The piNN vertex function in a meson-theoretical model
a r X i v :n u c l -t h /9905043v 2 10 A u g 1999
The πNN vertex function in a meson-theoretical model
R.B¨o ckmann 1,3,C.Hanhart 2,O.Krehl 1,S.Krewald 1,and J.Speth 1
1)Institut f¨u r Kernphysik,Forschungszentrum J¨u lich GmbH,
D-52425J¨u lich,Germany
2)Department of Physics and INT,University of Washington,Seattle,
WA 98195,USA
3)present address:MPI f¨u r biophysikalische Chemie,Am Fa?berg 11
37077G¨o ttingen (February 9,2008)
The πNN vertex function is calculated within a dispersion theoretical approach,including both πρand πσintermediate states,where the σmeson is an abbreviation for a correlated pion pair in a relative S-wave with isospin I=0.A strong cou-pling between the πσand πρstates is found.This leads to a softening of the πNN form factor.In a monopole parame-terization,a cut-o?Λ≈800MeV is obtained as compared to Λ≈1000MeV using πρintermediate states only.14.20.-Dh,13.75.Cs,13.75.Gx,11.10.St
I.INTRODUCTION
NT@UW-99-27,DOE/ER/40561-67,FZJ-IKP(TH)-1999-17
The pion nucleon-nucleon vertex function is needed in many di?erent places in hadron physics,such as the nucleon-nucleon interaction [1],pion-nucleon scattering and pion photoproduction [2,3],deep-inelastic scatter-ing [4],and as a possible explanation of the Goldberger-Treiman discrepancy [5–9].Commonly,one represents the vertex function by a phenomenological form factor.The cut-o?parameters employed in the various calcula-tions range from Λ(1)
πNN =300MeV in pion photoproduc-tion to Λ(1)
πNN =1700MeV in some One-Boson Exchange models of the nucleon-nucleon interaction,assuming a monopole representation (i.e.n =1in Eq.(10),see
below).The Skyrmion model [10,11]gives Λ(1)
πNN =860
MeV.A lattice gauge calculation gets Λ(1)
πNN =750±140MeV [12].
Unfortunately,the πNN form factor cannot be deter-mined experimentally.It is an o?-shell quantity which is inherently model-dependent.For a given model,the form factor –besides being a parametrization of the ver-tex function –summarizes those processes which are not calculated explicitly.
The simplest class of meson-theoretical models of the nucleon-nucleon interaction includes the exchange of one meson only.In these models,one needs ”hard”form factors for the following reason.In One-Boson Exchange potentials,the tensor force is generated by one-pion and one-rho exchange.A cut-o?below 1.3GeV would reduce the tensor force too strongly and make a description of the quadrupole moment of the deuteron impossible [13,1].
This situation changes when the exchange of two corre-lated bosons between nucleons is handled explicitly.The exchange of an interacting πρpair generates additional tensor strength at large momentum transfers.This im-plies softer cut-o?s for the genuine one-pion exchange [14,15].
Meson theory allows to undress the phenomenological form factors at least partly by calculating those processes which contribute most strongly to the long range part of the vertex functions [14,16].A physically very transpar-ent way to include the most important processes is given by dispersion theory.The imaginary part of the form factor in the time-like region is given by the unitarity cuts.In principle,one should consider the full three-pion continuum.A reasonable approximation is to reduce the three-body problem to an e?ective two-body problem by representing the two-pion subsystems by e?ective mesons [17].An explicit calculation of the selfenergy of the ef-fective meson incorporates the e?ects of three-body uni-tarity.In the case of the pion-nucleon form factor,one expects that πρand πσintermediate states are partic-ularly relevant.Here,”σ”is understood as an abbre-viation for the isoscalar two-pion subsystem.In many early calculations [7,18],the e?ect of the πσintermediate states was found to be negligible.In these calculations,a scalar σππcoupling has been used.Nowadays,such a coupling is disfavored because it is not chirally invariant.In the meson-theoretical model for pion-nucleon scatter-ing of Ref.[19],the exchange of two correlated mesons in the t-channel has been linked to the two-pion scattering model of Ref.[20].The resulting e?ective potential can be simulated by the exchange of an e?ective sigma-meson in the t-channel,if a derivative sigma two-pion coupling is adopted.
In the present work,we want to investigate the e?ect of the pion-sigma channel on the pion form factor using a derivative coupling.
