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Jet calculus: A simple algorithm for resolving QCD jets
Nuclear Physics B157 (1979) 45-107 © North-Holland Publishing Company
JET C A L C U L U S : A SIMPLE A L G O R I T H M F O R R E S O L V I N G QCD J E T S
K. K O N I S H I
Rutherford Laboratory, Chilton, Didcot, England A. U K A W A * and G. V E N E Z I A N O
CERN, Geneva, Switzerland Received 19 March 1979
It is argued that, in the leading logarithm approximation (LLA) of asymptotically free quantum chromodynamics (QCD), a simple alogrithm (jet calculus) can be justified for computing a large number of quantities related to the longitudinal and transverse structure of quark and gluon jets. We formulate the rules of jet calculus in two variables, x andy which are related to longitudinal (energy) and transverse momentum respectively. Throughout the paper, quark and gluon jets axe contrasted. After reviewing what has been done on longitudinal spectra, we turn to the transverse structure of jets down to relative transverse momenta of the order of a few GeV (independently of the total energy). The relation of our work to that of Dokshitzer, D'Yakonov and Troyan is elucidated. We also study jets generated by (almost) real photons and compare 15oth with standard "final" parton jets. We also discuss calorimeter measured quantities, compare our results with those of other authors and propose a new and easy way for measuring QCD-predictable anomalous dimensions. We finally give a Lagrangian formulation of jet calculus, as a non-local theory in 1 + 1 dimensions to be solved in the tree approximation.
1. I n t r o d u c t i o n and s u m m a r y
This past year has witnessed an impressive renewed interest in the use o f i m p r o v e d p e r t u r b a t i o n t h e o r y (IPT) for a s y m p t o t i c a l l y free theories and, in particular, for q u a n t u m c h r o m o d y n a m i c s (QCD). IPT has been applied in a variety o f new directions further and further away f r o m the classical use [1 ] in c o n j u n c t i o n w i t h o p e r a t o r p r o d u c t expansions for lightcone d o m i n a t e d processes. The o u t c o m e has been e x t r e m e l y rewarding, in the sense that a m u c h larger set o f QCD " p r e d i c t i o n s " is n o w b e c o m i n g available for confronting QCD w i t h present and very-near-future e x p e r i m e n t a l data. The general consensus is that these new predictions are n o t less reliable than the standard ones in e+e * Present address: Department of Physics, Princeton University, Princeton, New Jersey, USA. 45
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
46
hadrons and in deep inelastic lepton-hadron scattering. They all rely, of course, on the assumption that QCD confines (i.e., gives hadrons) and that the confinement mechanism is soft enough for not affecting appreciably the short-distance physics of hard processes. There are many processes which can be dealt with in this way (e+e - -+ n hadronic jets, e+e - -+ hadron + X, e+e - -+ hi + h2 + X (h I , h 2 are hadrons in two different jets), ep ~ e + h + X, h + h ~/~+/J- + X, h + h -+ t2+/./-(pT) +jets (--PT) + X; h + h 2 large PT jets + X, etc.). They are genuinely "hard" processes in the sense that a single large scale Q2 appears in the problem and that all ratios of large invariants are O(1). In this case, at sufficiently large Q2, %(Q2) l°g(Q21/Q 2) < < 1 ,
(1.1)
7r
where %(Q2) is the QCD running coupling constant. For these hard processes, the behaviour at Q2 ~ oo is related [ 2 - 4 ] to the behaviour at m ~ 0 (m being the typical mass of a patton inside an initial or a detected final hadron *). The Q2 ~ ,~, problem is thus resolved to one of studying mass singularities (as recognised long ago by Mueller [6]) and this can be done by using general cancellation theorems [7,8] and rather powerful power-counting techniques [9,10] especially in physical (e.g., axial) QCD gauges. The most interesting outcome [1 1 - 2 2 ] of this is the "derivation", in perturbation theory, of that particular version of the parton model which incorporates gluons as well as quarks and which includes predictable scaling violations in structure and fragmentation functions. This confirms fully an initial conjecture of Politzer [1 1]. In spite of the large number of processes which can be covered by these standard techniques, the limitation to genuine hard processes, as defined above, is somewhat frustrating [23]. Take, for instance, the process h I + h 2 -~ 2 large PT jets + X. Clearly, at high energies, the region p T / g = O(1) is only a tiny fraction of total phase space; it is, moreover, the region where the cross section is smallest. Even if we cannot possibly imagine that QCD-IPT can predict this cross section at PT < 1 GeV (soft hadron physics) it is not inconceivable that it can predict it for the (semi-hard) region 1 GeV < < IPTI < < E = X/s (this should already be hard physics!). The trouble is that large logarithms o f E / p T pop out and, in general now cq(a2) log(E2/p 2) ~ 1 ,
(1.2)
/r
so that such large logarithms have to be resummed. Since these are not related to mass singularities, the previously mentioned techniques are of no help and one has to resort to other methods of resummation. * The simplest case is, of course, the one in which no such hadrons appear in the process (e+e - ~ n jets). In this case absolute computation of the process is possible, as shown first by Sterman and Weinberg [4].
K. Kolzishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
47
Summarizing, besides the "classical" (e÷e - ~ hadrons, ep ~ e + X) and the "standard" (i.e., hard processes involving one large scale) applications of IPT, we can imagine "new" applications of IPT to processes involving several large ( > > 1 GeV) mass scales. These processes cover most of phase space and involve relatively large cross sections, hence they are phenomenologically very important. Their importance is also theoretical though: understanding semi-hard hadron physics could provide the long sought bridge between hard hadron physics (where parton degrees of freedom are relevant) and soft hadron physics (where resonances, reggeons and pomerons become the right degrees of freedom). It could eventually lead to important hints as to how confinement itself might originate. A first step towards understanding these intermediate (or semi-hard) kinematical regions was made by Dokshitzer, D'Yakonov and Troyan (DDT) [14,24] who examined processes like h I + h 2 ~ £ + £ - ( M , PT) + X in the region 1 GeV < < [PTI < < M and obtained a resummation of the leading logarithms of M2/p 2. Unfortunately, they were confronted with double logs (per power of as) and they could only obtain a simple result (DDT formula [14,24]) in the limit
as(p2) log(M2/p~)<< 1 .
(1.3)
7r
This puts severe limitations on the region of applicability of the DDT formula (for further discussion of the DDT case and its relation to the present work see sect. 5). The work described in this paper also goes in the direction of semi-hard processes. It deals with questions of multihadron spectra inside a single (quark or gluon initiated) jet. Longitudinal as well as transverse spectra are considered with relative transverse momenta chosen to be larger than a few GeV. Both cases have already been discussed by us in two short notes [25,26]. As explained in sect. 5, we obtain, unlike DDT, single logarithms (.per power of as) and, consequently, the applicability of our results is only limited by the constraint ~ ( p 2 ) < < 1. lr
(1.4)
This excludes the low-p T region, of course. On the other hand we are also limited to values ofpT < < Ejet by the fact that we use collinear kinematics throughout. Fortunately, as one goes to PT <~E, the process becomes genuinely hard and can be dealt with by standard lowest-order calculations. In the cases under consideration we find that our results for the semi-hard regions connect smoothly with those derived by other authors [27,28] in the hard region. Hence, the whole region 1 GeV < < PT d E is under control. It can be shown further that only an infinitesimal (as E ~ ~ ) . fraction of the cross section belongs to either the non-perturbative region PT <~ 1 GeV or to the hard region PT ~ O(E). The intermediate region described by our formalism essentially picks up the whole cross section. The outline of the paper is as follows: in sect. 2, we first review for completeness the diagramatic, perturbative approach to structure and fragmentation functions and
48
K. Konishi et aL / Jet calculus: a simple algorithm for resolving QCD lets
the Altarelli-Parisi (AP) evolution equations [29] appropriate for describing their Q= dependence in the LLA. We then introduce immediately our jet calculus rules (JCR), on the basis of the physical diagramatic picture for jets obtained by the perturbative calculations. The rules are given directly in terms of "physical" variables x and y (related to energy and transverse momentum respectively) and the equivalence to the previous formulation [25] in terms of moments is proven. A more rigorous derivation of the JCR is given in appendix A. Sect. 3 discusses several non-trivial consistency checks (sum rules, check with a result of Taylor [30] in ?~4~3, etc.). Details of some proofs are given in appendix B. In sect. 4 we discuss purely longitudinal spectra in quark and gluon jets, distinguishing the finite x from the small x region (where some subtle infrared problems still limit predictivity). A subsection is devoted to (quasireal) photon-initiated jets where results similar to some older ones of Witten [31 ] are obtained. The question of universality (i.e., independence of several quantities of the type of final partons) is briefly discussed in the text, with details presented in appendix C. Sect. 5 turns to complete (longitudinal and transverse) spectra. After a qualitative comparison with the DDT work [14,24] we extract some simple inclusiVe cross sections (essentially longitudinal moments) from our general formulae (whose computation can be done in principle by computer), and we describe the general features of the results, comparing them with the naive predictions obtained in lowest order. We also discuss calorimeter-measured cross sections and rederive from our general formulae the particular results obtained by other authors [32,33]. We then describe a new scaling behaviour for integrated cross sections and show how that allows a new way to measure the anomalous dimensions predicted by QCD. We close sect. 5 by presenting results for "initial-parton jets" and by contrasting them with "finalparton" ordinary jets. Sect. 6 describes a general formulation of jet calculus as a two-dimensional field theory, whose non-local Lagrangian is given, This is followed by some short conclusions.
2. General formulation of the jet calculus rules 2.1. Structure functions, fragmentation functions and their evolution
For completeness, and also in order to introduce our notations, we start by recalling some well-known results on structure and fragmentation functions as they come out of perturbative QCD calculations in the leading log Q2 approximation (LLA) [11-22]. Although these results have been generalized to include all non-leading logs [15,16,21], for the extensions considered in this paper we shall restrict ourselves to the LLA. Because of the property of factorization and process independence found at all
K. Konishiet al. / Jet calculus."a simplealgorithmfor resolvingQCDjets
49
orders, one can define a structure function G~(x, Q2) (a fragmentation function gives the probability of finding a parton of type a in the hadron h (a hadron h in the fragmentation of a parton of type a) with a fraction x of the parent momentum in a hard process characterized by one large scale, Q2. The actual physical cross section for the hard process under consideration is given in terms of G and D functions by a convolution of the type:
Dan(X, Q2)) which
1
°(Pi;qj;Q2)= ~fai oj ~i dxiG~i(xi' L. 0 • t h. X I-I dzj ~-Dg](zj,
Q2)
Q2)
oo(ai(xiPi); bj(qi/z/);
Q2),
(2.1)
where Pi(qj) refer to incoming (outgoing) hadrons, ai and bi are parton species, and where Oo is the lowest-order cross section at the parton level *. Since the process is hard, by definition o 0 is free of infrared problems and can be computed perturbatively in the running coupling constant. The Q2 dependence of Gh, a D ah can be obtained from that of more elementary parton-to-parton structure and fragmentation functions, through the relations
b
Dh(x,
x
102
Q2):
h
~ f __sDb (x/Z)Dba(Z ' Q2), b
(2.2)
x
where Gbh(x/z), D~ (x/z) refer to structure or fragmentation functions at a fixed reference value of Q2 (say Q2 = Qg > 1 GeV). The parton level functions Gab, Dba (and therefore also Gbn and Dan) satisfy the Altarelli-Parisi evolution equations [29]: 1
~yGab(X, ]I)= ~ f ~Pac(g) Gcb(X/Z' Y), C
d-~dyDba(X, Y)= ~
C
x
f
~Dbc(X/Z, Y)Pca(Z),
(2.3)
x
where Pba(X) are the Altarelli-Parisi elementary parton decay probability functions for the process a ~ b(x) + c(1 - x), which are given in table 1. The "evolution"vari* The hard process may also involve currents and/or hard photons. These will appear in a 0 but, of course, there is no structure or fragmentation function associated with them.
K. Konishi et al. /Jet calculus." a simple algorithm for resolving QCD jets
50
Table 1 Patton decay probability functions b
a
q
q
P(a --* b(x) + c(1 - x)) ~- Pba(X) /1 +x2 1 - -
CF
g
q
1 + (1 - x ) 2
CF g q
g g
1N f (x 2 + (1 - x ) 2 }
g
g
2CA
+
+x(1 - x) - l ~ ( x - 1)
- I N f s ( x - 1) (In comparing with ref. [29 ], notice our different use of the ( )+ symbol.) C F = ( N c2 - I)/2N c ,
CA=N c, 1
= f
ax
-
.
o
able Y ( r a p i d i t y , i m a g i n a r y t i m e ) replacing Q2 is given by y:
i
F~stU~)7
-2~b logL@ ] ,
(2.4)
where e s ( Q 2) = (b l o g ( Q 2 / A 2 ) -1 is the o n e - l o o p QCD r u n n i n g c o u p l i n g c o n s t a n t (12gb = 11N c 2Nf in QCD w i t h N c colours a n d G o i n g over to m o m e n t s b y the d e f i n i t i o n s "
Nf flavours).
1
a~(n, r) - f dx x"a~(x, r'), o 1
Oha (n, Y ) - f
dx xnDha(X, Y ) ,
(2.5)
o
* Our definitions differ slightly from the conventional ones where a weight x n-I is used. As seen below, our rules are a bit simpler with this definition.
K. Konishi et al. /Jet calculus. a Simple algorithm for resolving QCD /ets
51"
Table 2 The anomalous matrices A n for the flavour singlet component
An
\ Ag q
b
Agng ] A nb a
a
n+l q
q
CF[ - 1 + 1/{(n + 1) (n + 2) } - 2 ~ (l/j)] /=2
g
q
CF(n 2 + 3n + 4)/(n(n + 1) (n + 2) }
q
g
Nf (n 2 + 3n + 4)/{(n + 1) (n + 2) (n + 3) }
g
g
n+l CA [_1 + 2/(n(n + 1) } + 2/{(n + 2) (n + 3) } - 2 ~ (1/])] - INf i=2 Cv=(Nc2 - 1)/2Nc, C A = N c.
one finds immediately the well-known [1] behaviour:
G~(n, Y ) = e x p ( ( Y - Y o ) A n ) a b G b h ( n , YO), nba(n, Y) = D h (n, Yo) e x p ( ( Y - Yo)An)ba ,
(2.6)
where 1
ab_ An - . f dz znpab(Z). 0
(2.7)
For convenience a table o f A ab is given in table 2. The basic aim of this paper is to show that a number o f new and interesting measurable quantities pertaining to jet physics can be obtained, in the LLA, from a simple perturbative algorithm ("jet calculus") and from the phenomenological input provided by the G~(x) and Dha(x) functions at a reference point appearing in the r.h.s, of eqs. (2.2). Occasionally, we shall even be able to obtain results which are independent of such phenomenological inputs.
2.2. Jet calculus rules We give here an intuitive derivation o f our algorithm based on the diagramatic understanding [.12-20] of the origin of eqs. (2.3) in QCD perturbation theory in the axial gauge. Further details and a more rigorous derivation can be found in appendix A.
52
K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD jets 2
(a)
Dbo -- )- )-TREE, CHOICE OF b
2 =)--
a
b
(b)
I
-~ ~
a =~I-I
r
~
(c)
b I
oo •
I I
A
=
a
-
," b
(d)
Fig. I. (a) Jets as branching processes in the LLA. (b) Choosing one final parton. (c) Reconstructing a dressed ladder. (d) Evolution a ~ b ~ la Altarelli-Parisi.
The diagrams describing, for instance, the y-evolution of a fragmentation function Dba(X, y ) are shown in fig. lc. The initial parton a, which is typically far from its mass-shell (p] ~< Q2), decays into two lower mass partons, which in turn decay into even lower mass quanta until, eventually, one ends up with a set of final partons of mass comparable to a typical hadronic mass. This first, perturbative stage is followed by a second non-perturbative stage in which patrons convert into hadrons and which has to be described in terms of phenomenological (uncalculable) quantities like the factor of eq. (2.2). Conversely, the first stage can be understood perturbatively as we shall now explain. The process shown in fig. 1 is a typical branching process of the type encountered in many fields of applied mathematics [34]. The possible relevance of bran-
Dhb(X/Z)
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
53
I al )-
J
I
bI
,._,~~
(a)
I
~,
b1~ (b) a 1
02 Fig. 2. (a) Choosing two final partons in fig. la and reconstructing a triple ladder. (b) Evolution i ~ a 1 + a 2 described by eq. (2.8).
ching processes in the physics of jets had been already pointed out in the literature [35,36] * The particular structure of this branching process in the LLA of QCD is one in which masses are strongly ordered along the branches of the tree, so that each parent parton is always m u c h further off-shell than its offspring (i.e., if a -->b + c, q~, q2c <~ eq2a, with e fixed and < < 1 ) . The other important property in the LLA and in the axial gauge is that no interference term is allowed when squaring amplitudes (as already indicated in fig. 1) so that our trees have a direct probabilistic interpretation. In order to obtain eqs. (2.3), (2.6) from the above picture, it is enough to realize that, for every choice of b in the final set of partons, we can group the other final partons in a unique way into groups forming the dressed rungs of a ladder, whose discontinuity is taken (see figs. l b , c). It is these ladder diagrams which have been shown [ 1 3 - 2 0 ] to exponentiate in Y and to give the result (2.3), (2.6). By repeating this line of argument in the case o f two detected partons b, c, one can easily show that all contributions can be represented by the triple-ladder structure o f fig. 2a. Since each ladder provides an AP evolution one obtains the * The concept of branching processes is also implicit in the one of the frayed jets or jets as fractals, often emphasized by R.P. Feynman (private communication).
K. Konishi et al. / Jet calculus."a simple algorithmfor resolvingQCDlets
54
picture shown in fig. 2b which, in formulae, reads I
do(i~ala 2 +X) dxldx 2 --Dala2'i(Xl' X2' Q2)
Oi jet
Y
1
f
dy f
bl'b2'J 0
0
dx dg dwldW2Oalbl(Wl,y) Oa2b2(w2, Y)
^
XPJ~blb2(Z) Dji(X, Y'- y) Y
l - - x 2 dz 1
bl'b2'J 0
x1
XDa I bl(Xl/Zt,
t~(X 1 --
XZW1) (~(X 2
/~ - x l (tz 2 x2
-- X ( I -- Z) W2)
1
Z Zl+Z
Y) Dazbz(X2/z2, Y)
XlSj-~blb2(Zl/(Zl + z2))Dji(Zl +z2, Y - y ) ,
(2.8)
where the "intermediate" rapidity y will be given a physical interpretation in sect. 5 and the meaning of the remaining integration variables as momentum fractions is self-explanatory. Furthermore, the quantity Pj~b lbZ(z) coincides with Pb lJ (z) or Pb2j(1 - z) provided we drop their terms proportional to 6-functions. This corresponds physically to requiring that the vertex j ~ b lb2 in fig. 2b describes a real (and not a virtual) process. It then follows that P j ~ b 1 b2(Z ) = Pj--~b2bl (1 - z ) ,
(2.9)
which ensures a symmetric treatment of a I and a 2 in eq. (2.8). Notice that, whereas the fragmentation functions Dal bl (Wl, y), Da2b2(W2, y ) are just the same as those of eq. (2.3) (with Y ~ y ) , a new generalized fragmentation function Dji (x, Y - y ) appears, corresponding to an evolution from y to Y, which is completely calculable if both y and Y correspond to large values of Q2. When, later on, we consider final hadrons it will be only the D(w, y) functions which will be converted into hadroniclevel fragmentation functions, whereas D(x, Y - y)will still remain an evolution at the partonic level. It is a simple matter to transform eq. (2.8) into an equation for double moments and to make thus contact with our previous formulation. Defining Dala2,i(nl,
n n2, Y) - f dxldx2x nIl xz2Dala2,i(Xl, x2,
Y),
(2.10)
one immediately obtains: Y
Dala2,i(n 1, n 2,
=
Y) f 0
b2'i dy (eYAnl)albl(eYAn2)azb2 P bl nl,n 2 (e (Y-y)An I +n2 )ji (2.11)
55
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets Table 3 • be a The vernces Pm, n
pbc,a
b
c
,
a
-m, n
q
g
,
q
CF(B(m + 1, n) +B(m + 3, n ) )
q
q
,
g
Nf(B(rn+ 3, n+ l) +B(m + 1, n + 3))
g
g
,
g
2CA(B(m, n + 2) +B(m + 2, n) +B(m + 2, n + 2))
B(m,n) is the Euler's beta function.
where the A n are as given before (eq. (2.7), table 2) and 1
P blb2'jnl,n 2 :- : dz znl(1 - z)n2/OJ~bl b2(Z ). 0
(2.12)
• The various Pnb I b-2 , J can be found for convenience in table 3. Going finally through 1,u2 a Laplace transform to the variable E we find
d y e - YEDa I a 2 ,i(y~ n 1,n2"- :
Dal a~'i(E) = Ul,rt 2 o (ala2
E-
1
A n l - An2 PhI'n2 E -
1
Anl+n2
i>,
(2.13)
where we have gone over to a vector-matrix notation. Eq. (2.13) coincides with the proposal of ref. [25]. Clearly all the reasoning can be generalized to the case o f an arbitrary n-parton inclusive spectrum both in the (xi, Y ) description of eq. (2.8) and in the (ni, E ) description o f eq. (2.13). For completeness we restate here the general rules for the moments, D (nl' n2"'" n.m; E). These are as follows. ala 2 ... am,l (i) Consider all the tree diagrams for an effective ~3 theory, in which energy flows from the initial single particle i to the final set (a/, n / ) (see fig. 3a). (ii) For each tree write the contribution as in old-fashioned perturbation theory as a product of propagators and vertices and sum over all inequivalent trees and time orderings. There is conservation of moment at each vertex so that all internal lines have a well-defined moment in terms of the set ( n / ) (see again, for instance, the tree o f fig. 3a). (iii) The rules for propagators and vertices, depicted in figs. 3b and 3c are as follows: (a) For a p-particle intermediate state, write a factor ( E - Eint) -1 where gin t =
56
K. Konishiet al. /Jet calculus:a simplealgorithmfor resolvingQCDjets , bl j ~ ' C t l ' n l 5 F,,ni -
n1* n ~
k
~
.~
~z ~ °2'n2 .... n3 .n4U~l,/~'°3 ,n 3
b I n I 01
bn
bp-;
o.
bk%
° -"X
k=l
no C n'
(b)
(c)
Fig. 3. (a) Example of a contribution to a five-parton spectrum. (b) and (c): Feynman rules for intermediate states (b) and vertices (c).
~Pk=lAnk and An is given in eq. (2.7). Each Ank acts independently on the indices of the kth line. (b) For a vertex corresponding to a(p) ~ b(n 1) + c(n2), P = n I + n2, write a factor Pnl,nu which is given in eq. (2.12). The corresponding rules in (x i, y) variables are obvious *. We end this section by discussing how to incorporate hadrons in the scheme. To be definite, consider first the case of a two-particle inclusive cross section 1 do(i~h I +h 2+x) Opl,hz(x 1 ,x2, 0 2) - - Ojet dx 1dx2
(2.14)
Assuming the non-perturbative hadronization process to be soft (i.e., to involve large time scales) and not to interfere with the hard (i.e., short time scale) process described above, we can argue that, in analogy with eq. (2.2) for D~x, Q2), Dihl,h2 (Xl, X2, Q2) will be given in terms of the parton level quantities Dala2,i , Da, i by folding them into some phenomenological Q2.independent fragmentation functions for a parton to go (inclusively) into one or two hadrons, to read:
fdXal dxa2 Dhl~X11 Dh2 X(-~a2) al,a2J Xa1 Xa2 "2\Xal/
ohl'h2(xl, X2"02) = ~
XOala2'i(xal' Xa2;02)'1" ~af ~ a
dxa
h ,L I x 1 X 2
oal n21~'X-aa) oa'i(Xa' 02). (2.15)
* Actually, it is possible to use Feynman-type rules where energy is conserved at each vertex and one integrates over all final energies.
K. Konishi et al. /Jet calculus."a simple algorithm for resolving QCDlets (
~
hl h2
-
~
)
h1
=
_
%
)
h2
*
:
hl
C) _
~ h3
=
I~
-_
•
- ~ - ~
*
,
hl
h2
~
(o)
h1 h2
)
"*
."
57
)
h3
:-
hI
:
h2
)
h3
h1 h2 h3
PERM
* PERM
(b)
Fig. 4. (a) 2-hadron spectrum from 1-parton and 2-parton spectra. (b) 3-hadron spectrum from 1-parton, 2-parton and 3-parton spectra.
Eq. (2.15) is illustrated in fig. 4a, where the new phenomenological fragmentation function Dahl'h2(z l, z2) appearing on the r.h.s, is constrained by energy conservation to obey the equation
f dz2z2Dhal'h2(zi, z 2 ) = (1 - - z l ) D h a l ( Z l ) ,
(2.16)
h2
which we shall use later on. At first sight, eq. (2.15) does not look useful since it introduces a new phenomenological object. It is clear however that the second term only contributes if (pill + Ph2 )2 < A 2 ( 2 1 GeV2), whereas for 07hl + ph2) 2 > > A 2 only the first term should survive. In sect. 5 we shall consider the case of inclusive spectra at fixed PT (or at fixed invariant mass) and we shall then exploit the fact that, in the perturbative region, no new phenomenological input is needed besides that of Dab(z). Tire generalization of the above prescription to multihadron spectra is straightforward. In fig. 4b we give the explicit case of a three-hadron spectrum. Again, if all final momenta are such that Pi" Pj >> A2, the only surviving contribution will be the first one.
K. Konishi et al. /Jet calculus."a simple algorithm for resolving QCD ]ets
58
3. Consistency checks In this section several consistency checks of our rules are presented. These represent quite non-trivial tests of the hypothesis underlying our "derivation" of the jet calculus rules. They should also help the reader to become familiar with our algorithm.
3.1. Rederivation o f a result o f J. C Taylor in ¢36 Six-dimensional ~.~b3 theory is a useful laboratory for investigating consequences of asymptotic freedom in renormalizable field theories. Defining the interaction term in the Lagrangian as - ( 1 / 3 ! ) X¢3 one finds for the one-loop running coupling constant:
~-(Q2///2) =
au
_ X2
1 + 4aau log(Q2/p2) '
au
(470 3 .
(3.1)
This theory has no soft infrared divergence since
1 1 d6k k2 k ' P l k ' p 2
o
(oo,
(3.2)
i.e., does not diverge from the small Ikl integration. On the other hand the theory has collinear divergences, since
fdSk
1
ko (P' k) 2
03dOlklad[kl ...,fd_f Ikl Ikl f lpl21kl2(1 - cos 0) 2 d Ip-~Ip---[
(3.3)
diverges as log(Q2/m 2) for m -~ 0. It should also be noted that, as in axial-gauge QCD, only planar ladder diagrams give leading divergences since
Sk 1 1 ~ 0 (Pl k~-)~---2"k)
fd
(3.4)
i f p l and P2 are not parallel. The detailed analysis of the AP evolution equations for this theory lead again to eqs. (2.3) with
y= 4 lo_[a(m21u2)'~ 3 gt~(Q21ta2)},
P(z) = z(1 - z) - 1 6 ( 1 - z ) .
(3.5)
The rules of jet calculus for tp6a are also the same as for QCD, where from eq. (3.5), one finds 1
An=(n+2)(n+3)
1
Pn rn
1-2'
'
=
F(n + 2) F(m + 2) P(n+m+4)
(3.6)
K. Konishi et al. / Jet calculus."a simple algorithm for resolvingQCDlets
59
For example the average multiplicity of a ¢36 jet will be
(n) = D(n = 0, Y) = exp(A o Y) = [a(m21"2)]'/9 ~a(Q2/u2) ] '
(3.7)
as found by Taylor [30]. One can actually reconstruct the full multiplicity generating function: oo
G(u, Y)= ~ m=1
oo
Um
ore(Y) _ ~ Otot
(u - 1) k o k ( y ) + 1 ==-1 +F(u, Y ) ,
k= l
(3.8)
k !
where pk(Y) is the integrated k-particle inclusive cross section, that is, pk(r)
(3.9)
= D ( n l = n 2 = ... n k = 0 , Y ) .
In order to compute G(u, Y) it is then enough to evaluate the sum of all trees with zero moment in every leg and with each final particle multiplied with (u - l) - v. Because of a simple recurrence relation for trees one finds Y
F(V,
1 Y)=veAor+~Po,o f dyeAo(Y-y) F2(o, y)
(3.10)
o
which provides the differential equation d -~y F(V, Y) = A o F(v, Y) + I po,oF2(v, Y) ,
F(o, O) = v ,
A o = ½Po,o = 1"21
(3.11)
"
We recognise in eq. (3.11) the classical field equation for q~3 theory, which is the expected result since we are summing tree diagrams *. The solution of eq. (3.11) is simply (p -= 2Ao/Po,o):
G(v, Y) = 1 + F(v, Y ) -
(v + p) + v(p - 1) exp(A o Y)
(v+p)-vexp(AoY)
= p=l o=u-i
u
u+(1-u)exp(AoY)' (3.12)
which coincides with Taylor's result (ref. [30], eq. (6)). A couple of comments are in order about eqs. (3.11) and (3.12). First of all, writing the differential equation (3.11) is just one of the standard ways [38] of solving the population-growth problem of a single species by treating it as a Markov (branching) process. In the case at hand, A o and ½Po,o are the elementary probabilities for growth and birth, respectively. * A v e r y similar r e g g e o n c a l c u l u s p r o b l e m h a d b e e n solved b y S c h w i m m e r [37] b y t h i s same technique.
