A model of statistical quark-gluon plasma formation is considered.We look the dilepton production at critical temperature $T_{c}\sim170 Mev $ and completely free out temperature $T=150 MeV$ with the initial temperature as $T_{0}=570,400 (250) MeV$. Now we consider that quark mass is depending on the coupling value through parameterisation factor of the fireball formation and temperature. The rate of production is shown for invariant mass $M$ at the particular value of $ E=2.0,3.0 GeV$.It shows the significant production of leptons in this process for small value of invariant mass. However, the quark-hadron phase transition is a very weakly changed in the entropy of the system during this process of hadronisation.
We calculate the production of large mass dileptons from the passage of jets passing through the quark-gluon plasma. Using the relativistic kinetic theory, we rigorously derive the production rate for the jet-dilepton conversion in the hot medium. The jet-dilepton conversion is compared with the thermal dilepton emission and the Drell-Yan process. The contribution of the jet-dilepton conversion is not prominent for all values of the invariant mass $M$, and the Drell-Yan process is found to dominate over the thermal dilepton emission and the jet-dilepton conversion for $M>$2.5 GeV at RHIC. The …
We calculate the order α_s finite temperature correction to dilepton production in quark-gluon plasma arising from the two-loop photon self-energy diagrams for high invariant mass M >> T.
A model of cut-off momentum distribution functions in a Quark Gluon Plasma with finite baryon chemical potential is discussed. This produces a quark gluon plasma signature in Ultra Relativistic Nuclear Collisions with a specific structure of the dilepton spectrum in the transverse momentum region of $(1-4)~GeV$ and the dilepton production rate is found to be a strong decreasing function of the chemical potential.
This talk reviews the status of QCD calculations of photon and dilepton production rates in a Quark-Gluon plasma. Theses rates are known to order ${\cal O}(α_s)$. Their calculations involve various resummations to account for well identified physical effects that are briefly described. Lattice calculations of the spectral functions give also access to the dilepton rates. Comparison with perturbative results points to inconsistencies in both approaches when the dilepton energy becomes small.
We calculate the dilepton production rate from a thermalized quark-gluon plasma in heavy-ion collisions at RHIC energies. Higher-order QCD corrections are included by using an analytical correction factor $K^{(i)}$, which takes into account the $q\bar q$ initial-state interactions. We show that the analytic correction factor gives very good agreement with experimental Drell-Yan data and leads to large enhancement of the thermal dilepton emission rates. We compare the thermal dilepton yields with the expected production from open-charm decays and Drell-Yan background and assess the prospects of…
The production of the intermediate mass dileptons originating from the annihilation and Compton scattering of the jets and high energy photons (resolved photons) passing through the quark-gluon plasma is calculated. The contribution of the dilepton yield due to the jet-plasma and high energy photon-plasma interaction is pronounced compared to the thermal and Drell-Yan dilepton spectrum at intermediate mass. The ordinary spectrum of thermal and Drell-Yan processes is enhanced by the jet and photon-plasma mechanism. The numerical results match to the PHENIX data accurately in the intermediate ma…
The chemical equilibration of a highly unsaturated quark-gluon plasma has been studied at finite baryon density. It is found that in the presence of small amount of baryon density, the chemical equilibration for gluon becomes slower and the temperature decreases less steeply as compared to the baryon free plasma. As a result, the space time integrated yield of dilepton is enhanced if the initial temperature of the plasma is held fixed. Even at a fixed initial energy density, the suppression of the dilepton yields at higher baryo-chemical potential is compensated, to a large extent, by the slow…
We calculate the rapidity dependence of leading-order medium dilepton yields resulting from a quark-gluon plasma which has a local time-dependent anisotropy in momentum space. We present a phenomenological model which includes temporal evolution of the plasma anisotropy parameter, xi, and the hard momentum scale, p_hard. Our model interpolates between 1+1 dimensional collisionally-broadened expansion at early times and 1+1 dimensional ideal hydrodynamic expansion at late times. Using our model, we find that at LHC energies, forward high-energy medium dilepton production would be suppressed by …
We re-examine soft dilepton emission from a weakly coupled Quark-Gluon Plasma. We show that Braaten, Pisarski, and Yuan's result that the dilepton rate rises as E^-4 (and the spectral weight scales as 1/E) at small energy E<<gT is correct, but that the coefficient they found for this behavior is not correct, because their analysis was incomplete. At still smaller scales, the behavior moderates to ~1/E^2 for E <= g^4 T, consistent with a finite electrical conductivity. We evaluate the spectral weight in the E~g^4 T region by kinetic theory techniques and show that it satisfies a sum rule, which…
Using a Hagedorn resonance gas picture and quark-hadron duality we estimate the dilepton emission rate in the vicinity of the QCD deconfinement phase transition. The result is then used to calculate a dilepton spectrum in ultrarelativistic heavy ion collisions. We show that multibody contributions taken into account in the Hagedorn resonance gas approach provide an enhancement of the production rate of massive dileptons as compared to the previously considered sources.
