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\newcommand{\neff}{N_{\rm eff}}
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\begin{document}

\title{Helicity/Chirality Notes}


\maketitle

\section{Lepton Asymmetry Calculation}
$\mu_\alpha$ is the neutrino chemical potential for neutrinos of flavor $\alpha$ = e, $\mu$, $\tau$. \cite{Simha:2008mt} uses the dimensionless degeneracy parameter $\xi_\alpha \equiv \mu_\alpha /kT$. The asymmetry between neutrinos and antineutrinos of flavor $\alpha$ is described by the lepton asymmetry $\eta_L$, defined as
\begin{equation}
    \eta_L \equiv \Sigma_\alpha \frac{n_{\nu_\alpha}-n_{\bar{\nu}_\alpha}}{n_\gamma}=\frac{\pi^3}{12\zeta(3)}\Sigma_\alpha\bigg[\bigg(\frac{\xi_\alpha}{\pi}\bigg)+\bigg(\frac{\xi_\alpha}{\pi}\bigg)^3\bigg].
\end{equation}\label{eq:etaL}

As usual, we define the effective number of relativistic species $\neff = \neffv + \dneff$. \cite{Simha:2008mt} uses $\neffv$ =3.046 post $e^\pm$ annihilation, as is standard. They consider that $\dneff$ is due only to neutrinos ($\dneff = \dnv$) and comprises a sum over all neutrino species $\alpha$: $\dnv = \Sigma_\alpha \dnv(\xi_\alpha)$. For each species $\dnv(\xi_\alpha)$ can be written as (equation 2 of \cite{Simha:2008mt}):
\begin{equation}
    \dnv(\xi_\alpha)\equiv \frac{30}{7}\bigg(\frac{\xi_\alpha}{\pi}\bigg)^2 +\frac{15}{7}\bigg(\frac{\xi_\alpha}{\pi}\bigg)^4.\label{eqn:deltaN}
\end{equation}

We are interested in solving for $\eta_L$ in the case of a specific value of $\dneff$. In the main text, we use $\dneff = (2.862 + 0.306) - 3.046 = 0.122$. We are nearly equipped to solve for $\eta_\alpha$. The only detail remaining is the relationship between the dimensionless degeneracy parameters of the different species. \cite{Simha:2008mt} states that in general the neutrino degeneracies should equilibrate prior to BBN, assuming the 3 active neutrino flavors only mix with each other. Thus, we will consider it reasonable to introduce the parameter $\xi \equiv \xi_e=\xi_\mu=\xi_\tau$.

We now attempt to solve the equation
\begin{equation}
    3\bigg[ \frac{30}{7}\bigg(\frac{\xi}{\pi}\bigg)^2 +\frac{15}{7}\bigg(\frac{\xi}{\pi}\bigg)^4 \bigg] = 0.122.
\end{equation}

This equation is a solvable quadratic in $\xi^2$. The solutions are $\xi^2$ = +0.09 and -19.8. Assuming a real degeneracy pressure (real chemical potential), we find that $\xi$ = 0.305. (I have also assumed more neutrinos than antineutrinos in taking the positive solution--if there were more antineutrinos the sign would simply change). 

Finally, using equation \ref{eq:etaL} I find that $\eta_L = 0.63$.

Is this reasonable? Well, \cite{Simha:2008mt} found $\eta_L \simeq 0.072$ when they assumed $\nv$ = 3 and $\eta_L \simeq 0.115$ when they left $\nv$ free to vary. Given that our upper bound on $\neff$ is 3.168, which is less than the preferred value of $\nv =3.3$ found when $\nv$ was free to vary. I expect $\eta_L$ in our case to be less than in Simha's freely varying case since our $\dneff$ is smaller, so this seems to agree. At least the order of magnitude does not seem outrageous here.

\subsection{Lepton Asymmetry Comments - Jonathan}
I agree with all of the numerical calculations here, but I am not sure about the statement that the original paper authors assumed $\Delta \neff = \Delta N_\nu$.
%EC: Sure, I agree. I think what I should have written is that is what I assume in my calculations. 

First of all, our value of $\Delta \neff$ is smaller than theirs but their lepton asymmetry value (with $N_\nu$ allowed to change) is smaller than our new estimate ($0.63 > 0.115$). Also, if we plug in the $\Delta N_\nu$ from their average value of $N_\nu = 3.3^{+0.7}_{-0.6}$ (so $\Delta N_\nu \approx 1.0$), we get a value of $\eta_L \approx 1.9$, which is about an order of magnitude larger than their reported value of $\eta_L = 0.115$.
%EC: I think that the second value they give does not assume equilibrium of neutrino flavors. When I check other numbers in this paper that do assume flavor equilibrium I obtain similar values.

