#### A chiral covariant approach to \(\varvec{\rho \rho }\) scattering

Eur. Phys. J. C
A chiral covariant approach to ρρ scattering
D. Gülmez 2
U.-G. Meißner 1 2
J. A. Oller 0
0 Departamento de Física, Universidad de Murcia , 30071 Murcia , Spain
1 Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich , 52425 Jülich , Germany
2 Helmholtz-Institut für Strahlenund Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn , 53115 Bonn , Germany
We analyze vector meson-vector meson scattering in a unitarized chiral theory based on a chiral covariant framework restricted to ρρ intermediate states. We show that a pole assigned to the scalar meson f0(1370) can be dynamically generated from the ρρ interaction, while this is not the case for the tensor meson f2(1270) as found in earlier work. We show that the generation of the tensor state is untenable due to the extreme non-relativistic kinematics used before. We further consider the effects arising from the coupling of channels with different orbital angular momenta which are also important. We suggest to use the formalism outlined here to obtain more reliable results for the dynamical generation of resonances in the vector-vector interaction.
1 Introduction
It is now commonly accepted that some hadron resonances
are generated by strong non-perturbative hadron–hadron
interactions. Arguably the most famous example is the
(1405), which arises from the coupled-channel dynamics
of the strangeness S = −1 ground -state octet meson–baryon
channels in the vicinity of the π and K − p thresholds [1].
This resonance also has the outstanding feature of being
actually the combination of two near poles, the so-called two-pole
nature of the (1405). In a field-theoretic sense, one should
consider this state as two particles. This fact was predicted
theoretically [2,3] and later unveiled experimentally [4] (see
also the discussion in Ref. [5]). Another example is the scalar
meson f0(980) close to the K¯ K threshold, which is often
considered to arise due to the strong S-wave interactions in
the π π –K¯ K system with isospin zero [6–8]. A new twist was
given to this field in Ref. [9] where the S-wave vector–vector
(ρρ) interactions were investigated and it was found that due
to the strong binding in certain channels, the f2(1270) and
the f0(1370) mesons could be explained as ρρ bound states.
This approach offered also an explanation why the tensor
state f2 is lighter than the scalar one f0, as the leading order
attraction in the corresponding ρρ channel is stronger. This
work was followed up by extensions to SU(
3
) [10], to account
for radiative decays [11,12], and much other work; see e.g.
the short review in Ref. [13].
These findings concerning the f2(1270) are certainly
surprising and at odds with well-known features of the strong
interactions. It is a text-book result that the f2(1270) fits
very well within a nearly ideally mixed P-wave qq¯ nonet
comprising as well the a2(1320), f2(1525) and Ks∗(1430)
resonances [14–17]. Values for this mixing angle can be
obtained from either the linear or quadratic mass relations as
in Ref. [17]. Non-relativistic quark model calculations [18],
as well as with relativistic corrections [19], predict that the
coupling of the tensor mesons to γ γ should be
predominantly through helicity two by an E 1 transition. This
simple qq¯ picture for the tensor f2(1270) resonance has been
recently validated by the analyses performed in Ref. [20] of
the high-statistics Belle data [21–25] on γ γ → π π in both
the neutral and the charged pion channels. Another point
of importance in support of the qq¯ nature of the f2(1270)
is Regge theory, since this resonance lies in a parallel
linear exchange-degenerate Regge trajectory with a
“universal” slope parameter of around 1 GeV [26–30]. Masses and
widths of the first resonances with increasing spin lying on
this Regge trajectory (ρ, f2, ρ3, f4) are nicely predicted [31]
by the dual-hadronic model of Lovelace–Shapiro–Veneziano
[32–34].
One should stress that the results of Ref. [9] were obtained
based on extreme non-relativistic kinematics, pi2/m2ρ 0,
with p the rho-meson three-momentum and mρ the
vectormeson mass. This approximation, however, leads to some
severe simplifications:
• Due to the assumed threshold kinematics, the full ρ
propagator was reduced to its scalar form, thus enabling the
use of techniques already familiar from the pion–pion
interaction [8]. This was applied when considering the
iteration of the interactions in the Bethe–Salpeter
equation.
• Based on the same argument, the algebra involving the
spin and the isospin projectors of the two vector-meson
states could considerably be simplified.
However, as √sth = 2mρ = 1540 MeV, the lighter of the
bound states obtained in Ref. [9] is already quite far away
from the 2ρ threshold, with a nominal three-momentum of
modulus larger than a 60% of the ρ mass. It is therefore
legitimate to question the assumptions there involved. In this
work, we will reanalyze the same reactions using a fully
covariant approach. This is technically much more involved
than the formalism of the earlier work. However, as our aim
is to scrutinize the approximations made there, we stay as
much as possible close to their choice of parameters. Besides,
we also consider ρρ higher partial waves since the authors
of Ref. [9] only considered scattering in S-wave because of
the same type of near-threshold arguments. The inclusion of
these higher partial waves is also important when moving
away from threshold. As it will be shown, the near-threshold
approximation is only reliable very close to threshold.
There is one further issue not considered in Ref. [9]
concerning the multiple coupled-channel nature in the
energy region where these resonances (for the f0(1370) and
f2(1272)) appear. Regarding this point, a set of 13 coupled
channels were taken into account in the study of Ref. [35]
of the J PC = 0++ meson scattering. Of special significance
in their analysis among the channels considered are the
twopseudoscalar ones of π π , K K¯ , ηη, ηη and the two-resonance
modes (mimicking multi-meson states) of σ σ , ρρ and ωω. A
main conclusion of this reference is that the coupled-channel
dynamics is crucial to fully understand the meson spectrum
in the energy region around 1.5 GeV, which spans the ρρ
threshold region, as also concluded in Ref. [36]. In
particular Ref. [35] found that the f0(1370) was mostly a bare
state stemming from the SU (
3
) octet of bare 0++ scalar
resonances with mass around 1.4 GeV. This conclusions was
reached by the almost coincidence of the couplings of this
resonance to the two-pseudoscalar channels calculated both
from the tree-level Lagrangian and from the residue of the
meson–meson amplitudes at the resonance pole.
Furthermore, the mass from the pole position of the f0(1370) is also
very close to the bare mass of this octet of scalar resonances.
Reference [35] derived the couplings of the vector–vector
states to the pseudoscalar–pseudoscalar states and to σ σ by
invoking minimal coupling [37], although the direct
transitions between the vector–vector channels were not included.
We fill partially this gap here by considering the direct
scattering between the ρρ states but at the same time we keep
the strong assumption of Ref. [9] of neglecting any other
channels. In this sense it would be very valuable to improve
the work of Ref. [35] by incorporating this new formalism
to treat vector–vector scattering. This, however, goes beyond
the scope of this research, as our main emphasis here lies
on extending and critically reviewing the model of Ref. [9]
by taking into account the relativistic corrections and higher
partial waves.
Our work is organized as follows: in Sect. 2 we outline the
formalism to analyze ρρ scattering in a covariant fashion. In
particular, we retain the full propagator structure of the ρ,
which leads to a very different analytic structure of the
scattering amplitude compared to the extreme non-relativistic
framework. We also perform a partial-wave projection
technique, which allows us to perform the unitarization of the
tree-level scattering amplitudes using methods well
established in the literature. An elaborate presentation of our
results is given in Sect. 3, where we also give a detailed
comparison to the earlier work based on the non-relativistic
framework. Next, we consider the effect of the coupling
between channels with different orbital angular momentum.
We also improve the unitarization procedure by
considering the first-iterated solution of the N /D method in Sect. 4,
reinforcing our results obtained with the simpler
unitarization method. We conclude with a summary and a discussion
in Sect. 5. A detailed account of the underlying projection
formalism is given in Appendix A.
2 Formalism
The inclusion of vector mesons in a chiral effective
Lagrangian can be done in a variety of different ways, such
as treating them as heavy gauge bosons, using a tensor field
formulation or generating them as hidden gauge particles
of the non-linear σ -model. All these approaches are
equivalent, as shown e.g. in the review [37]. While in principle the
tensor field formulation is preferable in the construction of
chiral-invariant building blocks, we stick here to the hidden
symmetry approach as this was also used in Ref. [9].
To be specific, the Lagrangian for the interactions among
vector mesons is taken from the pure gauge-boson part of
the non-linear chiral Lagrangian with hidden local symmetry
[38,39],
1
L = − 4 Fμν F μν .
Here, the symbol . . . denotes the trace in SU(
2
) flavor space
and the field strength tensor Fμν is
Fμν = ∂μVν − ∂ν Vμ − i g[Vμ, Vν ],
(
1
)
(
2
)
ρ+
ρ−
ρ+
ρ−
ρ+
ρ−
ρ+
ρ0
ρ0
with the coupling constant g = MV /2 fπ and fπ ≈ 92 MeV
[5] the weak pion decay constant. The vector field Vμ is
Vμ =
√1 ρ0
2
ρ−
ρ+
− √12 ρ0
.
From the Lagrangian in Eq. (
1
) one can straightforwardly
derive the interaction between three and four vector mesons,
with the corresponding Lagrangians denoted as L3 and L4,
respectively. The former one gives rise to ρρ interactions
through the exchange of a ρ meson and the latter corresponds
to purely contact interactions. We did not include the ω
resonance in Eq. (
3
), since it does not contribute to the interaction
part (in the isospin limit).
Consider first the contact vertices for the 4ρ interaction.
These can be derived from Eq. (
2
) by keeping the terms
proportional to g2, leading to
g2
L4 = 2
VμVν V μV ν − VμV μVν V ν .
The three different isospin (I ) amplitudes for ρρ scattering
(I = 0, 1 and 2) can be worked out from the knowledge
of the transitions ρ+( p1)ρ−( p2) → ρ+( p3)ρ−( p4) and
ρ+( p1)ρ−( p2) → ρ0( p3)ρ0( p4) by invoking crossing as
well. We have indicated the different four-momenta by pi ,
i = 1, . . . , 4. The scattering amplitude for the former
transition is denoted by A( p1, p2, p3, p4) and the latter one by
B( p1, p2, p3, p4), which are shown in Figs. 1 and 2,
respectively.
