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Jet broadening measures in e+e− annihilation
Physics Letters B 295 (1992) 269-276 North-Holland
P H Y$1C $ L ETT ER $ B
Jet b r o a d e n i n g m e a s u r e s in e + e - a n n i h i l a t i o n S. Catani a,~, G. Turnock b and B.R. Webber a,2 a Theory Division, CERN, CH-1211 Geneva 23, Switzerland b Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB30HE, UK Received 11 September 1992
Predictions are presented for the distributions of two new e +e- event shape measures which involve the summed scalar transverse momentum relative to the thrust axis. The calculations include resummation of leading and next-to-leading logarithms to all orders in QCD perturbation theory, together with non-logarithmic terms up to second order. In the two-jet region the new event measures are complementary to the jet-mass-related quantities that were resummed earlier, and so they could be helpful in further refining the determination of the strong cou_plingconstant from e÷e- event shapes.
The perturbative calculations o f certain e+e - hadronic event shape variables have recently been extended from second o r d e r to include the r e s u m m a t i o n o f large logarithmic terms to all orders [ 1-4 ]. The r e s u m m a t i o n in these cases has been p e r f o r m e d with sufficient precision to p e r m i t i m p r o v e d measurements o f the strong coupling constant a~. Such m e a s u r e m e n t s have already been presented by several o f the collaborations working at LEP [ 5 ]. Their results have confirmed the theoretical expectations [ 1 ] that r e s u m m a t i o n can be helpful in reducing theoretical uncertainties associated with r e n o r m a l i z a t i o n scale dependence and h a d r o n i z a t i o n corrections. In particular, after r e s u m m a t i o n there is no longer a preference for very small scales, and the region o f event shapes in which the predictions are reliable is increased. So far, most o f the work on r e s u m m a t i o n has concentrated on the distributions of thrust [ 1 ] a n d heavy jet mass [2 ]. Both o f these quantities are sensitive mainly to the longitudinal structure o f jets and the r e s u m m e d expressions'for them turn out to be very similar. To check the consistency o f this m e t h o d o f a~-determination it would be very desirable to have r e s u m m e d predictions for a selection o f event shapes that were less correlated. F o r some o f the familiar event shape variables the leading logarithms can be resummed, but in most cases the full set o f next-to-leading terms required for a meaningful r e s u m m e d d e t e r m i n a t i o n o f as is not yet known. We present here the p r e d i c t e d distributions o f two event shape variables which are not very familiar but for which r e s u m m a t i o n can be p e r f o r m e d with the necessary precision. They are particularly sensitive to transverse jet d e v e l o p m e n t and so they should usefully c o m p l e m e n t the others that have been r e s u m m e d previously. A variable o f this type was p r o p o s e d some time ago for e+e - physics [6] but has so far been used m a i n l y for the study o f j e t b r o a d e n i n g in h a d r o n collisions [ 7,8 ]. F o r an e+e - a n n i h i l a t i o n event, we define the (total) j e t broadening B and the wide j e t broadening Bw as follows. First the thrust axis is found and the event is d i v i d e d into two hemispheres S+ and S_ by the plane p e r p e n d i c u l a r to this axis. F o r each hemisphere we define Research supported in part by the UK Science and Engineering Research Council and in part by the Italian Ministero della Universit& e Ricerca Scientifica. On leave of absence from INFN, Sezione di Firenze, Florence, Italy. 2 On leave of absence from Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
269
Volume 295, number 3,4
B+=
Z
PHYSICS LETTERSB
/ ]piAnT]/2~
i~S±
l
3 December 1992
IP~I,
(1)
i
where nT is the thrust axis ,1. Then
B=B+ + B _ ,
Bw=Max{B_,B_}.
(2)
Thus B is similar to the spherocity [ 9 ], except that it is defined with respect to the thrust axis instead of being minimized with respect to the choice of axis. The thrust and spherocity axes coincide to order a , but the former has better properties from several points of view. As far as we know, there is no convenient procedure for finding the spherocity axis and it does not permit the simple resummation of logarithmic contributions. In particular, the plane perpendicular to the spherocity axis can contain final-state momentum vectors (which is impossible for the thrust axis) and therefore it does not divide all events unambiguously into hemispheres. To order oq, B = B w = ½0where O is the oblateness [ 10], and the mean and maximum values of B are 0.65 or, [ 6 ] and 1/2x/~ = 0.29 respectively. The complete analytic one-loop result for the distribution of B is very cumbersome. Therefore we report here only its small-B expansion
B dcr _ _ a___~,Cv[4 In B + 3 + 4 B - 4 B 2 In B - 2 B 2 + O ( B 3 In B) ] o'dB
(3)
/r
where Cv= ~ in QCD. It turns out that this expression describes the first-order distribution within an accuracy of a few per cent out to B=0.26. In higher orders, the jet broadening and oblateness are not related. For a spherical event, for example, B - 2Bw = n/8, while O = 0. At present, no way is known to resum large logarithms in the oblateness distribution. The jet broadening distribution, on the other hand, can be easily resummed, because it exponentiates. By this we mean that the fraction of events with broadening less than B, denoted by R (B, oq), takes the form
R(B, ¢xs(Q 2) ) = C(as (Q 2) )2~(B, ¢x,(Q 2) ) +F(B, as(Q z) ),
(4)
where Q is the centre-of-mass energy and n+l
C(as)=l+
~ a~Cn, n=l
ln27(B, a s ) =
~ a~' E G,,, lnmB, n=l
F ( B , a , ) = ~. a~F,(B).
