(1) nCTEQ Nuclear Parton Distributionsand
(2) NLO Parton Shower Monte Carlo
A. Kusina
Southern Methodist University, Dallas, TX 75275, USA
Collaborators:F. I. Olness, K. Kovarık, I. Schienbein, S. Jadach, M. Skrzypek
10 January 2014
1 / 45
Education/Employment
Master’s studies: Jagiellonian University, Krakow, Poland;supervisor: Stanislaw Jadach;thesis: “Evolution of parton distributions in QCD as a two-levelMarkovian process and its Monte Carlo simulation”
PhD studies: Institute of Nuclear Physics PAN in Krakow, Poland;supervisor: Stanislaw Jadach;thesis: “Exclusive Kernels for NLO QCD Non-singlet Evolution”
Postdoc position: Southern Methodist University, Dallas, TX;working with Fred Olness
2 / 45
Topics/projects I have been working on
During PhD:
Evolution equations:DGLAP, CCFM
NLO parton shower Monte CarloDuring postdoc at SMU:
nCTEQ parton distribution functionsN3LO heavy quark production in DISWe combined the full massive NLO result with the N3LO masslesscalculation to obtain an approximate N3LO DIS structure functionswith heavy quark mass effects. This is being implemented inHeraFitter and QCDNUM programs.
W/Z production at LHCWe investigated theoretical uncertainties related with W/Z productionoriginating form the strange quark PDF, and observed how W/Zmeasurements at LHC can be used to constrain strange PDF.
Hybrid Variable Flavor Number Scheme:generalization of the variable-flavor-number scheme for heavy quarks
3 / 45
nCTEQ PDFs
nCTEQ nuclear PDFsin collaboration with:F. Olness, K. Kovarık, I. Schienbein and T. Jezo
4 / 45
Motivations
What are PDFs of boundprotons/neutrons?
Heavy ion collisions in LHC and RHIC
Differentiate flavors in free-proton PDFs (e.g. strange)charged lepton DIS
Fl±2 ∼
(13
)2[d + s] +
(23
)2[u + c]
neutrino DIS
Fν2 ∼
[d + s + u + c
]Fν
2 ∼[d + s + u + c
]Fν
3 ∼ 2[d + s − u − c
]Fν
3 ∼ 2[u + c − d − s
]5 / 45
Assumptions entering the nuclear PDF analysis
1 Factorization & DGLAP evolution
allow for definition of universal PDFsmake the formalism predictiveneeded even if it is broken
2 Isospin symmetry un/A(x) = dp/A(x)
dn/A(x) = up/A(x)
3 The bound proton PDFs have the same evolution equations and sumrules as the free proton PDFs provided we neglect any contributionsfrom the region x > 1 (it is expected to have negligible contribution[PRC 73, 045206 (2006), arXiv:hep-ph/0509241] )
Then observables OA can be calculated as:
OA = Z Op/A + (A − Z)On/A
With the above assumptions we can use the free proton framework toanalyze nuclear data
6 / 45
Schematics of Global Analysis
1 Parametrize PDFs at low initial scale µ = Q0 = 1.3GeV:
f (x,Q0) = f (x; a0, a1, . . . ) = a0xa1 (1 − x)a2 P(x; a3, . . . )
2 Use DGLAP equation to evolve f (x, µ) from µ = Q0 to µ = Q.
3 Define and minimize appropriate χ2 function(with respect to parameters a0, a1, . . . )
χ2(ai) =∑
experiments
wnχ2n(ai)
χ2n(ai) =
∑data points
(data − theory(ai)
uncertainty
)2
(by default wn = 1)
7 / 45
Uncertainties in global analysis
Experimental errors (included in PDFs error analysis)
Theoretical uncertainties (not included)
“Details” of Global Fits(e.g. parametrization, HF schemes; hard to estimate notincluded)
Propagating experimental errors to PDFs:
Hessian MethodEigenvector PDFsQuadratic approximationSimple computation of correlations
Lagrange Multipliers
Monte Carlo Methods
8 / 45
Hessian method [JHEP 07 (2002) 012, arXiv:hep-ph/0201195]
Expand χ2 function around minimum, a0i , and diagonalize
χ2 = χ20 +
∑ij
12
(ai − a0i )(aj − a0
j )(∂2χ2
∂yi∂yj
)0
= χ20 +
∑i
z2i
Choose ∆χ2 = χ2 − χ20 value (defining 1-σ uncertainty),
ideal case ∆χ2 = 1realistic global analysis ∆χ2 ∼ 10 − 100
Construct error PDFs corresponding to each eigenvector direction:
f ±i = f (zi) = f (0, . . . , zi = ±
√∆χ2, . . . , 0)
zi = ±
√∆χ2
Calculate errors of observable X:
∆X =12
√∑i
[X(f +
i ) − X(f −i )]2
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Available nuclear PDFs
Multiplicative nuclear correction factors
f p/Ai (xN , µ0) = Ri(xN , µ0,A,Z) f free proton
i (xN , µ0)
Hirai, Kumano, Nagai [PRC 76, 065207 (2007), arXiv:0709.3038]Eskola, Paukkunen, Salgado [JHEP 04 (2009) 065, arXiv:0902.4154]de Florian, Sassot, Stratmann, Zurita[PRD 85, 074028 (2012), arXiv:1112.6324]
Native nuclear PDFsnCTEQ [PRD 80, 094004 (2009), arXiv:0907.2357]
f p/Ai (xN , µ0) = fi(xN ,A, µ0)
fi(xN ,A = 1, µ0) ≡ f free protoni (xN , µ0)
10 / 45
nCTEQ framework [PRD 80, 094004 (2009), arXiv:0907.2357]
Functional form of the bound proton PDF same as for thefree proton (CTEQ6M, x restricted to 0 < x < 1)
xf p/Ai (x,Q0) = c0xc1 (1 − x)c2 ec3x(1 + ec4 x)c5 , i = uv, dv, g, . . .
d(x,Q0)/u(x,Q0) = c0xc1 (1 − x)c2 + (1 + c3x)(1 − x)c4
A-dependent fit parameters (reduces to free proton for A = 1)
ck → ck(A) ≡ ck,0 + ck,1(1 − A−ck,2
), k = 1, . . . , 5
PDFs for nucleus (A,Z)
f (A,Z)i (x,Q) =
ZA
f p/Ai (x,Q) +
A − ZA
f n/Ai (x,Q)
(bound neutron PDF f n/Ai by isospin symmetry)
11 / 45
Data sets
NC DIS & DYCERN BCDMS & EMC & NMCN = (D, Al, Be, C, Ca, Cu, Fe, Li, Pb, Sn,W)FNAL E-665N = (D, C, Ca, Pb, Xe)DESY HermesN = (D, He, N, Kr)SLAC E-139 & E-049N = (D, Ag, Al, Au, Be,C, Ca, Fe, He)FNAL E-772 & E-886N = (D, C, Ca, Fe,W)
Single pion production(not yet included)
RHIC - PHENIX & STAR
N = Au
Neutrino (to be included later)
CHORUS CCFR & NuTeV
N = Pb N = Fe
12 / 45
nCTEQ fits
Fit properties:
fit @NLO
using ACOT heavy quark scheme
kinematical cuts: Q > 2GeV,W > 3.5GeV
708 (1233) data points after (before)cuts
17 free parameters
7 gluon8 valence2 sea
χ2/dof = 0.87
Error analysis:
use Hessian method
χ2 = χ20 +
12
Hij(ai − a0i )(aj − a0
j )
Hij =∂2χ2
∂ai∂aj
tolerance ∆χ2 = 35 (every nucleartarget within 90% C.L.)
