# LOG#125. Basic Neutrinology(X).

The topic today is a fascinant subject in Neutrino Astronomy/Astrophysics/Cosmology. I have talked you in this thread about the cosmic neutrino background ($C\nu B$) and that the young neutrino Astronomy or neutrino telescopes will become more and more important in the future. The reasons are simple:

1st. If we want to study the early Universe, we need some “new” tool to overcome the last scattering surface as a consequence of the Cosmic Microwave Background (CMB). Neutrinos are such a new tool/probe! They only interact weakly with matter and we suspect that there are some important pieces of information related to the quark and lepton “complementarity” hidden in their mixing parameters.

2nd. Due to the GZK effect, we expect that the flux of cosmic rays will suffer a sudden cut-off at about $5\cdot 10^{19}eV=50\cdot 10^{18}eV=50EeV$, or about 8 joules. This Greisen–Zatsepin–Kuzmin limit (GZK limit) is a theoretical upper limit on the energy of cosmic rays, since at some high energy, that can be computed, they would interact with the CMB photons producing a delta particle ($\Delta$) which would spoil the observed cosmic rays flux as its decays would not be detected after “a long trip”. Then, it can only be approached when the cosmic rays travel very long distances (hundreds of million light-years or more). Here you are a typical picture of SuperKamiokande cosmic ray detection:

The limit is at the same order of magnitude as the upper limit for energy at which cosmic rays have experimentally been detected. There are some current experiments that “claim” to have observed this GZK effect, but evidence is not conclusive yet as far as I know. Some experiments claim (circa 2013, July) to have observed it, other experiments claim to have observed events well above the GZK limit. The next generation of cosmic ray experiments will confirm this limit from SM physics or they will show us interesting new physics events!

Inspired by the GZK effect, some people have suggested an indirect way to detect the existence of the cosmic relic neutrinos. Remember, cosmic neutrinos have a temperature about $1.9K$ if the SM is right, and their associated neutrino density now is about $110$ per cubic centimeter per species (neutrino plus antineutrino), or $330$ per cubic centimeter including the 3 flavors! Relic neutrinos are almost everywhere, but they are very, very feeble (neutral and weakly interacting) particles. While detecting the $C\nu B$ temperature is one of the most challenging tests of the standard cosmological model, we can try to detect the existence of these phantom neutrinos using a similar (quantum) trick than the one used in the GZK limit (there the delta particle resonance). If some ultra high energy cosmic ray (likely a neutrino coming from some astrophysical source) hits a “relic neutrino” with energy high enough to produce, say, a Z boson (neutral particle as the neutrino himself), then we should observe a “dip” in the cosmic ray spectrum corresponding to this “Z-burst” event! This mechanism is also called the ZeVatron or the Z-dip. It also shows the deep links between particle physics and Cosmology or Astrophysics. When an ultra-high energy cosmic neutrino collides with a relic anti-neutrino in our galaxy and annihilates to hadrons, this process proceeds via a (virtual) Z-boson:

$\nu_{UHE}+\bar{\nu}_{C\nu B}\longrightarrow Z\longrightarrow \mbox{hadrons}$

The cross section for this process becomes large if the center of mass energy of the neutrino-antineutrino pair is equal to the Z-boson mass (such a peak in the cross section is what we call “resonance” in High Energy physics). Assuming that the relic anti-neutrino is at rest, the energy of the incident cosmic neutrino has to be the quantity:

$\boxed{E_{eV}=\dfrac{m_Z^2}{2m_\nu}=4.2\cdot \left(\dfrac{eV}{m_\nu}\right)\cdot 10^{21}eV=42\left(\dfrac{0.1eV}{m_\nu}\right)\cdot 10^{21}eV}$

$\boxed{E_{ZeV}=4.2\left(\dfrac{eV}{m_\nu}\right)ZeV=42\left(\dfrac{0.1eV}{m_\nu}\right)ZeV}$

In fact, this mechanism based on “neutral resonances” is completely “universal”! Nothing (except some hidden symmetry or similar) can allow the production of (neutral) particles using this cosmic method. For instance, if this argument is true, beyond the Z-burst, we should be able to detect Higgs-dips (Higgs-bursts) or H-dips, since, similarely we could have

