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Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark mo

a r X i v :0808.0429v 1 [h e p -p h ] 4 A u g 2008

EPJ manuscript No.

(will be inserted by the editor)

Generalized parton distributions of pseudoscalar mesons in a

covariant constituent quark model

A.Van Dyck 1,T.Van Cauteren 1a ,J.Ryckebusch 1,and

B.

C.Metsch 2

1

Department of Subatomic and Radiation Physics,Ghent University,Proeftuinstraat 86,B-9000Gent,Belgium 2

Helmholz-Institut f¨u r Strahlen-und Kernphysik,Nu?allee 14-16,D-53115Bonn,Germany

Received:date /Revised version:date

Abstract.The isoscalar twist-two generalized parton distributions (GPDs)of the pion and the kaon are calculated in a Poincar′e covariant Bethe-Salpeter constituent quark model.Results are presented for several values of the parameters ξand t .The results satisfy the form factor constraints and the polynomiality condition.For the pion GPD,also the isospin symmetry constraint is ful?lled.The in?uence of kinematical variables and model parameters on the support of the GPDs is investigated.To this end,the strength parameters and quark masses of the constituent quark model are arti?cially varied.

PACS.11.10.St Bound and unstable states;Bethe-Salpeter equations –12.39.Ki Relativistic quark model –13.60.Fz Elastic and Compton scattering –14.40.Aq π,K and ηmesons

1Introduction

Unraveling the substructure of hadrons is a challenging issue.Whereas the theory of QCD provides us with the equations governing the quark-gluon dynamics,the ex-act way in which they form colorless bound states in the non-perturbative regime remains elusive to date.It is ex-pected that the endeavor of measuring and computing generalized parton distribution functions (GPDs)will con-

2 A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model

pion and kaon GPDs computed within the framework of the Bethe-Salpeter model developed in refs.[11,12,13,14]. The pseudoscalar ground-state mesons have a simple valence-quark substructure and possess only one helicity-conserving GPD,which makes the calculation less cumbersome than for nucleons.Yet,such a calculation provides insight in the dependence of the GPDs on hadron structure in the non-perturbative regime.

In order to compare the GPDs computed in a phe-nomenological model with the data from deeply inelas-tic scattering experiments,a Q2-evolution needs to be performed[15].The evolution equations depend highly on the kinematic region of the process.More speci?cally, one can distinguish between the DGLAP and ERBL re-gions,where the Q2-evolution is governed by the DGLAP and the ERBL equations,respectively.Parton model con-straints make GPDs vanish outside these two regions[16, 17].A model which resolves the ERBL and the DGLAP regions and has a vanishing GPD otherwise,is said to have the correct support.

In ref.[18],we have demonstrated that a support prob-lem may arise in dynamic quark models based on the Bethe-Salpeter approach if the interactions depend on the (light-cone)minus component of the relative momentum between constituent quark and antiquark.Furthermore, we provided representative results computed within the explicitly Poincar′e covariant constituent quark model de-veloped by the Bonn group[11],where a support problem did arise.

In this work,we focus on the in?uence of the meson binding energy on the support.The pion and the kaon are ideal for such a study because the former is a very deeply and the latter a moderately deeply bound state of quark and antiquark.Both the absolute binding en-ergy and meson mass are altered by changing the input quark masses and e?ective interaction strength.Yet the degree of support violation depends mostly on the rela-tive binding energy.Results will be shown for the isoscalar twist-two quark GPD of the pion and the kaon at di?er-ent relative binding energies and for di?erent values of the squared fourmomentum transfer t and relative plus-momentum transferξ.

This paper is organized as follows.Section2presents the de?nition of the GPD and the relevant kinematics. The Bonn model is introduced in sect.3,while sect.4is devoted to the calculation of GPDs in this model.In this section,we will also introduce the three model variants which will be used in sect.5.There,we show our results for the pion GPD.A summary and an outlook to future research are given in sect.6.

