qcd at the lhc · qcd at the lhc vittorio del duca ... to gluinos in susy decay chains, ... to...
TRANSCRIPT
Standard Model
The Standard Model has been a spectacular success,
weathering out all challenges
Past & Present:the LEP/SLD/Tevatron legacy
The comparison with the electroweak precision
measurements has not changed much in the last years
Precision tests of SM
Top quark Mass
Run I value: 178.0 ± 4.3 GeV
Run II value: 172.6 ± 0.8 ± 1.1 GeV
Tevatron
ICHEP 08
small effect on the quality of the SM fit and on the mH bounds
Precision tests of SM
Top quark Mass
Run I value: 178.0 ± 4.3 GeV
Run II value: 172.6 ± 0.8 ± 1.1 GeV
Tevatron
ICHEP 08
small effect on the quality of the SM fit and on the mH bounds
!m/m = 0.2 !"/" TH: δσ/σ = 9% Δm = 3 GeVAt the LHC the expected EXP error is Δm = 1 GeVso the TH cross section should be known at 3% level
Effects on global EW fits W-boson Mass March 08
tree level mW = mZ cos !W
W
t
b
W W W
H
W
one loopm2
W
!1! m2
W
m2Z
"=
!""2GF
(1 + !r)
!rt = ! 3!
16"
cos2 #W
sin4 #W
m2t
m2W
!rH =11!
48" sin2 #Wln
m2H
m2W
!mW ! m2
t!mW ! lnmH
δmt = 1 GeV ⇒ δmW(mt) = 6 MeV
so
ElectroWeak fits point to a light Higgs boson
use mt to estimate mH from EW corrections
as mt changes, large shifts in mH
mH = 87+36-27 GeVfrom EW fits
mH > 114.4 GeVfrom direct search at LEP
At 95% CLmH < 160 GeV from EW fitsmH < 190 GeV combined withdirect search at LEP
March 08
Effects on global EW fits Higgs boson Mass
... so, the Standard Model is very good, but that’s not enough
ElectroWeak Symmetry Breaking not tested
neutrino masses and mixings
dark matter
baryogenesis
?
??
foremost task of the LHC is to understand the EWSB:find the Higgs boson or whatever else is the cause of it
TOTEM
ALICE : ion-ion,p-ion
ATLAS and CMS :general purpose
27 km LEP ring 1232 superconducting dipoles B=8.3 T
TOTEM (integrated with CMS):pp, cross-section, diffractive physics
LHCb : pp, B-physics, CP-violation
Here: ATLAS and CMS
• pp √s = 14 TeV Ldesign = 1034 cm-2 s-1 (after 2009)
Linitial ≤ few x 1033 cm-2 s-1 (until 2009)
• Heavy ions (e.g. Pb-Pb at √s ~ 1000 TeV)
LHC
Event rate
On tape
Level-1SM processes are backgrounds to New Physics signals
LHC is a QCD machine
L = 1034 cm-2 s-1 = 10-5 fb-1 s-1
design luminosity
integrated luminosity (per year)
L ≈ 100 fb-1 yr-1
With 1 fb-1 we shall get ...
jets (pT > 100 GeV)
final state events
109
jets (pT > 1 TeV) 104
W ! e! 2⋅107
Z ! e+e! 2⋅106
bb 5⋅1011
tt 8⋅105
107 (Tevatron)
overall # of events (2008)
106 (LEP)
109 (BaBar, Belle)
104 (Tevatron)
even at very low luminosity, LHC beats all the other accelerators
LHC: the next future
calibrate the detectors, and re-discover the SMi.e. measure known cross sections: jets,
understand the EWSB/find New-Physics signals(ranging from Z’ to leptons, to gluinos in SUSYdecay chains, to finding the Higgs boson)
constrain and model the New-Physics theories
W, Z, tt
in all the steps above (except probably Z’ to leptons)
precise QCD predictions play a crucial role
B cross section in collisions at 1.96 TeVpp
d!(pp ! HbX, Hb ! J/" X)/dpT (J/")
CDF hep-ex/0412071
total x-sect is 19.4 ± 0.3(stat)+2.1!1.9(syst) nbFONLL = NLO + NLL
Cacciari, Frixione, Mangano, Nason, Ridolfi 2003
use of updated fragmentation functions by Cacciari & Nasonand resummation
good agreement with data no New Physics
an unbroken Yang-Mills gauge field theory featuring asymptotic freedom and confinement
in non-perturbative regime (low Q2) many approaches: lattice, Regge theory, χ PT, large Nc, HQET
in perturbative regime (high Q2) QCD is a precision toolkit for exploring Higgs & BSM physics
LEP was an electroweak machine
Tevatron & LHC are QCD machines
QCD
Precise determination of
strong coupling constantparton distributionselectroweak parametersLHC parton luminosity
Precise prediction for
Higgs productionnew physics processestheir backgrounds
!s
QCD at LHC
Goal: to make theoretical predictions of signals and backgrounds as accurate as the LHC data
Strong interactions at high Q2 Parton modelPerturbative QCD
factorisation
universality of IR behaviour
cancellation of IR singularities
IR safe observables: inclusive rates
jets
event shapes
Factorisation
pb
pa
PB
PA
!X =!
a,b
"
1
0
dx1dx2 fa/A(x1, µ2
F ) fb/B(x2, µ2
F )
! !ab!X
#
x1, x2, {pµi }; "S(µ2
R), "(µ2
R),Q2
µ2R
,Q2
µ2F
$
!X =!
a,b
"
1
0
dx1dx2 fa/A(x1, µ2
F ) fb/B(x2, µ2
F )
! !ab!X
#
x1, x2, {pµi }; "S(µ2
R), "(µ2
R),Q2
µ2R
,Q2
µ2F
$
}X X = W, Z, H, QQ,high-ET jets, ...
! = C"nS(1 + c1"S + c2"
2
S + . . .)
