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PQCD predictions for hadron collider physics: getting beyond the leading order

Kirill Melnikov Johns Hopkins University
QFTHEP 2011 September 25th 2011


Outline


The field of pQCD, as applied to hadron collider phenomenology, went through a remarkable transformation
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new on-shell technology for one-loop computations new phenomenological NLO QCD results for high-multiplicity processes automation of one-loop computations ­ Madgraph/Alpgen/Comphep@NLO? first NNLO results for fully differential quantities active search for a general subtraction scheme@NNLO

My goal in this talk is to describe ideas that lead to these developments and give examples of their phenomenological relevance. Please note that this is not a review talk on perturbative QCD, so that all examples are personally-biased and not inclusive


The need for higher orders


Experiments at the Tevatron and the LHC search for physics beyond the Standard Model in hard collisions, where all momentum transfers are large. Perturbative QCD is a systematic, improvable framework to describe such processes

Parton distribution functions are non-perturbative universal objects. Parton scattering crosssections are computable in perturbative QCD. Perturbative partons are evolved to hadrons using parton showers


The benefit of higher orders


Description of a particular process in higher orders of pQCD often leads to
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reduced sensitivity to unphysical renormalization/factorization scales ­ control of the normalization more realistic description of jets a possibility of ``fool-proof'' extrapolation between different kinematic regions (data driven background estimates) smaller PDF uncertainties and better compatibility between different PDF sets

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Apart from theoretical niceties, the validity of pQCD description of hard hadron collisions ­ and related benefits of going to higher orders ­ has been verified by the Tevatron and the early LHC data


Jet azimuthal correlations at the Tevatron and the LHC


Jet angular correlations allow us to trace how things work when additional jets are being created in the hard process


Z/gamma rapidity distributions


Rapidity distributions of dileptons in hadron collisions are known through NNLO. Remarkable consistency with Tevatron measurements. Input for PDF constraints.

Anastasiou, Dixon, Petriello, K.M.,


Loops

Loops and legs



Legs

Slide courtesy of Z. Kunszt


Loops and legs


For many years, progress in multi-loop computations was driven by the integration-byparts technique and the Laporta algorithm These are great tools that are applicable to single-scale (inclusive) problems such as R(s), tau-decays, QCD beta-function, g-2, quark masses from sum rules, muon decay and the Fermi constant, DGLAP evolution kernels etc They were also succesfully applied for computing two-loop scattering amplitudes for 2-> 2 scattering processes We have benefited a lot from this technology and continue to use it, but description of hadron collisions requires dealing with perturbative computations for processes with large number of external particles and large number of kinematic scales Such results should be applicable ``unintegerated'', fully differential distributions in order to be useful; certain complications and requires different approach to kinematic this leads to a somewhat










Anatomy of NLO computations


Lets recall how this is achieved in case of next-to-leading order (NLO) computations



The subtraction terms (Catani-Seymour, Frixione-Kunszt-Signer) are constructed to make the real emission matrix element squared integrable locally, provided a suitable infra-red safe definition of hadronic final state Large number of existing programs can handle real emission computations and generation of subtraction terms The problem ­ for a long time ­ was the computation of one-loop (!) matrix elements






One-loop computations


One loop computations become a problem if we try to turn a ``solution of principle'' to a ``solution of practice'' Practical one-loop computations are often performed along the following line
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each one-loop diagram is a linear combination of tensor integrals; each tensor integral can be expressed as a linear combination of scalar 1-point, 2point, 3-point and 4-point scalar integrals one-loop scalar integrals are known and have been tabulated

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Verdict : algorithm for one-loop computations exists, hence they are trivial This, of course, is almost right. The problem with this argumet is that
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number of Feynman diagrams grows factorially; number of terms produced by the tensor reduction grows very strongly; numerical instabilities (Gram determinant problem)



As the result ­ the standard procedure becomes hardly manageable if we go to higher multiplicity processes


Progress with NLO computations


Difficulties with one-loop computations lead to a very slow progress in NLO QCD computations for large number of external particles The LHC physics is high-multiplicity physics, so it is essential to go to 2->4 or even 2-> 5 processes As an example, typical searches for supersymmetry require 4 jets and misssing energy, so Z+4 jets is an irreducible background. A NLO prediction for Z+4 jets was absolutely impossible until very recently April 2001





G. Salam, talk at ICHEP 2010

In recent three to four years new technology for NLO computations appeared that allowed us to take on 2->4 and 2 5 computations


The change in the paradigm


The remarkable progress illustrated on the previous slide occurred ( at least partially) due to development of a radically new method for one-loop computations Instead of computing scattering amplitudes from Feynman diagrams, we construct them from on-shell gauge invariant tree-level scattering amplitudes The trick is a generalization of the old idea of unitarity where imaginary parts of scattering amplitudes are reconstructed from the unitairty cuts







Exploit the fact that large fraction of any one-loop computation is known

In the past few years, a procedure appeared that allows computation of the reduction coefficients directly from on-shell scattering amplitudes by-passing Feynman diagrams.


