

 31 Jul 1995

IHEP-95-90 of

V.I. Borodulin , R.N. Rogalyov , S.R. Slabospitsky

Protvino 1995

C O R E CO REmpendium lations

Version 2.1

Institute for High Energy Physics Protvino, Moscow Region, 142284

RUSSIA

1 2 3

1 2 3

E-mail: BORODULIN@mx.ihep.su E-mail: ROGALYOV@mx.ihep.su E-mail: SLABOSPITSKY@mx.ihep.su

PREFACE The present ( mpendium of lations, ) contains various formulas and relations used in the practical calculations in the Stan- dard Model.

The properties of the Pauli, Dirac, and Gell-Mann matrices, widely used in calculations in the Standard Model, are considered in details. Properties of the wave functions of free fermions and gauge bosons are also discussed.

We present the full Lagrangian of the Standard Model and the corre- sponding Feynman rules. The method of the evaluation of the Feynman (loop) integrals and calculations in non-covariant gauges is considered.

We discuss in brief the relativistic kinematic and present a large number of the matrix elements of the various processes in the Standard Model.

Almost all of the presented relations and formulas can be found in liter- ature. However, one can find different definitions, different normalizations, not widely used notations, etc. We try to collect various expressions in one place and make the notations and normalizations consistent.

We hope that the present will be useful for practical calculations in the Standard Model, especially for post-graduates and young physicists.

We wish to thank A.V. Razumov for multiple and useful discussions.

i

Version 2.1 Protvino, 1995

CORE 2.1 CO RE

CORE 2.1 2.1 4 2.2 4 2.3 5 2.4 5 2.5 6 2.6 7 2.7 8

3.1 9 3.2 10 3.3 4 4 10 3.4 12 3.5 13 3.6 14 3.7 15

4.1 = 2 + 1 16 4.2 = 2 21

5.1

22 5.2

24

2 6.1 26 6.2 27 6.3 2 2 27 6.4 28 6.5 29

ii

Main Properties Fiertz Identities

and Matrices Various Relations 4-dimensional Matrices Traces of Matrices Fiertz Identities for Matrices

Main Properties Representations of the Dirac Matrices Expansion of Matrices Products of the Dirac Matrices Fiertz Identities Traces of the -matrices Dirac Matrices Algebra in -dimensions

Odd-dimensional Case : Even-Dimensional Case:

Representation of 3-dimensional Vectors, Reflections and Ro- tations Using the Pauli Matrices Representation of 4-dimensional Vectors, Reflections and Ro- tations Using the Dirac Matrices

General Properties Spinors and Vectors Representation of Lorentz Transformations by Matrices Self-dual and Anti-self-dual Tensors Correspondence Between 2-spinors and 4-bivectors

+ Contents 1 NOTATION AND DEFINITIONS 1 2 PAULI MATRICES 4

3 DIRAC MATRICES 9 4 THEORY OF SPINORS IN DIMENSIONS 16 5 VECTOR ALGEBRA 22

6 -COMPONENT SPINORS 26

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : oe oe : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : oe : : : : : : : : : : : : : : : : : : : : oe : : : : : : : : : : : : : : : : : : : : : : :

oe : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : fl : : : : : : : : : : : : : : : : : : : : :

n : : : : : : : : : : : :

N n * : : : : : : : : : : : : : : :

n * : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : :

: : : : : :

\Gamma

_ _

_

\Theta

\Theta

6.6 30 7.1 33 7.2 35

8.1 37 8.2 38 8.3 38 8.4 39 8.5 40 8.6 (3) 41

10.1 52 10.2 53 10.3 55 10.4 59

10.4.1 59 10.4.2 59 10.4.3 60 10.4.4 61 10.4.5 63 10.4.6 65 10.4.7 66 10.4.8 67 10.4.9 68 10.4.10 70

11.1 72 11.2 73 11.3 74

iii

Isotropic Tetrads in Minkowski Space General Properties Bilinear Combination Decomposition

Main Properties Traces of the -matrices Fiertz Identity Products of the -matrices Convolutions of and with Invariant Tensors

General Remarks Propagators Some Popular Gauges Vertices

Gauge Boson Three-vertices Gauge Boson Four-vertices Gauge-boson-fermion Vertices Gauge-boson-Higgs Three-vertices Gauge-boson-Higgs Four-vertices Higgs Three-vertices Higgs Four-vertices Higgs-boson-fermion Vertices Gauge-boson-ghost Vertices Higgs-ghost Vertices

Dimensional Regularization Integrals Spence Integral (Dilogarithm)

7 DIRAC SPINORS 33 8 GELL-MANN MATRICES 37

9 STANDARD MODEL LAGRANGIAN 43 10 FEYNMAN RULES 52

11 INTEGRATION IN -DIMENSIONS 72

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : t : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : :

t : : : : : : : : : : : : : : : : : : : :

d f t : : : : : : : : : : : : : : SU : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

N

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : :

a

a

abc abc a

12.1 77 12.2 78 12.3 81 12.4 82 12.5 83 12.6 84

13.1 86 13.2 87 13.3 88 13.4 88 13.5 89 13.6 ( _) 90

14.1 91 14.2 92 14.3 93 14.4 94 14.5 95

15.1 97 15.2 98

15.2.1 _ 98 15.2.2 98 15.2.3 99 15.2.4 99 15.2.5 _ 100 15.2.6 _ 100 15.2.7 _ 100 15.2.8 ( ) _ 101 15.2.9 _ _ 101 15.2.10 101 15.2.11 _ _ 101

iv

Variables Event Shape Variables Two-body Final State Three-body Final State Lorentz Invariant Phase Space Width and Cross Section

Standard Model Higgs Decays Rates

and Decays Muon Decay Charged Meson Decay Quark Decay Heavy Quarkonia Decays

Annihilation Two-photon Process at Collisions

Reactions Cross Sections in the Parton Model Vector Boson Polarization Vectors

General Remarks Matrix Elements

+

+

+ + + + + + + +

12 KINEMATICS 77 13 DECAYS 86 14 CROSS SECTIONS 91 15 MATRIX ELEMENTS 97

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : W Z : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

QQ : : : : : : : : : : : : : : : : :

e e : : : : : : : : : : : : : : : : : : : : : : : : :

e e : : : : : : : : : : : : : l h : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : e e f f : : : : : : : : : : : : : : : : : : : : : : : : e e e e : : : : : : : : : : : : : : : : : : : : : : : e e flflfl : : : : : : : : : : : : : : : : : : : : : : : : e e l l fl : : : : : : : : : : : : : : : : : : : : : : : e e qqg : : : : : : : : : : : : : : : : : : : : : : : : e e qqfl : : : : : : : : : : : : : : : : : : : : : : : : gg qq : : : : : : : : : : : : : : : : : : : : : : : : : : flg flfl f f : : : : : : : : : : : : : : : : : : : : : : : qq QQ : : : : : : : : : : : : : : : : : : : : : : : : : qq qq : : : : : : : : : : : : : : : : : : : : : : : : : qq qq : : : : : : : : : : : : : : : : : : : : : : : : :

\Gamma

\Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

!! !! !! ! !

!! !

15.2.12 101 15.2.13 _ _ 101 15.2.14 _ _ 102

16.1 103 16.2 103 16.3 104

v Miscellanea Properties of Operators The Baker-Campbell-Hausdorff Formula

1 2 3 4 16 MISCELLANEA 103

REFERENCES 106

gg gg : : : : : : : : : : : : : : : : : : : : : : : : : f f W f f : : : : : : : : : : : : : : : : : : : : l * dug : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : :

\Lambda \Gamma

!! !

! Throughout this article units are used in which _ = = 1.Everywhere the indexes imply the summation. = = = ( = 1 ) is Kronecker symbol,

= 0 = = = = 1 Metric tensor in Minkowski space = ( = 0 1 ) equals:

= 0 = = 1 = = = = 1 The tensor is used for raising and lowering of the Lorentz subscripts and superscripts.

The scalar products of any two and vectors (both in Euclidean and in Minkowski spaces) is denoted as follows:

or ( ) i e = ( ) The scalar products of any two and Euclidean vectors would be also denoted as:

i e = = ( )

4-vector in Minkowski space is given by

( ) = ( ) = = ( ) The scalar product of any two vectors and in Minkowski space is given by

= = =

The products of the 4-vector with Dirac matrix denotes as usual

^ = =

. : ( = 1 2):

= = 1; = = 1; = 0 11 0 ;

= = = 2; = ;

= ; + + = 0

1

repeated -symbol in two dimensions

11 22 00 11 22

0 1 2 3

0 0 1 1 2 2 3 3

12 12 21 21

1 NOTATION AND DEFINITIONS

Totally antisymmetric tensor

h c ffi ffi ffi ffi i; k ; : : : ; n

ffi i k; ffi ffi : : : ffi :

g g _; * ; ; : : : ; n g _ *; g ; g g : : : g :

g

p q

p q pq ; : : p q pq :

p q

~p~q; : : ~p~q p q pq : p

p E; ~p p ; p ; p ; p ; p g p E; ~p :

p q

p g q p q p q p q p q p q p q :

p fl

p p g fl p fl p fl :

" " " A; B ;

" " " " " " " ; " " ; " " " " ffi " " ffi ffi ffi ffi " " " " " " :

ik ik i

k

k i

ik nn

_* _* _* nn

_*

_ _ _ _* *

_ _* * _ _ * *

_ _ _ _* * _ _ * *

AB:::N AB

AB AB AB BA AB AB AB AB BC C

A

AB CD CA DB DA CB AB CD AC DB AD BC

fflffl ffl

6

6 \Gamma

ffl

\Delta \Delta

\Delta ffl j \Gamma

\Gamma \Gamma \Gamma j fflffl

\Gamma \Gamma \Gamma \Gamma \Gamma

!

-symbol is used for rising and lowering of the spinor indexes (see Subsec- tion 2.5).

: ( = 1 2 3):

= = 1 = =

= = 2 = 6 . For any 3-vector one has:

+ = 0

: ( = 0 1 2 3): = = 1

= = = 2( ) = 6 = 24

. For any 4-vector one has:

+ + + + = 0

Sometimes one can make no difference between a vector and index. For example, one can write:

or ( ) instead of These notation can be used in operations with generalized Kronecker deltas:

or

2

-symbol in three dimensions Schouten identity

-symbol in four-dimensional Minkowski space

Schouten identity

Generalized Kronecker deltas

123 123

0123 0123

1 2 3 4 1 2 3 4

1 1 1 1

"

" " i; j; k ; ;

" " ; " " ; " "

ffi ffi ffi ffi ffi ffi

ffi ffi ffi

" " ffi ffi ffi ffi ; " " ffi ; " " :

p p " p " p " p " : " " ff; : : : * ; ; ;

" " :

" "

ffi ffi ffi ffi

ffi ffi ffi ffi ffi ffi ffi ffi

ffi ffi ffi ffi

; " "

ffi ffi ffi

ffi ffi ffi ffi ffi ffi

;

" " ffi ffi ffi ffi ; " " ffi ; " " :

p

p " p " p " p " p " :

" " p ; p ; p ; p " p p p p : ffi

ffi ::: ffi

::: ::: ::: ffi ::: ffi

; ffi

p q ::: p q

::: ::: ::: p q ::: p q

:

AB

ijk

ijk ijk ijk lmn

l i

l j

l k m i

m j

m k n i

n j

n k

ijk lmk li mj mi lj ijk ljk li ijk ijk

i

i i i i i i i i i i i i i i i i

fffi_*

_*fffi *aeoeo/

* _

* *

* ff

* fi ae _

ae *

ae ff

ae fi oe _

oe *

oe ff

oe fi o/ _

o/ *

o/ ff

o/ fi

_*fffi *aeoefi

* _

* *

* ff ae _

ae *

ae ff oe _

oe *

oe ff

_*fffi *aefffi *_ ae* ae_ ** _*fffi **fffi *_ _*fffi _*fffi

_

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

p p p p _*aeoe _ * ae oe j :::j i :::i

j i

j i

j i

j i

q :::q p :::p

n

n n n

ffl

\Gamma \Gamma \Gamma ffl

\Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

ffl

j j \Delta \Delta \Delta \Delta

fifi

fifi

fifi

fi

fifi

fifi

fifi fi

fifi

fifi

fifi fifi

fi

fifi

fifi

fifi fifi

fi

fifi

fifi

fifi

fi

fifi

fifi

fifi

fi

fifi

fifi fifi

fi

fifi

fifi

fifi fi

fifi

fifi

fifi

fi

fifi

fifi

fifi

fi

1 2 3 4 2 1 3 4 3 1 2 4 4 1 2 3 1 2 3 4 5 2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 5 1 2 3 4

1 2 3 4 1 1

1 1

1

1

1 1

n n

n

n

n n

n

n

In -dimensional Euclidean space one has:

= 1( )! In Minkowski space the minus sign appears:

= = 12

For any matrix = ( ) ( = 1 ) we use the following notation:

is the matrix, i.e. = (sometimes, the unit matrix will be denote just 1);

is the matrix, i.e. = = ; is the matrix, i.e. = ; is the matrix, i.e. ( ) = ( ) ; is the matrix, i.e. = ; - and - matrices should satisfy the following condi- tions:

= = ( ) hence = = =

det is the of matrix

det =

= 1!

Tr is the of matrix : Tr = (= ). The chief properties of the trace are as follows (below and are parameters):

Tr( + ) = Tr + Tr Tr = Tr Tr = Tr = (Tr ) Tr = Tr( ) = Tr( ) det( ) =

For any two matrices and the [ ] and

are denoted as usual:

[ ] +

3

Matrices

unit

inverse transposed complex conjugated Hermitian conjugated Hermitian unitary

determinant trace

commutator anticommutator

1 1 1

1 1

1 2

=1

Tr

n

ffi n m " " :

ffi " " ; ffi " " :

A a i; k ; : : : n I I ffi

A A A AA I A a a A a a A a a H U

H H; U U ; U U ; U U U U I:

A A

A " a a a

n " " a a a : A A A a a

* _

*A _B * A _ B; A A; A A A ; I n;

AB BA ; e e :

A B A; B A; B

A; B AB BA; A; B AB BA:

\Gamma \Gamma \Gamma ? ? \Lambda \Lambda \Lambda y y \Lambda

y

y \Gamma y \Gamma y y

? \Lambda y \Lambda

q :::q p :::p

q :::q ff :::ff p :::p ff :::ff

q q q p p p p p p _

q q q _ q q

p p p p _*

q q _*

ik

ik

ik ki

ik ik

ik ki

i i :::i i i i n

i i :::i k k :::k i k i k i k

ii ni ii

A A

\Gamma \Gamma \Gamma ffl

\Delta \Delta \Delta

\Delta \Delta \Delta

f g

j \Gamma f g j

P

1 1

1 +1

1 +1

1 2 3 1 2 3 1 2 3

1 2 3 1 2

1 2 1 2

1 2

1 2

1 2

1 2 1 2

1 1 2 2

m

m

m m n

m m n

n

n

n n

n n

The Pauli matrices ( = 1 2 3) are generators of the group (2). The

are equal [1, 2, 3, 4]:

= 0 11 0 = 0 0 = 1 00 1 The main properties of are as follows:

= Tr = 0 det = 1

= + (2.1)

Using relation (2.1), one gets:

= [ ] = 2 = 2

= + + Tr( ) = 2 Tr( ) = 2 Tr( ) = 2( + )

The Fiertz identities for the Pauli matrices have the form:

= 2 (2.2) = 32 12 (2.3)

Using (2.2), one can obtain the following relations:

= 12 [ + + ] = 12 [ + ] + = + = 12 [ + +

+ ]

4

1 2 3

2

2 PAULI MATRICES

oe i ; ; SU oe

oe ; oe ii ; oe :

oe oe oe ; oe ; oe ; oe oe i" oe ffi :

oe I; oe ; oe i" oe ; oe ; oe ffi ; oe oe oe i" I ffi oe ffi oe ffi oe ;

oe oe ffi ; oe oe oe i" ; oe oe oe oe ffi ffi ffi ffi ffi ffi :

oe oe ffi ffi ffi ffi ; oe oe ffi ffi oe oe :

ffi oe ffi oe oe ffi i" oe oe ; oe ffi ffi oe oe ffi i" oe oe ;

ffi oe oe ffi oe ffi ffi oe ; oe oe oe oe ffi ffi ffi ffi oe oe

i" oe ffi i" ffi oe :

y i i

i i i i i i k ikj j ik

i i k ikj j i k ik i k l ikl ik l il k kl i

i k ik i k l ikl i k l m ik lm im kl il km

i AB

i CD AD CB AB CD

i AB

i CD AD CB

i AD

i CB

AB iCD AD iCB iAD CB ikl kAD lCB

i AB CD AD

i CB

i AD CB

ikl k

AD

l CB

AB iCD iAB CD iAD CB AD iCB i AB

k CD

i AD

k CB

ik AD CB ik l

AD

l CB

ikl l

AD CB

ikl AD l

CB

\Gamma \Gamma

\Gamma f g\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma \Gamma

! ! ! 2.1

2.2

Main Properties Fiertz Identities The and matrices are defined as follows:

1 2 ( + ) =

0 1 0 0

1 2 ( ) =

0 0 1 0

The relations for these matrices are given by

( ) = Tr = 0 det = 0 [ ] = [ ] = [ ] = 2 [ ] =

= = = 0 = = = 0 = =

= 12 ( + ) = 12 ( ) ( ) = ( ) =

For any parameter one gets:

( 2 ) ( 2 ) = ( ) If ( ) (or ( )) is an arbitrary function of (or of ), and this function can be expanded into power series with respect to (or with respect to ), then

( ) = (0) + [ (1) (0)]

( ) = (0) + [ (1) (0)]

Any 2 2 matrix can be expanded over the set :

= + where = Tr , and = Tr( ). Let be the 3-vector. Then

= cosh + sinh ( ) = + (2.4)

5

+

+ 1 2 1 2

1 3 2 3 3 + 3

1 2 3 + 2+ 2 + 3 + 3 + +

+ 3 + 3

+ + + +

3 3 + + + +

+ +

+ +

+ +

0 0 12 12

2 2

2 0

+ oe oe

oe oe ioe ; oe oe ioe :

oe oe ; oe ; oe ; oe ; oe oe ; oe ; oe ioe ; oe ; oe oe ; oe ; oe oe ;

oe ; oe I; oe ; oe iI; oe ; oe ; oe ; oe I; oe oe ; oe oe oe ; oe oe oe ;

oe oe I oe ; oe oe I oe ;

oe oe oe oe ; oe oe oe oe :

, exp , oe oe exp , oe oe exp , : f oe oe f oe oe oe oe oe oe

oe oe oe oe

f oe oe f f f oe oe ;

f oe oe f f f oe oe :

A I; oe

A a I a oe ; a A a oe A ff

e ~ff ~ff~ff ff oe p p oe :

\Gamma

\Gamma

\Sigma y \Upsilon \Sigma \Sigma \Sigma \Sigma \Sigma \Sigma \Gamma

\Sigma \Sigma \Sigma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma

\Sigma \Sigma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma

n n

i i i i i i

ff oe i i i i

j j \Gamma

\Sigma \Upsilon f g f g \Sigma f g f g

\Gamma

\Gamma

\Gamma \Sigma

\Gamma \Gamma \Theta f g

p pp

! ! i i 2.3 2.4

and Matrices Various Relations oe oe\Gamma The components of the 4-vector equal:

= cosh = sinh = = 1 (2.5) and we have

= ln( + ) (2.6)

Let and be two 4-vectors, and = = 1, then

( + )( + ) = + = (2.7) where = + ( ) = + + , and the 3-vector in the relation (2.7) is expressed through and as in (2.6).

Here we present the various properties of 2 2 matrices and _ ( = 0 1 2 3):

( ); _ ( ) = 0 1 2 3 (2.8) where are Pauli matrices. With the help of -matrices any tensor in Minkowski space can be unam- biguously rewritten in spinorial form. In order to deal only with Lorentz- covariant expressions one should clearly distinguish between dot and undot, lower and upper Weyl indices. The -symbol (see Section 1) used here for rising and lowering indices. The chief properties of the matrices are as follows:

_ = = _ ( ) = (_ ) = _ det = det _ = 1( 1) for = 0(1 2 3)

For any 4-vector one has:

det = det _ =

6

0 2

2

2

20 2 2

2 0

2

2 2 0 0 0 0 0 0 0 0

0

.

.

. . .

. . . .

.

2

p p ~ff ; p ~ff~ff ff ; p ~p p ;

ff p~p p ~p : p q p q

p p oe q q oe a a oe e ; a p q ~p~q ; a p q p q i" p q fi

a ~a

oe oe _ ; ; ;

oe I; oe oe I; oe ; _ ; ; ; ; oe

oe

" oe oe " " oe ; oe " " oe ;

oe oe ; oe oe ; oe oe ; _ ; ; :

p

p oe p oe p :

y y

_ i i

i

i

i i k k l l fi oe

j j j jkl k l i

_ _ _ AB i

_AB i

i

_

_ _AA AB AB _

BB

_ AA AB AB

_BB

_ _ _ _ _ _

_

_ _ _ _

p pp \Gamma

p

\Theta j \Gamma j

\Gamma

q

i i

2.5 4-dimensional Matricesoe_

Various products of matrices have the form:

_ + _ = 2 _ + _ = 2

_ = 4 _ = 4

= 4 = 4 _ _ = 4 _ _ = 4

= _ _ = 2 _ = + _ _ = + +

= ( )

The commutators of and _ matrices have the special notation:

1 4 ( _ _ ) _

1 4 (_ _ )

The main properties of are as follows:

= 12 = 2 _ = 12 _ =

= _ = _ ( _ ) = + 2 (_ ) = + 2_

= _ = _ = 2 _ = 2 _

Tr = Tr_ = 2(0) for = 0 (1 2 3) Tr = Tr_ = 0 Tr( _ ) = Tr(_ ) = 2 Tr( _ _ ) = 2( + )

Tr( ) = Tr(_ _ ) = 12 ( )

7

.

.

.

.

.

.

.

.

. .

.

. . . . .

. .

. .

. . . .

. . . . . . . .

. .

. . . . . .

. . . . . . . . . . . .

.

.

.

. . . .

.

.

.

0 0

. . . . . . . . . . . . . .

oe oe oe oe oe g ffi ; oe oe oe oe g ffi ;

oe oe ffi ; oe oe ffi ; oe oe " " ; oe oe " " ; oe oe " " ; oe oe " " ;

oe oe " " oe oe " " g ; oe oe oe g oe g oe g oe i" oe ; oe oe oe g oe g oe g oe i" oe ;

" ioe oe oe oe " " " " " " " " :

oe oe

oe oe oe oe oe ; oe oe oe oe oe :

oe oe oe ; oe i " oe ; oe oe ; oe oe ; oe oe ; oe oe ;

oe oe g ffi oe ; oe oe g ffi oe ; oe " oe " ; oe " oe " ; " oe ioe ; " oe ioe :

oe oe _ ; ; ; oe oe ; oe oe oe oe g ;

oe oe oe oe g g g g g g i" ;

oe oe oe oe g g g g i" :

_ _ AC

*CB *

AC

_CB _* AB

_AC *

CB

*AC _

CB

_* A

B

_ AC

CB _ A

B _AC

_CB

A

B

_ AA _BB

AB AB _

AA _BB

AB

AB

_AA BB

_ AB

AB _AA BB

_ AB

AB

_ AA

* BB

AB AB _AA *BB AB

AB

_*

_ * * _* * ** _ _* * _**ae ae _ * * _* * ** _ _* * _**ae ae _*ae* _AA *BB aeCC *DD AC BD

AD BC AD BC AC BD

_ _

_*B A

_

AC

*CB *

AC

_CB _*A

B

_AC *

CB

*AC _

CB

_*

i i ik ikl l i i ik ik _* *_ _* *_

_ * AB _* AB _*B

A

_ * A

B

_* A

B

_*A

B

_*K A KB

_*K B KA

_*A

K

KB _*B

K

KA

_**ae *ae _* _**ae *ae _*

_ _ _* _* _ * _ * _*

_ * * ae _* *ae _ae ** _* *ae _**ae

_* *ae _* *ae _ae ** _* *ae _**ae

\Gamma \Gamma \Gamma

\Gamma

j \Gamma j \Gamma

\Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma

2.6 Traces of Matricesoe_

The Fiertz identities for equal:

_ = 2 = 2 _ _ = 2 (2.9) From the relations (2.9) one gets:

_ = 12 _ + + 2 _

= 12 + ( _ ) + ( )

2( )( _ ) _ _ = 12 + ( ) + (_ )

2( )(_ )

( )(_ ) = ( )(_ )

8

.

