A string-model case study: hep-th/0703280

[mitchell porter]

I am interested in string phenomenology - real-world model-building - but I've been held back by the large number of possibilities. I wanted to pick an existing model and study it until I really understood it from first principles; but which one should I choose? I've finally made a choice, and thought to create a thread about the model. It's http://arxiv.org/abs/hep-th/0703280" , where it was being promoted by one of the authors, Eric Mayes, as an example of a string model that makes detailed predictions.)

It's a Type IIA model - that's the string theory with even-dimensional D-branes. The model appears to be completely specified by two things: a choice of space-time manifold, and the positioning of intersecting stacks of D6-branes within that manifold. So you might say that the objective is to understand in detail how that gives rise to something a lot like observable reality. I intend to soldier on until the mechanics of that derivation are completely transparent to me. If anyone else wants to come along - or if people who already thoroughly understand this stuff want to join in - that would be great.

 

[JollyJoker]

I don't have the basics needed to do something like this, but it would be really nice if you could report on the progress here. I'm sure I'm not the only one who'd enjoy reading about it without being able to understand the details.

 

[mitchell porter]

The model isn't as worked out as I thought. I thought they had definite predictions for all SM and beyond-SM parameters, but these depend on geometric quantities (moduli) whose behavior they don't know how to predict. So in fact they're just fitting their model to the SM, and the resemblance to the SM might only last for a moment before the moduli changed, eventually stabilizing at completely different values. That's disappointing. However, they've gone on to study more rigid models where these parameters just can't change, so I might make one of those my case study... Or, I could just try to solve the moduli dynamics for the original model. :-)

 

[phyzguy]

I'm glad you pointed this out. It is along the lines of a question that I have had for a while. People seem concerned about the large number of possibilities in the string "landscape", but I hadn't seen where anyone had answered the question whether any of these possibilities can reproduce the standard model. It sounds like this question is still open, but this is a step in that direction.

 

[mitchell porter]

I'm going to persist with this, but the path ahead will be more complex than planned, because I don't intend to just reconstruct calculations in Type IIA string theory (though I must do at least that much). I also want to keep an eye on deeper issues, such as the relationship with M-theory. IIA string theory is M-theory compactified on a circle, and everything in the IIA theory has an M-theory description, but the relationship is complicated.

For example, in IIA language we might talk about "open strings ending on a Dp-brane", and it sounds the same whether p is 0, 2, 4, 6, or 8. In M-theory language, an open string is a cylinder-shaped M2-brane wrapped around the compactified dimension, but the IIA D-branes on which the M2-brane ends are http://www.sukidog.com/jpierre/strings/mtheory.htm" : a D2-brane is another M2-brane, but not wrapped around the 11th dimension; a D4-brane is a wrapped M5-brane; and a D6-brane is a type of defect in the space-time fields that couple to the M-branes. It's intriguing that the homogeneity of the IIA description of D-branes masks this diversity, and perhaps there is another perspective which conceptually re-unifies these diverse objects in an M-theoretic way.

Another deep issue is T-duality: a state in the IIA theory should have a T-dual description in the IIB theory. T-duality does a lot more than just replace even-dimensional D-branes with odd-dimensional D-branes, it mixes the metrical and non-metrical fields that couple to the strings, and ultimately should be understood as part of the U-duality group for a particular geometric background, something which must really be a manifestation of a symmetry of M-theory.

Returning to the model in the paper under study, it contains D6-branes and "orientifold planes", planes in the compact dimensions which act like mirrored boundaries that reflect the strings and branes which extend into them. Like D6-branes, these O-planes become geometric objects ("multi-centered Taub-NUT spaces") from the perspective of M-theory, so the whole construction corresponds to wrapped M2-branes propagating in 11-dimensional space compactified on a particular 7-manifold (a G2 manifold, actually). It might be a worthy challenge to study the M-theoretic counterpart of the IIA dynamics, and not just the IIA geometry.

However, first I need to understand IIA dynamics - open strings interacting with D-branes - and here I'm finding the right framework a little elusive, even after consulting Barton Zwiebach's textbook (which examines exactly this sort of IIA orientifold model). I can understand the quantization of a free string; I can understand the interaction of strings in space in terms of the path integral over string histories (the topological expansion); I can understand the quantization of an open string ending on a D-brane. But something about having lots of D-branes puzzles me. Do you treat the quantum theory of each "sector" separately (a sector here being defined by a pair of D-branes, so one is concerned with the strings stretching between those two D-branes), and then couple them by hand? Also, D-branes are supposed to have an explanation in terms of string field theory; does this provide a stringy (rather than M-theoretic) perspective in which strings and D-branes are on an equal footing? Normally D-branes are introduced as solitons from the supergravity limit of the string theory, but then consistency requires that they carry extra unexpected charges, with a justification coming from M-theory... It's all very interesting.

