A One-loop corrections to \phi_{\pi}(u)

# Modelling light-cone distribution amplitudes from non-relativistic bound states

TTP08-07

SFB/CPP-08-13

SI-HEP-2007-20

Modelling light-cone distribution amplitudes

[0.5em] from non-relativistic bound states

G. Bell and Th. Feldmann1

Institut für Theoretische Teilchenphysik,

Universität Karlsruhe, D-76128 Karlsruhe, Germany

Fachbereich Physik, Theoretische Physik I,

Universität Siegen, Emmy Noether Campus, D-57068 Siegen, Germany

Abstract

We calculate light-cone distribution amplitudes for non-relativistic bound states, including radiative corrections from relativistic gluon exchange to first order in the strong coupling constant. We distinguish between bound states of quarks with equal (or similar) mass, , and between bound states where the quark masses are hierarchical, . For both cases we calculate the distribution amplitudes at the non-relativistic scale and discuss the renormalization-group evolution for the leading-twist and 2-particle distributions. Our results apply to hard exclusive reactions with non-relativistic bound states in the QCD factorization approach like, for instance, or . They also serve as a toy model for light-cone distribution amplitudes of light mesons or heavy and mesons, for which certain model-independent properties can be derived. In particular, we calculate the anomalous dimension for the meson distribution amplitude in the Wandzura-Wilczek approximation and derive the according solution of the evolution equation at leading logarithmic accuracy.

## 1 Introduction

Exclusive hadron reactions with large momentum transfer involve strong interaction dynamics at very different momentum scales. In cases where the hard-scattering process is dominated by light-like distances, the long-distance hadronic information is given in terms of so-called light-cone distribution amplitudes (LCDAs) which are defined from hadron-to-vacuum matrix elements of non-local operators with quark and gluon field operators separated along the light-cone [1, 2] and [3, 4]. LCDAs appear in the so-called pQCD approach to hard exclusive reactions [5, 6, 7], in the QCD factorization approach to heavy-to-light transitions [8], in soft-collinear effective theory [9, 10], as well as in the light-cone sum rule approach to exclusive decay amplitudes [11, 12, 13] (for a recent review, see [14]).

Representing universal hadronic properties, LCDAs can either be extracted from experimental data, or they have to be constrained by non-perturbative methods. The most extensively studied and probably best understood case is the leading-twist pion LCDA, for which phenomenological constraints [15, 16, 17] from the transition form factor [18], as well as estimates for the lowest moments from QCD sum rules [2, 19, 20] and lattice QCD [21, 22] exist. On the other hand, our knowledge on LCDAs for heavy mesons [3, 23, 24], and even more so for heavy quarkonia [25, 26], had been relatively poor until recently.

Although LCDAs, in general, are not calculable in QCD perturbation theory, their evolution with the factorization scale (which is set by the momentum transfer of the hard process) can be calculated and is well understood, both, for light mesons [5] and for heavy mesons [23]. The situation becomes somewhat simpler, if the hadron under consideration can be approximated as a non-relativistic bound state of two sufficiently heavy quarks. In this case we expect exclusive matrix elements – like transition form factors [27] and, in particular, the LCDAs – to be calculable perturbatively, since the quark masses provide an intrinsic physical infrared regulator.

In this article, we are going to calculate the LCDAs for non-relativistic meson bound states including relativistic QCD corrections to first order in the strong coupling constant at the non-relativistic matching scale which is set by the mass of the lighter quark in the hadron. We discuss twist-2 and twist-3 LCDAs for 2-particle Fock states with approximately equal quark masses (for instance an meson), as well as 2-particle and 3-particle LCDAs for heavy mesons (like the ), where one of the quark masses is considered to be much larger than the second one (). Our results can also be viewed as a toy model for possible parameterizations of LCDAs for relativistic bound states, like the pion, kaon or meson at a low input scale, which may be evolved to the appropriate higher scales using the standard renormalization group equations in QCD (or HQET, respectively).

