Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics A 919 (2013) 15–31

www.elsevier.com/locate/nuclphysa

Study of the φπ0 transition form factor

Simone Pacetti

Physics Department and INFN, University of Perugia, 06123 Perugia, Italy

Received 19 June 2009; received in revised form 29 September 2013; accepted 4 October 2013

Available online 11 October 2013

Abstract

In 2008 the BaBar Collaboration published new data on the cross section for the annihilation e+e− →φπ0, obtained using the initial state radiation technique at a center of mass energy of 10.6 GeV.

Such a process represents an interesting test bed for the quark model. Indeed, since the φπ0 productionvia e+e− annihilation proceeds through a mechanism which violates the Okubo–Zweig–Iizuka rule, thecorresponding cross section could be characterized by contributions from non-qq̄ bound states, like hybridsor tetraquarks. However the nature of such possible non-standard contributions will be not investigated inthe following.

The φπ0 cross section is analyzed in connection with other data coming from different processes, thatinvolve the same mesons, using a method which implements the analyticity in the φπ0 transition formfactor by means of a dispersion relation procedure.© 2013 Elsevier B.V. All rights reserved.

Keywords: Electromagnetic form factors; Dispersion relations; Meson spectroscopy

1. Introduction

There exists only one measurement, performed by the BaBar Collaboration in 2008 [1], ofthe annihilation e+e− → φπ0. The data have been achieved using the initial state radiation tech-nique, that allows to scan energies, for the invariant mass of the hadronic final state, from theproduction threshold up to about 4.6 GeV.

There have been some previous attempts [2] to observe such a final state in e+e− annihilationbut only upper limits were given.

E-mail address: simone.pacetti@pg.infn.it.

0375-9474/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysa.2013.10.004

16 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

The process e+e− → φπ0 plays a crucial role in the understanding of the quark model. In-deed, the φπ0 channel, which is forbidden by the Okubo–Zweig–Iizuka (OZI) rule [3] for anyqq̄ vector meson, could be a likely decay mode for non-standard bound states as hybrids andtetraquarks [4].

The φπ0 cross section is analyzed in connection with other data coming from different pro-cesses, that involve the same mesons, using a procedure which implements the analyticity in theφπ0 transition form factor (TFF) by means of a dispersion relation procedure.

This study is based on the method reported in Ref. [5]. Such a method, which will be describedin the following, has the advantage of using TFFs instead of cross sections and decay rates.A TFF is defined in different energy regions and it is experimentally accessible through differentprocesses. In particular, the φπ0 TFF can be parametrized in terms of resonant contributions,whose coupling with the mesons under consideration is inferred from other measurements. In thisview, the OZI rule violation can be seen as a direct consequence of the large φ–ρ0π0 coupling(BR(φ → 3π) � 15% [6]).

In addition, since we use dispersion relations (DR) to implement analyticity, we need to knowthe asymptotic behavior of the TFF, for which we adopt the quark-counting rule (QcR) prescrip-tion [7]. As we will see in Section 2.3, this provides a further constraint on the quark structure ofthe involved mesons.

The paper is organized as follows. In Section 2 we introduce the main formulae for the crosssection and decay rates, and a general analytic description for the φπ0 TFF, whose free pa-rameters represent the properties of the vector meson contributions in the low-energy region.In Section 3 we discuss a first case where, for the description of the φπ0 TFF, we consideronly one contribution, the ρ0. Since such an intermediate state is completely known throughthe decay φ → ρ0π0, there are no degrees of freedom, and the description, in this case, ismodel-independent. In Section 4 we develop this procedure including further intermediate states,essentially ρ-recurrences, to achieve a global description of the TFF. Finally, the obtained resultsare summarized and discussed in the last section.

We would like to mention a recent study [8] of the φπ0 TFF, also based on a dispersivecomputation. However, the authors of Ref. [8] used a different procedure, in particular, theyparametrized the TFF in terms of pion-loop intermediate states, while we include directly vectormeson propagators.

2. φπ0 transition form factor

2.1. Cross section and decay rate formulae

To extract the TFF values, data on cross section and decay rates have to be compared withthe expressions for these quantities obtained under the assumption of pointlike mesons. Theconversion current for the vertex of Fig. 1, being φ a vector and π0 a pseudoscalar meson, hasthe general form [9]:

Jμ

φπ0 = eεμνρσ ενpρqσ Fφπ0

(q2), (2.1)

where e is the electron charge, εμνρσ is the fully antisymmetric Levi-Civita tensor, q is the4-momentum of the photon and, p and ε are the 4-momentum and the polarization vector ofthe φ. The tensor structure of this current follows from Lorentz and gauge invariance.

The TFF Fφπ0(q2) is an analytic function, defined in the q2 complex plane with the cut(4M2

π ,∞) along the real axis. Only real values of q2 are experimentally accessible. The TFF

S. Pacetti / Nuclear Physics A 919 (2013) 15–31 17

Fig. 1. Diagram of the conversion φγπ0.

