Helicity-Dependent Showers and Matching with VINCIA

1 MB·

by  Andrew J. Larkoski

CERN-PH-TH/2012-257
MIT-CTP4406
FTUAM-12-105
IFT-UAM/CSIC-12-96
Helicity-Dependent Showers and Matching with VINCIA
A.J.Larkoski1,J.J.Lopez-Villarejo2,3,andP.Skands2
1: Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge,
MA02139U.S.A.
2: TheoreticalPhysics,CERNCH-1211,Geneva23,Switzerland
3 3: Univ. AutonomadeMadridandIFT-UAM/CSIC,Madrid28049,Spain
1
0
2
Abstract — We present an antenna-shower formalism that includes helicity dependence

r for massless partons. The formalism applies to both traditional (global) showers and to
a
sector-based variants. We combine the shower with VINCIA’s multiplicative approach to
M
matrix-element matching, generalized to operate on each helicity configuration separately.
6 The result is a substantial gain in computational speed for high parton multiplicities. We
presentanimplementationofbothsectorandglobalshowers,withmin/maxvariations,and

] helicity-dependent tree-level matching applied through n ≤ 4 for V/H → qq¯+npartons
h
andthroughn≤5forH →ngluons.
p
-
p
e
h
[
1 Introduction

2
v
Multi-legamplitudesandtheircombinationwithparton-showerresummations,calledME/PSmatch-
3
3 ing, are among the most active topics in current high-energy phenomenology (see [1] for a review).
9 WhatME/PSmatchingprovidesisacalculationthatsmoothlyinterpolatesbetweenfixed-orderQCD
0
(andQED)amplitudesinthehigh-p regionandinfinite-orderlogarithmicapproximationsinthelow-
. ⊥
1
p one. Importantly,theoutputofsuchcalculationsisintheformoffullyhadronized“events”,which
0 ⊥
3 canbesubjectedtodirectanddetailedexperimentalcomparison.
1 However,currentstate-of-the-artmultilegME/PSmethods,suchasCKKW-L[2,3],MLM[4],and
:
v MENLOPS[5], arerathercomputationallyintensive, sothatthemostcomplexcalculationscanonly
Xi becarriedoutonlargeclusters. Theincreaseincomputationaltimewiththenumberoflegsispartly
duetotheamplitudesbecomingmorecomplicatedateachorder,butmoreimportantlythesemethods
r
a
(whichwerefertocollectivelyas“slicing”methods[6])algorithmicallytreateachmulti-legmatching
step as completely unrelated to all the others: a separate phase-space integration, event-generation,
and event unweighting step is needed for each multiplicity. As shown in [7], much faster algorithms
can be constructed by “nesting” the successive matching steps within each other, starting from the
Born level and using an overestimating (“trial”) parton shower as the only additional phase-space
generator. The matrix-element amplitudes can then be imprinted on the final answer by a simple
Monte-Carlo veto step. This approach, which we refer to as the “multiplicative” method, was first
developedforoneadditionallegin[8]andwasgeneralizedtomultiplelegsin[9].
In this paper, we develop an additional refinement of the multiplicative method, which further
increasesthealgorithmicspeedthatcanbeachievedwhenmatchingtolargepartonmultiplicities. As
abeneficialsideeffect,theintrinsicprecisionoftheunderlyingparton-showerformalismisincreased
aswell. Themainpointistoreplacetheordinary(helicity-summed)showerradiationfunctionswith
helicity-dependentones,suchasthosegivenin[10]. Thatis,weshalltreat(massless)quarksandglu-
onswithnegativeandpositivehelicitiesaseffectivelybeingdifferentparticles. Theresultingshower
generates a LL approximation to each individual multi-leg helicity amplitude (squared) separately,
and the resulting evolution can therefore be matched to one single such amplitude at a time. Thus,
insteadoftakingsumsandaveragesateachorder,wearenoweffectivelysamplinghelicityspaceby
MonteCarlo,savingsubstantialtimewhenmatchingtoseveralsuccessivelegs.
