Relation between the dipole polarizabilities of charged and neutral pions

Relation between the dipole polarizabilities of charged and neutral pions

L.V. Fil’kov Lebedev Physical Institute, Leninsky Prospect 53, Moscow 119991, Russia
Abstract

Using the fact that the contribution of the states with isospin in the difference of the amplitudes of the processes and is very small, we have analyzed the dispersion sum rules for the difference between the dipole polarizabilities of the charged and neutral pions as a function of the meson parameters. Then taken into account the current perturbation value of , we have found for values of the meson parameter within the region: MeV, , keV. It has been shown that the value of the decay width of the meson into can be found if the difference is reliably determined from the experiment. Estimation of the optimal value of the decay width has given keV.

polarizability, pion, meson, resonance, dispersion relations, chiral perturbation theory
pacs:
13.40.-f, 11.55.Fv, 11.55.Hx, 12.39.Fe

I Introduction

Pion polarizabilities are the fundamental structure parameters characterizing the behavior of the pion in an external electromagnetic field. Dipole polarizabilities arise as terms in the expansion of the non-Born amplitudes of Compton scattering in powers of the initial and final photon energies and . In terms of the electric and magnetic dipole polarizabilities, the corresponding effective interaction has the form:

 H(2)eff=−124π(α1→E2+β1→H2). (1)

The dipole polarizabilities measure the response of the hadron to quasistatic electric and magnetic fields. In what follows, these parameters are given in units .

The values of the pion polarizabilities are very sensitive to the predictions of different theoretical models. Therefore, an accurate experimental determination of them is very important for testing the validity of such models.

At present, the value of the difference of the charged pion dipole polarizabilities found from radiative meson photoproduction from protons mainz () is equal to and close to the value obtained from scattering of high energy mesons off the Coulomb field of heavy nuclei in Serpukhov antip () and equal to . On the other hand, these values differ from the prediction of the chiral perturbation theory (ChPT) (() gasser ()). The experiment of the Lebedev Physical Institute on radiative pion photoproduction from protons lebed () has given . This value has large error bars but nevertheless shows a large discrepancy with regard to the ChPT predictions, as well.

The preliminary result of the COMPASS collaboration has been found by studying the meson scattering off the Coulomb field of heavy nuclei compass (). This result is more close to the ChPT prediction. However to obtain this result the authors used very big values of momentum transfer (Gev/c). In this region an interference between the Coulomb and nuclear amplitudes should be taken into account fil4 (); fil-PS (); walch (). It should be noted that the authors of work antip () chose (GeV/c) to guarantee that the contribution of the strong interaction below the Coulomb peak is negligible.

The charged pion polarizabilities can be also found by studying the process . Investigation of the process at low and middle energies was carried out in the framework of different theoretical models and, in particular, in the frame of dispersion relations (DR). In Ref. fil1 (); fil2 (); fil3 () we have analyzed the processes and using DRs with subtractions for the invariant amplitudes and without partial-wave expansions. The subtraction constants have been uniquely determined in these works through the pion polarizabilities. The values of the polarizabilities have been found from the fit of the experimental data of the processes and up to 2500 MeV and 2250 MeV, respectively. As a result, we have found and . The result for is in good agreement with the values obtained in Ref.mainz (); antip (); lebed () whereas it is at variance with the ChPT prediction.

In the works bab (); holst (); kal (); garcia () the dipole polarizabilities of charged pions have been determined from the experimental data of the process in the full energy region MeV, (where is the square of the total energy in c.m. system). The results obtained in these works are close to ChPT predictions gasser (); burgi (). However, the values of the experimental cross section of the process in this region pluto (); dm1 (); dm2 (); mark () are very ambiguous, and, as has been shown in Ref. holst (); fil3 (), even changes of these values by more than 100% are still compatible with the present error bars.

Therefore, it is necessary to consider other additional possibilities of the determination.

Such an information could be obtained from the dispersion sum rules (DSR) for these parameters. However, the main contribution to DSR for is given by the meson, which is very wide and this causes additional uncertainties in the DSR calculation.

On the other hand, if we consider the DSR for the difference between the charged and neutral meson polarizabilities , then the contribution of mesons with isotopic spin in the channel to this difference would be equal 0 lvov1 (); fgr (), when the masses of charged and neutral mesons are equal each other. As a result, a model dependence for should be decreased essentially.

In the present work we investigate DSR for as a function of the decay width of , when the masses of the charged and neutral mesons are not equal each other. As will be shown, the contribution of the meson is small in this case too, and we can find a realistic limit on the value of .

It has been shown that the value of the decay width of the meson into can be found if the value is reliably determined from experiment.

Ii Dispersion sum rules for the pion polarizabilities

We will consider the helicity amplitudes and . These amplitudes have no kinematical singularities or zeros aber (). The relations between the amplitudes , and the ones with isotopic spins and read

 FC = √23(F0+1√2F2), FN = √23(F0−√2F2). (2)

The dipole ( and ) polarizabilities are defined rad (); fil2 () through expansion of the non-Born helicity amplitudes of Compton scattering on the pion in powers of at fixed

 M++(s=μ2,t) = 2πμ(α1−β1)+O(t), M+−(s=μ2,t) = 2πμ(α1+β1)+O(t), (3)

where is the meson mass (different for and ), .

