The analytic structure of conformal blocks
and the generalized Wilson-Fisher fixed points
We describe in detail the method used in our previous work arXiv:1611.10344 to study the Wilson-Fisher critical points nearby generalized free CFTs, exploiting the analytic structure of conformal blocks as functions of the conformal dimension of the exchanged operator. Our method is equivalent to the mechanism of conformal multiplet recombination set up by null states. We compute, to the first non-trivial order in the -expansion, the anomalous dimensions and the OPE coefficients of infinite classes of scalar local operators using just CFT data. We study single-scalar and -invariant theories, as well as theories with multiple deformations. When available we agree with older results, but we also produce a wealth of new ones. Unitarity and crossing symmetry are not used in our approach and we are able to apply our method to non-unitary theories as well. Some implications of our results for the study of the non-unitary theories containing partially conserved higher-spin currents are briefly mentioned.
The notion of criticality and its intimate relationship with phase transitions is central in our quests for understanding the physical world. Over the past few decades, significant progress in the study of criticality has been achieved for systems that can be described by quantum fields. In this case, critical behaviour is generally associated with the existence of conformal field theories (CFTs). The latter theories posses a large spacetime symmetry that allows the calculation of various physically relevant quantities such as scaling dimensions, coupling constants and central charges. This program has led to some remarkable results for two-dimensional systems where one can explore the infinite dimensional Virasoro algebra .
Nevertheless, critical systems are also abundant in dimensions and therefore we are forced to study higher-dimensional CFTs to understand them better. Higher-dimensional CFTs are much harder to explore than their two-dimensional counterparts, and that explains the relatively slow progress in their study up until the end of the millennium. This has changed dramatically by the advent of AdS/CFT [2, 3, 4] that put the focus back into higher-dimensional CFTs and their relevance not only for critical systems, but for quantum gravity and string theory as well. It is inside this fertile environment of general rethinking about CFTs in that the more recent significant progress of the higher-dimensional conformal bootstrap  was born. The latter can be described as a combination of analytic and numerical tools that give remarkably accurate results for critical points in diverse dimensions [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
One of the intriguing features of the conformal bootstrap approach is that some properties of the critical systems that are strictly related to the renormalization group description of phase transitions, can be seen instead as a direct consequence of conformal invariance. For instance, in the quantum field theory approach to the Wilson-Fisher (WF)  fixed point of the theory in (or its generalizations in other dimensions with different marginal perturbations111Strictly speaking they are relevant deformations for , we will often loosely call them marginal in the sense of .) there are two distinct small dimensionless parameters in the game: the coupling constant which turns on the interaction in the Lagrangian, and . One performs a (scheme-dependent) loop expansion in of some physical quantity like for instance the anomalous dimensions of local operators. When is slightly smaller than four the perturbation becomes slightly relevant at the Gaussian UV fixed point and the system flows to the infrared WF fixed point. The vanishing of the Callan-Zymanzik equation fixes the relation between and and gives scheme-independent -expansions for the anomalous dimensions.
The conformal bootstrap approach tells a parallel but different story. For instance, in the numerical bootstrap it suffices to ask for a consistent CFT with a scalar field having an operator product expansion (OPE) , in order to be able to select a unique solution of the bootstrap equations in the whole range . This procedure allows one to evaluate the low-lying spectrum of the primary operators of a theory  which compares well with strong coupling expansions and Monte Carlo simulations as well as with more recent conventional -expansions . If one restricts the search to unitary CFTs where convex optimization methods apply, one is led to a wealth of non-trivial results which are particularly impressive in the case of 3d Ising model, where very precise determinations of the bulk critical exponents are obtained [16, 22, 32]. However, the analytic approaches to bootstrap, if we exclude supersymmetric theories, have not yet reached the high level of accuracy of the conventional -expansion in quantum field theory, but nevertheless are in many cases much more simple to apply and sometimes give results that cannot be obtained by other analytic methods. For instance in the recent approach where the 4pt functions are expanded in terms of exchange Witten diagrams instead of the conventional conformal blocks [39, 40, 41], one obtains anomalous dimensions of some local operators to . Similarly in a study of CFTs with weakly broken higher-spin symmetry the spectrum of broken currents is obtained at the first non-trivial order of the breaking parameter [42, 43, 44, 45, 46, 47, 48, 49]. The Wilson-Fisher points with invariant symmetry have been studied in the limit of large global charges in [50, 51]. In , using the fact that in a theory in dimensions the equations of motion imply that becomes a descendant of at the WF fixed point, the anomalous dimensions of operators are obtained with no input from perturbation theory. Such an approach has also been generalized in various ways in [53, 54, 55, 46, 56, 57].
