Near Conformal Perturbation Theory in SYK Type Models

We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale $J$ and emergent reparametrization symmetry in the IR. This is given in the framework of the perturbation theory scheme of arXiv:1608.07567 based on specific (off-shell) breaking of conformal invariance in the UV, adjusted to yield the exact large-$N$ saddle point. While this breaking formally vanishes on-shell, it has a non-trivial effect on correlation functions and the effective action. In particular, it leads to the Schwarzian action with a specific coupling to bi-local matter. The method is applied to the evaluation of $O(1)$ corrections to the correlation function of bi-locals. As a byproduct we confirm precise agreement with the explicit, symmetry breaking procedure. We provide a verification in the large $q$ limit (Liouville theory), where the correlators can be calculated exactly at all length scales. In this case, our scheme illuminates how the enhanced $O(J)$ and the subleading $O(1)$ contributions originate from the Schwarzian dynamics of the soft mode and its interaction with $h=2$ (bi-local) matter.


Introduction
In the large N dynamics of models of SYK type [1][2][3][4][5][6][7][8][9][10][11][12] 1 (including tensor models [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]), a central role is played by the emergent Schwarzian mode which is dual to the gravitational mode in the dual theory. 2 While the origin of this mode can be traced to (time) reparametrization symmetry of the critical theory, its dynamics and its couplings to the matter degrees of freedom emerges in the near-critical region. Generally, the low energy dynamics of the soft modes arising from the spontaneous breaking of an approximate symmetry in a quantum field theory is an important problem which appears in many areas of physics. In quantum field theories with a set of global symmetries in many physical situations, one is interested in fluctuations around a classical solution which break some of these symmetries (e.g. a soliton solution [67,68]). Then a naive perturbation theory around this solution will be infrared divergent because of a zero mode associated with the broken symmetry. The way to deal with this is well known: one needs to introduce collective coordinates whose dynamics provides the essential low energy physics.
In many cases, however, the symmetry is approximate, and there is a parameter λ such that only when λ = 0 the action has the symmetry. This would cause the saddle point to shift and the zero mode will be lifted to an eigenvalue of order λ. In an expansion around this shifted saddle, there is no need for a collective coordinate and the answer for e.g. the connected correlation function would be then proportional to λ −1 which is large for small λ, hence "enhanced". Nevertheless, we would like to have an understanding of this enhanced contribution in terms of an effective low energy description.
In the present case, the bi-local collective picture offers a framework to study in detail the emergent zero mode dynamics coupled to bi-local matter. In particular, in [7] a framework was given for both evaluating the leading Schwarzian action, and also developing a systematic perturbation expansion near criticality. It relied on understanding the mechanism for (spontaneously) broken conformal symmetry in these nonlinear theories: which was introduced in [6] through an off-shell (source) mechanism. A related scheme was also used in [11].
In addition to the Schwarzian action, there is the action of fluctuations with conformal dimension h = 2 (which should not be confused with the infinite tower corresponding to operators with conformal dimensions h n > 2) which we call (h = 2) "matter" and interaction between this fluctuation field and the Schwarzian mode. The interpretation of these fluctuations in the gravity dual is not clear at this moment. The importance of a full understanding of the emergence of the Schwarzian mode (and even more of its interaction with matter) lies in the fact that this relates to the dual description of the gravitational mode in Jackiw-Teitelboim gravity in two dimensions [69][70][71][72][73][74], and its various possible interactions with matter. 3 In this paper we discuss further and complete the scheme of [6,7] developing fully the Schwarzian-matter coupled description. The key idea is to replace the explicit symmetry breaking term in the SYK action by a regularized O(1/J) source term. A systematic diagrammatic picture is then established, which will allow evaluation of perturbative effects. The singular nature of interactions following from [6] makes the explicit evaluations of contributions nontrivial. One of the improvements provided by the present work will be a clear, systematic procedure for their evaluation (where each diagram will reduce to well defined matrix elements of perturbation theory). Specifically, we prove that once the regularized source is chosen such that one gets the correct large N saddle, this diagrammatic procedure yields the correct 1/J expansion of the exact two point correlation function. These results receive a verification as follows.
In the q → ∞ limit, the SYK model can be solved with large N for any finite J. In this limit, the bi-local theory reduces to Liouville theory. 4 The situation here is a bit different from the finite q model where an explicit symmetry breaking term in the action for finite J is present. This breaking term plays an essential role in the large q limit. Nevertheless, as is well known, the resulting Liouville theory has an emergent conformal symmetry. The large-N saddle point breaks this symmetry. However, the finite J theory comes with a boundary condition -and the expansion around the saddle point which obeys this boundary condition does not lead to a zero mode since a candidate zero mode does not obey this boundary condition. Therefore a calculation for correlators of fluctuations around this saddle is well defined. This calculation was performed in [5] at finite temperature T and expanded in an expansion in powers of T /J. We perform this calculation at zero temperature in a different way and reproduce the zero temperature limit of the result of [5]. Instead of expanding around this saddle point one could have expanded around the saddle point appropriate to the large J (or long distance) limit. Now one gets a zero mode and the formalism developed above can be applied. We then compare an expansion of the exact result with this collective coordinate calculation and show a perfect agreement as expected.

