Note on ETH of descendant states in 2D CFT

We investigate the eigenstate thermalization hypothesis (ETH) of highly excited descendant states in two-dimensional large central charge $c$ conformal field theory. We use operator product expansion of twist operators to calculate the short interval expansions of entanglement entropy and relative entropy for an interval of length $\ell$ up to order $\ell^{12}$. Using these results to ensure ETH of a heavy state when compared with the canonical ensemble state up to various orders of $c$, we get the constraints on the expectation values of the first few quasiprimary operators in the vacuum conformal family at the corresponding order of $c$. Similarly, we also obtain the constraints from the expectation values of the first few Korteweg-de Vries charges. We check these constraints for some types of special descendant excited states. Among the descendant states we consider, we find that at most only the leading order ones of the ETH constraints can be satisfied for the descendant states that are slightly excited on top of a heavy primary state. Otherwise, the ETH constraints are violated for the descendant states that are heavily excited on top of a primary state.


Introduction
By eigenstate thermalization hypothesis (ETH) [1,2], a typical highly excited energy eigenstate in a quantum chaotic system behaves like a thermal state. It explains how various statistical ensembles emerge in quantum many-body systems. It was originally formulated in terms local operators in basis of energy eigenstates and can be called local ETH. The two-dimensional (2D) conformal field theory (CFT) with a large central charge c is dual to quantum gravity in three-dimensional (3D) anti-de Sitter (AdS) space with a small Newton constant [3], and it is the precursor of AdS/CFT correspondence [4][5][6]. ETH in a 2D large c CFT is interesting and is related to the information lost paradox of black hole in 3D AdS space, i.e. the Bañados-Teitelboim-Zanelli (BTZ) black hole [7]. In fact, both the diagonal and off-diagonal parts of the local ETH in 2D CFT are consistent with the coarse-grained results obtained from modular covariance of the one-point and multi-points functions on a torus [8][9][10][11].
As a generalization of local ETH, subsystem ETH was proposed in [12,13] and one directly compares the reduced density matrix (RDM) of the excited energy eigenstate with the RDM of the thermal state. By investigating the two-point functions of light operators and entanglement entropy (EE), it was found that ETH is satisfied for the heavy primary states at the leading order of large c expansion, and thus the heavy primary states behave like the canonical ensemble thermal states [14][15][16][17]. With 1/c corrections, it was found that the RDM's of the primary excited state and canonical ensemble thermal state are in fact different by comparing their Rényi entropy, entanglement entropy, relative entropy, trace square distance, and other quantities [12,[18][19][20][21][22]. A possible resolution is to replace the canonical thermal state with the generalized Gibbs ensemble (GGE) thermal state [23] by including an infinite number of Korteweg-de Vries (KdV) hierarchy conserved charges [24][25][26] and their corresponding chemical potentials in the ensemble [19][20][21][22]. We will briefly discuss about ETH of GGE in the conclusion part of this part.
Previous studies of ETH in 2D CFT focus only on the primary excited states. For each primary state, there exists an infinite tower of descendant states. Though the properties of descendant states are algebraically determined by those of primary states, it does not necessarily imply that they also satisfy ETH as the primary states do even at the leading order of large c expansion. In this paper we will investigate the subsystem ETH for a few of special descendant states, and find that some of them behave similarly to the primary states while some others behave differently.
The remaining part of the paper is organized as follows. In section 2 we obtain the constraints on the expectations values of the first few quasiprimary operators in the vacuum conformal family for a general state to satisfy ETH in terms of EE, relative entropy, and KdV charges at different orders of c. In section 3 we check these constraints for the primary states and various descendant states. We conclude with discussion in section 4. We collect calculation details in the appendices. In appendix A we give some useful details of the quasiprimaries and their correlation functions in 2D CFTs, including both reviews and new calculations. In appendix B we first review the short interval expansions of the EE and relative entropy of an interval with length with the details of enumerating the quasiprimaries, and then calculate the results up to order 12 , which is higher than the order 8 in literature. In appendix C we give the proof of a statement used in the previous appendix.

