Eur. Phys. J. A 34, 129–152 (2007)
DOI 10.1140/epja/i2007-10499-9
THE EUROPEAN
PHYSICAL JOURNAL A
Regular Article – Theoretical Physics
Baryon-baryon and baryon-antibaryon interaction amplitudes in
the spin-momentum operator expansion method
A.V. Anisovich1,2 , V.V. Anisovich2 , E. Klempt1,a , V.A. Nikonov1,2,b , and A.V. Sarantsev1,2
1
2
Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, Germany
Petersburg Nuclear Physics Institute, Gatchina, 188300 Russia
Received: 13 March 2007 / Revised: 30 October 2007
c Società Italiana di Fisica / Springer-Verlag 2007
Published online: 22 November 2007 – Communicated by A. Schäfer
Abstract. Partial-wave scattering amplitudes in baryon-baryon and baryon-antibaryon collisions and amplitudes for the production and decay of baryon resonances are constructed in the framework of the
spin-momentum operator expansion method. The approach is relativistically invariant and it allows us to
perform combined analyses of different reactions imposing analyticity and unitarity directly. The role of
final-state interactions (triangle and box diagrams) is discussed.
PACS. 11.80.Et Partial-wave analysis – 13.30.-a Decays of baryons – 13.60.Le Meson production –
14.20.Gk Baryon resonances with S = 0
1 Introduction
To understand strong interactions at low and intermediate energies is one of the important tasks when quantum
chromodynamics is being studied. At large momentum
transfer, QCD can be used efficiently due to the smallness of the strong-interaction coupling constant [1]; the
low-energy domain can be treated using effective field theories [2]. The resonance region is much more difficult to
access. Lattice gauge calculations are capable to reproduce
the masses of ground-state hadrons [3] but excited states
and their decay properties are difficult to extract from lattice data. For further progress, systematic experimental
information seems to be mandatory to identify the leading mechanisms responsible for the mass spectrum and for
the decay amplitudes of strongly interacting particles.
Recently, considerable progress has been achieved in
meson spectroscopy, even though a commonly agreed picture has not yet emerged. Recent reviews emphasizing different views can be found in [4–7]. The main sources of
recent progress were the study of reactions with multiparticle final states. The analysis of data on proton-antiproton annihilation at rest resulted in the discovery of a
number of particles in the region 1300–1800 MeV [8–14];
the investigation of the proton-antiproton annihilation
in flight led to a large set of new states over the region 1800–2500 MeV [15–18]. It appeared that the majority of the newly discovered states are lying on lina
b
e-mail: [email protected]
e-mail: [email protected]
ear trajectories against radial excitation number [19].
Such a pattern was not predicted by the classical quark
model of Godfrey and Isgur [20] using a linear confinement potential and additional interactions due to effective one-gluon exchange forces. More recent calculations
based on instanton-induced interactions [21,22] can, however, be tuned (by choosing an appropriate Dirac structure
of the confinement potential) to reproduce the observed
mass pattern very well. The pattern can be understood,
too, within a 5-dimensional theory holographically dual to
QCD (AdS/QCD) [23] which predicts masses proportional
to (N +L) where L is the intrinsic orbital angular momentum between quark and antiquark and N a radial quantum number. In the light-quark meson spectrum, practically all expected states are observed. However, there are a
few additional states which do not belong to these trajectories. These states are candidates to be of exotic nature,
e.g., they could be glueballs or hybrids.
The situation in the baryon sector is in some sense
reverse: for baryons, the quark model predicts a much
larger number of states than that observed experimentally. So far, the pattern seems to suggest that not all
degrees of freedom in the three-quark system are realised
in the spectrum of excited states. Instead, the pattern of
excited states follows the same (L + N ) pattern [24] which
is observed for mesons. If this is the case, the fact would
be an important phenomenon in the physics of highly excited states. Still, a detailed verification of this statement
is needed. On the other hand, the main information on
baryon resonances has come from the πN elastic scattering, and one may hope that many new states will be dis-
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The European Physical Journal A
covered in i) reactions involving strangeness in two hadron
final states and in ii) inelastic reactions induced by photons or protons with three or four particles in the final
states (for example, two-pion photoproduction).
The search for new baryon resonances is of topical
interest and several experiments like CB-ELSA, CLAS,
GRAAL, SAPHIR, and SPRING-8 pursue active searches
using photoproduction as a tool [25–52]. A few new resonances were suggested [53–55] in fits to these data sets.
Proton-proton collision experiments can provide an important source of information on baryon resonances including exotic states (e.g., pentaquarks [56–63]). COSY at
the Research Center Jülich is providing a wealth of data
on meson production in proton-proton inelastic scattering [64–115]. The experiments Anke and COSY-11 covered mainly the threshold region, while TOF covers the
full dynamical range. At present, the upgraded WASA
detector is installed at COSY and will provide highstatistics data on the production of neutral mesons in
N N interactions [116]. The data provide stringent information on nucleon-nucleon-meson vertices and on the formation of baryon resonances. Selected papers can be found
in [117–143].
The partial-wave analysis of such processes cannot be
carried out without taking into account the final-state interaction. In many processes, the inclusion of the protonproton interaction dramatically changes the description of
the data [144]. However, a number of important problems
for such analyses has not comprehensively developed yet.
Among these problems are a correct treatment of relativistic effects and of the contributions of triangle or box
diagrams.
In this paper, we present a relativistically invariant
approach for the partial-wave analysis of proton-proton
interactions. The method is based on the spin-momentum
operator expansion suggested in [145–148]. The contribution of triangle and box diagrams to the meson production
processes is discussed and certain examples are considered.
In sect. 2, we present the partial-wave expansion for
baryon-baryon and baryon-antibaryon scattering amplitudes. In sect. 3, the unitarity condition for fermionfermion partial-wave amplitudes is discussed. The angular momentum and spin operators for nucleon-nucleon
scattering are introduced in sect. 4, the nucleon-nucleon
partial-wave amplitude is constructed in sect. 5. In this
section, fermion-fermion one-loop diagrams and the crosssection for the two-fermion scattering are calculated.
The operators for N ∆ production are constructed in the
sect. 6. Some examples of amplitudes with multi-particle
final states are given in sect. 7. Properties of the triangle
and box diagrams are shortly discussed in sects. 8 and 9.
2 Selection rules for baryon-antibaryon and
baryon-baryon scattering amplitudes
2.1 Baryon pairs with isospin I = 0
First, consider the baryon-antibaryon scattering amplitude in a isospin singlet configuration, for example,
the ΛΛ̄ scattering amplitude. One can use two alternative representations of the baryon-antibaryon amplitude
Λ(p1 )Λ̄(p2 ) → Λ(p′1 )Λ̄(p′2 ).
In the t-channel representation the amplitude is the
sum of partial waves in the t-channel with definite quantum numbers: spin S, angular momentum L and total
momentum J (we define t = q 2 = (p′1 − p1 )2 ):
X M (s, t, u) =
ψ̄(p′1 )Q̃SLJ
(q)ψ(p
)
1
µ1 ...µJ
S,L,L′ ,J
µ1 ...µJ
′
(S,L′ L,J) 2
J
× ψ̄(p′2 )Q̃µSL
(q ).
(q)ψ(p
)
At
2
1 ...µJ
(1)
Here, Q̃ is the t-channel operator (the four-component spinors ψ(p) are given in appendix A) and µ1 , µ2 . . . µJ−1 , µJ
are the indices of the rank J operator.
Another representation is related to the s-channel (we
define s = (p1 + p2 )2 ):
X ′
J
′
c
′
ψ̄(p′1 )QµSL
M (s, t, u) =
(k
)ψ
(−p
)
⊥
2
1 ...µJ
S,L,L′ ,J
µ1 ...µJ
(S,L′ L,J)
× ψ̄ c (−p2 )QSLJ
(s).
µ1 ...µJ (k⊥ )ψ(p1 ) As
(2)
Here, ψ c (−p) are charge-conjugated four-component spinors (see appendix A) and QSLJ
µ1 ...µJ are the s-channel operators, where S, L, J are, correspondingly, spin, angular
momentum and total momentum of the partial wave in
the s-channel. The notations of momenta are as follows:
P = p1 + p2 = p′1 + p′2 ,
⊥
gνµ
= gνµ −
Pν Pµ
⊥P
≡ gµν
,
P2
k=
1
(p1 − p2 ),
2
⊥
k⊥ = kν gνµ
.
(3)
The representation (1) is suitable to consider the tchannel meson or Reggeon exchanges, while eq. (2) is
convenient for the s-channel partial-wave analysis. The
representations (1) and (2) are related to each other by
the Fierz transformation [149], with a corresponding reexpansion of the spin-momentum operators.
In terms of the SLJ representations, the states are
usually described as 2S+1 LJ . The P -parity can be calculated as P = (−1)L+1 and C = (−1)L+S . The states
with S = 0 are unambiguously defined and they form a
set of states with J P C = 0−+ , 1+− , 2−+ . . . The states
with S = 1 and L = J are also uniquely defined and
form the set J P C = 1++ , 2−− , 3++ . . . The states with
S = 1 and L = J − 1 and L = J + 1 have the same
J P C and can mix with each other, that are the states
J P C = 0++ , 1−− , 2++ , . . .
2.2 Nucleon-antinucleon scattering amplitude
Let us write the s-channel expansion for a pair of nucleons
where N = (p, n) forms an isodoublet. The systems pn̄
and np̄ have isospin I = 1, and the s-channel expansions
of their scattering amplitudes are determined by formulae
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
which are analogous to those for ΛΛ̄, eq. (2). The systems
pp̄ and nn̄ are a superposition of two states, with I = 0
and I = 1. The nucleon-antinucleon amplitudes read
a) p(p1 )n̄(p2 ) → p(p′1 )n̄(p′2 ) (I = 1):
2
11
C1/2
M1 (s, t, u) = M1 (s, t, u),
1/2, 1/2 1/2
b) p(p1 )p̄(p2 ) → p(p′1 )p̄(p′2 ) (I = 0, 1):
2
10
C1/2
M1 (s, t, u)
1/2, 1/2 −1/2
2
00
+ C1/2
M0 (s, t, u) =
1/2, 1/2 −1/2
1
1
M1 (s, t, u) + M0 (s, t, u),
2
2
(4)
(5)
10
C1/2
1
1
M1 (s, t, u) − M0 (s, t, u).
2
2
=
(6)
Note that, by writing the N N̄ (or N N ) scattering amplitudes, one√can use
√ alternatively either isotopic Pauli
matrices (I/ 2, τ / 2) or Clebsch-Gordan coefficients.
In (4), we use Clebsch-Gordan coefficients which allows
us to consider reactions in which states with I > 1/2 are
produced.
The s-channel operator expansion for N N̄ → N N̄ can
be written as
X ′
J
′
c
′
ψ̄(p′1 )QSL
MI (s, t, u) =
µ1 ...µJ (k )ψ (−p2 )
S,L,L′ ,J
µ1 ...µJ
(S,L′ L,J)
× ψ̄ c (−p2 )QSLJ
(s).
µ1 ...µJ (k)ψ(p1 ) AI
(7)
Since the two masses are equal, k = k⊥ holds. In eq. (7),
the summation is performed over all states (as well as
for the ΛΛ̄ scattering amplitude). The spin-momentum
operators QSLJ
µ1 ...µJ (k) for the states with J = 0, 1, 2 are
given in sect. 4.
2.3 Amplitude for pΛ → pΛ scattering
It is convenient to present the amplitude pΛ → pΛ precisely in the same technique which was used in the consideration of the s-channel fermion-antifermion system.
To this aim, we declare p being a fermion and Λ an antifermion. Then, the s-channel expansion for the pΛ → pΛ
scattering amplitude reads
Let us present the amplitude ΛΛ → ΛΛ in the technique
