Dilepton Decay of Low-mass Mesons K. GallmeisterU. Mosel and L. von Smekal Institut für Theoretische Physik Universität Giessen Giessen Germany and

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Dilepton Decay of Low-mass ρMesons

K. Gallmeister,∗U. Mosel, and L. von Smekal

Institut für Theoretische Physik, Universität Giessen, Giessen, Germany and

Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Giessen, Giessen, Germany

The HADES collaboration has extracted dilepton mass spectra for π−induced reactions on the

proton from a comparison of data taken on C and CH2targets. The spectra were interpreted in

terms of diﬀerent versions of vector meson dominance. Here we present results obtained from the

theory and generator GiBUU. We ﬁrst check the subtraction procedure used and then discuss the

obtained mass spectra for the proton target. We point out that any conclusions on the version of

VMD requires the knowledge of the ρspectral function in the interesting mass region.

I. INTRODUCTION

The radiative decay of nucleon resonances is governed

by electromagnetic transition form factors which take

the ﬁnite extension of the resonance and the nucleon

into account. A very successful description of such

photon-hadron couplings and has been achieved within

the vector-meson-dominance (VMD) model [1]. In this

model the coupling takes place through an intermediate

ρmeson. As discussed in some detail in [2] there are

two versions of VMD. In the more widely used version

VMD2 (in the notation of [2]) the coupling of the photon

takes place only through the ρmeson, whereas in ver-

sion VMD1 the coupling amplitude employs in addition

a coherently added term in which the photon couples di-

rectly to the hadron. Both methods are fully equivalent if

certain relations between the coupling constants involved

are met [2]. Since these relations in nature hold only ap-

proximately the HADES collaboration has recently tried

to ﬁnd experimental signatures for one or the other VMD

version [3].

The two VMD versions diﬀer in their predicted dilep-

ton invariant mass (Me+e−) distribution mainly at small

masses below the 2mπthreshold where the pion electro-

magnetic formfactor is not accessible. One possibility to

explore this low mass range is thus given by the Dalitz

decay of a nucleon resonance N∗→Ne+e−; this decay

is governed by the electromagnetic transition formfac-

tor in the time-like region. A conclusive comparison of

data with the VMD versions should be possible if the

ρ-spectral function in the mass range used for the com-

parison is known.

Therefore, merging the experimental information on

ρproduction in the π−+p→ρ+nreaction from

Ref. [4] with measured dilepton yields in the reaction

π−+p→e+e−+nfrom Ref. [3] seems to oﬀer an inter-

esting possibility to access also the low-mass region and

to explore the validity of VMD there.

The recent HADES experiment aims to determine the

dilepton yield in the reaction π−pat dilepton invariant

masses of about 100 MeV to 400 MeV, i.e. around and be-

low the 2πthreshold [3]. The experiment was performed

∗Contact e-mail: kai.gallmeister@theo.physik.uni-giessen.de

at an incoming pion momentum of pπ= 0.69 GeV, cor-

responding to √s= 1.49 GeV. It did not directly use a

proton target, but instead obtained data for CH2and

(with lower statistics) for C. A comparison of both then

leads to the dilepton mass spectrum for H. Invariant mass

cuts, assuming a quasi-free reaction process, are used to

isolate the n+e+e−ﬁnal state.

The further analysis uses results of an earlier publica-

tion [4] on a measurement and analysis of 2πproduction

in the π−+preaction. A partial wave analysis (PWA)

of these results employing the Bonn-Gatchina K-matrix

model [5, 6] then led to the cross section for ρproduction

primarily through the D13 N(1520) resonance.

The data obtained in [3] do not show the strong rise

(∝1/M3

e+e−) towards small dilepton invariant masses

contained in the VMD2 model [7]. On the other hand, a

good ﬁt to the data could be obtained by using VMD1

and ﬁtting the relative strength of the two components

with a free parameter.

For such a comparison of the VMD versions the ρspec-

tral function has to be known in the region of interest.

Crucial for this comparison is, therefore, the continuation

of the ρspectral function from an energy range, where

it has been measured by the 2πdecay, to lower masses.

The latter is inﬂuenced by the decay N(1520) →ρ+n,

i.e. by a hadronic interaction vertex independent of the

electromagnetic coupling of the ρto the virtual photon.

In VMD1 an additional assumption about the electro-

magnetic transition formfactor is required.

The purpose of this present paper is twofold. First,

we investigate the steps leading from the actual mea-

surements, which were performed on heavier targets (C,

CH2), to the spectrum on a proton. Second, we have a

closer look at the conditions necessary to decide between

VMD1 and VMD2.

II. MODEL

We describe the reactions π−+Cand π−+pwithin

the quantum-kinetic GiBUU theory and event generator.