Constraints of expectation values from ETH
We consider a 2D large c CFT on a cylinder with spatial period L, and choose the subregion A as a short interval with length L. In appendix B we calculate the EE and relative entropy for general translationally invariant states up to order 12  To define ETH, we also need to take the thermodynamic limit [12], i.e., taking the energy E and the total length L to infinity but keeping the energy density ε = E/L to be finite. In the thermodynamic limit, the inverse temperature β and the interval length satisfy β/L → 0, /L → 0, but there is no requirement for /β. All the constraints in this section should be understood as under the thermodynamic limit. To do short interval expansions of the EE and relative entropy we further require β. In summary we need β L, and the constraints should satisfy for all orders of the expansion of /β.

Constraints for all orders of large c
For two general states ρ, σ, requiring S A,ρ = S A,σ we get the constraints In fact S(ρ A σ A ) = 0 is equivalent to ρ A = σ A , and so all local operators {X } satisfy X ρ = X σ .
For the KdV charges Q ρ 2k−1 = Q σ 2k−1 , k = 1, 2, 3, · · · , we get Generally, the three sets of constraints are not equivalent, and their relations are summarized in figure 1. In fact, two states having the same KdV charges, i.e. satisfying (2.3), lead to non-vanishing relative entropy Figure 1: Relations of the three sets of constraints from the EE, relative entropy and KdV charges.
More specifically for requiring ETH, we choose the state σ to be the canonical thermal state ρ β and compare its RDM ρ A,β with the RDM ρ A of the state ρ. Because the modular Hamiltonian of RDM ρ A,β is a local integral of the stress tensor [27], requiring S(ρ A ρ A,β ) = 0 or S(ρ A,β ρ A ) = 0 is equivalent to requiring S A = S A,β [12], which then yields for all orders of c, with the expectation values in thermal state as given (A.9). One can then see the KdV charges of the thermal state as given in (A.10). On the other hand, By requiring that KdV charges Q ρ 2k−1 = Q β 2k−1 , k = 1, 2, 3, · · · , we get

Constraints for fixed orders of large c
In thermodynamical limit, the exact form of the thermal state EE is known [28] S A,β = c 6 log β π sinh π β . (2.7) We may relax the ETH condition up to different orders of c, i.e., by requiring S A − S A,β to be of different orders of c. If we just require S A − S A,β = O(c 0 ), then we get the leading order constraints of EE with t 0 , a 1 being arbitrary order c 0 constants. On the other hand, if we ask for more stringent ETH We call them the next-to-leading order constraints of EE.
Similarly, we can obtain the constraints by requiring the two relative entropies S(ρ A ρ A,β ) and get exactly the same results as the leading order constraints of EE (2.8). However, by requiring we get the next-to-leading order constraints of relative entropy which are different from (2.9) for the next-to-leading order constraints of EE.
As for the KdV charges, we require the following quantities to be small, Note that these quantities do not depend on normalization convention of the KdV charges. Requiring (2.11) to be at order 1/c, we get the leading order constraints of KdV charges (2.12) Requiring (2.11) to be at order 1/c 2 , we get the next-to-leading order constraints of KdV charges (2.13)

Checks for primary and various descendant eigenstates
We check the constraints obtained in the previous section for various highly excited energy eigenstates.
The states to be considered include the excited state |φ with φ a primary operator of conformal weight h φ , and descendant states of the same conformal family |φ , |∂ m φ , |∂ mφ . We have the definitioñ We also consider the vacuum conformal family descendant states |∂ m T , |∂ m A . Note that |∂ m T is just a special case of |∂ mφ with h φ = 0. One can see details of these states in appendix A.
Note that in this section we only check the aforementioned constraints up to level 6. If the constraints are violated, the results are conclusive. If the constraints are satisfied to level 6, we do not know whether the constraints will be violated at higher levels.
We take the thermodynamic limit for the CFT [12]. This requires that the level of the state is of order L 2 in L → ∞ and is of order c in large c limit. In the thermodynamic limit, the energy eigenstates we consider fall into three types depending on the values of the parameters.
The type I states include |φ and |φ with h φ = HL 2 + o(L 2 ), as well as |∂ m φ and |∂ mφ with Using results in appendix A, we get the expectation values Neither the all-order constraints (2.5) of EE/relative entropy nor (2.6) of the KdV charges are satisfied by the above expectation values for any H. However, the leading constraints of EE and relative entropy (2.8) are satisfied with the following identification of the parameter and t 0 = −4π 2 H 0 , a 1 = 4π 4 3β 2 H 0 . Similarly, the leading order constraints of KdV charges (2.12) are satisfied with the following identification of the parameter, and b 2 = 0. Despite that, none of the next-to-leading order constraints of EE (2.9), relative entropy (2.10), or KdV charges (2.13) can be satisifed.
The type II states include |∂ m φ and |∂ mφ with type state constraints to level 6 leading next-to-leading all Table 1: The three types primary and descendant states we consider in this section, and whether they satisfy the ETH constraints derived from EE, relative entropy, and KdV charges up to level 6. In the 2nd column we have definitions H = c all the constraints are violated. The 3rd, 4th, and 5th columns all apply to the constraints derived from EE, relative entropy, and KdV charges. We mark for constraints that are satisfied and mark × otherwise.

