Heavy quark multiplicities in gluon jets

Physics Letters B 285 (1992) 160-166 North-Holland

PHYSICS LETTERS 13

Heavy quark multiplicities in gluon jets M i c h e l a n g e l o L. M a n g a n o lstituto Nazionale di Fisica Nucleare, Scuola Normate ~uperiore and Dipartimento di Fisica, Pisa, Italy

and Paolo Nason Istituto Nazionale di Fisica Nucleare, Sezione di Milano, 1-20133 Milan, Italy and Gruppo CoLlegato di Parma, 1-43100 Parma, Italy

Received 19 March 1992

In view of recent theoretical developments we reanalyse the production of heavy quark pairs inside gluon jets, and compare the new results with the prediction of the HERWIG shower Monte Carlo program.

I. Introduction

In this paper we examine the Q C D predictions for heavy quark multiplicity in high PT jets. It was first pointed out in ref. [ 1 ] that perturbative Q C D alone can provide a fairly reliable prediction for this quantity. The problem can be stated as follows: to compute the number of heavy quark pairs contained in the average in a jet of high transverse m o m e n t u m Pv, in the limit when Pv >> m, m being the mass of the heavy quark. The calculability of the multiplicity is based upon two ingredients: - The calculability of heavy quark production cross sections at fixed order in QCD. This is a consequence of the fact that the mass of the heavy quark provides an infrared cutoff to the perturbative formulae, so that the relevant scale ofc~s is set to be of the order o f m . - The ability to sum an infinite set of terms in the perturbative expansion. In fact, because of the presence of two widely different scales, m and Px, we must worry about the appearance of large logarithms to all orders of the perturbative expansion. More precisely, we expect logarithmic terms of the form [c~s(m) × log 2 ( p ~ / m ) ] ~ in the nth order of the perturbative expansion. This resummation problem can be turned into an evolution equation for the multiplicity (see ref. [2] ). 160

Our lack of ability to compute light hadron multiplicities is due to the fact that the first requirement made above is no longer satisfied in this case. However, the Pv dependence of the light hadron multiplicity can be still obtained thanks to the evolution equation quoted above. It was shown in ref. [ 1 ] that the heavy quark multiplicity can be obtained by convoluting the probability for a gluon of virtuality K 2 to decay into a heavy quark pair, times the multiplicity of gluons of virtuality K 2 inside a gluon of given Px. Unfortunately, the gluon multiplicity formula used in ref. [ 1 ], which was taken from ref. [2], is incorrect in normalization. The error is quite subtle, and will be discussed in appendix A. Since only the energy dependence of light hadron multiplicities were considered in ref. [2 ], this fact has no effects upon the physical quantities considered there. It does however give an incorrect result for heavy quark multiplicities, since the normalization is fixed in that case. We find that the correct formula for the heavy quark multiplicity is

( 2m2j

1

dK2 o~(K 2) 1 +

P =

1 K

4m2

Xng(Q 2, K 2) ,

(1.1)

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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where ng(Q 2,K 2)

(l°g(Q2/A2)) a

(1.2) and

l+-syb-b \

'

b = 11CA--2NF 12~

(1.3)

The corresponding formulae given in ref. [ 1 ] have an exponential instead of a hyperbolic cosine in formula (1.2). Therefore, in the appropriate limit, that is to say for PT >> m, they are larger by a factor of two than the formulae given above. It is interesting to look at the form of the expansion of formulae ( 1.1 ) and ( 1.2 ) in the limit when log (m 2/A2) >> log ( Q 2 / m e). It is then convenient to rewrite the logarithms as Q2 1 l°gA2 - bas(It)

K2 log~-

1 + h a s ( p ) log.~_2

1 ( K 2) bas(/t) l + b a s ( # ) l ° g p 5 '

(1.4)

where/z is an arbitrary scale, to be chosen between K a n d Q, a n d the g l u o n m u l t i p l i c i t y f o r m u l a e can t h e n

be expanded according to the equation ng(Q 2, K 2 ) = [1 + P , (q, Z) ]

