This is a guest post from a unemployed theoretical physicist from New Zealand, Marni Dee Sheppeard. Marni is currently dabbling in quantum gravity research in Wellington. After a variety of jobs in experimental and particle physics, and also outside science, Marni completed a PhD in quantum gravity at Canterbury University in 2007. Jobs held since then include a Oxford University postdoc. She regularly discusses physics on her blog, Arcadian Pseudofunctor. I (T.D.) invited Marni to produce a guest post for this blog. Please find it below.

In the Standard Model of particle physics, there are three basic symmetries, namely

C: the interchange of positive and negative electric charge,

P: parity, associated to reflections of spatial coordinates,

T: the time reversal operation.

In quantum mechanics, each of these operations acts on a quantum state, and the reversal of the operation leads to a state that is closely related to the original one. In radioactive beta decay, a neutron always produces a proton, an electron and an electron antineutrino. A violation of CP symmetry was first observed in 1956 in the beta decay of cobalt-60, by Chien-Shiung Wu and her team. This demonstrates that our universe differentiates between a beta decay process and its mirror image, by favouring antineutrinos of a definite handedness in the weak interactions that we observe. But although the combined CP symmetry is not respected by nature, a combined CPT operation is supposed to behave nicely. This is because the CPT symmetry is understood in terms of Lorentz symmetry, which describes the behaviour of spacetime using the well tested theory of special relativity.

Although many theorists believe that the geometry of spacetime should emerge from more fundamental gravitational degrees of freedom, it is still generally thought that the CPT symmetry will be respected by nature. However, in a theory that does not begin with the classical description of spacetime, an apparent violation of CPT symmetry need not be directly associated to a breaking of the Lorentz symmetry, as we demonstrate.

A fundamental fermion is initially specified by a list (q,m,h) where m is its rest mass, q its charge and h its chirality, or handedness, which for a massless particle notes whether or not the spin is aligned with the direction of propagation. With massive particles we need to be more careful in describing propagation because their velocity is relative to the observer, as special relativity tells us. Still, chirality has two basic states: left and right. If we assume that all left and right states exist in nature, then the allowed values of the list (q,m,h) almost account for the fundamental fermions: the leptons and quarks. There remains a question about the antiparticles of neutral leptons. For a charged particle, the antiparticle has opposite charge, so the two particles cannot be confused. For the neutral neutrinos, however, it is not immediately clear that an antineutrino is a distinct state. Consider the possibility that all particles have distinct antiparticles. Charged particles have antiparticles of opposite charge, the same chirality and equal mass, but neutral antiparticles have like charge and like mass, so there could also be a neutral particle of like chirality with distinct mass. But by definition all particles have equal mass antiparticles with which they annihilate to photons, and one would like to claim that there is no CPT violation in the usual sense.

However, what we usually call a neutrino and an antineutrino might really be called a neutrino and a mirror neutrino, where the mirror neutrino has a distinct mass to the neutrino. The nomenclature is important, because in this case the neutrino and antineutrino can be the same particle!

The MINOS experiment has observed distinct masses for neutrinos and the so called antineutrinos, which we now call mirror neutrinos. In preliminary results released in 2010, MINOS claim that the total difference in masses squared for neutrinos is around 0.00232 eV2, while for mirror neutrinos it is 0.00336 eV2. Given our new mirror particle zoo, can we say anything about these numbers?

