Review of hadron-hadron interactions

Transcription

1 Chapter 10

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3 Deep inelastic scattering between leptons and nucleons Confirm quarks are more than mathematical objects High momentum transfer from leptons to hadron constituents QCD predicts small coupling between quarks and gluons at large momenta High p T processes computable with perturbation methods Review of hadron-hadron interactions Large cross-sections Low p T processes for most collisions Involvement of hadrons as whole, not only constituents No use of perturbation theory for low p T processes Various phenomenological models 28/04/16 F. Ould-Saada 3

4 DIS Information about structure of nucleon l+nàl +X Nucleon made of partons Valence quarks Sea of quarks and antiquarks gluons DIS High momentum transfer Q 2 vs short distances GeV 2 (HERA) à m

5 Stage 1 almost elastic l-q collision q carries fraction x of proton momentum P virtual boson absorbed by quark structure function F(x) describes momentum distribution of constituents within proton νn scattering depends on 3 SFs related to 3 helicity states of W bosons en depends on 2 SFs related to 2 helicity states of γ Stage 2 Parton fragmentation into two jets of hadrons: Hadronisation dresses naked quarks to form final state hadrons 1 st jet stems from parton interacting with lepton high-pt at large angle 2 nd spectator jet (or target jet) comes from spectator partons low-pt in direction of incident parton Fragmentation function D(z,Q 2 )) probability that a given hadron carries a fraction z of interacting parton energy

6 Proton P 0 = M! 0 ; electron P = E! p ; P' = E! ' p' Photon q =! ν = E E ' q = p! p'! ; Hadronic system W = E ' 0! p' 0 4 momentum transfer t = q 2 = ( P P' ) 2 = ( E E ') 2 ( p! p'! ) 2 high energy and small angles = 2m e 2 2EE +2 pp cosθ 4EE 'sin 2 (θ / 2) = Q 2 p p, sinθ θ t = q 2 p 2 θ 2 Laboratory system 4-momentum transfer to proton t = q 2 = ( M E ' 0 ) 2 0! p'! 0 centre of mass energy ( ) 2 = 2M 2 2ME ' 0 = 2MT p s = ( P + P 0 ) 2 = P 2 + P 2 + 2PP = m e + M 2 + 2EM M 2 + 2EM condition for elastic scattering P 0 q = Mν = M(E E ') Scattering Elastic: q 2 < 0 à space-like Annihilation process: q 2 > 0 à time-like Work out details 07/05/16 F. Ould-Saada 6

7 Elastic scattering of spin-0 point-like electron (m e,-e) by a fixed point-like nucleus (M,Ze) à Rutherford Electron spin through Dirac equation no proton spin à Mott Backscattering (θ=π) forbidden EM conserves helicity à anisotropy à angular dependency Recoil of target proton with finite mass à E /E term Màinfinity è NSàM Proton with spin At HE à magnetic moment of target in addition to electric charge Spatial extension of nucleus à form factor F à Experimental cross section " dσ % $ ' M " dσ % $ ' " = dσ % $ ' R ) " 1 β 2 sin 2 θ %, " + $ '. dσ % $ ' * # 2 &- R " cos 2 θ % $ ' # 2 & 28/04/16 F. Ould-Saada expt 7 NS " dσ % " $ ' = dσ % $ ' " dσ % $ ' " = dσ % $ ' " dσ % " $ ' = dσ % $ ' NS, -. " dσ % $ ' NS M ( * ) 1+ q2 R 1 " 1+ (2E 0 / M )sin 2 θ % $ ' # 2 & 1+ q2 ( 2(1+κ) 2 4M 2 ) * " θ % / + tan2 $ '+κ 2 0 # 2 & 1 µ N = (1+κ)µ 0 µ 0 = e! / 2M (Dirac point-like) = " = dσ % $ ' F( q)! 2 2 Z 2 α 2 " 4E 2 θ % 0 sin4 $ ' # 2 & " θ % + 2M 2 tan2 $ '- # 2 &,

