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Deep Inelastic Scattering (DIS)

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Deep Inelastic Scattering (DIS) Deep Inelastic Scattering (DIS) Ingo Schienbein Laboratoire de Physique UGA/LPSC Grenoble Campus Subatomique et de Cosmologie Universitaire CTEQ School on QCD and Electroweak Phenomenology Monday, October 8, 12 EAOM LPSC, Serge Kox, 20/9/2012 Mayaguez, Puerto Rico, USA 18-28 June 2018 Tuesday 19 June 18 Disclaimer This lecture has profited a lot from the following resources: • The text book by Halzen&Martin, Quarks and Leptons • The text book by Ellis, Stirling & Webber, QCD and Collider Physics • The lecture on DIS at the CTEQ school in 2012 by F. Olness • The lecture on DIS given by F. Gelis Saclay in 2006 Tuesday 19 June 18 Lecture 1 1. Kinematics of Deep Inelastic Scattering 2. Cross sections for inclusive DIS (photon exchange) 3. Longitudinal and Transverse Structure functions 4. CC and NC DIS 5. Bjorken scaling 6. The Parton Model 7. Which partons? 8. Structure functions in the parton model Tuesday 19 June 18 1. Kinematics of Deep Inelastic Scattering Tuesday 19 June 18 What is inside nucleons? ‣ Basic idea: smash a well known probe on a nucleon or nucleus in order to try to figure out what is inside ‣ Photons are well suited for that purpose because their interactions are well understood ‣ Deep inelastic scattering: collision between an electron and a nucleon or nucleus by exchange of a virtual photon Note: the virtual photon is spacelike: q 2 < 0 e- • q 2 2 2 2 • Deep: Q = -q >> MN ~1 GeV X 2≣ 2 2 p , A • Inelastic: W MX > MN ‣ Variant: collision with a neutrino, by exchange of a Z0 or W± Tuesday 19 June 18 18 Masses in deep inelastic scattering Kinematic variables Figure 2.1: Kinematics of DIS in the single exchange boson approximation. Let’s consider inclusive DIS where a sum over all • l′ brokenhadronic up such final that states the final X state is performed: consists of the scattered lepton andl a hadronic finalθ state X. Here, l and l0 are the four-momenta of the incoming and outgoing leptons, - - e (l)+N(p) → e (l’)+X(pX) q = l l0 is the four-momentum of the exchange boson (γ, Z,orW )andp theq four- − momentum of the hadron. The hadronic final state X carries the four-momentum pX . 2 2 2 2 2 •In DISOn-shell processes, conditions: an inclusive p =M sum, l =l’ over=m all hadronic final states is performed. Therefore, only the initial state momenta l and p and the final lepton momentum l0, which has to be X θ •measured,Measure are energy available and polar to describe angle of the scattered kinematics electron of DIS. (E’, ) P It is useful to introduce the following Lorentz-invariant quantities to describe the •kinematicsOther invariants of a DIS of process: the reaction: 2 2 2 Q = q = (l l0) > 0, the square of the momentum transfer, • − − − ⌫ = p q/M lab= E E , • · l − l0 0 x = Q2/(2p q)=Q2/(2M⌫) 1, the (dimensionless) Bjorken scaling variable, • · 0 y = p q/p l lab=(E E )/E 1, the inelasticity parameter, • · · l − l0 l * Heres ‘lab’=( designatesp + l)2 ,the the proton square rest frame of the p=(M,0,0,0) lepton–hadron which coincides energy