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1. Consider a field theory with Weyl fermion fields (ψ, ψ

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1. Consider a field theory with Weyl fermion fields (ψ, ψ

c

, λ) with electric charges respectively equal to (1. − 1, 0), with scalars (φ, φ

c

) with charges equal to (1, −1) and lagrangian

L = L

kin

+ h

g

1

ψλφ

c

+ g

2

ψ

c

λφ + m

2 λλ + m

23

φφ

c

+ h.c. i

(0.1) + m

21

|φ|

2

+ m

22

|φ

c

|

2

+ (g

3

(φφ

c

)

2

+ h.c.) + g

4

|φ|

4

+ g

5

|φ

c

|

4

+ g

6

|φφ

c

|

2

(0.2) Is the fermion ψ naturally massless in such a theory? Explain the result using symmetries.

What about the case m

3

= 0. (Assume the scalar potential is positive definite so that no scalar acquires a vacuum expectation value).

2. Consider now adding a neutral scalar η to the above model, modifying the lagrangian to L = L

kin

+ [g

1

ψλφ

c

+ g

2

ψ

c

λφ + g

7

ηλλ + m

3

η

∗

φφ

c

+ h.c.] (0.3) + m

21

|φ|

2

+ m

22

|φ

c

|

2

+ m

23

|η|

2

+ g

4

|φ|

4

+ g

5

|φ

c

|

4

+ g

6

|φφ

c

|

2

(0.4) Are fermions naturally massless in the above field theory? Explain the result as before (again the scalar potential is assumed positive definite).

1

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