Four-momentum Conservation Research Articles

Consistency of the pion form factor and unpolarized transverse momentum dependent parton distributions beyond leading twist in the light-front quark model

We investigate the interplay among the pion’s form factor, transverse momentum dependent distributions (TMDs), and parton distribution functions (PDFs) extending our light-front quark model (LFQM) computation based on the Bakamjian-Thomas construction for the two-point function [1,2] to the three-point and four-point functions. Ensuring the four-momentum conservation at the meson-quark vertex from the Bakamjian-Thomas construction, the meson mass is taken consistently as the corresponding invariant meson mass both in the matrix element and the Lorentz factor in our LFQM computation. We achieve the current-component independence in the physical observables such as the pion form factor and delve into the derivation of unpolarized TMDs and PDFs associated with the forward matrix element. We address the challenges posed by twist-4 TMDs and exhibit the fulfillment of the sum rule. Effectively, our LFQM successfully handles the light-front zero modes and offers insights for broader three-point and four-point functions and related observables. Published by the American Physical Society 2024

KinFit: A Kinematic Fitting Package for Hadron Physics Experiments

A kinematic fitting package, KinFit, based on the Lagrange multiplier technique has been implemented for generic hadron physics experiments. It is particularly suitable for experiments where the interaction point is unknown, such as experiments with extended target volumes. The KinFit package includes vertex finding tools and fitting with kinematic constraints, such as mass hypothesis and four-momentum conservation, as well as combinations of these constraints. The new package is distributed as an open source software via GitHub. This paper presents a comprehensive description of the KinFit package and its features, as well as a benchmark study using Monte Carlo simulations of the pprightarrow pK^+Lambda rightarrow pK^+ppi ^- reaction. The results show that KinFit improves the parameter resolution and provides an excellent basis for event selection.

Collinear factorization at subasymptotic kinematics and validation in a diquark spectator model

We revisit the derivation of collinear factorization for Deep Inelastic Scattering at subasymptotic values of the four-momentum transfer squared, where the masses of the particles participating in the interaction cannot be neglected. By using an inclusive jet function to describe the scattered quark final state, we can restrict the needed parton kinematic approximations just to the four-momentum conservation of the hard scattering process, and explicitly expand the rest of the diagram in powers of the unobserved parton transverse momenta rather than neglecting those. This procedure provides one with more flexibility in fixing the virtuality of the scattered and recoiling partons than in the standard derivation, and naturally leads to scaling variables that more faithfully represent the partonic kinematic at subasymptotic energy than the Bjorken's ${x}_{B}$ variable. We then verify the validity of the obtained factorization formula by considering a diquark spectator model designed to reproduce the main features of electron-proton scattering at large ${x}_{B}$ in Quantum Chromodynamics. In the model, the Deep Inelastic Scattering contribution to the cross section can be explicitly isolated and analytically calculated, then compared to the factorized approximation. Limiting ourselves to the leading twist contribution, we then show that use of the new scaling variables maximizes the kinematic range of validity of collinear factorization, and highlight the intrinsic limitations of this approach due to the unavoidably approximate treatment of four-momentum conservation in factorized diagrams. Finally, we briefly discuss how these limitations may be overcome by including higher-twist corrections to the factorized calculation.

Kinematics and Selection Rules for Light-by-Light Scattering in a Strong Magnetic Field

Selection rules that follow from parity and four-momentum conservation are listed for head-on light-by-light scattering in a strong magnetic field taking into account nontrivial dispersion laws of different photon eigen-modes. The wave-length shifts occur for certain transitions between photon eigen-modes.

Implementing NLO DGLAP evolution in parton showers

We present a parton shower which implements the DGLAP evolution of parton densities and fragmentation functions at next-to-leading order precision up to effects stemming from local four-momentum conservation. The Monte-Carlo simulation is based on including next-to-leading order collinear splitting functions in an existing parton shower and combining their soft enhanced contributions with the corresponding terms at leading order. Soft double counting is avoided by matching to the soft eikonal. Example results from two independent realizations of the algorithm, implemented in the two event generation frameworks Pythia and Sherpa, illustrate the improved precision of the new formalism.

