15) is now
I2 = 27N 2 4C~2C6Ps
ddx3ddx4ddx5ddx6
= d=2
1 + =2 and
= 2
the anomalous dimension of i. Expanding
in 1=N as
= 0 + 1 +
, the rst two
in (4.14). Since we are working on the leading
order in 1=N , we set
+ 1 +
as discussed in subsection 3.3.
{ 20 {
(x234)d=2 (x256)d=2 (x235)1
(x246)1
choose to shift as
C~k;l =
ddx3ddx4ddx5ddx6 @
(x234) +1+ (x256) +1 (x235) 1
Integration over x4 then gives
= v(d=2 +
ddx4 (x214)d=2+
+ 1 +
; d=2
We rewrite the propagator between x1 and x5 as
1 (x215)d=2
(a + d=2
(x215)d=2
) (d=2
= 0 from now on.
in (3.27). Noticing (4.36) we nd
1). We will
over x3 can be evaluated as
= v( ;
+ 1 +
(x245)d=2
1 (x215)d=2
(x214)d=2
Note that v( ;
1) =
d=2a( )a( + 1)a(
1) diverges at
= 0 as mentioned
( )s k(d=2
(d=2 +
(x214)d=2
+ 1 +
+ 1 +
(x214)d=2+
+1+ (x245)d=2
(x215)d=2
b + d=2
+ 1 +
+ 1 +
k + ) (k
a + d=2
1) (a) (s
b + d=2
a + d
2) (s
C~k;l =
1 (2s + d
3) (d
24N C4 2 (s)( ( ))2 (d=2)
H~2(k; l) =
a=1 b=0
Here H1(2) was de ned in (3.28).
I2 = 27N 2 4C~2C6Ps
Ps 2d (2s + d
3) (d
2 + s) (d
akal 12 C~k;l + O( 0
(s)( ( ))3 (d=2)
D~ 0s X
akalH~2(k; l) + O( 0) :
= 0. In the
residue at
+1 and 2
As in the scalar case, the shift of exponent by
changes the overall factor of the
factor is needed. The integral I2 thus becomes
H1(2)(b; l) = 0 :
{ 22 {
(s + 1)
(k + 1) (s
k + 1)
( 1)k = (1
1)s = 0 ;
= 0. We then
examine nite contributions at
variables b; l leads to
= 0.
Non-singlet currents
Js;ij = X ak@^s k i@^k j
The two point function of the currents are
Therefore, the anomalous dimensions are
2 ij kl) K1(xi) + K2(xi) ;
ik jl) K1(xi) + K2(xi) :
s(ij) = s[ij] = 2
1)(s + d
(s + d=2
2)(s + d=2
which reproduces the known results [34].
theory as
with a~k in (4.2). The two point functions are
Js;ij (x) =
X a~k@^k i^@^s k 1 j
four point functions are
ik jl) K~1(xi) + K~2(xi) :
s(ij) = s[ij] = 2
1)(s + d
(s + d=2
2)(s + d=2
as found in [37].
U(N~ ) scalar model
malization as
j (x2)i = C
Js =
X ak@^s k i ^k i ;
@
hJs (x1)Js (x2)i =
O(x) =
hO(x1)O(x2)i = N~ C2
We deform the system by (1.2) with the scalar operator
by N~ .
dimension of spin s current as s
we have the relation as
sU(N~ ) =
U(N~ ) = sGN.
U(N~ ) and that for the Gross-Neveu model in (4.17) as sGN,
deformation as
ddxO(x)O~(x)
large N as explained below.
! 1,
Conclusion and discussions
{ 25 {
this case, the introduction of
in (2.16) is not enough to regularize the divergence in a
to directly evaluate the bulk loop diagrams.
proofs, see also footnote 3.
{ 26 {
16H02182.
Alternative method for Bk;l
Instead of Bk;l in (3.18), we evaluate
Bk0;l =
ddx3ddx4 @
1 (x213) + =2
(x232) + =2
(x234)2
unless extra zeros or poles appear at
First we perform the integral over x3 as
@^1k@^2s l
= v( +
=2; +
=2; 2
d=2 ( (1
=2))2 (d=2
=2))2 (2
(x224)1
=2(x214)1 2 (x212)d=2 2+
(x224)1
(x214)1
@^1a@^2b (x212)d=2 2+
a=0 b=0
the integral over x4 becomes
1 4 d=2
(d=2) 1
(x212)d=2
)@^1k@^2s l
(x214)1
(x224)1
b + 1
=2) (s
=2(x224)d=2
{ 27 {
k + ) (k
a + 1
=2) (d=2
=2) (s
a + d=2
=2) ^s a
(x214)d=2
(l + ) (s
=2) (d=2
b + d=2
=2) ^s b
(x224)d=2
=2 = v(d=2
=2; )@^1s a ^s b
@2
=2 ;
=2 ;
(x212)d=2
Rewriting as
@^1a@^2b (x212)d=2 2+
(x212)d=2
) (2s
) (d=2
) (d=2
b + d=2
) (x^12)2s
C2N (s+1)(2
1+s)s
Bk0;l =
1 + s)
2C4N (s + 1)( ( ))2 (2
1 + 2s) D0sH1(2)0(k; l) + O( 0
we nd
H1(2)0(k; l) = ( 1)s X
a=0 b=0
k + ) (k
a + 1) (l + ) (s
b + 1)
a + d=2) (s
b + d=2)
(a + b + d=2
2) (2s
b + d=2) :
B00 =
2 B000 + O( 0) :
Bk:l =
1 + s)
4C4N (s + 1)( ( ))2 (2
This implies that
I(2) =
1 + s)
(s + 1)( ( ))2 (2
1 + 2s)
C D0s X akalH1(2)0(k; l) + O( 0) ;
Technical details for the theory of fermions
Two point function
The summation over k can be rewritten as
Xs1 (1
= 3F2
s)k (1
( + 1)k(1
+ 1; 1
n + a
n; a; b
= 2F1
Applying the formula (B.2), we nd the relation (4.10).
