NLO电弱红外奇点自动减法方法

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"NLO EW红外散度的自动减法" 在高能物理领域，特别是粒子物理学中，计算粒子碰撞过程的精确概率是至关重要的。这篇论文深入探讨了如何处理在Next-to-Leading Order (NLO) 电弱计算中的红外（IR）发散问题。红外发散通常出现在量子色动力学（QCD）和电弱理论的计算中，当粒子发射软光子或胶子时，可能会导致无穷大的贡献。Catani-Seymour偶极子减法方法是一种有效处理这些发散的技术，它通过引入特定的减因子来消除红外奇点。文章提出了该方法的一个推广，使其适用于次要阶电弱计算。这涉及到考虑无质量和有质量的partons（部分子）在无质量和有质量的spectator粒子（观察者粒子）背景下，因光子和胶子辐射产生的所有奇异行为。作者特别强调了同时减去源于QCD和电弱过程的奇异性，这些奇异性可能在Born级具有多个扰动阶次的过程的NLO校正中出现。此外，论文还讨论了非偶极类光子分裂的嵌入，这是对传统偶极子减法方案的扩展。非偶极类光子分裂指的是那些不遵循简单线性结构的光子分裂过程，它们在实际的物理过程中同样可能产生红外发散。这一扩展确保了减法方案的全面性和适用性。 Sherpa Monte-Carlo事件生成器被选为实施这个减法方案的平台。Sherpa是一个强大的工具，能够模拟高能物理实验中的粒子碰撞事件，包括NLO QCD计算。文中详细阐述了如何在Sherpa框架内实现这个新的减法方案，包括通过调整参数α来限制偶极相空间，以及扩展Sherpa原有的减法算法以适应电弱过程。为了验证这种方法的正确性，作者执行了一系列内部一致性检查，并展示了得到的结果。这篇开放获取的论文提供了NLO电弱计算中红外发散处理的新方法，这对精确预测高能物理实验结果，如大型强子对撞机（LHC）的数据分析，具有重要意义。这项工作不仅提升了理论计算的精度，也为未来更复杂的多阶效应计算奠定了基础。

Eur. Phys. J. C (2018) 78:119 Page 5 of 24 119

presence of photon splittings as their recoil partner assign-

ment is arbitrary and may involve information on initial or

ﬁnal state particles. Different possible choices are discussed

in Sect. 3.2, some of which may break this crossing invari-

ance.

The divergences of the I operator are encoded in the func-

tions V

and 

. While the former contains all soft-(quasi-

)collinear divergences the latter contains the pure (quasi-)

collinear ones. They do not only differentiate whether the

emitter is a photon or not, but also between different spins

of the emitter. Their precise form as well as the the ﬂavour

constants γ

and K

are given in Appendix B. A

encodes the

dependence on the phase space restriction of the individual

dipoles {α

dip

}. Finite terms originating in dipoles involving

initial state legs, however, can be pushed into the K opera-

tor. Thus, A

by convention only depends on α

. Its precise

form is given in Appendix C.

The K and P operators. The K and P operators collect all

pieces of the integrated dipole terms that are not collected in

the I operator and combines them with the collinear coun-

terterms C to give a ﬁnite result as  → 0. By construction

they contain only remainders of splittings where either the

emitter or the spectator is in the initial state. Thus, they are

comprised of terms arising due to the change of the ﬂavour

or the partonic momentum fraction x of an initial state due

to a splitting.

The K-operator is given by [41]



(x ;{α

dip

})

2π





(x ) − K

(x ) −





i,aa



(x )

−





(x ) − Q



(x ) + A



({α

dip

})



(2.9)

It depends on the partonic x, and the ﬂavour change from the

Born process initial state ﬂavour a to a



of the convolution

Eq. (2.6). Therein, the

K collect universal terms present in

all splitting involving an initial state as either emitter or spec-

tator. Then, while K contains solely remaining terms from

ﬁnal state splittings in the presence of initial state specta-

tors, the K

are their counterparts for initial state splittings

in the presence of a ﬁnal state spectator, i and k running over

all ﬁnal state partons in each case.

K contains solely related

correlations between both initial states, arising from dipoles

where both the emitter and the spectator are in the initial

state. The A

terms collect all ﬁnite terms arising when any

of α

, α

or α

is different from unity, thus restricting the

phase space of the respective dipoles. Again, the Q

are the

charge correlators of Eq. (2.5). Finally, K

contains the fac-

torisation scheme dependence. Currently, both only the

schemes is supported, setting these terms identically zero.

The P-operator now collects the remaining initial state

collinear singularity from all dipoles involving initial states

either as emitters or as spectators and cancels them against

the collinear counterterm. Through this counterterm a depen-

dence on the factorisation scale enters. The P operator is

given by [40,41]



(x,μ

)

2π



(x)







log

+ Q



log



(2.10)

Only initial state splittings are present, either in the presence

of a spectator in the ﬁnal state, which is encoded in the sum of

k, or with the opposite initial state b acting as the spectator.

It otherwise only depends on the Alterelli–Parisi splitting

function detailed in Appendix B.

3 Implementation

The implementation of the QED generalisation of the Catani–

Seymour dipole subtraction scheme in S

HERPA’s matrix ele-

ment generator A

MEGIC proceeds along the lines of [49]. As

in general real and virtual corrections of O(α

) contain

divergences of both QCD and QED origin, both cases are

included in this section. In the following, the general struc-

ture of the implementation is reviewed.

3.1 Identiﬁcation of dipoles

The starting point to construct the involved subtraction terms

in the Catani–Seymour subtraction formalism is a given

ﬂavour conﬁguration in the Born or the real emission phase

space and the perturbative order O(α

) in accordance with

the respective virtual or real correction to be computed. For

all parts, on-the-ﬂy variations of both the factorisation scale

and the renormalisation scale μ

are available through

an extension of the algorithm detailed in [56].

Differential subtraction terms. A given real emission con-

ﬁguration {ab}→{1, .., m + 1} at order O(α

) can in

general exhibit both QCD and QED divergences simultane-

ously. The following therefore describes the identiﬁcation of

both types of dipoles. Thus, all triplets {i, j, k} that can be

built from the external particles of the process are tested for

the presence of an infrared divergence, QCD or QED, by

checking for the existence of a dipole subtraction term. In

these triplets i and k may be in the initial or ﬁnal state while

j may be in the ﬁnal state only. Likewise, i = j, i = k, j = k

and triplets that only differ in a permutation of i and j are

considered identical. Then, the following steps are executed.

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