The properties of C-parameter and coupling constants

1.7 MB·English

by  R. Saleh-Moghaddam

ThepropertiesofC-parameterandcouplingconstants
R. Saleh-Moghaddam and M.E. Zomorrodian
Department of Physics, Faculty of physics,
Ferdowsi University of Mashhad, 91775-1436 Mashhad, Iran
R [email protected] , [email protected]
−
(Dated:March10,2015)
abstract:Inthisarticle,wepresentthepropertiesoftheC-parameterwhichisoneofeventshapevariables.
Weobtainthecouplingconstantsbothintheperturbativeandinthenon-perturbativepartoftheQCDtheory.
Toachievethiswefitthedispersivemodelaswellastheshapefunctionmodelwithourdata. Ourresultsare
consistentwiththeQCDpredictions.Weexplainmorefeaturesofourresultsinthemaintext.
PACSnumbers:12.38.-t;12.38.Bx;12.38.Lg
Keywords:Quantumchromodynamics;Perturbativecalculations;non-Perturbativetheory
5
1
0 I. INTRODUCTION mentumtensorθ .
jk
2
(cid:80)
pipi/|pi|
pr Eventshapevariablesine+e− annihilationprovideanideal θjk = i(cid:80)j|pki| . (1)
A testing ground to study Quantum Chromo-Dynamics (QCD) i
and have been measured and studied extensively in the last where pi arethespatialcomponents(j,k = 1,2,3)ofthei-th
2
threedecades.Inparticular,eventshapevariablesareinterest-
particlemomentuminthecentreofmassframe. Thesumoni
1
ing for studying the interplay between perturbative and non-
runsoverallthefinalstateparticles. Ifλi aretheeigenvalues
perturbativedynamics[1]. Oneofthemostcommonandsuc-
] ofthematrix,wehave:
h cessfulwaysoftestingQCDhasbeenbyinvestigatingthedis-
p tributionofeventshapesine+e− →hadrons,whichhavebeen C =3(λ λ +λ λ +λ λ ). (2)
1 2 2 3 3 1
-
p measuredaccuratelyoverarangeofcentre-of-massenergies,
Therealsymmetricmatrixθ haseigenvaluesλ with0 ≤
e and provide a useful way of evaluating the strong coupling jk i
h constant α . The main obstruction to obtaining an accurate λ3 ≤ λ2 ≤ λ1 ≤ 1. It describes an ellipsoid with orthogonal
s
[ value of α from distributions is not due to a lack of precise axes named minor, semi-major and major corresponding to
s
2 databuttodominanterrorsinthetheoreticalcalculationofthe thethreeeigenvalues. Themajoraxisissimilarbutnotiden-
v distributions. In particular, there are non-perturbative effects tical to nT. The C parameter varies in the range 0 ≤ C ≤ 1.
1 that cannot yet be calculated from first principles but cause C = 0 corresponds to a perfect two jet event (with massless
6 power-suppressedcorrectionsthatcanbesignicantatexperi- jets), while C = 1 characterizes a spherical event. Planar
7 events including in particular the O(α ) perturbative result,
mentallyaccessibleenergyscales[2]. S
1 aredistributedintherange0≤C ≤3/4.
0 Inthisarticle,weshowthecrosssectionforC-parameteras Figure 1 shows the distribution for C-parameter (0 ≤ C ≤
. aneventshapeobservable. Alsoweperusebothperturbative
1 1)byusingtheMonteCarlodata(PYTHIAprogram)indif-
andnon-perturbativetheoryforcalculatingcouplingconstants
0 ferentcenterofmassenergies. Weseethatthevaluesarebe-
5 using different models .It is mentioned that we have already tween0and1forallenergies. Weobserveadecreasingtrend
1 done some analyses on some event shape observables in our
for all distributions by increasing the C parameter. The fig-
v: previouspublications[3,4]. ure also indicates that there is not a significant difference on
i The outline of the paper is as the following: In sect.2 we thedistributionsbetweendifferentenergies. Weconcludethat
X
define and review the C-parameter and show the cross sec- thisparameterisinsensitivetothedataofdifferentenergies.
