# LOG#120. Basic Neutrinology(V).

Supersymmetry (SUSY) is one of the most discussed ideas in theoretical physics. I am not discussed its details here (yet, in this blog). However, in this thread, some general features are worth to be told about it. SUSY model generally include a symmetry called R-parity, and its breaking provide an interesting example of how we can generate neutrino masses WITHOUT using a right-handed neutrino at all. The price is simple: we have to add new particles and then we enlarge the Higgs sector. Of course, from a pure phenomenological point, the issue is to discover SUSY! On the theoretical aside, we can discuss any idea that experiments do not exclude. Today, after the last LHC run at 8TeV, we have not found SUSY particles, so the lower bounds of supersymmetric particles have been increased. Which path will Nature follow? SUSY, LR models -via GUTs or some preonic substructure, or something we can not even imagine right now? Only experiment will decide in the end…

In fact, in a generic SUSY model, dut to the Higgs and lepton doublet superfields, we have the same $SU(3)_c\times SU(2)_L\times U(1)_Y$ quantum numbers. We also have in the so-called “superpotential” terms, bilinear or trilinear pieces in the superfields that violate the (global) baryon and lepton number explicitly. Thus, they lead to mas terms for the neutrino but also to proton decays with unacceptable high rates (below the actual lower limit of the proton lifetime, about $10^{33}$  years!). To protect the proton experimental lifetime, we have to introduce BY HAND a new symmetry avoiding the terms that give that “too high” proton decay rate. In SUSY models, this new symmetry is generally played by the R-symmetry I mentioned above, and it is generally introduced in most of the simplest models including SUSY, like the Minimal Supersymmetric Standard Model (MSSM). A general SUSY superpotential can be written in this framework as

(1) $\mathcal{W}'=\lambda{ijk}L_iL_jE_l^c+\lambda'_{ijk}L_iQ_jD_k^c+\lambda''_{ijk}D_i^cD_j^cU_k^c+\epsilon_iL_iH_2$

A less radical solution is to allow for the existence in the superpotential of a bilinear term with structure $\epsilon_3L_3H_2$. This is the simplest way to realize the idea of generating the neutrino masses without spoiling the current limits of proton decay/lifetime. The bilinear violation of R-parity implied by the $\epsilon_3$ term leads by a minimization condition to a non-zero vacuum expectation value or vev, $v_3$. In such a model, the $\tau$ neutrino acquire a mass due to the mixing between neutrinos and the neutralinos.The $\nu_e, v_\mu$ neutrinos remain massless in this toy model, and it is supposed that they get masses from the scalar loop corrections. The model is phenomenologically equivalent to a “3 Higgs doublet” model where one of these doublets (the sneutrino) carry a lepton number which is broken spontaneously. The mass matrix for the neutralino-neutrino secto, in a “5×5” matrix display, is:

(2) $\mathbb{M}=\begin{pmatrix}G_{2x2} & Q_{ab}^1 & Q_{ab}^2 & Q_{ab}^3\\ Q_{ab}^{1T} & 0 & -\mu & 0\\ Q_{ab}^{2T} & -\mu & 0 & \epsilon_3\\ Q_{ab}^{3T} & 0 & \epsilon_3 & 0\end{pmatrix}$

and where the matrix $G_{2x2}=\mbox{diag}(M_1, M_2)$ corresponds to the two “gauginos”. The matrix $Q_{ab}$ is a 2×3 matrix and it contains the vevs of the two higgses $H_1,H_2$ plus the sneutrino, i.e., $v_u, v_d, v_3$ respectively. The remaining two rows are the Higgsinos and the tau neutrino. It is necessary to remember that gauginos and Higgsinos are the supersymmetric fermionic partners of the gauge fields and the Higgs fields, respectively.

