Symmetries and Conservation Laws

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Symmetries and Conservation Laws
1. Costituents of Matter 2. Fundamental Forces 3.
Particle Detectors 4. Symmetries and
Conservation Laws 5. Relativistic Kinematics 6.
The Static Quark Model 7. The Weak Interaction 8.
Introduction to the Standard Model 9. CP
Violation in the Standard Model (N. Neri)
Five Deities Mandala Tibet, XVIIth Century
The word is used also to indicate a circular
diagram, basically made by the association of
different geometric figures (the most used being
the dot, the triangle, the circle and the
square). The drawing has spiritual and ritual
meaning in both Buddhism and Hinduism.
Mandala from it.wikipedia.org
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Class Invariance Conserved quantity
Proper orthochronousLorentz symmetry translation in time  (homogeneity) energy
translation in space  (homogeneity) linear momentum
rotation in space  (isotropy) angular momentum
Discrete symmetry P, coordinate inversion spatial parity
C, charge conjugation charge parity
T, time reversal time parity
CPT product of parities
Internal symmetry (independent ofspacetime coordinates) U(1) gauge transformation electric charge
U(1) gauge transformation lepton generation number
U(1) gauge transformation hypercharge
U(1)Y gauge transformation weak hypercharge
U(2) U(1) SU(2) electroweak force
SU(2) gauge transformation Isospin
SU(2)L gauge transformation weak isospin
P SU(2) G-parity
SU(3) "winding number" baryon number
SU(3) gauge transformation quark color
SU(3) (approximate) quark flavor
S(U(2) U(3)) U(1) SU(2) SU(3) Standard Model
A classification of symmetries in particle physics
Wikipedia
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Symmetries of a physical system
Quantum system
Classical system
Lagrangian Formalism
Lagrangian Formalism
Hamiltonian Formalism
Hamiltonian Formalism

• Invariance of dynamical equations
• Invariance of commutation relations
• (Invariance of probability)

Invariance of Equations of Motion
E. Noethers Theorem (valid for any lagrangian
theory, classical or quantum) relates symmetries
to conserved quantities of a physical system
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A classical example
Let us do a translation
The equations of motion are translation Invariant
!
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If one calculates the forces acting on 1 and 2
In the classical Lagrangian formalism
L invariant with respect to q
p conserved
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In the Hamiltonian formalism
Commutation (Poisson brackets) with the
Hamiltonian ? conservation of dynamical quantities
Possible conservation of a dinamical quantity
Possible symmetry
This formalism can easily be extended to Quantum
Mechanics
In Quantum Mechanics, starting from the
Schroedinger Equation
Time evolution (unitary operator)
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Schroedinger and Heisenberg Pictures
Heisenberg
Schroedinger
for the operators in the Heisenberg picture
Taking the derivatives
Conserved quantities commute with H
In the case when there is an explicit time
dependence (non-isolated systems)
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Translational invariance a continuous spacetime
symmetry
The translation operator is naturally associated
to the linear momentum
For a finite translation
unitary
Self-adjoint the generator of space translations
If H does not depend on coordinates
The momentum is conserved
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Rotational invariance a continuous spacetime
symmetry
The rotation operator is naturally associated
with the angular momentum
Angular momentum (z-component) operator (angle
phi)
Self-adjoint rotation generator
A finite rotation
unitary
If H does not depend on the rotation angle f
around the z-axis
The angular momentum is conserved
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Time invariance (a continuous symmetry)
The generator of time translation is actually the
energy!
Using the equation of motion of the operators
If H does not dipend from t, the energy is
conserved
The continous spacetime symmetries Space
translation Space rotation Time translation
Linear momenum Angular momentum Energy
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Continuous symmetries and groups the case of
SU(2)
Combination of two transformations the result
depends on the commutation rules of the group
generators. For instance, in the case of space
translations
Commutative (Abelian) algebra of translations
Translation operator along x
(two translations commute). Moreover
and clearly
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In the case of rotations
Commutation rules for the generators
A non-commutative algebra Rotations about
different axes do not commute This is also the
case of SU(2), which we are now going to
introduce
Why SU(2) ?
In the case of a two level quantum system, the
relevant internal symmetries are described by the
SU(2) group, having algebraic structure similar
to O(3). SU(2) finds an application in the
Electroweak Theory. SU(3) can instead be applied
to QCD, the candidate theory of Strong
Interactions.
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Isospin Symmetry, SU(2)
Let us consider a two-state quantum system (the
original idea of this came from the neutron and
the proton, considered to be degenerate states of
the nuclear force). Since they were considered
degenerate, they could be redefined
Degeneration
Redefinition
Double degeneracy, similar to what happens in
s1/2 spin systems. The degeneration can be
removed by a magnetic field
One can introduce a two-components spinor
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The ridefinition now becomes
A symmetry for the Strong Interactions (broken by
electromagnetism)
SU(2) is a Lie group Properties can be deduced
from infinitesimal transformations
Which can be written in a general form
Pauli matrices
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Infinitesimal rotation of the p-n doublet
A finite rotation in SU(2)

