QT resummation portal: theory overview

Theory overview

Transverse momentum resummation generalizes the conventional collinear factorization for hadronic processes to calculate both normalization and shape of particle distributions. Since its conception in the late 1970's, Q_T resummation has been successfully applied to study all-order structure of hadronic differential distributions and provide excellent predictions for a variety of experiments. Analytical resummation should not be confused with Monte-Carlo showering models, which have different objectives, strengths, and weaknesses. In fact, Q_T resummation and parton showering methods are quite complementary, as shown by the comparison of their key features.

Analytical Q_T resummation	Parton showering programs (Pythia, MC@NLO, Sherpa...)
evaluate(s) effects of multiple parton radiation in hadronic scattering
applies to a restricted class of processes and observables (e.g., lepton distributions in Drell-Yan-like processes); inclusive with respect to hadronic radiation	apply to a wide range of observables; exclusive with respect to hadronic radiation
is proved to all orders in the QCD coupling by special factorization theorems devised for each qualified observable	no factorization proofs for individual observables
streamlined computation of higher-order corrections and high-p_T contributions	beyond the leading order, radiative contributions and high-p_T tails may be difficult to implement
NLO Q_T resummation formulated in 1979-1997; modern Q_T resummation approaches NNLO accuracy	modern showering programs approach NLO accuracy
resummation of all logarithms ln Q_T²/Q²	resummation of leading logarithms ln Q_T²/Q²
nonperturbative contributions are constrained by invoking their universality in the considered class of processes	nonperturbative scattering is evaluated in one of several available models
more strict and precise; relies on first principles of perturbative QCD	more flexible; more parameters to tune to describe various hadronic scattering effects

Structure of the resummed cross sections

The online plotter of resummed cross sections conveniently illustrates the generic structure of resummed cross sections by plotting several components contributing to the typical resummed cross section.

The fixed-order cross section includes only contributions up to a certain order in the QCD coupling strength $\alpha_{s}$ . This distribution is divergent at small values of the transverse momentum $Q_{T}$ . The presence of integrable singularities in $d\sigma/dQ_{T}$ at $Q_{T}\approx0$ indicates that summation of large contributions through all orders of perturbative QCD is needed in this region.
The asymptotic cross section is the most singular part of the fixed-order cross section in the limit $Q_{T}\rightarrow0$ . The ${\cal O}(\alpha_{S})$ asymptotic cross section consists of terms behaving as $Q_{T}^{-2}\left(1\mbox{\quad or\quad}\log\left(Q_{T}^{2}/Q^{2}\right)\right)\,$ . Higher powers of large logarithms $\log\left(Q_{T}^{2}/Q^{2}\right)$ arise at subsequent orders of $\alpha_{s}$ .
The resummed cross section is a combination of the fixed-order cross section with an all-order sum of large terms
$\begin{eqnarray*} \alpha_{S}^{n}\left(\frac{1}{Q_{T}^{2}}\ln^{m}\frac{Q^{2}}{Q_{... ...}_{T})\right), & & \quad n=1,\dots\infty;\\ & & m=0,\dots,2n-1.\end{eqnarray*}$

Explicitly, the resummed cross section for the process $AB\rightarrow VX\,$ is calculated as

$\begin{displaymath} \frac{d\sigma_{AB}}{dQ^{2}dydQ_{T}^{2}}\approx\int\frac{d^{2... ...^{2}}e^{i\vec{q}_{T}\cdot\vec{b}}\widetilde{W}_{AB}(b,\dots)+Y.\end{displaymath}$

The first term on the r.h.s. (the exponential term) is a Fourier-Bessel transform of the impact parameter space form-factor $\widetilde{W}_{AB}(b,\dots)$ . In one popular realization, $\widetilde{W}_{AB}(b,\dots)$ is given by

$\begin{displaymath} \widetilde{W}_{AB}(b,\dots)=\widetilde{W}_{AB}^{pert}(b_{*},\dots)\, e^{-{\cal F}_{AB}^{NP}(b,\dots)}.\end{displaymath}$

