Towards a bootstrap approach to higher orders of epsilon expansion

Journal of High Energy Physics, Feb 2018

Parijat Dey, Apratim Kaviraj

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Towards a bootstrap approach to higher orders of epsilon expansion

HJE Towards a bootstrap approach to higher orders of Parijat Dey 0 1 2 3 Apratim Kaviraj 0 1 2 3 0 24 rue Lhomond , 75231 Paris Cedex 05 , France 1 Ecole Normale Superieure, PSL Research University 2 C.V. Raman Avenue , Bangalore 560012 , India 3 Centre for High Energy Physics, Indian Institute of Science We employ a hybrid approach in determining the anomalous dimension and OPE coe cient of higher spin operators in the Wilson-Fisher theory. First we do a large spin analysis for CFT data where we use results obtained from the usual and the Mellin bootstrap and also from Feynman diagram literature. This gives new predictions at O( 4) and O( 5) for anomalous dimensions and OPE coe cients, and also provides a cross-check for the results from Mellin bootstrap. These higher orders get contributions from all higher spin operators in the crossed channel. We also use the bootstrap in Mellin space method for 3 in d = 6 CFT where we calculate general higher spin OPE data. We demonstrate a higher loop order calculation in this approach by summing over contributions from higher spin operators of the crossed channel in the same spirit as before. Conformal Field Theory; Renormalization Group - 1 Introduction 2.1 2.2 2.3 3.1 3.2 3.3 3.4 2 Higher orders of Wilson-Fisher from large spin Anomalous dimension OPE coe cient Alternative way of computing scalar OPE coe cient 3 3 theory with Mellin bootstrap OPE data at large spin A short review of Mellin bootstrap Higher spin OPE data A higher loop calculation 4 Discussion A Details of section 3.2 B Mack polynomial and continuous Hahn polynomial very successful in obtaining numerical results, in theories like Ising model [7{9]. The idea here is to constrain the space of CFTs starting with some given assumptions which are usually based on symmetries or unitarity of the theory (see [10]{[71] for related works.) Another goal of the boostrap program is to solve a CFT analytically. One aspect is using the bootstrap equation in a lightcone limit, introduced in [72, 73]. This assumes a higher spin sector of operators in a theory, whose anomalous dimensions and OPE coe cients can be computed in terms of a lower twist operators present in the spectrum. There has been many subsequent works [74]{[84], that has taken this approach further. In these works a systematic approach has been developed to compute the anomalous dimension of large spin double trace operators as an asymptotic expansion in inverse spin. { 1 { This brings us to the other aspect of analytic approach, which is to give an alternative way to Feynman diagrams, to calculate the OPE spectrum in CFTs with a perturbative parameter. In perturbative CFTs, such as the Wilson-Fisher xed point in 4 in d = 4 Gross-Neveu model [85{88] have shown various techniques of obtaining OPE data under expansion in a small parameter. The twist conformal blocks can be used e ciently to extract the leading order anomalous dimension in 4 dimensions [76]. In [89] a dispersion relation-based technique was used, that was inspired from the original work of Polyakov [4]. A new approach to bootstrap, also based on Polykov's work [4], was chalked out in [90, 91]. This involved replacing conformal blocks with a manifestly crossing symmetric basis of Witten diagrams in the Operator Product Expansion. The use of Witten diaof this new approach to bootstrap with usual one has been studied recently in [93]. All the analytic approaches mentioned above reach their limits at certain orders in perturbation. The large spin literature gives a systematic expansion in all orders of large spin. These results have then been used to obtain higher spin anomalous dimensions perturbatively, for example in Wilson-Fisher up to O( 2 ) order [76]. The Mellin bootstrap equations are successful in giving both higher spin OPE coe cients and