K. Konishi et al. / Jet calculus: a simple algorithmfor resolving QCD/ets
60
The other way to attack the same problem would be to write instead of (3.11) the equivalent linear partial differential equation
OY
F(v, Y)=(Aov+ ½Po,oV2) ~--~F(v, Y),
(3.13)
whose solution is again eq. (3.12). Second, the resulting distribution (3.12) is the one of an ideal Bose gas, for which
On/Ot°t
-
1
(n) + 1
(1
1
t~
(n) + 1
'
(n) = exp(Ao Y ) - 1 .
(3.14)
Since, for an ideal Base gas, (n> = (e cur - 1) -1 one finds that a ~ jet at large Q2/m2 is equivalent to an ideal Bose gas with k T ~ exp(A o Y) = (log(Q 2/m 2 )) 1/9 . E
(3.15)
Also notice that, in this theory On~01 Q e / m 2 ~ ~ 1 ,
ol Q2/m 2 --> ~> 0 ,
(3.16)
and still Y~nOn~ constant. Thus q~3 is an example of theory in which there is no enhancement due to bremsstrahlung and yet the cancellation theorem works. It would be quite interesting to know whether this is also the mechanism for QCD or not (see sect. 4 for a discussion of this point).
3.2. Momentum sum rules in QCD In a model for jets like ours in which inclusive cross sections are obtained as Mueller-like discontinuities, these constraints are equivalent to checks of unitarity, i.e., to the statement that our inclusive spectra can be indeed written as sum over exclusive processes. We start by a simple example of momentum sum rule at the parton level worked out directly in terms of the x, y variables in subsect. 2.2. The sum rule relates the two-parton and the single-parton spectra by the relation:
~ ;dXlX l Dala2,i(xl, x2, Y ) = ( 1 - x 2 ) D a 2 , i ( x 2 , Y) . al-Using eq. (2.8) the 1.h.s. can be easily reduced to: Y
[ x2 y) fo dy ~al f dx1xIDalbl(Xl' y) fdxdz "1-zz D a2b2[x(1-----z)'
(3.17)
K. Konishi et al. I Jet calculus." a simple algorithm for resolving QCD jets XPj~bl b2(Z) D~(x, Y - y ) .
61 (3.18)
By m o m e n t u m conservation at the single-particle spectrum level we know that:
f d X l X 1 Dalbl(Xl, y ) = 1 , any bl, y • (3.19) al Furthermore the sum over bl and the factor z in the integral allow the replacement
bl
P.i"+bl b2 (z) = Pb2j( 1 -- z ) .
(3.20)
The result is that the 1.h.s. of eq. (3.17)becomes:
fo of
dy
a
Da2b2 X(-1 - - z ) ' y
-
Ldy
\x
Pb2j(1 - z ) D j i ( x ' Y - Y )
'
+SdxSdyIDalj(~,Y) ~-~-~DIi(X, Y-Y)1 :fox f dyTy-y[Da2jt: Z
Y-
,
o where the AP equation (2.3) for D has been used twice. The integral o v e r y can now be done trivially and, using Dab(Z , y)ly=o = 8abS(Z -- 1),
(3.22)
one immediately verifies the validity of eq. (3.17). To prove eq. (3.17) at the hadronic level (i.e., with al, 2 -+ hl,2) is not as trivial. One obtains eq. (3.21), but then eq. (3.22) cannot be used if a is a hadron. Indeed, in the hadronic case, we have to add a second term (see eq. (2.15) and fig. 4a) in order to obtain the full two-hadron spectrum. Adding such a term and using eq. (2.16) it is rather easy to see that, eq. (3.17) is again recovered. The general form of m o m e n t u m conservation is a generalization of eq. (3.17) and reads m
f d x 1 x 1 Oala2...am,i(Xl, x2 "'Xm, Y ) = ( 1 - ~ a1
xi)
j= 2
XOa2...ara,i(x2 ... Xra, Y )
(3.23)
62
K. Konishi et aL /Jet calculus. a simple algorithm for resolving QCD jets
It is easiest to check this equation by going over to moments, for which eq. (3.23) becomes ~-J Dala2...am,i(1,n2 ... nm, Y ) = D a 2 . . . a m , i ( n 2 ... nm, Y ) al n -
~
]=2
(3.24)
Da2...am,i(n2 ... nj + I ... nm, Y )
The explicit proof of eq. (3.24) is given in appendix B(i) for the hadronic case (which contains, as an easy by product, the parton-level sum rule) using mathematical induction. Since proof by induction turns out to be very common in our framework we go into some details there explaining how they work out in this particular case. Notice again that, in order to satisfy momentum sum rules at the hadronic level, a whole set of new phenomenological quantities has to be introduced giving the decay function (at fixed Q2 = Q~) of a patton into m hadrons. 3.3. Flavour-conservation sum rules
There are sum rules expressing the fact that the flavour of the initial quark must be carried also by the jet. These sum rules, when written in terms of moments, take the form: Dala2...ami,i(r/l, n2 ... nm, 0; Y) m
=Dala2...am,i(nl, n2 ... nm, Y ) (1 - ~
j=l
6i,aj).
(3.25)
In this equation the final parton i on the 1.h.s. is supposed to represent the "valence" quark in the jet (i.e., the final quark whose line is connected with that of the initial quark). The meaning of 6i,aj on the r.h.s, which ensures that none of the final partons a I a 2 a m is the valence quark is similar. The most trivial case of eq. (3.24), m = 0, is . . .
Di,i(0, Y) = 1 ,
(3.26)
which follows from the fact that AoNs = 0, where NS stands for non-singlet. Eq. (3.26) expresses the fact that integrating the spectrum of the valence quark gives the total cross section (with no multiplicity factor). The general proof of eq. (3.25) by induction can be found in appendix B(ii).
4. Longitudinal spectra of QCD jets We now turn to the applications of our formalism to quark and gluon jets. This section will be devoted to longitudinal spectra (i.e., spectra in the momentum frac-
63
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets
tions along the jet axis, transverse momenta being integrated over). We find it convenient to split our discussion into two parts, finite x and very small x, also because, as we shall see, our predictions will not be as reliable in the second case. 4.1. Spectra at finite x 4.1. l. Single-particle spectra *. We start by considering the total m o m e n t u m fractions carried by quarks or gluons (Xq, Xig) in the two types (i = q, g) of jets. These quantities are just given by the 1st moment of the single-particle spectra:
x ia = Da, i(1, Y) = (alexp(A 1 Y)Ii) •
(4.1)
The explicit diagonalization o f A 1 then gives xq
=
3Nf P
+
16 e_(o/9)v P
Xgq = -16 - ( 1 - e_(O/9)y ) P xg = 3Nf (1 - e - ( ° / 9 ) Y ) , P xg
16
= __
0
+
3Nf
e-(,o/9)
Y ,
P
p = 16 + 3Nf.
(4.2)
We see that, asymptotically, Xq/X~ -- 3Nf/16 independently of i (the type of jet). On the other hand, at moderate values of Q2, the non-leading term is not negligible and one obtains the results shown in fig. 5. Nevertheless quarks and gluons share the total m o m e n t u m in roughly the same way in the two types of jets. We shall use this result later when discussing multiplicities. Let us now look at single-particle spectra near x = 1. These can be obtained through the study of large moments (which clearly emphasize x ~ !)- One obtains behaviours of the type: Oa, i(x; Y) ~ Ca,i(x, Y) (1 - x ) 2CvY+ba,i , x'-*l
(4.3)
where Ca, i depend logarithmically upon (1 - x) and in a complicated but known
way upon Y, whereas ha, i are just some integer numbers. Table 4 gives the values of Ca, i; ba, i for all cases. Notice that the ratio of quark to gluon spectra near x = 1 is independent of the * The results of this subsection partially overlap with those of Dokshitzer et al. [14,22], Cabibbo and Petronzio [39] and others.
64
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets 1.0
08
~x
0.6
04
0.2 q (valence)
/ ,
•
,
0.2
,
,
0.4
,
,
~
,
0.6
Y
1.0
08 g 06
0.4
0.2
I
I
I
0.2
I
0.4
I
I
I
I
0.6
Y
Fig. 5. Total momentum fractions carried by quarks and gluons in (a) quark and (b) gluon jets.
type of jet considered and is given by Dg, i(x; Y)/Dq,i(x; Y )
(1 - x ) ~1
4(CA - CF) Ylog(1/(1 - x ) )
'
i = q,g,
(4.4)
indicating that final gluons are softer than final quarks in each type of jet. Further-
K. Konishi et al. / Jet calculus."a simple algorithm for resolving QCD lets
65
Table 4 The behaviour n e a r x = 1 o f D a , i(x; Y) Da,i(x; Y)
~ C(1 1-x << 1
x) 2CFY+b
i
a
b
C
q
qvalence
- 1
A(Y)
q
g
0
A(Y)/(4(CA
q
qsea
1
NfA(Y)/(4(CA
g
q
0
NfA(Y)/(4CF(CA
g
g
1
NfA(Y)/(8(CA
- CF) Y log(1 - x ) - 1 ) -
CF)Y(2CFY + 1)
log(1 - x ) - 1 )
- CF) g log(1 - x ) - 1 )
- CF) 2
Y(2CFY+ 1)
(log(1 - x ) - l ) 2 )
A(Y)
= exp ((3/2 - 23') CFY)/F(2CFY), 3' = Euler's const (= 0 . 5 7 7 2 . . . ) .
more, if we take the ratio of quark spectra in the two types of jets we get: aq,g(X;
r)/Dq, q(X; Y)
(1 - x ) i f x-,1 4CF(CA -- CF) Ylog(1/(1 - x ) )
'
(4.5)
and the same ratio is also found for final gluons. Eq. (4.5) and its analog for final gluons are the first indication that gluon jets are softer than quark jets, a prediction which will be reinforced by our subsequent results. Finally, the property that the ratio given in eq. (4.5) does not depend upon the final parton will be a property common to many other quantities and is indeed a special case of a general "universality" feature of parton spectra discussed in appendix C.
4.1.2. Two-parton spectra. We now look at the two-parton spectrum Dab,i(Xl,X2; Y) in the region x 1, x2 finite. A few limits are particularly simple and interesting. We start with the case Xl + x2 ~ 1 by looking into large moments: f d x l d x 2 ( x l + x 2 ) n XlX2 Dab,i(Xl, x2; Y).
(4.6)
The result is a behaviour of the type: D a b , i ( X 1, X 2 ;
Y)
~ Aab,i(1 --x 1 x 1+x2--*1 Xl,X2~eO
--X2)2CFY+6ab,i ,
(4.7)
where A b a , i have at most a logarithmic dependence on (1 - x l - x2). The exponents ~ab,i are some integer numbers given in table 5. They show again that quark jets have an easier time producing a fast pair of partons than gluon jets; something that could be expected on the basis of single-particle results. It is also clear that in
K. Konishi et aL / Jet calculus. a simple algorithm for resolving QCD jets
66 Table 5
Dab,i(XlX2; Y)
A(1
~
- X 1 -- x2) 2CFY+fab,i
Xl+Xl-*l (x 1, x2 non-soft) i
a
b
q
qV
qSea
0
qSea
qSea
+2 (0 if q = q) a)
q
qV
g
-1
q
q
g
+1
q
g
g
0
g
qi
q ' / q i a)
q
g
0
g
g
+1
sea
+1
a) For the explanation of some symbols see table 6
a quark jet the leading pair is usually the valence quark plus a gluon, whereas in a gluon jet it is a (sea) quark plus a gluon. Finally we study the case 1--Xl <<1 ,
1--Xl-X2<
I ,
(4.8)
i.e., the analog of a double-Regge limit. This we can do by studying double moments Dab,i(m,n; Y ) for m > > n > > 1. By inverting the Mellin transform we then find Dab,i(X 1, X2; Y )
~
H(Y)
(1
-- x l ) k a b , i ( 1 - - X 1 -- X2)2CvY+hab, i ,
1--Xl<
1-Xl-X2<
(4.9)
up to logarithmic terms. The exponents kab,i, hab,i can be computed but, before commenting on the results, we wish to point out an amusing analogy with Regge theory. The Regge analog of eq. (4.3), (4.7) and (4.9) would be, respectively: Da,i(x ) ~
X-.~l
Dab,i(Xl, X 2 )
Dab,i(X 1, X 2 )
(1 - x)av(°)-2~a,
i ,
~ (1 - - X 1 -- X 2 ) ~ v ( 0 ) - 2 a a b ' i , Xl +x2.--~1 (1 - - X l ) 2 ( a a b ' i - a a ' i ) ---> x 1 ..-. 1 x2--* 1 --x I
1 (1 - - X 1 -- X 2 ) c ~ v ( 0 ) - 2 a a b ,
(4. l 0) i ,
K. Konishi et al. / J e t calculus. a simple algorithm for resolving QCD ]ets
67
Table 6 a O = 2CFY + I
i
a
b
C~a,b
aab,i
q
qv
qs, q-s
1
q
qs, q's
qs, qs
0
q
qi(q'i)
qi(qi)
0
q
qv
g
1
l
q
qs, q-s
g
0
0
q
g
qv
1
l
1
_1 1
q
g
qs, qs
21-
0
q
g
g
~1
1
g
qi(qi)
qi(qi)
~
0
g
q,~-
q,, ~-,
1
0
g
q
g
~1
1
g
g
q,q-
0
1
g
g
g
0
0
1
(a) When i = q, a subscript v in a (or b) indicates that a(b) in the final quark connected to i by a quark line (hence it carries the same flavour as i). (b) a = qs, q-s, b = q's, q-'smeans that both a and b are quarks or antiquarks from the sea and that they are not a single pair from the sea. (c) a = qi(q-i); b = q-i(qi) means that a, b are a single pair from the sea.
where av(0 ) is the v a c u u m trajectory ( p o m e r o n ) and O~a,i,O~ab,i are the leading trajectories exchanged in the ai, abi channels, respectively (in Regge theory they also d e p e n d on the respective m o m e n t u m transfers). If the analogy between jet physics and (multi) Regge theory is more than superficial then the e x p o n e n t s kab,i , hab,i of eq. (4.9) have to be related to ha, i of eq. (4.3) and to ~ab,i of eq. (4.7) b y hab,i = ~ab,i ,
kab,i = b a , i - ~ a b , i - 1 .
(4.11)
We have f o u n d that these equations are indeed satisfied for all a, b, i. Defining then (arbitrarily) av(0 ) = 2 C F Y + 1 (av(0) = 1 at Y : 0), we get the intercepts ¢Xa,i, 0lab,i shown in table 6. Notice a few amusing regularities in table 6: fermionic trajectories have half-integar intercepts; bosonic trajectories have integer intercepts; i f b is a gluon, O~a,i = ¢Xab,i ; for fixed i, the more exotic the trajectory the lower its intercept tends to be.
68
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets
4.2. Spectra at small x, multiplicity distributions, etc. 4.2.1. Single-parton spectra and multiplicities. When we go to the small x region, we encounter, in QCD, an infrared problem, unlike the case in ¢3 (see subsect. 3.1). As an example, let us consider the multiplicity of partons of type a in a jet of type i:
(na)i = Da, i(0, Y) = exp(Ao Y)ai.
(4.12)
Unfortunately A o is a matrix with divergent matrix elements (e.g., A gq, A gg = oo) and one has to see how to cut off this divergence. Clearly there is no outright divergence in the theory as long as Y is finite and as long as the final partons are not just on their zero-mass shell. The problem of an absolute prediction of (na) i is under study and is not an easy one. Fortunately it turns out that ratios of quark (and gluon) multiplicities in the two types of jets are finite and are simply given by the ratio of Casimir operators. One finds: ( n q ) q jet _ ( n g ) q jet _ CF/C A = 4 , (nq)g jet (r/g)g jet
in S U ( 3 ) .
(4.13)
This result comes from the fact that the (divergent) matrix A 0 has one (infinite) eigenvalue whose eigenvector projects onto quark and gluon states in a ratio CF/CA. This and other results will be seen to be stable against modifications of the IR cut off procedure used. Furthermore the fact that the same ratio holds again both for final quarks and for final gluons makes us believe the result at the hadronic level * There is indeed a simple intuitive interpretation of eq. (4.13) in the large Nc limit where CF/CA ~ ½. A quark carries a single colour index leading to a one-sided decay chain, whereas the gluon, having two colour indices, develops a two-sided chain ** as shown in fig. 6. The non-perturbative confining stage taking place later is not expected to make the two chains interfere appreciably, hence there will be twice as many hadrons in the gluon jet than in the quark jet. If we combine the result (4.13) with eq. (4.2) for the fractions of the total energy carried by quarks and gluons we see that the average x of a quark or of a gluon is CA/C F (9) times higher in a quark jet than it is in a gluon jet. Thus, gluon jets are softer than quark jets and yield higher multiplicities. We may also study directly the behaviour of spectra near x = 0 (insisting however that x is finite as QZ(Y) ~ oo). This has been done for quarks in a quark jet in refs. [ 14,22] by inverting the double transform in E, n back to Y and x. By this method we find Dq,q(X; Y)
Dg,q(X; Y)
~ ~1-6 N f ~""- ' F -1 y2e-WYI2(v)/v2 , x--*O X
~ x~O
4Cv y e_WYll (v)/v,
(4.14)
X
• Such a relation was first guessed, based on an analogy with QED, by Brodsky and Gunion [55]. • * The analogy with the topological expansion picture of the reggeon and the pomeron [40] is evident.
69
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD ]ets
Fig. 6. (a) Quark decays and (b) gluon decays.
where o = ~/8CAY log(I/x),
W= I-~CA + ~1U f (1 -
(4.15)
2CF/CA)
and I1,2(v) are the usual modified Bessel functions. Furthermore, distributions in a gluon jet are related to those in a quark jet by: Da,g(X; y )
~ CA Da, q(X; Y) x"*0 ~
(4.16)
for both a = q and a = g, in agreement with eq. (4.13) for the multiplicities. The reason for universality is again the same and is discussed in appendix C. 4.2.Z One soft, one hardparton. We now consider the case of one slow (Xl ~ 0) and one fast (x 2 = finite) parton and show that the two partons become uncorrelated in this limit. Consider indeed the (moment of the) correlation function Cab,i(x 1, X2 ; Y) = D a b , i(X IX2 ; Y) - Da,i(x 1, Y) Db,i(x2, Y) :
G~,i(e,
n; Y) =
,~ _
A~ - A .
aa'
~ ( E - A.+~) i'i
bb' - ( E - A e - An)[a 1 ,
(4.17)
bi
where e = 0 + in the moment relative to the slow particle and n = 0(1). In eq. (4.17) various factors are divergent since both A e and Pe, n contain divergent matrix ele-
70
K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD ]ets
ments. Since, however, pea'bn',i' ~
Aa',f ,
~---~0
An+ c ~ A n + O(e),
(4.18)
one find easily that Cab,i(• , /7; Y) is not divergent. In other words the divergent part in Dab,i(x1, x2; Y) for x 1 ~ 0, x z finite factorizes into Da,i(x1, Y) times Db,i(x2, Y). This result can be generalised to n-parton distributions, i.e., when one of the partons becomes wee its divergence factors out * 4.2.3. Multiparton spectra, correlations. This is the most delicate case, in which all partons are soft. In order to cure IR problems we can try various cut-off procedures and then see if there are finite quantities which do not depend on the cut-off procedure itself. We shall discuss here two regularization procedures ** (a) A o is given a fixed cut-off (e.g., by integrating between e and 1 e in eq. (2.7)). In this case the procedure followed in subsect. 3.1 and leading to eqs. (3.10), (3.11) can be repeated with the only extra complication due to the presence of two species of partons. Defining multiplicity generating functions for a quark and gluon jet by
l+Fq(uq, Ug, Y) = ~
UqUg
o(q --~nquarks + mgluons; Y)
n, m = 0
Otot
oo
1 + Fg(uq, Ug, Y ) = n=m=o uqug~
o-t-1o(g ~ nquarks + mgluons; Y) ,
(4.19)
one gets, instead of eq. (3.11), the coupled system: / dd Y - Jt?g + 1) 2 --(1 + F g ) ] , - =Aggpg(~K'g " ' 0 - v + 1) + 5Nf[(Fq 1 dd----yFq= AgqFg(F q + 1),
F q ( Y = O) = Uq
F g ( Y = 0) = Ug
Q
(4.20)
Furthermore A gg and A gq are divergent constants, but their ratio is simply CA/CF. The system (4.20) is just the standard way [38] of dealing with the problem of population growth of two competing species (here quarks and gluons). The general, as well as several particular cases of eqs. (4.20) are under study [41]. Here we just mention that, if we neglect q~ pair creation by setting Nf = 05 the problem is com-
Since ¢,~ does not have an IR problem this simple result is not valid there. ** A more rigourous treatment taking into account the actual kinematical boundaries is under study.
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets
71
pletely soluble and one gets (1 + F g) -
Ug Ug + (1 - Ug) exp(AggY) '
Uq (1 + F q) = {Ug + (1 - Ug) exp(AggY)
(4.21) }CF/C A "
Eqs. (4.21) give, for the gluon jet, an ideal Bose gas spectrum (i.e., a black-body radiation with kT/hv ~ exp(Agg Y) ~ o~ as we remove the cut-off), whereas the quark jet corresponds to CF/CA independent black bodies (giving the so-called Polya-type distribution *) except that Cv/CA is not an integer (for N c -+ oo, in particular the quark jet becomes half a black body ?). Vice versa, the QED limit can be recovered by letting C A ~ 0. In this case CF/CA -~ oo and the quark jet becomes equivalent to an infinite sum of independent black-body radiations which is well-known to give a Poisson spectrum, as it is known to be the case in QED. For completeness we give here the relevant exclusive cross sections as they follow from eqs. (4.21): oq..+q+ng
=
e_(CF/CA)YAo r(CF/CA + n) F(CF/CA)
a g'*ng
=
(1
- e-AoY) n n! '
e--4or(1 - e-AoY)n .
(4.22)
Notice that this method gives cross sections analogous to those o f ¢3 and in particular no enhancement (an/O o -~ constant). (b) The second IR regulafization procedure we wish to consider ** is of the "dimensional" type. We take small positive moments of each k-particle spectrum and obtain an e-dependent generating function (compare with eqs. (3.8), (3.9) in ¢3) for gluon multiplicities inside the two types of jets in the form (i = q, g):
Gi(v; Y)= 1 + Fie(v; Y), Fie(v;
Y)= ~
Ok ~.v Dgg...g,i(e, e .. e, Y)
k=l
k ~
.
-
dxixiDgg'"g'i(xl ""xk' Y)=- ~k f i k,e
(4.23)
where we have introduced v ----u - 1. * The phenomenological reievance and the possible theoretical origin of this type of distribution in loW-PT hadron physics is known [42]. We thank A. Giovannini for drawing our attention to this point. ** We are grateful to A. Bassetto and G. Marchesini for interesting discussions on the subject of this paragraph.
72
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
In accordance with the jet calculus rules, Fig,e satisfy the integral equations: k-1
Fik = V6kl(eAeY)gi+ll~__
f
y d r , bFe; F ~ _ l
o
X P~eC,t~_t)e (eA ke( Y- Y'))i,i
(4.24)
•
Taking the dominant contribution for e ~ 0, this gives F g "~
(ok/k!) C,(n)kg,
(4.25)
F g ~ (vk/k!)fk(7) (n)~,
(4.26)
(n)g
(4.27)
where =
e(2CA/e)Y,
7 ----CF/CA ,
and that Ck and fk('Y) satisfy the following simple recurrence formulae: d k -Ck/(k.
k!),
d I = 1 ,..
k-I dk = { k / 2 ( k 2 - 1)} ~
dtdk-t ,
(k~>2),
(4.28)
l= 1
and gk(~/) -=/k(T)/k!,
g, (T) = T ,
k-I g/(7) d , - t + dk } ,
gk(7) = (v/k) (~
(k >1 2 ) .
(4.29)
l= 1
We have been unable to solve the recurrence formulae (4.28) and (4.29) in a closed form. However, much can be said by. converting these into a set of differential equations for the generating functions Fe~(~") and their integrals D q or g(~-) - f
d~"F q or g(~.,),
(4.30)
--oo
where * ~"= log v ( n ) .
i * We notice that by keeping leading terms only, Fe(v, Y) depend ofl v and Y only through the combination v{n) = o exp AeY.
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
73
In fact we find d Dg = F g
d~ d Fg =Dg(1 + F g ) ,
d~"
d Fq = 7Dg(1 + F q ) . d~"
(4.31)
with the boundary conditions Fq'glo=o = O, __d Fgiv=0 = 1 , dv clFq I
do
v= o
= ,),= C F / C A .
(4.32)
From eqs. (4.31), (4.32) one easily gets the simple result: log(1 + F q) "" ( C F / C A ) log(1 + F g ) .
(4.33)
We notice that this same result follows in the previously discussed method for regularizing IR divergences, as it is easy to check from eqs. (4.20). The interpretation of eq. (4.33) is very simple if one recognizes in the quantity log0 + F ) the so-called Feynman pressure in his gas analog model [43]. In other words, the correlation coefficients [44] .t~l,g are all in the ratio f~t/ff = C F / C A , (l = 1, 2 ...),
(4.34)
which is finite in the limit e ~ 0. Although we have only discussed above final gluons, eqs. (4.33), (4.34) actually hold independently of the final parton species (q-g universality) as shown in appendix C. The generating functions Fq'g(~) contain in principle all the information about exclusive cross sections in QCD which are related [44] to the behaviour o f F near v = _ l (~ ~ oo). It is easy to show, by studying the differential equations (4.31), (4.32), that F q or g(o, Y)lo=- 1 = - 1 ,
(4.35)
in accord with eq. (4.19). Furthermore, we checked that the lowest exclusive cross sections, calculated from the divergent (for e ~ 0) inclusive cross sections, are indeed exponentially vanishing for e ~ 0, as we expect to be the case. Detailed results, however, are not consistent with known results such as the Sudakov form factor, showing that terms neglected for inclusive cross sections are important in computing exclusive cross sections. This is not surprising because a huge cancellation is involved in
74
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
obtaining exclusive cross sections from inclusive ones. It does not imply, however, that we cannot trust our predictions at the inclusive level. An interesting result which follows from eq. (4.31) is that (n(n - 1) ... (n - k + 1))e, i jet ~-/
Ildxi Xe (1/o) do/dx 1 ... dx k
is proportional to (n)ikjet , with the proportionality constant larger than unity. This indicates the presence of a long-range correlation, probably within the scheme of KNO scaling [45]. Another quantity of phenomenological interest is the dispersion defined by D = x/(n 2 ) - (n) 2 . Using eqs. (4.28), (4.29) one obtains Oq jet/(n)q jet ~ ½X/3 "" 0.87,
(4.36)
Dg jet/(n)g jet ~--N/ 1 .~ 0.58.
(4.37)
It is amusing to observe that the result, eq. (4.37), for the gluon jet, is numerically identical to the phenomenological value in soft hadronic processes found by Wrobrewski [46]. How much we should trust all these results is not presently clear. The prediction, eq. (4.34), looks to be very stable and, due to its simple physical meaning, we think it should be true in a more refined treatment. Also very stable is the presence of long-range correlations and KNO scaling. On the contrary, the exact values of the ratios ( n k ) / ( n ) k (or of the KNO function) do depend on the type of IR regularisation used. Finally, the fact that all these results do not depend upon the type of final particles we look at (universality) also has a very general origin in our framework and is independent of the IR regularization used. How universality arises in the scheme is explained in detail in appendix C. 4. 3. Photon ]ets
In this subsection we discuss the properties of jets initiated by (almost) real photons. For completeness, however, we shall first discuss the case of fragmentation into a photon, described by a fragmentation function D,r,i(x, Q2). This case has been already discussed in refs. [14, 47, 48].
Ca)
(b)
Fig. 7. (a) Fragmentation into a photon. (b) Photon fragmentation into hadrons.
K. Konishi et al. / Jet calculus: a simple algorithmfor resolvingQCDjets
75
From fig. 7a we see that D,r, i is related to Oj, i by the equation Q2 dq2 f dz 1 + (1 - z) 2
o~ ~i e2 f D~/'i(x' Q:)= 2n
q2 J z
z
Dj,i(x/z, Q2, q2).
(4.38)
Since, apart from a factor CF, the AP function [1 + (1 - z) 2 ]/z isP gq one gets
D'r,i(n,
o~ ~ e2. A~n" Q2 dq2 Q2 q2 = 27r / ' (CF) f -~-Dji(n, , ).
Q2)--f D(x' Q2)xndx
(4.39) The integration over q2 is now easily done and one gets a
1
(Anlg j
2Nf
D~/'i(n' 02) - 27rb ots(Q2) "/~1 e2
\CF]
- ~--~-~j
X 1 - A n/27rb
j/ii,
(4.40)
This result can be seen as the analog in jet physics of the structure function of a photon, first derived by Witten [31 ]. In the case of the fragmentation of 7 into a hadron (or a parton of type a) an equation similar to (4.38) can be written following now fig. 7b. One gets /~1f
Q2dq2 fdz q2 3 z (z: + (1 -
Da:/(x ' Q 2 ) = ~a .= e~N c f XDa,i(x/z q2).
