pretzelosityThe Wikitionary definition, while accurately identifying this as a term from mathematics and physics, when it says that: "the condition of having the knotted form of a pretzel", while not being completely misleading, is also not very helpful for purposes of understanding what the word means as it is used in practice in physics. In physics, at least, this term is used exclusively in a much more narrow and specific sense.

Basically, pretzelosity is one of a number of properties that describe the structure of a composite particle made up of more than one quark at the moment that something moving at close to the speed of light slams into it. These properties, collectively, are a hadron's parton distribution function.

*Background: What Are Parton Distribution Functions?*

A "parton distribution function" (PDF), in particle physics, is a formula or chart usually fitted with large empirical data sets, that sets forth the probability that a particle moving at close to the speed of light will hit a particular particle in a hadron when it flies into the hadron. This is more complicated than it seems because "virtual particles" within the hadron have a material probability of being hit, even though we don't normally think of those virtual particles as being present in a particular hadron at all.

(I spent a good part of my downtime while waiting to be a speech and debate tournament judge at a recent tournament hosted by my daughter's high school, perusing the open access 194 page "Handbook of perturbative QCD" that is basically the standard introduction to the world of parton distribution functions for people first learning about this obscure, but practically important, corner of the particle physics world, like incoming graduate assistants working at particle colliders for the first time.)

In an ideal world, one could use the Standard Model's QCD Lagrangian (i.e. the equation of that governs the strong force, one of the four fundamental forces of nature) and the fundamental constants of the Standard Model to calculate the parton distribution functions of a hadron exactly.

But, in reality, the math is too hard to get more than a qualitative idea about what patron distribution functions should look like from first principles, even for near genius physicists who got perfect math SAT scores and then spent many years studying advance math, even with the help of computers (although not for want of increasingly fruitful attempts to do so such as this recent study).

In practice, these PDFs are based instead upon voluminous data gathering from actual particle collisions at high speeds (similar in size to the student's t test reference chart at the back of your college statistics textbook) which is then fitted to smooth mathematical formulas that mere particle physics graduate students can understand. These formulas, in turn, are then used to predict how similar future collisions will behave with the exquisite precision that the Standard Model is famous for providing.

A PDF is a bit of a cheat when it comes to the question of the excellent accuracy of the Standard Model. But, PDF formulas still do leverage a comparatively modest amount of data into a much more vast number of potential predictions, even if they don't manage it with the comparative elegance of the one long, but compact formula, and the couple dozen fundamental physical constants, which give rise to them as the Standard Model should if we could really do all of the QCD calculations that it calls for properly in a reasonable amount of time.

*Back To Pretzelosity*

To a good approximation, under the kinds of conditions where this property of composite particles made up of quarks is studied, the following relationship holds true:

helicity − transversity = pretzelosityThe word pretzelosity is used in the following sentence from a recent physics preprint:

In addition, we examine the model relation between the orbital angular momentum and pretzelosity, and find it is violated in the axial-vector case.There are another fifteen other papers in the arxiv abstract database, most of which are published in respectable academic journals or physics conference proceedings, that use the term, the earliest of which was published in 2008 by Avakian, et al., which defines the term more exactly:

The leading twist transverse momentum dependent parton distribution function h⊥1T , which is sometimes called “pretzelosity,” is studied.

*A Couple More Background Concepts*

Helicity is a concept of a particle analogous to spin.

Traverse momentum is momentum at a right angle to the direction the incoming particle hitting the hadron is moving (which is the axial or longitudinal direction).

*Visualizing Pretzelosity*

The pretzelosity of a particle is basically the extent to which the probabilities that a relativistic particle hitting a composite particle made of quarks will hit some particular subparticle of it is influenced by the interaction between the target particle's spin and its momentum in the direction at right angles to the motion of the particle hitting it.

Imagine the hadron as a baseball being thrown from the pitchers mound to home base that is spinning as a fast moving electron zooms towards it from the direction of first base. The helicity of the baseball is related to the spinning of the baseball the electron moves towards it. The traverse momentum of the baseball is the direction and speed of the baseball as it moves towards home base multiplied by the baseball's mass. The extent to which the way the fast moving electron's interactions with the baseball/hadron are influenced by the interactions of these components of the baseball/hadron's motion is its pretzelosity.

To understand why someone might use the term pretzelosity for this property, imagine the line through space that a single point on the surface of the baseball/hadron would trace as it moved towards home base. For those of you who are spatial visualization challenged, this line would be basically pretzel shaped. This would be even more true in the case of a particle accelerator, like the LHC, where rather than moving in straight lines, high powered magnets cause the colliding particles move at extremely high speeds around a great big circular tube enclosing an area the size of a small city, many times each second, until they hit each other.

*What We Know About Pretzelosity*

Pretzelosity, in practice, turns out for reasons of fundamental physics, to have a much smaller impact on what will be hit in the target hadron than lots of other factors that influence that outcome, to an extent that is well defined mathematically. For example, in the case of one QCD observable measured at the HERMES experiment, the effect was possibly as large as 1%-2% of the total observed result, although the measured results were consistent with a 58% chance that pretzelosity actually had zero impact. Up quark components and down quark components of a hadron have different (and approximately opposite) pretzelosity.

The pretzelosity of a composite particle made of quarks is closely related to the nature of the motion of the subcomponents of a particle around the composite particle's center (their "orbital angular momentum"), although no one has described in a usable formula the exact relationship between pretzelosity and orbital angular momentum that is observed in nature.

Pretzelosity is one part of understanding the "proton spin puzzle." The proton spin puzzle, in turn, is the mystery of how the spin of the proton ends up being equal to the sum of the spin of its quarks, even though experiments seem to show that almost none of the overall proton's spin, which to oversimplify is made up of three quarks held together by gluons, actually comes from the quarks when their individual spins were measured one at a time and then averaged.

The proton spin puzzle is one of the important unsolved problems in physics, so pretzelosity is actually part of something that is kind of a big deal to physicists trying to understand the mysteries of the universe, even if it is a bit of an obscure and hard to understand and explain concept.

## 3 comments:

Burkardt develops the concepts without actually using the term itself in a 2006 paper.

http://arxiv.org/pdf/hep-ph/0611256.pdf

Every single usage in Google scholar where the term actually appears (19 pages of search results) is in the narrow particle physics PDF sense of the word and not in a mathematical sense and is in the period from 2008 to the present.

There are 16 Google scholar references in 2008. http://scholar.google.com/scholar?start=10&q=pretzelosity&hl=en&as_sdt=0,6&as_ylo=2008&as_yhi=2008

One article by Gerald A. Miller from February of that year interestingly defines it as synonomous with non-sphericity.

http://scholar.google.com/scholar?start=10&q=pretzelosity&hl=en&as_sdt=0,6&as_ylo=2008&as_yhi=2008

This is the earliest use of the word in Google scholar.

The Miller definition is interesting because is suggests that Prezelosity might be a good term to describe a central term of non-linear self-interaction giving rise to dark matter and dark energy effects in general relativity with a term that better captures graviton self-interaction than the conventional one.

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