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The P5 Report & The Future of Particle Physics (Part 1)

Particle physics is the epitome of ‘big science’. To answer our most fundamental questions out about physics requires world class experiments that push the limits of whats technologically possible. Such incredible sophisticated experiments, like those at the LHC, require big facilities to make them possible,  big collaborations to run them, big project planning to make dreams of new facilities a reality, and committees with big acronyms to decide what to build.

Enter the Particle Physics Project Prioritization Panel (aka P5) which is tasked with assessing the landscape of future projects and laying out a roadmap for the future of the field in the US. And because these large projects are inevitably an international endeavor, the report they released last week has a large impact on the global direction of the field. The report lays out a vision for the next decade of neutrino physics, cosmology, dark matter searches and future colliders. 

P5 follows the community-wide brainstorming effort known as the Snowmass Process in which researchers from all areas of particle physics laid out a vision for the future. The Snowmass process led to a particle physics ‘wish list’, consisting of all the projects and research particle physicists would be excited to work on. The P5 process is the hard part, when this incredibly exciting and diverse research program has to be made to fit within realistic budget scenarios. Advocates for different projects and research areas had to make a case of what science their project could achieve and a detailed estimate of the costs. The panel then takes in all this input and makes a set of recommendations of how the budget should be allocated, what should projects be realized and what hopes are dashed. Though the panel only produces a set of recommendations, they are used quite extensively by the Department of Energy which actually allocates funding. If your favorite project is not endorsed by the report, its very unlikely to be funded. 

An computer generate graphic showing two sprays of particles being injected from a single center point
The P5 report also created an awesome graphic that contains symbolism for the different aspects of the plan. The left  side depicts oscillating neutrinos and the shape of the Higgs potential. The right side depicts dark matter and the large scale structure of the universe. The central ball of light is supposed to represent discoveries of the unknown. Source

Particle physics is an incredibly diverse field, covering sub-atomic to cosmic scales, so recommendations are divided up into several different areas. In this post I’ll cover the panel’s recommendations for neutrino physics and the cosmic frontier. Future colliders, perhaps the spiciest topic, will be covered in a follow up post.

The Future of Neutrino Physics

For those in the neutrino physics community all eyes were on the panels recommendations regarding the Deep Underground Neutrino Experiment (DUNE). DUNE is the US’s flagship particle physics experiment for the coming decade and aims to be the definitive worldwide neutrino experiment in the years to come. A high powered beam of neutrinos will be produced at Fermilab and sent 800 miles through the earth’s crust towards several large detectors placed in a mine in South Dakota. Its a much bigger project than previous neutrino experiments, unifying essentially the entire US community into a single collaboration.

DUNE is setup to produce world leading measurements of neutrino oscillations, the property by which neutrinos produced in one ‘flavor state’, (eg an electron-neutrino) gradually changes its state with sinusoidal probability (eg into a muon neutrino) as it propagates through space. This oscillation is made possible by a simple quantum mechanical weirdness: neutrino’s flavor state, whether it couples to electrons muons or taus, is not the same as its mass state. Neutrinos of a definite mass are therefore a mixture of the different flavors and visa versa.

Detailed measurements of this oscillation are the best way we know to determine several key neutrino properties. DUNE aims to finally pin down two crucial neutrino properties: their ‘mass ordering’, which will solidify how the different neutrino flavors and measured mass differences all fit together, and their ‘CP-violation’ which specifies whether neutrinos and their anti-matter counterparts behave the same or not. DUNE’s main competitor is the Hyper-Kamiokande experiment in Japan, another next-generation neutrino experiment with similar goals.