r tion distribution for different energies. In sect.3, we present
a
thecalculationsoftheperturbativetheoryaswellasthenon-
perturbative theory up to the next to next to leading order
(NNLO). We also achieve similar calculations for both the III. POWERCORRECTIONSANDDIFFERENTMODELS
dispersiveandtheshapefunctionmodels. Andfinallyweex-
tractthecouplingconstantfromouranalysis. Thelastsection QCD calculations based on NLO perturbation theory as
summarizesourconclusions. well as NNLO theory are available for many observables in
high energy particle reactions, like the total hadronic cross
section in e+e− → hadrons and moments in deep inelastic
scattering (DIS) processes and also in hadronic decay. The
complicated nature of QCD, due to the process of gluon self
II. C-PARAMETER
couplingandtheresultinglargenumberofFeynmandiagrams
inhigherordersofperturbationtheory,sofarlimitedthenum-
The C parameter [5, 6] for electron−positron annihilation berofQCDcalculationsincompleteNNLO[7]. Weshallex-
eventsisderivedfromtheeigenvaluesλ ofthelinearizedmo- plainthemethodsofcalculationforNLOandNNLOtheories.
i
2
Allobservablesarechosensuchthattheytakevaluesbetween
0and1. Nowifweaddthethirdsentencetoaboveexpansion,
wewillhavetheNNLOtheory.
1 dσ α (µ) dA α (µ) dB α (µ) dC
=( s ) +( s )2 +( s )3 . (5)
σ dy 2π dy 2π dy 2π dy
tot
C gives the NNLO correction. The perturbative contribution
to (cid:104)yn(cid:105) is given up to NNLO in terms of the dimensionless
coefficientsA ,B andC as[1]:
y,n y,n y,n
α (µ) α (µ)
(cid:104)yn(cid:105) =( s )A +( s )2
PT 2π y,n 2π
µ2 α (µ)
(B +A β log )+( s )3(C +2B β
y,n y,n 0 E 2π y,n y,n 0
cm
FIG.1: ThecrosssectiondistributionsforC-parameterindifferent
energies.
µ2 µ2 µ2
log +A (β2log2 +β log ). (6)
E y,n 0 E 1 E
cm cm cm
Wealsogiveanexplanationofotherwaysforcalculatingthe
coupling constant such as the dispersive model and also the A , B and C are straightforwardly related to A , B
y,n y,n y,n y,n y,n
shapefunctionmodel. Thesemodelsincludeperturbativeand andC :
y,n
non-perturbativeregions. Oneoftheaspectswherethestudy
of event shapes has taught us general lessons about QCD is A = A ,
y,n y,n
the interplay between perturbative and nonperturbative cor-
rections. The phenomenological success of renormalization
based on models for power corrections has important conse- 3
B = B − C A ,
quences. y,n y,n 2 F y,n
Inevent-shapemoments(y),oneexpectsthehadronization
correctionstobeadditive,suchthattheycanbedividedintoa
3 9
perturbativeandanon-perturbativecontribution[8], C =C − C B +( C2 −K )A . (7)
y,n y,n 2 F y,n 4 F 2 y,n
that A , B and C are given in Table I also the constant
(cid:104)yn(cid:105)=(cid:104)yn(cid:105) +(cid:104)yn(cid:105) . (3) y,n y,n y,n
pt np K isgivenby[9,10]:
2
Nowweexplicatethesetheoriesandmodelsinthefollow-
1 3 123
ing: K = [− C2 +C C ( −44ζ )
2 4 2 F F A 2 3
A. NLOandNNLOcorrections +C T N (−22+16ζ )]. (8)
F R F 3
The perturbative expansion of a differential distribution Here Ecm denotes the centre of mass energy squared and
weightedoftheobservableycanbewrittenforanyinfrared- is the QCD renormalization scale. The NLO expression is
safe observable for the process e+e− → hadrons. The main obtained by suppressing all terms at order α3. The first two
S
theoreticalfeatureofjetobservablesisthattheycanbecom- coefficientsoftheQCDβ-functionare:
puted in QCD perturbation theory. As a result, jets observ-
11C −4T N
ablesarefinite(calculable)atthepartoniclevelorderbyorder β = A R F,
0
6
inperturbationtheory[7]. IfweassumetheNLOtheoryonly,
we can compute its corresponding perturbative expansion in
theform
TABLEI: ContributionstoC-parameteratLO,NLOandNNLO[1].
1 dσ =(αs(µ)) dA +(αs(µ))2 dB. (4) (cid:104)Cn(cid:105) A B C
σ dy 2π dy 2π dy n=1 8.6379 172.778±0.007 3212.2±88.7
tot
n=2 2.4317 81.184±0.005 2220.9±12.0
AgivestheLOresultandBtheNLOcorrection. σ denotes n=3 1.0792 42.771±0.003 1296.6±6.7
tot
the total hadronic cross section calculated up to the relevant n=4 0.5685 25.816±0.002 843.1±3.9
order. The arbitrary renormalization scale is denoted by µ. n=5 0.3272 16.873±0.001 585.0±3.2
3
17C2 −10C T N −6C T N
β = A A R F F R F. (9)
1
6
where C = N is color number, C = (N2 −1)/2N = 4/3,
A F
T =1/2andN isquarkflavors.
