Resummation of large logarithms in e+e− event shape distributions

NUCLEAR PHYSICS B

Nuclear Physics B407 (1993) 3-42 North-Holland

Resummation of large logarithms in e+e - event shape distributions S. Catani 1 Theory Division, CERN, CH-1211 Geneva 23, Switzerland L. Trentadue Dipartimento di Fisica, University di Roma II, Tor Vergata, I-00173 Rome, Italy G. Turnock Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK B.R. Webberg Theory Division, CERN, CH-1211 Geneva 23, Switzerland Received 29 January 1993 Accepted for publication 27 May 1993 We describe a method for the resummation of leading and next-to-leading large logarithms to all orders in QCD perturbation theory, applicable to e+e- event shape distributions that have the property of exponentiation near the two-jet region . After a general discussion of the conditions for exponentiation and the evaluation of matrix elements and phase space to next-to-leading logarithmic accuracy, we give details of the application of the method to the thrust and heavy jet mass distributions . We show how the resummed expressions can be matched with known second-order results to obtain improved predictions throughout the whole of phase space, and how to suppress spurious higher-order terms generated by resummation outside the physical region . We also give the necessary ingredients for the improvement of third-order predictions by resummation when they become available.

1. Introduction The measurement of hadronic event shape parameters in e+e- annihilation is one of the most important ways in which QCD can be tested and its coupling constant as determined . The recent flood of very accurate data from the LEP experiments has brought about a situation in which the theoretical uncertainties * Research supported in pan by the UK Science and Engineering Research Council and in part by the Italian Ministero dell'Università e Ricerca Scientifica. ~ On leave of absence from INFN, Sezione di Firenze, Italy. z On leave of absence from Cavendish Laboratory, University of Cambridge, UK 0550-3213/93/$06 .00 © 1993-Elsevier Science Publishers B.V . All rights reserved

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S. Catani et al. / Resummation

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in as-determination from event shapes are greater than the experimental ones [ 1 ] . Consequently theoretical progress in this area is urgently required. There are two major sources of theoretical uncertainty in QCD event shape predictions. One is the "hadronization" problem: how are perturbative calculations, even if carried out to very high order, to be related to the observed asymptotic final state, which consists not of free quarks and gluons but ofhadrons? Here there are some successful models but no deep theoretical understanding, and we have nothing new to say on this topic in the present paper. The best one can say in general is that, provided the hadronization process is local in phase space, i.e. involving limited momentum transfers, then hadronization corrections should decrease like inverse powers of the hard-scattering scale Q, which is the c.m . energy in e+e- annihilation . Unfortunately this means that accurate data over a wide range of energies are required in order to disentangle hadronization effects in a model-independent way. The other main source of uncertainty is the possible effect of higher-order terms in perturbation theory . At present the full perturbative calculation of event shapes is complete only up to second order in as [2 ] . In principle the terms of third order and beyond could take any value, but in practice we might expect them to be as times a number of order unity, that is, a few percent of the leading (first-order) term at LEP energies . The expectation that higher orders give corrections at the percent level is only reasonable, however, when we have taken into account all possible sources of anomalously enhanced higher-order terms . For such terms we would like a complete understanding to all orders, so that they can be resummed to leave a remainder that can be expected to be well-behaved . One source of enhancement in perturbative predictions of quantities that are not fully inclusive over the final state is the presence of logarithms of large ratios of momentum scales*. In particular for a generic dimensionless event shape variable y, well defined (i.e. infrared and collinear safe) in perturbation theory and vanishing in the twojet limit (the semi-inclusive region), the typical leading behaviour of the distribution in nth order at small y will be dQ _1 _ n 1 In2n- ~ 1 , as y rr dy ~ Y

(1)

or, defining the normalized event shape cross section R (y ) by R (Y ) =

Jov

dY o dy

(2 )

and taking account of the cancellation of real and virtual singularities at y = 0,

Non-logarithmic enhanced terms have also been considered in ref. [ 3 ].

S. Catani et al . / Resummation of large logs in e+e -

5

Thus even when as is small, a perturbatioe treatment of the shape distribution would only appear to make sense when asLZ is also small, which excludes a large region near the twojet configuration, where most of the events occur. Whenever L is large, even if a sLZ is still small, we can improve the range and accuracy of perturbatioe predictions by identifying these logarithmically-enhanced terms and systematically resumming them to all orders . A detailed understanding of logarithmically-enhanced terms now exists for a certain class of shape variables, namely those for which the leading logarithmic contributions exponentiate. By this we mean that at small y the logarithm of the shape cross section takes the form 1nR(y) ~ Lg~ (asL) ,

(4)

where the function g, has a power series expansion in asL. More precisely, the factorization of QCD matrix elements in the two-jet region implies that, provided the phase space by which R (y ) is defined also has factorization properties that we shall specify, then R(Y)

where

= C(as)~(Y~as) + D(Y>as) ,

C(as) - 1 + ~, CnaS n=1 0o n+~ ln~(Y,as) _ ~ ~ Gnm~xSLm n=l m=1

= Lgl (aSL ) + g2 (aSL ) + aSg3 (aSL ) + . . . ,

(5)

(6)

and D (y, as ) vanishes as y -> 0 order-by-order in perturbation theory . For later convenience we have defined â s = as/2n in the expansions . The word exponentiation refers to the fact that the terms aSLm with to > n + 1 are absent from In R (y ), whereas they do appear in R (y ) itself. The function g, resums all the leading contributions aSL"+ 1 , while ga contains the next-to-leading logarithmic (NLL) terms aSLn, and gs etc. represent the remaining subdominant logarithmic corrections as Lm with 0 < m < n. All the functions g; vanish at L = 0 since they resum terms with m > 0. Eq. (6) represents an improved perturbatioe expansion in the small-y region . If we can find the form of the functions gr, then we shall have a systematic perturbative treatment of the shape distribution throughout the region of y in which asL < 1, which is much larger than the domain asLZ « 1 in which the nonexponentiated perturbation series was applicable. Furthermore, the resummed expression (6) can be consistently matched with fixed-order calculations . In particular, one can evaluate the leading and next-to-leading functions gl and gz and combine them with the known order-as results on the shape cross section,

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after subtracting the resummed logarithmic terms, to obtain a description which is everywhere at least as good as the fixed-order result, and much better as y becomes small. In order to use the resummed result to measure the strong coupling constant as, we have to understand how the prediction depends on the renormalization scale used in the calculation. A change of scale from QZ to It2, say, will induce a change in as of the form as(uz ) = as(Qz ) - ßoasln(h z/QZ ) + . . . ,

(7)

where ßo = (33 - 2Nf) l 12~t is the one-loop coefficient of the QCD ß-function . However, the physical cross section R (y ) must be independent of ,u, and therefore the effect of the change in as on the resummed expression (6) must cancel amongst terms involving the same powers of L. This tells us that, whilst the leading function g, is renormalization group invariant, the non-leading functions gr (i 3 2 ) must have an explicit scale dependence . For instance, for the next-to-leading function ga we have a g2(asL>h z ) = gz(asL,Q 2 ) + ßo(asL) Zgi(asL)ln(,u /Q Z ) (8)

where as = as (l~2 ) and gl is the derivative of g, . It follows that a complete evaluation of both g, and gz is necessary before any meaningful measurement of as can be performed. In this paper we shall carry out an analysis along the above lines for two related event shape variables that have proved useful for as-determination : the thrust and heavy jet mass. The results of this analysis, in the form of explicit expressions for the functions we have called here g, and gz, as well as numerical results combining the resummed and fixed-order predictions, have been given in two recent Letters [4,5 ] . Here we give the detailed derivation of those results and a full discussion of the general procedures necessary to derive similar results for other event shape variables that can be shown to exponentiate [6,7] . We do not however make any further comparisons with data since that is already being done very thoroughly by the experimentalists [ 8 ] . The contents of the paper are as follows. In sect . 2 we present the general framework used to derive our results. This is based on the use of a coherent branching algorithm which generates the QCD matrix elements to NLL accuracy in the relevant phase space region . We also discuss the evaluation of phase space with the necessary precision. In sect. 3 this discussion is applied to the phase space calculations for the thrust and heavy jet mass. We show that to NLL accuracy these shape variables are directly related to the jet masses generated by the branching algorithm. Therefore in sect . 4 we provide a derivation of the jet mass distribution . This confirms the result of refs. [ 9,10 ] , using a simpler treatment based on our general analysis of phase space. The result is in the form of a Laplace transform, so in sect. 5 we give a simple general method for inverting

S. Catani et al. / Resummation of large logs in e+e-

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this and similar transforms to NLL accuracy . This yields the explicit forms of the leading and next-to-leading functions gi and gz defined above. In sect . 6 we make some general remarks on the resulting form of event shape distributions in the high-thrust region . As a consistency check on the resummed expressions, in sect . 7 we first compare the logarithmic coefficients obtained by expanding them to second order with those found by fitting numerical calculations based on the fixed-order matrix elements . We then discuss how to combine the resummed and fixed-order calculations, to obtain improved predictions throughout the whole of phase space. We investigate alternative prescriptions, which permit some estimation of the theoretical uncertainty involved. Finally in sect. 8 we present our conclusions. 2 . Shape variable distributions in the semi-inclusive limit In this section we describe a general formalism to compute a certain class of shape variable distributions to next-to-leading logarithmic accuracy in the semi-inclusive or Sudakov region. As in sect . 1, we consider a dimensionless shape variable y defined in such a way that the limit y -> 0 corresponds to the two-jet region in e+e- annihilation . The corresponding perturbative distribution and normalized cross section are respectively given by x 1 dQ 1 ~dQnxcl (pl~ . . . ~pn) b(Y-Y(pn . . . ,Pn)) ~ây -~~ n=2

R(Y)

