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New Tools for Forecasting
Old Physics at the LHC
Lance Dixon (CERN & SLAC)for the BlackHat Collaboration:
C. Berger, Z. Bern, LD, F. Febres Cordero, D. Forde,
T. Gleisberg, H. Ita, D. Kosower, D. Maitre
QFT2011, IISER, Pune
26 February 2011
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 2
The Large Hadron Collider
• Proton-proton collisions at 7 14 TeV center-of-mass energy,
3.5 7 times greater than previous (Tevatron)
• Luminosity (collision rate) 10—100 times greater
• New window into physics at shortest distances – just opened!
ATLAS
CMS
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 3
New Physics around the Corner
• Many theories predict a host of new massive particles in this mass
range, including a dark matter candidate
• supersymmetry
• new dimensions of space-time
• new forces
• etc.
• How to distinguish new physics from old (Standard Model)?
• From other types of new physics?
Expect new physics at the 100 GeV – 1 TeV mass scale, associated
with electroweak symmetry breaking. At least, a Higgs boson (or similar)
• Most new massive particles decay rapidly to old, ~massless particles: quarks, gluons, charged leptons, neutrinos, photons
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 4
Signals vs. Backgrounds
electron-positron colliders
– small backgrounds
vs.
hadron colliders
– large backgrounds
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 5
LHC Data Dominated by Jets
new physics L
HC
@ 7
Te
V
• Every process shown
also comes with
one more jet at
~ 1/5 the rate
• Should understand
Standard Model
production of
X + 1,2,3,… jetswhere
X = W, Z, tt,
WW, H, …
Jets come from
quarks and gluons.
• q,g from decay of
new particles?
• Or from old QCD?
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 6
Asymptotic Freedom
• Quantum fluctuations of massless virtual particles polarize vacuum
• Gluon self-interactions anti-screen color charge (gs smaller at short
distances) make QCD more calculable at high energies
Gross, Wilczek, Politzer (1973)
For Nc=3, Nf =5 or 6, gluons win
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 7
Asymptotic Freedom (cont.)
Running of as is only logarithmic,
slow at short distances (large Q or m).
confiningcalculable
Bethke
Hatekayama
jet
cone
Short-distance cross section
predictable using perturbative QCD
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 8
QCD Factorization & Parton Model
Asymptotic freedom: At short distances, quarks and gluons
(partons) in proton are almost free, and are sampled ―one at a time‖
infrared safe final state
Box separates ―femto-universe‖
from long-distance effects
like parton distributions.
size = factorization scale mF(―arbitrary‖)
Parton distribution
functions
(from experiment)
Renorm. scale
(―arbitrary‖)
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 9
Problem: Leading-order (LO) predictions only qualitative
due to poor convergence
of expansion in
(setting )
Short-Distance Cross Section
in Perturbation Theory
Example: Z production at Tevatron
- Distribution in rapidity Y
LO NLO NNLO
~50% corrections, LO NLO ADMP (2004)
(2007)
by NNLO, a precision observable
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 10
Uncertainty brought under much better control with NLO
corrections: ~ 50% or more ~ 15-20%
NLO really required for quantitative control of multi-jet
final states
LO uncertainty increases with njets
LO NLO NNLO
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 11
Example: Supersymmetry
• Symmetry between fermions (matter) and bosons (forces)
• For every elementary particle we have already seen, another one
should show up soon!
• Solution to the hierarchy problem:
fermion + boson corrections to Higgs mass cancel
• Lightest supersymmetric particle can be dark matter
spin 1/2
spin 0 spin 1/2
spin 0
spin 1
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 12
Backgrounds to Supersymmetry at LHC
• Decay from gluino to neutralino
(dark matter, escapes detector)
• 2 gluinos in event
Signal: missing energy (MET) + 4 jets
• SM background: Z + 4 jets,
Z neutrinos
Current state of art for
Z + 4 jets based on
LO approximation
normalization still
quite uncertain
cc
nn
n nLO
• Motivates goal of n nNLO
Recent SUSY Exclusion
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 13
CMS, 1101.1628
CMSSM
aT = MET + jets shape variable
• LHC off to an extremely promising start!
• As data increases rapidly this year, better SM theory can help
Reducing Background Systematics
Improves SUSY Search Sensitivity
L. Dixon New Tools for Forecasting Old Physics CERN 20 Jan. 2011 14
Conley, Gainer, Hewett,
Le, Rizzo, 1009.2539
70,000
SUSY
models
MET + 4 jet
search,
ATLAS-style
analysis
masked by
background
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 15
• NLO corrections require one-loop amplitudes, as well as
tree-level amplitudes with one additional parton.