• Generalization of a global phase transformation
• Three phase angles
• Non-commuting operators (Non-abelian phase
  invariance)

example
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The two nucleon system
A one-nucleon state can be described with the
base of the nuclear spinors
A two nucleon state can be constructed by
combining the Isospin states
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Isospin triplet
These states transform into one another in
Isospin rotations, similar to a 3-d vector for
ordinary rotations
Isospin singlet (scalar)
In an Isospin rotation
Isospin Invariance means there are two
amplitudes, I0 e I1 I1 states cannot be
internally distinguished by the Strong
Interactions Strong Interactions conserves I and
I3 (full I-spin invariance) Electromagnetic
interaction conserves I but not I3 (the charge is
different in different states of the same I-spin
multiplet, e.g. the proton and neutron)
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Isospin invariance at work pion-nucleon reactions
Let us consider the (strong) diffusion of pions
and nucleons. The diffusion amplitudes are the
element of the S matrix
which can be written as combinations of the
total isospin of the system
taking into account the conservation of I and
I3 Note, however, that S must commute also with
I1 and I2 which implies (Schurs Lemma) that S is
S(I). Therefore, the amplitude can be expressed
as a function of the allowed total isospin
amplitudes.
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Using the two invariant amplitudes (1/2 and 3/2),
one can write the amplitudes of the following
processes
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Some Isospin amplitudes that one can study by
making Pion-Nucleon diffusion experiments
p
p
N
N
Let us make the Pion-Nucleon experiment at an
energy of the Pion of about 200 MeV. This
generates a clear resonant peak in the pp cross
section, indicating the presence of a dominant
I3/2 resonance (the ?, m 1232 MeV)
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At the energy when the cross section resonates
with the ? state, the 3/2 amplitude is
dominant. Therefore S(1/2) can be neglected at
that energy and we have a prediction of relative
cross sections of
p p cross section dominates at the energy of the
Delta resonance
vs GeV
In agreement with the experimental observation of
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A little preview of a Lagrangian for a system of
particles Nucleon, Pions and some Interaction
(Yukawa style) between them. This effective
Lagrangian could describe reasonably well
hadronic physics at energies below 2 GeV
(assuming the proton and the pion to be
elementary)
Will produce the Dirac Equation A free non
interacting Nucleon (Kinetic Energy and Mass)
Will produce the Klein-Gordon Equation. This term
represents free Pions (Kinetic Energy and Mass
terms for each Pion)
A Yukawa interaction term (interaction term in
the simplest form, taking into account pion
parity through the Gamma-5 matrix)
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Nucleons and Quarks
I3 particle antiparticle particle antiparticle
1/2
-1/2
An Isospin triplet the Pion
I I3 Wave Function Q/e
1 1 1
1 -1 -1
1 0 o
0 o 0
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Building up strongly interacting particles
(hadrons) using Quarks as building blocks
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Will need to reconsider later on (Static Quark
Model section)
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More on the two nucleon state
S
Isospin part
The total wave function
A
(non relativistic decomposition)

• The Deuton case
• As it is known the deuton spin 1 ? a symmetric
• f has (-)l symm. It is known that the two
  nucleons are l0 or l2, f is symm.
• Then must be antisymmetric.
• This is because ? tot must be antisymmetric in
  nucleon exchange.

