Chapter 2
Theoretical Background

The total decay rate or width of the Z0, GZ0tot, must be corrected to include the secondary effects due to the electromagnetic (QED), weak, and quantum chromodynamics (QCD) interactions. The corrections are normally included by defining parameters to incorporate the differences from the ``bare'' calculation permitting the evaluation of the different partial decay rates. Such effects are called radiative corrections and naturally divide into three areas for the general process e+e-® Z0® f[`(f)] as follows:

(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%.

(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.

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

For example, the QED corrections are normally re-absorbed by defining the effective weak mixing angle, [`(q)]_W, while the weak corrections are re-absorbed in the r parameter. The agreement between the corrected predictions of the Standard Model and the most recent experimental results of today [22] gives strong support for the SM. This chapter presents an overview of the different methods for determining the number of elementary light neutrinos, N_n, with m_n £ M_Z⁰/2 in the framework of the electron-positron collider LEP (described in Chapter 3).

2.1  Theory of Neutrino Counting

(1) The Invisible Width Method: The method of measuring the invisible width of Z0, GZ0inv, from the total width, GZ0tot, the hadronic width, GZ0had, and the leptonic width, GZ0leptassuming all contributions from the invisible channels originate from light neutrinos, GZ0inv=NnGn[`(n)] [14].

(2) The Single Photon Counting Method: The method of measuring the total cross-section for the process e⁺e^-® n[`(n)]g which is directly proportional to the number of light neutrinos [19].

It is important to realize the complementarity between the different methods from the theoretical and experimental point of view which results in producing independent determinations of N_n. The direct and indirect methods use different theoretical inputs from the SM resulting in differing accuracy for the determination of the number of light neutrinos, N_n, according to the statistical and systematic constraints of the experiment. The single photon counting method has the advantage over the other methods in that only the neutrino partial width G_nn enters directly in the cross-section, resulting in a determination of N_n which is only weakly affected by experimental and theoretical uncertainties. Since the analysis presented in this thesis determines the number of light neutrinos using the single photon counting method, attention is placed on its theoretical discussion in Section 2.3.

2.2  The Invisible Width Method

The invisible width method precisely determines the number of neutrino families from the decomposition of the total width, GZ0tot, into its partial widths. In the SM, the total width is decomposed into the invisible width, GZ0inv, the hadronic width, GZ0had, and the leptonic width, GZ0lept, as follows:

GZ0 º GZ0tot=NnGZ0inv+GZ0had+GZ0lept

(2.3)

where

GZ0lept = GZ0e+e-+GZ0m+m- + GZ0t+t-

(2.4)

Assuming that all contributions from the invisible channels originate from neutrinos, the invisible width, GZ0inv, is given by

GZ0inv = GZ0tot - GZ0had - GZ0lept

(2.5)

and the number of elementary light neutrinos, N_n, is of the form

Nn = GZ0inv/ GZ0n[`(n)]

(2.6)

As a direct result of the radiative corrections on the G_Z⁰, the values of the top mass (m_t), Higgs mass (m_H), and the strong coupling constant (a_s) bear on the accuracy of the N_n measurement. If one varies m_H between 2 and 1000 GeV, m_t between 50 and 450 GeV, and a_s between 0.09 and 0.15, the theoretical uncertainty of G_Z⁰ amounts to 0.030 GeV where G_Z⁰ =2.492±0.017 GeV [23].

Experimental measurements of GZ0tot have been made with the high statistics samples of the Z0 collected from the LEP collider as shown in Table 1.3. The accuracy of the determination of the number of light neutrinos depends largely on the different experimental conditions and the theoretical inputs from the SM [15] [16] [17]  [18]. The experimental inputs important to the determination of Nn shown in Table 1.3 include MZ0, R¢, and L, with the theoretical input GZ0n[`(n)](theor.). Here, R¢ is defined as

R¢ =   GZ0had(exp.) GZ0mm(exp.) =    emm Nhad ehad Nmm

(2.7)

Here e_had and e_mm are the hadronic and leptonic efficiencies. N_had and N_mm are the numbers of hadronic and leptonic events respectively.

