Electron-positron annihilation into four charged pions and the a_1-rho-pi Lagrangian

Abstract

The excitation curve of e+e- annihilation into four charged pions in the rho(770) region is calculated using three existing models with rho mesons and pions in intermediate states supplemented by Feynman diagrams with the a_1(1260)pi intermediate states. A two-term phenomenological Lagrangian of the a_1-rho-pi interaction is used. The mixing angle is determined by fitting the e+e- --> pi+pi-pi+pi- cross section data of the Novosibirsk CMD-2 collaboration and also its combination with the low-energy part of the BaBar collaboration data. It is shown that the inclusion of the a_1-pi intermediate states succeeds in obtaining a good agreement with the data on both cross section and the rho^0 --> pi+pi-pi+pi- decay width. When moving to energies above 1 GeV, the rho(1450) and rho(1700) resonances are taken into account to get excellent agreement with the BaBar data over the full energy range up to 4.5 GeV.

Full text

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arXiv:hep-ph/0601234v2 5 Oct 2007
Electron–positron annihilation into four charged pions and the a1ρπ Lagrangian∗
Peter Lichard1,2and Josef Jur´ aˇ n1
1Institute of Physics, Silesian University in Opava, Bezruˇ covo n´ am. 13, 746 01 Opava, Czech Republic
2Institute of Experimental and Applied Physics, Czech Technical University, Horsk´ a 3/a, 120 00 Prague, Czech Republic
(Dated: February 2, 2008)
The excitation curve of e+e−annihilation into four charged pions in the ρ(770) region is calculated
using three existing models with ρ mesons and pions in intermediate states supplemented by Feyn-
man diagrams with the a1(1260)π intermediate states. A two-term phenomenological Lagrangian
of the a1ρπ interaction is used. The mixing angle is determined by fitting the e+e−→ π+π−π+π−
cross section data of the Novosibirsk CMD-2 collaboration and also its combination with the low-
energy part of the BaBar collaboration data. It is shown that the inclusion of the a1π interme-
diate states succeeds in obtaining a good agreement with the data on both cross section and the
ρ0→ π+π−π+π−decay width. When moving to energies above 1 GeV, the ρ(1450) and ρ(1700)
resonances are taken into account to get excellent agreement with the BaBar data over the full
energy range up to 4.5 GeV.
PACS numbers: 13.30.Eg, 13.66.Bc, 12.39.Fe, 13.25.Jx
I. INTRODUCTION
Task of describing the excitation curve of the e+e−→ π+π−π+π−reaction at low energies is closely related
to the investigation of the energy dependent decay width of the four pion decay of ρ(770). The validity of the
factorization of the cross section into the ρ(770) production and decay parts is generally assumed on the basis
of the vector meson dominance (VMD) hypothesis [1].
to determine a particular partial decay width at the nominal ρ(770) mass on the basis of measurement the e+e−
annihilation cross section into the corresponding final state at the corresponding energy. The current experimental
value Γ(ρ0→ π+π−π+π−) = (2.8 ± 1.4 ± 0.5) keV was obtained in that way [2]. In this work, we will also assume
a one-to-one correspondence between the cross section and the energy dependent decay width at the same energy,
ignoring the complications which may appear if some conditions are not met [3]. Needless to say that the evaluation
of the decay width is less demanding technically and computationally than that of the cross section (five-dimensional
quadrature instead of eight-dimensional one in the case of four-body final states).
The cross section of the e+e−annihilation into the 2π+2π−and π+π−2π0final states was considered by Decker,
Heiliger, Jonsson, and Finkemeier [4] in conjunction with the CVC-related decays of the τ lepton. The intermediate
states of their model contained ρ(770), a1(1260), and a scalar-isoscalar two-pion resonance. In the two-charged-two-
neutral case also the ω(782) was included. The a1ρπ vertex factors were adopted from the study of the three-pion
decay of the τ lepton by Isgur, Morningstar, and Reader [5]. Czy˙ z and K¨ uhn [6] constructed a Monte Carlo generator
of the reaction e+e−→ γ + 4π. The hadronic matrix elements were used in the form suggested in [4], corrected
only for some minor deficiencies. To check the soundness of their approach, Czy˙ z and K¨ uhn calculated also the
excitation curves of the nonradiative reactions e+e−→ π+π−π+π−and e+e−→ π+π−2π0in qualitative agreement
with available data. Ecker and Unterdorfer [7, 8] performed the first calculation of the processes e+e−→ 4π and
τ → ντ4π with the correct structure to O(p4) in the low energy expansion of the Standard Model extrapolated to
the resonance region. To get a good description of the e+e−→ π+π−π+π−cross section up to 1 GeV, they had to
include as additional contribution the a1exchange. They circumvented the a1ρπ Lagrangian ambiguity by choosing
a special relation among the individual coupling constants.
The four-pion decays of the ρ(770) are generally considered a convenient test ground of the low-energy effective the-
ories of the interactions of ρ mesons and pions. In the past, several papers appeared that calculated the corresponding
partial decay widths [9–13]. Moreover, Achasov and Kozhevnikov [12, 13] argued that the four-pion decay widths
of ρ(770) are not experimentally well defined because they require the averaging over a mass interval in which they
rise rapidly. They therefore calculated, in addition to the decay widths Γ(ρ0→ π+π−π+π−), Γ(ρ0→ π+π−π0π0),
The assumption of factorization allows experimentalists
∗This paper is dedicated to the late Julia Thompson, who drew the attention of one of us (P. L.) to the experimental program of the
Budker Institute of Nuclear Physics at Novosibirsk.

