Friday, June 22, 2012

SUSY broken by extra dimensions is viable

Technical: Because serving the old Echo comments started to randomly get stuck, preventing the TRF pages from being fully loaded (sometimes DISQUS didn't show up as a result: Echo may be deliberately trying to encourage the users to leave the platform earlier), I disabled the Echo comments counters everywhere.

You may still see the archived Echo discussions (which are empty for new blog entries since June 15th or so) if you click at the pirate icon anywhere at the bottom of the posts. Patience may sometimes be needed when you click at the pirates.

The Echo threads will be merged into the DISQUS ones soon. After lots of programming, the rough process has been verified to work in a testing "shortname". Sometime today, I should get an answer from DISQUS support whether or not there is a way to preserve the reply structure of the comment threads (which comments are nested replies to others) and the identity and avatars of some Echo posters. It's unfortunately likely that the answers to both questions will be No.

In that case, I feel that I can't or I shouldn't waste dozens of hours in a futile fight against the wind mills of the institutionalized dysfunction and I will start the conversion of those 73,842 Echo comments into DISQUS comments very soon – perhaps within days – and after October 1st, the Echo discussions will only be preserved without the reply structure, without any true recognizable identity (except for properly spelled names of the anonymous users) or avatars, and without all the yellow smileys as well, of course.
Supersymmetry hasn't been discovered yet but it remains the most likely candidate for the first "physics beyond the Standard Model" that the colliders may see on a sunny day in the future – and possibly not too distant one.

In many recent entries (see the category "string vacua and phenomenology"), I discussed various paradigms, new models, and regions in the parameter space of older models that remain attractive and compatible with all the upper bounds on the cross sections – i.e. with the fact that the experimenters haven't proved SUSY yet.

Dirac gluinos, maximal mixing in the stop sector, and various non-minimal models such as the Nanopoulos et al. aromatic flippons are examples. Chances are high that supersymmetry is there and one of its roles is to solve the hierarchy problem i.e. guarantee that the Higgs boson (and related particles) remain much lighter than the Planck scale.

But there is a new preprint on how SUSY may naturally avoid the LHC's attempts to find it – so far – which is very natural to read, especially for people who have internalized the stringy thinking. The paper by four Japanese-born physicists located at Berkeley, Kašiwa, and Tokyo is called
Compact supersymmetry
Hitoši Murayama, Yasunori Nomura, Satoši Širai, and Kohsaku Tobioka – the list includes some phenomenologist(s) whose name sounds very serious – claim that if supersymmetry is broken geometrically, i.e. by topologically nontrivial properties of the configuration that requires extra dimensions, one naturally explains why SUSY hasn't been found yet even though the framework is more constrained than some SUSY frameworks we are familiar with.

What are the models and how do they achieve the ambitious goal?

They basically employ the logic behind the Scherk-Schwarz compactification, argue that all superpartners have pretty much the same mass in this framework, and calculate that if this is the case, it's very hard to find them.

Scherk-Schwarz compactification

First, what is the Scherk-Schwarz compactification? It is a "somewhat mutated" yet simple way to make some extra dimensions compact. The simplest kind of compactification is the compactification on a circle and its higher-dimensional counterparts, the tori. All fields are periodic functions of the extra dimensions:\[

\phi(X^5) = \phi(X^5+2\pi R k),\quad k\in\ZZ.

\] You may impose such a periodicity for many coordinates \(X^5\), \(X^6\), and so on, and the general periodicity is given by a lattice. Such a compactification generally preserves all the supersymmetries of the higher-dimensional i.e. uncompactified theory.

The Scherk-Schwarz compactification adds a simple twist. It demands the fermionic fields to be antiperiodic rather than periodic::\[

\phi(X^5) = (-1)^k \phi(X^5+2\pi R k),\quad k\in\ZZ.

\] So if you shift \(X^5\) by \(2\pi R\), the action is equivalent to a rotation by 360 degrees. Consequently, the index \(m\) describing the Fourier modes is no longer integer, \(n\in\ZZ\), for the fermions: it is a half-integer, \(n\in\ZZ+1/2\).