Fig. 2. Real axis of q2 (not in scale). The three processes, involving the mesons φ and π0, in the corresponding q2

regions are also indicated. The lined band represents the cut, where the TFF is complex.

is real for q2 below the threshold 4M2π , while it is complex over the cut, i.e. for q2 > 4M2

π .It describes the photon–hadron vertex in terms of electromagnetic interaction with the quarkconstituents. The underlying physical process is the φ radiative decay: φ → π0γ , which occursat q2 = 0 (real photon). There are two other one-photon exchange processes, which involve thesame conversion, that can be used to investigate the φπ0 TFF in other kinematical regions (seeFig. 2):

• The decay: φ → π0e+e−, with 4m2e < q2 < (Mφ − Mπ0)2;

• The annihilation: e+e− → φπ0, with q2 � (Mφ + Mπ0)2.

Rates and cross section can be computed, as functions of the TFF, using the current of Eq. (2.1).The radiative decay rate is the constant quantity:

Γ = α

3

(M2

φ − M2π0

2Mφ

)3[Fφπ0(0)

]2, (2.2)

it is proportional to the squared value of the TFF at q2 = 0.The differential rate for the conversion decay φ → π0e+e− has the form

dΓ

dq2

(q2) = α2

9π

√1 − 4m2

e

q2

(1 + 2m2

e

q2

)1

q2

[(q2 + M2

φ − M2π0)

2 − 4M2φq2

(2Mφ)2

] 32 ∣∣Fφπ0

(q2)∣∣2

.

(2.3)

This is an energy-dependent quantity which gives information on the real value of the TFF up toq2 = 4M2

π and on its modulus in the region 4M2π < q2 < (Mφ − Mπ0)2.

Finally, the annihilation cross section:

σ(q2) = π

6

α2

(q2)3

q2 + 2m2e√

q2(q2 − 4m2e)

[(q2 + M2

φ − M2π0

)2 − 4M2φq2] 3

2∣∣Fφπ0

(q2)∣∣2

, (2.4)

from which we extract the modulus of the TFF above the physical threshold, i.e. for q2 >

(Mφ + Mπ0)2.

18 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

Fig. 3. (a) The e+e− → φπ0 cross section [1]. (b) The φπ0 TFF extracted from cross section (solid circles) and radiativedecay rate (triangle at

√q2 = 0).

2.2. The data

There are only two available sets of data on the φπ0 TFF, in addition to the BaBar cross sectionmeasurement [1], only the radiative decay rate Γ (φ → π0γ ) is known [6]. Unfortunately, thereare no data on φ → π0e+e− differential decay rate.1 The measurement of such a rate shouldcover a wide energy range below the physical threshold. The unphysical region (see Figs. 2and 3b) for this TFF, that is the energy interval not experimentally accessible, is very narrow:2Mπ0 � 270 MeV and no structure is expected in this energy range. From the radiative decayrate we extract the value at zero of the TFF as:

Fφπ0(0) ≡ ∣∣gφ

π0γ

∣∣ =√

3Γ (φ → π0γ )

α

(2Mφ

M2φ − M2

π0

)3/2

= (0.131 ± 0.004) GeV−1, (2.5)

this value corresponds to the modulus of the coupling gφ

π0γ.

Finally, from the annihilation cross section data (Fig. 3a), using the expression of Eq. (2.4),we obtain the TFF above the physical threshold, which is shown in Fig. 3b. Even though there

are no data on the differential rate dΓ (φ→π0e+e−)

dq2 , in Ref. [6] is reported the branching fraction

BR(φ → π0e+e−)

PDG = (1.12 ± 0.28) × 10−5, (2.6)

this value is the average of the only two existing sets of data collected by SND and CMD-2Collaborations [15,16]. In both cases the cuts applied to extract the events select small valuesof q2 and hence do not allow to study the energy dependence of the TFF as described in Eq. (2.3).The branching BR(φ → π0e+e−)PDG can be interpreted as the yield of the events with a smalle+e− invariant mass [15,16]. The only information that can be drawn is about the mean value of

1 The KLOE/KLOE-2 Collaboration will publish soon data on the differential rate dΓ2 (φ → π0e+e−) [10].

dq

S. Pacetti / Nuclear Physics A 919 (2013) 15–31 19

Fig. 4. Quark triangle diagram for the φγπ0 vertex, F̃φπ0 (q2) is the TFF obtained in terms on the only anomalous quark

triangle.

the modulus of the TFF over a small q2 interval, [(2me)2, (2me +�Ee)

2], from the threshold upto a not claimed maximum energy

√q2 = 2me +�Ee. More in detail this mean value, depending

on �Ee, can be defined as:

∣∣F�Ee

φπ0

∣∣2 = Γ (φ → π0e+e−)PDG

α2

9π

∫ (2me+�Ee)2

(2me)2

√1 − 4m2

e

q2

(1 + 2m2

e

q2

) 1q2

[ (q2+M2φ−M2

π0 )2−4M2φq2

(2Mφ)2

] 32 dq2

, (2.7)

for the phase-space integration, at denominator, we use the definition given in Eq. (2.3). With�Ee = 400 MeV [15] one gets:∣∣F�Ee=400 MeV

φπ0

∣∣ = 0.14 ± 0.02 GeV−1. (2.8)

This value has to be compared with the TFF at q2 = 0 of Eq. (2.5), extracted from the radiativedecay rate Γ (φ → π0γ ). They are perfectly compatible, since the higher value obtained for themean value reflects the rising behavior, towards the ρ0 peak, of the TFF. However, due to thelack of information on their q2-dependence, these data will not be used in the following analysis.

