Discussion about manuscript:
"The nonequivalence of Weyl and Majorana neutrinos
with standardmodel gauge interactions" by R.Plaga
among the author, Howard Haber, Phillip Mannheim and
an anonymous "world expert on neutrinos" (quote by H.Haber)
as "referee 2".
The material about the physics content is complete, editorial, personal
and preliminary (mainly some preliminary discussion in June 99 before
H.Haber had a chence to go through the
aper carefully) emails and parts of emails are no included.
Remarks in [] were added by me, if they contain a number it
refers to an equation number in the newest version of the
manuscript. Earlier manuscriptversions are no different in the physics
content.
14.10.99 from H.Haber
Dear Dr. Plaga:
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. For me, the only way to make
sure you have made no errors when dealing with Majorana fermions is to
go back to basics. Namely, consider the theory written in terms of
twocomponent fermions. A few years ago, I gave a series of lectures
in Trieste (unfortunately, these were not written up), in which I
derived the Standard Model of electroweak interactions using
twocomponent notation for the fermions. Starting from the
interactions of electrons and neutrinos in twocomponent notation, I
can derive the standard fourcomponent interaction of the electron and
massless Weyl neutrino. However, starting from the same twocomponent
Lagrangian, I can derive the interaction of the electron and the
fourcomponent Majorana neutrino. The resulting fourcomponent
Lagrangians represent precisely the same physics. There is no
difference between the theory of the Weyl neutrino and Majorana
neutrino. Moreover, I can give the Majorana neutrino mass m and see
that the m>0 limit is smooth.
I have posted a short description of this analysis on my web
site (http://scipp.ucsc.edu/~haber/majnu.ps) which you are free to
examine. The bottom line: if the massless neutrino is part of an
SU(2) doublet, then its coupling to the electron is fixed by gauge
invariance. This neutrino can be either viewed as a Majorana or Weyl
neutrino. The physics of both the Weyl and Majorana Lagrangians are
equivalent.
Sincerely,
ca. 19.10.99 from R.Plaga to H.Haber
Dear Professor Haber,
Thanks a lot for your very instructive analysis!
It is a very good idea to start with the SM
Lagrangian for 2 comp. spinors.
I have one question regarding "Massless Majorana and
Dirac neutrinos cannot be distinguished":
You convinced me that
(22) is the correct SM Lagrangian for
the Majorana field (20).
The Dirac Lagrangian (18) is based
on the same theory.
Is e.g. the CC neutrino production
cross section for \nu_M, calculated with
standard techniques from (22),
of the same magnitude as the one for the neutrino production
of \nu from (18)?
This is not obvious to me.
In spite of the fact that (18) and (22) are the
same Lagrangian, \nu and \nu_m are mathematically
different objects, so the same Lagrangian might in principle
yield different cross sections for them.
In fact I discussed this point with Boris
Kayser and he held the opinion (which seemed
reasonable to me) that one gets an
identical phenomenology if the Lagrangians
for \nu_M and \nu have *exactly* the same form
(i.e. the cross section would be equal
if there were no factor 1/2 in the kinetic term
of (22)).
If this WERE true then I think that (22) would
lead to \nu_M production cross section 4x higher then
the one calculated for \nu from (18)?
Thanks a lot again,
Rainer Plaga
22.10.99 from R.Plaga to H.Haber
Dear Professor Haber,
I attach a revised version of my manuscript
"Majorana neutrinos and standardmodel gauge interactions"
submitted for publication in EPJC.
I tried to make the point made in your email from Oct 14,1999
>I think the problem with your analysis can be traced back to an
>incorrect normalization of the Majorana fermion kinetic energy term
clearer than before.
For this I directly used some of the
results of the report on the subject you made
available to me. I discuss why your report
is not in contradiction with the revised version and
the detailed changes below.
My plea is to allow publication even you harbour some doubts about
the manuscript's correctness. It treats an elementary point of
considerable interest that is much
trickier than nearly all colleagues assume.
Not publishing it will prevent a much needed discussion.
I would be completely happy if it were published with some
editorial warning caveat of the sort:
"Published with doubt to stimulate discussion".
Before detailing the changes let me give the manuscript's bottom line
and why this bottom line is not contradicted by your report.
Your bottom line was:
>if the massless neutrino is part of an
>SU(2) doublet, then its coupling to the electron is fixed by gauge
>invariance.
I wholeheartedly agree. Following your derivation I give the Lagrangian
valid for massless Majorana neutrinos according to
the SM as eq (10) revised manuscript.
This eq is identical to (22) your report except for a
different overall normalisation constant of the Majorana
field (see below, the question of this constant has no
effect on my qualitative conclusions).
>This neutrino can be either viewed as a Majorana or Weyl neutrino.
>The physics of both the Weyl and Majorana Lagrangians are equivalent.
I do not immediately see how these facts follow from the fact that the
interaction of both Weyl and Majorana neutrino are fully determined
by the SM Lagrangian.
The mere existence of this Lagrangian does not seem to automatically
imply an identical phenomenology for massless Weyl vs Majorana
neutrinos.
Mathematically different objects which interaction is
fully specified by given Lagrangian
can in principle have a different phenomenology.
As an analogy to make my logic clear:
The interaction of electrons and positronium
is also fully determined by the SM Lagrangian.
We can't conclude that electrons and positronium have
the same phenomenology.
So the question if the SM Lagrangian gives the same
phenomenology for massless Weyl and Majorana neutrinos
needs careful discussion.
This discussion seems to be missing in your report.
The discussion of this point of section 1 revised manuscript identifies
eq(7) as the Majorana Lagrangian that gives identical
phenomenology and this one is not the SM one eq(10).
Sincerely
Rainer Plaga

