OPE selection rules for Schur multiplets in 4D N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} superconformal field theories

We compute general expressions for two types of three-point functions of (semi-)short multiplets in four-dimensional N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} superconformal field theories. These (semi-)short multiplets are called “Schur multiplets” and play an important role in the study of associated chiral algebras. The first type of the three-point functions we compute involves two half-BPS Schur multiplets and an arbitrary Schur multiplet, while the second type involves one stress tensor multiplet and two arbitrary Schur multiplets. From these three-point functions, we read off the corresponding OPE selection rules for the Schur multiplets. Our results particularly imply that there are non-trivial selection rules on the quantum numbers of Schur operators in these multiplets. We also give a conjecture on the selection rules for general Schur multiplets.


Introduction
The space of four-dimensional N = 2 superconformal field theories (SCFTs) has a rich structure. The best known N = 2 SCFTs are the SU(N c ) gauge theories with N f fundamental matter hypermultiplets where N f = 2N c is satisfied so that the beta function vanishes. While these theories are well-described by Lagrangian, there are many N = 2 SCFTs whose Lagrangian description is not known, such as Argyres-Douglas SCFTs [1][2][3], 1 Minahan-Nemeschanskey theories [14,15], and an infinite series of non-Lagrangian SCFTs of class S [16]. To study general N = 2 SCFTs including these non-Lagrangian theories, we need a technique that relies only on the symmetry and unitarity of SCFTs.
Recently there was important progress in this direction. The authors of [17] showed that the operator product expansions (OPEs) of a special class of BPS local operators are naturally encoded in a two-dimensional chiral algebra. These BPS operators are called "Schur operators" since they contribute to the Schur limit of the superconformal index [17][18][19]. We call the superconformal multiplets including a Schur operator "Schur multiplets." The existence of the associated chiral algebra implies, along with the four-dimensional unitarity and superconformal symmetry, that the "c central charge" of any interacting four-dimensional N = 2 SCFTs is constrained by c ≥ 11 30 [20], which is saturated by the minimal Argyres-Douglas SCFT. 2 Moreover, a similar analysis for N = 2 SCFTs with a flavor symmetry leads to a universal bound involving c and the flavor central charge [17,21]. For more recent works on the associated chiral algebras, see .
Let us briefly sketch how these bounds on the central charges were derived from the chiral algebra analysis. In four-dimensional N = 2 SCFTs, the superconformal symmetry and the unitarity impose strong constraints on superconformal multiplets appearing in

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OPEs, which we call "selection rules" in the following. In particular, the selection rules 2 ) + · · · , (1.1) were used to derive the central charge bounds mentioned above. Here, B R and C R(j,) are two types of Schur multiplets labeled by the SU(2) R charge R and the spin (j,) of the superconformal primary field, and the ellipses stand for non-Schur multiplets. 3 In particular, the B 1 multiplet is a Schur multiplet including a flavor current, and the C 0(0,0) multiplet is the stress tensor multiplet. The above selection rules (and unitarity) are crucial in deriving the central charge bounds. For example, the bound c ≥ 11 30 was derived by interpreting the reality of the OPE coefficients for the second selection rule in (1.1) in terms of the two-dimensional chiral algebra. 4 This implies that identifying the selection rules for Schur multiplets provides a powerful tool to reveal universal constraints on general N = 2 SCFTs.
Moreover, the selection rules are also important in recovering four-dimensional OPEs from the two-dimensional chiral algebra. Indeed, the 4d/2d correspondence of [17] implies that Schur operators with different quantum numbers could correspond to two-dimensional operators with the same quantum numbers. Therefore, it is generically non-trivial to recover four-dimensional OPEs from two-dimensional OPEs. The selection rules, however, strongly constrain Schur multiplets appearing in the four-dimensional OPEs and therefore will be useful for reconstructing the four-dimensional OPEs from the associated chiral algebra.
In this paper, we study the selection rules for up to non-Schur multiplets, where O Schur is an arbitrary Schur multiplet. Since the Schur operator in the stress tensor multiplet C 0(0,0) maps to the Virasoro stress tensor in the associated chiral algebra, the selection rules for (1.2) are particularly important in the study of the 4d/2d correspondence. In particular, they reveal how the four-dimensional operator associated with a Virasoro primary is related to those of the Virasoro descendants. Indeed we find that, when four-dimensional Schur operators O and O respectively correspond to a Virasoro primary and its descendant in the associated chiral algebra, the SU(2) R charge of O is always smaller than or equal to that of O (see for example (4.31)). Note that the selection rules for C 0(0,0) × C 0(0,0) and C 0(0,0) × B 1 were already identified respectively in [20] and [51], which we generalize to (1.2) for all Schur multiplets O Schur in this paper.

