A little toy jet

Volume 195, n u m b e r 4

PHYSICS LETTERS B

17 September 1987

A LITTLE TOY J E T ~" Howard GEORGI Department of Physws, Boston Umverstty, Boston, MA 02115, USA and Lyman Laboratory of Physws, Harvard Umverstty, Cambrtdge, MA 02138, USA Received 9 July 1987

I bmld and dtscuss a model of the energy flow m a high transverse energy hadromc jet. The model makes robust predictions for the four-momentum contained within a small circle m q and ~ about the jet axis

1. Particlets m the center of mass. Even in this age of Monte Carlo event generators, it is sometimes useful to have simplified models of complicated physics that can be solved explicitly. The model I discuss in this note grew out of my attempt to understand some "ISAJET" results on the invariant masses of QCD jets [ 1 ]. As I hope you will see, the model itself got rather interesting. I hope that it may be useful for back of the envelope calculations that complement more elaborate computer simulations. When a quark-antiquark is produced in e+e - annihilation (for example), it hadronizes by leaving behind a sort of string of color flux that eventually falls apart into particles and fills up the rapidity axis. The same sort of thing happens in hadronic collisions where a quark is thrown out at high transverse momentum. This quark is still "attached" to at least one of the outgoing proton jets by a string of color flux. To understand what this looks like in the lab, we should start in the center of mass of this quark-proton-jet system, and Lorentz transform to the lab frame. In this peculiar center-of-mass frame, the outgoing quark and the debris of the proton look rather like the outgoing quark and antiquark in a two-jet e+e - annihilation. Thus it may be interesting to take a very simple model of an e+e - annihilation jet and transform it into the frame appropriate to a high transverse momentum jet in a proton-proton collision. Suppose the lab energy of the outgoing quark in the lab is Eje t and the energy of the outgoing disrupted proton (before hadronization) is Eb . . . . Then, ignoring the masses of the quark and the proton, we can describe the Lorentz transformations from the center-of-mass frame of the string to the lab by the following mapping: (E, - E , O, 0)--~(Eb. . . .

Eb . . . .

0, 0),

(E, E, 0, 0) ~ (Eaet, cos 0jet Ej~t, sin 0jet Ejet, 0 ) ,

(1)

where E is the energy of the quark and beam-jet in the CM frame. Note that the rapidity of the jet in the lab is r/jet = l n ( c / s j ) ,

(2)

where C3-~-COs(IOjet),

Sj = S l n ( ½ O m )

•

(3)

Using the invariance of the scalar product of the two four-momenta in eq. (1), we find 2E 2 =~o~Ebo.m(1 --cos

0~e0,

~r Research supported m part by the National Science Foundation under Grant #PHY-82-15249.

0370-2693/87/$ 03.50 © Elsevier Science Pubhshers B.V. (North-Holland Physics Publishing Division)

581

Volume 195, number 4

PHYSICS LETTERSB

17 September1987

or

E=sj~jetEbeam

(4)

.

The string, in its CM frame, is an approximately uniform distribution in rapidity from --Ymax to

Ymax~ln(2E/p). I will model the energy and momentum distribution of the jet as a double integral over rapidity, y, and azimuthal angle, ~, of massless "particlets" with infinitesimal four-momentum (dE, dpx, dpy, dpz) = (p/2n)dy d~ PE(y)(cosh y, sinh y, cos ~, sin ~).

(5)

The particlet distribution function, PE(Y), has the following properties: PE(y)~I

for lYl <
(6)

Pz(y)'~O for lYl >>Y. . . .

(7)

P f PE(Y) cosh y d y = 2 E .

(8)

In words, the particlet distribution is about 1 for small y and goes to 0 when lYl is larger that Ymax.AS long as Ejet/p is very large, none of my results will depend on the precise functional form of Pz(y). We expect the dimensional parameter, p, to be of order 1 GeV. The main theoretical advantage of the particlet description is that it will give us an unambiguous picture of the energy flow in the jet while retaining maximum uniformity and symmetry of the distributions. A further practacal advantage is that, because the particlets are massless, there will be no difference between rapidity and pseudo-rapidity. We now simply have to transform our particlet distribution back into the lab frame and look at it.