Incidentally and amusingly enough, one may discuss the matrix model for such compactifications which we did with my ex-adviser more than a decade ago. The gauge group \(U(N)\) of the parent theory gets split to \(U(N)\times U(N)\). Somewhat similarly to matrix string theory, the two \(U(N)\) groups get interchanged if you make a trip around the matrix model's world volume spatial circle (but in this non-supersymmetric orbifold, the permutation is built-in and holds for everyone; in matrix string theory, it's one of the allowed branches of the spectrum). The bosonic degrees of freedom of the matrix model transform in the adjoint representation; but the fermions transform in the bifundamental \(({\bf N},\bar{\bf N})\) "off-block-diagonal" representation (like on a checkerboard). The model has interesting properties.

At any rate, if you have fermionic fields that are antiperiodic with the "antiperiod" \(2\pi R\) – the period including the same sign is \(4\pi R\) – their momenta must be \((n+1/2)/R\); the smallest momentum in the extra dimension is \(1/(2R)\).

The Japanese authors apparently replace \((-1)^{2J}\) which is negative for fermions by the R-parity; they make all the new superpartners antiperiodic. Supersymmetry is broken by the boundary conditions and is still preserved "locally" in the higher-dimensional spacetime which guarantees some soft enough behavior of the theory at energies higher than the compactification scale. The theory is very constrained, labeled by three parameters only at low energies, and it is able to escape detection because for equally heavy superpartners, the decays of one superpartner to another don't have enough energy to manifest themselves either as the missing transverse energy or jets with high enough transverse momentum – and the most constraining searches do rely on missing energy or jets' high transverse momentum. I suppose that's the reason.

The lyrics says: I shouldn't have read the paper. I shouldn't have. Perhaps not a single page. A hero was hiding in the paper. I love him; his name is Twist. Oliver SUSY-breaking Twist... (The Rebels version at 6:00-8:00 here.)

There's one technicality I don't understand about their paper. The compactification energy scale \(1/R\) can't really be comparable to the electroweak scale i.e. the Higgs mass (or vev). It must be much higher for various reasons; they talk about \(10\TeV\) or \(100\TeV\) scale. To make the superpartners light enough, close to the Higgs mass, in order to explain the hierarchy problem, they need to reduce the masses. They reduce them by saying that the supersymmetry twist isn't by \(1/2\) (antiperiodic). Instead, it is by a smaller number \(\alpha\) of order \(1/10\) or \(1/100\). So the fields' phase must shift by a fraction of a radian when you go around the compact circle.

The reason why I don't understand it is that many superpartner fields are naturally expected to be real fields – e.g. the Majorana gauginos and higgsinos – so they can't be rotated by a general phase! For the supercharges themselves, I may imagine a phase if you decide to describe the supersymmetry generators as a Weyl fermion and not a Majorana fermion. I am not sure whether they actually have an explanation for this simple "reality" objection. The objection becomes even more obvious if one talks – and they do – about the "line interval" i.e. \(S^1/ \ZZ_2\) Hořava-Fabinger-like orbifold which they want to talk about. (Just to be sure, Hořava-Fabinger differs from Hořava-Witten by antiperiodic conditions but none of them is viable for the Standard Model physics because in heterotic M-theory, the Standard Model is supposed to live at the end-of-the-world branes only.)

If you believe they really know what they're doing, the outcome of their model analysis is a nearly egalitarian spectrum of the superpartners. For example, their benchmark point 1 gives the Higgs mass \(125\GeV\). For this apparently correct Higgs mass, almost all Standard Model superpartners have masses between \(1384\) and \(1494\GeV\). The exceptions include stops (their \(1267\) and \(1557\GeV\) differ from each other a lot because the model naturally gives them a big mixing); and one lighter (out of two) chargino as well as two lighter (out of four) neutralinos; the masses of all of them are near \(770\GeV\). The masses of all the four remaining Higgs bosons are near \(820\GeV\).

If we tolerate a light Higgs mass near \(121\GeV\), which is marginally compatible with the LHC measurement within the error margins, they may make all the superpartner masses lighter than \(1\TeV\): see their benchmark point 2.