_ ~ 16 ~(1Qnxcl (pl~ . . . ~Pn) ©(Y-y(P1, . . . ~pn)) > n=2

(10)

where dQ~X°~ is the exclusive cross section for producing n final state partons p~, . . . , pn and the 8- and O-function constraints express the shape variable Y as a function of the parton momenta. In order to compute the shape cross section with a certain accuracy in In y, we need to compute both the exclusive cross sections and the phase space to the same level of accuracy . The present knowledge of QCD matrix elements (including virtual corrections) is sufficient to define a simple algorithm to obtain any e+e- exclusive cross section to NLL accuracy . We describe this algorithm in subsect. 2 .1, whilst in subsect. 2 .2 we discuss the accuracy needed in the evaluation of phase space. 2 .1 . THE COHERENT BRANCHING ALGORITHM TO NLL ACCURACY

Multiparton QCD matrix elements to leading infrared and collinear order have been known for a long time. In refs . [11-13] it was shown how the squares of

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e+e -

Fig. 1 . Coherent branching process in e+e - annihilation .

a

Fig. 2. Time-like splitting vertex .

these matrix elements can be generated by a branching algorithm, taking proper account of the coherence properties of soft gluon emission . In refs . [ 9,14,15 ] semi-inclusive form factors in the Sudakov region were evaluated to NLL accuracy . Combining these results, a consistent NLL coherent branching algorithm was set up in ref. [ 10 ] . It allows one to resum leading and next-to-leading logarithms to all orders in as for any one- or twojet dominated inclusive distribution . According to ref. [ 10], exclusive cross sections at the parton level in e+eannihilation are generated by a branching process in which the %rst step produces a quark-antiquark pair at the centre-of-mass energy Q and further steps represent consecutive parton splittings (fig . 1) . The parton kinematics can be described in terms ofparton masses k?, transverse momenta k it and longitudinal momentum fractions x~ with respect to two light-like and back-to-back momenta p and p p = ZQ(l,n) ,

p = ZQ(1~ -n) ,

(11)

where n is a unit 3-vector, which we shall normally assume to lie along the z-axis . Then for a parton in the hemisphere of p, as in fig. 1, we write z z + (12) ku = x~p~` + kir + kl` zk` pa , x~Q

where k1 = 0 for the primary quark with momentum k . At each time-like

9 S. Catani et al. / Resummation of large logs in e+e -

splitting vertex (fig. 2 ) four-momentum conservation gives k?

k'z

13

where

(14)

is the longitudinal momentum fraction involved in the branching and q2 is the modulus-squared of the resealed relative transverse momentum qi, given by 9i -

k1i

-

Zik1i-l

Zi(1 -

Zi)

(15)

A given splitting probability (see below) is associated with each branching diagram. In order for this probability to reproduce the infrared behaviour of the QCD matrix elements, the kinematically available phase space for parton emission must be reduced to the angular-ordered region in which the branching angles decrease as one moves from the hard vertex towards the final state [ 16 ] . Outside this angular-ordered region different parton emitters act coherently, leading to destructive interference in the matrix elements : the azimuthally integrated distribution (relevant for one- and twojet like quantities) vanishes to leading and next-to-leading order [ 17 ] . This phenomenon is called colour coherence . In terms of the kinematic variables {zi, qi}, the angular ordering constraints are given by [10] The probability weight associated with a branching diagram (fig . 2 ) is obtained by assigning the factor dz _ 4i dz .©Cz2(1 - z . ) z-z _ Qzl 7,ab ~ x z?(1 dPab ( , -z i ) z q'i 'z ) , z t ] (l~) S i i t [ qi 0 ~qi

to the vertex i involving the parton splitting a --~ be (a = q, q, g ) and the Sudakov form factor ratio db(z 2 g 2 ) (18) 2 2 d b( Z i+lqi+1) to the line connecting vertices i and i + 1 with momentum ki and parton label b. For external lines, the denominator in eq. (18 ) is set equal to unity. The Sudakov form factors are defined as follows: - z ~ _yz dqz i =exp{ 1 dz 4a(g z ) o q Jo x©(z z (1-z) z g z -Qô)P" b [as(zz (1-z)z gz ),z]~ (1 )

S. Catani et al. / Resummation of large logs in e+e -

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in terms of the vertex probabilities Pab in eq . (17 ) . These latter are given by [10] P99

[ceg,

Z]

P88

[aS~

Z]

a 1-~z z ra zl 1 27i CF 1-z + \~~ 2CFK1-z'

_ as z 1 1 ~ ~ + n 2 CAK z(1 - z) 2n 2

and the remaining ones are obtained by the symmetry relation for a --~ be (21)

The colour factors for a SU ( N ) gauge theory ( N = 3 in QCD) are CA =N,

CF=

Nz - 1 , 2N

(22)

and Nf is the number of active flavours (Nf = 5 at LEP energies) . The contributions of order as in eq . (20) are the unregularized one-loop Altarelli-Parisi splitting functions, whilst the O (as ) terms give the next-toleading infrared contributions in the Sudakov region . The coefficient K is_ dependent on the renormalization scheme . In the MS scheme it is given by [ 14]

The scale parameter Qô, introduced in the vertex probabilities (17) and in the Sudakov form factors (19 ), acts as an infrared and collinear cut-off for the exclusive parton distributions. From a formal point of view in QCD perturbation theory, the cut-off dependence cancels in infrared and collinear safe quantities, which are thus finite in the limit Qo -> 0. More physically, the cut-off Qo can be taken to define the borderline between the perturbative and non-perturbative components of the scattering process (see the discussion in sect . 4) . In e+e- annihilation we also have a parton cascade in the hemisphere of p, associated with the antiquark jet, in which the momenta are written as k;`=x;pt`+kL ;+

2 kzi` + k ~ u p . z x1Q

(24)

Here the momentum fractions are written as zl = zl /z ;_ 1 and then the branching process proceeds as for the quark jet. We shall consider in subsect. 3.3 the possibility of kinematic overlap between the two jets . The exclusive cross section dQnX~i for producing the n-parton final state is obtained by considering all angular-ordered branching diagrams with n final state

S. Catani et al. / Resummation of large logs in e+e -

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partons and summing the corresponding probability weights. Since the vertex probabilities (17 ) and the Sudakov form factors (19 ) are local in the branching variables { z;, q? }, it is a straightforward exercise to write down the corresponding equations for the generating functional of the exclusive distributions (see for instance ref. [ 17 ] ) . Concluding the description of the NLL branching algorithm, we should recall its range of validity . For one- and twojet-dominated inclusive distributions which exponentiate (see the Introduction), the algorithm allows one to compute the leading and next-to-leading functions g, and g2 in eq . (6) . In this case the evaluation of the subdominant contributions gs etc. would require the introduction of parton splitting probabilities Pab which are not azimuthally averaged and involve higher-order terms. The algorithm is less powerful for more exclusive or multijet distributions [ 18] . In these cases, since we are neglecting large-angle soft gluon contributions of order aSLz" - 2 , the algorithm can provide only the leading (cxsL2 n ) and nextto-leading (aSL 2n m )terms in the perturbatioe expansion of the cross section itself, which generally does not have the fully exponentiated form (6) . We should comment on the relationship between the NLL branching algorithm described above and that proposed in ref. [ 19 ] , which is the basis of the NLLj et simulation program [ 20 ] . The algorithm of ref. [ 19 ] provides a full treatment of next-to-leading collinear logarithms, which improves the description of inclusive distributions outside the semi-inclusive regions in which soft logarithms become important. In semi-inclusive regions, such as the high thrust region considered here, the resummation of next-to-leading collinear logarithms alone is not sufficient for the calculation of the shape cross sections .

2 .2. PHASE SPACE TO NLL ACCURACY

In deriving the resummed event shape distribution, it is important to understand what approximations can be made in the evaluation of the corresponding phase space. We can do this by means of the following power-counting procedure . The QCD parton distributions described in subsect. 2.1 have (at most) a double logarithmic spectrum dq 2 /92 dz/z (or dz/ [ 1 - z ] ) for each power of as. Let us denote by E~ a generic logarithmic variable bounded by 1 from above, like momentum fraction (Et = z;, 1 - z;) or rescaled angle (E~ = q+1 /z?q?) . A typical contribution to the cross section (10) at the nth order of perturbation theory has the form Rn(Y) ~ as

%t J0

dEl E1

~1 0

dE2 . .. E2

f i dE2n 0

E2n

O(Y-Y(Pi, . . . ,Pn)) ~

(25)

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Since the shape variable y is infrared and collinear safe, after the real-virtual cancellation of infrared and collinear singularities, the theta-function constraint leads to the small-y leading logarithmic behaviour Rn ~ aSL2n (L -1n 1/y) . Now let us now consider the effect of phase space approximations . Suppose we approximate the functional dependence of y ( p, , . . . , pn ) by neglecting terms of relative order E;: Y(IO~ . . . ~pn) =Y1 (1~1~ . . . ~Pn)[1 -~

O(Ei )] .

(26)

This approximation is equivalent to the rescaling y ~ y [ 1 + O (Ei ) ] in the O-function constraint of eq . (25) . It follows that after making the replacement (26) in eq. (25) and integrating over all variables other than Ei, one has a result of the form Rn (Y ) ^' as

f 1 E,dEi ln v

2n-1

[Y (1 + O (Ei ) ) ] ^' Rn (Y ) .f O (as 1n2n-2 y ) .

