Leading CFT constraints on multi-critical models in d > 2

Journal of High Energy Physics, Apr 2017

We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial ϕ m below their upper critical dimensions \( {d}_c=\frac{2m}{m-2} \), and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m ≥ 4 these theories coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d = 2, while for odd m the theories are non-unitary and start at m = 3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators ϕ k and of some families of structure constants in either the coupling’s or the ϵ-expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling’s expansion.

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Leading CFT constraints on multi-critical models in d > 2

Received: March Leading CFT constraints on multi-critical models in d Alessandro Codello 1 2 Mahmoud Safari 1 2 Gian Paolo Vacca 1 2 Omar Zanusso 1 2 W Symmetry 1 2 Open Access 1 2 c The Authors. 1 2 0 -Origins, University of Southern Denmark 1 via Irnerio 46 , 40126 Bologna , Italy 2 Campusvej 55 , 5230 Odense M , Denmark We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial m below their upper critical dimensions dc = m2m2 , and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d = 2, while for odd m the theories are non-unitary and start at m = 3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators k and of some families of structure constants in either the coupling's or the -expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling's expansion. cDipartimento di Fisica e Astronomia; Universita di Bologna Contents 1 Introduction Schwinger-Dyson consistency and CFT 2n-theory in d = d2n Anomalous dimensions Warm-up: 1 Climbing up: 2 The general case: k Structure constants Structure constants C1;2k;2l 1 Structure constants C1;1;2k Critical coupling g( ) Collecting the results: even potentials 2n+1-theory in d = d2n+1 Anomalous dimensions Structure constants Structure constants C1;k;l Structure constants C1;1;2k The special case of C111 for n > 1 Critical coupling g( ) for n = 1 Collecting the results: odd potentials Case n > 1 Conclusions A Free theory B Action of the Laplacian Introduction of the standard methods of quantum eld theory (QFT) are used, including perturbation when critical properties are under investigation. We will be interested in generalizing this idea to theories governed by the general potential. In a Ginzburg-Landau description their action is moment we shall simply ignore them. g is canonically dimensionless S[ ] = protected by a Z2 parity ( the 2n e ective potential describes a statistical system with a phase-transition that can be gk of (1.1) do not play a signi cant role in tuning the action to criticality, therefore for the The upper critical dimension of (1.1) is de ned as the dimension d at which the coupling dm = mension the statistical uctuations are weak and the physics of (1.1) is Gaussian and with an anomalous dimension. In the latter case a consistent expansion for the crit1We follow the convention that universality classes such as Ising's are denoted with typeset font, therefore the spin 1 Ising model at criticality is only one speci c realization of the Ising universality class and the two should not generally be confused. The paper will deal with universality classes to a greater extent. d = dm and that for n d2n of (1.2)! quantities in the form of a Taylor series in . The most important critical exponents of all the aforementioned special cases (Ising, let us point out that for n 3 the leading contributions arise from multiloop computations, 4 the divergences are subtracted as poles of the fractional dimensions Another interesting property is that the even models are known to interpolate in the sequence of minimal non-unitary multi-critical theories M(2; m + 2) studied in [13]. eld theories for any dimension 2 dm. The straightforward question that we will determining the critical properties of (1.1)? The paper is organized as follows: in section 2 we brie y summarize the main features the CFT correlators. Schwinger-Dyson consistency and