which was used for reaction pΛ → pΛ. So, we declare
the 1st Λ to be a fermion and the 2nd one to be an antifermion. One can distinguish between them, for example, in the c.m. system labeling a particle scattered into
the backward hemisphere as “antifermion”. Then the schannel expansion for the ΛΛ → ΛΛ scattering amplitude
reads
X ′
J
′
c
′
ψ̄Λ (p′1 )QµSL
MΛΛ→ΛΛ (s, t, u) =
(k
)ψ
(−p
)
...µ
Λ
2
1
J
(S,L′ L,J)
c
× ψ̄Λ
(−p2 )QSLJ
µ1 ...µJ (k)ψΛ (p1 ) AΛΛ→ΛΛ (s).
(9)
In this reaction, a selection rule for quantum numbers
caused by the Fermi statistics should be taken into account, such as
(−1)S+L+1 = −1.
(10)
Therefore, the following states contribute into (9) only:
S = 1:
S = 0:
(L = 1; J = 1), (L = 3; J = 2, 3, 4), . . .
(L = 0; J = 0), (L = 2; J = 2), . . .
(11)
2.5 Nucleon-nucleon scattering amplitude
The nucleon is an isodoublet with components p → (I =
1/2, I3 = 1/2) and n → (I = 1/2, I3 = −1/2). The systems pp and nn have total isospin I = 1, and the s-channel
expansions of their scattering amplitudes are determined
by formulae analogous to those for ΛΛ, eq. (9). The system pn is a superposition of two states, with total isospins
I = 0 and I = 1. The amplitudes read:
a) p(p1 )p(p2 ) → p(p′1 )p(p′2 ) (I = 1):
2
11
C1/2
M1 (s, t, u) = M1 (s, t, u), (12)
1/2, 1/2 1/2
b) p(p1 )n(p2 ) → p(p′1 )n(p′2 ) (I = 0, 1):
2
10
C1/2
M1 (s, t, u)
1/2, 1/2 −1/2
2
00
+ C1/2
M0 (s, t, u) =
1/2, 1/2 −1/2
1
1
M1 (s, t, u) + M0 (s, t, u),
(13)
2
2
c) n(p1 )n(p2 ) → n(p′1 )n(p′2 ) (I = 1):
2
1−1
M1 (s, t, u) = M1 (s, t, u). (14)
C1/2
−1/2, 1/2 −1/2
The s-channel operator expansion gives for MI (s, t, u) in
the reaction pn → pn (I = 0):
X ′
J
′
c
′
ψ̄p (p′1 )QµSL
(k
)ψ
(−p
M0 (s, t, u) =
)
n
2
1 ...µJ
S,L,L′ ,J
µ1 ...µJ
MN Λ→N Λ (s, t, u) =
X ′
J
′
c
′
ψ̄N (p′1 )QSL
µ1 ...µJ (k⊥ )ψΛ (−p2 )
S,L,L′ ,J
µ1 ...µJ
(S,L′ L,J)
c
(−p2 )QSLJ
× ψ̄Λ
µ1 ...µJ (k⊥ )ψN (p1 ) AN Λ→N Λ (s).
2.4 Amplitude for ΛΛ → ΛΛ scattering
S,L,L′ ,J
µ1 ...µJ
c) p(p1 )p̄(p2 ) → n(p′1 )n̄(p′2 ) (I = 0, 1):
10
1/2, 1/2 −1/2 C1/2 −1/2, 1/2 1/2 M1 (s, t, u)
00
00
+C1/2
1/2, 1/2 −1/2 C1/2 −1/2, 1/2 1/2 M0 (s, t, u)
131
(8)
S = 1:
S = 0:
(S,L′ L,J)
(s),
× ψ̄nc (−p2 )QSLJ
µ1 ...µJ (k)ψp (p1 ) A0
(L = 0; J = 1), (L = 2; J = 1, 2, 3), . . .
(L = 1; J = 1), (L = 3; J = 3), . . .
(15)
132
The European Physical Journal A
and for I = 1:
M1 (s, t, u) =
Finally, one has:
X S,L,L′ ,J
µ1 ...µJ
×
S = 1:
S = 0:
′
J
′
c
′
ψ̄p (p′1 )QSL
µ1 ...µJ (k )ψn (−p2 )
ψ̄nc (−p2 )QSLJ
µ1 ...µJ (k)ψ(p1 )
(S,LL,J)
(SLJ)
Im AΛΛ̄→ΛΛ̄ (s) = ρΛΛ̄
where
(S,L′ L,J)
A1
(s),
(L = 1; J = 0, 1, 2), (L = 3; J = 2, 3, 4), . . .
(L = 0; J = 0), (L = 2; J = 2), . . .
(16)
J (SLJ)
Oµµ′′1 ...µ
(s)
′′ ρ
1 ...µJ ΛΛ̄
=
Z
(S,LL,J)∗
(S,LL,J)
(s)AΛΛ̄→ΛΛ̄ (s)AΛΛ̄→ΛΛ̄ (s),
(18)
dΦ2 (p′′1 , p′′2 )
′′
′′
SLJ
′′
′′
×Sp QSLJ
.
′′ (k )(p̂1 + mΛ )
µ1 ...µJ (k )(−p̂2 + mΛ )Qµ′′
...µ
1
J
(19)
The selection rule for quantum numbers in (15) and (16)
is caused by the Fermi statistics.
Analogous partial-wave expansions can be written for
the reactions pp → pp and nn → nn (I = 1), with an
obvious replacing in (16): n → p for pp → pp and p → n
for nn → nn. Here, as for ΛΛ → ΛΛ, declaring one nucleon as a fermion and the second one as antifermion, one
distinguishes between them in the c.m. system labeling a
particle scattered into the backward hemisphere as “antifermion”.
3 Unitarity conditions and K-matrix
representations of baryon-antibaryon and
baryon-baryon scattering amplitudes
J
The projection operator Oµµ′′1 ...µ
′′ is presented in sect. 4.
1 ...µJ
The phase space is determined as
2π 4
dΦ2 (p1 , p2 ) =
2
d3 p2
d3 p1
1
(2π)4 δ (4) (P − p1 − p2 )
.
2
(2π)3 2p10 (2π)3 2p20
dΦ̃2 (p1 , p2 ) =
(20)
J
The projection operator Oµµ′′1 ...µ
′′ obeys the convolution
1 ...µJ
µ1 ...µJ
rule, Oµ1 ...µJ = 2J + 1, that gives
Z
1
(SLJ)
ρΛΛ̄ (s) =
dΦ̃2 (p′′1 , p′′2 )
2J + 1
′′
′′
SLJ
′′
′′
×Sp QSLJ
µ1 ...µJ (k )(−p̂2 + mΛ )Qµ1 ...µJ (k )(p̂1 + mΛ ) .
(21)
Here, we write down the unitarity conditions and give the
K-matrix representations of the baryon-antibaryon and
baryon-baryon scattering amplitudes suggesting that inelastic processes are switched off (for example, because
the energy is not large enough). Generalisation of the Kmatrix representations in case when inelastic channels are
switched on can be performed in a standard way.
The unitarity condition (18) results in the following Kmatrix representation of the amplitude ΛΛ̄ → ΛΛ̄:
(S,LL,J)
(S,LL,J)
AΛΛ̄→ΛΛ̄ (s)
=
KΛΛ̄→ΛΛ̄ (s)
(SLJ)
1 − iρΛΛ̄
(S,LL,J)
(s)KΛΛ̄→ΛΛ̄ (s)
.
(22)
3.2 ΛΛ scattering
3.1 ΛΛ̄ scattering
In this subsection, we consider the unitarity condition for
the amplitude with J = L. The generalisation for the J =
L ± 1 amplitude is considered in the last subsection. For
the amplitude ΛΛ̄ → ΛΛ̄ of eq. (2), the s-channel unitarity
(S,LL,J)
condition reads for J = L (we re-define As
(s) →
(S,LL,J)
AΛΛ̄→ΛΛ̄ (s)) as follows:
X
µ1 ...µJ
′
c
′
ψ̄(p′1 )QSLJ
µ1 ...µJ (k )ψ (−p2 )
(S,LL,J)
× ψ̄ c (−p2 )QSLJ
µ1 ...µJ (k)ψ(p1 ) Im AΛΛ̄→ΛΛ̄ (s) =
Z
X X
′
c
′
dΦ2 (p′′1 , p′′2 )
ψ̄(p′1 )QSLJ
µ1 ...µJ (k )ψ (−p2 )
j,ℓ µ1 ...µJ
(S,LL,J)
′′
′′
× ψ̄ℓc (−p′′2 )QSLJ
µ1 ...µJ (k )ψj (p1 ) AΛΛ̄→ΛΛ̄ (s)
X h
c
(k)ψ
(−p
)
ψ̄(p1 )QSLJ
×
′′
′′
2
µ1 ...µ
J
′′
µ′′
1 ...µJ
i+
(S,LL,J)
′′
′′
.
× ψ̄ℓc (−p′′2 )QSLJ
(s)
′′ (k )ψj (p1 )A
µ′′
...µ
Λ
Λ̄→Λ
Λ̄
1
J
(17)
Likewise, we consider the unitarity condition for the ΛΛ
scattering amplitude. The s-channel unitarity condition
for the amplitude ΛΛ → ΛΛ with J = L reads
(S,LL,J)
Im AΛΛ→ΛΛ (s) =
1 (SLJ)
(S,LL,J)∗
(S,LL,J)
ρ
(s)AΛΛ→ΛΛ (s)AΛΛ→ΛΛ (s),
2 ΛΛ
where the identity factor 1/2 is introduced. In this way,
we keep the definition (20) for dΦ̃2 (p′′1 , p′′2 ).
We have
Z
J (SLJ)
ρ
(s)
=
dΦ̃2 (p′′1 , p′′2 )
Oµµ′′1 ...µ
′′
ΛΛ
1 ...µJ
′′
′′
SLJ
′′
′′
×Sp QSLJ
(k
)(−p̂
+
m
)Q
+
m
)
.
(k
)(p̂
′′
′′
Λ
Λ
µ1 ...µJ
2
1
µ1 ...µ
J
(23)
...µJ
The convolution rule Oµµ11...µ
= 2J + 1, gives us
J
Z
1
(SLJ)
dΦ̃2 (p′′1 , p′′2 )
ρΛΛ (s) =
2J + 1
′′
′′
SLJ
′′
′′
×Sp QSLJ
µ1 ...µJ (k )(−p̂2 + mΛ )Qµ1 ...µJ (k )(p̂1 + mΛ ) ,
(24)
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
(SLJ)
thus leading to identical definitions for ρΛΛ (s) and
(SLJ)
ρΛΛ̄ (s), see (21). The unitarity condition (23) results
in the following K-matrix representation of the amplitude
ΛΛ → ΛΛ:
(S,LL,J)
KΛΛ→ΛΛ (s)
(S,LL,J)
AΛΛ→ΛΛ (s) =
(SLJ)
1 − 2i ρΛΛ
(S,LL,J)
(s)KΛΛ→ΛΛ (s)
.
(25)
Let us emphasize the appearance of the identity factor 1/2
in the denominator of (25).
3.3 Nucleon-antinucleon partial-wave amplitude
The K-matrix representation for N N̄ scattering amplitude is written precisely in the same way as for the ΛΛ̄
case. The only new aspect as compared to ΛΛ̄ is that the
N N̄ scattering is determined by two isotopic amplitudes,
see (5) and (6), with I = 0, 1:
1
1
M1 (s, t, u) + M0 (s, t, u),
2
2
1
1
b) pp̄ → nn̄ (I = 0, 1) : M1 (s, t, u)− M0 (s, t, u). (26)
2
2
a) pn̄ → pn̄ (I = 1) :
Being expanded over the s-channel operators QSLJ
µ1 ...µJ (k)⊗
′
SL′ J
Qµ1 ...µJ (k ), these amplitudes are represented through
(S,L′ L,J)
(S,L′ L,J)
partial-wave amplitudes A0
(s) and A1
(s).
The unitarity condition for these amplitudes leads again
to a K-matrix representation.
As above, we consider here the one-channel amplitude,
first for J = L. The two-channel amplitudes (S = 1, J =
L ± 1) are presented below in sect. 3.4.
(S,LL,J)
The imaginary part of the amplitude AI
(s) with
I = 0, 1 and J = L satisfying the s-channel unitarity
condition reads
(S,LL,J)
Im AI
(S,LL,J)
(s) = ρN N̄
(S,LL,J)∗
(s)AI
(S,LL,J)
(s)AI
(s),
(27)
where
(S,LL′ ,J)
ρN N̄
(s) =
1
2J + 1
Z
dΦ̃2 (p1 , p2 )
SL′ J
×Sp QSLJ
µ1 ...µJ (k)(−p̂2 + mN )Qµ1 ...µJ (k)(p̂1 + mN ) .
(28)
The unitarity condition (28) gives us the following Kmatrix representation:
133
and I = 1. The amplitudes read
a) pp → pp, nn → nn (I = 1) : M1 (s, t, u),
1
1
b) pn → pn (I = 0, 1) : M1 (s, t, u)+ M0 (s, t, u). (30)
2
2
The
expansion
over
s-channel
operators
′
SL′ J
(k
)
is
a
representation
of these
(k)
⊗
Q
QSLJ
µ1 ...µJ
µ1 ...µJ
(S,L′ L,J)
amplitudes through partial-wave amplitudes A0
(s)
(S,L′ L,J)
and A1
(s).
i) Partial-wave amplitudes N N → N N for J = L.
(S,LL,J)
For J = L, the amplitude AI
(s) with I = 0, 1
satisfying the s-channel unitarity condition are identical to those for nucleon-antinucleon scattering, eqs. (27)
and (28), except for the factor 12 in the amplitude originating from Fermi-Dirac statistics.
(S,LL,J)
AI
(S,LL,J)
(s) =
KI
1−
i
2
(s)
(S,LL,J)
(S,LL,J)
ρN N
(s)KI
(s)
(S,LL,J)
AI
(s)
=
1−i
.
(29)
3.4 Nucleon-nucleon scattering amplitude
The pp and nn systems are pure I = 1 states, while the pn
is a superposition of two states with total isospins I = 0
(31)
ii) Partial-wave amplitudes for S = 1, J = L ± 1.
In this case, four partial amplitudes form a 2 × 2 matrix
given by
b(S=1,L=J±1,J) (s) =
A
I
(S=1,J−1→J−1,J)
(S=1,J−1→J+1,J)
A
(s), AI
(s) I
.
(S=1,J+1→J−1,J)
(S=1,J+1→J+1,J)
A
(s), AI
(s) I
(32)
The K-matrix representation reads
b (S=1,L=J±1,J) (s)
b(S=1,L=J±1,J) (s) = K
A
I
I
−1
i (S=1,L=J±1,J)
(S=1,L=J±1,J)
b
× I − ρbN N
(s)KI
(s)
(33)
2
with the following definitions:
b (S=1,L=J±1,J) (s) =
K
I
(S=1,J−1→J−1,J)
(S=1,J−1→J+1,J)
K
(s), KI
(s) I
,
(S=1,J+1→J−1,J)
(S=1,J+1→J+1,J)
K
(s), KI
(s) I
(S=1,L=J±1,J)
ρbN N
(s) =
(S=1,J−1→J−1,J)
(S=1,J−1→J+1,J)
ρ
(s), ρN N
(s) NN
(S=1,J+1→J−1,J)
.
(S=1,J+1→J+1,J)
ρ
(s),
ρ
(s)
NN
NN
(S=1,L=J±1,J)
(S,LL,J)
KI
(s)
(S,LL,J)
(S,LL,J)
ρN N̄
(s)KI
(s)
.
Note that the matrices ρbI
b (S=1,L=J±1,J) (s) are symmetrical:
K
I
(S=1,J−1→J+1,J)
ρN N
(S=1,J−1→J+1,J)
KI
(S=1,J+1→J−1,J)
(s) = ρN N
and
(s),
(S=1,J+1→J−1,J)
(s) = KI
(s)
(34)
(s).
(35)
Let us emphasize that the definitions of the phase spaces
(S,L→L′ ,J)
for N N and N N̄ systems coincide: ρN N
(s) =
134
The European Physical Journal A
(S,L→L′ ,J)
ρN N̄
(s). In the equation imposing the unitarity condition (as well as in the K-matrix representation), the
identity of particles in the N N systems is taken into account directly by the factor 1/2. The unitarity conditions
for the ΛΛ̄, ΛΛ and N N̄ two-channel partial-wave amplitudes for S = 1 and J = L ± 1 are written similarly.
4 Nucleon-nucleon interaction operators
In this section, the proton-proton interaction operators are
constructed. These operators are constructed using angular momentum and spin operators, whose properties are
discussed below.
4.1 Angular-momentum operators
The angular-dependent part of the wave function of the
composite state is described by operators constructed using relative momenta of particles and the metric tensor.
(L)
Such operators (we denote them as Xµ1 ...µL , where L is the
angular momentum) are called angular-momentum operators; they correspond to irreducible representations of the
Lorentz group [145,147]. They satisfy the following properties [145]: i) Symmetry with respect to permutation of
any two indices:
Xµ(L)
= Xµ(L)
.
1 ...µi ...µj ...µL
1 ...µj ...µi ...µL
(36)
ii) Orthogonality to the total momentum of the system,
P = k1 + k2 :
Pµi Xµ(L)
= 0.
(37)
1 ...µi ...µL
The traceless property for the summation over two any
indices:
gµi µj Xµ(L)
= 0.
(38)
1 ...µi ...µj ...µL
Let us consider a one-loop diagram describing the decay of a composite system into two spinless particles which
propagate and then form again a composite system. The
decay and formation processes are described by angular
momentum operators. Due to the conservation of quantum numbers, this amplitude must vanish for initial and
final states with different spin. The S-wave operator is a
scalar and can be taken as a unit operator. The P -wave
operator is a vector. In the dispersion relation approach, it
is sufficient that the imaginary part of the loop diagram,
with S- and P -wave operators as vertices, is equal to 0. In