Our paper is organized as follows. In the following section we give a short introduction to the non-relativistic approximation and collect the definitions and properties of LCDAs for light and heavy mesons. The main result of our paper, the corrections from relativistic gluon exchange, are presented in Section 3. Here we also derive model-independent results for the meson LCDAs and , namely the cut-off dependence of positive moments and the anomalous dimension kernels, and investigate the impact of the 3-particle LCDAs to the Wandzura-Wilczek approximation beyond tree-level. We discuss the effect of QCD evolution above the non-relativistic matching scale in Section 4, including a new result for the meson LCDA , before we conclude. Some technical details of the calculation are collected in an appendix. Some of our results have already appeared in a proceedings article [28].

## 2 Light-cone distribution amplitudes and the non-relativistic limit

### 2.1 Non-relativistic approximation

The wave function for a non-relativistic (NR) bound state of a quark and an antiquark with respective masses and can be obtained from the resummation of NR (potential) gluon exchange as sketched in Figure 1. The solution of the corresponding Schrödinger equation with Coulomb potential yields

 ψC(→p) ∝κ5/2(κ2+|→p|2)2, (1)

where and is the reduced mass. The normalization of the wave function gives the (non-relativistic) meson decay constant

 fNR =2√Ncπκ3/2(m1+m2)1/2. (2)

For more details and references to the original literature, see e.g. [29] (also [27]).

In this approximation, the meson is entirely dominated by the 2-particle Fock state built from a bottom quark with mass and a charm antiquark with mass . Consequently to first approximation in the NR expansion, the meson consists of a quark with momentum and an antiquark with momentum , where is the four-velocity of the meson (). The spinor degrees of freedom for the meson are represented by the Dirac projector . Similarly, a pseudoscalar meson is interpreted as a bound state where both constituents have approximately equal momenta .

The non-relativistic approximation can also serve as a toy model for bound states of light (relativistic) quarks. We will in the following refer to “heavy mesons” as ”” (where we mean the realistic example of a meson, or the toy model for a meson) and “light mesons” as ”” (where the realistic example is , and the toy-model application would be the pion or also the kaon for ).

### 2.2 Definition of LCDAs for light pseudoscalar mesons

Following [1, 2] we define the 2-particle LCDAs of a light pseudoscalar meson via

 ⟨π(P)|¯q1(y)[y,x]γμγ5q2(x)|0⟩ =−ifπ∫10duei(up⋅y+¯up⋅x)[pμϕπ(u)+m2π2p⋅zzμgπ(u)], ⟨π(P)|¯q1(y)[y,x]iγ5q2(x)|0⟩ =fπμπ∫10duei(up⋅y+¯up⋅x)ϕp(u), ⟨π(P)|¯q1(y)[y,x]σμνγ5q2(x)|0⟩ =ifπ~μπ(pμzν−pνzμ)∫10duei(up⋅y+¯up⋅x)ϕσ(u)2D−2 (3)

with two light-like vectors and , and denoting the light-cone momentum fraction of the quark . The gauge link factor is denoted as

 [y,x]=Pexp[igs∫10dt(y−x)⋅A(ty+(1−t)x)]. (4)

is the twist-2 LCDA, while and are twist-3. For completeness, we have also quoted the twist-4 LCDA which, like the 3-particle LCDAs, will not be considered further in this work.2 All LCDAs are normalized to 1, such that the prefactors in (3) are defined in the local limit . In the definition of , we have included a factor , such that the relation between and from the equations of motion (see below) is maintained in dimensions.

#### Equations of motion

The equations of motion (eom) provide relations between the matrix elements defined in (3). Following [2] we obtain

 m2π2[ϕπ(u)+gπ(u)] =(m1+m2)μπϕp(u)+…, μπϕp(u)+~μπD−1[(2−D)ϕσ(u)+2u−12ϕ′σ(u)] =(m1+m2)ϕπ(u)+…, (2u−1)μπϕp(u)+~μπ2D−2ϕ′σ(u) =(m1−m2)ϕπ(u)+…, (5)

where the ellipsis denote contributions from 3-particle LCDAs which we do not specify here. In the local limit the contributions from the 3-particle LCDAs drop out and integration of (5) yields

 μπ =m2πm1+m2,~μπ=μπ−(m1+m2) (6)

and

 ∫10duuϕp(u) =12+m1−m22μπ. (7)

Notice that the relations (5,6,7) hold for the bare (unrenormalized) parameters and distribution amplitudes.

