**Guest blog by Gordon Kane**

I want to thank Luboš for suggesting that I explain the compactified M-theory predictions of the superpartner masses, particularly for the gluino that should be seen at LHC in Run II. I’ll include the earlier Higgs boson mass and decay branching ratio predictions as well. I’ll only give references to a few papers that allow the reader to see more details of derivations and of calculated numbers, plus a few of the original papers that established the basic compactification, usually just with arXiv numbers so the interested reader can look at them and trace the literature, because this is a short explanation only focused on the LHC predictions. I apologize to others who could be referenced. Before a few years ago it was not possible to use compactified string/M-theories to predict superpartner masses. All “predictions” were based on naturalness arguments, and turned out to be wrong.

String/M-theories must be formulated in 10 or 11 dimensions to give a consistent quantum theories of gravity. In order to examine their predictions for our 4D world, they obviously must be projected onto 4D, a process called “compactification”. Compactified string/M-theories exhibit gravity, plus many properties that characterize the Standard Model of particle physics. These include Yang-Mills gauge theories of forces (such as \(SU(3)_{\rm color} \times SU(2)_{\rm electroweak}\times U(1)\)); chiral quarks and leptons (so parity violation); supersymmetry derived, not assumed; softly broken supersymmetry; hierarchical quark masses; families; moduli; and more. Thus they are attractive candidates for exploring theories extending the Standard Model.

At the present time which string/M-theory is compactified (Heterotic or Type II or M-theory etc), and to what matter-gauge groups, is not yet determined by derivations or principles. Following a body of work done in the 1995-2004 era [1,2,3,4,5,6,7], my collaborators and I have pursued compactifying M-theory. The 11D M-theory is compactified on a 7D manifold of \(G_2\) holonomy, so 7 curled up small dimensions and 3 large space ones. We assume appropriate \(G_2\) manifolds exist – there has been a lot of progress via mathematical study of such manifolds in recent years, including workshops. For M-theory it is known that gauge matter arises from singular 3-cycles in the 7D manifold [3], and chiral fermions from conical singularities on the 7D manifold [4]. Following Witten [5], we assume compactification to an \(SU(5)\)-MSSM. Other alternatives can be studied later. Having in mind the goal of finding \({\rm TeV}\) physics arising from a Planck-scale compactification, and knowing that fluxes (the generalization of electromagnetic fields to extra dimensional worlds) have dimensions and therefore naturally lead to physics near the Planck scale but not near a \({\rm TeV}\), we compactify in a fluxless sector. With the LHC data coming we focused on moduli stabilization, supersymmetry breaking and electroweak symmetry breaking.

In order to calculate in the compactified theory, we need the superpotential, the Kähler potential and the gauge kinetic function. To learn the features characteristic of the theory, we take the generic Kähler potential and gauge kinetic function. The moduli superpotential is a sum of non-perturbative terms because the complex moduli have an axion imaginary part and it has a shift symmetry [8,9,10]. We do most of the calculations with two superpotential terms, since that is sufficient to guarantee that supergravity approximations work well, and we can find semi-analytic results. When it matters we check with numerical work for more terms in the superpotential. The signs of the superpotential terms are determined by axion stabilization [8,9,10]. We use the known generic Kähler potential [6] and gauge kinetic function [7]. By using the generic theory we find the natural predictions of such a theory, with no free parameters. This is very important – if one introduces extra terms by hand, say in the Kähler potential, predictivity is lost.

In addition to the above assumptions we assume the lack of a solution to the cosmological constant problem does not stop us from making reasonable predictions. Solving the CC problems would not help us learn the gluino or Higgs boson mass, and not solving the CC problems does not prevent us from calculating the gluino or Higgs boson mass. Eventually this will have to be checked.

We showed that the M-theory compactification stabilized all moduli and gave a unique de Sitter vacuum for a given manifold, simultaneously breaking supersymmetry. Moduli vevs and masses are calculable. We calculate the supersymmetry soft-breaking Lagrangian at the compactification scale. Then we have the 4D softly broken supergravity quantum field theory, and can calculate all the predictions of the fully known parameter-free soft-breaking Lagrangian. The theory has many solutions with electroweak symmetry breaking.

