On level crossing in conformal field theories

We study the properties of operators in a unitary conformal field theory whose scaling dimensions approach each other for some values of the parameters and satisfy von Neumann-Wigner non-crossing rule. We argue that the scaling dimensions of such operators and their OPE coefficients have a universal scaling behavior in the vicinity of the crossing point. We demonstrate that the obtained relations are in a good agreement with the known examples of the level-crossing phenomenon in maximally supersymmetric $\mathcal N=4$ Yang-Mills theory, three-dimensional conformal field theories and QCD.


Introduction
Let us consider a unitary conformal field theory (CFT) and examine the spectrum of scaling dimensions of conformal primary operators. The scaling dimensions ∆ i depend on various parameters of CFT such as coupling constants, Casimirs of internal symmetry group etc. In a close analogy with gauge theories, we assume that CFT depends on a small parameter 1/N and the scaling dimensions admit an expansion in powers of 1/N 2 , e.g. (1.1) Here the leading term ∆ i is an eigenvalue of the dilatation operator in the planar limit, the subleading terms can be obtained by diagonalizing the dilatation operator perturbatively in powers of 1/N.
In this note we ask the question what happens when the scaling dimensions of two operators collide in the planar limit for some value of the coupling constant, ∆ 2 . This occurs in particular in maximally supersymmetric N = 4 super Yang-Mills theory (SYM) for Konishi and double-trace operators (defined in (4.1) below). The scaling dimension of the double-trace operator is protected from corrections in the planar limit, whereas the scaling dimension of the Konishi operator is an increasing function of 't Hooft coupling constant that starts at 2 + O(λ) at weak coupling and increases as 2λ 1/4 at strong coupling [1]. As a consequence, the two levels cross each other at finite λ. We shall return to this example below.
It is well-known that, in the radial quantization, the dilatation operator in D−dimensional Euclidean CFT coincides with the Hamiltonian generating time translations on the cylinder R × S D−1 . In a unitary CFT, it is a hermitian operator with respect to the scalar product given by two-point correlation function of O i (x) on the cylinder and, therefore, its eigenvalues ∆ i have to satisfy von Neumann-Wigner non-crossing rule [2] stating that the levels of the dilatation operator with the same symmetry cannot cross.
How can we reconcile the non-crossing rule with the fact that the functions ∆ i providing the leading (planar) correction to (1.1) can cross each other? The situation here is very similar to that in quantum mechanics -the perturbative expansion (1.1) is well-defined if the energy level separation ∆ 1 − ∆ 2 is much larger compared to their 'interaction energy' defined by the leading nonplanar O(1/N) correction to the dilatation operator. In the opposite limit, for the perturbative expansion (1.1) becomes singular due to small denominators and needs to be resummed to all orders in 1/N. As we show below, the resummed expressions for ∆ i indeed respect the non-crossing rule.

Resummed scaling dimensions
Let us consider a pair of conformal operators O 1 and O 2 whose scaling dimensions depend on the coupling constant and admit the 1/N expansion (1.1). For the sake of simplicity we take O i (x) to be scalar operators normalized in such a way that their two-point functions are given by (2.1) We assume that for N → ∞ their scaling dimensions intersect at some value of the coupling constant, ∆ 2 , and remain separated from the rest of the spectrum of ∆ i 's by a gap that remains finite for N → ∞. We show in this section that for N large but finite, the scaling dimensions ∆ 1 and ∆ 2 do not intersect as one varies the coupling constant, the minimal distance between the two curves scales as O(1/N). 1 Let O 1 and O 2 be the eigenstates of the dilatation operator in the planar limit. The conformal symmetry implies that away from the crossing point, for ∆ can not be different from zero. For N large but finite it can receive corrections suppressed by powers of 1/N Requiring O + (x)O − (0) = 0 we find with a help of (2.3), (2.1) and (2.2) that, in the vicinity of the crossing point, the conformal symmetry restricts the possible form of the function ϕ( Here in the second relation we took into account (1.2) and neglected terms proportional to ǫ 2 ∼ 1/N 2 . Then, going to N → ∞ limit on the both sides of (2.4) and taking into account where γ does not depend on N and satisfies We can use this equation to express the coefficients c 1 and c 2 in terms of γ and obtain from (2.3) the conformal operators O ± that take into account the leading nonplanar correction. 2 To determine the scaling dimensions of the operators (2.3), we examine their two-point correlation functions where in the last relation we applied (2.6). Going through similar calculation of ∆ + we find The two solutions to (2.6) give rise to the same operators O ± (up to an overall normalization factor).