2De?nition and Conventions

GPDs are non-diagonal matrix elements of a bilocal?eld operator on the light cone.For partons with spin1/2in a pseudoscalar meson,the GPD associated with helicity

A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model3 conserving partons is de?ned as follows[16]:

H f(x,ξ,t)=

1

e ix?P+z? ˉP′|ˉψf(?z2)|ˉP z+=0,z⊥=0.(1)

In this equation,f refers to the?avor of the probed par-

ton,while?P=ˉP′+ˉP

2and

p⊥=(p1,p2).The skewednessξand the average plus-

momentum fraction of the struck parton x are de?ned in

?g.1:x denotes the fraction of the average meson plus-

momentum that is reabsorbed by the meson,whileξis a

measure for the plus-momentum that is lost in the process:

ˉP+?ˉP′+

ξ=

?t

(1+ξ)

andζ=2ξ

4 A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model

GPD is a polynomial inξof order≤n[21]:

d x x n?1H f(x,ξ,t)=n i=0a i(t)ξi,(6) wher

e the coe?cients a i o

f the polynomial depend on t.

The polynomiality condition can be regarded as a more

general form of the above form factor relation.Finally,for

pions,isospin invariance and charge conjugation lead to

the following relation:

H qπ(x,ξ,t)=?Hˉqπ(?x,ξ,t).(7) We will come back to these model constraints in sect.5. 3Formalism

In the Bonn model,mesons are described as bound states

of a constituent quark and antiquark.The model,based

on the Bethe-Salpeter equation,is Poincar′e covariant by

construction.Explicit covariance is important in the cal-

culation of the GPD of pseudoscalar mesons for two rea-

sons.First,the pion is a“deeply-bound”boson whose

static and dynamic properties are best reproduced with

a relativistic model[22].Second,in the calculation of dy-

namic quantities such as the GPD,recoil fourmomenta

have to be treated relativistically.The Bonn model is a

few-parameter model with only seven parameters which

are?tted to the meson mass spectrum[11].The fact that

no relativistic corrections need to be implemented is an

enormous asset to minimize the amount of parameters and

maximize the predictive power.In refs.[11,12,13,14],the

Bonn model is described in detail.In sects.3.1and3.2,

we give a brief summary to make this paper more selfcon-

tained.

=

Fig. 2.Diagrammatic representation of the Bethe-Salpeter equation(9).

3.1The model ingredients

A meson with on-shell fourmomentumˉP is described by the Bethe-Salpeter amplitude

χˉPαβ(x1,x2)= Θ|T{ψ1α(x1)ˉψ2β(x2)}|ˉP ,(8) where T is the time ordering operator acting on the Heisen-berg fermion?eld operatorsψα.The indices in Dirac,?avor and color space are combined in the multi-indices α≡(α,f,c).The variables x1and x2are the fourvectors denoting the spacetime positions of the quark and the an-tiquark,respectively.

The Bethe-Salpeter amplitude(8)is the solution of the Bethe-Salpeter equation,which in momentum space reads [12,23]:

χˉP=?iG0ˉP KˉPχˉP.(9) Here,indices and arguments have been suppressed for no-tational simplicity.It is tacitly assumed that one inte-grates over arguments and sums over indices that occur twice.G0ˉP is the product of the one-particle propagators, while KˉP denotes the interaction kernel.The pictorial rep-resentation of eq.(9)is shown in?g.2.The normalization of the Bethe-Salpeter amplitudes is given by[12]:ˉχˉP Pμ?

A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model 5

To transform the Bethe-Salpeter equation (9)into a solvable

integral equation,

two Ans¨a tze are made.The ?rst assumption is the instantaneous approximation of the in-teraction kernel K ˉP ,

K ˉP (p,p ′)≡V (p ⊥ˉP ,p ′

⊥ˉP ),

(11)

where p ⊥P ≡p ?(p ·P/P 2

)P is the fourvector perpen-dicular to P with p the relative fourmomentum between the constituent quark and the constituent antiquark in the meson.The second assumption is that the full quark

propagators S F

j (p )can suitably be approximated by free

fermion propagators with an e?ective constituent quark mass m j ,

S F

j (p )≡

i

χˉP (p 0

,p )|ˉP =(M,0).

