(1) ncteq nuclear parton distributions and (2) nlo parton … · 2014-01-13 · (1) ncteq nuclear...

(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

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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

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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 ∼

)2[d + s] +

)2[u + c]

neutrino DIS

Fν2 ∼

[d + s + u + c

2 ∼[d + s + u + c

3 ∼ 2[d + s − u − c

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

(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 +

(ai − a0i )(aj − a0

j )(∂2χ2

∂yi∂yj

= χ20 +

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

9 / 45

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) =

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

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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 +

Hij(ai − a0i )(aj − a0

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

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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

1.4 ( , )

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

1.4 ( , )

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

1.4 ( , )

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

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

( , )nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.0

10 3 10 2 10 1 10

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.1

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.0

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.0

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 1

0.8 ( , )

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.0

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

nCTEQ ±

0.01 0.1 1

nCTEQ ±

PRELIMINARY

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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

nCTEQ ±

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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

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NLO Parton Shower MC

NLO Parton Shower MCin collaboration with:S. Jadach, M. Skrzypek

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[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.

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[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

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Monte Carlo event generators

Complex finalstates in full detail

Arbitraryobservables andcuts from finalstates

23 / 45[JHEP 02 (2009) 007, arXiv:0811.4622]

LHC ”event”

• proton PDF• fits to data at

low scale (NNLO)

• parton shower• evolution eq.

(NNLO)•Monte Carlo

• hard process• fixed order

(NLO, NNLO)•Monte Carlo

• 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 =∑

∫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 =∑

∫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

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

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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

proton proton

(re-)constructed LO shower

Analytical integration ofthe LO MC distribution

over the multigluon phase-space gives the standard collinear formula:

0dxF dxB DF(t, xF) DB(t, xB) σB(sxFxB)

Wη-3 -2 -1 0 1 2 3

* Wη/ d

-610×

Wη-3 -2 -1 0 1 2 3

η∗W = 12 ln(xF/xB) – in

collinear limit – rapidity of Wboson

agreement of < 0.5%

29 / 45

Outline

1 Introduction

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NLO correction to hard process

Because LO shower covers thewhole phase space the NLOcorrections can be included bymeans of simple reweightingtechniques

∫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∑

kj; q1, q2)dσB

dΩ(sxFxB, θ) WNLO

31 / 45

NLO shower – proof of concept

Analytical integration ofthe NLO MC distribution

∫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

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

-910×

Wη-3 -2 -1 0 1 2 3

η∗W = 12 ln(xF/xB) – in

collinear limit – rapidity of Wboson

agreement of < 1%

32 / 45

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

(23π2 −

real correction (with subtraction)

β1(q1, q2, k) =[ (1 − β)2

dΩq(s, θF) +

(1 − α)2

dΩq(s, θB)

]−θα>β

1 + (1 − α − β)2

dΩq(s, θ) − θα<β

1 + (1 − α − β)2

dΩq(s, θ).

summation over all partons!

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Summation in NLO weight for hard process

∑j∈F

β1(q1, q2, kj)

∑j∈B

β1(q1, q2, kj)

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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

-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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1−st order corrections

proton proton

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

∑j∈F

β1(q1, q2, kj)

∑j∈B

β1(q1, q2, kj)

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.

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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)

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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

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kT–ordering & hardest emission

Comparison of the full NLO correction:∑

j WNLOj and its two hardest

(in kT ) components WNLOK=1 , WNLO

log-3 -2.5 -2 -1.5 -1 -0.5 0

0.014NLO total and K=1,2 contribs.Tot.

x =∏

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

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.

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Outline

1 Introduction

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NLO corrections in the MC ladder (gluons out of quarks)

Z1−st order corrections

proton proton

PVirt W(k2, k1) =

∣∣∣∣∣∣∣∣∣∣2

∣∣∣∣∣∣∣∣∣∣2 =

∣∣∣∣∣∣∣∣∣∣ +2

∣∣∣∣∣∣∣∣∣∣2

∣∣∣∣∣∣∣∣∣∣2 − 1

ρn(kl) = e−SISR

︸︷︷︸virtual + softcorrections

n∑p1=1

p1−1∑j1=1

︸︷︷︸1 NLO real correction(insertion)

n∑p1=1

p1−1∑p2=1

p1−1∑j1=1

p2−1∑j2=1

j2,p1 ,j2

︸︷︷︸2 NLO real corrections(insertions)

+ . . .

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ρn(kl) = e−SISR

n∑p1=1

p1−1∑j1=1

n∑p1=1

p1−1∑p2=1

p1−1∑j1=1

p2−1∑j2=1

j2,p1 ,j2

+ . . .

From the numerical point of view one or two

NLO insertions2

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.

