Appendix C
Radiative Corrections

(1) QED Corrections: The corrections due to initial or final state Bremsstrahlung consisting of those diagrams with an extra photon added as either a real or virtual photon loop. The initial state radiative corrections change the center-of-mass energy of the collision and consequently has a large effect on the line-shape normalization at LEP. The final state radiative corrections have a small effect at LEP of » (3aQ²_f)/4p £ 0.17% [99].

(2) Weak Corrections: The corrections due to all other electroweak diagrams including corrections due to the vector boson propagators g and Z⁰, the set of vertex corrections, and box diagrams with two massive boson exchange [100] [101].

(3) QCD Corrections: The corrections due the final state gluon radiation which changes the Z⁰ width by » a_s/4p = 4% [102].

QED Corrections

The QED Corrections are dependent on the details of the experimental setup but independent of the detailed structure of the underlying non-electromagnetic part of the theory. In fact, the only global parameters needed are M_Z, G_Z, v_f, a_f and m_f, without restriction to the SM. The corrections have coupling strengths which are determined by the low energy a with the associated O(a) diagrams shown in Figure C.1. The corrections are gauge invariant and renormalizable. At LEP, a realistic treatment of the QED corrections require higher order contributions than simply O(a). In general, the practical methods of dealing with higher order QED effects are broken down into four basic areas: i) soft photon resummation in the O(a) result, ii) explicit 2-loop calculations, iii) the structure function approach, and iv) the multi-photon radiation according to the method of Yennie, Frautschi, and Suura (YFS) [63]. The interested reader is referred to references [63] for an excellent discussion of these higher order QED contributions.

Weak Corrections

The Weak Corrections dependent on the detailed internal structure of the theory, e.g. dependence on the non-Abelian structure and the Higgs sector of the SM, but are independent of the experimental setup. In particular, they depend on the unknown parameters M_H, m_t and therefore include all the components necessary for the renormalizability of the SM. Due to the complex structure of the corrections, a detailed treatment of the corrections is required. The radiative corrections associated with the weak interaction involve corrections to the vector boson propagators g and Z⁰ shown in Figure C.2 [105], corrections to the vertex and corrections due to the box diagrams with two massive boson exchanged as shown in Figure C.3 [104], . In calculating the weak corrections, it is instructive to consider each of the above corrections separately.

Propagator Corrections

The propagator corrections involve all particles in the SM including the conjectured Higgs boson and top quark, thus depending on M_H and m_t. The propagator corrections are implemented in the e⁺e^- amplitude in terms of three 1PI self-energy functions [105]

Pgg(s) = ^ S   gg  (s) s PgZ(s) = ^ S   gZ  (s) s and PZ(s) labeleq:selfenergyfun

(C.1)

The renormalized self-energies are expressed in terms of the non-renormalized ones as follows [100] [101]:

^ S   gg  (s) = Sgg(s)- sSgg¢(0) ^ S   gZ  (s) = SgZ(s)- SgZ(0)+ s ì í î  2SgZ(0) MZ2 -  cos(qW) sW æ è  dMZ2 MZ2 -  dMW2 MW2 ö ø ü ý þ ^ S   ZZ  (s) = SZZ(s)- dMZ2 +dZ2Z (s-MZ2) ^ S   WW  (s) = SWW(s)- dMW2 +dZ2W (s-MW2)

(C.2)

with

dZ2Z = -Sgg¢(0)-2  cos2(qW)-sin2qW sinqWcos(qW)  SgZ(0) MZ2 +  cos2(qW)-sin2qW sin2qW æ è  dMZ2 MZ2  dMW2 MW2 ö ø dZ2W = -Sgg¢(0)-2  cos(qW) sW  SgZ(0) MZ2 +  cos2(qW) sin2qW æ è  dMZ2 MZ2  dMW2 MW2 ö ø dMW2 = Re ( SWW(MW2)) dMZ2 = Re æ ç ç è SZZ(MZ2)- ^ S   gZ  (MZ2)2 MZ2+ ^ S   gg  (MZ2) ö ÷ ÷ ø

(C.3)

with the corrected propagators containing the renormalized 1-particle irreducible (1PI) one-loop self-energies expressed by [105]:

^ D   W  (s) =  1 s-MW2 + ^ S   WW  (s) ^ D   Z  (s) =  1 s-MZ2 + ^ S   ZZ  (s)- ^ S   gZ  (s)2 s+ ^ S   gg  (s) ^ D   g  (s) =  1 s + ^ S   gg  (s)- ^ S   gZ  (s)2 s-MZ2+ ^ S   ZZ  (s) ^ D   gZ  (s) = ^ S   gZ  (s) [s+ ^ S   gg  (s)][s-MZ2 + ^ S   ZZ  (s)] - ^ S   gZ  (s)2

(C.4)

The leading terms of the quantity P_Z(s) are of the form:

PZ(s) = -Da+  cos2(qW)-sin2qW sin2qW D r   + ···

(C.5)

where

Da = -Re( Pgg(MZ2)), D r   =  a 4p ·  3 4sin2qWcos2(qW) æ è  mt MZ ö ø 2

(C.6)

where D[`(r)] is identified as the ``r parameter''  [107].

The O(a²) corrected Z⁰ width, DG_Z, is given by,

DGZ = å f  NCf  2 3 aMZ [ vf Re( FVZf (MZ2 )) +af Re( FAZf (MZ2 ))] + å f  NCf  a 3 MZ(vf2+af2) ·dQED + å f=q  NCf  a 3 MZ(vf2+af2) ·dQCD + å f  G(Z® Hf f   )

(C.7)

together with

dQED =  3a 4p Qf2 dQCD =  as(MZ2) p + 1.405 æ è  as(MZ2) p ö ø 2

(C.8)

and G(Z® Hf[`(f)]) [108]. Taking into account the mass dependence of the QCD corrections in the partial width for Z®b[`(b)] increases G_Z by 2 MeV with a_s(M_Z²)=0.11±0.01 [103]. The uncertainty from a_s induces an uncertainty in the total width of about ±6 GeV. The quantities are universal and can be applied to t-channel exchanges as well.

Photon Exchange

The diagonal g propagator diagram has the following form:

= Qe Qf  e2 1+Pgg(s)  1 s » Qe Qf  e2(s) s (C.9)

Z⁰ Boson Exchange

The diagonal Z⁰ propagator diagram has the following form:

=  e2 4sin2qWcos2(qW) ·  1 1+PZ(s) ·  J(e) ·J(f) s-MZ2 + i  Im( SZ(s)) 1+PZ(s) (C.10)

J(e) = v   e  [gm (I3e - 2sin2qWQe ) -gmg)5 I3e ] ue J(f) = v   f  [gm (I3f - 2sin2qWQf ) -gmg)5 I3f ] uf

(C.12)

g -Z⁰ Mixing

The gZ mixing diagram beyond the tree level has following form:

= gm(vf-afg5 ) + gmQf  PgZ(s) 1+Pgg(s) =  1 2sinqWcos(qW) ì í î gm é ë I3f -2Qf æ è sin2qW- sinqWcos(qW)  PgZ(s) 1+Pgg(s) ö ø ù û -I3f gmg5 ü ý þ (C.13)

Vertex Corrections

The vertex corrections refer to the gff and Z⁰ff 3-point functions in one-loop order after renormalization with the self-energy contributions to the external legs included. In contrast to the propagator corrections The vertex corrections, unlike the propagator corrections, are not universal, depending in general on the fermion species. For this reason, the vertex corrections must be listed separately for n, e, u, d type fermions (the b quark is exceptional due to the virtual top contributions in the vertex). Of the vertex correction diagrams shown in Figure C.3, only (a)-(c) have non-negligible contributions for f ¹ b, while (d)-(g) are negligible due to the small Yukawa couplings.

The values for the leptonic and hadronic form factors for Ös =M_Z² are listed in Table C.1 along with the Born coupling constants v_e, a_e. In addition to the contributions from the 2-point functions, the effective coupling constants near the Z peak are determined by

vf + Re( FVZf (MZ2) )      af + Re( FAZf (MZ2) )

(C.29)

These effective coupling constants allow an easy estimate of the quantitative effects in various measurable quantities. In general, the vertex corrections contribute a correction to the width and cross-section of the Z⁰from O(1%) to 4% (in the b case). Table C.2 lists the effects of the vertex corrections on the Born cross-section for the processes dG/G(Z®e⁺e^-) and dG/G(Z®b[`(b)]).