Modern unitarity techniques


Unitarity techniques in the contemporary context were introduced by Bern, Dixon and Kosower in 1990s and used for a number of high-profile computations. For a long time, this was a collection of tricks and brilliant guesswork. Solid computational method emerged in the past four years








Quadrupole cuts freeze loop momentum and give the box reduction coefficient directly; Britto, Cachazo, Feng The OPP tensor integral reduction technique; Ossola, Pittau, Papadopoulos The OPP procedure meshes well with unitarity; Ellis, Kunszt, Giele Generalized D-dimensional unitarity
Giele, Kunszt, Melnikov



From R. Britto talk, LoopFest 2008


OPP reduction


The OPP procedure is central for all existing implementations of the unitarity method. It is a novel approach to the reduction of one-loop tensor integrals to scalar integrals





OPP pointed out that computation of the reduction coefficients requires limited information about the function In fact, we need to know it only for such values of the loop momenta for which certain combinations of inverse propagators vanish (all combinations should be considered)




From OPP to generalized unitarity


The OPP procedure applied to full one-loop amplitudes leads to an unitarity-based framework for one-loop calculations
Ellis, Giele, Kunszt



The OPP procedure determines reduction coefficients from loop momenta for which combinations of inverse Feynman propagators vanish. If this occurs, some virtual particles go on their mass-shells and the one-loop amplitude factorizes into products of tree-amplitudes



Those tree amplitudes are conventional BUT, as a rule, have to be evaluated at a complex onshell momenta.


The power of unitarity: gluon amplitudes
1985

1993 2006

diagrams Giele, Zanderighi N-gluon amplitudes can be calculated for arbitrary N. Explicit numerical results available for N through 20. Factorial growth in the number of Feynman diagrams makes this computation impossible with traditional methods.


The algorithm: getting loops from trees


How to construct an algorithm that starts with tree scattering amplitudes and delivers oneloop scattering amplitudes? A unique way of writing the itegrand exists in non-abelian gauge quantum field theories
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Each color-ordered amplitude follows from a parent diagram Any diagram that contributes to a particular colorordered amplitude is obtained by pinching and pulling lines in the parent diagram Parent diagram possesses a well-defined set of propagators which is not changed by pinching and pulling


The algorithm


For numerical implementation
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specify all possible cuts that lead to non-vanishing contributions in dimensional regularization, starting with the quadruple cut loop momentum on the cut assumes complex values each cut produces a sum of products of certain number of tree amplitudes tree-amplitudes for complex on-shell momenta are computed using Berends-Giele recursion relations products of tree amplitudes provide reduction coefficients for master integrals For proper treatment of ultraviolet structure of the theory, one needs to perform this procedure in higher-dimensional (integer) space-time. For pure Yang-Mills, for example, D=5 and D=6 is sufficient to reconstruct the full one-loop scattering amplitude from on-shell unititarity cuts.

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The procedure allows us .to obtain an answer for a one-loop scattering amplitude without having to deal with Feynman diagrams AND off-shell degrees of freedom including ghosts !


The standard: MCFM
J. Campbell, R.K. Ellis


Automation and craftsmanship


It appears that new paradigm for NLO computations makes the automation of NLO computations possible.
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Madgraph to generate diagrams and OPP reduction procedure Automatic construction of FKS dipoles Parton shower (MC@NLO) is automatic

Craftsmanship still required to deal with highest multiplciity processes and processes with unusual features

MadLoop, Hirshi, et al

Bern, Dixon, Kosower, Berger, Forde, Maitre, Febres-Cordero, Gleisberg, Papadopoulos, Ossola, Pittau, Czakon, Worek, Bevilacqua, Ellis, Kunszt, Giele, Zanderighi, Melia, Rountsh, Denner, Dittmaier, Pozzorini, Kallweit


W/Z + jets @ NLO


D0 compares W+jets spectra with NLO QCD predictions Predictions for W+4jets at the LHC; transverse momenta distributions of four jets
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Blackhat + Sherpa collaboration
A computation of that complexity was unthinkable, just a few years ago


W+W-+jj @NLO


The background process to Higgs searches in weak boson fusion

Melia, K.M., Rontsch, Zanderighi


e+e- 5 jets at LEP


Outside fo the hadron collider physics context ­ production of 5 jets at LEP. Highest exclusive jet multiplicity studied at LEP. Great sensitivity to the strong coupling constant. Small hadronization effects if done properly.



Frederix, Frixione, Zandrighi, K.M.