R F
The perturbative coefficients (A , B andC ) are inde-
y,n y,n y,n
pendent of the centre-of-mass energy. They are obtained by
integrating parton-level distributions, which were calculated
recently to NNLO accuracy [1]. These values are shown in
TableI.
In order to define the non-perturbative part of the theory,
it is necessary to explain the dispersive as well as the shape
functionmodelasthefollowing.
B. Thedispersivemodel
Non-perturbative power corrections can be related to in-
fraredrenormalizationsintheperturbativeQCDexpansionfor
theevent-shapevariables.Thedispersivemodelforthestrong
couplingleadstoashiftinthedistributions[8]:
dσ dσ
(y)= pt(y−a .P) (10)
dy dy y
wherethenumericalfactorα dependsontheeventshapethat
y
for C-parameter it is a = 3π [11]. Then we obtain the non-
y
perturbativeparameterfromthismodelasthefollowing:
4C µ
(cid:104)yn(cid:105) =a .P=a . FM I
np y y π2 E
cm
µ k
.[α (µ )−α (µ)−(log( )+1+ )2β α2(µ)] (11)
0 I s µ 4πβ 0 s
I 0
In the MS renormalization scheme the constant k has the
valuek=(67/18−π2/6)C −5/9N ,TheMilanFactorMis
A F
knownintwoloopsasM = 1.49±0.20[11,12],fornumber FIG.2: FittingthedispersivemodelwithdatafortheC-parameter
offlavorN =3attherelevantlowscales[13].
F
AsaresultforMthatso-callednon-inclusiveMilanfactor,
wehad[14]: (cid:104)y3(cid:105)=(cid:104)y3(cid:105) +3(cid:104)y2(cid:105) (a .P)+3(cid:104)y1(cid:105) (a .P)2+(a .P)3.
NLO NLO y NLO y y
(15)
3.299C −0.862C −0.052N α defines the strong coupling constant and α defines the
M=1+ A +2× A F s 0
β β nonperturbativeparameteraccountingforthecontributionsto
0 0
the event shape below an infrared matching scale µ (cid:27) 2.
I
Therefore it is possible to obtain α and α in this model.
S 0
1.575C −0.104N
=1+ A F =1.49±0.20 (12) We are using the AMY data taken from TRISTAN at KEK,
β0 as well as DELPHI and ALEPH at CERN plus the PYTHIA
eventgenerator.Byfittingthedispersivemodelwithdiagrams
Nowwecanusethepowercorrectionstocalculatetheper-
depictedinFigure2,weobservethatthemodelisconsistent
turbativeandthenon-perturbativetheoriesforthemomentsof
withthedata.
y,wehave[8]:
We observe that all distributions for the dispersive model
(solid line) are in good agreement with the Monte Carlo
(cid:104)y1(cid:105)=(cid:104)y1(cid:105) +a .P, (13) (PYTHIA) and the experimental data when compared with
NLO y
NLO or NNLO prediction because the latters do not include
thepowercorrection. Inotherwordsthedispersivemodelin-
(cid:104)y2(cid:105)=(cid:104)y2(cid:105) +2(cid:104)y1(cid:105) (a .P)+(a .P)2, (14) cludes both perturbative and the non-perturbative part of the
NLO NLO y y
4
theory. Next step is to measure the strong coupling constant
TABLEIII:CouplingconstantsforC-parameterin[15].
andthenon-perturbativeparameterbyfittingtheabovemodel
Exp ALEPH DELPHI L3
with the data. The values for α (M ) and α (µ ) up to third
s Z0 0 I α (M ) 0.123±0.003 0.122±0.004 0.116±0.005
powercorrectionarecitedinTableII.Ourresultsarealsocon- s Z0
α (µ) 0.461±0.016 0.444±0.022 0.457±0.040
sistentwiththoseobtainedbyusingotherexperiments(Table 0 I
IIIisextractedfrom[15]).
C. Theshapefunctionmodel
KorchemskyandTafat[16]describepropertiesoftheevent
shapevariables1−T and M2 notincludedinNLOperturba-
H
tion theory so called shape function, which does not depend
onthevariablenorthecmsenergy. Thisismoregeneralthan
the dispersive model, considering both as a shift of the per-
turbative prediction and as a compression of the distribution
peak. We should mention that the shape function model in-
cludestheperturbativetheoryaswellasthenon-perturbative
partoftheory. ThismodelisacombinationofboththeNLO
prediction and the power correction terms (equation 3). To
calculate the strong coupling constant in perturbative theory,
weareusingequation4. Wealsousethefollowingexpansion
formeasuringthefreeparameterinnon-perturbativetheory.
