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JPsi strong couplings to the vector mesons

a r X i v :n u c l -t h /0502009v 1 2 F e

b 2005

BA-TH/2005-505,CERN-PH-TH/2005-009,FNT/T-2005/04

J/ψstrong couplings to the vector mesons

https://www.wendangku.net/doc/d614733784.html,porta and A.D.Polosa ?

Dipartimento Interateneo di Fisica,Universit`a di Bari and INFN Bari,via Amendola 173,I-70126Bari,Italy

F.Piccinini ?

Istituto Nazionale di Fisica Nucleare,Sezione di Pavia and

Dipartimento di Fisica Nucleare e Teorica,via A.Bassi 6,I-27100,Pavia,Italy

V.Riquer ?

CERN,Department of Physics,Theory Division,Geneva,Switzerland

(Dated:February 9,2008)

We present a study of the cross sections J/ψX →D (?)ˉD

(?)(X =ρ,Φ)based on the calculation of the e?ective tri-and four-linear couplings J/ψ(X )D (?)ˉD

(?)within a constituent quark model.In particular,the details of the calculation of the four-linear couplings J/ψXD (?)ˉD

(?)are given.The results obtained have been used in a recent analysis of J/ψabsorption by the hot hadron gas formed in peripheral heavy-ion collisions at SPS energies.

PACS numbers:12.38.Mh,12.39.-x,25.75.-q

I.INTRODUCTION

The problem of computing J/ψstrong couplings to π,ρand other pseudo-scalar and vector particles has its own interest because it opens the way to the calculation of cross sections of the kind (see Fig.1):

J/ψ{π,ρ,...}?→D (?)ˉD

(?)

.(1)Such cross sections are the basic ingredients to estimate the hadronic absorption background of J/ψin heavy-ion

collisions,as it is thoroughly discussed in [1,2].The description of processes like those in Eq.(1)is a hard task because no experimental test can be performed and moreover they are not amenable to ?rst principles calculations,so that one has to resort to build models and make approximations to describe their dynamics.

The dissociation process of the J/ψby hadrons has been considered in several approaches,but the predicted cross sections show very di?erent energy dependence and magnitude near threshold.Anyway,using di?erent approaches,one consistently ?nds non negligible cross section values (at least comparable with the nuclear one N J/ψ,N =nucleon)especially for the reactions with π’s and ρ’s,the most studied cases;for a review see for instance Ref.[3].This is certainly a clear indication that the picture of J/ψabsorption by nuclear matter,as an antagonist mechanism to the plasma suppression,is incomplete as long as interactions with the hadronic gas formed in nucleus-nucleus collisions are not considered.

The problem of calculating the J/ψdissociation by pseudo-scalar and vector mesons has been addressed in Refs.[1,2]within the Constituent-Quark–Meson model (CQM),originally devised to compute exclusive heavy-light meson decays and tested on a quite large number of such processes [4].The basic calculations refer to πand ρcontributions.The couplings to other mesons have been obtained under the hypothesis of ?avour/octet symmetry.

The typical e?ective Feynman diagrams contributing to the J/ψdissociation are depicted in Fig.1.The tri-linear

couplings ρD (?)ˉD

(?)have been calculated in [5]and the J/ψD (?)ˉD (?)couplings have been recently discussed in [6](we report these results at the end of Sect.II),where also four-linear couplings involving pions have been derived.The aim of the present note is to explain the method used and the results obtained in evaluating the four-linear couplings of the

kind J/ψρD (?)ˉD

(?)(see Fig.1,third diagram),since these are not calculated elsewhere within the CQM framework.The numerical values of the J/ψΦD (?)s ˉD (?)

couplings are also given.For completeness we report also the expressions for the tri-linear couplings discussed in Refs.[4,7].In the end,we present the cross section predictions,based on the

2ρ

J/ψ

ˉH

p2p3

J/ψ

ˉH

p2p3

J/ψ

ˉH

p2p3

p4 FIG.1:Tree level e?ective Feynman diagrams for the J/ψρ→HˉH reaction,H being D(?),with D(?)=D or D?.

complete set of contributing diagrams,for the processes J/ψρ→D(?)ˉD(?)and J/ψΦ→D(?)sˉD(?)s,together with an estimate of the associated theoretical uncertainties.

