ϵ-expansions near three dimensions from conformal field theory

We formally extend the CFT techniques introduced in arXiv: 1505.00963, to ϕ2d0d0−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \phi \frac{2{d}_0}{d_0-2} $$\end{document} theory in d = d0 − ϵ dimensions and use it to compute anomalous dimensions near d0 = 3, 4 in a unified manner. We also do a similar analysis of the O(N) model in three dimensions by developing a recursive combinatorial approach for OPE contractions. Our results match precisely with low loop perturbative computations. Finally, using 3-point correlators in the CFT, we comment on why the ϕ3 theory in d0 = 6 is qualitatively different.


Introduction
In a recent paper [1] Rychkov and Tan have demonstrated that non-perturbative arguments can be used to determine the low loop anomalous dimensions of critical Wilson-Fisher theory in d = 4 − dimensions. The argument is based purely on the idea that this theory is a conformal field theory, formalized via three (plus one 1 ) axioms. The fact that these results do not require perturbation theory is striking and worthy of further exploration.
In this paper, we will apply the techniques of [1] in d = 3− dimensions. In fact, we will begin with critical scalar field theory in d = d 0 − dimensions and find that the approach allows a formal extension to general d 0 with φ 2d 0 d 0 −2 potential. In the end, because of various constraints, we will find that d 0 gets narrowed down to 4 and 3 -φ 4 in four dimensions and φ 6 in three dimensions. For these cases the formalism allows a unified discussion. We will also find that φ 3 in six dimensions does not allow a simple generalization of this idea.
We further extend the analysis to the case of O(N ) model in three dimensions. One complication we have to deal with in d 0 = 3 O(N ) model is that the OPE contractions required for the computations become too cumbersome. We therefore develop the recursive combinatorics of these contractions using a diagrammatic formalism. This approach might have some mileage even beyond the specific problem that we tackle here.
In all the cases, we find indeed that our results for the anomalous dimensions match precisely with extant results in the literature, where they overlap. As far as we are aware, the only analytical path to these results before this paper were via perturbative loop computations. The φ 6 theories have been used to model multi-critical behavior, especially around tri-critical points. 1 We count the assumption that the anomalous dimensions are analytic in the → 0 limit, as a forth axiom.

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Our results are based purely on constraints from three point functions. It seems plausible that these axioms, together with four-point functions and bootstrap equations might be constraining enough to determine the theory (more) completely. 2 We hope to come back to this question in the future.

A formal -expansion from Wilson-Fisher CFT
We will consider scalar field theory in d = d 0 − dimensions with the action One reason why this class of theories is interesting is because when → 0, ie., when d = d 0 , the theory is renormalizable with a dimensionless coupling. Another (related) reason, which is crucial from our perspective is that the theory has a weakly coupled fixed point at a coupling proportional to , which we will call the Wilson-Fisher CFT. 3 When is finite, we have introduced the scale µ to make the coupling dimensionless. The action captures well-known φ 4 theory in four dimensions (this was the case considered in [1]), φ 6 theory in three dimensions and φ 3 theory in six dimensions. One goal of this paper is to present the discussion in a somewhat unified manner -we will see that the CFT formalism goes through without hitch for the d 0 = 3 case as well. The d 0 = 6 -expansion is known [2] to be significantly different from the other two in its structure, the origins of this difference are immediate from the CFT perspective, as we will see. However, our CFT considerations based on 3-pt functions will only be able to make qualitative predictions about d 0 = 6. The dimensionality of the scalar in d-dimensions can be used to define the following quantities: The Schwinger-Dyson equations of motion of the theory are given by Instead of viewing this as a dynamical equation, we will view this as a conformal multiplet shortening condition as in [1]: in the free theory, φ ν−1 is a primary, but in the interacting theory it is defined by the l.h.s. of the above equation, making it a descendant. As in [1] we will define our Wilson-Fisher theory by a set of three axioms. The first (Axiom I) of these says that the Wilson-Fisher theory is a conformal field theory. The second (Axiom 2 Some of the recent work on the conformal bootstrap is collected in . A pedagogical introduction can be found in [27]. 3 Typically, the case d0 = 4 is called the Wilson-Fisher fixed point, but this is a natural generalization.