The dispersion sum rules for the difference of the dipole polarizabilities was obtained in Ref. fil1 () using DRs at fixed without subtractions for the amplitude . In this case, the Regge-pole model allows the use of DR without subtractions aber (). Such a DSR is

 (α1−β1) = 12π2μ⎧⎪ ⎪⎨⎪ ⎪⎩∞∫4μ2 ImM++(t′,u=μ2) dt′t′ (4) +∞∫4μ2 ImM++(s′,u=μ2) ds′s′−μ2⎫⎪ ⎪⎬⎪ ⎪⎭.

As is evident from Eq.(2), the contribution of the isoscalar mesons to the difference equals 0 (if the masses of the charged and neutral pions are equal). We will study this difference when these masses do not equal each other.

The DSRs for the charged pions are saturated by the contributions of the , , , and mesons in the -channel and , , in the -channel. For the meson the contribution of the , , , , and mesons are considered in the -channel and the same mesons as for the charged pions in the -channel. Besides, we take into account a nonresonant S-wave contribution of two charged pions in the t channel.

The parameters of the , , , and mesons are given by the Particle Data Group pdg (). For the meson we took MeV pdg (), 425 MeV (the average value of the PDG estimate pdg ()), MeV zel ().

The parameters of the and mesons are taken as follows:

: MeV pdg (), MeV (the average of the PDG pdg () estimate), MeV, anis (), ;

: MeV, MeV, MeV morg (), bugg ().

The mass and the total decay width of the meson are taken from PDG: MeV, MeV. The decay has not yet been observed. Therefore we use this decay width according the work garcia ():

 Γh1→γπ0=e24πCh1(m2h1−m2π0)33m3h1, (5)

where the coefficient can be estimated using nonet symmetry garcia (); lvov2 ():

 Ch1(1170)≃9Cb1(1235)≃0.45. (6)

As a result we have MeV.

Recently, a lot of works have been devoted to the study of the meson (see, for example capr1 (); capr2 (); mous (); kamin (); pela ()). An average of the most advanced data on the meson gives (pela ())

 mσ=446±6,Γσ/2=276±5. (7)

In our analysis we use the values of the mass of the meson and its total decay width in the following intervals:

MeV,   MeV.

The values of the decay width of we consider from 0 up to 3 keV.

Expressions for the imaginary parts of the resonances under consideration are given in Appendix.

Besides the contribution of the , , and mesons we have taken into account a nonresonant contribution of the -waves with the isospin and 2 according to the diagrams of Fig. 1.

It is worth noting that the vertexes of the and meson poles in the dispersion approach include the full dynamics of the transitions on the mass shell. In this case there is no need to consider direct and rescattering mechanisms of transition separately.

According to the unitarity condition, the imaginary part of the amplitude for the -loop diagram in Fig. 1 can be written as

 ImM(s)++=BReTπ+π−→ππ, (8)

where is the contribution of the Born amplitude to the -wave of the amplitude and equal to

 B=16π(e24π)m2π±t2ln(1+q/q01−q/q0), (9)

is the momentum (energy) of the meson. The Born amplitude can be expressed in terms of the and isospin amplitudes as

 B=√23B(I=0)+√13B(I=2). (10)

Taking into account that

 B(γγ→π0π0)=−√13B(I=0)+√23B(I=2)=0 (11)

we have bogl ()

 B(I=0)=√23B,B(I=2)=√13B. (12)

The amplitudes of scattering are expressed through the amplitudes in the isotopic space and as follows:

 Tπ+π−→π+π− = 23(T(0)+12T(2)), Tπ+π−→π0π0 = 23(T(0)−T(2)). (13)

According to the relations (10) and (II) the imaginary parts of the loop contributions to the -wave of the amplitude are equal to

 ImM(s)++(γγ→π0π0)=49BRe(T(0)+T(2)), (14)
 ImM(s)++(γγ→π+π−)=19BRe(4T(0)+T(2)). (15)

The amplitudes and can be presented as

 ReT(I)=q0qηIcosδI0(t)sinδI0(t), (16)

where is the phase-shift of the -wave of scattering with isospin and is the inelasticity.

The expression for the phase-shift has been determined using the parameterization of Ref. garcia1 (). At low energy we have

 cotδ00(t) = √t2qμ2t−12μ2{μ√t+B0+B1w(t) (17) +B2w(t)2+B3w(t)3},

where

 w(t)=√t−√4m2k−t√t+√4m2k−t,

and is equal to 1.

For the energy (1.42 GeV) we use garcia1 ()

 δ00(t)=d0+Bq2km2k+Cq4km4k+Dθ(t−4m2η)q2ηm2η, (18)
 η00(t) = exp⎡⎣−qk√t(ϵ1+ϵ2qk√t+ϵ3q2kt)2 (19) −ϵ4θ(t−4m2η)qη√t],

where and ; and are the masses of and mesons, respectively.

The parameters in Eqs(17-19) are listed in Table 1.

