Hadron Production In Collisions Research Articles

Spin asymmetries for C -even quarkonium production as a probe of gluon distributions

Within the framework of transverse momentum dependent factorization in combination with nonrelativistic quantum chromodynamics (QCD), we study charmonium and bottomonium production in hadronic collisions. We focus on quarkonium states with even charge conjugation, for which the color-singlet production mechanism is expected to be also dominant in the small transverse momentum region, qT2≪4Mc,b2. It is shown that the distributions of linearly polarized gluons inside unpolarized, longitudinally, and transversely polarized protons contribute to the cross sections for scalar and pseudoscalar quarkonia in a very distinctive, parity-dependent way, whereas their effects on higher angular momentum states are strongly suppressed. We derive analytical expressions for single and double spin asymmetries, which would allow for the direct extraction of the gluon transverse momentum dependent distributions, mirroring the phenomenological studies of the Drell-Yan processes aimed at the extraction of their quark counterparts. By adopting Gaussian models for the gluon transverse momentum dependent distributions, which fulfill without saturating everywhere their positivity bounds, we provide numerical predictions for the transverse single-spin asymmetries. These observables could be measured at LHCSpin, the fixed target experiment planned at the LHC. Published by the American Physical Society 2024

The U-Matrix geometrical model for multi-particle production in high-energy hadronic collisions

Inspired by the picture portraying the KNO scaling violation as an extension of the geometrical scaling violation, the current study proposes a phenomenological model for multi-particle production in hadron collisions based on the geometrical approach and using the U-Matrix unitarization scheme of the scattering amplitude. The model has been fine-tuned and all parameters have been derived from optimal fits to various hadronic multiplicity distributions data in p + pp¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p\\left(\\overline{p}\\right) $$\\end{document} collisions across a broad range of energies. The results have revealed that our model furnishes a reasonable description of diverse multiplicity distributions at various energies. Besides, they have demonstrated a pronounced violation of the geometrical scaling, which eventually resulted in a significant violation of the KNO scaling. The study has also analyzed the higher-order moments of the multiplicity distribution. We have observed an unexpected overestimation of the fluctuations and correlations between final state particles with increasing energy, particularly above LHC energy. It is claimed that this overestimation is due to statistical fluctuations embedded in the U-matrix scheme. The findings of this study have shed light on the key role of the U-matrix scheme in the impact of collision geometry on multi-particle production processes at high energy.

Measurement of the fraction of jet longitudinal momentum carried by Λc+ baryons in pp collisions

Recent measurements of charm-baryon production in hadronic collisions have questioned the universality of charm-quark fragmentation across different collision systems. In this work the fragmentation of charm quarks into charm baryons is probed, by presenting the first measurement of the longitudinal jet momentum fraction carried by Λc+ baryons, z∥ch, in hadronic collisions. The results are obtained in proton-proton (pp) collisions at s=13 TeV at the LHC, with Λc+ baryons and charged (track-based) jets reconstructed in the transverse momentum intervals of 3≤pTΛc+<15 GeV/c and 7≤pTjet ch<15 GeV/c, respectively. The z∥ch distribution is compared to a measurement of D0-tagged charged jets in pp collisions as well as to 8 simulations. The data hints that the fragmentation of charm quarks into charm baryons is softer with respect to charm mesons, in the measured kinematic interval, as predicted by hadronization models which include color correlations beyond leading-color in the string formation. © 2024 CERN, for the ALICE Collaboration 2024 CERN

Towards higher-order calculations of quarkonia production with kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_T$$\\end{document}-factorization: P-wave charmonia

Inclusive P-wave charmonia production in hadronic collisions at high energies is discussed in the framework of non-relativistic QCD and kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_T$$\\end{document}-factorization formalism. We present two consistent approaches to merge the usual leading order kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_T$$\\end{document}-factorization calculations with tree-level next-to-leading order off-shell amplitudes. Using these prescriptions, we extracted long-distance matrix elements for χc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _c$$\\end{document} mesons from a combined fit to available Tevatron and LHC data. In contrast to previous (leading order) calculations, our fits do not rule out the equalness of the color singlet wave functions of χc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _{c1}$$\\end{document} and χc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _{c2}$$\\end{document} states. The extracted values of long-distance matrix elements are employed to analyse the χc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _c$$\\end{document} polarization data reported recently by the CMS Collaboration. Our predictions are in a reasonably good agreement with the Tevatron and LHC measurements within the theoretical and experimental uncertainties.

