Non-Linear Effects in Strong EM Field Alexander Titov Bogoliubov Lab. of Theoretical Physics, JINR, Dubna International conference on the structure of baryons BARYONS 10, Dec.7-11, 2010, Osaka, Japan Non-Linear Effects in Strong Electromagnetic Fields (Neutrino pair emission of electron in a strong EM wave fields) a strong EM wave fields) ( arXiv hep-ph: )

Non-Linear Effects in Strong EM Field 2 Outline Introduction: Volkov solution of Dirac equation in strong EM field Reaction in a strong EM field Weak decay of a neutron Reaction Summary (i) differential probability of partial multi-photon harmonics (ii) convergence problem at large field intensities in a strong EM field

3 Electron in a strong electromagnetic field D.M. Volkov, Z. Phys. 94, 250 (1935) Second order Dirac equation where and with is EM field tensor is four vector of electromagnetic field with the special part chosen as 4-componenrt spinor with Special case: plane wave

4 Solution: when Dirac solution for free electron with or spinor modification phase factor

5 Properties of Volkov’s solution Effective “quasi” momentum Effective electron mass with “quasi-momentum” and effective mass define momentum- energy conservation in processes with electrons

6 Neutrino Emission in a Strong Electromagnetic Field

7 Elementary vertices effective interactions where and

8 Neutrino emission in external EM field external field electron Volkov states

A.I. Baryon10, Osaka, Dec. 7-11, “Neutrino pair emission off electron in a strong EM fields” 9 Structure of matrix element Fourier series Each “harmonic” describes absorption (emitting) of n photons of external field A with wave vector k and emitting of outgoing neutrino pair with four-momentum Q with corresponding conservation low The amplitude is a sum of infinite numbers of “partial harmonics” with

10 Emission probability with

11 phase space with Properties of partial contributions Kinematical limit (phase space) increases (n>0 ) Results in decrease of the phase space even for one photon absorption 2 effects: Electron can interact with a few (n) photons, simultaneously (“cumulative” effect) Dressed electron mass exceeds free electron mass

12 Variation of in reaction in vacuum does not change shape of distribution. It changes overall normalization Differential emission probability In general, higher harmonics are not suppressed at large Increase of leads to decrease of kinematical limit (see case of n=1) “Cumulative effect” – reactions with n>1 increases the phase space and the kinematical limit Result is essentially non-perturbative even for small

13 Early works: N.~B.~Narozhnyi, A.~I.~Nikishov, and V.~I.~Ritus, Sov.\ Phys.\ JETP {\bf 20}, 622 (1965) [Zh.\ Eksp.\ Teor.\ Fiz.\ {\bf 47}, 930 (1964)]. V.~A.~Lyulka, Zh.\ Eksp.\ Teor.\ Fiz.\ {\bf 69}, 800 (1975). N.~P.~Merenkov, Yad.\ Fiz.\ {\bf 42}, 1484 (1985). V.~V.~Skobelev, Yad.\ Fiz.\ {\bf 46}, 1738 (1987). N.~D.~Sengupta, Bull. Calcutta Math. Soc. {\bf 44}, 175, (1952). I.~I.~Goldman, Phys. Lett. {\bf 8}, 103 (1964) L.~S.~Brown and T. W. B. Klibbe, Phys. Rev. A {\bf 133}, 705 (1964). H.~R.~Reiss, J. Math. Phys. {\bf 3}, 59 (1962). A.~I.~Nikishov and V.~I.~Ritus, Sov.\ Phys.\ JETP {( ) (i) (ii)

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15 Problem of convergence Higher harmonics dominate at large

16 Method of asymptotic summation

17 Asymmetry of production ofand

18 Neutron decay in the field of a strong electromagnetic wave L k e n p electron modification proton modification neutron modification

19 Amplitude

20 Weak decay in the field of a strong electromagnetic wave L k e n p

21 Summary Strong electromagnetic fields modify basic/fundamental interactions and result in non-trivial nonlinear non- perturbative effects. Modification of kinematical limits because of (a) electron dressing and (b) coherent interactions with several photons (cumulative effect) Elaborated method for calculation of the emission probabilities at arbitrary Leads to some not-trivial dynamical effects, like and asymmetries, decay widths, … Coherent interactions with n- field photons

22 Thank you very much for attention !

23 Backup

24 Emission of a photon by an electron in the field of a strong electromagnetic wave L k e(q) e’(q’)  ’  k’) electron  “dressed” electron Interaction of electron with an external field is considered non-perturbatively Interaction of electron with outgoing photon is consider in first order of perturbation theory e’(q) e(q)  ‘(k’)

25 Probability is a sum of partial contributions L nk e k’ e’

26 Photon emission in strong EM (results) At small field intensity  2 << 1 effect of mass modification is small, “cumulative” effect is large At large field intensity  2 >> 1 effect of mass modification is larger, than “cumulative” effect. However, the later one is also important. At  standard Klein-Nishina equation does not work even for n=1. Klein-Nishina

27 Problem of convergence

28 Asymptotic method

Non-Linear Effects in Strong EM Field 29 Compton  e   e Scattering (Klein-Nishina equation) with +

30 probability flux factor photon density ? relation between cross section d  to probability dW circularly polarized photon field energy density of EM field with reduced intensity of EM field

31 with “electron radius” and function

32 Final expression for the Compton scattering probability dW square of the total energy square of the momentum transfer in cross channel square of momentum transfer For further application it is convenient to use new invariant variable u

33 Compton  e   e scattering (results) Shape of differential distribution does not depend on field intensity because describes an overall factor BUT not matrix element!

34 Structure of matrix element “non-perturbative” outgoing electron “non-perturbative” incoming electron with In “regular” Compton scattering  e  >  e, one has

A.I. Baryon10, Osaka, Dec. 7-11, “Neutrino pair emission off electron in a strong EM fields” 35 Structure of matrix element (continuing) Fourier series Each harmonic describes absorption (emitting) of n photons of external field A with wave vector k and emitting of outgoing photon with the wave vector k’ with corresponding conservation low The amplitude is a sum of infinite numbers of “partial harmonics”

36 2 effects: Electron can interact with a few photons simultaneously (“cumulative” effect) Dressed electron mass exceeds free electron mass Results in decrease of the phase space even for one photon absorbtion Properties of partial contributions Kinematical limit (phase space) increases (n>0)

37 Summary of this part: Non-perturbative effects of QED may be seen even at small Perturbative QED does not work at finite values of Difference between predictions of perturbative QED and non- perturbative QED is large both qualitatively and quantitatively