Bound on asymptotics of magnitude of three point coefficients in 2D CFT

We use methods inspired from complex Tauberian theorems to make progress in understanding the asymptotic behavior of the magnitude of heavy-lightheavy three point coefficients rigorously. The conditions and the precise sense of averaging, which can lead to exponential suppression of such coefficients are investigated. We derive various bounds for the typical average value of the magnitude of heavy-light-heavy three point coefficients and verify them numerically. ar X iv :1 90 6. 11 22 3v 1 [ he pth ] 2 6 Ju n 20 19


Contents
The modular invariance plays a pivotal role in constraining the data of two dimensional conformal field theory. In two dimensions, a conformal field theory can be consistently defined on any Riemann surface, in particular, on a torus. The shape of the torus is determined by modular parameter τ ∈ H, where H is the upper half plane. The modular transformation acts on τ and maps it to another point in H, nonetheless the partition function of the conformal field theory defined on the torus remains invariant under such transformation. Physically, one cycle of the torus is interpreted as the spatial cycle while the other one is the thermal cycle. Modular transformation, for example, exchanges these cycles and thus can relate the low temperature behavior of a CFT with its high temperature behavior. This is how the universality in the low temperature behavior translates into a universal high temperature behavior, which is controlled by the asymptotic data of the CFT. Hence, one can get insight to the asymptotic behavior of the CFT data utilizing the modular invariance [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Recently, the implication of modular invariance in the density of high energy states has been analyzed rigorously in [15] leveraging the powerful machinery of complex Tauberian theorems. The fact that the Tauberian theorems can be extremely useful in context of CFT is pointed out in [16], subsequently, the same is emphasized in Appendix C of [4], where the authors have used Ingham's theorem [17]. The usefulness of making β complex while using Tauberian theorems and thereby gaining extra mileage in controlling the correction terms in various asymptotic quantities of CFT has been pointed out in [18] using the machinery developed in [19].
One of the pivotal ingredient in the proof and the error estimation in Cardy formula as done in [15] is the positive definiteness of the spectral density. Thus the method is not directly adaptable when estimating the asymptotic behavior of three point coefficients [3], since three point coefficients can be negative as well (see fig. 1 and 2). This has been emphasized repeatedly in [4,15] and in section 6.3 of [20], where they provided an explicit example to show why the positivity of the spectral density is really important in the context of the Tauberian theorems. While there is an obstruction in directly adapting the method, from a physical ground, we do expect the exponential suppression of average three point coefficients if the three point coefficients are not wildly fluctuating. This asymptotic behavior has implications in context of Eigenstate Thermalization Hypothesis [21][22][23][24][25][26][27][28][29][30][31][32][33][34] since the three point coefficients, probed in [3] are related to expectation value of some operator in a high energy eigenstate of a CFT on a cylinder. Recently, the generalization of KM result as done in [4] is utilized in [35]. To remind the readers, in [3], the modular covariance of torus one point function is used to estimate the asymptotic behavior of heavy-light-heavy three point coefficients, they found exponential suppression, which depends on dimension of heavy operator ∆, central charge of the CFT (c) and the dimension (∆ χ ) of the operator, χ such that it produces the light operator (with dimension ∆ O ) upon doing operator product expansion with itself and it has the least dimension among all such operators producing the light operator upon doing operator product expansion with itself. The result requires ∆ χ < c 12 . For completeness, here is the result (henceforth we will refer to it as "KM" result and the overline denotes some sense of averaging) from [3]: The aim of this paper is to make progress in understanding the asymptotic behavior of the magnitude of three point coefficients and investigating under which conditions one should expect such exponential suppression. In short, our objective is to provide a rigorous underpinning for the behavior of three point coefficients.
To motivate further why such characterization is indeed needed from a theoretical standpoint, let us consider two CFTs C 1 and C 2 with central charge c 1 and c 2 respectively, we tensor them together to construct a CFT C 1 ⊗ C 2 with 1 central charge c 1 + c 2 . Now consider an operator O ≡ O 1 ⊗ I, naively the average value of the three point coefficient f C 1 ⊗C 2 ∆O∆ as predicted by [3] would be controlled by the central charge c 1 + c 2 . On the other hand, using f C 1 ⊗C 2 ∆O∆ = f C 1 ∆O 1 ∆ , we see that f C 1 ∆O 1 ∆ can only possibly depend on central charge c 1 , not on c 2 . This example makes it very clear that one needs to be very precise about what is meant by the average value of three point coefficients. We will see that our result actually resolves this paradox. More examples follow to motivate why we might want to take up a rigorous approach in this direction.
