Thursday, February 27, 2014

We Don't Understand High Energy Hadronic Physics

Lately, the story of high energy physics has been one ultra-precise confirmation of Standard Model predictions after another, interrupted occasionally by improved measurements of neutrino physics parameters.

Today, a short and very understated high energy physics preprint from the Belle collaboration measuring the exclusively hadronic branching fractions in two kinds of systems is a rare exception to that trend.  The abstract doesn't even mention the deviation between theory and experiment, and no conclusions or analysis in the short four page paper discusses why this discrepancy could arise.

The Experiments Conducted

Upsilon Decays

One system studied was the decay of two kinds of upsilon mesons (i.e. mesons composed of bottom quark and anti-bottom quark pairs), one more excited than the other, into one of seven combinations of pions, kaons, rho mesons and omega mesons.

Electron-Positron Decays

The other measured decays from an electron-positron annihilation into five different sets of mesons:

(1) a spin-1 (i.e. vector) omega meson with a quark content of (up anti-up plus down anti-down)/sqrt(2) and spin-0 (i.e. pseudo-scalar) neutral pion, with a quark content of (up anti-up minus down anti-down)/sqrt(2).

(2) a neutral vector kaon (down quark and an anti-strange quark) with a ground state 892 MeV mass and a neutral anti-kaon (a linear combination of the short kaon with a quark content of (down anti-strange minus strange anti-down/sqrt(2)), and the long kaon with a quark content of (down anti-strange plus strange anti-down)/sqrt(2))

(3) a negatively charged vector kaon with a ground state 892 MeV mass (a strange quark and an anti-up quark) and a positively charged kaon (an up quark and an anti-strange quark),

(4) an excited spin-2 vector kaon with an excited state 1430 MeV mass (a strange quark and an anti-up quark) and a neutral anti-kaon  (a linear combination of the short kaon with a quark content of (down anti-strange minus strange anti-down/sqrt(2)), and the long kaon with a quark content of (down anti-strange plus strange anti-down)/sqrt(2)), and

(5) a negatively charged excited spin-2 vector kaon with an excited state 1430 MeV mass (a strange quark and an anti-up quark) and a positively charged kaon (an up quark and an anti-strange quark).

The Results

For the most part, these measured branching fractions of the upsilon decays aren't greatly different from the theoretical QCD predictions in decay paths where there is enough data to provide a statistically significant result, although there were some tensions between the predicted values and measured values for

But, in the case of the electron-positron decays, the experimental results were grossly at odds with the perturbative QCD predictions.

In a model where SU(3) flavor symmetry is perfect, the ratio of decay path (2) above to decay path (3) above should be 2-1, and it should be 6-1 in a model where SU(3) flavor symmetry is broken.  Instead, the ratio of these decay rates was greater than 9-1 at the sqrt(s)=3.67 GeV energy scale and greater than 33-1 at the 3.773 GeV energy scale.

Other measured decay cross sections, at energies of sqrt(s)=10.52 Gev, 10.58 GeV and 10.876 GeV were similarly grossly at odds with the predictions of both the SU(3) perfect flavor symmetry model and of the SU(3) broken flavor symmetry model.  At 10.52 GeV there is a 7 sigma excess in decay channel (2).  At 10.58 GeV there is a 6.7 sigma excess in decay channel (1), and a greater than 10 sigma excess in decay channels (2) and (5).  At 10.876 GeV there is a 7.2 sigma excess in decay channel (2) and a 4.5 sigma excess in decay channel (5).

On the other hand, none of the six measurements of decay channels (3) and (4) were more than 2.1 sigma from the predicted values, and the 10.52 GeV value for decay channel (1) and (5) were not more than 2.1 sigma (there was no measurement of decay channel (1) at 10.876 GeV).

Thus, six data points are 4.5 sigma or more from the predicted values, while five data points are 2.1 sigma or less from the predicted values.  If the theoretical prediction was correct, we would expect an average deviation of 1 sigma.


This result is grossly at odds with theoretical predictions.  This is particularly notable because the energy scales of these experiments are sufficiently high that they are in the range where perturbative QCD should not be materially inferior to lattice QCD predictions that are used at low (aka infrared) energy scales for pQCD breaks down.

The results have a "who ordered that?" character, as no leading beyond the Standard Model theory predicted this kind of gross discrepancy between theory and experiment.

Clearly, either something is seriously awry at the Belle experiment and causing huge systemic errors, or something is deeply flawed in the approach used to calculate the theoretical predictions for these decays.

This doesn't necessarily mean that new physics are necessary to solve the problem.  It could be that there is a flaw either in how perturbative QCD was applied, or in how perturbative QCD approximates the complete QCD calculations of the Standard Model in this context for some reason.

For example, perhaps there are intermediate decay paths to unknown linearly combined meson states that impact the decay cross-sections materially that QCD permits but which theorists have omitted in an oversimplifying assumption.

But, results like these do remind us that physicists have plenty to learn and discover, particularly when it comes to the hadronic physics of mesons, before they can claim to really understand QCD.

Thursday, February 20, 2014

What Mechanism Causes Neutrinos To Oscillate?


Neutrino Oscillation

There is compelling empirical evidence that the neutrinos oscillate from one of the three active neutrino flavors to another of the three active neutrino flavors according to a quite simple formula, and there is some (increasingly weak) evidence that a better fit to the data is secured by extending the model to include a very small probability of oscillation to a fourth sterile neutrino flavor with a mass greater than that of the heaviest neutrino mass, but less than 1 eV.*

The consensus formula that is used to describe, for example, the probability P that a muon neutrino (neutrino generation 2) with a given (kinetic) energy E (in GeV/c^2) will oscillate into a tau neutrino (neutrino generation 3) over a given distance L (in kilometers) is assumed, on a consensus basis that produces quite good fits to the data, to follow the formula below (from page 138 of the 2012 dissertation of Michael R. Dziomba):


where Θ23 is the empirically determined mixing angle between the second and third generation neutrinos, and Δm23 is the empirically determined difference in mass between the second and third generation neutrinos.  The current empirical best fit for sin2(2Θ23) is about .99 which implies that Θ23 is about 42 degrees, and the empirical best fit for Δm23 is about 0.048 eV/c2.

Thus, the probability of an oscillation to another flavor oscillates periodically as a function of distance for a neutrino of any given energy, and the higher the neutrino's energy, the more slowly this probability oscillates as a function of distance.

This general formulation is derived in depth, for example, in a 1998 review article on the proper calculations for particle oscillation in general (including neutral kaons and neutrinos). But, like the equation above, it essentially just assumes that oscillation happens in a black box without much concern for why a particle is actually able to display this behavior. The equation tells you how to predict what happens, but not why it does what it does at the critical moment when the oscillation takes place.

The formulas for the probability that neutrinos of one state will oscillate to another, in general, follow this form with the exception of the subscripts on the Θ mixing angle and Δm term. A full treatment of an individual source neutrino's probability of ending up in a particular end state would require consideration of all possible oscillations, rather than just one as in the example above.

Flavor Changing W Boson Emissions

There is only one other process in the Standard Model by which fermions change flavor involves weak force interactions.

In that process, up-type quarks transform themselves into down-type quarks (of a type determined by probabilities empirically measured in the pertinent constants of the CKM matrix) simultaneously with their emission of W+ bosons, down-type quarks transform themselves into up-type quarks simultaneously with their emission of W- bosons, charged leptons transform themselves into neutrinos with their emission of W- bosons, and neutrinos transform themselves into charged leptons with their emission of W+ bosons.  The probability of a particle emitting a W boson in a given time period is largely a function of the weak force coupling constant.

(Note in contrast, that strong force interactions are not flavor changing, although they do routinely create quark-antiquark pairs that permit the creation of new hadrons with more quarks than the source particle, a process which does not violate baryon number conservation, so long as this is permitted by mass-energy conservation.)

How does neutrino oscillation happen?

Neutrino Flavor Changes Via Two Step Virtual W Boson Emissions

If W boson emissions were involved in neutrino oscillation, this would involve, at least, a two step process.  First, a neutrino would emits a virtual W+ boson and becoming a virtual charged lepton.  Then, the virtual charged lepton would emit a virtual W- boson that would turn it into a neutrino of random flavor again.  In this virtual two step process, the virtual W+ and W- boson would annihilate each other and thus cancel out before either of these bosons can decay.

In the typical neutrino oscillation scenario, the probability of the intermediate charged lepton coming into existence, or of the virtual W bosons coming into existence and producing decay products, would be essentially nil because these possibilities would violate mass-energy conservation.

Of course, there could be additional contributions to the overall probabilities from four, six, etc. steps processes.

Of course, a neutrino cannot oscillate into a heavier neutrino flavor if its kinetic energy is less than the mass difference between the neutrino flavors of the end states, since mass-energy conservation forbids this possibility.  My assumption is that mass gained through neutrino oscillation results in a neutrino losing an equal amount of kinetic energy without altering the direction of its momentum, and that mass lost through neutrino oscillation results in a neutrino gaining an equal amount of kinetic energy without altering the direction of its momentum, on mass-energy and momentum conservation grounds.