The meson-meson scattering matrix T is an essential building block of our model.Formally,the scattering matrix is obtained by solving the Bethe-Salpeter equa-tion,T =V +V GT,starting from a pseudopotential V .Given the well-known di?culties in solving the four-dimensional Bethe Salpeter equation,one rather solves three-dimensional equations,such as the Blankenbecler-Sugar equation (BbS)[21]or related ones [22,23].The two-body propagator G is chosen to reproduce the two-
particle unitarity cuts in the physical region.The imag-inary part of G is uniquely de?ned in this way,but for the real part,there is complete freedom which leads to an in?nite number of
reduced equations [23].The energies of the interacting particles are well-de?ned for on-shell scattering only.For o?-shell scattering,there is an am-biguity.Di?erent choices of the energy components may a?ect the o?-shell behaviour of the matrix elements.As long as one is exclusively interested in the scattering of one kind of particles,e.g.only pions,one can compensate the modi?cations of the o?-shell behaviour by readjust-ing the coupling constants.This gets more di?cult as soon as one aims for a consistent model of many di?erent reactions.Moreover,in the calculation of the form fac-tor,the scattering kernel V may have singularities which do not agree with the physical singularities due to e.g.three-pion intermediate states [24].
In contrast to the Blankenbecler-Sugar reduction,time-ordered perturbation theory (TOPT)determines the o?-shell behaviour uniquely.Moreover,only physical sin-gularities corresponding to the decay into multi-meson intermediate states can occur [25].For the present pur-pose,we therefore will employ time-ordered perturbation theory.
II.THE MESON–MESON INTERACTION
MODEL
The Feynman diagrams de?ning the pseudopotentials
for πρand πσscattering are shown in Fig.1and Fig.2.We include both pole diagrams as well as t-channel and u-channel exchanges.The transition potential is given by one-pion exchange in the t-channel (see Fig.3).In Ref.[14],πρscattering has been investigated neglecting the A 1-exchange in the u-channel.
π
ρ
π
π
ρ
ρ
π, ω, Α1
π, ω, Α1
π
ρ
ρ
ρ
ρ
ππ
FIG.1.Diagrams describing the πρ→πρpotential
π
σ
π
π
πσ
σ
σ
σ
σσπ
π
ππ
FIG.2.Diagrams describing the πσ→πσpotential π
π
ρ
π
σ
FIG.3.Diagram describing the πρ→πσtransition poten-tial
The ππρand A 1πρinteractions are chosen according to the Wess-Zumino Lagrangian [26]with κ=1
2g ππρ(?μ ρν??ν ρμ)·( ρμ× ρν)
(2)
L A 1πρ=
g ππρ
2
π×(?μ A ν??ν A μ) (?μ ρ
ν??ν ρμ)(3)L ωπρ=
g ωπρ
2m π
?μ π·?μ πσ
(5)L σσσ
=g σσσm σσσσ.
(6)
The completely antisymmetric tensor has the component
?0123=+1.
In the presence of derivative couplings,the canonical momenta conjugate to the ?elds Φk ,
πk =
δL
2( ρ0× π)2
+
f 2
m π
σ˙ π·( ρ0× π)
+
g2ωπρ
m4
A1 ?ν π×(˙ ρν
??ν ρ0) 2
+
g2ππρ
Λ2?q2
n(10)
The cut-o?parametersΛ(n)and the coupling constants g2
4π=2.84can be determined from the decayρ→ππ.
We assume gπρA
1
=gππρ[26].The corresponding cut-o?
parametersΛ(1)ππρ=1500MeV andΛ(2)πρA
1=2600MeV
have been taken from Ref.[24].The decayω→πρ→πγgives the coupling constant g2πρω
4π
=0.25,Λ(1)ππσ=1300MeV,g2σσσ
s=1.2GeV.When the o?-shell momentum P is equal to the incoming on-shell momentum,the potentials V evaluated in the BbS re-duction and in TOPT are identical(see the arrow).In the BbS reduction,the zeroth component of the momen-tum vectors is not well-de?ned for o?-shell scattering.In Fig.5we have chosen on-energy shell components,fol-lowing Ref.[14].For the A1exchange(both in the s-and in the u-channel),both formalisms give fairly sim-ilar results.For the rho-exchange in the t-channel,the S11partial wave shows large di?erences:while TOPT predicts an attractive half-o?shell matrix element,the BbS-reduction gets repulsive for momenta larger than 650MeV.