Changes in the revised mansucript:
1. Prime's that were missing in the previous version
due to a TeX problem are reintroduced.
2. The derivation of the Lagrangian predicted by the SM
eq (10) for massless Majorana neutrinos
in sects 1,2 has been changed to the
one given in your report, because it is much clearer.
Consequently eq (10) is identical to eq (22) your report
except for the effect of a different neutrinofield normalisation:
Eq (22) your report is essentially eq (2) my manuscript,
except that I use a normalisation factor 1/sqrt(2)
(following the majority of authors) that you omit.
I explain in more detail than before on top p.2 why it should better
be included (standard 2ndquantised field normalisation) and at the
end of sect 2 why the issue is not of critical relevance
for the paper's basic conclusion.
Eq (9) revised manuscript is eq (21) your manuscript
modulo the normalisation factor.
3. A number of small corrections were applied in
the abstract and section 1,2.
8.12.99 from H.Haber
Dear Dr. Plaga:
My apologies for the long delay in responding. I have had a
chance to review your response, and I am still puzzled by a number of
points.
1. You display two different Lagrangians in eq. (7) and (10). In my
mind, if I assume that the neutrino and electron are related by an
SU(2) gauge symmetry, then I must conclude that eq. (10) is correct
and eq. (7) is not. I don't see how both of these equations can be
consistent with SU(2) gauge invariance. Recall that the SU(2) here
relates the lefthanded components of the neutrino and electron. In
the notation of my note, this is a relation between $\psi_{L_1}$ and
$\psi_{L_2}$. The normalization of these fields is fixed by the
requirement that the kinetic energy term in the twocomponent
Lagrangian is canonically normalized. In any case, I see no way of
changing the normalization of $\psi_{L_1}$ and $\psi_{L_2}$
differently without messing up the SU(2) invariance.
2. You seem to imply that if eq. (10) of your paper is the correct
theory of a Majorana field, then that field will not satisfy canonical
commutation relations. I am puzzled. In my twocomponent
formulation, all the twocomponent fields satisfy canonical
commutation relations. So, I do not see any way around the argument
that once you normalize the fields so that the coefficient of the
kinetic energy term is canonical, you are done.
In my mind, the first point above is essential. If you believe that
your eq. (7) is an SU(2)\times U(1) gauge invariant theory, please
exhibit precisely what the partner of the left handed electron field
is, and prove to me that your kinetic energy term is gauge invariant.
I would assert that with canonically normalized kinetic energy terms,
your eq. (7) is not an SU(2)\times U(1) invariant theory.
It should be easy to verify whether this is a correct statement or
not. I don't see where the subtlety lies. Unless you can enlighten
me further here, I cannot recommend publication of this paper in
European Physical Journal C.
Sincerely yours,
Howard Haber
editor, European Physical Journal C
11.12.99 from R.Plaga to H.Haber
Dear Professor Haber,
Thank you very much for your email from Dec 8,1999 concerning
my manuscript "Majorana neutrinos and standardmodel..." submitte to EPJC.
>1. You display two different Lagrangians in eq. (7) and (10). In my
>mind, if I assume that the neutrino and electron are related by an
>SU(2) gauge symmetry, then I must conclude that eq. (10) is correct
>and eq. (7) is not.
I COMPLETELY agree!
I state in section 2, l.34 that the Lagrangian L^A_Maj
(i.e. eq.(7)) is mathematically distinct from the SM one.
This difference is the value of the coupling constant the W,
and indeed this means that
the Langrangian displayed as eq.(7) is NOT SU(2) invariant.
>If you believe that
>your eq. (7) is an SU(2)\times U(1) gauge invariant theory,...
As I do NOT believe this, I cannot supply the proof
you asked for.
The impression that I claim eq(7) is SU(2) invariant probably
arose in section 1 where I go from eq(4) (which is of course SU(2)
invariant) to eq(7).
It is the "Pauli I" transformation U_1  eq(5)  which does
NOT respect SU(2) invariance. I imply that in the bold face
statement (first lines of sect.2) which asserts that
U_1 is not only a representation change of the theory. It would
be better to state the SU(2) noninvariance after a U_1 transformation
explicitely directly after eq(5), where the properties of U_1 are
discussed.
>2. You seem to imply that if eq. (10) of your paper is the correct
>theory of a Majorana field, then that field will not satisfy canonical
>commutation relations.
NO, I did not want to imply this. Eq.(10) is the correct theory
for a neutrino Majorana field with correct canonical commutation
relations. What I wanted to say is only that the INTRODUCTION
(i.e. definition) of the Majorana field in eq.(20) of your report
leads to a Majorana field that does not satisfy the canonical
commutation relations, and that
it is therefore preferable to define it with an additional factor
1/sqrt(2)
as in my eq.(2) which is used by many authors.
But as you already state this is really a side issue 
the important point your first one.
+++++++++++++++++++++++++++++++++++++++++++
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.