. . .

. . . .

. . . .

.

. . . . . . . . .

. . . . . .

. . . .

. . . . . . . . . . . . . .

. . . .

. . . . . . . .

oe oe oe ffi ffi ; oe oe " " ; oe oe " " :

oe oe g ffi ffi ffi oe oe ffi oe oe ;

oe oe g " " " " oe oe " "

oe " " oe ; oe oe g " " " oe " " oe "

" oe oe " ;

" oe oe " " oe oe " :

_ _ AA

BB _ A

B

A

B _

AA _BB AB AB

_AA BB

_

AB AB

_ AA

*BB _* AB

A

B AB _*B

A

_*B A A

B **B

A

_*B

A

_ AA

* BB

_* AB

AB AB AC

_*C

B

_*C A CB AB

_*C A CB AC

**C

B

_AA *BB _* AB AB AC _*B

C

AB AB _*A

C

CB

AC _*C

B

**A

C

CB

AC _*C

B

**A

C

CB AC **C

B

_*A

C

CB

\Gamma \Gamma \Gamma

2.7 Fiertz Identities for Matricesoe_ The main properties of the Dirac -matrices are as follows [1, 2, 3, 4, 5]:

+ = 2 (3.1) ( ) = ( ) = ( ) = ( ) = (3.2)

The commutator of the -matrices has a special notation:

1 2 ( ) = (3.3)

The matrix is defined as follows:

= 4! (3.4) The matrix has the following properties:

+ = 0 ( ) = ( ) = = 0 The of any 4 4-matrix is defined as follows:

_ (3.5)

From (3.5) one gets:

= = =

= ( ) =

In this Section for the string of the -matrices we shall use the special nota- tion:

= = (3.6) Odd- and even-numbered string of -matrices will be denoted as follows:

(3.7)

9

Dirac conjugation

0 2 2 0 0 5

5 0 1 2 3 5

5 5 5 2 5 5 5

0 0 5 5 5 5 5

3 DIRAC MATRICES

fl fl fl fl fl g ;

fl I; fl I; ; fl fl ; fl fl :

fl

oe fl fl fl fl ; oe oe fl

fl ifl fl fl fl i " fl fl fl fl : fl

fl fl fl fl ; fl I; fl fl ; fl ; fl :

A A fl A fl :

fl fl ; fl fl ; fl fl fl fl fl fl ; fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl :

fl

S S fl fl fl ; S S fl fl fl :

fl S fl fl fl ; S fl fl fl :

y y y y

_ * * _ _*

i i i

_

_* _ * * _ _* *_

fffi_* ff fi _ * _ _ _

_ _ ff fi * * fi ff ff fi _ * * _ fi ff * _ fi ff

n ff ff ff R n

R

ff ff ff

odd ff ff ff even ff ff ff

\Gamma \Gamma j \Gamma \Gamma j \Gamma

f g \Theta

j

\Gamma \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Gamma \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta

j \Delta \Delta \Delta j \Delta \Delta \Delta j \Delta \Delta \Delta j \Delta \Delta \Delta

1 2 2 1 1 2 2 +1 1 2 2

n n k k

3.1 Main Properties The non-singular transformation connects the different represen-tations of the -matrices (Pauli lemma). Here we present three representa- tions of the Dirac matrices.

representation

= 1 00 1 = 0 0 = 0 11 0

representation = = 12 ( + ) = 12 1 11 1

= 0_ 0 = 0 11 0 = 0 0 = 1 00 1 1 + = 2 1 00 0 1 = 2 0 00 1 where matrices and _ were defined in (2.8).

representation

= = 12 1 1 = 0 0 = 00 = 0 0 = 00 = 00

The following 16 matrices \Gamma ( = 1 16)

(3.8)

10

0 5

0 5

0 5

5 5

2 2

0 2

2

1 3

3

2 2

2

3 1

1

5 2

2

5 5

Dirac (standard) Chiral (spinorial)

Majorana

fl U flU fl

fl ; fl oeoe ; fl : fl U fl U ; U fl fl ;

fl oeoe ; fl ; fl oeoe ; fl ;

fl ; fl : oe oe

fl U fl U ; U oeoe ; fl oeoe ; fl ioe ioe ; fl oeoe ; fl ioe ioe ; fl oe oe :

A ; : : : ; I; fl ; fl ; fl fl ; oe

y y

y

D

i D

i i D

_ C C

_ D C C D D

_ _

_

i i

i

_ _ _ M M

_ D M M

A

_ _ _*

! ffl

\Gamma \Gamma ffl

p p \Gamma

\Gamma \Gamma \Gamma

ffl

p \Gamma

\Gamma

\Gamma \Gamma \Gamma

! ! !

! ! ! ! !

! !

! ! ! ! ! !

3.2

3.3

Representations of the Dirac Matrices Expansion of Matrices4 4\Theta are the full set of 4 4-matrices. The main properties of \Gamma are as follows:

Tr = 4 Tr = Tr = Tr = Tr = 0 (3.9) Any 4 4-matrix can be expanded over set of the \Gamma -matrices:

= + + + + (3.10) where the coefficients could be found from the following relations:

= 14 Tr = 14 Tr( ) = 14 Tr( ) = 14 Tr( ) = = 18 Tr( ) For the expansion of a matrix one can use another set of \Gamma (\Gamma =

):

= + = = ( + ) = ( )

= 2 = 2

These matrices have the following properties:

= = = = = = 0 Tr = Tr = 4 Tr = Tr = Tr = 0

The expansion of any 4 4-matrix over set of \Gamma -matrices has the form:

= + + + + where

= 18 Tr( ) = 18 Tr( ) = 18 Tr( ) = 18 Tr( )

= 18 Tr( )

11

5 5 0 5 5 5 0 5 5

5

5 5 5 5 2 2

2 2

I ; fl fl fl fl oe

A A a I a fl v fl a fl fl T oe ;

a A; a fl A ; v fl A ; a fl fl A ; T T oe A :

A X; Y; U ; V ; oe

X I fl ; Y I fl ; U I fl fl ; V I fl fl ; X X; Y Y:

U V XY Y X XU Y V ;

X Y ; U U oe :

A A a X a Y b U c V T oe ;

a XA ; a Y A ; b V A ; c U A ; T oe A :

0 0 0

A

_ _ _*

A _ _ _ _ _* _*

_ _ _ _ _* *_ _*

A A _ _ _*

_ _ _ _

_ _ _ _ _*

x y _ _ _ _ _* _* x y _ _ _ _

_* _*

\Theta \Theta

\Gamma \Gamma \Gamma

\Gamma \Gamma

\Theta \Gamma

= + = = + = + 2 = + 2 = ( ) = ( ) = ( ) = ( ) =

+( + ) + = 2( )

The totally antisymmetric tensor is defined as follows:

1 6 ( + + )

= + +

= = 6

The products of the type \Gamma \Gamma \Gamma are presented in the Table 3.1.

\Gamma \Gamma \Gamma \Gamma \Gamma

4 4 1 4 4 12 2 2 0 2 2 0

0 0 4

So-called Chisholm identities are given by [5]:

= 2 (3.11)

= = 2 + 2 (3.12)

12

5 5 5

5

5 5 5 5

5 5 5

5

5

5 5

16=1

5 5 5 5 5 5 5 5 5 5 5 5 5

Table 3.1.

fl fl g oe ; oe fl fl g fl fl g ; fl fl fl g fl i " oe ; fl oe i " oe ;

fl oe g fl g fl i" fl fl ; oe fl g fl g fl i" fl fl ; fl fl oe g fl fl g fl fl i" fl ; oe fl fl g fl fl g fl fl i" fl ;

oe oe g g g g i" fl

g g g g g g g g g g g g oe ; oe oe oe oe g g g g i" fl :

fl

fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl ; fl fl fl fl g fl g fl g fl ; fl i" fl fl ; fl fl i " fl :

fl fl fl fl fl fl fl fl oe oe I I fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl

oe oe oe

fl S fl S ; fl S fl fl fl S fl fl S S fl ;

0 0 0

_ * _* _* _* _ * _* * _ _* _ * _* _*fffi fffi _* _*fffi fffi * _* _* * ** _ *_*ff ff

_* * _* * ** _ *_*ff ff * _* _* * ** _ *_*ff ff _* * _* * ** _ *_*ff ff

fffi _* ff* fi_ ff_ fi* fffi_*

ff* fi* _oe ff_ fi* *oe fi* ff* _oe fi_ ff* *oe *oe fffi _* _* fffi ff* fi_ ff_ fi* fffi_*

_**

_** _ * * * * _ * _ * * _ * * * _ _ * * _ * * _** _* * _* * ** _ _** _**ff ff ff ff_** _**

A

A B A

B B * B * * B * _* B _*

ff ff ff ff

ff ff ff ff fffi fffi fffi

_ odd _ odd

R

_ even _ _ * odd _ * odd

R

odd *

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

j \Gamma \Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma

P

3.4 Products of the Dirac Matrices where in the last relation = . Using the relations (3.11) and (3.12), one gets:

= Tr( ) Tr( ) ^ ^ = + 12 Tr(^ ) ^ ^ ^ = ^ ^ for = 0

= 12 Tr( ) + 12 Tr( )

^ ^ = + 12 Tr(^ )^ + 12 Tr( ^ )^

= 14 Tr( ) + 14 Tr( ) + = 12 Tr( ) Using (3.11) and (3.12), one can write also the well known relations for ,

, , and :

= 2 = 2 = 4 = 2( + ) = 2( + ) = 0 = 2 = 2

Fiertz identities for -matrices could be obtained from the basic formula:

= 14 [ + + ( ) ( ) 12 ( ) ] (3.13) Using (3.13) one can obtain the well known relations:

\Gamma \Gamma = \Gamma \Gamma (3.14) The coefficients are presented in Table 3.2, where we use the traditional notations:

= = = = =

13

5 5 2

2

5 5

2 5 5

5 5

1 2 3 4

5 5 5 5

16

=1

5 5

S fl S fl S fl S I fl S fl ;

pS p p S pfl S fl p; pS p pS p; p ; fl S fl fl S fl fl fl S fl fl ;

pS p p S pS p fl pS pfl ;

S fl S fl fl fl S fl fl ; S S fl S fl :

S S S S

fl fl fl fl ; fl fl fl fl fl fl fl fl ; fl fl fl fl g ; fl fl fl fl fl fl fl fl fl fl fl fl fl fl

fl fl fl fl fl fl fl fl ; fl oe fl ; fl oe fl fl fl oe ; fl fl oe fl oe fl :

fl ffi ffi ffi ffi fl fl fl fl fl fl fl fl oe oe :

C ; C

S I; P fl ; V fl ; A fl fl ; T oe :

0even * odd _ even _ even even

even even

R

ff even

R ff

even even

R

_ odd _ ff odd ff ff odd ff

odd odd

R

odd R

odd R

odd ff odd ff ff odd ff

odd odd

R

ff odd ff

_ ff _ ff _ ff fi ffi _ ffi fi ff _ ff fi _ fffi _ ff ff ff ff _ ff ff ff ff ff ff ff ff

ff ff ff ff ff ff ff ff _ fffi _ _ fffi ffi _ ffi fffi _ ffi fffi _ fffi ffi

ij kl il kj il kj

_

il _ kj

_ il _ kj _*

il _* kj

M

ij

M

kl

N

MN Nil Nkj

MN

_ _ _*

\Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma X

1 2 3 4 4 1 2 3 3 2 1 4

1 4 3 2 2 3 4 1

3.5 Fiertz Identities

*

= =

1 0 1

3 0 0 3 1 0 1

Using relation (3.13) one gets:

(1 ) =

1 8 [2(1 ) (1 ) + 2((1 ) ) ((1 ) ) ((1 ) ) ]

(1 ) = 1 8 [2(1 ) (1 ) + 2((1 ) ) ((1 ) ) ((1 ) ) ]

(1 ) (1 ) = 14 [2(1 ) (1 ) ((1 ) ) ]

(1 ) (1 ) = 12 [(1 ) ] [(1 ) ] [(1 ) ] [(1 ) ] = [(1 ) ] [(1 ) ] [(1 ) ] [(1 ) ] = 2(1 ) (1 ) ( ) ( ) + ( ) ( ) = [( ) ( ) + ( ) ( ) ]

The trace of any odd-numbered string of -matrices (including any number of matrices) and trace of the product are equal to zero:

Tr = Tr ( ) = Tr( ) = 0 In this Subsection we use the following notation:

1 4 Tr( )

14

1

4

1

4

1 8

1

4

1

41

2

1

21

21 2

1

21

4

1

4

1 8

1

4

1

4

5

5 5 5 5 5 5

5 5 5 5 5

5 5 5 5 5 5 5 5 5

5 5 5 5 5 5 5 5

5 5 5 5

5 5

5 5

Table 3.2.

N S V T A P M I

V

T A P

fl ffi

fl fl fl fl fl fl fl oe oe ; ffi fl

fl fl fl fl fl fl fl oe oe ;

fl fl fl fl fl oe oe ; fl fl fl fl fl fl ;

fl fl fl fl fl fl fl fl ; fl fl fl fl fl fl ; fl fl fl fl fl fl fl fl fl fl fl fl :

fl fl fl fl fl

S S fl fl fl fl :

T fl fl ::: fl :

ij kl

il kj _ il _ kj _* il

_*

kj

ij kl

il kj _ il _ kj _* il

_*

kj

ij kl il kj _* il

_*

kj

ij kl _ il _ kj

_ ij _ kl _ il _ kj

_ ij _ kl il kj _ ij _ kl _ ij _ kl _ il _ kj _ il _ kj

_ * odd odd _ *

_ _ :::_ _ _ _

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

\Sigma

\Sigma \Sigma \Sigma \Upsilon \Gamma \Sigma \Sigma

\Sigma \Sigma \Upsilon \Sigma \Gamma \Sigma

\Sigma \Sigma \Sigma \Sigma \Gamma \Sigma \Sigma \Upsilon \Sigma \Upsilon

\Sigma \Sigma \Gamma \Sigma \Sigma \Sigma \Upsilon \Sigma \Upsilon

\Gamma

\Delta \Delta \Delta \Delta \Delta \Delta j

\Gamma \Delta 1 2 1 2n n

3.6 Traces of the -matricesfl

Then

= = +

= + + Tr( ) = 0 Tr( ) = 0 Tr( ) = 4 Tr( ) = 4 ( + + + ) Tr = 4( )

Using the relation (3.13), one can rewrite the trace of the product of two 4 4 matrices and as follows:

4Tr( ) = Tr( )Tr( ) + Tr( )Tr( ) + Tr( )Tr( )

Tr( )Tr( ) 12 Tr( )Tr( )

The additional equation can be obtained using the Chisholm identities (3.11) and (3.12):

Tr( )Tr( ) = 2 Tr( ) + Tr( )

In the framework of dimensional regularization one gets:

Tr = ( ) ; ; (4) = 4 =

+ = 2

= (2 )

= 4 + ( 4)

= 2 + (4 )

15

5 5 5 5

5 5 5 5

T g ; T g g g g g g ; T g T g T g T g T g T ;

fl ; fl fl fl ; fl fl fl fl fl i" ; fl fl fl fl fl fl fl i g " g " g " g " g " g " ;

oe oe g g g g :

A B AB A B fl A fl B fl A fl B

fl fl A fl fl B oe A oe B :

Afl B fl S AS B AS B :

I f n ; = = f ; g g n; fl fl fl fl g ; fl fl fl n fl ; fl fl fl fl g n fl fl ; fl fl fl fl fl fl fl fl n fl fl fl :

_* _* fffiffioe fffi ffioe ffoe fiffi ffffi fioe fffiffi*aeoe fffi ffi*aeoe ffffi fi*aeoe ff* fiffiaeoe ffae fiffi*oe ffoe fiffi*ae

_ * ff fi ffi * fffiffi* ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff

fffi _* ff* fi_ ff_ fi*

_ _ _ _ _* _*

_ _ odd odd odd

R

_* _* _ * * _ _*

_ ff _ ff _ ff fi _ fffi ff fi _ ff fi ffi _ ffi fi ff ff fi ffi

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma

\Theta

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

h i

1 2 3 4 5 6 1 2 3 4 5 6 1 3 2 4 5 6 2 3 1 4 5 6 4 5 1 2 3 6 4 6 1 2 3 5 5 6 1 2 3 4

3.7 Dirac Matrices Algebra in -dimensionsn

Let be -dimensional complex linear normalized space. The cases of odd and even should be considered separately.

Let vector has coordinates = ( ). The basis in can always be chosen so that the the norm squared of the vector looks as follows:

= + + + (4.1) A (hyper)plane is called isotropic if each vector in it has zero norm.

(4.2) So, one has the following natural decomposition of into the sum of three spaces:

= (4.3)

i.e., if then

= + + where (4.4)

= + and are isotropic -planes: = 0 and = 0. Let one consider Grassman algebra (algebra of external forms) on some

-dimensional vector space . An element has the form:

= (4.5) here are components of antisymmetric rank tensors. There are three important operations on :

16

Isotropic planes Grassman algebras

0 1 2 2 +1 20 1 +1 2

2 +1

2 +1

2 +1 2 +1

0 0

=1

20

=0

4 THEORY OF SPINORS IN DIMEN-

SIONS

Isotropic planes in have not more than dimensions

x x x x

x x x x x x

E n

n

x x x ; x ; :::; x E

x

x x x x x ::: x x :

E * :

E

E R E E ; x E

x x e ; E ; E :

x x x : E E *

G * F , G

, , e ::: e : , p

G

0 0

0

0 0

n

* * * * * *

*

* I I* II* *

I *

II *

*

i

i i

I *

II *

* * *

*

p

i :::i i i

i :::i

*

ffl

\Delta

\Phi \Phi 2

2 2 \Delta

\Delta \Delta ffl

2 ^ ^

X X

1

1

1

p

p

p

4.1 The Odd-dimensional Case : n *= 2 + 1

N External product. Let , then , the components of look as follows:

( ) = ( 1) + +

+( 1) + (4.6)

An element \Omega is called -form if its decomposition (4) has only -th term.

\Omega \Omega = ( 1) \Omega \Omega

there exists the dual element which has the components:

( ) = 1! (4.7) If then ( ) and has the following compo- nents:

( ( ) ) = ( 1) (4.8)

These operations satisfy the following relations [6]:

( ) = 0 (4.9) ( ) ( ( ) ) = 0 (4.10) ( ) ( ) + ( ( ) ) = ( ) (4.11)

here and are components of .

( ) = ( 1) ( ( ) (\Omega )) = ( 1) \Omega ( \Omega ) = ( 1) ( ) \Omega

It is also useful to introduce the operation :

= ( 1) (4.12)

It satisfies the properties:

( ) + ( ) = 0 (4.13) ( ( ) ) + ( ) ( ) = 0 (4.14)

17

( )

( ) ( ) ( ) ( )

1

=0

( )

( ) ( +1) 1 ( ) ( ) ( +1) ( )

0 0

=0

0 0 0 0

A. B. C.

x F ; , G x , G x ,

x , x , x ,

x , x , :

G p p

:

, G ,

, p " , : x F ; , G id x , G

id x , , x

x x , ; id x id x , ; id x y , y id x , x y ,;

x y x; y F

, , e ::: e ; id x x ; x id x :

H

H , , e ::: e :

H x , x H , ; H id x , id x H , :

\Gamma \Gamma

\Gamma

* * *

i :::i p i i :::i i i :::i i

p i i ::::i i i i i ::::i

p *

p q pq q p *

i ::::i i ::::i * ::::* * ::::* * * *

i ::::i p j i ::::i j

i i i i *

*

p

p * p i ::::i i i

p * p p p * p p

* p

p i ::::i i i

2 2 ^ 2 ^

^ \Gamma \Gamma

2

^ \Gamma ^8 2 \Lambda

\Lambda 2 2 ffi 2

ffi \Gamma

^ ^ffi ffi

ffi ^ ^ ffi

2

\Lambda \Lambda \Gamma ^ ^ \Lambda ffi \Gamma ^\Lambda ^ \Gamma ffi

\Gamma ^ ^ ^ ^ffi ffi

X X

1 +1 1 2 +1 2 3 +1 1

3 4 +1 1 2 +1 1

1 1 1 1

1 1 1 1

1

1

1

1

p p p

p p p

* p * p p p

p p

p

p

p

p

\Gamma \Gamma

\Gamma \Gamma

For any one can define the operation ( ) on Grassman algebra :

( ) = ( ) + + ( ) (4.15) where and are defined in (4.4). From the properties (4.9 - 4.11), (4.13), and (4.14) and the the expression for the norm squared (4.1) one can easily derive the following property of operation :

( ) ( ( ) ) = ( ) (4.16) Let now an element be written as the column vector:

=

scalar one-form (covector)

two-form (bivector)

-form ( -vector) Such 2 -component column vectors are called spinors in the space One can now construct the matrix representation of the operation : the column vector corresponding to ( ) should be written in the form ^ , where ^ is 2 2 matrix. So one has the representation

^ (4.17) of vectors by 2 2 matrices which satisfies the fundamental property:

^^ = (4.18)

or, if one substitutes vector + instead of here one can rewrite it in the form:

^^ + ^^ = 2 (4.19)

(2 + 1) Let us start with the observation that

= + 2 (4.20)

18

Definition of a spinor

-transformations

2 +1

0 0 0

0

2 +1

2 +1

x x x x

x E Clif f x G

Clif f x , id , , x H , ; ; x

Clif f Clif f x Clif f x , x x ,:

, G

,

, , ,

::: ,

:::::: * *

E Clif f Clif f x , x, x

x x x E

xx x x; x y x

xy yx x y: O *

q q q l l

0 0

0

* *

* i i i

i ::::i * *

* * * * *

_ _ _

ffl 2

ffi ffi ^

ffi ffi \Delta 2

ffl

ffi\Theta ! 2 \Theta

\Delta

\Delta ffl

\Delta

0B BBBB B@

1C CCCC CA 1 1 2

1 *

where is obtained from by the reflection with respect to the 2 -plane which is orthogonal to a unit space-like vector : = 1. The representa- tion (4.17) of this formula looks as follows:

^ ( ) = ^^^ (4.21) Let , where is the boost (or rotation) in ( ) plane ( = = 1), which transforms into . Since any rotation is a composition of an even number of reflections, one can easily derive the relations:

( ( )) = ^ ^ ^^ ^ = ^ (4.22) = ^ ^

Hence the action of reflections and rotations on spinors looks as follows:

= ^ = ^ ^ (4.23)

- .

^ = 0 (4.24)

Indeed, if ^ = 0 and ^ = 0 then, due to (4.19), = 0. Hence this hyperplane is isotropic. That it has not more than dimensions is seen from the (4.2). -dimensional plane can be manifestly constructed for the case when only is not equal to zero. (Strictly speaking, this plane has exactly

dimensions only for so called simple spinors (spinor is called simple, if rang of the system (4.24) is + 1). However, each spinor is a sum of simple spinors, for = 1 2 each spinor is simple. In general simple spinors in

lie in a manifold in the space of all spinors, which is determined by = 2 (2 + 1) equations).

One can also consider isotropic -forms instead isotropic -planes, that is equivalent. The -form corresponding to some isotropic -plane is the external product of the linearly independent vectors from this -plane.

Proofs of all the statements listed here can be found in [7].

