Towards a bootstrap approach to higher orders of epsilon expansion

We employ a hybrid approach in determining the anomalous dimension and OPE coefficient 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(ϵ4) and O(ϵ5) for anomalous dimensions and OPE coefficients, 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 ϕ3 in d = 6 − ϵ 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.


Introduction
Conformal Field theories (CFT) are interesting to Physicists for their roles in critical points of phase transition and studying RG flows. In the study of CFTs it is important to understand how to determine the operator spectrum of a theory. The conformal bootstrap is an approach, introduced in the pioneering works of [1][2][3][4][5] that is solely based on using the symmetries of a CFT. The last decade has seen a revival of the conformal bootstrap program following the work of [6]. This involved the equality (crossing symmetry) of the Operator Product Expansion (OPE) in the direct channel and crossed channel. It has been very successful in obtaining numerical results, in theories like Ising model [7][8][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 coefficients 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.

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The paper is organized as follows. In section 2 we begin by the computation of the anomalous dimension and OPE coeffient in the large spin limit for the φ 4 theory in 4 − dimension using the known results from large spin analytic bootstrap. Section 3 is dedicated to the study of φ 3 theory in 6 − 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
In this section we derive the anomalous dimensions and OPE coefficients of the operators in expansion for φ 4 theory in 4 − dimension in the large spin limit.
Let us take the OPE of the scalars φ × φ. We know that the operator content of this OPE consists of higher spin double-field operators of the schematic form, with conformal dimension, ∆ = 2∆ φ + 2m + + γ .

(2.2)
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 τ m . If we assume that the anomalous dimension at large spin has the following expansion, then we have [72][73][74]93], Note that there is a sum over m in case there are multiple operators with the same minimal twist τ m . The OPE coefficient of each operator is given by C m . In a similar way, the OPE coefficients of these operators at large spin, can be expressed in terms of the minimal twist operator(s) as follows, where we have,

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The subsequent orders of γ and δC 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.

Anomalous dimension
This section deals with the expansion of φ 4 theory in d = 4 − dimensions (Wilson-Fisher fixed point). We will use the above formulas to get the OPE data of double field operators at large spin. The dimension of the fundamental scalar φ reads For this theory the minimal twist operators are the double-field operators φ∂ φ themselves. This is because under the -expansion (eq (2.2)), their twists are τ m = τ = 2 + O( ) . For m = 0 we have the scalar φ 2 operator. Its twist and OPE coefficient are respectively given by, Substituting (2.8) in the first term of (2.4) for m = 0 we get, Here γ 0 | 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 O 2m, 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 φ-s) too. For O 0, = φ∂ φ, the OPE coefficients and twists are respectively,

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Substituting (2.10) in (2.4) we obtain the following, Here γ 0 | m>0 is the contribution of all spins m > 0 to γ 0 . This sum is over even spins m and can be performed exactly. For the O( 4 ) term it gives, Hence the anomalous dimension at O( 4 ) in the large limit is given by, (2.13) Now if we take the 3 dimension of φ 2 as an external input i.e. δ (2.14) 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 first 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, The remaining terms in (2.11) reads, (2.16)

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Now we focus on the O( 5 ) term. The sum in (2.22) can be done in two steps. First we sum over the terms without the Harmonic numbers. This is given by, Now the terms in (2.22) with the Harmonic numbers are given by, We can use the following identity for the Harmonic number, Hence adding the contribution from (2.21) and (2.22) we obtain, This completes the derivation of the anomalous dimension upto O( 5 / 3 ).

OPE coefficient
In this section we will compute the large spin correction to the OPE coefficients in . Substituting (2.8) in the first term of (2.6) for m = 0 we obtain, (2.29)

Alternative way of computing scalar OPE coefficient
One can compute the OPE coefficients of φ 2 using the known results from Feynman diagram as follows. We denote the twist and OPE coefficients of φ 2 as, We would like to compute the C i 0 's using the known results of the higher spin anomalous dimension. From the first term of (2.4) we obtain, Note that the anomalous dimension of the double trace operators ∆ is known from Feynman diagrams upto O( 4 ) [57,97],

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where H (2) is the generalized harmonic number of power 2 and ψ (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 coefficients of φ 2 , This gives an alternative way of computing the OPE coefficients of φ 2 using the known results from bootstrap and Feynman diagrams.

φ theory with Mellin bootstrap
In this section we will obtain the OPE data at large spin, for the φ 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 φ with conformal dimension, In the OPE of φ × φ the operator spectrum contains the operator φ itself with the OPE coefficient [91], There are also the double field higher spin operators φ∂ φ with dimension 2∆ φ + + γ.

