# I Majorana neutrinos, sphaleron reactions, baryon asymmetry

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1. Apr 12, 2018

### Kajagoogooooooo

Ive spent some time reading about the baryon asymmetry and the Sakharovs conditions, and there some things I didn't exactly get:
1. Interactions out of thermal equilibrium: isn't it trivial? our universe is expanding so, of course, it out of equilibrium.
2. CP violation: Ive read that Cronin and Fitch experiment proves that issue, the problem is that the amount of the CP violation there is too small to explain the baryon asymmetry in the universe.
3. Baryon number B violation.

My question is how can you solve problem 2. 3. using the lepton sector. Or more precisely by using the Majorana neutrinos in the lepton sector, making a lepton asymmetry.
And how do you convert it to a baryon asymmetry using the sphaleron reactions? (An explanation about those reactions would be nice since I'm not that familiar with them).

Thanks!

2. Apr 12, 2018

### Orodruin

Staff Emeritus
No, it is not trivial. Just like you need CP violation to be large enough, you need a system that is sufficiently far away from thermal equilibrium. For this to occur you essentially need the Hubble rate to be larger than the interaction rate of whatever process would keep your system in thermal equilibrium or, in other words, the timescale of interactions needs to be larger than the age of the universe.

More precisely, the known CP violation in the quark sector is to small for electroweak baryogenesis to produce the observed baryon asymmetry.
Baryon number is violated in the SM. Let us come back to this.

Let us go by steps. First, the creation of a lepton asymmetry is subject to the same Sakharov conditions as baryon number generation, but of course with L rather than B violation.

A popular way of introducing small neutrino masses into the SM is to extend it with right-handed (RH) neutrinos. This allows Yukawa couplings between the left-handed lepton doiblet, RH neutrinos, and the Higgs field that when the Higgs field takes a vev results in a Dirac mass term for neutrinos. However, since they are SM singlets, the RH neutrinos also allow for a Majorana mass term. A priori, the mass scale of this mass term is unknown as it is unrelated to any known scale. If it is chosen to be very large, it suppresses the masses of the LH neutrinos, which essentially go as $m^2/M$ where m is the Dirac mass and M the RH Majorana mass, giving a possible explanation of why SM neutrinos are so light. It is also quite natural to consider large M as a result of an extended symmetry broken at some high scale.

So far we have only wanted to explain neutrino masses, but let us consider the phenomenological implications in the early Universe. At the very early stage, the RH neutrinos would be held in thermal equilibrium with the SM. However, at some point the number of RH neutrino freezes out and are therefore out of equilibrium. The RH neutrinos decay through the very same Yukawa couplings as mentioned earlier. With CP violation in these Yukawa couplings, the branching ratio to final states containing a lepton will not be the same as that to final states containing an anti-lepton, thus providing violation of both CP and L and therefore all Sakharov conditions for producing a lepton asymmetry are satisfied.

They are typically referred to as sphaleron processes, not reactions, as they are a non-perturbative QFT effect. It is not easy to provide an I-level explanation (it is not easy to provide an A-level explanation either), but let us wave our hands around for a bit.

In the classical SM, both B and L are accidental symmetries and therefore conserved. However, on the quantum level, the B and L currents are anomalous, implying that their divergence is non-zero, leading to a B and L source term. Thus B and L are not conserved, but their non-conservation is proportional to an integer set by the different between different vacuum configurations of the electroweak gauge fields. The integer is proportional to the number of generations, ie, 3 in the SM, meaning that between different electroweak vacua, the baryon and lepton numbers changes by three units. However, the B-L current is non-anomalous and B-L is therefore conserved while B+L is not. At the present, these transitions are suppressed by an energy barrier, but in the early Universe they were very active as the temperature was higher. This means that part of whatever B or L symmetry will be transferred to the other sector as long as the sphaleron processes are in equilibrium. You can find out what this ratio is by considering the chemical potentials of all species in the early Universe. In other words, if you produce an asymmetry in L while sphaleron processes are active, part of that asymmetry will be converted into a baryon asymmetry.

For the baryon asymmetry, the Sakharov conditions of out of thermal equilibrium and C and CP violation are satisfied by the (extended) lepton sector, while baryon number violation occurs through the SM sphaleron processes.

Standard introductory references on leptogenesis are Leptogenesis for pedestrians or Baryogenesis via leptogenesis. You can probably find more information in those introductions.

Of course, there is quite a bit more going on than what I had time to discuss here and you can discuss many many more effects and possibilities to lower the scale at which leptogenesis occurs, but the above covers the main ideas.

Last edited: Apr 12, 2018
3. Apr 15, 2018 at 5:33 AM

### Kajagoogooooooo

Can you try to explain this so B.S. degree graduate would understand?
Ive tried reading about this on the net, but found a hard time understanding what is it: Dirac mass, SM singlets and what it has to do with vev results?
On what articles or models do you base those statements?
any references?