m=l
(5)
n=l
Here Cn and Gnm are constants, while Fn(B) are perturbatively computable functions that vanish at small B. Notice that although the perturbative expansion of R (B, as, (Q2)) itself contains terms of the form ot n Inm B for all values o f m up to 2n, its logarithm involves only rn~ n + 1. We call this property exponentiation. A similar property holds for the fraction Rw(Bw, as) with wide jet broadening less than Bw. In our calculations, the leading and next-to-leading terms in in 27, i.e. those with m >t n, are summed to all orders, and sub-dominant terms with m < n, as well as C and F, are included to second order only. If we now introduce a renormalization scale/~2 # Q2, taking into account that R is a renormalization group invariant quantity, then in place of as (Q2) in eq. (4) we find c~s(#2) together with an explicit Q2//z2 dependence of the next-to-leading logarithms and fixed-order terms. Thus we can study the scale dependence of the resummed expression and compare it with that of the fixed-order result. Let us now compute the jet broadening fractions R and Rw at small B. Determining the jet broadening distribution is a problem similar to that of finding the distribution of transverse energy emitted in Drell-Yan processes in hadron collisions [ 1 1,12 ]. The main difference is that here the transverse direction is defined relative to the jet itself rather than an external beam momentum. For this reason it is convenient to introduce a function Ta (Q, kt; Pt), which describes the distribution of the summed scalar transverse momentum Pt in a jet of type a "~ The factor of 2 in the denominator will be seen to simplifythe resummed expressions. 270
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3 December 1992
(a=q, g for a quark or gluon jet), produced with vector transverse momentum kt at scale Q. Following the general procedure described in ref. [4 ], we find that at small B each hemisphere contains a single quark jet and SO
2BQ
R(B, as(Q2))= B'~¢: 1
2BQ-ptl
[dptl
j"
, 0
0
Rw (Bw, a s ( Q 2 ) ) B w -
(
dpc2Tq(Q,O;pc,)Tq(Q,O;pt2),
(6)
dptTq(Q,O;pt)
(7)
0
The jet Pc distribution functions Ta may be calculated to next-to-leading logarithmic accuracy using the coherent branching formalism [ 13,14], from which one obtains the following integral equation: Q
Tq(Q,kt; Pc) = O ( P t -
1
Ikt I) + f -~¢z d2~ j[ dz 0 0
Pq,[°~s(z2(1-zlZ~2l,zl
oo
X O(pt, - Izkt +z( 1 - z ) ~ l )O(pt -Ptl - [ ( 1 -z)kt - z ( 1 - z ) ~ l ) -
\
Tq( I~1, kc; Pc)J.
(8)
Here Pqq [as, z] represents the splitting function which gives the distribution of the longitudinal momentum fraction z of the final quark in the parton branching q~qg. Its explicit form to next-to-leading accuracy in the MS scheme is
Pqq[as,z ] = ~ - ~ Cl÷z v~+
½CFK-, K=~-½7t2-~Nf,
(9,10)
where Nf is the number of flavours. The gluon jet distribution function Tg in eq. (8) satisfies a similar integral equation in terms of the gluon splitting functions P~ and Psq. Introducing the Laplace transform
Ta(a,/Ca; P)
----
i dot exp [u(Ik, I -Pc)]
Ta(Q,k,; Pc),
(11)
Iktl
we find that Q
Tq(Q, O; u) = 1 + _t 0
d2~ 1
~ dzPqq[OLs(Z2(1-z)zq2), z] 0
×{exp[-Zz(1-z)ul~l ]
Tq(zl~l,z(1-z)q;z,)T~((1-z)[~l,-z(1-z)~;
z,)-~q(l~l,0;z,)}.
(12)
We are interested in the small-pt region which corresponds to the large-u limit, where the exponential factor selects the region of small z( 1 - z ) ~ . Since the functions ~q and 2Ps in the integrand are regular in this region, and the splitting function Pqq is singular only at z = 1, to next-to-leading accuracy we can make the replacements
~Ps((1-z) lql,-z(1-z)~;v)~s(0 ,0;u)=l, Tq(z[~l,z(1--z)~;l,)~q(l~{,0;u).