Linear power corrections to top quark pair production in hadron collisions

We compute, in the framework of renormalon calculus, the OΛQCD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\Lambda}_{\ extrm{QCD}}\\right) $$\\end{document} corrections to the production of tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} $$\\end{document} pairs in hadron collisions under the assumption that qq¯→tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q\\overline{q}\ o t\\overline{t} $$\\end{document} is the dominant partonic channel. This assumption is not applicable to top quark pair production at the LHC but it is valid for the Tevatron where collisions of protons and anti-protons were studied. We show that the linear power correction to the total tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} $$\\end{document} production cross section vanishes provided one uses a short-distance scheme for the top quark mass. We also derive relatively simple formulas for the power corrections to top quark kinematic distributions. Although small numerically, these power corrections exhibit interesting dependencies on top quark kinematics.

Full top-quark mass dependence in diphoton production at NNLO in QCD

In this paper we consider the diphoton production in hadronic collisions at the next-to-next-to-leading order (NNLO) in perturbative QCD, taking into account for the first time the full top quark mass dependence up to two loops (full NNLO). We show selected numerical distributions, highlighting the kinematic regions where the massive corrections are more significant. We make use of the recently computed two-loop massive amplitudes for diphoton production in the quark annihilation channel. The remaining massive contributions at NNLO are also considered, and we comment on the weight of the different types of contributions to the full and complete result.

Production of {textrm{K}}_{textrm{S}}^0 , Λ ( overline{Lambda} ), Ξ±, and Ω± in jets and in the underlying event in pp and p–Pb collisions

The production of strange hadrons ( {textrm{K}}_{textrm{S}}^0 , Λ, Ξ±, and Ω±), baryon-to-meson ratios (Λ/ {textrm{K}}_{textrm{S}}^0 , Ξ/ {textrm{K}}_{textrm{S}}^0 , and Ω/ {textrm{K}}_{textrm{S}}^0 ), and baryon-to-baryon ratios (Ξ/Λ, Ω/Λ, and Ω/Ξ) associated with jets and the underlying event were measured as a function of transverse momentum (pT) in pp collisions at sqrt{s} = 13 TeV and p Pb collisions at sqrt{s_{textrm{NN}}} = 5.02 TeV with the ALICE detector at the LHC. The inclusive production of the same particle species and the corresponding ratios are also reported. The production of multi-strange hadrons, Ξ± and Ω±, and their associated particle ratios in jets and in the underlying event are measured for the first time. In both pp and p–Pb collisions, the baryon-to-meson and baryon-to-baryon yield ratios measured in jets differ from the inclusive particle production for low and intermediate hadron pT (0.6–6 GeV/c). Ratios measured in the underlying event are in turn similar to those measured for inclusive particle production. In pp collisions, the particle production in jets is compared with Pythia 8 predictions with three colour-reconnection implementation modes. None of them fully reproduces the data in the measured hadron pT region. The maximum deviation is observed for Ξ± and Ω± which reaches a factor of about six. The event multiplicity dependence is further investigated in p−Pb collisions. In contrast to what is observed in the underlying event, there is no significant event-multiplicity dependence for particle production in jets. The presented measurements provide novel constraints on hadronisation and its Monte Carlo description. In particular, they demonstrate that the fragmentation of jets alone is insufficient to describe the strange and multi-strange particle production in hadronic collisions at LHC energies.

New determination of the production cross section for γ rays in the Galaxy

The flux of $\gamma$ rays is measured with unprecedented accuracy by the $\textit{Fermi}$ Large Area Telescope from 100 MeV to almost 1 TeV. In the future, the Cherenkov Telescope Array will have the capability to measure photons up to 100 TeV. To accurately interpret this data, precise predictions of the production processes, specifically the cross section for the production of photons from the interaction of cosmic-ray protons and helium with atoms of the ISM, are necessary. In this study, we determine new analytical functions describing the Lorentz-invariant cross section for $\gamma$-ray production in hadronic collisions. We utilize the limited total cross section data for $\pi^0$ production channels and supplement this information by drawing on our previous analyses of charged pion production to infer missing details. In this context, we highlight the need for new data on $\pi^0$ production. Our predictions include the cross sections for all production channels that contribute down to the 0.5% level of the final cross section, namely $\eta$, $K^+$, $K^-$, $K^0_S$, and $K^0_L$ mesons as well as $\Lambda$, $\Sigma$, and $\Xi$ baryons. We determine the total differential cross section $d\sigma(p+p\rightarrow \gamma+X)/dE_{\gamma}$ from 10 MeV to 100 TeV with an uncertainty of 10% below 10 GeV of $\gamma$-ray energies, increasing to 20% at the TeV energies. We provide numerical tables and a script for the community to access our energy-differential cross sections, which are provided for incident proton (nuclei) energies from 0.1 to $10^7$ GeV (GeV/n).

