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Muon Flaws in the Standard Model

by John G. Cramer

Alternate View Column AV-214
Keywords: muon, mu-lepton, B decay, g-2, standard model violation, new physics
Published in the September-October-2021 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 05/11/2021 and is copyrighted ©2021 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.

The muon (or mu lepton or m-) is the famous "Who ordered that?" particle.  It is heavier brother of the electron.  It has the same negative charge and spin ½ as the electron but with a mass about 207 times larger.  Like the electron, it is subject to the weak and electromagnetic forces but not to the strong force.    Unlike the electron, however, the muon is unstable to decay into lighter particles, with a mean lifetime of 2.2 microseconds.  It undergoes a parity-violating 3-body decay into lighter particles: an electron, a mu neutrino, and an electron antineutrino, i.e., m- e- + nm + anti-ne.  The m+, the positive-charge antimatter twin of the m- similarly decays into a positron and two neutrinos, i.e., m+ e+ + anti-nm + ne.

When cosmic rays, mainly in the form of near-lightspeed protons, collide with atoms of the upper atmosphere, the collisions make many negative pi mesons, which decay with a mean lifetime of 26 nanoseconds into a muon and a mu anti-neutrino, i.e., p-m- + anti-nm.  If special relativity were not involved, the resulting muons would decay near where they were produced and never reach the surface of the Earth.  On average, they would travel only about 660 meters before decaying to electrons and neutrinos in the upper atmosphere.

However, because of relativistic time dilation, the internal decay clocks of near-lightspeed muons are slowed.  This greatly extends their lifetimes, allowing them to reach the surface well before they decay.  For this reason, we surface dwellers are bombarded by about one muon per minute for every square centimeter of surface area.  Every second, roughly one muon passes through your head.  Cosmic-ray muons reaching the Earth's surface produced unexpectedly stiff tracks in a large cloud chamber at Caltech, making possible their discovery in 1936 by Carl D. Anderson and my late friend and colleague Seth Neddermeyer.

We are focusing on muons in this AV column because they play an important role in two recent experimental results, both of which appear to be in conflict with the Standard Model (SM) of Particle Physics: (1) the decay of B mesons into muons as observed at the LHC, and (2) the precise measurement of the muon gyro-magnetic factor g, as measured at Fermi Lab.  Perhaps it is only a coincidence that both of these challenges to the SM involve muons, but the correspondence suggests that some significant property of muons, perhaps buried in the details of the weak interaction, is missing from the SM.  We will discuss these measurements in the order above.

The bottom (or beauty) quark is the heaviest member of the quark family having an electrical charge of -1/3, the other family members being the down and strange quarks.  An anti-bottom quark combined with an up quark form the B+ meson, which has a rest mass of 5,279.3 MeV/c2 and a mean lifetime of about 1.6 picoseconds.  With the very small probability of about once in two million decays, the B+ meson decays to a charged K+ meson (mass 493.7 MeV/c2) and two oppositely charged leptons, i.e., either B+K+ + e+ + e- or B+K+ + m+ + m-.  According to the SM, in this decay process the available energy (about 4.8 GeV) is so large that the difference between the e-masses and m-masses must have a negligible effect on the strengths of these two decay branches, and therefore they should be equal.  This SM prediction can be checked experimentally.

The Large Hadron Collider (LHC) at CERN has four major detectors, ATLAS, CMS, ALICE, and LHC-b.  The LHC-b detector, located at beam-intersection Region 8 deep under the town of Ferny-Voltaire, France , is devoted exclusively to the physics of the B meson.  On March 23, 2021, the LHC-b Collaboration announced that they had found in these two B+ meson decay branches a major discrepancy in the ratio of muon to electron decay strength.  The ratio, based on multiple years of data collection at LHC-b, was measured to be R=0.846±0.044 instead of 1.0.  In other words, for unknown reasons the B+ kaon decay into muons is observably weaker than the decay into electrons.  The SM principle of "lepton universality" dictates that the branches are equal.

The reported ratio has a statistical significance of 3.1 standard deviations.  In the physics community, a "solid" result requires a statistical significance of 5 standard deviations, so the result qualifies as an "observation" rather than as "evidence".  Nevertheless, it represents a strong indication that the muon is not behaving as the SM predicts that it should.

This brings us to the other anomaly, the muon's g-2 factor.  We will begin with the gyromagnetic factor g.  Consider a spherical particle with mass m and electric charge q.  If it spins on some axis with an angular momentum L, its moving charge will produce a magnetic field with a dipole moment m.  Classical electromagnetism tells us that the ratio L/m = g(q/2m) will be a constant, independent of spin rate.  Moreover, if the mass and charge are both distributed in the same way within the sphere, the gyromagnetic factor g will be exactly 1.  However, if the mass of the sphere is distributed uniformly within the volume, while the charge is localized on the surface, then g = 5/3 ~ 1.67.  If the mass is uniformly distributed while the charge is concentrated in a surface ring around the equator, then g = 5/2 = 2.5.  In other words, classical physics says that measuring g should tell us something about the particle's geometrical relation between mass and charge.