(
3
)
(
4
)
ρ0
ρ+
ρ−
ρ+
ρ−
ρ0
ρ+
ρ−
The contributions to those amplitudes from L4, cf. Eq. (
4
),
are indicated by the subscript c and are given by
Ac(k1, k2, k3, k4) = −2g2(2 (
1
)μ (
2
)ν (
3
)ν (
4
)μ
Bc(k1, k2, k3, k4) = 2g2(2 (
1
)μ (
2
)μ (
3
)ν (
4
)ν
In this equation, the (i )μ corresponds to the polarization
vector of the i th ρ. Each polarization vector is characterized
by its three-momentum pi and third component of the spin σi
in its rest frame, so that (i )μ ≡ (pi , σi )μ. Explicit
expressions of these polarization vectors are given in Eqs. (A.9) and
(A.10) of Appendix A. In the following, so as to simplify the
presentation, the tree-level scattering amplitudes are written
for real polarization vectors. The same expressions are valid
for complex ones by taking the complex conjugate of the
polarization vectors attached to the final particles.1
Considering the one-vector exchange terms, we need the
three-vector interaction Lagrangian L3. It reads
L3 =i g (∂μVν − ∂ν Vμ)V μV ν .
The basic vertex is depicted in Fig. 3 which after a simple
calculation can be written as
√
V3 = −
2g (qμ (
1
)ν − qν (
1
)μ) (
3
)μ (
2
)ν
− (kμ (
2
)ν − kν (
2
)μ) (
1
)μ (
3
)ν
− ( pμ (
3
)ν − pν (
3
)μ) (
2
)μ (
1
)ν .
In terms of this vertex, one can straightforwardly calculate
the vector exchange diagrams in Figs. 1 and 2. The expression
for the t -channel ρ-exchange amplitude, the middle diagram
1 The polarization vectors (p, σ ) in Appendix A are complex, so that
the polarization vectors associated with the final-state ρρ should be
complex conjugated in this case.
(
5
)
(
6
)
(
7
)
As ( p1, p2, p3, p4; 1, 2, 3, 4)
= At ( p1, − p3, − p2, p4; 1, 3, 2, 4).
The total amplitudes for ρ+ρ− → ρ+ρ− and ρ+ρ− →
ρ0ρ0 are
with the usual arguments ( p1, p2, p3, p4; 1, 2, 3, 4). By
crossing we also obtain the amplitude for ρ+ρ+ → ρ+ρ+
[that we denote C ( p1, p2, p3, p4; 1, 2, 3, 4)] from the
one for ρ+ρ− → ρ+ρ− by exchanging p2 ↔ − p4 and
2 ↔ 4, that is,
C ( p1, p2, p3, p4; 1, 2, 3, 4)
= A( p1, − p4, p3, − p2; 1, 4, 3, 2).
The amplitude C is purely I = 2, which we denote T (
2
). The
amplitude B is an admixture of the I = 0, T (0), and I = 2
amplitudes,
(k, σ2)
A = Ac + At + As ,
B = Bc + At + Au ,
(
10
)
(
11
)
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(p, σ3)
(q, σ1)
Fig. 3 Three-ρ vertex from L3
in Fig. 1, and denoted by At ( p1, p2, p3, p4; 1, 2, 3, 4),
is
At ( p1, p2, p3, p4; 1, 2, 3, 4)
2g2
= ( p1 − p3)2 − m2ρ + i 0+
× ( p1( p2 + p4) + p3( p2 + p4)) 1 · 3 2 · 4
+ 4( 1 · k3 4 · k2 2 · 3 + 1 · k3 2 · k4 3 · 4
+ 3 · k1 4 · k2 1 · 2 + 2 · k4 3 · k1 1 · 4)
where for brevity, we have rewritten (i ) → i , and
the scalar products involving polarization vectors are
indicated with a dot. The u-channel ρ-exchange amplitude
Au ( p1, p2, p3, p4; 1, 2, 3, 4) can be obtained from the
expression of At by exchanging p3 ↔ p4 and 3 ↔ 4. In
the exchange for the polarization vectors they always refer
to the same arguments of three-momentum and spin, that is,
(p3, σ3) ↔ (p4, σ4). In this way,
Au ( p1, p2, p3, p4; 1, 2, 3, 4)
= At ( p1, p2, p4, p3; 1, 2, 4, 3).
Notice that the second diagram in Fig. 2 is a sum of the
t -channel and u-channel ρ-exchange diagrams.
The s-channel exchange amplitude (the last diagram in
Fig. 1) can also be obtained from At by performing the
exchange p2 ↔ − p3 and 2 ↔ 3, with the same remark as
above for the exchange of polarization vectors. We then have
from which we find that
T (0) = 3B + C.
To isolate the I = 1 amplitude, T (
1
), we take the ρ+ρ−
elastic amplitude A, which obeys the following isospin
decomposition:
1 T (
2
)
A = 6
1 T (
1
)
+ 2
1 T (0).
+ 3
Taking into account Eqs. (
14
) we conclude that
T (
1
) = 2 A − 2B − C.
In terms of these amplitudes with well-defined isospin, the
expression in Eq. (A.48) for calculating the partial-wave
amplitudes in the S J I basis (states with well-defined total
angular momentum J , total spin S, orbital angular
momentum and isospin I ), denoted as T (SJ;I¯)S¯ (s) for the transition
( ¯S¯ J I ) → ( S J I ), simplifies to
Y 0¯(zˆ)
T (SJ;I¯)S¯ (s) = 2(2 J + 1) σ1,σ2,σ¯1
σ¯2,m
d pˆ Y m (p )∗(σ1σ2 M|s1s2 S)
× (m M M¯ | S J )(σ¯1σ¯2 M¯ |s¯1s¯2 S¯)(0M¯ M¯ | ¯S¯ J )
× T (I )( p1, p2, p3, p4; 1, 2, 3, 4),
(
17
)
with s the usual Mandelstam variable, p1 = |p|zˆ, p2 =
−|p|zˆ, p3 = p and p4 = −p , M = σ1 + σ2 and
M¯ = σ¯1 + σ¯2
The Mandelstam variables t and u for ρρ scattering in
the isospin limit are given by t = −2p2(1 − cos θ ) and
u = −2p2(1 + cos θ ), with θ the polar angle of the final
momentum. The denominator in At due to the ρ propagator,
cf. Eq. (
8
), vanishes for t = m2ρ and similarly the
denominator in Au for u = m2 . When performing the angular
proρ
jection in Eq. (
17
) these poles give rise to a left-hand cut
starting at the branch point s = 3m2 . This can easily be
ρ
seen by considering the integration on cos θ of the fraction
1/(t − m2ρ + i ε), which gives the same result both for the t
and the u channel exchange,
Here, G(s) is a diagonal matrix made up by the two-point
loop function g(s) with ρρ as intermediate states,
g(s) → i
d4q 1
(2π )4 (q2 − m2ρ )(( P − q)2 − m2 ) ,
ρ
where P2 = s and within our normalization, cf. Eq. (A.53),
Im g(s) = −|p|/8π √s. The loop function is logarithmically
divergent and it can be calculated once its value at a given
reference point is subtracted. In this way, one can write down
a once-subtracted dispersion relation for g(s) whose result
is3
1
g(s) = (4π )2
m2ρ
a(μ) + log μ2 + σ log(σ + 1) − log(σ − 1) ,
,
with ε → 0+. The argument of the log becomes negative
for 4p2 < −m2ρ , which is equivalent to s < 3m2ρ . Because
of the factor p2ε the imaginary part of the argument of the
log below the threshold is negative which implies that the
proper value of the partial-wave amplitude on the physical
axis below the branch point at s = 3m2ρ is reached in the
limit of vanishing negative imaginary part of s. The presence
of this branch point and left-hand cut was not noticed in
Ref. [9], where only the extreme non-relativistic reduction
was considered, so that the ρ propagators in the ρ-exchange
amplitudes collapsed to just a constant.
Once we have calculated the partial-wave projected
treelevel amplitude we proceed to its unitarization making use
of standard techniques within unitary chiral perturbation
theory [2,8,40]. This is a resummation technique that restores
unitarity and also allows one to study the resonance region.
It has been applied to many systems and resonances by now,
e.g. in meson–meson, meson–baryon, nucleon–nucleon and
W W systems. Among many others we list some pioneering
work for these systems [2,3,8,41–56]. In the last years this
approach has been applied also to systems containing mesons
and baryons made from heavy quarks, some references on
this topic are [57–61].
The basic equation to obtain the final unitarized T matrix
in the subspace of coupled channels S J I , with the same J I ,
is 2
T (J I )(s) = I − V (J I )(s) · G(s)
· V (J I )(s).
−1
(
19
)
(
20
)
(
21
)
(
22
)
(
23
)
(
24
)
2 In order to easier the comparison with Ref. [9] we take the same sign
convention for matrices V (s) and T (s) as in that reference.
3 It is the same result as calculating g(s) in dimensional regularization,
d = 4 + 2 , and replacing the 1/ divergence by a constant; cf. [62].
and μ is a renormalization scale typically taken around mρ ,
such the sum a(μ) + log m2ρ /μ2 is independent of μ. The
subtraction constant in Eq. (
21
) could depend on the quantum
numbers , S and J , but not on I due to the isospin symmetry
[3].
To compare with the results of Ref. [9], we also evaluate
the function g(s) introducing a three-momentum cutoff qmax,
the resulting g(s) function is denoted by gc(s),
0
qmax
dq
q2
w(s − 4w2 + i ε)
q2 + m2ρ . This integral can be done algebraically
1
gc(s) = 2π 2
with w =
[62]
1 ⎛ ⎡ ⎛ m2ρ ⎞
gc(s) = (4π )2 ⎝ σ ⎣ log ⎝ σ 1 + qm2ax + 1⎠
⎛
m2
− log ⎝ σ 1 + qm2ρax − 1⎠ ⎦
⎞ ⎤
⎧ ⎛
+2 log ⎨ qmmρax ⎝ 1 +
⎩
m2 ⎞ ⎫ ⎞
ρ ⎬
1 + qm2ax ⎠ ⎭ ⎠ .
