Higgs Rumors

It decayed, and there was light.  #HiggsRumors

Noncommutative Motives

As research into quantum gravity continues, it is wise to look at the mathematical issues at hand.  Most would agree that ordinary algebraic geometry is not sufficient to tackle the problem in its ultimate form.  The theory of motives, at least in its commutative form, is only recently finding applications in the study of scattering amplitudes and Calabi-Yau compactifications.  Noncommutative algebraic geometry and its generalized motives, as an extension of Grothendieck's dream of building a gateway between algebraic geometry and the assortment of Weil cohomology theories (de Rham, Betti, l-adic, crystalline, etc.) seems to be a more appropriate tool in the study of quantum geometry.  In the noncommutative framework, the role of algebraic varieties and classical Weil cohomologies is played by differential graded categories and numerous functorial invariants.  The gateway category Mot, through which all invariants factor uniquely, is the category of noncommutative motives and the different invariants (the Grothendieck, higher K-theory, and cyclic homology groups, etc.) are simply different representations of the motivic category.

M-theory 2023

In the recent Prometheus trailer (Alien prequel), Peter Weyland in his 2023 TED talk mentions M-theory as one of mankind's great accomplishments.  M-theory isn't yet complete, but I'm pretty certain it (or its completion) will be complete by 2023.  Either way, it's entertaining to imagine the status of high energy physics 11 years from now.

Majorana Fermion detected?

While all eyes are on the LHC to discover new, interesting particles species, condensed matter physicists have been working hard in attempts to detect Majorana fermions and Magnetic Monopoles in solids.  In today's talk at the American Physical Society’s March meeting in Boston, Massachusetts, Leo Kouwenhoven presented findings that Majorana fermions have been detected in an indium antimonide nanowire apparatus.  Indium antimonide nanowires are connected to a circuit with a gold contact at one end and a slice of superconductor at the other, and then exposed to a moderately strong magnetic field. Measurements of the electrical conductance of the nanowires showed a peak at zero voltage that is consistent with the formation of a pair of Majorana particles, one at either end of the region of the nanowire in contact with the superconductor. As a sanity check, the group varied the orientation of the magnetic field and checked that the peak came and went as would be expected for Majorana fermions.

Although other groups have previously reported circumstantial evidence for the appearance of Majorana fermions in solid materials, Jay Sau, a physicist at Harvard University in Cambridge, Massachusetts, who attended Kouwenhoven’s talk, says that this is a direct measurement. “I think this is the most promising-looking experiment yet,” he says. “It would be hard to argue that it’s not Majorana fermions.”

Multiple schemes have been proposed in which Majorana fermions act as the 'bits' in quantum computers, although Sau cautions that it’s not yet clear whether those created by Kouwenhoven will be long-lived enough to be used in that way. 

- Read more at Nature

The 5% Standard Model

As the LHC's Higgs hunt continues, it's worthwhile to reflect on the current state of cutting-edge experimental particle physics, as a whole.  Indeed the Higgs boson is a prediction of the Standard Model of particle physics, but as former CERN theoretical physicist John Ellis admitted, in the range 114-135 GeV the "present electroweak vacuum would be unstable for such a light Higgs in the Standard Model" forcing one to come up with new physics to stabilize it.  By "new", this means physics beyond the Standard Model, such as for example, supersymmetry.  Yet, let's pretend the Standard Model is nice and stable with a ~125 GeV Higgs particle.  Does it tell us about every type of matter in the universe?  Sadly, it doesn't.  Ordinary matter accounts for only 4.6% of the mass-energy content of the observable universe, while mysterious dark matter makes up about 23%.  The rest of the mass-energy content, about 72.4%, is in the form of dark energy.  So cosmologically, the Standard Model doesn't seem so standard after all. 

