Anomalous dimensions of higher spin currents in large N CFTs

Journal of High Energy Physics, Jan 2017

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. We 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 fields 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 \( \mathcal{N}=2 \) supersymmetry.

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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) Open Access. 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Yasuaki Hikida, Taiki Wada. Anomalous dimensions of higher spin currents in large N CFTs, Journal of High Energy Physics, 2017, 32, DOI: 10.1007/JHEP01(2017)032