So, using this assumption, we definitely get an upper bound, but the upper bound is much larger than the value they get in the paper. From what I understand, this is because the paper derives values of $\xi_e$ from BBN data directly. They note at the end of section 5 that the effects of neutrino degeneracy on $N_\nu$ are rather small compared to their average value of $N_\nu$. This is why I was worried about computing updated values for $\eta_L$, as it looked like the only way to do it would have been to work back through the data analysis and generate a figure that looks like Figure 2 in their paper. However, following a line from section 5, I think I have a reasonable way to get an estimate with the updated data from \cite{Fields:2019}.
%

\subsection{Estimate of $\eta_L$ using results from \cite{Fields:2019}}

We can use Equation (7) from the Simha paper to write $\xi_e$ in terms of the BBN parameters $Y_p, \eta_{10}, \neff$. That is,
\begin{align*}
	\eta_{\rm He} &= \left(\frac{Y_p - A}{0.0016}\right) + 6, \qquad A = 0.2482\pm0.0006 \\
    N_\nu &= 3 + \frac{43}{7}(S^2 - 1) \\
	S &= \sqrt{1 + \frac{7}{43}(\neff - 3)} \\
    \xi_e &= \frac{4}{574}\left(\eta_{10} + 100(S-1) - \eta_{\rm He}\right)
\end{align*}

Combining all of this and computing variables around the 95\% contour on Figure 18 of \cite{Fields:2019} we obtain a maximum value of
\begin{equation}
    \eta_L \lesssim 0.0355 \quad \text{(At 95\% certainty)}
\end{equation}

Note that we are assuming $Y_p$ is constant at its average value during this process. This actually seems like not a bad approximation as $Y_p$ seems to vary slowly in the small range of $\eta_{10}$'s we are using. This also assumes that the neutrinos are in equilibrium.

%EC: I suppose the question is really what we want to compare. My method is: (1) assume neutrino flavors in equilibrium, (2) assume DeltaNeff_neutrino accounts for the total DeltaNeff, (3) use the upper value of DeltaNeff from Fields' calculation, (3) Compute total lepton asymmetry. I think our methods differ on point1
%EC: I also think we need to be careful about mixing values, i.e. using the value for Neff that Fields et al. calculate with the abundances in the Simha paper
%EC: I guess which approach we take depends on if we want to find an approximate upper bound from a computed value of Neff or determine an upper bound from the electron neutrino chemical potential given by primordial abundances

%EC: all that being said, I'm still not sure why these 2 methods don't agree better.





\section{Maximal Chirality}
\cite{Brandenburg:2017rcb} finds that the maximal value of the chiral asymmetry is (equation 18):
\begin{equation}
    |\mu| << 4\alpha_{\rm em} \frac{k_B T}{\hbar c} \approx 1.5 \times 10^{14} T_{100}\; {\rm cm}^{-1}
\end{equation}
where $T_{100}$ is the temperature in units of 100 GeV. 
**When I tried to find the maximal chemical potential I got a different factor in front (instead of 4). --Emma

\cite{Brandenburg:2017rcb} defines the normalized chemical potential as 
\begin{equation}
    \mu = 24\alpha_{\rm em}(n_L - n_R) \bigg(\frac{\hbar c}{k_B T}\bigg)^2.
\end{equation}
This has units of inverse length. 

\cite{Simha:2008mt} defines the dimensionless degeneracy parameter as the neutrino chemical potential for neutrino species $\alpha$, $\mu_\alpha$, divided by the energy $kT$: $\xi_\alpha \equiv \mu_\alpha /kT$. 

\textit{Can $\mu$ and $\mu_\alpha$ be compared?} Where does normalized/expression in \cite{Brandenburg:2017rcb} come from? Does the maximal chemical potential in \cite{Brandenburg:2017rcb} just come from the assumption that fermions are in equilibrium with photons? \textit{What are we trying to compare with Brandenburg et al 2017?} 

For example, let's compare $\mu_{\alpha} = \xi_\alpha k T $ and $|\mu|$ (as in equation 4 above) at $T_{\rm BBN} = 0.1$ MeV. Defining $\mu_\nu = 3 \mu_{\alpha} = (3\xi_\alpha) k T$, for $\xi_\alpha = 0.3$ as calculated above, I find that  $\mu_\nu$/$|\mu| \sim 10$. So the maximal value estimated in \cite{Brandenburg:2017rcb} is within the bound from CMB+BBN constraints, assuming that these values are indeed comparable. 





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