Electroweak Measurements and Model Analysis Of Electroweak Data

abstract

The parameters of the Standard Model are accessed in many precision measurements at the LEP, SLD, and Tevatron colliders and in low-$Q^2$ experiments. The low-$Q^2$ range is meant to be relative to the square of the weak boson masses, $M_Z^2$ and $M_W^2$.

As mentioned in Chap. 1, of main interest are the electroweak coupling structure and the particle masses, especially those of the heavy particles, m_T, M_W, M_Z, and MH. In the following sections the different inputs to the combined analysis of electroweak data are discussed.

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The parameters of the Standard Model are accessed in many precision measurements at the LEP, SLD, and Tevatron colliders and in low-$Q^2$ experiments. The low-$Q^2$ range is meant to be relative to the square of the weak boson masses, $M_Z^2$ and $M_W^2$.

As mentioned in Chap. 1, of main interest are the electroweak coupling structure and the particle masses, especially those of the heavy particles, m_t, M_W, M_Z, and M_H. In the following sections the different inputs to the combined analysis of electroweak data are discussed.

4.1 W Boson Mass Measurements at LEP and Tevatron

The mass of the W boson is not only measured at LEP but also in $p\bar{p}$ collisions at the Tevatron . W bosons are produced in the parton-level process $q\bar{q^{\prime}}\to W$ and decays to $e\nu_e$ and $\mu\nu_\mu$ are selected. The observables sensitive to M_W [1] are the transverse lepton momentum ${p_T^\ell}$, the missing transverse momentum $p_{T}^{miss}=p_T^\nu$, and the transverse mass m_T, which is an approximation of the mass of the decaying W. It is calculated according to

$${m_T}=\sqrt{2 {p_T^\ell} p_{T}^{miss}(1-\cos\Delta\phi)}\;,$$

((4.1))

with the azimuthal angular difference, $\Delta\phi$, between the missing momentum and ${p_T^\ell}$. Examples of the measured m_T and ${p_T^\ell}$ distributions in the $W\to e\nu$ channel are shown in Fig. 4.1. The W mass is extracted from a binned log-likelihood fit to the m_T, ${p_T^\ell}$ and $p_{T}^{miss}$ spectra, in which the data in each bin is compared to predictions using different underlying M_W values, so-called templates . In case of the CDF measurement, the templated predictions are determined in a fast simulation procedure. The likelihoods are scanned in M_W steps of 1 MeV. The minima are determined for each data set and decay channel separately and the results are eventually combined, taking correlations into account.

Fig 4.1

Transverse mass (a) and transverse lepton momentum (b) distribution in $W\to e\nu$ events as measured by CDF [2]. The fit result using the two spectra in the range indicated by the arrows are also shown in the graphic

The m_T spectrum yields statistically and systematically the most precise M_W value, while the ${p_T^\ell}$ and $p_{T}^{miss}$ measurements have an about 20% larger error. The main sources of systematic uncertainties is from the tracker momentum scale in the muon and electron channel and the calorimeter energy scale for electrons. The former is calibrated in di-muon decays of $J/\psi$, $\Upsilon$ and Z bosons, while the latter relies on the precise calibration of the E/p ratio of electrons. It is also verified with $Z\to e^+ e^-$ events. In the latest CDF measurement, the lepton scale contributes with 23 MeV to the total systematic uncertainty of 34 MeV. Furthermore, uncertainties from variation of PDFs are determined with the standard CTEQ6 [3] error sets and comparison with the alternative MRST [4] parameterisation. They contribute with 13 MeV to the systematics. In the ${p_T^\ell}$ and $p_{T}^{miss}$ measurements the recoil scale and resolution play a larger role than for m_T, so that the corresponding uncertainty on the combined M_W value is 8–10 MeV each. The influence of the transverse momentum of the W, $p_T^W$, on the measured ${p_T^\ell}$ is well controlled by fits to Drell-Yann production data, exploiting the similarity to Z boson production. It actually affects the M_W value only by 4 MeV, even less than the uncertainties from lepton resolution, efficiency, backgrounds and simulation uncertainties, which each contribute between 2 and 6 MeV. The CDF measurement analysing $200 pb^{-1}$ of Run-II data yields

$$M_{\rm W}=80.413\pm0.034 {({stat.})}\pm 0.034 {({syst}.)} {GeV}.$$

((4.2))

The currently best ${\textrm{D\O}}$ measurement is based on $1 {fb}$ of Run-II data and the analysis of only the $W\to{\textrm e}\nu$ channel yields [5]:

$$M_{W}=80.401\pm 0.021 {({stat}.)}\pm 0.038 {({syst}.)} {GeV};.$$

((4.3))

with a total uncertainty smaller than in the CDF determination. The main source of systematics in this measurement is from the electron momentum scale and amounts to 34 MeV. The CDF and $D\phi$ results are combined taking correlations into account, which yields [1]:

$$M_{W} = 80.420 \pm 0.031 {GeV}$$

((4.4))

The uncorrelated uncertainties sum up to about 27 MeV, while the main correlated uncertainties are about 12 MeV from assumptions about the W and Z boson production, respectively the parton density functions applied in the analyses, and about 9 MeV from the description of radiative corrections.

The Tevatron result on M_W is compared to the LEP measurement in Fig. 4.2. The currently best M_W value combines all collider results and yields:

$$M_{W}=80.399\pm0.023 {GeV};.$$

((4.5))

Fig 4.2

Tevatron and LEP measurement of the W boson mass and their combination [13]

It agrees well with the W-mass derived in the analysis of other electroweak data of

$$M{W}=80.364\pm 0.020 {GeV};,$$

((4.6))

which mainly exploits the well-known relation between the precisely measured muon decay constant G_F and the electroweak gauge boson masses of Eq. (1.45). Radiative corrections and uncertainties on the other model parameters, including a variation of M_H between the current lower limit of 114.4 GeV and 1 TeV, are taken into account.

