It was reported a few days ago in the local Carlsbad, NM paper that the Waste Isolation Pilot Plant is slowly recovering after a minor radiation leak earlier this year. Most of the underground facility is accessible again, although the number of personnel allowed underground is still limited due to some hoist issues. Hopefully, this means that the science area will reopen soon so that experiments operating at the site (the largest is the EXO-200 double beta decay experiment) can start working again. It still may be quite some time before nuclear waste operations start again.
On Thursday, the EXO-200 experiment released a preprint on a search for exotic double beta decay modes involving no neutrinos but with emission of one or more “Majorons,” which are bosons related to the violation of lepton number required by these decay modes.
EXO-200 is an experiment at the Waste Isolation Pilot Plant (WIPP) near Carlsbad, NM. It consists of a time projection chamber (TPC) using highly enriched liquid xenon as the target material. The xenon is enriched with a higher fraction of a particular isotope that is expected to undergo double beta decay – which happens when regular beta decay is disallowed due to energy constraints but double beta decay, where two electrons (or positrons) are emitted at once is allowed. The main goal of EXO is to search for neutrinoless double beta decay, where the two neutrinos created along with the two electrons basically cancel one another. The existence of neutrinoless double beta decay requires that neutrinos be Majorana fermions, where they are their own antiparticles, instead of Dirac fermions like all the other known fermions. This paper looks at more complicated decay mode than the standard one of just two electrons and a nucleus in the final state.
The paper sets lower limits on the lifetime of xenon-136 using different models of neutrinoless double beta decays with Majorons in the final state. It also sets equivalent upper limits on the Majoron-electron coupling for some of these models.
Ed Brayton has a post today pointing to a particularly ignorant monologue by noted bigot Bryan Fischer of the American Family Association. In it Fischer says that
- The strong nuclear force is what holds atomic nuclei together (true),
- scientists don’t understand it (more or less false for decades), so therefore
- the strong nuclear force is Jesus.
This seems to be a version of the “god of the gaps” argument, arguing that things we don’t understand must be due to divine intervention. This is recognized by most people as a logical fallacy. Obviously, when this is applied to things that we actually do understand, it looks bad for religion. It’s also a dangerous argument for science because it encourages people to be incurious about the world. If we ascribe a supernatural origin to everything we don’t understand then there is no need for science; we already have the explanation for any problem.
Just in case anyone wants a brief explanation of the strong force:
Nuclei are made of protons and neutrons. Neutrons have no electric charge while protons are all positively charged. So, the electromagnetic forces between protons tend to try to push them apart – to cause the nucleus to break apart. The nucleus is held together because there is another, stronger force (unimaginatively called the strong force) that pulls the protons together more than electromagnetism pushes them apart. In quantum mechanics (the nucleons are nonrelativistic), this can be roughly modeled as a deep short-range square potential well that replaces the usual 1/r Coulomb potential from electrostatics. The potential is generated by the other nucleons in a nucleus, so this is only a very simple approximation.
In more advanced (but still not fundamental models) the forces between nucleons can be modeled as an exchange of mesons (typically pions), similarly to how electromagnetic interactions are caused by photon exchange between particles. The fundamental interaction comes from the local SU(3) color symmetry of quark fields in quantum chromodynamics. There are 3 colors and 8 bosons (called gluons) that allow for exchange of color charge between particles. The strong force is also what holds the nucleons together – they are made of quarks and gluons, which are all in turn believed to be elementary particles. Calculations of the properties of hadrons (protons, neutrons, pions, etc) from first principles requires the use of the world’s most powerful supercomputers. That is a field called Lattice QCD.
Nuclei are complicated objects made of complicated composite particles, so we can’t feasibly calculate anything we want to arbitrary precision. The strong force is also difficult to deal with it because of it’s large coupling constant (becoming nonperturbative in many problems) and it’s non-Abelian nature (a math term related to the properties of SU(3) that here means that gluons can interact directly with other gluons). We do still have a pretty good understanding of how they work. Fischer only had to look up the nuclear force on Wikipedia if he wanted to get some idea of what we know.
I’ve decided to start adding explanations of physics topics every once in a while. I’ll make a page for these so they’re easy to find. The idea is that I can compile a series of explanations for non-physicists.
My first post is on cross sections. I chose this because this is important in understanding nearly all measurements in high energy and nuclear physics.
Cross Sections in Classical Physics
The cross section in classical physics is closely related to the informal definition of a cross section.
Consider a three dimensional object, such as a sphere or a cylinder. If I rotate the object in some arbitrary way and look at it, I see that it covers a certain fraction of my field of vision. The effective size of that object as projected onto the plane perpendicular to my line of sight is its cross section. For a circle, this is always just πr2 where r is the radius of the sphere. For a cylinder, the size depends on how it’s been rotated. This kind of cross section is the informal definition.
