Integrability of the evolution equations for heavy-light baryon distribution amplitudes

We consider evolution equations describing the scale dependence of the wave function of a baryon containing an infinitely heavy quark and a pair of light quarks at small transverse separations, which is the QCD analogue of the helium atom.The evolution equations depend on the relative helicity of the light quarks. For the aligned helicities, we find that the equation is completely integrable, that is it has a nontrivial integral of motion, and obtain exact analytic expressions for the eigenfunctions and the anomalous dimensions. The evolution equation for anti-aligned helicities contains an extra term that breaks integrability and creates a"bound state"with the anomalous dimension separated from the rest of the spectrum by a finite gap. The corresponding eigenfunction is found using numerical methods. It describes the momentum fraction distribution of the light quarks in, e.g., $\Lambda_b$-baryon at large scales.


Precision tests of the flavor sector of the Standard Model may reveal new physics and remain
high on the agenda. Main attention has been so far focused on B-mesons but interest is developing to the heavy baryon decays as well. Such baryons are produced copiously at the LHC and, as more data are collected, studies of rare b-baryon decays involving flavor-changing neutral current transitions have to become quantitative in order to make an impact on the field.
Theoretical description of the b-hadron decays is based on factorization theorems that make use of the large mass of the b-quark in order to separate calculable effects of short distances from the nonperturbative large distance physics. The corresponding formalism is similar but much less developed for baryons as compared to mesons. For the exclusive decays involving large energy release in the final state, the relevant nonperturbative quantities are baryon wave functions at small transverse separations, dubbed light-cone distribution amplitudes (DA). Their study was started in Refs. [1,2,3] where the complete classification and renormalization group equations (RGE) that govern the scaledependence are presented.
In this work we point out that these equations have a hidden symmetry and completely integrable in the case that the light quarks have the same helicity. In other words, we identify a nontrivial quantum number that distinguishes heavy baryon states with different scale dependence and obtain exact analytic solution of the evolution equations. This phenomenon is similar to integrability of RGEs for the light baryons [4,5] and, in a more general context, to integrability in high-energy QCD [6,7,8] and in the N = 4 supersymmetric Yang-Mills theory [9,10,11] that attracted huge attention as a tool to check the AdS/CFT correspondence. The integrable model that we encounter in the present context is different; it has been discussed recently in [15].
The similar equation for the case that the two light quarks have opposite helicity contains an extra term that breaks integrability and creates a "bound state" with the anomalous dimension separated from the rest of the spectrum by a finite gap. The corresponding eigenfunction is found numerically. It describes the momentum fraction distribution of the light quarks in, e.g. Λ b , at large scales and can be called "asymptotic DA" in analogy to the accepted terminology for hadrons built of light quarks.
2. Consider at first the leading-twist DA of a baryon containing an infinitely heavy quark and a transversely polarized "diquark": a pair of light quarks with aligned helicities. It can be defined as [2] 0| Here q 1,2 = u, d, s are light quarks separated by a lightlike distance, h v (0) is the effective heavy quark field with four-velocity v, C is the charge conjugation matrix, u(v) is the Dirac spinor / vu(v) = u(v), and µ is the diquark polarization vector, v µ µ = 0. The transverse projections are defined with respect to the two auxiliary light-like vectors n andn which we choose such that v µ = (n µ +n µ )/2, v · n = 1, n ·n = 2: and similar for γ µ ⊥ . The Wilson lines connecting the quark fields are not shown for brevity. The heavy quark field h v can itself be related to the Wilson line as [16] 0|h so that the operator in Eq. (1) can be viewed as a pair of light quarks (a diquark), attached to the Wilson line with a cusp containing one lightlike and one timelike segment. Finally, the coupling f (2) B is defined as the matrix element of the corresponding local q 1 q 2 h v operator; it is inserted for normalization [2]. The parameter µ is the renormalization (factorization) scale. We tacitly imply using M S scheme.
