Event shapes and jet rates in electron-positron annihilation at NNLO

54 Pages·2009·0.51 MB·English

by  Stefan Weinzierl,03-126,25579,

MZ-TH/09-13
9 Event shapes and jet rates in electron-positron annihilation at
0
0 NNLO
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A
Stefan Weinzierl

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InstitutfürPhysik,UniversitätMainz,
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D-55099Mainz,Germany
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1
v
7
7 Abstract
0
1 This article gives the perturbative NNLO results for the most commonly used event shape
.
4 variables associated to three-jet events in electron-positron annihilation: Thrust, heavy jet
0 mass, wide jet broadening, total jet broadening, C parameter and the Durham three-to-two
9
jet transition variable. In addition the NNLO results for the jet rates corresponding to the
0
: Durham,Geneva,Jade-E0andCambridgejetalgorithmsarepresented.
v
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a
1 Introduction
Theexperimentsat LEP(CERN), SLC (SLAC) and PETRA (DESY) havecollected a wealth of
data from electron-positron annihilation with hadronic final state over a wide range of energies.
Of particular interest are three-jet events, which can be used to extract the value of the strong
coupling constant. Three-jet events are well suited for this task because the leading term in a
perturbative calculation of three-jet observables is already proportional to the strong coupling.
Incomparingexperimentstotheoryitisimportanttorestrictoneselftoinfra-redsafeobservables.
For three-jet events in electron-positron annihilation there is a well established set of infra-red
safe observables, which is widely used. Theseare the eventshape variables consistingof thrust,
heavy jet mass, wide jet broadening, total jet broadening and theC-parameter. In addition there
are observables related to a specific jet definition. First of all these are thejet rates associated to
the different jet definitions. In this paper the following jet algorithms are considered: Durham,
Geneva, Jade-E0 and Cambridge. As the Durham jet algorithmis the most popular one, for this
jet algorithmalsothethree-to-twojet transitionvariableis studied.
All these observables can be calculated in perturbation theory. In this article I present the
next-to-next-to-leading order (NNLO) results for these observables. For completeness I also
includethenext-to-leadingorder(NLO)and leadingorder(LO)results.
Another group published results for these observables earlier on in [1–3], but omitted cer-
tain subtraction terms related to soft gluons. The present calculation is based on the numerical
Monte-Carlo program reported in [4,5]. The additional soft gluon subtraction terms modify the
distributions in the peak region. Meanwhile the other group has added the missing subtraction
terms and the two programs are now for the main part of the observables in good agreement.
A few remaining differences are discussed in the numerical section. A detailed account of the
subtractiontermsusedin thiscalculationisgivenin acompanionpaper[6].
For the thrust distribution there exists an independent calculation of the logarithmic terms
based on soft-collineareffectivetheory [7]. Thiscalculation shouldagree withthe NNLO result
for small values of (1 T). For large values of (1 T) the logarithmic terms alone are not
− −
sufficient to givean accurate result. On the other hand the perturbativecalculation of this paper
givesthecorrectNNLOresultforlargeandintermediatevaluesof(1 T),buthasitslimitations
−
dueto the Monte-Carlo integrationmethod forvery small valuesof (1 T). Thisgivesa region
−
of overlap where the perturbative NNLO result and the one obtained from SCET should agree,
and indeedtheydo.
This article reports the pure perturbative results for the event shapes. Not included are soft-
gluonresummations[7–10]norpowercorrections [11,12].
It should be mentioned that the results of this paper rely heavily on research carried out in
the past years: Integration techniques for two-loop amplitudes [13–16], the calculation of the
relevant tree-, one- and two-loop-amplitudes [17–29], routines for the numerical evaluation of
polylogarithms [30–32], methods to handle infrared singularities [33–51] and experience from
theNNLOcalculationsofe+e 2jets andotherprocesses [40,52–68].
−
→
This paper is organised as follows: Section 2 gives the definition of the observables related
to three-jet events in electron-positron annihilation. Section 3 describes the perturbative calcu-
lation of these observables. Section 4 gives the numerical results. Finally, section 5 contains a
2
summary. Inanappendixausefulalgorithmforthedeterminationofthethrustaxisisdescribed.