! = C"nS [1 + (c11L + c10)"S + (c22L
2 + c21L + c20)"2
S + . . .]
is known as a fixed-order expansion in! !S
c1 = NLO c2 = NNLO
or as an all-order resummation
L = ln(M/qT ), ln(1 ! x), ln(1/x), ln(1 ! T ), . . .wherec11, c22 = LL c10, c21 = NLL c20 = NNLL
is the separation betweenthe short- and the long-range interactions
extracted from dataevolved through DGLAP
computed in pQCD
Factorisation-breaking contributions
underlying event
power corrections
double-parton scattering
diffractive events
MC’s and theory modelling of power corrections laid out and tested at LEP where they provide an accurate determination of models still need be tested in hadron collisions(see e.g. Tevatron studies at different )
!S
observed by Tevatron CDF in the inclusive sample
pp ! ! + 3 jets
potentially important at LHC !D ! !2
S
(see Rick Field’s studies at CDF)
!
s
predicted breakdown in dijet production at N3LO Collins Qiu 2007
Power corrections at TevatronRatio of inclusive jet cross sections at 630 and 1800 GeV
xT =2ET!
s
Bjorken-scaling variable
In the ratio the dependence on the pdf’s cancels
dashes: theory prediction with no power corrections
solid: best fit to data with free power-correctionparameter in the theory!
M.L. ManganoKITP collider conf 2004
Factorisation in diffraction ??
diffraction in DIS double pomeron exchange in pp
no proof of factorisation in diffractive eventsdata do not support it
LHC opens up a new kinematic range
LHC kinematic reach
Feynman x’s for the production of a particle of mass M x1,2 =M
14 TeVe±y
100-200 GeV physics is large x physics (valence quarks)at Tevatron, but smaller x physics(gluons & sea quarks) at the LHC
x range covered by HERAbut Q2 range must be provided by DGLAP evolution
rapidity distributions span widest x range
PDF evolution
factorisation scale is arbitraryµF
cross section cannot depend on µF
µF
d!
dµF
= 0
implies DGLAP equations
µF
dfa(x, µ2
F)
dµF
= Pab(x, !S(µ2
F )) ! fb(x, µ2
F ) + O(1
Q2)
µF
d!ab(Q2/µ2
F, "S(µ2
F))
dµF
= !Pac(x, "S(µ2
F )) " !cb(Q2/µ2
F , "S(µ2
F )) + O(1
Q2)
Pab(x, !S(µ2
F )) is calculable in pQCD
V. Gribov L. Lipatov; Y. DokshitzerG. Altarelli G. Parisi
d
d lnµ 2F
!
qi
g
"
=
!
PqiqjPqjg
PgqjPgg
"
!
!
qj
g
"
[a ! b](x) "
! 1
x
dy
ya(y) b
"
x
y
#
qi(x, µ2
F ) g(x, µ2
F )
factorisation for the structure functions (e.g. )
Fi(x, µ 2
F ) = Cij ! qj + Cig ! g
Fep2
, FepL
with the convolution
Cij , Cig coefficient functions
PDF’s
DGLAP evolution equations
perturbative series Pij ! !sP(0)ij + !
2sP
(1)ij + !
3sP
(2)ij
anomalous dimension !ij(N) = !
! 1
0
dx xN!1 Pij(x)
DGLAP evolution equations
d
d lnµ 2F
!
qs
g
"
=
!
Pqq Pqg
Pgq Pgg
"
!
!
qs
g
"
a list of PDF’sgeneral structure of the quark-quark splitting functions
Pqiqk= Pqiqk
=!ikPv
qq + Ps
Pqiqk= Pqiqk
=!ikPv
qq + Ps
flavour non-singlet
flavour asymmetryq±
ns,ik = qi ± qi ! (qk ± qk) P±
ns = Pv
qq ± Pv
sum of valence distributions of all flavours
qv
ns =
nf!
r=1
(qr! q
r) P
vns = P
vqq ! P
vqq + nf (P s
qq ! Ps
qq)
flavour singlet
qs =
nf!
i=1
(qi + qi)
withPqq = P
+ns + nf (P s
qq + Ps
qq)
Pqg = nf Pqig , Pgq = Pgqi
PDF historyleading order (or one-loop)anomalous dim/splitting functions Gross Wilczek 1973; Altarelli Parisi 1977
NLO (or two-loop)
F2, FL Bardeen Buras Duke Muta 1978
anomalous dim/splitting functions Curci Furmanski Petronzio 1980
NNLO (or three-loop)
F2, FL Zijlstra van Neerven 1992; Moch Vermaseren 1999
anomalous dim/splitting functions Moch Vermaseren Vogt 2004
the calculation of the three-loop anomalous dimension is the toughest calculation ever performed in perturbative QCD!
one-loop
two-loop
three-loop
!(0)ij /P (0)
ij
!(1)ij /P (1)
ij
!(2)ij /P (2)
ij
18 Feynman diagrams
350 Feynman diagrams
9607 Feynman diagrams
20 man-year-equivalents, lines of dedicated algebra code106
PDF global fitsJ. Stirling, KITP collider conf 2004
MRST, MSTW: Martin et al.
global fits
CTEQ: Pumplin et al.Alekhin (DIS data only)
methodPerform fit by minimising to all data,including both statistical and systematicerrors
!2
Start evolution at some , where PDF’s are parametrised with functional form, e.g.
Q2
0
xf(x, Q2
0) = (1 ! x)!(1 + !x0.5 + "x)x"
Cut data at and at to avoid higher twist contamination
Q2 > Q2
min W2
> W2
min
Allow as implied byE866 Drell-Yan asymmetry data
u != d
accuracyNNLO evolution
no prompt photon data included in the fits
PDF: recent developments
more accurate treatment of heavy flavours in the vicinity of the quark mass (few % effect on Drell-Yan at the LHC)
more systematic treatment of uncertainties in global fits
neural network PDF’s (NNPDF), based on unbiased priors(so far only DIS data, and only the non-singlet distributions)
MSTW, CTEQ, Alekhin
NNPDF Collaboration 08
note the larger uncertaintyin NNPDF
valence quark distribution
still far from global-fit analysisin NNPDF
matrix-elem MC’s fixed-order x-sect shower MC’s
final-state description
hard-parton jets.Describes geometry,
correlations, ...
limited access tofinal-statestructure
full information available at the hadron level
higher-order effects:loop corrections
hard to implement:must introduce
negative probabilities
straightforwardto implement
(when available)
included asvertex corrections
(Sudakov FF’s)
higher-order effects:hard emissions
included, up to highorders (multijets)
straightforwardto implement
(when available)
approximate,incomplete phase
space at large angles
resummation oflarge logs ? feasible
(when available)
unitarityimplementation
(i.e. correct shapesbut not total rates)
3 complementary approaches to !
