The Ritus-Narozhny conjecture:

history and re-summation of QED radiative

corrections in a strong constant crossed field

Physics Opportunities at a Lepton Collider in the Fully

Nonperturbative QED Regime

7-9 August, 2019, SLAC National Accelerator Laboratory, Stanford, USA

Alexander Fedotov1,2), Arseny Mironov1)

1)Institute for Laser and Plasma Technologies, National Research Nuclear University MEPhI,

Moscow, Russia2)Laboratory for Quantum Theory of Intense Fields, National Research Tomsk State University,

Tomsk, Russia

Ritus-Narozhny conjecture in a nutshell

Ordinary QED: lloop ' ~/mc

α =e2/4πlloopmc2

=e2

4π~c' 1

137� 1, α(k2) ' α

1− α3π log(k2

m2

) � 1Strong field QED (SFQED): g =

e2/4πcτloopmc2

' αχ2/3, χ� 1

{ε,p} {ε,p}

{ε′,p′}

{ω = k′,k′}

E

E

EE

E

NpQEDαχ2/3 & 1

χ

a0

1 α−3/2

1α−1/2

LCFAa0 & 1, χ1/3

• Ultrarelativistic particle (χ ' e~Eγm2c3

& 1):

c√p2‖ + e

2E2t2 +m2c2 ≈ cp‖ +ce2E2t2

2p‖

• Momentum conservation: p‖ = p′‖ + k′ and

energy mismatch: ∆ε = ε′ + k′ − ε ' e2E2t2

mγ

• Uncertainty principle: ∆ε · t ' ~

• Time scale: t ' ~γmc2

χ−2/3 (. 1ω

=⇒ LCFA),τloop = t/γ 1

Literature

Original papers:

• N. Narozhny, JETP 28, 371–374 (1969).• V. Ritus, JETP 30, 1181 (1970).• V. Ritus, Ann. Phys. 69, 555 (1972).• D. Morozov and V. Ritus, Nucl. Phys. B 86, 309 (1975).• D. Morozov and N. Narozhny, JETP 45, 23 (1977).• N. Narozhny, Phys. Rev. D 20, 1313 (1979).• N. Narozhny, Phys. Rev. D 21, 1176 (1980).• D. Morozov, N. Narozhnyj, V. Ritus, JETP 53, 1103 (1981).

Reviews:

• V.I. Ritus, Journ. Russian Laser Research 6, 584 (1985).• A.M. Fedotov, Journ. of Phys.: Conf. Series 826, 012027 (2017).

Experimental proposals:

• T.Blackburn et al, New J. Phys. 21, 053040 (2019).• V. Yakimenko et al, Phys. Rev. Lett. 122, 190404 (2019).• C. Baumann et al, Scientific Reports 9, 9407 (2019). 2

Historical review: Ritus (FIAN, 1970)

ibid, p.6:

3

Historical review: Ritus (FIAN, 1972)

ibid, p.2:

4

Historical review: Narozhny (Rochester, 1979-1980)

5

Summary of radiative corrections in CCF: polarization operator

Diagram Asymptotics (χ � 1) Reference1 loop

αχ2/3 Narozhny, 1969

2 loops

α2χ2/3 logχ Morozov&Narozhny, 1977

3 loops

α3χ2/3 logχ Narozhny, 1979

α3χ2/3 logχ Narozhny, 1979

α3χ log2 χ Narozhny, 1980

6

Summary of known radiative corrections in CCF: mass operator

Diagram Asymptotics (χ � 1) Reference1 loop

αχ2/3 Ritus, 1970

2 loops

α2χ logχ Ritus, 1972

α2χ2/3 logχ Morozov&Ritus, 1975

3 loops

α3χ2/3 log2 χ Narozhny, 1979

α3χ4/3 Narozhny, 1979

α3χ log2 χ Narozhny, 1980

α3χ5/3 Narozhny, 19807

1-loop polarization operator in CCF

After renormalization (Narozhny 1969; Ritus 1972):

=(Z−1 − 1

) (l2gµν − lµlν

)+

2∑i=1

πi(l2, χl)ε

(i)µ (l)ε

(i)ν (l)

where:

ε(1)µ (l) =eFµν l

ν

m3χl, ε(2)µ (l) =

eεµνλκFλκlν

2m3χl,

Z−1 − 1 = 4απ

∫ ∞4

dv

v5/2√v − 4

[f1(z)− log

(1− 1

v

l2

m2

)]