#### Tree-level result

At tree level, and in leading order of the expansion in the relative velocities, the quark and the antiquark in the NR wave function simply share the momentum of the meson according to their masses, . For ”light” mesons this implies3

 ϕπ(u) ≃ϕp(u)≃gπ(u)≃δ(u−u0), (8)

with and . Consequently, all positive and negative moments of the distribution amplitudes are simply given in terms of the corresponding power of . In particular, the Gegenbauer coefficients are given by

 an =2(2n+3)3(2+n)(1+n)∫10duϕπ(u)C(3/2)n(2u−1)→2(2n+3)3(2+n)(1+n)C(3/2)n(2u0−1). (9)

Notice that at tree-level and the corresponding LCDA can only be determined by considering the corresponding one-loop expressions (see below). The tree-level solutions (8) fulfill the eom-constraints from (5).

### 2.3 Definition of LCDAs for heavy pseudoscalar mesons

We define the 2-particle LCDAs of a heavy pseudoscalar meson following [3, 4],

 ⟨0|(¯q)β(z)[z,0](hv)α(0)|B(Mv)⟩ =−i^fB(μ)M4⎡⎣1+v/2⎧⎨⎩2~ϕ+B(t)+~ϕ−B(t)−~ϕ+B(t)tz/⎫⎬⎭γ5⎤⎦αβ, (10)

where is the heavy meson’s velocity, and . Here is the (renormalization-scale dependent) decay constant in HQET. The Fourier-transformed expressions, which usually appear in factorization formulas, are given through

 ~ϕ±B(t) =∫∞0dωe−iωtϕ±B(ω), (11)

where denotes the light-cone energy of the light quark in the meson rest frame.

#### Equations of motion

The equations of motion again provide relations between different LCDAs. Including the effect of the 3-particle LCDAs as defined in [31] (see also [32, 33]), we derive

 ωϕ−B(ω)−mϕ+B(ω)+D−22∫ω0dη[ϕ+B(η)−ϕ−B(η)] =(D−2)∫ω0dη∫∞ω−ηdξξ∂∂ξ[ΨA(η,ξ)−ΨV(η,ξ)]. (12)

The relation (12) is trivially fulfilled at tree-level and we will show below that it also holds after including the corrections to the NR limit. In [31], Kawamura et al. discuss a second relation which in the massive case reads

 (ω+m)ϕ−B(ω)+(ω−2¯Λ−m)ϕ+B(ω) \lx@stackrel?=−2ddω∫ω0dη∫∞ω−ηdξξ[ΨA(η,ξ)+XA(η,ξ)]−2(D−2)∫ω0dη∫∞ω−ηdξξ∂ΨV(η,ξ)∂ξ, (13)

with . We will show below that the equation (13) does not hold beyond tree level, since the integral on the right-hand side involving our result for the 3-particle LCDA does not converge. This confirms the criticism raised in [24, 34] that (13) is not consistent, since the renormalization prescription of light-cone operators in HQET and the expansion into local operators do not commute. Notice that in contrast to (12), the derivation of (13) involves derivatives with respect to .

If one neglects the 3-particle distribution amplitudes in (12), one arrives at the so-called Wandzura-Wilczek relation which has first been discussed for a massless light quark in [4]. The generalization to the massive case reads

 ∫ω0dη[ϕ−B(η)−ϕ+B(η)] ≃2D−2[ωϕ−B(ω)−mϕ+B(ω)], (14)

which again holds for the bare parameters and LCDAs in dimensions.

#### Tree-level result

By the same arguments as for light mesons, at tree-level the quark and the antiquark in a heavy meson just share the total momentum according to their masses, such that . In the NR limit, the 2-particle LCDAs of a ”heavy” meson are thus given by

 ϕ+B(ω) ≃ϕ−B(ω)≃δ(ω−m). (15)

Moreover, at tree level, the moments of the heavy meson LCDAs can be related to matrix elements of local operators in HQET [3]. The zeroth moment determines the tree-level normalization of the distribution amplitudes . For the first moment, one has the general decomposition

 ⟨0|(¯q)βi←Dμ(hv)α|B(v)⟩ ≃−iM^fB4[(avμ+bγμ)(1+v/)γ5]αβ. (16)

Multiplying by and taking into account the finite light quark mass in the NR set-up, the equation of motion for the light quark implies . The equation of motion for the heavy quark is obtained by multiplying with , from which one obtains , independent of the light-quark mass. This implies

 a=4¯Λ−m3,b=−¯Λ−m3.

