I have talked quite often in this blog about F-theory. This is partially due to "historical" reasons, that is, the F-theory GUT revolution happened recently, while this blog growth. Also the influence of a friend of mine to let learn algebraic geometry was a plus because F-theory relies a lot in that area of maths.

Of course another reason is that they are very good developed framework.

But that doesn't mean that there is not development in other areas of string theory. In particular from the eighties heterotic strings where the best candidate for a phenomenological model. Even today most books in string theory (such as the Becker- Becker-Swhartz one) use the heterotic to teach the math of compactification.

Today it has appeared an interesting paper in heterotic phenomenology so I will say a few things about the subject. Heterotic string/m-Theory models are mainly build by the compactification mechanism. In that aspect they differ from many advances in string theory phenomenology. The fact that a group of parallel N D-branes automatically give an U(n) gauge theory was an star point for building local models in which gravitational degrees of freedom can be ignored for many purposes. That derived in a lot of development of D-brane models in Type II A and type II B string models, and, later, the non-perturbative counterpart of type II-B, F theory, where in addition to D-branes one has (P,q) branes. In fact something is done about non local F-Theory models and some is made for M-theory from the duality among some F-theory and M-theory set ups. But here I am going to talk about the most conventional approach, full compactifications. I am not sure about it, but I think that the reason why not too much development in local M-theory models is because not too much is known for certain (despite the Bagger-Lambert minirevolution of two years ago) about the M-theory branes, although possibly that would apply better to type II A M-theory that to heterotic M-theory.

I am going now with some references. The article of today roots in his model of the year 2006: The Exact MSSM Spectrum from String Theory. I let here the abstract of the paper:

We show the existence of realistic vacua in string theory whose observable sector has exactly the matter content of the MSSM. This is achieved by compactifying the E_8 x E_8 heterotic superstring on a smooth Calabi-Yau threefold with an SU(4) gauge instanton and a Z_3 x Z_3 Wilson line. Specifically, the observable sector is N=1 supersymmetric with gauge group SU(3)_C x SU(2)_L x U(1)_Y x U(1)_{B-L}, three families of quarks and leptons, each family with a right-handed neutrino, and one Higgs-Higgs conjugate pair. Importantly, there are no extra vector-like pairs and no exotic matter in the zero mode spectrum. There are, in addition, 6 geometric moduli and 13 gauge instanton moduli in the observable sector. The holomorphic SU(4) vector bundle of the observable sector is slope-stable.

The observable sector of the theory has an SU(3)C × SU(2)L × U(1)Y × U(1)B−L gauge group. The B-L additional group is beyond the MSSM, but that is not as bad as it seems as they discuss in the paper of today. Additionally they have:

Matter spectrum:

– 3 families of quarks and leptons, each with a right-handed neutrino

– 1 Higgs–Higgs conjugate pair

– No exotic matter fields

– No vector-like pairs (apart from the one Higgs pair)

3 complex structure, 3 K¨ahler, and 13 vector bundle moduli

This sector is obtained by by two steps. First a Spin(10) group can arise from the

spontaneous breaking of the observable sector E8 group by an SU(4) gauge instanton

on an internal Calabi-Yau threefold. Later The Spin(10) group is then broken by discrete Wilson lines to a gauge group containing SU(3)C × SU(2)L × U(1)Y as a factor.

The structure of the hidden sector depends on the choice of a stable, holomorphic

vector bundle V ′. The topology of V ′, that is, its second Chern class, is constrained by two conditions: first, the anomaly cancellation equation:

$$ c2(V´) = c2(TX) - c2(V) - [W] $$

Here c2 means the second Chern class of the vector bundle V and [W] is a possible effective five-brane class. Ok, 'll stop writing the details that can be read in the paper. The important part is that they don't obtain in detail the aspects of the hidden sector (the sector of the other E8 group of the E8xE8 heterotic string). They simply assume it's existence.

