Single-spin asymmetries were long thought to vanish in high-energy scattering processes because of their specific time-reversal behavior. Time-reversal-odd phenomena, however, appear naturally when one includes effects of intrinsic transverse momenta of partons. The partons, quarks and gluons, enter the description of high-energy scattering processes in correlators which contain quark and gluon fields. The correlators, parameterized in terms of distribution and fragmentation functions, constitute matrix elements of non-local field configurations. For transverse momentum dependent (TMD) correlators the non-locality is along a light-front, in contrast to the non-locality along the light-cone for collinear correlators, integrated over transverse momenta. The TMD correlators require a careful treatment to assure color gauge invariance, leading to nontrivial gauge links connecting the parton fields. These give rise to time-reversal-odd phenomena, showing up as single spin and azimuthal asymmetries. The gauge links, arising from multi-gluon initial and final state interactions, depend on the color flow in the process, which has consequences for issues like universality and factorization.
We consider the various asymmetries, notably single spin asymmetries, that appear in leptoproduction as a consequence of the presence of time-reversal odd distribution functions. This could facilitate experimental searches for time-reversal odd phenomena.
We elaborate two different methods for extracting from data the relative phase of two amplitudes of some sequential decays of baryons. We also relate this phase to a particular physical angle. Moreover we suggest how to infer some time reversal odd observables from data.
It is shown that in the experiments for search of EDM of an electron (atom, molecule) the T-odd magnetic moment induced by an electric field and the T-odd electric dipole moment induced by a magnetic field will be also measured. It is discussed how to distinguish these contributions.
Longitudinal polarization of the HERA electron beam allows the study of spatial asymmetries in the final state of single inclusive hadron production in $\vec{e}p$-collisions. We estimate the size of a particular production asymmetry which is related to the time reversal properties of the underlying parton level subprocess.
We investigate the ''spontaneous'' hyperon transverse polarization in $e^+e^-$ annihilation and semi-inclusive deep inelastic scattering processes as a test of the universality of the naive-time-reversal-odd transverse momentum dependent fragmentation functions. We find that universality implies definite sign relations among various observables. This provides a unique opportunity to study initial/final state interaction effects in the fragmentation process and test the associated factorization.
The so-called time-reversal odd distribution functions are known to be non-vanishing in QCD due to the presence of the link operator in the definition of these quantities. I show that T-odd distributions can be non-vanishing also in chiral models, if vector mesons are introduced as dynamical gauge bosons of a hidden local symmetry. Moreover, since the flavor dependence of these distributions is different in chiral models respect to non chiral ones, the phenomenological analysis of experimental data will be able to distinguish between these two classes of models.
We study QCD evolution equations of the first transverse-momentum-moment of the naive-time-reversal-odd fragmentation functions - the Collins function and the polarizing fragmentation function. We find for the Collins function case that the evolution kernel has a diagonal piece same as that for the transversity fragmentation function, while for the polarizing fragmentation function case this piece is the same as that for the unpolarized fragmentation function. Our results might have important implications in the current global analysis of spin asymmetries.
We estimate the size of the azimuthal asymmetry of the semi-inclusive pion production in the deep-inelastic scattering of a polarized lepton beam off an unpolarized nucleon target for the produced hadron, arising due to both nonperturbative and perturbative effects.
We present results for leading-twist azimuthal asymmetries in semi-inclusive lepton-nucleon deep-inelastic scattering due to naively time-reversal odd transverse-momentum dependent parton distribution functions from the light-cone constituent quark model. We carefully discuss the range of applicability of the model, especially with regard to positivity constraints and evolution effects. We find good agreement with available experimental data from COMPASS and HERMES, and present predictions to be tested in forthcoming experiments at Jefferson Lab.
Time-Reversal-Invariance non-conservation has for the first time been unequivocally demonstrated in a direct measurement, one of the results of the CPLEAR experiment. What is the situation then with regard to time-reversal-invariance non-conservation in systems other than the neutral kaon system? Two classes of tests of time-reversal-invariance need to be distinguished: the first one deals with parity violating (P-odd)/time-reversal-invariance non-conserving (T-odd) interactions, while the second one deals with P-even/T-odd interactions (assuming CPT conservation this implies C-conjugation non…
We present some estimates of T-odd fragmentation and distribution functions, $H_1^\perp$ and $f_{1T}^{\perp}$, evaluated on the basis of a fit on experimental data in p$^\uparrow$p. Assuming the T-odd fragmentation function to be responsible for the single spin asymmetry in pion production in p$^\uparrow$p, we find the ratio $H_1^{\perp}/D_1$ to be in good agreement with the experimental results from DELPHI data on Z -> 2$-jet decay. We use our estimates to make predictions for ep$^\uparrow$.