II.THE MODEL

CQM is based on an e?ective Lagrangian which incorporates the heavy quark spin-?avor symmetries and the chiral symmetry in the light sector.In particular,it contains e?ective vertexes between an heavy meson and its constituent quarks(see the vertexes in the r.h.s.of Fig.2)whose emergence has been shown to occur when applying bosonization techniques to Nambu–Jona-Lasinio(NJL)interaction terms of heavy and light quark?elds[8].On this basis we believe that CQM can be considered as a quite reasonable approach to the computation of J/ψstrong couplings to be compared to the various methods available in the literature,often based on SU(4)symmetry[9].

In Fig.2we show the typical diagrammatic equation to be solved in order to obtain g4(g3),four(tri)-linear couplings, in the various cases at hand:on the l.h.s.it is represented the e?ective four-linear coupling to be used in the cross section calculation(to obtain one of the relevant tri-linear couplings we could discard either the J/ψor theρ);the e?ective interaction at the meson level(l.h.s.)is modeled as an interaction at the quark-meson level(r.h.s.of Fig.2). The J/ψis introduced using a Vector Meson Dominance(VMD)Ansatz:in the e?ective loop on the r.h.s.of Fig.2 we have a vector current insertion on the heavy quark line c while on the l.h.s.the J/ψis assumed to dominate the tower of1?,cˉc states mixing with the vector current(for more details see[6]).Similarly,vector particles coupled to the light quark component of the heavy mesonsρ,ω,when q=(u,d),or K?,Φ,when one or both light quarks involved are q=s,are also taken into account using VMD arguments.

The pion and other pseudo-scalar?elds have a derivative coupling to the light quarks of the Georgi-Manohar kind[10].

In this paper we will mainly focus on the reaction:

J/ψρ?→D(?)ˉD(?)(2) and in particular on the four-linear coupling J/ψρHˉH(third graph)in Fig.1.

In CQM,as in Heavy Quark E?ective Theory(HQET)[11],the heavy super-?eld H describes the charmed states D and D?,respectively associated to the annihilation operators P5,Pμ.H is written in the following way:

H(v)=

1+v/

FIG.2:Basic diagrammatic equation to compute the g 4couplings.The l.h.s.is the e?ective vertex J/ψρH ˉH

at meson level (?and ηare respectively the ρand J/ψpolarizations);while the r.h.s.contains the 1-loop process to be calculated in the CQM model.

In the following we will use the Feynman rules de?ned in

[4].

The

interaction terms relevant to this calculation are:

?ˉq ˉH

Q v +h.c.,(6)

which describes the vertex light quark (q ),heavy quark (Q v ),heavy meson (H ),and ˉ

q m 2ρ(l 2?m 2q )

?→

d 4l

1/μ2

1/Λ2

ds e ?s (l

+m 2q )

.(13)

4 The diagrammatic equation in Fig.2states that the e?ective vertex J/ψρHˉH is given by:

(?1)

(2π)4

Tr ?iˉH′(v′) i f J m Jη/i l/?m i m2ρl/+q/?m .(14)

H andˉH′represent the heavy-light external meson?elds labeled by their four-velocities v,v′while the √

f J m J m2ρ

Z H m H Z H′m H′ 10dx?(2π)4Tr ˉH′η/H(l/?q/x+m)?/(l/?q/x+q/+m)

2m H m H′

.(20)

By kinematic considerations the energy threshold of the reactions(2)for DˉD and D?ˉD channels is Eρ?0.77GeV whereas for D?ˉD?channel is Eρ?0.96GeV,withω≈1.We considerρparticles with energies in the range between 0.77and1GeV where the two?nal state mesons are almost at rest.