Outline:
In Section 2, we review the bilocal formalism leading to the coupled Schwarzian/(bilocal) matter system and a corresponding diagrammatic scheme. For correlation functions we describe techniques for evaluation of contributing Feynman diagrams using various Schwarzian identities. We apply this to the O(1) correction to the enhanced propagator and show agreement with a perturbative expansion around the exact saddle point solution of the theory. In Section 3 we consider the explicit example of large q SYK/ Liouville theory and perform explicit computations implementing the general method. We conclude with some open questions in Section 4.

The Method
In this subsection, we will give a brief review of our formalism [6,7]which was introduced to exhibit the Schwarzian mode and generate a perturbative expansion around the conformal point. The Sachdev-Ye-Kitaev model [2] is a quantum mechanical many body system with all-to-all interactions on fermionic N sites (N 1), represented by the Hamiltonian where χ i are Majorana fermions, which satisfy {χ i , χ j } = δ ij . The coupling constant J ijkl are random with a Gaussian distribution, The original model is given by this four-point interaction, with a simple generalization to analogous q-point interacting model [2,5]. In this paper, we follow the more general q model, unless otherwise specified. After the disorder averaging of the random couplings (which corresponds to annealed averaging at large N ), the effective action is written as where a, b denote the replica indices. Here τ is treated as an Euclidean time. We do not expect a spin glass state in this model [3] and we can restrict to replica diagonal subspace [6]. Therefore, introducing a (replica diagonal) bi-local collective field: with an appropriate O(N 0 ) measure µ and the collective action: where the trace log term comes from a Jacobian factor due to the change of pathintegral variable, and the trace is taken over the bi-local time. This action being of order N gives a systematic 1/N expansion, while the measure µ found as in [92] begins to contribute at one loop level (in 1/N ). At zero temperature, one can redefine the time τ to get rid of the energy scale J. This also shows that in the IR the theory is strongly coupled and the first term linear in the bi-local field can be dropped. At finite temperature T this redefinition rescales the thermal circle to have dimensionless size J/T , which then becomes the coupling. In this paper we will deal with the zero temperature limit.
In the IR with the strong coupling J|τ | 1, the collective action reduces to the critical action which exhibits an emergent conformal reparametrization symmetry τ → f (τ ) with The first term in (2.6) explicitly breaks this symmetry. The saddle point solution of the action (2.7) gives the critical classical solution, where τ ij ≡ τ i − τ j and b is given by (2.10) In general, for a bi-local primary field φ h (τ 1 , τ 2 ) with the conformal dimension h, the transformation is given by For an infinitesimal transformation f (τ ) = τ + ε(τ ), we get the variation of the field and 14) The critical saddle point spontaneously breaks the conformal reparametrization symmetry, leading to the appearance of zero modes in the strict IR critical theory. This problem was addressed in [6] in analogy with the quantization of extended systems with symmetry modes [67]. The above symmetry mode representing time reparametrization can be elevated to a dynamical variable introduced according to [68] through the Faddeev-Popov method which we summarize below. We insert into the partition function (2.5), the functional identity: where δ ∫ u·Ψ f projects out the zero mode. After an inverse change of the integration variable, it results in a combined representation with an appropriate Jacobian. Here where we have used the invariance of the critical action. The symmetry breaking term implies a modification of the critical, conformal theory, which we would like to calculate perturbatively (in 1/J) in a long distance expansion. However since the breaking term is singular and non-vanishing only at short distances, the corrected solution cannot be obtained by treating it as an ordinary perturbation. To see this more clearly, let the exact classical solution be Here Ψ (1) is the lowest order shift which should satisfy the linearized equation where the kernel is given by where the inverse Ψ (0) −1 is defined in the sense of the star product (i.e. matrix product): dτ A(τ 1 , τ ) A −1 (τ , τ 2 ) = δ(τ 12 ) and explicitly given by Since the kernel transforms under scaling with a dimension 1 − 1 q , i.e. K (0) ∼ |τ | −4+4/q , one might expect Ψ (1) to have the form Ψ (1) (τ 1 , τ 2 ) ∝ sgn(τ 12 )|τ 12 | −4/q which, however is in disagreement with the desired δ -source of (2.19). Indeed, one sees that the exact δsource could only be matched at the non-perturbative level, where the 1/J corrections are all summed up.
Nevertheless, it was shown in [7] that a consistent 1/J perturbation theory around the critical solution is possible. The key idea is to replace the explicit symmetry breaking term in (2.6) by a regularized source term which is determined as follows. One considers an off-shell extension (of Ψ (1) ): with a parameter s > 0, because the dimension of Ψ (1) needs to be less than the scaling dimension of Ψ (0) . At the end of the calculation we will take the on-shell limit s → 1/2.
With this ansatz, we have sgn(τ 12 ) (2.23) The coefficient γ is uniquely specified by the integral with γ(1/2, q) = 0 [7], so that the ansatz (2.22) reduces to the homogeneous (on-shell) equation in the limit s → 1/2. The overall coefficient B 1 obviously does not follow from the above, but must be taken from the numerical result found in [5]. In turn, this overall coefficient can be left arbitrary and its value is only fixed after summing the expansion. The RHS in Eq.(2.23) defines an off-shell regularized source term, which takes the form and replaces the non-perturbative source in (2.17) through We stress that on-shell (with the limit s → 1/2) so that Ψ 1/2 (τ 1 , τ 2 ) this vanishes. It is a highly nontrivial feature (of this breaking procedure) that it leads to systematic nonzero effects. In particular, separating the critical classical solution from the bi-local field: Ψ = Ψ (0) + Ψ we get where S[f ] is the action of the collective coordinate This can be evaluated by using the Q s given above. Taking s → 1/2 at the end of the calculation one gets the Schwarzian action [6,7].
where α is a dimensionless coefficient [7]. The limit s → 1/2 is non-trivial because the evaluation of the integral (2.27) leads to a simple pole at s = 1/2. This cancels the overall s − 1/2 in Q s to yield a finite result. In this sense we can view the source Q s as an off-shell extension. The above strategy can alternatively be described as follows. Suppose we knew the correction to the saddle point solution by e.g. solving the equation of motion following from the SYK action exactly and then performing a long distance expansion. Then we can define a source Q s using (2.24), and replace the symmetry breaking term in the SYK action by the source term dτ 1 dτ 2 Q s (τ 1 , τ 2 )Ψ(τ 1 , τ 2 ). In the rest of this section we will show that once this is done, a well defined perturbation expansion around the critical solution can be developed. The non-trivial aspect of this scheme is that a specification of this off-shell source is all that is needed to obtain a full near-conformal perturbation expansion. We will show for the example of sub-leading corrections for the two point correlation function that the method produces results which agree with a 1/J expansion of the exact answer.