The Deuton is an Isospin singlet
Now, since the Deuton has I0 and the Pion has
I1, considering the reactions
Isospin I 1 1
0,1 1
The reaction can proceed only via I1
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Gauge symmetries (global and local)
Gauge symmetries are continuous symmetries (a
continuous symmetry group). They can be global or
local. Global symmetries conserved quantities
(electric charge) Local symmetries new fields
and their transformation laws (Gauge theories)
Let us consider the Schoedinger equation
Let us consider a global phase transformation
the change in phase is the same everywhere
The Schroedinger equation is invariant for this
transformation. This invariance ia associated (E.
Noethers Theorem) to electric charge
conservation.
But then what happens if we consider a local
gauge tranformation ?
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How does one realize a local gauge invariance ?
Non invariant! And this is because
extra term !
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To solve the problem (and stick to the
invariance) we can introduce a new field. The
field would compensate for the extra term. The
new field would need to have an appropriate
transformation law.
Since the free Schroedinger Equation is not
invariant under
Let us modify the Equation
Compensating fields
Transformation laws
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This allows us to restore the invariance
To give the fields physical meaning
It is invariant, indeed

• The local gauge U(1) invariance of the free
  Schroedinger field
• Requires the presence of the Electromagnetic
  Field
• Dictates the field transformation law

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This gauge symmetry is the U(1) gauge symmetry
related to phase invariance There are, of
course, other possibilities.
In physics, a gauge principle specifies a
procedure for obtaining an interaction term from
a free Lagrangian which is symmetric with respect
to a continuous symmetry -- the results of
localizing (or gauging) the global symmetry group
must be accompanied by the inclusion of
additional fields (such as the electromagnetic
field), with appropriate kinetic and interaction
terms in the action, in such a way that the
extended Lagrangian is covariant with respect to
a new extended group of local transformations.
FREE
INTERACTING
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The U(1) Gauge Invariance and the Dirac field
Dirac Equation (1928) a quantum-mechanical
description of spin ½ elementary particles,
compatible with Special Relativity
Matrices 4x4 (gamma)
4 component spinor
Dirac probability current
Conjugate spinor
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The Dirac Lagrangian
Features a global gauge invariance
(phase transformations)
Let us require this global invariance to hold
locally. The invariance is now a dynamical
principle
Now the gauge transformation depends on the
spacetime points
Let us see how L behaves
Since
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This lagrangian is NOT gauge-invariant.
If we want a gauge-invariant L, one has to
introduce a compensating field with a suitable
transformation law
This new lagrangian is locally gauge-invariant.
This was made possible by the introduction of a
new field (the E.M. field).
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The gauge field A must however include a
(gauge-invariant) free-field term. This will be
the E.M. free field term (more on this in the
Standard Model lecture)
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Discrete symmetries P,C,T
Discrete symmetries describe non-continuous
changes in a system. They cannot be obtained by
integrating infinitesimal transformations. These
transformations are associated to discrete
symmetry groups
Parity P
Inversion of all space coordinates
The determinant of this transform is -1. In the
case of rotations, that would be 1
A unitary operator. Eigenvalues 1, -1 (if
definite-parity states) Eigenstates definite
parity states
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Parity is conserved in a system when
The case of the central potential
Bound states of a system with radial symmetry
have definite parity Example the hydrogen atom
Hydrogen atom wavefunction (no spin)
Radial part
Angular part
The effect of parity on the state is
Electric Dipole Transition ?l 1. This implies
a change of parity. Since, however, e.m.
interactions conserve parity, this indicates the
need to attribute to the photon an intrinsic
(negative) parity.
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The general parity of a quantum state
Let us consider a single particle a. The
intrinsic parity can be represented by a phase
Intrinsic
Spatial
Which is the physical meaning of the intrinsic
parity ? For instance, in a plane wave (momentum
eigenstates) representation
If one then lets go the momentum to zero, one can
see that the intrinsic parity has the meaning of
a parity in the p0 system
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The parity of the photon from a classical analogy
A classical E field obeys
Let us take the P
To keep the Poisson Equation invariant, we need
to have the following law for E
On the other hand, in vacuum (no charges)
And the parity operation would give
In order to make it consistent with the electric
field transformation
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The action of parity on relevant physical
quantities
Position
Time
Momentum
Angular Momentum
Charge
Current
E field
B field
Spin
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Symmetry at work the parity of the Pion
Parity of a complex system overall parity times
the product of the intrinsic parities of the
parts of the system. Let us illustrate this for
the Pion, using the reaction This is a Strong
Interaction process ? Parity is conserved
P conservation
J conservation
Pauli symmetry between neutrons
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Final considerations on parity
Some intrinsic parities cannot be observed.

