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Resummation of transverse momentum distributions in distribution space
Received: December
Resummation of transverse momentum distributions in distribution space
Markus A. Ebert
Frank J. Tackmann
D-
Hamburg
Germany
Open Access
c The Authors.
Di erential spectra in observables that resolve additional soft or collinear QCD emissions exhibit Sudakov double logarithms in the form of logarithmic plus distributions. Important examples are the total transverse momentum qT in color-singlet production, N -jettiness (with thrust or beam thrust as special cases), but also jet mass and more complicated jet substructure observables. The all-order logarithmic structure of such distributions is often fully encoded in di erential equations, so-called (renormalization group) evolution equations. We introduce a well-de ned technique of distributional scale setting, which allows one to treat logarithmic plus distributions like ordinary logarithms when solving these di erential equations. In particular, this allows one (through canonical scale to obtain the full distributional logarithmic structure from the solution's evolution kernel directly in distribution space. We apply this technique to the qT distribution, where the two-dimensional nature of convolutions leads to additional di culties (compared to one-dimensional cases like thrust), and for which the resummation in distribution (or momentum) space has been a long-standing open question. For the rst time, we show how to perform the RG evolution fully in momentum space, thereby directly resumming the logarithms [lnn(qT2 =Q2)=qT2 ]+ appearing in the physical qT distribution. The resummation accuracy is then solely determined by the perturbative expansion of the associated anomalous dimensions.
momentum; distributions; Resummation; Perturbative QCD
Contents
1 Introduction
Scale setting in distribution space
Toy example
Distributional scale setting
Integrating distributional di erential equations
Toy example in distribution space
Comparison to evolution in conjugate space
Cumulant space
Fourier space
2.6 Implementation of scale variations and pro le scales
Distributional scale setting in 2D
Overview and complications in qT resummation
Review of qT factorization
3.2 Implications of two-dimensional convolutions
Rapidity evolution in Fourier space
3.4 Illustration: e ects from energetic emissions 4 Resummation of the rapidity anomalous dimension Resummation of
in closed form
Iterative resummation of
Comparison to resummation in Fourier space
Turning o resummation using pro les
Nonperturbative modeling with the moment expansion
Resummation of soft and beam functions
Soft function
Iterative solution
Solution in closed form
Comparison to \naive" scale setting
Beam functions
Perturbativity of convolutions
The resummed transverse-momentum spectrum
6.1 Illustration at LL
Comparison to the literature
Resummation in Fourier space
Comparison to CSS formalism
Practical implementations
Early approaches for direct qT -space resummation
Coherent branching formalism
A Notation and conventions
A.1 Fourier transformations
A.2 Convolutions
A.3 Fixed-order perturbative expansions
One-dimensional plus distributions
De nition
B.2 Fourier transformation
B.3 Convolutions
C.1 De nition
C.2 Fourier transformation
C.3 Convolutions
C.4 Integral relations
C Two-dimensional plus distributions
D Fixed-order expansion of the soft function
Introduction
exhibit Sudakov double logarithms of the form
sn lnm(Q=k) with m
2n, where Q is the
section is usually contained in the regime k
Q, where the double logarithms can become
stable and reliable prediction in this regime.
(renormalization group) evolution equations of the form
dF (k; )
= F ( )F (k; ) ;
F (k; ) = F (k; 0) exp
provided that 0 is chosen of the order of k, 0
k. In this case, F (k; 0) is free of large
logarithms and can be reliably calculated in
xed-order perturbation theory, whereas all
the LO boundary term. In this case, eq. (1.2) reads
F (k; ) = (k) exp
(k) exp ln
such that it is obviously not possible to choose
0 = k to fully resum all logarithms
We then apply this technique to the resummation of the transverse momentum (~qT )
It has been a long-standing open question whether a direct resummation of the ~qT
disbrie y compare our ndings to those of ref. [24] in section 7.
In this paper, we derive the solution to perform the RG evolution entirely in
distriof momentum-space resummation.
An advantage of performing the resummation via the solution of the qT evolution
while we leave its numerical implementation to future work.
The remainder of the paper is organized as follows. In section 2, we introduce the
pieces get assembled into the
nal resummed ~qT spectrum in section 6 and provide some
and relations for plus distributions.
of the associated IR divergences between real and virtual contributions.
de ned predictions.
case relevant for qT in section 2.7.
Toy example
ln(k= ) and obeys the toy RGE
dF (k; )
eq. (2.1) yields the formal solution
F (k; ) = F (k; 0) U ( 0; ) ;
U ( 0 (...truncated)