Xcosh ( N / C~A as(/t)log2 Q~ K [I+Pz(q,Z)] ) , (1.5) where

q = a ~ ( # ) l o g ( K 2 / p 2) , Z = Ots(fl)log(Q2/It 2) ,

( 1.6 )

and P~ and P2 are analytic functions of the variables q and Z all vanishing for q and Z equal to zero. We will now make a few remarks about formula (1.5). Observe that the hyperbolic cosine has an expansion in even powers of its argument. Therefore, formula (1.5) correctly embodies all terms of the order

2 July 1992

[aslog2(Q/K)] n, the so called double logarithmic terms. The functions P~ and P2 contain powers of [as log ( Q / K ) ]n at most, since/1 is between K and Q. Terms of this form, when combined with any power of aslog 2 ( Q / K ) , will be called subleading. Consider now the case when log ( Q / m ) is large but as l o g 2 ( Q / m ) is small. In this case the multiplicity will be given by formula ( 1.1 ) with ng set to 1. The K 2 integration will yield a l o g ( Q / m ) (we are assuming that there is not enough K 2 variation in order for as to run appreciably). Next, let us consider the case when aslog ( Q / m ) is of order 1. In this case, we only need to resum all the leading terms, so that the gluon multiplicity can be approximated by the hyperbolic cosine alone, while the formula for p can be taken as in the previous case. Observed that in this region we cannot replace the hyperbolic cosine by one half of the exponential of its argument, since the argument itself is of order 1. The integral in K 2 is logarithmic also in this case, and terms which do not have a logarithmic enhancement are not correctly included. Finally, there is the case when c q ( m ) l o g ( Q / m ) is of order 1. In this case also the subleading terms are important, and one should use the complete formulae (1.1) and (1.2). One may also replace the hyperbolic cosine by a half of an exponential, since its argument is large. The exponential behaviour of the hyperbolic cosine gives a suppression of the large K region which is stronger than any inverse power of K 2. Therefore, the integral is controlled by small values of K 2. In order to see what is important in practice, we now plot in fig. 1 the multiplicity of a bottom quark pair, with m = 5 GeV, as a function of the energy of the jet using - the full formula ( 1.1 ), - formula ( 1.1 ) with ng set to 1, - formula ( 1.1 ) with ng given by formula ( 1.5 ) with P~ and 1'2 set to zero and/z = Q, - formula ( 1. l ) with ng given by formula ( 1.5 ) with P~ and P2 set to zero a n d / z = K , - f o r m u l a ( 1.1 ) with ng given by formula (1.5) without prefactor, i.e. with a = 0 . We see that the formula with r/g= 1 is below the full formula only for very large momenta. We also see that if we use the approximate double log formula (eq. ( 1.5 ) with P1 =/°2 = 0 ) for ng, choosing/z = Q or/z = K makes a factor of two difference at Q = 1 TeV. The full formula for ng, which includes subleading terms, 161

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2 July 1992

0.20

. . . . . . . .

I

. . . . . . .

mq = 5 GeV, A=200 MeV

///

0.15

/

Solid: Full Dots: n , = 1 Upper dashed: DLA, /z = K Middle dashed: a = 0 Lower dashed: D ~ , # = Q

0.10 Z

/d

/

/ ] / / / / /A / / // /~/ / / / /~ J /

/-

~~ /

'g

l

/

1

/

/////~//~////"

z 0.05

0.00

J 10

I

I

I

I

50

I

I

I

I

100 q (geV)

I

I

I

I

500

I~.

