Virtual particles - the ultimate source of any force

Transcription

1 Virtual particles - the ultimate source of any force Lecture notes Jan Rak Jyväskylä University, HIP, Finland October 23, 204 Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, 204 / 8

2 Special Relativity - time/space like 4-vectors The Lorentz-invariant square of a 4-vector x 2 = x µ x ν is said to be time-like, M 2 > 0 light-like, M 2 = 0 space-like, M 2 < 0. - x lightcone x ( x ) - (x ) time-like light-like (sinh(y), cosh(y)) y x + = e x + x + ξ =y M space-like ξ E x Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

3 Negative/positive invariant mass E [GeV] TL β<0 π 0 SL β>0 When P is the timelike 4-momentum, then the virtual particle has positive mass, as well as the real particle. E.g. Drell-Yan (DY) process. Virtual particle p [GeV/c] M m 0 Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

4 Drell-Yan process. q + q l + + l Time-like (s-channel) virtual photon propagates in space (β < ) and decays into l ±. How real is the virtual? Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

5 What is the lifetime of the virtual particle? The lifetime T of the virtual particle can be defined on the basis of the uncertainty principle: T = E where E is the distance from reality E = P 2 m 2 P 2 M 2 is the difference of energies of the real (m) and virtual (M) particles with the same 3-momentum P. Then the lifetime is T = E Ẽ m 2 M 2 where E = P 2 M 2 is the of the virtual particle, and Ẽ = E 2 + m 2 M 2 is the energy of the real particle. Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

6 What is the lifetime of the virtual particle? You can rewrite the previous equation as: ( ) E T = m 2 M 2 + m2 M 2 E 2 + () Analogously, the space dimensions L of the region of propagation of the virtual particle can be defined on the basis of the uncertainty principle for momenta and coordinates: L = / P, where P = E 2 m 2 E 2 M 2 is the difference of momenta of the real and virtual particles moving along the direction of the momentum vector P with the same energy E. Then here again, P = particle. P + P L = m 2 M 2 P 2 m 2 M 2 is the momentum of the real Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

7 What is the path length of the virtual particle? P L = m 2 M 2 + m2 M 2 P 2 + (2) At small differences of masses of the virtual and real particles, when follows m 2 M 2 E 2, m 2 M 2 P 2 T = and one can construct the 4-vector 2E m 2 M 2, L = 2 P m 2 M 2 {T, L} = 2P m 2 M 2 Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

8 What is the path length of the virtual particle? 2P {T, L} = m 2 M 2 valid for any 4-momentum of virtual particle timelike (β = P/E < ) and spacelike (β = P/E > ). k=(ω,k) k=(ω,k) p =(E, p ) P I =(E, p +k) 2 Time-Like Θ s Θ p = (E, p ) r q=(0,q) Space-Like p =(E, p ) Θ P II =(E, p -k) 2 p = (E, p ) q=(0,q) Space--Like N I. N N II. N Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

9 T and L of virtual e ± in Bremsstrahlung k=(ω,k) k=(ω,k) p =(E, p ) P I =(E, p +k) 2 Time-Like Θ s Θ p = (E, p 2 2 r q=(0,q) Space-Like ) 2 p =(E, p ) Θ P II =(E, p -k) 2 p 2 = (E, p q=(0,q) Space--Like 2 ) 2 N I. N N II. N 4-momentum of the virtual electron in diagram I, is P I = p 2 + k = {E, p 2 + k} and diagram II P II = p + k = {E 2, p + k} Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

10 T and L of virtual e ± in Bremsstrahlung It is evident that the square of mass of the virtual electron for diagram I has the form and hence M 2 I = (p 2 + k) 2 = m 2 e + 2p 2 k = m 2 e + 2E 2 ω( β 2 cos θ) T I = T II = E E 2 ω( β 2 cos θ) E 2 E ω( β cos θ) L I = L II = p 2 k E 2 ω( β 2 cos θ) p k E ω( β cos θ) Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

11 T and L of virtual e ± in Bremsstrahlung At ultrarelativistic energies E m e, E 2 m e and small angles θ, θ, one can write θ 2 β 2 cos θ β 2 + β 2 2 2γ2 2 ( + γ2 2 θ2 ) θ β cos θ 2 β + β 2 2γ 2 ( + γ 2 θ 2 ) where γ,2 = E,2 /m e. Furthermore, in the limit γ,2 then p 2 + k E and p k E 2. Then T I L I 2γ γ 2 ω( + γ 2 2 θ2 ) and T II L II 2γ γ 2 ω( + γ 2 θ 2 ) Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, 204 / 8

12 T and L of virtual e ± in Bremsstrahlung The main contribution to the process of the bremsstrahlung of an ultrarelativistic electron in the Coulomb field of a heavy nucleus arises from the region of small angles θ< /γ 2, θ< /γ. As a result, the effective path length of an ultrarelativistic electron in the bremsstrahlung in the Coulomb field has a magnitude of the order of high energy limit L L I L II c γ γ 2 ω For example for an electron with energy E E 2 00 GeV and photon energy ω = 0 MeV, the effective path length L of the virtual electron amounts to L cm mm (the virtual electron life-time T equals s). Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

13 Special Relativity - time/space like t channel q+q q+q s = s beam beam q q out FSR jet space like gluon q ISR jet q out q out q ISR jet q q out FSR jet Figure: Schematics of the t-channel parton elastic scattering event. Momentum is transferred but energy is not (in CM). Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

14 Formation time and ISR/FSR jets e γ * γ* e ISR jet FSR jet E t p t 2 Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

15 Special Relativity - 4-vectors With the metric tensor notation the 4-vector P of a particle with momentum p, energy E and invariant mass m is denoted as P = (E, p) or P = (E, p x, p y, p z ) The four-dot product of two 4-momentum vectors P and P 2 is then P P 2 = g µν P µ Pν 2 = E E 2 p p 2 = E E 2 p p 2 cos θ E E 2 ( cos θ) (3) where p p 2 is the dot product of the 3-momentum vectors, p and p 2 are the moduli of the 3-vectors and θ is the angle between them. The squared modulus of a 4-vector gives a famous Lorenz invariant P 2 = E 2 p 2 = m 2. (4) Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

16 The source of transverse momentum is the Invariant Mass This is the answer to my first question - how come that the two point-like object with no transverse dimension could ever scatter. P A x M inv = x x 2. s P B x 2 It is the invariant mass M inv = P P 2 = E E 2 p p 2 cos θ E E 2 ( cos θ) the source of any spatial >D propagation (opening angle). Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

17 Hard Scattering Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force October 23, / 8

18 Jan Rak (Jyväskylä University, HIP, Finland) VirtualOctober particles - the ultimate 23, 204 source of any force October 23, / 8 Thanks for your attention! Slides for this talk will be available at: https: //trac.cc.jyu.fi/projects/alice/wiki/lectures4 Virtual particles - the ultimate source of any force Lecture notes Jan Rak Jyväskylä University, HIP, Finland