Particle physics – Beyond the standard model

New physics with the muon – the anomalous muon magnetic moment

Dr Gavin Hesketh


Dr Hesketh is an associate professor of physics at University College London. As an experimental particle physicist, his research is focused on the search for physics beyond the standard model. With a background in the collider energy frontier, he now works on precision measurements of muons – including is anomalous magnetic moment. The most recent measurement of this quantity remains one of the most significant disagreements with the standard model, and the ongoing update may yield a definitive sign of new physics.

The talk covered the experimental method, interpretations of the result, and the near future for muon physics. 2016 saw the publication of Dr Hesketh’s first popular book, “The Particle Zoo” (Quercus).

My notes from the lecture (if they don’t make sense then it is entirely my fault)

The Standard Model seems to be complete but there are still lots of problems


1) (a) Hierarchy problem

In theoretical physics, the hierarchy problem is the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 1024 times stronger than gravity.

(b) Higgs mass

Why is the Higgs boson so much lighter than the Planck mass (or the grand unification energy, or a heavy neutrino mass scale)?

In physics, the Planck mass, denoted by mP, is the unit of mass in the system of natural units known as Planck units. It is approximately 0.02 milligrams. Unlike some other Planck units, such as Planck length, Planck mass is not a fundamental lower or upper bound; instead, Planck mass is a unit of mass defined using only what Max Planck considered fundamental and universal units.

(c) Naturalness

In physics, naturalness is the property that the dimensionless ratios between free parameters or physical constants appearing in a physical theory should take values “of order 1” and that free parameters are not fine-tuned. That is, a natural theory would have parameter ratios with values like 2.34 rather than 234000 or 0.000234.

It tends to suggest a possible area of weakness and future development for current theories such as the Standard Model, where some parameters may vary by many orders of magnitude, and which require extensive “fine-tuning” of their current values of the models concerned. The concern is that it is not yet clear whether these seemingly exact values we currently recognize, have arisen by chance or whether they arise from a more advanced theory not yet developed, in which these turn out to be expected and well-explained, because of other factors not yet part of particle physics models.


A neutrino (denoted by the Greek letter n) is a fermion (an elementary particle with half-integer spin) that interacts only via the weak subatomic force and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small (-ino) that it was long thought to be zero. The mass of the neutrino is much smaller than that of the other known elementary particles. The weak force has a very short range, the gravitational interaction is extremely weak, and neutrinos, as leptons, do not participate in the strong interaction. Thus, neutrinos typically pass through normal matter unimpeded and undetected.

The need for a particle like the neutrino was suggested by Pauli in his famous letter in 1930 to account for the continuous energy spectrum of the electrons emitted in nuclear beta-decay.


Although neutrinos were long believed to be massless, it is now known that there are also three discrete neutrino masses; each neutrino flavour is a combination of the three discrete masses. Although only differences of squares of the three mass values are known as of 2016, experiments have shown that these masses are tiny in magnitude. From cosmological measurements, it has been calculated that the sum of the three neutrino masses must be less than one millionth that of the electron.

The Standard Model of particle physics assumed that neutrinos are massless.

A Majorana fermion, also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesized by Ettore Majorana in 1937


Are neutrinos marjorana particles?

In physics, a Dirac fermion is spin ​½ particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category.

Are neutrinos dirac particles?


In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C symmetry) while its spatial coordinates are inverted (“mirror” or P symmetry).

Neutrinos and antineutrinos come in three different flavours—electron, muon, and tau—and can change from one flavour to another. If CP symmetry is respected, then neutrinos and antineutrinos should have the same probability of making this change.

In data collected from 2010 to 2017, 89 electron neutrinos and 7 electron antineutrinos were detected. These counts should have been closer to 68 and 9, respectively, if CP symmetry had been unbroken.

Whether the probability for electron neutrino appearance exceeds, or not, the electron antineutrino appearance probability depends on the value of the CP violating phase, δCP, introduced by Kobayashi and Maskawa. below left

image above right


The strong CP problem is a puzzling question in particle physics: Why does quantum chromodynamics (QCD) seem to preserve CP-symmetry?

