Phase-space Integrals Research Articles

Coherent-state overcompleteness, path integrals, and weak values

In the Hilbert space of a quantum particle the standard coherent-state resolution of unity is written in terms of a phase-space integration of the outer product zz. Because no pair of coherent states is orthogonal, one can represent the closure relation in non-standard ways, in terms of a single phase-space integration of the “unlike” outer product z′z, z′≠z. We show that all known representations of this kind have a common ground and that our reasoning extends to spin coherent states. These unlike identities make it possible to write formal expressions for a phase-space path integral, where the role of the Hamiltonian H is played by a weak energy value Hweak. Therefore, in this context, we can speak of weak values without any mention to measurements. The quantity Hweak appears as the ruler of the phase-space dynamics in the semiclassical limit.

Master integrals for splitting functions from differential equations in QCD

A method for calculating phase-space master integrals for the decay process $1 \to n$ massless partons in QCD using integration-by-parts and differential equations techniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G)HPLs as a series in the dimensional regulator $\epsilon$ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.

Automated event generation for loop-induced processes

We present the first fully automated implementation of cross-section computation and event generation for loop-induced processes. This work is integrated in the MadGraph5_aMC@NLO framework. We describe the optimisations implemented at the level of the matrix element evaluation, phase space integration and event generation allowing for the simulation of large multiplicity loop-induced processes. Along with some selected differential observables, we illustrate our results with a table showing inclusive cross-sections for all loop-induced hadronic scattering processes with up to three final states in the SM as well as for some relevant 2 → 4 processes. Many of these are computed here for the first time.

Soft expansion of double-real-virtual corrections to Higgs production at N3LO

We present methods to compute higher orders in the threshold expansion for the one-loop production of a Higgs boson in association with two partons at hadron colliders. This process contributes to the N$^3$LO Higgs production cross section beyond the soft-virtual approximation. We use reverse unitarity to expand the phase-space integrals in the small kinematic parameters and to reduce the coefficients of the expansion to a small set of master integrals. We describe two methods for the calculation of the master integrals. The first was introduced for the calculation of the soft triple-real radiation relevant to N$^3$LO Higgs production. The second uses a particular factorization of the three body phase-space measure and the knowledge of the scaling properties of the integral itself. Our result is presented as a Laurent expansion in the dimensional regulator, although some of the master integrals are computed to all orders in this parameter.

Removing phase-space restrictions in factorized cross sections

Factorization in gauge theories holds at the amplitude or amplitude-squared level for states of given soft or collinear momenta. When performing phase-space integrals over such states, one would generally like to avoid putting in explicit cuts to separate soft from collinear momenta. Removing these cuts induces an overcounting of the soft-collinear region and adds new infrared-ultraviolet divergences in the collinear region. In this paper, we first present a regulator-independent subtraction algorithm for removing soft-collinear overlap at the amplitude level which may be useful in pertubative QCD. We then discuss how both the soft-collinear and infrared-ultraviolet overlap can be undone for certain observables in a way which respects factorization. Our discussion clarifies some of the subtleties in phase-space subtractions and includes a proof of the infrared finiteness of a suitably subtracted jet function. These results complete the connection between factorized QCD and Soft-Collinear Effective Theory.

On the resolution of the phase space density required to obtain a specified accuracy of the solar wind bulk velocity

AbstractPlasma measurements with significantly improved accuracy and time resolution are needed to advance knowledge and understanding of plasma physical process in the magnetosphere and the solar wind. In this study, it is shown how small unresolved features in particle distribution functions can have significant impacts on the measurement of plasma bulk parameters such as the plasma bulk velocity. Errors are caused by the numerical integration scheme used to compute the phase space integrals and by errors or experimental uncertainties in measurements of the phase space distribution. Here the errors in the bulk velocity caused by the numerical integration procedure are investigated for both Gaussian (Maxwellian) velocity distributions and Kappa distributions. The particle distribution functions are perturbed by a small “bump” represented by a relatively small amplitude Gaussian with a thermal width that is much smaller than the thermal width of the unperturbed distribution. The goal is to find the grid spacing Δv in phase space required to obtain a specified accuracy of the mean flow velocity . For parameters encountered in the solar wind near the orbit of the Earth, it is shown that to determine the radial component of the solar wind bulk velocity with an accuracy less than 0.1 km/s requires a grid spacing less than Δv/v = 0.8% or, equivalently, an energy spacing less than ΔE/E = 1.6%. To accurately resolve the velocity components transverse to the heliocentric radial direction, it is found that the spacing of the angle bins must be less than Δθ = 0.4∘. The required spacing of the energy and angle bins are approximately 1 order of magnitude less than those of any solar wind ion instrument ever flown in space.