 

[smoit]

Just a couple of corrections: 1) an M-theory lift of an O6 plane is not a Taub-NUT but instead it is an Atiyah-Hitchin space. 2) once you include O6 planes, the U(1) isometry is broken so this is no longer M-theory on a circle times CY. In this case you also get N=1 SUSY in 4D. As you mentioned, the M-theory lift corresponds to a G2 compactification of M-theory, where the G2 manifold has co-dimension 4 singularities. In other words, locally the G2 manifold looks like an ALE fibration over a supersymmetric 3-cycle. The collapsed two-cycles in the fiber are in one-to-one correspondence with the simple roots of the corresponding gauge group. They are wrapped by the M2 branes (which have zero volume and their zero mode excitations correspond to massless particles) and if you carefully count the number of ways the membranes can wrap, you fill the adjoint representation of the corresponding gauge group. Furthermore, if the D6 branes intersect, you get charged chiral matter living at the intersections. These are then lifted to co-dimension seven singularities of the G2 manifold. In other words, in the M-theory picture, the charged chiral matter fields are localized at points on the supersymmetric three-cycle where the rank of the corresponding ADE singularity is enhanced by one, i.e. where an additional two-cycle in the ALE fiber has collapsed to zero size so you get extra states by wrapping the M2 branes.
So, effectively what you get in the low energy limit is 11 D sugra coupled to a 7D SYM, plus 4D charged chiral matter, transforming in fundamental rep of the corresponding gauge group.

 

[smoit]

If you are interested in this story you may find this paper by Jake Bourjaily very helpful: http://arxiv.org/abs/arXiv:0901.3785

 

[bobc2]

I'm not a string guy but have a couple of questions:

1) Are you saying that so far none of the string theories actually predict correct energies (masses) for all of the elementary particles of the Standard Model?

2) When I read the pedestrian books and watch the string videos on Discovery Channel, they always promote the concept of string theory by showing various modes of a vibrating string. And it's usually presented as analagous to a piano string, violin string (or cello), displaying the various orders of bending.

But, do the SM elementary particle representations actually correspond to those kinds of flexiblity modes (typcally associated with very high frequencies), or are the SM particles more analogous to coupled rigid body type modes?

The classical vibration case analogy would be something like a steel rod coupled to rigid boundaries through soft springs. Then you have six rigid body modes (three coupled rigid rod translations and three coupled rigid rod rotations). These give you six very low resonance frequencies. Then, at much higher frequencies, the rod begins a series of bending modes.

 

[smoit]

I'm not a string guy but have a couple of questions:

1) Are you saying that so far none of the string theories actually predict correct energies (masses) for all of the elementary particles of the Standard Model?

2) But, do the SM elementary particle representations actually correspond to those kinds of flexiblity modes (typcally associated with very high frequencies), or are the SM particles more analogous to coupled rigid body type modes?

1) The short answer is no. Otherwise, someone would have already received the Nobel prize for that. The SM particles correspond to the massless modes of the string and receive their masses via Yukawa-type interactions with the Higgs field. Hence, in order to compute their masses from ST, one needs to compute the Yukawa couplings. In String Theory these are functions of so-called moduli fields - scalar fields that describe the fluctuations of the internal metric, brane positions, etc. Once these fields obtain fixed vacuum expectation values (these are the values that minimize the energy density), the corresponding Yukawa couplings become fixed. One has to then run the Yukawa couplings from the unification scale down to the electroweak scale using the renormalization group equations and then compare them with the experimentally observed values. So far, the most sussessful approach to computing the Yukawa couplings has been coming from F-theory. They managed to parameterize all the dominant contribution to the Yukawas in terms of a single parameter. The pattern of Yukawa couplings they obtain in this way looks strikingly realistic! For more information on explicit computations see this paper: http://arxiv.org/abs/0910.0477
On the other hand, is seems like the masses of the supersymmetric partners of the Standard Model such as squarks, sleptons, gauginos would be "easier" to compute from string theory because they obtain masses from supersymmetry breaking, which is somewhat less dependent on the properties of individual compactification manifolds. In this way, one can obtain generic predictions for sparticle masses in entire classes of string compactifications.

2) No, the SM particles correspond to the zero energy, i.e. massless, modes of the string.

p.s. For any further questions about the basics of string theory, could you please refer to a non-technical thread. It would be nice to keep the discussion in this particular thread on point.

 

[bobc2]

p.s. For any further questions about the basics of string theory, could you please refer to a non-technical thread. It would be nice to keep the discussion in this particular thread on point.

Perhaps there should be notices on posts such as this one: Professional Specialists Only

But, your comments were quite good, and I did find your reference paper very helpful. Thanks.

 

[mitchell porter]

smoit, thanks very much for your remarks. I feel like I'm on the threshold of a vast expansion in understanding now, and will presently have a more substantive response, but I didn't want to let too many days pass without acknowledging your input.

 

[bobc2]

Thanks for starting your thread, Mitchell. I've been through the Zwiebach text some time back but haven't been touch with the math in quite a while. Also, it had been too long since thinking about QFT to work through the text without a lot of hammer and tonging and jumping back into QM, QED and Elementary Particles. I probably don't have the time in my present situation to keep up with your work (actually doing the homework as we go).