By parametrizing the observed hadronic cross-section s^obs_had as

shadobs(s) = ó õ s smin  shad(s¢) f(s,s¢) ds¢

(2.8)

with

shad(s) = spolehad  s MZ02 (s¢- MZ02)2 + s¢2G2Z0 / MZ02

(2.9)

With s_had^pole from measurement, the invisible width is given by

GZ0inv = GZ0tot é ê ë 1 - 3 æ ç è   æ Ö   MZ02 Nmm F 12 pemm L   ö ÷ ø -   Nhad ehad æ ç è   æ Ö   MZ02 emm F 12 pNmm L   ö ÷ ø ù ú û

(2.10)

where L is the integrated luminosity and F @ ([(G)/(M)])b with b = (( 2 a)/(p )) ln(( s)/(m2 )). The error on DGZ0inv is

DGZ0inv = (0.5 GZ0had-1.5 GZ0m+m-) æ è  DNmm Nmm Å  Demm emm ö ø ÅGZ0had æ è  DNhad Nhad Å  Dehad ehad ö ø Å ( 0.5 GZ0had+ 1.5 GZ0m+m- ) æ è  DL L ö ø ÅGZ0inv  DGZ0tot GZ0tot

(2.11)

where Å refers to the addition of errors in quadrature.

Experimental results using the invisible width method have achieved dNn =0.01 with DL /L = 1% and Dehad / ehad = Dem / em=1% with a L=10 pb-1. However, the method does not give the correct answer if Nn ¹ GZ0inv/ GZ0n[`(n)] , because of the initial assumption on GZ0inv. Hence, an independent determination of Nn is needed as a check against any possible new invisible particles.

2.3  The Single Photon Counting Method

2.3.1  The Lowest Order Cross-Section

The first order Feynman diagrams in the SM that contribute to the process e⁺e^-® n[`(n)]g are shown in Figure 2.1. The interaction Hamiltonian of the two Z<sup>0</sup> decay diagrams for the single photon process, e<sup>+</sup>e<sup>-</sup>® n[`(n)]g, is given by

Hint =  GF 2 si [ niga (1-g5)ni ][ e ga (gvi - gai g5)e ]

(2.12)

From this interaction Hamiltonian, the differential cross-section for the production of a photon forming an angle with respect to the beam axis, y=cosq_g, and energy, x=2E_g/Ös, is given by the following formula,

d2 s0 dx dy = G2F as(1-x)   [ (1- x/2)2 + x2y2/4 ] 6 p2 x (1-y2) · é ë 2+   Nn( gv2+ga2 )+ 2 ( gv+ga ) ( 1 - s(1-x) / MZ02 ) ( 1 - s(1-x) / MZ02 )2 + GZ0/ MZ02 ù û

(2.13)

where

x = the fraction of energy carried away by the emitted photon ( 2Eg / Ös) y = the emitted photon angle with respect to the beam axis (cosqg) gv = the vector coupling constants (-1/2+2sin2DqW » -0.04) ga = the axial coupling constants (-1/2) Nn = the number of light neutrinos with mn < MZ0/2

2.3.2  Single Photon Spectra

Figure 2.2 shows the center-of-mass dependence for the e⁺e^-® n[`(n)]g signal and the principal background, the radiative Bhabha process, e<sup>+</sup>e<sup>-</sup>® e<sup>+</sup>e<sup>-</sup>g. A feature of the general form of the spectra for the single photon signal is the the energy distribution which is soft divergent, i.e. d s/ d x µ 1 / x, and strongly forward/backward peaked angular distributions indicative of radiative processes, i.e. d s/ d y µ 1 / sin<sup>2</sup>q<sub>g</sub>. The selection criteria must attempt to utilize these attributes to accept as much of the single photon signal from the e<sup>+</sup>e<sup>-</sup>® n[`(n)]g process while reducing as much as possible any other type of event with the same topology and signature, namely the backgrounds. Among the most important backgrounds for the single photon signal is the e⁺e^-® e⁺e^-g  process shown in Figure 2.2. The selection criteria are discussed in greater detail in Chapter 6.