(27)

Thus eq. (26 ) introduces an error which is at most two logarithms down relative to the leading term . A similar argument shows that the phase space approximation (28) Y(pi, . . . ~Im) = Y2(p~, . . . ~pn)[1 +O(EiEj)] gives an error at most of order aSL 2n- s (three logarithms down) in the evaluation of the integral (25), and so on. The power counting just outlined is even stronger for the class of shape variables for which the leading logarithmic contributions exponentiate . In this case, by keeping terms of relative order Ei we can compute the leading and next-toleading exponent functions Lgl (as L) and g2 (a s L) in eq . (6). To NLL precision, we can neglect EiE1 terms, which will affect only the subdominant contribution asg3(asL) . As explained in sect . 1, we need to know g2 (a s L) in order to measure as . Therefore we must evaluate the phase space keeping terms of relative order Ei, but we can neglect those of relative order E i E~ . To see that the above two power-counting arguments are equivalent in the case ofexponentiating quantities, we expand the exponential function ~ in eqs. ( 5,6 ), and find that the leading contributions to the shape cross section R (y ) from gs are of the form as (asL)m (as L2 )n-m = as+1L2"-m with n ~ m , 1 . Thus any neglected contribution of g3 is indeed suppressed by three or more logarithms . In fact, by comparing with the exact two-loop matrix element (see sect . 7) we can determine the coefficient G21 which gives the leading ( m = 1) contribution of gs . After this improvement, the error in R (y ) is at most of order asL2 , i.e. four logarithms down. Exponentiation of the event shape cross section will occur when the corresponding phase space has appropriate factorization properties, like the matrix elements, either directly in terms of the variables of the coherent branching al-

S. Catani et al. / Resummation

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logs in e+e -

13

gorithm (q? and z~ ) or after some transformation of them, such as a Laplace or Fourier integral transformation . We have to establish that factorization is valid up to the required level of precision as defined above. 3. Kinematics and phase space for thrust and heavy jet mass In this section we apply the ideas of sect. 2 to the specific examples of the thrust and heavy jet mass event shape variables. In subsects . 3.1 and 3.2 we present the general kinematics for these quantities. Both are related to the hemisphere invariant masses, defined with respect to the thrust axis. Therefore in subsect. 3.3 we express these masses in terms of the variables of the coherent branching algorithm, within the necessary precision as discussed in subsect. 2.2. 3.1 . KINEMATICS OF THRUST

The definition of the thrust T is [21 ~ T = Max ~` ~` ~ n~ - Max Tn (29) ~~ ~Pr~ where the sum is over all final-state particles i and the maximum is with respect to the direction of the unit 3-vector n . At the maximum, n = nT, the thrust axis . It can be seen from this definition that the thrust is an infrared and collinear safe quantity, that is, it is insensitive to the emission of zero-momentum particles and to the splitting of one particle into two collinear ones. It approaches unity in the twojet limit. Thus the quantity T = 1 - T is of the type discussed in sect . 2. The normalized cross section i dT (30) RT(i) QdT - ~_ 7

is finite order-by-order in QCD perturbation theory, and a NLL exponentiation of In T-enhanced terms should be possible provided we can show that the corresponding phase space has the required factorization property . An essential part of the thrust definition (29 ) is the separation of the final state into two hemispheres Sn and Sn by the plane P orthogonal to the unit vector n . Each final-state momentum pt is assigned to Sn, S or Pn according to whether pt ~ n is positive, negative or zero . In fact we shall see shortly that no momenta can lie in Pn when n is the thrust axis, so we can ignore this possibility. Denoting the total momentum in the hemisphere Sn by k, we can use a representation of the form (12 ) to write k z QzkLz k = xp + k1 where xz > wl = , (31) 1 p x

+w

S. Catani et al. / Resummation of large logs in e+e -

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and similarly for the total momentum in hemisphere S

k = xP + ki

+ w1 z p

Then by momentum conservation

x=2(1+wi-wi+t),

z

z

ki zz > w1 = k + . Qz

where

(32)

k + k = p + p and we find

x=i(1-wi+wi+t)>

kl = -ki,

(33)

where

t =

1 - 2 (wl

~ + wl ) + (w1 - wl )z =

Q

`~ Tn .

(34)

We can show as follows that when n is the thrust axis then k 1 = 0, i .e . the hemisphere momenta are aligned with the thrust axis . This is not completely obvious because particle momenta are transferred between S and Sn as n is varied to maximize the expression Tn in eq . (29) . We can imagine n moving on a unit sphere whose surface is cut into 2m patches by the planes orthogonal to the m final-state momenta [ 22 ] . From eq . ( 34 ) we have that on each patch Tn is a smooth function which increases in the direction in which ki decreases. Hence either the patch contains a point where k 1 = 0 and T has a local maximum, or there is a point on its boundary where T is largest. Now as n crosses a patch boundary, a final-state momentum crosses from Sn to Sn , or vice versa. It is easy to see from eq . (29) that Tn is continuous but its gradient increases as the boundary is crossed. Thus a boundary point can never be a local maximum of Tn . It follows that the global maximum of Tn must be one of the non-boundary points where k1 = 0. Since the thrust axis always corresponds to an interior point of a patch, it is impossible for any momenta to lie in the plane orthogonal to it, as we asserted earlier* . Furthermore, since k 1 = k~ = 0 when n = nT we have wL

= kzlQz - w ,

wl = kz/Qz - w

(35)

~Qt

(36)

and T

~ ~Pt~

(x

+ ~-

1)_

_

~P t~

where t is given by eq . (34 ) in terms of the scaled jet masses ( 35 ) . For our purposes it will be legitimate to neglect final-state particle masses, which give rise to power corrections proportional to m?/Qz , and so we can set T = t. The main effect of the extra factor multiplying t in eq . (36) is to ensure that the kinematic upper limit on T is always unity, independent of the number and masses of final-state particles. * This statement, and those in the preceding paragraph, apply to the true thrust axis, and may not hold for the "thrust axes" found by approximate methods .

S. Catani et al. / Resummation of large logs in e+e -

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In the high-thrust region both w and 2u have to be small, and so we can expand eq. ( 36 ) to obtain 1-T-z-w+w+2w~w + . . . .

(37)

The term 2w~c is of relative order i and gives rise to corrections which vanish as T ~ 1 . Such terms will be included in the fixed-order non-logarithmic corrections to be added to the expression obtained by resumming logarithms of r to all orders . Therefore for the purposes of resummation we can simply write the thrust cross section ( 30 ) as RT(z) -

~ 1~

n=2

Q

dQnxcl(1~ 1 ~~ . .

~pn)

0(r-w-w) ~

(38)

In the low-thrust region, logarithms of r are no longer dominant and a full order-by-order evaluation of the thrust distribution using the exact expression (36 ) becomes necessary. One important aspect of the full result is the kinematic lower limit on T, which depends on the number of final-state particles. Neglecting masses, Tm;n = 2/3 for three particles, corresponding to a symmetric configuration. For four particles the minimum thrust corresponds to final-state momenta forming the vertices of a regular tetrahedron, each making an angle cos- ~ (1/~) with respect to the thrust axis. Thus Tm;n - 1/~ - 0.577 in this case . For more than four particles, Tm;n approaches 1 /2 from above as the number of particles increases. So far, the full perturbative evaluation of the thrust distribution has been performed only up to second order in as, corresponding to at most four partons in the final state, and so the perturbative prediction vanishes identically below T - 0.577 . One difficulty with resummation of logarithms is that, while it provides a reliable estimate of terms of third and higher orders in the high-thrust region, the logarithmic higher-order terms alone are not a reliable estimate at low thrust and in particular they may not vanish at T - T,in. Consequently if one wishes to use a resummed plus nth-order prediction throughout the whole of phase space then it may be necessary to suppress the terms of higher than nth order generated by resummation at low thrust . We shall discuss a simple recipe for doing this in subsect. 7.5 . 3 .2 . KINEMATICS OF HEAVY JET MASS

The definition of the heavy jet mass p that we shall adopt is simply i.e. the larger of the scaled masses-squared ( 35 ) of the two hemispheres separated by the plane orthogonal to the thrust axis. This is the definition that is normally

16

S. Catani et al. / Resummation of large logs in e+e-

used experimentally . The original proposal in ref. [23 ] was to separate the finalstate particles into two sets (not necessarily hemispheres) with scaled invariant masses-squared w and w such that w + iu is minimized. We shall show in subsect. 3.3 that in the low-mass region this differs from our definition only with respect to subleading terms. Thus the same resummed expression can be used for the two cases, but the fixed-order corrections will differ. The shape cross section for the heavy jet mass is Rx(P)

_ ~ 1~ n=2

d6n xcl (pl~ . . . ~Pn) ®(P - w) 0(P - w) ~

(40)

We see that this is closely related to eq . (38) for thrust . The difference is that we must integrate over a square rather than a triangular region of the (w, ~cv )plane, which will lead to different NLL terms, and of course different fixed-order non-logarithmic contributions. The kinematic upper limit on the heavy jet mass corresponds to having the minimum mass in the opposite hemisphere and therefore, neglecting particle masses, it is given by (1 - Tm ) where T, is the thrust of the corresponding final-state configuration. For three final-state particles, Amax = 1 - Tmin = 1/3, but for more than three particles Amax < 1 - Tmin because Tmin corresponds to a more equal distribution of momenta between the two hemispheres. Thus for four particles we saw that 1 - Tm;n = 1 - 1/~ = 0.423, corresponding to a tetrahedral configuration, whereas the maximum heavy jet mass is pmax = 1 - 2/ (1 + ~) = 0.420. This upper limit is attained when a single particle recoils along the thrust axis against three particles with momenta making an angle cos- ~ (1/~) to it. The value of pmax increases slowly with the number of particles, and is believed to approach a value near 0.477 asymptotically* . If the heavy jet mass is defined as in ref. [23], its kinematic upper limit is 1/3 independently of the number of final state particles [24] . This property may suggest that the corresponding QCD perturbative series has better convergence in the region of high heavy jet mass. 3.3 . HEMISPHERE AND JET MASSES IN THE COHERENT BRANCHING ALGORITHM

We have to consider next the evaluation of the hemisphere masses w and w within the precision of the phase space approximation (28 ) . Then, in sect . 4, we can derive the distribution of these masses by means of the coherent branching algorithm. What we shall show here is that the jet masses generated by the * Finding the value of pmax for an infinite number of partons is a difficult kinematic (geometrical) problem. It is trivial to show that pmax , 1/2, but we have not been able to find any parton configuration which saturates this upper limit. The maximum value pmax = 0.477 that we found corresponds to the kinematic configuration in which one particle lies in one hemisphere and all the others are disposed on the surface of a cone in the opposite hemisphere.

