neutrino + antineutrino annihilation...
What happens in neutrino + antineutrino annihilation?
The two particles meet at a single point and annihilate each other, producing a virtual Z boson, which is the neutral (i.e. no electric charge) carrier of the weak nuclear force. This Z boson then immediately decays to produce another particle/antiparticle pair, either a new pair of neutrinos, two charged leptons, or a quark/anti-quark pair:
\bar \nu + \nu \rightarrow Z^0 \rightarrow \bar e^+ + e^-
Conservation of classical weak leptonic current:
\partial_{\mu} J_{\mu}^L = \sum_j \partial_\mu(\bar \nu_j Z_\mu^0 \nu_j) = 0
Conservation of Energy states that what you can produce depends on how much energy there is available from the colliding neutrinos.
Violations of the lepton number conservation laws:
In the Standard Model, leptonic family number (LF) would be preserved if neutrinos were massless. Since neutrinos do have a tiny nonzero mass, neutrino oscillation has been observed, and conservation laws for LF are therefore only approximate. This means the conservation laws are violated by chiral anomalies, which results in anomalous leptonic current, although because of the smallness of the neutrino mass they still hold to a very large degree for interactions containing charged leptons. In other words, the classical leptonic current J_{\mu}^L is conserved under the Standard Model:
\partial_{\mu} J_{\mu}^L = \sum_j \partial_\mu(\bar e_j^+ \gamma_{\mu} e_j^-) = 0
Thus, it is possible to see rare muon decays such as:
\begin{matrix} & \mu^{-} & \rightarrow & e^{-} & + & \nu_e & + & \overline{\nu}_{\mu} \\ L: & 1 & = & 1 & + & 1 & - & 1 \\ L_e: & 0 & \ne & 1 & + & 1 & + & 0 \\ L_{\mu}: & 1 & \ne & 0 & + & 0 & - & 1 \end{matrix}
Because the lepton number conservation law in fact is violated by chiral anomalies, there are problems applying this symmetry universally over all energy scales. However, the quantum number (B − L) is much more likely to work and is seen in different models such as the Pati-Salam model.
Baryonic charge non-conservation:
Baryonic charge violation appears through the Adler-Bell-Jackiw quantum chiral anomaly of the U(1) group.
Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak Lagrangian conserves baryonic charge. Quarks always enter in bilinear combinations q \bar q, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current J_\mu^B is conserved under the Standard Model:
\partial_\mu J_\mu^B = \sum_j \partial_\mu(\bar q_j \gamma_{\mu} q_j) = 0
However, quantum corrections destroy this conservation law resulting in anomalous baryonic current non-conservation, instead of zero in the right hand side of this equation:
\partial_\mu J_\mu^B = \frac{g^2 C}{16\pi^2} G_{\mu\nu} \tilde{G}_{\mu\nu}
High energy physics (B − L):
In high energy physics, B − L is the difference between the baryon number (B) and the lepton number (L).
This quantum number is the charge of a global/gauge U(1) symmetry in some GUT models, called U(1)_{(B-L)}. Unlike baryon number alone or lepton number alone, this hypothetical symmetry is not broken by chiral anomalies or gravitational anomalies, as long as this symmetry is global, which is why this symmetry is often invoked. If B − L exists as a symmetry, it has to be spontaneously broken to give the neutrinos a nonzero mass if we assume the seesaw mechanism.
The chiral anomalies that break baryon number conservation and lepton number conservation individually cancel in such a way that B − L is always conserved.
One example is proton decay where a proton (B = 1; L = 0) decays into a pion (B = 0, L = 0) and positron (B = 0; L = −1).
\begin{matrix} & p^{+} & \rightarrow & \pi^{0} & + & \bar e^+ \\\ B - L: & 1 & = & 0 & + & 1 \end{matrix}
Proton decay:
\begin{matrix} & p^{+} & \rightarrow & \bar e^{+} & + & \pi^0 & \rightarrow & \bar e^+ & + & 2 \gamma \\\ B - L: & 1 & = & 1 & + & 0 & \rightarrow & 1 & + & 0 \end{matrix}
Neutron decay:
\begin{matrix} & n^{0} & \rightarrow & p^{+} & + & e^- & + & \bar \nu_e \\\ B - L: & 1 & = & 1 & - & 1 & + & 1 \end{matrix}
Muon decay:
\begin{matrix} & \mu^{-} & \rightarrow & e^{-} & + & \nu_e & + & \overline{\nu}_{\mu} \\ B - L: & -1 & = & -1 & - & 1 & + & 1 \end{matrix}
Neutrino + antineutrino annihilation:
\begin{matrix} & \bar \nu & + & \nu & \rightarrow & Z^0 & \rightarrow & \bar e^{+} & + & e^{-} \\ B - L: & 1 & - & 1 & = & 0 & = & 1 & - & 1 \end{matrix}
Hence proton decay conserves B - L, even though it violates both lepton number and baryon number conservation and neutron decay conserves baryon number B and lepton number L separately, so also the difference B - L is conserved.
Where Q is the electrical charge in elementary charge units and T_z is the third component of weak isospin. The weak hypercharge can be expressed as:
Y_W = 2(Q - T_z)
Weak hypercharge Y_W is related to B − L via:
X + 2Y_W = 5(B - L)
Integration via substitution:
X + 4(Q - T_z) = 5(B - L)
Where X is the U(1)_{(B-L)} symmetry GUT-associated conserved quantum number.
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Reference:
http://en.wikipedia.org/wiki/Neutrino"
http://en.wikipedia.org/wiki/Antineutrino"
http://en.wikipedia.org/wiki/Lepton_number#Violations_of_the_lepton_number_conservation_laws"
http://en.wikipedia.org/wiki/Chiral_anomaly"
http://en.wikipedia.org/wiki/Pati%E2%80%93Salam_model"
http://en.wikipedia.org/wiki/B%E2%88%92L"
http://en.wikipedia.org/wiki/Weak_hypercharge"
http://en.wikipedia.org/wiki/Weak_isospin"
http://en.wikipedia.org/wiki/Seesaw_mechanism"
