In this Ph.D. thesis, the primary goal is to present a recent investigation of the finite density thermodynamics of hot and dense quark-gluon plasma. As we are interested in a temperature regime, in which naive perturbation theory is known to lose its predictive power, we clearly need to use a refined approach. To this end, we adopt a resummed perturbation theory point of view and employ two different frameworks. We first use hard-thermal-loop perturbation theory (HLTpt) at leading order to obtain the pressure for nonvanishing quark chemical potentials, and next, inspired by dimensional reduction, resum the known four-loop weak coupling expansion for the quantity.
We present and analyze our findings for various cumulants of conserved charges. This provides us with information, through correlations and fluctuations, on the degrees of freedom effectively present in the quark-gluon plasma right above the deconfinement transition. Moreover, we compare our results with state-of-the-art lattice Monte Carlo simulations as well as with a recent three-loop mass truncated HTLpt calculation. We obtain very good agreement between the two different perturbative schemes, as well as between them and lattice data, down to surprisingly low temperatures right above the phase transition. We also quantitatively test the convergence of an approximation, which is used in higher order loop calculations in HTLpt. This method based on expansions in mass parameters, is unavoidable beyond leading order, thus motivating our investigation. We find the ensuing convergence to be very fast, validating its use in higher order computations.
We perform a detailed analysis of the predictions of resummed perturbation theory for the pressure and the second-, fourth-, and sixth-order diagonal quark number susceptibilities in a hot and dense quark-gluon plasma. First, we present an exact one-loop calculation of the equation of state within hard-thermal-loop perturbation theory (HTLpt) and compare it to a previous one-loop HTLpt calculation that employed an expansion in the ratios of thermal masses and the temperature. We find that this expansion converges reasonably fast. We then perform a resummation of the existing four-loop weak cou…
We calculate a resummed equation of state with lattice QCD simulations at imaginary chemical potentials. This work presents a generalization of the scheme introduced in 2102.06660 to the case of non-zero $μ_S$, focusing on the line of strangeness neutrality. We present results up to $μ_B/T \leq 3.5$ on the strangeness neutral line $\left\langle S \right\rangle = 0$ in the temperature range $130 \rm{MeV} \leq T \leq 280 \rm{MeV}$. We also extrapolate the finite baryon density equation of state to small non-zero values of the strangeness-to-baryon ratio $R=\left\langle S \right\rangle / \left\la…
We provide a brief review of the current status of resummed predictions for QCD final state observables such as event shapes and jet rates in a variety of different hard processes. Particular emphasis is given to more recent developments such as the study of non global observables, development of generalized resummation formulae and resummations for new types of event shape variables.
We review the proposal to resum the physics of hard thermal loops in the thermodynamics of the quark-gluon plasma through nonperturbative expressions for entropy and density obtained from a Phi-derivable two-loop approximation. A comparison with the recently solved large-N_f limit of hot QCD is performed, and some updates, in particular on quark number susceptibilities, are made.
We discuss the Hard Dense Loop resummation at finite quark mass and evaluate the equation of state (EoS) of cold and dense QCD matter in $β$ equilibrium. The resummation in the quark sector has an effect of lowering the baryon number density and the EoS turns out to have much smaller uncertainty than the perturbative QCD estimate. Our numerical results favor smooth matching between the EoS from the resummed QCD calculation at high density and the extrapolated EoS from the nuclear matter density region. We also point out that the speed of sound in our EoS slightly exceeds the conformal limit.
By making use of the finite-temperature real-time static potential that was introduced and computed to leading non-trivial order in Hard Thermal Loop resummed perturbation theory in recent work, and solving numerically a Schrödinger-type equation, we estimate the quarkonium (in practice, bottomonium) contribution to the spectral function of the electromagnetic current in hot QCD. The spectral function shows a single resonance peak which becomes wider and then disappears as the temperature is increased beyond 450 MeV or so. This behaviour can be compared with recently attempted lattice reconstr…
We present the elements of three applications of resummation methods in non-Abelian gauge theories: (1), QED X QCD exponentiation and shower/ME matching for LHC physics; (2), IR improvement of DGLAP theory; (3), resummed quantum gravity and the final state of Hawking radiation. In all cases, the extension of the YFS approach, originally introduced for Abelian gauge theory, to non-Abelian gauge theories, QCD and quantum general relativity, leads to new results and solutions which we briefly summarize.