CFT 2In the non unitary models, e.g. Lee-Yang universality class, the critical coupling and some structure m-theory S[ ] = ing (2.1) with (1.1). In (2.1) we introduced a reference (mass) scale which makes the almost marginal coupling g dimensionless for any d. The presence of the mass scale example, all the n di erent phases of a 2n theory coexist. Before diving more deeply into some technical details, it is worth noting that, with the More generally, after the displacement by all the theories will live in the arbitrarily real dimension d = dm . Theories living in continuous dimensions have already been realized for any value of the dimension d.3 The key idea of [1] is that all the CFT data of (2.1) must interpolate with that of the Gaussian theory in the limit dimensions for the eld and the composite operators in d dimensions. Let the canonical dimension of ! 0. We set some notation by de ning the scaling m of an interacting scalar theory 1 = m m = and the scaling dimensions of and k be respectively k = k The -terms represent the corrections from the canonical scaling dimensions and therefore must be proportional to some power of g or to ensure consistency of the Gaussian limit. 3Scale invariance seems to imply conformal invariance for several physically interesting critical models, supported at theoretical level [21]. The Schwinger-Dyson equations (SDE) generalize the notion of equations of motion zero. In practice, for any state of the CFT and for any list of operators Oi the relation (x) O1(y) O2(z) : : : = 0 order one can use the relation h2x (x)O1(y)O2(z) : : :i = m 1(x)O1(y)O2(z) : : :i teracting CFT the operator 1 are primaries, while the operator m 1 is a descendant.4 In other words, the interacting CFT enjoys one less independent operator, that is m 1, and a recombination of the conformal multiplets must take place. In particular, the scaling dimensions of m 1 must be constrained m 1 = =) m 1 = 1 + (m sides of the SDE. It is possible to nd a basis Oa of scalar primary operators with scaling a whose two point correlators are diagonal k with k 6= m hOa(x)Ob(y)i = factors which can in principle be set to one. However, for the moment, we will more constrained by conformal symmetry and reads hOa(x)Ob(y)Oc(z)i = are completely and uniquely speci ed by providing the scaling dimensions a and the reasons paramount target of any computation. Our goal is to extract the leading informations for a part of the conformal data of all 4A descendant operator in d > 2 is the derivative of a primary operator, which is annihilated by the of the d = 2 case. theories at an interacting xed point in dm with also conformal bootstrap techniques. 2n-theory in d = d2n dimensions as a power series in , eventually This section is dedicated to the investigation of the even potentials 2n in d = d2n that corresponds to the Ising universality class, the upper critical dimensions are d2n = = 4 ; 3 ; ; ; In the limit n ! 1 the critical dimensions tend to two, that is the dimensionality for which serves as a testing ground for the entire method. In the rst part of this section, we will kickstart the computation by obtaining the anomalous dimension for the eld by using a constraint which comes from the consistency of the two point function (2.7) with the SDE (2.5) in the limit ! 0. Then we will repeat scaling dimensions of all the composite operators k . We will see that 2, which is related on those that are not present at zeroth order in and are thus generated at quantum level. In the third part we will exploit the fact that the scaling dimension of the 2n 1 descendant operator can be computed in two di erent ways and use it to nd a critical value for the perturbation theory of [10]. All the results are summarized at the end of the section. Anomalous dimensions eld 1 and of the composite operators k with k 2. By LO we will generally mean leading order in g and in . Only when an explicit relation g( ) will be available (as for even potentials in section 3.3), leading order will mean leading order in . { 6 { We start with a simple analysis of the two point