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Γ(ρ+→ π+3π0), and Γ(ρ+→ 2π+π−π0), the e+e−→ π+π−π+π−reaction cross section as a function of incident
energy (excitation curve) and compared it to the CMD-2 data [2] from Novosibirsk.
With respect to the role of the axial-vector meson a1(1260) in the four-pion decays of ρ0, the situation is somewhat
controversial. On one side, the intermediate states containing the a1(1260) were either ignored [9–13] or shown [11]
to have little influence on the four-pion decay widths of ρ(770)1. On the other side, the analysis of the differential
distributions of charged pions coming from the e+e−annihilation in the energy range 1.05–1.38 GeV demonstrated
the dominance of the a1(1260)π intermediate states [14]. Given the large width of a1(1260) (Γa1=250 to 600 MeV
[15]) it would be surprising if the role of the a1-meson diminished so fast outside the above range, which is not too
far from the ρ(770) mass. Moreover, the a1(1260) meson was shown to be important in the four-pion decays of the
τ-lepton [4, 16, 17], which are in a sense isospin counterparts of the four-pion final states in the e+e−annihilation.
Recently, the paper of Achasov and Kozhevnikov has appeared [18] that included the intermediate states with the a1
meson using the generalized hidden local symmetry model [19]. This increased the ρ0→ 2π+2π−decay width from
0.94 keV to 1.59 keV assuming the nominal mass of the a1resonance, see Table I in [18]. Unfortunately, the authors
of [18] do not provide the comparison with the e+e−→ 2π+2π−cross section, which has a greater discriminatory
value than the decay width alone [12, 13].
The present work was triggered by our interest in the electromagnetic probes in relativistic heavy-ion collisions.
The production of prompt dileptons and photons is for a long time considered a powerful tool for investigating
the properties of dense systems created in the hadronic and nuclear collisions [20, 21]. The interest of the heavy
ion community in dilepton production has recently been boosted by very precise dimuon data by the CERN/NA60
collaboration [22]. The prompt dileptons and photons can, in principle, originate from two sources: (i) quark-gluon
plasma, (ii) hadron gas. The theoretical calculations of the dilepton and photon yield from the latter are hampered
by not uniquely known Lagrangian of the a1ρπ interaction [23, 24]. We suggest one way how to relieve this problem
and make the predictions of the dilepton and photon production from hadron gas more reliable. The results of the
present work have already been utilized [25] in the evaluation of the dimuon production rate in In-In collisions at 158
A GeV.
A way of narrowing the uncertainty interval in dilepton production from the hadron gas was advocated a long
time ago by Li and Gale [26]. They checked the feasibility of the dilepton rate calculation in, e.g., ωπ0collisions, by
comparing the inverse process (e+e−annihilation into the ωπ0final state) with the available experimental data. Li
and Gale also considered the e+e−annihilation into four pions in the narrow a1width approximation, i.e. as process
e+e−→ a1π. In the present work we handle it as a genuine four-final-pion process, considering not only the a1π
intermediate states but also other intermediate states following from various models based on the chiral perturbation
theory.
In this paper we investigate the role of the a1(1260) resonance in the e+e−annihilation into four charged pions and
in the four-charged-pion decays of ρ(770) in more detail. We supplement three existing models, which consider only
ρ and π in intermediate states (diagrams (a) and (b) in Fig. 1), with the a1 contribution (diagrams (d) in Fig. 1).
Those three models are (1) the model of Eidelman, Silagadze, and Kuraev [10], (2) one of the models considered by
Plant and Birse [11], and (3) the model of Achasov and Kozhevnikov [12, 13]. We will consider only the final state
with all charged pions, for which the experimental data are best. The difference between the approach of Achasov
and Kozhevnikov [18] and ours lies mainly in the a1ρπ Lagrangian. In [18], the generalized hidden local symmetry
Lagrangian was chosen, whereas we choose a more phenomenological two-term Lagrangian, which is often used in a
different branch of the particle physics. Its individual terms and their specific combinations appeared in many papers
computing the dilepton and photon production rate from a thermalized meson gas. In this paper we consider the
mixing angle between the two terms as a free parameter and will determine its value by fitting the excitation curve
of the e+e−→ π+π−π+π−reaction.
In order to evaluate the amplitude induced by eight diagrams Fig. 1(d) we have to choose a Lagrangian of the a1ρπ
interaction. For this choice, there are basically two approaches in the literature. One explores well defined theoretical
concepts to build dynamical models, the free parameters of which are then fixed by comparison with observed masses
and decay widths. See, e.g., Refs. [23, 27–34].
In other, more phenomenological, approaches the authors simply chose for the a1ρπ Lagrangian various expressions
built from the field operators and compatible with the fundamental conservation laws. Such Lagrangians, after
fixing their coupling constants, were then used to calculate various observable quantities, see, e.g., [35–38]. In some
approaches [5, 39], directly the vertex factors were written without showing the corresponding Lagrangian. From the
fact that different authors pick different Lagrangians one might get the impression that the choice of Lagrangian is
1Ecker and Unterdorfer [7, 8] also included the a1 contribution when calculated the ρ0→ 4π branching fractions, but it is impossible for
us to assess its role because they did not show results without it.