4.2 Top Mass Measurement at the Tevatron

The mass of the heaviest quark dominates radiative correction terms to the W propagator over other quark contributions, and a precise determination of m_t is therefore necessary for detailed model comparisons with electroweak data. The top quark mass is above the W+b threshold so that it practically always decays to this final state since W+s and W+d decays are CKM suppressed. The width of the top quark in the Standard Model is given at NLO by [6]:

$$\Gamma_t=\frac{G_{F}m_{t}^3}{8\pi\sqrt{2}} \left( 1-\frac{M_{W}^2}{m_t^2} \right)^2 \left(1+2\frac{M_{W^2}}{m_{t}^2}\right)\left\{ 1-\frac{2\alpha_{S}}{3\pi}\left(\frac{2\pi^2}{3}-\frac{5}{2}\right)\right\}\,$$

((4.7))

which numerically is equal to 1.35 GeV, for the currently best M _W and m _t values and $\alpha_{S}=0.118$. The corresponding lifetime is very short, $\tau_t=5\times 10^{-25}\textrm{s}$, so that the top decays before hadronisation starts or $t\bar{t}$ bound states can be formed. Experimentally, an upper limit of $1.8\times 10^{-13}{s}$ is derived at 95% C.L. by reconstructing decay vertices of top quarks and their corresponding decay lengths [7].

At the Tevatron, the dominant production of top quarks is in $t\bar{t}$ pairs with the following final states :

• 46.2% fully hadronic: $t\bar{t}\to W^+bW^-\bar{b}\to q\bar{q}^{\prime}b q^{\prime\prime}\bar{q}^{\prime\prime\prime}\bar{b}$

• 43.5% lepton+jets: $t\bar{t}\to W^+bW^-\bar{b}\to q\bar{q}^{\prime}b \ell^-\bar{\nu}_{\ell}\bar{b} + \ell^{\prime+}\nu_{\ell^{\prime}}b q^{\prime\prime}\bar{q}^{\prime\prime\prime}\bar{b}$

• 10.3% di-lepton: $t\bar{t}\to W^+bW^-\bar{b}\to \ell^+\nu_{\ell} b \ell^{\prime-}\bar{\nu}_{\ell^{\prime}} \bar{b}$

The branching fractions of the three decay types are identical to the corresponding W pair decay fractions (see Sect. 3.3).

The most precise m_t value is obtained in the analysis of the lepton+jets channel [8,9]. Top quark pairs are selected using the final state signatures, one lepton with high ${p_T^\ell}$, high $p_{T}^{miss}$, two light-quarks jets and two jets with a B hadron tag which is based on the fact the B hadrons travel several millimetres before they decay. The event kinematics of the two top quarks in the event are reconstructed from the final state leptons and jets, and where all jet-to-parton permutations are taken into account exploiting also the b-tag information. One of the potentially largest systematic uncertainty is due to the jet energy scale (JES) , because it enters directly the top mass estimator. The JES for light jets can however be determined in top decay events themselves by fixing the mass of the hadronically decaying W bosons to the externally measured M_W value. The JES for b-jets is then determined relative to the JES of light jets.

In the matrix-element weighting method, which is used in many top mass analyses, a likelihood is constructed from the event probability

$$P_{e}(x;m_{t},{JES})=f_{top} P_{s}(x;m_{\textrm t},{JES})+(1-f_{top})P_{b}(x;{JES}),$$

((4.8))

where x summarises the kinematic variables of the event, $f_{top}$ denotes the signal fraction, and P_s and P_b the signal and background probabilities. They both depend on the event kinematics and the JES. The signal probability is sensitive to m_t and can be written as [9] :

$$\begin{aligned} P_s(x; m_{t},{JES})&{}=\frac{1}{\sigma_{s,{obs}}(x;m_\textrm{t},{JES})}\\ \times \sum_i w_i \int_{q_1,q_2,y} dq_1 dq_2 f(q_1) f(q_2) \frac{(2\pi)^4|M(q\bar{q}\to t\bar{t}\to y)|^2}{2q_1q_2 s}\\ W(x,y;{JES}) d\Phi_6 \;.\end{aligned}$$

((4.9))

The matrix element squared, $|M(q\bar{q}\to t\bar{t}\to y)|^2$, is weighted by the PDFs, $f(q_i)$, and integrated over the parton momentum fractions q_i. A second integral is performed over the parton configurations y that correspond to the measured event kinematics x, where $W(x,y;{JES})$ is the transfer function that relates y to x and which depends also on the JES. The integral is normalised by the observed signal cross-section $\sigma_{s,{obs}}(x;m_{t},{JES})$. All parton-jet permutations are weighted with w_i and combined. In this way, a maximum of information is extracted from each event and m_t and the JES can be fitted simultaneously. The fit result for ${\textrm{D\O}}$ in the leptons+jets channel is shown in Fig. 4.3.

Fig 4.3

(a) Correlation of extracted top mass and jet energy scale (JES) in lepton+jets event analysis of ${\textrm{D\O}}$ [9]. (b) Top mass spectrum in fully hadronic top decays with two b-tagged jets, as determined by CDF [10]

Residual p_T or $\eta$ dependent uncertainties on the JES which can not be accessed by a global rescale factor are furthermore taken into account. Also the b-JES is further tested by varying b decay fractions, b fragmentation functions and detector response to b-jets. In the lepton+jets channel the total JES uncertainty is about 1.0 and 1.3 GeV for CDF and ${\textrm{D\O}}$, respectively. Other systematic error sources, like PDF uncertainties, ISR and FSR modelling, background uncertainties and modelling of the underlying event and pile-up are relatively smaller, so that the total systematics is at the 1.4–1.6 GeV level, compared to a 0.8–0.9 GeV statistical error that both Tevatron experiments obtain in about 3.2 (CDF) and $3.6 \textrm{fb}^{-1}$ (${\textrm{D\O}}$) [11].

The combined m_t and JES fit is also applied in the fully hadronic channel, while the di-lepton channel can obviously not rely on this technique. Here, the JES uncertainty is determined by varying jet energy scales within its uncertainty, in addition to the previously described JES systematics. The total JES error is therefore slightly larger, 2.0–2.6 GeV, than in the channels with hadronic top decays. An example of a mass spectrum measured in the hadronic channel is shown in Fig. 4.3.