From this we can read off the first moments at tree-level

 ⟨ω⟩+ ≃i^fBM⟨0|¯qγ5n/−(in−←D)hv|B⟩=a=4¯Λ−m3, (17) ⟨ω⟩− ≃i^fBM⟨0|¯qγ5n/+(in−←D)hv|B⟩=2b+a=2¯Λ+m3, (18)

where we introduced the light-like vectors and . In the non-relativistic limit , and we obtain

 ⟨ω⟩±≃m.

Notice that the light-quark mass drops out in the sum

 ⟨ω⟩++⟨ω⟩−=2¯Λ.

We stress that the relation between moments of and local matrix elements in HQET does not hold beyond the tree-level approximation [23, 24, 35].

## 3 Relativistic corrections at one-loop

The NR bound states are described by parton configurations with fixed momenta. Relativistic gluon exchange as in Figure 2 leads to modifications: First, there is a correction from matching QCD (or, in the case of heavy mesons, the corresponding low-energy effective theory HQET) on the NR theory. Secondly, there is the usual evolution under the change of the renormalization scale [5, 23]. In particular, the support region for the parton momenta is extended to for light mesons and for heavy mesons. In this section we collect the results for LCDAs for “light” and ”heavy” mesons including the first-order matching corrections from relativistic gluon exchange

 ϕM =ϕ(0)M+αsCF4πϕ(1)M+O(α2s). (19)

### 3.1 Light mesons

#### Local matrix elements

We first consider the leading-order relativistic corrections to the local matrix elements which are given by the vertex-correction and the wave-function renormalization of the quark fields. We find

 fπ =fNRπ[1+αsCF4π(−6+3 m1−m2m1+m2lnm1m2)+O(α2s)] (20)

and

 μπ=m2πZosm1m% os1+Zosm2mos2 =mπ[1+αsCF4π(3ε+3lnμ2m1m2−3 m1−m2m1+m2lnm1m2+4)+O(α2s)], ~μπ=μπ−m2πμπ =mπ[αsCF4π(6ε+6lnμ2m1m2−6 m1−m2m1+m2lnm1m2+8)+O(α2s)], (21)

where in the on-shell scheme. Our result for the decay constant is in agreement with [36] and the results for and are consistent with the eom-constraints in (6).

#### The twist-2 LCDA ϕπ(u)

Let us start with the case of equal quark masses, e.g. in case of a non-relativistic bound state, which may also serve as a toy-model for the pion LCDA.4

The first-order relativistic corrections arise from the collinear gluon exchange diagrams in Figure 2, where we also have to take into account the wave-function renormalization of the external quark lines (see Appendix A for details). The local limit of the light-cone matrix element (3) determines the relativistic corrections to the NR decay constant (20) (in this case, the diagrams with the gluon attached to the Wilson-line do not contribute). The remaining contributions to the NLO correction for the leading-twist LCDA contain an UV-divergent piece,

 ϕ(1)π(u)∣∣div. =2ε∫10dvV(u,v)ϕ(0)(v), (22)

which involves the well-known Brodsky-Lepage evolution kernel [5],

 V(u,v) =[(1+1v−u)uv θ(v−u)+(1+1¯v−¯u)¯u¯v θ(u−v)]+. (23)

The finite terms after -subtraction read

 ϕ(1)π(u;μ) =4{(lnμ2m2π(1/2−u)2−1)[(1+11/2−u)uθ(1/2−u)+(u↔¯u)]}+ +4{u(1−u)(1/2−u)2}++. (24)

Here the plus-distributions are defined as

 ∫10du{…}+f(u) ≡∫10du{…}(f(u)−f(1/2)), (25) ∫10du{…}++f(u) ≡∫10du{…}(f(u)−f(1/2)−f′(1/2)(u−1/2)). (26)

From this it follows that

 ∫10duϕ(1)π(u;μ)=∫10duuϕ(1)π(u;μ)=0,

such that the general normalization conditions and are not changed. Furthermore, our result for the distribution amplitude obeys the evolution equation

 ddlnμϕπ(u;μ) =αsCFπ∫10dvV(u,v)ϕπ(v;μ)+O(α2s). (27)

An independent calculation of the leading-twist LCDAs for the and meson has been presented in [25]. Our result is not in complete agreement with these findings. In particular, we find that the LCDA quoted in [25] is not normalized to unity as it should be.

