Study of resonant processes for multi-pion production in $\bar p
  +p\to\ell ^++\ell^- +n_\pi \pi$ annihilation
Study of resonant processes for multi-pion production in
p̄+ p→ ℓ+ + ℓ− + nππ annihilation
E. A. Kuraev,∗ C. Adamušč́ın,† and E. Tomasi-Gustafsson‡
DAPNIA/SPhN, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France
F. Maas
CNRS/IN2P3, Institut de Physique Nucléaire, UMR 8608
and Univ. Paris-Sud, Orsay, F-91405 France
(Dated: September 24, 2021)
Abstract
In frame of a phenomenological approach based on Compton-like Feynman amplitudes, we study
multi-pion production in antiproton nucleon collisions. The main interest of this reaction is related
to the possibility to study the properties of the presumable N̄N atom and of its resonances. For
the case of formation of a scalar or pseudoscalar resonant state, with IG(JP ) = 1−(0±), 0+(0−)
numerical results are obtained. The differential cross section in an experimental set-up where the
pions invariant mass is measured, is given with explicit dependence on the lepton pair and pions
invariant mass.
PACS numbers:
∗JINR-BLTP, 141980 Dubna, Moscow region, Russian Federation
†Department of Theoretical Physics, IOP, Slovak Academy of Sciences, Bratislava, Slovakia
‡Electronic address: etomasi@cea.fr
http://arxiv.org/abs/0704.1199v1
mailto:etomasi@cea.fr
In a previous paper [1] we calculated the differential cross section for single pion produc-
tion in the reactions p̄+N → π+ ℓ++ ℓ−, and showed that such reactions are measurable at
upcoming facilities, bringing unique information on electromagnetic and axial nucleon form
factors in the unphysical region of time-like momentum transfer squared, q2.
The purpose of the present work is to study the resonant production of a number nπ of
pions accompanied by a lepton pair, in the antiproton nucleon annihilation process. We limit
our discussion to quantum numbers of the excited resonances: IG(JP ) = 1+(0±) and 0+(0±).
The existence a N̄N bound state and of its resonances was predicted in a series of papers,
see Ref. [2, 3]. A partial wave analysis performed in [3] in the range 1960 < M∗ < 2410
combines data in different decay channels 3π0, π0η, π0ηη′ and reveals a series of resonant
states in the pp̄ system. Some of the states were precisely identified, in particular two
states with quantum numbers I(JPC) = 1(0−+), with masses M∗ = 2360 and 2070 MeV,
respectively and widths about 300 MeV. Higher spin states were also identified. Such states
show similar masses as dibaryon states, except that they exhibit large widths, which are
explained by their decay through hadronic states, which is impossible for dibaryons [4].
NN̄ state has hydrogen-like atomic structure. Similarly to positronium, in addition
to strong interaction, QED effects are present, due to the large masses (small distances)
involved. Special attention is payed to resonances with masses ∼ 2M , where M is the
nucleon mass. The resonances with zero orbital momentum are expected to have large
width, whereas higher values of orbital momentum lead to a smaller width given by [2]:
[(2ℓ+ 1)!!]2
MeV , (1)
which gives, for example, Γ = 100 MeV for ℓ = 0, Γ = 10 MeV for ℓ = 1, and Γ = 0.5 MeV
for ℓ = 2.
Experimental evidence of such series of dibaryon resonances exist [4], although it is still
controversial. An investigation of this problem will be possible at the upcoming FAIR facility
[5], where the planned antiproton beams render possible the measurement of the reactions
investigated in the present paper.
