leptogenesis and quantum · pdf filesteve blanchet, seminar lup, montpellier, 06/12/11 3 the...

Leptogenesis and quantum decoherenceLeptogenesis and quantum decoherence

Steve Blanchet

Theory Seminar

Laboratoire Univers et Particules de Montpellier, Dec. 6, 2011

With P. Di Bari, D. Jones and L. Marzola, 1112.XXXX

Steve Blanchet, Seminar LUP, Montpellier, 06/12/11 2

Outline Introduction

□ The matter-antimatter puzzle□ The neutrino mass puzzle□ An elegant solution: the seesaw mechanism and its cosmological

consequence, baryogenesis through leptogenesis Vanilla leptogenesis Adding flavor to vanilla leptogenesis

□ From unflavored to fully-flavored leptogenesis Density matrix formalism

□ Illustration with neutrino oscillations in vacuum and matter Density matrix for leptogenesis

□ The importance of decoherence General case: many heavy neutrino flavors

□ Phantom terms Summary and conclusion


The matter-antimatter puzzle

One can estimate the relic density of baryons, according to SM interactions (annihilations into pions):

Two independent sources of information, the temperature anisotropy in the CMB, as well as the synthesis of light elements in the early Universe (BBN) point to a much larger value:

To avoid the baryon annihilation catastrophe and to separate baryons from anti-baryons, a small asymmetry must be generated primordially.

The observable Universe is composed of matter (mainly baryons). Antimatter is only seen in particle accelerators and in cosmic rays.



1´000´000´002 1´000´000´000

Matter Antimatter

The annihilation occurs then very efficiently and one is left only with the small excess of matter!


EW baryogenesis in the SM is ruled out by LEP II, which puts the bound


The Standard Model contains all ingredients!

In order to produce a baryon asymmetry in the Early Universe, one needs to fulfill three conditions [Sakharov, 1967]

□ Baryon number violation

□ C and CP violation□ Departure from thermal equilibrium

ELECTROWEAK BARYOGENESIS ☺ To produce enough baryon asymmetry, the phase transition

needs to be strongly first order

Anomalous processes

Quark CKM matrix

At the electroweak phase transition

New physics is needed!

☺


The puzzle of neutrino masses

In the Standard Model neutrinos are massless.

Today, after 10 years of great successes in neutrino oscillation experiments, the evidence is overwhelming that neutrinos have masses and mix. One has measured quite precisely two mass-squared differences :

Sol.+ Reac.

Atm.+ Acc.

From the the measurement of the Z width at LEP, there should be 3 neutrino flavors, and thus 3 neutrino masses.

Neutrino oscillation experiments provide no information on the absolute neutrino mass scale! Fortunately, there are other probes possible…


□ Direct measurement (Tritium β-decay)[Mainz and Troitsk exps., 04]

□ Neutrinoless Double-β-Decay[CUORICINO exp., 10]

□ Cosmology (CMB+LSS) [WMAP,10]

The puzzle of neutrino masses

Bottom line: neutrinos involve a scale much smaller than all other mass scales in the SM!

New physics is required to explain it!


Resolution of the two puzzles

The seesaw mechanism

Small neutrino masses

Baryogenesis through leptogenesis

Matter-antimatter problem


The type I seesaw mechanism

The seesaw mechanism originates from the following extension of the SM Lagrangian:

This extension is clearly acceptable on grounds of gauge invariance and renormalizability, and is minimal in its particle content (here: 3 new particles).

Yukawa coupling Majorana mass term

where and are the Higgs and left-handed lepton doublets, respectively, and are RH neutrinos (gauge singlets).



The masses of the singlet neutrinos are essentially free parameters, and thus can be taken to be very large ( ).

Seesaw mechanism! [Minkowski, 77]

After spontaneous symmetry breaking, the vev of the Higgs leads to a Dirac mass term . The seesaw assumes

so that the neutrino mass term can be block-diagonalized as:

1st order

After diagonalization: 3 light Majorana neutrinos, mass

3 heavy Majorana neutrinos, mass


The three Sakharov conditions are fulfilled :□ Baryon number is violated in anomalous processes

□ CP is violated in the decay of the heavy neutrinos: interference between tree level and 1-loop diagrams

□ Decays are out of equilibrium at some point, parametrized by

Baryogenesis through leptogenesis

``decay parameter´´

CP asymmetry parameter

Baryogenesis through leptogenesis [Fukugita, Yanagida, 86] is the generation of a lepton asymmetry by the decay of heavy right-handed neutrinos, and the subsequent conversion into a baryon asymmetry by the anomalous sphaleron processes.


