1 Anomalies and the Standard Model

Transcription

1 1 Anomalies and the Standard Model The Glashow-Weinberg-Salam model of the electroweak interactions has been very successful in explaining a wide range of experimental observations. The only major prediction of the model that remains to be verified experimentally is the existence of the Higgs boson. The GWS is based on SU(2) L U(1) Y gauge symmetry. A weak isospin triplet of massless W bosons couples to left-handed massless quarks and leptons. A weak isospin singlet massless gauge boson B couples to weak hypercharge of massless fermions. The gauge covariant derivative acting on any field takes the form D µ = µ iga a µt a ig Y B µ, where T a is the SU(2) representation matrix of the field and Y is its hypercharge. The W and B bosons also couple to a complex weak isospin doublet scalar field called the Higgs field. The potential of the Higgs field is chosen so that the vacuum breaks the global SU(2) L U(1) Y symmetry. This potentially results in three massless Goldstone bosons, but because of the Higgs mechanism, the Goldstone bosons disappear and three of the gauge fields acquire mass as physical W ± and Z 0 states. The one remaining massless gauge boson is the photon of QED. The Higgs mechanism also allows for the fermions to acquire mass. However, the values of these masses are not fixed by the theory, which allows for one free parameter for each type of fermion. 1

2 1 ANOMALIES AND THE STANDARD MODEL Figure 1: Gross and Jackiw showed that anomalies spoil the renormalizability of gauge theories. 2

3 2 QUANTUM NUMBERS OF QUARKS AND LEPTONS 2 Quantum Numbers of Quarks and Leptons Unlike the gluons of QCD which couple only to quarks, the GWS gauge bosons couple to leptons and quarks. The lepton and quark fields are decomposed into left and right handed components. components assigned to weak isospin doublets: ( ) ( ) νe u E L = e, Q L =. d L L The left-handed The upper and lower components have weak isospoin ± 1 2. The right-handed fields are assigned to weak isospin singlets: e R, u R, d R. For a long time it was believed that the neutrinos were strictly massless and purely left-handed. Observations of neutrino oscillations have opened the possibility that neutrinos might have mass. Each of these particles has an electric charge that is a multiple Q of the charge of the proton. Hypercharge quantum number Y values are then assigned so that the electric charges come out right according to the relation Q = T z + Y. This requires the following assignments: Y EL = 1 2, Y Q L = + 1 6, Y e R = 1, Y ur = + 2 3, Y d R =

4 3 ANOMALY CANCELLATION REQUIREMENT 3 Anomaly Cancellation Requirement Because QCD is invariant under parity, the chiral current anomaly does not affect perturbative QCD. The neutral pion is a pseudo Goldstone boson of broken chiral symmetry. This is a non-perturbative problem in which the chiral anomaly plays and essential part. The GWS model is not symmetric under parity, and the W and Z bosons couple directly to chiral currents at the perturbative level. Soon after the model was introduced, Gross and Jackiw, among many others, realized such perturbative anomalies would make the GWS model non-renormalizable unless the quantum numbers of the fermions were such that all potential pertubative anomalies cancelled. The chiral anomaly is a disease of one-loop fermion triangle diagrams at least one of the three vertices involves the chiral Dirac matrix γ 5. Diagrams that can spoil renormalizablity are shown in Fig. 2. For each type, the sum of the two diagrams with the fermions circulating in opposite directions must be added. From the Feynman rules, the grouptheoretical factors involving the weak isopin and strong color matrices, and the weak hypercharges, can be factored out. The theory will be renormalizable if each such factor vanishes. For diagrams involving one U(1) current and 2 SU(3) currents, the overall factor is γ 5 Tr [ t a t b Y ] 1 = γ 5 2 δab ( ) L Y q, q where the sum runs over quarks q with a sign for the left-handed contributions. Evaluating the sum for 4

5 3 ANOMALY CANCELLATION REQUIREMENT Figure 2: From Peskin and Schroeder 5

6 4 ANOMALY CANCELLATION IN Z 0 PRODUCTION AT THE LHC the light u and d quarks gives ( 1) 2 1 ( ) ( 1 ) 3 = 0. Diagrams with a U(1) current and 2 SU(2) currents involve only left-handed fields ( ( 1) 1 ) ( + ( 1) 3 1 ) = 0, 2 6 where the quark contribution has been summed over the 3 colors. 4 Anomaly Cancellation in Z 0 Production at the LHC Huge numbers of W and Z bosons will be produced in pp collisions at the LHC. Because the masses of the W and Z bosons are large compared with the masses of the quark and gluon constituents in the colliding hadrons, it is generally a good approximation to assume that the quarks are massless in computing the production cross sections for these electroweak bosons at high energies. An exception, of course is the top quark, whose mass m t = 174 GeV, is approximately twice as large as that of the W or Z. Top quarks in loops pose a delicate problem because the Standard Model is renormalizable only if the left-handed quarks occur in SU(2) L doublets. Diagrams with quark triangle loops are shown in Fig. 3. 6

7 4 ANOMALY CANCELLATION IN Z 0 PRODUCTION AT THE LHC Figure 3: R. Gonsalves, J. Paw lowski, and C.-F. Wai, Phys. Rev. D 40, 2245 (1989). 7

8 4 ANOMALY CANCELLATION IN Z 0 PRODUCTION AT THE LHC It is important to evaluate the top-quark-mass contributions to measurable cross sections at LHC energies, in particular the W and Z transverse momentum distributions. Fig. 4 shows the Z transverse-momentum distribution at Tevatron and LHC energies including quark-triangle diagrams. Quark masses are taken into account using a simple threshold prescription by inserting a step function θ(q 2 T 4m2 f ) into sums over quark flavors f in the subprocess cross sections. There is a significant change in the shape of the distribution at LHC energies due specifically to the virtual triangle-anomaly diagrams. Because the triangle-anomaly contributions are comparable in magnitude to the O(α 2 s) QCD radiative corrections, it is essential to include the top and bottom quark masses in the virtual triangle diagram contributions. Homework Problem Compute the group theoretical factors and show that the anomalies cancel in each of the 10 types of triangle diagrams in the Standard Model. 8

9 4 ANOMALY CANCELLATION IN Z 0 PRODUCTION AT THE LHC 10 0 Z Transverse Momentum Distribution at the LHC and the Tevatron LHC: S 1/2 =14.0 TeV 10-1 LHC: No Virtual Triangle 10-2 Tevatron S 1/2 = 1.96 TeV Tevatron: No Virtual Triangle d!/dq 2 T (nb/gev2 ) Q T (GeV) Figure 4: Triangle diagram contributions to the Z 0 transverse momentum distribution at the Tevatron and the LHC 9