The top quark mass can however not only be measured by fully reconstructing its decay into W+b. Since at the Tevatron the top quarks are nearly produced at rest, the b-quark system is boosted by a relativistic factor of

$$\gamma_b=\frac{m_{t}^2+m_b^2-M_{W}^2}{2 m_t m_b}\approx 0.4\frac{m{t}}{m_b}\;,$$

((4.10))

which depends on m_t. Rather than determining the lifetime of the B hadron its two-dimensional decay length, $L_{2D}$, is measured. This is the distance between the primary vertex of the event and the reconstructed secondary vertex from the B decay in the plane transverse to the beam. The mean $\langle L_{2D}\rangle$ has a nearly linear relation with m_t. Similarly, the mean ${p_T^\ell}$ of the lepton is highly correlated with the top mass because also the W boson receives a boost proportional to m_t. CDF obtains the following results in $1.9 \textrm{fb}^{-1}$ of data [12]:

$$\begin{aligned} \langle L_{2D}\rangle = 0.596\pm0.017 {cm} \Rightarrow m_{t}=176.7^{+10.0}_{-8.9} {({stat}.)}\pm 3.4 {({syst}.)} {GeV}\\ \left\langle p_T^{\ell} \right\rangle = 55.2\pm 1.3 {\textrm GeV}\Rightarrow m{textrm t}=173.5^{+8.9}_{-9.1} {({\textrm stat}.)}\pm 4.2 {({syst}.)} {GeV} \end{aligned}$$

These measurements are by construction not directly affected by JES uncertainties. They are only indirectly influenced by a change of the event selection, mainly because of out-of-cone corrections to the JES. The total effect is however much smaller than other systematics. Dominating are the scale uncertainties on the $\langle L_{2D}\rangle$ and $\left\langle {p_T^\ell} \right\rangle$ and QCD ISR and FSR effects.

An overview over all top quark measurements by CDF and Dπ and their combination are shown in Fig. 4.4. The central value [11] of

$$m_{t}=173.1 \pm 0.6 {({stat}.)} \pm 1.1 {({syst}.)}{GeV}$$

((4.11))

is mainly influenced by the lepton+jet channel but all analyses yield very consistent results. The main systematic effects are from JES ($\pm 0.7 {GeV}$), and Monte-Carlo modelling of the ${t}\bar{t}$ signal used for calibrating the fit method ($\pm 0.5 {GeV}$). The uncertainty from the understanding of Colour Reconnection in $t\bar{t}$ events amounts to $\pm 0.4 {GeV}$, estimated by comparing two differently tuned Monte-Carlo parameter sets with CR. Similarly to the W-mass measurements at LEP, the CR systematics may become a dominating uncertainty since it is fully correlated among experiments and analysis channels, and because the JES uncertainty will decrease with increasing data statistics. The uncertainties on signal and background description are each contributing with $\pm 0.3 {GeV}$, which takes ISR, FSR and hadronisation effects into account. The remaining systematics from lepton momentum scale, multi-hadron interactions and finite Monte Carlo statistics are all smaller than $0.2 {GeV}$.

Fig 4.4

Tevatron measurements of the top quark mass and their combination [11]

A comparison with the indirect top quark mass of $m{t}=178.9^{+11.7}_{-8.6} {GeV}$ [13] determined from other electroweak data is only a rough test of the Standard Model relations between the electroweak parameters, which emphasises the important role of a precise direct measurement of the top quark mass.

4.3 Low-Q ² Measurements

In the general analysis of electroweak data, precision experiments at energies below the Z resonance are taken into account as well. Atomic parity violation, MØller scattering and Neutrino-nucleon scattering are discussed in the following. The hadronic cross-section measurement plays a separate, but important role, be0cause it is input to the determination of the running of the electromagnetic coupling $\alpha_{QED}$ from $Q^2=0$ to $Q^2=M_Z^2$, where many LEP and SLD measurements were performed.

4.3.1 Muon Lifetime Measurement

One of the most precise ingredients of the electroweak data set is the Fermi constant G_F, which is determined in measurements of the muon lifetime, $\tau_\mu$. The expression for the inverse lifetime

$$\frac{1}{\tau_\mu}=\frac{G_F^2m_\mu^5}{192\pi^3}(1+\Delta q)$$

((4.12))

includes phase space, QED and hadronic corrections to the lowest order formula, summarised into $\Delta q$. This term is calculated to second order QED [14] so that the relative theoretical uncertainty on G_F is less than 0.3 ppm. The two recent experiments by the MuLan [15] and FAST [16] collaborations analysed the decay time of muons that are brought to rest in a target. Both experiments measured the decay rate of positrons from the reaction $\mu^+\to e^+ \nu_e \bar{\nu}_\mu$ in scintillator detectors. MuLan is using a pulsed muon beam and scintillator detectors that are arranged spherically around the target, while FAST works with a continuous muon beam, and the scintillator pixel detector is the actual target. After background subtraction and noise correction, the muon lifetime is extracted from the exponential decay time spectrum with the following results [15,16]

$$\tau_\mu {({FAST})}=2.197083(32)(15)\upmu{{s}}$$

((4.13))

$$\tau_\mu {({MuLan})}=2.197013(21)(11)\upmu{{s}},$$

((4.14))

where the first error is statistical and the second systematic. The corresponding values of G_F improve on the current world average (WA) [17]:

$$G_{F} {({MuLan})} =1.166 371(6)\times 10^{-5}{GeV}^{-2}$$

((4.15))

$$G_{F} {({FAST})} =1.166 353(9)\times 10^{-5}{GeV}^{-2}$$

((4.16))

$$G_{F} {({WA})} =1.166 37(1) \times 10^{-5}{GeV}^{-2}.$$

((4.17))

For the current analysis of electroweak data this improvement is however not so relevant because the precision of G_F largely exceeds those of the other electroweak parameters. The last value of G_F is therefore currently used in combined data analyses. Still, it is nice to see that the more recent measurements are in good agreement with this value.

4.3.2 Atomic Parity Violation

Sensitivity to electroweak parameters is also given in atomic parity violation experiments. Z boson contributions in the interaction of electrons and nucleus lead to parity violation when atomic hyperfine amplitudes are studied. The currently most precise experiments [18] studied the parity violating, highly forbidden 6S–7S transition in Caesium, ${^{133}_{55}Cs}$. The experiments shine two polarised laser beams on Cs vapour to first pump the atoms into the excited state and then use the second beam as analyser. An external electric field is applied to create an additional Starck-induced component which interferes with the parity violating transition. The asymmetry between two perpendicular polarisation directions of the analysing laser beam is then proportional to the parity violation strength.

