Double real radiation corrections to gluon scattering at NNLO

Joao Piresa∗, E.W.N. Glovera,

aInstitute for Particle Physics PhenomenologyDepartment of PhysicsUniversity of DurhamDH1 3LEUK

We use the antenna subtraction method to isolate the double real radiation infrared singularities present in

the six-gluon tree-level process at next-to-next-to-leading order. We show numerically that the subtraction term

correctly approximates the matrix elements in the various single and double unresolved configurations.

1. Introduction

In proton-proton collisions, the factorised formof the inclusive cross section is given by,

dσ =∑i,j

∫dσijfi(ξ1)fj(ξ2)

dξ1

ξ1

dξ2

ξ2(1)

where dσij is the parton-level scattering cross sec-tion for parton i to scatter off parton j normalisedto the hadron-hadron flux2 and the sum runs overthe possible parton types i and j. The probabil-ity of finding a parton of type i in the protoncarrying a momentum fraction ξ is described bythe parton distribution function fi(ξ)dξ. By ap-plying suitable cuts, one can study more exclu-sive observables such as the transverse momen-tum distribution or rapidity distributions of thehard objects (jets or vector bosons, higgs bosonsor other new particles) produced in the hard scat-tering. The leading-order (LO) prediction is auseful guide to the rough size of the cross sec-

∗This research was supported in part by the UK Scienceand Technology Facilities Council and by the EuropeanCommission’s Marie-Curie Research Training Network un-der contract MRTN-CT-2006-035505 ‘Tools and PrecisionCalculations for Physics Discoveries at Colliders’. EWNGgratefully acknowledges the support of the Wolfson Foun-dation and the Royal Society. JP gratefully acknowledgesthe award of a Fundacao para a Ciencia e Tecnologia (FCT- Portugal) PhD studentship.2The partonic cross section normalised to the parton-parton flux is obtained by absorbing the inverse factorsof ξ1 and ξ2 into dσij .

tion, but is usually subject to large uncertaintiesfrom the dependence on the unphysical renormal-isation and factorisation scales, as well as possi-ble mismatches between the (theoretical) parton-level and the (experimental) hadron-level.

The theoretical prediction may be improved byincluding higher order perturbative predictionswhich have the effect of (a) reducing the renor-malisation/factorisation scale dependence and(b) improving the matching of the parton levelevent topology with the experimentally observedhadronic final state [1]. The partonic cross sec-tion dσij has the perturbative expansion

dσij = dσLOij +

(αs

2π

)dσNLO

ij +(αs

2π

)2

dσNNLOij

+O(α3s) (2)

where the next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) strong correc-tions are identified. For many processes, knowl-edge of the NLO correction is sufficient. However,for the main 2 → 1 or 2 → 2 scattering pro-cesses such as Drell-Yan, Higgs production, di-jetproduction, vector-boson plus jet, vector-bosonpair production or heavy quark pair production,the NLO corrections still have a large theoret-ical uncertainty and it is necessary to includethe NNLO perturbative corrections. In additionto reducing the renormalisation and factorisationscale dependence there is an improved matchingof the parton level theoretical jet algorithm with

Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 176–181

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the hadron level experimental jet algorithm be-cause the jet structure can be modeled by thepresence of a third parton. In this talk, we aremainly concerned with the NNLO corrections todi-jet production where the resulting theoreticaluncertainty is estimated to be at the few per-centlevel [1].

2. Antenna subtraction

Any calculation of these higher-order correc-tions requires a systematic procedure for extract-ing the infrared singularities that arise when oneor more final state particles become soft and/orcollinear. These singularities are present in thereal radiation contribution at next-to-leading or-der (NLO), and in double real radiation andmixed real-virtual contributions at next-to-next-to-leading order (NNLO).

There have been several approaches to builda general subtraction scheme at NNLO [2–12].We will follow the NNLO antenna subtractionmethod which was derived in [2] for processes in-volving only (massless) final state partons. Thisformalism has been applied in the computa-tion of NNLO corrections to three-jet productionin electron-positron annihilation [13–16] and re-lated event shapes [17–21]. It has also been ex-tended at NNLO to include one hadron in theinitial state relevant for electron-proton scatter-ing [22,23] while in refs. [24,25] the extension ofthe antenna subtraction method to include twohadrons in the initial state is discussed.