Anisotropic matching principle for the hydrodynamic expansion

Following the recent success of anisotropic hydrodynamics we propose a new, general prescription for the hydrodynamics expansion around an anisotropic background. The anisotropic distribution is fixing exactly the complete energy-momentum tensor, just like the effective temperature is fixing the proper energy density in the ordinary expansion around local equilibrium. This means that momen- tum anisotropies are already included at the leading order, allowing for large pressure anisotropies without the need of a next to leading order treatment. The first moment of the Boltzmann equation (local four-momentum conservation) provides the time evolution of the proper energy density and the four velocity. Differently from previous prescriptions, the dynamic equations for the pressure corrections are not derived from the zeroth or second moment of the Boltzmann equation, but they are taken directly from the exact evolution given by the Boltzmann equation. We check the effec- tiveness of this new approach by matching with the exact solution of the Boltzmann equation in the Bjorken limit with the collisional kernel treated in relaxation time approximation, finding an unprecedented agreement.

Axiomatization of Special Relativity in First Order Logic**Supported by the National Science Foundation of China under Grant Nos. 11235003 and 11475023, National Social Sciences Foundation of China under Grant No. 14BZX078 and the Research Fund for the Doctoral Program of Higher Education of China, and the Undergraduate Training Program of Beijing

The axiomatization of physical theories is a fundamental issue of science. The first-order axiomatic system SpecRel for special relativity proposed recently by Andréka et al. is not enough to explain all the main results in the theory, including the twin paradox and energy-mass relation. In this paper, from a four-dimensional space-time perspective, we introduce the concepts of world-line, proper time and four-momentum to our axiomatic system SpecRel+. Then we introduce an axiom of mass (AxMass) and take four-momentum conservation as an axiom (AxCFM) in SpecRel+. It turns out that the twin paradox and energy-mass relation can be derived from SpecRel+ logically. Hence, as an extension of SpecRel, SpecRel+ is a suitable first-order axiomatic system to describe the kinematics and dynamics of special relativity.

Diffraction from a moving grating

A first-principles tutorial account is given for the diffraction of either light or free, non-relativistic, massive particles from a moving grating. Discussion on the properties of the moving grating including non-reciprocity is followed by considering both Raman-Nath (thin grating) diffraction and Bragg (thick grating) diffraction in either transmission or reflection geometry. Use of four-momentum conservation with dispersion relations yield the angles of diffraction. The increase in the angle of diffraction for grazing incident by using a moving grating relative to using a static grating is calculated. It is shown that this increase is possible only for Raman-Nath geometry.

Mass-energy and momentum extraction by gravitational wave emission in the merger of two colliding black holes: The non-head-on case

We examine numerically the post-merger regime of two nonspining holes in non-head-on collisions in the realm of nonaxisymmetric Robinson-Trautman spacetimes. Characteristic initial data for the system are constructed and evolved via the Robinson-Trautman equation. The numerical integration is performed using a Galerkin spectral method which is sufficiently stable to reach the final configuration of the remnant black hole, when the gravitational wave emission ceases. The initial data contains three independent parameters, the ratio mass $\ensuremath{\alpha}$ of the individual colliding black holes, their initial premerger infalling velocity and the incidence angle of collision ${\ensuremath{\rho}}_{0}$. The remnant black hole is characterized by its final boost parameter, rest mass and scattering angle. The motion of the remnant black hole is restricted to the plane determined by the directions of the two initial colliding black holes, characterizing a planar collision. The net momentum fluxes carried out by gravitational waves are confined to this plane. We evaluate the efficiency of mass-energy extraction, the total energy and momentum carried out by gravitational waves and the momentum distribution of the remnant black hole for a large domain of initial data parameters. Our analysis is based on the Bondi-Sachs four-momentum conservation laws. The process of mass-energy extraction is shown to be less efficient as the initial data departs from the head-on configuration. Head-on collisions (${\ensuremath{\rho}}_{0}={0}^{o}$) and orthogonal collisions (${\ensuremath{\rho}}_{0}=90\ifmmode^\circ\else\textdegree\fi{}$) constitute, respectively, upper and lower bounds to the power emission and to the efficiency of mass-energy extraction. On the contrary, head-on collisions and orthogonal collisions constitute, respectively, lower and upper bounds for the momentum of the remnant. Distinct regimes of gravitational wave emission (bursts or quiescent emission) are characterized by the analysis of the time behavior of the gravitational wave power as a function of $\ensuremath{\alpha}$. In particular, the net gravitational wave flux is nonzero for equal-mass colliding black holes in non-head-on collisions. The momentum extraction and the patterns of the momentum fluxes, as a function of the incidence angle, are examined. The relation between the incidence angle and the scattering angle closely approximates a relation for the inelastic collision of classical particles in Newtonian dynamics.

