Pal, B K

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Indian J. Phytt. 52A, 2U2-206 (1978)Comments on the correlation function in deep inelastic scatteringB. K BalDrptniniatd of TIniiachal Pradenh (hviver.sify, Summer THU, Shnla~Tt{HeceAved 10 Dene,m,her 197(i, revised 14 June 1977)'rwo-par(jjcl(  ^ and thrcc-particlo con-elation functions in the central j'(5gion have been worked out in order to study the behaviour of multi- ])articl(  ^ productions at very high enej’gy The two-jiartieje con ela­tion function IS oxpj-i'ssed in terms ol relalivi' distance between the two particles in rapidity space. The three-pai tide correlation fuin- tion is also a function of relativet disianee in ra})ulity s])ace apart from containing individual vapidity terms1. JntrodtjctionThe striKitiire ol hadrons and their interactions a,re very little known evi-ii to-day Both theoretical and exjierirneiital efforts havi^  been going on lo intorprei, ih(‘ eoTistit.uents of hadrons in terms ol a few lundamental units (Gell-Mann ]9(i4 and Zwcjg 1904) wii.hout lUiKili success. Heeently. high energy electron has been listed to jirobe the structure of the hadronIn deep inelastic scattering, the compling constant involving the virtual jilioton is model dependent The virtual photon nucleon si^atteririg is similar to t.hat of the hadron-hadron scattering when th,e mass ol the virtual x>lit4.o is fixed. The obseivod sealing phenomena suggests that the coupling constant (Oalin & Colglaziei 197S) of t he virtual photon should be independent, on Q'^  when Q'^  t.ouds to inlinity In deei> inelastic scattei ing, physical (piaiititit's ari^  depondimt on Bjorken variable only (Bjor-ken iV Pasches I960). Our earlier work (T?al 1975) shows that the two particle eorrelation function is independent, on Q'“ and w when the jiartiiHes are not produced in the (central plateau. When both of them an‘ produced in the central region, their sub-iegion is much smaller than other sub-regions. This, of course, does not follow from the hadron-1ladron seatteriiig. We note that for large v and Q“ the sub-region of the virtual photon and the hadron produced nearer to it. is largei than other sub-regions. Using the t.eehiiique of Abarlmnel (Abaibonel 1970), the correlational function is pro- ]K>rti(mal to expl — (;Vj—ya)/L] whore v/j and aro the repidities ol the produced pailicles We note that it is mdependent of the virtual mass and the target. When thri'i? liadrons are produced iu the central legion, the correlation function is proportional to sum of the terms containing factors exp[—{y^  tmd„xp(—7/(/Z^ ), t — 1, 2, :i if i/i >  ;i/2 >  y/3 The first term is expected, for it con­tains the relative distance between f.he end particles in rapidity space.202Correlation function in deep inelastic scattering 203To now teaturo ol the Uu’ih' particle corrolaiion fiiuction is tliati it damps expjueutially mth a diaract.onKtic range for rapidity l/iCx[i-0Li^ {{))]. i 1,2.32, Kinematics and Coriielation FunctionsIjot the momenta ol the virtual photon, the tajget and the observed liadrons be P, pjj and p.^  respecti ’^-ely. The rapidity variable yi which is defined .^V Ui -  I ^^ {^oii~\-pLi)l{coi~~pT^ i) IS convenient, for oiir considorat urns at. high energy. Here u>i and pi^ i are the energy aiid longitudinalmonient.nin oftlu' i-th jiarticlo. At high energy, the transveise nionumtimi of any jmrtieh* is finite and does not play any role for various physical quantities of interest At. Jaboi'a.- tory frame, the square ol the total energy of the virtual jihoton and the t arget, s IS given by ,s M^~(^-\~2Mv Let and be the squaie of the sub-enei gios ol the virtual photon and i.-th particle, the vii'tuaJ photon and the target and tlie particles 1 and 2 respectively. Tlicrefore, we have•'iy ~  — 2 m if [ c exp( -  ?