In particle physics, CP stands for Charge+Parity or Charge-conjugation Parity symmetry: the combination of charge conjugation symmetry (C) and parity symmetry (P). According to the current mathematical formulation of quantum chromodynamics, a violation of CP-symmetry in strong interactions could occur. However, no violation of the CP-symmetry has ever been seen in any experiment involving only the strong interaction. As there is no known reason in QCD for it to necessarily be conserved, this is a “fine tuning” problem known as the strong CP problem.

The strong CP problem is sometimes regarded as an unsolved problem in physics.

In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion.

4) Structure of the standard model

The model does not explain gravitation

It requires 19 numerical constants whose values are unrelated and arbitrary.

The model is inconsistent with the emerging Lambda-CDM model of cosmology. Contentions include the absence of an explanation in the Standard Model of particle physics for the observed amount of cold dark matter (CDM) and its contributions to dark energy, which are many orders of magnitude too large. It is also difficult to accommodate the observed predominance of matter over antimatter (matter/antimatter asymmetry). The isotropy and homogeneity of the visible universe over large distances seems to require a mechanism like cosmic inflation, which would also constitute an extension of the Standard Model.

Issues of quantum triviality, which suggests that it may not be possible to create a consistent quantum field theory involving elementary scalar particles

The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda (Greek l) and associated with dark energy; second, the postulated cold dark matter (abbreviated CDM); and third, ordinary matter.

So the standard model is not a full description of the universe.

5) Matter/antimatter asymmetry

The Big Bang should have created equal amounts of matter and antimatter in the early universe. But today, everything we see from the smallest life forms on Earth to the largest stellar objects is made almost entirely of matter. Comparatively, there is not much antimatter to be found. Something must have happened to tip the balance. One of the greatest challenges in physics is to figure out what happened to the antimatter, or why we see an asymmetry between matter and antimatter.

6) Dark matter:

Dark matter is a form of matter that is thought to account for approximately 85% of the matter in the universe and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet undiscovered subatomic particles. Its presence is implied in a variety of astrophysical observations, including gravitational effects that cannot be explained by accepted theories of gravity unless more matter is present than can be seen. For this reason, most experts think dark matter to be abundant in the universe and to have had a strong influence on its structure and evolution. Dark matter is called dark because it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible to the entire electromagnetic spectrum, making it extremely difficult to detect using usual astronomical equipment.

7) Dark energy and gravity

In physical cosmology and astronomy, dark energy is an unknown form of energy which is hypothesized to permeate all of space, tending to accelerate the expansion of the universe. Dark energy is the most accepted hypothesis to explain the observations since the 1990s indicating that the universe is expanding at an accelerating rate.

A powerful repulsion between normal matter and hidden pockets of antimatter could be an alternate explanation for the mysterious force known as dark energy, according to a controversial new theory.

Possible solutions:

1) Supersymmetry:

In particle physics, supersymmetry (SUSY) is a principle that proposes a relationship between two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer spin. A type of spacetime symmetry, supersymmetry is a possible candidate for undiscovered particle physics, and seen as an elegant solution to many current problems in particle physics if confirmed correct, which could resolve various areas where current theories are believed to be incomplete. A supersymmetrical extension to the Standard Model would resolve major hierarchy problems within gauge theory, by guaranteeing that quadratic divergences of all orders will cancel out in perturbation theory.

In supersymmetry, each particle from one group would have an associated particle in the other, which is known as its superpartner, the spin of which differs by a half-integer. These superpartners would be new and undiscovered particles. For example, there would be a particle called a “selectron” (superpartner electron), a bosonic partner of the electron.

2) Extra dimensions:

In physics, extra dimensions are proposed additional space or time dimensions beyond the (3 + 1) typical of observed spacetime, such as the first attempts based on the Kaluza–Klein theory. Among theories proposing extra dimensions are:

Large extra dimension, mostly motivated by the ADD model, in an attempt to solve the hierarchy problem. This theory requires that the fields of the Standard Model are confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions that are large compared to the Planck scale.

Warped extra dimensions, based on warped geometry where the universe is a five-dimensional anti-de Sitter space and the elementary particles except for the graviton are localized on a (3 + 1)-dimensional brane or branes.

Universal extra dimension, proposed and first studied in 2000, assume, at variance with the ADD and RS approaches, that all fields propagate universally in the extra dimensions.

