Intro. Pair production Mass Infra-red High χ Conclusions
Nonperturbative calculations and open problems ofQED in strong external fields
Anton Ilderton
SLAC NPQED 2019
Ilderton, PRD 99 (2019) 085002 [arXiv:1901.00317]Ilderton, to appear
Intro. Pair production Mass Infra-red High χ Conclusions
Outline
1. Ritus-Narozhny conjecture
2. Constant fields and beyond: three examples
Sauter-Schwinger effect
Ritus mass
Infra-red emission
3. Ritus-Narozhny conjecture beyond constant fields
Intro. Pair production Mass Infra-red High χ Conclusions
Ritus-Narozhny conjecture
Furry expansion:QED coupling α runs, but typically small.
Treat perturbatively
Coupling to background: a0 “eE
mω
a0 ą 1: treat exactly
+ +
Furry expansion is a vital tool for theory and expt.
Ñ SLAC E144, recent & planned expt, all PIC codes, etc.
Conjecture: Furry expansion breaks down.Narozhnyi, JETP 28 (1969) 371; Ritus, JETP 30 (1970) 1181
Intro. Pair production Mass Infra-red High χ Conclusions
Ritus-Narozhny conjecture
Furry expansion:QED coupling α runs, but typically small.
Treat perturbatively
Coupling to background: a0 “eE
mω
a0 ą 1: treat exactly
+ +
Furry expansion is a vital tool for theory and expt.
Ñ SLAC E144, recent & planned expt, all PIC codes, etc.
Conjecture: Furry expansion breaks down.Narozhnyi, JETP 28 (1969) 371; Ritus, JETP 30 (1970) 1181
Intro. Pair production Mass Infra-red High χ Conclusions
Ritus-Narozhny conjecture
Furry expansion:QED coupling α runs, but typically small.
Treat perturbatively
Coupling to background: a0 “eE
mω
a0 ą 1: treat exactly
+ +
Furry expansion is a vital tool for theory and expt.
Ñ SLAC E144, recent & planned expt, all PIC codes, etc.
Conjecture: Furry expansion breaks down.Narozhnyi, JETP 28 (1969) 371; Ritus, JETP 30 (1970) 1181
Intro. Pair production Mass Infra-red High χ Conclusions
Ritus-Narozhny conjecture
a0 “eE
mωØ intensity b “
ω
mγ Ø energy χ “ a0b
Review: Fedotov J.Phys.: Conf.Ser. 826 (2017)
Constant crossed field E.B “ 0 “ E2 ´B2
Loops: power law scaling with αχ2{3
Ñ Furry breakdown as αχ2{3 Ñ 1 (χÑ 1600)
How to calculate? Physics unknown. (Need to resum all loops)
Quantum effects: χ ą 0.01. Recently: χ » 0.1 . . . 1.Cole et al PRX 8 (2018), Poder et al PRX 8 (2018)
Intro. Pair production Mass Infra-red High χ Conclusions
Ritus-Narozhny conjecture
a0 “eE
mωØ intensity b “
ω
mγ Ø energy χ “ a0b
Review: Fedotov J.Phys.: Conf.Ser. 826 (2017)
Constant crossed field E.B “ 0 “ E2 ´B2
Loops: power law scaling with αχ2{3
Ñ Furry breakdown as αχ2{3 Ñ 1 (χÑ 1600)
How to calculate? Physics unknown. (Need to resum all loops)
Quantum effects: χ ą 0.01. Recently: χ » 0.1 . . . 1.Cole et al PRX 8 (2018), Poder et al PRX 8 (2018)
Intro. Pair production Mass Infra-red High χ Conclusions
Non-perturbative pair production
A sufficiently strong electric field E can collapse into pairs:
PV T
„ peEq2 exp
„
´ πm2
eE
Non-perturbative effect. See talk by Gerald Dunne
Exponential supression below ES :“ m2{e
Assumes a constant & homogeneous field.
Schwinger effect: how to go beyond constant fields?
Intro. Pair production Mass Infra-red High χ Conclusions
Sauter-Schwinger effect in non-constant fields
Locally Constant Field Approximation?