[8]. One can also consider the Grassman algebra as the Fock space ( ) of a

19

Geometrical interpretation: spinors are forms in An analog with a fermion system

2 2 1 2 21 22 1 2

2 1 1 2 1 2 1

2 1

2 +1

0 2 +1

1 2 +1

The equation determines -dimensional isotropic plane for each spinor .

The components of can be interpreted as elements of the Grassman algebra on this -plane.

q q *

l l

q S q lql: R ae ae a ; a a a a a

R S S q a a qa a AqA ; A a a :

S , l,; R , l l ,:

* E

x * , x y x y

* * , * ,

* * ; E N C



*

* * * *

* *

G H F

0

0

\Gamma

\Gamma

l a a

a a a a

l l l

*

*

* * *

*

*

\Gamma j j\Gamma

j

\Sigma \Sigma ffl

\Delta

\Gamma ffl

1 2

1 2 2 1

2 1

fermion system with degrees of freedom. Let now and be the creation and annihilation operators which satisfy the relations:

= 0 = 0 = An arbitrary vector ( ) can be written in the form:

= 0 (4.25) The operators and can be represented by the action of creation and annihilation operators:

( ) where are components of vector . The -matricies in 2 + 1 dimensional space have the following form:

= + (4.26)

= (4.27)

= ( 1) where = (4.28) The = ( ) 2 are generators of (2 + 1), so one has natural realization of (2 + 1)-symmetry in ( ) in addition to usual whose generators are .

A generalization of the matrix of charge conjugation can be defined as the matrix representation of the operation in the space of spinors:

= ( 1) ( ) (4.29) Its main properties are as follows:

^ = ( 1) ^ = ( 1) Then

is scalar in (i.e. with respect to (2 +1)) if is even, and pseudoscalar (2 + 1-form) if is odd.

20

Invariant Forms

=0 + 2 +1

=1

2 2 +1

* a a a ; a ; a ; a ; a ; a ffi :

, ? H F

, ? , a :::a ? : id

x , x a , ? id x , x a , ?;

x x F fl *

fl a a

fl a ai fl ; N a a oe fl fl fl fl = SO *

SO * H F U

a a

C, , Cx x C; C :

j C, E O * * * *

y y y y

y y

y

y

y

y

y

\Gamma

i i i j i j i j ij

* p

i ::::i i i

i i

i i

i *

j j j j *

j j

* N F

*

i

i i

kl k l l k

* k l

p* * T

T *

f g f g f g

j 2

j j ^

^ ! jffi ! j

2

\Gamma \Gamma \Gamma

ffl

\Lambda \Gamma \Lambda

\Gamma \Gamma

X

X 1 1

( 1)

2

( +1)

2

p p F

p p

* * \Gamma

The space can be considered as 2 -dimensional hyperplane in , which is orthogonal to some vector . Each spinor can be repre- sented as the sum = + , where and are eigenspinors of ^:

^ = ^ = Only the -planes corresponding to eigenspinors of ^ lie in ; an arbitrary isotropic -plane intersects over some ( 1)-plane. In the language of external forms this idea can be formulated as follows. An arbitrary -form

on can be uniquely decomposed into two components:

= + where is -form, is ( 1)-form, both are defined on . The two forms

and give the geometrical interpretation of a spinor in .

The fact that there exist just the two types of isotropic -planes in (corresponding to spinors of the type and ) is the consequence of the theorem that the representation of (2 ) on -forms can be decomposed into two irreducible parts (self-dual and antiself-dual forms).

The most widely used case in high energy physics is = 2 ^ = = ( ). Concerning the vector (see (6.11 - 6.13)) one can say that it determines the line which is the intersection of the isotropic 2-plane corre- sponding to a spinor ( ) with (complexified) momentum space.

21

2 2 +1

2 +1

2 2

2 +1

2

2

2

5 +

E * E

a E ' O/ ' O/ a

a' '; aO/ O/: * a E * E *

* T E

T R a S; R * S * E R S E

* E ' O/ SO * *

* ; a fl ; , u p; n !

u p; n

* *

*

* *

*

*

*

*

2 \Gamma

\Gamma

^ \Gamma

4.2 The Even-Dimensional Case: n *= 2

5 VECTOR ALGEBRA Let be some basis and scalar products define a matrix :

= . The dual basis is the set of vectors , which satisfy the conditions:

= = ( )

Then

= \Delta

where \Delta = . Sometimes one needs to represent some vector in the form [9]:

= +

where is a linear combination of ( ), and = 0 for

= 1 .

= 1 = = = 2 = = +

= = 3 = = + +

=

Any vector in 3-dimensional Euclidean space can be represented in the matrix form:

^ = = + det ^ = = ( + + ) where is the Pauli matrices (see Section 2). The fundamental property of this representation is

(^) = hence ^^ + ^^ = 2 ( ) (5.1)

22

f g \Delta \Delta f g

\Delta \Delta

PP \Delta P P

P

ffl

j \Delta \Gamma \Gamma

\Gamma \Delta \Gamma

\Delta \Delta

p ; : : : ; p p p M M p p , ; :::; ,

, p ffi ; , , M :

, ffi = ; ffi Q

Q V ;

p ; :::; p m ! n V p i ; :::; m

m p V ffi m ffi ffi ffi p ffi p

V ffi m ffi ffi ffi p ffi p ffi p

V ffi :

~x

x ~x ~oe x oe x x ixx ix x ; x ~x ~x x x x ; oe

x ~x ~x I; xy yx ~x ~y I:

1

1

1

1

1 1 1 \Delta

1 \Delta 1 2

1 \Delta

1 2\Delta

1 \Delta 1 2 3

1 \Delta

3 1 2 1 2 3

21 22 23

2

n i j ij i j n

i j ij i j ij

ff i

p ;:::;p ;ff;p ;:::;p p ;: : : : : : : : : : ;p n

n p ;:::;pp ;:::;p

ff ff ff ff m i

ff p Qp p ff ff p p

p ff

p Q

ff Q_p p _ffp p Qpp p ff p Qp p ff

ff

p p ff

p p Q

ff Q_*p p p ff_*p p p Qp pp p p ff p Qpp p p ff p p Qp p p ff

ff

p p p ff

p p p Q

i i i

\Gamma \Delta \Delta \Delta

1 1 +1 1

1 1

1 1 1 1 1

1 1

2 1 2 1 2 2

2 1 2

1 1 2

2

1 2 1 2

3 1 2 3 1 2 3 3

2 3 1 2 3

1 3 1 2 3

1 2 1 2 3

3

1 2 3 1 2 3

i i n

n

n n

\Gamma

i j i j

! 5.1 Representation of 3-dimensional Vectors, Reflections and

Rotations Using the Pauli Matrices

Vectors

One should also note that

^^ ^^ = 2 If components of are real, then ^ = ^. However, in some practically important cases (^) = 0 and, hence, = 0, components of are complex, say, = . Then the matrix

^ = + represents a vector from 3-dimensional space-time, in that case ^ = ^. Let be an arbitrary vector and be the plane orthogonal to some unit vector . Then, vector which results from after the reflection in the plane is equal to:

= 2( )

or, in the considered matrix representation

^ = ^^^

Let and be the two unit vectors with the angle 2 between them:

= = 1 ( ) = cos( 2) Since any spatial rotation is a composition of two reflections, the rotation by the angle in the direction from to is given by the matrix

= ^^ i.e. an arbitrary vector transforms as follows:

^ = ^ = ^^^ ^^ The matrix can be rewritten in widely used form:

= ^^ = + ^ = cos( 2) sin( 2) (5.2) where sin( 2) = ,

. So, we get the two forms of representation of a spatial rotation:

23

\Gamma \Theta

\Delta

\Gamma \Gamma

6ffl

\Gamma

\Gamma ffl

\Delta \Gamma

\Theta \Gamma

xy yx i ~x ~y: ~x x x x ~x ~x ~x x ix

x x x xx x x

x x ~x S

~s ~x ~x S

~x ~x ~x~s ~s;

x sxs: ~p ~q `=

~p ~q ; ~p~q `= :

` ~p ~q

M qp; ~x

x M xM qpxpq: M M qp ~q ~p I i" r `= I i ~n~oe `= ; ~n `= ~p ~q; ss ! ` ! ss `

~n

2 2 0

3 1 0 1 0 3

2 2

1 qpr

y

y 0

0

0

0 \Gamma

d ! Reflections

Rotations

positive values of correspond to counterclockwise rotations if one sees from the head of vector

The rotation by angle about a unit vector is given by

= cos( 2) sin( 2)

The rotation in the plane of unit vectors and which transforms into is represented by

= + ^^2(1 + )

4 4 matrix ^, which represents the 4-vector in Minkowski space looks as follows:

^ =

0 0 + 0 0 +

+ 0 0 + 0 0

This matrix satisfies the fundamental property

(^) = ^^ + ^^ = 2 (5.3)

Using the relation (5.3) one can easily derive formulas for the reflections in 3-hyperplanes. Let be an arbitrary vector and be the 3-hyperplane orthogonal to some unit vector . Then, vector which results from after the reflection in the hyperplane is equal to

= 2 or, in the considered matrix representation

^ = ^^^

Let and be the two unit space-like vectors:

= = 1

24

ffl

\Gamma

ffl

\Delta

ffl \Theta

j

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma

\Delta ) \Delta ffl

\Gamma \Delta

\Gamma ffl

\Delta \Delta \Gamma

` ~n M `= I i ~n~oe `= :

~p ~q ~p ~q

M I qp~q ~p

x x x x fl

x x x ix x ix x x x x x ix x ix x x

:

x x xI xy yx x y:

x S

s x x

S

x x x s s;

x sxs: p q

p p q q :

0 3 1 2 1 2 0 3 0 3 1 2 1 2 0 3

2

_ _ _

_

0

0

0

q 0BBB@ 1CCCA 5.2 Representation of 4-dimensional Vectors, Reflections and

Rotations Using the Dirac Matrices

Vectors

Reflections Lorentz transformations

Lorentz transformation, which is a composition of the reflections in 3-hyper- planes determined by the vectors and is given by the matrix:

= ^^ i.e. an arbitrary vector transforms as follows:

^ = ^ = ^^^ ^^ The Lorentz transformation in the 2-plane (defined by the vectors and ), which transforms into , is represented by

= + ^^2(1 + )

25 p \Delta p q M qp; x

x M xM qpxpq:

p q p q

M I qpq p :

10 \Gamma

6 -COMPONENT SPINORS The representation (5.2) of spatial rotations acts on two-component column- vectors:

=

which are called (see [10] for details). So, the matrix in (5.2) can be written in the form

= cos( ) sin( ) ( ) sin( )( + ) sin( ) cos( ) + sin( ) =

= hence = ( ) The -tensor (see Section 1), which defines (2)-invariant scalar product on spinors has the form:

= 0 11 0 The main properties of the -tensor are as follows:

= = ^ = ^

For each matrix ^ corresponding to a vector and for each matrix corre- sponding to a rotation, one has

= ( ) (6.1) (2) ( ) = and this relation implies that this repre- sentation is equivalent to its complex conjugate, i.e. the conjugate spinor

= is transformed with the same matrix as . So, the bilinear form on spinors

( ) = = + (6.2) is Hermitian (2)-invariant form, while the form

= = (6.3) is also (2)-invariant, but not Hermitian. Both these forms are

with respect to reflections.

26

\Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma

2 )

h i \Gamma

u uu M M in n inn in in ff fifi ff ;

M M M M : " SU

" : " "" I; " I;

"x x ":

x x M

" M M ": M SU M M

u i"u M u

u; v u v u v u v SU

u; v u "v u v u v SU

1 2

2 3 2 2 1 2

2 1 2 2 3 2

1 1

2

1 1

1 1 2 2

0 1 1 0

` ` `

` ` `

T

T

T

T T

T

\Lambda \Lambda y \Gamma \Lambda \Gamma

\Gamma \Gamma \Lambda

0 \Lambda

y \Lambda \Lambda

! ! !

!

6.1 General Properties

spinors

2

not invari- ant

The object transforms as a vector under the reflections and rotations provided that and are spinors:

(6.4)

Hence, for any spinor the relation ^ = defines the vector . ( Tr ^ =

= = 0). The basic property which connects spinor and vector is ^

= 0 (6.5)

Since ^( ^ ) = ( ) = 0 = 0, vector should have complex components. So, this relation expresses

3 3 matrix of spatial rotations ( ) looks as follows:

= cos + (1 cos ) + sin

0

0

0

Eigenvalues of this matrix are 1 and . Eigenvector ( ), corre- sponding to the eigenvalue 1 directs the axes of the rotation. The other two eigenvectors have zero norm and correspond to eigenspinors of matrix (5.2).

4-vectors of Minkowski space can also be represented by 2 2 matrices:

= ++ _ _ = ++ det = det _ = ( ) ( ) ( ) ( ) = =

27

! !! \Delta ) \Delta \Theta

\Gamma \Gamma \Gamma \Gamma

\Theta j \Gamma \Gamma j \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Delta

uv "

u v

u M u; v M v; uv " M uv "M :

u , uu " ~, , uu " u "u u

~,

,u

, ,u , , u , , ~,

O n; ` O ` I `

n n n n n n n n n n n n n n n

`

n n n n

n n

:

e n ; n ; n

x oe x x x ixx ix x x ; x oe x x x ixx ix x x ;

x x x x x x; :

1 21 1 2 1 3 2 1 22 2 3 3 1 3 2 23

3 2 3 1

2 1

1 2 3

0 3 1 2 1 2 0 3

0 3 1 2

1 2 0 3

0 2 1 2 2 2 3 2

T

T T

T T T

i` _ _ _ _

\Gamma \Sigma

y

0B@ 1CA 0B@ 1CA

! !

6.2 6.3

Spinors and Vectors Representation of Lorentz Transformations by Ma- trices

equivalence between 2-component spinors and (complex) zero-norm vectors.

x x x x x x

2 2\Theta

For each : det = 1 matrix = is Hermitian if is Hermitian, and det = det . Hence = \Lambda , where \Lambda is some Lorentz transformation. So, 2 2 matrices

= : = 1

give the representation of Lorentz group on two-component spinors ( (3 1)

(2 ) , both and correspond to the same Lorentz transforma- tion). The relation

=

is also valid in four dimensions, as well as

= ( ) (2 ) However, in contrast to (2) ( ) = , and

(2 )

Subsequently, one should differentiate 2-spinors, which are transformed with the matrix from that which are transformed with the matrix . Here and below they will be denoted with the help of undot and dot indices, correspondingly:

The form

= =

is Lorentz ( (2 )) - invariant, while a Hermitian (2 )-invariant form on two-component spinors does not exist.

The representation of Lorentz group on antisymmetric second rank tensors is

and can be decomposed into two components. Any antisymmetric tensor can be written as

= + (6.6)

28

\Theta \Gamma , \Gamma

8 2

6

! ! h i \Gamma

L L L L

x x

L a bc d ad bc

SO ; SL ; C =Z L L

" " "L L " L SL ; C :

SU ; L L SL ; C

L L

u L u ; u L u

u; v u fflv u v u v SL ; C SL ; C

F

F F F ;

2

1

1

. .. .

0 1 1 0

+

T T

T

A BA B A BA B

T

_*

_* _* _*

0 y 0 0

\Gamma

\Gamma \Lambda

\Lambda \Lambda

\Gamma

! 6.4 Self-dual and Anti-self-dual Tensors

x x x x x

x x the representation of Lorentz group is not equivalent to its complex conjugate one.

reducible

where

= 12 2

The is self-dual, while is anti-self-dual:

1 2 = (6.7)

Each of these irreducible tensors has three independent components which transform through each other under Lorentz transformations. The corre- sponding second rank spinors look as follows:

= _ = _ (6.8) where and _ matrices are defined in Section 2. Note, that

_ = 0 = 0 For example, selfdual electromagnetic field tensor describes photons with negative helicity ( = ), while anti-selfdual electromagnetic field tensor describes photons with positive helicity ( = ). A selfdual tensor in matrix form looks as follows:

=

0

0

0

0

Using a 2-spinor one can construct symmetric traceless second rank spinor

= . Then one can transform it into the selfdual tensor with the help of -matricies:

= (6.9) This tensor can be represented in the form

= \Omega \Omega

29

\Sigma

\Upsilon

\Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma

F F i " F : F F

" F iF :

OE F oe ; OE F oe : oe oe

F oe ; F oe :

~E i ~B

~E i ~B

F

iF iF iF iF F F

iF F F iF F F

:

u OE u u OE

oe

OE oe OE :

OE k k

+

( ) ( )

+ . . . . + . .

23 13 12 23 12 13

13 12 23 12 13 12

_* _* _*aeoe

aeoe

_* _*

_*aeoe aeoe _*

AB _*

_*

AB AB _*

_*

AB

_* _*

_*

_* AB _*

_*

AB

_* A AB A B _*

_*

_* _*

AB

AB

_* _ * _ *

\Sigma

\Gamma

\Sigma \Sigma

\Gamma \Gamma

` '

0BBB@ 1CCCA 6.5 Correspondence Between 2-spinors and 4-bivectors

here ^ = ^\Omega = = \Omega \Omega = \Omega = 0, where the gauge spinor is determined by the requirement

= (6.10) The gauge spinor is defined up to the transformations + , hence the vector \Omega is defined up to the transformations \Omega \Omega + . Tensor does not depend on the arbitrariness in spinor and vector \Omega .

For example, \Omega can serve as a (gauge dependent) polarization vector of a photon with momentum , - as electromagnetic field tensor describing this photon.

Any ordered pair of light-like vectors and :

= 0 = 0 = 1 2 determines another pair of light-like (complex) vectors and , which are orthogonal to both and :

= 0 = 0 = 1 = 0 = 0 (6.11)

It is sufficient to require that the antisymmetric tensors

( ) and ( ) are self-dual:

^ ^ (1 ) = ^ ^ (1 ) = 0 (6.12)

while tensors

( ) and ( )

are anti-self-dual:

^ ^ (1 + ) = ^ ^ (1 + ) = 0 (6.13)

One should note that

( ) ( ) and ( ) ( )

30

) \Delta \Delta \Delta

\Gamma

!!

\Delta \Delta \Delta \Delta \Gamma \Delta \Delta

\Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

, ,

k uu ; uo/ k k k

o/

u o/ o/ u ffl : o/ o/ o/ *u

*k OE o/

k OE

k k k ; k ; k k = ;

! ! k k

! ! ; ! ! ; ! ! ;

! k ; ! k :

k ! k ! k ! k ! k ! fl k ! fl ;

k ! k ! k ! k !

k ! fl k ! fl : ! k ; k ! k ; k ! k ; k ! k ; k :

1 2 21 22 1 2

+ 1 2

+ + +

+

1 + 1 + 2 2 1 + 5 2 5

1 1 2 + 2 + 1 5 2 + 5

+ 2 1 1 2 2 1 + 1 2

A B A B AB A A A A _ _ _ _

_* A _

_

_*

_ _

_ _ _ _

i i

_ * * _ _ * * _

_ * * _ _ * * _

y y

\Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

) 6.6 Isotropic Tetrads in Minkowski Space

The vector ( ) can be interpreted as the polarization vector of the photon with momentum and ( ) helicity (in the gauge ( ) = 0). The explicit expression for can be obatained by using another arbitrary vector :

= 12\Delta ( ) (6.14) where

\Delta = = (2 )

and \Delta = 0. In spite of the presence of an arbitrary vector in (6.14) both and are almost independent on : if varies they acquire only a phase factor. Formula (6.14) can be rewritten in the equivalent form (compare with [11]):

^ = 12\Delta ^ ^^ 1 2 + ^ ^^ 1 2 (6.15) The following relations are also valid:

= + (6.16) ( ) ( ) ( ) ( ) = 2 (6.17) ^ ^ = 2^ ^ (6.18) For an arbitrary vector one has:

= 2 = 2 = =

or, in equivalent form

^ ^ ^ = ^ 1 +2 ^ ^ ^ = ^ 1 2 ^ ^ ^ = ^ 1 2 ^ ^ ^ = ^ 1 +2

31

ffl

p \Delta \Gamma \Delta \Gamma \Delta \Sigma

\Delta \Delta \Delta \Gamma \Delta \Delta 6

\Gamma p \Upsilon \Sigma

\Gamma \Delta \Gamma

\Gamma \Gamma \Gamma

\Gamma \Delta \Delta \Gamma \Delta \Delta

\Gamma \Gamma \Gamma \Gamma

! ! k ; k ! !

k k A

! q

! k k q q k k q k k i" ;

ffi k k k q k q q q k k ;

q ! ! q q

! k qk fl k qk fl :

! ! k k k k g k k i" ; " k k ! ! i ; ! ! fl I fl k k :

p

" i ! p; " i ! p; " ik p; " ik p;

k k ! ! fl ; k k ! ! fl ; k ! ! k fl ; k ! ! k fl :

+ 1 2 +

1 2

3

1 2 2 1 1 2

3 1 2 1 2 1 2 3

+

3

1 2

5

2 1

5

+ 1 2 2 1 1 2

1 2 + + 5 5 1 2

+ 1 2

1 2

5

1 2 + +

5

1 + 1

5

2 + 2

5

_ _ _ _ _qk k

k k q k k q

_ * _ * _ * _* k k _* fffi_* ff fi _ *

_ k k ! p k k ! p k ! ! p k ! ! p

\Gamma \Gamma

\Sigma

\Sigma

\Gamma \Sigma

\Gamma

\Gamma \Gamma

\Gamma

\Gamma \Gamma \Gamma \Gamma

1 2 1 2 1 2

1 2 1 2 1 2 + 1 + 2 +

\Gamma

\Gamma \Gamma

!

Vectors and are determined by and requirements (6.11 - 6.13) up to a factor.

positive negative

Let and are two arbitrary 2-component spinors which satisfy the relation

= Then the vectors

= = = 2 = 2 satisfy all the above mentioned properties. There always exists a basis, where all these vectors and spinors look as fol- lows:

= 10 = 01

= 12 (1 0 0 1) = 12 (1 0 0 1) = 12 (0 1 0) = 12 (0 1 0)

32

\Gamma

p p

\Gamma p p \Gamma

'

' ' ffl :

k ; k ' ' ; ! ' ; ! '

; ' ; k ; ; ; ; k ; ; ; ; ! ; ; i; ; ! ; ; i; :

1 . 2 . + . .