M.L. Mangano KITP collider conf 2004
Parton cross section
Parton shower MonteCarlo generators
HERWIG
PYTHIA
re-written as a C++ code (HERWIG++)
B. Webber et al. 1992
T. Sjostrand 1994
SHERPA F. Krauss et al. 2003
model parton showering and hadronisation
several automated codes to yieldlarge number of (up to 8-9) final-state partons
can be straightforwardly interfaced to parton-shower MC’s
ideal to scout new territory
large dependence on ren/fact scalesexample: Higgs (via gluon fusion) + 2 jets is αs4(Q2)
unreliable for precision calculations
Matrix-element MonteCarlo generators
Matrix-element MonteCarlo generators
multi-parton LO generation: processes with many jets (or V/H bosons)
ALPGEN M.L.Mangano M. Moretti F. Piccinini R. Pittau A. Polosa 2002
COMPHEP A. Pukhov et al. 1999
GRACE/GR@PPA T. Ishikawa et al. K. Sato et al. 1992/2001
MADGRAPH/MADEVENT W.F. Long F. Maltoni T. Stelzer 1994/2003
HELAC C. Papadopoulos et al. 2000
merged with parton showers
all of the above, merged with HERWIG or PYTHIA
processes with 6 final-state fermions
PHASE E. Accomando A. Ballestrero E. Maina 2004
MonteCarlo interfaces
CKKW S. Catani F. Krauss R. Kuhn B. Webber 2001
MLM L. Lonnblad 2002 M.L. Mangano 2005
procedures to interface parton subprocesses witha different number of final states to parton-shower MC’s
MC@NLO S. Frixione B. Webber 2002
POWHEG P. Nason 2004
procedures to interface NLO computations to parton-shower MC’s
at low pT, parton showermodels collinear radiation
Frixione Laenen Motylinski Webber 2005
Single top in MC@NLO
at high pT, NLO models hard radiation
Leading Order
Different methods (other than Feynman) to compute tree-level amplitudes
use recursively off-shell currents and go on-shell to get amplitudes
Berends-Giele relations (1988)
Cachazo Svrcek Witten (2004)
compute helicity amplitudes by sewing MHV amplitudes (++⋅⋅⋅+−−)
Britto Cachazo Feng (2004)
use on-shell recursive relationsby shifting to complex momenta
=∑
∑
+−
+− +
−−
−
−
−−
++
+++
= ++ −−
Leading Order
CSW and BCF are ideal to obtain compact analytic amplitudes... but numerically Berends-Giele seems to be the fastest
time [s] for 2 → n gluon amplitudes for 104 PS points
Duhr Höche Maltoni 2006
CO: colour orderedCD: colour dressed
Next to Leading OrderJet structure: final-state collinear radiation
PDF evolution: initial-state collinear radiation
Opening of new channels
Reduced sensitivity to fictitious input scales: µR, µF
predictive normalisation of observables
first step toward precision measurementsaccurate estimate of signal and backgroundfor Higgs and new physics
Matching with parton-shower MC’s: MC@NLO POWHEG
p p! t t Z at NLO
Lazopoulos McElmurry Melnikov Petriello 2008
NLO/LO ≡ K factor = 1.35
Reduced theoretical error: 25-35% at LO; 11% at NLO
dots: inclusive K factor dash: pT dependent K factor
probes EW properties of top quark
NLO cross sections: experimenter’s wishlist
2005 Les Houches
QCD, EW & Higgs working group hep-ph/0604210
High (EXP) demand for cross sections of Y + n jetsY = vector boson(s), Higgs, heavy quark(s), ...
Big TH community effort
To compute the cross section of Y + n jets, we need:1) tree-level amplitude for Y + (n+3) partons2) one-loop amplitude for Y + (n+2) partons3) a method to cancel the IR divergences and so to compute the cross section
3: until the mid 90’s, we did not have systematic methods to cancel the IR divergences2: until 2007-8, we did not have systematic methods to compute the one-loop amplitudes
NLO assembly kit
leading order
e+e!
! 3 jets
NLO virtual
NLO real
|Mtree
n|2
!
ddl 2(Mloopn )!Mtree
n =
"
A
!2+
B
!
#
|Mtreen |2 + fin.
= !
!
A
!2+
B
!
"
IR
d = 4 ! 2!
!
|Mtreen+1|
2 ! |Mtreen
|2 "
!dPS|Psplit|
2
example
NLO production rates Process-independent procedure devised in the 90’s
Giele Glover Kosower 1992-3
the 2 terms on the rhs are divergent in d=4
use universal IR structure to subtract divergences
the 2 terms on the rhs are finite in d=4
slicingsubtraction Frixione Kunszt Signer 1995; Nagy Trocsanyi 1996
dipole Catani Seymour 1996antenna Kosower 1997; Campbell Cullen Glover 1998
! = !LO
+ !NLO
=
!m
d!Bm Jm + !
NLO
!NLO
=
!m+1
d!Rm+1Jm+1 +
!m
d!Vm
Jm
!NLO
=
!
m+1
"
d!Rm+1Jm+1 ! d!
R,Am+1Jm
#
+
!
m
$
d!Vm
+
!
1
d!R,Am+1
%
Jm
NLO history of final-state distributionsK. Ellis Ross Terrano 1981e
+e!
! 3 jets
e+e!
! 4 jets Bern et al.; Glover et al.; Nagy Trocsanyi 1996-97
pp ! 1, 2 jets K. Ellis Sexton 1986 Giele Glover Kosower 1993
pp ! 3 jets Bern Dixon Kosower; Kunszt Signer Trocsanyi 1993-1995Nagy 2001
pp ! V + 1 jet Giele Glover Kosower1993
pp ! V + 2 jet Bern Dixon Kosower Weinzierl; Campbell Glover Miller 1996-97 Campbell K. Ellis 2003
pp ! V V Ohnemus Owens, Baur et al.1991-96; Dixon et al. 2000
pp ! V bb Campbell K. Ellis 2003
e+e!