The combination of transverse momentum and polarization effects in hard scattering processes, e.g polarized deep-inelastic leptoproduction, results in a rich variety of information on the hadronic structure. This information is encoded in a correspondingly large number of functions depending on both, longitudinal fractional momenta $x$ and transverse momenta $p_T$. Integration over transverse momenta establishes the connection to the usual distribution and fragmentation functions. Constraints from hermiticity and invariance under parity operation induce some relations between different functio…
In this talk, we examine the existing calculations of QCD evolution kernels for the scale dependence of two sets of twist-3 quark-gluon correlation functions, $T_{q,F}(x,x)$ and $T^{(σ)}_{q,F}(x,x)$, which are the first transverse-momentum-moment of the naive-time-reversal-odd Sivers and Boer-Mulders function, respectively. The evolution kernels at the leading order in strong coupling constant $α_s$ were derived by several groups with apparent differences. We identify the sources of discrepancies and are able to reconcile the results from various groups.
Time-Reversal-Invariance non-conservation has now been unequivocally demonstrated in a direct measurement at CPLEAR. What about tests of time-reversal-invariance in systems other than the kaon system? Tests of time-reversal-invariance belong to two classes: searches for parity violating (P-odd)/time-reversal-invariance-odd (T-odd) interactions, and for P-even/T-odd interactions (assuming CPT conservation this implies C-conjugation non-conservation). Limits on a P-odd/T-odd interaction follow from measurements of the electric dipole moment of the neutron (with a present upper limit of 6 x 10^-2…
In a bent crystal the $P$-odd effect of short-lived baryon spin rotation could imitate spin rotation caused by assumed EDM. Use of different behavior of $P$-odd and $T$-odd spin rotations at crystal turning around the direction of particle momentum makes it possible to exclude $P$-odd rotation contribution, when measuring short-lived baryons EDM. Subtraction of the measurement results for angle ranges $φ$ and $φ+ π$ from each other enables measuring $T$-odd rotation at scattering of negatively charged beauty and neutral baryons by axes of unbent crystal.
T-odd P-odd phenomenon of the photon polarization plane rotation (circular dichroism) is considered for an atomic (molecules) gas placed to an electric field. The expression for the T non-invariant polarizability of an atom (molecule) placed to an electric field is obtained. It is shown that the T-odd plane rotation angle increases when the interaction energy of an atom (molecule) with an electric field is the same order as the opposite parity levels spacing.
Generation of T-odd single spin asymmetries (SSA) by the various ingredients of QCD factorization is discussed. The possible use of SSA in studies of Generalized Parton Distribution (GPD) at HERA with the polarized lepton beam is suggested. The role of GPD in the investigation of orbital angular momenta of partons is discussed. The generalization of Equivalence principle, leading to the equipartition of momenta and total angular momenta, violated in perturbation theory, but possibly restored due to confinement and chiral symmetry breaking, is proposed. The T-odd fragmentation and fracture func…
Experiments with $Ω^{\pm}$ baryons moving in a crystal at LHC in addition to EDM measurement open possibility to study electric quadrupole moment, to observe birefringence effect (spin oscillations and spin dichroism) for $Ω^{\pm}$ baryons and to get information about constants describing P-odd, T-odd effects at $Ω^{\pm}$ baryon interactions with electrons and nuclei
Semi-inclusive deep inelastic scattering of polarized leptons off hadrons enables one to measure the antisymmetric part of the hadron tensor. For unpolarized hadrons this piece is odd under time reversal. In deep inelastic scattering it shows up as a $\langle \sin ϕ\rangle$ asymmetry for the produced hadrons. This asymmetry can be expressed as the product of a twist-three "hadron $\rightarrow$ quark" distribution function and a time reversal odd twist-two "quark $\rightarrow$ hadron" fragmentation function. This fragmentation function can only be measured for nonzero transverse momenta of the …
It is shown that in the experiments for search of the EDM of an electron (atom, molecule) the T-odd magnetic moment induced by an electric field and the T-odd electric dipole moment induced by a magnetic field will be also measured. It is discussed how to distinguish these contributions.