Typical values of the cutoff are around 1 GeV. The unitarity
loop function g(s) has a branch point at the ρρ threshold
(s = 4m2ρ ) and a unitarity cut above it (s > 4m2ρ ). The
physical values of the T -matrix T (J I )(s), with s > 4m2ρ , are
reached in the limit of vanishing positive imaginary part of s.
Notice that the left-hand cut present in V (J I )(s) for s < 3m2ρ
does not overlap with the unitarity cut, so that V (J I )(s) is
analytic in the complex s-plane around the physical s-axis
for physical energies. In this way, the sign of the vanishing
imaginary part of s for V (J I )(s) is of no relevance in the
prescription stated above for reaching its value on the real
axis with s < 3m2ρ according to the Feynman rules.
We can also get a natural value [2] for the subtraction
constant a in Eq. (
21
) by matching g(s) and gc(s) at
threshold where σ = 0. For μ = mρ , a usual choice, the final
expression simplifies to
qmax
a = −2 log mρ ⎝ 1 +
By varying qmax in this equation in the range of typical
values around 1 GeV (avoiding the explicit resolution of
hadrons into its quark–gluon constituents for shorter wave
lengths) one studies the generation of states within the
asymptotic hadronic degrees of freedom. In this way one
circumvents the fine tuning of subtraction constants, a
process that is associated with pre-existing (quark–gluon) states
[63].
It is also worth noticing that Eq. (
19
) gives rise to a T
matrix T (I J )(s) that is gauge invariant in the hidden local
symmetry theory because this equation just stems from the
partial-wave projection of a complete on-shell tree-level
calculation within that theory, which certainly is gauge
invariant.
3 Results
One of our aims is to check the stability of the results of
Ref. [9] under relativistic corrections, particularly
regarding the generation of the poles that could be associated with
the f0(1370) and f2(1270) resonances as obtained in that
paper. The main source of difference between our
calculated V (J I )(s) and those in Ref. [9] arises from the
different treatment of the ρ-meson propagator. The point is
that the authors of Ref. [9] take the non-relativistic limit
of this propagator so that from the expression 1/(t − m2ρ ),
cf. Eq. (
8
), or 1/(u − m2ρ ), only −1/m2ρ is kept. This is the
reason that the tree-level amplitudes calculated in Ref. [9]
do not have the branch-point singularity at s = 3m2ρ nor
the corresponding left-hand cut for s < 3m2ρ . It turns out
that, for the isoscalar tensor case, the resonance f2(1270)
is below this branch point, so that its influence cannot be
neglected when considering the generation of this pole within
this approach.
3.1 Uncoupled S-wave scattering
The issue on the relevance of this branch-point singularity
in the ρ-exchange amplitudes was not addressed in Ref. [9]
and it is indeed very important. This is illustrated in Fig. 4
where we plot the potentials V (J I )(s) in S-wave ( = 0)
(only S-wave scattering is considered in Ref. [9]).4 From
top to bottom and left to right we show in the figure the
potentials for the quantum numbers ( J, I ) equal to (0, 0),
(
2, 0
), (
0, 2
), (
2, 2
) and (
1, 1
). The red solid and black dotted
lines correspond to the real and imaginary parts of our full
covariant calculation of the V (J I )(s), respectively, while the
blue dashed ones are the results of Ref. [9]. The imaginary
part in our results for V (J I )(s) appears below s < 3m2ρ due
to the left-hand cut that arises from the t - and u-channel
ρexchanges.
It can be seen that our results and those of Ref. [9] are
typically similar near threshold (s = 4m2 ) but for lower
ρ
values of s they typically depart quickly due to the onset of
the branch-point singularity at s = 3m2 . The strength of
ρ
this singularity depends on the channel, being particularly
noticeable in the ( J, I ) = (
2, 0
) channel, while for the (0, 0)
channel it is comparatively weaker.
The strongest attractive potentials in the near-threshold
region occur for ( J, I ) = (0, 0) and (
2, 0
) and in every of
these channels Ref. [9] found a bound-state pole that the
authors associated with the f0(1370) and f2(1272)
resonances, respectively. For the (0, 0) quantum numbers the
pole position is relatively close to the ρρ threshold, while
for (
2, 0
) it is much further away. Two typical values of
the cutoff qmax were used in Ref. [9], qmax = 875 and
1000 MeV. We employ these values here, too, together with
qmax = mρ (so that we consider three values of qmax
separated by around 100 MeV), and study the pole positions for
our T (J I )(s) amplitudes in S wave. We only find a bound
state for the isoscalar scalar case, while for the tensor case
no bound state is found. In Table 1 we give the values of the
pole positions for our full calculation for qmax = mρ (first),
875 (second) and 1000 MeV (third row). For comparison we
also give in round brackets the bound-state masses obtained
in Ref. [9], when appropriate. As indicated above, the strong
differences for V (
20
)(s) between our full covariant
calculation and the one in Ref. [9] in the extreme non-relativistic
limit, cf. Fig. 4b, imply the final disappearance of the deep
bound state for the isoscalar tensor case. The nominal
threemomentum of a ρ around the mass of the f2(1270) has a
modulus of about 0.6mρ 460 MeV and for such high
values of three-momentum relativistic corrections are of
importance, as explicitly calculated here. On the contrary, the (0, 0)
pole is located closer to the ρρ threshold and the results are
4 Partial waves with = 0 are considered in Sect. 3.2.
more stable against relativistic corrections, though one still
finds differences of around 20 MeV in the bound-state mass.
In addition we also show in the third column of Table 1
the residue of T (00)(s) at the pole position sP . For a generic
partial wave T (J I ) (s), its residue at a pole is denoted by
S; ¯S
γ (SJ I )γ (J I ) and is defined as
¯S¯
γ (SJ I )γ ¯(S¯J I ) = − sl→imsP (s − sP )T (SJ;I¯)S¯ (s).
In terms of these couplings one can also calculate the
compositeness X (JS I ) associated with this bound state [64–66],
sP
,
which in our case determines the ρρ component in such
bound state. Notice that the derivative of g(s) from Eq. (
21
)
(which is negative below threshold) does not depend on the
subtraction constant; the dependence on the latter enters
implicitly by the actual value of the pole position sP . Of
course, if one uses a three-momentum cutoff then gc(s) must
be employed in the evaluation of X (JS I ). The compositeness
obtained for the pole positions in Table 1 is given in the fourth
column of the same table. As expected the ρρ component is
dominant, with X 0(000) > 0.5, and increases as the pole moves
closer to threshold, so that it is 73% for qmax = mρ and
√sP = 1516 MeV.
We can also determine the pole positions when g(s) is
calculated with exact analytical properties, Eq. (
21
), and
taking for a the values from Eq. (
25
) as a function of qmax. The
results are given Table 2, where we also give the residue
at the pole position and the calculated compositeness, in
the same order as in Table. 1. The results obtained are
quite close to those in this table so that we refrain of
further commenting on them. Nonetheless, we should stress
again that we do not find any pole for the isoscalar tensor
case.
We could try to enforce the generation of an isoscalar
tensor pole by varying qmax, when using gc(s), or by varying
a, if Eq. (
21
) is used. In the former case a much lower value
of qmax is required than the chiral expansion scale around
1 GeV (qmax 400 MeV), while for the latter a qualitatively
similar situation arises when taking into account the
relationship between a and qmax of Eq. (
25
). Even more serious are
two facts that happen in relation with this isoscalar tensor
pole. First, one should stress that such pole appears
associated to the evolution with qmax or a of a pole in the first
Riemann sheet, which violates analyticity. This is shown
in Fig. 5 where we exhibit the evolution of this pole as
a function of qmax. We start the series at a low value of
qmax = 300 MeV, where we have two poles on the real axis,
and increase the cutoff in steps of δqmax = 50 MeV. These
two poles get closer and merge for qmax = 403.1 MeV. For
larger values of the cutoff the resulting pole moves deeper
into the complex plane of the physical or first Riemann sheet.
Second, we find that X 0(220) is larger than 1. For example,
for qmax = 400 MeV, there are two poles at 1422.4 and
1463.4 MeV with X 0(220) = 2.7 and 3.8, in order, which of
course makes no sense as compositeness factors have to be
less or equal to one.
Next, we take into account the finite width of the ρ meson
in the evaluation of the unitarity two-point loop function g(s).
As a result the peak in the modulus squared of the isoscalar
scalar amplitude now acquires some width due to the width
itself of the ρ meson. To take that into account this effect we
convolute the g(s) function with a Lorentzian mass squared
distribution for each of the two ρ mesons in the
intermediate state [9,45,46]. The resulting unitarity loop function is
denoted by g(s) and is given by
1
g(s) = N 2
with (m) the width of the ρ meson with mass m. Due
to the P-wave nature of this decay to π π , we take into
account its strong cubic dependence on the decaying pion
three-momentum and use the approximation
with mπ the pion mass and ρ =∼ 148 MeV [5]. The function
g(s, m2, m22) is the two-point loop function with different
1
masses, while in Eq. (
21
) we give its expression for the equal
mass case. When evaluated in terms of a dispersion relation
it reads
1
g(s, m12, m22) = 16π 2
a(μ) + log
to the case without convolution. The resulting peak positions
are given in the last three rows of Tables 1 and 2. The effects
of the non-zero ρ width are clearly seen in Fig. 6, where
we plot |T (00)(s)|2 for the different values of qmax shown
in Table 1. The shape of the peaks follows quite closely a
Breit–Wigner form, though it is slightly wider to the right
side of the peak. We find that the width decreases with the
increasing value of qmax, being around 45, 65 and 95 MeV for
qmax = 1000, 875 and 775 MeV, respectively, of similar size
as those found in Ref. [9]. When using a subtraction constant
instead of qmax, relating them through Eq. (
25
), the picture
is quite similar. The peak positions are given in the last three
columns of Table 2 while the widths obtained are around 105,
70 and 50 MeV for a = −1.70, −1.94 and −2.14, in order.
