次导力N射流运营商的异常维度。第二部分资源-CSDN文库

Open

Access

156 浏览量 2020-04-01 17:32:12 上传评论收藏 1.31MB PDF 举报

资源推荐

资源详情

资源评论

JHEP11(2018)112

Published for SISSA by Springer

Received: August 16, 2018

Revised: October 24, 2018

Accepted: October 31, 2018

Published: November 19, 2018

Anomalous dimension of subleading-power N -jet

operators. Part II

Martin Beneke, Mathias Garny, Robert Szafron and Jian Wang

Physik Department T31, Technische Universit¨at M¨unchen,

James-Franck-Straße 1, D-85748 Garching, Germany

E-mail: mathias.garny@tum.de, robert.szafron@tum.de, j.wang@tum.de

Abstract: We continue the investigation of the anomalous dimension of subleading-power

N-jet operators. In this paper, we focus on the operators with fermion number one in

each collinear direction, corresponding to quark (antiquark) initiated jets in QCD. We

investigate the renormalization eﬀects induced by the soft loop and compute the one-loop

mixing of time-ordered products involving power-suppressed SCET Lagrangian insertions

into N-jet currents through soft loops. We discuss fermion number conservation in collinear

directions and provide explicit results for the collinear anomalous dimension matrix of the

currents. The Feynman rules for the power-suppressed SCET interactions in the position-

space formalism are collected in an appendix.

Keywords: Eﬀective Field Theories, Perturbative QCD

ArXiv ePrint: 1808.04742

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP11(2018)112

JHEP11(2018)112

B Soft master integral 56

C Auxiliary functions entering the collinear anomalous dimension 57

C.1 B-to-B mixing 57

C.2 B-to-C mixing 58

C.2.1 J

A∂χ

(x) → J

AAχ

, y

) 58

C.2.2 J

A∂χ

(x) → J

χ¯χχ

, y

) 62

D Anomalous dimension of hermitian conjugated operators 63

1 Introduction

The analysis of infrared (IR) divergences in QCD and gauge theories in general has always

been a fertile ﬁeld for exposing the universal structure of high-energy scattering ampli-

tudes and performing all-order resummations of the perturbative expansion in the gauge

coupling. What is commonly called the soft anomalous dimension of an amplitude of N

widely separated energetic particles refers in the framework of soft-collinear eﬀective the-

ory (SCET) to the simplest N-jet operator, where every jet is sourced by a single collinear

gauge-invariant quark or gluon ﬁeld [1, 2]. The increasing sophistication of multi-loop cal-

culations and the corresponding advance in precision has also triggered recent interest in

subleading-power eﬀects in the expansion in the scale 1/Q of the hard scattering [3–13]. In

a recent paper [14, 15] we began the systematic investigation of the one-loop anomalous di-

mension matrix of these subleading power operators. Previous relevant work on anomalous

dimensions of power-suppressed operators has been done in the context of heavy-quark de-

cay [16, 17] and for thrust [18, 19]. All-order resummations of subleading-power logarithms

can be found in refs. [16, 17, 20–22] covering cases with one or two collinear directions at the

leading logarithmic order (next-to-leading for heavy quark decay to one jet). The purpose

of our investigation is the complete analysis of one-loop infrared divergences of an arbitrary

subleading-power N-jet operator. This provides one of the ingredients in resumming or

generating at ﬁxed order subleading-power next-to-leading order logarithms for amplitudes

with any number of collinear directions or jets.

In the present paper, which follows upon ref. [14], we extend the calculation of the one-

loop anomalous dimension matrix from the case |F | = 2 to those with odd F , where F refers

to the fermion number of the product of collinear ﬁelds in a given collinear direction. This

contains as its simplest realizations the quark-antiquark initiated two-jet operators relevant

to the subleading-power resummation of thrust and other event shape variables in e

−

annihilation, and the threshold resummation of Drell-Yan type processes in hadron-hadron

collisions. In addition to the collinear renormalization kernels for the odd-fermion number

operators in a collinear sector, we discuss and calculate for the ﬁrst time the subleading-

power soft contributions to the anomalous dimension matrix, which did not appear for

the |F | = 2 operators. This involves a new contribution, which is not of the eikonal

– 1 –

JHEP11(2018)112

type, and arises instead from the mixing of power-suppressed soft-collinear interactions

in the SCET Lagrangian into power-suppressed N-jet operators with additional transverse

derivatives or collinear ﬁelds. The anomalous dimensions discussed here can be used to sum

subleading-power logarithms due to the evolution of the hard functions multiplying an N-jet

operator. Physical observables in general contain further logarithms from the evolution of

soft or jet functions already at the leading-logarithmic order (see the example of the thrust

distribution [22]). The soft and collinear kernels in the present paper contribute to the

next-to-leading logarithmic resummation, which yet has to be completed for an observable.

The outline of the paper is as follows. In section 2 we set up notation and conventions,

and discuss the form of the one-loop anomalous dimension matrix with respect to current

and time-ordered product operators, and its collinear and soft one-loop contributions. The

bulk of the paper is devoted to the calculation of the soft mixing contribution in section 3,

and the collinear kernels in section 4. We summarize in section 5. Translation rules from

positive to negative F, master integrals, and some auxiliary expressions for collinear kernels

are collected in appendices. We particularly note appendix A, which gives a complete list

of SCET Feynman rules in the position-space formalism [23, 24] up to the second order in

power-suppressed interactions and up to four-point vertices.

2 Set-up of notation and conventions

To make the paper self-contained we ﬁrst review some notation from ref. [14] and then

discuss the structure of the N-jet operator basis and the anomalous dimension matrix

relevant to the present work.

