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Is the neutrino mixing matrix an involution?

Off-topic and sad: Ray Munroe died. You may remember one of 42 comments, mostly on particle physics, he left on this blog. RIP.
Let me begin with the quarks.

There are three generations of quarks. The up-type quarks, \(u,c,t\), have three different Dirac masses which are well-defined for three "eigenstates" of the mass operator I will denote \(\ket u\), \(\ket c\), \(\ket t\).

Antineutrino detectors in the Daya Bay Reactor Neutrino Experiment in southern China, to be discussed below. The image shows the light sensors recording the neutrino interactions and the acrylic target vessels. (Photo by Roy Kaltschmidt, LBNL.)

The \(SU(2)_\text{weak}\) partners of these eigenstates are superpositions of the down-type quark Dirac mass eigenstates, \(\ket d\), \(\ket s\), \(\ket b\), but they are not the same states one-by-one. Instead, we have to rotate them by the \(V_{CKM}\) matrix as follows:\[

\pmatrix{ J^-_\text{weak} \ket u \\ J^-_\text{weak} \ket c \\ J^-_\text{weak} \ket t} =
\pmatrix{ V_{ud}&V_{us}&V_{ub} \\ V_{cd}&V_{cs}&V_{cb} \\ V_{td}&V_{ts}&V_{tb} }
\pmatrix{ \ket d \\ \ket s \\ \ket b }

\] The \(J^-\) operator on the left hand side is meant to pick the electroweak \(SU(2)\) partners. Because the matrix \( V = V_{CKM} \) above relates two orthonormal bases of a 3-dimensional complex space, it has to be unitary, \[

V_{CKM} V^\dagger_{CKM}={\bf 1} .

\] This Cabibbo-Kobayashi-Maskawa matrix is not too far from the unit matrix but it's still different and the difference is important for detailed nuclear processes. The \(V_{us},V_{cd}\) entries are close to \(0.2252\) and the entry is related to the Cabibbo angle, an angle from the era when the matrix was \(2\times 2\) and whose nonzero value allows Nature to violate the strangeness conservation law.

The other entries involving the third generation are smaller but nonzero, too. Without a loss of generality (using the freedom to change the phases of the six eigenstates, of which only five matter), the a priori 9-parameter \(U(3)\) matrix may be described by three "real' angles and one "complex" phase, i.e. by 4 real parameters. The complex phase violates the CP-symmetry.

At this point, it's the only truly established source of CP-violation we know in Nature although it's almost guaranteed that there must exist others, too. The main reason is that this known CP-violation by the CKM matrix seems too weak to be able to produce the matter-antimatter asymmetry shortly after the Big Bang that would be compatible with the mass ratio of baryons and photons in the current Universe.

What about the leptons?

You could say that the situation is totally equivalent in the lepton sector but it's not quite the case. There is a difference. The electrons, muons, and tau leptons have Dirac masses, much like the six quarks. However, only one-half of the degrees of freedom are known for the neutrinos. As far as experiments go, we only know left-handed neutrinos and right-handed antineutrinos. If that's so, their masses measured by the neutrino oscillations have to be Majorana masses: there can't be "Weyl masses".

The most usual story beyond what we observe is that there actually exist the other, right-handed neutrinos but their Majorana masses are huge, near the GUT scale. There are also Dirac masses (near the electroweak scale) that mix the known left-handed neutrinos with the unobserved superheavy right-handed neutrinos. By integrating out the heavy cousins, we obtain very low Majorana masses (near the observed ones) for the known left-handed neutrinos (well, the light states contain a tiny mixture of the original heavy spinors, too).

This process is known as the seesaw mechanism and there is a numerical argument why it may be right: the electroweak scale is close to the geometric average of the (huge) GUT scale and the (tiny) neutrino mass scale. This relationship is successfully predicted by the maths needed to integrate out the superheavy fields.

At any rate, the relevant low-energy lepton mass terms in the Lagrangian are:\[

\LL_\text{mass} = -\sum_{\alpha = e,\mu,\tau} m_\alpha \bar e_\alpha e_\alpha
- \frac 12 \nu_\alpha (M_\nu)_{\alpha\beta}\nu_\beta + \text{h.c.}

\] The first term is manifestly real: these are the usual Dirac mass terms for the three charged leptons. However, the second term isn't explicitly real which is why we have to manually add the Hermitian conjugate (the third term). The second term contains the Majorana mass terms for the neutrinos. Note that none of the two copies of \(\nu_\alpha\) is being complex-conjugated. That has many implications. For example, the mass terms always violate the lepton number by \(\Delta L=\pm 2\).

All the entries in the matrix \(M_\nu\) matter because the 2-component complex spinors \(\nu_\alpha\) also carry a 2-valued spinor index transforming under \(SL(2,\CC)\) which is being contracted by the antisymmetric \(\epsilon_{\kappa\lambda}\). The matrix may still be diagonalized – we may find the neutrino mass eigenstates – via \[

M_\nu = V^* D V^\dagger

\] where \(V^*\) is the complex conjugate and \(V^\dagger\) is the Hermitian conjugate of the neutrino mixing matrix, the PMNS matrix (both of the copies involve complex conjugation because the mass matrix is a form of a "quadratic form", not an "operator"), and \[

D = \diag(m_1,m_2,m_3)

\] is a diagonal matrix composed out of three real positive neutrino mass eigenvalues. Despite the different type of mass terms, the PMNS matrix depends on 3 real angles and 1 complex phase, just like the CKM matrix. Finally, I can get to the new hep-ph paper by Yoni BenTov, Woni BenZee, and Tony Zee:
The neutrino mixing matrix could (almost) be diagonal with entries \({\pm}1\)
The title says it all: the authors conjecture that the actual matrix \(V\) for the neutrinos is an involution. Aside from its unitarity, \(VV^\dagger={\bf 1}\), it apparently seems to satisfy \(V=V^\dagger\) which would also imply \(V^2={\bf 1}\): the matrix would be a matrix of a "reflection" with respect to some axes so the eigenvalues would have to be \(\pm 1\).

