Restriction on terms in the SM lagrangian

[Aethaeon]

Hello,
Just a few questions about a couple of terms in and not in the SM Lagrangian. I'll talk in particular about these fields, and their representations in SU(3) x SU(2) x U(1)

Q (3,2,1/6) (left-handed quarks, fermion)
U (3,1,2/3) (right-handed up-quarks, fermion)
$$\phi$$ (1,2,-1/2) (higgs, scalar)

1) Mass terms
There is a term in the Lagrangian of the form:

$$Z^a_b \overline{Q_a}\not{\nabla}Q^b$$

Why can't there be a term like

$$Z^a_b \overline{Q_a}Q^b$$ where a and b are the SU(3) indices

The diagonal of Z would act as mass terms for the Qs
Likewise for the other fermion fields.. I don't see an immediate reason why those terms should be excluded?

2) Generalizations of pre-mass terms
There is a term in the Lagrangian of the form

$$Z^a_b \overline{Q_a_{\alpha}} U^b \phi^{\alpha}$$

with a,b SU(3) indices and $\alpha$ the SU(2) index.
The reason this works out is because

$$\overline{Q_a_{\alpha}}$$

transforms under $(\bar{3},\bar{2},1/6)$, and the two fields together

$$U^b \phi^{\alpha}$$

transform under (3,2,-1/6) -- so the interaction conserves all the charges

Why can't there be a more general term that can mix up the SU(2) indices, like

$$Z^a_b^{\alpha}_{\beta} \overline{Q_a_{\alpha}} U^b \phi^{\beta}$$

Thanks so much for your time and help..

 

[lpetrich]

A very simple reason: such mass terms would break those symmetries. To see why, let's work it out in detail.

Let's consider a Lagrangian term with the form Y_i^*M_ijX_j

where superscript * means complex conjugate.

Now transform X and Y according to a gauge transformation, with small size a and Hermitian matrix T:

X_i -> X_i + i*a*T_ijX_j
Y_i -> Y_i + i*a*T_ijY_j
Y_i^* -> Y_i^* - i*a*T_ij^*Y_j^* = Y_i^* - i*a*Y_j^**T_ji
Y_i^*M_ijX_j -> Y_i^*M_ijX_j + i*a*Y_i^*(-T_ikM_kj + M_ikT_kj)X_j = Y_i^*M_ijX_j - i*a*Y_i^*([T,M]_ij)X_j

The commutator
[T,M] = T.M - M.T

To maintain the gauge symmetry, [T,M] must vanish for all possible T.

Likewise, for starless Y, this quantity must vanish:
M_ikT_kj + M_kjT_ik

or T^T.M + M.T = 0

Let's consider what will make the commutator [T,M] vanish for all T in an algebra. If T is in an irreducible representation of that algebra, then by definition, [T,M] will only vanish for all T if M is proportional to the identity matrix.

-

So are the quarks in an irreducible representation of the QCD gauge-symmetry group? Are the left-handed quarks and leptons, and the Higgses in an irreducible representation of the weak-isospin symmetry group?

Their representations are fundamental representations for those groups, and their algebra transformation matrices T are given by
T(a,b,1)_ij = d_aid_bj + d_ajd_bi
T(a,b,2)_ij = -i*d_aid_bj + i*d_ajd_bi
T(a)_ij = d_ij*(1 if i < a, -a if i = a, 0 if i > a)

It's evident that one can construct a SU(2) subgroup out of any two values of the indices, so I'll handle that case first. Note that the matrices here are the Pauli matrices:
T(1,2,1) = s₁, T(1,2,2) = s₂, T(2) = s₃

Let's start with s₃. It gives us
[s₃, M] = {{0, 2M₁₂}, {-2M₂₁, 0}}
So M must be diagonal.

[s₁,M] = {{0, -M₁₁ + M₂₂}, {M₁₁ - M₂₂, 0}}
[s₂,M] = {{0, i*(M₁₁ - M₂₂)}, {i*(M₁₁ - M₂₂), 0}}
Thus, M must be proportional to the identity matrix.

The fundamental representation of SU(n) is thus irreducible, and the mass terms that Aethaeon proposes for quarks are not allowed for unbroken QCD symmetry.

-

Let's now take on the case of
T^T.M + M.T
vanishing for SU(2)

For Pauli matrix s₃, we get {{2M₁₁,0},{0,-2M₂₂}}
So M must have the form {{0,M₁₂},{M₂₁,0}}

For Pauli matrices s₁ and s₂, we get
{{M₁₂+M₂₁,0},{0,M₁₂+M₂₁}}
and
{{i*(M₁₂+M₂₁),0},{0,-i*(M₁₂+M₂₁)}}
Thus, M must be proportional to the antisymmetric symbol
eps = {{0,-1},{1,0}}

Since there is no way to do this consistently for SU(3) or higher n, this sort of term exists only for SU(2). So Aethaeon's proposed mass terms cannot exist there either.

 

[Parlyne]

I'd just like to add that the proposed quark mass term is incoherent, even if the the coupling is left color diagonal. The issue here is that Q represents just the left-handed quark fields. Then,
$$\overline{Q}Q$$
is not a Lorentz scalar. Thinking in the two-component formalism, the two fields have different types of spin indices; so, this combination should really be written as
$$\overline{Q}_{\dot{\alpha}}Q_\alpha.$$
Since these can't be summed over, the proposed mass term actually has two free spin indices.