Evolution equation for the higher-twist B-meson distribution amplitudes
We find that the evolution equation for the three-particle quark-gluon B-meson light-cone distribution amplitude (DA) of subleading twist is completely integrable in the large limit and can be solved exactly. The lowest anomalous dimension is separated from the rest, continuous, spectrum by a finite gap. The corresponding eigenfunction coincides with the contribution of quark-gluon states to the two-particle DA so that the evolution equation for the latter is the same as for the leading-twist DA up to a constant shift in the anomalous dimension. Thus “genuine” three-particle states that belong to the continuous spectrum effectively decouple from to the leading-order accuracy. In turn, the scale dependence of the full three-particle DA turns out to be nontrivial so that the contribution with the lowest anomalous dimension does not become leading at any scales. The results are illustrated on a simple model that can be used in studies of corrections to heavy-meson decays in the framework of QCD factorization or light-cone sum rules.
pacs:12.38.Bx, 13.20.He, 12.39.Hg
B-meson light-cone distribution amplitudes (DAs) are the main nonperturbative input to the QCD description of weak decays involving light hadrons in the final state Beneke:1999br (); Beneke:2000wa (). In particular the leading-twist DA gives a dominant contribution in the heavy quark expansion and it received considerable attention already Grozin:1996pq (); Lange:2003ff (); Braun:2003wx (); Lee:2005gza (); Bell:2013tfa (); Braun:2014owa (); Feldmann:2014ika (). Utility of the QCD factorization techniques depends, however, on the possibility to control, or at least estimate, the corrections suppressed by powers of the -quark mass that involve higher-twist DAs. This task is attracting increasing attention and in the last years there have been several efforts to combine light-cone sum rules with the expansion in terms of B-meson DAs Khodjamirian:2006st (); DeFazio:2007hw (); Braun:2012kp (); Wang:2015vgv (). This technique allows one to tame infrared divergences which appear power-suppressed contributions in the purely perturbative framework and to calculate the so-called soft or end-point nonfactorizable contributions in terms of the DAs of increasing twist. One of the problems on this way is that higher-twist B-meson DAs involve contributions of multiparton states and are practically unknown.
In this letter we point out that the structure of subleading twist DAs is simpler as compared to what one may assume from their general partonic decomposition Kawamura:2001jm (); Nishikawa:2011qk (). This structure is revealed by considering the scale dependence of the DAs in the limit of large number of colors, , i.e. neglecting the corrections to the renormalization group equations. It turns out that the evolution equation for the three-particle DA in this approximation is completely integrable and can be solved exactly. The lowest anomalous dimension is separated from the rest, continuous, spectrum by a finite gap. The corresponding eigenfunction defines what can be called the “asymptotic” three-particle B-meson DA and has a relatively simple form. Most remarkably, it turns out that the higher-twist contribution to the two-particle B-meson DA that is related to the three-particle DA by QCD equations of motion (EOM), is expressed entirely in terms of this “asymptotic” state, the states that belong to the continuous spectrum do not contribute. As the result the DA evolves autonomously and does not mix with “genuine” three-particle contributions. The evolution equation for is the same as for the leading-twist DA up to a constant shift in the anomalous dimension. Finally, we discuss the evolution of the three-particle DA itself and its asymptotic behavior at small and large quark/gluon momenta which turns out to be nontrivial. This behavior is illustrated on the example of a simple model that can be used in phenomenological applications.
Ii Evolution equations
Following the established conventions Grozin:1996pq () we define the B-meson DAs as matrix elements of the renormalized nonlocal operators built of an effective heavy quark field , a light (anti)quark and gluons at a light-like separation:
Here is the heavy quark velocity, is the light-like vector, , such that , stands for an arbitrary Dirac structure, is the -meson state, is the factorization scale and is the B-meson decay constant in the heavy quark effective theory (HQET). Wilson lines connecting the fields are not shown for brevity; they are always implied.
The functions and are the leading- and subleading-twist two-particle B-meson DAs Beneke:2000wa (), and is the (lowest twist) three-particle DA that is the only one relevant for the present study. In notations of Kawamura:2001jm () . These three DAs are related by an EOM Beneke:2000wa (); Kawamura:2001jm ()
that can be solved to obtain as a sum of the so-called Wandzura-Wilczek (WW) term expressed in terms of Beneke:2000wa (), and a certain integral of the quark-gluon DA . The latter contribution is nontrivial because it involves a function of two variables. We will demonstrate, however, that this complication is to a large extent illusory as the integral appearing in the EOM essentially decouples from “genuine” quark-gluon correlations. This simplification is exactly analogous to what has been observed before Ali:1991em (); Balitsky:1996uh (); Braun:2000av (); Braun:2001qx () for the structure function in polarized deep-inelastic lepton-proton scattering.