First, does it seem to be right experimentally?

The answer at this moment is, remarkably, yes. The measured values of the matrix are\[

V_\text{exp} \approx \pmatrix{ 0.81\pm0.03 & 0.56\pm 0.06 & 0.15\pm 0.02\\
0.49\pm 0.09 & 0.52\pm 0.13 &0.69 \pm 0.12\\
0.31\pm 0.12 & 0.63\pm 0.11 & 0.70\pm 0.11

\] If you see too big or too small spaces in the equation above, right-click the equation and choose Math Settings, Math Renderer, HTML-CSS (confirm even if it is chosen). The equations on this page will quickly rebuild and fix themselves.

The newest entry we have learned is the value \(0.15\) at the end of the first line. A few days ago, it was announced in PRL by the Daya Bay (Chinese-American-Russian-Czech-HongKong-Taiwan collaboration) that this entry is nonzero. Equivalently, the value of an angle known as \(\theta_{13}\) satisfies\[

\sin^2 2 \theta_{13} = 0.092\pm 0.016 \text{ (stat) } \pm 0.005\text{ (syst) }

\] Comparing the mean value with the error margin, they finally got a 5-sigma evidence that this matrix entry is nonzero. It's of course not shocking. There's no reason why the entry should be zero so Gell-Mann's totalitarian principle (everything that isn't forbidden is mandatory) pretty much guarantees that there has to be a nonzero number over there.

Fine, if you look at the numerical values of the matrix \(V_\text{exp}\) above, you may check that it is real (so far but the expectation that a CP-violating phase may be found as well is natural at this point but we're not there yet) and symmetric, within the error margins.

Tony and his co-author call their observation "purely phenomenological" in character; one could perhaps find an even more accurate word, "purely numerological". Most of the "research" of patterns in the neutrino mass matrices are numerological guesses of a similar kind. At any rate, they don't offer any deeper explanation why the neutrino mixing matrix should have this particular property.

Note that they're saying that the eigenvalues are \((+1,-1,-1)\) which means that the matrix may be expressed as a rotation by 180 degrees around some axis. However, this axis doesn't coincide with any of the "eigenstates", neither with the charged lepton mass eigenstates nor with the neutrino mass eigenstates, so it still "nontrivially" mixes all the six privileged axes in the three-dimensional space.

Still, their statement is pretty bold and it should be falsified in a foreseeable future if it is wrong.

A generic real \(SU(3)\) matrix i.e. a generic \(SO(3)\) matrix may always be expressed as a rotation around some axis, so the eigenvalues are \((1,\exp(i\gamma),\exp(-i\gamma))\), but it's usually not the case that \(\gamma=\pi\). Their hypothesis only constrains one number, the angle \(\gamma\), but the value \(\pi\) is rather unlikely because the measure of the region of the group manifold near the rotation by \(0\) or \[\Huge \pi\] radians is suppressed; this big \(\pi\) was my gift to you for the PI 3/14 day. (Well, their Hermiticity conjecture also predicts that the CP-violating complex phase will remain absent.)

This was still elementary linear algebra. I don't know a physics reason why the \(3+3\) low-energy lepton eigenstates, i.e. two orthonormal bases of a 3-dimensional space should be related by a rotation by 180 degrees. They don't know such a reason, either. Most likely, the reason why they don't know the reason is that the statement is wrong. The low-energy mass matrices are derived parameters from some more fundamental high-energy parameters. The derivations are complicated and involve RG running. I just don't believe that these derived quantities exactly obey some very simple relationships.

Update: As Lukáš observed in the slow comments, the matrix written above isn't unitary/orthogonal at all: after all, all entries of \(V V^\dagger\) are manifestly and substantially positive, having only positive contributions, so you can't get the zeroes expected in the unit matrix. The fix requires to change some signs in the \(V\) matrix. Flip e.g. the sign of \(V_{11}\) and \(V_{33}\) above and the matrix will be orthogonal; it's their equation (II.5). These signs will differ from the conventional signs in a PMNS matrix by a redefinition of signs of the six eigenvectors (of the first column and the last row, to be specific). Thanks to Alejandro for pointing these signs out,

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reader Lukáš Chvátal said...

Hi there,

the \( V_{exp} \cdot V_{exp}^T \) is something like

0.99220 0.79160 0.70890
0.79160 0.98660 0.96250
0.70890 0.96250 0.98300

This remarkably is quite far from unity, despite the proposed formula authors' wish...

reader Luboš Motl said...

Dear Lukáši, good texing, good point. There must be some errors, such as missing signs, in their numerical form of the matrix. In their form, it's not unitary at all. Obviously, a matrix whose all entries are significant and positive is very far from the unit matrix because all the terms in all entries in \(V V^\dagger\) are positive (and cannot yield zero, as needed for most)...

reader Luboš Motl said...

Wikipedia makes it obvious that the entries 21 and 32 have to have a negative sign which of course means that the matrix isn't Hermitian or symmetric and the paper is shown to be wrong.