HERWIRI2: CEEX Electroweak Corrections in a Hadronic MC

0.1 MB·English

by  Scott Yost

HERWIRI2: CEEX Electroweak Corrections in a
Hadronic MC
2
1
0
2
ScottYost†
∗
n
TheCitadel
a
J Charleston,SC29409,USA

7 E-mail: [email protected]
2
ValerieHalyo‡

]
h PrincetonUniversity
p Princeton,NJ08544,USA
-
E-mail: [email protected]
p
e
h MiroslavHejna
[
PrincetonUniversity

1 Princeton,NJ08544,USA
v E-mail: [email protected]
6
0 B.F.L.Ward§
9
BaylorUniversityWaco,TX76798,USA
5
. E-mail: [email protected]
1
0
2 Reachingthe1%precisionlevelforW andZ productioncalculationsfortheLHCwillrequirea
1
mixtureofhigherorderQCD andelectroweakcorrections. Asafirststep towardimplementing
:
v
the combined QED QCD exponentiation proposed in previous work, we have implemented
Xi the O(a ) electrowea⊗kcorrectionsand YFS exponentiationstructure of the KK Monte Carlo in
r
a HERWIG. We discuss the current status of this program and sketch the further developments
neededtoreachthedesiredprecisionlevel.
10thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumFieldTheoryto
Phenomenology)
September26-30,2011
Mamallapuram,India BU-HEPP-11-06
Speaker.
∗
†ThisworkanditspresentationweresupportedinpartbyD.o.E.grantDE-PS02-09ER09-01andgrantsfromThe
CitadelFoundation.
‡WorksupportedinpartbyD.o.EgrantDE-FG02-91ER40671.
§WorksupportedinpartbyD.o.E.grantDE-FG02-09ER41600.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost
Vector boson production is one of the most important Standard Model processes observed at
the LHC, and electroweak radiative corrections will be needed for analysis at the percent level.
Previous studies [1] by some of the authors have found that electroweak corrections alone can
exceed 1% for some cuts of interest. These studies were based on HORACE,[2] which provides
state-of-the-art O(a ) radiative corrections with a final-state photon shower, and PHOTOS, [3]
whichaddsfinalstatephotonicradiation. Otherprogramsdevelopedforimplementingelectroweak
corrections for hadronic collisions include WINHAC [4] and ZINHAC[5]. Initial state photonic
radiation has been considered only in certain MRST parton distribution functions. [6] However,
noneofthemostrecentPDFsinclude QEDeffects.
For electron-positron colliders, precision electroweak corrections have been implemented in
theprogram KKMC[7]usingYFS[8]exponentiated multiple-photon radiation forboththeinitial
and final state, together with O(a ) electroweak corrections in the DIZET [9] package developed
for ZFITTER[10]. The DIZET corrections can be applied to any parton-level process, so there is
noobstacletoextendingthemtohadroniccollisions. YFSexponentiation providesthebasisforan
efficient representation of multi-photon phase space, which can also be implemented for hadronic
initial and final states. Some of the authors have proposed using a non-abelian extension of YFS
exponentiation as a basis for an integrated event generator implementing both multi-photon and
multi-gluon corrections inaunifiedframework,calledQCD QEDexponentiation. [11]
⊗
The collection of programs implementing QCD QEDexponentiation has been called HER-
⊗
WIRI, for High Energy Radiation with Infra-Red Improvements, with a version number distin-
guishing the class of corrections included. The name acknowledges that the initial versions build
upontheHERWIG[12]partonshowergenerator. Thefirsttobereleased, HERWIRI1,[13]imple-
mentedIR-improvedsplitting kernels[14]obtainedusingtheQCDanalogofYFSexponentiation.
This program is publicly available, and tests are in progress. The IR-improved kernels have also
been implemented [15] in MC@NLO[16] and POWHEG[17]. The structure of HERWIRIis not
tiedtoaparticular shower, andourultimategoalisacomplete showergenerator basedentirely on
QCD QEDexponentiation withexactO(a 2,a a ,a 2)residuals. [18]
s s
⊗
Thesecondversion, HERWIRI2,implementstheelectroweak radiativecorrections ofKKMC
in a hadronic shower generator. This note describes a version of HERWIRI2 which is presently