the case of spinless particles this requirement entails
Z
dΩ (1)
X = 0,
(39)
4π µ
where the integral is taken over the solid angle of the relative momentum. In general, the result of such an integration is proportional to the total momentum of the system
Pµ (the only external vector):
Z
dΩ (1)
X = λPµ .
(40)
4π µ
Convoluting this expression with Pµ and demanding λ =
0, we obtain the orthogonality condition (37). The orthogonality between the D- and S-waves is provided by the
traceless condition (38); conditions (37), (38) provide the
orthogonality for all operators with different angular momenta.
The orthogonality condition (37) is automatically fulfilled if the operators are constructed from the relative
⊥
momenta kµ⊥ and tensor gµν
. Both of them are orthogonal
to the total momentum of the system,
√ see eq. (3). In the
c.m. system, where P = (P0 , P) = ( s, 0), the vector k ⊥
is space like: k ⊥ = (0, k).
The operator for L = 0 is a scalar (for example a unit
operator), and the operator for L = 1 is a vector, which
can be constructed from kµ⊥ only. The orbital angularmomentum operators for L = 0 to 3 are:
X (0) = 1,
Xµ(2)
1 µ2
Xµ(1) = kµ⊥ ,
1 2 ⊥
⊥ ⊥
kµ1 kµ2 − k⊥ gµ1 µ2 ,
3
3
=
2
Xµ(3)
=
1 µ2 µ3
k2
5 ⊥ ⊥ ⊥
kµ1 kµ2 kµ3 − ⊥ gµ⊥1 µ2 kµ⊥3 + gµ⊥1 µ3 kµ⊥2 + gµ⊥2 µ3 kµ⊥1 .
2
5
(41)
(L)
The operators Xµ1 ...µL for L ≥ 1 can be written in the
form of a recurrent relation:
Xµ(L)
= kα⊥ Zµα1 ...µL ,
1 ...µL
Zµα1 ...µL =
L
X
2L − 1
L2
Xµ(L−1)
g⊥
1 ...µi−1 µi+1 ...µL µi α
i=1
!
L
X
2
⊥
(L−1)
−
g
X
. (42)
2L − 1 i,j=1 µi µj µ1 ...µi−1 µi+1 ...µj−1 µj+1 ...µL α
i<j
The convolution equality reads
2
Xµ(L)
k ⊥ = k⊥
Xµ(L−1)
.
1 ...µL µL
1 ...µL−1
(43)
Based on eq. (43) and taking into account the traceless
(L)
property of Xµ1 ...µL , one can write down the orthogonality-normalisation condition for orbital angular operators
Z
′
dΩ (L)
)
2L
X
(k ⊥ )Xµ(L1 ...µ
(k ⊥ ) = δLL′ αL k⊥
,
L′
4π µ1 ...µL
αL =
L
Y
2l − 1
l=1
l
.
(44)
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
Iterating eq. (42), one obtains the following expression for
(L)
the operator Xµ1 ...µL :
⊥
⊥
⊥ ⊥ ⊥ ⊥
(k
)
=
α
Xµ(L)
L kµ1 kµ2 kµ3 kµ4 . . . kµL
1 ...µL
k2
g ⊥ k ⊥ k ⊥ . . . kµ⊥L
− ⊥
2L − 1 µ1 µ2 µ3 µ4
+gµ⊥1 µ3 kµ⊥2 kµ⊥4 . . . kµ⊥L + . . .
4
k⊥
g ⊥ g ⊥ k ⊥ k ⊥ . . . kµ⊥L
+
(2L − 1)(2L − 3) µ1 µ2 µ3 µ4 µ5 µ6
+gµ⊥1 µ2 gµ⊥3 µ5 kµ⊥4 kµ⊥6 . . . kµ⊥L + . . . + . . . .
(45)
4.2 Projection operators and boson propagator
...µL
The projection operator Oνµ11...ν
is constructed from the
L
⊥
metric tensors gµν and it has the following properties:
...µL
Xµ(L)
Oνµ11...ν
= Xν(L)
,
1 ...µL
L
1 ...νL
α1 ...αL
µ1 ...µL
L
Oαµ11 ...µ
...αL Oν1 ...νL = Oν1 ...νL .
(46)
Taking into account the definition of the projection operators (46) and the properties of the X-operators (45), we
obtain
...µL
kµ1 . . . kµL Oνµ11...ν
=
L
1 (L)
X
(k ⊥ ).
αL ν1 ...νL
(47)
This equation presents the basic property of the projection
operator: it projects any operator with L indices onto the
partial-wave operator with angular momentum L.
For the lowest states,
⊥
Oνµ = gµν
1
2 ⊥
⊥
⊥
⊥
⊥
⊥
=
g
g
+ gµ1 ν2 gµ2 ν1 − gµ1 µ2 gν1 ν2 .
2 µ1 ν1 µ2 ν2
3
(48)
O = 1,
Oνµ11νµ22
For higher states, the operator can be calculated using the
recurrent expression
...µL
Oνµ11...ν
L
1
= 2
L
−
×
L
X
i<j
k<m
Let us introduce the positive value |k|2 :
2
|k|2 = −k⊥
=
[s − (m1 + m2 )2 ][s − (m1 − m2 )2 ]
. (51)
4s
In the c.m.s. of the reaction, k is the momentum of a
particle. In other systems,
we use this definition only in
p
2 ; clearly, |k|2 is a relativistically
the sense of |k| ≡ −k⊥
invariant positive value. Then, eq. (50) can be written as
Z
dΩ (L)
αL |k|2L
...µL
⊥
Xµ1 ...µL (k ⊥ )Xν(L)
(−1)L Oνµ11...ν
.
(k
)
=
L
1 ...νL
4π
2L + 1
(52)
The tensor part of the numerator of the boson propagator is defined by the projection operator. Let us write
it as
...µL
...µL
Fνµ11...ν
= (−1)L Oνµ11...ν
.
(53)
L
L
This definition guarantees that the width of a resonance
(calculated using the decay vertices) has a positive value.
4.3 Spin operators of two-fermion systems
The wave function for fermion particles with the momentum p is described as Dirac bispinor:
(p0 + m)ω
1
,
u(p) = √ √
(pσ)ω
2m p0 + m
ū(p) =
(ω ∗ (p0 + m), −ω ∗ (pσ))
√ √
.
2m p0 + m
(54)
To construct the operators for the two-fermion system,
one should also introduce the charge-conjugated bispinors:
i
(pσ)ω ′
u(−p) = √ √
,
2m p0 + m (p0 + m)ω ′
(ω ′∗ (pσ), −ω ′∗ (p0 + m), )
√ √
.
2m p0 + m
(55)
Here, the ω and ω ′ represent 2-dimensional spinors, ω ∗
and ω ′∗ are the conjugated and transposed spinors. The
normalisation condition can be written as
i,j=1
4
(2L − 1)(2L − 3)
L
X
metric tensor only. Therefore, it must be proportional
to the projection operator. After straightforward calculations, we obtain
Z
2L
dΩ (L)
αL k⊥
Xµ1 ...µL (k ⊥ )Xν(L)
Oµ1 ...µL . (50)
(k ⊥ ) =
1 ...νL
4π
2L + 1 ν1 ...νL
ū(−p) = −i
...µi−1 µi+1 ...µL
gµ⊥i νj Oνµ11...ν
j−1 νj+1 ...νL
135
...µi−1 µi+1 ...µj−1 µj+1 ...µL
gµ⊥i µj gν⊥k νm Oνµ11...ν
k−1 νk+1 ...νm−1 νm+1 ...νL
!
.
(49)
The product of two X-operators integrated over the
solid angle (which is equivalent to an integration over internal momenta) depends on external momenta and the
ū(p)u(p) = −ū(−p)u(−p) = 1,
X
m + p̂
u(p)ū(p) =
,
2m
polarisations
X
polarisations
where p̂ = pµ γµ .
u(−p)ū(−p) =
−m + p̂
,
2m
(56)
136
The European Physical Journal A
Let us consider a two-fermion system with the total momentum P = k1 + k2 and relative momentum
k = (k1 − k2 )/2, where k1 and k2 are their individual momenta, P 2 = s. For the sake of generality, let the fermions
have different masses, m1 and m2 . The two-fermion system can form two possible spin state, S = 0 (singlet
state) and S = 1 (triplet state). The spin operators for
these states act between bispinor and charge-conjugated
bispinor, ū(−k1 )S (i) u(k2 ) and have the following form:
S (0) = iγ5 ,
S (1) = γµ⊥ ,
⊥
γµ⊥ = γν gµν
.
(58)
It should be noted that u(−k1 ) and u(k2 ) have opposite
parities, so ū(−k1 )γ5 u(k2 ) is a scalar and ū(−k1 )u(k2 ) is
a pseudoscalar.
As is shown below, the γµ operator leads to the mixture
of states with total momentum L + 1 and L − 1. So, let us
introduce the operator for the pure S = 1 state:
!
4skα⊥ kβ⊥
(1)
⊥
⊥
√
, (59)
Spure = Γα = γβ gαβ −
M ( s + M )(s − δ 2 )
where M = m1 + m2 and δ = m1 − m2 . In the nonrelativistic limit, this operator is equal to the spin-1 operator
σ and satisfies the orthogonality of the triplet states with
the same parity.
4.4 Operators for 1 LJ states
In case of a singlet spin state, the total angular momentum
J is equal to the orbital angular momentum L between the
two particles. The ground state of such a system is 1 S0
(2S+1 LJ ) and the corresponding operator is just equal to
the spin-0 operator S (0) of eq. (57). For states with orbital
momentum L, the operator is constructed as a product
of the spin-0 operator S (0) and the angular-momentum
operator Xµ1 ...µJ :
Vµ1 ...µJ =
2J + 1
iγ5 Xµ(J)
(k ⊥ ).
1 ...µJ
αJ
(60)
The normalisation factor which is introduced here simplifies the expression for the loop diagram (see below).
4.5 Operators for 3 LJ states with J = L
The ground state in this series is 3 P1 , so one should make
(1)
(1)
a convolution of two vectors, Sµ and Xµ that creates
a J = 1 state (vector state). In this case the vertex operator is equal to εν1 ηξγ γη kξ⊥ Pγ . For states with higher orkξ⊥
(J)
...µJ
VµL=J
.
∼ εν1 ηξγ γη Xξν2 ...νJ Pγ Oνµ11...ν
J
1 ...µJ
(J)
...µJ
...µJ
= εν1 ηξγ kξ⊥ Xν(J−1)
εν1 ηξγ Xξν2 ...νJ Oνµ11...ν
Oνµ11...ν
J
2 ...νJ
J
×
2J − 1
.
J
(62)
Finally, using eq. (C.1) the vertex operator can be written
as
s
(2J + 1)J iεαηξγ γη kξ⊥ Pγ Zµα1 ...µJ
L=J
√
,
(63)
Vµ1 ...µJ =
(J + 1)αJ
s
where normalisation parameters are again introduced. Note that due to the property of the antisymmetric tensor
εαηξγ the vertex given by eq. (63) does not change, if one
replaces γη by the pure spin operator Γη .
4.6 Operators for 3 LJ states with L < J and L > J
To construct operators for 3 LJ states, one should multiply
the spin operator γα by the orbital momentum operator
for L = J + 1. So one has
...µJ
VµL<J
∼ γν1 Xν(J−1)
Oνµ11...ν
.
1 ...µJ
2 ...νJ
J
(64)
Using eq. (C.1) from appendix C, we write the vertex operator in the form
r
J
α
VµL<J
=
γ
Z
,
(65)
α µ1 ...µJ
1 ...µJ
αJ
and for the pure spin operator as
ṼµL<J
= Γα Zµα1 ...µJ
1 ...µJ
r
J
.
αJ
(66)
The normalisation constant is chosen to facilitate the calculation of loop diagrams containing such a vertex.
To construct such an operator for L > J one should
reduce the number of indices in the orbital operator by a
convolution with the spin operator:
VµL>J
1 ...µJ
= γα Xαµ1 ...µJ
r
J +1
,
αJ
(67)
r
J +1
.
αJ
(68)
and for pure spin state:
(J)
Xξν2 ...νJ
bital momenta, one needs to replace
and
by
perform a full symmetrisation over ν1 , ν2 , . . . , νJ indices,
which can be done by a convolution with the projection
(61)
Using eqs. (45) and (47), one has
(57)
where
r
...µL
operator Oνµ11...ν
. The general form of such a vertex is
L
hence given by
ṼµL>J
1 ...µJ
= Γα Xαµ1 ...µJ
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
5 Calculation of the NN → NN amplitude
where
Let us consider N (q1 )N (q2 ) → N (k1 )N (k2 ) transition amplitude with q1 , q2 , k1 , k2 being the nucleon momenta, and
k = (k1 − k2 )/2, q = (q1 − q2 )/2. In this section the structure of such amplitude for different initial and final states
is derived.
We start by considering N N → N N amplitudes for a
singlet state. For the 1 LJ state with L = J, the amplitude
is given by
As = ū(−q2 )Vµ1 ...µJ (q) u(q1 )
...µJ
×Fνµ11...ν
ū(k1 )Vν1 ...νJ (k) u(−k2 ).
J
As = −ū(−q2 ) γ5 u(q1 ) ū(k1 ) γ5 u(−k2 )
×(|k| |q|)n (2J + 1)PJ (z),
(70)
AL<J
= ū(−q2 )VµL<J
(q) u(q1 )
t
1 ...µJ
(71)
Using eq. (C.1), this amplitude can be written in the form
fi ai (|k||q|)J−1 ,
where
f1 = ū(−q2 ) γµ u(q1 ) ū(k1 ) γµ u(−k2 ),
f2 = ū(−q2 ) q̂ u(q1 ) ū(k1 ) q̂ u(−k2 ),
f3 = ū(−q2 ) k̂ u(q1 ) ū(k1 ) k̂ u(−k2 ),
f4 = ū(−q2 ) q̂ u(q1 ) ū(k1 ) k̂ u(−k2 ),
(73)
Here,
P ′′ (z)
PJ′ (z)
,
a2 = − J−1 2 ,
J
J|q|
1
′
a4 =
(P ′′ (z) − 2PJ−1
(z)),
J|k||q| J−2
a1 = −
′′
PJ−1
(z)
,
J|k|2
P ′′ (z)
.
a5 = J
J|k||q|
(74)
a3 = −
Likewise, the transition amplitude for the triplet state 3 LJ
with L > J is as follows:
AL>J
=
t
5
X
i=1
fi ai (|k||q|)J+1 ,
fi ai |k|J+1 |q|J−1 ,
5
X
fi ai |k|J−1 |q|J+1 ,
i=1
...µJ
Amix
= VµL>J
(q) (−1)J Oνµ11...ν
VνL<J
(k) =
t
1 ...µJ
J
1 ...νJ
where
r
a1 = −
a3 =
r
J PJ′ (z)
,
J + 1 2J − 1
′′
(z)
J PJ+1
,
J + 1 2J − 1
(75)
(77)
r
′′
(z)
J PJ−1
,
J + 1 2J − 1
r
J PJ′′ (z)
a4 = a5 = −
.
J + 1 2J − 1
(78)
a2 =
If one uses the operators based on the pure spin-1 operator
given by eqs. (66) and (68), the functions f1 , f2 , . . . , f5 are
substituted by the new functions f˜1 , f˜2 , . . . , f˜5 as follows:
f˜i = fj Mji ,
(72)
i=1
f5 = ū(−q2 ) k̂ u(q1 ) ū(k1 ) q̂ u(−k2 ).
5
X
i=1
where z = (kq)/(|k||q|) is the cosine of scattering angle
in c.m. system.
The transition amplitude for the triplet state 3 LJ with
L < J is the following:
...µJ
×Fνµ11...ν
ū(k1 )VνL<J
(k) u(−k2 ).
J
1 ...νJ
...µJ
Amix
= VµL<J
(q) (−1)J Oνµ11...ν
VνL>J
(k) =
t
1 ...µJ
J
1 ...νJ
(69)
For the sake of simplicity, we omit here and below the
invariant part of the amplitude, which will be considered
later on. Using eq. (60), the amplitude reads
5
X
′
P ′′ (z)
PJ+1
(z)
,
a2 = a3 = J+1
,
J +1
J +1
1
′
a4 = −
(P ′′ (z)+(2J + 1)PJ+1
(z)),
J +1 J
′′
P (z)
a5 = − J
.
(76)
J +1
If the spin-1 operator is defined as γν , there is a mixture
between two triplet amplitudes with L > J and L < J.
The corresponding transition amplitudes are given by
a1 = −
5.1 Structure of the amplitude
=
AL<J
t
137
where the transition matrix Mji is equal

