Complex Effective Action and Schwinger Effect

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tively electron background remarkable an field is of It netic interaction [5]. the action that effective Heisenberg- Schwinger the or as known Euler now the elec- is constant which found a field, in then tromagnetic electron electrons Euler an for of and action effective Heisenberg creation vacuumone-loop [4]. pair Dirac positrons the the and that against of found unstable discovery Sauter the became theory, after Soon Dirac theory. the ef- field quantum quantum spacetime nonperturbative in prominent expanding fect most an cos- the to the are and radiation due [3] [2] Hawking hole production black the particle a from [1], mic particles field of species electric all an of to due pairs ∗ lcrncades [email protected] address: Electronic nmdr unu edter,pril rpi pro- pair or particle theory, field quantum modern In and Heisenberg-Euler the of feature distinct most The charged of production of mechanism Schwinger The ffciems feeto n oei rpeeadDrco We or numbers: phe PACS Dirac (QED) and electrodynamics graphene quantum in of hole test a experimental and higher for electron even and of limit mass Schwinger effective the than production higher intense pair are of electron-positron surface technology for expl current limit action The Schwinger complex the production. the whi pair that formalism, sense also the but in-out in the theory in field reviewed quantum t is a In production of effects. pair radiation quantum Hawking and nonperturbative the of and consequence field the m electric are Schwinger strong The a theory. by field pairs quantum by predicted phenomena pnaeu arpouto rmbcgon ed rspace or fields background from production pair Spontaneous .INTRODUCTION I. eateto hsc,Kna ainlUiest,Kunsan University, National Kunsan Physics, of Department ope ffcieAto n cwne Effect Schwinger and Action Effective Complex Dtd oebr2,2016) 28, November (Dated: agPoKim Pyo Sang vroeCmtnwvlnt qaigt h etmass rest limit, the Schwinger electron to the equaling an gives wavelength of Compton and energy one energy-momentum potential over field. electrostatic of magnetic the conservation constant charge, the constant a of without a or Because in with polarization field vacuum electric the and mechanism [6]. out-vacua the and in- transformation the Bogoliubov la between a pairs mean produced the of measures am- number solution the negative frequency which negative of the of future, not also plitude remote the but into in solution potential solution frequency frequency remote nonstationary and positive the a electron a in by real only solution scattered of tunnels frequency is pair quantum positive past a a sea poten- become or Dirac to electrostatic positron the barrier the in mass to electron the due an sea and non- Dirac electric tial a the uniform and tilts A horizon, field a interaction. with gravitational spacetime stationary a field, magnetic w rtreodrsrne hnteShigrlimit, Schwinger the than stronger order surface B three the the on or stars, fields two magnetic 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TABLE I: Comparison of QED/QCD and condensed matter Worldline Instantons QED/QCD condensed matter Pair Production Dirac or Yang-Mills theory Dirac or Weyl theory & Vacuum Phase Integrals external gauge fields external electromagnetic fields Persistence Schwinger effect Landau-Zener tunneling Contour Integrals (Complex) Effective Actions FeynmanWorldline and the right-moving pseudo-spins, behaving like spin- Path Integrals 1/2 fermions. At the Dirac point, the particle has the zero mass and ultra-relativistic dispersion relation ǫ = Vacuum Dirac-Heisenberg- Polarization Wigner Formalism vF k with the Fermi velocity vF. Then, the Hamiltonian is described| | by the Weyl spinor. The recently discovered Schwinger-DeWitt In- Dirac semimetals [9, 10] and Weyl semimetals [11] are Out Formalism three-dimensional analogy of two-dimensional graphene. Including the time dimension, the effective theory for Dirac semimetals is prescribed by massless QED in the FIG. 1: The summary of different methods to the vacuum four-dimensional spacetime [12]. In this sense, the mass- polarization and the Schwinger effect. It does not exhaust all known methods in literature. less QED may be tested by the condensed matter analogy. The characteristic feature of QED is compared with the condensed matter analogy in Table I. By performing the path integral, the vacuum persis- In this paper, we review the effective QED action and tence amplitude for a general electromagnetic field is the Schwinger effect in a constant electric and magnetic given by field. To compute the QED action and to explore the relation between the QED action and the Schwinger effect, 1 S = exp Tr ln (ˆp γ m) , we shall employ the in-out formalism, which has been re- 0 − α α − pˆ γ qA γ m + iǫ h α α − α α − i cently elaborated since the seminal works by Schwinger (5) and DeWitt [13–15]. We also introduce a method, such as the reconstruction conjecture, which enables one to where ǫ is an arbitrary small number for the purpose of find the effective action from the knowledge of the vac- convergence. The vacuum persistence is uum persistence. There are different approaches to the 4 vacuum polarization and the Schwinger effect as summa- 2 −2W −2Im R d xLeff S0(A) = e = e , (6) rized in Fig. 1. | | where ∞ II. S-MATRIX FOR QED ACTION 1 ds is[(p−qA)2+ q σ F αβ ] isp2 W = Tr x e 2 αβ e x . (7) 2 Z0 s h | − | i A spin-1/2 fermion interacting with an external electromagnetic field is prescribed by the Lagrangian density The Heisenberg-Euler and Schwinger effective action for the Dirac equation per unit four volume is given by [1, 5] ∞ 1 ds 2 Re cosh(qXs) i ¯ ¯ ¯ −m s 2 = ψγαDαψ (Dαψ)γαψ mψψ. (1) eff = 2 3 e (qs) L 2 − − L −2(2π) Z0 s G Re sinh(qXs) h 2 2 where Dα = ∂α iqAα. The conventional method for 1 (qs) , (8) computing the QED− action is the S-matrix or the evolu- − − 3 Fi ˆ tion operator U, which is given by where is the Maxwell scalar and is the pseudo-scalar: F G ˆ 4 ¯ 1 1 S = Texp iq d xψγαψAα = F F αβ = B2 E2 , h− Z i F 4 αβ 2 − 4 1 = Texp i d x I . (2) = F ∗ F αβ = B E, (9) h Z L i G 4 αβ · Then, the out-state is expressed as and X = √2 +2i . The subtraction in Eq. (8) reg- ulates away thoseF termsG for renormalization of the vac- out = Sˆ† in , t = Uˆ(t, ) in , (3) | i | i | i −∞ | i uum energy and charge. In a pure electric field (B = 0), and the vacuum persistence amplitude is the Schwinger effect of pair production from the imaginary part of the QED action (8) would be dominated S (A)= 0, out 0, in = 0, in Sˆ 0, in . (4) by massless particle-antiparticle pairs if they exist since 0 h | i h | | i 3 the Compton wavelength for a massless particle is infi- nite and any electric field can produce pair of massless charged particles and antiparticles. The pair-production rate can be found from the Bogoli- ubov transformation between the in- and the-out vacua. The in-out formalism combined with the gamma-function ஶ ஶ regularization can recover the QED action (8), as will be shown in the next section. It can also be applied even to ෍ ෍ the massless QED by expressing the gamma function in ௡ୀ଴ ௠ୀ଴ terms of the zeta function [16], which will not be reviewed in this paper.