0.01000.02000.0
P [MeV]
?400.0
?200.0
0.0
200.0
400.0
?400.0
?200.0
0.0
?500.0
0.0
500.0
1000.0
1500.0
0.01000.02000.0
P [MeV]
?40.0
?30.0
?20.0
?10.0
0.0
50.0
100.0
150.0
?60.0
?40.0
?20.0
0.0
20.0
P01
S11P10
S11
H
a
l
f
o
f
f
?
s
h
e
l
l
p
o
t
e
n
t
i
a
l
V
S11
P10
FIG.5.Di?erent contributions to the half o?-shell poten-tials forπρscattering at
√
s=1.2 GeV.The solid line represents the scattering kernel V of TOPT,while the dashed line refers to the BbS reduction. The dotted line shows the TOPT result omitting the contact terms.The panels show contributions of speci?c diagrams of Fig.1;upper left:A1pole diagram,middle left:πpole dia-gram,lower left:ρ?exchange in the t-channel,upper right: A1u-channel exchange,middle right:ωpole diagram,lower right:ωu-channel exchange.
The singularities in the scattering kernel V due to uni-tarity cuts are handled by chosing an appropriate path in the complex momentum plane.The scattering equation, after partial wave decomposition,reads explicitly:
T(p,p′)=V(p,p′)+ dkk2V(p,k)G T OP T(E;k)T(k,p′)
(11) with
k=| k|e?iΦ,(12) whereΦis a suitably chosen angle[29].G T OP T(E;k) denotes the two-body propagator of time-ordered per-turbation theory.
The resulting T-matrix is shown in Fig.6for
√
and lower panel)which is larger than the diagonal πρpotential.The magnitude of the transition potential can be traced back to the interaction Lagrangian (see Fig.5).The derivative coupling favours large momentum trans-fers.The enhancement of the πρ?πρscattering matrix T due to these coupled channel e?ects will shift the max-imum of the spectral function to lower energies and thus lead to a softer form factor.
0.0
1000.0
2000.0Real(P) [MeV]?800.0
?600.0?400.0?200.00.0
?150.0
?100.0?50.00.0
?150.0
?100.0?50.00.0
0.0
1000.02000.0
Real(P) [MeV]
?600.0
?400.0?200.00.0200.0?100.0?50.00.050.0
?150.0?100.0?50.00.0F u l l o f f ?s h e l l T ?m a t r i x
πρ?>πρ
πρ?>σπ (k ρ=153 MeV)
σπ?>πρ (k σ=153 MeV)
FIG.6.The full o?-shell T -matrices for the transitions πρ→πρ(upper panels),πρ→σπ(central panels),and σπ→πρ(lower panels)are shown for E CM =1200MeV and k in CM =153MeV(solid lines)as functions of the real part of the complex o?-shell momentum P.The T -matrix for the tran-sition πρ→πρobtained without coupling to the σπchannel is given by the dashed line.For comparison,also the cor-responding scattering kernels V are displayed (dotted lines).The real parts of the T -matrices are shown on the left hand side,the imaginary parts on the right hand side.
III.THE PION–NUCLEON FORM F ACTOR
The present model for the
πNN vertex function F is
shown in Fig.7.
+
FIG.7.The πNN vertex function
Before coupling to the nucleon,the pion can disinte-grate into three-pion states which are summarized by both pion-sigma and pion-rho pairs.We ?rst evaluate
the vertex function in the N ˉN
channel.In this chan-nel,the πρand πσinteractions can be summed.After a decomposition into partial waves,one gets:
F N ˉN →π=F 0
N ˉN →π+
n =ρ,σ
dk k 2×f π←πn (t,k )G πn (t,k )V πn ←N ˉN (t ;k,p 0).
(13)
Here,p 0is the subthreshold on–shell momentum of the N ˉN –System [5].The bare vertex is called F 0N ˉN →π
.We have to include the self energies Σρand Σσof both the ρand the σinto the two-particle propagators G πρand G πσbecause the vertex function is needed in the time-like region.The annihilation potential V πn ←N ˉN has been worked out in Ref.[30].The form factor needed for the N ˉN
→πρ(σ)transition has not been determined self-consistently,but taken from Ref.[30].
The dressed meson–meson →pion vertex function f π←πn is given by
f π←πn (t,k )=f 0π←πn
(t,k )+ m =ρ,σ
dk ′k ′2
×f 0π←πm (t,k ′)G πm (t,k ′)T πm ←πn (t ;k ′,k ).