π1,2(l2, χl) =

4αχ2/3l m

2

3π

∫ ∞4

dv

v13/6v−1+2√v − 4f

′(z),

f(z) = i

∫ ∞0

dσ e−izσ−iσ3/3, f1(z) =

∫ ∞z

dz

(f(z)− 1

z

)z =

(v

χl

)2/3(1− l

2

vm2

)8

Graphical representation and high-χ asymptotics

Z−1−1 ' − α3π

log(χ

2/3l

), π1,2(0, χl � 1) '

[1

1.5

]·0.175(1−i

√3)αχ

2/3l m

2

100 101 102 103 104χl

10−4

10−2

100

|Reπ

1,2(

0,χl)|/m

2

|Re π1|0.18αχ

2/3l

|Re π2|0.26αχ

2/3l

0 5 10 15 20 25−2

0

2

×10−3

100 101 102 103 104χl

10−6

10−5

10−4

10−3

Z−

1−

1

|Re (Z−1 − 1)||Im (Z−1 − 1)|

100 101 102 103 104χl

10−4

10−2

100

|Imπ

1,2(

0,χl)|/m

2

|Im π1|0.30αχ

2/3l

|Im π2|0.46αχ

2/3l

0 5 10 15 20 25

−2

−1

0×10−2

−2 −1 0 1 2λ ×104

−1

0

1

2

π1(λ,χl

=10

4 )/m

2

Re π1Im π1

9

Dressed photon propagator Dcµν(l) (Narozhny 1969)

= + + + . . .

Dcµν(l) = D0(l2, χl)gµν +

2∑i=1

Di(l2, χl)ε

(i)µ (l)ε

(i)ν (l),

where

ε(1)µ (l) =eFµν l

ν

m3χl, ε(2)µ (l) =

eεµνλκFλκlν

2m3χl,

D0(l2, χl) =

−iZl2 + i0

,

D1,2(l2, χl) =

iZ2π1,2(l2 + i0) (l2 − Zπ1,2)

=−iZl2 + i0

− −iZl2 − Zπ1,2

• The only effect of Z(l2, χl) is passing to a running coupling α 7→α(l2, χl) = Z(l

2, χl)α (in what follows we always assume Z ≈ 1)• Effective photon mass m21,2 = Zπ1,2(l2, χl)

10

Similarity to the case of strong B-field

Let B � Bc = m2/e, then momentum of virtual e− on LLL:

p⊥ ∼√eB � m

Since motion on LLL is transverse to the field:

χ(eff)B ∼

B

Bc

p⊥m∼ eBm2

√eB

m=

(eB

m2

)3/2Hence

α(χ

(eff)B

)2/3m2 ' αeB

C.f. [e.g. A. E. Shabad, Tr. Fiz. Inst. Akad. Nauk SSSR 192, 5 (1988)]:

π(2)B (k

2 � eB) = 2απeB︸ ︷︷ ︸

from LLL

+α

3πk2 log

(eB

m2

)︸ ︷︷ ︸

from HLLs

Nice though not precisely the same (not CCF!). Hopefully more info

about this case in the talk by Igor Shovkovy!

11

Goal of this work

l

qp p′ = Γν Γµ

l

qp p′

In the Ritus Ep-representation [(i/∂ + e /A−m)Ep(x) = 0]:

−iM(p′, p) = (−ie)2∫

d4l

(2π)4d4q

(2π)4Γµ(l; p′, q)

i(q̂ +m)

q2 −m2 + i0Γν(−l; q, p)Dcµν(l)

• Dressed vertex:Γµ(l; p, q) =

∫d4xe−ilxEp(x)γ

µEq(x),

• Scattering amplitude [ūp′,sM(p′, p)|p2=m2up,s = (2π)4δ(4)(p−p′)M(χ)]:

M(χ) = M0(χ) + δM(χ), δM=

2∑i=1

δMi

• Residual renormalization: M 7→M(ren) = M−M|F=012

Part M0(χ)↔ D0 (c.f. Eq. (23) in Ritus, 1972)

M0(χ) =αm4

(2π)2

∫ +∞−∞

du

(1 + u)2

∫ +∞−∞

dλ

∫ +∞−∞

dµ

µ+ i0D0(m

2λ, χl)

×{

(2 + λ) Ai1(t) + 2u2 + 2u+ 2

1 + u

(χu

)2/3Ai′(t)

}• Notations: λ = l2/m2, µ = (q2 −m2)/m2, u = χl/χq• Divergent, renormalization:

Ai1(t) 7→ Ai(ren)1 (t) = −i∫ ∞−∞

dσ

2πσe−itσ

(e−iσ

3/3−1)

• M(ren)0 (χ� 1) ' 0.843(1− i√

3)αχ2/3m2 [Eq. (72) in Ritus, 1072]:

100 101 102 103 104χ

10−4

10−2

100

M0/m

2

|ReM0|0.84αχ2/3

|ImM0|1.46αχ2/3

0 5 10 15 20 25−0.1

0.0

0.1

13

Part δM(χ)↔ D1,2

δM1,2(χ) = −αm4

(2π)2

∫ +∞−∞

du

(1 + u)2

∫ +∞−∞

dλ

∫ +∞−∞

dµ

µ+ i0D1,2(m

2λ, χl)

×{[

1 + λu2 + 2u+ 2

2u2

]Ai1(t) +

(u2 + 2u+ 2

1 + u± 1)(χ

u

)2/3Ai′(t)

}Finite, no renormalization needed!

14

Part δM(χ)↔ D1,2

δM1,2(χ) = −αm4

(2π)2

∫ +∞−∞

du

(1 + u)2

∫ +∞−∞

dλ

∫ +∞−∞

dµ

µ+ i0D1,2(m

2λ, χl)

×{[

1 + λu2 + 2u+ 2

2u2

]Ai1(t) +

(u2 + 2u+ 2

1 + u± 1)(χ

u

)2/3Ai′(t)

}Finite, no renormalization needed! C.f. Eq.(42) in Narozhny, 1980:

1 we omit them too

2 T (χ) = −M(χ)/(2p0)3 extra factor (typo)

4 opposite metrics conventions

5 actually vanish (see below)

6 Φ(t) = πAi (t)

7 we do not use such perturbative expansion!14

Next step: integration over µ and λ

δM1,2(χ) = −iαm2

(2π)2

+∞∫−∞

du

(1 + u)2

+∞∫−∞

dλπ1,2/m2

(λ+ i0) (λ− π1,2/m2)

+∞∫−∞

dµ

µ+ i0

×{[

1 + λu2 + 2u+ 2

2u2

]Ai1(t) +

(u2 + 2u+ 2

1 + u± 1)(χ

u

)2/3Ai′(t)

}t =

(u

χ

)2/3(1 +

1 + u

u2λ+

1 + u

uµ

), χl =

χu

1 + u,

Ai1(t) = −i∫ ∞−∞

dσ

2π

1

(σ − i0)e−iσ3/3−itσ, Ai′(t) = −i

∫ ∞−∞

dσ

2πσe−iσ

3/3−itσ

Master integral over µ:∫ +∞−∞

dµe−iµz

µ+ i0= −2πiθ(z)

[if z is complex then θ(z) 7→ θ (Re(z))]. Note that∫ +∞−∞

dµµ e−iµz

µ+ i0= 2πδ(z) ∝ δ(σ)

can be neglected (hence we drop such terms). 15

Continuation: integration over λ

Integration over λ is more tricky. We have:

π(λ) =

∫ ∞0

dξ π̃(ξ) eiξλ, ξeff ' χ−2/3l

Consider the master integral:

J1(z) =

∫ +∞−∞

dλπ(λ)e−iλz

λ− π(λ) =∫ +∞−∞

dλ e−iλz∞∑n=0

(π(λ)

λ+ i0

)n+1

=

∞∑n=0

(n+1∏a=1

∫ ∞0

dξaπ̃(ξa)

)∫ +∞−∞

dλ

(λ+ i0)n+1exp

[i

(n+1∑a=1

ξa − z)λ

]

= −2πi∞∑n=0

(−i)nn!

(n+1∏a=1

∫ ∞0

dξaπ̃(ξa)

)(z −

n+1∑a=1

ξa

)nθ

(z −

n+1∑a=1

ξa

)Note that neff ∼ z, hencen+1∑a=1

ξa ∼ neff · ξeff 'z

χ2/3l

� z =⇒ J1(z) ≈ −2πi θ(z − ξeff)π(0)e−iπ(0)z

16

Validation of the idea by numerical calculation

Resulting approximation:

J1(z) =

+∞∫−∞

dλπ(λ, χl)e

−iλz

λ− π(λ, χl)≈

− 2πi θ(z − χ−2/3l

)π(0, χl)e

−iπ(0,χl)z

Similarly:

J2(z) =

+∞∫−∞

dλ

λ+ i0

π(λ, χl)e−iλz

λ− π(λ, χl)≈

− 2πi θ(z − χ−2/3l

)(e−iπ(0,χl)z − 1

) 100 101 102 103zχ

2/3l

10−3

10−2

|J1(z)|/χ

2/3

l

Exact Approx.