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To derive the above selection rules, we study three-point functions of the form C 0(0,0) O 1 O 2 , where O 1 and O 2 are arbitrary Schur multiplets. Our strategy is to write down the most general ansatz for the three-point functions and then impose the (semi-)shortening conditions corresponding to the Schur multiplets. The same strategy was employed in [20,51,52] to compute several three-point functions. We stress that, since our analysis relies only on the (semi-)shortening conditions which purely follow from the superconformal algebra, our results are applicable to any four-dimensional N = 2 SCFT.
Before studying the selection rules for C 0(0,0) × O Schur , we first apply our strategy to the selection rules for as a warm-up. While these rules were already identified in [53], 5 we believe it is worth showing an explicit derivation of the rules. Moreover, we evaluate the most general expressions for the three-point functions B R 1 B R 2 O with O being an arbitrary Schur multiplet, which contain more information than the selection rules. Let us here make an observation on our OPE selection rules. For some of the OPEs we study in this paper, the three-point function of the corresponding superconformal primary fields turns out to vanish even though three-point functions involving their descendants do not. This reflects the fact that the sum of the U(1) r charges of the superconformal primary fields in three-point functions is non-vanishing. On the other hand, we find that the sum of the U(1) r charges of Schur operators in these multiplets always vanishes, which suggests that the Schur operators play a central role in Schur multiplets. With this observation, we give a conjecture on the OPE selection rules for general Schur multiplets in section 5.
The outline of this paper is as follow. In section 2, we review the four-dimensional N = 2 superconformal algebra and the (semi-)shortening conditions for the Schur multiplets, and also introduce a useful formalism [52,54,55] to analyze the superconformal three-point functions. In sections 3 and 4, we derive the two types of three-point functions respectively. From these correlation functions, we present the B R 1 × B R 2 selection rule and the C 0(0,0) ×O Schur selection rules. section 5 is devoted to conclusions and discussions, where we conjecture more general selection rules between Schur multiplets as a natural generalization of our results. In appendix A, we summarize the nilpotent structure of the Grassmann variables what we call Fierz identities, and appendices B, C, and D are the details of our calculations.

(Semi-)shortening conditions and three-point functions
In this section, we review the four-dimensional N = 2 superconformal algebra and the short multiplets following [56] and introduce a useful formalism constructed in [52,54,55] for the computations of correlation functions of SCFTs. We follow the convention of [17] unless otherwise stated.
A general long multiplet of the four-dimensional N = 2 superconformal algebra is labeled by five eigenvalues of the Cartan subalgebra for the primary state, namely, the conformal dimension ∆, the Lorentz spin (j,), the irreducible representation R of SU(2) R , and the U(1) r charge r. 6 Here, the superconformal primary field is defined as a state annihilated by all conformal supercharges, S α i and Sα i . We denote the superconformal primary field by |∆, r where the parentheses in the scripts such as (i 1 · · · i 2R 1 ) denote the total symmetrization of the indices. Acting Q i α and Qα i on the primary, we can generate 256(2R + 1)(2j + 1)(2 + 1) components of the long multiplet. These long multiplets satisfy unitarity bounds, where E i and j i are defined by An N = 2 Poincaré supersymmetric field theory has a unitarity bound called the BPS bound, and if a long multiplet saturates the bound it becomes a short multiplet whose number of components is half the original one. The above unitarity bounds for N = 2 SCFTs play a similar role; when the eigenvalues (∆, R, r, j,) saturate the unitarity bounds (2.1) or (2.2), a long multiplet is shortened. If we consider the case of j = 0, ∆ = E 1 − 2, the superconformal primary field |∆ (i 1 ···i 2R ) (α 1 ···α 2 ) satisfies the condition Similarly, for = 0, ∆ = E 2 − 2, the superconformal primary field satisfies the condition These two conditions B 1 andB 2 are called shortening conditions. On the other hand, for ∆ = E 1 , the following condition is possible: (2.6) 6 We take R so that the Dynkin label for the irreducible SU(2)R representation is 2R.

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Multiplet Condition Conformal dimension and U(1) r charge of the primary Similarly for ∆ = E 2 , we can imposē (2.7) These two conditions C 1 andC 2 are called semi-shortening conditions.
The Schur multiplets are defined as multiplets satisfying a shortening condition or a semi-shortening condition for each of the chiralities. There are four types of Schur multiplet, which are denoted as B R , D R(0,) ,D R(j,0) , and C R(j,) in the notation of [56]. See table 1 for the definition of the four types and the relations among the quantum numbers of their superconformal primary states.
Some of these Schur multiplets contain important operators. For example, the C 0(0,0) multiplet contains the stress-tensor operator and the SU(2) R × U(1) r conserved current operator. We can regard the semi-shortening conditions as the conservation equations of these operators. In this paper, we assume that the theory we are considering has a unique stress-tensor multiplet C 0(0,0) . This assumption leads to a constraint on the three-point functions involving two stress tensor multiplets, as discussed later. The C 0(j,) multiplet is a higher-spin generalization of the stress-tensor multiplet C 0(0,0) and contains a higher spin current operator. It is expected that interacting (S)CFTs containing such a higher spin current have a decoupled free sector [57][58][59]. On the other hand, the B R multiplets are half-BPS multiplets whose superconformal primary field is annihilated by both Q α 1 and Qα 2 . As mentioned in section 1, the B 1 multiplet particularly contains a conserved flavor current. Finally, the D 0(0,0) ⊕D 0(0,0) multiplet is an N = 2 free vector multiplet, whose superconformal primary field has the conformal dimension 1, and its (semi-)shortening conditions imply the massless equation of motion.

