Appendix B
The Standard Model

(1) Three gauge couplings; a, a_s, and G_F.

(2) Two parameters representing the Higgs sector; < f > and m_f.

(3) Nine masses spread over five orders of magnitude for the six quarks and three charged leptons.

(4) Three Cabibbo angles and one phase factor for the mixing of the quark mass eigenstates.

The Glashow-Salam-Weinberg Model (GSW)

The symmetry group SU(2)_L is non-abelian due to the fact that its generator, T_i=[ 1/2]t_i, satisfies the non-Abelian algebra [T_i,T_j]=i f_ijkT_k. The non-Abelian nature of the SU(2)_L symmetry group leads the self-interaction of the gauge bosons associated with the group. The left-handed doublets, c_L^l and c_L^q, transform under the actions of the generators of the SU(2)_L symmetry group, T_i, while the right-handed singlets, f_R^l and f_R^q, transform only under the action of the hypercharge generator according to

Q = T3 +  Y 2

where Q is the electromagnetic charge, T³ is the third component of isospin, and Y is the weak hypercharge.

The Electroweak Interaction Energy

i) The invariance of the theory under the local gauge transformation:


cL ® c¢L = eia(x)·TcL and yR® y¢R = eib(x)YyR



ii) The inclusion of covariance derivatives, D_m, of the form


Dm = ¶m+igWm·T + ig¢  1 2 BmY



iii) The inclusion of kinetic terms for the vector bosons of the form


-  1 4 Wmn·Wmn -  1 4 BmnBmn



iv) The inclusion of a mechanism to generate fermion masses as well as the mass of the observed W^± and Z⁰ bosons, while keeping the photon massless. The ``Higgs Mechanism'' spontaneously breaks symmetry by introducing an SU(2) doublet of complex scalar fields with hypercharge Y = 1 of the form:


f = æ ç ç ç è f+ f° ö ÷ ÷ ÷ ø =   æ Ö  1 2   æ ç ç ç è f1+if2 f3+if4 ö ÷ ÷ ÷ ø =   æ Ö  1 2   æ ç ç ç è 0 v+h(x) ö ÷ ÷ ÷ ø

where the potential and kinetic energy contributions to the Lagrangian are of the form:


V(f) = m2fff+l(fff)2 T = (¶mf+ig  1 2 t·Wmf)f(¶mf+ig  1 2 t·Wmf)

The electroweak interaction energy is usually represented by the Lagrangian energy density, L, of fermions with the fields W, B_m as follows [91]

L = -  1 4 Wmn·Wmn-  1 4 BmnBmn

(B.2)

+ - c   L  gm æ è i ¶m-g  1 2 t ·Wm-g¢  Y 2 Bm ö ø cL

(B.3)

+ - y   R  gm æ è i ¶m-g¢  Y 2 Bm ö ø yR

(B.4)

+ ê ê æ è i ¶m-g  1 2 t·Wm-g¢  Y 2 Bm ö ø f ê ê 2   -V(f)

(B.5)

- (G1i - c   L  fyR+G2i - c   L  fcyR+ Hermite Complex Conjugate )

(B.6)

In general, the fundamental vector bosons are associated with the massless vector triplet in the expression, W_m = W_m⁽¹⁾, W_m⁽²⁾, W_m⁽³⁾ from SU(2)_L and a massless singlet B_m from U(1)_Y. As a result of spontaneous symmetry breaking, three bosons (denoted W_m⁺,W_m^-, and Z_m) acquire mass, while one (A_m, the photon) remains massless. The electroweak current incorporates the SU(2)_L and U(1)_Y currents Q = T³ + [(Y)/2]. Each of the terms in the Lagrangian energy density is the result of a specific energy density associated with the electroweak theory.

* Equation B.2 represents the kinematic energies and interactions of the intermediate vector bosons, W^±, Z⁰, and g.

* Equation B.3 represents the kinematic energies and their interactions of the intermediate vector bosons with the right-handed leptons and quarks.

* Equation B.4 represents the kinematic energies and their interactions of the intermediate vector bosons with the left-handed leptons and quarks.

* Equation B.5 gives the masses and couplings of the intermediate vector bosons and the Higgs particle.

* Equation B.6 gives the masses of leptons and quarks and their coupling to the Higgs particle where c_L, y_R are the fermion weak isospin doublets and singlets.

