Neutrino Physics Formulas

Below is a categorized list of important formulas used in neutrino theory, oscillations, interactions, astrophysics, cosmology, and detector physics. Natural units, (\hbar=c=1), are often used.

1. Relativistic Neutrino Energy

Energy–momentum relation

[E_\nu^2=p_\nu^2c^2+m_\nu^2c^4]

For an ultrarelativistic neutrino, (p_\nu c\gg m_\nu c^2):

[E_\nu \simeq p_\nu c+\frac{m_\nu^2c^3}{2p_\nu}]

In natural units:

[E_\nu \simeq p_\nu+\frac{m_\nu^2}{2E_\nu}]

Neutrino velocity

[v_\nu=\frac{p_\nu c^2}{E_\nu}]

For (m_\nu\ll E_\nu):

[v_\nu \simeq c\left(1-\frac{m_\nu^2c^4}{2E_\nu^2}\right)]

2. Neutrino Flavor and Mass Mixing

Flavor–mass relation

[|\nu_\alpha\rangle

\sum_{i=1}^{3}U_{\alpha i}^{*}|\nu_i\rangle
]

where

[
\alpha=e,\mu,\tau,
\qquad
i=1,2,3.
]

Inverse relation

[|\nu_i\rangle

\sum_{\alpha=e,\mu,\tau}U_{\alpha i}|\nu_\alpha\rangle
]

Antineutrino mixing

[|\bar{\nu}_\alpha\rangle

\sum_i U_{\alpha i}|\bar{\nu}_i\rangle
]

Unitarity of the PMNS matrix

[
U^\dagger U=UU^\dagger=I
]

or

[\sum_i U_{\alpha i}U_{\beta i}^{*}

\delta_{\alpha\beta}.
]

3. PMNS Mixing Matrix

The three-flavor Pontecorvo–Maki–Nakagawa–Sakata matrix is

[U_{\rm PMNS}

R_{23}R_{13}R_{12}.
]

A common parameterization is

[
U=
\begin{pmatrix}
c_{12}c_{13}
&
s_{12}c_{13}
&
s_{13}e^{-i\delta_{\rm CP}}
\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\rm CP}}
&
c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\rm CP}}
&
s_{23}c_{13}
\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\rm CP}}
&
-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\rm CP}}
&
c_{23}c_{13}
\end{pmatrix},
]

where

[
s_{ij}=\sin\theta_{ij},
\qquad
c_{ij}=\cos\theta_{ij}.
]

For Majorana neutrinos:

[U_{\rm Majorana}

U_{\rm PMNS}
\begin{pmatrix}
1&0&0\
0&e^{i\alpha_{21}/2}&0\
0&0&e^{i\alpha_{31}/2}
\end{pmatrix}.
]

4. Neutrino Mass-Squared Differences

[
\Delta m_{ij}^{2}=m_i^2-m_j^2
]

Important differences include

[
\Delta m_{21}^{2}=m_2^2-m_1^2
]

and

[
\Delta m_{31}^{2}=m_3^2-m_1^2.
]

Normal mass ordering

[
m_1<m_2<m_3
]

Inverted mass ordering

[
m_3<m_1<m_2
]

Sum of neutrino masses

[
\Sigma m_\nu=m_1+m_2+m_3
]

5. Neutrino Propagation Phase

For mass eigenstate (\nu_i):

[

|\nu_i(t)\rangle

e^{-iE_it}|\nu_i(0)\rangle
]

For relativistic neutrinos:

[
E_i\simeq E+\frac{m_i^2}{2E}.
]

The phase difference is

[

\Delta\phi_{ij}

\frac{\Delta m_{ij}^{2}L}{2E}.
]

The oscillatory term is commonly written as

[
\sin^2\left(\frac{\Delta m_{ij}^{2}L}{4E}\right).
]

In practical units:

[
\sin^2\left(
1.267,
\frac{\Delta m_{ij}^{2}(\mathrm{eV}^2)L(\mathrm{km})}
{E(\mathrm{GeV})}
\right).
]

6. Two-Flavor Neutrino Oscillation

Mixing equations

[

|\nu_\alpha\rangle

\cos\theta,|\nu_1\rangle
+
\sin\theta,|\nu_2\rangle
]