(4.41)
Now the factor 1(z2+ (1 - z) 2) is the AP function A qg up to a factor Nf and, going over to moments, one gets ct
Da,,r(n, Q2) = 7rb°q(Q2)
Nc ~ e ~
X { 1 ( 1+An/2nb ) [eYAn-eYOAn'(Q2)~]t --~Q~o)_]]ai (Aing),
(4.42)
showing again enhancement with respect to Da,g : Oa, g(n, 02) = (DYAn)ag.
(4.43)
Furthermore Da,.r is also enhanced by a factor R _-- N c ~,iei.2 It seems thus pos-
76
K. Konishi et al. / Jet.calculus: a simple algorithm for resolving QCD jets
sible that processes such as e+e - ~ ')'real + "/jet could be isolated and measured for instance at PETRA energies. Notice that for low moments (small x) the first term in the square bracket of eq. (4.42) dominates (An/2nb + 1 > 0) and the quasi-real photon is actually far (O(Q2)) off-shell. On the contrary, for large moments (large x) the second term dominates and the photon is only O(Qg) off-shell. In other words this experimental situation provides a source of e+e - like jets (Tvirtual -->hadrons) where the mass of the virtual photon can be varied and is actually optimized according to the momentum of the detected final parton. The particular limit Dq,7(x, Qz) for x --> 1 is worth noticing. We find
Dq,,y(x,
c~ N c ~ 1 Q2)x_,l as(Q2) CF _7.. e~ log(l/(1 --x)] '
(4.44)
to be compared with the gluon fragmentation behaviour
NfA(Y)(1 -
X)2CF Y Dq,g(X, Q 2 ) ~ 4CF(CA _ CF) Y log(l/(1 - x ) ) '
(4.45)
where the symbols are those of table 4. Clearly a photon jet is harder than a gluon jet and the shape of the distribution in x, together with its dependence of Y, are indeed quite different.
5. Combined longitudinal and transverse spectra of QCD jets
5.1. Two-particlespectra We now go back to eq. (2.8) and, as promised, we interpret the integration variable y in terms of the relative transverse momentum PT of the two final partons. From the way y enters in eq. (2.8) and from our heuristic derivation of that equation (as well as from the more rigorous one of appendix A) it is clear that y is related to the virtual mass ~ of the parton j (fig. 2b) by a relation analogous to eq. (2.4) i.e. y =TnblOg a s ~ 2 )
a s(4q2)'
(5.1)
2 1 2 where the factor of 4 has been introduced so t h a t y = Y for q 2 = qmax = ~Q • 2 1 2 The situations we shall consider here will refer to values of q < < ~Q since, for q2 ~ Q2, the process clearly involves an additional jet and one needs to describe it by the appropriate non-collinear kinematics. We shall come back later to the discussion of how smoothly our results connect with those obtained by other authors in the region q2 ~ Q2. We now try to estimate q2 in terms of the final momenta. Because of the strong
K. Konishi et al. / Jet calculus."a simple algorithm for resolving QCD]ets
77
ordering of masses (and OfkT) along each ladder [13-20] partons b l , b2 have masses much smaller than X / ~ , hence the final PT of a 1, a2 has to be comparable to the one O f b l , b 2 . Denoting by pbl,b2(26b 1,b2) the relative transverse momentum (angle) between bl and b2 and by PT(26) the that ofax, a 2 we find q2 =X2Z(I - - Z ) 6 2 Q 2 <~ 1Q2~2 ,
(5.2)
where we have used the fact that, for the LLA, the proportionality constant x2z(1 -- Z) can be set equal to one (if it is finite). Of course, at finite Q2, it will make some (non-leading) difference to use the more refined expression but this is not, strictly speaking, an effect that one can compute without a non-leading log calculation *. In terms of PT the relation with q2 is somewhat more complicated, but, essentially one has still q2 ~ O ( p ~ ) ,
(5.3)
with a proportionality constant of order one. Hence in the following we shall use eq. (5.1) with either Y = Y8 = ~
1
log[cq(P 2)/~(62Q 2)] ,
(5.4)
log [ a s ~ 2)/as(ap2)] .
(5.5)
or 1
Y = YT = ~
At this point one can immediately identify the integrand in eq. (2.8) with (1/o) da/dy. Since eqs. (5.4), (5.5) give d _ (Cq(4p2T)1-1 dy T
\~]
d
dlogp~
'
d _(Ots(~Q2).} -1 d dy~ 6 d-~'
(5.6)
one gets immediately the final result (see fig. 8) 1 d o ( i -~ al + a2 + X )
-O
d~ 2 dx i dx2
_- Dal a2,i(x 1' x2,~., Q2) -- tYs)(62 2- ~Q2
× jl~jj~-x2dglj~-Xldg2 I i" "2x1
zl x2
z2 (Zl + z 2 ) ^
g1
XDal ,JI (X1/z I, y6 ) Da2,J2(X2/z2, Y6 ) Pjl J2,j(z.-~~). XDji(z I +z2, Y - y 6 ) , * A similar problem of fixing Q2 exists in large-PT hadronic processes.
(5.7)
78
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets 200
I
f
I
O/A =100 /
I00
////e// NAIVE
RESULT NO N'.Z-
% ////////~
~c:;
PERTURBATIVE 3 JET REGION
/
~////~ OUR ////////~ RESULT /
20
q> o~
M~
~
10
PERTURBATIVE 2 JET REGION 4
2 NAIVE11 SULTI ee,
i 4
~
10
i 20
i 50
qTIA
S 8 (q~ Fig. 8. Naive versus our result for (0, 1) moment, ~aDqva,q (0, 0; q2, Q2).
where as in eq. (2.8) we have again eliminated two of the integration variables using the 6-functions. Exactly the same equation holds if we replace 62 by 4p 2. Going finally to the hadronic level, we trivially get from eq. (2.15)
Dhlh2,i(X l, X2
-2- %(82Q2) ~
62 ;
;~
)-
~
a d Xa d62
h
I' \Xa Xa
1 --x 2
J'Jl,J'2x I
×D~j~(xl/zl, ya)D~2(x2/z2, ya)Pjlj2,j
a•/dx
f
Y~t!
pl --x 1
dz 1 J
dz2
ZI x 2
Z2 (Z1 "l-Z2)
1
ji(gl +z2, Y - Y a )
(5.8)
The second term in eq. (5.8) involves now an unknown (phenomenological) fragmentation function for the decay a ~ hi + h2 + x at fixed 6. Fortunately, such a contribution is only important for small 6 = O(A/Q) and therefore, if we keep away from
K. Konishi et at / Jet calculus: a simple algorithm for resolving QCDlets
79
such non-perturbative region (PT > > A), only the first (perturbative) term contributes. This could be evaluated numerically once the input of single-particle fragmentation functions Dha(x) is given at some reference point [49] Qg and which could be measured in any convenient process *. We have not started yet any serious computer calculation of D~ I,h2, but fortunately one can already extract the basic physical properties of eq. (5.8) by either taking moments in x l , x2 (as done below) or by looking at more inclusive (calorimeter measured) quantities as explained in subsect. 5.2. Before discussing the detailed properties of our formulae we would like to compare qualitatively our results to those of Dokshitzer, D'Yakonov and Troyan (DDT) [14,24] who, in the particular case of e+e - -'-hadrons, look at the inclusive production of two hadrons h I and h 2 which are at 180 ° (within an uncertainty 2~), hence in opposite jets. Since the q,~- coming from the photon are necessarily back to back and thus define the jet axis, it follows that h I , h 2 should have a (small) angle 0(6) with respect to that axis. This automatically limits the possible real emissions along each ladder to have IkTI < 6Q. As a consequence real and virtual diagrams only cancel for the infrared region 0 < IkTI < 8Q and the region IkTI > ~iQ remains uncancelled. Since both soft and collinear divergences remain, double logs of Q2/p2 (62) appear at each power of a s. This can also be seen by the fact that the original q,-~ cannot be off-shell by O(Q2), otherwise they would give ~ = O(1). A careful study [14] shows that they are off-shell between O(p 2 ~- 62Q 2) and O(PTQ "" 6Q 2) (the latter limit being reached for emission of partons of Ikxl ~ Ikl ~ 6Q). This produces the appearance of a Sudakov-like form factor at the e.m. vertex which is known to exhibit double logarithms. In contrast, in our case, the angle between h I and h 2 is near zero (same side). First of all this implies no restriction on the opposite side jet. More important, on the same side, other unobserved partons are allowed to be emitted in directions other than that of hi and h2: any correlation (within ~)between the original jet axis and that of h i , h2 is now lost and the q,~- pair created by the photon can be as far off-shell as it likes. In other words, before the vertex j ~ b l , b2, the cancellation of real and virtual divergences simply gives the usual AP evolution ofD(x, Y - y). After the vertex all the bremsstrahlung kw's are necessarily much smaller than PT (strong ordering) and are included: the only uncancelled divergences are the hard (x v~ 0) collinear ones and they produce, in the usual way, the scaling violations of the remaining fragmentation functions. In conclusion, a single logarithm of 6(PT/Q) appears now for each power of a s provided we stay away from the soft region (x 1 or x 2 ~ 0). This we do by considering positive moments:
Dala2, i ( n l ' n 2 ; 6 ' a 2 ) - f d x l d x 2 x n l x n1 2 J2 1 d a ( i ~ a l +a2 + X ) o
d~2dxldX2
'
(5.9)
* Other semiphenomenologicalinputs are, of course, A and the exact forms ofy T and/or Y6-
K. Konishi et al. / Jet calculus."a simple algorithm for resolving QCD lets
80
for which we find, from eq. (5.7) Dala2,i(nl, n2; 6, Q2) _
1 as(62Q 2) ~ as(m 2) "](An1+An2)/27rbp [oq(62Q2)-].An1+n2/2~rb 26 2 nLa~)] nl'n2 L ~ -J ' (5.10)
and an identical one with Q2~2 _~ 4p~. Theoretically, the simplest example is the one in which i = q (quark jet), al is taken to be the final quark line coming from i itself (al = i = "valence" final quark), n 1 = 0 and one sums over a 2 setting n2 = 1. One thus finds that b o t h A n l andAn2 can be set to zero (charge and momentum conservation) and that Anl+n 2 and -Pnl,n 2 become both equal to A Ns , the non-singlet part of the anomalous dimension A1 : A1Ns- 16 The result is simply: 9" a
Vt ANS~jas(4P2)~ /as(4p~.)~A t-k'T 2 . _J ~Ots(a 2) ]
D.va,.(0, 1;p , 02)
(5.11)
Before commenting on eq. (5.11), let us mention that an explicit computation of the same quantity at order a s is consistent with it, since it gives:
a
Dqva, q(0, 1 ;p~-, Q2)
_ 1 F(-A Ns) sl naive 0(%) 2rr as L ~ + 6(p~) term_ , (5.12)
where the (negative) asg(p~) terms come from the one-loop virtual correction * Because of eq. (5.12), we shall refer to the first term (square brackets) in eq. (5.11) as the "naive" result. Before considering other, physically more interesting, choices of hi, 2 , al,2, we make the following general remarks on eq. (5.11). (a) Eq. (5.11) is a (fully normalized) QCD prediction only for p~- > > A 2. Yet, integrating it from p~- "" m 2 to p~- "" Q2 gives the momentum sum rule needed for consistency and normalization. (b) The average value ofp~- computed from eq. (5.11) is: (P~') = ( - A INs) as(Q2) Q2 ,
(5.13)
which is (up to the constant ( - A ~ s) = ~ coming from our definition of the inclusive cross section) the expected result. (c) The second factor (round brackets) on the r.h.s, of eq. (5.11) can be considered to be the correction to the naive result. It is similar, in spirit, to the T form factor of the DDT formula [14,24], although it has single, and not double, logs. • This is, strictly speaking, not a contribution to the two-particle spectrum, but can be considered as such for the purpose of satisfying sum rule constraints.
K. Konishi et al. /Jet
This correction by a factor
81
calculus: a simple algorithm for resolving QCD jets
factor goes to one at p+ N Q2 but it suppreses values of pi << Q2
(5.14) Its effect, as in the DDT formula [ 14,241, is to broaden the pT distribution with respect to the perturbative result. An example of this phenomenon is shown in fig. 8, for Q/A = 100 (Q z 50 GeV). (d) The (negative) o&p%) term of the perturbative result (5.12) has disappeared in eq. (5.11). One could think that this is due to having limited ourselves to pt >> A2, but indeed the integrated result mentioned in (a) shows that there is no rooms for 6 function terms (see also next point). (e) We can consider our result (5.11) as a smoothed version of the perturbative result (5.12). Such a type of smoothing is usually achieved by putting by hand an intrinsic (primordial) pT in structure and fragmentation functions. Here we have instead obtained it as a calculable perturbative effect. We expect that further, nonperturbative smoothing due to actual intrinsic pT will turn out to be smaller than usually assumed. (f) One can check that the integrated cross section up to 4p$ - Q2 is correctly obtained (up to logarithmically vanishing contributions) from the region p$ >KA2 (K> l), i.e. from the perturbative region. There is a residual (integrable) divergence at 4p$ = A2 due to the (unphysical?) blowing up of 0~~there. Clearly, we cannot penetrate into the region 4p$ < A2 by perturbative techniques, but it is reassuring that only a smaller and smaller fraction of the cross section lies there, as the energy goes up. This is to be contrasted with 10w-pT hadronic physics, or with what could be expected in softer theories, where essentially all the cross section lies at pT < A = 0.5 GeV, independently more or less of the total energy. (g) The integrated cross section up to p$ = P$~,,~ is even simpler than eq. (5.11) and, in the region A2 << pt,,,,
<< Q2 behaves like
We see here, and this feature persists for other moments (see below), that an amusing form of scaling emerges asymptotically in the logs of Q* and of 4pt,,, . In other words, in a log Q*, log 4p2T,max plane, straight lines out of the origin delimit regions of constant fraction of the cross section (see below for actual numbers). Before turning to our actual numerical predictions and to a discussion of them we have to consider (at least) two other interesting choices of i, al, a2, nl, ~22in eq. (5.10). Since one of our aims is to compare quark and gluon jets, and since for a gluon jet, the “valence” quark in the jet does not exist, we are led to consider the quan-
82
K. Konishi et al. /Jet calculus." a simple algorithm for resolving QCD jets
tities Dala2,i(1, 1, PT, Q ) ,
~
al a2
Dala2,i(1 , 1,6, Q ) ,
i - q, g. (5.16)
ala2
Such a case turns out to be also more advantageous than the previous one for translating parton-level results into hadronic-level results, as discussed later. Using momentum conservation constraints the final results, obtained from eq. (5.10), in the specific case (5.16), turn out to be only slightly more complicated than the ones already discussed. Furthermore, all the qualitative features discussed above remain unchanged. To be specific, eq. (5.11) gets replaced by
ala 2
%(4p2) Dala2,i(1, 1.p2, Q 2 ) _ p2 2rr ~
a,j
(-.42
) "Fcts(4p~')lA2i2~rb I r xl aJ L 0~s~,Q2)_I ji
,
(5.17)
and the same with 4p~ ~ 6 2 Q 2. Now A2 is a two-by-two matrix: its diagonalization and the projection of its eigenvectors onto quark and gluon states lead to our final results:
ala2
24 Dala2,i(l , 1;p~-, Q2)=p~. l°g(4p~/A2)
71pT
"qPT | '
L0.920J
e
(5.18)
[. 3.680 .J
and the same with 4p 2 -~ I52Q2 , where the upper (lower) figures are for a quark (gluon) jet and rIpT
_ log(2qT/A) log(Q/A) '
_ log(gQ/A) r/6 =-log(Q/A) "
(5.19)
A comparison of the two types of jets based on eqs. (5.18) clearly shows that the quark jet is much more narrow (in 6 or PT) than the gluon jet. To be more precise, consider the integrated cross sections up to p~aX (or 6max ). One then finds the anticipated "scaling" behaviour, i.e., dependence of the integrated cross sections upon p~aX, 6max , Q is only through r/p~ax and 66 max" Numerically one finds
rll61
/ _-0.6086 + /
f('~--<6max)= [0.363] 7/6max
/ 1.386 t'/'/6max , L 0.637]
(5.20)
where f("-<6max) is the fraction of events with 6 ~< 6max. A plot off(~<6max) as a function of r/6 max is shown in fig. 9 for both types of jets. We see for instance that, in order to have 70% of the events one needs, for a
K. Konishi et aL / Jet calculus: a simple algorithm for resolving QCD ]ets
83
1.o
o.~
-GLUON 0,1)
I
I
0.25
0.50
__
I
0.75
1.00
~m~x
Fig. 9. f(eq. (5.20)) versus n 6 mxa. quark jet * r/Smax = 0.5 and for a gluon jet ~16max = 0.7. In terms O f 6 m xa this is equivalent to
j(Q/A) 6 max
-
-°'s ,
quark j e t ,
( Q / A ) n~ m a x - 1 =
(5.21)
(Q/A) -°'3 ,
gluon j e t .
In numbers this means ~ max = 5 ° - - 6 ° (or p ~ a x "" 2.5 GeV) for a quark jet as compared t o 6 m a x = 14 ° - 15 ° (p~ax ~ 6 GeV) for a gluon jet at Q / A = 100 (Q = 50 GeV). Vice versa, at the same value of Q/A, an angle 26 = 10 ° includes 67% of the cross section for a quark jet, but only 45% for a gluon jet. We also remark that for f("-<6max)> 0.9 we have numerically seen that the logarithms Of 6max for a gluon and a quark jet are in a ratio near to CF/CA = 0.444, as recently found by Shizuya and Tye [60] following the Sterman-Weinberg method (for f approaching one we find the ratio to approach 0.436). As we have already remarked, our results are expected to make sense in the limited region of 6 ( p T ) : ( A / Q ) 2 < < 62 < < 1 ,
A z < < p ~ < < Qz .
(5.22)
We recall that the limitation to 62 < < 1 came from our use o f collinear kinematics. The region 6 "~ O(1) can be treated more appropriately by lowest-order calculations * For a comparison, in the case of the valence quark spectrum one finds, from eq. (5.14), nSmax = 0.433.
84
K. Konishi et a L / Jet calculus; a simple algorithm for resolving QCD jets 10
Q21A2
=
200 1000 5000 25000
I
I
I II
II
,i,!i
IIiI
I I. I:"".,
e~
%-
[., •." ~;/1.7~ ..."
c~b-
!/i/rl i
/ ///hl .. 1 / 1 1 / i //1 i I I
/}////II // ~/I'/
0.5
/ I'~I/ / iii//
,',,
\ \ \ \
,, ,,N
0.1
t 30 °
-~,. 7 .... t 60 °
...-
"~
t 90 ° Z6
~2
i
1 0°
150°
I 180 °
b
Fig. 10. Comparison between eq. (5.17) (full curves) and the formula from ref. [27] (dashed curves). The results from refs. [14, 24] are also shown (dotted curves). (justified since oq(Q 2) log ¢5 < < 1) which keep track however of the more complicated non-collinear kinematics, of certain interference terms (which are non-singular for ~ -+ 0), etc. For the double m o m e n t ~al,a2Oala2,i(1 , 1; ~, 0 2 ) , this region of~5 ~ O(1) has been investigated in detail by Basham et al. [ 2 7 ] , w h o have actually looked also at dependence on other angular variables*. Integrating over those variables we arrive at the * These authors emphasize rates with calorimeter detectors, which, however, for n 1 = n2 = 1 coincide with counter-measured cross sections.
85
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
result shown in fig. 10 (dotted curves). In the same figure we show our results (from eq. (5.18) in the case of a quark jet) and we see that the two curves agree quite well for all values of Q2 in some intermediate range of 6(6 ~ 50°). The results of ref. [27] lie below ours for small 6 if we keep using as(Q 2) in their formulae for all 6. On the other hand, if we use for instance as(Q 2 tg 216), their formulae lie above ours. Since our formulae can be seen as a resummation of all terms of the type (as(Q 2) log 6) n, we can say that our method resolves the above ambiguity as to whether to use as(Q 2), as Cts(62 Q2) or get something else in a lowest-order calculation. The correct leading log prediction should then be a smooth interpolating curve between our results for small 6 and those of ref. [27] at large 6. From fig. 10 it is obvious that such a smooth curve can be drawn. This indicates, in our opinion, the important fact that QCD gives predictions on the transverse structure of e+e - hadronic events which, with a smoothly varying cross section, go from narrow to broad two-jet events and finally to genuine three-jet events. It is only the approximations used in computing these various regimes that change while the underlying physics remains the same. Finally, when the results of Basham et al., are extrapolated further to angles near 180 °, they become again inadequate because of large logs of the non-collinearity angle. This is exactly the situation discussed by DDT and which we have briefly outlined above. We have checked that the formula of DDT gives a smooth extrapolation both in magnitude and in Q2 dependence * of the results of ref. [27]. 5. 2. Calorimeter-measured cross sections and a new way o f testing QCD
Another simple and interesting quantity which can be predicted is the generalization of the Sterman-Weinberg [4] expression for a two-jet cross section when one reduces very much the opening angle 6 of the calorimeter in which all but a fraction e of the energy must go. Consider values of 6 such that, again, % ( 0 2 ) log 6 ~ O(1),
(5.23)
while keeping cts(Q 2) log e < < 1 (i.e., e not too small). This quantity has been computed by Ellis and Petronzio [32] and, in a particular case, by Curci and Greco [33]. Their result can be expressed in terms of the parton level fragmentation function D(x, Y ) by the relation: 1 doi(6 ) _ ~ D i i ( x ' y _ y ~ ) , oi jet dx j
(5.24)
where Y~ = ~
1 lo-
as(m2) ~as(62Q 2) '
Y=
17r as(m2) 2 bl°gcq(a2)'
(5.25)
* To this purpose it is important to keep the PT dependence of the DDT form factor induced by the effective coupling constant as(p2).
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
86
i specifies the type of jet under consideration and x is the fraction of energy deposited in the calorimeter of opening angle 26. In the case 6 ~ A/Q, eq. (5.24) in the standard formula for a single-parton inclusive cross section which can be converted into eq. (2.6) for hadrons by folding in a phenomenological fragmentation function. On the other hand, for d > > A/Q, eq. (5.24) refers directly to calorimeter measurements. In appendix D we give a simple rederivation of eq. (5.24) starting from our eq. (5.7) for a two-parton inclusive cross section and integrating it in such a way as to avoid multiple counting. Notice that the result (5.24) depends on Q2 and 6 only through the combination Y y~, i.e., through r~8 = log(Q2/A2)/log(Q282/A 2) as we have seen to be the case for the moments of 2-parton spectra. Ellis and Petronzio have used eq. (5.24) in its integrated form in x between 1 - e and 1 (with neither e nor 1 - e too small). Another way to exploit eq. (5.24) is to consider positive * moments in x (i.e., in the energy deposited in the calorimeter). Indeed, defining Cin(6) =
1 f~(2E/q,-~) Ojet
n dE,
(5.26)
the prediction for c i ( 6 ) is simply:
cin(8) = ~
e x p ( A n ( Y - Y~))ii = ~
J
[as(62Q2)TAn/2~b
(5.27)
J [- °~s(a2) --] ji
Eq. (5.27) represents an absolutely normalized prediction and can be used, therefore, for direct tests of QCD. It can also be used, in the more familiar way of deep inelastic scattering physics, to measure anomalous dimensions by varying the angle 8 at fixed Q2 (or the other way around). Indeed, if we assume for a moment no quark-gluon mixing (A n is a number), we obtain d log Cin(6) = A--e-nd log Cim(6)
d~
A m d6
(5.28)
'
i.e., plotting log Cim versus log Ci as ~ is varied we should get a straight line with a predicted slope An/A m . In the general case with mixing, but with a definite type of jet, one can still diagonalize the matrix An and obtain the result .
,'~ 2 ~ 2 . \
hn/2rrb
ci(8, Q2) -_ ain\last°%~Q_g~.l{d )I
-i [as(62Q2)]] + ~n~ %--~Q~
h n/2rrb
-
,
(5.29)
• The zero moment has, as usual, an infrared divergence. This is the moment studied in refs. [32,33] where the divergence is avoided by restricting the range ofx.
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets Table 7 i/0~(62Q2)~ hn /2nb + fli {°~s(62Q2)'~hn-/21rb
2 3 4 5 6 7 8
1.1653 1.0937 1.0610 1.0440 1.0338 1.0272 1.0226
-0.1653 -0.0937 -0.0610 =0.0440 -0.0338 -0.0272 -0.0226
0.3628 0.1896 0.1271 0.0951 0.0756 0.0626 0.0532
0.6372 0.8104 0.8729 0.9049 0.9244 0.9374 0.9468
-0.6085 -0.8170 -0.9604 -1.074 -1.170 -1.252 -1.325
-1.386 -1.852 -2.192 -2.460 -2.683 -2.875 -3.043
~=4 q .q
C1,C 1 1.0
0.9
0.8
0.7
0.6
i'11//
0.5
0.4
0.3
0.2
0.1
I
i
I
i
I
10 o
20 °
30 °
40*
50*
6 Fig. 11. Ci(/i) versus ,5 at Q2/A2 = 2000.
I
87
88
K. Konishi et al. / J e t calculus." a simple algorithm for resolving QCD /ets 1.0 0.9 0.8 0.7
g g log C2/Iog C6
0.6 0.5 Cg m
0.4
0.3
Log c~/togC~
0z
o2. 6 : 0.423 o 0.537 2,4 =
a o
0.1
I
0.1
0.2
3,5
= 0.706 = O. 788
4,6
i
0.3
0.4
0.5
0.6 0.7 018 0'9 1'.0
c~ - - ~
Fig. 12. Plot of log Cgn venus log Cgm. am, n is the effective slope, defined by log Cg "" am, n log Cgn+ const. i i ~kn ~n where the values an, fin, +, _ are given in table 7 and some of the m o m e n t s are plotted in fig. t 1. One can see that for m o m e n t s of order 3 or higher the quark jet gets essentially a c o n t r i b u t i o n only from the first term in eq. (5.29) so that a relation like (5.28) still holds. For gluon jets instead the two terms in (5.29) tend to be comparable and the full form has to be used. Actually even in this case the expected curves in a log Cng, log Cgm plot come out most of the time to give a good a p p r o x i m a t i o n of straight lines, as shown in fig. 12. Our m e t h o d is clearly reminiscent of the one used to check QCD scaling violations in n e u t r i n o experiments, in particular by m e a s u r e m e n t o f x F 3 at different values of Q2. The m e t h o d we propose * (varying 6 at fixed Q2) is more suited for the case of e+e - ~ hadrons, b u t we wish to point out that its exploitation requires
* A similar technique for testing QCD, with special emphasis to large FT processes has been advocated independently by Furmanski [52]. We wish to thank him for discussions on his approach to this question.
K. Konishi et aL / Jet calculus: a simple algorithm for resolving QCD jets
89
use of calorimeters which measure total hadronic energies including neutrals. Turning our arguments around and assuming QCD to be valid, one can use eq. (5.29) to determine the nature of a given jet, i.e., whether it is a quark type (e.g., e÷e - off resonance) of gluon type (e.g., on top of a heavy quarkonium resonance) or of mixed nature (as expected in large-p T hadronic processes), in which case the proportion of quark to gluon jet could be estimated.
5.3. Jets produced by initial partons We now turn our attention to the existence in QCD of jets associated with "initial" partons, i.e., with partons which enter in the hard elementary process. Consider for instance, large-pT hadron-hadron collisions. These are described at the constituent level by a two-body process in which one parton from each hadron collide to produce two large-pT partons. The jets we have considered so far come from the evolution of these final partons, but, of course there will be also hadrons going in the forward and backward directions. Many of these hadrons are due to spectator partons and are expected to exhibit limited PT, but also the active initial quark through its evolution creates a jet of partons (and then of hadrons) with PT growing with energy. It is this last type of jets that we shall refer to as "initial parton jets". Since bremsstrahlung from the initial partons is associated with scaling viola-
b,y~O
~
y-O X
a Xb y~Y
Ca)
b ~------O
-"
(b) Fig. 13. (a) Initial patton jets. (b) Corresponding jet calculus diagram.
90
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
tions, the appearance of initial parton jets is expected to reveal itself when sizeable scaling violations occur, hence, presumably at the highest values of PT or Q2 presently attainable. Our techniques can be readily applied to this new situation. However, because of some crucial difference in kinematics, the resulting expressions will differ somewhat from the corresponding ones of ordinary final parton jets. Consider indeed the process drawn in fig. 13, in which a parton of type i in an initial hadron evolves and, before triggering the hard process as parton a, radiates other partons of which we study the inclusive spectrum (fig. 13a). The inclusive cross section for observing parton b with definite x b and a certain PT relative to the initial hadron momentum can be computed from the diagram of fig. 13b where PT now determines the mass of the patton a'* thus fixing the intermediate "time" of the vertex j ~ a'b'. It is then easy to obtain do(h~a + b + X)_ dxadXbdY
a',b',j
fdxdgdxldX
2 Gaa'(X 1, Y - y ) O b b ' ( X 2 ,
X P j ~ a , b , ( Z ) GJh(X, y ) t~(X a -- XZX 1) ~ ( x b -- x(1 - z ) x 2 ) ,
Y)
(5.30)
where 1 Y=2~
%(m 2 ) l°g as(4p 2) '
1 as(m 2) Y= 27rb log as(Q2)
(5.31)
As in the case of ordinary jets one can replace b with a hadron h' by just replacing Dbb' in eq. (5.30) with D~',. Actually, in this case, no further term such as the second one in eq. (5.8) is present. From eq. (5.30) we can go to moments in Xa, Xb and obtain
=(a,blexp((y_y)An)exp(YAm)Pn, m
(do(h ~ a + b + X ) ) dY
n~rn
Xexp(YAn+m)li) G~(n + m, y
= 0).
