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Discussion with Howard Haber and "world expert" about paper: "The non-equivalence of Weyl andMajorana neutrinos with standard-model gauge interactions" [ps][pdf] ,  by R.Plaga


Complete collection of all emails, concerning the physics [ascii] .
Note that the eq. numbers were not corrected to fit the newest manuscript version in this file.

An extract of the discussion is given below.
 Here eq. numbers were corrected to fit the newest manuscript version. "Short write up" refers to Haber´s manuscript "Massless Majorana and Weyl fermions cannot be distinguished"
(Klick here for remote copy [ps] ).

There seem to be 2 major points of contention.

Point 1.  The mathematical form of the Lagrangian of a massless Majorana neutrino, that has the identical phenomenology as the observed neutrino. (main criticism of H.Haber)

Point 2.  The correct normalisation factor in the mathematical definition of the Majorana neutrino. (main criticism of world expert, shared by Haber)

In both cases the contention is about a factor sqrt(2).
However, this factor sqrt(2) in point 1. and point 2. is not the same, i.e. even using the normalisation factor suggested by the world-expert and H.Haber in  2., I do not get the mathematical form of the Lagrangian suggested by H.Haber for point 1.

1. The mathematical form of the Lagrangian of a massless  Majorana neutrino, that has the identical phenomenology as the observed neutrino. (main criticism of H.Haber)

Email of R.Plaga from 11.12.99
 WHAT I do imply, however, is:
 --------------------------------------------------------------
 Basic assertion of manuscript (alternative formulation):

 part 1:
 IF AND ONLY IF the Lagrangian displayed as eq.(7) were valid
 for Majorana neutrinos, massless Majorana neutrinos
 would have the identical phenomenology as Weyl
 neutrinos.

 part 2:
 eq.(7) is not SU(2) invariant, therefore incorrect

 part 3:
 I conclude that neutrinos are Weyl and not Majorana fermions.
 --------------------------------------------------------------
end of  email

Answer of H.Haber 24.7.00
I would strongly disagree with your assertion of part 1.  Consider
first a theory of a single non-interacting massive Majorana fermion.
The Lagrangian for this theory is given by eq. 28 of my short write-up
("Massless Majorana and Weyl fermions cannot be distinguished") which
I sent to you.  The factors of 1/2 must be there.  The referee echos
this assertion, and I believe there cannot be an argument here.

You seem to believe that these arguments would somehow be modified if
I consider an interacting theory with gauge fields.  Of course, the
argument is slightly trickier since I cannot simply take the massive
Majorana theory and couple it to a gauge field (since the mass term is
not gauge invariant).  However, I can use the see-saw model to
construct a theory of a massive Majorana field with gauge
interactions.  As always, all one needs to do is to replace the
derivative of the kinetic energy term with a covariant derivative.
One is then assured of a gauge invariant theory.  In my short
write-up, I exhibit the see-saw model.  It is a simple matter to
convert the derivatives of the kinetic energy term to covariant
derivatives, and write out the interaction of the MASSIVE (but light)
Majorana neutrino with electrons and W's.  The result is an
interaction with coupling constant g as it appears in eq. 22 of my
short note.  There is no sqrt{2} ambiguity here.  [Actually, to be
more precise, the coupling g would be reduced by a mixing angle factor
that differs from 1 by a factor of order m_D^2/M^2 where m_D is
the Dirac mass and M is the see-saw mass---see eqs. 23--26 of my
short note.]  If one then takes the see-saw mass M to infinity, the
right-handed neutrino decouples and the light neutrino mass goes
smoothly to zero.  I can then rewrite this theory as one involving
massless Weyl fermions as I showed explicitly in my note.
end of email