These widths are significantly smaller than the PDG values
assigned to the f0(1370) resonance of 200–500 MeV [5].
Due to the coupling of the ρρ and π π , this pole could
develop a larger width. This is approximated in Ref. [9]
by considering the imaginary part of the π π box diagram,
with a ρ → π π vertex at each of the vertices of the box.
These vertices are also worked out from the non-linear chiral
Lagrangian with hidden gauge symmetry [38,39]. We refer
to Ref. [9] for details on the calculation of this contribution.
According to this reference one has to add to V0(00;00)0 and to
V0(22;00)2 the contribution V2(πJ I ), given by
with λ1/2(s) = s2 + m41 + m24 − 2sm21 − 2sm22 − 2m12m22.
The algebraic expression of this function when calculated
with a three-momentum cutoff for different masses can be
found in Ref. [62], to which we refer the interested reader.
When using the convoluted g(s) function we find similar
masses for the peak of |T (00)|2 in the (0, 0) channel compared
In the calculation of the function V!ππ , Ref. [9] introduces a
monopole form factor F (q) for each of the four ρ → π π
vertices in the pion box calculation,
F (q) =
2 − m2π
2 − (k − q)2
log λ1/2(s) + s − m22 + m21
,
(
31
)
V2(π00) = 20i ImV!ππ ,
V2(π20) = 8i ImV!ππ .
(
32
)
(
33
)
with k0 = √s/2, k = 0, q0 = √s/2 and q the
integration variables. This introduces a sizable dependence of the
results on the value of . Nonetheless, in order to compare
with Ref. [9] we follow the very same scheme of
calculation and take the same values for , that is, 1200, 1300 and
1400 MeV.5
The inclusion of the π π box diagram, on top of the
convolution with the ρ mass squared distribution for
calculating the g(s) function, does not alter the previous
conclusion on the absence of a pole in the isoscalar tensor channel.
However, the isoscalar scalar pole develops a larger width
around 200–300 MeV, which increases with , as can be
inferred from Fig. 7, where we plot |T (00)(s)|2. On the other
hand, the position of the peak barely changes compared to
the one given in the last two rows of Table 1. Tentatively
this pole could be associated to the f0(1370) resonance,
which according to Refs. [35,67] decays mostly to π π with a
width around 200 MeV. In the PDG [5] the total width of the
f0(1370) is given with a large uncertainty, within the range
200–500 MeV and the π π decay mode is qualified as
dominant. The nearby f0(1500) resonance has a much smaller
width, around 100 MeV, and its coupling and decay to π π is
suppressed. These properties of the f0(1500) are discussed
in detail in Ref. [35].
3.2 Coupled-channel scattering
We now consider the impact on our results when allowing for
the coupling between ρρ channels with different orbital
angular momenta, an issue not considered in Ref. [9]. In Table 3
we show the different channels that couple for given J I
quantum numbers and pay special attention to the ( J, I ) = (0, 0)
5 Another more complete scheme is two work explicitly with
coupledchannel scattering as done in Ref. [35], where ρρ and π π channels,
among many others, were explicitly included. In this way resonances
develop decay widths in a full non-perturbative fashion because of the
coupling between channels.
Table 3 Coupled channels with
different orbital angular
momentum
(J, I )
(0, 0)
(
2, 0
)
( , S) channels
(0, 0), (
2, 2
)
(0, 2), (
2, 0
), (
2, 2
)
and (2, 0) channels. Apart from the conservation of J and I ,
one also has to impose invariance under parity, which avoids
the mixing between odd and even ’s.
When including coupled-channel effects, one finds two
poles in the channels with ( J, I ) = (0, 0), which are reported
in Table 4 for various values of qmax (shown in the first
column). We give from left to right the pole mass (second
column), the residues (third and fourth ones) and compositeness
coefficients (fifth and sixth ones) of the different channels,
( , S) = (0, 0) and (
2, 2
), respectively. One of the poles is
heavier and closer to the ρρ threshold with similar properties
as the pole in the uncoupled case, compare with Table 1,
particularly for qmax = 775 MeV. Nonetheless, as qmax increases
the difference of the properties of this pole between the
coupled and uncoupled cases is more pronounced. In
particular let us remark that now X 0(000) is always 0.7 and for
qmax = 1 GeV the residue to the channel ( , S) = (0, 0) is
much larger than in the uncoupled case. Additionally, we find
now a lighter pole which lies above the branch-point
singularity at √3 mρ 1343 MeV. For lower values of the cutoff
qmax this pole couples more strongly to the ( , S) = (
2, 2
)
channel, but as the cutoff increases its residue for the channel
( , S) = (0, 0) also increases in absolute value and it is the
largest for qmax = 1 GeV. It is then clear that both channels
( , S) = (0, 0) and (
2, 2
) are relevant for the origin of this
pole. Note that the residues for this lighter pole are
negative, which is at odds with the standard interpretation of the
residue (γ (S00))2 of a bound state as the coupling squared.
This implies that the compositeness coefficients X (0S0) are all
negative, which is at odds with a probabilistic interpretation
as suggested in Refs. [64–66] for bound states. The moduli of
775
875
1000
Table 4 Bound-state poles in the partial-wave amplitudes of
quantum numbers ( J, I ) = (0, 0) with varying cutoff qmax, which is
indicated in the first column. The masses (second column), the residues to
( , S) = (0, 0) and (
2, 2
) (third and fourth columns) and the
compositeness coefficients X0000) and X2020) (sixth and seventh columns) are also
given. For the lighter poles the compositeness coefficients are small and
negative, so that they cannot be interpreted as physical states contrary
to common wisdom [64–66]
57.2
Table 5 Bound-state poles in the partial-wave amplitudes of
quantum numbers ( J, I ) = (
2, 0
) with different cutoffs qmax, indicated
in the first column. The masses (second column) and the residues to
( , S) = (
0, 2
), (
2, 0
) and (
2, 2
) (third, fourth and fifth columns) are
shown. Here, “−0.0” denotes a small but negative number
the | X (0S0)| are all small because this lighter pole lies quite far
from the ρρ threshold. The fact that its mass is not far from
the strong branch-point singularity at √3 mρ implies that this
pole is very much affected by the left-hand cut
discontinuity. In this respect, it might well be that the presence of this
pole with anomalous properties is just an artifact of the
unitarization formula of Eq. (19), which treats the left-hand cut
discontinuity of the potential perturbatively. One can answer
this question by solving exactly the N / D method [68–72], so
that the left-hand cut discontinuity of the potential is properly
treated and the resulting amplitude has the right analytical
properties. Let us recall that Eq. (
19
) is an approximate
algebraic solution of the N / D method by treating perturbatively
the left-hand cut discontinuities of the coupled partial waves
[40, 49, 52, 53]. For the uncoupled scattering such effects are
further studied in detail in the next section.
For the ( J, I ) = (
2, 0
) partial waves we have three
coupled channels, ( , S) = (
0, 2
), (
2, 0
) and (
2, 2
) and,
contrary to the uncoupled case, we now find a pole that lies
above the branch-point singularity. We give its mass and
residues for different qmax in Table 5, with the same
notation as in Table 4. Notice that these pole properties are very
stable under the variation of qmax. This pole couples by far
much more strongly to the channel with ( , S) = (
2, 2
) than
to any other channel. This indicates that it is mainly due to
the dynamics associated with the ( , S) = (
2, 2
) channel.
But the same comments are in order here as given above for
the lighter isoscalar scalar pole, because its residues shown in
Table 5 are negative and so are the corresponding
compositeness coefficients. Hence, the lighter pole for ( J, I ) = (0, 0)
and the one found for (
2, 2
) cannot be considered as robust
results of our analysis. This has to be contrasted to the case
of the heavier isoscalar scalar pole that is stable under
relativistic corrections, coupled-channel effects and has quite
standard properties regarding its couplings and
compositeness coefficients.
4 First-iterated solution of the N/ D method
In this section for definiteness we only consider
uncoupled scattering. We have in mind the ( J, I ) = (0, 0) and
( J, I ) = (
2, 0
) quantum numbers to which special
attention has been paid in the literature concerning the
generation of poles that could be associated with the f0(1370) and
f2(1270) resonances, as discussed above. Further
applications of the improved unitarization formalism presented in
this section are left for future work.
According to the N / D method [73] a partial-wave
amplitude can be written as
T =
N (s)
D(s)
,
D(si ) = 0.
where the function D(s) has only the unitarity or right-hand
cut (RHC) while N (s) only has the left-hand cut (LHC). The
secular equation for obtaining resonances and bound states
corresponds to look for the zeros of D(s),
(
34
)
(
35
)
Below threshold along the real s axis this equation is purely
real because D(s) has a non-vanishing imaginary part only
for s > sth.
However, with our unitarization procedure from
leadingorder unitary chiral perturbation theory (UChPT)6 we have
obtained the approximation
V (J I )(s)
T (J I )(s) = 1 − V (J I )(s)gc(s) ,
and the resulting equation to look for the bound states is
DU (s) = 1 − V (s)gc(s) = 0.
Notice that Eq. (
36
), contrary to the general Eq. (
35
), has
an imaginary part below the branch-point singularity at s =
3m2. Nonetheless, if the LHC were perturbative this
imaginary part would not have prevented having a strong peak in
the amplitude if there were already a bound state below the
branch-point singularity when the LHC is neglected.
However, this is by far not the case for the pole associated with the
f2(1270) in Ref. [9], which clearly indicates the instability of
this approach under the inclusion of relativistic corrections.
We can go beyond this situation by considering the
firstiterated solution to the N /D method. This is indeed similar
to Eq. (
36
) but improving upon it because it allows us to go
beyond the on-shell factorization employed in this equation.