An example involving tensored copies of CFT can be constructed where the applicability of Kraus-Maloney result [3] is very subtle. We consider 2 copies of 2D Ising model and one copy of a CFT with central charge 7 10 . The tensored CFT, C 1 , has total central charge c ef f = 17 10 . Focussing back to the 2D Ising model, we note that it has following primary operators: I with dimension 0, σ with dimension 1 8 and with dimension 1. The unnormalized torus expectation value (henceforth by torus one point function, we will mean unnormalized torus one point function unless otherwise mentioned) of is proportional to square of Dedekind eta (η 2 ) function [36]. We know that the three point coefficients that contribute to this one point function involves σ and its descendants. Upon doing a q expansion of η 2 : β ∝ η 2 (β) = n=∆− 1 8 a n e −β(n+ 1 8 − 1 24 ) , a n = Descendants at n th level one can deduce that the non-zero three point coefficients f ∆ ∆ are typically suppressed by P (n), the partition 2 of integer n, where n = ∆ − 1 8 . The suppression factor is basically the density of descendants of σ with dimension ∆. A list plot of a n looks as in fig. 1. At this point, one might argue that the lowest operator that contributes to the one point function of is σ, having dimension 1 8 , which is greater than cure this, we consider instead the aforementioned tensored CFT C and the following operator ⊗ I ⊗ I. The lowest operator that contributes to the torus one point function of ⊗ I ⊗ I is σ ⊗ I ⊗ I, having the dimension of 1 8 and this is clearly less than On the other hand, the three point coefficients of the tensored CFT still boils down to the three point coefficient of Ising Model and thereby suppressed by P (n). We remark that this exponential suppression coming from the factor of P (n) does not involve the effective central charge. Intuitively, the reason behind the departure from KM result is when we are looking at f C ∆O∆ of the tensored CFT, we are scanning over all possible operators with dimension ∆ 1 ≤ ∆ such that f Ising ∆ 1 ∆ 1 = 0. Thus the window over which "averaging" has actually been done to obtain what KM predicts contains widely fluctuating numbers, thus the applicability of KM is really very subtle. As mentioned, this indeed provides us with motivation of precisely defining what we are estimating. We will come back to this with a neat resolution at the end of the penultimate section §7 with a simplified example of 4 copies of 2D Ising model. Another example which is morally similar to the above would be to consider the 40 copies of 2D Ising model (call it CFT C 2 ) so that c ef f = 20. Now consider the operator We remark that even though the discussion that follows are morally same, the reason we include this is the qualitative difference of q expansion coefficients of η 2 and η 24 (see 1 and 2). While η 2 is a lacunary function, η 24 is a holomorphic function. Again, the three point coefficient of the tensored CFT depends only on the first 12 copies of the Ising CFT, as a result we have Thus the three point coefficients in this case is typically suppressed by b n / j ( i P (n i )) such that i n i = N . On the other hand, we note that is a modular cusp form of weight 12 and it can be shown using the properties of holomorphic modular form that |b n | = O(n 6 ) as seen in fig. 2. Thus we have suppression by a factor of j ( i P (n i )).  In what follows, we will consider a CFT with central charge c.
To make a precise sense of averaging, we consider an energy window of [∆ − δ, ∆ + δ] and probe the following quantity: where δ is an O(1) number. We can eventually relax this condition to δ ∆ κ with κ < 1/2. We let ∆ → ∞ and we define the functions 4 G and F in the following way: (1.9) 4 We remark that our analysis is sensitive to q expansion coefficient only, thus to be very precise, we should write The A can be thought of as the average absolute value of three point coefficients, where averaging is done over the energy window of width 2δ, centered at ∆. If we do not wish to average out by the number of states lying in that window, we should instead be considering the following quantity: Furthermore, the expectation of not wildly fluctuating three point coefficients is encapsulated as the following assumption: where O is a primary with dimension ∆ O and even spin 5 s. This is fairly a mild condition, satisfied by the primaries in the aforementioned examples. The condition is also satisfied by the quasi-primaries in the Identity module. Without loss of generality, we further assume 6 that (−1) s/2 f χOχ > 0, where χ is an operator with the lowest dimension (say ∆ χ ) such that f χOχ = 0. Like Kraus-Maloney [3], we will also require ∆ χ < c 12 . We define a parameter γ for future reference: We relax this condition in the penultimate section §7 in a restrictive scenario and obtain an estimate for three point coefficients.
It so turns out that, when δ O(1), the eq. (1.11) is not enough to prove something useful 7 , since this allows for the three point coefficients becoming arbitrarily negative. This motivates us to further assume some bound on how negative it can become. In particular, we will assume that where a ∆ is the coefficient of q ∆− c 12 in the q expansion of the torus one point function of O. Roughly, for large ∆, this is like 8 5 One can relax this condition to O being SL(2, R) primary (also known as quasi-primary) without much modification. We comment about the odd spin case in the concluding section. 6 If this assumption is wrong, we consider the operator −O and proceed. 7 One might hope to improve this because it is not clear whether Eq. (1.13) is necessary. 8 The a ∆ captures the sum of all the f wOw such the operator w has dimension ∆. Strictly speaking, we made a slight abuse of notation while writing f ∆O∆ since there can be multiple operators with dimension ∆ such the three point coefficient is non-zero.