For instance (p, n). They
are conventionally chosen to be 1. Because of
the Baryon Number conservation the actual P value
is not important as it cancels out in any
reaction.
Neutral Pion Parity.

It is deduced from the photon pair
distribution in
(study of the relative polarization of the two
photons)
Particles can be (and are) classified using J and
P (and C)
Transformation properties for rotations and space
reflections. Spin-parity
The intrinsic parities of Particles and
Antiparticles
P(particle) - P (antiparticle) FERMIONS P(pa
rticle) P (antiparticle) BOSONS
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Time Inversion T preliminary considerations
Time is a parameter characterizing the evolution
of a system
Special and General Relativity
Classical Physics

• The time flow depends on
• The motional status of the clock
• The spacetime structure

An absolute Cosmic Flow just one Clock for the
whole Universe
Is there a Time symmetry? Is there a possible
Arrow of Time based on asymmetry between past and
future? (A. Eddington , 1928)
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How can we define a time symmetry ? What do we
really need to run a movie backward? In Classical
Physics
Dynamical Law
State - A
State - B
Dynamical Law

1. The Initial and Final Conditions can be inverted
   (State ? - State)
2. The Dynamical Law is time-neutral t? -t leaves
   the law unchanged

Here is a time-neutral dynamical law
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Time Arrow in the MACROSCOPIC world
Conventional wisdom says that physical processes
at the MICROSCOPIC level are time-symmetric ? if
the time arrow is changed in sign, the
theoretical description would not change. On the
other hand, in the MACROSCOPIC world, one can
appreciate the existence of a preferential
direction of time from the past to the future.
Building up the Arrow of Time using the Second
Law of Thermodynamic. Entropy can only increase.
Ink dissolved in water in the left hand side of
the vessel. One has a trivial time sequence
corresponding to an increase of Entropy.
A
B
The only possible sequence is A?B?C?D ,even if
from the MICROSCOPIC viewpoint the opposite
sequence is possible as well (and does not
violate any other physical law).
C
D
There is no conflict between the MICRO
(T-reversible) laws and the MACRO
(T-irreversibile) behaviour. The key point is
that the macrostate D corresponds to a much
bigger number of microstates with respect to A
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What is the origin of the MACROSCOPIC Time Arrow?

Boltzmanns suggestion thermodynamic time can
arise from cosmic time.
If the Universe had begun from an equilibrium
condition, then (possibly) we would not have an
Entropy concept as we know it.
A
B
Analogy
C
D
Since all started from a relatively ordered
state, a very peculiar state indeed (the first
primi istanti del Big Bang), it is then natural
(emergent) a tendency to maximum Entropy ? it is
naturally emergent a concept of macroscopic
cosmic time
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Let us suppose a different Demiurgical Creation
If the Universe had begun from an equilibrium
condition (see below) then (likely) we would not
have a concept of Entropy like this. Likely, this
means no macroscopic time.
Creazione di Adamo Michelangelo Buonarroti
(Musei Vaticani - La Cappella Sistina)
Two species of perfect gases (the blue and the
red one). They do not interact except for
perfectly elastic scatterings. Time arrow
undefined
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Time Inversion T in Particle Physics
It changes the time arrow
T
Classical dynamical equations are invariant
because of second order in time
Classical microscopic systems T invariance is
fully respected.
Classical macroscopic systems time arrow
selected statistically (defined as the direction
of non decrease of entropy)
In the quantum case, the Schroedinger equation
Is not invariant for
T
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Let us now start from the Conjugate Schroedinger
Equation
And define a T-inversion operator
T
So, with this definition of T operator, we have
The operator representing T is an antilinear
operator. The square modulus of transition
amplitudes is conserved
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T
Let us take the complex conjugate
And we have the required result in terms of the
conjugate wave function
Wigner Theorem on Quantum Systems

• Any symmetry of a quantum system is given by
• either a unitary
• or an antiunitary operator

