What is the Boer-Mulders function/effect?

The formal definition of the Boer-Mulders function was given in the publication: D. Boer, P.J. Mulders, "Time reversal odd distribution functions in leptoproduction,'' Physical Review D 57 (1998) 5780, preprint version (november 1997)

The Boer-Mulders function (commonly denoted by h1) describes the net polarization of quarks inside an unpolarized proton. Protons and quarks are both spin-1/2 particles that can be polarized in specific directions (they can have spin up or spin down in a specific direction). If one averages over the polarization states of the proton, then we call it an unpolarized proton (the net polarization averaged over many protons in a proton beam for instance is then zero). It turns out that the quarks can be polarized on average even inside such an unpolarized proton, as long as they are not moving exactly along the proton direction. If the proton moves along the z direction say, then the quarks can have some transverse momentum kT with respect to the proton momentum, which together define a plane. The quarks can then have a net polarization orthogonal (or transverse) to that plane. A nonzero Boer-Mulders function means that there is such a net quark polarization. Pictorially this can be displayed as follows:

If this function is nonzero, then it reflects the presence of a handedness inside the proton (P.(kT x sT)). So the Boer-Mulders function is a quark distribution that quantifies a particular spin-orbit correlation. It is similar to the Sivers function (D. Sivers, "Single spin production asymmetries from the hard scattering of point-like constituents", Physical Review D 41 (1990) 83) which refers to unpolarized quarks inside a transversely polarized proton.

The Boer-Mulders function requires that the quarks have a transverse momentum with respect to the proton, therefore, it is referred to as a transverse momentum dependent parton distribution or a TMD. Partons is what Feynman called the constituents of the proton: the quarks, the antiquarks and the gluons. There is an analogous effect for gluons (P.J. Mulders and J. Rodrigues, "Transverse momentum dependence in gluon distribution and fragmentation functions", Physical Review D 63 (2001) 094021), even though they are spin-1 particles like the photon and one is dealing with so-called linear polarization. And there is actually an analogous effect for photons inside a proton, but also for photons "inside" an electron. A recent experimental demonstration of such a QED analogue was obtained with the Relativistic Heavy Ion Collider at BNL: the effect of linear polarized photons from unpolarized gold nuclei were observed in ultra-peripheral gold-gold collisions (STAR Collaboration, "Measurement of e^+e^- momentum and angular distributions from linearly polarized photon collisions", Physical Review Letters 127 (2021) 052302).

Is there any evidence that the quark Boer-Mulders function is nonzero?

Yes, there is a first-principles demonstration using lattice QCD: B.U. Musch et al., "Sivers and Boer-Mulders observables from lattice QCD", Physical Review D 85 (2012) 094510

Furthermore, this distribution can offer an explanation for an anomalously large asymmetry in the Drell-Yan process which was measured in 1986 and 1989 independently at CERN and Fermilab, as demonstrated in the article: D. Boer, "Investigating the origins of transverse spin asymmetries at RHIC,'' Physical Review D 60 (1999) 014012. There also have been measurements of asymmetries in semi-inclusive deep inelastic scattering by the COMPASS, HERMES and Jefferson Lab experiments that are indicative of a nonzero Boer-Mulders function. 

In addition, the effect is allowed by all the symmetries of the microscopic theory of quarks and gluons (Quantum Chromodynamics or QCD), so there is no reason why it would have to be zero (like the linear polarization of photons inside an electron is fully allowed and generally nonzero in QED). Based on the parity (or space inversion) symmetry of QCD one can conclude that the quarks can only have a net transverse polarization though, not a net longitudinal polarization. The latter would be a sign of the weak nuclear force in action, but that is expected to be a much smaller effect. 

Formal definition      

For those familiar with Quantum Field Theory language, the formal definition of the quark Boer-Mulders function is given by:

The state |P> denotes a proton with a momentum P and the right hand side is the expectation value of a particular quark operator inside the proton. This quark operator is nonlocal and contains a Wilson line LC that makes it color gauge invariant. The Dirac structure projects out a specific quark polarization state (a helicity flip state or transversely polarized state). F.T. stands for a three-dimensional Fourier transform and C is a specific path of the Wilson line LC that depends on the process in a calculable and therefore predictable way. So the magnitude of the function is process dependent which leads to the natural question to ask:

Is it a property of the proton?

It requires a proton (or any other hadron for that matter) and interactions with that proton. In a sense it is a property of the proton "in action", so it is not a static property (not a part of the proton wave function in that sense), but rather a dynamic one. This does not make it less fundamental, in my opinion. This is simply how one sees the proton in high energy scattering processes. In a sense this is true for any quark distribution: only when one views the proton with a sufficiently high energetic probe, one can speak of quark distributions inside the proton in the first place. One needs high energy to look inside a proton and to see or probe the quarks. Think of it as with an X-ray picture: that displays what someone looks like when viewed by X-rays. The X-rays are not part of the person being "photographed", but the interaction with the X-rays does produce a picture of the person, even if it looks quite different from the more familiar pictures in the visible light spectrum. Just like the picture of the sun in the neutrino spectrum is still a picture of the sun. Likewise the Boer-Mulders function is different when viewed with photons or with other protons for instance, but all will show nonzero net quark polarization inside unpolarized protons.  

Is it related to the electric dipole moment?

No, the electric dipole moment (EDM) operator of quarks is a local operator, the one that one would get by integrating over all quark momenta, which for the Boer-Mulders function gives identically zero (as it is an odd function of kT). Moreover, the Boer-Mulders function is parity conserving, whereas the EDM violates P and CP. It is also not related to the EDM of the proton which is the expectation value of a (local) vector operator, not of a tensor operator.

Rather, the Boer-Mulders function is related to a kind of anomalous magnetic moment, see M. Burkardt, "Transverse deformation of parton distributions and transversity decomposition of angular momentum,'' Physical Review D 72 (2005) 094020

Further scientific references 

Research papers by other people about the Boer-Mulders function

D. Boer, 1-2-2022