function that will directly uncover a precise leading order relation between 1 and the coupling g. The determination of 2 requires the analysis of three point function h 2i and is a bit more involved for n > 2. Finally we shall be able to obtain the anomalous dimensions k with k n from the k k+1i. In these rst computations we will proceed step by step in order to correlators as detailed in appendix A. Warm-up: 1 Let us consider in d dimensions the propagator of the interacting theory h (x) (y)i = in eq. (A.2). Thus we will make the replacement C ! c everywhere from now on. On applying rst the SDE in one point one shows that 1 is at least of order g2. Then applying the SDE also to the second point gives the leading expression for 1 in terms of and recalling 1 = + 1 gives 2x h (x) (y)i = 2x x = c tions 4( + 1) 1 ! 4 2n 1 in the numerator and 1 + + 1 ! 1 + 2n in the denominator, expression using the SDE one nds h2x (x) (y)i = 2n 1(x) (y)i = O(g2) : 1 is at least of order g2. To obtain another useful relation one acts with a 2x2y h (x) (y)i = 2x2y x = c 2 1(2 1 +2)(2 1 2 )(2 1 +2 2 ) 16( + 1)( + 1 + 1) 1(1 + 1) ! 16 2n( 2n + 1) 1 in the numerator and 2 1 + 4 ! 2 2n + 4 two point function of eq. (A.3) of appendix A gives instead h2x (x)2y (y)i = (2n 1)! 2n 1(x) 2n 1(y)i L=O anomalous dimension Using the fact that we nd the explicit formula g2 + O(g3) : c = 1 = h2x (x) (y) 2(z)i = 2n 1(x) (y) 2(z)i L=O g which agrees with the perturbative result [10]. Climbing up: 2 where it appears is To determine 2 we need to consider the three point functions. The simplest correlator h (x) (y) 2(z)i = Laplacian can be easily obtained from eq. (B.3) given in appendix B by setting 2 = 2 1 2 and 2 = 3 = 2 = 2 + 2 From this expression we easily determine the leading order contributions 2x h (x) (y) 2(z)i L=O should match the one obtained by applying the SDE 1 = following expression 2 = n = 2 : In order to nd the leading value of 2 in the general case n > 2 we act with the second one nds (we skip the intermediate steps) which we should compare with the leading order result obtained applying the SDE, 2n 1(x) 2n 1(y) 2(z)i L=O so that by comparison we obtain 2 1 = 16(n 2)(2n 1)!2 Using the explicit expression for 1 given in eq. (3.9) we nd 2 = 8 (n 2)(2n)! This quantity has not been reported in the perturbative results given in [10]. The general case: k To determine k at rst we could think to consider h the free theory whenever k > 2. To investigate all k ki, but this correlator is zero in 2 we instead consider the following three point function h (x) k(y) k+1(z)i = k 6= 2n subsection 3.3. 1. Indeed for k = 2n 2; 2n 1 other terms are present. Nevertheless, if one restrict the analysis to the lowest order, these extra terms which are subleading can from the general expression (A.8) and reads C1fr;kee;k+1 = (k +1)! ck+1 : The main recursion relation can then be derived for k n 1 applying a Laplacian in x the same reasoning of the previous subsections, we nd the following LO expression On the other hand using the SDE one gets h2x (x) k(y) k+1(z)i = C2frnee 1;k;k+1 = k!(k +1)!(2n 1)! (k n+1)!(n Vice versa when k n 2 the free correlator is zero and the the full correlator in eq. (3.21) is at least of order O(g2). The expression obtained from the SDE in eq. (3.21) has a leading the rst term in eq. (3.20) is negligible and that k+1 k = O(g). Then by comparing eqs. (3.20) and (3.21) one nds the recurrence relation k = (n 2)! n! (k n+1)! O(g) ceases to exists for k 2 and is substituted by some relation involving O(g2) solve the recurrence relation to obtain n 1. With this condition one can k = we correctly reproduce eq. (3.13). The above relation says that for k 1 the leading contribution to the anomalous next to leading order O( 2), and for k in the next subsection. ourselves to h for this to be nonzero is family of even universality classes. In order to get some information from the three point functions using the SchwingerDyson equations we need to have one of the elds to appear with power one. The h are already explored and give information on the scaling dimensions i. In the following we therefore concentrate on the rest of these correlation functions. Structure constants C1;2k;2l 1 The remaining correlation functions consist of h lj 6= 1. These vanish in the these are at least proportional to the coupling or smaller. Now if h in the free theory it implies that C1kl are at least of order O(g2) and to nd their value at leading order we need to know h 2n 1 k li beyond free theory. Therefore we will not be 2n 1 k li not to vanish in the free theory we must have the following conditions. 