Gluon fragmentation into quarkonium at next-to-leading order

We present the first calculation at next-to-leading order (NLO) in $\alpha_s$ of a fragmentation function into quarkonium whose form at leading order is a nontrivial function of $z$, namely the fragmentation function for a gluon into a spin-singlet S-wave state at leading order in the relative velocity. To calculate the real NLO corrections, we introduce a new subtraction scheme that allows the phase-space integrals to be evaluated in 4 dimensions. We extract all ultraviolet and infrared divergences in the real NLO corrections analytically by calculating the phase-space integrals of the subtraction terms in $4-2\epsilon$ dimensions. We also extract the divergences in the virtual NLO corrections analytically, and detail the cancellation of all divergences after renormalization. The NLO corrections have a dramatic effect on the shape of the fragmentation function, and they significantly increase the fragmentation probability.

On the calculation of soft phase space integral

The recent discovery of the Higgs boson at the LHC attracts much attention to the precise calculation of its production cross section in quantum chromodynamics. In this work, we discuss the calculation of soft triple-emission phase space integral, which is an essential ingredient in the recently calculated soft-virtual corrections to Higgs boson production at next-to-next-to-next-to-leading order. The main techniques used this calculation are method of differential equation for Feynman integral, and integration of harmonic polylogarithms.

Four-body contributions to B ¯ → X s γ $$ \overline{B}\to {X}_s\gamma $$ at NLO

Ongoing efforts to reduce the perturbative uncertainty in the B -> Xs gamma decay rate have resulted in a theory estimate to NNLO in QCD. However, a few contributions from multi-parton final states which are formally NLO are still unknown. These are parametrically small and included in the estimated error from higher order corrections, but must be computed if one is to claim complete knowledge of the B -> Xs gamma rate to NLO. A major part of these unknown pieces are four-body contributions corresponding to the partonic process b -> s qbar q gamma. We compute these NLO four-body contributions to B -> Xs gamma, and confirm the corresponding tree-level leading-order results. While the NLO contributions arise from tree-level and one-loop Feynman diagrams, the four-body phase-space integrations make the computation non-trivial. The decay rate contains collinear logarithms arising from the mass regularization of collinear divergences. We perform an exhaustive numerical analysis, and find that these contributions are positive and amount to no more than 1% of the total rate in the Standard Model, thus confirming previous estimates of the perturbative uncertainty.

Interaction energy and itinerant ferromagnetism in a strongly interacting Fermi gas in the absence of molecule formation

We investigate the interaction energy and the possibility of itinerant ferromagnetism in a strongly interacting Fermi gas at zero temperature in the absence of molecule formation. The interaction energy is obtained by summing the perturbative contributions of Galitskii-Feynman type to all orders in the gas parameter. It can be expressed by a simple phase space integral of an in-medium scattering phase shift. In both three and two dimensions (3D and 2D), the interaction energy shows a maximum before reaching the resonance from the Bose-Einstein condensate side, which provides a possible explanation of the experimental measurements of the interaction energy. This phenomenon can be theoretically explained by the qualitative change of the nature of the binary interaction in the medium. The appearance of an energy maximum has significant effects on the itinerant ferromagnetism. In 3D, the ferromagnetic transition is reentrant and itinerant ferromagnetism exists in a narrow window around the energy maximum. In 2D, the present theoretical approach suggests that itinerant ferromagnetism does not exist, which reflects the fact that the energy maximum becomes much lower than the energy of the fully polarized state.

Fragmentation in jets at NNLO

Beam and jet functions in Soft-Collinear Effective Theory describe collinear initial- and final-state radiation (jets), and enter in factorization theorems for N-jet production, the Higgs pT spectrum, etc. We show that they may directly be calculated as phase-space integrals of QCD splitting functions. At NLO all computations are trivial, as we demonstrate explicitly for the beam function, the transverse-momentum-dependent beam function, the jet function and the fragmenting jet function. This approach also highlights the role of crossing symmetry in these calculations. At NNLO we reproduce the quark jet function and calculate the fragmenting quark jet function for the first time. Here we use two methods: a direct phase-space integration and a reduction to master integrals which are computed using differential equations.

