Renormalization Scale Dependence Research Articles

Nonperturbative renormalization of domain wall fermions: Quark bilinears

We find the renormalisation coefficients of the quark field and the flavour non-singlet fermion bilinear operators for the domain wall fermion action, in the regularisation independent (RI) renormalisation scheme. Our results are from a quenched simulation, on a 16^3x32 lattice, with beta=6.0 and an extent in the fifth dimension of 16. We also discuss the expected effects of the residual chiral symmetry breaking inherent in a domain wall fermion simulation with a finite fifth dimension, and study the evidence for both explicit and spontaneous chiral symmetry breaking effects in our numerical results. We find that the relations between different renormalisation factors predicted by chiral symmetry are, to a good approximation, satisfied by our results and that systematic effects due to the (low energy) spontaneous chiral symmetry breaking and zero-modes can be controlled. Our results are compared against the perturbative predictions for both their absolute value and renormalisation scale dependence.

Parton densities and structure functions at next-to-next-to-leading order and beyond

We summarize recent results on the evolution of unpolarized parton densities and deep-inelastic structure functions in massless perturbative QCD. Due to the last year's extension of the integer-moment calculations of the three-loop splitting functions, the next-to-next-to-leading order (NNLO) evolution of the parton distributions can now be performed reliably at momentum fractions x ≳ 10−4, facilitating a considerably improved theoretical accuracy of their extraction from data on deep-inelastic scattering. The NNLO corrections are not dominated, at relevant values of x, by their leading small-x terms. At large x the splitting-function series converges very rapidly; hence, employing results on the three-loop coefficient functions, the structure functions can be analysed at next-to-next-to-next-to-leading order (N3LO) for x > 10−2. The resulting values for αs do not significantly change beyond the NNLO, their renormalization scale dependence reaches about ±1% at N3LO.

Three-jet cross sections in hadron-hadron collisions at next-to-leading order.

We present a new QCD-event generator for hadron colliders which can calculate one-, two-, and three-jet cross sections at next-to-leading order accuracy. We study the transverse energy spectrum of three-jet hadronic events using the k(radially) algorithm. We show that the next-to-leading order correction significantly reduces the renormalization and factorization scale dependence of the three-jet cross section.

Calculations of two-loop virtual corrections tob→sl+l−in the standard model

We present in detail the calculation of the virtual O(alpha_s) corrections to the inclusive semi-leptonic rare decay b --> s l+ l-. We also include those O(alpha_s) bremsstrahlung contributions which cancel the infrared and mass singularities showing up in the virtual corrections. In order to avoid large resonant contributions, we restrict the invariant mass squared s of the lepton pair to the range 0.05 < s/mb^2 < 0.25. The analytic results are represented as expansions in the small parameters s/mb^2, z = mc^2/mb^2 and s/(4 mc^2). The new contributions drastically reduce the renormalization scale dependence of the decay spectrum. For the corresponding branching ratio (restricted to the above s-range) the renormalization scale uncertainty gets reduced from +/-13% to +/-6.5%.

QCD corrections to associatedtt¯hproduction at the Fermilab Tevatron

We present in detail the calculation of the O(alpha_s^3)inclusive total cross section for the process (p pbar-> t tbar h) in the Standard Model, at the Tevatron center-of-mass energy sqrt(s_H)=2 TeV. The next-to-leading order QCD corrections significantly reduce the renormalization and factorization scale dependence of the Born cross section. They slightly decrease or increase the Born cross section depending on the values of the renormalization and factorization scales

Photoproduction of the isolated photon at DESY HERA in next-to-leading order QCD

The NLO QCD calculation for the photoproduction of the isolated photon with a large pT at the HERA ep collider is presented. The single resolved photon contribution and the QCD corrections of order alpha_s to the Born term are consistently included. The NNLO contributions, the box and the double resolved photon subprocesses, are sizeable and are taken into account in addition. The importance of the isolation cut, as well as the influence of other experimental cuts on pT and final photon rapidity distributions are discussed in detail. The investigation of the renormalization scale dependence is performed in order to estimate the size of missing higher order QCD corrections. Results are compared with experimental data and with the prediction of a different NLO calculation.