2.3. Parametrization of the transition form factor

It has been shown [11] that the conversion amplitude φγπ0, at quark level, is dominated bythe triangle anomaly, see Fig. 4. Assuming only this contribution, the φπ0 TFF can be computedas superposition of loops with quarks of different flavors and it turns out to have the same formas the one computed assuming the vector meson dominance (VMD), i.e. the sum of vector mesonpropagators (in this simple example only single real poles have been considered)

F̃φπ0(q2)

F̃φπ0(0)=

∑V

GV

1 − q2/M2V

, with:∑V

GV = 1,

where MV and GV stand for the mass and the normalized coupling of the V vector meson. Inother words, the microscopic description of the TFF, in terms of quark loops, is equivalent to thephenomenological one based on VMD. Such a correspondence between the two interpretationsis usually referred to as “quark-hadron duality” [12]. It was initially introduced to establish aconnection between the low-q2 resonant behavior of the hadronic total cross section, in e+e−annihilation, and the QCD scaling. In this view, the concept of duality is related to the twofold,non-perturbative and perturbative, representation of QCD.

Hereafter we will consider, for the φπ0 TFF, a VMD-based description at low q2 and a per-turbative QCD power law at high momentum transferred. In particular, using the prescriptionsgiven in Ref. [5], the TFF is parametrized by means of the threefold expression:

20 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

Fig. 5. Schematic representation of the parametrization for the TFF in the resonance region.

Fφπ0

(q2) =

⎧⎪⎪⎨⎪⎪⎩

F anφπ0(q

2), q2 < 4M2π ,

F Resφπ0(q

2), 4M2π � q2 < sasy,

Fasyφπ0(q

2), q2 � sasy,

(2.9)

where sasy, the momentum squared from which we assume the QcR power law behavior, is afree parameter. The three definitions refer to three q2-intervals which cover the whole time-likeregion.

• The resonance region (superscript “Res”): 4M2π � q2 < sasy, where the TFF is described in

terms of intermediate vector meson resonances (Vj , with j = 1, . . . ,N ). Such resonancesare assumed to couple to the virtual photon (see Fig. 5), so that the TFF is parametrized as asum of propagators weighted by the corresponding coupling constants:

F Resφπ0 =

N∑j=1

M2j

eFVj

gVj

φπ0

M2j − q2 − iΓjMj

, 4M2π � q2 � sasy, (2.10)

where Mj and Γj are the mass and the width of the resonance Vj , FVjand g

Vj

φπ0 are thecouplings to the photon and to the hadronic final state. The number and species of interme-diate resonances to be considered are established following two criteria: the quantum numberconservation and the OZI rule. For this channel, being IG(JPC) = 1+(1−−), we expect onlyρ-like contributions even though they are OZI-suppressed. In particular we introduce threeresonances: below threshold the ρ0(770), whose parameters are completely fixed (see Sec-tion 3), above the physical threshold two additional ρ-recurrences, visible in the data (Fig. 3):a broad structure around 1.6 GeV and a narrower one at ∼ 1.9 GeV [1].Other possible threshold effects, e.g.: 4π , KK , etc., besides the π+π− one, which is ac-counted for by the vector meson propagators of the parametrization given in Eq. (2.10), havebeen not included because they are suppressed by phase space and large masses [8,13].

• The asymptotic region (superscript “asy”): q2 � sasy, where we use the power law behaviorfrom QcR, i.e.:

∣∣F asyφπ0

(q2)∣∣ = ∣∣F Res

φπ0

(sasy)∣∣( sasy

q2

)nh+ nλ+lq−12 =2

, (2.11)

nh = 2 is the number of external hadronic fields, nλ = 1 is the hadronic helicity, and lq = 0is the quark–antiquark angular momentum in the π0 (see Ref. [5]).The applicability of QcR, introduced for space-like transferred momenta, in time-like re-gion as well, follows from the analyticity of TFFs. Indeed, assuming that Fφπ0(q2) isbounded in the whole Im(q2) > 0 half-plane and that it vanishes as q2 → ∞, uniformlyin 0 � Arg(q2) � π , then, from the Phragmén–Lindelöf theorem [14], we have

S. Pacetti / Nuclear Physics A 919 (2013) 15–31 21

lim|q2|→∞

Fφπ0(|q2|eiθ1)

Fφπ0(|q2|eiθ2)= 1, ∀θ1, θ2 ∈ [0,π].

In particular, with θ1 = π and θ2 = 0, the space- and time-like limits, respectively, coincideand so the power law, obtained for q2 → −∞, can be assumed also in the time-like region(q2 → ∞).

