Minkowski空间中费米离子绑定态：光锥奇点解析与结构探索
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更新于2024-07-16
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本文主要探讨了在Minkowski空间中处理费米离子束缚态的一种新颖方法，特别是在自旋1/2构成的两体约束系统中。作者们应用了贝瑟-萨尔佩特方程（Bethe-Salpeter equation，简称BS方程），这是一种量子场论中用于描述粒子绑定状态的重要工具。通过采用纳卡尼西积分表示形式，他们成功地将BS方程的求解直接置入Minkowski时空框架下，这个框架对于相对论性量子力学至关重要。
在处理过程中，关键的一步是利用精确的零平面投影技术，它有助于处理端点奇异性问题。端点奇异性是BS方程在某些情况下会出现的数学难题，它涉及到在四维时空坐标中的奇异行为。通过这种方式，研究者得以在保持物理意义的同时，克服了这些技术上的挑战。
该研究特别关注了费米子-费米子的0+状态，即两个费米子组成的束缚态，其稳定性受制于与相互作用核的耦合常数。作者们通过数值分析，揭示了耦合常数和光锥振幅（light-cone amplitudes）对结合能的影响，这些振幅对应于标量、伪标量和矢量玻色子的交互作用。这种梯形近似方法使得理论分析更加具体且适用。
进一步，作者们将这种方法扩展到了夸克-反夸克系统，特别是考虑零状态下的情况。他们利用了先进的数值技术，从晶格计算中获得了费米子质量子和交换玻色子的质量数据。值得注意的是，这些计算结果显示了光锥振幅中自旋自由度的独特性质，这可能对理解基本粒子间的相互作用机制提供了新的洞察。
总结来说，这篇文章的核心贡献在于提供了一种有效的方法，可以直接在Minkowski空间中解决带有自旋1/2的费米离子束缚态问题，特别是在处理奇异性并得到可靠结果方面。这不仅有助于深化我们对强相互作用的理解，还为未来的粒子物理计算和理论研究开辟了新的路径。同时，开放访问的出版模式确保了这一成果的广泛共享和学术界的讨论。
764 Page 4 of 20 Eur. Phys. J. C (2017) 77 :764
with i, j = 1, 2, 3, 4. The coefficients c
ij
are obtained by
performing the convenient traces:
c
ij
(k, k
, p) =
1
N
i
(k, p)
Tr
S
i
(k, p)
/p
2
+ /k + m
× Γ
1
S
j
(k
, p)
Γ
2
/p
2
− /k − m
(11)
and are explicitly given in Appendix A (see also Ref. [16]).
Notice that the coefficients for a pseudoscalar exchange and
a vector can easily be obtained from the coefficients for a
scalar interaction, as explained in Appendix A.
3 Nakanishi integral representation and the light-front
projection
In the 1960s, for a bosonic case, Nakanishi (see Ref. [31]
for all the details) proposed and elaborated a new integral
representation for perturbative transition amplitudes, relying
on the parametric expression of the Feynman diagrams. The
key point in his formal approach is the possibility to express
a n-leg transition amplitude as a proper folding of a weight
function and a denominator that contains all the independent
scalar products of the n external four-momenta. Noteworthy,
by using NIR, the analytic properties of the transition ampli-
tudes are dictated by the above mentioned denominator. As
a final remark, one should recall that the weight function is
unique, as demonstrated by Nakanishi exploiting the analyt-
icity of the transition amplitude, expressed through NIR (see
[31]).
A step forward of topical interest was carried out in Ref.
[16], where the generalization to the fermionic ground state
was presented. It should be recalled that originally NIR was
established only for the bosonic case, with some caveat about
a straightforward application to the fermions, as recognized
by Nakanishi himself, who was aware of the possible tricky
role of the numerator in the Dirac propagator.
In the spirit of Ref. [16], one can apply NIR to each scalar
function in Eq. (7), tentatively generalizing the Nakanishi
approach to the fermionic case. Let us recall that the denom-
inator of a generic Feynman diagram contributing to the
fermionic transition amplitudes has the same expression as
in the boson case analyzed by Nakanishi [2], and this is the
main feature leading to NIR. In conclusion, one can write for
each φ
i
(k, p) in Eq. (7)
φ
i
(k, p) =
1
−1
dz
∞
0
dγ
g
i
(γ
, z
;κ
2
)
k
2
+ z
p · k − γ
− κ
2
+ i
3
,
(12)
where κ
2
= m
2
− M
2
/4. For each scalar function of the BS
amplitude it is associated one weight function or Nakanishi
amplitude g
i
(γ
, z
;κ
2
), which is conjectured to be unique
and encodes all the non-perturbative dynamical information.
The power of the denominator in Eq. (12) can be chosen as
any convenient integer. Actually, the power 3 is adopted fol-
lowing Ref. [16]. The scalar functions φ
i
(k, p) must have
well-defined properties under the exchange k →−k:even
for i = 1, 2, 4 and odd for i = 3. Those properties can
be straightforwardly translated to the corresponding proper-
ties of the Nakanishi weight function g
i
(γ
, z
;κ
2
) under the
exchange z
→−z
, i.e. they must be even for i = 1, 2, 4
and odd for i = 3.
By inserting Eq. (12)inEq.(10), one can write the
fermionic BSE as a system of coupled integral equations,
given by
1
−1
dz
∞
0
dγ
g
i
(γ
, z
;κ
2
)
k
2
+ z
p · k − γ
− κ
2
+ i
3
= g
2
j
1
−1
dz
∞
0
dγ
K
ij
(k, p;γ
, z
) g
j
(γ
, z
;κ
2
),
(13)
where the kernel that includes also the Nakanishi denomina-
tor of the BS amplitudes on the r.h.s. is
K
ij
(k, p;γ
, z
) = i(μ
2
− Λ
2
)
2
×
d
4
k
(2π)
4
c
ij
(k, k
, p)
(
p
2
+ k)
2
− m
2
+ i
(
p
2
− k)
2
− m
2
+ i
×
1
(k − k
)
2
− Λ
2
+ i
2
(k − k
)
2
− μ
2
+ i
×
1
k
2
+ z
p · k
− γ
− κ
2
+ i
3
. (14)
It is necessary to stress that the validity of the NIR for the BS
amplitude is verified a posteriori. Namely, if the generalized
eigen-equation in Eq. (13) admits eigen-solutions then NIR
can be certainly applied to the scalar function φ
i
(k, p).Let
us recall that Eq. (10) formally follows from the BSE in Eq.
(1).
One can perform the four-dimensional integration on k
in Eq. (14), obtaining
K
ij
(k, p;γ
, z
)
=
1
8π
2
M
2
(μ
2
− Λ
2
)
2
3
n=1
P
(n)
ij
(k, p;γ
, z
), (15)
where
P
(1)
ij
(k, p;γ
, z
)
=
a
0
ij
+ a
1
ij
( p · k) + a
2
ij
( p · k)
2
+ a
3
ij
k
2
[(1 − z)(k
−
− k
−
d
) + i ][(1 + z)(k
−
− k
−
u
) − i ]
×
1
0
dvv
2
(1 − v)
2
F(k
−
,γ,z;γ
, z
;v) (16)
123
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