Calculating Higher Order Calculating Higher Order Corrections: An UpdateCorrections: An Update

Lance DixonLance DixonALCPG04 Theorist’s Rally for the LCALCPG04 Theorist’s Rally for the LC

SLAC, 1/7/04SLAC, 1/7/04

OutlineOutline

NLO (EW+QCD) corrections to Higgs productionNLO (EW+QCD) corrections to Higgs production

NNLO QCD corrections to event shapesNNLO QCD corrections to event shapes

(prospects)(prospects)

HttHZHee , ,

... EEC, thrust,jets,-3 ee

NLO corrections to Higgs productionNLO corrections to Higgs production

Important for extracting Higgs couplingsImportant for extracting Higgs couplings

Electroweak corrections at 500 GeV can be ~ 10% Electroweak corrections at 500 GeV can be ~ 10%

Processes Considered:Processes Considered:

ZHee

Hvee

Httee

Higgstrahlung: gHZZ

WW fusion: gHWW

Top-associated production: gHtt

NLO NLO ZHZH and and HH production production

Belanger et al., hep-ph/0212261

H process is particularlychallenging due to:1) large number of virtual diagrams (1350 in all; “only” 249 lacking e)2) complicated 5-point integrals(98 in all; “only” 15 lacking e)

First H calculation used GRACE-LOOPto generate, evaluate loop diagrams;nonlinear gauge for diagram checking.

Second independent calculation:

One-loop corrections are all electroweak.

Denner et al., hep-ph/0301189

NLO NLO ZHZH and and HH production results production results

Denner et al., hep-ph/0301189

Belanger et al., hep-ph/0211268, hep-ph/0212261

IBA scheme (?): set to (MZ);similarly for g2(MW)

“Blips” due to 2-particle intermediatestate rescatterings (WW, ZZ, tt).

Largest corrections to IBA are nearend of phase space, where Borncross section varies rapidly (and is tiny anyway).

ZHZH experimental accuracy experimental accuracy

Conclusion: Theoretical corrections to (e+e- -> ZH) are under good control;they only get large (vs. IBA) whereexperimental accuracy suffers.

Is the same true of WW fusion process?Probably, at least at 500 GeV…

hep-ph/0311092

NLO NLO ttHttH production production

Dawson, Reina, hep-ph/9808443QCD corrections were computed first.“Standard” NLO QCD formalism, butalso IR divergent 5-point integrals.

They are modest away from threshold.

Electroweak corrections turn out to be comparable

Now there are QCD and electroweak corrections.

Dittmaier et al., hep-ph/9808433

Belanger et al., hep-ph/0307029

Denner et al., hep-ph/0307193, hep-ph/0309274

You et al., hep-ph/0306036

(latter two agree well)

-

ttHttH experimental accuracy experimental accuracy

hep-ph/0311092

Conclusion: Theoretical corrections to (e+e- -> ttH) are also now under good control: 10-20% NLO corrections are ~ experimental uncertainty in g2

ttH

-

ProspectsProspects for NNLO corrections to for NNLO corrections to ee++ee-- event shapes event shapes

Important for extracting a preciseImportant for extracting a precise ss

at large momentum transferat large momentum transfer QQ

Many fewer hadronic events than at the Many fewer hadronic events than at the ZZ resonance resonance

Also more backgroundsAlso more backgrounds

However, nonperturbative correctionsHowever, nonperturbative corrections ~~ 1/Q1/Q

MotivationMotivation

ee++ee- - annihilation to hadrons is experimentallyannihilation to hadrons is experimentally cleanclean– AtAt full energy LCfull energy LC, not, not quitequite as clean as atas clean as at MMZZ::

– UseUse eeRR-- to suppressto suppress WWWW backgroundbackground

– UseUse anti-b-tagging anti-b-tagging to suppressto suppress tttt background background

(Schumm, hep-ex/9612013; Schumm, Truitt, hep-ex/0102010; (Schumm, hep-ex/9612013; Schumm, Truitt, hep-ex/0102010;

Burrows, hep-ex/0112036)Burrows, hep-ex/0112036)

Excellent opportunity forExcellent opportunity for high high QQ22, clean, clean ss(Q)(Q)

measurementmeasurement– ss(M(MZZ)) is biggestis biggest uncertainty uncertainty now in testing GUT coupling now in testing GUT coupling

constant unification.constant unification.

