Leading Log Approximation Research Articles

Semiclassical solution to the BFKL equation with massive gluons

In this paper we proceed to study the high energy behavior of a scattering amplitudes in the Yang-Mills theory with the Higgs mechanism for the gauge boson mass. The spectrum of the $j$-plane singularities of the $t$-channel partial waves, and corresponding eigenfunctions of the BFKL equation in leading log approximation (LLA), were previously calculated by us numerically in arXiv:1401.4671 . Here we develop a semiclassical approach and investigate the influence of the impact parameter exponential decrease, existing in this model, on the high energy asymptotic behaviour of the scattering amplitude. This approach is much simpler than the numerical calculations, and reproduces our earlier numerical results. The analytical (semianalytical) solutions which have been found in this paper, can be used to incorporate correctly the large impact parameter behavior in the framework of CGC/saturation approach. This behaviour is interesting as provides the high energy amplitude for the electroweak theory, which can be measured experimentally.

Rare decay $$\pi ^0\rightarrow e^+e^-$$ π 0 → e + e - : on corrections beyond the leading order

The preceding experimental and theoretical results on the rare decay $\pi^0 \to e^+e^-$ are briefly summarized. Already computed two-loop QED corrections are reviewed and the bremsstrahlung contribution beyond the soft-photon approximation is analytically calculated. The possible further contribution of QCD loop corrections is estimated using the leading logarithm approximation. The complete result can be used to fit the value of the contact interaction coupling $\chi^r$ to the recent KTeV experiment with the result $\chi^r(M_\rho) = 4.5 \pm 1.0$.

In-medium jet shape from energy collimation in parton showers: Comparison with CMS PbPb data at 2.76 TeV

We present the medium-modified energy collimation in the leading-logarithmic approximation (LLA) and next-to-leading-logarithmic approximation (NLLA) of QCD. As a consequence of more accurate kinematic considerations in the argument of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) fragmentation functions (FFs) we find a new NLLA correction $\mathcal{O}({\ensuremath{\alpha}}_{s})$ which accounts for the scaling violation of DGLAP FFs at small $x$. The jet shape is derived from the energy collimation within the same approximations and we also compare our calculations for the energy collimation with the event generators pythia6 and YaJEM for the first time in this paper. The modification of jets by the medium in both cases is implemented by altering the infrared sector using the Borghini-Wiedemann model. The energy collimation and jet shapes qualitatively describe a clear broadening of showers in the medium, which is further supported by YaJEM in the final comparison of the jet shape with CMS PbPb data at center-of-mass energy 2.76 TeV. The comparison of the biased versus unbiased YaJEM jet shape with the CMS data shows a more accurate agreement for biased showers and illustrates the importance of an accurate simulation of the experimental jet-finding strategy.

Multi-Regge form of gluon-exchange amplitudes in supersymmetric Yang-Mills theories

Within supersymmetric Yang-Mills theories, all effective vertices for the interaction of Reggeized gluons with particles were found both in the leading-logarithm approximation and in the next-to-leading-order approximation. The contributions of scalars to the eigenfunction of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel for the adjoint representation of the gauge group and to impact factors similar to those in QCD were calculated. The impact factors that arise in supersymmetric Yang-Mills theories, but which are not present in QCD, are also found. Fulfillment of all bootstrap conditions, which guarantees the multi-Regge form of multiparticle amplitudes in supersymmetric Yang-Mills theories in the next-to-leading-order approximation, was proven. A method for testing bootstrap relations without calculating explicit expressions for vertices and impact factors was developed. This method was used to prove fulfillment of bootstrap conditions in theories featuring any number of fermions and scalar particles transformed according to any representations of the color group and involved in interactions with one another via Yukawa coupling.

Evolution equation for 3-quark Wilson loop operator

The evolution equation for the 3 quark Wilson loop operator has been derived in the leading logarithm approximation within Balitsky high energy operator expansion.

The Multi-Regge limit of NMHV amplitudes in N=4 SYM theory

We consider the multi-Regge limit for N=4 SYM NMHV leading color amplitudes in two different formulations: the BFKL formalism for multi-Regge amplitudes in leading logarithm approximation, and superconformal N=4 SYM amplitudes. It is shown that the two approaches agree to two-loops for the 2->4 and 3->3 six-point amplitudes. Predictions are made for the multi-Regge limit of three loop 2->4 and 3->3 NMHV amplitudes, as well as a particular sub-set of two loop 2 ->2 +n N^kMHV amplitudes in the multi-Regge limit in the leading logarithm approximation from the BFKL point of view.