1 2 +

A B A B AB A A A A A A A A

y y y

\Gamma

y

\Gamma ! !

o

o o

o o o o

o

7 DIRAC SPINORS Dirac spinors ( ) and ( ) describe the solutions of the Dirac equation with positive and negative energy:

(^ ) ( ) = 0 (^ + ) ( ) = 0 (7.1) They are functions of 4-momentum on the mass shell = + . The spinors are defined as follows:

_ = _ = _( )(^ ) = 0 _( )(^ + ) = 0 and the normalization condition is chosen, so that:

_( ) ( ) = +2

_( ) ( ) = 2

Symbol stands for the polarization of the fermion. The axial-vector of the fermion spin is defined by the relations:

_( ) ( ) =

= 1 = 0

The spinor ( ) describes a fermion with momentum and the vector of spin . The spinor ( ) describes an antifermion with momentum and the vector of spin . (One should note, that axial vector describing spin of a fermion has only spatial non-zero components in the rest frame of this fermion. However, it transforms together with the vector under Lorentz transformations.) Spinors ( ) and ( ) satisfy the following relations:

( ) _( ) = (^ + ) (1 + ^)2 (7.2)

( ) _( ) = (^ ) (1 + ^)2 (7.3) ^ ( ) = ( ) ^ ( ) = ( ) (7.4)

33

\Gamma p

\Gamma

\Gamma

\Delta \Gamma \Delta \Gamma

\Gamma

u p; n v p; n

p m u p; n ; p m v p; n :

p p ~p m

u u fl ; v v fl ; u p; n p m ; v p; n p m ;

u p; n u p; n m;

v p; n v p; n m:

n n

u p; n fl fl u p; n m n ;

n n ; n p :

u p; n p n v p; n p

n n

p u p; n v p; n

u p; n u p; n p m fl n ;

v p; n v p; n p m fl n ; nfl u p; n u p; n ; nfl v p; n v p; n ;

0 2 2 0 0

5

5

5 5 5

_

_ _ _

y y 7.1 General Properties

conjugated

as well as the Gordon identities:

_( ) ( )

= 12 _( ) ( + ) + ( ) ( ) _( ) ( )

= 12 _( ) ( ) + ( + ) ( )

Both + and are light-like vectors. Another couple of light- like (complex) vectors and determined by + and , are described in the Subsection 6.6. Using the vectors one can obtain the following identities:

^ ( ) = 2 ( ) ^ ( ) = 2 ( ) (7.5)

^ ( ) = 2 ( ) ^ ( ) = 2 ( ) (7.6) ^ ( ) = ^ ( ) = ^ ( ) = ^ ( ) = 0 (7.7)

Let us now introduce the spinors:

( ) ( ) ( ) ( ) ( ) ( )

They satisfy the following relations:

^ + ^ = 2 ^ ^ = 2 ^ + ^ = ^ ^ = 0 ^ + ^ = 2 ^ ^ = 2 ^ + ^ = ^ ^ = 0

^ = ^ = 0 ^ = 2 ^ = 2

^ = ^ = 0 ^ = 2 ^ = 2 (1 + ) = (1 ) = 0 (1 + ) = (1 ) = 0

Since + and are light-like vectors, spinors and satisfy the massless Dirac equation. and describe its right-handed solutions,

34

\Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma p \Gamma \Gamma pp \Gamma p \Gamma

\Gamma \Gamma

j \Sigma \Gamma j \Gamma \Sigma

\Gamma \Gamma \Gamma \Gamma

p \Gamma p \Gamma p p\Gamma

\Gamma \Gamma

u p ; n fl u p ; n

m u p ; n p p oe p p u p ; n ; u p ; n fl fl u p ; n

m u p ; n p p fl oe p p fl u p ; n ; p mn p mn

! ! p mn p mn

! ! u p; n v p; n ; ! v p; n u p; n ;

! u p; n v p; n ; ! v p; n u p; n ; ! u p; n ! v p; n ! u p; n ! v p; n :

' p; n u p; n v p; n ;

p; n u p; n v p; n :

p mn ' ' ; p mn ' ' ; p mn ' p mn ' ;

p mn ; p mn ; p mn p mn ;

! ' ! ' ; ! ' ; ! ' ;

! ! ; ! ' ; ! ' ; fl ' fl ' ;

fl fl :

p mn p mn '

'

1 1 2 2

1 1 1 2 1 2 2 2 1 1 5 2 2

1 1 1 2 5 1 2 5 2 2

+

+ +

+ +

+ +

+ + +

+ + + +

+ + + + + + + 5 + 5

5 5 +

+

_

_ _* * _

_ _* *

_ _ _ _

_ _ _ _

_ _ _ _

\Gamma \Sigma

\Gamma \Gamma

\Gamma \Gamma

\Sigma \Sigma

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma

\Sigma \Sigma \Gamma

\Theta \Lambda h i

\Gamma \Delta \Gamma \Delta \Gamma \Delta \Gamma \Delta \Gamma \Delta \Gamma \Delta

\Gamma \Delta \Gamma \Delta

and - left-handed solutions. In chiral representation they look as follows:

= 0 0 = 0 = 0

where and are two-component spinors.

Using the vectors one can write down the expressions for the bilinear spinor combinations of the type:

( )_( ) = + + + + (7.8) There is very important case, when all the four vectors , , and lie in a single plane (this case corresponds to spin-flip and spin-non-flip amplitudes).

Let us denote the polarization vectors which lie in the 2-plane of the vectors and through and . They satisfy the conditions

= = 0 = = 1 and look as follows:

= 1\Delta = 1\Delta (7.9) where

\Delta = ( )

A fermion with the momentum and an arbitrary polarization vector , in general, can be represented as a superposition

( ) = ( ) + ( ) where the coefficients and depend on the polarization . So, to describe the full set of helicity amplitudes one can restrict the consideration to the following four cases:

= =

35

\Delta \Delta \Gamma \Delta \Gamma \Delta \Gamma

\Delta \Gamma

\Gamma

\Sigma \Sigma

'

' ' ; ; ' ' ; :

' ; ; '

! u p ; n u p ; n S V fl T oe A fl fl P fl :

p p n n

p p N N

N p N p ; N N ;

N p pm p m p ; N p pm p m p ;

p p m m : p n

u p ; n ff u p ; N fi u p ; N ;

ff fi n

n N ; n N :

+ + .

+

= . + +

. +

. +

1 1 2 2 5 5

1 2 1 2

1 2 1 2

1 1 2 2 21 22

1

1 2

1 1 1 2 2

1 2

2 2 2 1

1 2 2 21 22

1 1 2 2

B B

A A

B B A A

_ _ _* _* _ _

_ _ _ _ _ _

i i i i i i

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma \Gamma

\Sigma

! ! ! !

` ' ` '

q

7.2 Bilinear Combination Decomposition Then one has:

_( ) = 1 2 1 2 + ^ ^ ^ 2 (7.10)

_( ) = ( ^ + )^ 1 2 + ( ^ + )^ 1 2 (7.11) where

= + 2

= 12\Delta = 12\Delta = 12 Left hand sides of the equations (7.10) and (7.11) are given in a brief form: for instance, the symbol _(+ ) denotes ( = + )_( = ). The vectors and are determined by and with the help of relations (7.5 - 7.7) [12, 13].

36

\Sigma \Sigma \Sigma \Gamma \Upsilon \Gamma p

\Sigma \Upsilon \Sigma \Upsilon

p \Delta \Sigma p \Delta \Gamma p

\Gamma \Gamma \Delta

\Gamma \Gamma

uu ; j fl j fl m k m k ! ;

uu ; j k m k fl j k m k fl ;

j p p m m p p m m ; k j p mm j p ; k j p mm j p ; k k :

uu ; u p ; n N u p ; n N ! ! p N

+

5 5

1 2 2 1

+ 1 1 2

5

2 2 1

5

1 2 1 2 1 2 1 2 1 + 1

1

2 2 2 + 2

2 1 1 1 2

1 1 1 2 2 2 + 1 1

_ _ _ _ _ _

\Gamma

\Sigma

\Gamma

\Sigma

\Gamma \Gamma

\Gamma

! ` ' ` '

8 GELL-MANN MATRICES The Gell-Mann 3 3 matrices ( = 1 8) are generators of the group

(3). Their properties were presented elsewhere [3, 4, 14, 15, 16]. Usually in QCD instead of one deals with matrices :

1 2

Eight matrices equal:

=

0 1 0 1 0 0 0 0 0

=

0 0

0 0 0 0 0

=

1 0 0 0 1 0 0 0 0

=

0 0 1 0 0 0 1 0 0

=

0 0 0 0 0

0 0

=

0 0 0 0 0 1 0 1 0

=

0 0 0 0 0 0 0

= 13

1 0 0 0 1 0 0 0 2

The main properties of (or ) are as follows:

= det = 0 ( = 1 7) det = 112 3 [ ] = = 13 + (8.1) where ( ) is totally symmetric (anti-symmetric) tensor. The non-zero elements of and are equal to:

= 1 = = = = = = 12 = = 32

= = = = = = = = 12 = = = = 13 = = = = 12 3

37

\Theta

j

\Gamma \Gamma

\Gamma

\Gamma p \Gamma

\Gamma p f g

\Gamma \Gamma p \Gamma \Gamma \Gamma

\Gamma p \Gamma p

* i ; : : : ; SU

* t

t * : *

* ; *

i i ; * ;

* ; *

i

i

; * ;

* i

i

; * :

t * t t ; t ; i ; : : : ; ; t ;

t ; t if t ; t ; t ffi d t ; d f

f d

f ; f f f f f f ; f f ; d d d d d d d d ; d d d d ; d d d d :

1 2 3

4 5 6

7 8

8

123 147 156 246 257 345 367 458 678

146 157 247 256 344 355 366 377 118 228 338 888 448 558 668 778

i i i

i i i

i i i i i

a b abc c a b ab abc c

abc abc

abc abc

y

0B@ 1CA 0B@ 1CA 0B@ 1CA

0B@ 1CA 0B@ 1CA 0B@ 1CA

0B@ 1CA 0B@ 1CA

8.1 Main Properties Throughout this Section we use two additional notations:

= + = = = 0 ( ) ( )

Thus from (8.1) one has:

= 16 + 12 ( + ) = 16 + 12 (8.2)

Trace of any string of matrices can be evaluated recursively using the relation (8.2):

Tr ( ) = 16 Tr ( ) (8.3)

+ 12 Tr ( ) Using (8.1) and (8.3) one gets:

Tr( ) = 0 Tr( ) = 12 Tr( ) = 14 ( + ) = 14 Tr( ) = 112 + 18 Tr( ) = 124 + 124 + 116

The Fiertz identity for has the form:

= 12 ( 13 ) (8.4) Any 3 3 matrix can be expanded over set :

= + where = 13 Tr = 2 Tr( ) Decomposition of the two and _ color spinors products into color-singlet and color-octet parts has the form:

_ = 3 + 2 =

38

j j

\Gamma \Theta f g

p p

h d if ; h h h ; h ; S a a : : : a t t : : : t ; S a a : : : a t : : : t t :

t t ffi d if t ffi h t :

t S a a : : : a ffi S a a : : : a

h S a a : : : a k :

t ; t t ffi ; t t t d if h ; t t t t ffi ffi h h ; t t t t t h ffi ffi h h h h :

t t t ffi ffi ffi ffi : A I; t A a I a t ; a A; a t A :

u u

u u ffi " t ; " " ffi :

1 2 1 2

1 2 1 2 2

1 2 2

0 0

abc abc abc abc bca cab aab

n a a a R n a a a

a b ab abk abk k ab abk k

a

n a a n

a a k n

a a b ab a b c abc abc abc a b c d ab cd abn ncd a b c d e abc de ab cde abn nck kde

a a ik

a jl il kj ik jl

a

i i i i

i i

i j

ij k k

ij

k l kl

\Gamma

\Gamma

1 2 2 1

1

1

n n

n n

n n

\Gamma

\Gamma

8.2

8.3

Traces of the -matrices Fiertz Identity

t

a

The product of matrices could be written in the form + using the following relations (see (8.2)):

( ) = 16 ( ) + 12 ( ) (8.5) Thus, the products of two, three, and four matrices equal:

= 16 + 12 ( + ) = 16 + 12 = 16 + 112 + 14 = 136 + 124 + 112 [ + ]

+ 18 The products of the type have the form:

= 12 Tr( ) 16 = 43 = 16 = 14 16

= 18 16

= 16 + 124 + 116 The products of the type (here is denoted any permutation of the

-matrices) are given by

= 43 = 29 = 169 The products of the ( ) ( ) and ( ) ( ) are presented on the following tables (in these tables symbol ( ) stands for , etc).

39

\Gamma

\Gamma \Gamma \Gamma \Gamma

\Gamma

n t a a t S a a : : : a ffi S a a : : : a h S a a : : : a k :

t t ffi d if t ffi h t ; t t t ffi t h h h t ; t t t t ffi ffi h h h ffi ffi h t

h h h t :

t St t St S S; t t I; t t t t ; t t t t ffi t t ; t t t t t h t t t ; t t t t t t t t t t ffi ffi h h :

SS S t

t t : : : t t : : : t t ; t t t t I; t t t t I: S abc S abc S abcd S abcd

abc t t t

0 1 2 1 2 2 1 2 2

\Pi \Pi

\Pi \Pi

a i i n a a n a a k n

a b ab abk abk k ab abk k a b c ab c abc abk kcn n a b c d ab cd abk kcd abk cd ab cdk k

abn cdk nkp p

k k k k k k k a k a k a b k ab a b k a b c k abc a b c k a b c d k a b c d ab cd abn ncd

a

a a a a a a n a b a b a b b a

a b c

\Gamma \Gamma 1 1

1 2 2 1 n n n n i

n n

\Gamma \Gamma

i j

8.4 Products of the -matrices

Table 8.1.

t

a The products of the ( ) on the ( ) . All products are contain the com-

mon factor .

( ) 10 ( ) 1 ( ) 8 ( ) 1 ( ) 8 ( ) 64

The products of the ( ) on the ( ) . All products are contain the common factor .

( ) 14 ( ) +31 ( ) 5 ( ) +40 ( ) +31 ( ) + ( ) ( ) +4 ( ) +31 ( ) 5 ( ) ( ) +4 ( ) 5 ( ) +40 ( ) +4 ( ) 32 ( ) 5 ( ) ( ) +4 ( ) 32 ( ) ( ) +4 ( ) 32 ( ) +256

The convolutions of the coefficients and with the -matrices equal:

= + 13 = ( )

= 2 12

= ( + + + ) 13 23

= ( + + ) = ( + + ) 13

= ( + + ) = 109 = 2 = 89

= 56 = 32 = 23 The Jacobi identities for the coefficients and equal:

+ + = 0 + + = 0

40

\Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma

abc abc I

abc bac cab acb bca cba

abcd abcd I

abcd bacd cabd dabc abdc badc cadb dacb acbd bcad cbad dbac acdb bcda cbda dbca adbc bdac cdab dcab adcb bdca cdba dcba

d f t d t t t t t ffi ; f t i t t t t ;

h t t t ffi ; d d t t t t t t t t t t t t t d I ffi t ; d f t i t t t t t t t t t t t t ; f d t i t t t t t t t t t t t t f I; f f t t t t t t t t t t t t t ; d t t t I; f t t t iI; h t t t I;

d t t t ; f t t it ; h t t t :

f d f f f f f f ; d f d f d f :

\Pi 1 27

\Pi 1 81

71

2

1

21

2

1

21

2

abc abc a abc c a b b a ab abc c b a a b

abc c a b ab abk kcl l a b c b a c c a b c b a abc ab c abk kcl l a b c b a c c a b c b a

abk kcl l a b c b a c c a b c b a abc abk kcl l a b c b a c c a b c b a abc a b c abc a b c abc a b c

abc a b c abc a b c abc a b c

abc abc abk kcl bck kal cak kbl abk kcl bck kal cak kbl

8.5 Convolutions of and with

Table 8.2.

d f tabc abc a

The various relations of a such type were presented in [15]:

+ + = 13 ( + + ) = 23 ( ) + 3 = + + +

= = = 0

= 3 = 53 = 32 = 32 = 56 = 12 = 136 (13 7 + 13 ) = 112 ( 7 + + 9 ) = 136 ( 21 + 19 ) + 16 ( 4 ) = 34 ( + ) = 14 (5 + + 5 6 ) = = 24

= =

= = 0

= 20 = 20

= 0 = 36 =

Following the results of [15] we present here the (3)-invariant tensors of the third, forth, and fifth ranks.

41

\Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma

d d d d d d ffi ffi ffi ffi ffi ffi ; f f ffi ffi ffi ffi d d d d ;

d d ffi ffi ffi ffi ffi ffi f f f f ; d f d f :

f f ffi ; d d ffi ; f f f f ; d f f d ; d d f f ; d d d d ; d d d d ffi ffi ffi ffi ffi ffi d d ;

d d d f d f d f d f ; d d d d ffi ffi ffi ffi ffi ffi

d d d d ; d f f d d f d f ;

f f f f ffi ffi ffi ffi ffi ffi d d :

d d ; f f ; h h ; h h ; d d d d ; d d d f ; d d f f ; d f d f ; d f f f ; f f f f ; h h h h :

SU

40

3 32

3

112

320

3

32

3

abk kcl bck kal cak kbl ab cl ac bl al bc abk kcl ac bl al bc ack blk alk bck

abk kcl ac bl al bc ab cl ack blk alk bck aac aac abc abm

akl bkl ab akl bkl ab pak kbl lcp abc pak kbl lcp abc pak kbl lcp abc pak kbl lcp abc piq qjm mkt tlp ij kl ik jl il jk ikm mjl

piq qjm mkt tlp ijm mkl ikm mjl ilm mjk piq qjm mkt tlp ij kl ik jl il jk

ikm mjl ilm mjk piq qjm mkt tlp ikm mil ilm mkj

piq qjm mkt tlp ij kl ik jl il jk ikm mjl

abc abc abc abc

abc abc abc bac abk klc cbn nla abk klc cbn nla abk klc cbn nla abk klc cbn nla abk klc cbn nla abk klc cbn nla

abk klc cbn nla

8.6 Invariant Tensors(3)SU

There are two independent tensors of the third rank ( ):

= and = There are eight independent tensors of the forth rank:

= = = = = = = =

The convolutions of these tensors of the type = are presented by following matrix:

=

64 8 8 0 0 0 0

8 64 8 0 0 0 0 8 8 64 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 40 20 20 0 0 0 0 0 20 40 20 0 0 0 0 0 20 20 40

There are 32 independent tensors of the fifth rank ( ):

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

42

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma

H H d H f :

H ffi ffi ; H ffi ffi ; H ffi ffi ; H d d ; H d d ; H d f ; H d f ; H d f :

Q H H

Q :

H H ffi d ; H ffi d ; H ffi d ; H ffi d ; H ffi d ; H ffi d ; H ffi d ; H ffi d ; H ffi d ; H ffi d ; H ffi f ; H ffi f ; H ffi f ; H ffi f ; H ffi f ; H ffi f ; H ffi f ; H ffi f ; H ffi f ; H ffi f ; H f f f ; H f f f ; H f f f ; H f f f ; H f f f ; H f f f ; H f d f ; H d d d ; H d d d ; H d d d ; H d d d ; H d d d :

(1) (2) (1) (2) (3) (4) (5) (6) (7) (8)

( ) ( )

40

340

340 3

40

340

3

40

3

200

9

20

340

3

40

3

20

3

200

9

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

ikl ikl ikl ikl ikl

ijkl ij kl ijkl ik jl ijkl il jk ijkl ijm klm ijkl ikm jlm ijkl ijm klm ijkl ikm jlm ijkl ilm jkm

fffi ff

ijkl

fi ijkl

fffi

ijklm ijklm ij klm ijklm ik jlm ijklm ik ilm ijklm il jkm ijklm jl ikm ijklm im jkl ijklm jm ikl ijklm kl ijm ijklm km ijl ijklm lm ijk ijklm ij klm ijklm ik jlm ijklm jk ilm ijklm il ikm ijklm jl ikm ijklm im jkl ijklm jm ikl ijklm kl ijm ijklm km ijl ijklm lm iik ijklm ijp pkt tlm ijklm ikp pjt tlm ijklm ijp plt tkm ijklm ikp plt tjm ijklm kjp plt tim ijklm ilp pjt tkm ijklm ijp pkt tlm ijklm ijp pkt tlm ijklm ikp pjt tlm ijklm ijp plt tkm ijklm ikp plt tjm ijklm ilp pjt tkm

0BBBB BBBBBBB BB@

1CCCC CCCCCCC CCA

9 STANDARD MODEL LAGRANGIAN In this Section we present the basic Lagrangian of the Standard Model(SM), corresponding to the (3) (2) (1) local gauge symmetry (see, for example, [3, 4, 17]). The algebra of the semisimple group (3) (2)

(1) is generated by Gell-Mann matrices = (a =1,...8) (Section 8), Pauli matrices = 2 (Section 2) and hypercharge with the following commutation relations

= = = = = 0

The full SM Lagrangian has the form [3, 4]:

= + + + + + (9.1) Here is the Yang-Mills Lagrangian without matter fields

= 14 ( ) ( ) 14 ( ) ( ) 14 ( ) ( ) (9.2) where ( ) ( ) ( ) are given by

( ) = + ( ) =

( ) = +

with the (2) (1) original gauge fields and the gluon fields. The infinitesimal gauge transformations of these fields are given by

= ( ) = = ( ) = = ( )

Here ( ) and ( ) stand for the covariant derivatives, and are the

(3) and (2) gauge coupling constants, respectively, and are an

43

SU SU U

SU SU U t *

o/ oe = Y

t ; t i f t ; o/ ; o/ i ffl o/ ;

o/ ; Y t ; o/ t ; Y :

: F W F W F W F W F G F G ; F W ; F G ; F W

F W @ W @ W g ffl W W ; F W @ W @ W ;

F G @ G @ G g f G G ;

W ; W SU U G

ffiW @ ` x ;

ffiW @ ` gffl ` W W `

ffiG @ ffl g f ffl G G ffl

W G g g SU SU ffl `

\Theta \Theta \Theta \Theta L L L L L L L L

L \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Theta

\Gamma D\Gamma D D D

a a i i

a b abc c i j ijk k

i a j a

i _*

_* i

_* _* a

_*

_* a

i _*

a _* _*

i _* _

i * *

i _

ijk j

_

k *

_* _ * * _

a _* _

a * *

a

_ s

abc b

_

c *

i _ _

a

_

_ _

i _ _

i ijk j k

_

ij _

j

a _ _

a s abc b c

_

ab _

b

ij _

ab _ s a i

1

2

0 0 0

0 0 0 0

0

( )

h ih i h i h i \Theta \Lambda

G F H M GF FP G

G

arbitrary functions depending on the space-time coordinates. It can be easily checked that Lagrangian (9.2) is invariant under these gauge transformations.

Lagrangian describes coupling of fermions with gauge fields. For simplicity we shall consider one lepton generation, say and , and three quark generations. Fermions constitute only doublets and singlets in (2)

(1)

= =

= = = where and denote left- and right-handed components of the spinors, respectively:

= 1 2 The neutrino is assumed to be left-handed, while right-handed components of both up- and down-quarks enter in the . Indices and numerate three quark generations: = 1 2 3, and ( ) refers to the up (down) quarks. A possible mixing of quark generations was taken into account by introduction of Kobayashi-Maskava matrix (see, for example, [4, 18] for details). The infinitesimal gauge transformations of fermion fields looks as follows:

= 2 ( ) + 2 ( ) = 2 ( ) + 2 ( ) + ( ) where is (1) gauge coupling constant. Obviously, lepton and quark fields belong to the fundamental representation of the (3) (2) (1). Under the requirements of the (3) (2) (1) local gauge symmetry and renormalizability of the theory, the Lagrangian acquires the following expression

= _ ^ + _ ^ + _ ^ + _ ^ + _ ^ (9.3)

44

e *

SU U

R e ; L *e

R q ; R q ; L

q

V q L R

e fl e:

I i I; i ; ; I i

V

ffi ig Y ` x ig oe ` x ; ffi ig Y ` x ig oe ` x ig t ` x ; g U

SU SU U SU SU U

iLD L iRD R i L D L R D R i R D R ;

L

\Theta

\Sigma

L

\Theta \Theta \Theta \Theta L

L

e R

L

L

I I R i i R I

I

Ii i

R;L

iI lep

i

i lep

quark

i

i s a a quark

L R

I

I

q

L I I

q R I

i

i

q

R i

1 5

! \Gamma \Delta \Gamma \Delta 0B@ 1CA

! !

X i j X

F

\Gamma

\Gamma

\Gamma

\Gamma

F 0 0 0

F F

where covariant derivatives are given by

= 2 2 = 2 = 2 2 = 2 We remind that the value of hypercharge is determined by the following relation = + 2 with being the charge operator.