nearingcompletion.1 HERWIRI2ismotivatedbythesuccessfulapplicationofYFSexponentiation
in BHLUMI[19], BHWIDE[20], KoralZ[21], KoralW[22], YFSWW3[23], KKMC, and related
programsforLEPphysics. Alloftheseprogramsbenefitfromaveryefficientrepresentation ofN-
photon phasespace, withcomplete control overthesoftandcollinear singularities foranarbitrary
number ofphotons. Realand virtual IR singularities cancel exactly toall orders. Thenon-abelian
extension toQCD QEDshouldhavesimilaradvantages.
⊗
WhilebasedonHERWIG,HERWIRI2islargelyindependentoftheunderlyingshower. HER-
WIGgeneratesthepartonmomentaandshower,andHERWIRI2passesthegeneratedhardprocess
momenta to KKMC routines to add photons and electroweak corrections. Although KKMC was
developed for e+e collisions, it was designed to be extended to more general processes, so the
−
ability to select quarks as the incoming state already exists in all but the lowest-level generation
routines.
1HERWIRI2doesnotincorporateHERWIRI1,althoughthetwoprogramscanbeusedincombination.
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HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost
KKMC is a precision generator for e+e f f +ng , f = m ,t ,d,u,s,c,b for CMS energies
−
→
from 2mt to 1 TeV. The precision tag for LEP2was 0.2%. ISR and FSR g emission is calculated
uptoO(a 2),including interference. TheMCstructure isbased onYFSexponentiation, including
residuals calculated perturbatively to the relevant orders in a kLl. (L=ln(s/m2)). Exact collinear
e
bremsstrahlung is implemented for up to three photons. Electroweak corrections [24, 25, 26] are
included viaDIZET6.21[9],andbeamstrahlung canbemodeledoverawiderangeofenergies via
abuilt-inoruser-defined distribution.
Therearetwomodesofoperation: exclusiveexponentiation(EEX)andcoherentexclusiveex-
ponentiation (CEEX).EEXappliesYFSexponentiation todifferentialcross-sections, whileCEEX
applies it at the amplitude level. The CEEX mode is written in a manner that is most readily
extended to quark scattering, so it is taken as the basis for HERWIRI2. The orders of residuals
included inCEEXmodearea ,a L,a 2L2,anda 2L.
CEEX was introduced for pragmatic reasons, because the traditional (EEX) exponentiation
of spin-summed differential cross sections suffered from a proliferation of interference terms in
processes with multiple diagrams, limiting its utility in reaching the desired 0.2% precision tag
for LEP2.[27] CEEX works at the level of spinor helicity amplitudes, greatly facilitating the cal-
culation of effects such as ISR-FSR interference, which are included in KKMC, and therefore in
HERWIRI2. CEEXis maximally inclusive: all real photons radiated are kept in the event record,
no matter how soft or collinear. There is no need to integrate out a region of soft phase space,
because theexponentiated amplitudes arewell-behaved atk=0.
TheCEEXcrosssectionforqq f f hastheform[7]
→
¥
s = 1 (cid:229) dPSr (n) (~p,~k) (1)
flux CEEX
n=0Z
where
2
r (n) = 1 eY(~p,Emin)1 (cid:229) M ~p ~k (2)
CEEX n! 4~l ,~m (cid:12)(cid:12) ~l ~m !(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
TheYFSformfactoris (cid:12) (cid:12)
Y(~p,E ) = Q2Y(p ,p ,E )+Q2Y(p ,p ,E )+QQ Y(p ,p ,E )
min i 1 2 min f 3 4 min i f 1 3 min
+QQ Y(p ,p ,E ) QQ Y(p ,p ,E ) QQ Y(p ,p ,E ), (3)
i f 2 4 min i f 1 4 min i f 2 3 min
− −
whereQ,Q areinitialandfinalpartoncharges, and
i f
Y(p,p ,E )=2a B(p,p ,E )+2a ReB(p,p ), (4)
i j min i j min i j
withrealandvirtualformfactors definedreespectively by
d3k p p 2
i j
B = , (5)
−Zk0<Emin 8p 2k0(cid:18)pi·k− pj·k(cid:19)
i d4k 2p +k 2p k
e i j
B = − . (6)
(2p )3 k2 2p k+k2 −2p k k2
Z (cid:18) i· j· − (cid:19)
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HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost
Then-photon helicity-spinor amplitudecanbeexpandedintermsofordera r havingtheform
Mn(r)=(cid:229)P (cid:213)i=n1Si(Pj)b 0(r) ~~lp; XP!+(cid:229)j=n1b 1(r)(cid:18)S~l~pmj(Pk ;j)XP(cid:19)+···+1<j1(cid:229)<···<jnSb n(jr(1)P(cid:18)j1~l~)p··~~mk·S; Xj(nPPj(cid:19)n)
 