[

|\nu_\beta\rangle

-\sin\theta,|\nu_1\rangle
+
\cos\theta,|\nu_2\rangle.
]

Appearance probability

[

P(\nu_\alpha\rightarrow\nu_\beta)

\sin^2(2\theta)
\sin^2\left(
\frac{\Delta m^2L}{4E}
\right)
]

or, in practical units,

[

P(\nu_\alpha\rightarrow\nu_\beta)

\sin^2(2\theta)
\sin^2\left[
1.267,
\frac{\Delta m^2(\mathrm{eV}^2)L(\mathrm{km})}
{E(\mathrm{GeV})}
\right].
]

Survival probability

[

P(\nu_\alpha\rightarrow\nu_\alpha)

1-
\sin^2(2\theta)
\sin^2\left(
\frac{\Delta m^2L}{4E}
\right).
]

Probability conservation

[
P_{\alpha\alpha}+P_{\alpha\beta}=1.
]

7. Oscillation Length

[

L_{\rm osc}

\frac{4\pi E}{|\Delta m^2|}
]

In practical units:

[
L_{\rm osc}(\mathrm{km})
\simeq
2.48,
\frac{E(\mathrm{GeV})}
{|\Delta m^2|(\mathrm{eV}^2)}.
]

First oscillation maximum

[

\frac{\Delta m^2L}{4E}

\frac{\pi}{2}.
]

Therefore,

[

\frac{L}{E}

\frac{2\pi}{|\Delta m^2|}.
]

In practical units:

[

1.267\frac{|\Delta m^2|L}{E}

\frac{\pi}{2}.
]

8. General Three-Flavor Oscillation Probability

The vacuum transition probability is

[

P_{\alpha\beta}

\delta_{\alpha\beta}

4\sum_{i>j}
\operatorname{Re}
\left(
U_{\alpha i}^{}U_{\beta i}
U_{\alpha j}U_{\beta j}^{}
\right)
\sin^2\Delta_{ij}
]

[
\quad
+
2\sum_{i>j}
\operatorname{Im}
\left(
U_{\alpha i}^{}U_{\beta i}
U_{\alpha j}U_{\beta j}^{}
\right)
\sin(2\Delta_{ij}),
]

where

[

\Delta_{ij}

\frac{\Delta m_{ij}^{2}L}{4E}.
]

For antineutrinos:

[
U\rightarrow U^{*},
]

so the sign of the CP-odd term changes.

9. CP Violation in Neutrino Oscillations

CP asymmetry

[

A_{\rm CP}^{\alpha\beta}

\frac{

P(\nu_\alpha\rightarrow\nu_\beta)

P(\bar{\nu}\alpha\rightarrow\bar{\nu}\beta)
}{
P(\nu_\alpha\rightarrow\nu_\beta)
+
P(\bar{\nu}\alpha\rightarrow\bar{\nu}\beta)
}.
]

Jarlskog invariant

[

J_{\rm CP}

c_{12}s_{12}
c_{23}s_{23}
c_{13}^{2}s_{13}
\sin\delta_{\rm CP}.
]

Equivalently,

[

J_{\rm CP}

\frac{1}{8}
\sin2\theta_{12}
\sin2\theta_{23}
\sin2\theta_{13}
\cos\theta_{13}
\sin\delta_{\rm CP}.
]

CP-odd probability difference

[
P_{\alpha\beta}-P_{\bar{\alpha}\bar{\beta}}
\propto
J_{\rm CP}
\sin\Delta_{21}
\sin\Delta_{31}
\sin\Delta_{32}.
]

10. Neutrino Oscillation in Matter

The electron-neutrino matter potential is

[
V_e=\sqrt{2}G_FN_e.
]

For antineutrinos:

[
V_{\bar e}=-\sqrt{2}G_FN_e.
]

Here,

[
N_e=Y_e\frac{\rho}{m_N},
]

where (\rho) is the matter density and (Y_e) is the electron fraction.