with the in thelab frame center-of-mass for fixed target system,experiments • Tuesday 19 June 18 S =2p l = s M 2 m2, where M is the hadron mass and m the lepton mass, • · − − W 2 = p2 =(p + q)2, the square of the invariant mass of the hadronic final state. • X Kinematic variables • There are two independent variables to describe the kinematics of inclusive DIS (up to trivial φ dependence): l′ l θ (E’,,θ) or (x,Q2) or (x,y) or ... q • Relation between Q2, x, and y: Q2 p q X Q2 = (2p l)( )( · ) S =2p l · 2p q p l · · · =(p + l)2 p2 l2 P = Sxy = 2MExy − − • Invariant mass W of the hadronic final state X: (also called missing mass since only outgoing electron measured) W 2 M 2 =(p + q)2 = M 2 + 2p q+q2 elastic scattering: W =MN, x=1 ⌘ X N · Q2 Q2 = M 2 + Q2 = M 2 + (1 x) N x − N x − inelastic: W ≥ MN + m�, x<1 Tuesday 19 June 18 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 The ep→eX cross section as function of W Halzen&Martin, Quarks&Leptons, Fig. 8.6 - - e ki kf e Data from SLAC; The elastic peak at W=M has been reduced by a factor 8.5 q 1 Invariant 2 Elastic a resonance Inelastic ... mass W n } + 0 peak ep→ea →epé region p Elastic peak: W=M, x=1 (proton doesn’t break up: ep→ep) The missing mass of the• hadronic system can be derived from measuring electron 2 2 2 2 2 2 2 • WResonances:=(p + W=Mq) =R, ωM=1/x=1+(M+2M⌫ R+-Mq )/Q (Note that there is also a non-resonant background in the resonance region!) B. Kilminster (Uni. Zürich) – Phenomenology of Particle Physics, FS2015 13 • ‘Continuum’ or ‘inelastic region‘: W>~1.8 GeV complicated multiparticle final states resulting in a smooth distribution in W (Note there are also charmonium and bottonium resonances at W~3 and 9 GeV) Tuesday 19 June 18 Kinematic phase-space Phase Space in (ν,Q2) plane Elastic scattering Halzen&Martin, x2=1 q2 =2M⌫ W 2 = M 2 Q =(2MEx)! y ! Quarks&Leptons, Fig. 9.3 Allowed kinematic region 2ME Line of constant invariant mass ME Line of constant momentum fraction 2 2 2 2 2 2 2 2 Fixed W=MR: Q =2MExy, x=Q /(Q +MR -M ) → Q = M -MR + 2MEy Hence: fixed W curves are parallelB. Kilminster to W=M(Uni. Zürich) curve! – Phenomenology of Particle Physics, FS2015 14 Tuesday 19 June 18 Kinematic phase-space Phase Space in (ν,Q2) plane Elastic scattering Qx2=1=(2MEx)q2 =2yM⌫ W 2 = M 2 ! ! • The phase space is separated into a resonance region (RES) and the inelastic region at W~1.6 ... 1.8 GeV Allowed kinematic region 2ME (red line) • LineThe phaseof constant space is separated into a W~1.8 GeV invariantdeep and amass shallow region at Q2 ~1 GeV2 (blue horizontal line) • In global analyses of DIS data often the ME DIS cuts Q2>4 GeV2, W>3.5 GeV are employed Resonances Line of constant momentum fraction deep inelastic • The W-cut removes the large x region: ~1 GeV2 W2= M2 + Q2/x (1-x) > 3.5 GeV shallow inelastic • The Q-cut removes the smallest x: Q2 = S x y > 4 GeV2 B. Kilminster (Uni. Zürich) – Phenomenology of Particle Physics, FS2015 14 Tuesday 19 June 18 Kinematic phase-space Phase Space in (ν,Q2) plane Elastic scattering x2=1 q2 =2M⌫ W 2 = M 2 Q =(2MEx)! y ! Allowed kinematic region 2ME W~1.8 GeV Line of constant invariant mass With increasing energy E the