We also need to have the \(\mu\) parameter in the theory. That is done following the method of Witten [5] who pointed out a generic discrete symmetry in the compactified M-theory that implied \(\mu=0\). We recognized that stabilizing the moduli broke that symmetry, so \(\mu\approx 0\). Since \(\mu\) would vanish if either supersymmetry were unbroken or moduli not stabilized, its value should be proportional to typical moduli vevs (which we calculated to be about \(1/10\) or \(1/20\) of the Planck scale) times the gravitino mass, so \(\mu\approx 3\TeV\). Combining this with the electroweak symmetry breaking conditions gives \(\tan\beta\approx 5\).

The resulting model (let’s call it a model even though it is a real theory and has no adjustable parameters, since we made the assumptions about compactifying to the \(SU(5)\)-MSSM, using the generic Kähler potential and gauge kinetic function, and estimating \(\mu\)) has a number of additional achievements. The lightest modulus can generate both the matter asymmetry and the dark matter when it decays, and thus their ratio. The moduli dominate the energy density of the universe soon after the end of inflation, so there is a non-thermal cosmological history. Axions are stabilized and there is a solution to the strong CP problem. There are no flavor or CPV problems, and EDMs are predicted to be small, below current limits, since the soft-breaking Lagrangian at the high scale is real at tree level, and the RGE running is known [14]. I mention these aspects to illustrate that the model is broadly relevant, not only to LHC predictions.

The soft-breaking Lagrangian contains the terms for the Higgs potential, \(M_{H,u}\) and \(M_{H,d}\) at the high scale. At the high scale all the scalars are about equal to the gravitino mass, about \(40\TeV\) (see below). All the terms needed for the RGE running are also calculated, so they can be run down to the \({\rm TeV}\) scale. \(M_{H,u}\) runs rapidly, down to about a \({\rm TeV}\) at the \({\rm TeV}\) scale. One can examine all the solutions with electroweak symmetry breaking, and finds they all have the form of the well-known two Higgs doublet “decoupling sector”, with one light Higgs and other physical Higgs bosons whose mass is about equal to the gravitino mass. For the decoupling sector the Higgs decay branching ratios are equal to the Standard Model ones except for small loop corrections, mainly the chargino loop. The light Higgs mass is calculated by the “match and run” technique, using the latest two and three loop contributions for heavy scalars, etc., and the light Higgs mass for all solutions is \(126.4\GeV\). This was done before the LHC data (arXiv:1112.1059 and reports at earlier meetings), though that doesn’t matter since the calculation does not depend on anything that changes with time. The RGE calculation has been confirmed by others.

The value of the gravitino mass follows from gaugino condensation and the associated dimensional transmutation. The M-theory hidden sectors generically have gauge groups (and associated matter) of various ranks. Those with the largest gauge groups will run down fastest, and their gauge coupling will get large, leading to condensates, analogous to how QCD forms the hadron spectrum but at a much higher energy scale. This scale, call it \(\Lambda\), is typically calculated to be about \(10^{14}\GeV\). The superpotential \(W\) has dimensions of mass cubed, so \(W\sim\Lambda^3\). The gravitino mass is \[

M_{3/2}=\frac{e^{K/2}W}{M_{pl}^2}\approx\left(\frac{\Lambda}{M_{pl}}\right)^3\cdot \frac{M_{pl}}{V_3}

\] since \(e^{K/2}\sim 1/V_3\). The factor \((\Lambda/M_{pl})^3\) takes us from the Planck scale down a factor \(10^{-12}\), and including the calculable volume factor gives \(M_{3/2}\approx 50\TeV\). This result is generic and robust for the compactified M-theory. It predicts that scalars (squarks, sleptons, \(M_{H,u}\), \(M_{H,d}\)) are of order \(50\TeV\) at the high scale, before RGE running.

