# Beyond the Standard Model

After the discovery of a Higgs boson, all the particles included in the SM have been experimentally observed. Despite its extraordinary phenomenological success, the SM is considered an incomplete theory. It manifests some theoretical underlying problems which suggest the need for a “deeper” theory of which the Standard Model could be just an effectual theory.

Furthermore, there are some unsolved questions that suggest the demand to extend this theory to describe new phenomena, like:

• The Dark Energy/ Dark Matter problem. Several experimental evidences highlight the necessity to introduce at least two physical quantities, the Dark Energy (DE) and Dark Matter (DM), in the calculation of the total mass-energy of the Universe. Some of the most reliable theories explain the presence of the DM assuming the existence of a very massive (at least of the order of the mass of the $Z^0$) stable neutral particle that interacts only weakly with matter, while for the DE the framework is more troublesome at this time. The SM does not provide any candidate with the characteristics asked for the DM, and neutrinos, of course, are not massive enough.
• The matter-antimatter asymmetry in the Universe. The Big Bang should have created equal amounts of matter and antimatter in the early Universe, but at the present the Universe is almost entirely made of matter and this can not be explained only by CP violation, too small. One of the greatest challenges in physics is to figure out what happened to the antimatter, or how the matter/antimatter asymmetry arises.
• The neutrinos mass problem. Experimental evidences of neutrinos oscillation require them to have a mass, but in the SM it is not possible to assign a mass to neutrinos through the Yukawa coupling without providing for the existence of right-handed neutrinos, not yet observed.

In addition to these aspects of Nature not explained by the SM, there are also some unclear features of the model that need a better understanding in order to develop a more general theory:

• The great number of free parameters.
The Standard Model is dependent on 19 free parameters. This makes the theory dependent on too many arbitrary variables to retain that the SM could be a fundamental theory. Furthermore, the presence of so many parameters to be determined considerably reduces the predictive power.

• 3 coupling constants, $g$, $g'$ (or $sin\ \theta_W$) and $g_S$, for the three interactions;
• 9 Yukawa coupling constants which determine the masses of fermions;
• 2 parameters for the scalar potential: $v$ (or $m_H$, or $\mu^2$) and $\lambda$, as well determinable after the discovery of the Higgs boson;
• 3 mixing angles of the CKM matrix and the complex phase that parameterize the $CP$ violation;
• the $\theta_{QCD}$ angle for the $CP$ violation in the Lagrangian of the Strong Interaction.
• The problem of coupling constants. The comparison of the measured values of all the observable quantities with the corresponding theoretical predictions (the global electroweak Standard Model fit) shows that the three fundamental coupling constants (for strong, weak and electromagnetic interactions) do not converge to a single common value with the increasing of energy, as it would be desirable and expected by the Grand Unification Theories (GUT).
• The number of quark-lepton families. The SM does not explain the existence of three families for leptons and quarks, nor why there are such differences between their masses. However, experimental evidences in $e^+e^-$ collisions at $\textsf{LEP}$ indicate that the number of these families should be right three.
• Renormalizability and hierarchy problem. The hierarchy problem arises because the SM cannot be considered a valid theory for infinite energy scales and it becomes necessary to consider a new phenomenology over a certain energy threshold (e.g. the GUT or unification with the gravitational force). One of the free parameters that determinates the SM is $\mu^2$, which has the dimensions of an energy. In the SSB mechanism, the term $\mu^2$ is negative in order to ensure the presence of a ring of minima in the complex plane for the VEV of the Higgs field, which determinates as expression for the mass of the Higgs boson the $m_H=\sqrt{-2\mu^2}$. For further orders of perturbative expansion, the renormalization of $\mu^2$ includes corrections $\delta\mu^2$ relative to fermionic, bosonic and auto-interactive loops, $\mu^2=\mu^2_0+\delta\mu^2$, making the mass of the Higgs boson not renormalizable.

The mass of the Higgs, then, takes the form $m^2_H(phys)\sim m_H^2+c\Lambda^2$ where $\Lambda$ indicate the cut-off parameter depending on the scale parameter to which the SM is no longer valid.

The renormalization allows to reabsorb the divergences and to obtain a well-defined theory that is extendible to each order of the perturbative expansion.
Among the first order (one-loop) corrections, the one associated with the top quark is certainly the most important of those possible in the SM. So, by calculating the first order corrections to the term $\mu^2\Phi^\dag\Phi$ in the SSB Lagrangian, a self-interaction contribution for the four bosons can be obtained:

$\delta\mu^2 \propto \lambda \int^{\Lambda}\frac{1}{k^2-m_H^2} \ d^4 k.$

The renormalizability of the theory assures that no inconsistency will arise if the cut-off term $\Lambda$ would go to the infinity. Even so, here $\Lambda$ could be considered as the energy scale to which the new Physics appears and where the SM must be modified. Anyway, the integral in the equation above clearly diverges quadratically (there are four powers of $k$ in the numerator and two in the denominator), but by assuming $\Lambda\equiv\Lambda_{GUT}$ (or $\Lambda\equiv\Lambda_{Plank}$), the correction $\delta\mu^2$ makes $\mu^2$ positive and of the order of $10^{16}GeV$ (or $10^{19}GeV$).
This result seems rather surprising, since the equation for the mass of the Higgs boson $m_{H}=\sqrt{-2\mu^2}=\sqrt{\frac{\lambda}{2}}v$ with the value $v\approx 246\ GeV$ leads to the conclusion that $|\mu|$ can hardly assume a value greater than a few hundred GeV.

• The gravity. The SM does not incorporate gravity. Quantum gravitational effects are expected to become effective at the Planck scale, $M_P\sim 10^{19}GeV$. It would be desirable to have a theory that includes all the four fundamental interactions.

The GUTs are an answer to the need to define a more fundamental theory that allows the effective unification of the three forces described in the SM by identifying a single (instead of the actual three) coupling constant and a single group of symmetry. The Standard Model should then become the limit for low energies for this new theory. The key point is that the coupling constants are not really constants but they vary as a function of the considered energy.
This has been experimentally confirmed by measures performed at the $\textsf{LEP}$, where the coupling constants were found to asymptotically converge on the same values for energies of the order of $10^{15\div16}\ GeV$.

This energy scale is designated as the scale of grand unification $\Lambda_{GUT}$ and for the three coupling constants it results:

$\alpha_3=\alpha_2=\frac{5}{3}\alpha_1\equiv\alpha_{GUT}$,

with the three constants defined as

$\alpha_1 = \frac{5}{3}\frac{g'^2}{4\pi} = \frac{5\alpha}{3cos^2\theta_W}$,
$\alpha_2 = \frac{g^2}{4\pi}=\frac{\alpha}{sin^2\theta_W}$,
$\alpha_3 = \frac{g^2_S}{4\pi}$,

where $\alpha$ is the fine structure constant and $\frac{5}{3}$ is a renormalization factor.

Several gauge groups candidates for the description of the Unitary Interaction
were proposed: the first and the simplest was SU(5). The theory obtained was perfectly consistent, but its phenomenological predictions for $sin\theta_W$ and for the average lifetime of the proton are incompatible with the experimental evidences.

Other most accredited candidate groups are SO(10) and E6 (the latter inspired by the Superstring Theory):  their predictions on experimental data are roughly distant from those of SU(5). This is due to how the symmetry is broken and to the way in which these gauge groups are included into the SM. The interest for these two generalizations derives from the fact that they both predict the existence of massive and right-handed neutrinos as it seems necessary to explain the oscillations of neutrinos, recently confirmed by experimental evidences.