Note that, up to the matched orders, interference effects between amplitudes with different in-
ternal helicity structures are still fully taken into account, since the last matched helicity amplitude
does contain a sum over all contributing internal-line helicities. At subsequent orders, however, the
helicity-dependentshowerdoesnotgeneratetheequivalentofafullspin-densitymatrixtreatment(see
e.g.[11]). Norareanyexplicitazimuth-dependentcorrelationsamongsuccessiveemissionsmanifest
in the helicity basis as would be the case with linearly polarized Altarelli-Parisi kernels [12]. The
finalprecisionisneverthelessstillimproved,sinceunphysicalhelicityassignmentsarenotallowedto
contribute.
In section 2, we generalize the VINCIA shower and matching formalism [13] to include helicity
dependence. Insection3, wederiveexplicithelicity-dependentQCDantennafunctions, considering
both sector and global antenna types. Finally, in section 4 we present a set of comparisons to ma-
trix elements, speed benchmarks, and validations on selected LEP distributions, and in section 5 we
summarizeandprovideabriefoutlook.
2 The Shower and Matching Algorithm
Thehelicityofaparticleistheprojectionofthespinoftheparticleontoitsmomentum. Formassless
particles,helicityisLorentzinvariantandtakesthevalues±sforparticleswithtotalspins. Typically,
incomputingmatrixelements,onesumsoverhelicitiesoftheincomingandoutgoingparticlesbecause
theyarenotdirectlyobserved. However,beginningwithobservationsbyParkeandTaylor[14]inthe
1980s, it was discovered that individual helicity amplitudes are significantly simpler in form than
are helicity-summed matrix elements. In addition, chirality or handedness is important in processes
mediated by the weak interaction. Thus, we expect that a Monte Carlo parton shower based on the
helicitystructureofQCD,ratherthansummedoverhelicities,isbothfasterwhenmatchingtomatrix
elementsaswellasmoreaccurate,especiallyinweakdecays.
In this section, we discuss the modifications to the VINCIA shower and matching algorithms
requiredtotakehelicityintoaccount. Theantennafunctionsthemselveswillbethetopicofsection3.
2.1 Helicity-DependentShowering
Thehelicity-dependentshoweralgorithmfollowstheunpolarizedonequiteclosely[13], withdiffer-
encesenteringonlyinafewveryspecificplaces,asfollows.
If VINCIA is asked to shower an event that contains unpolarized partons, e.g. an unpolarized
Z → qq¯event,itfirsthandstheeventtoapolarizer functionwhichsearchestheVINCIAlibraryfor
helicity-dependent matrix elements corresponding to the given process (see section 2.2 below for a
full list). If this search is successful, helicities for the final-state partons are assigned based on their
relativematrix-elementweights. Intheexampleofon-shellZ → dd¯(Z → uu¯)decay,aphase-space-
dependent average of 97% (83%) of the events will end up as q q¯ , with the other being q q¯ . For
L R R L
off-shellZ bosons,thefulle+e− → γ∗/Z → qq¯matrixelementswouldbeusedinstead.
Trial branchings are generated as in the unpolarized shower, according to the unpolarized trial
functions. Theunpolarizedtrialfunctionsareessentiallyanoverestimatingeikonalterm,see[9],with
additionalcollinear-singulartermsforsectorshowers[7]. Aftertheselectionofbranchinginvariants,
the probability that the branching will be accepted at all, summed over all possible post-branching
helicities,iscomputed:
(cid:80)
P = aphys = hi,hj,hka(hA,hB → hi,hj,hk) , (1)
accept
a a
trial trial
with h the (fixed) helicities of the parent partons, h the helicities of the daughter ones, and
A,B i,j,k
a(h ,h → h ,h ,h ) a helicity-dependent antenna function, the precise forms of which will be
A B i j k
discussed in section 3. This accept probability is not exactly identical to the unpolarized one, since
thehelicitiesoftheparentpartonsAandB arenotaveragedover. Notealsothat,forsectorshowers,
thetrialfunctionappearinginthedenominatorshouldbethefullonecontainingboththesoft-eikonal
andtheadditionalcollinear-singulartrialterms.