In a recent short paper  we have extended some of the above ideas to the wide class of generalized free CFTs. More specifically our method is based on considering two of the three axioms of , but we did not have to assume any Lagrangian equations of motion (eom). We pointed out that the mechanism of recombination of conformal multiplets222Conformal multiplet recombination has been discussed in the context of the holography of higher-spin theories in [59, 60]. For a more recent work see . can be directly read from the analytic properties of the conformal blocks without further assumptions. We considered in particular the set of scalar states with dimensions where ; corresponds to the dimension of a canonical scalar field, while corresponds to the subclass of generalized free CFTs with kinetic term that are coupled to the stress tensor [62, 63]. These states have generically scalar null descendants with dimensions , which is equivalent to saying that their conformal blocks are in principle singular at . However, since the 4pt functions of the free theory are always finite, the states with must either decouple in the free field theory limit or have their singularities somehow removed. In the first case it is the descendants that emerge as regular primary fields in the free theory limit. In the second case, as it was firstly discussed in , the would-be singular blocks with do arise in the OPE of free nonunitary CFTs but their singularities cancel out due to the presence of corresponding null states in the spectrum. The fate of singular blocks in the conformal OPE is briefly summarized in Appendix B.
In this work we will heavily use the first of the above two mechanism in order to calculate explicitly various critical quantities in nontrivial CFTs. As we switch-on the interaction by assuming non-vanishing anomalous dimensions in , we have and nothing is singular anymore. In a CFT language the corresponding scalar multiplet becomes long. Using then the explicit results for the residues of the conformal multiplets, and the known OPE coefficients of the free theory, we will be able to calculate analytically the leading corrections to the anomalous dimensions for wide classes of operators in many nontrivial CFTs. In particular, we find that for any pair of positive integers the (generically fractional) space dimension is an upper critical dimension, in the sense that there is a consistent smooth deformation of the free theory at representing a generalized WF fixed point associated with the marginal perturbation . At such WF fixed points we can calculate the anomalous dimensions of composite operator of the form with , as well as non-trivial OPE coefficients. We are also able to present -symmetric generalizations of the above WF fixed points, and calculate the anomalous dimensions, at the first non-trivial order in , of the corresponding scalar composite operators as well as of fields carrying symmetric traceless tensors representations of , having general form , for any and spin . In this work we describe in more detail our calculations presented in  and present some new results.
The content of this paper is as follows. In Section 2 we study with a new method the singularities of generic conformal blocks as a function of the scaling dimension of the exchanged operators in a 4pt function of four arbitrary scalars. Requiring the cancellation of singularities in the expansion in conformal blocks of a suitable function we find that conformal blocks have simple poles in . The position and corresponding residues of these poles coincide with those dictated by the null states and obtained in [22, 65] by completely different methods. Here we also point out the intimate relationship of the poles in generic conformal blocks with partially conserved higher-spin currents and their corresponding generalized Killing tensors. In Section 3 we review and improve the analysis of  regarding the definition and the OPE of generalized free CFTs. Also, generalizing our earlier results, we give a remarkably simple formula for the total central charge of nonunitary generalized free CFTs in arbitrary even dimensions. In Section 4 we study in detail the generalized Wilson-Fisher fixed points near generalized free CFTs in arbitrary dimensions for single-scalar theories and give analytic results for the anomalous dimensions of large classes of operators and OPE coefficients to leading order in the -expansion. In sections 5 and section 6 we study respectively nontrivial CFTs with O global symmetry and theories with multiple marginal deformations. We give there too results for anomalous dimensions and OPE coefficients that match earlier calculations, but we also give many new ones. Finally in Section 7 we draw some conclusions. Technical details of our calculations are presented in the Appendices.