A fireball of QGP is evoluted at temperature dependent chemical potential by a statistical model in the pionic medium. We study the dilepton emission rate at temperature dependent chemical potential (TDCP) from such a fireball of QGP. In this model, we take the dynamical quark mass as a finite value dependence on temparature and parametrization factor of the QGP evolution. The temperature and factor in quark mass enhance in the growth of the droplets as well as in the dilepton emission rates. The emission rate from the plasma shows dilepton spectrum in the intermediate mass region (IMR) of (1.…
We discuss the possibility of same-sign dileptons as a signature for Majorana neutrinos. The production mechanism is given by a single heavy neutrino production and decay proton-proton -> l^{\pm} N X -> l^{\pm} l^{\pm} X'. Cross section and distributions are presented for the LHC energies.
We discuss the double-spin asymmetry \aqt in the transversely polarized Drell-Yan process at small transverse-momentum Q_T of the produced dilepton. The soft gluon corrections relevant for small Q_T are resummed to all orders in α_s, up to the next-to-leading logarithmic accuracy. We show that the soft gluon corrections largely cancel in the spin asymmetry, but the significant corrections still remain.The asymmetries \aqt are calculated for pp collision at RHIC and J-PARC, and for p\bar{p} collision at GSI. A novel asymptotic formula for \aqt at small Q_T is presented, which provides a new app…
The pion electromagnetic form factor at finite temperature is studied using an effective chiral lagrangian that includes explicitly vector mesons. We find that in the time-like region around the rho meson resonance it decreases with increasing temperature and leads to a suppression of dilepton production from pion-pion annihilation in a hot hadronic matter. Effects on dilepton production in high energy heavy ion collisions and its relevance to the phase transition in a hot hadronic matter are discussed.
A calculation of the thermal quark propagator is presented taking the gluon condensate above the critical temperature into account. The quark dispersion relation and the dilepton production following from this propagator are derived.
The dilepton production rate from the quark-gluon plasma is calculated from the imaginary part of the photon self energy using a quark propagator that contains the gluon condensate. The low mass dilepton rate obtained in this way exhibits interesting structures (peaks and gaps), which might be observable at RHIC and LHC.
We calculate leading-order dilepton yields from a quark-gluon plasma which has a time-dependent anisotropy in momentum space. Such anisotropies can arise during the earliest stages of quark-gluon plasma evolution due to the rapid longitudinal expansion of the created matter. A phenomenological model for the proper time dependence of the parton hard momentum scale, p_hard, and the plasma anisotropy parameter, xi, is proposed. The model describes the transition of the plasma from a 0+1 dimensional collisionally-broadened expansion at early times to a 0+1 dimensional ideal hydrodynamic expansion …
We discuss the emission of high momentum lepton pairs in central Au+Au collisions at RHIC ($\sqrt{s_{NN}}=$200 GeV) and Pb+Pb collisions at the LHC ($\sqrt{s_{NN}}=$5500 GeV). Yields of dileptons produced through interactions of jets with thermal partons have been calculated, with next-to-leading order corrections through hard thermal loops (HTL) resummation. They are compared to thermal dilepton emission and the Drell-Yan process. A complete leading order treatment of jet energy loss has been included. Jet-plasma interactions are found to dominate over thermal dilepton emission for all values…
Recently there was a significant change of views on physical properties and underlying dynamics of Quark-Gluon Plasma at $T=170-350 MeV$, produced in heavy ion collisions at RHIC. Instead of weakly coupled gas of quasiparticles, it is rather a liquid-like matter with multiple bound states. In this paper we discuss how one can test these ideas experimentally, using the ``penetrating probes'' and looking for certain peaks at some invariant masses. In dileptons the most promising are modified $ρ,ω$, with $M(T\approx T_c)\sim .5 GeV$ and also near zero binding at $M(T\approx (1.5-2)T_c)=1.5-2 GeV$…
We compute dilepton production from the deconfined phase of the quark-gluon plasma using leading-order (3+1)-dimensional anisotropic hydrodynamics. The anisotropic hydrodynamics equa- tions employed describe the full spatiotemporal evolution of the transverse temperature, spheroidal momentum-space anisotropy parameter, and the associated three-dimensional collective flow of the matter. The momentum-space anisotropy is also taken into account in the computation of the dilepton production rate, allowing for a self-consistent description of dilepton production from the quark-gluon plasma. For our…
Assuming that rho-mesons exist in a quark-gluon plasma close to the QCD phase transition, we calculate the dilepton production rate from quark-antiquark annihilation via a rho-meson state using Vector Meson Dominance. The result is compared to the rates from direct quark-antiquark annihilation and from pion annihilation. Furthermore we discuss the suppression of low mass dileptons if the quarks assume an effective mass in the quark-gluon plasma.