Multiple time dimensions, i.e. the possibility that there might be more than one dimension of time, has occasionally been discussed in physics and philosophy, although those models have to deal with the problem of causality.

String theory has one notable feature that requires extra dimensions for mathematical consistency. Spacetime is 26-dimensional in bosonic string theory, 10-dimensional in superstring theory, and 11-dimensional in supergravity theory and M-theory.

In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the usual four of space and time and considered an important precursor to string theory.

3) New particles:

(a) WIMPs

Weakly interacting massive particles (WIMPs) are hypothetical particles that are thought to constitute dark matter. There exists no clear definition of a WIMP, but broadly, a WIMP is a new elementary particle which interacts via gravity and any other force (or forces), potentially not part of the standard model itself, which is as weak as or weaker than the weak nuclear force, but also non-vanishing in its strength. A WIMP must also have been produced thermally in the early Universe, similarly to the particles of the standard model according to Big Bang cosmology, and usually will constitute cold dark matter.

(b) leptoquarks

Leptoquarks are hypothetical particles that carry information between quarks and leptons of a given generation that allow quarks and leptons to interact. They are colour-triplet bosons that carry both lepton and baryon numbers. They are encountered in various extensions of the Standard Model, such as technicolour theories or GUTs based on Pati–Salam model, SU(5) or E6, etc. Their quantum numbers like spin, (fractional) electric charge and weak isospin vary among theories.

(c) vector-like quarks

Vector-like quarks are hypothetical spin ½ particles that transform as triplets under the colour gauge group and whose left- and right-handed components have the same colour and electroweak quantum numbers. They are by nature not affected by the Higgs bounds. A fermion is defined as Vector-like (VL) under a specific gauge group G if its left and right-handed projections belong to the same representation of such a group. Like other quarks, vector-like quarks would be spin-½ particles that interact via the strong force. While all spin-½ particles have left- and right-handed components, the weak force only interacts with the left-handed components of Standard Model particles. However, vector-like quarks would have “ambidextrous” interactions with the weak force, giving them a bit more leeway in how they decay. While the Standard Model top quark always decays to a bottom quark (b) by emitting a W boson (t→Wb), a vector-like top can decay three different ways: T→Wb, T→Zt or T→Ht


VLQs increase corrections to Higgs mass. Interactions at the quantum level between the Higgs boson and the top quark ought to lead to a huge Higgs boson mass, possibly as large as the Planck mass (>1018 GeV). So why is it only 125 GeV?

4) New forces/bosons

Yet-to-be-discovered bosons may solve several outstanding puzzles. The axion (a spin-0 boson) may explain the apparent absence of CP violation in strong interactions. The observed dark matter and dark energy may also be explained by the existence of new bosonic particles. The possibility to solve such central questions motivates numerous searches for new bosons.

In particle physics, W′ and Z′ bosons (or W-prime and Z-prime bosons) refer to hypothetical gauge bosons that arise from extensions of the electroweak symmetry of the Standard Model. They are named in analogy with the Standard Model W and Z bosons.

The dark photon (also hidden, heavy, para-, secluded photon, or phaeton) is a hypothetical hidden sector particle, proposed as a force carrier similar to the photon of electromagnetism but potentially connected to dark matter. In a minimal scenario, this new force can be introduced by extending the gauge group of the Standard Model of Particle Physics.

Higgs Triplet Models Þ Can provide Majorana neutrinos with masses

5) Quark substructure

In particle physics, preons are point particles, conceived of as sub-components of quarks, and leptons. The word was coined by Jogesh Pati and Abdus Salam, in 1974. Interest in preon models peaked in the 1980s but has slowed, as the Standard Model of particle physics continues to describe the physics, mostly successfully, and no direct experimental evidence for lepton and quark compositeness has been found.

In the hadronic sector, some effects are considered anomalies within the Standard Model. For example, the proton spin puzzle, the EMC effect, the distributions of electric charges inside the nucleons, as found by Hofstadter in 1956, and the ad hoc CKM matrix elements.

The EMC effect is the surprising observation that the cross section for deep inelastic scattering from an atomic nucleus is different from that of the same number of free protons and neutrons (collectively referred to as nucleons).

