The Seesaw Mechanism

                                                                                    By Robert D. Klauber

1        Background

It may seem unusual to have such low values for masses of neutrinos, when all other particles like electrons, quarks, etc are much heavier, with their masses relatively closely grouped.  Given that particles get mass via the Higgs mechanism, why, for example, should the electron neutrino be 10⁵ times or more lighter than the electron, up and down quarks.  That is, why would the coupling to the Higgs field be so many orders of magnitude less?

One might not be too surprised if the Higgs coupling were zero, giving rise to zero mass.  One might likewise not be too surprised if the coupling resulted in masses on the order of the Higgs, or even the GUT, symmetry breaking scale.

Consider the quite reasonable possibility that after symmetry breaking, two neutrinos of a certain class (see “Majorana neutrino” below) exist, with one having zero mass (no Higgs coupling) and the other having (large) mass of the symmetry breaking scale.  As we will see, it turns out that reasonable superpositions of these fields can result in two neutrinos, one very light (but not zero, and like that observed) and one very heavy (of symmetry breaking scale, and unobserved).

2        Dirac vs Majorana Neutrino Mass Terms

We don’t know a great deal, experimentally, about neutrino mass, but on general theoretical grounds, two distinct classes of neutrino mass terms are allowed in the Lagrangian of electroweak interactions.  These are called Dirac and Majorana mass terms.

The Dirac mass terms, which are the usual terms dealt with in introductory quantum field theory (QFT), have form

                                                               ,                                                          (1)

 and Majorana mass terms, which may look unfamiliar to the uninitiated, have form

                                          ,                                     (2)

where sub/superscripts L and R designate left or right hand chirality, and the superscript c represents charge conjugation.  That is,

  destroys a LH chiral neutrino and creates a RH antineutrino,

  creates   “   “       “         “        and  destroys “   “        “  ,

 creates   “   “       “         “        and  destroys “   “        “   (does same as  ),

 destroys “  “       “         “         and creates   “    “        “  (does same as  ),

and for R subscript, interchange L ↔ R everywhere above.

Note that the subscript always refers to particles.  For a non conjugated field, no overbar means destroys particles, overbar means creates particles, and antiparticle actions for the same field are just reversed from particle actions (particle ↔ antiparticle, LH ↔ RH, destroy ↔ create).

Charge conjugating a field has the same effect on particle/antiparticle and creation/destruction as an overbar (overbar is effectively a complex conjugate transpose [plus a γ₀ multiplication]).  That is, the overbar and the superscript “c” have the same effect.  The charge conjugation merely lets us have the overbar (row) operator effect in a non overbar (column) vector.  In fact, the symbol  is used by some for the  term of (2), with similar changes for other terms, where one must keep in mind for such notation that inner product in spinor space is implied, even though there is no obvious transpose term (row vector on left) in .

Note that the first term in (1) destroys a RH particle and creates a LH one.  The Feynman diagram for this term shows a RH particle disappearing at a point and a LH particle appearing.  Thus weak (chiral) charge is not conserved, as a LH neutrino has +1/2 weak charge and a RH neutrino has zero weak charge.

Somewhat similarly, the first term in (2) creates two LH neutrinos out of the vacuum and thus also does not conserve weak charge.  But, importantly, it also does not conserve lepton number (which the Dirac terms do.)

Mathematically, charge conjugation of the field, where C is the charge conjugation operator, can be expressed as

                                               ,                                         (3)

which needs some study in spinor space to fully understand, but doing so would lead us astray from the task at hand.

With all this in mind, we can then express (1) and (2) in terms of a mass matrix  as

                                                                                              (4)

with

                                                                .                                                           (5)

3         See-sawing

Suppose, as suggested at the beginning, that Higg’s or GUT symmetry breaking gave only Majorana mass to neutrinos.  That is, coupling to the Higgs (or Higges) was not done in a way that led to Dirac mass terms.  That is, the mass matrix would be diagonal, unlike (5), of form

                                                                 ,                                                            (6)

and our Lagrangian mass terms would look like

                                                   ,                                              (7)

where we have represented the fields directly coupled to the Higgs by (ν  N)^T.  In other words, ν and N are the mass eigenstates for our neutrinos.

On the other hand, the weak eigenstates ν_L andν_R (and their conjugates) of (4), which are linear superpositions of ν and N, interact directly via the weak force, and represent what we detect in weak interaction experiments (ignoring in this context the fact that ν_R has zero weak charge and does not so interact.)

Finding (6) from (5) is just an eigenvalue problem, with m_ν and M the eigenvalues.  That is, we could think of our fields in two different, but essentially equivalent, ways: 1) a mix of Majorana and Dirac particles with the column vector of fields in (4), or 2) pure Majorana fields associated with the mass matrix of (6), represented by the different vector of fields (ν  N)^T.

Heuristically, finding (ν  N)^T from (ν_L^c ν_R)^T can be thought of as “rotating” our basis vectors in an abstract space until we find an alignment giving the fields vector the components (ν  N)^T.

So far in our world, we may have only seen Dirac particles with mass m_D, though tests may yet show we also have Majorana mass terms such as (4) and (5) suggest.  Assuming that is the case, what would the mass matrix (6) look like in order to give us the kind of Dirac masses m_D that we see?  Remember we are looking for a reason why that mass is so much lower than that of other particles.

