Heavy quark fragmentation to polarised quarkonium

Physics Letters B 312 ( 1993 ) 486-490 North-Holland

PHYSICS LETTERS B

Heavy quark fragmentation to polarised quarkonium A d a m F Falk a , l M i c h a e l L u k e b 2, M a r t i n J Savage b,3,4 a n d M a r k B W i s e ~ Stanford Lmear Accelerator Center, Stanford, CA 94309, USA b Department ofPhystcs, Untverstty of Cahforma at San Dtego, 9500 Gdman Drtve, La Jolla, CA 92093, USA c Cahforma Instttute of Technology, Pasadena, CA 91125, USA Received 17 May 1993 Edxtor H Georgl

We calculate the polarlsatmn of ~"s and F ' s produced by the fragmentatmn of heavy quarks We find that fragmentation to transversely ahgned q u a r k o m u m is shghtly enhanced relative to longitudinally polansed quarkonlum This net ahgnment corresponds to a ~ 5% asymmetry m the angular d~strlbutmn ofleptons produced m the subsequent decay of the quarkonlum We point out that the leading gluon c o n t n b u t m n to q/and T productmn arises from the evolutmn of the charm quark fragmentatmn functmn, rather than from &rect gluon fragmentatmn

Q u a r k o n m m production in high energy processes 1s dominated by the fragmentation of heavy quarks and gluons [ 1 - 4 ] For example, m Z ° decay the short distance process Z°~q/gg ~s suppressed relative to the fragmentatton process Z ° - ~ u c e b y a factor of mc/mz, z 2 correspondmg to the small overlap of the q/wavefunctlon wtth a c¢ pair that has relauve m o m e n t u m of order m z Recently tt has been shown by Braaten and Yuan [ 3 ] and Braaten, Cheung and Yuan [4 ] that the process independent fragmentation functmns for quarkonlum production m htgh energy processes are calculable For example, in Z ° decay to a ~uo f energy E>> m., the differential decay rate may be written m the factorased form 1

dF(ZO-~,(E)+X)= ~ ~ dzd~(Z°ot(E/z)+X,l~)D,~,(z, lt)

(1)

0

Here dl~(Z°~l(E/z) + X , ll) as the differential decay rate for a Z ° to decay to an on-shell parton of type t, while the fragmentatmn function D,~(z, lt) g~ves the probability for the patton to split into a ~, with m o m e n t u m fracnon z and another parton with m o m e n t u m fraction 1 - z The advantage of thts factonsed form is that the fragmentation functton ts independent of the production process Q C D corrections of the form log(mz/mc) necessitate the mtroductlon of a factortsatton scale/~, large logartthms of#/rnc m D,~(z, It) may be summed usmg the Altarelh-Partst e q u a t t o n s [ 5,6 ] The fragmentatton functions Dc~(z, l~) and Da~r(z, 11) were calculated in ref [4 ], exphcxtly summing over quarkonlum polartsattons However, the fragmentation functtons for transverse and longitudinally polartsed quarkonlum are lndtvldually calculable and are lnterestmg m their own right In this work we compute the fragmentatmn functton for productton of transversely polartsed ~"s from c quarks and F ' s from b quarks The Feynman dmgrams responsible for the fragmentatton of a heavy quark into quarkontum are shown m fig 1, where the black orcle represents some quark-ant~quark productton process For defintteness we constder Z ° l 2 3 4

Address after 1 October 1993 Department of Physics, UC San Diego, La Jolla, CA 92093, USA Address after 1 September 1993 Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 Address after 1 September 1993 Department of Physics, Carnegie Mellon Umverslty, Pittsburgh, PA 15213, USA SSC Fellow

486

0370-2693/93/$ 06 00 © 1993 Elsevier Science Pubhshers B V All rights reserved

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C C

(a)

F~g 1 Feynman dmgrams responsible for the fragmentation

(b)

C--)~C

decay to v/ce, however the fragmentation function we will derive is independent of the production process and is trivially extended to b fragmentation to 7" Factorlsatlon is only manifest in # A = 0 gauge (where ~] is the antlquark four-momentum), in which the diagram in fig lh vanishes to leading order In molE Following the manipulations In eqs ( 9 ) - ( 11 ) of ref [ 4], we write the decay rate F ( Z °-, gt(E)ce), averaging over initial spins and summing over colours, as 1 ~

r(z°~(e)ce)= ~m~

dz

[dq] [d~/] (2~)46(4)(pz-q-q)

ds

~ o

~0

s-

4m~

T

rn~

i--z]

IA(e)

12

,

(2)

o

where A(e) is the amplitude to produce a ~, with polarlsatlon e, Pz is the Z ° four-momentum, s=q 2 ts the lnvarlant mass of the original c quark, [dq ] -d3q/16g 3qo, and z is the longitudinal momentum fraction of the ~, Choosing a frame such that q = (q°, 0, 0, q3), we have z = (Po-t-P3)/( qo-t-q3), where p is the ~, four-momentum Squaring the result from fig 1 and using the identity ~ ' %T