The demonstration of part 1 of this assertion  supplied in section 1
of the mansucript  is actually nothing more than
a review a paper by Pauli (Ref. 4).
Boris Kayser  with whom I discussed this
point in Aspen  agreed with me on part 1 (for him it seemed
even more or less trivial).
Only he was not convinced about part 2 (the one you stated in your
last email)!
**************************************
Do you disagree with assertion part 1?
**************************************
IF part 1 and 2 are correct, the conclusion in part 3 is
unavoidable.
>I don't see where the subtlety lies.
I think the subtlety lies here:
>From your report I got the impression (perhaps wrongly!)
that for you from statement 1:

1. A well defined Lagrangian can derived from the SM
for two different massless fermions "A" and "B".

automatically follows statement 2:

2. Fermions "A" and "B" have the same phenomenology.

If this WERE true just by deriving the correct SM Lagrangian for
Majorana neutrinos (i.e. eq.(10)) and setting: A=Weyl neutrino
B=Majorana neutrino
the equivalence of Weyl and Majoran neutrinos were proven.
However, I think this statement is NOT true in general.
Counterexample:
A=massless electron
B=electron neutrino
The analogy is close  in both cases the SM
model Lagrangian for A and B
(Eq(4) and eq(10) for the Weyl/Majorana case)
yields different electroweak couplings for
fermions A and B, respectively.
I would be very interested in knowing your
answer to the question in *** above.
This seems to be the only point where we
perhaps disagree.
Thanks again for your time
Rainer Plaga
12.7.00 from H.Haber to R.Plaga
Dear Dr. Plaga:
I have received your email. I know this has been excessively
delayed. Although in the final analysis, I cannot accept your
arguments, to be fair I have consulted one of the world experts in
neutrinos when he visited SLAC in June. He promised me a response in
a week, and that was more than a month ago. I sent him another email
reminder yesterday (before receiving your most recent note). So I am
hoping to have the final word shortly.
I am presently at CERN until July 17, after which I return to Santa
Cruz. If I have had no word by then, I will review the file once more
and try to present my final arguments.
Sincerely,
Howard Haber
editor, European Physical Journal C
24.7.00 from H.Haber to R.Plaga
Dear Dr. Plaga,
I have received the opinion of the referee, and I have reviewed
all your correspondence to me on your paper.
I am enclosing a copy of the referee report below. The referee
believes as I do that there is a fundamental error in your work, which
is basically associated with the normalization of the kinetic energy
term of your Majorana theory.
I shall comment briefly on some of your email arguments. Here
is one of your main points:
> +++++++++++++++++++++++++++++++++++++++++++
>
> 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.
> 
I would strongly disagree with your assertion of part 1. Consider
first a theory of a single noninteracting massive Majorana fermion.
The Lagrangian for this theory is given by eq. 28 of my short writeup
("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. If I
take the m>0 limit, I get a theory of a massless Majorana fermion,
which is identical to that of the Weyl fermion as demonstrated in my
note.
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 seesaw 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
writeup, I exhibit the seesaw 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 seesaw masssee eqs. 2326 of my
short note.] If one then takes the seesaw mass M to infinity, the
righthanded 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.
To summarize, the theory of a massive Majorana neutrino interacting
with the electron and Wboson (say, in a seesaw model), is a
welldefined unambiguous theory. By the way, I do not need to use the
seesaw mechanism. I can introduce a Higgs triplet, and give the
lefthanded neutrino a Majorana mass in a completely gauge invariant
way. (In this case, there is no need to introduce a righthanded
neutrino field at all.) Again, I would have a theory of a massive
Majorana neutrino interacting with the electron and Wboson. Again,
taking the neutrino mass to zero (by taking the triplet Higgs vacuum
expectation value to zero), I would end up with a theory equivalent to
that of a massless Weyl neutrino. The limit is again smooth.
So for me, the definition of a massless Majorana neutrino is most
easily understood as the limiting case of a massive Majorana neutrino
as its mass goes to zero. Since the limit is a smooth one, I see no
reason to define a massless Majorana neutrino in some other way (which
would probably not have a smooth limit from the massive case). In
the final analysis, there does not seem to be a choice here. The
normalization of the kinetic energy term is dictated by the principles
of quantum field theory, while gauge interactions are dictated by the
principles of gauge invariance. These principles are not in conflict.
Surely, you would agree that it is possible to have a theory of
MASSIVE Majorana neutrinos interacting in a gauge invariant way with
the electron and Wboson (I have specified two possible models
abovethe seesaw model and the Higgs triplet model). If you accept
this, then the massless Majorana neutrino limit is clear. If not, you
would have to tell me what is the theoretical inconsistency of the
seesaw or Higgs triplet model. These are definitely theories of
massive Majorana neutrinos, since the neutrinos of these models are
clearly not Dirac fermions. In the limit of the Majorana mass going
to zero, I think it would be appropriate to call these massless
Majorana neutrinos.
Finally, you suggest some possible counterexample in the case of a
massless electron:
> Counterexample:
>
> A=massless electron
> B=electron neutrino
>
> The analogy is close  in both cases the SM
> model Lagrangian for A and B
> (Eq(4) and eq(10) for the Weyl/Majorana case)
> yields different electroweak couplings for
> fermions A and B, respectively.
I must admit that I do not see the conflict. If you take the mass of
the electron to zero in the Standard Model, you still must retain both
the left and righthanded electron fields in the theory if you want
these fields to be charged under electromagnetism (which is a
conserved symmetry of the theory). The limit of m_e>0 is smooth
again, and this does not affect the nature of the gauge interactions of the
electron.
As one of the editors of European Physical Journal C, it is my job to
consider papers submitted for publication. This is a peerreviewed
journal, so I am obligated to consider the advice of referees.
Sincerely yours,
Howard Haber
editor, European Physical Journal C
enclosure: referee report given below
2.8.00 from R.Plaga (appendices are response to referee 2 below) to H.Haber
Dear Professor Haber,
Thank you very much for your email from 24.7.00
concerning my manuscript "Majorana neutrinos and
standardmodel gauge interactions" submitted for EJPC.
1.
The referee claims my manuscript is wrong because of an incorrect
normalisation factor 1/sqrt(2), however, the independent
reference quoted by HER/HIM (Mannheim) confirms me! (see Appendix 1 for
more details, including a list of quotes from
further references confirming my normalisation factor
and attachments with scanned pages from Mannheim´s article).
As you seem to agree with the referee on this point both in the
manuscript and your
last email, with all due respect, I have to conclude that your point of
view
cannot be entirely correct.
Mannheim also concludes that Majorana and Weyl neutrios are equivalent
without
taking into account interactions, a view with which I agree.
My paper is about factors of \sqrt(2) and
the world specialist got entangled in an obvious
selfcontradiction about a very basic (called "trivial" by her/him)
such factor.
In view of this I see only two possible ways to proceed:
further review or acceptance.
You treated me very fairly, now go all the way and accept the paper.
2.
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 standardmodel expression
for Weyl neutrinos) by directly replacing all Weyl spinors with Majorana
spinors. So by just replacing Weyl spinor with  in your view equivalent