eigenvalues span 10 orders ofmagnitude→ require numericalprecision
use noise reducing derivatives
13 / 45
nCTEQ results
A-dependence of bound proton PDFs
x f Ai (x,Q) A = (1, 2, 4, 9, 12, 27, 56, 108, 207)
PRELIMINARY
uval dval
u g
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nCTEQ results
Nuclear correction factors(Q = 10GeV)
Ri(Pb) =f Pbi (x,Q)
f pi (x,Q)
different solution for d-valence& u-valence compared to EPS09& DSSZ
sea quark nuclear correctionfactors similar to EPS09
nuclear correction factorsdepend largely on underlyingproton baseline
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
PRELIMINARY
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nCTEQ results
Nuclear PDFs (Q = 10GeV)
x f Pbi (x,Q)
nCTEQ dval & uval solutionbetween HKN07 & EPS09
nCTEQ features largeruncertainties than previousnPDFs
better agreement betweendifferent groups (nPDFs don’tdepend on proton baseline)
in EPS09 and DSSZ analysisuval & dval are tied togetherwhereas in nCTEQ they aretreated as independent
10 3 10 2 10 1 10.0
0.1
0.2
0.3
0.4
0.5
( , )nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
( , )nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10
5
10
15
20
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.2
0.4
0.6
0.8
1.0
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.2
0.4
0.6
0.8
1.0
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8 ( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
( , )
nCTEQHKN07EPS09DSSZ
PRELIMINARY
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nCTEQ results
Structure function ratio
R =FFe
2 (x,Q)
FD2 (x,Q)
good data description
despite different u-valence &d-valence ratios are similar toEPS09
0.01 0.1 1
0.80
0.85
0.90
0.95
1.00
1.05
1.10
[±
]
=
nCTEQ ±
HKN07
EPS09
0.01 0.1 1
0.80
0.85
0.90
0.95
1.00
1.05
1.10
[±
]
=
nCTEQ ±
HKN07
EPS09
PRELIMINARY
17 / 45
Main difference with the free-proton PDFs
More parameters & less data→ less constraining power→ moreassumptions (fixing) about ai parameters
Assumptions replace uncertainties!(error bands underestimates true uncertainties)
0.01 0.1 1
0.80
0.85
0.90
0.95
1.00
1.05
1.10
[±
]
=
nCTEQ ±
HKN07
EPS09
18 / 45
Summary
We have first nCTEQ error PDFs (still preliminary, single piondata is being included)
nCTEQ PDFs features larger (more realistic) uncertainties.
Plans for future:finish current analysisredo analysis of neutrino nuclear correctionshigh x region, deuterium nuclear corrections – CJ collaboration inJLABinclude new data (LHC, JLAB, Minerνa, ???)different methods for minimizing and estimating errors
19 / 45
NLO Parton Shower MC
NLO Parton Shower MCin collaboration with:S. Jadach, M. Skrzypek
20 / 45
[arXiv:1310.6090]
[Acta Phys.Polon. B43 (2012) 2067, arXiv:1209.4291]
[PRD 87 (2013) 034029, arXiv:1103.5015]
[Acta Phys.Polon. B42 (2011) 2433, arXiv:1111.5368]
[JHEP 1108 (2011) 012, arXiv:1102.5083]
What are parton shower Monte Carlo event generators?
Parton shower Monte Carlo event generatorComputer program designed to simulate the final states of high-energycollisions in full detail down to the level of individual stable particles.
The aim is to generate a large number of simulated collision events,each consisting of a list of final-state particles and their momenta,such that the probability to produce an event with a given list isproportional (approximately) to the probability that the correspondingactual event is produced in the real world.
21 / 45
[Bryan Webber, scholarpedia.10662]
Why do we need Monte Carlo event generators?
LHC experiments requiredetailed understanding ofsignals and backgrounds
full detector simulationfull exclusive hadronicevents
can only be done withMonte Carlo eventgenerators
22 / 45
Monte Carlo event generators
Complex finalstates in full detail
Arbitraryobservables andcuts from finalstates
23 / 45[JHEP 02 (2009) 007, arXiv:0811.4622]
LHC ”event”
hn
h1
W/Z
p
p h2
• proton PDF• fits to data at
low scale (NNLO)
• parton shower• evolution eq.
(NNLO)•Monte Carlo
(LO)
• hard process• fixed order
(NLO, NNLO)•Monte Carlo
(NLO)
• hadronization• modeling
24 / 45
Factorization vs. Monte Carlo
Every Parton Shower Monte Carlo (MC) is based on the concept offactorization. For pp collisions it can be written as:
σ =∑
ij
∫dx1dx2 fi(x1, µ)fj(x2, µ) σi(p1, p2, αS(µ),Q2/µ2)
In MC we want to generate exclusive distributions (full 4-momentadependence)
Above formula is inclusive!
It violates 4-momentum conservation (k⊥ component);
Features huge oversubtractions (time ordered exponential isconstructed by geometrical series).
25 / 45
Factorization vs. Monte Carlo
Every Parton Shower Monte Carlo (MC) is based on the concept offactorization. For pp collisions it can be written as:
σ =∑
ij
∫dx1dx2 fi(x1, µ)fj(x2, µ) σi(p1, p2, αS(µ),Q2/µ2)
In MC we want to generate exclusive distributions (full 4-momentadependence)
These problems have been solved only in LO approximation;
Big progress towards NLO (POWHEG, MC@NLO);
Still no solution for full NLO in the shower.
25 / 45
Aim
Ultimate aimConstruct MC generator with NLO shower and NLO/NNLO hardprocess
Steps on the way
reformulate collinear factorization to exclusive (unintegrated)form;
construct new LO shower;
include NLO corrections to hard process;
include NLO corrections to the shower itself (ladders);
26 / 45
Outline
1 Introduction
2 Our projectRebuilding LO MC showerIncluding NLO corrections to hard processRelation to other methodsNLO corrections in the shower
27 / 45
LO shower
Main features of the new LO shower:based on modified collinear factorization;
conserving 4-momentum (also transverse component)giving perturbative series directly in terms of exponent notgeometrical series (avoiding oversubtractions)real emissions regularized geometrically in 4 dimensionssubtractions defined in unintegrated form (corresponding to MCcounterterms)
angular ordering;
no gaps (cover thewhole phase space)
28 / 45
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
(re-)constructed LO shower
Analytical integration ofthe LO MC distribution
over the multigluon phase-space gives the standard collinear formula:
σ0 =
∫ 1
0dxF dxB DF(t, xF) DB(t, xB) σB(sxFxB)
*
Wη-3 -2 -1 0 1 2 3
* Wη/ d
σ
d
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-610×
*
Wη-3 -2 -1 0 1 2 3
Ratio
CM
C/Co
ll.