Both the theoretical foundations as well as all the ele-

mentary reaction input and the numerical algorithms are

described in detail in Ref. [8]. The source code used for

the present calculations can be obtained from [9]. All

arXiv:2210.04788v1 [nucl-th] 10 Oct 2022

the special equations for dilepton production are given

in Ref. [7]. As described there we use the VMD2 variant

for the dilepton-decay of a ρmeson.

Essential input for the calculations are the properties

of the hadronic interaction channels; for the problem at

hand the relevant channel is π−+p→n+ρ. In the

present version of GiBUU these are taken from the partial

wave analysis of Manley and Saleski [10].1Contrary to

the non-relativistic spectral function employed there in

the present calculations we use a relativistic form which

– when integrated over m– is normalized to 1 for constant

widths

A(m) = 2

m2Γ(m)

(m2−M2)2+m2Γ(m)2.(1)

Here Mstands for the peak mass of either the ρmeson

or the N(1520) resonance. The width Γis either that

equal to the sum of dilepton and pion decay widths for

the ρmeson or that of the nucleon resonance. We use

this Breit-Wigner form for the spectral function of the

ρmeson also below the 2πthreshold even though the

physical character of this meson becomes doubtful in this

mass range.

In the Manley-Saleski analysis the decay width of a

nucleon resonance N∗with mass Minto Nρ is given

by the product of the phase-space element with a Blatt-

Weisskopf (BW) formfactor Bl,

Γ∗(W, m)=Γ∗

qρ(W, m)

WB2

l(qρ(W, m)R).(2)

Here qρis the c.m. momentum of the decay products,

qρ(W, m) = (3)

2Wp(W2−(MN+m)2)(W2−(MN−m)2),

Wis the invariant mass of the decaying N∗,mis the run-

ning mass of the ρmeson and MNis that of the nucleon.

The BW formfactor regularizes the width at large m. It

does not, however, limit the high-momentum (low-mass)

behavior.

In Fig. 1 we show the width for the decay N(1520) →

ρN at the invariant mass at which the HADES ex-

periment was performed. Being dominated by phase-

space this width obviously favors the production of high-

momentum, very-low-mass ρmesons. This is in contrast

to usually used formfactors which are used to mimick

the ﬁnite extension of hadrons and thus cut oﬀ the high-

momentum parts of a transition.

The Manley-Saleski analysis does not describe explic-

itly the secondary decay of primary decay daughters. For

1The HADES analysis of the 2πcross sections in [4] used the

Bonn-Gatchina analysis which leads to somewhat diﬀerent decay

widths.

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6

W = 1.49 GeV

Γ(W,m) / Γ0

∗

m [GeV]

FIG. 1. Decay width of Eq. (2) for the decay N(1520) →Nρ

as a function of the ρmass mat ﬁxed W= 1.49 GeV.

the present calculations we use the expression [11]

Γρ→π+π−(m)=Γ0

π(m)

π(Mρ)

1+[qπ(Mρ)R]2

1+[qπ(m)R]2

×Θ(m−2mπ),(4)

with

qπ(m) = 1

2pm2−4m2

π,(5)

being the c.m. momentum of the pions.2Here mis the

’running mass’ of the ρmeson and Mρis its nominal mass

at the peak of the spectral function. Finally, the dilepton

decay width is given by

Γρ→e+e−(m) = Cρ

m3(1 + 2m2

e/m2)p1−4m2

e/m2

×Θ(m−2me).(6)

Here meis the electron mass and C= 9.078 ×10−6is an

empirical coupling constant [7].

III. RESULTS

We start our discussion with pointing out that the

beam energy in this experiment is so low that only the

low-mass tail of the ρmeson is populated in the reaction

π−+p→n+ρand even for the N(1520) the full peak

cannot be reached. This is illustrated in Fig. 2 showing

the spectral functions for the N(1520) resonance and the

ρmeson, the latter shifted by the nucleon mass.

The experiment was run at an πN invariant mass of

1.46 GeV (indicated by a vertical bar in Fig. 2). For the

present experiment even the very low mass region below

the 2πthreshold of the ρmeson (m < 1.22 GeV) is essen-

tial because it is here that the predictions of VMD1 and

2Note that this deﬁnition of the ρdecay width from [11] diﬀers by

a factor Mρ/m from the one used in [7]. For the dilepton spectra

from heavy-ion collisions this diﬀerence is irrelevant.

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DileptonDecayofLow-massMesonsK.Gallmeister,U.Mosel,andL.vonSmekalInstitutfürTheoretischePhysik,UniversitätGiessen,Giessen,GermanyandHelmholtzResearchAcademyHesseforFAIR(HFHF),CampusGiessen,Giessen,GermanyTheHADEScollaborationhasextracteddileptonmassspectraforinducedreactionsontheprotonfromacompar...

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Dilepton Decay of Low-mass Mesons K. GallmeisterU. Mosel and L. von Smekal Institut für Theoretische Physik Universität Giessen Giessen Germany and

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