Vanilla leptogenesis


□ Strong washout when . Weak washout when

Leptogenesis is a well-posed problem which has been extensively studied ever since the discovery of neutrino masses in 1998. The dynamics can be described by a set of Boltzmann equations:

CP violation Out-of-equilibrium conditionSphalerons conserve B-L !

□ The crucial parameters are Ki and "i. The dynamics in particular is described by Ki, which enters in Di and W.



The baryon-to-photon ratio obtained from leptogenesis can be expressed in the convenient form

where is the final efficiency factor.

The result should be compared to the measured value [WMAP,10]


Vanilla leptogenesis follows from the following assumptions:

□ Hierarchical RH neutrinos

□ Negligible contribution from the heavier RH neutrinos

□ No flavor effects

□ Natural seesaw mechanism


One can do the exercise and compute the efficiency for each value of K, obtaining


Ksol Katm


The fact that the Yukawa couplings are involved in the neutrino mass matrix leads to an upper bound on the CP asymmetry [Asaka et al., 01; Davidson, Ibarra, 02]

from which one obtains a lower bound on M1 and on the reheat temperature [Davidson, Ibarra, 02; Buchmüller, Di Bari, Plümacher, 02] :


Assuming hierarchical RH neutrinos, both graphs contribute roughly equally to the CP asymmetry

Vertex correction Self-energy correction


Adding flavor to vanilla leptogenesis


However, when , the τ charged lepton Yukawa interactions are in equilibrium, i.e. .

These interactions are then fast enough to ‘measure’ if the flavor of the state produced in the decay of the heavy neutrino is τ or not; a 2-flavor basis (‘� ‘ and ‘eμ‘ in the following) is defined. The good quantum states are then [Abada, Davidson, Josse-Michaux, Losada, Riotto, 06 ; Nardi, Nir, Racker, Roulet, 06]

Adding flavor to leptogenesis

In the previous discussion, we implicitly assumed that the good quantum state for the leptons is a coherent superposition of flavor eigenstates

defined as the state in which the heavy neutrino N1 decays thanks to the Yukawa interaction


l1

NO FLAVOR EFFECTS

N1

Φ

Φ


WITH FLAVOR EFFECTS (equal branchings)

lμl�

l1 N1

Φ

Φ

τR

τR


Second type of effect: additional contribution to the individual CP asymmetries:

First type of effect: the rates of decay and inverse decay in each flavor are suppressed by the branchings

The evolution is again described by (classical) Boltzmann equations:

Same as before!

[Barbieri et al., 99; Nardi et al., 06]

[Barbieri et al. 99; Nardi et al., 06]



Pictorially, the two sources of CP violation can be seen as follows


e+μ

e+μτ

τ

1)

2)

This new source of CP violation gives rise to the possibility of having a dramatic enhancement compared to the unflavored case [SB, Di Bari, 06] :

One-flavor dominance

and

Different flavor composition of leptons and anti-leptons!


In general, the Boltzmann description is only valid in the limit where the lepton states are well-defined, i.e. either conventional flavor states or heavy lepton flavor states, .


Unflavored Boltzmann equations are valid in the limit where decays and inverse decays are fast compared to the charged lepton Yukawa interaction:

It is NOT enough that the interaction is in thermal equilibrium, ! This is the quantum Zeno effect [SB, Di Bari, Raffelt, 06]

Flavored Boltzmann equations are valid in the limit where charged lepton flavor eigenstates are the good quantum states

The intermediate case where , there is only a partial loss of coherence and the problem has to be described within a quantum kinetic treatment.


Density matrix calculation required!

[SB, Di Bari, Raffelt, 06]



Density matrix


The most appropriate formalism to describe the problem of an ensemble of mixed particles propagating including decoherence effects is the density matrix one.

Density matrix

For one particle which is a superposition of flavor eigenstates � , �, the density matrix is defined as the expectation value of the number operator:

for particles and anti-particles, respectively.