1 is odd, either k or l must be even while the other must be odd, so we restrict 1; n > 1. As previously discussed, the condition These are equivalent to k +l otherwise we will be back to the case h for k; l satisfying the conditions k k+1i which is already studied. In summary, k 6= 0 or 1 ; we can nd the leading order (O(g)) structure constants C1;2k;2l 1. One can use the SDE h x (x) 2k(y) 2l 1(z)i = x to the correlation function h (x) 2k(y) 2l 1(z)i one nds L=O C1;2k;2l 1 x l + 1)(d2n One readily sees, using the relation 2n 1 = 1 + 2, that the denominators in the above two expressions are equal. Comparing the coe cients we nd C1;2k;2l 1 = 2n 1;2k;2l 1 1)! 4(k C2frnee 1;2k;2l 1 = (n+l k 1)!(k +n l)!(k +l n)! and c is the normalization of the free propagator given in eq. (A.2). Structure constants C1;1;2k O(g) for other coe cients of the form C1;1;2k with k in the range 2 1, which turn out to be of order O(g2). These can be extracted from the analysis of the family of correlators considered in the previous subsection h (x) (y) 2k(z)i = where k > 1. Clearly the coe cients C1;1;2k for k > 1 vanish in the free theory. We proceed so that we obtain h2x (x)2y (y) 2k(z)i = C1;1;2k = (n 1)4Cfree 2n 1;2n 1;2k 1)!2 16k(k 1)(k For higher values of k one needs to know the correlation function h range of validity for this formula is therefore 2 1 and k 6= n 1; n. As mentioned before, 1; n 1; n were excluded from the possible values k can take in this subsection, and 1; n provide a di erent way to compute 2(n 1); 2n, which can be shown to be consistent with the results of the previous subsections. Critical coupling g( ) 1) , only if we knew the anomalous dimension 2n 1. The general formula for the anomalous dimension k was excluded there because the correlation function h k 2ni in these cases would involve a (2n 1)! g (n 1) 2y h (x) (y) 2n(z)i one notices that h (x) (y)2n 1 2n(z)i = (2n 1)! (2n 1)! LO (2n 1)! C1;1;2n 4 The structure constant on the r.h.s. evaluated in the free theory is nonzero for k C2frnee 1;2n 1;2k = (2k)!(2n 1)!2 k!2(2n k 1)! C1;1;2k = (2k)!(n 1)4 c2n+k 1 16k(k 1)(k n)(k n+1)k!2(2n k 1)! since 2 1 +2 d = 2 1 = O(g2), and C1;1;2n = (2n 1)! n(n 1)(d2n 2)2 on the the scaling dimension of the descendant operator 2n 1 from the equation of motion, (n 1) + O(g2) = 2n 1 = which gives the linear relation 2(n 1) (2n a non primary operator, h (x) (y)2n 1 2n(z)i L=O We notice that the expressions (3.43) and (3.45), for 1 and k, are in agreement with the results obtained in [10] with a perturbative computation. + O( 2) = is linear in , it is possible to uniquely determine g = 1) g + (n 1) which shows that the non trivial xed point of the CFT is IR attractive (g > 0). Collecting the results: even potentials Anomalous dimensions. The anomalous dimensions k for 1 1 are found to be of O(g2) but only the rst two, 1 and 2, are determined at leading order. The rest are of O(g) and their leading values, together with 1 and 2 are summarized here 2 = 8 k = 2(n 1) (n 1)3(n+1) n!6 (2n)! (k n)! One may write a generating function for the anomalous dimensions of all these multiat O( ), which gives k for any k n in eq. (3.45), can be written as F (even)(x; y; ) = ex (pxy sinh pxy 2 cosh pxy) + O( 2) ; k(n; ) = @yn @xk so that one has We nd C1;2k;2l 1 = C116 = C114 = + O( 2) ; (3.48) k 6= 0; 1, and (2k)!(2l 1)! l + 1) (n+l k 1)!(k +n l)!