btphysics said:
Hello,
why one can use a bi-doublet scalar field (2,2) under SU(2)L x SU(2)R ? In terms of group theory, we should have only triplets (3,1) or (1,3) since 2 x 2=3+1 ? But in left right symmetric models, indeed yukawa coupling are formed with bi-doublet scalars.
Best regards
This is very much SM question. So, this thread should be moved to particle physics sub-forums.
As for the answer, recall that in the interacting theory of massless nucleon and mesons, we introduce the following 8 \times 8 matrix of meson fields
\Phi ( x ) = I_{ 8 \times 8 } \ \sigma ( x ) + i \gamma_{ 5 } \tau_{ i } \pi_{ i } ( x ) ,
where \sigma ( x ) is an iso-scalar in the [1] representation of SU(2) and \pi_{ i } ( x ) is an iso-vector in the [3] representation of SU(2). We couple this to the nucleon field N ( x ) \in \ [2] by (Yukawa)
\mathcal{ L }_{ \mbox{int} } = \bar{ N } ( x ) \Phi ( x ) N ( x ) .
a) The requirement that \mathcal{ L }_{ \mbox{int} } be invariant under the (vector) SU(2) transformation U, implies that
N ( x ) \rightarrow U N ( x ) , \ \ \ \Phi ( x ) \rightarrow U \Phi ( x ) U^{ \dagger } .
b) Since the nucleon in the model is massless, we also demand that \mathcal{ L }_{ \mbox{int} } be invariant under the axial iso-spin transformation U_{ 5 } = \exp ( i \gamma_{ 5 } \alpha_{ i } \tau_{ i } / 2 ). This implies
N \rightarrow U_{ 5 } N , \ \ \ \Phi \rightarrow U^{ \dagger }_{ 5 } \Phi U^{ \dagger }_{ 5 } .
With a bit of algebra we can combine the transformations in (a) and (b) to form the invariance group SU_{ L } (2) \times SU_{ R }(2) of \mathcal{ L }_{ \mbox{int} } as follows
N_{ R } \rightarrow R N_{ R } ,
in the (1 , 2) representation of SU_{ L } (2) \times SU_{ R }(2),
N_{ L } \rightarrow L N_{ L },
in the (2 , 1) representation, and
\Phi \rightarrow L \Phi R^{ \dagger },
in the (2 , 2) representation of SU_{ L } (2) \times SU_{ R }(2).
So, in short, it is the massless (chiral) fermions that require mesons from (2 , 2 ) representation.
See:
https://www.physicsforums.com/showpost.php?p=3819325&postcount=6
See also pages 115-121 in the textbook by Ta-Pei Cheng & Ling-Fong Li:
“Gauge Theory of Elementary Particle Physics, Problems and Solutions” , Oxford University Press, 2000.
Sam