We reexamine the derivation of the leading order QCD evolution equations of twist-3 quark-gluon correlation functions, $T_{q,F}(x,x)$ and $T^{(σ)}_{q,F}(x,x)$, which are the first transverse-momentum-moment of the naive-time-reversal-odd parton distribution functions - the Sivers and Boer-Mulders function, respectively. The evolution equations were derived by several groups with apparent differences. We identify the sources that are responsible for the differences, and are able to reconcile the results from various groups.
We study the transverse momentum dependent quark-gluon-quark correlation function. Using a spectator diquark model, we calculate the eight time-reversal-odd interaction-dependent twist-3 quark distributions appearing in the decomposition of the transverse momentum dependent quark-gluon-quark correlator. In order to obtain finite results, we assume a dipole form factor for the nucleon-quark-diquark coupling, instead of a point-like coupling. The results are compared with the time-reversal-odd interaction-independent twist-3 TMDs calculated in the same model.
The T-reversal-violating optical gyrotropy is discussed. An expression for dielectric permittivity describing T-reversal-violating phenomena in a medium exposed to external electric and magnetic fields is presented. Optical phenomena of birefringence and polarization plane rotation of photon in a medium exposed to external electric and magnetic fields are considered.
In this paper we will describe a parallel between phenomenological and symmetry-based descriptions of the process dependence of the time-reversal-odd phenomena in QCD, such as the Sivers effect, with the goal of defining the essential elements that lead to such process dependence in QCD. This will then serve as a starting point to explore possible generalizations in a different gauge theory.
Spin precession of channelled particles in bent crystals at the LHC gives unique possibility for measurements as electric and magnetic moments of charm, beauty and strange charged baryons so and constants determining CP ($T_{odd}, P_{odd}$) violation interactions and $P_{odd}, T_{even}$ interactions of baryons with electrons and nucleus (nucleons). For a particle moving in a bent crystal a new effect caused by nonelastic processes arises: in addition to the spin precession around the direction of the effective magnetic field (bend axis), the direction of electric field and the direction of the…
Time-reversal breaking and parity-conserving millistrong interactions, suggested in 1965, still remain a viable mechanism of CP-violation beyond the Standard Model. One of its possible manifestations is the T-odd asymmetry in the transmission of tensor-polarized deuterons through a vector-polarized hydrogen gas target. Upon the rotation of the deuteron polarization from the vertical direction into the ring plane, the T-odd asymmetries, odd against the reversal of the proton polarization in the target, will continuously oscillate with first or second harmonics of the spin precession frequency. …
The origin of the large CP-odd and T-odd asymmetry observed in the decay K_L -> pi^+ pi^- e^+ e^- is traced to the polarization properties of the photon in the decay K_L -> pi^+ pi^- gamma. The Stokes vector of the photon S = (S_1, S_2, S_3) is studied as a function of the photon energy and found to possess CP-violating components S_1 and S_2 which are sizeable over a large part of the phase space, despite being proportional to the epsilon parameter of the K_L wave function. The component S_2 is T-even and manifests itself as a circular polarization of the photon, while S_1 is T-odd and gives …
A new possibility for the study of time-reversal violation is described. It consists in measurement of nonpolarized neutron transmission through nonpolarized nuclear target placed into electric field
The T-reversal-violating optical gyrotropy is discussed. A new optical phenomenon of photon polarization plane rotation in a medium exposed to an electric field caused by the Stark mixing of atom (molecule) levels and T-odd phenomenon of photon polarization plane rotation appearing at photon passing through non-center-symmetrical diffraction grating are considered. The experimental methods to observe these phenomena are proposed.
We re-examine the jet probes of the nucleon spin and flavor structures. We find for the first time the time-reversal odd (T-odd) component of a jet, conventionally thought to vanish, can survive due to the non-perturbative fragmentation and hadronization effects and could be testable. This additional contribution of a jet will lead to novel jet phenomena relevant for unlocking the access to several spin structures of the nucleon, which were thought to be impossible by using jets. As examples, we show how the T-odd constituent can couple to the proton transversity at the Electron Ion Collider (…
It is shown that the T- and P-odd weak interactions yield to the existence of both electric field and magnetic (directed along the electric field) field around an electric charge. Similarly the assotiated magnetic field is directed along the vector of strength of stationary gravitational field.