Helicitiesforthethreedaughterpartonsarethenassignedaccordingtotherelativeprobabilities
a(h ,h → h ,h ,h )
A B i j k
P(h ,h → h ,h ,h ) = . (2)
A B i j k (cid:80)
a(h ,h → h ,h ,h )
hi,hj,hk A B i j k
Notethatitisimportantthatthedenominatorherebeexactlythesameasthenumeratorineq.(1).
All other aspects of the showering remain unmodified (for matching, see below). For complete-
ness, we also note that, when helicity dependence is switched off, the unpolarized antenna functions
are obtained as direct helicity sums over the helicity-dependent ones, averaging over the parent he-
licities. The treatments with and without helicity dependence are thus intimately related and it is
straightforwardtogofromonetotheother.
For one showering step, there should therefore also be little difference between the helicity-
dependentandunpolarizedtreatments,uptodifferencescausedbythehelicity-dependentfiniteterms
not being equal to their averages. However, the chosen helicities at one step then become the input
helicities for the next step. This excludes some unphysical configurations from the effective helicity
average/suminthenextstep,yieldinganimprovementinaccuracyovertheunpolarizedcase.
2.2 MatchingtoMatrixElements
Theprocedureformatchinghelicitymatrixelementstothehelicity-dependentpartonshowerinVIN-
CIA is similar for matching to spin-summed matrix elements [9]. At each step in the shower, the
partonshowerprovidesanestimatefortheratio
|M |2
n
P = , (3)
n |M |2
n−1
where |M |2 is the matrix element after n additional quark or gluon emissions from the Born-level
n
matrixelement. Tomatchthepartonshowertotheexactmatrixelementatthisstage,theemissionof
thenthpartonisacceptedwiththeprobability
aPhys
Pn = P . (4)
accept aTrial ME
aPhys is the antenna for the current branching and aTrial is an overestimate antenna that is strictly
largerthanaPhys. ThematchingfactorP is
ME
|M |2
n
P = , (5)
ME |M |2
n shower
where|M |2 istheapproximationtothematrixelementprovidedbythepartonshower:
n shower
(cid:88)
|M |2 = a |M |2 , (6)
n shower i n−1i
i
with i running over the possible clusterings and helicities and a the corresponding antenna for the
i
ithconfiguration. Notethattheformofthisapproximationdependsontheformulationoftheshower
(globalvs.sector);see[7]fordetailsonthedifferencesbetweenglobalandsectormatching.
TheratioP inahelicity-dependentshowerhasespeciallyniceandgeneralproperties. Fordecays
n
of colorless resonances to massless quarks, P is independent of the CP structure of the resonance
n
for helicity matrix elements. This allows the use of, for example, matrix elements for Z decay to
massless quarks for use in matching processes which include W bosons or photons. Only the total
spinoftheresonanceisrelevantformatchingtohelicitymatrixelements.
Theproofofthisisstraightforward. Considertheratioofamplitudes
M
n
ρ ≡ . (7)
M
n−1
Thiscanbewrittengenericallyas
Mn u¯(qf)OgnJ/(a−bγ5)Ognu(q¯f)
ρ = = , (8)
Mn−1 u¯(qi)Ogn−1J/(a−bγ5)Ogn−1u(q¯i)
where q ,q (q¯,q¯ ) are the (anti)quark before and after additional radiation, respectively, J/ is the
i f i f
currentcarriedbytheresonance,(a−bγ )parametrizestheCP structureofthecurrent,andOnisan
5 g
unspecifiedoperatorwhichproducesadditionalQCDradiationuptoO(gn). Irrespectiveoftheform
s
of the current J/, the radiation operators On and On−1 contain an even number of gamma matrices:
g g
oneforeachquark-gluonvertexandoneforeachquarkpropagator. Then,itistruethat
[On,γ ] = [On−1,γ ] = 0, (9)
g 5 g 5
whichallowsustofreelyinterchangethepositionof(a−bγ )andO . Wethenhave
5 g
Mn u¯(qf)OgnJ/Ogn(a−bγ5)u(q¯f)
ρ = = . (10)
Mn−1 u¯(qi)Ogn−1J/Ogn−1(a−bγ5)u(q¯i)
Helicityspinorsareeigenvectorsofγ witheigenvalueequaltotheirhelicity. Thehelicityofmassless
5
quarksispreservedinQCD;thatis,
γ u(q¯) = h u(q¯) and h = h . (11)
5 q¯ q¯i q¯f
Wethenfindthat
u¯(q )OnJ/Onu(q¯ )(a−bh ) u¯(q )OnJ/Onu(q¯ )
f g g f q¯ f g g f
ρ = = , (12)
u¯(q )On−1J/On−1u(q¯)(a−bh ) u¯(q )On−1J/On−1u(q¯)
i g g i q¯ i g g i
provingthatP isindependentoftheCP structureoftheresonance. Notethatbycrossingsymmetry,
n
thisproofalsoholdsforparticlescreatedinthefinalstatefromparton-partonscattering.