(13)

Making use of eqs.(11)and (13),the p 0-integration in eq.(9)can be carried out,leading to the well-known Salpeter

equation [24]:Φ(p )=

d 3p ′

M +ω1+ω2

?

d 3p ′

M ?ω1?ω2

,(14)

with the energy projection operators Λ±j =(ωj ±H j )/(2ωj ),the Dirac Hamiltonian H j (p )=γ0(γ·p +m j )and the

energy ωj =

2

(a c +b c |x q ?x ˉq |)

×(1I ?1I ?γ5?γ5?γμ?γμ),(15)

where x q (ˉq )is the position of the constituent (anti-)quark.The instanton interaction accounts for the mass splittings in the pseudoscalar and the scalar sectors [22].In momen-tum space,it can be written as

d 3p ′

(2π)3

R Λ(p ,p ′)(1I T r [Φ(p ′)]+γ5T r [γ5Φ(p ′)]),

(16)

where G (g,g ′)includes the ?avour dependent couplings g

and g ′,and R Λis a Gaussian regulating function with cut-o?Λ[26,27].The ’t Hooft instanton induced interaction can account for the low mass of the pion.The con?ne-ment and the instanton interaction contain ?ve additional

6 A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model

model parameters,bringing the total to seven.These seven model parameters are?tted to the Regge trajectories and the pseudoscalar ground state masses and are kept?xed in the calculation of dynamic observables.In this sense, the results in sect.5are predictions.

The Salpeter amplitudesΦare calculated by solving eq.(14).Imposing the normalization condition,written in terms of the Salpeter amplitudes,then yields the mass spectrum[12].

4GPDs in the Bethe-Salpeter Quark Model

The?rst step in the calculation of the generalized par-ton distribution de?ned by eq.(1)is the calculation of the bilocal current matrix element ˉP′|ˉψ(?z2)|ˉP . In a Bethe-Salpeter based approach,the matrix element of any dynamic variable can be calculated with the Man-delstam formalism[28].This formalism acts as a starting point for the derivation of the GPD in terms of the Bethe-Salpeter amplitudes.A straightforward calculation yields the bilocal current matrix element in lowest order[27]:

ˉP′|ˉψ(?z2)|ˉP

= d4x′2i Tr ˉχˉP′(x′2,z2) γ+ + d4y′1i Tr (i?y′1?m2)ˉχˉP′(?z2,y′1) .(17) The second term in eq.(17)(coupling to the quark)is depicted in?g.3.Inserting eq.(17)in eq.(1)gives

H(x,ξ,t)=H q(x,ξ,t)+Hˉq(x,ξ,t),(18)

Fig.

sion(17)for the plus-component of the bilocal current matrix element.

with the term originating from the quark-current coupling, H q(x,ξ,t)=

i

(2π)4

δ x?P+?ˉP′+

2

+p?m2 ˉχˉP′(p+ˉP′2)γ+χˉP(p) ,(19)

and the term originating from the antiquark-current cou-pling,

Hˉq(x,ξ,t)=

i

(2π)4

δ x?P++ˉP′+

2?

ˉP′

2

+p?m1 χˉP(p)γ+ .(20)

Equations(17)-(20)make use of the full Bethe-Salpeter amplitudeχ.This amplitude can be reconstructed from the Salpeter amplitudes as follows:once the Salpeter equa-

tion is solved,the vertex functionsΓˉP=G?1

0ˉP

χˉP can be calculated.In the meson rest frame,these vertex functions read:

ΓˉP(p⊥ˉP)|ˉP=(M,0)=Γ(p)=?i d3p

A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model7 latter frame is found:

χˉP(p)=SΛ

ˉP χ(M,0)(Λ?1ˉ

P

p)S?1Λ

ˉP

.(22)

In this equation,ΛˉP denotes the Lorentz transformation and SΛ

ˉP

denotes the corresponding boost operator acting on the fermion?eld operators in eq.(9).