So what kind of model explains the physics of dark energy and dark matter (the 95.4% of the universe), along with the 4.6% nicely described by the Standard Model?  Many theorists would agree such a model must come from a complete theory of quantum gravity.  The leading contender for such a theory is M-theory, the theory underlying the 10-dimensional string theories and 11-dimensional supergravity.  There also exist other theories, such as Loop Quantum Gravity, which essentially aims to "quantize" space via Wilson loop operators.  Ultimately, the goal is unification of all forces and matter in the universe, using just a single theory.  And this theory, in turn, should describe 100% of the universe.

Are we close to figuring out a complete theory of quantum gravity, hence a theory of all matter and forces?  There are hints that we are, but as always, many hurdles are mathematical.  Historically, Newton had to invent Calculus to describe motion properly.  Einstein had to invoke the tools of Riemann's Differential Geometry to describe space-time curvature.  And will it now be Connes' Noncommutative Geometry that will serve as the magic bullet for quantum gravity?  There is evidence that it might, as the coordinates of branes in string/M-theory are naturally noncommutative.  Erik Verlinde has even proposed a model for dark energy and dark matter, which as a matrix model, is an application of noncommutative geometry.  From a historical perspective, the use of new geometrical mathematics has proven fruitful, so we may very well be on the verge of a new physics revolution.

Unofficial Higgs combined plots are in

As promised, Philip Gibbs has produced combined plots for the Higgs mass, which includes data from LHC, Tevatron and LEP. Notice that nice peak centered at 124-125 GeV!

If this is a light E6 GUT Higgs, we'd expect to see a light isosinglet quark such as the D quark at the LHC very soon. We should even be able to predict its mass (~>250 GeV). A Z' boson would also be nice. For more info, see slides here and here. On to 2012!

Higgs rumors were correct

The official results are in, at least for the LHC's 2011 data, and it appears the rumors were quite accurate. See TRF and QDS for further details. To quote CMS member Dorigo, who stated there is now "Firm Evidence" with the current data,

So the summary is that ATLAS has a 3.6-sigma significance at 126 GeV, by combining their three most sensitive channels; CMS has a 2.4-sigma significance at 124 GeV, by combining all the meaningful search channels -even less sensitive ones.

After an ATLAS and CMS combined plot is produced, there might very well be over 4 sigma evidence for a light Higgs. I'm certain Philip Gibbs is working on a combined plot at this very moment. There is still a Christmas present to be delivered! Stay tuned.

Why a light Higgs is cool

As to why many theorists are excited over news of a possible light Higgs boson with 125 GeV mass, here's a memorable excerpt from a September 2011 interview with Clerk Maxwell Professor of Theoretical Physics and former CERN staff member John Ellis:

In the first scenario (114-135 GeV), we could be looking at a Standard Model Higgs boson. This range has been refined experimentally: recent LHC results presented in Mumbai excluded the Standard Model Higgs from about 135 GeV to about 500 GeV, while LEP had previously excluded it up to 114GeV. That leaves a narrow low-mass range of about 20 GeV where it could lie. But if found in this range, the Standard Model theory would still be incomplete; the present electroweak vacuum would be unstable for such a light Higgs in the Standard Model, so we would have to come up with new physics to stabilise it.

Higgs Candidate Events

While there has been no official announcement on the possible Higgs mass from CERN, there are some nice images available on possible candidate events where the Higgs might have appeared, as mentioned at TRF.

Candidate events in the CMS Standard Model Higgs Search using 2010 and 2011 data (Click images to enlarge)

A typical candidate event including two high-energy photons whose energy (depicted by red towers) is measured in the CMS electromagnetic calorimeter. The yellow lines are the measured tracks of other particles produced in the collision.

A typical candidate event including two high-energy photons whose energy (depicted by red towers) is measured in the CMS electromagnetic calorimeter. The yellow lines are the measured tracks of other particles produced in the collision. The pale blue volume shows the CMS crystal calorimeter barrel.

