顶夸克横截面研究：两圈增强与硬匹配

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"这篇研究论文详细探讨了在高能粒子物理学中，特别是在电子-正电子(e+e-)碰撞中，如何处理和分析顶夸克(top quark)的横截面，尤其是在顶夸克被强烈提升至高动量转移时的情况。文章强调了顶夸克横截面对于探测新物理学现象的重要性，因为它们对新物理非常敏感，同时由于顶夸克的重质量，使得相关计算在可控制的量子色动力学（Quantum Chromodynamics, QCD）框架内是可行的。作者Andrè H. Hoang、Aditya Pathak、Piotr Pietrulewicz和Iain W. Stewart在文中提出了一种在两个循环水平上进行硬匹配的方法，以精确计算顶夸克的喷注质量截面微分。硬匹配是理解高能量过程的关键，因为它允许我们分离出不同物理尺度下的动态效应。在这个过程中，他们特别关注了μ约为顶夸克质量mt的匹配系数，这是实现对数重新求和至N^3LL（近似对数领先阶）所必需的最后一部分，如果忽略四环尖瓣异常维数的影响。为了处理包含虚拟顶夸克回路的贡献，研究人员采用了速度重新归一化组（velocity renormalization group, VRG）方法。VRG允许他们在不同的能量尺度上系统地追踪和合并对数项，从而揭示了在μ ≃ mt附近两个有效理论之间匹配系数的快速RG演变。这种快速的演化现象表明了在重夸克质量尺度上的复杂动力学，并提供了对强相互作用更深入的理解。论文发表在JHEP12(2015)059，由Springer代表SISSA发布，于2015年8月27日提交，11月17日接受，12月10日正式发表。这项工作不仅加深了对顶夸克物理特性的认识，也为未来高能实验的分析提供了重要的理论工具和技术支持。"

JHEP12(2015)059

space S

†

(x) = P exp



∞

ds n·A

(ns+x)



, and ultrasoft Wilson lines Y

n,¯n

are the analogs

with ultrasoft gluon ﬁelds in bHQET. The diﬀerence between the SCET ﬁelds and bHQET

ﬁelds is that SCET still contains soft and collinear ﬂuctuations at the top mass scale, i.e.

the SCET ﬁelds contain mass mode ﬂuctuations which scale as (p

, p

−

, p

⊥

) ∼ (m, m, m)

and (Q, m

/Q, m) or (m

/Q, Q, m) which are absent in bHQET. This makes our EFT

above the top mass scale an SCET

type theory. There are six ﬂavors in the MS running

coupling in QCD and SCET, and ﬁve ﬂavors in bHQET.

The notation above diﬀers from ref. [14] which used a hybrid of SCET

and SCET

where the current operator was written as

SCET

= ¯χ

†

¯n

. (2.2)

Here the Wilson lines S

n,¯n

describe exclusively soft mass mode ﬂuctuations and have ultra-

soft zero-bin subtractions. In eq. (2.1) the SCET operator only describes soft ﬂuctuations

above and of order of the mass scale m, and not far below m. This simpliﬁes the setup for

the matching coeﬃcient calculation, which in particular can be viewed as going from a six

ﬂavor theory to a ﬁve ﬂavor theory.

The matching coeﬃcients between these eﬀective theories are deﬁned by

+1)

QCD

= C

+1)

SCET



1 + O(m/Q)



, (2.3)

+1)

SCET

= C

)

bHQET



1 + O(ˆs/m)



. (2.4)

Here both the currents and Wilson coeﬃcients refer to the renormalized quantities. When

we refer to the bare objects we will indicate this explicitly as e.g. in J

(bare,n

+1)

SCET

. For

all quantities we consider we use the renormalized coupling constant. When we want to

separate the color structures of the matching coeﬃcients we will do so in the following way:

+1)

= 1 + C

(1, n

+1)

+ C

, n

+1)

+ C

, n

+1)

+ C

, n

+1)

+ C

, n

+1)

)

= 1 + C

(1, n

)

| {z }

O(α

)

+ C

, n

)

+ C

, n

)

+ C

, n

)

+ C

, n

)

| {z }

O(α

)

. (2.5)

In all the objects above the coupling is renormalized in the MS scheme with the number

of dynamical ﬂavors, n

, being either n

or (n

+ 1) as indicated by the superscript. Here

is the number of light quarks, and the additional ﬂavor indicates the heavy quark (here

the top quark). The choice for the number of ﬂavors in each of the expressions above is

motivated by the scales at which these objects live compared to the top mass. Note that

we have kept the number of ﬂavors appearing in C

unspeciﬁed, as it can be expressed in

either the n

- or the (n

+ 1)-ﬂavor scheme. We will be explicit about which scheme we are

using in the equations below.

The hard functions in eq. (1.2) are related to the Wilson coeﬃcients via

(Q, µ) = |C

, H



, µ



= |C

. (2.6)

Here the dependence on Q in the hard function H

appears due to the boost factor Q/m.

– 6 –

JHEP12(2015)059

In eq. (1.2) all the functions live at their respective scales and are evolved to a common

scale µ

ﬁnal

through renormalization group running. While the jet and the soft functions

have convolution running [14], the large logarithms of the hard matching coeﬃcients are

summed by multiplicative evolution factors,

evol

(Q, m, µ

ﬁnal

; µ

, µ

, ν

) ≡H

+1)

(Q, µ

) U

+1)

(Q, µ

, µ

) (2.7)

× H

)



, µ

; ν

, ν



)



, µ

ﬁnal



for µ

' Q, µ

' m and µ

ﬁnal

< µ

. On the l.h.s. the dependence on µ

and µ

only

comes from higher order corrections when the objects in eq. (2.7) are truncated at a given

order in resummed perturbation theory. The same is true for the rapidity scales ν

and

, which are induced by the rapidity RGE that will be discussed further below and in

section 5.1. We will frequently drop these arguments that appear after the semicolon. The

evolution factors here obey the RG equations

dµ

+1)

(Q, µ

, µ) = −γ

+1)

(Q, µ) U

+1)

(Q, µ

, µ) ,

dµ

)



, µ, µ

ﬁnal



= +γ

)



, µ



)



, µ, µ

ﬁnal



, (2.8)

where γ

)

is the anomalous dimension for the squared current in bHQET.

Eqs. (2.3) and (2.4) suggest two diﬀerent methods that one can use to calculate the

O(α

) piece of C

or equivalently H

1) Indirect calculation using the known result for C

and the matrix elements for the

QCD and bHQET current operators in pure dimensional regularization.

Using eq. (2.3) and (2.4), and taking matrix elements of the operators with onshell

top-quark states as in [13], we have

+1)

QCD

i = C

+1)

)

bHQET

i. (2.9)

Using the relation between bare and renormalized bHQET currents

)

bHQET

i = Z

)

bHQET

(bare, n

)

bHQET

i, (2.10)

we get

)

+1)

QCD

+1)

)

bHQET

(bare, n

)

bHQET

. (2.11)

Note that the terms on the r.h.s. involve objects with diﬀerent ﬂavor number schemes

for the strong coupling, which must all be converted to n

-ﬂavors to get C

)

. Here

we work in dimensional regularization for both UV and IR divergences and renormalize

the quantities in the MS scheme. With this regulator we can use the known two loop

result for the heavy form factor hJ

QCD

i given in refs. [30, 31]. The result for C

is also

– 7 –

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