– ss(M(MZZ)) an important input into other LC physics;an important input into other LC physics;

e.g.e.g. extraction of extraction of mmtt fromfrom tttt threshold scan.threshold scan.

With With Giga-Giga-ZZ (or (or LEP/SLCLEP/SLC archivalarchival data), compare data), compare

full energy results tofull energy results to MMZZ results to fit out nonperturbative results to fit out nonperturbative

((1/Q1/Q) corrections) corrections

Need for NNLONeed for NNLO

Measurements ofMeasurements of s s in 3-jet rate, other ein 3-jet rate, other e++ee- - event event

shapes, dominated byshapes, dominated by theoretical uncertainty theoretical uncertainty due due mainly to truncation of perturbative series atmainly to truncation of perturbative series at NLONLO::

2

0

)( )/ln(4

)(

s

iis

ii AQBAO

Fit toFit to NLONLO theory leads to:theory leads to:

large scatterlarge scatter inin s s values from different event shapes,values from different event shapes,

large renormalization scale dependencelarge renormalization scale dependence

Burrows, hep-ex/9709010

3

)( ...

s

iC

What about NLO EW corrections?What about NLO EW corrections?Sudakov effects lead to growth withSudakov effects lead to growth with QQ::

Result for Thrust distribution at 1 TeV:Result for Thrust distribution at 1 TeV: W

W MQ /ln2

However, it depends on how result isHowever, it depends on how result is normalizednormalized;; normalizing tonormalizing to instead ofinstead of will givewill give much smallermuch smaller EW EW

corrections, except atcorrections, except at very large 1-Tvery large 1-T. . – True momentum scale isTrue momentum scale is ~~ (1-T)(1-T)22QQ..

Maina et al., hep-ph/0210015

NNLO QCD Diagram TypesNNLO QCD Diagram Types

Status of NNLO Diagrams (Amplitudes)Status of NNLO Diagrams (Amplitudes)

Remaining Obstacle: Remaining Obstacle: Phase-space integrationPhase-space integration

3 types of terms, each contributes3 types of terms, each contributes 1/1/4 4 + … + … singularitiessingularities

using dimensional regularization,using dimensional regularization, D=4-2D=4-2, for all integrals., for all integrals.– 5 partons5 partons, tree x tree, tree x tree

– 4 partons4 partons, 1-loop x tree, 1-loop x tree

– 3 partons3 partons, 2-loop x tree and 1-loop x 1-loop, 2-loop x tree and 1-loop x 1-loop

3-parton3-parton phase-space integrationsphase-space integrations trivialtrivial (all partons visible)(all partons visible)

5-partons5-partons (doubly unresolved) the(doubly unresolved) the hardesthardest. .

Several types of singular regions, behavior ofSeveral types of singular regions, behavior of understood:understood:– double softdouble soft Berends, Giele (1989) Berends, Giele (1989)

– 2 collinear + 1 soft2 collinear + 1 soft Campbell, Glover, hep-ph/9710255 Campbell, Glover, hep-ph/9710255

– 3 collinear3 collinear Catani, Grazzini, hep-ph/9810389, 9908523 Catani, Grazzini, hep-ph/9810389, 9908523

– 2 collinear, twice2 collinear, twice Del Duca, Frizzo, Maltoni, hep-ph/9909464 Del Duca, Frizzo, Maltoni, hep-ph/9909464

– plus overlapsplus overlaps

Recent Advances in Recent Advances in Phase-space integrationPhase-space integration

Analytic behavior ofAnalytic behavior of 5-parton5-parton cross section incross section in individual individual singular regions understood for a while;singular regions understood for a while;

more recently formulasmore recently formulas interpolatinginterpolating between the differentbetween the different

regions have been found:regions have been found: – Kosower, hep-ph/0301069; Weinzierl, hep-ph/0302180, 0306248.Kosower, hep-ph/0301069; Weinzierl, hep-ph/0302180, 0306248.

If such formulae were simple enough to integrate If such formulae were simple enough to integrate analyticallyanalytically, they could be, they could be subtractedsubtracted from the exactfrom the exact

integrand, and the finiteintegrand, and the finite difference difference integrated numericallyintegrated numerically

over phase-space in 4-dimensions.over phase-space in 4-dimensions.

However, this has not yet been done.However, this has not yet been done.

Recent Advances (cont.)Recent Advances (cont.)