anomalous dimensions up to O( 3), with signi cant ease [90{92]. There are certain complications that arise beyond this order which makes it di cult to get higher loops results. Even though such di culties are expected at higher orders, it is desirable to know how much can be done with the present bootstrap-driven techniques and without further intricacy. The goal of this paper is to present some calculations for the higher spin double trace operators with the tools of the known methods. The calculations are simple but will take us to some high orders in perturbation. The results presented are mostly unknown in the Feynman diagram literature. In the rst half of the paper we have used the large spin analysis from the usual bootstrap approach to compute anomalous dimensions and OPE coe cients of up to O( 5) and O( 4) respectively for the Wilson-Fisher theory in d = 4 This computation takes in information of in nite higher spin minimal twist operators. This comes from existing -expansion results obtained with Mellin bootstrap as well as those from Feynman diagram literature. In [93] it is shown that the di erence between the usual and Mellin bootstrap starts at O( `2) for the double trace operators having dimensions = d 2 + ` + `. For higher spin operators in 4 theory, ` `12 in the large spin limit. Hence the large spin expressions from usual and Mellin approach should agree uptil O(1=`4). Since we calculate up to O( 5=`3) we can safely use these formulae. The second half of the paper uses the ideas of Mellin bootstrap. The theory used is The paper is organized as follows. In section 2 we begin by the computation of the anomalous dimension and OPE coe ent in the large spin limit for the 4 theory in 4 dimension using the known results from large spin analytic bootstrap. Section 3 is dedicated to the study of dimensions using the ideas of Mellin bootstrap. We conclude in section 4 with a brief discussion of the future directions. The appendices give the calculational details of the paper. Higher orders of Wilson-Fisher from large spin 2 in In this section we derive the anomalous dimensions and OPE coe cients of the operators expansion for 4 theory in 4 dimension in the large spin limit. Let us take the OPE of the scalars . We know that the operator content of this OPE consists of higher spin double- eld operators of the schematic form, where we have, C0 = X C`m 2 `m C1 = X C`m 2 2 `m with conformal dimension, O2m;` We consider such operators at large spin, for which it was shown in [72, 73] that the leading anomalous dimension is determined from the operator(s) having the minimum nonzero twist m. If we assume that the anomalous dimension at large spin has the following expansion, then we have [72{74, 93], Note that there is a sum over `m in case there are multiple operators with the same minimal twist m. The OPE coe cient of each operator is given by C`m . In a similar way, the OPE coe cients of these operators at large spin, can be expressed in terms of the minimal twist operator(s) as follows, C` = C(0) 1 + C0 + ` ; 1 ` C1 + ` m 2 m ; m 2 ; 1 ` `m + + E log( 2 ) (log(4) m +2)+2 (2 1) m +(2+log(4)) m +3 m : { 3 { (2.1) (2.2) (2.3) (2.4) (2.5) m 2 + E (2.6) 1 ` m : 1 ` m ; The subsequent orders of and C` in 1=` can also be computed easily using the techniques in [74, 93]. However for simplicity we will focus only on the leading order terms. This section deals with the expansion of 4 theory in d = 4 dimensions (Wilson-Fisher xed point). We will use the above formulas to get the OPE data of double eld operators at large spin. The dimension of the fundamental scalar reads This is because under the -expansion (eq (2.2)), their twists are m = = 2 + O( ) . For `m = 0 we have the scalar 2 operator. Its twist and OPE coe cient are respectively given by, Substituting (2.8) in the rst term of (2.4) for `m = 0 we get, 432 (`m +1) `mH`m +109 (`m +2) `3m +373`2m 384`m 324 5832`2m (`m +1) 2 +O( 4) : (2.10) { 4 { 0j`m=0 = + 2 9`2 4 + 5 472392`2 3 ( 18 log(`) 18 E +11) 243`2 26244`2 629856 (4) +72 log(`) 2187 0(3) +3 log(`)(12 log(`)+36 E 41) +27 2 +6 E(18 E 41) 202 +5832(27 E 29) 0(3) +314928 0(4) +26244C0(3) Here 0j`m=0 is what one gets from (2.4) with only `m = 0. Now let us consider the contribution from higher spin minimal twist operators of (2.1). Note that the operators O2m;` with m > 0 can contribute at orders suppressed as ` 4 or beyond, and hence would not contribute at ` 2 or ` 3, which we consider in this paper. It is the same for other higher twist operators (like those composed of four or more -s) too. H2`m ) +C`(m2) 2 +C`(m3) 3 +O( 4); For O0;` = C`m = m = 2 2 (`m!) 2 (2`m)! 