Double handbag description of proton-antiproton annihilationinto a heavy meson pair

# Double handbag description of proton-antiproton annihilation into a heavy meson pair

Alexander T. Goritschnig    Bernard Pire Centre de Physique Théorique, École Polytechnique, CNRS, 91128 Palaiseau, France    Wolfgang Schweiger Institut für Physik, Universität Graz, A-8010 Graz, Austria
July 10, 2019
###### Abstract

We propose to describe the process in a perturbative QCD motivated framework where a double-handbag hard process factorizes from transition distribution amplitudes, which are quasiforward hadronic matrix elements of operators, where q denotes light quarks and c denotes the heavy quark. We advocate that the charm-quark mass acts as the large scale allowing this factorization. We calculate this process in the simplified framework of the scalar diquark model and present the expected cross sections for the PANDA experiment at GSI-FAIR.

###### pacs:
13.75.-n, 13.85.Fb, 25.43.+t

## I Introduction

The collinear factorization framework allows us to calculate a number of hard exclusive amplitudes in terms of perturbatively calculable coefficient functions and nonperturbative hadronic matrix elements of light-cone operators. The prime example is the calculation of the deeply virtual Compton-scattering amplitude in the handbag approximation with generalized parton distributions, nonforward matrix elements of a quark-antiquark nonlocal operator between an incoming and an outgoing baryon state. Strictly speaking, this description is only valid in a restricted kinematical region, called the generalized Bjorken scaling region, for a few specific reactions, and in the leading-twist approximation. It is, however, suggestive to extend this framework to the description of other reactions where the presence of a hard scale seems to justify the factorization of a short-distance dominated partonic subprocess from long-distance hadronic matrix elements. Such an extension has, in particular, been proposed in Ref. Goritschnig:2009sq () for the reaction with nucleon to charmed baryon generalized parton distributions. The extension of the collinear factorization framework to the backward region of deeply virtual Compton-scattering and deep exclusive meson production Pire:2004ie (); Lansberg:2012ha () leads to the definition of transition distribution amplitudes (TDAs) as nonforward matrix elements of a three-quark nonlocal operator between an incoming and an outgoing state carrying a different baryon number. Here, too, this description is likely to be valid in a restricted kinematical region, for a few specific reactions, and in the leading twist approximation. We propose here to extend the approach of Ref. Goritschnig:2009sq () to the reaction which will be measured with the PANDA PANDA () detector at GSI-FAIR. For this process the baryon number exchanged in the t-channel implies that hadronic matrix elements with operators enter the game. Let us stress that we have no proof of the validity of this approximation but take it as an assumption to be confronted with experimental data. For this approach to be testable, one needs to model the occurring nucleon to charmed meson TDAs. In contrast to the TDAs, which have been much discussed PSS (), we do not have any soft meson limit to normalize these TDAs. We will rather use an overlap representation in the spirit of Ref. Diehl:2000xz ().

The kinematical situation for scattering is sketched in Fig. 1. The momenta and helicities of the incoming proton and antiproton are denoted by , and , and the momenta of the outgoing and by and , respectively.  The mass of the proton is denoted by and that of the by . We choose a symmetric center-of-momentum system (CMS) in which the longitudinal direction is defined by the average momentum of  the incoming proton and the outgoing , respectively. The transverse momentum transfer is symmetrically shared between the incoming and outgoing hadrons.

In light-cone coordinates the hadronic momenta are parameterized as follows,

 p=[(1+ξ)¯p+,m2+Δ2⊥/42(1+ξ)¯p+,−Δ⊥2],p′=[(1−ξ)¯p+,M2+Δ2⊥/42(1−ξ)¯p+,+Δ⊥2],q=[m2+Δ2⊥/42(1+ξ)¯p+,(1+ξ)¯p+,+Δ⊥2],q′=[M2+Δ2⊥/42(1−ξ)¯p+,(1−ξ)¯p+,−Δ⊥2], (1)

where we have introduced sums and differences of the hadron momenta,

 ¯p := 12(p+p′),¯q := 12(q+q′)andΔ := p′−p=q−q′. (2)

