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DEEP INELASTIC SCATTERING Electron Scattering Off Nucleons (Fig 7.1)

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DEEP INELASTIC SCATTERING Electron Scattering Off Nucleons (Fig 7.1) DEEP INELASTIC SCATTERING Electron scattering o® nucleons (Fig 7.1): 1) Elastic scattering: E0 = E0(θ) 2) Inelastic scattering: No 1-to-1 relationship between E0 and θ Inelastic scattering: nucleon gets excitet nucleon resonances ) E0; p0 E; p Ep0 ; P 0 Invariant mass of nucleon resonance: 2 2 2 2 2 2 2 2 2 2 W c = P 0 = (P + q) = M c + 2P q + q = M c + 2Mº Q ¡ Here the Lorentz-invariant quantity º is de¯ned by: P q º = M Since in lab. frame 4-mom of incomming proton: P = (Mc; 0) 4-mom of exchanged photon : q = ((E0 E)=c; q) ¡ the energy transfer by the photon to the proton in this frame is º = E E0 ¡ 1 THE ¢(1232) RESONANCE Pronounced resonance observed with mass W = 1232 MeV/c2. ¢+-resonance P 0 p0 q = (º; q) P = (M; 0) p proton electron ++ + 0 + ¢-resonance exists in 4 charge states: ¢ ; ¢ ; ¢ ; ¢¡. Here ¢ Resonance width ¡ 120 MeV ' ~ 24 lifetime: ¿ = = 5:5 10¡ s: ) ¡ £ ☞ Typical lifetime for strong interaction processes (c¿ 1:5 fm!) ' ¢+ decays: ¢+ p + ¼0 ! ¢+ n + ¼+ ! + 0 Pions (¼ ; ¼ ; ¼¡): ☞ lightest hadrons (strongly interacting particles): 140 MeV/c2 » ☞ bosons: spin-0 2 STRUCTURE FUNCTIONS For W & 2:5 GeV/c2 individual resonances not distinguishable: ☞ many hadrons produced: Hadrons P 0 p0 q = (º; q) P = (M; 0) p proton electron Describe dynamics by form factors: W1 and W2 structure functions ☞ In elastic scattering, only one free variable: E0 = E0(θ) Since W = M, we have relationship 2Mº Q2 = 0: ¡ ☞ In inelastic scattering, one extra degree of freedom: Exitation energy of proton Since now W > M, we have relationship 2Mº Q2 > 0: ¡ 2 Cross section is function of two variables: (E0; θ) or (Q ; º) Cross section formula (a la Rosenbluth formula for elastic case): 2 d σ dσ ¤ 2 2 2 θ = W2(Q ; º) + 2W1(Q ; º) tan d­dE0 µd­¶Mott · 2¸ W2 describes electric interaction; W1 describes magnetic interaction 3 DEEP INELASTIC SCATTERING EXPERIMENTS First experiments in late 60's at SLAC: 25 GeV/c electron beam. At ¯xed scattering angle (4±) measure scattered electron energy E0 Experimental observation by varying beam energy (Fig. 7.2): ☞ Rapid fall-o® of cross-section with Q2 in resonance region ☞ Slower fall-o® of cross-section with Q2 above resonance region Behaviour above resonance region was surprising! Remember for elastic scattering: ☞ O® pointlike particle (Mott formula): dσ=dQ2 = constant ☞ O® real nucleon (dipole ¯t): dσ=dQ2 1=Q8 / Here, in Deep Inelastic Scattering (Fig. 7.3): Above resonances dσ=dQ2 constant point-like constituents . ' ) 4 Change of variables: 1) Introduce Lorentz-invariant dimensionless Bjorken scaling variable Q2 Q2 x = ´ 2P q 2Mº x gives a measure of the inelasticity of the process: For elastic scattering: 2Mº Q2 = 0 x = 1 ¡ ) For inelastic scattering: 2Mº Q2 > 0 0 x 1 ¡ ) · · 2) Change to dimensionless structure functions: 2 2 2 F1(x; Q ) = Mc W1(Q ; º) 2 2 F2(x; Q ) = º W2(Q ; º) From experimental data: Observation: Structure functions F1 and F2 depend only weakly on Q2 for ¯xed x (Fig. 7.4) Conclusion: Electrons scatter o® a point charge. But since nucleon is an extended object: Nucleons have a sub-structure made up of point-like constituents F1 results from magnetic interaction vanishes for spin-0 particles 1 ) For spin- 2 particles one can show the relation 2xF1(x) = F2(x): Since data con¯rm this (Fig. 7.5): 1 The point-line constituents of the nucleon have spin- 2 . 