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图书名称：《标准模型以及超越 第2版 英文版》

【作　者】（美）保罗·兰盖克作

【页 数】 636

【出版社】 世界图书出版有限公司 , 2022.08

【ISBN号】978-7-5192-9614-8

【价 格】169.00

【分 类】标准模型-物理模型-研究-英文

【参考文献】 （美）保罗·兰盖克作. 标准模型以及超越 第2版 英文版. 世界图书出版有限公司, 2022.08.

图书封面：

图书目录：

《标准模型以及超越 第2版 英文版》内容提要：

本书提供了标准模型和其他非阿贝尔规范理论的物理和形式主义的高级介绍。它为理解超对称、弦论、额外维度、动力学对称破缺和宇宙学提供了坚实的背景。除了更新第一版的所有实验和唯象结果外，它还包含了对撞机物理的新章节；对希格斯粒子、中微子和暗物质物理学的进一步讨论；还有许多新问题。 这本书首先回顾了场论中的计算技术和量子电动力学的现状。然后重点讨论全局和局部对称性以及非阿贝尔规范理论的构造。对量子色动力学、对撞机物理、弱电相互作用和理论以及中微子质量和混合物理的结构和测试进行了深入的探索。最后一章讨论了扩展标准模型的动机，并考察了超对称性、扩展规范群和大统一。 本文全面涵盖规范场理论、对称性和标准模型之外的主题，为读者提供了理解标准模型的结构和唯象后果、构造扩展以及在树级执行计算的工具。它为读者在粒子物理学方面进行更深入的研究奠定了必要的背景。

《标准模型以及超越 第2版 英文版》内容试读

CHAPTER T

Notation and Conventions

D0:10.1201/b22175-1

In this chapter we briefly survey our notation and conventions.

Conventions

We generally follow the conventions used in (Langacker,1981).In particular,(u,v,p,o)are Lorentz indices;(i,j,k 1...3)are three-vector indices;(i,j,k 1...N)are alsoused to label group generators or elements of the adjoint representation;(a,b,c)run overthe elements of a representation,while (a,B,y)and (r,s,t)refer to the special cases ofcolor and flavor,respectively.(a,B)are also occasionally used for Dirac indices.(m,n)areused as horizontal (family)indices,labeling repeated fermions,scalars,and representations.

The summation convention applies to all repeated indices except where indicated.Opera-tors are represented by capital letters (T,Q,Y),their eigenvalues by the same symbols orby lower case letters!(t,q,y),and their matrix representations by (L',LQ,Ly).In Feyn-man diagrams,ordinary fermions are represented by solid lines;spin-0 particles by dashedlines;gluons by curly lines;other gauge bosons by wavy lines;and gluinos,neutralinos,and charginos by double lines.Experimental errors are usually quoted as a single num-ber,with statistical,systematic,and theoretical uncertainties combined in quadrature andasymmetric errors symmetrized.

Units and Physical Constants

We take h=c=1,implying that E,p,m,,have "energy units,"such as electronvolts (eV).2 Related energy units are

1 eV =103 meV 10-3 keV 10-6 MeV 10-9 GeV

=10-12TeV=10-15PeV=10-18EeV,

(1.1)

where the prefixes represent milli,kilo,Mega,Giga,Tera,Peta,and Exa,respectively.Onecan restore.conventional units at the end of a calculation using the values of h,c,and hclisted in Table 1.1.We use Heaviside-Lorentz units,in which the fine structure.constant isa =e2/4m,where e >0 is the charge of the positron.

1Or sometimes er for the electric charge of the rth quark.

2Most likely only dimensionless quantities,such as a or ratios of masses,are fundamental.

1

2

The Standard Model and Beyond

TABLE 1.1 Conversions and physical constants.i~6.6×10-22MeV-s

c3.0×1010cm/shc~197 MeV-fm

a-1137.04

a-1(M2)128.9

sin20w(M2)~0.2313

ag(M)~0.034

ag(M2)~0.010

a(M2)0.118

Gr~1.17×10-5GeV-2Mw 80.39 GeV

Mz ~91.19 GeV

me ~0.511 MeV

mu~105.7 MeV

m,~1.78GeV

mp ~938 MeV

mr±~140MeV

mK±~494MeV

Mp1.22×1019GeV

MH ~125 GeV

1g~5.6×1023GeV

k~1.16×104oK/eV

1barn=10-24cm21yr~3.16×107s~π×107s

For more precise values,see(Patrignani,2016).The Planck constant isMp=G where GN is the

gravitational constant.

Operators and Matrices

The commutator and anti-commutator of two operators or matrices are

[A,B=AB-BA,(A,B)=AB +BA.

(1.2)

The transpose,adjoint,and trace of an n x n matrix M are

transpose:MT (Mab =Mba),

adjoint:Mt =MT+

(1.3)

trace Tr M=

Ir(MM2)=Tr (M2M1),

Tr M=Tr MT.

