On 1/N diagrammatics in the SYK model beyond the conformal limit

In the present work we discuss aspects of the 1/N expansion in the SYK model, formulated in terms of the semiclassical expansion of the bilocal field path integral. We derive cutting rules, which are applicable for all planar vertices in the bilocal field diagrams. We show that these cutting rules lead to novel identities on higher-point correlators, which could be used to constrain their form beyond the solvable conformal limit. We also demonstrate how the cutting rules can simplify the computation of amplitudes on an example of the six-point function.


Introduction and summary of results
The SYK model [1][2][3][4] is a maximally chaotic quantum mechanical many-body model that is solvable at large N. It also displays emergent conformal symmetry in the strong coupling limit, that is broken both spontaneously and explicitly. Maximal chaos and emergent conformal symmetry at strong coupling motivate using the SYK model as a holographic toy model for quantum near-extremal black holes [2,5,6].
The SYK is a model of N interacting Majorana fermions, living in (0 + 1) (Euclidean) dimensions. The Hamiltonian is given by [2]: The couplings j i jkl are randomized with respect to the Gaussian distribution P( j i jkl ) = N 3 12πJ 2 e − N 3 j 2 i jkl 12J 2 . The correlation functions and other physical quantities in the model are supposed to be disorder-averaged, using the rules At finite temperature the constant βJ admits the role of the dimensionless coupling; in the IR limit the model is in the strong coupling regime βJ 1. At large N limit in the language of * e-mail: arefeva@mi.ras.ru * * e-mail: khramtsov@mi.ras.ru * * * e-mail: tikhanovskaya@mi.ras.ru Feynman diagrams in terms of Majorana fermions all correlation functions of the model are dominated by the melonic graphs and their cuts [2,7,8].
An alternative formulation of the model can be obtained by averaging over the disorder in the partition function, introducing colorless bilocal auxiliary fieldsG(τ 1 , τ 2 ) andΣ(τ 1 , τ 2 ) and integrating out the fermions (see [4] for detailed derivation): where the action in terms of G, Σ fields has the form: (here and henceforth the notationF means the integral operator with the kernel F(τ, τ )). This path integral describes accurately describes the first few orders in 1/N, where the replicadiagonal assumption is valid 1 . The factor N in the action justifies the usage of the steepest descent method to evaluate the path integral at large N. Thus the large N expansion of SYK in terms of the bilocal fields is a semiclassical expansion for the path integral (3). The fermionic correlators of the SYK model are expressed in terms of the correlators of the fieldG: In the strong coupling limit ∂ τ → 0 the action (4) is conformally invariant. The conformal parts of all correlation functions in SYK can be computed analytically [2,3,8], however on the level of 4− and higher-point functions there are contributions that break the conformal symmetry and dominate in the βJ → ∞ limit. The non-conformal contributions come from the reparametrization soft mode [3,4,[9][10][11], which is dual to the gravitational mode in the bulk theory. Because of this, the studies of the non-conformal corrections to the correlation functions in SYK can be useful to gain insight into the full bulk dual of SYK.
In this work we discuss the technical tools which could be used to perform such calculations. Our main result are the cutting rules for planar diagrams. These cutting rules are exact in βJ, and can be applied at any order of the 1/N expansion to either obtain additional constraints on the higher-point correlators beyond the conformal limit, or to simplify the calculations, as we demonstrate on the example of the non-conformal six-point function.

Systematic 1/N expansion in SYK
Suppose G, Σ is the large N saddle point of the path integral (3). We define new variables and perform the semiclassical expansion in the bilocal fluctuation fields g, σ. The action (4) in terms of these fluctuation fields reads We consider the diagrammatics for this action. Every propagator of a bilocal field carries a factor of 1/N, whereas every vertex carries a factor of N. Therefore, the degree of a bilocal graph, contributing to a correlator, in the 1/N expansion is given by where L is the amount of lines in the diagram and V is the number of vertices.

Propagators
The two-point functions of the bilocal fields are determined by the quadratic part of the action (7) [3]: HereK is the symmetric ladder kernel [3]: Inverting the quadratic form, we find the propagators: The geometric series of the inverse ladder kernel in (11) is precisely the sum of the ladder diagrams, contributing to the fermionic four-point function. It can be computed explicitly by diagonalizing the ladder kernel in the conformal limit [2,3,11]. Schematically, one obtains • Conformal part: F conformal ∼ h m c ∆,m F ∆,m (1, 2, 3, 4)-has the form of the conformal partial wave expansion in a 1D CFT.
• Non-conformal part: Here k(h, n) is the ladder kernel eigenvalue, and ψ 2,n are the h = 2 eigenfunctions of the symmetric ladder kernel (10).