When the term “preon” was coined, it was primarily to explain the two families of spin-½ fermions: leptons and quarks. More recent preon models also account for spin-1 bosons, and are still called “preons”. Each of the preon models postulates a set of fewer fundamental particles than those of the Standard Model, together with the rules governing how those fundamental particles combine and interact. Based on these rules, the preon models try to explain the Standard Model, often predicting small discrepancies with this model and generating new particles and certain phenomena, which do not belong to the Standard Model.

6) Sterile neutrinos

Sterile neutrinos (or inert neutrinos) are hypothetical particles (neutral leptons – neutrinos) that interact only via gravity and do not interact via any of the fundamental interactions of the Standard Model. The term sterile neutrino is used to distinguish them from the known active neutrinos in the Standard Model, which are charged under the weak interaction.

This term usually refers to neutrinos with right-handed chirality, which may be added to the Standard Model. Occasionally it is used in a general sense for any neutral fermion, instead of the more cautiously vague name neutral heavy leptons (NHLs) or heavy neutral leptons (HNLs).

The existence of right-handed neutrinos is theoretically well-motivated, as all other known fermions have been observed with both left and right chirality, and they can explain the observed active neutrino masses in a natural way. The mass of the right-handed neutrinos themselves is unknown and could have any value between 1015 GeV and less than 1 eV.

The number of sterile neutrino types (should they exist) is not yet theoretically established. This is in contrast to the number of active neutrino types, which has to equal that of charged leptons and quark generations to ensure the anomaly freedom of the electroweak interaction.

The search for sterile neutrinos is an active area of particle physics. If they exist and their mass is smaller than the energies of particles in the experiment, they can be produced in the laboratory, either by mixing between active and sterile neutrinos or in high energy particle collisions. If they are heavier, the only directly observable consequence of their existence would be the observed active neutrino masses. They may, however, be responsible for a number of unexplained phenomena in physical cosmology and astrophysics, including dark matter, baryogenesis or dark radiation. In May 2018, physicists of the MiniBooNE experiment reported a stronger neutrino oscillation signal than expected, a possible hint of sterile neutrinos.

MiniBooNE is an experiment at Fermilab designed to observe neutrino oscillations (BooNE is an acronym for the Booster Neutrino Experiment). A neutrino beam consisting primarily of muon neutrinos is directed at a detector filled with 800 tons of mineral oil (ultrarefined methylene compounds) and lined with 1,280 photomultiplier tubes.

7) axions/ALPS

The axion is a hypothetical elementary particle postulated by the Peccei–Quinn theory in 1977 to resolve the strong CP problem in quantum chromodynamics (QCD). If axions exist and have low mass within a specific range, they are of interest as a possible component of cold dark matter.

Any Light Particle Search (ALPS) experiment is helping DESY researchers to track down such lightweights of the subatomic world.

The QCD axion was originally predicted as a dynamical solution to the strong CP problem. Axion like particles (ALPs) are also a generic prediction of many high energy physics models including string theory.

8) Sphalerons, majorons and familions

A sphaleron is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and is involved in certain hypothetical processes that violate baryon and lepton numbers. Such processes cannot be represented by perturbative methods such as Feynman diagrams, and are therefore called non-perturbative. Geometrically, a sphaleron is a saddle point of the electroweak potential (in infinite-dimensional field space).

Since a sphaleron may convert baryons to antileptons and antibaryons to leptons and thus change the baryon number, if the density of sphalerons was at some stage high enough, they could wipe out any net excess of baryons or anti-baryons. This has two important implications in any theory of baryogenesis within the Standard Model

Mathematically, majorons may be modelled by allowing them to propagate through a material while all other Standard Model forces are fixed to an orbifold point.

I haven’t been able to find out what a familion actually is. Scientific papers include them but the only things I have been able to find out are that it is a very light and weakly coupled pseudo-Goldstone boson and it could be massless.

In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries.

If the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then the Nambu–Goldstone bosons are not massless, though they typically remain relatively light; they are then called pseudo-Goldstone bosons or pseudo-Nambu–Goldstone bosons (abbreviated PNGBs).

9) String theory and loop quantum gravity

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.

Extensions, revisions, replacements, and reorganizations of the Standard Model exist in attempt to it. String theory is one such reinvention, and many theoretical physicists think that such theories are the next theoretical step toward a true Theory of Everything.