2 Definition of the observables
Theeventshapevariablethrust[69,70]isdefined by
max(cid:229) ~p ~n
j
~n j ·
T = , (1)
(cid:229) ~(cid:12)(cid:12)pj (cid:12)(cid:12)
j
(cid:12) (cid:12)
(cid:12) (cid:12)
where ~p denotes the three-momentum of particle j and the sum runs over all particles in the
j
final state. Thethrustvariablemaximisesthetotallongitudinalmomentumalongtheunit vector
~n. The value of~n, for which the maximum is attained is called the thrust axis and denoted by
~n . Thevalueofthrustrangesbetween1/2and1,whereT =1correspondstoanidealcollinear
T
two-jet event and T = 1/2 corresponds to a perfectly spherical event. Usually one considers
insteadofthrustT thevariable(1 T),suchthatthetwo-jetregioncorrespondsto(1 T) 0.
− − →
Forthree-parton eventswehave(1 T) 1/3.
− ≤
The plane orthogonal to the thrust axis divides the space into two hemispheres H and H .
1 2
These are used to define the following event shape variables: The hemisphere masses [71] are
defined by
2
M2 = (cid:229) p , i=1,2, (2)
i j
j Hi !
∈
where p denotes the four-momentum of particle j. The heavy hemisphere mass M and the
j H
lighthemispheremassM arethen defined by
L
M2 =max M2,M2 , M2 =min M2,M2 . (3)
H 1 2 L 1 2
(cid:0) (cid:1) (cid:0) (cid:1)
Thelighthemispheremassisafour-jet observableand vanishesforthree partons. Fortheheavy
jet massitisconvenienttointroducethedimensionlessquantity
M2
r = H, (4)
Q2
whereQisthecentre-of-massenergy. Forthree-partoneventswehaver 1/3. Inleadingorder
≤
thedistributionoftheheavyjet massisidenticalto thedistributionof(1 T).
−
Thehemispherebroadenings[72,73]aredefined by
(cid:229) ~p ~n
j T
×
Bi = j∈H2i(cid:12)(cid:229) ~p (cid:12), i=1,2, (5)
(cid:12) k (cid:12)
| |
k
3
where the sum over j runs over all particles in one of the hemispheres, whereas the sum over k
is overall particles in thefinal state. Thewidejet broadening B , the narrow jet broadening B
W N
and thetotaljet broadeningB are defined by
T
B =max(B ,B ), B =min(B ,B ), B =B +B . (6)
W 1 2 N 1 2 T 1 2
The narrow jet broadening is a four-jet observable and vanishes for three partons. For three-
partoneventswehaveB ,B 1/(2√3).
W T
≤
TheC-and D-parameters [74,75]are obtainedfromthelinearisedmomentumtensor
q ij = 1 (cid:229) pikpkj, i, j=1,2,3, (7)
(cid:229) ~p ~p
| l| k | k|
l
where the sum runs over all final state particles and pi is the i-th component of the three-
k
momentum~p of particle k in the c.m. system. The tensorq is normalised to have unit trace. In
k
termsoftheeigenvaluesoftheq tensor,l , l , l , withl +l +l =1,onedefines
1 2 3 1 2 3
C = 3(l l +l l +l l ),
1 2 2 3 3 1
D = 27l l l . (8)
1 2 3
The range of values is 0 C,D 1. The D-parameter is again a four-jet observable and van-
≤ ≤
ishes for three partons. For three-parton events we haveC 3/4. TheC-parameter exhibits in
≤
perturbationtheoryasingularityatthethree-parton boundaryC=3/4 [76].
Theproductionrateforthree-jeteventsisdefinedbytheratioofthecrosssectionforthree-jet
eventsby thetotalhadroniccross section
s
3 jet
R3 = s − . (9)
tot
Thisdependsonthedefinitionofthejetalgorithm,usuallyspecifiedthrougharesolutionvariable
andarecombinationprescription. Theclusteringprocedureofajetalgorithminelectron-positron
annihilationisin mostcases defined throughthefollowingsteps:
1. Definea resolutionparametery
cut
2. Foreverypair(p ,p )offinal-stateparticlescomputethecorrespondingresolutionvariable
k l
y .
kl
3. If y is thesmallest valueof y computedaboveand y <y then combine (p,p ) into
ij kl ij cut i j
asinglejet(’pseudo-particle’)withmomentum p accordingtoarecombinationprescrip-
ij
tion.