Written in terms of these vertex functions,the quark GPD arising from the Mandelstam formalism contains three quark propagators S F:

H qπ(x,ξ,t)=?1

(2π)4

δ 2x+ξ?1

2

)S1F(

ˉP′

2

)γ+ S1F(

ˉP

2

+p) ,(23)

These propagators can be directly linked with the inter-mediate quark lines in the diagram of?g.3.It turns out that the denominators of these three propagator terms en-sure the correct support region for the GPDs,x∈[?ξ,1] (ERBL and DGLAP regions)for p?independent vertex functions.For p?dependent vertex functions,it is a priori unclear whether the GPD will be con?ned to the support region.In previous work,we have shown that the Bonn model is prone to a support problem[18].In the next para-graphs,we will quantitatively investigate which physical parameters in?uence the support behavior of the model. The role of the variablesξand t will be analyzed,and the dependence on the binding strength will be examined through a comparison of the GPD of the kaon and the pion in three di?erent model variants.4.1Model variants

In the forthcoming section,the GPDs of pseudoscalar mesons will be shown in three di?erent model variants:the full model,the reduced model and the increased quark mass (IQM)model.The parameters of these models are pre-sented in table1.Notice that the parameters a c and b c of the con?nement interaction remain?xed for all three model variants.

4.1.1Full model

The full model is the one referred to as Model B in ref.

[26].It provides an accurate description of the pion and other meson properties such as its mass,electromagnetic form factor,electroweak decay widths,etc.In the full

model,the pion mass is calculated as M full

π

=141MeV

which implies a large mass defect of?full

=(2m n?

M full

π

)=619MeV;accordingly we shall call the pion “deeply bound”.The kaon is moderately bound in this

model,with a mass of M full

K

=506MeV and a mass de-

fect of?full

M K

=(m n+m s?M full K)=424MeV.

4.1.2Reduced model

In the reduced model,the’t Hooft instanton induced in-teraction is omitted.As we have mentioned in sect.3.2, the instanton interaction accounts for the deep binding of the pion(and to a lesser degree also of the kaon).Ac-cordingly,neglecting the’t Hooft interaction will provide insight into the importance of binding e?ects in the GPD

8 A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model

Parameter Full model IQM model

m n[MeV]380800

m s[MeV]550-

a c[MeV]-1135-1135

b c[MeV/fm]13001300

g[GeV?2] 1.62 1.62

g′[GeV?2] 1.35 1.35

Λ[fm]0.420.42

572

188

0.33

M K[MeV]506-

?K M[MeV]424-

?K M/M K0.84-

Table1.Overview of the parameters of the three models that

were used in this work:the constituent quark masses,the con-

?nement o?set and slope,the’t Hooft interaction range and

the’t Hooft interaction strengths.Also presented is a summary

of the masses M,binding energies?M and relative binding en-

ergies?M/M of the pion and the kaon in the di?erent models.

results.As a matter of fact,the calculated pion mass in-

creases to M red

π=572MeV in this approach(?red

=

188MeV).The kaon mass increases to M red

K

=728MeV

(?red

M K

=202MeV).

4.1.3Increased quark mass(IQM)model

Not only the’t Hooft instanton induced interaction has an e?ect on the pion binding energy.Also the non-strange constituent quark mass m n a?ects the pion mass and mass

defect.To investigate the in?uence of the binding energy on the support of the generalized parton distributions, both mechanisms must be studied.

In the IQM model,we will only show results for the pion GPD.After combining the IQM model results with the kaon results from the full model,one can determine the in?uence of the heavy quarks.Increasing the non-strange quark mass by more than a factor of2to m n=800MeV

yields a pion mass of M IQM

π

=1095MeV.The mass defect

in this model is?IQM

=505MeV.