Vertices
From (2.16) and (2.26), near the critical theory, we have the partition function where S[f ] is given by the Schwarzian action (2.28). This leads to the interaction between the Schwarzian mode and the bi-local matter Ψ. First taking an infinitesimal transformation f (τ ) = τ + ε(τ ), we expand the Schwarzian action (2.28) to obtain which gives the Schwarzian propagator and the cubic vertex Because of the projector δ ∫ u · Ψ f which projects out the zero mode, the matter bi-local propagator is given by the non-zero mode propagator D c (2.51). Finally the interaction between the Schwarzian mode and the bi-local matter is given by the interaction term Ψ f · Q s . Taking an infinitesimal transformation τ + ε(τ ) with (2.12), we have the interaction term where we have used a shorthand notation d τ ≡ d1 q ,τ . This leads to the (τ )Ψ(τ 1 , τ 2 ) vertex: the (τ ) (τ )Ψ(τ 1 , τ 2 ) vertex: and so on. The diagrammatic expressions of these Feynman rules are shown in Figure  1.

Critical SYK eigenvalue problem
Now we focus on the eigenvalue problem of the critical theory. We have the conformal symmetry: τ → f (τ ) in the critical theory (i.e. strict large J): where S[Ψ (0) + J −1 Ψ (1) + · · · ] = S c [Ψ (0) ] + · · · . This leads to a zero mode in the strict large J limit: for the second equation is fixed by the orthonormality condition of the eigenfunction χ (0) (2.41). This zero mode leads to the enhanced contribution of the Green's function G (−1) , which will be discussed in Section 2.4 as well as in Appendix A. Also for the notation we will use below for the perturbation theory, readers are referred to Appendix A.
For general modes, the second term in (2.20) is already diagonal, but contains an extra factor Ψ q−2 (0) . Because of this extra factor, the eigenvalue problem of this kernel must be like The extra factor appearing in the RHS can be compensated by defining new eigenfunctions by so that the new kernel and eigenfunctions obey the standard eigenvalue problem We note that the eigenvalue does not change under this new definition. Since the new eigenfunctionsχ (0) obey the standard eigenvalue problem, we impose the orthonormality and the completeness n,ωχ For the eigenvalue problem (2.38), we denote that χ q (h) by using the fact that the complete eigenfunctions are spanned by the conformal dimension h containing the continuous modes (h = 1/2 + ir, 0 < r < ∞) and the discrete modes (h = 2n, n = 0, 1, 2, · · · ) with ω (−∞ < ω < ∞) [4,5]. The eigenvalue is known to be independent of ω and we can parametrize it as which excludes the trivially diagonalized second term in (2.20). Therefore, the nontrivial eigenvalue problem becomes where the explicit form of the inverse is given in (2.21). Next we note that the eigenvalue problem of the following form was solved in [5]: where the eigenfunctions are independent of q with 46) and the eigenvalue .
The h = 2 contributions has a single-pole D and a double-pole D contributions. Detailed evaluation of these are given in Section 3.4 and Appendix D. The former is given by while the latter contribution is given by where we used the change of variables in (3.14). The other contributions are given by single-poles located at h