1
0
0
0
√

s

−κ
0
0
M

√

s

−κ
0
0

M

√
√
⊥ ⊥
⊥ ⊥
 κ2 (k ⊥ q ⊥ ) − sκ(k q ) − sκ(k q ) s

M
M
M2
0
0
0
0
(79)
to
0


−κ(k ⊥ q ⊥ ) 


−κ(k ⊥ q ⊥ ) 
.

κ2 (k ⊥ q ⊥ )2 

1
(80)
Then, the transition amplitudes for the 3 LJ triplet state
with L < J and L < J are
AL<J
=
t
5
X
fj Mji ai (|k||q|)J−1 ,
i,j=1
AL>J
=
t
5
X
fj Mji ai (|k||q|)J+1 .
(81)
i,j=1
The transition amplitude for the 3 LJ triplet state with
L = J is given by
AL=J
= ū(−q2 )VµL=J
(q) u(q1 )
t
1 ...µJ
...µJ
×Fνµ11...ν
ū(k1 )VνL=J
(k) u(−k2 ).
J
1 ...νJ
(82)
138
The European Physical Journal A
Using expressions given in appendix C, this amplitude can
be written in the form
AL=J
= (f1 a1 + f5 a5 + f6 a6 )(|k||q|)J ,
t
(83)
where
f3 = ū(−q2 ) γµ u(q1 ) ū(k1 ) γν u(−k2 ) nµ nν ,
nµ =
εµαβγ kα qβ Pγ
√
,
s |k| |q|
a5 = −
a1 = −
2J + 1
P ′ (z),
(J + 1)J|k| |q| J
2J + 1
zPJ′ (z),
(J + 1)J
a6 =
2J + 1
P ′′ (z).
(J + 1)J J
(84)
5.2 One-loop diagrams
The calculation of one-loop diagrams for different vertex
operators is an important step in the construction of a
unitary N N amplitude. Let us start from the loop diagram for the singlet state and derive all expressions for
the case of different particle masses, m1 and m2 . Taking
into account that
h
i
Sp γ5 (m1 + k̂1 )γ5 (m2 − k̂2 ) = 2(s − δ 2 ),
(85)
where δ = m1 − m2 , the one-loop diagram for the singlet
state is equal to
Z
i
dΩ h
−
Sp Vµ1 ...µJ (k ⊥ )(m1 + k̂1 )Vν1 ...νJ (k ⊥ )(m2 − k̂2 )
4π
...µJ
= 2(s − δ 2 )|k|2J Oνµ11...ν
(−1)J .
(86)
J
The factor (−1) is related to the fermionic nature of the
baryon in the loop.
To calculate one-loop diagrams for different triplet states, the following relations are helpful:
h
i
⊥
Sp γµ⊥ (m1 + k̂1 )γν⊥ (m2 − k̂2 ) = 2(s − δ 2 )gµν
+ 8kµ⊥ kν⊥ ,
h
i
⊥
Sp Γµ (m1 + k̂1 )Γν (m2 − k̂2 ) = 2(s − δ 2 )gµν
.
(87)
Using these relations and eqs. (C.6)-(C.12) given in appendix C, we obtain the following results for the L < J
and L > J states:
Z
i
dΩ h L<J
Sp Vµ1 ...µJ (m1 + k̂1 )VνL<J
(m
−
k̂
)
=
−
2
2
1 ...νJ
4π
8J|k|2
...µJ
2(s − δ 2 ) −
|k|2(J−1) Oνµ11...ν
(−1)J ,
J
2J + 1
Z
i
dΩ h L>J
−
(m
−
k̂
)
=
Sp Vµ1 ...µJ (m1 + k̂1 )VνL>J
2
2
...ν
1
J
4π
8(J + 1)|k|2
...µJ
|k|2(J+1) Oνµ11...ν
(−1)J . (88)
2(s−δ 2 )−
J
2J +1
In case of spin-1 operators, the two triplet states with the
same parity are not orthogonal to each other; the interference loop diagram is equal to
Z
i
dΩ h L<J
Sp Vµ1 ...µJ (m1 + k̂1 )VνL>J
(m
−
k̂
)
=
−
2
2
1 ...νJ
4π
p
J(J + 1) 2(J+1) µ1 ...µJ
|k|
Oν1 ...νJ (−1)J .
(89)
8
2J + 1
The one-loop diagram for the L = J triplet state is given
by
Z
i
dΩ h
Sp Vµ1 ...µJ (m1 + k̂1 )(k ⊥ )Vν1 ...νJ (k ⊥ )(m2 − k̂2 ) =
−
4π
...µJ
2(s − δ 2 )|k|2J Oνµ11...ν
(−1)J .
(90)
J
Direct calculations show that the transition loop diagrams
between the triplet state with L = J and the triplet states
with L > J and L < J vanish. For vertex operators describing pure spin states (66) and (68), one has the following one-loop diagrams:
Z
i
dΩ h L<J
(m
−
k̂
)
=
Sp Ṽµ1 ...µJ (m1 + k̂1 )ṼνL<J
−
2
2
...ν
1
J
4π
−
Z
...µJ
2(s − δ 2 )|k|2(J−1) Oνµ11...ν
(−1)J ,
J
i
dΩ h L>J
Sp Ṽµ1 ...µJ (m1 + k̂1 )ṼνL>J
(m2 − k̂2 ) =
1 ...νJ
4π
...µJ
2(s − δ 2 )|k|2(J+1) Oνµ11...ν
(−1)J .
J
(91)
5.3 Cross-sections
The expressions for one-loop diagrams can be used to calculate cross-sections for different spin-orbital momentum
states. The cross-section is given by
Z
|k| ρ(s)
dΩ 2
(2π)4 |A|2
√ dΦ =
|A| .
(92)
dσ =
|q| 16πs
4π
4|q| s
To calculate the amplitude squared |A|2 , one can use the
expressions for the one-loop diagram given by eqs. (86),
(91), (89) and (90).
For the 1 LJ state, one has
dσ =
2J + 1
|k|2J+1 |q|2J−1 s.
64πm2
(93)
For the 3 LJ state (L = J), the result is
dσ =
2J + 1
|k|2J+1 |q|2J−1 s.
64πm2
(94)
The decay of the 3 LJ states with (L < J) and 3 LJ (L > J)
is determined by the sum of two vertices:
=J
AL6
= λ1 VµL<J
+ λ2 VµL>J
tr
1 ...µJ
1 ...µJ
(95)
Then, the cross-section is equal to
dσ = λ21 dσ11 + λ22 dσ22 + λ1 λ2 (dσ12 + dσ21 ),
(96)
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
where
dσ11
dσ22
dσ12
dσ21
p(k )
p(q )
1
2J + 1
|k|2J−1 |q|2J−3
=
2
256πsm
8J|q|2
8J|k|2
2s −
,
× 2s −
2J + 1
2J + 1
2J + 1
=
|k|2J+3 |q|2J+1
2
256πsm
8(J + 1)|q|2
8(J + 1)|k|2
2s −
,
× 2s −
2J + 1
2J + 1
J(J + 1)
1
|k|2J+3 |q|2J+1 ,
=
2
4πsm 2J + 1
1
J(J + 1)
=
|q|2J+3 |k|2J+1 .
(97)
4πsm2 2J + 1
For pure spin-1 operators, VµL<J
and VµL>J
, the cross1 ...µJ
1 ...µJ
section reads
dσ =
λ21 dσ11
+
λ22 dσ22
,
(98)
where
139
1
+
K (q )
2
R
p(k2)
Λ(q )
3
+
Fig. 1. Reaction pp → pK Λ: pp scattering with production
of a resonance R in the intermediate state.
for 5 LJ (J = L − 2) by
Wµ(5)
= ψ̄α1 (k1 )γα2 Xα(J+2)
u(−k2 ),
1 ...µJ
1 α2 ν1 ...νJ
1
Vµ(5)α
= γα2 Xα(J+2)
,
1 ...µJ
1 α2 ν1 ...νJ
(104)
for 5 LJ (J = L) by
(J)
dσ11
dσ22
ν1 ξ
J
Wµ(6)
= ψ̄α (k1 )γβ Oαβ
Xξν2 ...νJ Oµν11...ν
...µJ u(−k2 ),
1 ...µJ
(2J + 1)s
=
|k|2J−1 |q|2J−3 ,
64πm2
(2J + 1)s
|k|2J+3 |q|2J+1 .
=
64πm2
6 Decay into 3/2
+
and 1/2
+
(J)
ν1 ξ
J
Vµ(6)α
= γβ Oαβ
Xξν2 ...νJ Oµν11...ν
...µJ ,
1 ...µJ
(99)
(105)
for 5 LJ (J = L − 1) by
Wµ(7)
=
1 ...µJ
particles
(J)
α1 α2
J
ψ̄α1 (k1 )γα2 Xξν2 ...νJ Oµν11...ν
iεν1 βτ η kτ Pη Oβξ
...µJ u(−k2 ),
+
Let k1 be the momentum of the 3/2 particle and k2 the
momentum of 1/2+ . In this case, there are two spin states,
S = 1 and S = 2. Let us start from the S = 1 states. Such
states are constructed using the vector spinors ψα for 3/2spin particle and spin operators.
For 3 LJ (J = L − 1), that corresponds to
− + − +
(0 , 1 , 2 , 3 . . .) states, and the operators read
Wµ(1)
= ψ̄α (k1 )Vµ(1)α
u(−k2 ),
1 ...µJ
1 ...µJ
(J+1)
.
= iγ5 Xαµ
Vµ(1)α
1 ...µJ
1 ...µJ
(100)
...αJ
u(−k2 ),
= ψ̄α1 (k1 )iγ5 Xα(J−1)
Oµα11...µ
Wµ(2)
J
1 ...µJ
2 ...αJ
(101)
where the projection operator is needed for index symmetrisation. For 3 LJ (J = L) (1− , 2+ , 3− , 4+ . . .), the operators can be expressed as
...αJ
Wµ(3)
= γ5 εα1 βξη ψ̄β (k1 )kξ Pη Xα(J−1)
Oµα11...µ
u(−k2 ),
1 ...µJ
2 ...αJ
J
...αJ
Vµ(3)β
= γ5 εα1 βξη kξ Pη Xα(J−1)
Oµα11...µ
.
1 ...µJ
2 ...αJ
J
(102)
In case of S = 2, there are five operators. For 5 LJ
(J = L + 2) the operators are given by
J
Wµ(4)
= ψ̄α1 (k1 )γα2 Oνα11να22 Xν(J−2)
Oµν11...ν
...µJ u(−k2 ),
1 ...µJ
3 ...νJ
1
J
Vµ(4)α
= γα2 Oνα11να22 Xν(J−2)
Oµν11...ν
...µJ ,
1 ...µJ
3 ...νJ
and for 5 LJ (J = L + 1) by
Wµ(8)
=
1 ...µJ
α1 α2
J
iεν1 βτ η kτ Pη Oβν
ψ̄α1 (k1 )γα2 Xν(J−2)
Oµν11...ν
...µJ u(−k2 ),
3 ...νJ
2
α1 α2
1
J
Vµ(8)α
= iεν1 βτ η kτ Pη Oβν
γα2 Xν(J−2)
Oµν11...ν
...µJ . (107)
1 ...µJ
3 ...νJ
2
For 3 LJ (J = L + 1), the operators are given by
...αJ
1
Vµ(2)α
= iγ5 Xα(J−1)
Oµα11...µ
,
1 ...µJ
2 ...αJ
J
(J)
α1 α2
1
J
Vµ(7)α
= iεν1 βτ η kτ Pη Oβξ
γα2 Xξν2 ...νJ Oµν11...ν
...µJ , (106)
1 ...µJ
(103)
The one-loop diagram amplitudes for the corresponding operators are calculated in appendix D.
7 Example: amplitude for the reaction
pp → pK+ Λ
Let us start from pp scattering with the production of a
resonance R in the intermediate state which decays into
K + Λ. The diagram for the process is shown in fig. 1. Consider the partial-wave amplitude for the pp having quantum numbers J = n, L and S in the initial state. The
general form of the angular dependent part of this partial
amplitude is
ū(−k1 )Q(S,L,J)
ν1 ...νn u(k2 ) ū(q3 )Ñα1 ...αm (R → KΛ)
(S,L,J)
...αm
×Fβα11...β
(q2 + q3 )Qβ1 ...βm ν1 ...νn u(−q1 )
m
−{k1 ⇔ k2 } ,
(108)
140
The European Physical Journal A
p(q )
p(k )
where
1
1
(n+1)
Vα(1−)µ
(k ⊥ ) = γξ γµ⊥ Xξα1 ...αn (k ⊥ ) ,
1 ...αn
π, ρ (kt)
p(k )
R
2
+
(n+1)
Vα(2−)µ
(k ⊥ ) = Xµα
(k ⊥ ) .
1 ...αn
1 ...αn
K (q2)
Here, kt = Q23 − k2 is the ρ-meson momentum. In case of
a spin-1/2 resonance in the intermediate state, one should
use eq. (112) with n = 0. For the upper operator, one has
Λ (q )
3
+
Fig. 2. Reaction pp → pK Λ: t-channel exchange diagram.
where P = q1 + q2 + q3 = k1 + k2 . The resonance R with
spin J = m + 1/2 is produced in the intermediate state
and decays into a final-state meson and a nucleon.
(i−)µ ⊥
A(i−)
(q1 )u(k1 )ρµ ,
i = 1, 2,
upper = ū(q1 )V
k
k
1
1µ 1ν
⊥
.
q1µ
