Parton Model. 2 q Q, 1

Transcription

1 Parton Model How do we exect the structure functons w and w to behave? It s nstructve to consder some secal cases: ) e e elastc scatterng from a ont-lke roton w, q q q w, q q m m m Defnng and notng that then q, x, M m m, w x Mw x x If one lots νw versus, then one would observe that νw M corresonds to a sngle lne located at M νw νw x M

2 Also, the value of νw would be ndeendent of. ) e e Elastc Scatterng from a Comoste Proton In ths case, we have or w, q GE G M q M q w, q GM M q 4M GE G M w, q M Mw, q GM M M νw and Mw are no longer related. If one lots νw as a functon of would stll observe a sngle eak at. M, one M νw νw M

3 3 However, νw would dro radly versus snce as a functon of. G and G both fall off radly E M Smlarly, for e en* nelastc exctaton of a nucleon resonance N*, one also observes a rad fall-off of νw versus. 3) eq eq Elastc Scatterng q sgnfes a ont-lke artcle (quark) whch carres a charge e and has a mass m q = x M. The cross secton for eq eq elastc scatterng s (for sn-½ q) d q q e cos sn 4 de d 4E sn mq m q The structure functons n ths case are q w, q e e m q Mx w, q e M x Mx νw lotted as a functon of would eak at x M νw νw x M

4 4 whle νw s ndeendent of. 4) e e x where conssts of ont-lke charged artons Ths s a generalzaton of 3). We defne f (x ) as the robablty for arton of tye to have a fracton x of nucleon s momentum (mass). Equaton 6.50 then generalzes to w q dxf xe, 0 e f M M M x where we use x x Mx M Snce s defned as x, we also have M Smlarly, one can show that, w q e x f x Mw q e x f x x, The above two equatons show that νw and Mw exhbt the scalng behavor. Namely, they only deend on a sngle arameter x. One therefore defnes the followng structure functons: F Mw F x x F w e x f x

5 5 The scalng behavor of νw and Mw was observed n the oneerng deenelastc scatterng exerments. It was called dee nelastc snce the nvarant mass w of the hadronc system s much larger that the roton mass w >> M. Note that the relaton F( x) F( x)/ x s a result of the assumton that the arton has sn-½. If one assumes that the artons are sn-0 object, then the relevant scatterng cross sectons would be the e - π - e - π - : In ths case, we have d ded cos 4 4E sn m q whch mles q w m w 0 F (x) = 0 Exerments favor x F (x) = F (x), rather than F (x) = 0. Therefore, they suort the nterretaton that charged artons have sn-½. The henomenon of scalng was frst observed at SLAC n the late 960 s. As the followng fgure shows, as the beam energy ncreases the nelastc cross sectons rse as w ncreases. At large w, the underlyng rocess s the eq eq elastc scatterng.

6 Fgure 7. Vsual fts to sectra showng the scatterng of electrons from hydrogen at 0 o for rmary energes, E, from 4.88 GeV to 7.65 GeV. The elastc eaks have been subtracted and radatve correctons aled. The cross sectons are exressed n nanobarns er GeV er steradan. 6

7 7 A smlar attern was also observed when the nelastc cross secton was measured at a fxed beam energy, but wth the sectrometer angle (scatterng angle) varyng from very small (θ =.5 o ) to larger (θ = 8 o ) angles. As the momentum transfer * * ncreases, the exctatons of the nucleon resonances N, fall off radly. However, the cross sectons for deely nelastc scatterng (large w) reman szeable. Fgure 8. Vsual fts to sectra showng the scatterng of electrons from hydrogen at a rmary energy E of aroxmately 3.5 GeV, for scatterng angles from.5 o to 8 o.

8 8 The scalng behavor of νw (= F ) was observed as shown n the followng fgure: In ths fgure, νw s shown to be ndeendent of for a fxed value of w w x. The SLAC data also rovded the frst drect evdence for the exstence of sea quarks. As shown below, M w, whch s F (x), was observed to rse as w ncreases w x. The large arton densty at small-x s due to the gluon slttng nto quark-antquark sea.

9 9 The SLAC exerments have used both the hydrogen and deuterum targets to extract the F structure functons for roton and for neutron. Recall that 4 F / x u u d d (gnorng heaver quarks) 9 9 n 4 F / x un un dn dn d d u u 9 9 where sosn symmetry s assumed n order to relate the arton dstrbutons n neutron to those n roton. n The rato, F / F, s shown to be close to at x 0, and t aroaches ~ 0.3 as x.

10 0 The above equatons mly that n F / F 4, where the ¼ lmt s reached 4 n n when d = 0, and the lmt F / F 4 s obtaned when u = 0. The F / F data suggests that down quark dros more radly than the u quark as x. Indeed, d( x)/ u( x) x as x. one fnds n Another nterestng SLAC result s that the F F data can be used to reveal the valence quark dstrbuton. It can be readly shown that where we assume u x d x n x F F uvx dv x 3 and the valence quarks u v, d v are defned as u x u x u x v d x d x d x v The early SLAC data shows that the valence quarks dstrbuton eak at x ~ ⅓, as one mght exect f the effectve quark mass s roughly ⅓ of roton s mass. w F w n F n