Indeed, in the Standard Model, this process must exist and make some contribution to observed neutrino oscillations.

Does neutrino oscillation take place via the same mechanism as flavor changing in quarks? 

As I understand the matter, however, the neutrino oscillation formula noted above is independent of the mechanism by which neutrino oscillation occurs.  So far as I know, this formula does not assume that the process involves this two step weak force process involving a virtual charged lepton and the emission of a virtual W+ and W- boson, at rates dictated by the strength of the coupling of the weak force to ordinary left handed neutrinos and ordinary right handed anti-neutrinos.

The PMNS matrix that governs the transition probabilities of oscillating neutrinos, however, is closely analogous to the CKM matrix that governs the transition probabilities of flavor changing quarks that emit W bosons.

But, while the CKM matrix is relevant only in the event that a quark emits a W boson, which happens at a rate that is a function of the strength of the quark's coupling to the weak force, nothing about the PMNS matrix implies that the oscillation event that triggers its application involves an emission of a boson by the neutrino or is related to the weak force in any way.

I assume that if the probability of a neutrino flavor change happened matched the prediction from first principles of the weak force, that this aspect of the neutrino oscillation process would be widely discussed, when, in fact, I've never seen anyone assert that this is the case.  So, my assumption without actually doing the calculations myself (which I don't have the expertise to do) is that virtual pairs of weak force interactions contribute only minimally to the overall neutrino oscillation rates that are observed.

When are particles prone to state changing and mixing?

Looking at the matter from a forest level view, as opposed to a detailed Standard Model explanation of each individual case, a pattern emerges.  Certain kinds of physical systems that share key properties with neutrinos seem to be particularly lightly tied to a particular state and prone to easily take on new or mixed states.

As particles (fundamental and composite alike) become charged and gain higher mass, their particle character tends to be very well defined and if they are transformed into something else with equal or lesser mass-energy, the transformation tends to take places decisively in a single direction (e.g. from higher rest mass to lower rest mass, rather than visa versa).

In contrast, particles with neutral electric charge, particularly when they are less massive, tend to be less well defined as distinct particle states and become more prone to oscillating or mixing different particle states that are similar in mass.

1.  As I noted in my previous post at this blog, many neutral pseudo-scalar mesons that have no charm or bottom quarks are proposed to be linear combinations of electric and color charge and flavor neutral pseudoscalar quarkonium pairs of up, down and/or strange quarks (each of which is much lighter than mesons containing charm or bottom quarks), to the point where the only observable state is the blended linear combination.

In particular, it appears that scalar mesons and axial vector mesons (aka pseudovector mesons) also arise from the mixing of pseudoscalar quarkonium pairs involving lighter quark types (up, down and strange), but on a much more extreme scale.  In contrast, less neutral vector quarkonium mesons (with spin and isospin 1) made of lighter quarks tend to be well defined and distinct particle states that aren't as prone to mixing. Similarly, mixing may explain why free glueballs are not observed; they mix with similar mesons.

2.  Electroweak theory supposes that the four electroweak bosons (the photon, W+, W- and Z boson) have their source in four massless bosons (the W+, neutral W, W- and neutral B bosons) which acquire mass and experience spontaneous symmetry breaks that causes the neutral W and neutral B bosons to coalesce into the massless photon (γ) and Z boson through the Higgs mechanism.  The photon and Z boson in this model are a linear combination of the neutral W and neutral B bosons that is a function of the trigonometric functions of the weak mixing angle (the cosine of which is equal to the mass of the real world W boson divided by the mass of the Z boson).  The W and Z bosons then "eat" three of the four Goldstone bosons leaving the Higgs boson as the sole remaining Goldstone boson of the weak force.

Notably, for these purposes, in electroweak theory the neutral W and B bosons that are linearly combined are both electrically neutral and massless prior to becoming a photon and Z boson via spontaneous symmetry breaking.

3. I have previously noted the manner in which the Higgs boson appears in some ways to resemble some sort of composite of the four electroweak bosons (the W+, W-, Z and photon), with the same aggregate electric charge, their aggregate mass divided by the square root of four (the number of particles involved), and so on.

4.  CP violation tends to be seen in the decays of relatively light particles with neutral electric charge such as the neutral kaon, although it is also observed in the much heavier neutral B meson and neutral D meson.  There are early experimental indications that CP violation exists in the mixing of the highly chiral active neutrinos as well.

5. In theory, a neutral fermion that is its own anti-particle (something that is impossible for a charged fermion), which some physicists argue could include the neutrino (something I think is highly unlikely because of the important of distinction between particle v. antiparticle character for neutrinos), can acquire what is called Majorana mass in a process akin to state mixing of similar particles.  Quasi-particles in superconducting systems seem to behave in this manner.

6. The search is on for any evidence of neutron and anti-neutron oscillation (although I suspect that this simply doesn't happen absent neutron annihilation class energies).

7. Up and down quarks, which have nearly degenerate mass, are particularly prone to mutate into each other via flavor changing weak force process.  Indeed, these transitions are the only flavor changing processes routinely observed in nature today.

8. Supersymmetry theories, generically, have a sparticle spectrum that does not match one to one with the Standard Model particle spectrum as a result of mixing that is predicted to occur in the sparticle spectrum.

Towards Oscillation As A Highly Suppressed Default Rule, Rather Than An Exception

Conceptually, it isn't entirely unfair to think of particles with neutral electric charge (and even more so, those with true scalar or otherwise even parity spins) as being less firmly tied to the direction of time than charged particles (particularly fermions) with odd parity.

Put another way, perhaps it is useful to think about everything that changes between one particle (be it composite or fundamental) state and another as a barrier to that transition that requires energy to change.  So, the less a particle has to change to reach a new state, the more likely it is to happen.

In the case of neutrinos, the absolute scale of the mass differences between neutrino mass states is so tiny relative to the differences between any other two particles with distinct masses (just dozens of meV or less, in each case), and there is never any change in electric charge or weak coupling strength or color charge or parity between neutrino mass states, so the barriers to oscillation are lowest for these particles and are so low that their mixings with each other are particularly great.

It might make sense to think of neutrino oscillation as the norm and default rule, and of all of the other particles as exceptions to the general principle that particles freely oscillate into each other, with their oscillations suppressed by their electric charge, color charge and much greater mass differences.

Or, perhaps one might go so far as to say that each of these features makes them more wave-like and less point-like than other particles in this context, and that these aspects involve a continuum between extremes rather than a discrete either or choice (something that is indeed true in general in quantum mechanics). Since they are fermions rather than bosons, neutrinos can not react the absolute extreme of literally mixing with each other at the same time and place, as bosons do, but they can do the next best thing, by freely and frequently oscillating from one flavor state into another.

In this view, hypothetical massless fermions (none of which have ever been observed outside theoretical physics models) would all maximally oscillate with each other infinitely rapidly to the point that all possible kinds of massless electric charge, color charge, parity neutral fermions would be indistinguishable from each other as distinct particles entirely, and neutrinos escape that extreme circumstance only as a result of their mass which is intimately connected to the fact that they have, at least, a weak force charge.

A corollary of this hypothesis would be that hypothetical sterile right handed neutrinos that do not mix with left handed neutrinos by some newly postulated force, would necessarily have zero mass as well. Hence, they would also not interact even via gravity. And, if they had no interactions via any force with anything, they would, by definition, not exist. In this view, massless photons and gluons can exist only because they have electromagnetic and strong force interactions, respectively, with particles that have electric and color charge, respectively, which connect them to the world and endow them with energy.

The "Standard Model" Line

What I have articulated isn't the same as the more technically minded "technical" Standard Model description of what is going on from a mechanism point of view, although it is not equivalent and is also more poetic, less mechanical, and gives goes a bit more deeply than the not terribly introspective typical description of how the Standard Model following the lead of Pontecorvo of PMNS matrix fame describes the process.