• Both terms are infrared divergent; use dimensional
regularization with
• After adding terms, renormalizing q(x), all poles cancel.
• One-loop amplitudes were the bottleneck, for a long time
• Simplest example – Z production:
Need for Loop Amplitudes
tree + 1 parton1 loop
16
Loops get difficult quickly!
For W + n jets (maximum number of external gluons only)
# of jets # 1-loop Feynman diagrams
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 17
A Better Way to Compute?
• Backgrounds (and many signals) require detailed
understanding of scattering amplitudes for
many ultra-relativistic (―massless‖) particles
– especially quarks and gluons of QCD
• However, Feynman diagrams, while very general and powerful, are not optimized for these processes
• There are more efficient methods for multi-gluon + quark processes!
• Feynman told
us how to do this
– in principle
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 18
The Analytic S-MatrixBootstrap program for strong interactions: Reconstruct scattering
amplitudes directly from analytic properties (on-shell information):
Chew, Mandelstam;
Eden, Landshoff,
Olive, Polkinghorne;
Veneziano;
Virasoro, Shapiro;
… (1960s)
Analyticity fell out of favor in 1970s with the rise of QCD & Feynman rules
Now resurrected for computing amplitudes in perturbative QCD
– as alternative to Feynman diagrams!
Perturbative information now assists analyticity. “QFT without the details”
• Poles
• Branch cuts
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 19
Helicity Formalism Exposes
Tree-Level Simplicity in QCD
Many [color-ordered] helicity amplitudes either vanish or are very short
Parke-Taylor formula (1986)
Analyticity
makes it possible
to recycle this
simplicity into
loop amplitudes
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 20
Basic variables: two-component
spinors & spinor products
If momenta ki are real, they are complex square roots of sij:
Lorentz products:
Which always obey:
Better to use
spinor products:
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Special Complex Momenta
real (singular)
complex (nonsingular)
• Make sense of most basic process with all 3 particles massless
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For Efficient Computation
Reduce the number of ―diagrams‖
Reuse building blocks over & over
Recycle lower-point (1-loop) & lower-loop (tree)
on-shell amplitudes
Recurse
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Amplitudes Are ―Plastic‖
They fall apart – factorize – into simpler ones in special limits
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Explore Limits in Complex PlaneBritto, Cachazo, Feng, Witten, hep-th/0501052
Inject complex momentum at leg 1, remove it at leg n.
special limits poles in z
Cauchy:
residue at zk = [kth factorization limit] =
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BCFW (On-shell) Recursion Relations
Ak+1 and An-k+1 are on-shell tree amplitudes with fewer legs,
and with momenta shifted by a complex amount
Britto, Cachazo, Feng, hep-th/0412308
An
Ak+1
An-k+1
Trees recycled into trees
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All Gluon Tree Amplitudes Built From:
(In contrast to Feynman vertices, it is on-shell, gauge invariant.)
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One-loop Plasticity = Generalized Unitarity
Ordinary unitarity:put 2 particles on shell
Generalized unitarity:put 3 or 4 particles on shell
trees get simplerTrees recycled into loops!
cuts require complex loop momenta
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One-Loop Amplitude Decomposition
rational part
When all external momenta are in D=4, loop momenta in D=4-2e(dimensional regularization), one can write: BDDK (1994)
known scalar one-loop integrals,
same for all amplitudes
coefficients are all rational functions – determine algebraiclally
from products of trees using (generalized) unitarity
Missing from the old, nonpertubative analytic S-matrix
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Generalized Unitarity for Box Coefficients di
Britto, Cachazo, Feng, hep-th/0412308
Number of dimensions = 4 = number of constraints (li2 = mi
2)
discrete solutions (2, labeled by ±)
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Each box coefficient uniquely
isolated by a ―quadruple cut‖
given simply by a product of 4
tree amplitudes
Britto, Cachazo, Feng,
hep-th/0412103
bubble coefficients come from ordinary
double cuts, after removing contributions of
boxes and triangles
triangle coefficients come from triple cuts,
product of 3 tree amplitudes, but these are also
―contaminated‖ by boxes
Ossola, Papadopolous, Pittau, hep-ph/0609007;
Mastrolia, hep-th/0611091; Forde, 0704.1835;
Ellis, Giele, Kunszt, 0708.2398; Berger et al., 0803.4180;…
Unitarity method
– numerical implementation
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Triangle coefficients
Solves
for suitable definitions of
Box-subtracted
triple cut has poles
only at t = 0, ∞
Triangle coefficient c0
plus all other coefficients cj
obtained by discrete Fourier
projection, sampling at
(2p+1)th roots of unity
Forde, 0704.1835; BH, 0803.4180
Triple cut solution depends on one complex parameter, t
Bubble similar
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Bottom Line:
Trees recycled into loops!
