Structure of Hadrons and the parton model

Transcription

1 Structure of Hadrons and the parton model Ben Kilminster University of Zürich - Physik Institut Phenomenology of Particle Physics - PPP2 Lecture: 10/3/2015

2 Topics in this lecture How do we study the structure of composite particles? Is the proton an elementary particle? If not, what do we see inside the proton? Are there only charged partons inside the proton? 2

3 A little background Proton has an anomalous magnetic moment - Stern (1933) 3x higher than expected for spin 1/2 particle Pions (1947), Kaons (1947), and a host of other hadrons had been found Similar to Periodic table of elements of Mendeleev, 1869 Hinting at some fundamental structure This was not accepted generally - most theories did not explain growing number of hadrons as combinations of fundamental entities Eight-fold way or SU(3) flavor symmetry by Gell-Man & Ne eman (1961) Essentially, a periodic table of hadrons grouped by spin 8: Spin 1, 8 : Spin 1/2, 10 Spin 3/2 Quarks had been proposed in 1964 by Gell-Mann and Zweig Provided deeper explanation of SU(3) Three quarks for Muster Mark - James Joyce, Finnegans Wake Electrical charge 1/3, 2/3 e 3 Quarks: up, down, strange But no evidence of particle with fractional charge had been found Gell-Man : A search for [quarks] would help to reassure us of the nonexistence of quarks Most people at the time considered quarks mathematical constructs 3

4 Probing a charge distribution To probe a charge distribution in a target we can scatter electrons on it and measure their angular distribution Rutherford Scattering / Coulomb scattering : Can be derived classically Assumptions : Scatterer is very massive (no recoil) Target is thin (only one scatter occurs) Particle and scatterer are small (point-like) Only Coulomb force matters For derivation, need : Rutherford scattering differential cross-section : Number of target nuclei in some area k = p / h 4

5 Probing a charge distribution To probe a charge distribution in a target we can scatter electrons on it and measure their angular distribution The measurement can be compared with the expectation for a point charge distribution e - e - ki kf d d = d d point Form factor F (q) 2 Momentum transfer: Charge distribution q = k i d d point k f Structureless target / spin-1/2 scatterer : = (Z )2 E 2 4k 4 sin 4 ( /2) 1 k 2 E 2 sin2 /2 Second term comes from interaction of electrons magnetic moment with magnetic field of target (target has B-field in rest frame of electron) As β 0, E m, this reduces to Rutherford formula 5

6 Form factor F (q 2 ) Form factor is Fourier transform of the probability density of charge 6

7 Proton is not a point Electron Scattering from Hydrogen 188 MeV (LAB) 2 Cross Section (cm /sterad) Mott Experimental Anomalous Moment McAllister, Hofstadter, Phys. Rev. 102, d = (Z )2 E 2 d point 4k 4 sin 4 1 ( /2) Mott formula accounts for spin of electron k 2 E 2 sin2 / Scattering Angle (deg) Structureless point-like target does not describe the data! Phenomenological form factor determined RMS radius of proton 0.7 fm (radius when deviations from Coulomb force occur) 7

8 LAB experiment Elastic e-p and e-he experiment in MeV electron beam Gas target (He or H) at 2000 psi Small slits : δ(e) ~ 1 MeV Cerenkov counter measures energy Lining up - CsBr crystal & telescope Multiple scattering & radiation considered Test absorber used to study effects δ(θ) ~ 2.4 8

9 Electron-proton scattering The scattering picture used so far needs to be extended for a composite object e - e - ki kf q n Problem is that the proton breaks up - so not a simple AB CD interaction like e- µ scattering p 9

10 e - -ã scattering in the lab frame k =(E,k ) 1) Matrix element k=(e,k) q=(å,q) ì Ö p=(m,0) p M 2 = e4 q 4 Lµ e L muon µ M 2 = 8e4 q 4 2M 2 E 0 E apple cos 2 2 2) Transferred momentum q 2 ' 2k k 0 ' 4EE 0 sin 2 2 q 2 2M 2 sin2 2 d d = 2 4E 2 sin 4 2! E 0 E apple cos 2 2 q 2 2M 2 sin2 2 Target recoil 10