Here g(∆) is a positive polynomial, defined as where H and are O(1) positive numbers. If we set α = γ, then 2 has to be strictly less than ∆ O 2 − 3 4 . Again this is clearly satisfied by the quasi-primaries in the Identity module. The aforementioned examples involving the primary of 2D Ising model also conform to this condition. In the case of quasi-primaries in the Identity module, we have ∆ χ = 0 and ∆ O ≥ 2, then one can further relax and assume: When δ ∆ κ with κ > 0, we can relax the condition given in (1.13) by setting α = γ, without imposing any condition on . Below, we state our final result in terms of δ, where it is to be understood that δ ∆ κ with 0 ≤ κ < 1/2. The κ = 0 means that we are looking at an O(1) window. We remark at this point that having a condition on how negative a ∆ can get does also appear in the context of Tauberian theorems. We will expound on this in appendix §A with a simple toy example with a hope of possible extension of the result that follows.
Results at Finite Central charge-Average Three Point Coefficient: Under the assumption as stated in Eq. (1.11) and Eq. (1.13), we show that typical average value of magnitude of f ∆O∆ in ∆ → ∞ limit is lower bounded by a universal funtion in ∆, which depends only on the central charge, ∆ χ and ∆ O .
where a, b are O(1) numbers which depends on the dimension of the lowest excited state.
Here T (∆) is given by and C ∆O∆ is given by where ∆ χ is the dimension of the lowest lying nontrivial operator χ with non-zero f χOχ and N O , N O are given by This lower bound matches with the result obtained in [3] by using saddle point approximation and inverse Laplace transformation of the high temperature behavior of torus one point function of the light operator O. We reemphasize that we require ∆ χ ≤ c 12 for the proof to go through. This is same as the requirement in [3]. We have assumed finite c through out the first part of our calculation. The scenario ∆ χ > c/12 is dealt with in §7.
One can easily see that how the lower bound resolves the issue with tensored CFT. Considering the single CFT with central charge c 1 , the lower bound that we obtain (see the first of the eqs. (1.16)) is greater than the lower bound obtained by considering the tensored CFT with central charge c 1 + c 2 . Thus there is no contradiction/paradoxical situation when considering three point coefficients involving the operator O ≡ O 1 ⊗ I. In fact, the discussion at the end of the §7 gives more insight to this paradox. In particular we show that only in the integrated form, it makes sense to derive asymptotic behavior rather than making a statement about typical behavior of three point coefficients.
As mentioned, the above result can be generalized to an operator with even spin, in particular, one can obtain results for the operator O = T +T and thus verify against the known result for f ∆O∆ . We also verify it for the tensored CFTs, as elucidated in §6. The implications and further possible extensions of the above is discussed in concluding section §8. The appendix §A expounds on some idea regarding how to possibly prove an upper bound in a more generic set up, thus exactly deriving the asymptotic behavior.
Results under milder assumption: Technically we will see that the ineq. (1.13) is only needed for ∆ > ∆ * for some order one number ∆ * . Thus if the condition is violated, it needs to be violated for an infinite number of ∆. Thus one can always form an increasing subsequence {∆ k } such that we have for some order one number u. In that scenario, we have found an increasing subsequence ∆ k such that for δ < δ gap (assuming that there is a minimal gap in the operator spectra given by 2δ gap , recently it has been shown that 2δ gap ≤ 1 [15,37]), the asymptotic behavior (k → ∞) of B is bounded below by g(∆ k ) exp 2πγ c∆ k 3 for some polynomial g(∆). This also helps us to gain more insight to the condition (1.13). The condition is a statement that all the negative a ∆ forms a subsequence such that {|a negative ∆ |} is asymptotically upper bounded by T (∆).
Results at large central charge-Average Three Point Coefficient: The lower bound can also be obtained for infinite central charge. In this scenario, the lower bound has explicit dependence on the width of energy window. The precise result is obtained by keeping ∆/c finite while taking c → ∞. In particular, we assume that We further assume that ∆ χ ∼ O(c), this is consistent with HKS sparseness condition [5] while estimating the density of states. For ∆ > c 6 , we derive where a, b are O(1) numbers which depend on the ratio of dimension of the lowest excited state and central charge c and on . The T (∆) and C ∆O∆ are given by We remark that if ∆ χ ≤ c, in the leading order, one obtains (1.28) The above expression is same as the one obtained in [3] in the following limit ∆ → ∞ and then taking c → ∞.
The above result is consistent with Eigenstate thermalization hypothesis (ETH) in the sense that the ETH predicts the average value of the expectation value of a primary operator in heavy state is same as its thermal expectation value. Since the magnitude of a number is always bigger than the number, we expect A to be greater than or equal to the thermal expectation value. One can easily verify that the right hand side of Eq. (1.28) is indeed the thermal expectation value for β ET H = π √ 3 . The β ET H perfectly matches with the β obtained by solving the following equation for β In short, we show that The large central charge behavior reveals that the three point function can not fall off exponentially unless ∆ χ is of the order of central charge. In particular, this rules out the possibility of suppression of f ∆T ∆ at large central charge, since in that case ∆ χ=I = 0. Similar behavior is expected for KdV charges as well [30][31][32][33]. We remark that the result can easily be generalized for operators O with even spin. We comment about the odd spin case in the conclusion. Here again, the statement that we made about finding subsequence holds true if the condition in the ineq. (1.13) is not satisfied.
For the rest of the paper, we assume that the ineq. (1.13) is satisfied unless otherwise mentioned.