O(αs) CORRECTIONS TO B→ Xse+e-DECAY IN THE 2HDM

O(αs) QCD corrections to the inclusive B→Xse+e-decay are investigated within the two-Higgs doublet extension of the standard model (2HDM). The analysis is performed in the so-called off-resonance region; the dependence of the obtained results on the choice of the renormalization scale is examined in details. It is shown that O(αs) corrections can suppress the B→Xse+e-decay width up to 1.5–3 times (depending on the choice of the dilepton invariant mass s and the low-energy scale μ). As a result, in the experimentally allowed range of the parameter space, the relations between the B→Xse+e-branching ratio and the new physics parameters are strongly affected. It is also found that though the renormalization scale dependence of the B→Xse+e-branching is significantly reduced, higher order effects in the perturbation theory can still be non-negligible.

Higgs radiation off top quarks at the Tevatron and the LHC.

Higgs bosons can be searched for in the channels pp macro/pp-->tt macro H + X at the Fermilab Tevatron and the Cern Large Hadron Collider (LHC). We have calculated the QCD corrections to these processes in the standard model at next-to-leading order. The higher-order corrections reduce the renormalization and factorization scale dependence considerably and stabilize the theoretical predictions for the cross sections. At the central scale mu = (2m(t)+M(H))/2 the properly defined K factors are slightly below unity for the Tevatron (K approximately 0.8) and slightly above unity for the LHC (K approximately 1.2).

Next-to-leading order results for tt macro h production at the Tevatron.

We compute the Omicron(alpha(3)(s)) inclusive total cross section for the process pp macr0 -->tt macro h in the standard model, at the square root of s(H) = 2 TeV. The next-to-leading order corrections drastically reduce the renormalization and factorization scale dependence of the Born cross section and slightly decrease the total cross section for renormalization and factorization scales between m(t) and 2m(t).

DIRECT CP ASYMMETRY IN INCLUSIVE RARE B-DECAYS IN 2HDM

The direct CP asymmetry in the inclusive B → Xdγ and B → Xde+e-decays is investigated in the two-Higgs doublet extension of the Standard Model (2HDM). The investigation is performed in the lowest nonvanishing order of the perturbation theory using the existing restrictions on the 2HDM parameters space. It is shown that the direct CP asymmetry in the B → Xdγ decay can deviate significantly from the Standard Model predictions. In the presence of only one source of CP violation (the CKM matrix weak phase) aCP(B → Xdγ) can have the sign opposite to that in the SM. The new source of CP violation can make |aCP(B → Xdγ)| arbitrary small (unlike the SM case) and hence unmeasurable. Quantitatively, the obtained results suffer from the uncertainty of the choice of renormalization scale. As for the B → Xde+e-rate asymmetry, its renormalization scale dependence in the lowest nonvanishing order does not allow to conclude if this quantity is efficient for testing new physics beyond the Standard Model.

Renormalon-inspired resummations for vector and scalar correlators — estimating the uncertainty in αs( M2τ) and α( M2Z)

We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients containing the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known next-to-leading order (NLO), and next-NLO (NNLO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach in which all RG-predictable ultraviolet logarithms are resummed to all-orders, removing all dependence on the renormalization scale. We can also obtain fixed-order CORGI results. Including suitable weight-functions we can numerically integrate these results for the D-function in the complex energy plane to obtain so-called “contour-improved” results for the ratio R and its tau decay analogue R τ . We use the difference between the all-orders and fixed-order (NNLO) results to estimate the uncertainty in α s ( M 2 Z ) extracted from R τ measurements, and find α s ( M 2 Z )=0.120±0.002. We also estimate the corresponding uncertainty in α( M 2 Z ) arising from hadronic corrections by considering the uncertainty in R( s), in the low-energy region, and compare with other estimates. Analogous resummations are also given for the scalar correlator. As an adjunct to these studies we show how fixed-order contour-improved results can be obtained analytically in closed form at the two-loop level in terms of the Lambert W-function and hypergeometric functions.

