Form Factors and Structure Functions

Transcription

1 Form Factors and Structure Functions Yury Kolomensky Physics 129, Fall 2010

2 Standard Model Primer 3 fundamental interactions Electromagnetic: U=q 1 q 2 /r Vector couplings Fine structure constant α=e 2 /4π Strong: U=λ 1 (r)λ 2 (r) /r Vector couplings Coupling constant αs=λ 2 /4π depends strongly on distance Weak: U=Q w1 Q w2 exp(-m W(Z) r)/r Axial (W ±,Z 0 ) and vector (Z 0 ) couplings Coupling constant αw=qw 2 /4π

3 Running of Coupling Constants Generic property of any field theory: higher order corrections (loops) induce momentum (distance) dependence of coupling constants. This is known as an effect of vacuum polarization The magnitude and the direction of the change depends on the type of the interaction α S (Q) Q (GeV) = /Δx

4 Running of Coupling Constants QED (electromagnetic interactions): photons have no charge, so to first order include only fermion loops Coupling constant depends on momentum transfer Q 2 ~ 2 /Δx 2, the number of active fermions nf (with mass (2mf) 2 <Q 2 ), and the value of the coupling measured at some physical scale μ (typically, an electron mass). α(q 2 )= α(µ 2 ) 1 n f α(µ 2 ) 3π log (Q2 /µ 2 )

5 Fine Structure Constant Low energies: α 1 = (51) L3 High Energies: ˆα(M 2 Z) 1 = (15)! -1 (Q 2 ) small angle large angle Radiative Bhabha scattering data e + e γe + e log(-q 2 /GeV 2 )

6 Strong Interactions Both quarks and gluons contribute to loops Boson and fermion loops have opposite signs (statistics) There exists a value of momentum Λ~300 MeV where the coupling constant is ~infinite Perturbative expansion breaks down α S (Q 2 )= α(1 GeV 2 ) 0.5 α(mz 2 )=0.1184± π (33 2n f ) log(q 2 /Λ 2 )

7 !") α &* +,-!"(!"% Strong Coupling Quark confinement K"+;)4LLM 0..B)1'.+#<,(-)2-#,,.$('J. >.?)) I''(A(+#,(&' G.#H;)!"#$%&'(# Two regimes: quarks are asymptotically free at short distances (high energies), but are confined to hadrons when pulled apart (low energies)!"$!"#!e0 α (Μ ) = ± < F # #! #!!,*./012 Asymptotic freedom

8 Hadronic Structure e(k) e(k ) Goal: quantitative understanding of hadronic structure How does proton mass arise from its constituents (quarks and gluons) How does proton spin arise from its constituents? Complications: Strong interactions, quark confinement Main tool: lepton (electron, muon) scattering High energy probe, no strong interactions between target and

9 Accelerator Probes of Matter A better magnifying glass Need to look deeper into the structure of matter Spatial resolution is limited by wavelength of the probe Microscope: visible light λ 1 µm cell structure Higher energy particles: λ = 2π/p X-rays: λ nm atomic, crystal structure Charged particles: Rutherford experiment: p ~ 10 kev, λ m Discovery of quarks: p ~ 10 GeV, λ 0.1 fm LHC: p ~ 1 10 TeV, λ m quark

10 Inclusive Fixed-Target Scattering }W 2 Scatter lepton off nucleon (proton or neutron) and measure only the momentum of the scattered lepton 2 kinematic variables: scattered momentum magnitude k and angle θ Define Lorentz-invariant quantities: Q 2 q 2 = (k k ) 2 =4EE sin 2 (θ/2) ν p q M = E E W 2 (p + q) 2 = M 2 +2Mν Q 2

11 Elastic Scattering No extra particles created, target is not broken apart: W 2 =M 2 Only one free variable left: Q 2 Can express the cross section in terms of the cross section for a point-like particle times a form-factor, which says something about the structure: dσ dω = dσ F dω (Q 2 ) 2 point