A depiction of the DUNE experiment. A high intensity proton beam at Fermilab is used to create a concentrated beam of neutrinos which are then sent through 800 miles of the Earth’s crust towards detectors placed deep underground South Dakota. Source

Construction of the DUNE experiment has been ongoing for several years and unfortunately has not been going quite as well as hoped. It has faced significant schedule delays and cost overruns. DUNE is now not expected to start taking data until 2031, significantly behind Hyper-Kamiokande’s projected 2027 start. These delays may lead to Hyper-K making these definitive neutrino measurements years before DUNE, which would be a significant blow to the experiment’s impact. This left many DUNE collaborators worried about its broad support from the community.

It came as a relief then when P5 report re-affirmed the strong science case for DUNE, calling it the “ultimate long baseline” neutrino experiment. The report strongly endorsed the completion of the first phase of DUNE. However, it recommended a pared-down version of its upgrade, advocating for an earlier beam upgrade in lieu of additional detectors. This re-imagined upgrade will still achieve the core physics goals of the original proposal with a significant cost savings. With this report, and news that the beleaguered underground cavern construction in South Dakota is now 90% complete, was certainly welcome holiday news to the neutrino community. This is also sets up a decade-long race between DUNE and Hyper-K to be the first to measure these key neutrino properties.

Cosmic Implications

While we normally think of particle physics as focused on the behavior of sub-atomic particles, its really about the study of fundamental forces and laws, no matter the method. This means that telescopes to study the oldest light in the universe, the Cosmic Microwave Background (CMB), fall into the same budget category as giant accelerators studying sub-atomic particles. Though the experiments in these two areas look very different, the questions they seek to answer are cross-cutting. Understanding how particles interact at very high energies helps us understand the earliest moments of the universe, when such particles were all interacting in a hot dense plasma. Likewise, by studying the these early moments of the universe and its large-scale evolution can tell us about what kinds of particles and forces are influencing its dynamics. When asking fundamental questions about the universe, one needs both the sharpest microscopes and the grandest panoramas possible.

The most prominent example of this blending of the smallest and largest scales in particle physics is dark matter. Some of our best evidence for dark matter comes analyzing the cosmic microwave background to determine how the primordial plasma behaved. These studies showed that some type of ‘cold’, matter that doesn’t interact with light, aka dark matter, was necessary to form the first clumps that eventually seeded the formation of galaxies. Without it, the universe would be much more soup-y and structureless than what we see to today.

The “cosmic web” galaxy clusters from the Millenium simulation. Measuring and understanding this web can tell us a lot about the fundamental constituents of the universe. Source

To determine what dark matter is then requires an attack from two fronts: design experiments here on earth attempting directly detect it, and further study its cosmic implications to look for more clues as to its properties.

The panel recommended next generation telescopes to study the CMB as a top priority. The so called ‘Stage 4’ CMB experiment would deploy telescopes in both the south pole and Chile’s Atacama desert to better characterize sources of atmospheric noise. The CMB has been studied extensively before, but the increased precision of CMS-S4 could shed light on mysteries like dark energy, dark matter, inflation, and the recent Hubble Tension. Given the past fruitfulness of these efforts, I think few doubted the science case for such a next generation experiment.

A mockup of one of the CMS-S4 telescopes which will be based in the Chilean desert. Note the person for scale on the right (source)

The P5 report recommended a suite of new dark matter experiments in the next decade, including the ‘ultimate’ liquid Xenon based dark matter search. Such an experiment would follow in the footsteps of massive noble gas experiments like LZ and XENONnT which have been hunting for a favored type of dark matter called WIMP’s for the last few decades. These experiments essentially build giant vats of liquid Xenon, carefully shield from any sources of external radiation, and look for signs of dark matter particles bumping into any of the Xenon atoms. The larger the vat of Xenon, the higher chance a dark matter particle will bump into something. Current generation experiments have ~7 tons of Xenon, and the next generation experiment would be even larger. The next generation aims to reach the so called ‘neutrino floor’, the point as which the experiments would be sensitive enough to observe astrophysical neutrinos bumping into the Xenon. Such neutrino interactions would look extremely similar to those of dark matter, and thus represent an unavoidable background which would signal the ultimate sensitivity of this type of experiment. WIMP’s could still be hiding in a basement below this neutrino floor, but finding them would be exceedingly difficult.