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
η∗W = 12 ln(xF/xB) – in
collinear limit – rapidity of Wboson
agreement of < 0.5%
29 / 45
Outline
1 Introduction
2 Our projectRebuilding LO MC showerIncluding NLO corrections to hard processRelation to other methodsNLO corrections in the shower
30 / 45
NLO correction to hard process
Because LO shower covers thewhole phase space the NLOcorrections can be included bymeans of simple reweightingtechniques
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ) WNLO
MC
31 / 45
NLO shower – proof of concept
Analytical integration ofthe NLO MC distribution
σ0 =
∫dx0Fdx0B d0(t0 , x0F )d0(t0 , x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−12CFαs
π2 P(zFi))δ
xF =x0F∏n1
i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−12CFαs
π2 P(zBj))δ
xB=x0B∏n2
j=1 zBj
× dτ2(P −n1+n2∑
j=1kj; q1 , q2)
dσBdΩ
(sxFxB , θ) WNLOMC
over the multigluon phase-space gives the standard collinear formula:
σ1 =
1∫0
dxF dxB dz DF(t, xF) DB(t, xB) σB(szxFxB)δz=1(1 + ∆S+V ) + C2r(z)
*
Wη-3 -2 -1 0 1 2 3
NLO(
CMC)
, NL
O(Co
ll.)
0
5
10
15
20
25
-910×
*
Wη-3 -2 -1 0 1 2 3
NLO:
CM
C/Co
ll.
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
η∗W = 12 ln(xF/xB) – in
collinear limit – rapidity of Wboson
agreement of < 1%
32 / 45
NLO correction to hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
soft+virtual NLO correction (kinematics independent!)
∆S+V =CFαs
π
(23π2 −
54
)
real correction (with subtraction)
β1(q1, q2, k) =[ (1 − β)2
2dσB
dΩq(s, θF) +
(1 − α)2
2dσB
dΩq(s, θB)
]−θα>β
1 + (1 − α − β)2
2dσB
dΩq(s, θ) − θα<β
1 + (1 − α − β)2
2dσB
dΩq(s, θ).
summation over all partons!
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Summation in NLO weight for hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
34 / 45
NLO weight distribution
h_wt1rEntries 100000
Mean 0.9851
RMS 0.04095
wt0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.05
0.1
0.15
0.2
0.25
0.3
-310×h_wt1r
Entries 100000
Mean 0.9851
RMS 0.04095
NLO weight
WNLOMC
Correcting to NLO requiresonly reweighting with WNLO
MCweight:
strongly peaked
positive
without long-ranged tails
Good behavior of WNLOMC is a
profit form reformulating thecollinear factorization andrebuilding LO shower.
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Relation to other methods
POWHEG
First generate hardest emission with NLO accuracy (independentof the shower)
Additional complications of vetoed/truncated shower (emissionsfrom the shower can not be harder then the first emission)
MC@NLO
Generate, in a separate MC branch, events according to anon-positive NLO correcting distribution
Negative weights
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Relation to other methods
gluon
Born
Z
1−st order corrections
quar
k
proton proton
P
P
P
P
P
P
P
P
Virt
Main differences with respect to POWHEG and MC@NLO are:“Democratic” summation over all emitted gluons, withoutdeciding explicitly which gluon is involved in hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
differences related to modified factorization scheme (like absence of(1/(1 − z))+ distributions in the real part of the NLO corrections)
Here I concentrate only on the first point.
37 / 45
Relation to other methods
LO
NLO
Inclusive distribution of gluons on thelog Sudakov plane of:
rapidity t = ξ
v = ln(1 − z)
NLO correction located near therapidity of the hard process t = ξ = tmax
complete phase space near (z = 0,t = tmax) corner
standard LO MCs feature empty“dead zone” (used by POWHEGand MC@NLO )
We can anticipate that the dominantcontribution to
∑j∈F WNLO
j is from thehardest in kT gluon (from LO shower)
38 / 45
kT–ordering & hardest emissionhardest in kT gluon rest
K = 1 component reproducesthe LO distribution over allthe region where NLOcorrection is non-negligible(hard process corner).
This represents the hardestemission of POWHEG.
NLO correction
39 / 45
kT–ordering & hardest emission
Comparison of the full NLO correction:∑
j WNLOj and its two hardest
(in kT ) components WNLOK=1 , WNLO
K=2 :
(x)10
log-3 -2.5 -2 -1.5 -1 -0.5 0
(x)
QN
LO
xD
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014NLO total and K=1,2 contribs.Tot.
K=1
K=2
x =∏
j zj
K = 1 saturates the entiresum very well
K = 2 component is small(additional NNLO terms)
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Ordering & hardest emission – summary
We can keep the sum∑
j WNLOj or restrict to WNLO
K=1 .
We use angular ordering but no vetoed/truncated showers areneeded (we just relabel already generated gluons)
In POWHEG scheme K = 1 gluon is generated separately in thefirst step, requiring the use of vetoed/truncated showers in caseof angular-ordered LO MC shower.
41 / 45
Outline
1 Introduction
2 Our projectRebuilding LO MC showerIncluding NLO corrections to hard processRelation to other methodsNLO corrections in the shower
42 / 45
NLO corrections in the MC ladder (gluons out of quarks)
gluon
quar
k
Z1−st order corrections
proton proton
P
P
P
P
P
P
P
PVirt W(k2, k1) =
∣∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣∣2
∣∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣∣2 =
∣∣∣∣∣∣∣∣∣∣ +2
1
2
1
∣∣∣∣∣∣∣∣∣∣2
∣∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣∣2 − 1
ρn(kl) = e−SISR
2
1
n
2
n−1
p
︸ ︷︷ ︸virtual + softcorrections
+
n∑p1=1
p1−1∑j1=1
2
p
n
1
1
j1
︸ ︷︷ ︸1 NLO real correction(insertion)
+
n∑p1=1
p1−1∑p2=1
p1−1∑j1=1
j1,p2
p2−1∑j2=1
j2,p1 ,j2
2
p
n
j2
j1
2p
1
︸ ︷︷ ︸2 NLO real corrections(insertions)
+ . . .
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NLO corrections in the MC ladder (gluons out of quarks)
ρn(kl) = e−SISR
2
1
n
2
n−1
p+
n∑p1=1
p1−1∑j1=1
2
p
n
1
1
j1
+
n∑p1=1
p1−1∑p2=1
p1−1∑j1=1
j1,p2
p2−1∑j2=1
j2,p1 ,j2
2
p
n
j2
j1
2p
1
+ . . .