2 The analytic structure of the conformal OPE
2.1 Singularities of generic conformal blocks
We begin by studying the analytical structure of generic conformal blocks as functions of the conformal dimension of the exchanged operator. This will play the central role in our study of WF fixed points in this paper. For our purposes we will consider conformal blocks with general external scalar operators. In a generic CFT in dimensions the 4pt function of arbitrary scalars can be parametrised as 
where , and is the scaling dimension of , while and are the cross ratios. The function can be expanded in terms of conformal blocks , i.e. eigenfunctions of the quadratic (and quartic) Casimir operators of :
with and being the scaling dimensions and the spin of the primary operators contributing to the channel. The scalar 2pt functions are normalized as
and we normalize here the conformal blocks such that putting and in the limit we have
These conformal blocks form a complete basis which can be used to expand a general function of and . Our starting point is the expansion of into conformal blocks , with and arbitrary parameters. We will use such an expansion to extract the positions and the residues of the poles of the conformal blocks in the variable. The same expansion will be used to find the OPE of generalized free field theories, however the results of this section are valid for any CFT in arbitrary dimensions.
A systematic method of finding the conformal block expansion of is described in detail in the Appendix A and here we briefly sketch its salient features. We begin with the following expansion,
and the goal is to find the coefficient . The idea is to apply the following differential operator
to both sides of (5). is the quadratic Casimir operator of while is its eigenvalue. The definitions and explicit formulae of and can be found in Appendix A. projects out all the conformal blocks corresponding to the eigenvalues appearing in the product. Applying recursively , as we show in Appendix A, we obtain the following identity
where is the Pochhammer symbol. The coefficient is given by
with . Clearly it has three families of simple poles at , with
The left-hand side of (7), , is a regular function of , thus such singularities must cancel on the right-hand side. Since the ’s are linearly independent, the only way for such a cancellation to happen is that another conformal block must appear in the sum, having , and most importantly it becomes singular with the opposite sign residue, namely the residue should be proportional to yet another conformal block . Thus we can write
with . We note and actually differ by an integer, namely, .
Now another key observation is to recall that the conformal blocks are eigenfunctions not only of the quadratic Casimir but also of the quartic Casimir operator of with corresponding eigenvalue . Therefore and share the same eigenvalues and . We look for the possible solutions of the system of algebraic equations
where the Casimir eigenvalues are respectively,
By solving these equations we discover a precise link between the position of the poles of a generic conformal block and the scaling dimension and spin of the conformal block contributing to the residue in (12). We explicitly find three possible families of solutions shown in table 1. In the first and the third rows of the table the condition that follows automatically from the solution of the equations. In the second row, namely when , the constraint from quartic Casimir is actually redundant, so the condition that =integer is necessary to find the solutions. It is amusing to see that we re-obtain in such a purely algebraic way the spectrum of the representations listed in (9). In other words, the requirement of the cancellation of singularities of the coefficients of the expansion (7) with the poles of the conformal blocks can be equivalently reformulated as the search of solutions of the algebraic system (14) combined, in some cases, with the condition that be an integer. Furthermore requiring the complete cancellation of the singularities in (7) allows us to fix the factor in the residue for each case. In the following we will study each case separately and compute the corresponding .
In the first case, namely and , the OPE coefficient in (7) becomes singular for , and its residue is given by
with . The cancellation of the singularity requires that
It thus yields
We then move to the second case, the OPE coefficient becomes singular when for , and the residue is given by
Now we have
which leads to for this case,
Finally, we will consider the third case, . Now the residue is
we have finally,
It is well known that poles in a conformal block associated with a primary occur at special scaling dimensions and spin where some descendant of the state created by becomes null [22, 65], which generalizes the original results of Zamolodchikov for two-dimensional conformal blocks . This null state and its descendants form together a sub-representation, thus the residue of the associated pole is proportional to a conformal block, as we found in (12). It is interesting to notice that requiring the cancellation of singularities in the conformal block expansion of reproduces exactly the complete list of null states and their residues of any CFT in arbitrary space dimensions. Actually our table 1 coincides exactly with a similar table in [22, 65] and our residues (17),(20) and (22) coincide, apart form a different normalization of the conformal blocks, with those calculated there with a completely different method.