(1.4)

Vectors,Metric,and Relativity

Three-vectors and unit vectors are denoted by and i=,respectively.We do notdistinguish between upper and lower indices for three-vectors;e.g.,the inner (dot)product

.may be written asy,,or iyi.The Levi-Civita tensorjk,where i,j,=1...3,is totally antisymmetric,with e123 =1.Its contractions are

EijkEijk=6,

EijkEijm=2δkm,

EijkEimn =8jmokn -8jnkm,

(1.5)

where the Kronecker delta function is

1,i=j

10,i≠j

(1.6)

eiik is useful for vector cross products and their identities.For example,

(A×B)i=Eijk Aj Bk

(1.7)

(A×B)·(C×D)=Eijkeilm AjBkCDm=(A.C)(B.D)-(A·D)(B.C)

(1.8)

Notations for four-vectors and the metric are given in Table 1.2.

The four-momentum of a particle with mass m is p4=(E,p)with p2=E2-p2 =m2.

Notation and Conventions

3

(The symbol p is occasionally used to represent p rather than a four-vector,but themeaning should always be clear from the context.)The velocity B and energy are given by

E

1

E

V1-B2

(1.9

m

Under a Lorentz boost by velocityBL(the relativistic addition of B toB,which is equivalentto going to a new Lorentz frame moving with -BL)

p→p=(E,'),

(1.10)

where

E=Yz(E+B·),'=1+YL(可+BE),

(1.11)

with

p=BB·元，

1

PL=P-P,

YL=

(1.12)

√1-

TABLE 1.2 Notations and conventions for four-vectors and the metric

Contravariant four-vector A#=(A0,A),#=(t,)

Covariant four-vector

A4=gA=(A0,-A)

x4=(t,一)

Metric

guw=gw=diag(1,-1,-1,-1)

1,4=y

9站三g09uo=6张=

0,4≠y

Lorentz invariant

A·B≡ALBH=guA“B"=A0BO-A·B

Derivatives

8三品=（品，），三品=（品，-0三0,8=品-2

0.A=0uA=驶+7.A

a分46=aa4b-(0a)b

Antisymmetric tensor

eHuPo,with E0123 =+1 and e0123=-1

Contractions

eHvpo Euvpo =-24EHvpO EuvpT =-6ggeHupo euvTw =-2(gfgo-gogg)

Translation Invariance

Let P4 be the momentum operator,i)and f)momentum eigenstates,

P)=1),Pf=p1f)

(1.13)

and let O(x)be an operator defined at spacetime point a,so that

O(z)=eiP.0(0)e-iP..

(1.14)

Then the z dependence of the matrix element (fO(x)i)is given by

(f()i)=ei(p-)(f()i).

(1.15)

The combination of Lorentz and translation invariance is Poincare invariance.

4

The Standard Model and Beyond

The Pauli Matrices

The 2x2 Pauli matrices=(01,2,03)(also denoted by especially for internal symme-tries)are Hermitian,oi=o,and defined by

Oi,Oj=2ieijkak.

(1.16)

A convenient representation is

(1.17

There is no distinction between o;and o'.Some useful identities include

2

Tri=>iaa=0,

Tr(o0j）=26j

a=1

(1.18)

{0,0}=26I→02=I,diaj dijI+ieijkok.

The last identity implies

(A·(B.)=ABI+i(A×)·,

(1.19)

where A and B are any three-vectors (including operators)and A.is a 2x2 matrix.Thus,(A.G)2=A2I for an ordinary real vector A with A =A,and

iA=(cos A)I+i(sin A)A.G.

(1.20)

Any 2 x 2 matrix M can be expressed in terms of o and the identity by

M=Tr(M)I+Tr(Ma).d.

(1.21)

The SU(2)Fierz identity is given in Problem 1.1.

The Delta and Step Functions

The Dirac delta function 6(x)is defined (for our purposes)by

6(x-a)g(x)dx=g(a)

(1.22)

-00

for sufficiently well-behaved g(x).Useful representations of 6(x)include

十●

6(x-a）=

ek(x-a)dk=lim

2

(x-a)2+y2

(1.23)

2rJ-00

The derivative of 6(x)is defined by integration by parts,

'(x-a)g()d三

g(e)

+odo(x-a)

dg

(1.24)

Suppose a well-behaved function f(x)has zeroes at zoi.Then

e-∑路

(1.25)

Notation and Conventions

The step function,e(x),is defined by

O(z-z')

x>x'

10x

(1.26

from which 6(x)=de/da.

Useful Integrals

Gaussian:

e-ax2+Bz dx =e82/4a

for Rea>0

Va

dzee4-00

(1.27)

Yukawa:

4π

2+2

1.1 PROBLEMS

1.1 Let xn,n =1...4,be arbitrary Pauli spinors (i.e.,two-component complex columnvectors).Then the bilinear formnis an ordinary number.Prove the Fierz identity

(x4X3)·(X2X1）=2mr(x4X1)(x2X3)-(X4X3)(x2X1),

where ng =+1.(The identity also holds for anticommuting two-component fields if onesets np =-1.)Hint:expand the 2 x 2 matrix x1x2 in (x4x1)(x2x3)using (1.21).

1.2 Justify the result (1.25)for 6(f(x)).

1.3 Calculate the surface area fdon of a unit sphere in n-dimensional Euclidean space,so that fd=fdnkdk.Show that the general formula yields

d21=2,

d22=2π，

d23=4π，

d24=2π2.

Hint:Use the Gaussian integral formula(1.27)to integratefein both Euclidean

and spherical coordinates.

1.4 Show that the Lorentz boost in (1.11)can be written as

E

coshyLsinhyL

E

sinhyr

coshyL、PI

where

1,1+BL

班=21n1-B配

is the rapidity of the boost.

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