Real CMS proton-proton collision events in which 4 high energy electrons (green lines and red towers) are observed. The event shows characteristics expected from the decay of a Higgs boson but is also consistent with background Standard Model physics processes.

Real CMS proton-proton collision events in which 4 high energy electrons (green lines and red towers) are observed. The event shows characteristics expected from the decay of a Higgs boson but is also consistent with background Standard Model physics processes.

Real CMS proton-proton collision events in which 4 high energy muons (red lines) are observed. The event shows characteristics expected from the decay of a Higgs boson but is also consistent with background Standard Model physics processes.

All images and descriptions copyrighted property of © 2011 CERN and used for educational purposes.

Higgs rumors at 124.6 GeV

December 13 comes ever closer and the rumors about the Higgs mass get more detailed. Lubos Motl has commented on a recent post at QDS by Tommaso Dorigo in which he seems to hint at a possible Higgs mass from diphoton Higgs decay channels

- gamma: a gamma-ray is a photon, i.e. a quantum of light. A very energetic one, to be sure: a gamma ray is such only if it carries significantly more energy than a x-ray, so above a Mega-electron-Volt or so. The gammas we will be hearing about are those directly coming from a Higgs boson decay, and these have an energy of 62.3 GeV, equivalent to the kinetic energy of a mosquito traveling at 9 centimeters per second.

Here, the Higgs mass 124.6 GeV = 62.3 GeV x 2, from a process that can be written as H -> gamma gamma - where the Higgs decays to two high energy photons. Of course, Tommaso admits

I teased my most gullible readers with a (wrong) covert give-away of the Higgs mass ...

Either way, it is fun to speculate when the actual announcement is only a few days away. So let's see how close this 124.6 GeV is to the official (statistical) CMS value on Monday.

Higgs rumors at 125 GeV

As we all await CERN's official CMS and ATLAS results for the 2011 Higgs hunt, rumors about its mass have surfaced at notable blogs such as Philip Gibbs' viXra log, Peter Woit's Not Even Wrong and Tommaso Dorigo's Quantum Diaries Survivor. As mentioned by "Alex" in the viXra comment section,

Today rumour is: Higgs at 125 Gev around 2-3 sigma…

Such a rumor, if true, would not only indicate evidence for the existence of the Higgs boson, but is evidence for a light Higgs boson (115-135 GeV), which popular models such as E6 GUTs and M-theory on G2-manifolds predict. Of course, 2-3 sigma evidence isn't really conclusive but it does favor physics beyond the Standard Model. These are exciting times and by December 12 and 13 we'll all get to see if the rumors are true. Moreover, Philip Gibbs has also promised everyone a combined CMS and ATLAS plot once the data is released. How's that for an early Christmas present?

Update: Over at Lubos Motl's TRF blog, a commenter "azerty13" said he received the following email from CERN Director General Rolf Heuer:

Dear colleagues,

I would like to invite you to a seminar in the main auditorium on 13 December at 14:00, at which the ATLAS and CMS experiments will present the status of their searches for the Standard Model Higgs boson. These results will be based on the analysis of considerably more data than those presented at the Summer conferences, sufficient to make significant progress in the search for the Higgs boson, but not enough to make any conclusive statement on the existence or non-existence of the Higgs. The seminar will also be webcast.

Rolf Heuer

Such an email, if genuine, definitely supports the 2-3 sigma portion of the 125 GeV Higgs mass rumor. Stay tuned.

Update: As mentioned at viXra log, the latest incarnation of the rumor at Woit's blog gives 3.5 sigma in ATLAS and 2.5 sigma in CMS which amounts to about 4.3 sigma combined for the 10/fb. Keep in mind 5 sigma evidence is what is required at this stage of the Higgs hunting game.

M-theory 11/11/11

With so many 11's around today, it seems fitting to mention some M-theory related material. A few days ago, Hisham Sati updated his On the geometry of the supermultiplet in M-theory paper which argues that the massless supermultiplet of D=11 supergravity can be generated from the decomposition of reps of the exceptional Lie group F4 and its maximal compact subgroup Spin(9). The dynamical origin of this is proposed to result from Cayley plane bundles over eleven-dimensional spacetime.