 
 (7) 
withresidualspinoramplitudes b (r) andcomplexsoftphotonfactorsS withtheproperty
i j
2
S(Pj) = 2pa Q2 pa pb (8)
j − p k − p k
a j b j
(cid:12) (cid:12) (cid:18) · · (cid:19)
(cid:12) (cid:12)
whereQ,p ,p belong tothe(cid:12)initial(cid:12)orfinalfermionsdepending onthepartition P .
a b j
KKMCincorporates the DIZETlibrary (version 6.21) from the program ZFITTER.[10] The
g andZ propagators aremultiplied byvacuum polarization factors:
Hg = 2 1P g , HZ =4sin2(2q W)r E8Wpa G√m M2Z2. (9)
−
Vertex corrections are incorporated into the coupling of Z to f via form factors in the vector cou-
pling:
(f)
T
g(Z,f) = 3 Q F(f)(s)tanq . (10)
V sin(2q )− f v W
W
Box diagrams contain these plus a new angle-dependent form-factor in the doubly-vector compo-
nent:
(i) (f) (i) (f) (f) (i) (i,f)
T T 2T Q F (s) 2QT F (s)+4QQ F (s,t)
g(Z,i)g(Z,f) = 3 3 − 3 f v − i 3 v i f box . (11)
V V sin2(2q )
W
Thecorrection factorsarecalculated atthebeginning ofarunandstoredintables.
The Drell-Yan cross section with multiple-photon emission can be expressed as an integral
overtheparton-level processq(p )q(p ) f(p )f(p )+ng (k),integratedoverphasespaceand
i 1 i 2 3 4
→
summedoverphotons. Thepartonmomenta p ,p aregeneratedusingpartondistributionfunctions
1 2
givingaprocess atCMSenergyqandmomentumfractions x ,x suchthatq2=x x s:
1 2 1 2
s = dx1dx2(cid:229) f(q,x )f(q,x )s (q2)d (q2 x x s), (12)
DY x x i 1 i 2 i − 1 2
Z 1 2 i
wherethefinalstatephase spaceincludes p ,p andk,i=1, ,nandmultiplegluon radiation +
3 4 i
···
hadronization isincluded through ashower.
The parton-level cross section s (q2) can be calculated by KKMC, which integrates over a
i
finalspacephasespacewithtwofermionsandanarbitrary numberofphotons:
¥
s (q2)= (cid:229) dPS s (~p,~k) (13)
i 2+n i
n=0Z
HERWIRI2 uses HERWIG 6.5 as the shower generator, which creates the hard process first,
at Born level, in subroutine HWEPRO (HWHDYP), and then passes it to the cascade generator
HWBGEN.HERWIRI2findstheZ/g andthepartons interacting withitintheeventrecord. The
∗
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HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost
initial partons define p , p , which are transformed to the CM frame and projected on-shell to
1 2
create a starting point for KKMC,which generates the final fermion momenta p ,p and photons
3 4
k (bothISRandFSR.)Thegeneratedparticlesaretransformedbacktothelabframeandplacedin
i
theeventrecord.
InadditiontothebasicDYprocess,HERWIGgenerates“Compton”eventsg+q q+Z/g .
∗
→
About 10% of the events have this form. This is factorized into gluon emission times a hard
EW process at a shifted value of q2. These have a different profile in the event record, but can be
processedbyKKMCaswell. Thereisalsoathirdclassofeventswiththeemissionofanadditional
hardgluon. About1%oftheeventshavethisform,andalsohaveasignificantshiftoftheZenergy
fromitsgeneration scale.
Withachangeofvariables, theDrell-Yancrosssection inHERWIGcanbeexpressed as
s = dx1dx2(cid:229) f(q,x )f(q,x )s (q2)d (q2 x x s)
DY x x i 1 i 2 i − 1 2
Z 1 2 i
= qmaxdqP(q) 1 dx1(cid:229) P W(i) (q2,x )= W (14)
Zqmin Zq2/s x1 i i HW 1 h HWi
where P(q) is a normalized, integrable, crude probability distribution for q, P is the crude proba-
i
bilityofgenerating partoni,andW istheHERWIGeventweight. Thisweightdepends onlyon
HW
thehardBorncrosssectionandisnotalteredbytheshower.
Thecrudeprobability distributions usedbyHERWIGare
1 Ng N2q
P(q)= 2[Pg(q)+PZ(q)], Pg(q)= q4, PZ(q)= (q2 M2)+G 2M2 (15)
− Z Z Z
TheHERWIGeventweightis
W =(cid:229) W(i) , W(i) = 1 f(q,x )f(q,x )ln s s (i) (q2) (16)
HW HW HW P(q) i 1 i 2 q2 HW
i (cid:18) (cid:19)
andthecorresponding probability forselectingpartoniis
(i)
P =W /W (17)
i HW HW
Wehavechosentointroduceelectroweakcorrectionsinaminimallyinvasiveway,incorporat-
ingtheminaformfactor
s (q2)
F(i)(q2)= i (18)
EW s (i) (q2)