A photo of the current XENONnT experiment. This pristine cavity is then filled with liquid Xenon and closely monitored for signs of dark matter particles bumping into one of the Xenon atoms. Credit: XENON Collaboration

WIMP’s are not the only dark matter candidates in town, and recent years have also seen an explosion of interest in the broad range of dark matter possibilities, with axions being a prominent example. Other kinds of dark matter could have very different properties than WIMPs and have had much fewer dedicated experiments to search for them. There is ‘low hanging fruit’ to pluck in the way of relatively cheap experiments which can achieve world-leading sensitivity. Previously, these ‘table top’ sized experiments had a notoriously difficult time obtaining funding, as they were often crowded out of the budgets by the massive flagship projects. However, small experiments can be crucial to ensuring our best chance of dark matter discovery, as they fill in the blinds pots missed by the big projects.

The panel therefore recommended creating a new pool of funding set aside for these smaller scale projects. Allowing these smaller scale projects to flourish is important for the vibrancy and scientific diversity of the field, as the centralization of ‘big science’ projects can sometimes lead to unhealthy side effects. This specific recommendation also mirrors a broader trend of the report: to attempt to rebalance the budget portfolio to be spread more evenly and less dominated by the large projects.

A pie chart comparing the budget porfolio in 2023 (left) versus the projected budget in 2033 (right). Currently most of the budget is being taken up by the accelerator upgrades and cavern construction of DUNE, with some amount for the LHC upgrades. But by 2033 the panel recommends a much more equitable balance between different research area.

What Didn’t Make It

Any report like this comes with some tough choices. Budget realities mean not all projects can be funded. Besides the pairing down of some of DUNE’s upgrades, one of the biggest areas that was recommended against were ‘accessory experiments at the LHC’. In particular, MATHUSULA and the Forward Physics Facility were two experiments that proposed to build additional detectors near already existing LHC collision points to look for particles that may be missed by the current experiments. By building new detectors hundreds of meters away from the collision point, shielded by concrete and the earth, they can obtained unique sensitivity to ‘long lived’ particles capable of traversing such distances. These experiments would follow in the footsteps of the current FASER experiment, which is already producing impressive results.

While FASER found success as a relatively ‘cheap’ experiment, reusing detector components from and situating itself in a beam tunnel, these new proposals were asking for quite a bit more. The scale of these detectors would have required new caverns to be built, significantly increasing the cost. Given the cost and specialized purpose of these detectors, the panel recommended against their construction. These collaborations may now try to find ways to pare down their proposal so they can apply to the new small project portfolio.

Another major decision by the panel was to recommend against hosting a new Higgs factor collider in the US. But that will discussed more in a future post.

Conclusions

The P5 panel was faced with a difficult task, the total cost of all projects they were presented with was three times the budget. But they were able to craft a plan that continues the work of the previous decade, addresses current shortcomings and lays out an inspiring vision for the future. So far the community seems to be strongly rallying behind it. At time of writing, over 2700 community members from undergraduates to senior researchers have signed a petition endorsing the panels recommendations. This strong show of support will be key for turning these recommendations into actual funding, and hopefully lobbying congress to even increase funding so that more of this vision can be realized.

For those interested the full report as well as executive summaries of different areas can be found on the P5 website. Members of the US particle physics community are also encouraged to sign the petition endorsing the recommendations here.

And stayed tuned for part 2 of our coverage which will discuss the implications of the report on future colliders!