! 4 fermions Denner Dittmaier Roth Wieders 2005
pp ! V V + jet Dittmaier Kallweit Uwer; Campbell K. Ellis Zanderighi 2007
pp ! V V V Lazopoulos Melnikov Petriello; Hankele Zeppenfeld 2007Binoth Ossola Papadopoulos Petriello 2008
pp! V + 3 jet K. Ellis Melnikov Zanderighi; Berger et al. (BlackHat) 2009
pp ! !!
pp ! !! + 1 jetBailey et al. 1992; Binoth et al. 1999Bern et al. 1994 Del Duca Maltoni Nagy Trocsanyi 2003
NLO history of final-state distributionspp ! QQ
pp ! QQ + 1 jet
Dawson K. Ellis Nason 1989 Mangano Nason Ridolfi 1992
Brandenburg Dittmaier Uwer Weinzierl 2005-7Dittmaier Kallweit Uwer 2007-8
pp ! H + 1 jet
pp ! HQQ
(GGF; mt → ∞) C. Schmidt 1997 De Florian Grazzini Kunszt 1999
Beenakker et al. ; Dawson et al. 2001
(GGF; mt → ∞) Campbell K. Ellis Zanderighi 2006(WBF) Campbell K. Ellis; Figy Oleari Zeppenfeld 2003 Ciccolini Denner Dittmaier 2007 (includes s channel)
pp ! H + 2 jets
pp! t t b bBredenstein Denner Dittmaier Pozzorini 2008
(only qq annihilation)
Harris Laenen Phaf Sullivan Weinzierl 2002Campbell K. Ellis Tramontano 2004; Cao Yuan et al. 2004-5W: Campbell Tramontano 2005
pp! t (+ W )
t: Maltoni Paul Stelzer Willenbrock 2001b: Campbell, K. Ellis Maltoni Willenbrock 2002pp! H Q
(WBF) Figy Hankele Zeppenfeld 2007pp! H + 3 jets
Lazopoulos McElmurry Melnikov Petriello 2008pp! t t Z
... summarising
long time span to add one more jet to a cross section
2 → 2 and 2 → 3 processes: almost all computed and included into NLO packages
2 → 4 processes: very few computede+e!
! 4 fermions Denner Dittmaier Roth Wieders 2005
pp! t t b b (only qq annihilation) Bredenstein Denner Dittmaier Pozzorini 2008
pp! V + 3 jet
2 → 5 processes: none
K. Ellis Melnikov Zanderighi; Berger et al. (BlackHat) 2009
Summary of NLO K factors
NLO multileg working group hep-ph/0803.0494
all: CTEQ6M at NLOK: CTEQ6L1 at LOK’: CTEQ6M at LO
gluon-initiated processes: large K factors
more jets: smaller K factors
NLO progress: cross sections
automated subtraction of IR divergencesautomating dipoles Gleisberg Krauss 2007
Frederix Gehrmann Greiner (MadGraph) 2008Seymour Tevlin (TeVJet); Hasegawa Moch Uwer2008
dipoles proliferate with the spectators:may not be adapt for multi-jet environment
subtraction for multi-jet environmentantennanew standard subtraction
Daleo Gehrmann Maitre 2006Somogyi Trocsanyi 2006; Somogyi 2009
for each dipole, one must considerall possible spectators ...result: in W + 2 jets (MCFM)24 counterterm events for each real event
Catani Seymour 1996
DipolesTo gain exact factorisation of phase space, modify momenta
pµir = pµ
i + pµr !
yir,n
1! yir,npµ
n
=1
1! yir,n(pµ
i + pµr ! yir,nQµ
irn)
pµn =
11! yir,n
pµn
Catani Seymour 1996
yir,n =2pi · pr
Q2irn
emitter momentum
spectator momentum
momentum is conserved Qµirn = pµ
ir + pµn = pµ
i + pµr + pµ
n
A |M0m+1|2 =
!
n !=i,r
Dir,n(p1, . . . , pm+1) + . . .
counterterm
New standard subtraction
pµir =
1
1 ! !ir
(pµi + pµ
r ! !irQµ) , pµ
n =1
1 ! !ir
pµn , n "= i, r
!ir =1
2
!
y(ir)Q !
"
y2(ir)Q ! 4yir
#
yir =2pi · pr
Q2
pµir +
!
n
pµn = p
µi + p
µr +
!
n
pµnmomentum is conserved
mapping à la Catani-Seymour, but all momenta but the unresolved ones are treated simultaneously like spectators
1
i
r
Q
!ir
!m + 2
m+1
···
· · ·
!Q
1
!ir
···
···
m+1
!m + 2
irr
Somogyi Trocsanyi 2006
one-loop amplitudes
↓ IR finite terms are process dependent: many final-state particles → many scales → lengthy expressions
Kunszt Signer Trocsanyi 1994; Catani 1998↑ IR divergences are universal
An can be reduced to boxes, triangles and bubbleswith rational coefficients
one-loop n-point amplitudes An are IR divergent
I: master integralsb, c, d: rational functions of kinematic variables
51
42
6
3
An =!
i1i2i3i4
di1i2i3i4 IDi1i2i3i4 +
!
i1i2i3
ci1i2i3 IDi1i2i3 +
!
i1i2
bi1i2 IDi1i2
i4
i3i2
i1 i3
i2
i1
i1 i2
higher polygons contribute only to O(ε)
use unitarity cuts: Cutkovsky rule
An =!
i1i2i3i4
di1i2i3i4 I4i1i2i3i4 +
!
i1i2i3
ci1i2i3 I4i1i2i3 +
!
i1i2
bi1i2 I4i1i2 + Rn
Bern Dixon Dunbar Kosower 1994
NLO progress: unitarity method 1
p2 + i0! 2!i "+(p2)
to factorise coefficients of An into products of tree amplitudes
compute b, c, d with D=4: cut-constructible termsO(ε) part of b, c, d: rational term Rn
Rn computable through on-shell recursionBerger Bern Dixon Forde Kosower 2006
generalized unitarity:quadruple cuts with complex momenta
box coefficient di determined byproduct of 4 tree amplitudes
however, ci, bi still difficult to extract from tripleand double cuts because terms already includedin di must be subtracted
Britto Cachazo Feng 2004
Ossola Papadopoulos Pittau 2006reduction of one-loop amplitude at integrand level
NLO progress: OPP method
for quadruple cuts, similar to BCF, but algorithmic: can be automated
A(q!) =N(q)
D1 · · · Dm
q’ lives in D dimensions; q lives in 4 dimensionsε part of numerator treated separately
Di = (q! + pi)2 !m2i q! = q + q q · q = 0
partial fraction the numerator into terms with 4, 3, 2, 1 denominator factors
N(q) =!
