Table Of ContentHadron Mass Effects in Power Corrections to Event
Shapes∗
3
1
0
2
n
a
J VicentMateu†
9 CenterforTheoreticalPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,
1
USA
IFIC,UVEG-CSIC,ApartadodeCorreos22085,E-46071,Valencia,Spain
]
h E-mail:[email protected]
p
-
Iain W.Stewart
p
e CenterforTheoreticalPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,
h
USA
[
E-mail:[email protected]
1
v JesseThaler
5
CenterforTheoreticalPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,
5
USA
5
4 E-mail:[email protected]
.
1
0
Westudytheeffectofhadronmassesontheleadingpowercorrectionofdijetevent-shapedistri-
3
1 butions.Wedefinethetransversevelocityoperator,thatdescribestheeffectsofhadronmasses. It
:
v dependsonthe“transversevelocity”r,whichisdifferentfromoneonlyfornon-vanishinghadron
i
X masses. We find that hadron-mass effects in general break universality. However we provide
r a simple method to identify universality classes of event shapes with a common power correc-
a
tion. Wealsocomputetheanomalousdimensionofthepowercorrectionandthestructureofthe
correspondingWilsoncoefficient,findinganontrivialresult.
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗IFIC/13-01,LPN13-009,MIT-CTP4432
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
HadronMassEffectsinPowerCorrectionstoEventShapes VicentMateu
1. Introduction
Eventshapeshavebeencrucialtopindownthestructure ofQCD.Intherecentyearsthesub-
jecthasattractedalotofattentionduetotheverypreciseextractionofthestrongcouplingconstant
a from fits to the tail of the thrust distribution [1, 2, 3] and moments of the thrust distribution
s
[4, 5]1. Excellent reviews on event shapes are [7, 8], where the definition of the most commonly
usedcanbefound.
Eventhough eventshapes areinfrared safe observables, theyreceivesizable corrections from
hadronization effects. In the tail of the distribution these effects are known as power corrections
and are suppressed by inverse powers of the center-of-mass energy Q. The first studies of power
corrections were inspired on renormalon techniques. The dispersive approach of Dokshitzer and
Webber [9, 10, 11] replaces a by an effective coupling below some cutoff scale. Within this
s
approachitwasfoundthattheleadingpowercorrectionfordifferenteventshapeswereproportional
tooneanother,withacalculablecoefficient[9,12]. Later,SalamandWicke[13]pointedoutusing
theflux-tubemodelthathadron masseffectsbreakthatuniversality.
A different approach to power corrections is based on the factorization properties of QCD
at very high energies. The shape function introduced in [14, 15] parametrizes nonperturbative
corrections and describes the tail power corrections to any order. Moreover, within this approach
nonperturbative parameters are expressed asmatrixelements ofQCDoperators. LeeandSterman
[16]showedthatthefactorizationapproachpredictsthesameuniversalityrelationsasthedispersive
model. Recently it has been shown in the Soft-Collinear Effective Theory framework (SCET for
short)[18,19,20,21,22]frameworkthathadron massesbreakuniversality [17].
2. Event shapes inthe dijet limit
An event shape e is an observable defined on the kinematical properties of the final-state
hadrons (energy and three-momentum). Forour purposes we willuse the dimensionful quantities
p =|~p |(transverse momentum)andm⊥= p2 +m2 (transversemass),andthedimensionless
⊥ ⊥ ⊥
variables h =−lntan(q /2)(pseudorapidity) aqndy=1/2ln[(E+p )/(E−p )](rapidity). Hereq
z z
refers tothe polarangle from the zˆ axis, which wetake tobe aligned withthethrust axis. Theve-
locityofaparticleisv=|~p|/E,andwedefinethetransversevelocityasr= p /m⊥. Formassless
⊥
particles p =m⊥,y=h andr=v=1,buttheequalities nolongerholdfornon-zero masses.
