PLB30540S0370-2693(14)00720-510.1016/j.physletb.2014.09.062The AuthorsTheory

Fig. 1

The spectrum of eigenvalues E(xn)≡En of the discretized version of Eq. (51) x↦xn=(2n+1)/200, n=0,1,…,99 (blue dots) compared to the “unperturbed” spectrum E(xn) (red solid curve). In order not to overload the plot, only every second eigenvalue is shown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2

The x-even eigenfunctions ηk+(xn)=ηk+(−xn) for k=0 (the ground state), and k=10,50,100,150 (from left to right), for L=5 and N=500. Normalization is arbitrary.

Fig. 3

Asymptotic Λb distribution amplitude for the simplest choice (60) of the profile function ξ0(s,μ0).

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

V.M.

Braun

⁎

S.E.

Derkachov

A.N.

Manashov

aInstitut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, GermanyInstitut für Theoretische PhysikUniversität RegensburgRegensburg

D-93040

GermanybSt. Petersburg Department of Steklov Mathematical Institute, 191023 St. Petersburg, RussiaSt. Petersburg Department of Steklov Mathematical InstituteSt. Petersburg

191023

RussiacSt. Petersburg State Polytechnic University, 195251 St. Petersburg, RussiaSt. Petersburg State Polytechnic UniversitySt. Petersburg

195251

RussiadDepartment of Theoretical Physics, St. Petersburg University, 199034 St. Petersburg, RussiaDepartment of Theoretical PhysicsSt. Petersburg UniversitySt. Petersburg

199034

Russia⁎Corresponding author.

Editor: B. Grinstein

Abstract

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., Λb-baryon at large scales.

1Precision 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 become quantitative in order to make an impact on the field. In particular, the Λb→Λμ+μ− decays are receiving a lot of attention, see e.g. Ref. [1] and references therein.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. A recent discussion using SCET formalism can be found in Ref. [2]. 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. [3–5] where the complete classification and renormalization group equations (RGE) that govern the scale-dependence 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 [6,7] and, in a more general context, to integrability in high-energy QCD [8–10] and in the N=4 supersymmetric Yang–Mills theory [11–13] 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 [17].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. Λb, at large scales and can be called “asymptotic DA” in analogy to the accepted terminology for hadrons built of light quarks.2Consider 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 [4](1)〈0|[q1T(z1n)Cn̸γ⊥μq2(z2n)]hv(0)|Bj=1(v)〉=13ϵ⊥μu(v)fB(2)(μ)Ψ⊥(z1,z2;μ). Here q1,2=u,d,s are light quarks separated by a lightlike distance, hv(0) is the effective heavy quark field with four-velocity v, C is the charge conjugation matrix, u(v) is the Dirac spinor v̸u(v)=u(v), and ϵμ is the diquark polarization vector, vμϵμ=0. The transverse projections are defined with respect to the two auxiliary light-like vectors n and n¯ which we choose such that vμ=(nμ+n¯μ)/2, v⋅n=1, n⋅n¯=2:(2)ϵ⊥μ=g⊥μνϵν,g⊥μν=gμν−(nμn¯ν+nνn¯μ)/(n⋅n¯), and similar for γ⊥μ. The Wilson lines connecting the quark fields are not shown for brevity. The heavy quark field hv can itself be related to the Wilson line as [18](3)〈0|hv(0)|h,v〉=Pexp[ig∫−∞0dαvμAμ(αv)], 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 fB(2) is defined as the matrix element of the corresponding local q1q2hv operator; it is inserted for normalization [4]. The parameter μ is the renormalization (factorization) scale. We tacitly imply using MS¯ scheme.The product ϵ⊥μu(v) on the right-hand-side (r.h.s.) of Eq. (1) can be expanded in irreducible representations corresponding to physical baryon states with JP=1/2+ and JP=3/2+ using suitable projection operators, see [4]. These (ground) states form the SU(3)F multiplets (sextets), Σb,Ξb,Ωb and Σb⁎,Ξb⁎,Ωb⁎, respectively, which are degenerate in the heavy b-quark limit. The double-strange Ωb baryon is of special interest for flavor physics as it only decays through weak interaction. The DA Ψ⊥(z1,z2;μ) is written usually in terms of its Fourier transform(4)Ψ⊥(z̲;μ)=∫0∞dω1∫0∞dω2e−i(ω1z1+ω2z2)Ψ⊥(ω1,ω2;μ)=∫0∞ωdω∫01due−iω(uz2+u¯z1)Ψ˜⊥(ω,u;μ), where z̲={z1,z2} and in the second line we redefine ω1=uω, ω2=u¯ω with u¯=1−u. The variables ω1 and ω2 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 to the leading power accuracy in the heavy quark mass.The scale dependence of Ψ˜⊥(ω,u,μ) or, equivalently, Ψ⊥(z̲;μ), is governed by the renormalization group equation [3,4](5)(μ∂∂μ+β(αs)∂∂αs+2αs3πH)fB(2)(μ)Ψ⊥(z̲;μ)=0. The evolution kernel H is an integral operator which can be decomposed as(6)H=H12+H1h+H2h−4. The kernels Hkh are due to heavy–light quark interactions,(7)H1hf(z̲)=∫01dαα[f(z̲)−α¯f(α¯z1,z2)]+ln⁡(iz1μ)f(z̲),H2hf(z̲)=∫01dαα[f(z̲)−α¯f(z1,α¯z2)]+ln⁡(iz2μ)f(z̲). They are identical to the Lange–Neubert kernels [19–21]. The remaining contribution(8)H12f(z̲)=∫01dαα[2f(z̲)−α¯f(z12α,z2)−α¯f(z1,z21α)] 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(9)z12α=α¯z1+αz2,α¯=1−α. The evolution kernels (7), (8) can be written in terms of the generators of the collinear subgroup of conformal transformations(10)S+=z2∂z+2jz,S0=z∂z+j,S−=−∂z, where j=1 is the conformal spin of the light quark. The generators satisfy the standard SL(2) commutation relations(11)[S+,S−]=2S0,[S0,S±]=±S±. One can show that [23]H1h=ln⁡(iμS+(1))−ψ(1),H2h=ln⁡(iμS+(2))−ψ(1), where S+(1), S+(2) act on the first, z1, and the second, z2, light-cone coordinate, respectively. The last kernel (8) is written in terms of the two-particle Casimir operator S122