−1 0 1 2 3 40

1

2×10−2

PQED regime:

σeff ' 1, ueff ' 1, χl ∼ χ =⇒ zeff '(u

χ

)2/31 + u

u2σ ∼ χ−2/3l

NPQED regime: zeff ∼1

π(0, χl)' 1αχ

2/3l

� χ−2/3l17

Overlap of occurred θ-functions

Re u2/3 (1+u)σu

> 0

Re1+u

3 σu2/3

> 1

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

u

σ

Overlap: u > 0, σ > σ0(u) =

(u2

1 + u

)1/318

Midway result and plan for further steps

δM(χ) =

2∑i=1

δMi(χ), πi(0, χl) = αm2Πi(χl), Πi(χl � 1) = Kiχ2/3l

δM1,2(χ) =αm2

2π

∫ ∞0

du

(1 + u)2

∫ ∞σ0(u)

dσ e−iσ3/3−iσ(u/χ)2/3

×{[

1

σ+ σ

(χu

)2/3(u2 + 2u+ 21 + u

± 1)](

e−igσϕ1,2(u) − 1)

+g

2σ

(χu

)2/3 2 + 2u+ u21 + u

ϕ1,2(u)e−igσϕ1,2(u)

},

where ϕi(u) =(1 + u)Πi(χl)

(χu)4/3, χl =

χu

1 + uand g = αχ2/3.

• For u� 1 we have χl ≈ uχ; if also χl � 1 then ϕi(u) ≈ Ki/χ2/3l .

• Let us split: δMi = δM(I)i + δM(II)i + δM

(III)i according to different

effective ranges of u and σ

• It is convenient to swap integration order:∫du dσ 7→

∫dσ du in

δM(I,II)i

19

Calculation of δM(I)i

Here σeff ' 1 and ueff ' 1, hence χl ∼ χ� 1 and gσϕi(u) ∼ αKi � 1.

δM(I)i =

αm2

2π

∫ ∞0

dσ

σe−iσ

3/3

∫ σ3/20

du

(1 + u)2

(e−igσϕi(u) − 1

)≈ −iα

2χ2/3m2

2π

∫ ∞0

dσ e−iσ3/3

∫ σ3/20

duϕi(u)

(1 + u)2= −C1Kiα2m2,

C1 =i

2π

∫ ∞0

dσ e−iσ3/3

∫ σ3/20

du

u2/3(1 + u)5/3= 0.242 + 0.310i

δM(I) =

2∑i=1

δM(I)i = (−0.341 + 0.0478i)α2m2 = O

(α2)

This correction is subleading (actually perturbative) and can be neglected

20

Calculation of δM(II)i

Here σeff ' 1, ueff ' α3/2 � 1 (hence the upper limit in the integral overu can be set to infinity) and χl ≈ uχ ' g3/2 � 1.Thus zeff '

(uχ

)2/31+uu2 σ ' (αg)−1 � χ

−2/3l ∼ g−1.

δM(II)1,2 ≈

(2± 1)αχ2/3m22π

∫ ∞0

dσ σe−iσ3/3

∫ ∞0

du

u2/3

(e−iασK1,2/u

2/3 − 1)

= −e−iπ6 (2± 1) · 356

2√π

Γ

(5

6

)√K1,2 α

3/2χ2/3m2

where we used ∫ ∞0

du

u2/3

(e−iζ/u

2/3 − 1)

= −3√iπζ,∫ ∞

0

dσ σ3/2e−iσ3/3 = e−

5iπ12 3−

16 Γ

(5

6

)

δM(II) =

2∑i=1

δM(II)i = (−0.995 + 1.72i)α3/2χ2/3m2

21

Calculation of δM(III)i

Here σeff ' 1/g � 1, (χl)eff ' 1 and ueff ' 1/χ� 1.Thus zeff '

(uχ

)2/31+uu2 σ ' α−1 � χ

−2/3l ∼ 1.

