kirill melnikov johns hopkins university march 5th...
TRANSCRIPT
QCD for hard scattering processes at hadron colliders
Kirill Melnikov
Johns Hopkins University
March 5th 2012
Outline
● Lecture I
– QCD Lagrangian, degrees of freedom
– Deep inelastic scattering; evidence for quarks
– Parton distributions function, Altarelli-Parisi evolution equation
– Final states in electron-positron annihilations, total cross-section, average number of gluons
– Concept of infra-red and collinear safety, jets and jet rates
● Lecture2
– Hadron collisions
– Parton showers
– Fixed -order computations
– Matching parton showers and leading order computations: MLM and CKKW
– New ideas in next-to-leading order computations: QCD needs no ghosts
– Next-to-next-to-leading order – the next frontier
Outline
● Clarifications
– My goal is to give you a sense of main ideas and concepts behind perturbative QCD
– I will not discuss latest phenomenological results in any details
– I will try to discuss some (recent) theoretical advances
– There is not a single topic I will cover comprehensively due to lack of time
Useful references:
1) G. Salam “Perturbative QCD for the LHC”, hep-ph 1103.1318
2) G. Sterman “ Some basic concepts in perturbative QCD” hep-ph/0809.5118
3) G. Sterman “ QCD and jets”, hep-ph/0412013
4) J. Campbell, J. Huston and W.J. Stirling, “ Hard interactions of quarks and gluons: a primer for LHC physics”, hep-ph/0611148
R.K. Ellis, J.W. Stirling and B.R. Webber, “ QCD and Collider Physics”, Cambridge University Press, 1996
QCD Lagrangian
● QCD is a a gauge field theory based on the group SU(3); it is described by the following Lagrangian
are generators of the group SU(3), they are 3x3 matrices
Each quark flavor comes in three colors and there are eight gluons
Degrees of freedom
● QCD Lagrangian contains quarks and gluons as degrees of freedom but those ``particles'' are not observed expeimentally – ``color confinement''. Lattice gauge theory provides a connection to the real world by defining QCD through the path integral that can be computed numerically
Ratio of lattice results by HPQCD collaboration to experimental results; simulation with 3 active flavors
S. Durr et al., BMW collaboration
Limitations of lattice QCD and perturbation theory for QCD
● In its current form, lattice QCD can not be applied to physics at the LHC:
– it deals only with static quantities for which Wick rotation to Eucledian space can be justified ( path integral must converge absolutely)
– it is computationally very expensive to simulate physics up to energy scales of a few TeV, much larger than the QCD scale
● Another way to approach QCD is to to treat it as weakly interacting theory, e.g. quantize it and compute Green's functions of quark and gluon fields perturbatively. This can be done – but what do results mean?
Deep inelastic scattering
● Consider scattering of an electron on the proton through a single photon exchange, into any hadronic state X. The cross-section is a function of two variables – energy and scattering angle of the final state electron.
● It is convenient to introduce a different set of variables
Deep inelastic scattering
● The amplitude for a deep inelastic scattering event is proportional to the matrix element of the electromagnetic current between the proton and the hadronic state X
● Summing over X, we find that the cross-section is expressed in terms of the hadronic tensor
SLAC-MIT experiment that observed scaling
In the limit of large Q, the structure functions depend on x only !
Bjorken scaling
Not an elastic scattering
● To appreciate how non-trivial this result is, imagine computing the hadronic tensor by restricting hadronic state X to be just a recoiled proton
Assumption of the elastic scattering or – similarly – that the process goes through any finite number of hadronic final states leads to structure functions that fall off with Q, in contrast to observations
To understand how Q-independence is possible note that in the non-relativistic Quantum Mechanics the form-factor is a Fourier transform of the charge density
Suggests that DIS occurs because protons have elementary constituents
Quark parton model
● Imagine that a proton is a ``beam'' of nearly collinear massless quarks and that deep inelastic scattering amounts to an interaction of one of the quarks from a proton with the off-shell photon
The scattering cross-section is proportional to the number of quarks with a definite momentum fraction
Partons have spin 1/2
Quark parton model
● The quark-parton model explains the observation of scaling in DIS, but ...