Superspace formalism
In this subsection, we review a useful superspace formalism following [52,54,55]. We denote by z := (x µ , θ α i ,θα i ) the coordinate of the N = 2 superspace. We then define JHEP04(2019)060 chiral/anti-chiral variables x µ ± and derivatives D i α ,Dα i as 7 In the superspace formalism, the (semi-)shortening conditions reviewed above are expressed in terms of the covariant derivatives D i α andDα i . For example, let L (i 1 ···i 2R ) (z) be a superfield for the B R multiplet. Then the shortening conditions for B R imply that any correlation function involving L (i 1 ···i 2R ) (z 1 ) satisfies , · · · are superfields for general superconformal multiplets. In the rest of this section, we review basic techniques to solve differential equations of this form. We use them in sections 3 and 4 to derive selection rules for the Schur multiplets.

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Next, for given different three points z 1 , z 2 , and z 3 in superspace, we introduce the superconformally covariant variable Z 1 = (X 1αα , Θ iα 1 ,Θα 1i ) by 14) The variable Z 1 transforms similarly to z 1 . In particular, the following identity will be important in our calculations below: Similar variables Z 2 and Z 3 are defined as cyclically permuting z 1 , z 2 and z 3 in the above definition. 9 These Z variables play central role in expressing three-point functions. We also define SU(2) R matrices using Z 1 as

(Semi-)shortening conditions for three-point functions
Let us now consider a three-point function of three quasi superfields Φ I i (z i ) for i = 1, 2, 3. We denote their conformal dimension and U(1) r charge by (q i +q i ) and (q i −q i ) respectively. The subscript I i expresses SU(2) R and SL(2, C) indices collectively. A general three-point function Φ I 1 (z 1 )Φ I 2 (z 2 )Φ I 3 (z 3 ) is written as where T J 1 I 1 and T J 2 I 2 are some functions composed of (2.9) and (2.12) in the representation of SU(2) R × SL(2, C) specified by I 1 and I 2 . On the other hand, the function H(Z 3 ) satisfies following homogeneity property where a andā are fixed by The conformal dimension of (Xαα, Θ iα ,Θα i ) is (1, 1 2 , 1 2 ), and the U(1)r charge is (0, 1 2 , − 1 2 ), respectively.

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Therefore H(Z 3 ) has the conformal dimension q 3 +q 3 − (q 1 +q 1 ) − (q 2 +q 2 ) and the U(1) r charge (q 3 − q 3 +q 1 − q 1 +q 2 − q 2 ). The function H(Z 3 ) is not fully determined by the global superconformal symmetry. When some of the Φ I i (z i ) correspond to a (semi-)short multiplet, their shortening conditions restrict the form of H(Z 3 ). For example in [20,51,52], H(Z 3 ) is determined in such cases, up to an overall constant. The formalism (2.20) is very useful when we consider the (semi-)shortening conditions such as (2.8). Since the prefactor in (2.20) , it trivially satisfies the (semi-)shortening conditions. This implies that the (semi-)shortening conditions only constrain the function H(Z 3 ). Therefore, hereafter, we focus on H(Z 3 ).
It is easy to find that D 1 ,D 1 , D 2 andD 2 act on H(Z 3 ) as where derivativesD jβ , D β j , Q α i andQα i are defined respectively as Moreover, quadratic derivatives such as D Therefore, the (semi-)shortening conditions are now translated into partial differential equations of H(Z 3 ) with respect to Z 3 . While the (semi-)shortening conditions of the first and second superfields, Φ I 1 (z 1 ) and Φ I 2 (z 2 ), are easily expressed as partial differential equations for H(Z 3 ), it is not straightforward to translate the conditions for the third superfield Φ I 3 (z 3 ) into a similar equation for H(Z 3 ). To consider the (semi-)shortening conditions of the third superfield, we change the variable from Z 3 to Z 2 . 10 Indeed, using the cyclicity of z 1 , z 2 and z 3 , the correlation function (2.20) is also expressed as (2.27) 10 Here, we can also use Z1 instead of Z2.