The six-point remainder function to all loop orders in the multi-Regge limit

Abstract We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric $ \mathcal{N}=4 $ Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem — the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w * , which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w| → 0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log |w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log |w| is given in terms of modified Bessel functions.

Single-valued harmonic polylogarithms and the multi-Regge limit

Abstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar $ \mathcal{N}=4 $ super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w, w ∗). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν, n) which are related to (w, w ∗) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N3LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.

BFKL approach and2→5maximally helicity violating amplitude inN=4super-Yang-Mills theory

We study MHV amplitude for the 2 -> 5 scattering in the multi-Regge kinematics. The Mandelstam cut correction to the BDS amplitude is calculated in the leading logarithmic approximation (LLA) and the corresponding remainder function is given to any loop order in a closed integral form. We show that the LLA remainder function at two loops for 2 -> 5 amplitude can be written as a sum of two 2 -> 4 remainder functions due to recursive properties of the leading order impact factors. We also make some generalizations for the MHV amplitudes with more external particles. The results of the present study are in agreement with all leg two loop symbol derived by Caron-Huot as shown in a parallel paper of one of the authors with collaborators.

Radiative effects in the processes of exclusive photon electroproduction from polarized protons

Radiative effects in the electroproduction of photons in polarized $ep$ scattering are calculated in the leading log approximation and analyzed numerically for kinematical conditions of current measurement at Jefferson Lab. Radiative corrections to the cross sections, their azimuthal distributions, and Fourier coefficients are in particular focus. Kinematical regions where the radiative corrections are considerable are identified.

A REVIEW OF NATURALNESS AND DARK MATTER PREDICTION FOR THE HIGGS MASS IN MSSM AND BEYOND

Within a two-loop leading-log approximation, we review the prediction for the lightest Higgs mass (mh) in the framework of constrained MSSM (CMSSM), derived from the naturalness requirement of minimal fine-tuning (Δ) of the electroweak scale, and dark matter consistency. As a result, the Higgs mass is predicted to be just above the LEP2 bound, mh = 115.9±2 GeV , corresponding to a minimal Δ = 17.8, the value obtained from consistency with electroweak and WMAP (3σ) constraints, but without the LEP2 bound. Due to quantum corrections (largely QCD ones for mh above LEP2 bound), Δ grows ≈ exponentially on either side of the above value of mh, which stresses the relevance of this prediction. A value mh>121 (126) GeV cannot be accommodated within the CMSSM unless one accepts a fine-tuning cost worse than Δ>100 (1000), respectively. We review how the above prediction for mh and Δ changes under the addition of new physics beyond the MSSM Higgs sector, parametrized by effective operators of dimensions d = 5 and d = 6. For d = 5 operators, one can obtain values mh as large as 130 GeV with an acceptable Δ<10. The size of the supersymmetric correction that each individual operator of d = 6 brings to the value of mh for points with Δ<100 (<200), is found to be small, of few ≤4 GeV (≤6 GeV) respectively, for M = 8 TeV where M is the scale of new physics. This value decreases (increases) by approximately 1 GeV for a 1 TeV increase (decrease) of the scale M. The relation of these results to the Atlas/CMS supersymmetry exclusion limits is presented together with their impact for the CMSSM regions of lowest fine-tuning.

Bootstrapping the three-loop hexagon

We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar $ \mathcal{N} = 4 $ super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol’s entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3 → 3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

Generalized event shape and energy flow studies in e+e− annihilation at $ \sqrt {s} = 91.2{ - }208.0 $ GeV

We present results from a study of hadronic event structure in high energy e+e− interactions using the L3 detector at LEP. A new class of event shape distributions are measured at and above the Z boson pole for light quark (u, d, s, c) flavours. Energy flow correlations are studied for all hadronic events. Next-to-leading-log QCD calculations and QCD models with improved leading-log approximations are compared to data and good agreement is found at the Z-pole whereas some discrepancies are observed at higher centre-of-mass energies.