(5.32)
The differences between this equation and eq. (5.10) stem from the fact that, whereas for final parton jets energy and qT decrease together from the hard vertex to the final partons, here they decrease in opposite directions. This fact will be reflected in some phenomenologically interesting differences between the two cases. To be definite let us consider again some simple moments which are free of infrared difficulties. For instance, we take a to be a valence quark (as measured, e.g., ~' Notice again the difference with DDT where it is parton a in fig. 13b which must have a fixed mass of order x/p~Q 2 < < Q2.
91
K. Konishi et al. / Jet ealcuhts: a simple algorithm for resolving QCD /ets
in xF3) with n = 0 and we sum over b with m = 1. Using charge and momentum conservation one gets =~as(4p~) /{-~n ANS\)G~al(1,Yo)J"1 t °q(m2) IANlSl2~rb L ~TT [as(4P~) )
(~)o,1
( \S~-lG~a(1,y ).
-- p~
(5.33)
As in eq. (5.11), we recognise in the square brackets of eq. (5.33) the naive result, but now the correction factor decreases with PT instead of increasing. In other words, one gets a PT distribution which is narrower (and not broader) than the naive one (see fig. 14). Concerning the normalization of eq. (5.32), we notice that here not all the energy is available to partons a and b since a fraction of it is carried away by the spectators. Because of this the cross section is normalized to Q2
dp T d(~T)
f
0,1
qo
=~[GhVal(1,Yo)-G~a(1, Y)] ,
(5.34)
val
and not to ~val G~al(0, Y) = 3 (for a proton). In order to illustrate further the characteristics of this type of jet we have computed Qz
dPx Pr(do/dPT)O,1
J
=
and found
Q2
,
(5.35)
f (_~s]
(P~) = ~ !
¢
Z~c~:'(1,v,r-)
l°gfQZ/A2) ~G~ al(I,Yo) - G~al(l, Y) val
XYlarge(-A'~ Q2 Fas(m2)]A~s/2"b ~2,b: log(aVA2) L~(-(-~T/
(5.36)
This result can be contrasted with that of er~ (5.13) for a final jet which does not exhibit the last damping factor (log Q 2 )a I ]21rb , typical of scaling violations.
92
K. Konishi et al. / J e t calculus." a simple algorithm for resolving QCD jets 10 S
I
Q/A: 100 (
INSIDE 1 JET A •
TRANSITION REGION r
2 JETS A
100
% x
% ,-:." E
O o W~
10
1
2
10
5
50
PT/A Fig. 14. Our results for initial patton jets, eq. (5.32) compared to the naive result, eq. (5.12), and final parton jets, eq. (5.11).
The physical origin for these different behaviours is to be found in the fact that in the "initial jet", a large Pa" (requiring a large mass) can only occur after a long evolution which has degraded its energy by repeated emission of quanta whereas in the "final jet" the original parton emits large-pT quanta (i.e., has a large mass) right at the beginning of its evolution. An interesting way to test QCD emerges from its prediction of initial parton jets. Since the (PT) of such jet (relative to the beam axis) grows with Q2 (the scale of
K. Konishi et aL /Jet calculus: a simple algorithm for resolving QCD jets
93
the hard process) one predicts that, in deep hadronic reactions, the average PT of beam and target fragmentation products should increase with the PT of the largeangle jet. Some experimental indications supporting such effects have been recently reported [53].
6. Jet field theory: outlook From the discussion in sects. 2, 4 and 5 we have seen how one can give a physical meaning to jet calculus amplitudes representing inclusive cross sections for an initial parton o f x = 1 a n d y = Y giving rise to a set of final partons characterized by (xi, Yi) (fig. 15a). The values x i correspond of course to the energy fractions carried by the "detected" partons whereasy/correspond to how much off-shell the final partons are allowed to go. For Yi ~ 0 the ith parton distribution has to be folded
(o)
= l (yx :=Y)
(b)
~
(Xl ' Yl )
~ ~ " ~
( xn' Yn)
(x2, Y2 )
~
(x,Y) (xi,Y i )
(x=l,y 1)
(c)
01
a2 = 0 (~i-~2)
[ O ( y I -Y2 ) 0 o 2 o I ( ~ 2 - ~ I ; Yl-Y2 )
•.. O ( y 2 - Y l ) O o 2 0 1 ( ~ 2 - ~ 1 ;
(d) ~ Y - 3
y2-Yl)]
= o(e~ -e~2-e~3)O(yl-y2)O(y2-y3) P - ¢~.,-L) aI ~
0203
-
,
Fig. 15. (a), (b) General Green functions in the jet calculus. (c), (d) Feymnan rules of jet field theory (?), propagators (c) and vertices (d).
94
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
into a phenomenological D h fragmentation function, whereas for Yi large no such factor is needed and one is already computing the cross section for depositing an energy Ei(xiE ) into a calorimeter of angle 8 i given by 1 lo- as(P2) Y i = ~ b Sas(Si~Q2 ) •
(6.1)
Furthermore, considering initial parton jets (as in subsect. 5.3), one can also have Green functions such as G(x 1 = 1 , y I " 0 ;
x2, Y2; x3, Y 3 ~ Y ) ,
(6.2)
where (x 1 = 1,y I = 0) is the initial quark, (x2, Y2) is a forward jet detected by a calorimeter and x a = XBj. Hence, in general, the Green function G(X, y ; {xi, Y i ) ) ~ G ( z ; z i )
(6.3)
has a physical meaning with z = (x, y ) (fig. 15b). One can ask at this point if a general Lagrangian formulation is possible in order to reproduce our jet calculus rules. The answer is yes, but the resulting Lagrangian (defining jet field theory) is non-local. Its structure is easily found to be given by the following action A =Aki n +Ain t ,
Akin =
d ~ ' d y ~+(~', y ) [Sat~ 0~y 8 ( ~ - ~ j ' ) - Pa,t3(~' - ~)] qJt3(~,Y ) ,
Aint = f d ~ l d ~ 2 d ~ 3 d y
~;(~2, Y) ~ ( ~ 3 , . F ) / 6 a ~ 3 7 ( ~ 2 - ~1) ~ a ( ~ l , Y)
X 6(e ~1 -- e ~2 -- e~3),
(6.4)
where ~ = log x, ~, ~+ are destruction and creation operators of partons and P, fa are the same quantities we defined in sect. 2. The Feynman rules derived from eq. (6.4) are straightforward (fig. 15c, d): (i) for each propagator from z I = (~1, Yl ) to z 2 = (~2, Y2) write a factor Oa2al(Zl, z2) = 0(~1 - ~2) [0(,Vl - Y2)Oa2al(~2 - ~1 ;YI - Y 2 ) + 0(Y2 - y l ) D a 2 a l ( ~ 2 - ~1 ;Y2 - Y l ) ] ;
(6.5)
(ii) for each vertex a l ( z l ) ~ a2(z2) ~ a3(z3) write a factor 6( e~l -- e~3)/13al--~a2a3(~2 -- ~1 ) ~(V1 -- Y2) 6(Y2 -- Y3) ;
(iii) perform flat integration over all intermediate z i.
(6.6)
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
95
Of course this is a complicated non-local theory. There are however two points which may lead us to hope that it is not completely untractable: first the theory is in 2 dimensions, second it is only to be solved in the tree (classical) approximation. It is amusing to notice at this point that, in the case of ~b36, the theory can be made local by a suitable reshuffling of factors between vertices and propagators. Unfortunately this trick does not work in QCD, due to the a more singular infrared structure. We would like to conclude this paper with an optimistic outlook based on the fact that we have kept asking more and more ambitious and detailed questions about perturbative QCD (e.g., from one- to two-particle spectra, from longitudinal to transverse structure ...) and we have kept getting sensible and (probably) reliable answers to be confronted with data. Which is the limit beyond which we are forbidden to ask questions to perturbation theory? The answer seems to be that, within jet physics, we can ask questions up to the stage in which, through jet evolution, masses and relative transverse momenta of partons have been degraded down to values a few times larger than A "" 0.5 GeV. Up to there, the jet calculus formalism we have outlined seems to be applicable and powerful. Moreover, it has been possible recently [54] to follow in this way not only the pattern of momenta as produced in jets but also that of colour. Surprisingly perhaps, one seems to be able to identify finite-mass colourless clusters of quarks and gluons in the jet's final state (preconfinement). There seems to be a contradiction between this intuitive picture of hadronization in which bunches of final partons in colour singlet configurations give the final hadrons, and one in which each individual final parton gives rise to a bunch of hadronic products. We wish to point out, however, that there is not necessarily a contradiction between the two pictures: indeed any perturbative calculation has to stop at a stage in which the final partons are still quite far from their mass-shell (q2parton > > A 2), whereas the partons which make up hadrons are closer to it (qp2arton ~ A2). It is conceivable that, whereas far off-shell partons yield hadrons separately, almost on-shell ones have to group in a colour singlet system before binding into hadrons. This important issue clearly deserves further study. If this hint is confirmed by further, more rigorous investigations, we may end up with the surprising and exciting conclusion that perturbative QCD, when translated in the precise and simple rules of jet calculus, can lead to a powerful tool for predicting the properties of final hadronic clusters in jets, where the only (but by no means trivial) task left to non-perturbative confinement will be the statement that such low-mass colourless dusters cannot be further resolved into their constituents. We thank D. Amati for discussions. One of us (K.K.) would like to acknowledge interesting discussions with D. Duke, M. Fukugita, N.A. McCubbin, P. Scharbach, J.C. Taylor and J. Wosiek and thank CERN for the very warm hospitality enjoyed at the Theory Division at various stages of this work.
96
K. Konishi et al. /Jet calculus." a simple algorithm for resolving QCD jets
Appendix A An induction proof of jet calculus rules We present here an induction proof of the jet calculus rules, (i)-(iii), of subsect. 2.2 for computing Dal ...am,i(n 1, n2 .... nm ; E). In the cases rn = 1 and m = 2, we obtain, by computing the (dressed) ladder diagrams of fig. lc and fig. 2a, yl Da,i(n; Y) = (eanY)a i = ~ . l
(An) 1 ,
(A.1)
and oo
Dala2,i(nl, n2; Y)=
(ll + 12)! yl+ll+12+l
~
bl,b2,J l, ll,12=O ll! 12! (ll +12 + l + 1)!
X (An)lalb1(Zn)la22b 2 pblb2,J l nl,n 2 ( A n l +n2)ii ,
(A.2)
respectively, where the factorials arise because of the ordering in the k2's over which we integrate to obtain the maximum number of logarithms (LLA), while the structures containing An and Pnl,n 2 appear as a result of integration over the longitudinal momentum fractions. When a Laplace transform is made from variable Y to E, eqs. (A. 1), (A.2) clearly agree with the jet calculus rules. Before going into the case of general m, we introduce some notations. Let us call J-diagrams the tree diagrams which can be obtained from the original Feynman diagrams by omitting all unobserved patton lines. (Our jet calculus diagrams such as in figs. ld, 2b, 3a can be regarded as J-diagrams). A J-diagram G for k-parton distributions has k - 1 vertices labelled as v 1v2, ... vk, and 2 k - 1 lines labelled by a = 1, 2 ..... 2k - 1. In the original Feynman diagrams (such as figs. lc, 2a), each line has additional la vertices (a = 1, 2 ..... 2k - 1). Consider the contribution to Dk(Y) from a Feynman diagram specified by a Jdiagram G and a set of integers (la). (Here and below we omit parton indices i, al ... ak and moments nl, n2 .... n k for simplicity). This has the form, k--1
2k--1
×({la}) t=ll--I P v , t a=ll--I AtaY k-l+zala .
(A.3)
It is convenient to consider the total contribution to Dk(Y) from G with l = Y.la fixed, D~'t(Y). According to the jet calculus rules, it is given by
DG,t(y)
yl+k-I (l + k -
1)!
~
f°+i** dE Et+k_ l 1 PE-1 p ,-' 2n--iE - s~, s~2 o--i ~
K. Konishi et al. / Jet calculus: a simple algorithmfor resolvingQCD/ets 1
•" E _
~k
97 (A.4)
,
while the total contribution of the diagram G is
D~(Y)- ~
D~'I(Y)
l=O o + i*o
= ~f a-i,*
2nidEeEr'E 1.~IPE 1 P... P - -1 , - ~2 E- ~k
(A.5)
where -~i represents an appropriate sum over A's (see fig. 3b), and the summation is over the different time orderings of vl, ... vk_ 1 . Now assume that eq. (A.4) or eq. (A.5) (equivalent to the jet calculus rules) is true for k ~< m - 1, and I = 0, 1, 2 .... Any J-diagram G for m observed partons can be uniquely decomposed into two J-diagrams G 1 and G2 having kl and k2 observed partons (kl + k2 = m, 1 ~< kl ~< k2 ~< m - 1) and a line O which are joined at a vertex v (see fig. 16). Denote by ll, 12 and l the vertices representing the emission of unobserved partons i in the original Feynman diagrams of G1, G 2 and the line O, respectively. We find then, oo
D6m(y)=
~ GI,I l G ,12 l l+1 (11+12+m--2) ! ttl,t2=o Dtq (Y)Dk~ (Y)Pv A Y (l+ll +12 + m 1)! ' (A.6)
where the last combinatorial factor arises (as before) from the transverse momentum integrations with the ordering among k2's along the line O.
t ffl kl
Fig. 16. Decomposition of a J-diagram G into two J-diagrams G 1 and G2 and the line O.
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
98
We substitute eq. (A.4) into eq. (A.6), go over to the variable E by a Laplace transform, and then perform the summation over/, l I and 12. This gives,
DCm(E) ? dYe-EYDG(y) =
0
°l+i~ (]E l
1
1 s~l p,.. p. E l - 61~ 1 kl
I'j 271i E l o 1 ...i ~
1
1
x ~ f:+i= de2
1
E_EI_E2 PVE~ A'
1
1
2 o2-i~ 2hi E 2 - ~ P " ' P E z -
(A.7)
s~k22
where we have dropped vanishing terms such as N~1o2 E -1 (E1/E~ (1 - E2/E)-(L÷I).Our aim is to show that eq. (A.7) is equivalent to eq. (A.5) for k=m. For this purpose consider a fixed time ordering of vertices in G 1 and G 2 separately, and label them from left to right as vl, ..., v~ 1 and v I ..... v? 1" The rela1-~2tive ordering of these two sets is specified by saying that there are qi vertices of G 1 between v 2 1 and v 2 of G 2, where i = 1,2 .... k2, and qi are non-negative integers such that Nf2qi = kl - 1. Then the desired relation (the equivalence of (A.7) and (A.5)) can be expressed in the form,
:. 577-.~J~
= ~, - ~ , k2
:H Xql~=kl--1
= sr~ --~J
sI-I, s>s' ,= P~ ~ - e ,
P[3
k 1-1
,
13=1 °t-=Pfl--1 E -
H
~1 _ ~
-s~
k2-1
.,N
(A.S)
]='*=*l "] ~=li
where Po -1) - p1,3s - Ef=lqi + 1, and:" represents the time (normal) ordering of P ' s and ( E - M . We prove eq. (A.8) by induction with respect to kl. For kl = 1, it can be easily checked. Assume then that it holds for kl ~< n - 1. For kl = n, we use an identity, kl
FI
1
a:lE 1 - s~
-
kt-2
1
(
CtH1
.
.
1
.
1
)
1
. (A.9)
99
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
on the left-hand side of eq. (A.8). But then by the induction hypothesis, the 1.h.s. becomes
1
-i-I
I-]
1
k2 Z q'6 =kl --2
6=1
1
X .9{11_1
__
.~lkl [IPI1P : ,
(A.10)
where Po - 1, p~ ~;i=lqi + 1, and in the second term in the curly bracket, the replacement s~ 1 . ~ M~ should be made. Consider a term in the sum and denote 1--* 1 t by/3o the smallest/3 such that Pao = kl - 1. Then this term can be written as B°[~i1
~
1
,
E-
× {
)(
1 E-
l 1_ -
~ 0 E-
) _
1
_
-
q
~o
W. - al
: o
s~ll_ 1 - s ~
-
1 -
,:qE--~l
s~l-1
1- s~
-
liP l i P : . (A.11)
Substitution of this into eq. (A. 10)evidently gives the right-hand side of eq. (A.8). This completes our derivation of jet calculus rules. Appendix B Details o n
induction
proofs
of various
sum rules
B. 1. M o m e n t u m sum rules (eq. (3.24))
We present .the proof here for hadron distributions; the one at the parton level can be obtained by making a formal replacement, hl k
at the end.
5kl
100
K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD ]ets
Elementary relations needed are * / .....
°
.
nol
,
\
~h j/dXhXhdh(xh)= I)
1
C -Xb0 o-y
.....
~
n
=
n~1
,
(B.1)
(B.2) (B.3)
(this follows from a Ward-Takahashi type identity, Pnb,Cia-- A nba - A nba + 1, and eq. (B.2)), and Pnbe,a
,re+l-=
+ pbc, a
n+l,m
= pbc, a n,m
(B.4)
"
The h a d r o n distributions (see fig. 4) satisfy a recursion formula,
hl,nl
k-I ~"~T~.. I--ik I
where ~div. means a s u m m a t i o n over all possible ways of dividing (h I ... hk) into two groups, consisting o f l and k - I lines. Furthermore, the non-perturbative decay functions d a' {hi }(Z 1 ... zj) satisfy themselves a set o f m o m e n t u m sum rules,
=
!
}-hknk=I""-
hk-1nk-1
(B.6)
z
hk.1nk-~
The p r o o f of the m o m e n t u m sum rule for k = 1 is trivial if one uses eqs. (B.1) and (B.2). The case k = 2 is also easy, as
n.1~ h,n =
n
h,n
e 0----
n;*1C)_~n.i
(B.7)
Next, we assume that the sum rule is valid for k ~< m. Then for k = m + 1 we find, using eqs. * The general proof is easiest in the E-description of moments, as is done below, since each distribution is a simple (matrix) product of various vertices, propagators and (non perturbatire) decay functions.
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
101
(B.1)-(B.3), (B.5), (B.6) rri
=
.
rr,, :
÷
.. div. I=I
--~[~.
;
h,,',,
m
~'}r.-i
"'..
Z.~:L@C._. h, o, . -
~"~hn~nm
"~hn-~nrn
m-1
~,~ hrnr oI
)" r~ *'
'
~
=
'
hr nr*'
~ h m . n
(B.8)
fl,I
where we have used a shorthand notation, m
~l-
~rn =- C hi, i= 1
C
upper blob
nj,
~ m - , =-
C
lower blob
ni •
Consider the third term of eq. (B.8) first. For a fixed division (hi l ... hh/hh+ 1 ... hi m ) of (h 1 ..- hm ), there are two terms, having the vertices Pzl, Z m - l + l and P z t+ l , z m - t ' respectively (but otherwise identical), which can be combined using eq. (B.4). The third term can now be summed with the fifth term, which together cancel exactly the second term by use of the recursion formula, eq. (B.5). The rest, the 1st, 4th and 6th terms, in which all the moments are conserved everywhere, can again be simplified by using eq. (B.5), giving Zm
~1
hl,nl
~
~'m*1 ~ ,
-
hm71m
r=1
hlnl hr nr÷l
.t... hm,nm
which proves the desired result for k = rn + 1. B.2. Charge sum rules (eq. (3.25)) We introduce a new propagator,
o
(B.9)
-=(E -/T,n)-' ; ~'n--0
describing the propagation of a "valence" quark. Denoting by i the given flavour quantum numbers of a quark, elementary relations necessary are given by (a wavy
102
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
line denoting the operation of integrating over the momentum fraction of the valence quark),
~1
i
f
i0 i ~
b,n
=
dx i
i
ioi
i f
Di" (.x i Y ) n
b
)
(B.10)
= 1
- 8bi
i
n
(B.1 1)
i
i ° jr ~
•
(B.12)
The lowest sum rule is nothing but eq. (B.11) which is easily derived from A0qq = 0. We next assume that the charge sum rule, eq. (3.25) is proven for k ~< m. Then, for k = m + 1, we have (by using the recursion formula),
b1
o~ - t
[:1 div
div
1=~ div
r
bri b
where use was made of eqs. (B.I0)-(B.12). The 2nd and 3rd terms ofeq. (B.13) are seen to cancel by using eq. (B. 12) and the recursion formula, while the last term can be simplified again by use of the recursion formula, thus the result becomes
=
~
w h i c h is t h e s u m r u l e f o r k = m + l .
m-
ZSbri
br
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
103
Appendix C Quark,ghon universality We present first two sufficient conditions for quark-gluon universality and then show, using Nachtman's positivity constraints, that one of these sufficient conditions is satisfied for n i/> 2 (i = all). (i). The first sufficient condition. A sufficient condition is that the distribution is dominated by a single pole (the largest eigenvalue) coming from the initial propagator, (E - A N ) -1 , where N - ~ini . In order to see this, let us write
(AN)ha = ~k~V(~Ar)ba + ~kN ( ~ N ) b a , where XN are the two eigenvalues o f A N and and 5aN respective projection operators. If X~v dominates the whole distribution, then one can make a replacement, +
(E - AN)~a1 -+ (E - X+N)-I ( ~N)ba •
Using the fact that ~ v is a matrix of rank 1, +
+
( ~ N)jq/(~)N)jg = rN ,
(indep of j ) ,
one then easily derives (indep of ( b i } ) .
O q~(bi)rvxtr~g'~{bi) (Y)~rN (ni } t " )/L" (ni )
This condition is satisfied in the examples, eqs. (4.5), (4.7) and (4.9). (if} The second suffiicent condition. Another sufficient condition is that the largest pole comes from the final propagator, (E - ~,iZni )-1 as in the case of multiplicity correlation coefficients. In this case we make a replacement (E
_
~
Ani)-I "~ (E
i
_
~
X+.)-' [I i
t
i
+
(~ni)bib
,i .
Again, using the fact that ( g n )+q j / ( 9
n+) g j = S n ,
(indep of j)
we easily derive that D q-~ ( b i ) ( y a l D g~ (bi ~ (hi} " " ( n i )
(Y)=f(ni}'
(indep of {b i })
(iii) Proof that for n i ~ 2 (i = all) the largest pole comes from the initial propagator. The Nachtman's positivity condition gives +
+
+
2Xn+ 1 ( X n +Xn+ 2 •
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD /ets
104
This, together with the relation 2X; < X,],
(e.g. f o r N f = 4, X; = - 2 . 5 4 , X,~ = - 4 . 0 0 ) ,
leads to the inequality +
)k+ + )k+ < ~'m+n
for all m/> 2 ,
'
n t> 2 .
There is a conservation of moment at each vertex, therefore the above inequality immediately leads to the conclusion that the largest pole comes from the initial propagator, (E - X~v)-1 , where N = Zin i.
Appendix D A jet calculus derivation of eq. (5.23) We wish to show here that eq. (5.7) for a 2-parton spectrum implies eq. (5.23) for calorimeter-measured cross sections. The latter equation has been argued [32] to be valid in a range e < x < 1 - e', but we shall see fror~ our derivation that such limitations are less severe than conjectured and do not prevent the possibility of taking (positive) moments of eq. (5.23) to yield the simple result eq. (5.26) whose applications we discussed in the text. It is actually convenient to go back to eq. (2.8), from which (5.7) follows if one fixesy. For our purpose, we have to fix b o t h y =y~ a n d x and then use a sum rule in order to get a unit branching ratio for j to decay into anything. This we can do by performing the sum rule: ffdXldx2Xl ala2,d
X
x2 X--
1 do(i-*a I +a 2 + X ) dXldX 2
X2 U
x,y
fixed
which avoids multiple counting and takes care of the case of fig. 2b in which al, a2 are coming from bl and b2 respectively. Since, however, a I and a2, as arbitrary decay products of j, could happen to be both products o f b 1 or b2, we see that we still have to allow an evolution from y down to y' before the splitting, and integrate overy'. The correct formula is then:
f
X1
X2
v ff
1 d°i(6)- ~ dxldX2 -3 Oijet dx ala2 ~ x x-x 2 0
×
~ b 1 ,b2,j,j'
dy'
f d z d w d w l d w 2 D a l b l ( W l , Y')Da2b2(w2, Y')Pj'-*blb2(Z)
K. Konishi et al. / Jet calculus." a simple algorithm ]'or resolving QCD ]ets
105
X Dj'j(w, y - y ' ) Dji(x, Y - y ) 8 (x 1 - x w z w I ) 6 (x 2 - xw(1 - z) W 2 )
y f
dy, f d x 2 d z d w z d w z
az,bl 0 b 2 ,jj'
w
(1-z)
ww2
1 - - ( 1 - - Z ) WW2
Da2b2(W2,y ) Pj'~blb2(Z) Dj j ( w , y - y ) Dji(x, Y - y ) , where the momentum sum rule for Dal bl has been used. Expanding the denominator (1 - (1 - z) ww2) -1 and replacing P with P because of the factor z(1 - z) one easily gets in the vector-matrix notation of eq. (2.13) and ref. [25]: y
oo
~
f
a2 n = o
de'
- y)li)
o 0o
dy' a2
[(a2 lD(n + 1, y') D(n + 2, Y - y ' ) ] D(x, Y - y)[i)
n=0
oo
(azl[D(n + 1 , y ) - D ( n + 2 , y ) ] D(x, Y - y ) a2
n= 0
= (a21D(1,y)D(x, Y - y ) l i ) = ~
Da2i(x, Y - y ) , a2
which is the desired result. Notice that, from this derivation, Dji(x, Y - y ) represents the number of parton of type j, energy fraction x and mass m(y) produced one jet of type i at Q2 = Q2(y). It can be easily argued however that, within the LLA, this is also, after summing over], the number of colorimeters of angle 5 getting an amount of energy xEjet deposited in them. It is also obvious from this derivation that the result holds for any x not infinitesimaUy (as Q2 ~ o0) close to zero and one so that (positive) moments can be taken).
References
[1] [2] [3] [4] [5] [6]
H.D. Politzer, Phys. Reports 14 (1974) 129; and references therein. H.D. Politzer, Phys. Lett. 70B (1977) 430. D. Amati, R. Petronzio and G. Veneziano, Nuel. Phys. B140 (1978) 54. G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436. A.H. Mueller, Phys. Rev. D9 (1974) 963. C.G. Callan and M.L. Goldberger, Phys. Rev. Dll (1975) 1553.
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K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD lets
[7] T. Kinoshita, J. Math. Phys. 3 (1962) 650. [8] T.D. Lee and M. Nauenberg, Phys. Rev. 133 (1964) •549. [9] V.N. Gribov and L.N. Lipatov, Yad. Fiz. 15 (1972) 781 (Sov. J. Nucl. Phys. 15 (1972) 438). [10] J. Frenkel and J.C. Taylor, Nucl. Phys. BI16 (1976) 185. [11] H.D. Politzer, Nucl. Phys. B129 (1977) 301. [12] C. Sachrajda, Phys. Lett. 73B (1978) 185; 76B (1978) 100. [13] C. Llewellyn Smith, Schladming Lectures, 1978, Oxford Univ. preprint, 47/78; C.H. Craig and C.H. Llewellyn Smith, Phys. Lett. 72B (1978) 349. [14] Yu. L. Dokshitzer, D.I. D'Yakonov and S.I. Troyan, Proc. 13th Winter School of LNPI, Leningrad (1978), vol. 1; (English Translation, SLAC-TRANS-183 (1978). [15] D. Amati, R. Petronzio and G. Veneziano, Nucl. Phys. B146 (1978) 29. [16] R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.C. Ross, Phys. Lett. 78B (1978) 281; Nucl. Phys. B152 (1979) 285. [17] J. Frenkel, M.J. Shailer and J.C. Taylor, Nucl. Phys. B148 (1979) 228. [18] S.B. Libby and G. Sterman, Phys. Lett. 78B (1978) 618; G. Sterman, Phys. Rev. DI 7 (1978) 2778. [19] W.R. Frazer and J.F. Gunion, Univ. of California preprint, UCSD-10P10-194 (1978). [20] W.J. Stifling, Nucl. Phys. B145 (1978) 477. [21] A.H. Mueller, Phys. Rev. D18 (1978) 3705. [22] Yu. L. Dokshitzer, Leningrad preprint, 330 (1977); JETP (Soy. Phys.) 72 (1977) 1216; Y. Kazama and Y.-P. Yao, Phys. Rev. Lett. 41 (1978) 611; Univ. Michigan preprints UM HE 78-53, 54. [23] G. Veneziano, Proc. 19th Int. Conf. on High-energy physics, Tokyo, 1978, p. 725. [24] Yu. L. Dokshitzer, D.I. D'Yakonov and S.I. Troyan, Phys. Lett. 78B (1978) 290; 79B (1978) 269. [25] K. Konishi, A. Ukawa and G. Veneziano, Phys. Lett. 78B (1978) 243; K. Konishi, Physica Scripta 19 (1979) 195. [26] K. Konishi, A. Ukawa and G. Veneziano, Phys. Lett. 80B (1979) 259. [27] C.L. Basham, L.S. Brown, S.D. Ellis and S.T. Love, Washington preprint, RLO-1388-759 (1978) and RLO-1388-761 (1978). [28] P.M. Stevenson, Nucl. Phys. B150 (1979) 357. [29] G. AltareUi and G. Parisi, Nucl. Phys. B126 (1977) 298; G. Parisi, Proc. 11 th Rencontre de Moriond (1976), ed. J. Tran Thanh Van. [30] J.C, Taylor, Phys. Lett. 73B (1978) 85. [31] E. Witten, Nucl. Phys. B120 (1977) 189. [32] R.K. Ellis and R. Petronzio, Phys. Lett. 80B (1979) 249. [33] G. Curci and M. Greco, Phys. Lett. 79B (1979) 406. [34] S. Karlin and H.M. Taylor, A first course in stochastic processes, (Academic Press, N.Y., S.F., London, 1966). [35] A.M. Polyakov, JETP (Sov. Phys.) 32 (1971) 296; 33 (1971) 850. [36] S.J. Orfanidis and V. Rittenberg, Phys. Rev. D10 (1974) 2892. [37] A. Schwimmer, Nucl. Phys. B94 (1975) 445. [38] M.S. Bartlett, An introduction to stochastic processes, (Cambridge University Press, 1978) p. 75, eq. (4). [39] N. Cabibbo and R. Petronzio, Nucl. Phys. B137 (1978) 395. [40] G. Veneziano, Proc. 9th Ecole d'Ete de Physique des Particules, Gif-sur-Yvette, (1977) vol. II, p. 23, and references therein. [41] A. Giovannini, private communication. [42] A. Giovannini, Nuovo Cim. 10A (1972) 713; A. Giovannini and H.B. Nielsen, Niels Bohr preprint NBI/HE/73/71.
