Pocket partonometer

Volume 172, n u m b e r 3,4

PHYSICS LETTERS B

22 May 1986

POCKET PARTONOMETER John P. RALSTON Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA Received 24 February 1986

We introduce a fast analytic algorithm producing asymptotic estimates for the small-x sea-quark and gluon distributions relevant at supercollider energies. Numerical comparisons are made between the analytic solution with boundary conditions and currently accepted numerical output in the region Q2 >/100 GeV 2, 10 < 1 / x < 108.

In one hour's running the 20 + 20 TeV supercollider will produce more than a thousand W+- pairs, ten thousand 100 GeV heavy quarks, as well as uncountable bottom, charmed and lighter objects [1]. At even higher energies [2,3], production cross sections for objects with fixed mass increase even further. Huge rates occur because of the rapid increase of sea-quark and gluon distributions [3-5] as the momentum fraction x decreases. To keep pace with high rates there is obviously a need for e~perimentalists and theorists to be able to make fast calculations using parton distributions. Folklore has grown up that parton distributions and scaling violations are either non-perturbative or a problem for extensive computer processing. But in the region of interest for very high energies, Q2 ~ 100 GeV 2 and x ,~ 10 -1 , the important distributions in QCD turn out to be very simple. The distribution for gluons and sea quarks are so simple, in fact, that the reader over 30 can ,a find them himself using the device introduced in this paper: the Pocket Partonometer. Assuming the reader is in a hurry, he will build the partonometer and find directions for use on its back. If the reader is one of many physicists who cannot build or operate machinery without knowing how it works, here is how it works: ,1 Readers under 30 m a y have to learn h o w to use a slide rule. These are n o t sold in stores b u t can be f o u n d at m a n y garage sales.

430

The small-x quark distributions qi(x, Q2) and gluon distribution G(x, Q2) are analytically tractable problems [3]. The coupled Altarelli-Parisi (GLAP) equations [6] ,2 1

2nf

dG _ as(t) dY( r~. dt 27r f 7 qi(Y' t)PGq(X/Y) x

+ G(y, t)PGG(X/y)) , dq i as(t ) 1 dy d---i-= 21r f y [qi(Y' t)Pqq(X/y) x

(1)

+ G(v, t)PqG(x/q)],

can be reduced to simple differential equations for the gluon and sea-quark qs(X, Q2) distributions. In the crudest approximation of symmetrically distributed sea-quarks, qs(X, Q2) ~ nfqi(x ' Q2), one finds

b2 xG(y' ~) - ½xG(y, ~), aya~ aXqs(Y, ~) at "= ~ cxG(y, ~),

c = 2nf/b,

(2)

for large Q2 and 1Ix. Herey = 8N[b in(l/x) and = In ln(Q2/A~cD) , b = 11 - -~nf, and we assume Xqs xG and y~ >~ 1. A complete solution to (2) for gen*2 The original reference to this approach is ref. [7].

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 172, number 3,4

PHYSICS LETTERS B

eral boundary conditions can be given [8] in terms of Bessel functions and powers ofx/~ and x/~, but is not needed here. For definiteness we discuss the upquark sea distribution Us(X, Q2) in the following. Because the GLAP equations have traditionally been used to study Q2-dependence, boundary conditions at Go = In ln(Q2/A~cD) are often imposed by fitting x-dependence. However, it is convenient to fit the Q2-dependence at fixed x so the system agrees with data, and then to evaluate the evolution in x. A suitable solution to (2) with boundary conditions (K(Q2)) imposed is

xG(x, 0 2) = K(Q 2) exp ([2y(~ - G0)] 1/2~,

(3)

XUs(X, Q2)

(4)

22 May 1986 5OO

i0~

ff ~.~

lO

mL.,,.,.-I I0

.,,..I

I0 z

10"~

. . . . . .d

. . . . . .d

IO(

,,,.,~.d

I0~

Jl,,.,~

10 6

1

,Ju~3

10 7

I08

Q2(GeV2) 10

=

(1/b)[2(~

-

~0)/y] 1/2xG(x, Q2),

71

K(Q 2) = 50.36 [exp (~ - G0) - 0.957] X exp[--7.597(~ - ~0)1/2].