Four-jet production at LHC and Tevatron in QCD

We demonstrate that in the back-to-back kinematics the production of four jets in the collision of two partons is suppressed in the leading log approximation of pQCD, compared to the \^Ehard processes involving the collision of four partons. We derive the basic equation for four-jet production in QCD in terms of the convolution of generalized two-parton distributions of colliding hadrons in the momentum space representation. Our derivation leads to geometrical approach in the impact parameter space close to that suggested within the parton model and used before to describe the four-jet production. We develop the independent parton approximation to the light-cone wave function of the proton. Comparison with the CDF and D0 data shows that the independent parton approximation to the light-cone wave function of the proton is insufficient to explain the data. We argue that the data indicate the presence of significant multiparton correlations in the light-cone wave functions of colliding protons.

Asymptotic behavior of double parton distribution functions

The double parton distribution functions are investigated in the region of small longitudinal momentum fractions in the leading logarithm approximation of perturbative QCD. It is shown that these functions have the factorization property in the case of one slow and one fast parton.

Parton Cascades, Small x, Multiple Interactions, and Saturation

In this paper I discuss the dynamics of particle production in high energy reactions. It includes parton cascades and hadronization in e(+)e(-)-annihilation, small x evolution including the Double Leading Log approximation and the BFKL equation, saturation at high densities and the BK equation, and finally the Lund Dipole Cascade model for high energy collisions, which is implemented in the DIPSY MC.

Current non-conservation effects in ultra-high energy neutrino interactions

The overall hardness scale of the ultra-high energy neutrino-nucleon interactions is usually estimated as Q 2 ∼ m 2 The effect of non-conservation of weak currents pushes this scale up to the top quark mass squared and changes dynamics of the scattering process. The Double Leading Log Approximation provides simple and numerically accurate formula for the top-bottom contribution to the total cross section σνN . Corresponding correction to σνN appears to be numerically large. It is comparable with the leading contribution evaluated in the massless quark approximation.

Spectral densities for hot QCD plasmas in a leading-log approximation

We compute the spectral densities of $T^{\mu\nu}$ and $J^{\mu}$ in high temperature QCD plasmas at small frequency and momentum,\, $\omega,k \sim g^4 T$. The leading log Boltzmann equation is reformulated as a Fokker Planck equation with non-trivial boundary conditions, and the resulting partial differential equation is solved numerically in momentum space. The spectral densities of the current, shear, sound, and bulk channels exhibit a smooth transition from free streaming quasi-particles to ideal hydrodynamics. This transition is analyzed with conformal and non-conformal second order hydrodynamics, and a second order diffusion equation. We determine all of the second order transport coefficients which characterize the linear response in the hydrodynamic regime.

Electroweak corrections to hadronic event shapes and jet production in [formula omitted] annihilation

We present a complete calculation of the electroweak O ( α 3 α s ) corrections to three-jet production and related event-shape observables at electron–positron colliders. The Z-boson resonance is described within the complex-mass scheme, rendering the calculation valid both in the resonance and off-shell regions. Higher-order initial-state radiation is included in the leading-logarithmic approximation. We properly account for the corrections to the total hadronic cross section and for the experimental photon isolation criteria. To this end we implement contributions of the quark-to-photon fragmentation function both in the slicing and subtraction formalism. The effects of the electroweak corrections on various event-shape distributions and on the three-jet rate are studied. They are typically at the few-percent level, and remnants of the radiative return are found even after inclusion of appropriate cuts.

Higgs-regularized three-loop four-gluon amplitude in $ \mathcal{N} = 4 $ SYM: exponentiation and Regge limits

We compute the three-loop contribution to the \( \mathcal{N} = 4 \) supersymmetric Yang-Mills planar four-gluon amplitude using the recently-proposed Higgs IR regulator of Alday, Henn, Plefka, and Schuster. In particular, we test the proposed exponential ansatz for the four-gluon amplitude that is the analog of the BDS ansatz in dimensional regularization. By evaluating our results at a number of kinematic points, and also in several kinematic limits, we establish the validity of this ansatz at the three-loop level.We also examine the Regge limit of the planar four-gluon amplitude using several different IR regulators: dimensional regularization, Higgs regularization, and a cutoff regularization. In the latter two schemes, it is shown that the leading logarithmic (LL) behavior of the amplitudes, and therefore the lowest-order approximation to the gluon Regge trajectory, can be correctly obtained from the ladder approximation of the sum of diagrams. In dimensional regularization, on the other hand, there is no single dominant set of diagrams in the LL approximation. We also compute the NLL and NNLL behavior of the L-loop ladder diagram using Higgs regularization.