$B$-meson Ioffe-time distribution amplitude at short distances

We propose the approach for a lattice investigation of light-cone distribution amplitudes (LCDA) of heavy-light mesons, such as the $B$-meson, using the formalism of parton pseudo-distributions. A basic ingredient of the approach is the study of short-distance behavior of the $B$-meson Ioffe-time distribution amplitude (ITDA), which is a generalization of the $B$-meson LCDA in coordinate space. We construct a reduced ITDA for the $B$-meson, and derive the matching relation between the reduced ITDA and the LCDA. The reduced ITDA is ultraviolet finite, which guarantees that the continuum limit exists on the lattice.

Introduction.-The B-meson physics plays a remarkable role in particle physics, both in a detailed examination of the Standard Model and in the search of new physics beyond the Standard Model. One of the most important functions describing the structure of the B-meson is its light-cone distribution amplitude (LCDA) [1]. It is an inherent part of hard-collinear factorization theorems for many exclusive B decay reactions [2][3][4][5][6][7][8], where the amplitude is factorized into a convolution of the hard scattering kernel and the B-meson LCDA. It is also an essential element in the light-cone sum-rule studies [9][10][11][12] of the B-meson decays.
The perturbative structure of the B-meson LCDA may be studied in a model-independent way, e.g., using the renormalization group equation [13][14][15][16] and constraints on the perturbative tail of the leading-twist LCDA φ + B (ω, µ) [17,18]. On the other hand, the nonperturbative aspects of B-meson LCDA has been mainly explored within models based on QCD sum rules [9,19].
A first-principle approach to study the nonperturbative aspects of the B-meson LCDA may be provided by lattice gauge simulations. However, there was not much work in this direction. The difficulties arise from the fact that, in the heavy quark effective theory (HQET), the Bmeson LCDA is defined through the matrix element of a nonlocal operator in which the heavy and light quarks are separated along the light-cone.
Thus, it cannot be calculated directly on the Euclidean lattice. Moreover, unlike in the case of the parton distribution functions of the nucleon, it is impossible to access B-meson LCDA by computing its moments, just because the operator product expansion (OPE) does not exist in this case [9]. One might propose to calculate instead the inverse moments of LCDA which are more relevant to phenomenology. However, they are not related to matrix elements of local operators.
The recent developments of parton physics on the lattice (e.g., quasi-PDFs [20][21][22], pseudo-PDFs [23][24][25], lattice cross sections [26,27]) provide the possibility of studying light-cone parton distributions directly with lattice simulations. In particular, there were attempts of accessing the leading twist B-meson LCDA within the quasi-distribution amplitude (quasi-DA) approach, either in coordinate space [28] or momentum space [29]. Although the matching relation that links quasi-DA and LCDA has been investigated, it is still not clear how one can approach the continuum limit because of the existence of ultraviolet (UV) singularities.
In this Letter, we propose to deal with the UV singularities using the pseudo-PDF approach [23]. Its essential idea is that, if the operator is multiplicatively renormalizable, one can choose a proper ratio that defines an UV finite reduced Ioffe-time distribution. To this end, we will study the short-distance behavior of the B-meson Ioffe-time distribution amplitude (ITDA) and construct a reduced ITDA.
Using the results of the one-loop calculation, we will show that the reduced-ITDA can be factorized into the position-space LCDA and a hard function. Furthermore, the UV finiteness allows the reduced ITDA calculated on the lattice to approach its continuum limit. This result is crucial for building a practical method of accessing Bmeson LCDA on the lattice.
B-meson Ioffe-time distribution amplitude.-Our starting object is a nonlocal heavy-light operator O µ (z, 0; v) ≡q(z)S(z, 0)γ µ γ 5 h v (0) in HQET, where S(z, 0) ≡ P exp[igz ν 1 0 dtA ν (tz)] is a Wilson line, and h v is the heavy quark field in HQET, with v denoting its velocity, v 2 = 1 and h v satisfying / vh v = h v . By Lorentz covariance, the meson-to-vacuum matrix element can be parametrized as where M B,v (ν, µ) and M B,z (ν, µ) are two scalar functions and ν ≡ v · z will be referred to as the "Ioffe-time" of the B-meson (note that in the QCD case the Ioffe-time is the inner product of momentum p and z [30,31]). F (µ) is the decay constant of B-meson defined by the matrix element of the local current Unlike the QCD case, decay constant in HQET is scale dependent.