All the couplings that we can extract by direct computation can be written in terms of7basic expressions which we call:L5,A,B,C,D,E,F.The latter are linear combinations of the I i,L i integrals listed in the Appendix and are de?ned by:

(2π)4

lμ

??m2

iN c d4l(l2??m2)(v·l+δ)(v′·l+δ′)=C gμν+D vμvν+E v′μv′ν+F(vμv′ν+v′μvν).(23) The?nal expression of the loop integral can the be reduced to a sum of terms of the general form:

S S(H,H′)C 10dx g(S)4(x,Eρ)(24)

where S(H,H′)represent the scalar combinations of momenta and polarizations of H and H′occurring in the

calculation;g(S)

,are the corresponding couplings.Here C is given by:

C=m2J

fρ

2π2 ∞E0dE p Eσ(E)

6 J/ψXDˉD X=ρX=Φ

D Ax4±21.5±0.5GeV?4

g22

m D

(A+B+2xA(ω?1)?mL5)27±413±1GeV?2 g42

D ((m2+m2ρx(1?x))L5?2Am?2C+D?E+2F(1?ω))?8±3?7±1GeV?2

g61

D ((m2+m2ρx(1?x))L5?2Bm?2C?D+E+2F(1?ω))?6±2?5±1GeV?2

g82

h1(mL5+(A?B)x)(ω?1)1±20.1±0.6GeV?1 h2B(x?1)?9±4?5±1GeV?1 h3mL5?Bx?6±12?6±3GeV?1 h4mL5+A(x?1)?B?35±15?20±4GeV?1 h5A35±1115.7±3GeV?1 h6Ax15±86±2GeV?1 h7B?mL516±1611±4GeV?1 h8(m2+m2ρx(1?x))L5?2C?D?E?2Fω1.3±21.1±0.8

h9D+F?mA?19±7?17±3

h10E+F?mB?15±6?13±2

J/ψXD?ˉD?X=ρX=Φ

m D?

(A+B?mL5)25.5±0.613.6±0.1GeV?2 f21

D?Ax4±21.5±0.5GeV?4

f41

D?B(x?1)?2.2±0.7?1.2±0.1GeV?4

f6(?m2L5+2C+D+E+2Fω+m2ρ(x2?x)L5)(1?ω)0.03±0.10.03±0.03 f71

D?(D+F?mA)?8±2?8.2±0.3GeV?2

f91

D?(?m2L5+2mB+2C+D?E+2F(ω?1)+m2ρ(x2?x)L5)7.3±0.86.3±0.2GeV?2

f112

analogously βand λfor the L HH Φare

β=?0.48

(32)λ=+0.14GeV ?1.

(33)

As for the couplings J/ψD (?)ˉD

(?),they have been extensively discussed in [6].Here we just report the main results.Observe that

L J/ψHH =ig J/ψHH Tr[ˉHγ

μH ]J μ,(34)

where H can be any of the pairs D D ?or D s D ?

s (neglecting SU (3)breaking e?ects).As a consequence of the spin symmetry of the HQET we ?nd:

g J/ψD ?D ?=g J/ψDD ,g J/ψDD ?=

g J/ψDD

s of the process for the three ?nal states under

consideration (DD,DD ?

,D ?

D ?

).This calculation has been made by using the tri-and four-linear couplings quoted above,assuming their validity in the energy range

√

s ,showing values of the same order as the cross sections for

J/ψπ→D (?)

ˉD (?).This,given also the higher spin multiplicity of the ρmeson with respect to pions,demonstrates

the importance of the ρcontribution to the J/ψabsorption in the hot hadron gas,formed in peripheral heavy-ion collisions at SPS energy,as discussed thoroughly in Ref.[2].Aiming at calculating thermal averages with T ≈170MeV,we didn’t discuss in the present paper the introduction of any arbitrary form factors since the exponential statistical weight acts as a cut o?in the high energy tail.

Acknowledgments

We wish to thank L.Maiani for the stimulating collaboration and encouragement.

Appendix

In this Appendix are listed the I i and L i integrals occurring in the calculation and their linear combinations A,B,...,F .These integrals have been computed adopting the proper time Schwinger regularization prescription,with cut-o?μ=0.3GeV (0.5GeV when is present a strange quark),Λ=1.25GeV.In the following N c =3.

I 1=iN c

d 4l

(l 2??m 2)

N c

Λ2

,?m 2(2π)41

16π3/2 1/μ2

1/Λ2

(37)

I 5(δ,δ′

,ω)=iN c

d 4l (l 2??m 2)(v ·l +δ)(v ′·l +δ′)

16π3/2

1/μ2

1/Λ2

ds σe ?s (?