*T

~ P'Ps =~u-p2

(3)

tO sum over transverse q/polarlsatlons, we find ~2 T ,A(,),2 = (S-- 1m ~)" DT(s' z) ( - - g " a + ~ )

tr (Fa~Fp0) '

(4)

where

Dr(s, z) ~

2 5 6 _ _ 2 _ 2 g'2 " ~ - 1/, O~s J q~

X[8m~s(z_2+2)_8ma(z+4

6+ z

We have dropped terms suppressed by

(01CT~cl~u(p, ~ ) ) = f . m . e

8) )-7 +

16mZ(s--m2)z2+8(s--mZ)2z2(1--z)] 2-z

mc/E in our result

(2-z) 2

(5)

fv, is the ~t decay constant, defined by

u

(6)

This is related to the nonrelatavlsnc radml wavefunctlon at the origin, R~,(0), by f ~ , - - - ~ R v , ( 0 ) Numerically, from the ~' leptonlc width, f ~ 4 1 0 MeV The total decay rate to transversely aligned q/s may now be written in the factorlsed form

/'(Z0---~Ip'Tce)----

-F'(Z°---)CC)i dz i dsO ( s o

4m2 z

.m2") _ _ (s_m2)4DT(s,z), 1 {~-5,]

(7)

o

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where the decay rate to free quarks is

F(z°-'ce)= ~m~

[dq] [&]] (2n)4d(4)(pz-q-gl)

( . ) PzP~ tr(F~rN) _gap+ --~z

19 August 1993

(8)

Performing the integral over s in (7), we find the expression for the fragmentation function to transversely aligned ~'s

DTc~'(z'lt=3mc)= ~

16

as( 3mc)2f 2 23 z((21---zz)) 26 (16- 32z+ 76z2- 36z3 +6z4)

(9)

Adding the longitudinal polarlsatlon, we recover the total fragmentation function D cX. +L (z) given in ref [ 4], DT+Ltz / t = 3 m c ) = _ 1 6 ols(3m~)2f2z(1--z) 2 (16_32z+72z2_32z3+5z4) c~v,, , 81m,~ (2--2) 6

(10)

In experiments performed at a hadron colhder, the gluon fragmentation function also plays an important role The Altarelh-Parlsl evolution of D T(T+L) c~ ~, (z, ~) from the scale/t = 3m~ to a high energy scale ~t = M (such as M= mz) will induce a gluon fragmentation function ~g~D T(T+L) (z, M) via the gluon splitting function Pg~ce(Z) That is, a high energy gluon may split into a quark-antlquark pair of lower energy, either of which may then fragment to quarkonlum In the leading logarithmic approximation this effect, of order a 3 ln(M/m~), dominates over the direct contribution to gluon fragmentation computed in ref [3 ], which is of order a 3 without the In (M/m~ ) enhancement # It is equally straightforward to evolve the quark fragmentation functions (9) and (10) to high energies using the Altarelh-Parlsl equations and the quark splitting functions Pc~cg(Z) and Pc~gc(2)=Pc~cg(1- z ) Because f~ dzP,.~g(z)=0, this evolution softens the z distributions of DV,~,(z, It) and ~ ,T+L r ) (z,/~) but leaves their integrals f~ ~ d./~T(T+L) --~_~ (Z, /1) unchanged [4] We now define ( t o be the ratio of transverse to total fragmentation probabilities,

f~ dzDX~,(z,l~) (----- f~ d z D~T,+~L,(/_~ )

(11)

As discussed in the previous paragraph, ( is Independent of~t Evaluating ( 11 ), we find ( = 0 69, to be compared with ( = 2 for production o f unaligned ~,'s Therefore a small excess of transversely aligned ~/s will be produced Since the dependence onf~ and rnc drops out of ( 11 ), the ratio ( is the same for ~,'s and ~ ' s In leading logarithmic approximation, the corresponding ratio for gluon fragmentation to ~,'s has the same value and is/z independent, since the gluon fragmentation functions are induced only by quark fragmentation Hence, at a hadron colhder, where ~u's are produced both by quark and gluon fragmentation, the fraction of transversely ahgned ~"s is also ( = 0 69 The asymmetry (is measurable through the angular distribution of the leptons in the decay ~u~ l + l - Defining 0 to be the angle between the ahgnment axis and the lepton momentum, the angular distribution d F / d cos 0 in the ~' rest frame has the form

dF(~u~l+ l - ) d cos 0

oc 1 + a c o s 2 0 ,

(12)

where 3(-2 a= - 2-(

(13)

~ This apphes only to ~uand V production, gluon fragmentation to r/o'sand r/b'SlS of order a~, and so the &rect gluon fragmentation calculated m ref [3] dominates 488