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FIG. 1: Selected Feynman diagrams describing decay ρ0→ π+π−π+π−
not very important and that various Lagrangians lead to identical, or at least similar, results. This is not true, and
the observable quantities may be very sensitive to the choice of the a1ρπ Lagrangian, as was demonstrated, e.g., by
Song [23]. The discussion of the a1phenomenology with emphasis on the photon and dilepton production from a hot
meson gas can be found in Ref. [24].
With so many a1ρπ effective Lagrangians, it is interesting to learn which Lagrangian is preferred experimentally.
This would require constructing a general Lagrangian with a set of free parameters and fixing them by comparing all
possible observables with the existing data. Of course, such a program is very ambitious. In this paper we are going
to do something much simpler. Below, we choose a two-component Lagrangian and determine its two free parameters
by requiring that the decay width of a1(1260) be reproduced and the best possible fit obtained for the excitation
curve of the e+e−→ π+π−π+π−reaction. Even this restricted program cannot be accomplished completely. Firstly,
the width of a1(1260) is not known reliably. We will consider three values from the range given in the Particle Data
Group tables [15], namely 250, 400, and 600 MeV. Secondly, the result of the fit will depend also on the basic ρ and π
intermediate state model to which we add the a1π contribution. Nevertheless, we will show that the inclusion of the
a1π intermediate states is necessary for obtaining good agreement with the e+e−→ π+π−π+π−excitation curve.
Following the suggestion of T. Barnes [40] we also include the ratio of the D-wave and S-wave amplitudes of the
a1→ ρπ decay as a fitted quantity. The importance of the D/S ratio for selecting among the a1ρπ Lagrangians was
stressed in a different context in [24]. We use the experimental value D/S = −0.14±0.11 [41], which was obtained by
genuine partial wave analysis of the reaction π−p → π+π−π−p. We consider the other values which exist in literature
strongly model biased. They were obtained by fitting the three-pion mass spectrum in the decay τ → ντ3π using the
model of Isgur, Morningstar, and Reader [5] and then calculating the D/S ratio from the optimal parameters of the
model.
II.ORIGINAL MODELS, MODIFICATIONS, AND ADDITIONS
As we already stated, we will complement three existing models of the four-pion decays, which consider only
intermediate states with ρ mesons and pions, with the intermediate states containing the axial-vector resonance
a1(1260). We must admit that our choice of models is rather arbitrary. Moreover, we took the models as they
appeared in the literature and did not check their compatibility with chiral symmetry and with the constraints on
coupling of resonances to the pions [42, 43].
Those three models are characterized below.

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A. Model of Eidelman, Silagadze, and Kuraev (ESK)
This model [10] is based on the effective chiral Lagrangian by Brihaye, Pak, and Rossi [44], which was investigated
also in [45]. It follows from that Lagrangian that all (a) and (b) diagrams depicted on Fig. 1 contribute to the
ρ0→ π+π−π+π−decay rate. Their amplitudes (in the notation slightly different from ours) are shown in the paper.
Our usage of this model will differ from the original paper in three respects: (1) We add a1diagrams Fig. 1(d). (2)
We use a different value of the parameter αk, defined in [10]. Instead of 0.55 we set αk = 0.5, which follows from
the KSRF relation [46], to be in conformity with other two models. (3) We replace the scalar part of the ρ-meson
propagator with fixed mass and fixed width by the prescription
Pρ(s) = −
i
s − M2
ρ(s) + imρΓρ(s), (2.1)
which uses the running mass squared M2
The last point deserves more comments. The denominator of our propagator (2.1) is an analytic function in the
s-plane with a cut running from 4m2
πto infinity, as required by general principles. This property differs (2.1) from
most of the formulas used in the literature. The real function M2
dispersion relation, which guarantees that the condition M2
ρ(s) and the energy dependent total width Γρ(s) from Ref. [47].
ρ(s) is calculated from Γρ(s) using a once subtracted
ρ(m2
ρ) = m2
ρis satisfied. Further condition
dM2
ρ(s)
ds
?????
s=m2
ρ
= 0(2.2)
is not fulfilled automatically and serves as a test that all important contributions to the total ρ-meson width Γρ(s)
have properly been taken into account. See [47] for details. If we replace the mρaccompanying Γρ(s) in Eq. (2.1) by
√s, as it is done in some existing formulas, the condition (2.2) cannot be satisfied for any reasonable choice of Γρ(s).
The running mass approach [47] takes into account, in addition to the basic two-pion decay channel, several channels
(ωπ0, K+K−, K0¯K0, and ηπ+π−) which open as the ρ resonance goes above its nominal mass. It also considers
structure effects described by the strong form factors. In these two respects it differs from other approaches that
appeared in the literature [48–50]. Gounaris and Sakurai [48] considered only the two-pion contribution to the total
width of the ρ0resonance and ignored structure effects. Vaughn and Wali [49] took into account the strong form
factor, but again ignored higher decay channels. Melikhov, Nachtmann, Nikonov, and Paulus [50] included the K+K−
and K0¯K0channels, but did not consider the strong form factors.
We will use the ρ propagator (2.1) not only in conjunction with the ESK model, but also with the other ones. This
is the main reason why our results calculated within the original models (i.e., without the a1π intermediate states)
and presented below differ slightly from the results quoted in the original papers.
B. One of the models of Plant and Birse (PB/HG)
Plant and Birse [11] investigated several models of the four-pion decays of ρ0(770). One of them (labeled HG) is a
corrected version of the model by Bramon, Grau, and Pancheri [9], which was based on the hidden gauge theory of
Bando et al. [51]. The 2π2ρ contact terms, see diagram (b1) in Fig. 1, are missing in this approach. The amplitude
of the (a1) diagram is different from that in work by Eidelman, Silagadze, and Kuraev [10] by a factor (−1/2). The
amplitudes of (a2) and (b2) diagrams are equal to their ESK counterparts.
C.Model of Achasov and Kozhevnikov (AK)
Achasov and Kozhevnikov [12, 13] studied the four-pion decays of ρ(770), five-pion decays of ω(782), and the
processes related to them. Namely, the e+e−annihilation into the four- and five-pion final states and the four-pion
decays of the τ lepton. They used the Weinberg Lagrangian [52] obtained upon the nonlinear realization of chiral
symmetry. From their rather extensive work we adopt their prescriptions for the amplitudes (a1), (a2), and (b2)
of the ρ0→ π+π−π+π−decay. The contact amplitude (b1) is again vanishing. Achasov and Kozhevnikov used a
fixed-mass, variable-width formula for the ρ-meson propagator.