● If we take the equivalence of partons and quarks seriously, we can draw complicated Feynman diagrams that provide corrections to this result. We have neglected them and still obtained a sensible answer. Why did this happen?
● Bjorken scaling works very well – but it is not exact and structure functions show some (weak) dependence on Q. How can we understand it?
● We need more than just collinear quarks in the proton; quarks carry only about 70 per cent of the proton momentum
● Is it possible that those observations are related and that neglected ``strong interaction effects'' provide small Q-dependent corrections to structure functions and explain violation of the Bjorken scaling?
Asymptotic freedom
● Two related ideas justify the quark-parton model as a reasonable starting point for developing systematic description of deep inelastic scattering. One of them is the asymptotic freedom that allows us to argue that reasonable perturbative explansion is possible for short-distance observables since the coupling constant is small
Factorization
● The second idea is factorization.
● The key point is that it takes a much longer time for the hadronic interactions to change proton wave function than the time it takes for the virtual photon to collide with proton constituents
● Therefore, in each DIS event we take a ``photograph'' of the ``frozen'' proton wave function squared; we use these ``photographs'' to infer the probability distribution that a parton with particular fraction of the proton momentum participates in the collisions.
● Factorization implies that parton distributions do not depend on the probe with which proton is struck and on the particular hadron collision process
How to understand the violation of the Bjorken scaling
● Quarks are not massless; they are not even asymptotic states. Since they are ``inside'' the proton, they are off-shell by a small amount
● We can imagine a quark with the off-shellness large enough to justify perturbative considerations but, at the time, still sufficiently small so that quark can be treated as massless in the hard scattering process
● Off-shell quark radiates gluons, increasing its off-shellness and changing at the same time. The change is perturbative and leads to the DGLAP evolution equation
● Can take any value of to stop the PDF evolution – artificial boundary between the PDF and the hard scattering
Violation of the Bjorken scaling
● Although we said that quarks are constituents of protons, it is obvious that this can not be the full truth – gluon distributions must be generated dynamically. The evolution equation becomes more complicated
The DIS cross-section
● The final result for the DIS cross-section reads
Higher-order diagrams change the DIS cross-section by a small amount since the strong coupling constant is relatively small
The hard scattering cross-section depends on the parton off-shellness in a predictible way, that allows DIS cross-section to be independent of it
is often referred to as the ``factorization scale; it corresponds to the transverse momentum of the parton that participates in the hard scattering
Taking large(r) values of the factorization scale allows us to avoid having large logarithmic corrections in hard parton scattering cross-sections
Parton distribution functions
● Parton distribution functions are non-perturbative objects that can not be computed from first principles. They must be extracted from measured cross-sections, kinematic distributions etc. These data are taken at different values of Q and x. Data points are used in the fits by assuming functional form of pdfs at some initial value of Q. Parameters are determined by fits to various data points accessible through DGLAP evolution
Final states
● We will talk about final states in hadro-production reactions. For simplicity we consider electron-positron annihilation. The cross-section is a lovely function of the center-of-mass energy, exhibiting clear resonance structure, but it flattens out at higher energies.
● Remarkably, values of R can be understood in a very easy way within the quark-parton model
Final states ● For relatively low-energy collisions hadrons are distributed (almost) uniformly in their
production angles; there are no prefered directions. At high energies, this changes.
● It turns out that at high energies hadrons fly out in narrow streams that are referred to as jets. There are more two-jet events than three-jet events, more three-jet events than four-jet events etc.
Observation of two and three jet events by TASSO collaboration
Thrust measurement by L3 collaboration
Final states in perturbative QCD
● We will try to understand these features of final hadron states in perturbative QCD
● First diagram in the figure below gives the result for the cross-section which describes data well. The remaining three diagrams should give small corrections to the description of physics provided by first diagram
We will study contribution of those diagrams to the electron-positron annihilation cross-section working in the approximation where the gluon momentum is small (soft approximation)
Real emission diagrams
● The soft-gluon approximation is defined by and we are only interested in contributions to the amplitude that scale as
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Eikonal current
The amplitude for soft gluon emission factorizes into the eikonal current and the amplitude of low partonic multiplicity
Real emission diagrams
● Squaring the matrix elements and taking spin sums over all polarizations and exploiting factorization of the phase-space, we obtain
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Gluon radiation is dominated by events with small gluon energies and small relative angles between gluon momentum and quark (anti-quark) momentum
Does it hold up? How many gluons are emitted?