with ǫ = ∆ 1 − ∆ 2 . We recall that this relation holds up to corrections suppressed by powers of 1/N. The following comments are in order. An attentive reader will likely notice that ∆ ± coincide with energies of a two-level system with a Hamiltonian (2.10) Comparison with (1.1) shows that ∆ (p) ± ∼ γ 2p /ǫ 2p−1 and, therefore, the expansion (1.1) becomes singular for ǫ = O(1/N), in agreement with our expectations. The relation (2.8) takes into account an infinite class of the leading corrections O (1/(ǫ 2p−1 N 2p )) to all orders in 1/N and it remains finite for ǫ → 0. At the crossing point, for ǫ = 0, we find from (2.8) that ∆ ± = ∆ 1 ± γ/N, so that the leading nonplanar correction to the resummed scaling dimensions scales as O(1/N) (and not O(1/N 2 ) as one would expect from (1.1)). 4 Let us assume that the scaling dimensions ∆ 1 and ∆ 2 are continuous functions of the coupling constant g, intersecting in the planar limit at g = 0. Then, the relation (2.8) defines two functions ∆ ± (g) that satisfy the non-crossing rule. Namely, the two levels ∆ + (g) and ∆ − (g) approach each other for g = 0 but remain separated by a finite gap (see Figure 1) We recall that the parameter γ defines the leading nonplanar correction to the two-point correlation function (2.2) and (2.5). As can be seen from Fig. 1, the level crossing leads to a change of behavior of the scaling dimensions in the vicinity of the crossing point. Away from the crossing point, the resummed scaling dimensions approach their values in the planar limit, e.g. ∆ + ≈ ∆ 1 at large negative g goes into ∆ + ≈ ∆ 2 at large positive g.
3 I would like to thank Sergey Frolov for suggesting this interpretation. 4 Analogous phenomenon has been previously observed for critical dimensions of composite operators in the nonlinear O(N ) sigma-model [3]. At large N their expansion runs in powers of 1/N but the leading nonplanar correction scales as O(1/ √ N ) due to the level crossing.

Resummed OPE coefficients
In the previous section, we defined the conformal primary operators O ± in the transition region (1.2) and determined their scaling dimensions (2.8). Let us examine the properties of the OPE coefficients C φφO ± defining the correlation function φφO ± involving scalar conformal operator φ with the scaling dimension ∆ φ separated from ∆ ± by a finite gap, Let us consider a four-point function of the operators φ and decompose it over the conformal blocks describing the contribution of different conformal multiplets . For x 12 → 0, or equivalently u → 0 and v → 1, the contribution of the operators O ± to (3.1) takes the form where dots denote terms suppressed by powers of u and 1 − v.
The OPE coefficients in (3.2) are given by C 2 Replacing O ± with their explicit expressions (2.3), we find using (2.6) where C φφO i are the OPE coefficients for the operators O i in the planar limit and for ǫ > 0. The relation (3.3) defines two smooth functions of ǫ that cross each other for ǫ = 0 and take the values C 2 φφO ± = (C φφO 1 ± C φφO 2 ) 2 /2 at the crossing point ǫ = 0. The four-point correlation function (3.1) is a well-defined function of the parameters of the underlying CFT. In particular, at large N it admits an expansion that runs in powers of 1/N 2 . It is natural to require that each term of this expansion should be finite for ǫ → 0. As we will see in the moment, this requirement leads to nontrivial consequences for the OPE coefficients C φφO i .
The conformal blocks entering the right-hand side of (3.1) depend on the scaling dimensions of the operators φ i . Since ∆ φ i are well-separated from each other in the transition region (1.2), the corresponding conformal blocks F φ i automatically satisfy the regularity condition mentioned above. This is not the case however for the conformal blocks F O ± . According to (2.10) and (3.4), the large N expansion of ∆ ± and c 1 is singular in the region (1.2) and, as a consequence, the coefficients of the expansion of (3.2) in powers of 1/N contain poles in ǫ. They cancel however in the sum F O + + F O − that remains finite for ǫ → 0. Namely, Notice that the second term on the right-hand side is finite for ǫ → 0 but it scales as O(1/N).