4. Repeat untilall pairsofobjects(particles and/orpseudo-particles)havey >y .
kl cut
The various jet algorithms differ in the precise definition of the resolution variable and the re-
combinationprescription. Thevariousrecombinationprescriptionsare:
4
1. E-scheme:
E =E +E , ~p =~p +~p . (10)
ij i j ij i j
The E-scheme conserves energy and momentum, but for massless particles i and j the
recombinedfour-momentumisnot massless.
2. E0-scheme:
E +E
i j
E =E +E , ~p = ~p +~p . (11)
ij i j ij i j
~p +~p
i j
(cid:0) (cid:1)
TheE0-schemeconservesenergy,butnotmome(cid:12)ntum. F(cid:12)ormasslessparticlesiand j isthe
(cid:12) (cid:12)
recombinedfour-momentumagainmassless.
3. P-scheme:
~p +~p
i j
E = E +E = ~p +~p , ~p =~p +~p . (12)
ij i j i j ij i j
E +E
(cid:12) i j(cid:12)
(cid:12) (cid:12)(cid:0) (cid:1) (cid:12) (cid:12)
(cid:12) (cid:12)
The P-scheme conserves momentum, but not energy. For massless particles i and j is the
recombinedfour-momentumagainmassless,as intheE0-scheme.
For the Durham [77], Geneva [78] and Jade-E0 [79] jet algorithms the resolution variables and
therecombinationprescriptionsaredefined as follows:
2min(E2,E2) 1 cosq
i j ij
Durham: y = − , E-scheme,
ij Q2
(cid:0) (cid:1)
8 2E E 1 cosq
i j ij
Geneva: y = − , E-scheme,
ij 9 · E(cid:0)i+Ej 2 (cid:1)
2E E 1 cosq
i j (cid:0) i(cid:1)j
Jade-E0: y = − , E0-scheme, (13)
ij Q2
(cid:0) (cid:1)
where E and E are the energies of particles i and j, and q is the angle between ~p and ~p .
i j ij i j
Q is the centre-of-mass energy. The jet transition variable y is the value of the jet resolution
23
parameter y , for which the event changes from a three-jet to a two-jet configuration. Similar,
cut
y is defined as the value of the resolution parameter, where the event changes from a four-jet
34
toathree-jetconfiguration. Thethree-jetrateisrelated tothejettransitionvariablesy andy :
23 34
1 1
ds ds
R (y ) = dy dy . (14)
3 cut 23 34
Z dy −Z dy
23 34
ycut ycut
The Cambridge algorithm [80] distinguishes an ordering variable v and a resolution variable
ij
y . TheCambridgealgorithmisdefined as follows:
ij
5
1. Select apairofobjects(p ,p ) withtheminimalvalueoftheordering variablev .
i j ij
2. Ify <y theyarecombined,onerecomputestherelevantvaluesoftheorderingvariable
ij cut
and goesback to thefirst step.
3. Ify y andE <E theni isdefined as aresolvedjet and deletedfrom thetable.
ij cut i j
≥
4. Repeat until only one object is left in the table. This object is also defined as a jet and
clusteringisfinished.
As orderingvariable
v =2 1 cosq (15)
ij ij
−
isused. Theresolutionvariableis as inthe(cid:0)Durhamalg(cid:1)orithm
2min(E2,E2) 1 cosq
i j ij
y = − (16)
ij Q2
(cid:0) (cid:1)
and theE-schemeisused as recombinationprescription.