Scheme of the proof
We consider a CFT on a torus, where the spatial cycle is of length 2π and the thermal cycle is of length β. We will be looking at the torus one point function of O, a primary operator of the CFT. For simplicity, we consider a spinless operator 9 . The main tool we are going to use is modular covariance property of torus one point function of this operator, O and modular invariance of partition function Z(β). In several of steps below, we will be heavily using triangle inequality in the form | f (x)dx| ≤ |f (x)|dx and the inequality −|r| ≤ r ≤ |r| for any real number r.
We will do the proof in several steps: 1. First of all we will assume f χOχ > 0 for the first nontrivial operator χ , that produces operator O upon doing operator product expansion with itself. Under this assumption, we warm up with the restrictive scenario (in stead of considering Eq. (1.11) ) where f ∆O∆ ∈ [−M, M ] for some positive integer M . We arrive at (1.16). This is done in §3. We relax this condition afterwards in §4 and allow for polynomial growth.
2. The large central charge analysis is done in the §5.
3. In §6, we sketch out an extension of the lower bound for operators O with even spin (and not necessarily primary, rather a quasi-primary) and provide explicit verifications of the bound.
The basic idea is to have an approximation for weighted three point coefficient B, which reproduces the leading behavior of torus expectation value in high temperature limit. Then we show that for suitable choice of β, this is the leading behavior i.e. the contribution to the torus one point function coming from the heavy states are really suppressed. One way to do this is bound the heavy contribution by a part of partition function (we shall name it Z H in what follows), which gets contribution from the heavy states (or derivatives of Z H ). This, in turn, can be estimated to be bounded by some subleading term. We will see, the HKS bound [5] and its modified version will play a crucial role in this step.
One more technical remark is in order whose importance will become clear as we go along. Even though, we will be heavily using triangle inequality in the form f ∆O∆ ≤ |f ∆O∆ |. One would like to integrate both hand side over the interval [∆ − δ, ∆ + δ] to have a lower bound. This can not be done directly, rather one has to introduce a bandlimited function 10 φ − , which is less than or equal to the indicator function of the mentioned interval: this is required to make sure the contribution from the heavy states are cut off. So schematically we have something like: To relate this to the torus one point function , we have to add to the both sides of the inequality the contribution coming from the states, not in the interval. Our final job would then be to estimate this extra contribution and show they are subleading and does not matter in large ∆ limit.
The readers who want to circumnavigate the technical details for their first read can now skip directly to the §6.

