ϵ-expansion in the Gross-Neveu

Journal of High Energy Physics, Oct 2016

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 + ϵ dimensions. To do this, we extend the “cowpie contraction” algorithm of arXiv:​1506.​06616 to theories with fermions. Our results match perfectly with Feynman diagram computations.

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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)


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Avinash Raju. ϵ-expansion in the Gross-Neveu, Journal of High Energy Physics, 2016, pp. 97, Volume 2016, Issue 10, DOI: 10.1007/JHEP10(2016)097