From the numerical point of view one or two
NLO insertions2
1
are sufficient (restrict number of pi sums).
Sums over jk can be also restricted (like in the case of hardprocess) using variable
upj = ηp − ηj + λ ln(1 − zj)
and choosing only the hardest (or two hardest) in upj emissions.
44 / 45
Summary/Prospects
The “proof of concept” of the methodology is done!
3 we extended collinear factorization to be suitable for MC;
3 we constructed a new LO Parton Shower;
3 method for implementing NLO corrections in hard process is working;
3 we have a method for including NLO corrections in the showerladders;
but... this is done for non-singlet case (gluonstrahlung), now we need somemore work to go to the practical level
7 include singlet diagrams (easier)
7 optimize technique of adding NLO corrections to the ladders;
7 work on implementation of W/Z production at LHC(first: LO shower + NLO hard process);
45 / 45
BACKUP SLIDES
NLO Parton Shower
46 / 45
LO shower
Main features of new LO shower:based on modified collinear factorization;
conserving 4-momentum (also transverse component)giving perturbative series directly in terms of exponent notgeometrical series (avoiding oversubtractions)real emissions regularized geometrically in 4 dimensionssubtractions defined in unintegrated form (corresponding to MCcounterterms)
angular ordering;
cover the whole phase space – no gaps;
47 / 45
(re-)constructed LO shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ(C(0)0 Γ
(1)F Γ
(1)B ) =
∞∑n1=1
∞∑n2=1
σ[C(0)
0 (P′K(1)0F )n1(P′′K(1)
0B )n2]
T .O.
48 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
d0(t0, x0F) – initial PDF (preferably in MC scheme)
SF & SB – Sudakov formfactors
LO DGLAP kernel P(zBj) = 12 (1 + z2)
49 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
d0(t0, x0F) – initial PDF (preferably in MC scheme)
SF & SB – Sudakov formfactors
LO DGLAP kernel P(zBj) = 12 (1 + z2)
49 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
d0(t0, x0F) – initial PDF (preferably in MC scheme)
SF & SB – Sudakov formfactors
LO DGLAP kernel P(zBj) = 12 (1 + z2)
49 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
d0(t0, x0F) – initial PDF (preferably in MC scheme)
SF & SB – Sudakov formfactors
LO DGLAP kernel P(zBj) = 12 (1 + z2)
49 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
Ξ – Z-boson rapidity (division between F and B hemispheres)
ηi = 12 ln k+
k− – rapidity of emitted gluons (η0F > · · · > ηi−1 > ηi · · · > Ξ)
ki – rescaled 4-momenta
50 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
Ξ – Z-boson rapidity (division between F and B hemispheres)
ηi = 12 ln k+
k− – rapidity of emitted gluons (η0F > · · · > ηi−1 > ηi · · · > Ξ)
ki – rescaled 4-momenta
50 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
Ξ – Z-boson rapidity (division between F and B hemispheres)
ηi = 12 ln k+
k− – rapidity of emitted gluons (η0F > · · · > ηi−1 > ηi · · · > Ξ)
ki – rescaled 4-momenta
50 / 45
(re-)constructed LO MC shower
gluon
Born
quar
k
Z
proton proton
P
P
P
P
P
P
P
P
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ)
d3E(kj) = d3k2k0
1k2 = π
dφ2π
dk+
k+ dη – eikonal phase-space for real gluondσBdΩ
– LO hard process matrix-element
51 / 45
(re-)constructed LO shower
Analytical integration of LO MC distribution
over the multigluon phase-space gives the standard collinear formula:
σ0 =
∫ 1
0dxF dxB DF(t, xF) DB(t, xB) σB(sxFxB)
52 / 45
LO MC shower vs. collinear factorization
*
Wη-3 -2 -1 0 1 2 3
* Wη/ d
σ
d
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-610×
*
Wη-3 -2 -1 0 1 2 3
Rat
io C
MC
/Co
ll.
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
η∗W = 12 ln(xF/xB)
η∗W – in collinear limit –rapidity of W boson
agreement of < 0.5%
53 / 45
NLO correction to hard process
54 / 45
NLO correction to hard process
σ0 =
∫dx0Fdx0B d0(t0, x0F)d0(t0, x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−1
2CFαs
π2 P(zFi))δxF=x0F
∏n1i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−1
2CFαs
π2 P(zBj))δxB=x0B
∏n2j=1 zBj
× dτ2(P −n1+n2∑
j=1
kj; q1, q2)dσB
dΩ(sxFxB, θ) WNLO
MC
55 / 45
NLO correction to hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
soft+virtual NLO correction (kinematics independent!)
∆V+S =CFαs
π
(23π2 −
54
)
real correction (with subtraction)
β1(q1, q2, k) =[ (1 − β)2
2dσB
dΩq(s, θF) +
(1 − α)2
2dσB
dΩq(s, θB)
]−θα>β
1 + (1 − α − β)2
2dσB
dΩq(s, θ) − θα<β
1 + (1 − α − β)2
2dσB
dΩq(s, θ).
summation over all partons!
56 / 45
NLO correction to hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
soft+virtual NLO correction (kinematics independent!)
∆V+S =CFαs
π
(23π2 −
54
)
real correction (with subtraction)
β1(q1, q2, k) =[ (1 − β)2
2dσB
dΩq(s, θF) +
(1 − α)2
2dσB
dΩq(s, θB)
]−θα>β
1 + (1 − α − β)2
2dσB
dΩq(s, θ) − θα<β
1 + (1 − α − β)2
2dσB
dΩq(s, θ).
summation over all partons!
56 / 45
NLO correction to hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
soft+virtual NLO correction (kinematics independent!)
∆V+S =CFαs
π
(23π2 −
54
)
real correction (with subtraction)
β1(q1, q2, k) =[ (1 − β)2
2dσB
dΩq(s, θF) +
(1 − α)2
2dσB
dΩq(s, θB)
]−θα>β
1 + (1 − α − β)2
2dσB
dΩq(s, θ) − θα<β
1 + (1 − α − β)2
2dσB
dΩq(s, θ).
summation over all partons!
56 / 45
NLO correction to hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
soft+virtual NLO correction (kinematics independent!)
∆V+S =CFαs
π
(23π2 −
54
)
real correction (with subtraction)
β1(q1, q2, k) =[ (1 − β)2
2dσB
dΩq(s, θF) +
(1 − α)2
2dσB
dΩq(s, θB)
]−θα>β
1 + (1 − α − β)2
2dσB
dΩq(s, θ) − θα<β
1 + (1 − α − β)2
2dσB
dΩq(s, θ).
summation over all partons!
56 / 45
Summation in NLO weight for hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1, q2, kj)
P(zBj) dσB(s, θ)/dΩ
57 / 45
NLO weight distribution
h_wt1rEntries 100000
Mean 0.9851
RMS 0.04095
wt0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.05
0.1
0.15
0.2
0.25
0.3
-310×h_wt1r
Entries 100000
Mean 0.9851
RMS 0.04095
NLO weight
WNLOMC
Correcting to NLO requiresonly reweighting with WNLO
MCweight:
strongly peaked
positive
without long-ranged tails
Good behavior of WNLOMC is a
profit form reformulating thecollinear factorization andrebuilding LO shower.