Matter parameter

[

A=2EV_e

2\sqrt{2}G_FN_eE.
]

Effective mixing angle in matter

[

\sin^2(2\theta_m)

\frac{
\sin^2(2\theta)
}{
\left(
\cos2\theta-\frac{A}{\Delta m^2}
\right)^2
+
\sin^2(2\theta)
}.
]

Effective mass-squared difference

[

\Delta m_m^2

\Delta m^2
\sqrt{
\left(
\cos2\theta-\frac{A}{\Delta m^2}
\right)^2
+
\sin^2(2\theta)
}.
]

Oscillation probability in constant-density matter

[

P_m(\nu_\alpha\rightarrow\nu_\beta)

\sin^2(2\theta_m)
\sin^2\left(
\frac{\Delta m_m^2L}{4E}
\right).
]

11. MSW Resonance

The Mikheyev–Smirnov–Wolfenstein resonance condition is

[

2\sqrt{2}G_FN_eE

\Delta m^2\cos2\theta.
]

Resonance electron density

[

N_e^{\rm res}

\frac{
\Delta m^2\cos2\theta
}{
2\sqrt{2}G_FE
}.
]

Resonance energy

[

E_{\rm res}

\frac{
\Delta m^2\cos2\theta
}{
2\sqrt{2}G_FN_e
}.
]

At resonance:

[
\theta_m=\frac{\pi}{4}
]

and therefore

[
\sin^2(2\theta_m)=1.
]

12. Neutrino Evolution Equation

In the flavor basis:

[

i\frac{d}{dt}|\nu_f\rangle

H_f|\nu_f\rangle.
]

For three-flavor propagation in matter:

[

H_f

\frac{1}{2E}
U
\begin{pmatrix}
m_1^2&0&0\
0&m_2^2&0\
0&0&m_3^2
\end{pmatrix}
U^\dagger
+
\begin{pmatrix}
V_e&0&0\
0&0&0\
0&0&0
\end{pmatrix}.
]

13. Coherence and Wave-Packet Formulas

Coherence length

[
L_{\rm coh}^{ij}
\simeq
\frac{4\sqrt{2}E^2\sigma_x}
{|\Delta m_{ij}^2|},
]

where (\sigma_x) is the spatial width of the neutrino wave packet.

Wave-packet separation

[
\Delta x_{ij}
\simeq
\frac{|\Delta m_{ij}^2|L}{2E^2}.
]

Oscillations remain coherent when

[
\Delta x_{ij}\lesssim \sigma_x.
]

14. Weak Interaction Lagrangian

Charged-current interaction

[

\mathcal{L}_{\rm CC}

-\frac{g}{\sqrt{2}}
\bar{\ell}\alpha
\gamma^\mu P_L
U{\alpha i}\nu_i
W_\mu^{-}
+\mathrm{h.c.}
]

Neutral-current interaction

[

\mathcal{L}_{\rm NC}

-\frac{g}{2\cos\theta_W}
\bar{\nu}i
\gamma^\mu P_L
\nu_i
Z\mu.
]

The left-handed projection operator is

[
P_L=\frac{1-\gamma^5}{2}.
]

15. Fermi Constant Relation

[

\frac{G_F}{\sqrt{2}}

\frac{g^2}{8M_W^2}.
]

The weak interaction strength at low energies is controlled by (G_F).

16. Neutrino Interaction Rate

Event rate

[

N_{\rm events}

N_T
\int
\Phi_\nu(E)
,\sigma_\nu(E)
,\epsilon(E)
,P_{\alpha\beta}(E,L)
,dE,dt.
]

For a narrow energy interval:

[
N_{\rm events}
\simeq
\Phi_\nu
,\sigma_\nu
,N_T
,\epsilon
,T.
]

Here:

• (\Phi_\nu): neutrino flux,

• (\sigma_\nu): interaction cross section,

• (N_T): number of targets,

• (\epsilon): detection efficiency,

• (T): exposure time.

Number of targets

[

N_T

\frac{M_{\rm det}}{M_{\rm mol}}
N_A n_{\rm target},
]

where (N_A) is Avogadro’s constant.

17. Neutrino Mean Free Path

[

\lambda_\nu

\frac{1}{n_T\sigma_\nu}.
]

Using matter density:

[

\lambda_\nu

\frac{1}
{\rho N_A\sigma_\nu},
]

with an appropriate target-number conversion.

