Effective operator and allowed loop level interactions

[Safinaz]

Hi all,

Some processes can not happen at the tree level, but it happen via loops, like for Higgs decay to pair of glouns or pair of photons, (h -> gg), (h -> y y) . For instance, effectively h -> gg written as $~ h~ G^a_{\mu\nu} G_a^{\mu\nu}~$ which is Lorentz and gauge invariant ..
Now if we considered instead of SM- h, a coloured triplet scalar, T= (3,2), then on the loop level i could mediate this decay $T^0 \to yy$ by a loop of charged SU(3) triplet scalars $T^\pm$ such that the coluor will be conserved in that process, but still effectively there is no an operator as $T^a B_{\mu\nu} B^{\mu\nu}$ (where a is the SU(3) index) because it is not gauge invariant,

The question can this process happen or no ?
In short, is it a condition for a process to be allowed on the loop level to has an effective operator or it's just sufficient that for this process to conserve colour or charge via loop mediator ?

Best.

 

[RGevo]

Dear safinaz,

Please see http://arxiv.org/abs/1505.02646.

The operator h G^a_munu G_a^munu is not gauge invariant. The Higgs is part of a double and is charged under su(2). The correct operator has H^dag H before the field strengths, this operator leads to an interaction of the sort you write with Wilson coefficients and vev there too.

The operators arise when you integrate out degrees of freedom. If the fields you have in mind generate the correct operator then yes, the interactions can occur. But they have to be possible (at tree or loop level) in the full theory.

The gauge invariant operator you say that does not exist is, indeed does and is listed in the class 4 operators in the paper I cited.

I suggest some lectures on effective field theory, try those by manohar as an example.

 

[Safinaz]

Hello,

The operator $h G^a_{\mu\nu} G_a^{\mu\nu}$ is not gauge invariant. The Higgs is part of a double and is charged under su(2).

I meant by "h" here the SU(2) singlet (after SU(2) symmetry breaking and acquiring vev ), so that the couplings here as you said will be of dimension mass to be re-normalizable term. Also i meant by being gauge invariant that it's now singlet under SU(2), and ( 8 $\otimes$ 8 = 1 )
under SU(3).

The operators arise when you integrate out degrees of freedom.

Do you mean by integrating out degrees of freedom: like in$~ H^\dagger ~H~ G^a_{\mu \nu} G_a^{\mu \nu}$, when the Higgs fileds acquire vevs. and we got at last an operator : $v~ h G^a_{\mu \nu} G_a^{\mu \nu}$ ?

If I take that for the operator I'm looking for, consider: $S_a^\dagger ~S^a B_{\mu \nu} B^{\mu \nu}$, where S is $(8,2,1/2)$ colored scalar. So it's doublet under SU(2) like SM Higgs, but it dosn't acquire vev. at SSB., so that we can't get an operator as:
$v~ s^a B_{\mu \nu} B^{\mu \nu}$ (s^a now in a neutral scalar, singlet under SU(2)). Also it's not SU(3) invariant because there is a loss SU(3) index "a"

Please see http://arxiv.org/abs/1505.02646.

Thanks for the reference, but i can't find coloured scalars contributions to the effective operators of $h \to \gamma \gamma$, is it talking only in the domain of the SM ..

 

[RGevo]

Hello,
I meant by "h" here the SU(2) singlet (after SU(2) symmetry breaking and acquiring vev ), so that the couplings here as you said will be of dimension mass to be re-normalizable term. Also i meant by being gauge invariant that it's now singlet under SU(2), and ( 8 $\otimes$ 8 = 1 )
under SU(3).

I know what you meant. You want to expand around the VEV in the broken phase of the theory, where a combination of the Higgs doublet H (charged under SU2) and another doublet field also charged under SU2 generates a gauge invariant interactions which contain the field h. But my point is that you should consider the operators of the quark doublets, Higgs doublets etc. or else you will write down interactions which can't actually be generated in a gauge invariant way.

Do you mean by integrating out degrees of freedom: like in$~ H^\dagger ~H~ G^a_{\mu \nu} G_a^{\mu \nu}$, when the Higgs fileds acquire vevs. and we got at last an operator : $v~ h G^a_{\mu \nu} G_a^{\mu \nu}$ ?

The Higgs paper I cited does the calculation for H->gamma gamma generically in the dimension-6 baryon number conserving EFT up to one-loop. Therefore, the calculation involve the operators which enter the tree level expression in the EFT, as well as the one-loop contributions.

Consequently, if you have an idea of a BSM theory which can generate any of these operators (appearing in either the tree-level or the loop-level result), then your theory will alter this rate.

Therefore, the job which you have is to check whether your model does generate these interactions.

You should note that ONLY BSM physics generates these operators, since the calculation is performed in the full SM appended by the operators. Since you have some idea for new physics (a new Scalar with $(8,2,1/2)$ +...etc.), if you write down the possible interactions it has with SM particles, then it should be easier for you to see what operators it will generate.

If I take that for the operator I'm looking for, consider: $S_a^\dagger ~S^a B_{\mu \nu} B^{\mu \nu}$, where S is $(8,2,1/2)$ colored scalar. So it's doublet under SU(2) like SM Higgs, but it dosn't acquire vev. at SSB., so that we can't get an operator as:
$v~ s^a B_{\mu \nu} B^{\mu \nu}$ (s^a now in a neutral scalar, singlet under SU(2)). Also it's not SU(3) invariant because there is a loss SU(3) index "a"

Thanks for the reference, but i can't find coloured scalars contributions to the effective operators of $h \to \gamma \gamma$, is it talking only in the domain of the SM ..

As mentioned above, that's because its in the effective theory of operators involving SM fields. YOU must figure out which operators your particular theory generates.

Edit: I can't see how you can generate the operator you need, since the only interaction I see working with SM fields is $S_a^\dagger ~S^a H^\dagger ~H$. So have H-> SS, or some one-loop contributions to the higgs mass and wave function renormalisation... Do you agree?