We show that respecting the underlying Lorentz structure in the parton model has very strong consequences. Failure to insist on the correct Lorentz covariance is responsible for the existence of contradictory results in the literature for the polarized structure function g_2(x), whereas with the correct imposition we are able to derive the Wandzura-Wilczek relation for g_2(x) and the target-mass corrections for polarized deep inelastic scattering without recourse to the operator product expansion. We comment briefly on the problem of threshold behaviour in the presence of target-mass corrections. Careful attention to the Lorentz structure has also profound implications for the structure of the transverse momentum dependent parton densities often used in parton model treatments of hadron production, allowing the k_T dependence to be derived explicitly. It also leads to stronger positivity and Soffer-type bounds than usually utilized for the collinear densities.
Spin observables allow a deeper understanding of the nature of the underlying dynamics and positivity reduces substantially their allowed domains. We will present some new positivity constraints for spin observables and their implications for parton distributions. We will also make some comparisons with recent data.
We briefly recall the main physical features of the parton distributions in the quantum statistical picture of the nucleon. Some predictions from a next-to-leading order QCD analysis are successfully compared to recent unpolarized and polarized experimental results. We will discuss the extension to the transverse momentum dependence of the parton distributions and its relevance for semiinclusive deep inelastic scattering. Finally, we will present some new positivity constraints for spin observables and their implications for parton distributions.
Double hard scattering in proton-proton collisions is described in terms of double parton distributions. We derive bounds on these distributions that follow from their interpretation as probability densities, taking into account all possible spin correlations between two partons in an unpolarized proton. These bounds constrain the size of the polarized distributions and can for instance be used to set upper limits on the effects of spin correlations in double hard scattering. We investigate the stability of the bounds under leading-order DGLAP evolution to higher scales.
We investigate the positivity of double parton distributions with a non-trivial dependence on the parton colour. It turns out that positivity is not preserved by leading-order evolution from lower to higher scales, in contrast to the case in which parton colour is summed over. We also study the positivity properties of the distributions at small distance between the two partons, where they can be computed in terms of perturbative splitting kernels and ordinary parton densities.
The violation of the so-called Lorentz invariance relations between parton distribution functions is considered in a model independent way. It is shown that these relations are not violated in a generalized Wandzura-Wilczek approximation, indicating that numerically their violation may be small.
A parton-model description of high-energy hadronic interactions in the presence of Lorentz violation is presented. This approach is used to study lepton-hadron and hadron-hadron interactions at large momentum transfer. Cross sections for deep inelastic scattering and the Drell-Yan process are calculated at first order for minimal and nonminimal Lorentz violation. Estimated bounds are placed using existing LHC and future US-based Electron-Ion Collider data.
As with parton distributions, flexible phenomenological parameterizations of generalized parton distributions (GPDs) are essential for their extraction from data. The large number of constraints imposed on GPDs make simple Lorentz covariant models viable; but, such models are often incomplete in that they employ the impulse approximation. Using the GPD of the pion as a test case, we show that the impulse approximation can lead to violation of the positivity bound required of GPDs. We focus on a particular model of the pion bound-state vertex that was recently proposed and demonstrate that sati…
Tif Lab, Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano（米兰大学物理系TIF实验室和INFN米兰分部）；University of Jyvaskyla, Department of Physics（于韦斯屈莱大学物理系）
Role of Positivity Constraints in Determining Polarized Parton Densities
我们已经重新分析了世界上的包容性极化DIS的数据，在NLO和LO QCD，包括非常精确的JLab Hall A中子数据，并研究了正性约束的作用，并证明了它们的重要性，在确定奇怪的胶子密度。我们已经表明，较高的扭曲修正是必不可少的，在分析目前的数据上的结构函数g1。实现了一致的QCD分析，并给出了在MSbare和JET方案的极化部分子密度的结果。
A framework is presented for the factorization of high-energy hadronic processes in the presence of Lorentz and CPT violation. The comprehensive effective field theory describing Lorentz and CPT violation, the Standard-Model Extension, is used to demonstrate factorization of the hadronic tensor at leading order in electroweak interactions for deep inelastic scattering and for the Drell-Yan process. Effects controlled by both minimal and nonminimal coefficients for Lorentz violation are explored, and the equivalent parton-model description is derived. The methodology is illustrated by determini…
In the paper we focus on the study of the functional complexity of the Lorentz parametrizing functions in connection with the time-reversal transformations. We argue that the interactions encoded in the corresponding correlators of non-local quark(-gluon) operators generate additional sources of functional complexity for the parametrizing functions which are not discussed in the literature. We also revisit the Lorentz parametrization of different correlators given by the hadron matrix elements of the non-local operators. The evidences for the new parametrizing function existence have been pres…
There have been recent debates about whether $\overline{\text{MS}}$ parton densities exactly obey positivity bounds (including the Soffer bound), and whether the bounds should be applied as a constraint on global fits to parton densities and on nonperturbative calculations. A recent paper (JHEP ${\bf \text{11}}$ (2020) 129) appears to provide a proof of positivity in contradiction with earlier work by other authors. We examine their derivation and find that its primary failure is in the apparently uncontroversial statement that bare pdfs are always positive. We show that under the conditions u…
Photon structure functions in supersymmetric QCD are investigated in terms of the parton model where squark contributions are evaluated. We calculate the eight virtual photon structure functions by taking the discontinuity of the squark massive one-loop diagrams of the photon-photon forward amplitude. The model-independent positivity constraints derived from the Cauchy-Schwarz inequalities are satisfied by the squark parton model calculation and actually the two equality relations hold for the squark contribution. We also show that our polarized photon structure function $g_1^γ$ for the real p…
We derive new Lorentz Invariance and Equation of Motion Relations between twist-three Generalized Parton Distributions (GPDs) and moments in the parton transverse momentum, $k_T$, of twist-two Generalized Transverse Momentum-Dependent Distributions (GTMDs), as a function of the parton longitudinal momentum fraction $x$. Although GTMDs in principle define the observables for partonic orbital motion, experiments that can unambiguously detect them appear remote at present. The relations presented here provide a solution to this impasse in that, e.g., the orbital angular momentum density is connec…
We present a study of the parton distributions in transverse position or impact parameter space using a recent parametrization of u and d quark generalized parton distributions (GPDs) $H(x,t)$ and $E(x,t)$ at zero skewness. We make a comparative study between different parametrizations and discuss the region of validity of the positivity condition.