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II) says that operators V n and correlators between them in the Wilson-Fisher theory tend to operators φ n and their correlators in the → 0 (ie., free theory) limit. The third axiom is the most non-trivial one, and in our case it formalizes the multiplet shortening condition via the equality (Axiom III) where α( ) is a-priori unknown. This means that the dimension of these operators are protected by the conformal algebra to be Note that in many of these statements, we need various integrality conditions on various functions of d 0 (like the subscript d 0 +2 d 0 −2 above) in order for them to make sense. The most stringent of them will turn out to be the condition that 2/(d 0 − 2) is a positive integer. Together with the condition that d 0 is an integer, it leaves only d 0 = 3, 4 as the solutions. We will discuss this when it arises, but we will proceed formally for now, for the simple reason that we can.
The two-point function in the interacting CFT is which in the free limit goes to The scaling dimensions of V n is given by ∆ n = nδ +γ n where γ n is the anomalous dimension of V n . Axiom II demands that the latter tend to the former in the free limit. We will assume further that the anomalous dimensions are analytic at = 0 and admit a Taylor expansion 4 in : γ n = y n,1 + y n,2 2 + . . . (2.9) Now using and applying 2 x 2 y on (2.7), then using (2.5) and demanding that the result should tend to 2 x 2 y acting on (2.8), we get the relation

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where we have extracted a sign σ = ± for the square root which will be fixed eventually via further CFT arguments. In arriving at the above result, we have used (2.5) and the fact that (2.13) The Γ(ν) arises because the full contraction of φ k (x)φ k (y) gives rise to a k!, and for k = d 0 +2 d 0 −2 , this can be written as Γ(ν). We follow [1] closely in these steps. These further constraints arise from 3-pt correlators involving V n and V n+1 [1]. In the free theory limit, we can write which follows essentially from dimensional analysis. We will first determine the coefficients f and ρ that show up in this expression because we will need them.

Counting contractions
The OPE coefficient f can be trivially determined by direct contraction to be The coefficient ρ requires a bit more work because it depends on d 0 . To determine it, we first note that the number of contractions (n − r) that one needs between φ n and φ n+1 , so that one is left with φ d 0 +2 d 0 −2 after the contractions, is given by Now, of these (n − r) contractions that need to be done, the first can be done by starting with φ's in φ n and contracting with the φ's in φ n+1 . A little thought shows that the choice of the φ's in φ n can be made in n C n−r ways, and the contractions with the φ n+1 can be done in (n + 1) × n × (n − 1) × . . . × (r + 2) ways. So the net result for the number of contractions is This quantity is equal to ρf because f = (n + 1)! is a common factor, so in the end we have In the case d 0 = 4 where r = 1 this reduces to n/2 as was found in [1], and for the case d 0 = 3 where r = 2, this yields which we will soon use to compute anomalous dimensions.

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Note also the crucial role that the integrality of r plays in these arguments. One might hope to generalize the conclusions to generic r by re-writing the factorials in terms of Gamma functions, but the meaning of such an operation is unclear. This is because the arguments for the contractions were combinatorial. Indeed for d 0 = 6 were r = 1/2, we will see that the situation is qualitatively different. This is the first indication from the CFT approach that the d 0 = 6 case where r is no longer integral is bound to have a conceptually different -expansion compared to the d 0 = 3, 4 cases. In particular, we will see that the latter theories have an anomalous dimension γ φ that starts at O( 2 ) while the six dimensional theory it starts at O( ).

Matching with the free theory
The idea now is to take 3-pt correlators involving the V n × V n+1 OPEs and get constraints on the anomalous dimensions by demanding that they have a smooth free theory limit. The crucial point, as we emphasized in the discussion before (2.5), is that at finite , We are rather telegraphic in the discussion of this section (even though it is technically complete): we refer the reader to [1] for more context and elaborations, this section is a direct generalization of their work.
The relevant terms in the OPE are [1] (see also the original work of [28][29][30]): We will demand that the (leading behavior of the) 3-pt correlators of this object tend to the corresponding free field 3-pt correlators We are working here in the |x| |z| limit. The first line follows immediately from (2.21). To evaluate the l.h.s. of the second line we use (2.21) and the fact that .
(2.24) where we have used (2.10). The presence of √ γ 1 suggests that this object vanishes in the → 0 limit. Therefore, to reproduce (2.23) we need q 1 and q 2 to stay finite in that limit. Noting that the box acting on the argument of V 1 (0) brings out a factor of α due to Axiom III (together with producing the requisite V d 0 +2 d 0 −2 (z) inside the leftover 2-pt correlator), we find that for the correct free field match we need lim →0 q 3 α = ρ(n). (2.25)