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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

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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

proton proton

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]

48 / 45

(re-)constructed LO MC shower

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

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

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

kj; q1, q2)dσB

dΩ(sxFxB, θ)

49 / 45

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

kj; q1, q2)dσB

dΩ(sxFxB, θ)

49 / 45

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

kj; q1, q2)dσB

dΩ(sxFxB, θ)

49 / 45

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

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

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

kj; q1, q2)dσB

dΩ(sxFxB, θ)

ηi = 12 ln k+

50 / 45

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

kj; q1, q2)dσB

dΩ(sxFxB, θ)

ηi = 12 ln k+

50 / 45

proton proton

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

kj; q1, q2)dσB

dΩ(sxFxB, θ)

d3E(kj) = d3k2k0

1k2 = π

dφ2π

k+ dη – eikonal phase-space for real gluondσBdΩ

– LO hard process matrix-element

51 / 45

Analytical integration of LO MC distribution

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

-610×

Wη-3 -2 -1 0 1 2 3

η∗W = 12 ln(xF/xB)

η∗W – in collinear limit –rapidity of W boson

agreement of < 0.5%

53 / 45

54 / 45

∞∑n1=0

∞∑n2=0

∫dxF dxB

× e−SF

∫Ξ<ηn1

( n1∏i=1

2CFαs

π2 P(zFi))δxF=x0F

∏n1i=1 zFi

× e−SB

∫Ξ>ηn2

( n2∏j=1

2CFαs

π2 P(zBj))δxB=x0B

∏n2j=1 zBj

× dτ2(P −n1+n2∑

kj; q1, q2)dσB

dΩ(sxFxB, θ) WNLO

55 / 45

∑j∈F

β1(q1, q2, kj)

∑j∈B

β1(q1, q2, kj)

∆V+S =CFαs

(23π2 −

β1(q1, q2, k) =[ (1 − β)2

dΩq(s, θF) +

(1 − α)2

dΩq(s, θB)

]−θα>β

1 + (1 − α − β)2

dΩq(s, θ).

56 / 45

∑j∈F

β1(q1, q2, kj)

∑j∈B

β1(q1, q2, kj)

∆V+S =CFαs

(23π2 −

β1(q1, q2, k) =[ (1 − β)2

dΩq(s, θF) +

(1 − α)2

dΩq(s, θB)

]−θα>β

1 + (1 − α − β)2

dΩq(s, θ).

56 / 45

∑j∈F

β1(q1, q2, kj)

∑j∈B

β1(q1, q2, kj)

∆V+S =CFαs

(23π2 −

β1(q1, q2, k) =[ (1 − β)2

dΩq(s, θF) +

(1 − α)2

dΩq(s, θB)

]−θα>β

1 + (1 − α − β)2

dΩq(s, θ).

56 / 45

∑j∈F

β1(q1, q2, kj)

∑j∈B

β1(q1, q2, kj)

∆V+S =CFαs

(23π2 −

β1(q1, q2, k) =[ (1 − β)2

dΩq(s, θF) +

(1 − α)2

dΩq(s, θB)

]−θα>β

1 + (1 − α − β)2

dΩq(s, θ).

56 / 45

Summation in NLO weight for hard process

∑j∈F

β1(q1, q2, kj)

∑j∈B

β1(q1, q2, kj)

57 / 45

Mean 0.9851

RMS 0.04095

wt0 0.2 0.4 0.6 0.8 1 1.2 1.4

-310×h_wt1r

Entries 100000

Mean 0.9851

RMS 0.04095

NLO weight

WNLOMC

MCweight:

strongly peaked

positive

58 / 45

Mean 0.9851

RMS 0.04095

wt0 0.2 0.4 0.6 0.8 1 1.2 1.4

-310×h_wt1r

Entries 100000

Mean 0.9851

RMS 0.04095

NLO weight

WNLOMC

MCweight:

strongly peaked

positive

58 / 45

As in LO, the NLO MC distribution can be integrated analyticallyover the multigluon phase space giving standard collinear formula:

“coefficient function”: C2r(z) = −CFαsπ

(1 − z)

is different then the MS one: CMS2r (z) =

CFαsπ

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

As in LO, the NLO MC distribution can be integrated analyticallyover the multigluon phase space giving standard collinear formula:

“coefficient function”: C2r(z) = −CFαsπ

(1 − z)

is different then the MS one: CMS2r (z) =

CFαsπ

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)

∫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

60 / 45

Wη-3 -2 -1 0 1 2 3

-910×

Wη-3 -2 -1 0 1 2 3

agreement of < 1%

61 / 45

Wη-3 -2 -1 0 1 2 3

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

64 / 45

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

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

∑j∈F

β1(q1 , q2 , kj)

∑j∈B

β1(q1 , q2 , kj)

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

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

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

We can anticipate that the dominant contribution to

WNLOMC = 1 +

∑j∈F

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

Comparison of the full NLO correction:∑

j WNLOj and its two hardest

(in kT ) components WNLOK=1 , WNLO

log-3 -2.5 -2 -1.5 -1 -0.5 0

0.014NLO total and K=1,2 contribs.Tot.

x =∏

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

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

Z1−st order corrections

proton proton

PVirt W(k2, k1) =

∣∣∣∣∣∣∣∣∣∣2

∣∣∣∣∣∣∣∣∣∣2 =

∣∣∣∣∣∣∣∣∣∣ +2

∣∣∣∣∣∣∣∣∣∣2

∣∣∣∣∣∣∣∣∣∣2 − 1

ρn(kl) = e−SISR

n∑p1=1

p1−1∑j1=1

n∑p1=1

p1−1∑p2=1

p1−1∑j1=1

p2−1∑j2=1

j2,p1 ,j2

+ . . .