deep inelastic region ME dominates the phase space! deep inelastic Line of constant momentum fraction ~1 GeV2 B. Kilminster (Uni. Zürich) – Phenomenology of Particle Physics, FS2015 14 Tuesday 19 June 18 Neutrino cross sections at atmospheric ν energies NeNeWithuu trtrincreasingiinnoo crcr energyoossss Ess theeectct deepiioonn inelasticss aatt aatmtm regionooss dominatespphheerriicc nn the ee nnphaseeerrgg space!iieess Paschos,JYY,PRD65(2002)033002 P. Lipari, hep-ph/0207172 Tuesday 19 June 18 Homework Problems 1. Recap that the allowed kinematic region for ep→eX is 0≤x≤1 and 0≤y≤1. Construct the phase space in the (ν,Q2)-plane yourself. [Ex. 8.11 in Halzen] 2. Show that Q2 = 2 E E‘ (1 - cos(θ)) = 4 E E‘ sin2(θ/2) neglecting the lepton mass. Here, the z-axis coincides with the incoming lepton direction and θ is the polar angle of the outgoing lepton with respect to the z-axis 3. Show that in the target rest frame x= [2 E E‘ sin2(θ/2)]/[M(E-E’)] still neglecting the lepton mass and the energies E, E’ are now in the target rest frame Tuesday 19 June 18 II. Cross section for inclusive DIS (photon exchange) Tuesday 19 June 18 The cross section for inclusive ep→eX • Let’s consider inclusive DIS where a sum over all l′ hadronic final states X is performed: l θ - - e (l)+N(p) → e (l’)+X(pX) q • The amplitude (A) is proportional to the interaction of a leptonic current (j) with a hadronic current (J): X P 1 A jµJ ⇠ q2 µ The leptonic current is well-known µ ˆµ µ • j = l0,sl0 j l, sl =¯u(l0,sl0 )γ u(l, sl) perturbatively in QED: h | | i The hadronic current is non-pert. • J µ = X, spins Jˆµ p, s and depends on the multi-particle final h | | pi state over which we sum: Tuesday 19 June 18 The cross section for inclusive ep→eX The cross section which is proportional to the amplitude squared can be factored into a leptonic and a hadronic piece: dσ A 2 L W µ⌫ ⇠ | | ⇠ µ⌫ MMA A** l l γµ l' γν µν Leptonic tensor L calculable in pert. theory Q2 Q2 µ ν µν Hadronic tensor J J W not calculabe in pert. theory pX p p Tuesday 19 June 18 The cross section for inclusive ep→eX 3 4 3 1 2 d l0 1 e µ⌫ d l0 dσ = AX spindQX 3 = 2 2 Lµ⌫ W 4⇡ 3 F h| | i (2⇡) 2E0 F (q ) (2⇡) 2E0 XX • With the Møller flux: F =4 (l p)2 l2p2 =4 (l p)2 m2M 2 2S · − · − ' p p • The phase space of the hadronic final state X with NX particles: NX 3 4 (4) d pk 4 (4) dQX =(2⇡) δ (p + q pX ) 3 = (2⇡) δ (p + q pX )dΦX − (2⇡) 2Ek − kY=1 • The amplitude with final state X: 2 2 e µ e ⌫ A = [¯u(l0)γ u(l)] X J (0) N(p) A⇤ = [¯u(l)γ u(l0)] N(p) J †(0) X X q2 h | µ | i X q2 h | ⌫ | i Tuesday 19 June 18 The cross section for inclusive ep→eX 3 4 3 1 2 d l0 1 e µ⌫ d l0 dσ = AX spindQX 3 = 2 2 Lµ⌫ W 4⇡ 3 F h| | i (2⇡) 2E0 F (q ) (2⇡) 2E0 XX AMMA** l l In the amplitude squared appears the leptonic tensor: γµ l' γν 1 µν L = u¯(l0)γ u(l)¯u(l)γ u(l0) L µ⌫ 2 µ ⌫ Q2 Q2 sl s X Xl0 1 µ ν J J µν = Tr[γ (l/ + m)γ (/l0 + m)] W 2 µ ⌫ p 2 X = 2[lµl⌫0 + l⌫ lµ0 gµ⌫ (l l0 m )] p p − · − The hadronic tensor is defined as: 4 (4) 4⇡Wµ⌫ = dΦX (2⇡) δ (p + q pX ) N(p) J⌫†(0) X X Jµ(0) N(p) − h | | ih | | i spin statesX XZ D E Note that the factor 4� is a convention.
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