3 + + 54 and can be performed exactly. For the O( 4) term it gives, Here 0j`m>0 is the contribution of all spins `m > 0 to 0. This sum is over even spins `m Hence the anomalous dimension at O( 4) in the large ` limit is given by, 0jO( 4) 4 17496 0(3) 162 log(`)(4 log(`)+8 E 7) 162 2 +162 E(7 4 E)+1421 + 972 `2 26244`2 Now if we take the 3 dimension of 2 as an external input i.e. 0(3) = 17943976 (computed using Feynman diagrams), then the anomalous dimension at O( 4) in the large 2592 (3) 162 log(`)(4 log(`)+8 E 7)+162 E (7 4 E ) 135 2 +160 This matches precisely with the ` 2 term known in literature [57, 97].1 Now we will compute the O( 5) anomalous dmension of these operators in the large ` limit. In order to do that we need to perform the `m sum in (2.11) at O( 5). We will rst focus on the terms in (2.11) without the Harmonic numbers. This sum (over even spins only) can be easily done resulting in the following expression, `m (`m +1) ((54 E 44)`m +27 E +59) 27`m (`m +1) (2`m +1) log(`) 27 2 12 log(`)+243 (3)+27 E The remaining terms in (2.11) reads, 1 1There is a typo in [97] and the correct expression can be read o from [57]. Adding (2.9), (2.15) and (2.16) we obtain the following contribution at O( 5), 0jO( 5) 5 Here we have used the 3 order OPE coe cient of 2 i.e. C(3) = 23 (3) computed using the 3 anomalous dimension of 2 as the external input [91]. Now we take the values of 0(4) and (4) as an external input from [94{96, 98] Plugging these values in (2.17) we obtain the large spin anomalous dimension at O( 5), 0jO( 5) 5 41)) (2.17) (2.18) (2.20) (2.21) (2.23) 2592 (3)+162 log(`)(4 log(`)+8 E 23)+162 E(4 E 23)+162 2 +2297 3779136 (4) +1889568 0(4) 33858 2 +54 E 24 E(12 E 113)+216 2 +3941 Adding the O( 4) contribution from (2.21) and (2.22) we get, 1 ` O( 4) 4 2592 (3)+162 log(`)(4 log(`)+8 E 23)+162 E(4 E 23)+135 2 +2621 : Now we will consider the anomalous dimension term subleading in `, 1 = X C`m 22`m+ m 1 (2 p 1) m 2 ( 2 m 2 ) 2m + `m 2m + `m + 12 ` m 1 Substituting (2.8) in the (2.20) for `m = 0 we get, 3(18 log(`)+18 E 47) 243`3 135 log(`) 24 log(`)(12 log(`)+36 E 41)+48 E (18 E The higher spin exchanges give, 1 ` `m>0 = X `m>0 + 4 (2`m +1) we sum over the terms without the Harmonic numbers. This is given by, 1 Now the terms in (2.22) with the Harmonic numbers are given by, 1 X `m=2 5 (2`m + 1) 486`3`2m (`m + 1) 2 H`m 6H2`m + 3H`m+ 21 We can use the following identity for the Harmonic number, to write, H2`m = H`m + H`m 21 + log 2 and H`m+ 21 H`m 21 = 2 2`m + 1 (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) (2`m + 1) `m=2 `2m (`m + 1) 2 1 X (2`m + 1) `m=2 `2m (`m + 1) 2 H`m 6H2`m + 3H`m+ 21 2H`m 6( 1 + log 2 + 2`m log 2) 2`m + 1 (3) 2 2 Hence adding the contribution from (2.21) and (2.22) we obtain, 1 ` O( 5) 14171760`3 10935 (3)(64 log(`)+64 E 211)+4665600 (5) +270 log(`) 24 log(`)(12 log(`)+36 E 113)+48 E (18 E 113)+162 2 +4589 +54 5 E (24 E (12 E This completes the derivation of the anomalous dimension upto O( 5=`3). 2.2 OPE coe cient In this section we will compute the large spin correction to the OPE coe cients in . Substituting (2.8) in the rst term of (2.6) for `m = 0 we obtain, C0j`m=0 = 2 log( 2 ) 9 `2 + 4 52488`2 3 + 486 `2 36 log( 2 ) log(`)+3 2 +(18 E 11) log(4) 4(81 E (4 E 7) 242) log( 2 )+27 2(8 E 7+log(4096)) + 648 (3)(1 8 log( 2 ))+108 log(`) 12 log( 2 ) log(`)+2 2 +(8 E 7) log(8) to the OPE coe cients can also be computed easily. We are stopping at O( 4) since general spin OPE coe cients are known only till O( 3) . 