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The T operator in Quantum Mechanics
Action on position and momentum
Then considering
and calculating
The operator has to be antilinear
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Particle Physics what do we really measure ?
Non-interacting particles in the initial state
Non-interacting particles in the final state
Quantum interference. Unitary evolution.
Performing a momentum measurement of a final
states involves (in general) a selection process
that traces away the other quantum mechanical
degrees of freedom (exception Dalitz Plots).
A measurement on a particle can be done during a
relatively long time ? a momentum eigenstate can
be built with (almost) arbitrary accuracy
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We see the consequence of T-invariance at the
microscopic level by analizing the the
transition amplitudes when no interference is
present between i and f. One single transition
amplitude has to be involved. Otherwise the
measurement process would generate phase
cancellation and irreversibility ! The selection
is made on momentum eigenstate of the final
state. If these conditions are met, one can use
the detailed balance principle on the initial and
final momentum eigenstates
Note the detailed balance DOES NOT imply the
equality of the reaction rates
A classical test, the study of the (Strong)
reaction
T is violated at the microscopic level il the
Weak Nuclear Interactions
Physical Review Letters 109 (2012) 211801. BaBar
experiment at SLAC Comparing the reactions
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Charge Conjugation C
An internal discrete symmetry
It changes the sign of the charges (and magnetic
moments)
In the case of a quantum state
The C eigenstates are the neutral states
For the photon case
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The charge conjugation of the photon from a
classical analogy
A classical E field obeys
Let us take the C
To keep the Poisson Equation invariant, we need
to have the following law for E
On the other hand, for a system of charges
And the charge conjugation operation would give
In order to make it consistent with the electric
field transformation
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The C-parity reverses the signs of charges and
magnetic moments. The electromagnetic
interaction is not affected. For interactions
that are C-invariant
Let us distinguish between particles that have
antiparticles And particles that dont
Spin, momentum
The action of the C operator
Mass, type
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The C-parity of a state can be calculated for a
neutral state if we know the wave function of the
state. It is the product between the parities of
the total wavefunction and the parities of the
constituents.
Since charge conjugation of two particles of
opposite charge, swaps the identify of the
particles, one has to account for the proper
quantum statistics
True also in general for spin zero particles
For a couple of femions, instead
In the pi-zero decay
This decay in 3 photons
Is forbidden if C is conserved in electromagnetic
interactions. In fact
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Action of C,P,T
C P
T CPT
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Positronium
Similar to the H atom. Actually, the true atom.
We require the total wavefunction to be
antysimmetric, considering the electron and the
positron as different C-states of the same
particle
The space part
The spin part
Triplet
Singlet
The C conjugation for the Ps state
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Positronium in the l0 (fundamental) state Let us
now calculate C
This state has C1 since in decays in 2 ?s
This state has C - 1 since in decays in 3 ?s
Singlet Triplet
Antisymmetry by electron/positron exchange
C-parity conservation determines the Ps decay
modes
Singlet Triplet
See the vertices in the relevant Feynman diagrams
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Photons, Spin, Helicity
Gauge - invariant
Coulomb Gauge
Free propagation
Plane wave solution
Transversality condition in the Coulomb Gauge
Plane polarization Circular polarization
For instance
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Circular polarization
Which can be expressed by using the rotating
vectors
The polarization vectors can be associated to the
photon spin states
If the wave propagates along z
Jz only due to spin
Let us make a rotation around the z axis
Eigenstates of Jz with eigenvalues 1,-1,0
Transversality the Jz0 state does not exist
e
e
Photons with Jz 0 are virtual (longitudinal
photons). They have m?0
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Helicity
Projection of the spin in the momentum direction
An approximate quantum number for massive
particles So much better inasmuch the particle
is relativistic Exact for photons
The advantages of a description over a
description

• The Helicity is unchanged by rotation
• Since , the helicity can be
  defined in a relativistic context

Invariance laws in action
An example E.M. interactions conserve Parity One
can then build a Parity-violating quantity, like
Then, in E.M. interactions this quantity must be
zero. And one can test this! In E.M. Interactions
right-handed and left-handed photons appear with
equal amplitudes. In this way they compensate to
the result and conserve Parity.
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The Neutrino
C, P are violated in Weak Nuclear Interactions
Neutrinos takes part only in Weak Nuclear
Interactions In the massless neutrino
approximation
Experimental evidence indicates that in Weak
Interactions Neutrinos are always left-handed.
Antineutrinos are always right-handed !
P
C
CP
To a very good approximation, Weak interactions
conserve CP (not C, not P)
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The CPT Theorem
In a local, Lorentz-invariant quantum field
theory, the interaction (Hamiltonian) is
invariant with respect to the combined action of
C,P,T (Pauli, Luders, Villars, 1957)
A few consequences