(13,14)
The solution is then straightforward, viz. 271
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3 December 1992
1
~r(Q, 0; .)=exp (!d'2fdzPqq[Oq(z2(1-z)aq2),z]{exp[-2z(1-z)vlq]]-l}). 7.o
(15)
The exponentiated result in eq. (15) follows from the next-to-leading approximations (13) and (14). In order to clarify their physical meaning, let us first notice that the pt-distribution 2rg in eq. (12) describes the soft gluon cascades produced in the jet evolution. Due to the coherence properties of soft radiation in QCD, these cascades develop in an angular region 0 < ( 1 - z ) l # l / Q , smaller than the jet angular size 1/2 v. Therefore multiple (non-abelian) gluon correlations are completely integrated over the relevant two-jet region and cancel by unitarity as in eq. (13). On the other side, the approximation (14) amounts to neglecting the small angular recoil z( 1 - z ) # / Q of the quark which thus radiates soft gluons independently. The summed scalar transverse momentum Pt is therefore approximately conserved in the jet evolution, leading to the exponentiated result ( 15 ) in the space of the Laplace index v. To predict the jet broadening at small B we write eq. (6) in the form ~+ioo
R(B, oq(Q2) ) = -J dvexp(2vBQ) B'~ 1
[~q(Q, 0;v)]2
(16)
2hi
1/
where the integration contour is to the right of all singularities of Tq. This gives the exponentiated factor 27in eq. (4). Similarly for the wide jet broadening ~+ioo
2
~-i~
Both the integrals in eq. ( 15 ) and eqs. (16), ( 17 ) can be performed analytically to next-to-leading logarithmic accuracy as explained in detail in ref. [ 4 ]. Including the renormalization scale dependence in the way outlined earlier, we find
X(B, a~(112),Q2/la2)= e x p [ ~ ( l n
B, cq(ctz), Q2/#2)1 F ( 1 - 2 : ( l n B, oq(/~2)) ) '
Zw(Bw, as(#z),
Q2/1~2)=
(18)
exp [ ~ ( l n Bw, oq(/12), Q2/I.t2) ] [ F ( l _ 5 ~ ( l n B w ' oq(/~2)))]2
(19)
where F represents the Euler gamma function and (defining co= 2/~oa~ In B for brevity) A(l) [ l n ( 1 + o 9 ) - o 9 ] + ~(lnB'a~'Q2/Pz)- zfl~a~ B (~) " 2A(l)yE ( O ) +-~o In(l+°))+ n---~o ~
Q2_A(2 ) A(l)nfloln-~
) + A(1)/~1( ln(1 +o9) ~ ½1n2(l+r°)+ 1+o9
)(
ln(l+co)-
°)
co) l+-oJ + O ( a T * ' l n ' B ) ,
(20)
)
(21)
6e(lnB, a s ) = rC]~o\ l + e ) / ' A(I)=CF'
B(I)=--3CF'
A(u)=½CF(~-½zc2-~Nf)'
'80=
33-2Nf 1 5 3 - 19Nf 12n , i l l 24rr2
(22)
Here YE= 0.57 72... is the Euler constant and Nf, the number of flavours, is 5 at LEP energies. Having evaluated 27 to next-to-leading accuracy, we can obtain the non-logarithmic coefficient C1 in eq. (5) by comparison with the one-loop analytical result for R (B, as). We find 272
Volume 295, n u m b e r 3,4
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CF ( n z _ ~ z ) .
C~ = ~
(23)
The solid curves in figs. 1 and 2 show the resummed expressions (4), using eqs. ( 18 ), (19) and including C~, but setting other sub-dominant terms (i.e. those with fewer powers of in B than of as) and the remainder functions F to zero. For definiteness we have assumed the (five-flavour) value A n = 200 MeV, corresponding to as (QZ) = 0.116, with scale #2 = Q2. Expanding the resummed results ( 18 ) - (20) to second order in oq gives 4
R(B, oq)=l-ots Cv ( 2 1 n Z B + 3 1 n B ) + a ~ ~ K m l n ' B + .... 7~
(24)
m=2
where the dots represent sub-dominant terms, and the leading and next-to-leading second-order coefficients K,, are as listed in table 1. To this order, the full fixed-order prediction for all values of B can be calculated numerically using the program EVENT [ 15 ], based on the matrix elements of ref. [ 16 ]. The corresponding fixedorder predictions are shown by the dashed curves in figs. 1 and 2. Extrapolating to very small B and fitting the logarithmic terms, we obtain the fitted coefficients given in table 1, which are seen to be consistent with those obtained by resummation. Next, by fixing these coefficients and fitting the single-logarithmic term we can estimate the first sub-dominant coefficient G21 in eq. (5): Gz1=-5.1_+0.4
forR(B, a s ) ,
-5.5_+0.2
forRw(Bw, a s ) .
(25)
Finally from the non-logarithmic term we can estimate C2: C2=-3.2_+0.4
forR(B, a s ) ,
-4.6_+0.2
forRw(Bw, as) •
(26)
To obtain a prediction for the full second-order range orB (i.e. up to the four-parton phase space boundary) a number of procedures can be adopted [ 1-5 ] for matching the resummed and fixed-order expressions. The ....
10
,I
.
.
.
.
~. . . .
[
'
'
'
\ a
-
6
--
\
--
b b
2
0 0.0015
0.01
0.05 B
0.1
0.15
Fig. 1. Total jet broadening distribution in e÷e - annihilation at 91 GeV. Solid: r e s u m m e d expression (see text). Dashed: second-order prediction without resummation.
273
Voiume 295, number 3,4
PHYSICS LETTERS B 12
,111
I
,,
1
I
3 December 1992
,I,
\
\
10 -
\
B-
Fig. 2. Wide jet broadening distribution in e+e- annihilation at 91 GeV. Solid: resummed expression (see text). Dashed: second-order prediction without resummation. Table 1 Fitted versus calculated O( & ) log coeffkients for Nr= 5 Coefficient
& 4 &
kv(Bw, cu,)
R(B, a,) fitted
theory
fitted
theory
- 1.4kO.6 1.610.6 Q.4fO.S
- 1.50 1.77 0.36
-0.4rto.3 1.61tO.4 0.4kO.3
-0.31 1.77 0.36
differences between the results obtained with various procedures may provide an estimate of the theoretical unce~ainty due to missing higher order contributions. For illustration, we show in fig. 3 the total jet broadening distribution using the simplest type of matching procedure, in which one writes g
=[ 1 -f(g)]
where!(x)
(resummed)+f(z)
(second-order)
,
(27)
increases rapidly from zero to one as x passes through the point x= 1; for example
f(x)=(l+x_5)-l.
(28)
The matching point B.should be chosen such that CX, In *BoS 1. In fig. 3, the value Bo=BL= 0.09 has been used, where B, is the value of B at which the resummed and fixed order results coincide, corresponding to CX, in2 B,= 0.67. When comparing with data, an estimate of the matching ambiguity for this procedure could be obtained by varying B.within a reasonable range. More sophisticated matching procedures are discussed in ref. t41. An important aspect of resummation is its effect on the renormalization scale dependence of the predictions. 274
Volume 295, n u m b e r 3,4
PHYSICS LETTERS B
'
'
'
'
I
. . . .