4.1.4Model summary

The masses,binding energies and relative binding energies (de?ned as the binding energy devided by the mass)cal-culated in the di?erent models are summarized in table 1.The deep binding of the pion in the full model is re-?ected in the high relative binding energy.?M/M in the full model is much smaller for the kaon than for the pion. Further,?M/M of the kaon in the full model is larger than in the reduced model,and also larger than?M/M of the pion in the reduced and the IQM models.In sect. 5,we will come back to these(relative)binding energies.

4.2Model constraints

In sect.2,we introduced three constraints that serve as stringent tests for any GPD calculation.These are the isospin symmetry relation(7)for the pion GPD,the form factor relation(4)and the polynomiality condition(6).In

A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model9

this section,we elaborate on these constraints,and show

that they are ful?lled in our model.

The pion up and down quark GPDs must ful?ll the isospin symmetry relation of eq.(7).The equality is ex-act in our calculations.The strange quark content of the kaon prevents an isospin symmetry relation of the type (7).The results show a small di?erence between the quark and antiquark GPDs at opposite x(e.g.in the full model, H u

K+

(x=0.5,ξ=0,t=?0.5GeV2)=0.764versus

H s

K+

(x=?0.5,ξ=0,t=?0.5GeV2)=?0.784).We will elaborate on these di?erences in sect.5.

The GPD is related to the electromagnetic form factor through relation(4).Taking into account eq.(5),which relates the partial form factors with the meson form factor, and the isospin symmetry relation(7)for the pion,one ?nds that

+∞?∞d xH uπ+(x,ξ,t)=Fπ+(t).(24)

Note that,due to the support properties of the GPDs in the Bonn model,the integration domain is x∈(?∞,+∞). For the kaon,the relation becomes

e u +∞?∞d xH u K+(x,ξ,t)

+e s +∞?∞d xH s K+(x,ξ,t)=F K+(t),(25) where e u=2/3and e s=?1/3.

We have compared the results of eqs.(24)and(25) with a direct computation of the electromagnetic form fac-tors Fπ+and F K+in the Bonn model[11].These numeri-cal calculations were performed independently and yielded results which were compatible at the few-%level[27].

An even more stringent test than the form-factor con-dition of eq.(4),is the polynomiality condition of eq.(6). This condition was veri?ed numerically for di?erent values of t up to order n=5[27].

5Results and Discussion

Theπ+up-quark GPD results in the three di?erent model variants are shown in?gs.4-6,the K+up-quark and strange-antiquark GPDs in?gs.7-8.A three-dimensional picture of the(full model)π+up-quark GPD as a function of x and t with skewednessξ=0is presented in?g.9. For all cases,the results are shown for a representative selection of t andξvalues.

It was shown in ref.[18]that GPDs in the Bonn Model violate the support condition.This can also be deducted from?gs.4-8.In general,the curves tend to have longer tails as|t|decreases.Especially for the deeply bound pion in the full model,this e?ect is clearly visible(see?g.9).As the relative binding energy decreases,the e?ect becomes smaller.Furthermore,whereas the maximum of the GPD curve lies outside the support region for the deeply bound pion in the full model(peak value at|x|>1),it lies within the support interval for the other pion and kaon calcula-tions.

Another observation which can be made,is that the pion GPD in the full model displays a clear shoulder in the region x∈[0,1]at|t| 0.1GeV2(see?g.4).This shoul-der is much less pronounced or even absent in the other results.In this respect,it is interesting to note that the computed pion form factor shows a bump at low Q2=?t,

10 A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model

Fig.4.The pion GPD H u

π+in the full model for di?erent

values of t .Values for ξshown are ξ=0(red,solid line),ξ=0.4(green,dashed line)and ξ=0.7(blue,dotted line).

which can be ascribed to the instantaneous approxima-tion

[11,29,30].

The observed shoulder in the pion GPD might therefore be an artefact of this Ansatz .