A quite comprehensive treatment of a variety of neutrino physics and neutrino oscillation issues from the issues at the foundation of trying to integrate massive neutrinos into the Standard Model from 1987 articulates essentially the same conceptual framework for neutrino oscillation at page 24 that is the consensus view today (by getting into the mud of the material I stated in an earlier footnote that I would disregard):
The oscillations of neutrinos are analogous in their quantum-mechanical nature to K°<=±K° oscillations. Suppose that the state vectors of the neutrinos taking part in the weak interactions (ve,vM,vT,. . . ) are superpositions of the state vectors of neutrinos (Dirac or Majorana) with different masses. 
What would be the behavior of a neutrino beam in this case? 
It is clear that at some distance from the source of neutrinos of a given type, the state vectors of neutrinos with different masses (because of the difference in the masses) would acquire different phase factors. The state vector of a neutrino would then be a superposition of the state vectors of neutrinos of different (all possible, in principle) types. It is obvious that the probability of finding a neutrino of a given type would be a periodic function of the distance between the source and the detector. This phenomenon was called neutrino oscillations (Pontecorvo, 1957, 1958). 
In order for oscillations of neutral kaons, neutrinos, etc. to be possible, the following conditions have to be realized: (i) the particle interaction Lagrangian should contain terms that preserve some quantum numbers (strangeness in the case of kaons, lepton numbers in the case of leptons, etc.); (ii) the total Lagrangian (and the mass term) should not be diagonal with respect to these quantum numbers, and the relevant quantum-number non-conserving couplings should be much weaker than those preserving the quantum numbers. The states with definite mass (and width) would then be superpositions of states possessing definite strangeness in the case of neutral kaons, definite lepton numbers in the case of neutrinos, etc.
In other words, neutrino oscillation happens because neutrino flavors for purposes of the weak force and neutrino mass states are two separate things with separate sources in the Standard Model equations that will get out of synch with each other in an oscillating fashion in a way that leads neutrinos to change flavor now and then in a predictable way over time.

Oscillation is a fundamental property of neutrinos because the three weak interaction states of a neutrino at its origin are fundamentally constructed in the Standard Model as a mix of the three different mass states that are inherently indeterminate.

This oscillation is conceptually more like the process of the wave function of probability amplitudes regarding other properties of a quantum particle (such as its location) collapsing when a particle is measured that produces random outcomes regarding its final destination which are distributed proportionately to the square of the probability amplitudes of a particle being located at a particular place, as opposed to the kind of flavor changing process seen in flavor changing W bosons interactions. It is a part of the equation of how a neutrino particle propagates rather that an interaction.

Later in the same treatment from 1987, at footnote 22, the paper also notes that in that case of higher generation neutrinos with masses of under 100 eV, which could experience "radiative decay" just as the the charged leptons do into a lighter lepton of the same electric charge and lepton number together with a photon, but that they are metastable because they have predicted mean lifetimes longer than the age of the universe (i.e. 13.6 billion years or so). So, the neutrino oscillation process is distinct in source from the flavor changing decays of heavy charged leptons into lighter charged leptons.

This analysis is also partially stymied by the same uncertainty regarding the origin of neutrino mass (Dirac, Majorana or Pseudo-Dirac) that still remains an open question today, 27 years later, despite all of the experimental advances that we have made since then.  Each leading approach has its own problems and is an awkward fit to the overall Standard Model framework, although the Dirac mass fix is somewhat more minimalist.

A New (or Old) Neutrino Oscillation Boson?

Another possibility is that neutrino oscillation actually does involve flavor changing interactions triggered by the emission of a boson much like that of flavor changing charged W boson interactions, with some boson other than the W boson. Perhaps a flavor changing neutral current is mediated by virtual Z bosons in a manner that is somehow suppressed completely or nearly completely in charged particles, but not in the electrically neutral fundamental particles that are neutrinos.

Or, perhaps there might be some undiscovered, possibly charge-phobic, Z' boson, either massive or without mass, that mediates this interaction in a way analogous to the W boson, which might or might not be part of the electroweak forces. Perhaps this neutrino oscillation Z' boson, if one existed, would have a mass relative to the regular Z boson similar to the mass of the charged leptons relative to their neutrino partners. Thus, the Z' boson might have a mass on the order of 45 eV as opposed to the approximately 90 GeV of the Z boson.

Perhaps the charge phobic Z' boson could also be the boson that mediates a force that operates between neutral dark matter particles.  A charge-phobic boson would also not be impaired in its ability to mediate interactions between electrically neutral dark matter particles.  The sweet spot for a "dark photon" mass is about 1 MeV to 100 MeV in the kind of sophisticated self-interacting dark matter model that comes closest to reproducing the kind of dark matter halo shapes inferred from astronomy observations of galaxies.  This is well within the workable range for a Z' particle that would mediate neutrino oscillations as well, however. (A recent study by Matt Strassler, et al. has established that any such Z' boson cannot produce significant numbers of charged leptons if produced in Higgs boson decays, but this wouldn't be expected in the charge-phobic Z' case.)

More radically, in the case of a massless Z', one could imagine the Z' boson as a neutral W boson that somehow escaped becoming a part of the weak mixing angle linear combination that creates the photon and Z boson of electroweak unification.  This would not, however, be a good fit to a dark photon in a sophisticated self-interacting dark matter model of the kind necessary to come close to reproducing the dark matter halo shapes that are inferred from astronomy observations.

At 45 eV, it would ordinarily be prevented by mass-energy conservation from decaying into anything other than pairs of neutrinos and anti-neutrinos, but at anywhere from about 10 to 100 MeV, it would be massive enough to also decay to pairs of first generation quarks or charged leptons, and even at 1.2 MeV it would be massive enough to decay to pairs of electrons.  But, if the boson was truly charge-phobic, it would not do either of these things because it does not interact with charged particles since it is charge phobic so it would create no quark pairs or charged lepton pairs even if had enough mass-energy to do so.

Experimental Tests For Boson Mediated Neutrino Oscillations

There would be tell tale sign of neutrino oscillation mediated by any form of boson, as opposed to neutrino oscillations that is not mediated by another particle. For example, in the unmediated scenario, all changes in rest mass due to neutrino oscillations would be converted into or out of kinetic energy while maintaining momentum in the same direction. But, in the case of boson facilitated neutrino oscillations, the aggregate changes in neutrino beam mass due to neutrino oscillations would differ from the aggregate changes in neutrino beam velocity due to that adjustment in its kinetic energy.

Another experimental signature of boson mediated neutrino oscillations, particular in the light Z' boson case, would be a bump in the number of neutrinos and antineutrinos that is equal in magnitude that have combined mass-energy equal to half of the Z' boson mass.

A third experimental signature of boson mediated neutrino oscillations that would be more model specific would be a suppression of neutrino oscillations within strong electromagnetic fields, if the Z' boson was charge-phobic. This is more of a long shot and not very general, but has the virtue of being quite easy to test experimentally by placing a powerful electromagnetic field along the path of the neutrino beam leaving a nuclear reactor for which there is already a baseline level of neutrino events in place. This could even be scheduled to be turned off and on so that the experiment would use the reactor's own electricity during times when the demand for electricity from the grid was low and turn off during periods of high electrical grid demand, providing data sets that were naturally robust to effects associated with seasonal change during the year or differences in neutrino production from year to year for a variety of hard to determine reasons (e.g. a natural aging cycle of the nuclear fuel used at the reactors).


* There is a technical distinction between the three "weak force" neutrino flavors and the three neutrino mass eigenstates. This footnote acknowledges that I am deliberately ignoring that distinction in this discussion for those who understand the distinction, in the interest of clarity for a lay reader. I do not believe that this intentional imprecision does great injustice to the gist of the idea discussed in this post, except in setting for the "Standard Model" explanation as I do later on.

Wednesday, February 19, 2014

Hadron Nomenclature

Hadrons are composite particles made up of fundamental particles that are bound together by the strong force. The two kinds of hadrons that have been observed experimentally are mesons (made up of pairs of quarks and anti-quarks) and baryons (made up of three quarks each).

No observed hadrons include top quarks, but names for such particles exist. Hypothetical hadrons with four quarks and no constituent mesons are called tetraquarks (and would be bosons just like mesons, but with potentially higher spin). Hypothetical hadrons with five quarks and no constituent mesons or baryons are called pentaquarks (and would be fermions just like baryons but with potentially higher spin).

Hypothetical hadrons with no quarks are called glueballs aka gluonium.  All glueballs would be bosons. Glueballs ground states are distinguished from each other by their quantum numbers, and they are usually categorized by total angular momentum which is always an integer value since they are bosons made up of spin-1 bosons called gluons combined together into a composite particle (spin-0 is scalar or pseudo-scalar, spin-1 is vector, and spin-2+ is tensor).

Issues related to modeling these hypothetical hadron states and fitting them to data are discussed, for example, in this power point presentation.

A more authoritative, but less fully spelled out and contextualized summary of hadron nomenclature can be found in a review article by the Particle Data Group.

Meson Nomenclature

A meson is a boson with spin-0 (i.e. total angular momentum which is quantum number J) is called pseduo-scalar in the ground state (which implies that the quarks in the meson have oppositely aligned spins) or spin-1 called vector (which implies that the quarks in the meson have spins aligned in the same direction).

A meson made of a quark with a particular color charge, and an anti-quark of the anti-color charge of the same type, or of a linear combination of these color charge neutral quark and anti-quark pairs.

Meson Types By Spin and Quark Content.

In a nutshell, pions (pseudo-scalar) (Greek letter pi), rho mesons (vector) (cursive p) and omega mesons (vector) (cursive lowercase w) have only up and down quarks.

Kaons (pseudo-scalar or vector) (K) and eta mesons (pseudo-scalar) (cursive n) have strange quarks and light quarks (i.e. up and/or down quarks). Phi mesons (vector) (three pronged Y) are made of strange quark pairs. Charged kaons have strange and up quarks, neutral kaons have strange and down quarks.