3 Perturbative expansion
O
The perturbative expansion of a differential distribution for any infrared-safe observable for
theprocesse+e 3 jetscan bewrittenup toNNLO as
−
→
On ds (µ) a s(µ)OndA¯O(µ) a s(µ) 2OndB¯O(µ) a s(µ) 3 OndC¯O(µ)
= + + .(17)
s (µ) dO 2p dO 2p dO 2p dO
tot
(cid:18) (cid:19) (cid:18) (cid:19)
A¯O gives the LO result, B¯O the NLO correction and C¯O the NNLO correction. The variable
n denotes the moment of the distribution. Unless stated otherwise, the value n = 1 is used as
default. s denotes the total hadronic cross section calculated up to the relevant order. The
tot
arbitrary renormalisationscaleis denotedbyµ. Then-thmomentisgivenby
1 ds
On = On dO. (18)
h i s Z dO
tot
In practisethenumerical programcomputesthedistribution
On ds (µ) a s(µ)OndAO(µ) a s(µ) 2OndBO(µ) a s(µ) 3 OndCO(µ)
= + + , (19)
s (µ) dO 2p dO 2p dO 2p dO
0
(cid:18) (cid:19) (cid:18) (cid:19)
normalisedtos ,whichistheLOcrosssectionfore+e hadrons,insteadofthenormalisation
0 −
to s tot. There is a simple relation between the two distri→butions: The functions AO, BO andCO
are related tothefunctionsA¯O, B¯O andC¯O by
A¯O = AO,
B¯O = BO AtotAO,
−
C¯O = CO AtotBO Btot At2ot AO,
− − −
(cid:0) (cid:1)
6
where
3(N2 1)
A = c − ,
tot
4N
c
N2 1 243 3
B = c − 44z N + +(8z 11)N .
tot 3 c 3 f
8N 4 − 4N −
c c
(cid:20)(cid:18) (cid:19) (cid:21)
N denotes the number of colours and N the number of light quark flavours. A and B are
c f tot tot
obtainedfrom theperturbativeexpansionofs [81–83]:
tot
a a 2
s = s 1+ sA + s B +O(a 3) . (20)
tot 0 2p tot 2p tot s
(cid:18) (cid:19)
(cid:16) (cid:17)
The perturbativecalculation of the inclusivehadronic cross section s (tot) is actually known to
O(a 3)[84,85], althoughweneed here onlythecoefficients up toO(ha 2i).
s s
It is sufficient to calculate the functions A¯O, B¯O andC¯O for a fixed renormalisation scale µ0,
which can be taken conveniently to be equal to the centre-of-mass energy: µ = Q. The scale
0
variationcan berestored fromtherenormalisationgroupequation
d a 1 a 2 1 a 3 1 a 4
µ2 S = b S b S b S +O(a 5), (21)
dµ2 2p −2 0 2p −4 1 2p −8 2 2p s
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
11 4
b = C T N ,
0 A R f
3 −3
34 5
b = C2 4 C +C T N ,
1 3 A− 3 A F R f
(cid:18) (cid:19)
2857 1415 205 158 44
b = C3 C2+ C C 2C2 T N + C + C T2N2.
2 54 A− 27 A 9 A F− F R f 27 A 9 F R f
(cid:18) (cid:19) (cid:18) (cid:19)
Thecolourfactors are defined as usualby
N2 1 1
C =N , C = c − , T = . (22)
A c F R
2N 2
c
The values of the functions A¯O, B¯O and C¯O at a scale µ are then obtained from the ones at the
scaleµ by
0
A¯O(µ) = A¯O(µ0),
1 µ2
B¯O(µ) = B¯O(µ0)+2b 0ln µ2 A¯O(µ0),
(cid:18) 0(cid:19)
µ2 1 µ2 µ2
C¯O(µ) = C¯O(µ0)+b 0ln µ2 B¯O(µ0)+4 b 1+b 20ln µ2 ln µ2 A¯O(µ0). (23)
(cid:18) 0(cid:19) (cid:20) (cid:18) 0(cid:19)(cid:21) (cid:18) 0(cid:19)
Finally,an approximatesolutionofeq. (21)fora is givenby
s
a (µ) 2 b lnL b 2 1 2 b b 5
s = 1 1 + 1 lnL + 0 2 , (24)
2p b L −b 2 L b 4L2 2− b 2 −4
0 ( 0 0 "(cid:18) (cid:19) 1 #)
7
whereL=ln(µ2/L 2).