• The analytic region (superscript “an”): q2 < 4M2π . In this region the TFF is reconstructed

using the previous two expressions [Eqs. (2.10) and (2.11)] in the dispersion relation inte-gral [5,14]:

F anφπ0

(q2) = exp

[√4M2

π − q2

π

( sasy∫4M2

π

ln |F Resφπ0(s)|ds

(s − q2)√

s − 4M2π

+∞∫

sasy

ln |F asyφπ0(s)|ds

(s − q2)√

s − 4M2π

)]. (2.12)

3. Model-independent description with only ρ0 contribution

The ρ0 contribution, due to the process e+e− → γ ∗ → ρ0 → φπ0, represents an importantOZI-violating intermediate state. The presence of such a kind of contribution is inferred by thelarge φ − ρ0π0 coupling. The rate Γ (φ → ρ0π0) for the single neutral channel ρ0π0 can beobtained from the total rate Γ (φ → ρπ), correctly accounting for the interference among thethree isospin channels, for details see Appendix A. In particular, we find that only the 18.6% ofthe BR(φ → ρπ) = 0.140 ± 0.004 [6] is due to ρ0π0. Using the general expressions:

Γ(V → V ′P

) = |gVV ′P |212π

{[M2V − (MV ′ − MP )2][MV − (MV ′ + MP )2]}3/2

(2MV )3,

Γ(V → e+e−) = αMV

3|FV |2 , (3.1)

for the decay of a vector meson V into the vector meson V ′ and the pseudoscalar P , and theelectromagnetic coupling of a vector V to e+e−, respectively, we extract the couplings in termsof the corresponding rates and we get (V = φ, V ′ = ρ0, and P = π0):∣∣gφ

ρ0π0

∣∣ = (0.865 ± 0.013) GeV−1 and |Fρ0 | = 16.6 ± 0.2. (3.2)

The knowledge of these couplings completely determines the ρ0 contribution in terms of theparametrization of Eq. (2.10). We are now ready to establish whether this contribution is ableto describe the cross section measured by the BaBar Collaboration. Looking at Fig. 6, wherethe cross section and the corresponding TFF as expected in case of only ρ0 contribution (solidcurves) are shown in comparison with the data, we note that:

• Above the physical threshold (Mπ0 + Mφ), vertical dashed line in Figs. 6a and 6b, the ρ0

contribution alone is not enough to describe the data, indeed, even though the measuredφπ0 cross section is very small, less than 0.2 nb, this contribution gives solely few pb (seeFig. 6a); similar conclusions can be drawn for the TFF of Fig. 6b;

22 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

Fig. 6. Data (points) on the e+e− → φπ0 cross section (a) and the corresponding φπ0 TFF (b) compared with theexpectations in case of only ρ0 contribution (solid curves). The vertical dashed lines indicate the (M

π0 +Mφ) threshold.

• Below the theoretical threshold q2 = 4M2π , and in particular at q2 = 0, the prediction for

the TFF, assuming only the ρ0 contribution, is Fφπ0(0) = (0.177 ± 0.003) GeV−1 and it ishigher than the experimental value extracted from the radiative decay rate Γ (φ → π0γ ) andreported in Eq. (2.5).

Both these observations point to the presence of additional resonant contributions that, in lightof the discrepancy between the predicted cross section and data, Fig. 6a (similarly for the TFF inFig. 6b), should lie around 1.6 and 1.9 GeV. Moreover, following the vector meson dominancerelation:

gφ

π0γ≡ g

γ

φπ0 �∑V

gVφπ0

eFV

=g

ρ0

φπ0

eFρ0+

gρ′φπ0

eFρ′+

gρ′′φπ0

eFρ′′, (3.3)

where two additional ρ-recurrences are included, to reconcile the value of the TFF at q2 = 0 withthat of Eq. (2.5), ρ′ and ρ′′ must give a global negative contribution, i.e.:

gρ′φπ0

eFρ′+

gρ′′φπ0

eFρ′′� −0.046. (3.4)

This means that, at least the coupling of the ρ′, which looking at the data of Fig. 6 appears as thedominant contribution besides the ρ0, must be negative.

4. Global description including additional resonances

In this section we extend the analysis of the φπ0 TFF including, in addition to the ρ0, otherresonant contributions as suggested by the arguments previously discussed. However, since theexperimental information on the ρ-recurrences as well as their classification are quite uncertainand unstable, we consider masses, widths, and couplings of the additional contributions as free

S. Pacetti / Nuclear Physics A 919 (2013) 15–31 23

parameters to be fit to the data and to the theoretical constraints. Therefore, contrary to the previ-ous case where, with the only ρ0 contribution, we had a fully frozen parametrization, this globaldescription is now model-dependent.