8 Spatial form factor to remove point like approximation describe electric charge spatial distribution F(q) is Fourier transform of f(r) f(r) in momentum space ρ(r) = e f (r) = dq dv F(! q) 1 Ze ; f (r)dv =1 ρ(r)dv = e e i! q!x f (r)d 3 x

9 Form Factors, Electric charge distribution and Magnetisation " dσ % " $ ' = dσ % $ ' NS, F 1 (q 2 )+ q2 ( 2(F 4M 2 ) 1 (q 2 )+κf 2 (q 2 )) 2 * " θ % / - + tan2 $ '+κ 2 F 2 2 (q 2 ) 0. # 2 & 1 Low q 2 à F 1 p (0) = F 2 p (0) = F 2 n (0) =1 ; F 1 n (0) = 0 More convenient, Linear combinations à Electric and Magnetic Form Factors (normalised) distributions of electric charge and magnetic dipole moment G p,n E (q 2 ) = F p,n 1 (q 2 ) q2 4M κf p,n 2 2 (q 2 ) G p,n M (q 2 ) = F p,n 1 (q 2 )+κf p,n 2 (q 2 ) G p E (0) =1; G p M (0) = +2.79; G n E (0) = 0; G n M (0) = 1.91 Rosenbluth formula for ep scattering No interference between electric and magnetic form factors " dσ % $ ' Ros " = dσ % $ ' NS ( * ) * + G 2 E + q2 4M G 2 2 M 1+ q2 4M 2, + q2 2M G " 2 2 M tan 2 θ %* $ '- # 2 &*. Read more in book + study slides 28/04/16 F. Ould-Saada 9

10 Summary ep scattering From Rutherford to Rosenbluth

11 Parameterisation: linear dependency for constant q 2 " dσ % $ ' Ros " dσ % / $ ' NS " = A(q 2 )+ B(q 2 )tan 2 θ % $ ' # 2 & Experimental proof that Scattering mediated by single photon exchange Figure next slide Measurement of proton and neutron Form Factors Parameterisation - Scaling law Parameterisation - Dipole formula! G(q 2 ) = G p E (q 2 ) = G p M(q 2 ) = G n M(q 2 ) # " µ p µ n # $ G n E (q 2 ) = 0 % ( ' * G(q 2 1 ) = ' * '! 1+ q2 "(GeV / c) 2 # ' $ * * & 0.71 ) 2 28/04/16 F. Ould-Saada 11

12 Electric and magnetic form factors of the proton and magnetic form factor of the neutron. f(r)- Fourier transform of G HE: elastic FF very small, inelastic scattering much more likely! f (r) dipole = 3.06 e 4.25r small momentum transfer $ G p E (q 2 1 ) f (0)& 1 %& 6q 2 r 2 ' ) () ; r2 = 0.81fm 28/04/16 F. Ould-Saada 12

13 W: invariant mass final-state hadrons > M Q 2 : squared energymomentum transfer Proton P 0 = M! 0 ; Photon q =! ν = E E ' q = p! p'! W 2 = ( P 0 + q) 2 = M 2 + 2P 0 q + q 2 = M 2 + 2M ν Q 2 > M 2 2 independent variables Q 2, ν OR x, ν Compare elastic (ES) e-parton (m) and inelastic (IS) scattering e-proton (M) W1, W2: structure functions 2Mν W 2 +Q 2 M 2 ; x Q2 2Mν IS : 2Mν > Q 2 ; x <1 ES : W 2 = M 2 ; 2Mν = Q 2 ; x =1 DIS : Q 2 >> M 2 ; ν = E E ' >> M! d 2 σ $ # & " dq 2 dν % IS d 2 σ dq 2 dν ES = 4πα 2 Q 4 = 4πα 2 Q 4 E ' E cos2 E ' E cos2! θ $ '! # & W 2 (Q 2,ν)+ 2W 1 (Q 2,ν)tan 2 θ $ * ) # &, " 2 %( " 2 % + θ 1+ 2 Q2 θ 2 4m 2 tan2 δ ν Q2 2 2m 28/04/16 F. Ould-Saada 13