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[54] [55]
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K.G. Wilson, CorneU Univ. preprint CLNS-131 (1970). A.H. Mueller, Phys. Rev. D4 (1971) 150. Z. Koba, H.B. Nielsen and P. Olesen, Nucl. Phys. B40 (1972) 317. A.K. Wrobrewski, Proc. 4th Int. Symp. on Multiparticle hadrodynamics, Pavia, 1973, ed. F. Duimio, A. Giovannini and S. Ratti, p. 625. C.H. Llewellyn Smith, Phys. Lett. 79B (1978) 83. S.J. Brodsky, T.A. DeGrand, J.F. Gunion and J.H. Weis, Phys. Rev. Lett. 41 (1978) 672. R.P. Feynman, R.D. Field and G.C. Fox, CALT-68-651 (1978); R.D. Field and R.P. Feynman, Nuct. Phys. B136 (1978) 1. K. Shizuya and S-H. Tye, Phys. Rev. Lett. 41 (1978) 787. M.B. Einhorn and B.G. Weeks, Nucl. Phys. B146 (1978) 445. W. Furmanski, Talk at Symp. on Jets in high-energy collisions, NBI-Nordita, 1978; R. Sosnowski, Proc. 19th Int. Conf. on High-energy physics, Tokyo, 1978; CERN-TH2572; M.G. Albrow et al., Nucl. Phys. B135 (1978) 461. D. Amati and G. Veneziano, CERN preprint TH-2620 (1979); A. Bassetto, M. Ciafaloni and G. Marchesini, preprint SNS 4/1979. S.J. Brodsky and J.F. Gunion, Phys. Rev. Lett. 37 (1976) 402.
JET C A L C U L U S : A SIMPLE A L G O R I T H M F O R R E S O L V I N G QCD J E T S
K. K O N I S H I
Rutherford Laboratory, Chilton, Didcot, England A. U K A W A * and G. V E N E Z I A N O
CERN, Geneva, Switzerland Received 19 March 1979
It is argued that, in the leading logarithm approximation (LLA) of asymptotically free quantum chromodynamics (QCD), a simple alogrithm (jet calculus) can be justified for computing a large number of quantities related to the longitudinal and transverse structure of quark and gluon jets. We formulate the rules of jet calculus in two variables, x andy which are related to longitudinal (energy) and transverse momentum respectively. Throughout the paper, quark and gluon jets axe contrasted. After reviewing what has been done on longitudinal spectra, we turn to the transverse structure of jets down to relative transverse momenta of the order of a few GeV (independently of the total energy). The relation of our work to that of Dokshitzer, D'Yakonov and Troyan is elucidated. We also study jets generated by (almost) real photons and compare 15oth with standard "final" parton jets. We also discuss calorimeter measured quantities, compare our results with those of other authors and propose a new and easy way for measuring QCD-predictable anomalous dimensions. We finally give a Lagrangian formulation of jet calculus, as a non-local theory in 1 + 1 dimensions to be solved in the tree approximation.
1. I n t r o d u c t i o n and s u m m a r y
This past year has witnessed an impressive renewed interest in the use o f i m p r o v e d p e r t u r b a t i o n t h e o r y (IPT) for a s y m p t o t i c a l l y free theories and, in particular, for q u a n t u m c h r o m o d y n a m i c s (QCD). IPT has been applied in a variety o f new directions further and further away f r o m the classical use [1 ] in c o n j u n c t i o n w i t h o p e r a t o r p r o d u c t expansions for lightcone d o m i n a t e d processes. The o u t c o m e has been e x t r e m e l y rewarding, in the sense that a m u c h larger set o f QCD " p r e d i c t i o n s " is n o w b e c o m i n g available for confronting QCD w i t h present and very-near-future e x p e r i m e n t a l data. The general consensus is that these new predictions are n o t less reliable than the standard ones in e+e * Present address: Department of Physics, Princeton University, Princeton, New Jersey, USA. 45
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
46
hadrons and in deep inelastic lepton-hadron scattering. They all rely, of course, on the assumption that QCD confines (i.e., gives hadrons) and that the confinement mechanism is soft enough for not affecting appreciably the short-distance physics of hard processes. There are many processes which can be dealt with in this way (e+e - -+ n hadronic jets, e+e - -+ hadron + X, e+e - -+ hi + h2 + X (h I , h 2 are hadrons in two different jets), ep ~ e + h + X, h + h ~/~+/J- + X, h + h -+ t2+/./-(pT) +jets (--PT) + X; h + h 2 large PT jets + X, etc.). They are genuinely "hard" processes in the sense that a single large scale Q2 appears in the problem and that all ratios of large invariants are O(1). In this case, at sufficiently large Q2, %(Q2) l°g(Q21/Q 2) < < 1 ,
(1.1)
7r
where %(Q2) is the QCD running coupling constant. For these hard processes, the behaviour at Q2 ~ oo is related [ 2 - 4 ] to the behaviour at m ~ 0 (m being the typical mass of a patton inside an initial or a detected final hadron *). The Q2 ~ ,~, problem is thus resolved to one of studying mass singularities (as recognised long ago by Mueller [6]) and this can be done by using general cancellation theorems [7,8] and rather powerful power-counting techniques [9,10] especially in physical (e.g., axial) QCD gauges. The most interesting outcome [1 1 - 2 2 ] of this is the "derivation", in perturbation theory, of that particular version of the parton model which incorporates gluons as well as quarks and which includes predictable scaling violations in structure and fragmentation functions. This confirms fully an initial conjecture of Politzer [1 1]. In spite of the large number of processes which can be covered by these standard techniques, the limitation to genuine hard processes, as defined above, is somewhat frustrating [23]. Take, for instance, the process h I + h 2 -~ 2 large PT jets + X. Clearly, at high energies, the region p T / g = O(1) is only a tiny fraction of total phase space; it is, moreover, the region where the cross section is smallest. Even if we cannot possibly imagine that QCD-IPT can predict this cross section at PT < 1 GeV (soft hadron physics) it is not inconceivable that it can predict it for the (semi-hard) region 1 GeV < < IPTI < < E = X/s (this should already be hard physics!). The trouble is that large logarithms o f E / p T pop out and, in general now cq(a2) log(E2/p 2) ~ 1 ,
(1.2)
/r
so that such large logarithms have to be resummed. Since these are not related to mass singularities, the previously mentioned techniques are of no help and one has to resort to other methods of resummation. * The simplest case is, of course, the one in which no such hadrons appear in the process (e+e - ~ n jets). In this case absolute computation of the process is possible, as shown first by Sterman and Weinberg [4].
K. Kolzishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
47
Summarizing, besides the "classical" (e÷e - ~ hadrons, ep ~ e + X) and the "standard" (i.e., hard processes involving one large scale) applications of IPT, we can imagine "new" applications of IPT to processes involving several large ( > > 1 GeV) mass scales. These processes cover most of phase space and involve relatively large cross sections, hence they are phenomenologically very important. Their importance is also theoretical though: understanding semi-hard hadron physics could provide the long sought bridge between hard hadron physics (where parton degrees of freedom are relevant) and soft hadron physics (where resonances, reggeons and pomerons become the right degrees of freedom). It could eventually lead to important hints as to how confinement itself might originate. A first step towards understanding these intermediate (or semi-hard) kinematical regions was made by Dokshitzer, D'Yakonov and Troyan (DDT) [14,24] who examined processes like h I + h 2 ~ £ + £ - ( M , PT) + X in the region 1 GeV < < [PTI < < M and obtained a resummation of the leading logarithms of M2/p 2. Unfortunately, they were confronted with double logs (per power of as) and they could only obtain a simple result (DDT formula [14,24]) in the limit
as(p2) log(M2/p~)<< 1 .
(1.3)
7r
This puts severe limitations on the region of applicability of the DDT formula (for further discussion of the DDT case and its relation to the present work see sect. 5). The work described in this paper also goes in the direction of semi-hard processes. It deals with questions of multihadron spectra inside a single (quark or gluon initiated) jet. Longitudinal as well as transverse spectra are considered with relative transverse momenta chosen to be larger than a few GeV. Both cases have already been discussed by us in two short notes [25,26]. As explained in sect. 5, we obtain, unlike DDT, single logarithms (.per power of as) and, consequently, the applicability of our results is only limited by the constraint ~ ( p 2 ) < < 1. lr
(1.4)
This excludes the low-p T region, of course. On the other hand we are also limited to values ofpT < < Ejet by the fact that we use collinear kinematics throughout. Fortunately, as one goes to PT <~E, the process becomes genuinely hard and can be dealt with by standard lowest-order calculations. In the cases under consideration we find that our results for the semi-hard regions connect smoothly with those derived by other authors [27,28] in the hard region. Hence, the whole region 1 GeV < < PT d E is under control. It can be shown further that only an infinitesimal (as E ~ ~ ) . fraction of the cross section belongs to either the non-perturbative region PT <~ 1 GeV or to the hard region PT ~ O(E). The intermediate region described by our formalism essentially picks up the whole cross section. The outline of the paper is as follows: in sect. 2, we first review for completeness the diagramatic, perturbative approach to structure and fragmentation functions and
48
K. Konishi et aL / Jet calculus: a simple algorithm for resolving QCD lets
the Altarelli-Parisi (AP) evolution equations [29] appropriate for describing their Q= dependence in the LLA. We then introduce immediately our jet calculus rules (JCR), on the basis of the physical diagramatic picture for jets obtained by the perturbative calculations. The rules are given directly in terms of "physical" variables x and y (related to energy and transverse momentum respectively) and the equivalence to the previous formulation [25] in terms of moments is proven. A more rigorous derivation of the JCR is given in appendix A. Sect. 3 discusses several non-trivial consistency checks (sum rules, check with a result of Taylor [30] in ?~4~3, etc.). Details of some proofs are given in appendix B. In sect. 4 we discuss purely longitudinal spectra in quark and gluon jets, distinguishing the finite x from the small x region (where some subtle infrared problems still limit predictivity). A subsection is devoted to (quasireal) photon-initiated jets where results similar to some older ones of Witten [31 ] are obtained. The question of universality (i.e., independence of several quantities of the type of final partons) is briefly discussed in the text, with details presented in appendix C. Sect. 5 turns to complete (longitudinal and transverse) spectra. After a qualitative comparison with the DDT work [14,24] we extract some simple inclusiVe cross sections (essentially longitudinal moments) from our general formulae (whose computation can be done in principle by computer), and we describe the general features of the results, comparing them with the naive predictions obtained in lowest order. We also discuss calorimeter-measured cross sections and rederive from our general formulae the particular results obtained by other authors [32,33]. We then describe a new scaling behaviour for integrated cross sections and show how that allows a new way to measure the anomalous dimensions predicted by QCD. We close sect. 5 by presenting results for "initial-parton jets" and by contrasting them with "finalparton" ordinary jets. Sect. 6 describes a general formulation of jet calculus as a two-dimensional field theory, whose non-local Lagrangian is given, This is followed by some short conclusions.
2. General formulation of the jet calculus rules 2.1. Structure functions, fragmentation functions and their evolution
For completeness, and also in order to introduce our notations, we start by recalling some well-known results on structure and fragmentation functions as they come out of perturbative QCD calculations in the leading log Q2 approximation (LLA) [11-22]. Although these results have been generalized to include all non-leading logs [15,16,21], for the extensions considered in this paper we shall restrict ourselves to the LLA. Because of the property of factorization and process independence found at all
K. Konishiet al. / Jet calculus."a simplealgorithmfor resolvingQCDjets
49
orders, one can define a structure function G~(x, Q2) (a fragmentation function gives the probability of finding a parton of type a in the hadron h (a hadron h in the fragmentation of a parton of type a) with a fraction x of the parent momentum in a hard process characterized by one large scale, Q2. The actual physical cross section for the hard process under consideration is given in terms of G and D functions by a convolution of the type:
Dan(X, Q2)) which
1
°(Pi;qj;Q2)= ~fai oj ~i dxiG~i(xi' L. 0 • t h. X I-I dzj ~-Dg](zj,
Q2)
Q2)
oo(ai(xiPi); bj(qi/z/);
Q2),
(2.1)
where Pi(qj) refer to incoming (outgoing) hadrons, ai and bi are parton species, and where Oo is the lowest-order cross section at the parton level *. Since the process is hard, by definition o 0 is free of infrared problems and can be computed perturbatively in the running coupling constant. The Q2 dependence of Gh, a D ah can be obtained from that of more elementary parton-to-parton structure and fragmentation functions, through the relations
b
Dh(x,
x
102
Q2):
h
~ f __sDb (x/Z)Dba(Z ' Q2), b
(2.2)
x
where Gbh(x/z), D~ (x/z) refer to structure or fragmentation functions at a fixed reference value of Q2 (say Q2 = Qg > 1 GeV). The parton level functions Gab, Dba (and therefore also Gbn and Dan) satisfy the Altarelli-Parisi evolution equations [29]: 1
~yGab(X, ]I)= ~ f ~Pac(g) Gcb(X/Z' Y), C
d-~dyDba(X, Y)= ~
C
x
f
~Dbc(X/Z, Y)Pca(Z),
(2.3)
x
where Pba(X) are the Altarelli-Parisi elementary parton decay probability functions for the process a ~ b(x) + c(1 - x), which are given in table 1. The "evolution"vari* The hard process may also involve currents and/or hard photons. These will appear in a 0 but, of course, there is no structure or fragmentation function associated with them.
K. Konishi et al. /Jet calculus." a simple algorithm for resolving QCD jets
50
Table 1 Patton decay probability functions b
a
q
q
P(a --* b(x) + c(1 - x)) ~- Pba(X) /1 +x2 1 - -
CF
g
q
1 + (1 - x ) 2
CF g q
g g
1N f (x 2 + (1 - x ) 2 }
g
g
2CA
+
+x(1 - x) - l ~ ( x - 1)
- I N f s ( x - 1) (In comparing with ref. [29 ], notice our different use of the ( )+ symbol.) C F = ( N c2 - I)/2N c ,
CA=N c, 1
= f
ax
-
.
o
able Y ( r a p i d i t y , i m a g i n a r y t i m e ) replacing Q2 is given by y:
i
F~stU~)7
-2~b logL@ ] ,
(2.4)
where e s ( Q 2) = (b l o g ( Q 2 / A 2 ) -1 is the o n e - l o o p QCD r u n n i n g c o u p l i n g c o n s t a n t (12gb = 11N c 2Nf in QCD w i t h N c colours a n d G o i n g over to m o m e n t s b y the d e f i n i t i o n s "
Nf flavours).
1
a~(n, r) - f dx x"a~(x, r'), o 1
Oha (n, Y ) - f
dx xnDha(X, Y ) ,
(2.5)
o
* Our definitions differ slightly from the conventional ones where a weight x n-I is used. As seen below, our rules are a bit simpler with this definition.
K. Konishi et al. /Jet calculus. a Simple algorithm for resolving QCD /ets
51"
Table 2 The anomalous matrices A n for the flavour singlet component
An
\ Ag q
b
Agng ] A nb a
a
n+l q
q
CF[ - 1 + 1/{(n + 1) (n + 2) } - 2 ~ (l/j)] /=2
g
q
CF(n 2 + 3n + 4)/(n(n + 1) (n + 2) }
q
g
Nf (n 2 + 3n + 4)/{(n + 1) (n + 2) (n + 3) }
g
g
n+l CA [_1 + 2/(n(n + 1) } + 2/{(n + 2) (n + 3) } - 2 ~ (1/])] - INf i=2 Cv=(Nc2 - 1)/2Nc, C A = N c.
one finds immediately the well-known [1] behaviour:
G~(n, Y ) = e x p ( ( Y - Y o ) A n ) a b G b h ( n , YO), nba(n, Y) = D h (n, Yo) e x p ( ( Y - Yo)An)ba ,
(2.6)
where 1
ab_ An - . f dz znpab(Z). 0
(2.7)
For convenience a table o f A ab is given in table 2. The basic aim of this paper is to show that a number o f new and interesting measurable quantities pertaining to jet physics can be obtained, in the LLA, from a simple perturbative algorithm ("jet calculus") and from the phenomenological input provided by the G~(x) and Dha(x) functions at a reference point appearing in the r.h.s, of eqs. (2.2). Occasionally, we shall even be able to obtain results which are independent of such phenomenological inputs.
2.2. Jet calculus rules We give here an intuitive derivation o f our algorithm based on the diagramatic understanding [.12-20] of the origin of eqs. (2.3) in QCD perturbation theory in the axial gauge. Further details and a more rigorous derivation can be found in appendix A.
52
K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD jets 2
(a)
Dbo -- )- )-TREE, CHOICE OF b
2 =)--
a
b
(b)
I
-~ ~
a =~I-I
r
~
(c)
b I
oo •
I I
A
=
a
-
," b
(d)
Fig. I. (a) Jets as branching processes in the LLA. (b) Choosing one final parton. (c) Reconstructing a dressed ladder. (d) Evolution a ~ b ~ la Altarelli-Parisi.
The diagrams describing, for instance, the y-evolution of a fragmentation function Dba(X, y ) are shown in fig. lc. The initial parton a, which is typically far from its mass-shell (p] ~< Q2), decays into two lower mass partons, which in turn decay into even lower mass quanta until, eventually, one ends up with a set of final partons of mass comparable to a typical hadronic mass. This first, perturbative stage is followed by a second non-perturbative stage in which patrons convert into hadrons and which has to be described in terms of phenomenological (uncalculable) quantities like the factor of eq. (2.2). Conversely, the first stage can be understood perturbatively as we shall now explain. The process shown in fig. 1 is a typical branching process of the type encountered in many fields of applied mathematics [34]. The possible relevance of bran-
Dhb(X/Z)
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
53
I al )-
J
I
bI
,._,~~
(a)
I
~,
b1~ (b) a 1
02 Fig. 2. (a) Choosing two final partons in fig. la and reconstructing a triple ladder. (b) Evolution i ~ a 1 + a 2 described by eq. (2.8).
ching processes in the physics of jets had been already pointed out in the literature [35,36] * The particular structure of this branching process in the LLA of QCD is one in which masses are strongly ordered along the branches of the tree, so that each parent parton is always m u c h further off-shell than its offspring (i.e., if a -->b + c, q~, q2c <~ eq2a, with e fixed and < < 1 ) . The other important property in the LLA and in the axial gauge is that no interference term is allowed when squaring amplitudes (as already indicated in fig. 1) so that our trees have a direct probabilistic interpretation. In order to obtain eqs. (2.3), (2.6) from the above picture, it is enough to realize that, for every choice of b in the final set of partons, we can group the other final partons in a unique way into groups forming the dressed rungs of a ladder, whose discontinuity is taken (see figs. l b , c). It is these ladder diagrams which have been shown [ 1 3 - 2 0 ] to exponentiate in Y and to give the result (2.3), (2.6). By repeating this line of argument in the case o f two detected partons b, c, one can easily show that all contributions can be represented by the triple-ladder structure o f fig. 2a. Since each ladder provides an AP evolution one obtains the * The concept of branching processes is also implicit in the one of the frayed jets or jets as fractals, often emphasized by R.P. Feynman (private communication).
K. Konishi et al. / Jet calculus."a simple algorithmfor resolvingQCDlets
54
picture shown in fig. 2b which, in formulae, reads I
do(i~ala 2 +X) dxldx 2 --Dala2'i(Xl' X2' Q2)
Oi jet
Y
1
f
dy f
bl'b2'J 0
0
dx dg dwldW2Oalbl(Wl,y) Oa2b2(w2, Y)
^
XPJ~blb2(Z) Dji(X, Y'- y) Y
l - - x 2 dz 1
bl'b2'J 0
x1
XDa I bl(Xl/Zt,
t~(X 1 --
XZW1) (~(X 2
/~ - x l (tz 2 x2
-- X ( I -- Z) W2)
1
Z Zl+Z
Y) Dazbz(X2/z2, Y)
XlSj-~blb2(Zl/(Zl + z2))Dji(Zl +z2, Y - y ) ,
(2.8)
where the "intermediate" rapidity y will be given a physical interpretation in sect. 5 and the meaning of the remaining integration variables as momentum fractions is self-explanatory. Furthermore, the quantity Pj~b lbZ(z) coincides with Pb lJ (z) or Pb2j(1 - z) provided we drop their terms proportional to 6-functions. This corresponds physically to requiring that the vertex j ~ b lb2 in fig. 2b describes a real (and not a virtual) process. It then follows that P j ~ b 1 b2(Z ) = Pj--~b2bl (1 - z ) ,
(2.9)
which ensures a symmetric treatment of a I and a 2 in eq. (2.8). Notice that, whereas the fragmentation functions Dal bl (Wl, y), Da2b2(W2, y ) are just the same as those of eq. (2.3) (with Y ~ y ) , a new generalized fragmentation function Dji (x, Y - y ) appears, corresponding to an evolution from y to Y, which is completely calculable if both y and Y correspond to large values of Q2. When, later on, we consider final hadrons it will be only the D(w, y) functions which will be converted into hadroniclevel fragmentation functions, whereas D(x, Y - y)will still remain an evolution at the partonic level. It is a simple matter to transform eq. (2.8) into an equation for double moments and to make thus contact with our previous formulation. Defining Dala2,i(nl,
n n2, Y) - f dxldx2x nIl xz2Dala2,i(Xl, x2,
Y),
(2.10)
one immediately obtains: Y
Dala2,i(n 1, n 2,
=
Y) f 0
b2'i dy (eYAnl)albl(eYAn2)azb2 P bl nl,n 2 (e (Y-y)An I +n2 )ji (2.11)
55
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets Table 3 • be a The vernces Pm, n
pbc,a
b
c
,
a
-m, n
q
g
,
q
CF(B(m + 1, n) +B(m + 3, n ) )
q
q
,
g
Nf(B(rn+ 3, n+ l) +B(m + 1, n + 3))
g
g
,
g
2CA(B(m, n + 2) +B(m + 2, n) +B(m + 2, n + 2))
B(m,n) is the Euler's beta function.
where the A n are as given before (eq. (2.7), table 2) and 1
P blb2'jnl,n 2 :- : dz znl(1 - z)n2/OJ~bl b2(Z ). 0
(2.12)
• The various Pnb I b-2 , J can be found for convenience in table 3. Going finally through 1,u2 a Laplace transform to the variable E we find
d y e - YEDa I a 2 ,i(y~ n 1,n2"- :
Dal a~'i(E) = Ul,rt 2 o (ala2
E-
1
A n l - An2 PhI'n2 E -
1
Anl+n2
i>,
(2.13)
where we have gone over to a vector-matrix notation. Eq. (2.13) coincides with the proposal of ref. [25]. Clearly all the reasoning can be generalized to the case o f an arbitrary n-parton inclusive spectrum both in the (xi, Y ) description of eq. (2.8) and in the (ni, E ) description o f eq. (2.13). For completeness we restate here the general rules for the moments, D (nl' n2"'" n.m; E). These are as follows. ala 2 ... am,l (i) Consider all the tree diagrams for an effective ~3 theory, in which energy flows from the initial single particle i to the final set (a/, n / ) (see fig. 3a). (ii) For each tree write the contribution as in old-fashioned perturbation theory as a product of propagators and vertices and sum over all inequivalent trees and time orderings. There is conservation of moment at each vertex so that all internal lines have a well-defined moment in terms of the set ( n / ) (see again, for instance, the tree o f fig. 3a). (iii) The rules for propagators and vertices, depicted in figs. 3b and 3c are as follows: (a) For a p-particle intermediate state, write a factor ( E - Eint) -1 where gin t =
56
K. Konishiet al. /Jet calculus:a simplealgorithmfor resolvingQCDjets , bl j ~ ' C t l ' n l 5 F,,ni -
n1* n ~
k
~
.~
~z ~ °2'n2 .... n3 .n4U~l,/~'°3 ,n 3
b I n I 01
bn
bp-;
o.
bk%
° -"X
k=l
no C n'
(b)
(c)
Fig. 3. (a) Example of a contribution to a five-parton spectrum. (b) and (c): Feynman rules for intermediate states (b) and vertices (c).
~Pk=lAnk and An is given in eq. (2.7). Each Ank acts independently on the indices of the kth line. (b) For a vertex corresponding to a(p) ~ b(n 1) + c(n2), P = n I + n2, write a factor Pnl,nu which is given in eq. (2.12). The corresponding rules in (x i, y) variables are obvious *. We end this section by discussing how to incorporate hadrons in the scheme. To be definite, consider first the case of a two-particle inclusive cross section 1 do(i~h I +h 2+x) Opl,hz(x 1 ,x2, 0 2) - - Ojet dx 1dx2
(2.14)
Assuming the non-perturbative hadronization process to be soft (i.e., to involve large time scales) and not to interfere with the hard (i.e., short time scale) process described above, we can argue that, in analogy with eq. (2.2) for D~x, Q2), Dihl,h2 (Xl, X2, Q2) will be given in terms of the parton level quantities Dala2,i , Da, i by folding them into some phenomenological Q2.independent fragmentation functions for a parton to go (inclusively) into one or two hadrons, to read:
fdXal dxa2 Dhl~X11 Dh2 X(-~a2) al,a2J Xa1 Xa2 "2\Xal/
ohl'h2(xl, X2"02) = ~
XOala2'i(xal' Xa2;02)'1" ~af ~ a
dxa
h ,L I x 1 X 2
oal n21~'X-aa) oa'i(Xa' 02). (2.15)
* Actually, it is possible to use Feynman-type rules where energy is conserved at each vertex and one integrates over all final energies.
K. Konishi et al. /Jet calculus."a simple algorithm for resolving QCDlets (
~
hl h2
-
~
)
h1
=
_
%
)
h2
*
:
hl
C) _
~ h3
=
I~
-_
•
- ~ - ~
*
,
hl
h2
~
(o)
h1 h2
)
"*
."
57
)
h3
:-
hI
:
h2
)
h3
h1 h2 h3
PERM
* PERM
(b)
Fig. 4. (a) 2-hadron spectrum from 1-parton and 2-parton spectra. (b) 3-hadron spectrum from 1-parton, 2-parton and 3-parton spectra.
Eq. (2.15) is illustrated in fig. 4a, where the new phenomenological fragmentation function Dahl'h2(z l, z2) appearing on the r.h.s, is constrained by energy conservation to obey the equation
f dz2z2Dhal'h2(zi, z 2 ) = (1 - - z l ) D h a l ( Z l ) ,
(2.16)
h2
which we shall use later on. At first sight, eq. (2.15) does not look useful since it introduces a new phenomenological object. It is clear however that the second term only contributes if (pill + Ph2 )2 < A 2 ( 2 1 GeV2), whereas for 07hl + ph2) 2 > > A 2 only the first term should survive. In sect. 5 we shall consider the case of inclusive spectra at fixed PT (or at fixed invariant mass) and we shall then exploit the fact that, in the perturbative region, no new phenomenological input is needed besides that of Dab(z). Tire generalization of the above prescription to multihadron spectra is straightforward. In fig. 4b we give the explicit case of a three-hadron spectrum. Again, if all final momenta are such that Pi" Pj >> A2, the only surviving contribution will be the first one.
K. Konishi et al. /Jet calculus."a simple algorithm for resolving QCD ]ets
58
3. Consistency checks In this section several consistency checks of our rules are presented. These represent quite non-trivial tests of the hypothesis underlying our "derivation" of the jet calculus rules. They should also help the reader to become familiar with our algorithm.
3.1. Rederivation o f a result o f J. C Taylor in ¢36 Six-dimensional ~.~b3 theory is a useful laboratory for investigating consequences of asymptotic freedom in renormalizable field theories. Defining the interaction term in the Lagrangian as - ( 1 / 3 ! ) X¢3 one finds for the one-loop running coupling constant:
~-(Q2///2) =
au
_ X2
1 + 4aau log(Q2/p2) '
au
(470 3 .