2.2 Singularities of conformal blocks and higher-spin theories
It should not be surprising that the analytic properties of generic conformal blocks are intimately connected with the rich and nontrivial structure of higher-spin gauge theories  that underlies generic free CFTs and, as we will see, dictates also their nearby critical points. The first and third sets of dimensions shown in table 1 are in one-to-one correspondence with the dimensions of the partially (or better: multiply) conserved higher-spin currents and their corresponding generalized Killing tensors of generic non-unitary theories, such as those of scalars with kinetic terms [63, 69, 70, 71], but also fermionic ones that have not been studied yet. Indeed, in such theories one can construct partially conserved higher-spin currents which are symmetric and traceless spin- operators satisfying equations such as
The dimensions of these currents are . The positive integer is the depth of partial conservation and the maximal depth case corresponds to the usual conserved higher-spin currents with dimensions . We then observe that coincide with the positions of the singularities in the third line of table 1 if we identify . When interactions are turned-on one generally expects that the conservation equation (23) is modified as
The RHS represent operators with spin and dimensions which coincide with the dimension and spin of the would-be null states in the third line in table 1.
The existence of the above partially conserved currents is connected with existence of corresponding generalized Killing tensors. These have been discussed extensively in [72, 73, 74, 75, 76, 77, 78, 79] as well as more recently in [63, 69, 70, 71]. For our limited purposes here, however, it is simpler to just generalize the discussion of the usual higher-spin conserved currents in . Consider the marginal deformations of the form
Following  one expects that under relatively mild assumptions, such as the existence of a large- expansion, can be considered as spin- partially massless gauge field in a possible nontrivial UV fixed point (induced partially massless higher-spin gauge theory) of the theory. The existence of the gauge fields implies the presence of corresponding Killing tensors that generalize the conformal Killing equation for gravity. For example, in the case of the usual conserved higher-spin currents and their corresponding higher-spin gauge fields, the Killing equation takes the form 
where the parentheses denote total symmetrization and trace subtraction. Notice that in this case, to a spin- gauge fields corresponds a spin- Killing tensor. For partially conserved higher-spin currents of depth one expects a generalization of (26) with more derivatives, such that to a spin- partially massless gauge field of depth corresponds a generalized Killing tensor of spin . From (25) we can read the dimensions of to be and these coincide with the dimensions of the would-be null states in the first line of table 1. The Killing equation (26) and its generalization sets to zero the unphysical gauge degrees of the free partially massless gauge field . When the extended higher-spin gauge symmetry is broken, however, one generally expects that the theory is no longer free333The arguments for the case of the usual higher-spin gauge theories were presented e.g. in [81, 82], but one expects that they generalize to the case of partially massless higher-spin gauge theories as well. and those degrees of freedom enter the spectrum of the interacting theory. Schematically one can write
where the spin- field is no longer a partially massless gauge field. This is of course a sketch of the expected Higgsing mechanism for partially massless gauge fields, or equivalently the corresponding multiplet recombination. From the above we can read the dimension of the spin Killing tensors to be . We then recognise these operators as the singularities in the first line of table 1. Finally, the second set of dimensions in table 1 corresponds to states and their shadows irrespective of their spin. The generalized multiplet recombination that we are describing here is the explicit realization of the algebraic analysis of [72, 73, 74, 75, 76, 77, 78, 79].