One-loop single-real-emission contributions topp→H+Xat next-to-next-to-next-to-leading order

I compute the contributions of the one-loop single-real-emission amplitudes, $gg\ensuremath{\rightarrow}Hg$, $qg\ensuremath{\rightarrow}Hq$, etc., to inclusive Higgs boson production through next-to-next-to-next-to-leading order (${\mathrm{N}}^{3}\mathrm{LO}$) in the strong coupling ${\ensuremath{\alpha}}_{s}$. The next-to-leading and next-to-next-to-leading order terms are computed in closed form, in terms of $\mathrm{\ensuremath{\Gamma}}$ functions and the hypergeometric functions $_{2}{F}_{1}$ and $_{3}{F}_{2}$. I compute the ${\mathrm{N}}^{3}\mathrm{LO}$ terms as Laurent expansions in the dimensional regularization parameter through order $({ϵ}^{1})$. To obtain the ${\mathrm{N}}^{3}\mathrm{LO}$ terms, I perform an extended threshold expansion of the phase space integrals and map the resulting coefficients onto a basis of harmonic polylogarithms.

Semiclassical initial-value representation of the transfer operator

The ability of semiclassical initial-value representation (IVR) methods to determine approximate energy levels for bound systems is limited due to problems associated with long classical trajectories. These difficulties become especially severe for large or classically chaotic systems. This work attempts to overcome such problems by developing an IVR expression that is classically equivalent to Bogomolny's formula for the transfer matrix [E. B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)] and can be used to determine semiclassical energy levels. The method is adapted to levels associated with states of desired symmetries and applied to two two-dimensional quartic oscillator systems, one integrable and one mostly chaotic. For both cases, the technique is found to resolve all energy levels in the ranges investigated. The IVR method does not require a search for special trajectories obeying boundary conditions on the Poincaré surface of section and leads to more rapid convergence of Monte Carlo phase space integrations than a previously developed IVR technique. It is found that semiclassical energies can be extracted from the eigenvalues of transfer matrices of dimension close to the theoretical minimum determined by Bogomolny's theory. The results support the assertion that the present IVR theory provides a different semiclassical approximation to the transfer matrix than that of Bogomolny for ℏ≠0. For the chaotic system investigated the IVR energies are found to be generally more accurate than those predicted by Bogomolny's theory.

Carlomat, version 2 of the program for automatic computation of lowest order cross sections

Version 2 of carlomat, a program for automatic computation of the lowest order cross sections of multiparticle reactions, is described. The substantial modifications with respect to version 1 of the program include: generation of a single phase space parameterization for the Feynman diagrams of the same topology, an interface to parton density functions, improvement of the color matrix computation, the Cabibbo–Kobayashi–Maskawa mixing in the quark sector, the effective models including scalar electrodynamics, the Wtb interaction with operators of dimension up to 5 and a general top–Higgs coupling. Moreover, some minor modifications have been made and several bugs in the program have been corrected. Program summaryProgram title: carlomat, version 2.0Catalogue identifier: AEDQ_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDQ_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 49058No. of bytes in distributed program, including test data, etc.: 25748755Distribution format: tar.gzProgramming language: Fortran 90/95Computer: AllOperating system: LinuxClassification: 4.4, 11.2Catalogue identifier of previous version: AEDQ_v1_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 1671Does the new version supersede the previous version?: YesNature of problem:Leading order predictions for reactions of two particle scattering into a final state with up to 10 particles within the Standard Model and some effective models.Solution method:As in version 1 of the program, the matrix element in the helicity basis and multichannel Monte Carlo phase space integration routine are generated automatically for a user specified process. The color matrix is divided into smaller routines and written down as a stand alone program that is calculated prior to compilation and execution of the Monte Carlo program for computation of the cross section. The phase space integration routine is substantially shortened in order to speed up its compilation. The code generation part of the program is modified to incorporate the scalar electrodynamics and effective Lagrangians of the top quark interactions with the W and Higgs bosons. Routines necessary for computing the helicity amplitudes of new couplings are added.Reasons for new version:The main reasons for the revision are: 1.To adjust the program for the description of hadron collisions.2.To facilitate computation of the color matrix that is usually much more involved for processes of the hadron–hadron collision than for processes of electron–positron annihilation.3.To shorten compilation time of the generated kinematical routines.4.To implement some extensions of the standard model in the program.Summary of revisions:A few substantial modifications are introduced with respect to version 1.0 of the program. First, a single phase space parameterization is generated for the Feynman diagrams of the same topology taking into account possible differences in mappings of peaks in the individual diagrams, which speeds up a compilation time of the Monte Carlo program for multiparticle reactions by a factor 4–5 with respect to the previous version. Second, an interface to parton density functions is added that allows predictions to be made for hadron collisions. Third, calculation of the color matrix is facilitated. Fourth, the Cabibbo–Kobayashi–Maskawa mixing in the quark sector is implemented. Fifth, the effective models including scalar electrodynamics, the Wtb interaction with operators of dimension up to 5 and a general top–Higgs coupling are implemented. Moreover, some minor modifications have been made and several bugs in the program have been corrected.Restrictions:Although the compilation time has been shortened in the current version, it still may be quite long for processes with 8 or more final state particles. Another limitation is the size of the color matrix that, if too big, may prevent compilation or result in a very long execution time of the color compilation program. This actually may happen already for some QCD processes with 7 partons such as gg→5g, the commutation time of the color matrix of which, is about 200 h.Running time:Depends strongly on the selected process and, to a lesser extent, on the Fortran compiler used. The following amounts of time are needed at different computation stages of the top quark pair production parton level process gg→bud̄b̄μ−ν̄μ, to produce the appended test output files on a PC with the Pentium 4 3.0 GHz processor with Absoft (GNU, Intel) Fortran compilers: code generation takes 3.7 s (3.7 s, 2.4 s), compilation, computation and simplification of the color matrix takes about 1 s (1 s, 1 s), compilation of all the generated routines takes just a few seconds and execution of the Monte Carlo program takes about 44 s (41 s, 23 s).