In the antenna subtraction method the sub-traction term is constructed from products ofantenna functions with reduced matrix elements(with fewer final state partons than the origi-nal matrix element), and integrated over a phasespace which is factorised into an antenna phasespace (involving all unresolved partons and thetwo radiators) multiplied with a reduced phasespace (where the momenta of radiators and un-resolved radiation are replaced by two redefinedmomenta). The full subtraction term is obtainedby summing over all antennae required for theproblem under consideration. In the most generalcase (two partons in the initial state, and two or

more hard partons in the final state), this sumincludes final-final, initial-final and initial-initialantennae.

The relevant antenna is determined by both theexternal state and the pair of hard partons it col-lapses to. In general we denote the antenna func-tion as X . For antennae that collapse onto a hardquark-antiquark pair, X = A for qgq. Similarly,for quark-gluon antenna, we have X = D for qggand X = E for qq′q′ final states. Finally, wecharacterise the gluon-gluon antennae as X = Ffor ggg, X = G for gqq final states. At NNLO wewill need four-particle antennae involving two un-resolved partons and one-loop three-particle an-tennae.

In all cases the antenna functions are de-rived from physical matrix elements: thequark-antiquark antenna functions from γ∗ →qq + (partons) [27], the quark-gluon antennafunctions from χ → g + (partons) [28] and thegluon-gluon antenna functions from H → (par-tons) [29].

A key element in any subtraction method, isthe ability to add back the subtraction term inte-grated over the unresolved phase space. In the an-tenna approach, this integration needs to be per-formed once for each antenna. Although no prob-lem in principle has been identified, at present,not all of the necessary integrals have been com-pleted. The current state of play is summarizedin Table 1.

3. Double real radiation gluonic contribu-

tions

We now consider the six-gluon contributionto the NNLO di-jet cross section. At leadingcolour3, the double real radiation contribution isgiven by,

dσRNNLO = N

(αsN

2π

)2 ∑σ∈S6/Z6

A06(σ(1), . . . , σ(6))

× 1

4!dΦ4(p3, . . . , p6; p1, p2)J

(4)2 (p3, . . . , p6).(3)

where the normalisation factor N includes allQCD-independent factors as well as the LO de-

3To simplify the discussion, we systematically drop thesub-leading colour terms.

J. Pires, E.W.N. Glover / Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 176–181 177

Table 1Integrated antenna functions

final-final final-initial initial-initial

X03 ✓[2] ✓[26] ✓[26]

X04 ✓[2] ✓[22] (in progress) [24]

X03 ⊗ X0

3 ✓[2] (in progress) (in progress)

X13 ✓[2] ✓[22] (in progress)

pendence on the renormalised QCD coupling con-stant αs. Sn/Zn is the group of non-cyclic per-mutations of n symbols and A0

6(σ(1), . . . , σ(6))is the square of a colour ordered amplitude. p1

and p2 are the momenta of the ingoing gluons,while p3, . . . , p6 are the momenta of the four finalstate gluons and where the phase space integra-tion dΦ4 is over the 4-parton final state providedthat precisely 2-jets are observed by the jet algo-

rithm J(4)2 . Here singularities occur in the phase

space regions corresponding to two gluons becom-ing simultaneously soft and/or collinear and thesubtraction term must successfully regularize alldouble unresolved and single unresolved singular-ities.

The antenna subtraction term dσSNNLO for the

six-gluon process was derived in Ref. [25].

4. Numerical checks

In this section we will test how well the sub-traction term dσS

NNLO approaches the double realcontribution dσR

NNLO in the single and doubleunresolved regions of phase space so that theirdifference can be integrated numerically over thefour-dimensional phase space. We will do thisnumerically by generating a series of phase spacepoints that approach a given double or single un-resolved limit. For each generated point we com-pute the ratio,

R =dσR

NNLO

dσSNNLO

(4)

where dσRNNLO is the matrix element squared

given in equation (3) and dσSNNLO is the sub-

traction term of Ref. [25]. The ratio of the ma-trix element and the subtraction term should ap-proach unity as we get closer to any singularity

and the quality of this convergence represents thefirst sanity check on the NNLO antenna subtrac-tion method for the six-gluon process.