58 / 45
NLO weight distribution
h_wt1rEntries 100000
Mean 0.9851
RMS 0.04095
wt0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.05
0.1
0.15
0.2
0.25
0.3
-310×h_wt1r
Entries 100000
Mean 0.9851
RMS 0.04095
NLO weight
WNLOMC
Correcting to NLO requiresonly reweighting with WNLO
MCweight:
strongly peaked
positive
without long-ranged tails
Good behavior of WNLOMC is a
profit form reformulating thecollinear factorization andrebuilding LO shower.
58 / 45
NLO correction to hard process
As in LO, the NLO MC distribution can be integrated analyticallyover the multigluon phase space giving standard collinear formula:
σ1 =
1∫0
dxF dxB dz DF(t, xF) DB(t, xB) σB(szxFxB)δz=1(1 + ∆S+V ) + C2r(z)
“coefficient function”: C2r(z) = −CFαsπ
(1 − z)
is different then the MS one: CMS2r (z) =
CFαsπ
1+z2
1−z [2 ln(1 − z) − ln(z)]
singular logarithmic terms ln(1−z)1−z of MS are absent
we have a build-in resummation of lnn(1−z)1−z terms
59 / 45
NLO correction to hard process
As in LO, the NLO MC distribution can be integrated analyticallyover the multigluon phase space giving standard collinear formula:
σ1 =
1∫0
dxF dxB dz DF(t, xF) DB(t, xB) σB(szxFxB)δz=1(1 + ∆S+V ) + C2r(z)
“coefficient function”: C2r(z) = −CFαsπ
(1 − z)
is different then the MS one: CMS2r (z) =
CFαsπ
1+z2
1−z [2 ln(1 − z) − ln(z)]
singular logarithmic terms ln(1−z)1−z of MS are absent
we have a build-in resummation of lnn(1−z)1−z terms
59 / 45
NLO correction to hard process – proof of concept
To test concept of our method we compare numerically inclusivedistributions obtained from the full scale MC generation (4-momentaconserving)
σ0 =
∫dx0Fdx0B d0(t0 , x0F )d0(t0 , x0B)
∞∑n1=0
∞∑n2=0
∫dxF dxB
× e−SF
∫Ξ<ηn1
( n1∏i=1
d3E(ki)θηi<ηi−12CFαs
π2 P(zFi))δ
xF =x0F∏n1
i=1 zFi
× e−SB
∫Ξ>ηn2
( n2∏j=1
d3E(kj)θηj>ηj−12CFαs
π2 P(zBj))δ
xB=x0B∏n2
j=1 zBj
× dτ2(P −n1+n2∑
j=1kj; q1 , q2)
dσBdΩ
(sxFxB , θ) WNLOMC
and from the collinear formula
σ1 =
1∫0
dxF dxB dz DF(t, xF) DB(t, xB) σB(szxFxB)δz=1(1 + ∆S+V ) + C2r(z)
60 / 45
NLO correction to hard process – proof of concept
*
Wη-3 -2 -1 0 1 2 3
NL
O(C
MC
), N
LO
(Co
ll.)
0
5
10
15
20
25
-910×
*
Wη-3 -2 -1 0 1 2 3
NL
O:
CM
C/C
oll.
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
η∗W = 12 ln(xF/xB)
η∗W – in collinear limit –rapidity of W boson
agreement of < 1%
61 / 45
NLO correction to hard process – proof of concept
*
Wη-3 -2 -1 0 1 2 3
LO
an
d N
LO
-810
-710
-610
*
Wη-3 -2 -1 0 1 2 3
Rat
io
NL
O/L
O
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
η∗W = 12 ln(xF/xB)
η∗W – in collinear limit –rapidity of W boson
NLO correction is only∼ 1.5% of LO!
62 / 45
Summary of presented scheme
NLO corrections to hard process added on top of the LO MCwith a simple, positive weight;
There is a built-in resummation of lnn(1−x)1−x terms;
Virtual+soft corrections ∆V+S are completely kinematicsindependent (all the complicated dΣc± contributions ofMC@NLO scheme are absent);
There is no need to correct for the difference in the collinearcounter-terms of the MC and MS scheme
but there is a price for it:
we needed to rebuild LO shower
deviations from MS factorization scheme means differentcoefficient function and PDFs (this is well controlled)
63 / 45
Summary of presented scheme
NLO corrections to hard process added on top of the LO MCwith a simple, positive weight;
There is a built-in resummation of lnn(1−x)1−x terms;
Virtual+soft corrections ∆V+S are completely kinematicsindependent (all the complicated dΣc± contributions ofMC@NLO scheme are absent);
There is no need to correct for the difference in the collinearcounter-terms of the MC and MS scheme
but there is a price for it:
we needed to rebuild LO shower
deviations from MS factorization scheme means differentcoefficient function and PDFs (this is well controlled)
63 / 45
Outline
1 Introduction
2 Our projectRebuilding LO MC showerIncluding NLO corrections to hard processRelation to other methodsNLO corrections in the shower
64 / 45
Relation to other methods
There are two well established methods for including NLOcorrections to hard process in MC shower:
POWHEG
MC@NLO
There is also an effort of Japanese group of H. Tanaka (arXiv:1106.3390)
In the following I show some of the conceptual differences betweenour scheme and POWHEG and MC@NLO methods (concentratingon POWHEG).
65 / 45
Relation to other methods
Main differences with respect to POWHEG and MC@NLO are:“Democratic” summation over all emitted gluons, withoutdeciding explicitly which gluon is involved in hard process
WNLOMC = 1 + ∆S+V +
∑j∈F
β1(q1 , q2 , kj)
P(zFj) dσB(s, θ)/dΩ+
∑j∈B
β1(q1 , q2 , kj)
P(zBj) dσB(s, θ)/dΩ
differences related to modified factorization scheme (like absence of(1/(1 − z))+ distributions in the real part of the NLO corrections)
Here I concentrate only on the first point.
66 / 45
Relation to other methods
For better visualization we restrict to one “ladder” and a simplifiedweight:
WNLOMC = 1 +
∑j∈F
β1(q1, q2, kj)
P(zFj) dσB(s, θ)/dΩ
67 / 45
Relation to other methods
LO
NLO
Inclusive distribution of gluons onthe log Sudakov plane of:
rapidity t = ξ
v = ln(1 − z)
NLO correction located near therapidity of the hard processt = ξ = tmax
complete phase space near(z = 0, t = tmax) corner
standard LO MCs featureempty “dead zone” (used byPOWHEG and MC@NLO )
68 / 45
kT–ordering & hardest emission
We can anticipate that the dominant contribution to
WNLOMC = 1 +
∑j∈F
WNLOj
comes from the gluon with the highest
ln kTj ∼ tj + ln(1 − zj)
that is the closest to the hard process corner (z = 0, t = tmax).
