2 GeV ππ散射全局参数化：实轴与复数域分析
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本文主要探讨的是"散射高达2 $$\mathrm{\,GeV}$$的$$\pi\pi$$（介子-介子）过程的全局参数化"。全球参数化是一种在理论物理中常用的方法，它将复杂的现象简化为一系列可处理的数学函数，用于描述粒子间的相互作用，如在本例中的ππ散射。作者们，来自西班牙马德里大学和瑞士伯尔尼大学，提供了一种S0（虚标量波）和P（标量波）局部波的全面参数化，直到能量约为2 $$\mathrm{\,GeV}$$。
这个参数化涵盖了重要的部分波色散分析结果，这些分析是基于实轴上的实验数据进行的，上限为1.12 $$\mathrm{\,GeV}$$. 他们还确保了在复杂平面上的适用性，并且在前向色散关系的限制下，达到了1.43 $$\mathrm{\,GeV}$$。这意味着参数化不仅考虑了能量较低的物理过程，还考虑了更高的能量区域，尽管在此以上范围，他们主要依赖于实验数据来指导。
此外，文中特别强调了这些全球参数化在解析连续性的表现，它们能够准确地描绘出三个关键共振态：$$\sigma/f_0(500)$$、$$f_0(980)$$和$$\rho(770)$$的极参数特性。这些极参数是描述粒子性质的重要物理量，它们反映了这些共振结构在散射过程中对能量和动量转移的响应。
这篇研究为物理学家们提供了一个实用工具，使得他们在处理ππ散射现象时，可以利用这些参数化进行预测和理论分析，同时考虑到色散原理和实验观察的约束。这对于理解和模拟强相互作用，尤其是低能区的QCD行为，具有重要意义。文章发表在《欧洲物理杂志C》上，是开放获取的，便于学术界共享和进一步研究。
1008 Page 4 of 18 Eur. Phys. J. C (2019) 79 :1008
Let us now present separately the parameterizations we
have used to describe the two partial waves of interest for
this work.
3.1 S0-wave parameterization
As explained above, our parameterizations will be consistent
with the dispersive data analysis of [28], which extends up
to 1.43 GeV. Above that we will only provide three phe-
nomenological fits to three sets of incompatible data, care-
fully matched to our parameterizations below. Let us discuss
both regions separately.
3.1.1 S0-wave parameterization below 1.4 GeV
The σ/f
0
(500) and f
0
(980) resonances dominate the behav-
ior of the S0 partial wave in this region. The somewhat con-
troversial f
0
(1370) couples very weakly to two pions and its
effect in this region can be treated as background. For our
purposes it is important to remark that the σ/f
0
(500) has an
associated pole very deep in the complex plane that produces
a wide structure increasing monotonously from threshold up
to roughly 900 MeV, reaching a phase-shift of 90
◦
around 800
MeV, as seen in Fig. 1. It is known [40] that the σ/f
0
(500)
pole can be generated in the S0 partial wave by a simple trun-
cated conformal expansion, that we will call t
0
0,conf
(s).How-
ever, above 900 MeV the f
0
(980) pole adds a further sharp
increase that makes the phase larger than 200
◦
right below
the K
¯
K threshold. The phase then keeps growing slower but
monotonously up to 2 GeV.
It is worth noticing that the interplay between the
σ/f
0
(500) and f
0
(980) poles produces a sharp dip in the
modulus of the amplitude and the elasticity right above the
K
¯
K threshold. In order to describe the f
0
(980) effects accu-
rately and consistently with the dispersive results, we will
factorize in the S matrix the f
0
(980) shape separately from
the conformal expansion that contains the σ/f
0
(500) pole.
In other words, we will use S
0
0
= S
0
f
0
S
0
0,conf
, where
S
0
0
= 1 + 2iσ(s)t
0
0
,
S
0
0,conf
= 1 + 2iσ(s)t
0
0,conf
,
S
0
f
0
= 1 + 2iσ(s)t
0
f
0
. (3)
This factorization ensures elastic unitarity for the S0 wave,
i.e., |S
0
0
|=1, when both the conformal and the f
0
(980)
contributions fulfill elastic unitarity independently, i.e.,
|S
0
0,conf
|=|S
0
f 0
|=1. This will be the case in the elastic
region below the
¯
KK threshold, s < 4m
2
K
.
For our purposes, we are interested in the amplitude
partial-wave
t
0
0
(s) = t
0
0,conf
(s) + t
0
f
0
(s) + 2i σ(s)t
0
0,conf
(s)t
0
f
0
(s). (4)
Now, the conformal factor of the partial wave is con-
structed by analogy to the elastic formulation in Eq. (1)
t
0
0,conf
(s) =
1
σ(s)
1
0
0
(s) − i
, s < 1.4GeV (5)
where, building on [40]
0
0
(s) =
√
s
2q(s)
m
2
π
s − z
2
0
/2
z
2
0
m
π
√
s
+
N
n=0
B
n
ω(s)
n
. (6)
Let us remark that
0
0
(s) = cot δ
0
0
(s) in the elastic region
s ≤ 4m
2
K
, implying |S
0
0,conf
|=1. The s −z
2
0
/2 denominator
provides the so-called Adler zero at s
Ad ler
= z
2
0
/2 required
by chiral symmetry [59]. For z
0
= m
π
one would recover
the Current-Algebra result, namely the leading order ChPT
value. However, for us it will be a free parameter, fundamen-
tal to describe the subthreshold region. As we will see, it
comes out from the fits consistent with the dispersive evalua-
tion, which in turn is consistent with higher order ChPT eval-
uations (see [28,60]). Note also that if we did not include the
∼ 1/
√
s term added to the conformal series then, for the val-
ues of B
n
obtained from the fit, we would find in the unphys-
ical region that
0
0
(s
Ad ler
) →+i ∞ and
0
0
(0) → i 0
+
,so
that necessarily
0
0
(s) = i somewhere between those two
subthreshold points. This would yield a spurious pole, i.e., a
ghost, in the partial wave. As explained in [40] these ghosts
are mostly harmless and have little relevance in the fit quality
and the pole positions, but as a matter of principle it is bet-
ter to remove them. Actually, as explained there, adding the
1/
√
s term to the conformal series simply shifts the value
0
0
(0) so that the spurious pole disappears. The 1/
√
s cut
does not introduce any additional singularity in the partial
wave since its square-root cut falls right on the left cut. At
the same time this term is suppressed in the physical region
and thus barely affects the fit and resonance pole positions.
As shown below, for this wave it will be enough to set N = 5
to obtain a good overall χ
2
/d.o. f. in the elastic region.
The conformal variable is defined as
ω(s) =
√
s − α
√
s
0
− s
√
s + α
√
s
0
− s
, (7)
where s
0
corresponds to the highest value of s where the
expansion is real and then α sets the center of the conformal
expansion. We have found in practice that the S0 wave is
more conveniently described if the conformal expansion, by
becoming imaginary, introduces some inelasticity above the
K
¯
K threshold [50]). Thus, we choose s
0
= 4m
2
K
with α =
1 for simplicity, so that the expansion center lies near 0.7
GeV. Hence, between K
¯
K threshold and 1.4 GeV, the
0
0
(s)
function will be complex, which effectively introduces an
inelasticity. We tried otherwise, with a higher s
0
, so that no
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