Hi Biswajoy,

I send you two proofs:

First a mathematical one. The 2x2 Dirac mass matrix has 4 fields, namely n1L n2L n1R n2R. The Lagrangian density has 4 mass terms i.e. bar-n1R n1L, bar-n2R n1L, bar-n1R n2L, bar-n2R n2L, each one multiplied by a complex constant, plus hermitian conjugate terms. You can redefine away all 4 phases, then. In the case of Majorana instead you have 2 fields only, namely n1L n2L and 3 terms in the Lagrangian density, n1L C-1 n1L, n1L C-1n2L, n2L C-1 n2L each one multiplied by a complex constant, plus hermitian conjugate terms (C=charge conjugation matrix). No way of getting rid of all 3 phases, 1 of them is physical.

Then a more physical proof. Suppose that the 2x2 Majorana mass matrix concerns electron and muon neutrinos. The mass term

mee neL C-1 neL 

leads to neutrinoless double beta decay. It can be thought of as the result of exchanging two light massive neutrinos 

mee = U2ei mi, with i=1,2 

where mi are by definition real; however the two amplitudes sum with a phase, that can lead to partial or total cancellation. Thus there is a physical phase, that plays no role in oscillations but does play a role in neutrinoless double beta decay rate.

Hi Francesco. Thanks for the beautiful answer. In the 3x3 case I have the following argument: please see if you agree. A 3x3 complex matrix has 18 parameter. Nine of them are gone by the unitarity condition U^dagger U=1. Now I am left with 9. Three of them are removed by rotating charged lepton fields. Now I am left with six. Out of six there are three angles. Now I have three phases. Now out of this three phases why do I choose one for Dirac and two for Majorana. Why not all three Majorana  for example?

Hi Biswajoy

The name "Dirac" or "Majorana" is not immediately transparent, so maybe it is better to speak of phases that matter in this or that observable quantity.

The usual convention is as follows: The phase that matters for oscillations of light neutrinos is called Dirac phase (since it is present also for Dirac neutrino masses), whereas the so called Majorana phases matter only for neutrinoless double beta decay and potentially for analogous processes where muon is captured and e+ is emitted  (but its importance for what I know is just academic).

These phases obey some selection (or ordering) rules. The usual oscillations (i.e. oscillations of ultra-relativistic neutrinos) do not break total lepton number. Instead, neutrinoless double beta (and the other processes alluded above) do, and the special feature of neutrinoless double beta decay is to break electronic lepton number by 2 units. 

Now, concerning measurements. It is surely possible to measure the Dirac phase (actually it is already hinted by the data) whereas it seems hopeless to reconstruct the Majorana phases by measurements: This is at least what we find in the recent couple of papers with Dell'Oro and Marcocci.

But the phases which are absorbed in the redefinition of charged leptons should reappear in a right handed mixing matrix of a left-right symmetric theory. Notice that when we absorb phases in left handed charged lepton (or quark in the CKM case) fields, we actually push it to the right handed fields which are connected by the mass term and then say that they will never be observed because there is no right handed mixing in standard model. 

Dear Biswajoy, since you talk about Majorana neutrino, and its existence has been questioned, I think that the first think to consider is an experiment proving that such a particle exists. With this respect, it is useful to know that I proved Majorana neutrino existence like the supersymmetric partner of Z^0. (See in http://arxiv.org/abs/1205.2894 (part II)) ..

Then by using reactions therein considered you could answer to your question with respect to a CKM matrix type ....

Dear Biswajoy

we do not ADD Majorana phases, there ARE THERE in general theories of neutrino mass. The discovery ofMajorana phases and full analysis and parametrization of lepton mixing matrix was first given in 

http://journals.aps.org/prd/abstract/10.1103/PhysRevD.22.2227

see also

http://journals.aps.org/prd/abstract/10.1103/PhysRevD.84.073011

http://journals.aps.org/prd/abstract/10.1103/PhysRevD.23.1666

Cheers, JOSE

the point is that you can NOT eliminate these phases in the general case of Majorana neutrinos, this is explained in the rererences I gave CHEERS

Dear Prof. Valle, Thank you very much for the references. Yes, I can see that in a general 3x3 complex mixing matrix, which is sandwiched between charged leptons and Majorana neutrinos, we will end up with three phases which cannot be removed by possible phase redefinitions.  Out of which one is a Dirac phase and two are Majorana phase differences. The Majorana phase differences are written as  α2-α1 and α3-α1.

Also, as Francesco mentioned that in the two generation mixing matrix case Majorana phase should remain, I have done the following counting.  In the 2x2 complex matrix we have 8 parameters. Out of which four goes by the condition U^dagger U=1, and two are absorbed by two charged leptons. That means we are left with (8-4-2) two parameters. One is the angle  θ of 2x2 rotation and another is a Majorana phase which remains physical.