For each unresolved configuration, we will de-fine a variable that controls how we approach thesingularity subject to the requirement that thereare at least two jets in the final state with pT >50GeV. The centre-of-mass energy

√s is fixed to be

1000 GeV. All of the single and double unresolvedregions of phase space were analysed separately.In each case, we generated 10000 random phasespace points and computed the average value of Rtogether with the standard deviation. The resultsof the numerical study for all limits (apart fromthe single collinear limit we shall discuss below)are summarized in Table 2.

The numerical results show that the combina-tion of the antennae in the subtraction term cor-rectly describes the infrared singularity structureof the matrix element. All of the double unre-solved and single soft singularities present in thematrix elements are cancelled in a point-by pointmanner.

Finally we generate points corresponding to thefinal and initial state single collinear regions ofthe phase space. For the final-final collinear sin-gularity, fig. 1(a) shows the R-distribution for arange of values of x = sij/s. Similarly in theinitial-final collinear limit, we show the distribu-tions of R for small values of x = s1i/s in fig. 2(a).We see that the subtraction term, which is basedon azimuthally averaged antennae functions doesnot accurately describe the azimuthal correlationspresent in the matrix elements and antenna func-tions when a gluon splits into two collinear glu-ons. This is the reason why the distributions in

J. Pires, E.W.N. Glover / Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 176–181178

Table 2Results for the average and standard deviation of R for 10000 phase space points close to singular limits.

small parameter < R > σ

double soft (s − sij)/s = 10−6 0.9999994 4 × 10−5

triple collinear (final) sijk/s = 10−9 1.0000004 4.2 × 10−5

triple collinear (initial) s1jk/s = −10−10 0.99954 0.04

soft/collinear (final) (s − sijk)/s = 10−6, sij/s = 10−6 0.99999993 0.0001

soft/collinear (initial) (s − sijk)/s = 10−7, s1i/s = −10−7 0.99999998 1.6 × 10−7

double collinear (final-final) sij/s = 10−8, skl/s = 10−8 0.9999995 0.00037

double collinear (final-initial) sjk/s = 10−10, s1i/s = −10−10 1.00012 0.018

double collinear (initial-initial) s1i/s = −10−10, s2j/s = −10−10 1.00001 0.004

single soft (s − sijk)/s = 10−7 0.999998 1.9 × 10−5

figs. 1(a) and 2(a) have such a broad shape. Itis clear that as we approach the collinear limitsx → 0, the azimuthal terms are not suppressedand the subtraction term is not, point by point,an adequate representation of the matrix element.

Nevertheless, the azimuthal terms coming fromthe single collinear limits do vanish after an az-imuthal integration over the unresolved phasespace. Here we are performing a point-by-pointanalysis on the integrand defined by the ma-trix element squared and the subtraction term.One possible strategy is to introduce a correc-tion term ΘF 0

3

(i, j, z, k⊥) to the F 04 four-gluon

antenna functions which reconstructs the angu-lar terms [25]. Subtracting ΘF 0

3

(i, j, z, k⊥) fromthe final-final and, initial-final and initial-initialconfigurations (by crossing momenta to the initialstate) produces a subtraction term that is locallyfree of angular terms.

With this azimuthally modified subtractionterm, we recompute the distributions in figs. 1(b)and 2(b). For 10000 final state single collinearphase space points and x = 10−12 we obtainedan average R = 0.99994 and a standard deviationof σ = 0.015. We repeated the same analysis forthe initial state collinear configuration for x =−10−12 and obtained an average of R = 1.00007and a standard deviation of σ = 0.012. For bothcases, the distributions now peak around R = 1with a more pronounced peak as the limit is ap-proached, just as in the double unresolved andsingle soft limits discussed earlier. This demon-

strates the convergence of the counterterm to thematrix element.

However, the azimuthal correction termΘF 0

3

(i, j, z, k⊥) has the unfortunate side effect ofgenerating new divergences which are not presentin the matrix element.