$\widetilde{W}_{AB}^{pert}(b_{*},\dots)$ denotes the perturbatively calculable part of $\widetilde{W}_{AB}(b,\dots)$ . It depends on the perturbative Sudakov exponent $\exp\left(-{\cal S}^{pert}(b,Q)\right)$ and convolutions of parton distribution functions $f_{a/A}(x,\mu)$ with coefficient functions ${\cal C}_{ab}(x,\mu)$ :

$\begin{displaymath} \widetilde{W}_{AB}^{pert}(b,\dots)=\sum_{a,b,j}e^{-{\cal S}^... ...times f_{b/B})(x_{b},b)({\cal C}_{ja}\otimes f_{a/A})(x_{a},b);\end{displaymath}$

$\begin{eqnarray*} {\cal S}^{pert}(b,Q) & = & \int_{C_{1}^{2}/b^{2}}^{C_{2}^{2}Q^... ...t_{x}^{1}\frac{d\xi}{\xi}{\cal C}(x,b\mu)\, f(\frac{x}{\xi},\mu).\end{eqnarray*}$

The functions ${\cal S}^{pert}(b,Q)\,$ and ${\cal C}_{ab}(x,b\mu)\,$ can be calculated order-by-order in perturbative QCD.

The variable $b_{*}$ is introduced to ``freeze'' the perturbative form-factor $\widetilde{W}_{AB}^{pert}(b_{*},\dots)\,$ when b $\,$ exceeds $b_{max}\sim1\mbox{\quad GeV}^{-1}$ :

$\begin{displaymath} b_{*}\equiv\frac{b}{\sqrt{1+\left(\frac{b}{b_{max}}\right)^{2}}}.\end{displaymath}$

The region of large b is dominated by non-perturbative physics. The nonperturbative contribution is parametrized by a phenomenological function $e^{-{\cal F}_{AB}^{NP}(b,\dots)}\,$ which is found from the comparison with experimental data.

The Y-term is introduced to combine the all-order sum of large logarithmic terms with the fixed-order cross section. The Y-term is simply the difference between the fixed-order and asymptotic cross sections:

$\begin{displaymath} Y=\left(\frac{d\sigma_{AB}}{dQ^{2}dydQ_{T}^{2}}\right)_{\beg... ...eft(\frac{d\sigma_{AB}}{dQ^{2}dydQ_{T}^{2}}\right)_{asymptotic}\end{displaymath}$

At small $Q_{T}$ , the fixed-order cross section is approximated well by the asymptotic cross section, so that the Y-term vanishes, and the resummed cross section is completely determined by the exponential term. At large $Q_{T}$ , the exponential term cancels with the asymptotic cross section (up to higher-order corrections), so that the resummed cross section is determined mostly by the fixed-order cross section.

Bibliography

In addition to our publications, we list a few prominent papers on Q_T resummation in Drell-Yan-like processes. This list is absolutely not inclusive. Other important studies exist and can be found by search for citations of the listed papers in SPIRES database.