Ppairs „ V TE2 exp”
´πESE
ı
?ÝÑ
ż
d4xE2pxq exp”
´πESEpxq
ı
Q. When does this work?
A1. Often a good approximation. Schneider et al PRD 98 (2018)
A2. Not exact in general. Nikishov NPB 21 (1970)Narozhnyi & Nikishov Sov.J.Nucl.Phys. 11 (1970), Dunne et al PRD73 (2006)
A3. Only exact for E “ Ept` zq.Tomaras et al PRD 62 (2000)
Ilderton JHEP 1409 (2014)
Intro. Pair production Mass Infra-red High χ Conclusions
Sauter-Schwinger effect in non-constant fields
Locally Constant Field Approximation?
Ppairs „ V TE2 exp”
´πESE
ı
?ÝÑ
ż
d4xE2pxq exp”
´πESEpxq
ı
Q. When does this work?
A1. Often a good approximation. Schneider et al PRD 98 (2018)
A2. Not exact in general. Nikishov NPB 21 (1970)Narozhnyi & Nikishov Sov.J.Nucl.Phys. 11 (1970), Dunne et al PRD73 (2006)
A3. Only exact for E “ Ept` zq.Tomaras et al PRD 62 (2000)
Ilderton JHEP 1409 (2014)
Intro. Pair production Mass Infra-red High χ Conclusions
Sauter-Schwinger effect in non-constant fields
Locally Constant Field Approximation?
Ppairs „ V TE2 exp”
´πESE
ı
?ÝÑ
ż
d4xE2pxq exp”
´πESEpxq
ı
Q. When does this work?
A1. Often a good approximation. Schneider et al PRD 98 (2018)
A2. Not exact in general. Nikishov NPB 21 (1970)Narozhnyi & Nikishov Sov.J.Nucl.Phys. 11 (1970), Dunne et al PRD73 (2006)
A3. Only exact for E “ Ept` zq.Tomaras et al PRD 62 (2000)
Ilderton JHEP 1409 (2014)
Intro. Pair production Mass Infra-red High χ Conclusions
Semiclassical approximation
Semiclassical approx: P „ exp”
´ iSpxclqı
Instanton: periodic classical path. Complex-valued1.
m:xµcl “ eFµνpxclq 9xνcl
Example:
E ” E0 sech2pωqq with q “ λ t`?1´ λ2 z
Interpolating spacetime dependence
´iSpxclq „
¿
dq 9q 9q2 “ λ` a20 tanh2pωqq
Contour integral over the instanton in the complex plane.
1Lavrelashvili et al NPB 329 (1990), Kim & Page PRD 75 (2007), Bender et al PRL 104 (2010), Dumlu& Dunne PRD 84 (2011), Ilderton Torgrimsson Wårdh PRD 92 (2015)
Intro. Pair production Mass Infra-red High χ Conclusions
Semiclassical approximation
Semiclassical approx: P „ exp”
´ iSpxclqı
Instanton: periodic classical path. Complex-valued1.
m:xµcl “ eFµνpxclq 9xνcl
Example: E ” E0 sech2pωqq with q “ λ t`?1´ λ2 z
Interpolating spacetime dependence
´iSpxclq „
¿
dq 9q 9q2 “ λ` a20 tanh2pωqq
Contour integral over the instanton in the complex plane.
1Lavrelashvili et al NPB 329 (1990), Kim & Page PRD 75 (2007), Bender et al PRL 104 (2010), Dumlu& Dunne PRD 84 (2011), Ilderton Torgrimsson Wårdh PRD 92 (2015)
Intro. Pair production Mass Infra-red High χ Conclusions
Interpolating instantons
Complex qpτq plane E ” Epqq , q “ t` 0z
-1 1
- Π2
Π2 Time-dependent.
Instanton circles a branch.
Extended / nonlocal.
Exact results in Eptq:not LCFA.
Intro. Pair production Mass Infra-red High χ Conclusions
Interpolating instantons
Complex qpτq plane E ” Epqq, q “ 0.9t` 0.4z
-1 1
- Π2
Π2
Time & space dependent.
Branch points move closer.
Extended / nonlocal
Intro. Pair production Mass Infra-red High χ Conclusions
Interpolating instantons
Complex qpτq plane E ” Epqq , q “ 0.7pt` zq
-1 1
- Π2
Π2
q: lightfront time.Brodsky et al Phys.Rept. 301 (1998)
Branch cut Ñ pole.
Instantons Ñ points (Cauchy)Ilderton Torgrimsson Wårdh PRD 92 (2015)
Localisation when LCFA exact.
LCFA: exception, not rule.