4.1. ρ(1450), ρ(1700) and C(1480)

Since 1988 the Particle Data Group [6] decided to replace the ρ(1600), that were the onlyexcited ρ below 2 GeV, with two states: ρ(1450) and ρ(1700). This is a consequence of theclear theoretical and experimental evidence that the 1.6 GeV energy region contained more thanone ρ-recurrence. Furthermore, also the so-called C(1480) observed in the charge exchangereaction π−p → φπ0n as a clear peak in the φπ0 invariant mass distribution [17], has to beincluded in the list of all possible contributions. The observation of this structure generated agreat deal of theoretical and experimental works [18]. In particular, there has been a theoreticaleffort to find the best observable to identify the nature of this resonance: qq̄-ρ-like meson orexotic with a hidden strangeness content. The electronic width Γ (C(1480) → e+e−) and thebranching BR(C(1480) → φπ0) seemed good candidates for this purpose. The leptonic widthfor a tetraquark state should be suppressed with respect to that for a qq̄ by a factor � 1, repre-senting the price to be paid in creating an extra quark pair from the vacuum. Instead, the couplingof an exotic C(1480) to the φπ0 final state should be favored compared with that of a standardmeson, which is OZI-suppressed. Unfortunately the cross section σ(e+e− → C(1480) → φπ0)

gives information only on the product of the two couplings, that, without data on other indepen-dent channels, cannot be disentangled. It follows that the cross section alone does not allow todistinguish between the two possible resonant contributions: the exotic C(1480) and the stan-dard ρ(1450), if both exist. Moreover, there are authors claiming that the C(1480) is actually theρ(1450) [19].

The overpopulation of this energy region makes the description of TFFs in terms of resonancesdifficult. The lack of information on partial widths for these ρ-recurrences and the impossibilityof distinguishing their contributions in certain decay channels puzzle the extraction of the cou-plings, which is essential for this description. In the following we will refer to the first excited ρ

of the φπ0 TFF simply as the ρ′, avoiding any a priori identification with ρ(1450), C(1480), orρ(1700) and gaining precious information from the one or the other according to the case.

For instance, the first important hint on the coupling with the actual final state φπ0 can be

obtained for the ρ(1450). Even though there is no direct information on the gρ′φπ0 , as in the ρ0

case (Section 3), the known upper limit [20]:

Γ(ρ′ → e+e−)

BR(ρ′ → φπ

) ≡ Γ (ρ ′ → φπ)Γ (ρ′ → e+e−)

Γρ′< 70 eV (4.1)

helps in understanding whether ρ(1450), here called ρ′, can contribute to the annihilatione+e− → φπ0.

Multiplying the two expressions of Eq. (3.1), with V = ρ′, V ′ = φ, and P = π0 we obtain

Γ (ρ′ → φπ)Γ (ρ′ → e+e−)

Γρ′

=|gρ′

φπ0 |212πΓρ′

{[M2ρ′ − (Mφ − Mπ0)2][M2

ρ′ − (Mφ + Mπ0)2]} 32

(2Mρ′)3

αMρ′

3|Fρ′ |2 , (4.2)

which, using the values for Mρ′ and Γρ′ given in Ref. [6], provides the upper limit:

24 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

∣∣∣∣M2ρ′g

ρ′φπ0

Fρ′

∣∣∣∣ < 0.19 GeV. (4.3)

This value leaves room to a possible important contribution from the ρ′, in fact it is quite highwith respect to the corresponding coupling for the ρ0: |M2

ρgρ

φπ0/Fρ | � 0.03 GeV. In particular,the peak values are [see Eq. (2.10)]:

∣∣∣∣Mρgρ

φπ0

eFρΓρ

∣∣∣∣ � 0.9 GeV−1 and

∣∣∣∣Mρ′gρ′φπ0

eFρ′Γρ′

∣∣∣∣ < 1.1 GeV−1. (4.4)

4.2. Once more the ρ(1900)

The cross section data, as well as the data for the TFF reported in Fig. 3, show an accumulationof events around 1.9 GeV. Following the line of thought of Ref. [1], we identify this structureas the second excited ρ, called ρ′′ or ρ(1900). In the same reference it is found a statisticalsignificance, for such a resonance, of 2 × 10−3, which means there is a 0.2% chance that thepeak at 1.9 GeV was an accidental fluctuation of the data.

This ρ′′ appears as an already known structure, previously observed in six [21] and four-pion [22] final states. As suggested in Ref. [23] it could be the manifestation of a cryptoexoticJPC = 1−− tetraquark state, whose coupling with a multi-particle final state should be favoredby a milder suppression factor due to the creation of a lower number of quark pairs from thevacuum.

If the bump in the BaBar φπ0 cross section is the already known ρ(1900), this should be thefirst observation of ρ(1900) decaying in this channel and also its first manifestation as a peakinstead of a dip.

4.3. χ2 definition

We are now ready to define the χ2. We work directly with the TFF instead of the cross section,this allows us to take advantage from important regularities, as analyticity, and constraints fromdifferent sets of data. The χ2 is defined as the sum of two contributions. The first one is fromthe data on both, the annihilation cross section σ(e+e− → φπ0) and the radiative decay rateΓ (φ → π0γ ), and the second comes from the theory, i.e. the analyticity requirement, which isimposed by means of the DRs.