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for some function G(Z 2 ). The action of D i 3α andD 3αi on the G(Z 2 ) are given by where the derivatives are now defined by As shown in [52,55], Z 3 and Z 2 are related as Using these relations, we see that the function It is important to consider the third superfield conditions since it is insufficient to fix the function H(Z 3 ) only considering the first and second superfields of the (semi-)shortening conditions in section 4.2.
In the following sections, we will use the above formalism and techniques to study the three-point functions of Schur multiplets.

B R 1 × B R 2 fusion
In this section, we study the most general expressions for three-point functions of two half-BPS Schur multiplets B R and an arbitrary Schur multiplet O I . 11 Our result is particularly consistent with the fusion rules for B R 1 × B R 2 which were first obtained in [53].
The general expression for the three-point function B R 1 B R 2 O I is given by where L (i 1 ···i 2R ) (z) is the superfield of B R multiplet, and the parentheses denote the total symmetrization of the indices. Hereafter, we will often omit the parentheses with the understanding that the indices associated with the same Latin and Greek alphabet letters are always totally symmetrized.

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Each of the B R 1 and B R 2 multiplets satisfies two shortening conditions as shown in table 1. As we have mentioned in the previous section, the shortening conditions are translated into differential equations for H(Z 3 ). For the two B R multiplets, the differential equations are written as It is easy to solve (3.3) and (3.4), since these are merely first-order linear equations for Θ orΘ. In contrast, (3.2) and (3.5) contain both X and Θ (orΘ) derivations and therefore are more complicated. However, if we use theZ 3 , the two equations (3.2) and (3.5) become simpler, because D α i andQα i are expressed in terms ofZ 3 as Indeed, the most general solution to (3.3) and (3.4) is simply expressed in terms of Z 3 as while that of (3.2) and (3.5) is written in terms ofZ 3 as Here, we use the short-hand notation The above two expressions, (3.8) and (3.7), must be equal under the relation (2.16). Therefore, our strategy is to rewrite (3.8) in terms of Z 3 by using (2.16) and restrict the parameters in the expression to be consistent with (3.7). This gives us the most general solution to the equations (3.2)-(3.5). After solving (3.2)-(3.5), we have to check it also satisfies the (semi-)shortening conditions of the third superfield O I (z 3 ). In this process, we relate H(Z 3 ) to G(Z 2 ) using (2.32) and see if the G(Z 2 ) satisfies the differential equations corresponding to the third set of (semi-) shortening conditions. Below, we apply this strategy to evaluate the most general expression for the three- Let us first consider the case of O I in the B R 3 multiplet. In this case, the function H(Z 3 ) has dimension −2R := 2R 3 − 2R 1 − 2R 2 and vanishing U(1) r charge. Note here that R 1 , R 2 , and R 3 are constrained by the inequalities 0 In other words, R 3 must be such that Moreover, since the primary of B R is a scalar, so is the H(Z 3 ). Therefore, the function H(Z 3 ) only carries SU(2) R indices. The most general ansatz for H(Z 3 ) is then written as 12 where R ∈ Z ≥0 , and k 1 · · · k 2R 3 are the SU(2) R indices associated with B R 3 . As mentioned above, indices associated with the same Latin and Greek alphabet letters, such as l 1 · · · l 2R 1 , are totally symmetrized. On the other hand, the same function should also be written in terms ofZ 3 . The most general expression in terms ofZ 3 is given by For the above two expressions to be consistent, the coefficients A, B and C have to satisfy some conditions. To identify the conditions, we change the variables fromZ 3 to Z 3 in (3.12) by using (2.16). Using the Fierz identities summarized in appendix A, we see that the conditions are for an arbitrary constant A. Up to an overall constant, the function H(Z 3 ) is written as (3.14) This is the most general expression for H(Z 3 ) satisfying (3.2)-(3.5).
Although it satisfies the shortening conditions of the B R 1 and B R 2 multiplets, it is non-trivial whether the expression (3.14) satisfies the shortening conditions of the third multiplet B R 3 . To check the shortening conditions for B R 3 , let us relate the H(Z 3 ) to G(Z 2 ) using (2.32). Indeed, as reviewed above, the correlation function (3.1) is also written as
In the rest of this paper, we omit the subscript 3 of X 3 , Θ 3 andΘ 3 in the expression for H(Z 3 ). Similarly, we omit the subscript 2 of X 2 , Θ 2 andΘ 2 in the expression for G(Z 2 ).
Let us turn to the case of O I in theD R(j,0) multiplet. 13 In this case, the function H(Z 3 ) has dimension −2R + j + 1 and U(1) r charge −j − 1. Since the highest possible degree ofΘ in H(Z 3 ) is two (see (3.7)), the only possible value of j is j = 0, and therefore the , we see that the most general expression for such H(Z 3 ) is given by where k 1 · · · k 2R 3 are the SU(2) R indices forD R 3 (0,0) . As mentioned at the end of the previous subsection, (X, Θ,Θ) stands for (X 3 , Θ 3 ,Θ 3 ) here. However, when we change the variables from Z 3 toZ 3 , this expression cannot be written in the form of (3.8). 14 This means that there are no solutions to (3.2)-(3.5). Therefore, theD R(j,0) multiplet does not appear in the B R 1 × B R 2 selection rule. Note that its conjugate implies that the D R(0,) multiplet also does not appear in the B R 1 × B R 2 fusion. By using the same argument, we can extend our results to that H(Z 3 ) for a non-vanishing correlation function Let us finally consider the case of O I in the C R 3 (j,) . By using the similar argument in previous section 3.2, = j is necessary for the three-point function to be non-vanishing. Therefore the function H(Z 3 ) has dimension 2 − 2R + 2j and U(1) r neutral, where we 13 Note here that the result for O I in the D R(0,j) multiplet follows from this case by CPT. 14 In particular,Θ l 1Θ l 2 cannot be mapped toΘm 1Θ m 2 .