[24]

(12)H12=2[ψ(J12)−ψ(1)], where S122=S+S−+S0(S0−1)=J12(J12−1), S+=S+(1)+S+(2) etc., and ψ(x) is Euler's digamma function. Thus, the complete evolution kernel takes a very compact form(13)H=ln⁡(iμS+(1))+ln⁡(iμS+(2))+2ψ(J12)−4ψ(2). The evolution equation for the DA in momentum space, Ψ⊥(w1,w2;μ), is given by the same expression with the SL(2) generators in the momentum space representation [22]. Eigenfunctions of H correspond to the states that have autonomous scale dependence and the corresponding eigenvalues define anomalous dimensions.3Our main result is that this evolution equation can be solved explicitly. To this end we consider the following operators(14)Q1=i(S+(1)+S+(2)),Q2=S0(1)S+(2)−S0(2)S+(1). It is possible to show that Q1 and Q2 commute with each other and with the evolution kernel H:(15)[Q1,Q2]=[Q1,H]=[Q2,H]=0. The first two relations are trivial, the last one can be verified using the explicit expressions for Q2 and H.If H is interpreted as a Hamiltonian of a certain quantum-mechanical model, the operators Q1 and Q2 correspond to the conserved charges. In the formalism of the quantum inverse scattering method (QISM) the charges Q1,Q2 appear in the expansion of the element C(u) of the monodromy matrix,C(u)=uQ1+Q2. The commutation relation [C(u),H]=0, and its generalization to more degrees of freedom then follow directly from the QISM [14–16]. 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 Q1, Q2 are self-adjoint operators with respect to the SL(2,R) invariant scalar product(16)〈Φ|Ψ〉=1π2∫C−d2z1∫C−d2z2(Φ(z̲))⁎Ψ(z̲), where the integration goes over the lower complex half-plane, Imzi<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 [25]. They are labeled, for the present case, by two real numbers: s>0 and x∈R such that(17)C(u)ϕs,x(z1,z2)=s(u−x)ϕs,x(z1,z2), with(18)ϕs,x(z̲)=sz12z22∫01dα(αα¯)ixexp⁡[is(α¯/z1+α/z2)]=sρ(x)eis/z1z12z22F11(1+ix,2,is(z2−1−z1−1)), where(19)ρ(x)=πx/sinh⁡(πx). The eigenfunctions ϕs,x(z̲) form a complete system in the Hilbert space defined by the scalar product (16)(20)〈ϕs′,x′|ϕs,x〉=2πsδ(s−s′)δ(x−x′). Since the conserved charges Q1 and Q2 commute with the Hamiltonian H, they share the same set of eigenfunctions,(21)Hϕs,x(z̲)=γ(s,x)ϕs,x(z̲). 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(22)γ(s,x;μ)=2ln⁡(μs/s0)+E(x),E(x)=ψ(1+ix)+ψ(1−ix)+2γE where s0=e2−γE. Going back to the RGE equation (5), we expand the DA Ψ⊥(z̲,μ) over the eigenfunctions of H(23)Ψ⊥(z̲,μ)=∫0∞dss∫−∞∞dx2πη⊥(s,x;μ)ϕs,x(z̲). The expansion coefficients η⊥(s,x;μ)=〈ϕs,x|Ψ⊥〉 evolve autonomously,(24)fB(2)(μ)η⊥(s,x;μ)=fB(2)(μ0)η⊥(s,x;μ0)(μμ0)−83β0(μ0ss0)83β0ln⁡LL43β0[E(x)−4πβ0αs(μ0)], where L=αs(μ)/αs(μ0), β0=113Nc−23nf. For large scales, the coefficients η⊥(s,x;μ) slowly drift towards smaller values of both