The ordinary neutrinos come in three possible mass states. In 2006, Carl Brannen fitted these three masses to a simple formula, resembling the Koide formula for the three charged lepton masses. First discovered by Yoshio Koide in 1981, this exact relation between the masses of the electron, muon and tau particles was used to predict the tau mass from the more accurately known electron and muon masses. The prediction was correct, and the formula remains in agreement with experimental results. Brannen’s prediction for the neutrino masses was based on the observation that the Koide formula is expressed in terms of a certain matrix. This matrix map is a quantum Fourier transform of the square roots of the three masses. In quantum information theory, this map is a three dimensional analogue of the qubit Hadamard gate, which is built from two elementary Pauli spin operators. The neutrino scale is selected so that the sum of mass squares is around 0.00232 eV2, as observed. Ignoring the choice of mass units, there are two remaining parameters in the neutrino matrix. One parameter is identical to the charged lepton case. The single remaining parameter appears to differ from the charged lepton case by π/12. That is, there is a complex phase parameter of exp(2i/9) for the charged leptons, and exp(2i/9+ πi/12) for the neutrinos. Both of these phases were then used to fit hadron masses into triplets, according to a theoretical analysis based on measurement algebras rather than traditional wave functions.

In the Fourier matrix, a conjugation of phases leaves the set of three masses unchanged. For the equal mass charged leptons and antileptons, the phases exp(±2i/9) give equal masses. Are the phase signs associated to elementary quantum data, such as the charge q? After the release of the MINOS results, many theorists were wondering about the possibility of CPT violation in the neutrino sector. Let us choose phases so that particle and antiparticle phases sum to zero, which is the phase associated to a boson set of three identical masses. The neutrino phase now has two sign components, so we may double the allowed mass sets for the neutral particles by choosing a sign mismatch, as in exp(2i/9- πi/12). With this phase, at the same scale as the neutrinos, the difference of mass squares for the mirror neutrinos is around 0.00336 eV2, as observed.

The mirror mass prediction corresponding to the lightest neutrino mass is now 0.00117 eV. Very soon after I posted this calculation on my blog last June, keen astronomer Graham Dungworth pointed out that the thermal equivalent of 0.00117 eV under the law for black body radiation is 2.73K, which is precisely the temperature of the cosmic microwave background. But that is a subject for another day.

If the CPT symmetry is respected, CP violation may be paired to T violation as it is in the working Standard Model. Neutrinos come in left and right handed states, just like the other leptons. Ordinary beta decay, however, produces mirror neutrinos, introducing a pervasive coupling between ordinary matter and the mirror world.

March 5, 2011 7:53 pm

Cool to see you here, Kea.
Congratulations.

March 6, 2011 1:53 am

I have been following Marni Sheppeard’s work for some time and it was rather difficult to even half-understand what was going on. So here is an instant synopsis for newcomers.

In 2005, Bilson-Thompson proposed to map the first Standard Model generation (and the gauge bosons) onto three-strand braids. It is often anticipated that there will be states in quantum gravity related to knot theory (e.g. made of knotted Wilson lines), so this is an idea about how the observed particle spectrum might arise in such a theory.

Brannen, cited above for his work on the Koide relation, has a paper proposing mysteriously that if you examine the propagation of a spin 1/2 particle from a quantum-information perspective, you get three propagators and these are the three generations. (I did say I only half-understand…) Apparently this meshes with his explanation of the Koide formula too – that is, it is part of the same conceptual synthesis. So Sheppeard is developing an approach to quantum gravity to which the single-generation Bilson-Thompson mapping applies, and in which Carl Brannen’s work will explain the existence of three generations, as well as offering a Koide-inspired approach to the particle masses. In particular, she posits a mapping between Bilson-Thompson’s braids and Brannen’s 3×3 matrices, a mapping which ultimately ought to be derived (from a cohomology theory) rather than just postulated.

However, Bilson-Thompson did not use all possible braids, and under Sheppeard’s mapping (of braids to matrices) the unused braids are assigned specific quantum numbers, including their mass. The “mirror neutrino” with a mass of 0.00117 eV corresponds to one of these unused braids.

It’s probably important to emphasize how much “mathematical” speculation is occurring in this model. By this I mean speculation that a model with the desired properties actually exists. Bilson-Thompson’s mapping and Sheppeard’s mapping (of braids to quantum numbers and/or matrices) ought to follow from some underlying quantum-gravitational theory. But as things stand, there isn’t even a demonstration in any properly specified model that Bilson-Thompson’s braids propagate like particles, let alone that they behave like particles bearing the quantum numbers he assigns to them.