2. Particlets m the lab. I show in the appendax that the particlet four-momentum in the lab has the form (dE, dpx, dpy, dpz) =2P-z

exp(~At/) r 3

PE(Y) dr/d~ (cosh 1/, sinh r/, cos ~, sin ~) ,

(9)

where At/= r / - r/jet,

(10)

and r = 2~/sinh 2(1A~/) + sin 2( ½q~),

( 11 )

and the connection of 1/and ~ with the rapidity in the center-of-mass frame is ey = [exp( - ½Ar/)/cjr]~,/Ebeam/Ej~t.

(12)

Notice that the particlet distribution,

D( r/, O) = (P/2g)[ exp( ~Ar/)/r3]pE(Y) ,

(13)

directly gives the transverse energy flow in the jet as a function of r/and 0I promised you that the properties of the jet would not depend in any important way on the functional form of PE(Y), and yet this function appears explicitly in eq. (9). Nevertheless, for energetic jets, my statement is true, and the reason is worth understanding. Notace that r goes to zero when r/=tl.let and ~=0, that is, at the position of the jet in r/and 0. So the particlet distribution in the lab appears to blow up at the jet. But PE(Y) ~ 1 for y<> [exp( --Ym,x)/C~]JEbeam/Ejet~p/Ejet sin 0j~t • 582

(14)

Volume 195, number 4

PHYSICS LETTERS B

17 September 1987

In the lab, if Ejet >>p, then because the right-hand side of eq. (14) is very small, PE(Y) is close tO 1 except for the particlets very close to the jet direction in rapidity and angle. Because the integral Of PE(y) is also fixed by eq. (8), I can determine all the important properties of my toy jet without reference to the explicit form of PE(y). It is amusing to note that the particlet distribution in the lab depends on the jet energy only through PE(Y), where the effect on increasing Eje t does not change the value Of PE(Y) at small y, but moves the limiting value of y to larger y. This, in turn, moves the limiting value of r, below which the physics depends on the details of PE(Y), to smaller r, because of eq. (12). The particlet distribution looks like the cone of a volcano, with a small section in the very center that depends on the details of the shape OfPE (y). Increasing Ej~tdoes not effect the shape of the base of the cone, but it allows the center to grow higher. For large t/, the distribution becomes uniform (with particlet density p/2rc), the maximum rapidity is determined by the lab frame translation of -Ymax
eq <2Ebeam/P,

(15)

as expected for the beam jet [see eq. (8)]. 3. Parttclets near the jet. We can now discuss the question that onginally motivated this investigation: What is the jet mass? To do that, we must first decide what we mean by the four-momentum of the jet. Near the high transverse momentum jet in t/and ~, the variation of particlet distribution is dominated by PE(Y) and by the factor of 1/r 3, which for small At/and ~ is (At/2 ~_~2)-3/2 .

(16)

Notice that the jet as quite symmetrical in At/and ~, despite the fact that the t/direction is picked out by the beam jet. The correlations with the beam direction do not show up in the leading behavior of the peak at the jet. Thus a reasonable definition of the jet four-momentum would seem to be the sum of the particlet fourmomenta in a circular region around the jet in At/and ~b. Call the radius of this region a. If a satisfies 1 >> a >> piEjet ,

( 17 )

then we can calculate the leading contributions to the four-momentum of the jet using only eqs. ( 6 ) - ( 8 ) . The largest contribution comes from keeping only the r dependence in eq. (9) and in PE(Y), setting At/and to zero elsewhere in the integrand. Then, because PE (Y) ~ 1 except where it vanishes for small r [ by eq. (14)], we can set it equal to one and take the integral over an annular region that looks approximately like plEje, sin 0~et< r < a .

(18)

This gives the following contribution to the jet four-momentum: (E~et-p/~ sin 0jet)(1, COS 0je~, sin 0jet, 0) .

(19)

In this order, the jet mass is still zero, but we see the effect of the finite size of the jet in the fact that not all of the energy of the initial quark is contained within the cone r < ~. The next-order contributions come from the non-trivial At/and ~ dependence of the factor exp(3At/) and of the particlet four-momentum. These give

pa( ~2q/s, + ~s,/c,, ~c/s, - ~s,/c,, ~, 0).

(20)

The higher-order terms in r do not contribute in this order, so long as the region used for the definition of the jet is circular, because the At/dependence and the # dependence give canceling contributions. Furthermore, if 583

Volume 195, number 4

PHYSICS LETTERSB

17 September ! 987

there are additional contributions to the particlet distribution from other jets in the process, they will contribute only in order j2 because the area of the jet in the A~/-0 plane is noV. There are also, of course, order j2 contributions from eq. (9). These depend on the details of PE(Y). Only the contributions in eqs. (19) and (20) are model independent. The sum of the two contributions in eqs. (19) and (20) gives a jet four-momentum with a non-zero mass given by the simple expression

x/pJEjet sin O,¢t .