parameters: s→0, thanks to the factor s8ln⁡L/3β0, 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(25)ϕ˜s,x(ω,u)=〈e−iω(u¯z1+uz2)|ϕs,x(z1,z2)〉. Using that 〈e−ikz|z−2eis/z〉=−(1/ks)J1(2ks) [23] one obtains(26)ϕ˜s,x(ω,u)=1ωuu¯∫01dααα¯αixα¯−ixJ1(2wsα¯u¯)J1(2wsαu). The eigenfunctions ϕ˜s,x(ω,u) are orthogonal and form a complete set:(27)s2π∫0∞ω3dω∫01duuu¯ϕ˜s,x(ω,u)ϕ˜s′,x′⁎(ω,u)=δ(s−s′)δ(x−x′),(28)ω3uu¯∫0∞sds∫−∞∞dx2πϕ˜s,x(ω,u)ϕ˜s,x⁎(ω′,u′)=δ(ω−ω′)δ(u−u′). Making use of Bateman's expansion for the product of two Bessel functions we obtain a series representation(29)ϕ˜s,x(ω,u)=1ω∑n=0∞inϰn−1Cn3/2(1−2u)Hn(x)1sωJ2n+3(2sω). Here Cn3/2(x) are the Gegenbauer polynomials, J2n+3(x) are Bessel functions, and(30)ϰn=(n+1)(n+2)4(2n+3). The functions Hn(x) are given by the continuous Hahn polynomials up to the prefactor ρ(x):(31)Hn(x)=in∫01du(uu¯)ixCn3/2(1−2u)=(n+1)(n+2)2inρ(x)F23(−n,n+3,1+ix2,2|1), e.g.(32)ρ−1(x)H0(x)=1,ρ−1(x)H1(x)=3x,ρ−1(x)H2(x)=5x2−1, etc. Hahn polynomials are real functions, have the symmetry Hn(x)=(−1)nHn(−x), and form a complete orthogonal system. In our normalization(33)∫−∞∞dx2πHn(x)Hm(x)=ϰnδmn. Collecting everything we obtain the final result:(34)Ψ˜⊥(ω,u;μ)=ω2uu¯∫−∞∞dx2π∫0∞sdsϕ˜s,x(ω,u)η⊥(s,x;μ), where the scale dependence of η⊥(s,x;μ) is given in Eq. (24). The expansion coefficients of η⊥(s,x;μ) in Hahn polynomials are related to the expansion coefficients of Ψ˜⊥(ω,u;μ) in Gegenbauer polynomials,(35)η⊥(s,x;μ)=∑ninηn⊥(s;μ)Hn(x)↦Ψ˜⊥(ω,u;μ)=uu¯∑nψn⊥(ω;μ)Cn3/2(2u−1), by the Bessel transform (cf. Eq. (30) in Ref. [23])(36)ψn⊥(ω;μ)=∫0∞dssωJ2n+3(2sω)ηn⊥(s;μ). Making use of the asymptotic expansion for the Bessel function one finds that the small-s behavior ηn⊥(s)∼spn translates into the large-ω asymptotics of the function ψn⊥(ω)∼ω−1−pn unless there is some cancellation, see below.The expansion coefficients at the reference (low) scale can be calculated from a given model of the DA as(37)η⊥(s,x;μ0)=∫0∞ωdω∫01duϕ˜s,x⁎(ω,u)Ψ˜⊥(ω,u;μ0). In the existing studies it is usually assumed that Ψ˜⊥(ω,u;μ0) is decreasing exponentially at large energies ω. For a rather general model of this type(38)Ψ˜⊥(ω,u;μ0)=ω2uu¯∑ncn(ωϵn)κne−ω/ϵnϵn4Cn3/2(2u−1) one obtains(39)η⊥(s,x;μ0)=s∑nincn(sϵn)nHn(x)Γ(n+4+κn)Γ(2n+4)F11(n+4+κn2n+4|−sϵn). In particular, for the simplest phenomenologically acceptable model [3–5](40)Ψ˜⊥(ω,u;μ0)=ω2uu¯e−ω/ϵ0ϵ04↦η⊥(s,x;μ0)=ρ(x)se−sϵ0. Exponential decrease ∼e−ω/ϵn of each Gegenbauer harmonics in (38) amounts, from the view point of the relation in Eq. (36), to the fine tuning such that all leading power terms in the asymptotics ω→∞ drop out. This fine tuning is, however, destroyed by the evolution so that a power-like asymptotics is always generated.To see this, consider the simplest model in (40) corresponding to the