S. Catani et al. / Resummation oflarge logs in e+e -

I l

K~

17

I

Fig. 3. Jet branching kinematics .

algorithm are equal to the hemisphere masses, and the thrust axis lies along the initial parton axis, within the required precision. First we show that the jets generated by coherent branching are confined with sufficient accuracy to their respective hemispheres. Consider a final state parton with momentum kn emitted in the branching of the forward jet J (fig. 3 ), i.e. the jet aligned along the momentum pin eq . (11) z kLn (41 ) kn = Xnpt` + k1n + 2Pu ~ xn Q The ratio between the (-)- and (+)-components of the momentum of this parton is kn = + kn

(42)

k1n

zQz

.xn

If this quantity is less than unity, the particle is travelling in the forward hemisphere . Introducing the coherent branching variables, i.e. the momentum fractions z; in (14 ) and resealed transverse momenta q t in (15 ), and using momentum conservation, we obtain k1n = zn(1 - zn)4n + znzn-1(1 - zn-1)4n-t + . . .

+

znzn-i . . .z1(1 - zi)4i

(43)

Xn = Zn Zn -1 . . . Z1X .

Therefore, taking into account the angular ordering constraint (16 ), it is straightforward to see that ki n = ~Z n Zn _1 . . .

zl (1 - zOl z 4i ( 1 + O(Et))

~

(44)

where E, = q2/z? ~qz i . We shall show below that x = 1 + O(EtEj), where the O (E,Ej ) term is proportional to the invariant mass kz/Qz of the backward jet J . Using these results we have (45)

18

S. Catani et al. / Resummation of large logs in e+e -

where the equality is valid apart from subdominant terms of order EtE; and the inequality comes from the coherence constraint (16 ) . A similar argument shows that k /kn < 1 for particles from jet J, within the same precision. Thus all partons produced by the forward jet J (backward jet J ) can be taken to be emitted into the forward (backward) hemisphere . From our discussion of thrust kinematics in subsett. 3.1, it follows that the thrust axis is aligned along the jet axes p and p, and the thrust is given by

provided no other partition of the final state partons into two subsets J', J' gives a larger thrust, i.e. a lower value of kz + kz . Actually it would suffice to show this for any two other hemispheres, but let us consider more generally transferring any set of partons with resultant momentum k~ from J to J and vice versa for a set with momentum k;. We can write

where wi ; < x? and wi; < Xi since the sets lie within opposite hemispheres originally. After the transfer, we have kz +k z ~ kz+kz+2(k; -k;)z-2(k-k)~ (k;-k;) = kz+kz+2(kc-k1)z x~

z; J

(48)

which represents an increase*. In conclusion, we have shown that to NLL accuracy we can assume that the thrust axis lies along the initial parton (jet) directions and that the jet masses generated by coherent branching are the corresponding hemisphere masses, in terms of which the thrust is given by eq . (46) . Moreover, since eq . (48) is valid independently of the fact that (k - k; + k; ) and (k + k - k; ) belong to disjoint hemispheres, we have also shown that to NLL accuracy the heavy jet mass as defined in [23] actually coincides with that defined with respect to the thrust axis. We are now in a position to write down simple expressions for the thrust and heavy jet mass cross sections in terms of jet distributions generated by the coherent branching algorithm. Considering for instance the thrust cross section (38), we should first integrate the distributions d6nX°~ over the phase space of the final state partons keeping the momenta k and k of the quark and antiquark * The O(e;e ;) term in eq . (48) comes from contributions proportional to k2 lQZ or kz /Q2 .

S. Catani et al. l Resummation of large logs in e+e -

jets fixed, to obtain i

RT(i)

_ ~ dxdz 0

19

f~

kz kz dkzdkz8Cl-x- xQz ~ 8C1-x- xQz Jo x Jq (Qz, kz ) J9 (Qz > k z ) O (TQz - kz - kz ) . (49 )

Here JQ (Qz, kz ) is the jet mass distribution, de%ned as the probability of producing a %nal state jet with invariant mass kz from a parent parton a produced in a hard process at the scale Qz . It obeys the normalization condition

za z z

~~dk J (Q ,k ) = 1 . 0

(50)

The kinematic correlation between the two jets in eq. (49) is only due to longitudinal momentum conservation, which gives (as anticipated earlier) x = 1 -~ O(k /Q ) and z = 1 -~ O(k lQ ) . In the small-i limit this correlation is of relative order T and can be neglected, leading to the factorized expression

z z

z z

Rr(T) _

~W dkz dkzJg(Qz,k z ) Jq(Qz >k z ) O(zQ z -k z - kz ) . 0

(51)

Proceeding in the same way we arrive at the following factorization formula for the heavy jet mass cross section Rx (P ) =

~

J

°

dkz dk zJq (Qz, kz ) Je (Qz , k z ) O (PQz - kz ) ~ (PQz - kz ) .

(52)

Eqs. ( 51) and ( 52 ) give the thrust and heavy jet mass cross sections to NLL order provided the jet mass distribution is evaluated to the same accuracy, as we shall do in sect . 4. 4. Jet mass distribution Applying the coherent branching algorithm together with the kinematics (13 ) at the %rst vertex of fig. 1, one obtains the following evolution equation for the quark jet mass distribution [ 9,10] Q z dgz Jq (Qz, kz ) = 8 (k z ) + dz O (z z (1 - z )z qz - Qô) 0 9 0

f

f1

x [ f ~dRz r ~dk,z 8 J0

J00

(kz-z(1-z)qz-

k~ _ zz lgzz

J9 x Jq ( zzgz, k~z ) Jg ( (1 - z )zqz, 9z ) (9z, kz )~ (53)

20

S. Catani et al. / Resummation of large logs in e+e -

For the antiquark jet we have Jq (Qz, kz ) = Jq (Qz, kz ), whilst the gluon distribution Jg satisfies an equation similar to (53) in terms of the gluon splitting functions Pgg and Pgq . Let us consider the kinematics, expressed by the 8-function in the integrand of eq. (53), in the light of the discussion of phase space approximations in subsect. 2.2. The secondary quark and gluon jet masses will be of the form

where (z', q~z ) and (z", q~~z ) are the branching variables for the next vertices on the corresponding branches . Thus, neglecting terms of order EtE~, the 8-function implies that z ~~ ( 1 - z ~~ ) q~z q~~z z ~ (1 - z ~ ) kz - z(1 - z)q z 1 + 1 + (1 - z)zgz z zz9z z

I . (55)

Now, due to the angular ordering constraint (16), the quantity q~~zl (1 - z )zqz is of order E;. Thus the third term in square brackets, representing the contribution from the mass of the gluon jet, is O (EIEi ) and can be neglected. The integration over qz may then be done trivially by applying the normalization condition ( 50 ), to obtain Qz d ~z i z JQ (Q ~ ~ dz0(zz(1-z)zgz-Qôl z, k ) - ~(k z )+ J0

9

0

x Pqg[as(z z (1-z)z g z ),

zl

x [zJq ( zz g z , zk z - z z (1 - z ) q z ) © ( k z - z (1 - z )

qzl

- Jq (qz>

kz )] . (56)

Eq. (56) can be solved order-by-order by successive substitution into the integration on the right-hand side, which gives to one-loop order Jq (Qz ,k z )-8(k z )+

~CF[kzClnQz-4/J

R

+O(~s)~ {1+0(kzlJ Q (5%)

As expected, terms of nth order are of the form [kz lnm-1 Qz

(58)

R

with m 5 2n; the R-distribution is defined by %~ dk z h (kz)

Ckz

lnm

~ z IR

~~ 2

dkz h(kz)kz

h (t))

lnm Qz .

(59)

S. Catani et al. l Resummation of large logs in e+e -

21

In order to evaluate JQ (Qz , kz ) with next-to-leading accuracy to all orders, eq. ( 56 ) can be diagonalized by taking its Laplace transform J~(Q z ) --_

J0

~dkze-~kz J(Qz kz) .

(60)

Differentiating with respect to ln Qz, we find dJ9(Qz) %ldzPg9[as(zz(1-z)zQz),z]O(zz(1-z)zQz-Qô~ d ln Qz o x [ e -~Zti-Z)QZ j9~ (zzQz ) - je (Qz ) ] . (61) Z We are interested in the region kz « Qz and thus we have to solve eq . (61) for » 1 . To the required accuracy we can replace z by 1 in all slowly varying functions of z since the q ~ qg Altarelli-Parisi splitting function is singular only for z -> 1 . We then have v Qz

-Jo1dzP99[as((1-z)zQz),z]©r(1-z)zQz-Qô~ ddlnQz) \z [ e -v(l-z)Q Z - ] J4 (62) x 1 (Q2 ) , with the solution -~Q z c~-x~ - 1 1nJ9(Qz) dx e 1 -x ~i 0

fX dzP44 [as((1-z)(1-x)Qz )>z] p((1-z)(1-x)Qz-Qô~ . 0

(63)

Eq. (63 ) is the resummed expression for the jet mass distribution we were looking for. This result was already obtained in refs . [ 9,10] ; here we have repeated its derivation using a simplified treatment based on our general analysis of phase space in subsect. 2.2. The factor [exp (-vQz (1 - x ) ) - 1 ] in (63 ) is due to a combination of real and virtual contributions and regularizes the collinear (x -+ 1) and infrared ( z -~ 1) singularities of the QCD matrix element. Therefore if we consider the case of fixed coupling constant, the expression ( 63 ) is finite when the cut-off Qo vanishes. However, the running of the coupling constant provides a relevant class of leading logarithmic contributions and in order to keep as ( (1 - z ) (1 - x ) Qz ) in the perturbative regime the regulator Qo has to be chosen larger than the QCD scale ~1MS . In this respect, the resummation of the perturbative series itself suggests that in the region of very small jet mass, kz < ~1MS, non-perturbative contributions are dominant . They may be introduced on a phenomenological basis by keeping Qo fixed in eq . (63) and multiplying Jq by a non-perturbative ( Qo-dependent ) hadronization distribution [9, 25 ] .