Similar methods work for multiple loops
– especially in theories with lots of supersymmetry
like N=4 super-Yang-Mills and N=8 supergravity
Rocket: Giele, Zanderighi, 0805.2152
Ellis, Giele, Kunszt, Melnikov, Zanderighi, 0810.2762
NLO W + 3 jets (large Nc), W+W+ + 2 jets EMZ,
0901.4101, 0906.1445; Melia, Melnikov, Rontsch, Zanderighi,1007.5313
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CutTools: Ossola, Papadopolous, Pittau, 0711.3596
NLO WWW, WWZ, ... Binoth+OPP, 0804.0350
NLO ttbb, tt + 2 jets Bevilacqua, Czakon, Papadopoulos,
Pittau, Worek, 0907.4723; 1002.4009; now going into MadGraph
Blackhat: Berger, Bern, LD, Febres Cordero, Forde, H. Ita, D. Kosower, D. Maître;
T. Gleisberg, 0803.4180, 0808.0941, 0907.1984, 0912.4927, 1004.1659, 1009.2338
+ Sherpa NLO production of W,Z + 3,4 jets
Method for
rational part:
specialized
Feynman
rules
_ _ _
SAMURAI: Mastrolia, Ossola, Reiter, Tramontano, 1006.0710
D-dim’l
unitarity
Automated On-Shell Methods at One-loop
NGluon: Badger, Biedermann, Uwer, 1011.2900
D-dim’l
unitarity
+ on-shell
recursion
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 34
Besides virtual corrections,
also need real emission
• General subtraction methods for integrating real-emission contributions developed in mid-1990s
Frixione, Kunszt, Signer, hep-ph/9512328;
Catani, Seymour, hep-ph/9602277, hep-ph/9605323
• Recently automated by several groups
Gleisberg, Krauss, 0709.2881;
Seymour, Tevlin, 0803.2231;
Hasegawa, Moch, Uwer, 0807.3701;
Frederix, Gehrmann, Greiner, 0808.2128;
Czakon, Papadopoulos, Worek, 0905.0883;
Frederix, Frixione, Maltoni, Stelzer, 0908.4272
Infrared
singularities
cancel
As a result…
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There has been a dramatic increase
recently in the rate at which NLO
predictions for new processes are
being made!
Les Houches Experimenters’ Wish List
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Feynman
diagram
methods
now joined
by
on-shell
methods
based on
analyticity
(unitarity)
table courtesy of
C. Berger
BCDEGMRSW; Campbell, Ellis, Williams
Berger,
Melia, Melnikov, Rontsch, Zanderighi
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W + 3 jets at Tevatron
Leading-color adjustment procedure Exact treatment of color
Rocket
Ellis, Melnikov,
Zanderighi,
0906.1445
Berger et al.,
0907.1984
• Much smaller uncertainties than at LO.
• Agrees well with data; more data available ~now from Tevatron and LHC
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Total Transverse Energy HT at LHC
often used in supersymmetry searches
0907.1984
flat LO/NLO ratio
due to good
choice of
scale m = HT
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Leptonic Variables in W + 3 jets at LHC
rapidity
distributions
remember
u(x)/d(x) large
as x 1
W+W- transverse
ratios trace a
remarkably large
left-handed
W polarization
– may be useful
to separate it from
top, new physics
0907.1984
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• Strong left-handed polarization for W+ and W-
• Increases slowly with W boson pT and with number of jets, due to increasing parton x
• Stable against QCD corrections
BlackHat+Sherpa
0912.4927
W polarization fractions at moderate pT
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NLO pp W + 4 jets now availableBerger et al., 1009.2338
First hadron collider process known at NLO with 5 objects in final state.
Also important SUSY background.
NLO vs. LHC W + jets data (ATLAS)
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1012.5382
• Already compared
to NLO for up to 2 jets.