1000

Fig. 1. Multiplicity of a 5 GeV heavy quark versus the jet energy. Comparison between the full result, the double logarithmic approximation for different choices of scale, the full result without prefactor and the O ( a s) formula.

fixes the scale choices, so that this uncertainty is no longer there. As one can see, the effect of the prefactor is of the order of 50% at Q = l TeV. From fig. 1 we learn that the intermediate region is in fact much larger than one expects. At this point, a word of caution is necessary about the reliability of the subleading terms in the multiplicity formula. They all originate from the evolution of the jet cascade. Subleading terms (i.e. terms of order aslog(Q/m) ) arising from other sources, like the hard vertex, are not correctly included. Furthermore, they are in general process dependent. We cannot therefore maintain a universal multiplicity formula which is correct at subleading level. A related ambiguity has to do with the precise meaning of the scale Q. In refi [ 1 ], which deals with gluon production via a fictitious tr[F*'~F,,~] term, Q is well defined, and corresponds to the total energy available in the process. In a more complex process this correspondence may no longer be valid. In hadroproduction, for example, if all the available energy of the incoming gluons went into the heavy quark pair, the quarks would come out back to back in transverse momentum, and the competing process gg--~QO, which is certainly not embodied in formula ( 1.1 ), would come into play. Nevertheless, since the Q value is tied to 162

the momentum exchanged between the coloured particles involved in the hard collision, we expect Q to be of the order of PT of the jet. We cannot decide, however, whether we should put Q=PT or Q = ½pT or Q=2pv in formula (1.1). The related ambiguity is again a process dependent subleading effect. In principle, subleading terms arising from evolution are of the same order of the process dependent ones. One may question then whether it is consistent to include the former and not the latter. This issue has no complete answer. However, the soft terms arising from evolution are those that produce particles which are collimated with the jet, while all the other terms are in general not collimated, or may be collimated to the beam jets in the case of hadronic production. Therefore, in some sense, the multiplicity formula we are considering applies to the heavy quark pairs belonging to the jet. In ref. [3] a calculation was carried out for the multiplicity of jet clusters, where the clusters are defined in terms of the so called kv algorithm [4]. The formulae they obtain for the multiplicity of jets inside gluons are similar to ours except for their subleading terms, which are specific to their definition of clusters. For heavy quark production it is more appropriate to use a definition of the multiplicity based

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upon the virtuality of the gluon, instead of a clustering algorithm. In fact, formula ( 1.1 ) represents a direct decay of an off-shell gluon into a heavy quark pair. Therefore, we de not expect the subleading terms of their formula to be correct also in our case. We we will also show, however, results obtained with their formula, in order to assess the importance of subleading terms.

definition of the mass of confined objects in QCD. As one can see, the variation due to the uncertainty in the value of the b mass is generally negligible. In fig. 3 we show a comparison of the analytical calculation of charm multiplicity with the results obtained from the Monte Carlo program HERWIG [ 5 ] (version 5.4). The input parameters for the MC calculation are me= 1.5 GeV, and A = 2 0 0 MeV. The version of Herwig used in this study incorporates the two-loop expression of ~,, and it is argued in ref. [ 6 ] that in some regions of phase-space the value of A can be interpreted as Ah-~. For comparison, we also show the results obtained from the MC using a one-loop o~s, in the evolution of the shower. The MC adds to the quark masses a non-perturbative contribution of 480 MeV which offsets the threshold for the decay of a virtual gluon into the heavy quark pair. For a meaningful comparison with the MC results the mass chosen for the analytical calculation is therefore m e - 2 GeV. The solid lines in fig. 3 represent the result of eq. (1.1) with the gluon multiplicity given in eq. (1.2). The upper line corresponds to the choice A = 2 0 0 MeV and the lower curve to A = 100 MeV. This last one reproduces rather well the result of the one-loop MC. The dashed lines are obtained instead by using the gluon multiplicity given in eq. (24) of ref. [3] (upper curve: A = 2 0 0 MeV; lower curve:

2. Numerical results

A significant variation in the results originates both from the uncertainty in the choice of A and in the value of the heavy quark mass. To some extent, A is also undetermined from a theoretical point of view, since the multiplicity formulae do not include any renormalization scale dependence and are not sensitive to the renormalization scheme. The A that appears there is related to the A ~ only up to an unknown factor of order one. In fig. 2 we show the variation of the charm and bottom multiplicity due to changes in A and in the value of the masses. The range between 100 and 300 MeV for A is chosen for illustrative purposes only. The uncertainty in the quark masses is always around a few hundred MeV, because of the intrinsic limitation in the theoretical

0.50

2 July 1992

...... I ........ I ........ Charm (upper) and Bottom (lower) multiplicities Solid: h=200 MeV, me=l.5 (rob=5) GeV ~ ; / / :i / ~ Dots: me= 1.3 and 1.8 (mb=4.75 a n d Dashes: h=lO0 and 300 MeV L

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102

. . . . . . .