Box Corrections

The electroweak box diagrams include the two gauge boson diagrams as well as the QED box diagrams. The contribution of the QED boxes is smaller than that of the gauge boson boxes due to the cancellations between real and virtual photon contributions to the phase space. The weak box diagrams consist of the Z⁰Z⁰ and W^±W^± exchange diagrams

with contributions to the weak amplitude typically of the from

= e2  JNC,CC(e) ·JNC,CC(f) s ·  a 2p [ I(s,t,MZ) -,+ ±I(s,u,MZ) ] e2  JNC,CC(e)5 ·JNC,CC(f)5 s ·  a 2p [ +,- ±I5 (s,t,MZ) + I5(s,u,MZ) ]

(C.30)

where

JNC(e,f) = v   e,f  gm (ve,f2 +ae,f2 -2ve,f ae,f g5 ) ue,f JNC(e,f)5 = v   e  gmg5 (ve,f2 +ae,f-2ve,f ae,f g5 ) ue,f JCC(e,f) =  1 4sin2qW v   e,f  gm(1-g5 ) ue,f

(C.31)

The contribution of the gauge boson box diagrams near the Z⁰ peak is of the order the a/p to the photon exchange amplitude. The contribution to the differential cross-section at Ös = M_Z² is smaller than 0.02%. This is because these box diagrams are non-resonant and their contribution to the Z⁰ cross-section is suppressed by the large resonance factor.

Improved Born Approximation

As an example of the implementation of radiative effects in the Standard Model, the cumulative effects of the QED, Weak, and QCD radiative corrections are presented for the general process e⁺e^-® Z⁰® f[`(f)]. The approach requires correcting the Born approximation for the process by including effective values of parameters defining the coupling strengths and normalizations. It is sufficient to include the leading light fermion and heavy top contributions along with the structure suggested from the consideration of higher orders in the leading terms. The contributions from the propagators, vertices and boxes are incorporated through the coupling strengths defined in terms of the original Born parameters as follows,

e2 s ®  e2(s) s a ® a(MZ2) =  a 1-Da = 1.064a MZGZ0 ®  s MZ2 ·MZGZ sin2qW ® sin2 q   W     (in J(e) and J(f)) =  1 2 æ ç è 1-   æ Ö 1-  4A rMZ2 (1-Da)   ö ÷ ø

(C.32)

where the r parameter serves as the normalization of the Z⁰ amplitude and is given by

r =  1 1-Dr     Dr = 3  Gmmt2 8p2Ö2

(C.33)

and the effective mixing angle parameter, A, is

A =  pa Ö2GF º (37.2802±0.0003 GeV)2

(C.34)

Using these effective parameters, the improved Born invariant amplitude for e⁺e^-® Z⁰® f[`(f)] near the Z⁰ resonance is given by

M   Born  = Qe Qf  4pa(MZ2) s Jem(e) ·Jem(f) +Ö2GmMZ2r J   (e) em  · J   (f) em  s-MZ2+i  s MZ2 MZGZ

(C.35)

where the matrix elements of the electromagnetic current are

Jem(e) = v   e  gmue and Jem(f) = u   f  gmvf

(C.36)

or redefined in terms of the effective parameters as

Jefff = ( Ö2GmMZ2rf)[ 1/2] f   gm[I3f (1-g5 ) -2Qf sin2qf] f

(C.37)

The total Z⁰ width obtained by including the radiative corrections is given by

GZ = å f  GZ(Z®f f   ) = NfC  MZ 12p Ö2GFMZ2r Ö   1-4mf   é ë (If3-2Qfsin2 - q   W  )2(1+2mf) + (1-4mf) ù û

(C.38)

The improved Born approximation yields values which agree with experimental results to within 1%, except for the G(Z⁰® b[`(b)]). In the case of G(Z<sup>0</sup>® b[`(b)]), non-leading terms in the vertex corrections, which are numerically more significant than in the self-energies, give rise to the deviations from the correct results. Notice that the improved Born approximation does not contain any information about the Higgs mass.

More involved treatment of G(Z⁰® b[`(b)]) as well as other processes with similar maladies yield closer agreement to experimental results. In the case of the G(Z<sup>0</sup>® b[`(b)]), Additional corrections from the vertex corrections for the G(Z⁰® b[`(b)]) involving the top quark leads to the modification of the improved Born approximation parameters r®r_b and k® k_b where

r ® r æ è 1-  4 3 Dr ö ø º rb sin2 q   W  ® sin2 q   W  æ è 1+  2 3 Dr ö ø º sin2 q   W  ·kb .

(C.39)

The single Z⁰b[`(b)] vertex in the Born amplitude is renormalized by replacing r with

r ® Ö   rrb

(C.40)

With this parameterization, G(Z⁰® b[`(b)]) is determined to be 362 MeV, which agrees with the complete order calculations to within 1% citebib:omicronalphares.