More recently, the technique ofMore recently, the technique of sector decompositionsector decomposition,,

Binoth, Heinrich, hep-ph/0004013Binoth, Heinrich, hep-ph/0004013

originally developed for originally developed for loop integralsloop integrals, has been applied , has been applied

toto doubly unresolved phase-space integralsdoubly unresolved phase-space integrals::– Gehrmann-De Ridder et al., hep-ph/0311276Gehrmann-De Ridder et al., hep-ph/0311276– Anastasiou et al., hep-ph/0311311Anastasiou et al., hep-ph/0311311

Anastasiou et al. Anastasiou et al. have been able to compute integrals for thehave been able to compute integrals for the NNLONNLO ee++ee-- -> 2- -> 2-jetjet raterate in this way.in this way.

To expand a multi-dimensional singular integralTo expand a multi-dimensional singular integral

such assuch as

first split intofirst split into sectorssectors:: x > y x < yx > y x < y

Sector decompositionSector decomposition

One-dimensional singular integrals are easily One-dimensional singular integrals are easily expanded in expanded in using “plus” distributions:using “plus” distributions:

wherewhere

x

x

nxxdxI

n

n

n ln

!)(

1

0

11

0

0

)( 111

0

yxyxdydxI

1

0

1

0

)(ln

)0()( )(ln

)( x

xfxfdx

x

xxfdx

nn

)( 11

0

1

0

1 yxyxdydxIx

)( 11

0

1

0

2 yxyxdxdyIy

Now that the singularities are Now that the singularities are separatedseparated, expansion in, expansion in can be carried out as in 1-dimensional case. can be carried out as in 1-dimensional case.

Anastasiou et al. found a parametrization ofAnastasiou et al. found a parametrization of

4-parton phase space4-parton phase space (5 variables) (5 variables)

which is amenable to which is amenable to automated sector decompositionautomated sector decomposition::

Sector decomposition (cont.)Sector decomposition (cont.)

Let Let y = y’xy = y’x in in II11 x = x’yx = x’y inin II22

)1()( '1'31'1

0

1 yyxdydxI )1()( '1'311

0

'2 xxydydxI

As an example, they numerically integrated the As an example, they numerically integrated the 4-parton 4-parton contribution to the contribution to the NNff-dependent part of the -dependent part of the

NNLO NNLO 2-jet rate2-jet rate (JADE algorithm, y=0.1)(JADE algorithm, y=0.1) order-by-order in order-by-order in ::

Sector decomposition (cont.)Sector decomposition (cont.)

Adding the (2,3)-parton contributions, the Adding the (2,3)-parton contributions, the poles cancel, poles cancel,

and the right answer is obtained (and the right answer is obtained (tottot is known is known))

Most important lessons of this technique:Most important lessons of this technique:– It is It is not necessarynot necessary (for a (for a humanhuman) to understand the behavior of the) to understand the behavior of the

integrand in the various singular regions.integrand in the various singular regions.– It is It is not necessarynot necessary to integrate singular terms to integrate singular terms analyticallyanalytically to verify to verify

cancellations.cancellations.

Sector decomposition (cont.)Sector decomposition (cont.)

Method looks generalizable to the Method looks generalizable to the NNLONNLO 3-jet3-jet problem problem– Either by a direct attack on the Either by a direct attack on the 5-particle5-particle phase space phase space

((5-variables5-variables -> -> 8-variables8-variables))– Or by using it to integrate Or by using it to integrate subtraction terms subtraction terms over 4-particle over 4-particle

subspaces. subspaces.

Much recent progress in computing NLO Much recent progress in computing NLO QCDQCD and and particularly particularly electroweak electroweak corrections for important corrections for important multi-body multi-body final-state final-state electroweak electroweak processes at a linear collider. processes at a linear collider.

Here, surveyed a few relevant for precision extraction of Here, surveyed a few relevant for precision extraction of Higgs Higgs couplings. With the one-loop EW corrections,couplings. With the one-loop EW corrections,

theoretical precision generally appears adequate.theoretical precision generally appears adequate.

For precision measurement of For precision measurement of ss, , NNLONNLO QCDQCD corrections corrections

to to hadronic event shapeshadronic event shapes are required, but not yet available. are required, but not yet available.

However, I predict they will be within a year, due to recentHowever, I predict they will be within a year, due to recent

progress and the upcoming KITP workshop in collider progress and the upcoming KITP workshop in collider

physics.physics.

ConclusionsConclusions