A new method for combining QCD matrix elements and parton showers in Monte
Carlo simulations of hadronic final states is outlined. The aim is to provide
at least a leading-order description of all hard multi-jet configurations
together with jet fragmentation to next-to-leading logarithmic accuracy, while
avoiding the most serious problems of double counting.Comment: 6 pages, 8 figures. Talk at XXXV Rencontres de Moriond, Les Arcs,
  France, March 200

Webber, B. R.

English

arXiv

A new method for combining QCD matrix elements and parton showers in Monte Carlo simulations of hadronic final states is outlined. The aim is to provide at least a leading-order description of all hard multi-jet configurations together with jet fragmentation to next-to-leading logarithmic accuracy, while avoiding the most serious problems of double counting

Webber, B.R.

CERN Document Server

COMBINING QCD MATRIX ELEMENTS AND PARTON SHOWERS B.R. WEBBER Theory Division, CERN, 1211 Geneva 23, Switzerland, and Cavendish Laboratory, University of Cambridge, Cambridge CB3 OHE, U.K. A new method for combining QCD matrix elements and parton showers in Monte Carlo simulations of hadronic final states is outlined. The aim is to provide at least a leading-order description of all hard multi-jet configurations together with jet fragmentation to next-to­leading logarithmic accuracy, while avoiding the most serious problems of double counting. 1 Introduction The Monte Carlo simulation of multi-jet final states is a challenging problem in QCD and im­portant for new physics searches. Two extreme approaches to this problem can be formulated as follows. One can use the corresponding matrix elements with bare partons representing jets. Then one must add a model for conversion of the partons into hadrons; any realistic model will include parton showering, and hence extra jet production and potential double counting. Alternatively, one can use the parton model to generate the simplest relevant final state (e.g. e+e- -> qij) and produce addition jets by parton showering. However, this gives a poor simula­tion of configurations with several widely separated jets. For earlier work on combining these approaches see refs. 1•2•3•4•5. Here I outline a method 6 in which the matrix element and parton shower domains are separated at some value y1 of the kr (Durham) jet resolution 7 Yij = 2min{Ef,EJ}(l -cosfJ;j )/Q2 The proposed method has the following features: At Yij > Yi multi-jet cross sections and distributions are given by matrix elements modified by Sudakov form factors. At Yij < Yl they 3 are given by parton showers subjected to a 'veto' procedure, which cancels the Y1 dependence of the modified matrix elements to next-to-leading logarithmic (NLL) accuracy. Note that the procedure does not aim at a complete description of any configuration to n�xt-to-leading order (NLO) in a3, although this might be possible after subtracting NLO terms froin the Sudakov form factors (see ref. 5). The main objective is to describe all hard multi-jet configurations to leading order, i.e. O(a�-2) for n jets, together with jet fragmentation to NLL accuracy ,  while avoiding major problems of double counting and/or missed phase-space regions. 2 Modified Matrix Elements The exclusive e+e- n-jet fractions at c.m. energy Q and kr-resolution Y1 = QI/Q2 are given to NLL accuracy by 8 R2(Q1, Q) [D.q(Q1, Q)]2 R3(Q1,Q) 2 [D.q(Q1,Q)]2 {q dqrq(q,Q)D.9(Q1,q) }qi R4(Q1,Q) 2 [D.q(Q1,Q)]2 { fq dqrq(q,Q)D.9(Q1,q) fq dq'rq(q',Q)D.9(Q1,q') }qi }q, + rq dq r q(q, Q)D.g(Q1, q) rq dq' r g(q', q)D.g(Q1, q') }qi }q, + rq dqrq(q,Q)D.g(Q1,q) r dq1r,(q1)D.1(Q1,q') } }qi }q, etc., where r q,g,f are q -. qg, g -. gg and g -. qq branching probabilities and Aq,g are the quark and gluon Sudakov form factors exp (- l: dqrq(q,Q)) exp (- l: dq [rg(q, Q) + r J(q)]) with The Sudakov form factor A;(Q1, Q) represents the probability for a parton of type i to evolve from scale Q to scale Q1 without any branching (resolvable at scale Q1). Thus R2 is the probability that the produced quark and antiquark both evolve from Q to Qi without branching. More generally, the probability to evolve from Q to q ?: Q1 without branching (resolvable at scale Q1) is A;(Q1, Q)/ A;(Q1, q). In R3, the quark q (or antiquark q) evolves from Q to Qi without branching, while the q (or q) evolves from Q to q, branches, and the resulting partons evolve from q to Qi without branching. 