Toward massless and massive event shapes in the large-β0 limit

Abstract

We present results for SCET and bHQET matching coefficients and jet functions in the large-β0 limit. Our computations exactly predict all terms of the form $$ {\alpha}_s^{n+1}{n}_f^n $$ α s n + 1 n f n for any n ≥ 0, and we find full agreement with the coefficients computed in the full theory up to $$ \mathcal{O}\left({\alpha}_s^4\right) $$ O α s 4 . We obtain all-order closed expressions for the cusp and non-cusp anomalous dimensions (which turn out to be unambiguous) as well as matrix elements (with ambiguities) in this limit, which can be easily expanded to arbitrarily high powers of αs using recursive algorithms to obtain the corresponding fixed-order coefficients. Examining the poles laying on the positive real axis of the Borel-transform variable u we quantify the perturbative convergence of a series and estimate the size of non-perturbative corrections. We find a so far unknown u = 1/2 renormalon in the bHQET hard factor Hm that affects the normalization of the peak differential cross section for boosted top quark pair production. For ambiguous series the so-called Borel sum is defined with the principal value prescription. Furthermore, one can assign an ambiguity based on the arbitrariness of avoiding the poles by contour deformation into the positive or negative imaginary half-plane. Finally, we compute the relation between the pole mass and four low-scale short distance masses in the large-β0 approximation (MSR, RS and two versions of the jet mass), work out their μ- and R-evolution in this limit, and study how their implementation improves the convergence of the position-space bHQET jet function, whose three-loop coefficient in full QCD is numerically estimated.

Small-momentum expansion of heavy-quark correlators in the large-$β_0$ limit and $α_s$ extractions

“…(6.5) of Ref. [33]. Finally, the expanded exponential is easily combined with the (already expanded) accompanying finite polynomials into a single expansion using…”

“…[32], which was employed in the original calculation of the small-momentum expansion of the vector correlator in this limit [24]. (This formalism was recently generalised to the case of quantities with cusp anomalous dimension in [33]. )…”

New analyses of event shape observables in electron-positron annihilation and the determination of αs running behavior in perturbative domain

“…It is noted that many studies on renormalons in softcollinear effective theory (SCET) are given in refs. [43][44][45][46][47]. Since the PMC scale Q is independent of the choice of renormalization scale µ r and the conformal coefficient B(y, µ r ) con is also renormalization scale-independent, the PMC prediction in eq.…”

Linear power corrections to e+e– shape variables in the three-jet region

“…In order for it to vanish also in the collinear limit, the function h V should be bounded for large η and arbitrary ϕ. In fact, since the rapidity is limited by a logarithm of Q/l t , an exponential behaviour in η may ultimately lead to powers of Q/l t (where Q is the hard scale in the process) thus canceling the l t 12 The definition of linearity implies V ({p}, l, l) = V ({p}) + V µ 2 ({p}, l) lµ + V µ 3 ({p}, l) lµ. 13 For ease of notation, we omit to indicate these higher order terms in the following.…”

“…For typical e + e − collider energies these corrections are expected to be of the order of few percent; therefore, they must be included in the calculation of EESV. Power corrections can be estimated either using Monte Carlo event generators [2][3][4], or with the help of analytic models [5][6][7][8][9][10][11][12]. The Monte Carlo approach is certainly the most practical, but it is also very difficult to justify it theoretically in a convincing way.…”