III. BOGOLIUBOV TRANSFORMATION FIG. 2: The Feynman diagrams for the one-loop effective action. The wavy lines denote the virtual photons and the dashed lines denote the virtual gravitons. The in-state of a charge under the interaction with an external electromagnetic field and/or a curves spacetime evolves into the out-state. These states are related to IV. IN-OUT FORMALISM FOR EFFECTIVE each other through the Bogoliubov transformation ACTION ∗ˆ† ˆ ˆ † aôutκ = µκaînκ + νκbinκ = UκaînκUκ, The Schwinger variational principle leads to the ef- ˆ ˆ ∗ † ˆ ˆ ˆ † boutκ = µκbinκ + νκaôutκ = UκbinκUκ, (10) fective action for a particle in an electromagnetic field and/or curved spacetime [13, 14] wherea ˆκ and ˆbκ are the annihilation operators for a particle and antiparticle carrying quantum number κ for the eiW = 0, out 0, in , (17) field, such as energy-momentum and spin etc. The quan- h | i tization rule based on the CTP theorem leads to the where W is the integrated action bosonic commutation relation † † 4 ′ ˆ ˆ ′ W = √ gd x eff . (18) [âoutκ, aôutκ′ ]= δκκ , [boutκ, boutκ′ ]= δκκ , (11) Z − L and the fermionic commutation relation The energy-momentum tensor is the variation of W with † † ′ ˆ ˆ ′ respect to the spacetime metric [6, 15]. In the in-out for- aôutκ, aôutκ′ = δκκ , boutκ, boutκ′ = δκκ . (12) { } { } malism, the integrated action can be expressed in terms The mean number of produced pairs is given by of the scattering matrix as N = ν 2. (13) κ | κ| W = i ln( 0, out 0, in ) = ( 1)2|σ|i ln(µ∗ ), (19) − h | i − V κ The Bogoliubov coefficients satisfy the relation Xκ 2 2|σ| 2 µκ ( 1) νκ =1. (14) where a factor is introduced to make the summation | | − − | | dimensionless. TheV one-loop effective action is equivalent The relation (14) lies at the root of the vacuum persis- to the sum of all Feynman diagrams in Fig. 2. In fact, the tence in the next section. summation of all Feynman diagrams in a constant electric For a bosonic field, the out-vacuum is the multi- field was attempted in Ref. [18], but it is not practical to particle states but unitary inequivalent representation of proceed in this direction. The one-loop effective action the in-vacuum [17] may become complex due to particle production from 0; out = Uˆ 0; in the background field. The most advantageous point of | i κ| i Yκ the in-out formalism is the consistent relation between ∗ nκ the complex effective action and the vacuum persistence 1 νκ = nκ, n¯κ; in . (15) for particle production µκ −µκ | i Yκ Xnκ 0, out 0, in 2 = e−2ImW , For a fermionic field, the out-vacuum is given by |h | i| 2|σ| 2|σ| 2ImW = ( 1) ln(1 ( 1) Nκ). (20) ˆ − V − − 0; out = Uκ 0; in Xκ | i | i Yκ = ν∗ 1 , 1¯ ; in + µ 0 , 0¯ ; in . (16) Kim, Lee and Yoon have introduced the so-called − κ| κ κ i κ| κ κ i gamma-function regularization method [17, 19, 20] Yκ The Pauli blocking suppresses production of more than W= i ln µ∗ = i ln Γ(a + ib (κ)). (21) ± κ ± λ λ one pair in the same state. Xκ Xλ Xκ 4