It is straightforward to verify that the subleading corrections to (3.5) involve both even and odd powers of 1/N. This seems to be in contradiction with the requirement that large N expansion of the correlation function (3.1) should run in even powers of 1/N. We can eliminate all terms with odd powers of 1/N by imposing the additional condition on the OPE coefficients 5 This relation implies that if the scaling dimensions of two operators O 1 and O 2 approach each other and satisfy (1.2), their OPE coefficients in the planar limit have to differ by a large factor of N. Combining together (3.6) and (3.3), we obtain the following expression for the resummed OPE coefficients where ǫ = ∆ 1 − ∆ 2 and C φφO 1 are the OPE coefficients in the planar limit. Let us examine the expansion of (3.7) at large N and ǫ > 0 Notice that the leading correction to C φφO − contains a pole at ǫ = 0. Its appearance is an arfifact of the large N expansion. As follows from (3.7), the resummed expression for C φφO − is finite and it takes the value C φφO 1 / √ 2 for ǫ = 0. Away from the crossing point, for |ǫN|≫ 1, the structure constants C φφO ± coincide with those in the planar limit, C φφO 1 and C φφO 2 .
The dependence of the OPE coefficients (C φφO ± /C φφO 1 ) 2 on the coupling constant in the vicinity of the crossing point is shown in Fig. 1. We demonstrate in the next section that the relations (3.7) and (3.8) are in a good agreement with the known properties of threeand four-dimensional CFT's.
We recall that the relation (3.7) was obtained from the requirement for the four-point correlation function (3.1) to have a regular 1/N 2 expansion as ǫ → 0. We substitute (3.7) into (3.5) and verify that this condition is indeed verified A distinguished feature of this relation as compared with the analogous asymptotic behaviour of the conformal blocks is the appearance of terms enhanced by powers of ln u. For ǫ = 0 the 1/N−expansion on the right-hand side of (3.9) can be resummed leading to We recognize the two terms on the right-hand side as describing the leading u → 0 asymptotics of conformal blocks corresponding to two conformal primary operators with the scaling dimensions ∆ 1 ± γ/N.

Examples of level-crossing
In this section, we review the known examples of the level-crossing phenomenon in three-and four-dimensional CFT's and make a comparison with the results obtained in the previous sections.

Level-crossing in N = 4 SYM
Our first example concerns the mixing of the Konishi and double-trace operators in fourdimensional N = 4 SYM theory. As was mentioned in the Introduction, the scaling dimensions of these operators collide in the planar limit at finite value of the coupling constant. In planar N = 4 SYM, the Konishi and double-trace operators take the form where Φ I (with I = 1, . . . , 6) are real scalar fields and tr[Φ (I 1 . . . Φ I J ) ] is a symmetric traceless SO(6) tensor. The normalization factors on the right-hand side of (4.1) were introduced for the two-point correlation functions of the operators to scale as O(N 0 ) at large N. At Born level, for zero coupling constant, the scaling dimensions of the operators (4.1) are ∆ K = 2 and ∆ D = 2J, respectively. At strong coupling, the scaling dimensions can be computed using the AdS/CFT correspondence, see e.g. [4,5] where dots denote terms suppressed by powers of 1/N 2 and λ −1/2 . We observe that, at strong coupling, for J ≈ λ 1/4 the two levels cross each other.
To establish the connection with the results of the previous sections, we examine threepoint correlation functions of the operators (4.1) and half-BPS operators where Y I is an auxiliary six-dimensional null vector, Y 2 I = 0. This operator carries the R−charge J and it scaling dimension is protected from quantum corrections, ∆ O J = J. The form of the three-point functions is fixed by conformal symmetry The correlation function in the second line of (4.4) factors out in the planar limit into the product of two-point correlation functions of half-BPS operators (4.3) leading to The OPE coefficient C JJK was computed at strong coupling in Ref. [6]. It was found that C JJK develops a pole at J = ∆ K /2 ≈ λ 1/4 6 where M = (J + 1)(J + 2) 2 (J + 3)/12 ≈ J 4 /12 is the dimension of the J−symmetric traceless representation of the SO(6). As was argued in [6], the appearance of a pole in Substituting these expression into (3.8), we can determine the subleading terms in (4.5) and (4.6). Moreover, we can apply (2.8) and (3.7) to obtain the resummed expressions for the scaling dimensions and the OPE coefficients in the transition region |J − λ 1/4 |= O(1/N) where the subscripts +/− correspond to the double-trace and Konishi operators, respectively. At the crossing point, for J ≈ λ 1/4 , the level splitting is This expression is in agreement with the one given in Ref. [6] (see the latest version). The structure constants (4.8) are finite in the transition region and do not require the additional renormalization of operators advocated in [6].