11 The arton model requres that the total momentum of the nucleon s carred by all artons. In other words, hence, 0 xpf x dx P 0 xf x dx where P s the momentum of the nucleon, and sgnfes varous tyes of artons. Agan, mortant nformaton can be extracted by combnng the F data wth the n F data: 4 F x dx u x u x xdx d x d x xdx 9 9 n 4 F xdx d x d xxdx ux u xxdx One obtans, from Equaton 6.64 and Equaton 6.65, the momentum fracton carred by the u and the down quarks. x u x u x dx x d x d x dx Therefore, one concludes that the u quarks carry roughly twce the momentum fracton of roton comared wth the down quarks. Ths s consstent wth the quark model (assumng sea quarks carry a small fracton of the momentum). Equaton 6.66 also mles that ~ 50% of the roton momentum s carred by neutral artons. Indeed, t s now qute well establshed that gluons are resonsble for ~50% of the nucleon s momentum. The gluon dstrbuton functon, g(x), can be determned from rocesses such as jet roducton, or from the scalng volaton n DIS, whch s due to the coulng between quarks and gluons. The structure functon measured n DIS s a sum of quarks and antquarks wth varous flavors. It s useful to solate the quark dstrbuton of a artcular flavor. Ths flavor decomoston can be made by usng a varety of technques:

12 n a) Snce F s mostly senstve to u(x) whle F s senstve to d(x), a comarson n between F and F can rovde nformaton on u(x) and d(x). b) Sem-nclusve DIS In ths tye of measurement, a hadron (usually an energetc on or kaon) s detected n concdence wth the nelastcally scattered electron. If the vrtual hoton struck a u(d) quark, ths quark s most lkely to hadronze nto a π + (π - ) meson. Therefore, the flavor of the struck quark can be reasonably well determned. c) The Drell-Yan Process The Drell-Yan rocess s bascally an electromagnetc rocess occurrng n hadronhadron nteracton. The underlyng mechansm s the annhlaton of the quarkantquark ar nto a vrtual hoton, whch q μ - γ* q μ + subsequently decays nto a ar of charged letons (μ + μ -, or e + e - ). It has been demonstrated that roton-roton and roton-nucleus nduced Drell-Yan rocess s very senstve to the antquark dstrbuton n the nucleon and/or nucleus. Furthermore, one can use the Drell-Yan rocess to measure the quark dstrbutons of mesons. Indeed, the N x and K N x, as well as the N x reactons have been measured, and they rovde the rare nformaton we have so far on the arton dstrbutons n π, K and. Note that these artcles cannot be studed n Dee Inelastc Scatterng exerments snce they are not avalable as targets.

13 3 d) Generalzed Drell-Yan Process: W/Z roducton u μ + u μ - w + z o d ν μ u μ + The Drell-Yan rocess can be generalzed to descrbe W/Z roducton n hadronhadron collder. The underlyng rocesses nvolve a quark-antquark annhlaton nto the charge W boson or the neutral Z boson. By comarng W + and W - roducton, one can solate the contrbutons from u versus d quarks (and u versus d too). e) Neutrno-Induced DIS As wll be exlaned n the next chater, neutrno nduced DIS reactons are very effectve n searatng the quark dstrbutons from the antquark dstrbutons. Of artcular nterest s the ossblty of usng sem-nclusve ν-nduced DIS to determne the strange (and ant-strange) quark dstrbutons n the nucle. Ths s accomlshed by detectng the μ + μ - ar from the ν-nduced DIS. The mechansm for roducng a μ + μ - ar n ν-nduced DIS s as follows: ν μ μ - w + μ + s c w + ν μ s The μ - s from the (ν μ, μ - ) rocess, and the μ + s from the charm decay. In a smlar fashon, beam can be used to robe the s dstrbuton n the nucleon. Although sx s xdx 0, t s ossble that s(x) and 0 on x such that at certan x, sx s x 0. s x have dfferent deendences

14 4 Symmetres n Parton Dstrbutons It s useful to use symmetry to connect varous arton dstrbutons. Some examles are: a) Isosn Symmetry Ths allows us to make connecton between the arton dstrbutons n hadrons whch are sosn artners of each other. The roton and neutron form an sosn doublet, and one can use sosn symmetry to relate ther arton dstrbutons: u x d x d x u x n n u x d x d x u x n n Smlarly, π + and π - are members of the sosn trlet, and ther arton dstrbutons are related: u x d x d x u x u x d x d x u x b) Charge-Conjugaton Symmetry For two hadrons whch are related by artcle-antartcle oeraton (chargeconjugaton), ther arton dstrbutons are also related. For examle, the arton dstrbutons n and are connected: u x u x d x d x u x u x d x d x Smlarly, π + and π - are related by charge-conjugaton oeraton (they are antartcles of each other). Hence,

15 5 From these equatons, we obtan u x u x d x d x u x u x d x d x u x d x d x u x v x u x d x u x d x s x We conclude that there are only two arton dstrbutons, v π (x) and s π (x), the valence and sea quark dstrbutons, requred to descrbe π + and π -. c) SU(3) Symmetry The SU(3) symmetry can be used to relate the arton dstrbutons of varous SU(3) multlets. For examle, the Σ + (uus) arton dstrbutons are related to roton (uud) s dstrbutons as follows u x u x s x d x s x d x Snce the SU(3) symmetry s known to be broken, the above equaton s only aroxmately true. Exermentally, the arton dstrbutons of the hyeron (lke Σ + ) can be measured usng the Drell-Yan rocess wth a hyeron beam. However, n ractce, these are dffcult measurements and have not been done yet. Sn-Deendent Structure Functons Polarzed DIS usng olarzed electron or muon beams scatterng off olarzed hydrogen or deuterum targets has been used to determne the sn-deendent structure functons g (x) and g (x). Ths s an actve area of research. From all exerments carred out so far, t was found that only 30% of roton s sn s carred by the u and down quarks. The other 70% must resde n sea quarks, gluons, or/and orbtal angular momentum. However, t s not yet clear how roton s sn s dstrbuted nto these varous comonents.