The Dipole-Antenna approach to Shower

Monte Carlo's

W. Giele, HP2 workshop, ETH Zurich, 09/08/06

• Introduction

• Color ordering and Antenna factorization

• Constructing the Dipole-Antenna based Sudakov

• Shower definition and Algorithmic implementation

• Matching to Fixed Order

• Conclusions

David Kosower, Peter Skands and W.G.

Introduction• Our main goal is to develop a shower which

has a straightforward matching to fixed order matrix elements at LO and NLO.

• A secondary goal is to make explicit uncertainties within the shower.

• To start we implemented a gluonic cascade shower (Hgluons) with LO/NLO matching to fixed order matrix elements.

• In the remainder of the talk the term leading logs is used for(all other are called sub-leading logs).

122 log&log nnS

nnS

Introduction

• Current matching schemes adapt the matrix elements to accommodate the existing shower MC’s:• MC@NLO (S. Frixione & B.R. Webber)• See talk P. Nason…• Add first branching analytic (D. Soper & Z. Nagy)• CKKW (S. Catani, F. Krauss, R. Kuhn & B.R. Webber)• “Mangano”-matching (M. Mangano)

• We want to construct a shower which will take the matrix elements “as-is” (i.e. no modifications needed).

Color Ordering• Both the shower and ordered amplitudes use

ordered amplitudes in the large Nc limit

• (At a later stage we will include the color suppressed terms into the hard matrix elements.)

2

)!1(

)23(

22

21

)!1(

)23(21

1)12(~)(

)12()(~)( 21

C

n

nPn

aan

nP

an

NnmgggHM

nmTTTTrgggHM n

M.Mangano, S. Parke & Z. XuF.A. Berends & W.G.Z. Bern & D.A. Kosower (@1-loop)

Antenna Factorization

• The antenna function combines both soft and collinear behavior for each dipole

• The collinear behavior is shared with neighboring dipoles: this lead to two types of antenna formalisms for ordered amplitudes

2

)1(211

2

11

2ˆˆˆ();;()12(

n

n

iiii pppmpppAnm

D.A. KosowerJ. Campbell & E.W.N. GloverT. Gehrmann, A. Gehrman-De Ridder & E.W.N. Glover

Sector Antenna functions• Sector Dipole-Antenna functions:

• Each antenna function has full soft and collinear contribution

• Phase space sectors split the collinear contribution over the neighboring dipoles

• Leads to an exact invertible shower in color ordered space (complete phase space coverage)

n

iiniiiiii

n

iin

iiiiiiiiiiiiiiiiiiiiii

ggggggiiii

ppppmyyAgggHM

ssyssyyyzyyy

zz

zzzzzPtermsfinitezP

yyyA

1

2111

21,,1

1

221

1,,11,1,1,,1,1,11,,111,

1,1

1

22222

1,,1

1;|)ˆˆˆ(||),(||)(|

/,/;,

)1(

)1()1()(;)(

1|),(|

D.A. Kosower

Global Antenna functions

• Global Dipole-Antenna functions (used in the remainder of this talk): • Each antenna function has the full soft and partial

collinear contribution• The sum over the neighboring dipoles has the

correct collinear behavior

2111

21,,1

1

221

22

1,,1

|)ˆˆˆ(||),(||)(|

)1()()(

2

)2)(1()(;)(

1|),(|

niiiiii

n

in

ggggggggg

ggggggiiii

ppppmyyAgggHM

zHzHzP

z

zzzHtermsfinitezH

yyyA

T. Gehrmann, A. Gehrman-de Ridder & E.W.N. Glover

G. Gufstafson & U. Peterson

Dipole-Antenna Sudakov’s• Given an antenna function we can construct

a dipole Sudakov which measures the probability no new dipole is resolved at a certain (ordered) resolution scale

• The resolution scale can be chosen in many ways:

“pt-ordering”

babaI yyyyR 1111 4),( ),min(2),( 1111 baba

II yyyyR abbabaIII yyyyyR 1111 27),(

“Virtuality-ordering”

Dipole-Antenna Sudakovs

• We can now calculate the probability of not resolving a new dipole:

• And the probability not to resolve any new dipole in the event

n

iiiRn

y

bababa

dipoleRdipoleR

QQn

yyARyyRdydyR

QQRQQ

a

s

1

21,

2

1

0

1

0

2

111111

2222

;)12(

),(),(exp)(

/;

1

2/

Dipole-Antenna Sudakov’s

(log scale)

In the shower MC the Sudakov is calculated at initialization phase from the numerically implemented antenna function and resolution scale.

Dipole-Antenna Shower• We can now define the Dipole-Antenna

based shower (in a MC@NLO notation):

n

iRisriiinRn

n

iRiriiinRn

nnnn

QppppppPOOQQ

QpppppPOOQQ

OOQQpOS

1

2112

2202

1

2111

2201

220

)|ˆ̂ˆ̂ˆ̂ˆ̂()ˆ̂

();(

)|ˆˆˆ()ˆ();(

)();()}{|(

Dipole-Antenna Shower

• We can formulate the leading log shower as a simple Markov chain suitable for numerical implementation (this is the master formula on which the shower is based):

""

)}ˆ{|()|ˆˆˆ(

)();()}{|(

11

211

220

Logsleadingnon

pOSQpppppP

OOQQpOSn

inRiriii

nnnn

Dipole-Antenna Shower

21,,

21,1,,

21,,

2

21,

22

11

|)ˆ,ˆ(|);ˆ,ˆ(ˆˆ

);()|ˆˆˆ(

22

irriniiirriRirriS

QQ

iiRiriii

yyAQyyQydyd

Q

QQQpppppP

R

• The branching probability density is simply the rate of change in the probability of not resolving a new dipole.

