XI SERC School on Experimental High-Energy Physics National Institute of Science Education and Research 13th November 2017

Standard Model & Beyond Lecture III

Sreerup Raychaudhuri TIFR, Mumbai

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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 gauge invariance, by writing

where as before. Invariance is now guaranteed. 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:







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c.c. interactions

n.c. interactions

This leads to vertices

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Comparing with the IVB hypothesis for the , we should be able to identify

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, and consider them to be QED vertices, we must identify

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 will then look like

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

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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

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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

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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

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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 16

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)

Glashow theory: include the field in the gauge field matrix, i.e.

where is the hypercharge of the field. 17

Thus,

and

Multiplying these

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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...

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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

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Eigenvalues of the mass matrix:

Determinant = 0 ; trace = , i.e.

and



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Determination of parameters:

Experimental measurements show that

GeV and GeV

It follows that  We can now calculate:

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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, e.g. pions and Kaons have intrinsic parity This led to the famous 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 25

Yang & Lee (1956) Maybe parity is not conserved in weak interactions, i.e. is really parity-violating channel is really 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)...

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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

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If we can measure the chirality of neutrinos emitted in beta decay, then we should have

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 .

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Thus the weak interactions do have the form V – A and the W boson vertex for electrons is of the form

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

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Mass Generation Trouble with the mass terms…

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

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Now, so we get

i.e.

⇒ quark mass is created by Yukawa coupling and Higgs vev

…and Yukawa coupling is

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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

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Full mass terms

Mass matrices

These are, in general, complex 3 x 3 matrices with no special symmetry

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Consider -quark term:

Theorem: Any square complex matrix can be diagonalised by a bi-unitary transfn

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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

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The physical quarks are related to the gauge basis by

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Rewrite the charged current interaction terms...

where is the Cabibbo-Kobayashi-Maskawa (CKM) matrix Thus, the physical charged currents can have mixed flavours… (FCCC)

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What about the electromagnetic current?

Now, , so that

No flavour-mixing in the e.m. current. Similarly no FCNC. 40

Flavour-mixing occurs solely through the charged current, and the CKM matrix . This can be parametrised as

neglecting .

Experimentally

CKM matrix has hierarchical structure about diagonal 41

CP Violation Charged current interactions

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