term n=0 in (38) as an example. As the result of the evolution (24) all harmonics with n>0 become excited(41)ηn⊥(s,μ)=cn(μ)s(μ0s)−δe−sϵ0 where δ=−8/3β0ln⁡L and(42)cn(μ)∼∫dxH0(x)L4/3β0E(x)Hn(x). For the corresponding coefficients in the Gegenbauer expansion (35) one obtains using (36)(43)ψn⊥(ω,μ)=cn(μ)ϵ0−2(ϵ0μ0)δ(ωϵ0)n+2Γ(n+4−δ)Γ(2n+4)F11(n+4−δ2n+4|−ωϵ0). The confluent hypergeometric function F11(a,b|ω) decreases as a power of ω at ω→∞, cf. Eq. (62) below, unless a−b is a nonnegative integer, in which case the asymptotic behavior is exponential. Thus, unless δ=0 and n=0, we obtain(44)ψn⊥(ω,μ)∼(ω/ϵ0)−2+δ. Note that the asymptotic behavior is the same for any n.4Next, 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 jP=0+ SU(3)F triplet and all longitudinal DAs of heavy baryons in the jP=1+ sextets, see [4]. For definiteness, consider the Λb-baryon DA [3,4] defined as(45)〈0|[uT(z1n)Cγ5n̸d(z2n)]hv(0)|Λ(v)〉=fΛ(1)(μ)ΨΛ(z1,z2;μ)uΛ(v). The evolution equation for ΨΛ(z1,z2;μ) contains an additional term corresponding to the gluon exchange between the light quarks (in Feynman gauge)(46)H12↦H12−δH12,δH12f(z̲)=∫01dα∫0α¯dβf(z12α,z21β) that corresponds to H↦H−1/J12(J12−1) in the SL(2)-invariant representation of the evolution kernel in Eq. (13). Expanding ΨΛ(z1,z2;μ) in terms of the eigenfunctions (18) of the integrable Hamiltonian (13)(47)ΨΛ(z̲,μ)=∫0∞dss∫−∞∞dx2πηΛ(s,x;μ)ϕs,x(z̲) one obtains the RGE equation for the expansion coefficients ηΛ(s,x,μ)(48)(μ∂∂μ+β(αs)∂∂αs+2αs3π[2ln⁡(μss0)+E(x)])fΛ(1)(μ)ηΛ(s,x,μ)=2αs3πfΛ(1)(μ)∫−∞∞dx′V(x,x′)ηΛ(s,x′,μ), where E(x) is defined in Eq. (22) and the kernel V(x,x′) is given by the matrix element(49)〈ϕs′,x′|δH12|ϕs,x〉=δ(s−s′)V(x,x′). We obtain(50)V(x,x′)=12π∑n=0∞ϰn−1Hn⁎(x)Hn(x′)(n+1)(n+2)=12sinh⁡π(x−x′)[x−x′xx′−πsinh⁡π(x−x′)sinh⁡πxsinh⁡πx′]. It is easy to see that V(x,x)∼1/(2πx2) for large x, and decreases exponentially in |x−x′|.In order to solve (48) one needs to find the eigenfunctions of the integral equation(51)E(x)ηE(x)−[VηE](x)=EηE(x). If V→0, obviously all eigenfunctions are localized in x, ηa(x)∼δ(x−a). The spectrum of eigenvalues is continuous, Ea=E(a)≥0, and double degenerate since E(a)=E(−a). In order to understand the effect of the “perturbation” V we consider the discretized version of this equation: x→xn=(n+1/2)Δx, Δx=L/N, n=−N,…,N−1. The unperturbed eigenfunctions, ηk(xn)=δnk, correspond to local excitations at the k-th site. Discretizing the integral in (48) one replaces the original eigenvalue problem (51) by the eigenvalue problem for the matrix Vnk=δnkE(xn)−ΔxV(xn,xm). Since V(x,x′)=V(−x,−x′), 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. Diagonalizing 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 ηk(xn) remain well localized around the point xk, see Fig. 2