BasedontheuniversalityofP ,thecurrentVINCIAimplementationincludesmatchingtothehe-
n
licityamplitudeslistedintab.1,withV andS denotingagenericcolorlessspin-1andspin-0particle,
respectively. The corresponding massless helicity amplitudes squared were obtained by modifying
Decaying OrderBeyondBorn
Particle 0 1 2 3 4
V,S qq¯ qq¯g qq¯gg,qq¯q(cid:48)q¯(cid:48) qq¯ggg,qq¯qq¯(cid:48)g qq¯gggg,qq¯q(cid:48)q¯(cid:48)gg,qq¯q(cid:48)q¯(cid:48)q(cid:48)(cid:48)q¯(cid:48)(cid:48)
S gg ggg gggg ggggg -
Table 1: Matrix elements available in VINCIA for helicity-dependent matching corrections, with V (S) a
genericcolorlessspin-1(spin-0)bosonwitharbitrarycouplings.
MADGRAPHv.4.4.26[18]toextractindividualhelicityconfigurationsandareevaluatedatruntime
usingtheHELASlibraries[19]. Forprocessesinvolvingdecayingmassivevectorbosons,thespinof
thevectorissummedoverincomputingthehelicitymatrixelement. Inaddition,matchingisdonefor
on-shellparticlesandsothemassofthedecayingresonanceisartificiallysettothecenterofmassen-
ergyQ. NotethattheHELASlibrariesaresofarnotincludedintheVINCIApackageitselfandmust
bedownloadedseparately. ThedefaultVINCIAmaketargetwillautomaticallyattempttodownload
theMADGRAPHpackageandcompiletheHELASlibraryfromthere. Alternatively,theuserhasthe
optionofprovidingaprecompiledHELASlibrary.
2.3 MassivePartons
Thehelicity-dependentformalismpresentedinthispaperislimitedtomasslesspartons. Formassive
partons, VINCIA reverts to the unpolarized massive framework presented in [20]. By default, this
applies to charm and heavier quarks. User options are provided that allow the massive treatment to
be applied also to s quarks, or to force c and/or b ones to be treated as massless. For completeness,
wedescribeherewhatthecodedoeswhenhelicity-dependenceisswitchedon(asitisbydefault)and
one or more massive partons are present either in the Born-level event (e.g., via a Z → b¯b decay) or
arecreatedduringtheshowerevolution(e.g.,viaag → b¯bsplitting).
If a massive parton is present already in the Born-level event, the entire event is treated as un-
polarized. That is, all parton helicities are ignored, including those of massless partons. The mas-
sive shower algorithm described in [20] is applied, and hence mass corrections are included, even if
helicity-dependenceisnot.
If no massive parton is present in the Born-level event, the helicity-dependent shower described
inthispaperisappliedtotheevent,buttrialsplittingsofgluonstomassivequarksarestillallowed. If
such a branching is accepted, all helicities are then ignored from that point onwards, and the further
eventevolutionproceedsaccordingtotheunpolarizedalgorithm[20],asabove.