D mesons (pseudo-scalar or vector) (D) are made of charm quarks and a lighter quark and are called "strange" if the other quark is a strange quark. D mesons with up quarks are electrically neutral, D mesons with down type quarks are charged. Charmed eta mesons (pseudo-scalar) and J/Psi mesons (vector) are made of charm quark pairs. Both of these states are also called "charmonium".

B mesons (pseudo-scalar or vector) (B) are made of bottom quarks and a lighter quark, and are called "charmed" if the other quark is a charm quark, and "strange" if the other quark is a strange quark. B mesons with up or charm quarks type quarks are charged; B mesons with down or strange quarks are electrically neutral. Bottom eta mesons (pseudo-scalar) and Upsilon mesons (vector) (Y) are made of bottom quark pairs. Both of these states are also called "bottomonium".

Other Meson Types

In addition to the familiar quantum number L even mesons discussed elsewhere in this section (the pion, eta and eta prime pseudo-scalar mesons; the rho, omega or φ, J/psi, upsilon and theta vector mesons; D mesons, B mesons and T mesons), there are corresponding quantum number odd mesons with the same total angular momentum J, and in the case of neutral mesons, the same quantum number C, as their more familiar counterparts.  In JPC notation, + means "even" and - means "odd".

Neither the true scalar mesons nor the pseudoscalar and/or axial vector mesons surrender easily to simple construction as a simple single quark and anti-quark pair.

Scalar Mesons

The symbols "a" (isospin 1) and "f" or "f'" (isospin 0) apply to mesons with ground state JPC quantum numbers 0++ which are also known as (true) scalar mesons.  The Greek letter capital Chi (which looks like a cursive capital X) is used together with subscripts to describe heavy quarks that the scalar meson contains.  Wikipedia notes in part with regard to scalar mesons that:
The light (unflavored) scalar mesons may be divided into three groups; those having a mass below 1 GeV/c2, those having a mass between 1 GeV/c2 and 2 GeV/c2, and other radially-excited unflavored scalar mesons above 2 GeV/c2. The heavier scalar mesons containing charm and/or bottom quarks all occur well over 2 GeV/c2. Many attempts have been made to determine the quark content of the lighter scalar mesons; however, no consensus has yet been reached. 
The scalar mesons in the mass range of 1 GeV/c2 to 2 GeV/c2 are generally believed to be conventional quark-antiquark states with orbital excitation L = 1 and spin excitation S = 1,[1] although they occur at a higher mass than one would expect in the framework of mass-splittings from spin-orbit coupling.[2] The scalar glueball [3] is also expected to fall in this mass region, appearing in similar fashion to the conventional mesons but having very distinctive decay characteristics. The scalar mesons in the mass range below 1 GeV/c2 are much more controversial, and may be interpreted in a number of different ways. 
Since the late 1950s, the lightest scalar mesons were often interpreted within the framework of the linear sigma model, and many theorists still choose this interpretation of the scalar mesons as the chiral partners of the pseudoscalar meson multiplet.[4] Ever since Jaffe first suggested the existence of tetraquark multiplets in 1977,[5] the lightest scalar mesons have been interpreted by some theorists to be possible tetraquark or meson-meson molecule states. The tetraquark interpretation works well with the MIT Bag Model of QCD,[6] where the scalar tetraquarks are actually predicted to have lower mass than the conventional scalar mesons. This picture of the scalar mesons seems to fit experimental results well in certain ways, but often receives harsh criticism for ignoring unsolved problems with chiral symmetry breaking and the possibility of a non-trivial vacuum state as suggested by Gribov.[7] 
In-depth studies of the unflavored scalar mesons began with the Crystal Ball and Crystal Barrel experiments of the mid 1990s, focusing on the mass range between 1 GeV/c2 and 2 GeV/c2. With the re-introduction of the sigma meson as an acceptable candidate for a light scalar meson in 1996 by Tornqvist and Roos,[8] in-depth studies into the lightest scalar mesons were conducted with renewed interest. The "Particle Data Group" provides current information on the experimental status of various particles, including the scalar mesons.
Pseudovector Mesons aka Axial Vector Mesons

The symbols "b" (isospin 1) and "h" or "h'"(isospin 0) apply to pseudovector mesons aka axial vector mesons with ground state JPC quantum numbers 1++.

A recent (October 2013) paper on the subject of these mesons suggests that their nature is dreadfully complicated and that they are the subject of multiple distinct theoretical approaches to explaining them.  An extended linear sigma model such as one proposed in 2012 seems to explain them (and scalar mesons).  A power point presentation such a model can be found here.  Another which is somewhat more clear is here.

In a nutshell, scalar mesons and pseudovector or axial vector mesons are interpreted as a rather elaborate mixes of quarkonium states of pairs of up quarks and anti-up quarks, down quarks and anti-down quarks, and strange quarks and anti-strange quarks.

Unclassified Mesons

The symbol X applies to mesons when its properties cannot be determined completely enough to assign a meson type to it.

Hypothetical Top Quark Mesons

Hypothetical mesons including top quarks would follow the pattern for D and B mesons and presumably would be called T mesons and top eta mesons, and a hypothetical vector meson made of a top quark and anti-top quark pair would be a theta meson. Both of these states would also be called "toponium."

Suppose that such mesons did exist (and no such mesons have been observed to date), even ever so briefly and ever so infrequently, because top quarks decay via the weak force on average about twenty times as fast as the speed with which hadronization occurs on average.  The hypothetical T mesons could be no lighter than about 173 GeV (the mass of a single top quark compared to which the mass of a up, down or strange quark, or the minimum binding energy of a pseudo-scalar meson might be negligible), and any form of toponimum would have a mass in excess of 346 GeV.

Also, even if a T meson or top eta meson or theta meson were created by physicists, it might be difficult to distinguish this decay signature from the decays of a pair of bare top quarks that immediately decayed hadronically.

Mesons That Are Linear Combinations

Certain neutral mesons are linear combinations of pairs of quarks and anti-quarks of the same type.

The combinations of spin-0 quark-anti-quark pairs are the neutral pion (uu-dd/sqrt(2)), the K-short neutral kaon (d-anti-s minus s anti-d/sqrt(2)), the K-long neutral kaon (d-anti-s plus s anti-d/sqrt(2)), the eta meson (uu+dd-2ss/sqrt(6)), the eta prime meson (uu+dd+ss/sqrt(3)). The two neutral kaon types mix with each other.

The combinations of spin-1 quark-anti-quark pairs are the neutral rho (uu-dd./sqrt(2)), and the omega meson (uu+dd/sqrt(2)).

Models of scalar mesons and axial vector mesons suggest that many resonances that currently cannot be easily fit into simple quark models are linear combinations of quarkonia.

Anti-particles of mesons

Mesons that are linear combinations or are made of a quark and an anti-quark of the same type are their own anti-paticles. Charged mesons and neutral mesons with mixed quark flavors that are not linear combinations have antiparticles.

The State of Experimental Meson Observation

Which mesons have we observed so far?

The masses of the ground states of the mesons predicted by the Standard Model range from the 0.135 GeV of the neutral pion to the upsilon meson which has a mass of 9.46 GeV is the heaviest meson ground state that is possible in the Standard Model.

Many excited meson states of mesons whose quark content and spin have been determined experimentally have also been observed, with the heaviest being one of the excited states of the upsilon meson with a mass of 11.02 GeV.  This is about 1.56 GeV heavier than the ground state of the upsilon meson which is more than twice the 1.1 GeV of mass in excess of the bottom quark and anti-bottom quark in the ground state that it is made of due to the mass-energy of the gluons that bind the ground state of the upsilon together.

This heaviest excited upislon mesons is the heaviest hadrons ever observed experimentally to date.  No meson ground state, excited meson, or baryon with a mass in excess of 6.3 GeV, other than bottomonium ground states and excited states, have ever been observed.   The Standard Model predicts, however, that it is possible to create some baryons with two bottom quarks, and certainly the triple bottom omega baryon (which the Standard Model predicts is the heaviest possible hadron that does not involve a top quark) that are heavier than the upsilon meson.

Consider the case of mesons composed of pairs of the two heaviest flavors of quarks that form bosons. There are two ground states of charmonium (the charmed eta meson and the J/Psi meson) and the two ground states of bottomonium (the bottom eta meson and the upsilon meson) that have been observed.  But, high energy physicists have also observed about twenty-three excited charmonium states and thirteen different excited bottomonium states that have observed precisely enough to measure their masses (not all of which have been fully characterized in terms of quantum numbers), and described in academic publications.  Not all of these observations are definitive or complete discoveries, however.  For example, physicists have not yet fully characterized the quantum numbers of eleven of the twenty-three excited charmonium states and three of the thirteen excited bottomonium states which have been experimentally observed.