Intro. Pair production Mass Infra-red High χ Conclusions
Field-dependent mass shift
“Mass operator” (elastic scattering) in constant E.Ritus, ZhETF 75 (1978) & 76 (1979)
F0 = iF1 = N0 =`F0 = iF1 = N0 =„ m
ˆ
1´α
2
E
ES
˙
m “ renormalised. Field-dependent mass correction.
Classical! – survives ~Ñ 0 limit..
Two-loop effective action; ImÑ pair productionLebedev & Ritus, Sov.Phys.JETP 59 (1984)
P „ e2E2 exp”
´ πm2
eE
ı´
1` απ¯2
Conjecture: loops exponentiate? Beyond constant fields?
Intro. Pair production Mass Infra-red High χ Conclusions
Field-dependent mass shift
“Mass operator” (elastic scattering) in constant E.Ritus, ZhETF 75 (1978) & 76 (1979)
F0 = iF1 = N0 =`F0 = iF1 = N0 =„ m
ˆ
1´α
2
E
ES
˙
m “ renormalised. Field-dependent mass correction.
Classical! – survives ~Ñ 0 limit..
Two-loop effective action; ImÑ pair productionLebedev & Ritus, Sov.Phys.JETP 59 (1984)
P „ e2E2 exp”
´ πm2
eE
ı´
1` απ¯2
Conjecture: loops exponentiate? Beyond constant fields?
Intro. Pair production Mass Infra-red High χ Conclusions
Field-dependent mass shift
“Mass operator” (elastic scattering) in constant E.Ritus, ZhETF 75 (1978) & 76 (1979)
F0 = iF1 = N0 =`F0 = iF1 = N0 =„ m
ˆ
1´α
2
E
ES
˙
m “ renormalised. Field-dependent mass correction.
Classical! – survives ~Ñ 0 limit..
Two-loop effective action; ImÑ pair productionLebedev & Ritus, Sov.Phys.JETP 59 (1984)
P „ e2E2 exp”
´ π1
eEm2
´
1´α
2
E
ES
¯2ı
Conjecture: loops exponentiate ùñ same mass appears.? Beyond constant fields?
Intro. Pair production Mass Infra-red High χ Conclusions
The infra-red in the LCFA
Bremsstrahlung: σ „
ż
ωIR
dk
k„ logωIR
Constant crossed fields: σ „
ż
ωIR
dk
k2{3„ 0
F 0=
iF1
=N
0=
k
E.B “ 0 “ E2 ´B2
Softening of IR divergence: feature or bug?
Same behaviour in PIC codes using LCFA.Harvey Ilderton King PRA 91 (2015), DiPiazza et al PRA 98 (2018)
Exact plane wave calculation: Ą crossed field as limit.
Intro. Pair production Mass Infra-red High χ Conclusions
The infra-red in the LCFA
Bremsstrahlung: σ „
ż
ωIR
dk
k„ logωIR
Constant crossed fields: σ „
ż
ωIR
dk
k2{3„ 0
F 0=
iF1
=N
0=
k
E.B “ 0 “ E2 ´B2
Softening of IR divergence: feature or bug?
Same behaviour in PIC codes using LCFA.Harvey Ilderton King PRA 91 (2015), DiPiazza et al PRA 98 (2018)
Exact plane wave calculation: Ą crossed field as limit.
Intro. Pair production Mass Infra-red High χ Conclusions
The infra-red in the LCFA
Bremsstrahlung: σ „
ż
ωIR
dk
k„ logωIR
Constant crossed fields: σ „
ż
ωIR
dk
k2{3„ 0
F 0=
iF1
=N
0=
k
E.B “ 0 “ E2 ´B2
Softening of IR divergence: feature or bug?
Same behaviour in PIC codes using LCFA.Harvey Ilderton King PRA 91 (2015), DiPiazza et al PRA 98 (2018)
Exact plane wave calculation: Ą crossed field as limit.
Intro. Pair production Mass Infra-red High χ Conclusions
The infra-red in plane waves
Photon emission in plane wave IR divergent ðñ Ep0q “ 0
e.g. constant for finite duration
σ „ Ep0q logωIRloooooomoooooon
` finite termsloooooomoooooon
contains CCF limit
Missed if assumed constant from the start.Ò
Dinu et al, PRD 86 (2012)
CCF result does not generalise: no softening of IR.Ilderton Torgrimsson PRD 87 (2013)
LCFA at odds with QED in the IR.