In the partial cross-section for the single photon signal, the Z0 pole at x=(s-MZ02)/s leads to a pronounced enhancement in the energy distribution for center-of-mass energies above the MZ0, e.g. Ecm » MZ0+7 GeV. In fact, the total cross-section given by Equation 2.13 increases by about 30% as a function of each additional neutrino family. This large change in the total cross-section, however, must be modified due to the inclusion of the O(a) virtual and real contributions from the the electroweak radiative corrections to the single photon signal. Since this analysis uses data taken at center-of-mass energies near the MZ0, e.g. Ecm » MZ0±3 GeV, attention is focused on the signal and corrections to the signal for Ecm » MZ0±3 GeV. The partial cross-section for the single photon signal from the process e+e-® n[`(n)]g has been determined using the point interaction approximation in the limit of MW ® ¥. For the center-of-mass energies considered in the analysis, i.e. MZ0-3.0 GeV £ Ös £ MZ0+3.0 GeV, this approximation is justified since it affects the total cross-section by is less than 1%.

The Electroweak Radiative Corrections

The electroweak radiative corrections affect both the shape and the position of the peak around the Z⁰ [24]. The weak corrections can be absorbed in the single photon total cross-section through the r parameter, the effective sin²[`(q)]<sub>W</sub>, and the modified Z<sup>0</sup> propagator [25]. The total shift of the peak of the M<sub>Z<sup>0</sup></sub> due to the weak corrections is 35 MeV over the expression with constant G<sub>Z</sub> [14]. The electromagnetic radiative corrections have the same type of corrections as those for the Z<sup>0</sup> line shape, but are much simpler in structure, since the e<sup>+</sup>e<sup>-</sup>® n[`(n)]g signal has no charged final states. The formalism of structure functions has been used to calculate the corrected single photon cross-section. In this scheme, s₀(s¢) is the ``reduced'' cross-section for the process e<sup>+</sup>e<sup>-</sup>® n[`(n)]g in the new center of mass system given by s<sup>¢</sup>= (1-x)s = s-2E<sub>g</sub>Ös. Introducing functions with transverse degrees of freedom renders the ``reduced'' partial cross-section in the form [26]:

d2 s0 dx dy = H(a)(x,y;s)  s0 ( (1-x)s ) = H(a)(x,y;s¢)  s0 ( s¢)

(2.14)

where the ``reduced'' cross-section has the form

s0(s¢) =   GF2 s¢ 12 p é ë   2 + Nn( gv2+ga2 ) + 2 ( gv+ga ) ( 1 - s¢/ MZ02 ) ( 1 - s¢/ MZ02 )2 + GZ0/ MZ02 ù û »   12p MZ02 é ë   Nns¢GZ0n[`(n)] ( s¢- MZ02 )2 + s¢2GZ0tot2 / MZ02 ù û + W terms

(2.15)

and the radiator function, H^(a)(x,y;s^¢), given by the expression  [27]

H(a)(x, y; s¢) =  a p    1+(1-x)2 x    1 1-y2 Q æ è 1-  4 m2(1-x) s¢ -|y | ö ø

(2.16)

In the structure function formalism, the total cross-section for producing a real photon can be written by integrating Equation 2.16.

s(s) = ó õ 1 xmin  dx ó õ +ymin -ymin  dy  H(a)(x,y;s) s0 ( (1-x)s )

(2.17)

where the minimum detectable energy and angle of the photon is given by x_min = 2E^min_g/Ös and y_min=cos(q^min_g) as shown in Figure 2.3. Although the resummation of the collinear and soft photon affects the distributions by a few percent, the total cross-section of the observed single photon is reduced to 75% of the Born approximation as a result of the O(a) radiative corrections.

The total cross-section is also affected by the choice of the single photon selection criteria (refer to Figure 2.3). The selection criteria aim at carefully selecting as much of the single photon events from the signal process, while removing as much of the background as possible. Among the most important background to the single photon signal is the low Q² radiative Bhabha scattering, e⁺e^-® e⁺e^-g, where the photon is observed in the acceptance volume while the electron and positron escape detection down the beam pipe. A natural choice for vetoing such events would be to cut as high as possible with the veto angle, but for center-of-mass energies near the M_Z⁰ this also eliminates much of the signal. Consequently, the elimination of the low Q² radiative Bhabha background, along with the other possible backgrounds, requires the use of Monte Carlo event generators or analytical programs to correctly study and determine the best method for selecting the single photon events from the process e⁺e^-® n[`(n)]g. This will be discussed in detail in Chapter 6.