The Quark-Parton Model Before uarks and gluons were generally acceted Feynman roosed that the roton was made u of oint-like constituents artons Both Bjorken Scaling and the Callan-Gross relationshi can be exlained by assuming that Dee Inelastic Scattering is dominated by the scattering of a single virtual hoton from oint-like sin-half constituents of the roton. X X Scattering from a roton with structure functions Scattering from a oint-like uark within the roton
Elastic Scattering at Very High 2 «At high 2 the Rosenbluth exression for elastic scattering becomes From elastic scattering, the roton magnetic form factor is at high 2 Due to the finite roton size, elastic scattering at high 2 is unlikely and inelastic reactions where the roton breaks u dominate. X M.Breidenbach et al., Phys. Rev. Lett. 23 (1969) 935
Kinematics of Inelastic Scattering X For inelastic scattering the mass of the final state hadronic system is no longer the roton mass, M The final state hadronic system must contain at least one baryon which imlies the final state invariant mass M X > M ë For inelastic scattering introduce four new kinematic variables: «Define: Bjorken x (Lorentz Invariant) Here where Note: in many text books W is often used in lace of M X hence inelastic elastic Proton intact
«Define: In the Lab. Frame: (Lorentz Invariant) X So y is the fractional energy loss of the incoming article In the C.o.M. Frame (neglecting the electron and roton masses): «Finally Define: for (Lorentz Invariant) In the Lab. Frame: n is the energy lost by the incoming article
Relationshis between Kinematic Variables Can rewrite the new kinematic variables in terms of the suared centre-of-mass energy, s, for the electron-roton collision Neglect mass of electron For a fixed centre-of-mass energy, it can then be shown that the four kinematic variables are not indeendent. i.e. the scaling variables x and y can be exressed as Note the simle relationshi between y and n and For a fixed centre of mass energy, the interaction kinematics are comletely defined by any two of the above kinematic variables (excet y and n) For elastic scattering there is only one indeendent variable. As we saw reviously if you measure electron scattering angle know everything else.
Inelastic Scattering Examle: Scattering of 4.879 GeV electrons from rotons at rest Place detector at 10 o to beam and measure the energies of scattered Kinematics fully determined from the electron energy and angle! e.g. for this energy and angle : the invariant mass of the final state hadronic system Elastic Scattering roton remains intact Inelastic Scattering roduce excited states of roton e.g. Dee Inelastic Scattering roton breaks u resulting in a many article final state DIS = large W
Inelastic Cross Sections Reeat exeriments at different angles/beam energies and determine 2 deendence of elastic and inelastic cross-sections M.Breidenbach et al., Phys. Rev. Lett. 23 (1969) 935 Elastic scattering falls of raidly with due to the roton not being oint-like (i.e. form factors) Inelastic scattering cross sections only weakly deendent on 2 Dee Inelastic scattering cross sections almost indeendent of 2! i.e. Form factor 1 Scattering from oint-like objects within the roton!
In the Lab. frame it is convenient to exress the cross section in terms of the angle,, and energy,, of the scattered electron exerimentally well measured. Inelastic Cross Sections X jet X-section in the Lab. frame (3) Electromagnetic Structure Function Pure Magnetic Structure Function
Measuring the Structure Functions «To determine and for a given and need measurements of the differential cross section at several different scattering angles and incoming electron beam energies (see Q13 on examles sheet) Examle: electron-roton scattering F 2 vs. Q 2 at fixed x J.T.Friedman + H.W.Kendall, Ann. Rev. Nucl. Sci. 22 (1972) 203 s Exerimentally it is observed that both and are (almost) indeendent of
Bjorken Scaling and the Callan-Gross Relation «The near (see later) indeendence of the structure functions on Q 2 is known as Bjorken Scaling, i.e. It is strongly suggestive of scattering from oint-like constituents within the roton «It is also observed that and are not indeendent but satisfy the Callan-Gross relation sin ½ As we shall soon see this is exactly what is exected for scattering from sin-half uarks. Note if uarks were sin zero articles we would exect the urely magnetic structure function to be zero, i.e. sin 0
In the arton model the basic interaction is ELASTIC scattering from a uasi-free sin-½ uark in the roton, i.e. treat the uark as a free article! The arton model is most easily formulated in a frame where the roton has very high energy, often referred to as the infinite momentum frame, where we can neglect the roton mass and In this frame can also neglect the mass of the uark and any momentum transverse to the direction of the roton. Let the uark carry a fraction of the roton s four-momentum. After the interaction the struck uark s four-momentum is Bjorken x can be identified as the fraction of the roton momentum carried by the struck uark (in a frame where the roton has very high energy)
The arton model redicts: Bjorken Scaling ë Due to scattering from oint-like articles within the roton Callan-Gross Relation ë Due to scattering from sin half Dirac articles where the magnetic moment is directly related to the charge; hence the electro-magnetic and ure magnetic terms are fixed with resect to each other. «At resent arton distributions cannot be calculated from QCD Can t use erturbation theory due to large couling constant «Measurements of the structure functions enable us to determine the arton distribution functions! «For electron-roton scattering we have: Due to higher orders, the roton contains not only u and down uarks but also anti-u and anti-down uarks (will neglect the small contributions from heavier uarks)
Scaling Violations In last 40 years, exeriments have robed the roton with virtual hotons of ever increasing energy Non-oint like nature of the scattering becomes aarent when l g ~ size of scattering centre e - e - µ g Scattering from oint-like uarks gives rise to Bjorken scaling: no 2 cross section deendence IF uarks were not oint-like, at high 2 (when the wavelength of the virtual hoton ~ size of uark) would observe raid decrease in cross section with increasing 2. 10-15 m To search for uark sub-structure want to go to highest 2 10-18 m HERA 188
HERA e ± Collider : 1991-2007 «DESY (Deutsches Elektronen-Synchroton) Laboratory, Hamburg, Germany e ± 27.5 GeV 820 GeV Ös = 300 GeV H1 2 km ZEUS «Two large exeriments : H1 and ZEUS «Probe roton at very high Q 2 and very low x
Examle of a High Q 2 Event in H1 ëevent kinematics determined from electron angle and energy e + jet ëalso measure hadronic system (although not as recisely) gives some redundancy
Proton-Proton Collisions at the LHC «Measurements of structure functions not only rovide a owerful test of QCD, the arton distribution functions are essential for the calculation of cross sections at and colliders. Examle: Higgs roduction at the Large Hadron Collider LHC ( 2009-) The LHC will collide 7 TeV rotons on 7 TeV rotons However underlying collisions are between artons Higgs roduction the LHC dominated by gluon-gluon fusion 7 TeV Cross section deends on gluon PDFs t t t H 0 Uncertainty in gluon PDFs lead to a ±5 % uncertainty in Higgs roduction cross section Prior to HERA data uncertainty was ±25 % 7 TeV
Summary s At very high electron energies : the roton aears to be a sea of uarks and gluons. s Dee Inelastic Scattering = Elastic scattering from the uasi-free constituent uarks Bjorken Scaling Callan-Gross oint-like scattering Scattering from sin-1/2 s Describe scattering in terms of arton distribution functions which describe momentum distribution inside a nucleon s The roton is much more comlex than just uud - sea of anti-uarks/gluons s Quarks carry only 50 % of the rotons momentum the rest is due to low energy gluons