The Cayley plane, OP^2, is a projective plane over the octonions and its isometries form the group F4. Lines in OP^2 are 8-spheres and given any two points in OP^2 there is a unique 8-sphere passing through them. Given any three distinct points, if we apply an F4 transformation that fixes one of the points, we get a Spin(9) transformation.

In matrix parlance, F4 is the automorphism group of the algebra of 3x3 Hermitian matrices over the octonions, the exceptional Jordan algebra J(3,O). We can construct OP^2 using the rank one projectors of J(3,O). It can actually be defined as the space of all such rank one projectors. Normalizing the rank one projectors turns them into primitive idempotents, that is, matrices P that satisfy P^2=P which cannot be decomposed as an orthogonal sum of other idempotents. As the identity matrix of J(3,O) is just a 3x3 matrix with ones on the diagonal, its straightforward to see that it decomposes into an orthogonal sum of three primitive idempotents. This is called the capacity and is why J(3,O) is an algebra of degree three.

Going back to the geometry of OP^2, the three distinct points mentioned earlier can be interpreted as three orthogonal primitive idempotents of J(3,O) with orthogonality being a result of these matrices satisfying P1.P2=0 under regular matrix multiplication. To simplify the picture, let's just imagine applying an F4 transformation on the identity matrix where we want to keep one of the diagonal ones fixed. This can be done with a Spin(9) transformation. Since we can fix any of the three diagonal ones of the identity matrix, there are three copies of Spin(9) inside F4 we can use. This freedom of choice we have is what some people refer to as triality.

Creation of Matter and E7

Recently, Ferrara and Kallosh posted a paper on arxiv entitled Creation of Matter in the Universe and Groups of Type E7. The abstract is as follows:

We relate the mechanism of matter creation in the universe after inflation to a simple and universal mathematical property of extended N > 1 supergravities and related compactifications of superstring theory. We show that in all such models, the inflaton field may decay into vector fields due to a nonminimal scalar-vector coupling. This coupling is compulsory for all scalars except N=2 hyperscalars. The proof is based on the fact that all extended supergravities described by symmetric coset spaces G/H have duality groups G of type E7, with exception of U(p,n) models. For N=2 we prove separately that special geometry requires a non-minimal scalar-vector coupling. Upon truncation to N=1 supergravity, extended models generically preserve the non-minimal scalar-vector coupling, with exception of U(p,n) models and hyperscalars. For some string theory/supergravity inflationary models, this coupling provides the only way to complete the process of creation of matter in the early universe.

As mentioned in the article, one can consider d=4, N=8 supergravity arising from M-theory on T^7, with duality group G=E7(7) acting on a 56-dimensional Freudenthal triple system (FTS) over the split-octonions. Another type arises from the N=2 magic supergravity based on the FTS over the octonions, with duality group G=E7(-25). At an algebraic level, E7(7) and E7(-25) are unified in the complexified duality group G=E7(C) acting on the FTS over the bioctonions. There is, however, no corresponding E7(C) supergravity theory at this time.

Dilogarithm Motives in Physics

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About this episode

Spencer Bloch

"Dilogarithm Motives Arising in Physics"

Talk given at "Algebraic Geometry, K-theory, and Motives" (a conference dedicated to Andrei Suslin's 60th birthday) St. Petersburg, Russia June 25-29, 2010.

In this very clear talk, Bloch explains how one is often writing down interesting periods on n-dimensional projective space when doing high energy physics. One can start with a hypersurface defined by the vanishing of a certain configuration polynomial F of degree d. F determines the hypersurface X, and X determines the motive.