A first measurement of time-reversal (T) asymmetries that are not also CP asymmetries has been recently achieved by the BaBar collaboration. We analyze the measured asymmetries in the presence of direct CP violation, CPT violation, wrong strangeness decays and wrong sign semileptonic decays. We note that the commonly used S_{ψK} and C_{ψK} parameters are CP-odd, but have a T-odd CPT-even part and a T-even CPT-odd part. We introduce parameters that have well-defined transformation properties under CP, T and CPT. We identify contributions to the measured asymmetries that are T conserving. We exp…
We show that a system of quarks interacting with chiral fields provides a physical representation of a ``non-standard'' time reversal for particle multiplets proposed by Weinberg. As an application, we argue that, if the internal structure of hadrons is described by a chiral lagrangian, the so-called time-reversal-odd quark distribution functions might not be forbidden by time-reversal invariance.
We discuss the effects of so-called gluonic poles in twist-three hadronic matrix elements, as first considered by Qiu and Sterman, in the Drell-Yan process. These effects cannot be distinguished from those of time-reversal odd distribution functions, although time-reversal invariance is not broken by the presence of gluonic poles. Both gluonic poles and time-reversal odd distribution functions can lead to the same single spin asymmetries. We explicitly show the connection between gluonic poles and large distance gluon fields, identify the possible single spin asymmetries in the Drell-Yan proce…
Recent measurements in paramagnetic molecules improved the limit on the electron electric dipole moment (EDM) by an order of magnitude. Time-reversal (T) and parity (P) symmetry violation in molecules may also come from their nuclei. We point out that nuclear T,P-odd effects are amplified in paramagnetic molecules containing deformed nuclei, where the primary effects arise from the T,P-odd nuclear magnetic quadrupole moment (MQM). We perform calculations of T,P-odd effects in the molecules TaN, ThO, ThF$^+$, HfF$^+$, YbF, HgF, and BaF induced by MQMs. We compare our results with those for the …
Motivated by the recent experimental announcements for direct measurements of time-reversal non-invariance in the neutral kaon system, we make a comparative discussion of the CPLEAR and KTeV measurements. The most suitable way to consistently incorporate the mixing, the time evolution and the decays of kaons, is to describe the neutral kaon system as a system with a non-Hermitean Hamiltonian. In this framework, the physical (decaying) incoming and outgoing states are distinct and belong to dual spaces. Moreover, since they are eigenstates of the full Hamiltonian, they never oscillate. This is …
We give a general overview of the role and interpretation of fragmentation functions in hard processes. Transverse momentum dependence gives rise to time-reversal odd fragmentation functions contributing at leading order. Final state interactions are necessary to have non-zero time reversal odd functions. We present a model calculation for the special case of interference fragmentation functions in two-hadron inclusive lepton hadron scattering.
A system of quarks interacting with chiral fields is shown to provide a physical realization of a "non-standard" time reversal for particle multiplets which mixes the multiplet components. We argue that, if the internal structure of the nucleon is governed by a chiral dynamics, the so-called T-odd quark distribution functions are not forbidden by time-reversal invariance and hence might be non vanishing. This agrees with some other recent results. From a phenomenological point of view, this would give rise to single-spin asymmetries in inclusive processes involving a transversely polarized nuc…
The isovector time-reversal- and parity-violating pion-nucleon coupling $\bar{g}_π^{(1)}$ is uniquely sensitive to dimension six interactions between right-handed light quarks and the Standard Model Higgs doublet that naturally arises in left-right symmetric models. Recent work has used the $\bar{g}_π^{(1)}$-induced one-loop contribution to the neutron electric dipole moment $d_n$, together with the present experimental $d_n$ bound, to constrain the CP-violating parameters of the left-right symmetric model. We show that this and related analyses are based on an earlier meson theory $d_n$ compu…
We have scrutinized the transverse momentum dependent quark-gluon-quark correlation function. We have utilized the light-front quark-diquark model to study the time-reversal-odd interaction-dependent twist-3 gluon distributions which we have obtained from the disintegration of the transverse momentum dependent quark-gluon-quark correlator. Specifically, we have studied the behavior of $\tilde{e}_{L}$ and $\tilde{e}_{T}$ while considering our diquark to be an axial-vector.
We present a general formalism for the evaluation of time-reversal odd parton distributions. This formalism is applied to the evaluation of the two T-odd TMDs allowed at the twist-2 level, i.e., the Sivers and the Boer-Mulders functions. We have performed the calculation for two different models of proton structure: a non relativistic constituent quark model and the MIT bag model. The results obtained in both models are compared. So are the two T-odd functions. We comment on the fulfilment of the Burkardt sum rule as well as on the current status of T-odd TMDs in phenomenology.
