Dear Biswajoy, I agree with what you write. In Majorana Model neutrino and antineutrino are the same particle but in this model they have different  from zero mass. In Dirac Model neutrino and antineutrino are two different  particles but they have zero mass. So between Majorana Model and Dirac Model there are important differences. It seems to me Standard Model accepts more Dirac Model. In the new Non-Standard Model neutrino and antineutrino are the same particle, like in Majorana Model, but both have zero mass, like in Dirac Model. Consequently the Non-Standard Model is different whether from the Dirac Model or from the Majorana Model. It isn't the consequence of a trivial adjustment  but it is due to fundamentals of the Non-Standard Model in which the "Theorem of Charge and Spin" is valid and neutrino isn't a fermion but it is a boson.

Hola Victor,

I would say that the simplest option is to have an effective operator, that uses only the fields of the "standard model" of elementary particles, as described by Weinberg. This has canonical dimension 5 and is not renormalizable, but it does the job and leads to the interesting conclusion that the neutrinos have Majorana mass.

If you want instead a renormalizable model (for whatever reason) I guess that you would like to add right handed neutrinos to the "standard model" spectrum, or you may want to enhance the gauge symmetry. There are many options unfortunately, and as Wolfenstein put it "having too many models is like having none".

In a sense, one should ask what we want from a "model". If you want to explain what you see, again I would prefer to stick to Weinberg, just as was done by Fermi for the theory of beta decay: the simplest option that describes the observed facts. If you want to derive predictions, you can use some model; but you should be aware of the arbitrariness of your choices and if possible you should justify it  somehow.

Best regards, Francesco

Victor,

Francesco has very correctly said that the simplest scenario is to have a dimension 5 operator. Let me tell you what a dimension five operator for the neutrino mass is. Neutrinos are a part of a weak doublet, in terms of the SU(3) x SU(2) x U(1) quantum numbers you can write that doublet as L=(1,2,-1/2). Then using the charge relation Q=T3L+ Y, you can see that the upper element of the doublet has Q=0, which is the neutrino. Now suppose I write a term like, LL, then that will have a mass dimension 3/2+3/2=3, but this term is not gauge invariant, as it will transform as   (1,1+3,-1). I have used the SU(2) representations 2 x 2 = 1+3. But the LL term, though super-renormalizable is disallowed by gauge invariance, simply because total hypercharge is -1.

I would like to keep terms of mass dimension less than equal to 4, hence now I can attach a scalar to LL term, such as  LLH, where H is (1,2,1/2). But alas, this term is also not gauge invariant. Next, I can attach another H to the operator, but this time I have mass dimension 5, therefore I need to divide the operator by a mass M. I don't know what is M, therefore this is new dimension full parameter. The term looks like (1/M) L L H H. This term is gauge invariant. When H acquires VEV, the mass is of the form /M. Now you can take your question to the next level, namely, what is the simplest model which can generate this dimension 5 operator? Note that this term violates lepton number by two units.   

Reference:  "Baryon and lepton non-conserving processes", S. Weinberg,  Phys. Rev. Lett. 43, 1566 (1979).

My own proposal is based on p-adic thermodynamics, which would provide a description of massivation for all particles: also neutrinos. It has nothing to do with Higgs mechanism. One can speak of thermal massivation.

Generalization of conformal invariance is essential. Virasoro scaling generator L_0  to which mass squared operator  is proportional would take the role of energy in p-adic thermodynamics and p-adic temperature would be from the existence of Boltzmann weights p^(L_0/T) inverse of an integer: T=1/n, n=1 holds true for fermions.

 p-adic primes  characterising particles are postulated to be primes near but below power of 2: p =about 2^k,   to get the mass scales correctly (there is exponential sensivity to the power of 2 since real mass square is proportional to 1/p.  

 The p-adic valued mass squared is mapped continuously to real number by what I call canonical identification: x= sum_n x_n^p^n--->sum x_np^(-n).   Thermal expansion converges extremely rapidly for the primes involved: Electron corresponds to Mersenne prime M_127=2^127-1 so that the calculations are practically exact.

 2 decades after first calculations  I finally  understand from number theoretical argument why some primes characterize elementary particles and  why primes near but below powers of prime p are favored. p=2 corresponds to p-adic length scale hypothesis.

This is however a long story . It relies on   the vision that physics is number theoretically universal - adelic -  and scattering amplitudes exist in all number fields both reals and p-adic number fields. This implies rather refined number theory: algebraic extensions of rational numbers characterising particles  possess preferred primes, which are "ramified" (see Wikipedia) :  these would be p-adic primes characterising elementary particle. Besides this Negentropy Maximization  Principle which is behind TGD based quantum measurement theory is involved: this leads to a generalisation of p-adic  length scale hypothesis.  

In the Non-Standard Model neutrino real mass is exactly zero.

There is a different answer in my latest paper. The theory is relatively unknown, but there is a precise answer. Kindly review the paper and let me know if my answer makes sense or not.