2.3 Alternative way of computing scalar OPE coe cient One can compute the OPE coe cients of 2 using the known results from Feynman diagram as follows. We denote the twist and OPE coe cients of 2 as, m = 2 2 3 162 + We would like to compute the C0i 's using the known results of the higher spin anomalous dimension. From the rst term of (2.4) we obtain, 2 3 0 = C(0) 0 18`2 486`2 18C0(0) log(`)+2(9 E 1)C0(0) +27C0( 1 ) + 4 52488`2 + `2 4 ( 2 12) 972 162 log(`) 4C0(0) log(`)+(8 E 3)C0(0) +12C0( 1 ) 108 2(9 E 1)C0( 1 ) +27C0( 2 ) 1 X `m=2 4 +O( 5) ; 157464 4 8 `2 +` 4 3 54 Note that the anomalous dimension of the double trace operators is known from Feynman diagrams upto O( 4) [57, 97], 432 (`+1) `H` +109 (`+2) `3 +373`2 384` 324 Similarly, the higher spin contribution (2.10) gives, C0j`m>0 = 4 (2`m + 1) (log( 2 ) Adding (2.29) and (2.30) we nally obtain the correction to the OPE coe cient at O( 4), { 8 { + + (2.32) (2.33) ! 65 (2.34) ` where H( 2 ) is the generalized harmonic number of power 2 and ( 1 )(x) is the polygamma function. Now we take the large spin limit of (2.34) and compare it with (2.33). This results in the following solution for the OPE coe cients of 2 , 0 C(0) = 2; 0 This gives an alternative way of computing the OPE coe cients of 2 using the known results from bootstrap and Feynman diagrams. 3 theory with Mellin bootstrap In this section we will obtain the OPE data at large spin, for the 3 theory in d = 6 dimension and compare them with the general spin OPE data, that can be obtained using Mellin bootstrap techniques [91]. In this theory there is a fundamental scalar the operator spectrum contains the operator itself with the OPE with dimension 2 + ` + . ` 3.1 OPE data at large spin First we will evaluate the anomalous dimension and OPE coe cients at large spin using (2.4). Note that the minimal twist operator here is the operator . Using (3.1) with `m = 0 in (2.4) one gets, Similarly using (2.6) one gets the large spin OPE coe cients, We can also obtain the terms subleading in 1=` . They are given by, 1 = ` 4 `3 ; C1 = ` (1 + 6 log 4) `3 : To compute the subleading terms with 1=`4 suppression we have to incorporate the contribution of the higher spin operators themselves. This is easy to see since their twists start with 4 + O( ). We have only computed the O( ) term. Before we discuss the O( 2 ) order, we will compute the general spin anomalous dimensions and OPE coe cients using Mellin bootstrap. Then we will discuss how to use that information to obtain O( 2 ) data, and also discuss the possible di culties. 5 9 { 9 { with (3.1) (3.2) (3.3) (3.4) (3.5) Let us start with a quick review of the bootstrap in Mellin space, introduced in [90{92]. The idea is to write a 4-point of scalars as the sum over Witten diagrams. For identical scalars we have, conveniently written in Mellin space as, where the notations are given by, (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) There are also poles that come from the measure of the Mellin integral (3.7). Such These poles are called the physical poles. poles occur at, s = + n where n = 0; 1; 2 : These poles do not correspond to operators present in the OPE, and hence they are called unphysical poles. The idea of Mellin bootstrap is to equate their residues to 0. These and Here P (s) respectively. h;`(s; t) is a Mack polynomial of degree `. Their form is shown explicitly in appendix B. Also 2 = (h + `)=2 and 2 = (h `)=2. The t and u channel Mellin amplitudes are obtained by interchanging (s ! t + ; t ! s ) and (s ! s t) The operator content of the OPE comes from the poles of the Mellin amplitude M ;`(s; t). In order to have the correct u, v dependencies of a certain channel, the Mellin amplitude must have certain poles. In particular for the s-channel OPE we require poles at, 2s = ( `) + 2n where n = 0; 1; 2 ; (s;)`( ) = 2 i(( h)2 (s;`)(s) = equations can be summarized as, and X 6=0 c ;`q(2;;`s) + 2 X c ;`0q `0 (2;t) ;`j`0 ! The notations are de ned in appendix A. Here we point out that q(1;s); q(2;s) are s-channel contributions and q(1;t); q(2;t) denote the crossed channel contributions. The