Structure of subatomic particles Introduction Rutherford and Mott scattering Sizes of particles Form factors Deep inelastic scattering March 2011

Introduction This chapter describes what led to the present picture of partons in the proton, the result of the SLAC deep inelastic scattering experiment. To get there we first study the elastic scattering case of two point-like particles, like e- - scattering, through the kinematic variables, inclusion of spin and other experimental facts. We go from simple Rutherford scattering, to Mott scattering, to the inclusions of form factors in the Rosenbluth formula. We also discuss the important measurement by Hofstadter of the radius of the proton. All the above will be an introduction to the actual Deep Inelastic Scattering (DIS) formalism and experimental results, with their physical interpretation. March 2011

e- - elastic scattering p’ Treat as spinless particles. Use the laboratory frame (initial  at rest). The four momentum transfer between initial and final electrons carried by the exchanged photon: q = k – k’. Neglecting the electron mass (k2=k’2≈0) If p’ is the four-vector of the scattered , q+p=p’. March 2011

Order of cross section According to the Feynman rules, amplitude will have a factor of  from the couplings at the two vertexes, and a propagator. Since a real photon has mass zero, the propagator has the form of 1/q2. March 2011

Rutherford Scattering Formula Rutherford assumed a scattering of two point-like spinless particles, where the target is infinitely heavy and thus doesn’t recoil. In this case the energy of the scattered electron, E’, is the same as that of the initial electron, E. March 2011

Mott Scattering Formula One can use the Dirac equation to take into account the spin of the electron. If one still considers the target to be point-like, spinless and infinitely heavy, one gets the Mott formula: (Actually, the factor is 1-2sin2(/2), which, for 1, turns into cos2(/2)) March 2011

Form factors So far, particles treated as point-like particles. Charge was concentrated in one point. If the particle is an extended object, one can assign a charge distribution to that object. Usually one talks about a normalized charge distribution, or a probability density (r), so that a particle with charge Q has a charge density of Q(r), with The probability density enters in the theoretical calculations through the scattering potential V(x), and the outcome of the calculation is F(q2) is the form factor, obtained from the Fourier transform of the probability density. March 2011

Form factors (2) Note, form factor is function of q2 only because assumed spherical symmetric system. In that case one can write March 2011

Form factors (3) In principle, the radial charge distribution (r) could be determined from the inverse Fourier transform if the q2 dependence of the form factor is known: Hofstadter measured electron scattering from 12C in 1957. Dashed line: plane wave scattering from homogenous sphere with diffuse surface. March 2011

proton charge radius Low-q2 behaviour F(q2): Measure F(q2) at low q2 mean square radius: Fourier expansion: Measure F(q2) at low q2 Yielding March 2011

Experimental Results McAllister and Hofstadter obtained a first approximation to the structure and size of a proton – a mean squared radius of the proton was found to be: (0.78±0.20)x10-13cm at 236MeV. (0.70±0.24)x10-13cm at 188MeV. Together, the best result was: <r2>1/2=(0.74±0.24)x10-13cm. March 2011

Nuclear charge distributions Using measurements of form factors, get information on charge density. The density is described by a two-parameters Fermi function: c=1.07 fm A1/3; a=0.54 fm Almost constant charge density in interior. c is the radius at which density decreases by one half. March 2011

Dirac Scattering Formula If one takes into account that the target has a finite mass and thus recoils, the energy of the elastically scattered electron, E’, is: In addition, one treats both particles as Dirac particles, namely as point-like particles with spin ½, having magnetic moments: The terms in front of the square brackets are from the Rutherford formula, with a recoiling target. The first term in the square brackets is from the Mott formula, while the second terms includes the magnetic spin flip. March 2011

Nucleon magnetic moment Dirac formula assumes that the nucleon has a magnetic moment corresponding to the nuclear magneton (n.m.):  (Dirac)  (observed) Proton 1 n.m. +2.79n.m. Neutron -1.91n.m. March 2011

Rosenbluth Scattering Formula All the previous equations assume point-like particles. As protons are not point-like and the magnetic moments differ in reality to the Dirac , we have to include the Form Factors GE and GM and a factor to correct for . Rosenbluth: March 2011

Experimental Results Three theoretical curves: a) Mott cross-sec – a spinless point charge proton. b) Dirac cross-sec – a point-like proton with mass M, and (Dirac). c) Rosenbluth cross-sec – a point-like proton with mass M and corrected . Results deviate from theory due to a “structure factor” – the proton is not point-like but has finite size. Form Factors introduce this deviation. March 2011

Test of Rosenbluth formula The form factors are function of Q2 only, with the normalizations The formula can be tested by doing an experiment at fixed Q2 and varying the angle : March 2011

Dipole form factor Form factors have been measured to higher energies for protons and for neutrons. The latter have been obtained from scattering on deuterium target, from which the measured cross sections on the proton were subtracted (+ some corrections). The resulting form factors obey a simple law: The universal form factor, called the dipole form factor, can be described by an empirical formula: MV2=(0.84GeV)2=0.71 GeV2 March 2011

Nucleon form-factor data March 2011

High Q2 f.f. data What is the physical meaning of a form factor? It measures the amplitude that under an impact the proton remains intact. As the impact gets larger, the probability that the proton remains a proton gets smaller. The elastic scattering cross section gets small. For example, for Q2=20 GeV2, the elastic cross section is reduced by 6 orders of magnitude. March 2011