Smoit has made a valid point about getting off point with attempts to satisfy inquiries of the nontechnical types like myself. Would it make any sense to try to run a parallel thread that attempts to follow your progress with more nontechnical descriptions of what is going? I'm not sure what format would be appropriate. Perhaps timely comments on the parallel thread from you and/or others making the journey with you along with Q&A with the nontechnical guys.

I'm not sure whether it could work. When you do even the most basic things, like quantizing a boson string, you realize how totally removed the object of your study is from those objects in classical mechanics. Trying to find familiar analagous classical mechanics examples to enlighten one about what goes on in the 10^-33 domain with many dimensions and CY spaces, etc., might be completely misleading and miss the point of ST completely. But, if you ST folks feel like it could be fruitful to somehow include us nontechnical guys (even have time for the diversion it would cost you) --well, you will just have to decide.

In any case, I think you have a very interesting and worthwhile project going here. From what I recall from the Zwiebach text (I may not be remembering accurately), I was disappointed that he did calculations only for the bosons and, although making general remarks about Superstrings, did not actually do calculations for the leptons and quarks.

 

[smoit]

Dear Mitchell,
You are welcome! Good luck with your research! I'm not sure what exactly you are interested in but if you want to work on string pheno I highly recommend learning some algebraic geometry, at least some basics. You may find Brian Greene's TASI lectures helpful: http://arxiv.org/abs/hep-th/9702155 after which you can take a stab at working through Denef's lectures: http://arxiv.org/abs/0803.1194. You also need to get some background on N=1 D=4 supergravity. I think that the appropriate chapter in Weinberg's SUSY volume may be a good source. Wess and Bagger is not very illuminating but is an exellent reference if you need to look something up.

Dear bobc2,
Starting a non-technical string pheno thread may work but I'm afraid that any sensible discussion may quickly turn into string theory bashing. I'd be happy to contribute my two cents as long as people don't diverge too far from the main topic. You are very welcome to try and we'll see what happens.

 

[bobc2]

String bashing would be a very unfortunate outcome. I would see nothing fruitful in that for sure.

 

[mitchell porter]

bob, the 2009 edition of Zwiebach has more about superstrings.

 

[mitchell porter]

Now I want to make a really lateral move and talk about Koide's mass relation. The important facts are summarized in the first few pages http://arxiv.org/abs/0812.2103" : The relation (between electron, muon, tauon masses) appears to be exact, but if it derives from the unification scale, something must be cancelling the radiative corrections that would otherwise spoil it. I can't find a single string phenomenology paper which talks about it, which is remarkable - you'd think that such a striking fact would be a standard part of the beyond-SM model-building lore - but perhaps comment #9 tells us why: the SM particle masses are the hardest to calculate; they are the model output furthest removed, deductively and computationally, from the assumptions which define the model.

Nonetheless, this has to be a potentially important constraint and guide for model-builders, if only they can find a way to use it. So that's my next focal point. I suppose I'll be looking at the specifics of lepton mass generation in intersecting-braneworld models like the "case study", trying to see if something like a unification-scale Koide relation which remains protected at lower energies is possible.

 

[atyy]

Now I want to make a really lateral move and talk about Koide's mass relation. The important facts are summarized in the first few pages http://arxiv.org/abs/0812.2103" : The relation (between electron, muon, tauon masses) appears to be exact, but if it derives from the unification scale, something must be cancelling the radiative corrections that would otherwise spoil it. I can't find a single string phenomenology paper which talks about it, which is remarkable - you'd think that such a striking fact would be a standard part of the beyond-SM model-building lore - but perhaps comment #9 tells us why: the SM particle masses are the hardest to calculate; they are the model output furthest removed, deductively and computationally, from the assumptions which define the model.

Nonetheless, this has to be a potentially important constraint and guide for model-builders, if only they can find a way to use it. So that's my next focal point. I suppose I'll be looking at the specifics of lepton mass generation in intersecting-braneworld models like the "case study", trying to see if something like a unification-scale Koide relation which remains protected at lower energies is possible.

That's fascinating. Wikipedia links to http://arxiv.org/abs/hep-ph/0505220.

Would using Koide's formula as a constraint rule out 4 or more generations, ie. models like Mayes's later http://arxiv.org/abs/1012.3465v1?

 

[mitchell porter]

The three-generation Koide formula is logically independent of its four-generation generalization. The three-generation formula is concerned with the angle between the symmetric vector (1,1,1) and the "sqrt-of-mass" vector (√(m_e),√(m_mu),√(m_tau)). A four-generation formula would specify the angle between (1,1,1,1) and (√(m_e),√(m_mu),√(m_tau),√(m_tau')). These angles are independent degrees of freedom.

If you're just asking whether the mechanism which produces the three-generation relation ("angle = 45 degrees") can operate when there are four or more generations - that's impossible to know when we don't know what the mechanism is. For all I know, it might somehow require extra generations. That latest paper by Mayes et al talks about "flavor democracy" favoring four generations, and as Alejandro Rivero's paper points out, the symmetric vector and the democratic matrix are related.