Conclusion and discussion
We calculated the short interval expansions of EE and relative entropy to order 12 . Using the results to require ETH for a highly excited state, we got the leading order and next-to-leading order constraints in the large c expansion on the expectation values of the first few quasiprimary operators in the vacuum conformal family with respect to the state considered. We also obtained the constraints from the first few KdV charges.
We checked the constraints for the primary and various descendant states. We found that these constraints can only be satisfied for at most the leading order of large c, and even the leading order constraints are violated for descendant states that are far away from their primary states. Note that when we say a descendant state is close to or far away from its primary state, we refer to just the primary state in the same conformal family of the descendant state.
In [29], we have investigated the conditions for a state to have classical gravity dual, and we call a state satisfying these conditions a geometric state. Based on the holographic EE and Rényi entropy [30][31][32], the geometric state conditions are obtained by requiring the Rényi entropy to be at most of order c in large c limit. There are some differences between the geometric state conditions and the ETH state constraints in this paper as listed below. (1) For the geometric state conditions, the states does not necessarily have translational symmetry and so the one-point functions are not necessarily constants. For the ETH state constraints, we consider the globally excited energy eigenstate, and the one-point functions are constants. (2) In the geometric state conditions we do not require the state to be at high energy, and so we focus on the order of c and do not care about the order of L. In the ETH state constraints we need to firstly take thermodynamic limit L → ∞ and then do expansion of large c.
Although there are differences mentioned above, still it is interesting to compare the results. We stress that the comparison in this paragraph is under the thermodynamic limit. For translationally invariant states, the geometric state conditions in [29] can be recast as The states satisfying the ETH state constraints of EE and relative entropy (2.8), (2.9), (2.10) also satisfy the geometric state constraints (4.1). However, the states satisfying the ETH constraints of KdV charges (2.12), (2.13) do not necessarily satisfy the geometric state constraints. In fact the geometric state conditions (4.1) are not only consistent with but also equivalent to the leading order constraints of EE (2.8). Requiring the energy density be nonnegative in the thermodynamic limit, we get t 1 in (4.1) is nonpositive t 1 ≤ 0. We obtain that (4.1) and (2.8) are equivalent with the identification t 1 = − π 2 6β 2 . The equivalence of the geometric conditions and the leading order constraints is remarkable. The geometric conditions come from S (n) The leading order ETH constraints of EE come from S A = S A,β + O(c 0 ) with (2.7). A possible way to understand this is that a translationally invariant geometric state is dual to a Bañados metric [33] with constant stress tensor and in thermodynamic limit the classical Bañados geometry metric is exactly the same as the BTZ black hole metric with some identification of parameters. By the holographic EE [30,31], it ensures that the translationally invariant geometric state satisfies the leading order ETH constraints of EE (2.8).
The ETH state constraints we obtained are about how close a state ρ and the canonical ensemble can be in the thermodynamic limit. Although the RDM of a primary state is the same as RDM of the thermal state ρ β in the leading order of c, the RDM's of the considered descendant states that are far away from the primary state are very different from the thermal state RDM. Recently we have proved that the RDM of the canonical ensemble thermal state ρ β and RDM of the microcanonical ensemble thermal state ρ E = 1 are the same in thermodynamic limit [34]. In both of the canonical and microcanonical ensembles, one needs to average over all the states, including both the primary and descendant states. In fact, there are far more descendant states than primary states at high energy. The density of all the states is given by Cardy formula [35] and the density of primary states is [8] Ω Note that in the thermodynamic limit one has E → ∞, L → ∞, and ε = E/L is finite. We get the ratio of the number of primary states and the number of all states is  [19][20][21][22]. In this paper we obtained the same results for some descendant states that are close to their primary states. A possible resolution to the mismatch of the excited states and the thermal state was proposed in [19][20][21][22], and it is to replace the canonical ensemble thermal state by the GGE [23].