The corresponding probability amplitude is calculated precisely in atomic many-body theory. It is proportional to the weak charge of the atomic nucleus

$$Q_W(Z,N) = -2\left\{C_{1u}(2Z+N)+C_{1d}(Z+2N) \right\},$$

((4.18))

for an atom with Z protons and N neutrons. The weak charges of the up and down quark, $C_{1u}$ and $C_{1d}$, are proportional to the axial and vector couplings of Eq. (1.28):

$$C_{1q}=2g_{Ae}g_{Vq}.$$

((4.19))

The atomic weak charge can therefore also be written as

$$Q_W(Z,N)\approx Z(1-4\sin^2\theta_w)-N$$

((4.20))

The most recent calculation [19] yields $Q_W(Cs)=-72.74\pm0.46$.

4.3.3 MØller Scattering at EA316

The MØller scattering experiment EA316 at the End Station A [20] at SLAC used a high-intensity pulsed and polarised electron beam of 45 and 48 GeV energy which passed through a cylinder filled with liquid hydrogen of 1.57 m length. The scattered electrons were detected in a magnetic spectrometer in an angular range $4.4<\theta_{{lab}} <7.5 {{mrad}}$ in the laboratory frame. MØller scattering electrons of 13–24 GeV are measured and separated from background of ep scattering in a radially segmented calorimeter. The average beam polarisation is $P_b=0.89\pm 0.04$.

In presence of neutral charge currents a parity violating asymmetry in the measured electron rate is expected between right-handed and left-handed incident beam:

$$A_{PV}=\frac{\sigma_R-\sigma_L}{\sigma_R+\sigma_L} \;.$$

((4.21))

After correcting for the polarisation P_b, calorimeter linearity, beam characteristics and backgrounds, the result obtained is

$$A_{PV}=-131\pm 14 {({stat}.)} \pm 10 {({syst}.)} \times 10^{-9}$$

((4.22))

The relation to the effective weak mixing angle is given by

$$A_{PV}=-A(Q^2,y) \rho^{(e;e)}\left\{1-4\sin^2\theta_{eff}(Q)+\Delta\right\},$$

((4.23))

where $\rho^{(e;e)}$ is the low-energy ratio of the weak neutral charge and the charged couplings, and $\Delta$ contains residual $O(\alpha)$ corrections. The effective analysing power is given by

$$A(Q^2,y)=\frac{G_F Q^2}{\sqrt{2}\pi\alpha_{{QED}}(Q)}\frac{1-y}{1+y^4+(1-y)^4}F_{{QED}}$$

((4.24))

where the momentum transfer is $Q^2=0.026{GeV}^2$, $y=Q^2/s\approx 0.6$. The factor $F_{{QED}}=1.01\pm 0.01$ takes QED radiative corrections into account, like ISR/FSR, box and vertex corrections. Numerically, A is determined from simulations to be $A=(3.25\pm 0.05) \times 10^{-9}$. The electroweak mixing angle derived from this measurement is

$$\sin^2\theta_{eff}(Q^2)=0.2397\pm 0.0010 {({stat}.)}\pm 0.0008 {({syst}.)}$$

((4.25))

at the experimental $Q^2=0.026{GeV}^2$, in agreement with a Standard Model expectation of $\sin^2\theta_{eff}(Q^2)=0.2381\pm 0.0006$. After evolution to M_Z one obtains

$$\sin^2\theta_w\left(M_Z^2\right)=0.2330\pm 0.0011 {({stat}.)}\pm 0.0009 {({syst}.)} \pm 0.0006 {({evolution})}$$

((4.26))

which is used as input to the electroweak data analysis.

4.3.4 Neutrino-Nucleon Scattering at NuTeV

A measurement of the weak mixing angle is also performed in $\nu$ and $\bar{\nu}$ nucleon interactions in the NuTeV detector [21], which was built as a 690 ton steel-scintillator target. The very pure muon neutrino and anti-neutrino beams undergo charged (CC) and neutral current (NC) interactions. Both NC and CC reactions create short-range hadronic cascades in the detector, only the CC reaction however produces a final state muon that penetrates the detector over a longer distance. The length of the events measured from the interaction vertex in units of traversed scintillator counters is therefore a characterisation of their NC and CC nature.

By counting those events the NC/CC ratio of the cross-section differences of $\nu$ and $\bar{\nu}$ reactions can be determined:

$$R^- = \frac{\sigma(\nu_\mu N\to \nu_\mu X)-\sigma(\bar{\nu}_\mu N\to \bar{\nu}_\mu X)}{\sigma(\nu_\mu N\to \mu^- X)-\sigma(\bar{\nu}_\mu N\to \mu^+ X)}$$

((4.27))

According to Paschos-Wolfenstein [22] this ratio is related to the left and right handed couplings of the neutrinos and the u and d valence quarks in the target by

$$R^-=4g_{L\nu}^2\sum_{u,d}\left\{g_{Lq}^2-g_{Rq}^2\right\} =\rho_\nu\rho_{ud}\left\{\frac{1}{2}-\sin^2\theta_w\right\} \;,$$

((4.28))

from which the on-shell value of $\sin^2 \theta_w$ can be extracted. NuTeV determines [21]

$$\begin{aligned}\sin^2\theta_w &{}= 1-\frac{M_W^2}{M_Z^2}\\ &{}= 0.22773 \pm 0.00135 {({stat}.)} \pm 0.00093 {({syst}.)}\\ &- 0.00022 \left(\frac{m_{t}^2-(175 {GeV})^2}{(50 {GeV})^2}\right) + 0.00032 \ln\left(\frac{M_{H}}{150 {GeV}}\right)\end{aligned}$$

with the given dependence on m_t and M_H. This measured value disagrees from the Standard Model prediction by about 3$\sigma$. Figure 4.5 compares the low-$Q^2$ measurements to the Standard Model prediction, after conversion to $\sin^2\theta_w$. However, $R^-$ is a derived quantity obtained from the individual NC/CC ratios for neutrinos and anti-neutrinos, $R^\nu$ and $R^{\bar{\nu}}$, which are measured by NuTeV directly. The deviation can be isolated to be mainly in $R^\nu$ which is also much more sensitive to $\sin^2\theta_w$. The ratios $R^\nu$ and $R^{\bar{\nu}}$ are displayed separately in Fig. 4.5.