In more detail we have:

χ2 = χ2exp + τ · χ2

th, (4.5)

where τ is a weighting factor for the theoretical part, that will be described in the following. Theexperimental contribution is:

χ2exp =

N∑j=1

( |Fφπ0(sexpj )| − F

expj

δFexpj

)2

+( |Fφπ0(0)| − F

exp0

δFexp0

)2

. (4.6)

The first term concerns the annihilation data {sexpj ,F

expj }, with: j = 1,2, . . . ,N and s

expj �

(Mφ + Mπ0)2, while the second is for the radiative decay rate. The theoretical component χ2th

forces the analyticity of the TFF through the DR for the logarithm [5,24]

S. Pacetti / Nuclear Physics A 919 (2013) 15–31 25

lnF(q2) =

√4M2

π − q2

π

∞∫4M2

π

ln |F(s)|ds

(s − q2)√

s − 4M2π

, (4.7)

that gives the real value of the TFF below the theoretical threshold (q2 < 4M2π ) in terms of an in-

tegral of the logarithm of its modulus over the cut (4M2π ,∞). The relation of Eq. (4.7) guarantees

the continuity of the parametrization for the TFF (zero-th derivative) across the threshold 4M2π .

Instead, the continuity of the first derivative has to be imposed to connect the last two definitionsof Eq. (2.9), in the resonance and asymptotic regions, to the first one in the analytic region. Informulae we get:

χ2th =

(dFφπ0

ds

∣∣∣∣s=sth

−∞∫

sth

K(q2 → s+

th , s)

ln∣∣Fφπ0(s)

∣∣ds

)2

, (4.8)

where sth = 4M2π and K(q2, s) is the derivative of the kernel of Eq. (4.7):

K(q2, s

) = 2sth − s − q2

2π√

sth − q2√

s − sth(s − q2)2. (4.9)

Since this condition has to be exactly verified, the value of the parameter τ , which weights theχ2

th in Eq. (4.5), must be chosen large enough to force the vanishing of the χ2th itself.

4.4. Fit results

Before giving the result of the fit, we summarize the key points of the parametrization usedfor the φπ0 TFF. We adopted the threefold definition of Eq. (2.9) given in Ref. [5]. The time-like region, where we parametrize the TFF, has been divided in three intervals. The asymptoticregion, from a certain energy sasy (free parameter in the fit) up to infinity, where we used theQcR prescription which gives the power-law behavior: |Fφπ0(q2)| ∝ (q2)−2 as q2 → ∞. Theresonance region (4M2

π , sasy), where the TFF is described in terms of vector meson propagators.More in detail in that region we consider the three resonances ρ0, ρ′, and ρ′′, and the TFF is:

F Resφπ0(s) = Mρ

eFρ

gφ

ρπ0

Γρ

MρΓρ

M2ρ − s − iΓρMρ

+ Mρ′

eFρ′

gφ

ρ′π0

Γρ′Mρ′Γρ′

Mρ′ 2 − s − iΓρ′Mρ′eiδ

+ Mρ′′

eFρ′′

gφ

ρ′′π0

Γρ′′Mρ′′Γρ′′

Mρ′′ 2 − s − iΓρ′′Mρ′′eiδ′ + An.r., (4.10)

where we include also additional relative phases δ′ and δ′′ to account for possible rescatteringeffects, and a non-resonant constant amplitude An.r.. Finally, the analytic region q2 < 4M2

π ,where the real TFF is obtained using the previous two definitions in the integral of the DR forthe logarithm of Eq. (4.7).

We will consider two fits: with real couplings, i.e.: the phases are set to zero and the gVφπ0 ’s are

allowed to take negative and positive values (gray bands in Fig. 7), and with complex couplings,i.e. with free relative phases (hatched bands in Fig. 7).

The fit is shown in Fig. 7 and the corresponding best values for masses, widths, phases, andtotal couplings are reported in Table 1. The error bands have been obtained with the followingprocedure: we generate many samples of data by Gaussian fluctuation of the original ones, we fit

26 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

Fig. 7. The fit of the φπ0 TFF (a) and of the corresponding cross section (b). The error bands have been obtained witha Monte Carlo technique (see the text). The solid gray band corresponds to the case where we use real couplings, whilethe hatched one is for complex couplings.

Table 1Best fit parameters. For each parameter we give two values corresponding to the cases where we use real couplings,upper rows, and complex couplings (additional relative phases), lower rows. The value of the constant amplitude is alsogiven in terms of the TFF at the ρ0 peak, and, in both cases it represents about the 2.5% of this peak value.

Cont.∣∣M2

VFV

gφ

V π0

∣∣ (MeV) Γ Ve+e− BRV

φπ0 (eV) M (MeV) Γ (MeV) Phase (rad)

ρ′ (−)15 ± 6 2.8 ± 0.9 1640 ± 46 143 ± 85 π(0)

16 ± 7 3.1 ± 1.4 1617 ± 58 150 ± 99 3.0 ± 0.5

ρ′′ (−)6.8 ± 2.9 2.3 ± 0.8 1895 ± 22 62 ± 46 π(0)

7.3 ± 3.5 2.6 ± 1.0 1895 ± 29 67 ± 55 3.5 ± 0.8

An.r. −0.024 ± 0.006 GeV−1 � 0.025 · |Fφπ0 (M2

ρ)|−0.023 ± 0.006 GeV−1

all these samples obtaining sets of parameters and curves for the TFF, we treat these parametersand curves with the usual statistical procedures. We take the mean and the standard deviation asthe best value and the corresponding error for the parameters and, as the central value and thehalf-width of the band at each

√q2 for the TFF and cross section curves (see Fig. 7).