λ
(cid:104)C1(cid:105)=(cid:104)C1(cid:105) + 1 . (16)
NLO E
cm
Analogously for the second moments and the third mo-
ments,wefind[17]:
λ λ
(cid:104)C2(cid:105)=(cid:104)C2(cid:105) +2 1 (cid:104)C1(cid:105) + 2 . (17)
NLO E NLO E2
cm cm
λ λ λ
(cid:104)C3(cid:105)=(cid:104)C3(cid:105) +3 1 (cid:104)C2(cid:105) +3 2 (cid:104)C1(cid:105) + 3 . (18)
NLO E NLO E2 NLO E3
cm cm cm
Inordertocalculatethestrongcouplingconstantconcerned
withtheperturbativetheory,weperformafittingprocedureby
includingjustthefirstpartofequation16. Ontheotherhand
tofindthecouplingconstantinthenon-perturbativeregionwe
doafittingprocedurebytakingintoaccountbothparameters
inequation16. Weachieveasimilaranalysisforequations17
and 18. The coefficient λ is interpreted as the first moment
1
and λ as the second moment of the shape function. These
2
parameters are also known as the so called universal scales
[18].
Figure 3 shows the distributions obtained from the exper- FIG.3: FittingtheC-parameterinshapefunctionmodelwithdata.
imental data as well as PYTHIA. By fitting the shape func-
tionmodelwiththecorrespondingdistribution,weobtainthe
strongcouplingconstantα (M )andthenon-perturbativepa-
s Z0
rameter(λ ). OurresultsareindicatedinTableIV.
1
TABLEIV:Measurementsofthecouplingconstantsusingtheshape
functionmodel.
TABLEII:Measurementsofthecouplingconstantsusingthedisper-
n 1 2 3
sivemodel.
α (M ) 0.149±0.003 0.143±0.003 0.140±0.0064
n 1 2 3 s Z0
λ 0.578±0.037 0.521±0.039 0.449±0.033
1
α (M ) 0.149±0.003 0.117±0.001 0.112±0.003
s Z0
α (µ) 0.460±0.015 0.471±0.007 0.499±0.017
0 I
5
These values are in good agreement with the results from
othereventshapevariablescarriedoutinourpreviouspubli-
cations [3, 4]. Our results are also consistent with the QCD
predictions[13].
D. Thevariance
Thesimplepredictionforthevarianceofeventshapevari-
ables(y)onhadronlevelis[19]:
Var(y)=(cid:104)y2(cid:105)−(cid:104)y(cid:105)2. (19)
Wecanobtainapurelyperturbativeexpressionforthevari-
FIG.4: Fittingvariancebyusingthedispersivemodel.
ance in the dispersive model and also in the shape function
model,uptostronglysuppressedcorrectionsO(α (Q2)). The
S
first and second orders give the identical prediction in NLO
andNNLOmoments.
However the NLO and NNLO do not include non- pertur-
bativepart.
ThusincaseoftheC-parameter,wehave:
Var(C)=(cid:104)C2(cid:105) −(cid:104)C(cid:105)2 . (20)
NLO NLO
On the other hand, if we take into account the non-
perturbativepartofthemodel,wewillhave:
Var(C)=(cid:104)C2(cid:105) −(cid:104)C(cid:105)2 . (21)
total total
where the subscript indicates the perturbative as well as the
non-perturbativepartsofourcalculationsinbothmodels. In
Figures 4 and 5 we show the variance for dispersive as well
asshapefunctionmodel. Theobtainedresultsbyfittingvari-
ance with models are summarized in Table V and Table VI
respectively.
Both tables indicate that our values for the coupling con-
stant in perturbative and in the non-perturbative regions are
in good agreement with the QCD predictions. They are also
consistentwiththoseobtainedfromotherexperiments[13].
FIG.5: Fittingvariancebyusingtheshapefunctionmodel.
TABLEV:Measurementsofthecouplingconstantsbyusingthevari-
anceofdispersivemodel.
Observable α (M ) α (µ)
s Z0 0 I
C-parameter 0.1039±0.0059 0.5183±0.0211
TABLEVI:Measurementsofcouplingconstantsbyusingthevari-
anceofshapefunctionmodel.
Observable α (M ) λ
s Z0 1
C-parameter 0.1052±0.0031 0.7164±0.0342
6
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