. At the same time the lowest x-even eigenstate changes drastically: It becomes delocalized, see Fig. 2, and separated from the rest of the spectrum by a finite gap11

The size of the gap coincides with the gap in the spectrum of anomalous dimensions of three-light-quark operators in the large-N limit [7], indicating that these problems are related. Unraveling this connection goes beyond the tasks of this letter.

(52)ΔE=E0≃−0.3214. In the continuum limit (Δx→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:(53)η0(x)≃2E02+x2ρ(x)[E0−E(x)],η0(0)=1. It can be convenient to expand this function in Hahn polynomials(54)η0(x)=∑n=0,2,…∞χnHn(x), where the first few coefficients read(55)χ0≃0.612,χ2≃−0.126,χ4≃0.0574,χ6≃−0.0338,χ8≃0.0226,χ10≃−0.0163. The normalization is given by(56)∫−∞∞dx2πη02(x)=∑n=0∞ϰnχn2≃0.0758. Coming back to the representation of the DA in the form (47) we can separate the contribution of the discrete (lowest) level as(57)ηΛ(s,x,μ)=ξ0(s,μ)χ0−1η0(x)+ηΛ′(s,x,μ). where the function ηΛ′(s,x,μ) accounts for the contribution of the continuum spectrum and must be orthogonal to η0(x),(58)∫dxη0(x)ηΛ′(s,x,μ)=0. Going over to the momentum space we obtain for the contribution of the lowest state (asymptotic DA)(59)fΛ(1)(μ)Ψ˜Λ(0)(ω,u;μ)=fΛ(1)(μ0)χ0−1ω2uu¯(μμ0)−83β0L43β0[E0−4πβ0αs(μ0)]∫−∞∞dx2πη0(x)∫0∞sdsϕ˜s,x(ω,u)ξ0(s,μ0)(μ0ss0)83β0ln⁡L, 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” ξ0(s,μ0) at the reference scale, which can be arbitrary. For the simplest ansatz(60)ξ0(s,μ0)=se−sϵ0, cf. (40), we obtain (at the scale μ0)(61)Ψ˜Λ(0)(ω,u)=1ϵ02uu¯∑n=0,2,…∞in(χnχ0)Γ(n+4)Γ(2n+4)Cn3/2(1−2u)(ωϵ0)n+2F11(n+42n+4|−ωϵ0)=1ϵ02uu¯{(ωϵ0)2e−ω/ϵ0−142(χ2χ0)(ωϵ0)4C23/2(1−2u)F11(68|−ωϵ0)+…}, where the first few Hahn expansion coefficients χn are given in Eq. (55). The function Ψ˜Λ(0)(ω,u) is plotted in Fig. 3

. Note that the contributions of higher Gegenbauer polynomials Cn3/2(1−2u) in (61) are accompanied by increasing powers of ω so that the deviation from the “naive” ∼uu¯ shape is increasing with the energy: the distribution becomes broader. Taking into account that for large ω(62)ωn+2F11(n+42n+4|−ω)≃ω−2Γ(2n+4)Γ(n),n>0, we see that all terms in this expansion except for the first one have a power-like asymptotics at ω→∞.22

The exponential falloff of the first term is tightened to the particular choice of the profile function (60) and in this sense accidental.

One formally gets(63)Ψ˜Λ(0)(ω,u)≃ω→∞ω−2uu¯∑n=2,4…∞in(χnχ0)Γ(n+4)Γ(n)Cn3/2(1−2u). The series in (63) is divergent indicating that for large ω the function ΨΛ(0)(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 ξ0(s,u) provided it vanishes sufficiently fast at s→0 and s→∞.

