Announcements 11/26 Tomorrow:10.1, 10.3 Wednesday: 10.4, 10.5, 10.7 Problem 10.5: diagram only Friday: 10.8, Error in eq Find the decay rate for top decay, t  W + b, neglect- ing the bottom mass, but including the W-mass

Announcements 11/28 Today: 10.4, 10.5, 10.8 Problem 10.5: diagram only Friday: 10.7, Monday: read 11A-11C, Quiz T A muon neutrino with energy E = 10.0 MeV is attempting to scatter off of a stationary electron. What is s? What is the cross-section? How far would the neutrino have to travel on average before scattering if it were traveling through water? Determine the decay rate and branching ratio for the Z.

Announcements 11/30 Today: 10.7, Monday: read 11A-11D, Quiz T Wednesday: read 11E-11H, Quiz U, problems 11.1, 11.3

Questions from the Reading Quiz “Could we go over the CKM matrix?” “What is its role in the decay rate of the W-boson?”

The Quarks, the W, and their Masses And God said: Let there be three left-handed doublets of quarks And three right-handed up quarks And three right-handed down quarks And God said: Let there be W-exchange within the doublet W And God said: Let the up quarks acquire mass MUMU mDmD And down quarks too But God chose not to make the mass matrices diagonal

Straightening out the Mass Matrices We can always straighten out the up mass matrices by replacing u RA and u LA by linear combinations to make M diagonal W Similarly, you can straighten out the down mass matrices MUMU mDmD But this will leave the W-couplings mixed up If you wanted to, you could just straighten out one of the mass matrices (say M) and leave the other one alone Then the down quark mass matrix is complicated, but the W- coupling is simple

W-decays in two bases Let’s pretend we are calculating the W-decay to quarks There are nine combinations on the right Suppose we consider the down-type masses negligible Then we could diagonalize only the up-type quarks The complication is that d” actually represents more than one particle Hence A = B = 1 actually represents ud*, us*, and ub*

Questions from the Reading Quiz “Is there a reason that the W-particle decays to hadrons 2/3 of the time, but only 1/9 to everything else?” Let’s work in the basis where only the up-quark mass matrix is diagonal Look at all the possible decays: Top is too heavy Quarks come in three colors The e, , , u, d, s, and c quarks are pretty light compared to the W The b quark is kind of light, and almost completely absent anyway All these decays have the same coupling, and therefore the same rate What is the branching ratio for any one of them?

Review of Leptons and Weak Stuff The electron field is actually a combination of left- and right-handed parts Split them up The left-handed electron and the left handed neutrino go together into an SU(2) L doublet Let’s pretend there’s just the electron and its corresponding neutrino Neutrino has only a left-handed part So there is really just a left-handed doublet with Y = -1/2 and a right-handed singlet with Y = -1. They are also singlets under color

The Dirac Equation for Leptons Each of the fields has its corresponding Dirac Equation The derivative terms can be shown to be invariant under SU(2) L and U(1) Y The mass terms are not

Announcements 12/3 Today: read 11A-11D, Quiz T Wednesday: read 11E-11H, Quiz U, problems 11.1, 11.3

Questions from the Reading Quiz “Is a fermion essentially massless until it interacts with the Higgs field? If the Higgs field is responsible for all fermion masses, how do the fermions exist before they couple to the Higgs?” “What were fermions like before the spontaneous symmetry breaking occurred?”

Questions from the Reading Quiz “It almost seems as though the Higgs field and Higgs boson are used interchangeably sometimes (like the top of page 178). Is there a difference? If so, what is it and why does it seem to be so subtle?” “It almost seems as though the electromagnetic field and photon are used interchangeably sometimes. Is there a difference? If so, what is it and why does it seem to be so subtle?”

Questions from the Reading Quiz “Why do we say that the Higgs can't decay into W or Z's when, according to Wikipedia, a Higgs with a mass of 126 GeV will decay to W's 23.1% of the time and Z's 2.9% of the time?.” “For masses below 130 GeV/c², the Higgs decays mainly in a pair of b quarks. The decay into a pair of tau leptons is also important in that region of mass. The search in the 2 photons final state is relevant for a Higgs mass lower than a 150 GeV/c². For larger masses, the decay is almost entirely through the H→WW*/WW and H→ZZ*/ZZ processes.”

Virtual Particles in Z-decay For low mass Higgs: Platinum plated event Gold plated event For low-intermediate mass Higgs ( ) we can’t make real WW pairs For intermediate mass Higgs: But we can make one real and one virtual W And then later

Announcements 12/5 Today: read 11E-11H, problems 11.1, 11.3 Friday: Quiz U, problems 11.4, 5, 7

Questions from the Reading Quiz “I'm still a little unclear regarding the local gauge transformations that assure the correct PHI0 direction of the minimized potential?” A simplified theory: the Abelian Higgs model Gauge group U(1) One scalar field  with charge +1 The constant a is irrelevant This doesn’t have a minimum unless c > 0 If b > 0 then the minimum is at  = 0 If b < 0 then the minimum is at

Making the phase go away The minimum is at The phase of this field is, in fact, arbitrary We aren’t necessarily at the minimum Write the field in general as: This theory is gauge invariant under the transformation: Pick the gauge choice: Then we have: The remaining degree of freedom gets “eaten” by the photon, which becomes massive. Something similar happens in superconductivity

And now, back to the real theory: In general, the field looks like: A general SU(2) gauge transformation looks like: With a suitable choice of  a (x), we can make this of the form Rewrite this in the form:

The Standard Model – Minimal Description What do we need to describe the standard model? We need the gauge group: SU(3) c  SU(2) L  U(1) Y We need a list of how the scalar particles transform under this group: We need a list of how the left- and right-handed fermions transform under this group: One quark One Lepton Times three: Now simply include every possible coupling

Another way of doing the fermions It is arbitrary what you call a particle and what you call an anti-particle For example, we have a right-handed electron and a left-handed positron If we add either one, the other is implied By convention, usually add only left-handed fields: Gauge group: Scalars: Left-handed fermions: Note that the first two numbers in each triplet tell you how many particles there are of that type So, for example, there are 3  3  2 particles with charge +1/6 But the last number tells us Y, and doesn’t count as a multiplier

Let’s count parameters How many relevant real parameters are there in the standard model? Gauge couplings: 3 Can pick g s, g and g’ Normally pick  s, , and sin 2  W Scalar potential, 2 Can pick v and Usually pick (v or G F ) and m H Fermion masses: 9 Six quarks Three charged leptons Weak mixings from CKM matrix: 4 Three angles One phase Total: 18 real parameters

Anomalies Certain diagrams called triangle diagrams cause problems They ruin gauge invariance unless we get really lucky Several quantities, if summed over left-handed fermions, must vanish, otherwise we are dead: For T 3 2, note that this is ¼ for SU(2) doublets and 0 for singlets

Flexibility of the Standard Model Are we sure we have it right No, but … Can we just add another left-handed quark pair? This would be a new massless pair of quarks Easily detected Okay, we’ll add a whole new set of quarks We will make them heavy so no one notices But this ruins anomaly cancellation Fine, we’ll throw in some leptons too Make the leptons and the quarks all heavy But this increases the number of neutrinos and changes the Z-width

Flexibility of the Standard Model (cont.) What other changes can we make? Can we change the scalar sector? Certainly possible Can we add neutrino masses? Many ways to do this N = 3 is not even demanded by anomaly cancellation