I should explain a little more the supersymmetric terminology. The neutralino is a hypothetical particle predicted by supersymmetry. There are some neutralinos that are fermions and are electrically neutral, the lightest of which is typically stable. They can be seen as mixtures between binos and winos (the sparticles associated to the b quark and the W boson) and they are generally Majorana particles. Because these particles only interact with the weak vector bosons, they are not directly produced at hadron colliders in copious numbers. They primarily appear as particles in cascade decays of heavier particles (decays that happen in multiple steps) usually originating from colored  supersymmetric particles such as squarks or gluinos. In R-parity conserving models, the lightest neutralino is stable and all supersymmetric cascade-decays end up decaying into this particle which leaves the detector unseen and its existence can only be inferred by looking for unbalanced momentum (missing transverse energy) in a detector. As a heavy, stable particle, the lightest neutralino is an excellent candidate to comprise the universe’s cold dark matter. In many models the lightest neutralino can be produced thermally in the hot early Universe and leave approximately the right relic abundance to account for the observed dark matter. A lightest neutralino of roughly $10-10^4$ GeV is the leading weakly interacting massive particle (WIMP) dark matter candidate.

Neutralino dark matter could be observed experimentally in nature either indirectly or directly. In the former case, gamma ray and neutrino telescopes look for evidence of neutralino annihilation in regions of high dark matter density such as the galactic or solar centre. In the latter case, special purpose experiments such as the (now running) Cryogenic Dark Matter Search (CDMS)  seek to detect the rare impacts of WIMPs in terrestrial detectors. These experiments have begun to probe interesting supersymmetric parameter space, excluding some models for neutralino dark matter, and upgraded experiments with greater sensitivity are under development.

If we return to the matrix (2) above, we observe that when we diagonalize it, a “seesaw”-like mechanism is again at mork. There, the role of $M_D, M_R$ can be easily recognized. The $\nu_\tau$ mass is provided by

$m_{\nu_\tau}\propto \dfrac{(v_3')^2}{M}$

where $v_3'\equiv \epsilon_3v_d+\mu v_3$ and $M$ is the largest gaugino mass. However, an arbitrary SUSY model produces (unless M is “large” enough) still too large tau neutrino masses! To get a realistic and small (1777 GeV is “small”) tau neutrino mass, we have to assume some kind of “universality” between the “soft SUSY breaking” terms at the GUT scale. This solution is not “natural” but it does the work. In this case, the tau neutrino mass is predicted to be tiny due to cancellations between the two terms which makes negligible the vev $v_3'$. Thus, (2) can be also written as follows

(3) $\begin{pmatrix}M_1 & 0 & -\frac{1}{2}g'v_d & \frac{1}{2}g'v_u & -\frac{1}{2}g'v_3\\ 0 & M_2 & \frac{1}{2}gv_d & -\frac{1}{2}gv_u & \frac{1}{2}gv_3\\ -\frac{1}{2}g'v_d & \frac{1}{2}gv_d & 0 & -\mu & 0\\ \frac{1}{2}g'v_u& -\frac{1}{2}gv_u& -\mu & 0 & \epsilon_3\\ -\frac{1}{2}g'v_3 & \frac{1}{2}gv_3 & 0 & \epsilon_3 & 0\end{pmatrix}$

We can study now the elementary properties of neutrinos in some elementary superstring inspired models. In some of these models, the effective theory implies a supersymmetric (exceptional group) $E_6$ GUT with matter fields belong to the 27 dimensional representation of the exceptional group $E_6$ plus additional singlet fields. The model contains additional neutral leptons in each generation and the neutral $E_6$ singlets, the gauginos and the Higgsinos. As the previous model, but with a larger number of them, every neutral particle can “mix”, making the undestanding of the neutrino masses quite hard if no additional simplifications or assumptions are done into the theory. In fact, several of these mechanisms have been proposed in the literature to understand the neutrino masses. For instance, a huge neutral mixing mass matris is reduced drastically down to a “3×3” neutrino mass matrix result if we mix $\nu$ and $\nu^c$ with an additional neutral field $T$ whose nature depends on the particular “model building” and “mechanism” we use. In some basis $(\nu, \nu^c,T)$, the mass matrix can be rewritten

(4) $M=\begin{pmatrix}0 & m_D & 0\\ m_D & 0 & \lambda_2v_R\\ 0 & \lambda_2v_R & \mu\end{pmatrix}$

and where the $\mu$ energy scale is (likely) close to zero. We distinguish two important cases:

1st. R-parity violation.