The most general solution to (3.3) and (3.4) is written as
where k i and (β i ,β i ) are the SU(2) R and SL(2, C) indices associated with C R 3 (j,j) . On the other hand, the most general solution to (3.2) and (3.5) is written as For the above two expressions to be consistent, the coefficients have to satisfy some conditions. Unless R = 0, the conditions are On the other hand, for R = 0 or equivalently R 3 = R 1 + R 2 , all the coefficients have to vanish. Therefore, the three-point function Next, we consider the semi-shortening conditions for C R 3 (j,) . For that purpose, we relate H(Z 3 ) to G(Z 2 ) via (2.32). Indeed, the three-point function (3.1) can be rewritten as ( 3.22) and the explicit form of G(Z 2 ) becomes

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In terms of G(Z 2 ), the semi-shortening conditions for C R 3 (j,j) are written as for j > 0 and for j = 0. It is straightforward to check (3.23) satisfies these conditions. In summary, the function H(Z 3 ) in B R 1 B R 2 C R 3 (j,j) is given by, up to an overall constant for any integer or half-integer j ≥ 0 and

Selection rule
In the above subsections, we have computed the most general expression for non-vanishing three-point functions of the form B R 1 B R 2 O . From these results, we see that the selection rules for two B R multiplets are written as up to non-Schur multiplets. This is particularly consistent with eq. (3.44) of [53]. Especially, for R 1 = R 2 = 1, the selection rule is written as which is consistent with the harmonic superspace analysis in [61].
In this section, we turn to the selection rules for C 0(0,0) × O Schur for an arbitrary Schur multiplet O Schur . These selection rules are important in studying the corresponding twodimensional chiral algebra, since the highest weight component of the SU(2) R current operator in the stress-tensor multiplet C 0(0,0) is mapped to the Virasoro stress-tensor operator in the chiral algebra [17]. To derive the selection rules, we compute the three-point functions of the form C 0(0,0) O I 1 O I 2 for two Schur multiplets O I 1 and O I 2 .

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Recall here that the stress-tensor multiplet C 0(0,0) has two semi-shortening conditions The most general solution to these two equations are respectively written as For the above two expressions to be consistent, the functions f, g, h,f ,g, andh have to satisfy some conditions. Moreover, they are also constrained by the (semi-)shortening conditions associated with O I 1 and O I 2 . Below, we solve all these conditions to find general expressions for H(Z 3 ). Since the concrete calculations are highly involved, we here write the results and, details of the computations are in appendices B, C, and D.
Let us first consider the three-point function C 0(0,0) B R O I . We denote by J (z) the superfield of the stress-tensor multiplet C 0(0,0) . The three-point function is then written as The (semi-)shortening conditions for C 0(0,0) and B R are encoded in (3.4), (3.5), (4.1), and (4.2). The three-point function is consistent with these four conditions only when O I is B R , C R(j,j) , or C R−1(j,j) . Up to an overall constant, the expressions for H(Z 3 ) in these three cases are written as C R(j,j) : where k i and (β i ,β i ) are respectively the SU(2) R and SL(2, C) indices associated with the third multiplet. The derivations of these functions are given in appendix B. We see that these are also consistent with the (semi-)shortening conditions for the third Schur multiplet. Indeed, using (2.32), we see that the function G(Z 2 ) corresponding JHEP04(2019)060 to the above H(Z 3 ) is written as C R(j,j) : Here n i and (δ i ,δ i ) are respectively the SU(2) R and SL(2, C) indices associated with the third multiplet O I . These equations are all consistent with the (semi-)shortening conditions for the third multiplet. Let us briefly comment on the case of O I in C 0(0,0) . When we assume that there is only one stress tensor in the theory, the corresponding three-point function C 0(0,0) B 1 C 0(0,0) has to be symmetric under the action of Z 2 exchanging the first and the third multiplets. This Z 2 symmetry implies that the function G(Z 2 ) is invariant under (X 2 , Θ 2 ,Θ 2 ) ↔ (−X 2 , −Θ 2 , −Θ 2 ). However, the expression (4.12) is not invariant under this Z 2 action. Therefore, in an SCFT with unique stress tensor multiplet, the three-point function C 0(0,0) B 1 C 0(0,0) must vanish [52].
Before closing this subsection, let us also make a quick comment on the correlation function C 0(0,0) B R C R−1(j,j) . In CFTs, any correlation function of conformal descendant fields is obtained by differentiating the correlation function of the conformal primary fields. This particularly implies that, when a correlation function of conformal primary fields vanishes, the corresponding descendant correlators also vanish. This, however, is not the case for superconformal descendants in SCFTs. Indeed, when we set all Grassmann variables, θ 1,2,3 ,θ 1,2,3 , to zero in (4.9), the correlation function vanishes. This shows that the correlator of the three superconformal primary fields in C 0(0,0) , B R , and C R−1(j,j) vanishes, while there are non-vanishing correlators involving superconformal descendants. This is a common feature of SCFTs [62].