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The Search for Simplicity : The Higgs Boson’s Self Coupling

When students first learn quantum field theory, the mathematical language the underpins the behavior of elementary particles, they start with the simplest possible interaction you can write down : a particle with no spin and no charge scattering off another copy of itself. One then eventually moves on to the more complicated interactions that describe the behavior of fundamental particles of the Standard Model. They may quickly forget this simplified interaction as a unrealistic toy example, greatly simplified compared to the complexity the real world. Though most interactions that underpin particle physics are indeed quite a bit more complicated, nature does hold a special place for simplicity. This barebones interaction is predicted to occur in exactly one scenario : a Higgs boson scattering off itself. And one of the next big targets for particle physics is to try and observe it.

A feynman diagram consisting of two dotted lines coming merging together to form a single line.
A Feynman diagram of the simplest possible interaction in quantum field theory, a spin-zero particle interacting with itself.

The Higgs is the only particle without spin in the Standard Model, and the only one that doesn’t carry any type of charge. So even though particles such as gluons can interact with other gluons, its never two of the same kind of gluons (the two interacting gluons will always carry different color charges). The Higgs is the only one that can have this ‘simplest’ form of self-interaction. Prominent theorist Nima Arkani-Hamed has said that the thought of observing this “simplest possible interaction in nature gives [him] goosebumps“.

But more than being interesting for its simplicity, this self-interaction of the Higgs underlies a crucial piece of the Standard Model: the story of how particles got their mass. The Standard Model tells us that the reason all fundamental particles have mass is their interaction with the Higgs field. Every particle’s mass is proportional to the strength of the Higgs field. The fact that particles have any mass at all is tied to the fact that the lowest energy state of the Higgs field is at a non-zero value. According to the Standard Model, early in the universe’s history when the temperature were much higher, the Higgs potential had a different shape, with its lowest energy state at field value of zero. At this point all the particles we know about were massless. As the universe cooled the shape of the Higgs potential morphed into a ‘wine bottle’ shape, and the Higgs field moved into the new minimum at non-zero value where it sits today. The symmetry of the initial state, in which the Higgs was at the center of its potential, was ‘spontaneously broken’  as its new minimum, at a location away from the center, breaks the rotation symmetry of the potential. Spontaneous symmetry breaking is a very deep theoretical idea that shows up not just in particle physics but in exotic phases of matter as well (eg superconductors). 

A diagram showing the ‘unbroken’ Higgs potential in the very early universe (left) and the ‘wine bottle’ shape it has today (right). When the Higgs at the center of its potential it has a rotational symmetry, there are no preferred directions. But once it finds it new minimum that symmetry is broken. The Higgs now sits at a particular field value away from the center and a preferred direction exists in the system. 

This fantastical story of how particle’s gained their masses, one of the crown jewels of the Standard Model, has not yet been confirmed experimentally. So far we have studied the Higgs’s interactions with other particles, and started to confirm the story that it couples to particles in proportion to their mass. But to confirm this story of symmetry breaking we will to need to study the shape of the Higgs’s potential, which we can probe only through its self-interactions. Many theories of physics beyond the Standard Model, particularly those that attempt explain how the universe ended up with so much matter and very little anti-matter, predict modifications to the shape of this potential, further strengthening the importance of this measurement.

Unfortunately observing the Higgs interacting with itself and thus measuring the shape of its potential will be no easy feat. The key way to observe the Higgs’s self-interaction is to look for a single Higgs boson splitting into two. Unfortunately in the Standard Model additional processes that can produce two Higgs bosons quantum mechanically interfere with the Higgs self interaction process which produces two Higgs bosons, leading to a reduced production rate. It is expected that a Higgs boson scattering off itself occurs around 1000 times less often than the already rare processes which produce a single Higgs boson.  A few years ago it was projected that by the end of the LHC’s run (with 20 times more data collected than is available today), we may barely be able to observe the Higgs’s self-interaction by combining data from both the major experiments at the LHC (ATLAS and CMS).

