Eur. Phys. J. C (2014) 74:2763 Page 3 of 14 2763
Fig. 1 Mandelstam plane (ν, t). The Mandelstam triangle is the region
contoured by the s = (m
N
+ M
π
)
2
, u = (m
N
+ M
π
)
2
and t = 4M
2
π
lines. Our region of study R is the trapezium formed by the three pre-
vious lines and t = 0,whichismarkedinred
part of the Mandelstam triangle bounded by t ≥ 0 corre-
sponds to the region R (marked in red in Fig. 1) where the
positivity conditions are considered.
In terms of the (ν, t) variables the Mandelstam diagram
is given by t ≤ 4M
2
π
and |ν|≤νth(t) = M
π
+t/(4m
N
).In
order to obtain the region R one should add the restriction
t ≥ 0.
3 Partial-wave decomposition and positive-definite
spectral function
It is well known that the full isospin amplitude can be written
in terms of the partial-wave (PW) amplitudes as [33]
A
I
(s, t)
t=t (s,z
s
)
=
∞
=0
S
(s, z
s
)
F
I
(s) (7)
with
A
I
≡
A
I
B
I
,
F
I
≡
f
I
+
f
I
(+1)−
and
S
(s, z
s
) = 4π
⎛
⎝
W+m
N
E+m
N
P
+1
(z
s
) +
W−m
N
E−m
N
P
(z
s
)
−
W+m
N
E+m
N
P
(z
s
) +
W−m
N
E−m
N
P
+1
(z
s
)
1
E+m
N
P
+1
(z
s
) −
1
E−m
N
P
(z
s
)
−
1
E+m
N
P
(z
s
) −
1
E−m
N
P
+1
(z
s
)
⎞
⎠
, W =
√
s. (8)
Here P
(z
s
) are the conventional Legendre polynomials and
z
s
= 1 +
2st
λ(s,m
2
N
,M
2
π
)
with λ(s, m
2
N
, M
2
π
) =[s − (m
N
+
M
π
)
2
][s −(m
N
−M
π
)
2
],istheK¨all ´en function. The kernel
matrices of this set, S
(s, z
s
), are always analytical functions,
real for real values of the Mandelstam variables (s, t, u).
Thus, in the case s ≥ s
th
the whole analytic discontinuity
is due to the partial waves f
I
k
(s):
Im
A
I
(s + i, t) =
∞
=0
S
(s, z
s
(s, t)) Im
F
I
(s + i). (9)
Since a fixed-t dispersion relation for the analysis of the sub-
threshold amplitude will be used in Sect. 4, our interest is
focused on obtaining a positive-definite spectral function in
the physical region s ≥ s
th
. On the right-hand side of Eq. (9),
the imaginary part of each PW is positive due to unitarity, i.e.,
Im f
I
k
(s) ≥ 0fors ≥ s
th
, but the kernel matrices always con-
tain negative elements. Therefore, it is proper to construct a
combination of A
I
and B
I
in the form
D
I
α
(s, t) ≡ α A
I
(s, t) + ν B
I
(s, t) = α D
I
(s, t)
+ (1 − α)νB
I
(s, t) (10)
such that its imaginary part satisfies
ImD
I
α
(s, t) ≥ 0. (11)
In order to guarantee Eq. (11), it is proven in great detail in
Appendix that the validity region for the combination factor
α should be α
min
(t) ≤ α ≤ α
max
(t) with
α
min
(t) =
1 +
t
4m
N
M
π
1 −
t
4m
2
N
1 +
t
2m
N
M
π
+
t
4m
2
N
= 1 −
t
4m
N
M
π
+ O
p
2
m
2
N
,
α
max
(t) = 1 +
t
4m
N
M
π
, (12)
where M
π
= O( p) and t = O( p
2
) [25–27].
It is worth noting that here the Mandelstam variable t must
be greater than zero, t ≥ 0, due to the application of Eq. (56)
and the fact of P
k
(z
s
) ≥ 0forz
s
≥ 1 in the appendix. This
is the reason why our analysis of the positivity constraints
is restricted to the upper part of the Mandelstam triangle R
(see Fig 1).
So far, the s-channel positive-definite spectral function
above threshold is clear. The corresponding u-channel one is
easily obtained by crossing symmetry:
123
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