• In progress:
comparison to NLO
for up to 4 jets, and
for ~ 30pb-1
One indicator of NLO progress
pp W + 0 jet 1978 Altarelli, Ellis, Martinelli
pp W + 1 jet 1989 Arnold, Ellis, Reno
pp W + 2 jets 2002 Campbell, Ellis
pp W + 3 jets 2009 BH+Sherpa
Ellis, Melnikov, Zanderighi
pp W + 4 jets 2010 BH+Sherpa
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Conclusions
• New and efficient computational approaches to one-loop QCD amplitudes now used to forecast important Standard Model backgrounds at the LHC– exploit analyticity/unitarity: build loop amplitudes out of trees
– implemented numerically in several programs: BlackHat, CutTools, Rocket, Samurai, …
• Long and growing list of complex processes computed at NLO with these techniques:
• VVV (V=W or Z)
• ttbb, ttj, ttjj
• W+W+jj
• Wjjj, Zjjj, Wjjjj
• Success here an essential ingredient for optimal exploitation of LHC data!
Extra slides
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 45
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 46
On-Shell Recursion at One LoopBern, LD, Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005;
Berger, et al., hep-ph/0604195, hep-ph/0607014, 0803.4180
• New features compared with tree case,
especially branch cuts
• Determine cut terms efficiently using
(generalized) unitarity
• Same techniques work for one-loop QCD amplitudes
Trees recycled into loops!
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 47
Generic analytic properties of shifted 1-loop amplitude,
Loop amplitudes with cuts
Cuts and poles in z-plane:
But if we know the cuts (via unitarity in D=4),
we can subtract them:
full amplitude cut-containing partrational part
Shifted rational function
has no cuts, but has spurious poles in z
because of Cn:
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 48
No cuts in D=4 – can’t get from D=4 unitarity
However, can get using D=4-2e unitarity:
Bern, Morgan (1996); Bern, LD, Kosower (1996);
Brandhuber, McNamara, Spence, Travaglini hep-th/0506068;
Anastasiou et al., hep-th/0609191, hep-th/0612277;
Britto, Feng, hep-ph/0612089, 0711.4284;
Giele, Kunszt, Melnikov, 0801.2237;
Britto, Feng, Mastrolia, 0803.1989; Britto, Feng, Yang, 0803.3147;
Ossola, Papadopolous, Pittau, 0802.1876;
Mastrolia, Ossola, Papadopolous, Pittau, 0803.3964;
Giele, Kunszt, Melnikov (2008); Giele, Zanderighi, 0805.2152;
Ellis, Giele, Kunszt, Melnikov, 0806.3467;
Feng, Yang, 0806.4106; Badger, 0806.4600;
Ellis, Giele, Kunszt, Melnikov, Zanderighi, 0810.2762
Rational function R
49
Recent analytic application: One-loop amplitudes
for a Higgs boson + 4 partons
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011
Badger, Glover, Risager, 0704.3914
Glover, Mastrolia, Williams, 0804.4149
Badger, Glover, Mastrolia, Williams, 0909.4475
Badger, Glover, hep-ph/0607139
LD, Sofianatos, 0906.0008
Badger, Campbell, Ellis, Williams, 0910.4481
by parity
H = f + f†Unitarity for cut parts, on-shell
recursion for rational parts (mostly)
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Color organization
• Divide into leading-color terms, such as:
and subleading-color terms, such as:
The latter include many more terms, and are much more time-consuming for
computer to evaluate. But they are much smaller (~ 1/30 of total cross
section) so evaluate them much less often (or drop them, sometimes)
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Like (W,Z) + jets, top quarks + jets
also a very important background
• Not only inclusive t t production, where recent results are
approaching full NNLO accuracyMoch, Uwer, 0804.1476; Ahrens et al., 1003.5827
• But also in association with jets, which can boost the t t
system, increasing missing ET, or provide jets to pass
various signal cuts. Cross sections large. • NLO tt + 1 jet: Dittmaier, Uwer, Weinzierl, hep-ph/0703120,…
• + top decays: Melnikov, Schulze, 1004.3284
• NLO tt + bb: Bredenstein, Denner, Dittmaier, Pozzorini, 0905.0110,
1001.4006; Bevilacqua, Czakon, Papadopoulos, Pittau, Worek, 0907.4723
• NLO tt + 2 jets: Bevilacqua, Czakon, Papadopoulos, Worek, 1002.4009
_
_
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NLO at LHC
Background to t t + Higgs, H bb at LHC (for lt)_
Bredenstein et al.,
0807.1248, 0905.0110First done using Feynman diagrams
Recomputed using unitarity (CutTools) Bevilacqua et al., 0907.4723
much improved
scale uncertainties at NLO
shape changes in bb distributions
from LO to NLO (K=NLO/LO)
_
Like , a background to
L. Dixon Forecasting Physics at LHC QFT11 Pune 26 Feb. 2011 53
Bevilacqua, Czakon,
Papadopoulos,
Worek, 1002.4009
Again large reduction in scale dependence from LO NLO
Only computed via unitarity (CutTools)