Fortunately, thanks to sophisticated new data analysis techniques, LHC experimentalists are currently significantly outpacing the projected sensitivity. In particular, powerful new machine learning methods have allowed physicists to cut away background events mimicking the di-Higgs signal much more than was previously thought possible. Because each of the two Higgs bosons can decay in a variety of ways, the best sensitivity will be obtained by combining multiple different ‘channels’ targeting different decay modes. It is therefore going to take a village of experimentalists each working hard to improve the sensitivity in various different channels to produce the final measurement. However with the current data set, the sensitivity is still a factor of a few away from the Standard Model prediction. Any signs of this process are only expected to come after the LHC gets an upgrade to its collision rate a few years from now.

Limit plots on HH production in various different decay modes.
Current experimental limits on the simultaneous production of two Higgs bosons, a process sensitive to the Higgs’s self-interaction, from ATLAS (left) and CMS (right). The predicted rate from the Standard Model is shown in red in each plot while the current sensitivity is shown with the black lines. This process is searched for in a variety of different decay modes of the Higgs (various rows on each plot). The combined sensitivity across all decay modes for each experiment allows them currently to rule out the production of two Higgs bosons at 3-4 times the rate predicted by the Standard Model. With more data collected both experiments will gain sensitivity to the range predicted by the Standard Model.

While experimentalists will work as hard as they can to study this process at the LHC, to perform a precision measurement of it, and really confirm the ‘wine bottle’ shape of the potential, its likely a new collider will be needed. Studying this process in detail is one of the main motivations to build a new high energy collider, with the current leading candidates being an even bigger proton-proton collider to succeed the LHC or a new type of high energy muon collider.

Various pictorial representations of the uncertainty on the Higgs potential shape.
A depiction of our current uncertainty on the shape of the Higgs potential (center), our expected uncertainty at the end of the LHC (top right) and the projected uncertainty a new muon collider could achieve (bottom right). The Standard Model expectation is the tan line and the brown band shows the experimental uncertainty. Adapted from Nathaniel Craig’s talkhere

The quest to study nature’s simplest interaction will likely span several decades. But this long journey gives particle physicists a roadmap for the future, and a treasure worth traveling great lengths for.

Read More:

CERN Courier Interview with Nima Arkani-Hamed on the future of Particle Physics on the importance of the Higgs’s self-coupling

Wikipedia Article and Lecture Notes on Spontaneous symmetry breaking

Recent ATLAS Measurements of the Higgs Self Coupling

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Three Birds with One Particle: The Possibilities of Axions

Title: “Axiogenesis”

Author: Raymond T. Co and Keisuke Harigaya

Reference: https://arxiv.org/pdf/1910.02080.pdf

On the laundry list of problems in particle physics, a rare three-for-one solution could come in the form of a theorized light scalar particle fittingly named after a detergent: the axion. Frank Wilczek coined this term in reference to its potential to “clean up” the Standard Model once he realized its applicability to multiple unsolved mysteries. Although Axion the dish soap has been somewhat phased out of our everyday consumer life (being now primarily sold in Latin America), axion particles remain as a key component of a physicist’s toolbox. While axions get a lot of hype as a promising dark matter candidate, and are now being considered as a solution to matter-antimatter asymmetry, they were originally proposed as a solution for a different Standard Model puzzle: the strong CP problem. 

The strong CP problem refers to a peculiarity of quantum chromodynamics (QCD), our theory of quarks, gluons, and the strong force that mediates them: while the theory permits charge-parity (CP) symmetry violation, the ardent experimental search for CP-violating processes in QCD has so far come up empty-handed. What does this mean from a physical standpoint? Consider the neutron electric dipole moment (eDM), which roughly describes the distribution of the three quarks comprising a neutron. Naively, we might expect this orientation to be a triangular one. However, measurements of the neutron eDM, carried out by tracking changes in neutron spin precession, return a value orders of magnitude smaller than classically expected. In fact, the incredibly small value of this parameter corresponds to a neutron where the three quarks are found nearly in a line. 