To describe the phase-shift we use Schenk’s parameterization schenk () in the energy region up to 1.5 GeV, assuming that ananth ()

 tanδ20=qq0{A02+B20q2+C20q4+D20q6}(4μ2−s20t−s20), (20)

where

 A20=−0.044,B20=−0.0855μ2,C20=−0.00754μ4, D20=0.000199μ6,s20=−11.9μ2.

Iii Calculation of (α1−β1)π±

The value of has been determined from the investigation of the process in the works fil1 (); kal (); garcia (): , , .

These values are in good agreement with the prediction of ChPT bell () . Therefore, in order to determine we have added the value of to the results of the calculations of with help of DSR (4) at different values of the decay width of , when the mass and the total decay width of the meson vary within the following values: MeV, MeV.

The results of the calculation are shown in Fig.2. Line (1) corresponds to calculations with . Lines (2) and (3) correspond to 400 MeV, 600 MeV and 550 MeV, 400 MeV, respectively. As is evident from this Figure the values of weakly depend on the mass and the total decay width of the meson in the region under consideration. The values of obtained are within .

The greatest contribution to is given by the and mesons. The parameters of the meson and so its contribution are well known. On the other hand the experimental data on the meson are very poor. In particular, the decay of this meson into was not observed yet still in the experiment. Therefore, a reliable experimental determination of will allow to determine the real value of the decay width . For example, if the result of work mainz () is confirmed, then MeV.

Line (4) in Fig.1 is the result of the calculations of DSR (4) for at MeV and MeV. This result strongly depends on the decay width and indicates that realistic values of can be obtained if keV.

The influence of the upper integration limit () in the DSR (4) on the results of the calculation was investigated. They are not practically changed for more than (6 GeV). In the present work we performed the integration up to (20 GeV).

Iv Conclusions

Using the fact that the contribution of the state with isospin to the difference is very small we have analyzed DSR for this difference at the real values of the pion masses. DSR has been calculated for the meson parameters within the intervals: MeV, , keV. In order to determine we have added =-1.9 to . The values of found weakly depend on the meson parameters and are in the range . This result is in agreement with the experimental values obtained in work mainz (), whereas it is at variance with the calculations in the framework of ChPT gasser ().

It has been shown that further experimental investigation of can be an opportunity to determine the decay width .

Besides, the analysis of DSR for showed that more realistic values of this parameter () can be obtained with help of DSR (4) if the decay width keV. The values keV were obtained early in works fil1 (); fil2 (); dubn (); achas () also. Results with keV quoted in the resent literature are listed in mous (); hofer ()

Acknowledgements

The author would like to thank H.J. Arends, A. Thomas, Th. Walcher, V.L. Kashevarov, and A.I. L’vov for useful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 443).

Appendix A

The contributions of the vector and axial-vector mesons , and to are calculated with the help of the expression

 ImM(V)++(s,u=μ2)=∓4g2VsΓ0(m2V−s)2+Γ20 (A1)

where is the meson mass, the sign ”+” corresponds to the contribution of the and mesons and

 g2V=6π√m2Vs(mVm2V−μ2)3ΓV→γπD1(m2V)/D1(s), (A2)
 Γ0=(q2i(s)q2i(m2V))32m2V√sD1(m2V)/D1(s)ΓV (A3)

Here is connected with the centrifugal potential and equal to blatt (), fm is an effective interaction radius, and are the total decay width and the decay width into of these mesons. The momentums for , and mesons are equal to , , , , and , respectively.

Appendix B

The amplitude of the contribution of a scalar meson to the process can be written as

 T=gs√t−Ms−i12Γs. (B1)

Then it is easy to show that the imaginary part of the amplitude of the meson contributions to the process under consideration could be presented as

 ImMσ++(t)=gσ(√t+Ms)Γσ0(t)(t−M2σ)2+(Γσ0(t))2, (B2)

where

 gσ=8πt[23MσΓγγΓσ√M2σ−4μ2], (B3)
 Γσ0=Mσ(√t+Mσ)2√t(t−4μ2M2σ−4μ2)1/2Γσ. (B4)

These expressions (B2)-(B4) can be very useful to describe scaler mesons with large decay widths.

As the two mesons give a big contribution to the decay width of the meson and the threshold of the reaction is very close to the mass of the meson, we consider Flatté’s expression flatte () for the meson contribution to the process .

For :

 ImMf0++=gf0Γ0f0(m2f0−t)2+Γ20f0, (B5)

where

 Γ0f0 = ⎡⎢⎣Γf0→ππ⎛⎝t−4μ2m2f0−4μ2⎞⎠1/2 (B6) +Γf0→kk⎛⎝t−4m2km2f0−4m2k⎞⎠1/2⎤⎥⎦mf0.

For :

 ImM++=gf0Γ0f0([m2f0−t −⎛⎝4m2k−tm2f0−4m2k⎞⎠1/2mf0Γf0→kk⎤⎥⎦2+Γ20f0⎞⎟ ⎟⎠−1, (B7)
 Γ0f0=Γf0→ππmf0⎛⎝t−4μ2m2f0−4μ2⎞⎠1/2. (B8)

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