Majorana spinors
you seem to be claiming to get a Lagragian with a DIFFERENT
phenomenology!
3.
Please note that I say eq(7) is DIFFERENT from the SM Lagrangian.
So in my view the SM Lagrangian (the one you attempt to derive in your
manuscript)
must be DIFFERENT from eq(7)!!
Your argument that the Majorana Lagrangian contains a additional factor
1/2
is:
a. therefore not in contradiction with the basic point of my paper
b. no longer holds once you add the normalisation factor 1/sqrt(2) in
eq(20) your
manuscript
4.
>I must admit that I do not see the conflict. If you take the mass of
>the electron to zero in the Standard Model, you still must retain both
>the left and righthanded electron fields in the theory if you want
>these fields to be charged under electromagnetism (which is a
>conserved symmetry of the theory). The limit of m_e>0 is smooth
>again, and this does not affect the nature of the gauge interactions of the
>electron.
I feel I did not manage to get my basic point through to you at all
(my mistake).
The massless electron and neutrino, though there are perfectly possible
within gauge theory BEHAVE DIFFERENTLY. e.g. their cross sections on
protons are very different.
I claim that massless Majorana and Weyl neutrinos are different ONLY in
this sense.
Both have perfectly fine SM Lagrangians in every respect
(so I do not contend most of what you wrote), but this is
not the same
as phenomenological equivalence. In your manuscript "Massless Majorana
and Weyl Neutrinos cannot be distinguished" you are EXCLUSIVELY
concernced
with deriving the SM Lagrangian for Majorana neutrinos.
You seem to think after this has been done, the equivalence is
established.
It was the intention of the massless electron example to demonstrate
that this is not true in general.
In other words, when you write (p.1) "The two formulations are
indistinguishable,
as they arise from exactly the same Lagrangian ..."
would also hold for the massless electron and neutrino Lagrangian.
Still they are of course phenomenologically different.
24.7.00 from referee 2
REFEREE REPORT

"Majorana Neutrinos and Standard Model Gauge Interactions"
by R. Plaga
I have read the author's article and I have a number of critical remarks
regarding the first two pages.
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.
For further details see Rev. Mod. Phys. 59 (1987) 671 and the
discussion of the Majorana mass term (subsection C) on pages 681682.
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.
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}.
By the way, the latter IS NOT THE CHARGE CONJUGATED field of \nu_L;
it is just a field which carries opposite additive quantum numbers
with respect to \nu_L; it has nothing to do with the CHARGE CONJUGATION
operation.
To define the operation of CHARGE CONJUGATION, the presence of a
RH field \nu_R (which differs from C (\bar{\nu_L})^{T}) is necessary.
In the standard theory with LH neutrino fields only, the CHARGE
CONJUGATION OPERATION for the neutrino fields \nu_L CANNOT BE DEFINED.
See also Appendix B, page 747, in Rev. Mod. Phys. 59 (1987) 671
for another proof of the equivalence of massless \nu_M and \nu_L (if
the theory contains LH weak charged currents only).
2.8.00 from R.Plaga to referee2 (via H.Haber)
Appendix A.
****************************************************************************
Further discussion of my "trivial normalisation error"
****************************************************************************
The third referee says the manuscript contains a
> trivial normalization error that he makes when he expresses
> \nu_M in terms of \nu_L and C (\bar{\nu_L})^{T}
Namely:
> \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.
She/he then cites
>P. Mannheim in Int. J. Theor.
>Phys. 23 (1984) 643 in which
>all the necessary
>normalization factors are rigorously derived
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.
(see attached sanned cover page and page containing eq(19) from
Mannheim´s article).
It also seems obvious to me that the factor 1/sqrt(2) must be
included to ensure the standard field normalisation
anticommutation rules (eq(3) my mansucript),
if \Psi_L are properly field normalised (as they must be be).
With the field normalisation used by the referee and Haber
(eq.(20) Haber´s manuscript, Haber´s \Psi_L1 must already be
properly field normalised because of eq(17) second eq)
one would obtain the anticommutation relation
[\nu_M,\eta]=2i\delta(xx´), a factor 2 larger than for
all other fields, which would lead to absurdities likes
Fermi field number operators which are 2 etc.
This view is shared by all standard textbook/review
references (except Bilenky/Petcov see below).
I selected 5 references in Appendix B which
include the 1/sqrt(2) factor. All these references also
discuss second quantisation of the neutrino field, I guess
once this is considered, one is led to the 1/sqrt(2) factor.
The other reference given by the referee
>Rev. Mod. Phys. 59 (1987) 671 by Bilenky,Petcov
indeed gives the same normalisation as the referee,
but again quotes Mannheim as more BASIC Majorana reference
(beginning of Appendix A of Bilenky,Petcov).
Second quantisation is not included at all in this review.
APPENDIX B
*******************************************************************
References confirming the normalisation of the Majorana
field used by me:
*******************************************************************