● Let us check if the overall picture makes sense. We argued that the total cross-section is obtained from the quark-anti-quark final state with no gluons. We can try to validate this statement using our computation for ee → qqg production.
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Probability to emitt a single gluon
Probability to produce a quark pair
This result seems to be incompatible with the observation that the total cross-section is given by the annihilation into a quark pair without any gluons.... So what is going on?
Where are the gluons?
● Gluons follow the direction of a quark and an antiquark. Large number of gluons appear because of collinear and soft emissions that do not change the direction of the energy flow
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Number of gluons is a bad observable: it changes because of soft and collinear, non-perturbative splitting. Energy flow is a good observable, it is immune to soft and collinear smplittings
Generalization #1 : good observables are infra-red and collinear safe, i.e. they are not affected by soft and collinear splittings of final state partons
Generalization #2: energy flows → notion of jets and jet algorithms
IRC safe observable: total cross-section
● As the first step in discussing IRC safe observables, we will talk about total the cross-section
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Cancellation of infra-red and collinear- enhanced corrections between real emission diagrams and virtual diagrams
IRC safe observable: jets
● We would not spoil the infra-red cancellation if we restrict the integration of the real emission contributions to regions around quark and anti-quark angular directions – Sterman-Weinberg jets
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kt-jet algortihm for electron-positron colliders
Perturbative computations with jets
● Processes with different number of jets in the final state have different ``leading order'' contributions
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Leading order contribution and the virtual correction for 2 jet production
Leading order contribution to 3 jet production and real emission correction to 2 jet
And there is still a lot of soft and collinear activity inside each of the jets that smoothly connects perturbative and non-perturbative domains
Back to the hadron collisions
● We can now put all this together, to attempt the description of hadron collisions
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proton proton
Collinear evolution of the initial parton, PDF
Hard partonic scattering
Collinear evolution of the final state parton
Hadronization
Lecture 2
● We will talk about parton showers and calculation of hard scattering cross-sections
● Useful references
– L. Dixon, “ Calculating scattering ampitudes efficiently”, hep-ph/9601359
– R.K. Ellis, Z. Kunszt, K. Melnikov, G. Zanderighi, “One-loop calculations in QFT: from Feynman diagrams to the unitarity cuts”, hep-ph/11054319
– A. Buckley et al. “ General purpose event generators for the LHC physics”, hep-ph/11012599
– B. Webber “Monte Carlo simulation of hard hadronic processes”, Annual Review of Nucl. Part. Science 36 (1986), 253
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Hadron collisions
● We will now summarize the fate of the struck parton that participates in hadronic collision :
– it emerges from the proton with hadronic-scale virtuality
– it increases its virtuality to (almost) Q by emitting gluons along the collision axis
– it undergoes hard scattering with another parton
– after emerging from hard scattering, it decreases its virtuality by emitting gluons collinear to its direction
– after virtuality decreases to hadronic-scale, it turns into a hadron without significant changes in kinematics
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Therefore we need four ingredients to describe physics at hadron colliders in all detail:
Hard scattering partonic cross-sections
Parton distribution functionsCollinear evolution, both in initial and final states
Hadronization ( I will not discuss it)
Collinear evolution and parton shower
● We have seen that the dominant effect of gluon radiation contains double logarithmic enhacement from emissions of soft and collinear gluons. It is instructive to re-visit how this comes about paying attention to issues of gauge invariance
● The eikonal current is transverse , we can compute it in any gauge we want. In Feynman (unphysical) gauge, the dominant contribution comes from the interference term between gluon emissions by a quark and an anti-quark.
● In physical gauges, enhanced emissions by quark and anti-quark are independent
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The double logarithmic enhancement occurs if the softest gluon is emitted along the direction of the harder particle. The softest gluon must be emitted from one of the external lines. All external lines emit independently.