The dependence of ∆ ± on the coupling constant in the vicinity of λ 1/4 ≈ J follows the same pattern as shown in Fig. 1. In particular, examining ∆ − as a function of the coupling constant, we find that it grows for λ 1/4 ≪ J as the scaling dimension of the Konishi operator ∆ K and, then, approaches the scaling dimension of the double-trace operator ∆ D for λ 1/4 ≫ J. 7 As a consequence, it satisfies the relation ∆ − < ∆ D < 2J, so that the correlation function O J (1)O J (2)O − (0) can not be extremal for finite N (see footnote 6). 8 As the second example, we consider the mixing of leading-twist operators carrying a nonzero Lorentz spin S. The scaling dimension of twist-two operators grows at large spin as ∆ 2 = 2 + S + γ S , where γ S = 2Γ cusp (λ) lnS + O(S 0 ) with Γ cusp (λ) being the cusp anomalous dimension andS = S e γ E (with γ E being the Euler constant) [12]. For twist-four operators, the lowest scaling dimension scales at large spin as ∆ 4 = 4 + S − c(N)/S 2 with positive c(N) [13][14][15]. Comparing ∆ 2 (S) and ∆ 4 (S) we find that the two functions cross each other The calculation of such correlators in the AdS/CFT turns out to be a nontrivial task [8][9][10][11].
at γ S = 2 + O(1/S 2 ), or equivalently Γ cusp (λ) ∼ 1/lnS. The important difference with the previous example is that the crossing point is located at weak coupling and, therefore, it can be analyzed using perturbation theory.

Level-crossing in three-dimensional CFT
Recently an important progress has been achieved in understanding the properties of threedimensional unitary CFT's within the conformal bootstrap approach. It was found that the constraints of crossing symmetry and unitarity restrict the possible values of the scaling dimensions of the operators ∆ i in a highly nontrivial way [17,18]. For our purposes we shall examine the extremal solutions to these constraints living on the boundary of the allowed region of ∆ i 's. The spectrum of the corresponding threedimensional unitary CFT's can be uniquely reconstructed and it is parameterized by the Figure 2. The scaling dimensions of the three lowest spin−2 operators and their OPE coefficients (multiplied by 4 ∆ i ) as found in Ref. [18].
dimension ∆ σ of the leading Z 2 −odd scalar operator σ. As was shown in [18], the resulting scaling dimensions reveal a dramatic transition at ∆ ⋆ σ = 0.518154 (15). The values of ∆ i 's at the transition point are remarkably close to the expected values of the critical exponents of three-dimensional Ising model.
Another intriguing feature of the 3d Ising point is that some operators disappear from the spectrum. As an example, consider the leading spin−2 operators that appears in the OPE σ×σ. The lowest lying operator is the stress-energy tensor T whose conformal weight ∆ T = 3 is protected and is independent on ∆ σ . For the next-to-lowest spin−2 operator T ′ , the scaling dimension and the OPE coefficient at the 3d Ising point are given by ∆ ⋆ T ′ = 5.500(15) and (C ⋆ σσT ′ ) 2 = 2.97(2) × 10 −4 , respectively. However, slightly above this point, for ∆ σ = 0.5182, the OPE coefficient decreases by the order of magnitude C 2 σσT ′ = 2.21 × 10 −5 and the scaling dimension changes to ∆ T ′ = 4.334 (see Figure 16 in Ref. [18]).
A close examination of the spectrum of lowest spin 2 operators (see Figure 15 in Ref. [18]) shows that in the vicinity of the 3d Ising point the levels approach each other in a pairwise manner. Moreover, the variation of the scaling dimensions and the OPE coefficients in the transition region follows the same pattern as the one shown in Fig. 1. This suggests that the spectrum in this region can be described by the relations (2.8) and (3.3). Important difference with the previous case is that the three-dimensional CFT's under consideration do not have a natural parameter like 1/N and the arguments leading to (3.6) and (3.7) do not apply. In what follows we shall apply (2.8) and (3.3) and interpret the parameter γ/N as 'interaction energy' v between levels.