The product µ ⊥ u(v) on the right-hand-side (r.h.s.) of Eq. (1) can be expanded in irreducible representations corresponding to physical baryon states with J P = 1/2 + and J P = 3/2 + using suitable projection operators, see [2]. These (ground) states form the SU (3) F multiplets (sextets), , Ω * b , respectively, which are degenerate in the heavy b-quark limit. The doublestrange Ω b baryon is of special interest for flavor physics as it only decays through weak interaction. The DA Ψ ⊥ (z 1 , z 2 ; µ) is written usually in terms of its Fourier transform where z = {z 1 , z 2 } and in the second line we redefine ω 1 = uω, ω 2 =ūω withū = 1 − u. The variables ω 1 and ω 2 correspond to the energies (up to a factor two) of light quarks in the baryon rest frame. The DA (4) is the most important nonperturbative input in QCD calculations of exclusive heavy baryon decays.
The scale dependence of Ψ ⊥ (ω, u, µ) or, equivalently, Ψ ⊥ (z; µ), is governed by the renormalization group equation [1,2] The evolution kernel H is an integral operator which can be decomposed as The kernels H h k are due to heavy-light quark interactions, They are identical to the Lange-Neubert kernels [17,18,19]. The remaining contribution takes into account the interaction between the light quarks; it is similar to the standard Efremov-Radyushkin-Brodsky-Lepage evolution kernel for the pion DA. Here and below we use the notation The evolution kernels (7), (8) can be written in terms of the generators of the collinear subgroup of conformal transformations where j = 1 is the conformal spin of the light quark. The generators satisfy the standard SL(2) commutation relations One can show that [21] H h 1 = ln iµS + act on the first, z 1 , and the second, z 2 , light-cone coordinate, respectively. The last kernel (8) is written in terms of the two-particle Casimir operator S 2 12 [22] where S 2 12 = (S (1) + S (2) ) 2 = J 12 (J 12 − 1) , and ψ(x) is the Euler's ψ-function. Thus, the complete evolution kernel takes a very compact form The evolution equation for the DA in momentum space, Ψ ⊥ (w 1 , w 2 ; µ), is given by the same expression with the SL(2) generators in the momentum space representation [20]. Eigenfunctions of H correspond to the states that have autonomous scale dependence and the corresponding eigenvalues define anomalous dimensions.

3.
Our main result is that this evolution equation can be solved explicitly. To this end we consider the following operators It is possible to show that Q 1 and Q 2 commute with each other and with the evolution kernel H: The first two relations are trivial, the last one can be verified using the explicit expressions for Q 2 and H.
If H is interpreted as a Hamiltonian of a certain quantum-mechanical model, the operators Q 1 and Q 2 correspond to the conserved charges. In the formalism of the quantum inverse scattering method (QISM) the charges Q 1 , Q 2 appear in the expansion of the element C(u) of the monodromy matrix, The commutation relation [C(u), H] = 0, and its generalization to more degrees of freedom then follow directly from the QISM [12,13,14]. Note that in classical applications of integrable models one encounters Hamiltonians that commute with the sum of diagonal elements, A(u) + D(u), of the monodromy matrix. In our case the Hamiltonian commutes with C(u), which corresponds to a new, nonstandard integrable model.
The conserved charges Q 1 , Q 2 are self-adjoint operators with respect to the SL(2, R) invariant scalar product where the integration goes over the lower complex half-plane, Im z i < 0. The eigenfunctions of C(u) provide the basis of the so-called Sklyanin's representation of Separated Variables and are known in explicit form [23]. They are labeled, for the present case, by two real numbers: s > 0 and x ∈ R such that with where The eigenfunctions φ s,x (z) form a complete system in the Hilbert space defined by the scalar product (16) Since the conserved charges Q 1 and Q 2 commute with the Hamiltonian H, they share the same set of eigenfunctions, The simplest way to calculate the eigenvalues is to compare the large-z asymptotics of the expressions on the both sides of this equation. In this way one obtains the anomalous dimensions γ(s, x; µ) = 2 ln(µs/s 0 ) + E(x) , where s 0 = e 2−γ E . In order to calculate the scale dependence we expand the DA Ψ ⊥ (z, µ) over the eigenfunctions of H The expansion coefficients η ⊥ (s, x; µ) = φ s,x |Ψ ⊥ evolve autonomously, where L = α s (µ)/α s (µ 0 ), β 0 = 11 3 N c − 2 3 n f . For large scales, the coefficients η ⊥ (s, x; µ) slowly drift towards smaller values of both parameters: s → 0, thanks to the factor s 8 ln L/3β 0 , and |x| → 0, taking into account that ψ(1 + ix) ∼ ln |x| for x → ±∞.