Collinear evolution and parton shower
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We seem to find that soft/collinear real emission corrections change the production cross-section by huge amount. But this is clearly wrong, as we saw even in case of quark pair production where corrections to total cross-section did not contain large logarithms due to cancellation between real and virtual correciotns
Collinear evolution and parton shower
● It is easiest to implement this cancellation by writing
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This equation offers an opportunity to describe complicated partonic final states with arbitrary multiplicity and implement real-virtual cancellation on the fly
The Sudakov form factor
Collinear evolution and parton shower
● To turn previous formula into a useful tool to generate correct probability distribution of histories of parton evolutions, we need to change variables and make all histories equally probable
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Collinear evolution and parton shower
● We can use this result to compute unit probability as the sum over ``events'' with different probabilities/histories
● To understand how this is done note that, given a set , we can reconstruct the kinematics of n-gluon events
● Also, one can verify that the probability to have a particular values for does not depend on values of and variables after a particular emission took place..
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Parton showers
● This procedure, with small but important refinements, is in the heart of the so-called parton shower event generators (PYTHIA, HERWIG, SHERPA, etc.).
● Refinements take into account
– possibilities of branchings of partons of different type into different final state
– extension of soft/collinear approximation to collinear approximation
– inclusion of running coupling constant
● As the result, the Sudakov form factor changes, but the general strategy remains
● The inclusion of soft radiation is tricky since soft large angle emissions by different particles interfere. It is a wave effect that can not be dealt with in a simple picture, where emissions occur independently.
● The solution employs ideas of color coherence and angular ordering. It amounts to a particular choice of variables (emission angles) in which to develop the parton shower
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Parton showers● Parton showers are used everywhere. Among other things, they give an easy way to
describe radiation of arbitrary multiplicty, at very little cost. How well do they do depends on the observable at hand – in general, they work well if observable is dominated by collinear radiation
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Parton shower does not do too well for high (hard) jet multiplicity: hard jets are not produced in the soft/collinear regime
0 5 10 15 20 GeV
Hard scattering cross-sections● Partonic cross-sections are computed as power-series in the strong coupling constant
● We want to know hard scattering cross-sections because they define the most basic properties of hadronic events – energy flow – in the bulk of the phase-space. There are many ways, all based on some approximations to perturbative QCD, that are used to determine hard scattering cross-sections
– fixed order computations (leading order, next-to-leading order, next-to-next-to-..)
– resummations
– parton showers in the perturbative domain
● The simplest approach is based on employing leading order matrix elements
– ``exact'' matrix elements - all diagrams → more physics (interference)
– spin correlations - estimates of uncertainties
● But : fixed order computations can be prohibitively hard due to the factorial growth of Feynman diagrams. Even a leading order for gluon scattering, we see explosive growth
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Hard scattering cross-sections● Leading order computations are often unreliable for two different reasons
– rates for processes with n-jets depend on . . This leads to strong dependence on the renormalization and factorization scales. Renormalization scale dependence must disappear if ``exact'' partonic cross-sections are used
– rates for processes with large scale hierarchy are described incorrectly by leading order computations – they do not include Sudakov suppression that must be present
● To address those shortcomings we can
– extend fixed-order computations to higher orders in the strong coupling constant
– combine fixed-order computations with parton showers since parton showers do work well when hierarchy of energy scales is large
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Putting together parton showers and fixed orders: MLM procedure
● Parton shower programs do not describe hard radiation correctly; fixed-order matrix element corrections do not describe radiation in the hierarchical situations well. Why not to put those techniques together?
● Suppose we want to have a sample of events that can be used to analyze Z+2jets. Naively, we can compute leading order prediction for Z, Z+1jet and Z+2 jets and shower them independently.
● But – since parton shower can produce relatively hard gluons – there is a problem of counting same contributions twice (once in the shower, once in the matrix element computation). MLM procedure allows to remove this double counting by requiring that each of the hard jets reconstructed after Z+n jet matrix elements are showered, matches the direction of one of the hard parton.