Article Nuclear Lattices, Mass and Stability

I think that in UV Wilsonian approach for Standard Model' completion, the simpler way is through RH neutrini, as a see-saw type I mechanism. This is also connected to a Leptogenesis mechanism through RH neutrini decays. However, UV Wilsonian approach is successfully right for the Fermi model of weak interaction, where 4-fermion interactions are completed with W,Z bosons in propagators, but we don't know if it is the only possible one! For example, in my recent studies, I have seen how an operator like HHLL/M can be directly generated by non-perturbative effects, a new class of instantons called "exotic instantons". These can also generate other operators like (UDD)^2/M^5 (Majorana mass for the neutron!) without the needing of mediator-fields. These effects are perfectly calculable and controllable in a lot of realistic examples. 

In the Non-Standard Model neutrino and anti-neutrino are the same particle.

@ Daniele: Actually if neutrino and anti neutrino are same particles then we say that the neutrino is a Majorana particle. It is an open question, nobody knows whether the neutrino is a Dirac particle or a Majorana particle. There are some experiments which will tell us in future whether neutrino is a Dirac particle or a Majorana particle. Some of these experiments are  (i) neutrinoless double beta decay (ii) magnetic moment of the neutrino.  

Dear Biswajoy, I agree with what you write. In Majorana Model neutrino and antineutrino are the same particle but in this model they have different  from zero mass. In Dirac Model neutrino and antineutrino are two different  particles but they have zero mass. So between Majorana Model and Dirac Model there are important differences. It seems to me Standard Model accepts more Dirac Model. In the new Non-Standard Model neutrino and antineutrino are the same particle, like in Majorana Model, but both have zero mass, like in Dirac Model. Consequently the Non-Standard Model is different whether from the Dirac Model or from the Majorana Model. It isn't the consequence of a trivial adjustment  but it is due to fundamentals of the Non-Standard Model in which the "Theorem of Charge and Spin" is valid and neutrino isn't a fermion but it is a boson.

Dear Daniele,

just a couple of remarks:

1) The spin of the neutrino is ordinarily measured in the beta decay processes, since we  can measure the spin of all other particles taking place in the decay.

2) Also, Goldhaber most famous experiment probed it. In short, it is 1/2. Thus "spin-statistics theorem" prescribes that it is a fermion.

It is also possible to test this character, to some extent. Few years ago,  some colleagues tried to speculate that neutrinos are  spin 1/2 particles, but (without much justification) have bosonic character:

http://arxiv.org/abs/hep-ph/0501066

But as I told them in Venice conference

http://arxiv.org/pdf/hep-ph/0504238.pdf

in their assumption, neutrinos should not interact with electrons, while they do. Apparently these ideas are not any more pursued.

So, to the best of my knowledge, the conclusion is that neutrinos are spin 1/2 particles and also fermions, according to conventional ideas and calculable theories at least. 

Best regards

Dear Vissani, I understand your reasoning that is fully in concordance with the Standard Model, also with regard to the Beta decay process. Like I wrote in my preceding comment the "Theorem of Charge and Spin" is valid in the Non-Standard Model  and it relates electric charge to spin and vice versa. I don't know what your colleagues speculated, without justification, but in the Non-Standard Model there is no contradiction and whether positive semi-integer spins or negative semi-integer spins are predicted for fermions. It is in concordance just with the electric charge that can be positive and negative. In this context  neutrinos and all bosons have zero spin. I understand it can seem strange but the Non-Standard Model implies just a critical review of some fundamental ideas. It is possible to find those ideas for more details in my contributions.

In the Non-Standard Model the physical speed of neutrino is exactly the speed of light.

Dear all contributors: thank you so much for all your comments. Btw, today I just found a rather old paper written by Raby and Slansky, discussing how to add neutrino mass in the Standard Model, one way for doing that is including Dirac mass term. Of course, this is one alternative, the other alternatives are to go beyond Standard Model. What is your comment?

Dear Victor, the Dirac neutrino mass term is surely a possibility, however, I would like to raise the issue whether this is part of the "standard model" or not.

Indeed, the Dirac mass term requires to introduce the right handed neutrino, that has no gauge interaction: it is neutral, colourless and without SU(2)L charge.

In the formulation of Glashow, Weinberg, Salam, the right handed neutrino is absent. So I would dare to say that Dirac neutrino is not a part of the standard model, but rather one possible extension of the standard model. Immediately one asks: is it a reasonable extension?

Now, as soon as you introduce the right handed neutrinos, you can introduce a Majorana mass term for this field. In fact, this is an gauge invariant and renormalizable operator, namely a legitimate one. If this Majorana mass is large, when you integrate the field away, you are back with dim.5 operators we already mentioned. (But in this case, you derive it from a renormalizable model). This is called in jargon seesaw model -- of type I, to be precise. Other cases are possible that boil down into the dim.5 operator (that, to be sure,  contains only left neutrinos, thus it is minimal in this sense).