Among the numerous variants of string theory, M-theory, whose mathematical existence was first proposed at a String Conference in 1995, is believed by many to be a proper “ToE” candidate, notably by physicists Brian Greene and Stephen Hawking.

An important property of string theory is its supersymmetry, which together with extra dimensions are the two main proposals for resolving the hierarchy problem of the standard model, which is (roughly) the question of why gravity is so much weaker than any other force. The extra-dimensional solution involves allowing gravity to propagate into the other dimensions while keeping other forces confined to a four-dimensional spacetime, an idea that has been realized with explicit stringy mechanisms.

Loop quantum gravity (LQG) is a theory of quantum gravity, attempting to merge quantum mechanics and general relativity, while incorporating the standard model particles. It takes seriously the key insight from general relativity that space-time is a dynamic entity, not a fixed framework. It competes with string theory, another candidate for a theory of quantum gravity. However, unlike string theory, LQG is not a candidate for a theory of everything – the goal of which is to explain all of particle physics, unifying gravity with the other forces at the same time. In contrast to LQG, string theory (for the most part) is background-dependent (built on a fixed framework) and doesn’t account for the dynamic nature of space-time at the heart of relativity.

10) Some new particles still to be discovered!!!!!!!!!!!!!!!!!!!!!!!!!

All this search for new physics does find its way into the press.

With so many BSM theories which do you search for?

The traditional approach is to target the search for maximum sensitivity to the specific modes


Ask a slightly different question – Does the standard model still work using data?

Recast Contour Gambit

Recast the analysis

How to validate new theories – use a global statistical fit

To put it simply does the standard model still work

The LHC is good for energies up to 14TeV however to look for the new physics we need a future collider that would require energies up to 100TeV

For neutrinos we can use astronomical data

The muon

The muon (from the Greek letter mu (μ) used to represent it) is an elementary particle similar to the electron, with an electric charge of −1 e and a spin of 1/2, but with a much greater mass. It is classified as a lepton. As is the case with other leptons, the muon is not believed to have any sub-structure—that is, it is not thought to be composed of any simpler particles.

Muons were discovered by Carl D. Anderson and Seth Neddermeyer at Caltech in 1936, while studying cosmic radiation. Anderson noticed particles that curved differently from electrons and other known particles when passed through a magnetic field. They were negatively charged but curved less sharply than electrons, but more sharply than protons, for particles of the same velocity. It was assumed that the magnitude of their negative electric charge was equal to that of the electron, and so to account for the difference in curvature, it was supposed that their mass was greater than an electron but smaller than a proton. Thus Anderson initially called the new particle a mesotron, adopting the prefix meso- from the Greek word for “mid-“. The existence of the muon was confirmed in 1937 by J. C. Street and E. C. Stevenson’s cloud chamber experiment.


Carl David Anderson (September 3, 1905 – January 11, 1991) was an American physicist. He is best known for his discovery of the positron in 1932, an achievement for which he received the 1936 Nobel Prize in Physics, and of the muon in 1936.


Seth Henry Neddermeyer (September 16, 1907 – January 29, 1988) was an American physicist who co-discovered the muon, and later championed the Implosion-type nuclear weapon while working on the Manhattan Project at the Los Alamos Laboratory during World War II.

Jabez Curry Street (May 5, 1906 – November 7, 1989) was an American physicist, a co-discoverer of atomic particles called muons.

A cloud chamber, also known as a Wilson cloud chamber, is a particle detector used for visualizing the passage of ionizing radiation.


Paths of a muon

A muon is a second generation fundamental fermion. It is 207 times heavier than an electron and like the electron it is a lepton.

Muons are unstable charged particles with a mean life time of 2.2 µs (the second longest after the neutrons). They can be negatively and positively charged and decay to ≈100% into an electron and two neutrinos.

Muon g−2 (pronounced “gee minus two”) is a particle physics experiment at Fermilab to measure the anomalous magnetic dipole moment of a muon to a precision of 0.14 ppm, which will be a sensitive test of the Standard Model. It might also provide evidence of the existence of entirely new particles.