What mesons remain to be observed and/or characterized?

The only kind of meson ground state predicted by the Standard Model that has not yet been observed is the vector charmed B meson (made up of a charm quark and a bottom anti-quark or visa versa) which should have a mass of about 6.29-6.35 GeV (which is slightly more than the mass of the pseudo-scalar charmed B meson which has a mass of 6.28 GeV and the same quark content based on the mass differences between other pseudo-scalar and vector mesons with the same quark conduct).

A discovery of this last Standard Model meson ground state at the LHC is probably right around the proverbial corner after the LHC powers up again in 2015, because there is only one other hadron, the pseudo-scalar charmed B meson, and there are no other fundamental particles or pairs of fundamental particles, in that mass vicinity, to provide background noise, the signal physicists are looking for is very well understood (which allows searches to use very tight data cuts), and another new hadron mass state was just discovered in 2012 at 5.945 GeV, so the LHC has sufficient power to make observations in this mass range.

More interestingly, there are many unclassified resonances which are possibly mesons that have not been fully characterized or have quantum numbers different from the meson ground states (e.g. about fifteen excited meson states that have at least one heavy quark have been observed to have tensor spins of J=2 to J=5).  There are many unclassified meson states made up only of light quarks with masses from 0.5 GeV to about 2.5 GeV which may reflect not yet described linear combinations of mesons, tetraquarks, "meson molecules", or particularly high order excited states of light mesons that experiments have not had enough energy to produce for mesons with heavier ground states.

This is some of the most interesting unexplored territory of meson physics that is within the realm of what can be investigated experimentally.  But, only a quite small share of theoretical and phenomenological physics scholarship is devoted to exploring exotic hadronic states like these, either within the framework of QCD or some beyond the Standard Model modification of Standard Model QCD.

Why is it so hard to observe and characterize relatively light meson resonances?

The masses at which high energy physicist can consistently and completely observe and characterize heavy hadrons lags an order of magnitude behind their ability to observe and characterize heavy fundamental particles such as the W boson, Z boson, Higgs boson and top quark which are seven to fifteen times as heavy as the heaviest experimentally observed hadron.

There is still one suspected meson resonance lighter than a proton, with a mass of 0.5 GeV, that have not been fully described and characterized.  So, the fact that there are light hadron resonances that have not yet been fully described isn't as surprising as it seems.

All but six of the masses (only one of which, f(500), is not well characterized yet) of the scores and scores of exotic hadrons and excited hadron states observed to date (all of them other than fifteen bottomonium states) lie in the crowded mass range between the proton mass of 0.938 GeV and 6.4 GeV.  This mass range with a peak mass less than 7 times as great as the low end mass where all the action is in hadronic physics, grows more crowded at the low end of that range.  So, teasing out the single of a particular suspected hadron state from the thicket of all of the other hadron resonances with similar masses that emerge from a particle collider operating at a given energy scale is not an easy task.  The limited precision of QCD calculations to date makes it even harder to separate signals from backgrounds, or to properly characterize a resonance that is observed.

In contrast, the fundamental particles have masses spread out over a much more vast range, from electron neutrinos with a mass of about 1 meV to top quarks with a mass of about 173 GeV.  The top of this range is about 100,000,000,000,000 times as great as the bottom of this range.

Simlarly, the only two fundamental particles that have masses that are within experimental error bars of each other are the strange quark and the muon.  In contrast, there are a dozen or so sets of hadrons with the same total angular momentum and same number of quarks that have distinct masses that differ by 1% or less (which is about the accuracy of current theoretical calculations of hadron masses from first principles in QCD).

On the other hand, measuring the mass of an observable composite hadron or fundamental charged lepton that can actually be produced in a laboratory is much easier than determining the mass of a light quark which can never be directly observed even in ideal conditions due to the fact that quarks are confined within hadrons.

Baryon Nomenclature

Baryons, which have three quarks each and can have total angular momentum (J) commonly called spin of 1/2 or 3/2 in the ground state. They can be made of:

* three identical flavor quarks (in which case they must have spin-3/2),
* two quarks of one flavor and one of another (in which case there is one combination each of spin-1/2 or spin-3/2), or
* three quarks of different flavors (in which case there are two possible spin-1/2 combinations and one possible spin-3/2 combination) (this is not possible for particles made only of the two light quark flavors).

Ordinary baryons have three quarks, one with each color charge, and no antiquarks. Every baryon has a corresponding antibaryon made of three antiquarks, one of each anticolor charge, with an opposite electric charge and parity.

While there are several meson resonances that are linear combinations of multiple quark pairs, there are no known baryons are linear combinations of different baryon combinations.  This may be because they are fermions rather than bosons, although I don't really know for sure why this is the case.

The name of a baryon is a function of its isospin, i.e. the total angular momentum J, after disregarding spin attributable to quarks other than the up and down quarks in the baryon. Up and down quarks have isospin 1/2 which can be positive or negative for each such quark. Strange, charm, bottom and top quarks have isospin 0. The existing categories of baryons are as follows:

1.  A nucleon (isospin 1/2) (always J=1/2) and delta baryons (isospin 3/2) (triangle) (always J=3/2) each have three light quarks.

All six of these ground states which are predicted by the Standard Model have been observed.

2.  A lambda baryon (isospin 0) (upside down V) (J=1/2) with an up and a down quark of opposite spins, and one heavy quark, and a sigma baryon (isospin 1) have two light quarks and one heavy quark (J=1/2 or 3/2).

All of these except the bottom sigma (which has not been observed in either J=1/2 and J=3/2 form) have been observed experimentally.  Thus, nineteen of the twenty-one ground states of these kinds of baryons which are predicted by the Standard Model have been observed.  These are expected to be two of the heaviest sigma baryons.

3.  A chi baryon (isospin 1/2) (three horizontal lines) (J=1/2 or 3/2) have one light quark and two heavy quarks.

All four chi baryons without charm and bottom quarks, and all four chi baryons with one strange quark and one charm quark have been observed experimentally.  Both of the J=1/2 bottom chi baryons and one of the two J=3/2 bottom chi baryons have been observed experimentally.  One of the two J=1/2 double charmed chi baryon has also been observed.

Scientists have not yet observed one of the J=3/2 bottom chi baryons, one of the J=1/2 double charmed chi baryons, both of the J=3/2 double charmed baryons, any of the four of the charmed bottom chi baryons, or any of the four of the double bottom chi baryons.

Thus, exactly half of the twenty-four kinds of chi baryon ground states have been observed so far.  In general, the baryons not yet observed are the heaviest ones.

4.  An omega baryon (isospin 0) (J=1/2 or 3/2) has three heavy quarks.

Two of the eight omega baryon ground states predicted by the Standard Model with J=1/2 have been observed (the charmed omega which has two strange quarks and a charm quark, and the bottom omega with two strange quarks and a bottom quark).   Two of the ten omega baryon ground states predicted by the Standard Model with J=3/2 (the plain omega baryon made of three strange quarks and the charmed omega with two strange quarks and one charm quark) have been observed.  Thus, four of the eighteen kinds of omega baryon ground states have been observed.

The State of Experimental Baryon Observations

Which baryons have been observed?

The project of experimentally observing all baryon ground states predicted by the Standard Model is significantly less far along than the parallel project for mesons.

Forty-one of the sixty-nine baryon ground states predicted by the Standard Model have been observed experimentally. The lighest observed baryon, the proton, has a mass of 0.938 GeV.  The heaviest observed baryon, the bottom chi meson observed at the LHC in 2012, has a mass of 5.945 GeV.

All twenty-eight of the unobserved ground states involve baryons with at least one bottom quark or two charm quarks.  Physicists have observed and characterized the ground state of every single hadron composed of two or three quarks that contain only up, down or strange quarks that is predicted to exist by the Standard Model, and every hadron with a single charm quark and also doesn't have a bottom quark.

Physicists have also observed other excited baryon states and have observed some resonances that are probably unobserved baryons of some type which have not yet been definitively classified.  But, I get the impression that there are considerably more unexplained and excited meson states than there are excited and unclassified baryon states that have been observed to date.

Which baryons have not yet been observed?

All of the undiscovered hadron ground states contain (1) at least one bottom quark (the vector charmed B meson, and twenty-two kinds of baryons containing bottom quarks), and/or (2) two or more charm quarks (the two kinds of double charmed Omega baryons and three of the four kinds of double charmed Xi baryons, and the triple charmed omega baryon).

We are making progress on these frontiers of hadronic physics, however.  We have observed the ground states of eight of the thirty baryon ground states predicted to exist in the Standard Model that contain at least one bottom quark (although no double bottom quark baryons have been observed yet), and nine out of the ten meson ground states that contain at least one bottom quark (including both of the meson ground states that contain two bottom quarks).  We have observed both of the meson ground states that contain two charm quarks and one of the seven baryons ground states predicted to exist in the Standard Model that contain two or more charm quarks but no bottom quarks.