Electroweak Mass Terms

The mass term led by the gauge invariant Higgs-field interaction in the Lagrangian energy density of the electroweak theory gives rise to the masses of the three bosons (denoted W_m⁺, W_m^-, and Z_m), while rendering the the photon, A_m, massless.

ê ê æ è -g  t 2 ·Wm-  i 2 g¢Bm ö ø f ê ê 2   =  1 8 ê ê ê æ ç è gWm(3)+g¢Bm g(Wm(1)-iWm(2)) g(Wm(1)+iWm(2)) gWm(3)+g¢Bm ö ÷ ø æ ç è 0 v ö ÷ ø ê ê ê 2   =  1 2 æ è  1 2 v g ö ø 2   ( Wm(1) 2 + Wm(2) 2 ) +  v2 8 (Wm(3),Bm) æ ç è g2 -gg¢ -gg¢ g¢2 ö ÷ ø æ ç è W(3)m Bm ö ÷ ø

(B.7)

The physical content becomes transparent by performing the transformation from the fields W and B_m to the ``physical'' fields:

Wm± =  1 Ö2 (Wm(1) ±Wm(2))

(B.8)

Zm =  cosqWWm(3) + sinqWBm

(B.9)

Am = -sinqWWm(3) + cosqWBm

(B.10)

In these fields the mass terms are the diagonals of the matrix form of Equation B.9 and are given by

MW =  1 2 vg and MZ =  1 2 v Ö   g2+g¢2

(B.11)

if the mixing angle in Equation B.8 B.9  B.10 is chosen as

cosqW =  MW MZ =  g¢ Ö g2+g¢2

(B.12)

Identifying A_m with the photon field which couples via the electric charge e=Ö{4pa} to the electron, e can be expressed in terms of the gauge couplings in the following way

e =  gg¢ Ö g2+g¢2 and  g g¢ =  sinqW cosqW

(B.13)

Identifying the coupling constant g with the Fermi constant G_F from the low energy regime

g2/8 = GFMW2/Ö2

(B.14)

fixes the vacuum expectation parameter v=(G_FÖ2)^-1/2 @ 246 GeV. The weak mixing angle, q_W, relates the weak isospin coupling constant g and the hypercharge coupling constant g^¢ through the relation g^¢=g tanq_W.

Electroweak Feynman Rules

The Z⁰ Coupling to Fermion Pairs

The differential cross-section for the coupling of the Z⁰ to fermions is obtained for the electroweak theory with the aid of the Feynman Rules derived from the Lagrangian energy density. The invariant amplitude for the process e⁺e^-® f[`(f)] can be expressed as

M = -i   æ Ö  GFMZ2 Ö2   ela - u   ga(vf+afg5)v

(B.18)

where e is the polarization vector of the Z⁰, [`(u)] and v are the spinor for the fermion, f, and the Z propagator normalized to the photon propagator is taken as,

c0 =  s s-MZ2 + iMZGZ0

The partial width for the decay process Z⁰® f[`(f)] is the square of the invariant amplitude and can be expressed in the lowest order as

GZ0 = å f  G0(Z0® f - f   ) = å f  NfC  a 3 MZ Ö   1-4mf   [ vf2 (1+2mf) + af2 (1-4mf) ]

(B.19)

where N_C^f = 1, 3 for f = lepton, quark, m_f=m_f²/s, a = (Ö2G_FM_W²sin²q_W)p, and the Z-fermion vector and axial couplings are given by Equations B.18. The lowest order partial widths for various fermions are listed in Table B.1 [24].

The differential cross-section s( e⁺e^-® Z⁰® f[`(f)]) can be written in the form

ds dW =  a2 4s NCf Ö   1-4mf   · é ë G1(s) (1+cos2q) + G3(s) 2cosq Ö   1-4mf   + G2(s) sin2 q4mf ù û

(B.20)

where

G1(s) = Qe2 Qf2 + 2Qe Qf ve vf Re( c0(s) ) + (ve2+ae2)(vf2+af2-4mfaf2) | c0(s) | 2 G2(s) = Qe2 Qf2 + 2Qe Qf ve vf Re( c0(s) ) + (ve2+ae2)vf2 | c0(s) | 2 G3(s) = 2Qe Qf ae af Re( c0(s) ) +4ve ae vf af | c0(s) | 2

In the region where s » MZ02, this expression can be reduced in terms of the of the partial widths GZ0e+e- and GZ0f[`(f)] as follows

s( e+e-®Z0® f - f   ) =   12 pGZ0e+e-GZ0f[`(f)] (s - MZ02)2 + MZ02 GZ0tot2

(B.21)

which is a special case of the first order Breit-Wigner resonance formula.