In the known cases aλ = 0, 1/2, 1. The gamma- function regularization originates± in± part from the proper-time integral

∞ ds e−as ln Γ(a + ib)= e−ibs . (22) Z s 1 e−s − · · · 0 − The final expression of the effective action should involve the renormalized quantities, which can be done by the Schwinger substraction scheme [1], which discards any divergent terms in Eq. (21). The contour integral or the wick-rotation in the complex plane results in the vacuum persistence (imaginary part) as well as the vacuum polar- FIG. 3: The contour for Eq. (28) in the first quadrant, which ization (real part) [21]. In the next section we shall ap- gives the principal value and the imaginary part for the vac- ply the gamma-function regularization to find the QED uum persistence in Eq. (30). action in a constant electric with or without a parallel magnetic field [17, 19, 20]. After summing over spin states, the logarithm of the gamma function, (22), takes the form

V. Γ-FUNCTION REGULARIZATION ∞ m2+p2 ∗ ds s i ⊥ s ln Γ( κ )= coth e 2qE . (28) − Z0 s 2 In this section we apply the gamma-function regularization method to compute the QED action in a constant Applying the Cauchy theorem along the contour in the electric field. The QED action in an electric field is in- Fig. 3, one obtains the following result teresting since the vacuum becomes unstable against the ∞ m2+p2 ∗ ds s − ⊥ s pair production and the quantum field theory should be ln Γ( κ ) = iP cot e 2qE − − Z s 2 able to properly handle the Schwinger effect. The in- 0 ∞ m2+p2 out formalism has proven a consistent field theoretical 1 −π ⊥ n + e qE . (29) method in this sense. n nX=1 Let us consider the Dirac equation for a spin-1/2 fermion in an electric field E(t) with a fixed direction. Finally, the effective action (19), after taking into account It has the momentum- and spin-state component equa- the density of states, gives the complex action tion 2 ∞ 2 1 qE ds − m s eff = P e qE cot s f 1 (s) 2 2 2 2 2 2 ∂t + m + p⊥ + (pk + qAk) +2iσE(t) φσp⊥ =0.(23) L 2 2π Z0 s − 2 m2+p2 qE d p⊥ −π ⊥ In the case of a constant electric field, the vector potential i ln 1 e qE . (30) − 2π (2π)2 − is A = Et. The asymptotic positive frequency solu- Z k − tions at t = are given by the parabolic function Here, P denotes the principal value and f 1 =1/s s/3 → ±∞ 2 − [17] subtracts two divergent terms in the principal integral