Systematic studies on the nuclear parton distribution with photon and hadron productions in nuclear collisions at the LHC* *Supported by the National Key Research and Development Program of China (2018YFE0104700, 2018YFE0104800), the National Natural Science Foundation of China (12061141008, 12175085), and the Fundamental Research Funds for the Central Universities ( CCNU220N003)

A systematic study on the impact of widely-used nuclear-modified parton distribution function (nPDF) parameterizations on the production of direct photons and charged hadrons is performed by employing a next-to-leading order Monte Carlo event generator JETPHOX in hadronic collisions at LHC energies. The nuclear modification factors of photon and charged hadron productions are studied in three types of collision systems, i.e., small ( and ), medium , and large , in a wide rapidity acceptance. The results illustrate that the direct photon production process is a sensitive probe to decipher the difference of the nPDF parameterization implementations, which provides new insights for the experimental measurements to refine the nuclear modifications of the parton distributions. To provide a benchmark for searching for quark gluon plasma in the upcoming small system measurements at LHC energies, we carry out detailed studies on the cold nuclear matter effect in collisions. Our outcomes show that the cold nuclear matter effects are negligible for the charged hadron production in collisions, which can be used as a baseline to subtract the initial state contribution.

Quarkonium production and elliptic flow in small systems measured with ALICE

Abstract. Quarkonium production in hadronic collisions provides a unique testing ground for understanding quantum chromodynamics (QCD) since it involves both perturbative and non-perturbative regimes of this theory. A variety of experimental observables, such as the production cross-section, polarization, particle correlations, serve as insights into the phenomenology of the quarkonium production and help constrain the theoretical models. Measurements of quarkonia as a function of multiplicity probe multiple parton interactions (MPI), i.e., several parton-parton interactions occurring in a single hadron-hadron collision. Finally, measurements of the azimuthal correlation structure of emitted particles in high multiplicity pp collisions can probe the existence of collective behaviour in small systems. The ALICE detector can reconstruct inclusive quarkonia over a broad kinematical range, spanning from mid-rapidity up to forward rapidity, and down to zero transverse momentum. In addition, at midrapidity, the non-prompt charmonium contribution can be separated from the prompt contribution. In these proceedings, we present new results on the inclusive, prompt and non-prompt J/ψ production cross sections measured by ALICE in pp collisions at different collision energies. The self-normalized ψ(2S)-to-J/ψ yield ratio measurement is investigated as a function of the charged particle multiplicity in pp collisions at √S = 13. TeV. Finally, the first measurement of the J/ψ elliptic flow (v2) in high multiplicity pp collisions at √S = 13. TeV is presented. Results will be compared to theoretical models.

Rivet and the analysis preservation in heavy-ion collisions experiments

The comparison of experimental data and theoretical predictions is important for our understanding of the mechanisms for interactions and particle production in hadron collisions, both at the Large Hadron Collider and at the Relativistic Heavy-Ion Collider experiments. Several tools were ideated to help with that. Rivet (Robust Independent Validation of Experiment and Theory) is a framework that facilitates the comparison between measurements from high-energy physics experiments and Monte Carlo event generators able to produce outputs using the HepMC package. Rivet contains a repository with analysis algorithms developed by experiments, providing analysis documentation and preservation. The recent developments for the implementation of centrality and multiplicity classes in Rivet are presented in this contribution.