4.2
C 0(0,0)DR(j,0) O I Let us next consider C 0(0,0)DR(j,0) O I . We denote byN (j 1 ···j 2R )(α 1 ···α 2j ) (z) the superfield of aD R(j,0) multiplet. The three-point function is written as 2 ) multiplets. Moreover, considering the (semi-)shortening conditions for each of these third multiplets, we find that the only possible third multiplets O I which can have a non-vanishing H(Z 3 ) are D R(0,j) , C R+ 1 2 (j 1 ,j+j 1 + 1 2 ) , and C R− 1 2 (j 1 ,j+j 1 + 1 2 ) . The explicit expressions for H(Z 3 ) for these three cases are written, up to an overall constant, as where j 1 ≥ 0 is an integer or a half-integer, and k i and (β i ,β i ) are respectively the SU(2) R and SL(2, C) indices for the third multiplet. Note that the first expression for the case of O I in the D R(0,j) multiplet is only for j = 1/2. In the case j = 1/2, the function H(Z 3 ) has two independent terms as where A and B are arbitrary constants. For the detail of derivations of (4.16) and (4.17), see appendix C.
Note here that the second and third lines of (4.16) are proportional toΘ, which means that the three-point functions of the superconformal primaries vanish. This can also be seen from the fact that the sum of the U(1) r charges of the superconformal primaries does not vanish. study here is O I = C R (j 2 , 2 ) , in which case the three-point function is written as (z) is the superfield in the C R(j,) multiplet. In this case, the H(Z 3 ) has to satisfy the semi-shortening conditions (4.1) and (4.2) associated with C 0(0,0) and the semi-shortening conditions for the other two C R(j 1 , 1 ) multiplets. The semi-shortening conditions for the second multiplet, C R(j 1 , 1 ) , are written as while the semi-shortening conditions for the third multiplet, C R (j 2 , 2 ) , are similarly expressed in terms of G(Z 2 ).
We have found that a non-vanishing H(Z 3 ) in (4.18) satisfying all these semishortening conditions is possible only for the following two types of correlator: 15 C 0(0,0) C R(j+ 1 ,j) C R(j+ 2 ,j+ 1 + 2 ) , (4.21) up to charge conjugation, where 1 and 2 are non-negative (half-)integers. Note that the function H(Z 3 ) is U(1) r neutral for all these cases. Since general solutions for H(Z 3 ) in these two cases are highly involved, it is beyond the scope of this paper to identify the most general expression for the allowed H(Z 3 ). However, we find a special solution for each of the above two types of correlators, 16 which is sufficient to identify the selection rule.

Selection rules
We here write down the selection rules for C 0(0,0) × O Schur read off from the threepoint functions we computed above. Note that all the following rules are only up to non-Schur multiplets. 17 It has very recently been shown in [63] that the square of the OPE coefficient of C 0(0,0) × C n−1( n−1 2 , n−1 2 ) ⊃ C n( n 2 , n 2 ) for n ∈ N is proportional to n i=1 (c−ci) with cn ≡ n(6n+5) 6(2n+3) . This implies that the constant prefactor of H(Z3) for this channel vanishes when c = ci for i = 1, 2, · · · , n.
• For R > 1, the selection rule is (4.31) • For R = 1, because of the Z 2 -symmetry, the selection rule is which is consistent with [51].
• For R = 1 2 , the rule is The selection rules for C 0(0,0) ×D R(j,0) are written as follows.
• For R > 0, (4.34) • For R = 0, Note here that, for the C R,(j,) type multiplets on the right-hand sides, the corresponding three-point functions of the superconformal primaries vanishes. This reflects the fact that the sum of the U(1) r charges of the primaries in three-point function is non-vanishing. However, the sum of the U(1) r charges of the Schur operators in the same multiplets vanishes, which implies that the three-point functions of the Schur operators can be nontrivial. Note also that the selection rules for C 0(0,0) × D R(0,j) are obtained by taking the charge conjugate of (4.34) and (4.35).
• For j − = 1 ≥ 1 2 , R > 1 2 , (4.38) • For = j > 0, R > 1, When j is an integer, the stress-tensor multiplet C 0(0,0) in the last term on the righthand side must be excluded by the Z 2 -symmetry, under the assumption of uniqueness of the stress tensor.