103

Q (GeV) Fig. 2. Sensitivity of the charm and bottom multiplicities to the values of ,4 and of their mass.

163

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0.4

........

[

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........

I

....

Charm multiplicity

/

":,'-:HERWIG two loop, h = 200 MeV _

0.3

o: HERWlG one loop,

/

A = 200 MeV

/

0.2

-- Dashes: n~ from ref. 3 upper: h = 200 MeV lower: A = 100 MeV

Z ~8

/

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Solid: Full, ng from this work upper: A = 200 MeV lower: A = 100 MeV

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--

Z

0.1

. . . . . . . .

0.001

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. . . . . . . .

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102

. . . .

10 3 Q (GeV)

Fig. 3. Charm pair multiplicity: comparison between the HERWIG Monte Carlo and the analytic formulae, with the gluon multiplicity given by formula ( 1.2 ) (solid) and eq. ( 24 ) of ref. [ 3 ] (dashed). The charm mass was taken equal to 1.5 GeV in HERWIG and 2 GeV in the analytic formulae. cidental, and would rather interpret the differences b e t w e e n t h e t w o c h o i c e s f o r g l u o n m u l t i p l i c i t i e s as r e p r e s e n t i n g a p l a u s i b l e r a n g e o f u n c e r t a i n t y d u e to the ignorance of the sub-leading contributions. An analogous comparison and similar comments

A = 100 M e V ) . I n t h i s case t h e a n a l y t i c c a l c u l a t i o n s e e m s t o b e t t e r a g r e e w i t h t h e t w o - l o o p M C result. I n v i e w o f t h e c o m m e n t s m a d e in t h e p r e v i o u s s e c t i o n concerning the non-universality of the sub-leading l o g a r i t h m s , we e x p e c t t h i s a g r e e m e n t to b e r a t h e r ac0.25

f

....... multiplicity I Bottom

I

'

'

,~: HERWIG two loop, h = 200 MeV o: HERWIG one loop, h = 200 MeV

0.20

0.15

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_

/ /

Solid: Full, n~ from this work upper: A = 200 MeV lower: A = 1 0 0 M e V

/ k / / ~ ~ //

/ /

! Z

0.10

Dashes: ng from ref. 3 MeV upper: ^ : lower:

A =

100

'4

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Aa Z

MeV

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J

i:

Q (GeV)

Fig. 4. Same as fig. 3 for bottom multiplicity. The bottom mass was taken equal to 5 GeV in HERWIG and 5.5 GeV in the analytic formulae. 164

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hold for fig. 4, where we show MC and analytic multiplicities for pairs of bottom quarks. In this case the mass is chosen to be mb = 5 GeV for the MC calculation and rnb= 5.5 GeV for the analytic one.

Acknowledgement

We wish to thank B. Webber for useful discussions and comments. Upon completion of this work we were informed that an early calculation of charm multiplicity in e+e - annihilation was performed in ref [ 7 ]. Their asymptotic formula has the correct hyperbolic cosine behaviour. It does not however incorporate correctly coherence effects, which were not well understood at that time. The argument of the hyperbolic cosine is therefore incorrect by a factor of x/2. The hyperbolic cosine behaviour of the gluon multiplicity was also found in ref. [ 8 ].