Interaction probability

[

P_{\rm int}(L)

1-e^{-L/\lambda_\nu}.
]

For (L\ll\lambda_\nu):

[
P_{\rm int}\simeq \frac{L}{\lambda_\nu}.
]

Survival probability

[

P_{\rm surv}(L)

e^{-L/\lambda_\nu}.
]

18. Neutrino Optical Depth

[

\tau_\nu

\int n_T(s)\sigma_\nu(E),ds.
]

The transmission probability is

[

P_{\rm transmission}

e^{-\tau_\nu}.
]

Using column density (X):

[

\tau_\nu

N_A,\sigma_\nu(E),X.
]

19. Neutrino–Electron Elastic Scattering

For

[
\nu_\ell+e^-\rightarrow\nu_\ell+e^-,
]

the differential cross section can be written as

[

\frac{d\sigma}{dT}

\frac{2G_F^2m_e}{\pi}

\left[

g_L^2

+

g_R^2

\left(1-\frac{T}{E_\nu}\right)^2

g_Lg_R\frac{m_eT}{E_\nu^2}
\right],
]

where (T) is the electron recoil kinetic energy.

Maximum electron recoil energy

[

T_{\max}

\frac{2E_\nu^2}
{m_e+2E_\nu}.
]

For antineutrinos, the roles of (g_L) and (g_R) are interchanged.

20. Inverse Beta Decay

The principal reactor-antineutrino reaction is

[
\bar{\nu}_e+p\rightarrow e^++n.
]

Threshold energy

[

E_{\nu}^{\rm th}

\frac{
(m_n+m_e)^2-m_p^2
}{
2m_p
}
\simeq 1.806\ \mathrm{MeV}.
]

Approximate neutrino-energy reconstruction

[
E_{\bar{\nu}e}
\simeq
E{e^+}
+
(m_n-m_p).
]

If (E_{e^+}) is the total positron energy:

[
E_{\bar{\nu}e}
\simeq
E{e^+}+1.293\ \mathrm{MeV}.
]

Using positron kinetic energy (T_{e^+}):

[
E_{\bar{\nu}e}
\simeq
T{e^+}+1.806\ \mathrm{MeV}.
]

Approximate cross section

[
\sigma_{\rm IBD}
\simeq
\frac{
G_F^2\cos^2\theta_C
}{
\pi
}
(1+3g_A^2)
E_ep_e.
]

21. Charged-Current Quasielastic Scattering

For neutrinos:

[
\nu_\ell+n\rightarrow\ell^-+p.
]

For antineutrinos:

[
\bar{\nu}_\ell+p\rightarrow\ell^++n.
]

An approximate reconstructed neutrino energy is

[

E_\nu^{\rm QE}

\frac{

2m_NE_\ell

\left(
m_\ell^2+m_N^2-m_{N'}^2
\right)
}{
2\left(
m_N-E_\ell+p_\ell\cos\theta_\ell
\right)
}.
]

Nuclear binding-energy corrections are normally included in realistic analyses.

22. Deep-Inelastic Neutrino Scattering

Bjorken variables

[
Q^2=-q^2
]

[

x

\frac{Q^2}{2p\cdot q}
]

[

y

\frac{p\cdot q}{p\cdot k}.
]

For a stationary nucleon:

[

y

\frac{E_\nu-E_\ell}{E_\nu}.
]

Hadronic energy

[

E_{\rm had}

yE_\nu.
]

Neutrino-energy reconstruction

[
E_\nu
\simeq
E_\ell+E_{\rm had}.
]

At sufficiently high energies, the total neutrino–nucleon cross section is approximately proportional to energy:

[
\sigma_{\nu N}\propto E_\nu.
]

23. Coherent Elastic Neutrino–Nucleus Scattering

For CE(\nu)NS:

[
\nu+A\rightarrow\nu+A.
]

The differential cross section is approximately

[

\frac{d\sigma}{dT}

\frac{G_F^2M}{4\pi}
Q_W^2
\left(
1-\frac{MT}{2E_\nu^2}
\right)
F^2(Q^2),
]

where

[

Q_W

N-
(1-4\sin^2\theta_W)Z.
]

Maximum nuclear recoil

[

T_{\max}

\frac{2E_\nu^2}
{M+2E_\nu}.
]

Coherence condition

[
qR\lesssim1.
]

24. Beta Decay and Neutrino Mass

For nuclear beta decay:

[
(A,Z)\rightarrow(A,Z+1)+e^-+\bar{\nu}_e.
]

Near the endpoint, the electron spectrum contains

[
\frac{dN}{dE_e}
\propto
F(Z,E_e),
p_eE_e
(E_0-E_e)
\sqrt{
(E_0-E_e)^2-m_\beta^2
}.
]

Effective electron-neutrino mass

[

m_\beta^2

\sum_i
|U_{ei}|^2m_i^2.
]

25. Neutrinoless Double-Beta Decay

The process is

[
(A,Z)\rightarrow(A,Z+2)+2e^-.
]

Its inverse half-life is

[

\left[

T_{1/2}^{0\nu}

\right]^{-1}

G^{0\nu}
|M^{0\nu}|^2
\left|
\frac{m_{\beta\beta}}{m_e}
\right|^2.
]

Effective Majorana mass

[

m_{\beta\beta}

\left|
\sum_i
U_{ei}^{,2}m_i
\right|.
]

For three neutrinos:

[

m_{\beta\beta}

\left|
m_1c_{12}^2c_{13}^2
+
m_2s_{12}^2c_{13}^2e^{i\alpha_{21}}
+
m_3s_{13}^2e^{i(\alpha_{31}-2\delta_{\rm CP})}
\right|.
]

26. Dirac Neutrino Mass

A Dirac mass term is

[

\mathcal{L}_{D}

-m_D\bar{\nu}_L\nu_R+\mathrm{h.c.}
]

After electroweak symmetry breaking:

[

m_D

\frac{y_\nu v}{\sqrt{2}},
]

where (y_\nu) is the neutrino Yukawa coupling and (v) is the Higgs vacuum expectation value.

27. Majorana Neutrino Mass

A Majorana mass term has the form

[

\mathcal{L}_{M}

-\frac{1}{2}
m_M
\overline{\nu_L^c}\nu_L
+\mathrm{h.c.}
]

A Majorana particle satisfies

[
\nu=\nu^c.
]

28. Type-I Seesaw Mechanism

The neutrino mass matrix can be written as

[

\mathcal{M}_\nu

\begin{pmatrix}
0&m_D\
m_D^T&M_R
\end{pmatrix}.
]

When (M_R\gg m_D), the light-neutrino mass matrix is

[
m_\nu
\simeq
-m_DM_R^{-1}m_D^T.
]

For one generation:

[
m_{\rm light}
\simeq
\frac{m_D^2}{M_R}.
]

The heavy-neutrino mass is approximately

[
m_{\rm heavy}\simeq M_R.
]

29. Weinberg Operator

The dimension-five neutrino-mass operator is

[

\mathcal{L}_5

\frac{c_{\alpha\beta}}{\Lambda}
\left(
\overline{L_\alpha^c}\tilde{H}^{*}
\right)
\left(
\tilde{H}^{\dagger}L_\beta
\right)
+\mathrm{h.c.}
]

After electroweak symmetry breaking:

[
m_\nu
\sim
\frac{cv^2}{\Lambda}.
]

30. Neutrino Magnetic Moment

The magnetic-moment interaction is

[

\mathcal{L}_{\mu}

\frac{1}{2}
\mu_\nu
\bar{\nu}
\sigma_{\mu\nu}
\nu
F^{\mu\nu}.
]

For a minimally extended Standard Model Dirac neutrino:

[
\mu_\nu
\simeq
\frac{3eG_Fm_\nu}
{8\sqrt{2}\pi^2}.
]

It is commonly expressed in units of the Bohr magneton:

[
\mu_B=\frac{e\hbar}{2m_e}.
]

Spin-precession probability in a transverse magnetic field

[

P(\nu_L\rightarrow\nu_R)

\sin^2(\mu_\nu B_\perp L)
]

for a simple constant-field approximation.