(k +l n)! within the limits of eq. (3.26), namely k + l C1;1;2k = k(k 1)(k n)(k n+1) (2n)!2 k!2(2n k 1)! for k 6= n 1; n and 2 1. In this scheme all the factors are absent. We note, however, that for comparison with results obtained from perturbation theory other normalizations may prove more convenient. set of leading order structure constants that we have found we report all the ones of O( 2) and only a few of the in nite sequence of order O( ). For the Ising universality class: C114 = C125 = C136 = 20 ; C116 = For the Tricritical universality class: C114 = And nally for the Tetracritical universality class: C118 = C136 = C127 = C125 = ; C1;1;10 = ; C1;1;12 = ; C1;1;14 = C116 = C118 = C125 = 6 ; C136 = 54 ; C1;1;10 = 555984 2 odd potentials 2n+1 for n a natural number n 1 which arise as particular cases of (2.1) P T -symmetry [23{25]. On a general action S[ ] as in (1.1) P T -symmetry acts as d2n+1 = 2 + = 6 ; which similarly to d2n tend to two in the limit n ! 1. In a Ginzburg-Landau description fact these models seem to be non-unitary for all d The well-known upper critical dimension of the Lee-Yang universality class is six. < 1 in a physically interesting scenario. The models with odd potentials are much quantities in the process. As for the content of this section, it will mostly follow the development of section 3, C1;1;1. In the third part we will show that the possibility to x the coupling to its critical value as a function of only occurs for the Lee-Yang universality class. All the results will be summarized in the nal part of this section. Anomalous dimensions 1 and the coupling g by acting with two Laplacians on the propagator and using the SDE, which now gives the operatorial relation 2 so that 2n is a descendant of . Taking into account that the results of section 3 must be shifted as n ! n + 12 , so that 1 = c2ondd 1 ((22nn+11))2! 3g22 + O(g3) = g2 + O(g3) ; codd is obtained from (A.2) after the shift n ! n + 12 , codd = correlator h we can directly infer i when all the operators are primary. Therefore from expression (3.17) 2 = c2ondd 1 ((22nn + 3)(2n 3)(2n + 1)! 16 + O(g3) = (2n + 3)(2n 3)(2n + 1)! g2 + O(g3) ; 2n = 1 + dimensions. From the study of the correlator of primary operators Unfortunately we are not able to nd a closed expression for the other anomalous h (x) k(y) k+1(z)i = one which involves the descendant operator '2n 2n(x) k(y) k+1(z)i = Acting on (4.7) with a Laplacian in x and keeping only leading order terms gives h2x (x) k(y) k+1(z)i L=O 2C1fr;kee;k+1 2n k k+1i is zero in the free theory we corrections are expressed in terms of integer powers of g we conclude that k+1 least of order O(g2) and thus k = O(g2) ; Structure constants potentials, one can consider the correlation functions h 2ki with the action correlation function h i with the action of a triple Laplacian also gives some leading order information on the structure constants. Below, we consider each case in turn. Structure constants C1;k;l for the correlator h 2n k li to acquire a contribution in the free theory is di erent. Here k; l have to be either both even or both odd. Furthermore they must satisfy (2n)!k!l! codd ; (4.12) This is equivalent to k+l free theory is 2n(x) k(y) l(z)i L=O 2n. In this case the correlator h Let us therefore consider for k; l 6= 2n h (x) k(y) l(z)i = This tells that the correlator involving the descendant operator 2n gets three contributions depends on the C1;k;l, the scaling dimensions 2n = k, l and the dimension d. In the following we shall restrict to few considerations based on this relation. Acting 2n(x) k(y) l(z)i = one nds 4 C1;k;l 2x h (x) k(y) l(z)i = C1;k;l 1 (d 2)(1+k l) + 2( 1 + k (d 2)(1+l k) + 2( 1 + l [(d 2)(k +l 1) + 2( k + l 2)(1+k l) + 2( 1 + k + : : : (4.16) the same coordinate dependence of the expression in eq. (4.12), so that n and gives C1;k;k = k!2(2n 1)2 Structure constants C1;1;2k Let us nally consider the correlator h 2k . Again, the analysis in this case follows function obtained using the SDE twice 2n(x) 2n(y) 2k(z)i L=O This gives the structure constants 26k(k 1)(4(k n)2 1) k!2(2n C2frne;e2n;2k : (4.20) nonzero if k 2n, therefore the range of validity of this equation is 2 2n. For k = n the correlation function under study h 2ki involves a descendent operator and therefore this