Harmony of Scattering Amplitudes

The KITP program The Harmony of Scattering Amplitudes is still underway and there have been many wonderful talks on the geometry of scattering amplitudes in twistor space. The twistor approach allows one to look at scattering processes in a more algebraic geometrical fashion where as Ed Witten noted (arXiv:hep-th/0312171), one should focus on holomorphic curves.

At the most basic level, one is interested in degree one genus zero curves. In the complex case, such curves are copies of CP^1, 2-spheres. Witten argued that n-particle MHV amplitudes with two particles of negative helicity and n-2 with positive helicity localize on such degree one genus zero curves. This is the special case of Witten's more general conjecture that the twistor version of the n particle scattering amplitude is nonzero only if the points are supported on an algebraic curve in twistor space of degree d=q-1+l (where q is number of negative helicity particles and l is the number of loops). So for example, the tree level ++--- amplitude is nonzero on a curve of degree d=2-1+0=1, a degree one genus zero curve, a 2-sphere, as expected (see diagram above).

More recently, there is a more combinatorial way to view the MHV (and N^kMHV) amplitudes. This approach allows one to use associahedra, bubble diagrams and chorded polygons, for example. Below is a chorded polygon for the ++--- amplitude, and up to rotation and CPT transformation, is the only one contributing to the amplitude. For the --+++ amplitude there is another such polygon, so for the n=5 MHV amplitudes only (2(n-3))!/(n-3)!(n-2)!=2 total chorded polygons contribute.


















In such chorded polygons the chords physically correspond to twistor fields exchanged between degree one genus zero instantons. So given an n-point N^kMHV amplitude, one can draw many different chorded polygons (given by Catalan number C_{n-2}), but those which contribute are those that have no internal twistor field triangles, and describe configurations where each genus zero curve in the process has at least two points with different helicities. Below is a diagram for a non-contributing chorded polygon for the n=8 NNMHV amplitude (note the "illegal" internal twistor field triangle).


















As Nima Arkani-Hamed has noted, the twistor approach to scattering amplitudes is revealing a deeper mathematical unity that Feynman diagrams obscure. The mathematics so far involves the Riemann moduli space of surfaces of genus g with n marked points, Gromov-Witten invariants, symplectic geometry, quantum cohomology and motives.

QM over Split Composition Algebras

Over at viXra log, Philip Gibbs had a nice post on quantum mechanics and non-locality. In traditional quantum mechanics, it is often assumed one is constructing projective spaces over the complex field. However, as John Baez has noted at the n-category cafe, one can always formulate quantum mechanics over the quaternions and octonions as well. In order for octonionic quantum mechanics to be properly formulated, the Jordan formulation must be used in order to define projective spaces. Even then, one is limited to constructing a projective plane in the best case, due to algebraic topological constraints.

Back in my undergrad days, I was interested in studying quantum mechanics over arbitrary division algebras, which inevitably leads to the study of Jordan algebras as normed spaces over the reals. In the octonionic case, first studied by Jordan, Wigner and von Neumann back in the 1920's, one can have an algebra of 3x3 Hermitian operators in the maximal case. This case yields the exceptional Jordan algebra, with its corresponding projective space OP^2, the Cayley-Moufang plane. Even in this somewhat pathological case, it is possible to construct a 27-dimensional normed vector space over the reals. This is done by defining an inner product on the exceptional Jordan algebra, (X,Y)=tr(XoY), which induces a positive definite form, the norm, (X,X)=tr(X^2)=|X|^2. This norm also works for any nxn Jordan algebras over R,C,H. In all these cases, the length of a Hermitian operator is zero if and only if it's the zero vector (zero matrix). This means, in particular there are no rank one operators with zero length, and hence our projective spaces as manifolds, are easily described with the number of charts given by the degree of the Jordan algebra. In quantum mechanics this means we can normalize our rank one operators and the norm squared acquires a nice probabilistic interpretation.