Assuming that gluon condensates are important even above the deconfining phase transition, we develop a model for the dilepton yield from a quark gluon plasma. Using a simple fire ball description of a heavy ion collision, and various estimates of the strengths of the gluon condensates, we compare our predicted dilepton yields with those observed in the CERES and HELIOS experiments at CERN. The simple model gives an adequate description of the data, and in particular it explains the observed considerable enhancement of the yield in the low mass region.
We consider the $a_{1} (1260)$ contribution to photon and dilepton productions from hot hadronic matter. The effective lagrangian with $π, ρ$ and $a_{1}$ discussed by Ko and Rudaz is employed, which successfully reproduces the $Γ(a_{1} \rightarrow πγ) = 670$ keV. Our results for the photon production rate are comparable to those obtained by Xiong {\it et al.}, while the dilepton production rates come out smaller than those obtained by Gale {\it et al.} by an order of magnitude.
We calculate the yield and elliptic flow of mid-rapidity dileptons emitted from the quark-gluon plasma generated in Pb-Pb collisions at LHC. We use relativistic anisotropic hydrodynamics for the 3+1 dimensional evolution of the quark-gluon plasma and convolve this with the momentum-anisotropic local rest frame production rate for dileptons. The effects of momentum anisotropy of the quark distribution functions, viscosity to entropy density ratio, centrality of the collisions, and initial momentum anisotropy on the results are investigated and discussed.
We consider the dilepton emission from a dense and hot hadron gas near a critical point. The hadron gas is described as a resonance one. The dilepton spectrum generated by vector resonances is shown to be equivalent to the one generated by quarks.
We investigate the effects of a baryon-antibaryon asymmetry on the spectrum of dileptons radiating from a quark gluon plasma. We demonstrate the existence of a new set of processes in this regime. The dilepton production rate from the corresponding diagrams is shown to be as important as that obtained from the usual quark-antiquark annihilation.
Dilepton production from the quark-gluon plasma (QGP) phase of ultra-relativistic heavy-ion collisions is computed using the leading-order (3+1)-dimensional anisotropic hydrodynamics. It is shown that high-energy dilepton spectrum is sensitive to the initial local-rest-frame momentum-space anisotropy of the QGP. Our findings suggest that it may be possible to constrain the early-time momentum-space anisotropy in relativistic heavy-ion collisions using high-energy dilepton yields.
It has been predicted that dilepton production may be used as a quark-gluon plasma probe. We calculate the rapidity distributions of thermal dileptons produced by an evolving quark-gluon plasma assuming a longitudinal scaling expansion with initial conditions locally determined from the hadronic rapidity density. These distributions are compared with Drell-Yan production and semileptonic charm decays at invariant mass $M = 2$, 4, and 6 GeV.