4 Q Thus the overall NLL probability is which gives R3(Qi, Q) after integration over Q1 < q < Q. We can improve the description of 3-jet distributions throughout the region Yqg, Yqg > Yl by using the full tree-level matrix element squared Mqqg in place of the NLL branching prob­ability rq(q, Q). We first generate qij9 momentum configurations according to Mqqg, with kr­resolution cutoff Yii > Yl = QUQ2, then weight each configuration with an extra factor of [Llq(Q1,Q)]2Ll9(Q1,q) where q2 = min{yq9,yq9}Q2. For consistency we also use a8(q) in Mqqg· For four or more jets, there are several branching configurations with different colour factors. For example there is a contribution from q _..... q9 branching at scale q followed by 9 _..... 99 at scale q': The probability of this is which contributes to the term with colour factor CpCA. The product rq(q,Q)r9(q',q) is an approximation to the full matrix element squared Mqqgg in the kinematic region where y99 is smallest interparton separation. Thus it is legitimate in NLLA to replace it by Mqqgg in that region. The remaining factor of (Llq(Q1, Q)]2Ll9(Q1, q)Ll9(Q1, q') is the extra Sudakov weight to be applied. In general, the proposed procedure for generating n-parton configurations is thus as follows: • First distribute the parton momenta according to the relevant n-parton matrix element squared Mm using a fixed value a8(Q1) for the strong coupling. • Use the kr-clustering algorithm to determine the resolution values y2 = 1 > y3 > ... , > Yn > Yl at which 2; 3, . . . , n jets are resolved. These give the nodal values of Qj = Q ../fij for a tree diagram that specifies the kr-clustering sequence for that configuration. • Apply a coupling-constant weight factor of a8(q3)a8(q4) · · · a8(qn)/(a8(Q1)r-2 < l. • For each internal Jine of type i from a node at scale qi to Qk < qi, apply a Sudakov weight factor Ll;(Q1,Qj)/b..;(Q1,Qk) < l. For an external line from a node at scale qj, the weight factor is b..; ( Q1, Qj). 5 Since the weight factors are all less than unity, unweighted events can be generated by rejecting those for which the product of weights is less than a random number-. As an example, the following clustering sequence for e+e- _, qijggg has Sudakov weight A9(Q1, Q) A9(Q1, Q) A9(Q1, q3) A9(Q1, q3) Aq(Q1, q4) A9(Q1, q5) x Aq(Q1, q3)nq(Q1, q4)n9(Q1, q4) [n9(Q1, q5)]2 [nq(Q1, Q)]2n9(Q1, q3)n9(Q1, q4)n9(Q1, q5) Note that the weight factor is actually independent of the structure of the clustering tree and is the same as that for the Abelian (QED-like) graph with the same nodal scale values {qj}· The clustering of partons will sometimes be 'wrong' but this should not affect LL and NLL terms. Other clustering procedures can be envisaged 4 which should be equivalent in the dominant regions. 3 Vetoed Parton Showers Having generated multijet distributions above the resolution value y1 according to matrix ele­ments modified by form factors, it remains to generate distributions at lower values of Yij by means of parton showers. This should be done in such a way that the dominant (LL and NLL) dependence on the arbitrary parameter Yl cancels. Any residual dependence on Y1 could be useful for tuning less singular terms to obtain optimal agreement with data. Note that Y1 must set an upper limit on interparton separations Yij generated in the showers. Otherwise the exclusive jet rates at resolution Yl could be changed by showering. At first sight, this might suggest that we should evolve the showers from the scale Q1 = Qy1fh instead of Q. However, this would not lead to cancellation of dependence on log Yl . Consider, for example, the 2-jet rate at resolution Yo = Q6/Q2 < Yl· If we start from R2 at scale Q1 and then evolve from Q1 to Qo, we obtain a 2-jet rate of [Aq(Q1, Q)n9(Qo, Qi)]2 instead of the correct result R2(Qo, Q) = [Aq(Qo, Q)j2. This is because, although Yij values in the showers are limited by Y1, the angular regions in which they evolve should still correspond to scale Q rather than Q1. Consequently we should allow the showers to evolve from scale Q but veto any branching with scale q > Q1 - i.e., the selected parton branching is forbidden but that parton has its scale reset to q for subsequent branching. The 2-jet rate at any scale Qo < Q1 is now given by the sum of probabilities of 0, 1, 2, . . .  