In the case of a particle ensemble with a distribution of momenta, one can easily generalize the above definition:

This time, the expectation value is on an entire statistical ensemble, such that we are really talking about occupation numbers.


As an illustration, neutrino oscillations can be very neatly described with the density matrix formalism.

Density matrix – Polarization vector

For neutrinos propagating in vacuum, we have the following Heisenberg equation generalizing the usual Schrödinger-type equation

:

where are the eigenvalues of the free Hamiltonian.

For a 2-flavor problem, it is very insightful to use the following decomposition into Pauli matrices:

which leads to the (flavor) polarization vector evolution:

Spin precession in a magnetic field!


e (z)

μ

B

P(t)

x

y

Neutrino oscillations in vacuum:

2θ

Vacuum mixing angle

Coherence is encoded in the length of the polarization vector:Length 1 = pure state / Length zero = incoherent mixture

P(0)



Neutrino oscillations in matter:

Neutrino of different flavors feel a different index of refraction within the medium. This effect can be accounted for by calculating the neutrino self-energy (within thermal field theory).

The real part of the self-energy can be thought about as an effective or thermal mass. It will drive flavor oscillations in the same way as a vacuum mass term.

The imaginary part of the self-energy describes collisions and therefore damping of the coherent evolution

ReNC

CC

Im =



Collisions occur in matter (e.g. ) and affect propagation by causing incoherence, i.e. the damping of flavor oscillations [Stodolsky,

1987]:

The degree of coherence is encoded in the length of the polarization vector.

μ

B

P(t)

x

y

e (z)

μ

B

P(t +2)

x

y

2θ

P(t +3)

After some time, completely incoherent state, i.e. prob. ½ to be e or μ

P(t+1)


Strong damping limit :


Density matrix for leptogenesisWith P. Di Bari, D. Jones and L. Marzola, 1112.XXXX


We will be using the density matrix formalism to describe the asymmetry between leptons and anti-leptons. We want to track the loss of coherence between decay

Density matrix for leptogenesis

and the subsequent inverse decay

The differences with the neutrino case are

1. We will be interested in the difference between the lepton and anti-lepton number densities. This will have an important implication for flavor oscillations.

2. There is no vacuum mass term, and therefore no mixing. The ¨magnetic field¨ will have a different orientation in flavor space.

3. The initial orientation of the lepton is not given by the weak flavor eigenstate, and the main flavors will not be e and μ but and


The equation describing the evolution of the lepton asymmetry in the flavor basis is given by

where

is the generalized CP asymmetry.

are the generalized branching ratios; at tree level

The anti-commutator structure is generic for terms changing the number of leptons.

This equation does not contain yet any new ingredient: just a change of basis!



Adding now the effect from charged lepton Yukawa couplings,

one finds [Abada et al, 06; Blanchet, 08; Beneke et al., 10]:

Oscillatory term driven by the real part of the self energy.

Double commutator representing damping of the coherent evolution of the lepton state since production. It destroys the off-diagonal elements of the density matrix.

Depends on the total lepton number because leptons and anti-leptons oscillate in different directions in flavor space.



τ (z)

eμ

eτ

Refractive index Damping rate

x

y

When the lepton state is (on average or effectively) fully projected on the z-axis, the fully-flavored Boltzmann equations can be used.

When the lepton state remains in its original direction ( ), then the unflavored Boltzmann equations can be used. (Quantum Zeno effect)

11

2

2

Using the polarization vector decomposition, we have:



Damping of flavor oscillations When considering flavor oscillations, since leptons and anti-leptons oscillate

in opposite directions, an equation for is needed in order to close the system.

Gauge interaction (flavor blind)

These gauge interactions are very fast and damp flavor oscillations [Beneke, et al., 2010]

The reason is that gauge interaction push to be diagonal in flavor space, and therefore the commutator structure describing oscillations vanishes!

We can neglect flavor oscillations and consider only the damping or loss of coherence through this interaction:


General case: many heavy flavorsWith P. Di Bari, D. Jones and L. Marzola, 1112.XXXX


General case: Many heavy flavors The seesaw mechanism requires at least 2 heavy Majorana neutrinos. This

means that, at least in principle, one should calculate the contribution from the heavier RH neutrinos to the lepton asymmetry.