In the first-iterated N /D solution one identifies the
numerator function N (s) to the tree-level calculation V (J I )(s) and
employs the exact dispersive expression for D(s). Namely,
it reads7
N (s) = V (J I )(s),
D(s) = γ0 + γ1(s − sth ) + 21 γ2(s − ssth )2
+
(s − sth)s2
π
∞
sth
ds
ρ(s )V (J I )(s )
(s − sth)(s − s)(s )2
,
(
36
)
(
37
)
(
38
)
N (s)
TN D(s) = D(s) ,
with the phase space factor ρ(s) given by ρ(s) = σ (s)/16π .
We have taken three subtractions in the dispersion relation
for D(s) because V (J I )(s) diverges as s2 for s → ∞.
From our present study we have concluded that T (J I )(s) is
stable in the threshold region under relativistic corrections as
6 In spite of the fact that the input for the unitarization procedure comes
from the hidden local symmetry Lagrangians and not from chiral
perturbation theory, we keep the standard notation of UCHPT to refer to
the unitarization procedure employed and its degree of sophistication
[2,40,49].
7 A comprehensive introduction to the N /D method is given in
Refs. [40,70–72,74–76].
1.5
1
0.5
()s 0
D
-0.5
-1
-1.51
well as under the addition of coupled ρρ higher partial waves.
Because of the stability of the results in this region under
relativistic corrections and by visual inspection of the potentials
in Fig. 4 one concludes that the near-threshold region is quite
safe of the problem related to the branch-point singularity of
the LHC associated with one-ρ crossed-channel exchanges.
We then determine the subtractions constants, γ0, γ1 and γ2 in
D(s) by matching TN D(s), Eq. (
38
), and T (J I )(s), Eq. (
36
),
around threshold (sth). At the practical level it is more
convenient to match 1/ T (s), so that in the threshold region up
to O(s3) one has
γ0 + γ1(s − sth ) + 21 γ2(s − ssth )2
= 1 − V (s)gc(s) −
(s − sth)s2
π
ρ(s )V (s )
(s − sth)(s − s)(s )2
×
sth
≡ ω(s).
∞
ds
In this way,
γ0 = 1 − V (sth)gc(sth),
γ1 = ω (sth),
γ2 = ω (sth).
(
39
)
(
40
)
The dependence of our present results on the cutoff used
in T (J I )(s) stems from the matching conditions of Eq. (
40
).
However, let us stress that the analytical properties of D(s)
and N (s) are correct, they have the RHC and LHC with
the appropriate extent and branch-point singularities,
respectively, and the resulting amplitude is unitarized.
(J,I)=(0,0) D(s)
qmax=0.7 GeV
qmax=1 GeV
qmax=1.3 GeV
0
1.4
s (GeV)
1.1
1.2
1.3
1.5
1.6
1.7
1.8
Fig. 8 D(s) function, Eq. (
38
), for (J, I ) = (0, 0). Above threshold
only the real part is shown
(J,I)=(0,0) ReD(s)
ReD, qmax=1 GeV
ReDU, qmax=1 GeV
0
1.1
1.2
1.3
1.5
1
))0.5
s
(
D
(
e
R 0
We plot D(s) for ( J, I ) = (0, 0) in Fig. 8 for qmax = 0.7,
1 and 1.3 GeV by the red solid, green dashed and blue
dashdotted lines, in order. The crossing with the zero line
(dotted one) indicates the mass of the bound state. This mass
decreases with increasing qmax, being around 1.4 GeV for
the largest cutoff and very close to threshold for the
smallest. In Fig. 9 we compare the real (left) and imaginary parts
(right panel) of D(s) and DU (s) = 1 − V (s)gc(s) for a
cutoff of 1 GeV. We do not show more values of the
cutoff because the same behavior results. The function D(s)
and DU (s) match up to around the branch-point
singularity at √s = 3m2ρ . Below it DU (s) becomes imaginary, cf.
Eq. (
37
), while D(s) remains real and has this right property
by construction, cf. Eq. (
38
). Above threshold the imaginary
parts of both functions coincide as demanded by unitarity.
We see that for these quantum numbers our new improved
unitarization formalism and the one used to derive Eq. (
19
)
agree very well. The bound-state mass remains the same
as given in Table 1 because the functions D(s) and DU (s)
match perfectly well in the region where these poles occur,
as is clear from Fig. 9. This should be expected because for
( J, I ) = (0, 0) the branch-point singularity is much weaker
than for other cases, e.g. ( J, I ) = (
2, 0
), as discussed above.
4.2 Results J = 2, I = 0
We plot D(s) for ( J, I ) = (
2, 0
) in Fig. 10 for qmax = 0.7,
1 and 1.3 GeV by the red solid, green dashed and blue
dashdotted lines, respectively. One can see that in this region the
function D(s) is large and negative so that there is by far no
pole in the f2(1270) region. In order to show the curves more
clearly we use only qmax = 1 GeV in Fig. 11, for other values
(J,I)=(0,0) ImD(s)
ImD, qmax=1 GeV
ImDU, qmax=1 GeV
0
s (GeV)
(J,I)=(
2,0
) D(s)
1.4
s (GeV)
Fig. 10 D(s) function, Eq. (
38
), for (J, I ) = (
2, 0
). Above threshold
only the real part is shown
of qmax the behavior is the same. In the left panel we
compare the real parts of TN D(s) (black solid) and T (
20
)(s) (red
dashed) while in the right panel we proceed similarly for the
real parts of D(s) and DU (s) = 1 − V (
20
)(s)gc(s), with the
same type of lines in order. All the functions match near the
threshold region and above it, but they strongly depart once
we approach the LHC branch-point singularity at s = 3m2
ρ
and beyond (for smaller values of s). Notice that D(s), which
has no such branch-point singularity, follows then the smooth
decreasing trend already originating for 3m2 < s < 4m2. For
( J, I ) = (0, 0) the corresponding smooth trend is that of a
decreasing function; cf. Fig. 8. The branch-point
singularity is clearly seen in T (
20
)(s) because it is proportional to
V (
20
)(s).
In summary, the conclusions obtained in Sect. 3.1
regarding the generation of the pole that could be associated with
1.4
s (GeV)
1.4
s (GeV)
1.1
1.2
1.3
1.5
1.6
1.7
1.8
1.1
1.2
1.3
1.5
1.6
1.7
1.8
(J,I)=(
2,0
) ReD(s)
D, Q=1 GeV
DU, Q=1 GeV
2
0
-2
-4
)) -6
s
(D -8
(
eR-10
-12
-14
-16
600
the f0(1370) and the absence of that associated with the
f2(1270) as claimed in Ref. [9] fully hold. As a matter of
fact, they get reinforced after considering the more
elaborated unitarization process that is obtained here by taking the
first-iterated N /D solution.
5 Summary and conclusions
In this paper, we have revisited the issue of resonance
generation in unitarized ρρ scattering using a chiral covariant
formalism. The main results of our study can be summarized
as follows:
(i) We have developed a partial-wave projection formalism
that is applicable to the covariant treatment of ρρ
scattering. In particular, we point out that accounting for
the full ρ-meson propagator leads to a branch point in
the partial-wave projected amplitudes at s = 3m2ρ
1.8 GeV2, about 208 MeV below the 2ρ threshold.
This branch point does not appear in the extreme
nonrelativistic treatment of the propagator.
(ii) Evaluating the T -matrix using the standard form, see
Eq. (
19
), which treats the left-cut perturbatively, we
find a pole in the scalar isoscalar channel close to the ρρ
threshold that can be associated with the f0(1370)
resonance, in agreement with the findings of Ref. [9], though
there are relatively small quantitative differences.
(iii) In contrast to Ref. [9], we do not find a tensor
state below the scalar one. This can be traced back
to the influence of the aforementioned branch point.
We therefore conclude that the state that is
identified in Ref. [9] with the f2(1270) is an artifact of
the non-relativistic approximation and its generation
does not hold from the arguments given in that
reference.
(iv) We have also worked out the effects of the
coupling between ρρ channels with higher orbital
angular momenta, which soon become strong when one
departs from threshold and lead to additional states.
These, however, have negative composite coefficients
and are thus not amenable to a simple bound-state
interpretation. As these states are close to the branch
point at s = 3m2ρ , the perturbative treatment of
the left-hand cut, as employed in Sect. 3.2, is
certainly not sufficient to decide as regards their
relevance.
(v) We have improved the treatment of the left-hand cut
by employing the first-iterated N /D method, in
particular this method avoids the factorization approach of
leading order UChPT. We worked the solutions that
follow for uncoupled scattering in the ( J, I ) = (0, 0) and
(
2, 0
) channels. The outcome fully agrees with the
conclusions already obtained from UChPT and, notably,
the absence of a pole that could be associated with the
f2(1270) is firmly reinforced.
A lesson from points (iii) and (v) is clear. A strongly
attractive interaction around the threshold of a given channel is by
far not sufficient condition to generate a multi-hadron bound
state. This argument, as used in Ref. [9], is in general terms
too naive because it does not take into account the possible
rise of a singularity in the true potential between the range of
validity of the approximation used and the predicted
boundstate mass from the latter. It could be rephrased as trying to
deduce the values of the function 1/(1 + x ) for x < −1 by
knowing its values for x around 0.
We conclude that the approach presented here should be
used to investigate the possible generation of meson
resonances from the interaction of vector mesons. In the next
steps, we will investigate how the relativistic effects affect
the conclusions of the SU(
3
) calculation of Ref. [10] and will
further sharpen the framework along the lines mentioned, in
particular by solving exactly the N /D equations [70–72]. On
top of this, a realistic treatment of the meson spectroscopy
in the region above 1 GeV definitely requires the inclusion
of more channels than just ρρ (or more generally vector–
vector ones). However, the formalism developed here could
be very valuable to accomplish this aim phenomenologically
if implemented in multi-channel studies, e.g. by extending
the work of Ref. [35].