The minus momentum components can be obtained by using the on-mass shell conditions and . The skewness parameter gives the relative momentum transfer in the plus direction, i.e.,

 ξ := p+−p′+p++p′+=−Δ+2¯p+. (3)

The Mandelstam variable is given by

 s= (p+q)2= (p′+q′)2. (4)

In order to produce a pair, must be larger than . The remaining Mandelstam variables, and , read

 t= Δ2= (p′−p)2= (q−q′)2 (5)

and

 u= (q′−p)2= (p′−q)2, (6)

so that . For later convenience we also introduce the abbreviations

 Λm:= √1−4m2/sandΛM:= √1−4M2/s. (7)

For further relations between the kinematical quantities, see Appendix A.

## Iii Double Handbag Mechanism

The double handbag mechanism which we use to describe is shown in Fig. 2.

It is understood that the proton emits an diquark with momentum and the antiproton a -diquark with momentum . They undergo a scattering with each other, i.e. they annihilate in our case into a gluon which subsequently decays into the heavy pair. Those produced heavy partons, characterized by , and , , are reabsorbed by the remnants of the proton and the antiproton to form the and the , respectively. One could, of course, also think of vector-diquark configurations in the proton and annihilation to produce the pair. But in common diquark models of the proton it is usually assumed that the probability to find a diquark is smaller than the one for the diquark. Further suppression of diquarks as compared to diquarks occurs in hard processes via diquark form factors at the diquark-gluon vertices Jakob:1993th (). We thus expect that our final estimate of the cross section will not be drastically altered by the inclusion of vector-diquark contributions and we stick to the simpler scalar diquark model.

The whole hadronic four-momentum transfer is also exchanged between the active partons in the partonic subprocess

 S[ud](k1)¯¯¯¯¯¯¯¯¯¯¯¯¯S[ud](k2)→¯c(k′1,λ′1)c(k′2,λ′2). (8)

In Eq. (8) we neglect the mass of the (anti)diquark, but take into account the heavy (anti-) charm-quark mass . In order to produce the heavy pair, the Mandelstam variable of the partonic subprocess has to be

 ^s≥4m2c, (9)

where . We have taken the (central) value for the charm-quark mass from the Particle Data Group PDG (), which gives . Thus, the heavy-quark mass is a natural intrinsic hard scale which demands that the intermediate gluon has to be highly virtual. This allows us to treat the partonic subprocess perturbatively, even at small , by evaluating the corresponding Feynman diagram. All the other non-active partons inside the parent hadrons are unaffected by the hard scattering and thus act as spectators.

For the double handbag mechanism the hadronic amplitude can be written as

 Mμν=∑a(′)i∑α′i∫d4¯k1θ(¯k+1)∫d4z1(2π)4ei¯k1z1∫d4¯k2θ(¯k−2)∫d4z2(2π)4ei¯k2z2×⟨¯¯¯¯¯¯¯D0:p′|TΨca′1α′1(−z1/2)ΦS[ud]a1(+z1/2)|p:p,μ⟩~Ha(′)iα′i(¯k1,¯k2)×⟨D0:q′|TΦS[ud]†a2(+z2/2)¯Ψca′2α′2(−z2/2)|¯p:q,ν⟩, (10)

where the assignment of momenta, helicities, etc., can be seen in Fig. 2. and denote color and spinor indices, respectively. In analogy to the hadronic level we have introduced the average partonic momenta , , of the active partons. We note once more that the full hadronic momentum transfer is also transferred between the active partons, i.e. . The hard scattering kernel, denoted by , describes the hard subprocess. The soft part of the transition is encoded in the Fourier transform of a hadronic matrix element which is a time-ordered, bilocal product of a quark and a diquark field operator:

 ∫d4z1(2π)4ei¯k1z1⟨¯¯¯¯¯¯¯D0:p′|TΨca′1α′1(−z1/2)ΦS[ud]a1(+z1/2)|p:p,μ⟩. (11)