We calculate the yield of lepton pair production from jet-plasma interaction where the plasma is anisotropic in momentum space. We compare both the $M$ and $p_T$ distributions from such process with the Drell-Yan contribution. It is observed that the invariant mass distribution of lepton pair from such process dominate over the Drell-Yan up to $3$ GeV at RHIC and up to $10$ GeV at LHC. Moreover, it is found that the contribution from anistropic quark gluon plasma (AQGP) increases marginally compared to the isotropic QGP. In case of $p_T$-distribution we observe an increase by a factor of $3-4$…
We study dilepton production from a thermally equilibrating quark-gluon plasma expected to be formed in ultra relativistic heavy-ion collisions. The pre-equilibrium dynamics of quark-gluon plasma is studied within the color flux-tube model by solving non-abelian relativistic transport equations. This dilepton rate crucially depends on the collision time of the plasma. The results are compared with the Drell-Yan productions. We suggest that a measurement of this rate at RHIC and LHC will determine the initial field energy density.
A fireball model with time evolution based on transport calculations is used to examine the dilepton emission rate of an ultra-relativistic heavy-ion collision. We assume a transition from hadronic matter to a quark-gluon plasma at a critical temperature T_C between 130-170 MeV. We include thermal corrections to the hadronic spectra below T_C and use perturbation theory above T_C. The sensitivity of the spectra with respect to the freeze-out temperature, the initial fireball temperature and the critical temperature is investigated.
The observational consequences of certain broken symmetries in a thermalised quark gluon plasma are elucidated. The signature under study is the spectrum of dileptons radiating from the plasma, through gluon fusion. Being a pure medium effect, this channel is non-vanishing only in plasmas with explicitly broken charge conjugation invariance. The emission rates are also sensitive to rotational invariance through the constraints imposed by Yang's theorem. This theorem is interpreted in the medium via the destructive interference between various multiple scattering diagrams obtained in the specta…
Under very general assumptions we show that the quark dispersion relation in the quark-gluon plasma is given by two collective branches, of which one has a minimum at a non-vanishing momentum. This general feature of the quark dispersion relation leads to structures (van Hove singularities, gaps) in the low mass dilepton production rate, which might provide a unique signature for the quark-gluon plasma formation in relativistic heavy ion collisions.
We review various production mechanisms of photons and small mass dileptons at large transverse momentum in heavy ion collisions at the LHC. Their relevance as a signal for quark-gluon plasma formation is discussed.
The resummation technique of Braaten and Pisarski is used for a complete calculation of the $α^2 α_s$-contribution to the production rate of dileptons with energy E >> T and invariant mass M <~ T from a quark-gluon plasma. In particular the cancellation of the infrared singularity by medium effects is discussed in detail.
We calculate the medium dilepton yield from a quark-gluon plasma which has a time-dependent local momentum-space anisotropy. A phenomenological model for the hard momentum scale, p_hard(tau), and plasma anisotropy parameter, xi(tau), is constructed which interpolates between free streaming behavior at early times (tau << tau_iso) and ideal hydrodynamic behavior at late times (tau >> tau_iso). We show that high-energy dilepton production is sensitive to the assumed plasma isotropization time, tau_iso, and can therefore be used to experimentally determine the time of onset for hydrodynamic expan…
In a recent Letter, Geiger presents calculations of the dilepton emission from the early stage of ultrarelativistic heavy ion collisions using the parton cascade model (PCM). He shows that the $M_\perp$ scaling is not observed. In this Comment, we point out that this is largely due to a defect in the PCM.
We calculate leading-order medium photon and dilepton yields from a quark-gluon plasma using (3+1)-dimensional anisotropic hydrodynamics. In the case of dileptons, at leading-order it is sufficient to take into account non-equilibrium corrections to the rate through the use of anisotropic distribution functions. In the case of photons, non-equilibrium corrections to the rate are taken into account using a self-consistent modification of the distribution functions and the corresponding anisotropic hard-loop quark self-energies. We present predictions for the high-energy photon and dilepton spec…
We compute the production rate of lepton pair by static photons at finite temperature at two-loop order. We treat the infrared region of the gluon phase space carefully by using a hard thermal loop gluon propagator. The result is free of infrared and collinear divergences and exhibits an enhancement which produces a result of order $\sim e^2 g^3$ instead of $\sim e^2 g^4$ as would be expected from ordinary perturbation theory.