two di erent equations together determine the anomalous dimensions and OPE coe cients of operators Now let us use the above equations for the higher spin operators in 3 theory. As an input we will use the dimension and OPE coe cient of given by (3.1) and (3.2) respectively. Let us write the unknowns as = `( 1 ) + O( 2 ) and C` = C(0) + C( 1 ) + O( 2 ) : ` ` Using this in (3.13) we obtain for the s-channel, 2+` (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) and in the crossed channels we get, c ;`q(2;;`s) = 2 1 `C`(0)(3 + 2`) `( 1 ) 2(3 + 2`) 4(2 + `) (3 + `) (2;t) c ;`0=0q ;`j`0=0 = ( 2 ) ` (4 + 2`) 2(2 + `) (3 + `) 2 3 + O( 2 ) ; + O Again in the second equation (3.14) we get from the s-channel, c ;`q(1;;`s) = 24+3`C(0)(3+2`) 2 32 +` ` 2(2+`) (3+`) + 24+3` 2 32 +` 2(2+`) (3+`) 0 @C`(0)(3+2`) 10 9 + `( 1 ) log(4)+H 12 +` 2H1+` + and in the crossed channels, c ;`0q (1;t) ;`j`0=0 = X ` C`( 1 )(2+`)(3+2`) C(0) (1+`) `( 1 ) 190 ((2+`)(3+2`) E 1 `) 1 ` A+O( 2 ) : q=0 3 (2+q) (3+q) 2(2+`) (3+`) (1 q+`) 1+q 1+q 2+` ( 1 ) q+121 ` (3+q+`) (3+2`) q 2 2q` + + 2 2(3+2`)H2+q 4(3+2`)H1+` +2(3+2`)H2+q+` +2(3+2`)H2+2` +O( 2 ) ; and nally the disconnected piece, q0(1;`;jt0) = With the above we can solve (3.13) and (3.14) to obtain the anomalous dimension [76, 99], and the OPE coe cients, = 4 3(` + 2)(` + 1) they are given by, The C( 1 ) can be expressed as a nite sum and obtained for any `. For the rst few spins, 2 Let us now elaborate on why it was simple to obtain the O( ) results and what comes in the way for the next order. First note that in the crossed channels, (3.17) and (3.19) only the spin 0 exchange i.e the contributes at O( ). The other operators such as the higher spin operators, or operators with twists greater than 2 contribute from O( 2 ) or beyond. Even in the s-channel the higher twist operators contribute from a higher order. Thus we easily obtain the O( ) results. In section 5 of [91] the di culties in going beyond O( 3) for the 4 in 4 are described. The problems one faces in getting O( 2 ) for the 3 theory in 6 are similar to the ones faced in getting O( 4) of the 4 theory. There are two main problems at these higher orders. One is the involvement of in nitely many operators as discussed above. The other problem is the in nite number of poles of that can contribute. In (A.7) there are only two poles that contribute at O( ) in the crossed channels, which makes the calculation simple. The former problem is one that is related to the intrinsic di culty that one expects at higher orders in perturbation. However the latter is a calculational hurdle, that can supposedly be bypassed. In the following section, we demonstrate a calculation that does precisely that, by writing the Witten diagrams di erently and avoiding the sum dimension dimension over in nite poles of . 3.4 A higher loop calculation We have seen in section 2 that in lightcone bootstrap higher orders in perturbation are obtained from a sum over contributions from in nite number of operators. In lightcone bootstrap, for the , the higher spin operators of the type O` contribute from O( 2 ) in the crossed channel. So it is not surprising to expect the same from Mellin bootstrap, and here too these operators contribute from O( 2 ) in the t and u channels. In this section we will demonstrate a calculation that systematically computes these contributions in the crossed channels. For simplicity we will take only the spin 0 operator in the s-channel, and the calculation would correspond to its OPE Instead of (3.8) we will use the following expression for the Mellin amplitude of the The Q`;m(t) is de ned in appendix B. The last term is a polynomial ambiguity, which is present in the de nition of the exchange Witten diagram. It comes from how one chooses the scalar-scalar-spin vertex. In the calculations of this section we will simply drop it and come back to it in the end. We will put this Mellin amplitude in (3.7) and compute the coe cient of u log u(1 v)0. This term will get contribution from only the spin 0 operators. In all the three channels, we will simply