Let us examine the ∆ σ −dependence of the scaling dimensions and the OPE coefficients of three lowest levels shown in Fig. 2. As was already mentioned, the lowest state with ∆ T = 3 (black line) corresponds to the stress-energy tensor. For the next-to-lowest level (red line), the scaling dimension ∆ − and the OPE coefficient C σσ− vary slowly with ∆ σ to the left from the 3d Ising point, for ∆ σ < ∆ * σ , and approach the following values for Figure 3. Dependence of the scaling dimensions of the spin−2 operators and their OPE coefficients C 2 σσ± = C 2 σσ± /C 2 σσ1 on the level splitting ǫ. Dots stand for the exact values found in [18], solid lines are described by (4.17) and (4.18). (4.14) For the next-to-next-to-lowest level (blue line), ∆ + and C σσ+ vary slowly with ∆ σ to the right from the 3d Ising point, for ∆ σ > ∆ * σ , and approach the same values (4.14) for ∆ σ ≫ ∆ * σ . To describe the variation of ∆ ± and C σσ± in the transition region, for 0.51815 < ∆ σ < 0.51820 , (4.15) we introduce the level splitting function ǫ = ǫ(∆ σ ) analogous to (1.2) It is straightforward to verify that away from the transition region ǫ measures the difference between the scaling dimensions: ǫ = ∆ − − ∆ + < 0 for ∆ σ ≪ ∆ * σ and ǫ = ∆ + − ∆ − < 0 for ∆ σ ≫ ∆ * σ . We find that ǫ is a smooth monotonic function of ∆ σ in the transition region (4.15) and it vanishes for ∆ σ = 0.51817. Then, we can invert the function ǫ = ǫ(∆ σ ) and obtain the dependence of ∆ ± and C σσ± on ǫ as shown in Fig. 3.
Applying (2.8), we expect that the scaling dimensions ∆ ± (ǫ) in the transition region (4.15) are given by where ∆ 1 is defined in (4.14) and v is the interaction energy. For the OPE coefficients, we apply (3.3) and (3.4) to get where c 1 (ǫ) = √ ǫ 2 + 4v 2 − ǫ /(2v) and C σσ1 is given by (4.14). To fix v and C σσ2 , we compare (4.17) and (4.18) with the results of Ref. [18] shown by dots in Fig. 3. We find a good agreement for v = 0.126 , C σσ2 /C σσ1 = 0.057 . These values depend on the number of the crossing-symmetry constraints n comp = 231 used in Ref. [18] and they are expected to decrease as n comp → ∞. An immediate consequence of (4.18) and (4.19) is that the OPE coefficient C 2 σσ− (ǫ) decreases by the order of magnitude in the transition region (4.15). This result is in a quantitative agreement with the expected properties of the operator T ′ mentioned in the beginning of the subsection.
We observe from Fig. 3 that the OPE coefficients C 2 σσ− given by (4.18) start to deviate from their exact values for ǫ > 0.5. We recall that the relations (4.18) were obtained under the assumption that the levels ∆ ± are separated from the rest of the spectrum of ∆ i by a finite gap, e.g. |∆ + − ∆ − |≪ |∆ − − ∆ T |. Since ∆ − (ǫ) decreases linearly with ǫ, this condition is not fulfilled for ǫ ≫ v and, as a consequence, C 2 σσ− receives the additional corrections due to mixing of the operator O − with the stress-energy tensor.
It is interesting to note that, according to (4.19), the ratio of the OPE coefficients C σσ2 /C σσ1 is anomalously small. This makes the properties of three-dimensional CFT's in the vicininity of critical 3d Ising model similar to those of four-dimensional N = 4 SYM theory in which case the OPE coefficients satisfy (3.6) at large N. Notice that for the operators with sufficiently large scaling dimensions in unitary CFT's, their OPE coefficients have to fall off exponentially fast [19]. Yet another surprising feature of 3d Ising point is that the same asymptotic behaviour also holds for the lowest spin−2 operators.
We would like to emphasize that the above analysis relies on the assumption that the extremal solutions living on the boundary of the allowed region of ∆ i 's admit an interpretation in terms of three-dimensional CFT's, so that the curves shown in Fig. 2 can be interpreted as describing a continuous flow of the conformal data. 9 This assumption is nontrivial and it does not hold e.g. for two-dimensional CFT's. In the latter case, the spectrum of the scaling dimensions of the extremal solutions to the left of the two-dimensional Ising point is inconsistent with Virasoro symmetry making a CFT interpretation impossible [18].