A subtlety arises concerning the matching to matrix elements. As described above, the helicity-
dependentformalismallowsustousematrixelementsforZ decaytorepresentanyvectorboson,and
onesforH decaytorepresentanyscalar. Forspin-summedmatrixelements,however,thisuniversality
breaksdown. Theusershouldthereforebeawarethat,whilethefullrangeofmatchedmatrixelements
arestillavailableforZ andH0decaystounpolarizedmassiveparticles,thecorrespondingcorrections
forW andH+ decaystomassivepartonshavesofarnotbeenimplemented.
Forfuturereference, wenotethatphase-spacemapsforantennaeinvolvingmassiveparticlesare
availablein[20]andasetofspin-dependentantennafunctionsweredefinedin[21].
++ −+ +− −−
g → gg : 1/z(1−z) (1−z)3/z z3/(1−z) 0
+
g → qq¯: - (1−z)2 z2 -
+
q → qg : 1/(1−z) - z2/(1−z) -
+
q → gq : 1/z (1−z)2/z - -
+
Table 2: Helicity-dependent Altarelli-Parisi splitting functions P(z) for splittings a → bc, with z defined as
theenergyfractiontakenbypartonb. Thelabelsinthetoprowdenotethehelicitiesofthetwofinalparticlesin
theordertheyappear: (h ,h ). Theemptycolumnsareforbiddenbyquarkchiralsymmetry. BythePandC
b c
invarianceofQCD,thesameexpressionsapplyafterexchanging−↔+orq ↔q¯.
3 Helicity-Dependent Antenna Functions
Parton showers (including the dipole/antenna varieties) are governed by the properties of soft and
collinear emissions in QCD. The soft and collinear limits of massless QCD matrix elements are the
universalAltarelli-Parisisplittingfunctions[15]andtaketheschematicform
1
siljim→0|M(1,...,i,j,...,n)|2 = sijgs2CijPi,j←i(cid:98)j(z)|M(1,...,i(cid:98)j,...,n)|2 , (13)
whereCij isthecolorfactor,gs2 = 4παs istheQCDcouplingandparticlesiandj arereplacedbyi(cid:98)j
inthematrixelementontheright. P (z)isthesplittingfunctionrepresentingthedistributionof
i,j←i(cid:98)j
energyfractionz carriedbyparticlei. TheAltarelli-Parisisplittingfunctionsformasslessquarksand
gluons of definite helicity were given in their original paper and are reproduced in tab. 2. Note that
rowsintab.2sumtothefamiliar,unpolarized,Altarelli-Parisisplittingfunctions.
The VINCIA Monte Carlo is a dipole-antenna shower [13] based on nested 2 → 3 splitting
processes. ThissplittingcanberepresentedasIK → ijk,forinitialpartonsI,K andfinalpartonsi,
j, k. As VINCIA works in the color-ordered limit of QCD, the initial and final partons are assumed
to be in color order, as well. We will also assume that all partons are massless, unless otherwise
specified. Thephasespaceforemissionisdefinedbythedimensionlessvariablesy andy where
ij jk
2p ·p 2p ·p
i j j k
y = , y = , (14)
ij jk
s s
ands ≡ (p +p +p )2 = (p +p )2 istheinvariantmassofthedipoleantennasystem. Thephase
i j k I K
spaceoftheemissionisdefinedbythetriangley ,y ≥ 0,y +y ≤ 1.
ij jk ij jk
Theprobabilityofemissionisgovernedbytheantennafunctionwhichisafunctionofallrelevant
momenta, quantum numbers and the formulation of the shower. For the splitting IK → ijk, the
antennafunctioncanbeexpressedintheform
atype(order)(p ,p ,p ), (15)
j/IK i j k
where type refers to global or sector antennae and order is the order in α to which the antennae
s
are computed. When obvious from context, the superscripts will be omitted. In this paper, we will
considerexclusivelythelowestorderantennafunctionsandsowecandefinethecolor-andcoupling-
strippedantenna
a (p ,p ,p ) = g2C a¯ (p ,p ,p ). (16)
j/IK i j k s j/IK j/IK i j k
For simplicity, we will work with the color- and coupling-stripped antenna in the following. For
masslesspartons,a¯ (p ,p ,p )isafunctionofthekinematicinvariantsy andy only.