12 Meaning of the Form Factor For purely Coulomb interaction (spinless target, e.g. a 12 C nucleus) the form-factor is a Fourier transform of charge distribution: F (q) = Angular dependence is trivial. Can expand in terms of small values of Q 2 =-q 2 : F (Q 2 )= d 3 rρ(r) d 3 rρ(r)e iq r 1+iq r 1 2 (q r) q2 r 2 Charge radius

13 Nucleon Form Factors For nucleons, have to take into account electric (Coulomb) and magnetic (spin-flip) interactions: dσ dω = α 2 4E 2 sin 4 θ/2 E E point-like target Electric form-factor G 2 E + τg 2 M cos 2 θ/2+2τg 2 M sin 2 θ/2 1+τ τ Q2 4M 2 Magnetic form-factor Measure form factors by measuring dependence of the cross section on the scattering angle θ

14 Elastic Form Factor Determine the average charge radius of the proton: r 2 dge (q 2 ) =6 dq 2 q 2 =0 =(0.81 fm) 2

15 Inelastic Scattering }W 2 >M 2 Since the lepton and the hadronic system do not interact after scattering, can factorize the cross section into leptonic and hadronic tensors (c.f. calculation for the eµ scattering cross section) dσ L µν (k, k )W µν (p, p ) L µν (k, k )=2 k µk ν + k νk µ (k k m 2 )g µν

16 Inelastic Scattering (cont.) Summing over spins, and assuming parity conservation, we can write the most generic form of the hadronic tensor W µµ (p, q = p p) = W 1 g µν + W 2 M 2 pµ p ν + W 4 M 2 qµ q ν + W 5 M 2 (pµ q ν + q µ p ν ) Conservation of the EM current leads to two conditions q µ W µν = q ν W µν =0 which means only two functions, W1 and W2, survive. These functions describe the internal structure of the proton and its possible excitations.

17 Kinematics Reminder: two independent Lorentz-invariant variables Q 2 q 2 = (k k ) 2 =4EE sin 2 (θ/2) ν p q M = E E W 2 (p + q) 2 = M 2 +2Mν Q 2 Inelastic cross section depends on two structure functions: d 2 σ de dω = α 2 4E 2 sin 4 θ 2 W 2 (ν,q 2 ) cos 2 θ 2 +2W 1(ν,Q 2 ) sin 2 θ 2

18 Stanford Linear Accelerator Center

19 End End Station Station A A

20 End Station A: How It All Started SLAC-R-090 (08/1968) Experiment E-4: SLAC-MIT-CIT (precursor to discovery of quarks) First high-power LH2 target at SLAC!

21 SLAC 8-GeV Spectrometer P.N. Kirk et al, PRD8, 63 (1973) SLAC-PUB

22 Inelastic Scattering Cross Section J.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972). Elastic peak Δ pπ resonance Deep Inelastic Scattering regime

23 Structure Functions in DIS J.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972). x=1/ω=0.25

24 Scaling in DIS Consider scattering off point-like target (e.g. muon) Scattering is elastic, so p =p + q E =E Q2 2M ν =E E = Q2 2M νw point 2 =δ 1 Q2 2M 2MW point 1 = 1 Q2 2Mν δ Q2 2M

25 Bjorken Scaling Define two dimensionless variables: x = For point-like targets, Q2 2p q = Q 2 2M(E E ) y = p q p k = E E E =1 E E νw point 2 =δ (1 x) 2MW point 1 =xδ (1 x) Bjorken x (scaling var) Fraction of incident energy carried away by hadronic remnants If the structure functions W1,2 do not depend on Q 2, but only on Bjorken x, it hints at scattering off point-like objects

26 J.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972). Scaling In DIS x=1/ω=0.25

27 Nobel Prize in Physics, 1990

28 Parton Model of DIS dσ i dx i q 2 2 x p q DIS cross section is an incoherent sum of elastic scattering cross sections off individual constituents (partons). We now these partons as asymptotically free quarks. Parton model should work perfectly for Q 2 where the strong coupling constant is zero. In this limit MW 1 F 1 (x) νw 2 F 2 (x)

29 Parton Distribution Functions Define parton momentum distribution f i (x) dp i dx probability to find a parton with momentum fraction x of the proton Then where F 2 (x) = i F 1 (x) = 1 i 1 0 q 2 i xf i (x) dxf i (x) =1 2x F 2(x) Callan-Gross relation