ON THE TRANSVERSE MOMENTUM DISTRIBUTION OF MASSIVE LEPTON PAIRS.
By Yuri L. Dokshitzer, Dmitri Diakonov, S.I. Troian.
Phys.Lett.B79:269-272,1978.
SMALL TRANSVERSE MOMENTUM DISTRIBUTIONS IN HARD PROCESSES.
By G. Parisi & R. Petronzio.
Nucl.Phys.B154:427,1979.
SUMMING SOFT EMISSION IN QCD.
By Jiro Kodaira & Luca Trentadue.
Phys.Lett.B112:66,1982.
BACK-TO-BACK JETS IN QCD.
By John C. Collins & Davison E. Soper.
Nucl.Phys.B193:381,1981,ERRATUM-ibid.B213: 545,1983.
BACK-TO-BACK JETS: FOURIER TRANSFORM FROM b TO k-transverse.
By John C. Collins & Davison E. Soper.
Nucl.Phys.B197:446,1982.
PARTON DISTRIBUTION AND DECAY FUNCTIONS.
By John C. Collins & Davison E. Soper.
Nucl.Phys.B194:445,1982.
VECTOR BOSON PRODUCTION AT COLLIDERS: A THEORETICAL REAPPRAISAL.
By Guido Altarelli, R.K. Ellis, M. Greco, G. Martinelli.
Nucl.Phys.B246:12,1984.
TRANSVERSE MOMENTUM DISTRIBUTION IN DRELL-YAN PAIR AND W AND Z BOSON PRODUCTION.
By John C. Collins, Davison E. Soper, George Sterman.
Nucl.Phys.B250:199,1985.
NONLEADING CORRECTIONS TO THE DRELL-YAN CROSS-SECTION AT SMALL TRANSVERSE MOMENTUM.
By C.T.H. Davies & W.James Stirling.
Nucl.Phys.B244:337,1984.
DRELL-YAN CROSS-SECTIONS AT SMALL TRANSVERSE MOMENTUM.
By C.T.H. Davies, B.R. Webber, W.James Stirling.
Nucl.Phys.B256:413,1985.
W and Z production at next-to-leading order: From large q(t) to small.
By Peter B. Arnold & Russel P. Kauffman.
Nucl.Phys.B349:381-413,1991.
Fragmentation of transversely polarized quarks probed in transverse momentum distributions.
By John C. Collins.
Nucl.Phys.B396:161-182,1993. [hep-ph/9208213]
Dispersive approach to power behaved contributions in QCD hard processes.
By Yuri L. Dokshitzer, G. Marchesini, B.R. Webber.
Nucl.Phys.B469:93-142,1996. [hep-ph/9512336]
Vector boson production in hadronic collisions.
By R.K. Ellis, D.A. Ross, Sinisa Veseli.
Nucl.Phys.B503:309-338,1997. [hep-ph/9704239]
W and Z transverse momentum distributions: Resummation in q(T) space.
By R.K. Ellis & Sinisa Veseli.
Nucl.Phys.B511:649-669,1998. [hep-ph/9706526]
Power corrections to event shapes and factorization.
By Gregory P. Korchemsky & George Sterman.
Nucl.Phys.B555:335-351,1999. [hep-ph/9902341]
Universality of nonleading logarithmic contributions in transverse momentum distributions.
By Stefano Catani, Daniel de Florian, Massimiliano Grazzini.
Nucl.Phys.B596:299-312,2001. [hep-ph/0008184]
On the resummation of subleading logarithms in the transverse momentum distribution of vector bosons produced at hadron colliders.
By Anna Kulesza & W.James Stirling.
JHEP 0001:016,2000. [hep-ph/9909271]
Role of the nonperturbative input in QCD resummed Drell-Yan Q(T) distributions.
By Jian-wei Qiu & Xiao-fei Zhang.
Phys.Rev.D63:114011,2001. [hep-ph/0012348]
Joint resummation in electroweak boson production.
By Anna Kulesza, George Sterman, Werner Vogelsang.
Phys.Rev.D66:014011,2002. [hep-ph/0202251]
Resummed event shape variables in DIS.
By Mrinal Dasgupta & Gavin P. Salam.
JHEP 0208:032,2002. [hep-ph/0208073]
Differential cross-section for Higgs boson production including all orders soft gluon resummation.
By Edmond L. Berger & Jian-wei Qiu.
Phys.Rev.D67:034026,2003. [hep-ph/0210135]
QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum.
By Xiang-dong Ji, Jian-ping Ma, Feng Yuan.
Phys.Rev.D71:034005,2005. [hep-ph/0404183]
Universality of soft and collinear factors in hard-scattering factorization.
By John C. Collins & Andreas Metz.
Phys.Rev.Lett.93:252001,2004. [hep-ph/0408249]
Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC.
By Giuseppe Bozzi, Stefano Catani, Daniel de Florian, Massimiliano Grazzini.
Nucl.Phys.B737:73-120,2006. [hep-ph/0508068]

	Q_T resummation portal at Michigan State University A collection of resources on transverse momentum resummation
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