Intro. Pair production Mass Infra-red High χ Conclusions
A closer look at high χ behaviour.
All for constant crossed fields.
E.B “ 0 “ E2 ´B2
Look at pulsed plane waves.
Depends only on χ
Processes depend on a0 & b
χ “ a0b compositeDifferent ways to reach high χ
Expect logs. Why 2{3? Ñ Airy
No Airy functions?Airy reappears in LCFA
Ritus, J.Russ.Laser Res. 6 (1995) 497
No exact results. . . see below. . .
Intro. Pair production Mass Infra-red High χ Conclusions
A closer look at high χ behaviour.
All for constant crossed fields.
E.B “ 0 “ E2 ´B2 Look at pulsed plane waves.
Depends only on χ
Processes depend on a0 & b
χ “ a0b compositeDifferent ways to reach high χ
Expect logs. Why 2{3? Ñ Airy
No Airy functions?Airy reappears in LCFA
Ritus, J.Russ.Laser Res. 6 (1995) 497
No exact results. . . see below. . .
Intro. Pair production Mass Infra-red High χ Conclusions
A closer look at high χ behaviour.
All for constant crossed fields.
E.B “ 0 “ E2 ´B2 Look at pulsed plane waves.
Depends only on χProcesses depend on a0 & b
χ “ a0b compositeDifferent ways to reach high χ
Expect logs. Why 2{3? Ñ Airy
No Airy functions?Airy reappears in LCFA
Ritus, J.Russ.Laser Res. 6 (1995) 497
No exact results. . . see below. . .
Intro. Pair production Mass Infra-red High χ Conclusions
A closer look at high χ behaviour.
All for constant crossed fields.
E.B “ 0 “ E2 ´B2 Look at pulsed plane waves.
Depends only on χProcesses depend on a0 & b
χ “ a0b compositeDifferent ways to reach high χ
Expect logs. Why 2{3? Ñ Airy
No Airy functions?Airy reappears in LCFA
Ritus, J.Russ.Laser Res. 6 (1995) 497
No exact results. . . see below. . .
Intro. Pair production Mass Infra-red High χ Conclusions
Same χ, different high.
Look at existing pulse (plane wave) calculations.
π 2π 3π 4π
Emax
π 2π 3π 4π
Emax
Different ways to reach the same high χ “ a0b.
Behaviour of physical processes is not universal.Baier & Katkov Sov.Phys.JETP 26 (1968), Khokonov TPL 31 (2005)
Dinu, Harvey, Ilderton, Marklund, Torgrimsson PRL 116 (2016)
Compare high-χ behaviour as a0 Ñ8 or bÑ8.
Podszus & DiPiazza PRD 99 (2019)
Ilderton PRD 99 (2019)
Intro. Pair production Mass Infra-red High χ Conclusions
Summary: nonlinear Compton & spin flip
F0 = iF1 = N0 = andF0 = iF1 = N0 =
High intensity, a0 Ñ8 gap High energy, bÑ8
P „αa
2{30
b1{3
„αχ2{3
bP „
αa20b
log b „ αa30χlogχ
(χ „ a0 Ñ8) (χ „ bÑ8)
‚ Same from LCFA ‚ High χ (energy) scaling: log.
‚ Cannot reproduce bÑ8
Intro. Pair production Mass Infra-red High χ Conclusions
Summary: nonlinear Compton & spin flip
F0 = iF1 = N0 = andF0 = iF1 = N0 =
High intensity, a0 Ñ8 gap High energy, bÑ8
P „αa
2{30
b1{3„αχ2{3
b
P „αa20b
log b „ αa30χlogχ
(χ „ a0 Ñ8)
(χ „ bÑ8)
‚ Same from LCFA
‚ High χ (energy) scaling: log.
‚ Cannot reproduce bÑ8
Intro. Pair production Mass Infra-red High χ Conclusions
Summary: nonlinear Compton & spin flip
F0 = iF1 = N0 = andF0 = iF1 = N0 =
High intensity, a0 Ñ8 gap High energy, bÑ8
P „αa
2{30
b1{3„αχ2{3
bP „
αa20b
log b
„ αa30χlogχ
(χ „ a0 Ñ8)
(χ „ bÑ8)
‚ Same from LCFA
‚ High χ (energy) scaling: log.