31. Neutrino Decay

For a rest-frame lifetime (\tau_i), the survival probability is

[

P_i

\exp\left(
-\frac{L}{\gamma_i c\tau_i}
\right).
]

Since

[
\gamma_i=\frac{E_i}{m_i},
]

we obtain

[

P_i

\exp\left(
-\frac{Lm_i}{E_i c\tau_i}
\right).
]

In natural units:

[

P_i

\exp\left(
-\frac{Lm_i}{E_i\tau_i}
\right).
]

32. Neutrino Time of Flight

For (m_\nu\ll E_\nu):

[
v_\nu
\simeq
c\left(
1-\frac{m_\nu^2c^4}{2E_\nu^2}
\right).
]

The delay relative to a massless particle is approximately

[
\Delta t
\simeq
\frac{L}{2c}
\frac{m_\nu^2c^4}{E_\nu^2}.
]

For two mass eigenstates:

[
\Delta t_{ij}
\simeq
\frac{L}{2c}
\frac{\Delta m_{ij}^2c^4}{E_\nu^2}.
]

33. Neutrino Flux

For isotropic emission from a source with neutrino luminosity (L_\nu):

[

F_\nu

\frac{L_\nu}{4\pi d^2}.
]

If (L_\nu) is an energy luminosity, the approximate number flux is

[

\Phi_\nu

\frac{L_\nu}
{4\pi d^2\langle E_\nu\rangle}.
]

Spectral flux

[

\Phi_\nu(E)

\frac{dN_\nu}
{dE,dA,dt}.
]

Fluence

[

\mathcal{F}_\nu(E)

\int \Phi_\nu(E,t),dt.
]

34. Power-Law Astrophysical Neutrino Spectrum

A common model is

[

\frac{d\Phi_\nu}{dE}

\Phi_0
\left(
\frac{E}{E_0}
\right)^{-\gamma}.
]

The energy-weighted flux is

[
E^2\frac{d\Phi_\nu}{dE}.
]

For an (E^{-2}) spectrum:

[

E^2\frac{d\Phi_\nu}{dE}

\text{constant}.
]

35. Flavor Ratios of Astrophysical Neutrinos

A standard pion-decay source approximately produces

[

\nu_e:\nu_\mu:\nu_\tau

1:2:0.
]

After propagation over astrophysical distances, oscillations approximately give

[
\nu_e:\nu_\mu:\nu_\tau
\approx
1:1:1.
]

The averaged transition probability is

[

\overline{P}_{\alpha\beta}

\sum_i
|U_{\alpha i}|^2
|U_{\beta i}|^2.
]

The observed flux is

[

\Phi_\beta^\oplus

\sum_\alpha
\overline{P}{\alpha\beta}
\Phi\alpha^{S}.
]

36. Pion-Decay Neutrino Production

Charged-pion decay

[
\pi^+
\rightarrow
\mu^+
+
\nu_\mu
]

[
\pi^-
\rightarrow
\mu^-
+
\bar{\nu}_\mu.
]

Muon decay

[
\mu^+
\rightarrow
e^+
+
\nu_e
+
\bar{\nu}_\mu
]

[
\mu^-
\rightarrow
e^-
+
\bar{\nu}e
+
\nu\mu.
]

For a two-body decay (A\rightarrow B+\nu), in the rest frame of (A):

[

E_\nu

\frac{m_A^2-m_B^2+m_\nu^2}
{2m_A}.
]

For negligible neutrino mass:

[
E_\nu
\simeq
\frac{m_A^2-m_B^2}{2m_A}.
]

37. Neutron Beta-Decay Neutrino Production

[
n\rightarrow p+e^-+\bar{\nu}_e.
]

Energy conservation gives

[

E_n

E_p+E_e+E_{\bar{\nu}_e}.
]

The available decay energy is

[

Q

(m_n-m_p-m_e)c^2.
]

38. Solar Neutrino Production

The net proton–proton chain reaction is

[
4p
\rightarrow
{}^4\mathrm{He}
+
2e^+
+
2\nu_e
+
\text{energy}.
]

Approximately two electron neutrinos are produced per completed chain.

The solar-neutrino flux at Earth is approximately

[

\Phi_\nu

\frac{\dot N_\nu}
{4\pi d_{\rm ES}^2}.
]

39. Reactor Antineutrino Flux

A reactor spectrum can be modeled as

[

\frac{dN_{\bar{\nu}}}{dE}

\sum_i
f_iS_i(E),
]

where (f_i) is the fission rate of isotope (i) and (S_i(E)) is its antineutrino spectrum.

