ϵ-expansion in the Gross-Neveu
Received: October
-expansion in the Gross-Neveu
Avinash Raju 0 1 2
0 Open Access , c The Authors
1 Bangalore 560012 , India
2 Center for High Energy Physics, Indian Institute of Science
We use the recently developed CFT techniques of Rychkov and Tan to compute anomalous dimensions in the O(N ) Gross-Neveu model in d = 2 + this, we extend the \cowpie contraction" algorithm of arXiv:1506.06616 to theories with fermions. Our results match perfectly with Feynman diagram computations.
Conformal and W Symmetry; Field Theories in Lower Dimensions
-
O(N ) Gross-Neveu model in 2 +
Matching with the free theory
A OPE coe cients from 3-point function
B Computing f2p 2p and f2p+1 2p+1 from cow-pies
Contents
1 Introduction
Counting contractions
3.4 f2p+1 2p+1
B.2 f2p+1 2p+1
Introduction
uses only conformal symmetry and analyticity in
as inputs.
dimensions [3].2
results are unchanged for the U(N ) model as well.
where they overlap.
interacting theory in the
various composite operators.
! 0 limit which help us determine anomalous dimensions of
Note added.
The paper [12] also discusses the same problem, and even though the
details of the algorithm are di erent, our results agree.
O(N ) Gross-Neveu model in 2 +
dimensions is given by
S =
coupling constant is proportional to
and hence this theory describes a weakly coupled
xed point for small values of . We have introduces a scale
to make the coupling constant
The engineering dimension of the elds is xed by the action
The equations of motion for this theory are given by
) A = 0
) A = 0
The interacting theory enjoys conformal symmetry.
free theory, which the interacting theory operator approaches to in the
! 0 limit.
For de niteness, we call the interacting theory operators as V2n, V2An+1 a and V 2An+1 a,
which in the free limit goes to
@ V1A =
3 =
Operators V3Aa and V 3Aa are not primaries, instead they are related to the primaries
by the multiplet shortening conditions
This puts restrictions on the dimensions of these operators
hV1Aa(x1)V 1Bb(x2)i =
AB
In the free limit this becomes
of the operator and the engineering dimension, i.e,
n = n + n. We also make the crucial
a power series expansion
n = yn;1 + yn;2 2 + : : :
The two-point function of two primaries of same dimension
Our rst task is to x
in (2.6). Di erentiating (2.8) and substituting appropriate factors
matrices, we obtain
(2 1 + 1)(2 1 + 1
3A word on notations: small latin indices a, b,
are the spinor indices whereas A, B, etc stand for
O(N ) indices.
Left hand side of (2.11) takes the form
which in the free limit evaluates to
Comparing both sides, we obtain
2hV3Ac(x)V 3Bd(y)i
1. The exact sign will be determined later. Following [1, 2], we consider
correlators of the form
which in the free limit goes to
hV2n(x1)V2An+1 a(x2)V 1Bb(x3)i;
hV2n(x1)V2An+1 a(x2)V 3Bb(x3)i
h 2n(x1) 2An+1 a(x2) 1Bb(x3)i;
h 2n(x1) 2An+1 a(x2) 3Bb(x3)
where we have introduced operators
a . The reason we are interested in these correlators is because of its sensitivity
to multiplet recombination. To see this, we notice that in the free theory, 2n
) aA whereas in the interacting theory V2n
computable and by Axiom:2, we expect them to match in the limit
A
2n+1 a as a shorthand for ( bB bB)n and
In the free case, we have following OPE
for arbitrary n. This is matched with the interacting theory OPE
V2n(x1)
V2An+1 a(x2)
+ q2(6 x126 @2)ac V1Ac(x2)
Counting contractions
we also need OPE's of the form
We now turn our attention to computing f and
coe cients in (2.17). Apart from (2.17),
2n+2(x2)
f2n+1(x122) (n+1)h(6 x12)ab bA + 2n+1x122(
where p is the number of upper double cow-pies which stand for
, r+ is the number of
type , m
is the number of uncontracted
s and s respectively. A contraction is always
between an upper + and a lower
or vice-versa.
The various coe cients f s and s in our notation becomes
f2p = Fpp;;00;;10;0;1
f2p+1 = Fpp+;01;1;0;0;0;1
f2p 2p = Fpp;;00;;10;1;2
f2p+1 2p+1 = Fpp+;01;0;0;0;1;2
cow-pie to two di erent kernels of lower double cow-pie resulting in a factor of
as following recursion equation
Fpp;;00;;10;0;1 = (N p
and we obtain
f2p = p!(N
The recursion equation can therefore be written by inspection
Fpp+;11;0;0;1 = (N (p + 1)
p(p + 1))Fpp;0;10;;00;;11
f2p+1 = (p + 1)!(N
lower cow-pies analogous to the computation of f2p. This gives a factor of N p p(p
in the lower row. This gives a factor of
) that their
contribution is given by
pFpp 11;;00;;10;1;2. Notice that the coe cient is di erent from the naive
avoid over-counting and to keep track of the index structure.
Thus we get the recursion equation
Fpp;;00;;10;1;2 = p [N
1] Fpp 11;;00;;10;2
Fpp;;00;1;1 = (p + 1)(N
p)Fpp;0;11;;01
f2p+1 2p+1
is contracted.
order, we can see that its contribution is (p + 1)Fpp;01;0;0;1;1;2.
Case (a) is similar to the computation of f2p+1 and gives a factor of (p + 1)(N
2p =
So we have following recursion equation
Fpp+;01;1;0;0;1;2 = (p + 1)(N
1)Fpp;0;10;;01;;12
recursion equations above, we get
2p+ (...truncated)