3 Generalized free field theories and their central charges
Before studying generalized Wilson-Fisher fixed points, namely interacting theories, we will briefly review the generalized free field theories in the context of conformal bootstrap . Scalar generalized free conformal field theories (GFCFTs) can be defined as the CFTs generated by a single elementary scalar field with scaling dimension and 2pt function normalized to be . All other correlation functions of the theory, either of or its composites as well as currents built from , are given by simple Wick contractions. As a consequence all the correlation functions with an odd number of ’s vanish, a condition that may be called elementariness. The simplest nontrivial example of a correlation function is the 4pt function of the elementary fields which is given by
Taking advantage of the expansion of given in (7) and putting , it is easy to obtain the conformal block expansion of . It suffices to note that the exchange in (28) entails and a change of sign of the with odd, while those with even stay unchanged, then
As is an arbitrary (real) number, generalized free CFTs do not necessarily admit a Lagrangian description. For the latter to be true one would require a non-vanishing coupling of the theories to the energy momentum tensor . According to (29) the subclass coupled to , i.e. to the primary of scaling dimension and spin has with ; corresponds to the canonical free theory, while all cases with describe non-unitary theories as the fundamental scalar lies below the unitarity bound. According to table 1 the dimension of this field coincides with the subclass of scalar states having a null scalar descendant. A specific property of the conformal block expansion (29) in this subclass of theories, as first noted in , is that some of the OPE coefficients are singular for some space dimension depending on the scale dimensions of . Since the expanded function is regular these singularities must cancel with corresponding singularities of the conformal blocks. This corresponds exactly to the cancellation mechanism that has been explained in full generality in the previous section.
A particular class of generalized free CFTs coupled to the energy momentum tensor appears to play an important role in the study of the expansions of vector-like theories. One quite intriguing observation made in  is that the highly nontrivial results for the corrections to the central charges444By that we colloquially mean the coefficient in front of the 2pt function of the energy momentum tensor. of vector-like theories in simplify considerably when is even. There, the nontrivial result is simply given by the sum of two terms, each one of them being the contribution of a free CFT. The general formula can be presented as
The case with is relevant only for the fermions. The first term is the contribution of the free vector-like CFTs of canonical scalars or Dirac spinors in dimensions, with
where the corresponding spinor representation has dimension . The second term in (30) is the contribution of the free CFT of a single generalized free field, the field. In the scalar case (for ) this is the non-unitary scalar with fixed scaling dimension in any , while in the fermionic case it is the scalar with fixed dimension . In a Lagrangian description and can be described by higher derivative actions. The explicit results read
The observation above indicates that for even dimensions (for fermions and scalar respectively), there exist two types of free CFTs that are naturally connected to each other. On the one hand we have the canonical free CFTs of scalars and fermions whose corresponding scaling dimensions are and . On the other hand, we have the associated CFTs of and . The latter can be consistently defined as generalized free CFTs in any even dimension. Their OPE structure was studied in  and their corresponding central charges were calculated to be exactly (32), either by using the OPE or by the direct evaluation of their energy momentum tensor .
It is also interesting to note that having at hand the general results (7) and (8) we can give a general formula for the central charge of the whole class of scalar GFCFTs that are coupled to the energy momentum tensor. The result follows using the fact that the coefficient in front of the energy momentum conformal block is determined by a Ward identity , and then taking properly into account the normalization of the conformal blocks that we use here (see e.g. ) we have that
where clearly we have .
For a given even we may define the normalized total central charge of the theory as the sum of the central charges for all free scalars with . Except for , these are all nonunitary operators which in general correspond to ghost states. This is a delicate process because for even we hit the poles of in the numerator of (34). However, if we first do the sum for general , Mathematica gives a remarkably simple answer
Notice that this is positive despite the fact that the underlying theory contains negative norm states. Now we could take the even and obtain a finite result. Apparently, this result involves an underlying regularization that we do not yet understand. Nevertheless the result is consistent with the observation that
In other words, in the limit the sum (35) becomes the Euler alternating sum which is evaluated to after appropriate regularization. This is still positive despite having summed over an infinity of ghost states. It would be interesting to unveil the possible physical interpretation behind this simple result.