The detector flux is

[

\Phi_{\bar{\nu}}(E,L)

\frac{1}{4\pi L^2}
\sum_i f_iS_i(E).
]

Including oscillations:

[

\Phi_{\bar{\nu}_e}^{\rm detected}

\Phi_{\bar{\nu}e}^{0}
P{\bar e\bar e}.
]

40. Supernova Neutrino Luminosity

For approximately thermal emission:

[

L_\nu

4\pi R_\nu^2
\left(\frac{7}{8}\right)
\sigma_{\rm SB}T_\nu^4
]

for one fermionic species under an idealized blackbody approximation.

Fermi–Dirac spectrum

[
\frac{dN_\nu}{dE}
\propto
\frac{E^2}
{\exp(E/T_\nu-\eta_\nu)+1}.
]

Number flux at Earth

[

\Phi_\nu(E)

\frac{1}{4\pi d^2}
\frac{d\dot N_\nu}{dE}.
]

41. Neutrino Cosmological Density

For standard relic neutrinos:

[

T_\nu

\left(\frac{4}{11}\right)^{1/3}T_\gamma.
]

The neutrino number density per flavor, including neutrino and antineutrino, is approximately

[

n_{\nu_i+\bar{\nu}_i}

\frac{3}{11}n_\gamma.
]

Neutrino contribution to cosmological density

[
\Omega_\nu h^2
\simeq
\frac{\sum_i m_{\nu_i}}
{93.14\ \mathrm{eV}}.
]

Effective relativistic energy density

[

\rho_{\rm rad}

\rho_\gamma
\left[
1+
\frac{7}{8}
\left(\frac{4}{11}\right)^{4/3}
N_{\rm eff}
\right].
]

42. Effective Number of Neutrino Species

[

N_{\rm eff}

\frac{
\rho_{\rm rad}/\rho_\gamma-1
}{
\frac{7}{8}
\left(\frac{4}{11}\right)^{4/3}
}.
]

This quantity parameterizes the relativistic energy density beyond photons.

43. Neutrino Free-Streaming

An approximate thermal neutrino velocity is

[
v_\nu
\simeq
\frac{\langle p_\nu\rangle}{m_\nu}
]

for nonrelativistic neutrinos, where

[
\langle p_\nu\rangle
\simeq
3.15,T_\nu.
]

The free-streaming scale depends on the ratio

[
\lambda_{\rm FS}
\sim
\frac{v_\nu}{H}.
]

44. Atmospheric Neutrino Path Length

For a neutrino produced at altitude (h), detected at Earth radius (R_\oplus), and zenith angle (\theta_z):

[

L

\sqrt{

(R_\oplus+h)^2

R_\oplus^2\sin^2\theta_z

}

R_\oplus\cos\theta_z.
]

For an upward-going neutrino produced close to the surface:

[
L\approx -2R_\oplus\cos\theta_z
]

for (\cos\theta_z<0).

45. Neutrino Chord Length Through Earth

For nadir angle (\theta_n):

[

L_{\rm chord}

2R_\oplus\cos\theta_n.
]

For an emergence angle (\alpha) above the local horizon:

[
L_{\rm chord}
\simeq
2R_\oplus\sin\alpha.
]

This is important for Earth-skimming tau-neutrino calculations.

46. Tau-Lepton Decay Length

A tau produced by a (\nu_\tau) interaction has decay length

[

L_\tau

\gamma_\tau c\tau_\tau.
]

Using

[
\gamma_\tau=\frac{E_\tau}{m_\tau c^2},
]

we obtain

[

L_\tau

\frac{E_\tau}{m_\tau c^2}
c\tau_\tau.
]

In natural units:

[

L_\tau

\frac{E_\tau}{m_\tau}\tau_\tau.
]

47. Tau-Neutrino Regeneration

A schematic transport equation is

[

\frac{\partial\Phi_{\nu_\tau}(E,X)}{\partial X}

-N_A\sigma_{\rm tot}(E)\Phi_{\nu_\tau}(E,X)
+
S_{\rm NC}(E,X)
+
S_{\tau\rightarrow\nu_\tau}(E,X).
]

The first term describes attenuation, while the source terms describe neutral-current down-scattering and tau-decay regeneration.

48. Charged-Particle Energy Loss

For a high-energy muon or tau traveling through matter:

[

-\frac{dE}{dX}

a(E)+b(E)E.
]

Here:

• (a(E)): ionization loss,

• (b(E)E): radiative loss.

