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Realistic fermion masses from D6-branes

The most interesting hep-ph paper today is

All Fermion Masses and Mixings in an Intersecting D-brane World
by Van Mayes of Houston. Well, it's a string phenomenology paper so it's more interesting than a dozen of average hep-ph preprints combined. Since my childhood, I wanted to calculate the "constants of Nature". It took some time to understand that one may only calculate the dimensionless ones – those don't depend on a social convention, the choice of units. Mass ratios of elementary particles were the first constants I was obsessed with – even before the fine-structure constant.

Well, at the beginning, I also failed to appreciate that the proton wasn't quite elementary so the proton-to-electron mass ratio, \(m_p/m_e\approx 1836.15\), was interesting enough. I figured out it was equal to \(6\pi^5\). Good numerology proves one's passion. ;-) I still think that the numerical agreement between this simple formula and the measured ratio is rather impressive.



OK, in more adult terms, the Standard Model has some 29 parameters or so. Most of them describe the mass matrices of the quarks and leptons. Wouldn't it be great to calculate them? String theory in principle allows you to calculate all the dimensionless constants encoded in the masses – once you insert a finite number of bits that describe the string compactification, you may calculate all such constants with an unlimited accuracy, at least in principle and after you figure out a calculational framework that really allows you any accuracy in principle.



String theory's realistic vacua require supersymmetry, for many reasons, and the maximum decompactified number of spacetime dimensions is 10, 11, or (if you promise to undo the decompactification of two soon) 12. The realistic classes of vacua in string/M/F-theory include
  • \(E_8 \times E_8\) heterotic strings on a Calabi-Yau three-fold
  • their strongly coupled limit, Hořava-Witten M-theory on a line interval times the Calabi-Yau
  • M-theory on a singular manifold of \(G_2\) holonomy
  • type IIA braneworlds with D6-branes
  • F-theory on Calabi-Yau four-folds or perhaps \(Spin(7)\) holonomy manifolds, perhaps with lots of fluxes
There are various dualities between these five groups. The second is the strong coupling limit of the first. The \(G_2\) holonomy manifolds of M-theory may be obtained from type IIA string theory with D6-branes – if those branes are replaced by Kaluza-Klein monopoles (which are ultimately smooth vacuum solutions of Einstein's equations). Also, if the \(G_2\) holonomy manifolds or the Calabi-Yau four-folds used for F-theory are written as certain fibrations, one may find a dual heterotic string theory. And there are a few more.

Mayes discusses some developments in type IIA string theory with D6-branes. There's some sense in which I love the five classes above equally. The IIA braneworlds with D6-branes are a great class of semi-realistic string compactifications. Incidentally, you may understand those braneworlds rather well from Barton Zwiebach's undergraduate textbook of string theory!

Just to be sure, I am not 100% sure that our world has to be described by a string vacuum from at least one of the five classes above (the number of correct classes may be higher than one due to dualities – equivalences across the groups). There may be other classes we have overlooked due to our incomplete understanding of string theory or string theory may be wrong as a theory of our Universe, in principle. But even if the confidence were just in "dozens of percent" that our Universe belongs to one of those groups, I would view it as a moral imperative for a sufficiently intelligent person to dedicate some time to get closer to such a TOE – or to find a viable alternative to string theory (which seems extremely unlikely to me).

Mayes uses type IIA string theory on an orbifold of the six-torus, \(T^6/\ZZ_2\times \ZZ_2\). The hidden six dimensions aren't too complicated – they are flat, in fact. You just make all six flat dimensions periodic, using a lattice. That's what a six-torus is. The orbifold means the "division by the group", in this case \(\ZZ_2\times \ZZ_2\): you identify points (or physical configurations, more generally) that are related by the geometric (or generalized) transformations representing the group elements. Here the orbifold group has \(2\times 2 =4\) elements. One of them is the identity element and the other three are "analogous to each other". So although \(\ZZ_2\times \ZZ_2\) looks like a group that is "all about the number two and its powers", there is a triplet hiding underneath it.