14 Comparing e-p and e-p à Condition on structure functions (SFs): Bjorken, 1967 W 2 (Q 2,ν) 1 ν δ ν Q2 ; W 1 (Q 2,ν) 2m In DIS, SFs depend on dimensionless variables 2m Q2 4m 2 ν δ ν Q2 have only very weak dependence on Q 2, on ν and on nucleon size, as in ES case Bjorken scaling law Q 2,ν x Q2 2Mν finite Q 2 = 2mν (elastic scattering) x = m / M x: fraction of nucleon mass carried by parton interacting with lepton è structure function only depends on x, not on Q 2 and ν νw 2 (x,q 2 Q ) 2,ν F 2 (x) ; MW 1 (Q 2 Q,ν) 2,ν F 1 (x) 07/05/16 F. Ould-Saada 14

15 SLAC, 1968 Data Linear accelerator, 3km E e =20 GeV on H and D targets Measure E and θ E ',θ Q 2,ν,W d 2 σ dωde ' = f (W ) Elastic peak W=M (removed) W~ GeV: excitation of baryonic resonances (Δ(1230MeV) W>1.8GeV: continuum with no resonances At fixed W Cross section decreases rapidly with increasing Q 2 (Form factor figure slide 12) Inelastic Scattering ~constant for Q 2 >1 à Bjorken scaling

16 e-p interaction ratio between elastic (solid curve) and inelastic crosssections (points) and Mott cross-section (on a point-like and spin-less target) as a function Q 2 IS increasingly more important than ES: Q 2 >1 At fixed W IS ~constant for Q 2 >1 Bjorken scaling

17 At fixed x, structure functions F 1,2 have very weak dependence on Q 2 as shown by data for Q 2 from 2 to 18 GeV 2 à Scaling behaviour As Fourier transform of a spherically symmetric pointlike distribution is a constant à proton has a sub-structure of point-like charge constituents More in chap 14 28/04/16 F. Ould-Saada 17

18 (a) λ γ >> d N : the photon "sees" a point-like nucleon (b) λ γ ~ d N : the cross section depends on q 2 through a form factor, F(q 2 /M 2 N ), corresponding to the charge density of the nucleon. To keep the form factor dimensionless, a mass scale is necessary, taken to be the mass of the nucleon, M N. (c) λ γ << d N : the photon interacts directly with a parton, independently on the rest of the nucleon. The cross section becomes simpler. 28/04/16 F. Ould-Saada 18

19 e-scattering on spin ½ particle m=xm e-p with structure " W 1 F 1 M ; W 2 F % 2 # $ ν & ' ν M Q 2 = 2mν m = Q2 2ν = xm 2xF 1 = F 2 Callan-Gross relation " dσ % " $ ' = dσ % $ ' NS d 2 σ dωde' = " dσ % $ ' F 1 F 2 = Q2 4m 2 ) + * +, + ( " 1+ 2τ tan 2 θ % +. * $ '- ; τ = Q2 ) # 2 &, 4m 2 0 / ( " W 2 (Q 2,ν)+ 2W 1 (Q 2,ν)tan 2 θ % + 0 * $ '- ) # 2 &, 1 0 NS 2W 1 W 2 = 2τ F 1 (x,q 2 )=0 spin = 0 2xF 1 (x,q 2 ) = F 2 (x,q 2 ) spin = /04/16 F. Ould-Saada 19

20 Work out details in book F 2 (x) = 2xF 1 (x) x = Q2 2Mν dx dν = x ν d dx = x d ν dν d 2 σ dq 2 dx = x d 2 σ ν dq 2 dν = 4πα 2 E ' 1 " θ %( " Q 4 E x cos2 $ ' νw 2 (Q 2,ν)+ 2νW 1 (Q 2,ν)tan 2 θ % + * $ '- # 2 &) # 2 &, = 4πα 2 Q 4 d 2 σ dq 2 dx = 4πα 2 Q 4 d 2 σ dq 2 dx = 4πα 2 Q 4 E ' E E ' E E ' E 1 " θ %( x cos2 $ ' F 2 (x) + 2 ν # 2 & M F " 1(x)tan 2 θ % + * $ '- ) # 2 &, 1 x cos2 F 2 (x) x! θ $ ' # &) F 2 (x) + 2xF 1 (x) " 2 %( " cos 2 θ %( $ '* 1+ # 2 &) Q 2 4M 2 x 2 tan2 Q2 4M 2 x 2 tan2 " θ % + $ '- # 2 &,! θ $ * # &, " 2 % + Infinite (nucleon) momentum reference frame: p >>M Neglect masses and transverse momenta Nucleon consists of point-like particles, partons 4-momentum P 0 of nucleon distributed among partons IS with (Q 2,ν) result of ES on parton of 4- p= xp 0 4-momentum P 0 of nucleon distributed among partons Structure Function F 2 (x)/x is distribution function of partons in nucleon