These considerations do not mean that it is impossible to have plain Dirac neutrino mass. The point is just that, in order to have it, one should introduce right handed neutrinos as discussed, but also, one needs postulating a global symmetry to forbid the above mentioned term: Majorana mass term for the right handed neutrinos.

To go beyond Standard Model, from my viewpoint, means to go toward Non-Standard Model. Giving a different from zero mass to neutrino, in concordance with Majorana's intuition, doesn't  solve the problem and indeed it introduces other problems like for instance the question of the physical speed of neutrino. Therefore on the basis of these and other considerations I have to confirm physical properties of neutrino in Non-Standard Model:

1. Neutrino mass is zero

2. Neutrino and antineutrino are the same particle, in concordance with Majorana

3.The physical speed of neutrino is the speed of light as it has been confirmed by CERN-LNGS experiment (2011)

4. Neutrino is a boson and not a fermion

5. Neutrino spin is zero and not 1/2

6. Neutrino respects the Planck relation.

I agree with F. Vissani on Type-I seesaw mechanism being a simple mechanism for neutrino mass.

The question isn't which is the most simple mechanism for giving mass to neutrino but if neutrino has a real mass.

Let me add a few words on what is a type I see-saw mechanism, which will display its simplicity. As Francesco has very rightly said, that if you want to add a right handed neutrino, then by default you are also adding a large Majorana mass to it, unless you forbid it by introducing as global symmetry such as the lepton number. Then, the neutrino mass matrix is a 2x 2 matrix of the form in the νL   νR basis,

0      mD

mD    M

The zero in the left hand corner is a νL νL term, which is forbidden because left handed  neutrinos are a part of a SU(2) doublet. To allow this term, one needs to have a triplet Higgs, which contributes to ρ parameter, when it gets a VEV. So let us not include it: However we see that ,  M νRνR mass term is gauge invariant as right handed neutrinos are singlets.  The off-diagonal terms are natural Dirac masses, that is they are of the order of 100 GeV or so. Therefore you see that I have not added small Yukawa couplings to make Dirac masses small artificially. Then we diagonalize this mass matrix to go from flavor basis to mass basis.  The two eigenvalues are approximately,

M and mD2/M.

If mD is 100 GeV and M is 1013 GeV, then, one eigenvalue is 1 eV. In the reverse way you can say that if neutrino mass is of the order of 1 eV then M should be of the order of 1013 GeV.

Reference:

M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, ed. by D. Freedman and P. Van Nieuwenhuizen, North Holland, Amsterdam (1979), pp. 315-321.

T. Yanagida (1980). "Horizontal Symmetry and Masses of Neutrinos". Progress of Theoretical Physics 64 (3): 1103–1105.

Anyway if you think giving mass to neutrino you solve all your problems, yes let you choose the most simple mechanism.

To maybe add a new point to the discussion, there are also models which generate the neutrino mass only radiatively, i.e., as quantum corrections. In these settings, the tree level neutrino mass is identically vanishing, while loop corrections involving new particles generate a non-zero neutrino mass. These models tend to be relatively simple in what concerns the particle content, however, in practice they may be difficult because the loop-contributions to the neutrino mass may be hard to calculate.

If you are after mere simplicity only, the simplest approach mentioned here was indeed the Weinberg operator. However, if you want to keep renormalisability, what has not been mentioned and what is even simpler than a type I seesaw setting is generating a neutrino mass with a Higgs triplet obtaining a vacuum expectation value. This is often referred to as "seesaw type II" and it generates a tree-level neutrino mass by adding only *one* field to the SM, which is less than the at least two necessary for seesaw type I.

In general, you will always need spontaneous symmetry breaking to generate active neutrino masses. While right-handed neutrinos are total singlets and may have a direct mass term, this is impossible for active neutrinos since they do share SM quantum numbers.

The most disconcerting aspect of modern physics is the existence of one single type of mass for which for instance mass of lead ball and electron mass have the same physical nature. In actuality Einstein demonstrated the mass variation with the speed only for slowly accelerated electron, but then he applied that property to all masses.  Question is now: does exist in the universe really one single type of mass? In General Relativity Einstein established an equivalence for which inertial mass and gravitational mass are the same thing. The true problem nevertheless regards massive elementary particles: is particle's mass and mass of ordinary bodies the same thing? In the Theory of Reference Frames massive elementary particles have a different type of mass: electrodynamic mass that changes with the speed. Does neutrino could have an electrodynamic mass? Also electrodynamic mass like ordinary mass involves two effects:

1. neutrino would have to have variable speed because massive particles with constant speed don't exist in nature.

2. neutrino's electrodynamic mass would have to change with the speed.

The most simple mechanism is that neutrino has null real mass because " all what is real is also rational".

Anyway if you think giving mass to neutrino you solve all your problems, yes let you choose the most simple mechanism.