On the other hand, at the non-relativistic scale , the distribution amplitude shows a singular behaviour at . As a consequence, the convergence of the Gegenbauer expansion is not very good at the non-relativistic scale, with the Gegenbauer coefficients in (9) only falling off as (and alternating signs). A better characterization of the LCDA at is given in terms of the moments

 ⟨ξn⟩π(μ)≡ ∫10du(2u−1)nϕπ(u;μ), (28)

which are linear combinations of Gegenbauer coefficients of order . This corresponds to an expansion of the LCDA in terms of a delta-function and its derivatives,

 ϕπ(u;μ) =2∞∑n=0(−1)nn!δ(n)(2u−1)⟨ξn⟩π(μ). (29)

Results for the first few moments are shown in Table 1 for the strict non-relativistic limit, including the NLO corrections from (24) and comparing with the non-relativistic corrections of order discussed by Braguta et al. in [26]. Keeping first-order corrections in only, this formally amounts to the replacement

 ϕNRπ(u) = δ(u−1/2)+v2NR24δ′′(u−1/2)+O(v4NR). (30)

In particular, this fixes the moment . The authors [26] propose a resummed formula,

 ϕNRπ(u) → 1vNRθ(u−1−vNR2)θ(1+vNR2−u). (31)

The comparison in Table 1 shows that for , the effect of the corrections is qualitatively and quantitatively similar to the corrections from (24).

It is also interesting to determine the correction to the first inverse moment of the LCDA which appears in QCD factorization formulas

 ⟨u−1⟩(1)π(μ) ≡∫10duϕ(1)π(u;μ)u≃3(2.73+1.08lnμ2m2). (32)

Finally, we quote the result for the derivative of at the endpoints

 ϕ′π(0;μ)=−ϕ′π(1;μ) =αsCF4π(4+12lnμ2m2)+O(α2s), (33)

which is sometimes discussed in the context of non-factorizable contributions to hard exclusive reactions [37, 38].

For non-equal quark masses, the NLO corrections to the -renormalized twist-2 LCDA are given by

 ϕ(1)K(u;μ) =2{(lnμ2m2K(u0−u)2−1)[(1+1u0−u)uu0θ(u0−u)+(u↔¯uu0↔¯u0)]}+ +4{u(1−u)(u0−u)2}+++2δ′(u−u0)(2u0(1−u0)lnu01−u0+2u0−1). (34)

The first moment now becomes

 ∫10duuϕK(u;μ) =u0+αsCF4π[(−43lnμ2u20m2K−7(1−u0)3lnu20−389)u0−(u0↔¯u0)]. (35)

#### 2-particle LCDAs of twist-3

The twist-3 LCDAs for the 2-particle Fock states are obtained in the same way as the twist-2 one. After absorbing the corrections to the local matrix elements into the renormalized values for and , we obtain a UV-divergent piece

 ϕ(1)p(u)∣∣div. =2ε[(1+1u0−u)θ(u0−u)+(u↔¯uu0↔¯u0)]+ (36)

and a finite NLO contribution to the twist-3 LCDA associated to the pseudoscalar current

 ϕ(1)p(u;μ) =2{(lnμ2m2K(u0−u)2−1)[(1+1u0−u)θ(u0−u)+(u↔¯uu0↔¯u0)]}+ +4u0(1−u0)({1(u0−u)2}+++δ′(u−u0)lnu01−u0)+2{2u0−1(u0−u)}+. (37)

In particular, the first moment of now reads

 ∫10duuϕp(u;μ) =u0+αsCF4π[(−3lnμ2m2K+6u0lnu0−4)u0−(u0↔¯u0)], (38)

which is in agreement with the eom-constraint from (7). At the endpoints we now have

 ϕp(0;μ) =αsCF4π(2+2u0u0lnμ2m21−2)+O(α2s) (39)

and similar for with , i.e. . For the twist-3 LCDA associated to the pseudotensor current (whose normalization factor starts at order ), we simply have

 ϕσ(u) =2[uu0θ(u0−u)+(u↔¯uu0↔¯u0)]+O(αs). (40)

In contrast to the other 2-particle LCDAs in (8), we find that is not given by a delta-like distribution in the NR limit and has support for .