Three point function
1 (x231)d=2
(x212)d=2
+ 1; 1
s + l)s 1 (1
s + l)s 1 (1
( + 1)l
Xs1 (1
s)l (1
= 2F1
( + 1)
As a result, left hand side of (4.10) becomes
Ts(xi) = 2N~ (d
Using (4.7), we nd
Moreover, we have
x^23x12 x31 + x^31x12 x23
= x212x^13
x213x^12 = x212x213
Using the formulas
we obtain
x^12(x122 x23 x231) x^23(x223 x12 x231)+x^31(x123 x12 x223)
2 2 2
1 (x231)d=2
2s 1 (s + d=2
(x212)d=2
(x213)d=2 (x212)d=2
2s 1 (s + d=2
(s + 1) (s + d=2
1) (2x^31)k (2x^21)s k
(k + 1) (s
k + 1) ( ) (x231) +k (x221) +s k
(x213) (x212)
tain (4.42).
Elements for integral I1(2)
of the type K1(2).
ddqddq0
q)2q2q02(p
q0)2((q
q0)2)d=2 1+
q)2q2((q
q0)2)d=2 1+
(d=2 + 1 +
yd=2 2+
yq0)2 + x(1
x)p2 + y(1
2xyp q0)d=2+1+
Integration over q then leads to
d=2 (1 +
d=2 (1 +
n=0 k=0
n (x(1
x)p2 + y(1
yd=2 2+
y)q02)1+
x)p2)n k
( 2xyp q0)k
y)q02)n
yd=2 2+
2xyp q0)1+
We then rewrite as
(q02)2+n+ (p
(3 + n +
(2 + n +
q0)2 =
(3 + n +
(2 + n +
(w2 + z(1
z)p2)3+n+
(w2 + z(1
z)p2)3+n+
(3 + n +
(3 + n +
(p2)l=2
z)p2)3+n+
The integrals over x; y; z are
y)n+m+1
n + m + 1
n + k) (m
n + k
+ d=2
(d=2 2 n
+k l=2) (d=2 1+l=2)
integral involving q as
dzzd=2 3 n
+k l=2(1 z)d=2 2+l=2 =
over q and q0. Therefore we arrive at
Kk;l;m;n =
d=2) Xk
a a + l +
in the momentum representation.
(a + l + m + ) (n + )
(a + l + m + n + 2 )
q0)2) +
(d=2 +
(1) ( +
xq^0)k(xq^0)lxd=2 2+
x)q02)
The integral over q0 is
ddq0 (q^0)a+l+m(p^
q^)k(q^)lxd=2 2+
xq0)2 + x(1
x)q02)d=2+
( 1)a(p^)k a(q^0)a+l Z 1
dxxa+l+d=2 2
d=2 k
( ) a=0
k ( 1)a(p^)k a(q^0)a+l
a + l +
(q^0)a+l+m(p^
yp)2 + y(1
y)p2)2
dy d=2 (2
y)p2)2 d=2 (yp^)a+l+m(p^
d=2 (2
(a + l + m + n + 2 )
d=2 l=2 :
order to do so we compute
ddqddq0
q0)2q2(p
q)2((q
q0)2)d=2 1+
we can see there is no pole at
nd no 1= -pole term.
these integrals are related to Kk;l;m;n as
There are similar integrals with p
q replaced by q , p
q0 and q0. We can show that
ddqddq0
ddqddq0
q^)k(q^)l(q^0)m(p^
q)2q02(p
q0)2((q
q0)2)d=2 1+
ddqddq0
q^)k(q^)l(q^0)m(p^
q)2q2q02((q
q0)2)d=2 1+
q^)k(q^)l(q^0)m(p^
q)2q2(p
q0)2((q
q0)2)d=2 1+
(p^)k+l+m+n
(p^)k+l+m+n
= Km;n;k;l (p2)2 d=2
(p^)k+l+m+n
by exchanging q and q0 or q and p
Bk;l;m;n =
d=2) Xk
2) a=0 b=0
b a + b + d=2
(k + b + d=2
b + 1)
(k + l + d=2)
1 + d=2) (m + a + b
1 + d=2)
(n + m + a + b
2 + d)
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