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1. Discuss fermion mass in a model with one fermion λ, one scalar φ and lagrangian

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The first two exercises are about naturalness and spurions, the third is about the ‘accidental’

properties of renormalizable lagrangians. A second set of questions on exercise 3 will be given next time.

1. Discuss fermion mass in a model with one fermion λ, one scalar φ and lagrangian

L = L

kin

+ [g

1

λλφ + g

2

φ

4

+ h.c.] + V (|φ|) (0.1)

2. Do the same for a model with fermions λ, ψ, N , scalar φ and lagrangian

L = L

kin

+ [g

1

λψφ + g

2

ψφ

∗

N + mN

2

+ g

3

φ

4

+ h.c.] + V (|φ|) (0.2) 3. Consider a field theory with gauge symmetry U (1)

L

× U (1)

R

× U(1)

B

with the following

spectrum of left handed Weyl spinors

ψ

i

= (1, 0, 1) N = (0, 0, 1) (0.3)

ψ

ic

= (0, 1, −1) N

c

= (0, 0, −1) (0.4) where i is a flavor index that runs from 1 to 3 (more generally one could even think of i = 1, . . . , n). In the scalar sector the fields are

ϕ = (−1, 0, 0) Φ = (2, 0, 0) (0.5)

ϕ

c

= (0, −1, 0) Φ

c

= (0, 2, 0) (0.6)

T = (−2, 0, −2) T

c

= (0, −2, −2) . (0.7) It is further assumed that ϕ, ϕ

c

, Φ and Φ

c

acquire an expectation value while T and T

c

do not. The gauge symmetry is thus broken according to: U (1)

L

× U (1)

R

× U (1)

B

→ U (1)

B

. Write down the most general renormalizable lagrangian and discuss the features of the fermion mass spectrum at tree level and beyond (notice you do not need to work out all the details of the scalar potential, but only a few couplings. assume all mass parameters in the scalar potential are comparable and much smaller than the cut-off scale...forget naturalness).

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