The proton's transverse polarization structure is examined in terms of the Lorentz-covariant Pauli-Lubanski vector in QCD. We find that there are contributions from leading, subleading, and next-to-subleading partonic contributions in the light-front system of coordinates. The subleading and next-to-subleading contributions are related to the leading one through Lorentz symmetry. And the leading contribution obeys a simple partonic angular momentum sum rule that gives a clear physical interpretation to a relation known previously.
We study the space-time structure of polynomiality and positivity---the most important properties which are inherent to the generalized parton distributions (GPDs). In this connection, we re-examine the issue of the time- and normal- ordering in the operator definition of GPDs. We demonstrate that the contribution of the anti-commutator matrix element in the collinear kinematics, which was previously argued to vanish, has to be added in order to satisfy the polynomiality condition. Furthermore, we schematically show that a new contribution due to the anti-commutator modifies likewise the so-ca…
We investigate the one-gluon-exchange ($αα_s$) corrections to the real photon structure functions $W_{TT} $, $W_{LT}$, $W_{TT}^{a} $ and $W_{TT}^τ$ in the massive parton model. We employ a technique based on the Cutkosky rules and the reduction of Feynman integrals to master integrals. We show that a positivity constraint, which is derived from the Cauchy-Schwarz inequality, is satisfied among the unpolarized and polarized structure functions $W_{TT}$, $W_{TT}^a$ and $W_{TT}^τ$ calculated up to the next-to-leading order in QCD.
Generalizing the Soffer Bound: Positivity Constraints on Parton Distributions of Spin-3/2 Particles
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所南方核科学理论中心）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Quark Matter Research Center, Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所重离子研究中心）；KEK Theory Center, Institute of Particle and Nuclear Studies, KEK（日本高能加速器研究机构粒子与核研究部KEK理论中心）；Heavy Ion Science and Technology Key Laboratory, Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所重离子科学与技术重点实验室）；School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）
As in the case of the hydrogen atom, bound-state wave functions are needed to generate hadronic spectra. For this purpose, in 1971, Feynman and his students wrote down a Lorentz-invariant harmonic oscillator equation. This differential equation has one set of solutions satisfying the Lorentz-covariant boundary condition. This covariant set generates Lorentz-invariant mass spectra with their degeneracies. Furthermore, the Lorentz-covariant wave functions allow us to calculate the valence parton distribution by Lorentz-boosting the quark-model wave function from the hadronic rest frame. However,…
We derive the first positivity bounds for low-energy Effective Field Theories (EFTs) that are not invariant under Lorentz boosts. "Positivity bounds" are the low-energy manifestation of certain fundamental properties in the UV -- to date they have been used to constrain a wide variety of EFTs, however since all of the existing bounds require Lorentz invariance they are not directly applicable when this symmetry is broken, such as for most cosmological and condensed matter systems. From the UV axioms of unitarity, causality and locality, we derive an infinite family of bounds which (derivatives…
Structure functions of polarized spin-1 hadrons will be measured at various accelerator facilities in the near future. Recently, transverse-momentum-dependent and collinear parton distribution functions were theoretically proposed at twist 3 and twist 4 in addition to the twist-2 ones, so that full investigations became possible for structure functions of spin-1 hadrons in the same level with those of the spin-1/2 nucleons. Furthermore, twist-3 tensor-polarized multiparton distribution functions were also recently found for spin-1 hadrons. In this work, we show relations among the collinear pa…
The nonperturbative parton distributions, obtained from the Lorentz contracted wave functions, are analyzed in the formalism of many-particle Fock components and their properties are compared to the standard perturbative distributions. We show that the collinear and IR divergencies specific for perturbative evolution treatment are absent in the nonperturbative version, however for large momenta< $ \vep^2_i \gg σ$ (string tension), the bremsstrahlung kinematics is restored. A preliminary discussion of possible nonperturbative effects in DIS and high energy scattering is given, including in part…