A Quasiprimaries and their correlation functions
In this appendix, we give some useful details of 2D CFT we need in this paper, including both reviews and new calculations. The basics of 2D CFT can be found in [36][37][38].

A.1 Vacuum conformal family
Many details of the quasiprimary operators in the vacuum conformal family can be found in the papers [19,39,40]. We only consider the holomorphic sector, and the anti-holomorphic sector is similar. We count the number of independent holomorphic operators at each level in the vacuum conformal family as among which the holomorphic quasiprimary operators are counted as We list these quasiprimary operators in table 2, and to level 9 their explicit forms can be found in [19,39,40]. We do not need the explicit forms of other quasiprimary operators in this paper. The conformal transformation rules of the quasiprimary operators to level 8 can be found in [19].
In the vacuum conformal family, one can define an infinite number of mutually commuting conserved KdV charges Q 2k−1 , k = 1, 2, · · · [24][25][26]. In terms of the quasiprimary operators, we can write the first four KdV charges on a cylinder as [29] For the CFT on a circle with period L in the ground state, which is just a vertical cylinder with spatial period L, we have the one point functions For the CFT on a circle with period L in thermal state with inverse temperature β L, which is a torus with module τ = iβ/L, we use the results in [41] and get the one point functions expanded by Note that, as stated in [41], the above one point functions can be expanded to an arbitrary order of q.
For the CFT on an infinite straight line in thermal state with inverse temperature β, which is just a horizontal cylinder with temporal period β, we have [39] T β = − π 2 c 6β 2 , A β = π 4 c(5c + 22) 180β 4 , We have orthogonalized the operators, such that the correlation function of two holomorphic quasiprimary operators X 1,2 on a complex plane C with coordinate f takes the form where we have defined f 12 ≡ f 1 − f 2 . The correlation function of three holomorphic quasiprimary operators X 1,2,3 takes the form with C X 1 X 2 X 3 being the structure constant. In this paper, we need the structure constants We also need the four-point functions (A. 16) In the above multi-point functions, we have used the notation in the form where · · · denotes permutation terms of * and # is the total number of terms in the parentheses.

A.2 Non-vacuum conformal family
We consider the non-vacuum conformal family of the primary operator φ, with conformal weight h φ > 0 and normalization α φ . Without loss of generality and for simplicity, we choose it to be holomorphic, and the the generalization to a non-chiral primary operator is straightforward. The next quasiprimary operator in the conformal family of φ is with conformal weight hφ = h φ + 2, and normalization factor Using state operator correspondence in 2D CFT, on a cylinder with spatial period L we construct the ket and bra states Similarly we define |φ , φ |, |∂ mφ , ∂ mφ |. The state |φ is primary, and states |φ , |∂ m φ , |∂ mφ are descendant. We have the normalization factors Using the states, we have the normalized density matrices As in [19], we get the structure constants From the structure constants and the conformal transformation rules of T , A, B, D, we get the expectation values Similarly, we get the structure constants Then we get the expectations values For general quasiprimary operators X , Y, we have the three point function on a complex plane Using the three point function we get the expectations values X ∂ m φ , X ∂ mφ with X = T, A, B, D. Similarly, we obtain the expectations values X ∂ m T , X ∂ m A with X = T, A, B, D. We will not give the explicit forms of these expectations values.

B Short interval expansion of EE and relative entropy
We review the short interval expansion of EE and relative entropy from OPE of twist operators. We also obtain the EE and relative entropy to higher orders than the ones in literature. The method twist operators was proposed in [28] to calculate Rényi entropy in 2D CFT. The OPE of twist operators was formulated in [42][43][44]. Without loss of generality, we only include the contributions from the holomorphic sector, and the contributions from the anti-holomorphic sector can be added easily.

B.1 OPE of twist operators and enumerating quasiprimaries
We consider one short interval A = [0, ] on a general Riemann surface R with translational symmetry, and the constant time slice is in a state with density matrix ρ. Tracing the degrees of freedom of the complement of A that we callĀ, we get the RDM ρ A = trĀρ. To get the EE we use the replica trick and first calculate the Rényi entropy and then take the n → 1 limit. We calculate the partition function tr A ρ n A of the CFT on an n-fold Riemann surface R n , and it equals the two-point function of twist operators σ,σ in the n-fold CFT on single copy of the Riemann surface R [28] Note that in this paper we just focus on the holomorphic sector of the twist operators.