We calculate the production of large mass dileptons from the jet-dilepton conversion in the expanding quark-gluon plasma at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies. The jet-dilepton conversion exceeds the thermal dilepton production and Drell-Yan process in the large mass region of 3.9 GeV$<M<$5.8 GeV and 6.3 GeV$<M<$8.7 GeV in central Pb+Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV and 5.5 TeV, respectively. We present the numerical solution of ideal fluid hydrodynamics. We find that the transverse flow leads to a rapid cooling of the fire ball. The suppre…
It is investigated under which conditions an adiabatic adaption of the dynamic and spectral information of vector mesons to the changing medium in heavy ion collisions, as assumed in schematic model calculations and microscopic transport simulations, is a valid assumption. Therefore time dependent medium modifications of low mass vector mesons are studied within a non-equilibrium quantum field theoretical description. Timescales for the adaption of the spectral properties are given and non-equilibrium dilepton yields are calculated, leading to the result that memory effects are not negligible …
The production of large $p_{T}$ dileptons and photons originating from photoproduction processes in p-p collisions at Large Hadron Collider energies is calculated. The comparisons between the exact treatment results and the ones of the equivalent photon approximation approach are expressed as the $Q^{2}$ (the virtuality of photon) and $p_{T}$ distributions. The method developed by Martin and Ryskin is used for avoiding double counting when the coherent and incoherent contributions are considered simultaneously. The numerical results indicate that, the equivalent photon approximation is only ef…
In this paper we present a comprehensive formalism for dilepton production from the collision of two polarized spin-$\tfrac{1}{2}$ hadrons by identifying the general angular distribution of the cross section in combination with a complete set of structure functions. The various structure functions are computed in the parton model approximation where we mainly consider the case when the transverse momentum of the dilepton pair is much smaller than its invariant mass. In this kinematical region dilepton production can be described in terms of transverse momentum dependent parton distributions.
Using the Landau-level representation for the imaginary part of the photon polarization tensor, we derive an explicit expression for the dilepton production rate from a hot quark-gluon plasma in a quantizing background magnetic field. We study in detail the dependence of the production rate on the dilepton invariant mass and the transverse momentum at mid-rapidity. We also investigate the angular dependence and ellipticity of dilepton emission. By comparing the result with the zero-field Born approximation, we find that the magnetic field leads to a strong enhancement of the dilepton rate at s…
Theories of quantum gravity suggest the existence of a minimal length scale. It is interesting to speculate what the consequences of the existence of such a length scale would be for models of large extra dimensions, in particular, the ADD model. When ADD model is conflated with the minimal length scale scenario, processes involving virtual exhange of gravitons cease to be ulraviolent divergent. We study the production of dileptons at hadron colliders as an example of a process mediated by virtual gravitons. We find that the bounds we derive on the effective string scale are significantly diff…
We calculate the double transverse-spin asymmetries, $\aqt$, in transversely polarized Drell-Yan process at the transverse-momentum $Q_T$ of the produced lepton pair. We perform all-order resummation of the logarithmically enhanced contributions in the relevant Drell-Yan cross sections at small $Q_T$, which are due to multiple soft gluon emission in QCD, up to the next-to-leading logarithmic accuracy. The asymmetries $\aqt$ to be observed in polarized experiments at RHIC and J-PARC are studied numerically as a function of $Q_T$. We show that the effects of the soft gluon resummation to the pol…
The dilepton production by the constituents of the strongly interacting quark-gluon-plasma (sQGP) is addressed. In order to make quantitative predictions at realistically low plasma temperatures (O(T_c)), experimentally relevant low dilepton mass (O(1 GeV)) and strong coupling (alphaS=0.5-1), we take into account not only the higher order pQCD reaction mechanisms, but also the non-perturbative spectral functions (off-shellness) and self-energies of the quarks, anti-quarks and gluons thus going beyond the leading twist. For this purpose, our calculations utilize parametrizations of the non-pert…
Dilepton production from hot hadronic matter is studied in an effective chiral Lagrangian with pions, $ρ$-mesons, and $a_1$ mesons, We find that the production rates from reactions that involve axial-vector mesons dominate over contributions from all other reactions when the dilepton invariant mass is above 1.5 GeV.