2nd. R-parity conservation and a “mixing” with the singlet.

In both cases, the sneutrinos, superpartners of $\nu^c$ are assumed to acquire a v.e.v. with energy size $v_R$. In the first case, the $T$ field corresponds to a gaugino with a Majorana mass $\mu$ than can be produced at two-loops! Usually $\mu\approx 100GeV$, and if we assume $\lambda v_R\approx 1 TeV$, then additional dangerous mixing wiht the Higgsinos can be “neglected” and we are lead to a neutrino mass about $m_\nu\sim 0.1eV$, in agreement with current bounds. The important conclusion here is that we have obtained the smallness of the neutrino mass without any fine tuning of the parameters! Of course, this is quite subjective, but there is no doubt that this class of arguments are compelling to some SUSY defenders!

In the second case, the field $T$ corresponds to one of the $E_6$ singlets. We have to rely on the symmetries that may arise in superstring theory on specific Calabi-Yau spaces to restric the Yukawa couplings till “reasonable” values. If we have $\mu=0$ in the matrix (4) above, we deduce that a massless neutrino and a massive Dirac neutrino can be generated from this structure. If we include a possible Majorana mass term of the sfermion at a scale $\mu\approx 100GeV$, we get similar values of the neutrino mass as the previous case.

Final remark: mass matrices, as we have studied here, have been proposed without embedding in a supersymmetric or any other deeper theoretical frameworks. In that case, small tree level neutrino masses can be obtained without the use of large scales. That is, the structure of the neutrino mass matrix is quite “model independent” (as the one in the CKM quark mixing) if we “measure it”. Models reducing to the neutrino or quark mass mixing matrices can be obtained with the use of large energy scales OR adding new (likely “dark”) particle species to the SM (not necessarily at very high energy scales!).

# LOG#110. Basic Cosmology (V).

## Recombination

When the Universe cooled up to $T\sim eV$, the neutrinos decoupled from the primordial plasma (soup). Protons, electrons and photons remained tighly coupled by 2 main types of scattering processes:

1) Compton scattering: $e+\gamma \leftrightarrow e+\gamma$

2) Coulomb scattering: $e^-+p\leftrightarrow H+\gamma$

Then, there were little hydrogen (H) and though $B_H>T$ due to small baryon fraction $\eta_b$.

The evolution of the free electron fraction provided the ratio

$X_e\equiv =\dfrac{n_e}{n_e+n_H}=\dfrac{n_p}{n_p+n_H}$

where $n_p+n_H\approx n_b$ and the second equality is due to the neutrality of our universe, i.e., to the fact that $n_e=n_p$ (by charge conservation). If $e^-+p\longrightarrow H+\gamma$ remains in the thermal equilibrium, then

$\dfrac{n_en_p}{n_H}=\dfrac{n_e^0n_p^0}{n_H^0}\longrightarrow \dfrac{X_e^2}{1-X_e}=\dfrac{1}{n_e+n_H}\left[\left(\dfrac{m_eT}{2\pi}\right)^{3/2}e^{-\left[m_e+m_p-m_H\right]/T}\right]$

where we have

$\dfrac{1}{n_e+n_H}\left(\dfrac{m_eT}{2\pi}\right)^{3/2}=n_p+n_H=n_b-4n(He)\approx n_p+n_H=n_b=\eta_b\eta_\gamma$

It gives $\eta_b\eta_\gamma\sim 10^{-9}T^3\approx 10^{15}$

and the last equality is due to the fact we take $T\sim E_0$. It means that $X_e\approx 1$ at $T\sim E_0$. As we have $X_e\longrightarrow 0$, we are out of the thermal equilibrium.

From the Boltzmann equation, we also get

$a^{-3}\dfrac{d(n_ea^3)}{dt}=n_e^0n_p^0\langle \sigma v\rangle \left( \dfrac{n_Hn_\gamma}{n_H^0n_\gamma^0}-\dfrac{n_e^2}{n_e^0n_p^0}\right)$

or equivalently

$a^{-3}\dfrac{d(n_ea^3)}{dt}=n_b\langle \sigma v\rangle \left(\dfrac{n_H}{n_b}\dfrac{n_e^0n_p^0}{n_H^0}-\dfrac{n_e^2}{n_b}\right)$

i.e.

