Distinguishing the SM and SUSY With Running Coupling Constants At The LHC

The chart above via Lubos Motl's blog, illustrates the running of the the inverse of the Standard Model (SM) and Minimal Supersymmetric Model (MSSM) coupling constants with energy scale for the electromagnetic force, i.e. U(1), the weak force, i.e. SU(2) and the strong force, i.e. SU(3).

Distinguishing the Standard Model From SUSY Via The Running Of Gauge Coupling Constants

One of the clear and generic differences between supersymmetry theories and the Standard Model of particle physics is that the beta functions of the three fundamental forces, electromagnetism, the weak force and the strong force, are very different.

These differences cause the MSSM to experience gauge constant unification (i.e. a point at which all three forces have equal strength) at the grand unification theory (GUT) scale of about 10^15 GeV (in contrast, interactions at the LHC are in the general vicinity of 10^3 GeV aka 1 TeV).  Gauge coupling unification is one of the features of SUSY theories generically that makes them attractive and the GUT scale doesn't vary all that much between variations on the SUSY theme.

Supersymmetric theories generally presuppose that above the GUT scale spontaneously breaks the symmetries of the three gauge couplings that there is only a single unified force and that high energy physics above the GUT scale are essentially different in kind from the low energy effective SUSY theory - conveniently affording cosmology theorists freedom to effectively change the laws of physics during the first 10^-32 seconds or less after the Big Bang (by way of comparison it takes roughly a million times as long as that for hadronization of quarks to occur, or for a top quark, W boson or Z boson to decay once emitted).  By 10^-6 seconds after the Big Bang, the laws of physics would start to take approximately their current form.

In contrast, the three Standard Model beta functions for its gauge couplings, unmodified by new physics, do not converge at a single point.  The electroweak gauge coupling constants converge to identical values in the Standard Model at about 10^12 GeV, and at all energy scales before that point, the strong force coupling constant is larger than the weak force coupling constant which is in turn larger than the electromagnetic coupling constant, even though their relative strengths start to converge.

But, after that point, the strength of the electromagnetic coupling constant and weak force coupling constant are inverted, then the electromagnetic force coupling constant becomes identical to the strong force coupling constant after which their strengths are also inverted, and finally, the weak force coupling constant and strong force coupling constant converge around 10^18 GeV while the electromagnetic force is stronger than either of the other two forces.

In both the Standard Model and supersymmetric models, at higher energies the electromagnetic force gets stronger and the strong force gets weaker.  But, in supersymmetric models, rate of change in the electromagnetic force per unit change in the logarithm of energy is about 50% greater, and and the rate of change in the strong force per unit change in logarithm of energy is about 33% smaller.

The running of the weak force coupling constant differs even more dramatically between the two models. Starting around 1 TeV, i.e. at energies accessible before the LHC completes its run, the direction in which the weak force coupling constant runs is different between the SM and Supersymmetric models.  In the Standard Model, the weak force is weaker at higher energies, while in supersymmetric models, generically, the weak force is stronger at higher energies.  Also, the weak force coupling constant in the Standard Model runs about 500% as fast as in the MSSM.

Experimental Prospects For Discriminating Between SM and SUSY Gauge Coupling Constant Running At The LHC

Some analysis of our prospects for distinguishing experimentally between the running of particular gauge coupling constants as predicted by the Standard Model and the SUSY prediction, accompanied by some back of napkin estimates of the numbers involved based on the literature follows.

In a nutshell, distinguishing experimentally between SUSY and the SM via the running of the strong force coupling constant looks challenging, but prospects for making that distinction via the observed running of the fine structure constant and weak force coupling constant looks potentially quite fruitful.

Of course, SUSY models are not the only new physics models that would change the running of the gauge coupling constants at TeV scale energies relative to the Standard Model prediction, so experiments measuring these parameters at the LHC, in general, serve as a good model independent way to search for new physics at the LHC.  These measurements could quite possibly discern new physics at energies far lower than any of the other observable consequences of the new physics from the Standard Model prediction.

The Strong Force Coupling Constant - Prospects Weak At LHC

The running of the strong force is very difficult to measure with precision and the differences between its strength at easily attained energies like the Z boson mass, and its strength at the highest energies attainable by the LHC, which differ by a bit more than one order of magnitude, are easily calculated theoretically.

But, these differences are quite small relative the the precision with which the strong force coupling constant can be measured at all in a particular experiment.  And, the difference between the running of the strong force coupling constant in the SM and MSSM is smaller than the differences between the running of the other two coupling constants.  So, it is unlikely that the LHC will be able to use the running of the strong force coupling constant to distinguish between the Standard Model and supersymmetric models.

The strong force coupling constant, which is 0.1184(7) at the Z boson mass, would be about 0.0969 at 730 GeV and about 0.0872 at 1460 GeV, in the Standard Model and the highest energies at which the strong force coupling constant could be measured at the LHC is probably in this vicinity.

