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)

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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 •

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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 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

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4 KINETIC FORMULATION OF PAIR PRODUCTION

Kinetic equation for the single particle distribution function f(P¯, t) =< 0 a† (t)a ¯(t) 0 > | P¯ P | 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)ε (t) = ⊥ , /exp(− 2 − 2

τ) W ω (t) ( − f 2 2 2 2 1 where ω(t) = ε + P (t), with ε = 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) = dt0ω(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 − → ∞ E 0

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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 2 dt1 ε(p, t1), = dp1dp2 σ(p1, p2)f(p1, t)f(p2, t) × Z dV dt Z t0 2 2   (v1 v2) v1 v2 , where × p − − | × | cross-section σ of two-photon annihilation m2 + p2 ∆(p, t) = eE(t) ⊥ , πe4 pε2(p, t) σ(p , p ) = (τ 2 + τ 1/2) 1 2 2m2τ 2(τ 1) − p 2 2 2 − ε( , t) = m + p + [p3 eA(t)] √τ + √τ 1 q ⊥ − ln − (τ + 1) τ(τ 1) , The particle number density × √τ √τ 1 − − − − p dp t-channel kinematic invariant n(t) = 2 f(p, t) Z (2π)3 (p + p )2 1 τ = 1 2 = [(ε + ε )2 (p + p )2]. 4m2 4m2 1 2 − 1 2

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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 ωt, 0 t NT, T = (t/τL) ≤ ≤ ω E(t) = Eme− sin ωt.

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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 ω/m = 2.84 10− ∼ ≈ · · size: z0 = 9µm B. Liesfeld et al: “Single-shot autocorrelation at relativistic intensity”, Jena Preprint (2004)

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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 distribution 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)   dt1 2 exp 2i dt2ε(t2) × Z ε (t1) Z t0  t1   

Mean pair density (eE )2 m2 + 2ε2 < n > = m dp 3ω2 + 4ε2 12(2π)3 Z ε4(ω2 4ε2)2 − εω3 + sin (T ε) cos [(2N + 1)T ε] π(ω2 4ε2) − + 3 Low frequency limit ω 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 ω 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).

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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 ω = 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 = √µ 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~ωE × 6 × × Photon energy/ rest energy 2 e mec 2 10− 0.006 0.006 0.02 ~ω × 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

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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

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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 2 dt1 ω(p, t1), × Z  t0  where m π ω˙ (p, t) m(t)m˙ (t) ∆(p, t) = = , ω(p, t) ω2(p, t) 2 2 ω(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.

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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) ω2(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 ω (t) Z m (t0) −∞ 2 mm˙ p ∆ = 2 , ∆ = ∆ 2 . D.B., A. Prozorkevich, S. Smolyansky, arXiv:hep-ph/0411383 ω m − m t 2 m(t) ω(t0) f3(p, t) = Q(p, t)f1(p, t), Q(p, t) = exp 2 ∆m(t0)dt0 = , − Z m(t ) ω(t) t0 0 1 f˙ = ∆u , u˙ = ∆(1 + 2f ) 2ωv , v˙ = 2ωu . k 2 k k k − k k k

3 ∞ ∞ 1 2 1 2 ntot(t) = 2 ni(t) = p dp [2f1(p, t) + f3(p, t)] = p dp f1(p, t)[2 + Q(p, t)], π2 Z π2 Z Xi=1 0 0

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16 VECTOR BOSON PRODUCTION IN CONFORMAL COSMOLOGY (I)

0,30 n 1 Conformal transformation relates n 0,24 3 ]

2 3

- Einstein frame g˜µν(x) = Ω (x)gµν Jordan frame 0,18 m c

[

y t i

s 0,12

Conformal invariance requires a “conformal mass” n e D 0,06 m(x) = Ω(x)m˜ (x) a(t˜)mobs −→ α = 1/2 For homogeneous, isotropic FRW space-time, 0,00 1,0x10-11 3,0x10-11 5,0x10-11 p = (γ 1)ε , Time [ s ] ph − ph α m(t) = (t/tH) mW , α = 2/(3γ 2) − 106

1 n 1 tH = [(1 + α)H]− is the age of the universe for the 4 ]

10 3

n -

3 m

Hubble constant H and m is the present-day W- c 2

W [ 10

y t i s

boson mass. n 0

e 10 d

l a n

i -2 Example stiff ﬂuid: γ = 2, α = 1/2; F 10 α = 1/2 radiation: γ = 4/3, α = 1 10-4

3 10-6 Compare: density of CMB photons, nCMB 430 cm− 10-18 10-16 10-14 10-12 10-10 10-8 ∼ Initial time [ s ]

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17 VECTOR BOSON PRODUCTION IN CONFORMAL COSMOLOGY (II)

6 energy density pressure

EoS of the evolving massive vector boson ] mean pressure 3

4 m c gas *

e 2 G

d3p [ ε(t) = 2 ω(2 + Q)f1, Z (2π)3 0 3 2 4 d p p -2 p(t) = 3 (2 + Q)f1 + δpvac, 3 Z (2π) ω -4 1,82x10-10 1,84x10-10 1,86x10-10 1,88x10-10 1,90x10-10 Time [ s ] where δpvac(t) is the pressure contribution due to vacuum polarization, 2 d3p 2ω2 + m2 1 p2 4 ]

- δpvac(t) = 3 1 + Q + 2 f1. 3 m

−3 Z (2π) ω 2 m c

e 2 G

EoS evolves from phantom (p ε) to dust [ energy density ≤ − 1 mean pressure (p = 0).

0 D.B., M. Dabrowski, arXiv: hep-th/0407078 -1 D.B., A. Prozorkevich, S. Smolyansky, -2 arXiv: hep-ph/0411383 3,0x10-10 6,0x10-10 9,0x10-10 1,2x10-9 Time [ s ]

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18 CONCLUSIONS

Kinetic approach gives important new insights to pair production • Simplest laser ﬁeld model predicts production of dense electron-positron plasma • in the focus of counter-propagating laser ﬁelds Observable manifestation: several gamma-pairs per laser pulse Inertial mechanism formulated. Applications to kinetics of chiral phase transition in HIC • and particle production in conformal cosmology suggested

OUTLOOK: WORK TO BE DONE

More realistic laser ﬁelds and Background estimation • Application of the approach to other situations, e.g. in • – heavy-ion collisions, – compact stars,

BLTP DUBNA, NOVEMBER 11, 2005