1. B.Kayser, The Physics of Massive neutrinos,
World Scientific,Singapore,1989 p.89:
\nu´=fM_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(1))


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(1))


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(1))

>By the way, the latter IS NOT THE CHARGE CONJUGATED field of \nu_L;
>it is just a field which carries opposite additive quantum numbers
>with respect to \nu_L; it has nothing to do with the CHARGE CONJUGATION
>operation.
The operation C (\bar{\nu_L})^{T}
is universally called "charge conjugation" in the
literature e.g.:

D.Bailin,A.Love
Supersymmetric Gauge Field Theory and
String Theory
Institute f Physics,Bristol,1994
p.6
"We define the "chargeconjugate spinor" \Psi_D^c
by putting \Psi_D^c = C (\bar{\Psi_D})^{T}"

If the referee´s criticism was linguistic, I agree of course
to use the term "particleantiparticle conjugation" in place
of "charge conjugation" in a revised version.
11.8.00 from R.Plaga to H.Haber
Dear Professor Haber,
Concerning the equivalence of Weyl and Majorana fields,
I did not give an direct answer to your argument
that a factor 1/2 should appear in the Majorana Lagrangian
(quoted from your report p.4):
>like in scalar field theory, if one uses real fields, the corresponding
>coefficient of the kinetic term is 1/2 while for complex fields it is 1.
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.
First two general remarks:
1. In the case of the scalar field
the definition of a neutral scalar field
in terms of a complex field DOES contain the factor 1/sqrt(2),
I maintain is necessary for the Majorana field definition (in
concordance with nearly all authors) (for the
scalar case, see Bjorken,Drell 12.5 first eq from which follows
immediately:
\psi(real)= 1/sqrt(2)(\Psi+\Psi^+)
2. As a result of the
factor 1/2 in the kinetic term, the complex and real
scalar field CAN be distinguished on the KINEMATCAL 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 also 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).
Now the demonstration that there is no factor 1/2 in the Majorana case:
The most direct way to see this is: Contrary to the scalar case
Weyl and Majorana fields have the SAME number of degrees of freedom
(namely 2) and from the argument in 2. above
it is clear that their kinetic terms should then be equal.
FORMALLY this follows from eq(21) your report, IF you insert
a factor 1/sqrt(2) in eq(20) that is necessary for proper field
normalisation.
The worldexpert recommended
Mannheim article, using the proper 1/sqrt(2) normalisation,
indeed gets this result: IDENTICAL kinetic terms for Weyl
(his eq(34))
and Majorana (his eq(33)) fields (NO additional factor 1/2 for the
Majorana kinetic term).
(Mannheim has a common factor 1/2 in BOTH the Majorana/Weyl Lagrangian
which stems from his use of the antisymmetrised differential operator).
>From the identical kinetic (and mass) terms
Mannnheim concludes (correctly I think) the full KINEMATICAL (i.e. with
no interactions considered) equivalence of Weyl and Majorana fields.
This is in
concordance with what I wrote under 2. above (identical
kinetic/mass terms lead to identical kinematic phenomenology).
If we can agree on this, there is only one more tiny step that leads to
my manuscript´s main conclusion: if you accept the
additional factor 1/sqrt(2) in eq(20)
your report, clearly the Majorana interaction term in eq(22) your report
obtains an additional factor sqrt(2) and is thus no longer equal
to the Weyl interaction term in eq(18) your report, leading
to a measurable difference between Weyl and Majorana fields.
Up to the development of the SM, the interaction term was
not calculable from neutrinoindependent quantities.
So it was always possible to introduce a factor 1/sqrt(2) in the
interaction term for the Majorana neutrino
to obtain complete MajoranaWeyl equivalence. This is how
the "complete equivalence" idea got born in the late 1950s.
What went wrong is that it got not "updated" around 1980.
Sincerely Rainer Plaga
7.9.00 from referee 3
Referee's Report to "Majorana ...." by Plaga, September 7, 2000.
Because of sharp disagreements between the author on one hand and the editor
and referee on the other, I have been consulted to provide fresh input,
and I am sorry to have to inform the author that I concur with the judgments
of both the editor and the earlier referee that the paper is not recommendable
for publication in European Physical Journal C. Moreover, I do not even
believe the paper to be correct, with it apparently being in disagreement
with the discussion presented by Mannheim, Int. Jour. Theor. Phys. 23, 643
(1984), a paper explicitly referred to in the earlier correspondence on the
author's paper.
I believe the author to be incorrect in his identifying his Eq. (2) and his
Eq. (6), and it is this faulty identification which gives him the
contentious $\surd 2$ factor in his Eq. (8). Specifically, what Mannheim
shows and comments on his Eq. (49) is that the Majorana field
$\chi=[\psi^W_R+(\psi^W_R)^C]/\surd 2=\chi^C$
constructed from the right handed Weyl spinor $\psi^W_R$ has the same
independent number of degrees of freedom (viz. two) as the original
$\psi^W_R$ and constitutes only a rewriting of the fields, one for which
the chosen normalization factor of $1/\surd 2$ then correctly normalizes the
ensuing $\chi$ field kinetic energy operator. As such, this definition of
$\chi$ is not a transformation of the fields but only a decomposition of a
Majorana spinor into the conjugate pair $D(0,1/2)+D(0,1/2)^C=
D(0,1/2)+D(1/2,0)$, i.e. into a Weyl spinor and its own conjugate, to
thereby show that a real Majorana spinor is reducible under the complex
Lorentz group.
However, a PauliGursey transformation of the form (Mannheim Eq. (52))
$\psi \rightarrow a\psi +b\gamma_5\psi^C$
(where $a^2+b^2=1$ preserves the norm of the kinetic energy operator)
is an entirely different object being a true transformation between
different degrees of freedom, yielding for a righthanded spinor
(Mannheim Eq. (54))
$\psi_R \rightarrow a\psi_R +b(\psi_L)^C$
with all terms transforming as $D(0,1/2)$ only, as they must in order to
preserve Lorentz invariance, and with the two component $\psi_R$ then
being mixed with two whole new degrees of freedom associated with
a whole other twocomponent spinor $(\psi_L)^C$. As we thus see, the
identification the author makes in his Eq. (6) in which he claims
to identify $\chi$ with $a\psi_R +b(\psi_L)^C$ is not valid, nor even
Lorentz invariant for that matter.
Moreover, not only does the author err in this regard (as least as best I can
tell), this whole analysis is anyway irrelevant to his attempt to compare the
Weyl and Majorana theories in the event of gauge interactions. Specifically,
if one wishes to compare the two cases, they must both have the same
independent number of degrees of freedom (viz. two). Thus one cannot even
make a PauliGursey transformation in the first place, since it requires the
presence of not one but two separate Weyl spinors (viz. four degrees of
freedom in total).
On a further issue, it was noted by the earlier referee that there is
already a proof in the literature of the equivalence of Weyl and Majorana
spinors in the presence of gauge currents (Bilenky and Petcov, Rev. Mod.
Phys. 59, 671 (1987), see appendix page 747), and if the author wishes to
make a claim to the contrary, it is his job in his paper to not
merely present his own result, but also to carefully analyze the
result of Bilenky and Petcov and show why it should now be set aside.
25.9.00 from R.Plaga to referee 3
Reply to report of referee 3 from Sept 7,00