Both the gauge fields and fermion ones described above have zero mass, while in the reality all charged fermions are massive and intermediate bosons are known to be very heavy. To make the weak bosons massive one can use Higgs mechanism of spontaneous breakdown of the (2) (1) symmetry to the (1) symmetry. The widely accepted way to do that consists in the introduction of the Higgs (2) doublet \Phi (with = 1). This doublet acquires the nonzero vacuum expectation value:

\Phi =

0

2 The potential term (\Phi ), which can give rise to the symmetry violation, reads

(\Phi ) = \Phi \Phi + \Phi \Phi One can easily verify that the vacuum expectation value satisfies to the con- ditions:

\Phi = 12 \Phi = 0 \Phi = 12 + = 0 It means, that only the symmetry generated by is not broken on this vacuum. Let us choose the Lagrangian for the Higgs field interaction with

45

D @ ig Y W ig oe W ; D @ ig Y W ;

D @ ig Y W ig oe W ig t G ; D @ ig Y W ig t G :

Y Q o/ Y = Q

SU U U

SU Y

! ? v : V

V _ * :

o/ ! ? oe ! ? ;

Q ! ? oe Y :

Q

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Theta p \Gamma

6

L _ _ _

i

i _

R _ _ _

q L _ _ _

i

i _ s

a a

_

q R _ _ _ s

a a

_

i i

0 0 0 0

3

2 + + 2

3

0BBB @

1CCC A

i j

\Gamma \Delta

0 0 0 0

gauge fields in the form:

= \Phi \Phi (\Phi ) (9.4) Then one finds that only gauge boson coupling to ( i.e. photon) remains massless, while other bosons acquire masses. Diagonalization of the mass matrix gives

= 12 ( ) = 12 (9.5) = 1+ ( ) = 12 + (9.6) = 1+ ( + ) = 0 (9.7) where are charged and neutral weak bosons, is the photon. It is suitable to introduce rotation angle between ( ) and ( ), which is called the

sin + (9.8) The relation of constants with electromagnetic coupling constants fol- lows from (9.3). Since the photon coupling with charged particles is pro- portional to + , we should identify this quantity with the electric charge :

= + (9.9)

In order to find mass spectrum in the Higgs sector, let us express doublet \Phi in the form

\Phi = 1

2 +

One can verify that Nambu-Goldstone bosons have zero masses and may be cancelled away by suitable choice of the (2) (1) rotation. The only physical component of the Higgs doublet is , which acquires mass

= 2

46

D D V :

Q

W W iW ; M gv;

Z g g gW g W ; M g g v; A g g g W gW ; M ; W ; Z A

# W ; W Z; A

# g = g g : g; g e

gg = g g e

e ggg g :

i! v OE iz :

! ; z

SU U

OE

m _:

L \Gamma p \Upsilon p \Gamma p

j p

p

p \Gamma

\Theta p

L _

_

L

_ _ _ W

_ _ _ Z

_ _ _ A _ _ _

W

W

H

+ 1 2 2 2

3 0 2 2

2 2

3 0

3 0 2 2

2 2

2 2

+

\Gamma \Delta \Gamma \Delta

q q

0B@ \Gamma \Delta 1CA

H \Sigma

0

0 0

0

0

\Sigma

0 0 0

0 0

0

0

\Sigma

Weinberg angle The Lagrangian generates fermion mass terms. Supposing the neu- trinos to be massless, we write the Yukawa interaction of the fermions with Higgs doublet in the form

= _\Phi _ \Phi _ \Phi + (9.10)

Here we introduced doublet related with by

= and are the Yukawa coupling constants. Then the masses of fermions in the tree approximation are given by

= 2 (9.11) It is well known that quantization of dynamical systems is governed by Lagrangians having local gauge symmetry requires an additional care. Free- dom of redefining gauge and matter fields without changing the Lagrangians leads to the vanishing of some components of the momenta, canonically con- jugate to the gauge fields, say

= = 0 (for = 0) To perform the quantization procedure, one should add to the Lagrangian a gauge fixing terms, breaking explicitly the local symmetry. In the functional integral formulation it leads, in the case of non-Abelian gauge symmetry, to modification of the path integral measure [19]. As a result, the measure of the path integral will be multiplied by functional determinant \Delta ( ). In order to apply the well known methods of perturbation theory, one may exponentiate \Delta ( ) and redefine the initial Lagrangian. It can be made by introducing auxiliary fields and _ which are scalar fields anticommut- ing with themselves and belonging to the adjoint representation of the Lie algebra. The fields and _ are called Faddeev-Popov ghosts (FP ghosts).

The gauge fixing terms are usually chosen in the form

= ( ) + 2

47

f L R f L R f L ioe R h:c:

L L

L V L ; f

m f v :

ffiL ffi@ A F _ :

W W

c c

c c

L B F W , B ;

L L \Gamma \Gamma \Gamma

p \Gamma

e

i

i i i

I

I I I

i I

i i I I I; i

I; i

I; i

_

_

a _

a _ a a

a a

GF a a a

2 0

0

2

X X i j

\Gamma \Delta M M

\Lambda

where are auxiliary fields introduced to linearize this expression, is the gauge parameter, = . Then FP ghosts enter in the Lagrangian in the following way

= _ ( ) (9.12) As it was pointed above, these additional terms violate local gauge invariance, but the final Lagrangian becomes invariant under the global transformations mixing the gauge fields and FP ghosts. This symmetry, found by by Bec- chi, Rouet, and Stora, was called BRS symmetry. The BRS infinitesimal transformations are defined by the following relations

( ) = ( ) ( ) ( ) = ( ) _ ( ) = ( ) ( ) = 2 ( ) ( )

( ) = 0 Here denotes any matter field, the parameter does not depend on and anticommutes with and _ , as well as with all fermion fields. Using these relations, the formula (9.12) can be written in the brief form:

= _ (9.13) where means left differentiation.

In our case we choose the gauge fixing part of the Lagrangian in the form

= ( + ) + ( + ) (9.14) + ( + ) + ( ) + ( )

+ + 2 + 2 + 2

then FP-ghost Lagrangian looks as follows:

= (9.15) _ + + _ +

+_ + + _ + _

48

B ,

F @ W

L c @F@W D W c :

ffi x ifigc x t x ; ffi W x fiD c x ; ffi c x fiB x ; ffi c x fi gf c x c x ; ffi B x : fi x

c c

L ffiffifi c ffi @ W ; ffi=ffifi

B @ W , M ! B @ W , M ! B @ Z , M z B @ A B @ G

, B B , B B , B B , B B ;

ffi ffifi c ffi @ W , M ! c ffi @ W , M !

c ffi @ Z , M z c ffi @ A c ffi @ G ;

\Gamma

\Gamma

L L

a

a _ a

_

F P a

a

c_

cb _

b

BRS a a BRS a

_

ab _

b

BRS a a BRS a abc b c BRS a

a a

GF a BRS _ a_

_

_ W W

_

_ W W

Z _ _ Z Z A _ _ a _ a

_

W

Z Z Z A A A G a

G

a G

BRS _

_ W W

BRS _

_ W W

Z BRS _ _ Z Z A BRS _ _ a BRS _ a

_

+ + +

+

+ + +

i i jj n i j i j\Gamma \Delta \Gamma \Delta i jo GF

\Gamma \Gamma \Gamma

\Gamma

FP

\Gamma \Gamma \Gamma

where the fields are constructed from original ghosts just like the bosons from initial fields .

Now, we are ready to present the total Lagrangian of the

rewritten in the terms of physical fields [17].

= 12 14 ( ) 14 ( ) 14 ( ) (9.16) + cot + + + + + + + cot ) + cot + 2

+ 2 sin

1 4 where the field sthrenghtes are given by

=

=

= _^ + _ ^ + _ ^ (9.17) + 2 sin _ ^ + _ ^ + sin 2 _ ^ + sin 2 _ ^(2 sin 1 2 ) _ ^ + 2 sin _ ^ ( ) + _ ^

+ sin 2 _ ^( 1 2 2 sin )

49

c ; c c ; c Z ; A W ; W

F F F F G ie # g g g g W Z @ W Z W @ W W W @ Z ie g g g g W A @ W A W @ W W W @ A g f G G @ G e g g g g W W A A e # g g g g W W Z Z e # g g g g g g W W A Z

e

# g g g g W W W W

g f f G G G G ;

G ; F ; : : :

F @ W @ W ; G @ G @ G ;

ie@e i* @* i q @q

e

# * W e e W *

e

# * Z* e

# eZ #

fl e eeAe

e

# q W q V q W q V e

# q Z

fl Q # q

L \Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma

\Gamma \Gamma

\Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma \Delta \Delta \Delta

L

p

\Gamma \Gamma \Gamma p

\Gamma \Gamma

A Z _ _ _ _

_*

_* Z

_*

A _*

a _*

W fffl fiffi ffffi fifl fl ffi ff fi fl ffi ff fi fl ffi ff fi

fffl fiffi ffffi fifl

fl ffi ff fi

fl ffi ff fi fl ffi ff fi s abc a_ b* _ c *

fffl fiffi fffi flffi

ff fi fl ffi

W fffl fiffi fffi flffi ff fi fl ffi W ffffi fifl fffl fiffi fffi flffi ff fi fl ffi

W

fffi flffi fffl fiffi

ff fi fl ffi

s

rab rcd a

_

b *

c _ d *

a

_* _*

_* _ * * _

a

_* _

a * *

a

_

L L

n

n n

W

L L L L

W

L L

W

W

W I;i

I i L Ii i I L iI

W I

I I W I

0 3 0 3

+ 2 2 2

+ + + +

+ + 2 + 2 2 + 2 +

2 2

+ +

2

+ + + +

+

2 5

+ +

5 2

i j ij i j i

ji j i ji j

i j

Xi j ` '

X i j X ` '

G

\Gamma

\Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

\Gamma

\Gamma

\Gamma

\Gamma \Gamma

F

\Gamma

\Gamma

Standard Model

+ sin 2 _ ^( 1 +2 2 sin ) + _ ^ + _

= 12 ( ) 2 + 12 ( ) + (9.18) + + 12 +

+ sin + sin 2 + 2 sin ( ) + 2 sin ( + ) + ( + cot 2 ) + sin 2 + sin ( ) + ( )

+ 4 sin ( + 2 ) + 2 cos ( )

+ 2 sin ( ) + 4 sin ( + 2 ) + 2 cos ( + ) 2 sin ( + ) + 2 sin + cot 2 + + 2 cot(2 ) 4 sin 2 sin

4 sin 32 sin 32 sin 8 sin ( + ) 16 sin 8 sin ( ) Here symbol is used as usual: ( ) .

50

e

# q Z

fl Q # q

e Q q Aq g q G fl t q

@ OE m OE @ z @ ! @ ! M W W M Z M W @ ! W @ ! M Z @ z eM# OEW W eM# OEZ

e

# W ! @ OE iz

e

# W ! @ OE iz

ie A # Z ! @ ! e# Z z @ OE

ieM # Z W ! W ! ieM A W ! W !

e

# OE W W Z

ie

# OEZ W ! W ! ie

# OEA W ! W !

e

# z W W Z e

# zZ W ! W !

e

# zA W ! W ! e

# ! ! W W e # ! ! Z e ! ! A

e # ! ! A Z emM # OE emM # ! ! OE

em M # z OE

e m M # OE

e m M # z e m M # ! ! OE z

e m M # z OE e m M # ! !

f @ g f @ g f @ g @ f g

\Gamma \Gamma L \Gamma

\Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma

\Gamma

\Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

j \Gamma

W i

i i W i

n

n n n s

n

n a_ _ a

_ H _ _ _ W _

_

Z _ W _

_

_

_

Z _ _

W

W _

_ Z

W _

W

_ _

W

_ _

_ W _ _

W

_ _

Z W _ _ _ W _ _ _

W _

_

_ W

_

_ _

W

_

_ _ W _

_

_

W

_

_ _ W

_

_ _

W _

_ W

_ _

_ _ H

W W

H W W H W W

H W W

H W W H W W

H W W H W W

_ _ _ _

5 2 2 2 2 2 + 2 + 2 2 + +

+ 2 + +

+ + + + + 2 2

2 + 2 2 + +

2 + + 2

2

2 + 2

2 + + 2 + +

2 2

+ + 2 2 + 2 2 + 2

2 + 2 3 2 +

2 2 2 2

2 2

4 2 2

2 2

4

2 2 2 2

+ 2 2 2 2

2 2

2 2

2 2 2 2

+ 2

X ` 'X X

i j ` ' ` '` ' ` '

H

\Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma $ \Gamma $

\Gamma $ $ \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma

\Gamma

\Gamma $ $

= sin 2 _ sin 2 _ (9.19) + 2sin 2 _ _ + sin 2 _ + 2 sin 2 ( ) _ ( + )

+ 2 sin 2 ( ) _ ( + )

sin 2 _ + sin 2 _ = _ ( + ) _ ( + ) _ (9.20)

_ ( + ) _ + cot _ _

+ _ _ cot _ _

_ _ + cot _ _ + _ _

+ cot 2 _ _ + 2 sin _

+ cot 2 _ + _ 2 sin _ + 2 cot _ _ 2 sin _ + _

sin 2 _

51

em M # OEee

e M # m OEq q

ie m M # ! e* ! * e

iem M # zefl e ie

M # ! V q m m m m fl q

ie M # ! V q m m m m fl q

ie M # m q fl q

ie M # m q fl q

c @ , M c c @ , M c c @ c c @ , M c c @ c ie # W @ c c @ c c

ieW @ c c @ c c ie # W @ c c @ c c ieW @ c c @ c c ie # Z @ c c @ c c ieA @ c c @ c c

i! , eM # c c , eM c c , e# M c c

i! , eM # c c , eM c c , e# M c c

i, e M # z c c c c , e

# M OE c c c c , e

# M OEc c

L \Gamma \Gamma p

\Gamma p \Gamma \Gamma

p \Gamma \Gamma \Gamma

L \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma

e Z W Z W n

n n n

e Z W

L L

e

Z W

Z W I;i

Ii I I i I i i

Z W I;i

iI i i I I i I

Z W I

I I I

Z W i

i i i

W W W W A A Z Z

Z

Z a a W _ _ Z _ Z

_ _ A _ A W _ _ Z _ Z _ _ A _ A W _ _ _ _ _ _

W W W Z W W A

Z

W

Z Z

W W W Z W W A

Z

W

Z Z

W

Z W

W

W

W

Z

W

Z Z Z

+ 5 + + 5

5 5 5

+ 2 2 2 2 + 2

2 2 2 +

+ + +

+ + + + + +

+

+ + + + + + +

X i j

X \Gamma \Delta X \Gamma \Delta X X

i ji j i j i j i ji j ! !

i j i j

M

\Gamma

\Gamma

FP

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma \Gamma \Gamma

10 FEYNMAN RULES In this Section we present the complete list of Feynman rules corresponding to the Lagrangian of SM (see (9.16 - 9.20)).

First of all we define the propagators by the relation

\Delta ( ) = 0 ( ( ) (0) 0 (10.1) where presents any field. Curly, wavy and zigzag lines denote gluons, pho- tons and weak bosons respectively, while full, dashed and dot lines stand for fermions (leptons and quarks), Higgs particles and ghosts fields, respectively. Arrows on the propagator lines show : for the and fields the flow of the positive charge, for the fermion that of the fermion number, and for the ghost that of the ghost number.

The vertices are derived using , instead of usual usage of . All the momenta of the particles are supposed to flow in. The only exception was made for the ghost fields, where direction of momentum coincides with the direction of ghost number flow. This convention permits to minimize the number of times when the imaginary unit appears.

It should be noted ones more, that all fields can be "divided" into two parts:

: (photon), , , (gluon), , (Hiigs).

: , (pseudogoldstones), , , , ( ghosts).

Charged fermions have the electric charge (in the positron charge units) as follows:

( ) = ( ) = ( ) = ( ) = ( ) = ( ) = +

( ) = ( ) = ( ) = + 23

( ) = ( ) = ( ) = 13

52

k i d x e ! T OE x OE ?; OE

W !

i A W Z G H ! z c c c c

e

Q e Q _ Q o/ e; Q e Q _ Q o/ e;

Q u Q c Q t e;

Q d Q s Q b e:

j j ffl ffl

\Gamma \Gamma

ij ikx i j i

z A a

4

+ +

+ + +

Z

\Gamma

\Sigma \Sigma \Sigma

\Gamma \Gamma \Gamma

L i L physical fields non-physical fields

10.1 General Remarks

I I

The electric charge (or strong coupling constant in QCD) is related to the fine structure constant (or in QCD) as follows:

= 4 = 4 The electric charge, the sin , and Fermi constant are related as follows:

2 2 sin = 2 (10.2) Finally, every loop integration is performed by the rule

(2 ) (10.3) and with every fermion or ghost loop we associate extra factor ( 1).

1

+ 1 +

1

+ 1 +

1 + 1 +

1 + 1 +

k _ ^ +

+ =

^ +

+

53

e g

ff ff

ff ff ess ; ff ff gss :

# G

e

# M

G :

d k i ss ;

W W

k M i" g ,

k k k , M i"

Z Z

k M i" g ,

k k k , M i"

A A

k i" g ,

k k k i"

G G ffi

k i" g ,

k k k i"

k m

k m i"

k m m k i"

j j p p

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma

s s

QED QCD s s

W F

W

W

F

d

d

_ *

W

_* W

_ *

W W

_ *

Z

_* Z

_ *

Z Z

_ *

_* A

_ *

a _

b * ab

_* G

_ *

2 2 +

2 2 2 2

2 2 2 2 2 2

2 2 2 2 2 2

s Z

\Gamma \Delta ! \Gamma \Delta ! \Gamma \Delta !

\Gamma \Delta !

\Gamma 10.2 Propagators

1

+

1

+

1

+

_ 1

+

_ 1

+

_ 1

+

_ 1

+

_ 1

+

54

OE OE

k m i"

! !

k , m i"

z z

k , m i"

c c

k , m i"

c c

k , m i"

c c

k , m i"

c c

k i"

c c ffi

k i"

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

H W W Z Z W W W W Z Z

Z Z

A A

a b

ab

2 2 +

2 2

2 2 +

2 2

+

2 2

2 2 2

2

\Gamma \Gamma \Gamma

Here we discuss the explicit forms of the propagators for some popular gauges. Let us consider a theory with free boson Lagrangian:

= 14 = One can fix a gauge in one of three ways [3, 20]:

to impose a gauge condition,

to add a (GFT) to the Lagrangian

to fix a form of the Hamiltonian. In a rigorous theory one should impose two gauge conditions. However, as it is usually accepted, we write only one condition. It should be considered rather as a symbol which denotes acceptable for a given gauge procedure of quantization, described somewhere in literature.

In practical calculations one needs an explicit form of a propagator with satisfactory prescription for poles (which plays a key role in the loop calcu- lations). For this purposes it is sufficient to fix a gauge as mentioned in and . Polarization vectors of physical bosons and ghosts should be chosen in accordance with a detailed quantization procedure applicable for a given gauge.

1.

Notation ( ) = ( )

GFT = 12 ( ) Propagator

= 1+ [ (1 ) + ]

Comments

= 1 is Feynman gauge, while = 0 is Landau gauge. For the photon (gluon) propagator one should write ( ) (see Subsection 10.2).

55

F ; F @ A @ A : @ A x B x L , @ A

D k i" g , k kk i" : , ,

, ,

L \Gamma \Gamma ffl ffl ffl

ffl ffl \Gamma ffl \Gamma \Gamma

ffl

_* _* _ * * _ _ _ GF _ _

_* _*

_ *

G A

2 2

2 2

Gauge Fixing Term Generalized Lorentz gauge.

i ii iii

ii iii

Covariant gauges.

10.3 Some Popular Gauges 2.

Notation ( ) ( ) = ( )

GFT = 12 ( ) Propagator

= + [ (1 ) + ]

Comments The gauge parameter = ( ) for the case of ( ) boson (see Subsection 10.2), 2 is the vacuum expectation value of the gauge field, are generators, is the vector boson mass. = 1 is 't Hooft- Feynman gauge, = 0 is Landau gauge, corresponds to gauge. Non-physical gauge bosons should also be taken into account in loop calculations. They also have gauge-dependent propagator, see Subsection 10.2.

3.

Notation = 0 = 1 2 3

GFT = 12 ( ) Propagator

= 1+ [ ]

Comments The proper corresponds to the case = 0.

4.

56

R @ A x i, v; o/ OE B x

L , @ A

D ffik M i" g , k kk ,M i" :

, , , W Z v= o/ M ,

, ,

~@ ~A ; k ; ; : L , @ A :

D k i" g k k k k g k k g~k ,k k q~k :

,

ffl \Gamma ffl \Gamma ffl

\Gamma \Gamma \Gamma \Gamma ffl

p

! 1

ffl ffl \Gamma ffl

\Gamma \Gamma \Gamma j j \Gamma j j

ffl

, _ a

_

a a

GF _ a_

ab

_*

ab

_*

_ *

W Z a

GF k k

_* _*

_ * _ * * _ _ *

2 2 2 2 2

2 2

0 0 0 0

2

2

4

't Hooft gauges ( -gauges).

unitary Coulomb gauge.

Coulomb gauge The general axial gauge.

Non-covariant gauges

Notation ( ) = ( ) GFT = 14 [ ] Propagator

= 1+ [ ( + )+ ( ) ( )( + ) ]

Comments Feynman rules in this gauge usually do not contain ghosts. As it has been shown in [20] one has to consider an additional gauge vector in order to have a correct prescription for poles. The quantization in this gauge was considered, for example, in [21, 22].

The gauge vector has the form:

= ( ; ) = ( ; ) = ( ; ) The explicit form of the component structure of and should be considered separately in the cases 0 = 0 and 0. The following widely used gauges are obtained in the limit = 0:

4a. 0.

= 1+ [ ( + )+ + ( )( + ) ]

= ( ; ); = ( ; ) 4b. = 0.

= 1+ [ ( + )+ ]

= ( ; ); = ( ; ) 4c. 0.

= 1+ [ ( + )+ + ( )( + ) ]

= ( ; ); = ( ; ) 5.

57

n A x B x : L , n @ n A :

D k i" g n k k n n kn k n k i" ,k n n kn k n k i" k k :

n n n n ~n n ~n ; n n n ; n ; n :

n n n ? ; n n !

,

n ?

D k i" g n k k n n kn k n k i" n n kn k n k i" k k ;

n n ~n ; i ~n n n ~n ; i ~n :

n

D k i" g n k k n n kn k n k i" ;

n ~n ~n n ~n ~n :

n !

D k i" g n k k n n kn k n k i" n n kn k n k i" k k ;

n ~n ~n n ~n ~n :

ffl ffl \Gamma \Delta \Delta ffl

\Gamma \Delta \Delta \Delta \Gamma \Gamma \Delta \Delta \Delta

ffl

\Gamma \Delta \Delta \Delta \Delta \Delta \Delta

\Gamma j j j j

\Gamma \Delta \Delta \Delta j j j j \Gamma

\Gamma \Delta \Delta \Delta \Delta \Delta \Delta

j j j j \Gamma

_ _ GF

_* _*

_ * _ *

_ *

_ _ _

_ _

_* _*

_ * _ *

_ *

_ _

_* _*

_ * _ *

_ _

_* _*

_ * _ *

_ *

_ _

2 2

2 2 2

2

0 0 3 0 1 2 3

2 2 2

2 2

2 2

2

0 0

2

2

2 2

2 2

2

\Lambda

\Lambda \Lambda

\Lambda \Lambda

\Lambda

?

\Lambda

\Lambda \Lambda

\Lambda \Lambda

? ?

\Lambda

? ?

\Lambda \Lambda \Lambda

\Lambda \Lambda

\Lambda \Lambda

?

\Lambda

?

Temporal gauge:

Light-cone gauge: Proper axial gauge: Planar gauge.

Notation ( ) = ( ) = 0 GFT = [ ( )] Propagator

= 1+ ( + )+ = ( ; ) = ( ; ) if 0;

= ( ; ) = ( ; ) if 0

Comments Yang-Mills theory is not multiplicatively renormalizable in this gauge. Quantization in this gauge is also poorly understood. This gauge has the same denotation as the axial gauge, that is not suitable. However, that should not lead to confusion (see the beginning of this Subsection).

58

n A x B x ; n : L @ n A :

D k i" g n k k n n kn k n k i" ; n n ~n ; i ~n ; n n ~n ; i ~n ; n ?

n ~n ~n ; n ~n ~n n ! :

ffl 6 ffl \Delta ffl

\Gamma \Delta \Delta \Delta \Gamma j j j j j j j j \Gamma

ffl

_ _ GF n _

_* _*

_ * _ *

_ _

_ _

2 1 2

2

2 0 0 2

2

" #\Lambda

\Lambda

? ?

