Domenico Fiorenza Interactions

Interactions

In this section we will consider the three possible interactions between the elementary particle in the standard model.

Symmetry breaking

The central part of the theory of the standard model are the symmetry breaking mechanisms. Essentially, given a lagrangian density $\mathcal{L}$ , one is interested in finding the conserved currents, according to the Noether's Theorem. There are two fundamental mechanisms of simmetry breaking:

Explicit breaking.
Spontaneous breaking.

In the first case, given a Lagrangian density $L$ and a group of symmetries $G$ , one proceeds in eliminating the lagrangian terms that are symmetry breaking and develops the results for a new symmetrical lagrangian. Finally one considers the corrections carried by the symmetry breaking terms and tries to control them. In the second case we have a perfectly symmetrical lagrangian density $\mathcal{L}$ under a group $G$ of transformations, but a solution $\phi$ for the Euler-Lagrange equations problem is not symmetrical respect to the group transformation. We will see when such a mechanism is possible in quantum mechanics.

Higgs model with global symmetries

A typical model of symmetry breaking mechanism is the Higgs lagrangian. In the global symmetry case we have the following lagrangian density:

\mathcal{L}= \partial_{\mu} \phi \partial^{\mu} \phi^{*} - \mu^2 \phi \phi^{*} + \lambda (\phi \phi^{*})^2

where $\phi : \mathbb{R}^4 \to \mathbb{C}$ is a quantum field. An equation for $\phi$ is:

(\Mbox + \mu^2) \phi + 2\lambda \phi ( \phi \phi^{*}) = 0.

Remark that the lagrangian density has a global symmetry for phase transformations:

\phi^{'} = e^{i \alpha} \phi \quad \phi^{*'} = e^{-i \alpha} \phi^{*}.

Through the Noether’s theorem we obtain the following expression for the current:

J = i [(\partial^{\mu} \phi)\phi^{*} - \phi ( \partial^{\mu} \phi^{*})]

It is convenient to introduce the real fields $\phi_1,\phi_2$ as:

\phi = \frac{\phi_1 + i \phi_2}{\sqrt{2}}

The potential $V$ is written explicitly in the new fields:

V = \frac{\mu^2}{2} ( \phi_1^2 \phi_2^2 ) + \frac{\lambda}{4}( \phi_1^2 \phi_2^2 )^2.

For $\lambda = 0$ we obtain the theory for the free case and consequently a field satisfying the K-G equation. Then we assumpt $\lambda \ne 0$ . In particular, looking forward for a stability study, we want a theory with a potential bounded from the bottom, then, since the $\lambda$ term is dominant to the infinite, we want $\lambda \gt 0$ . Once the $\lambda$ is fixed, we have two possible theories:

Case $\mu^2 \gt 0$ . The potential is a quartic with a minimun in zero (not interesting todo)
** Case $\mu^2 \le 0$ . ** The potential takes a “mexican hat” form: the origin is no more a minimum point, but a local maximum. The minimum configurations are a circle centered in the origin. We have to operate a choice between infinetely many points over the circle: this is done by adding a “driving term” to $\mathcal{L}$ :

\mathcal{L}_{\epsilon}= \partial_{\mu} \phi \partial^{\mu} \phi^{*} - \mu^2 \phi \phi^{*} + \lambda (\phi \phi^{*})^2 + \epsilon^{*} \phi + \epsilo \phi^{*}.

The critical values are obtained by posing to zero the values of the derivative of $\phi$ and $\phi^{*}$ . Explicitly:

\eta_{\epsilon} = \eta_{\epsilon}(\eta_{\mu^2 + 2 \lambda \eta_{\epsilon} \eta_{\epsilon}^{*}}) = \epsilon

Taking a real $\epsilon$ we have :

\eta_{\epsilon} = \eta + \delta

\eta = \sqrt{\frac{- \mu^2}{2 \lambda}}

\delta = \frac{\epsilon}{2 \mu^2}

It is now explicit that a stable solution $(\phi_1, \phi_2) = ( \eta, 0 )$ is not symmetrical according to the phase transformation. Now we can come up with a definition:

Definition. Given a lagrangian density $\mathcal{L}$ with a symmetry group $G$ , we say that the symmetry is spontaneously broken if the group of the symmetries of the solution of the Euler-Lagrange equations is strictly contained in $G$ .

In the spontaneous symmetry breaking mechanism it is crucial to consider systems with infinitely many degrees of freedom, when we are working in the quantum mechanics behaviour. Example.

Consider a particle confined in a potential with a double hole, i.e. two stability points symmetrical respect to the origin. It is a system with only one degree of freedom. Call $\psi_1,2$ the states of equilibrium, respectevely in the two holes. Recall that a state is an element of an Hilbert space $\mathcal{H}$ with a certain scalar product $(,)$ . Let $H$ be the hamiltonian of the system, then the the amplitude of transitioning to $\psi_1 \to \psi_1$ and \psi_2 \to \psi_2 $is the same, say$ A $. The problem is that in a system with finite degree of freedom the amplitude of transiction from$ \psi_1 \to \psi_2 $is$ \delta $and eventually$ \delta$ is non zero. This is called the tunnel effect, and depends on the uncertain principle of localization of the quantum mechanics. The configuration of minimum energy is then the eigenvector of the matrix:

\left( \begin{matrix} A & \delta \\ \delta & A \end{matrix} \right),

and results

\psi_0 = \frac{\psi_1 + \psi_2 }{\sqrt{2}}

Again we can introduce a driving term $\mathcal{L}_\epsilon$ and force a state of minimum energy, but in the limit $\mathcal{L}_\epsilon \to 0$ it happens that the state of minimum energy results again $\psi_0$ . It is crucial that $\delta \ne 0$ . The systems, where $\delta = 0$ , have necessarily infinte degrees of freedom, conversely non every system with infinite degrees of freedom is a model of spontaneous symmetry breaking: the Ising model is an example.

Stability study

It is worth to indentify the particles in the model presented so far. Remark that the lagrangian $\mathcal{L}$ is for a theory with interaction, that is out of our control so far. So it is convenient to do the stability study of the equations, around the minumum point $\eta$ . First introduce two new fields $\sigma_{1,2}$ such that:

\phi = \eta + \frac{\sigma_1 + i \sigma_2}{\sqrt{2}}.

We call $\sigma_{1,2}$ the fluctuations nearby $\eta$ . We substitute the new expression for $\phi$ in $\mathcal{L}$ . It results, by considering terms until order $2$ :

\mathcal{L}= \partial^{\mu} \sigma_1 \partial_{\mu} \sigma_1 + \partial^{\mu} \sigma_2 \partial_{\mu} \sigma_2 + \sum_{i,j} M_{ij} \sigma_i \sigma_j

M_12 = M_21 = M_22 = 0

M_11 = 4 \lambda \eta^2

For each fluctuation $\sigma_i$ we have a K-G lagrangian, remark that by computations, in a theory reaching order 2, they are not coupled together, since the mixed term in matrix $M$ are zero, and the mixed term in the derivatives is zero. We can now state that this model describes the interaction between a mass $M_11$ particle and a massless one. This is a general result occurring in the symmetry breaking mechanism, stated by the Goldstone theorem: a massless particle appears in the theory each time we have a symmetry breaking mechanism.

Revised on December 20, 2010 at 15:56:41 by Anonymous