A second more successful approach is to can-cel the angular terms by combining phase spacepoints related to each other by a rotation of thesystem of unresolved partons [30,13]. When bothcollinear partons are in the final state, this canbe achieved by considering pairs of phase spacepoints which are related by rotating the collinearpartons by an angle of π/2 around the resul-tant parton direction. Similarly, for the initial-final state collinear configurations produced whenpμ

i → pμ+pμj for i = 1, 2 and with i||j and p2 = 0,

the phase space points should again be related byrotations of π/2 about the direction of pμ. Thishas the consequence of rotating pμ

i off the beamaxis and therefore has to be compensated by aLorentz boost.

The effect of combining pairs of phase spacepoints is shown in Figs. 1(c) and 2(c). Wesee that the distributions for both final-final andinitial-final collinear limits have a very sharppeak around R = 1. For the final state sin-gularity and x = 10−12 we obtained an aver-age of R = 0.999996 and a standard deviationof σ = 0.00015. Similarly, for the initial statecollinear singularity and x = −10−12 we foundR = 0.9999991 and σ = 0.00011. There is an

J. Pires, E.W.N. Glover / Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 176–181 179

0

50

100

150

200

250

300

350

0.92 0.96 1 1.04 1.08

# ev

ents

R

#PS points=10000

x=sjk/s

x=10-8

x=10-10

x=10-12

3108 outside the plot3067 outside the plot3065 outside the plot

0

1000

2000

3000

4000

5000

0.92 0.96 1 1.04 1.08

# ev

ents

R

#PS points=10000

x=sjk/s

x=10-8

x=10-10

x=10-12

98 outside the plot11 outside the plot0 outside the plot

0

2000

4000

6000

8000

10000

0.92 0.96 1 1.04 1.08

# ev

ents

R

#PS points=10000

x=sjk/s

x=10-8

x=10-10

x=10-12

73 outside the plot7 outside the plot0 outside the plot

Figure 1. Distribution of R for 10000 single finalstate collinear phase space points for x = sij/sand x = 10−8 (red), x = 10−9 (green) andx = 10−10 (blue) for (a) the uncorrected matrixsubtraction term (b) the azimuthally correctedsubtraction term and (c) pairs of phase spacepoints related by a rotation of π/2 about the axisof the collinear pair.

enormous improvement compared to the raw dis-tributions shown in Figs. 1(a) and 2(a) and a sig-

0

50

100

150

200

0.92 0.96 1 1.04 1.08

# ev

ents

R

#PS points=10000

x=s1i/s

x=10-8

x=10-10

x=10-12

5173 outside the plot5160 outside the plot5159 outside the plot

0

1000

2000

3000

4000

0.92 0.96 1 1.04 1.08

# ev

ents

R

#PS points=10000

x=s1i/s

x=10-8

x=10-10

x=10-12

34 outside the plot2 outside the plot0 outside the plot

0

2000

4000

6000

8000

10000

0.92 0.96 1 1.04 1.08

# ev

ents

R

#PS points=10000

x=s1i/s

x=10-8

x=10-10

x=10-12

8 outside the plot0 outside the plot0 outside the plot

Figure 2. Distribution of R for 10000 singleinitial state collinear phase space points for (a)the uncorrected matrix subtraction term (b) theazimuthally corrected subtraction term and (c)pairs of phase space points related by a rotationof π/2 about the axis of the collinear pair andthen boosted so the initial direction is preserved.

nificant improvement compared to adding an az-imuthal correction term shown in Figs. 1(b) and2(b). The phase space point averaged subtrac-

J. Pires, E.W.N. Glover / Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 176–181180

tion term clearly converges to the matrix elementas we approach the single collinear limit and cor-rectly subtracts the azimuthally enhanced termsin a point-by-point manner.

5. Conclusions

In this contribution, we have discussed the ap-plication of the antenna subtraction formalism toconstruct the subtraction term relevant for thesix-gluon double real radiation contribution todi-jet production. The subtraction term is con-structed using four-parton and three-parton an-tennae. We showed that the subtraction termcorrectly describes the double unresolved limitsof the gg → gggg process. In particular, by com-bining phase space points, the subtraction termavoids the problems associated with angular cor-relations produced when a gluon splits into twocollinear gluons. The final goal is the constructionof a numerical program to compute NNLO QCDestimates of di-jet production in hadron-hadroncollisions.

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