**The Strong CP Problem Stated**

The idea behind parity symmetry is that the equations of particle physics are invariant under mirror inversion. This leads to the prediction that the mirror image of a reaction (such as a chemical reaction or radioactive decay) occurs at the same rate as the original reaction.Parity symmetry appears to be valid for all reactions involving electromagnetism and strong interactions. . . . The first test based on beta decay of Cobalt-60 nuclei was carried out in 1956 by a group led by Chien-Shiung Wu, and demonstrated conclusively that weak interactions violate the P symmetry or, as the analogy goes, some reactions did not occur as often as their mirror image. . . . charge conjugation is a simple symmetry between particles and antiparticles, and so CP-symmetry was proposed in 1957 by Lev Landau as the true symmetry between matter and antimatter. In other words a process in which all particles are exchanged with their antiparticles was assumed to be equivalent to the mirror image of the original process.

In 1964, James Cronin, Val Fitch with coworkers provided clear evidence (which was first announced at the 12th ICHEP conference in Dubna) that CP-symmetry could be broken, too, winning them the 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetry C between particles and antiparticles and the P or parity, but also their combination. The discovery shocked particle physics and opened the door to questions still at the core of particle physics and of cosmology today. The lack of an exact CP-symmetry, but also the fact that it is so nearly a symmetry, created a great puzzle.

Only a weaker version of the symmetry could be preserved by physical phenomena, which was CPT symmetry. Besides C and P, there is a third operation, time reversal (T), which corresponds to reversal of motion. Invariance under time reversal implies that whenever a motion is allowed by the laws of physics, the reversed motion is also an allowed one. The combination of CPT is thought to constitute an exact symmetry of all types of fundamental interactions.Because of the CPT symmetry, a violation of the CP-symmetry is equivalent to a violation of the T symmetry.CP violation implied nonconservation of T, provided that the long-held CPT theorem was valid. In this theorem, regarded as one of the basic principles of quantum field theory, charge conjugation, parity, and time reversal are applied together. . . .

In particle physics, the strong CP problem is the puzzling question why quantum chromodynamics (QCD) does not seem to break the CP-symmetry.

QCD does not violate the CP-symmetry as easily as the electroweak theory; unlike the electroweak theory in which the gauge fields couple to chiral currents constructed from the fermionic fields, the gluons couple to vector currents. Experiments do not indicate any CP violation in the QCD sector. For example, a generic CP violation in the strongly interacting sector would create the electric dipole moment of the neutron which would be comparable to 10−18 e·m while the experimental upper bound is roughly a trillion times smaller.

This is a problem because at the end, there are natural terms in the QCD Lagrangian that are able to break the CP-symmetry.

For a nonzero choice of the θ angle and the chiral quark mass phase θ′ one expects the CP-symmetry to be violated. One usually assumes that the chiral quark mass phase can be converted to a contribution to the total effective angle, but it remains to be explained why this angle is extremely small instead of being of order one; the particular value of the θ angle that must be very close to zero (in this case) is an example of a fine-tuning problem in physics, and is typically solved by physics beyond the Standard Model.

There are several proposed solutions to solve the strong CP problem. The most well-known is Peccei-Quinn theory, involving new scalar particles called axions. A newer, more radical approach not requiring the axion is a theory involving two time dimensions first proposed in 1998 by Bars, Deliduman, and Andreev.

Also, strong CP problem may be solved in one of quantum gravity theories.

From Wikipedia (emphasis added, references omitted) (see also, for example, here and here).

There are no charge parity symmetry violations, which are equivalent to time symmetry violations (i.e. the physical laws involved have an arrow of time), that have been empirically discovered in the strong nuclear force, the electromagnetic force, or gravity. All observed CP violations involve the weak nuclear force.

This isn't terribly surprising for the electromagnetic force or gravity, neither of which has any equations that contain a natural term which would lead to CP violations. But, it is more surprising in the case of the strong force, because it has such terms. As of 2008, Lattice QCD efforts had not resolved this problem.