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D.
a1ρπ Lagrangian and the amplitude of the diagrams containing the a1 meson
We choose the following interaction among the a1, ρ, and π fields
L =ga1ρπ
√2
(L1cosθ + L2sinθ), (2.3)
where ga1ρπand θ are yet undetermined parameters,
L1 = Aµ· (Vµν× ∂νφ),
L2 = Vµν· (∂µAν× φ),
(2.4)
(2.5)
and Vµν= ∂µVν− ∂νVµ. The isovector composed of the ρ-meson field operators is denoted by Vµ, similar objects
for π and a1are φ and Aµ, respectively. We write
φ1 =
1
√2
i
√2
?φc+ φ†
?φc− φ†
c
?,
?,
φ2 =
c
φ3 = φn,
and assume that φc contains the annihilation operators of the positive pion and creation operators of the negative
pion. φnis the operator of neutral pion field.
A specific combination of terms (2.4) and (2.5) appeared in the pioneering work by Wess and Zumino [27]. Term
(2.4) alone was used by Xiong, Shuryak, and Brown [39] in their study of the photon production from meson gas.
Janssen, Holinde, and Speth [37] picked the term (2.5) when they evaluated the amplitude of the πρ scattering.
Another combination of (2.4) and (2.5) appeared in the calculation of dilepton production from meson gas by Song,
Ko, and Gale [53].
Lagrangian (2.3) leads to the following factor for the vertex in which an incoming a+
(index µ), and an outgoing π+meet
1(index α), an outgoing ρ0
Vαµ(pa1,pρ,pπ) =
ga1ρπ
√2
?cosθ?pα
ρpµ
ρpµ
π− (pπpρ)gαµ?
a1− (pa1pρ)gαµ??.
− sinθ?pα
(2.6)
The a−
is straightforward.
1ρ0π−vertex acquires an extra minus sign. The evaluation of the decay rate of a+
1→ ρ0π+using vertex (2.6)
Γa+
1→ρ0π+ =
g2
a1ρπ
192πm3
× R(m2
a1
λ1/2(m2
a1,m2
ρ,m2
π+)
a1,m2
ρ,m2
π+) , (2.7)
where
λ(x,y,z) = x2+ y2+ z2− 2xy − 2xz − 2yz
and
R(x,y,z) =
?
(x − y − z)2+
− 2?(x − z)2+ y(x + z − 2y)?cosθsinθ
+
?(x + y − z)2+ 2xy?sin2θ .
y
2x(x − y + z)2?
cos2θ
If we assume the charge independent a1ρπ coupling constant and masses of ρ and a1, the width of the decay a+
is obtained from (2.7) by changing just the pion mass. Formula
1→ ρ+π0
Γa+
1= Γa+
1→ρ0π++ Γa+
1→ρ+π0
(2.8)
enables us to find the coupling constant ga1ρπfor given Γa+
two a1ρπ vertices, the overall sign of the Lagrangian (2.3) is not important and we can assume a non-negative cosθ.
1and sinθ. Because each of the diagrams Fig. 1(d) contains

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A question arises whether the narrow ρ-width approximation used above is accurate enough for the purpose of
determination the ga1ρπcoupling constant. Achasov and Kozhevnikov [18] showed that the a1decay width calculated
as Γ(a1→ ρπ) came out larger than that calculated as Γ(a1→ 3π) using the same coupling constant ga1ρπ. We have
examined this issue, too, using our two-component Lagrangian (2.3) and got that the Γ(a1→ 3π)/Γ(a1→ ρπ) ratio
is smaller than unity (like in [18]) for sinθ<
∼0.2 and sinθ>
values of sinθ that will be met in our calculations, this ratio is greater than one. Moreover, we have found that if one
takes into account the strong form factors in the a1ρπ and ρππ vertices, the results of both approaches become almost
identical. This can be explained as follows. According to Kokoski and Isgur [54] formula, which will be shown later,
the strong form factor in a particular decay vertex is a decreasing function of the three-momentum of an outgoing
particle in the rest frame of the parent particle. In the a1→ 3π decay the intermediate mass of ρ is mostly smaller
than its nominal mass, what means higher momenta of particles emerging from the a1ρπ vertex. In addition, the
a1→ 3π width is further reduced by the form factor in the ρππ vertex.
Let us now turn to the amplitude of the a1diagrams Fig. 1(d) for the decay ρ0(p) → π−(p1)π+(p2)π+(p3)π−(p4).
We first introduce the notation
∼0.65. In the remaining interval, which contains all the
qi = p − pi,
rij = pi+ pj,
sij = r2
ij,(2.9)
and then write the amplitude in the form
M(λ)
d
= ǫµ
λJd,µ,
where ǫµ
λis the polarization vector of the decaying ρ0and
Jd,µ = −(1 − P12P34)(1 + P14)(1 + P23)Vαµ(−q4,−p,p4)
× Pαβ
a1(q4)Vβν(q4,r12,p3)gρ(p2− p1)νPρ(s12).
Here, Pij denotes the operator that interchange four-momenta piand pj. The axial-vector meson propagator
Pαβ
a1(q) = i
−gαβ+
q2− m2
1
a1qαqβ
m2
a1+ ima1Γa1
(2.10)
is chosen in a simple fixed-mass, fixed-width form. Here, we are going to consider the four-pion system with invariant
energies less than 1 GeV. The invariant mass of the three-pion system, which is equal to
0.86 GeV. Achasov and Kozhevnikov [18] showed that the a1decay width is negligible in that energy range. Referring
to their finding we set Γa1= 0 in Eq. (2.10). The scalar part of the ρ propagator is again used in the form (2.1).
?q2, is thus limited by
E. Technicalities
The complete amplitude of the ρ0→ π+π−π+π−decay is
M(λ)= ǫµ
λJµ,(2.11)
where ǫµ
λis the polarization vector of the decaying ρ0and
Jµ= Ja,µ+ Jb,µ+ Jd,µ.
Four-vectors Ja,µand Jb,µdescribe the contributions from (a) and (b) diagrams in a particular model and Jd,µis the
contribution from (d) diagrams. The sum over the ρ-meson polarizations of the amplitude (2.11) squared is given by
?
λ
???M(λ)???
2
=
?
−gµν+pµpν
m2
ρ
?
JµJ∗
ν. (2.12)
This formula is more complicated than that used in [10], because the four-vectors Ja,µand Jb,µof PB/HG and AK
models do not satisfy the transversality condition Jµpµ= 0. We used the algebraic manipulation program REDUCE
[55] to express the sum (2.12) in terms of six invariants sij, i < j, j = 2,3,4 defined in (2.9). Of course, only five of