Born
KKMCwillcalculate theEWformfactor, andmultiplyitbytheHERWIGBorncrosssection.
s (i)(q2)
s = W , W =F(i)(q2)W =W KK . (19)
HW+EW h Toti Tot EW HW HWs (i) (q2)
Born
TheKKMCcrosssectioniscalculated usingaprimarydistribution
ds P(dir)i(vs,v) =s B(io)rn(s(1−v))21 1+√11 v g ivgi−1vmgi−ingi (20)
(cid:18) − (cid:19)
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HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost
with
2a s 2a s
g = Q2 ln 1 , g = Q2ln (21)
i p i m2 − i p i m2
(cid:20) (cid:18) i (cid:19) (cid:21) (cid:18) i (cid:19)
togeneratethefactorvgivingthefractionofsremainingafterISRphotonemission,s =s(1 v).
X
−
TheKKMCcrosssectionis
ds ds
s (q2)= ds Cru Mod =s W W . (22)
Prids ds Prih Cru Modi
Z Pri Cru
W iscalculated duringISRgeneration andW isgenerated afters isavailable.
Cru Mod X
TheHERWIGandKKMCweightsarecombinedtocalculatethetotalHERWIRI2weight,
s (q2) W(i)W(i)
s = W i = W s (i)(q2) Cru Mod , (23)
Tot * HWs (i)⋆(q2)+ * HW Pri s (i)⋆(q2)+
Born Born
This average could be calculated using a joint probability distribution for q and v, D(q,v)=
P(q)ds /dv,withP(q)fromHERWIG.AnadaptiveMC(S.Jadach’sFOAM[28])couldcalculate
Pri
thenormalizationofthedistributionatthebeginningoftherun,inasimilarmannertohowKKMC
presently integrates the one-dimensional primary distribution. To account for beamsstrahlung,
KKMC already permits such a user-defined distribution, in up to three variables. However, as a
first step, we have tried to run HERWIRI2 using KKMC’s one-dimensional primary distribution.
Thisrequiresfixinganoverallscaleq toinitialize KKMC(e.g.,q =M ).
0 0 Z
Thebuilt-inprimarydistribution forelectronsatscaleq canbeusedforthelow-levelgenera-
0
tion ofv. Thetransformation from thisdistribution toadistribution atHERWIG’sgenerated scale
qforquarkiisthenobtained byachangeofvariables:
ds (i)(q2,v) W(i) W(i)
s =s (e) W Pri Crud Mod (24)
Tot Pri * HW ds (e)(q2,v)! s (i)⋆(q2) !+
Pri 0 Born
with
ds (i)(q2,v) s (i) (q2(1 v))
Pri =Wg(i) Born − , (25)
ds (e)(q2,v) s (e) (q2(1 v))
Pri 0 Born 0 −
where
Wg = gg i vv gi−gevmgi−inge. (26)
e(cid:18) min(cid:19)
Theg factorsarecalculated usingq2/m2 forpartoniandq2/m2 fortheelectron.
i 0 e
Shufflingthenumeratorsanddenominators aboutgivestheexpression usedinHERWIRI2:
s Tot= WHWWModWKarlWFFWg (27)
(cid:10) (cid:11)
withtwonewweights
s (e)W(i) s (i) (q2(1 v))
W = Pri Crud , W = Born − . (28)
Karl FF
s (e) (q2(1 v)) s (i)⋆(q2)
Born 0 − Born
HERWIRI2 is still under development, so any numerical results must be treated as prelimi-
nary. Arunfor ppcollisions at10TeVwiththeZ/g invariant massbounded by40GeVand140
∗
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HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost
GeV, using HERWIG 6.520 default parameters and PDFs, yields a cross-section of 1183.7 1.3
±
pb, compared to1098.8 1.0 pbforHERWIGalone, givinganelectroweak contribution of7.7%.
±
Turning on ISR gives a much wider weight distribution and consequently, greatly reduced effi-
ciency. Preliminary results give a cross-section of 1212 109 pb, showing an additional 2.4%
±
contribution fromISR.
Work is in progress to optimize MC generation in the presence of ISR. Once HERWIRI2
is complete and thoroughly tested, it will be compared to other available hadronic/electroweak
generators. Inparticular, itwillbeinterestingtoseetheeffectofinitialstateradiation,whichisnot
presentintheotherprograms,butappearstoenteratthe2–3%level,makingitcrucialtoprecision
calculations.
Acknowledgments
S.Yostthankstheorganizers ofRADCOR2011fortheinvitation topresenttheseresults,and
D.MarlowandthePrincetonPhysicsDepartmentfortheirsupportandhospitality duringacritical
period ofitsdevelopment. S.YostandB.F.L.Wardalsoacknowledge thehospitality oftheCERN
theorydivision, whichcontributed greatlytotheprogress ofthiswork.
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