A Relativistic Description of Hadronic Decays of the Meson $

a r X i v :h e p -p h /0507239v 1 20 J u l 2005A relativistic description of hadronic decays of the meson π1

Nikodem Pop l awski ?

Nuclear Theory Center,Indiana University,Bloomington,IN 47405This article is based on the author’s Ph.D.thesis currently being written under the supervision of Prof.Adam Szczepaniak.Abstract The subject of this work is analysis of hadronic decays of exotic meson π1in a fully relativistic formalism,and comparison with the nonrelativistic results.The relativistic spin wave functions of mesons and hybrids are constructed based on unitary representations of the Lorentz group.The radial wave functions are obtained from phenomenological considerations of the mass op-erator.We ?nd that decay channels π1→πb 1and π1→πf 1are favored,in agreement with results obtained using other models,thus indicating some model independence of the S +P selection rules.We will also report on e?ects of meson ?nal state interactions in exotic channels.1Introduction In a region around 2GeV a new form of hadronic matter is expected to exist in which the gluonic degrees of freedom are excited.In mesons these can result in resonances with exotic J P C quantum numbers.The adiabatic potential calculations show π1(1?+)as the lowest energy excited gluonic con?guration [1].The present models of hybrid decays (for instance [2,3])are nonrelativistic and therefore one should investigate corrections arising from fully relativistic treatment.The case of π1

is of a special interest also because its evidence has been reported by the E852collaboration and new experimental searches are planned for JLab and GSI.

2Relativistic spin wave function for mesons and hybrids

For a system of non-interacting particles the spin wave function is constructed as an element of an irreducible representation of the Poincare group.We will assume m u =m d =m .In the rest frame of a quark-antiquark pair

l μq =(E (m q ,q ),q ),l μˉq =(E (m ˉq

,?q ),?q ),the normalized spin-1wave function (J P C =1??)is given by the Clebsch-Gordan coe?cients and can be written in terms of Dirac spinors as

Ψλq ˉq q ˉq (q ,l q ˉq

=0,σq ,σˉq )=12m q ˉq ˉu (q ,σq ) γi ?2q i ?c

2004American Institute of Physics 1

where m qˉq is the invariant mass and?i(λqˉq)are polarization vectors corresponding to spin1quan-tized along the z-axis.The wave function of a qˉq system moving with a total momentum l qˉq=l q+lˉq is given by

Ψλqˉq qˉq(q,l qˉq,λq,λˉq)= σq,σˉqΨλqˉq qˉq(q,lˉq=0,σq,σˉq)D?(1/2)λqσq(q,l qˉq)D(1/2)λˉqσˉq(?q,l qˉq),(2) where the Wigner rotation matrix

D(1/2)

λλ′(q,P)= (E(m,q)+m)(E(M,P)+M)+P·q+iσ·(P×q)

2(E(m,q)+m)(E(M,P)+M)(E(m,q)E(M,P)+P·q+mM) λλ′

corresponds to a boost withβγ=P/M.One can show

Ψλqˉq qˉq(q,l qˉq,λq,λˉq)=?

1

2m qˉq

ˉu(l q,λq) γμ?lμq?lμˉq

√

√

√m

qˉq+2m

v(lˉq,λˉq)?μ(l qˉq,λ)

·Y1l(ˉq)<1,λ;1,l|J,λqˉq>,(6)

with q=Λ(l qˉq→0)l q.In order to construct meson spin wave functions for higher orbital angular momenta L one need only to replace Y1l with Y Ll.

In the rest frame of the3-body system corresponding to a qˉq pair with momentum?Q and transverse gluon with momentum Q,the total spin wave function of the hybrid is obtained by coupling the qˉq spin-1wave function(3)and the gluon wave function(J P C=1??)to a total spin S=0,1,2and J P C=0++,1++,2++states respectively,and then with one unit of the orbital angular momentum to the exotic state1?+:

Ψλex qˉq g(S)(λq,λˉq,λg)=

λqˉq,σ=±1,M,l

Ψλqˉq qˉq(q,l qˉq=?Q,λq,λˉq)<1,λqˉq;1,σ|S,M>·D(1)σλg(ˉQ)Y1l(ˉQ).

The spin-1Wigner rotation matrix D(1)relates the gluon helicityσto its spinλg quantized along the z-axis.The corresponding normalized wave functions are then given by:

Ψλex

qˉq g(S=0)

= 8π λqˉqΨλqˉq qˉq(q,?Q,λq,λˉq)[??(λqˉq)·??c(Q,λg)][ˉQ·?(λex)],

2

Ψλex

qˉq g(S=1)

= 8π λqˉqΨλqˉq qˉq(q,?Q,λq,λˉq)[??(λqˉq)×??c(Q,λg)]·[ˉQ×?(λex)],

Ψλex

qˉq g(S=2)

= 104π λqˉqΨλqˉq qˉq(?Q,λq,λˉq)ˉQ·[??(λqˉq)???c(Q,λg)]·?(λex),(7) whereˉQ=Q/|Q|,(A?B)ij=2A(i B j)?2

(2π)32E(m,p q)

d3pˉq

√√N(P)

·ψL(m qˉq(p q,pˉq)

√

1?|c|2.

