HIGH ENERGY PHYSICS DEPARTMENT
UNIVERSIDAD NACIONAL AUTONOMA DE MEXICO



 PRECISION ANALYSIS OF THE ELECTROWEAK STANDARD MODEL.


THE STANDARD MODEL.

The Standard Model (SM) of strong and electroweak interactions is a mathematically consistent quantum field theory which predicts or is consistent with all known aspects of the elementary particles and their interactions over an enormous range of probes and scales. In particular, it is now clear that, with the possible exception of the Higgs sector, the SM is the correct theory of nature to an excellent approximation down to a distance scale of 10-16 cm, corresponding to energies of about 100 GeV.

Weak neutral currents are a primary prediction and a direct test of electroweak unification , a key feature of the SM. They were discovered in 1973 by the Gargamelle experiment at CERN, and confirmed by the HPWF detector at Fermilab. Subsequently, neutrino-nucleon and neutrino-electron scattering experiments improved to the per cent level, testing the weak interaction quantitatively. Electron-deuteron and electron-positron scattering, as well as atomic parity violation experiments, are sensitive to weak-electromagnetic interference effects, and were crucial for the confirmation of the electroweak SM. The W and Z bosons, the very massive mediators of the weak force, were finally discovered directly by the UA1 and UA2 Collaborations at CERN in 1982 and 1983, respectively.

With the basic structure of the SM established, CERN's Large Electron Positron accelerator (LEP) and SLAC's Stanford Linear Collider (SLC) were designed to test it on the quantum level. With these machines it was possible to determine many Z properties with per mille accuracy, including the outstanding measurement of the Z boson mass at LEP 1 with a relative precision of 2 parts in 100,000. Run I of Fermilab's Tevatron (CDF and DØ) and the second phase of LEP (ALEPH, DELPHI, L3, and OPAL), contribute per mille determinations of the W boson mass. With the mass of the top quark as determined at the Tevatron, other high precision observables can also be calculated within the SM. The agreement with the measurements establishes the SM as a spontaneously broken renormalizable gauge theory, and verifies the gauge group and representations. It predicts at least one extra state, the Higgs boson, with a mass below or about 1 TeV (1000 GeV). Combining all direct and indirect data in a likelihood fit, it is possible to extract more precise information on its mass leading to upper bounds of at most a few hundred GeV.

The high accuracy of theory and experiment allows severe constraints on possible TeV scale physics, such as unification or compositeness. For example, the ideas of technicolor and non-supersymmetric Grand Unified Theories (GUT's) are strongly disfavored. On the other hand, supersymmetric unification, as generically predicted by string theories, is supported by the observed approximate gauge coupling unification at an energy slightly below the Planck scale.

Scientists at the University of Pennsylvania and at the Universidad Nacional Autónoma de México (UNAM) have long been involved with these questions. Members of the Department of Physics and Astronomy played a crucial role in the discovery of the top quark at Fermilab, in the discovery of CP violation in B mesons, and in many neutrino experiments. The theoretical and phenomenological efforts include a long-term ongoing project to analyze precision data, and are summarized on this web site.

The standard electroweak model (SM) is based on the gauge group SU(2)xU(1) [S. Weinberg , A. Salam , S.L. Glashow, J. Iliopoulos, and L. Maiani], with gauge bosons Wμi, i = 1,2,3, and B&mu for the SU(2) and U(1) factors, respectively, and the corresponding gauge coupling constants g and g'. The left-handed fermion fields $\displaystyle \psi_i =\left( \begin{array}{c} \mbox{$\nu_i$}\\ \ell_{i}^- \end{array} \right)\hspace{-.2cm}\raisebox{-1.8ex}{$_L$}$ and $\displaystyle \left( \begin{array}{c} \mbox{$u_j$}\ \mbox{$d^{'}_i$} \end{array} \right)$ of the ith fermion family transform as doublets under SU(2), where di'≡∑jVijdj, and Vij is the Cabibbo-Kobayashi-Maskawa mixing matriz. (Constrains on Vij and tests of universality are discussed in H. Abele et al and in the Section on the Cabibbo-Kobayashi-Maskawa mixing matrix of the Particle Data Group.) The right handed fields are SU(2) singlets. In the minimal model there are three fermion families and a single complex Higgs doublet $\displaystyle \phi \equiv \left( \begin{array}{c} \phi^{+}\ \phi^{o} \end{array} \right) $.