arXiv:2006.05663v1 [hep-ph] 10 Jun 2020
When z 2 → 0, M B,v term gives the twist-2 distribution while M B,z is a higher-twist contribution. Because we are only interested in the leading-twist distribution at present, we rename M B,v as M B for short, and call M B (ν, −z 2 , µ) the Ioffe-time distribution amplitude (ITDA) of the B-meson.
If z is a lightlike vector, e.g., only the minus component of z is nonzero, then ITDA will reduce to the light-cone ITDA I + B (ν, µ), i.e., M B (ν, 0, µ) = I + B (ν, µ), which is actually the LCDA in position space. The B-meson LCDA that appears in the factorization theorems of B-meson exclusive decay is defined by the Fourier transform of There are no light-cone separations on the Euclidean lattice, but as proposed in Refs. [20,32], one can study equal-time separations z = (0, 0, 0, z 3 ). The same idea may also be applied for the B-meson LCDA. In this case, ν = −v 3 z 3 and z 2 = −z 2 3 . One can choose the Lorentz index µ = 0 in Eq. (1), so that the higher-twist part disappears. In the quasi-PDF-based approaches [28,29] one deals with the B-meson quasi-DAφ + B (ω, v 3 , µ) that can be expressed in terms of ITDA as A matching relation linking the quasi-DA and LCDA was derived in Ref. [29]. However, integration over the parameter z 3 present in both arguments of the ITDA M B (−v 3 z 3 , z 2 3 , µ) mixes two distinct phenomena: the ν-dependence that governs the ω-shape of the LCDA, and the z 2 3 -dependence that corresponds to the probing scale for the LCDA. For this reason, we propose to proceed along the lines of the pseudo-PDF approach [23,25] in which these phenomena are clearly separated.
Hard contribution at one loop.-Formally, the LCDA in coordinate space I + B (ν, µ) can be approached by taking z 2 → 0 limit of ITDA M (ν, −z 2 , µ). However, logarithmic dependence on z 2 will be generated when hard corrections of ITDA are included. As a result, the z 2 → 0 limit cannot be approached directly, and a perturbative matching is needed.
Under quantum correction, the hard part will be generated by gluon exchanges. As indicated in Refs. [23,33], the hard contribution can be determined at operator level. The Feynman diagrams are presented by Fig. 1. A calculation has been performed in Ref. [28], where the UV and IR singularities are regularized by dimensional regularization (DR). To distinguish the UV and IR singularities for ITDA, we will adopt Polyakov regularization [34] for UV singularities, in which the gluon propagator in coordinate representation is replaced by . Collinear singularities are regularized by the mass of light quark m; the soft singularity is regularized by DR. We will work in Feynman gauge but the results are gauge invariant. According to Eq. (1), to study the hard contribution of M B (ν, −z 2 , µ), one should consider the one-loop correction to both the nonlocal operator and decay constant. We consider the decay constant first. Note that the Bmeson decay constant in HQET is UV divergent and scale dependent, which is different from the pion decay constant case. Under Polyakov regularization, the one-loop correction to decay constant is where γ E is the Euler-Mascheroni constant and F (a) (0) denotes the decay constant before the correction. Now we turn to the one-loop hard contribution of the nonlocal operator. To begin with, we consider the heavy quark and light quark self energies. Up to one-loop, we have for heavy and light quarks, respectively. Here d = 4−2 IR is the dimension of space-time in DR, and µ IR denotes the infrared (IR) scale associated with the soft singularity 1/ IR . The self-energy of the gauge link has already been calculated in PDF case. The result reads [33,35] The heavy quark-Wilson line vertex is presented in Fig. 1(a). In HQET, the heavy quark can be expressed as a Wilson line along the v-direction. At one-loop level, the exchange of gluon between Wilson lines along v and n directions contributes × ln a 2 (ln 2iv · z − 1 2 ln(−z 2 )) Note that the exchange of the gluon between the two Wilson lines generates a cusp singularity [34], which is represented by ln a 2 . Another interesting feature here is that, because of the existence of cusp singularity, there is a double logarithmic dependence on z 2 , which is very different from the nucleon Ioffe-time distribution function case.