In Eq. (11) takes out an diquark from the proton state at the space-time point . The diquark then takes part in the hard partonic subproces. The reinserts the quark at into the remnant of the proton which gives the desired final hadronic state . At this stage the appropriate time-ordering of the quark field operators (denoted by the symbol ) has to be taken into account. The remnant of the proton, which does not participate in the hard partonic subprocess, constitutes the spectator system. For the transition we have the Fourier transform

 ∫d4z2(2π)4ei¯k2z2⟨D0:q′|TΦS[ud]†a2(+z2/2)¯¯¯¯Ψca′2α′2(−z2/2)|¯p:q,ν⟩, (12)

which can be interpreted in a way analogous to Eq. (11). The amplitude (LABEL:ampl) is thus a convolution of a hard scattering kernel with hadronic matrix elements Fourier transformed with respect to the average momenta and of the active partons.

For the active partons we can now introduce the momentum fractions

 x1 := k+1p+andx′1 := k′+1p′+. (13)

For later convenience we also introduce the average fraction

 ¯x1=k+1+k′+1p++p′+=¯k+1¯p+, (14)

which is related to and by

 x1=¯x1+ξ1+ξandx′1=¯x1−ξ1−ξ, (15)

respectively.

As for the processes in Refs. Goritschnig:2009sq (); DFJK1 (), due to the large intrinsic scale given by the heavy quark mass , the transverse and minus (plus) components of the active (anti)parton momenta in the hard scattering kernel are small as compared to their plus (minus) components. Thus, the parton momenta can be replaced by vectors lying in the scattering plane formed by the parent hadron momenta. For this assertion one only has to make the physically plausible assumptions that the momenta are almost on mass-shell and that their intrinsic transverse components [divided by the respective momentum fractions (15)] are smaller than a typical hadronic scale of the order of GeV. We thus make the following replacements:

 k1 → [k+1,x21Δ2⊥8k+1,−x1Δ⊥2]withk+1=x1p+, k′1 → [k′+1,m2c+x′21Δ2⊥/42k′+1,x′1Δ⊥2]withk′+1=x′1p′+, k2 → [x22Δ2⊥8k−2,k−2,−x2Δ⊥2]withk−2=x2q−, k′2 → [m2c+x′22Δ2⊥/42k′−2,k′−2,−x′2Δ⊥2]withk′−2=x′2q′−. (16)

As a consequence of these replacements it is then possible to explicitly perform the integrations over , , and . Furthermore, the relative distance between the (anti-)-diquark and the (anti-)-quark field operators in the hadronic matrix elements is forced to be lightlike, i.e., they have to lie on the light cone and thus the time ordering of the field operators can be dropped. After these simplifications one arrives at the following expression for the amplitude:

 Mμν=∑a(′)i,α(′)i∫d¯k+1θ(¯k+1)∫dz−12πei¯k+1z−1∫d¯k−2θ(¯k−2)∫dz+22πei¯k−2z+2×⟨¯¯¯¯¯¯¯D0:p′|Ψca′1α′1(−¯z1/2)ΦS[ud]a1(+¯z1/2)|p:p,μ⟩~Ha(′)iα′i(¯k1,¯k2)×⟨D0:q′|ΦS[ud]†a2(+¯z2/2)¯¯¯¯Ψca′2α′2(−¯z2/2)|¯p:q,ν⟩. (17)

From now on we will omit the color and spinor labels whenever this does not lead to ambiguities and replace the field-operator arguments and by their non-vanishing components and , respectively. Furthermore, if one uses and to rewrite the and integrations in the amplitude (LABEL:eq:integration-1) as integrations over the longitudinal momentum fractions and , respectively, one arrives at,