⊥
Dijet event shapes tend to zero for a configuration of two narrow back-to-back jets plus soft
radiation (that is, foradijet configuration) and viceversa. Inthe dijet limite≪1one canexpand
e=e¯+O(e2),wheree¯isingeneralsimplerthan e. Inparticularonecanalwayswrite
e¯= 1 (cid:229) m⊥f (r,y), (2.1)
Q i e i i
i∈X
where the function f isspecific fora given event shape. In this limitone can derive factorization
e
theorems for the differential cross section, which are very convenient to perform resummation of
singularlogarithmstoallordersinperturbation theory, andtoidentify powercorrections [23].
1Seealso[6]foradeterminationusingtheHeavyJetMassdistribution.
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HadronMassEffectsinPowerCorrectionstoEventShapes VicentMateu
3. PowerCorrections
UsingSCETonecanderiveafactorization formulaforthesingularcrosssection:
ds dsˆ ℓ
s = dℓ s e− F(ℓ) 1+O(e) . (3.1)
e
de Z de (cid:18) Q(cid:19)
(cid:2) (cid:3)
Here dsˆ /de refers to the partonic singular distribution whereas ds /de is the nonperturbative
s s
singulardistribution. dsˆ /dedivergesaslogi(e)/efore→0andhencedominatesinthedijetlimit.
s
F istheshapefunction, whichcontainsnonperturbative corrections (plussomeperturbative terms
e
[17]).
Inthe tail ofthe distribution, defined by the condition Qe≫L theshape function canbe
QCD
expanded ininversepowersofℓ≫L :
QCD
a L L 2
F(ℓ)=d (ℓ)−d ′(ℓ)W e(m )+O s QCD +O QCD , (3.2)
e 1 ℓ2 ℓ3
(cid:16) (cid:17) (cid:16) (cid:17)
HereW e(m )isanonperturbative matrixelementdefinedby
1
W e =h0|Y†Y†(Qeˆ)Y Y |0i. (3.3)
1 n¯ n n n¯
andeˆistheevent-shape operatordefinedas
eˆ|Xi=e(X)|Xi, (3.4)
with |Xi the state of a configuration of particles in the final state of a given event, and e(X) the
value oftheeventshape forthatconfiguration. Y andY¯ areWilsonlines ofsoftgluonfieldsinthe
light-like directions nandn¯.
UsingEq.(3.2)in(3.1)onefindsatleadingorderthattheeffectofthepowercorrections isto
shiftthedistribution:
ds dsˆ W e d dsˆ dsˆ W e
= − 1 +... = e− 1 +.... (3.5)
de de Q de de de(cid:18) Q (cid:19)
Asimilarresultisfoundinthedispersive model[9,10,11].
4. Universality
In order to study the effects of hadron masses on power corrections we need to express the
event-shape operator eˆ in terms of quantum fields. Following the approach of Refs. [16, 23] we
findthatitcanbewrittenintermsoftheenergy-momentum tensor. Letusstartbyintroducing the
“transverse velocityoperator”, definedbyitsactiononastate|Xi:
Eˆ (r,y)|Xi= (cid:229) m⊥d (r−r)d (y−y)|Xi. (4.1)
T i i i
i∈X
It is important to use rapidity y and not pseudorapidity h as only the former transforms in an
additive wayunderalongitudinal boost. InRef.[17]itwasshownthatEˆ canbeexpressed solely
T
intermsoftheenergy-momentum tensor. Nowtheevent-shape operatorcanbewrittenas
1 +¥ 1
eˆ≡ dy dr f (r,y)Eˆ (r,y). (4.2)
e T
QZ−¥ Z0
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HadronMassEffectsinPowerCorrectionstoEventShapes VicentMateu
EˆT(r,y) δv δη
v(r,y)
η(r,y)
tˆ
Figure1:Graphicalrepresentationofthetransversevelocityoperator.Measurementsaremadewithrespect
tothethrustaxistˆ. Thearrowscorrespondtoparticleswithlengthsgivenbytheparticlevelocities.Shading
indicates which particles are measured by the operator. Note that the velocity v(r,y) and pseudo-rapidity
h (r,y)arefunctionsofthetransversevelocityrandrapidityy.