Enhanced contribution: G (−1)
In this and next subsections, we consider the bi-local two-point function: where the expectation value is evaluated by the path integral (2.5). After the Faddeev-Popov procedure and changing the integration variable as we discussed in Section 1, this two-point function becomes where now the expectation value is evaluated by the gauged path integral (2.16). We expand the bi-local field around the critical saddle-point solution Ψ = Ψ (0) + Ψ, so that the two-point function is now Taking an infinitesimal transformation f (τ ) = τ + ε(τ ) with (2.12), we find the order O(J) contribution Following the standard perturbation theory, (which is described in detail in Appendix A), the exact Green's function can be written as (2.58) In the following, we will be interested in the contribution coming from the zero mode of the lowest order kernel, n = 0. Since λ Here we suppressed all τ (and ω) dependence since they don't play any crucial role here. The expression of the perturbative kernels are obtained as (see Appendix A) where we used the short-hand notation for S (n) c defined in (A.17). The first and second order eigenvalue and eigenfunction corrections are given by where for the first order shift of the zero-mode eigenfunction χ (1) 0 we used (A.22). In the rest of this subsection, we relate the diagrammatic expression of G (−1) in (2.57) to the eigenvalue perturbation expression (2.59). Now we note that the Schwarzian action is given by Expanding f (τ ) = τ + (τ ) we know that in the s → 1/2 limit, the first nonzero term in an expansion in (τ ) is the term which is quadratic in (τ ) and given by For the zero mode (2.37) with λ (0) 0,ω = 0, the eigenvalue problem is written as . (2.69) Taking one more variation respect to f (τ ), we obtain where after the variation we set f (•) = • and then Fourier transformed τ to ω (and τ to ω ). Multiplying Ψ (1) (τ 1 , τ 2 ) and integrating, the first term becomes K (1) (2.62). Therefore, now we find Then, the first order zero-mode eigenvalue correction λ Now this equation immediately shows that 0 is the quadratic kernel of the Schwarzian action. Hence, using this relationship, we relate the diagrammatic expression (2.57) to the perturbation expression (2.59) as (2.74)

Diagrams for G (0)
Next we consider the subleading O(J 0 ) contribution G (0) . Besides the non-zero mode contribution D c , the contributions to G (0) given by four diagrams (in Figure 2) that arise from interactions between the Schwarzian mode and the bilocal field, and also the Schwarzian redefinition of the field itself: The corresponding Feynman diagrams are shown in Figure 2 and they are given by 5 where we defined d h,ω as the Fourier transform of d h,τ defined in Eq.(2.13). For b 1 and b 2 , there are also (τ 1,2 ↔ τ 3,4 ) contributions we did not write explicitly above. We note that the integration by parts of this operator is given by a shadow transform: From the definition of the operator (2.13), the action of this operator onto a product can also be written as Now we evaluate the diagrams. For b 1 , using (2.24) and (2.73), we find In this subsection, we always omit the normalization factor N (0) 0,ω introduced through (2.37) in order to simplify the notation. This factor does not play any crucial role here and the final expressions are independent of this factor.
For b 2 , first using (2.60) and (2.73) we have (2.80) Now following the same manipulation as in (2.69) -(2.71), but now for non-zero mode, we find For the RHS, we can move d onto Ψ (1) by the shadow transform (2.77). Using this relation, b 2 is now written as where for the first term, we used the completeness ofχ (0) (2.42). The first term cancels with b 1 , while the second term combines into the first order shift of the eigenfunctions (2.65). Therefore, we find where we implemented the explicit symmetrization of the external legs. For b 3 , using (2.24) and (2.73), we find Now we consider the equation of motion of Ψ (1) (2.24) with the on-shell limit (s → 1/2): Taking variations respect to f (τ ) and f (τ ), then multiplying another Ψ (1) , we find Therefore, we can write b 3 as (2.86) For b 4 , first using (2.59) and (2.73) we have Again using the relation (2.81), this is written as For the first term, we use the shadow transform (2.77) to move d onto Ψ (1) . After using the completeness ofχ (0) (2.42), we obtain dω 2π The first term precisely cancels with b 3 , while the second term combines with the other term in b 3 to give the second order eigenvalue shift. Therefore, we finally obtain (2.90) The first and second order eigenvalue and eigenfunction corrections which will be needed below are given by (2.64) -(2.66). We now found that the diagrammatic expressions denoted by b 1 -b 4 shown in Figure  2 are given by the matrix elements (2.83) and (2.90). These are in agreement with the standard perturbative evaluations summarized in Appendix A. In the next section, we will explicitly evaluate these matrix elements in the large q SYK model as an example. The total O(J 0 ) contribution to the correlation function is given by the following It is seen that the contribution of diagrams b 3 + b 4 is precisely of the same form, and cancels the double pole term in the propagator D . The result is therefore given by the single pole propagator D and the result from diagrams b 1 + b 2 which as we have shown above involves the first order correction to the zero mode eigenfunction. We evaluate this (and the other matrix elements) in the large q case in the following section. In this limit (with the higher massive modes decoupling) the result reads: This result consists of two terms: the first one due to Schwarzian interaction with the bi-local and the second representing the contribution of the h = 2 matter. For general q one would have the further contribution of massive mode propagators. We also mention that a similar evaluation in [11] gave a different result. We are not sure why the method of [11] disagrees at O(J 0 ), however we have compared our result in the large q limit, where a finite J evaluation is possible, and observed disagreement (this will be seen in the next Section). We also note that the contributions of diagrams b 3 + b 4 represents the O(J 0 ) correction to the O(J) Schwarzian action (at the quadratic level). This again differes from the correction found in [11] .