i1i2i3i4
(di1i2i3i4 + di1i2i3i4)"
i !=i1i2i3i4
Di
+!
i1i2i3
(ci1i2i3 + ci1i2i3)"
i !=i1i2i3
Di +!
i1i2
(bi1i2 + bi1i2)"
i !=i1i2
Di
d, c, b vanish upon integration
reduced to problem of fitting d, c, b by evaluating N(q) at different values of q,e.g. singling out choices of q such that 4, 3, 2, 1 among all Di vanish, and theninverting the system. First find all possible 4-pt functions, then 3-pt functions, etc.
NLO progress: integer dimensions Giele Kunszt Melnikov 2008
An =!
i1i2i3i4i5
ei1i2i3i4i5 IDi1i2i3i4i5 +
!
i1i2i3i4
di1i2i3i4 IDi1i2i3i4
+!
i1i2i3
ci1i2i3 IDi1i2i3 +
!
i1i2
bi1i2 IDi1i2
in D dimensions the one-loop amplitude An can be reduced to pentagons, boxes, triangles and bubbles
take Ds = # of spin states of internal particlesin Ds dimensions, gluons have Ds-2 spin states, quarks have 2(Ds-2)/2 spin states
A(D,Ds)n =
!dDq
N (Ds)(q)d1 · · · dn
with D ≤ Ds
dependence of A on Ds is linear N (Ds)(q) = N0(q) + (Ds ! 4)N1(q)
- compute N0, N1 numerically separately through 2 different integer values of Ds
- on Ds-dimension cuts, spin density matrix is well defined- choose basis of master integrals with no explicit D dependence in the coefficients- after reduction to master integrals, continue D to 4-2ε
procedure can be completely automated
RocketGiele Zanderighi 2008
A Fortran 90 code based on OPPand integer dimensions
computes- up to one-loop 20-gluon amplitudes- one-loop W + 5-parton amplitudes- (leading-colour) NLO W + 3-jet cross section
Giele Zanderighi 2008K. Ellis Giele Kunszt Melnikov Zanderighi 2008
K. Ellis Melnikov Zanderighi 2009
W + 3-jet cross section at NLOK. Ellis Melnikov Zanderighi 2009
NLO very stable under scale variations
K factor even smaller than in W + 2 jets
BlackHatA C++ code based on generalised unitarity,and on-shell recursion for the rational parts
Berger Bern Dixon Febres-Cordero Forde Ita Kosower Maitre 2008
computes- one-loop 6-gluon (and MHV 7- and 8-gluon) amplitudes- (leading-colour) one-loop W + 5-parton amplitudes- (leading-colour) NLO W + 3-jet cross section
BlackHat 2008BlackHat 2009
Inclusive jet cross section at Tevatron
good agreement between NLO and data over several orders ofmagnitude
constrains the gluondistribution at high x
Is NLO enough to describe data ?
pT
Is NLO enough to describe data ?Drell-Yan cross section at LHC with leptonic decay of the W W
|MC@NLO ! NLO| = O(2%)
NNLO useless without spin correlations
S. Frixione M.L. Mangano 2004
Precisely evaluated Drell-Yan cross sections could be used as ``standard candles’’ to measure the parton luminosity at LHC
W, Z
Drell-Yan W acceptances at LHC with leptonic decay of the W
K. Melnikov, F. Petriello 2006
μ = mW/2, mW, 2mW
NNLO corrections are large for pe,⊥ = 40, 50 GeVbut are at the percent level for pe,⊥ = 20, 30 GeV
At LO, pe,⊥ ≤ mW/2
Is NLO enough to describe data ?
Total cross section for inclusive Higgs production at LHC
µR = 2MH µF = MH/2
lowercontour bands are
upperµR = MH/2 µF = 2MH
scale uncertaintyis about 10%
NNLO prediction stabilises the perturbative series
Higgs production at NNLOa fully differential cross section:bin-integrated rapidity distribution, with a jet veto
jet veto: requireR = 0.4
for 2 partons
|pjT | < p
vetoT = 40 GeV
|p1
T |, |p2
T | < pveto
T
|p1
T + p2
T | < pveto
T
R2
12 = (!1 ! !2)2 + ("1 ! "2)
2
if
if
R12 > R
R12 < R
MH = 150 GeV (jet veto relevant in the decay channel)H ! W+W
!
K factor is much smaller for the vetoed x-sect than for the inclusive one:average increases from NLO to NNLO: less x-sect passes the veto|pj
T |
C. Anastasiou K. Melnikov F. Petriello 2004
Drell-Yan production at LHCZ
Rapidity distribution foran on-shell bosonZ
NLO increase wrt to LO at central Y’s (at large Y’s)30%(15%)NNLO decreases NLO by 1 ! 2%
scale variation: ! 30% at LO; ! 6% at NLO; less than at NNLO1%
C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003
Scale variations in Drell-Yan productionZ
solid: vary and together
dashed: vary only
dotted: vary only
µR
µR
µF
µF
C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003
Drell-Yan production at LHCW
Rapidity distributionfor an on-shell boson (left) boson (right) W
+
W!
C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003
distributions are symmetric in Y
NNLO scale variations are at central (large) 1%(3%) Y
World average of !S(MZ)
S. Bethke hep-ex/0606035
!S(MZ) = 0.1189 ± 0.0010
Rightmost 2 columns give the exclusive mean value of calculated without that measurement, and the number of std. dev. between this measurement and the respective excl. mean
!S(MZ)
NNLO corrections may be relevant ifthe main source of uncertainty in extracting info from data is due to NLO theory: measurements!S
NLO corrections are large: Higgs production from gluon fusion in hadron collisions
NLO uncertainty bands are too large to testtheory vs. data: b production in hadron collisions
NLO is effectively leading order:energy distributions in jet cones
NNLO state of the art Drell-Yan W, Z production total cross section Hamberg van Neerven Matsuura 1990
Harlander Kilgore 2002
Higgs productiontotal cross section
fully differential cross section
Harlander Kilgore; Anastasiou Melnikov 2002Ravindran Smith van Neerven 2003
Anastasiou Melnikov Petriello 2004Catani de Florian Grazzini 2007
e+e!