Nonetheless, at least we can say Koide’s relation is an experimental fact, that it is being neglected by theorists, and that it makes sense to seek explanations for it which would embed it in a larger pattern extending to properties of the other fermions, such as the parameters of the mixing matrices, and this is what Brannen and Sheppeard are doing.

3. March 6, 2011 7:43 am

Mitchell, you are right that the matrices should be derived using something like algebraic topology, but this is about a universal cohomology theory that does not yet exist. The interplay between physics and mathematics works both ways. Mathematicians are not making satisfactory progress on this problem while they ignore the physical input, and the physical input will always appear mathematically speculative when it is not defined in trendy, existing terms.

March 6, 2011 4:33 pm

The topic of the post is very interesting and I am glad someone wrote about it because there is an argument that I do not know how to clarify.
The CPT theorem implies that particles and antiparticles must have equal mass: $m=\bar{m}$. The idea that the recent MINOS result could suggest CPT violation is based on the inverse of the previous statement ‘CPT violation implies that $m \neq \bar{m}$‘; nevertheless, the theorem goes in one direction only. This is because it has been proved that CPT violation can only occur (in realistic, local, causal field theory) if Lorentz symmetry is also broken (hep-ph/0201258); therefore, the description of CPT violation in nature requires a theory that also includes Lorentz violation. Since the analysis used by MINOS to determine the oscillation parameters is Lorentz invariant any conclusion about CPT violation is meaningless. I am not saying that MINOS result (if real) is not CPT violation, it is just that the data should be analyzed using a Lorentz-violating theory to identify the CPT-violating component that has a meaning in field theory.

5. March 6, 2011 5:37 pm

Rose, if you read the post again you will see that we can get around the CPT issue by inventing a particle zoo where all particles DO have masses equal to antiparticles. I am saying that CPT IS conserved in this sense, but only through the renaming of the ‘antineutrino’ to ‘mirror neutrino’ … which has a distinct mass.

March 8, 2011 2:40 pm

“Inventing a particle zoo” only to explain one experimental anomaly seems very ‘expensive’ to me. I don’t think that is the right way to go. I mean, how many parameters will your model have? The massive description uses six (two mass-squared differences, three mixing angles, and one CP-violating phase), if you can construct a model with a number of parameters not far from six that also gives an idea about other anomalous results such as LSND or MiniBooNE I would take it as an interesting possibility.

• March 8, 2011 6:17 pm

We have fewer parameters than the SM.

• March 8, 2011 8:37 pm

By the way, we have ideas on the MiniBooNE/LSND results. Let us stick to the antineutrino case. These analyses assume a P depending on sin^2(f.L/E) for f=f(m), such that at small L the probability of seeing the electron antineutrinos drops off. In the new cosmology, however, there should be a P cutoff given by the behaviour of our neutrino/antineutrino gases, providing a minimum P for electron antineutrino observation. This kind of solution makes far more sense than postulating delta m squares that are far higher than the known particle masses.

March 6, 2011 6:10 pm

The particle production of charged lepton pairs in a radiation bath at equilibrium, eg. for the first generation ie. electron and positron of identical masses namely 511,000eV, implies equiabundances of photons and electron positron pairs at an appearance threshold temperature of 5.93*10^9 Kelvin from the Boltzman Constant 0.00008617eV K^-1.

However, the Brannen Sheppeard interpretations of neutrino mass states for the first generation of these neutral leptons revealed non identical eigenmass states for the electron neutrino and electron antineutrino at 0.00038eV and 0.00117eV. The significance of this latter mass energy magnitude was quickly apparent-namely the radiation bath at an equilibrium temperature of 2.73K was almost identical to that of the socalled relic photons of the Cosmic Microwave Background for the photon/neutrino ensemble with 0.00117/2 eV half masses.