The emission of a lepton pair permits to select the appropriate kinematics adapted to
the excitation of such resonances, when the total p̄N center of mass (CMS) energy exceeds
the mass of the resonances, due to a known mechanism, called ’return to resonances’ [6]:
the lepton pair carries away the extra energy and momentum, providing the condition of
π1(q1)
π2(q2)
πn(qn)
p̄(−p1)
ℓ−(p−)
ℓ+(p+)
p(p2)
Jp = 0±
γ∗(q)
q − p1
π1(q1)
π2(q2)
πn(qn)
p̄(−p1)
p(p2)
Jp = 0±
ℓ+(p+)
ℓ−(p−)
γ∗(q)
p2 − q
(b)(a)
FIG. 1: Feynman diagrams for the reaction p̄+ p → ℓ+ + ℓ− + nππ: lepton pair emission form the
antiproton (a) and from the proton (b).
exciting the resonances. Experimentally it manifests through a deformation of the Breit
Wigner distribution, yielding a ’radiative’ tail. For the excitation of a narrow resonance, the
emission of soft quanta alters the shape of the resonance curve: the height of the resonance
curve decreases and a radiative tail arises. Indeed, for total energy W , higher than the
resonance mass M ′, the most favourable mechanism is when the energy ”excess” W −M ′ is
absorbed by the photon emission (with energy ω =W −M ′). Here the small extra factor α
is compensated by the resonance denominator.
Let us consider the annihilation of an antiproton and a proton in a lepton pair, and a
resonant state which subsequently decays into a set of nπ pions: p̄(p1) + p(p2) → γ∗(q) +
R(JP ,M ′) → ℓ+(p+) + ℓ−(p−) + π1(q1) + ... + πnπ(qnπ). Such state can have the quantum
number of scalar, pseudoscalar, axial or vector particle. The Feynman diagrams for such
reactions are plotted in Fig. 1. We neglect direct photon emission from the resonant state,
assuming that it is (indirectly) included via the width of the resonance.
To describe the kinematics of the process it is convenient to introduce two kinematical
variables: the invariant mass squared of the lepton pair
q2 = (p+ + p−)
2 (2)
and the one of the pions:
s1 = (
2 = (p1 + p2 − q)2. (3)
One can write the phase volume as:
dΓx =
(2π)−2δ4
p1 + p2 − p+ − p− −
Πxi=1
2ǫi(2π)3
dΓQq (4)
with ∫
dΓq =
2ǫ+(2π)3
2ǫ−(2π)3
δ4(q − p+ − p−),
Πnπi=1
2ǫi(2π)3
(2π)4,
dΓQq =
d4qd4Qδ4(p1 + p2 − q −Q).
It is convenient to calculate this last quantity in CMS, where ~q = −~Q, ~p1 = −~p2. In this
reference frame we have:
d4q =
dqO|~q|dq2dOq =
dΓQq, (5)
where dOq = 2πd cos θq is the element of solid angle, integrated on the azymuthal angel φ.
Then, using the relations
s = q0 + Q0; (s1 − q2)/
s = Q0 − q0, and s = (p1 + p2)2, one
finds:
dΓQq =
∫ √s−√q2
dq2d cos θq
Λ(s, s1, q2), Λ(a, b, c) = a
2+ b2+ c2−2(ab+ac+ bc),
with θq = ~̂q, ~p1. The function Λ(s, s1, q
2) must be positive in the physical region.
Let us restrict our considerations to scalar JP = 0+ and pseudoscalar JP = 0− states.
The corresponding matrix elements are:
Mires =
2)Jµv̄(p1)Oiµu(p2)
gi1g2
s1 −M ′2 + iM ′Γ′
where M ′ and Γ′ are the mass and the width of the resonance, Vρ(q
2) = m2ρ/(m
ρ − q2)
corresponds to the coupling of the photon to the ρ meson (mρ is the ρ meson mass), when
interacting with a nucleon, Jµ(q) = v̄(p+)γµu(p−) is the leptonic current and Oiµ
Oiµ = Fpµ(q)
−p̂1 + q̂ +M
(p1 − q)2 −M2
γ(i) +
p̂2 − q̂ +M
(p2 − q)2 −M2
Fpµ(q) (8)
(where γ(i) = γ5 for a pseudoscalar resonant state and γ
(i) = 1 for a scalar state) contains
the hadronic structure:
Fpµ(q) = F
2)γµ +
(q̂γµ − γµq̂), (9)
which is parametrized in terms of the Pauli and Dirac proton form factors, F1 and F2 (in
time-like region).
Let us note that the hadron current J hµ = v̄(p1)Oiµu(p2) obeys the gauge invariance:
J hµ qµ = 0.