Quantum Chromodynamics (QCD)

Incorporating the covariant derivative,

Dm = ¶m+igBm where Bm=  1 2 llblm

(B.22)

the locally gauge invariant Lagrangian [94] for the strong interaction has the form

L = - y   (igmDm-m)y-  1 2 tr(GmnGmn)

(B.23)

where B_m is a 3×3 matrix in color space, b^l_m are the eight color gauge fields (gluons) and l^l are the generators of the SU(3) gauge group. The gluon field-strength tensor is given by

Gmn = (ig)-1[Dn,Dm] = ¶nBm -¶mBn+ig[Bn,Bm]

(B.24)

In the framework of QCD, the Feynman rule for the quark-gluon vertex in terms of the parameters g and l^l is of the form

The Running Strong Coupling Constant, a_s

The ``running'' of a<sub>s</sub>(q<sup>2</sup>) has different implications for the two different regimes of momentum transfer. For low momentum transfers, a<sub>s</sub>(q<sup>2</sup>) is very large preventing the use of perturbative techniques in calculations for QCD as used in QED and leads to the confinement of free quarks into colorless singlet state. For large momentum transfers, a<sub>s</sub>(q<sup>2</sup>)® 0 as q<sup>2</sup>® ¥ permitting the use of perturbative techniques to be applied to QCD calculations and resulting in ``asymptotic freedom'' of quarks. Asymptotic freedom refers to the phenomenon of quarks to behave as if they are free, although no quark has been observed in isolation.

Fragmentation

The approach of fragmentation functions is applied to the differential cross-section for a particular hadron, h, as follows,

ds(e+e-® hX) dz = å q  s(e+e-® q - q   )[ Dhq(z)+Dh[`(q)](z)]

(B.26)

where z = E_h/E_q = 2E_h/s is the fraction of the quark's energy carried by the hadron, h. The fragmentation function is constrained to satisfy conservation of probability and momentum as follows,

å q  ó õ 1 0  zDqh(z)dz = 1

(B.27)

å q  ó õ 1 zmin  [Dhq(z)+Dhq(z)]dz = nh

(B.28)

where z_min is the threshold energy for producing the hadron of type h and n_h is the average multiplicity of hadrons of type h. The fragmentation function D^h_q(z) can be parameterized in many different ways testimony to the large number of different Monte Carlo generators which have been developed to model fragmentation.

One common parameterization [96] used in the independent fragmentation picture is the following,

Dhq(z) = f  (1-z)n z

(B.29)

This parameterization agrees with the logarithmic increase in the mean multiplicity of fragmentation with energy observed in experimentation, i.e.

ánhñ @ fln(Eq/mh)

as E_q®¥. Another model [97] fragments the heavy quarks (c, b, t) into hadrons which carry away relatively high fractions of the initial energy. This model is referred to as the Peterson model and is expressed as

Dhq(z) = Kz-1 é ë 1-  1 z -  eQ 1-z ù û -2

(B.30)

where the parameter e_Q=ám²_qi+p²_Tqiñ/ám²_Q+p²_TQñ regulates the ``hardness'' of the fragmentation function.

The Z⁰ Coupling to Quark Pairs

Present measurements of R confirm the three colors of quarks and as a function of the center-of-mass energy give an indication of the threshold for the production of a new quark flavor. Including O(a_s) corrections to R, yields a ~ 5% correction which is still not yet distinguishable within present experimental errors,

R = 3 å q  e2q é ë 1 +  as(q2) p ù û

(B.33)

Three jet events in e⁺e^- collisions have also yielded a wealth of information about the energy dependence of the strong interaction coupling constant, a_s(q²). The rate of three-jet events is directly proportional to a_s(q²) and arises from the hadrons which fragment from an initial q[`(q)] pair and a gluon radiated from the initial q or [`(q)]. The measured rates at Ös = 30 GeV indicate that a_s @ 0.2. At the higher center of mass energies of the LEP collider where Ös  @ 90 GeV, a_s has been determined to be 0.124±0.008 [18] consistent with the ``running'' of the coupling constant. The angular distribution of the gluon jets is also consistent with the gluon being a spin-1 particle as required by QCD.

The Future of the Standard Model