2 July 1992

Normally the solution with the minus sign is neglected, because it does not give the correct perturbative anomalous dimension. On a more solid ground one can say that the anomalous dimension with the - sign contributes a higher twist effect, which was introduced by putting the phase space constraints in the evolution equation, and can therefore be neglected. However this is only true for N > 1. For N = 1 it is not a higher twist effect, and therefore, for the multiplicity, there are two anomalous dimensions

y~+~=+ C ~ . Xt

(A6)

The evolution equation can be written directly for N=I 1

d

log Q2

-

it

o

CAas(Q2) Qf ~~.2 dkz DI (k2) ~Z

Appendix. Derivation of the multiplicity formula

"

(A7)

0

The evolution equation with phase space restrictions in the small x region can be written as [2 ]

d

log Q2

-

Q2

1

1

t=log~- 5 ,

x

the equation becomes

c~s(t)=~,

r~b

c = C~"

(A8)

-

Defining the Mellin transform

ctdD'(t~)- i dt'D~(t') dt

1

(A2) or

the evolution equation becomes

d

I

dDN(Q2) CA°~s(Q2)~zX-IDN(zQ2). d log Q2 -

7f

Assuming the ansatz fixed, we get

(A3)

Du(k 2) = (k2) yN,and

taking c~

(A4)

and the solutions are

N-1 + /(N-1"]2+ 4 J

dt

c t dD~ - - (t) -D~(t) . dt

(A10)

The solutions are easily obtained:

0

2CA O~s 1 27r N - - I + T N '

~'N=--T-~/\

(A9)

-co

ON(Q2) = I d x x N - ' D ( Q 2 ' x) , o

~)N=

Defining

CaO~s

zc

(A5)

Dl +-)( t)=t-'/4 exp ( +_2 N/~) ,

(All)

which agree with refs. [ 1,2 ] when nf= 0, except for an extra factor x/2 in the exponent, due to the fact that we have not considered coherence effects here. We now require that for Q2= Q2 the multiplicity is one, and that, as Q2 starts to grow, our solution is given by a power expansion in c~s. It is easy to show that in this case we must have 165

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which in the Q >> Qo limit coincides with the result of ref. [2] up to a factor of ½ in the overall normalization.

D} ±)(t) = (t/to)-~/4 ×~

exp

2

-2

+ e x p ( 2 tN/~ - 2 N / ~ ) 1 . In fact, we can write

~-~°= (t/to)-l/4=

,{j

~boq(Qo)

(A12)

o2)

l+b~l°g~oo -1

( 1 + b a s ( Q o ) l o g Q~o2o)-1/4,

'

(A13)

and one can see immediately that the only combination o f the two solutions for which the half integer powers of as, do cancel is the one given above. Including coherence and the light flavours, we therefore get ~2,

{l°g(a 2/A2)'~a

×cosh(N/2C" 7g

166

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(~-N/logQ°2~

)) ' (A14)

References

[1 ] A.H. Mueller and P. Nason, Phys. Left. B 157 (1985) 226; Nucl. Phys. B 266 (1986) 265. [2] A. Bassetto, M. Ciafaloni, G. Marchesini and A.H. Mueller, Nucl. Phys. B 207 (1982) 189; A. Bassetto, M. Ciafaloni and G. Marchesini, Phys. Rep. 100 (1983) 201, and references therein; A.H. Mueller, Nucl. Phys. B 213 (1983) 85; B 228 (1983) 351;B241 (1984) 141. [3] S. Catani, Yu.L. Dokshitzer, F. Fiorani and B.R. Webber, preprint CERN-TH.6328/91, Cavendish-HEP-91/ 15, and CERN-TH.6328/91, Cavendish-HEP-91/ 15 (E). [4] S. Catani, Yu.L. Dokshitzer, M. Olsson, G. Turnock and B.R. Webber, Phys. Lett. B 269 ( 1991 ) 432. [5] G. Marchesini and B.R. Webber, Nucl. Phys. B 310 (1988) 461. [6] S. Catani, G. Marchesini and B.R. Webber, Nucl. Phys. B 349 (1991) 635. [ 7 ] W. Furmanski, R. Petronzio and S. Pokorski, Nucl. Phys. B 155 (1979) 253. [ 8 ] A. Basseno, M. Ciafaloni and G. Marchesini, Nucl. Phys. B 163 (1980) 477.