(21 )

4. Conclusions. The main results of this study are eqs. (19)-(21 ). I repeat that these are independent of the precise shape of the center-of-mass frame particlet distribution, so long as eqs. ( 6 ) - ( 8 ) , (17) are satisfied. Eqs. (19)-(21 ) nicely illustrate the dilemma of jet definition. If J is too small, then because of eq. (19), we do not pick up all of the four-momentum of the original quark. But if J is too large, because of eq. (20), we misidentlfy the direction of the original quark. But, it seems clear that it makes sense to scale to smaller J as the energy of the jet increases. For example, if one takes J a ~

(22)

then the jet mass associated with the color stnng goes like E~/O as E~etincreases, while the misidentification of the quark energy goes like x/~J~t- One could even use eq. (19) to correct for thc difference between the jet energy and the quark energy. These issues deserve further study. This study could be extended in a variety of other ways: (1) gluon jets could be included by considering the gluon to be attached to both of the beam jets by strings; (2) a picture of a complete P - P scattering event could be put together by combining two or more of these toy jets; (3) the decay of a heavy particle (even a W) into quark-antiquark could be studied by transforming the center-of-mass decay into the frame in which both particles have high momentum transverse to the beam direction; (4) the shape of the jet could be studied by defining the jet in terms of an elliptical region in A,/-~ space.

I am very grateful to the SSC group at Boston University for stimulating this idea. I particularly thank R. Sekhar Chivukula, Ken Lane and Scott Whitaker for help and encouragement. Junegone Chay read the manuscript with characteristic care and uncovered many errors.

Appendix. From center of mass to lab. To transform the center-of-mass frame particlet distribution, eq. (5), to the lab, I will decompose the particlet four-momentum into a canonical set of momenta that can be easily transformed. Two of these, I can take immediately from eq. (1) to be the four-momenta of the unhadronized initial particles. I will take the other four-momenta to be (0,0,0, 1 ) ~ ( 0 , 0 , 0 , 1),

(0,0, 1 , O ) ~ ( c J s j , c J s j , 1 , 0 ) .

(23)

It is easy to see that these transformations just leave the z axis unchanged and transform the x, y and t axes appropriately. When we write the particlet four-momentum in terms of the four-vectors on the left-hand sides of eqs. (1) and (23), transform them into the lab, we find

(p/2n)PE(y) dyd~ [(eY/2sj)

%/~jet/Ebeam(1, cos 0jet, sin 0jet, 0) "~ (e-Y/2sj) ~

(1, 1, 0, 0)

+cos ~ (cj/sj, cj/sj, 1, 0) +sin ~ (0, 0, 0, 1 )] =D(~/, 0) dr/d~ (cosh ~/, sinh 0, cos 0, sin 0) • 584

(24)

Volume 195, number 4

PHYSICS LETTERS B

17 September 1987

Taking the ratios of the space components to the time components, in eq. (24) we find

( eY/2sj) N/Ejet/Ebearn COS 0jet "q- (e -Y/2s~) N/Ebeam/Ejet -q-( Cj/Sj) cos ( e V/2sj) N/Ejet/Ebeam -q- ( e - Y/2sj) N/Ebeam/Ejet"~ ( Cj/Sj) COS

= tanh r/,

(eY/2sj) N//Ejet/Ebeam sin 0j~t +cos cos (eY/2sj) N/Ejet/Ebeam + (e -Y/2s~) N/Ebeam/Ejet-[- (Cj/Sj) cos ~ - cosh r/' + sin ~ (eY/2sj) N/Ejet/Ebeam + (e -Yl2sj) ~/Ebeam/Eje t +

sin ¢

(cj/sj)

cos ¢ - cosh r/"

(25)

These can be solved for y and ~ in terms of r/and 0, yielding eq. (12) and cos ¢ - ~

exP(½r/) cos O - cx/~j/sj exp( - ½r/) r

(26)

The volume element, then transforms as follows: dy d( - exp(Ar/) r2 dr/d~.

(27)

Inserting eqs. (12), (26) and (27) into eq. (24), gives eq. (13). Reference [ 1] ISAJET, F.E. Paige and S.D. Protopopescu, BNL 31987.

585