SOFT GLUON RESUMMATIONvs

PARTON SHOWER SIMULATIONS

Massimiliano Grazzini (INFN, Firenze)

MCWS workshop

LNF may 2006

Fixed order calculations

Parton Shower simulations

Predictions for the hard cross section can be obtained through:

We limit ourselves to considering hard scattering events:

Introduction: QCD hard scattering

the starting point is the factorization theorem

σ(p1, p2;Q2) =

∑a,b

∫ 1

0

dx1

∫ 1

0

dx2 fh1,a(x1, µ2

F ) fh2,b(x2, µ2

F )

×σab(x1p1, x2p2, αS(Q2), µ2

F )

All-order resummed calculations

Fixed order calculations

Truncate perturbative expansion at a given order in

It provides a systematic framework to compute the partonic cross section for an inclusive enough hard scattering process

+α2S(µR) σ(NNLO)(p1, p2;Q{Q1, ...};µF , µR) + ....

)

+αS(µR) σ(NLO)(p1, p2;Q{Q1, ...};µF , µR)

σ(p1, p2;Q{Q1, ...};µF ) = αkS(µR)

(

σ(LO)(p1, p2;Q{Q1, ...})

It is reliable only when all the scales are of the same order:if large logarithmic contributions of the formQ1 ! Q

(αSL2)n L ≡ lnQ1/Qwith arise that may spoil the

perturbative expansion

αS

In general: even if KLN theorem guarantees the cancellation of IR singularities, soft-gluon effects can still be large when real and virtual contributions are kinematically unbalanced

Look for an improved perturbative expansion when

αSL2 α

n

SL

2nα

2

SL

4

αSL α2

SL

3α

n

SL

2n−1

Leading Logs (LL)

Next to Leading Logs (NLL)

.....

.....

Soft-gluon resummations

αSL2 : one soft and one collinear log for each power of αS

αSL ∼ 1

................

Resummation is possible if the observable fulfills the property known as exponentiation

This implies two basic conditions:

Matrix element factorization

Phase space factorization

The first is a consequence of gauge invariance and unitarity: in soft and collinear limits the singular structure of QCD matrix elements can be factored out in a (universal) process independent manner

The second condition regards kinematics and depends on the cross section consideredIf this condition can be fulfilled (typically working in a conjugate space) resummation is feasible

Well known examples:

Event shapes and jet rates in e

+e−

Resummation of soft gluons near threshold

} Q2

z = Q2/s → 1

distributions in hadron collisions or Energy-Energy Correlation in

Work in Mellin N-space

Work in impact parameter b-space

s

Typically use Laplace transform

qT

e+e−

General structure of resummed cross section in conjugate space :

The functions control LL, NLL, NNLL contributions, respectively

Note: NLL terms formally suppressed by one power of with respect to LL The expansion is as systematic as the ordinary perturbative expansion

αS

Note: The terms coming from are of the same order as thosecoming from the combined effect of and

g1, g2, g3

H(1) g1

gi(λ = 0) = 0The are defined such that

H(αS) =∑

n

αn

SH

(n)

At NLL we have to include , but also g1 g2 H(1)

σres.ν ∼ αk

Sσ(LO)

ν H(αS) exp{L g1(αSLν) + g2(αSLν) + αSg3(αSLν) + ...}

Lν = ln ν

ν

g2

-independenthard coefficientν

How to combine resummed cross section to fixed order ?

matching procedure σ = σres.

+ σfin.

σfin. = σ

f.o.

− [σres.]f.o.

Start from resummed contribution which includes all the logarithmically enhanced terms

obtained by subtracting from the fixed order result the truncation of the resummed result at the same order: it does not contain large logarithmic terms

σres.

Define:

standard fixed order result

In this way the calculation is everywhere as good as the fixed order result but much better in the region where soft gluon effects are important

-

Parton showersProvide an all-order approximation of the partonic cross section in the soft and collinear regions

somewhat similar to resummed calculations

Make possible to include hadronization effects

What about logarithmic accuracy ?