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them are independent and we used the identity?
the decay width. When calculating the excitation curve, m2
When calculating the decay width of an unpolarized parent particle, we may take advantage of the spherical
symmetry of the problem and choose the following kinematic configuration: (1) The parent particle a is at rest. (2)
The summed momentum p12of particles 1 and 2 points in the direction of the z axis. (3) The individual momenta
p1and p2lie in the xz plane. Then the following formula, written in a general case with arbitrary masses and spins,
is valid
i<jsij= m2
ρ+ 8m2
ρis replaced by s, the square of the incident energy.
πfor the checks in the process of evaluation of
Γ =
N
16(2π)6m2
a
?ma−m3−m4
m1+m2
?1
dm12p∗
1
?ma−m12
m3+m4
dm34
× p12p∗
3
?1
−1
dcosθ∗
1
−1
dcosθ∗
3
?2π
0
dϕ3|M|2. (2.13)
The last quantity is the amplitude squared, averaged over the initial spin states, and summed over the final spin
states. The factor N takes into account the identity of the final particles and equals 1/4 in our case. The asterisk
denotes the momentum in the corresponding rest frame (1-2 or 3-4), p = |p|, and mij=√sij=?(pi+ pj)2.
For evaluation of the integrals in (2.13) we used a sequence of the five one-dimensional Gauss-Legendre quadratures
of the sixteenth order. We prefer this method to Monte-Carlo integration because we use the result of the integration
in a minimization procedure and therefore we require that the same value of the optimized variable (Lagrangian mixing
angle) yield always the same value of the minimized function, which would not be satisfied with the Monte-Carlo
integration. Nevertheless, we checked our computer code by evaluating the decay width for a particular value of the
mixing angle using a completely independent code based on the Monte-Carlo method.
To convert the calculated decay width into the cross section, we start with the formula
σ4π(s) =σπ+π−(s)
Γπ+π−(s)Γ4π(s).
Using
σπ+π−(s) =πα2
3s
?
1 −4m2
π
s
?3/2
|Fπ(s)|2,
where Fπ(s) is the contribution of the ρ resonance to the pion form factor, and
Γπ+π−(s) =g2
ρW
48π
?
1 −4m2
π
s
?3/2
with W =√s, we arrive at
σ4π(s) =
?4πα
gρ
?2
1
W3|Fπ(s)|2Γ4π(s). (2.14)
We further use the VMD expression for the dielectron decay width of ρ0
Γe+e− =4πmρ
3
?α
gρ
?2
(2.15)
and get
σ4π(s) =12πΓe+e−
mρW3
|Fπ(s)|2Γ4π(s). (2.16)
If we set, following Achasov and Kozhevnikov [13],
Fπ(s) =
m2
Dρ(s),
ρ
(2.17)
where the inverse ρ-meson propagator Dρ(s) is defined in Eq. (2.2) of [13], we reproduce their Eq. (3.1).

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TABLE I: χ2/NDF of the fits to the CMD-2 cross section data (11 data points)
Γa1
(MeV)
250
400
600
Only ρ, π
ESK
[10]
1.60
1.53
1.61
17.6
PB/HG
[11]
1.34
1.37
1.41
15.0
AKonly a1
[12, 13]
1.28
1.30
1.31
14.8
1.68
1.82
1.94
/
In our opinion, Eq. (2.16) overestimates the cross section if the experimental value of Γe+e− is used. The reason is
that the dielectron decay width calculated from (2.15) is smaller than the experimental value. We will therefore stick
with formula (2.14).
We utilize the scalar part of the ρ-meson propagator (2.1) to write our Ansatz for the ρ-meson contribution to the
pion form factor
Fπ(s) =
M2
ρ(0)
M2
ρ(s) − s − imρΓρ(s). (2.18)
As shown in [47], this formula gives the correct value of the mean square radius of the pion. The form factor (2.17)
fails in this test.
For the ρ coupling constant we use the same value as in [11–13], namely gρ = 5.89. This value is compatible
with what follows from the KSRF relation [46] (5.900 ± 0.011). Both values are little lower than gρ= 6.002 ± 0.015
calculated from the ρ-meson width.
III.LOW-ENERGY RESULTS (W < 1 GeV)
We deal with the excitation curves of the reaction e+e−→ π+π−π+π−calculated in three different models (ESK,
PB/HG, AK) supplemented with the a1diagrams Fig. 1(d). We first fit them to the CMD-2 data [2] by varying the
sine of the mixing angle θ, defined in (2.3), for the three fixed values of the width of the a1(1260) meson. We did
not consider the first two points in the CMD-2 data, because they give only upper bounds of the cross section. The
ratios of the usually defined χ2to the number of degrees of freedom (NDF), which characterize the quality of the
fit, are shown in Table I. The last row in Table I shows the values of χ2/NDF that indicate how well (or badly) the
original models without a1agree with the data. No free parameter is involved in the latter case. Table I shows that
the ratio χ2/NDF is always greater than one. In what follows, we shall therefore multiply the statistical errors of the
quantities obtained in the process of minimization by the square root of that ratio.
The inspection of Table I shows that the inclusion of the a1contribution greatly improves the agreement with the
data. The interference between the original diagrams and the new ones is important, the results of the combined
model are better than those of the a1diagrams alone. The best results (lowest χ2) are obtained with the AK model
supplemented with the a1π intermediate states.
To investigate the sensitivity of our results to the input data, we combine the CMD-2 data [2] and the low-energy
(s < 1 GeV2) part of the BaBar data [56] (BaBar-LE in what follows) into a new set and repeat the calculations.
The results are shown in Table II. Their comparison with the results obtained from the CMD-2 data alone shows
two important differences: (i) The agreement of all models with data, characterized by χ2/NDF, is now better. It
indicates that the CMD-2 and BaBar-LE data are compatible, so the increased number of data points does not bring
proportional increase of χ2. (ii) Whereas merging of the a1diagrams with the PB/HG or AK model improves the fit,
adding the ESK model diagrams to the pure a1contribution leads to the opposite effect.
Next, we add the D/S ratio [41] to the set of fitted experimental values and repeat the calculations. The results are
shown in Table III. The salient feature of those results is a clear preference of the highest assumed value (600 MeV)
of the total a1width.
In Figs. 2, 3, and 4 we show the comparison of the combined set of data with the excitation curves calculated in
all three models supplemented with the a1 diagrams Fig. 1(d). The same comparison for the a1 diagrams alone is
depicted in Fig. 5.
In Table IV we can see the values of sinθ together with their errors (defined in the usual way [57]) obtained from
the fit to the CMD-2 & BaBar-LE cross section data and to the D/S ratio. As we mentioned above, three different
values of the a1(1260) width are assumed. Table V compares the experimental value of the ρ0→ π+π−π+π−decay