### 3.2 Heavy mesons

The calculation of the LCDAs for a meson (which again can be considered as a toy model for LCDAs of mesons with ) goes along the same lines as for the case. However, important differences arise because the heavy -quark is to be treated in HQET which modifies the divergence structure of the loop integrals (notice that in our set-up, a charm quark in a meson is treated as ”light”). As a consequence, the evolution equations for the LCDAs of heavy mesons [23] differ from those of light mesons.

#### The LCDA ϕ+B(ω)

Let us first focus on the distribution amplitude which enters the QCD factorization formulas for exclusive heavy-to-light decays. In the local limit we derive the corrections from soft gluon exchange to the decay constant in HQET. We find

 ^fM(μ) =fNRM[1+αsCF4π(3lnμm−4)+O(α2s)]. (41)

Notice that the decay constant of a heavy meson exhibits the well-known scale dependence in HQET [39]. The remaining NLO corrections to the distribution amplitude contain an UV-divergent piece (details of the derivation can be found in Appendix B)

 ϕ(+,1)B(ω;μ)∣∣div. =2ωϵ[θ(m−ω)m(m−ω)+θ(ω−m)ω(ω−m)]+−δ(ω−m)[1ϵ2−1ϵ(1−lnμ2m2)] (42)

and a finite piece

 ϕ(+,1)B(ω;μ)ω =2[(ln[μ2(ω−m)2]−1)(θ(m−ω)m(m−ω)+θ(ω−m)ω(ω−m))]++4[θ(2m−ω)(ω−m)2]++ +4θ(ω−2m)(ω−m)2−δ(ω−m)m(12ln2μ2m2−lnμ2m2+3π24+2) (43)

with an analogous definition of plus-distributions as in (25,26). Notice that, in order to separate the UV divergence coming from the longitudinal momentum integration, we have introduced an auxiliary parameter to split the support region of the LCDA into two parts. The distribution amplitude in (43) obeys the evolution equation

 ddlnμϕ+B(ω;μ) =−αsCF4π∫∞0dω′γ(1)+(ω,ω′;μ)ϕ+B(ω′;μ)+O(α2s), (44)

where the anomalous dimension can be read off the UV-divergent terms in (42) and is given by [23]

 γ(1)+(ω,ω′;μ) =(Γ(1)cusplnμω−2)δ(ω−ω′)−Γ(1)cuspω[θ(ω′−ω)ω′(ω′−ω)+θ(ω−ω′)ω(ω−ω′)]+ (45)

with .

In contrast to the light meson case, the normalization of the heavy meson distribution amplitude is ill-defined. Imposing a hard cutoff and expanding to first order in , we find

 ΛUV∫0dωϕ+B(ω;μ) ≃1−αsCF4π[12ln2μ2Λ2UV+lnμ2Λ2UV+π212]+O(α2s)+O(m/ΛUV) (46)

and similarly for the first moment

 1ΛUVΛUV∫0dωωϕ+B(ω;μ) ≃αsCF4π[2lnμ2Λ2UV+6]+O(α2s)+O(m/ΛUV). (47)

The last two expressions provide model-independent properties of the distribution amplitude which have been studied within the operator product expansion in [35]. Our results are in agreement with these general findings.

We finally quote our result for two phenomenologically relevant moments in the factorization approach to heavy-to-light decays [24, 35]

 (λB(μ))−1≡∞∫0dωϕ+B(ω;μ)ω =1m(1−αsCF4π[12ln2μ2m2−lnμ2m2+3π24−2])+O(α2s), (48)

and

 σB(μ) ≡σ(1)B(μ)=lnμm+αsCF4π[8ζ3]