We expect that the model of the Λb DA in Eq. (61), or a more general one in Eq. (59), will be sufficient for phenomenological applications. If necessary, the contributions of the continuum spectrum (57) can be added, in which case the scale dependence of ηΛ′(s,x,μ) can be approximated using Eq. (24). In this approximation the orthogonality condition (58) will not hold at all scales, which is, however, unlikely to be numerically significant.The evolution equation for the Λb DA has also been discussed in Ref. [5] using a different representation ηΛ(s,x)↦ρˆ2(wr,ϰ) where we use the notation ϰ for the variable called u′ in [5] to avoid confusion with the momentum fraction. The relation is simply s=1/wr for the first variable, whereas going over from the x- to ϰ-representation corresponds to the Fourier transform(64)ηˆ(ϰ)=∫−∞∞dxη(x)(ϰϰ¯)ix=∫−∞∞dxη(x)eipx,p=ln⁡ϰϰ¯,ϰ¯=1−ϰ. In other words if our variable x (17) is interpreted as a quasimomentum, then p=ln⁡(ϰ/ϰ¯) is the corresponding generalized coordinate.33

We thank the referee for suggesting to make this comparison.

For the ground state η0(x)=η0(−x) implies ηˆ0(ϰ)=ηˆ0(1−ϰ).The end-point behavior of ηˆ0(ϰ) is determined by the position of the (nearest) singularity of η0(x) in the complex x plane. A singularity (simple pole) at x0=±ia corresponds to ηˆ0(ϰ)∼[ϰϰ¯]a for ϰ→0,ϰ→1. The position of the singularity can be related to the value of energy E0, alias the lowest anomalous dimension. One can show that the term in V in Eq. (51) does not contribute close to the singularity so that the following exact relation holds:(65)E0=E(x0)=ψ(1+a)+ψ(1−a)+2γE. Using an (approximate) value E0=−0.3214, Eq. (52), we obtain, to the same accuracy(66)a=0.3460. Assuming that the asymptotic ∼[ϰϰ¯]a behavior can be extrapolated to the whole interval ϰ∈[0,1] one obtains a model for the eigenfunction of the ground state(67)ηˆ0(ϰ)=[ϰϰ¯]a↦η0(x)=Γ[a+ix]Γ[a−ix]/Γ2[a], which turns out to be in good agreement numerically with a more complicated parametrization in Eq. (53).This result agrees well with the approximation for the asymptotic Λb DA in the ϰ-space ρˆ2(wr,ϰ)∼[ϰϰ¯]1/3 found in Ref. [5] (see Fig. 4 there) by expanding the eigenfunction in Gegenbauer polynomials and retaining the first few terms. Numerical convergence of this expansion (away from the end-points) observed in [5] is in fact directly related to our result that the lowest eigenstate of the evolution equation is separated from the continuum spectrum by a finite gap (52).The above discussion cannot be directly applied to all other eigenstates of the evolution equation that belong to the continuum spectrum. In particular for the integrable case these solutions are given by plane waves, ηˆ(ϰ)∼eipx, and possess an essential singularity at ϰ→0,ϰ→1. This singularity is only seen in extreme proximity to the end-points and cannot be found using the Gegenbauer expansion. Whether such singularities are important for phenomenological applications remains to be studied.5To 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 jP=1+ sextets (Σb,Ξb,Ωb and Σb⁎,Ξb⁎,Ωb⁎), 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 (jP=0+ SU(3)F triplet and all longitudinal DAs of heavy baryons in the jP=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 (52). The corresponding eigenfunction is delocalized in the x-space and can be found using numerical methods (53) (see also (67)). It can be interpreted as the asymptotic DA at large scales (59), (61) and deviates significantly from the naive ∼u(1−u) shape at large quark energies, cf. (63) and Fig. 3.We expect that evolution equations for the higher-twist DAs of heavy baryons [3,4] and for the three-particle quark–gluon DAs of B-mesons in the large-Nc limit [21] have similar properties and can be studied using the same methods. These can be important for practical applications since heavy hadron decay form factors for physical values of the b-quark mass are likely to be dominated by soft contributions that can be related to higher-twist DAs using light-cone sum rules, see e.g. [1,26].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 [27–30].

Acknowledgements

This work was supported by RFBR grants 13-01-12405, 14-01-00341 (S.D.) and by the DFG, grant BR2021/5-2 (A.M.).

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