That reason posits that the field components of the vector in (7) are the ones directly coupled to the Higgs field.  It works best if the mass m_ν = 0, as that means there is no Higgs coupling for the ν field, but there is such coupling for the N.  Note that if we took m_ν ≠ 0, but m_ν << M, we would still be left with our original problem, which is “why is one mass so much smaller than the others?”.  Having zero mass is easier to explain (no coupling) than extremely low mass (extremely small coupling.)

The characteristic equation for the eigenvalue problem solution of (5) is then

                                                     ,                                                 (8)

with eigenvalues,

                                 .                           (9)

For λ₁ = m_ν = 0, we must have the minus sign in (9) and

                                                                    .                                                            (10)

Then, we would have, with the plus sign in (9) for λ₂ = M,

                                                                                                                      (11)

We’ll work out the eigenvector N (i.e., for λ₂) expressed in the (ν_L^c ν_R)^T basis and leave the simpler case ν eigenvector (i.e., for λ₁) for the reader.

From the eigenvalue problem for (4) and (5), with the eigenvalue λ₂ of (11) we get the two equations

                                                                                       (12)

This yields

                                                                     ,                                                              (13)

and an eigenvector

                                                                    .                                                           (14)

Some care is needed to note that the top component here is really the ν_L^c field with the fractional factor indicating the size of the ν_L^c field compared to the ν_R field.  That is, N is really a superposition of the two fields, such that if ν_R has a coefficient of one in that superposition, then the ν_L^c field has a coefficient of .   In other words, in (12), the symbol ν_R really stands for the coefficient (effectively, the magnitude) of theν_R field, not the field itself (which the location in the column vector denotes.)

Note also that, up to here, we have ignored the Hermitian conjugate half of (4), which we will have to include.  So our true N will also include that, and is, in terms of the fields themselves, rather than as a two component vector, expressed as

                                                       .                                                (15)

Similarly, the other eigenvector is found to be

                                                        .                                                (16)

If we now assume (to be justified later)

                                                                      ,                                                              (17)

then N is composed almost entirely of ν_R (and its similar sibling ν_R^c), from (11) and (17) is very heavy, and is thus effectively sterile.  Conversely, ν_R can be thought of as composed almost entirely of N.  Similarly, ν is composed almost entirely of ν_L (and ν_L^c), and conversely, ν_L is almost entirely composed of the weightlessν.

The Name “See-Saw Mechanism”

Note, from (10), that for given value of m_D, a higher value for  means a lower the value for , and vice versa.  This is the reason for the name “see-saw mechanism”.  Further, from (15) and (16), the higher the value for , the more ν_R → N and ν_L → ν.

Approached in a different way, given  and , m_D will be the geometric mean of those two masses, and will generally be closer to the lower of the two.  (If  and , then m_D = 10.)  Further, if (17) holds, from (10), we have

                                                              ,                                                      (18)

and from (11),

                                                                                                                                       (19)

Mass Hierarchy

Thus, the mass hierarchy appears naturally as

                                                          .                                                  (20)

4        The

 Assumption

The astute reader might question if we have gained anything.  We originally sought a reason why the known Dirac neutrino mass m_D is so small compared to other masses.  We got that via the analysis above, but in the process, we had to make another, seemingly arbitrary, assumption (17).  With this assumption, we appear merely to be substituting one mass hierarchy problem for another.  That is, we now have to ask why m_D turns out to be so much smaller than .

The answer is this.  If we start with the mass matrix purely in terms of Majorana particles, i.e., (6) with one field having zero mass (uncoupled to Higgs particle(s)),

                                                                  ,                                                            (21)

and do a slight “rotation” in the 2D space of (ν  N)^T, we end up with a matrix like (5) with the characteristic (17), our initial assumption.  As an aside, we also find the relations (18) and (19), as well as the “see-saw” relationship (10).

5        Summary

The treatment of Sections 1-3 began with a general, non-diagonal mass matrix, looked at finding the mass eigenvalues of that matrix, and examined the relationships engendered between the masses.  Having done that, we find that looking at it somewhat in reverse, as in Section 4, can be helpful pedagogically.

Start with the mass eigenstates fields ν and N, the ones coupled directly to the Higgs field (with ν having zero coupling), and the diagonal mass matrix (21).  The weak eigenstates fields ν_L and ν_R (and their charge conjugation fields) are superpositions of the ν and N fields.

We then ask “If we transform (ν  N)^T into the (ν_L^c ν_R)^T, what would the transformed mass matrix look like?”  Well, if nature has chosen to make this a slight transformation (think of a small “rotation” in the 2D space of the fields), which is reasonable, then we would get a mass matrix with a very small upper left diagonal term , a very large lower right diagonal term , and off diagonal terms m_D which are each the geometric mean of the diagonal ones, as in (10).  We would have a “see-saw” relation between the masses, and could readily have Dirac neutrino masses m_D of the order observed.  For a greater “rotation” in 2D fields space, the greater would be the “see-saw” effect (bigger  and lower  ), and also the greater the value of m_D.

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