Based upon the masses of the doubly charmed mesons and mesons containing bottom quarks observed to date and the mass of the bottom quark, the undiscovered doubly charmed baryons without bottom quarks have masses in a bit excess of 3.5 GeV, the undiscovered baryons that contain one bottom quark have masses in excess of 5.8 GeV, double bottom baryons (none of which have been observed yet) have masses in excess of 10 GeV, and the triple bottom omega baryon (which should be the heaviest possible baryon that does not include a top quark) should have a mass in excess of 14.2 GeV, but probably not much more than 24 GeV.

It is reasonable to expect that only the double and triple bottom baryons will remain unobserved experimentally by the time that the LHC completes its run, and with luck, even some of those may be detected at the LHC.

Why is it harder to create exotic baryons in the lab than it is to create exotic mesons?

I do not know, but suspect, that this may be a product of the fact that exotic hadrons of all kinds are extremely unstable and that it may be harder to form baryons (which must be made of either three quarks or of three antiquarks) than it is to form mesons (which must be made of one quark and one anti-quark) in a particle collider environment where particles and anti-particles are produced in almost identical quantities.

For example, for a double bottom omega baryon to form, the collider must have generated at least two bottom quark pairs and a strange or charm quark pair, and realistically lots of other stuff as well in multiple jets of decay products.  In contrast, production of a single bottom quark pair can generate the quark content necessary to create bottomonium.

Subscripts and superscripts and further specifications in hadron nomenclature


Particles with charmed in the name have the subscript lower case c added to their symbol. Mesons with strange in their name have the subscript lower case s added to their symbol. Short kaons have subscript upper case S added to the K symbol, while long kaons have subscript upper case L added to their symbol. Baryons with charm or bottom quarks have one subscript for each charm or bottom or (hypothetically) top quark that is present (lower case c, b or t respectively).

Particles with total angular momentum J that is a subscript for all mesons except pseudoscalar and vector mesons.


Charged mesons have a superscript with a + or - indicating their electric charge. Neutral pions, kaons, D mesons and B mesons have superscript zero. The superscript zero is omitted for omega mesons, eta mesons, phi mesons, J/Psi mesons, and upsilon mesons which always have neutral electrical charge.

All baryons but neutrons (which always have neutral electric charge) carry a superscript identifying their electric charge. Baryons with electric charge +2 have superscript ++, while baryons with electric charge -2 have superscript --.

Mesons that can be either pseudo-scalar or vector (i.e. kaons, D mesons and B mesons) have an asterisk superscript if they are vector rather than pseudo-scalar (i.e. if they have quarks with aligned spins and thus spin-1, rather than quarks with complementary spins and thus spin-0).

Baryons of a type that can have total angular momentum J=1/2 or 3/2 (i.e. sigma, chi or omega baryons) have an asterisk superscript is they have total angular momentum J=3/2.

Additional Specification After A Particle Symbol

Sometimes a symbol for a kind of hadron is followed by a number in parentheses. This number is its mass in units of MeV/c^2. When quantum numbers are known for a resonance whose classification has not been determined it is typically included after the mass number, if not otherwise described, in JPC format.

Every hadron, in principle, also has an infinite number of higher mass, excited, higher energy states in addition to a ground state which are sometimes described with additional notation that (to be perfectly frank) I don't fully understand as a mere amateur science enthusiast.

Footnote on Exotic Hadrons

In this post, I have used the term "exotic hadrons" in a broad sense to encompass all hadrons which are not found in nature at this point in the history of the universe.  In other words, hadrons which would be exotic to you and me.  This includes almost all hadrons except the proton, the neutron, the pion, and maybe a handful of other mesons with light quarks or strange quarks.  This includes all of the highly unstable hadrons that contain charm or bottom quarks.

There is also a narrower sense of the phrase "exotic hadrons", which means hadronic resonances whose properties cannot be explained as a composite particle made up of two or three quarks, depending upon its total angular momentum J.  There are stronger indications of exotic meson resonances in this sense than there are of exotic baryons in this sense.  With regard to "exotic meson" reasonances, the Particle Data Group discussion referenced at the start of this post states:
Gluonium states or other mesons that are not qq states are, if the quantum numbers are not exotic, to be named just as are the qq mesons. Such states will probably be difficult to distinguish from qq states and will likely mix with them, and we make no attempt to distinguish those “mostly gluonium” from those “mostly qq.”
An “exotic” meson with JPC quantum numbers that a qq system cannot have, namely JPC
0−−, 0+−, 1−+, 2+−, 3−+, · · · , would use the same symbol as does an ordinary meson with all the same quantum numbers as the exotic meson except for the C parity. But then the J subscript may still distinguish it; for example, an isospin-0 1−+ meson could be denoted ω1.
Wikipedia discusses exotic mesons here.

The Particle Data Group discussion referenced at the start of this post has the following discussion of ""exotic baryons" of that type:
8.5. Exotic baryons
In 2003, several experiments reported finding a strangeness S = +1, charge Q = +1 baryon, and one experiment reported finding an S = −2, Q = −2 baryon. Baryons with such quantum numbers cannot be made from three quarks, and thus they are exotic. The S = +1 baryon, which once would have been called a Z, was quickly dubbed the Omega(1540)+, and we proposed to name the S = −2 baryon the Omega (1860). However, these “discoveries” were then completely ruled out by many experiments with far larger statistics: See our 2008 Review [2].
As of early 2014, no such exotic hadrons have been observed and characterized.  The most exotic resonances discovered so far (using the term in an intermediate and relative sense) are suspected "meson molecules" which I have discussed in previous posts at this blog.

Tuesday, February 18, 2014

Abstract of the Week

Three families of quarks and leptons, one Higgs to rule them all, and in the darkness bind them.
From here (the allusion, of course, is to J.R.R. Tolkein's Lord of the Rings series).

Monday, February 17, 2014

To Do: QCD States That Violate The Constituent Quark Model

In some of my recent reading I've come across a fascinating possibility for exotic hadrons that hasn't gotten enough attention.  But, I don't have time at the moment to fully discuss it.

The core idea is that there a hadron states that are not inconsistent with the Standard Model QCD Lagrangian which are inconsistent with the "constituent quark model".  In other words, there might be some hadrons that are neither made up of a specific combination of an integer number of quarks, nor a linear combination of such integer numbers of quarks, and that are also not glueballs.  Thus, some hadrons might not be possible to describe either in terms of quark components, or as glueballs.  Or, at any rate, might not have a straight forward connection to their quark components, for example in the case of hypothetical hybrid mesons (see also Section 14.3 of this PDG review).

There are no hadron resonances definitively identified with such states, although there are some suspected resonances that might fit this explanation which are being studied further at this point.

While this isn't strictly speaking "beyond the Standard Model" new physics since it simply draws on existing, consensus QCD equations, it would be a huge paradigm shift in terms of what kinds of exotic particles are possible in the universe that is almost completely unexplored so far.

I'll post more when I have it doped out better and I am in a better position to cite to a coherent set of scholarly papers on the subject.  For example, this power point seems to clarify that the constituent quark model is really just a low energy simplification of QCD that omits the impact of gluons and "sea quark-antiquark pairs."  These arise in more sophisticated "parton" models.

The Reactor Anomaly Might Not Be Real

Last Thursday (February 13, 2014), Boris Kayser at the Fermi laboratory published a very informative and brief (three pages including caption, abstract and end notes) pre-print entitled simply "Are There Sterile Neutrinos."

This paper summarized the state of the evidence for one or more light massive sterile neutrinos in addition to the three massive "fertile" neutrinos that are known to exist.

The Theoretical Background Regarding Sterile Neutrinos

Precision electroweak experiments rule out the existence of more than three light "fertile" neutrinos that interact via the weak force.

But, any positive evidence, even if indirect, of beyond the Standard Model particles is always notable, and these data do not rule out particles that do not couple to either photons or weak force bosons (the W or Z bosons) called "sterile neutrinos" because of their lack of interaction with other particles of nature via any of the Standard Model forces.

This isn't a terribly radical idea, because even in the Standard Model, right handed particles don't interact via the weak force (and likewise, left handed antiparticles don't interact via the weak force).  But, because quarks and charged leptons oscillate between left and right parity modes, and Standard Model neutrinos come only in parties that allow for weak force interactions, the lack of weak force interactions with right parity particles doesn't have many notable consequences until you get into the thick of quantum mechanical calculations.

Massive neutrinos naturally oscillate between different neutrino flavors in a manner that is a function of their respective masses and the probabilities set forth in the PMNS matrix (which can be parameterized into three mixing angles and one CP violation phase with Dirac neutrinos, plus two additional parameters in the case of Majorana mass neutrinos).

The mechanism of neutrino oscillation is not well understood.  For example, it isn't clear if neutrino oscillation is mediated by W bosons or some other kind of boson, even though the relative masses and oscillation frequencies in a three neutrino model are increasingly well understood.  So, it is possible given what we know now about the mechanism of neutrino oscillations that a neutrino that does not interact via the weak force could still oscillate back and forth with fertile neutrino flavors.