includes several terms as can be seen by writing Using eq. (B.3) of appendix B, the leading term in this expression can be shown to be h (x) (y) 2n(z)i = 2z h (x) (y) (z)i = It turns out that the coe cient of this leading term which we can now call C1;1;2n satis es structure constant C111 which we compute in the next subsection. The special case of C111 for n > 1 Let us now consider the action of a triple Laplacian on h i for n > 1, which lies outside box operator three times one nds the following leading contribution the SDE has been used three times 2n(x) 2n(y) 2n(z)i L=O C2frne;e2n;2n Comparing the two, we obtain the following expression of order O(g3) for the structure C111 = LO C2frne;e2n;2n (2n 1)6 (2n)!3 28n(n 1) g3 = 28n(n 1)n!3 dimension 1. Following [15], one may evaluate at leading order 2x2y2z h (x) (y) (z)i L=O 32( On the other hand, in this case C111 is already known, because eq. (4.18) is still valid for k = 1 and gives5 C111 = corresponding equation found from the SDE 8 32C111 8 codd = we nd the relation Lee-Yang universality class g at the xed point is proportional to p 5This is also in agreement with the OPE coe cient found in [16]. which shows again that the interacting xed point is IR attractive. Collecting the results: odd potentials stants in the normalization obtained by rescaling the elds the propagator to unity. cod1d=2 which normalizes 1 in terms of , and nally using the relation (4.6), which links the anomalous scaling of the descendant operator 2 to the one of summary, for the Lee-Yang universality class we get one obtains the leading -dependence of 2. In g2 = 1 = 2 = g2 = g = , we nd that Moreover, the fact that 1 + k k = O( ) : p2=3 + O( ) ; In fact one can restrict to l repetition. Some of these structure constants are listed as follows C122 = ; C111 = ; C113 = ; C133 = : (4.35) C114 = Case n > 1 constraints. It is not possible to xed point g( ) so the results are expressed in terms of the coupling g, which always appears through the combination g condd 1=2, with codd given in eq. (4.4). We start from the anomalous dimensions. The leading order constraints give xed point coupling g( ) and therefore we expect both negative 1 and 2 (which is instead 1 = 2 = (2n+3)(2n 1)3 (codd2 g)2 (2n 3)(2n+1)! + O(g3) ; + O(g3) ; 2 = 2 (2n+3)(2n 1) + O(g) : k = O(g2) ; 2n = 1 + from which we can deduce a well determined leading order result for their ratio While for k > 2, all one can get is Furthermore, from the relation between the scaling dimension of and 2n one nds C1;1;2k = C111 = 26k(k 1)(4(k n)2 1) k!2(2n 28n(n 1)n!3 (codd2 g)3 + O(g4) : For the structure constants we have, at the leading order approximation: + O(g2) ; Conclusions value d = dm . What renders our analysis unique is that for most values of m, the upper The sequence of models for m even enjoys Z2 parity and encodes the scale invariant independent computation based on perturbation theory [22]. The extent of our results di ers between even and odd models, and the strength of the 2n, for which we could obtain the anomalous dimensions 1 and 2 and k n, two entire families of structure constant C1;2k;2l 1 and C1;1;2k, as well as a relation between critical coupling g( ). In section 4 we studied the odd potentials 2n+1, for which we could determine 1 and 2 together with the structure constants C1;k;l, C1;1;2k and C1;1;1. Only for the cubic potential 3, corresponding to the Lee-Yang universality class, we could to re-express all critical quantities in terms of 1, which yields some simpli cation. All Our analysis is very encouraging in that it can be considered as a rst step in the bridge from criticality in dimension d 2 to the well known minimal models in CFT perturbation theory [41, 42], which may prove useful in this direction. A special comment must be made on unitarity of the spectrum. In fact, the -expansion Furthermore, almost all the m potentials have a purely rational upper critical dimension. the presence of negative norm states should be investigated. The possible non-unitarity of the spectrum should be distinguished from the nonall odd models