21 Interpretation of scaling simplest in a reference frame where target is moving with very high velocity (Infinite RF) p T and rest masses of constituents (partons) may be neglected Parton Model (partons=quarks+gluons) Target nucleon= stream of partons with 4-momentum xp x= fraction of nucleon 3-momentum carried by parton in infinite RF If one parton (mass: m) scattered by photon (4-momentum: q) P = ( p, p! ) ; p = p! (M 0) ( ) = m 2 c 2 0 ( xp + q) 2 = x 2 P 2 + 2xP q + q 2 If x 2 P 2 = x 2 M 2 c 2 << Q 2 x = q2 2P q = Q2 2Mν Invariant P.q evaluated in lab 28/04/16 F. Ould-Saada 21

22 Parton 4-momentum after the interaction ( q + xp 0 ) 2 = Q!## 2 + " 2xM ## $ ν + x2 M 2 = m 2 << W 2 0 x= fraction of nucleon 3-momentum carried by parton in infinite RF Equivalent to parton of mass m stationary in lab System, with elastic relation x = Q2 2Mν Q Q = 2mν ; if Q >> M c x = = 2Mν x= fraction of nucleon mass carried by struck parton m M To identify constituent partons with quarks, need to know spin and electric charge

23 q f (x) : quark densities / momentum distribution of quark of flavour f q f (x) dx: probability of finding in a nucleon a quark of flavour f with momentum fraction in interval x to x+dx Nucleon= valence quarks (carry observed quantum numbers)+ sea quarks (q-qbar pairs from radiated gluons) 2 F 2 (x) = x e f [ q f (x) + q f (x)] F en (x) 2 2 f = e f x # F lp 2 (x) = x 1 ( 9 d p + d p ) + 4 ( 9 u p + u p ) + 1 f & ( 9 sp + s p ) $ % ' ( F ln 2 2 (x) = xq f (x)e f # F ln 2 (x) = x 1 9 d n + d n $ % Isospin symmetry :u d n p ( ) + 4 ( 9 un + u n ) + 1 ( 9 sn + 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) Isoscalartarget : N p = N n [ ] = 5 F ln 2 (x) = 1 2 F lp 2 (x) + F ln 2 (x) 18 [ ] x q(x) + q (x) 28/04/16 F. Ould-Saada q =u,d & ' ( [ ] x s(x) + s (x) f =1,N q

24 F 2 (x)= x e f 2 F 2 ln (x) = 5 18 f "# q f (x)+ q f (x) $ % [ ] x q(x)+ q(x) q=u,d [ ] F en x s(x)+ s (x) 1 2(x) 0.14 x[ q(x)+ q(x) ] dx (0.50 ± 0.05) 0 5 F 2 ν p (x)= x [ q(x)+ q(x) ] q=u,d ν F N 2 (x) 18 5 F ln 2 (x) % Data 18 5 F ln ν ( 2 (x) F N 2 (x) & ' ) * Parton Charges:+ 2 3 and /04/16 F. Ould-Saada 24