22

S. Catani et al. / Resummation

of large logs in e+e-

In this paper we choose a different (but physically equivalent) way of dealing with hadronization effects. As long as as In ( vQ2 ) < 1, the resummed expression (63 ) has a formally convergent power-series expansion in ag In ( v QZ ) to NLL accuracy . Therefore in the following we shall compute this expression in the limit Qo --> 0 . As explained in refs. [4,5 ], we may then introduce non-perturbative effects directly for the thrust and heavy jet mass cross sections, by convoluting the resummed perturbative component with phenomenological hadronization corrections . Inserting into eq . (63) the next-to-leading splitting function from eq . (20), we find (defining u = 1 - x ) du (e _uvQz_ 1)[~Qz 1nJq(QZ ) _~~ gq2 A(as(q2 )) + zB(as(uQ2))] + O(as(aslnvQ2 )") with where

2 A(as) __ ~ A( 1 > ~. ( ) A(2) n A O) = CF

1CFK A~2) -z

(64) B(as) __ ~ Bn)

(65)

B(l) _ -3CF z

(66)

and K is given by eq . (23) . To evaluate the integrals in eq . (64), we note first that the a integration may be performed with NLL accuracy by setting [ 9,10 ] e-uUQz

(67 - 1 - -4 (u - v ) , ) where v = 1 /erwQ 2 and yE = 0.5772 . . . is the Euler constant. The q2 integration also becomes simple ifwe use the renormalization group equation to change the integration variable to as : \ dq 2 - - 1 d + O(as)' (68) q ßo aa ßoas with 11CA - 2Nf Nf - 3CF Nf . = 17CÂ - SCA (69) ßo ° ßi 12n 24n2 In this way we end up with a function of as (vQ2 ) and as (v2Q2 ), which can be expanded in turn as a power series in as ( QZ ) In (vQ2 ) by using the solution of the renormalization group equation (68 )

ss Cl -

as(QZ) - ßi as(QZ) ln(1 + ßoas(QZ ) lnv) as(vQ2 ) = 1 + ßoas(Q Z ) lnv ~l ßo 1 + ßoas(Q Z ) lnv (70) +O(as(QZ )(ag(QZ )lnv)")] .

23 S. Catani et al. / Resummation of large logs in e+e -

We find to NLL precision 1nJ4(QZ )

=

ln(vQ2 ) .h[ßoasln(vQ 2 )J + .f2[ßoasln(vQ 2 )J + O [as lnn- (vQ2 ) J ~

where

as (QZ ) and A(i) fi(1~) _ - 2nßo7~L(1-2~,)ln(1-2~,)-2(1-~,)ln(1-~)],

(71)

cxg =

(72)

Acz) B~ 1 ) A~ 1 ) + ln(1 - .i) [ln(1 -~,) -ln(1 -2~)] nßoE 2nßo Znß

1

[ln(1-2.1)-21n(1-~,) + ZInZ(1-2~,)-lnZ(1-~,)] . (73)

To express this result in terms of as (,u2 ) for an arbitrary value of the renormalization scale h2 , we must take into account the scale dependence of the next-to-leading function f2 , which is given by a formula analogous to eq . (8), .i2(1~,Ft Z )

_

.fz(~>Q Z )

+ ~ Z .fi

(~) ln(~c2/QZ),

(74)

where f2 (~, QZ ) is f2 (.1) as defined above and

Eqs. (71)-(74) resum all the leading and next-to-leading logarithmic contributions to Jq (Qz ) and provide an exponentiated expression for the Laplace transform of the jet mass distribution . In order to find the jet mass distribution itself, we have to perform the inverse transformation (Qz> k2 ) = 27L1 ,fr dU evkz .J9 (QZ ) where the contour of integration C runs parallel to the imaginary axis, to the right of all singularities of the integrand. This type of integral occurs in the expressions for all the event shapes that we shall study, and so we consider its evaluation to next-to-leading accuracy more generally in sect . 5. J9

5. Evaluation of integral transforms Instead of the differential jet mass distribution Jq, it is more convenient to deal with the mass fraction Rq (w ), which gives the fraction of jets with masses

S. Catani et al. / Resummation of large logs in e+e -

24

less than wQ2 Ra (w

~ ) _ .f0

dk2 Jq (Q2 , k 2 ) 0 (wQ 2 - k2 ) .

(77)

Using the representation (78)

where the integration contour C is the same as in eq . (76), we see that Rq(w)

2~ci f_ c

dve~u,QzJ9(Q2)

where N = vQ2 and .~ has the logarithmic expansion .~(as,1nN) = fi(ßoaslnN)1nN+ .iz(ßoaslnN) +ßoasÎ3(ßoaslnN) + . . . ,

(80)

whose leading and next-to-leading functions fi and fz are given in eqs. (72,73). Since the function .~ in the exponent varies more slowly with N than wN we can Taylor expand with respect to In N around In N = L - In (1 /w ) . More precisely, introducing the integration variable u = wN we have

°° 1 + ~ ~~(n) (aS, L,) lnn u} , n=2 n ~ (81) where Y(as,L) _ .~ W (as L) ,

(82)

n ~(n)(~S L) = Ln~(as L) .

(83)

Then, using the result 1

2ni ,fc

dulnk uexp[u-(1-y)lnu] =

dk 1 dyk T (1 -Y)

(84)

where T is the Euler T-function, and noticing that the nth derivative .~~n ~ (as, L ) is (at most) of logarithmic order as - ~ (asL )k , we can see that the series of terms

S. Catani et al. / Resummation of large logs in e+e -

25

with n , 2 in the exponent of eq . (81) give only subdominant contributions. They can therefore be neglected to NLL accuracy and we obtain __ exp[L. fi (~.) + .Îz(ß) + ~ zfi (~) ln(~cz/Q z ) ]

(85)

where

.1 = ßoas(Izz )L, L = ln(1/w) , (86) and in the last equality of eq . ( 85 ) we have neglected terms of higher than nextto-leading order and used the formula (74) to insert the dependence on the renormalization scale pz . In the case of the thrust cross section, using the Laplace representation ( 78 ) for the ©-function in eq . (51), we have _ dv e~ tQz RT( (87) [jq(Q z )] z ~ ~ ) - 2ni .f_ c Thus the exponent of the Laplace transform is simply double that for R9 (w ) and we obtain - exp [2Lfi(~) _ + 2.ÎZ0) + 2~,zf; (~,)ln(pz/Qz)] (88) RT(T) r[1-2.%(~)-2 .1f;(ß)] where now ~ = ßoas(,u z )ln(1/r) . This means that the leading and next-toleading exponent functions in eq . ( 6 ) are given explicitly by gi(asL) = 2fi(ßoasL)~ gz(asL) = 2.iz(ßoasL) -1nT[1 -2f (ßo~sL) - 2ßoasL.ii(ßaasL)] . (89) Finally for the heavy jet mass from eq. ( 52 ) we have exp[2Lfi(~) + 2fz(~) + 2~zf1 (.1)ln(Iz z / Qz )] z q (90)

to be evaluated in this case at ~ = ßoa s (Iz z ) L with L = In (1 / p ) . Comparing the thrust and heavy jet mass cross sections (88) and (90), we see that the difference is confined to the T-functions, which give rise to only next-to-leading differences between the distributions of these two event shapes . These differences, as expected, are due to the next-to-leading region of phase space in which the two hemisphere masses are of the same order of magnitude. The results given above are valid to NLL precision. Note however that the method described here for evaluating the inverse integral transform can be systematically extended to higher logarithmic orders . For instance, once the nextto-next-to-leading function fs in (80) is known, one can compute the jet mass

S. Catani et al. / Resummation of large logs in e+e -

26

fraction R9 to the same accuracy . Considering for simplicity the case ,uz = Qz, a straightforward calculation shows that the NLL result ( 8 5 ) has to be multiplied by the subdominant factor (91)

where f = f (~ ) and ~ and 1 are respectively the Euler yr-function and its first derivative . Similar subdominant factors apply to the thrust and heavy jet mass cross sections . ~ ~

~

6. Behaviour of shape distributions at high thrust The expressions (88) for the thrust and (90) for the heavy jet mass resum all the leading and next-to-leading logarithms, within the definition given in the Introduction . Thus in terms of the expression ( 5 ) for the shape cross section they provide the exponential form factor E. Before considering the inclusion of non-logarithmic corrections, we discuss the qualitative behaviour of the shape distributions resulting from the resummation of the logarithmic terms. We shall consider in detail only the thrust cross section. Similar comments apply to the heavy jet mass case . The thrust distribution is obtained by taking the derivative of the thrust cross section RT as follows 1 dQ _ 1 d (92) Q dT 1 - T d1nT RT In the large-T region, is given to NLL accuracy by eq. (88 ) . Before discussing the NLL result, let us consider the so-called double logarithmic (DL) approximation, which is obtained by summing only the contributions as 1n2" T in This corresponds to neglecting g2 in eq . (6) and to approximating gl by its first-order term in the as-expansion (thus we are considering the limit asL « 1) . Using eqs. (88), (89) we have 1 dQ 1 __ __ -2An) as ln ( - T) exp{-A~~> as lnz (1 - T)} (DL) .(93) n 1-T n QdT This result was first obtained in refs. [26] . To DL accuracy, in the nth order of the perturbative expansion the thrust distribution is proportional to (-)"+las 1n2"-1 (1 _ T)/ (1 - T), thus diverging to foo (depending on whether n is even or odd) as T -> 1 . The DL resummation cures this sick behaviour leading to a physically meaningful result . The resummed expression ( 93 ) is indeed positive definite and f nite. More precisely, it vanishes at T - 1 and has a peak at the value T = Tp given by (T )

(T

RT (T )

RT .

