Particle Physics WS 2012/13 (20.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101

Electron Proton Scattering Electron Proton Scattering are probing the structure of the proton! In e-p → e-p scattering, the nature of the IA strongly depends on the wave length of the virtual particle, thus on the momentum transfer |q| At very low |q| values λ >> rp: scattering is equivalent to that from „point-like“ spin-less objects elastic scattering At low |q| values λ ~ rp: Scattering is equivalent to that from an extended charged object At high |q| values λ < rp: The wave length is sufficiently short to resolve sub-structure → scattering from constituent quarks inelastic scattering At very high |q| values λ << rp the proton appears to be sea of quarks and gluons

Elastic vs. Inelastic Scattering p3 e p3 e e p1 p1 p4 p2 p2 invariant mass W, p4 p p p inelastic scattering exitation of proton deep inelastic scattering proton gets split up in many pieces elastic scattering need a single variable to describe scatter process for given p1,p2 e.g. scatter angle ϴ, E3, q2, … need two independent variables to describe scatter process for given p1,p2

Inelastic Scattering e p3 q2 <0 for t-channel process, e thus define Q2 = -q2 p1 p2 invariant mass W, p4 p work in lab-frame: p2 = (M,0,0,0) 4 momentum transfer: q = p1 – p3 energy loss: ν = E1-E3= qp2/M fractional energy loss: y = (E1-E3)/E1 = qp2/p1p2 invariant mass: W2 = (q+p2)2 = q2 + 2qp2 + p22 = q2 + 2qp2 + M2 due to baryon number conservation W2 ≥ M2 (proton is lightest baryon) = 0 elastic scattering > 0 inelastic scattering W2-M2 = q2 +2qp2 = 1 elastic scattering 0 < x < 1 inelastic scattering Bjorken variable x: x = −𝑞2 2𝑞𝑝2 = 𝑄2 2𝑞𝑝2 =

Inelastic Scattering Need to distinguish carefully between lab frame and CMS and LI form of variables! Lab frame: proton at rest p2 = (M,0,0,0) LI form energy loss of incoming particle Bjorken x fraction energy loss of incoming particle 4 momentum transfer square ν = E1-E3 ν = qp2/M x = −𝑞2 2𝑀ν x = 𝑄2 2𝑞𝑝2 y = (E1-E3)/E1 y = qp2/p1p2 q = (p1-p3)2 q = (p1-p3)2

Inelastic Scattering Note: most DIS (deep inelastic scattering) results from HERA collider which collides electron beams on proton beams (proton not at rest before collisions; lab frame ≠ CMS) example of inelastic ep scattering in H1 detector more about the HERA collider and its experiments later in the lecture ….

Inelastic Scattering e p3 e p1 p2 invariant mass W, p4 p Produce excited states e.g. Δ+ (1232) deep inelastic scattering (DIS); proton splits up in many final state particles

Double differential x-section for inelastic scattering Tow variables used to describe inelastic scattering process → need to quote double differential cross-section, e.g. 𝑑2σ 𝑑Ω𝑑𝐸3 (in a given lab frame) or (LI) 𝑑2σ 𝑑𝑥𝑑𝑄2 observation: inelastic cross-section drops off significantly less than elastic one as function of q2

Q2 dependence of (in)elastic ep cross-section Rosenbluth formular for elastic scattering process (in lab frame: p2 = (M,0,0,0)): 𝐺𝐸2 𝑞2 + 𝜏𝐺𝑀2 𝑞2 1+τ +2 τ 𝐺𝑀2 𝑞2 𝑡𝑎𝑛2 ϴ/2 𝑑σ 𝑑Ω = α2 4𝐸12𝑠𝑖𝑛4ϴ/2 𝐸3 𝐸1 τ = 𝑄2 4𝑀2 Inelastic scattering 𝑑σ 𝑑Ω Mott Saw last time from fit to data: G(q2) = 1 1+𝑄2 /0.71 𝐺𝑒𝑉2 2 ~ Q-4 elastic scattering for Q2 >> M (τ >> 1): 𝑑σ 𝑑Ω / ~ (2τ tan2ϴ/2) GM2 ~ Q-6 𝑑σ 𝑑Ω Mott Inelastic cross-section almost independent of q2

(In)elastic scattering x-section formular Rosenbluth formular [in lab frame: p2 = (M,0,0,0)]: 𝑑σ 𝑑Ω = α2 4𝐸12𝑠𝑖𝑛4ϴ/2 𝐸3 𝐸1 𝐺𝐸2 𝑞2 + 𝜏𝐺𝑀2 𝑞2 1+τ 𝑐𝑜𝑠2 ϴ 2 +2 τ 𝐺𝑀2(𝑞2)𝑠𝑖𝑛2 ϴ/2 f2(Q2) f1(Q2) Rosenbluth formular in LI form: 𝑑σ 𝑑𝑄2 = 4π α2 𝑄4 [ 𝒇𝟐 𝑸𝟐 1−𝑦− 𝑀2𝑦2 𝑄2 + 1 2 𝑦2 𝒇𝟏(𝑸𝟐)] elastic cross section inelastic cross section Inelastic cross section [in lab frame: p2 = (M,0,0,0)]: 𝑑σ 𝑑𝐸3𝑑Ω = α2 4 𝐸12𝑠𝑖𝑛4ϴ/2 [ 𝑭𝟐 𝒙,𝑸𝟐 ν 𝑐𝑜𝑠2ϴ/2 + 2 𝑀 𝑭𝟏(𝒙,𝑸𝟐)] Inelastic cross section in LI form: 𝑑σ 𝑑𝑥𝑑𝑄2 = 4π α2 𝑄4 [ 𝑭𝟐 𝒙,𝑸𝟐 𝑥 1−𝑦− 𝑀2𝑦2 𝑄2 + 1 2 𝑦2 𝑭𝟏(𝒙,𝑸𝟐)] Form factors have been replaced by structure functions, which depend on two paramters, which cannot anymore interpreted as Fourier transforms of electric charge and magnetic moment distribution.

Bjorken Scaling Hypothesis (1967) „if scattering is caused by point-like constituents (partons) structure functions for fixed x must be independent of Q2“ Experimental observation: F1(x,Q2) → F1(x) F2(x,Q2) → F2(x) defined as Bjorken saling! Proton consists out of point-like partons!