 Mμν=∫d¯x1¯p+∫dz−12πei¯x1¯p+z−1∫d¯x2¯q−∫dz+22πei¯x2¯q−z+2×⟨¯¯¯¯¯¯¯D0:p′|Ψc(−z−1/2)ΦS[ud](+z−1/2)|p:p,μ⟩~H(¯x1¯p+,¯x2¯q−)×⟨D0:q′|ΦS[ud]†(+z+2/2)¯¯¯¯Ψc(−z+2/2)|¯p:q,ν⟩. (18)

As in Ref. Goritschnig:2009sq () for (), the () transition matrix element is expected to exhibit a pronounced peak with respect to the momentum fraction. The position of the peak is approximately at

 x0=mcM=0.68. (19)

From Eq. (9) one then infers that the relevant average momentum fractions and have to be larger than the skewness . This means that the convolution integrals in Eq. (18) have to be performed only from to 1 and not from 0 to 1.

In the following section we will analyze the soft hadronic matrix elements in some more detail.

## Iv Hadronic Transition Matrix Elements

Compared to Eq. (LABEL:ampl) the Fourier transforms of the hadronic matrix elements for the and transitions are rendered to Fourier integrals solely over and , respectively. Hence we have to study the integral

 ¯p+∫dz−12πei¯x1¯p+z−1⟨¯¯¯¯¯¯¯D0:p′|Ψc(−z−1/2)ΦS[ud](+z−1/2)|p:p,μ⟩, (20)

over the transition matrix element and the integral

 ¯q−∫dz+22πei¯x2¯q−z+2⟨D0:q′|ΦS[ud]†(+z+2/2)¯¯¯¯Ψc(−z+2/2)|¯p:q,ν⟩, (21)

over the transition matrix element instead of Eqs. (11) and (12), respectively.

We will first concentrate on the transition (20) and investigate the product of field operators . For this purpose we consider the quark field operator in the hadron frame of the outgoing , cf. e.g., Refs. Kogut:1969xa (); Brodsky:1989pv (), where the has no transverse momentum component. It can be reached from our symmetric CMS by a transverse boost Dirac:1949cp (); Leutwyler:1977vy () with the boost parameters

 b+=(1−ξ)¯p+andb⊥=Δ⊥2. (22)

In this hadron-out frame we write the field operator in terms of its “good” and “bad” light cone components,

 Ψc=12(γ−γ++γ+γ−)Ψc≡Ψc++Ψc−, (23)

by means of the good and bad projection operators and , respectively. After doing that we eliminate the appearing in and by using the quark energy projector

 ∑λ′1v(k′1,λ′1)¯v(k′1,λ′1)=k′1⋅γ−mc. (24)

In the hadron-out frame it explicitely takes on the form

 ∑λ′1v(^k′1,λ′1)¯v(^k′1,λ′1)=k′+1γ−+m2c2k′+1γ+−mc, (25)

since there the quark momentum is

 ^k′1=[k′+1,m2c2k′+1,0⊥]. (26)

With those replacements the quark field operator becomes

 Ψc = 12k′+1∑λ′1{v(^k′1,λ′1)(¯v(^k′1,λ′1)γ+Ψc) (27) +

As in the case of in Ref. Goritschnig:2009sq (), one can argue that the contribution coming from dominates over the one in the square brackets, and thus the latter one can be neglected. Since this dominant contribution can be considered as a plus component of a four-vector, one can immediately boost back to our symmetric CMS where it then still holds that

 Ψc(−z1/2)=12k′+1∑λ′1v(k′1,λ′1)(¯v(k′1,λ′1)γ+Ψc(−z1/2)). (28)

Furthermore, one can even show that in on the right-hand side of Eq. (28) only the good component of is projected out, since

 ¯v(k′1,λ′1)γ+Ψc(−z−1/2)=¯v(k′1,λ′1)γ+P+Ψc(−z−1/2). (29)

Finally, we note that such manipulations are not necessary for the scalar field operator of the diquark.

