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Schwinger Effect: Pair Creation in Strong Fields

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Schwinger Effect: Pair Creation in Strong Fields SCHWINGER EFFECT: PAIR CREATION IN STRONG FIELDS BLTP DUBNA, NOVEMBER 11, 2005 1 PAIR CREATION IN STRONG FIELDS: EXAMPLES Heavy Ion Collisions • Magnetars: B 1015G • ∼ Optical and X-Ray Lasers: • E 1015 1017 V/m ∼ ÷ ARTIST VIEW OF A MAGNETAR (NASA) ILC PROJECT (DESY HAMBURG) BLTP DUBNA, NOVEMBER 11, 2005 2 DYNAMICAL SCHWINGER EFFECT IN THE FOCUS OF AN OPTICAL LASER David Blaschke (University Bielefeld & JINR Dubna) Craig Roberts (Argonne National Laboratory) Sebastian Schmidt (Helmholtz-Gemeinschaft Bonn) Alexander Prozorkevich, Stanislav Smolyansky (Saratov State University) Introduction: Schwinger Effect • Kinetic formulation of pair production • + Application to e e− production in optical lasers • Example: Jena experiment • Outlook: XFEL and other systems • Inertial mechanism for particle production • Application to conformal cosmology: massive vector bosons • BLTP DUBNA, NOVEMBER 11, 2005 3 SCHWINGER EFFECT: PAIR CREATION IN STRONG FIELDS Pair creation as barrier penetration + To “materialize” a virtual e e− pair in a in a strong constant field • constant electric field E the separation d must be sufficiently large positive levels positive levels xE eEd = 2mc2 ε 2 T Probability for separation d as quantum • ε fluctuation 2ε 2 T/E T d 2m2c3 2E P exp = exp = exp crit / −λc − e~E − E negative levels negative levels Emission sufficient for observation when E • ∼ Ecrit m2c3 Schwinger result (rate for pair production) E 1:3 1018V=m crit ≡ e~ ' × dN (eE)2 1 1 E = exp nπ crit For time-dependent fields: Kinetic Equation d3xdt 4π3 n2 − E • Xn=1 approach from Quantum Field Theory J. Schwinger: “On Gauge Invariance and Vacuum Polarization”, Phys. Rev. 82 (1951) 664 BLTP DUBNA, NOVEMBER 11, 2005 4 KINETIC FORMULATION OF PAIR PRODUCTION Kinetic equation for the single particle distribution function f(P¯; t) =< 0 ay (t)a ¯(t) 0 > j P¯ P j df (P¯; t) @f (P¯; t) @f (P¯; t) 5 = + eE(t) dt @t @P (t) k E = 0.7 t 0 1 ¯ 4 E = 1.0 = (t) dt0 (t0)[1 2f (P ; t0)] cos[x(t0; t)] 0 2W Z W E0 = 1.5 −∞ ) 0 E0 = 3.0 ¯ /E 3 Kinematic momentum P = (p1; p2; p3 eA(t)), π − eE(t)" (t) = ? ; /exp(− 2 − 2 τ) W ! (t) ( − f 2 2 2 2 1 where !(t) = " + P (t), with " = m + p¯ q ? k ? ? and x(t0; t) = 2[Θ(t) Θ(t0)]. p − 0 t −3 −2 −1 0 1 2 3 4 5 Θ(t) = dt0!(t0) τ Z Schmidt, Blaschke, et al: “Non-Markovian effects −∞ Constant field: Schwinger limit reproduced in strong-field pair creation” π Phys. Rev. D 59 (1999) 094005 f(τ ) = exp − ! 