2.1 Operator basis

We consider N copies of the collinear SCET Lagrangian L

[24], i = 1, . . . , N, furnished

with corresponding collinear ﬁelds ψ

, as well as one set of soft ﬁelds ψ

that interact with

all collinear ﬁelds and with themselves according to the soft Lagrangian L

, in total

SCET

i=1

(ψ

, ψ

) + L

(ψ

) . (2.1)

The collinear ﬁelds are characterized by N pairs of light-like reference vectors n

i±

with

i−

· n

= 2, n

i−

· n

j−

= O(1), deﬁning N widely separated directions. We are interested

in current operators of the form

J =

dt C({t

}) J

(0)

i=1

, t

, . . . ) , (2.2)

characterized by one soft and N collinear contributions with certain transformation prop-

erties under soft- and collinear gauge transformations [14]. The collinear contributions are

composed of n

collinear building blocks ψ

, t

, . . . ) =

k=1

) , (2.3)

– 2 –

JHEP11(2018)112

that are oﬀset along direction n

by an amount t

from the origin, which is chosen to be at

the position of a hard interaction generating the N-jet current. Furthermore, dt =

and C({t

}) denotes a Wilson coeﬃcient.

Up to O(λ

), the soft building block J

(0) ≡ 1 is trivial, and soft ﬁelds do not enter

via J

as well [14, 15]. Therefore, the most general current basis contains only collinear

ﬁelds at this order. The complete operator basis can be constructed from the elementary

collinear building blocks χ

= W

†

, its conjugate ¯χ

, and A

⊥i

= W

†

[iD

⊥i

], as well as

currents obtained by acting with one or several derivatives i∂

⊥i

on the elementary building

blocks [14]. Here ξ

is the collinear quark ﬁeld, and W

a collinear Wilson line. At the

leading power only a single building block in each direction contributes (i.e. n

= 1 for

i = 1, . . . , N ) and no extra derivatives appear. Each additional building block or extra

derivative supplies a relative power suppression of order λ.

Every collinear factor J

in the current operator J can be characterized by its fermion

number F

, equal to the diﬀerence of the number of χ

and ¯χ

building blocks.

In the

present paper we consider the case of collinear directions with odd fermion number, which

implies |F

| = 1, 3 at O(λ

). In the following, we write down explicitly the basis of operators

for F

= 1. The leading power operator is J

) = χ

iα

), where we indicate the

open Dirac index α. At O(λ) there are two operators,

∂

) = i∂

⊥i

iα

) ,

, t

) = A

⊥i

)χ

iα

) . (2.4)

At O(λ

) there are further possibilities,

∂

) = i∂

⊥i

i∂

⊥i

iα

) ,

∂

, t

) = A

⊥i

)i∂

⊥i

iα

) ,

∂

)

, t

) = i∂

⊥i

)χ

iα

)) ,

, t

) = A

⊥i

)χ

iα

) ,

¯χ

, t

) = χ

iα

)¯χ

iβ

)χ

iγ

) . (2.5)

The superscript denotes the number of building blocks (A, B, C for one, two and three,

respectively) and the power suppression relative to A0.

When expanding in powers of λ, time-ordered products of currents J

with subleading-

power O(λ

) Lagrangian insertions L

(n)

need to be considered (n > 0). In contrast to the

current operators above, these insertions contain the soft ﬁeld explicitly. For each collinear

sector we discriminate interactions involving only collinear quarks, both soft and collinear

quarks, or no quarks, denoted by V = ξ, ξq, YM, respectively. Soft and collinear gluons

may be contained in all three contributions. The Lagrangians L

(n)

are given in ref. [24] for

n = 1, 2, see also appendix A. Consequently, for F

= 1 the basis needs to be complemented

by the time-ordered product operators

T 1

χ,V

) = i

x T

), L

(1)

(x)

, (2.6)

As we will see, F

is conserved under operator mixing up to O(λ

) in the absence of a mass term for

the quark ﬁeld.

– 3 –

剩余69页未读，继续阅读

评论收藏

内容反馈

weixin_38692707

粉丝: 8
资源: 901

次导力N射流运营商的异常维度。 第二部分

次引力N射流运营商的异常维度

N $$ \ mathcal {N} $$ = 4个超级Yang-Mills和QCD的完整四环尖点异常维度

行政区域维度表、时间维度表、日期维度表数据.rar

运营商数据指标体系梳理建设方案.pptx

三大运营商力保沪交通枢纽通信网络安全、畅通.pdf

电信运营商的渠道一体化运营思路.pdf

电信运营商行业大数据应用考试试题及答案.docx

从软异常维度超导Regge限制

互联网时代运营商产业发展规划.pdf

高三次理论中异常维数的谱

云计算助力电信运营商数字化转型中关键策略.pdf

通信行业周报2021年1月第2期：运营商投资价值凸显.zip

移动运营商课程体系.doc

通信：5G套餐即将面世，运营商迎来平台化经营时代.pdf

移动互联网下的运营商大数据应用探究.zip

运营商大数据变现实践.pptx

国际运营商数字化转型发展模式探究.docx

移动运营商日志文件

数据仓库维度建模笔记

电信运营商投资效益关联分析模型.docx

QCD中低非单一异常尺寸的五环贡献

5G时代的运营商计费模式探讨.docx

IBM关于某运营商的客户流失管理案例

中国第三方数据中心运营商分析报告（2022年）.pdf

最新资源

次导力N射流运营商的异常维度。第二部分