After spontaneous symmetry breaking the Lagrangian for the fermion fields is

$\displaystyle \mathcal{L}_F$ $\displaystyle =$ $\displaystyle \sum_i \bar{\psi}_i \bigg{(}i{\partial \kern-0.19cm/}-m_i-\frac{gm_iH}{2M_W}\bigg{)}\psi_i$
    $\displaystyle -\frac{g}{s\sqrt{2}}\sum_i \bar{\psi}_i \gamma^\mu (1-\gamma^5)(T^+W_\mu^+ + T^- W_\mu^-)\psi_i$
    $\displaystyle -e\sum_i q_i \bar{\psi}_i \gamma^\mu \psi_i A_\mu$
    $\displaystyle -\frac{g}{2\cos \theta_W} \sum_i \bar{\psi}_i \gamma^\mu (g_V^i-g_A^i \gamma^5)\psi_i Z_\mu.$  (1)

&thetaW &equiv tang-1 (g' / g) is the weak angle; e = g sin θW is the positron electric charge; and A ≡ B cos θW + W3 sin θW is the (massless) photon field. W± ≡ (W 1 - i W 2) / √2 and Z ≡ - B sin &thetaW + W 3 cos &thetaW are the massive charged and neutral weak boson fields, respectively. T + and T - are the weak isospin raising and lowering operators. The vector and axial-vector couplings are


gVi ≡ t3L(i) - 2 qi sin2 θW ,     (2.a)
gAi &equiv t3L(i)     (2.b)

where t3L(i) is the weak isospin of fermion i (+1/2 for &mui and &nui; -1/2 for di and ei) and qi is the charge of &psii in units of e .

The second term in $ \mathcal{L}_F$ represents the charged-current weak interaction [J. M. Conrad, M. H. Shaevitz, and T. bolton, P. Langacker]. For example, the coupling of a W to an electron and a neutrino is

$\displaystyle -\frac{e}{s\sqrt{2}\sin\theta_W}\bigg{[}W_\mu^-~\bar{e}~\gamma^\mu~(1-\gamma^5)~\nu~+~W_\mu^+~ \bar{\nu}~\gamma^\mu ~(1-\gamma^5)~e\bigg{]}$     (3)  

For momenta small compared to MW, this term gives rise to the efective four-fermion interaction with the Fermi constant given (at tree level, i.e., lowest order in perturbation theory) by GF / √2 = g 2 / 8MW2 . CP violation is incorporated in the SM by a single observable phase in Vij. The third term in $ \mathcal{L}_F$ describes electromagnetic interactions (QED), and the last is the weak neutral-current interaction.

In Eq. (1), mi is the mass of the ith fermion ψi. For the quarks these are the current masses. For the light quarks, as described in the Particle Listings, mu ≈ 1.5 - 4.5 MeV, md ≈ 5 - 8.5 MeV and ms ≈ 80 - 155 MeV. These are running MS masses evaluated at the scale μ = 2 GeV . (In this section we denote quantities defined in the MS scheme by a caret; the exception is the strong coupling constant, &alphas, which will always correspond to the MS definition and where the caret will be dropped.) For the heavier quarks we use QCD sum rule constraints [Jens Erler and M. Luo] and recalculate their masses in each call of our fits to account for their direct &alphas dependence. We find, mc(&mu = mc) = 1.290 +0.040 -0.045 GeV, with a correlation of 29%. The top quark "pole" mass, mt = 177.9 ± 4.4 GeV, is an average of CDF results from run I [CDF: T. Affolder et. al.] and run II [E. Thomson for the CDF Collaborationet. al.], as well as the DØ dilepton [DØ : B. Abbott et al.] and lepton plus jets [F. Canelli for the DØ Collaboration.] channels. The latter has been recently reanalized, leading to a somewhat higher value. We computed the covariance matrix accounting for correlated systematic uncertainties between the different channels and experiments according to [CDF: T. Affolder et. al.] and [L. Demortier et. al.]. Our covariance matrix also accounts for a common 0.6 GeV uncertainty (the size of the tree-loop term [K. Melkinov and T. v. Ritbergen] ) due to the conversion from the pole mass to the MS mass. We are using a BLM optimized [S. J. Brodsky, G. P. Lepage, and P. B. Mackenzie] version of the two-loop perturbative QCD formula [N. gray et al.] which should correspond approximately to the kinematic mass extracted from the collider events. The tree-loop formula [K. Melkinov and T. v. Ritbergen] gives virtually identical results. We use MS masses in all expressions to minimize theoretical uncertainties. We will use above value for mt (together with MH = 117 GeV) for the numerical values quoted in section 2 and 4. See "The Note on Quark Masses" in the particle listings for more information. In the presence of right-handed neutrinos, Eq. (1) gives rise also to Dirac neutrino masses. The posibility of Majorana masses is discussed in "Neutrino mass" in the Particle Listings [J. Gunion, H. E. Haber, G.L. Kane, and S. dawson, The Hunter's Guide, M. Sher, M. Carena and H. E. Haber] .

H is the physical neutral Higgs scalar which is the only remaining part of &Phi after spontaneous symmetry breaking. The Yukawa coupling of H to &psii, which is flavor diagonal in the minimal model, is gmi / 2MW . In non-minimal models there are additional charged and neutral scalar Higgs particles.

If you are interested to continue reading about the Electroweak Standard Model, we recommend the following items:

Jens Erler
Instituto de Fisica-UNAM
e-mail: erler@fisica.unam.mx 		Paul Langacker
Department of Physics, University of Pennsylvania
e-mail: pgl@electroweak.hep.upenn.edu 		Marcial Sanchez
Instituto de Fisica - UNAM
e-mail: mar@fisica.unam.mx