Derivation of the result: Warm up
This section deals with a restrictive scenario where we assume |f ∆O∆ | < M for some O(1) number M . Later we relax this condition and allow for power law growth, which requires more sophistication and a modified version of a bound derived by Hartman, Keller and Stoica in [5], henceforth called as HKS bound.

A Lemma
Let us divide the contribution from light states and heavy states towards the torus one-point function separately: and ∆ H > c 12 . The aim is to bound the O H by a part of partition function on which one can apply the HKS bound. Assuming |f ∆O∆ | < M ∼ O(1), the lemma states that The proof follows from the following observation: The Lemma: Bound on one point function We will be using the following result later:

Main Proof
The main idea of the proof stems by giving a lower bound to the indicator function for an interval by a band limited function. We remind the readers that the role of this function is to facilitate cutting off the contribution coming from the heavy states to torus one point function. In particular, following [15], we consider a function φ − (∆ ) such that we have From this one can arrive at At this point, we define the following positive definite measures using the density of states and the weighted density of states (weighted by absolute value of three point coefficients): such that (3.7) can be integrated against the measure dG to obtain: We remark that this is not in contradiction with Eq. (3.6). Rather it puts more constraint on the possible functions that we can choose. Combining the last two inequalities, we have Now one obtains from the above: Had it been the case that f ∆O∆ is positive everywhere 11 , we would have obtained the following: . This is precisely where the problem lies. We know that f ∆O∆ need not be positive for all ∆. Nonetheless, as we are interested in ∆ → ∞ limit, we are required to show that the second term in the right hand side of the ineq. (3.13) is bounded above by some term, subleading compared to the two other terms in the ineq. (3.13). For now, we assume that it can be done and we proceed. We will come back to this at the end of this section.
We also define Laplace transform of density of states and weighted density of states via following: where Z(β) is the partition function and O β is the one-point function of the primary operator O. Furthermore, we define Fourier transform of φ − via φ − (∆) = ∞ −∞φ − (t)e −ı∆t to rewrite the inequality (3.15) in the following form: Now let us focus to the quantity Y ρ (β + ıt). This is the point where we are going to use modular covariance, which states that One can actually see that the positivity is not really a necessary condition, all we require for the Eq. (3.15) to be true for finite β is the following condition: Let us denote the region ∆ / ∈ ([∆ − δ, ∆ + δ] ∪ (−∞, 0)) as S. Certainly if all the f ∆ O∆ is negative definite for ∆ ∈ S, this is not true. But we can relax this condition further more as mentioned in the main text.
where τ is the modular parameter of the torus 12 . Hence we have where we have defined We also note that if O L 4π 2 /β is dominated by the first excited state with dimension ∆ χ , which is true in 4π 2 /β → ∞ limit, we have (3.23) Here T (∆) is given as which, in ∆ → ∞ limit, goes as: Here N O is a ∆ independent factor, given by The expression (3.25) is what is obtained in [3] by doing naive saddle point approximation.
Now we use (3.20) and the inequality −|r| ≤ r for any real number r. In particular, we choose , and the reality of r is guaranteed by choosing φ − (∆ ) to be a real valued function. 12 We remark that we are treating τ andτ to be independent variables and set them to ± ı(β+ıt) 2π Using −|r| ≤ r, we rewrite the inequality (3.18) as At this point we assume thatφ − has bounded support on The objective of having a bounded support is to cut off the contribution from heavy states in a nice way, as we will see. The bandlimited function has also been used in the analysis of Cardy formula in [15,37]. Then we move in the absolute value under the integral given in the eq. (3.28) to obtain: where in the second inequality, we have used (3.5) and in the last inequality we have used the boundedness of the is achieved. Now we recast (3.27) using the above inequality: which can then be turned into the following using (3.25): Now our aim is to see that in the large ∆, the second term in the first line is subleading; we will achieve this by making sure Λ * is below compared to some threshold value.
In order to do that, let us look at the first term. The first term in the integral can be obtained by saddle point method (this discussion is similar to that of Section. 4.1 in [15], the only difference is that here we choose β = πγ c 3∆ and γ = 1 − 12∆χ c to make sure that the saddle is at ∆ = ∆ 13 .) and given by where we have assumed that φ − is chosen in a way c − exists and is an O(1) number.
One can always achieve that by choosing an even integer ν such that ν > 2 (so that the function goes to zero as ∆ → ∞) and φ − is given by (more detailed choice of useful functions can be found in [15,37]): The second term in the first line of (3.31) is already estimated in [15], but unlike them, here we are choosing β = πγ c 3∆ . So we can not directly use their estimate. Upon restimating the second piece, we obtain: 13 Ideally, we should have chosen We also remark that the value of β chosen here matches with the saddle point β * in [3] only at large ∆, the subleading correction are different.
Hence the second piece goes like Thus the second term is subleading in large ∆ limit if the exponential growth in T (∆) is bigger than the growth of ρ 0 (∆) , this boils down to some upper bound on Λ * , which depends on ∆ χ , to be specific we have The existence of Λ * requires ∆ χ < c 12 . In [15] when they were dealing with density of states, ∆ χ is effectively 0 for them, hence (Λ * ) max = 2π. Here we have (Λ * ) max < 2π for treating the lower bound. Hence if choose φ − in a way such that this condition (3.36) is satisfied, we have On the other hand we also know that [15]: where we have used c ± to denote that one can in principle choose different function for estimating the density of states. In leading order ρ 0 (∆) is given by Thus combining everything, we have As a last step, one can basically optimize the ratio c − c + by choosing different functions, but the crucial point is that they will depend on lowest excited state ∆ χ . We also remark that the upper bound is trivial and follows directly from the assumption that |f ∆O∆ | < M .
As promised, we come back to the task of showing that the third piece in the the eq. (3.13) is in fact subleading.
For δ O(1), we have to bring in the extra condition on how negative the three point coefficient can get: This implies where we have used β = πγ c 3∆ followed by a clever use of saddle point approximation and the constraint α < γ. If α = γ, then the suppression is by a polynomial piece, for which we needed the extra constraint on . The case for which ∆ χ = 0, we have where we have used |φ − (∆ )| < K for some order one number K and chosen β = πγ c 3∆ , we let ∆ → ∞. The term becomes subleading in ∆ → ∞ limit provided ∆ O > 3 2 and this is guaranteed since O has to be in the Identity module and it is not the Identity operator itself.

Allowing for Power Law growth
In this section, we relax the condition (1.11) on f ∆O∆ to the following: The rational behind doing this is to include the quasi-primaries in the Identity module under the umbrella of our result. For example, for stress-energy tensor T , we have f ∆T ∆ = (h − c/12).
Here again we are required to show that the contribution coming from the heavy states to the torus one point function is suppressed. The trick is to show that this is bounded above by derivative of Z H . In order to estimate derivative of Z H , we need to have a modified version of the HKS bound [5] involving derivatives of partition function with respect to β. The following subsection achieves this. Similar discussion regarding light state dominance of derivative of partition function can be found in section 5.1 of [38].