= (q1 − kt )ν gµν −
2
k12
ѵ+1 ...µn = Xµ(n)
,
1 ...µn
1/2− , 3/2+ , 5/2− , . . . ,
(n+1)
ѵ−1 ...µn = iγν γ5 Xνµ
,
1 ...µn
1/2+ , 3/2− , 5/2+ , . . . .
(109)
Let us write down the amplitude for the 1/2+ resonance in the intermediate state. In this case, one finds
X
− ⊥
ū(−k1 )Q(S,L,J)
ν1 ...νn u(k2 ) ū(q3 )Ñ (q23 )
(113)
Summing over the polarisations yields
X
(S,L,J)
The initial pp state operator Qν1 ...νn is defined by
eq. (60) for S = 0 and eqs. (63), (65), (67) for S = 1.
If the resonance in the intermediate state has the spin
1/2 (m = 0), the same expressions define the Rp state operator. For the spin-3/2 resonance in the intermediate state,
(S,L,J)
the operator Qβ1 ...βm ν1 ...νn is defined by eqs. (100)–(107).
The operators for the R → 0− + 1/2+ transitions were
defined in [147]:
M=
(112)
polarisations
ρα ρβ = −gαβ +
ktα ktβ
.
kt2
(114)
Finally, we arrive at the following amplitude for the ρ
exchange:
(i−)µ ⊥
⊥
A(i)
(q1 )u(k1 )ū(q3 )Ñ − (q23
)
ρ = ū(q1 )V
√
q̂2 + q̂3 + s23
BW (s23 )V (i−)ν (k2⊥ )u(k2 )
×
√
2 s23
ktµ ktν ,
i = 1, 2.
(115)
× − gµν +
kt2
In case of t-channel exchange of a pseudoscalar meson, π,
one should substitute the operator V (i−)µ by Ñ − , so we
have
⊥
Aπ = ū(q1 )Ñ − (q1⊥ )u(k1 )ū(q3 )Ñ − (q23
)
√
q̂2 + q̂3 + s23
×
BW (s23 )Ñ − (k2⊥ )u(k2 ).
√
2 s23
(116)
S,L,J
√
q̂2 + q̂3 + s23
(S,L,J)
BW (s23 )Q(S,L,J)
(s, s23 )
√
ν1 ...νn u(−q1 )A
2 s23
−{k1 ⇔ k2 },
Q23µ Q23ν ⊥
q23
= (q2 − q3 )ν gµν −
, Q23 = q2 + q3 ,
s23
(110)
×
where A(S,L,J) (s, s23 ) is the partial amplitude for pp → Rp
and BW (s23 parameterises the resonance R).
Another type of processes, which may contribute to
this reaction, is the t-channel exchange of pseudoscalar
and vector particles (π and ρ). This diagram is shown in
fig. 2.
First, consider the exchange of a ρ meson. The vertex operators for the transition baryon → vector meson +
baryon are given in [147]. Thus the pρ → R operators are
given by
(i−)
Alower = ū(Q23 )V (i−)µ (k2⊥ )u(k2 )ρµ ,
i = 1, 2,
Q23µ Q23ν
1
⊥
,
(111)
k2µ
= (k2 − kt )ν gµν −
2
s23
8 Triangle diagram amplitude with
pion-nucleon rescattering: logarithmic
singularity
In the amplitudes describing production of three-particle
final states, the unitarity condition is fulfilled automatically when the final-state rescattering is properly taken
into account. However, rescattering may lead to singularities where the amplitude tends to infinity. The triangle
diagram with the ∆ in the intermediate state gives us an
example of this type of process: it has logarithmic
singu√
larity which under certain conditions ( s ∼ mN + m∆ )
can be near the√physical region.
Because of s ∼ mN +m∆ , we consider the amplitude
pp → N ∆ with L′ = 0 (the produced N ∆ system is in the
S-wave). The quantum numbers of the final state are then
restricted to
J P = 1+ , 2+ .
(117)
The initial pp system (I = 1) has
S = 0:
S = 1:
L = 0, 2, 4, . . . ,
L = 1, 3, 5, . . . ,
J P = 0+ , 2+ , 4+ , . . . ,
J P = 0− , 1− , 2− , 3− , . . . . (118)
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
Here,
π(pπ)
p(p2)
∆(p )
p(p’’ )
′⊥p′
∆
2
p(p )
1
Fig. 3. Triangle diagram with final-state pion-nucleon rescattering.
We thus consider the transition
pp (S = 0, L = 2, J
=2 )→
(S=0,S ′ =2,L=2,L′ =0,J=2)
pole
Apole
(s)
N N →N N π = CN N →N N π Gpp→N ∆
∆µν (p∆ )
′⊥p∆
× ū(p′1 )g∆ k1µ
γν ′ u(−p′2 )
2
m∆ − p2∆ − im∆ Γ∆
(2)
× ū(−p2 )iγ5 Xνν ′ (k)u(p1 ) .
(119)
pole
Here, the factor CN
N →N N π refers to the isotopic ClebschGordan coefficients, and
(120)
The numerator of the 3/2-spin fermion propagator is written in the form used in [146,147]:
∆µν (k) =
(S=0,S ′ =2,L=2,L′ =0,J=2)
(s)
−p̂′′ + m
∆µ′ ν (p′∆ )
γν ′ 2 2 ′′2
′2
− p∆ − im∆ Γ∆
m − p2 − i0
∆µ′′ ν ′′ (−p∆ )
′′⊥p′′
′⊥p∆
′
×g∆ k2µ′′ ∆ 2
g
k
u(−p
)
∆ 2ν ′′
2
m∆ − p2∆ − im∆ Γ∆
(2)
× ū(−p2 )iγ5 Xνν ′ (k)u(p1 ) .
(122)
′⊥p′
×g∆ k1µ′ ∆
m2∆
One can simplify (122) by extracting the numerator in
the singular point that corresponds to
m2 = p′′2
2 ,
m2π = kπ2 .
(125)
triangle
Atriangle
(s)
N N →N N π = CN N →N N π Gpp→N ∆
′⊥p′
× ū(p′1 )g∆ k1µ′ ∆ (tr)∆µ′ ν (p′∆ (tr))γν ′ (−p̂′′2 (tr) + m)g∆
∆µ′′ ν ′′ (−p∆ )
′⊥p∆
′
g∆ k2ν ′′ u(−p2 )
m2∆ − p2∆ − im∆ Γ∆
(2)
× ū(−p2 )iγ5 Xνν ′ (k)u(p1 )
Z
d4 kπ
1
1
×
4
2
2
2
i(2π) mπ − kπ − i0 m − (p∆ − kπ )2 − i0
1
× 2
.
(126)
m∆ − (P − p∆ + kπ )2 − im∆ Γ∆
′′⊥p′′
×k2µ′′ ∆ (tr)
′⊥p′
′′⊥p′′
The momenta k1µ′ ∆ (tr), p′∆ (tr), p′′2 (tr), k2µ′′ ∆ (tr) obey
the constraints (125). The integral in (126) corresponds to
the triangle diagram with spinless particles. Its calculation
is performed in appendix E.
(121)
The decay vertex g∆ is determined by the imaginary part
of the loop diagram ∆ → N π → ∆. For the sake of sim′
plicity, we change in (119) Γν ′ (k⊥
) → γν ′ ; however, using
definition (59) one can easily rewrite eq. (119) in a more
expanded form.
Taking into account the rescattering process in the amplitude (119), πN → ∆ → πN , one has the following
triangle diagram amplitude (see fig. 3):
triangle
Atriangle
N N →N N π = CN N →N N π Gpp→N ∆
Z
d4 kπ
1
′
× ū(p1 )
4
2
i(2π) mπ − kπ2 − i0
(124)
(S=0,S ′ =2,L=2,L′ =0,J=2)
The corresponding pole amplitude reads
γµ⊥
p∆ = p′2 + pπ = p′′2 + kπ ,
Then, eq. (122) reads
N ∆ (S ′ = 2, L′ = 0, J P = 2+ ).
1
k̂ + M∆ ⊥
− gµν
+ γµ⊥ γν⊥ ,
2M∆
3
kµ kν
⊥
⊥
= gµν
γν , gµν
= gµν −
2 .
M∆
p′∆ = p′1 + kπ ,
P = p′∆ + p′′2 .
m2∆ = p′2
∆,
+
′⊥p∆
⊥p∆ ′
k1µ
= gµµ
′ p1µ′ .
(123)
and
p(p’1)
P
′′⊥p′′
∆ ′′
k2µ′′ ∆ = gµ⊥p
′′ α p2α ,
′⊥p∆
∆ ′
k2ν
= gν⊥p
′′
′′ α p2α ,
p(p’2)
∆
⊥p′
k1µ′ ∆ = gµ′ α∆ p′1α ,
π(kπ)
∆(p’ )
141
9 Box diagram singularities in the reaction
NN → ∆∆ → NNππ
The primary aim of a partial-wave analysis is to extract
the pole singularities of amplitudes, thus determining resonances. Of course, the existence of other singularities like
threshold singularities should be taken into account. This
is possible using the K-matrix technique, see [150–152]
and references therein. Singularities due to resonances in
the intermediate state need a more sophisticated treatment.
The existence of triangle diagram singularities, which
may be located near the physical region of a three-particle
production reaction, was proven in [153,154]: these singularities diverge as ln(s − s0 ). Stronger singularities
(with a (s − s0 )−1/2 behaviour) are related to box diagrams [155,156].
Here, we present box diagram and triangle diagram
singular amplitudes for the reaction N N → ∆∆ →
N N ππ taking into account the spin structure in a way
142
The European Physical Journal A
p(p )
p(p )
3
p(P )
3
p(P )
1
1
∆(k )
∆(k’ )
π(k )
1
π(p )
1
1π
π(p )
1
1
π(p )
∆(k )
p(P )
p(P2)
p(p )
2
4
π(p )
2
2π
2
2
2
π(k )
∆(k’ )
p(p )
4
Fig. 5. Box diagram with pion-pion rescattering.
Fig. 4. Pole diagram for the reaction N N → ∆∆ → N N ππ.
9.2 Box diagram amplitude with pion-pion rescattering
which allows us to include these singular amplitudes into
partial-wave analyses (this was not yet done in [155,156]).
Let us introduce the following notations for the twopole and box diagrams in the reactions N N → ∆∆ →
N N ππ (see figs. 4 and 5).
The initial-state momenta are:
P1 + P2 = P,
1
(P1 − P2 ) = q. (127)
2
P 2 = W 2,
The final-state momenta:
2
(p1 + p3 ) = s13 ,
p1 + p3 = k1 ,
(p2 + p4 )2 = s24 ,
p2 + p4 = k2 ,
(p1 + p2 )2 = s,
2
(p1 + p3 + p2 ) = s4 ,
1
1
= −p3⊥k1 ,
k1⊥ = (p1 − p3 )⊥k1 = p⊥k
1
2
(p2 + p4 + p1 )2 = s1 ,
1
2
k2⊥ = (p2 − p4 )⊥k2 = p⊥k
= −p4⊥k2 ,
2
2
p1 + p2 = p.
(128)
Here, the symbol ⊥ki means the component of a vector
perpendicular to ki :
pµ⊥ki = pµ − kiµ
(ki p)
.
ki2
(129)
9.1 (NN)S-wave state with JP = 0+ , two-pole diagram
In pp collision with I = 1, the S-wave ∆∆ state is produced. First, consider the two-pole diagram of fig. 4. The
amplitude for the production and decay of two ∆-isobars,
N N → ∆∆ → N N ππ, omitting charge indices and corresponding Clebsch-Gordan coefficients, reads
= ū(−P2 )u(P1 ) GN N →∆∆ (W )
AN N →∆∆→(N π)(N π)
∆µν ′ (k1 )
⊥
× ū(p3 )g∆ k1µ
2
M∆ − s13 − iM∆ Γ∆
∆ν ′ ν (−k2 )
⊥
× 2
(−)k2ν
g∆ u(−p4 ) .
M∆ − s24 − iM∆ Γ∆
(130)