I . . . .
I'
3 December 1992
' ' ' 1
. . . .
I''
10.0 9.0 8.0 7.0 6.0 5.0 4.0 ~3 "0
3.0
b
2.0 b
0.8 0.7 0.6 0.5
N
0
0.05
0.1
0.15 B
x .
0.2
0.25
0.3
Fig. 3. Predicted distribution of total jet broadening in e + e - annihilation at 91 GeV. Solid: perturbative prediction including resummation to all orders. Dashed: r e s u m m e d prediction with H E R W I G hadronization corrections.
I
10.0
........
I
........
I
........
I
V ......
_- -_-_- ...............
--
B = 0.05
........
5.0 --
m
_ _ _ --_
B=O.1
b b
1.0
0.5 10-2
B = 0.21
........
I
10-1
........
I
10 0 ~2/Qe
i i
i
|
i
i i I l l
i01
i
i
i
i
i i t i
102
Fig. 4. Dependence on renormalization scale ~2 of total jet broadening at various values of B. Solid: r e s u m m e d expression (see text). Dashed: second-order prediction without resummation. 275
Volume 295, number 3,4
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For other event shape variables, it has been found that r e s u m m a t i o n reduces the scale dependence and eliminates the preference for small scales which is found in fixed order [ 1,5 ]. As illustrated in fig. 4, this is also the case for the jet b r o a d e n i n g distribution. The fixed-order and r e s u m m e d expressions are shown at discrete values of B for a wide range of~2/Q2. For estimating the theoretical uncertainty associated with the scale dependence, a smaller range, say ½< ,u2/Q 2 < 2, should be sufficient. Note that the variation with f12/Q2 is quite small (of the order of 2%) over this range. Having correctly treated the small transverse m o m e n t u m region, we do not see any valid argument for using scales much different from O 2. Another i m p o r t a n t c o n t r i b u t i o n to the theoretical uncertainty in event shape predictions is the effect of hadronization. Here one can only use models to estimate the difference between the predictions at the partonic and hadronic levels. We have used the simulation program H E R W I G [ 17 ] to construct a hadronization correction matrix for jet broadening, in the same way as was done earlier for the thrust distribution [ 1 ]. As shown by the dashed curve in fig. 3, the resulting corrections are comparable to those that were found for the thrust distribution. In conclusion, we find that jet b r o a d e n i n g should be a good event shape measure for a , d e t e r m i n a t i o n in e + e annihilation. O u r predictions also provide a novel " b l i n d test" of QCD: as far as we are aware, no measurement of the distribution of this quantity has ever been presented, at any energy. We await the data with interest and trepidation. One of us ( G . T . ) is most grateful for the hospitality of the CERN Theory G r o u p while part of this work was carried out.
References [ 1] S. Catani, G. Turnock, B.R. Webberand L. Trentadue, Phys. Lett. B 263 ( 1991 ) 491. [2] S. Catani, G. Turnock and B.R. Webber, Phys. Lett. B 272 ( 1991 ) 368. [3] G. Turnock, Cambridge preprint Cavendish-HEP-92/3. [4 ] S. Catani, L. Trentadue, G. Turnock and B.R. Webber,Cambridgepreprint Cavendish-HEP-91/ 11, CERN preprint CERN-TH.6640/ 92. [5] OPAL Collab., P.D. Acton et al., Z. Phys. C 55 (1992) 1; ALEPH Collab., D. Decamp et al., Phys. Lett. B 284 (1992) 163; L3 Collab., O. Adriani et al., Phys. Lett. B 284 (1992) 471; H. Furstenau, in: Proc. XXVII Recontres de Moriond, QCD and high energyhadronic interactions, to appear. [6] P.E.L. Rakow and B.R. Webber, Nucl. Phys. B 191 (1981 ) 63. [7] R.K. Ellis and B.R. Webber, in: Proc. 1986 Summer Study on the Physics of the SuperconductingSupercollider (Snowmass, CO, 1986 ), Division of Particles and Fields of the APS, eds. R. Donaldson and J. Marx (APS, New York, 1987). [8] CDF Collab., F. Abe et al., Phys. Rev. D 44 ( 1991 ) 601. [9 ] H. Georgi and M. Machacek, Phys. Rev. Lett. 39 ( 1977) 1237, [ 10] Mark J Collab., D.P. Barber et al., Phys. Rep. 63 (1980) 337. [ 11 ] C.T.H. Davies and B.R. Webber, Z. Phys. C 24 (1984) 133. [ 12] G. Altarelli, G. Martinelli and F. Rapuano, Z. Phys. C 32 (1986) 369. [ 13 ] S. Catani and L. Trentadue, Phys. Lett. B 217 ( 1989) 539; Nucl. Phys. B 327 ( 1989) 323; B 353 ( 1991 ) 183. [ 14] S. Catani, G. Marchesini and B.R. Webber,Nucl. Phys. B 349 ( 1991 ) 635. [ l 5 ] Z. Kunszt, P. Nason, G. Marchesini and B.R. Webber, in: Z physics at LEP1, CERN report CERN 89-08, Vol. 1 (CERN, Geneva, 1989) p. 373; P. Nason, private communication. [ 16] R.K. Ellis, D.A. Ross and A.E. Terrano, Nucl. Phys. B 178 ( 1981 ) 421. [171 B.R. Webber, Nucl. Phys. B 238 (1984) 492; G. Marchesini and B.R. Webber, Nucl. Phys. B 310 (1988) 461; G. Abbiendi, I.G. Knowles,G. Marchesini, M.H. Seymour, L. Stanco and B.R. Webber, Comput. Phys. Commun. 67 (1992) 465.
