Announcements

1. Announcements • Today: 9.6, 9.8 • Friday: 10A – 10E • Monday: 10F – 10H 9.6 Only do differential cross-section See problem 7.7 to do most of the work for us 11/14

2. Weak Interactions Neutrinos • First discovered in -decay • Energy spectrum of electron: • Must be some particle carrying off the rest of the energy • We now know it is neutron decay • Probably an invisible, neutral particle • Must be a fermion to conserve angular momentum • Must be neutral • Must be very light (< 1 eV or so)

3. Other Neutrino Interactions • Muon – like a heavy electron, but unstable • Decay requires two invisible particles • Pion – strongly interacting particle • Easily produced in proton-nucleus collisions • Decays to muon plus neutrino • Neutrinos can be converted back to the corresponding charged particles • Electron neutrinos make electrons, muon neutrinos make muons

4. The Leptons • We have already discussed the quarks – there are six of them • They each come in three colors, and have strong interactions • There are six other spin ½ fermions in the standard model, called leptons • There are anti-particles for each of these as well • Neutrino physics is currently evolving better names are 1 2 3 • When weak interactions were first researched, quarks weren’t known • We will focus on leptons first

5. Fermi Theory of Muon Decay • First attempt – Fermi Theory – assumed this was a basic matrix element • Original guess was something like this: • New fundamental constant: • This is not the only form that respects Lorentz invariance: • These five combinations (and linear combinations) are only objects that respect Lorentz invariance includingP, T, and PT • No combination fit all the available date

6. V – A Theory of Muon Decay • In 1956, it was proposed that parity might be violated in weakinteractions • In early 1957, this was quickly experimentally verified • Suddenly there were other possibilities: • A coupling is called a vector coupling, and a 5 is called axial vector • We will call this the V – A theory of weak interactions • Only left-handed fields participate in these interactions

7. Muon decay rate calculation • Treat electron as massless: • The  terms are anti-symmetric under , the other terms are symmetric • The cross terms will automatically vanish

8. Muon decay rate calculation (2) • This amplitude squared was solved in problem 4.11 • Note decay rate rises rapidly as mass increases • Weak interactions get stronger as you go up in energy • Eventually, get probabilities >1  no good

9. Announcements • Today: 10A – 10E • Monday: 10F – 10H • Monday: 11/16

10. The W - particle • This interaction is not renormalizable, since GF ~ GeV-2 • Maybe this is not really what is going on? • To get this to work, we need W coupling something like: • The factor of 22 is for convenience later • The index  implies the W particle must havepolarization vector, like a photon • Spin 1, like a photon • The W must be charged, unlike a photon • The W must be massive, or it would have already been discovered

11. Dealing With Spin-1 Massive Particles • Polarization vector satisfies same equations as before: • But this time there are three such polarizations • For example, if • Then the three polarizations are: • We need to find • For propagator • For summing over initial/final states • The propagator:

12. Questions from the Reading Quiz “I have no idea what's going on with the groups and the electroweak coupling/interaction. I understand that the SU(2) and U(1) aren't really independent. But it's all confusing and it's making my head hurt.” • Spin 1 particles run into trouble with renormalization unless they are gauge-type couplings • What we think is going on so far is:

13. A Toy Model – The Two Photon Model • Surprisingly, it is sometimes ambiguous which are the actual particles • Consider the following toy model: The Carlson two-photon model • Classically, if you shake a particle with both types of charge, you would make both types of fields • Quantum mechanically, you would create states that are superpositions of each type of field • Unless there is something logically picking out particular directions in A1A2-space, it is not obvious which ones you want to think of as the “real” fields.

14. Rotating Fields Arbitrarily • We can change the fields in any arbitrary way, for example • We can just as easily work with these fields

15. Announcements • Today: 10F – 10H • Monday: 10.1, 10.3 • Wednesday: 10.4, 10.5, 10.8 11/16

16. Weak Interactions with One Lepton Pair • Naively, there is one charged lepton field and one neutrino, • The left- and right-handed pieces of the massive electron have different weak interactions, and should be divided • Without mass, only the left-handed neutrino has weak interactions • There is no reason to even believe there is a right-handed neutrino • Weak interactions connect the left-handed neutrino and electron • Masses connect the left- and right-handed electrons

17. Mass and Couplings with One Lepton • The Feynman rule for W-coupling for one lepton: The Vertex • There is no reason, in principle, that the mass can’t be apparently complex • This can easily be fixed, for example, by redefining the field eR by a phase: • Hence the phase is irrelevant • We work with eL and eR’, and drop the primes

18. Weak Interactions with Multiple Leptons • Naively, there are three charged lepton fields and neutrinos, • The left- and right-handed pieces of the massive electron have different weak interactions, and should be divided • Without mass, only the left-handed neutrino has weak interactions • There is no reason to even believe there is a right-handed neutrino • Weak interactions connect the left-handed neutrino and lepton • Masses connect the left- and right-handed leptons

19. The Weak coupling with Many leptons • The Feynman rule for W-coupling for many leptons: The Vertex

20. Complicated mass? • There is no reason, in principle, that the mass can’t be apparently complicated • This matrix is completely arbitrary • We can nonetheless always “change basis” to straighten it out • For example, suppose the mass matrix looked like this: • Define new states: • The new mass matrix is then:

21. Complicated Couplings? • We originally had The Vertex • But we now defined new states • This makes our W-couplings complicated • In the leptons, this can be fixed simply by similarly redefining the neutrinos • Drop the irrelevant primes

22. Weak Interactions with One Quark Pair • Naively, there is one up quark and one down quark • The left- and right-handed pieces of the massive quarks have different weak interactions, and should be divided • All four of these exist in the Standard Model • Weak interactions connect the up and down quarks • Masses connect the left- and right-handed quarks

23. The Weak coupling with many quarks • The Feynman rule for W-coupling for many quarks: The Vertex • Warning: This is actually incorrect! • This is different from the leptons • As I will explain soon (I hope)

24. Complicated mass? • There is no reason, in principle, that the masses can’t be apparently complicated • These matrices are completely arbitrary • We can nonetheless always “change basis” to straighten them out • For example, suppose the mass matrices looked like this: • Define new states: • The new mass matrix is then:

25. Complicated Couplings? • We originally had • But we now defined new states • This makes our W-couplings complicated • In the quarkss, can this can fixed simply by similarly redefining the up quarks? • No! This messes up the mass matrix M • The couplings really are complicated in the quark sector

26. The Weak coupling with many quarks • By appropriate redefinition of the various fields, the mass matrices for the up- and down-type quarks can always be made diagonal and real • Such a redefinition will, however, introduce a unitary matrix V into the charged current interactions • This matrix is called the Cabibbo-Kobayashi-Maskawa matrix, or CKM matrix • Some, but not all, of the parameters of V can be eliminated by appropriate redefinition of the corresponding fields.