we would like to point out that the quintic model 5 has upper critical dimension dc = 130 > 3, implying that model further in the future [30]. Note added. After the completion of this work we became aware of the two works [43, 44] here. In particular the leading anomalous dimensions k for k > n. Moreover coincides with our eq. (3.48), once the composite operators k are rescaled by pk! in order nd (the square of) a family of leading OPE coe cients (see eq. (4.36) of [44]) which family of O( 2) structure constants C1;1;2k that we have reported in eq. (3.49). Free theory A.C. and O.Z. are grateful to INFN Bologna for hospitality and support. one can rescale the elds to obtain two point functions normalized to one. We nally consider a generic three point correlator of the form n1, n2 and n3 connected by l12, l23 and l31 propagators, in cyclic order respectively. One ni = lij + lki () lij = (ni + nj i 6= j 6= k : The correlator is non zero when there exists a solution such that lij are non negative integers (lij 0). Then the number of all possible con gurations (contractions) is given by Here Sdm is the area of the dm-dimensional sphere. A generic two point correlator for the k is given by h (x) (y)i f=ree c = k(x) l(y)i f=ree Nn1;n2;n3 = n1! n2! n3! l12! l23! l31! so that, with the above normalization, the explicit form of the correlator is given by Cnfr1e;en2;n3 = n1! n2! n3! n1+n2 n3 ! n2+n3 n1 ! n3+n1 n2 ! ( +2)( + 2 d)( +4 d) correlators and nd some lengthy expressions. The action of one Laplacian 2x is: Action of the Laplacian the simple relations @x jx y one rst derives The action of two Laplacians 2x2y is: 1 2( 1 + 3 +2 d)( 1 + 2 1 2(2 + 1)( 1 + 3 +2 d) 2 3( 1 + 2 +2 d)( 1 + 3 +2 d) 1 3( 1 + 2 +2 d)( 1 + 3 1(2 + 1) (2 2 3 + ( 1 +2 d)( 1 + 2 + 3 +4 d)) 1 3(2 + 1)( 1 + 2 +2 d) Open Access. This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. (2015) 29FT01 [arXiv:1505.00963] [INSPIRE]. [arXiv:1302.0884] [INSPIRE]. 1819] [INSPIRE]. River Edge U.S.A. (2001). [3] F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE]. [Erratum ibid. A 13 (1980) 3785] [INSPIRE]. Percolation Problem and the Yang-lee Edge Singularity, J. Phys. A 14 (1981) 2391 [INSPIRE]. (2015) 025012 [arXiv:1506.03357] [INSPIRE]. Nucl. Phys. B 77 (1974) 91 [Erratum ibid. B 86 (1975) 548] [INSPIRE]. [INSPIRE]. Phys. Rev. Lett. 54 (1985) 1354 [INSPIRE]. [INSPIRE]. [18] P. Basu and C. Krishnan, -expansions near three dimensions from conformal eld theory, JHEP 11 (2015) 040 [arXiv:1506.06616] [INSPIRE]. point in 4- dimensions, Phys. Rev. D 93 (2016) 125025 [arXiv:1512.00013] [INSPIRE]. [INSPIRE]. to quantum eld theory with cubic interaction, Phys. Rev. D 70 (2004) 025001 [Erratum ibid. [24] C.M. Bender, V. Branchina and E. Messina, Ordinary versus PT-symmetric 3 quantum eld theory, Phys. Rev. D 85 (2012) 085001 [arXiv:1201.1244] [INSPIRE]. eld theory, Phys. Rev. D 87 (2013) 085029 [arXiv:1301.6207] [INSPIRE]. models, Phys. Lett. B 266 (1991) 363 [INSPIRE]. Criticality Behavior of the Blume-capel Quantum Chain as a Check of Zamolodchikov's Conjecture, Nucl. Phys. B 330 (1990) 741 [INSPIRE]. hep-th/9402143 [INSPIRE]. [29] A. Mossa and G. Mussardo, Analytic properties of the free energy: The Tricritical Ising model, J. Stat. Mech. 0803 (2008) P03010 [arXiv:0710.0991] [INSPIRE]. the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical bootstrap, arXiv:1611.08407 [INSPIRE]. [INSPIRE]. from the Conformal Operator Product Expansion, Phys. Rev. Lett. 118 (2017) 061601 blocks and the generalized Wilson-Fisher xed points, JHEP 04 (2017) 056 [1] S. Rychkov and Z.M. Tan , The -expansion from conformal eld theory , J. Phys. A 48 [2] Y. Nakayama , Scale invariance vs conformal invariance , Phys. Rept . 569 ( 2015 ) 1 [4] A.B. Zamolodchikov , Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory, Sov . J. Nucl. Phys . 