(B.4)
We can write the OPE of twist operators as [42][43][44] with c n being the normalization factor, the summation K being over all the orthogonalized holomorphic quasiprimary operators Φ K in CFT n , and h K being the conformal weight of Φ K . We have definition with C y x denoting the binomial coefficient. The OPE coefficient d K can be calculated as [43] with α K being the normalization of Φ K . We have used R n,1 to denote the n-fold Riemann surface R n that results from the replica trick for one interval A = [0, ] on the complex plane C. To calculate the right-hand side of (B.7), we map R n,1 with coordinate z to a complex plane with coordinate f by the transformation [28,43] f (z) = z − z with the summation of K being over all the CFT n holomorphic quasiprimary operators Φ K that are direct products of the holomorphic quasiprimary operators in each copy of the original CFT. In this paper we only consider the contributions of the holomorphic part of the vacuum conformal family, and the relevant CFT n quasiprimary operators are counted as [x + (1 − x)trx L 0 ] n = 1 + nx 2 + n(n + 1) 2 x 4 + n(n 2 + 3n + 8) 6 x 6 + n(n + 1)(n 2 + 5n + 30) 24 x 8 + nx 9 + n(n + 1)(n + 2)(n 2 + 7n + 72) 120 x 10 + n(n + 1)x 11 + n(n + 3)(n 4 + 12n 3 + 169n 2 + 438n + 640) 720 x 12 + n(n + 1)(n + 2) 2 Note the definition of trx L 0 in (A.1). The direct product quasiprimary operators take the form and we list these operators to level 13 in table 3.
The one-point function of X j 1 1 · · · X j k k on R is independent of the replica indices X j 1 1 · · · X j k k R = X 1 R · · · X k R , (B.12) level quasiprimary ?  Table 3: The CFT n holomorphic quasiprimary operators that are direct products of the vacuum conformal family holomorphic quasiprimary operators in each copy of the original CFT. We have omitted the replica indices for these operators. For example, at level 8, In the third column, we mark for nonidentity operators with generally non-vanishing contributions to the single interval EE in a translationally invariant state, i.e., with non-vanishing coefficients a K defined in (B.15), and we mark × for operators with vanishing coefficients a X 1 ···X k . Note that for k = 0, i.e. the identity operator, we do not need to calculate the coefficient a X 1 ···X k . We count the number of operators in the fourth and fifth columns, with the notation n k = n(n − 1) · · · (n − k + 1). The counting is consistent with (B.10).
We sum the replica indices of the OPE coefficient d j 1 ···j k X 1 ···X k and define [41] b X 1 ···X k ≡ j 1 ,··· ,j k d j 1 ···j k X 1 ···X k with some constraints for 0 ≤ j 1 , · · · , j k ≤ n − 1. (B.13) The constraints are to avoid overcounting of the quasiprimary operators. For example, the constraints for d j 1 j 2 j 3 T T A are j 1 < j 2 , j 1 = j 3 , j 2 = j 3 . Except the identity operator, all the coefficients b X 1 ···X k are vanishing in the limit n → 1. We get the Rényi entropy From the coefficient b X 1 ···X k , we further define [21,45] and get the EE written as In this paper we focus on the entanglement entropy, instead of the Rényi entropy. To calculate the coefficients a X 1 ···X k we do not need the full forms of b X 1 ···X k or d j 1 ···j k X 1 ···X k . A general holomorphic quasiprimary operator X transform under a general map z → f (z) as with · · · denoting terms with the Schwarzian derivative For the transformation (B.8) we have When we calculate d j 1 ···j k X 1 ···X k using (B.7), the contributions from · · · terms in (B.17) would be of order O(n − 1) or of higher orders. For k ≥ 2, when we compute b X 1 ···X k using (B.7), the summation of the replica indices would lead to another order O(n − 1) or higher order factor for each term. For k ≥ 2, the · · · terms in (B.17) only contribute order O(n − 1) 2 or higher order terms to b X 1 ···X k , and so would not contribute to a X 1 ···X k defined in (B.15). For k = 1, a X = 0 for h X > 2. We will prove it the next appendix.
To level 13 the nonidentity CFT n quasiprimary operators with non-vanishing a X 1 ···X k are marked with in the 3rd column of table 3. For k = 1 we only need to calculate a T . For k ≥ 2, we use the various multi-point functions in appendix A and get d j 1 ···j k X 1 ···X k with some irrelevant O(n − 1) terms.
Summing the replica indices, we get b X 1 ···X k with some irrelevant O(n−1) 2 terms. Then we get a X 1 ···X k .