11 Proton structure : Hadronic tensor d L e µ L µ muon! d L e µ W µ proton electron-muon scattering parametrizes the current at the proton vertex The most general form of the tensor W depends on g ãå and on the momenta p and q W µ = W 1 g µ + W 2 M 2 pµ p + W 4 M 2 qµ q + W 5 M 2 (pµ q + q µ p ) From current conservation ãj ã =0 p ã q å W ãå p =p+q W 5 = W 4 = p q q 2 W 2 2 p q q 2 W 2 + M 2 q 2 W 1 11

12 Hadronic tensor - 2 Only two independent inelastic structure functions (W1 and W2) W µ = W 1 g µ + qµ q 1 q 2 + W 2 M 2 p µ p q q 2 qµ p p q q 2 Each structure function has two independent variables q [ q 2 p q M square of transferred four-momentum energy transferred to the nucleon by the scattering electron Dimensionless variables [ x = q2 2p q = y = p q p k q2 2M 0 apple x apple 1 0 apple y apple 1 Bjorken scaling variable The invariant mass of the hadronic system in the final state is W 2 =(p + q) 2 = M 2 +2M + q 2 12

13 Electron-proton scattering The scattering picture used so far needs to be extended for a composite object The invariant mass spectrum shows the elastic peak, excited baryons followed by an inelastic smooth distribution e - e - ki kf q p... } 2 1 n Invariant mass W Elastic peak a resonance ep ea + epé 0 The missing mass of the hadronic system can be derived from measuring electron W 2 =(p + q) 2 = M 2 +2M + q 2 Inelastic region 13

14 Kinematic phase-space Elastic scattering x =1! q 2 =2M! W 2 = M 2 Allowed kinematic region Line of constant invariant mass Line of constant momentum fraction 14

15 Cross section We can now use the hadronic tensor to calculate the matrix element In the laboratory frame: (L e ) µ W µ =4EE applecos W 2(,q 2 )+sin 2 2 2W 1(,q 2 ) Using the flux factor and phase-space factor dq de 0 d = 2 4E 2 sin 4 2 Integrating on the outgoing electron energy dq d = 2 4E 2 sin 4 2 E 0 E apple cos 2 2 W 2(,q 2 )+sin 2 2 2W 1(,q 2 ) apple cos 2 2 W 2(,q 2 )+sin 2 2 2W 1(,q 2 ) 15

16 Increasing spatial resolution The key factor for understanding the proton substructure is the wavelength of the probing photon p 1 1 Fermi q 2 p 1 q 2 1 Fermi 16

17 SLAC-MIT experiment 17

18 SLAC-MIT experiment 20 Gev SPECTROMETER 1.6 Gev SPECTROMETER BEAM MONITORS TARGET POSITION 8 Gev SPECTROMETER 18

19 Bjorken Scaling In 1968 J.Bjorken proposed that in the structure functions should depend only on the ratio å/q 2 (proportional to x) in the limit q 2 and å In other words: at large Q 2 -q 2 the inelastic e-p scattering is viewed as elastic scattering of the electron on free partons within the proton Ö* Ö* quark lim Q 2!1, /Q 2 lim Q 2!1, /Q 2 fixed W 2(Q 2, ) =F 2 (x) fixed MW 1(Q 2, ) =F 1 (x) Bjorken

20 Parton distributions Partons carry a different fraction x of the proton s momentum and energy Ö* Sum over partons = X i Z Ö* dxe 2 i [ ] i xe,xp Integrate over momentum fractions E,p (1-x)p The probability that the struck parton carries a fraction x of the proton momentum is usually called parton distribution or parton density function Total probability must be equal to one: f i (x) = dp i dx X Z i dx xf i (x) =1 R.Feynman