25 Weak cross section (G F instead of α QED ) 3 structure functions F 1,F 2,F 3, F 3 takes into account Parity violation in WI x = Q2 2Mν y = ν E = Q2 2MxE y = E µ (1 cosθ) Mx dq 2 = 2MExdy d 2 σ dq 2 dx = 4πα 2 Q 4 E ' E 1 x cos2! θ $ ' # &) F 2 (x) + 2xF 1 (x) " 2 %( Q 2 4M 2 x 2 tan2! θ $ * # &, " 2 % + d 2 σ ν,ν dxdy = G 2 FME ν 1 y Mxy π 2E ν F (x)+ xy 2 F (x) xy 1 y F 3 (x) E ν >> M Mxy 2E ν 0 d 2 σ ν,ν dxdy = σ 0 2 E ν ( F 2 (x) xf 3 (x))(1 y) 2 + ( F 2 (x)± xf 3 (x)) d 2 σ dxdy = A(x)+ (1 y)2 + B(x) d 2 σ dxdy = 2MEx d 2 σ σ 0 = dxdq 2 G 2 FM π = cm 2 GeV 1 Work out details in book

26 v v e e e v e e e v e e Purely leptonic processes J=0 à isotropy J=1 à angular dependency Scattering through 180 o forbidden amplitude with factor (1+cosθ) dσ dω (ν ee ) = G 2 Fs 4π 2 dσ dω (ν e e ) G 2 Fs $ 1+ cosθ ' & ) 4π 2 % 2 ( 2 2 HE : E CM dy dω = 1 4π 2m e c 2 E ν σ tot (ν e e ) = 1 3 σ tot (ν e e ) dσ dy = G F π 2 s dσ dy = G F π 2 s ( 1 y ) 2

27 Compare νe and νn J=0 à isotropy J=1 à angular dependency Valence and sea quarks in nucleon! v µ d µ u J = 0 v µ d µ + u J = 0 v µ u µ d J =1 v µ u µ + d J =1 (25) dσ νe dy dσ ν e dy = G 2 Fs π = G 2 Fs π ( 1 y ) 2 d 2 σ ν N dx dy = 2G 2 FME ν " xq(x)+ xq(x) 1 y π # d 2 σ ν N dx dy = 2G 2 FME ν " xq(x) 1 y π # ( ) 2 ( ) 2 + xq(x) d 2 σ ν,ν dxdy = σ 0 2 E " ν #( F 2 (x) xf 3 (x))(1 y) 2 + ( F 2 (x)± xf 3 (x)) $ % $ % $ % 1 2 Fν (x) 2 xfν (x) Fν 2(x)+ xf 3(x) ν ( ) = 2xq(x) ( ) = 2xq(x) Fν 2 Work out details in book " $ # $ % $ ( ) ( ) ' $ (x) = 2x q(x)+ q(x) ( )$ xf 3(x) ν = 2x q(x) q(x)

28 Possible interactions u,d, and s F2 depends on valence and sea quarks F3 depends on valence quarks only Isoscalar target F ν 2 (x) = 2x q(x)+ q(x) ( ) ( ) F ν p 2 (x) = 2x d(x)+ u(x) xf ν p 3 (x) = 2x d(x) u(x) ( ) ( ) F ν p 2 (x) = 2x u(x)+ d (x) xf ν p 3 (x) = 2x u(x) d (x) ( ) ( ) F ν N 2 (x) = F ν N 2 (x) = x u(x)+ u(x)+ d(x)+ d (x)+ s(x)+ s (x) xf ν N N 3 (x) = Fν 3 ( ) ( ) xf ν 3 (x) = 2x q(x) q(x) (x) = x u(x)+ d(x)+ s(x) u(x) d (x) s (x) ( ) = x u v (x)+ d v (x) Measurement of F2 & F3 è extract distribution functions

29 Number of valence quarks in the nucleon: 3 n v = 1 xf ν N (x) 1 3 dx= [ u 0 v (x)+ d v (x)] dx 2.8± x 50% of nucleon momentum carried by non EW particles gluons 1 ν F N 2 (x) dx F 2 ln (x)dx = 0.49 ± 0.06 No strangeness in nucleon n s = 1 1# x[ s(x)+ s (x)] dx = 9F ln F & 2ν N (x) 0$ % ' ( dx = 0.05± 0.18 Sea quarks carry no proton momentum n sea = 1 1 x! " u(x)+ d (x)# ν $ dx = F N ν! " 2 + xf N 3 (x)# $ dx = 0.02 ±