(3.1)
This theory has no soft infrared divergence since
1 1 d6k k2 k ' P l k ' p 2
o
(oo,
(3.2)
i.e., does not diverge from the small Ikl integration. On the other hand the theory has collinear divergences, since
fdSk
1
ko (P' k) 2
03dOlklad[kl ...,fd_f Ikl Ikl f lpl21kl2(1 - cos 0) 2 d Ip-~Ip---[
(3.3)
diverges as log(Q2/m 2) for m -~ 0. It should also be noted that, as in axial-gauge QCD, only planar ladder diagrams give leading divergences since
Sk 1 1 ~ 0 (Pl k~-)~---2"k)
fd
(3.4)
i f p l and P2 are not parallel. The detailed analysis of the AP evolution equations for this theory lead again to eqs. (2.3) with
y= 4 lo_[a(m21u2)'~ 3 gt~(Q21ta2)},
P(z) = z(1 - z) - 1 6 ( 1 - z ) .
(3.5)
The rules of jet calculus for tp6a are also the same as for QCD, where from eq. (3.5), one finds 1
An=(n+2)(n+3)
1
Pn rn
1-2'
'
=
F(n + 2) F(m + 2) P(n+m+4)
(3.6)
K. Konishi et al. / Jet calculus."a simple algorithm for resolvingQCDlets
59
For example the average multiplicity of a ¢36 jet will be
(n) = D(n = 0, Y) = exp(A o Y) = [a(m21"2)]'/9 ~a(Q2/u2) ] '
(3.7)
as found by Taylor [30]. One can actually reconstruct the full multiplicity generating function: oo
G(u, Y)= ~ m=1
oo
Um
ore(Y) _ ~ Otot
(u - 1) k o k ( y ) + 1 ==-1 +F(u, Y ) ,
k= l
(3.8)
k !
where pk(Y) is the integrated k-particle inclusive cross section, that is, pk(r)
(3.9)
= D ( n l = n 2 = ... n k = 0 , Y ) .
In order to compute G(u, Y) it is then enough to evaluate the sum of all trees with zero moment in every leg and with each final particle multiplied with (u - l) - v. Because of a simple recurrence relation for trees one finds Y
F(V,
1 Y)=veAor+~Po,o f dyeAo(Y-y) F2(o, y)
(3.10)
o
which provides the differential equation d -~y F(V, Y) = A o F(v, Y) + I po,oF2(v, Y) ,
F(o, O) = v ,
A o = ½Po,o = 1"21
(3.11)
"
We recognise in eq. (3.11) the classical field equation for q~3 theory, which is the expected result since we are summing tree diagrams *. The solution of eq. (3.11) is simply (p -= 2Ao/Po,o):
G(v, Y) = 1 + F(v, Y ) -
(v + p) + v(p - 1) exp(A o Y)
(v+p)-vexp(AoY)
= p=l o=u-i
u
u+(1-u)exp(AoY)' (3.12)
which coincides with Taylor's result (ref. [30], eq. (6)). A couple of comments are in order about eqs. (3.11) and (3.12). First of all, writing the differential equation (3.11) is just one of the standard ways [38] of solving the population-growth problem of a single species by treating it as a Markov (branching) process. In the case at hand, A o and ½Po,o are the elementary probabilities for growth and birth, respectively. * A v e r y similar r e g g e o n c a l c u l u s p r o b l e m h a d b e e n solved b y S c h w i m m e r [37] b y t h i s same technique.
K. Konishi et al. / Jet calculus: a simple algorithmfor resolving QCD/ets
60
The other way to attack the same problem would be to write instead of (3.11) the equivalent linear partial differential equation
OY
F(v, Y)=(Aov+ ½Po,oV2) ~--~F(v, Y),
(3.13)
whose solution is again eq. (3.12). Second, the resulting distribution (3.12) is the one of an ideal Bose gas, for which
On/Ot°t
-
1
(n) + 1
(1
1
t~
(n) + 1
'
(n) = exp(Ao Y ) - 1 .
(3.14)
Since, for an ideal Base gas, (n> = (e cur - 1) -1 one finds that a ~ jet at large Q2/m2 is equivalent to an ideal Bose gas with k T ~ exp(A o Y) = (log(Q 2/m 2 )) 1/9 . E
(3.15)
Also notice that, in this theory On~01 Q e / m 2 ~ ~ 1 ,
ol Q2/m 2 --> ~> 0 ,
(3.16)
and still Y~nOn~ constant. Thus q~3 is an example of theory in which there is no enhancement due to bremsstrahlung and yet the cancellation theorem works. It would be quite interesting to know whether this is also the mechanism for QCD or not (see sect. 4 for a discussion of this point).
3.2. Momentum sum rules in QCD In a model for jets like ours in which inclusive cross sections are obtained as Mueller-like discontinuities, these constraints are equivalent to checks of unitarity, i.e., to the statement that our inclusive spectra can be indeed written as sum over exclusive processes. We start by a simple example of momentum sum rule at the parton level worked out directly in terms of the x, y variables in subsect. 2.2. The sum rule relates the two-parton and the single-parton spectra by the relation:
~ ;dXlX l Dala2,i(xl, x2, Y ) = ( 1 - x 2 ) D a 2 , i ( x 2 , Y) . al-Using eq. (2.8) the 1.h.s. can be easily reduced to: Y
[ x2 y) fo dy ~al f dx1xIDalbl(Xl' y) fdxdz "1-zz D a2b2[x(1-----z)'
(3.17)
K. Konishi et al. I Jet calculus." a simple algorithm for resolving QCD jets XPj~bl b2(Z) D~(x, Y - y ) .
61 (3.18)
By m o m e n t u m conservation at the single-particle spectrum level we know that:
f d X l X 1 Dalbl(Xl, y ) = 1 , any bl, y • (3.19) al Furthermore the sum over bl and the factor z in the integral allow the replacement
bl
P.i"+bl b2 (z) = Pb2j( 1 -- z ) .
(3.20)
The result is that the 1.h.s. of eq. (3.17)becomes:
fo of
dy
a
Da2b2 X(-1 - - z ) ' y
-
Ldy
\x
Pb2j(1 - z ) D j i ( x ' Y - Y )
'
+SdxSdyIDalj(~,Y) ~-~-~DIi(X, Y-Y)1 :fox f dyTy-y[Da2jt: Z
Y-
,
o where the AP equation (2.3) for D has been used twice. The integral o v e r y can now be done trivially and, using Dab(Z , y)ly=o = 8abS(Z -- 1),
(3.22)
one immediately verifies the validity of eq. (3.17). To prove eq. (3.17) at the hadronic level (i.e., with al, 2 -+ hl,2) is not as trivial. One obtains eq. (3.21), but then eq. (3.22) cannot be used if a is a hadron. Indeed, in the hadronic case, we have to add a second term (see eq. (2.15) and fig. 4a) in order to obtain the full two-hadron spectrum. Adding such a term and using eq. (2.16) it is rather easy to see that, eq. (3.17) is again recovered. The general form of m o m e n t u m conservation is a generalization of eq. (3.17) and reads m
f d x 1 x 1 Oala2...am,i(Xl, x2 "'Xm, Y ) = ( 1 - ~ a1
xi)
j= 2
XOa2...ara,i(x2 ... Xra, Y )
(3.23)
62
K. Konishi et aL /Jet calculus. a simple algorithm for resolving QCD jets
It is easiest to check this equation by going over to moments, for which eq. (3.23) becomes ~-J Dala2...am,i(1,n2 ... nm, Y ) = D a 2 . . . a m , i ( n 2 ... nm, Y ) al n -
~
]=2
(3.24)
Da2...am,i(n2 ... nj + I ... nm, Y )
The explicit proof of eq. (3.24) is given in appendix B(i) for the hadronic case (which contains, as an easy by product, the parton-level sum rule) using mathematical induction. Since proof by induction turns out to be very common in our framework we go into some details there explaining how they work out in this particular case. Notice again that, in order to satisfy momentum sum rules at the hadronic level, a whole set of new phenomenological quantities has to be introduced giving the decay function (at fixed Q2 = Q~) of a patton into m hadrons. 3.3. Flavour-conservation sum rules
There are sum rules expressing the fact that the flavour of the initial quark must be carried also by the jet. These sum rules, when written in terms of moments, take the form: Dala2...ami,i(r/l, n2 ... nm, 0; Y) m
=Dala2...am,i(nl, n2 ... nm, Y ) (1 - ~
j=l
6i,aj).
(3.25)
In this equation the final parton i on the 1.h.s. is supposed to represent the "valence" quark in the jet (i.e., the final quark whose line is connected with that of the initial quark). The meaning of 6i,aj on the r.h.s, which ensures that none of the final partons a I a 2 a m is the valence quark is similar. The most trivial case of eq. (3.24), m = 0, is . . .
Di,i(0, Y) = 1 ,
(3.26)
which follows from the fact that AoNs = 0, where NS stands for non-singlet. Eq. (3.26) expresses the fact that integrating the spectrum of the valence quark gives the total cross section (with no multiplicity factor). The general proof of eq. (3.25) by induction can be found in appendix B(ii).
4. Longitudinal spectra of QCD jets We now turn to the applications of our formalism to quark and gluon jets. This section will be devoted to longitudinal spectra (i.e., spectra in the momentum frac-
63
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets
tions along the jet axis, transverse momenta being integrated over). We find it convenient to split our discussion into two parts, finite x and very small x, also because, as we shall see, our predictions will not be as reliable in the second case. 4.1. Spectra at finite x 4.1. l. Single-particle spectra *. We start by considering the total m o m e n t u m fractions carried by quarks or gluons (Xq, Xig) in the two types (i = q, g) of jets. These quantities are just given by the 1st moment of the single-particle spectra:
x ia = Da, i(1, Y) = (alexp(A 1 Y)Ii) •
(4.1)
The explicit diagonalization o f A 1 then gives xq
=
3Nf P
+
16 e_(o/9)v P
Xgq = -16 - ( 1 - e_(O/9)y ) P xg = 3Nf (1 - e - ( ° / 9 ) Y ) , P xg
16
= __
0
+
3Nf
e-(,o/9)
Y ,
P
p = 16 + 3Nf.
(4.2)
We see that, asymptotically, Xq/X~ -- 3Nf/16 independently of i (the type of jet). On the other hand, at moderate values of Q2, the non-leading term is not negligible and one obtains the results shown in fig. 5. Nevertheless quarks and gluons share the total m o m e n t u m in roughly the same way in the two types of jets. We shall use this result later when discussing multiplicities. Let us now look at single-particle spectra near x = 1. These can be obtained through the study of large moments (which clearly emphasize x ~ !)- One obtains behaviours of the type: Oa, i(x; Y) ~ Ca,i(x, Y) (1 - x ) 2CvY+ba,i , x'-*l
(4.3)
where Ca, i depend logarithmically upon (1 - x) and in a complicated but known
way upon Y, whereas ha, i are just some integer numbers. Table 4 gives the values of Ca, i; ba, i for all cases. Notice that the ratio of quark to gluon spectra near x = 1 is independent of the * The results of this subsection partially overlap with those of Dokshitzer et al. [14,22], Cabibbo and Petronzio [39] and others.
64
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets 1.0
08
~x
0.6
04
0.2 q (valence)
/ ,
•
,
0.2
,
,
0.4
,
,
~
,
0.6
Y
1.0
08 g 06
0.4
0.2
I
I
I
0.2
I
0.4
I
I
I
I
0.6
Y
Fig. 5. Total momentum fractions carried by quarks and gluons in (a) quark and (b) gluon jets.
type of jet considered and is given by Dg, i(x; Y)/Dq,i(x; Y )
(1 - x ) ~1
4(CA - CF) Ylog(1/(1 - x ) )
'
i = q,g,
(4.4)
indicating that final gluons are softer than final quarks in each type of jet. Further-
K. Konishi et al. / Jet calculus."a simple algorithm for resolving QCD lets
65
Table 4 The behaviour n e a r x = 1 o f D a , i(x; Y) Da,i(x; Y)
~ C(1 1-x << 1
x) 2CFY+b
i
a
b
C
q
qvalence
- 1
A(Y)
q
g
0
A(Y)/(4(CA
q
qsea
1
NfA(Y)/(4(CA
g
q
0
NfA(Y)/(4CF(CA
g
g
1
NfA(Y)/(8(CA
- CF) Y log(1 - x ) - 1 ) -
CF)Y(2CFY + 1)
log(1 - x ) - 1 )
- CF) g log(1 - x ) - 1 )
- CF) 2
Y(2CFY+ 1)
(log(1 - x ) - l ) 2 )
A(Y)
= exp ((3/2 - 23') CFY)/F(2CFY), 3' = Euler's const (= 0 . 5 7 7 2 . . . ) .
more, if we take the ratio of quark spectra in the two types of jets we get: aq,g(X;
r)/Dq, q(X; Y)
(1 - x ) i f x-,1 4CF(CA -- CF) Ylog(1/(1 - x ) )
'
(4.5)
and the same ratio is also found for final gluons. Eq. (4.5) and its analog for final gluons are the first indication that gluon jets are softer than quark jets, a prediction which will be reinforced by our subsequent results. Finally, the property that the ratio given in eq. (4.5) does not depend upon the final parton will be a property common to many other quantities and is indeed a special case of a general "universality" feature of parton spectra discussed in appendix C.
4.1.2. Two-parton spectra. We now look at the two-parton spectrum Dab,i(Xl,X2; Y) in the region x 1, x2 finite. A few limits are particularly simple and interesting. We start with the case Xl + x2 ~ 1 by looking into large moments: f d x l d x 2 ( x l + x 2 ) n XlX2 Dab,i(Xl, x2; Y).
(4.6)
The result is a behaviour of the type: D a b , i ( X 1, X 2 ;
Y)
~ Aab,i(1 --x 1 x 1+x2--*1 Xl,X2~eO
--X2)2CFY+6ab,i ,
(4.7)
where A b a , i have at most a logarithmic dependence on (1 - x l - x2). The exponents ~ab,i are some integer numbers given in table 5. They show again that quark jets have an easier time producing a fast pair of partons than gluon jets; something that could be expected on the basis of single-particle results. It is also clear that in
K. Konishi et aL / Jet calculus. a simple algorithm for resolving QCD jets
66 Table 5
Dab,i(XlX2; Y)
A(1
~
- X 1 -- x2) 2CFY+fab,i
Xl+Xl-*l (x 1, x2 non-soft) i
a
b
q
qV
qSea
0
qSea
qSea
+2 (0 if q = q) a)
q
qV
g
-1
q
q
g
+1
q
g
g
0
g
qi
q ' / q i a)
q
g
0
g
g
+1
sea
+1
a) For the explanation of some symbols see table 6
a quark jet the leading pair is usually the valence quark plus a gluon, whereas in a gluon jet it is a (sea) quark plus a gluon. Finally we study the case 1--Xl <<1 ,
1--Xl-X2<
I ,
(4.8)
i.e., the analog of a double-Regge limit. This we can do by studying double moments Dab,i(m,n; Y ) for m > > n > > 1. By inverting the Mellin transform we then find Dab,i(X 1, X2; Y )
~
H(Y)
(1
-- x l ) k a b , i ( 1 - - X 1 -- X2)2CvY+hab, i ,
1--Xl<
1-Xl-X2<
(4.9)
up to logarithmic terms. The exponents kab,i, hab,i can be computed but, before commenting on the results, we wish to point out an amusing analogy with Regge theory. The Regge analog of eq. (4.3), (4.7) and (4.9) would be, respectively: Da,i(x ) ~
X-.~l
Dab,i(Xl, X 2 )
Dab,i(X 1, X 2 )
(1 - x)av(°)-2~a,
i ,
~ (1 - - X 1 -- X 2 ) ~ v ( 0 ) - 2 a a b ' i , Xl +x2.--~1 (1 - - X l ) 2 ( a a b ' i - a a ' i ) ---> x 1 ..-. 1 x2--* 1 --x I
1 (1 - - X 1 -- X 2 ) c ~ v ( 0 ) - 2 a a b ,
(4. l 0) i ,
K. Konishi et al. / J e t calculus. a simple algorithm for resolving QCD ]ets
67
Table 6 a O = 2CFY + I
i
a
b
C~a,b
aab,i
q
qv
qs, q-s
1
q
qs, q's
qs, qs
0
q
qi(q'i)
qi(qi)
0
q
qv
g
1
l
q
qs, q-s
g
0
0
q
g
qv
1
l
1
_1 1
q
g
qs, qs
21-
0
q
g
g
~1
1
g
qi(qi)
qi(qi)
~
0
g
q,~-
q,, ~-,
1
0
g
q
g
~1
1
g
g
q,q-
0
1
g
g
g
0
0
1
(a) When i = q, a subscript v in a (or b) indicates that a(b) in the final quark connected to i by a quark line (hence it carries the same flavour as i). (b) a = qs, q-s, b = q's, q-'smeans that both a and b are quarks or antiquarks from the sea and that they are not a single pair from the sea. (c) a = qi(q-i); b = q-i(qi) means that a, b are a single pair from the sea.
where av(0 ) is the v a c u u m trajectory ( p o m e r o n ) and O~a,i,O~ab,i are the leading trajectories exchanged in the ai, abi channels, respectively (in Regge theory they also d e p e n d on the respective m o m e n t u m transfers). If the analogy between jet physics and (multi) Regge theory is more than superficial then the e x p o n e n t s kab,i , hab,i of eq. (4.9) have to be related to ha, i of eq. (4.3) and to ~ab,i of eq. (4.7) b y hab,i = ~ab,i ,
kab,i = b a , i - ~ a b , i - 1 .
(4.11)
We have f o u n d that these equations are indeed satisfied for all a, b, i. Defining then (arbitrarily) av(0 ) = 2 C F Y + 1 (av(0) = 1 at Y : 0), we get the intercepts ¢Xa,i, 0lab,i shown in table 6. Notice a few amusing regularities in table 6: fermionic trajectories have half-integar intercepts; bosonic trajectories have integer intercepts; i f b is a gluon, O~a,i = ¢Xab,i ; for fixed i, the more exotic the trajectory the lower its intercept tends to be.
68
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets
4.2. Spectra at small x, multiplicity distributions, etc. 4.2.1. Single-parton spectra and multiplicities. When we go to the small x region, we encounter, in QCD, an infrared problem, unlike the case in ¢3 (see subsect. 3.1). As an example, let us consider the multiplicity of partons of type a in a jet of type i:
(na)i = Da, i(0, Y) = exp(Ao Y)ai.
(4.12)
Unfortunately A o is a matrix with divergent matrix elements (e.g., A gq, A gg = oo) and one has to see how to cut off this divergence. Clearly there is no outright divergence in the theory as long as Y is finite and as long as the final partons are not just on their zero-mass shell. The problem of an absolute prediction of (na) i is under study and is not an easy one. Fortunately it turns out that ratios of quark (and gluon) multiplicities in the two types of jets are finite and are simply given by the ratio of Casimir operators. One finds: ( n q ) q jet _ ( n g ) q jet _ CF/C A = 4 , (nq)g jet (r/g)g jet
in S U ( 3 ) .
(4.13)
This result comes from the fact that the (divergent) matrix A 0 has one (infinite) eigenvalue whose eigenvector projects onto quark and gluon states in a ratio CF/CA. This and other results will be seen to be stable against modifications of the IR cut off procedure used. Furthermore the fact that the same ratio holds again both for final quarks and for final gluons makes us believe the result at the hadronic level * There is indeed a simple intuitive interpretation of eq. (4.13) in the large Nc limit where CF/CA ~ ½. A quark carries a single colour index leading to a one-sided decay chain, whereas the gluon, having two colour indices, develops a two-sided chain ** as shown in fig. 6. The non-perturbative confining stage taking place later is not expected to make the two chains interfere appreciably, hence there will be twice as many hadrons in the gluon jet than in the quark jet. If we combine the result (4.13) with eq. (4.2) for the fractions of the total energy carried by quarks and gluons we see that the average x of a quark or of a gluon is CA/C F (9) times higher in a quark jet than it is in a gluon jet. Thus, gluon jets are softer than quark jets and yield higher multiplicities. We may also study directly the behaviour of spectra near x = 0 (insisting however that x is finite as QZ(Y) ~ oo). This has been done for quarks in a quark jet in refs. [ 14,22] by inverting the double transform in E, n back to Y and x. By this method we find Dq,q(X; Y)
Dg,q(X; Y)
~ ~1-6 N f ~""- ' F -1 y2e-WYI2(v)/v2 , x--*O X
~ x~O
4Cv y e_WYll (v)/v,
(4.14)
X
• Such a relation was first guessed, based on an analogy with QED, by Brodsky and Gunion [55]. • * The analogy with the topological expansion picture of the reggeon and the pomeron [40] is evident.
69
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD ]ets
Fig. 6. (a) Quark decays and (b) gluon decays.
where o = ~/8CAY log(I/x),
W= I-~CA + ~1U f (1 -
(4.15)
2CF/CA)
and I1,2(v) are the usual modified Bessel functions. Furthermore, distributions in a gluon jet are related to those in a quark jet by: Da,g(X; y )
~ CA Da, q(X; Y) x"*0 ~
(4.16)
for both a = q and a = g, in agreement with eq. (4.13) for the multiplicities. The reason for universality is again the same and is discussed in appendix C. 4.2.Z One soft, one hardparton. We now consider the case of one slow (Xl ~ 0) and one fast (x 2 = finite) parton and show that the two partons become uncorrelated in this limit. Consider indeed the (moment of the) correlation function Cab,i(x 1, X2 ; Y) = D a b , i(X IX2 ; Y) - Da,i(x 1, Y) Db,i(x2, Y) :
G~,i(e,
n; Y) =
,~ _
A~ - A .
aa'
~ ( E - A.+~) i'i
bb' - ( E - A e - An)[a 1 ,
(4.17)
bi
where e = 0 + in the moment relative to the slow particle and n = 0(1). In eq. (4.17) various factors are divergent since both A e and Pe, n contain divergent matrix ele-
70
K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD ]ets
ments. Since, however, pea'bn',i' ~
Aa',f ,
~---~0
An+ c ~ A n + O(e),
(4.18)
one find easily that Cab,i(• , /7; Y) is not divergent. In other words the divergent part in Dab,i(x1, x2; Y) for x 1 ~ 0, x z finite factorizes into Da,i(x1, Y) times Db,i(x2, Y). This result can be generalised to n-parton distributions, i.e., when one of the partons becomes wee its divergence factors out * 4.2.3. Multiparton spectra, correlations. This is the most delicate case, in which all partons are soft. In order to cure IR problems we can try various cut-off procedures and then see if there are finite quantities which do not depend on the cut-off procedure itself. We shall discuss here two regularization procedures ** (a) A o is given a fixed cut-off (e.g., by integrating between e and 1 e in eq. (2.7)). In this case the procedure followed in subsect. 3.1 and leading to eqs. (3.10), (3.11) can be repeated with the only extra complication due to the presence of two species of partons. Defining multiplicity generating functions for a quark and gluon jet by
l+Fq(uq, Ug, Y) = ~
UqUg
o(q --~nquarks + mgluons; Y)
n, m = 0
Otot
oo
1 + Fg(uq, Ug, Y ) = n=m=o uqug~
o-t-1o(g ~ nquarks + mgluons; Y) ,
(4.19)
one gets, instead of eq. (3.11), the coupled system: / dd Y - Jt?g + 1) 2 --(1 + F g ) ] , - =Aggpg(~K'g " ' 0 - v + 1) + 5Nf[(Fq 1 dd----yFq= AgqFg(F q + 1),
F q ( Y = O) = Uq
F g ( Y = 0) = Ug
Q
(4.20)
Furthermore A gg and A gq are divergent constants, but their ratio is simply CA/CF. The system (4.20) is just the standard way [38] of dealing with the problem of population growth of two competing species (here quarks and gluons). The general, as well as several particular cases of eqs. (4.20) are under study [41]. Here we just mention that, if we neglect q~ pair creation by setting Nf = 05 the problem is com-
Since ¢,~ does not have an IR problem this simple result is not valid there. ** A more rigourous treatment taking into account the actual kinematical boundaries is under study.
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD lets
71
pletely soluble and one gets (1 + F g) -
Ug Ug + (1 - Ug) exp(AggY) '
Uq (1 + F q) = {Ug + (1 - Ug) exp(AggY)
(4.21) }CF/C A "
Eqs. (4.21) give, for the gluon jet, an ideal Bose gas spectrum (i.e., a black-body radiation with kT/hv ~ exp(Agg Y) ~ o~ as we remove the cut-off), whereas the quark jet corresponds to CF/CA independent black bodies (giving the so-called Polya-type distribution *) except that Cv/CA is not an integer (for N c -+ oo, in particular the quark jet becomes half a black body ?). Vice versa, the QED limit can be recovered by letting C A ~ 0. In this case CF/CA -~ oo and the quark jet becomes equivalent to an infinite sum of independent black-body radiations which is well-known to give a Poisson spectrum, as it is known to be the case in QED. For completeness we give here the relevant exclusive cross sections as they follow from eqs. (4.21): oq..+q+ng
=
e_(CF/CA)YAo r(CF/CA + n) F(CF/CA)
a g'*ng
=
(1
- e-AoY) n n! '
e--4or(1 - e-AoY)n .
(4.22)
Notice that this method gives cross sections analogous to those o f ¢3 and in particular no enhancement (an/O o -~ constant). (b) The second IR regulafization procedure we wish to consider ** is of the "dimensional" type. We take small positive moments of each k-particle spectrum and obtain an e-dependent generating function (compare with eqs. (3.8), (3.9) in ¢3) for gluon multiplicities inside the two types of jets in the form (i = q, g):
Gi(v; Y)= 1 + Fie(v; Y), Fie(v;
Y)= ~
Ok ~.v Dgg...g,i(e, e .. e, Y)
k=l
k ~
.
-
dxixiDgg'"g'i(xl ""xk' Y)=- ~k f i k,e
(4.23)
where we have introduced v ----u - 1. * The phenomenological reievance and the possible theoretical origin of this type of distribution in loW-PT hadron physics is known [42]. We thank A. Giovannini for drawing our attention to this point. ** We are grateful to A. Bassetto and G. Marchesini for interesting discussions on the subject of this paragraph.
72
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
In accordance with the jet calculus rules, Fig,e satisfy the integral equations: k-1
Fik = V6kl(eAeY)gi+ll~__
f
y d r , bFe; F ~ _ l
o
X P~eC,t~_t)e (eA ke( Y- Y'))i,i
(4.24)
•
Taking the dominant contribution for e ~ 0, this gives F g "~
(ok/k!) C,(n)kg,
(4.25)
F g ~ (vk/k!)fk(7) (n)~,
(4.26)
(n)g
(4.27)
where =
e(2CA/e)Y,
7 ----CF/CA ,
and that Ck and fk('Y) satisfy the following simple recurrence formulae: d k -Ck/(k.
k!),
d I = 1 ,..
k-I dk = { k / 2 ( k 2 - 1)} ~
dtdk-t ,
(k~>2),
(4.28)
l= 1
and gk(~/) -=/k(T)/k!,
g, (T) = T ,
k-I g/(7) d , - t + dk } ,
gk(7) = (v/k) (~
(k >1 2 ) .
(4.29)
l= 1
We have been unable to solve the recurrence formulae (4.28) and (4.29) in a closed form. However, much can be said by. converting these into a set of differential equations for the generating functions Fe~(~") and their integrals D q or g(~-) - f
d~"F q or g(~.,),
(4.30)
--oo
where * ~"= log v ( n ) .
i * We notice that by keeping leading terms only, Fe(v, Y) depend ofl v and Y only through the combination v{n) = o exp AeY.
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
73
In fact we find d Dg = F g
d~ d Fg =Dg(1 + F g ) ,
d~"
d Fq = 7Dg(1 + F q ) . d~"
(4.31)
with the boundary conditions Fq'glo=o = O, __d Fgiv=0 = 1 , dv clFq I
do
v= o
= ,),= C F / C A .
(4.32)
From eqs. (4.31), (4.32) one easily gets the simple result: log(1 + F q) "" ( C F / C A ) log(1 + F g ) .
(4.33)
We notice that this same result follows in the previously discussed method for regularizing IR divergences, as it is easy to check from eqs. (4.20). The interpretation of eq. (4.33) is very simple if one recognizes in the quantity log0 + F ) the so-called Feynman pressure in his gas analog model [43]. In other words, the correlation coefficients [44] .t~l,g are all in the ratio f~t/ff = C F / C A , (l = 1, 2 ...),
(4.34)
which is finite in the limit e ~ 0. Although we have only discussed above final gluons, eqs. (4.33), (4.34) actually hold independently of the final parton species (q-g universality) as shown in appendix C. The generating functions Fq'g(~) contain in principle all the information about exclusive cross sections in QCD which are related [44] to the behaviour o f F near v = _ l (~ ~ oo). It is easy to show, by studying the differential equations (4.31), (4.32), that F q or g(o, Y)lo=- 1 = - 1 ,
(4.35)
in accord with eq. (4.19). Furthermore, we checked that the lowest exclusive cross sections, calculated from the divergent (for e ~ 0) inclusive cross sections, are indeed exponentially vanishing for e ~ 0, as we expect to be the case. Detailed results, however, are not consistent with known results such as the Sudakov form factor, showing that terms neglected for inclusive cross sections are important in computing exclusive cross sections. This is not surprising because a huge cancellation is involved in
74
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
obtaining exclusive cross sections from inclusive ones. It does not imply, however, that we cannot trust our predictions at the inclusive level. An interesting result which follows from eq. (4.31) is that (n(n - 1) ... (n - k + 1))e, i jet ~-/
Ildxi Xe (1/o) do/dx 1 ... dx k
is proportional to (n)ikjet , with the proportionality constant larger than unity. This indicates the presence of a long-range correlation, probably within the scheme of KNO scaling [45]. Another quantity of phenomenological interest is the dispersion defined by D = x/(n 2 ) - (n) 2 . Using eqs. (4.28), (4.29) one obtains Oq jet/(n)q jet ~ ½X/3 "" 0.87,
(4.36)
Dg jet/(n)g jet ~--N/ 1 .~ 0.58.