We show that role of nonperturbative input in the $b$-space QCD resummation formalism for heavy boson transverse momentum ($Q_T$) distribution strongly depends on collision energy $\sqrt{S}$. At collider energies, the larger $\sqrt{S}$ is, the weaker role nonperturbative input plays, and better predictive power the $b$-space resummation formalism has.
Based on the hard-thermal-loop resummed improved ladder Dyson-Schwinger quation for the fermion mass function, we propose a procedure how we can get the gauge invariant solution in the sense it satisfies the Ward-Takahashi identity. Results of the numerical analysis are shown and properties of the ``gauge-invariant'' solutions are discussed.
We calculate and resum a perturbative expansion of nuclear enhanced power corrections to the structure functions measured in deeply inelastic scattering of leptons on a nuclear target. Our results for the Bjorken $x$-, $Q^2$- and $A$-dependence of nuclear shadowing in $F_2^A(x,Q^2)$ and the nuclear modifications to $F_L^A(x,Q^2)$, obtained in terms of the QCD factorization approach, are consistent with the existing data. We demonstrate that the low-$Q^2$ behavior of these data and the measured large longitudinal structure function point to a critical role for the power corrections when compare…
After almost two decades of investigation, jet quenching has become a fundamental tool to study the properties of the QCD matter produced in high-energy nuclear collisions. Despite the large progress in both experimental and theoretical tools, several unknowns remain to be solved. In particular the systematics of the QCD matter parameters extracted at different energies and centralities is still puzzling. On the other hand, a lot of attention has been put in the last few years to a correct resummation of the multiple scatterings that the particles of the jet suffer when traversing the QCD matt…
We analyze the role of the nonperturbative input in the Collins, Soper, and Sterman (CSS)'s $b$-space QCD resummation formalism for Drell-Yan transverse momentum ($Q_T$) distributions, and investigate the predictive power of the CSS formalism. We find that the predictive power of the CSS formalism has a strong dependence on the collision energy $\sqrt{S}$ in addition to its well-known $Q^2$ dependence, and the $\sqrt{S}$ dependence improves the predictive power at collider energies. We show that a reliable extrapolation from perturbatively resummed $b$-space distributions to the nonperturbativ…
The pinch technique (PT) is applied to obtain the gauge-independent resummed gluon self-energy in a hot Yang-Mills gas. Calculation is performed at one- loop level in linear gauges which preserve rotational invariance and the res ummed propagators and vertices are used. The effective gluon self-energy, w hich is obtained as the sum of the resummed gluon self-energy and the resumm ed pinch contributions, is not only gauge-independent but also satisfies the transversality relation. Using this gauge-independent effective gluon self- energy, we calculate the damping rate for transverse gluons in t…
We perform the OPE analysis of the contribution of colour-singlet compound states of reggeized gluons to a generic hard process in QCD and calculate the spectrum of the corresponding higher twist anomalous dimensions in multi-colour limit. These states govern high energy asymptotics of the structure functions and their energies define the intercept of the Regge singularities both in the Pomeron and the Odderon sectors. We argue that due to nontrivial analytical properties of the energy spectrum, the twist expansion does not hold for the gluonic states with the minimal energy generating the lea…
Using high-statistics datasets generated in (2+1)-flavor QCD calculations at finite temperature we construct estimators for the radius of convergence from an eighth order series expansion of the pressure as well as the number density. We show that the estimator for pressure and number density will be identical in the asymptotic limit. In the vicinity of the pseudo-critical temperature, $T_{pc}\simeq 156.5$~MeV, we find the estimator of the radius of convergence to be $μ_B/T \gtrsim\ 3$ for strangeness-neutral matter. We also present results for the pole structure of the Padé approximants for t…
Using the framework of soft-collinear effective theory (SCET), we factorize and calculate $e^+e^-$ angularity distributions, including perturbative resummation and the incorporation of a universal model for the nonperturbative soft function. Angularities $τ_a$ are a class of event shapes varying in their sensitivity to the substructure of jets in the final state, controlled by a continuous parameter $a<2$. We calculate the jet and soft functions in factorized angularity distributions for all $a<1$ to first order in the strong coupling $α_s$ and resum large logarithms to next-to-leading logarit…
We calculate the production rate of soft real photons from a hot quark -- gluon plasma using Braaten -- Pisarski's perturbative resummation method. To leading order in the QCD coupling constant $g$ we find a logarithmically divergent result for photon energies of order $gT$, where $T$ is the plasma temperature. This divergent behaviour is due to unscreened mass singularities in the effective hard thermal loop vertices in the case of a massless external photon.