Conclusions and discussions
In  [53]. We emphasize that our analysis relies only on the shortening conditions for Schur multiplets and therefore does not depend on any detail of four-dimensional N = 2 SCFTs. Let us here discuss an interesting constraint appearing in the selection rules for C 0(0,0) × O Schur . Suppose that O and O are two Schur multiplets so that O appears in the OPE of C 0(0,0) × O, i.e., C 0(0,0) × O ⊃ O . We denote by R (s) and R (s) the SU(2) R charges of the Schur operators in O and O , respectively. We also denote respectively by h and h the holomorphic dimensions of the two-dimensional operators associated with the Schur operators in O and O . See table 2 for the relation between the two-dimensional holomorphic dimension and four-dimensional quantum numbers of operators. Now, we see that our selection rules imply In the 4d/2d correspondence of [17], this means that the SU(2) R charge of the fourdimensional ancestor of a two-dimensional operator is always smaller than or equal to those of its Virasoro descendants. Since the SU(2) R symmetry is broken in the associated chiral algebra [17], this relation between the SU(2) R charge and the holomorphic dimension is surprising. 18 See also [64] for a remarkable discussion on reconstructing the SU(2) R -filtration of the chiral algebra.

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Multiplet Schur operator Table 2. Schur operators in Schur multiplets with their U(1) r and SU(2) R charges. Here Φ is the superconformal primary field of the multiplet. The rightmost column shows the holomorphic dimension of the corresponding operator in the two-dimensional chiral algebra.
Another interesting observation is that, in some of the OPE channels allowed by our selection rules, the three-point function of the superconformal primaries vanishes even though those of their descendants do not. As mentioned already, this is a common feature of SCFTs [62]. Indeed, the vanishing of the three-point function of the primaries reflects the fact that the sum of their U(1) r charges is non-vanishing. Therefore, our selection rules for Schur multiplets do not imply the non-vanishing of the three-point function of the corresponding superconformal primaries.
On the other hand, when we focus on the Schur operator in each Schur multiplet, we see that the sum of their U(1) r charges vanishes whenever the corresponding three Schur multiplets are allowed by the selection rules. This seems to suggest that the threepoint functions of Schur operators are always non-vanishing whenever the corresponding Schur multiplets have non-vanishing three-point functions. 19 This observation leads us to a conjecture on BPS selection rules of general Schur multiplets. Suppose that a Schur multiplet O Schur i for i = 1, 2, and 3. Then we conjecture that the following two conditions are satisfied: Note that these conditions are necessary for the three-point functions of the Schur operators to be non-vanishing. Recognizing (5.1) and (5.2) as principle of selection rule related to four-dimensional N = 2 SCFT whose stress tensor is unique, we recover our all selection rules in section 3.4 and 4.4. We leave the detailed study of this conjecture to future work.

A Fierz identities
In this appendix, we summarize useful identities for Grassmann variables Θ iα andΘα i , which we call Fierz identities. We first introduce the following variables: From the nilpotent structure of Θ andΘ, we see that the following Fierz identities hold 20 Moreover, by using the variables m l , we can expand powers ofX 2 as [20] We can also derive 21 The following Fierz identities are derived by using the Mathematica package grassmann.m [65]: 21M i j := Θ iαX ααΘαj ,m l := trM l .

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In this appendix, we describe a derivation of the most general expression for the correlation function Let us first consider the case of O I =D R (j,0) . In this case, H(Z 3 ) has dimension 2(R − R) + j − 1 and U(1) r charge −j − 1. Since the absolute value of the U(1) r charge is at most one, the spin j must be zero. Then the only possibility for R is R = R − 1, which implies However, this is not consistent with (4.3) unless A = 0, and therefore the correlation function C 0(0,0) B RDR (j,0) has to vanish. From charge conjugation, we see that C 0(0,0) B R D R (0,j) also vanishes.

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In this case, the only possible candidate for H(Z 3 ) is This is not consistent with (4.4) unless A = 0.
In this case, the only possible candidate is which is not consistent with (3.5) unless A = 0.
In this case, there are three possible terms in H(Z 3 ); However, this is not consistent with (4.4) unless A = B = C = 0.
The constraint (3.4) is then expressed as which implies B = 0. Moreover, for (B.5) to be consistent with (4.4), we should set C = −4i(2j)A. Indeed, with this condition imposed, the function (B.5) can be rewritten as and therefore satisfies (4.2). We also see that this expression satisfies (3.5).