The light quark-Wilson line vertex is presented in Fig. 1(b). This contribution is the same as the vertex contribution in the PDF operator (see, e.g., Ref. [33]). Direct calculation gives whereū ≡ 1 − u. Fig. 1(c) represents the contribution from interaction between light and heavy quarks. To calculate this contribution, we adopt the non-relativistic approximation, where the momentum of light quark is given by p = mv. Under this approximation, we have The Lorentz structure of the type γ µ / zγ 5 has been neglected because it yields a higher-twist contribution to the ITDA. One may notice that the box diagram has no ln z 2 dependence, so that it gives the same contribution to the light-cone ITDA. This means that the box diagram does not contribute to the matching relation. This has been confirmed by the calculation in momentum space [29].
Adding all contributions together, the soft IR singularities 1/ IR , as well as the logarithmic dependence on the soft scale µ IR , are canceled. According to Eqs. (1) and (5), one can derive the one-loop hard contribution of the ITDA M B (ν, −z 2 , µ): Reduced Ioffe-time distribution amplitude.-The hard contributions above involves UV singularities that are regularized by a. In lattice computations, the matrix elements are calculated on discrete space-time. The UV singularities corresponds to the singularities in the continuum limit (i.e., the lattice spacing a → 0). Although the ITDA can be computed on the lattice, however, the UV divergences obstruct to approach the result in continuum space-time from lattice data. Thus one should renormalize the UV singularities for a practical lattice evaluation.
Based on the auxiliary field formalism [36], it has been shown that the off-light-cone operator defining the Bmeson quasi-DA is multiplicatively renormalizable [29]. So, the bare and renormalized operators are related by (12) where Z is a renormalization factor and Λ denotes a cutoff. A similar equation can be written down for the decay constant. The operator with a superscript "R" denote the renormalized operator while the operator without it denotes a bare one.
Since the renormalization relation holds at operator level, it is valid for any matrix element of the operator. For example, one can replace B-meson state with the leading Fock state of B-meson. Similar to the definition of B-meson ITDA, such matrix element can be parameterized as where f (µ) and m B (ων, z 2 ) are the "decay constant" and ITDA of the Fock state |b(v)q(ωv) , respectively; ωv is the momentum carried by the light quark. Note that the higher-twist contribution that is proportional to z α has been neglected. f (µ) is defined through matrix element of local operator As discussed above, the UV divergence only depends on the operator, i.e., the matrix elements of hadron state and its Fock state should involve the same UV structure. Furthermore, the HQET operator is multiplicatively renormalizable, so the ratios of hadron and Fockstate matrix elements should be UV finite: .
These relations indicate that for the ratio of meson and Fock-state ITDAs, the continuum limit exists on the lattice, therefore the ratio can be evaluated with lattice simulations. For the sake of simplicity, we define a reduced ITDA M (ν, −z 2 ) by dividing Fock-state ITDA at ω = 0: Because Eq. (11) is an operator relation, one can immediately get the one-loop correction to the denominator by taking matrix element. Then, the one-loop correction of the reduced ITDA is It is easy to see that the linear and logarithm divergences that related to the link are canceled; the cusp singularity that arises from the gluon exchange between heavy quark and link is canceled as well. This indicates the UV finiteness of the reduced ITDA. Thus it can be evaluated on the lattice, and the continuum limit can be approached. Furthermore, the one-loop correction of the reduced ITDA is a plus-distribution. If we define a quasi-DA from the reduced ITDA by taking Fourier transform with z 3 , this will lead to a rapidly decreasing behavior of the corresponding quasi-DA φ(ω, v 3 , µ) at large ω.