**The Conjecture: the Strong CP problem is due to the gluon's lack of mass**

My personal conjecture is that the strong CP problem can be explained by the mass of the force carrying bosons involved, although why particular bosons have the masses that they do, of course, like all of the other masses in the Standard Model, is a mystery. What follows is a statement of this conjecture which I do not claim as anything but my own and I am not attempting to state is a scientifically authoritative statement of an expert. Instead, it is a speculation of an educated layman, although I would like to think that this speculation is grounded enough to be something other than a totally crackpot idea, at least at a heuristic level, even if it may, in fact, be flawed in some respect.

The electromagnetic force is carried by the massless, spin one particle called the photon, which travels at the speed of light. Indeed, special relativity implies that massless particles must always travel at the speed of light. As Wikipedia explains in its article on special relativity.

For massless particles, m is zero. The relativistic energy-momentum equation still holds, however, and by substituting m with 0, the relation E = pc is obtained; when substituted into Ev = c2p, it gives v = c: massless particles (such as photons) always travel at the speed of light.

A particle which has no rest mass (for example, a photon) can nevertheless contribute to the total invariant mass of a system, since some or all of its momentum is cancelled by another particle, causing a contribution to the system's invariant mass due to the photon's energy. For single photons this does not happen, since the energy and momentum terms exactly cancel.

Likewise, special relativity implies that massive particles can never travel at the speed of light.

The gluons of the strong force are also canonically massless spin one particles, but instead of interacting only with charged particles, like photons, they interact only with particles that have color charge (i.e. quarks and other gluons), which comes in types called red, green and blue, rather than the electromagnetic force's positive or negative.

In extensions of the Standard Model in which gravity is also a force with a mediating boson, this force is carried by a massless, spin two particle called the graviton, which interacts with matter-energy (in general relativity (and unlike classical Newtonian gravity), massless energetic particles, like photons are affected by gravity, not just massive particles.

Thus, the carrier bosons of all of the forces that do no exhibit CP symmetry violations, which are equivalent to T symmetry violations, travel at the speed of light.

In contrast, the weak force, which does exhibit CP violations, which are equivalent to T violations, are the W+, the W- and the Z particle, the former having masses of about 80 MeV and the latter having a mass of about 91 MeV, and hence, as a result of special relativity (which unlike general relativity, is integrated seamlessly with quantum mechanics), these massive, spin 1 carrier bosons do not travel at the speed of light.

Why should this matter? Because, special relativity provides that the rate at which time passes for a particle is a function of its velocity as a fraction of the speed of light. The faster a particle moves, the slower time passes for it. For example (per Wikipedia on special relativity): "Time dilation explains a number of physical phenomena; for example, the decay rate of muons produced by cosmic rays impinging on the Earth's atmosphere."

In the limiting case, where something is traveling at the speed of light, special relativity implies that time stops moving in its frame of reference. From the point of view of an observer inside a photon, gluon, or hypothetical graviton, its departure and arrival take place at precisely the same moment in time.

So, why do electromagnetism, the strong force and gravity all obey time symmetry?

Because their carrier bosons are massless and do not experience time. One can't violate time symmetry, I propose, in an event that does not take place, within the frame of reference of the carrier particle, over a period of time and is instead, instantaneous in that frame of reference. In contrast, the weak force has a carrier boson which is massive and thus does experience time, so it may violate time symmetry. It experiences the difference between before and after and thus has the capacity to treat them differently.

**When could the CP violating term of the QCD Lagrangian matter?**

Why would the QCD Lagrangian's natural form have a CP violating term at all then?

One possibility is that while the spin one gluon, which is a massless boson and carries the strong force is massless, that integer spin glueballs, which are massive composite bosons composed of entirely of gluons experience time, and hence could conceivably be time symmetry violating.