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In this case, the possible H(Z 3 ) for j > 0 is of the form For this to be consistent with (3.5) and (4.2), we must impose B = −2jA and C = −4ijA.
On the other hand, for j = 0, H(Z 3 ) is given by Using the Fierz identity (A.6), we see that this satisfies all the (semi-)shortening conditions.
In this appendix, we describe the details of our computations of C 0(0,0)DR (j,0) O I . We solve the equations (4.1), (4.2), (3.4), and (4.14)/(4.15) together with the (semi-)shortening conditions associated with the third Schur multiplet O I . We note here that the most general solution to (4.14) or (4.15) is written as are totally symmetric, which is implicit in the first term in the bracket in (C.1). 22 The function H(Z 3 ) has dimension ∆ 3 − 2R − j − 3 and U(1) r charge r 3 − j − 1, where ∆ 3 and r 3 are the dimension and the U(1) r charge of the third multiplet O I respectively. The case of O I = B R has already been studied in section B.1. Moreover, H(Z 3 ) turns out to vanish for O I =D R (j ,0) . Indeed, for O I =D R (j ,0) , the U(1) r charge of H(Z 3 ) is −j − j − 2, which is not possible since the U(1) r charge is bounded from below by −1. Therefore,D R (j ,0) dose not appear in the OPE of C 0(0,0) ×D R(j,0) . In the rest of this appendix, we consider the remaining cases O I = D R (0,) and C R (j , ) .
Up to charge conjugation, the possible values of the U(1) r charge is 0, 1 2 , and 1. Therefore 22 For example in the case of j = 1, f7 is defined with some function f by f using the function f7 we have just defined, and SU(2)R indices are symmetrized similarly.
In this case, the only possible H(Z 3 ) is of the form 23 For this to be consistent with (4.1) and (4.14), we must set B = − 4i j+1 A , C = − 2i(2j−1) j+1 A. Then the function G(Z 2 ) given by (2.32) is now written as However, we see that this does not satisfy the shortening condition (C.2) unless A = 0.
In this case, the possible solution is given by which is not consistent with (4.14) unless A = 0. 23 Recall here thatΘαγ 2j =Θα i ikΘγ 2j k .

In this case, H(Z 3 ) is given by
It is straightforward to show that this satisfies (4.2). Let us next consider the semishortening condition for the second multiplet. For j > 0, the condition (4.14) reads On the other hand, for j = 0, the condition (4.15) implies C = −4iA. Note that the term proportional to B in (C.10) does not exist for j = 0.
To solve the conditions associated with the third multiplet, let us relate the above H(Z 3 ) to G(Z 2 ) via (2.32). The result is generally written as where C is a free parameter that is present only in the case of j > 0. We see that this expression satisfies the shortening condition (C.2) for arbitrary C. On the other hand, the semi-shortening condition implies C = 0 unless j = 1 2 . For j = 1 2 , the semi-shortening condition does not restrict the value of C.
Note that this expression vanishes if j = 0. Therefore there is no non-trivial solution for j = 0. For j > 0, the above expression is consistent with (4.2) and (4.14) if and only if C = A−B 2 and D = i(A + B). With these conditions imposed, the function G(Z 2 ) given by (2.32) is written as It is then straightforward to see that A = B = 0 is necessary for this to be consistent with (C.2) and (C.3). Therefore, no non-trivial solution exists in this case.

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We see that this is of the form of (4.4) and (C.1) and therefore satisfies (4.14)/(4.15) and (4.2). On the other hand, the function G(Z 2 ) is now evaluated as We see that this expression satisfies the semi-shortening conditions associated with the third multiplet for any j ≥ 0, j 1 ≥ 0 and R ≥ 0.
The only possible H(Z 3 ) is of the form In the case of j > 0, the constraints (4.2) and (4.14) imply that B = −4iA + C and D = −C. On the other hand, for j = 0, the constraints (4.2) and (4.14) imply B = −4iA.

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for some i ≥ 0. When we change the variables from (X, Θ,Θ) to (X, Θ,Θ), it gives rise to a term proportional to one of the following 10) which is prohibited by (4.2) since the most general solution to (4.2) is written as (4.4). 25 Therefore we must impose λ k = 0 for (D.7) to satisfy the semi-shortening conditions. Since k is arbitrary, this means that g = 0 as a function of X. Therefore, there is no non-trivial solution to the semi-shortening conditions in the case of C 0(0,0) C R(j+ 2 ,j+ 1 + 2 ) C R+1(j+ 1 ,j) for j, 1 ≥ 0 and 2 > 0.
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