$\nu_{UHE}+\bar{\nu}_{C\nu B}\longrightarrow H\longrightarrow \mbox{hadrons}$

or more generally, with some (likely) “dark” particle, we should also expect that

$\nu_{UHE}+\bar{\nu}_{C\nu B}\longrightarrow X\longrightarrow \mbox{hadrons}$

In the H-dip case, taking the measured Higgs mass from the last LHC run (about 126GeV), we get

$\boxed{E_{eV}(H-dip)=\dfrac{m_H^2}{2m_\nu}=7.9\left(\dfrac{eV}{m_\nu}\right)\cdot 10^{21}eV=79\left(\dfrac{0.1eV}{m_\nu}\right)\cdot 10^{21}eV}$

$\boxed{E_{ZeV}(H-dip)=7.9\left(\dfrac{eV}{m_\nu}\right)ZeV=79\left(\dfrac{0.1eV}{m_\nu}\right)ZeV}$

In the arbitrary “dark” or “weakly interacting” particle, we have (in general, with $m_X= x GeV$) the formulae:

$\boxed{E_{eV}(X-dip)=\dfrac{m_X^2}{2m_\nu}=\dfrac{(x GeV)^2}{2m_\nu}=\left(\dfrac{x^2}{2m_\nu}\right)\cdot 10^{18}eV^2=\left(\dfrac{x^2}{2000}\right)\left(\dfrac{1eV}{m_\nu}\right) 10^{21}eV}$
or equivalently
$\boxed{E_{ZeV}(X-dip)=\dfrac{m_X^2}{2m_\nu}=\left(\dfrac{x^2}{200}\right)\left(\dfrac{0.1eV}{m_\nu}\right) ZeV=\left(\dfrac{x^2}{2000}\right)\left(\dfrac{1eV}{m_\nu}\right) ZeV}$

Therefore, cosmic ray neutrino spectroscopy is a very interesting subject yet to come! It can provide:

1st. Evidences for relic neutrinos we expect from the standard cosmological model.

2nd. Evidence for the Higgs boson in astrophysical scenarios from cosmological neutrinos. Now, we know that the Higgs field and the Higgs particle do exist, so it is natural to seek out this H-dips as well!

3rd. Evidence for the additional neutral weakly interacting (and/or “dark”) particles from “unexpected” dips at ZeV (1ZeV=1Zetta electron-volt) or even higher energies! Of course, this is the most interesting part from the viewpoint of new physics searches!

Neutrino telescopes and their associated Astronomy is just rising now! IceCube is its most prominent example…

Moreover, following one of the most interesting things in any research (expect the unexpected and try to explain it!) from the scientific viewpoint, I am quite sure the neutrino astronomy and its interplay with cosmic rays or this class of “neutrino spectroscopy” in the flux of cosmic rays open a very interesting window for the upcoming new physics. Are we ready for it? Maybe…After all, the neutrino mixing parameters are very different (“complementary”?) to the quark mixing parameters. You can observe it in this mass-flavor content plot:

Neutrino oscillations are a purely quantum effect, and thus, they open a really interesting “new channel” in which we can observe the whole Universe. Yes, neutrinos are cool!!! The coolest particles in all over the world! We can not imagine yet what neutrino will show and teach us about the current, past and future of the cosmological evolution.

Remark: When I saw the Fermi line and the claim of the Dark Matter particle “evidence” at about 130 GeV, I wondered if it could be, indeed, a hint of a similar “resonant” process in gamma rays, something like

$\gamma \gamma\longrightarrow H (resonance)$

since the line “peaked” close to the known Higgs-like particle mass ($126GeV\sim 130GeV$). Anyway, this line is controversial and its presence has yet to be proved with enough statistical confidence (5 sigmas are usually required in the particle physics community). Of course, the issue with this resonant hypothesis would be that we should expect that this particle would decay into hadrons leaving some indirect clues of those events.  The Fermi line can indeed have more explanations and/or be a fluke in the data due to a bad modeling or a bad substraction of the background. Time will tell us if the Fermi line is really here as well.

Final (geek) remark: I wonder if the Doctor Who fans remember that the reality bomb of Davros and the Daleks used “Z-neutrinos“!!! I presently do not know if the people who wrote those scripts and imagined the Z-neutrino were aware of the Z-bursts…Or not… LOL The Z-neutrino powered crucible was really interesting…

And the reality bomb concept was really scaring…

However, neutrinos are pretty weakly interacting particles, at least when they have low energy, so we should have not fear them. After all, their future applications will surprise us much more. I am quite sure of it!