\Lambda

? ?

?

\Lambda

?

2 cot ( ) + ( ) + ( )

( ) + ( ) + ( )

( ) + ( ) + ( )

( + 2 ) cot ( + 2 )

59

ZW W

p

WW

q

Zk

e # k p g p q g q k g

AW W

p

WW

q

Ak

e k p g p q g q k g

GGG

G p

G

q

Gk

ig f p k g q p g k q g AAW W

W W

A A

AZW W

W W

A Z

e g g g g g g e # g g g g g g

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma

flfi ff

W fl fffi ff fifl fi flff

flfi ff

fl fffi ff fifl fi flff

b fi

c fl

a ff

s abc fl fffi ff fifl fi flff

ff fi

ffi fl

ff fi

ffi fl

fffl fiffi ffffi fifl fffi flffi W fffl fiffi ffffi fifl fffi flffi

+

+

+

+

+ +

+ +

2 2

i j i j

i j

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

Gauge Boson Three-vertices Gauge Boson Four-vertices 10.4.1 10.4.2

10.4 Vertices

cot ( + 2 )

sin ( + 2 )

( ( ) + ( ) + ( ) )

_

_

_

_

2 2 sin (1 ) 2 2 sin (1 )

60

ZZW W

W W

Z Z

e # g g g g g g

W W W W

W W

W W

e

# g g g g g g

GGGG

G G

G G

g f f g g g g

f f g g g g f f g g g g

W



W

W



W

e

# V fl fl

e

# V fl fl

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma

p \Gamma p \Gamma

ff fi

ffi fl

W fffl fiffi ffffi fifl fffi flffi

ff fi

ffi fl

W

fffl fiffi ffffi fifl fffi flffi

a ff

b fi

d ffi

c fl

s

rab rcd fffl ffifi ffffi fifl

rac rdb ffffi fifl fffi flffi rad rbc fffi flffi fffl ffifi

I i

i I

ff

i I

I i

ff

W

iI

ff

W

iI ff

+ +

2 2

+ +

+ +

2 2

2 +

+

+ 5 5

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma

\Gamma

Gauge-boson-fermion Vertices10.4.3

_

_

sin 2

1 2 (1 ) 2 sin

_

_

sin 2

1 2 (1 ) 2 sin

_

_

_

_ ( = )

2 sin ( ) 2 sin ( )

61

Z



Z

e

# fl fl Q #

Z



Z

e

# fl fl Q #

A



A

qqG

q q

G

eQ fl n i; I g t fl OE ! W

! p q

OE

W

OE ! W

! p q

OE

W

i e

# p q

i e

# p q

\Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

I I

I I

ff

W

ff I W

i i

i i

ff

W

ff i W

n n

n n

ff

A B

a

ff

n ff s a BA ff

ff ff W

ff

W

ff

5 2

5 2

+

+

+

+

` ' ` '

\Gamma \Delta \Gamma \Gamma

\Gamma

\Gamma

Gauge-boson-Higgs Three-vertices10.4.4

2 sin ( ) 2 sin ( )

cot 2 ( ) ( )

2 ( ) cot 2sin 2

62

z ! W

! p

z

q

W

z ! W

! p

z q

W

e

# p q

e

# p q

! ! Z

! !

Z

p q

! ! A

! !

A

p q e # p q e p q

OE zZ

z OE

Z

p q

i e sin # p q

OE W W

W W

OE

OE Z Z

Z Z

OE

e # M g e# M g

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma

ff ff W

ff

W

ff

ff ff W ff ff

ff

W

ff

ff fi ff fi

W Z fffi

W

Z fffi

+

+

+

+

+

+

+

+

+ +

\Gamma \Gamma

\Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma

\Gamma

sin

sin

2 sin

2 sin 2

2 cos 2 cos

63

! W Z

W Z

!

! W A

W A

!

i e # M g i e M g ! W Z

W Z

!

! W A

W A

!

i e # M g i e M g OE OE W W

W W

OE OE

OE OE Z Z

Z Z OE OE

e

# g

e

# g

OE ! W Z

W Z

OE !

OE ! W Z

W Z

OE !

i e

# g

i e

# g

\Gamma \Gamma

\Gamma

ff fi ff fi

W Z fffi W fffi

ff fi ff fi

W Z fffi W fffi

ff fi ff fi

W

fffi

W

fffi

ff fi ff fi

W

fffi

W

fffi

+ +

+ +

+

+

+

+

+ +

2 2

2 2

+

+

+

+

2 2

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

Gauge-boson-Higgs Four-vertices10.4.5

2 sin 2 sin 2 sin

2 sin 2

2 cos 2 cos

2 sin 2 sin

64

OE ! W A

W A

OE !

OE ! W A

W A

OE !

i e

# g

i e

# g

z z W W

W W

z z

z z Z Z

Z Z z z

e

# g

e

# g

z ! W Z

W Z

z !

z ! W Z

W Z

z !

e

# g

e

# g

z ! W A

W A

z !

z ! W A

W A

z !

e

# g

e

# g

\Gamma \Gamma \Gamma ff fi ff fi

W

fffi

W

fffi

ff fi ff fi

W

fffi

W

fffi

ff fi ff fi

W

fffi

W

fffi

ff fi ff fi

W

fffi

W

fffi

+ +

+

+

2 2 +

+

2 2

2 2

+

+

+

+

2 2 +

+

+

+

2 2

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

2 sin 2 cot 2

2 2 cot 2

3 sin 2

sin 2 sin 2

65

! ! W W

W W

! !

! ! Z Z

Z Z

! !

e

# g e # g

! ! A A

A A

! !

! ! A Z

A Z

! !

e g e # g

OE OE OE

OE OE

OE

e

#

m

M

z z OE

z z

OE

! ! OE

! !

OE

e

#

m

M

e

#

m

M

\Gamma \Gamma \Gamma

ff fi ff fi

W

fffi W fffi

ff fi ff fi

fffi W fffi

W

H

Z

W

H

Z W

H

Z

+ +

+ +

+

+

2 2 2 2

+

+

+

+

2 2

2

+

+ 2 2

\Gamma \Gamma

\Gamma \Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

Higgs Three-vertices10.4.6

3 sin 2

3 sin 2

sin 2 sin 2 sin 2

2 sin 2

66

OE OE OE OE

OE OE OE OE

z z z z

z z z z

e

#

m

M

e

#

m

M

! ! OE OE

! ! OE OE

! ! z z

! ! z z

e

#

m

M

e

#

m

M

OE OE z z

z z OE OE

! ! ! !

! ! ! !

e

#

m

M

e

#

m

M

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

W

H

Z W

H

Z

W

H

Z W

H

Z

W

H

Z W

H

Z

2 2

2

2

2 2

2

2

+

+

+

+ 2 2

2

2

2 2

2

2

+ +

+ +

2 2

2

2

2 2

2

2

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma

Higgs Four-vertices10.4.7

_

_ sin 2 _

_ 2 sin 2 + +

_

_ 2 sin 2 + + _

_

_

_ sin 2 sin 2

67

OE



OE

e

#

m M

!



!

i e

# M V m m m m fl

!



!

i e

# M V m m m m fl

z



z

z



z

i e

#

m M fl

i e

#

m M fl

\Gamma \Gamma p \Gamma

\Gamma p \Gamma

\Gamma

n n

n n

W

n Z

I i

i I W Z

Ii i I i I

i I

I i W Z

iI I i i I

I I

I I

i i

i i W

I Z W

i Z

+

+

+ 5

5 5 5

i\Gamma \Delta \Gamma \Delta j i\Gamma \Delta \Gamma \Delta j \Gamma

\Gamma

Higgs-boson-fermion Vertices10.4.8

_

_

_

_ cot cot

_

_

_

_

_

_

_

_ cot cot

68

c c W

c

p

c

W

c c W

c

p

c

W

e # p e # p c c W

c

p

c

W

c c W

c

p

c

W

e p e p c c W

c

p

c

W

c c W

c

p

c

W

e # p e # p

\Gamma

\Gamma

\Gamma

Z

Z ff

Z

Z

ff

W ff W ff A

A ff

A

A

ff

ff ff Z

Z ff

Z

Z

ff

W ff W ff

+

+

+

+

+

+

+

+

+

+

+

+

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

Gauge-boson-ghost Vertices10.4.9 _

_

_

_

_

_

_

_ cot cot

_

_

_

_

_

_

69

c c W

c

p

c

W

c c W

c

p

c

W

e p e p c c Z

c

p

c

Z

c c Z

c

p

c

Z

e # p e # p c c A

c

p

c

A

c c A

c

p

c

A

e p e p

c c G

c

p

c

G

ig f p

\Gamma \Gamma

\Gamma

\Gamma

A

A ff

A

A

ff

ff ff

ff ff

W ff W ff

ff ff

ff ff

a b

c ff

s abc ff

+

+

+

+

+

+

+

+

+

+

+

+

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

_

_

_

_ cos 2 2 sin

_

_

_

_ 2 sin

cos 2 2 sin

_

_

_

_

2 sin

70

c c !

c c

!

c c !

c c

!

i, e #

# M i, e M

c c !

c c

!

c c !

c c

!

i, e

# M

i, e #

# M

c c !

c c

!

c c !

c c

!

i , e M i, e# M

\Gamma \Gamma

\Gamma

Z

Z

A

A W W

W

Z W W

Z

Z

Z

Z Z

W

Z

W W

W

Z

A

A

Z

Z W W

Z

W

Z

+

+

+

+

+

+

+

+

+

+

+

+

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

Higgs-ghost Vertices10.4.10 _

_

_

_ 2 cot 2 cot

_

_

_

_ 1 2 cot

1 2 cot

_

_

sin 2

71

c c z

c c

z

c c z

c c

z

i , e # M i , e # M

c c OE

c c

OE

c c OE

c c

OE

, e # M , e # M

c c OE

c c

OE

, e

# M

\Gamma \Gamma \Gamma

\Gamma

W W Z W W Z W W Z W W Z

Z Z

Z Z

Z

W

Z

+

+

+

+

+

+

+

+

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

\Gamma

11 INTEGRATION IN -DIMENSIONSN 11.1 Dimensional Regularization A powerful method of the evaluation of the loop integrals (which very often are divergent) is (DR) [23]. The idea of DR is to consider the loop integral as an of - number of dimensions. Then one can calculate this integral in that region of the complex plane, where this function is convergent.

A typical loop integral looks as follows:

(2 )

( ) ( ( ) )

where ( ) are 4-momenta (masses) of external particles; ( ) is a function of masses and momenta and .

To use the DR method one needs to transform the product of denom- inators into expression such as : + ( ) + , where is the linear combination of momenta and is a combination of , ( ), and . That can be done by using of :

1 = \Gamma ( + )

\Gamma ( )\Gamma ( )

(1 ) [ + (1 )] 1 1 = ( )

[( ) + ] 1 = \Gamma ( + + )

\Gamma ( )\Gamma ( ) \Gamma ( )

( ) (1 ) [ + ( ) + + (1 )]

where \Gamma ( ) is the Euler .

Using the Wick rotation and replacement 4 , one can obtain a typical integral in -dimensional Euclidean space:

= ( )( + 2( ) + ) Re 0 The differential has the form:

= \Omega \Omega = \Omega = 2 \Gamma ( 2 ) (11.1)

72

Dimensional Regularization

analytic function

Feynman parameterization Gamma function

n

n

d p

ss

P q ; m ; p

m p k ;

q m P q ; m ; p

m q p

p pk M k q M q q q m

a b

ff fi ff fi dx

x x ax b x ;

a b

n a b dx a b x b ;

b b : : : b

ff : : : ff ff ff : : : ff dx dx : : : dx

x x x : : : x b x b x x : : : b x ;

z

p ip n n

J d p P q ; m ; pp pk M ; ff ? :

d p d p p dp d ; d ss = n ;

Z Q

Z Z

Z Z Z

Z

Z

4

4 =1 2 2

2 2

2 2

1 0

1 1

+ 1

0 +1

1 2

1 1 2

1 0 1 0 2 0 1 11 2 1 1 1 1

1 1 2 2 1 1 + +

0 0

2 2 1

*i i * li i i

i i *i i *

i i

* i i i j i

ff fi

ff fi

ff fi

n n n ff ff ff

m

m

m

x x

m

ffm m m ff ff

m m m m ff ::: ff

n *i i *

ff

n

n n n n n

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma ! !

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma

1 2

1 2

1 2

1

2

m

m

m

m

n

\Gamma

11.2 Integrals

\Omega = (sin ) \Omega (11.2)

= (sin )(sin )

where 0 0 2 (The last equality in (11.2) obeys for the integer .)

Let us introduce the following notation:

^ ( ) 1( + 2( ) + ) ( ) (11.3)

( ) \Gamma ( ) (11.4) Then :

= ^ 1 = ( ) \Gamma ( )\Gamma ( ) = \Gamma ( 2 ) = ^ = ( ) = ^ = \Gamma ( 2 ) + 2 \Gamma ( 1 2 )( )

= ^ = \Gamma ( 2 ) + 12 \Gamma ( 1 2 ) ( ) = ^ = \Gamma ( 2 )

1 2 \Gamma ( 1 2 )( )( + + )

= ^ = \Gamma ( 2 ) + + 22 \Gamma ( 1 2 )( )

For calculation of the basic integral one can use the well-known relation [24]:

( + ) =

\Gamma ( )\Gamma ( )

2\Gamma ( )( )

73

d # d# d

# d# # d# : : : d# ;

# ss; # ss: n

J f p d p p pk M f p ;

J iss iM k ff :

I J issM k ff ff ff n J : I J p k I ;

I J p J k ff n n ff n M k ;

I J p p J k k ff n ff n g M k ; I J p p p J k k k ff n

ff n M k g k g k g k ; I J p p J k k ff n n ff n M k :

I x x M dx ff M :

Z Z

2 1 1 1 2 1 1 3 2 2 1

1

2 2 0

2

2 2

0

2

2 2 2

2 0

0 2 2 0 2 2 2

0 2 2

0

2 2

2 2 0 2 2 2

0

0 2 2

+1

2

2 1

2

2

n n n n n

n n n n n n

i

n

ff

n= n

ff

n=

ff n=

n

_ _ _

_* _ * _ * _* _** _ * * _ * *

_* * _* * ** _ _ _ _

fi

ff

fi ff fi

ff

^ ^ ^ ^

j j \Gamma

\Gamma \Gamma \Gamma \Gamma f \Gamma \Gamma \Gamma \Gamma g

f \Gamma \Gamma \Gamma \Gamma g

f\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma g

\Gamma f \Gamma \Gamma \Gamma \Gamma g

\Gamma

\Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma

1 \Gamma \Gamma

\Gamma

2

+1

2

n

fi

11.3 Spence Integral (Dilogarithm) As a rule the final expressions for the loop integrals include so-called

or [24, 25, 26]:

Li ( ) = Li( ) ln(1 ) = ln [ (1 ) ] (11.5) Dilogarithm is a special case of the polylogarithm [24, 25, 26]:

Li ( ) [ 1 or = 1 Re 1] (11.6) The main properties of Li( ) are as follows:

Li ( ) + Li ( ) = 2 Li ( ) Li ( ) + Li ( ) = 4 Li ( ) 2 Li ( )

Li ( ) Li ( ) = 2 ( 1)(2 + 1) [ 1]

Li ( ) = Li ( ) ( = 1 2 ) Li ( ) = 1 Li ( ) = ln(1 ) The Riemann sheet of the Li ( ) has a cut along the real axes when 1, and

Im Li ( ) = \Theta ( 1) ln( )

where the \Theta ( ) is the step function (see Subsection 16.1). The equation ReLi ( ) = 0 has two solutions on the real axes

= 0 and 12 6 ReLi ( ) achieves its maximum at = 2:

Li (2) = 4 and at this point the Li ( ) has the expansion as follows [9]:

Li (2 ) = 4 4 6 548 15

74

Spence integral Euler dilogarithm

z z tt dt tt z dt arg z ! ss :

z zk z ! ; z ; * ? :

z z z z ; iz iz z z ;

iz iz i zk z ! ;

z tt dt n ; ; : : : ; z z z ; z z :

z z ? z i" ss z z ; x

z

z ; z : : z z

ss ;

z ffi ss ffi ffi ffi ffi : : :

Z Z X

XZ

2 0

1

0 1

=1

1 2

1 4 1 2

=0

2 +1

0

1

0 1

2 2

2

1 2 2

2

2

2 2

2 2 3 4 5

z *

k

k

*

n n n n n n n n n n

n n

k

k k

n

n

z n

j \Gamma \Gamma \Gamma \Gamma

j j j j j

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma j j

\Gamma \Gamma \Gamma

\Sigma \Sigma \Gamma

ss

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma 1

\Gamma

\Gamma \Gamma

1

\Gamma

One easily gets:

Li (0) = 0 Li (1) = 6 Li ( 1) = 12 Li ( 12 ) = 12 12 ln 2 Li ( ) = 48 = ( 1)(2 + 1) = 0 915965594 The various relations with Li are as follows [24, 25, 26]:

Li ( ) = Li (1 ) + 6 ln ln(1 ) [ (1 ) ] Li ( ) = Li ( 1 ) 12 ln + ln + 3 [ ( ) ] Li ( ) = Li ( 11 ) + 12 ln (1 ) ln( ) ln(1 ) 6

[ ( ) ] The Hill identity has the form [9, 25]:

Li ( ) = Li ( ) + Li ( ) Li 1 Li 1

ln 11 ln 11

1 11 ln 1 11 ln where the function compensates for the cut in the Riemann sheet of the logarithm [9]:

ln = ln + ln + ( )

A typical integral, which can be expressed via the dilogarithm, is, for exam- ple:

ln( + ) = ln ln Li ( ) + Li ( )

The Euler \Gamma ( ) is given by the integral representation [24]:

\Gamma ( ) Re 0

75

Gamma function

; ss ; ss ; ss ;

i ss i ; k : : : :

z z ss z z argz ; arg z ! ss ; z z z iss z ss arg z ! ss ; z z z z z ss

arg z ! ss :

!z ! z ! !z!z z !z!z

! !z

z !z

j !; !z ! j z; !z z;

j

xy x y j x; y :

p qt

t dt p

b a b

q p a

q p :

z

z dt t e ; z ? :

X

` ' ` '` ' ` ' ^ * ^ *

Z

Z

2 2

2

2

2

2

2 2

2

2

=0 2 2

2 2

2

2 2 2

2

2 2 2

2

2 2 2 2 2

2 2 0

1

k

k

b a

z t

\Gamma \Gamma \Gamma \Sigma \Gamma \Sigma \Gamma

\Gamma \Gamma \Gamma \Gamma j j j \Gamma j \Gamma \Gamma j \Gamma j

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma j \Gamma j

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma j

1 1

\Gamma \Gamma

G G The main properties of the \Gamma ( ) are as follows [24]:

\Gamma (1 + ) = \Gamma ( ) \Gamma ( + 1) = ! \Gamma ( )\Gamma ( ) = sin( ) \Gamma ( )\Gamma (1 ) = sin( )

\Gamma ( 12 + )\Gamma ( 12 ) = cos( ) \Gamma (2 ) = 2 \Gamma ( )\Gamma ( 12 + ) \Gamma (1) = \Gamma (2) = 1 \Gamma 12 = \Gamma ( ) 1 + \Gamma (1); \Gamma (1) = \Gamma (1)\Psi (1) = \Psi (1) = = 0 577 where is Euler constant.

76

z z z z ; n n ; z z ssz ssz ; z z ssssz ;

z z ssssz ; z ss z z ;

; ss; z z fl : : : : ; fl

` '

(2 1)

0

z z

\Gamma \Gamma \Gamma

\Gamma p

p

j ' \Gamma \Gamma

\Gamma !

0 0

12 KINEMATICS 12.1 Variables The nice book by E. Byckling and K. Kajantie [27] contains a lot of infor- mation about relativistic kinematics. Here we present a brief description of relativistic kinematics following the Review of Particle Properties [18].

Initial (final) particles total momentum (energy) squared will be denoted by:

= (12.1)

Let and be energy and momentum of a particle. The energy and mo- mentum of this particle ( ) in the frame moving with the velocity are given by the Lorentz transformation:

= ( + ) = ( + ) = (12.2) where = 1 1 and ( ) are the components of perpendicular (parallel) to . The beam direction choose along the -axes. 4-momentum of a particle

= ( ) can be written as:

= = ( )

= cos sin = sin sin = cos (12.3)

where is the azimuthal angle (0 2 ); is the polar angle (0

). Another parameterization of looks as follows:

= cosh = sinh (12.4) where = + is the transverse mass ("old" definition), is the rapidity.

is defined by

1 2 ln

+ = ln + = tanh (12.5)

77

Rapidity

s p p : E ~p

E ; ~p ~fi

E fl E fip ; p fl p fiE ; ~p ~p ; fl = fi ~p p ~p

~fi

z p E; ~p

E p ; ~p p ; p ; p ; p ~p OE #; p ~p OE #; p ~p #;

OE OE ss # # ss

p

E m y; p ; p ; p m y; m m p y

y

y E pE p E pm pE :

i X j i X j i j i j i j

2 2 2 0

2 2 2

1

initial

i

final

j

_

x y z x y z

_

x y z

z

z

z z

j p \Gamma

j j j j j j

^ ^ ^^

j \Gamma

0 0 0 jj 0jj jj 0? ?

? jj

?

? ? ? ?

?

\Gamma

12.2 Event Shape Variables

Under a boost along -direction to a frame with velocity ,

+ tanh is defined by:

ln(tan( 2)) (12.6) sinh = cot cosh = 1sin tanh = cos

For and 1 one has : . Feynman's = variable is given by

= max ( + )( + )maxin cms = 2 (12.7) The last equation is valid for two particles collisions, and here is total energy squared (see (12.1)). In the collider's experiments the following additional variables are used:

= sin transverse energy = (\Sigma ) "missing" transverse momentum

= (\Sigma ) "missing" transverse energy

where sum is performed over all detected particles. The "distance" in ( )-plane between two particles (clusters) 1 and 2 is given by

\Delta (\Delta ) + (\Delta ) \Delta = \Delta = The "transverse" mass of the particle (cluster) with momentum and the "missing" transverse momentum (energy) ( ) is given by:

( ) ( + + ) ( + )

In this Subsection we describe in brief event shape variables for -particle final state (for details, see, for example [28]). None of the variables presented in this Subsection are Lorentz invariant.

78

Pseudorapidity

z fi

y y fi: j

j #= ;

j #; j # ; j #:

p m # =fl j y

x x

x pp E pE p ; x ps :

s

E E # ;

~p ~p ;

~E ~E

j; OE R OE j ; OE OE OE ; j j j :

c ~p ~p ~E

M c; ~p m p p ~p ~p :

n

q

q

1 2 2 1 2 1 2 2 2 2 2 2

F

z z

z z

z

mis

mis

c mis mis

mis c c mis c mis

! j \Gamma AE AE ss

ss p

\Gamma \Gamma \Gamma \Gamma \Gamma

j \Gamma \Gamma

j \Gamma

\Gamma ? ? ?

? ?

? ? ? ? ? ? ? ?

The tensor is defined as [28, 29]:

(12.8) where = 1 2 3 corresponds to the and components. By standard diagonalization of one can find three eigenvalues

with + + = 1 Then, the is defined as:

3 2 ( + ) 0 1 (12.9)

Eigenvectors can be found that correspond to the three eigenvalues . The eigenvector is called the , while the

is spanned by and . Sphericity is essentially a measure of the summed with respect to spheric- ity axes. So, one can use another definition of the sphericity:

= 32 min (12.10) where is a component of perpendicular to . So, the sphericity axes given (12.10) by the vector for which minimum is attained. A 2-jet event corresponds to 0 and isotropic event to 1.