21 10OooOOO First hints of Bjorken s scaling First data at 6 : F2 plotted as function of the scaling variable å/q 2 is roughly independent of Q 2 and v More data at different angles: F2 for a fixed î does not depend on the transferred momentum Q 2! = 2q p Q 2 = 2M Q νw ω = 4 6 O 18 O Q 2 [(GeV/c) 2 ] 7145A5 O 26 O Friedman, Kendal and Taylor

22 Structure function revisited In Feynman s parton model the structure functions are sums of the parton densities constituting the proton W 2 (,Q 2 )! F 2 (x) = X i e 2 i xf i (x) 2W 1 W 2 MW 1 (,Q 2 )! F 1 (x) = 1 2x F 2(x) The result 2xF1=F2 is known as Callan- Gross relation and is a consequence of quarks having spin 1/2 Comparing e-p with e-ã scattering cross sections (with m quark mass): = Q2 2m 2, W 1 = F 1 /M, W 2 = F 2 / [m = xm, Q 2 =2m ]! F 1 F 2 = 1 2x The parton carries a fraction x of the proton mass M 22

23 Parton looks like a reduced Proton Proton E p m Parton xe xp xm m = sqrt(x 2 E 2 - x 2 p 2 ) Scaling! 23

24 Proton/Neutron parton densities Gluons Valence quarks Quark-antiquark pairs from gluon splitting ( sea ) 1 x F ep 2 = 2 2 [u p +ū p ] [d p + 3 d p ]+ 2 1 [s p + s p ]. 3 We write the equivalent structure function for the neutron as 1 x F en 2 = [u n +ū n ]+ 2 1 [d n + 3 d n ]+ 1 3 Proton and neutron parton densities are correlated 2 [s n + s n ]. u p (x) =d n (x) u(x) d p (x) =u n (x) d(x) s p (x) =s n (x) s(x) 24

25 Constraints to parton densities We assume the three lightest quark flavours (u,d,s) occur with equal probability in the sea u s = ū s = d s = d s = s s = s s = S(x) u(x) =u v (x)+u s (x) d(x) =d v (x)+d s (x) Combining all constraints: 1 x F ep 2 = 1 9 [4u v + d v ]+ 4 3 S 1 x F 2 en = 1 9 [u v +4d v ]+ 4 3 S At small momenta (x~0) the structure function is dominated by low-momentum quark pairs constituting the sea. For x~1 the valence quarks dominate and the ratio F2 en /F2 ep becomes u v +4d v 4u v + d v

26 Ratio of structure functions low momentum fractions (sea) Partons carry most of the hadron momentum (valence) 26

27 Summary of F2 proton 27

28 Something is missing Summing over the momenta of all partons we should reconstruct the total proton momentum: u Z 1 0 Z 1 0 dx x(u +ū + d + d + s + s) =1 p g p =1 Neglecting the small fraction carried by the strange quarks we have and using the results of experimental data dx x(u +ū) g ' 1 u d = = 0.46 g Experimental data indicate that about 50% of the proton momentum is carried by neutral partons, not by quarks! 28

29 Gluons and the parton model Ö* quark gluon Ö* i xe,xp + Initial Gluon emission E,p O(~em) (1-x)p O(~s ~em) Ö* quark antiquark + crossed diagram 29

30 Gluon emission: contribution to F2 Proton Ö* (z=x/y) p pi=yp zpi=xp Ö* i 2 F 2 (x, Q 2 ) x = X i e 2 i Z 1 x dy y f i(y) (1 x/y) = X i e 2 i f i (x) Scaling: no Q2 dependence Ö* + Initial Gluon emission quark gluon 2 F q!qg 2 (x, Q 2 ) x P qq (z) = 4 3 = X i 1+z 2 1 z e 2 i Z 1 x apple dy y f s i(y) 2 P qq(x/y) log Q2 µ 2 Cut-off for soft gluons Scaling violation! Splitting function: probability of a quark to emit a gluon and reduce momentum by a fraction z N.B.: divergent for soft gluons (z 1) 30