! 3 jetsde Ridder Gehrmann Glover Heinrich 2007Weinzierl 2008
Melnikov Petriello 2006fully differential cross section
event shapes, αs
!s(M2Z) = 0.1240± 0.0008 (stat)± 0.0010 (exp)± 0.0011 (had)± 0.0029 (theo)
TH uncertainty almost halved from NLO to NNLOno log resummation is included
discrepancy found in logarithmic contributions to thrust distribution at N3LL by Becher Scwartz 2008 using SCET. Confirmed error in wide-angle soft emissions through independent NNLO calculation by Weinzierl 2008
e+e!
! 3 jets de Ridder Gehrmann Glover Heinrich 2007
CO = N2c Clc
O + CscO +
1N2
c
CsscO + NfNc Cnf
O +Nf
NcCnfscO + N2
f CnfnfO
O =!s
2"AO +
!!s
2"
"2BO +
!!s
2"
"3CO
any IR observable to NNLO
with
disagreement in N2c , N0
c , NfNc
traced back to soft counterterm missing in RR contributionWeinzierl 2008
NNLO cross sectionsAnalytic integration
Sector decomposition
Hamberg, van Neerven, Matsuura 1990Anastasiou Dixon Melnikov Petriello 2003
first method
flexible enough to include a limited class of acceptance cutsby modelling cuts as ``propagators’’
cancellation of divergences is performed numerically
Denner Roth 1996; Binoth Heinrich 2000Anastasiou, Melnikov, Petriello 2004
flexible enough to include any acceptance cuts
can it handle many final-state partons ?
Subtractionprocess independent
cancellation of divergences is semi-analyticcan it be fully automatised ?
de Ridder Gehrmann Glover Heinrich 2007
Two-loop matrix elements
two-jet production qq!! qq
!, qq ! qq, qq ! gg, gg ! gg
C. Anastasiou N. Glover C. Oleari M. Tejeda-Yeomans 2000-01
Z. Bern A. De Freitas L. Dixon 2002
photon-pair production qq ! !!, gg ! !!
C. Anastasiou N. Glover M. Tejeda-Yeomans 2002Z. Bern A. De Freitas L. Dixon 2002
e+e!
! 3 jets
L. Garland T. Gehrmann N. Glover A. Koukoutsakis E. Remiddi 2002
!!! qqg
V + 1 jet productionT. Gehrmann E. Remiddi 2002
Drell-Yan productionV
R. Hamberg W. van Neerven T. Matsuura 1991
Higgs productionR. Harlander W. Kilgore; C. Anastasiou K. Melnikov 2002
qq ! V
qq ! V g
gg ! H (in the limit)mt ! "
universal collinear and soft currents
3-parton tree splitting functions
Campbell Glover 1997; Catani Grazzini 1998; Frizzo Maltoni VDD 1999; Kosower 2002
universal IR structure
Collinear and soft currents process-independent procedure
2-parton one-loop splitting functions
Bern Dixon Dunbar Kosower 1994; Bern Kilgore C. Schmidt VDD 1998-99;Kosower Uwer 1999; Catani Grazzini 1999; Kosower 2003
universal subtraction countertermsseveral ideas Kosower; Weinzierl; de Ridder Gehrmann Heinrich 2003
Frixione Grazzini 2004; Somogyi Trocsanyi VDD 2005
but devised only for e+e!
! 3 jetsde Ridder Gehrmann Glover 2005; Somogyi Trocsanyi VDD 2006
back to NLO IR limits
collinear operatorCir|M
(0)m+2(pi, pr, . . .)|
2 !1
sir"Mm+1(0)(pir, . . .)|P
(0)fifr
|Mm+1(0)(pir, . . .)#
soft operator
Sr|M(0)m+2(pr, . . .)|
2 !sik
sirsrk
"Mm+1(0)(. . .)|Ti · Tk|Mm+1(0)(. . .)#
counterterm!
r
"
#
!
i!=r
1
2Cir + Sr
$
% |M(0)m+2(pi, pr, . . .)|
2
performs double subtraction in overlapping regions
Why is NNLO subtraction so complicated ?
A1|M(0)m+2|
2 =!
r
"
#
!
i!=r
1
2Cir +
$
%Sr !!
i!=r
CirSr
&
'
(
) |M(0)m+2(pi, pr, . . .)|
2
the NLO counterterm
contains spurious singularities when parton becomes unresolved, but they are screened by
s != r
Jm
Cir (Sr ! CirSr) |M(0)m+2|
2 = 0
can be used to cancel double subtractionCirSr
Sr (Cir ! CirSr) |M(0)m+2|
2 = 0
NLO overlapping divergences
has the same singular behaviour as SME, and is free of double subtractions
Cir (1 ! A1) |M(0)m+1|
2 = 0 Sr (1 ! A1) |M(0)m+1|
2 = 0
NNLO subtraction
!NNLO
=
!m+2
d!RRm+2Jm+2 +
!m+1
d!RVm+1Jm+1 +
!m
d!VVm
Jm
the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences
NNLO subtraction
!NNLO
=
!m+2
d!RRm+2Jm+2 +
!m+1
d!RVm+1Jm+1 +
!m
d!VVm
Jm
the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences
!NNLO
=
!
m+2
"
d!RRm+2Jm+2 ! d!
RR,A2
m+2 Jm
#
takes care of doubly-unresolved regions,but still divergent in singly-unresolved ones
NNLO subtraction
!NNLO
=
!m+2
d!RRm+2Jm+2 +
!m+1
d!RVm+1Jm+1 +
!m
d!VVm
Jm
the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences
!NNLO
=
!
m+2
"
d!RRm+2Jm+2 ! d!
