The muon anomaly and the Higgs mass – part I June 10, 2008Posted by dorigo in physics, science.
Tags: Higgs boson, muon anomaly, QCD
Note: despite the technical nature of the matter, I have made an effort to keep this post to a level simple enough that non-scientists should be able to handle. Feedback is welcome!
Nowadays when you are presented with a statement about the inner consistency of the Standard Model of particle physics (SM), and on the range of mass values that a neutral Higgs boson may possess in order to
fit the observed value of several fundamental quantities -all related in a non-trivial way among each other- you are entitled to raise both eyebrows.
Indeed, theorists today speak of the SM as a still-dead entity, because they know it cannot be the ultimate theory. They say it only describes things so well because we have not tested it at energies and forces large
enough. They are quick to point out, if requested (or even without any prodding) that the SM is just “an effective theory”, meaning the opposite (with theorists this happens, at times -it is a lingo barrier rather than backward thinking). They may add that the SM fails to explain the smallness of the mass of the Higgs boson (which has in earnest no apparent reason to be as light as the model wants it), and it does not grant a high-energy unification of fundamental forces in a way which is pleasing to their eye, with the three coupling constants meeting at a single, very large energy scale.
The Standard Model does have those shortcomings. But it has survived more than thirty years of painstaking scrutiny. So your eyebrows have to come down once you realize that, despite all caveats, the predictive power of the combination of existing theory and excellent determination of its free parameters is astonishing. It ain’t no string theory!
There are dozens, but one might say hundreds, of experimental predictions that can be worked out, only to find the SM in exceedingly good health. It is thus not surprising that a handful of these predictions has shown some nagging disagreement with the data in the past. Among them, one might quote a few that are still unresolved today (each of them representing a deviation of measurement from theory by roughly two to three standard deviations): if you accept a list without explanation, I may quote the inconsistency of the measured value of the W boson mass with the observed ratio between neutrino charged-current and neutral-current interactions measured by the NuTeV experiment; the Z boson asymmetry measured by LEP, which shows a difference when measured in leptonic versus hadronic decays; the branching ratio of decays; and the anomalous magnetic moment of the muon.
The anomalous magnetic moment of leptons
A magnetic moment is a property of charged particles with a non-zero value of spin. Although quantum mechanics prevents us from drawing a perfect analogy, a spinning charged sphere develops a magnetic field, and so do charged elementary particles. For them the magnetic moment is easily computed as the product of charge by spin, divided by mass.
The so-called gyromagnetic ratio is then a pure number defined as the magnetic moment computed in units of its charge divided by twice its mass: for electrons (where is the electron charge). The magnitude of describes the magnetic properties of the electron. All charged leptons have gyromagnetic ratios very close to 2, but not quite equal to it. They are exactly 2 in Dirac’s theory of charged fermions, but quantum corrections cause a shift. The deviation of g from 2 -its residual from it- is called anomaly, and it is universally recognized as a very important number, . It is a crucial quantity in electrodynamics, and in particle physics in general, because it is a very small number which can be measured directly, and small non-zero numbers can usually be measured with high accuracy.
Indeed, the electron anomaly is measured with exquisite precision: it is found that , or to within 0.24 parts per billion! It is by its measurement that we know the value of the fine structure constant, -the fundamental quantity of quantum electrodynamics. Theoretical predictions for can be computed to fractions of a billionth too, so a direct comparison of the two provides a spectacular test of our understanding of the underlying physics.
For muons, has been measured with accuracy better than five parts in ten millions, and here theory and measurement differ by 3.4 standard deviations. A paper by Massimo Passera and collaborators, which I will describe in detail tomorrow, discusses the discrepancy and critically analyzes it in terms of the consistency of electroweak fits and low-energy measurements used as input. What I want to do today is to give some preliminary information about the problem, so that I have a chance of explaining to outsiders the details tomorrow.