With the prospect of large numbers of $J/Ψ$ decay events becoming available in the near future, it is interesting to search for symmetry violating effects as probes of new physics and tests of the standard model. $J/Ψ$ decay events could provide the first observation of weak effects in otherwise strongly decaying particles. We calculate a T odd asymmetry in the $J/Ψ$ decay into photon plus lepton pair due to Z boson exchange. Extensions to hadronic final states are also discussed.
We compute a T-odd triple vector correlation for the $μ\rightarrow eγ$ decay and the $μ\rightarrow e$ conversion process. We find simple results in terms of the CP violating phases of the effective Hamiltonians. Then we focus on the minimal Left-Right symmetric extension of the Standard Model, which can lead to an appreciable correlation. We show that under rather general assumptions, this correlation can be used to discriminate between Parity or Charge-conjugation as the discrete Left-Right symmetry.
We study the decays of $Λ_b \to Λ(\to p π^-) \ell ^+ \ell^-$ with $\ell = (e, μ, τ)$. In particular, we examine the full angular distributions with polarized $Λ_b$ and identify the time-reversal asymmetries or T-odd observables. By using the homogeneous bag model, we find that the decay branching fractions of $Λ_b \to Λ\ell^+\ell^-$ are $(9.1\pm 2.5,7.9\pm 1.8, 2.1\pm 0.2)\times 10^{-7}$ for $\ell =(e, μ,τ)$, respectively. In addition, we obtain that $A_{FB}^{\ell} = -0.369\pm 0.007$ and $A_{FB}^{h}=-0.333\pm 0.004$, averaged in the range of $15 \leq q^2 \leq 20~ \text{GeV}^2$. These results a…
A complete study of teh angualr distributions of the process Lambda_b to Lambda V(1-) with lambda to p pi- and V(J/Psi) to l+l- or V(rho0,omega, phi) to pi+ pi-, K+K- is performed. Emphasis is put on the initial Lambda-b polarization produced in the proton-proton collisions and, without any dynamical assumption, polarized density-matrices of the vector-mesons V are derived and help to construct T-odd observables which allow us to perform tests of both Time-Reversal and CP violation.
Nuclear forces violating parity and time reversal invariance P,T-odd produce P,T-odd nuclear moments, for example, the nuclear Schiff moment. In turn, this moment can induce the electric dipole moment (EDM) in the atom. The contribution to the Schiff moment from the soft collective quadrupole and octupole vibrations in spherical nuclei is calculated in the framework of the quasiparticle random phase approximation with separable quadrupole and octupole forces. The values of nuclear Schiff moments predicted for odd 217-221Ra and 217-221Ra isotopes indicate a possibility of enhancement by a facto…
A channelled particle, which moves in a crystal, alongside with electromagnetic interaction also experiences weak interaction with electrons and nuclei, as well as strong interaction with nuclei. Measurements of polarization vector and angular distribution of particles scattered by axes (planes) of unbent crystal enable to obtain limits for the EDM value and for values of constants describing P- and T-odd interactions. The same measurements also allow studying magnetic dipole moment of charged and neutral particles. Investigation of left-right asymmetry by the use of two unbent crystals makes …
Parity and time invariance violating (${\cal P},{\cal T}$-odd) nuclear forces produce ${\cal P},{\cal T}$-odd nuclear moments, for example, the nuclear Schiff moment. In turn, this moment can induce electric dipole moments (EDMs) in atoms. We estimate the contribution to the Schiff moment from the soft collective octupole vibrations existing in many heavy nuclei. Two possible mechanisms are considered, mixing of the ground state of an odd-$A$ nucleus with the octupole phonon state, and ${\cal P},{\cal T}$-odd admixture to the single-particle wave function of the valence nucleon. We found pract…
We demonstrate that polarization measurements in exclusive semileptonic B decays are powerful tools for unraveling non-standard model sources of T-violation. Measurements of the transverse polarization of the tau lepton in the B to D tau nu and B to D* tau nu decays probe separately effective scalar and pseudoscalar CP-violating four-Fermi interactions, whereas the D* polarization in the B to D* l nu (l=e,mu) decay is sensitive to an effective right-handed quark current interaction. Two T-odd polarization structures exist involving the D* polarization and they can be isolated and studied separ…
The Weyl fermions with a well defined chirality are known to demand that the dimension of space which they inhabit must be odd. It is shown here, however, that not all odd dimensional spaces are equally good hosts: in particular, an arbitrary number of chiral Weyl fermions can acquire a Majorana mass only in three (modulo eight) dimensions. The argument utilizes a) the precise analogy that exists between the Majorana mass term and the Cooper pairing of time-reversed Weyl fermions, and b) the conditions on the requisite time-reversal operator, which are implied by the Clifford algebra. The theo…
Internal target experiments with high quality proton beams allow for a new class of experiments providing null tests of time reversal symmetry in forward scattering. This could yield more stringent limits on T-odd P-even observables. A excellent candidate for such experiments is the proton deuteron system. This system is analyzed in terms of effective T-violating P-conserving nucleon-nucleon interactions and bounds on coupling strengths that might be expected are given.