5 THE PARTON MODEL Model by Feynman and Bjorken: Partons constituents of proton ´ Today: ☞ charged partons = quarks ☞ neutral partons = gluons The model: look at the proton in a fast moving system transverse momentum and rest masses of partons can be neglected ) proton structure given by longitudinal momenta of partons ) Impulse approximation: proton decomposed into free moving partons ☞ valid for short duration process: neglect inter-parton interactions ☞ valid in DIS since short distance inter-parton interactions are weak (asymptotic freedom!) ☞ electron interaction with proton viewed as incoherent sum of interaction with individual partons ☞ Parton/electron interaction regarded as elastic scattering process Interpretation of Bjorken x when neglecting parton masses: ☞ x is fraction of proton 4-momentum carried by struck parton. Thus, DIS in lab-frame: ☞ Parton with 4-mom. xP interacts with photon with q = (º=c; q) Interpretation of Q2 variable (in Breit frame (Fig. 7.6)): ¸ ~ ~ = = 2¼ q Q2 j j ¸: wavelength of virtual photon. p Thus, Q2 ¯xes spatial resolution with which structures can be studied 6 STRUCTURE FUNCTIONS IN THE PARTON MODEL Assume nucleon is build from di®erent types of quarks f with electric charges zf e Probability for photon interacting with quark z2. / f Partons inside nucleon: ☞ Valence quarks: responsible for quantum numbers ☞ Sea quarks: virtual quark-antiquark pairs ☞ Gluons: bind quarks together Distribution function of momentum: - quarks qf (x) - antiquarks q¹f (x) - gluons g(x) Structure function F2 (electric interaction) is then 2 F2(x) = x zf [qf (x) + q¹f (x)] Xf Also DIS experiments based on alternate beam particles: muons: as electrons, sensitive to electric charges of quarks neutrinos: sensitive to the weak charges of quarks Combining DIS results (electrons, muons, neutrinos, antineutrinos): separate F2 into contributions from valence and sea quarks. ) Observations (Fig. 7.7): ☞ Sea quarks contribute only at small x; negligible above x = 0:35; ☞ Valence quarks contrinute maximally at x 0:20; ' distribution goes to zero for x 0 and x 1 ! ! ☞ For large x, F2 become small: unlikely that one quark alone carries major fraction of momentum 7 THE QUARK STRUCTURE OF NUCLEONS From DIS: Nucleons have constituents (quarks) which are ✔ electrically charged ✔ point-like ✔ spin-1/2 Should be able to explain quantum numbers of nucleons from those of quarks u d p n (uud) (udd) Charge z +2=3 1=3 1 0 ¡ Isospin I 1=2 1=2 I3 +1=2 1=2 +1=2 1=2 ¡ ¡ Spin s 1=2 1=2 1=2 1=2 In total six di®erent quarks { six “flavours". ☞ Three generations (families) of doublets with increasing masses: u c t Q=e = 2=3 0 1 0 1 0 1 