One of the positive side-effects of preparing a seminar is being forced to get up-to-date with the latest experimental and theoretical developments on the topic. And this is of particular benefit to lazy bums like myself, who prefer to spend their time playing online chess than reading arxiv preprints.

It happened last week, in the course of putting together a meaningful discussion of the state of the art in global electroweak fits to standard model observables, and their implications for the unknown mass of the Higgs boson: by skimming the hep-ph folder I found a very recent paper by a colleague in Padova, which I had shamefully failed to notice in the last couple of careless visits.

The paper, titled "The muon g-2 discrepancy: new physics or a relatively light Higgs ?" by Massimo Passera, W.Marciano, and A.Sirlin, discusses the current status of the discrepancy between the experimentally measured value of the muon magnetic moment and the state-of-the-art of theoretical predictions, and then moves to determine what the difference would imply for the mass of the Higgs boson in case one were to attribute it entirely to a mismeasurement of QCD corrections. In order to explain what connects the two things, I need to provide a minimum of background on the topic. If you know the details, you may well skip the following section.

Determinations of the Muon Anomaly

I have discussed the basic physics of the muon magnetic moment in a couple of recent posts. My wish to repeat myself is limited, so please visit those pages for a zeroth-order introduction, and to figure out what the muon magnetic moment is.

The muon magnetic moment receives significant corrections from quantum loop effects due to all the three forces of nature affecting elementary particles: electromagnetic, weak, and strong interactions. These effects are calculable within the standard model with great precision, thanks to the development of very sophisticated calculation tools, plus an ounce or two of black magic.

The magnetic moment of muons can also be determined experimentally with great precision, by studying the precession of the muon spin when the particle is made to orbit in a magnetic field. Muons live only two microseconds on average, but if they are energetic their lifetime as observed in our laboratory gets multiplied by the factor $\gamma = E/m$. For a 3.09-GeV muon, the observed lifetime is thus on average 64.4 microseconds: this allowed the experimenters at Brookhaven (see the ring in the picture above) to let these particles make several turns in a circular ring. When the muons finally decay, the direction of their spin vector with respect to the direction of motion can be determined from the direction of the electron they decayed into, thanks to the characteristics of weak interactions governing the $\mu \to e \nu_e \nu_\mu$process.

You might have noticed that I picked a very particular value for the momentum of a muon in the example above. Indeed, the momentum of 3.09 GeV/c is "magic", because of a peculiarity in the formula which connects the spin precession frequency to the muon anomaly and the electric and magnetic field in which the particle is traveling. The difference between precession frequency and cyclotron frequency, $\omega_a$, is given by the formula

$\vec \omega_a = \frac{e}{mc} \left[a_\mu \vec B - \left(a_\mu - \frac{1}{\gamma^2-1} \right) \vec \beta \times \vec E\right]$

where B and E are magnetic and electric field vectors, and $\vec \beta$is the particle velocity vector. Now it so happens that for gamma=29.3 (which corresponds to the magic momentum of 3.09 GeV/c), the term proportional to the electric field vanishes! This simplifies the measurement considerably.

The Discrepancy and QCD Effects

The experimental measurement of the muon anomalous moment performed at the Brookhaven laboratories a few years ago yielded $a_\mu = 116592089 \times 10^{-11}$. Instead, the sum of all contributions computed by theorists amounts to $a_\mu^{th.} = 116591773 \times 10^{-11}$, or$a_\mu^{th.2} = 116591960 \times 10^{-11}$, depending on how you compute the corrections due to quantum chromodynamics (there are in fact two ways to compute the contribution of virtual loops involving quarks and gluons to the muon magnetic moment -more on that below). Accounting for uncertainties, the first estimate is off by three to four standard deviations from the Brookhaven measurement, depending on some details of the computation; the second corresponds instead to just a two-standard-deviation discrepancy.

These QCD effects are non calculable with perturbation theory, because they occur at very low energy, where the value of the strong coupling constant is very large: a series of terms in powers of $\alpha_s$ -corresponding to the evaluation of the simplest diagrams and then correcting the estimate by evaluating the effect of more and more virtual loops (which get increasing factors of $\alpha_s$ as the number of strong-interaction vertices increases) - will not converge.

Given the trouble with perturbation series, theorists revert to quantities that are closely connected to the QCD correction to the diagrams describing the interaction of the muon with the electromagnetic field, which allow to determine its magnetic moment. There are two ways to do this: extrapolating from the measured value of the cross section of electron-positron annihilations yielding pion pairs (a simple-to-measure process) or taking into account tau-lepton decays involving hadrons in the final state. These are the methods that provide the two different theoretical estimates quoted above.

Despite the fact that the electron-positron cross section method provides a much more discrepant value from the experimental estimate than the one based on hadronic tau decays, the two in some way "agree to disagree" with experiment; in addition, recent investigations have demonstrated that the two theoretical calculations are not as discrepant with one another as once thought.

Blaming it on the Hadronic Cross Section

There are many hypotheses to explain the discrepancy between experiment and theory on the muon anomaly. However, one can take the stand that it is entirely due to a underestimate of the hadronic cross section, which is at the heart of the QCD correction.

This hypothesis has a very interesting consequence. The standard model, in fact, is a complex system, in which a small modification anywhere brings consequences everywhere else. Increasing the hadronic cross section produces an increase of the hadronic contribution to the fine-structure constant, and this -in case some not unreasonable assumptions are made- may make the upper limit on the allowed mass range for the Higgs boson from global electroweak fits become smaller than the 114.4 GeV lower limit experimentally established by LEP II! Other assumptions instead move the 95% CL upper limit on the Higgs mass up to a more reasonable 135 GeV.

The authors consider a number of different possibilities to try and bridge the discrepancy between theory and experiment on the muon anomaly. If tau-decay data are used as a basis, the shift in the hadronic parameters bring the upper limit on the Higgs boson to 138 GeV, still much smaller than the one from the "official" one of 158 GeV.

The authors conclude, evidently showing they caress the idea that new physics is actually the source of the discrepancy on the muon anomaly:

If the $\Delta a_\mu$discrepancy is real, it points to "new physics", like low-energy supersymmetry where Da_mu is reconciled by the additional contributions of supersymmetric partners and one expects $M_H < 135 GeV$ for the mass of the lightest scalar. If, instead, the deviaton is caused by an incorrect leading-order hadronic contribution, it leads to reduced upper bounds for the Higgs mass. This reduction, together with the LEP lower bound, leaves a narrow window for the mass of this fundamental particle.

It seems to me that we are in a risky game: either there is supersymmetry -which we may soon discover at the LHC, bringing a new renaissance in particle physics- or we are going to have to wait for at least four years for a Higgs boson discovery, since the Higgs boson, if it is there, is light -exactly what makes it hard to be seen by the ATLAS and CMS experiments.