The g−2 blue storage-ring magnet at Fermilab, which was originally designed for the Brookhaven g−2 experiment. The geometry allows for a very uniform magnetic field to be established in the ring. The diameter is 15m

The muon, like its lighter sibling the electron, acts like a spinning magnet. The parameter known as the “g-factor” indicates how strong the magnet is and the rate of its gyration. The value of g is slightly larger than 2, hence the name of the experiment. This difference from 2 (the “anomalous” part) is caused by higher-order contributions from quantum field theory. In measuring g−2 with high precision and comparing its value to the theoretical prediction, physicists will discover whether the experiment agrees with theory. Any deviation would point to as yet undiscovered subatomic particles that exist in nature

Fermilab is continuing an experiment conducted at Brookhaven National Laboratory to measure the anomalous magnetic dipole moment of the muon. The Brookhaven experiment ended in 2001, but ten years later Fermilab acquired the equipment, and is working to make a more accurate measurement (smaller σ) which will either eliminate the discrepancy or confirm it as an experimentally observable example of physics beyond the Standard Model.

In physics, Larmor precession is the precession of the magnetic moment of an object about an external magnetic field. Objects with a magnetic moment also have angular momentum and effective internal electric current proportional to their angular momentum; these include the muon


Direction of precession for a negatively-charged particle. The large arrow indicates the external magnetic field, the small arrow the spin angular momentum of the particle. The muon behaves like a spinning top/a bit like a gyroscope

The angular momentum vector J precesses about the external field axis with an angular frequency known as the Larmor frequency ws = -gqB/2m where g is the g-factor of the system, -q is the charge of the muon, B is the magnitude of the applied magnetic field and m is the mass of the precessing system.

The spin angular momentum of a muon precesses counter-clockwise about the direction of the magnetic field. A muon has a negative charge, so the direction of magnetic moment is opposite to that of its spin.

Dirac predicted that g would be 2 in 1928.


The magnetic moment of charged leptons: – exactly 2 at tree level (Dirac’s prediction)

Kush and Foley in 1948 found that g for an electron was not 2 but ge = 2.00238(6)] below




In the framework of relativistic quantum mechanics, ws = -gqB/2m                                          = (2 + 2a)qB/2m where g becomes 2 + 2a

In physics, the fine-structure constant, also known as Sommerfeld’s constant, commonly denoted by (the Greek letter alpha), is a dimensionless physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles.

First loop calculated by Schwinger in 1948 g = 2 + a/2+ ….

Revised and Improved Value of the QED Tenth-Order Electron Anomalous Magnetic Moment – Tatsumi Aoyama, Toichiro Kinoshita, Makiko Nio

State of the art: O(5) in QED


The most current value for an electron is 2.00231930436256(35) with an uncertainty of 1.7×10−13

For the muon g is 2.0023318418(13) with an uncertainty of 6.3 x 10−10

A recent measurement of a 1/a = 137.035999046(27)

→ new prediction of ae = 0.00115965218161(23) compared to measured ae = 0.00115965218073(28)

→ 2.5σ difference

Electrons are dominated by QED

Muons, having a higher mass than electrons also have QCD and electroweak loops contributing as well as QED

In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion.

Anomalous magnetic moment of the muon with dynamical QCD + QED

Electroweak Fermion-loop Contributions to the Muon Anomalous Magnetic Moment

An observable is a physical quantity that can be measured and compared with a prediction. For the present global analysis, 30 observables have been calculated and then measured in particle physics experiments on B meson decays.

In 2013, LHCb decided to measure for the first time these observables, and found a 3.7 sigma discrepancy with the Standard Model. LHCb confirmed this tension in 2015 with more data.

– a = 0.00116592089(63) (measured)

– a ~ 0.00116591821(36) (prediction)

PRD 73(2006)072003, KNT18, PRD97, 114025

→ 3.7σ difference

Are there more particles in the loops? Supersymmetry etc.

As written earlier the first muon g-2 experiments were carried out at Brookhaven. The move occurred in 2013 and it took a month to fit the apparatus into its new home. It is always a good idea to do experiments again but better. First run occurred in 2018.

Alexander Keshavarzi, Daisuke Nomura and Thomas Teubner

This work presents a complete re-evaluation of the hadronic vacuum polarisation contributions to the anomalous magnetic moment of the muon.

