XI SERC School on Experimental High-Energy Physics National Institute of Science Education and Research 13 th November 2017 Standard Model & Beyond Lecture III Sreerup Raychaudhuri TIFR, Mumbai
2 Fermions in the SU(2) W Gauge Theory If fermions are to interact with the, and bosons, they must transform as doublets under, just like the scalar doublet Consider a fermion doublet where the and are two mass-degenerate Dirac fermions. Taking construct the free Lagrangian density Sum of two free Dirac fermion Lagrangian densities, with equal masses.
3 Now, under a global gauge transformation, if then It follows that the Lagrangian density must be invariant under global gauge transformations. As before, we try to upgrade this to a local invariance, by writing gauge where Invariance is now guaranteed. as before.
4 Expand the covariant derivate and get the full Lagrangian density free fermion free gauge interaction term Expand the interaction term... are charged currents (c.c.) is a neutral current (n.c.)
5 Write the currents explicitly:
6 c.c. interactions n.c. interactions This leads to vertices
7 Comparing with the IVB hypothesis for the able to identify, we should be or or Q. Can we identify the with the photon (forgetting its mass)? If the are charged, we will have, under Now, if the term is to remain invariant, we must assign charges and to the A and B, s.t. term transforms as To keep the Lagrangian neutral, we require
8 But if we look at the vertices, we must identify vertices, and consider them to be QED i.e.. Now solve the equations: and... result is Two alternatives: A and B cannot be the Fermi-IVB particles defeats whole effort... cannot be the photon... (already hinted by the mass)
9 Electroweak Unification Why not just include the group as a direct product with the group? The can be a new (neutral) gauge boson. The transformation matrix on a fermion of charge look like will then where is the generator of and the direct product means that The gauge field matrix should expand to and give us interaction terms as before...
10 i.e. to the interaction terms with the W boson we must now add interaction terms with the photon: Working back, we can write this as where
11 This generator of can be rewritten Paradox! which contradicts Glashow (1961) : We cannot treat weak interactions and electromagnetism as separate (direct product) gauge theories i.e. they must be considered together electroweak unification
12 SU(2) W x U(1) y Model Introduce a new which is different from and exists as a direct product with the... The gauge transformation matrix will become where, which, by construction, commutes with all the We now expand the gauge field matrix as is a new gauge field and is a new quantum number which is clearly same for both the A and B component of the fermion doublet.
13 We now construct the gauge-fermion interaction term as before Expanding as before
14 Glashow (1961): for some reason, the and mix, i.e. the physical states are orthonormal combinations (demanded by gauge kinetic terms) of the and... In terms of this, the neutral current terms come out to be If we now wish to identify with the photon, we require to set
15 Solving for and we get Recall that. It follows that Choose. Then Note that is some arbitrary angle... it must be nonzero, else We can also obtain Now, these are precisely the eigenvalues of the operator i.e. (Looks like Gell-Mann-Nishijima relation for quarks...) Call the weak isospin and the weak hypercharge
This gauge theory works pretty well and can give the correct couplings of all the gauge bosons... up to the angle, which is not determined by the fermion sector... Determination of Back to the gauge boson mass term... (Salam 1966, Weinberg 1967) 16 Glashow theory: include the field in the gauge field matrix, i.e. where is the hypercharge of the field.
17 Thus, and Multiplying these
18 Consider only the neutral bosons: One cannot have mass terms of the form and in a viable field theory, since our starting point is always a theory with free fields. Thus, it is essential to transform to orthogonal states and choose to cancel out cross terms...
19 Rewrite the neutral boson mass terms as The diagonalising matrix will be where How to determine?
20 Write out the interaction terms for the gauge bosons with the scalar doublet. One finds that once again, to match the couplings to the charges of the W bosons, we get the Gell-Mann-Nishijima relation, i.e. Now, the lower component develops a vacuum expectation value, so it must be neutral, i.e. It follows that Weinberg angle
21 Eigenvalues of the mass matrix: Determinant = 0 ; trace =, i.e. and
22 Determination of parameters: Experimental measurements show that GeV and GeV It follows that We can now calculate:
23 Parity Violation The puzzle Consider the Fermi form of the current-current interaction: Under parity: i.e. parity is conserved in the Fermi theory Before the 1950s, it was thought that parity conservation is as sacred as conservation of energy, momentum and angular momentum... But it was known that some particle are pseudoscalars, This led to the famous e.g. pions and Kaons have intrinsic parity puzzle
24 Through the early 1950s, cosmic ray experiments showed the existence of two degenerate particles and, each with mass around 483 MeV and lifetime around 12 ns. However, it was seen that indicating that and. Note that the phase space for these decays is very different: MeV MeV Since the lifetimes are identical, the strength of weak interactions must be different for these different decays universality is violated
Yang & Lee (1956) Maybe parity is not conserved in weak interactions, i.e. is really is really parity-violating channel parity-conserving channel They also showed that none of the earlier experiments had really tested intrinsic parity violation... Suggested that if the mean value of a parity-odd variable was nonzero e.g. if could be found to be nonzero This would be a smoking gun signal for parity violation Experiment was actually performed by Wu et al (1957)... 25
26 Maximal parity violation In 1956, Marshak & Sudarshan, and separately, Feynman & Gell-Mann, assumed the parity-violating weak interactions to be of the form Parity is conserved when (V current), (A current) Parity is maximally violated when (V-A currents) Parity is partially violated for other values of... Rewrite the leptonic current as
If we can measure the chirality of neutrinos emitted in beta decay, then we should have 27 and hence Goldhaber et al did an experiment in 1957 with the electron capture process and found that the neutrino is always left-chiral... It follows that.
Thus the weak interactions do have the form V A and the W boson vertex for electrons is of the form 28 We will have similar interactions for the muon and for the nucleons and for the quarks i.e. couple only to left-handed quarks and leptons
29 Hence, the interaction must be of the form i.e. we must have doublets whereas the right-handed partners must be singlets of and there are no right-handed neutrinos X X
30 Trouble with the mass terms Mass Generation Now, for example, is not gauge invariant, because transforms under but does not gauge invariance demands Appeal to spontaneous symmetry-breaking Write a Yukawa interaction term Now, after symmetry-breaking, we have to shift
31 Now, so we get i.e. quark mass is created by Yukawa coupling and Higgs vev and Yukawa coupling is
32 For the -quark, Here is the charge conjugate of When, we will have where Now, we will have i.e. quark mass is created by Yukawa coupling and Higgs vev
33 Flavour Mixing Above picture is too naïve assumes only and quarks Full Yukawa Lagrangian
34 Full mass terms Mass matrices These are, in general, complex 3 x 3 matrices with no special symmetry
35 Consider -quark term: Theorem: Any square complex matrix can be diagonalised by a bi-unitary transf n
36 We say that the and are in the gauge basis (unphysical), while the and are in the mass basis (physical). Similarly, if we define and, or, and we can get
The physical quarks are related to the gauge basis by 37
38 Rewrite the charged current interaction terms... where is the Cabibbo-Kobayashi-Maskawa (CKM) matrix Thus, the physical charged currents can have mixed flavours (FCCC)
39 What about the electromagnetic current? Now,, so that No flavour-mixing in the e.m. current. Similarly no FCNC.
Flavour-mixing occurs solely through the charged current, and the CKM matrix. This can be parametrised as 40 neglecting. Experimentally CKM matrix has hierarchical structure about diagonal
41 Charged current interactions CP Violation Under, we can show that i.e. If, then is not invariant, i.e. is not conserved (violated). Since and we see that the CKM matrix does have violated. Cause of matter-antimatter asymmetry our very existence