The group acts on the three complex coordinates labeling the six-torus, \(z_1,z_2,z_3\), by changing signs of two of the three coordinates (or by doing nothing). You may check that these operations are closed under composition and the group is isomorphic to \(\ZZ_2\times \ZZ_2\). These orbifolds were considered interesting since the mid 1980s, the First Superstring Revolution, and with the D6-branes added, they have been known to be damn promising in phenomenology since 2000 or so. Note that there also exist interesting orbifolds of the torus \(T^6\) that involve the group \(\ZZ_3\) – but the six-torus must split to two-tori defined with the angle of 120 degrees. The angles in Mayes' tori may be arbitrary.

Most of the key papers that Mayes uses are about one decade old – papers by Cvetič, Shiu, Uranga; Chen, Li, Mayes, Nanopoulos, and others. Type IIA string theory is great for braneworlds because the fermions and the Higgs doublet emerge really naturally from the branes.

Note that D6-branes are filling the 3+1-dimensional spacetime and they have 3 extra dimensions along the compactified directions. Those latter 3 are exactly equal to 1/2 of the number of the compactified dimensions which means that two generic D6-branes intersect at one point of the extra dimensions. The intersection is where some extra fields may live – fields arising from open strings stretched between two different D6-branes.

On top of that, the cubic couplings such as the Yukawa couplings may be calculated from "open world sheet instantons", triangular (=topologically a disk) fundamental world sheets stretched between the three intersections where the three fields involved in the cubic coupling live! That's wonderful because such "open world sheet instantons" effects are naturally suppressed with \(\exp(-AT)\) where \(A\) is the area and \(T\) is the string tension. That braneworld has a natural "exponential" explanation why the Yukawa couplings may be very small – and why they may differ by orders of magnitude from each other.

In another subfield, pure phenomenology, people have been playing with the fermion mass matrices for some time. The masses of quarks have been mostly understood by the 1970s – the top quark mass was really the newest and only new added parameter, in the mid 1990s. On the other hand, the neutrino masses – only seen through neutrino oscillations – have only been increasingly clearly measured since the late 1990s or so.

By now, the lepton masses plus the (squared) mass differences of the neutrinos and the mixing angles have been measured analogously precisely and completely as their quark counterparts. So in the quark sector, you basically need to know the masses of six quarks, the mass eigenvalues (three upper, three lower quarks), and the CKM matrix depending on four angles.

The story in the lepton sector is almost the same except that the upper and lower quarks are replaced by charged leptons and their neutrinos; the mixing matrix is called the PMNS matrix; there is one "overall" parameter labeling the neutrino masses that is unknown (only the differences of squared masses are known, as I mentioned, because only the differences affect the oscillations which is how the neutrino mass parameters are being measured – we haven't seen a neutrino in its rest frame yet); and there is a possibility that the neutrino masses aren't really Dirac masses but Majorana masses – in which case their fundamental origin could be unequivalent to the quark masses.

The neutrino mass-or-mixing matrices have been measured. One can see that the neutrinos are much lighter than the charged leptons and all the quarks. On top of that, they are apparently much more mixed than the quarks. All the angles in the CKM matrix are "rather small". On the other hand, many angles in the PMNS matrix seem to be "very far from zero or all the multiples of 90 degrees". That means that they're close to things like 45 degrees.

OK, a bit quantitatively. The CKM matrix is a \(3\times 3\) unitary matrix in \(U(3)\) – which encodes the transformation you have to do with the 3 upper-type quark mass eigenstates to get the upper \(SU(2)\) partners of the 3 lower-type quark mass eigenstates. Five of the phases may be thrown away by redefining six phases of the quark mass eigenstates (one of those phases which rotates all 6 quarks equally doesn't affect the CKM matrix so it's one parameter that has to be "subtracted from the subtraction"). It means that out of 9 parameters in the \(U(3)\) matrix, four are left – basically three real angles of an \(SO(3)\) matrix and one CP-violating "complex angle".