The anomalous magnetic moment of the muon stands as an enduring test of the Standard Model (SM), where the ∼ 3.5σ (or higher) discrepancy between the experimental measurement and the SM prediction could be an indication of the existence of new physics beyond the SM.

There are 185 people involved in this research. Muons are put in a magnetic field and the precession frequency is measured.

A beam of muons with aligned spins is directed into a storage ring that has a very precisely known magnetic field. As the beam goes around this storage ring, the muons’ spins wobble, or precess. Scientists measure the rate that they precess very precisely. The magnitude of that precession is directly related to the difference of g from 2, or g-2.

Cyclotron frequency:


Use “magic momentum” 3.09 GeV

Muon g-2 begins second run MARCH 26, 2019

The Current Status of the Fermilab Muon g–2 Experiment

Muons are unstable elementary particles and are heavier than electrons and neutrinos but lighter than all other matter particles. They decay via the weak interaction. Because leptonic family numbers are conserved in the absence of an extremely unlikely immediate neutrino oscillation, one of the product neutrinos of muon decay must be a muon-type neutrino and the other an electron-type antineutrino (antimuon decay produces the corresponding antiparticles, as detailed below). Because charge must be conserved, one of the products of muon decay is always an electron of the same charge as the muon (a positron if it is a positive muon). Thus all muons decay to at least an electron, and two neutrinos. Sometimes, besides these necessary products, additional other particles that have no net charge and spin of zero (e.g., a pair of photons, or an electron-positron pair), are produced.

The dominant muon decay mode (sometimes called the Michel decay after Louis Michel) is the simplest possible: the muon decays to an electron, an electron antineutrino, and a muon neutrino. Antimuons, in mirror fashion, most often decay to the corresponding antiparticles: a positron, an electron neutrino, and a muon antineutrino. In formulaic terms, these two decays are:


Observation of such decay modes would constitute clear evidence for theories beyond the Standard Model.

COMET (COherent Muon to Electron Transition) is currently a funded experiment in J-PARC, Tokai, Japan. In contrast to the usual muon decay to an electron and neutrino, COMET seeks to look for neutrinoless muon to electron conversion, where the electron flies away with an energy of 104.8 MeV.

Mu2e, or the Muon-to-Electron Conversion Experiment, is a particle physics experiment at Fermilab in the US. The goal of the experiment is to identify physics beyond the Standard Model, namely, the conversion of muons to electrons without the emission of neutrinos, which occurs in a number of theoretical models.

As the spin precesses the positron momentum will vary

Count positrons above a certain threshold – produce a wiggle plot



Improve the wiggle by using Fourier transforms

The Fourier transform (FT) decomposes a function of time (a signal) into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes.


Include vertical and horizontal beam motion, pile-up, muon losses and energy scale

The process is like doing a thousand piece jigsaw puzzle with one piece missing.

The Fermilab target is to reduce experimental uncertainty to 140 ppb. The precision relies on mastering the system.

Runs in 2019/20 will accumulate ~20 x BNL

→ could push significance to ~5-10σ


The above graph shows the target for the end of the run

Before the Higgs discovery there were obvious signs of a new particle – mass couplings.

With the muon g-2 there is an obvious sign of something new – but what?

Planned g-2 experiment at J-PARC

– provide completely independent measurement

Muon g-2 Theory Initiative underway


In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic field. It is also sometimes referred to as the self-energy of the gauge boson (photon).

The hadronic vacuum polarisation contributes to the anomalous magnetic moment of the muon


Light-by-light scattering is a very rare phenomenon in which two photons – particles of light – interact, producing another pair of photons. … These interactions are known as ultra-peripheral collisions. Hadronic light-by-light scattering contribution to the muon magnetic anomaly: Constituent quark loops and QCD effects



– Needs μ > 0, `light’ SUSY-scale Λ and/or large tan b – …excluded by LHC for simplest (like CMSSM)

– causes large χ2 in simultaneous SUSY-fits with LHC data and g-2

– However, SUSY does not have to be minimal

– could have large mass splittings (with lighter sleptons), be hadrophobic/leptophilic

Many other ideas out there, eg:

– 2 Higgs doublet model, Stockinger et al., JHEP 1701 (2017) 007

The data seems to be matching well with the Standard Model (SM) predictions. However there is a strong belief due to many unanswered questions like the Dark matter, Neutrino masses and mixings, Hierarchy problem, Strong CP-problem that physics beyond SM must exist. Two-Higgs-doublet model (2HDM) is one of the simplest extensions of the SM. 2HDM models are one of the natural choices for beyond SM models containing two Higgs doublets instead of just one. There are also models with more than two Higgs doublets example three Higgs doublet models etc.