Level-crossing in QCD
As another example of level-crossing phenomenon, we consider the spectrum of scaling dimensions of composite operators that appeared in the study of the QCD evolution equations for three-particle distribution amplitudes [20,21]. In the simplest case of baryon distribution amplitude of helicity 1/2, these operators involve covariant derivatives acting on three quark fields and have the following form B n 1 n 2 n 3 = ǫ ijk D n 1 q ↑ i D n 2 q ↓ j D n 3 q ↑ k , S = n 1 + n 2 + n 3 , (4.20) 9 I would like to thank Slava Rychkov for discussing this point.
As was shown in [20], the one-loop dilatation operator acting on the operators B n 1 n 2 n 3 can be mapped into a Hamiltonian of a spin chain of length 3 where spin operators at each site are identified with the generators of the collinear SL(2) subgroup of the conformal group 10 acting on quark fields in (4.20). Here H 1 coincides with a Hamiltonian of a completely integrable Heisenberg SL(2) spin chain and the operator H 2 is given by where S 2 ij stands for the sum of two SL(2) spins at sites i and j. The eigenvalues of (4.21) define the spectrum of anomalous dimensions of baryonic operators (4.20) at one loop.
The Hamiltonian (4.21) depends on the parameter ǫ. For baryonic operators (4.20), its value is uniquely fixed ǫ = 1. To understand the properties of (4.21), it is convenient to treat ǫ as a new coupling constant and examine the dependence of the eigenvalues of (4.21) on ǫ. For ǫ = 0 the Hamiltonian (4.21) is completely integrable and its exact eigenspectrum can be found with a help of Bethe ansatz technique. For ǫ = 0 the Hamiltonian (4.21) is neither integrable, nor cyclic symmetric. Nevertheless, its eigenspectum can be determined for abritrary ǫ using the technique described in [20,21].
The Hamiltonian (4.21) is invariant under the exchange of two sites 1 ↔ 3 and its eigenstates can be classified according to the parity under this transformation. For ǫ = 0 the eigenstates with different parity have the same energy in virtue of integrability, but the degeneracy is lifted for ǫ = 0. The flow with ǫ of energy levels with different parity is independent from one another and the non-crossing rule is not applicable. At the same time, the levels with the same parity are not allowed to cross.
As an example, we show on Fig. 4 the ǫ−dependence of the energies of parity-even states with the total spin S = 20. We observe that for sufficiently large negative ǫ the energy levels approach each other. As was shown in [20], in the upper part of the spectrum, the crossing points correspond to the collision of levels of the Hamiltonians H 1 and H 2 with ψ 1 |ψ 2 = 0 and ψ i |ψ i = 1. Assuming that the rest of eigenstates is irrelevant in the vicinity of the crossing point, we can look for the eigenstate of (4.21) on a linear space spanned by the states |ψ 1 and |ψ 2 . Then, it is convenient to define the state  such that ψ 1 |ψ 2 = 0 and ψ 1 |ψ 1 = 1, and evaluate the matrix elements of H 1/2 with respect to |ψ 1 and |ψ 2 . In the standard manner, the diagonal matrix elements define corrections to the energies of two states It is proportional to the overlap of the two eigenstates ψ 1 |ψ 2 and does not depend on ǫ.
As follows from (4.25), the levels could cross at E 1 (ǫ) = E 2 (ǫ), or equivalently The crossing does not happen due to nonzero interaction energy |v| = 0. The eigenvalues of H 1/2 in the vicinity of (4.27) are given by (4.28) To make this expression more explicit, we consider the limit of large spin S ≫ 1. In this limit, E ± (ǫ) describe the one-loop anomalous dimensions of baryonic operators (4.20) with large number of derivatives S.
The Schrödinger equations (4.23) have been thoroughly studied in [20]. For the eigenstates with large total spin S, their energies can be expanded in powers of 1/S leading to 11 where ψ(x) = (ln Γ(x)) ′ is Euler's digamma function and nonnegative integers ℓ 1 and ℓ 2 enumerate the energy levels. The eigenstates with the maximal |e i | correspond to ℓ 1 = ℓ 2 = 0.