We can easily relabel gluons generated in MC (∑
j →∑
K), so they areordered in transverse momentum
κK+1 < κK , κK ∼ ln kTK
with K = 1 being the hardest one.
69 / 45
kT–ordering & hardest emissionhardest in kT gluon rest
K = 1 component reproducesthe LO distribution over allthe region where NLOcorrection is non-negligible(hard process corner).
This represents the hardestemission of POWHEG.
NLO correction
70 / 45
kT–ordering & hardest emission
Comparison of the full NLO correction:∑
j WNLOj and its two hardest
(in kT ) components WNLOK=1 , WNLO
K=2 :
(x)10
log-3 -2.5 -2 -1.5 -1 -0.5 0
(x)
QN
LO
xD
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014NLO total and K=1,2 contribs.Tot.
K=1
K=2
x =∏
j zj
K = 1 saturates the entiresum very well
K = 2 component is small(additional NNLO terms)
71 / 45
Ordering & hardest emission – summary
We can keep the sum∑
j WNLOj or restrict to WNLO
K=1 .
We use angular ordering but no vetoed/truncated showers areneeded (we just relabel already generated gluons)
In POWHEG scheme K = 1 gluon is generated separately in thefirst step, requiring the use of vetoed/truncated showers in caseof angular-ordered LO MC shower.
72 / 45
Angular ordered shower
73 / 45
Mapping (rescaling)
We order all gluons by the distance form Ξ – the rapidity of theproduced Z boson. We define permutation π:|ηπi − Ξ| > |ηπi−1 − Ξ|, i ∈ F,B
Mapping is now defined recursively as:
kπi = λikπi , λi =s(xi−1 − xi)
2(P −∑i−1
j=1 kπj) · kπi
, i = 1, 2, ..., n1 + n2.
Features of the mapping:
pure rescaling
preserves angels
preservers soft factors (dα/α)
no gaps in phase-space
74 / 45
LO MC algorithm
1 variables zF and zB are generated by the FOAM MC Sampler;2 four-momenta kµi are generated separately in the F and B parts of
the phase space with the constraints∑
j∈F αj = 1 − zF and∑j∈B βj = 1 − zB;
3 double-ordering permutation π is established;4 rescaling parameter λ1 is calculated; kπ1 = λ1kπ1 is set, such that
(P − kπ1)2 = sx1;5 parameter λ2 is calculated and kπ2 = λ2kπ2 is set, such that
(P − kπ1 − kπ2)2 = sx2 = szπ1zπ2 and so on... ;6 in the rest frame of P = P −
∑j kπj four-momenta qµ1 and qµ2 are
generated according to the Born angular distribution.
75 / 45
NLO corrections in the MC ladder (gluons out of quarks)
gluon
quar
k
Z1−st order corrections
proton proton
P
P
P
P
P
P
P
PVirt W(k2, k1) =
∣∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣∣2
∣∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣∣2 =
∣∣∣∣∣∣∣∣∣∣ +2
1
2
1
∣∣∣∣∣∣∣∣∣∣2
∣∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣∣2 − 1
ρn(kl) = e−SISR
2
1
n
2
n−1
p+
n∑p1=1
p1−1∑j1=1
2
p
n
1
1
j1
+
n∑p1=1
p1−1∑p2=1
p1−1∑j1=1
j1,p2
p2−1∑j2=1
j2,p1 ,j2
2
p
n
j2
j1
2p
1
+ . . .
76 / 45
NLO corrections in the MC ladder (gluons out of quarks)
h_lxWTn1Entries 9.162e+08Mean -2.332RMS 0.568
log10(x)-3 -2.5 -2 -1.5 -1 -0.5 0
-510
-410
-310
-210
-110
1
h_lxWTn1Entries 9.162e+08Mean -2.332RMS 0.568
NLO DGLAP
log10(x)-3 -2.5 -2 -1.5 -1 -0.5 00.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
DGLAP NLO Excl/Incl
Numerical results for D(x,Q) from two Monte Carlos inclusive and exclusive.Blue curve is single NLO insertion, red curve is double insertion component.Evolution 10GeV→1TeV starting from δ(1 − x).The ratio demonstrates 3-digit agreement, in units of LO.
77 / 45
NLO corrections in the MC ladder (gluons out of quarks)
ρn(kl) = e−SISR
2
1
n
2
n−1
p+
n∑p1=1
p1−1∑j1=1
2
p
n
1
1
j1
+
n∑p1=1
p1−1∑p2=1
p1−1∑j1=1
j1,p2
p2−1∑j2=1
j2,p1 ,j2
2
p
n
j2
j1
2p
1
+ . . .
From the numerical point of view one or two
NLO insertions2
1
are sufficient (restrict number of pi sums).
Sums over jk can be also restricted (like in the case of hardprocess) using variable
upj = ηp − ηj + λ ln(1 − zj)
and choosing only the hardest (or two hardest) in upj emissions.
78 / 45
hardest emission in upj
distribution of LO gluons NLO correction
hardest in upj gluon (LO) rest of LO gluons
79 / 45
Gluon pair component of the NLO kernel, ∼ CFCA (FSR)
Virt
Straightforward inclusion of gluon pair diagram in the previousmethod would ruin Monte Carlo weight due to presence of Sudakovdouble logarithmic +SFSR in 2-real correction:∣∣∣∣∣∣ 2
1
∣∣∣∣∣∣2 =
∣∣∣∣∣∣ + +
∣∣∣∣∣∣2 −∣∣∣∣∣∣ 2
1
∣∣∣∣∣∣2and −SFSR in the virtual correction:∣∣∣∣∣∣
∣∣∣∣∣∣2 =(1 + 2<(∆ISR + VFSR)
) ∣∣∣∣∣∣ 1−z
z∣∣∣∣∣∣2 .
80 / 45
Gluon pair component of the NLO kernel, ∼ CFCA (FSR)
SOLUTION:
Resummation/exponentiation of FSR, see next slides for details of thescheme and numerical test of the prototype MC.
81 / 45
NLO FSR correction at the end of the ladder, ∼ CFCA
Additional NLO FSR correction at the end of the ladder:
e−SISR−SFSR
∞∑n,m=0
m∑r=1
21
n−1
n−2
r m
2
where Sudakov SFSR is subtracted in the virtual part:∣∣∣∣∣∣∣∣∣∣∣∣∣∣2
=(1 + 2<(∆ISR + VFSR−SFSR )
) ∣∣∣∣∣∣∣ 1−z
z∣∣∣∣∣∣∣2
.
and FSR counterterm is subtracted in the 2-real-gluon part:∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣2
=
∣∣∣∣∣∣∣ + +
∣∣∣∣∣∣∣2
−
∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣2
−
∣∣∣∣∣∣∣∣∣∣∣∣∣∣2
.