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TABLE II: χ2/NDF of the fits to the combined CMD-2 & BaBar-LE cross section data (27 data points)
Γa1
(MeV)
250
400
600
Only ρ, π
ESK
[10]
1.42
1.48
1.55
9.4
PB/HG
[11]
1.20
1.21
1.22
10.4
AK only a1
[12, 13]
1.19
1.18
1.19
10.3
1.32
1.39
1.44
/
TABLE III: χ2/NDF of the fits to the CMD-2 & BaBar-LE cross section data and to the D/S ratio (28 data points)
Γa1
(MeV)
250
400
600
ESK
[10]
3.66
1.98
1.65
PB/HG
[11]
1.95
1.34
1.20
AK only a1
[12, 13]
1.99
1.33
1.18
1.96
1.48
1.41
width [2] with the results obtained from various models under the same conditions. The results for both quantities
obtained with other data sets (CMD-2 only, CMD-2 & BaBar-LE) are very similar.
IV. HIGH-ENERGY RESULTS (W UP TO 4.5 GeV)
When we want to get a good description of the data on the e+e−annihilation to four charged pions at energies
above 1 GeV, we should consider also the contribution from diagrams where higher ρ resonances couple to the virtual
photon and then convert into four pions. We include two resonances: ρ′= ρ(1450) and ρ′′= ρ(1700). We assume
that the decay of those resonances into four pions is governed by the same Feynman diagrams as that of ρ(770), with
all coupling constants scaled by the same factor (different for ρ′and ρ′′). This assumption implies that the four-pion
FIG. 2: Excitation curves calculated in the original (without a1 meson) and expanded ESK model compared to the CMD-2
and BaBar-LE data. The D/S ratio was also used in fit.

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10
FIG. 3: Excitation curves calculated in the original (without a1 meson; dash-dotted curve close to the abscissa) and expanded
PB/HG model compared to the CMD-2 and BaBar-LE data. The D/S ratio was also used in fit.
FIG. 4: Excitation curves calculated in the original (without a1 meson; dash-dotted curve close to the abscissa) and expanded
AK model compared to the CMD-2 and BaBar-LE data. The D/S ratio was also used in fit.
TABLE IV: Values of sinθ from the fit to the CMD-2 & BaBar-LE data and to the D/S ratio.
Γa1
(MeV)
250
400
600
ESK
[10]
PB/HG
[11]
0.4278(32)
0.4624(34)
0.5046(44)
AK only a1
[12, 13]
0.4267(32)
0.4608(32)
0.5022(41)
0.4092(33)
0.4352(24)
0.4659(27)
0.4312(35)
0.4679(39)
0.5132(55)

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11
FIG. 5: Excitation curves calculated from the a1π diagrams only, compared to the CMD-2 and BaBar-LE data. The D/S ratio
was also used in fit.
TABLE V: Decay width Γ(ρ0→ π+π−π+π−) (keV) calculated in various models using sinθ from the fits to the CMD-2 &
BaBar-LE data and to the D/S ratio. Experimental value is (2.8 ± 1.4 ± 0.5) keV [2].
Γa1
(MeV)
250
400
600
Only ρ, π
ESK
[10]
4.28(01)
2.81(01)
1.94(02)
16.2
PB/HG
[11]
3.16(25)
3.55(28)
3.77(30)
0.59
AKonly a1
[12, 13]
2.70(23)
3.03(26)
3.22(27)
0.89
4.52(30)
5.08(32)
5.39(37)
/
decay widths of ρ′and ρ′′have the same shape in W as that of ρ(770). They only differ from it by constant factors.
The simplifying assumption we have made allows us to use the same cross section formula (2.14) as in the low energy
case, with Fπ(s) replaced by
F(s) = Fρ(s) + δFρ′(s) + ǫFρ′′(s), (4.1)
where Fρ(s) differs from (2.18) by including also Γρ0→π+π−π+π−(s) into the total decay width of ρ(770),
Fρ′(s) =
m2
ρ′
m2
ρ′ − s − imρ′Γρ′, (4.2)
and similar expression holds also for Fρ′′(s). Constants δ and ǫ not only include the coupling constants modification
factors mentioned above, but also account for the fact that the couplings of ρ′and ρ′′to photon differ from that of
ρ(770), which is fixed by VMD. Unknown complex parameters δ and ǫ will be determined, together with the masses
and widths of ρ′and ρ′′and other two parameters mentioned later on, by fitting the experimental excitation function2.
2When we replaced (2.18) by (4.1) in the low-energy region and kept the form-factor parameters as determined in the high-energy fit,
we got the results that differed slightly from that in Sec. III. If we varied also those parameters when fitting the low-energy data, they
acquired unphysical values (masses of ρ′and ρ′′around 1 GeV). It may signify that some contribution important at low energies (scalar
resonances) is still missing in our approach.