Recoil effects in multiphoton electron-positron pair creation

Triply differential probability rates for electron-positron pair creation in laser-nucleus collisions, calculated within the $S$-matrix approach, are investigated as functions of the nuclear recoil. Pronounced enhancements of differential probability rates of multiphoton pair production are found for a nonzero momentum transfer from the colliding nucleus. The corresponding rates show a very dramatic dependence on the polarization of the laser field impinging on the nucleus; only for a linearly polarized light are the multiphoton rates for electron-positron pair production considerably large. We focus therefore on this case. Our numerical results for different geometries of the reaction particles demonstrate that, for the linearly polarized laser field of an infinite extent (which is a good approximation for femtosecond laser pulses), the pair creation is far more efficient if the nucleus is detected in the direction of the laser-field propagation. The corresponding angular distributions of the created particles show that the high-energy pairs are predominantly produced in the plane spanned by the polarization vector and the laser-field propagation direction, while the low-energy pairs are rather spread around the latter of the two directions. The enhancement of differential probability rates at each energy sector, defined by the four-momentum conservation relation, is observed with varying the energy of the produced particles. The total probability rates of pair production are also evaluated and compared with the corresponding results for the case when one disregards the recoil effect. A tremendous enhancement of the total probability rates of the electron-positron pair creation is observed if one takes into account the nuclear recoil.

Mass, momentum and kinetic energy of a relativistic particle

A rigorous definition of mass in special relativity, proposed in a recent paper, is recalled and employed to obtain simple and rigorous deductions of the expressions of momentum and kinetic energy for a relativistic particle. The whole logical framework appears as the natural extension of the classical one. Only the first, second and third laws of non-relativistic classical dynamics are postulated, in an axiomatic form which does not employ the concept of force. The axiomatic statements of the second and third laws of relativistic dynamics, which yield the relativistic definitions of mass and four-momentum and the conservation of four-momentum for an isolated pair of relativistic particles with a small relative velocity, are proved as simple consequences of the classical ones and of the Lorentz transformation of coordinates. Then, relativistic four-force and three-force are defined, and the expression of relativistic kinetic energy is deduced. Finally, a simple proof of the Lorentz invariance of the conservation of the sum of four-momenta for any set of particles, with arbitrary relative velocities, is presented.

NEUTRINO OSCILLATIONS AND ENERGY–MOMENTUM CONSERVATION

A description of neutrino oscillation phenomena is presented which is based on relativistic quantum mechanics with four-momentum conservation. This is different from both conventional approaches which arbitrarily use either equal energies or equal momenta for the different neutrino mass eigenstates. Both entangled state and source dependence aspects are also included. The time dependence of the wave function is found to be crucial to recovering the conventional result to second order in the neutrino masses. An ambiguity appears at fourth order which generally leads to source dependence, but the standard formula can be promoted to this order by a plausible convention.

Next-to-leading order hard scattering using fully unintegrated parton distribution functions

We calculate the next-to-leading order fully unintegrated hard scattering coefficient for unpolarized gluon-induced deep inelastic scattering using the logical framework of parton correlation functions developed in previous work. In our approach, exact four-momentum conservation is maintained throughout the calculation. Hence, all nonperturbative functions, like parton distribution functions, depend on all components of parton four-momentum. In contrast to the usual collinear factorization approach where the hard scattering coefficient involves generalized functions (such as Dirac {delta} functions), the fully unintegrated hard scattering coefficient is an ordinary function. Gluon-induced deep inelastic scattering provides a simple illustration of the application of the fully unintegrated factorization formalism with a nontrivial hard scattering coefficient, applied to a phenomenologically interesting case. Furthermore, the gluon-induced process allows for a parametrization of the fully unintegrated gluon distribution function.

Axiomatizing Relativistic Dynamics without Conservation Postulates

A part of relativistic dynamics (or mechanics) is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein's famous $E=mc^2$. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.