The miracle: both are free of any collinear or soft divergences!!!
82 / 45
ISR+FSR NLO corrections at end of the ladder
D[1]NS(x,Q) =
e−S∞∑
n,m=0
n−1
n−2
1
m21
2
+
n−1∑j=1
1 2n−1
n−2
1
j
m
2
+
m∑r=1
21
1
r m
2
n−1
n−2
= e−SISR
δx=1 +
∞∑n=1
( n∏i=1
∫Q>ai>ai−1
d3ηi ρ(1)1B (ki)
)e−SFSR
∞∑m=0
( m∏j=1
∫Q>anj>an(l−1)
d3η′j ρ(1)1V (k′j )
)
×
[β(1)
0 (zn) +
n−1∑j=1
W(kn, kj) +
m∑r=1
W(kn, k′r)]δx=
∏nj=1 xj
β(1)0 ≡
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣2
∣∣∣∣∣∣∣∣∣ 1−z
z∣∣∣∣∣∣∣∣∣2 , W(k2, k1) ≡
∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣2
∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣2
+
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣2 =
∣∣∣∣∣∣∣∣∣ + +
∣∣∣∣∣∣∣∣∣2
∣∣∣∣∣∣∣∣∣2
1
∣∣∣∣∣∣∣∣∣2
+
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣2 − 1.
83 / 45
BACKUP SLIDES
nCTEQ
84 / 45
85 / 45
Hessian method [JHEP 07 (2002) 012, arXiv:hep-ph/0201195]
Expand χ2 function around minimum, a0i ,
χ2 = χ20 +
∑ij
12
(ai − a0i )(aj − a0
j )(∂2χ2
∂yi∂yj
)0
86 / 45
Hessian method [JHEP 07 (2002) 012, arXiv:hep-ph/0201195]
Expand χ2 function around minimum, a0i , and diagonalize
χ2 = χ20 +
∑ij
12
(ai − a0i )(aj − a0
j )(∂2χ2
∂yi∂yj
)0
= χ20 +
∑i
z2i
87 / 45
Hessian method [JHEP 07 (2002) 012, arXiv:hep-ph/0201195]
Expand χ2 function around minimum, a0i , and diagonalize
χ2 = χ20 +
∑ij
12
(ai − a0i )(aj − a0
j )(∂2χ2
∂yi∂yj
)0
= χ20 +
∑i
z2i
Choose ∆χ2 = χ2 − χ20 value (defining 1-σ uncertainty),
ideal case ∆χ2 = 1realistic global analysis ∆χ2 ∼ 10 − 100
Construct error PDFs corresponding to each eigenvector direction:
f ±i = f (zi) = f (0, . . . , zi = ±
√∆χ2, . . . , 0)
zi = ±
√∆χ2
Calculate errors of observable X:
∆X =12
√∑i
[X(f +
i ) − X(f −i )]2
88 / 45
nCTEQ results
Nuclear correction factors(Q = 10GeV)
Ri(Pb) =f Pbi (x,Q)
f pi (x,Q)
different solution for d-valence& u-valence compared to EPS09& DSSZ
sea quark nuclear correctionfactors similar to EPS09
nuclear correction factorsdepend largely on underlyingproton baseline
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 ( , )
nCTEQEPS09DSSZ
10 3 10 2 10 1 1
( , )
nCTEQEPS09DSSZ
PRELIMINARY
89 / 45
nCTEQ results
Nuclear PDFs (Q = 10GeV)
x f Pbi (x,Q)
nCTEQ d-valence & u-valencesolution between HKN07 &EPS09
nCTEQ features largeruncertainties than previousnPDFs
better agreement betweendifferent groups (nPDFs don’tdepend on proton baseline)
10 3 10 2 10 1 10.0
0.1
0.2
0.3
0.4
0.5
( , )nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
( , )nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10
5
10
15
20
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.2
0.4
0.6
0.8
1.0
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.2
0.4
0.6
0.8
1.0
( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 1
0.0
0.2
0.4
0.6
0.8 ( , )
nCTEQHKN07EPS09DSSZ
10 3 10 2 10 1 10.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
( , )
nCTEQHKN07EPS09DSSZ
PRELIMINARY
90 / 45
nCTEQ vs. EPS09
nCTEQ
xup/Av (Q0) = xcu
1 (1 − x)cu2 ecu
3x(1 + ecu4 x)cu
5
xdp/Av (Q0) = xcd
1 (1 − x)cd2 ecd
3 x(1 + ecd4 x)cd
5
cuvk = cuv
k,0 + cuvk,1
(1 − A
−cuvk,2
)cdv
k = cdvk,0 + cdv
k,1
(1 − A
−cdvk,2
)
EPS09
up/Av (Q0) = Rv(x,A,Z) u(x,Q0)
dp/Av (Q0) = Rv(x,A,Z) d(x,Q0)
Rv =
a0 + (a1 + a2x)(e−x − e−xa ) x ≤ xa
b0 + b1x + b2x2 + b3x3 xa ≤ x ≤ xe
c0 + (c1 − c2x)(1 − x)−β xe ≤ x ≤ 1
10 3 10 2 10 1 1x0.0
0.1
0.2
0.3
0.4
0.5
(,
)
nCTEQNP14EPS09
10 3 10 2 10 1 1x0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7nCTEQNP14EPS09
we set: cdv1 = cuv
1
cdv2 = cuv
291 / 45
nCTEQ results
Structure function ratio
R =FFe
2 (x,Q)
FD2 (x,Q)
good data description
despite different u-valence &d-valence ratios are similar toEPS09
0.01 0.1 1
0.80
0.85
0.90
0.95
1.00
1.05
1.10
[±
]
=
nCTEQ ±
HKN07
EPS09
0.01 0.1 1
0.80
0.85
0.90
0.95
1.00
1.05
1.10
[±
]
=
nCTEQ ±
HKN07
EPS09
PRELIMINARY
92 / 45
K. Kovarık, DIS2013
K. Kovarık, DIS2013
K. Kovarık, DIS2013
Compatibility of nuclear corrections for νA and l±A DIS
Fit to l±A DIS and DY dataχ2/dof = 0.89
Fit to νA DIS data onlyχ2/dof = 1.33
96 / 45
PRL106, 122301 (2011), arXiv:1012.0286PRD80, 094004 (2009), arXiv:0907.2357
Compatibility of nuclear corrections for νA and l±A DIS
x
−210
−110 1
]A−
l 2R
[F
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.202
=5 GeV2
QA=56, Z=26
w = 0
w = 1/7
w = 1/2
w = 1
∞w =
x
−210
−110 1
]A
ν 2R
[F
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.202
=5 GeV2
QA=56, Z=26
w = 0
w = 1/7
w = 1/2
w = 1
∞w =
χ2 =∑
l±A data
χ2i +
∑νA data
wχ2i , w = 0, 1
7 ,12 , 1,∞
97 / 45
PRL106, 122301 (2011), arXiv:1012.0286PRD80, 094004 (2009), arXiv:0907.2357
Compatibility of nuclear corrections for νA and l±A DIS
It was argued in the literature that νA DIS data is compatiblewith l±A data:Paukkunen, Salgado, JHEP 07, 32 2010, arXiv:1004.3140
de Florian, Sassot, Stratmann, Zurita, PRD85, 074028 (2012), arXiv:1112.6324
However these results were obtained using uncorrelated errorsfor NuTeV experiment, which were crucial for the nCTEQconclusions PRL106, 122301 (2011).