Naïvely one would expect the mass and charge of leptons to have about the same distribution, leading to g = 1.  Therefore, it is somewhat surprising that for both electrons and muons the experimentally measured value of g is slightly larger than 2.  Does this mean that these particles have some weird relation between their mass and charge distributions?  No.  It means that we cannot use naïve geometrical arguments and classical electromagnetism to describe particles that are quantum-mechanical in nature.

For all charged spin ½ particles, the Dirac equation predicts that g=2.  This is one of several peculiarities of the half-integer-spin particles called "fermions": they all have antimatter twins with opposite mirror-symmetry, they must be rotated by twice 360 degrees to return to their original state, and they have about twice the value of g that one would expect from classical physics.

But that is not the whole story.  For the g of both the electron and the muon, there are many additional higher-order magnetic contributions that together add up to a bit over a 0.2% increase.  Physicists like to characterize this anomalous extra contribution to the gyromagnetic factor as ai = (g-2)/2, where i=e for the electron and i=m for the muon.  The measured value of ae for the electron is ae=0.001 159 652 180 73(28).  It agrees with Standard Model predictions (a summation of 12,672 Feynman diagrams) to more than 10 significant figures, making it the most accurately verified theoretical prediction in all of physics.

  However, it should be noted that the magnetic contribution to ae from processes involving a particle of mass M scales as (me/M)2.  Therefore, contributions for M greater than about 100 electron masses will be very small.  For a m, with mm 207 times larger than me, these contributions will be larger and more significant, providing more sensitivity to new physics.

The measurement of the muon g has been done at both Brookhaven National Lab (2004 publication) and at Fermi Lab (2021 publication), both experiments using the same dedicated magnetic storage ring, which was transported from BNL to Fermi Lab.  In the experiment, an intense accelerated proton beam collides with a target to produce p+ mesons that decay into positive muons.  Muons of a selected momentum range are injected into the storage ring and "kicked" electrically into stable orbits.  They circle the ring before undergoing weak-interaction decays into two neutrinos and a positron.  The positrons are deflected out of the ring and detected by scintillation counters arrayed around the ring.

Because the g of the muon is not precisely 2, as the muons orbit in the storage ring's magnetic field, their spin vectors will precess, changing direction at a rate proportional to am.  Because the weak-interaction decay of the muon violates conservation of parity, the positron tends to be emitted along the changing spin direction, so the preferred decay direction varies as the muons orbit.  This gives the positron rate of detection a sinusoidal variation with time, which the experimenters call a "wiggle plot".  The value of am can be extracted from the wiggle frequency of the plot.  And it is a maxim in physics that the most accurate results are done with frequency measurements.

Accurate extraction of this frequency, which sounds simple, is actually a very complicated process, because it depends slightly on many characteristics of the beam of muons, the storage ring, and the detection apparatus, and these must be corrected.  The experimenters, in the recent announcement, have provided a detailed account of all the many calibrations and corrections that went into their final result.

And what they found was that a m=0.001 165 920 40(54).  When combined with the previous 2004 Brookhaven experimental result, the overall result is a m= 0.001 165 920 61(41).  The best theoretical predictions from the Standard Model give am = 0.001 165 918 10(43).  Therefore the Standard Model and experiment differ by 4.2 standard deviations.  As I said above, a "solid" result requires a statistical significance of 5 standard deviations, so the muon a m discrepancy is also classified as an "observation" rather than "evidence".

Nevertheless, these two "observations" involving the muon, taken together, suggest that the Standard Model of Particle Physics is on the verge of being falsified in the muon weak-interaction sector.  There are already suggestions that what is missing may be an additional "5th force" that has previously been missed, or that some unknown massive particle that couples to leptons but not quarks awaits discovery.

But I am hoping that these experimental results lead to more than just the discovery of a new particle, or even a new force.  The present Standard Model of Particle Physics is rather a paste-up, with two dozen adjustable parameters that lack any fundamental basis. Further, as I have said in previous columns, it has become increasingly apparent that the workhorse theories of QED and QFT, pillars of the Standard Model, have infinities that must be subtracted, are incompatible with General Relativity, can give ridiculously wrong predictions, and are in urgent need of replacement.  I hope that these emerging cracks in the Standard Model will lead to its replacement with a better and more fundamental theory that can explain all of the experimental results while avoiding the pitfalls of the current approach.

Watch this AV column for further developments.

John G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of quantum mechanics, The Quantum Handshake - Entanglement, Nonlocality, and Transactions, (Springer, January-2016) is available online as a hardcover or eBook at: or

SF Novels by John Cramer:  Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at and .  His new novel, Fermi's Question is coming soon from Baen Books.

Alternate View Columns Online: Electronic reprints of 212 or more "The Alternate View" columns by John G. Cramer published in Analog between 1984 and the present are currently available online at: .


B Meson Decay into Muons:

The LHCB Collaboration, "Measurement of CP -averaged observables in the B0→ K0μ+μ decay", preprint ArXiv 2003.04831.

Measurement of the Muon g-2:

B. Abi, et al. (Muon g−2 Collaboration), "Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm", Phys. Rev. Lett. 126, 141801 (2021), LINK.

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