Radiative parton processes in perturbative QCD: An improved version of the Gunion and Bertsch cross section from comparisons to the exact result

In this work we compare the Gunion-Bertsch approximation of the leading order perturbative QCD radiation matrix element to the exact result. To this end, we revisit the derivation of the Gunion-Bertsch approximation as well as perform extensive numerical comparisons of the Gunion-Bertsch and the exact result. We find that when employing the matrix elements to obtain rates or cross sections from phase space integration, the amplitude by Gunion and Bertsch deviates from the correct result in characteristic regions of the phase space. We propose an improved version of the Gunion-Bertsch matrix element which agrees very well with the exact result in all phase space regions.

Soft triple-real radiation for Higgs production at N3LO

We present the two first terms in the threshold expansion of Higgs production partonic cross-sections at hadron colliders for processes with three partons in the final state. These are contributions to the inclusive Higgs cross-section in gluon fusion at N3LO. We have developed a new technique for the expansion of the squared matrix-elements around the soft limit and for the reduction of the required phase-space integrals to only ten single-scale master integrals. We compute the master integrals building upon modern techniques for the integration of multidimensional integrals in dimensional regularization. Our results constitute an important step towards a systematic computation of the Higgs boson cross-section as an expansion around the threshold limit.

Anharmonic state counts and partition functions for molecules via classical phase space integrals in curvilinear coordinates

An algorithm is presented for calculating fully anharmonic vibrational state counts, state densities, and partition functions for molecules using Monte Carlo integration of classical phase space. The algorithm includes numerical evaluations of the elements of the Jacobian and is general enough to allow for sampling in arbitrary curvilinear or rectilinear coordinate systems. Invariance to the choice of coordinate system is demonstrated for vibrational state densities of methane, where we find comparable sampling efficiency when using curvilinear z-matrix and rectilinear Cartesian normal mode coordinates. In agreement with past work, we find that anharmonicity increases the vibrational state density of methane by a factor of ∼2 at its dissociation threshold. For the vinyl radical, we find a significant (∼10×) improvement in sampling efficiency when using curvilinear z-matrix coordinates relative to Cartesian normal mode coordinates. We attribute this improved efficiency, in part, to a more natural curvilinear coordinate description of the double well associated with the H2C-C-H wagging motion. The anharmonicity correction for the vinyl radical state density is ∼1.4 at its dissociation threshold. Finally, we demonstrate that with trivial parallelizations of the Monte Carlo step, tractable calculations can be made for the vinyl radical using direct ab initio potential energy surface evaluations and a composite QCISD(T)/MP2 method.