• That is the branching probability density is simply the derivative of the Sudakov at the resolution scale.

• With a simple Metropolis Algorithm one can pick the pairs of invariants according to this probability density.

Algorithmic Implementation(Vincia: VIrtual Numerical Collider Interactions

with Antennae) • Provide an antenna function, 23 momentum mapping scheme and a resolution function.

• For each dipole make a “trial” branching according to two dimensional probability distribution:

• Select the dipole which will resolve a new dipole first (i.e. the largest )

• Construct its branching momenta (imposing momentum conservation and on-shell conditions):

22/ |)ˆ,ˆ(|))ˆ,ˆ(())ˆ,ˆ(()ˆ,ˆ( rbarnrbarrbarSrbar yyAyyRssQssP s

)ˆ,ˆ(2rbarabR yyRsQ

11 ˆˆˆ araaa ppppp

Algorithmic Implementation• Angular distributions between the 3 leading jets (angle(j1,j2), angle(j1,j3), angle(j2,j3)) in 3,4,5,6,7,8 exclusive jet events.

• Distribution of the sum of the 3 angles.

• Kt-jet algorithm used with Yr=0.001; M=500 GeV

• 1,000,000 showered events (30 min to generate on laptop).

• (stacked histograms)

• (logarithmic vertical scale)

• Distributions rich in structure (which are all explainable…)

Algorithmic Implementation

• Same for energy of 3 leading jets

• (“standard” global antenna function)

• (Type I evolution variable, “Pt-ordering”)

Matching to Fixed Order

)()()(||)(

)(||);()();(

);()()}{|(

21

2

222211

21

201

220

2

20

2

20

Sdipoles

nn

Q

Q

nSn

Sdipoles

Q

Q

nnnnnnnS

nnnn

OOOOAOO

AQQOOQQ

QQOOpOS

m

m

22

21

22

21

)(||1)||exp();( 22222

21

Q

Q

iS

Q

Q

iSm AAQQ

• First we need to expand the shower in• The expanded event sudakov is given by

• Now we can expand out the shower function:

S

Matching to Fixed Order• An observable in the matched shower is given by

shower off the hard subtracted matrix element.• (we replace and the

matrix elements with the modified matrix element)• To find what the modified matrix elements are we

expand out the shower

)|(|)(~|

)|()(~

|)(|

112

111

112

1

nnLOSn

nnSnLOn

ppSggHmPSd

ppSggHVggHmPSdd

d

)|())(( 11 nn ppOSppOO

Matching to Fixed Order• We now expand the shower in the strong coupling

constant.• In the (n+1)-gluon contribution the shower function

is replaced by a delta function• In the n-gluon amplitude we get an additional term

from the expansion proportional to the Born term.

dipolesSnnnnLOnS

nnLOSnS

nnSnLOn

OOOOAggHmPSd

OOggHmPSd

OOggHVggHmPSdd

d

)()()(|||)(|

)(|)(~|

)()(~

|)(|

21

221

12

111

12

1

Matching to Fixed Order• After the expansion of the shower function we

recover the (unmodified) subtraction formalism for NLO calculations.

• We see that the small limit of the shower is identical to the NLO subtraction scheme.

)(|)(|

)()(|)(|

12

111

12

1

nnLOSnS

nnSnLOn

OOggHmPSd

OOggHVggHmPSdd

d

S

Matching to Fixed Order• The modified matrix elements are the usual

subtracted matrix elements.

dipolesnLOnnLOnLO

abbaba

y

baba

dipolesnnLOnn

ggHmAggHmggHm

syyAyyyyydyd

AggHmggHVggHV

a

21

2211

211

2

2

21

0

21111

ˆ1

0

1111

22111

|)ˆˆ(||||)(||)(~|

212

67

6

1114

)1(

1|ˆ,ˆ(|)]ˆˆ1(ˆˆ[ˆˆ

|||)(|)()(~

1

Matching to Fixed Order

• The matrix elements can be inserted in the shower program using any subtraction scheme.(The difference between 2 subtraction functions is finite and numerical calculable).

• Sub-leading logarithms (double unresolved at LO and unresolved at NLO) need to be regulated in a leading-log shower (by cut or Sudakov inspired regulation function)

Matching to Fixed Order

• 2, 3,… exclusive jet fractions as a function of the Kt-jet resolution parameter.

• Matching shower with fixed order strongly reduces the dependence on the (arbitrary) hard part (non soft/collinear) of the antenna function.

• Being able to change the shower hardness we can see the importance of matching

• We can also estimate the residual uncertainties within the leading log approximation

H2 gluons + shower H2,3 gluons + shower

Conclusions

• We implemented a timelike gluonic shower based on antenna-dipole functions

• The shower:• resums all leading logarithms . • the shower is a 23 brancher which keeps the partons on-shell

and maintains energy-momentum conservation at each stage of the shower evolution.

• Phase space is covered completely in a process independent way.

• numerical implementation of antenna function, evolution variable and 23 mapping allow for an estimate of the uncertainties within the leading log shower.

• in the small coupling constant limit the shower is the NLO subtraction scheme; this allows “as-is” insertion of matrix elements.

122 log&log nnS

nnS

Future: To-Do…

• Constructing of space-like showers for hadron colliders. (Crossing is straightforward because of Lorentz invariant formulation.)

• Adding in the quarks (massless and massive).• Adding in the color suppressed logarithms.• Both a stand-alone version and a PYTHIA

module will be provided.• In this shower implementation it seems very

promising to go beyond leading logs. This needs to be explored…