OPE data at large spin
First we will evaluate the anomalous dimension and OPE coefficients at large spin using (2.4). Note that the minimal twist operator here is the operator φ. Using Similarly using (2.6) one gets the large spin OPE coefficients, We can also obtain the terms subleading in 1/ . They are given by, (3.5) 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 coefficients using Mellin bootstrap. Then we will discuss how to use that information to obtain O( 2 ) data, and also discuss the possible difficulties.

A short review of Mellin bootstrap
Let us start with a quick review of the bootstrap in Mellin space, introduced in [90][91][92]. The idea is to write a 4-point of scalars as the sum over Witten diagrams. For identical scalars we have, Here W (s,t,u) ∆, (u, v) are the s, t and u channel Witten diagrams respectively. They are conveniently written in Mellin space as, (3.7) Here M ∆, (s, t) is called the Mellin amplitude of the Witten diagram W ∆, (u, v). In the s-channel it is given by, where the notations are given by, .
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 · · · , (3.11) These poles are called the physical poles.
There are also poles that come from the measure of the Mellin integral (3.7). Such poles occur at, s = ∆ φ + n where n = 0, 1, 2 · · · . (1,t) 14) The notations are defined 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 different equations together determine the anomalous dimensions and OPE coefficients of operators in the s-channel.

Higher spin OPE data
Now let us use the above equations for the higher spin operators in φ 3 theory. As an input we will use the dimension and OPE coefficient of φ given by (3.1) and (3.2) respectively. Let us write the unknowns as Using this in (3.13) we obtain for the s-channel, 16) and in the crossed channels we get, Again in the second equation (3.14) we get from the s-channel, and in the crossed channels,

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and finally the disconnected piece, (3.20) With the above we can solve (3.13) and (3.14) to obtain the anomalous dimension [76,99], . (3.21) and the OPE coefficients, The C (1) can be expressed as a finite sum and obtained for any . For the first few spins, they are given by, 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 φ contributes at O( ). The other operators such as the higher spin operators, or operators with twists greater than 2∆ φ 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 difficulties in going beyond O( 3 ) for the φ 4 in 4 − dimension are described. The problems one faces in getting O( 2 ) for the φ 3 theory in 6 − dimension are similar to the ones faced in getting O( 4 ) of the φ 4 theory. There are two main problems at these higher orders. One is the involvement of infinitely many operators as discussed above. The other problem is the infinite number of poles of ν 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 difficulty 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 differently and avoiding the sum over infinite poles of ν.

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 infinite number of operators. In lightcone bootstrap, for the φ 3 theory in d = 6 − , the higher spin operators of the type O ∼ φ∂ µ 1 ∂ µ 2 · · · φ 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 JHEP02(2018)153 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 coefficient at O( 2 ) order.
Instead of (3.8) we will use the following expression for the Mellin amplitude of the Witten diagram [91,100], (3.24) The Q ∆ ,m (t) is defined in appendix B. The last term is a polynomial ambiguity, which is present in the definition 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 coefficient of u ∆ φ 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 = ∆ φ in (3.7) and getting the coefficient of the log term from the s-channel we get, (3.25) For the t-integral we simply put v = 1 and use Barnes Lemma. Then carrying out the sum over m we get, (3.26) Now we put ∆ = ∆ φ = 2 − 5 9 + δ Now let us come to the crossed channels. We substitute s → ∆ φ + t, t = s − ∆ φ and s → ∆ φ − 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 first and then integrate over t and sum over m. For example, in the u-channel we get for the spin = 0 exchange, (3.28)

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Expanding the m = 0 term from this in we get, (3.29) The m > 0 terms give 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, This precisely cancels the O( ) term of the s-channel (3.27). The O( ) anomalous dimension of φ 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, 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 = 2 in u-channel and following the same route as above we get, . (3.33) Here δ (1) 2 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 = 4, 6, · · · giving the general result, This sum over even spins can be carried out and it gives 1 32 3 − π 2

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The above demonstrates an example of how one would calculate contributions from other operators in the crossed channels. For the φ 3 in 6− such contributions come at O( 2 ) from only the higher spin operators bilinear in φ . Now, we leave the result undetermined because of the presence of the polynomial ambiguity of (3.24), which we cannot fix. 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 fix 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(γ 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 fix these ambiguities or find an alternative way to deal with them [101]. It is yet unknown how to do this. We leave this problem for future work.

Discussion
We analysed the higher spin operators at the Wilson-Fisher fixed point for φ 4 theory in 4 − dimensions and φ 3 theory in 6 − diemnsions.
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 infinite 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 fix the ambiguity following [93].

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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, We sum up the coefficients 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],

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