B.2 EE
Using the above coefficients, we get EE of a short interval A in state ρ on a Riemann surface R [21] To check the EE formula and the coefficients a X 1 ···X k in (B.20), we consider several examples. The first case is that R is a vertical cylinder with spatial period L. We denote the state density matrix as ρ L , we have the expectation values (A.7). We get the entanglement entropy which matches the exact result [28] S A,L = c 6 log L π sin π L . (B.23) It is similar for the CFT on an infinite straight line in thermal state with inverse temperature β, which is just the 2D CFT on a horizontal cylinder with temporal period β. We use the expectation values (A.9) and get a result that matches the EE (2.7).
On a torus with low temperature, we have spatial and temporal periods L and β that satisfy L β. We denote the density matrix as ρ L,q with q = e −2πβ/L which is valid as long as the interval length is not comparable with total length L. The order c 0 part of (B.25) was calculated to order q 2 in [46], to order q 3 in [47], and to order q 4 in [48]. The order 1/c part of (B.25) is new, and we calculate it using the method in [46][47][48].
For the CFT on a cylinder with spatial period L in the primary state |φ , we denote the density matrix as ρ L,φ . We have the expectation values (A.26), from which we get the EE The result to order 8 has been calculated in [19]. Setting h φ = c 24 L 2 β 2 + 1 in (B.26), we get a result that matches (2.7) to order c in large c limit, and this is consistent with the result in [16,17].

B.3 Relative entropy
Similarly, for two translationally invariant states ρ and σ, which correspond to the Riemann surface R and S respectively, we have the relative entropy [21] . C Proof of a X = 0 for h X > 2 In this appendix, we give a proof of a X = 0 with X being a quasiprimary operator in the holomorphic sector of the vacuum conformal family and h X > 2. Note that h X > 2 is equivalent to h X ≥ 4.
General a X 1 ···X k is defined in (B.15), and a X is the special k = 1 case.
Under a general conformal transformation z → f (z), the operator X transforms formally as We have for the conformal transformation (B.8). Note that (B.19), s(z) = O(n − 1). To prove a X = 0 for h X > 2, we only need to show F X = O(s 2 ) for a small s.
All the operators in holomorphic sector of the vacuum conformal family can be constructed from T by derivatives, normal orderings, and linear combinations. We can recursively organize all general holomorphic quasiprimary operators {X } as linear combinations of operators in the forms (∂ p T X ), ∂ q X with integers p = 0, 1, 2, · · · , q = 1, 2, 3, · · · . Note the relation (∂ p T ∂ r X ) = ∂(∂ p T ∂ r−1 X ) − (∂ p+1 T ∂ r−1 X ), we do not need the include (∂ p T ∂ r X ) with r ≥ 1. For examples, at level 2 we have T , at level 4 we have (T T ), ∂ 2 T and get A, and at level 6 we have (∂ 2 T T ), (T A), ∂ 4 T , ∂ 2 A and get B, D. Explicitly, we can recursively write a quasiprimary operator X with h X > 2 as with the summation of all nonidentity quasiprimary operators Y with h Y ≤ h X − 1. In fact the constants u X Y = 0, for h Y ≥ h X −1, v X Y = 0 for h Y ≥ h X . Generally for a fixed X , the decomposition (C.6) may not be unique. Writing in terms of states, we have ]|Y (C.7) with L k being modes of the stress tensor T . We multiply it with the bra state Using the orthogonality of the quasiprimary operators and the Virasoro algebra we get v X T = − 12u X T (h X − 3)(h X − 2)h X (h X + 1) . (C.9) We write X as We get for p ≥ 2 ψ p = 1 (p + 1)! s (p−2) + O(s 2 ). (C. 19) Then we obtain F (∂ p T T ) = c (p + 1)(p + 2)(p + 4)(p + 5) s (p+2) + O(s 2 ). (C.20) Note that From (C. 15), (C.20), (C.21) and (C.10), we get for h X ≥ 4 From F A = O(s 2 ), we get by induction F X = O(s 2 ) for all holomorphic quasiprimary operators in the vacuum conformal family with h X ≥ 4. Thus we prove that a X = 0 for h X > 2.