30 Proton Structure Functions sea quarks gluons Valence quarks Structure functions are a charge-weighted sum of the quark momentum distributions. Have to account for valence quarks (which are always in the proton) and sea quarks (which pop out of the gluon field)

31 Proton and Neutron Structure Functions F p 2 (x) x 2 2 = [u(x)+ū(x)] [d(x)+ 3 d(x)] [s(x)+ s(x)] 3 (ignore charm and bottom quarks, which do not contribute much, except at very high energies, e.g. LHC) Isospin symmetry implies interchange u p (x) d n (x): F n 2 (x) x 2 2 = [d(x)+ 3 d(x)] [u(x)+ū(x)] [s(x)+ s(x)] 3

32 Valence Quark Distributions u V (x) u(x) ū(x) d V (x) d(x) d(x) Expect maximum to be near x~1/3 Sum rules: dxu V (x) dxd V (x) dxs V (x) dxu(x) ū(x) =2 dxd(x) d(x) =1 dxs(x) s(x) =0

33 Valence Quark Distribution J.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972). 33

34 Sea dominates Proton vs Neutron F 2 n/f 2 p Fermi motion correction u V dominates J.I. Friedman and H.W. Kendall, Ann.Rev.Nucl.Part.Sci.22, 203 (1972). 34

35 Sea Quarks Sea quarks are produced by splitting of the gluons into quarkantiquark pairs. Gluons are in turn radiated off valence quarks Both of these processes make sea quarks very soft (small momentum, which means small x). Expect probability to find a sea quark to increase steeply at low x

36 Photon-Nucleon Cross Section

37 Unpolarized Structure Functions From PDG08 Scaling violations! sea quarks dominate valence quarks dominate 37

38 Parton Distributions

39 Perturbative QCD QCD is based on the SU(3) symmetry of color Non-abelian group: gauge carriers charged under group 3 quark colors, 8 colored gluons Predicts asymptotic freedom Gross, Wilczek, Politzer (1973) Perturbative QCD: perturbative expansion in terms of αs Log dependence on Q2 : scaling violations in DIS Confirmed by: Measuring running of αs(q 2 ) QCD sum rules Q 2 evolution of parton distributions e + e annihilation into hadrons, jets, etc.

40 Nobel Prize in Physics, 2004

41 Momentum Sum Rule dx x[u +ū + d + d + s + s] ε q Basic momentum sum rule: F p 2 (x)dx =4 9 ε u ε d 0.18 F n 2 (x)dx = 4 9 ε d ε u 0.12 ε q + ε g =1 Naively, would expect εq 1. However, DIS measures Momentum fraction carried by all quarks } ε q=0.54 at low Q 2 (ignoring strange quarks) with error of about 10%. εq=0.48 predicted in 1974 by Gross and Wilczek from perturbative QCD (PRD 9, 980 (1974)]: consequence of the color structure of SU(3). First indirect evidence for the gluon

42 Gottfried Sum Rule 1 0 F p 2 (x) F n 2 (x) x dx = [ū(x) d(x)]dx = 1 3 Valid for symmetric sea, i.e. ū(x) = d(x) Experimentally: F p 2 (x) F n 2 (x) x dx = ± (NMC, Q 2 =4 GeV 2 ) Interpretation: sea quark distributions are asymmetric, i.e. ū(x) = d(x) = s(x) Most likely consequence of small quark mass difference

43 Scaling Violations At finite Q 2, the strong coupling constant αs is not small Corrections of order O(α S) can be important For example, here are some contributions to DIS A.D. Martin Acta Physica Polonica B39, 2025 (2008) The virtual diagrams typically lead to O(1+α S (Q 2 )/π) factor in the cross section (log dependence of F 2 on Q 2 ) First term is qualitatively different. It changes the momentum fraction x of the parton. In other words, it evolves parton PDF f i (x) To find the distribution sampled by the photon, need to integrate over all y, or equivalently, all gluon momenta