(14)
The bare vertex function is called f 0.The vertex func-tion f requires the o?-shell elements of the T -matrix for meson-meson scattering T πm →πn discussed in the previ-ous chapter.Only the partial wave with total angular
momentum J π=0?of the N ˉN
→πvertex function is needed.
The form factor Γ(t)is de?ned as
Γ(t )=
F N ˉN →π
π
∞
9m 2π
Im Γ(t ′)dt ′
0.0
60.0120.0
180.0
t[m π2
]
0.01.02.03.04.0
5.06.0I m (Γ(t ))
0.0
30.060.0
90.0
?t[m π2
]
0.00.20.40.6
0.81.0Γ(t )
FIG.8.Real (right panel)and imaginary (left panel)parts of the πNN form factor as functions of the momentum trans-fer t.Solid line:coupled πρand πσchannels;dotted line:only the πρchannel is considered;dashed line:rescattering in the πρchannel is omitted;dashed-dotted line:only the πσchannel is included.The short-dashed line in the right panel refers to a monopole form factor with Λ(1)=800MeV.
We con?rm earlier ?ndings [7,18]that the pion-sigma states by themselves,even if iterated,do not generate an appreciable contribution to the spectral function.The πρintermediate states clearly dominate.Including rescat-tering processes in a model with only πρstates,one ?nds a large shift of the spectral function to smaller en-ergies,which emphasizes the importance of the corre-lations.The new aspect of our work is the large shift induced by the coupling between πρand πσstates.
In Ref.[31]Holinde and Thomas introduced an e?ec-tive π′exchange contribution to the One-Boson Exchange NN interaction in a phenomenological way in order to shift part of the tensor force from the πinto the π′ex-change.This allowed them to use a rather soft cut-o?Λ(1)
πNN =800MeV to describe the NN phase shifts.The maximum of the spectral function shown in Fig.8is lo-cated at 1.2GeV (t ≈75m 2π)which coincides with the
mass of the π′
.Thus the π′used in Ref.[31]can be interpreted in terms of a correlated coupled πρand πσexchange.Note,that the form factor derived here must not be used in Meson-Exchange models of the nucleon-nucleon interaction,such as discussed in Ref.[1],but only in models which include the exchange of correlated πρand πσpairs.
The form factors obtained via the dipersion relation are shown in the right part of Fig.8.The numerical re-sults can be parameterized by a monopole form factor.The inclusion of an uncorrelated πρexchange leads to a
relatively hard form factor of Λ(1)
πNN =2100MeV.In the present model,using πρintermediate states only,the cut-o?momentum is reduced to Λ(1)
πNN =1500MeV.If one treats the singularities of the scattering kernel V by ap-proximating the propagator of the virtual pion by a static one (see the u-channel exchange in Fig.1),the resulting πρinteraction becomes more attractive and produces a
much softer form factor corresponding to Λ(1)
πNN =1000MeV [14],employing only πρintermediate states.The
present model,including both πρand πσintermediate
states,leads to Λ(1)
πNN =800MeV.
IV.SUMMARY
Microscopic models of the πNN vertex function are re-quired in order to understand why the phenomenological form factors employed in models of the two-nucleon inter-action are harder than those obtained from other sources.In Ref.[14],a meson-theoretic model for the πNN ver-tex has been developped.The inclusion of correlated πρ
states gave a form factor corresponding to Λ(1)
πNN =1000MeV.This is still harder than the phenomenological form factors required in the description of many other physi-cal processes.Within the framework of Ref.[14],a fur-ther reduction of the cut-o?Λ(1)
πNN is impossible.Corre-lated πσstates were not considered in Ref.[14]because of the results obtained in Refs.[7,18].In the present work we have shown that the ?ndings of Refs.[7,18]have to be revised.Meson-theoretic analyses of πN scatter-ing strongly suggest a derivative σππcoupling.This is shown to enhance the o?-shell coupling between πρand πσintermediate states in the dispersion model for the πNN form factor.A softening of the πNN form factor is obtained which largely removes the remaining discrep-ancies between the phenomenological form factors.Acknowledgments
C.H.acknowledges the ?nancial support through a Feodor-Lynen Fellowship of the Alexander-von-Humboldt Foundation.This work was supported in part by th U.S.Department of Energy under Grant No.DE-FG03-97ER41014.
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