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Because of rejecting too hard a radiation in the shower, MLM procedure – effectively – introduces Sudakov suppression of exclusive configurations with too distinct kinematic scales
MLM/CKKW procedures
● MLM/CKKW procedure works well. But since it is still a leading order procedure, the overal normalization is uncertain. To settle this, requires higher-order computations.
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No MCFM result for W+3jets at that time
because it used to be too complicated to obtain
Higher-order computations
● Higher-order computations have been a subject of interest since very early development of Quantum Field Theory. They have important impact on our understanding of particle physics
– Schwinger's explanation of the electron anomalous magnetic moment
– prediction of the top quark mass from B-mixing and electroweak corrections at Z-pole
– constraints on the Higgs mass from precision electroweak physics
– constraints on exotica from rare decays of B-mesons
– consistency of the CKM picture
– value of the W-mass
– proof that particle physics is well-described by the gauge SU(2)xU(1) theory
In hadron collider physics, perturbative computations have been extended to one-loop processes with up to five particles in the final state and to two-loop processes with just one particle in the final state
Anatomy of NLO computations● To perform next-to-leading order (NLO) computation for a given process, one needs to
consider two sources of radiative corrections – virtual (one-loop) and real (additional
parton in the final state)
● The subtraction terms are constructed to make the real emission matrix element squared integrable locally, provided a suitable infra-red safe definition of the final state. The subtraction terms are known in full generality and can written down for an arbitrary process
● The problem that was holding NLO computations back for a long time was the computation of one-loop (!) matrix elements for high multiplicity processes
One-loop computations can't be so difficult !
● Traditionally, one-loop computations employ Passarino-Veltman reduction. This is general, algorithmic solution based on the following observations
– 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, 2-point, 3-point and 4-point scalar integrals
– one-loop scalar integrals are universal and can be tabulated
● Verdict : algorithm for one-loop computations exists, hence they are trivial
● This, of course, is almost right. The problem with this argumet is that
– number of Feynman diagrams grows a factorial of the number of external particles
– 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
● These problems manifested itself in the following fact: just three years ago, no NLO QCD computation for any 2 → 4 process at a hadron collider existed. On the other hand, 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
In recent three to four years new technology for NLO computations appeared that allowed us to take on 2->4 and 2 → 5 computations
G. Salam, talk at ICHEP 2010
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 unitarity 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.
Traditional unitarity in perturbative computations
● Recall how Dirac or Pauli form factors of a massive lepton can be computed through one-loop in QED
● Dispersion relation for the Dirac form factor reads
● Two features of this formula are relevant for what follows:
– the integrand is computed from the product of on-shell four-dimensional tree scattering amplitudes;
– subtraction term (the rational part) requires additional input.
Modern unitarity techniques
● Unitarity techniques in the contemporary context were introduced by Bern, Dixon and Kosower in 1990s. Solid computational method – Ossola-Pittau-Papadopoulos reduction algorithm and generalized D-dimensional unitarity – emerged in the past four years
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 (go one shell). Of course, all possible combinations of inverse propagators must 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
● 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 on-shell momenta.
The power of unitarity: gluon amplitudes
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.
1993 20061985
Giele, Zanderighi
diagrams
The algorithm: getting loops from trees
● How to construct an algorithm that starts with tree scattering amplitudes and delivers one-loop scattering amplitudes?
● A unique way of writing the itegrand exists in non-abelian gauge quantum field theories .
Each color-ordered amplitude follows from a parent diagram
Any diagram that contributes to a particular color-ordered 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: numerical implementation
● For numerical implementation
– 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 !
W/Z + jets @ NLO
● D0 compares W+jets spectra with NLO QCD predictions due to Blackhat/Sherpa and MCFM/Rocket
● 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/Z+jets at NLO : choosing the scales
● When NLO computations are available, it is good to use them. But can we learn from the existing computations if leading order results can be improved in an approximate manner to catch the main features of NLO?