Renormalon resummation and exponentiation of soft and collinear gluon radiation in the thrust distribution

The thrust distribution in e+e- annihilation is calculated exploiting its exponentiation property in the two-jet region t = 1-T << 1. We present a general method (DGE) to calculate a large class of logarithmically enhanced terms, using the dispersive approach in renormalon calculus. Dressed Gluon Exponentiation is based on the fact that the exponentiation kernel is associated primarily with a single gluon emission, and therefore the exponent is naturally represented as an integral over the running coupling. Fixing the definition of Lambda is enough to guarantee consistency with the exact exponent to next-to-leading logarithmic accuracy. Renormalization scale dependence is avoided by keeping all the logs. Sub-leading logs, that are usually neglected, are factorially enhanced and are therefore important. Renormalization-group invariance as well as infrared renormalon divergence are recovered in the sum of all the logs. The logarithmically enhanced cross-section is evaluated by Borel summation. Renormalon ambiguity is then used to study power corrections in the peak region Qt \gsim Lambda, where the hierarchy between the renormalon closest to the origin (~1/Qt) and others (~1/(Qt)^n) starts to break down. The exponentiated power-corrections can be described by a shape-function, as advocated by Korchemsky and Sterman. Our calculation suggests that the even central moments of the shape-function are suppressed. Good fits are obtained yielding alpha_s^{MSbar} (M_Z) = 0.110 \pm 0.001, with a theoretical uncertainty of ~5%.

QCD factorization forB→PP

In this work, we give a detailed discussion for QCD factorization involved in the complete chirally enhanced power corrections in the heavy quark limit for B decays to two light pseudoscalar mesons, and present some detailed calculations of radiative corrections at the order of ${\ensuremath{\alpha}}_{s}.$ We point out that the infrared finiteness of the vertex corrections in the chirally enhanced power corrections requires twist-3 light-cone distribution amplitudes (LCDAs) of the light pseudoscalar symmetric. However, even in the symmetric condition, there is also a logarithmic divergence from the end points of the twist-3 LCDAs in the hard spectator scattering. We point out that the decay amplitudes of $\stackrel{\ensuremath{\rightarrow}}{B}\mathrm{PP}$ predicted by QCD factorization are really free of the renormalization scale dependence, at least at the order of ${\ensuremath{\alpha}}_{s}.$ At last, we briefly compare the QCD factorization with the generalized factorization and PQCD method.

Two-loop virtual corrections to B→ Xsℓ +ℓ − in the standard model

We calculate O( α s ) two-loop virtual corrections to the differential decay width dΓ(B→X sℓ +ℓ −)/d s ̂ , where ŝ is the invariant mass squared of the lepton pair, normalized to m b 2. We also include those contributions from gluon bremsstrahlung which are needed to cancel infrared and collinear singularities present in the virtual corrections. Our calculation is restricted to the range 0.05⩽ s ̂ ⩽0.25 where the effects from resonances are small. The new contributions drastically reduce the renormalization scale dependence of existing results for dΓ(B→X sℓ +ℓ −)/d s ̂ . For the corresponding branching ratio (restricted to the above ŝ-range) the renormalization scale uncertainty gets reduced from ∼±13% to ∼±6.5%.