According theleadingpowercorrection iswrittenintermsofadouble integral:
+¥ 1
W e = dy dr f (r,y)h0|Y†Y†E (r,y)Y Y |0i. (4.3)
1 e n¯ n T n n¯
Z−¥ Z0
AsdepictedinFig.1,thetransversevelocityoperatorEˆ (r,y)involvesaspheroidthatexpands
T
inbothspaceandtimewithafinitevelocityv,anditmeasuresthetotaltransversemassforparticles
in an infinitesimal interval in both h and the velocity v (or equivalently an infinitesimal interval
in y and r). Following Ref. [16] one can apply boost transformations along the thrust axis to
figure out inwhich cases universality ispreserved. Both the vacuum |0iand the Wilson lines are
boost invariant, howeverunderaboostofrapidity y′ thetransverse velocityoperatortransforms as
follows:
U(y′)Eˆ (r,y)U(y′)†=Eˆ (r,y+y′). (4.4)
T T
Therefore, choosing y′=−yinEq.(4.3)wecanwritetheleadingpowercorrection as
1
W e =c drg (r)W (r), (4.5)
1 e e 1
Z0
with
W (r)≡h0|Y†Y†Eˆ (r,0)Y Y |0i (4.6)
1 n¯ n T n n¯
auniversalnonperturbative function, and
+¥ 1 +¥
c = dyf (1,y), g (r)= dyf (r,y). (4.7)
e e e e
Z−¥ ceZ−¥
Thefunctiong (r)encodes allhadronmasseffects. Defining
e
1
W ge ≡ drg (r)W (r), (4.8)
1 Z0 e 1
onecanwriteW e =c W ge,whichimpliesthattheleadingpowercorrectionfortwoeventshapese
1 e 1 1
and e are proportional to each other if g (r)=g (r). Wewill denote the set ofall event shapes
2 e1 e2
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HadronMassEffectsinPowerCorrectionstoEventShapes VicentMateu
c t t t C r
e 2 (a) ±
2
Common 2 2 3p 1
1−a
Table1: Expressionforthec coefficientsforvariousdijeteventshapes.Sincec aredefinedusing f (1,y),
e e e
theyhavethesamevalueineachuniversalityclass.Heret referstothrust,t to2-Jettiness,t toangularities,
2 a
CtotheC-parameterandr tothehemispheremasses.
±
with the same g (r) function as a universality class. All event shapes with the same universality
e
classhavethesamepowercorrection uptoacalculable factor.
The coefficients c match exactly the classic universality prefactors obtained when hadron
e
masses are neglected [24, 9, 10, 25, 14, 26, 27, 28, 16]. Table 1 summarizes the values of the c
e
coefficients forthemostcommoneventshapes.
5. Massschemes anduniversality classes
The standard definition of an event shape involves in general both the energy and the mag-
nitude of the momentum of the final state hadrons (on top of the directions). In an experimental
environment one has access to a limited amount of information. Although directions are easily
measured, in general one has information on the energy deposited by the particle in the detector,
butnotonitsmomentum. Iftheparticleisidentifiedthemomentumcanbeofcoursereconstructed,
butthatisnotalwayspossible.
The E-scheme is an alternative definition for any event shape in a way that only the experi-
mentally accessible information is used. Specifically one makes the following replacement in the
event-shape definition:
E
i
~p → ~p . (5.1)
i i
|~p|
i
Itiseasytoshow[13,17]thatalleventshapesdefinedintheE-schemebelongtothesameuniver-
salityclass: theE-schemeclass.
Analogously onecandefinetheP-schemeclassbythereplacement E →|~p|. Althoughevent
i i
shapes defined in that way do not belong to the same class, they have nevertheless similar power
corrections [13,17].
Wedefinetwoadditionalschemes: theR-scheme,inwhichonereplacesh byyintheP-scheme
expression of e¯and thenuses eR =e¯R;andthe J-scheme, inwhichonesets r=1inthe R-scheme
expression.
Table2summarizeswhichuniversality classeventshapes(invariousschemes)belongto.