/<) — Mjw sinh jy, ] -andvdicu^piovidcd/Sbu exp(//JiS’ia -  c^ xpl - ( / / , -  /y^ )|-  — r/‘“ andI' -- ciUTgy ol the virtual jihoton,I/l >  !/z /fi 'xt/M <  //i <  hi .‘'/TIfaij.0 )To .start, with, avi‘ have got the virtual jirojectile y  (tlu' plioton) and 1.Jk‘ taiget. proton P. Tlu‘ one particle distribut ion i.s defined a.s the iiivci iant. (plant ity ol disl.ribution ol one jiartieular tyjie of partuh'S due tlio collision bidAvt^ en t.Jie viitual particle and the targ(d The particle distribution can be defined similarly when we ar(i interested in distiibiition ol two particular types of paitieles due 1.0 collision b(dwoen the virtual projectile and t.lie target, atot i« tin' total eross- seotioii of the virtual jirojeetile and the target. With the holji of giuuualized optical theorem ol Mueller (Muelhu* 1970), Ihe total cros.s-seetujn o-fof, particle], particle 2 and partiede 1 and 2 xu'odiiced together distiilnitions aic given by the I'ollowiug equations :(2)This IS obtained in (figure la) The liist t(M’ms jii eq. (3) is due to Poineronelujn exidiango. Here ftp{Q“) is the coupling constant of the virtual photon with the PomeroneJion trajectory and is i.liat of tlie target with the romoronchoii trajectory. The second tei in in eq. (2) is due to other Irajeetoiios 7? jiR{Q}) and are the eorresjiondmg eoujiliug constants due to excliange of ti ajoctories E.204 B .  K .  P a lI. II. 1.fl-),11 ,.l , 1U) (OFijf (I'l-)- 'I’olal (.Irui'S-fciisi-tum oi viil.iuil jirnjool.ilii mjirdnontccl l)y wuvy liiu^  iiiul llio Uii^c-l i '  l)y lliick liiir. Tlitj ihui liiiu ( rnpruHOiilh F’oiiu'i'oncliuii uinl Ijiijc'U oryn'tclm.iigo lirl.WLiuii tbn ju(ijnctilo and tint taigulP'lg. (ll») Oun mcliiMiv(' c‘i(js,s-at-( Imii fi[, Tlio iliiii lint / i('pK'si tits I’luiiciomlionand Koggc Irajoi'tory oxuluijjgr, hi'twctou thct virtual ]ii oj('c lil(' and tlio olisnivod hadron p,, 'rho thin him j  I'oiirdsoiiK I’oiimroimhoii and Jirggc tuijoclui v ('xdiango Ix'lwooii (hr taigol, and lh(' ohsi-rvod liadioii />^ l'’ig (!<')• tSinnlai iiitm protal-iou liohli? good a,s in Fig (Id). Only onr lias to irjtlarr llio olisrrvrd hadron pi by jjj.Fig (Id) 'J'\vo-]}ai lirjr inrliihivr ci oss-sootion dim lo rolliKion brlwri'ii ihr virlaialjn'o|oc,tilr y and tlm targol, J*. Thi’ intni'])rrlatiou of linn Ihirs ( and L ib simdai' to that ol  ^ and j  in Fig, (lb). Tho thm hnr  ^ Imir lopirbunls I'omrionchon and Koggo Trajratory oxohango botivoen iJio obHOivod hadrons jii,/ and Fig (lit) 'I’ho tlnor-\iartH!lo muhisivo rross-HORtion due to colliHion bol/Worn thovirtual piojoutilo y and tho target P. Tho thin lino 7 roprosonts roinoionohon and Koggi' tiaji'otory oxrhango botwoon viitual ]ii'ojoctilo y  and obsoivod hadron j)i. Tho linos ) mid Ic I'oprosout I’omoronclum and lioggo trajootory oxohango botwoon ohsi'rvod hadrons i>x and and obsorvod hadions p2 and 'p^  jospootivoly. Tho thin I n io rop ioson l/S  I’onioronrhon and Jtoggo tiajootory oxohungr botwoon ubaorvod hadion jOj and tho fmgot F.f W  -  /Jpm iiA^fpp{P u)+M Q ^)liA ‘^fl-RiVu) I ■ '•.111 ‘(This is giv(*u lu fifjuic (Jb))PijA^ ^ /hiQ^ )/iA‘\lrp{p±2)-r/^ p{(/^ )/U^ fpli{p±2) I(Soti ligure (Ic*))-(!“ ay^ (()))(3)i^ )Correlation function in deep inelastic scattering ^  l^ p{Q^ )ftA^ j'pp{p^ x)Jpp{Pi,^ .) I-+/h-{Q )^liA^JpR{Pi,x)j'Rp{Pi.i) (Soo figure (Id))-(]-a7,(0)|205(■"))In sUtisticaJ moclianics (Mueller J971), eurjelaiioj) t'unetioiiri are defined to measure, the difl'eroiice between lire i)artiele distributions acliiaJly obsei vi tl in an interaction and ilic jiartiele distiibiitions (‘xpeided i1' there rii(‘ no coik'Iii- Ljous present. The two particle correlation tunclion t'h(//v y/:>, yv_Lj) rs rlefiiiedby the following equation~{f>gA^ lf^ tot){f>aA~(f^ tot) .. (fi)'Hus IS zero when the total eneigy including all sub-iMieigies are allowtal t(» tend to iiiliiuty. The sub-energy is the .smallest of all. \^ ’e note, tlieieloie,how the correlation Innctiou teixls ti> zero with tie st'paiation between obsei ved hadrons in rapidity is given by<^ Ayi^  !