1 E 0 BLTP DUBNA, NOVEMBER 11, 2005 5 PAIR PRODUCTION IN MULTI-TW LASER FIELDS (I) + Observable: photon pair (e + e− 2 γ) Kinetic formulation for E(t) = A_(t) in ! − the Hamiltonian gauge Aµ = (0; 0; 0; A(t)) − γ e p 1 t p df(p; t) 1 + 2 γ = ∆(p; t) dt0 ∆(p; t0) [1 2f(p; t0)] e dt 2 Z − t0 t dν cos 22 dt1 "(p; t1)3; = dp1dp2 σ(p1; p2)f(p1; t)f(p2; t) × Z dV dt Z t0 2 2 4 5 (v1 v2) v1 v2 ; where × p − − j × j cross-section σ of two-photon annihilation m2 + p2 ∆(p; t) = eE(t) ? ; πe4 p"2(p; t) σ(p ; p ) = (τ 2 + τ 1=2) 1 2 2m2τ 2(τ 1) − p 2 2 2 − "( ; t) = m + p + [p3 eA(t)] pτ + pτ 1 q ? − ln − (τ + 1) τ(τ 1) ; The particle number density × pτ pτ 1 − − − − p dp t-channel kinematic invariant n(t) = 2 f(p; t) Z (2π)3 (p + p )2 1 τ = 1 2 = [(" + " )2 (p + p )2]: 4m2 4m2 1 2 − 1 2 BLTP DUBNA, NOVEMBER 11, 2005 6 PAIR PRODUCTION IN MULTI-TW LASER FIELDS (II) Time dependence of the density n(t) for Time dependence of the density n(t) for a a monochromatic field Gaussian wave packet 2π 2 E(t) = Em sin !t; 0 t NT; T = (t/τL) ≤ ≤ ! E(t) = Eme− sin !t: BLTP DUBNA, NOVEMBER 11, 2005 7 APPLICATION TO JENA MULTI-TW LASER Laser diagnostic by nonlinear Thomson scat- tering off e− in a He-gas jet Pulse intensity: I = 1018 W/cm2, duration: Colliding laser pulses of a Ti:sapphire laser , wavelength: nm, cross- 5 6 τL 80fs λ = 795 with Em=Ecrit 3 10− and !=m = 2:84 10− ∼ ≈ · · size: z0 = 9µm B. Liesfeld et al: “Single-shot autocorrelation at relativistic intensity”, Jena Preprint (2004) BLTP DUBNA, NOVEMBER 11, 2005 8 APPLICATION TO JENA MULTI-TW LASER Momentum distribution of created e+e − Stratum-like structure of the momentum dis- pairs: p stands for transversal (solid line) and tribution at the time of zero field amplitude longitudinal (dotted line) momentum. t = T=2, determining the residual density. Blaschke, Prozorkevich, Smolyansky, Tarakanov, Proc. ’Kadanoff-Baym Eqs. III’, Kiel (2005) BLTP DUBNA, NOVEMBER 11, 2005 9 APPLICATION TO JENA MULTI-TW LASER Equilibrium-like momentum distribution at the time of maximal field amplitude t = T=4. Stratum-like structure of the momentum dis- tribution at the time of zero field amplitude t = T=2, determining the residual density. BLTP DUBNA, NOVEMBER 11, 2005 10 APPLICATION TO JENA MULTI-TW LASER (II) Analytic estimate for E E , f 1 m crit 1 n(t) = dp (m2 + p2 ) 2(2π)3 Z ? 2 t t eE(t1) 0 1 dt1 2 exp 2i dt2"(t2) × Z " (t1) Z t0 B t1 C @ A Mean pair density (eE )2 m2 + 2"2 < n > = m dp 3!2 + 4"2 12(2π)3 Z "4(!2 4"2)2 − "!3 + sin (T ") cos [(2N + 1)T "] π(!2 4"2) − + 3 Low frequency limit ! m Number of e e− pairs in the volume λ for a 2 2 weak field (Jena Ti:AlO3 laser, solid line) and 3 (eE) m n 1:6 10− = nr for near-critical field Em=Ecrit = 0:24, λ = 0:15 h i ≈ × m ! nm (X-FEL, dashed line). Several gamma-pair events per laser pulse ! BLTP DUBNA, NOVEMBER 11, 2005 11 APPLICATION TO JENA MULTI-TW LASER (III) Time dependence of the γ quanta num- + Wavelength dependence of the mean density ber from e e− annihilation for the case of + 5 of e e− pairs (solid line) and their annihila- a monochromatic field with E =E = 3 10− . m crit · tion rate (dotted line). BLTP DUBNA, NOVEMBER 11, 2005 12 LASER PARAMETERS AND DERIVED QUANTITIES Laser Parameters Optical X-ray FEL Focus: Design Focus: Focus: Diffraction limit DESY Available Goal Wavelength λ 1 µm 0.4 nm 0.4 nm 0.15 nm ~ hc Photon energy ! = λ 1.2 eV 3.1 keV 3.1 keV 8.3 keV Peak power P 1 PW 110 GW 1.1 GW 5 TW Spot radius (rms) σ 1 µm 26 µm 21 nm 0.15 nm Coherent spike length (rms) t 500 fs 20 ps 0.04 fs 0.04 fs 0.08 ps 4 ÷ Derived Quantities P 26 W 19 W 23 W 31 W Peak power density S = 2 3 10 2 5 10 2 8 10 2 7 10 2 πσ × m × m × m × m Peak electric field = pµ c S 4 1014 V 1 1011 V 2 1013 V 2 1017 V E 0 × m × m × m × m 4 7 5 Peak electric field/critical field = c 3 10− 1 10− 1 10− 0.1 E~!E × 6 × × Photon energy/ rest energy 2 e mec 2 10− 0:006 0:006 0:02 ~! × 3 4 2 Adiabaticity parameter γ = e λ 9 10− 6 10 5 10 0.1 E− × × × A. Ringwald, Phys. Lett. B 510 (2001) 107; hep-ph/0112254 BLTP DUBNA, NOVEMBER 11, 2005 13 ACCUMULATION EFFECT IN NEAR-CRITICAL FIELDS 2 Particle number density n(T ; E0) = a0(E0) sin (2πT ) + ρ(T; E0)T ; T = t/λ 10−10 10−11 Results are nicely fitted with 10−12 ) −1 10−13 ρ(T; E0) = ρ(E0) + ρ0(E0)T : −14 11 10 For E = 0:5 E0, a0 = 1:2 10− fm 3, period 12 3 × − −3 −15 ρ = 5:4 10− fm− /period, ρ0/ρ = 0:0033/period. 10 × (fm −16 ρ 10 Comparison with Schwinger rate 10−17 10−18 4 0.1 0.2 0.3 0.4 0.5 m λ E0 bπEcr=E0 ρ = a 3 e− E0/Ecr 4π Ecr Attention: Accumulation rate ρ(0; E0) (solid), E0 0:35 Ecr backreactions become important! Schwinger rate a = 1; b = 1 (dashed), ∼ a = 0:305; b = 1:06 (dot-dashed) Roberts, Schmidt, Vinnik: “Quantum effects with an X-Ray Free-Electron Laser”, Phys. Rev. Lett (2002) 153901 BLTP DUBNA, NOVEMBER 11, 2005 14 INERTIAL MECHANISM OF PAIR PRODUCTION µ 2 Scalar field [@µ@ + m (t)]φ(x) = 0 Example: Cooling hadronic fireball T (t) produces pion excess at chiral transition T Kinetic equation, non-Markovian source χ via scalar meson decay: σ 2 π t ! df(p; t) 1 = ∆(p; t) dt0 ∆(p; t0) [1+2f(p; t0)] dt 2 Z t0 t mσ cos 22 dt1 !(p; t1)3; × Z 4 t0 5 where m π !_ (p; t) m(t)m_ (t) ∆(p; t) = = ; !(p; t) !2(p; t) 2 2 !(p; t) = m (t) + p T T p χ The particle number density dp Low-momentum pion excess observed at n(t) = 2 f(p; t) CERN-SPS, detailed comparison: Z (2π)3 DB, Prozorkevich, Smolyansky, in prep. BLTP DUBNA, NOVEMBER 11, 2005 15 INERTIAL MECHANISM FOR VECTOR BOSON PRODUCTION µ 2 µ Massive vector boson [@µ@ + m (t)]φ(x) = 0 , constraint @µu = 0 Kinetic equation, transversal (fi; i = 1; 2) and longitudinal (f3) distributions t 1 f_i(p; t) = ∆(p; t) dt0∆(p; t0)[1 + 2fi(p; t0)] cos 2θ(p; t; t0); 2 Z −∞ t 1 m2(t) !2(t ) _ p p p p p 0 p p p f3( ; t) = 2∆m( ; t)f3( ; t) + ∆( ; t) 2 dt0∆( ; t0) 2 [2f3( ; t0) + Q( ; t0)] cos 2θ( ; t; t0): − 2 ! (t) Z m (t0) −∞ 2 mm_ p ∆ = 2 ; ∆ = ∆ 2 : D.B., A.
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