Difficult matching with fixed order

Much more flexible, since they can give a fully exclusive description of the final state

No analytical information

+

+

-

The effect of quantum interferences is thus approximated by angular ordering constraint

The logarithmic accuracy achievable by parton showers is instead limited by quantum mechanics

Parton showers are essentially probabilistic: quantum interference cannot be taken into account

This problem is overcome by using colour coherence: soft gluon radiated at large angles distructively interfere

The extension to higher logarithmic accuracy is not necessarily feasible

Angular ordering allows to reach “almost” NLL accuracy(for inclusive enough observables)

Resummed calculations can be in principle performed to arbitrary logarithmic accuracy

Resummation: an explicit example

Consider the production of a vector boson at small transverse momentum

Two-scale problem:

The recoiling gluon is forced to be either soft or collinear to one of the incoming partons

large logarithmic contributions of the form appear

dσ

dqT

→ +∞

dσ

dqT

→ −∞

qT → 0 LO:

NLO:

as

Real radiation strongly inhibited: KLN cancellation still at work but

αn

Slog2n Q2/q2

T

spoiling the perturbative expansion

qT ! Q = MZ

qT

qT1

Single-gluon contribution in the small limitqT

ω = Eg/Q

=αS

2π

∫

dq2T1

q2T1

(

δ2(qT − qT1) − δ2(qT))

CF

∫ 1−qT1/Q

0

dz

(

2

1 − z− (1 + z)

)

A(1)

= CF

B(1)

= −

3

2CF

Write delta functions in b-space

∫

d2b

(2π)2eibqT

∫ Q2

0

dq2T1

q2T1

(

J0(qT1b) − 1)

(

A(1) logQ2

q2T1

+ B(1)

)

Performing irrelevant azimuthal integration

Real Virtual

z = 1 − ω

qT1 = θEg

=αS

2π

∫ Q2

0

dq2T1

q2T1

(

δ2(qT − qT1) − δ2(qT))

(

A(1) logQ2

q2T1

+ B(1) + . . .

)

O(qT1/Q)Neglect terms

Universal coefficients

=αS

2π

∫

d2b

(2π)2eibqT

∫ Q2

0

dq2T1

q2T1

(

e−ibqT1− 1

)

(

A(1) logQ2

q2T1

+ B(1)

)

αS

2π

qq splitting kernel

αS

2π

∫

dωdθ2

θ2

(

δ2(qT − qT1) − δ

2(qT ))

CF

1 + z2

1 − z

More gluons: in b-space kinematics fulfill exact factorization

δ2(qT − qT1 − . . .qTn) → e

ib(qT −qT1−...qT n) = eibqT

∏

i

e−ibqT i

Adding a 1/n! symmetry factor thesingle gluon contribution exponentiates

∫

d2b

(2π)2eibqT exp

{

∫ Q2

0

dq2T1

q2T1

(J0(bqT1) − 1)αS

2π

(

A(1) logQ2

q2T1

+ B(1)

)

}

Replacing αS → αS(q2

T1) we can control subleading effects

The resummed cross section is now finite as qT → 0

This is what we observe in the data !

G. Parisi, R. Petronzio (1979)G. Gurci, M.Greco, Y.Srivastava(1979)

J. Kodaira, L. Trentadue (1982) J. Collins, D.E. Soper, G. Sterman (1985)

F. Landry e al. (2003)

Z production at the Tevatron Run 1

with

An improved b-space formalismWe use b-space resummation and introduce some novel features

G. Bozzi, S. Catani, D. de Florian, MG (2005)

Parton distributions are factorized at

where the large logs are organized

as:

αS = αS(µR)

Unitarity constraint enforces correct total cross section

and

dσ(res.)ac

dq2T

=1

2

∫∞

0db b J0(bqT ) Wac(b, M, s;αS(µ2

R), µ2R, µ2

F )

WN (b, M ;αS(µ2

R), µ2

R, µ2

F ) = HN

(

αS(µ2

R);M2/µ2

R, M2/µ2

F

)

× exp{GN (αS(µ2

R), bM ;M2/µ2

R, M2/µ2

F )}

GN (αS, bM ;M2/µ2R, M2/µ2

F ) = L g(1)(αSL)

+g(2)N

(αSL;M2/µ2R) + αS g(3)

N(αSL;M2/µ2

R, M2/µ2F ) + . . .