31 F2 results from HERA Ö* F p 2+c i (x) x= (i=24) x= x= x= x= x= x= NMC BCDMS SLAC H1 (i=20) x= H1 96 Preliminary (ISR) H1 97 Preliminary (low Q 2 ) H Preliminary (high Q 2 ) Scaling violation (logarithmic dependence) x< x= x= x=0.002 x= NLO QCD Fit H1 Preliminary c i (x)= 0.6 (i(x)-0.4) 8 x=0.005 x= x=0.013 (i=10) x=0.02 x=0.032 Ö* 4 x=0.05 x=0.08 x=0.13 i 2 0 x=0.18 x=0.25 x=0.40 x= (i=1) Q 2 /GeV 2 Approximate scaling, x~

32 Parton distributions from 0.2 HERA 0.05) xs ( xg ( 0.05) x xf 1 H1 and ZEUS 2 2 Q = 1.9 GeV xf 1 H1 and ZEUS 2 2 Q = 10 GeV 0.8 HERAPDF1.0 exp. uncert. 0.8 HERAPDF1.0 exp. uncert. 0.6 model uncert. parametrization uncert. xu v 0.6 model uncert. parametrization uncert. xg ( 0.05) xu v 0.4 xd v 0.4 xs ( 0.05) xd v 0.2 xs ( 0.05) xg ( 0.05) x x xf H1 and ZEUS HERAPDF1.0 exp. uncert. 2 Q Figure218: The parton distribution functions from HERAPDF1.0, xu v, xd v, xs = 2x(Ū = 10 GeV at Q 2 = 1.9GeV 2 (top) and Q 2 = 10 GeV 2 (bottom). The gluon and sea distributions a down by a factor 20. The experimental, model and parametrisation uncertainties ar separately. model uncert. 0.6 parametrization uncert. xu B. v Kilminster (Uni. Zürich) Phenomenology of Particle Physics, FS

33 Experimental techniques 33

34 HERA accelerator complex 34

35 Kinematic region Larger momentum transfers (large electron scattering angles) Larger sensitivity to sea quarks (gluons) 35

36 HERA experiments ZEUS Experiment 36

37 Typical DIS event S e Electromagnetic calorimeter Energy (GeV) Quark jet Scattered electron E Electron (30 GeV) Scattered electron ìe Quark jet Proton (820 GeV) Phi (degrees) Theta (degrees) Hadronic calorimeter 37

38 Events with two jets Muon Jet 1 Jet 1 Muon Jet 2 Jet 2 Transverse view Longitudinal view e + p e + q + anti-q + proton-remnant muon from decay of heavy quark? 38

39 Kinematic reconstruction To fully characterize a deep inelastic scattering event kinematics both Q 2 and x (or y) have to be measured Q 2 = q 2 = (k k ) 2, e(k) +p(p ) e(k )+X x = Q2 2P q, of the scatter y = Q 2 /(sx) collision (s = Both variables can be measured e.g. detecting only the scattered electron y e = 1 E e (1 cos θ e ), 2E e Q 2 e = 2E e E e(1 + cos θ e ). More precise methods combine the measurements (energy and polar angle) of both electron and hadronic system 39

40 Measured kinematic range ZEUS 1994 Q 2 (GeV 2 ) a) b) x Figure 7: a)the distribution of the events in the (x,q 2 )plane.b)the(x,q 2 )-bi structure function determination. Also indicated are lines ofconstantyando The γ H B. values Kilminster for this (Uni. figure Zürich) are Phenomenology calculated directly of Particle fromphysics, x and QFS

41 References F.Halzen, A.Martin, Quarks and Leptons, Wiley, Sections 8/9/10 C.Amsler, Kern- und Teilchenphysik, UTB Thanks to Prof. Vince Chiochia for slides 41