Large q: Liouville Theory
The scheme described in the previous section ensures that once we have an expression for the first correction to the critical saddle point, Ψ (1) , an unambiguous perturbation theory (which is really a derivative expansion) can be developed by replacing the actual symmetry breaking source term by a new source term which would give rise to this Ψ (1) .
This procedure is necessary since the symmetry breaking term is singular and nonzero only at short distances, which implies that this cannot be used as a perturbation in a long distance expansion. This means one has to make an ansatz for Ψ (1) given in (2.22).
In the large q limit, the SYK model simplifies considerably and at the leading order the action is that of Liouville theory on the bilocal space. At large N this model can be now solved exactly at all scales, i.e. for any finite J. Even though the symmetry breaking term of the SYK model needs to be included to obtain this large q limit, the resulting Liouville action acquires an emergent reparametrization symmetry. The large N saddle point breaks this symmetry. However, this does not lead to a zero mode, and a calculation of the exact bilocal function proceeds without any obstruction. In particular, in this exact calculation there is no emergent Schwarzian dynamics.
Nevertheless, we would like to understand the long distance derivative expansion in this limit, and extract the dynamics of the soft mode, since the Schwarzian action is the most direct link to a dual description in terms of JT gravity. In this section we develop this expansion using the scheme of the previous section. However, now we can simply use the leading correction to the critical saddle point to determine the regularized source Q s . The result of the previous section then ensures that the scheme using this source reproduces the large J expansion of the exact answer for correlation functions.

Bilocal theory at large q
Let us begin with the collective action for SYK model (2.6) This J is kept fixed as q → ∞. At large q, we can do the following field redefinition Using (3.3) in (3.1), and performing a 1/q expansion we get a field independent O(1) term and the next contribution is O(1/q 2 ), given by the Liouville action The details of the derivation of (3.4) is given in Appendix B. Note that there is no O(1/q) term in this expansion. The kinetic term of SYK -the first term of (3.1)provides a 1/q piece which cancels with a 1/q piece coming from the second term. The inclusion of the symmetry breaking term is crucial. Nevertheless the action S L [Φ] has an emergent reparametrization symmetry for (3.5) At finite temperature and finite J , we need to impose the physical requirement that the expectation value of the bilocal field Ψ(τ 1 , τ 2 ) should be equal to the free fermion two point function 1 2 sgn(τ 12 ) in the short distance limit. This means we need to impose a boundary condition Φ(τ, τ ) = 0 . At zero temperature the expansion is really in J |τ 12 |: this means that we cannot really access the point τ 1 = τ 2 . However the zero temperature theory should be really thought of as a limit of the finite temperature theory. Accordingly we should impose the condition (3.6) even at zero temperature. In fact, as is well known, the Liouville action on an infinite plane has a symmetry which has two copies of Virasoro, i.e. τ 1 and τ 2 can be reparametrized by different functions. The restriction to the same function as in (3.5) comes from this boundary condition (3.6). The above derivation is in a 1/q expansion. The fact that the resulting action is a standard two derivative action signifies that to leading order of this expansion there is a single pole of the two point correlation function of the bi-local field. This appears to be in conflict with the well known fact that even for q = ∞ there are an infinite number of poles in the conformal limit, i.e. an infinite number of solutions ofg(q; h) = 1 wherẽ g(q; h) is defined in (2.50). In fact at large q the solutions to this equation are given by h = 2 and the tower [9] h n = 2n + 1 + 2 q However the residues of these poles all vanish as q → ∞ except for h = 2 [79]. The 1/q expansion of the functiong(q; h) is given bỹ where ψ(x) denotes the digamma function and γ is the Euler-Mascheroni constant.
Since ψ(−n) for integer n has a pole, the coefficient of the 1/q term is singular for h = 2n + 1. This is the signature of the infinite number of solutions of the spectral equation in a 1/q expansion. In the following, however, we will restrict our attention to values of h close to 2. Therefore these other solutions will not be relevant for us. Note, however, on-shell modes corresponding to these infinite tower of solutions do have non-trivial higher point correlation functions [9].

Quantum fluctuations at leading order
The equation of motion which follows from the action (3.4) is given by We should really view the zero temperature theory as a limit of the finite temperature theory. At finite temperature, periodicity in both τ 1 and τ 2 , together with the boundary condition (3.6) determines the solution uniquely [5] and its zero temperature limit is given by Φ cl (τ 12 ) = − 2 log J |τ 12 | + 1 , Using the bilocal map we can write, using (3.10) The operator which appears in the square bracket is the expression for K ex which is defined in (A.2). In the following it will be convenient to definẽ It is convenient to define a conformally covariant operator K L (similar to (2.39)) and redefine the fieldη The eigenvalue problem we need to solve is These eigenfunctions are given bỹ where 22) and the eigenvalue λ (ex) ν is given by The eigenfunctions satisfy the following orthonormality condition which follows from self-adjointness of the operator K L in the interval [0, ∞) with the boundary conditioñ Here N ν,ω appears in the orthonormality relation We do not have an analytic expression for N ν,ω , but we can determine it perturbatively in ω/J to get for the discrete modes For the continuous modes, we will not need to know the ω/J corrections since there is no enhancement. So, we use We now perform a mode expansion of the fluctuation field in terms of these exact eigenfunctions as follows 6  In the following we will not need the correction to the continuous series. The coefficients a 1 and a 2 for the n = 0 case are given by 1/π and zero respectively (see Appendix C for details). Using (3.25) − (3.28) in (3.15), we get Note that the eigenvalue λ (ex) ν arises from the large q limit of the eigenvalue of the finite q bilocal kernel (see (2.47), (2.50)), with h = ν + 1/2 From (3.30), we can read off the momentum space bilocal correlator Since the solution (3.10) breaks the symmetry (3.5), one might expect that there is a zero mode given by its variation δΦ cl . This would be of the form As expected this solves the equation of motion which follows from (3.15), but does not satisfy the boundary condition (3.12) at any finite J. Indeed, (3.29) shows that there is no eigenfunction with ν = 3/2. Since there is no zero mode, a calculation of the bilocal two point function proceeds in a straightforward fashion. Using (3.28) and (3.32), we can write down the exact bilocal two point correlator in position space as (3.34)