)~t-1-T

~

S. Catani et al. / Resummation of large logs in e+e -

27

The fact that the thrust distribution vanishes at T -_ 1 has a simple physical interpretation [27] . At the parton model level the e+e- annihilation process produces a quark-antiquark pair . Since both the quark and the antiquark carry colour charge they are necessarily accompanied by their own bremsstrahlung field ( soft gluons ) and the ensuing configuration has T ~ 1 . The vanishing of the thrust distribution for T = 1 thus corresponds to the impossibility of producing a bare charge in gauge theories . As a further consequence, the expression (94) for the peak position is not analytic in as : in fact, this thrust configuration is built up by a non-trivial colour field involving an infinite number ofgluons . Note however that the DL approximation ( 93 ) is not quantitatively reliable in the vicinity of the peak, since it predicts that the latter occurs at a s In 1 / (1 - T ) _ n/2CF = 1 .2, which is inconsistent with the DL assumption asL « 1 . The qualitative features of the DL approximation remain valid to leading logarithmic and NLL order: only the quantitative results are changed. The leading logarithmic (LL) result for the thrust distribution is obtained by considering the g, function in the form factor expression (6) . Using eq . (88) we have 1 dQ 2 +~1fi(~l)]exp { -2fi(~)ln(1-T)} (LL)~ 6âT = 1-T [.ÎO~ ) (9s> where, _ -2ßoa s ln(1 - T) . A first attempt to evaluate the LL expression ( 9 s ) was performed in ref. [ 28 ] and the complete LL result was obtained in [29], where a part of the NLL corrections was also taken into account. Note that the result (9s) reduces to the DL expression (93) in the limit ßo -> 0. In fact, the function f is simply obtained (see eq . (64) ) by integrating the DL contribution du/u dqZ/q2 as (qz ) over qz using the running coupling constant. However, the argument of the coupling constant is given by the relative transverse momentum z (1 - z ) q in the branching process (see eq . (17 ) ) and this scale is not related in any simple way to the thrust variable . This implies that no trivial resealing as (QZ ) -> as ( (1 - T )vQ z ) (where p is any fixed power) in the DL result (93) is able to reproduce the LL expression (9s) . Note also that the function fi (~, ) has a singularity at ~, = 2 . As discussed in sect . 4, this singularity is related to the infrared behaviour of the running coupling constant and signals the region 2ßoas In 1 / (1 - T ) ~ 1 in which the resummed perturbative expansion breaks down and non-perturbative contributions are dominant . Finally, the NLL expression for the thrust distribution is obtained by inserting eq. ( 88 ) into ( 92 ). The form factor ~ in eq . ( 6 ) thus acquires the g2 contribution in eq. (89) . This consists of two terms with different physical origins. The A~2 ~- and B~~>-contributions in f2 represent NLL dynamical effects, whilst the remaining part of f2 and the T-function in ( 89 ) take into account the kinematic

28

S. Catani et al. / Resummation

of large logs in e+e -

effects of energy-momentum conservation on the LL contributions . The position of the peak in the thrust distribution also moves in going from the DL to the NLL approximation. In general we have 1 a ° In 1 b + O(as(Q2 )) , (96) ßoas(QZ) + 7,p where the LL and NLL coefficients a and b can be computed from the corresponding distributions. The calculation shows that with increasing logarithmic accuracy the position of the peak moves towards smaller values of the thrust . One can also check that TP is still in the region 2ßoas In 1 / (1 - T) < 1 in which the resummed perturbative expansion is consistent . Nonetheless, this does not mean that the region close to the peak of the thrust distribution can be accurately described by the resummed calculation . In fact, using 2ßoas (QZ) ~ (ln Q/CMS)-~, eq. (96) gives 1 - Tp ~ GMs/Q, whilst one expects that hadronization effects produce corrections of the order dT ~ (pl) /Q, where (p1 ) is a typical hadronic scale. In the peak region perturbative and non-perturbative contributions are thus of similar magnitude, independent of the value of Q. 7. Combining with fixed order Having obtained a resummed expression such as (88) for the shape cross section at small values of a shape variable y, one could simply introduce a `switching" procedure to combine the resummed and fixed-order calculations and thus to obtain a prediction for all y. In the spirit of ref. [ 30 ] the following formula could be used dy

Yo

dY Resummed

yo

dy Fixed-order ( 97 )

where the switching function f (x) increases rapidly from zero to one as x passes through the point x = 1 . The switching point yo should be chosen such that as 1n2 yo < 1 and in practice one can use yo - yr,, where yL is the value of y at which the resummed and fixed order results coincide [ 6 ] . Such a procedure is very simple to implement but introduces large uncertainties, due to the choice of the function f (x ) and the switching point yo, in the region around y ti yL . This region is likely to be relevant for as determination because it is far away from both the extreme twojet region and the multijet region, which are affected respectively by large non-perturbative effects and by large higher-order corrections. To overcome this problem and understand its origin, one has to rely on the general exponentiation formula ( 5 ) . The expression in eq . ( 5 ) can in fact be used to predict the shape cross sections throughout the whole of the phase space. For this purpose, once the resummed form factor ~ is known, the coefficient function C and the remainder D can be

S. Catani et al. l Resummation of large logs in e+e -

29

evaluated by comparing with fixed-order results. In subsect. 7.1, we first check the consistency of the resummed and fixed-order calculations by comparing the coefficients of the logarithmic terms. Then we discuss the evaluation of the nonlogarithmic remainder and the general features of the resulting prediction . The procedure for extracting the remainder is not unique : methods corresponding to different assumptions about subdominant terms in third and higher order are equally valid in the absence of information on such terms. This non-uniqueness (called the matching ambiguity) is discussed in subsect. 7 .3 and then exploited to give predictions with improved behaviour in the low-thrust region. The alternative "log R" matching scheme is also discussed and shown to give very similar predictions. Finally we discuss the correct inclusion of renormalization scale dependence in the different matching procedures and present results on this dependence for the thrust distribution . 7 .1 . LOGARITHMIC COEFFICIENTS

Expanding eq . ( 6 ) to second order, we can write where

R (Y ) = 1 + âsR i (Y ) + âsR2 (Y ) , 2n

Rn(Y) _ ~ m=0

RnmLm

+ Dn(Y) ~

(98) (99)

the remainder functions Dn (y ) vanish as y -> 0, and Rio = Ci , Rzo = Cz ,

R 11 = Gii ,

Riz = Giz

Rzi = Gzi+C,G,1, Rzz = Gzz + zGi~ + CiGiz , Rzs = Gz3 + G1z G 11 , i Gz Rza - z iz

(100)

From the resummed expressions (88), (90) we obtain the coefficients G m for m = n, n + 1 presented in table 1 . For future reference we give results also for third order, to be compared with three-loop calculations when they become available. The coefficients Gn for the heavy jet mass are the same as those Gn for the thrust except for

GZZ = GzZ + 2nz CF/3 , Gx = Gs3 + ~cz (11CA -2Nf )CF/3- 16~(3)CF . (101) Thrust and heavy jet mass coincide to first order in perturbation theory . The logarithmic coefficients G 1 , and G, z we have computed agree with the analytic

30

S. Catani et al. / Resummation of large logs in e+e TABLE 1 Logarithmic coefficients Gn ,n for thrust ; ~(3) = 1 .202057 . . . Gi,

_

+3CF

Giz

=

-2CF

G zz

=

-CF[48nzCF + (169 - 12nz)Cp - 22Nf]/36 -CF (11Cp-2Nf )/3

Gz3 G33

=

+CF[2304~(3)CF-792nzCFCp-(3197 - 132nz)CÂ +(108+ 144~cz)CFNf + (1024-24nz)CA Nf -68Nf]/108

G3a

-7Cp(11Cp - 2Np)z/108

first-order calculation . The constant term Cl is found from the first-order normalization condition R ( 3 ) = 1, which gives Using this result and those in table 1, we can predict the coefficients RZ~ for m = 2, 3, 4 from eqs. (100) . We may then compare these predictions with coefficients extracted by fitting the numerically calculated second-order terms in the thrust and heavy jet mass cross sections to eq . (99) . For the numerical calculations the Monte Carlo integration program EVENT [2,31 ], based on the matrix elements of ref. [ 32 ] , was used . The fits were performed by initially allowing all the Rzm to vary and extracting Rz4, then fixing R24 to be equal to its predicted value and varying the other RZm to extract RZ3, and so on. We used high statistics (2 x 10~ four-parton and 10~ three-parton events) binned logarithmically into 99 bins for y , 1 .5 x 10-~ . Errors were estimated by a X z test. Unfortunately in addition to statistical errors there are also systematic errors associated with the part of the second-order contribution that vanishes as y ~ 0. We have chosen, for the sake of clarity, not to estimate the latter but to assess the agreement between the resummed and fixed-order calculations in the following way. Selecting a lower limit to the fit at y = 10 -3 we reduce the upper limit, yu, until the x z for the fit falls approximately to unity and is stable under further reduction. For yu less than this, the value obtained from a fit including a generic term estimating the vanishing part of the coefficient would clearly be consistent with the value we obtain . Since we find good agreement between the predicted and fitted value in any case this could not modify our conclusion that the resummed and second-order calculations are consistent. The results of a fit to the coefficients RZm for both thrust and heavy jet mass* * In this section we consider only the heavy jet mass defined with respect to the thrust axis . As

S. Catani et al. / Resummation of large logs in e+e Fitted and calculated Fit

Thrust

31

TABLE 2

O(~S) logarithmic coefficients (Nf = 5) . Theory

Fit

Heavy jet mass Theory

RZO

+34 f 22

CZ

+40 f 20

CH

RZi

+34 f 12

Gzl + 4.21

+40 f 11

GH + 4.21

RzZ

-19 f 14

-19.75

-10 f 12

-8 .05

Rz3

-19 f 11

-20.89

-20 f 9

-20.89

Rz4

+4 f 4

3.56

+4 + 4

3.56

with yu = 5 x 10 -2 are presented in table 2 (further details can be found in ref. [33 ] ) . We see that the fixed-order and resummed calculations are consistent to second order. We may therefore read off estimates for GZ 1 . The corresponding single-logarithmic term in the exponent of the form factor ~ is a part of the subdominant correction as g3 (a s L) and as such is not generated by the NLL coherent branching algorithm. Nevertheless we can now include it in our resummed prediction with the estimated coefficient. As explained in subsect. 2.2, this corrects our prediction of the shape cross sections to the level of terms suppressed by three powers of L . Finally the second-order term in the coefficient function C (ag ) may also be estimated from the constant term Rzo, with the results shown in table 2. 7.2 . REMAINDER FUNCTIONS

The first- and second-order remainder functions Dn (y ) are found from the fixed-order predictions Rn (y ) after subtracting off the logarithmic terms as specified by eqs. (99) . From the first-order result we find Di (y) = DH(y) - CF[6y(1 + lny) + 4lnyln(1 -y) -21n 2 (1 -y) + 3(1-2y)ln(1-2y)+Zy z -4Li z ( l y y )], where L12(Z) _

°° Zn n=1

(103)

(104)

discussed in subsect. 3.2, the alternative definition in ref. [23] gives the same NLL resummed expression, but the two-loop coefficients C, GZ~ and the remainder function DZ will be different.