The parameters achieved with this technique are in agreement with those obtained in Ref. [1]with a different fit procedure, that does not exploit information coming from the radiative decayrate and the QcR for the asymptotic behavior. This is the reason why, in case of real couplings,we get the same sign for ρ′ and ρ′′ while they are opposite in Ref. [1], where there is a lowersensitivity to the complex structure of the fit function.

The additional flat non-resonant contribution turns out to be only a small fraction (about 2.5%of the ρ0 peak value) of the TFF in the resonance region.

S. Pacetti / Nuclear Physics A 919 (2013) 15–31 27

4.5. gρ′φπ0 coupling

Since ρ0, ρ′, and ρ′′ are intermediate resonances produced through an e+e− annihilationprocess, the cross section can give information only on the product between the electromagneticand the mesonic couplings, e.g.: the modulus of (M2

V /FV )gVφπ0 for a generic vector meson V (see

Table 1 for ρ′ and ρ′′). To know the pure mesonic coupling gVφπ0 , which describes the transition

φ → π0 with emission of a virtual vector meson V , we need to know the coupling FV , whichinstead gives the amplitude of the decay V → e+e−.

The situation appears quite interesting in the case of the ρ′ resonance. This structure is mainlyobserved decaying in four-pion final states, hence, assuming such final states as the dominantdecay channels we can give an estimate, or at least an upper limit, for Fρ′ . Using the usualBreit–Wigner formula for a generic process: e+e− → V → [final state], which proceeds throughan intermediate vector resonance V , we have at the vector meson peak:

σ(V peak) = 12πΓ (V → e+e−) · BR(V → [final state])

M2V ΓV

, (4.11)

where MV and ΓV are the mass and total width of the resonance. Being the branching BR(V →[final state]) lower or equal to one (it is � 1 when the final state is the dominant decay channelof the vector meson V ), from Eq. (4.11) we get the upper limit

Γ(V → e+e−)

�σ(V peak)M2

V ΓV

12π. (4.12)

It remains to relate the decay rate Γ (V → e+e−) to the coupling FV , by means of Eq. (3.1) weobtain:

|FV | =√

αMV

3Γ (V → e+e−)�

√4πα

σ(V peak)MV ΓV

= e√σ(V peak)MV ΓV

. (4.13)

If we identify the ρ′ with the ρ(1700) of Ref. [6], we may take advantage from the fact that itdecay predominantly in four pions, hence we put BR(ρ′ → 4π) = 1. There are two channels:ρ′ → 2(π+π−) and ρ′ → 2π0π+π− and the peak values of the cross sections are [22]:

σ2(π+π−)(ρ′ peak) = 29.2 ± 0.7 nb

σπ+π−2π0(ρ′ peak) = 18.2 ± 0.7 nb

}�⇒ σ4π

(ρ′ peak

) = 47.4 ± 1.0 nb. (4.14)

From the formula of Eq. (4.13) and using the value for |M2ρ′g

ρ′φπ0/Fρ′ | reported in Table 1, we

get:

|Fρ′ | �{

65 ± 20

64 ± 18and

∣∣gφ

ρ′π0

∣∣� {0.32 ± 0.07 GeV−1,

0.36 ± 0.07 GeV−1,(4.15)

where, as in the following, the upper and lower values refer to the cases with real and complexcouplings respectively. The electromagnetic branching ratios are:

BR(ρ′ → e+e−)

�σ4π (ρ′ peak)M2

ρ′

12π=

{(8.7 ± 0.5) × 10−6,

(8.5 ± 0.6) × 10−6.(4.16)

Finally, by using the value for Γ (ρ′ → e+e−)BR(ρ′ → φπ0), we can extract upper limits forthe rates of the decay ρ′ → φπ0 as

28 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

Γ(ρ′ → φπ0) � {330 ± 115 keV,

360 ± 150 keV.(4.17)

These values are lower than the OZI-violating width of the φ meson in its decay in ρπ + 3π ,which is:

Γ (φ → ρπ + 3π) = (0.153 ± 0.004) · (4260 ± 50) keV

= 650 ± 20 keV. (4.18)

5. Conclusions

The φπ0 TFF has been studied by means of a dispersive procedure which collects all theexperimental information under the aegis of the analyticity. The main low-energy contributionto this TFF is provided by the ρ0 meson, whose coupling to φπ0 is well known. Consider-ing only this contribution we get a model-independent description without free parameters.However, as shown in Fig. 6, this single-resonance TFF is unable to describe neither the datafrom the cross section σ(e+e− → φπ0), nor the point at q2 = 0 from the radiative decay rateΓ (φ → π0γ ).