Fig 4.5

(a) Evolution of the electroweak mixing angle $\sin^2\theta_w$ as a function of momentum transfer Q, compared to the measurements in atomic parity violation, MØller scattering, neutrino-nucleon scattering and at the Z pole [20]. (b) NC/CC ratios as measured by NuTeV separately in neutrino and anti-neutrino nucleon scattering, $R^\nu$ and $R^{\bar{\nu}}$ [23]. The anti-neutrino ratio $R^{\bar{\nu}}$ is in agreement with the Standard Model prediction, shown as a point, while $R^\nu$ is not. In a recent re-analysis of the data [17] the difference of the to the Standard Model prediction is reduced to 1.9 standard deviations

A complication of the NuTeV measurement arises however from the fact that also strange sea-quark contributions need to be taken into account in the parameter extraction. A detailed analysis of the strange/anti-strange asymmetry component in the PDFs yields a negative, but small correction of $-0.0014\pm 0.0010$ [17,24] to $\sin^2\theta_w$, reducing the Standard Model difference to 1.9 $\sigma$. Furthermore, new evaluations of the radiative correction terms involving QED hard photon emission [25] may lead to additional corrections of the measurement value and proper treatment of PDFs is needed [17,26]. Scrutinising the NuTeV result is therefore still ongoing. New measurements with higher resolution are proposed [27] and may resolve the situation experimentally.

4.3.5 Running of $\alpha_{QED}$

The running of the electromagnetic coupling $\alpha_{QED}$ is a consequence of the screening of the electromagnetic charge by polarisation of the vacuum. The running is experimentally established in the measurement of Bhabha scattering ${e}^+ {e}^-$ at LEP, where the differential cross-section $d\sigma/dt$ is determined. The dependence on the electromagnetic coupling is [28]:

$$\frac{d\sigma}{dt}=\frac{4\pi \alpha_{\rm QED}(t^2)}{t^2}(1+ \varepsilon)(1+ \delta_{\gamma})(1+\delta_{z})\;,$$

((4.30))

where the s-channel contributions $\delta_\gamma$ and $\delta_Z$ are much smaller than the radiative corrections $\varepsilon$. The overall scale of the cross-section is however not a good measure, because it is normalised by the beam luminosity which is derived from low-angle Bhabha scattering which again assumes $\alpha_{QED}(t^2)$ to be known. On the other hand the angular distribution contains information on $\alpha_{\textrm QED}(t^2)$ since

$$Q^2=t\approx -s\frac{1-\cos\theta}{2},$$

((4.31))

and s is known precisely at LEP. Such a measurement is performed by OPAL at low $Q^2$ [28] in the very forward luminosity monitor with its very high angular resolution and by L3 in both low and high $Q^2$ regions [29]. Figure 4.6 summarises the LEP results [30] and shows clearly that data are incompatible with a constant value of $\alpha_{QED}$ and that the running follows the QED prediction.

Fig 4.6

(a) Electromagnetic coupling constant measured by LEP in Bhabha scattering as a function of $Q^2$ compared to a constant value of $\alpha_{QED}$ and the QED prediction [30]. (b) Data and corresponding uncertainty used to determine R(s) in regions where perturbative QCD can not be safely applied [32]

For the electroweak data analysis a more precise method is however necessary. As mentioned in Chap. 1, the leptonic and top quark part of $\Delta \alpha(M_Z^2)$ are calculable with small theoretical uncertainty. Experimental input is then used to derive the hadronic contributions, $\Delta \alpha_{had}^{(5)}$, applying the dispersion relation [31]:

$$\Delta \alpha_{had}^{(5)}(Q^2)=-\frac{Q^2\alpha_{QED}}{3\pi}\int_0^{\infty}ds\frac{R(s)}{s(s-Q^2)}$$

((4.32))

with the hadronic to leptonic ${e}^+{e}^-$ annihilation cross-section ratio

$$R(s)=\frac{\sigma^0(e+e^-\to {{hadrons}})}{\sigma^0(e^+ e^-\to \mu^+\mu^-)},{{and}}\; \sigma^0(e^+e^-\to \mu^+\mu^-)=\frac{4\pi\alpha_{\textrm QED}^2}{3s}.$$

((4.33))

The ratio R(s) is calculable in perturbative QCD only in regions away from $q \bar{q}$ thresholds and resonances. The approach of [32] is therefore to apply theoretical calculations for $\sqrt{s}>12 {GeV}$ and experimental data for lower $\sqrt{s}$, which is shown in Fig. 4.6. This yields the already mentioned value of [32]:

$$\Delta \alpha_{had}^{(5)}=0.02758\pm 0.00035\;.$$

((4.34))

A more theoretically driven value is obtained in [31], where perturbative QCD is used to calculate R(s) also for $\sqrt{s}>1.4 {GeV}$, with the result

$$\Delta \alpha_{had}^{(1)}=0.02749\pm 0.00012.$$

((4.35))

The two estimations agree well, and also with other independent determinations [17].

4.3.6 Anomalous Magnetic Moment of the Muon

Charged elementary particles with half-integer spin have a magnetic dipole moment $\mu$ parallel to the spin ${\textbf s}$ :

$$\boldsymbol{\mu}=g\frac{q}{2m}{\textbf s}\;.$$

((4.36))

with the Landé g factor. For leptons, g has a value of about 2, and the exact value depends on radiative corrections , summarised in the anomaly:

$$a_\ell=\frac{g-2}{2}.$$

((4.37))

The measurements of these parameters are sensitive to higher order quantum corrections from QED, electroweak theory, hadronic contributions and possibly new physics beyond the Standard Model. The sensitivity of $a_\ell$ to high mass scales $\Lambda$ is proportional to

$$\frac{\delta a_\ell}{a_\ell}=\frac{m_\ell^2}{\Lambda^2}$$

((4.38))

so that it is more advantageous to measure $g\!\!-\!\!2$ with muons than with electrons. The most precisely measured value of $a_\mu$ is obtained by the E821 experiment [33]. Polarised muons from a pion beam were stored in a superconducting magnet ring producing a highly uniform ${\textbf B}$ field of 1.45 T. The degree of polarisation reached 95%. For vertical focusing of the muon beam, electric quadrupoles were arranged around the ring which had a central orbital radius of 7.11 m. The cyclotron frequency taking both electrical and magnetic fields into account is given by [34]

$$\omega_C=-\frac{q}{m}\left\{\frac{{\textbf B}}{\gamma} - \frac{\gamma}{\gamma^2-1}{\beta}\times{\textbf B}\right\},$$

((4.39))

and the spin precession frequency is equal to

$$\omega_S=-\frac{q}{m}\left\{\left(\frac{g-2}{2}+\frac{1}{\gamma}\right){{\textbf B}}-\frac{g-2}{2}\frac{\gamma}{\gamma+1}({\beta}\cdot{{\textbf B}}){\beta}-\left(\frac{g}{2}-\frac{\gamma}{\gamma+1}\right){\beta}\times{{\textbf E}}\right\}$$