Deep inelastic scattering (DIS) At high energies, elastic scattering becomes relatively unlikely (elastic form factor falls rapidly with q2). Instead: proton breaks up into hadrons: e- p  e- X ‘Deep’ – high q2, ‘Inelastic’ – proton breaks up. At HERA, reached already Q2=40,000 GeV2, probing structures down to ~ 10-18 m. March 2011

Review of elastic scattering E3 and  are related: only have to measure one of the (say ) March 2011

Bjorken x In DIS, mass of system X not fixed: q2 and  no longer related  G(q2) replaced by F(,q2) equivalently: E3 and  no longer related  have to measure both E3 and  Final state contains at least one baryon  Bjorken x: (x=1 for elastic scattering e-pe-p) March 2011

SLAC DIS experiment Most general form of DIS cross section is: Measured DIS cross section by detecting the scattered electron and measuring its energy E3 for different scattering angles  March 2011

Result of SLAC DIS experiment Contrary to the elastic case, the cross section is found to depend very weakly on q2, reminiscent of the Rutherford experiment. Measurements presented in the figure are for two different values of the invariant mass W of the hadronic final state X. From cross section can obtain the structure functions F1 and F2, which can be functions of  and q2, or alternatively, functions of x and Q2: Fi(x,Q2). March 2011

Bjorken scaling F1 and F2 are found to be approximetly independent of Q2 for fixed x. This suggests that the virtual photon is scattering off pointlike constituents within the nucleon. March 2011

Bjorken scaling (2) March 2011

Scaling violation Latest measurements from HERA presented at the Moriond Electroweak workshop, March 2004 March 2011

Callan-Gross relation F1 and F2 are found to be closely related: known as Callan-Gross relation. March 2011

The Parton Model Can obtain Bjorken scaling and Callan-Gross relation by assuming that the virtual photon scatters elastically from a point-like object within the proton, named by Feynman as partons. The other partons are spectators. Assume interaction with parton takes place sufficiently fast that can be treated as free particle of mass m: relates E’ and  March 2011

e-parton elastic scattering elastic scattering cross section for e-partone-parton can be written as: z – charge of partons in units of e. Assume from here that partons are quarks. proton made up of 3 quarks: uud. Can extract F1 and F2: March 2011

Consequensces of parton model F1 and F2 are related as follows: Callan-Gross relation using: thus F2 is only function of x, independent of q2 Bjorken scaling F1, F2 are functions of one variable only because underlying scattering is elastic and pointlike, giving extra constraint between E’ and . partons are quarks with spin ½. Only one unknown function, because charge and magnetic moment are related for Dirac particles:  = e/2m. But measured F2 not delta function! March 2011

Expectations for F2 proton is pointlike proton consists of only 3 free quarks quark-parton model (QPM) proton consists of only 3 bound quarks (Fermi motion) proton consists of 3 bound ‘valence’ quarks + other partons (QCD) March 2011

Physical meaning of Bjorken x use frame where proton momentum is very large (‘infinite momentum frame’). In this frame, momenta of partons collinear with parent proton. Each parton carries a fraction z of the proton’s momentum, and are approximately massless (p2≈0). q In addition, Q2-q2»M2 March 2011

Bjorken x (2) neglecting z2M2 : Thus, in the infinite momentum frame, x is the fraction of the proton momentum carried by the (massless) parton. This also means that measured cross section at a given x is proportional to the probability of finding a parton with a fraction x of the proton momentum. since can measure parton’s momentum from scattered electron alone (E’,) March 2011

Quark distribution functions proton structure F2 is not a delta function, since not all quarks have x=1/3. We thus define quark distribution functions as follows: probability that u quark in proton has momentum x probability that d quark in proton has momentum x March 2011

qdfs (2) neutron structure isospin symmetry (see later): can extract u(x) and d(x) from measurement of the proton and neutron structure function. These function can not yet be predicted from theory which needs a better understanding of the non-perturbative regime. March 2011

inclusion of ‘sea’ quarks need to extend picture by inclusion of ‘sea’ quarks in addition to the valence quarks : In order to find the fraction of the proton’s momentum carried by u and anti-u quarks, denoted fu, and the one of fd, need to integrate F2 over x: from experimental measurements (eg page 26) Only 54% of proton’s momentum is carried by quarks  rest, by gluons. March 2011

Picture of the proton March 2011

Wavelength of probe : q² klein  lange WW: nur Valenz-Quarks sichtbar (fermi-verschmiert) q² groß  kurze WW: See-Quarks sichtbar (Stroboskop-Aufnahme) March 2011

QCD and scaling violation Existence of gluon proven experimentally (3-jets events). Opened field of Quantum Chromodynamics (QCD – see later). Closer picture of F2 vs Q2 showed scaling violation, which can be explained as follows: As the momentum of the probe increases and the distance it resolves decreases, it begins to see the detailed quantum mechanical subprocesses of QCD in the environment of the struck quark. What may have appeared to be a quark with a given x at low Q2, may be revealed as a quark and a gluon, at higher Q2, with the quark having lower x. March 2011

Scaling violation (2) As the momentum of the probe increases, the average fraction of the total proton momemtum carried by the quarks appears to decrease. As the momentum of the probe increases still further, it may see the gluon radiated be the valence quark dissociating into a quark-antiquark pair. So there will appear to be even more quarks carrying very low fractions of the total proton momentum. Number of low x partons increases as q2 increases, while at high x, the number decreases as q2 increases. March 2011

So after all, F2 is not just function of x, but also of Q2: Scaling violation Latest measurements from HERA presented at the Moriond Electroweak workshop, March 2004 So after all, F2 is not just function of x, but also of Q2: Curves in the figure: result of a perturbative QCD (pQCD) calculation. March 2011