Acknowledgements We thank Maxim Mai for useful discussions and
for his contribution during the early stages of this investigation. We
would also like to thank E. Oset for some criticism which led us to
add some additional material to the manuscript. This work is supported
in part by the DFG (SFB/TR 110, “Symmetries and the Emergence
of Structure in QCD”). JAO would like to acknowledge partial
financial support from the MINECO (Spain) and ERDF (European
Commission) grants FPA2013-40483-P and FPA2016-77313-P, as well as
from the Spanish Excellence Network on Hadronic Physics
FIS201457026-REDT. The work of UGM was supported in part by The Chinese
Academy of Sciences (CAS) President’s International Fellowship
Initiative (PIFI) Grant No. 2015VMA076.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
A Partial-wave projection formalism
In this appendix we detail the projection formalism used in
this work to calculate the different ρρ partial waves. First, we
give the expression for the polarization vectors for a massive
spin-one particle with a three-momentum p and third
component of spin σ in the z axis of its rest frame, which we
denote by ε(p, σ ). In the rest frame they are given by
Notice that one could also include any arbitrary rotation
around the zˆ axis to the right end of Eq. (A.5). Of course,
this does not have any affect on either Eqs. (A.4) and (A.6)
(for the latter one let us note that Bz (|p|) commutes with a
rotation around the zˆ axis).
The action of U (p) on (0, σi ) gives us the polarization
vectors with definite three-momentum p, whose expressions
are
that takes zˆ to pˆ ,
R( pˆ)zˆ = pˆ.
with E p = $m2 + p2. We also introduce the rotation R( pˆ)
In terms of the polar (θ ) and azimuthal (φ) angles of pˆ, this
rotation is defined as
R( pˆ) = Rz (φ)Ry (θ ),
with the subscripts z and y indicating the axis of rotation. For
later convenience we write the Lorentz transformation U (p)
as
U (p) = R( pˆ)Bz (|p|)R( pˆ)−1,
−β and β = |p|/E p. Namely,
where Bz (|p|) is a boost along the zˆ axis with velocity v =
Bz (|p|) = ⎜⎜
⎝
and
γ =
1
$1 − β2
ε(0, σ ) =
with
" 0 #
εσ
,
⎛ 0 ⎞
ε0 = ⎝ 0
1
⎛ 1 ⎞
∓1
⎠ , ε±1 = √2 ⎝ ±i ⎠ .
0
(A.1)
(A.2)
Next, we take a Lorentz transformation U (p) along the vector
p that takes the particle four-momentum at rest to its final
value,
U (p)
" m #
0
=
" E p #
p
,
ε(p, σ ) =
γ β pˆ · εσ
εσ + pˆ (γ − 1) pˆ · εσ
#
with the corresponding similar expressions for the ρ±(x )
fields. Here a(p, σ ) and a(p, σ )† refer to the annihilation and
creation operators, with the canonical commutation relation
[a(p , σ ), a(p, σ )†] = δσ σ (2π )32E pδ(p − p ).
(A.12)
In order to check the time-reversal and parity-invariance
properties of the vector–vector scattering amplitudes worked
out from the chiral Lagrangians in Eq. (
1
) we notice that the
polarization vectors in Eq. (A.9) satisfy the following
transformation properties:
ε(−p, σ )∗ = (−1)σ (−ε(p, σ )0, ε(p, σ )),
ε(−p, σ ) = (−ε(p, σ )0, ε(p, σ )).
A one-particle state |p, σ is obtained by the action of the
creation operators on the vacuum state,
|p, σ
=a(p, σ )†|0, σ .
From Eq. (A.12) it follows the following normalization for
such states:
p , σ |p, σ
= δσ σ (2π )32E pδ(p − p).
1
4π
Next, we consider a two-body state characterized by the CM
three-momentum p and the third components of spin σ1 and
σ2 in their respective rest frames. This state is denoted by
|p, σ1σ2 . Associated to this, we can define the two-body state
with orbital angular momentum with its third component
of orbital angular momentum m, denoted by | m, σ1σ2 as
| m, σ1σ2 = √
dpˆ Y m ( pˆ)|p, σ1σ2 .
(A.16)
Let us show first that this definition is meaningful because the
state | m, σ1σ2 transforms under the rotation group as the
direct product of the irreducible representations associated
to the orbital angular momentum and the spins s1 and s2 of
the two particles.
Every single-particle state |p, σ under the action of a
rotation R transforms as
× D(s2)(R)σ2σ2 | m , σ1σ2 .
(A.24)
8 For a general Lorentz transformation these manipulations give rise to
the Wigner rotation [76].
(A.10)
R|p, σ
= RU (p)|0, σ
= U (p )U (p )−1 RU (p)|0, σ ,
(A.17)
and p = Rp.8 It is straightforward to show that
R = U (p )−1 RU (p).
For that we explicitly write the Lorentz transformations
U (p ) and U (p) as in Eq. (A.6) so that
U (p )−1 RU (p) = R( pˆ )Bz (|p|)−1 R( pˆ )−1 R R( pˆ)
× Bz (|p|)R( pˆ)−1.
Next, the product of rotations R( pˆ )−1 R R( pˆ) is a rotation
around the z axis, Rz (γ ), since it leaves invariant zˆ. Thus,
R( pˆ )−1 R R( pˆ) = Rz (γ ),
or, in other terms,
R( pˆ ) = R R( pˆ)Rz (γ )−1,
(A.11)
(A.13)
(A.14)
(A.15)
(A.18)
(A.19)
(A.20)
(A.21)
(A.23)
Taking into account Eqs. (A.20) and (A.21) in Eq. (A.19) it
follows the result in Eq. (A.18) because Bz (|p|) and Rz (γ )
commute. Then Eq. (A.17) implies that
R|p, σ
= U (p )R|0, σ =
D(s)(R)σ σ |p , σ , (A.22)
σ
with D(s)(R) the rotation matrix in the irreducible
representation of the rotation group with spin s.
Now, we can use this result to find the action of the rotation
R on the state |p, σ1σ2 , which is the direct product of the
states |p, σ1 and | − p, σ2 (once the trivial CM movement
is factorized out [76]). In this way,
R|p, σ1σ2 =
D(s1)(R)σ1σ1 D(s2)(R)σ2σ2 |p , σ1σ2 .
We are now ready to derive the action R on | m, σ1σ2 ,
R| m, σ1σ2 =
D(s1)(R)σ1σ1 D(s2)(R)σ2σ2
dpˆ Y m (R−1 pˆ )|p , σ1σ2
D( )(R)m m D(s1)(R)σ1σ1
σ1,σ2
=
σ1,σ2
1
× √4π
σ1,σ2,m
In this equation we have made use of the property of the
spherical harmonics
Y m (R−1 pˆ ) =
D( )(R)m m Y m ( pˆ ).
(A.25)
Equation (A.24) shows that under rotation the states defined
in Eq. (A.16) have the right transformation under the action
of a rotation R, and our proposition above is shown to hold.
Now, because of the transformation in Eq. (A.24),
corresponding to the direct product of spins s1, s2 and , we can
combine these angular momentum indices and end with the
L S J basis. In the latter every state is labeled by the total
angular momentum J , the third component of the total
angular momentum μ, orbital angular momentum and total spin
S (resulting from the composition of spins s1 and s2). Namely,
we use the notation | J μ, S for these states which are then
given by
|J μ, S =
(σ1σ2 M|s1s2 S)(m Mμ| S J )| m, σ1σ2 ,
σ1,σ2,m,M
(A.26)
where we have introduced the standard Clebsch–Gordan
coefficients for the composition of two angular momenta.9
Next we introduce the isospin indices α1 and α2
corresponding to the third components of the isospins τ1 and τ2. This
does not modify any of our previous considerations since
isospin does not transform under the action of spatial
rotations. Within the isospin formalism the ρρ states obey Bose–
Einstein statistics and these symmetric states are defined by
|p, σ1σ2, α1α2 S
1
= √2 (|p, σ1σ2, α1α2 + | − p, σ2σ1, α2α1 ) ,
with the subscript S indicating the symmetrized nature of the
state under the exchange of the two particles. One can invert
Eq. (A.16) and give the momentum-defined states in terms
of those with well-defined orbital angular momentum,
|p, σ1σ2, α1α2 =
Y m ( pˆ)∗| m, σ1σ2, α1α2
=
√4π
√4π
,m
J,μ, ,m
S,M,I,t3
Y m ( pˆ)∗
× (σ1σ2 M |s1s2 S)(m M μ| S J )
× (α1α2t3|τ1τ2 I )| J μ, S, I t3 ,
(A.28)
9 The Clebsch–Gordan coefficient (m1m2m3| j1 j2 j3) is the
composition for j1 + j2 = j3, with mi referring to the third components of the
spins.
with I the total isospin of the particle pair and t3 is the third
component. Taking into account this result we can write the
symmetrized states as
|p, σ1σ2, α1α2 S =
√4π
J,μ, ,m
S,M,I,t3
We can also express the state | J μ, S, I t3 in terms of
the states |p, σ1σ2, α1α2 without symmetrization by
inverting Eq. (A.28). We would obtain the same expression as
Eq. (A.31), but with a factor 1/√4π instead of 1/√8π ,
namely
| J μ, S, I t3 = √
dpˆ Y m ( pˆ)(σ1σ2 M |s1s2 S)
× (m M μ| S J )(α1α2t3|τ1τ2 I )|p, σ1σ2, α1α2 .