We consider the limiting case where l1 and l2 are oriented in the same plane as τ.

Effective two flavor case!

This still does look pretty horrible. However, when we stay away from the transition region we can use a Boltzmann description.


General case: many heavy flavors

Case

1 Heavier neutrino N2 produces an asymmetry when T ~ M2 according to the Boltzmann equation:

which is nothing else than the usual unflavored equation in the flavor basis

The solution for the diagonal elements can be conveniently written in the form

These terms stem from the different flavor composition of leptons and anti-leptons. They escape washout at production!

Phantom terms ! [Di Bari, et al. 2010]

At this point, this rewriting is purely formal and unphysical since no interaction defines what the tau direction is.


Phantom terms Phantom terms are peculiar, in that

2. They strongly depend on the initial RH neutrino abundance.

1. They only affect the final asymmetry when there is a subsequent washout which distinguishes flavor (asymmetric washout) .

No effect on the total asymmetry in N1 leptogenesis, as theycancel exactly!

2 We now turn to the washout of the asymmetry from the lightest RH neutrino N1, which occurs in the flavor regime as M1 < 1012 GeV

However, large flavor asymmetries possible (100% efficient)! Effect on BBN?

If

Asymmetric!

Phantom term potentially dominant!


Phantom terms

To summarize this example

N2 produces the asymmetry

T

1012 GeV

109 GeV

Tau lepton Yukawa interactions

N1 washout

DecoherenceFlavors are physical

Charged lepton flavors not physical

Phantom terms emerge

Phantom terms would also appear if both N1 and N2 were in the same flavor regime.


Most general case

The master equation describing the generation of asymmetry from any configuration of RH neutrino masses is given by

Charged muon Yukawa interactions turn on when


Summary Leptogenesis, as the cosmological consequence of the seesaw

mechanism, provides an attractive explanation to the BAU.

Within leptogenesis, charged lepton Yukawa interactions (mainly τ)break the coherence of the lepton state produced in the RH neutrino decay.

On the other hand, when these rates are comparable, the density matrix for leptogenesis must be solved.

Unflavored leptogenesis with hierarchical RH neutrinos leads typically to a N1-dominated scenario (vanilla lep.) and there is a lower bound on M1 for successful leptogenesis. A strong washout with no dependence on the initial conditions is naturally obtained.

The density matrix formalism allows to describe a mixed system of particles with partial coherence.

When they dominate over inverse decay rates, the off-diagonal elements of the density matrix are damped, and fully flavored leptogenesis emerges.


Summary The density matrix can be written for more than one RH neutrino.

Typically, the contributions from the heavier neutrinos are not negligible (e.g. phantom terms!).

We are now able to calculate the asymmetry for any RH neutrino mass configuration, i.e. in any of these 10 cases:


º

N

m ~ 0.1 eV

M ~ 1014 GeV

Conventional picture




□ A sample of the processes that can play a role:

Decays and inverse decays

ΔL = 1 scatterings involving top quarks

ΔL = 1 scatterings involving gauge bosons

ΔL = 2 scatterings with intermediate N



This can be easily understood by plotting the inverse decay rate as a function of z.

Note that the out-of-equilibrium condition is NOT Γ< H, which would lead to K < 1. Leptogenesis is indeed perfectly viable in the strong washout regime, for K > 1.

The strong washout is actually favored for two main reasons: 1. No dependence on the initial conditions

2. The measured neutrino masses: K atm = 50, K sol = 9

Boltzmann suppression


WEAK WASHOUT STRONG WASHOUT


Neutrino oscillations in the Sun:

The ¨magnetic field¨ is an evolving function of the electron density ne . Electron neutrinos propagate in the Sun differently from muon or tau neutrinos because they can interact with electrons inside the Sun. The corresponding effective potential is .

e (z)

μ

B

P(t)

x

y

2θm

P(0)



Neutrino oscillations in the Sun:

Complete flavor conversion! This is the MSW effect!Mikheyev, Smirnov, Wolfenstein, 1985

e (z)

μ

B

P(t)

x

y

P(0)


The polarization vector follows the magnetic field when precession is faster than the change in B: adiabatic limit!

leptogenesis and quantum · pdf filesteve blanchet, seminar lup, montpellier, 06/12/11 3 the...

Documents