The classical picture of the neutron (left) looks markedly different from the picture necessitated by CP symmetry (right). The strong CP problem is essentially a question of why our mental image should look like the right picture instead of the left. Source: https://arxiv.org/pdf/1812.02669.pdf

This would not initially appear to be a problem. In fact, in the context of CP, this makes sense: a simultaneous charge conjugation (exchanging positive charges for negative ones and vice versa) and parity inversion (flipping the sign of spatial directions) when the quark arrangement is linear results in a symmetry. Yet there are a few subtleties that point to the existence of further physics. First, this tiny value requires an adjustment of parameters within the mathematics of QCD, carefully fitting some coefficients to cancel out others in order to arrive at the desired conclusion. Second, we do observe violation of CP symmetry in particle physics processes mediated by the weak interaction, such as kaon decay, which also involves quarks. 

These arguments rest upon the idea of naturalness, a principle that has been invoked successfully several times throughout the development of particle theory as a hint toward the existence of a deeper, more underlying theory. Naturalness (in one of its forms) states that such minuscule values are only allowed if they increase the overall symmetry of the theory, something that cannot be true if weak processes exhibit CP-violation where strong processes do not. This puts the strong CP problem squarely within the realm of “fine-tuning” problems in physics; although there is no known reason for CP symmetry conservation to occur, the theory must be modified to fit this observation. We then seek one of two things: either an observation of CP-violation in QCD or a solution that sets the neutron eDM, and by extension any CP-violating phase within our theory, to zero.

This term in the QCD Lagrangian allows for CP symmetry violation. Current measurements place the value of \theta at no greater than 10^{-10}. In Peccei-Quinn symmetry, \theta is promoted to a field.

When such an expected symmetry violation is nowhere to be found, where is a theoretician to look for such a solution? The most straightforward answer is to turn to a new symmetry. This is exactly what Roberto Peccei and Helen Quinn did in 1977, birthing the Peccei-Quinn symmetry, an extension of QCD which incorporates a CP-violating phase known as the \theta term. The main idea behind this theory is to promote \theta to a dynamical field, rather than keeping it a constant. Since quantum fields have associated particles, this also yields the particle we dub the axion. Looking back briefly to the neutron eDM picture of the strong CP problem, this means that the angular separation should also be dynamical, and hence be relegated to the minimum energy configuration: the quarks again all in a straight line. In the language of symmetries, the U(1) Peccei-Quinn symmetry is approximately spontaneously broken, giving us a non-zero vacuum expectation value and a nearly-massless Goldstone boson: our axion.

This is all great, but what does it have to do with dark matter? As it turns out, axions make for an especially intriguing dark matter candidate due to their low mass and potential to be produced in large quantities. For decades, this prowess was overshadowed by the leading WIMP candidate (weakly-interacting massive particles), whose parameter space has been slowly whittled down to the point where physicists are more seriously turning to alternatives. As there are several production-mechanisms in early universe cosmology for axions, and 100% of dark matter abundance could be explained through this generation, the axion is now stepping into the spotlight. 

This increased focus is causing some theorists to turn to further avenues of physics as possible applications for the axion. In a recent paper, Co and Harigaya examined the connection between this versatile particle and matter-antimatter asymmetry (also called baryon asymmetry). This latter term refers to the simple observation that there appears to be more matter than antimatter in our universe, since we are predominantly composed of matter, yet matter and antimatter also seem to be produced in colliders in equal proportions. In order to explain this asymmetry, without which matter and antimatter would have annihilated and we would not exist, physicists look for any mechanism to trigger an imbalance in these two quantities in the early universe. This theorized process is known as baryogenesis.

Here’s where the axion might play a part. The \theta term, which settles to zero in its possible solution to the strong CP problem, could also have taken on any value from 0 to 360 degrees very early on in the universe. Analyzing the axion field through the conjectures of quantum gravity, if there are no global symmetries then the initial axion potential cannot be symmetric [4]. By falling from some initial value through an uneven potential, which the authors describe as a wine bottle potential with a wiggly top, \theta would cycle several times through the allowed values before settling at its minimum energy value of zero. This causes the axion field to rotate, an asymmetry which could generate a disproportionality between the amounts of produced matter and antimatter. If the field were to rotate in one direction, we would see more matter than antimatter, while a rotation in the opposite direction would result instead in excess antimatter.