S. Catani et al. / Resummation of large logs in e+e -

32

o.s o.a F

v

0.3

b É

0.2

bT

0.0

0.01

0.05 1-T

0.1

Fig. 4. Thrust distribution at Q = mZ = 91 .2 GeV with renormalization scale le = Q. Dot-dashed : fixed-order prediction (98) . Dotted: resummed prediction of eq . (105).

For the second-order functions Dz (y ) and Dz (y ) we again obtain numerical values by comparison with the matrix element integration program EVENT. Putting everything together, the prediction is given by eq . ( 5 ) with C = 1 + ~sCt + ~sCz , ln~ = Lgi (asL) + gz(asL) + ~xsGziL D = ~sDl (Y ) + âsDz (Y ) ~

(105 )

The corresponding form of the differential thrust distribution at LEP energy is shown by the dotted curve in fig. 4, to be compared with the dot-dashed curve showing the fixed-order result . We have used the two-loop expression* for the running coupling as ( Qz ) as a function of the QCD scale rlMS and for definiteness we have taken CMS = 250 MeV, corresponding to as ( mZ ) = 0.120. Resummation is seen to have the expected effect of strongly suppressing the distribution at very low values of 1 - T, but it also has a significant effect at higher values, tending in fact to enhance the distribution for 1 - T > 0.04 . 7.3 . MATCHING AMBIGUITY

The procedure described so far for matching the resummed and fixed-order calculations is not unique . In order to clarify this point let us consider the usual * We use the same expression as in ref. [2], i.e . eq. (B .2) in ref. [34] .

33 S. Catani et al. l Resummation of large logs in e+e -

perturbative expansion for the shape cross section R (Y ) = 1 + ~ ~SRn (Y ) ~ n=1

(106 )

As discussed in sect . 1, the exponentiation formula ( 5 ) represents an improved perturbative expansion for R (y ) in the small y region since the large logarithmic corrections in Rn (y ) have been resummed in the form factor ~. This resummation necessarily introduces higher-order logarithmic contributions also in the large-y region, where they are no longer the dominant terms due to multijet production. Moreover these contributions are not uniquely determined if the shape cross section R (y ) is truncated at a fixed order in the improved perturbative expansion (i.e. the C and D functions at a given order in ag and In ~ at the corresponding logarithmic order) . In fact, the exponentiation theorem says nothing about non-logarithmic corrections: they may exponentiate or not exponentiate . The only rigorous statement of the theorem is that as y -~ 0 the shape cross section behaves as

(107) 1nR(Y) -1nC(cxs) + 1nE(Y,as) + . . . (y ~ 0) ,

where the dots stand for contributions which vanish order-by-order in perturbation theory . In other words, at a given order in the improved perturbative expansion one can rewrite eq . ( 5 ) in the equivalent form R(Y) = C(Y,as)~(Y~as) + D(Y,as) ,

(I08)

where the functions C and In É have expansions similar to eq . (6 ) with ydependent coefficients Cn (y), Gn , (y) such that (109) {Cn(Y)~ Gnm(Y)Î ~ lCn~ GnmÎ as y -> 0 , and that the perturbative expansion in as of eq. (108 ) reproduces (106 ) . The freedom in defining the functions C, ~, D in eq . (108 ) has been called the matching ambiguity by the ALEPH Collaboration [8 ] and it is actually an uncertainty due to higher-order contributions induced in the exponentiation formula by the resummation procedure. In this sense the matching ambiguity (the use of different matching formulae) can be used to check whether the supposed (asymptotic) convergence of the improved perturbative expansion is consistent. Note however that this freedom does not concern the logarithmic and constant coefficients Cn and Gn , . In fact, (i) any sub-dominant logarithmic term must be exponentiated in the form factor, like the contribution of G2 ,L in eq. (105 ) and (ü ) any constant term Cn âs must be factorized with respect to the form factor . In this way, the dominant behaviour of the form factor will force R (y ) to vanish for y -> 0, as must happen for the physical reasons discussed in sect . 6.

34

S. Catani et al. / Resummation of large logs in e+e -

7 .4. LOG-R MATCHING

A good example of a matching scheme which is equivalent to that defined by eq . (105 ) in the sense discussed above, but simpler to use, is the "log-R" scheme defined by recasting eq . ( 5 ) in the form 1nR(y) = K(as) + ln~(Y,~s) + H(y,as) ,

(110)

where again K (~xs ) represents a power series in as and H(y, a s ) is a remainder that vanishes as y ---> 0. Now to second order we can equally well write the fixed-order prediction (98) as 1nR(Y) _ ~sROY) + ~xs{R2(Y) - z fROY)l z } ~

(111)

To resum logarithmic terms of order cxs and beyond, we should add the NLL expression for In ~ to this after subtracting its first- and second-order parts, which are already included in R i (y ) and RZ (y ) . Hence the resummed prediction with K (cxg ) and H (y, cxg ) evaluated to second order is 1nR(Y) = LgO~sL) + g2(~xsL) +~s(ROY) - GuL_G~aL z ) (112) +~xs(Ra(Y) - ZfROY)l z - GZZL z- Ga3L 3 ), where the functions gl and gz for the thrust distribution are given in eqs . (89) and the coefficients G, are as in table 1 . The advantage of this matching scheme is that since all terms are exponentiated it is not necessary to separate out the constant and remainder parts K(cxs) and H(y,crs), or the subdominant logarithmic term ~SGzi L, explicitly : they are all implicit in the unsubtracted parts of R 1 and RZ in eq . (1 12 ) . There is thus no need to calculate or fit quantities analogous to Ci, C2, D,, DZ and GZ, as was the case in the "R matching" scheme (105) . For the same reason it is particularly simple to extend the log-R matching scheme to incorporate information from improved fixed-order calculations as it become available. For example, once the third-order correction ~s R3 (y ) is calculated for a given shape cross section, then the corresponding correction to the resummed prediction (112 ) for In R (y ) is ~s{R3(Y) - ROY)R2(Y) + 3~ROY)] 3 -Gs3L 3-

G3aL a

},

(113)

with G33 and G3a (for the thrust distribution ) as given in table 1 . The error in the prediction, due to the largest neglected subdominant term, then becomes of order as L3 , i.e . five logarithms below the leading term . Obviously, the logarithmic accuracy might be further improved as described at the end of sect . 5 . Note that the R matching and log-R matching schemes, respectively given by eqs. (105 ) and (112 ), are in a sense two extreme procedures for combining fixed-order and resummed calculations. In the log-R matching all the known perturbative contributions are exponentiated, whilst in the case of R matching

S. Catani et al. / Resummation

of large logs

in e+e -

35

one exponentiates the least. In particular, potentially large ~z terms [ 3 ] appearing in the coefficients Cl and Cz of eq. (105 ) are completely exponentiated in the log-R matching scheme (112 ) . 7 .5 . MATCHING KINEMATIC CONSTRAINTS

The matching ambiguity can be also be exploited to impose physical constraints which would not otherwise be satisfied by the resummed prediction . A particularly important constraint is the vanishing of the cross section beyond the upper kinematic boundary y = ymax, which implies that R (Y =

Ymax ) =

1 >

d-R

(Y =

Ymax ) = 0 ~

(114 )

This is not ensured in eq . (105 ) or (1 12 ) because the remainder and subdominant terms of third and higher order are not available to cancel the logarithmic terms generated by resummation . Although these logarithmic terms are not enhanced at large y, they become important because the full cross section becomes very small. Their quantitative effect can be seen from the small excess at large 1-Tinfig .4 . The constraints (114 ) are particularly simple to enforce in the log-R matching scheme (112 ) . Since the fixed-order coefficients R n (y ) satisfy the constraints order-by-order, we have only to modify the logarithmic expression 2 3 Lgi (asL) + gz(~sL) -~s(GaL + G izLz ) -~xs(GzzL + Gz3L ) (115)

so that it vanishes, together with its first derivative, at y = ymax BY definition this expression vanishes like L3 at small L, and so the required behaviour can be achieved by replacing L by L where (116 L = ln (1 /y - 1 /ymax + 1) ,

)

which is equivalent to L up to terms that vanish linearly as y -> 0. The improved prediction with log-R matching is thus 1nR(Y) = LgOasL) + Sz(~sL) + ~ s(ROY) - GuL - y2L z ) (117) +~s(Rz(Y) - 2[ROY)] z - GzzLz- Gz3L3 ) .