((4.40))

with the relativistic $\gamma$ factor and the velocity ${\beta}$ of the muons. For a magnetic field perpendicular to the muon momentum, $\beta\cdot{\textbf B}=0$, the spin precession relative to the momentum occurs at a frequency

$$\begin{aligned}\omega_a &{}= \omega_S-\omega_C\\ &{}=-\frac{q}{m}\left\{ a_\mu {{\textbf B}} - \left(a_\mu-\frac{1}{\gamma^2-1}\right) \beta \times{{\textbf E}}\right\}.\end{aligned}$$

The dependence on the electric field is eliminated for the “magic” $\gamma=29.3$ [35], which corresponds to a muon momentum of $p=3.09 {GeV}$. Due to this simplification, $a_\mu$ can be determined from a measurement of $\omega_a$ and B. The latter is measured with proton-NMR based on water probes, using the proton Larmor frequency

$$\omega_p=g_p\frac{eB}{2m_p}\;.$$

((4.42))

The muon anomalous magnetic moment is therefore derived from the relation

$$a_\mu=\frac{\omega_a/\omega_p}{\lambda-(\omega_a/\omega_p)}$$

((4.43))

with $\lambda=\mu_\mu/\mu_p=3.18334539(10)$ [36]. The value of $\omega_p$ is calibrated to a spherical ${H_2O}$ probe with a small systematic uncertainty of $0.17\times 10^{-9}$.

What remains to be measured is $\omega_a$, which is done by detecting the decay electron rate of the parity violating muon decay $\mu\to e\nu_e\nu_\mu$. The preferred direction of the decay electron with respect to the muon spin depends on the electron energy [34]. Applying a calorimetric selection of electrons above a certain energy threshold, $E_{th}$, the measured electron rate follows the time dependence

$$N(t,E_{th})=N_0(E_{th})e^{-t/\gamma\tau_\mu}\left[1+A(E_{th})\cos(\omega_a t + \phi(E_{th})\right]\;,$$

((4.44))

with the asymmetry parameter

$$A(E_{th})=P\frac{y_{th}(2 y_{th}+1)}{-y_{th}^2+y_{th}+3}\;.$$

((4.45))

This parameter depends on the polarisation $P\approx 95\%$ of the muon beam and the energy ratio $y_{th}=E_{th}/E_{\max}$, where the maximal electron energy in the laboratory frame is $E_{\max}=3.09 {GeV}$. Since the momenta of the decay electrons are smaller than the muon momenta, the electrons are swept to the inside of the storage ring and are detected in 24 scintillating fibre calorimeters, evenly placed around the ring. A typical energy threshold is 1.8 GeV and the corresponding asymmetry, A, is about 35%. Figure 4.7 shows the time dependence of the detected number of electrons with the typical oscillation structure overlaid on an exponential decay curve.

Fig 4.7

(a) Total number of electron above 1.8 GeV as a function of time (modulo $100\,\upmu{s}$) from the 2001 $\mu^-$ data set of E821 (from [34])

The detailed data analysis takes the beam dynamics, electrical field corrections, pitch effects, magnetic field systematics into account. The final experimental value is

$$a_\mu({{exp}.}) = 11659208.0(6.3)\times 10^{-10},$$

((4.46))

with a statistical uncertainty of $5\times 10^{-10}$ and a systematic uncertainty of $4\times 10^{-10}$. The total relative uncertainty is $5\times 10^{-10}$.

This high experimental precision is a great challenge for the theoretical calculation of the expected value of $a_\mu$. The first order QED result [37]

$$a_\mu=\frac{\alpha_{QED}}{2\pi}$$

((4.47))

is known since a long time already. The recent calculations cover:

• complete QED 4-loop results, 5-loop leading logarithmic corrections, and an estimation of the remaining 5-loop corrections, which would involve 12672 Feynman diagrams but is beyond the accuracy needed [34,38]

  $$a_\mu({{QED}})=(1165847180.9 \pm 1.4_{5-{{loops}}} \pm 0.8_{\alpha_{QED}} \pm 0.4_{{masses}})\times 10^{-10}$$

  ((4.48))

• hadronic vacuum polarisation [39]

  $$a_\mu({{had}.}) = (690.8 \pm 4.4)\times 10^{-10}$$

  ((4.49))

• hadronic light-by-light scattering [40,41]

  $$a_\mu({{lbl}.}) = (+10.5 \pm 2.6) \times 10^{-10}$$

  ((4.50))

• higher order hadronic vacuum polarisation [40]

  $$a_\mu({{had}.,{ho}.}) = (-9.8 \pm 0.1)\times 10^{-10}$$

  ((4.51))

• electroweak contribution in leading 2-loop and 3-loop order [42,40]

  $$a_\mu({{EW}}) = (15.2 \pm 0.2)\times 10^{-10},$$

  ((4.52))

The result of these complex calculations is the Standard Model prediction of [40]

$$a_\mu({{SM}}) = (11659178.5 \pm 5.1)\times 10^{-10}\;.$$

((4.53))

The deviation of the theory from experiment

$$\Delta a_\mu=a_\mu({{exp}.})-a_\mu({{SM}})=(29.5\pm 8.1) \times 10^{-10}$$

((4.54))

is at the level of 3.6 standard deviations. The theoretical uncertainty is slightly smaller than the experimental one and all recent theoretical approaches [17,34,40] are showing a discrepancy in the order of 3 standard deviations or more. Some uncertainty is however still in the treatment of the hadronic contributions to $a_\mu({{SM}})$, which are usually derived from ${e}^+{e}^- \to {{hadrons}}$ data. If $\tau$-spectral functions corrected for isospin-breaking effects are used instead, the deviation from the Standard Model is only about 2 sigma [43]. Preliminary data from $\pi^+\pi^-+\gamma_{ISR}$ production with initial state photons taken at the $\Upsilon(4S)$ resonance by the BaBar collaboration point in the same direction [44]. More experimental and theoretical understanding of the different ways to extract the hadronic corrections to $a_\mu({{SM}})$ is thus necessary.