The team’s findings can be summarized in the plot above. Regions in purple, red, and above the orange lines (dependent upon a particular constant \xi which is proportional to weak scale quantities) signify excluded portions of the parameter space. The remaining white space shows values of the axion decay constant and mass where the currently measured amount of baryon asymmetry could be generated. Source: https://arxiv.org/pdf/1910.02080.pdf

Introducing a third fundamental mystery into the realm of axions begets the question of whether all three problems (strong CP, dark matter, and matter-antimatter asymmetry) can be solved simultaneously with axions. And, of course, there are nuances that could make alternative solutions to the strong CP problem more favorable or other dark matter candidates more likely. Like most theorized particles, there are several formulations of axion in the works. It is then necessary to turn our attention to experiment to narrow down the possibilities for how axions could interact with other particles, determine what their mass could be, and answer the all-important question: if they exist at all. Consequently, there are a plethora of axion-focused experiments up and running, with more on the horizon, that use a variety of methods spanning several subfields of physics. While these results begin to roll in, we can continue to investigate just how many problems we might be able to solve with one adaptable, soapy particle.

Learn More:

  1. A comprehensive introduction to the strong CP problem, the axion solution, and other potential solutions: https://arxiv.org/pdf/1812.02669.pdf 
  2. Axions as a dark matter candidate: https://www.symmetrymagazine.org/article/the-other-dark-matter-candidate
  3. More information on matter-antimatter asymmetry and baryogenesis: https://www.quantumdiaries.org/2015/02/04/where-do-i-come-from/
  4. The quantum gravity conjectures that axiogenesis builds upon: https://arxiv.org/abs/1810.05338
  5. An overview of current axion-focused experiments: https://www.annualreviews.org/doi/full/10.1146/annurev-nucl-102014-022120
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A new anomaly: the electromagnetic duality anomaly

Article: Electromagnetic duality anomaly in curved spacetimes
Authors: I. Agullo, A. del Rio and J. Navarro-Salas
Reference: arXiv:1607.08879

Disclaimer: this blogpost requires some basic knowledge of QFT (or being comfortable with taking my word at face value for some of the claims made :))

Anomalies exists everywhere. Probably the most intriguing ones are medical, but in particle physics they can be pretty fascinating too. In physics, anomalies refer to the breaking of a symmetry. There are basically two types of anomalies:

  • The first type, gauge anomalies, are red-flags: if they show up in your theory, they indicate that the theory is mathematically inconsistent.
  • The second type of anomaly does not signal any problems with the theory and in fact can have experimentally observable consequences. A prime example is the chiral anomaly. This anomaly nicely explains the decay rate of the neutral pion into two photons.
    Fig. 1: Illustration of pion decay into two photons. [Credit: Wikimedia Commons]

In this paper, a new anomaly is discussed. This anomaly is related to the polarization of light and is called the electromagnetic duality anomaly.

Chiral anomaly 101
So let’s first brush up on the basics of the chiral anomaly. How does this anomaly explain the decay rate of the neutral pion into two photons? For that we need to start with the Lagrangian for QED that describes the interactions between the electromagnetic field (that is, the photons) and spin-½ fermions (which pions are build from):

\displaystyle \mathcal L = \bar\psi \left( i \gamma^\mu \partial_\mu - i e \gamma^\mu A_\mu \right) \psi + m \bar\psi \psi

where the important players in the above equation are the \psis that describe the spin-½ particles and the vector potential A_\mu that describes the electromagnetic field. This Lagrangian is invariant under the chiral symmetry:

\displaystyle \psi \to e^{i \gamma_5} \psi .