(A.32)
The extra factor of 1/√2 in Eq. (A.31) is a
symmetrization factor because of the Bose–Einstein symmetry
properties of the two-particle state in the symmetrized states
|p, σ1σ2, α1α2 S, which disappears when employing the
nonsymmetrized states. In order to obtain the normalization of
the states | J μ, S, I t3 it is indeed simpler to use Eq. (A.32)
1
8π σ1,σ2
M,m
α1,α2
1
4π σ1,σ2
M,m
α1,α2
(A.27)
× (m M μ| S J )(α1α2t3|τ1τ2 I )|p, σ1σ2, α1α2 S. (A.31)
though, of course, the same result is obtained if starting from
Eq. (A.31). The two-body particle states with definite
threemomentum satisfy the normalization
invariant under the transformation Tˆ → R( pˆ)Tˆ R( pˆ)†,
which implies at the level of the matrix elements that
S p , σ1σ2, α1α2|Tˆ |p, σ¯1σ¯2, α¯ 1α¯ 2 S
The total energy conservation guarantees that the modulus
of the final and initial three-momentum in Eq. (A.34) is
the same, which we denote by |p|. In terms of this result
and Eq. (A.32) it follows straightforwardly by taking into
account the orthogonal properties of Clebsch–Gordan
coefficients and spherical harmonics that
We are interested in the partial-wave amplitude
corresponding to the transition between states with quantum numbers
J ¯S¯ I to states J S I , which corresponds to the matrix
element
T (J I )
S; ¯S¯ = J μ, S, I t3|Tˆ | J μ, ¯S¯, I t3 ,
with Tˆ the T -matrix scattering operator. Here we take the
convention that the quantum numbers referring to the initial
state are barred. Of course, the matrix element in Eq. (A.35) is
independent of μ and t3 because of invariance under rotations
in ordinary and isospin spaces, respectively. We can calculate
this scattering matrix element in terms of those in the basis
with definite three-momentum by replacing in Eq. (A.35) the
states in the J S basis as given in Eq. (A.31). We then obtain
in a first step
Here we have not shown the explicit indices over which the
sum is done in order not to overload the notation.10 We use
next the rotation invariance of the T -matrix operator Tˆ to
simplify the previous integral so that, at the end, we have just
the integration over the final three-momentum angular solid.
There are several steps involved that we give in quite detail.
The referred rotational invariance of Tˆ implies that it remains
10 They correspond to those indicated under the summation symbol in
Eq. (A.31) both for the initial and final states.
(A.37)
Under the action of the rotation R( pˆ)† (R( pˆ)† pˆ = zˆ and
R( pˆ)†pˆ = pˆ ) the final and initial states transform as,
cf. Eq. (A.23),
R( pˆ)†|p, σ¯1σ¯2, α¯ 1α¯ 2 S
R( pˆ)†|p , σ1σ2, α1α2 S
=
=
with the convention that R inside the argument of the rotation
matrices refers to R( pˆ). We insert Eqs. (A.37) and (A.38)
into Eq. (A.36), and next transform pˆ → pˆ as integrations
variables, take into account the invariance of the solid angle
measure under such rotation and use Eq. (A.25) for
Y ¯m¯ ( pˆ) = Y m¯ (R( pˆ)zˆ) =
¯
Dm(¯¯)m¯ (R†)Y ¯m¯ (zˆ),
Y m ( pˆ ) = Y m (R( pˆ) pˆ ) =
Dm( )m (R†)Y m ( pˆ ).
(A.39)
m
¯
m
Then Eq. (A.36) for T (J I ) can be rewritten as
S; ¯S¯
dpˆ
dpˆ (σ1σ2 M |s1s2 S)(m M μ| S J )
× (α1α2t3|τ1τ2 I )Dσ(s11σ)1 (R†)∗ D(s2) (R†)∗
σ2σ2
× Dm( )m (R†)∗Y m ( pˆ )∗(σ¯1σ¯2 M¯ |s¯1s¯2 S¯)
× (m¯ M¯ μ| ¯S¯ J )(α¯ 1α¯ 2t3|τ¯1τ¯2 I )
× D(s¯1) (R†)Dσ(¯s¯22σ¯)2 (R†)Dm(¯¯)m¯ (R†)Y ¯m¯ (zˆ)
σ¯1σ¯1
× S p , σ1σ2, α1α2|Tˆ | |p|zˆ, σ¯1σ¯2, α¯ 1α¯ 2 S. (A.40)
Let us recall that from the composition of two rotation
matrices one has [76,77]
m1,m2
=
M
(m1m2 M | 1 2 L)Dm(11m)1 (R)Dm(22m)2 (R)
(m1m2 M | 1 2 L)D(ML)M (R).
(A.41)
We apply this result first to two combinations in Eq. (A.40):
σ1,σ2
The same relation in Eq. (A.41) is applied once more to the
following combinations in Eq. (A.43):
(m M μ| S J )D(MS)M (R†)Dm( )m (R†)
(m M μ | S J )Dμ(J μ)(R†),
(m¯ M¯ μ| ¯S¯ J )D(S¯) (R†)Dm(¯¯)m¯ (R†)
M¯ M¯
(m¯ M¯ μ¯ | ¯S¯ J )Dμ(¯J μ)(R†).
We take Eq. (A.44) into Eq. (A.43) which now reads
Now, the partial-wave amplitude T (SJ;I¯)S¯ is independent of μ
so that we have
μ=−J
(A.46)
(A.42)
(A.44)
Sˆ = I − i Tˆ
The same result in Eq. (A.45) is obtained with the product
Dμ(J μ)(R†)∗ Dμ(¯J μ)(R†) replaced by
1
2 J + 1
J
μ=−J
Dμ(J μ)(R†)∗ Dμ(¯J μ)(R†) = 2δJμ¯+μ 1 ,
(A.47)
as follows from the unitarity character of the rotation
matrices. As a consequence any dependence in pˆ present in
the integrand of Eq. (A.45) disappears in the average of
Eq. (A.46), the integration in the solid angle pˆ is trivial and
it gives a factor 4π . Taking into account the Kronecker delta
from Eq. (A.47) in the third component of the total
angular momentum and a new one that arises because Y m¯ (zˆ) is
¯
enxoptrzeesrsoioonnfloyr fTor(J mI¯) := 0, we then end with the following
S; ¯S¯
dpˆ Y m ( pˆ )∗(σ1σ2 M |s1s2 S)
T (J I ) Y ¯0(zˆ)
S; ¯S¯ = 2(2 J + 1)
σ1,σ2,σ¯1
σ¯2,α1,α2
α¯1,α¯2,m
where we have removed the primes on top of the spin and
orbital angular momentum third-component symbols and in
the previous sum M = σ1 + σ2 and M¯ = σ¯1 + σ¯2.
Next, we derive the unitarity relation corresponding to
our normalization for the partial-wave projected amplitudes
T (SJ;I¯)S¯ . We write the Sˆ matrix as
which satisfies the standard unitarity relation
Sˆ · Sˆ† = I,
with I the identity matrix. In terms of the T -matrix, cf. (A.49),
this implies that
Tˆ − Tˆ † = −i Tˆ Tˆ †.
Expressed with the matrix elements in the basis S J this
relation becomes
2ImTˆ (J I )
S; ¯S¯ = − J μ, S, T t3|Tˆ Tˆ †| J μ, ¯S¯, T t3 .
In deriving the left-hand side of this equation we have
taken into account that because of time-reversal symmetry
T (J I )
S; ¯S¯ = T (J I ) . On the right-hand side we introduce now
¯S¯; S
a two-body resolution of the identity of states | J μ, S, I t3
(A.49)
(A.50)
(A.51)
(A.52)
(we have restricted our vector space to the one generated by
these states) such that, taking into account their
normalization in Eq. (A.34), one ends with
ImT (J I )
S; ¯S¯ = −
,S
|p | T (J I )
8π √s ,S; ,S T (J I )∗ ,S ; ¯S¯ .
The phase space factor is included in the diagonal matrix
such that now the diagonal matrix elements of the identity
operator I and S are just 1 and ηi e2iδi , in order, where ηi is
the inelasticity for channel i and δi its phase shift.
Finally, let us mention that a partial-wave expansion of
the helicity amplitudes (not S J partial waves as here) for
vector–vector scattering was developed in Ref. [78]. The
main point in this reference is to provide partial-wave
amplitudes in the helicity basis which are free of kinematical
singularities, while here we are interested in deriving
nonperturbative ρρ scattering amplitudes from the hidden local
gauge Lagrangian.
(A.53)
(A.54)
(A.55)
52. U.-G. Meißner, J.A. Oller, Nucl. Phys. A 673, 311 (2000).
arXiv:nucl-th/9912026
53. J.M. Alarcon, J. Martin Camalich, J.A. Oller, Ann. Phys. 336, 413
(2013). arXiv:1210.4450 [hep-ph]
54. M. Albaladejo, J.A. Oller, L. Roca, Phys. Rev. D 82, 094019 (2010).
arXiv:1011.1434 [hep-ph]
55. P.C. Bruns, M. Mai, U.-G. Meißner, Phys. Lett. B 697, 254 (2011).
arXiv:1012.2233 [nucl-th]
56. M. Mai, U.-G. Meißner, Eur. Phys. J. A 51(
3
), 30 (2015).
arXiv:1411.7884 [hep-ph]
57. D. Gamermann, E. Oset, Eur. Phys. J. A 33, 119 (2007).
arXiv:0704.2314 [hep-ph]
58. J.M. Dias, F. Aceti, E. Oset, Phys. Rev. D 91(
7
), 076001 (2015).
arXiv:1410.1785 [hep-ph]
59. O. Romanets, L. Tolos, C. Garcia-Recio, J. Nieves, L.L.
Salcedo, R.G.E. Timmermans, Phys. Rev. D 85, 114032 (2012).
arXiv:1202.2239 [hep-ph]
60. X.W. Kang, J.A. Oller, Phys. Rev. D 94(
5
), 054010 (2016).