276
P H Y$1C $ L ETT ER $ B
Jet b r o a d e n i n g m e a s u r e s in e + e - a n n i h i l a t i o n S. Catani a,~, G. Turnock b and B.R. Webber a,2 a Theory Division, CERN, CH-1211 Geneva 23, Switzerland b Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB30HE, UK Received 11 September 1992
Predictions are presented for the distributions of two new e +e- event shape measures which involve the summed scalar transverse momentum relative to the thrust axis. The calculations include resummation of leading and next-to-leading logarithms to all orders in QCD perturbation theory, together with non-logarithmic terms up to second order. In the two-jet region the new event measures are complementary to the jet-mass-related quantities that were resummed earlier, and so they could be helpful in further refining the determination of the strong cou_plingconstant from e÷e- event shapes.
The perturbative calculations o f certain e+e - hadronic event shape variables have recently been extended from second o r d e r to include the r e s u m m a t i o n o f large logarithmic terms to all orders [ 1-4 ]. The r e s u m m a t i o n in these cases has been p e r f o r m e d with sufficient precision to p e r m i t i m p r o v e d measurements o f the strong coupling constant a~. Such m e a s u r e m e n t s have already been presented by several o f the collaborations working at LEP [ 5 ]. Their results have confirmed the theoretical expectations [ 1 ] that r e s u m m a t i o n can be helpful in reducing theoretical uncertainties associated with r e n o r m a l i z a t i o n scale dependence and h a d r o n i z a t i o n corrections. In particular, after r e s u m m a t i o n there is no longer a preference for very small scales, and the region o f event shapes in which the predictions are reliable is increased. So far, most o f the work on r e s u m m a t i o n has concentrated on the distributions of thrust [ 1 ] a n d heavy jet mass [2 ]. Both o f these quantities are sensitive mainly to the longitudinal structure o f jets and the r e s u m m e d expressions'for them turn out to be very similar. To check the consistency o f this m e t h o d o f a~-determination it would be very desirable to have r e s u m m e d predictions for a selection o f event shapes that were less correlated. F o r some o f the familiar event shape variables the leading logarithms can be resummed, but in most cases the full set o f next-to-leading terms required for a meaningful r e s u m m e d d e t e r m i n a t i o n o f as is not yet known. We present here the p r e d i c t e d distributions o f two event shape variables which are not very familiar but for which r e s u m m a t i o n can be p e r f o r m e d with the necessary precision. They are particularly sensitive to transverse jet d e v e l o p m e n t and so they should usefully c o m p l e m e n t the others that have been r e s u m m e d previously. A variable o f this type was p r o p o s e d some time ago for e+e - physics [6] but has so far been used m a i n l y for the study o f j e t b r o a d e n i n g in h a d r o n collisions [ 7,8 ]. F o r an e+e - a n n i h i l a t i o n event, we define the (total) j e t broadening B and the wide j e t broadening Bw as follows. First the thrust axis is found and the event is d i v i d e d into two hemispheres S+ and S_ by the plane p e r p e n d i c u l a r to this axis. F o r each hemisphere we define Research supported in part by the UK Science and Engineering Research Council and in part by the Italian Ministero della Universit& e Ricerca Scientifica. On leave of absence from INFN, Sezione di Firenze, Florence, Italy. 2 On leave of absence from Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
269
Volume 295, number 3,4
B+=
Z
PHYSICS LETTERSB
/ ]piAnT]/2~
i~S±
l
3 December 1992
IP~I,
(1)
i
where nT is the thrust axis ,1. Then
B=B+ + B _ ,
Bw=Max{B_,B_}.
(2)
Thus B is similar to the spherocity [ 9 ], except that it is defined with respect to the thrust axis instead of being minimized with respect to the choice of axis. The thrust and spherocity axes coincide to order a , but the former has better properties from several points of view. As far as we know, there is no convenient procedure for finding the spherocity axis and it does not permit the simple resummation of logarithmic contributions. In particular, the plane perpendicular to the spherocity axis can contain final-state momentum vectors (which is impossible for the thrust axis) and therefore it does not divide all events unambiguously into hemispheres. To order oq, B = B w = ½0where O is the oblateness [ 10], and the mean and maximum values of B are 0.65 or, [ 6 ] and 1/2x/~ = 0.29 respectively. The complete analytic one-loop result for the distribution of B is very cumbersome. Therefore we report here only its small-B expansion
B dcr _ _ a___~,Cv[4 In B + 3 + 4 B - 4 B 2 In B - 2 B 2 + O ( B 3 In B) ] o'dB
(3)
/r
where Cv= ~ in QCD. It turns out that this expression describes the first-order distribution within an accuracy of a few per cent out to B=0.26. In higher orders, the jet broadening and oblateness are not related. For a spherical event, for example, B - 2Bw = n/8, while O = 0. At present, no way is known to resum large logarithms in the oblateness distribution. The jet broadening distribution, on the other hand, can be easily resummed, because it exponentiates. By this we mean that the fraction of events with broadening less than B, denoted by R (B, oq), takes the form
R(B, ¢xs(Q 2) ) = C(as (Q 2) )2~(B, ¢x,(Q 2) ) +F(B, as(Q z) ),
(4)
where Q is the centre-of-mass energy and n+l
C(as)=l+
~ a~Cn, n=l
ln27(B, a s ) =
~ a~' E G,,, lnmB, n=l
F ( B , a , ) = ~. a~F,(B).