Unlike photons, which when they collide simply make a more energetic massless photon rather than interacting electromagnetically since they are not themselves electrically charged, gluons are self-interacting because they themselves have color charge (two colors per gluon, rather than the single color of quarks), so composite gluons, called glueballs are theoretically expected. This self-interaction is also not merely a curiosity, it is the main thing that makes the mathematics of QCD calculations so much profoundly more complicated than QED calculations, and is what makes it a non-abelian gauge theory, and gives rise to both the asymptotic freedom at short distances and the confinement of quarks at greater distances, that are the main practical implications of QCD. (An in depth treatment of glueballs from a 2007 thesis can be found here.)

Glueballs are theoretically predicted in low energy QCD systems, in which they remain confined and not subject to easy direct observation, and are massive. Their existence is inferred from lattice approximations of the QCD equations and are also motivated by a modest number of suggestive particle accelerator observations. Multiple different lattice QCD researchers attempting to approximate the same underlying Yang-Mills QCD equations of the Standard Model in different ways have independently confirmed the theoretical prediction of various types of massive glueballs and have reached reasonably consistent inferences about their masses.

But, glueballs have never been definitively observed experimentally (although there have been some suggestive data to indicate their detection) and aren't succeptible to observation by anything nearly as straightforward as the high energy atom smashing environment (or the high energy quark-gluon plasma environment) in which we observe QCD in the high energy limit. As Wikipedia, on the subject of glueballs, explains: "Glueballs are extremely difficult to identify in particle accelerators, because they mix with ordinary meson states. Theoretical calculations show that glueballs should exist at energy ranges accessible with current collider technology. However, due to the aforementioned difficulty, they have (as of 2011) so far not been observed and identified with certainty." The GlueX experiment at the Thomas Jefferson National Accelerator Facility (JLab) accelerator, which starts in 2014, is designed to directly observe glueballs and exotic mesons, tetraquarks, and hybrid mesons (made up of both one or more gluons and two or more quarks).

As an aside, "normal" mesons are "composed of one quark and one antiquark." One of the main accomplishments of quark theory was to systematize from five components, in three color variants, and their antiparticles, 52 kinds of normal mesons (quark-anti-quark combinations),, and the 150 kinds of baryons (three quark combinations), most of which, but not all of which, have been observed experimentally, counting antiparticles separately, but excluding parity distinctions and excluding theoretically imaginable combinations involving top quarks, as top quarks appear to decay too quickly to form mesons or baryons. Of the 202 combinations in the Standard Model, only one, the proton, is stable, and only one more, the neutron, which has a mean lifetime of 14 minutes and 46 seconds, has a mean lifetime of more than 5x10^-8 seconds (about twenty millionths of a second), and the vast majority are much more short lived.

Still, it isn't unreasonable, given the natural form of the QCD Lagrangian and the sensitivity of the strong force to chirality (despite the fact that it observes parity symmetry in all experiments conducted to date), to hypothesize that QCD may exibit CP violations in the low energy limit in cases involving massive glueballs that are not present in high energy systems where gluons are massless and hence timeless, if glueballs are carriers of the strong nuclear force when they combine into these composite particles.

On the other hand, if glueballs cease to fulfill the strong nuclear force carrier role filled by individual gluons when they form composite particles, then CP violations would be absent from QCD, because it would have no massive force carriers.

I don't understand QCD well enough to know if gluons within composite force carriers can carry the strong nuclear force, but my understanding is that QCD theorists do, so determining if CP violations in low energy glueball systems if my hypothesis is correct should be a relativity straightforward matter.

Assuming that there is a distinction, I'm also not entirely clear how one would modify the equations of QCD to distinguish between those where CP violating terms are non-physical, because the strong nuclear force is carried by a massless gluon, such as the case of strong nuclear force interactions that could give rise to an electric dipole moment in the neutron, and those where CP violating terms would be physical because the strong nuclear force itneractions are carried by a massive boson such as composite glueball.

I am, of course, not the first person to suggest that glueballs might be theta dependent, and hence CP violating (and more obliquely here and here). Others scientists note that the QCD Lagrangian itself has been modified since it was originally proposed, and given that the experimental bounds on QCD are only on the order of a single digit percentage, it is entirely possible that it may need to be modified further in subtle ways in the future.

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