We derive a set of new formulas for various distributions in dilepton production via virtual photon bremsstrahlung from pseudoscalar mesons and unpolarized spin-one-half fermions. These formulas correspond to the leading and sub-leading terms in the Low-Burnett-Kroll expansion for real photon bremsstrahlung. The relation of our leading-term formulas to previous works is also shown. Existing formulas are examined in the light of Lorentz covariance and gauge invariance. Numerical comparison is made in a simple example, where an "exact" formula and real photon data exist. The results reveal large…
In this paper, we propose an observable counting a weighted difference between right-handed and left-handed lepton pairs, which is coined dilepton helical rate. The weight is the momentum difference of the lepton pairs projected onto an auxiliary vector. We derive the helical rate in a quark-gluon plasma with a vorticity in the limit when the quark and lepton masses are ignored. We find the helical rate is maximized when the auxiliary vector is parallel to the vorticity, in which case it has a nearly spherical oblate ellipsoidal distribution. We propose that it can be used as a vortical-meter …
We investigate in detail the effects of the R-parity lepton number violation in the minimal supersymmetric standard model (MSSM) on the parent process $pp \to e^+ e^- + X$ at the CERN Large Hadron Collider (LHC). The numerical comparisons between the contributions of the R-parity violating effects to the parent process via the Drell-Yan subprocess and the gluon-gluon fusion are made. We find that the R-violating effects on $e^+ e^-$ pair production at the LHC could be significant. The results show that the cross section of the $ e^+ e^-$ pair productions via gluon-gluon collision at the LHC ca…
We discuss dilepton production from a quark-gluon plasma which has a time-dependent anisotropy in momentum space. A phenomenological model for the hard momentum scale, p_hard(tau) and the plasma anisotropy parameter, xi(tau), is presented. The model interpolates between early-time 1+1 free streaming behavior (tau << tau_iso) and late-time ideal 1+1 hydrodynamical behavior (tau >> tau_iso). Using this model, we find that the dilepton rate arising from electromagnetic annihilations of quarks in the kinematic range 3 <p_T< 8 GeV is sensitive to the assumed isotropization time of the system, tau_i…
We discuss the impact on the study of gluon transverse momentum dependent distributions (TMDs) of the associated production of a lepton pair and a $Υ$ (or a $J/ψ$) in unpolarised proton-proton collisions, $pp\to {\cal Q} \, \ell \bar{\ell} X$, at LHC energies, where one can assume that such final states are dominantly induced by gluon fusion. If the transverse momentum of the quarkonium-dilepton system - namely, the transverse momentum imbalance of the quarkonium state and the lepton pair - is small, the corresponding cross sections can be calculated within the framework of TMD factorisation. …
The D0 collaboration has recently introduced new variables, a_T and phi* to more accurately probe the low Q_T domain of Z boson production at hadron colliders than had been previously possible through a direct study of the Q_T distribution. The comparison of such accurate data to precise theoretical predictions from QCD perturbation theory will yield important information on the ability of resummed QCD predictions as well as parton shower models to describe the low Q_T domain and should enable more stringent constraints on non-perturbative effects. In the present paper we provide analytical pr…
We consider the thermal production of dileptons and photons at temperatures above the critical temperature in QCD. We use a model where color excitations are suppressed by a small value of the Polyakov loop, the semi Quark-Gluon Plasma (QGP). Comparing the semi-QGP to the perturbative QGP, we find a mild enhancement of thermal dileptons. In contrast, to leading logarithmic order in weak coupling there are far fewer hard photons from the semi-QGP than the usual QGP. To illustrate the possible effects on photon and dileptons production in heavy ion collisions, we integrate the rate with a realis…
We show that new physics can show up in dileptonic events through its radiative contributions to the dilepton invariant mass, leading to unique "monocline" features in $m_{\ell\ell}$, as well as the angular distribution of the leptons. We focus in particular on the case of dark matter with scalar messengers coupling it to the quarks and leptons. Consistent thermal models require the dark matter to have masses of 100's GeV and have $\gsim 1$ couplings to the Standard Model (SM), implying that radiative corrections to the SM Drell-Yan rate can be sizeable. We consider the case of Majorana, Dirac…
It is expected that hadron collider measurements of the Higgs boson mass using the decay h -> W^+W^-, followed by the leptonic decay of each W-boson, will be performed by fitting the shape of a distribution which is sensitive to the Higgs mass. We demonstrate that the variable most commonly used to measure the Higgs mass in this channel is not optimal as it contains an unnecessary and even counter-productive approximation. We remove that approximation, without introducing any cost in complexity, and demonstrate that the new variable is a clear improvement over the old: its performance is never…