j/IK i j k ij jk
The unpolarized global and sector antennae used in VINCIA were defined in [7,9,13]. We wish
to extend the global and sector antennae to include full helicity dependence of all partons in the an-
tenna. Ourdiscussionwillonlyincludeantennaeinwhichallparticlesaremassless. Antennasplitting
functions including helicity dependence were defined in [10] as ratios of matrix elements, but here,
wewillpresentageneraltreatmentoftheformoftheantennae. Therearemanyconstraintsthatmust
beimposedontheantennaetodeterminethesingularterms;mostimportantly,thehelicity-dependent
antennafunctionsmustappropriatelyreproducethehelicity-dependentAltarelli-Parisisplittingfunc-
tionsinthecollinearlimits. Notethatthisonlyconstrainsthesingulartermsoftheantenna;thenon-
singular terms are unconstrained and can be interpreted as uncertainties in higher log-order terms.
Also, when summed over final parton helicities, the antenna functions should reproduce the unpo-
larized antennae functions, up to terms that are non-singular. In the following subsections, we will
discusstheconstructionofglobalandsectorhelicity-dependentantennae.
Beforediscussingtheglobalandsectorantennae, wewilldistinguishthedefinitionandutilityof
helicitytodefineamasslessparticle’sspinfromotherdefinitionsintheliteratureorusedinsimulation
code. The Les Houches Accords of 2001 [16] outlined a set of variables by which to define the
properties of particles in Monte Carlo event simulations. The variable SPINUP was introduced to
quantifythespinofaparticleandisdefinedtobe:
double SPINUP(I):cosineoftheanglebetweenthespin-vectorofparticleIand
the3-momentumofthedecayingparticle,specifiedinthelabframe
This definition of spin for particles in Monte Carlos is unfortunately complicated and not widely
applicable. Itsusehasbeenmainlyrestrictedtotreatingpolarizedτ decays. Inthisandfuturework,
we propose using chirality as the basis for defining spin for massive or massless fermions. Chirality
isLorentzinvariant,relevantinweakdecaysandreducestohelicityformasslessfermions.
Also, polarized splitting functions [12] should be distinguished from helicity splitting functions.
It is first an issue of semantics. Helicity is the handedness of circular polarization of a particle with
respecttoitsmomentum. Formasslessparticles,thisisLorentzinvariant,asmentionedearlier. Polar-
izedsplittingfunctionsinsteadreferencethelinearpolarizationofaparticlewithrespecttotheplane
of the splitting. They are thus not Lorentz invariant, even for massless particles. However, polarized
splitting functions can be used to approximate the azimuthal correlations between subsequent emis-
sionsandtheeffectontheenergydistributionoftheshower. Helicitysplittingfunctionsdonothave
thisproperty;however,theazimuthalcorrelationsdoofcoursereappearwhentheshowerismatched
tomatrixelements.
3.1 GlobalAntennae
Theformsoftheglobalantennaearefoundbyenforcingseveralrequirements. Globalantennaecon-
tainthefullsoftlimitofemittedgluonsbutneighboringantennaeshareacollinearlimitofgluons. To
constructthehelicity-dependentglobalantennae,then,everypossiblehelicityconfigurationofneigh-
boringantennaemustreproducethecorrectcollinearlimits. Alsothehelicitydependentantennaecan
become negative over a significant region of phase space. For the use of the antennae functions as
probabilitydistributionsonphasespacetheymustbepositiveonallofphasespace.