● The important issue is the question of renormalization and factorization scales; choosing them right is always difficult . A few years ago, Bauer and Lange showed that such choices can have important consequences – when scales are properly chosen, shapes of leading and next-to-leading order kinematic distributions match quite well
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Bauer, Lange, 14 TeV LHC, W+2jets
Learning from the parton shower
● The Bauer-Lange analysis works well because it respects a well-known feature of QCD partons branchings
● The CKKW/MLM procedure respects this choice and, in fact, does more careful scale adjustment. The scales are chosen on an event-by-event basis by identifying most probable ``history'' of an event
● iteratively cluster particles that are closest according to some measure (usually, algorithm is used).
● for each node, choose the relative momentum of the daughters as the scale for the strong coupling constant – this is the parton shower choice.
Scale setting and W+3 jets at NLO● CKKW/MLM procedure does a very good jobs in describing NLO shapes
S. Hoche, J. Huston, D. Maitre, J. Winter, G. ZanderighiBlackhat/Rocket/Sherpa comparison
NLO → NNLO
● In the past decade, we became quite good in dealing with NLO QCD computations . What about computations of yet one order higher fo hadron collider physics?
● Such computations are not needed for every process, of course, but they are important in certain cases ! The case in point are searches for the Higgs boson, both direct and indirect
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
– experimental searches 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, including its interference with the signal
– spin correlations of leptons are used to discriminate against the background
NNLO computations for unintegrated kinematics of the final state are required – similarly to the NLO computations
Indirect constraints on the Higgs boson
● Indirect constraints on the Higgs boson require precise knowledge of the top mass and the W-boson mass
● Hadron colliders are getting to the point where both of these measurements are claimed to give fantastic precision
– the top quark mass – better than 1 GeV (about 0.5%)
– The W-boson mass – better than 15 MeV (about 0.02%).
● A lot of this accuracy is coming from fitting kinematic edges of distributions where no radiative corrections are possible at the first approximation
● But, of course edges aren't really sharp and radiative corrections – including the NNLO ones – are highly desirable
Processes for which there is some reason to study them at NNLO include top pair production, 2jet production, H+jet, V+jet, top decay . For all these processes we require flexible implementation to allow arbitrary cuts
Anatomy of NNLO computations
● For 2 → 2 @ NNLO we need to put together
– 2 → 2 scattering amplitudes for at two loops
– 2 → 3 scattering amplitudes at one-loop
– 2 → 4 scattering amplitude at tree level
All of these ingredients – including the two-loop matrix element – are available. Yet, it is non-trivial to put them together. The reason is the complicated structure of divergences. These divergences must cancel at the end – due to infra-red safe nature of jet observables and the Kinoshita-Lee-Nauenberg theorem but it is difficult to achieve this explicitly.
Anatomy of NNLO
● To understand the challenge of NNLO, lets recall the divergence structure of the three contributions
● Phase-space integration is finite for two-loop graphs, produces 1/ep^2 for real-virtual graphs and produces 1/ep^4 for real-real graphs !
Therefore, the phase-space integration must be done to produce divergences yet – at the same time – it can not be done because we want to keep kinematics of final state arbitrary. This is a problem that we routinely face with those computations
How we deal with this in practice
● In practice, there are two approaches
– ``subtract and add'' such that the difference is integrable numerically over full n+2 phase-space and, at the same time, the subtraction term can be analytically integrated over the unresolved phase-space
– Phase-space partitioning / sector decomposition
Known IR/collinear limits of amplitudes
An example: Z → qqg
● Here is what one can do using an oversimplified example
Non-singular along quark direction
Non-singular along anti-quark direction
We choose the phase-space parametrization different in the two terms, simplifying extraction of singularities at the singular direction
Finite quantity in the singular limits
Conclusions
● During the past ten years the field of pQCD – and its applications to hadron collider physics – went through a remarkable development
● Powerful theoretical tools appeared that improve our ability to describe all stages of hadron collisions from parton distributions in the proton, to hard parton scattering cross-sections, to evolution of partons to final state hadrons
● Using these tools, we establish general consistency of perturbative QCD predictions with the Tevatron and the LHC data, with a healthy fraction of less-than-three sigma deviations
● In the future, theoretical progress in the field will come from
– re-summations for exclusive realistic (jetty) observables
– CKKW@NLO
– general scheme for NNLO and its verification on multi-particle processes
– automation of NLO (Madgraph@NLO, Alpgen@NLO etc.)