(4.37)
It is amusing to observe that the result, eq. (4.37), for the gluon jet, is numerically identical to the phenomenological value in soft hadronic processes found by Wrobrewski [46]. How much we should trust all these results is not presently clear. The prediction, eq. (4.34), looks to be very stable and, due to its simple physical meaning, we think it should be true in a more refined treatment. Also very stable is the presence of long-range correlations and KNO scaling. On the contrary, the exact values of the ratios ( n k ) / ( n ) k (or of the KNO function) do depend on the type of IR regularisation used. Finally, the fact that all these results do not depend upon the type of final particles we look at (universality) also has a very general origin in our framework and is independent of the IR regularization used. How universality arises in the scheme is explained in detail in appendix C. 4. 3. Photon ]ets
In this subsection we discuss the properties of jets initiated by (almost) real photons. For completeness, however, we shall first discuss the case of fragmentation into a photon, described by a fragmentation function D,r,i(x, Q2). This case has been already discussed in refs. [14, 47, 48].
Ca)
(b)
Fig. 7. (a) Fragmentation into a photon. (b) Photon fragmentation into hadrons.
K. Konishi et al. / Jet calculus: a simple algorithmfor resolvingQCDjets
75
From fig. 7a we see that D,r, i is related to Oj, i by the equation Q2 dq2 f dz 1 + (1 - z) 2
o~ ~i e2 f D~/'i(x' Q:)= 2n
q2 J z
z
Dj,i(x/z, Q2, q2).
(4.38)
Since, apart from a factor CF, the AP function [1 + (1 - z) 2 ]/z isP gq one gets
D'r,i(n,
o~ ~ e2. A~n" Q2 dq2 Q2 q2 = 27r / ' (CF) f -~-Dji(n, , ).
Q2)--f D(x' Q2)xndx
(4.39) The integration over q2 is now easily done and one gets a
1
(Anlg j
2Nf
D~/'i(n' 02) - 27rb ots(Q2) "/~1 e2
\CF]
- ~--~-~j
X 1 - A n/27rb
j/ii,
(4.40)
This result can be seen as the analog in jet physics of the structure function of a photon, first derived by Witten [31 ]. In the case of the fragmentation of 7 into a hadron (or a parton of type a) an equation similar to (4.38) can be written following now fig. 7b. One gets /~1f
Q2dq2 fdz q2 3 z (z: + (1 -
Da:/(x ' Q 2 ) = ~a .= e~N c f XDa,i(x/z q2).
(4.41)
Now the factor 1(z2+ (1 - z) 2) is the AP function A qg up to a factor Nf and, going over to moments, one gets ct
Da,,r(n, Q2) = 7rb°q(Q2)
Nc ~ e ~
X { 1 ( 1+An/2nb ) [eYAn-eYOAn'(Q2)~]t --~Q~o)_]]ai (Aing),
(4.42)
showing again enhancement with respect to Da,g : Oa, g(n, 02) = (DYAn)ag.
(4.43)
Furthermore Da,.r is also enhanced by a factor R _-- N c ~,iei.2 It seems thus pos-
76
K. Konishi et al. / Jet.calculus: a simple algorithm for resolving QCD jets
sible that processes such as e+e - ~ ')'real + "/jet could be isolated and measured for instance at PETRA energies. Notice that for low moments (small x) the first term in the square bracket of eq. (4.42) dominates (An/2nb + 1 > 0) and the quasi-real photon is actually far (O(Q2)) off-shell. On the contrary, for large moments (large x) the second term dominates and the photon is only O(Qg) off-shell. In other words this experimental situation provides a source of e+e - like jets (Tvirtual -->hadrons) where the mass of the virtual photon can be varied and is actually optimized according to the momentum of the detected final parton. The particular limit Dq,7(x, Qz) for x --> 1 is worth noticing. We find
Dq,,y(x,
c~ N c ~ 1 Q2)x_,l as(Q2) CF _7.. e~ log(l/(1 --x)] '
(4.44)
to be compared with the gluon fragmentation behaviour
NfA(Y)(1 -
X)2CF Y Dq,g(X, Q 2 ) ~ 4CF(CA _ CF) Y log(l/(1 - x ) ) '
(4.45)
where the symbols are those of table 4. Clearly a photon jet is harder than a gluon jet and the shape of the distribution in x, together with its dependence of Y, are indeed quite different.
5. Combined longitudinal and transverse spectra of QCD jets
5.1. Two-particlespectra We now go back to eq. (2.8) and, as promised, we interpret the integration variable y in terms of the relative transverse momentum PT of the two final partons. From the way y enters in eq. (2.8) and from our heuristic derivation of that equation (as well as from the more rigorous one of appendix A) it is clear that y is related to the virtual mass ~ of the parton j (fig. 2b) by a relation analogous to eq. (2.4) i.e. y =TnblOg a s ~ 2 )
a s(4q2)'
(5.1)
2 1 2 where the factor of 4 has been introduced so t h a t y = Y for q 2 = qmax = ~Q • 2 1 2 The situations we shall consider here will refer to values of q < < ~Q since, for q2 ~ Q2, the process clearly involves an additional jet and one needs to describe it by the appropriate non-collinear kinematics. We shall come back later to the discussion of how smoothly our results connect with those obtained by other authors in the region q2 ~ Q2. We now try to estimate q2 in terms of the final momenta. Because of the strong
K. Konishi et al. / Jet calculus."a simple algorithm for resolving QCD]ets
77
ordering of masses (and OfkT) along each ladder [13-20] partons b l , b2 have masses much smaller than X / ~ , hence the final PT of a 1, a2 has to be comparable to the one O f b l , b 2 . Denoting by pbl,b2(26b 1,b2) the relative transverse momentum (angle) between bl and b2 and by PT(26) the that ofax, a 2 we find q2 =X2Z(I - - Z ) 6 2 Q 2 <~ 1Q2~2 ,
(5.2)
where we have used the fact that, for the LLA, the proportionality constant x2z(1 -- Z) can be set equal to one (if it is finite). Of course, at finite Q2, it will make some (non-leading) difference to use the more refined expression but this is not, strictly speaking, an effect that one can compute without a non-leading log calculation *. In terms of PT the relation with q2 is somewhat more complicated, but, essentially one has still q2 ~ O ( p ~ ) ,
(5.3)
with a proportionality constant of order one. Hence in the following we shall use eq. (5.1) with either Y = Y8 = ~
1
log[cq(P 2)/~(62Q 2)] ,
(5.4)
log [ a s ~ 2)/as(ap2)] .
(5.5)
or 1
Y = YT = ~
At this point one can immediately identify the integrand in eq. (2.8) with (1/o) da/dy. Since eqs. (5.4), (5.5) give d _ (Cq(4p2T)1-1 dy T
\~]
d
dlogp~
'
d _(Ots(~Q2).} -1 d dy~ 6 d-~'
(5.6)
one gets immediately the final result (see fig. 8) 1 d o ( i -~ al + a2 + X )
-O
d~ 2 dx i dx2
_- Dal a2,i(x 1' x2,~., Q2) -- tYs)(62 2- ~Q2
× jl~jj~-x2dglj~-Xldg2 I i" "2x1
zl x2
z2 (Zl + z 2 ) ^
g1
XDal ,JI (X1/z I, y6 ) Da2,J2(X2/z2, Y6 ) Pjl J2,j(z.-~~). XDji(z I +z2, Y - y 6 ) , * A similar problem of fixing Q2 exists in large-PT hadronic processes.
(5.7)
78
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets 200
I
f
I
O/A =100 /
I00
////e// NAIVE
RESULT NO N'.Z-
% ////////~
~c:;
PERTURBATIVE 3 JET REGION
/
~////~ OUR ////////~ RESULT /
20
q> o~
M~
~
10
PERTURBATIVE 2 JET REGION 4
2 NAIVE11 SULTI ee,
i 4
~
10
i 20
i 50
qTIA
S 8 (q~ Fig. 8. Naive versus our result for (0, 1) moment, ~aDqva,q (0, 0; q2, Q2).
where as in eq. (2.8) we have again eliminated two of the integration variables using the 6-functions. Exactly the same equation holds if we replace 62 by 4p 2. Going finally to the hadronic level, we trivially get from eq. (2.15)
Dhlh2,i(X l, X2
-2- %(82Q2) ~
62 ;
;~
)-
~
a d Xa d62
h
I' \Xa Xa
1 --x 2
J'Jl,J'2x I
×D~j~(xl/zl, ya)D~2(x2/z2, ya)Pjlj2,j
a•/dx
f
Y~t!
pl --x 1
dz 1 J
dz2
ZI x 2
Z2 (Z1 "l-Z2)
1
ji(gl +z2, Y - Y a )
(5.8)
The second term in eq. (5.8) involves now an unknown (phenomenological) fragmentation function for the decay a ~ hi + h2 + x at fixed 6. Fortunately, such a contribution is only important for small 6 = O(A/Q) and therefore, if we keep away from
K. Konishi et at / Jet calculus: a simple algorithm for resolving QCDlets
79
such non-perturbative region (PT > > A), only the first (perturbative) term contributes. This could be evaluated numerically once the input of single-particle fragmentation functions Dha(x) is given at some reference point [49] Qg and which could be measured in any convenient process *. We have not started yet any serious computer calculation of D~ I,h2, but fortunately one can already extract the basic physical properties of eq. (5.8) by either taking moments in x l , x2 (as done below) or by looking at more inclusive (calorimeter measured) quantities as explained in subsect. 5.2. Before discussing the detailed properties of our formulae we would like to compare qualitatively our results to those of Dokshitzer, D'Yakonov and Troyan (DDT) [14,24] who, in the particular case of e+e - -'-hadrons, look at the inclusive production of two hadrons h I and h 2 which are at 180 ° (within an uncertainty 2~), hence in opposite jets. Since the q,~- coming from the photon are necessarily back to back and thus define the jet axis, it follows that h I , h 2 should have a (small) angle 0(6) with respect to that axis. This automatically limits the possible real emissions along each ladder to have IkTI < 6Q. As a consequence real and virtual diagrams only cancel for the infrared region 0 < IkTI < 8Q and the region IkTI > ~iQ remains uncancelled. Since both soft and collinear divergences remain, double logs of Q2/p2 (62) appear at each power of a s. This can also be seen by the fact that the original q,-~ cannot be off-shell by O(Q2), otherwise they would give ~ = O(1). A careful study [14] shows that they are off-shell between O(p 2 ~- 62Q 2) and O(PTQ "" 6Q 2) (the latter limit being reached for emission of partons of Ikxl ~ Ikl ~ 6Q). This produces the appearance of a Sudakov-like form factor at the e.m. vertex which is known to exhibit double logarithms. In contrast, in our case, the angle between h I and h 2 is near zero (same side). First of all this implies no restriction on the opposite side jet. More important, on the same side, other unobserved partons are allowed to be emitted in directions other than that of hi and h2: any correlation (within ~)between the original jet axis and that of h i , h2 is now lost and the q,~- pair created by the photon can be as far off-shell as it likes. In other words, before the vertex j ~ b l , b2, the cancellation of real and virtual divergences simply gives the usual AP evolution ofD(x, Y - y). After the vertex all the bremsstrahlung kw's are necessarily much smaller than PT (strong ordering) and are included: the only uncancelled divergences are the hard (x v~ 0) collinear ones and they produce, in the usual way, the scaling violations of the remaining fragmentation functions. In conclusion, a single logarithm of 6(PT/Q) appears now for each power of a s provided we stay away from the soft region (x 1 or x 2 ~ 0). This we do by considering positive moments:
Dala2, i ( n l ' n 2 ; 6 ' a 2 ) - f d x l d x 2 x n l x n1 2 J2 1 d a ( i ~ a l +a2 + X ) o
d~2dxldX2
'
(5.9)
* Other semiphenomenologicalinputs are, of course, A and the exact forms ofy T and/or Y6-
K. Konishi et al. / Jet calculus."a simple algorithm for resolving QCD lets
80
for which we find, from eq. (5.7) Dala2,i(nl, n2; 6, Q2) _
1 as(62Q 2) ~ as(m 2) "](An1+An2)/27rbp [oq(62Q2)-].An1+n2/2~rb 26 2 nLa~)] nl'n2 L ~ -J ' (5.10)
and an identical one with Q2~2 _~ 4p~. Theoretically, the simplest example is the one in which i = q (quark jet), al is taken to be the final quark line coming from i itself (al = i = "valence" final quark), n 1 = 0 and one sums over a 2 setting n2 = 1. One thus finds that b o t h A n l andAn2 can be set to zero (charge and momentum conservation) and that Anl+n 2 and -Pnl,n 2 become both equal to A Ns , the non-singlet part of the anomalous dimension A1 : A1Ns- 16 The result is simply: 9" a
Vt ANS~jas(4P2)~ /as(4p~.)~A t-k'T 2 . _J ~Ots(a 2) ]
D.va,.(0, 1;p , 02)
(5.11)
Before commenting on eq. (5.11), let us mention that an explicit computation of the same quantity at order a s is consistent with it, since it gives:
a
Dqva, q(0, 1 ;p~-, Q2)
_ 1 F(-A Ns) sl naive 0(%) 2rr as L ~ + 6(p~) term_ , (5.12)
where the (negative) asg(p~) terms come from the one-loop virtual correction * Because of eq. (5.12), we shall refer to the first term (square brackets) in eq. (5.11) as the "naive" result. Before considering other, physically more interesting, choices of hi, 2 , al,2, we make the following general remarks on eq. (5.11). (a) Eq. (5.11) is a (fully normalized) QCD prediction only for p~- > > A 2. Yet, integrating it from p~- "" m 2 to p~- "" Q2 gives the momentum sum rule needed for consistency and normalization. (b) The average value ofp~- computed from eq. (5.11) is: (P~') = ( - A INs) as(Q2) Q2 ,
(5.13)
which is (up to the constant ( - A ~ s) = ~ coming from our definition of the inclusive cross section) the expected result. (c) The second factor (round brackets) on the r.h.s, of eq. (5.11) can be considered to be the correction to the naive result. It is similar, in spirit, to the T form factor of the DDT formula [14,24], although it has single, and not double, logs. • This is, strictly speaking, not a contribution to the two-particle spectrum, but can be considered as such for the purpose of satisfying sum rule constraints.
K. Konishi et al. /Jet
This correction by a factor
81
calculus: a simple algorithm for resolving QCD jets
factor goes to one at p+ N Q2 but it suppreses values of pi << Q2
(5.14) Its effect, as in the DDT formula [ 14,241, is to broaden the pT distribution with respect to the perturbative result. An example of this phenomenon is shown in fig. 8, for Q/A = 100 (Q z 50 GeV). (d) The (negative) o&p%) term of the perturbative result (5.12) has disappeared in eq. (5.11). One could think that this is due to having limited ourselves to pt >> A2, but indeed the integrated result mentioned in (a) shows that there is no rooms for 6 function terms (see also next point). (e) We can consider our result (5.11) as a smoothed version of the perturbative result (5.12). Such a type of smoothing is usually achieved by putting by hand an intrinsic (primordial) pT in structure and fragmentation functions. Here we have instead obtained it as a calculable perturbative effect. We expect that further, nonperturbative smoothing due to actual intrinsic pT will turn out to be smaller than usually assumed. (f) One can check that the integrated cross section up to 4p$ - Q2 is correctly obtained (up to logarithmically vanishing contributions) from the region p$ >KA2 (K> l), i.e. from the perturbative region. There is a residual (integrable) divergence at 4p$ = A2 due to the (unphysical?) blowing up of 0~~there. Clearly, we cannot penetrate into the region 4p$ < A2 by perturbative techniques, but it is reassuring that only a smaller and smaller fraction of the cross section lies there, as the energy goes up. This is to be contrasted with 10w-pT hadronic physics, or with what could be expected in softer theories, where essentially all the cross section lies at pT < A = 0.5 GeV, independently more or less of the total energy. (g) The integrated cross section up to p$ = P$~,,~ is even simpler than eq. (5.11) and, in the region A2 << pt,,,,
<< Q2 behaves like
We see here, and this feature persists for other moments (see below), that an amusing form of scaling emerges asymptotically in the logs of Q* and of 4pt,,, . In other words, in a log Q*, log 4p2T,max plane, straight lines out of the origin delimit regions of constant fraction of the cross section (see below for actual numbers). Before turning to our actual numerical predictions and to a discussion of them we have to consider (at least) two other interesting choices of i, al, a2, nl, ~22in eq. (5.10). Since one of our aims is to compare quark and gluon jets, and since for a gluon jet, the “valence” quark in the jet does not exist, we are led to consider the quan-
82
K. Konishi et al. /Jet calculus." a simple algorithm for resolving QCD jets
tities Dala2,i(1, 1, PT, Q ) ,
~
al a2
Dala2,i(1 , 1,6, Q ) ,
i - q, g. (5.16)
ala2
Such a case turns out to be also more advantageous than the previous one for translating parton-level results into hadronic-level results, as discussed later. Using momentum conservation constraints the final results, obtained from eq. (5.10), in the specific case (5.16), turn out to be only slightly more complicated than the ones already discussed. Furthermore, all the qualitative features discussed above remain unchanged. To be specific, eq. (5.11) gets replaced by
ala 2
%(4p2) Dala2,i(1, 1.p2, Q 2 ) _ p2 2rr ~
a,j
(-.42
) "Fcts(4p~')lA2i2~rb I r xl aJ L 0~s~,Q2)_I ji
,
(5.17)
and the same with 4p~ ~ 6 2 Q 2. Now A2 is a two-by-two matrix: its diagonalization and the projection of its eigenvectors onto quark and gluon states lead to our final results:
ala2
24 Dala2,i(l , 1;p~-, Q2)=p~. l°g(4p~/A2)
71pT
"qPT | '
L0.920J
e
(5.18)
[. 3.680 .J
and the same with 4p 2 -~ I52Q2 , where the upper (lower) figures are for a quark (gluon) jet and rIpT
_ log(2qT/A) log(Q/A) '
_ log(gQ/A) r/6 =-log(Q/A) "
(5.19)
A comparison of the two types of jets based on eqs. (5.18) clearly shows that the quark jet is much more narrow (in 6 or PT) than the gluon jet. To be more precise, consider the integrated cross sections up to p~aX (or 6max ). One then finds the anticipated "scaling" behaviour, i.e., dependence of the integrated cross sections upon p~aX, 6max , Q is only through r/p~ax and 66 max" Numerically one finds
rll61
/ _-0.6086 + /
f('~--<6max)= [0.363] 7/6max
/ 1.386 t'/'/6max , L 0.637]
(5.20)
where f("-<6max) is the fraction of events with 6 ~< 6max. A plot off(~<6max) as a function of r/6 max is shown in fig. 9 for both types of jets. We see for instance that, in order to have 70% of the events one needs, for a
K. Konishi et aL / Jet calculus: a simple algorithm for resolving QCD ]ets
83
1.o
o.~
-GLUON 0,1)
I
I
0.25
0.50
__
I
0.75
1.00
~m~x
Fig. 9. f(eq. (5.20)) versus n 6 mxa. quark jet * r/Smax = 0.5 and for a gluon jet ~16max = 0.7. In terms O f 6 m xa this is equivalent to
j(Q/A) 6 max
-
-°'s ,
quark j e t ,
( Q / A ) n~ m a x - 1 =
(5.21)
(Q/A) -°'3 ,
gluon j e t .
In numbers this means ~ max = 5 ° - - 6 ° (or p ~ a x "" 2.5 GeV) for a quark jet as compared t o 6 m a x = 14 ° - 15 ° (p~ax ~ 6 GeV) for a gluon jet at Q / A = 100 (Q = 50 GeV). Vice versa, at the same value of Q/A, an angle 26 = 10 ° includes 67% of the cross section for a quark jet, but only 45% for a gluon jet. We also remark that for f("-<6max)> 0.9 we have numerically seen that the logarithms Of 6max for a gluon and a quark jet are in a ratio near to CF/CA = 0.444, as recently found by Shizuya and Tye [60] following the Sterman-Weinberg method (for f approaching one we find the ratio to approach 0.436). As we have already remarked, our results are expected to make sense in the limited region of 6 ( p T ) : ( A / Q ) 2 < < 62 < < 1 ,
A z < < p ~ < < Qz .
(5.22)
We recall that the limitation to 62 < < 1 came from our use o f collinear kinematics. The region 6 "~ O(1) can be treated more appropriately by lowest-order calculations * For a comparison, in the case of the valence quark spectrum one finds, from eq. (5.14), nSmax = 0.433.
84
K. Konishi et a L / Jet calculus; a simple algorithm for resolving QCD jets 10
Q21A2
=
200 1000 5000 25000
I
I
I II
II
,i,!i
IIiI
I I. I:"".,
e~
%-
[., •." ~;/1.7~ ..."
c~b-
!/i/rl i
/ ///hl .. 1 / 1 1 / i //1 i I I
/}////II // ~/I'/
0.5
/ I'~I/ / iii//
,',,
\ \ \ \
,, ,,N
0.1
t 30 °
-~,. 7 .... t 60 °
...-
"~
t 90 ° Z6
~2
i
1 0°
150°
I 180 °
b
Fig. 10. Comparison between eq. (5.17) (full curves) and the formula from ref. [27] (dashed curves). The results from refs. [14, 24] are also shown (dotted curves). (justified since oq(Q 2) log ¢5 < < 1) which keep track however of the more complicated non-collinear kinematics, of certain interference terms (which are non-singular for ~ -+ 0), etc. For the double m o m e n t ~al,a2Oala2,i(1 , 1; ~, 0 2 ) , this region of~5 ~ O(1) has been investigated in detail by Basham et al. [ 2 7 ] , w h o have actually looked also at dependence on other angular variables*. Integrating over those variables we arrive at the * These authors emphasize rates with calorimeter detectors, which, however, for n 1 = n2 = 1 coincide with counter-measured cross sections.
85
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
result shown in fig. 10 (dotted curves). In the same figure we show our results (from eq. (5.18) in the case of a quark jet) and we see that the two curves agree quite well for all values of Q2 in some intermediate range of 6(6 ~ 50°). The results of ref. [27] lie below ours for small 6 if we keep using as(Q 2) in their formulae for all 6. On the other hand, if we use for instance as(Q 2 tg 216), their formulae lie above ours. Since our formulae can be seen as a resummation of all terms of the type (as(Q 2) log 6) n, we can say that our method resolves the above ambiguity as to whether to use as(Q 2), as Cts(62 Q2) or get something else in a lowest-order calculation. The correct leading log prediction should then be a smooth interpolating curve between our results for small 6 and those of ref. [27] at large 6. From fig. 10 it is obvious that such a smooth curve can be drawn. This indicates, in our opinion, the important fact that QCD gives predictions on the transverse structure of e+e - hadronic events which, with a smoothly varying cross section, go from narrow to broad two-jet events and finally to genuine three-jet events. It is only the approximations used in computing these various regimes that change while the underlying physics remains the same. Finally, when the results of Basham et al., are extrapolated further to angles near 180 °, they become again inadequate because of large logs of the non-collinearity angle. This is exactly the situation discussed by DDT and which we have briefly outlined above. We have checked that the formula of DDT gives a smooth extrapolation both in magnitude and in Q2 dependence * of the results of ref. [27]. 5. 2. Calorimeter-measured cross sections and a new way o f testing QCD
Another simple and interesting quantity which can be predicted is the generalization of the Sterman-Weinberg [4] expression for a two-jet cross section when one reduces very much the opening angle 6 of the calorimeter in which all but a fraction e of the energy must go. Consider values of 6 such that, again, % ( 0 2 ) log 6 ~ O(1),
(5.23)
while keeping cts(Q 2) log e < < 1 (i.e., e not too small). This quantity has been computed by Ellis and Petronzio [32] and, in a particular case, by Curci and Greco [33]. Their result can be expressed in terms of the parton level fragmentation function D(x, Y ) by the relation: 1 doi(6 ) _ ~ D i i ( x ' y _ y ~ ) , oi jet dx j
(5.24)
where Y~ = ~
1 lo-
as(m2) ~as(62Q 2) '
Y=
17r as(m2) 2 bl°gcq(a2)'
(5.25)
* To this purpose it is important to keep the PT dependence of the DDT form factor induced by the effective coupling constant as(p2).
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
86
i specifies the type of jet under consideration and x is the fraction of energy deposited in the calorimeter of opening angle 26. In the case 6 ~ A/Q, eq. (5.24) in the standard formula for a single-parton inclusive cross section which can be converted into eq. (2.6) for hadrons by folding in a phenomenological fragmentation function. On the other hand, for d > > A/Q, eq. (5.24) refers directly to calorimeter measurements. In appendix D we give a simple rederivation of eq. (5.24) starting from our eq. (5.7) for a two-parton inclusive cross section and integrating it in such a way as to avoid multiple counting. Notice that the result (5.24) depends on Q2 and 6 only through the combination Y y~, i.e., through r~8 = log(Q2/A2)/log(Q282/A 2) as we have seen to be the case for the moments of 2-parton spectra. Ellis and Petronzio have used eq. (5.24) in its integrated form in x between 1 - e and 1 (with neither e nor 1 - e too small). Another way to exploit eq. (5.24) is to consider positive * moments in x (i.e., in the energy deposited in the calorimeter). Indeed, defining Cin(6) =
1 f~(2E/q,-~) Ojet
n dE,
(5.26)
the prediction for c i ( 6 ) is simply:
cin(8) = ~
e x p ( A n ( Y - Y~))ii = ~
J
[as(62Q2)TAn/2~b
(5.27)
J [- °~s(a2) --] ji
Eq. (5.27) represents an absolutely normalized prediction and can be used, therefore, for direct tests of QCD. It can also be used, in the more familiar way of deep inelastic scattering physics, to measure anomalous dimensions by varying the angle 8 at fixed Q2 (or the other way around). Indeed, if we assume for a moment no quark-gluon mixing (A n is a number), we obtain d log Cin(6) = A--e-nd log Cim(6)
d~
A m d6
(5.28)
'
i.e., plotting log Cim versus log Ci as ~ is varied we should get a straight line with a predicted slope An/A m . In the general case with mixing, but with a definite type of jet, one can still diagonalize the matrix An and obtain the result .
,'~ 2 ~ 2 . \
hn/2rrb
ci(8, Q2) -_ ain\last°%~Q_g~.l{d )I
-i [as(62Q2)]] + ~n~ %--~Q~
h n/2rrb
-
,
(5.29)
• The zero moment has, as usual, an infrared divergence. This is the moment studied in refs. [32,33] where the divergence is avoided by restricting the range ofx.
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets Table 7 i/0~(62Q2)~ hn /2nb + fli {°~s(62Q2)'~hn-/21rb
2 3 4 5 6 7 8
1.1653 1.0937 1.0610 1.0440 1.0338 1.0272 1.0226
-0.1653 -0.0937 -0.0610 =0.0440 -0.0338 -0.0272 -0.0226
0.3628 0.1896 0.1271 0.0951 0.0756 0.0626 0.0532
0.6372 0.8104 0.8729 0.9049 0.9244 0.9374 0.9468
-0.6085 -0.8170 -0.9604 -1.074 -1.170 -1.252 -1.325
-1.386 -1.852 -2.192 -2.460 -2.683 -2.875 -3.043
~=4 q .q
C1,C 1 1.0
0.9
0.8
0.7
0.6
i'11//
0.5
0.4
0.3
0.2
0.1
I
i
I
i
I
10 o
20 °
30 °
40*
50*
6 Fig. 11. Ci(/i) versus ,5 at Q2/A2 = 2000.
I
87
88
K. Konishi et al. / J e t calculus." a simple algorithm for resolving QCD /ets 1.0 0.9 0.8 0.7
g g log C2/Iog C6
0.6 0.5 Cg m
0.4
0.3
Log c~/togC~
0z
o2. 6 : 0.423 o 0.537 2,4 =
a o
0.1
I
0.1
0.2
3,5
= 0.706 = O. 788
4,6
i
0.3
0.4
0.5
0.6 0.7 018 0'9 1'.0
c~ - - ~
Fig. 12. Plot of log Cgn venus log Cgm. am, n is the effective slope, defined by log Cg "" am, n log Cgn+ const. i i ~kn ~n where the values an, fin, +, _ are given in table 7 and some of the m o m e n t s are plotted in fig. t 1. One can see that for m o m e n t s of order 3 or higher the quark jet gets essentially a c o n t r i b u t i o n only from the first term in eq. (5.29) so that a relation like (5.28) still holds. For gluon jets instead the two terms in (5.29) tend to be comparable and the full form has to be used. Actually even in this case the expected curves in a log Cng, log Cgm plot come out most of the time to give a good a p p r o x i m a t i o n of straight lines, as shown in fig. 12. Our m e t h o d is clearly reminiscent of the one used to check QCD scaling violations in n e u t r i n o experiments, in particular by m e a s u r e m e n t o f x F 3 at different values of Q2. The m e t h o d we propose * (varying 6 at fixed Q2) is more suited for the case of e+e - ~ hadrons, b u t we wish to point out that its exploitation requires
* A similar technique for testing QCD, with special emphasis to large FT processes has been advocated independently by Furmanski [52]. We wish to thank him for discussions on his approach to this question.
