Harmonic Polylogarithms Research Articles

Three loop QCD corrections to the heavy-light form factors in the color-planar limit

We present the analytic expressions for the color-planar contributions to the heavy-light form factors at three loops in perturbative QCD. These form factors play an important role in the precision predictions of various observables in top quark and flavour physics. We compute the master integrals using the method of differential equations. We perform the ultraviolet renormalization for all the appearing fields and parameters. The analytic results for the renormalized form factors are expressed in terms of generalized harmonic polylogarithms. We also study the Sudakov behaviour of these form factors in the asymptotic limit, which enables us to obtain the complete logarithmic three-loop and partial four-loop contributions.

Three-loop helicity amplitudes for photon+jet production

We present three-loop helicity amplitudes for the production of a single photon in association with one jet in Quantum Chromodynamics, a final state which provides a standard candle of the Standard Model at the Large Hadron Collider. We employ a recently-proposed variation of the so-called tensor projection method in the 't Hooft-Veltman scheme (tHV) which avoids the computation of contributions due to unphysical (-2$\epsilon$)-dimensional polarisations of the external states. We obtain compact analytic results expressed in terms of harmonic polylogarithms.

Drinfel’d associator and relations of some special functions

We observe the differential equation dG(z) / dz = (x0 / z + x1 / (1- z))G(z) in the space of power series of noncommutative indeterminates x0, x1 , where the coefficients of G(z) are holomorphic functions on the simply connected domain ℂ \ [(-∞,0)∪(1,+∞)].Researches on this equation in some conditions turn out different solutions which admit Drinfel'd associator as a bridge. In this paper, we review representations of these solutions by generating series of some special functions such as multiple harmonic sums, multiple polylogarithms and polyzetas. Thereby, relations in explicit forms or asymptotic expansions of these special functions from the bridge equations are deduced by identifying local coordinates.

Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: chemical potentials and deformations

We describe how to calculate the Hagedorn temperature of mathcal{N} = 4 SYM theory and type IIB superstring theory on AdS5 × S5 via the Quantum Spectral Curve (QSC) — providing further details on our previous letters [1] and [2]. We solve the QSC equations perturbatively at weak ‘t Hooft coupling λ up to seven-loop order and numerically at finite coupling, finding that the perturbative results can be expressed in terms of single-valued harmonic polylogarithms. Moreover, we generalize the QSC to describe the Hagedorn temperature in the presence of chemical potentials. Finally, we show that the Hagedorn temperature in certain deformations of mathcal{N} = 4 SYM theory (real-β and γi deformation) agrees with the one in mathcal{N} = 4 SYM theory at any value of λ.

Three-Loop Gluon Scattering in QCD and the Gluon Regge Trajectory.

We compute the three-loop helicity amplitudes for the scattering of four gluons in QCD. We employ projectors in the 't Hooft-Veltman scheme and construct the amplitudes from a minimal set of physical building blocks, which allows us to keep the computational complexity under control. We obtain relatively compact results that can be expressed in terms of harmonic polylogarithms. In addition, we consider the Regge limit of our amplitude and extract the gluon Regge trajectory in full three-loop QCD. This is the last missing ingredient required for studying single-Reggeon exchanges at next-to-next-to-leading logarithmic accuracy.

The two-loop massless off-shell QCD operator matrix elements to finite terms

We calculate the unpolarized and polarized two–loop massless off–shell operator matrix elements in QCD to O(ε) in the dimensional parameter in an automated way. Here we use the method of arbitrary high Mellin moments and difference ring theory, based on integration-by-parts relations. This method also constitutes one way to compute the QCD anomalous dimensions. The presented higher order contributions to these operator matrix elements occur as building blocks in the corresponding higher order calculations up to four–loop order. All contributing quantities can be expressed in terms of harmonic sums in Mellin–N space or by harmonic polylogarithms in z–space. We also perform comparisons to the literature.

Boomerang webs up to three-loop order

Webs are sets of Feynman diagrams which manifest soft gluon exponentiation in gauge theory scattering amplitudes: individual webs contribute to the logarithm of the amplitude and their ultraviolet renormalization encodes its infrared structure. In this paper, we consider the particular class of boomerang webs, consisting of multiple gluon exchanges, but where at least one gluon has both of its endpoints on the same Wilson line. First, we use the replica trick to prove that diagrams involving self-energy insertions along the Wilson line do not contribute to the web, i.e. their exponentiated colour factor vanishes. Consequently boomerang webs effectively involve only integrals where boomerang gluons straddle one or more gluons that connect to other Wilson lines. Next we classify and calculate all boomerang webs involving semi-infinite non-lightlike Wilson lines up to three-loop order, including a detailed discussion of how to regulate and renormalize them. Furthermore, we show that they can be written using a basis of specific harmonic polylogarithms, that has been conjectured to be sufficient for expressing all multiple gluon exchange webs. However, boomerang webs differ from other gluon-exchange webs by featuring a lower and non-uniform transcendental weight. We cross-check our results by showing how certain boomerang webs can be determined by the so-called collinear reduction of previously calculated webs. Our results are a necessary ingredient of the soft anomalous dimension for non-lightlike Wilson lines at three loops.