The box diagram amplitude with pion-pion rescattering
in the Feynman technique (see fig. 5) is equal to
-wave (s)
AN N →∆∆→N N +(ππ→ππ)S = ASππ→ππ
Z
×GN N →∆∆ (W ) ū(−P2 )u(P1 ) ū(p3 )
d4 k ′
i(2π)4
′⊥
′⊥
g∆ k1µ
∆µν ′ (k1′ )∆ν ′ ν (−k2′ )(−)k2ν
g∆
2
′
2
′
(M∆ − s13 − iM∆ Γ∆ )(M∆ − s24 − iM∆ Γ∆ )
1
× 2
2 − i0)(m2 − k 2 − i0) u(−p4 ) .
(mπ − k1π
π
2π
×
(131)
-wave (s) is the S-wave ππ-scattering ampliThe factor ASππ→ππ
tude. Here we take into account the low-energy ππ interaction only. In the K-matrix representation, it is written
in the form
r
K(s)
s − 4m2π
1
S -wave
Aππ→ππ (s) =
, ρ(s) =
.
1 − iρ(s)K(s)
16π
s
(132)
In (132), we take into account the full S-wave as observed
experimentally, including the so-called sigma-meson, independently of its existence. Generally speaking, it is possible to account for higher waves as well, but the box diagram with two ∆’s leads to singularities
√ near the physical
region of the production process at s . 0.6 GeV only.
The approximation used in the calculation of the box
diagram (131) is related to the extraction of the leading
terms of the singular amplitude. To this aim, we fix the
numerator of the integrand in the propagator poles by
setting
2
k1′2 → M∆
,
2
k2′2 → M∆
,
2
k1π
→ m2π ,
2
k2π
→ m2π ,
(133)
which leads in (131) to the substitution
⊥k1 (box)
,
⊥k2 (box)
,
′⊥
⊥
k1µ
→ k1µ
(box) = −p3
′⊥
⊥
k2ν
→ k2ν
(box) = −p4
k1′ → k1 (box),
k2′ → k2 (box).
(134)
Now, in the c.m. system, the momenta ka (box) read
q
2
2
k1 (box) = W/2, 0, 0, W /4 − M∆ ,
k2 (box) =
q
2
2
W/2, 0, 0, − W /4 − M∆ .
(135)
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
Here, we denote the four-momentum as k
=
(k0 , kx , ky , kz ). Under the constraints of eq. (133),
the numerator of the integrand does not depend on the
integration variables, and it can be written separately for
the leading singular (LS) term:
(LS)
-wave (s)G
AN N →∆∆→N N +(ππ→ππ)S = ASππ→ππ
N N →∆∆ (W )
⊥k (box)
× ū(−P2 )u(P1 ) ū(p3 )g∆ (−p3µ 1
)∆µν ′ (k1 (box))
⊥k box)
×∆ν ′ ν (−k2 (box))p4ν (
g∆ u(−p4 )
Z
d4 k ′
1
×
1
2
4
′
i(2π) (M∆ − ( 2 p + k + p3 )2 − iM∆ Γ∆ )
×
×
1
2 − ( 1 p − k ′ + p )2 − iM Γ )
(M∆
4
∆ ∆
2
where
1
p − k ′ = k2π ,
2
p1 + p2 = p . (137)
The box diagram integral of eq. (136) is calculated in appendix F: in this appendix, we demonstrate the effects of
the box diagram on the ππ spectra.
In the Feynman technique, the box diagram amplitude
with pion-nucleon rescattering in the resonance state (I =
3/2, J = 3/2) reads (see fig. 6):
AN N →∆∆→N π+(N π→N π)∆ = GN N →∆∆ (W )
× ū(−P2 )u(P1 )
1
∆µµ′ (p∆ )
× ū(p3 )g∆ (p2 − p3 )µ⊥p∆ 2
2
M∆ − p2∆ − iM∆ Γ∆
Z
′
k̂1N
+ mN
d4 k ′ 1 ′
⊥p∆
′
(k
−
k
)
g
×
′
∆
2π
1N
µ
2
′2 − i0 g∆
i(2π)4 2
mN − k1N
×
⊥k′
⊥k2′
′
′
− k1N
)µ′ 1 ∆µ′ ν ′ (k1′ )∆ν ′ ν (−k2′ ) 12 (−k2π
+ p4 )ν
2 − k ′2 − iM Γ )(M 2 − k ′2 − iM Γ )
(M∆
∆ ∆
∆ ∆
1
2
∆
1
(138)
× 2
′2 − i0) g∆ u(−p4 ) ,
(mπ − k2π
where p∆ = p2 + p3 . By fixing the numerator of (138) at
2
k1′2 → M∆
,
2
k2′2 → M∆
,
2
k1π
→ m2π ,
1
1
∆(k’ )
1
∆(k’ )
p(k’ )
p(p )
p(P )
3
1N
π(k’ )
2π
2
π(p )
2
p(p )
2
4
Fig. 6. Box diagram with pion-nucleon rescattering.
we write the leading singular (LS) terms of the box diagram amplitude as follows:
AN N →∆∆→N π+(N π→N π)∆ = GN N →∆∆ (W )
× ū(−P2 )u(P1 )
1
∆µµ′ (p∆ )
⊥p
× ū(p3 )g∆ (p2 − p3 )µ ∆ 2
2
M∆ − p2∆ − iM∆ Γ∆
1
⊥p
× (k1 (box) − p1 − k2 (box) + p4 )µ′ ∆
2
⊥k (box)
×g∆ k̂1 (box) − p̂1 + mN g∆ p1µ′1
⊥k (box)
×∆µ′ ν ′ (k1 (box))∆ν ′ ν (−k2 (box))p4ν 2
×
9.3 Box diagram amplitude with pion-nucleon
rescattering
1
2 (p1
π(p )
p(P )
(LS)
1
, (136)
(m2π − ( 21 p + k ′ )2 − i0)(m2π − ( 12 p − k ′ )2 − i0)
1
p + k ′ = k1π ,
2
143
2
k1N
→ m2N ,
(139)
×
×
Z
g∆ u(−p4 )
d4 kπ
1
2
4
i(2π) (mN − (p∆ − kπ2 )2 − i0)
1
2 − (p − k + p )2 − iM Γ )
(M∆
∆
π
1
∆ ∆
2
(M∆
− (kπ + p4
)2
1
.
− iM∆ Γ∆ )(m2π − kπ2 − i0)
(140)
9.4 (NN)D-wave state with JP = 2+ , two-pole and box
diagrams
The production of ∆∆ near the threshold in the S-wave
leads to a J P = 2+ state as well and, correspondingly,
to a strong box diagram singularity in this wave. In the
J P = 2+ wave, the transition (N N )D-wave → (∆∆)S -wave
is related to the two-pole amplitude
A(N N )D →(∆∆)S →(N π)(N π) = GN N →∆∆ (W )
(2)
× ū(−P2 )Xν ′ ν ′′ (q)u(P1 )
∆µν ′ (k1 )
⊥
× ū(p3 )g∆ k1µ
2 −s
M∆
13 − iM∆ Γ∆
∆ν ′′ ν (−k2 )
⊥
× 2
(−k2ν )g∆ u(−p4 ) .
M∆ − s24 − iM∆ Γ∆
(141)
144
The European Physical Journal A
The box diagram amplitude with the pion-pion rescattering is given by
(LS)
-wave (s)G
AN N →∆∆→N N +(ππ→ππ)S = ASππ→ππ
N N →∆∆ (W )
(2)
× ū(−P2 )Xν ′ ν ′′ (q)u(P1 )
⊥k (box)
× ū(p3 )g∆ (−p3µ 1
)∆µν ′ (k1 (box))
⊥k box)
×∆ν ′′ ν (−k2 (box))p4ν (
g∆ u(−p4 )
Z
1
d4 k ′
×
2 − ( 1 p + k ′ + p )2 − iM Γ )
i(2π)4 (M∆
3
∆ ∆
2
×
×
2
(M∆
−
( 12 p
−
k′
1
+ p4 )2 − iM∆ Γ∆ )
1
. (142)
(m2π − ( 21 p + k ′ )2 − i0)(m2π − ( 12 p − k ′ )2 − i0)
In the leading singular-term approach, the box diagram
amplitude with the pion-nucleon rescattering can be written in the form
(LS)
AN N →∆∆→N π+(N π→N π)∆ = GN N →∆∆ (W )
(2)
× ū(−P2 )Xν ′ ν ′′ (q)u(P1 )
1
∆µµ′ (p∆ )
⊥p
× ū(p3 )g∆ (p2 − p3 )µ ∆ 2
2
M∆ − p2∆ − iM∆ Γ∆
1
⊥p
× (k1 (box) − p1 − k2 (box) + p4 )µ′ ∆ g∆
2
⊥k (box)
∆µ′ ν ′ (k1 (box))
× k̂1 (box) − p̂1 + mN g∆ p1µ′1
⊥k2 (box)
×∆ν ′′ ν (−k2 (box))p4ν
g∆ u(−p4 )
×
×
Z
4
d kπ
1
i(2π)4 (m2N − (p∆ − kπ2 )2 − i0)
1
2 − (p − k + p )2 − iM Γ )
(M∆
∆
π
1
∆ ∆
1
×
. (143)
2
2
(M∆ − (kπ + p4 ) − iM∆ Γ∆ )(m2π − kπ2 − i0)
10 Conclusion
We have developed a new method for the partial-wave
analysis of data on the baryon-baryon and baryon-antibaryon collision. The method is based on the operator decomposition approach which was successfully applied before to a number of meson-induced reactions. The article
emphasises the analysis of reactions with three or four
particles in the final state, where triangle and box singularities might play an important role. A full set of partialwave amplitudes is constructed for nucleon-nucleon elastic
scattering and for N ∆ and ∆∆ production. With these
amplitudes, expressions for partial widths and for reaction cross-sections are presented. Some examples how to
calculate contributions from triangle and box diagrams in
simple cases are explicitly given. The application of the
methods developed here to the analysis of new data obtained and expected from COSY should provide valuable
information about the hadron spectrum and properties of
hadron interaction.
We would like to thank L.G. Dakhno for helpful discussions
and critical reading of the manuscript. The work was supported by a FFE grant of the Research Center Jülich and
by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich SFB/TR16. We would like to thank the
Alexander von Humboldt foundation for generous support in
the initial phase of the project, A.V.A. for a AvH fellowship
and A.V.S. for the Friedrich-Wilhelm Bessel award. A.V.S.
gratefully acknowledges the support from Russian Science Support Foundation. This work is also supported by Russian Foundation for Basic Research and RSGSS 5788.2006.2 (Russian
State Grant Scientific School).
Appendix A.
The baryon wave functions ψ(p) and ψ̄(p) = ψ + (p)γ0 obey
the Dirac equation
(p̂ − m)ψ(p) = 0,
ψ̄(p)(p̂ − m) = 0.
The γ-matrices were used in the form
I 0
0 σ
γ0 =
, γ=
,
0 −I
−σ 0
0 I
,
γ5 = iγ0 γ1 γ2 γ3 =
I 0
γ0+ = γ0 ,
γ + = −γ,
with the standard Pauli matrices
0 1
0 −i
σ1 =
, σ2 =
,
1 0
i 0
σa σb = iεabc σc .
σ3 =
(A.1)
(A.2)
1 0
0 −1
,
(A.3)
The Dirac equation gives four wave functions
!
ϕj
√
j = 1, 2:
ψj (p) = p0 + m (σp)
,
p0 +m ϕj
√
+ (σp)
, (A.4)
,
−ϕ
ψ̄j (p) = p0 + m ϕ+
j
j
p0 + m
!
(σp)
√
χj
p
+m
0
j = 3, 4:
ψj (−p) = i p0 + m
,
χj
√
(σp)
+
,
,
−χ
ψ̄j (−p) = −i p0 + m χ+
j
j
p0 + m
(A.5)
where ϕj and χj are two-component spinors
ϕj1
χj1
ϕj =
,
χj =
,
ϕj2
χj2
(A.6)
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
which are normalized as follows:
ϕ+
j ϕℓ = δjℓ ,
In the c.m. system, we have
χ+
j χℓ = δjℓ .
(A.7)
The solutions with j = 3, 4 refer to antibaryons. The corresponding wave function is given by
ψjc (p) = C ψ̄jT (−p),
j = 3, 4:

0
0 −σ2
 0
C = γ2 γ0 =
=
0
−σ2 0
−i
0
0
i
0
0
−i
0
0
j, j = 1, 2: ψN j (p1 ) ≃
(A.8)
Equation (A.8) reads:

i
0
.
0
0
j = 3, 4:
=
0 −σ2
−σ2 0
√
−i p0 + m
√
i p0 + m
σ2 χ∗j
(σp)
p0 +m
σ2 χ∗j
!
=
T
p)
∗
− (σ
p0 +m χj
χ∗j
√
p0 + m
!
ψ̄N j ′ (p′1 ) ≃
ℓ, ℓ = 3, 4:
(A.9)
c
ψ̄Λℓ
(−p2 )
(σp)
p0 +m
!
.
(A.10)
In (A.10), we have used the commutator
σ2 (σ T p) = −σ1 p1 σ2 − σ2 p2 σ2 − σ3 p3 σ2 = −(σp)σ2 .
(A.11)
We define the two-component spinor for antibaryons as
−χ∗j2
0 −1
.
(A.12)
χ∗j =
ϕcj = −iσ2 χ∗j =
1 0
χ∗jℓ
The wave functions defined in eqs. (A.4)-(A.5) are normalized as follows:
ψ̄j (p)ψℓ (p) = 2m δjℓ ,
j, ℓ = 1, 2:
ψ̄j (−p)ψℓ (−p) = −2m δjℓ .
(A.13)
j, ℓ = 3, 4:
They obey the completeness relation
X
ψjα (p) ψ̄jβ (p) = (p̂ + m)αβ ,
j=3,4
ψjα (−p) ψ̄jβ (−p) = (p̂ − m)αβ .
√
2mN
(σk)
2mN
ϕN j
,
+
ϕ+
N j ′ , −ϕN j ′
(σk′ )
2mN
!
−(σk′ )
c
2mΛ χΛℓ′
χcΛℓ′
,
,
√
c+
c+ −(σk)
, −χΛℓ ,
≃ −i 2mΛ χΛℓ
2mΛ
(B.2)
c
ψ̄N (p′1 )Q̂000 (k ′ )ψΛ
(−p′2 )
L = 0, J = 0:
(0,00,0)
c
(−p2 )Q̂000 (k)ψN (p1 ) AN Λ→N Λ (s) ≃
× ψ̄Λ
√
√
c
4mN mΛ ϕ+
4mN mΛ
χ
χc+
′
′
N j Λℓ
Λℓ ϕN j
(0,00,0)
×AN Λ→N Λ (s),
′
c
′
ψ̄N (p′1 )Q̂101
L = 0, J = 1:
µ (k )ψΛ (−p2 )
(1,00,1)
c
× ψ̄Λ
(−p2 )Q̂101
µ (k)ψN (p1 ) AN Λ→N Λ (s) ≃
√
√
c
i 4mN mΛ ϕ+
σχ
χc+
′
′
Λℓ
Nj
Λℓ σϕj i 4mN mΛ
(1,00,1)
×AN Λ→N Λ (s) .
(A.14)
Appendix B.
We consider the operators with L = 0 from eqs. (60)
and (66) in the c.m. system (p1 = −p2 = k and p′1 =
−p′2 = k′ ). For L = 0, we have the following operators in
the nonrelativistic approach:
0 I
000
Q (k) = iγ5 = i
,
I 0
0 σ
101
Q (k) = Γµ ≃
.
(B.1)
−σ 0
ϕ↑ (N j)
ϕ↓ (N j)
,
ϕ+
N j = (ϕ↑ (N j), ϕ↓ (N j)) .
(B.4)
For the Λ, we determine the bispinor to be given by
χcΛℓ
i) The S-wave terms in the the nonrelativistic limit.
(B.3)
Let us consider bispinors with real components. For nucleons, we write
ϕN j =
j=1,2
X
2mN
!
where ϕN j and χcΛℓ are two-component spinors. For the
waves with J = 0, 1 we have
=
ϕcj
ϕN j
√
≃ i 2mΛ
′
c
ψΛℓ
′ (−p2 )
′
ϕcj
√
′
where
ψjc (p)
145
= iσ2
ϕ↑ (Λℓ)
ϕ↓ (Λℓ)
=
ϕ↓ (Λℓ)
−ϕ↑ (Λℓ)
.
(B.5)
Within this definition, we can re-write (B.3) in terms of
the traditional technique which uses the Clebsch-Gordan
coefficients. For J = 0, we have
I
+ I
c
√
√
ϕ
=
ϕ
χc+
χ
=
Nj
Λℓ
Nj
Λℓ
2
2
1
√ (ϕ↑ (N j)ϕ↓ (Λℓ) − ϕ↓ (N j)ϕ↑ (Λℓ)) =
2
X
00
C1/2α,
1/2−α ϕα (N j)ϕ−α (Λℓ),
α
(B.6)
146
The European Physical Journal A
...µn
Zµα1 ...µn (q)(−1)n Oνµ11...ν
Xβν1 ...νn (k) =
n
and for J = 1, J3 = 0,
+ σ3 c
c+ σ3
χΛℓ √ ϕN j = ϕN j √ χΛℓ =
2
2
1
√ (ϕ↑ (N j)ϕ↓ (Λℓ) + ϕ↓ (N j)ϕ↑ (Λℓ)) =
2
X
10
C1/2α,
1/2−α ϕα (N j)ϕ−α (Λℓ).
(B.7)
ii) The D-wave component in the operator γµ⊥ .
Equations (B.2) and (B.3) allow one to see easily the existence of the D-wave admixture in the operator γµ⊥ . By
⊥
using the operator Q̂101
µ (k) = γµ in (B.3), one has the
following next-to-leading term in the (J = 1)-wave:
′
′
√
+ (σk ) (σk ) c
σ
χ ′
− 4mN mΛ ϕN j ′
2mN 2mΛ Λℓ
√
(σk) (σk)
(1,00,1)
σ
ϕN j
4mN mΛ AN Λ→N Λ (s). (B.8)
× χc+
Λℓ
2mΛ 2mN
The spin operators in (B.8) can be presented as
k(σk)
(σk) (σk)
σ
≃
+σO
2mΛ 2mN
2mΛ mN
k2
mΛ m N
,
(B.9)
where the first term in the right-hand side refers to
the D-wave, while the second one gives the correction
to the S-wave term. In the operator Γα (k⊥ ), the Dwave admixture is canceled√due to the second term:
−[4sk⊥α (k⊥ γ)]/[(mN + mΛ )( s + mN + mΛ )(s − (mN −
mΛ )2 )].
Appendix C. Useful relations for Zα
µ1 ...µn and
(n−1)
Xν2 ...νn
2n − 1
,
n
q
q
We now consider some further expressions used in the
one-loop diagram calculations. In our case, the operators
(n+1)
Xαµ1 ...µn and Zµβ1 ...µn are constructed, where α and β indices are convoluted with tensors. Let us start with the
loop diagram with a Z-operator:
Z
dΩ α
...µn
Z
(k ⊥ )Tαβ Zνβ1 ...νn (k ⊥ ) = λOνµ11...ν
(−1)n .
n
4π µ1 ...µn
(C.5)
For different tensors Tαβ , one has the following λ’s:
Tαβ = gαβ ,
Tαβ = kα⊥ kβ⊥ ,
αn 2n−2
|k|
,
n
αn
|k|2n .
λ=
2n + 1
λ=−
(C.1)
αn
...µn β
Zµα1 ...µn (q)(−1)n Oνµ11...ν
Zν1 ...νn (k) = 2 (−1)n
n
n
!
q q n−1 "
⊥ ⊥
⊥ ⊥
k
k
q
q
α
α
β
β
′′
⊥
′
2
2
×
Pn−1
+
gαβ Pn −
k⊥
q⊥
2
2
q⊥
k⊥
#
kα⊥ qβ⊥
qα⊥ kβ⊥
′′
′′
′
+ p 2 p 2 Pn−2 − 2Pn−1 + p 2 p 2 Pn , (C.2)
k⊥ q⊥
k⊥ q⊥
αn
...µn
Xαµ1 ...µn (q)(−1)n Oνµ11...ν
Xβν1 ...νn (k) =
(−1)n
n
(n + 1)2
!
q q n+1 "
kα⊥ kβ⊥
qα⊥ qβ⊥
′′
⊥
′
2
2
×
Pn+1
+
gαβ Pn+1 −
k⊥ q⊥
2
2
q⊥
k⊥
#
kα⊥ qβ⊥
qα⊥ kβ⊥
′′
′′
′
+ p 2 p 2 Pn+2 − 2Pn+1 + p 2 p 2 Pn , (C.3)
k⊥ q⊥
k⊥ q⊥
(C.6)
(C.7)
Equation (C.6) can be easily obtained using eqs. (C.1)
and (50), while eq. (C.7) can be obtained using eqs. (42)
and (50). For the X operators, one has
Z
dΩ (n+1)
(n+1)
...µn
X
(k ⊥ )Tαβ Xβν1 ...νn (k ⊥ ) = λOνµ11...ν
(−1)n ,
n
4π αµ1 ...µn
(C.8)
where
αn
Tαβ = gαβ ,
λ=−
|k|2n+2 ,
n+1
αn
|k|2n+4 .
(C.9)
Tαβ = kα⊥ kβ⊥ ,
λ=
2n + 1
To derive eq. (C.8), the properties
αµ1 ...µn
Oαν
=
1 ...νn
In this appendix, we list a few useful expressions.
2 ...νn
Zµα1 ...µn = Xν(n−1)
Oµαν1 ...µ
2 ...νn
n
n+1 "
qα⊥ qβ⊥ ′′
⊥
2
2
gαβ
Pn′ − 2 Pn−1
k⊥
q⊥
q⊥
#
kα⊥ qβ⊥
kα⊥ kβ⊥ ′′
qα⊥ kβ⊥
′′
′′
− 2 Pn+1 + p 2 p 2 Pn + p 2 p 2 Pn . (C.4)
k⊥
k⊥ q⊥
k⊥ q⊥
2
×(−k⊥
)
α
αn−1
(−1)n
n(n + 1)
2n + 3 µ1 ...µn
O
2n + 1 ν1 ...νn
(C.10)
of the projection operator and eq. (43) are used. The interference term between X and Z operators is given by
Z
dΩ (n+1)
...µn
X
(k ⊥ )Tαβ Zνβ1 ...νn (k ⊥ ) = λOνµ11...ν
(−1)n ,
n
4π αµ1 ...µn
(C.11)
with
Tαβ = gαβ ,
λ = 0,
Tαβ = kα⊥ kβ⊥ ,
λ=−
αn
|k|2n+2 .
2n + 1
(C.12)
Equation (C.12) is calculated using eq. (C.1) and the orthogonality properties (44) of the X operators.
Appendix D. N∆ one-loop diagrams
The calculation of the one-loop diagram for different vertex operators is an important step in the construction of
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
the unitary N ∆ amplitude. Consider the loop diagram for
S = 1 and derive all expressions in case of different particle masses (m1 is mass of ∆ and m2 is nucleon mass).
Let us start with 3 LJ (J = L − 1) states. Using the
expression
γα⊥k1 γβ⊥k1
⊥k1
iγ5 (m2 − k̂2 ) =
Sp iγ5 (m1 + k̂1 ) gαβ −
3
kα⊥ kβ⊥
4
− gαβ −
(s − δ 2 ),
(D.1)
3
m21
where δ = m1 − m2 , the one-loop diagram for the operator (100) is given by
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
Sp Vµ(1)α
(m
+
k̂
)
g
−
1
1
αβ
1 ...µn
4π
3
(m2 − k̂2 ) =
×Vν(1)β
1 ...νn
(D.2)
Here, eqs. (C.8) and (C.9) were used.
For 3 LJ (J = L + 1) states, one has
...µn
×Oνµ11...ν
(−1)n .
n
3/2
a3 = −
(D.3)
(D.4)
i = 1, 2, (D.5)
where
3/2
Γαβ = gαβ +
4skα⊥ kβ⊥
√
√
.
(s + M δ)( s + M )( s + δ)
(D.8)
γα⊥k1 γβ⊥k1
⊥k1
Oµα11µα22 Sp γµ1 (m1 + k̂1 ) gαβ
−
3
×γν2 (m2 − k̂2 ) Oβν11νβ22 =
a1 = 2(s − δ 2 ),
One can also introduce the pure spin operator in a way
that the transition loop diagram is equal to zero. Then
eqs. (100)-(102) can be rewritten in the following way:
u(−k2 ),
= ψ̄α (k1 )Γαβ Vµ(i)β
Wµ(i)
1 ...µn
1 ...µn
γα⊥k1 γβ⊥k1
dΩ
⊥k1
Sp Vµ(3)α
(m
+
k̂
)
g
−
1
1
αβ
1 ...µn
4π
3
×Vν(3)β
(m2 − k̂2 ) =
1 ...νn
(D.9)
where
γα⊥k1 γβ⊥k1
dΩ
⊥k1
Sp Vµ(1)α
(m
+
k̂
)
g
−
1
1
αβ
1 ...µn
4π
3
×Vν(2)β
(m2 − k̂2 ) =
1 ...νn
4
αn−1 |k|2n+2 µ1 ...µn
(s − δ 2 )
Oν1 ...νn (−1)n+1 .
3
2n + 1 m21
Z
(2)
Direct calculations also show that transition loop diagrams between 3 LJ (J = L − 1) and 3 LJ (J = L + 1)
states are equal to
Z
Thus, the transition loop diagram vanishes identically.
For 3 LJ (J = L) states, one has
a1 Oβα11βα22 + a2 Zαξ 1 α2 Zβξ1 β2 + a3 Xα(2)
Xβ1 β2 ,
1 α2
γα⊥k1 γβ⊥k1
dΩ
⊥k1
Sp Vµ(2)α
(m
+
k̂
)
g
−
1
1
αβ
1 ...µn
4π
3
×Vν(2)β
(m2 − k̂2 ) =
1 ...νn
4
αn−1
|k|2 n
(s − δ 2 )
1+ 2
|k|2n−2
3
2n − 1
m1 (2n + 1)
1 ⊥k1 γα⊥k
γβ ′
′
3/2
3/2
1
Γββ ′
−
Sp iγ5 (m1 + k̂1 )Γαα′ gα⊥k
′ β′
3
4
(D.7)
×iγ5 (m2 − k̂2 ) = − gαβ (s − δ 2 ).
3
To calculate loop diagrams with S = 2, the following expression is used:
αn
|k|2 (n + 1)
4
(s − δ 2 )
1+ 2
|k|2n+2
3
n+1
m1 (2n + 1)
Z
Then, it is easy to find that
n+1
4
...µn
(s − δ 2 )sαn−1 2
|k|2n Oνµ11...ν
(−1)n .
n
3
4n − 1
...µn
×Oνµ11...ν
(−1)n .
n
147
(D.6)
a2 =
16
32δ
−
(s−(m1 + m2 )2 ),
9m1 27m21
64
.
27m21
(D.10)
For 5 LJ (J = L + 2), the operator one-loop diagram is
equal to
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
