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3. Structure Functions and Quark Distributions 2

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3. Structure functions and quark distributions Elastic electron-parton scattering: Parton i, charge z , mass m , parton density f (x) i i i Elastic scattering dσ 4πα 2 E′ ⎛ θ Q2 θ ⎞ = ⋅ z2 ⋅ f (x)⎜cos2 + sin2 ⎟ Q2 Q2 2 4 i i ⎜ 2 ⎟ 1= and x = ⇒m = xM dQ Q E ⎝ 2 2mi 2 ⎠ i 2miν 2Mν Inelastic ep→eX scattering in parton model: dσ 4πα 2 E′ ⎛ θ Q2 θ ⎞ = ⋅ z2 ⋅ f (x)⎜cos2 + sin2 ⎟ 2 4 ∑ i i ⎜ 2 2 ⎟ dQ dx Q E i ⎝ 2 2M x 2 ⎠ Structure functions Compared to the macroscopic expression: 2 2 2 dσ 4πα E′ F (x) ⎛ θ 2xF (x) Q θ ⎞ F2(x) = x ⋅ zi fi (x) = ⋅ 2 ⎜cos2 + 1 sin2 ⎟ ∑ 2 4 ⎜ 2 2 ⎟ i dQ dx Q E x ⎝ 2 F2(x) 2M x 2 ⎠ 2xF1(x) = F2(x) Proton model F2(x) 1 3.1 Parton Spin Callan-Gross relation 2xF 1 = 1 for spin ½ partons F2 Proton constituents are spin ½ partons e.g.: for spin 0 partons, sin2θ/2 term in cross section disappears: F1(x)=0 3.2 Sea and valence quarks Parton distribution function fi(x) = Probability to find parton with xi∈[x, x+dx] Partons = Quarks and Gluons u u valence quarks d q sea quarks q Quark composition of the proton uv + uv + dv + (us + us ) + (ds + ds ) + (ss + ss ) u(x), d(x) (anti) quark Sea: Heavy quark contribution strongly suppressed densities of u and d ep F2 (x) 2 = ∑zi ⋅ fi (x) x i 4 1 1 = (u p (x) + u p (x)) + (d p (x) + d p (x)) + (s p (x) + s p (x)) 9 9 9 2 Quark composition of the neutron en F2 (x) 2 = ∑zi ⋅ fi (x) x i 4 1 1 = (un (x) + u n (x)) + (d n (x) + d n (x)) + (sn (x) + s n (x)) 9 9 9 F en(x) 4 1 2 = (d(x) + d (x)) + ()u(x) + u(x) + s(x) + s(x) x 9 9 Iso-spin symmetry un (x) = d p (x) = d(x) d n (x) = u p (x) = u(x) In total 6 unknown quark distributions sn (x) = s p (x) = s(x) q n (x) = q p (x) = q(x) Sum rules 1 u (x)dx = 2 ∫ v 0 1 valence q(x) = qv (x) + qs (x) d (x)dx = 1 ∫ v 0 q(x) = qs (x) sea 1 (q (x) − q (x))dx = 0 ∫ s s 0 p n For F2 and F2 the following relation can be obtained (Nachtmann): 4 1 en (d d ) u u s s 1 F (x) + + ()+ + + < 2 = 9 9 < 4 4 F ep (x) 4 1 2 (u + u ) + ()d + d + s + s 9 9 (for all x) n p The ratio F2 / F2 allows to probe the sea and valence quark contributions. Valence quarks dominate Sea quarks dominate For x→0: For x→0: (no momentum left for sea) • Created by gluon splitting en F2 (x) uv + 4dv 1 • Gluon spectrum ~1/x ep → ≥ en F2 (x) 4uv + dv 4 F2 (x) ep → 1 F2 (x) Observation for protons: use us ≈ ds ≈ ss ≈ us ≈ ds ≈ ss uv >> dv for x→ 1 3 Valence quarks F ep (x) 4 1 2 = (u + u ) + ()d + d + s + s x 9 9 1 4 = []4u + d + S 9 v v 3 F en (x) 4 1 2 = (d + d ) + ()u + u + s + s x 9 9 1 4 = []u + 4d + S 9 v v 3 1 1 []F ep (x) − F en (x) = []u (x) − d (x) x 2 2 3 v v As uv >> dv: uv –dv is a measure for valence quark distribution Valence quarks mostly at large x Sum of quark momentum Scattering at an iso-scalar target: #p = #n (e.g. C, Ca) 1 5 1 F eN = []F ep + F en = x ⋅[u + u + d + d ] + x ⋅[s + s] 2 2 2 2 18 9 5 5 ≈ x ⋅[u + u + d + d ] = [Sum of all quark momenta] 18 18 Small s quark distribution neglected 1 Naively one expects: 18 eN ⋅ F2 (x)dx ≈ 1 5 ∫0 1 18 eN Experimental observation: ⋅ F2 (x)dx ≈ 0.5 5 ∫0 • probed quarks and anti-quarks carry only 50% of nucleon momentum • Remaining momentum carried by gluons (see later) 4 3.3 Neutrino nucleon scattering µ ± ν µ ν µ N → µ X W • More information on quark distribution N X • Separation between quarks / anti-quarks Q2 ν x = y = ν = E − E′ QPM: 2Mν E µ − ν µ − dσ(ν µN → µ X) = d Fνp (x) and Fνn (x) dy → i i u + ν µ µ dσ(ν N → µ + X) µ = u → Fν p (x) and Fν n (x) dy i i d Structure functions for neutrino scattering F νn = F ν p i i Equal because of Charge symmetry νp ν n Fi = Fi 1 1 F νN = (F νp + F νn ) = (F ν n + F ν p ) = F ν N for i = 1,2 i 2 i i 2 i i i ν N νN