Iintermediate summary: the NLO revolution


Last three years ­ spectacular progress in NLO QCD computations for collider physics (mostly for the LHC and the Tevatron) Refinements of traditional diagrammatic techniques and new, unitarity-based technology, allowed us to obtain NLO QCD results for processes with large multiplicity final states for a variety of processes of importance for the LHC It appears that new methods for one-loop computations are sufficiently robust to allow automation, similar to what has happened with leading order computations at the end of 1990s. First results look very encouraging






Adding one order in pQCD : NNLO


There are cases when NLO computations are insufficient because of either achievable precision (W,Z production) or because NLO effects are large (Higgs production) In the former case ­ the NNLO QCD effects influence the Higgs boson exclusion indirectly, through the W-mass in precision electroweak; in the latter case ­ directly, by affecting production cross-sections




The search for the Higgs boson: pp H WW


NNLO QCD corrections to this process, in the large top mass approximation, were computed nearly ten years ago. Both NLO and NNLO QCD effects are large. Usefulness of corrections to the total cross-section unclear
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experimental results are divided into 0-jet, 1-jet, 2-jet bins a cut on the transverse mass of the W-bosons is introduced to suppress the background spin correlations of leptons are used to discriminate against the background

NNLO computations for unintegrated kinematics of the final state are required


The search for Higgs boson : pp H WW


Such computations have been done; the results are used in the experimental analysis and allow us to draw serious conclusions ( bump significance )

Anastasiou, Dissertori, Grazzini, Stoeckli, Webber


Anatomy of NNLO


We have flexible tools to describe 2 1 processes (pp W, pp H) through NNLO in perturbative QCD. We would like to extend those results to cover 2 2 processes as well. For a variety of reasons, we may be interested in pp jj, pp tt, pp Zj, pp Hj etc. Some 2 2 processes, such as pp W+W- and pp gamma gamma do not require the full power of the NNLO technology How far are we from first physics results on 2 2 scattering @ NNLO ? For 2 2 @ NNLO we require
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2 2 scattering amplitudes for at two loops 2 3 scattering amplitudes @ one-loop, integrated over the phase-space of the unresolved parton 2 4 scattering amplitude integrated over the phase-space of two unresolved partons

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Large number of 2 2 scattering amplitudes at two-loops is available since 2001, we definitely can compute 2 3 amplitudes at NLO and clearly 2 4 scattering amplitudes for most basic processes are well-known ­ so what is the problem ?


Why NNLO is non-trivial if loop contributions are known?


The reason we have not done that are infra-red / collinear divergencies



Each of these contributions leave in a different phase-space and each is infra-red divergent. They must be combined before numerical integration is attempted, but how to do this efficiently is unknown ­ it is a matter of active research Two main lines of thought Subtractions; NLO analog: Catani-Seymour Applied at NNLO to e+e- 3j Sector decomposition NLO analog: FKS Applied to pp H, pp W,Z





Subtractions terms are (still) very difficult to construct Sector decomposition ­ difficult to keep phase-space parametrization ``local'', i.e. original applications of sector decompositions attempted to find nice global parametrization of the final state particles phase-space




Sector decomposition and FKS


The approach to NLO computations by Frixione, Kunszt and Signer (FKS) is an efficient procedure to deal with infra-red divergencies at NLO. It is based on two simple observations
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a phase-space for N+1 final states particles that contributes to a N-jet observable can be partitioned into sectors in such a way that, at any sector, one and only one identified particle can become soft or at most two identified particles can become collinear; for each such sector, a phase-space parametrization that trivializes extraction of singularities, is obvious

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A recent suggestion to apply similar considerations to NNLO computations seems very promising ! Czakon
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pre-partitioning of the phase-space choice of a suitable parametrization in each of the pre-sectors sector decomposition and the extraction of singular limits


NNLO beyond 2 2 ?


If the program of developing suitable infra-red/collinear divergences extraction algorithms succeeds, it will be very general and, in fact, applicable to higher multiplcities The bottleneck for getting physics will be the two-loop high-multiplicity diagrams (2 3 and higher) Does the OPP/generalized unitarity algorithm for tensor reduction exist in higher-loops?





Very first studies of the maximal cut for a 2->2 two-loop box diagram appeared recently
Mastrolia, Ossola, Kosower, Larsen


Conclusions


During the past ten years the field of pQCD computations for hadron collider phsyics went through a remarkable transformation A unitarity-based paradigm for NLO computations has been developed; using amplitudes (i.e. only physical degrees of freedom) and produces tangible physics results that were unthinkable before Automation of NLO computations appears to be within reach NNLO results for fully differential computations became a reality and are heavily used in the experimental studies Parton showers are combined with NLO QCD computations and with high-multipliticy leading order computations










Conclusions


Many advances were driven by simple ideas


Berend-Giele recursion Integration by parts, Laporta algorithm Asymptotic expansions Sector decomposition Britto-Cachazo-Feng-Witten OPP













Highly non-trivial ideas are being developed in the context of scattering amplitude in N=4 SYM


new symmetries recursion relations for the integrands General solutions for tree-level scattering amplitudes twistors and other unusual mathematical structures







Can any of these ideas turn into something that is useful in practice?


Conclusions


Top quark forward-backward asymmetry Feature in Wjj Demise of the CKM Proton charge radius in muonic hydrogen Muon anomalous magnetic moment