‚ Cannot reproduce bÑ8
Intro. Pair production Mass Infra-red High χ Conclusions
Summary: nonlinear Compton & spin flip
F0 = iF1 = N0 = andF0 = iF1 = N0 =
High intensity, a0 Ñ8 gap High energy, bÑ8
P „αa
2{30
b1{3„αχ2{3
bP „
αa20b
log b „ αa30χlogχ
(χ „ a0 Ñ8) (χ „ bÑ8)
‚ Same from LCFA ‚ High χ (energy) scaling: log.
‚ Cannot reproduce bÑ8
Intro. Pair production Mass Infra-red High χ Conclusions
Summary: nonlinear Compton & spin flip
F0 = iF1 = N0 = andF0 = iF1 = N0 =
High intensity, a0 Ñ8 gap High energy, bÑ8
P „αa
2{30
b1{3„αχ2{3
bP „
αa20b
log b „ αa30χlogχ
(χ „ a0 Ñ8) (χ „ bÑ8)
‚ Same from LCFA ‚ High χ (energy) scaling: log.
‚ Cannot reproduce bÑ8
Intro. Pair production Mass Infra-red High χ Conclusions
Summary: photon helicity flip at high χ
ϵ′µ ϵµ
detector
High intensity, a0 Ñ8 gap High energy, bÑ8
P „α2χ4{3
b2P „ α2 a
60
χ2
`
1`# logχ˘
pχ „ a0 Ñ8q pχ „ bÑ8q
LCFA cannot reproduce high-energy behaviour.
Taking a0 Ñ8 precludes taking bÑ8.
Intro. Pair production Mass Infra-red High χ Conclusions
Summary: photon helicity flip at high χ
ϵ′µ ϵµ
detector
High intensity, a0 Ñ8 gap High energy, bÑ8
P „α2χ4{3
b2P „ α2 a
60
χ2
`
1`# logχ˘
pχ „ a0 Ñ8q pχ „ bÑ8q
LCFA cannot reproduce high-energy behaviour.
Taking a0 Ñ8 precludes taking bÑ8.
Intro. Pair production Mass Infra-red High χ Conclusions
Discussion: high energy
1. “High χ” can be reached by high energy.
2. High χ-energy behaviour; logs as usual.
3. High energy limit not recovered from constant field / LCFA.(e.g. impossible to obtain log terms)
4. Corollary: LCFA (& PIC!) has wrong high energy behaviour.(But tough to reach this regime.)
Intro. Pair production Mass Infra-red High χ Conclusions
Discussion: high energy
1. “High χ” can be reached by high energy.
2. High χ-energy behaviour; logs as usual.
3. High energy limit not recovered from constant field / LCFA.(e.g. impossible to obtain log terms)
4. Corollary: LCFA (& PIC!) has wrong high energy behaviour.(But tough to reach this regime.)
Intro. Pair production Mass Infra-red High χ Conclusions
Discussion: high intensity
1. αχ2{3 scaling appears to persist in pulses.
2. R-N conjecture applies beyond constant fields.. . . if LCFA holds . . .
3. What if LCFA does not hold?
-2 -1 1 2ϕ
-2 -1 1 2ϕ
a0
sech2 plane wave Ñ delta function in limit.Ilderton, to appear
Intro. Pair production Mass Infra-red High χ Conclusions
Ultra-short pulses
Exact, closed form results for pair production, γ emission,& related one-loop processes
High-intensity behaviour: Ilderton, to appear
Ppair „4
3
α
πlog a0 , PNLC „
12α
πlog2 a0
(Leading-log only.)
No power law! Slow log increase instead.
Scaling different when LCFA does not hold.
Difference in behaviour: comes from the infra-red...
Intro. Pair production Mass Infra-red High χ Conclusions
Ultra-short pulses
Exact, closed form results for pair production, γ emission,& related one-loop processes
High-intensity behaviour: Ilderton, to appear
Ppair „4
3
α
πlog a0 , PNLC „
12α
πlog2 a0
(Leading-log only.)
No power law! Slow log increase instead.
Scaling different when LCFA does not hold.
Difference in behaviour: comes from the infra-red...
Intro. Pair production Mass Infra-red High χ Conclusions
Conclusions
New physical effects in strong background fields.
Ritus-Narozhny conjecture: uncharted strong field regime?
Many open questions re RN conjecture.
High χ not universal: not at high energy or in short pulse.
LCFA? (Fails in the IR and the UV).
If there: effects to improve Furry expansion?
Resummation? Inclusion of back-reaction? Depletion?For the future!
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