See you in my next neutrinological post!

May the Z(X)-burst induced superGZK neutrinos be with you!

# LOG#108. Basic Cosmology (III).

The current Universe has evolved since its early phase of thermal equilibrium until the present state. The departure from thermal equilibrium in the early Universe made a fossil record we can observe at current time!

There are some easy rules for thermal equilibrium. The easiest one, is that coming from the “interaction rate” $\Gamma_{int}$. It can be expressed in the following way:

$\Gamma_{int}>\mbox{Expansion rate H}$

and then, at a given temperature T, we get

$\Gamma_{int}(T)=n(T)\langle \sigma \vert v \vert \sigma \rangle^T$

and where $H\approx \dfrac{T^2}{M_p}$

Remark: If $\Gamma_{int}=aT^n$ $\forall n>2$, then

$N_{int}=\int_t^\infty T_{int}(t')dt'=\dfrac{\Gamma (H)\vert_t}{n-2}<1$

and it implies that a particle interacts less than once after the time $\Gamma =H$.

Moreover, we can understand roughly the so-called decoupling era:

1st. Any interaction mediated by a massless gauge boson provides

$\sigma\sim \dfrac{\alpha^2_X}{s}$ with $s\sim E^2$ and $\alpha_X=\dfrac{g^2_X}{4\pi}$

and this implies that

$\Gamma \sim n\langle \sigma v\rangle\sim T^3\dfrac{\alpha_X^2}{T^2}=\alpha^2_XT$

and

$\dfrac{\Gamma}{H}\sim \alpha_X^2\dfrac{M_p}{T}$

so the equilibrium temperature is found whenever $T\leq \alpha_X^2M_p$!

2nd. Interactions mediated by any massive gauge boson provides

$\sigma\sim G_X^2s$ with $G_X\sim \dfrac{\alpha_X}{m_X^2}$

and this implies that

$\Gamma \sim T^3G_X^2T^2=G_X^2T^5$

and

$\dfrac{\Gamma}{H}\sim G_X^2M_pT^3$

and then

$\left( G_X^2M_p\right)^{-1/3}\leq T\leq m_X\longrightarrow \mbox{Equilibrium temperature (E.T.)}$

Moreover,

$T\leq \left( G_X^2M_p\right)^{-1/3}\sim \left(\dfrac{m_X}{100\mbox{GeV}}\right)^{4/3}\mbox{MeV}\longrightarrow \mbox{Freeze out}$

As a consequence, we can realize that the out-of-equilibrium phenomena in the early and current Universe are very important processes! In particular:

1) They provide the formation of light elements during the Big Bang Nucleosynthesis (BBN), also known as primordial nucleosynthesis, i.e., the formation of the first light elements after the Big Bang (circa 300000 years after the Universe “birth”).

2) They provide the path of recombination of electrons and protons into hydrogen atoms.

3) They imply the  (likely) production of dark matter (or equivalently the presence of some kind of “modified gravity” or/and modified newtonian dynamics).

Boltzmann’s equation for annihilation of particles in equilibrium

There is a beautiful equation that condenses the previous physical process of equilibrium at a given temperature and the particle production it yields. Conceptually speaking, we have

$\begin{pmatrix}\mbox{Boltzmann}\\ \mbox{Equation}\end{pmatrix}:$

$\begin{pmatrix}\mbox{Rate of change}\\ \mbox{in the abundance}\end{pmatrix}=\begin{pmatrix}\mbox{Rate of}\\ \mbox{particle production}\end{pmatrix}-\begin{pmatrix}\mbox{Rate of}\\ \mbox{particle erasing/annihilation}\end{pmatrix}$