W±Z production at NNLO QCD and NLO EW matched to parton showers with MiNNLOPS

We consider W±Z production in hadronic collisions and present high-precision predictions in QCD and electroweak (EW) perturbation theory matched to parton showers. To this end, we match next-to-next-to-leading order QCD corrections to parton showers using the MiNNLOPS method and consistently combine them with next-to-leading order EW corrections matched to parton showers. This is the first time such accuracy in the event generation is achieved for any collider process, and we study in detail the impact of different choices in the combination of QCD and EW corrections as well as QCD and QED showers. Spin correlations, interferences and off-shell effects are retained by considering the full leptonic processes ppto {ell}^{+}{ell}^{-}{ell}^{prime pm }{nu}_{ell}^{prime } with ℓ′ ≠ ℓ and ℓ′ = ℓ without approximations, and the matching to QED radiation is performed preserving the resonance structure of the process. We find that NNLO QCD predictions including QCD and QED shower effects provide a very good approximation in the bulk-region of the phase space, while EW effects become increasingly important in the high-energy tails of kinematic distributions. Our default predictions are in excellent agreement with recent ATLAS data.

PanScales showers for hadron collisions: all-order validation

We carry out extensive tests of the next-to-leading logarithmic (NLL) accuracy of the PanScales parton showers, as introduced recently for colour-singlet production in hadron collisions. The tests include comparisons to (semi-)analytic NLL calculations of a wide range of hadron-collider observables: the colour-singlet boson transverse momentum distribution; global and non-global hadronic energy flow variables related to jet vetoes and analogues of jettiness distributions; (sub)jet multiplicities; and observables sensitive to the DGLAP evolution of the incoming momentum fractions. In the tests, we also include an implementation of a standard transverse-momentum ordered dipole shower, to establish the size of missing NLL effects in such showers, which, depending on the observable, can reach 100%. This paper, together with [1], constitutes the first step towards process-independent NLL-accurate parton showers for hadronic collisions.

Multiplicity dependence of charged-particle jet production in pp collisions at mathbf {sqrt{s}}=mathbf {13~TeV}

The multiplicity dependence of jet production in pp collisions at the centre-of-mass energy of sqrt{s} = 13 {mathrm {TeV}} is studied for the first time. Jets are reconstructed from charged particles using the anti-k_mathrm {T} algorithm with resolution parameters R varying from 0.2 to 0.7. The jets are measured in the pseudorapidity range |eta _{mathrm{jet}}|< 0.9-R and in the transverse momentum range 5<p_mathrm {T,jet}^{mathrm{ch}}<140 {mathrm {GeV}}/c. The multiplicity intervals are categorised by the ALICE forward detector V0. The p_{mathrm T} differential cross section of charged-particle jets are compared to leading order (LO) and next-to-leading order (NLO) perturbative quantum chromodynamics (pQCD) calculations. It is found that the data are better described by the NLO calculation, although the NLO prediction overestimates the jet cross section below 20 {mathrm {GeV}}/c. The cross section ratios for different R are also measured and compared to model calculations. These measurements provide insights into the angular dependence of jet fragmentation. The jet yield increases with increasing self-normalised charged-particle multiplicity. This increase shows only a weak dependence on jet transverse momentum and resolution parameter at the highest multiplicity. While such behaviour is qualitatively described by the present version of PYTHIA, quantitative description may require implementing new mechanisms for multi-particle production in hadronic collisions.

Measurement of Prompt D0 , Λc+ , and Σc0,++(2455) Production in Proton-Proton Collisions at s=13 TeV

The p_{T}-differential production cross sections of prompt D^{0}, Λ_{c}^{+}, and Σ_{c}^{0,++}(2455) charmed hadrons are measured at midrapidity (|y|<0.5) in pp collisions at sqrt[s]=13 TeV. This is the first measurement of Σ_{c}^{0,++} production in hadronic collisions. Assuming the same production yield for the three Σ_{c}^{0,+,++} isospin states, the baryon-to-meson cross section ratios Σ_{c}^{0,+,++}/D^{0} and Λ_{c}^{+}/D^{0} are calculated in the transverse momentum (p_{T}) intervals 2<p_{T}<12 and 1<p_{T}<24 GeV/c. Values significantly larger than in e^{+}e^{-} collisions are observed, indicating for the first time that baryon enhancement in hadronic collisions also extends to the Σ_{c}. The feed-down contribution to Λ_{c}^{+} production from Σ_{c}^{0,+,++} is also reported and is found to be larger than in e^{+}e^{-} collisions. The data are compared with predictions from event generators and other phenomenological models, providing a sensitive test of the different charm-hadronization mechanisms implemented in the models.