m=l
(5)
n=l
Here Cn and Gnm are constants, while Fn(B) are perturbatively computable functions that vanish at small B. Notice that although the perturbative expansion of R (B, as, (Q2)) itself contains terms of the form ot n Inm B for all values o f m up to 2n, its logarithm involves only rn~ n + 1. We call this property exponentiation. A similar property holds for the fraction Rw(Bw, as) with wide jet broadening less than Bw. In our calculations, the leading and next-to-leading terms in in 27, i.e. those with m >t n, are summed to all orders, and sub-dominant terms with m < n, as well as C and F, are included to second order only. If we now introduce a renormalization scale/~2 # Q2, taking into account that R is a renormalization group invariant quantity, then in place of as (Q2) in eq. (4) we find c~s(#2) together with an explicit Q2//z2 dependence of the next-to-leading logarithms and fixed-order terms. Thus we can study the scale dependence of the resummed expression and compare it with that of the fixed-order result. Let us now compute the jet broadening fractions R and Rw at small B. Determining the jet broadening distribution is a problem similar to that of finding the distribution of transverse energy emitted in Drell-Yan processes in hadron collisions [ 1 1,12 ]. The main difference is that here the transverse direction is defined relative to the jet itself rather than an external beam momentum. For this reason it is convenient to introduce a function Ta (Q, kt; Pt), which describes the distribution of the summed scalar transverse momentum Pt in a jet of type a "~ The factor of 2 in the denominator will be seen to simplifythe resummed expressions. 270
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3 December 1992
(a=q, g for a quark or gluon jet), produced with vector transverse momentum kt at scale Q. Following the general procedure described in ref. [4 ], we find that at small B each hemisphere contains a single quark jet and SO
2BQ
R(B, as(Q2))= B'~¢: 1
2BQ-ptl
[dptl
j"
, 0
0
Rw (Bw, a s ( Q 2 ) ) B w -
(
dpc2Tq(Q,O;pc,)Tq(Q,O;pt2),
(6)
dptTq(Q,O;pt)
(7)
0
The jet Pc distribution functions Ta may be calculated to next-to-leading logarithmic accuracy using the coherent branching formalism [ 13,14], from which one obtains the following integral equation: Q
Tq(Q,kt; Pc) = O ( P t -
1
Ikt I) + f -~¢z d2~ j[ dz 0 0
Pq,[°~s(z2(1-zlZ~2l,zl
oo
X O(pt, - Izkt +z( 1 - z ) ~ l )O(pt -Ptl - [ ( 1 -z)kt - z ( 1 - z ) ~ l ) -
\
Tq( I~1, kc; Pc)J.
(8)
Here Pqq [as, z] represents the splitting function which gives the distribution of the longitudinal momentum fraction z of the final quark in the parton branching q~qg. Its explicit form to next-to-leading accuracy in the MS scheme is
Pqq[as,z ] = ~ - ~ Cl÷z v~+
½CFK-, K=~-½7t2-~Nf,
(9,10)
where Nf is the number of flavours. The gluon jet distribution function Tg in eq. (8) satisfies a similar integral equation in terms of the gluon splitting functions P~ and Psq. Introducing the Laplace transform
Ta(a,/Ca; P)
----
i dot exp [u(Ik, I -Pc)]
Ta(Q,k,; Pc),
(11)
Iktl
we find that Q
Tq(Q, O; u) = 1 + _t 0
d2~ 1
~ dzPqq[OLs(Z2(1-z)zq2), z] 0
×{exp[-Zz(1-z)ul~l ]
Tq(zl~l,z(1-z)q;z,)T~((1-z)[~l,-z(1-z)~;
z,)-~q(l~l,0;z,)}.
(12)
We are interested in the small-pt region which corresponds to the large-u limit, where the exponential factor selects the region of small z( 1 - z ) ~ . Since the functions ~q and 2Ps in the integrand are regular in this region, and the splitting function Pqq is singular only at z = 1, to next-to-leading accuracy we can make the replacements
~Ps((1-z) lql,-z(1-z)~;v)~s(0 ,0;u)=l, Tq(z[~l,z(1--z)~;l,)~q(l~{,0;u).
(13,14)
The solution is then straightforward, viz. 271
Volume 295, number 3,4 Q2
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3 December 1992
1
~r(Q, 0; .)=exp (!d'2fdzPqq[Oq(z2(1-z)aq2),z]{exp[-2z(1-z)vlq]]-l}). 7.o
(15)
The exponentiated result in eq. (15) follows from the next-to-leading approximations (13) and (14). In order to clarify their physical meaning, let us first notice that the pt-distribution 2rg in eq. (12) describes the soft gluon cascades produced in the jet evolution. Due to the coherence properties of soft radiation in QCD, these cascades develop in an angular region 0 < ( 1 - z ) l # l / Q , smaller than the jet angular size 1/2 v. Therefore multiple (non-abelian) gluon correlations are completely integrated over the relevant two-jet region and cancel by unitarity as in eq. (13). On the other side, the approximation (14) amounts to neglecting the small angular recoil z( 1 - z ) # / Q of the quark which thus radiates soft gluons independently. The summed scalar transverse momentum Pt is therefore approximately conserved in the jet evolution, leading to the exponentiated result ( 15 ) in the space of the Laplace index v. To predict the jet broadening at small B we write eq. (6) in the form ~+ioo
R(B, oq(Q2) ) = -J dvexp(2vBQ) B'~ 1
[~q(Q, 0;v)]2