put v = 1 in order to have this particular term. Note that this is essentially same as expanding in terms of Q`(t), since we are looking at spin 0 in s-channel. Taking the residue at s = in (3.7) and getting the coe cient of the log term from (3.24) the s-channel we get, For the t-integral we simply put v = 1 and use Barnes Lemma. Then carrying out the sum 2C ;` ( ) (1 h+ ) 4 ( ) ( h+2 ) ( 2 ) 4 2 ( 2 ) 2 2 + h+ 2 + 1 h+ 2 + Now we put d = 2h = 6 = = 2 9 5 + ( 2 ) 2 + O( 3) for the spin 0 and C ;0 = C(0) + C( 1 ) 2 + O( 3) , we get from the above, 0 0 exchange. Also with A (s)(u; v) u log u = C(0) + 0 Now let us come to the crossed channels. We substitute s ! + t; t = s s t in (3.24) to get the t and u channel Mellin amplitudes respectively. Here the calculation becomes tricky if one follows the same route as in s-channel. This is because after these substitutions, the t-integral and sum over m is not straightforward. So, what we do is to expand the expression in rst and then integrate over t and sum over m. For example, in the u-channel we get for the spin `0 = 0 exchange, A `0=0 : (3.25) : (3.26) (3.27) and Expanding the m = 0 term from this in we get, Z dt C0(0)vt 2 2 i 1+t The m = 0 term above is the only term that contributes at O( ) . The t channel gives an equal contribution. Carrying out Barnes Lemma for the O( ) part of (3.29) we get from the crossed channels, A (crossed)(u; v) u log u = C(0) : 0 This precisely cancels the O( ) term of the s-channel (3.27). The O( ) anomalous dimension of is hence consistent with this alternative approach. Let us now go one step further and look at the O( 2 ) terms. This order gets contribution from the m = 0 of (3.29) as well as the m > 0 terms as shown in (3.30). The latter can be summed to give, A The last step has been done putting v = 1 and then using Barnes Lemma to integrate over t. However there are more contributions to O( 2 ), and these come from the higher spin operators in crossed channels. We will now compute these contributions systematically. Taking the spin `0 = 2 in u-channel and following the same route as above we get, A Here 2( 1 ) is the O( ) anomalous dimension of the spin 2 (3.15). Summing over m and followed by integrating t we get, These steps can be repeated for `0 = 4; 6; giving the general result, A = This sum over even spins `0 can be carried out and it gives 312 3 3 (3.29) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) The above demonstrates an example of how one would calculate contributions from other operators in the crossed channels. For the 3 in 6 such contributions come at O( 2 ) from only the higher spin operators bilinear in . Now, we leave the result undetermined because of the presence of the polynomial ambiguity of (3.24), which we cannot x. So, our calculation will possibly give a part of the correct result. Let us now say a few words on what role the polynomial pieces can play. In [90, 91] it has been commented that this piece might be important to x in order to make the Witten diagrams a convergent basis, which is important for numerical analysis. The polynomial ambiguity might also contribute to higher orders in perturbation, which in our case is the O( 2 ). In [93] it was shown that they will contribute from the order of O( `2) which is indeed O( 2 ) for this case (and O( 4) in Wilson-Fisher d = 4 ). So in order to get the complete results at higher orders one needs to x these ambiguities or nd an alternative way to deal with them [101]. It is yet unknown how to do this. We leave this problem for future work. 4 4 Discussion We analysed the higher spin operators at the Wilson-Fisher xed point for 4 theory in dimensions and For the former we have employed a hybrid method- we have done a large spin analysis but at higher orders of perturbation. In doing so we used results from both the newly introduced Mellin bootstrap and also from Feynman diagram literature. The higher order terms hence obtained are unknown in literature, and moreover the technique also provides a cross-check for the results of Mellin bootstrap. For the latter