(5)

The values Q2 = 5 GeV 2 and AQC D = 200 MeV were used. Some discussion of a form similar to (3) already exists [3,9] but (3) is quite convenient and accurate [5] for our purposes. To show that (3)--(5) are indeed solutions, we must have y~ >> 1 ; there are corrections of order 1/X/~ f o r y >> 1. While ~ is never large,y is about 29 at x = 10 - 4 , so we have corrections that are small indeed. Nevertheless, aware that no amount of large-logarithmic argumentation(LLA) will convince experienced workers in these days, we must compare honest numbers with numbers. We will show that eq. (3) is not only an asymptotic estimate, but quite a good numerical fit. Moreover, the crude approximation to Us(X' Q2) is also accurate enough if a simple change of normalization is made [5]. In fig. la we compare the analytic solution for xG(x, Q2) with the numerical work of EHLQ [4]. The plots agree well because EHLQ use the GLAP equations and we have chosen the same boundary conditions. The plots cannot be compared for x < 10 -4 because the numerical work has not been extended into that region ,3. The asymptotic character of the approximate solution, however, implies that it becomes increasingly accurate for smaller x, at fixed Q2. :¢3 Numerical instability is known to be a problem for x < 10 -4. Recent numerical checks confirm that ( 3 ) - ( 5 ) continue to be quite accurate for smaller x.

lO-t

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Q2(GeV2) Fig. 1. Comparison of LLA distribution function x-dependences (solid lines) with the numerically integrated distributions of ref. [4 ] (dashed lines), at x = 10 --4, 10-3, 10-2, 10 -1 (top to bottom). (a) the gluon distribution xG(x, Q2) (eq. (3)); (b) the up-quark sea distribution XUs(X, Q2) (eq. (8)). The analytic approximations do break down for the other end, x/> 10 -1 , as shown by fig. 1. At large Q2 ~ 100 GeV 2, non-perturbative and higher-order QCD effects not incorporated in the GLAP equations are postponed. Present understanding [9] constrains the region of validity of (1)-(5) to one for which Q2 ~ few GeV 2, 1 0 < 1/x < 1/x 0 where

Xo C(xo ' Q2) ~ asQ2(1 fm2),

(6)

for Q2 = M 2 , e.g., one can use (3)-(5) down to l/x ~< 108. The corresponding energy for a collider process involving a W-+, given a typical x ~ O(Mw/V~), is x/~ ~< 1010 GeV, which seems high enough for most purposes. Heavy quark thresholds have not been included: 431

,Oll ~o)1

zOl

Place down-arrow at 1/x. Read N on top scale above right Q2-scale. Transfer N to xG(x, Q2) scale. Place Q2 on left-Q 2 scale below N. Read xG(z, Q2) at up-arrow.

xG(x, q~):

To find

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I

xua(x,qs): x I = x/lO

• ~, ~ ='c(=,,Q,)Jioo

To find

Ol

.

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,

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to to

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o

Volume 172, number 3,4

22 May 1986

PHYSICS LETTERS B

for t- or heavier particle distributions one must model thresholds numerically such as done in ref. [4] ,4 However, departures from a symmetrically distributed sea, which complicate the boundary conditions, can be reproduced by the ratios u s ~ d s ~ s s ~ 2c s ~ 2b s. The analytic gluon solution (3), boundary conditions included, is calculated by the partonometer (fig. 2). The scales automate the logarithms of certain functions of 1Ix and Q2 left to the reader. In systematic testing the accuracy of the gizmo is at the 1 0 - 2 0 % level depending on the operator's ability to read logarithmic scales. It is much better than interpolating between graphs such as fig. 1 a. The speed is even faster than adding a new card ,4 to an existing program that runs. Gluon distributions are read off directly; see the example below. Quark sea distributions can be evaluated using the identity

XUs(X,Q2) = (2[b)axG(x, Q2)/Oy,

i

(7) 0

and evaluating the derivative numerically. But wait! To minimize reading errors, one finds that the derivative above and the normalization change are roughly represented by

XUs(X,QZ)~x'G(x',Q2)[lOO,

x'=x/lO.