On the lattice, the denominator in the reduced ITDA will be evaluated nonperturbatively. However, at short distances, it can be calculated in perturbation theory. Thus, this ratio defines a nonperturbative renormalization scheme for the B-meson ITDA.
Matching relation.-The reduced ITDA and the MS LCDA can be linked by a matching relation. Similar to the PDF case, one can use the nonlocal light-cone OPE.
To determine the hard function, we also need the oneloop correction to the light-cone ITDA, which can be extracted from the one-loop correction to the light-cone operator [37]. The result reads One can find that the singularities regularized by ln m 2 are the same for the reduced-ITDA and light-cone ITDA. Then, a matching formula for reduced ITDA and lightcone ITDA can be written down: whereμ ≡ µe γ E . We have chosen z = (0, 0, 0, z 3 ) so that the reduced ITDA can be computed on the lattice. This relation allows one to convert the reduced ITDA calculated on the lattice to the LCDA in coordinate representation. The Fourier transformation of the latter enters the factorization theorems of B-meson exclusive decay. Finally, let us take a look at the evolution equations for the ITDAs. Since the light-cone ITDA does not depend on z 2 , one can write down the z 2 -evolution equation for the reduced ITDA: On the other hand, the reduced ITDA does not depend on µ. Thus, by taking the derivative with respect to ln µ on both sides, one can get the renormalization group equation for light-cone ITDA: By including the anomalous dimension of the decay constant, one will find that the above equation reproduces the RGE for heavy-light light-cone operator (see, e.g., Ref. [37,38]). Implementations on the lattice.-In recent lattice calculations, the typical lattice spacing a is around 0.1 fm.
The Compton wavelength of the b-quark is much smaller than the lattice spacing, m b 1/a. Hence HQET is a natural framework to study B-meson on the lattice.
The renormalization of full HQET is complicated, even in continuum theory. In fact, the full HQET Lagrangian is not renormalizable. However, since the operator we are going to measure is taken in the m b → ∞ limit, we can restrict ourselves to the static approximation of HQET. At the lowest order of 1/m b expansion, there will be no higher dimensional operators getting mixed in, and the renormalization property is simple. The higher dimensional operators in lattice theory can also been excluded under static approximation, hence the continuum limit of the reduced ITDA can be approached, without considering the operator mixing.
Similar to the lattice calculation of hadron Ioffe-time distribution functions [24], a possible way to get the ITDA is to calculate M B (−v 3 z 3 , z 2 3 ) for several values of v 3 , and then to fit the data by a function of ν and z 2 3 . A proper framework might be the leading-order moving HQET [39] or moving NRQCD [40]. The meson and Fock state decay constants should also be calculated on the lattice, or using the phenomenological result. The Fock state ITDA at ω = 0, i.e., m B (0 · ν, z 2 3 ) necessary for the construction of reduced ITDA should be calculated on the lattice as well.
We note that, in practical lattice HQET, a result with large noise-to-signal ratio might be expected. Still, a rough evaluation of the B-meson LCDA with lattice methods is of great value because there is very little knowledge on the B-meson LCDA, even from first principle calculations.
Summary.-To access B-meson leading-twist lightcone distribution amplitude from lattice QCD computations, we have proposed the approach based on the strategy of reduced Ioffe-time distributions. The reduced Ioffe-time distribution amplitude of B-meson is constructed by the ratio of meson ITDA and the ITDA of meson's leading Fock state, in which the light-quark momentum is zero. According to the multiplicative renormalizability of the off-light-cone operator, the ratio is UV finite, hence one can approach its continuum limit from the lattice data. A matching relation that maps LCDA to the reduced ITDA is also derived. These results provide a basis for a practical computation of B-meson LCDA with lattice methods. This work is supported by Jefferson Science Associates, LLC under U.S. DOE Contract #DE-AC05-06OR23177 and by U.S. DOE Grant #DE-FG02-97ER41028.