The suppression of the gaugino masses from the gravitino scale to the \({\rm TeV}\) scale is completely general (Acharya et al, hep-th/0606262; Phys.Rev.Lett 97(2006)191601). The supergravity expression for the gaugino masses, \(M_{1/2}\), is a sum of terms each given by an F-term times the derivative of the visible sector gauge kinetic function with respect to each F-term. The visible sector gauge kinetic function does not depend on the chiral fermion F-terms, so the associated derivative vanishes, and \(M_{1/2}\) is proportional to the moduli F term generated by gaugino condensation in the hidden sector 3-cycles. The ratio of the gaugino condensate F-term to the chiral fermion F-term is approximately the ratio of volumes, \(V_3/V_7\), of order 1/40, for appropriate dimensionless units. \(V_7\) determines the gravitino mass but not \(M_{1/2}\). Let’s finally turn to the gaugino masses. The reader should understand now that the prediction is not just a “little above the limits”, but follows from a generic, robust calculation. Semi-quantitatively, the gluino mass is \([(\Lambda/M_{pl})^3/V_7]M_{pl}\).

Then the gaugino masses with the suppression described above are generically about \(1\TeV\). Detailed calculation, using the Higgs boson mass to pin down the gravitino mass more precisely (giving \(M_{3/2}=35\TeV\)) then predicts the gluino mass to be about \(1.5\TeV\), the wino mass \(614\GeV\), and the LSP bino about \(450\GeV\) [12]. These three states can be observed at LHC Run II but none of the other superpartners should be seen in Run II (also an important prediction). The higgsinos and squarks can be seen at an \(\sim 100\TeV\) collider via squark-gluino associated production [12,13].

The LHC gluino production cross section is \(10\)-\(15\,{\rm fb}\) [12]. Note that for squarks and gluinos having equal masses the squark exchange contribution to gluino production is significant, so the usual cross section claimed for gluino production is larger than our prediction when squarks are heavy. Simplified searches using larger cross sections will overestimate limits. Surprisingly, experimental groups and many phenomenologists have reported highly model dependent limits much larger than the correct ones for the compactified M-theory as if those limits were general. The wino pair production cross section is also of order \(15\,{\rm fb}\). The wino has nearly 100% branching ratio to bino + higgs, which is helpful for detection. Gluinos decay via the usual virtual squarks about 45% into first and second family quarks, 55% into 3^{rd} family quarks, so simplified searches will overestimate limits. Branching ratios and signals are explained in [12]. The LHC t-tbar cross section is about \(4500\,{\rm fb}\), so it gives the main background (diboson production gives the next worse background). Background study should of course be done by experimenters, for realistic branching ratios to not be misleading. We estimate that to see a \(3\sigma\) signal for a \(1.5\TeV\) gluino will take over \(40\,{\rm fb}^{-1}\) integrated luminosity at LHC, so perhaps it can be seen by or during fall 2016 if the luminosity accumulates sufficiently rapidly.

- E.Witten, hep-th/9503124; NuclPhysB443
- Papadoupoulos, P. Townsend hep-th/9506150
- B.Acharya, hep-th/9812205
- B.Acharya and E.Witten, hep-th/0109152
- E.Witten, hep-ph/0201018
- C.Beasley and E. Witten, hep-th/0203061
- A.Lukas and D.Morris, hep-th/0305078
- B.Acharya, K.Bobkov, G.Kane, P.Kumar, D. Vaman, hep-th/0606262; PhysRevLett 97(2006)191601
- B.Acharya, K.Bobkov, G.Kane, P.Kumar, J.Shao hep-th/0701034
- B.Acharya, K.Bobkov, G.Kane, P.Kumar, J.Shao, arXiv:0801.0478
- B.Acharya, K.Bobkov, P.Kumar, arXiv:1004.5138
- S.Ellis, G.Kane, and B.Zheng, arXiv:1408.1961; JHEP 1507(2015)081
- S.Ellis and B.Zheng, arXiv:1506.02644
- S.Ellis and G.Kane, arXiv:1405.7719.

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