Vertices
The action (7) contains two types of vertices. We have two vertices for the field g and an infinite series of vertices from the field σ coming from the logarithm. One can show [12] that the g-vertices correspond to the fermionic graphs labeled as contact diagrams by Gross and Rosenhaus [7,8]. Meanwhile, the σ-vertices correspond to the planar diagrams. Based on this, we will refer to g-vertices as the contact vertices, and to σ-vertices as the planar vertices.

Contact vertices
From the action (7) we obtain two contact vertices: They are drawn on the Fig.1.

Derivation
Let us consider a vertex V of order 4 or higher. Then we can always cut it in such a way that the cut crosses two solid lines and nothing else, see Fig.3. Suppose that we cut through propagators G(τ a , τ b ) and G(τ c , τ d ), which connect two vertices U 1 (τ 1 , τ a , τ d ) and U 2 (τ 2 , τ b , τ c ).   Here we denote all the time variables which are untouched by cut propagators by bold letters. In this case, we can write the vertex as Now we can rewrite the product of propagators through the ladder kernel using the definition (10): We can diagonalize the ladder kernel using the system of its (normalized) eigenfunctions Substituting (19) into (18) and then into (17), we obtain Thus, we the vertex V has a form of sum over products of smaller vertices U 1 and U 2 , which are connected by the "partial waves" ψ h .

Cutting symmetry identities
Let us consider the 1PI planar four-point function of the bilocals. It is obtained using the quartic vertex shown on Fig.2B. The important property of the cutting rule is that it represents an equivalent way to rewrite a diagram. Therefore, different patterns of cuts on the diagrams should give the same result. In principle, one can cut the diagrams along the pairs of solid lines as much as needed, as long as each solid line is cut no more than once. The equivalence of different ways to cut gives us a set of relations reminiscent of the crossing symmetry equations. The simplest cutting symmetry equation for the 4-point function can be obtained from the equivalence of one vertical cut and one horizontal cut, see Fig.4. We can write the equation for the integrands (which in this case holds up to full derivatives): We emphasise that this relation is exact in βJ, and thus can be used to constrain correlation functions beyond the conformal limit. e.g. with non-conformal contributions of the form (15) to the external lines. One can obtain similar relations for higher-point functions.

The six-point function example
Let us consider the planar 3-point function of bilocal fields to the leading order of 1/N. The conformal contribution was computed in [7], and it was used as one of the building blocks to construct the all point correlation functions in the conformal limit of SYK in [8]. In the leading 1/N order the planar 3-point function is given by the 3-point planar vertex (see Fig.2A) dressed with gσ external lines: S 2 := g(τ 1 , τ 2 )g(τ 3 , τ 4 )g(τ 5 , τ 6 ) planar = g(τ 1 , τ 2 )g(τ 3 , τ 4 )g(τ 5 , τ 6 )V (3) σ ; where on r.h.s. the fields are averaged with respect to the quadratic action (9). We assume that one of the external lines is determined by the nonconformal contribution to the propagator, whereas the other two are conformal, as shown on the left side of the Fig.5. The Wick contractions give a factor 3! = 6. The vertex gives the factor of N J 6 /6, whereas the external propagators give a factor 2 N J 2 each. The expression for the planar diagram reads  Our approach is to cut the vertex in such a way that the We cut the diagram through the propagators G(t 23 ) and G(t 61 ), as shown on Fig.5. Using (20), we write the correlator as and we can evaluate U-s separately. So, h,n (τ 1 , τ 2 ) = dt 1 dt 2 g(τ 1 , τ 2 )σ(t 1 , t 2 ) |G(t 12 )| |G(t 12 )| ψ * h,n (t 1 , t 2 ) (23) = 2 N J 2 dt 1 dt 2 1 K − 1 non-conformal (τ 1 , τ 2 ; t 1 , t 2 )ψ * h,n (t 1 , t 2 ) = 2 N J 2 1 k(2, n) − 1 ψ * 2,n (τ 1 , τ 2 ) .
The external lines of this piece are conformal, so one can take the OPE limit in them, analogously to the calculation of the fully conformal six-point function in [7]. In that case the U (2) reduces to a 3-loop integral, which is much simpler to evaluate. This method of cutting off the external non-conformal lines can be applied to reduce the integral in any hugher-point planar 1PI correlation function.