– 1 TeV Leptoquark Bauer + Neubert, PRL 116 (2016)

Leptoquarks are hypothetical particles that carry information between quarks and leptons of a given generation that allow quarks and leptons to interact.

More recent studies, performed at the LHC, have raised the excluded range to about 1 TeV. For leptoquarks to be proven to exist, the missing energy in particle collisions attributed to neutrinos would have to be excessively energetic. It is likely that the creation of leptoquarks would mimic the creation of massive quarks – single new scalar could solve g-2, B-factory anomalies and still satisfy limits from LEP and LHC…

– axion-like particle contributing like p0 (neutral pion) in HLBL Marciano et al, PRD 94 (2016) 115033

– inevitably, a dark photon e.g. Feng et al, PRL 117 (2016) 071803

The dark photon (also hidden, heavy, para-, secluded photon, or phaeton) is a hypothetical hidden sector particle, proposed as a force carrier similar to the photon of electromagnetism but potentially connected to dark matter.

If the tension is resolved it will set tight limits on these new physics scenarios.

Complementary experiments are needed:

Electric dipole moments

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system’s overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, the most commonly used unit in atomic physics and chemistry is the debye (D).

Not to be confused with spin which refers to the magnetic dipole moments of particles, much experimental work is continuing on measuring the electric dipole moments (EDM) of fundamental and composite particles, namely those of the electron, muon and neutron, respectively. As EDMs violate both the parity (P) and time-reversal (T) symmetries, their values yield a mostly model-independent measure of CP-violation in nature (assuming CPT symmetry is valid). Therefore, values for these EDMs place strong constraints upon the scale of CP-violation that extensions to the standard model of particle physics may allow. Current generations of experiments are designed to be sensitive to the supersymmetry range of EDMs, providing complementary experiments to those done at the LHC.

Indeed, many theories are inconsistent with the current limits and have effectively been ruled out, and established theory permits a much larger value than these limits, leading to the strong CP problem and prompting searches for new particles such as the axion.

If EDM exists then it is an additional source of evidence to improve the standard model

Charged lepton flavour violation

A Lepton Flavour Violation (LFV) is a transition between e, m, t sectors that don’t conserve lepton family number.

In the charged lepton sector Lepton Flavour Violation is heavy suppressed in the Standard Model

Example of lepton flavour conservation is a muon decay


Example of CLFV: neutrinoless muon decay


Many BSM models include charged lepton flavour violation

– leptoquarks, compositeness, Higgs doublets, heavy neutrinos…

…or invoke it for leptogenesis of matter-antimatter asymmetry

Heavy mediator → low rate process

– like beta decay with the massive W boson


Neutrino oscillations violate lepton flavour conservation → technically possible in charged lepton sector …but suppressed by ~10-50 Put one of these models in a loop, rate may increase… There is no “floor”! – current limits ~10-12 – sensitivity purely experimental limitation → any observation of cLFV is new physics!

One CLFV interaction in 107 muon decay is like looking for one specific grain of sand

Mu3e @ PSI is 2m long

– phase 1 (2020) & 2 (2026) – aiming for x 104 on limit

→ 10-16 after phase 2

11 institutes, 60 collaborators – Liverpool, Bristol, Oxford, UCL



Central tracker: Four layers; Re-curl tracker: Two layers


Pixel dimension: 80 x 80 μm2, Thickness: 50 μm,

The LHC will run until about 2040. The idea is to produce smaller, quicker experiments. Push the sensitivity further. Maintain diversity of research skills in the field.

Muon physics is one of the most exciting areas for the coming decade.

Dirac said the g factor for a fundamental particle should be 2 so the fact that for a muon it is greater than 2 indicates that the muon has an internal structure.

A proton has g = 5 which indicated that it wasn’t fundamental before experiments proved it wasn’t.

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