My eq.(7) and Haber´s eq.(22) (eq. (28) is the same with mass term and no
interaction term) differ by a factor 1/2 in the kinetic
term. Had I used Haber´s normalisation in the definition of the
Majorana neutrino, the expressions would differ by a factor sqrt(2)
appearing in eq(7)´s interaction term.  My equation (10) is identical
to Haber´s eq.(22) if one uses the same normalisation factor in the Majorana neutrino definition.
Thus,  correcting for the contentious normalisation factor,
Haber and I agree on the Lagrangian predicted by the SM for the Majorana
neutrino. However, Haber says this Lagrangian gives the observed phenomenology and I say it doesn´t. This is really the basic
disagreement as already realised by Haber earlier:

email of H.Haber from 14.10.99
 I finally had a chance to go through your paper carefully and
write a detailed response.  You seem to be claiming that one can
distinguish a massless Majorana neutrino and a massless Weyl neutrino
based on its charged current coupling strength.  I disagree, and I
think the problem with your analysis can be traced back to an
incorrect normalization of the Majorana fermion kinetic energy term.
However, this is a tricky business.
end of email

My Lagrangian eq.(7) is derived in section 3 of my manuscript, but
a more elementary and immediate argument for its apparent
correctness than given there is:

Email from 2.8.00 from R.Plaga
Your statement that you DISAGREE with my 
statement that eq.(7) manuscript is the
one which give phenomenological equivalence for massless Majorana and
Weyl neutrinos is in square contradiction with the point of view
of Boris Kayser (THIS point we discussed repeatedly over several days).
Please note: Eq(7) obtains from eq (4) (usual standard-model expression
for Weyl neutrinos) by DIRECTLY replacing all Weyl spinors with Majorana
spinors.
end of email

 

Indirectly H.Haber answers to this in his short write up where he
notes after eq.(28):

Haber short write up p.4
The factor 1/2 that appears in eq.(22) in front of the kinetic energy term
is correct. Just like in scalar field theory, if one uses real fields, the corresponding
coefficient of the kinetic energy term is 1/2 while for complex field it is 1.
end of quote

I disagree:

email  11.8.00 from R.Plaga
You are of course right that when going from a complex to a
real scalar field, a factor 1/2 appears in the kinetic term
(e.g. Bjorken,Drell Relativistic Quantum fields, chapter 12.5).
However it is not correct to conclude the same happens when going
from Weyl to Majorana fields. As a result of the
factor 1/2 in the kinetic term, the complex and real
scalar field CAN be distinguished on the KINEMATICAL level.
That this must be true can be seen from
the fact that the neutral scalar field has 1 degree of freedom and the
complex 2, which leads of course to observable differences.
(see Landau,Lifshitz Vol4, § 12 where the factor 1/2 is explained
by noting that real scalar fields have 1/2 as many degrees of freedom).
Contrary to the scalar case Weyl and Majorana fields have the SAME
number of degrees of freedom (namely 2) and [...]
it is clear that their kinetic terms should then be equal.
end if email
 

2. The correct normalisation factor in the mathematical definition of the Majorana neutrino. (main criticism of world expert, shared by Haber)

Email by world expert from 12.7.00
There is one main point: if you have a LH neutrino field \nu_L,
then the Majorana neutrino field you can form from this is

  \nu_M = \nu_L +  C (\bar{\nu_L})^{T}, where the \bar = Dirac conjugation.

One does not need 1/\sqrt{2} as a normalization factor in the RH side of
this equality.
 The kinetic term for a Majorana field is

  0.5 \bar{\nu_M} \gamma_{\mu} d/dx_{\mu} \nu_M,

the factor 0.5 in front being very important. The physical reason for the
factor 0.5 is the same as the reason for which the same factor appears in
the kinetic term of a REAL scalar field.
As it is not difficult to show, one has (neglecting a term which
represents a total derivative)

  0.5 \bar{\nu_M} \gamma_{\mu} d/dx_{\mu} \nu_M =
   \bar{\nu_L} \gamma_{\mu} d/dx_{\mu} \nu_L,

and there are no additional factors of \sqrt{2}, which form the basis of
the author's arguments.
 In my opinion, the author's article is just wrong because of this
trivial normalization error that he makes when he expresses

  \nu_M in terms of \nu_L and    C (\bar{\nu_L})^{T}.
end of email

Haber uses the same normalisation as the world expert in the definition
of the Majorana neutrino in his short write up, eq.(20).
I quote some references to bolster the derivation of
the normalisation given in section 2 of my manuscript