$a^{-3}\dfrac{d(n_ea^3)}{dt}=n_b\langle \sigma v\rangle \left( (1-X_e)\left(\dfrac{m_eT}{2\pi}\right)^{3/2}e^{-E_0/T}-X_e^2n_b\right)$

Using that $n_e=n_bX_e$ and $\dfrac{d}{dt}(n_ba^3)=0$, we obtain

$\dfrac{dX_e}{dt}=\left[(1-X_e)\beta -X_e^2n_b\alpha^{(2)}\right]$

with

$\beta\equiv \langle \sigma v\rangle \left(\dfrac{m_eT}{2\pi}\right)^{3/2}e^{-E_0/T}$, the ionization rate, and

$\alpha^{(2)}\equiv \langle \sigma v\rangle$ the so-called recombination rate. It is taken the recombination to the n=2 state of the neutral hydrogen. Note that the ground state recombination is NOT relevant here since it produces an ionizing photon, which ionizes a neutral atom, and thus the net effect is zero. In fact, the above equations provide

$\alpha^{(2)}=9\mbox{.}78\dfrac{\alpha^2}{m_e^2}\left(\dfrac{E_0}{T}\right)^{1/2}\ln \left(\dfrac{E_0}{T}\right)$

The numerical integration produces the following qualitative figure

The decoupling of photons from the primordial plasma is explained as

$\mbox{Compton scattering rate}\sim\mbox{Expansion rate}$

Mathematicaly speaking, this fact implies that

$n_e\sigma_T=X_en_b\sigma_T$

where $\sigma_T$ is the Thomson cross section. For the processes we are interesting in, it gives

$\sigma_T=0\mbox{.}665\cdot 10^{-24}cm^2$

and then

$n_e\sigma_T=7\mbox{.}477\cdot 10^{-30}cm^{-1}X_e\Omega_bh^2a^{-3}$

Thus, we deduce that

$\dfrac{n_e\sigma_T}{H}=0\mbox{.}0692a^{-3}X_e\Omega_bh\dfrac{H_0}{H}$

$\dfrac{n_e\sigma_T}{H}=113X_e\left(\dfrac{\Omega_bh^2}{0\mbox{.}02}\right)\left(\dfrac{0\mbox{.}15}{\Omega_mh^2}\right)^{1/2}\left(\dfrac{1+z}{1000}\right)^{3/2}\left[1+\dfrac{1+z}{3600}\dfrac{0\mbox{.15}}{\Omega_mh^2}\right]^{-1/2}$

and where $X_e\leq 10^{-2}$ implies that the decoupling of photons occurs during the time of recombination! In fact, the decoupling of photons at time of recombination is what we observe when we look at the Cosmic Microwave Background (CMB). Fascinating, isn’t it?

## Dark Matter (DM)

Today, we have strong evidences and hints that non-baryonic dark matter (DM) exists (otherwise, we should modify newtonian dynamics and or the gravitational law at large scales, but it seems that even if we do that, we require this dark matter stuff).

In fact, from cosmological observations (and some astrotronomical and astrophysical measurements) we get the value of the DM energy density

$\Omega_{DM}\sim 0\mbox{.}2-0\mbox{.}3$

The most plausible candidate for DM are the Weakly Interacting Massive Particles (WIMPs, for short). Generic WIMP scenarios provide annihilations

$X_{DM}+\bar{X}_{DM}\leftrightarrow l+\bar{l}$

where $X_{DM}$ is some “heavy” DM particle and the (ultra)weak interaction above produces light particles in form of leptons and antileptons, tighly couple to the cosmic plasma. The Boltzmann equation gives

$a^{-3}\dfrac{d(n_Xa^3)}{dt}=\langle \sigma_X v\rangle \left( n_X^{(0)2}-n_X^2\right)$

Define the yield (or ratio) $Y_X=\dfrac{n_X}{T^3}$. It is produced since generally we have