?σ2)

s ?1/2

1+Erf σ

√16π2

1/μ2

1/Λ2

ds e ?sσ2

s ?1

(38)

in the last expression we have de?ned

σ≡σ(δ,δ′,y,ω)=

δ(1?y )+δ′y 1+2(ω?1)y +2(1?

ω)y 2

.(39)

In the previous equations ?m 2,δand δ′are given by

?m 2=m 2+x m 2ρ(x ?1)

(40)δ=??x q ·v =??x E ρ

(41)δ′=??x q ·v ω=??x ωE ρ,

(42)

with m =0.3GeV the constituent mass for light quark u ,d .The expression of ω=v ·v ′in the rest frame of J/ψis

ω=

m 2J/ψ+m 2ρ?m 2H ?m 2

H ′+2E ρm J/ψ

√

The L i integrals are de?ned in the following way:

L i =?

??m 2iN c

d 4l

(l 2??m 2)=

N c

Λ2

,?m 2??m 2Γ

?1,

?m 2

μ2

(47)

L 3(δ)=?

(2π)4

16π3/2

1/μ2

1/Λ2

ds e ?s (?m 2?δ2)

?1/2

1+Erf δ

√

??m 2iN c

d 4l

(l 2??m 2)(v ·l +δ)(v ′·l +δ′)

1+2y 2(1?ω)+2y (ω?1)

×× 1/μ2

1/Λ2

ds σe

?s (?m 2?σ2) ?s 1/2

1+Erf σ√ω2?1

B =

L 3(δ)+δ′L 5(δ,δ′,ω)?(L 3(δ′)+δL 5(δ,δ′,ω))ω

2(ω2

?1)

L 5(δ,δ′,ω)δ′2+(L 3(δ)?(L 3(δ′)+2δL 5(δ,δ′,ω))ωδ′+δ(L 3(δ′)+δL 5(δ,δ′,ω))?δL 3(δ)ω+I 5(δ,δ′,ω)(ω2?1)+L 5(δ,δ′,ω)?m 2(ω2?1)

D =

2(ω2

?1)2

2(L 1+δ′L 3(δ′))ω3+(I 5(δ,δ′,ω)+2δ(L 3(δ′)+δL 5(δ,δ′,ω))+L 5(δ,δ′,ω)?m 2)ω2?(2L 1+5δ′L 3(δ′)

+3δ(L 3(δ)+2δ′

L 5(δ,δ′

,ω)))ω?I 5(δ,δ′

,ω)+3δ′

L 3(δ)+δL 3(δ′

)+δ2

L 5(δ,δ′

,ω)+3δ′2

L 5(δ,δ′

,ω)?L 5(δ,δ′

,ω)?m 2

F =

10 S(DD)g L

iεαβγδqα?βηγηδ1h1?εαβγδ?αρβJγˉD?δD

iεαβγδqα?βηγpδ1η1·p21

D h2εαβγδ?αρβJγ?δ?μDˉD?μ

iεαβγδqα?βηγ1pδ1η·p1?1

m D?m D h3εαβγδ?αρβ?δˉD?γJ·?D

iεαβγδqα?βηγ1pδ2η·p11

D h2εαβγδ?αρβˉD?γ?δ?μDJμ

iεαβγδqα?βpγ1pδ2η·η11

D?(h5?h6)εαβγδ?αρμJβ?δ?μˉD?γD

iεαβγδqαηβηγ1pδ1?·p21

m D?m D h6?εαβγδ?αρμJβ?μˉD?γ?δD

iεαβγδqαηβηγ1pδ2?·p21

m D?m D h7?εαβγδ?αρμJβ?γˉD?μ?δD

iεαβγδ?αηβηγ1pδ1q·p1?1

m D?m D (h5?h6)?εαβγδ?μραJβ?δˉD?γ?μD

iεαβγδ?αηβηγ1pδ11

m D?m D h2?εαβγδ?μραJβ?μˉD?γ?δD

iεαβγδ?αηβηγ1pδ2q·p2?1

m D h8?εαβγδραJβˉD?γ?δD

iεαβγδ?αηβpγ1pδ2q·η11

m D?m D

(h2+h6)?εαβγδ?μρα?γˉD?β?δDJμiεαβγδηαηβ1pγ1pδ2?·p1?2

m D?m2D

h10εαβγδJα?γˉD?β?δ?μDρμ

11 S(D?D?)f L

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