RR,A2
m+2 Jm
#
takes care of doubly-unresolved regions,but still divergent in singly-unresolved ones
+
!
m+1
"
d!RVm+1Jm+1 ! d!
RV,A1
m+1 Jm
#
still contains poles in regions away from 1-parton IR regions1/!
NNLO subtraction
!NNLO
=
!m+2
d!RRm+2Jm+2 +
!m+1
d!RVm+1Jm+1 +
!m
d!VVm
Jm
the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences
+
!
m
"
d!VVm
+
!
2
d!RR,A2
m+2 +
!
1
d!RV,A1
m+1
#
Jm
!NNLO
=
!
m+2
"
d!RRm+2Jm+2 ! d!
RR,A2
m+2 Jm
#
takes care of doubly-unresolved regions,but still divergent in singly-unresolved ones
+
!
m+1
"
d!RVm+1Jm+1 ! d!
RV,A1
m+1 Jm
#
still contains poles in regions away from 1-parton IR regions1/!
2-step procedure
G. Somogyi Z. Trocsanyi VDD 2005
construct subtraction terms that regularise the singularities of the SME in all unresolved parts of the phase space, avoiding multiple subtractions
G. Somogyi Z. Trocsanyi VDD 2006
perform momentum mappings, such that the phase space factorises exactly over the unresolved momenta and such that it respects the structure of the cancellations among the subtraction terms
A2 countertermconstruct the 2-unresolved-parton counterterm using the IR currents
Cirs (1 ! A2) |M(0)m+2|
2 = 0
Cir;js (1 ! A2) |M(0)m+2|
2 = 0
Srs (1 ! A2) |M(0)m+2|
2 = 0
CSir;s (1 ! A2) |M(0)m+2|
2 = 0
performing double and triple subtractions in overlapping regions
Constructing d!RR,A2
m+2
! Use IR factorization formulae valid in the doubly-unresolved regions Catani-Grazzini, Campbell-Glover
! To account for double subtraction in overlappingregions: subtract the collinear limit of the soft one
A2|M(0)m+2|
2 =!
r
!
s !=r
"!
i !=r,s
#1
6Cirs +
!
j !=i,r,s
1
8Cir;js +
1
2Srs
+1
2
$
CSir;s ! CirsCSir;s !!
j !=i,r,s
Cir;jsCSir;s
%&
!!
i !=r,s
#
CSir;sSrs + Cirs
$1
2Srs ! CSir;sSrs
%
+!
j !=i,r,s
Cir;js
$1
2Srs ! CSir;sSrs
% &'
|M(0)m+2|
2
Easy to check:
Cirs(1!A2)|M(0)m+2|
2 = 0
Cir;js(1!A2)|M(0)m+2|
2 = 0
CSir;s(1!A2)|M(0)m+2|
2 = 0
Srs(1!A2)|M(0)m+2|
2 = 0
Zoltán Trócsányi: Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD Squared Matrix Elements ETH Zürich, March 8, 2005 – p.16/22G. Somogyi Z. Trocsanyi VDD 2005
Triple-collinear mappingpµ
irs =1
1 ! !irs
(pµi + pµ
r + pµs ! !irsQ
µ) , pµn =
1
1 ! !irs
pµn , n "= i, r, s
!irs =1
2
!
y(irs)Q !
"
y2(irs)Q ! 4yirs
#
pµirs +
!
n
pµn = p
µi + p
µr + p
µs +
!
n
pµnmomentum conservation
straightforward extension of NLO collinear mapping
···
· · ·
m
Q
1
!m + 2
!irs
ir
s
···
···
m
!m + 2
1
!irs ! irs
r
sQ
phase space
The subtraction scheme at NNLO
!NNLO = !NNLO{m+2} + !NNLO
{m+1} + !NNLO{m}
!NNLO{m+2} =
!
m+2
"
d!RRm+2 Jm+2 ! d!RR,A2
m+2 Jm
!d!RR,A1
m+2 Jm+1 + d!RR,A12
m+2 Jm
#
d=4
!NNLO{m+1} =
!
m+1
"
d!RVm+1 Jm+1 ! d!RV,A1
m+1 Jm
+
!
1
$
d!RR,A1
m+2 Jm+1 ! d!RR,A12
m+2 Jm
%#
d=4
!NNLO{m} =
!
m
"
d!VVm +
!
2
d!RR,A2
m+2 +
!
1
d!RV,A1
m+1
#
d=4
Jm
has to be finitein the doubly-unresolvedregions!
RestrictsA1 and A12
severely
Zoltán Trócsányi: Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD Squared Matrix Elements ETH Zürich, March 8, 2005 – p.13/22
!NNLO{m+1} =
!
m+1
"#
d!RVm+1 +
!
1
d!RR,A1
m+2
$
Jm+1
!
#
d!RV,A1
m+1 +
%
!
1
d!RR,A1
m+2
&
A1
$
Jm
'
!=0
remainder is finite by KLN theorem
!NNLO{m} =
!
m
"
d!VVm
+
!
2
#
d!RR,A2
m+2 !d!RR,A12
m+2
$
+
!
1
#
d!RV,A1
m+1 +
%
!
1
d!RR,A1
m+2
&
A1
$'
!=0Jm
NNLO counterterms
G. Somogyi Z. Trocsanyi VDD 2006
Resummationscoefficients of αs expansion of an observable Θ may contain logs
in some kinematic regions, logs may become large, spoiling convergence
if resummed to all orders, series convergence improves
!(n)0 (pi,", g0) =
n!
i=1
Z1/2i ("/µ, g(µ)) !(n)
R (pi, µ, g(µ)),
Θ0 independence of μ implies
d!(n)0
dµ= 0 ! d ln !(n)
R
d lnµ= "
n!
i=1
!i(g(µ))
renormalisation of Θ is
RG evolution resums !ns (µ2) lnn(Q2/µ2) into !n
s (Q2)
example: e+e- → hadrons, where Q2 = s
Example: Higgs production from gluon fusion
gluon density is rapidly increasing as x → 0 ⇒ Higgs production occurs near partonic threshold
total energy of gluons in the final state is EX =s!m2
H
2mH" 0
need to resum soft-gluon emission
Higgs qT distribution Bozzi Catani de Florian Grazzini 2005
At small qT, NLO blows up
!µ(p1, p2;µ2, !) !< 0|Jµ(0)|p1, p2 >= v(p2)"µu(p1) !!