$Y=\dfrac{n_X}{s}$

and since $sa^3=constant$, then $s\propto T^3$. Thus,

$\dfrac{dY}{dt}=T^3\langle \sigma v\rangle \left( Y_{EQ}^2-Y^2\right)$

and

$Y_{EQ}=\dfrac{n_X^0}{T^3}$

Now, we can introduce a new time variable, say

$x=\dfrac{m}{T}$

Then, we calculate

$\dfrac{dx}{dt}=-\dfrac{m}{T^2}\dfrac{dT}{dt}=-\dfrac{m}{T^2}\left(-\dfrac{\dot{a}}{a}T\right)=xH$

For a radiation dominated (RD) Universe, $\rho\propto T^4$ implies that $H\propto T^2$ and $H(x)=-\dfrac{H(m)}{x^2}$

In this case, we obtain

$\dfrac{dY}{dx}=\dfrac{\lambda}{x^2}\left(Y^2-Y_{EQ}^2\right)$

with $\lambda=\dfrac{m^3\langle \sigma v\rangle}{H(m)}$

The final freeze out abundance is got in the limit $Y_\infty=Y(x\longrightarrow \infty)$. Typically, $\lambda >>1$, and when $Y_{EQ}\sim 1$ and $Y\approx Y_{EQ}$, for $x>>1$, and there, the yield drops exponentially

$\dfrac{dY}{dx}\approx \dfrac{\lambda Y^2}{x^2}$

or

$\dfrac{dY}{Y^2}\approx \dfrac{\lambda dx}{x^2}$

Integrating this equation,

$\displaystyle{\int_{Y_f}^{Y_\infty}\dfrac{dY}{Y^2}=\int_{x_f}^\infty \dfrac{\lambda}{dx}{x^2}}$

and then

$\dfrac{1}{Y_\infty}-\dfrac{1}{Y_f}=\dfrac{\lambda}{x_f}$

Generally, $Y_f>>Y_\infty$ and the freeze out temperature for WIMPs is got with the aid of the following equation

$Y_\infty=\dfrac{x_f}{\lambda}$

Indeed, $n\langle \sigma v\rangle= H\longrightarrow x_f\sim 10$

A qualitative numerical solution of the “WIMP” miracle (and its freeze out) is given by the following sketch

The present abundance of heavy particle relics gives

$\rho_X=mY_\infty T_0^3\left(\dfrac{a_1T_1}{a_0T_0}\right)^3\approx \dfrac{mY_\infty T_0^3}{30}$

and where the effect of entrpy dumping after the freeze-out is encoded into the factor

$\left(\dfrac{a_1T_1}{a_0T_0}\right)^3$ with $\left(\dfrac{g_\star (0)}{g_\star (f)}\right)^3\approx \dfrac{1}{30}$

Moreover, the DM energy density can also be estimated:

$\Omega_X=\Omega_{DM}=\dfrac{x_f}{\lambda}\dfrac{mT_0^3}{30\rho_c}=\dfrac{H (m) x_fT_0^3}{30m^2\langle \sigma v\rangle\rho_c}$

so

$\Omega_X=\left[\dfrac{4\pi^3Gg_\star (m)}{45}\right]^{1/2}\dfrac{x_fT_0^3}{30\langle \sigma v\rangle \rho_c}=0\mbox{.}3h^{-2}\left(\dfrac{x_f}{10}\right)\left(\dfrac{g_\star (m)}{100}\right)^{1/2}\dfrac{10^{-39}cm^2}{\langle \sigma v\rangle}$

The main (current favourite) candidate for WIMP particles are the so called lightest supersymmetric particles (LSP). However, there are other possible elections. For instance, Majorana neutrinos (or other sterile neutrino species), Z prime bosons, and other exotic particles. We observe that here there is a deep connection between particle physics, astrophysics and cosmology when we talk about the energy density and its total composition, from a fundamental viewpoint.

Remark: there are also WISP particles (Weakly Interacting Slim Particles), like (superlight) axions and other exotics that could contribute to the DM energy density and/or the “dark energy”/vacum energy that we observe today. There are many experiments searching for these particles in laboratories, colliders, DM detection experiments and astrophysical/cosmological observations (cosmic rays and other HEP phenomena are also investigated to that goal).

See you in a next cosmological post!