44 Evolution of Parton PDFs Taking into account gluon radiation, we can write F 2 (x, Q 2 ) = qi 2 [f i (x)+δf i (x, Q 2 )] x i where δf i (x, Q 2 )= α S 2π log Q2 µ 2 1 x dy y f i(y)p qq (x/y) some arbitrary momentum scale PDF measured at Q 2 =µ 2 Can rewrite it as a differential equation: quark-quark splitting function df i (x, Q 2 ) d log Q 2 = α S 2π 1 x dy y f i(y)p qq (x/y) DGLAP equation

45 DGLAP Equations Yuri Dokshitzer Vladimir Gribov Lev Lipatov Guido Altarelli Giorgio Parisi Sov.J.Nucl.Phys. 15, 438 (1972) Nucl. Phys. B126, 298 (1977) Яд. Физ. 15, 781(1972) Sov.Phys. JETP 46, 641 (1977) ЖЭТФ 73, 1216 (1977) Nuclear Physics B126 (1977) North-Holland Publishing Company Describe evolution of the quark and gluon distribution functions with Q 2 ASYMPTOTIC FREEDOM IN PARTON LANGUAGE G. ALTARELLI * Laboratoire de Physique Th~orique de l'ecole Normale Sup~rieure **, Paris, France G. PARISI *** Institut des Hautes Etudes Scientifiques, Bures-sur- Yvette, France Received 12 April 1977 A novel derivation of the Q2 dependence of quark and gluon densities (of given helicity) as predicted by quantum chromodynamies is presented. The main body of predictions of the theory for deep-inleastic scattering on either unpolarizedor polarized targets is re-obtained by a method which only makes use of the simplest tree diagrams and is entirely phrased in parton language with no reference to the conventional operator formalism.

46 logq 2 ( Σ g DGLAP Equations q ( x, Q 2) V logq 2 ) = α s 2π (P qq q = α s 2π V ( ) Pqq 2n f P qg P gq P gg ( Σ g ) Σ q + q dy where q and P q x y P (x/y)q(y) q V q q Interpretation: quark and gluon PDFs feed off each other, increasing at larger Q 2 and smaller x. This introduces logarithmic scaling violations to DIS. DIS is sensitive to gluon distributions at O(αS). Can factorize the cross section: 1 A.D. Martin Acta Physica Polonica B39, 2025 (2008) Non-perturbative PDFs cannot be computed Hard-scattering cross sections: can be computed perturbatively

47 Parton Distributions

48 Spin (Polarized) Structure Functions Scattering of polarized beam off polarized targets Spin sum rule: Experiment: 48 (this used to be known as Spin Crisis of parton model) The rest (70%) of the spin has to come from gluons and orbital angular momentum

49 Polarized Parton Distributions !"$ '()*(+ +*,!"#,-*./++! ! %!"#!"&! %!"&!"&! %!"& !"&! &! %# &! %& 0 Figure 16.5: Distributions of x times the polarized parton distributions q(x)

50 50

51 e + e - Annihilation into Hadrons Recall from QED e e + e e µ µ + σ(e + e µ + µ )= 4πα 2 87 nb 3s s (GeV) 2 What is the difference for quarks? Charge, and the fact that they are confined (not free). If they were free, cross section would be: e e + q e q _ count 3 colors q σ(e + e q q) = 4πα 3s q2 3 How do we account for hadronization, i.e. production of hadrons from quarks? Quark-hadron duality theorem: the sum over all possible hadronic final states away from bound states is equivalent to quark cross section. Basically, it means that probability for a quark to hadronize into something is unity. 2

52 e + e - Annihilation into Hadrons e e + e q hadrons hadrons σ(e + e q q) = 4πα 3s 2 i 3q 2 i Taking into account corrections of O(αS) due to gluon radiation, R σ(e+ e q q) σ(e + e µ + µ ) =3 i q 2 i 1+ α S π Here the sum is over all quarks that can be produced at the CM energy of the collision, i.e. (2mq) 2 < s Ratio R measures the number of colors, and quark masses, i.e. fundamental parameters of QCD

53 Hadronic R Ratio (u,d,s): R 2 (u,d,s,c): R 10/3 (u,d,s,c,b): R 11/3