(8)

I

This estimate is actually quite close to the re-scaled XUs(X,Q2) of ref. [5] and is not too bad a match to ~:4 Private communication with well known phenomenologist.

!

J/F,7

Fig. 2. The partonometer. To assemble: cut on solid lines, fold on dotted lines.

I A

433

Volume 172, number 3,4

PHYSICS LETTERS B

the output o f ref. [4]. A comparison o f the rough estimate (8) and EHLQ [4] is given in fig. lb. The agreement is quite satisfactory, notwithstanding a factor o f ~ in the region x ~ 10 - 2 , Q2 ~ 102 GeV 2. In fact, one can read the scales as accurately as one can trust the formalism. The errors [4,10] from different boundary conditions, e.g. a factor of two for x o f 10 - 2 or smaller, are larger than the error o f the partonometer. Let us now present some examples. First, get a technician or your kid to put together a copy o f the partonometer. (Do not cut up the paper.) We want xG(x, Q 2 ) a t x = 10 - 4 and Q2 = 104: (1) Line up the downward pointing arrow on the slider at l/x = 104. (2) Find Q2 = 104 on the right Q2-scale o f the slider. Read 1.5 X 103 on the top scale. It helps to attach a paper clip to the slider (only) at your value o f Q2 in order to line up the scales. (3) Transfer 1.5 X 103 to the xG(x, Q2) scale, and put Q2 = 104 on the left Q2.scale o f the slider under this. (4) Read offxG(x, Q2) ~ 80 at the upward pointing arrow on the xG(x, Q2) scale. F o r the up-quarks, at Q2 = 104, 1Ix = 104, set 1Ix = 105 and repeat the operations above: then XUs(X, Q2) ,~ 200/100 = 2.0. Charm and b o t t o m quarks will have distributions roughly ~ o f this. Remember this means, e.g., b(10 - 4 , 104) ~ 104. While distributions are useful, most questions of production rates usually require integrations over the distributions. At the level o f 30%, one can again automate [ 11 ] much o f this to calculate the differential luminosity (z[s')~Z[~z,producing cross section estimates in seconds.

434

22 May 1986

Many hours o f wholesome and pleasant time can be spent using your new partonometer. Never leave it around carelessly where a graduate student can steal it! Maintenance is simpler than for most high energy machines: an occasional ironing will keep up its appearance. G o o d luck! This work was supported in part under Department of Energy Grant No. DE-FG02-85ER40214.

References [1 ] F. Paige, in: PP options for the Supercollider, Proc. University of Chicago and Argonne Workshop (1984), eds. J. Pilcher and A.R. White; see also R. Donaldson and J. Morfin, eds., Proc. 1984 Summer Study on the Design and utilization of the SSC (Snowmass, CO, 1984). [2] W.-K. Tung, Proc. 1978 DUMAND Summer Workshop (1979), ed. A. Roberts; Yu. Andreev, I. Berezinsky and A. Smirnov, Phys. Lett. B 84 (1979) 247. [3] T.A. deGrand, Nuel. Phys. B151 (1979) 485. [4] E. Eichten, I. Hinehliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579. [5] D.W. McKay and J.P. Ralston, Phys. Lett. B 167 (1986) 103. [6] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [7] V. Gribov and L. Lipatov, Yad. Fiz. 15 (1972) 78, 1218. [8] D.W. McKay and J.P. Ralston, in preparation. [9] L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Rep. 100 (1983) 1; J.C. Collins, in: Proc. 1984 Summer Study on the Design and utilization of the SSC (Snowmass, CO, 1984), eds. R. Donaldson and J. Morfin; and IAS preprint (1986). [10] P.W. Johnson and W.-K. Tung, in: Proc. 1984 Summer Study on the Design and utilization of the SSC (Snowmass, CO, 1984), eds. R. Donaldson and J. Morfin. [11 ] J.P. Ralston, Pocket SupercoUider, in preparation.