The following discussion is based on properties of the renormalization group equations for heavy-light operators under collinear conformal transformations. The corresponding generators read
where is the conformal spin, for the light quark and for the gluon. The generators satisfy the standard commutation relations We distinguish the generators acting on quark and gluon coordinates by the subscript and , respectively.
The starting observation is that both the one-loop renormalization group equations (RGE) for the DAs and the EOM relations are invariant under special conformal transformations Knodlseder:2011gc (); Braun:2014owa (). It is therefore natural to expand the DAs in terms of the eigenfunctions of the corresponding generator Braun:2014owa ()
They form a complete orthonormal set
with respect to the invariant scalar product Gelfand ()
where the integration goes over the complex coordinates in the lower half-plane and the integration measure is defined as
Going over from quark/gluon coordinates to the corresponding momenta
can be done easily making use of the following expressions Braun:2014owa ():
Staying in coordinate space for the time being, we write the two-particle DAs as
and the three-particle DA
Here and below . Inserting these expressions in the EOM relation (3) one derives for the expansion coefficients
Invariance under special conformal transformations implies that terms with different values of cannot get mixed by the RGE. Thus the leading twist contributions must have autonomous scale dependence:
The RGE for the three-particle DA is more complicated,
where the “Hamiltonian” to the one-loop accuracy is given by a sum of two-particle kernels
Note that in difference to Braun:2014owa (); Knodlseder:2011gc (); Braun:2009mi () we include the QCD coupling in the definition of the quark-antiquark-gluon operator, . This redefinition affects the constant terms in the kernels.
where is defined in terms of the corresponding quadratic Casimir operator . This representation makes manifest that the Hamiltonian commutes with the generator of special conformal transformations
and therefore the RGE (14) is “diagonal” in . This symmetry alone is not sufficient, however, to find the solution since the problem has two degrees of freedom — the light-cone coordinates of the light quark and the gluon. It turns out, however, that for the leading contribution for a large number of colors
there is an additional “hidden” symmetry. Namely, it is possible to construct one more “conserved charge”, , that commutes both with and the large- Hamiltonian :
Having two conserved charges for a problem with two degrees of freedom allows one to diagonalize the Hamiltonian, i.e. in our case find the multiplicatively renormalizable operators, without the need to solve the RGE equation (14) explicitly. This property is known as complete integrability.
The explicit expression for can be found using the formalism of the quantum inverse scattering method (QISM) Faddeev:1979gh ():
In this approach the charges appear in the expansion of the element of the monodromy matrix for an open spin chain, The commutation relation can be verified by a direct calculation using the coordinate-space representation for the kernels as given in Eq. (16). or, more elegantly, with the help of the QISM techniques. This derivation will be given elsewhere BM ().
The “conserved charges” , and the “Hamiltonian” are self-adjoint operators with respect to the scalar product (7):
It follows that they have real eigenvalues and can be diagonalized simultaneously:
Note that we write the eigenvalues of as a product where is an eigenvalue of and is a real number (but not necessarily positive), . This structure is motivated by QISM BM (). The eigenfunctions are labeled by two “quantum numbers”, and , and provide the basis of the so-called Sklyanin’s representation of Separated Variables Sklyanin:1991ss (). They can be found using the method developed in Derkachov:2003qb (),
The functions are symmetric under reflection . Since the eigenvalue has to be real, can take real or imaginary values. It is possible to show that for imaginary there exists only one normalizable solution corresponding to the particular value . For this special solution the hypergeometric function disappears and the eigenfunction becomes very simple:
This solution has the lowest energy
and can be interpreted as the ground state of the large- Hamiltonian. It describes the “asymptotic” quark-gluon DA with the lowest anomalous dimension. The state is normalized as
The eigenfunctions corresponding to real values of belong to the continuous spectrum. They are orthogonal to the ground state, , and normalized as
The corresponding eigenvalue (energy) is
The gap between the ground state and the continuous spectrum
coincides with the gap in the spectrum of anomalous dimensions of twist-three light quark-antiquark-gluon operators with large number of derivatives, see Ref. Derkachov:1999ze ().