Email 2.8.00 from R.Plaga
References confirming the normalisation of the Majorana
field used by eq.(2) my manuscript.
*******************************************************************
------------------------------------------------------------------
1. B.Kayser, The Physics of Massive neutrinos,
World Scientific,Singapore,1989 p.89:
\nu´=f-M_D/M_R F
f= 1/sqrt(2) (\Psi_L+(\Psi_L)^c)

\nu´ is identified on p.90 l.4 as Majorana neutrino, in the massless
limit (here:M_R --> \inf) eq.(6) my manuscript occurs
-------------------------------------------------------------------

----------------------------------------------------------------
2. M.Fukugita,A.Suzuki (eds.)
Physics and Astrophysics of Neutrinos, Springer,1991
Article: "Physics of Neutrinos" by M.Fukugita,T.Yanagida
section 7, eq(8):
"\chi_{+-} = 1/{sqrt(2)}(\psi +- \psi^c)
where \psi^c is the charge conjugation of \psi" (my eq(2))
------------------------------------------------------------------

------------------------------------------------------------------
3. T.D.Lee, Particle Physics and Introduction to Field Theory,
Harwood,Chur,1981,p.54 (Excercise (ii) of section 3):
"Hint for (ii):Define the Majorana field operator
\Psi_M = 1/sqrt(2) (\Psi_L + \Psi_L^c) " (my eq(6))
------------------------------------------------------------------

-------------------------------------------------------------------
4. P.Roman,Theory of Elementary Particles,North Holland,1960
Chapter 4g, Majorana Theory,p.307. eq 4.37
"\chi = 1/sqrt(2) (\Psi + \Psi^c)"

as definition for Majorana field.
4.39b gives the ususal anticommutation relations for \chi.(my eq(2))
-------------------------------------------------------------------

------------------------------------------------------------------
5. W.Pauli, Relativistic Field Theories, Rev.Mod.Phys.13,203(1941)
p.225,eq.100
v = 1/sqrt(2) (u+C^*u*)
(where u is the usual Dirac spinor and the second term charge
conjugation in Pauli´s notation).(my eq(2))
------------------------------------------------------------------
end of email

The world expert gave the following references to bolster
his normalisation factor.

Email 12.7.00 from world expert
For further details see Rev. Mod. Phys. 59 (1987) 671 and the
discussion of the Majorana mass term (subsection C) on  pages 681--682.
In particular, please note footnote 12 on page 682.

Another useful article might be that of P. Mannheim in Int. J. Theor.
Phys. 23 (1984) 643. I have read it many years ago -- all the necessary
normalization factors are  rigorously derived, if I remember correctly, in
this article.
end of email

Email 2.8.00  from R.Plaga
He then cites
>P. Mannheim in Int. J. Theor.
>Phys. 23 (1984) 643 in which
as basic reference.
This reference contains the above definition of the Majorana
neutrino \nu_M AS WRITTEN BY ME *WITH* THE FACTOR 1/sqrt{2}!!!
equation (19):
\Psi_M= 1/sqrt{2} (\Psi_L^W + (\Psi_L^W)^C)
is identical to equation (6) my manuscript.
Rev. Mod. Phys. 59 (1987) 671 by Bilenky,Petcov
indeed gives the same normalisation as the referee,
but quotes Mannheim as more BASIC Majorana reference
(beginning of Appendix A of Bilenky,Petcov).
end of email

To be absolutely sure that there is no misunderstanding I asked
P.Mannheim personally, if my normalisation is wrong.

Email 15.11.00 from R.Plaga
Using standard text-book normalisations and notations
(to be absolutely definite: the ones of Bjorken-Drell vol.2)
everywhere, is the following decomposition
of a Majorana spinor field "\chi" into a
Weyl spinor field \Psi^W_L and its own charge conjugate (\psi^W_L)^C WRONG?

   \chi=[\psi^W_L+(\psi^W_L)^C]/\sqrt 2$)

(  ) yes, the decomposition is WRONG  ( ) no, this decomposition is not WRONG
end of email

Email 17.11.00 from P.Mannheim
 As regards your question, the
relation $\chi=[\psi^W_L+(\psi^W_L)^C]/\sqrt 2$ as given in my Majorana
paper as Eq. (49) is certainly correct.
end of email