Consider a process like

$\mbox{particle type 1}+\mbox{particle type 2}\leftrightarrow \mbox{particle type 3}+\mbox{particle type 4}$

and where the particle 1 is the one we are interested in. Then, we deduce that

$\underbrace{\dfrac{1}{a^3}\dfrac{d(n_1a^3)}{dt}}_\text{change in comoving volume}=\underbrace{\int\dfrac{d^3p_1}{(2\pi)^32E_1}\int\dfrac{d^3p_2}{(2\pi)^32E_2}\int\dfrac{d^3p_3}{(2\pi)^32E_3}\int\dfrac{d^3p_4}{(2\pi)^32E_4}}_\text{phase space invariant}\times A$

where A is certain complicated facter involving “delta functions” of the energies and momenta of the particles 1,2,3,4 and an additional term depending on the statistics of the particle. Explicitly, it takes the form

$A=\left[(2\pi)^4\delta^3(p_1+p_2-p_3-p_4)\delta (E_1+E_2-E_3-E_4)\vert M\vert^2\right]\times S$

with $S=\left[ f_3f_4(1\pm f_1)(1\pm f_2)-f_1f_2(1\pm f_3)(1\pm f_4)\right]$

and where

$f_i=\dfrac{1}{e^{(E_i-\mu_i (t))/T}\pm 1}$

is the Fermi-Dirac (FD, -)/Bose-Einstein (BE,+) distribution. In fact, the above FD/BE factors provide the so-called Pauli blocking/”Bose-Einstein” enhancement effects for the particle production in the processes $3+4\rightarrow 1+2$ and $1+2\rightarrow 3+4$. Indeed, particle physics enter into the game here (see above formulae again) and we assume

$M(1+2\rightarrow 3+4)=M(3+4\rightarrow 1+2)$

Do you recognize the principle of detailed balance in this equation?

We can simplify the assumptions a little bit:

1st. The kinetic equilibrium is taken to be a rapid elastic scattering and we input the FD/BE statistics without loss of generality.

2nd. The annihilation in thermal equilibrium will be calculated from the sum of the chemical potential in any balanced equation.

3rd. Low temperature approximation. Suppose that

$T<< (E-\mu)$

then we obtain the Maxwell-Boltzmann approximation to the FD/BE statistics

$f\approx e^{-(E-\mu)/T}$ and $1+f\approx 1$, so, since $E_1+E_2=E_3+E_4$, we get

$f_3f_4(1\pm f_1)(1\pm f_2)-f_1f_2(1\pm f_3)(1\pm f_4)\approx e^{-(E_1+E_2)/T}\left( e^{\frac{(\mu_3+\mu_4)}{T}}-e^{\frac{(\mu_1+\mu_2)}{T}}\right)-\star$

What is $\star$? After a change of variable

$\mu_i (t)\longrightarrow n_i(t)=g_ie^{\mu_i/T}\int \dfrac{d^3p}{(2\pi)^3}e^{-E_i/T}$

$\star$ is the “equilibrium number density” deducen from the expression

$n_i^{0}\equiv g_i\int {d^3p}{(2\pi)^3}e^{-E_i/T}=\begin{cases}g_i\left(\dfrac{m_iT}{2\pi}\right)^{3/2}e^{-m_i/T},\;\; \mbox{if}\;\; m_i>>T\\ g_i\dfrac{T^3}{\pi^2},\;\;\mbox{if}\;\; m_i<

and it yields

$n_i=e^{\mu_i/T}n_\gamma^{0}$

and then we finally get that

$\star$ equals $e^{-(E_1+E_2)/T}\left[\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\right]$

Now, we can define the thermally averaged cross section

$\langle \sigma v\rangle\equiv \dfrac{1}{n_1^0n_2^0}\int \dfrac{d^3p_1}{(2\pi)^32E_1}\cdots \dfrac{d^3p_4}{(2\pi)^32E_4}e^{-(E_1+E_2)/T}(2\pi)^4\delta^3 (p_1+p_2-p_3-p_4)\times$

$\times \delta (E_1+E_2-E_3-E_4)\vert M\vert^2$

The Boltzmann equation becomes with these conventions

$\dfrac{1}{a^3}\dfrac{d(n_1a^3)}{dt}=n_1^0n_2^0\langle \sigma v\rangle \left(\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\right)$

Remark (I): LHS is similar to $\dfrac{n_1}{t}\sim n_1H$ and the RHS is similar to $n_1n_1\langle \sigma v\rangle$

Remark (II): If the reaction rate is $n_1\langle \sigma v\rangle >> H$, then it provides the chemical equilibrium condition well known in the nuclear statistical equilibrium as the Saha equation, i.e.,

$\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\approx 0$