Sphericity is not an infrared safe quality in QCD perturbation theory. Sometimes one can use the generalization of the sphericity tensor, given by

(12.11)

The is define as [28, 30]:

3 2 0

1 2 (12.12)

79

sphericity

sphericity

sphericity axes sphericity event plane

aplanarity

S p p~p ; a; b ; ; x; y z

S

* * * ; * * * :

S * * ; S : ~s * ~s

~s ~s

~p

S ~p~p ; ~p ~p ~n ~s

~n S S

S ~p p p~p ; A

A * ; A :

PP

PP P P

2 1 2 3 1 2 3

2 3 1

1 2

2

2

( ) 2

2

ab i ai bi

i i

ab

i i

~n

i i

i i

i i i

r ab i i r ai bi

i i r

ffl

j j j

* *

j ^ ^

j j ss ss

j j jj j ffl

j ^ ^

? ? ?

\Gamma

Sphericity

Aplanarity The aplanarity measures the transverse momentum component out of the event plane. A planar event has 0 and isotropic one .

The is given by [28, 31]

max ( ) 12 1 (12.13) and the is given by the vector for which maximum is attained. 2-jet event corresponds to 1 and isotropic event to .

In the plane perpendicular to the thrust axes, a and

may be defined in just the same fashion as thrust [28], i.e.

max ( ) (12.14) Finally, a third axes, the , is defined perpendicular to the thrust ( ) and major ( ) axes. The is calculated just as thrust and major values.

The is given by [28]

In general, 0, corresponds to an event symmetrical around the thrust axes and high to aplanar event.

The , = 0 1 2 are defined by [28, 32]:

(cos ) (12.15) where is the opening angle between hadron and , and is the total visible energy of the event. ( ) are the Legendre polynomials [24]:

( ) = 1 ( ) = ( ) = 12 (3 1)

( ) = 1 (2 1) ( ) ( 1) ( )

80

thrust

thrust axes

major axes major value

minor axes

minor value

oblatness

Fox-Wolfram moments

A A T

T ~n~p~p ; T :

~t ~n

T T

~m M

M ~n~p~p :

~t ~m M

O

O M M : O ~t O

H l ; ; ; : : : ; H ~p ~pE P # ; # i j E

P z

P z ; P z z; P z z ; : : : P z k k zP z k P z :

PP

PP

X \Theta \Lambda

1

2

=1

1

2

=1 ( )=0 1

1

=1 2 0 1 2 2

1 2

~n

i i

i i

i

a a

a ~n ; ~n~t i

i

i i

a i

a i l l

i;j

i j

vis

l ij

ij vis

l

k k k

ss ss ffl

j j jj j ^ ^

ss ssffl

j j jj j ffl

j \Gamma ss

ffl

j j jj j

\Gamma \Gamma \Gamma \Gamma

j j

j j

\Gamma \Gamma 1 Thrust Major and minor values

Oblatness Fox-Wolfram moments

12.3 Two-body Final State

Neglecting the masses of all the particles, one gets = 1. If momentum is balanced, then = 0. 2-jet events tend to give 1 for even and

0 for odd. The summary of the discussed quantities are presented in Table 12.1.

Summary of event shape variables.

all = 0 isotropic 1 - 1 -

2-jet 0 - 1 0 1 = 01 0 planar - 0 - 0 1 -

In the collision of two particles of mass and and momenta and

= ( + ) = + + 2 where the last equation is valid in the frame, where one particle (second one) is at rest (Lab frame).

The energies and momenta of the particles 1 and 2 in their center-of-mass system (cms) are equal to:

= + 2 = +2 (12.16)

= = [ ( + ) ][ ( ) ]2 (12.17) or

= 12 ( )

where ( ) is the so-called [27]:

( ) ( ) 4 (12.18)

= + + 2 2 2 = ( + ) ( ) (12.19)

81

kinematical function

H H H l H l

S A T O Hm H

H H ; H

m m p p s p p m m E m ;

E s m ms ; E s m ms ;

~p ~p ; ~p s m m s m ms ;

~p s * s; m ; m ; * x; y; z

* x; y; z x y z yz

x y z xy yz zx

x y z x y z :

q

0 1

0 1 2

1

2

1

2 2 +1

1 2 1 2 1 2 2 21 22 1 2

1

21 22

2

21 22

1 2 1

1 2 2 1 2 2

1 1 2 21 22

2 2 2 2

2 2

l l

i l

k k

Lab

=

ssss ss ssAE

\Gamma p \Gamma p \Gamma j j \Gamma \Gamma \Gamma p

j j p

j \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

f \Gamma p p gf \Gamma p \Gamma p g

\Lambda \Lambda \Lambda \Lambda \Lambda

\Lambda

Table 12.1. 12.4 Three-body Final State

Let us now consider the two-body reaction (4-momenta of the particles are presented in the parentheses):

( ) + ( ) 1( ) + 2( )

+ = +

The Lorentz-invariant Mandelstam variables for reaction 2 2 are defined by:

= ( + ) = ( + ) = ( ) = ( ) (12.20) = ( ) = ( )

and they satisfy

+ + = + + +

Two limits of t (corresponding to = 0 and ) equal:

= + 2 2 = (12.21) = + 12 ( + )( + )

1 2 ( ) ( )

Let us consider three-body decay of particle with mass

( ) 1( ) + 2( ) + 3( ) Defining

+ (12.22)

then

+ + = + + + and = ( ) = + 2

The 1 3 decay is described by two variables (for example, and ). If is fixed, then limits of variation are equal to:

82

a p b p p p

p p p p

s p p p p ; t p p p p ; u p p p p ;

s t u m m m m :

# ss

t m m E E ~p ~p

m m s s m m s m m

s * s; m ; m * s; m ; m :

a M a P p p p :

p p p ; m p ;

m m m M m m m ; m P p M m M E :

m m m m

1 2 1 2

2 1 2 2 1 2 2 2 2 2 1 2

2 2 21 22

2 21 1 1 2 21 2 2 21 22

1 2 2 2 1 2 21 22

1 2 3

2 2

212 223 213 2 21 22 23 2 2 2 2

12 13 12 213

a b

a b

a b a b a b

a b cm

a a a a a b

= a b =

ij i j ij ij ij k k k

!

! \Gamma \Gamma \Gamma \Gamma

\Gamma \Sigma j jj j \Gamma \Gamma \Gamma

\Sigma

!

j j

\Gamma \Gamma !

\Sigma

\Lambda \Lambda \Lambda \Lambda

12.5 Lorentz Invariant Phase Space

= + 12 ( + )( + )

1 2 ( ) ( ) =

= ( + ) ( ( )

where = and = . 2 3 scattering is described by five independent variables. For example,

= ( + ) = ( ) = ( )

Lorentz invariant phase space (LIPS) of particles with 4-momenta ( = 1 2 ) and the total momentum = is given by:

( ; ) ( ) (2 ) 2 (12.23) Through of this Subsection we use the following notation:

This LIPS can be generated recursively as follows [18, 27]:

= ( ; )(2 ) ( ; ) (12.24) where = and ( + + + ) ( ) , or:

= ( ; )(2 ) ( ; ) (12.25) here = and

( + + ) ( )

83

m m m m m M m m m m

m * m ; M ; m * m ; m ; m E E E m E m ;

E E

s p p ; m ; m ; t p p ; t p p :

n p j ; ; : : : n P p

dR P p ; p ; : : : p ffi P p d pss E :

s P : dR dR P p ; q ss dq dR q p ; : : : p ; q p m m : : : m q P m dR dR P q; p ; : : : p ss dq dR q p ; : : : p ; q p

m : : : m q P m :

i j

q q

P

X Y

P P

X

213 21 23

212

212 2 23 212 21 22

212

1 2 212 2 23 1 2 212 21 22

1 3 2 21 21 23 23 2 1

+

2 3 2

2 212 223 1 1 2 2 3 2

=1 1 2 (4)

=1 =1

3

3

2

2 3 2 1 1 1 1=1 1 2 1 2 2 2 2

+1 +1 3 2 1 =1

1 2 2 2

= +1

2

= = m m m

m

M m m

m

a b q b

jn j j

n n

n

j

j

n

j

j

j

n n n n ni i n n

n n j j n j j

j l l

j

n

l j

l

\Gamma \Gamma \Gamma \Sigma

\Gamma \Gamma \Upsilon \Gamma

!

\Gamma \Gamma

j \Gamma

j

^ ^ p \Gamma ^ ^ p \Gamma

\Sigma

\Lambda \Lambda \Lambda \Lambda \Lambda \Gamma \Lambda \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma

\Gamma

2 12

2 1

2 2

12

2 2

12

2 3

12

12.6 Width and Cross Section

The integrated LIPS for = = = = 0 equals:

(0) = 1(2 ) ( 2)( 1)!( 2)! ( ) Two-particle LIPS equals:

= 1(2 ) 4 \Omega = 1(2 ) = 1(2 ) where is momentum of first (second) particle in cms (see (12.17)). Different choice of and leads to:

= 1(2 ) [ ( + ) ][ ( ) ]2 ( + ) = 1(2 ) 2 1 4 = = = 1(2 ) 2 (1 ) = 0 = 1(2 ) 2 = = 0 Three body decay final state LIPS equals:

= 1(2 ) 4 = 1(2 ) where and are defined in (12.22), ( ) is the energy of the first (second) particle in rest frame. This is the standard form of the Dalitz plot.

The partial decay rate ( ) of a particle of mass into bodies in its rest frame is given in terms of the Lorentz-invariant matrix element

by:

\Gamma = (2 )2 ( ; ) (12.26)

84

partial width

m m : : : m R ss ss=n n P :

R ss pP d ss sspP ss ssps ; p

m m

R ss ss s m m s m ms ; m m s; R ss ss ms ; m m m; R ss ss ms ; m ; R ss ss ; m m :

dR ss sss dm dm ss ss dE dE ; m m E E

P

M n M

d ssM M dR P p ; p ; : : : ; p :

Z q

s

1 2

3

1 2 2

2 6 1 2 1 6 12 6 1 1

1 2

2 6

1 2 2 1 2 2

1 2

2 6

2

1 2

2 6

21

2

2 6 1 2

3 9

2 2

12 213 9 2 1 2

12 13 1 2

4 2

1 2

n n n

n n

fi

fi n n

\Gamma \Gamma p p p

\Gamma \Gamma \Gamma ^ p \Gamma \Gamma

j j

\Gamma

\Gamma

\Lambda

\Lambda

\Lambda \Lambda

\Lambda

The differential cross section of the reaction + 1 + 2 + + ( + ) is given by:

= (2 )2 ( ; ) (12.27) = [ ( + ) ][ ( ) ] = 4[( ) ]

85

a b : : : n p p P

doe ssI M dR P p ; p ; : : : ; p ; I s m m s m m p p m m :

4 2

1 2 2 2 2 2 2 2

a b

fi n n a b a b a b a b

!j j j \Gamma \Gamma \Gamma \Gamma

13 DECAYS 13.1 Standard Model Higgs Decays Rates Standard Model Higgs is expected to have a mass between 45 Gev and 1 TeV, and, since it couples directly to the masses of other particles, to decay into heaviest available particles. The SM Higgs decay rates, calculated without radiative corrections are as follows (see [33] and references therein):

_ \Gamma =

4 2

where = 1 4 and = 1(3) for = lepton (quark).

( ) \Gamma = 8 2 1 (3 4 + 4) where = 4 , is ( )-boson mass.

Higgs decay into two photons or two gluons proceeds via loops. Its decay rates are equal [34]:

\Gamma = 8 2 where = + + + , and (1).

\Gamma = 4 2 9 where 3 is the sum of contributions from quarks = 1 2 given by [35]:

= 1 41 = 2 + (4 1) ( )

where , and

( ) = 2[arcsin( 12 )] 14

( ) = 12 ln 1 + 1 41 1 4 2 + ln 1 + 1 41 1 4 14

86

H f f ; N G mss m fi ; fi m =m N f

H W W ZZ ; G M mss xx x x ; x M =m M W Z

H flfl; ff Gss m I ; I I I I : : : I O

H gg; ff Gss m N ; N N N j ; ; : : : ;

N dx dy xyxy i" * * * G * ; * m =m G z z ; z ;

G z zz ss iss zz ; z :

q

P Z Z

" # " #

2

3

2 2 + 2 2 2 2 2

2

3

3 2

2

3

3 2

1 0

1 0

2 2

2

2 2

c F f

H

f H c

F H

H

F

H

lepton hadron W

s F H

j j j

j

x

m

m

j j j j

j j H

! p \Gamma ! p p \Gamma \Gamma

! p j j

j j ss ! p j j j

\Gamma \Gamma \Gamma \Gamma

j

\Gamma p *p

\Gamma \Gamma p \Gamma \Gamma p \Gamma \Gamma p \Gamma ^

\Gamma

\Sigma

\Gamma

2

2Hj

13.2 and Decays

vanishes for and 1 3 for .

_ \Gamma = 307 ( ) = _ (except _)

\Gamma = 3512 ( ) = _ \Gamma = 2048 cos

(7 403 sin + 1609 sin ) ( ) = ( ) = 3(1 8 + 20 )4 1 arccos( 3 12 )

(1 )( 272 132 + 1 ) 3(1 6 + 4 ) ln

The partial decay widthes for gauge bosons to decay into massless fermions_

are equal to [4, 18]:

\Gamma = 6 2 227 1 MeV _ \Gamma =

6 2 (707 3) MeV

_ \Gamma =

6 2 [ + ]

=

167 1 0 3 MeV ( _) 83 9 0 2 MeV ( ) 298 0 0 6 MeV ( _) 384 5 0 8 MeV ( _)

375 2 0 4 MeV ( _)

For lepton = 1, while for quarks = 3(1 + + 1 409 12 77 ), where 3 is due to color and the factor in parentheses is a QCD correction.

87

N m m N = m m

H W f f ; g mss F ffl ; ffl mm ; H W f f ; W tb

g m

ss F ffl ; ffl

m

m ;

H Z f f ; g mss #

# # F ffl ; ffl mm ; F z z zz z z

z z z z z z:

f f

W e * ; G Mss ;

W u d ; C G Mss V V ;

Z ; C G Mss g g

: : ** ; : : e e ; : : uu ; : : dd ;

: : bb ;

C C : :

X

X

8?!? :

4

3

+

4

3

4

3 4

2 4

2 4

2

2

3

2 2

2

2 4

1 2

+ + 3

+ 3 2 2

3 2 2

+

( )

q q H q q H

H W

H

H W

H H

W

W W

Z

H

e

F W

i i

F W

ij ij

i i

F Z

iV iA

ff M

ss

ff

ss

ff

ss

o/ ! AE ! ! !

!

\Theta \Gamma

\Gamma p \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma

! p ss \Sigma

! p j j ss \Sigma j j ! p ss

\Sigma \Sigma \Sigma \Sigma

\Sigma

\Gamma

\Sigma 0 \Sigma 0

0 0

\Gamma 2 2

3 3 s V s s

W Z

13.3

13.4

Muon Decay Charged Meson Decay In the SM the total muon decay width is equal (up to 100% accuracy) to the width of the decay

_

The matrix element squared for this decay equals [4]:

= 128 ( ) ( ) Then the total muon width is given by [36]:

\Gamma = 192 ( )(1 + 35 )[1 + ( )2 ( 254 )] (13.1) where ( ) = 1 8 + 8 12 ln , and

( ) = 23 ln( ) + 16 136 For pure coupling (and neglecting of the electron mass) in the rest frame of the polarized muon ( ) the differential decay rate is:

\Gamma ( ) = 192 [3 2 (1 2 ) cos ] (cos ) where is the angle between the electron momentum and the muon spin, and = 2 .

The decay constant for pseudoscalar meson is defined by [18]

0 (0) ( ) = The state vector is normalized by ( ) ( ) = (2 ) 2 ( ). The annihilation rate of the ( _ ) _ decay is given by

\Gamma ( _ ) = 8 (1 ) (13.2) where = 1 for decay and = (3 ) for _ one, and

is the heaviest final fermion mass.

88

_ e * * : M G p p p p : G m

ss F

m m

m M

ff m

ss ss ;

F x x x x x x

ff m ff ss mm ss : V A

_

d _ G mss x x # x dxd # ; # x E =m

f P

! A P k ? if k :

! P k P k ? ss E ffi ~k ~k P q q f f

P f f C G Vss f m M mM ; C P l* C V P q q m

2 2 2 5

3

2 2

2 2

2

3 4 2 1 1

2 5

3

2

3 3 1 2

2 2

2 2 2

2

2 3 4

e _ f _ * e * tot_ F _ e

_

_ W

_

_

_

e

F _ _ _

P

_ P _

q

F q q

P f P

f

P

l q q f

! j j

\Gamma \Gamma \Gamma \Gamma

\Gamma ss \Gamma

\Gamma \Sigma \Gamma

j j

j \Gamma !

! j j \Gamma

! j j !

\Gamma \Gamma \Gamma \Gamma

\Upsilon \Upsilon

0 0 0 0

0

0 1 2

3 4

e _ 0

0

13.5 Quark Decay In the region the total quark width is given by [4]:

\Gamma ( _ ) = 64 (13.3) For the case of + the width of the heavy quark decay

+ equals [37]:

\Gamma ( + ) = 8 2 2 ( ) (13.4)

180 MeV where

( ) = (1 ) + (1 + ) 2

= 12 [ ( + ) ][ ( ) ]

here is (or ) momentum in the -quark rest frame. The width of the heavy decay

+ ( ) is given by [37]:

\Gamma ( + ( ) = 192 ( ; ; \Gamma ) where

( ) = 2 ( ) 1 + + 2( + + )[(1 ) + ] ( 0 ) = 2 [ 3(1 )] + 2 (1 ) [3(2 ) (2 + )(1 ) ]

= ln + 1+ (1 ) = 1 [arctan( 1 ) arctan( 1 )]

89

m M

Q q q q G mss V V : m m m Q W q

Q W q G mss V km f mm ; Mm

V mm ;

f x; y x x y y ;

k m m m m m m m ;

k W q Q

Q

Q q W l*

Q q W l* G mss V F mm mm m ;

F a; b; c dt f b; t b t b bt tat c ; F a; ; c

a c a A a a a a c a a B; A cc a ; B a c c ac :

` ' q

Z q

2 3 4

2 5

3

2 2

3 2 2

2

2 2

2 3

2 2

2 2 2 2

2 5

3

2 2

2 2

2

2

(1 ) 0

2 2

2

4

2

2

q W

F Q Qq q q

Q W q

F Q

Qq

Q

Q q

Q

W

Q

Qq

Q

W

Q

Q Q

W q Q W q

F Q Qq Q

W

q Q

W

W

b Q

o/

! j j j j AE !

! p j j

ss j j

\Gamma \Gamma

\Gamma \Gamma \Gamma

! ! ! ! j j

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma p p \Gamma \Gamma p

0 \Gamma p

2 3 4 2

0 13.6 Heavy Quarkonia Decays Suppose that the matrix element of the vector state decay is given by

= _( ) ( )

Then

\Gamma ( ) = 12 = 12 \Gamma ( ) Denote 4 (0) , where (0) is bound state wave function in the origin. The width of the decay of the quark antiquark vector state 1 equals:

\Gamma (1 ) = 43 where = 1(3) for colorless (color) quarks, is the effective charge:

= ( _ _) = ( + ) = = ( _ + _) = ( ) = = _ = = _ = \Upsilon = _ =

For positron annihilation (with = 1) one has:

\Gamma (0 ) = 4 \Gamma (1 ) = 169 ( 9) For quarkonia annihilation one gets:

\Gamma (0 ) = 12 For the two (three) gluon annihilation one need to change : 2 9 ( 5 18):

\Gamma (0 ) = 83 \Gamma (1 ) = 4081 ( 9)

90

V V l l M g e u l fl u l :

V l l g ss M ; g ss V l lM : R ss

l l N ff QM R : N Q

ae uu dd Q ; ! uu dd Q ;

OE ss Q ; J= cc Q ;

bb Q :

Q

flfl ffM R ;

flflfl ss ss ffM R :

flfl ff QM R :

ff Q ff = ff ff =

gg ffM R ;

ggg ss ss ffM R :

s

+ +

+ 2 + 20 2

+

2 2

2

20

1

2

2 1

2

2

3

1 3

2 1

21

2

2 1

2

2

3

1 3

2 1

182 1

92 4 92 1 9

2 2

20

2 3

2

20

2 4

2

20

2 4 2 3 3

2

2

20

2 3

2

20

V *V * V V V

V

c q c q

q q q q q

e

q

q s s

s

s

! \Gamma

! ! j j j

! \Gamma ) j j) j \Gamma j

)) )

!

! \Gamma

!

!!

!

! \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma

p p

p p

\Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma \Gamma

QQ( _) 14 CROSS SECTIONS 14.1 Annihilation For pointlike spin- fermions the differential cross section in the cms for_

via single photon and -boson (with mass and total width \Gamma ) is given by [18]:

\Omega = 4 1 + cos + (1 ) sin (14.1)

+ 4 (1 + )[1 + cos + (1 ) sin ] (14.2)

+ (1 + )[1 + cos ] 8 cos

4 2 [1 + cos + (1 ) sin ] (14.3)

2 cos

where = 1 4 is the velocity of the final state fermion in the center of mass, is the charge of the fermion in units of the proton charge,

= 116 sin cos ( )( ) + \Gamma = 1256 sin cos ( ) + \Gamma = 1 + 4 sin = 2 4 sin = 2 here the subscript refers to the particular fermion and

= + 12 for = 12 for The first (14.1), second (14.2), and third (14.3) terms correspond to the_

process via single photon annihilation, via -boson exchange, and photon - -boson interference, respectively. For (i.e. 1) the annihilation via single photon exchange (14.1)

91

e e f f Z M

doe d

ff

s fiQ # fi # ff

s fiO/ V V # fi #

fi a V # fiV V a #

ff

s fi Q O/ V V # fi #

a fi # ;

fi m =s

Q

O/ # # s s Ms M M ; O/ # # ss M M ;

V # ; V T Q # ; a T ;

f

T *; u; c; t; T l ; d; s; b:

e e f f Z

Z s m fi

n on

on o q

1

2+

2 2 2 2 2 2

2 2 2 2 2 2

2 2 2 2

2

1 2 2 2

2

1 2 2

2

2 2 2 2

2 4 4

2

2 2 2 2 2 3 2 3

3 3

+

2

Z Z

f

f f f f

f f f

f f

W W

Z Z Z Z

W W Z Z Z

W f f f W f f

f

!

\Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma !

AE !

\Gamma

\Gamma \Gamma

e e+

\Gamma

14.2 Two-photon Process at Collisions

tends to:

= 4 3 86 3(GeV ) nb (14.4)

When an and collide with energies and , they emit and

virtual photons with energies and and 4-momenta and . In the equivalent photon approximation (EPA) [38], the cross section for the reaction

(14.5) is related to the cross section for by:

( ) ( ) = ( ) (14.6) where = 4 , = 4 and

= 1 + 2 ( ) After integration (including that over in the region ( )

( ) ), the cross section is

( ) = ( ) ln ( ) 1 ln3 ( ) (14.7) where = , and

( ) = (1 + 2 ) ln 1 12 (1 )(3 + ) The value ( ) depends on properties of the produced system . For example, ( ) for hadron production ( = ), and ( )

for the lepton pair production ( = ). For the production of a resonance of mass and spin = 1 one has:

( ) = (2 + 1) 8 \Gamma ( ) (14.8)

( )(ln 1) 13 (ln )

92

oe ssffs Q : Qs : e e E E dn dn ! ! q q

e e e e X

flfl X doe s doe s dn dn doe W ; s E E W ! !

dn ffss !E !E m !q E d!! dqq :

q m ! =E E ! q q

oe s ffss dzz f z qm z z oe zs ;

z W =s

f z z z z z : q X q m X h q M X l l

M J

oe s J ff R flflM

f Ms sMm M sM ;

h i Z " ! #

" #

2 2 2

2

+ 1 2 1 2 1 2 1 2

+ +

1 2 2 1 2 2 1 2

2

2

2 2

2 2

2 2

2 2 2 2 2

2 2

1 2

2

2 3

2

2 2 2 2 2 2 +

2

3 2 20

2 2

2

2

3

f

f

EP A e e e e X flfl X

i

i

i

i

i

e i

i i

i i

i i

i e i i i i i max

EP A z

max

e

flfl X

max max ae max ll

R

EP A

R

R

e R R

ss !