76 / 45

h_lxWTn1Entries 9.162e+08Mean -2.332RMS 0.568

log10(x)-3 -2.5 -2 -1.5 -1 -0.5 0

h_lxWTn1Entries 9.162e+08Mean -2.332RMS 0.568

NLO DGLAP

log10(x)-3 -2.5 -2 -1.5 -1 -0.5 00.8

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

ρn(kl) = e−SISR

n∑p1=1

p1−1∑j1=1

n∑p1=1

p1−1∑p2=1

p1−1∑j1=1

p2−1∑j2=1

j2,p1 ,j2

+ . . .

From the numerical point of view one or two

NLO insertions2

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)

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

∣∣∣∣∣∣2 =

∣∣∣∣∣∣ + +

∣∣∣∣∣∣2 −∣∣∣∣∣∣ 2

∣∣∣∣∣∣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

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

∣∣∣∣∣∣∣2

∣∣∣∣∣∣∣ + +

∣∣∣∣∣∣∣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−1∑j=1

1 2n−1

m∑r=1

= 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

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣2 =

∣∣∣∣∣∣∣∣∣ + +

∣∣∣∣∣∣∣∣∣2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣2 − 1.

83 / 45

BACKUP SLIDES

84 / 45

85 / 45

Expand χ2 function around minimum, a0i ,

χ2 = χ20 +

(ai − a0i )(aj − a0

j )(∂2χ2

∂yi∂yj

86 / 45

χ2 = χ20 +

(ai − a0i )(aj − a0

j )(∂2χ2

∂yi∂yj

= χ20 +

87 / 45

χ2 = χ20 +

(ai − a0i )(aj − a0

j )(∂2χ2

∂yi∂yj

= χ20 +

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

1.4 ( , )

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

1.4 ( , )

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

1.4 ( , )

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

nCTEQEPS09DSSZ

10 3 10 2 10 1 1

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

10 3 10 2 10 1 10

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.1

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.0

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.0

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 1

0.8 ( , )

nCTEQHKN07EPS09DSSZ

10 3 10 2 10 1 10.0

nCTEQHKN07EPS09DSSZ

PRELIMINARY

90 / 45

nCTEQ vs. EPS09

xup/Av (Q0) = xcu

1 (1 − x)cu2 ecu

3x(1 + ecu4 x)cu

xdp/Av (Q0) = xcd

1 (1 − x)cd2 ecd

3 x(1 + ecd4 x)cd

cuvk = cuv

k,0 + cuvk,1

(1 − A

−cuvk,2

k = cdvk,0 + cdv

(1 − A

−cdvk,2

up/Av (Q0) = Rv(x,A,Z) u(x,Q0)

dp/Av (Q0) = Rv(x,A,Z) d(x,Q0)

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

nCTEQNP14EPS09

10 3 10 2 10 1 1x0.0

0.7nCTEQNP14EPS09

we set: cdv1 = cuv

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

nCTEQ ±

0.01 0.1 1

nCTEQ ±

PRELIMINARY

92 / 45

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

−210

−110 1

=5 GeV2

QA=56, Z=26

w = 1/7

w = 1/2

∞w =

−210

−110 1

=5 GeV2

QA=56, Z=26

w = 1/7

w = 1/2

∞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

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

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)

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) =

f p/Ai (x,Q) +

A − ZA

f n/Ai (x,Q)

(bound neutron PDF f n/Ai by isospin symmetry)

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DS04 framework

Parametrization (GRV98 baseline, FFNS, Q20 = 0.4GeV2)

f p/Ai (xN ,Q2

Wi(y,A,Z) fi(xN

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)

)αv (1 −

)βv+ ns

)αs (1 −

Ws(y,A,Z) = Aδ(1 − y) +asNs

)αs (1 −

Wg(y,A,Z) = A δ(1 − y) +ag

)αg (1 −

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.

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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

NMCE-139

He/D Be/D

C/D Al/D

Ca/DF2 /

Be/C Al/C

Pb/C Sn/C

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 .

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e e′

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 =

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.

1.2 CTEQ6.6

CTEQ6.1

si(x)s0(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:

x104 103 102 101 1

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

TevatronW+

y4 2 0 2 4

10 LHC 7W+

108 / 45

(1) ncteq nuclear parton distributions and (2) nlo parton … · 2014-01-13 · (1) ncteq nuclear...

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