The HotQCD Collaboration performed Taylor expansion calculations in 2017 for the pressure, energy density, and entropy density at non-zero chemical potentials up to the $6^{th}$ order. Since then, they have significantly improved the statistics for lattices with temporal extents of $N_τ=8$ and $12$, and have also included results for $N_τ=16$ that were not previously available. They have also calculated the $8^{th}$-order expansion coefficients for $N_τ=8$. These calculations showed that the Taylor series expansion for the pressure is accurate up to $μ_B / T \leq 2.5$. In this study, we use th…
The resummed expression for the quark form factor illustrates the fact that dimensional continuation provides a regularization not only for ultraviolet and infrared singularities of fixed order QCD amplitudes, but also for the Landau pole arising in resummed calculations. Explicit renormalization group invariant analytic expressions for the logarithm of the form factor are given up to two-loop order in the QCD beta function.
Chiral phase transition in thermal QCD is studied by using the Dyson-Schwinger (DS) equation in the real time hard thermal loop approximation. Our results on the critical temperature and the critical coupling are significantly different from those in the preceding analyses in the ladder DS equation, showing the importance of properly taking into account the essential thermal effects, namely the Landau damping and the unstable nature of thermal quasiparticles.
Some of the most arduous and error-prone aspects of precision resummed calculations are related to the partonic hard process, having nothing to do with the resummation. In particular, interfacing to parton-distribution functions, combining various channels, and performing the phase space integration can be limiting factors in completing calculations. Conveniently, however, most of these tasks are already automated in many Monte Carlo programs, such as MadGraph, Alpgen or Sherpa. In this paper, we show how such programs can be used to produce distributions of partonic kinematics with associated…
We present threshold enhanced QCD corrections to inclusive processes such as Deep inelastic scattering, Drell-Yan process and Higgs productions through gluon fusion and bottom quark annihilation processes using the resummed cross sections. The resummed cross sections are derived using renormalisation group invariance and mass factorisation theorem that these hard scattering cross sections satisfy and Sudakov resummation of QCD amplitudes. We show how these higher order threshold QCD corrections improve the theoretical predictions for the Higgs production through gluon fusion at hadron collider…
We present predictions for the gluon-fusion Higgs $p_T$ spectrum at third resummed and fixed order (N$^3$LL$'+$N$^3$LO) including fiducial cuts as required by experimental measurements at the Large Hadron Collider. Integrating the spectrum, we predict for the first time the total fiducial cross section to third order (N$^3$LO) and improved by resummation. The N$^3$LO correction is enhanced by cut-induced logarithmic effects and is not reproduced by the inclusive N$^3$LO correction times a lower-order acceptance. These are the highest-order predictions of their kind achieved so far at a hadron …
We report the results of including resummed splitting functions in the QCD evolution equations at small x, and discuss the predictions that follow for the deep inelastic structure functions. *Contribution at XXX Rencontres de Moriond, Les Arcs, March 1995
We present a leading evaluation of the resummed coefficient function for the shape function. It is also shown that the coefficient function is short-distance-dominated. Our results allow relating the shape function computed on the lattice to the physical QCD distributions.