Therefore we considered, in addition to the ρ0, two further resonances defined as ρ-recur-rences, that, having free parameters, make the description model-dependent. The first excitedρ-recurrence of our analysis, the ρ′, has mass and width compatible (�M , �Γ < two standarddeviations) with those of the ρ(1700) of Ref. [6]:

Mρ(1700) = 1720 ± 20 MeV, Γρ(1700) = 250 ± 100 MeV.

For the ρ′, assuming the four-pion final state as the dominant decay channel, we have estimated

the electromagnetic coupling Fρ′ and the gρ′φπ0 [Eq. (4.15)]. The obtained values are compatible

with the expectation for an excited ρ-like state, given those of the ground state ρ0 reported inEq. (3.2).

The second excited state, the ρ(1900), is perfectly compatible with the “dip” observed forthe first time by the DM2 Collaboration in the six-charged-pion final state and subsequentlyconfirmed by other experiments in the same final state [21] and also in the four-pion channel [22].In all these cases it appeared as a dip, this could be the first manifestation as a peak. There are noclear interpretations for this structure, but the most interesting, that could explain the couplingwith the OZI-forbidden φπ0 final state and the multi-particle decay channels, is a vector mesoncryptoexotic state [25], i.e.: hybrid [qq̄g] or tetraquark [qqqq]. However, we only mentionedthese possible interpretations without any quantitative treatment which, actually, goes beyondthe aim of this work.

We may conclude that, with the present statistics, there is no evidence of the C(1480), i.e. theφπ0 TFF is well described by the three main contributions: ρ0, ρ(1700), and ρ(1900). Moreover,the absence of further structures is also theoretically enforced by the dispersive procedure usedto describe the data.

On the experimental side, the KLOE/KLOE-2 Collaboration will publish soon [10] the firstmeasurement of the q2-dependent differential decay rate dΓ

dq2 (φ → π0e+e−). These data, cover-

ing the still empty region: 4m2e � q2 � (Mπ − Mπ0)

2, will represent an interesting test bed forthe results, or better predictions, of this analysis in that region.

S. Pacetti / Nuclear Physics A 919 (2013) 15–31 29

Acknowledgements

I would like to express my gratitude to Rinaldo Baldini, Lia Pancheri, Yogi Srivastava andAdriano Zallo for the precious and instructive discussions which helped me in developing thiswork.

Appendix A. The rate Γ (φ → π0ρ0)

The φ meson decays into π+π−π0 with a probability of ∼ 15.5%, however this decay isdominated by the ρπ intermediate states. The three channels: ρ0π0, ρ+π− and ρ−π+ participatewith equal amplitudes.

This process has been studied by the KLOE Collaboration [26] and we will use their fit pro-cedure in order to reconstruct the Dalitz plot and then to extract the rate for the single channelφ → ρ0π0.

The observed decay is φ(pφ) → π0(p0)π+(p+)π−(p−), where the 4-momenta in parenthe-

ses are in the center of mass of the φ. The Dalitz plot density distribution is:

D(x,y) ∝ | �p+ × �p−|2 · |Aρπ + Adir + Aωπ |2, (A.1)

where x and y variables are:

x = E+ − E−, y = Mφ − Mπ0 − E+ − E−, (A.2)

and the three amplitudes Aα (α = ρπ, “dir”, ωπ ) are:

Aφπ =0,+,−∑

k

ak

M2k

q2k − M2

k + iqkγk(q2k )

, γk

(q2k

) = Γk

(q2k − s2

0k

M2k − s2

0k

) 32 M2

k

q2k

,

Aωπ = aωeiφωM2

ω

q20 − M2

ω + iq0Γω

,

Adir = adireiφdir . (A.3)

The quantity q2k (k = 0,+,−) is the invariant mass of the pion pair with electric charge k:

q20 = (p+ + p−)2 = (pφ − p0)

2 = (Mφ − Mπ0)2 − 2Mφy,

q2+ = (p0 + p+)2 = (pφ − p−)2 = M2π + Mφ(Mπ0 + x + y),

q2− = (p0 + p−)2 = (pφ − p+)2 = M2π + Mφ(Mπ0 − x + y), (A.4)

then Mk and Γk are masses and widths of the meson ρk , and the values s0k represent the corre-sponding thresholds:

s00 = 4M2π , s0+ = s0− = (Mπ + Mπ0)

2. (A.5)

We consider now the fractions:

Rαβ =

∫dx dy|Aα(x, y)|2∫dx dy|Aβ(x, y)|2 , (A.6)

where α and β are either single channels or sums of channels. By taking the parameters directlyfrom the “Fit(c)” of Ref. [26], we get:

30 S. Pacetti / Nuclear Physics A 919 (2013) 15–31

Rρ0π0

ρπ =∫

dx dy|Aρ0π0(x, y)|2∫dx dy|Aρπ(x, y)|2 = 0.187. (A.7)

This number is quite far from the natural expectation: Rρ0π0

ρπ = 1/3, this is due to large interfer-ence effects among the three ρ’s in the Dalitz plot density [see Eq. (A.1)].

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