The corrections to the ground state energy can be calculated in a standard quantum-mechanical perturbation theory, evaluating the matrix element . The answer can be written as
where is the anomalous dimension for the leading twist DA , Eq. (13), and is a constant that does not depend on :
The value of coincides exactly with the gap between the spectrum of anomalous dimensions of twist-three light quark-antiquark-gluon operators and the leading-twist quark-antiquark operators for a large number of derivatives, cf. Braun:2000av ().
A generic three-particle DA can be expanded in the eigenfunctions of the large- Hamiltonian
where the coefficient functions and can be calculated using the scalar product as
They have autonomous scale dependence up to corrections:
where , ,
Explicit solution of the evolution equation for the DA in the large- limit presents our main result.
Remarkably, all terms involving the hypergeometric function vanish thanks to the identity
that is related to the orthogonality condition . Thus only the ground state (with the lowest anomalous dimension) contributes to the EOM relation (3). One finds
leading to the following very simple relation in the -space:
Going over from quark/gluon coordinates to the corresponding momenta
can be done easily making use of Eq. (9). The scalar product in momentum space reads
and the eigenfunctions of the evolution equation take the form
The scale-dependence of the coefficients and differs by a simple overall factor
where is defined in Eq. (LABEL:Rfactor) and is a constant, see Eq. (31). In other words, the subleading twist contribution to is suppressed at large scales as compared to the WW contribution by the universal factor that does not depend on the light quark momentum. To the accuracy there is no mixing with “genuine” quark-gluon degrees of freedom.
It is tempting to define the “asymptotic” quark-gluon DA as the contribution with the lowest anomalous dimension (for a given ):
The corresponding expression in momentum space reads
Iii Asymptotic behavior at small and large momenta
One of the reasons why the renormalization group evolution is interesting is that it gives insight in the expected behavior of the DAs at large and small momenta, which is important for the status of factorization theorems. Although one cannot make any rigorous statements on the shape of the DAs at low scales, it is usually assumed that the “true” DAs have the same asymptotic behavior as in perturbation theory. This assumption proved to be successful for modeling of parton distributions and DAs of light hadrons, so that it is natural to use the same logic for heavy-light systems.
For small momenta there are no surprises. Using explicit expressions we find
respectively. This behavior is in agreement with arguments based on quark-gluon duality Khodjamirian:2006st (). If both quark and gluon momenta are small one obtains
The large-momentum asymptotics is much more interesting. An inspection of the the asymptotic DA (49) reveals that it does not decrease for large gluon momenta (because of the last term that is -independent). As a consequence, integral over all momenta is ill-defined, and the normalization of the asymptotic DA to a matrix element of a local operator even at a single scale is not possible. This problem is seen even better in coordinate space. Using the definition in (48) and explicit expression for one obtains
The behavior of at is determined by the small- asymptotics of . Assuming a power-law behavior one obtains
This function is not analytic at the origin : the limit depends on the way the variables approach zero and exists only if . For one gets
If the gluon coordinate and the quark position is kept constant, the singularity cannot be avoided. It translates to the constant behavior at large gluon momentum, as seen explicitly from the momentum space representation.
The singular behavior , corresponding in physics terms to the instability due to gluon falling to the center of the color-Coulomb field, is not a special pathology of the asymptotic DA: the contributions of the continuum spectrum are even more singular, , so that the corresponding momentum space DAs are increasing (and oscillating) functions of the gluon momentum. We are able to show that all such singularities are, however, spurious and cancel in the sum of contributions of the asymptotic DA and the corrections. Most importantly, this cancellation is not spoiled by the evolution: The singularity is not generated at higher scales provided it is not present already in the nonperturbative ansatz at a reference low scale. This result implies that for small , alias large , the hierarchy of contributions with increasing anomalous dimensions is lost; the leading large- asymptotics of the “asymptotic” DA is exactly cancelled by the contributions with larger anomalous dimensions. This pattern appears to be unconventional, we are not aware of examples of similar behavior for light quark systems.
To this end we rewrite the expansion (33) as the integral over imaginary axis
Note that the contribution of the asymptotic DA is taken into account in the second line of Eq. (56) by moving the integration contour to the left of the singularity at , due to . This representation remains valid after the scale dependence (35) of the coefficients is taken into account: The anomalous dimension (36) is an analytic function in the strip and gives the anomalous dimension of the asymptotic DA.
In order to study the limit we write
and use that the coefficient in (56) is symmetric under so that one can replace without changing the value of the integral. For small
so that the asymptotic behavior of the DA