Sp Vµ(4)α
−
(m
+
k̂
)
g
1
1
αβ
1 ...µn
4π
3
(m2 − k̂2 ) =
×Vν(4)β
1 ...νn
αn−2
...µn
|k|2n−4 (−1)n Oνµ11...ν
n
2n − 3
n
9 n−1
− a2 |k|2 +a3
|k|4
.
× a1 +
4 2n−1
2n+1
(D.11)
For 5 LJ (J = L − 2), the one-loop operator is given by
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
(5)α
Sp Vµ1 ...µn (m1 + k̂1 ) gαβ −
4π
3
×Vν(5)β
(m2 − k̂2 ) =
1 ...νn
...µn
αn |k|2n+4 (−1)n Oνµ11...ν
n
a2 |k|2 a3 |k|4
9
(2n+3)a1
,
+ −
+
×
(n + 1)(n + 2) 4
n + 1 2n + 1
(D.12)
148
The European Physical Journal A
while for 5 LJ (J = L), the operator is written as
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
(6)α
Sp Vµ1 ...µn (m1 + k̂1 ) gαβ −
4π
3
×Vν(6)β
(m2 − k̂2 ) =
1 ...νn
αn−1
...µn
|k|2n (−1)n Oνµ11...ν
n
2n(2n + 1)
(2n + 3)(n + 1)a1
2n + 5
9 2
2n + 1
×
− |k| a2
+
3n
8
9
n(2n − 1)
a3 |k|4 (n + 1)2
+
.
(D.13)
2(2n − 1)
The one-loop transition diagram between 5 LJ (J = L + 2)
and 5 LJ (J = L − 2) states can be expressed as
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
(4)α
Sp Vµ1 ...µn (m1 + k̂1 ) gαβ −
4π
3
9
α
n−2
...µn
×Vν(5)β
a3 |k|2n+4 (−1)n Oνµ11...ν
,
(m2 − k̂2 ) =
n
1 ...νn
4 2n + 1
(D.14)
and the one-loop transition diagram between 5 LJ (J =
L + 2) and 5 LJ (J = L) states as
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
−
(m
+
k̂
)
g
Sp Vµ(4)α
1
1
αβ
1 ...µn
4π
3
×Vν(6)β
(m2 − k̂2 ) =
1 ...νn
3αn−2 (n + 1)
...µn
|k|2n (−1)n Oνµ11...ν
n
8(2n + 1)(2n − 1)
2n + 3
a2 − 2|k|2 a3 .
×
n
(D.15)
For the one-loop transition diagram between 5 LJ (J =
L − 2) and 5 LJ (J = L) we get
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
Sp Vµ(5)α
(m
+
k̂
)
g
−
1
1
αβ
1 ...µn
4π
3
×Vν(6)β
(m2 − k̂2 ) =
1 ...νn
n+1
3αn
2n+4
n µ1 ...µn
2
,
|k|
(−1) Oν1 ...νn a2 − 2|k| a3
8(2n + 1)
2n − 1
(D.16)
and, for 5 LJ (J = L − 1), we find to
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
(7)α
Sp Vµ1 ...µn (m1 + k̂1 ) gαβ −
4π
3
×Vν(7)β
(m2 − k̂2 ) =
1 ...νn
sαn−1
...µn
|k|2n+2 (−1)n Oνµ11...ν
n
2(2n + 1)
a1 (n + 1)(2n2 + n − 2) 9 2 n + 1
×
. (D.17)
− |k| a2
n2 (2n − 1)
8
2n − 1
Finally, the operator one-loop diagram For 5 LJ (J = L+1)
is equal to
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
(8)α
Sp Vµ1 ...µn (m1 + k̂1 ) gαβ −
4π
3
×Vν(8)β
(m2 − k̂2 ) =
1 ...νn
sαn−2 (n + 1)
...µn
|k|2n−2 (−1)n Oνµ11...ν
n
2(2n − 1)(2n − 3)
n−1
9
,
× a1 − |k|2 a2
8
2n + 1
(D.18)
and the one-loop transition diagram between 5 LJ (J =
L − 1) and 5 LJ (J = L + 1) can be written as
Z
γα⊥k1 γβ⊥k1
dΩ
⊥k1
(7)α
Sp Vµ1 ...µn (m1 + k̂1 ) gαβ −
4π
3
(m2 − k̂2 ) =
×Vν(8)β
1 ...νn
9
sαn−2
2n
n µ1 ...µn n + 1
2
|k|
(−1)
O
a
+
|k|
a
(n+1)
.
1
2
ν1 ...νn
4n2 − 1
n
16
(D.19)
Appendix E. Amplitude of the triangle
diagram
In the last two appendices, we give results on triangle diagrams in numerical form. First, we calculate the triangle
diagram integral which enters eq. (126):
Z
d4 kπ
1
2
Aspinless
(W
,
s)
=
(E.1)
triangle
4
2
i(2π) mπ − kπ2 − i0
×
1
1
.
m2∆ −(p−p∆ +kπ )2 − im∆ Γ∆ m2N − (p∆ − kπ )2 − i0
Notations of the momenta are illustrated by fig. 7. Here,
p = p1 + p2 ,
p2 = W 2 ,
p2∆ = s.
(E.2)
The physical region is located in the interval
(mN + mπ )2 ≤ s ≤ (W − mN )2 .
(E.3)
2
The triangle diagram amplitude Aspinless
triangle (W , s) determined by (E.1) is shown in the physical region (E.3) in
p∆
k
p ∆− π
kπ
p−p
∆ +k
π
Fig. 7. Triangle diagram.
p−p∆
A.V. Anisovich et al.: Baryon-baryon and baryon-antibaryon interaction . . .
149
GeV
0.02
0.01
2
Wmin + 50 MeV
Im s, GeV
−2
p3
p
k′+ 3
0.5
0
Physical
1
−
2 p+k′
p1
1
2− p−
1
−
2 p−k′
p2
k′+
p
region
4
-0.5
0
1 p+
−2
p4
Wmin + 125 MeV
0.03
Fig. 9. Box diagram.
0.5
0.02
0
fig. 8 (left column). In the right column, there are positions of the logarithmic singularities on the second sheet
of the complex-s plane. The physical region of the reaction
is also shown (thick solid line): it is located on the lower
edge of the cut related to the threshold singularity (thin
solid line). The positions of logarithmic singularities read
Physical
0.01
region
-0.5
0
Wmin + 200 MeV
0.03
0.5
2
2
(W 2 − M∆
− m2N )(M∆
+ m2π − m2N )
(tr)
s(±) = m2π +m2N +
2
2M∆
h
2
2
2
± (mπ − (M∆ − mN ) )(mπ − (M∆ + mN )2 )
i1/2
. (E.4)
×(W 2 − (M∆ − mN )2 )(W 2 − (M∆ + mN )2 )
0.02
0
Physical
0.01
region
-0.5
0
Wmin + 275 MeV
0.03
0.5
2
Here M∆
= m2∆ − im∆ Γ∆ .
In the left column of fig. 8, the real and imaginary
parts of the amplitude (E.1) at different total energies
W are shown by solid and dashed curves, respectively.
In the right column, one sees the singularity positions,
(tr)
(tr)
(tr)
s(−) (black circles) and s(+) (black squares). When s(+)
moves into the third sheet, its position is shown as an open
square.
0.02
0
Physical
0.01
region
-0.5
0
Wmin + 350 MeV
0.03
0.5
0.02
0
Physical
0.01
region
Appendix F. Amplitude of the box diagram
-0.5
0
Here, we calculate the box diagram integral which enters
eq. (136), the notations of momenta are given in fig. 9.
Wmin + 425 MeV
0.03
0.5
Aspinless
(W 2 , s3 , s4 , s12 ) =
box
Z
d4 k ′
1
1
2
4
′
i(2π) (m∆ − ( 2 p + k + p3 )2 − im∆ Γ∆ )
0.02
0
Physical
0.01
region
-0.5
0
1.2
1.6
2.0
2.4
2
s, GeV
0
1
2
3
2
×
Re s, GeV
×
Fig. 8. Triangle diagram amplitude. In the left columns, real
and imaginary parts of the amplitude are shown by solid and
dashed curves, respectively. The initial energy, W , is shown
on the top of each panel. In the right columns, singularity
(tr)
positions, s(±) , eq. (E.4), are shown on the 2nd sheet of the
(tr)
complex-s plane. When s(+) moves to the 3rd sheet, its position is shown by the open square.
1
(m2∆ − ( 21 p − k ′ + p4 )2 − im∆ Γ∆ )
1
(m2π −( 12 p+k ′ )2 −i0)(m2π −( 12 p−k ′ )2 −i0)
.
(F.1)
Remind that s3 = (p−p3 )2 , s4 = (p−p4 )2 , s12 = (p1 +p2 )2 ,
W 2 = p2 .
In fig. 10, we show the results of our calculation of
Aspinless
(W 2 , s3 , s4 , s12 ) as a function of pion-pion energy
box
squared s12 at different total energies W , under the following constraint on s3 and s4 :
√
s3 = s4 = s12 W + m2N .
(F.2)
The European Physical Journal A
2.4 m∆
0.04
Im s12, GeV
GeV
2
−4
150
s3=s4, GeV
1
1.71
3
1.91
5
2.36
7
2.9
9
3.48
11
4.07
11
Physical region
0
-0.5
0
−0.04
1
0.08
11
2.5 m∆
0
Physical region
0.04
-0.5
0
1
−0.04
3 m∆
0.08
1
Physical region
-0.5
−0.04
3.5 m∆
0.08
0
1 Physical region
-0.5
−0.04
4.5 m∆
0.08
-0.5
−0.04
5.5 m∆
0.08
0 1
Physical region
-0.5
−0.04
−1
10
0
1
10
0
10
2
s12, GeV
1
3
5
7
9
11
2.08
3.29
5.22
7.25
9.31
11.37
1
3
5
7
9
11
2.43
4.7
8.04
11.47
14.94
18.41
1
3
5
7
9
11
2.77
5.55
9.62
13.83
18.06
22.31
5
10
2
Re s12, GeV
Fig. 10. Box diagram amplitude as a function of s12 under
the constraint (F.2) (corresponding magnitudes of s3 and s4 are
shown in the right column). In the left columns, real and imaginary parts of the amplitude are shown by solid and dashed
curves, respectively. The initial energy, W , is shown on the top
of each panel. On the right columns singularity positions, sbox
12 ,
eq. (F.4), are shown on the 2nd sheet of the complex-s12 plane.
This constraint corresponds to the following kinematics in the c.m. system:
p = (W ; 0; 0; 0),
q
m2π + p21z ; 0; 0; p1z ,
p1 =
p2 =
q
m2π
+
p21z ; 0; 0; −p1z
,
(F.3)
The positions of the box diagram singularities are given
by the formula
2
sbox
12 = 2mπ +
1
(s3 − m2N )(s4 − m2N )
2W 2
2
2
(2W 2 M∆
−W 2 (s3 − m2N ))(2W 2 M∆
−W 2 (s4 − m2N ))
2
4)
2
2
2
2W ((W − 2M∆ ) − 4M∆
2
(s3 −m2N )2
(2W 2 M∆
−W 2 (s3 −m2N ))2
2
−
−2m
−
π
2 )2 −4M 4 )
2W 2
2W 2 ((W 2 −2M∆
∆
+
×
2
(s4 −m2N )2
(2W 2 M∆
−W 2 (s4 − m2N ))2
−2m2π −
2 )2 −4M 4 )
2
2
2
2W
2W ((W −2M∆
∆
12
.
(F.4)
At s3 = s4 , eq. (F.4) reads
2
sbox
12 = 4mπ +
2
− s3 + m2N )2
W 2 (2M∆
2 )2 − 4M 4 .
2
(W − 2M∆
∆
(F.5)
2
is given by
Recall that in (F.4) and (F.5) M∆
2
M∆
= m2∆ − im∆ Γ∆ .
11
0.04
0
1.91
2.56
3.73
4.99
6.29
7.6
p3 =
11
0 1
Physical region
0.04
0
1
3
5
7
9
11
11
0.04
0
1.74
2
2.56
3.2
3.89
4.58
11
0
0.04
0
1
3
5
7
9
11
q
m2N + p23z ; 0; 0; p3z ,
q
2
2
p4 =
mN + p3z ; 0; 0; −p3z ,
q
q
m2π + p21z + m2N + p23z = W/2.
2
(F.6)
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
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