! j

\Gamma \Gamma \Gamma

^\Gamma ^ \Gamma

\Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma \Gamma , \Gamma ,

6!

\Theta \Gamma \Gamma

\Gamma

\Gamma \Gamma ! !

! \Gamma

+ + th

\Gamma \Gamma

e e+

\Gamma 14.3 Reactions

where is the mass that enters into the from factor of the tran- sition: for = and for = _ or _ resonances.

The reaction of the lepton hadron deep inelastic scattering (DIS)

( ) ( ) ( ) (14.9) is described by the following invariant kinematic variables (the 4-momenta and masses of the particles are denoted in the parentheses) [18]:

= is four-momentum transferred by exchanged particle ( , , or

) to the target,

= = is the lepton's energy loss in the lab frame, and are

the initial and final lepton energies in the lab,

= = 2( ) if sin ( 2) then

4 sin ( 2), where is the lepton's scattering angle in the lab,

= = in the parton model, is the fraction of the target hadron's

momentum carried by the struck quark,

= = is the fraction of the lepton's energy lost in the lab,

= ( + ) = + 2 is the mass squared of the system recoil-

ing against the lepton,

= ( + ) = + The differential cross section of the reaction (14.9) as a function of the dif- ferent variables is given by

= ( ) = 2 \Omega = ( )

93

M flfl R

M m R ss ; ae ; !; OE; : : : M M R cc bb

l k; m h P; M l k ; m X; q k k fl Z

W

* E E E E

Q q EE ~k ~k m m ; EE #= m ; m ;

Q EE #= #

x ; x y ; W P q M M * Q ;

s P k M :

d oe dxdy * s M

d oe d*dQ

ssM *

E

d oe d dE x s M

d oe dxdQ :

0

0 0 0 0

2 2 2 2 2 2 2

2 2

2 2

2 2 2 2

2 2

2 2 2

2

2 2 2

2

ae R

l l q P

M

l l l l

Q

M*

Q

q P

q P k P

* E

Q

xy

lab

!, , ! \Gamma

\Gamma \Gamma \Gamma \Delta \Gamma \Gamma AEss

\Gamma \Gamma \Gamma

0 0 0 \Sigma

\Delta 0 0

0 0 0 0

\Delta \Delta

\Delta

0 0

2 2

2

0 0 0

l h

14.4 Cross Sections in the Parton Model

Parity conserving neutral current process, , can be written in terms of two structure functions ( ) and ( ):

= 4 ( ) (14.10)

(1 ) + ( ) Parity violating charged current processes, and , can be written in terms of three structure functions ( ), ( ), and

( ):

= ( )2 ( + ) (14.11)

[(1 ( ) ] + ( 2 ) where the last term is positive for and reactions and negative for and

_ reaction.

In the framework the reaction

(14.12) where is a particle (or group of the particles) with large mass (invariant mass) or with high can be considered as a result of the hard interaction of the one -parton from hadron with -parton from hadron. Then the cross section of the reaction (14.12) can be written as follows:

( ) = ( ) ( )^( ) (14.13)

where sum is performed over all partons, participating in the subprocess

; ( ) is in -hadron; is a typical momentum transfer in partonic process and ^ is partonic cross section.

94

parton model

parton distribution

l h l X F x; Q F x; Q

d oe dxdy

ssff s M

Q

y F y xF Ms M xyF :

lh *X *h lX

F x; Q F x; Q F x; Q

d oe dxdy

G s M

ss

M Q M

y M xys M F y xF y y xF ;

l * l *

h h C X; C

p i h j h

oe h h CX f x ; Q f x ; Q oe ij C dx dx ;

ij C f x; Q h Q

ij C oe

h i n o

X Z

1 2 2 2 2 2 2

4

2 2 1

2

2 2

1 2 2 2 3 2

2 2 2 4

2 2 2 2

2 2

2 1 2 3

+

1 2 1 2 1 2 1 2 2 2 1 2

2

NC NC NC NC NC

CC CC CC

F W

W

CC CC CC

ij

h i

h j

hi

! \Gamma \Theta \Gamma \Gamma \Gamma

! !

\Gamma \Theta \Gamma \Gamma \Gamma \Sigma \Gamma

! ! !

! !

\Sigma \Sigma \Gamma ?

1 2

14.5 Vector Boson Polarization Vectors Let us consider a vector boson with mass and 4-momentum ( = ). Three polarization vectors of this boson can expressed in terms of ,

= ( ) = + as folows [39]:

( = 1) = (0 ) ( = 2) = (0 0) ( = 3) = ( )

(14.14)

It is easy to verify that

( ) = 0 ( ) ( ) = (14.15) For = 0 (i.e. = ( 0 0 )) these polarization vectors can be chosen as follows:

( = 1) = (0 0 1 0) ( = 2) = (0 1 0 0) ( = 3) = ( 0 0 )

(14.16)

Massless vector boson has only two polarization states, = 1 and 2, on its mass-shell. In the axial gauge for the polarization vectors of the gluon there appears an additional condition (see Subsection 10.3):

( ) = 0 where is axial gauge fixing vector. For this case polarization vectors ( = 1 2) can be chosen as follows:

( ) = ( ) ( ) (14.17) where ( ) are given in (14.14) or (14.16).

95

m p p m

p

p E; p ; p ; p ; p p p

" p; * ; p p ; p p ; p ; " p; * ; p ; p ; ; " p; * ; p ; p ; p :

p " p; * ; " p; * " p; * ffi : p p E; ; ; p

" p; * ; ; ; ; " p; * ; ; ; ; " p; * p; ; ; E :

*

ffl

ffl p; * n ; n

ffl p; * ;

ffl p; * " p; * " p; * np n p ; " p; *

q

9??= ??;

9?=? ;

2 2 2 2

1 2 1

1

_

_

_ x y z x y

_

~p p x z y z_

p y x _ E

m ~p

~p

E x y z

_ _ _ _ **

_

_ _ _

m

_g _g _

_g _g _ _ _

\Gamma \Gamma

\Gamma

\Gamma \Delta \Delta

? j j ?

j j

0 ?

2 ? ?

0

For the system of two photons (gluons) with momenta and the polarization vectors can be written in the explicitly covariant form:

( ) = 12\Delta ( ) ( ) ( ) (14.18) where sign +( ) corresponds to positive (negative) helicity, is any arbitrary vector, which is independent on and (it may be a momentum of some particle), and

\Delta = = ( )(2 ( )( ) ( )) These vectors were considered also in Subsection 6.6. Projectors on various combinations of the helicity states look as follows:

1 2 (+) ( ) + ( ) (+) =

1 2( ) ( ( ) ) 1

2 (+) ( ) ( ) (+) = 2 ( )

1

2 (+) (+) + ( ) ( ) =

1 2\Delta 2[( )( )( ) + ( ) ( )(( ) + ( ) )]

+ ( )( ( ) )

1

2 (+) (+) ( ) ( ) =

2\Delta (( ) ( ) ) + (( ) ( ) ) = ( )2\Delta + ( ) + ( )

96

p p "

" p p q qp p qp p i" q p p :

q p p

ffi p p qp qp q p p :

" " " " p p p p p p g ; " " " " ip p " ; " " " " p p qp qp g q q p p p p qp p q qp p q q p p p p p p g ;

" " " " u i p p q qp p " p p q qp p " ;

i p p q " q " qp " qp " :

\Theta \Lambda \Gamma \Delta \Gamma \Delta \Gamma \Delta

\Gamma \Delta

\Gamma \Delta

1 2 1(2)

3

1 2 2 1 1 2 1 2

1 2 3 1 2 1 2 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 1 2 1 2

1 2 2 1 2 1 2 2 1 2 1 2 1 2 1 2

1 2 1 2

3 1 2 1 2 1 2 2 1

1 2

3 1 2

_ _ i _

_ _ _*fffi * ff fi

qp pqp p _ * _ * * _ _* _ * _ * p p _* _ * _ * _* _ * _ * * _

* _ _*

_ * _ *

_ _ *qp p * * _qp p _ *qp p * _qp p p q_* p q_*

ffl

\Sigma p \Gamma \Gamma \Sigma

\Gamma

\Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma f \Gamma \Gamma g

\Gamma \Gamma \Gamma f \Gamma \Gamma g

1 2 1 2

1 2

1 2 1 2 1 2 1 2 2 1

Two Photons (Gluons) System 15 MATRIX ELEMENTS 15.1 General Remarks In this Section we present the matrix elements squared for various processes in the Standard Model. Almost all of these were presented in the book by R. Gastmans and Tai Tsun Wu [14]. The symbol is used to denote the square of the absolute value of the matrix element summed over the and degrees of freedom (polarization and color), but

averaging over the state degrees of freedom. So, one can use the well-known crossing relations to obtain for pro- cesses differing from each other by repositioning the final and/or initial par- ticles. The averaged over the initial state degrees of freedom matrix element squared can be obtained from by trivial procedure:

: 12 12 (spin) = 14 _ __ : 12 12 (spin) 13 13 (color) = 136

_ : 12 12 (spin) 18 13 (color) = 196

: 12 12 (spin) 18 18 (color) = 1256

For the 2 2 processes the differential cross section is related to the as follows:

(2 2) =

16 (15.1)

where and are defined in (12.20) and (12.27).

The notations, used through of this Section, are the same as in Section 10:

is the electric charge of the positron, = ,

is the charge of the quark in units of the positron charge, is the QCD coupling constant, = ,

is the Fermi constant.

97

M M

M

M

M M M e e ; e fl; flfl M M ;

qq; qq; qq M M ;

gq; gq M M ;

gg M M :

M

doe

dt

M

ssI ; I s ;

t I

e ff ff Q g ff ff G

2 2

2

2 2 2 + 2 2

2 2 2 2 2 2

2

2

2

2 2

4

1 137

4 QED ess f s QCD s gss

F

j jj j

j j

j j j j j j

\Delta ) j j j j \Delta \Delta ) j j j j \Delta \Delta ) j j j j \Delta \Delta ) j j j j

! j j

! j j ss

j ss j

\Gamma \Sigma

2 2 s

initial final without initial

15.2 Matrix Elements

As in Section 12 for the reaction 2 2

( ) + ( ) 1( ) + 2( )

+ = +

the Lorentz-invariant Mandelstam variables for reaction are given by

= ( + ) = ( + ) = ( ) = ( )

= ( ) = ( ) + + = + + +

_ ( = , = ).

= 8 1 + + ( + )(2 ) (15.2) = 8 + for = = 0 _

= 3

The detailed description of the process _ with -boson exchange is presented in Subsection 14.1.

= 8 1 + + 8 ( ) (15.3)

+ 2 ( 2 )( 6 ) = 8 + + for = 0

98

(no -boson exchange)

(no -boson exchange)

a p b p q q

p p q q

s p p q q ; t p q p q ; u p q p q ; s t u m m m m :

e e f f Z e e l l l e l _; o/

M e s t u m m s m m ;

e t us ; m m : e e qq

M Q M :

e e f f Z

e e e e Z

M e s t u m s m

st u m u m ; e s t us t ; m :

\Theta \Lambda

n \Theta \Lambda o

1 2 1 2

1 2 1 2

1 2 2 1 2 2 1 1 2 2 2 2

1 2 2 2 1 2

2 2 21 22

+ + +

2 4

2

2 2 2 2 2 2

4 2 2

2

+

2 2 2

+

+ +

2 4

2

2 2 2 2

2 2 4 4 4 4

2 2

a b e e f e f

e f

q f e

!

!

\Gamma \Gamma \Gamma \Gamma

! ffl ! 6

j j \Gamma \Gamma

ffl ! j j j j

!

!

j j \Gamma

\Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma

\Gamma

\Gamma \Gamma

15.2.1 15.2.2

( ) + ( ) ( ) + ( ) + ( ) = 0

= 8

( )( ) ( ) + ( )

( )( ) = = 0 For the case of = ( + ) 4 , i.e. in the limit

= = ( 0) the is given by [2]:

= 64 + + (15.4) where is -photon energy in cms.

( ) + ( ) ( ) + ( ) + ( ) = = 0 Invariants:

= 2( ) = 2( ) = 2( ) (15.5) = 2( ) = 2( ) = 2( )

= , for example,

= 4 ( ) + + + (15.6) = , i.e.

= 4 ( ) ( + ) + ( + ) + ( + ) (15.7) where in (15.6) and (15.7) we use:

( ) ( ) ( ) ( ) (15.8)

99

e e flflfl e p e p fl k fl k fl k ; m :

M e

p k p k p k p k

p k p k

:

e e flflfl; m m :

s p p m

p p m; ; M

M e m !! ! m !! ! m !! ! ; ! i

e e l l fl

e p e p l q l q fl k ; m m :

s p p ; t p q ; u p q ; s q q ; t p q ; u p q :

l e e e _ _ fl

M e v v t t u uss : l e e e e e fl

M e v v ss s s tt t t uu u uss tt :

v pp k pp k ; v qq k qq k :

P \Theta \Lambda Q ^i j i j i j *

+ + 1 2 1 2 3

2 6

3 =1 1 2 1

2 2 2

3 =1 1 2 +

2 2

2

2 6 1

2 3

2 2

1 3

2 3

1 2

2

+ +

+ 1 2 + 1 2

1 2 1 1 1 2 1 2 2 2 2 1 + +

2 6 2 2 2 2 2 + + 2 6 2 2 2 2 2 2 2

1 1

2 2

1 1

2 2

e i i i i i

i i i e

e e

e e

i

e l

p q p q _p

_ _

_q

_ _

! ffl !

j j ffl ! 6 !

j j

j j \Gamma \Gamma \Gamma

!

!

\Gamma \Gamma \Gamma \Gamma ffl 6 !

j j \Gamma \Gamma ffl !

j j \Gamma \Gamma

j \Gamma j \Gamma

\Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma

0 0 0

\Gamma \Gamma

0 0

0 \Gamma \Gamma

0 0 0 0 0 0

0 0

+

+ \Gamma

\Gamma

15.2.3

15.2.4

_ For this reaction the invariants are the same as in (15.5).

= 16 + + +( )( ) (15.9)

_ For this reaction the invariants are the same as in (15.5).

= 12 ( + ) + + + (15.10) where and are defined in (15.8).

_ = = 0 The final _-pair can be in color or color final states.

= 16 13 = 16 73 6 = + = 16 83 6 (15.11) where

= + + 4 + (15.12)

4 + = ( )( ) (15.13) For = 0,

= + =

100

,

singlet octet

e e qqg

M e Q g t t u us q k q k : e e qqfl

M e v Q v t t u uss ; v v

gg qq m m

qq

M g O/ ; M g O/ O/ ;

M M M g O/ O/ ;

O/ m tm u m um t mm t mm u

m m t

m m u ;

O/ m t m us :

m

O/ t uut ; O/ uts :

^ * ^ *^ *

\Gamma \Delta \Gamma \Delta

+

2 4 2 2 2 2 2 2

1 2

+

2 6 2 2 2 2 2

2 4 0 2 4 0 1

2 2 2 4 0 1

0

2

2

2

2 2

2 2

2

2 2

2

1

2 2

2

0

2 2

1 2

f s p f q p q

q

singl s oct s

tot singl oct s

q

!

j j !

j j \Gamma ! 6 j j j j \Gamma

j j j j j j \Gamma

\Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma

\Gamma \Gamma

\Gamma

0 0 \Gamma

0 0

0

15.2.5 15.2.6

15.2.7

( ) _ _

= 32

_

= 24

= 8 _ _ = 0 = = 0

= 16 + + 2 (2 ) (15.14)

= 0

= 16 + + 83 (15.15) _ _ = 0

= 16 1 [ + + 83 ] (15.16)

= 288 ( + + )( + + ) (15.17) _ _

( ) + _ ( ) ( ) + _ ( ) = 0

= 128 ( )( )( ) + \Gamma (15.18) where = 1 for _ _ , = 3 for _ _ ( _ _), and = 9 for _ _ , and \Gamma are the mass and total width of the -boson.

101

, , , ,

flg flfl f f gfl qq

M g e Q O/ :

flfl qq

M e Q O/ :

flfl e e

M e O/ :

qq QQ m m m

M g t u m s ms : qq qq m

M g s t ut u stu : qq qq m

M g s t s t u stu : gg gg

M g s t u s t us t u : f f W f f

f p f p f p f p ; m :

M C G M p p p ps M M ; C l * l * C l * qq qq l * C q q q q M W

" #

2 2 2 2 0

2 4 4 0 +

2 4 0

2 4 2 2 2 2

2

2 4 4 4 4

2 2

2

2 4

2 2

4 4 4 2

2 4 4 4 4 2 2 2

2 2 2

1 2 3 4

1 1 2 2 3 3 4 4 1 2 3 4

2 2 4 1 4 2 3

2 2 2 2

1 2 3 4

s f

f

q Q

s q

s q

s

s

; ; ; F W

W W W

W W

! ffl ! j j

ffl ! j j ffl ! j j

! 6

j j \Gamma !

j j \Gamma !

j j \Gamma !

j j ! !

! 6 j j \Gamma

! ! !!

\Gamma

\Lambda \Gamma 0\Gamma 0 \Gamma 0 0 \Gamma

15.2.8 15.2.9 15.2.10 15.2.11 15.2.12 15.2.13

_ _

( ) + _( ) ( ) + _( ) + ( ) = 0

= 256 ( ) + \Gamma (15.19) = 1( )( ) ( ) + ( )

( + ) 2 (( ) + ( )) ( )( ) ( )( ) = 2( ) ( ) ( ) + ( )

= 2( ) ( ) ( ) + ( )

102

l * dug

l p * p d p u p g k ; m :

M G M g A A As M M ;

A kp kp s p p p p

m m s p p p p p p p p p p p p ; A mkp p p p k p p ;

A mkp p p p k p p :

n \Theta \Lambda \Theta \Lambda o\Theta \Lambda

\Theta \Lambda

1 2 3 4 2 2 4 2 1 2 3

2 2 2 2

1 3 4 1 4 2 2 3 2

2 2 1 4 2 3 1 3 2 3 1 4 2 4

2

2

4 2 2 3 1 1 4 3

2

3 2 1 4 2 2 3

d;u F W s

W W W

u d

u

d

!

! 6

j j \Gamma \Gamma \Gamma

\Gamma \Gamma \Gamma

\Gamma

\Gamma 15.2.14 16 MISCELLANEA 16.1

16.2

Miscellanea Properties of Operators Let us consider the recursion = + for given and . Then

= +

= 2 [1 1 + 4 ] = =

Various representations of the -function:

( ) 1(2 ) ( ) = ( + 1) ; ( ) = ( + ) 0 ( ) = ( ) = sin( )( )

1 = 1 ( )

-functions \Theta ( ) and ( )

\Theta ( ) 1(2 ) = 1 00 0

( ) 1( ) = 1 01 0

1 ( ) = \Gamma ( ) 0

( ) = ln ++

The various properties of the operators can be found, for example, in [5, 40, 41]. Let ( ) be any function from the operator (matrix) , which can

103

Dirac Step

A aA bA A A A ffz fiz ; z a b=a ; ff A z Az z ; fi z A Az z :

ffi ffi x ss e dt; ffi x; ff ffss ff x ; ff ffi x; fi fiss x fi ; fi ; ffi x; ff ffss e ; ff ; ffi x; ff ffss ffxffx ; ff ;

x i" x issffi x :

x " x

x ssi et i" dt ; x ?; x ! " x iss et dt ; x ?; x !

a i"

i

k e ff dff; k ;

e e dtt b i"a i" :

f A A

q Z

Z ( Z (

Z Z ` '

1 2 0 1 1 2 1 2 2 1 2 01 2 1 0 11 2

2 2 2 2

0

( + ) 1

0

n n n n n n

;

ixt

ff x

ixt ixt

k

k iff a i" k

ita itb

ffl

\Sigma \Gamma \Gamma \Gamma \Gamma ffl

j

! 1 ! p ! 1 ! 1

\Sigma P \Upsilon ffl

j \Gamma j P \Gamma ffl

\Gamma *

\Gamma

\Gamma \Gamma 1 \Gamma 1

\Gamma

1 \Gamma 1

1

\Gamma 1

1

\Gamma \Gamma

1

2 2 16.3 The Baker-Campbell-Hausdorff Formula

expanded into series with respect to operators (matrices) :

( ) = Let be a parameter, then:

= 1 = = ( ) = ( )

Let and be noncommuting operators, and be parameters ( inte- ger). Then:

= ( ) ( ) = ( )

= + [ ] + 2! [ [ ]] + 3! [ [ [ ]]] +

Let be an operator and there exists the inverse operator . Then for any integer :

= ( ) ( ) = ( ) Let ( ) be an operator, depending on the scalar variable , then

( ) = ( ) ( ) ( )

= ( )

Let and be non-commuting operators, then :

= (16.1) where

= + ; (16.2)

104

A f A c A : ,

e e ; e Ae A; e A e A ; e f A e f A :

A B , n n

e B e e Be ; e F B e F e Be ;

e Be B , A; B , A; A; B , A; A; A; B

A A

n

AB A ABA ; Af B A f ABA : A x x

dA x

dx A x

dA x

dx A x ; de

dx e

dA t

dt e dt:

A B

e e e ;

Z A B

X Z

P

=0 2 3

1 1 1 1 1

1 1 1 ( ) 1

0

(1 ) ( ) ( )

1

n n

n n

,A ,A ,A ,A ,A n ,A n ,A ,A

,A n ,A ,A ,A n ,A ,A ,A ,A

,A ,A

n n

A x t A t tA t

A B Z

ffl ffl

\Delta \Delta \Delta ffl

ffl

\Gamma

1 \Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma

\Gamma

\Gamma \Gamma \Gamma \Gamma

\Gamma

\Gamma \Gamma

\Gamma

=1i i 1

= 12 [ ]; (16.3) = 112 [ ] + 112 [ ] ; (16.4) = 148 [ ] + 148 [ ] ; (16.5) = 1120 [ ] + 1120 [ ] (16.6)

1 360 [ ]

1 360 [ ] 1

720 [ ]

1 720 [ ]

The other terms can be evaluated from the relation (see [5, 41]):

1 ( + 1)! [[ ]] = +

[[ ]]

! (16.7)

where = ; = ; = . The repeated commutator bracket is defined as follows

[[ ]] = [[ ]] = [[ ]] Since relation (16.7) must be satisfied identically in , one can equate the coefficients of on the two sides of this relation. In particular, = 0 1 2 3 4 gives (16.2, 16.3, 16.4, 16.5, 16.6), respectively.

105

Z A; B Z A; A; B A; B ; B Z A; A; B ; B A; A; B ; B Z A; A; B ; B ; B A; A; A; B ; B

A; A; B ; B ; B A; A; A; B ; B A; A; A; A; B A; B ; B ; B ; B ; : : :

k Z ; Z A ,

A ; B

j ;

e e e Z , Z Z n, Z

A ; B B; A ; B A; A ; B :

, , j ; ; ; ;

\Theta \Lambda \Theta \Lambda h \Theta \Lambda i h\Theta \Lambda i

^h \Theta \Lambda i * ^ h\Theta \Lambda i*^

h\Theta \Lambda i* ^h \Theta \Lambda i *^ h

\Theta \Lambda i* ^h\Theta \Lambda i *

X X

P P

^ *

2 3 4 5

=0 =0

=1 =1 1

0 +1

k

k

j

j j

Z ,A ,B n n n n n n

n n j

\Gamma \Gamma \Gamma \Gamma

1

0

1

1 0 1 \Gamma

REFERENCES

[1] Bogolyubov N.N. and Shirkov D.V.,

(Chichester: Wiley-Interscience, 3rd edn, 1980).

[2] Berestetskii V.B., Lifshits E.M., and Pitaevskii L.P.,

(Moscow: Nauka, 1980) (in Russian).

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