46 ( 1987 ) 1090 [Yad . Fiz. 46 ( 1987 ) [5] H. Kleinert and V. Schulte-Frohlinde , Critical properties of 4-theories , World Scienti c, [6] O.F. de Alcantara Bon m , J.E. Kirkham and A.J. McKane , Critical Exponents to Order 3 for 3 Models of Critical Phenomena in Six -dimensions , J. Phys. A 13 ( 1980 ) L247 [7] O .F. de Alcantara Bon m , J.E. Kirkham and A.J. McKane , Critical Exponents for the [8] J.A. Gracey , Four loop renormalization of 3 theory in six dimensions , Phys. Rev. D 92 [9] A.J. Macfarlane and G. Woo , 3 Theory in Six Dimensions and the Renormalization Group , [10] J. O'Dwyer and H. Osborn , -expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations , Annals Phys . 323 ( 2008 ) 1859 [arXiv:0708.2697] [11] A.A. Belavin , A.M. Polyakov and A.B. Zamolodchikov , In nite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl . Phys . B 241 ( 1984 ) 333 [INSPIRE]. [12] L. Zambelli and O. Zanusso , Lee-Yang model from the functional renormalization group , Phys. Rev. D 95 ( 2017 ) 085001 [arXiv:1612.08739] [INSPIRE]. [13] A.A. Belavin , V.A. Belavin , A.V. Litvinov , Y.P. Pugai and A.B. Zamolodchikov , On correlation functions in the perturbed minimal models M (2; 2n + 1), Nucl . Phys . B 676 [14] J.L. Cardy , Conformal Invariance and the Yang-lee Edge Singularity in Two-dimensions , [16] C. Hasegawa and Yu. Nakayama, - Expansion in Critical 3-Theory on Real Projective Space from Conformal Field Theory, Mod. Phys. Lett. A 32 ( 2017 ) 1750045 [arXiv:1611.06373] [17] Y. Nakayama , Bootstrapping critical Ising model on three-dimensional real projective space , [19] S. El-Showk , M. Paulos , D. Poland , S. Rychkov , D. Simmons-Du n and A. Vichi , Conformal Field Theories in Fractional Dimensions, Phys. Rev. Lett . 112 ( 2014 ) 141601 [21] B. Delamotte , M. Tissier and N. Wschebor , Scale invariance implies conformal invariance for the three-dimensional Ising model , Phys. Rev . E 93 ( 2016 ) 012144 [arXiv:1501.01776] [22] A. Codello , M. Safari , G.P. Vacca and O. Zanusso, to appear. [23] C.M. Bender , D.C. Brody and H.F. Jones , Extension of PT symmetric quantum mechanics [25] C.M. Bender , V. Branchina and E. Messina , Critical behavior of the PT-symmetric i 3 [26] A. Koubek and G. Mussardo , ( 1 ; 2) deformation of the M ( 2 ; 2n + 1) conformal minimal [27] G. von Gehlen , O [30] A. Codello , M. Safari , G.P. Vacca and O. Zanusso, to appear. [31] M.E. Fisher , Yang-Lee Edge Singularity and 3 Field Theory, Phys. Rev. Lett . 40 ( 1978 ) [32] A. Raju , -expansion in the Gross-Neveu CFT , JHEP 10 ( 2016 ) 097 [arXiv:1510.05287] [33] A. Codello , Scaling Solutions in Continuous Dimension, J. Phys . A 45 ( 2012 ) 465006 [34] T. Hellwig , A. Wipf and O. Zanusso , Scaling and superscaling solutions from the functional renormalization group , Phys. Rev. D 92 ( 2015 ) 085027 [arXiv:1508.02547] [INSPIRE]. [35] S. El-Showk , M.F. Paulos , D. Poland , S. Rychkov , D. Simmons-Du n and A. Vichi , Solving the 3D Ising Model with the Conformal Bootstrap , Phys. Rev . D 86 ( 2012 ) 025022 [36] S. El-Showk , M.F. Paulos , D. Poland , S. Rychkov , D. Simmons-Du n and A. Vichi , Solving Exponents , J. Stat. Phys . 157 ( 2014 ) 869 [arXiv:1403.4545] [INSPIRE]. [37] R. Gopakumar , A. Kaviraj , K. Sen and A. Sinha , Conformal Bootstrap in Mellin Space, Phys. Rev. Lett . 118 ( 2017 ) 081601 [arXiv:1609.00572] [INSPIRE]. [38] R. Gopakumar , A. Kaviraj , K. Sen and A. Sinha , A Mellin space approach to the conformal [39] F. Gliozzi and A. Rago , Critical exponents of the 3d Ising and related models from Conformal Bootstrap , JHEP 10 ( 2014 ) 042 [arXiv:1403.6003] [INSPIRE]. [40] A.L. Pismenskii , Calculation of the critical index bootstrap method, Theor. Math. Phys . 185 ( 2015 ) 1516 [INSPIRE]. [41] L.F. Alday , Large Spin Perturbation Theory, arXiv:1611 .01500 [INSPIRE]. [42] L.F. Alday , Solving CFTs with Weakly Broken Higher Spin Symmetry , arXiv:1612.00696 [43] F. Gliozzi , A. Guerrieri , A.C. Petkou and C. Wen , Generalized Wilson -Fisher Critical Points [44] F. Gliozzi , A.L. Guerrieri , A.C. Petkou and C. Wen , The analytic structure of conformal

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Alessandro Codello, Mahmoud Safari, Gian Paolo Vacca, Omar Zanusso. Leading CFT constraints on multi-critical models in d > 2, Journal of High Energy Physics, 2017, 127, DOI: 10.1007/JHEP04(2017)127