In general it is however interesting to observe that the electroweak corrections are small compared to the hadronic uncertainties, so that they cast dependencies on the Standard Model parameters, including M_H. The electroweak term can be written as [34,42]

$$\begin{aligned}a_\mu({{EW}})&{}=\frac{G_F}{\sqrt{2}}\frac{m_\mu^2}{8\pi^2}\left\{ \frac{5}{3}+\frac{1}{3}\left(1-4\sin^2\theta_w\right)^2-\frac{\alpha_{QED}}{\pi}[155.5(4)(1.8)]\right\}\nonumber\\ &{}=(15.4\pm 0.2 \pm 0.1)\times 10^{-10}\;.\end{aligned}$$

((4.55))

The $a_\mu$ dependence on the mass of the Standard Model Higgs boson is only of the order $\frac{G_F}{\sqrt{2}}\frac{m_\mu^2}{4\pi}\frac{m_\mu^2}{M_H^2}\log\frac{M_H^2}{m_\mu^2}$ [34], and determines the first $\pm 0.2\times 10^{-10}$ error, while the second is from higher-order hadronic effects.

On the other hand, new physics may enter the game through loop contributions. Super-symmetric particles, for example, would give rise to an additional term [45]

$$a_\mu({SUSY})\approx 13 \times 10^{-10} \left(\frac{100 {GeV}}{M_{SUSY}}\right)^2{tan}\beta\,{{sign}}(\mu),$$

((4.56))

where $M_{SUSY}$ denotes the common mass scale of SUSY particles, $\tan\beta$ the ratio of the two Higgs vacuum expectation values of the two Higgs doublets, and $\mu$ the Higgsino mass parameter. Assuming SUSY masses in the order of 200 GeV, i.e. in the near reach of the LHC, a value of $\beta\approx 6$ would compensate the full $\approx 3 \sigma$ difference between experiment and theoretical prediction.

4.4 Model Analysis of Electroweak Data

The data set of precision electroweak measurements consists of

• precise M_Z and M_W measurements at LEP, SLD, and Tevatron,

• precise Z line-shape data from LEP and SLD, including Z width, branching fractions, and decay asymmetries,

• the top mass determination at the Tevatron,

• measurements at low $Q^2$.

The Standard Model predictions for the electroweak observables are calculated using the TOPAZ0 [46] and ZFITTER [47] programs. They contain higher order QCD corrections, two-loop corrections for M_W, complete fermionic two-loop corrections for $\sin^2\theta_{eff}^{\ell}$, and three-loop top-quark contributions to the $\rho$ parameter [13] . The theoretical uncertainties on M_W are 4 MeV and 0.000049 on $\sin^2\theta_{eff}^{\ell}$. As stated in [13], the latter dominates the theoretical uncertainty in Standard Model fits and the indirect M_H extraction, which could be improved by a more accurate, two-loop calculation of the partial Z decay widths.

The measured high-$Q^2$ observables are compared to the theoretical predictions in Fig. 4.8. The free parameters in the fit to data are $\Delta\alpha_{{had}}^{(5)}$, $\alpha_s(M_z^2)$, M_Z, m_t, M_H. The agreement is good, and expressed in terms of $\chi^2/{{DoF}}$ a value of $17.3/13$ is obtained, which corresponds to a fit probability of 18%. Adding the low-$Q^2$ measurements, excluding $a_\mu$, one gets a $\chi^2/{{DoF}}=27.5/17$ and a probability of only 5.2%. This is mainly due to the NuTeV neutrino-nucleon scattering result, as discussed above. The muon anomalous magnetic moment has only negligible sensitivity to electroweak parameters, but should still be added to the global $\chi^2$. Doing this, one finally obtains a $\chi^2/{{DoF}}$ of $40.5/18$ or a probability of 0.2%. The “outliers” are easily identified:

• the forward-backward asymmetry of b quarks at the Z pole, $A_{{FB}}^{0,b}$, with 2.9 $\sigma$, and

• the muon anomalous magnetic moment, $a_\mu$, with 3.6 $\sigma$.

Both gave rise to numerous theoretical speculations and explanations. An attractive one exists for the latter, $a_\mu$. In the Minimal Super-symmetric Standard Model (MSSM) additional contributions from elementary super-partner particles in the first and second order loops lead to additional radiative corrections that shift the expectation value closer to the measurement [45].

A global data analysis in the constrained MSSM (CMSSM) was performed in [48], with the result shown in Fig. 4.8. Apart from $a_\mu$, the SUSY-sensitive heavy flavour observables $Br(b\to s\gamma)$, $Br(B_s\to \mu^+\mu^-)$, and the cold dark matter (CDM) density in the universe, $\Omega_c h^2$.

Fig 4.8

Compatibility of electroweak data with the SM [13] (left) and the CMSSM [48] (right). The left table compares most recent data, including 2009 results, while the right table is recent, but not updated with latest results. The level of agreement with electroweak data is however practically unchanged

The first two are rare b-decay processes that are induced by penguin loops in the Standard Model. They get enhanced by additional SUSY loop-contributions which are proportional to ${tan}\beta$ and ${tan}^6\beta$ [45], respectively. The most recent measurements [49] are $Br(b\to s\gamma)=(352\pm 23\pm 9)\times 10^{-6}$, and $Br(B_s \to \mu^+\mu^-)< 2.3 \times 10^{-8}$. The CMSSM analysis uses the experimental to theory ratio $R(b\to s\gamma)=Br(b\to s\gamma)/Br_{SM}(b\to s\gamma)$ in the fit.

The dark matter density is measured from a scan of the cosmic microwave background (CMB) by WMAP [50]. A fit to the angular power spectrum of temperature and polarisation data, adding further information from small and large scale cosmic structures, yields a matter density of $\Omega_m h^2=0.128\pm0.008$ and a baryonic density of $\Omega_b h^2=0.0223\pm0.007$, so that the cold dark matter density is $\Omega_c h^2=\Omega_m h^2-\Omega_b h^2=0.106\pm0.009$. Assuming super-symmetry with R-parity conservation, i.e. with a Lightest Super-symmetric Particle (LSP) that can not decay, this particle contributes to the CDM density. The predicted value of $\Omega_c h^2$ actually is proportional to the mass of the LSP and to the thermally averaged annihilation cross-section of the LSP to Standard Model particles. Theoretical models (e.g. [51]) take details of the annihilation and co-annihilation processes into account in the determination of the temperature dependent abundances of dark matter.