K. Konishi et aL / Jet calculus: a simple algorithm for resolving QCD jets
89
use of calorimeters which measure total hadronic energies including neutrals. Turning our arguments around and assuming QCD to be valid, one can use eq. (5.29) to determine the nature of a given jet, i.e., whether it is a quark type (e.g., e÷e - off resonance) of gluon type (e.g., on top of a heavy quarkonium resonance) or of mixed nature (as expected in large-p T hadronic processes), in which case the proportion of quark to gluon jet could be estimated.
5.3. Jets produced by initial partons We now turn our attention to the existence in QCD of jets associated with "initial" partons, i.e., with partons which enter in the hard elementary process. Consider for instance, large-pT hadron-hadron collisions. These are described at the constituent level by a two-body process in which one parton from each hadron collide to produce two large-pT partons. The jets we have considered so far come from the evolution of these final partons, but, of course there will be also hadrons going in the forward and backward directions. Many of these hadrons are due to spectator partons and are expected to exhibit limited PT, but also the active initial quark through its evolution creates a jet of partons (and then of hadrons) with PT growing with energy. It is this last type of jets that we shall refer to as "initial parton jets". Since bremsstrahlung from the initial partons is associated with scaling viola-
b,y~O
~
y-O X
a Xb y~Y
Ca)
b ~------O
-"
(b) Fig. 13. (a) Initial patton jets. (b) Corresponding jet calculus diagram.
90
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
tions, the appearance of initial parton jets is expected to reveal itself when sizeable scaling violations occur, hence, presumably at the highest values of PT or Q2 presently attainable. Our techniques can be readily applied to this new situation. However, because of some crucial difference in kinematics, the resulting expressions will differ somewhat from the corresponding ones of ordinary final parton jets. Consider indeed the process drawn in fig. 13, in which a parton of type i in an initial hadron evolves and, before triggering the hard process as parton a, radiates other partons of which we study the inclusive spectrum (fig. 13a). The inclusive cross section for observing parton b with definite x b and a certain PT relative to the initial hadron momentum can be computed from the diagram of fig. 13b where PT now determines the mass of the patton a'* thus fixing the intermediate "time" of the vertex j ~ a'b'. It is then easy to obtain do(h~a + b + X)_ dxadXbdY
a',b',j
fdxdgdxldX
2 Gaa'(X 1, Y - y ) O b b ' ( X 2 ,
X P j ~ a , b , ( Z ) GJh(X, y ) t~(X a -- XZX 1) ~ ( x b -- x(1 - z ) x 2 ) ,
Y)
(5.30)
where 1 Y=2~
%(m 2 ) l°g as(4p 2) '
1 as(m 2) Y= 27rb log as(Q2)
(5.31)
As in the case of ordinary jets one can replace b with a hadron h' by just replacing Dbb' in eq. (5.30) with D~',. Actually, in this case, no further term such as the second one in eq. (5.8) is present. From eq. (5.30) we can go to moments in Xa, Xb and obtain
=(a,blexp((y_y)An)exp(YAm)Pn, m
(do(h ~ a + b + X ) ) dY
n~rn
Xexp(YAn+m)li) G~(n + m, y
= 0).
(5.32)
The differences between this equation and eq. (5.10) stem from the fact that, whereas for final parton jets energy and qT decrease together from the hard vertex to the final partons, here they decrease in opposite directions. This fact will be reflected in some phenomenologically interesting differences between the two cases. To be definite let us consider again some simple moments which are free of infrared difficulties. For instance, we take a to be a valence quark (as measured, e.g., ~' Notice again the difference with DDT where it is parton a in fig. 13b which must have a fixed mass of order x/p~Q 2 < < Q2.
91
K. Konishi et al. / Jet ealcuhts: a simple algorithm for resolving QCD /ets
in xF3) with n = 0 and we sum over b with m = 1. Using charge and momentum conservation one gets =~as(4p~) /{-~n ANS\)G~al(1,Yo)J"1 t °q(m2) IANlSl2~rb L ~TT [as(4P~) )
(~)o,1
( \S~-lG~a(1,y ).
-- p~
(5.33)
As in eq. (5.11), we recognise in the square brackets of eq. (5.33) the naive result, but now the correction factor decreases with PT instead of increasing. In other words, one gets a PT distribution which is narrower (and not broader) than the naive one (see fig. 14). Concerning the normalization of eq. (5.32), we notice that here not all the energy is available to partons a and b since a fraction of it is carried away by the spectators. Because of this the cross section is normalized to Q2
dp T d(~T)
f
0,1
qo
=~[GhVal(1,Yo)-G~a(1, Y)] ,
(5.34)
val
and not to ~val G~al(0, Y) = 3 (for a proton). In order to illustrate further the characteristics of this type of jet we have computed Qz
dPx Pr(do/dPT)O,1
J
and found
Q2
,
(5.35)
f (_~s]
(P~) = ~ !
¢
Z~c~:'(1,v,r-)
l°gfQZ/A2) ~G~ al(I,Yo) - G~al(l, Y) val
XYlarge(-A'~ Q2 Fas(m2)]A~s/2"b ~2,b: log(aVA2) L~(-(-~T/
(5.36)
This result can be contrasted with that of er~ (5.13) for a final jet which does not exhibit the last damping factor (log Q 2 )a I ]21rb , typical of scaling violations.
92
K. Konishi et al. / J e t calculus." a simple algorithm for resolving QCD jets 10 S
I
Q/A: 100 (
INSIDE 1 JET A •
TRANSITION REGION r
2 JETS A
100
% x
% ,-:." E
O o W~
10
1
2
10
5
50
PT/A Fig. 14. Our results for initial patton jets, eq. (5.32) compared to the naive result, eq. (5.12), and final parton jets, eq. (5.11).
The physical origin for these different behaviours is to be found in the fact that in the "initial jet", a large Pa" (requiring a large mass) can only occur after a long evolution which has degraded its energy by repeated emission of quanta whereas in the "final jet" the original parton emits large-pT quanta (i.e., has a large mass) right at the beginning of its evolution. An interesting way to test QCD emerges from its prediction of initial parton jets. Since the (PT) of such jet (relative to the beam axis) grows with Q2 (the scale of
K. Konishi et aL /Jet calculus: a simple algorithm for resolving QCD jets
93
the hard process) one predicts that, in deep hadronic reactions, the average PT of beam and target fragmentation products should increase with the PT of the largeangle jet. Some experimental indications supporting such effects have been recently reported [53].
6. Jet field theory: outlook From the discussion in sects. 2, 4 and 5 we have seen how one can give a physical meaning to jet calculus amplitudes representing inclusive cross sections for an initial parton o f x = 1 a n d y = Y giving rise to a set of final partons characterized by (xi, Yi) (fig. 15a). The values x i correspond of course to the energy fractions carried by the "detected" partons whereasy/correspond to how much off-shell the final partons are allowed to go. For Yi ~ 0 the ith parton distribution has to be folded
(o)
= l (yx :=Y)
(b)
~
(Xl ' Yl )
~ ~ " ~
( xn' Yn)
(x2, Y2 )
~
(x,Y) (xi,Y i )
(x=l,y 1)
(c)
01
a2 = 0 (~i-~2)
[ O ( y I -Y2 ) 0 o 2 o I ( ~ 2 - ~ I ; Yl-Y2 )
•.. O ( y 2 - Y l ) O o 2 0 1 ( ~ 2 - ~ 1 ;
(d) ~ Y - 3
y2-Yl)]
= o(e~ -e~2-e~3)O(yl-y2)O(y2-y3) P - ¢~.,-L) aI ~
0203
-
,
Fig. 15. (a), (b) General Green functions in the jet calculus. (c), (d) Feymnan rules of jet field theory (?), propagators (c) and vertices (d).
94
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
into a phenomenological D h fragmentation function, whereas for Yi large no such factor is needed and one is already computing the cross section for depositing an energy Ei(xiE ) into a calorimeter of angle 8 i given by 1 lo- as(P2) Y i = ~ b Sas(Si~Q2 ) •
(6.1)
Furthermore, considering initial parton jets (as in subsect. 5.3), one can also have Green functions such as G(x 1 = 1 , y I " 0 ;
x2, Y2; x3, Y 3 ~ Y ) ,
(6.2)
where (x 1 = 1,y I = 0) is the initial quark, (x2, Y2) is a forward jet detected by a calorimeter and x a = XBj. Hence, in general, the Green function G(X, y ; {xi, Y i ) ) ~ G ( z ; z i )
(6.3)
has a physical meaning with z = (x, y ) (fig. 15b). One can ask at this point if a general Lagrangian formulation is possible in order to reproduce our jet calculus rules. The answer is yes, but the resulting Lagrangian (defining jet field theory) is non-local. Its structure is easily found to be given by the following action A =Aki n +Ain t ,
Akin =
d ~ ' d y ~+(~', y ) [Sat~ 0~y 8 ( ~ - ~ j ' ) - Pa,t3(~' - ~)] qJt3(~,Y ) ,
Aint = f d ~ l d ~ 2 d ~ 3 d y
~;(~2, Y) ~ ( ~ 3 , . F ) / 6 a ~ 3 7 ( ~ 2 - ~1) ~ a ( ~ l , Y)
X 6(e ~1 -- e ~2 -- e~3),
(6.4)
where ~ = log x, ~, ~+ are destruction and creation operators of partons and P, fa are the same quantities we defined in sect. 2. The Feynman rules derived from eq. (6.4) are straightforward (fig. 15c, d): (i) for each propagator from z I = (~1, Yl ) to z 2 = (~2, Y2) write a factor Oa2al(Zl, z2) = 0(~1 - ~2) [0(,Vl - Y2)Oa2al(~2 - ~1 ;YI - Y 2 ) + 0(Y2 - y l ) D a 2 a l ( ~ 2 - ~1 ;Y2 - Y l ) ] ;
(6.5)
(ii) for each vertex a l ( z l ) ~ a2(z2) ~ a3(z3) write a factor 6( e~l -- e~3)/13al--~a2a3(~2 -- ~1 ) ~(V1 -- Y2) 6(Y2 -- Y3) ;
(iii) perform flat integration over all intermediate z i.
(6.6)
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
95
Of course this is a complicated non-local theory. There are however two points which may lead us to hope that it is not completely untractable: first the theory is in 2 dimensions, second it is only to be solved in the tree (classical) approximation. It is amusing to notice at this point that, in the case of ~b36, the theory can be made local by a suitable reshuffling of factors between vertices and propagators. Unfortunately this trick does not work in QCD, due to the a more singular infrared structure. We would like to conclude this paper with an optimistic outlook based on the fact that we have kept asking more and more ambitious and detailed questions about perturbative QCD (e.g., from one- to two-particle spectra, from longitudinal to transverse structure ...) and we have kept getting sensible and (probably) reliable answers to be confronted with data. Which is the limit beyond which we are forbidden to ask questions to perturbation theory? The answer seems to be that, within jet physics, we can ask questions up to the stage in which, through jet evolution, masses and relative transverse momenta of partons have been degraded down to values a few times larger than A "" 0.5 GeV. Up to there, the jet calculus formalism we have outlined seems to be applicable and powerful. Moreover, it has been possible recently [54] to follow in this way not only the pattern of momenta as produced in jets but also that of colour. Surprisingly perhaps, one seems to be able to identify finite-mass colourless clusters of quarks and gluons in the jet's final state (preconfinement). There seems to be a contradiction between this intuitive picture of hadronization in which bunches of final partons in colour singlet configurations give the final hadrons, and one in which each individual final parton gives rise to a bunch of hadronic products. We wish to point out, however, that there is not necessarily a contradiction between the two pictures: indeed any perturbative calculation has to stop at a stage in which the final partons are still quite far from their mass-shell (q2parton > > A 2), whereas the partons which make up hadrons are closer to it (qp2arton ~ A2). It is conceivable that, whereas far off-shell partons yield hadrons separately, almost on-shell ones have to group in a colour singlet system before binding into hadrons. This important issue clearly deserves further study. If this hint is confirmed by further, more rigorous investigations, we may end up with the surprising and exciting conclusion that perturbative QCD, when translated in the precise and simple rules of jet calculus, can lead to a powerful tool for predicting the properties of final hadronic clusters in jets, where the only (but by no means trivial) task left to non-perturbative confinement will be the statement that such low-mass colourless dusters cannot be further resolved into their constituents. We thank D. Amati for discussions. One of us (K.K.) would like to acknowledge interesting discussions with D. Duke, M. Fukugita, N.A. McCubbin, P. Scharbach, J.C. Taylor and J. Wosiek and thank CERN for the very warm hospitality enjoyed at the Theory Division at various stages of this work.
96
K. Konishi et al. /Jet calculus." a simple algorithm for resolving QCD jets
Appendix A An induction proof of jet calculus rules We present here an induction proof of the jet calculus rules, (i)-(iii), of subsect. 2.2 for computing Dal ...am,i(n 1, n2 .... nm ; E). In the cases rn = 1 and m = 2, we obtain, by computing the (dressed) ladder diagrams of fig. lc and fig. 2a, yl Da,i(n; Y) = (eanY)a i = ~ . l
(An) 1 ,
(A.1)
and oo
Dala2,i(nl, n2; Y)=
(ll + 12)! yl+ll+12+l
~
bl,b2,J l, ll,12=O ll! 12! (ll +12 + l + 1)!
X (An)lalb1(Zn)la22b 2 pblb2,J l nl,n 2 ( A n l +n2)ii ,
(A.2)
respectively, where the factorials arise because of the ordering in the k2's over which we integrate to obtain the maximum number of logarithms (LLA), while the structures containing An and Pnl,n 2 appear as a result of integration over the longitudinal momentum fractions. When a Laplace transform is made from variable Y to E, eqs. (A. 1), (A.2) clearly agree with the jet calculus rules. Before going into the case of general m, we introduce some notations. Let us call J-diagrams the tree diagrams which can be obtained from the original Feynman diagrams by omitting all unobserved patton lines. (Our jet calculus diagrams such as in figs. ld, 2b, 3a can be regarded as J-diagrams). A J-diagram G for k-parton distributions has k - 1 vertices labelled as v 1v2, ... vk, and 2 k - 1 lines labelled by a = 1, 2 ..... 2k - 1. In the original Feynman diagrams (such as figs. lc, 2a), each line has additional la vertices (a = 1, 2 ..... 2k - 1). Consider the contribution to Dk(Y) from a Feynman diagram specified by a Jdiagram G and a set of integers (la). (Here and below we omit parton indices i, al ... ak and moments nl, n2 .... n k for simplicity). This has the form, k--1
2k--1
×({la}) t=ll--I P v , t a=ll--I AtaY k-l+zala .
(A.3)
It is convenient to consider the total contribution to Dk(Y) from G with l = Y.la fixed, D~'t(Y). According to the jet calculus rules, it is given by
DG,t(y)
yl+k-I (l + k -
1)!
~
f°+i** dE Et+k_ l 1 PE-1 p ,-' 2n--iE - s~, s~2 o--i ~
K. Konishi et al. / Jet calculus: a simple algorithmfor resolvingQCD/ets 1
•" E _
~k
97 (A.4)
,
while the total contribution of the diagram G is
D~(Y)- ~
D~'I(Y)
l=O o + i*o
= ~f a-i,*
2nidEeEr'E 1.~IPE 1 P... P - -1 , - ~2 E- ~k
(A.5)
where -~i represents an appropriate sum over A's (see fig. 3b), and the summation is over the different time orderings of vl, ... vk_ 1 . Now assume that eq. (A.4) or eq. (A.5) (equivalent to the jet calculus rules) is true for k ~< m - 1, and I = 0, 1, 2 .... Any J-diagram G for m observed partons can be uniquely decomposed into two J-diagrams G 1 and G2 having kl and k2 observed partons (kl + k2 = m, 1 ~< kl ~< k2 ~< m - 1) and a line O which are joined at a vertex v (see fig. 16). Denote by ll, 12 and l the vertices representing the emission of unobserved partons i in the original Feynman diagrams of G1, G 2 and the line O, respectively. We find then, oo
D6m(y)=
~ GI,I l G ,12 l l+1 (11+12+m--2) ! ttl,t2=o Dtq (Y)Dk~ (Y)Pv A Y (l+ll +12 + m 1)! ' (A.6)
where the last combinatorial factor arises (as before) from the transverse momentum integrations with the ordering among k2's along the line O.
t ffl kl
Fig. 16. Decomposition of a J-diagram G into two J-diagrams G 1 and G2 and the line O.
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
98
We substitute eq. (A.4) into eq. (A.6), go over to the variable E by a Laplace transform, and then perform the summation over/, l I and 12. This gives,
DCm(E) ? dYe-EYDG(y) =
0
°l+i~ (]E l
1
1 s~l p,.. p. E l - 61~ 1 kl
I'j 271i E l o 1 ...i ~
1
1
x ~ f:+i= de2
1
E_EI_E2 PVE~ A'
1
1
2 o2-i~ 2hi E 2 - ~ P " ' P E z -
(A.7)
s~k22
where we have dropped vanishing terms such as N~1o2 E -1 (E1/E~ (1 - E2/E)-(L÷I).Our aim is to show that eq. (A.7) is equivalent to eq. (A.5) for k=m. For this purpose consider a fixed time ordering of vertices in G 1 and G 2 separately, and label them from left to right as vl, ..., v~ 1 and v I ..... v? 1" The rela1-~2tive ordering of these two sets is specified by saying that there are qi vertices of G 1 between v 2 1 and v 2 of G 2, where i = 1,2 .... k2, and qi are non-negative integers such that Nf2qi = kl - 1. Then the desired relation (the equivalence of (A.7) and (A.5)) can be expressed in the form,
:. 577-.~J~
= ~, - ~ , k2
:H Xql~=kl--1
= sr~ --~J
sI-I, s>s' ,= P~ ~ - e ,
P[3
k 1-1
,
13=1 °t-=Pfl--1 E -
H
~1 _ ~
-s~
k2-1
.,N
(A.S)
]='*=*l "] ~=li
where Po -1) - p1,3s - Ef=lqi + 1, and:" represents the time (normal) ordering of P ' s and ( E - M . We prove eq. (A.8) by induction with respect to kl. For kl = 1, it can be easily checked. Assume then that it holds for kl ~< n - 1. For kl = n, we use an identity, kl
FI
1
a:lE 1 - s~
-
kt-2
1
(
CtH1
.
.
1
.
1
)
1
. (A.9)
99
K. Konishi et al. / Jet calculus." a simple algorithm for resolving QCD jets
on the left-hand side of eq. (A.8). But then by the induction hypothesis, the 1.h.s. becomes
1
-i-I
I-]
1
k2 Z q'6 =kl --2
6=1
1
X .9{11_1
__
.~lkl [IPI1P : ,
(A.10)
where Po - 1, p~ ~;i=lqi + 1, and in the second term in the curly bracket, the replacement s~ 1 . ~ M~ should be made. Consider a term in the sum and denote 1--* 1 t by/3o the smallest/3 such that Pao = kl - 1. Then this term can be written as B°[~i1
~
1
,
E-
× {
)(
1 E-
l 1_ -
~ 0 E-
) _
1
_
-
q
~o
W. - al
: o
s~ll_ 1 - s ~
-
1 -
,:qE--~l
s~l-1
1- s~
-
liP l i P : . (A.11)
Substitution of this into eq. (A. 10)evidently gives the right-hand side of eq. (A.8). This completes our derivation of jet calculus rules. Appendix B Details o n
induction
proofs
of various
sum rules
B. 1. M o m e n t u m sum rules (eq. (3.24))
We present .the proof here for hadron distributions; the one at the parton level can be obtained by making a formal replacement, hl k
at the end.
5kl
100
K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD ]ets
Elementary relations needed are * / .....
°
.
nol
,
\
~h j/dXhXhdh(xh)= I)
1
C -Xb0 o-y
.....
~
n
=
n~1
,
(B.1)
(B.2) (B.3)
(this follows from a Ward-Takahashi type identity, Pnb,Cia-- A nba - A nba + 1, and eq. (B.2)), and Pnbe,a
,re+l-=
+ pbc, a
n+l,m
= pbc, a n,m
(B.4)
"
The h a d r o n distributions (see fig. 4) satisfy a recursion formula,
hl,nl
k-I ~"~T~.. I--ik I
where ~div. means a s u m m a t i o n over all possible ways of dividing (h I ... hk) into two groups, consisting o f l and k - I lines. Furthermore, the non-perturbative decay functions d a' {hi }(Z 1 ... zj) satisfy themselves a set o f m o m e n t u m sum rules,
=
!
}-hknk=I""-
hk-1nk-1
(B.6)
z
hk.1nk-~
The p r o o f of the m o m e n t u m sum rule for k = 1 is trivial if one uses eqs. (B.1) and (B.2). The case k = 2 is also easy, as
n.1~ h,n =
n
h,n
e 0----
n;*1C)_~n.i
(B.7)
Next, we assume that the sum rule is valid for k ~< m. Then for k = m + 1 we find, using eqs. * The general proof is easiest in the E-description of moments, as is done below, since each distribution is a simple (matrix) product of various vertices, propagators and (non perturbatire) decay functions.
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
101
(B.1)-(B.3), (B.5), (B.6) rri
=
.
rr,, :
÷
.. div. I=I
--~[~.
;
h,,',,
m
~'}r.-i
"'..
Z.~:L@C._. h, o, . -
~"~hn~nm
"~hn-~nrn
m-1
~,~ hrnr oI
)" r~ *'
'
~
=
'
hr nr*'
~ h m . n
(B.8)
fl,I
where we have used a shorthand notation, m
~l-
~rn =- C hi, i= 1
C
upper blob
nj,
~ m - , =-
C
lower blob
ni •
Consider the third term of eq. (B.8) first. For a fixed division (hi l ... hh/hh+ 1 ... hi m ) of (h 1 ..- hm ), there are two terms, having the vertices Pzl, Z m - l + l and P z t+ l , z m - t ' respectively (but otherwise identical), which can be combined using eq. (B.4). The third term can now be summed with the fifth term, which together cancel exactly the second term by use of the recursion formula, eq. (B.5). The rest, the 1st, 4th and 6th terms, in which all the moments are conserved everywhere, can again be simplified by using eq. (B.5), giving Zm
~1
hl,nl
~
~'m*1 ~ ,
-
hm71m
r=1
hlnl hr nr÷l
.t... hm,nm
which proves the desired result for k = rn + 1. B.2. Charge sum rules (eq. (3.25)) We introduce a new propagator,
o
(B.9)
-=(E -/T,n)-' ; ~'n--0
describing the propagation of a "valence" quark. Denoting by i the given flavour quantum numbers of a quark, elementary relations necessary are given by (a wavy
102
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD jets
line denoting the operation of integrating over the momentum fraction of the valence quark),
~1
i
f
i0 i ~
b,n
=
dx i
i
ioi
i f
Di" (.x i Y ) n
b
)
(B.10)
= 1
- 8bi
i
n
(B.1 1)
i
i ° jr ~
•
(B.12)
The lowest sum rule is nothing but eq. (B.11) which is easily derived from A0qq = 0. We next assume that the charge sum rule, eq. (3.25) is proven for k ~< m. Then, for k = m + 1, we have (by using the recursion formula),
b1
o~ - t
[:1 div
div
1=~ div
r
bri b
where use was made of eqs. (B.I0)-(B.12). The 2nd and 3rd terms ofeq. (B.13) are seen to cancel by using eq. (B. 12) and the recursion formula, while the last term can be simplified again by use of the recursion formula, thus the result becomes
=
~
w h i c h is t h e s u m r u l e f o r k = m + l .
m-
ZSbri
br
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD lets
103
Appendix C Quark,ghon universality We present first two sufficient conditions for quark-gluon universality and then show, using Nachtman's positivity constraints, that one of these sufficient conditions is satisfied for n i/> 2 (i = all). (i). The first sufficient condition. A sufficient condition is that the distribution is dominated by a single pole (the largest eigenvalue) coming from the initial propagator, (E - A N ) -1 , where N - ~ini . In order to see this, let us write
(AN)ha = ~k~V(~Ar)ba + ~kN ( ~ N ) b a , where XN are the two eigenvalues o f A N and and 5aN respective projection operators. If X~v dominates the whole distribution, then one can make a replacement, +
(E - AN)~a1 -+ (E - X+N)-I ( ~N)ba •
Using the fact that ~ v is a matrix of rank 1, +
+
( ~ N)jq/(~)N)jg = rN ,
(indep of j ) ,
one then easily derives (indep of ( b i } ) .
O q~(bi)rvxtr~g'~{bi) (Y)~rN (ni } t " )/L" (ni )
This condition is satisfied in the examples, eqs. (4.5), (4.7) and (4.9). (if} The second suffiicent condition. Another sufficient condition is that the largest pole comes from the final propagator, (E - ~,iZni )-1 as in the case of multiplicity correlation coefficients. In this case we make a replacement (E
_
~
Ani)-I "~ (E
i
_
~
X+.)-' [I i
t
i
+
(~ni)bib
,i .
Again, using the fact that ( g n )+q j / ( 9
n+) g j = S n ,
(indep of j)
we easily derive that D q-~ ( b i ) ( y a l D g~ (bi ~ (hi} " " ( n i )
(Y)=f(ni}'
(indep of {b i })
(iii) Proof that for n i ~ 2 (i = all) the largest pole comes from the initial propagator. The Nachtman's positivity condition gives +
+
+
2Xn+ 1 ( X n +Xn+ 2 •
K. Konishi et al. / Jet calculus: a simple algorithm for resolving QCD /ets
104
This, together with the relation 2X; < X,],
(e.g. f o r N f = 4, X; = - 2 . 5 4 , X,~ = - 4 . 0 0 ) ,
leads to the inequality +
)k+ + )k+ < ~'m+n
for all m/> 2 ,
'
n t> 2 .
There is a conservation of moment at each vertex, therefore the above inequality immediately leads to the conclusion that the largest pole comes from the initial propagator, (E - X~v)-1 , where N = Zin i.
Appendix D A jet calculus derivation of eq. (5.23) We wish to show here that eq. (5.7) for a 2-parton spectrum implies eq. (5.23) for calorimeter-measured cross sections. The latter equation has been argued [32] to be valid in a range e < x < 1 - e', but we shall see fror~ our derivation that such limitations are less severe than conjectured and do not prevent the possibility of taking (positive) moments of eq. (5.23) to yield the simple result eq. (5.26) whose applications we discussed in the text. It is actually convenient to go back to eq. (2.8), from which (5.7) follows if one fixesy. For our purpose, we have to fix b o t h y =y~ a n d x and then use a sum rule in order to get a unit branching ratio for j to decay into anything. This we can do by performing the sum rule: ffdXldx2Xl ala2,d
X
x2 X--
1 do(i-*a I +a 2 + X ) dXldX 2
X2 U
x,y
fixed
which avoids multiple counting and takes care of the case of fig. 2b in which al, a2 are coming from bl and b2 respectively. Since, however, a I and a2, as arbitrary decay products of j, could happen to be both products o f b 1 or b2, we see that we still have to allow an evolution from y down to y' before the splitting, and integrate overy'. The correct formula is then:
f
X1
X2
v ff
1 d°i(6)- ~ dxldX2 -3 Oijet dx ala2 ~ x x-x 2 0
×
~ b 1 ,b2,j,j'
dy'
f d z d w d w l d w 2 D a l b l ( W l , Y')Da2b2(w2, Y')Pj'-*blb2(Z)
K. Konishi et al. / Jet calculus." a simple algorithm ]'or resolving QCD ]ets
105
X Dj'j(w, y - y ' ) Dji(x, Y - y ) 8 (x 1 - x w z w I ) 6 (x 2 - xw(1 - z) W 2 )
y f
dy, f d x 2 d z d w z d w z
az,bl 0 b 2 ,jj'
w
(1-z)
ww2
1 - - ( 1 - - Z ) WW2
Da2b2(W2,y ) Pj'~blb2(Z) Dj j ( w , y - y ) Dji(x, Y - y ) , where the momentum sum rule for Dal bl has been used. Expanding the denominator (1 - (1 - z) ww2) -1 and replacing P with P because of the factor z(1 - z) one easily gets in the vector-matrix notation of eq. (2.13) and ref. [25]: y
oo
~
f
a2 n = o
de'
- y)li)
o 0o
dy' a2
[(a2 lD(n + 1, y') D(n + 2, Y - y ' ) ] D(x, Y - y)[i)
n=0
oo
(azl[D(n + 1 , y ) - D ( n + 2 , y ) ] D(x, Y - y ) a2
n= 0
= (a21D(1,y)D(x, Y - y ) l i ) = ~
Da2i(x, Y - y ) , a2
which is the desired result. Notice that, from this derivation, Dji(x, Y - y ) represents the number of parton of type j, energy fraction x and mass m(y) produced one jet of type i at Q2 = Q2(y). It can be easily argued however that, within the LLA, this is also, after summing over], the number of colorimeters of angle 5 getting an amount of energy xEjet deposited in them. It is also obvious from this derivation that the result holds for any x not infinitesimaUy (as Q2 ~ o0) close to zero and one so that (positive) moments can be taken).
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K. Konishi et al. / Jet calculus. a simple algorithm for resolving QCD lets
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