For constant (a) and (b), the range is approximately

[

R(E,E_{\rm min})

\frac{1}{b}
\ln
\left(
\frac{a+bE}
{a+bE_{\rm min}}
\right).
]

49. Cherenkov Radiation

Cherenkov emission occurs when

[
\beta n>1,
]

where

[
\beta=\frac{v}{c}.
]

Cherenkov angle

[

\cos\theta_C

\frac{1}{\beta n}.
]

Energy threshold

[

E_{\rm th}

\frac{mc^2}
{\sqrt{1-\frac{1}{n^2}}}.
]

Frank–Tamm photon yield

[

\frac{d^2N}{dx,d\lambda}

\frac{2\pi\alpha}{\lambda^2}
\left(
1-\frac{1}{\beta^2n^2}
\right).
]

50. Detector Effective Area

The expected event rate can be written as

[

N

T
\int
\Phi_\nu(E,\Omega)
A_{\rm eff}(E,\Omega)
,dE,d\Omega.
]

The effective area may be expressed schematically as

[

A_{\rm eff}

A_{\rm geom}
P_{\rm int}
P_{\rm trigger}
P_{\rm selection}.
]

51. Detector Effective Volume

[

V_{\rm eff}

V_{\rm gen}
\frac{N_{\rm selected}}{N_{\rm generated}}.
]

The effective target mass is

[

M_{\rm eff}

\rho V_{\rm eff}.
]

A corresponding effective area may be estimated using

[
A_{\rm eff}
\sim
V_{\rm eff}\rho N_A\sigma_\nu.
]

52. Exposure

[

\mathcal{E}(E)

A_{\rm eff}(E)T.
]

For a direction-dependent experiment:

[

\mathcal{E}(E,\Omega)

\int
A_{\rm eff}(E,\Omega,t),dt.
]

Expected events:

[

N

\int
\Phi(E,\Omega),
\mathcal{E}(E,\Omega)
,dE,d\Omega.
]

53. Statistical Significance

For a simple counting experiment with signal (S) and background (B):

[
Z\simeq\frac{S}{\sqrt{B}}
]

when (B) is sufficiently large.

A more accurate Asimov significance is

[

Z_A

\sqrt{
2\left[
(S+B)\ln\left(1+\frac{S}{B}\right)-S
\right]
}.
]

Poisson probability

[

P(n|\mu)

\frac{\mu^ne^{-\mu}}{n!}.
]

54. Likelihood in Neutrino Analysis

For binned Poisson data:

[

\mathcal{L}(\boldsymbol{\theta})

\prod_i
\frac{
\mu_i(\boldsymbol{\theta})^{n_i}
e^{-\mu_i(\boldsymbol{\theta})}
}{
n_i!
}.
]

The log-likelihood is

[

\ln\mathcal{L}

\sum_i
\left[
n_i\ln\mu_i-\mu_i-\ln(n_i!)
\right].
]

A likelihood-ratio statistic is

[

-2\ln\lambda

-2\ln
\left[
\frac{
\mathcal{L}(\boldsymbol{\theta})
}{
\mathcal{L}(\hat{\boldsymbol{\theta}})
}
\right].
]

55. Chi-Square Function

A simple chi-square statistic is

[

\chi^2

\sum_i
\frac{
\left(
N_i^{\rm obs}-N_i^{\rm pred}
\right)^2
}{
\sigma_i^2
}.
]

With nuisance parameters (\eta_k):

[

\chi^2_{\rm total}

\chi^2_{\rm data}
+
\sum_k
\left(
\frac{\eta_k}{\sigma_{\eta_k}}
\right)^2.
]

56. Neutrino Effective Mass Quantities

Three distinct mass observables are commonly used:

Beta-decay mass

[

m_\beta

\sqrt{
\sum_i|U_{ei}|^2m_i^2
}.
]

Neutrinoless-double-beta-decay mass

[

m_{\beta\beta}

\left|
\sum_iU_{ei}^{2}m_i
\right|.
]

Cosmological mass sum

[

\Sigma m_\nu

m_1+m_2+m_3.
]

These observables probe different combinations of neutrino masses and mixing parameters.