In Eq. (7), the coupling constant g2 is related to the width of the decay channel:
Γ′nπ =
Q , (10)
where Γ′nπ is the width of the decay of the pp̄ state to nπ pions, g1 represents the coupling
constant at the p̄p vertex to the resonant state. When the resonance is heavier than twice
the nucleon mass its value can be related to the p̄p decay width:
(gi1)
8πΓp̄p
M ′Ai
, (11)
with Ai = β for J
P = 0−, and Ai = β
3 for JP = 0+ and β =
1− (4M2/M ′2).
The corresponding differential cross sections are:
dq2ds1
Γ′nπΓ
(s1 −M ′2)2 +M ′2Γ2i
Λ(s, s1, q2)
Di, for M ′ > 2M, (12)
dq2ds1
α2M ′Γ′nπ
12π2q2
(s1 −M ′2)2 +M ′2Γ2i
Λ(s, s1, q2)
Di, for M ′ < 2M, (13)
with r =
1− (4M2/s) and
d cos θq
gµν −
S(i)µν , (14)
S(i)µν =
Tr(p̂1 −M)Oiµ(p̂2 +M)Oi∗ν . (15)
Two dimensionless variables, X = q2/s and Y = s1/s can been defined, using the invariant
mass of the lepton pair, q2, Eq. (2), and the invariant mass of the pions, s1, Eq. (3). The
kinematical region scanned by the reaction, is defined by the conditions Λ(s, s1, q
2) > 0 and
it corresponds to Y ≤ (1−
X)2, as it is shown in Fig. 2.
FIG. 2: Allowed kinematical region, as a function of X = q2/s and Y = s1/s (shaded area).
The angular integration entering in Di, Eq. (14), can be calculated with the help of the
following relations:
d cos θq
d cos θq
q2ss1 +M2Λ(s, s1, q2)
d cos θq
d cos θq
Λ(s, s1, q2)
∣∣∣∣∣
q2 + s1 − s+ r
Λ(s, s1, q2)
q2 + s1 − s− r
Λ(s, s1, q2)
∣∣∣∣∣ , (16)
with d1,2 = q
2 − 2p1,2q and d1 + d2 = s1 + q2 − s. The integration is performed in the center
of mass of the initial particles.
The differential cross sections for heavy scalar and pseudoscalar states have been cal-
culated for the reaction p̄ + p → ℓ+ + ℓ− + nππ, as a function of the s and of the total
momentum carried by the pions. Since the coupling constant g1 and the partial width Γp̄p
are not known, the results for the differential cross section are rescaled by these quantities.
For the numerical application, we focus on the scalar resonance f0(980) (Figs 3) and on the
pseudoscalar resonance, with η(1405) (Fig. 4). The electromagnetic nucleon form factors
have been parametrized according to [8].
The two dimensional plots show a structured landscape, limited by the kinematical cut
due to the physical accessible region. Peaks appear in the projection to the q2 axis (inte-
gration over s1), and are due to the vector meson resonances coming from enhanced photon
Graph2D_y
Entries  290
Mean   0.9608
RMS    0.2355
]2 [GeV1s
0 1 2 3 4 5
Graph2D_y
Entries  290
Mean   0.9608
RMS    0.2355
 [GeV1
-1010
Graph2D_x
Entries  290
Mean   0.6289
RMS     0.111
]2 [GeV2q
0 1 2 3 4 5 6
Graph2D_x
Entries  290
Mean   0.6289
RMS     0.111
FIG. 3: (Color online) Double differential cross section for the reaction p̄ + p → M∗ + nππ,
M∗ ≡ f0(980), as a function of q2 and s1.
vector meson coupling included in the FFs model. The projection to the s1 axis (integration
over q2) shows a bump which is due to the excitation of the pp̄ resonant state.
The absolute value of the cross section is in general about two orders of magnitude smaller
than in the case of one pion production, as calculated in Ref. [1], but still measurable in the
resonance region.
To avoid double counting, we do not consider direct emission of a virtual photon from
the resonance, as well as from charged component from its decay, as we imply that it is
implicitly taken into account in the resonance width.