4 The smooth deformations of generalized free CFTs
After introducing the generalized free CFTs in the previous section, we now define and study their smooth deformations using conformal invariance as our only input. The results obtained include the calculation of the anomalous dimensions of an infinite class of scalar operators as well as OPE coefficients at the first non-trivial order in the -expansion. Our approach is similar in spirit to the one of Rychkov and Tan , even if we do not use the equations of motion to obtain our results. This way of reasoning suggests a possible definition of Wilson-Fisher fixed point and its extension to generalized free theories without any dynamical notion related to Lagrangians.
Let us start by considering a generalized free CFT in dimensions. We say that this theory is close to a WF fixed point in dimensions if it admits a smooth deformation in , i.e. if there is a one-to-one mapping to another CFT in which any local operator of the free theory corresponds to an operator of the deformed theory with the same spin, but with scaling dimension and relevant 3pt couplings analytic functions of yielding the free results in the limit
where is the anomalous dimension of at the th order in the expansion. Some, but not necessarily all, of the above -corrections should be different from zero. Note that the above definition does not imply that all primary operators of the free theory correspond to primary operators of the interacting one. Actually, the main ingredient of our calculation is the fact that some operator which is primary in free theory becomes a descendant when the interaction is turned on.
4.1 A simple example
We begin by applying in detail our method in the simple case of the deformation of a canonical free theory in dimensions before generalizing it to the wide class of generalized free CFT with . Consider the following OPE in a free theory
The OPE coefficients are computed using Wick contractions with normalization . Using the above, we calculate the mixed 4pt function in the form (1) and obtain555From now on we will suppress for simplicity the dependence of the conformal blocks.
where , , and
Notice that according to (12) and the table 1 the conformal block appears to be singular having a simple pole exactly at , however one can see that the corresponding residue computed in (20) is zero. Let us investigate the viability of a deformed theory by setting
where denotes the anomalous dimension of at -th order in . In the deformed (thus interacting) theory the first contributing conformal block is with
Now the residue is no longer vanishing:
thus eq. (12) becomes
It follows that in an interacting theory with there exists a scalar with scaling dimension which is a descendant of . Precisely the primary contains a sub-representation (a null state) with the same Casimir eigenvalue ( apart from -corrections) and scaling dimension . It is important to stress that according to (41) this mechanism holds true only in the interacting theory; in the free theory the primary does not contain that sub-representation, so eq.(44) may be viewed as playing the analog of the classical equation of motion of the Wilson-Fisher fixed point.
Let us analyze the deformed theory at . There are two cases to be considered:
If we take we would have
At the descendant operator of the deformed theory becomes degenerate with , therefore, according with the assumed one-to-one mapping between free and deformed theory, also is a descendent and its contribution should match in the limit with the coefficient of in (39), however this is impossible because of the different dependence in .
Hence the only consistent case is:
and forms in the limit another eigenfunction of having the same eigenvalue with .666 It is worth noting that the decomposition (46) in two terms survives in the limit , while the first term is absent if we directly put in (43). Now the coefficient of the descendant is finite, hence there is a perfect matching with the contribution in the free theory if and only if
which is consistent with the classic results for the corresponding anomalous dimensions evaluated using other methods (see e.g. ).
Let us stress that eq.s (46) and (47) rigorously show, using no other assumptions than conformal invariance, that the only consistent smooth deformation of the canonical free theory in is an interacting theory in which is a descendant of . Clearly, this corresponds to the interacting theory generated by the marginal perturbation and the conformal block expansion of the deformed theory becomes
Now, new results on OPE coefficients can be obtained by considering the deformations of suitable OPEs in which a contribution appears on the RHS, for instance
Both the above OPE expansions contain a contribution which is no longer a primary field in the deformed theory, hence it should be replaced by the conformal block of . Repeating the calculation described previously and using the corresponding values for the residues we obtain the following OPE coefficients
where the second equalities are obtained using (47).
4.2 The critical points
In this section we will apply the method explained in detail above to the much more general class of generalized free CFTs in various spacetime dimensions. Furthermore, we will demonstrate that we are able calculate precise values of anomalous dimensions instead of their ratios as in (47). We begin by considering the free OPE, but for a more general class of operators,
The corresponding OPE coefficients are calculated to be