## Higgs mechanism is essential

As the readers of my blog know, I have developed, in a series of papers, the way to manage massive solutions out of massless theories, both in classical and quantum cases. You can check my latest preprints here and here. To have an idea, if we consider an equation

$\Box\phi+\lambda\phi^3=0$

then a solution is

$\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)$

provided

$p^2=\mu^2\left(\frac{\lambda}{2}\right)^\frac{1}{2}$

being $\mu$ and $\theta$ two arbitrary constants and $\rm sn$ a Jacobi elliptical function. We see that a massless theory has massive solutions arising just from a strong nonlinearity into the equation of motion. The question one may ask is: Does this mechanism work to give mass to particles in the Standard Model? The answer is no and this can already be seen at a classical level. To show this, let us consider the following Yukawa model

$L=\bar{\psi}(i\gamma\cdot\partial-g\phi)\psi+\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}\phi^4$

being $g$ a Yukawa coupling. Assuming $\lambda$ very large, one is reduced to the solution of the following Dirac equation that holds at the leading order

$\left(i\gamma\cdot\partial-g\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)\right)\psi=0$

and this equation is exactly solved in a closed form, provided the fermion has exactly the same mass of the boson, that is $g=\sqrt{\lambda/2}$. So, we see that the massless fermion acquires mass too but it must be degenerate with respect to the bosonic field. This would imply that one needs a different scalar field for each fermion and such bosons would have the same masses of the fermions. This is exactly what happens in a supersymmetric theory but the theory we are considering is not. So, it would be interesting to reconsider all this with supersymmetry, surely something to do in the near future.

This means that Higgs mechanism is essential yet in the Standard Model to understand how to achieve a finite mass for all particles in the theory. We will see in the future what Nature reserved us about.

### 3 Responses to Higgs mechanism is essential

1. Luboš Motl says:

“One of the key ingredients to build up a quantum field theory is to have a set of solutions of classical equations of motion to start with. … [W]e have to know how to solve the theory in a limit we are not so familiar: strong coupling limit.”

Marco, this is an inconsistent, misguided combination of topics. Classical field equations, pretty much by definition, are only directly relevant in the classical limit of a quantum theory.

Quantum theory may “conventionally” (but not necessarily) be constructed as a quantum perturbation or deformation of a classical theory. When it is so, the classical equations and their solutions matter for the 0th approximation and perturbations are being added.

But this strategy becomes arbitrarily useless at strong coupling. Classical field equations associated with a particular classical limit have no relevance at strong coupling. Strong coupling means to send “g” to infinity. The proper dimensionless definition of “infinity” is such that the statement is equivalent to sending “hbar” to infinity. In that limit, the classical theory is completely wrong.

You might find different degrees of freedom that become classical at the strong coupling – e.g. the AdS/CFT ones – but the original gauge-theoretical degrees of freedom simply can’t be treated as classical variables at the strong coupling.

Cheers
LM

• mfrasca says:

Hi Lubos,

After some questioning we are back here talking about physics. A couple of questions about your comment.

You are right: $g\rightarrow\infty$ is equivalent to have a classical theory. There is a theorem by Barry Simon in quantum mechanics about this matter and I have got published a paper on this issue http://arxiv.org/abs/hep-th/0603182 . The same happens in quantum field theory. This can be seen by formally rescaling the time variable (check my paper http://arxiv.org/abs/hep-th/0511068 ).

Here I am saying something different: I have a set of solutions to start with that hold in the limit $g\rightarrow\infty$. Could I use them to do perturbation theory in QFT?

Cheers,

Marco

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