When one attempts to give a similar normed space construction for Jordan algebras over the split composition algebras, it turns out the story isn't so nice. The first property that goes out the window is positive definiteness. So in quantum mechanics over split composition algebras there are a bunch of rank one projectors that have zero length. To this, one may say, "so what?" Well, for one, one can't assign a probabilistic interpretation to pure states described by these vectors. Now one may reply, "so just mod these out and define your projective space accordingly" Sure, we can try to do this but what if the physics actually requires the use of these pathological rank one operators?

Quantum mechanics over split composition algebras has already found use in M-theory compactifications, especially in describing extremal black hole charge vectors. In M-theory on T^5 and T^6, the charge vector spaces are actually Jordan algebras over the split octonions. In the black hole context, rank one operators describe 1/2 BPS states with zero entropy. This can be seen by noting rank one operators are those with zero determinant. So what does the (semi)norm mean in this context? I'm not really sure yet. In a literal sense, it gives the distance squared of an operator from the zero matrix. If one borrows some terminology from D-brane constructions, perhaps the norm can be interpreted as giving a type of tension, proportional to some theoretical mass. This would give an interpretation to the non-trivial charge vectors with zero norm: they describe some type of "massless" 1/2 BPS black holes. The other non-zero norm, rank one charge vectors describe "massive" 1/2 BPS black holes. The spectral decomposition of a full rank 3x3 Hermitian operator in the charge space then says that a 1/8 BPS black hole can be viewed as a bound state of elementary massive 1/2 BPS black holes, in some sense.

FBI can't crack code. Can you?

For those interested in a cryptographical challenge, the Feds are asking for help on the cracking of a code written by Ricky McCormick, 41, three days before he was found dead on June 30th, 1999 in a St. Louis field.

"We are really good at what we do," said Dan Olson, the chief of the FBI's Cryptanalysis and Racketeering Records Unit. "But we could use some help with this one."

Read more on the challenge at MSNBC.

Is N=8 Supergravity Finite?

In a recent paper arXiv:1103.4115 [hep-th], Renata Kallosh shows that E7(7) U-duality predicts the all-loop UV finiteness of perturbative N=8 supergravity.

Automorphic Instanton Partition Functions on Calabi-Yau Threefolds

Persson posted a nice paper recently, exploring the relationships between Calabi-Yau threefolds, U-duality groups and automorphic instanton partition functions. The paper discusses recent attempts at describing the moduli space in type IIA/B string theory on X×S^1, or the hypermultiplet moduli space in type IIB/A on X. It is well known that in certain classes of N=2 supergravities (e.g. magic supergravities) one can use automorphic techniques to constrain quantum corrections. Given a D=3 U-duality group G_3(Z), BPS-degeneracies are recovered from the Fourier coefficients of its related automorphic forms. From this stems the conjecture that the instanton partition function in N=2 supergravity on R^3×S^1 should correspond to an automorphic representation in the quaternionic discrete series of G_3.

arXiv:1103.1014[hep-th]

Abstract: We survey recent results on quantum corrections to the hypermultiplet moduli space M in type IIA/B string theory on a compact Calabi-Yau threefold X, or, equivalently, the vector multiplet moduli space in type IIB/A on X x S^1. Our main focus lies on the problem of resumming the infinite series of D-brane and NS5-brane instantons, using the mathematical machinery of automorphic forms. We review the proposal that whenever the low-energy theory in D=3 exhibits an arithmetic "U-duality" symmetry G(Z) the total instanton partition function arises from a certain unitary automorphic representation of G, whose Fourier coefficients reproduce the BPS-degeneracies. For D=4, N=2 theories on R^3 x S^1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function can be realized as a holomorphic section on the twistor space Z over M. We also offer some comments on the close relation with N=2 wall crossing formulae.

Ferrara: Black Holes and Supergravity

Part I: The Attractor Mechanism


Part II: N=8 Supergravity and Black Hole Charge Orbits


November 10, 2010
Lectures given for 25th Anniversary of ICTP's Dirac Medal, ICTP, Trieste, Italy.