Discrepancies between the number of neutrinos observed a certain distances from nuclear reactors (the reactor anomaly) and the theoretically predicted number have been argued to be a better fit to a 3 fertile + 1 sterile neutrino model, in which the fourth neutrino flavor with a mass on the order of 1 eV, can oscillate into other neutrino flavors but does not have weak force interactions.

Data from cosmic background radiation surveys like WMAP and Planck have long pointed to the possibility of one (or, until recently) even two additional massive sterile neutrinos that oscillate as fourth or fifth neutrinos flavors with the three kinds known to the Standard Model.

To be clear, however, these light sterile neutrinos are not themselves dark matter candidates, even though heavier sterile neutrinos (particularly those with masses on the order of 2 keV, are attractive dark matter candidates.  Light sterile neutrinos with masses on the order of 1 eV are too light and too "hot" to fill the role of dark matter in astronomy, although some theories propose three generations of sterile neutrinos, like the other Standard Model fermions, with one or two light sterile flavors enough to count as neutrinos for cosmology purposes and at least one other that serves as a dark matter candidate.

The Latest Evidence Tends To Disfavor A Reactor Anomaly Light Sterile Neutrino

On balance, however, the evidence for a light sterile neutrino flavor that oscillates back and forth with fertile neutrino flavors is getting weaker.

(1) The statistical strength of the reactor anomaly has been reduced to a statistically insignificant 1.4 sigma by the discovery that PMNS matrix neutrino mixing angle θ13 is higher than previously assumed.
The hints of sterile neutrinos include the “reactor anomaly”. This is the observation that the reactor ν¯e flux measured by detectors that are only (10 – 100) m from reactor cores is ∼ 6% below the theoretically expected value. If the missing flux has disappeared by oscillating into another flavor or flavors, this behavior, like that observed in LSND and MiniBooNE, points to a splitting ∆m2 larger than ∼ 0.1 eV2. A recent analysis finds that if one uses the now-known value of the mixing angle θ13, and takes into account ν¯e flux measurements at detectors that are about 1 km from their reactors, the missing flux is reduced to ∼4% of the theoretically expected value, and the significance of the discrepancy is reduced to 1.4 σ. The story of the reactor anomaly no doubt will continue. It would be desirable to see if ν¯e flux that is produced by a radioactive source, rather than a reactor, disappears as well.
The PMNS matrix mixing angle θ13 was once widely assumed to be nearly zero, but is now known to be closer to 9 degrees.  Generally speaking, discrepancies between experimentally data and Standard Model theoretical predictions of two sigma or more are ignored as random noise.

If there is a signal, and not just noise, however (which could be due to weak experimental power because neutrinos are so hard to detect, rather than because a sterile neutrino does not exist) at least one sterile neutrino mass flavor must have a mass that is at least the square root of 0.1 ev2, which is 0.316 eV. Thus, the lightest possible sterile neutrino flavor in a 3+1 neutrino flavor scenario that fits the reactor data with three fertile Standard Model neutrinos and one heavier sterile neutrino, would be at least five times as heavy as the heaviest of the three fertile neutrino mass eigenstates in a normal neutrino mass hierarchy. With respect to a mixing angle into a fourth neutrino flavor, that data are as follows:
The appearance probability . . . is reported to be 0.0026 by LSND. If the disappearance probability . . . is ∼ 0.06, as suggested by the reactor data, the constraint is very comfortably satisfied.
(2) Non-Reactor Source Experiments Disfavor The Reactor Anomaly Sterile Neutrino Hypothesis

The ICARUS and OPERA experiments (among other things) look at neutrino sources other than nuclear reactors, which is useful because nuclear reactor neutrino sources are hard to model accurately because they are so complex.  Their data disfavor a source for the reactor anomaly that is due to neutrino oscillations with a sterile neutrino flavor.
Some evidence that the low-energy νe excess may not be due to oscillation has come from the ICARUS and OPERA experiments. These experiments have searched for νµ → νe at L/E ∼ 35 m/MeV, an L/E larger than those of LSND and MiniBooNE, but such that one is still not very sensitive to oscillations driven by the squared-mass splittings among ν1,2,3, while being sensitive to those driven by splittings larger than ∼ 0.03 eV2 (i.e. 0.173 eV). From negative results, ICARUS and OPERA disfavor a νµ → νe origin of the low-energy νe excess reported by MiniBooNE, although the strength of this disfavoring has been questioned.
UPDATE: OPERA constraints an a light sterile neutrino that oscillates with "active" neutrino flavors is found in figure 5 of this paper (which honestly is less clear than one might hope).

(3) The sum of neutrino masses with a sterile neutrino is ruled out by Plank data using model dependent assumptions.

Planck and WMAP cosmic background radiation data is mixed on this possibility. It is not inconsistent with there being four rather than three neutrino flavors, although three flavors are more likely than four given the data which produces a value for "Neff" that has a value about a quarter of a neutrino species in excess of the value predicted if there are exactly three neutrino species with a significant margin of error.

Five neutrino flavors, however, are strongly disfavored by the cosmic background radiation data, unless the heaviest sterile neutrino has a mass of more than about 10 eV.  A sterile neutrino that heavy would  be treated as dark matter rather than a neutrino for the purposes of cosmic background radiation cosmology models.

The most recent recent Plank data, however, excludes at roughly a 95% probability, a sum of the masses of all neutrino species combined of 0.23 eV or more, which rule out a sum of three neutrino mass species of 0.377 eV or more at a roughly 3.2 standard deviation level, which is a more than 99% probability that this hypothesis is not true.  Yet, 0.377 eV is the minimum sum of the four neutrino masses with sum of masses of 0.06 eV which is the minimum sum of the three fertile neutrino masses, plus the minimum sterile neutrino mass that can fit reactor anomaly data.

Still, Kayser's paper notes that this measurement of the sum of all of neutrino species masses is based on model dependent assumptions about how neutrinos oscillate between flavors and mass states that fit all existing data (including simple 3+1 models) but might not be true if sterile neutrinos don't oscillate as naturally into fertile neutrinos as fertile neutrinos do into each other.

(4) A fifth neutrino species is more strongly disfavored.

As noted above, Planck and WMAP data strongly disfavor a fifth neutrino flavor (e.g. a 3+2 model with two sterile neutrino flavors that are heavier than the fertile neutrino flavors, or 1+3+1 model with one sterile neutrino that is lighter than the lightest fertile neutrino and one that is heavier than the heaviest fertile neutrino), keeping in mind that a sterile neutrino with mass in excess of about 10eV does not count for these purposes.

According to Kayser's paper, most of the reactor data can be fit almost as well with four neutrino species as with five.


On balance, it is more likely than not, given the evidence available right now, that all evidence in favor of a light sterile neutrino of the kind that would address the perceived reactor anomaly is really just experimental uncertainty and systemic error in prior experiments. But, the evidence is not so overwhelming that the possibility can be definitively overruled.

Numerous experiments are underway or planned, to get to the bottom of this question in the next few years to the next decade or so.

I personally expect that these experiments will reveal three fertile massive Dirac neutrinos with a normal mass hierarchy, significant CP violation, and no light sterile neutrinos.  But, time will tell.

Off Topic Footnote: Pion Polarization Anomaly Was Probably Due To Experimental Error

What is a pion?

A pion is meson, which is a type of composite particle made out of a quark and an anti-quark.  Charged pions have an up quark and an anti-down quark or visa versa.  Neutral pions have a linear combination of up quarks and anti-up quarks, and down quarks and anti-down quarks.  Pions are the lightest particles that include quarks, with masses of about 138 MeV for neutral pions and 140 MeV for charged pions, and are the longest lived hadrons (i.e. composite particles bound by gluons) other than protons and neutrons.

Charged pions, at least, can be polarized, since they are made up of electrically charged components that are not distributed homogeneously.  Their polarization sheds light on the internal structure of mesons, thus confirming (or alternatively in tension with) the predictions of QCD (quantum chromodynamics aka strong force physics) regarding this internal structure.

New Pion Polarization Experimental Data Confirms Standard Model Predictions

In recent experimental physics news with a similar bottom line to recent reactor anomaly experimental data, a new high precision measurement of the polarization of pions at the COMPASS experiment (predominantly negatively charged pions) is consistent with the theoretical prediction derived from Standard Model QED and QCD calculations of this fairly simple system (background on the COMPASS experiment is available here).

If correct, this resolves tensions between the theoretically predicted result and the experimentally measured result in several previous, less precise experiments.  Thus, another set of data points that supported possible beyond the Standard Model physics has once again been tentatively quashed by using more accurate experimental procedures.

FWIW, however, I'm not personally aware of any particular beyond the Standard Model theories that had been based upon, or relied upon, this anomaly, although the need for new physics to explain the old results which were in tension with the theoretical prediction is discussed, for example, here.  Pion polarization is also relevant to parity conservation (or the lack thereof) in hadrons.