á d s b Q=e = 1=3 @ A @ A @ A á ¡ Mass pattern (constituent masses): 0:3 1:3 175 0 1 0 1 0 1 [GeV=c2] 0:3 0:45 4:2 @ A @ A @ A Quarks inside nucleons: ☞ Valence quarks: u, d ☞ Sea quarks: u + u,¹ d + d,¹ s + ¹s; (c, b, t quarks too heavy) 8 Quark Charges Structure functions of proton and neutron in electron scattering: 1 4 1 F e;p(x) = x (dp + dp + d¹p) + (up + up + u¹p) + (sp + s¹p) 2 ¢ ·9 v s s 9 v s s 9 s s ¸ 1 4 1 F e;n(x) = x (dn + dn + d¹n) + (un + un + u¹n) + (sn + s¹n) 2 ¢ ·9 v s s 9 v s s 9 s s ¸ Assume isospin symmetry: Neutron and proton related through quark symmetry operation: u d: $ p n p n u (x) = d (x) ui(x) and d (x) = u (x) di(x) i i ´ i i ´ Also drop small s-quark terms, thus e;p 1 4 F (x) = x (dv + ds + d¹s) + (uv + us + u¹s) 2 ¢ ·9 9 ¸ e;n 1 4 F (x) = x (uv + us + u¹s) + (dv + ds + d¹s) 2 ¢ ·9 9 ¸ De¯ne for each quark and anti-quark q qv + qs. Then ´ 1 4 F e;p(x) = x (d + d¹) + (u + u¹) 2 ¢ ·9 9 ¸ 1 4 F e;n(x) = x (u + u¹) + (d + d¹) 2 ¢ ·9 9 ¸ Now, \average" nucleon form factor: F e;p(x) + F e;n(x) 5 F e;N(x) 2 2 = x [q (x) + q¹ (x)] 2 ´ 2 18 f f fX=u;d In neutrino scattering, no dependence on quark charges: º;N F2 (x) = x [qf (x) + q¹f (x)] Xf Thus, measure (sum of squares of) quark charges through comparison Assignment of charges +2/3 and 1/3 for u- and d-quarks con¯rmed ¡ 9 Quark momentum distributions (Fig. 8.1) Integration of experimental momentum distribution: 1 1 º;N 18 e;N F2 (x) dx F2 (x) dx 0:5: Z0 ' 5 Z0 ' Only half of nucleon momentum carried by quarks! Rest by gluons ) n p Consider ratio of neutron and proton structure functions, F2 =F2 : ☞ Goes to 1 for low x: sea quarks dominate (Fig. 8.2) ☞ Goes to 1/4 for x 1 ! 2 2 2 2 - would have expected 2=3 = (2zd + zu)=(2zu + zd) 2 2 - observation looks like zd=zu 10 Constituent quarks DIS experiments: ☞ half of nucleon momentum carried by quarks ☞ other half carried by gluons DIS: probing quarks deep inside hadrons ☞ large Q2 small length scale (¸ h= Q2) ) ' p Quark masses estimated from DIS (current quark masses) are small: mcur = 1:5 5 MeV=c2 u ¡ mcur = 3 9 MeV=c2 d ¡ For spectroscopy (magnetic moments, hadron exitation energy, . ): ☞ Combine sea quarks and gluons with valence quarks ☞ Constituent quarks (or e®ective quarks) ☞ Each valence quark drags along a lot of \junk" larger masses ) me® 300 MeV=c2 u ' me® 300 MeV=c2 d ' d-quark heavier than u-quark: ☞ proton (uud) mass: 938.3 MeV/c2 ☞ neutron (udd) mass: 939.6 MeV/c2 In addition Coulomb repulsion larger for proton than neutron 11 Quarks in hadrons Hundreds of unstable hadrons known. From their study: information on the strong interaction. Baryons Lowest mass states (ground states): neutron and proton 3 quarks ➩ half-integral spins Many exited states: ☞ Parallel to atomic spectra ☞ BUT: energy gap between states of same order as nucleon mass Classify exited states as individualo particles ) Baryons produced in pairs: baryon + anti-baryon Conserved quantum number: baryon number B ) Baryon: B = +1 Quark: B = +1/3 Anti-baryon: B = 1 Anti-quark: B = 1/3 ¡ ¡ Experiment: B conserved in all reactions.
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