we have used the Mellin bootstrap technique. We calculated OPE data for general spin operators. We have also demonstrated how one can approach a higher perturbative order in this approach, that involves contributions from in nite operators in the crossed channels. There are several interesting future directions one can pursue: One can use these techniques for other theories too, like large N CFTs or theories in other dimensions. It would be interesting to systematically extend the ideas to higher orders in and ` 1. This would require knowledge of higher twist operators too. This brings us to the question of obtaining higher twist OPE data from bootstrap. One expects to get these from other kinds of correlators. It is an interesting open problem how to use Mellin bootstrap for other kinds of correlators. Finally it is important to understand the role of the polynomial ambiguities in Witten diagrams [101]. Developing the systematics of the epsilon expansion in usual bootstrap is one of the most exciting future directions. It may shed light to the polynomial ambiguities of the Witten diagram basis and can be used to x the ambiguity following [93]. HJEP02(18)53 Acknowledgments We acknowledge useful discussions with Kausik Ghosh, Rajesh Gopakumar and Aninda Sinha. We thank Aninda Sinha for comments on the manuscript. We also thank Fernando Alday for discussions and pointing out some typos. A Details of section 3.2 The residues from 2 ( ) give u +n log u and u +n dependence which are unphysical because they do not occur in the s-channel OPE. The residues can be expanded in the basis of the continuous Hahn polynomials Q`2;s0+`, in the following way, M (s)(s ! M (t)(s ! M (u)(s ! ; t) = X c ;`q(s;)` Q`2;0 +`(t) + ; t) = ; t) = ;` X ;`;`0 X ;`;`0 c ;`q(t);`j`0 Q`;0 2 +`(t) + (u) c ;`q ;`j`0 Q`;0 2 +`(t) + )2) +(s (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) denote contributions from the physical and other spurious poles. The Hahn polynomials Q`;0(t) are de ned in terms of the Mack polynomaials P (;s`)(s; t) as Q`;0(t) = 4 ` ( 1 )`(2h h;` s = 2 ` ; t : If we Taylor expand M (s ; t) around s = 1)` P , M (s ; t) = M ( ; t) + (s )M 0( ; t) : The rst term gives the logarithmic unphysical term and the second gives the nonlogarithmic one (or the power law). This can be applied to (A.1). In the s-channel one has, q(s;)`(s) = 41 ` (2s + ` 1)` (2h 2s ` 1)` (h ( s)2 ` 2s) (s;)`( ) : If we write this as, qi;(;s`)(s) = q(2;;`t) +(s )q(1;;`s) +O((s = 41 ` (2 +` h) (` +2 )(`+ +2 2h) ) (` 42 ` (2 +2 +` h+1) )2(`+ +2 2h)2 ; the rst term in the second line, is related to the u log u term , and the second term is related to the coe cient of the power law term u . t t 2 The continuous Hahn polynomials are de ned as [100, 102{104], The continuous Hahn polynomials Q`;0(t) are orthogonal polnomials, and their properties are detailed in appendix B. We can use this in the crossed channels to get, (B.2) (B.3) (B.4) (B.5) (B.6) P ;`(s; t) = (h + 1)` (h 1)` (m`);n = 2 ` ( 1 )m+n`! m!n!(` m n)! (` + h 4F3 1) m(` + + ` 2 1)n ` m 2 ` ` m X X m=0 n=0 (`) m;n h + ` 2 s ( t)n ; (B.1) m m 2 + n ` n 2 + m + n ` m n The more general Q`;+m`(t) are given by, m; 1 h + ; 1 h + ; n 1 + ; 2 2h + ; + ` 2 2 m; + n; 1 Q`2;s0+`(t) = 2` ((s)`)2 (2s + ` 1)` 3F2 `; 2s + ` 1; s + t s ; s ; 1 : These are orthogonal polynomials which satisfy the following orthogonality condition, qi;(;t`)j`0 (s) = qi;(;u`j)`0 (s) = 1 Z 1 Z `(s) `(s) Z Z d d dt We sum up the coe cients of the log and the power law terms individually from all the three channels and for each ` equate them to 0. B Mack polynomial and continuous Hahn polynomial The Mack polynomials are given by [100, 102, 103], dt 2(s + t) 2( t)Q`2;s0+`(t)Q`0;0 2s+`0 (t) = ( 1 )` `(s) `;`0 ; `(s) = (2s + ` 1)`2 (2s + 2` 4(` + s) 1) (2s + ` ( + ` 1)`(2h ` 1)` P +` h;` s = + m; t : 1) : 2 4``! 4 ` Open Access. 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Parijat Dey, Apratim Kaviraj. Towards a bootstrap approach to higher orders of epsilon expansion, Journal of High Energy Physics, 2018, 153, DOI: 10.1007/JHEP02(2018)153