Things are more complicated in the R matching scheme. In this case making the replacement (116 ) in eq . (105 ), as well as in eqs. (99 ) for the remainder, ensures that 1-R, but not its derivative, vanishes at y = ymax . The non-vanishing part is of course of order as: YmaxR~(Ymax) _ - ~ s(C~G2i

+ CzGn) - ~sC2G21 ~

(118)

Recall, however, that any of the coefficients C and G1 can be modified according to eq. (109), so as to vanish at ymax . For simplicity we choose to modify

36

S. Catani et al.

/

Resummation of large logs in e+e-

only G and Gz i ,

(119) (n = 1,2) . Gnl -' Gnl(Y) _ (1 - Y~Ymax)Gn1 Equally valid options would be to require C, = Cz = 0 or Cz = Gz, = 0 at Y = Ymax~ For the choice (119 ), the coefficient C in eq . (105 ) is unchanged but the exponentiated factor becomes ln~ = Lg~ (asL) + gz(asL) - (Y~Ymax)~sG11L + (1 - Y~Ymax)~gCJz1L . (120)

Note also that the corresponding changes have to be made in eqs. ( 99,100 ) for the remainder, so that D, is no longer given by (103 ) but has an additional contribution : z D1(Y)=Dt(Y ) +G  [L - (1 - Y~Ymax)L ] +G,z(L -L z ),

and analogously for Dz . Then the final R-matched prediction is

(121)

(122) R(Y) _ (1 + âsC, + âsC2)E + âsD, (Y) + âsD2(Y) ~ In the case of shape variables like thrust, for which the kinematic limit depends on the number of particles, there is some ambiguity in the definition of ymax . Since the terms we are seeking to remove are of order as, it would seem sensible in perturbation theory to choose y max between the four- and five-parton limits . As explained in subsect. 3 .1, for thrust we have Tm;n = 0.577 for four partons. Thus we have taken ymax = 1 - Tmin = 0.43 in this case . The predictions for the thrust distribution resulting from these modifications to the log-R and R matching schemes are shown in fig. 5 by the solid and dashed curves respectively . We see that both improved schemes remove the small lowthrust tail whilst having very little other effect on the distribution . These curves represent our best current estimate of the QCD prediction at the parton level, i.e. before correction for hadronization. Note that for the thrust distribution the matching ambiguity as measured by the difference between the R and log-R schemes defined here is very small. The larger discrepancies sometimes seen in the literature [8 ] result from the use of schemes in which there are residual non-exponentiated logarithms and/or non-factorized constant terms after matching. It is of interest to study how resummation affects the apparent convergence of the predictions. In fixed order, the apparent convergence may be estimated by the change in going from first to second order, divided by the first-order result . This is indicated by the dot-dashed curve in fig. 6. For the resummed predictions, we compare the LL resummation matched to the first-order result with the full NLL resummation matched to second order. We see that in both the R and log-R matching schemes (dashed and solid curves, respectively) the apparent convergence is improved by about a factor of two relative to the fixedorder results over a wide range of thrust, including the low-thrust region . At very

S. Catani et al. l Resummation of large logs in e+e-

37

0 .5

0 .4

F v

0.3

F

0 .2

b b b

0 .0

0 .01

0 .05 1-T

0 .1

Fig. 5. Thrust distribution as in fig. 4. Solid: resummed prediction with log-R matching [eq. (117 ) ] . Dashed: resummed prediction with R matching [eq. (122) ]. For comparison we also reproduce (dotted) the "unimproved" R-matched prediction (105) from fig. 4.

0 .5

0 .0

-0 .5

-1 .0

0 .01

0 .05 1-T

0 .1

Fig. 6. Convergence of predictions for thrust distribution . fixed-order predictions at first and second order, divided by ratio with (LL terms + first order) and (NLL -1- second order) (log-R matching) . Dashed : same with R

Dot-dashed : difference between first order. Solid: corresponding in place of first and second order matching.

high thrust the resummed predictions tend to zero, while the fixed-order ones diverge, so the comparison of relative changes becomes less meaningful .

S. Catani et al. / Resummation of large logs in e+e -

38

o.so 0.50 0.40 0 .30 Fb

bb

\ Ê

0 .90

0 .20

0. 9 0 .08 0 .07 0 .08 0 .05 0 .04 0 .03 0.02 2 10_

0 .70

kz/4Z

10 2

Fig. 7 . Dependence of thrust predictions on renormalization scale . Solid : resummed prediction with log-R matching [eq . (117 ) ] . Dashed : resummed prediction with R matching [eq . (122 ) ] . Dot-dashed : fixed-order prediction (98) . 7 .6 . RENORMALIZATION SCALE DEPENDENCE

Figure 7 show the renormalization scale dependence of the resummed and fixed-order predictions for a wide range of values of the renormalization scale ratio ,uz / Qz. For values of uz different from Qz , every second-order coefficient Fz acquires an explicit ,uz dependence of the form Fz(,~ z ) = Fz(Qz ) + 2nßoln(lc z /Q z ) FOQ z ),

(123)

where F, is the corresponding first-order coefficient. We have to make this change in R z (y ) in eq . (98 ) for the fixed-order prediction, and Cz , Gzl and Dz (y ) in the resummed formula (105 ) or (122 ), where the next-to-leading exponent function gz is also modified according to eq . (8) . For the range of thrust and scale values shown in fig. 7, the resummed prediction for the thrust distribution has a slightly reduced scale dependence compared with the fixed-order result . The difference between the R and log-R matching schemes remains small for all values of the scale. An important aspect of the scale dependence issue is illustrated in fig. 8. The second-order prediction for a very small scale ~z « Qz is similar to the resummed prediction with a scale of the order of the physical scale Q z, especially if a somewhat smaller value of as is used in the former case. Thus in fig. 8 we see that the second-order expression with ,u z = 0.001 Qz and ~1MS = 100 MeV [as (mZ ) = 0.105 ] reproduces fairly well the resummed result with ~z = Qz and CMS = 250 MeV [as (mZ ) = 0 .120] . The large logarithms of ,uz /Qz introduced in this way are able to mimic the resummed dynamical logarithms

S. Catani et al. l Resummation of large logs in e+e -

39

0.5

0.4 F v b b

0.3

b

F

0.01

0.05 1-T

0.1

Fig . 8 . Comparison between resummed thrust prediction (117 ) with scale factor 1e 2 /QZ = 1 and 2 ~ms = 250 MeV (solid), and fixed-order prediction (98) with scale factor u /Q z = 0 .001 and 100 MeV (dot-dashed) . `lMS =

over a range of T. This empirical observation, and the fact that the resummed prediction agrees well with experiment, accounts for the preference for small scales in many second-order event shape analyses . Presumably this is an accident which will not remain true in higher orders, since for example the behaviour of the third-order prediction at very small scales should be very different . In any case, it is important to emphasise that, once the important dynamical logarithms have been resummed as recommended in this paper, there is no valid argument for the use of renormalization scales much differentfrom the physical scale Q2 . 8. Conclusions In this paper we have described the theoretical basis for the evaluation and resummation of leading and next-to-leading large logarithms to all orders in QCD perturbation theory, for any e+e- observable which is (a) infrared and collinear safe and (b ) dominated at small values by the twojet configuration . The tools for evaluating the logarithmic contributions to any fixed perturbatioe order are provided by the NLL coherent branching algorithm discussed in subsect. 2.1, but their resummation to all orders is actually possible in closed form only for quantities having simple kinematic properties . Among these, the first one to be resummed to NLL accuracy was the energy-energy correlation function in the back-to-back configuration [ 7,14 ] . Here we have concentrated on shape variables whose phase space in the two-jet region has factorization properties leading to the exponentiation of the logarith-

40

S. Catani et al. / Resummation

of large logs

in e+e -

mit contributions. The shape variables which have so far been fully resummed to NLL precision are the thrust [4], the heavy jet mass [5 ] (defined either with respect to the thrust axis or as originally proposed by Clavelli et al. [23] ), and the wide and total jet broadening [6] . One can also consider the sum of the heavy and light jet masses MS - MH + ML: this shape variable is related to the thrust in the twojet limit (the corresponding resummed expression being given by eq. (88) with the replacement r ~ MslQz ) but differs from it for a multijet configuration. Note that simple exponentiated expressions are not available for variables like the jet mass difference MD - MH-ML or the oblateness O [2,35 ] . The reason for this is that in the relevant kinematic limit (MD/Q 2, O -~ 0) twojet and multijet configurations contribute equally, and at present we know the QCD matrix elements to NLL accuracy in the twojet region only. Among the remaining shape variables commonly used in e+e- physics, a NLL resummation for the C-parameter [ 2,36 ] looks promising: we have checked that the leading double logarithms do exponentiate and a NLL analysis is in progress . The resummation of logarithmic corrections for jet cross sections is also feasible, provided a suitable clustering algorithm is used. In ref. [ 18 ], all the logarithmic terms aSL2" and aSL2" - ~ were resummed for any jet rate within the k1 (or Durham) algorithm. In particular, the differential twojet rate exponentiates to that accuracy and we hope to report on a complete NLL calculation elsewhere. The resummed calculations described in this paper can be matched with fixedorder results and so they have the advantage of taking into account all current knowledge of the corresponding quantity in perturbation theory . When further fixed-order calculations are performed, or improved resummation techniques are developed, they can readily be included within the framework of the exponentiation formula (6), thereby increasing the precision of the predictions and extending their range of validity . The main residual source of uncertainty in comparing QCD predictions with e+e- data is hadronization . The presence of a singularity at ßoas In (1 /y ) - z in the resummed results presented in this paper, due to the infrared behaviour of the running coupling constant, is evidence of the importance of non-perturbative effects in the twojet region . Optimistically, one may hope that hadronization will eventually be predicted from first principles using non-perturbative techniques . Alternatively, if sufficiently precise data were available over a range of energies, one might be able to parametrize the hadronization process in terms of a limited number of energy-independent functions. At present, however, we have to rely on models in order to estimate the hadronization corrections to be applied to perturbative predictions. The most successful hadronization models [ 37, 38 ] are those incorporated in parton-shower event simulation programs [ 39,40] , which are based on the assumption that hadronization is a local process in phase space, involving a rather large number of partons at a low momentum-transfer scale. Such models are not

S Catani et al. / Resummation of large logs in e+e -

41

well suited for estimating corrections to fixed-order perturbative calculations, which involve at most four partons in second order . In contrast, the resummation procedure described in this paper corresponds very closely to what is done, at a less precise level, in parton shower simulations. Therefore resummed calculations can be combined more reliably with hadronization corrections estimated from models based on parton showers, always bearing in mind the possibility that the estimated corrections could be biased by shortcomings common to all models . One of us (L .T. ) is grateful for the hospitality of LPTHE Orsay, and another (G.T. ) for that of the CERN Theory Group, while part of this work was carried out. We have benefited from discussions with S. Banerjee, S. Bethke, T. Hebbeker, R. Miquel, M. Schmelling and D.R . Ward .

[ 1] [2] [3] [4] [5] [6] [7] [8]

[9] [ 10] [ 11 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ] [16] [ 17 ]

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