We are now ready to compare the relations (4.28), (4.32) and (4.33) with the exact energy spectrum of the Hamiltonian (4.21) shown in Fig. 4 for S = 20. We remind that for ǫ = 0 we have H 1/2 = H 1 and the energy levels are given by e 1 , Eq. (4.29), for ℓ 1 = 0, 1, . . . starting from the maximal one. For ǫ = 0 away from the crossing point, we find from (4.25) and (4.30) that E 1 (ǫ) = e 1 + O(1/S 2 ), so that the variation of the energy with ǫ is very slow at large S. We observe from Fig. 4 that at large negative ǫ the maximal energy grows linearly with (−ǫ). According to (4.25), (4.29) and (4.30), it is given by 34) 11 Expression for e 1 in this relation only holds in the upper part of the spectrum close to the maximal energy level, the general expression for e 1 can be found in [20].
These results are in a good agreement with properties of the exact spectrum shown in Fig. 4. As an example, we consider the flow of two highest levels shown by red and blue lines in Fig. 4. The energies of these levels follow (4.28). They approach each other for ǫ ⋆ = −7.215 and the level splitting is 2|v|= 4.46 × 10 −4 . Applying (4.32) and (4.33) for ℓ 1 = ℓ 2 = 0 and S = 20, we find ǫ ⋆ = −7.229 and 2|v|= 4.32 × 10 −4 which are close to the exact values.
Notice that the flow of energy levels in the lower part of the spectrum in Fig. 4 is more complicated and it deviates from (4.28). The reason for this is that the interaction between the lowest energy eigenstates is much stronger compared to that for the highest energy levels. Namely, the eigenstates in the lower part of the spectrum mix strongly with few other eigenstates and finding their energies requires diagonalization of the mixing matrix whose size grows as ln S for large spin. This problem was solved in [20,21].

Concluding remarks
In this note, we examined the properties of conformal operators whose scaling dimensions approach each other for some values of the parameters of a unitary CFT and satisfy von Neumann-Wigner non-crossing rule. We argued that the scaling dimensions of such operators and their OPE coefficients have a universal scaling behavior in the vicinity of the crossing point. We demonstrate that the obtained relations are in a good agreement with the known examples of the level-crossing phenomenon in maximally supersymmetric N = 4 Yang-Mills theory, three-dimensional conformal field theories and QCD.
In gauge theories, the scaling dimensions do not verify the non-crossing rule in the planar limit, but the level crossing leads to breakdown of 1/N expansion. Namely, the nonplanar corrections develop singularities close to the crossing point that have to be resummed to all orders in 1/N. We argued that their resummation is very similar to resolving the mixing of operators in gauge theories at weak coupling with 1/N playing the role of the coupling constant. The resulting expressions for the scaling dimensions and the OPE coefficients, Eqs. (2.8) and (3.7), respectively, remain finite in the transition region and obey the noncrossing rule. Away from this region their values coincide with those in the planar limit.
The scaling dimensions of twist-two operators are known to be increasing functions of the Lorentz spin S. For sufficiently large values of the coupling constant and/or the spin, they cross the scaling dimensions of the leading twist-four double-trace operators. For small (large) spin S, the crossing happens at strong (weak) coupling. We used the known results for Konishi operators (S = 0) and for twist-two operators with large spin (S ≫ 1) to show that, due to the level crossing, the scaling dimensions and the OPE coefficients of twist-two and twist-four operators get swapped at the crossing point leading to a dramatic change of their asymptotic behaviour. 12 This effect is believed to play an important role in verifying the conjectured S−duality in N = 4 SYM [23,24]. In particular, for finite N, the spectrum of the scaling dimensions should be invariant under the weak/strong coupling duality transformation h → 1/h with h = g 2 /(4π). In the special case of Konishi operator, the S−duality implies the existence of a nonperturbative operator K ′ whose scaling dimension at weak coupling is related to that of the Konishi operator as ∆ K ′ (h) = ∆ K (1/h) [25]. Let us examine the flow of the scaling dimensions ∆ K ′ (h) and ∆ K (h) and ignore, for the sake of simplicity, the mixing with other operators. Obviously, the functions ∆ K ′ (h) and ∆ K (h) could cross each other at h = 1. As before, this can not happen due to a mixing between the operators K and K ′ . Replacing ∆ 1 = ∆ K (h) and ∆ 2 = ∆ K (1/h) in (2.8), we obtain the expressions for ∆ ± (h) which verify the non-crossing rule and are invariant under the S−duality transformation, ∆ ± (h) = ∆ ± (1/h). Adding more operators does not change this result but makes the corresponding picture of levels flow more complicated due to appearance of the additional crossing points. 13 It would be interesting to test this mechanism using a lattice formulation of N = 4 SYM [26].