Perturbative expansion
In this subsection, we calculate the various ingredients needed to obtain a perturbative expansion of the bi-local propagator, which are the matrix elements given by (2.64) − (2.66) for Liouville theory. The full (or "exact") kernel K L can be expanded in powers of (|ω|J −1 ) as follows. This is the analog of the expansion (A.7).
The kernel K (0) now has a zero mode, The eigenvalue and eigenfunction for this mode will be corrected by the perturbations. The corresponding exact eigenvalue is λ 0,ω which has the expansion Similarly, for the first order correction to the eigenfunction, we get While we have written these down using the expansion of the known exact eigenvalue and eigenfunction, one can of course calculate these directly in perturbation theory. The first order correction follows easily from (3.37) and (3.40) in (2.64). For the first order correction to the eigenfunction, we were unable to perform the sum analytically in (2.66). The expression (3.43) can be used to verify that standard perturbation theory indeed leads to the correct second order correction to the eigenvalue. The details of this calculation have been included in Appendix F. These calculations are in agreement with the results of [5].

Evaluation of bilocal two point function
We now have all the necessary ingredients to evaluate the bilocal two point function (3.34) perturbatively in (|ω|J −1 ) using the formalism of Section 2 and Appendix A. First, we note that the values of ν to be integrated over are given by (3.29). The imaginary and discrete values give rise to a continuous and discrete contribution respectively.
The discrete sum receives an enhancement from the zero mode so it needs to be treated separately. We separate it from the non zero modes in order to write η(t, z)η(t , z ) = G (−1) (t, z; t , z ) + G (0) (t, z; t , z ) , (3.45) where G (−1) and G (0) are given by equations (2.59) and (2.60) respectively. The perturbative corrections to the zero mode eigenvalue and eigenfunction can be substituted in these expressions to write down the bilocal propagator explicitly. Using (3.42) and (3.40) in (2.59), we get the enhanced propagator (3.46) Next, let us consider the O(1) contribution G (0) . It receives a contribution from D c , which for the Liouville case is given by The details of evaluation of D c is given in Appendix D. The result is D c (t, z; t , z ) = 4q 2 (zz ) (3.48) We identify the first term in (3.48) with the single pole term D and the second term with the double pole term D in (2.51) respectively. The remaining terms in (2.60) can also be evaluated using (3.39) − (3.43) to get (3.49) From (3.48) and (3.49), we see that the double pole precisely cancels. This is an explicit illustration of the claim made in Section 2 that the final answer for the bilocal propagator receives contribution only from the simple pole in D c (which corresponds to h = 2 matter) and from the b 1 + b 2 diagrams. Collecting all the terms, we get the bilocal propagator for Liouville theory up to O(1) to be (3.50)

Comparison with zero temperature limit of four-point function
We now show that the zero temperature limit of the finite temperature four point function calculated in [5] agrees with (3.50). The SYK four point function at finite temperature up to O(1) is given by 7  2 v 2 ∂ 2 y + 4 sin 2x 2 ∂ 2 x + 1 4 which is the finite temperature version of the Bessel operator we consider. They, of course, lead to the same result since at finite temperature the periodicity conditions ensure a unique Green's function.
Using (3.53) in (3.51) and replacing the sum over the discrete Fourier index n by a continuous integral over ω, we get Now, we use the following integrals expressed in terms of the quantity ξ ≡ −(t−t ) 2 +z 2 +z 2 2zz (zz ) The details of the derivation of these integrals are given in Appendix E. Using these integrals, we see from (3.50) and (3.54), η(t, z)η(t , z ) = 4q 2 F(t, z; t , z ) . (3.55) The proportionality factor of 4q 2 arises as a consequence of the field redefinition in (3.3).

