## Tuesday, December 16, 2008

### CKM matrix from F-theory

See these related articles:
F-theory GUTs
F-theory and experiments
Sweet SUSY from F-theory
Strings 2008: Tuesday (Vafa's talk)
Less than a month ago, I forgot to highlight the following cute paper:
Heckman, Vafa: Flavor hierarchy from F-theory
There exists a nice large collection of unexplained numbers in the Standard Model: the fermion masses or the Yukawa couplings, if you wish. They are pretty hierarchical, vastly different from each other, and have a lot of unexplained detailed features.

As far as the hierarchical character goes, people have been explaining them e.g. in terms of worldsheet instantons in string-theoretical braneworlds. The couplings are naturally exponentials of some negative areas (of triangular "disks"), and for reasonable distributions of the areas, you obtain an exponentially hierarchical distribution of the couplings.

In phenomenology, outside string theory, people often talked about Yukawa textures. These are guesses about the form of the mass terms in a particular basis where many elements of the matrices are strictly zero. Of course, in reality, they will never be strictly zero because they're not protected by any complete symmetries. But there might exist approximations in which they're zero and the structure of the couplings is then "qualitatively" explained.

The F-theory bottom-up phenomenology by Vafa et al. is a very new and cool sector of the "landscape" of phenomenologically viable string-theoretical models. While the gravity always lives in the "bulk" of the higher-dimensional space, the whole Standard Model is concentrated at some special places of the compactification manifolds - described by F-theory.

The gauge fields live on type IIB 7-branes, i.e. real codimension-two surfaces (del Pezzo...). The fermions live on intersections of similar 7-branes, i.e. real codimension-four surfaces. And the Yukawa couplings arise from double/triple intersections of such surfaces, i.e. real codimension-six points in the hidden manifolds.

In the new flavor hierarchy paper, they realize that this brane picture generates a new U(1) that is locally conserved, up to some point: it is literally the geometric rotation of the internal dimensions around the brane's locus, i.e. a phase added to some natural complex coordinates on the curves.

That restricts the Yukawa coupling matrices in the zeroth approximation to be rank-one matrices. There are corrections from pieces of the wave functions that depend on non-uniform background fields etc. When they estimate these things and calculate the CKM matrix (the unitary matrix relating the three lower-quark mass eigenstates and the low-quark SU(2) partners of the upper-quark mass eigenstates), they obtain the following Ansatz:

 1 ε ε³ ε 1 ε² ε³ ε² 1

That's great because ε can be argued to be the square root of the GUT fine structure constant, about 0.2. So the matrix above is

 1 0.2 0.008 0.2 1 0.04 0.008 0.04 1

That's called a prediction. For the CKM matrix, the prediction is more likely to be accurate than for the absolute fermion masses because several factors determining the "absolute normalization" may cancel in the CKM matrix.

This kind of a prediction that can be derived from more fundamental and more constrained principles would be unthinkable in the non-stringy models of particle physics. String theory is clearly the most predictive framework in high-energy physics - and the only framework whose predictions go beyond the general methods of ordinary quantum field theory where numbers such as the Yukawa couplings are inevitably adjustable, independent parameters.

You know, in quantum field theory, new matter fields and their couplings are cookies that may be thrown into a bag arbitrarily (or people who are only represented by some dull bureaucratic numbers written on a tax form for the IRS). In string theory, all these concepts are manifestations of real objects that must actually co-exist "somewhere" in the extra dimensions of space, according to well-defined laws. And their interactions are determined by the universal laws of string theory: all the diverse particles are ultimately made out of the same "stringy stuff". These laws still admit many solutions but locally in the hidden geometry, they determine "everything" which is why the allowed models are much more constrained (although not unique).

Some careful readers may also care whether the Heckman-Vafa prediction in this approximation is fine. Well, here's the real CKM matrix (with the CP-odd angle set to zero), originally invented by Kobayashi and Maskawa who extended the work by Cabibbo:

 0.97 0.23 0.004 0.23 0.97 0.04 0.008 0.04 0.99

Incidentally, their model doesn't exclude a higher number of generations a priori: a "richer" version of the model can be constructed. However, they can calculate the CKM matrix for a four-generation model and the ε^3 entry disappears from the 3x3 block of the lightest three generations. That effectively means a disagreement with observations which means that their model, combined with the known data, directly predicts that there exist no additional generations. It is not yet known whether this prediction of F-theory is correct.

Well, we report, you decide. ;-)

You may also see an extensive, 138-page paper about IIB/F-theory GUTs that was released one day later: Blumenhagen et al. See also Cosmology of F-theory GUTs by Heckman, Tavanfar, and Vafa where all things seem to work!

#### 1 comment:

1. antonio carlos mottaJan 21, 2013, 10:41:00 PM

the predictions of THEORY-F ARE CORRECT.THE VIOLATIONS OF CP APPEAR TO CP-EVEN SYMMETRY,WHILE TO CP-ODD APPEAR AS CONSERVED.THE TIME IS SPLITTED AS TWO DIMENSIONS CURVING THE SPACE THROUGH OF TWO OPPOSITE TORSIONS.THERE APPEAR THE STR.THE CKM MATRIX IMPLY THAT STR AND GTR ARE JUST EMMERSION OF THEORY MORE AMPLIED