Fortheunpolarizedglobalantennae,itisastraightforwardexercisetoincorporateallconstraintsto
determinetheantennae. WepresentanexampleofthisinappendixA.1. Theconstructionofhelicity-
dependentglobalantennae ismoresubtle, but weemploythefollowingrequirements tosimplifythe
analysis:
× 1 1 1 yjk yij yj2k yi2j 1 y y
yijyjk yij yjk yij yjk yij yjk ij jk
qq¯→ qgq¯
++ → +++ 1 0 0 0 0 0 0 0 0 0
++ → +−+ 1 −2 −2 1 1 0 0 2 0 0
+− → ++− 1 0 −2 0 1 0 0 0 0 0
+− → +−− 1 −2 0 1 0 0 0 0 0 0
qg → qgg
++ → +++ 1 0 −α+1 0 2α−2 0 0 0 0 0
++ → +−+ 1 −2 −3 1 3 0 −1 3 0 0
+− → ++− 1 0 −3 0 3 0 −1 0 0 0
+− → +−− 1 −2 −α+1 1 2α−2 0 0 0 0 0
gg → ggg
++ → +++ 1 −α+1 −α+1 2α−2 2α−2 0 0 0 0 0
++ → +−+ 1 −3 −3 3 3 −1 −1 3 1 1
+− → ++− 1 −α+1 −3 2α−2 3 0 −1 0 0 0
+− → +−− 1 −3 −α+1 3 2α−2 −1 0 0 0 0
qg → qq¯(cid:48)q(cid:48)
++ → ++− 0 0 0 0 0 0 1 0 0 0
2
++ → +−+ 0 0 1 0 −1 0 1 0 0 0
2 2
+− → ++− 0 0 1 0 −1 0 1 0 0 0
2 2
+− → +−+ 0 0 0 0 0 0 1 0 0 0
2
gg → gq¯q
++ → ++− 0 0 0 0 0 0 1 0 0 0
2
++ → +−+ 0 0 1 0 −1 0 1 0 0 0
2 2
+− → ++− 0 0 1 0 −1 0 1 0 0 0
2 2
+− → +−+ 0 0 0 0 0 0 1 0 0 0
2
Table 3: Table of coefficients for helicity-dependent global antenna functions. By the C and P invariance
of QCD, the same expressions apply with + ↔ −, q ↔ q¯. All other antennae are zero. The parameter
α determines the form of the spin-summed global antennae. The default choice in VINCIA is α = 0 which
correspondstotheGehrmann-DeRidder,Gehrmann,andGlover(GGG)spin-summedantennae[17].Thefinite
termsarechosensothattheantennaearepositiveonalloffinal-statephasespace.
1. Bose-Einstein symmetry. The antenna functions must be symmetric when gluons of the same
helicityareexchanged.
2. C and P symmetry of QCD. The expressions for the antennae are unchanged with + ↔ −,
q ↔ q¯.
3. Neighboringantennaesumtoreproducethefullcollinearlimits.
4. Thesingulartermsofthehelicity-dependentglobalantennaemustsumtoreproducethesingular
termsoftheunpolarizedglobalantennae.
5. Positivity of global antennae. Because the collinear limits of gluons are constructed from the
sumofneighboringantennae,theantennaearenotguaranteedtobepositiveeveninthesingular
regions of phase space. The positivity requirement must be enforced in the singular as well as
non-singularregionsofphasespace.
Acarefulaccountingoftheserequirementsproduceshelicity-dependentglobalantennaethatdepend
on three arbitrary parameters. One of these parameters fixes the form of the spin-summed or unpo-
larized antennae, which we call α, while the other two are artifacts of the proliferation of helicity-
dependent antennae. The latter two parameters can consistenly be set to zero, which we choose to
do in the following. The complete procedure, with fully general expressions, is described in ap-
pendix A.2. Here, we just give the forms of the single-parameter antenna functions implemented in
VINCIA,whicharedefinedintab.3.
Toestimateshoweruncertaintiesduetotheambiguouschoiceofnon-singularterms,wedefinea
set of MIN and MAX antenna functions which are smaller and larger, respectively, over all of phase
space than the default antennae. For the MAX antennae, the finite terms of all helicity-dependent
antennae are fixed to a large constant value; we choose to set the constant to be 5.0. This is large
enough to guarantee that all antennae are positive on phase space. MIN antennae are more subtle
because some helicity-dependent antennae cannot be decreased and remain positive on phase space.
To assuage this, we choose constants to subtract from those helicity-dependent antennae which are
large and positive enough to allow this. (Specifically, we subtract the minimum value on the 2 →
3 phase-space of the singular pieces of the given antenna.) This procedure guarantees that, when
summed over helicities, the MIN antennae are smaller than the default antennae in VINCIA. We
presentthechoiceofMINantennaeintab.4.