w l±A χ2(/pt) νA χ2(/pt) total χ2(/pt)1-corr 708 736 (1.04) 3134 4246 (1.36) 4983 (1.30)1-uncorr 708 809 (1.14) 3110 3115 (1.00) 3924 (1.02)
98 / 45
Compatibility of nuclear corrections for νA and l±A DIS
Recent paperPaukkunen, Salgado, PRL110, 212301 (2013), arXiv:1302.2001suggests a method to renormalize the NuTeV data so that the tensionbetween νA and l±A data sets are removed
Instead of looking at the ratio of cross-sections Rν(x, y,E) ≡σνexp(x,y,E)
σνCTEQ6.6(x,y,E)
they suggest considering
Rν(x, y,E) ≡
σνexp(x, y,E)/Iνexp(E)
σνCTEQ6.6(x, y,E)/IνCTEQ6.6(E)
where
Iνexp(E) ≡∑
i∈fixed E
σexp,i(x, y,E) × Bi(x, y)
Bi(x, y) is size of the experimental (x, y)-bin
However this analysis still do not take into account correlated errorsprovided by NuTeV collaboration.
99 / 45
K. Kovarık, DIS2013
EPS09 framework [JHEP 04 (2009) 065, arXiv:0902.4154]
Parametrization (using CTEQ6.1M baseline)
f Ai (xN , µ0) = Ri(xN , µ0,A,Z) fi(xN , µ0)
Ri(x,A,Z) =
a0 + (a1 + a2x)(e−x − e−xa ) x ≤ xa
b0 + b1x + b2x2 + b3x3 xa ≤ x ≤ xe
c0 + (c1 − c2x)(1 − x)−β xe ≤ x ≤ 1
PDFs for nucleus (A,Z)
f (A,Z)i (x,Q) =
ZA
f p/Ai (x,Q) +
A − ZA
f n/Ai (x,Q)
(bound neutron PDF f n/Ai by isospin symmetry)
101 / 45
DS04 framework
Parametrization (GRV98 baseline, FFNS, Q20 = 0.4GeV2)
f p/Ai (xN ,Q2
0) =
∫ A
xN
dyy
Wi(y,A,Z) fi(xN
y,Q2
0).
allows for 0 < xN < AWeight functions Wi parameterize nuclear effects for valence, sea and gluon
Wv(y,A,Z) = A[avδ(1 − εv − y) + (1 − av)δ(1 − εv′ − y)
]+ nv
( yA
)αv (1 −
yA
)βv+ ns
( yA
)αs (1 −
yA
)βs
Ws(y,A,Z) = Aδ(1 − y) +asNs
( yA
)αs (1 −
yA
)βs
Wg(y,A,Z) = A δ(1 − y) +ag
Ng
( yA
)αg (1 −
yA
)βg
Nuclear structure functions are defined as usual
A FA2 (x,Q2) = Z Fp/A
2 (x,Q2) + (A − Z) Fn/A2 (x,Q2)
X In principle takes into account x > 1 region.X Convenient for working in Mellin space.? Unfortunately in the actual grids 0 < x < 1, and sum rules are consistent with (0, 1) range.
102 / 45
De Florian, Sassot, PRD 69, 074028 (2004)arXiv:hep-ph/0311227
DS04 framework
DS04 feature very good data description (DIS, DY, 420 data points) withχ2/ndof = 0.76
0.8
1
0.8
1
0.8
1
0.8
1
10-2
10-1
1
NMCE-139
He/D Be/D
C/D Al/D
Ca/DF2 /
FA
F2
FD
Fe/D
Ag/D
xN
Au/D
10-2
10-1
1
0.8
1
0.8
1
0.8
1
10-2
10-1
1
NMC
Be/C Al/C
Ca/C
F2 /
FA
F2
FC
Fe/C
Pb/C Sn/C
xN
10-2
10-1
1
103 / 45
Variables: DIS of nuclear target eA→ e′X
DIS variables in case on nucleons
in nucleus
Q2 ≡ −q2
xA ≡Q2
2 pA·q
pA – nucleus momentumxA ∈ (0, 1) – analog of Bjorken variable(fraction of the nucleus momentum carried by a nucleon)
Analogue variables for partons:
pN =pAA – average nucleon momentum
xN ≡Q2
2 pN ·q= A xA – parton momentum fraction with respect to the
average nucleon momentum pN
xN ∈ (0,A) – parton can carry more than the average nucleonmomentum pN .
104 / 45
e e′
q
A X
Motivations: proton Strange PDF
Before CTEQ6.6 proton PDFs it was assumedP. Nadolsky et al. PRD 78, 013004 (2008), arXiv:0802.0007
s(x) = s(x) ∼ κu(x) + d(x)
2, κ =
12
Underestimating s PDFuncertainty, as u, d are muchbetter constrained.
Neutrino-nucleon dimuondata (CCFR, NuTeV)
allowed to fit sPDF independently of u, dsea.
0.001 0.01 0.1 1.
0.9
1.
1.1
1.2 CTEQ6.6
CTEQ6.1
si(x)s0(x)
x
105 / 45
ATLAS strange measurement
ATLAS has used W/Z production to infer constraints on the strangequark distribution (2010 data, 35pb−1)G. Aad et al. (ATLAS Collaboration) Phys. Rev. Lett. 109, 012001 (2012), arXiv:1203.4051
for Q2 = 1.9 GeV2 and x = 0.023:
rs = 0.5(s + s)/d = 1.00+0.25−0.28
106 / 45
CMS Wc final states measurement
CMS measured ratios of cross-sections using 36pb−1 of dataCMS-PAS-EWK-11-013
arXiv:1310.1138
R±c =σ(W+c)σ(W−c)
= 0.92 ± 0.19(stat.) ± 0.04(sys.)
see also:W. J. Stirling, E. Vryonidou, Phys. Rev. Lett. 109, 082002 (2012), arXiv:1203.6781
107 / 45
Motivations: proton Strange PDF
large proton s-quark PDFuncertainty:
105
x104 103 102 101 1
0
2
4
6
8
10
s+su+d
NNPDF2.1
much more important in LHC thanin Tevatron
heavy quarks contributions moreimportantlarger rapidity rangehigh energy→ probes PDFs atsmall x
y3 2 1 0 1 2 3
/dy [
nb]
σ d
210
110
1
TevatronW+
du
dc
sc
su
y4 2 0 2 4
/dy [
nb]
σ d
210
110
1
10 LHC 7W+
du
dc
sc
su
108 / 45