We analyze the electromagnetic pion and kaon form factor by including radiative and higher-twist effects within the framework of resummed pQCD in the space-like region. We focus on the transition from the perturbative to non-perturbative behavior in the phenomenological intermediate energy regime. Using a modified k_T factorization scheme with transverse degrees of freedom, we evaluate the non-perturbative soft contributions as distinct from the hard contributions, ensuring no double counting via the Ward identity at Q^2=0. The soft contributions are obtained via local quark-hadron duality whi…
The cold and dense QCD equation of state (EoS) at high baryon chemical potential $μ_B$ involves at order $α^2_S$ an all-loop summation of the soft mode $m_E\sim α_S^{1/2} μ_B$ contributions. Recently, the complete soft contributions at order $α^3_S$ were calculated, using the hard thermal loop (HTL) formalism. By identifying {\em massive} renormalization group (RG) properties within HTL, we resum to all orders $α_S^p, p\ge 3$ the leading and next-to-leading logarithmic soft contributions. We obtain compact analytical expressions, that show visible deviations from the state-of-the art results, …
The one-loop QCD effective charge $α_s^{eff}$ for quark-quark scattering is derived by diagrammatic resummation of the one-loop amplitude using an arbitary covariant gauge. Except for the particular choice of gauge parameter $ξ= -3$, $α_s^{eff}$ is found to {\it increase} with increasing physical scale, $Q$, as $\ln Q$ or $\ln^2 Q$. For $ξ= -3$, $α_s^{eff}$ decreases with increasing $Q$ and satisfies a renormalisation group equation. Also, except for the case $ξ= 19/9$, convergence radii of geometric series are found to impose upper limits on $Q$.
We evaluate the second and fourth order quark number susceptibilities in hot QCD using two variations of resummed perturbation theory. On one hand, we carry out a one-loop calculation within hard-thermal-loop perturbation theory, and on the other hand perform a resummation of the four-loop finite density equation of state derived using a dimensionally reduced effective theory. Our results are subsequently compared with recent high precision lattice data, and their agreement thoroughly analyzed.
We present a next-to-leading evaluation of the resummed coefficient function for the shape function. The results confirm our previous leading order analysis, namely that the coefficient function is short-distance-dominated, and allow relating the shape function computed with a non-perturbative technique to the physical QCD distributions.
Scattering amplitudes of partons in QCD contain infrared divergences which can be resummed to all orders in terms of an anomalous dimension. Independently, in the limit of high-energy forward scattering, large logarithms of the energy can be resummed using Balitsky-Fadin-Kuraev-Lipatov theory. We use the latter to analyze the infrared-singular part of amplitudes to all orders in perturbation theory and to next-to-leading-logarithm accuracy in the high-energy limit, resumming the two-Reggeon contribution. Remarkably, we find a closed form for the infrared-singular part, predicting the Regge lim…
We suggest that at any given order of Feynman diagram calculation all renormalization group (RG)-predictable terms should be resummed to all-orders. This ``complete'' RG-improvement (CORGI) serves to separate the perturbation series into infinite subsets of terms which when summed are renormalization scheme (RS)-invariant. Crucially all ultraviolet logarithms involving the dimensionful parameter, Q, on which the observable depends are resummed, thereby building the correct Q-dependence. There are close connections with the effective charge approach of Grunberg.