Another point is that Majorana mass is a lepton number violating term. Will the Majorana phase appear in oscillation probability of neutrino to anti-neutrino ? For example this possibility is discussed by Xing, in  arXiv:1301.7654

Dear Biswajoy,

- I agree with your counting above

- I agree that if you have right handed currents you cannot get rid of the additional phases (but to the best of my knowledge we do not have evidence for right handed currents)

- The theoretical result of ZZ Xing that you mention seems reasonable to me (but I am not sure it has a practical application because this kind of process is suppressed m/E.)

On another matter: here a simple manner to do the counting in general.

Everybody knows the theorem that any complex matrix M can be written as M=U diag(m) V with U and V unitary and diag(m) diagonal non-negative with entries mi. [This theorem is used routinely in setting the Yukawa couplings in standard form, one can also calculate these matrices by the Mathematica package, using the command SingularValueDecomposition]

In the case of Majorana, M is symmetric. With few manipulations the theorem implies,

M = U diag(m eia) Ut

So there are n phases, that can be attached to the masses (if some masses are zero, this decreases). However, it is quite evident that one of the phases can be plugged into a redefinition of U, then it goes away in the usual manner. The phases ai are the so called Majorana phases, their number is just n-1. The matrix U is the conventional mixing matrix with the conventional number of phases (n-1)(n-2)/2 (just as for quarks).

If you like it better, you can also rewrite

M = V diag(m) Vt  where V=U diag( eia/2 )

the physics is unchanged. E.g. in neutrinoless double etc due to light neutrinos, you have two times the vertex (thus two times the mixing) and one time the mass; the result for mee is the same. Instead, in ordinary oscillations you have combinations such as

Hvacuum=M M+ /2E = V diag(m2) V+/2 E=U diag(m2) U+/2 E

where Majorana phases play no role. 

Dear Biswajoy,

the neutrino to anti neutrino oscil is helicity suppressed, was first described in http://journals.aps.org/prd/abstract/10.1103/PhysRevD.23.1666 Regarding the counting and parametrization was exhaustively desrbed in http://journals.aps.org/prd/abstract/10.1103/PhysRevD.22.2227 and the symmetric presentation given there is BETTER than PDG for LNV processes. The PDG form is more convenient for standard oscillations only  CHEERS

Dear Prof. Valle: I have included your references in the statement of the question above. Because now we are sure that neutrinos are massive, helicity will not be exactly conserved, even though chirality will be exactly conserved.                                              

Therefore a nu_e --> anti nu_nu oscillation probability including the effects of the Majorana phase and the effects of non-zero θ13, may perhaps be comparable to 13  oscillation probability for some region of parameter space. Therefore such an analysis is perhaps worth doing.

No, Biswajoy

neutrino to anti-neutrino is helicity suppressed as shown in our 1981 paper above

CHEERS

The reasons how many phases you can cancel is that You can absorb it to the Fermions fields. 

@ Sebastian

Correct. As UPMNS matrix is sandwitched between left handed currents, these phases are first absorbed in left handed fields. But then because the mass term connects left to right, phases are transferred to right handed fermions. They never reappear because there is no right handed mixing in the standard model.

Yes. 

In Dirac fields We have charge !!! and We don't have C - symmetry 

So from lorentz symmetry You have SU(2) X SU(2) where you can construct Wayl Fermion Fields (2 dim - with mass zero). First repr SU(2) cannot be the same as second  - LEFT and RIGHT fields. But if You have field that charge is zero You can show that You can derive right handed fields from left handed. In this place is difference and this is reason why We have diffrent phase in field transformations. 

SOME TIME AGO Dr. Brahmachari asked a question on the arrow of time: Does the CP violation in K-meson decays determine the arrow of time ?

I wanted to answer it then, but I can not find it, and am answering it now.  To determine the direction of time in decays of particles you need a measurable quantinty. This is a different topic but Dr Brahmachari will see it.

1.One such quantity  is the difference

               Γ{K(long) --> π(- ) μ(+) ν } -  Γ{ K(long) --> π(+ )μ(-) ν},

implying at the end of the process an excess of μ(+)  over μ(-) in the universe. There are other processes as well.

2. When the universe is in equilbrium the inverse processes ( recombinations) also take place and wash out the difference.

3. Thus we need a deviation from equilibrium. This is provided by the expansion of the Universe and therefore an intitial condition in the Universe.

Conclusion: CP violation in particle deceays is not enouph to determine the direction  of the future and we need all there conditions of Sacharov to create an asymmetry and an arrow of time.

Dear Prof. Paschos, Here is the link of the previous question I asked on the arrow of time.

https://www.researchgate.net/post/We_know_that_CP_violation_leads_to_T_violation_as_CPT_is_conserved_Does_it_also_imply_the_existence_of_an_arrow_of_time_in_particle_interactions

I have always believed that the types of neutrinos are determined by the spatial dimension, and that the positive and negative directions of neutrinos are determined by the upper and lower directions，so I couldn't give you what you want,just state my viewpoint.