vetoed branchings (represented b y  crosses) and n o  actual resolved branchings: 6 The sum of these probabilities for the quark line is The series sums to 1/ i:i.q(Qi, Q), cancelling the Yi dependence and giving i:i.q(Qo, Q). Similarly for the antiquark line. For the 3-jet rate at scale Qo < Qi there are two possibilities: either the event is a 2-jet at scale Qi and then has one branching resolved at scale Qo, or it is a 3-jet at scale Qi and remains so at scale Qo. The probability of the first case is 2[i:i.q(Qi.Q)]2 [�:���:��r1o:1 dqfq(q,Q)i:i.g(Qo,q) while that of the second case is The sum is indeed Yi-independent and equal to R3(Qo, Q). Similarly for higher jet multiplicities. Notice that evolution after a branching at scale q > Qi starts at scale q rather than Q or Qi. In general, vetoed showers should evolve in the phase space for angular-ordered branching of each parton 9. This depends on the colour structure of the matrix element. As illustrated below, the angular region for parton i is a cone bounded by the direction of parton j (and vice-versa), where i and j are colour-connected. · j If the colour structure is not unique, colour connections must be selected according to their relative contributions to the matrix element squared, which are well-defined in the limit that 7 the number of colours Ne is large. Corrections to the large-Ne limit are normally of relative order l /N'; , so this approximation is adequate to� 103.  For high parton multiplicity, when the colour structure is not known even at large Ne, it may be possible to use the clustering scheme discussed above as a first approximation in assigning colour connections. 4 Comments/Conclusions • Modified matrix elements plus vetoed parton showers, interfaced at some value YI of the kT-resolution parameter, should provide a convenient way to describe simultaneously the hard multi-jet and jet fragmentation regions. • The matrix element modifications are coupling-constant and Sudakbv weights computed directly from the kT-clustering sequence. • Dependence on YI is cancelled to NLLA by vetoing Yij > YI in the parton showers. • This prescription avoids double-counting problems and missed phase-space regions. • In principle one needs the matrix elements Mn for Yij > YI at all values of n. In practice, if we haven::; N, then YI must be chosen large enough for Rn>N(QI, Q) to be negligible. • This approach is being implemented (with N = 5) in the e+e- event generator APACIC++10. • It should be possible to extend it to lepton-hadron and hadron-hadron collisions. • Extension to NLO along the lines of ref. 5 may also be possible. Acknowledgments It is a pleasure to acknowledge my collaborators in this project, S. Catani, F. Krauss and R. Kuhn. This work was supported in part by the UK Particle Physics and Astronomy Research Council and by the EU Fourth Framework Programme 'Training and Mobility of Researchers', Network 'Quantum Chromodynamics and the Deep Structure of Elementary Particles', contract FMRX-CT98-0194 (DG 12 - MIHT). References 8 1. M. H. Seymour, Comput. Phys. Commun. 90 (1995) 95 [hep-ph/9410414]. 2. J. Andre and T. Sjostrand, Phys. Rev. D57 (1998) 5767 [hep-ph/9708390). 3. G. Corcella and M. H. Seymour, Phys. Lett. B442 (1998) 417 [hep-ph/9809451). 4. C. Friberg and T. Sjostrand, hep-ph/9906316. 5. J. Collins, JHEP 0005 (2000) 004 [hep-ph/0001040) . 6. S. Catani, F. Krauss, R. Kuhn and B. Webber, CERN preprint in preparation. 7. Y.L. Dokshitzer, in Workshop on Jet Studies at LEP and HERA, Durham 1990, see J. Phys. Gl7 (1991) 1572 ff. 8. S .  Catani, Y. L. Dokshitzer, M. Olsson, G .  Turnock and B. R. Webber, Phys. Lett. B269 (1991) 432. 9 .  G. Marchesini and B. R. Webber, Nucl. Phys. B310 (1988) 461. 10. F. Krauss, R. Kuhn and G. Soff, Acta Phys. Polon. B30 (1999) 3875 [hep-ph/9909572); R. Kuhn, F. Krauss, B. Ivanyi and G. Soff, hep-ph/0004270. 

Combining QCD Matrix Elements and Parton Showers

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