It's similar with the neutrinos' PMNS matrix. There's some CKM-like unitary matrix \(U\). A funny observation was that this matrix was close to\[

U_{TB} = \pmatrix {\sqrt{2/3} & \sqrt{1/3} & 0 \\ -\sqrt{1/6} & \sqrt{1/3} & - \sqrt{1/2} \\ -\sqrt{1/6}&\sqrt{1/3}&\sqrt{1/2} }.

\] All the matrix entries are (plus minus) square roots of small integer multiples of \(1/6\). You may check that it's a unitary matrix: all pairs of rows are orthogonal to each other, all rows have length equal to one, and to make a check, the same two types of conditions hold for columns or their pairs, too.

This Ansatz for the PMNS matrix is very close to the observed one and only a decade ago or so, this form of the PMNS matrix was actually falsified, primarily by seeing that the entry "zero" isn't quite zero. A new transformation involving neutrinos of 1st and 3rd generations (because the vanishing entry is in the 1st row and 3rd column) was observed for the first time – at some moment a decade ago.

The matrix \(U_{TB}\) is an intriguing piece of numerology but is there any reason why this form should be the right one (or close to the right one)? The answer is that such reasons were found in the flavor symmetries. There are three generations of quarks and leptons. The generations have different masses but there may still be some symmetries acting on the three generations that constrain the form of the mass matrices – in a nontrivial but not "complete" way, so that different eigenvalues are still allowed.

This "more serious level of neutrino matrix numerology" has led the people to realize that the special form of the unitary matrix above, the "tribimaximal mixing", may be derived from the assumption of flavor symmetries, either \(A_4\) or \(\Delta(27)\), two finite groups. The first is just the group of even permutations of four elements. The second one is more complex and I discussed it in a similar blog post six years ago and e.g. this 8-year-old one.

Mayes localized his pet D-braneworld model and argued that it produces a close-to-tribimaximal mixing matrix – which is non-trivial – and with some choice of some vevs of the many Higgses in the model, all the parameters determining the fermion masses and mixing seem to be OK, too. He seems to assume many values of the parameters. At the end, I think that he can't calculate a single combination of them from the first principles – although I am not sure, maybe he claims that he can.

But even if the "nominal" predictive power of his construction is zero, he has done some non-trivial reverse engineering of the fermion mass parameters. The braneworld he has apparently can rather naturally – in some colloquial sense, but maybe also a technical sense – explain the hierarchy between the fermion masses and the nearly maximal mixing of some neutrino species, among other things.

There are many qualitative choices one can make while choosing a type IIA D-braneworld. Ideally, we would want the number of predictions that arise from his model to be greater than the number of choices that had to be made – imagine both credits and debits are counted in bits or nats. But even if that comparison indicates that he hasn't produced more than he inserted, it's still true that the number of detailed microscopic theories that have a reasonable chance to explain the spectrum of the Standard Model, approximate values of the masses and/or their hierarchies, and the approximate values of the mixing angles, is extremely limited.

Grand unification can do something but it's always limited because grand unified theories still have some parameters. His string compactification ends up being a Pati-Salam theory which is strictly speaking not a grand unified theory because the gauge group has two factors. But his Pati-Salam theory behaves much like a grand unified theory, exhibits the gauge coupling unification, and other things. There's also the \(U(1)_{B-L}\) gauge group in it.

It seems plausible to me that models like that are so amazingly on the right track that a few weeks or months or years of work by some folks could have a chance to "nearly prove" that the model is actually right – that it predicts something. I think it's just terribly painful for this Earth with more than 7 billion humans to only produce "several" people who work at the D-braneworld string phenomenology at this moment (and similarly "several" \(G_2\) holonomy phenomenologists, and analogously with the other three – I guess that the F-theory researcher class is most numerous right now), a truly fascinating subfield. Individuals who would like to reduce this number and similar numbers of researchers further are simply animals. I will never consider them full-blown human beings.

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