黎曼曲面上的全纯形式与CHY积分：两圈理论探索
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更新于2024-07-16
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"全纯形式的CHY回路被积"
在量子场论中，散射振幅是理解和计算粒子相互作用的关键工具。Cachazo-He-Yuan（CHY）方法是一种创新的计算散射振幅的方式，它最初被提出用于树级水平的理论。然而，随着对更复杂物理过程的研究深入，该方法已经超越了树级，现在也被应用到循环级别，即包含了闭合回路的散射过程。这篇研究论文由Humberto Gomez、Sebastian Mizera和Guojun Zhang共同撰写，发表于JHEP03(2017)092，并在2017年3月被Springer出版。
文章的核心贡献在于介绍了如何从黎曼曲面上的全纯形式构建Φ3理论的CHY积分，这个理论处理的是含有三次相互作用项的量子场。作者提出了一套简单规则，可以将传统的费曼图直接转换为对应的CHY积分表达式。这不仅简化了复杂的计算，也为理论物理学家提供了直观且实用的工具来处理涉及多个回路的散射问题。
在CHY方法中，散射振幅被表示为黎曼曲面上的积分。对于Φ3理论，这些积分通常涉及到多环结构，而本研究中，作者成功地将这种构造扩展到了最多两个环的情况。他们扩展了原先在arXiv:1604.05373中提出的Λ算法，这一算法现在能够处理双循环的情况。通过这个新算法，研究人员能够对包含七个外部粒子的CHY积分进行解析验证，这是对CHY方法在循环级别的一个显著验证。
此外，该方法还为未来进一步扩展到更高循环顺序提供了自然的途径。这意味着CHY方法不仅适用于低阶计算，而且有潜力处理更复杂的高阶散射过程，这对于理解更深层次的物理现象，如强相互作用和引力效应等，具有重要意义。
这篇论文通过引入全纯形式的CHY回路被积，以及Λ算法的扩展，为量子场论的散射振幅计算提供了一个强大的新工具，特别是在处理循环散射问题上。这标志着CHY方法在理论物理学中的应用进一步深化，为未来的研究开辟了新的方向。
JHEP03(2017)092
Let us consider a Riemann surface of genus 2 as a hyperelliptic curve embedded in
CP
2
, namely
y
2
= (z − a
1
)(z − a
2
)(z − λ
1
)(z − λ
2
)(z − λ
3
), (2.7)
where (λ
1
, λ
2
, λ
3
) parametrize the curve, and (a
1
, a
2
) are two fixed branch points such that
a
1
6= a
2
. Since we will be ultimately interested in the degeneration of the curve near λ
1
= a
1
and λ
2
= a
2
, we denote A
1
-cycle the one that goes around λ
1
= a
1
in the degeneration
limit. The A
2
-cycle is defined analogously.
Note that this curve has several singular points, but we are only interested in those
where the two A-cycles are pinching at different points, i.e., singularities where the Riemann
surface degenerates to a sphere with four extra punctures. Furthermore, as it will be shown
below, many of the others singularities cancel out after computing the CHY integrals.
It is well-known that over the algebraic curve given in (2.7) there are just two global
holomorphic forms, which can be written as
Ω
1
(z)dz :=
z − a
2
y
dz, (2.8)
Ω
2
(z)dz :=
z − a
1
y
dz. (2.9)
In order to pinch the A-cycles, we take, without loss of generality, the parameters λ
1
= a
1
and λ
2
= a
2
. Thus the curve in (2.7) becomes y = (z − a
1
)(z − a
2
)y
t
where y
t
is a double
cover sphere given by (y
t
)
2
= z − λ
3
. Under this degeneration of the Riemann surface the
holomorphic forms turn into
(a
1
− λ
3
)
1/2
Ω
1
(z)
λ
1
=a
1
λ
2
=a
2
dz =
(a
1
− λ
3
)
1/2
(z − a
1
) y
t
dz =: ω
1
z
dz, (2.10)
(a
2
− λ
3
)
1/2
Ω
2
(z)
λ
1
=a
1
λ
2
=a
2
dz =
(a
2
− λ
3
)
1/2
(z − a
2
) y
t
dz =: ω
2
z
dz, (2.11)
where we have included normalization factors. Note that ω
1
z
dz and ω
2
z
dz are now meromor-
phic forms over a sphere defined by the quadratic curve (y
t
)
2
= z −λ
3
. It is straightforward
to see that ω
1
z
dz has only two simple poles (one on upper sheet and the other one on the
lower sheet), which are associated with the A
1
-cycle, i.e.,
I
A
1
ω
1
z
dz = ±1,
I
A
2
ω
1
z
dz = 0, (2.12)
where “ + ” is the residue on the upper sheet and “ − ” is the residue on the lower sheet.
In an analogous way, for ω
2
z
dz one has
I
A
1
ω
2
z
dz = 0,
I
A
2
ω
1
z
dz = ±1. (2.13)
Therefore, as it is shown in figure 2, the holomorphic form Ω
1
(z) is related with the A
1
-
cycle and so its CHY interpretation means that it connects a puncture with itself by a
line around the B
1
-cycle. In a similar way, the holomorphic form Ω
2
(z) is related with the
A
2
-cycle and so it connects a puncture with itself by a line around of the B
2
-cycle.
– 7 –
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