Modified HKS bounds
Following the derivation of the usual HKS bound [5], we separate the contribution from the light and the heavy states towards partition function Z(β) as where ∆ H > c 12 . Modular invariance implies that where prime denotes that the partition function is evaluated at β = 4π 2 β . For notational simplicity let us define We note that if β > 2π, we have Upon using the fact that ∆ H > c 12 , in β → ∞ limit 14 , we arrive at Modular invariance tells us that in β → ∞ limit, we have Thus we arrive at , β → ∞ (4.11) Exchanging β and β we obtain , β → 0 (4.12) 14 We choose β large enough so that β 2 In particular, we will be needing the following estimate:

Estimation of three point coefficient
Instead of repeating all the basic details, we start with the inequality as given in the ineq. (3.29) and do the estimation in the following manner: where we have used and fact that Again, we need to show that the above piece is subleading compared to the first piece in the ineq. (3.27). We do that by estimating Z Hk 4π 2 β β 2 +Λ 2 * via (4.13) and choosing β = πγ c 3∆ → 0 (this is similar to the step done in [15] while estimating the contribution to the partition function from heavy states): Hence the second piece goes like This differs from the eq. (3.35) by a polynomial piece. Hence the rest of the analysis from the earlier section goes through in a straightforward manner.

Large central charge
In this section, we will be looking at states with dimension: In this scenario, the asymptotic behavior of the B is given by Eq. (1.26). If we divide by the density of states at large central charge, the result is given by Eq. (1.27). The basic scheme of the proof stays more or less same. Below, we sketch out the salient points for the large central charge analysis.

HKS Bound:
The large central charge analysis requires a careful reconsideration of the modified HKS bound. Earlier, one of the crucial input has been the following approximation: This approximation remains true even in the finite β but c → ∞ limit as we get the maximum power of c only when all the derivatives act on Z H or Z L . Thus we have The rest of the calculation proceeds in a similar manner and we obtain In particular, we have Z Hk c→∞ 4π 2 β 2 2k Z Lk , β < 2π (5.5) and the right hand side of the above is same as in Eq. (4.13).

Sparseness Condition:
The second crucial assumption that we have to make here is ∆ χ ∼ O(c). One can relax this assumption but then there would not be any exponential suppression in the central charge. For example, if one tensors really large number of copies of 2D Ising model, we have a large central charge, but ∆ χ would be still be an order one number. Furthermore, we have to assume that for β > 2π, the following sparseness condition holds: The above sparseness condition is the analogue of HKS [5] sparseness condition, which states that While the Eq. (5.7) condition is satisfied if and only if the density of states for low lying spectra satisfy the sparseness condition (5.6) that we require here is stricter when ∆ χ = 0: When ∆ χ = 0, we see that our result requires more sparseness in the low lying spectrum at large central charge. The above guarantees that T (∆) in large central charge dominated by the lowest nontrivial state with dimension ∆ χ only.
The rest of the analysis follows in a similar manner should we choose β = πγ 1 3 (5.10) and Λ * to satisfy the following condition:

Extension and Verification
The objective of this section is to provide with numerous verification of the results that we derived earlier.
Here we work at finite c.

Verification I: Identity Module
In this section we will be considering the operator Q ≡ T +T . We remark that T,T being SL(2, R) primaries (also known as quasi-primaries), under modular transformation, we have Now the lowest state is the identity instead of χ as we have and we should set ∆ χ=I = 0. The operator has a spin 2, thus i s f IQI > 0, hence we can proceed with this operator. Therefore, we conclude that We emphasize that all the prefactors in (1.19) involving c cancels out neatly. This precise cancellation is required for exact matching onto known result for three point coefficient. We know that Q is an operator corresponding to energy density: Hence, a direct computation of the quantity A using Eq. (6.4) would provide At this point we see that, Eq. (6.3) is consistent with Eq. (6.5) since b < 1 15 . This bound on b is guaranteed because of the inequality (3.6).
We further point out that one can derive an upper bound using the methodology if f ∆O∆ ≥ 0 for all ∆ > ∆ * . This happens because then one can write for ∆ > ∆ * + δ (this is true eventually as we let ∆ → ∞).
where φ + (∆ ) is above the indicator function, as done in [15]. Now we note that and hence it is subleading at large ∆ after setting β = πγ c 3∆ .
Here we have used Now one can proceed in a similar way as done for the lower bound and obtain an upper bound. In fact we know that this is the case for three point coefficients involving quasi-primaries in the identity module, which grows as a polynomial in ∆.