(−∞) and renormalizes the vacuum energy and charge. φσp⊥ = Dκ(ζ), We now consider spin-1/2 fermion in a constant electric

(∞) −iπκ √2π −i π (κ+1) and a constant parallel magnetic field. The Dirac equa- φ = e D ( ζ)+ e 2 D (iζ), σp⊥ κ − Γ( κ) −κ−1 tion projected into spin-eigenstates gives the Bogoliubov − coefficients [22] (24) ∗ √2πe−iπ(2κ +κ+1) ∗ where µ(σ) = , ν(σ) = e−iπκ , (31) (r)n Γ( κ) (r)n 2 2 − 2 i π 2σ 1 m + p⊥ ζ = e 4 (p qEt), κ = − i .(25) where n is the Landau level, σ = 1/2 is the spin eigen- r k qE − 2 − 2qE value along the magnetic field, r±= 1 is another spin ± Then, the Bogoliubov coefficients are eigenvalue along the electric field, and 2 √ −iπ(κ+1) r +1 m + qB(2n +1 2σ) 2πe −iπκ κ = + i − . (32) µp⊥ = , νp⊥ = e . (26) − 2 2qE Γ( κ) − The effective action and the Schwinger effect follow from The mean number of produced pairs is the spin-averaged Bogoliubov coefficients

2 2 m +p⊥ (σ) (σ) 1 (σ) (σ) 1 2 −π qE (σ) 2 (σ) 2 Np⊥ = µp⊥ = e . (27) µn = µ(1)nµ(−1)n , νn = ν(1)nν(−1)n . (33) | | 5

Then, the mean number of produced pairs in the n-th Kim and Schubert seriously considered the reconstruc- Landau level and the spin σ is tion of the effective action from the pair production [23]. If the argument of the logarithmic function of the imag- 2 − −π m +qB(2n+1 2σ) N (σ) = ν(σ) 2 = e qE . (34) inary part of the effective action can be factorized into n | n | a product of one plus or one minus exponential factors, The effective action follows from Eq. (19) as then the structure of simple poles and their residues of these factors uniquely determine the analytical structure qE qB of the proper-time integrand of the effective action: = i ln µ(σ)∗ . (35) Leff − 2π 2π n Xσn 2Im = ( 1)2|σ| ln 1 + ( 1)2|σ|N . (39) Leff − − κ Xκ After summing over the spin and Landau levels, one obtains the unrenormalized effective action The distribution function of produced particles in known physics has the general form ∞ 2 1 qE qB ds s Bs −i m s eff = coth cot e 2qE . 1 L 2 2π 2π Z s 2 2E Nκ = (40) 0 eβωκ + cos θ (36) where cos θ = 1 for the Fermi-Dirac (FD) distribution, cos θ = 1 for the Bose-Einstein (BE) distribution and Finally, the contour integral in the fourth quadrant gives − the complex effective action [22] cos θ = 0 for the Boltzmann (B) distribution. Then, the vacuum persistence becomes [24, 25] ∞ 1 ds 2 −m s 2|σ|+1 2|σ|+1 −βωκ eff = 2 P 3 e 2Im FD/BE = ( 1) ln 1 + ( 1) e , L −2(2π) Z0 s L − − Xκ (qBs)(qEs) (qs)2 1 (B2 E2) (41) ×htanh(qBs) tan(qEs) − − 3 − i ∞ and 2 i qE qB 1 − πm k πB qE 2|σ| 2|σ| −βωκ + e coth k . (37) 2Im B = ( 1) ln 1 + ( 1) e . (42) 2 2π 2π k E L − − kX=1 Xκ One can show that the vacuum persistence still holds The conjecture by Kim and Schubert is that the effective action may be reconstructed by the vacuum persis- qE qB (σ) tence as [23–25] 2Im eff = ln 1 Nn . (38) L − 2π 2π − ∞ Xσn 2σ+1 ds − βωκ s = ( 1) P e π LFD/BE − Z s Xκ 0 VI. RECONSTRUCTION OF EFFECTIVE (cos s)1−2|σ| ACTION f 1 −|σ|(s) , (43) ×h sin s − 2 i and One can obtain the pair-production rate from the ef- ∞ 2|σ| ds βωκ (cos s) fective action provided that the action has an imaginary 2|σ| − π s B = ( 1) P e f (s) , L − Z s sin s − |σ| part due to the instability of the Dirac vacuum. The Xκ 0 h i QED action by Heisenberg, Euer and Schwinger, for in- (44) stance, has poles in the proper-time integral. It is a consequence of the Cauchy theorem. In fact, the mean num- where the function f|σ| corresponds to the Schwinger sub- ber of produced pairs from the vacuum is related to the traction scheme to make the action finite. In QED the vacuum persistence as shown in Eq. (20). Now, one may subtracted terms correspond to the renormalization of raise another question whether one can find the effec- the vacuum energy and charge. tive action from the pair-production rate. The optical The argument leading to Eqs. (43) and (44) is the dispersion relation in particle physics is the most well- Mittag-Leffler’s theorem [26]. Simply stated, the theo- known example. There are various methods to compute rem reads that let bn, (n = 1, 2, ) be a sequence of ··· the pair-production rate or the mean number under the complex numbers satisfying bn as n and | | → ∞ → ∞ influence of a strong electric field in QED, chromelectric let g and h be two meromophic functions having sim- field in QCD, black holes and expanding universe etc. ple poles at b1,b2, such that Res[g,bn] = Res[h,bn] ··· Therefore, the inverse procedure will provide strong field for n = 1, 2, . Then, g h is an entire function. As ··· − 2 physics with a field theoretical method useful for under- the Mittag functions (entire functions, such as ez ) are standing the vacuum polarization in strong field, which independent of renormalization, it is highly likely that belongs to nonperturbative quantum regime. So, any sys- they are removed through renormalization and only those tematic method in this direction will open a new window functions with simple poles are physically relevant ones to the regime of nonperturbative quantum phenomena. for the vacuum polarization [23]. 6