Inorder to salvage the tenet of equi mass particle pairs the electron antineutrino was proposed as a chiral pair and concomitantly, so also for the electron neutrino which would compromise a photon neutrino chiral mix at a radiation equilibrium temperature of 0.89Kelvin. Effectively, with chiral forms and differing masses there would be two species per generation of two forms and twelve in total.

Prior to the preliminary Minos result we were familar with only the role of the left hand chiral antineutrino that relates to the 0.00017 eV species or rather half masses with two chiral forms of Normal Matter . We are left then to attribute the R chiral form to the vacuum state, mirror and/or shadow matter states. It would be remarkable and extremely coincidental were the CMB temperature and neutrino creation annihilation not causally linked and independent .

To what form of matter do we link the 0.00038 species/2 eV half mass electron neutrino R and L chiral forms? Symmetry implies that were one to ascribe a mathematical form, eg. the Standard Model as applicable to the matter zoo we know of that associates with the L electron antineutrino it would equivalently apply to the 0.00038eV species, where the radiation bath at 0.89K is wholly subsumed within that which we observe at 2.73K.

The Dark Matter abundance by mass overwhelms that of the Normal Matter mass abundance ca. 5/1 but the numerical abundances remain unknown other than those given by neutrino mass ratios. Without doubt the electron neutrino 0.00038eV species represents what we call Dark Matter. The total mass states for the three neutrino flavours cumulatively represent no more than a total mass fraction of total matter, mass fraction ~0.003 (dark and normal). Yet, in the like manner that we attribute the L electron antineutrino with its cousins of charged leptons and quarks (as hadrons and mesons) of Normal Matter so also must we place the 0.00038eV neutrinos into a new phyllum of matter we have associated as the unknown form of Dark or Mirror Matter. A new phyllum of alloparticles emerges with different mass states in a 1:1 correspondence to Normal Matter. This new form of matter at a lower mass scale may be of either L or R form and by analogy I imply L, leaving the vacuum state corespondingly depleted and chiral so as to conserve parity.

If the Minos results are vindicated for the antineutrino masses and I presume no one quibbles about the neutrino masses then the mathematical description should conform to the physics and cosmological description I present and not vice versa.

7. March 6, 2011 7:09 pm

“… then the mathematical description should conform to the physics and cosmological description I present and not vice versa.”

Exactly. Mathematicians like Connes and Marcolli have put a great deal of effort into understanding QFT from a traditional viewpoint, because they know it is relevant to their efforts to understand theorems like the Riemann hypothesis. If they ignore a greatly improved physical picture, they are almost certainly blocking their own progress on the mathematical problems.

The latter 20th century led too many people into a dark corner, where physics and maths were considered separate. But when Riemann wrote his only paper on Number Theory, he was working on hydrodynamics.

8. March 6, 2011 11:13 pm

Marni, “electron anti-neutrinos”, for example, are detected because they are the SU(2) partner of the right-handed positron. In other words, by introducing your “mirror neutrinos” to explain anti-neutrino oscillations, you also introduce a new charged lepton, with the same/similar mass to the positron. Obviously this is ruled out, so the picture you present here doesn’t make any physical sense.

9. March 7, 2011 8:55 am

Still having a problem with reading comprehension, Rhys?

10. March 7, 2011 10:10 am

I don’t think so, but if I have misunderstood your suggestion, perhaps you should try to explain it again, rather than resorting immediately to snide comments.

For somebody who complains constantly that her work isn’t taken seriously, you seem very reticent to actually discuss it.

11. March 7, 2011 5:55 pm

Rhys, this is completely off topic, because this post discusses the achiral braid set, for which the mirror neutrinos are the only extra states. Representation theory is not the starting point for quantum gravity. But, briefly: as Graham and I have discussed for many months now (on a blog which you clearly ‘read’, and in other written work) there is something called Dark Matter. As many physicists have considered before, Graham has suggested that the mirror matter is a component of DM.