(16)
2hi
1/
where the integration contour is to the right of all singularities of Tq. This gives the exponentiated factor 27in eq. (4). Similarly for the wide jet broadening ~+ioo
2
~-i~
Both the integrals in eq. ( 15 ) and eqs. (16), ( 17 ) can be performed analytically to next-to-leading logarithmic accuracy as explained in detail in ref. [ 4 ]. Including the renormalization scale dependence in the way outlined earlier, we find
X(B, a~(112),Q2/la2)= e x p [ ~ ( l n
B, cq(ctz), Q2/#2)1 F ( 1 - 2 : ( l n B, oq(/~2)) ) '
Zw(Bw, as(#z),
Q2/1~2)=
(18)
exp [ ~ ( l n Bw, oq(/12), Q2/I.t2) ] [ F ( l _ 5 ~ ( l n B w ' oq(/~2)))]2
(19)
where F represents the Euler gamma function and (defining co= 2/~oa~ In B for brevity) A(l) [ l n ( 1 + o 9 ) - o 9 ] + ~(lnB'a~'Q2/Pz)- zfl~a~ B (~) " 2A(l)yE ( O ) +-~o In(l+°))+ n---~o ~
Q2_A(2 ) A(l)nfloln-~
) + A(1)/~1( ln(1 +o9) ~ ½1n2(l+r°)+ 1+o9
)(
ln(l+co)-
°)
co) l+-oJ + O ( a T * ' l n ' B ) ,
(20)
)
(21)
6e(lnB, a s ) = rC]~o\ l + e ) / ' A(I)=CF'
B(I)=--3CF'
A(u)=½CF(~-½zc2-~Nf)'
'80=
33-2Nf 1 5 3 - 19Nf 12n , i l l 24rr2
(22)
Here YE= 0.57 72... is the Euler constant and Nf, the number of flavours, is 5 at LEP energies. Having evaluated 27 to next-to-leading accuracy, we can obtain the non-logarithmic coefficient C1 in eq. (5) by comparison with the one-loop analytical result for R (B, as). We find 272
Volume 295, n u m b e r 3,4
P H Y S I C S LETTERS B
3 December 1992
CF ( n z _ ~ z ) .
C~ = ~
(23)
The solid curves in figs. 1 and 2 show the resummed expressions (4), using eqs. ( 18 ), (19) and including C~, but setting other sub-dominant terms (i.e. those with fewer powers of in B than of as) and the remainder functions F to zero. For definiteness we have assumed the (five-flavour) value A n = 200 MeV, corresponding to as (QZ) = 0.116, with scale #2 = Q2. Expanding the resummed results ( 18 ) - (20) to second order in oq gives 4
R(B, oq)=l-ots Cv ( 2 1 n Z B + 3 1 n B ) + a ~ ~ K m l n ' B + .... 7~
(24)
m=2
where the dots represent sub-dominant terms, and the leading and next-to-leading second-order coefficients K,, are as listed in table 1. To this order, the full fixed-order prediction for all values of B can be calculated numerically using the program EVENT [ 15 ], based on the matrix elements of ref. [ 16 ]. The corresponding fixedorder predictions are shown by the dashed curves in figs. 1 and 2. Extrapolating to very small B and fitting the logarithmic terms, we obtain the fitted coefficients given in table 1, which are seen to be consistent with those obtained by resummation. Next, by fixing these coefficients and fitting the single-logarithmic term we can estimate the first sub-dominant coefficient G21 in eq. (5): Gz1=-5.1_+0.4
forR(B, a s ) ,
-5.5_+0.2
forRw(Bw, a s ) .
(25)
Finally from the non-logarithmic term we can estimate C2: C2=-3.2_+0.4
forR(B, a s ) ,
-4.6_+0.2
forRw(Bw, as) •
(26)
To obtain a prediction for the full second-order range orB (i.e. up to the four-parton phase space boundary) a number of procedures can be adopted [ 1-5 ] for matching the resummed and fixed-order expressions. The ....
10
,I
.
.
.
.
~. . . .
[
'
'
'
\ a
-
6
--
\
--
b b
2
0 0.0015
0.01
0.05 B
0.1
0.15
Fig. 1. Total jet broadening distribution in e÷e - annihilation at 91 GeV. Solid: r e s u m m e d expression (see text). Dashed: second-order prediction without resummation.
273
Voiume 295, number 3,4
PHYSICS LETTERS B 12
,111
I
,,
1
I
3 December 1992
,I,
\
\
10 -
\
B-
Fig. 2. Wide jet broadening distribution in e+e- annihilation at 91 GeV. Solid: resummed expression (see text). Dashed: second-order prediction without resummation. Table 1 Fitted versus calculated O( & ) log coeffkients for Nr= 5 Coefficient
& 4 &
kv(Bw, cu,)
R(B, a,) fitted
theory
fitted
theory
- 1.4kO.6 1.610.6 Q.4fO.S
- 1.50 1.77 0.36
-0.4rto.3 1.61tO.4 0.4kO.3
-0.31 1.77 0.36
differences between the results obtained with various procedures may provide an estimate of the theoretical unce~ainty due to missing higher order contributions. For illustration, we show in fig. 3 the total jet broadening distribution using the simplest type of matching procedure, in which one writes g
=[ 1 -f(g)]
where!(x)
(resummed)+f(z)
(second-order)
,
(27)
increases rapidly from zero to one as x passes through the point x= 1; for example
f(x)=(l+x_5)-l.
(28)
The matching point B.should be chosen such that CX, In *BoS 1. In fig. 3, the value Bo=BL= 0.09 has been used, where B, is the value of B at which the resummed and fixed order results coincide, corresponding to CX, in2 B,= 0.67. When comparing with data, an estimate of the matching ambiguity for this procedure could be obtained by varying B.within a reasonable range. More sophisticated matching procedures are discussed in ref. t41. An important aspect of resummation is its effect on the renormalization scale dependence of the predictions. 274
Volume 295, n u m b e r 3,4
PHYSICS LETTERS B
'
'
'
'
I
. . . .
