Quarks Leptons And All That

In this Appendix we summarize the current model of the fundamental forces and particles of nature and compute some color factors needed in Chapter 17.

D.l FUNDAMENTAL PARTICLES AND FORCES

There are three kinds of fundamental particles. These are the fermions, which have spin i, the gauge bosons, which have spin 1 and are the carriers of the fundamental forces, and a scalar particle called the Higgs. As of the spring of 1993 (the time this book was finished), the fundamental forces were believed to be three in number: the strong forces mediated by gluons, the electroweak forces mediated by the photon and the three intermediate vector bosons {W±, and Z), .nd gravity, which is not discussed at all in this book. (There are expectations that the strong and electroweak forces, and possibly gravity, are really a single unified force.)

The fundamental particles are summarized in Table D.l. The only particles in this table which have not been observed to date are the Higgs and the top quark t. Discovery of these two particles is one of the missions of the Superconducting Super Collider (SSC). The Higgs is an essential part of the verification of the standard electroweak model, which predicts its existence (see Sec. 15.4), and the top quark is expected on the basis of recent measurements which suggest that there are only three generations (denoted by G in the table) of particles.

The fermions can be classified according to which forces they experience. The leptons interact only through the electroweak (EW) forces described by the Standard Model, while the quarks also experience the strong forces described by QCD. The leptons in the Standard Model consist of the three neutral (and maybe massless) neutrinos and three charged leptons which include the electron, muon, and r. All of these particles have distinct antiparticles, for a total of 2 x 6 = 12 particles. Because these particles do not see the strong forces, which are sensitive to an intrinsic property referred to as color, they can be said to be colorless. The quarks are colored; each comes in three colors corresponding to the three degrees

Table D.l The fundamental particles of nature. • Fermions: (spin 1/2, charge q)

G

quarks (QCD and EW)

leptons (EW only)

q

mass

q

mass

1

uR

UG

uB

2/3

~ 5MeV

e

-l

~ 0.5 MeV

dR

do

dB

-1/3

~ 7MeV

0

< 18 eV

o

CR

CG

CB

2/3

~ 1500

-l

~ 105 MeV

SR

SG

SB

-1/3

~ 200

"M

0

< 0.25 MeV

o

tR

tG

tB

2/3

?

r

-1

~ 1780 MeV

o

bR

bG

bB

"1/3

~ 4500

VT

0

< 35 MeV

7, W±, Z° — photon and intermediate vector bosons — EW

of freedom of the SU(3) gauge group. The quarks also experience the standard electroweak interactions, which (except for electromagnetic) are not described in this book [see Cheng and Li (1984)].

It has been found that the strong color "charge" goes to zero as the momentum flowing through the interaction vertex becomes very large (a property known as asymptotic freedom; see Chapter 17). This means that perturbation theory can be used to study high energy interactions. Conversely, at low momenta (large distances) the strong forces become very strong, and colored particles (quarks and gluons) are therefore confined to colorless clusters. All observed strongly interacting particles, or hadrons, are believed to be composites of quarks and gluons.

There are two major types of hadrons: mesons, which are bosons with integral spin and believed to be colorless composites of gluons and quark-antiquark pairs, and baryons, which are fermions with half-integral spin and are colorless composites of three quarks (one of each color, denoted by qR, qB, or qG for red,

Note that these matrices satisfy the relations tr (AaAb) = 2 6ab

The space of all traceless 3x3 Hermitian matrices, A, is an eight-dimensional linear vector space spanned by the eight matrices AQ. A scalar product can be defined by A • B = tr {A B}, in which case Eq. (D.2a) shows that the basis "vectors" A are orthogonal and normalized to 2. Since the commutator —i [aA0, is also Hermitian and traceless, it can be expanded in terms of the complete set Aa. The expansion is written iAaj — if abc ïAc

The fabc are the structure constants of the group and, from the above definition, are antisymmetric in the first two indices fabc — - fbac■ They are also antisymmetric in the last two indices (and hence are fully antisymmetric in all indices). This can be shown from the orthogonality property (D.2) and the commutation relations (D.3):

= tr {AeAaAb — AeA&Aa} = tr {A(,AeAa - AbAaAe} = 4ifeab

Yhe explicit values of fabc can be found by direct computation:

The structure constants fabc can be used to construct another representation of SU(3). Define the 8 x 8 matrices

These matrices are clearly Hermitian and traceless, and we will show that they satisfy the commutation relations

Therefore, they have the correct algebraic properties to represent the group. This representation is referred to as the regular representation, and because it is eight dimensional, it is the correct representation for color transformations of the gluons.

614 QUARKS, LEPTONS, AND ALL THAT which gives

To evaluate the second identity (D.8b), we first prove that tr {Fa [FbFc + FcFbl}

= FadeFbefFcfd. + FadeFcejFbfd FaedFtif Fcef

= - (FdbeFaef + FbaeFde}) Fcfd ~ {FebdFadf + FbadFedf) Fcef = tr {F(,FaFc} + Fbae tr {FeFc} - tr {F6FaFc} - Fbadtr{FdFc} = 0 , where the Jacobi identity was again used in the second step and in the last step we used (D.8a). Armed with this result it is an easy matter to prove (D.8b) by multiplying both sides of the commutation relation (D.5) by F^ and taking the trace:

= 2ifaeffbfgfdge •

Dividing by 2 gives the result (D.8b).

Because the Jacobi identity holds for all matrices, these results can be generalized to higher dimensional representations.

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