6. Anomalous Dimensionand Matching Coefficient
TheexpressionofW (r)inEq.(4.6)isonlyaformaldefinition. Ingeneral,matrixelementsin
1
aquantumfieldtheorywhichdonotdirectlycorrespondtoanobservablehavetobedefinedwithin
ascheme. InRef.[17]theanomalous dimension ofW (r)intheMSschemewascomputedatone
1
loop. The diagrams giving a non-vanishing contribution are shown in Figs. 2 and 3. It turns out
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HadronMassEffectsinPowerCorrectionstoEventShapes VicentMateu
Class g(r) Eventshape
JetMassclass(W 0 orW r ) 1 r ,t ,t J,t J ,CJ
1 1 ± 2 (a)
E-schemeclass(W 1 orW E) r t ,t E =t E,CE,r E,t R,t R ,CR,r R
1 1 (a) 2 ± (a) ±
rn class(W n) rn generalized angularities t [17]
1 (n,a)
Thrustclass(W 1gt ) gt (r) t ,r ±P,t2P
C-parameterclass(W gC) g (r) C
1 C
r2 class(W 2) r2 t ,t P
1 (2,a) (a→−¥ )
Table2: EventshapeclasseswithauniversalfirstpowercorrectionparameterW ge. Foragiveneventshape,
1
thefullpowercorrectionis W e =c W ge. SuperscriptsE,P,J,andRcorrespondtoeventshapesmeasured
1 e 1
intheE-,P-,J-,andR-schemes,respectively.
thatonlynon-abelian termscontribute, andonefinds
d a C
m W (r,m )= − s Aln(1−r2) W (r,m ). (6.1)
dm 1 p 1
h i
Interestingly, the anomalous dimension is r-dependent, although there is no mixing for different
valuesofr. Thisimpliesthathadronmassesplayanessential role.
FromEq.(6.1),fortworenormalization scales m and m ofcomparable sizeonehas
0
a (m )C m
W e(m )=W e(m )+ s 0 Aln W e,ln(m ), (6.2)
1 1 0 p m 1 0
(cid:16) 0(cid:17)
with
W e,ln(m )≡− drln(1−r2)c g (r)W (r,m ). (6.3)
1 0 Z e e 1 0
Given that W (r,m ) runs, one expects that the expansion of the shape function in Eq. (3.2) should
1
involve anon-trivial matching coefficient. Hencewewrite
L 2
F(ℓ)=d (ℓ)+ drCe(ℓ,r,m )c g (r)W (r,m )+O QCD . (6.4)
e Z 1 e e 1 ℓ3
(cid:16) (cid:17)
ConsistencywithEq.(6.1)requiresthatthematchingcoefficientatoneloophasthefollowingform
C a (m ) d 1 m
Ce(ℓ,r,m )=−d ′(ℓ)+ A s ln(1−r2)
1 p dℓ(cid:18)m hℓi+(cid:19)
a (m )
+ s d ′(ℓ)de(r)+O(a 2). (6.5)
p 1 s
ThestructureofEq.(6.5)waschecked byanexplicitcalculation inRef.[17].
7. Conclusions
We have studied hadron-mass effects for event shapes in the SCETformalism. These effects
have been expressed in terms of QCD matrix elements. Our results show that hadron masses
break universality although within certain classes itisstill preserved. Wehave computed the one-
loop running of the power correction, finding a nontrivial anomalous dimension and matching
coefficient. Welargely confirmtheresultsofRef.[13].
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HadronMassEffectsinPowerCorrectionstoEventShapes VicentMateu
Figure2: IndependentemissiondiagramswithAbelianandnon-Abeliancontributions.Thefouradditional
diagramsobtainedbyahorizontalfliporcomplexconjugationarenotshown.
Figure 3: Triple gluon Y-diagrams for the O(a 2) correction to W (r). The twelve additional diagrams
s 1
obtainedbyahorizontalfliporcomplexconjugationarenotshown.Diagramswithallthreegluonscoupled
toWilsonlinesofthesamedirectionvanish.
Acknowledgments
ThisworkwassupportedbytheofficesofNuclearandParticlePhysicsoftheU.S.Department
of Energy (DOE) under grant numbers DE-FG02-94ER-40818 and DE-FG02-05ER-41360, the
European Community’s Marie-Curie Research Networks under contract PITN-GA-2010-264564
(LHCphenOnet) and by MISTI global seed funds. VM is supported by a Marie Curie Fellowship
undercontract PIOF-GA-2009-251174.
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