Of course, I still don't know whether the model is viable or suffers from some childish diseases or errors. In string theory, we would normally expect the Scherk-Schwarz compactification to be utterly unrealistic. First, it requires a "non-chiral" higher-dimensional theory to start with (which doesn't easily allow the left-right asymmetric electroweak interactions of leptons and quarks); second, it creates an instability; third, the "smaller first" doesn't seem to be allowed. But maybe those three problems are related and solved by something I don't see, perhaps some generalization of the simple circular compactification into something more realistic.

It's surely an exciting possibility at least as a vague idea even if the model isn't quite right.

Mack Sisters (JP) play Bedřich Smetana's "The Moldau". They almost manage to emulate the strength of a string orchestra...

Reply from authors

The authors have quickly sent me an incredibly detailed answer to my questions. It argues that the \(S^1/\ZZ_2\) nature of their orbifold makes it more possible, and not less, to introduce the generic twist angle. It also allows the chirality (probably for the Hořava-Witten-like reason but in that case, they would need matter at fixed points which wouldn't have KK modes). Also, the gauginos are actually Dirac ones in this picture. With this doubling of spinors, I can imagine that there is some room for the phase to act.

What they pointed out that I neglected was the \(SU(2)\) R-symmetry of the 5D supersymmetric parent theory. This has some freedom for internal phases that may be rotated. I think that this R-symmetry group arises because \(SO(5,1)\sim USp(2,2)\sim U(1,1,\HHH)\). These quaternionic matrices may multiply the pseudoreal but not real columns/vectors (the spinors) from the left but there may be a non-commuting \(U(1,\HHH)=SU(2)\) action/multiplication from the right (of the column/vector) as well, and that's probably the R-symmetry. I've never appreciated this extra maneuvering space of 5D phenomenology.

The chirality comes from the \(\ZZ_2\) orbifolding except that their answer seems to suggest the proximity to the stringy vacua which seems tighter than the real one to me. In stringy braneworlds and heterotic M-theory etc., all non-gravitational matter is carried at branes or fixed points of these orbifolds so it can't carry the (unlimited fractional) momentum: you wouldn't have KK modes of such fields. So they treat all the MSSM fields like gravitinos in heterotic M-theory which may be OK but it's just not full-fledged proper string/M-theory, I think.


I was trying to verify the statement above. Can't one take M-theory compactified on \(CY_3\times S^1\) and twist the circle by an element of the \(SU(2)\) R-symmetry group that the vacuum should still possess (exactly)? By a general angle? Why hasn't this been considered in string phenomenology? A problem could be that locally in five dimensions, the spectrum carries the \(\NNN=2\) SUSY. So there would have to be a "natural" extension of the Standard Model to \(\NNN=2\). But this would require either whole hypermultiplets or even gauge bosons for all flavors of quarks and leptons, right? What the gauge group could be? And if they were hypermultiplets, how would the model be protected against pairing to heavy Dirac spinors?


Another update: after a sequence of e-mails, I understand everything what they're saying about my confusing points. In the 5D bulk, the theory indeed has 8 supercharges and extends the Standard Model to an extended SUSY theory. The chiral multiplets become non-chiral hypermultiplets. A theory of this sort is OK; one just can't get the Yukawa coupling directly from the more supersymmetric theory. They obtain them otherwise. A half of the extended SUSY multiplets are projected out by the \(\ZZ_2\) projection and due to the \(SU(2)\) R-symmetry in five dimensions, they are able to impose a twist on the circle labeled by a continuous \(\alpha\). The model isn't quite crazy; it differs from the normal stringy scenario, too. I wish it could be right. I will be looking whether this model may be explicitly embedded in string theory.


  1. Can they accommodate the 140 GeV or so gamma rays in their dark matter?

  2. The answer is probably no. The spectrum of the models they present doesn't allow any superpartner as light as 130-140 GeV. Assuming at least remotely right Higgs mass, near 125 GeV, the neutralino LSPs etc. are around 700 GeV. So they either predict it's not there or they must get it from somewhere else.

  3. Just from reading this interesting TRF article I got the impression that this model draws from certain pieces nicely put together of but is not directly derived from full string/M-theory ...? I always like the top down models better...

  4. The Smetana music tugs my heart-strings!