Let us discuss now the possibility for the creation of an atomic state of p̄N type. In the
case of a charged atom (N = n) the situation is similar to the deuteron case, with a weakly
bound state. One may expect several narrow states with masses ≤ 2M and binding energy
of ≃ 2− 3 MeV. The experimental observation of such states has been reported [4].
The coupling constant g1 related to the creation of the bound state (p̄n) can be related
(following [7]) to the wave function of this state:
g1 =M
−3/2|Ψ(0)|,
|ψ|2d3r = 1, Ψ(0) = −M
V (r)rΨ(r)dr, (17)
where Ψ(r) and r0 are the wave function and the radius of the p̄n bound state and V (r) is
Graph2D_y
Entries  403
Mean    1.947
RMS    0.2783
]2 [GeV1s
0 1 2 3 4 5
Graph2D_y
Entries  403
Mean    1.947
RMS    0.2783
Graph2D_x
Entries  403
Mean   0.6166
RMS    0.09974
]2 [GeV2q
0 1 2 3 4 5 6
-1010
Graph2D_x
Entries  403
Mean   0.6166
RMS    0.09974
 [GeV1
-1210
-1110
-1010
FIG. 4: (Color online) Double differential cross section for the reaction p̄ + p → M∗ + nππ,
M∗ ≡ η(1405) as a function of q2 and s1.
the potential of the p̄− n interaction.
The reaction p̄ + n → π1 + ... + πnπ can be calculated using a similar formalism with
evident modifications, replacing the proton FFs with the neutron FFs and adding a contact
term proportional to (F n1 − F
1 )qµ/q
2. One may expect that the general behavior of the
distributions will be similar to the case of p̄p collisions. The size of the relevant resonance
should be be much smaller than for p̄p one (several fermi), as for a deuteron-like object.
Neutral atoms, (N = p) are of electromagnetic nature, their typical size can be estimated
to 25 fm, and their binding energy of the order of 20 keV. The width of these resonances
in S state is rather large. Such resonances, with orbital momentum L = 0 have typically
strong interaction width of order of 100 MeV. For L 6= 0, the width should be smaller, see
Eq. (1).
In conclusions, we have calculated the double differential cross section for the multipion
production in proton antiproton collision, with emission of a leptonic pair. Numerical esti-
mations show that the cross section is measurable in the kinematical region which will be
accessible at FAIR and will allow to investigate the formation and the resonant structure of
a possible NN̄ system.
[1] C. Adamušč́ın, E. A. Kuraev, E. Tomasi-Gustafsson and F. Maas, arXiv:hep-ph/0610429, to
appear in Phys. Rev. C.
[2] L. N. Bogdanova, O. D. Dalkarov and I. S. Shapiro, Annals Phys. 84 (1974) 261.
[3] A. V. Anisovich et al., Phys. Lett. B 517 (2001) 273; Phys. Lett. B 517 (2001) 261.
[4] B. Tatischeff et al., Phys. Rev. C 59 (1999) 1878; B. Tatischeff and E. Tomasi-Gustafsson,
arXiv:nucl-ex/0411044.
[5] An International Accelerator Facility for Beams of Ions and Antiprotons, Conceptual Design
Report, http://www.gsi.de.
[6] V.Baier and V. S. Fadin, Phys. Lett. B 27 (1968) 223; G. Bonneau and F. Martin, Nucl. Phys.
B 27 (1971) 381; F. Berends and R. Gastmans, ”Electromagnetic Interactions of hadrons, II,
Plenum Press, New York-London, 1978.
[7] A.I Akhiezer and I. Pomeranchuk, ’Some questions in hadron theory’, Chapter I, Moscow 1950
(in russian).
[8] F. Iachello, A. D. Jackson and A. Lande, Phys. Lett. B 43, 191 (1973); F. Iachello, eConf
C0309101, FRWP003 (2003) [arXiv:nucl-th/0312074]; F. Iachello and Q. Wan, Phys. Rev. C
69, 055204 (2004).
http://arxiv.org/abs/hep-ph/0610429
http://arxiv.org/abs/nucl-ex/0411044
http://www.gsi.de
http://arxiv.org/abs/nucl-th/0312074
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