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Using the expression (A.3) from [1] for the q i we find that the q 1 , q 2 finiteness conditions are automatically satisfied. Further, the leading behavior of q 3 in → 0 limit comes from and so for q 3 to blow up, it is clearly a necessary condition that y 1,1 = 0, Putting them together we obtain the recursion relation Summing the telescoping series, we get which is the final answer, once we fix the numerical value of K (which is the same as fixing the numerical value of y 1,2 ). This can be accomplished via (2.6), which can be written as To leading power in , this translates to Now this can be used to fix K by setting n = d 0 +2 d 0 −2 in (2.30). For d 0 = 4, this gives K = 2/3 and using this one fixes σ = + and y 1,2 = 1/108, reproducing the results of [1]. 5 For d 0 = 3, using (2.20) we get K = 6/5. The final answers written directly in terms of anomalous dimensions are The formulas for n m=1 m and n m=1 m 2 are useful in getting these results. The final result agrees with the perturbative results in (for example) [31] where they overlap.

Fixing loose ends
In obtaining the above result, we summed the telescoping series, and for doing that we implicitly assumed that the recursion relations arising from the OPEs involving the descendants has the same form as the ones arising from primaries. This needs an explicit check for n = 4, 5, because these are the only cases where the contractions involve descendants as well. This check can be done using relations (A.4-A.7) in [1].
Another assumption we made is that y 1,2 = 0. To prove this, we first note that the q 3 ∼ 1/ √ γ 1 due to (2.25). Using (2.26) for n = 1, 2, 3, 4, this gives It is straightforward to check that these relations can all hold together at the same time, only if γ 1 ∼ 2 . (Note that when one adds the last three conditions above, the resultant relation together with the first, gives rise to a system that is identical to that discussed near eq. (3.39) in [1].) The arguments in this subsection apply without any further subtleties to the O(N ) model that we discuss in the next section, so we will not repeat this discussion there.

Generalization to O(N ) model
Now we will consider generalization of the previous discussion to the O(N ) model in d 0 = 3. The Lagrangian is of the form where φ ≡ φ a stands for a collection of N scalar fields indexed by a. The theory has an O(N ) symmetry. We will use the techniques of [1] to compute the anomalous dimensions of two series of operators in this CFT W a 2p+1 and W 2p (3.2) which tend to the free field operators in the → 0 limit. Apart from the relation

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independently in the free limit, parallel to the discussion in the previous section, we find that , (3.6) where σ is again a sign that will soon be determined. The N -dependence arises from the various ways that Φ a 5 (x) can be contracted with Φ a 5 (y) in the free theory. This is the first in a series of contractions that we will need -in this particular case it can be done by inspection.
We will fix the anomalous dimensions by constructing telescoping series as in the last section. The relevant relations that can be used to determine these series are To determine the coefficients ρ which are crucial for proceeding further, it behooves us to develop a formalism which can accomplish contractions systematically. This formalism might be of some use/interest in and of itself, so this is what we turn to next.