Renormalizable Abelian-Projected Effective Gauge Theory Derived from Quantum Chromodynamics

We show that an effective Abelian gauge theory can be obtained as a renormalizable theory from QCD in the maximal Abelian gauge. The derivation improves in a systematic manner the previous version that was obtained by one of the authors and was referred to as the Abelian-projected effective gauge theory. This result supports the view that we can construct an effective Abelian gauge theory from QCD without losing characteristic features of the original non-Abelian gauge theory. In fact, it is shown that the effective coupling constant in the resulting renormalizable theory has a renormalization-scale dependence governed by the $\beta$-function that is exactly the same as that of the original Yang-Mills theory, irrespective of the choice of gauge fixing parameters of the maximal Abelian gauge and the parameters used for identifying the dual variables. Moreover, we evaluate the anomalous dimensions of the fields and parameters in the resultant theory. By choosing the renormalized parameters appropriately, we can switch the theory into an electric or a magnetic theory.

Renormalization and -improvement of the static axial current

A systematic treatment of O(a)-improvement in lattice theories with static quarks is presented. The Schr\"odinger functional is discussed and a renormalization condition for the static axial current in the SF-scheme is introduced. Its relation to other schemes is computed to 1-loop order and the 2-loop anomalous dimension is derived. In finite volume renormalization schemes such as the SF-scheme, the renormalization scale dependence of the renormalized quantities is described by the step scaling function which can be computed by MC- simulations. We evaluate its lattice spacing effects in perturbation theory.

Updated analysis ofɛ′/ɛin the standard model with hadronic matrix elements from the chiral quark model

We discuss the theoretical and experimental status of the CP violating ratio eps'/eps. We revise our 1997 standard-model estimate-based on hadronic matrix elements computed in the chiral quark model up to O(p^4) in the chiral expansion-by including an improved statistical analysis of the uncertainties and updated determination of the Cabibbo-Kobayashi-Maskawa elements and other short-distance parameters. Using normal distributions for the experimental input data we find Re eps'/eps = (2.2 \pm 0.8) x 10^{-3}, whereas a flat scanning gives 0.9 x 10^{-3} < Re eps'/eps < 4.8 x 10^{-3}. Both results are in agreement with the current experimental data. The key element in our estimate is, as before, the fit of the Delta I=1/2 rule, which allows us to absorb most of the theoretical uncertainties in the determination of the model-dependent parameters in the hadronic matrix elements. Our semi-phenomenological approach leads to numerical stability against variations of the renormalization scale and scheme dependence of the short- and long-distance components. The same dynamical mechanism at work in the selection rule also explains the larger value obtained for \ratio with respect to other estimates. A coherent picture of K -> pi pi decays is thus provided.

K0– [formula omitted] mixing in the 1/ Nc expansion

We present the result for the invariant B ̂ K factor of K 0– K ̄ 0 mixing in the chiral limit and to next-to-leading order in the 1/ N c expansion. We explicitly demonstrate the cancellation of the renormalization scale and scheme dependences between short- and long-distance contributions in the final expression. Numerical estimates are then given, by taking into account increasingly refined short- and long-distance constraints of the underlying QCD Green's function which governs the B ̂ K factor.

Complete renormalization group improvement — avoiding factorization and renormalization scale dependence in QCD predictions

For moments of leptoproduction structure functions we show that all dependence on the renormalization and factorization scales disappears provided that all the ultraviolet logarithms involving the physical energy scale Q are completely resummed. The approach is closely related to Grunberg's method of effective charges. A direct and simple method for extracting Λ MS from experimental data is advocated.

Photonic penguins at two loops and mt-dependence of BR[B→Xsl+l−

We calculate two-loop matching conditions for all the operators that are relevant to B→Xsl+l− decay in the Standard Model. In effect, we are able to remove the ±16% uncertainty in the decay spectrum, which was mainly due to the renormalization-scale dependence of the top-quark mass. We find 1.46×10−6 for the branching ratio integrated in the domain ml+l−2/mb2∈[0.05,0.25], for l=e or μ. There remains around 13% perturbative uncertainty in this quantity, while the non-perturbative effects are expected to be smaller.