The total cross section described above only gives a very rough description of the size of the object. A cross section can also be used to glean information about the shape of the object. In physics, this is the differential cross section.
Again imagine your field of vision as a plane with the object at some rotation with respect to this plane. Now shoot tiny particles uniformly from that plane (and perpendicular to it) toward the object. Mathematically, I will quantify this as the intensity, I, which is just the number of particles passing through the plane per unit area per unit time.
If the particles are too far from the center of the object (assuming it is finite in size) they will miss. If they hit the object, they will be deflected at some angle related to the orientation of the surface of the object at the intersection between the object and the path of the particle. Thus, the position at which I fire the particle and its deflection angle tell us about the shape of the object’s surface. This is described mathematically by the differential cross section:
The d in dΩ indicates that we look at an infinitesimally small region of scattering direction space, although we can get a good estimate as long as the region is small enough for whatever we’re trying to measure. The total cross section is then the sum over all scattering directions,
Now add in some long range force like gravity. Gravity pulls particles toward the object, so their paths are deflected. Some particles will get pulled into the object, so clearly the effective cross section has increased. Particles with more momentum will be harder to deflect, so this effective cross section will also depend on the properties of the incoming particles; a generic cross cross section is a function of properties such as particle mass, particle momentum, magnetic moment, etc.
What if I leave in gravity but reduce the physical size of my object to nothing? The incoming particles can no longer fall into the object, but they will still be deflected. But remember that the differential cross section is defined only by the properties of the scattered particles (and maybe the orientation of the object) – not on anything related to the physical size of the object. In fact, we could even model an object with a physical size as simply being a point object that happens to have an infinitely strong repulsive force at what would normally be called the object’s surface. So, we can retain our original definition of the differential cross section but then define the total cross section as the integral of the differential cross section (hence the term “differential”). This gives us a definition of cross section based on the object’s interactions with particles. This definition is also a convenient one for comparing to experimental data.
Cross Sections in Quantum Physics
Quantum mechanics and quantum field theory (QFT – basically relativistic quantum mechanics) have different definitions of the cross section, although they should reduce to the classical one in the classical limit of the theories.
In quantum theories, the scattering process described by a cross section occurs in a probabilistic manner. Our classical definition of the differential cross section is basically probabilistic too. Ignoring the overall normalization giving the total cross section it describes the probability that a scattered particle will scatter in a particular way. However, the classical definition does not describe a probabilistic process. If we shoot a particle at our classical object, we can calculate exactly what it would do. In quantum physics, we can only calculate the probability that the given type of scattering will happen. Additionally, we can scatter particles off of a field localized in some region of space or scatter particles off of each other; objects as we know them won’t really exist at the scales of quantum theories.
The cross section in quantum theories basically just tells us the probability that a given process (what given outgoing particles come out of scattering of some incoming particles with known properties) will happen. The differential cross section tells us the details (what directions, orientations, momenta, etc. the particles in the final state end up with). More formally,
The rate of interactions for the cross section σ that we want to measure is the product of the cross section and the luminosity – the analog of the intensity used in QFT calculations. (The quantum mechanics definition is based on waves rather than particles so I won’t discuss here). The luminosity just quantifies the strength of the “beam” (or beams for a collider). This definition tells us how to get the cross section from a measurement: we know the properties of our beams (or beam and target) and we measure an event rate. The differential cross section just requires us to determine the rate of events with the given kinematics instead of the total rate.
By defining everything through probabilities, it is also easy to treat inelastic scattering, where the outgoing particle(s) (final state) are different from the incoming particle(s) (initial state). Even if there are dozens of particles in the final state, the cross section is really just the probability that that final state will occur (with an extra normalization factor making it an area).
Some Concluding Thoughts
Cross sections are a fundamental piece of much of modern physics, particularly high energy (particle) and nuclear physics. Event rates or numbers of events are very basic kinds of measurements, and these can be turned into cross sections by relating to the expected or measured luminosity.
In some cases just measuring the existence (or nonexistence) of a process is interesting. It could show us the existence of a new particle or a new interaction. Quantum field theories such as the Standard Model predict cross sections (total and differential) for many kinds of interactions and disallow others. Comparing measured cross sections to theoretical models lets us test the validity of those models.
There are many examples of cross sections in measurements that have been in the news over the past few years.
In dark matter direct detection, experiments seek to place upper bounds on the possible interaction cross sections between dark matter particles and Standard Model particles. This requires theoretical modeling of the dark matter distribution near Earth to determine the correct “luminosity” since the “beam” is just particles flying around in space and through detectors on Earth.
At the LHC, physicists are trying to determine whether or not the particle believed to be the Higgs boson really is a Standard Model Higgs by studying its interactions. The cross sections (often ratios of cross sections rather than absolute cross sections to reduce uncertainties) for many initial and final states are being compared to Standard Model predictions (and even non-Standard Model ones to test other models) to see if they are consistent.