In contrast, in the MSSM, we would expect a strong force coupling constant of about 0.1024 at 730 GeV (about 5.7% stronger) and about 0.0952 at 1460 GeV (about 9% stronger).

Current individual measurements of the strong force coupling constant at energies of about 40 GeV and up (i.e. without global fitting or averaging over multiple experimental measurements at a variety of energy scales), have error bars of plus or minus 5% to 10% of the measured values.  But, even a two sigma distinction between the SM prediction and SUSY prediction would require a measurement precision of about twice the percentage difference between the predicted strength under the two models, and a five sigma discovery confidence would require the measurement to be made with 1%-2% precision (with somewhat less precision being tolerable at higher energy scales).

The Fine Structure Constant - Prospects Good At LHC

The differences between the running of the electromagnetic force coupling constant (aka the fine structure constant) in the Standard Model relative to supersymmetric models are also fairly modest over one order of magnitude, but they are still a moderate more distinct than the differences between the two models in the running of the strong force coupling constant.

But, because strength of electromagnetic interactions can be measured with a precision approximately 100,000 times as great as the strong force interactions, the prospects of being able to distinguish between the Standard Model and supersymmetric models based upon the running of this coupling constant at the LHC is much greater.

Even back in 2000, experimenters were able to measure differences in magnitude of the fine structure constant in experiments spanning energies ranges from about 2 GeV^2 to 3434 GeV^2 with a precision equal to about a third of the observed differences in force strength at the different energy levels (which was consistent with the Standard Model prediction).  The energies at the LHC are five to eight times as great as those made in 2000, and the precision of the measurements at the LHC on a percentage basis are probably at least somewhat improved from those made a decade and a half earlier.  The amount by which the fine structure constant should run under SUSY models at peak LHC energies should be on the same order as the amount by which it should run in the Standard Model at energies two and a half to four times as great as those made in 2000.

By 2011, measurements of the running of the fine structure constants at low GeV scale energies at the BES experiment were far more precise, measuring the differences in coupling strength due to its running with a precision of 1.2% or so.  The 2011 study also illustrates the capacity of relatively low energy scale precision electroweak measurements to shed light on phenomena that actually appear at one or two orders of magnitude greater energies.  The measurements in the 2011 paper increased the high end of the two sided one sigma confidence interval electroweak prediction of the Higgs boson mass from 115 GeV to 128 GeV, finally extending this range to masses that encompassed the ultimately discovered Higgs boson mass.  Yet, the study itself only measured events taking place at energies ranging from 2.6 GeV to 3.65 GeV.

Naively then, it ought to be possible to discriminate experimentally between the running of the fine structure constant in SUSY models and in the Standard Model at something on the order of 3-4 sigma by the time that the LHC's run is complete.

But, there may be material model dependence in an experiment based upon the running of the fine structure constant.

While the slope of the running of the fine structure constant in the Standard Model relative to the logarithm of the energy scale involved is fairly flat, in SUSY models that slope of the running of the fine structure constant is typically kinked, becoming more pronounced at masses ca. 1-2 TeV as the impact of supersymmetric particles somewhat below those masses kick in.  For whatever reason, visually at least, in charts of the running of SUSY gauge coupling constants, this kink appears more pronounced for the electroweak forces than it does for the strong force.

Thus, distinguishing between the SM and SUSY based upon the running of the fine structure constant may be more difficult than it seems if the lightest supersymmetric particle (LSP) has a mass that is close to or beyond the ability of the LHC to discern.  And, the fact that no supersymmetric particles have been observed so far at the LHC strongly favors SUSY models with a relatively heavy LSP, if indeed SUSY exists at all.

So, while the LHC may meaningfully constrain SUSY parameter space via its measurements of the running of the fine structure constant, this constraint may be less powerful than one would hope.

Perhaps the most important way in which constraints from the measured running of the fine structure constant at the LHC may constrain SUSY parameter space will be to rule out SUSY models in which there is a significantly sub-TeV superpartner that is not easily observed at the LHC.  For example, this could exclude SUSY models with an unexpectedly long lived LSP which passes beyond the range of existing detectors before it decays.  Searches for missing transverse energy already serve this purpose to some extent.  But, confirmation from the measured running of the fine structure constant at the LHC would make the conclusion drawn from missing transverse energy much more robust because the measurements in these two experimental tests are almost completely independent of each other.

The Weak Force Coupling Constants - Prospects Decent At LHC

The precision with which the weak force coupling constant's running can be measured at the LHC is intermediate between the precision with which the running of the electromagnetic fine structure constant, and the QCD strong force coupling constant can be measured.  And, as noted above, even distinguishing between Standard Model and SUSY predictions regarding the running of this constants may be challenging and model dependent.

But, the differences between the beta function of the weak force coupling constant in the Standard Model and in supersymmetric models is so great that the signal discriminating between the two theories should be so intense that it may reveal itself even if the measurements of the running of the weak force coupling constant aren't terribly precise and the theoretical differences between the Standard Model and SUSY values for it manifest only a few hundred GeV from the peak energy scales at which the LHC can measure these effects.