Using abbreviations πi(0, χl) = αm2Πi(χl) and E1(ζ) =

∫∞1dt e−ζt/t we

have:

δM(III)i =

α2m2χ

2π

∫ ∞0

dχlχ2l

Πi(χl)

∫ ∞(χlχ )

2/3

dσ

σe−igσΠi(χl)/χ

4/3l = C2,iα

2χm2,

C2,i =1

2π

∫ ∞0

dχlχ2l

Πi(χl) E1

(iαΠi(χl)/χ

2/3l

)=

{−0.0395− 0.472i−0.0634− 0.703i

δM(III) =

2∑i=1

δM(III)i = −(0.103 + 1.18i)α2χm2

22

Justification by numerical calculation

10−3

10−1

101

|δM(I

I)|/m

2

|Re δM(II)||Im δM(II)|0.995 · α3/2χ2/31.72 · α3/2χ2/3

103 104 105 106 107 108χ

10−1

101

103

|δM(I

II) |/m

2

|Re δM(III)||Im δM(III)|0.103 · α2χ1.18 · α2χ

23

Calculation of δM: Summary

Mass radiative correction: M(χ) = M(ren)0 +δM, δM≈ δM(II)+δM(III)

Lowest-order PQED

correction M(ren)0

0.843(1− i√

3)αχ2/3m2

NPQED correction

due to photon

emission δM(II)(−0.995 + 1.72i)α3/2χ2/3m2

NPQED correction

due to trident pair

production∗ δM(III)−(0.103 + 1.18i)α2χm2

∗ Cf. 2-loop PQED result [Eq.(76) in Ritus 1972]:

δM(2−loop) = −[0.208 + (0.133 lnχ− 0.725)i]α2χm2

24

More on physical meaning of δM (II)

C.f. ’nutshell’ picture on slide 1:

• Ultrarelativistic electron in a strong transverse field: ε ≈ p‖ + e2E2t2

2p‖• Momentum conservation: p‖ = p′‖ + k′• Energy mismatch for massless photon (notation: u = k′‖/p′‖):

∆ε(0) ≈ ε′ + ω − ε = e2E2t2

2p′‖− e

2E2t2

2p‖=e2E2t2k′‖

2p‖p′‖=e2E2t2u

2mγ

• Energy due to photon mass m2eff ' αχ2/3l m

2 [χl ' uχ1+u , k′‖ =p‖u1+u ]:

∆ε(ph) ≈ m2eff

2k′‖' αχ

2/3m2

2mγ

(1 + u

u

)1/3• ∆ε = ∆ε(0) + ∆ε(ph) = min (assuming u� 1):

u '(αχ2/3m2

e2E2t2

)3/4, ∆ε '

√eEtm

γ

(αχ2/3

)3/425

• Uncertainty principle ∆ε · t ' 1 (recall χ ' eEγm2 ):

t ' γmχ2/3

√α, u ' α3/2 � 1

Modification due to photon effective mass!

• Proper time loop scale τloop = t/γ and

e2/4πτloopm

' α3/2χ2/3

• Alternatively (z = (u/χ)2/3 � 1):

dW(CCF)rad (u)

du= − αm

γ(1 + u)2

{Ai1(z) +

2

z

[1 +

u2

2(1 + u)

]Ai′(z)

},∫ α3/2

0

dW(CCF)rad (u)

dudu ' αχ

2/3m

γ

∫ α3/20

du

u2/3' α

3/2χ2/3m

γ

26

χ-dependence

0

2000

4000

6000

Re

(M)/m

2

M0 + δM(II) M0 + δM(II) + δM(III) M0

0.0 0.2 0.4 0.6 0.8 1.0

χ ×109

0

20000

40000

60000

−Im

(M)/m

2

1200 1350 15000.7

0.8

1200 1350 15001.2

1.3

1.4

27

Conclusion

• NOVELTY: first truly NPQED calculation (re-summed bubble-typecorrections to on-shell mass operator).

• TECHNICAL INNOVATIONS:• correct account for renormalization in NPQED expression;• reasonable approximation for an integral over photon virtuality λ in

NPQED regime;

• identification of terms differently localized in the integration region +hint for their physical interpretation.

• OBSERVATIONS:• g = αχ2/3 ' 1: effect is small (reduction by . 3% w.r.t. 1-loop)• g = αχ2/3 � 1:

• electron mass essentially reduces (may even vanish at χ ' 109);• trident pair production dominates over photon emission.

28

Prospects for further studies

• further justification by direct numerical evaluation of∫du dλ [

∫dσ];

• detailed study of lowest-order correction to add into PIC-QED codes;• diagrams with higher multiplicity in virtual channel

are of great potential interest!

• vertex issue [Morozov et al, 1981 vs Gusynin, Miransky, and Shovkovy,PRL 83, 1291 (1999)]:

= O(αχ2/3

)or O(1)?

29

Questions?

29