Going over to the DA in momentum space, Ψ ⊥ (ω, u; µ), we define the corresponding eigenfunctions as Using that The eigenfunctions φ s,x (ω, u) are orthogonal and form a complete set: Making use of the Bateman's expansion for the product of two Bessel functions we obtain another representation Here C The functions H n (x) are given by the continuous Hahn polynomials up to the prefactor ρ(x): e.g.
etc. Hahn polynomials are real functions, have the symmetry H n (x) = (−1) n H n (−x), and form a complete orthogonal system. In our normalization Collecting everything, we obtain the final result: where the scale dependence of η ⊥ (s, x; µ) is given in Eq. (24). The expansion coefficients at a reference (low) scale can be calculated from a given model of the DA as For example, choosing In particular, for the simplest phenomenologically acceptable model [1,2,3] Ψ ⊥ (ω, u; µ 0 ) = ω 2 uū e −ω/ 0 In general, the large-ω behaviour of the DA Ψ ⊥ (ω, u; µ 0 ) is governed by the the small-s asymptotic of η ⊥ (s, x; µ 0 ) and vice versa. However, the O(1/ω 2 ) power tail at ω → ∞ is always generated for large scales µ > µ 0 by the evolution.
4. Next, we consider heavy baryons with the light quarks having opposite helicity. The scale dependence of the leading twist DAs does not depend on the spin of the light quark pair and is the same for the j P = 0 + SU (3) F triplet and all longitudinal DAs of heavy baryons in the j P = 1 + sextets, see [2]. For definiteness, consider the Λ b DA [1,2] defined as The evolution equation for Ψ Λ (z 1 , z 2 ; µ) contains an additional term corresponding to the gluon exchange between the light quarks (in Feynman gauge) that corresponds to H → H − 1/J 12 (J 12 − 1) in the SL(2)-invariant representation of the evolution kernel in Eq. (13). Expanding Ψ Λ (z 1 , z 2 ; µ) in terms of the eigenfunctions (18) of the integrable Hamiltonian (13) where E(x) is defined in Eq. (22) and the kernel V (x, x ) is given by the matrix element We obtain It is easy to see that V (x, x) ∼ 1/(2πx 2 ) for large x, and decreases exponentially in |x − x |. In order to solve (42) one need to find the eigenfunctions of the integral equation If V → 0, obviously all eigenfunctions are localized in x, η a (x) ∼ δ(x − a) and the spectrum of eigenvalues is continuous, E a = E(a) ≥ 0, and double degenerate since E(a) = E(−a). In order to understand the effect of due to the "perturbation" V we consider a discretized version of this equation: x → x n = (n + 1/2)∆x, ∆x = L/N , n = −N, . . . , N − 1. The unperturbed eigenfunctions, η k (x n ) = δ nk , correspond to the local excitation at the k − th site. Discretizing the integral in (42) one replaces the original eigenvalue problem (45) by the eigenvalue problem for the matrix , all eigenstates have definite parity with respect to x → −x; the double degeneracy is lifted and one can study x-even and x-odd eigenstates separately. Diagonalising this matrix numerically we find that the shift of eigenvalues as compared to the unperturbed spectrum is surprisingly small, δE = E − E ≤ 0.003, for all eigenstates except for the lowest one, cf. Fig. 1, and the corresponding eigenfunctions η k (x n ) remain well localized around the In the continuum limit (∆x → 0, L → ∞) this phenomenon can be understood as creation of a bound state in addition to the continuum spectrum that remains to be largely unperturbed. The "wave function" of this (lowest) state can be approximated to a good accuracy (better that 1% for |x| < 3) by the following expression: , η 0 (0) = 1 .