Thank you very much for thetime and effort to write
a report about my manuscript.
I find your criticism does not apply, however I have to admit
that my manuscript was not organised very well, which is
why I sent a revised, new version.
> Moreover, I do not even
> believe the paper to be correct, with it apparently being in disagreement
> with the discussion presented by Mannheim, Int. Jour. Theor. Phys. 23, 643
> (1984), a paper explicitly referred to in the earlier correspondence on the
> author's paper.
It was the previous referee and NOT ME who brought
up Mannheim´s article to bolster her/his case. I never
stated that I find Mannheim´s article 100% correct.
You indeed found one point where my manuscript disagrees
with Mannheim´s. However here Mannheim disagrees not only
with me but also two other well known references and I will
show below where he errs and give these references.
Moreover this disagreement does NOT touch the basic disagreement
of the factor 1/sqrt(2) as the
following paragraph from you seems to show:
> Specifically, what Mannheim
> shows and comments on his Eq. (49) is that the Majorana field
>
> $\chi=[\psi^W_R+(\psi^W_R)^C]/\surd 2=\chi^C$
>
> constructed from the right handed Weyl spinor $\psi^W_R$ has the same
> independent number of degrees of freedom (viz. two) as the original
> $\psi^W_R$ and constitutes only a rewriting of the fields, one for which
> the chosen normalization factor of $1/\surd 2$ then correctly normalizes the
> ensuing $\chi$ field kinetic energy operator.
If the abaove equation is rewritten for left handed components _L
(as Manheim does in eq.(19)) you get exactly my eq.(6) (eq.(5) in new
revised
version) second,third term. From this follows directly my eq.(8) ((7) in
new revised
mansucript) WITH the factor 1/sqrt(2).
( with P_L \nu_L = \nu_L, P_L \nu_L^c = 0 ).
Therefore I cannot understand your remark:
> and it is this faulty identification which gives him the
> contentious $\surd 2$ factor in his Eq. (8).
Faulty identification or not, eq(8) follows from what you yourself
wrote!
> However, a PauliGursey transformation of the form (Mannheim Eq. (52))
>
> $\psi \rightarrow a\psi +b\gamma_5\psi^C$
>
> (where $a^2+b^2=1$ preserves the norm of the kinetic energy operator)
> is an entirely different object being a true transformation between
> different degrees of freedom, yielding for a righthanded spinor
> (Mannheim Eq. (54))
>
> $\psi_R \rightarrow a\psi_R +b(\psi_L)^C$
>
> with all terms transforming as $D(0,1/2)$ only, as they must in order to
> preserve Lorentz invariance, and with the two component $\psi_R$ then
> being mixed with two whole new degrees of freedom associated with
> a whole other twocomponent spinor $(\psi_L)^C$. As we thus see, the
> identification the author makes in his Eq. (6) in which he claims
> to identify $\chi$ with $a\psi_R +b(\psi_L)^C$ is not valid, nor even
> Lorentz invariant for that matter.
What you write here is indeed what Mannheim wrote on p.655657 and
I think it is incorrect.
If \Psi_R is INSERTED in the term b \gamma_5 \Psi^c
one obviously gets b \gamma_5 (\Psi_R)^c =  \gamma_5 (Psi_R)^c
because (\Psi_R)^c is left handed
((e.g. Itzykson,Zuber, Quantum field theory, Mc Graw Hill, 1980) p.88:
"charge conjugation connects the two chiralities...").
In the original reference (Pauli, Nuovo Cim.,6,204 (1957)) it is
completely clear that an insertion is really meant.
The view that the PauliGuernsey transformation connects the Weyl
with the Majorana neutrino in the way described in my manuscript
can also be found in great detail in:
 C.Ryan,S.Okubo Nuovo Cimento Suppl. {\bf 2}
(1964) 234.
 M.Fukugita, T.Yanagida, in: Physics and Astrophysics
of Neutrinos,M.Fukugita,A.Suzuki(eds.) (Springer,Tokyo,1991) p.1
(section 17).
The remark about Lorentz non invariance puzzles me: even if the
definition of the Majorana neutrino would not follow from the Pauli
transformation, the definition I use is standard. Do you want to imply
that
the Majorana neutrino is Lorentz non invariant?
> Thus one cannot even
> make a PauliGursey transformation in the first place, since it requires the
> presence of not one but two separate Weyl spinors (viz. four degrees of
> freedom in total).
The discussion of the PauliGursey was always made in the presence of
interactions (see papers by Pauli,Ryan etc). Do you want to imply that
GAUGE interactions pose special problems for this transformation?
If yes, I would be grateful for telling me about the nature of these
problems.
> On a further issue, it was noted by the earlier referee that there is
> already a proof in the literature of the equivalence of Weyl and Majorana
> spinors in the presence of gauge currents (Bilenky and Petcov, Rev. Mod.
> Phys. 59, 671 (1987), see appendix page 747), and if the author wishes to
> make a claim to the contrary, it is his job in his paper to not
> merely present his own result, but also to carefully analyze the
> result of Bilenky and Petcov and show why it should now be set aside.
What Bilenky and Petcov show on p. 747 is that massless Majorana
neutrinos do not violate lepton number. Not only do I not claim
otherwise
in my manuscript, I even discuss this proof in section 3 (this section
is removed in the new revised version).
Clearly this is no equivalence proof. An "equivalence theorem" is only
MENTIONED
quoting the Ryan,Okubo paper mentioned above.
18.10.00 from referee 3 (now identified as P.Mannheim)
The author appears to have totally misunderstood my remarks of September 7,
and even goes as far to assert that they are incorrect and even irrelevant to
his discussion. Since this is not the case, I am again unable to recommend
his paper for publication in European Physical Journal C.
The key point that the author misses is as follows.
The PauliGursey transformations for 4component spinors take the form
$\psi \rightarrow a\psi +b\gamma_5\psi^C ...(A)$.
Thus multiplying this equation on the left by the left handed helicity
projector operator yields
$\psi_L \rightarrow a\psi_L b(\psi_R)^C ...(B)$