/i^  Pxi) =Lfpit{pxi)lRp{Px‘^  I - ( l - a j { ( 0 ) )wheri‘^  JpR{P±i)fjiPiPx-i){P>x\fPxAl~^^'^V\ — il/i —^.-[l-cx7t(0)]-^:=r:2( 7 )and PgA^ '^  *^T’e invariant distiibiitions o f  |)artj(*l(‘ 1, jiarliele :2 aixljiaitudes 1 and 2 together lespectivcly The three-particle correlation linu'tron ^hiyi> Pxi'> H ' l ' y ‘.\ Px-i) defined by tlie lolJoAMiig eipration .G-AVi^  Pxi  ^y ‘2> Pxi\ Va, Pxa)1 l<^ toApgr'^ PoA^ -VpgA^ '^ PgA^ A-pQA^ '^pgA^ J I M^^ to^ PgA^ PoA^ PgA:^ («)where pgA^ '“''^  invariant distribution oI'particles 1, 2 and II togethei (sis-figure le). This is zcjo when the total energ}  ^ in<‘liiding all snb-eneigies an allowed to tend to infinity The sub-energies and smallest oi allThe measure of the rate of approach to the scaling limit is given by tin* ii('\t leading trvxieetory A* The three-particle correlation function expressial m Uims of the seiraration between obsei A''ed hadrons in rapidity is given byAlii y/2: Aiai :V3. Axs) ——JpP{Pi.A)hlp{Pxa)fpRiVxi)^nixpn^^)-'-Q^p{-{yy~if.A)lL\—^fpp{l^xz)}Rp{pxa)fpRiPxy)(hil(ip\:2.nm .^;}-'-cxp{-ij,iL)-:V>j^(Aii)>i’(Aia)y>ie(Ai3)/^ii//^p(2^wiip)-iexp(-7/3/A)-^fppiPxi)fppiPxi)fPRiPxi)fiBlM^^^iz)-^^^V{~y2m ■ • • ■ (9 )206 B. K. PalTJius wn olw^ivo that tho charactoristic distanco over which Iho correlation hiuct.iou IS elfectivo is given by 1/[1—aji(0)J, where R is the next to l^imoronchon trajectory which controls tho approach to sacling, for p, w, A^ , and /  trajectories, we liave a/i(0) J and 2 We have used our earlier work (Pal 1975). The notations used hero are properly explained there. Expressions for various pliysical quantities o.g., total cross-soction, correlation functions etc. have been properly interpreted on the basis of tho assumed model there. For the sake of completeness, wo are reproducing some of our earlier works. We have used smooth functions fpR{pn) and/flp (pi2) coirelation lunction<'9- (7). Here and 7^ x2 transverse momentum ol theobserved hadrons of energy-momentum and p  ^ respectively The lower suffix P  and R denote pomeronchon exchange from one side and Regge trajectory (5X(!hange from tlu^  other side The/'s are the coupling constants of the observed hadrons with Poineionchon and ilegge trajecf.ory. Other smooth functions 111 eq. (9) hav(i similar interpietation.3. ConclusionWhen hadrons are prodiieed in file central reihm, the eorrelation lunction becouKis independent ol the beam and the taigef.. The two-partiele ( orielation (’unetion depends only on tho relative distanee of the two jiartides in rapidity space. The f,hree-pait,jcle comdation fiiiietiou hehavos siiniJarly bid' it also eoid.ains terms as function of eaih mdividual rapidity Howiwer, when any of the hadrons is produced in the current Iragmentation region, tho eorrelalion fiiuef iou highly model-dependent and dojiends on the mass ol the virtual photon (Pal 1975). We hope that it will be (Nisiei- to visualize the mauy-body system at high energies with the help of higbei eorrelation functionR efekbnoesAbai'baiid H. D. T 1970 J^ hy^ . Rev. D3 2963 Chan H M. 197J Phya. Rev. Letters 26 672.Calm K. N. tSt E W. 1973 Rw 8D 3019.Hjorhi'u J. 1). k  Pasdios E. A. 1969 Phys. Rev, 185 1975. (jioll-Maun M. 1904 Phys Rev. Letters 8. 214,Muollor A. H. 1970 Rhya Rev. D2. 2963.Miiollor A. H. 1971 Phys. Rev, D4 150.L’al 13. K. 1975 Acta Physica Austnaca 43. 191. l^al 13 K  (to bo pLiblishod, Proq, Theor. Phys ).Zwoij  ^a  1964 CEKN 8l82/Th. 401.

Comments on the Correlation Function in Deep Inelastic Scattering

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Comments on the Correlation Function in Deep Inelastic Scattering

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