The CMSSM (or mSUGRA) analysis [48] assumes the soft SUSY-breaking scalar masses to be universal at the GUT scale, with a value of M_0, as well as the soft SUSY-breaking gaugino masses, $M_{1/2}$, and the trilinear couplings, A_0. Furthermore, the ratio of the two vacuum expectation values of the Higgs doublets, ${tan} \beta$, and the sign of the Higgs mixing parameter $\mu$ are varied. The parameters at the electroweak scale are derived from renormalisation group equations (RGE). As can be seen in Fig. 4.8, the CMSSM fit describes the observables used in the previous analysis as well as the Standard Model fit. The constraints from heavy flavour decays and the CDM density are also fulfilled and in agreement with the measured values of $R(b\to s\gamma)$, $Br(B_s\to \mu^+\mu^-)$ and $\Omega_c h^2$. The main difference to the Standard Model analysis is the much better agreement with the observed value of $a_\mu$. In terms of $\chi^2$ the CMSSM fit reaches of 17.34 per 14 of freedom, which corresponds to a fit probability of 24%. The CMSSM is therefore removing parts of the discrepancy between theory and experiment, but can not explain all deviations, like the 2.7 $\sigma$ effect in $A_{FB}^{0,b}$.

Another comparison of Standard Model and MSSM predictions [52] with data is shown in Fig. 4.9, where the theory is confronted with the current W and top mass measurements. As can be observed, the Standard Model is in agreement with these data if the Higgs boson mass is small and close to the LEP exclusion limit of ${\mathrm{MHz}}>114.4 {GeV}$ [53]. On the other hand, SUSY is fitting well if the SUSY mass scale is not too light.

Fig 4.9

Direct measurement of W and top mass, compared to Standard Model calculations with varying Higgs mass and to MSSM predictions with different SUSY mass scales [52]

4.5 Electroweak Constraints on New Particles

The last missing particle to complete the Standard Model is the Higgs boson, which up to now has not been discovered yet. Theoretical arguments (see Chap. 1) indicate that its mass should be in the sub- TeV region. The currently most stringent search limits are from the LEP and Tevatron experiments. The final LEP result is shown in Fig. 4.10, from which a lower limit of ${\mathrm{MHz}}>114.4 {GeV}$ at 95% C.L. [53] is derived . Very recent search results by CDF and $D\phi$ using up to $4.2 \textrm{fb}^{-1}$ of $p\bar{p}$ data per experiment, shown as well in Fig. 4.10, furthermore exclude a Higgs mass between 160 and 170 GeV at 95% C.L. [54] . With the full data set of projected $2\times 10 \textrm{fb}^{-1}$ until the end of 2011, a larger mass interval will be covered.

Fig 4.10

(a) Confidence level ratio of the signal+background hypothesis, CL_s, as observed and expected in LEP data as a function of M_H [53]. The lower limit on M_H is derived from the crossing point with the $CL_s=0.95$ line. The observed limit of $114.4 {GeV}$ is slightly lower than expected. The bands indicate the 65 and 95% probability regions of the expected limit in absense of any signal. (b) Observed and expected 95% C.L. upper limits on the ratio of the Standard Model cross-section of Higgs production, as a function of M_H for the combined, preliminary Tevatron data. The bands indicate the 65 and 95% probability regions of the expected limit in absense of any signal. The observed limit corresponds to an exclusion of a Higgs mass between 160 and 170 GeV at 95% C.L. [54]

Fig 4.11

(a) Electroweak measurements and their individual constraints on the Higgs mass [12]. (b) The sensitivity of the SM and CMSSM Higgs boson h expressed in terms of $\Delta\chi^2$ of the combined fit to the electroweak data [48]

However, as long as the Higgs boson or super-symmetric particles are not discovered, the analysis of current electroweak data can give hints about the possible mass range of these particles.¹ Figure 4.11 shows the sensitivity of some of the electroweak parameters to the Higgs mass in terms of the corresponding constraints on M_H that can be derived. All data combined are used in the Standard Model analysis which yields the left $\Delta\chi^2$ curve of Fig. 4.11. Theoretical uncertainties are indicated by the band, but they do not change the general behaviour of a clear minimum at rather low M_H. Not taking into account the low-$Q^2$ data, the $\chi^2$ minimum is at a Higgs mass of $90^{+36}_{-27} {GeV}$. Since the dependence of radiative corrections on M_H is logarithmic (see Eq. 1.50), the one-sided 95% confidence level upper limit is found at 163 GeV, increasing to $191 {GeV}$ when the lower direct Higgs mass bound is included [13].

In the CMSSM analysis [48] the mass of the lightest CP-even Higgs boson is even more constrained, because there is a theoretical upper bound of about 135 GeV. On the other hand, the minimal $\chi^2$ is at $M_h= 110^{+8}_{-10} {({exp}.)} \pm 3 {({theo}.)} {GeV}$, and higher than the Standard Model Higgs mass. This is in much better agreement with the lower bounds on M_h obtained by LEP for the lightest super-symmetric Higgs boson which are in the order of 90 GeV [56].

The CMSSM fit [48] gives further indications on preferred parameter values, which are best compatible with data:

$$\begin{aligned}M_0&{}=85^{+40}_{-28}{GeV}\\ M_{1/2}&{}=280^{+140}_{-30} {GeV}\\ A_0&{}=-360^{+300}_{-140} {GeV}\\ \tan\beta&{}=10^{+9}_{-4} {GeV}\\ {sign}(\mu)&{}=+1 ({fixed})\end{aligned}.$$

With this parameter set, the super-symmetric particle mass spectrum consists of a light Higgs boson at $M_h \approx 100 {GeV}$, charged and heavy Higgs bosons at $M_{H^{\pm}},M_H,M_A \approx 450\hbox{--}500 {GeV}$, and the lightest neutralino as LSP at $M_{\tilde{\chi}^0_1} \approx$ $100\hbox{--}120 {GeV}$.

The prospects for the Large Hadron Collider to discover the Standard Model Higgs boson are therefore excellent, although challenging in the low M_H region, as will be discussed in the Chap. 7. If the electroweak and Higgs sector of the Standard Model is not realised in its current form and if super-symmetry is the extension nature has chosen, the LHC is as well the ideal machine to push the discovery frontier far beyond Tevatron and LEP. Assuming, for example, the CMSSM scenario and the above described parameter set, only $1 \textrm{fb}^{-1}$ of pp data will be sufficient to see clear evidence for super-symmetric particle production [57].