Verification II: Non Identity module
Tensored CFT-I We consider 4 copies of 2D Ising model and tensor them. We take O = ⊗ I ⊗ I ⊗ I with dimension 1. The torus one point function of O is given by The q expansion of the above can numerically provide us with We have checked for various δ, the above quantity is bounded below by some order one number times T (∆), as given in the ineq. (1.16). The central charge appearing in T (∆) should be given by c ef f = 2. In the fig. 3, we plot the ratio of B and T (∆) and verify that it is an order one number as we vary ∆. Since the absolute value is larger, we have B ≥ B and we verify the lower bound as in the ineq. (1.16) for this model. If we want to be very careful about the order one number we should divide B by 2δ and then compare with T (∆). Tensored CFT-II We consider 2 copies of 2D Ising model and one copy of Monster CFT. The tensored CFT has effective central charge of c ef f = 13. We look at the operator ⊗ I ⊗ I. Once again we plot the ratio of weighted three point coefficients B and T (∆) and find that it is bounded below by an order one number for large enough ∆, as can be seen in fig. 4. In fact it seems that curve asymptotes to a constant value; the value being greater than 1 might seem contradictory at first glance, nonetheless we should divide by 2δ if we really want to careful about the order one number. One can easily see that dividing by 2δ fixes the apparent contradiction.  In this section we prove two results pertaining to the case ∆ χ > c 12 . We note that neither the Kraus-Maloney analysis nor the analysis that we did above is applicable when ∆ χ > c 12 . Naively if one tries to extend the result, one would have obtained a power law behavior. Below in the second part, we will see that the qualitative picture of power law behavior is correct.
The result I: The first one states that if ∆ χ > c 12 + ∆ O 4π , then there would always be at least two three point coefficients with opposite sign. This follows from Hellerman [2] like argument applied on the following quantity The modular covariance for O β translates into The result then follows from considering N = 1 case 16 .
The result II: The second result states that the q expansion coefficient of g(sβ) ≡ O (sβ) is of the order of ∆ ∆ O /2 under the following assumptions • There exists s ∈ N such that O (sβ) , has an expansion in integer powers of q or it is an expansion of the form q n+α , where n is an integer and α is a fixed number. The first one can always be arranged when the set A, defined below is a finite set and ∆ is a rational number.
One should contrast this with the estimated q expansion coefficient for the case ∆ χ < c/12, where we have a exponential growth given by the function in (1.18).
In order to prove this let us consider the q expansion of g(sβ) ≡ O (sβ) : g(sβ) ≡ O (sβ) = n a n q n , n ∈ N (7.4) where the fact n ∈ N comes from the condition ∆ χ > c 12 . The q expansion coupled with the modular invariance of f (β) tells us that 16 Another way to state this result would be: if the all three coefficients are positive (or are of same sign), there has to be one operator χ with dimension below c 12 + ∆ O 4π such that f χOχ = 0. But we don't know of any example where all the three point coefficients are of same sign.
for some order number M . Now we use the fact that the powers of q that appears in the q expansion are intergers, hence we have a n = dq q n+1 g (sβ + ısΩ) = 1 0 dΩ q −n g (sβ + ısΩ) . (7.7) Subsequently one can write Now we choose β = 1 n to prove that We can verify our results by considering integer powers of η function. In fig. 6,7,8 we have plotted the q expansion coeffiecient as a function of ∆ to show that they satisfy the bound given in (7.9). The analysis above can be made much more precise when ∆ O is an integer [39]. Especially theorem 1 and theorem 2 stated there is of direct relevance here. The eq. 1.1 embodies both the condition mentioned above, which says that previous notation, s = N . In particular, it is shown that a n = O n ∆ O /2−1/5 , (7.11) which is consistent with what we have shown earlier in this section. In [40], it has further been shown that n≤X a n = O X ∆ O /2−1/10 (7.12) Below in the fig. 9, we verify the eq. (7.11) and in the fig. 10, we verify the eq. (7.12). is suppressed by P (n) ("fermionic" partitioning), where n = ∆ 1 − 1 8 and P (n) is partition of integer. Nonetheless, if we want to compute B we have to integrate over f Ising