Three-loop helicity amplitudes for four-quark scattering in massless QCD

We compute the three-loop corrections to the helicity amplitudes for q overline{q} → Q overline{Q} scattering in massless QCD. In the Lorentz decomposition of the scattering amplitude we avoid evanescent Lorentz structures and map the corresponding form factors directly to the physical helicity amplitudes. We reduce the amplitudes to master integrals and express them in terms of harmonic polylogarithms. The renormalised amplitudes exhibit infrared divergences of dipole and quadrupole type, as predicted by previous work on the infrared structure of multileg scattering amplitudes. We derive the finite remainders and present explicit results for all relevant partonic channels, both for equal and different quark flavours.

Bootstrapping octagons in reduced kinematics from A2 cluster algebras

Multi-loop scattering amplitudes/null polygonal Wilson loops in mathcal{N} = 4 super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an 1 + 1 dimensional subspace of Minkowski spacetime (or boundary of the AdS3 subspace). Since the edges of a 2n-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of G(2, n)/T ∼ An−3. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping A2 functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters v, 1 + v, w, 1 + w of A1 × A1, there are two letters v − w, 1 − vw mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping A2 functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to 2n-gons in terms of A2 functions and beyond.

A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the overline{mathrm{MS}} -pole quark mass relation

We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter x and the dimension d, in the whole kinematic range of x. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the overline{mathrm{MS}} and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy.

Three-loop vertex integrals at symmetric point

This paper provides details of the massless three-loop three-point integrals calculation at the symmetric point. Our work aimed to extend known two-loop results for such integrals to the three-loop level. Obtained results can find their application in regularization-invariant symmetric point momentum-subtraction (RI/SMOM) scheme QCD calculations of renormalization group functions and various composite operator matrix elements. To calculate integrals, we solve differential equations for auxiliary integrals by transforming the system to the ε-form. Calculated integrals are expressed through the basis of functions with uniform transcendental weight. We provide expansion up to the transcendental weight six for the basis functions in terms of harmonic polylogarithms with six-root of unity argument.

Master integrals for bipartite cuts of three-loop photon self energy

We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

Diphoton Amplitudes in Three-Loop Quantum Chromodynamics.

We consider the three-loop scattering amplitudes for the production of a pair of photons in quark-antiquark annihilation in QCD. We use suitably defined projectors to efficiently calculate all helicity amplitudes. We obtain relatively compact analytic results that we write in terms of harmonic polylogarithms or, alternatively, multiple polylogarithms of up to depth three. This is the first calculation of a three-loop four-point scattering amplitude in full QCD.

Gravitational scattering at the seventh order in G : Nonlocal contribution at the sixth post-Newtonian accuracy

A recently introduced approach to the classical gravitational dynamics of binary systems involves intricate integrals (linked to a combination of nonlocal-in-time interactions with iterated $\frac{1}{r}$-potential scattering) which have so far resisted attempts at their analytical evaluation. By using computing techniques developed for the evaluation of multiloop Feynman integrals (notably harmonic polylogarithms and Mellin transform) we show how to analytically compute all the integrals entering the nonlocal-in-time contribution to the classical scattering angle at the sixth post-Newtonian accuracy, and at the seventh order in Newton's constant, $G$ (corresponding to six-loop graphs in the diagrammatic representation of the classical scattering angle).

Exact results for {Z}_m^{mathrm{OS}} and {Z}_2^{mathrm{OS}} with two mass scales and up to three loops

We consider the on-shell mass and wave function renormalization constants {Z}_m^{mathrm{OS}} and {Z}_2^{mathrm{OS}} up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters sqrt{1-{tau}^2} and sqrt{1-{tau}^2}/tau which extends the findings from ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order mathcal{O} (ϵ2) and mathcal{O} (ϵ) at two- and three-loop order which are crucial ingredients for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation.