It can be convenient to expand this function in Hahn polynomials where the first few coefficients read Coming back to the representation of the DA in the form (41) we can separate the contribution of the discrete (lowest) level as where the function η Λ (s, x, µ) accounts for the contribution of the continuum spectrum and must be orthogonal to η 0 (x), Going over to the momentum space, for the contribution of the lowest state (asymptotic DA) we obtain cf. Eq. (24). Note that the restriction to the contribution of the discrete level implies a certain relation between the momentum fraction distribution between the two light quarks and their total momentum ω, the remaining freedom is encoded in the "profile function" ξ 0 (s, µ 0 ) at the reference scale, which can be arbitrary. For the simplest ansatz where the Hahn expansion coefficients χ n are given in Eq. (49). The function Ψ (0) Λ (ω, u) is plotted in Fig. 3. Note that the contributions of higher Gegenbauer polynomials C 3/2 n (1 − 2u) in (55) are accompanied by increasing powers of ω so that the deviation from the "naive" ∼ uū shape is increasing with the energy: the distribution becomes broader. Indeed, taking into account that for large ω ω n+2 one formally gets The series in (57) is divergent indicating that for large ω the function Ψ (0) Λ (ω, u) develops end-point singularities at u → 0, 1. This feature is rather robust and does not depend on the precise choice of the profile function ξ 0 (s, u) provided it vanishes sufficiently fast at s → 0 and s → ∞. The large -ω region is, however, irrelevant for the applications to heavy baryon decays in the framework of QCD factorization as the contributions of light quark energies of the order of the heavy quark mass have to be subtracted [24].
We expect that the model of the Λ b distribution amplitudes in Eq. (55), or a more general one in Eq. (53), will be sufficient for phenomenological applications. If necessary, the contributions of the continuum spectrum (51) can be added, in which case the scale dependence of η Λ (s, x, µ) can be approximated using Eq. (24). In this approximation the orthogonality condition (52) will not hold at all scales, which is, however, unlikely to be numerically significant.

5.
To summarize, in this letter we have studied the evolution equations that determine the scale dependence of leading-twist DAs of heavy-light baryons, containing one infinitely heavy and two light quarks. The evolution equations are different for the cases that the light quarks have the same, or opposite, chirality. For the first case, which corresponds to transverse DAs of j P = 1 + sextets (Σ b , Ξ b , Ω b and Σ * b , Ξ * b , Ω * b ), the evolution equation turns out to be completely integrable, that is it has a nontrivial integral of motion. The anomalous dimensions form a continuum spectrum parametrized by two real numbers, s and x (22), and the corresponding eigenfunctions are known exactly (18), (26). For the second case (j P = 0 + SU (3) F triplet and all longitudinal DAs of heavy baryons in the j P = 1 + sextets), integrability is broken by an additional contribution to the evolution kernel that effectively corresponds to an attractive interaction between the light quarks and creates a bound state. As the result, the lowest anomalous dimension becomes separated from the rest of the spectrum (that remains continuous) by a finite gap (46). The corresponding eigenfunction is delocalized in the x-space and can be found using numerical methods (47). It can be viewed as an asymptotic DA at large scales (53), (55) and deviates significantly from the naive ∼ u(1 − u) shape at large quark energies, cf. (57) and Fig. 3.
We expect that evolution equations for the higher-twist DAs of heavy baryons [1,2] and for the three-particle quark-gluon DAs of B-mesons in the large-N c limit [19] have similar properties and can be studied using the same methods. Analogous unconventional integrable models with the Hamiltonian commuting with the diagonal entry D(u) of the monodromy matrix have appeared recently in the studies of high-energy scattering amplitudes in the N = 4 supersymmetric Yang-Mills theory [25,26,27,28].