arXiv:1606.06665 [hep-ph]
61. L. Roca, M. Mai, E. Oset, U.-G. Meißner, Eur. Phys. J. C 75(
5
),
218 (2015). arXiv:1503.02936 [hep-ph]
62. J.A. Oller, E. Oset, J.R. Pelaez, Phys. Rev. D 59, 074001 (1999)
((E) Phys. Rev. D 60, 099906 (1999); (E) Phys. Rev. D 75, 099903
(2007). arXiv:hep-ph/9804209)
63. T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C 78, 025203 (2008).
arXiv:0803.2550 [nucl-th]
64. T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C 85, 015201 (2012).
arXiv:1108.5524 [nucl-th]
1. R.H. Dalitz , S.F. Tuan , Phys. Rev. Lett. 2 , 425 ( 1959 )
2. J.A. Oller , U.-G. Meißner, Phys. Lett. B 500 , 263 ( 2001 ). arXiv:hep-ph/0011146
3. D. Jido , J.A. Oller , E. Oset , A. Ramos , U.-G. Meißner, Nucl. Phys. A 725 , 181 ( 2003 ). arXiv:nucl-th/0303062
4. H.Y. Lu et al. [CLAS Collaboration], Phys. Rev. C 88 , 045202 ( 2013 ). arXiv: 1307 . 4411
5. C. Patrignani et al. (Particle Data Group), Chin Phys. C 40 , 100001 ( 2016 )
6. J.D. Weinstein , N. Isgur , Phys. Rev. D 41 , 2236 ( 1990 )
7. G. Janssen , B.C. Pearce , K. Holinde , J. Speth , Phys. Rev. D 52 , 2690 ( 1995 ). arXiv:nucl-th/9411021
8. J.A. Oller , E. Oset, Nucl. Phys. A 620 , 438 ( 1997 ). ((E) Nucl . Phys. A 652 , 407 ( 1999 ) arXiv:hep- ph/9702314])
9. R. Molina , D. Nicmorus , E. Oset, Phys. Rev. D 78 , 114018 ( 2008 ). arXiv: 0809 .2233 [hep-ph]
10. L.S. Geng , E. Oset, Phys. Rev. D 79 , 074009 ( 2009 ). arXiv: 0812 .1199 [hep-ph]
11. H. Nagahiro , J. Yamagata-Sekihara , E. Oset , S. Hirenzaki , R. Molina , Phys. Rev. D 79 , 114023 ( 2009 ). arXiv: 0809 .3717 [hepph]
12. J.J. Xie , E. Oset , L.S. Geng , Phys. Rev. C 93 , 025202 ( 2016 ). arXiv: 1509 .06469 [nucl-th]
13. E. Oset , L.S. Geng , R. Molina , J. Phys. Conf. Ser . 348 , 012004 ( 2012 )
14. D.B. Lichtenberg , Unitary Symmetry and Elementary Particles , 2nd edn . (Academic, New York, 1978 )
15. M. Koll , R. Ricken , D. Merten , B.C. Metsch , H.R. Petry , Eur. Phys. J. A 9 , 73 ( 2000 ). arXiv:hep-ph/0008220
16. R. Ricken , M. Koll , D. Merten , B.C. Metsch , H.R. Petry , Eur. Phys. J. A 9 , 221 ( 2000 ). arXiv:hep-ph/0008221
17. Review on 'Quark Model' in PDG( 2016 ) [5] by C. Amsler , T. DeGrand and B . Krusche
18. M. Krammer , H. Krasemann, Phys. Lett. B 73 , 58 ( 1978 )
19. Z.P. Li , F.E. Close , T. Barnes, Phys. Rev. D 43 , 2161 ( 1991 )
20. L.Y. Dai , M.R. Pennington , Phys. Rev. D 90 ( 3 ), 036004 ( 2014 ). arXiv: 1404 .7524 [hep-ph]
21. T. Mori et al. [Belle Collaboration], Phys. Rev. D 75 , 051101 ( 2007 ). arXiv:hep-ex/0610038
22. T. Mori et al. [Belle Collaboration], J. Phys, Soc. Jap . 76 , 074102 ( 2007 ). arXiv: 0704 .3538 [hep-ex]
23. K. Abe et al. [Belle Collaboration]. arXiv:0711 . 1926 [hep-ex]
24. S. Uehara et al. [Belle Collaboration], Phys. Rev. D 78 , 052004 ( 2008 ). arXiv: 0805 .3387 [hep-ex]
25. S. Uehara et al. [Belle Collaboration], Phys. Rev. D 79 , 052009 ( 2009 ). arXiv: 0903 .3697 [hep-ex]
26. J.A. Carrasco , J. Nebreda , J.R. Pelaez , A.P. Szczepaniak , Phys. Lett. B 749 , 399 ( 2015 ). arXiv: 1504 .03248 [hep-ph]
27. J.T. Londergan , J. Nebreda , J.R. Pelaez , A. Szczepaniak , Phys. Lett. B 729 , 9 ( 2014 ). arXiv: 1311 .7552 [hep-ph]
28. J.R. Pelaez , F.J. Yndurain , Phys. Rev. D 69 , 114001 ( 2004 ). arXiv:hep-ph/0312187
29. R. Garcia-Martin , R. Kaminski , J.R. Pelaez , J. Ruiz de Elvira, F.J. Yndurain , Phys. Rev. D 83 , 074004 ( 2011 ). arXiv: 1102 .2183 [hepph]
30. A.V. Anisovich , V.V. Anisovich , A.V. Sarantsev , Phys. Rev. D 62 , 051502 ( 2000 ). arXiv:hep-ph/0003113
31. B. Ananthanarayan , G. Colangelo, J. Gasser , H. Leutwyler, Phys. Rep . 353 , 207 ( 2001 ). arXiv:hep-ph/0005297
32. G. Veneziano, Nuovo Cim. 57 , 190 ( 1968 )
33. C. Lovelace, Phys. Lett. B 28 , 264 ( 1968 )
34. J.A. Saphiro , Phys. Rev . 179 , 1345 ( 1969 )
35. M. Albaladejo , J.A. Oller , Phys. Rev. Lett . 101 , 252002 ( 2008 ). arXiv: 0801 .4929 [hep-ph]
36. S.J. Lindenbaum , R.S. Longacre , Phys. Lett. B 274 , 492 ( 1992 )
37. U.-G. Meißner, Phys. Rept . 161 , 213 ( 1988 )
38. M. Bando , T. Kugo , S. Uehara , K. Yamawaki , T. Yanagida, Phys. Rev. Lett . 54 , 1215 ( 1985 )
39. M. Bando , T. Kugo , K. Yamawaki , Phys. Rep . 164 , 217 ( 1988 )
40. J.A. Oller , E. Oset, Phys. Rev. D 60 , 074023 ( 1999 ). arXiv:hep-ph/9809337
41. N. Kaiser , P.B. Siegel , W. Weise, Nucl. Phys. A 594 , 325 ( 1995 ). arXiv:nucl-th/9505043
42. N. Kaiser , P.B. Siegel , W. Weise, Phys. Lett. B 362 , 23 ( 1995 ). arXiv:nucl-th/9507036
43. Z.H. Guo , J.A. Oller , J. Ruiz de Elvira, Phys. Rev. D 86 , 054006 ( 2012 ). arXiv: 1206 .4163 [hep-ph]
44. L. Roca , E. Oset , J. Singh , Phys. Rev. D 72 , 014002 ( 2005 ). arXiv:hep-ph/0503273
45. L. Alvarez-Ruso , J.A. Oller , J.M. Alarcon , Phys. Rev. D 80 , 054011 ( 2009 ). arXiv: 0906 .0222 [hep-ph]
46. L. Alvarez-Ruso , J.A. Oller , J.M. Alarcon , Phys. Rev. D 82 , 094028 ( 2010 ). arXiv: 1007 .4512 [hep-ph]
47. S. Sarkar , E. Oset, M.J. Vicente Vacas , Nucl. Phys. A 750 , 294 ( 2005 ). (Erratum: [Nucl . Phys. A 780 , 90 ( 2006 ) ]) arXiv:nucl-th/0407025
48. J.A. Oller , Nucl. Phys. A 725 , 85 ( 2003 )
49. J.A. Oller , Phys. Lett. B 477 , 187 ( 2000 ). arXiv:hep-ph/9908493
50. R.L. Delgado , A. Dobado , F.J. Llanes-Estrada , Phys. Rev. Lett . 114 , 221803 ( 2015 ). arXiv: 1408 .1193 [hep-ph]
51. A. Dobado , M.J. Herrero , J.R. Pelaez , E. Ruiz Morales , Phys. Rev. D 62 , 055011 ( 2000 ). arXiv:hep-ph/9912224
65. F. Aceti , E. Oset, Phys. Rev. D 86 , 014012 ( 2012 ). arXiv: 1202 .4607 [hep-ph]
66. T. Sekihara, T. Hyodo , D. Jido , PTEP 2015 , 063D04 ( 2015 ) arXiv: 1411 .2308 [hep-ph]
67. D.V. Bugg , Eur. Phys. J. C 52 , 55 ( 2007 ). arXiv: 0706 .1341 [hep-ex]
68. M. Albaladejo , J.A. Oller , Phys. Rev. C 84 , 054009 ( 2011 ). arXiv: 1107 .3035 [nucl-th]
69. M. Albaladejo , J.A. Oller , Phys. Rev. C 86 , 034005 ( 2012 ). arXiv: 1201 .0443 [nucl-th]
70. Z.H. Guo , J.A. Oller , G. Ríos, Phys. Rev. C 89 ( 1 ), 014002 ( 2014 ). arXiv: 1305 .5790 [nucl-th]
71. J.A. Oller , Phys. Rev. C 93 , 024002 ( 2016 ). arXiv: 1402 .2449 [nuclth]
72. D.R. Entem , J.A. Oller , arXiv: 1610 .01040 [nucl-th]
73. G.F. Chew , S. Mandelstam, Phys. Rev . 119 , 467 ( 1960 )
74. M. Albaladejo , J.A. Oller , Phys. Rev. C 84 , 054009 ( 2011 ). arXiv: 1107 . 3035
75. M. Albaladejo , J.A. Oller , Phys. Rev. C 86 , 034005 ( 2012 ). arXiv: 1201 .0443 [nucl-th]
76. A.D. Martin , T.D. Spearman , Elementary Particle Theory (NothHolland Publishing Company , Amsterdam, 1970 )
77. M.E. Rose , Elementary Theory of Angular Momentum (Dover , New York, 1995 )
78. M.F.M. Lutz , I. Vidana , Eur. Phys. J. A 48 , 124 ( 2012 ). arXiv:1111 . 1838 [hep-ph]