Since 2006 that model has been further developed and has lead to this paper today: The Mass Spectra, Hierarchy and Cosmology of B-L MSSM Heterotic Compactifications

The two papers even share one co-author, Burt A. Ovrut. The abstract reads:

The matter spectrum of the MSSM, including three right-handed neutrino supermultiplets and one pair of Higgs-Higgs conjugate superfields, can be obtained by compactifying the E_{8} x E_{8} heterotic string and M-theory on Calabi-Yau manifolds with specific SU(4) vector bundles. These theories have the standard model gauge group augmented by an additional gauged U(1)_{B-L}. Their minimal content requires that the B-L gauge symmetry be spontaneously broken by a vacuum expectation value of at least one right-handed sneutrino. In previous papers, we presented the results of a quasi-analytic renormalization group analysis showing that B-L gauge symmetry is indeed radiatively broken with an appropriate B-L/electroweak hierarchy. In this paper, we extend these results by 1) enlarging the initial parameter space and 2) explicitly calculating all renormalization group equations numerically, without approximation. The regions of the initial parameter space leading to realistic vacua are presented and the B-L/electroweak hierarchy computed over these regimes. At representative points, the mass spectrum for all sparticles and Higgs fields is calculated and shown to be consistent with present experimental bounds. Some fundamental phenomenological signatures of a non-zero right-handed sneutrino expectation value are discussed, particularly the cosmology and proton lifetime arising from induced lepton and baryon number violating interactions.

Since the 2006 paper math sophistication has grown and in the way of the theory the have used things such as monads, spectral covers or cohomological methods to calculate the texture of Yukawa couplings and other parameters. The key ingredient is still a Calabi-Yau manifolds with Z3xZ3 homotopy and a vector bundle with SU(4) structure group. The observable matter spectrum is basically the same of the previous paper. As I said before they state that: The existence of the extra U(1)BL gauge factor, far from being being extraneous or problematical, is precisely what is required to make a heterotic vacuum with SU(4) structure group phenomenologically viable. The reason is the following. As is well-known, four-dimensional N = 1 supersymmetric theories generically contain two lepton number violating and one baryon number violating dimension four operators in the superpotential. The former,

if too large, can create serious cosmological di culties, such as in baryogenesis

and primordial nucleosynthesis , as well as coming into conflict with direct measurements of lepton violating decays.

Well, the details can be read in the paper. The important thing is that the model is mature enough to allow explicit and accurate renormalization group analysis of the effective field theory and do precise predictions of some aspects.

This is not the only line of investigation in heterotic string theory. As I have stated this gives mainly an MSSM but no unification group scheme is followed. But there are such kind of constructions. As early as in 2005 there is a paper doing such a thing from heterotic M-theory: An SU(5) Heterotic Standard Model-

The authors are Vincent Bouchard, Ron Donagi. and the abstract says:

We introduce a new heterotic Standard Model which has precisely the spectrum of the Minimal Supersymmetric Standard Model (MSSM), with no exotic matter. The observable sector has gauge group SU(3) x SU(2) x U(1). Our model is obtained from a compactification of heterotic strings on a Calabi-Yau threefold with Z_2 fundamental group, coupled with an invariant SU(5) bundle. Depending on the region of moduli space in which the model lies, we obtain a spectrum consisting of the three generations of the Standard Model, augmented by 0, 1 or 2 Higgs doublet conjugate pairs. In particular, we get the first compactification involving a heterotic string vacuum (i.e. a {\it stable} bundle) yielding precisely the MSSM with a single pair of Higgs.

If one reads the paper one can see that it cites the papers in heterotic string that are the basic of the other models. This can look a bit surprising since one article is about heterotic string (which has 10 dimensions)and the other about heterotic M-theory (which has 11 dimensions). Nut they actually work with compactifications in a calaby-Yau threfold. That is because the eleventh dimension of the heterotic M-theory has an special character and no compactification of is is done. On the contrary the type II A M-theory has a more conventional eleventh dimensions and it requires compactification on G2 holonomy bundles, which are harder to work. Well, I am far from being an expert in the heterotic phenomenology, but I thought that the today paper was a good occasion to say some things about it.

Besides this paper today there has been many other interesting papers. Fortunately Lubos has written an entry doing a brief comments on them and I prefer link you to that entry to get the info: A generous hep-th Tuesday

Update: In the Lubos entry (where very gently this post is linked, thanks Lubos ;)) the papers about heterotic phenomenology have been discussed and someone posted two papers about G2 heterotic compactifications, concretelly : http://arxiv.org/abs/0810.3285 and http://arxiv.org/abs/0905.1968.

It has also been discussed an issue about the prediction of the paper considered here saying that the mass of the Higgs boson was around 101 - 106 GeV. while the LEP had excluded with a 95% confidence level a Higgs mass minor than 114 GeV. Lubos argues that 95% is not enought to exclude the value f there are goo theoretical reasons to do s. I would add that Jester (resonance) wrote a post stating that the LEP exclusion ny worked for conventional Higgs. I don't remember the details so I can't say if that is relevant, but I recommend the readers of this blog to make a search in resonances.

## Tuesday, June 01, 2010

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