12. March 8, 2011 8:56 am

Aren’t you trying to explain the MINOS data? I was completely on-topic.
If CPT is conserved, then any mixings and interactions in the anti-neutrino sector have to be the same as those in the neutrino sector. In particular, the interactions responsible for beta-decay have to be the same as those responsible for a(n anti-)neutrino bouncing off a nucleus and producing a charged lepton. I don’t understand how you’re proposing to get around this.

I’m not sure why dark matter suddenly came up, but anyway…
Such light particles, with small scattering cross-sections, can’t be thermal dark matter, because they would be ‘hot’. Hot dark matter is ruled out because it screws up structure formation in the early universe. So you need to play some fancy games if you want to argue that any such particles form a sizeable chunk of the dark matter.

Finally, why did you mention quantum gravity? Neutrinos can only get mass after electroweak symmetry breaking, the scale of which is ~100 GeV. So even leading order gravitational effects would give neutrino masses of ~(100 GeV)^2/M_P ~ 10^-5 eV, which is far too small. As such, neutrino masses are unlikely to tell us anything about gravity.

You just don’t seem to be presenting any physically coherent picture. I’m willing to be shown to be wrong.

13. March 8, 2011 6:16 pm

Rhys, you think quantum gravity has nothing to do with particle masses? Really? Just because some text book tells you otherwise?

• March 8, 2011 7:38 pm

Why *would* quantum gravity have anything (important) to do with particle masses? If you claim that the masses of known particles are somehow tied to properties of gravity, you need to provide some evidence, because the Planck scale doesn’t seem to be connected in any way to electroweak symmetry breaking or fermion masses (not even to the neutrino masses, as I mentioned above).

And I’m still curious to hear your take on the other issues I raised, which seem critically important to the topic of the post.

14. March 8, 2011 7:45 pm

Rhys, people who demonstrate such a profound lack of listening ability waive their rights to have their questions immediately answered, especially when their questions make no sense in the context of the discussion. When I was at school, gravity was a theory about mass. Neither the SM nor conventional string theory say anything about fundamental particle masses, which are specified by parameters. I should not have to remind you that physics at ‘the Planck scale’ is a purely speculative matter for which there is no experimental basis whatsoever. If the LHC finds a fairy field (Higgs boson), then I might start taking the deeply flawed mainstream views on EW symmetry breaking more seriously. But it won’t.

15. March 11, 2011 7:13 pm

“Brannen’s prediction for the neutrino masses was based on the observation that the Koide formula is expressed in terms of a certain matrix. This matrix map is a quantum Fourier transform of the square roots of the three masses. In quantum information theory, this map is a three dimensional analogue of the qubit Hadamard gate, which is built from two elementary Pauli spin operators. The neutrino scale is selected so that the sum of mass squares is around 0.00232 eV2, as observed. Ignoring the choice of mass units, there are two remaining parameters in the neutrino matrix. One parameter is identical to the charged lepton case. The single remaining parameter appears to differ from the charged lepton case by π/12. That is, there is a complex phase parameter of exp(2i/9) for the charged leptons, and exp(2i/9+ πi/12) for the neutrinos. Both of these phases were then used to fit hadron masses into triplets … For the equal mass charged leptons and antileptons, the phases exp(±2i/9) give equal masses. Are the phase signs associated to elementary quantum data, such as the charge q? After the release of the MINOS results, many theorists were wondering about the possibility of CPT violation in the neutrino sector. Let us choose phases so that particle and antiparticle phases sum to zero, which is the phase associated to a boson set of three identical masses. The neutrino phase now has two sign components, so we may double the allowed mass sets for the neutral particles by choosing a sign mismatch, as in exp(2i/9- πi/12).”

Very exciting. I didn’t understand the 2/9 factors you were referring to, but I understand complex exponent phases so this makes it simple.

You’re right that if it turns out that there is a gauge theory of gravity, mass will be the charge of quantum gravity, not just a Higgs field introduced to break an unobserved but well-believed electroweak symmetry into the observed, broken-but-mixed electroweak theory.