L = lnM2b2/b2

0L = ln

(

1 + M2b2/b2

0

)

µF ∼ M

The qt spectrum of the Higgs

We applied the formalism to compute the Higgs spectrum at the LHC

G. Bozzi, S. Catani, D. de Florian, MG (2003,2005)

NLL+LO and NNLL+NLO results with consistent study of theoretical uncertainties and high quality matching to fixed order

Integral of resummed spectra reproduces the correct NLO and NNLO total cross sections

Calculation implemented in the fortran code HqT available at

http://theory.fi.infn.it/grazzini/codes.html

Higgs spectrum: comparison of different approaches

C. Balazs et al., Les Houches 2003

RESBOS: basically NLL+LO accuracy: NLO at large included though a K-factor

Kulesza et al.: joint resummation of transverse momentum and threshold corrections

qT

Berger et al.: basically (N)NLL+LO

Reasonable agreeement in shape but Pythia 6.2 considerably softer !

HERWIG 6.3 (no ME correction)

PYTHIA 6.2 (ME correction included)

Higgs spectrum: comparison of different approaches

C. Balazs et al., Les Houches 2003

RESBOS: basically NLL+LO accuracy: NLO at large included though a K-factor

Kulesza et al.: joint resummation of transverse momentum and threshold corrections

qT

Berger et al.: basically (N)NLL+LO

Reasonable agreeement in shape but Pythia 6.2 considerably softer !

HERWIG 6.3 (no ME correction)

PYTHIA 6.2 (ME correction included)

normalized to the same area

Nice agreement with MC@NLO NNLL effect tends to make the spectrum harder

NLO result (not shown) diverges to as+∞ pWWT → 0

WW : Resummation vs MC@NLOMG (2005)

Resummation effects generally small on leptonic observables

Effects seen only when hard cuts are applied

Recent progress: automated resummation

Resummed calculations usually worked out (when possible) analytically for each observable generally painful

Fixed order NLO calculations typically implemented in observable-independent parton levels MC codes

A. Banfi, G. Salam, G. Zanderighi (2003)

Automatic resummation of a large class of event-shape variables

Avoids the need to find the conjugate space in which the observable factorizes

To be done: matching with fixed order

CAESAR

Price to pay: more limited range of applicability and accuracy

Recent progress: non-global logsNon global observables: sensitive to emissions in only a part of phase space

These observables are affected by previously neglected single logarithmic contributions

M. Dasgupta, G. Salam (2001)

These effects are due to soft-gluon radiation at large angles difficult to take them into account in MC parton

shower

Recently: even “superleading” terms discovered in gap-between jets cross section at hadron colliders

J.R.Forshaw, A. Kyrieleis, M.H.Seymour (2006)

Resummed in closed form only in the large limitNc

Summary

Resummed calculations allowed us to push the validity of QCD perturbation theory to the boundary of the available phase space where fixed order predictions are not reliable

Resummed predictions are automatically provided by standard MC:

Much more flexible, since they can give a fully exclusive description of the final state

Make possible to include hadronization effects

Difficult matching with fixed order

Logarithmic accuracy often unclear

Difficult to estimate uncertainties

+

+

---

Analytical resummations provide the most advanced theoretical accuracies available at present

Easier to estimate uncertainties

Up to NNLL in some cases (threshold, , EEC)

Easy matching with fixed order

qT

Have to be worked out for each observable (but progress in automatization is being made)

Bottom line:

MC and analytical resummations are complementary !Analytical resummed calculation will be particularly helpful in the validation of MC simulation tools

-

+++

EXTRA SLIDES

MWWT

50 GeV

The distribution is completely off at NLO !The position of the peak is shifted by about

plTmin > 25 GeV 35 GeV < p

lTmax

< 50 GeV

pmiss

T > 20 GeV mll < 35 GeV∆φ < 45o

But try now with cuts used for Higgs search in theH → WW → lνlν

pjetT

< 30 GeV

channel