30 Integral of PDFs Extraction from combination of cross sections In proton, u & d quarks have largest probability density at large x residual memory of x~1/3 for valence quarks reduction due to gluon emission Gluons and sea anti-quarks have large probability at low x. gluons carry ~ 50% of proton momentum

31 1 σ 0 E ν 1 σ 0 E ν d 2 σ ν N dx dy = # xq(x)+ xq(x) ( 1 y) 2 % dx dy $ & dxdy = / 3 = 0.32 d 2 σ ν N dx dy = # xq(x) ( 1 y) 2 + xq(x) % dx dy $ & dxdy = 0.3 / = 0.16 Make use of experimental values of previous slide to estimate ratio of cross sections σ E σ (ν N) σ (ν N) 2

32 Q 2 dependence of Structure Functions Free quarks Gluon exchange Gluon emission

33 Scaling is approximately correct but not exact Deviations from scaling due to QCD corrections to Quark Parton Model Quark can radiate gluon Gluon can split into qqbar or gg Analysis of data with QCD corrections à α s and Λ 28/04/16 F. Ould-Saada 33

34 QCD nicely explains data

35 Cross-section measurements in pp and ppbar collisions data include high energy cosmic ray interactions Transverse momentum High p t : rapid increase with s, low cross sections, perturbative methods (QCD) Low p t : ln(s) increase, high cross sections, nonperturbative methods (α s too large) Phenomenological models based on many-body QCD pomeron, a pseudo-particle with the quantum numbers of the vacuum

36 Schematic representation of the status of two hadrons before collision for (a) peripheral and (b) central collisions. Due to relativistic effects, the two hadrons contract along the direction of motion. (c) Sketch of the increase with energy of the hadron size and opacity proton darker (relativistic contraction not shown) Opacity:

37 Example of measurements entering model building

38 HE nucleus-nucleus (A-A) collisions studied since end of 1980s using ion beams 8 O, 16 S, 82 Pb at 15GeV/nucleon at Brookhaven at 200 (158 for 82 Pb) GeV/nucleon at the CERN SPS Results show that the A-A collision can be explained as a series of hadron nucleus collisions (superposition model). Only few nucleons of the projectile (or target) interact inelastically, producing a heavy nuclear fragment, some light fragments and several spectator nucleons. About 20% of these nucleons reinteracts inside the target nucleus. The hadron nucleus interaction is thus considered as a superposition of hadronhadron interactions. One of the most used models in Monte Carlo simulations of A-A interactions: Glauber model of multiple nuclear scattering: a hadron crossing a nucleus can undergo multiple interactions in each interaction, hadrons are produced which may in turn interact within the same nucleus, giving rise to a intra-nuclear cascade. A-A interactions are important in the study of cascades induced by HE cosmic rays (p, He and heavier nuclei) with nuclei in the upper atmosphere

39 HE A-A collisions used to search for possible state of matter denoted - QGP The very high temperatures and densities achieved in the collisions should, for a very short time, allow quarks and gluons to exist in a free state, i.e., no longer confined in hadrons, in a kind of soup or plasma. State of matter might have existed around 10-6 s after Big Bang. Predicted by QCD The Brookhaven Relativistic Heavy Ion Collider (RHIC), 2000, devoted to these studies gold ions which colliding at GeV/nucleon. RHICmay have indications for a QGP behaving more like a liquid than a gas. At LHC, ALICE (as well as ATLAS and CMS) studies in detail 82 Pb ion interactions up to 5.5 TeV/nucleon. The study of the properties of QGP can help to understand the origin of particles (p, n) may also have important implications for our understanding of cosmology.

40 10.9 Read the Slides presented at CERN on heavy ion collisions and ALICE The LHC and the Search for the Higgs Read this section in the book we already discussed the Higgs in the previous chapter Read the Slides presented at CERN on ATLAS, Higgs and other searches In particular it is important to know how the Higgs is produced in e+e- and hadron colliders, how it decays depending of its mass and how it is discovered! Solve problems 10.2, , 10.7