Wednesday, February 12, 2014

Graphing Koide's Rule Solutions

If you take three masses M1, M2 and M3, from smallest to greatest, and then redefine them in units of M1 (which you can do without loss of generality) such that you have 1, M2/M1=x^2, M3/M1=y^2, then Koide's rule can be stated as follows:

(1)        x^2+y^2+1/(x+y+1)^2=2/3

This is equivalent to:

(2)      3x^2+3y^2+3=2x^2+2y^2+4x+4y+4xy+2

which is equivalent to:

(3)    x^2+y^2-4x-4y-4xy+1=0

This is an equation for a hyperbola (the determinant of the coefficients of the equation in the general quadratic curve equation called delta is -27, determinant J of some of those coefficients is negative, and the quantity I=2).

In standard form, this would be written:

(4)   (x+2)^2-4(x+2)(y+2)+(y+2)^2=-9

Graphed (at this link), it looks like:

But, since x>0 and y>0, only the portions of the solution in the top right quadrant of the graph are solutions to the Koide relationship, and since we can arbitrarily define x>y or x=y, only half of the solutions in that quadrant apply.

Examination of the graph reveals that there are two basic classes of Koide relationship solutions.  One group of solutions is near the origin where M1>>M2 and M1>>M3 and M2 and M3 are somewhat similar in size.  In the other group of solutions, M3>>M2 and M3>>M1.  This is far out on the x-axis.

An example of a solution of the first type is the charm-bottom-top quark triple, which is a very close approximation of the Koide triple.

Another solution of this type can be constructed using the particle data group values of the up and down quark masses (2.3 MeV and 4.8 MeV respectively).  The positive mass that would complete this triple using the unmodified Koide's formula is 232.14 MeV (holding the down quark mass fixed and considering the range of up quark masses consistent with the 0.38-0.58 up/down mass ratio of PDG implies a range of 210.96 MeV-252.20 MeV; looking at those ratios while keeping the sum of the light quark masses in the PDG range tweaks the outcome 5-8% or so).  This also isn't particularly close to a hypothetical "pole mass" for the strange quark, which is heavier than this in a naive extrapolation, but is somewhat ill defined in any case as perturbative QCD really isn't applicable to this extreme infrared energy scale.

Of course, this isn't a good fit for reality at all.  The low end is twice the muon mass and more than twice the best estimate of the strange quark mass.  The high end is about a fifth of the charm quark mass and a seventh of the tau lepton mass.  But, there is a solution to the formula in this mass range for these light quark masses.

The original electron-muon-tau triple of 0.511 MeV, 105 MeV and 1776 MeV more or less, is an example of the other class of solutions.

It is also worth noting the solution for the special case in which one of the masses is zero.  In that case, the ratio of greater of the other masses to the lesser of the other masses must be approximately 13.92820.  This corresponds to the x-axis intercept on the graph above if it extended that far.

In general, when all three masses are non-zero, the greatest ratio of two masses in the triple must be greater than 13.92820.

Tuesday, February 11, 2014

The First Great Post-Neolithic Depression

From here citing Stephen Shennan et al., Regional population collapse followed initial agriculture booms in mid-Holocene Europe. Nature Communications (2013). (Open access).

The raw number of archaeological finds in Western and Northern Europe, aggregated suggest a deep (more than 50% from peak to trough), roughly 500 year long slump between the surge in population associated with the arrival of farming in the Atlantic area (the Atlantic Neolithic) and the immediate emergence of Megalithism from the Atlantic Neolithic, and the advent of the parallel Copper Age cultures known as "Corded Ware" (associated with Indo-Europeans) and "Bell Beaker" (associated with a non-Indo-European substrate in the region).

This slump rivals or exceeds subsequent major slumps in the region around 2000 BCE, around 1200 BCE, and in the early Middle Age and again about 800 years later.  Climate events and documents plagues are reasonable well correlated with these slumps.

We understand far less well what caused the first one.  There are some indications that this may be a case of primitive early Neolithic farming techniques leading to a boom-bust cycle as fertile land is depleted that moved like an echoing wave following the initial wave of the Neolithic.  But, at this point, merely establishing that there was a major slump between the initial Neolithic and the Copper Age almost everywhere in West Eurasia, represents major progress relatively to previous models that assumed steady growth more or less continuously from the early Neolithic to the present.  Climate could also be a suspect, but it if was we would expect the impact to be fairly uniform across West Eurasian, rather than being staggered, and it isn't at all clear that this is the case.

As further charts in the linked post indicate, however, the clear picture for the region as a whole is rather less clear when broken down by sub-region.  The Atlantic Neolithic surge is pretty much universal.  But, the subsequent Copper Age patterns vary considerably by region.  Corded Ware was pretty much absent from the modern British Isles, Ireland and France, for example.  Very different sample sizes in different regions (which may reflect genuine differences in population by region rather than simply differences in sampling strategy, or not) also influence how the regional results feed into the overall result.

The big surge in population with the Atlantic Neolithic is associated with a dramatic change in population genetics between the preceding Mesolithic era (i.e. late hunter-gather) and the Neolithic.  The second Copper Age peak is associated with another significant change in population genetics in the region.  Not later than about 3000 BCE (and possibly sooner), the population genetics of this part of Europe are substantially similar to those found in Europe today.

The results exclude Southern Europe and Eastern Europe where Corded Ware played a prominent role and Bell Beaker played a much more secondary role.

Pre-Neolithic Climate Conditions

Data from another fairly recent open access paper in PNAS by Samuel Bowles (of the interdisciplinary Sante Fe Institute) and Jung-Kyoo Choi (a Korean academic economist) entitled "Coevolution of farming and private property during the early Holocene" has pretty iffy analysis but good data.  It indicates that during the Upper Paleolithic era immediately prior to the Neolithic, that climate was subject to dramatic shifts in temperature.  The key paleoclimate data are presented in the chart below.

Ignoring their simulated data in light gray bar graph form (which flows from a problematic model):
Estimated dates of some well-studied cases of the initial emergence of cultivation are on the horizontal axis (85455). Climate variability (Left) is an indicator of the 100-y maximum difference in surface temperature measured by levels of δ18O from Greenland ice cores (SI Appendix). A value of 4 on the vertical axis indicates a difference in average temperature over a 100-y period equal to about 5 °C.
Pre-Holocene temperatures were also much colder, on average.

The Supplemental materials section linked above states:
Differences in temperature (Centigrade) are about 1.2 times the difference in the δ18O signal shown in Fig. S1 (2). The data indicate that changes in mean temperature as great as 8 degrees (C) occurred over time spans as short as two centuries. By way of comparison, the Little Ice Age that devastated parts of early modern Europe experienced a fall in average temperatures of one or two degrees, and the dramatic warming of the last century raised average temperatures by one degree, comparing the unprecedentedly hot 1990s with a century earlier (3, 4). The variability of climate during the late Pleistocene required high levels of geographical mobility, which was an impediment to any substantial investments in tree crops or field preparation or even stores and storage facilities. The scale and pace of climate change is truly extra ordinary: for example δ18O signals from sea cores indicate that between 25 and 60 ka, variations in sea surface temperature of 3–5 degrees Centigrade occurred over periods of 70 years or less in the  Santa Barbara Basin, California (5) (sea surface temperatures today are about this different between the Santa Barbara Basin and northern Vancouver Island). Think about the frequency of  moves and the distances that early humans may have traveled. A change of 9 degrees Centigrade in the course of a millennium appears to have been common prior to the Holocene. That's the  difference in the average daily temperature in Cape Town and Mombasa 4 thousand kilometers  to the north. While humans and the wild species on which they depended could of course adapt  to a few degrees change in temperature, we infer that the distances covered and frequency of  moves must discouraged the kinds of investment that farming requires.  

As the chart indicates, intermittent periods of wild temperature variation over the span of just a few generations was the norm for the entire Upper Paleolithic era (from about 40,000-50,000 years ago until about 10,000 years ago), after which temperatures became much more stable starting at the beginning of the Holocene era about 10,000 years ago when farming first emerged in the Fertile Crescent and China and the middle latitudes of the Americas (farming arose independently at later times in the New Guinea Highlands, Sub-Saharan Africa and the Eastern United States).

So, a key part of the answer to the question of why farming emerged when it did is that the climate was too unpredictable for farming to be a viable means of food production during almost the entire Upper Paleolithic era.  Farming didn't emerge before the Holocene because it couldn't in the climate conditions at the time.

Thus, a last post-Neolithic hurrah of Upper Paleolithic temperature variability illustrated in the second chart, that would have wrecked havoc on the crops of early farmers in Europe, could explain the slump that the experienced after the Atlantic Neolithic peaked.

Also, what has been interpreted by some as a behavioral revolution in the Upper Paleolithic that distinguishes anatomically modern humans from behaviorally modern humans, could simply have been a necessary cultural reaction to much more adverse climate conditions.