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TABLE VI: Results of the fit to the BaBar cross section data and to the D/S ratio (145 data points) for Γa1= 600 MeV.
ModelESK
[10]
1.21
PB/HG
[11]
1.12
0.4592(28)
0.3665(97)
1.439(13)
0.568(21)
0.1145(51)
-0.019(12)
1.923(24)
0.284(44)
-0.0002(17)
-0.0054(12)
AKonly a1π
[12, 13]
1.12
0.4588(27)
0.3657(97)
1.438(13)
0.568(21)
0.1144(51)
-0.021(12)
1.922(24)
0.283(44)
-0.0003(17)
-0.0054(12)
χ2/NDF
sinθ
β (GeV)
mρ′ (GeV)
Γρ′ (GeV)
Re(δ)
Im(δ)
mρ′′ (GeV)
Γρ′′ (GeV)
Re(ǫ)
Im(ǫ)
1.12
0.4474(22)
0.3505(89)
1.419(12)
0.564(20)
0.1038(41)
-0.039(11)
1.903(21)
0.247(38)
-0.0016(11)
-0.00373(94)
0.4603(28)
0.3695(98)
1.442(13)
0.566(21)
0.1149(53)
-0.015(13)
1.926(24)
0.290(45)
0.0002(18)
-0.0056(13)
Another effect that has to be taken into account when dealing with higher energies is connected with the structure of
the strongly interacting particles. Our decay amplitudes have been derived under the assumption that the pions, ρ’s,
and a1’s are elementary quanta of the corresponding quantum fields. But this assumption is justified only when their
mutual interaction is soft. When the momenta of the mesons which enters a specific interaction vertex get higher,
the contribution of that vertex to the amplitude becomes smaller than in the case of the point-like participants.
This effect is usually described by strong form factors. Given the present status of the strong interaction theory, we
have to turn to models. For example, in the chromoelectric flux-tube breaking model of Kokoski and Isgur [54], the
vertex describing a two-body decay is modified by the factor exp{−p∗2/(12β2)}, where p∗is the three-momentum
magnitude of the decay products in the parent particle rest frame and β ≈ 0.4 GeV. For decay of the ρ meson with a
(non-nominal) mass W into two on-mass-shell pions, this form factor can be written as
FKI(s) = exp
?
−s − s0
48β2
?
,(4.3)
where s = W2and s0is the threshold value of s (4m2
decay of ρ0contains many vertices, some of them with more than three incoming/outgoing particles. Applying the
Kokoski-Isgur factor to each of them would be cumbersome and would require additional assumptions in the case of
more complicated vertices. We will therefore assign an “effective” strong form factor of the form (4.3) to the complete
amplitude of the decay ρ0→ π+π−π+π−, but with s0= 16m2
will be considered as another parameter. With the masses and widths of ρ′and ρ′′, complex parameters δ and ǫ, and
with the sine of the mixing angle θ there are ten real parameters to be determined by fitting the e+e−→ π+π−π+π−
cross section data of BaBar collaboration [56].
In the following, we assume the a1decay width of 600 MeV, for which the results of all models at energies below
1 GeV were best. The same value was used in the a1propagator (2.10).
The resulting optimized values of the parameters listed above and their MINUIT [57] errors are shown in Table VI for
the three different models supplemented with the a1π intermediate states and for the latter alone. Mutual comparison
of the χ2/NDF ratios clearly shows that the presence of the a1π intermediate states is crucial for obtaining good
agreement of the calculated excitation curve with data. They provide a good fit even if taken alone. Adding the
diagrams with π’s and ρ’s in the intermediate states does not change the quality of the fit if their amplitudes are taken
from the PB/HG and AK models. On the other hand, the inclusion of the amplitudes of the ESK model brings some
deterioration of the fit. The values of parameter β do not differ very much from the value advocated in [54] for the
three-line vertices, what indicates that the effective-strong-form-factor approach we have chosen (4.3) is reasonable.
The values of the sine of the mixing angle θ are somewhat lower than those at low energies. The central values of
the masses and widths of ρ′and ρ′′a little different from those listed in [15], but differences are acceptable keeping
in mind relatively large errors.
The graphical comparison of the excitation curve calculated from the model containing only the diagrams with the
a1π intermediate states, shown in Fig. 1(d), with experimental data [56] is presented in Fig. 6. The excitation curves
of other models (ESK, PB/HG, AK) combined with the a1π contribution differ only slightly and are not shown.
πin the two-pion decay). The complete amplitude of the four-pion
π. When fitting the experimental excitation curve, β