ZZ production at nNNLO+PS with MiNNLOPS

We consider ZZ production in hadronic collisions and present state-of-the-art predictions in QCD perturbation theory matched to parton showers. Next-to-next-to-leading order corrections to the quark-initiated channel are combined with parton showers using the MiNNLOPS method, while next-to-leading order corrections to the loop-induced gluon fusion channel are matched using the Powheg method. Their combination, dubbed nNNLO+PS, constitutes the best theoretical description of ZZ events to date. Spin correlations, interferences and off-shell effects are included by calculating the full process pp → ℓ+ℓ−ℓ(′)+ℓ(′)−. We show the crucial impact of higher-order corrections for both quark- and gluon-initiated processes as well as the relevance of the parton shower in certain kinematical regimes. Our predictions are in very good agreement with recent LHC data.

Λ_{c}^{+} Production and Baryon-to-Meson Ratios in pp and p-Pb Collisions at sqrt[s_{NN}]=5.02 TeV at the LHC.

The prompt production of the charm baryon Λ_{c}^{+} and the Λ_{c}^{+}/D^{0} production ratios were measured at midrapidity with the ALICE detector in pp and p-Pb collisions at sqrt[s_{NN}]=5.02 TeV. These new measurements show a clear decrease of the Λ_{c}^{+}/D^{0} ratio with increasing transverse momentum (p_{T}) in both collision systems in the range 2<p_{T}<12 GeV/c, exhibiting similarities with the light-flavor baryon-to-meson ratios p/π and Λ/K_{S}^{0}. At low p_{T}, predictions that include additional color-reconnection mechanisms beyond the leading-color approximation, assume the existence of additional higher-mass charm-baryon states, or include hadronization via coalescence can describe the data, while predictions driven by charm-quark fragmentation processes measured in e^{+}e^{-} and e^{-}p collisions significantly underestimate the data. The results presented in this Letter provide significant evidence that the established assumption of universality (colliding-system independence) of parton-to-hadron fragmentation is not sufficient to describe charm-baryon production in hadronic collisions at LHC energies.

W+W\u2212 production at NNLO+PS with MINNLOPS

We consider W+W− production in hadronic collisions and present the computation of next-to-next-to-leading order accurate predictions consistently matched to parton showers (NNLO+PS) using the MiNNLOPS method. Spin correlations, interferences and off-shell effects are included by calculating the full process pp → e+νeμ− overline{nu} μ. This is the first NNLO+PS calculation for W+W− production that does not require an a-posteriori multi-differential reweighting. The evaluation time of the two-loop contribution has been reduced by more than one order of magnitude through a four-dimensional cubic spline interpolation. We find good agreement with the inclusive and fiducial cross sections measured by ATLAS and CMS. Both NNLO corrections and matching to parton showers are important for an accurate simulation of the W+W− signal, and their matching provides the best description of fully exclusive W+W− events to date.

Advancing M\u0131NNLOPS to diboson processes: Z\u03b3 production at NNLO+PS

We consider Zγ production in hadronic collisions and present the first computation of next-to-next-to-leading order accurate predictions consistently matched to parton showers (NNLO+PS). Spin correlations, interferences and off-shell effects are included by calculating the full process pp → ℓ+ℓ−γ. We extend the recently developed MiNNLOPS method to genuine 2 → 2 hard scattering processes at the LHC, which paves the way for NNLO+PS simulations of all diboson processes. This is the first 2 → 2 NNLO+PS calculation that does not require an a-posteriori multi-differential reweighting. We find that both NNLO corrections and matching to parton showers are crucial for an accurate simulation of the Zγ process. Our predictions are in very good agreement with recent ATLAS data.

Prompt photon production in high-energy pA collisions at forward rapidity

Prompt photon production in hadronic collisions at the RHIC and the LHC energies is investigated within the QCD color dipole approach. Predictions for the nuclear modification factor in $pA$ collisions are evaluated based on parton saturation framework and the results are compared to the experimental measurements as a function of the photon transverse momentum at different rapidity bins. The reliability of the models is performed with the data from PHENIX, ATLAS, and ALICE Collaborations. Moreover, we show that the observed $x_T$-scaling of prompt photon production in $pp$ and $pA$ collisions can positively be addressed in the QCD color dipole formalism.