Semiclassical Ehrenfest paths

Trajectories are a central concept in our understanding of classical phenomena and also in rationalizing quantum mechanical effects. In this work we provide a way to determine semiclassical paths, approximations to quantum averages in phase space, directly from classical trajectories. We avoid the need of intermediate steps, like particular solutions to the Schroedinger equation or numerical integration in phase space by considering the system to be initially in a coherent state and by assuming that its early dynamics is governed by the Heller semiclassical approximation. Our result is valid for short propagation times only, but gives non-trivial information on the quantum-classical transition.

Grassmann phase space theory and the Jaynes–Cummings model

The Jaynes–Cummings model of a two-level atom in a single mode cavity is of fundamental importance both in quantum optics and in quantum physics generally, involving the interaction of two simple quantum systems—one fermionic system (the TLA), the other bosonic (the cavity mode). Depending on the initial conditions a variety of interesting effects occur, ranging from ongoing oscillations of the atomic population difference at the Rabi frequency when the atom is excited and the cavity is in an n-photon Fock state, to collapses and revivals of these oscillations starting with the atom unexcited and the cavity mode in a coherent state. The observation of revivals for Rydberg atoms in a high-Q microwave cavity is key experimental evidence for quantisation of the EM field. Theoretical treatments of the Jaynes–Cummings model based on expanding the state vector in terms of products of atomic and n-photon states and deriving coupled equations for the amplitudes are a well-known and simple method for determining the effects.In quantum optics however, the behaviour of the bosonic quantum EM field is often treated using phase space methods, where the bosonic mode annihilation and creation operators are represented by c-number phase space variables, with the density operator represented by a distribution function of these variables. Fokker–Planck equations for the distribution function are obtained, and either used directly to determine quantities of experimental interest or used to develop c-number Langevin equations for stochastic versions of the phase space variables from which experimental quantities are obtained as stochastic averages. Phase space methods have also been developed to include atomic systems, with the atomic spin operators being represented by c-number phase space variables, and distribution functions involving these variables and those for any bosonic modes being shown to satisfy Fokker–Planck equations from which c-number Langevin equations are often developed. However, atomic spin operators satisfy the standard angular momentum commutation rules rather than the commutation rules for bosonic annihilation and creation operators, and are in fact second order combinations of fermionic annihilation and creation operators. Though phase space methods in which the fermionic operators are represented directly by c-number phase space variables have not been successful, the anti-commutation rules for these operators suggest the possibility of using Grassmann variables—which have similar anti-commutation properties. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of phase space methods in quantum optics to treat fermionic systems by representing fermionic annihilation and creation operators directly by Grassmann phase space variables is rather rare. This paper shows that phase space methods using a positive P type distribution function involving both c-number variables (for the cavity mode) and Grassmann variables (for the TLA) can be used to treat the Jaynes–Cummings model. Although it is a Grassmann function, the distribution function is equivalent to six c-number functions of the two bosonic variables. Experimental quantities are given as bosonic phase space integrals involving the six functions. A Fokker–Planck equation involving both left and right Grassmann differentiations can be obtained for the distribution function, and is equivalent to six coupled equations for the six c-number functions.The approach used involves choosing the canonical form of the (non-unique) positive P distribution function, in which the correspondence rules for the bosonic operators are non-standard and hence the Fokker–Planck equation is also unusual. Initial conditions, such as those above for initially uncorrelated states, are discussed and used to determine the initial distribution function. Transformations to new bosonic variables rotating at the cavity frequency enable the six coupled equations for the new c-number functions–that are also equivalent to the canonical Grassmann distribution function–to be solved analytically, based on an ansatz from an earlier paper by Stenholm. It is then shown that the distribution function is exactly the same as that determined from the well-known solution based on coupled amplitude equations.In quantum–atom optics theories for many atom bosonic and fermionic systems are needed. With large atom numbers, treatments must often take into account many quantum modes—especially for fermions. Generalisations of phase space distribution functions of phase space variables for a few modes to phase space distribution functionals of field functions (which represent the field operators, c-number fields for bosons, Grassmann fields for fermions) are now being developed for large systems. For the fermionic case, the treatment of the simple two mode problem represented by the Jaynes–Cummings model is a useful test case for the future development of phase space Grassmann distribution functional methods for fermionic applications in quantum–atom optics.

Collision Rate and Symmetry Factor in Gluon Plasma

The elastic and inelastic collision rates in a gluon gas are calculated. The symmetry factor and the phase space integral are discussed in detail. With a symmetry factor and well constrained phase space, the same result can be obtained as that of the full phase space without the factor. Such an equivalence is illustrated by analytic and numerical calculations for gg → gg and gg → ggg processes.