Putting everything together gives for the transition matrix element (20)

 ¯p+∫dz−12πei¯x1¯p+z−1⟨¯D0:p′|Ψc(−z−1/2)ΦS[ud](+z−1/2)|p:p,μ⟩=¯p+2k′+1∑λ′1∫dz−12πei¯x1¯p+z−1⟨¯D0:p′|v(k′1,λ′1)(¯v(k′1,λ′1)γ+Ψc+(−z−1/2))ΦS[ud](+z−1/2)|p:p,μ⟩. (30)

Proceeding in an analogous way for the transition matrix element, where the role of the and components are interchanged, we get for Eq. (21)

 ¯q−∫dz+22πei¯x2¯q−z+2⟨D0:q′|ΦS[ud]†(+z+2/2)¯¯¯¯Ψc(−z+2/2)|¯p:q,ν⟩=¯q−2k′−2∑λ′2∫dz+22πei¯x2¯q−z+2⟨D0:q′|ΦS[ud]†(+z+2/2)(¯¯¯¯Ψc+(−z+2/2)γ−u(k′2,λ′2))¯u(k′2,λ′2)|¯p:q,ν⟩. (31)

Also here only the good components of the quark field are projected out on the right-hand side.

Using now Eqs. (30) and (31) and attaching the spinors and to the hard subprocess amplitude by introducing

 (32)

we get for the amplitude (18)

 Mμν=14(¯p+)2∑λ′1,λ′2∫d¯x1∫d¯x2Hλ′1,λ′2(¯x1,¯x2)1¯x1−ξ1¯x2−ξ×¯v(k′1,λ′1)γ+¯p+∫dz−12πei¯x1¯p+z−1⟨¯D0:p′|Ψc+(−z−1/2)ΦS[ud](+z−1/2)|p:p,μ⟩×¯q−∫dz+22πei¯x2¯q−z+2⟨D0:q′|ΦS[ud]†(+z+2/2)¯¯¯¯Ψc+(−z+2/2)|¯p:q,ν⟩γ−u(k′2,λ′2). (33)

Introducing the abbreviations

 H¯cSλ′1μ:=¯v(k′1,λ′1)γ+¯p+∫dz−12πei¯x1¯p+z−1⟨¯D0:p′|Ψc+(−z−1/2)ΦS[ud](+z−1/2)|p:p,μ⟩ (34)

and

 Hc¯Sλ′2ν:=¯q−∫dz+22πei¯x2¯q−z+2⟨D0:q′|ΦS[ud]†(+z+2/2)¯¯¯¯Ψc+(−z+2/2)|¯p:q,ν⟩γ−u(k′2,λ′2), (35)

for the pertinent projections of the hadronic transition matrix elements, we can write the hadronic scattering amplitude in a more compact form:

 Mμν=14(¯p+)2∑λ′1,λ′2∫d¯x1∫d¯x2Hλ′1,λ′2(¯x1,¯x2)1¯x1−ξ1¯x2−ξH¯cSλ′1μHc¯Sλ′2ν. (36)

## V Overlap Representation of H¯cSλ′1μ

In the following section we will derive a representation for the hadronic and transition matrix elements as an overlap of hadronic light-cone wave functions (LCWFs) for the valence Fock components of and Diehl:2000xz (). Since we only need them for , i.e., in the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi region, the hadronic transition matrix elements admit such a representation. For doing that we will make use of the Fock expansion of the hadron states and the Fourier decomposition of the partonic field operators in light-cone quantum field theory.

At a given light-cone time, say , the good independent dynamical field components and of the diquark and the quark, respectively, have the Fourier decomposition

 ΦS[ud](+z−1/2)=∫dk+1k+1∫d2k1⊥16π3θ(k+1)[a(S[ud]:k+1,k1⊥)e−ık+1z−12+b†(S[ud]:k+1,k1⊥)e+ık+1z−12] (37)

and

 Ψc+(−z−1/2)=∫dk′+1k′+1∫d2k′1⊥16π3θ(k′+1)∑λ′1[c(c:k′+1,k′1⊥,λ′1)u+(k′1,λ′1)e+ık′+1z−12+d†(c:k′+1,k′1⊥,λ′1)v+(k′1,λ′1)e−ık′+1z−12]. (38)

The spinors