ThefunctionCO can bedecomposedintosixcolourpieces
N2 1 1 N
CO = 8cN− Nc2COlc+COsc+N2COssc+NfNcCOnf + NfCOnfsc+N2fCOnfnf . (25)
(cid:0) c (cid:1)(cid:20) c c (cid:21)
In addition, there are singlet contributions, which arise from interference terms of amplitudes,
wheretheelectro-weakbosoncouplestotwodifferentfermionlines. Thesesingletcontributions
are expected to be numerically small [28,86,87] and neglected in the present calculation. We
define
N2 1 N2 1 N2 1 1
CO|lc = 8cN− Nc2COlc, CO|sc = 8cN− COsc, CO|ssc = 8cN− N2COssc, (26)
(cid:0) c (cid:1) (cid:0) c (cid:1) (cid:0) c (cid:1) c
N2 1 N2 1 N N2 1
CO|nf = 8cN− NfNcCOnf, CO|nfsc = 8cN− NfCOnfsc, CO|nfnf = 8cN− N2fCOnfnf,
(cid:0) c (cid:1) (cid:0) c (cid:1) c (cid:0) c (cid:1)
e.g. thefunction CO includesthecolourfactors.
lc
|
The functions AO, BO and CO are calculated for a fixed renormalisation scale equal to the
O
centre-of-mass energy: µ =Q. They depend only on the valueof the observable . Since only
0
QCD corrections with non-singlet quark couplings are taken into account and singlet contribu-
tionstoCO areneglected,thefunctionsAO,BO andCO donotdependonelectro-weakcouplings.
4 Numerical results
InthissectionIpresentthenumericalresultsfortheeventshapevariablesandjetratesatnext-to-
next-to-leadingorder. AllresultshavebeenobtainedbynumericalMonte-Carlointegration. The
Monte-Carlo integration introduces a statistical error, which will be quoted with all results. For
theinfraredsingularitiesahybridmethodbetweensubtractionandslicinghasbeenused. Unless
statedotherwiseallresultshavebeen obtainedwiththeslicingparameter
s
h = min =10 5. (27)
−
Q2
It should be noted that the slicing procedure introduces in addition a systematic error. The size
of this error can be estimated by varying theslicing parameter h . However, lowering theslicing
parameterwillincreasethestatisticalerror. Apracticalcriteriaistorequirethatthevariationdue
to the slicing parameter is smaller than the statistical error, with the possible exception for the
boundaries of the distributions. Imposing this criteria h =10 5 turns out to be a good compro-
−
mise between accuracy and efficiency. The boundaries of the distributions close to the two-jet
region deserve special attention: There the value of the observable is comparable to h and one
expects sizable corrections. I will discuss the dependence on the slicing parameter in the close-
to-two-jetregionindetailfor afewexamples.
In a first series of plots I show the comparison of the NNLO results with the NLO and LO
8
results for the LEP I centre-of-mass energy Q2 = m with a = 0.118. The results for the
Z S
event shape variables thrust, heavy jet mass, wide jet broadening, total jet broadening, C pa-
p
rameter and three-to-two jet transition variable y are shown in figs. 1 to 6. For each of these
23
observablethedistributionweightedbytheobservableandnormalisedtothetotalhadroniccross
sectionisshown,e.g. forthrustthedistribution
1 T ds
− (28)
s d(1 T)
−
is shown. The corresponding plots for the jet rates for the jet algorithms of Durham, Geneva,
Jade-E0 and Cambridge are shown in figs. 7 to 10. In all these plots is the leading-order pre-
diction shown in light blue, the next-to-leading-order prediction is shown in pink and the next-
to-next-to-leading order prediction is shown in dark blue. The bands give the range for the
theoretical prediction obtainedfrom varyingthe renormalisationscale from µ=Q/2 to µ=2Q.
In addition the experimental data points [88] from the Aleph experiment (where available) are
also shown in these plot. Note that the theory predictions in these plots are the pure perturba-
tive predictions. Power corrections or soft gluon resummation effects are not included in these
results.
Numerical results for centre-of-mass energies different from Q2 =m are also easily ob-
Z
tained. As an example I show in fig. 11 the thrust distribution at centre-of-mass energies of
p
Q2 = 91.2GeV, 133GeV, 161GeV, 172GeV, 183GeV, 189GeV, 200GeV, 206GeV, again
withexperimentaldatapointsfromtheAlephexperiment. AsfortheLEPIIenergiesthebinsize
p
oftheexperimentaldataisrather large, Ishowinfig. 11 thedistribution
1 ds
(29)
s d(1 T)
−
withoutan additionalfactor(1 T).