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13
FIG. 6: Theoretical excitation curve compared with the BaBar data [56]. The D/S ratio was also used in fit. The result of the
pure a1 model is shown. The other models combined with a1π intermediate states provide almost identical curves.
V. CONCLUSIONS AND COMMENTS
Our low-energy results show that the inclusion of the a1π intermediate states is of vital importance for obtaining
a good agreement with the experimental data on the cross section of the reaction e+e−→ π+π−π+π−as a function
of the incident energy (see Tables I, II, and III or Figs. 2, 3, and 4). The χ2/NDF ratio gets much smaller if a
particular model is supplemented with the diagrams containing the a1resonance in the intermediate states. Viewing
from another perspective, the pure a1model provides relatively good agreement with the cross section data, much
better than each of the three models without the a1π intermediate states. See Tables mentioned above and Fig. 5.
Adding the diagrams from the original models to the pure a1model improves the fit in the case of the PB/HG and
AK models, but worsens it in the case of the ESK model. Unfortunately, the χ2/NDF ratio remains greater than one
everywhere. It may be the consequence of our ignoring some important contributions, perhaps those with a scalar
resonance considered in [4, 6]. The original models ESK, PB/HG, and AK do not contain any scalar resonances. We
also have not included them as our main concern was the role of the a1resonance. It is also possible that the a1ρπ
Lagrangian should contain more terms than considered in (2.3).
Using the D/S ratio as an additional data point is important. It can discriminate among various models. In our
case it increases the separation of the ESK model from the others. It also strongly prefers larger values of the assumed
a1 width. In calculations without the D/S ratio we were able to find a value of sinθ for each Γa1that led to an
acceptable fit to the e+e−→ 2π+2π−cross section. However, the lowest value of Γa1is excluded if we add the D/S
ratio to the fitted data (see Table III).
As to the partial decay width of ρ0→ π+π−π+π−, the conclusion is not so categorical. Two models (PB/HG and
AK) in their original forms provided results that were a little smaller, but did not contradict strongly the experimental
value with its large errors. Only the original ESK model gave too large figure, which was in a clear disagreement
with the experimental value. The inclusion of the a1π intermediate states brought all values into the interval given
by the experimental value and its errors summed linearly, see Table V. It must be said that the pure a1model gives
the decay widths that are close to the one-sigma upper limit or even beyond it.
We originally hoped that our study would tell us the form of the a1ρπ Lagrangian. But with respect to the a1ρπ
Lagrangian, no clear picture can be inferred from the low-energy results yet. The optimal values of the sine of the
mixing angle, see Table IV, are from a broad interval and depend not only on the choice of the original model to which
the a1diagrams are added, but also on the assumed value of the a1width. The optimized values of sinθ squeeze into
interval (0.40,0.51).
The quality of the fit over the whole energy range of the BaBar experiment [56], as measured by the χ2/NDF ratio,
seems to be better than that at low energies. But a more careful investigation in terms of the confidence level, which

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TABLE VII: Survey of the mixing parameter sinθ for various versions of two-component a1ρπ Lagrangian (2.3) that appeared
in literature.
No.
1
2
3
4
5
sinθ
0
0.2169
0.5582
0.6308
1
0.40–0.51
0.41–0.47
Reference
[38, 39]
[23, 24, 53]
[58]
[23, 24, 59]
[37]
our low-energy fits
our all-energy fits
takes into account χ2and NDF separately, shows equal quality of those two fits. The values of sinθ are compatible
with those found at low energies, sinθ ∈ (0.41,0.47) (lower boundary is obtained from the fits where Γa1= 250 MeV
was used). They occupy a narrower interval, what can be explained by lesser importance of sub-dominant diagrams
with only ρ and π mesons in the intermediate states.
It is interesting to compare our estimates of the mixing parameter sinθ, defined in Eq. (2.3), with its values that
have been used so far, see Table VII. The values in the first and fifth row simply reflect the fact that Lagrangians in
Refs. [37–39] contained just one term. The remaining rows refer to various sets of four fundamental parameters (m0,
g, σ, and ξ) of the model in which the vector and axial-vector mesons were included as massive Yang-Mills fields of
the SU(2)×SU(2) chiral symmetry [23]. The model built on previous works [28–30]. Our mixing parameter is related
to the parameters η1and η2of that model, which can be expressed in terms of the four fundamental parameters using
Eqs. (2.9) and (2.10) in [23]. The formula is very simple
sinθ =
η2
1+ η2
?η2
2
.
In [23], the fundamental parameters of the massive Yang-Mills model were determined using the experimental values of
the masses and width of the ρ and a1mesons. This procedure is not unique, there are two solutions. The corresponding
mixing parameter is shown in the second and fourth row of Table VII. Row 3 corresponds to an ad hoc choice of
fundamental parameters made in [58]. Last two rows show the range of our results obtained from various models
(ESK, PB/HG, AK) and various assumed values of Γa1(250, 400, and 600 GeV). Unfortunately, there is no overlap
with the rows above. The issue definitely requires more attention. The phenomenological models may be improved by
including other intermediate states, perhaps those with scalar resonances. The parameters of the massive Yang-Mills
model may be tuned by using richer experimental input (Γ(a1→ πγ) and D/S ratio as in [24, 32]) and more realistic
formulas for relating theoretical parameters to experimental quantities, e.g., calculating the a1 width as a1 → 3π
instead of a1→ ρπ.
Our failure to obtain a more precise value of the mixing angle of the a1ρπ Lagrangian suggests that it is necessary
to make a simultaneous fit to data about several physical processes. The natural candidates are the e+e−annihilation
into various four-pion final states, the decay of the τ lepton into neutrino and three or four pions, and the exclusive
hadronic reactions of the type investigated, e.g., in [37].
Acknowledgments
One of us (P. L.) is indebted to David Kraus for useful discussions. We thank Prof. T. Barnes for useful cor-
respondence. This work was supported by the Czech Ministry of Education, Youth and Sports under contracts
MSM6840770029, MSM4781305903, and LC07050.
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