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( = 0 69 corresponds to a = 0 053, about a 5% a s y m m e t r y It is useful to c o m p a r e this with the polarisatlon o f v/'s p r o d u c e d from nonleptonlc b decay Assuming the four-quark a m p l i t u d e factorlses (which should give a reasonable estimate, as the V/is a c o m p a c t object on the hadronlc scale AQCD), a straightforward calculation o f the decay b - , v/s-, l + l - s gives

or=

rn~ - m ~ 2,

m~,+m~ ~ - 0 4 6

(14)

This is in nice agreement with a recent CLEO inclusive measurement, which finds [7] (see also ref [8] ) or=-0

4 4 + 0 16

(15)

(where we have a d d e d the systematic and statistical errors hnearly) Since the inclusive branching ratio for B--, v/X is ( 1 02 + 0 05 + 0 09)% [ 7 ], this decay m a y p r o v i d e a significant background to 9' production by fragm e n t a t i o n at high energy colhders There is no comparable decay process to compete with b quark fragmentatton for ~ production, therefore this m o d e should allow a cleaner m e a s u r e m e n t o f ( Fragmentatton Into radially excited q u a r k o m u m states V/' is also described by eqs (9) and (10) Since (f~,~2s)/ f~,)2 _~0 4, fragmentation to these states is not substantially suppressed, and the subsequent decay 9" --, v/nzc is an a d d m o n a l source o f v/'s However, data from e ÷ e - ~ v/(2 S) ~ v/zczc--,1÷ l - rczcis conststent with a 1 + cos20 angular distribution for the leptons (where 0 ~s the c o m angle between the lepton m o m e n t u m and the b e a m axis), indicating that the matrix element for 9,(2S)--, v/z~r has the form Ae~,~2s) ~,, where A is a function o f the plon energies and m o m e n t a [9 ] F o r all low-lying radial excitations, this form for the decay matrix element follows from heavy quark spin s y m m e t r y The a h g n m e n t o f the p r o d u c e d v/'s ~s therefore the same as that o f the v/"s a n d our result for ( stdl apphes S l m d a r remarks hold for fragmentation to radmlly excited T" states [ 10 ] We note that v/ (or ~') production can also arise from the fragmentation into P-wave q u a r k o n l u m which subsequently decays electromagnencally to 9' (or T'), as menttoned m ref [ 4 ] F o r weakly b o u n d nonrelatlvlStlC q u a r k o m u m , the fragmentation into P-wave states ~s suppressed, however, m the c h a r m o n l u m system this may be an i m p o r t a n t a d d l t t o n a l source o f v/'s In summary, we have calculated the leading contribution to the transverse f r a g m e n t a u o n funcnon for quark o n l u m production from heavy quarks We find that the 9' and T" are essennally unaligned, with only a ~ 5% angular anlsotropy m the lepton d l s t n b u n o n p r o d u c e d m their electromagnenc decays We also point out that the d o m i n a n t contribution to gluon f r a g m e n t a n o n into 9' or ~ in high energy processes arises from A l t a r e l h Pans1 e v o l u n o n o f heavy quark fragmentation functions, rather than from d~rect gluon fragmentation This work was supported in part by the Department o f Energy under contracts DE-AC03-76SF00515 (SLAC), D E - F G 0 3 - 9 0 E R 4 0 5 4 6 ( U C San Diego) and DE-FG03-92ER40701 (Caltech) M J S acknowledges the support o f a Superconducting Supercollider N a t i o n a l Fellowship from the Texas National Research L a b o r a t o r y C o m m i s s i o n u n d e r grant F C F Y 9 2 1 9

References [ 1] J H Kuhn and H Schneider, Phys Rev D 24 ( 1981 ) 2996 [ 2 ] V Barger, K Cheung and W Y Keung, Phys Rev D 41 (1990) 1541 [3] E Braaten and T C Yuan, NUHEP-TH-92-93, UCD-92-25 (1992) [4] E Braaten, K Cheung and T C Yuan, NUHEP-TH-93-2, UCD-93-1 (1993) [5] G Altarelh and G Pansl, Nucl Phys B 126 (1977)298 [6] R B Field, Apphcatlons ofperturbanve QCD (Addison-Wesley, Reading, MA, 1989) [7] CLEO Collab, A Bean et al, Decay rates and polarization m inclusive and exclusive B ~ , decays, Talk presented at the 1992 Intern Conf on High energy physics (Dallas, TX) 489

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[8 ] H Schroder, ARGUS results on B decays via b--,c transmons, m Proc 25th Intern Conf on High energy physics, eds K K Phua and Y Yamaguchl (World Scientific, Singapore, 1991 ) [9] G S Abrams (Mark I Collab ), m Proc 1975 Intern Symp on Lepton and photon Interactions at high energies, ed W Kirk (Stanford Linear Accelerator Center, CA), L S B r o w n a n d R N Cahn, Phys Rev Lett 35 (1975) 1 [ 10] ARGUS Collab, H Albrecht et al, Z Phys C 35 (1987) 283, CUSB Collab, V Fonseca et al, Nucl Phys B 242 (1984) 31, CLEO Collab, private commumcatlon

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