Due to this symmetry the current density j^\mu = \bar{\psi} \gamma_5 \gamma^\mu \psi is conserved: \nabla_\mu j^\mu = 0. This then immediately tells us that the charge associated with this current density is time-independent. Since the chiral charge is time-independent, it prevents the \psi fields to decay into the electromagnetic fields, because the \psi field has a non-zero chiral charge and the photons have no chiral charge. Hence, if this was the end of the story, a pion would never be able to decay into two photons.

However, the conservation of the charge is only valid classically! As soon as you go from classical field theory to quantum field theory this is no longer true; hence, the name (quantum) anomaly.  This can be seen most succinctly using Fujikawa’s observation that even though the field \psi and Lagrangian are invariant under the chiral symmetry, this is not enough for the quantum theory to also be invariant. If we take the path integral approach to quantum field theory, it is not just the Lagrangian that needs to be invariant but the entire path integral needs to be:

\displaystyle \int D[A] \, D[\bar\psi]\, \int D[\psi] \, e^{i\int d^4x \mathcal L} .

From calculating how the chiral symmetry acts on the measure D \left[\psi \right]  \, D \left[\bar \psi \right], one can extract all the relevant physics such as the decay rate.

The electromagnetic duality anomaly
Just like the chiral anomaly, the electromagnetic duality anomaly also breaks a symmetry at the quantum level that exists classically. The symmetry that is broken in this case is – as you might have guessed from its name – the electromagnetic duality. This symmetry is a generalization of a symmetry you are already familiar with from source-free electromagnetism. If you write down source-free Maxwell equations, you can just swap the electric and magnetic field and the equations look the same (you just have to send  \displaystyle \vec{E} \to \vec{B} and \vec{B} \to - \vec{E}). Now the more general electromagnetic duality referred to here is slightly more difficult to visualize: it is a rotation in the space of the electromagnetic field tensor and its dual. However, its transformation is easy to write down mathematically:

\displaystyle F_{\mu \nu} \to \cos \theta \, F_{\mu \nu} + \sin \theta \, \, ^\ast F_{\mu \nu} .

In other words, since this is a symmetry, if you plug this transformation into the Lagrangian of electromagnetism, the Lagrangian will not change: it is invariant. Now following the same steps as for the chiral anomaly, we find that the associated current is conserved and its charge is time-independent due to the symmetry. Here, the charge is simply the difference between the number of photons with left helicity and those with right helicity.

Let us continue following the exact same steps as those for the chiral anomaly. The key is to first write electromagnetism in variables analogous to those of the chiral theory. Then you apply Fujikawa’s method and… *drum roll for the anomaly that is approaching*…. Anti-climax: nothing happens, everything seems to be fine. There are no anomalies, nothing!

So why the title of this blog? Well, as soon as you couple the electromagnetic field with a gravitational field, the electromagnetic duality is broken in a deeply quantum way. The number of photon with left helicity and right helicity is no longer conserved when your spacetime is curved.

Physical consequences
Some potentially really cool consequences have to do with the study of light passing by rotating stars, black holes or even rotating clusters. These astrophysical objects do not only gravitationally bend the light, but the optical helicity anomaly tells us that there might be a difference in polarization between lights rays coming from different sides of these objects. This may also have some consequences for the cosmic microwave background radiation, which is ‘picture’ of our universe when it was only 380,000 years old (as compared to the 13.8 billion years it is today!). How big this effect is and whether we will be able to see it in the near future is still an open question.

 

 

Further reading 

  • An introduction to anamolies using only quantum mechanics instead of quantum field theory is “Anomalies for pedestrians” by Barry Holstein 
  • The beautiful book “Quantum field theory and the Standard Model” by Michael Schwartz has a nice discussion in the later chapters on the chiral anomaly.
  • Lecture notes by Adal Bilal for graduate students on anomalies in general  can be found here.