Similarly theρ(I=1)andω,φ(I=0)states(J P C=1??)are given by(8),but instead of Ψλqˉqδλ0one must use(3).The b1(I=1)and h1(I=0)states(J P C=1+?)contain the wave function(5)with q=Λ(P→0)p q.Finally,the a(I=1)and f(I=0)states(J P C=0,1,2++) correspond to(6).

The hybrid state in its rest frame is given by

|π1(I3,λex)>= allλ,c,f1(2π)32E(m,p q)d3pˉq(2π)32E(m g,Q)

·(2π)32(E(m,p q)+E(m,pˉq)+E(m g,Q))δ3(p q+pˉq+Q)λc g

c q cˉq√

μex ,

m qˉq g(p q,pˉq,Q)

where the spin wave function Ψq ˉq g was given in (7)for S =0,1,2and the orbital wave function ψ′L depends only on m q ˉq and the invariant mass of the three-body system.Here m g denotes the e?ective mass of the gluon coming from its interaction with virtual partcles,and λc g

c q c ˉq are the

Gell-Mann matrices.Constants N are ?xed by normalization (m M is mass of meson)

=(2π)32E (m M ,P )δ3(P ?P ′)δλλ′δI 3I ′3

.4Decays of π1and nonrelativistic limit

We will assume that a transverse gluon in π1creates a quark-antiquark pair and therefore the hybrid decays into two mesons.The Hamiltonian of this process in the Coulomb gauge is given by

H = all c,f

d 3x ˉψc 1f 1(x )(gγ·A c g (x ))ψc 2f 2(x )δf 1f 2

1(2π)32E (m,k )

[u (k ,λ)b k λcf +v (?k ,λ)d ??k λcf ]e i k ·x and

A c g (x )= λ

d 3k

2Γπη,Γπf 1=1

5Orbital wave function

The orbital angular momentum wave function for a meson or a hybrid depends on the potential between quark and antiquark or for a qˉq g system.An explicit form of such a potential is not known exactly and such a function must be modeled.Because of the Lorentz invariance it may depend on momenta only through the invariant mass of particles.Moreover,it must tend to zero for large momenta fast enough to make the amplitude convergent.The most natural choice is the exponential function

ψL(m qˉq(p q,pˉq)/μ)=e?m2qˉq(p q,pˉq)/8μ2

for a meson,and

ψ′L(m qˉq(p q,pˉq)/μex,m qˉq g(p q,pˉq,Q)/μ′ex)=e?m2qˉq(p q,pˉq)/8μ2ex e?m2qˉqg(p q,pˉq,Q)/8μ′2ex

for a hybrid.The integrals for the decay amplitude are not elementary and must be computed numerically.In a nonrelativistic limit,however,they can be expressed in terms of the error function.

The free parameters of the presented model are m,m g,μ’s and g.The pion form factor constants fπand Fπ(whose behaviour is experimentally known)de?ned by

<0|Aμ,i(0)|πk(p)>=fπpμδik,<πi(p′)|Vμ,j(0)|πk(p)>=Fπ(pμ+p′μ)i?ijk,(11) allow us to?t m andμπ(with theπstate given by(8)).The axial and the vector currents are de?ned by

Aμ,i(0)=ˉψcf(0)γμγ5σi

2

ψcf(0),(12)

withψcf(x)given in(10).By virtue of the Lorentz invariance fπis a constant,whereas Fπis a function of Q2=?(p?p′)2.Impossibility of?nding the generators H and M0i of the Poincare group in presence of interaction together with normalization of states violate the Lorentz covariance between spatial and time components but do not break a rotational symmetry.The resulting form factors will depend on the frame of reference.

6Results and summary

Taking mπ=(3?140+770)/4=612MeV(in normalization)and m=306MeV givesμπ= 220MeV.Assuming also g2=10,m g=500MeV[5],m ex=1.6GeV and equality of all parameters

μleads to the following values:Γπb

1=150MeV,Γπf

1

=20MeV,Γπρ=3MeV.In the nonrelativistic

limit one obtains respectively:230,31and0MeV.

Two important conclusions come from this work.Firstly,numerical results show that relativis-tic corrections arising from a Wigner rotation are signi?cant.Therefore,models with no Wigner rotation are either nonrelativistic or inconsistent.These corrections decrease in general(for rea-sonable values of m)the width rates forπ1or make them di?erent from zero if they vanished for the NR case.Secondly,theπ1prefers to decay into two mesons,one of which has no orbital angular momentum and the other has L=1(S+P selection rule).This is in agreement with other models,for instance[6].Calculations involving decay rates ofπ1into strange mesons and ?nal state interactions(b1→π+ω,ω+π→ρ)are in preparation.

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References

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[2]N.Isgur,R.Kokoski,J.Paton,Phys.Rev.Lett.54,869(1985)

[3]F.E.Close,P.R.Page,Nucl.Phys.B443,233(1995)

[4]B.Bakamjian,L.H.Thomas,Phys.Rev.92,1300(1953)

[5]A.P.Szczepaniak,E.S.Swanson,arXiv:hep-ph/0308268(2003)

[6]P.R.Page,E.S.Swanson,A.P.Szczepaniak,Phys.Rev.D59,034016(1999)

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