Towards quantum noncommutativeκ-deformed field theory

We introduce a new {kappa}-star product describing the multiplication of quantized {kappa}-deformed free fields. The {kappa} deformation of local free quantum fields originates from two sources: noncommutativity of space-time and the {kappa} deformation of field oscillators algebra; we relate these two deformations. We demonstrate that for a suitable choice of {kappa}-deformed field oscillators algebra, the {kappa}-deformed version of the microcausality condition is satisfied, and it leads to the deformation of the Pauli-Jordan commutation function defined by the {kappa}-deformed mass shell. We show by constructing the {kappa}-deformed Fock space that the use of the {kappa}-deformed oscillator algebra permits one to preserve the bosonic statistics of n-particle states. The proposed star product is extended to the product of n fields, which for n=4 defines the interaction vertex in perturbative description of the noncommutative quantum {lambda}{phi}{sup 4} field theory. It appears that the classical four-momentum conservation law is satisfied at the interaction vertices.

Theoretical analysis for the apparent discrepancy between [formula omitted] and pp data in charged particle forward–backward multiplicity correlations

The strength of charged particle forward–backward multiplicity correlation in p¯+p and p+p collisions at s=200GeV is studied by PYTHIA 6.4 and compared to the UA5 and STAR data correspondingly. It is turned out that a factor of 3–4 apparent discrepancy between UA5 and STAR data can be attributed to the differences in detector acceptances and observing bin interval in both experiments. A mixed event method is introduced and used to calculate the statistical correlation strength and the dynamical correlation strengths stemming from the charge conservation, four-momentum conservation, and decay, respectively. It seems that the statistical correlation is much larger than dynamical one and the charge conservation, four-momentum conservation and decay may account for most part of the dynamical correlation. In addition, we have also calculated the correlation strength by fitting the charged particle multiplicity distribution from PYTHIA to the Negative Binomial Distribution and found that the result agrees well with the correlation strength calculated by mixed events.

Gaining analytic control of parton showers

Parton showers are widely used to generate fully exclusive final states needed to compare theoretical models to experimental observations. While, in general, parton showers give a good description of the experimental data, the precise functional form of the probability distribution underlying the event generation is generally not known. The reason is that realistic parton showers are required to conserve four-momentum at each vertex. In this paper we investigate in detail how four-momentum conservation is enforced in a standard parton shower and why this destroys the analytic control of the probability distribution. We show how to modify a parton shower algorithm such that it conserves four-momentum at each vertex, but for which the full analytic form of the probability distribution is known. We then comment how this analytic control can be used to match matrix element calculations with parton showers, and to estimate effects of power corrections and other uncertainties in parton showers.

Of Mass and Reference Frames

Wilczek replies: Ian Durham is correct in observing that four-momentum is conserved under very general circumstances. It is not the case, however, that this entails conservation of mass. Although the mass of an isolated particle can be determined from its four-momentum, that connection does not apply to multiparticle states. And since particles do interact, and even decay, a result that is only valid for single-particle states does not allow one to draw general conclusions. That is why the many specific manifestations of nonconservation of mass I discussed in my column can be—and are—perfectly consistent with special relativity in general, and with the conservation of four-momentum in particular.© 2005 American Institute of Physics.

Gravitational slingshot

The slingshot effect is an intriguing phenomenon that has been used effectively by NASA to send spacecraft to outer edges of the solar system. This phenomenon can be satisfactorily explained by Newtonian physics. However, if it is presented as a problem involving four-momentum conservation, the methods of relativistic kinematics easily lead to the conditions necessary for an accelerating as well as a retarding scenario. This problem provides an example that showcases the frequent utility of relativistic methods to analyze problems of Newtonian mechanics.

Radiative corrections in symmetrized classical electrodynamics

The physics of radiation reaction for a point charge is discussed within the context of classical electrodynamics. The fundamental equations of classical electrodynamics are first symmetrized to include magnetic charges: a double four-potential formalism is introduced, in terms of which the field tensor and its dual are employed to symmetrize Maxwell's equations and the Lorentz force equation in covariant form. Within this framework, the symmetrized Dirac-Lorentz equation is derived, including radiation reaction (self-force) for a particle possessing both electric and magnetic charge. The connection with electromagnetic duality is outlined, and an in-depth discussion of nonlocal four-momentum conservation for the wave-particle system is given.