Counting contractions using cow-pies
We will develop a recursive approach to compute the coefficients f and ρ. To do this we first introduce some graphical notation. We first define to stand for the total number of contractions between (φ 2 ) p φ µ 1 . . . φ µr and (φ 2 ) p+q φ µ 1 . . . φ µs such that m of the φ's are left uncontracted. Graphically, we describe this using a cow-pie diagram, as shown in the next figure. F p,r p+q,s;m , in this language, stands for the total number of ways in which the kernels in the upper array of cow-pies in the figure can be contracted (aka connected by line-segments) with the kernels in the lower array of cow-pies -but with the restriction that one has a total of m leftover un-contracted JHEP11(2015)040 kernels. We use the following terminology in what follows -in the figure, the upper array contains p double cow-pies and r single cow-pies, while the lower array contains p+q double cow-pies and s single cow-pies. The rationale behind the introduction of this notation is that the quantities we want to compute can be seen to be (3.10) We will evaluate these quantities by setting up a descending iteration in p. We will start by evaluating F p,0 p,1;1 . There are three distinct kinds of contractions one encounters when starting from F p,0 p,1;1 and trying to reduce p recursively. The idea is that we try to count the number of ways in which the p'th upper double cow-pie (PUDC, for short) can be contracted with the lower array. These can be symbolized by the following three figures: It is easy to see that there are 2 × p × N ways of contracting the PUDC the first way, 6 while there are 2p × 2(p − 1) ways of doing the second type of contractions, and there are JHEP11(2015)040 2p×2 ways of doing the contractions the third way. Note that in each case, a bit of thought reveals that the result of each type of contraction is simply F p−1,0 p−1,1,1 . So we get a recursion relation Together with the knowledge that F 0,0 0,1;1 = 1 (which follows trivially upon inspection) this immediately lets us evaluate An entirely similar recursion can be constructed for F p,1 p+1,0;1 , Tand a closely related result follows: The launching condition for the iteration is seen by inspection to be F 0,1 1,0;1 = 2. This yields (3.14) The results for f 's are sufficiently simple that it is possible to guess these answers by doing the contractions explicitly (if somewhat painfully) for low p's. So our recursive formalism might seem like an overkill. However, the usefulness of the formalism becomes clear in evaluating the ρ's (or equivalently F p,0 p,1;5 and F p,1 p+1,0;5 ) for which we have not been able to come up with an alternate way to count the contractions without using the recursion relations. 7 We will start with F p,0 p,1;5 . There are three distinct types of contractions one needs to take care of in this case. The first corresponds to the case where both kernels in the PUDC are contracted (Type I), the second corresponds to only one of the PUDC kernels being contracted (Type II), and the third corresponds to none of the PUDC kernels being contracted (Type III). Type I follows a very similar structure as the previous cases we considered and contributes 2p × (2p + N ) × F p−1,0 p−1,1;5 to the right hand side of the iteration equation, we will skip the details and the associated figures. Type II on the other hand splits into two subcases which can be captured by the following figures: In hindsight, it seems plausible that one can perhaps guess the right expressions for ρ by matching with the N = 1 case, as well as some general arguments about the order of polynomials that one can expect (in p and N ) and explicitly working out the low order cases to match undetermined coefficients. This is ugly and feels like cheating, so we will stick to our systematic combinatorial approach, which has its own elegance. This enables us to use the match with the N = 1 case as a sanity check on our results.

(3.15)
Unlike in the previous case of f 's we see that now there are new structures arising on the right hand side. So we need to come up with recursion relations for them as well. When we have a closed system of recursion relations, we will have enough information to solve for all of them. So now we turn to the recursion relations for F p,0 p,2;4 , F p,0 p+1,0;4 and F p,0 p+1,1;3 . For F p,0 p,2;4 there are two types of contractions for the PUDC with the lower layer cowpies. Type I, which has both kernels of PUDC contracted, and Type II which has only one kernel of PUDC contracted. There is no Type III because it is easy to convince oneself that when both kernels of PUDC are un-contracted, the result must give zero.
Type I gets contributions from four types of figures. Of these the first two are familiar structures that we have seen before leading to the contribution 2N p + 4p(p − 1)) × F p−1,0 p−1,2;4 , and the third one also works along similar lines adding a contribution 8p × F p−1,0 p−1,2;4 . The

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forth figure takes the form: It gives rise to a new structure equal to 2 × F p−1,0 p,0;4 . Turning to Type II there are two relevant figures: The first contributes 4p × F p−1,0 p−1,3;3 and the second 4 × F p−1,0 p−1,1;3 . Altogether we get the recursion relation At this point, we have covered a fairly representative sample of the various kinds of contractions involved in the computations of this section. So now we will merely write down the rest of the recursion relations that are relevant in the determination of F p,0 p,1;5 , without belaboring the details.

Anomalous dimensions
Now we have all the ingredients necessary to set up the telescoping series and compute the anomalous dimensions along the lines of the previous section. The relevant q's take the form We have checked that this result matches with perturbative loop computations, for example, in Hager [31], at two loop level. 8 For completeness we also present the anomalous dimensions of general operators W using our telescoping series: Summing these expressions, we get the anomalous dimensions both of which reduce (for even and odd n respectively) to n(n − 1)(n − 2)/30 + O( 2 ) that we found in the previous section, when N = 1.

Comments on d 0 = 6 theory
Our discussion in the previous section was formally in generic d 0 , but as we emphasized at various points, in practice there are restrictions arising from the fact that r = 2/(d 0 − 2) 8 To make the comparison with Hager [31], we make a few comments about notation. We are using the Peskin&Schroeder conventions for beta functions and anomalous dimensions. In particular, (19) in [31] should be divided by two to match our anomalous dimension conventions. Moreover, (19) is written in terms of the coupling (wR in [31]), which we can solve in terms of at the fixed point, by setting the beta function (18) to zero and solving forwR at leading order. Plugging the resulting expression forwR into (19) and dividing by the factor of two mentioned above, we find a precise match with (3.33).
We will not explore this case further here, but this preliminary observation is enough to see why the case of d 0 = 6 is likely to have qualitative differences from the d 0 = 3, 4 cases.