3.2 SectorAntennae
The helicity-dependent sector antennae are defined by reproducing the appropriate Altarelli-Parisi
splitting functions as the emitted particle becomes collinear with respect to either of the initial parti-
cles. This requirement uniquely fixes the singular components of all sector antennae but still allows
forfreedomofthechoiceofnon-singulartermsintheantennae. Thenon-singulartermscanbecho-
sensothatthesectorantennaereproducematrixelementsforparticularprocesses,forexample,which
was done in [10]. Positivity of the sector antenna functions is guaranteed in the singular regions be-
cause the antennae reproduce the universal Altarelli-Parisi functions. However, for some antennae,
non-singularpiecesmustbeaddedtokeeptheantennaepositiveinthenon-singularregionsofphase
space.
Defining the antenna functions by a ratio of matrix elements is one prescription for choosing the
non-singular terms that are necessary to enforce positivity of the antenna on all of phase space. Our
prescription for the choice of non-singular terms for the sector antennae is to add only the minimal
× 1 1 1 yjk yij yj2k yi2j 1
yijyjk yij yjk yij yjk yij yjk
qq¯→ qgq¯
++ → +++ 1 0 0 0 0 0 0 -4
++ → +−+ 1 -2 -2 1 1 0 0 2
qg → qgg
++ → +++ 1 0 1 0 -2 0 0 -3
++ → +−+ 1 -2 -3 1 3 0 -1 3
gg → ggg
++ → +++ 1 1 1 -2 -2 0 0 -4
++ → +−+ 1 -3 -3 3 3 -1 -1 3.7
Table4: TableofcoefficientsforMINhelicity-dependentglobalantennafunctions. Inthistable, αhasbeen
setto0whichisthedefaultchoiceinVINCIA.Onlythoseantennaewithnon-zerofinitetermsareshown.
terms necessary. For antennae whose singular terms are positive on all of phase space, we choose
to set the non-singular terms to 0. For those antennae which require the addition of non-singular
terms for positivity, we choose to add constants where possible and only include higher order terms
iny andy ifnecessaryforsimplicity. Anexampleoftheconstructionofsectorantennaefromthe
ij jk
collinear limits and positivity is given in appendix A.3 and the coefficients of the terms in the sector
antennaearegivenintab.5.
Toestimateshoweruncertaintiesduetotheambiguouschoiceofnon-singularterms,wedefinea
setofMINandMAXantennafunctions,asintheglobalshowercase. Forsimplicity,thefiniteterms
forthesectorMINandMAXantennaearechosentobethesameasthoseintheglobalcase.
In the VINCIA code, the sector antennae are derived from the global antennae. Note from tab. 3
andtab.5thatmuchofthestructureofthesectorantennaeiscapturedbytheglobalantennaeifα = 1.
To construct a sector antenna, the corresponding global antenna with the same helicity and flavor
structureisevaluatedwithα = 1andthemissingtermsaddedtorecoverthefullsectorantenna. The
preciserelationshipbetweenthe sector(a¯sct)andglobal (a¯gl)antennaeforα = 1forgluonemission
is:
(cid:40) (cid:32) (cid:33)
1+y +y2
a¯sct (y ,y ) = a¯gl (y ,y ) + δ δ δ δ jk jk
j/IK ij jk j/IK ij jk Ig hKhk hIhi hIhj y
ij
(cid:32) (cid:33)(cid:41)
1 1+yjk +yj2k
+ δ −
hIhj y (1−y ) y
ij jk ij
(cid:40) (cid:32) (cid:33)
1+y +y2
ij ij
+ δ δ δ δ
Kg hIhi hIhj hKhk y
jk
(cid:32) (cid:33)(cid:41)
1 1+yij +yi2j
+ δ − .
hKhj y (1−y ) y
jk ij jk
Here, δ is one if I is a gluon and zero otherwise and δ is one if the helicity of particles i and j
Ig hihj
arethesameandzerootherwise. Forantennaewithgluonssplittingtoquarks,thesectorantennaeare
twicetheglobalantennae.