Expansion around the critical saddle point
In the J |τ 12 | 1 limit (3.10) becomes We will call this the "critical saddle point". This is in fact what follows from the large q limit of Ψ (0) in (2.9) and plays its role. There is, however, an important difference. For finite q the critical solution J |τ 12 | is not a solution of the full theory because of the term which breaks the reparametrization symmetry. In this case (3.56) is also a solution of the classical equations of motion. Φ (0) , however, does not satisfy the boundary condition (3.6). In fact, in an expansion around the critical limit, there is no reason to impose this boundary condition. Keeping this point in mind we will call Φ cl the "exact" solution and develop a perturbative expansion around the critical solution following the same steps as in the finite q SYK model. An expansion around this solution would in fact lead to a normalizable zero mode given by (3.40) which needs to be treated properly in precisely the same way as the finite q SYK model. In order to deal with this, we introduce a source term given by 8 where the source is again a regularized version of (2.24). In this case, the kernel K (0) L can be read off from the first line of (3.15) as The source is then related to the O(1/J ) correction to the classical solution by The O(1/J ) correction to the classical solution is given by 9 As expected, the source in (2.24) with this non-regularized Φ (1) (z) vanishes. Following the treatment in finite q SYK, we therefore introduce a regulator s to define This results in a regularized version of the source Q L s given by (3.62) Since the exact classical solution is now known, the coefficient of the resulting Schwarzian action for the soft mode can be now determined unambiguously. The Schwarzian action is given by The fact that this source term goes as J −1 is reminiscent of the conformal breaking term in finite q SYK. It should be noted that if we add other source terms which are suppressed by higher powers of J , that would be inconsistent with the finite q picture. 9 We have stripped off the power of J in the expression for Ψ (1) in order to maintain consistency in notation with Section 2. Also, since the domain of integration of the z coordinate is always positive, we drop the absolute sign.
where Φ f 0 corresponds to the transformed classical solution (3.64) To determine the coefficient one could proceed by writing f (τ ) = τ + ε(τ ), evaluate (3.63) in an expansion in ε(t), perform the limit s → 1/2 and resum to obtain the full expression. Since we know that the answer should be proportional to the Schwarzian it is adequate to expand to second order in ε(t). We have performed this calculation, but will omit the details here. Alternatively we can calculate the propagator of the Fourier transform of ε(τ ), which we call ε(ω) given in (2.73). For the Liouville theory this is given by The main ingredient is contained in the set of conformal Ward identities which are described for the arbitrary q model in equations (2.69)-(2.71). These equations can be explicitly verified in our case with the expressions for K (3.67) It is straightforward to see that Comparing with (3.40), we get (from the definition) To apply (2.73) we need to note that while the source term for finite q was defined by (2.24) while in the Liouville theory we defined the source term by (3.57). Thus in Liouville theory we have where we have used (3.69) and (3.42). This propagator can be easily seen to follow from the Schwarzian action The coefficient is in precise agreement with the large q limit of the action given in [5]. The calculation of the bilocal two point function now follows the diagrammatic technique of Section 2.

Conclusion
In this work, we concentrated on the development of a complete understanding of systematic near conformal perturbation expansion in SYK type models. It develops further the initial work of [6,7] where the soft mode with Schwarzian dynamics is extracted from the bi-local field in a systematic fashion and arises as an emergent degree of freedom. This mode interacts with the remaining "matter" degrees of freedom which include a component dual to an operator with conformal dimension h ∼ 2, in a manner which is completely determined. The non-triviality of exhibiting this interacting representation lies in the fact that, as we explain, the symmetry breaking effects responsible both for the Schwarzian and Schwarzian-matter interactions comes from a very subtle off-shell regularization which produces non-zero effects when removed in the limiting procedure. This leaves a series of interacting vertices that are determined explicitly. Representing these corrections in a diagrammatic picture provides a complete and transparent scheme. This allows for a concrete perturbation calculation of corrections to leading (conformal) correlation functions and other physical quantities. We do this for the bi-local two point functions by considering the evaluation of the first correction (in a low energy expansion) to the leading (enhanced) answer. The evaluation of these is facilitated by a series of conformal identities. We applied this formalism to the large q limit, where the model is reduced to the Liouville theory which is exactly solvable. In this case, expansion around the correct saddle-point does not result in a zero mode and the correlators can be calculated exactly. We can nevertheless expand around a saddle point appropriate for a long distance expansion and apply the perturbation scheme described above. In addition, it is instructive to see the workings of the method in this example of a prototype conformal field theory. We note that it is of interest to consider the complex SYK model already, since issues related to applicability of perturbation theory have been observed recently [93]. Indeed we believe that the applicability of the scheme goes beyond the SYK model, applying generally to perturbations in conformal field theory and more generally quantum field theory.
Returning to the case of SYK theories, the present systematic reformulation of the model as a bi-local matter coupled Schwarzian theory might offer the needed insight into the outstanding question of its gravity dual. The Liouville theory in particular provides the simplest limiting case. In this limit the higher h modes decouple and we find an interacting picture of the soft mode with bi-local h = 2 matter. We emphasize this fact since most studies of the gravity dual focus on the dilaton gravity sector of the dual theory. In [71][72][73][74], the enhanced part of the bi-local two point function has been reproduced in a dual theory which contains bulk fields which can be thought to be dual to the SYK fermions. Our preliminary results indicate that some parts of the subleading correction may be obtainable by considering the effect of a coupling to the non-trivial dilaton background [94].
However, additional ingredients, related to the contribution of h = 2 bi-local matter, are probably necessary to understand the low energy sector completely. We hope to return to this problem in future studies. various stages of this work. AG acknowledges support from the KITP, University of California, Santa Barbara and the Dean's Arts and Sciences fellowship at the University of Kentucky while part of this work was in progress. The work of S.R.D

A Perturbative Expansions
In this appendix, we summarize several relations for the expansions (A.6) -(A.10) and introduce some short-hand notations. Formally we suppose to have the exact action S and the exact classical solution Ψ cl . Expanding the action around the exact solution, we have .
Then, introducing K (ex) from K (ex) using the analogous definition as in (2.39) where we replace all Ψ (0) by Ψ cl , formally we can consider the exact Green's function G (ex) which is determined by the Green's equation In order to invert the kernel K (ex) in this Green's equation, we consider the eigenvalue problem of the kernel K (ex) : n,ω and the eigenvalues λ n,ω , where n, ω are some quantum numbers. We normalize the eigenfunctions by requiring Now we consider a perturbative expansion in 1/J of the above eigenvalue problem. We expand each of the quantities of interest as and similarly for the redefined kernel (2.39), and the bi-local propagator and eigenfunctions corresponding to it. In this paper, the superscript in a round bracket denotes the order of 1/J expansion, while the subscript denotes quantum numbers.
The exact Green's function can be then written as In the following, we will be interested in the contribution coming from the zero mode of the lowest order kernel, n = 0. Since λ (0) 0 = 0, we have Here we suppressed all τ (and ω) dependence since they don't play any crucial role here.