since bringing the lefthanded projector through the conjugation sign
involves an anticommutation of $\gamma_5$ and $\gamma_2$. Similarly,
multiplying the PauliGursey transformation by the righthanded helicity
projector yields
$\psi_R \rightarrow a\psi_R +b(\psi_L)^C ...(C)$.
Both of these two transformations are Lorentz invariant since the one on
$\psi_L$ only involves $D(0,1/2)$, while that on $\psi_R$ only involves
$D(1/2,0)$. Now in his response to my report, even while supplying no proof,
the author asserts (insists even) that in the above PauliGursey
transformation one can replace $\psi$ by $\psi_L$, viz. (the left hand
portion of the author's new Eq. (5))
$\psi_L \rightarrow a\psi_L +b\gamma_5(\psi_L)^C ...(D)$.
This transformation is purely an invention of the author, and, unlike my Eq.
(B), does not follow from the initial Eq. (A). Moreover, the author's
transformation of his Eq. (D) could not be valid since it mixes $D(0,1/2)$
with $D(1/2,0)$ and thus, unlike my Eq. (B), does not preserve Lorentz
invariance. Thus we see that it is Eq. (B) which follows from Eq. (A), and
not Eq. (D); and regardless of the possible status of Eq. (D), the author
cannot dispute that Eq. (B) does follow from Eq. (A) and that it must thus
be valid.
As regards the issue of Lorentz invariance, since the author asked me, I
would note in passing that a Majorana spinor remains a Majorana spinor under
Lorentz transformations, while, equally, a Weyl spinor remains a Weyl spinor
under Lorentz transformations. Consequently, the reducible decomposition of a
Majorana spinor into a Weyl spinor and its own conjugate (viz.
$\chi=[\psi^W_R+(\psi^W_R)^C]/\surd 2$) is Lorentz invariant, as are
the transformations of my Eqs. (B) and (C) on Weyl spinors of definite
handedness. It is only the author's Eq. (D) which mixes differing
handednesses of Weyl spinors which is not Lorentz invariant. Moreover, since
gauge interactions can be broken up into pieces with definite handednesses,
my Eqs. (B) and (C) are completely compatible with left and right handed
current interactions.
I must admit to being completely mystified by the author's response to my
report. Specifically, in my report I presented my Eq. (B) and advised the
author that it was this equation rather than his Eq. (D) which followed
from Eq. (A). Then in his response to me the author attempted to argue that
he could derive his Eq. (D) from Eq. (A), but made no effort at all to show
that my Eq. (B) did not follow from Eq. (A). In fact in both his reply and
his revised manuscript he completely ignored my Eq. (B), and acted as though
it never even existed.
Thus, just as I had noted in my earlier report, it continues to be the case
that the author obtains his contentious root two factor through use of his
invalid Eq. (D) (both in the earlier and recently revised manuscripts), and
that this is the origin of his subsequent invalid claims regarding the
status of the Weyl and Majorana theories.
15.11.00 from R.Plaga to P.Mannheim
In the "Majorana issue" I really only need one cross from you
(but this I really need):
Using standard textbook normalisations and notations
(to be absolutely definite: the ones of BjorkenDrell 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]/\surd 2$)
( ) yes, the decomposition is WRONG ( ) no, this decomposition is not WRONG
thanks a lot rainer
17.11.00 from P.Mannheim
Rainer: nice to hear from you, and sorry that I could not reply earlier, I
am just overwhelmed with things to do. As regards your question, the
relation $\chi=[\psi^W_L+(\psi^W_L)^C]/\surd 2$ as given in my Majorana
paper as Eq. (49) is certainly correct. However, you could if you want use
$\chi=[\psi^W_L+(\psi^W_L)^C]$ instead and then put the $\surd 2$ factor in
the canonical commutator. Essentially, what is required is that for any
definition of the fields you choose, you should normalize their canonical
commutators so that the energymomentum tensor defined from the Lagrangian
leads to operators $P_{\mu}$ and $M_{\mu \nu}$ which close on the correctly
normalized Poincare algebra. Then rewriting the fields in terms of creation
and annihilation operators will lead to a Hamiltonian whose normalized
particle eigenstates are to be the in and out states for the Smatrix. With
such a definition the resulting Smatrix will not be sensitive to where you
put the factor of $\surd 2$. Thus while I agree that the choice of the
particular canonical commutators that you use is axiomatic (in the sense
that taking all commutators to be zero gives a different theory), they are
not completely arbitrary, since you do need to recover Poincare invariance.
Thus in my Majorana paper, when I work in two component notation, the field
commutators needed in my Eq. (75) to obtain compatibility with the
equations of motion are far from obvious (and do not appear in standard
quantum field theory text book discussions). However, you can, in your
case, avoid any of that discussion by simply using the
$\chi=[\psi^W_L+(\psi^W_L)^C]/\surd 2$ and the asasociated action given as
my Eq. (51), an action which must however be varied in the presence of the
$\chi =\chi^C$ constraint given in Eq. (50) in order to get the correct
equations of motion (which is why I instead worked with unconstrained two
component notation spinors).
1.12.00 from R.Plaga to P.Mannheim
Dear Philip,
Thanks for your further, detailed helpful remarks on Majorana fermions.
However, on the whole issue I´m quite desperate.
>As regards your question, the
>relation $\chi=[\psi^W_L+(\psi^W_L)^C]/\surd 2$ as given in my Majorana
>paper as Eq. (49) is certainly correct.
Howard Haber told me that my manuscript was refereed by
a "world expert in neutrinos visiting SLAC" who was personally asked
to overtake this task by him.
Before, Howard Haber had refereed the manuscript in excruciating detail
(writing several reports, for which I am very grateful!).
The ONLY reason to reject the manuscript given
by the world expert was (I attach her/his short report
below) was:

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.
...
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}.

What she/he writes is direct contradiction with your statement above (and thus wrong).
> However, you could if you want use
>$\chi=[\psi^W_L+(\psi^W_L)^C]$ instead and then put the $\surd 2$ factor in
>the canonical commutator.
This is the form the referee prefers.
I agree with what you say. But as long as my choice
of the normalisation factoris VIABLE
(in fact it is standard) the referee´s conclusion is wrong.
It is ironic indeed, when the referee further writes:

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.

Clearly she/he is convinced that only one prefactor is "correct".
The only way in which this view can be upheld is if one
tacitly assumes standard
anticommutator relations. This I find very reasonable.
But then the prefactor she/he thinks should be used is wrong.
I realise that a wrong referee report does not make
my manuscript correct.
However, this was an elementary error by a world expert having
been asked by another world expert to look at this manuscript (so she/he did itwith some care).
Moreover, Howard Haber expressedly agreed with the world expert´s report.
It makes me quite sad if a paper is rejected also based on a referee report
which is so obviously, glaringly wrong.
Your report is a different matter. The crucial thing is the validity
of eq.(49) of your 1984 paper.
Whether this eq. is the PauliGuersey transformation
or not does not touch the central argument of my manuscript.
Why did you not include weak interactions in the equivalence proof of your
1984 paper?
All I´m saying is that if you make up for this now,
you´ll find that equivalence no
longer holds.
27.12.00 from Howard Haber
Dear Dr. Plaga:
I am unaware of any policy at any of the peerreviewed journals
that forbid an author to quote the words of an anonymous referee of
his or her paper in a physics discussion outside the context of the
reviewing process. As long as the anonymous referee remains
anonymous, I see no inherent conflict.
Sincerely,
Howard Haber
31.12.00 from R.Plaga
Dear Prof. Haber,
As a "discourse among collagues" below I convey to you
Philip Mannheim´s statement that is in square contradiction
with the central, elementary yet critical claim of referee 2
(cited first for your ease of reference).
I say "critical" because a quantitative MajoranaDirac neutrino
equivalence with interactions can hold only for one certain normalisation.
The disagreement between referee 2
and Philip Mannheim (who was called upon as authority FOR THIS CLAIM
by referee 2 NOT me!)
on a most elementary, yet absolutely critical issue is obvious.
For me it is difficult to eascape the conclusion that a further clarification/discussion
of some basic issues would be useful to at least a part of the community
(to which I certainly belong!).

Excerpt from report of referee 2 (the anonymous "world expert") (conveyed to
me in your email from July 24,2000)

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.
...
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}.

The only substantiation given is Mannheim´s reference (as Bilenky/Petcov
also quotes it for the normalisation):

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.

Accordingly I asked P. Mannheim (email from Nov 15,2000):

Excerpt, my email to P.Mannheim from Nov 15,2000

Using standard textbook normalisations and notations
(to be absolutely definite: the ones of BjorkenDrell 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

P. Mannheim´s answer (email from Nov 17,2000):

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.

I attach his full email below[email from P.Mannheim
from Nov 17,00 above]. He continues to point out
that referee 2´s form for the decomposition can be correct
if the normalisation of the commutation relations are changed
accordingly. In other words:
referee´s 2 normalisation is not incorrect if nonstandard normalisations
are used at other places.
While this is certainly correct it does not change the
fact that referee 2´s verdict
is in square contradiction with Philip´s statement.