−
ThemostimportantresultsofthispaperaretheperturbativecoefficientsAO,BO andCO. For
thesixeventshapevariables(thrust,heavyjetmass,widejetbroadening,totaljetbroadening,C-
parameter and thethree-to-twojet transitionvariable)thenumericalvalues fortheLO functions
O
AO, theNLO functionsBO and theNNLOfunctionsCO, allweightedby , aregivenintables1
to 7. The corresponding values for thejet rates defined according to the Durham, Geneva, Jade-
E0 or Cambridge algorithms are given in tables 8 to 11. These values are obtained by Monte-
Carlo integration and the statistical Monte-Carlo integration error is indicated in these tables.
The systematic error due to the slicing parameter h is not included in these tables, but will be
discussedbelow. ThevaluesfortheDurhamjetratehavealreadybeengiveninref.[4]. Duetoa
typo in the numerical program which was corrected after publication of [4] the corrected values
are repeated here.
ForthesixeventshapevariablestheLOfunctionsAO,theNLOfunctionsBO andtheNNLO
functions CO are also shown in in figs. 12 to 17. The leading-order function AO and the next-
to-leading order function BO can be computed with high precision and the graphs are shown
simply with solid lines in figs. 12-17. For the next-to-next-to-leading order function CO the
Monte-Carlointegrationerrorsaretypicallyatthepercentlevel,andthisfunctionisshownwith
errorbars correspondingto theMonte-Carlo integrationerrors in figs.12-17.
9
The NNLO function CO can be split up into the contributions from the individual colour
factors
CO , CO , CO , CO , CO , CO (30)
lc sc ssc nf nfsc nfnf
| | | | | |
defined in eq. (26). For the event shape variables the contributions from the individual colour
factors toCO are given in tables 12 to 18. For the jet rates the contributions from the individual
colour factors toCO are given in tables 19 to 20. In addition the individual colour contributions
are plottedforthesixeventshapevariablesinfigs. 18-23.
In fig. 24-26 the results for the thrust distribution are compared with the calculation of
Becher and Schwartz based on soft-collinear effective theory [7]. The leading-order, next-to-
leading-orderandnext-to-next-to-leadingordercoefficientsA ,B andC arecompared
1 T 1 T 1 T
− − −
in fig. 24. The individual colour factors of the coefficient C are compared in fig. 25. The
1 T
−
effective theory gives a good description of the thrust distribution for small values of (1 T).
−
For values of (1 T) close to 1 the effective theory is not valid and deviations from the pertur-
−
bative result can be seen. On the other hand the perturbative result obtained from Monte-Carlo
integrationhas itslimitationthroughtheslicingparameter introducedto handletheinfrared sin-
gularities. ThedeviationsofthenumericalMonte-CarloresultsfortheC coefficientfromthe
1 T
−
SCET results for values of (1 T)< 0.003 are an artefact of the slicing parameter. In figs. 24
−
and 25 the value h = 10 5 was used for the slicing parameter. To study the situation in more
−
detail,Ishowinfig.26thevariationofthenumericalresultfortheleadingcolourfactorofC
1 T
with the slicing parameter h . The numerical results for h =10 5, h =10 7 and h =10 9 a−re
− − −
plotted. Forsmallervaluesofh theSCET resultisapproached.
As a further example I also show the dependence on the slicing parameter h for the colour
factor N2 for the next-to-next-to-leadingorder coefficientC for the three-to-two jet transition
c y23
distribution in fig. 27. In this plot the numerical results for h =10 5, h =10 7 and h =10 9
− − −
are shown.
Fig. 28 compares the results of this calculation to the updated ones from ref. [1]. For the
six event shape distributions one observes good agreement with the exception of the three-to-
two jet transition distribution at very small values of y . The discrepancy at small values of
23
y can be traced back to the colour factor N2 and could have an explanation in terms of a
23 c
systematic error due to the slicing parameter. The study of the h -dependence in fig. 27 gives
hintsinthisdirection,butdoesnotgiveaconclusiveanswer. Althoughinfig.27thethreeresults
for h = 10 5, h = 10 7 and h = 10 9 are consistent with each other, using h = 10 9 instead
− − − −
of h = 10 5 for the comparison with ref. [1] would reduce the discrepancy significantly (but
−
also enlarge the errorbars). Given the complexity of the calculation the agreement of the two
numericalprogramsin allotherdistributionsis remarkable.
5 Conclusions
In this article I reported on the NNLO calculation of observables associated to three-jet events
in electron-positron annihilation. I provided NNLO results for the event shape variables thrust,
10