The observed model dependence of the GPD properties points at a signi?cant relative binding energy dependence of the support of the generalized parton distributions in the Bonn model.To quantify the support problem,a sup-port parameter (φ)is introduced.For the pion,the gen-eralized quark and antiquark distributions are related via the isospin symmetry relation (7),so that the knowledge of one of them implies the knowledge of the other.The

Fig.5.The pion GPD H u

π+in the reduced model for di?erent

values of t .Values for ξshown are ξ=0(red,solid line),ξ=0.4(green,dashed line)and ξ=0.7(blue,dotted line).

support parameter is therefore de?ned through the quark GPD:

φ=

1

|H u

π+(x,ξ,t )|d x

?∞

|H u K +(x,ξ,t )|d x (27)

A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model11

in the IQM model for di?erent

Fig.6.The pion GPD H u

π+

values of t.Values forξshown areξ=0(red,solid line),

ξ=0.4(green,dashed line)andξ=0.7(blue,dotted line).

and

φˉq= ξ?1|H s K+(x,ξ,t)|d x

12 A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model

t(GeV2)

Modelξ?0.01?0.1?1.0

0.110.210.17

0.4-0.140.22

-0.170.12

Reduced model00.600.640.69

-0.700.61

0.7---

0.670.710.68

0.4--0.75

--0.64

Table2.Values of the pion support parameterφof eq.(26)corresponding to?gs.4-6.

increasing|t|except for very large values(|t| 5.0GeV2). For a?xed|t|,theφexhibits hardly any dependence onξ.

Also for the kaon GPD,the support improves when the instanton induced interaction is switched o?.Notice that the values of the two support parametersφq andφˉq are similar.This result is compatible with the?nding that the quark and antiquark GPD have similar shapes and shows that the small di?erences between both GPDs do not alter the support signi?https://www.doczj.com/doc/803110909.html,bining these observations with the fact that the support does improve signi?cantly in the IQM model with respect to the full model,we conclude that concerning the support the binding strength is more important than the particular constituent quark mass or ?avor.6Summary and Conclusions

In this work,the isoscalar twist-two generalized parton distributions of the pion and the kaon were calculated in the Poincar′e covariant Bethe-Salpeter constituent quark model developed by the Bonn group.The?rst moment of the GPD in the full x-region equals the electromag-netic form factor.Results were shown in the full x-region for several values ofξand t.It turns out that the Bonn model violates the support condition.We have illustrated the strong correlation between the support and the(rel-ative)binding energy of the meson.Therefore,the deep binding of the pion induces strong support violations.We have found thatξhardly in?uences the support,whereas a moderate dependence on t is predicted.The constituent quark masses have an impact on the support,but only through the(relative)binding energy.A mass di?erence

A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model13

t(GeV2)

Parameter Modelξ?0.01?0.1?1.0

0.480.580.47

0.4--0.62

--0.56

Reduced model00.680.700.73

-0.760.62

0.7---

0.450.560.42

0.4--0.61

--0.57

Reduced model00.650.670.73

-0.730.64

0.7---

14 A.Van Dyck et al.:Generalized parton distributions of pseudoscalar mesons in a covariant constituent quark model

Fig.8.The kaon GPDs H u

K+and H s

K+

in the reduced model

for di?erent values of t.Values forξshown areξ=0(red line),ξ=0.4(green line)andξ=0.7(blue line).The solid line refers to the u GPD,the dashed line to theˉs GPD.

Acknowledgements

The authors wish to thank F.LLanes Estrada,S.Scopetta and D.Van Neck for enlightening discussions.AVD and TVC are grateful to the Research Foundation-Flanders (FWO)for?nancial support.BCM acknowledges the sup-port of the European Community-Research Infrastructure activity under the FP6“Structuring the European Re-search Area”programme(Hadron Physics,contract num-ber RII3-CT-2004-506078)and the support within the

Fig.9.The H u

π+

(x,0,t)GPD in the full model as a function of x and?t forξ=0.

DFG SFB/TR16“Subnuclear Structure of Matter-Elek-tromagnetische Anregung subnuklearer Systeme”.

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