For example, suppose that the differences between the SM and a SUSY model's fine structure constant and weak force coupling constant both start to arise at 900 GeV and that the SUSY impact on the running of the fine structure constant is sufficiently slight that it is only definitively capable of being observed with current experimental apparatus at energy scales of 1900 GeV, at which the LHC may not be powerful enough to measure coupling constant strength.

The corresponding weak force coupling constant strength ought to change by an equal amount, and in a highly noticeable opposite direction from its previous running with the energy scale of the experiments at 1000 GeV.  Even if this running can't be measured as precisely as the running of the fine structure constant, by a presumed peak measureable energy scale at the LHC of 1400 GeV, the signal should be a strong as the fine structure constant running signal would be at 5900 GeV or more (since the difference in the direction of the running of the coupling constant would make it easier to see even when the precision is modest).

So, even if the measurements of the running of the weak force coupling constant at the LHC are a few orders of magnitude less precise than the measurements of the running of the fine structure constant at the LHC, there is still a very good chance that the LHC could measure this running with sufficient precision to discriminate between the predictions of the two models.

Slight Tweaks To The Standard Model Could Permit Gauge Coupling Unification

Of course, most people recognize that it is not at all reasonable to be confident that the low energy effective theory called the Standard Model really holds without modification all of the way up to energies in excess of 10^12 GeV which are far beyond those that can ever be created in man made experiments.

Even if we don't discover any entirely new forces or particles between the electroweak scale and the GUT scale, that doesn't necessarily imply that the Standard Model will perform perfectly over the additional nine orders of magnitude without even slight modifications.

As it happens, a very subtle tweak to one or more of the Standard Model beta functions could make a gauge coupling constant unification in the Standard Model possible.

For example, in the Standard Model the strong force coupling constant gets about 75% weaker between 1 TeV and 10^12 GeV.  But, if it instead it declined by 78% over those nine orders of magnitude, the three Standard Model coupling constants would converge at 10^12 GeV.  Given the immense complexity and numerous assumptions that go into the QCD and renormalization group calculations that ultimately help determine the strength of the strong force coupling constant at 10^12 GeV in the Standard Model, it would hardly be shocking to learn that some factor that could change its value by 4%-5% at such high energies was omitted or miscalculated using existing methodologies.

Similarly, the weak force coupling constant gets about 29% weaker between 1 TeV and 10^14 GeV.  But, if it got only 25% weaker over those eleven orders of magnitude, the three Standard Model coupling constants would converge at 10^14 GeV.  This would be a bigger percentage adjustment relative to the canonical Standard Model expectation, but would still be quite modest.

There are admittedly solid theoretical reasons for the beta functions of the Standard Model gauge couplings to have the forms that they do, and those equations have not been contradicted by experiments to date over energy scales that span three orders of magnitude.

But, one can easily imagine new physics that give rise to such subtle effects in the running of these gauge couplings at extremely high energy scales, such as quantum gravity considerations, in at least one or two of the gauge couplings, even if there is otherwise a new physics desert between the electroweak scale studies at the LHC and the GUT scale.

My personal conjectures

For what it is worth, my own personal conjecture as an educated layman is that by the end of its run, the LHC will not see any statistically significant deviation from the Standard Model expectation in the observed running of the three gauge coupling constants.  If this happens, this fact, as much as the mere non-discovery of superpartners at the LHC, may be the nail in the coffin of supersymmetry, and as a consequence of that, of string theory.  Pretty much every SUSY theory that serves any of the purposes that originally motivated it (e.g. the hierarchy problem and gauge unification), even if it has an LSP with a mass on the order of 10 TeV or so, for example, should have some observable consequences at the energy scales of 1 TeV or so probed at the LHC.

Failure to discover any deviations at all from the Standard Model at the LHC, even those like the running of coupling constants that may derive from key elements of the theory at much higher energies, may not rule out every last bit of the SUSY parameter space.  But, this may be sufficient to relegate SUSY theorists and M-theory to a role in the theoretical physics world comparable to other notable theories that are disfavored but have not been completely ruled out in every possible permutation such as Technicolor, preon theories, non-SUSY GUT theories, and the like.

However, I do personally believe that someday, not necessarily soon or within the time frame of the current LHC run, that it is likely that some new physics that leads to gauge coupling unification of the three Standard Model forces at an energy scale within an order of magnitude or three of the SUSY GUT scale will be discovered.  And, I further suspect that the value of the unified gauge coupling, should one be discovered, will be closer to the 0.0250 value around which the individual interactions of the Standard Model gauge coupling beta functions naively extrapolated appear, rather than around the 0.0400 value where the MSSM gauge couplings converge at around 10^15 GeV.  Simply put, the Standard Model comes so close to such a beautiful result that it is hard to believe that we aren't actually just missing a little something that prevents it from doing that.