BaBar collaboration announced that they observed time reversal (T) asymmetry through $B$ meson system. In the experiment, time dependencies of two distinctive processes, $B_- \rightarrow \bar{B^0}$ and $\bar{B^0}\rightarrow B_-$($-$ expresses CP value) are compared with each other. In our study, we examine event number difference of these two processes. In contrast to the BaBar asymmetry, the asymmetry of events number includes the overall normalization difference for rates. Time dependence of the asymmetry is more general and it includes terms absent in one used by BaBar collaboration. Both o…
Over the past years a lot of progress has been made in the understanding of single spin asymmetries in hard scattering processes. We briefly review this subject, covering the non-vanishing of time-reversal odd parton distributions, universality of fragmentation functions, and the discovery of previously unknown time-reversal odd parton distributions at subleading twist. Interestingly enough, all these developments were initiated by simple model calculations.
We have calculated the atomic electric dipole moments (EDMs) induced in ^{199}Hg, ^{129}Xe, ^{223}Rn, ^{225}Ra, and ^{239}Pu by their respective nuclear Schiff moments S. The results are (in units 10^{-17}S(e {fm}^{3})^{-1}e cm): d(^{199}Hg)=-2.8, d(^{129}Xe)=0.38, d(^{223}Rn)=3.3, d(^{225}Ra)=-8.5, d(^{239}Pu)=-11. We have also calculated corrections to the parity- and time-invariance-violating (P,T-odd) spin-axis interaction constant in TlF. These results are important for the interpretation of atomic and molecular experiments on EDMs in terms of fundamental P,T-odd parameters.
The CALTECH gamma-transition experiment testing time reversal symmetry via the E2/M1 mulipole mixing ratio of the 122 keV gamma-line in Fe-57 has already been performed in 1977. Extending an earlier analysis in terms of an effective one-body potential, this experiment is now analyzed in terms of effective one boson exchange T-odd P-even nucleon nucleon potentials. Within the model space considered for the Fe-57 nucleus no contribution from isovector rho-type exchange is possible. The bound on the coupling strength phi_A from effective short range axial-vector type exchange induced by the exper…
We compute the dynamical relaxation times for chiral transport phenomena in strongly coupled regime using the AdS/CFT correspondence. These relaxation times can be a useful proxy for the dynamical time scale for achieving equilibrium spin-polarization of quasi-particles in the presence of magnetic field and fluid vorticity. We identify the Kubo relations for these relaxation times and clarify some previous issues regarding time-dependence of the Chiral Vortical Effect. We study the consequences of imposing time-reversal invariance on parity-odd thermal noise fluctuations that are related to ch…
Parity and time invariance violating forces produce collective P- and T- odd moments in nuclei with static octupole deformation. Collective Schiff moment, electric octupole and dipole and also magnetic quadrupole appear due to the mixing of rotational levels of opposite parity and can exceed single-particle moments by more than a factor of 100. This enhancement is due to two factors, the collective nature of the intrinsic moments and the small energy separation between members of parity doublets. The above moments induce T- and P- odd effects in atoms and molecules. Experiments with such syste…
We show possible transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and 4 functions in addition to the leading twist-2 ones by investigating all the possible decomposition of a quark correlation function in the Lorentz-invariant way. The Hermiticity and parity invariance are imposed in the decomposition; however, the time-reversal invariance is not used due to an active role of gauge links in the TMDs. Therefore, there exist time-reversal-odd functions in addition to the time-reversal even ones in the TMDs. We list all the functions up to tw…