Transverse momentum dependent PDFs at N3LO

We compute the quark and gluon transverse momentum dependent parton distribution functions at next-to-next-to-next-to-leading order (N3LO) in perturbative QCD. Our calculation is based on an expansion of the differential Drell-Yan and gluon fusion Higgs production cross sections about their collinear limit. This method allows us to employ cutting edge multiloop techniques for the computation of cross sections to extract these universal building blocks of the collinear limit of QCD. The corresponding perturbative matching kernels for all channels are expressed in terms of simple harmonic polylogarithms up to weight five. As a byproduct, we confirm a previous computation of the soft function for transverse momentum factorization at N3LO. Our results are the last missing ingredient to extend the qT subtraction methods to N3LO and to obtain resummed qT spectra at N3LL′ accuracy both for gluon as well as for quark initiated processes.

N-jettiness beam functions at N3LO

We present the first complete calculation for the quark and gluon N-jettiness ( {mathcal{T}}_N ) beam functions at next-to-next-to-next-to-leading order (N3LO) in perturbative QCD. Our calculation is based on an expansion of the differential Higgs boson and Drell-Yan production cross sections about their collinear limit. This method allows us to employ cutting edge techniques for the computation of cross sections to extract the universal building blocks in question. The class of functions appearing in the matching coefficents for all channels includes iterated integrals with non-rational kernels, thus going beyond the one of harmonic polylogarithms. Our results are a key step in extending the {mathcal{T}}_N subtraction methods to N3LO, and to resum {mathcal{T}}_N distributions at N3LL′ accuracy both for quark as well as for gluon initiated processes.

Two-parton scattering amplitudes in the Regge limit to high loop orders

We study two-to-two parton scattering amplitudes in the high-energy limit of perturbative QCD by iteratively solving the BFKL equation. This allows us to predict the imaginary part of the amplitude to leading-logarithmic order for arbitrary t-channel colour exchange. The corrections we compute correspond to ladder diagrams with any number of rungs formed between two Reggeized gluons. Our approach exploits a separation of the two-Reggeon wavefunction, performed directly in momentum space, between a soft region and a generic (hard) region. The former component of the wavefunction leads to infrared divergences in the amplitude and is therefore computed in dimensional regularization; the latter is computed directly in two transverse dimensions and is expressed in terms of single-valued harmonic polylogarithms of uniform weight. By combining the two we determine exactly both infrared-divergent and finite contributions to the two-to-two scattering amplitude order-by-order in perturbation theory. We study the result numerically to 13 loops and find that finite corrections to the amplitude have a finite radius of convergence which depends on the colour representation of the t-channel exchange.

Quark mass effects in two-loop Higgs amplitudes

We provide two two-loop amplitudes relevant for precision Higgs physics. The first is the two-loop amplitude for Higgs boson production through gluon fusion with exact dependence on the top quark mass up to squared order in the dimensional regulator ε. The second result we provide is the two-loop amplitude for the decay of a Higgs boson into a pair of massive bottom quarks through the Higgs-to-gluon coupling in the infinite top mass limit. Both amplitudes are computed by finding canonical bases of master integrals, which we evaluate explicitly in terms of harmonic polylogarithms. We obtain the bare, renormalized and IR-subtracted amplitude and provide the results in terms of building blocks suitable for changing renormalization schemes.

Subleading logarithmic QED initial state corrections to [formula omitted] to O(α6L5)

Using the method of massive operator matrix elements, we calculate the subleading QED initial state radiative corrections to the process e+e−→γ⁎/Z⁎ for the first three logarithmic contributions from O(α3L3),O(α3L2),O(α3L) to O(α5L5),O(α5L4),O(α5L3) and compare their effects to the leading contribution O(α6L6) and one more subleading term O(α6L5). The calculation is performed in the limit of large center of mass energies squared s≫me2. These terms supplement the known corrections to O(α2), which were completed recently. Given the high precision at future colliders operating at very large luminosity, these corrections are important for concise theoretical predictions. The present calculation needs the calculation of one more two–loop massive operator matrix element in QED. The radiators are obtained as solutions of the associated Callen–Symanzik equations in the massive case. The radiators can be expressed in terms of harmonic polylogarithms to weight w = 6 of argument z and (1−z) and in Mellin N space by generalized harmonic sums. Numerical results are presented on the position of the Z peak and corrections to the Z width, ΓZ. The corrections calculated result into a final theoretical accuracy for δMZ and δΓZ which is estimated to be of O(30keV) at an anticipated systematic accuracy at the FCC_ee of ∼100keV. This precision cannot be reached, however, by including only the corrections up to O(α3).