Q2
µ2, #s(µ2), !
"
Resummation: Sudakov form factorSudakov (quark) form factor as matrix element of EM current
obeys evolution equation
Q2 !
!Q2ln
!!
"Q2
µ2, "s(µ2), #
#$=
12
!K
%"s(µ2), #
&+ G
"Q2
µ2, "s(µ2), #
#$
RG invariance requires
µdG
dµ= !µ
dK
dµ= !K("s(µ2))
γK is the cusp anomalous dimension
K is a counterterm; G is finite as ε→ 0
Korchemsky Radyushkin 1987
!!Q2, !
"= exp
#12
$ !Q2
0
d"2
"2
%G
!!1, #s("2, !), !
"! 1
2$K
!#s("2, !)
"ln
&!Q2
"2
'()solution is
cusp anomalous dimensionloop expansion of the cusp anomalous dimension
K =!
6718! !2
"CA !
109
TF Nf
!(i)K = 2Ci
"s(µ2)#
+ KCi
!"s(µ2)
#
"2
+ · · ·
with
Factorisation of a multi-leg amplitude
Mueller 1981Sen 1983Botts Sterman 1987Kidonakis Oderda Sterman 1998Catani 1998Tejeda-Yeomans Sterman 2002Kosower 2003Aybat Dixon Sterman 2006Becher Neubert 2009Gardi Magnea 2009
to avoid double counting of soft-collinear region (IR double poles), Ji removes eikonal part from Ji, which is already in SJi/Ji contains only single collinear poles
MN (pi/µ, !) =!
L
SNL("i · "j , !) HL
"2pi · pj
µ2,(2pi · ni)2
n2i µ
2
# $
i
Ji
"(2pi · ni)2
n2i µ
2, !
#
Ji
"2("i · ni)2
n2i
, !
#
pi = !iQ0/!
2 value of Q0 is immaterial in S, J
N = 4 SUSY in the planar limit
colour-wise, the planar limit is trivial:can absorb S into Ji
each slice is square rootof Sudakov form factor
Mn =n!
i=1
"M[gg!1]
#si,i+1
µ2, !s, "
$%1/2
hn({pi}, µ2, !s, ")
β fn = 0 ⇒ coupling runs only through dimension
ln!!
"Q2
µ2, !s(µ2), "
#$= !1
2
!%
n=1
"!s(µ2)
#
#n "!Q2
µ2
#"n!&
$(n)K
2n2"2+
G(n)(")n"
'
⇒ IR structure of N = 4 SUSY amplitudes
Sudakov form factor has simple solution
!s(µ2)µ2! = !s("2)"2!
- introduce auxiliary vector ni (ni2 ≠ 0) to separate collinear region
- define a jet using a Wilson line along ni
Jet definition
u(p) J
!(2p · n)2
n2µ2, !
"=< p|"(0)!n(0,!")|0 >
!n(!2, !1) = P exp
!ig
" !2
!1
d!n · A(!n)
#
Wilson line
partonic jet
eikonal jet J!
2(! · n)2
n2, "
"=< 0|!!(!, 0)!n(0,"!)|0 >
Eikonal jet & cusp anomalous dimension
eikonal jet does not depend on any kinematic scale
single poles carry (! · n)2/n2 dependence,
thus violate classical rescaling symmetry wrt β ⇒ cusp anomalous dim
δj is a constant
double poles and kinematic dependence of single poles are controlled by cusp γK, as for quark form factor
J!
2(! · n)2
n2, "
"= exp
#14
$ µ2
0
d#2
#2
%$Ji
&%s(#2, ")
'! 1
2&K
&%s(#2, ")
'ln
!2(! · n)2µ2
n2#2
"()
Soft function S(cN )ijklSNL
!!a · !b, "s(µ2), #
"
=#
i!j!k!l!
< 0|!k,k!
!!2(0,!)!i,i!
!1(!, 0)!j,j!
!3(0,!)!l,l!
!!4(!, 0)|0 > (cL)i!j!k!l!
matrix evolution equation
µd
dµSJL
!!a · !b, "s(µ2), #
"
= !#
N
[!S ]JN
!!a · !b, "s(µ2), #
"SNL
!!a · !b, "s(µ2), #
"
formal solution
S!!a · !b, "s(µ2), #
"= P exp
#!1
2
$ µ2
0
d$2
$2!S
!!a · !b, "s(µ2), #
"%
ΓS soft anomalous dimension
!S =!!
n=1
!(n)S
"!s(µ2)
"
#n
Soft anomalous dimensionfor an amplitude with an arbitrary # of legs
!(2)S =
K
2!(1)S Aybat Dixon Sterman 2006
K is 2-loop coefficient of cusp anomalous dimension
ΓS has cusp singularities like γJ
Reduced soft function
S does not have cusp singularities ⇒ must respect rescaling βi → κi βi
S depends only on !ij =("i · "j)2
2("i · ni)2
n2i
2("j · nj)2
n2jfactorisation becomes
MN (pi/µ, !) =!
L
SNL("ij , !) HL
"2pi · pj
µ2,(2pi · ni)2
n2i µ
2
# $
i
Ji
"(2pi · ni)2
n2i µ
2, !
#
Dixon Magnea Sterman 2008
S has only single poles due to large-angle soft emissions
SJL (!ij , ") =SJL (#i · #j , ")
n!
i=1
Ji
"2(#i · ni)2
n2i
, "
#
Reduced soft anomalous dimension
!
j !=i
!
! ln "ij!S("ij , #s) =
14$(i)
K (#s)
evolution equation for reduced soft anomalous dimension
Becher Neubert; Gardi Magnea 2009
solution
with
!(i)K = Ci!K("s) !K = 2
"s(µ2)#
+ K
!"s(µ2)
#
"2
+ K(2)
!"s(µ2)
#
"3
+ · · ·
only 2-eikonal-line correlations
!S(!ij , "s) = !18#K("s)
!
i !=j
ln(!ij)Ti · Tj +12
$S("s)n!
i=1
Ci
generalises 2-loop solution
Are there any 3(or more)-line correlations ?Gardi Magnea say maybe; Becher Neubert say no