We study the splitting functions for the evolution of fragmentation distributions and the coefficient functions for single-hadron production in semi-inclusive electron-positron annihilation in massless perturbative QCD for small values of the momentum fraction and scaling variable x, where their fixed-order approximations are completely destabilized by huge double logarithms of the form alpha_s^n 1/x ln^(2n-a) x. Complete analytic all-order expressions in Mellin-N space are presented for the resummation of these terms at the next-to-next-to-leading logarithmic accuracy. The poles for the first…
Estimates of Lee-Yang zeros and a possible critical point on the pion condensate boundary in the QCD isospin phase diagram using an unbiased exponential resummation on the lattice
We study next-to-leading-power (NLP) threshold corrections in colour-singlet production processes, with particular emphasis on Drell-Yan (DY) and single-Higgs production. We assess the quality of the partonic and hadronic threshold expansions for each process up to NNLO. We determine numerically the NLP leading-logarithmic (LL) resummed contribution in addition to the leading-power next-to-next-to-leading logarithmic (LP NNLL) resummed DY and Higgs cross sections, matched to NNLO. We find that the inclusion of NLP logarithms is numerically more relevant than increasing the precision to N$^3$LL…
Taylor expansion in powers of baryon chemical potential ($μ_B$) is an oft-used method in lattice QCD to compute QCD thermodynamics for $μ_B>0$. Based only upon the few known lowest order Taylor coefficients, it is difficult to discern the range of $μ_B$ where such an expansion around $μ_B=0$ can be trusted. We introduce a resummation scheme for the Taylor expansion of the QCD equation of state in $μ_B$ that is based on the $n$-point correlation functions of the conserved current ($D_n$). The method resums the contributions of the first $N$ correlation function $D_1,\dots,D_N$ to the Taylor exp…
We elaborate on the fact that quarkonium in hot QCD should not be thought of as a stationary bound state in a temperature-dependent real potential, but as a short-lived transient, with an exponentially decaying wave function. The reason is the existence of an imaginary part in the pertinent static potential, signalling the ``disappearance'', due to inelastic scatterings with hard particles in the plasma, of the off-shell gluons that bind the quarks together. By solving the corresponding Schrödinger equation, we estimate numerically the near-threshold spectral functions in scalar, pseudoscalar,…
We apply the closed-time path integral formalism to study the medium effects of non-equilibrium gluon matter. We derive the medium modified resummed gluon propagator to the one loop level in non-equilibrium in the covariant gauge. The gluon propagator we derive can be used to remove the infrared divergences in the secondary parton collisions to study thermalization of minijet parton plasma at RHIC and LHC.
In this talk I present the results of two calculations that make use of Non-Relativistic QCD and the newly developed Soft-Collinear Effective Theory. The first process considered is inclusive radiative $Υ$ decay. The second process considered is the leading color-octet contribution to $e^+ e^- \to J/ψ+ X$.
We present first results on the resummation of Next-to-Soft Virtual (NSV) logarithms for the threshold production of pseudoscalar Higgs boson through gluon fusion at the LHC. These results are presented after resumming the NSV logarithms of the kind ${\log}^{i}(1-z)$ to $\overline{\text{NNLL}}$ accuracy and matching them systematically to the fixed order NNLO cross-sections. These results are obtained using collinear factorization, renormalization group invariance and recent developments in the NSV resummation techniques. The phenomenological implications of these NSV resummed results for 13 T…
We show that the resummation of large perturbative corrections in QCD leads to ambiguities in high energy cross sections that are suppressed by powers of large momentum scales. These ambiguities are caused by infrared renormalons, which are a general feature of resummed hard-scattering functions in perturbative QCD, even though these functions are infrared safe order-by-order in perturbation theory. As in the case of the operator product expansion, the contributions of infrared renormalons to coefficient functions may be absorbed into the definition of higher-dimensional operators, which induc…
The gauge dependence of the hot gluon self energy is examined in the context of Pisarski's method for resumming hard thermal loops. Braaten and Pisarski have used the Ward identities satisfied by the hard corrections to the n-point functions to argue the gauge fixing independence of the leading order resummed QCD plasma damping rate in covariant and strict Coulomb gauges. We extend their analysis to include all linear gauges that preserve rotational invariance and display explicitly the conditions required for gauge fixing independence. It is shown that in covariant gauges the resummed damping…
We present a new framework for computing resummed and matched distributions in processes with many hard QCD jets. The intricate color structure of soft gluon emission at large angles renders resummed calculations highly non-trivial in this case. We automate all ingredients necessary for the color evolution of the soft function at next-to-leading-logarithmic accuracy, namely the selection of the color bases and the projections of color operators and Born amplitudes onto those bases. Explicit results for all QCD processes with up to $2\to 5$ partons are given. We also devise a new tree-level mat…
We present an accurate theoretical prediction for the production of Higgs boson through bottom quark annihilation at the LHC up to next-to-next-to-next-to leading order (N$^3$LO) plus next-to-next-to-next-to-leading logarithmic (N$^3$LL) accuracy. We determine the third order perturbative Quantum Chromodynamics (QCD) correction to the process dependent constant in the resummed expression using the three loop bottom quark form factor and third order quark soft distribution function. Thanks to the recent computation of N$^3$LO corrections to this production cross-section from all the partonic ch…
We use AdS/QCD to analyze the quark and gluon scalar and pseudo-scalar condensates around static color sources described by a circular Wilson loop. We also derive the static dipole-dipole interactions between rectangular Wilson loops in AdS/QCD and discuss their relevance for static string interactions in QCD at strong coupling.