B Details of Derivation of Liouville from Bilocal
In this Appendix, we give a detail derivation of the action (3.4) from (3.1). For the logarithm term, we use the following fact that sgn(τ 12 ) 2 = θ(τ 12 ) − where the last term is defined for example for n = 2 as For the interaction term we have We consider only q = even integer case and the sign function disappeared here. We want to rewrite the RHS as the following form where function c 1 and c 2 are to be determined. Taking logarithm of both-hand sides and expanding the logarithm, one finds Comparing the both sides, the unfixed functions are determined as Combining everything, the collective action is now written as The O(q 0 ) and O(q −1 ) order terms are identically vanish due to the free field equation of motion. Therefore, the first non-trivial order is O(q −2 ). Lower order terms can be explicitly written down as

C Details of Exact Eigenfunctions
In this Appendix, we elaborate on various properties of the exact eigenfunctions (3.21).

C.1 Zero temperature limit
Consider the exact eigenfunctions at finite temperature at large q [5] which can be written as This allows us to writex = ω nJ (2J z + 1) + O 1 (nJ ) 2 (C.5) Taking the zero temperature limit is equivalent to taking n → ∞. In this limit, sinx 2 x 2 (C.6) and lim n→∞ P ±ν On doing so, we can write the exact eigenfunction in the zero temperature but finite coupling limit as  Figure 3: Plot of the absolute difference between the allowed values of ν, where we have defined δν = ν MS −ν exact . We see that the difference is almost zero for small values of ω 2J , which corresponds to the conformal limit.

C.2 Quantization Condition
Let us impose the following boundary condition on the exact eigenfunctions Using equation (C.8), imposing the above boundary condition requires us to solve the following equation in order to get the quantized values of ν We can solve this equation numerically and do a polynomial fit of order two for the curve given by Figure 3 to get where a 1 = 0.318048 and a 2 = 0.0046914. This is in good agreement with the results obtained in [5]. This can be seen as follows We have used equation (2.31) of [5], to get from the first to second line of (C. 19). The following Table 1 gives the difference in quantized values for ν from the exact and approximate methods as a function of ω 2J . It provides numerical data in support of the above claim (see also Figure 3)

C.3 Perturbative determination of exact normalization
We derive (3.26) here. Let us consider ν = 2n + 3 2 + |ω| πJ where ξ ω (ν) vanishes. So, the integral we need to perform is  with ν = ir. The ν integral can be written as We have extended the limits over the entire imaginary axis and defined ν = ir in going from the first to second line of (D.2) using (C.14). For z > z we close the contour clockwise in the right half plane, which gives T 1 only has a simple pole at ν = 3 2 ,whereg(3/2) = 1. 10 We can evaluate the residue to get (D.5) The T 2 term is a little more complicated. It has simple poles at ν = 2n + 3 2 where ξ ν = 0. We can evaluate the residue there to get Let us consider the integrand in the second term of (D.6). It has singularities coming from ξ(−ν) as well asg(ν) − 1. Let us expand the following quantity around ν = 3 2 ξ(−ν) g(ν) − 1 = 1 (D.8) We have a double pole coming from the first term of (D.8) and a simple pole from the second term. Using (D.8) in (D.6), and evaluating the residues, we get The first term in (3.47) exactly cancels with I d in (D.9). Usingg ( 3 2 ) = 3 2 ,g ( 3 2 ) = 1, ξ 3 2 = − π 2 and ξ 3 2 = 0 in (D.9), we get (3.48) We have dropped the ω subscript while writing ξ here because we are working in the conformal limit.

E Integrals of Products of Bessel Functions
In this appendix, we consider the following integrals involving a product of two Bessel functions: ∞ −∞ dω e −iω(t−t ) J ±ν (|ω|z > ) J ν (|ω|z < ) = 2 ∞ 0 dω cos(ω(t − t )) J ±ν (ωz > ) J ν (ωz < ) . (E.1) The plus sign case was already evaluated in Appendix D of [4]. The result is integral can be performed as π 0 dφ (sin φ) 2ν (ξ − cos φ) s (E.8) where s is a general real number. One can see that for any value of ξ (and for any value of s), the analytical continuation α = ae iθ with θ = 0 → ±π/2 does not hit the branch cut of the hypergeometric function. Therefore, we can perform a naive analytical continuation for this φ-integral part. We note that for the cosine integral (E.1), the contribution from the first term in Eq.(E.6) vanishes due to the α −1 factor in Eq.(E.5) after the analytical continuation. Therefore, for the cosine integral (E.1), the contribution solely comes from the second term in Eq.(E.6), which is written as where