• Mass of the particle Mass of the antiparticle
• (Magnetic moment of the particle) -- (Magnetic
  moment of antiparticle)
• 3) Lifetime of particle Lifetime of
  antiparticle

Proton Antiproton Electron Positron
Q e -e -e e
B o L(e) 1 -1 1 -1
µ
s
Protons, electrons
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CPT Theorem (wikipedia) In quantum field theory
the CPT theorem states that any canonical (that
is, local and Lorentz-covariant) quantum field
theory is invariant under the CPT operation,
which is a combination of three discrete
transformations charge conjugation C, parity
transformation P, and time reversal T. It was
first proved by G.Lüders, W.Pauli and J.Bell in
the framework of Lagrangian field theory. At
present, CPT is the sole combination of C, P, T
observed as an exact symmetry of nature at the
fundamental level.
CPT and the Fundamental Interactions Concern
systems for which a quantum field theory has been
developed (i.e. systems subject to Strong,
Electromagnetic and Weak Interactions). The
relative Lagrangian is CPT-invariant.
Note CPT is a flat-spacetime theorem (it does not
concern Gravity)
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Invariance of Physical Laws and Invariance of
Physical Systems
Let us consider the quantity
It is an Electric Dipole Moment (EDM)
How this quantity P,T transforms ?
Now let us consider a system bound by a P
conserving interaction, like the Electromagnetic
Interaction. Since P is conserved, one can expect
that this quantity should be zero. We will
consider the Ammonia Molecule
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This system of minimum energy does not display
the full symmetry of the Electromagnetic
Interaction There are in fact two degenerate
states
The full ammonia wavefunction has the form
Which displays the symmetry of the underlying
interaction. This is an example of a Broken
Symmetry The minimum energy configuration does
not display the symmetries of the theory

• Other examples
• A ferromagnet (when T is below the Curie
  temperature)
• The Higgs potential

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A recipe to search for violation of fundamental
laws
Consider a system S that obeys some interaction
I. Build up a quantity Q that is violated by
said interaction. Q is violated by I. Find a
(non-degenerate) s state of the system S. Check
that Q is zero on the state s.
Example the neutron Electric Dipole Moment
(EDM) The magnetic moment (like a spin) changes
sign with the T inversion The EDM changes sign
with the P inversion Current best limit on
neutron EDM
"The Neutron EDM in the SM  A Review".
arXivhep-ph/0008248
dn lt 1032ecm
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Particle Numbers baryonic, flavor, and leptonic
Flavor
The flavor is the quark content of a hadron
Massa (MeV) Quark U D S C B
p uud 2 1 0 0 0
n udd 1 2 0 0 0
? uds 1 1 -1 0 0
?c udc 1 1 0 1 0
p u-dbar 1 -1 0 0 0
K- s-ubar -1 0 -1 0 0
D- d-cbar 0 1 0 -1 0
Ds c-sbar 0 0 1 1 0
B- b-ubar -1 0 0 0 -1
? b-bbar 0 0 0 0 0
938 940 1116 2285 140 494 1869 1970 5279 9460
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Favor quantum numbers refer to quark content of
hadrons They are conserved in Strong and
Electromagnetic Interactions They are violated in
Weak Interaction
Strangeness
Charm
Beauty
Top
In a Stong Nuclear (or E.M.) process, all flavors
are conserved
In Weak Interactions instead
Baryon Number
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Baryon Number
The Baryon Number is equivalent to
Baryons have B1 while Antibaryons have B
-1 Mesons have B 0 This law follows from the
conservation of the Quark Number. Quarks
transform into each other. They disappear (or
appear) in pairs.
Flavor quantum numbers refer to the identity of
quarks
(Isospin 1/2 o -1/2 in doublets) Strangeness
-1 for the s quark Charm 1 for the c
quark Bottom -1 for the b quark Top 1 for the
t quark
Violated in Weak Interactions
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The Leptonic Numbers
Electronic Lepton Number
Muonic Lepton Number
Numero leptonico tauonico
The Leptonic Numbers are conserved in any known
interaction WITH THE EXCEPTION OF Neutrino
Oscillations. In Neutrino Oscillations, they are
violated.
This process does not involve neutrino
oscillations
So, it does not take place .
However, we can define a total lepton number
To the best of our knowledge the Total Lepton
Number (sum of the three leptonic numbers) is
conserved in every interaction.
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