I . . . .
I'
3 December 1992
' ' ' 1
. . . .
I''
10.0 9.0 8.0 7.0 6.0 5.0 4.0 ~3 "0
3.0
b
2.0 b
0.8 0.7 0.6 0.5
N
0
0.05
0.1
0.15 B
x .
0.2
0.25
0.3
Fig. 3. Predicted distribution of total jet broadening in e + e - annihilation at 91 GeV. Solid: perturbative prediction including resummation to all orders. Dashed: r e s u m m e d prediction with H E R W I G hadronization corrections.
I
10.0
........
I
........
I
........
I
V ......
_- -_-_- ...............
--
B = 0.05
........
5.0 --
m
_ _ _ --_
B=O.1
b b
1.0
0.5 10-2
B = 0.21
........
I
10-1
........
I
10 0 ~2/Qe
i i
i
|
i
i i I l l
i01
i
i
i
i
i i t i
102
Fig. 4. Dependence on renormalization scale ~2 of total jet broadening at various values of B. Solid: r e s u m m e d expression (see text). Dashed: second-order prediction without resummation. 275
Volume 295, number 3,4
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3 December 1992
For other event shape variables, it has been found that r e s u m m a t i o n reduces the scale dependence and eliminates the preference for small scales which is found in fixed order [ 1,5 ]. As illustrated in fig. 4, this is also the case for the jet b r o a d e n i n g distribution. The fixed-order and r e s u m m e d expressions are shown at discrete values of B for a wide range of~2/Q2. For estimating the theoretical uncertainty associated with the scale dependence, a smaller range, say ½< ,u2/Q 2 < 2, should be sufficient. Note that the variation with f12/Q2 is quite small (of the order of 2%) over this range. Having correctly treated the small transverse m o m e n t u m region, we do not see any valid argument for using scales much different from O 2. Another i m p o r t a n t c o n t r i b u t i o n to the theoretical uncertainty in event shape predictions is the effect of hadronization. Here one can only use models to estimate the difference between the predictions at the partonic and hadronic levels. We have used the simulation program H E R W I G [ 17 ] to construct a hadronization correction matrix for jet broadening, in the same way as was done earlier for the thrust distribution [ 1 ]. As shown by the dashed curve in fig. 3, the resulting corrections are comparable to those that were found for the thrust distribution. In conclusion, we find that jet b r o a d e n i n g should be a good event shape measure for a , d e t e r m i n a t i o n in e + e annihilation. O u r predictions also provide a novel " b l i n d test" of QCD: as far as we are aware, no measurement of the distribution of this quantity has ever been presented, at any energy. We await the data with interest and trepidation. One of us ( G . T . ) is most grateful for the hospitality of the CERN Theory G r o u p while part of this work was carried out.
References [ 1] S. Catani, G. Turnock, B.R. Webberand L. Trentadue, Phys. Lett. B 263 ( 1991 ) 491. [2] S. Catani, G. Turnock and B.R. Webber, Phys. Lett. B 272 ( 1991 ) 368. [3] G. Turnock, Cambridge preprint Cavendish-HEP-92/3. [4 ] S. Catani, L. Trentadue, G. Turnock and B.R. Webber,Cambridgepreprint Cavendish-HEP-91/ 11, CERN preprint CERN-TH.6640/ 92. [5] OPAL Collab., P.D. Acton et al., Z. Phys. C 55 (1992) 1; ALEPH Collab., D. Decamp et al., Phys. Lett. B 284 (1992) 163; L3 Collab., O. Adriani et al., Phys. Lett. B 284 (1992) 471; H. Furstenau, in: Proc. XXVII Recontres de Moriond, QCD and high energyhadronic interactions, to appear. [6] P.E.L. Rakow and B.R. Webber, Nucl. Phys. B 191 (1981 ) 63. [7] R.K. Ellis and B.R. Webber, in: Proc. 1986 Summer Study on the Physics of the SuperconductingSupercollider (Snowmass, CO, 1986 ), Division of Particles and Fields of the APS, eds. R. Donaldson and J. Marx (APS, New York, 1987). [8] CDF Collab., F. Abe et al., Phys. Rev. D 44 ( 1991 ) 601. [9 ] H. Georgi and M. Machacek, Phys. Rev. Lett. 39 ( 1977) 1237, [ 10] Mark J Collab., D.P. Barber et al., Phys. Rep. 63 (1980) 337. [ 11 ] C.T.H. Davies and B.R. Webber, Z. Phys. C 24 (1984) 133. [ 12] G. Altarelli, G. Martinelli and F. Rapuano, Z. Phys. C 32 (1986) 369. [ 13 ] S. Catani and L. Trentadue, Phys. Lett. B 217 ( 1989) 539; Nucl. Phys. B 327 ( 1989) 323; B 353 ( 1991 ) 183. [ 14] S. Catani, G. Marchesini and B.R. Webber,Nucl. Phys. B 349 ( 1991 ) 635. [ l 5 ] Z. Kunszt, P. Nason, G. Marchesini and B.R. Webber, in: Z physics at LEP1, CERN report CERN 89-08, Vol. 1 (CERN, Geneva, 1989) p. 373; P. Nason, private communication. [ 16] R.K. Ellis, D.A. Ross and A.E. Terrano, Nucl. Phys. B 178 ( 1981 ) 421. [171 B.R. Webber, Nucl. Phys. B 238 (1984) 492; G. Marchesini and B.R. Webber, Nucl. Phys. B 310 (1988) 461; G. Abbiendi, I.G. Knowles,G. Marchesini, M.H. Seymour, L. Stanco and B.R. Webber, Comput. Phys. Commun. 67 (1992) 465.
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