F3 = −F3 Additional structure function to account for parity violation Double differential cross section: Scattering at iso-scalar target 2 2 2 d σ(νN,νN) GF ⎡ νN y νN y νN ⎤ = 2ME ⎢(1− y)F2 (x) + 2xF1 (x) ± y(1− )xF3 (x)⎥ dxdy 2π ⎣ 2 2 ⎦ 2 2 to account for 4πα GF parity violation Q4 a 2π 5 Structure functions in QPM 1 F νN = F νN 1 2x 2 νp νp F2 = 2x[d + u ] xF3 = 2x[d − u ] νn n n νn n n F2 = 2x[d + u ] xF3 = 2x[d − u ] = 2x[u + d ] = 2x[u − d ] Iso-scalar νN νN F2 = x[u + u + d + d ] xF3 = x[(u + d ) − (u + d )] target νN νN F2 = x[Q(x) + Q (x)] xF3 = x[Q(x) − Q (x)] Measures sum of Measures valence quarks and anti-quarks quarks νN νN Measurement: F2 + xF3 = 2xQ (x) Sea and valence quarks νN νN F2 − xF3 = 2xQ (x) Sea quarks 6 Parton distribution in eN and νN scattering Question: Do the parton distribution seen in electro- eN magnetic (F2 ) and in weak interaction eN (F2 ) agree ? F νN (x) x[Q(x) + Q (x)] 18 2 = = F eN (x) 5 5 2 ⋅ x[Q(x) + Q (x)] 18 Factor from fractional charge Answer: • e.m. and weak quark structure is the same • Factor 18/5 → fractional quark charge Summary: eN and νN scattering eN scattering d 2σ eN 2πα 2 5 = x[]1+ (1− y)2 ⋅ []Q(x) + Q(x) dxdy Q4 18 5 FeN (x) = x[]Q(x) + Q (x) 2 18 νN + ν N scattering d 2σ νN G2 = F xs[]Q(x) + (1− y)2 ⋅Q (x) dxdy 2π d 2σ νN G2 = F xs[]Q(x) + (1− y)2 ⋅Q(x) dxdy 2π νN νN F2 (x) = x[Q(x) + Q(x)] F3 (x) = x[Q(x) − Q(x)] νN νN F2 (x) = x[Q(x) + Q(x)] F3 (x) = x[Q(x) − Q(x)] 7 Summary: nucleon structure • Nucleons contain point-like constituents: evidenced by approximate scale invariance of the structure function F2(x) • Constituents have spin 1/2: 2xF1≈F2 • Same quark distribution for e.m. and weak interaction But • Quarks amount only to ~50% of nucleon momentum → remainder carried by gluons 2 • Logarithmic Q dependence of F2 for very small/large x and large Q2 observed Understood within perturbative QCD 4. Scaling-violation in deep-inelastic lepton-nucleon scattering 2 Scaling violation: F2,3 = F2,3(x, Q ) ? 8 Scaling violation in a qualitative QCD picture Quarks q(x) probed by the photon have history: z x z < x Gluon-emission Gluon-absorption z x z > x The smaller the wavelength of the probe (Q2 of the photon) the more quantum fluctuations can be resolved : Small Q2 Large Q2 2 effects: Boson-gluon fusion QCD Compton process e q(x) q(x) xP zP xP g(z) q(z) zP P x < z < 1 Increase of sea quarks will soften the valence quark distribution as Q2 increases 2 F2 will rise with Q at small x where the sea quarks dominate and will fall with Q2 at large x where valence quarks dominate. 9 Quantitative description of scaling violation: DGLAP evolution (Dokshitzer, Gribov, Lipatov, Altarelli, Parisi, 1972 – 1977) of quark and gluon density functions q(x) and g(x) for large Q2: evolution of quark density with lnQ2 z z x ji x i Quark Gluon density density Splitting functions Pij and Pig Probability that a parton j (quark or gluon) emits a parton i with momentum fraction ε=x/z of the parent parton. Evolution of gluon density x zx z Splitting functions are calculated as power series in αs up to a given order: qq α P (z,α ) = P 0(z) + s P1(z) + ij s ij 2π ij K 10 0 In leading order: Pij (z,αs ) ≡ Pij (z) 4 1+ z2 4 1+ (1− z)2 P (z) = P (z) = qq 3 1− z gq 3 z 2 2 ⎛ z 1− z ⎞ z + (1− z) P (z) = 6 + + z(1− z) P (z) = gg ⎜ ⎟ qg 2 ⎝1− z z ⎠ Deep-inelastic muon-nucleon scattering: NMC Experiment µ beam Eµ=90, 120, 200, 280 GeV p 2 n 2 F2 (x,Q ) F2 (x,Q ) 11 Deep-inelastic electron nucleon scattering: H1 and ZEUS at HERA ep 27.6 GeV 890 GeV 920 GeV s ≈ 318 GeV e p 12 F2(x) Large increase of F2(x) for very small x: unexpected 2 F2(x,Q ) Described by QCD evolution 13 Experimental determination of the gluon density 2 Using the DGLAP evolution eq.
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