Discussion and Outlook
In this work, we have proved a rigorous lower bound on the asymptotic behavior of the magnitude of the heavy-light-heavy three point coefficients. One might wonder whether considering the negative of the light operator O would turn the lower bound into an upper bound. First of all, this would not be the case, since we are estimating magnitude of three point coefficients. Second of all, we remark that our prove requires f χOχ (ı) s > 0 for the lowest dimensional operator χ contributing to the three point functions. If f χOχ (ı) s < 0, then our results would apply for the operator −O if the (1.13) holds true for −f ∆O∆ . We emphasize that among the operator A 1 = O and A 2 = −O we want the condition (1.13) to be true for the one for which f χA i χ (ı) s > 0. We also proved that if the inequality given in (1.13) is not satisfied, one can find an increasing subsequence such that the bound holds, provided we restrict ourselves to the subsequence while taking ∆ → ∞ limit. We have also shown that an upper bound can be obtained if the three point coefficients are positive except for a finite number of ∆. Some thoughts over proving the upper bound in a more generic set up is presented in appendix §A. We hope to fill in the details in future.
The result continues to hold at a large central charge under a certain sparseness condition on the spectra. The sparseness condition, as it turns out, is stronger than the sparseness condition [5] required for Cardy like behavior of the density of states at large c with ∆/c being finite. Furthermore, if one does not assume that ∆ χ ∼ O(c), the three point coefficients does not fall off exponentially in central charge.
In this paper we have considered operators O with even spin. If we only have integer spin operators with dimension ∆ in the spectrum, then f ∆O∆ = 0 if O has odd spin, this motivates our choice of assuming that O has even spin. We remark that the partition function is not invariant under T modular transformation if we have operators with half-integer spin. Nonetheless, if one considers S modular transformation only, it is possible to have operators with half integer spin, in that scenario, O can possibly carry odd spin with f ∆O∆ being non-zero, in which case, the operator with dimension ∆ must carry half integer spin. The odd spin introduces a factor of i in the analysis, but this gets compensated precisely by the same factor appearing in f ∆O∆ 17 [41].
We have verified our result for the quasi-primaries belonging to the Identity module and for nontrivial primary involving operator of 2D Ising model. It is not clear whether similar bound can rigorously be obtained specific for heavy primaries. This would require knowledge about the torus conformal block. Another obvious extension of this work is to consider extended Virasoro algebra and putting the results of [4] on rigorous footing. It would also be interesting to investigate the torus two point function [27][28][29] using similar methods.
We have shown that in the large central charge limit, the average value A ≥ O β ET H , this raises the question of whether the relative sign of three point coefficients are important to match up with the ETH prediction, since ETH predicts behavior of three point coefficients whereas we are probing the absolute value of the 17 The author thanks Ken Intriligator for discussion on this point. three point coefficients for heavy states.
We have derived the order of magnitude of q expansion coefficients of one point function when ∆ χ > c 12 . This behavior is sharply different from the case where ∆ χ < c 12 . In the former, we have polynomial growth while in the later we have exponential growth of q expansion coefficient. We remark that the case when ∆ χ = c We end with a paragraph speculative in nature. If the inequality (1.13) is satisfied, then it is tempting to conjecture similar existential result. In particular, one can hope to show that there exists a subsequence of ∆ k s such that B for a small interval centered around ∆ k asymptotes to the function T (∆) (which earlier appeared as a lower bound) with an error that goes like polynomial or one over a polynomial. If there is only finite number of negative a ∆ , we can prove an upper bound as outlined earlier in section where we discuss T +T . If there is an infinite number of negative a ∆ , then we form a sequence ∆ k such that magnitude of each of the term in the sequence a ∆ k is upper bounded. One can hope to extend this to B for small energy window centered around ∆ k . This can for example be done by making the energy window δ small enough via suitable choice of a function φ − . We refer to [37] for the optimal φ, but it would cost us a relative error suppressed by ∆ −y for some positive number y. The second speculation is regarding a curious observation that we make: in all of the examples involving tensored CFTs, where ∆ χ < c/12, the q expansion coefficient turns out to be positive except possibly for a finite number of ∆. If this is really a generic feature and can be proven, then one can simply get rid of the eq. (1.13) and find asymptotics of three point coefficients without considering the absolute value.

An anecdote
I have been curious about the applicability of Kraus-Maloney result to the tensored CFT since 2017 when I started working on extending the result for extended Virasoro algebra with Diptarka Das and Shouvik Datta. The issue was brought up then by John McGreevy in a different context. The initial guess was that the tensored CFT has more symmetry than Virasoro, hence KM would not be applicable. Nonetheless, nowhere in KM's analysis had it ever mentioned that the only symmetry algebra is Virasoro. Hence the issue was not really resolved and I visited the problem on and off. This work happens to resolve this paradox by carefully looking at the window over which the averaging is being done.

A More on Tauberian theorems & upper bound
In general, the Tauberian theorems requires positivity, for example the famous Hardy-Littlewood theorem (it was first proven by Hardy and Littlewood, later a simplified proof was given by Karamata.). It states: If a n ≥ 0 and f (x) = n a n x n ∼ C 1 − x , x → 1 (A.1) the following holds true: A more detailed exposition of the above can be found in the chapter 7 of the book [42] by G.H.Hardy. It would be interesting to see whether similar idea can be applied to the discussion of heavy light heavy three point coefficients. If possible, it would then provide us with an upper bound as well.
Below we provide the readers with a sketch of how a proof of the above kind would look like. We hope to fill in the details in future. In stead of individual heavy light heavy three point coefficient, we consider a(∆), which is in fact the sum of all the heavy light heavy three point coefficients for a particular ∆. In other words, we have In order to sketch the method of our proof, we consider the following condition: We further note that the extra term exp (−α∆ ) does not alter the behavior of the integrated three point coefficient by that much, since we have |Σ ± | ≤ Ke πγ and we recall that φ ± approximates the indicator function from the above and the below and |φ ± | is bounded by an order one number.
On the other hand, since b(∆) is positive definite, the analysis of asymptotics of the b(∆) falls naturally under the umbrella of the Tauberian theorems. In particular, one can set up an inequality of the following form e β(∆−δ)  fig. 3 and 4 since the curve asymptotes/saturates to a constant value. For the fig. 5, one might need to probe high enough value of ∆ to see the saturation. We will come back to this in the future.