We present threshold enhanced QCD corrections to the bottom quark energy distribution in Higgs boson decay and to hadroproduction in $l^+l^-$ annihilation beyond leading order in the strong coupling constant. This is achieved using the resummed decay distribution obtained using renormalisation group invariance and the mass factorisation theorem that they satisfy and Sudakov resummation of soft gluons.
We have computed the hard dilepton production rate from a weakly magnetized deconfined QCD medium within one-loop photon self-energy by considering one hard and one thermomagnetic resummed quark propagator in the loop. In the presence of the magnetic field, the resummed propagator leads to four quasiparticle modes. The production of hard dileptons consists of rates when all four quasiquarks originating from the poles of the propagator individually annihilate with a hard quark coming from a bare propagator in the loop. Besides these, there are also contributions from a mixture of pole and Landa…
NLO and LPM-resummed computations of thermal dilepton production from a hot QCD plasma are reviewed. An interpolation applicable for all positive photon invariant masses is presented, and possibilities for comparisons with lattice and experimental data are pointed out.
We apply QCD resummation techniques to study the transverse momentum distribution of Higgs bosons produced via gluon-gluon fusion at the LHC. In particular we focus on the joint resummation formalism which resums both threshold and transverse momentum corrections simultaneously. A comparison of results obtained in the joint and the standard recoil resummation frameworks is presented.
We construct a set of parton distribution functions (PDFs) in which fixed-order NLO and NNLO calculations are supplemented with soft-gluon (threshold) resummation up to NLL and NNLL accuracy respectively, suitable for use in conjunction with any QCD calculation in which threshold resummation is included at the level of partonic cross sections. These resummed PDF sets, based on the NNPDF3.0 analysis, are extracted from deep-inelastic scattering, Drell-Yan, and top quark pair production data, for which resummed calculations can be consistently used. We find that, close to threshold, the inclusio…
Parton showers in Monte Carlo event generators reflect to a certain accuracy our understanding of QCD radiation at all orders. For observables sensitive to interjet energy flow in well defined regions of phase space, it has been known for some time that relevant all-order dynamics is substantially more complicated than that encoded via angular ordering in parton shower algorithms, even to leading logarithmic accuracy. Here we investigate the extent of the numerical mismatch between leading logarithmic analytical estimates (resummation) and parton showers in an effort to better understand the a…
The small $x$ behavior of the flavor non-singlet $g_{1}$ structure function is analysed numerically by taking into account the all-order resummation of $α_{s} \ln^{2}x $ terms. We include a part of the next-to-leading logarithmic corrections coming from the resummed ``coefficient function'' which are not considered in the literatures to respect the factorization scheme independence. The resummed coefficient function turns out to give unexpectedly large suppression effects over the experimentally accessible range of $x$ and $Q^{2}$. This fact implies that the higher order logarithmic correction…
We use the scalar model with quartic interaction to illustrate how a nonperturbative variational technique combined with renormalization group (RG) properties efficiently resums perturbative expansions in thermal field theories. The resulting convergence and scale dependence of optimized thermodynamical quantities, here illustrated up to two-loop order, are drastically improved as compared to standard perturbative expansions, as well as to other related methods such as the screened perturbation or (resummed) hard-thermal-loop perturbation, that miss RG invariance as we explain. Being very gene…
Burgess and Marini have recently pointed out that the leading contribution to the damping rate of energetic gluons and quarks in the QCD plasma, given by $γ=c g^2\ln(1/g)T$, can be obtained by simple arguments obviating the need of a fully resummed perturbation theory as developed by Braaten and Pisarski. Their calculation confirmed previous results of Braaten and Pisarski, but contradicted those proposed by Lebedev and Smilga. While agreeing with the general considerations made by Burgess and Marini, I correct their actual calculation of the damping rates, which is based on a wrong expression…