Calculating the muon anomaly
So, what is it exactly that goes in the calculation of the muon anomaly? Well, it boils down to adding together the contributions of different processes which modify the Dirac picture of a muon (whose momentum is labeled “p” and then “p'”) emitting a photon, as in the graph on the right. At “leading order” -that is, when ignoring everything else but the bare-bone electromagnetic process of photon emission- the gyromagnetic ratio is 2 and the anomaly is zero. However, in subatomic physics every process that is not forbidden will happen, with a certain probability which is the square of the “amplitude” of the corresponding particle diagram. Looking beyond the bare-bone process, at “higher order” one needs to consider a huge number of other processes, such as the emission of a second photon by the muon line, with the former subsequently reabsorbed by the muon after the emission of the main outgoing photon, as in the top left graph of the figure below.
As the number of allowed vertices increases, there may be two photon emissions, and fancier things may start happening, as shown below:
Here we have to count at the same order (because they have the same number of vertices -points where three lines meet) diagrams where a single photon is emitted and reabsorbed, but the photon spends some time in the form of a virtual loop of charged leptons, as in the two graphs shown in the lower right above. At still higher orders the diagrams to consider are many more, but they respect the general structure with lines and blobs like those shown in the figures above.
Similarly, we can imagine that the incoming muon emits and reabsorbs a virtual Z boson; or the muon may emit a W boson and temporarily turn into a muon neutrino, as in the diagram in the center in the figure below. It is only by computing each and every possible virtual diagram, with all known particle interactions, that the total quantity we have to compute -the muon anomaly- comes out right.
Of course, the number of diagrams diverges as the number of “vertices” (points where particle lines intersect) increases. But physicists are good at performing approximations: by organizing the “higher order” corrections in series of the number of vertices, they can prove that each additional term in the series is a small correction to the former. So they just continue calculating more and more complex diagrams until they have to give up (since the number of diagrams to be computed grows factorially with the number of vertices), and their final result will be good to within the estimated contribution of the first neglected term in the series.
From the “nuts and bolts” description I gave above you have by now realized that in the “master” diagram we considered -photon emission from a charged muon line-, strong and electroweak physics enter by necessity at higher order in the perturbation series. Thanks to their electrical charge, even quarks may be produced by a photon fluctuation, and quarks are subject to strong interactions: what they may do, while they are alive in the red blob shown in the graph on the right, needs to concern us. The strength of those interactions will affect the final result for the muon anomaly despite the virtual nature of the quarks! The same goes with W and Z bosons which a muon line can emit (W bosons also connect to photon lines, thanks to their electric charge).
Thus,in the calculation of higher orders of the muon anomaly, there enter not only electrodynamics (which we claim to know inside-out), but also weak and strong interactions: the former are those mediated by the exchange of weak vector bosons (W and Z), the latter are instead those governing the dynamics of quarks -the constituents of nuclear matter- and gluons, the carrier of strong force.
The weakness of an interaction means that as we go to higher orders in the perturbation series diagrams with more vertices become very improbable, and the corrections they cause become small very quickly: the
series converges, and we can calculate it [Post-scriptum: The series does not actually converge – this is a mistake I prefer to not correct, see the comments thread below- but the calculation does work for quantum electrodynamics!]. But for quantum chromodynamics -QCD, the theory of strong interactions- this unfortunately does not happen! Alas, the basic QCD processes we need to compute are
“non perturbative”: higher-order contributions are large and cannot be neglected, no matter how you reorganize your perturbation series.
QCD is a wonderful theory, and high-energy processes can be computed with it with great precision, because at high energy the strong coupling constant (a number by which the probability of any QCD process has to be multiplied once for every particle vertex in the diagram describing the process) is small; but at low energy is large, and perturbation series diverge.
Because of that nasty property of QCD, a calculation of the muon anomaly needs to rely on approximations, modeling, and the knowledge of low-energy QCD. Some processes that help derive the quantities which are used in these approximations are those we measure in low-energy electron-positron scattering experiments: the cross section of these reactions determine how strong an impact some QCD virtual diagrams have in the muon g-2 calculation.
Ok, I think I have dumped above the preliminary information one needs to have in order to read the next, I hope enlightening, post, which will discuss the recent analysis by Massimo Passera and collaborators. In their paper they explain how the upper theoretical bound on the mass of the Higgs boson depends on the amount of uncertainty in low-energy hadronic cross sections one is willing to allow. Those who can’t wait for the post, and can read a hep-ph paper without assistance, are encouraged to get it here.
UPDATE: This link will bring you to the second part of this post.