Last edited: Apr 15, 2018 at 6:08 AM
4. Apr 15, 2018 at 6:17 AM

### Orodruin

Staff Emeritus
Generally, massive fermions can be Dirac or Majorana particles. However, they can only be Majorana particles if they are not charged under any of the interactions of your model. This is what it means to be a SM singlet, it has no charge and can therefore have a Majorana mass, so if you introduce right-handed neutrinos, they can have a Majorana mass.

In addition, when you introduce right-handed neutrinos, you can add a Dirac mass term, which is just the same way as you give quarks and charged leptons Dirac masses. A property of Dirac masses is that they are mass terms that involve both left- and right-handed particles. Overall, this means that you have a Majorana mass matrix of the form
$$\begin{pmatrix} 0 & m \\ m & M \end{pmatrix},$$
where $m$ is the Dirac mass between the left- and right-handed neutrinos and $M$ is the Majorana mass of the right-handed neutrinos. Note that the upper left element is zero because left-handed neutrinos are charged under the weak interactions of the SM and therefore cannot have a Majorana mass terms (this is a truth with modification, but let us assume it for now). The absolute values of the eigenvalues of this mass matrix are the masses of the particles. In the case that $M = 0$, you would recover a single Dirac fermion with mass $m$ (a Dirac fermion has twice the number of degrees of freedom as compared to a Majorana fermion). On the other hand, if you have $m \ll M$ then the absolute values of the eigenvalues of the mass matrix are $m_1 \simeq m^2/M$ and $m_2 \simeq M$, i.e., one very heavy Majorana fermion (essentially the right-handed neutrino) and one very light Majorana fermion (essentially the left-handed neutrino).

The vacuum expectation value (vev) of the Higgs field enters into the picture through the Dirac mass term. A bare Dirac mass term is not allowed on the Lagrangian level as it would break electroweak gauge symmetry. However, an interaction term involving the Higgs field, the LH neutrino, and the RH neutrino is allowed. Once the Higgs field takes a vev, a remnant of this interaction is a Dirac mass term. (This is the source for all Dirac fermion mass terms in the SM, i.e., those of quarks and charged leptons.)

I am not sure it can be put on a more basic level than that without losing a lot of the content.

5. Apr 15, 2018 at 6:27 AM

### Orodruin

Staff Emeritus
This should be discussed in the introduction articles mentioned in #2.

6. Apr 15, 2018 at 9:26 AM

### Kajagoogooooooo

and a left-handed lepton doublet as you have said, are particles that do interact in my model, and as a result, I cannot define them by having a mass (Majorana neither Dirac)?

7. Apr 15, 2018 at 9:49 AM

### Orodruin

Staff Emeritus
Not before electroweak symmetry breaking, no.

8. Apr 15, 2018 at 10:00 AM

### Kajagoogooooooo

Thanks a lot!
Just one more thing,
I have read that there are mechanisms that generate the asymmetry directly from the baryons like Affleck-Dine baryogenesis.
http://iopscience.iop.org/article/10.1088/1367-2630/14/12/125013
This article was the best I found from a quick search but there are too many concepts that are out of my hands.
Can you please explain this mechanism a bit? or explain a good mechanism you know that goes directly from baryogenesis?
A good article will be good too (even though an explanation will be better of course)

Again thanks a lot!

BTW:

Is it important to ask for C and CP violation?
CP violation alone is not enough?

Last edited: Apr 15, 2018 at 1:37 PM
9. Apr 16, 2018 at 3:14 AM

### Orodruin

Staff Emeritus
What do you want to know that is not covered by the article?

Yes. Both are important.

10. Apr 16, 2018 at 4:26 AM

### Kajagoogooooooo

What I understood about the AD from the article it that is generated from the Super Symmetry Theory,
which means that we take every boson`s superpartner (bosonino) or the superpartner of the fermion (sfermion)
and by inflation fields from CP violation (don't know what it is) we cause the superpartners to become their original partners.
This process leads to a baryon asymmetry.

11. Apr 16, 2018 at 5:29 AM

### Orodruin

Staff Emeritus
Unfortunately, this is a rather garbled up version. I would suggest learning more about cosmology and particle physics in order to understand it better.

How Affleck-Dine would actually occur is roughly along the lines:
• In the early Universe, after inflation, the superpartners interact with the inflaton field. The inflaton field is a hypothetical field that would drive an inflationary phase of the early Universe. After inflation ends, the Universe would be essentially empty with only the inflaton field left.
• The energy stored in the inflaton field would reheat the Universe through its interactions with other fields. This is what eventually leads to the hot Big Bang of standard cosmology.
• The interactions of the inflaton field with the supersymmetric partners of the SM baryon number carrying particles could violate C and CP as well as baryon number, thereby possibly satisfying the corresponding Sakharov condition and creating a net baryon number.
• The decays of those particles to standard model particles would then lead to a non-zero baryon number in the standard model sector.
Note that these are all rather complex issues and in order to understand them properly you need several years of studying cosmology and particle physics.