VII. QED ACTIONS FROM field with or without a constant parallel magnetic field, RECONSTRUCTION CONJECTURE the information of the mean number straightforwardly leads to the one-loop effective action. We reconstruct the effective action from the conjecture by Kim and Schubert in section VI. As will be shown below, the reconstruction of the effective action is simple VIII. CONCLUSION and useful provided that the pair production is known by some means. First, in the case of a constant electric field, the com- Physics in a strong field exhibits nonperturbative ef- plex action gives the vacuum persistence fects, such as the Schwinger effect of pair production in an electric field and the Hawking radiation of particles in 2 2 2 qE d p m +p⊥ a black hole. Spontaneous particle production requires a ⊥ −π qE 2Im eff = 2 2 ln 1 e . (45) consistent framework for quantum field theory since the L − 2π Z (2π) − vacuum becomes unstable against particle production. Then, Eqs. (42) and (44) imply the effective action To explore the nonperturbative effect, one should be able to calculate the one-loop effective action, whose real part 2 ∞ m2+p2 qE d p⊥ ds − ⊥ s is the vacuum polarization, and whose imaginary part is = P e qE eff 2 the vacuum persistence. We have shown that the in-out L 2π Z (2π) Z0 s 1 s formalism provides complex one-loop effective actions, cot(s) + . satisfying the consistent requirement. In particular, the × − s 3 h i in-out formalism combined with the gamma-function ex- (46) plicitly gives the exact effective action once the Bogoli- Second, in the case of a parallel electric and magnetic ubov coefficient is known in terms of the gamma func- field, the vacuum persistence (38) implies the effective tions. We have also advanced a conjecture for recon- action struction of the effective action from the knowledge of the pair production. We have illustrated the QED action in ∞ 2 − 1 qE qB ds − m +qB(2n+1 2σ) s a constant electric and magnetic field. The reconstruc- eff = e qE L −2 2π 2π Z s tion conjecture will provide a very useful field theoretical Xσn 0 method in various areas of physics. 1 s cot s + ×h − s 3i ∞ 2 1 qE qB ds − m s = e qE Acknowledgements −2 2π 2π Z0 s B E B2 E2 coth s cot s + − . (47) This work was supported in part by Basic Science Re- × E − Bs2 3 h i search Program through the National Research Founda- Here, the dots denote the Schwinger substraction scheme tion of Korea(NRF) funded by the Ministry of Educa- for renormalization of the vacuum energy and charge. As tion(2015R1D1A1A01060626) and by Institute for Basic illustrated for the Schwinger effect in a constant electric Science (IBS) under IBS-R012-D1.

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