One-particle reducible contribution to the one-loop scalar propagator

Prepared for submission to JHEP

One-particle reducible contribution to the one-loop scalar propagator in a constant eld

James P. Edwards

a

a and Christian Schuberta

Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edicio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, México E-mail:

[email protected], [email protected]

Abstract: Recently, Gies and Karbstein showed that the two-loop Euler-Heisenberg La-

grangian receives a nite one-particle reducible contribution in addition to the well-known one-particle irreducible one.

Here, we demonstrate that a similar contribution exists for

the propagator in a constant eld already at the one-loop level, and we calculate this contribution for the scalar QED case.

We also present an independent derivation of the

Gies-Karbstein result using the worldline formalism, treating the scalar and spinor QED cases in a unied manner.

Contents

1 Introduction

1

2 Worldline rederivation of the Gies-Karbstein addendum to the EHL

3

2.1

Scalar QED

3

2.2

Spinor QED

6

3 An analogous addendum for the scalar propagator

6

3.1

The propagator in conguration space

7

3.2

The propagator in momentum space

9

4 Summary and outlook

1

9

Introduction

The Euler-Heisenberg Lagrangian (`EHL'), one of the rst serious calculations in quantum electrodynamics [1], describes the one-loop amplitude involving a spinor loop interacting non-perturbatively with a constant background electromagnetic eld. It is also the rst, and prototypical, example of an eective Lagrangian in eld theory. Euler and Heisenberg found for it the following well-known integral representation:

(1)

Lspin (a, b) = −

1 8π 2

Z

∞

0

dT −m2 T e2 ab e . T tanh(eaT)tan(ebT) (1.1)

Here

T

denotes the (Euclidean) proper-time of the loop fermion, and

2 two invariants of the Maxwell eld by a

− b2

=

B2

− E 2 , ab

= E · B.

a, b

are related to the

The superscript `(1)'

stands for `one-loop'. The EHL contains the information on nonlinear QED eects [2] such as photonphoton scattering [1, 36], photon dispersion [7, 8], and photon splitting [811]. Combined with spinor helicity methods, it allows one to explicitly compute the low-energy limit of the photon amplitudes for arbitrary

N

N-

in the helicity decomposition [12]. Its imaginary part

holds the information on Sauter-Schwinger pair production [13]. See [14] for a review of the uses of the EHL. For scalar QED, an analogous result was obtained by Weisskopf and Schwinger [13, 15]:

(1) Lscal (a, b)

1 = 16π 2

Z 0

∞

dT −m2 T e2 ab e . T sinh(eaT ) sin(ebT ) (1.2)

 1 

However, we will call this Lagrangian scalar EHL for simplicity. The Lagrangians (1.1),(1.2) require renormalization, but we will not bother here to include the corresponding counterterms, since they will not play a role as far as the present paper is concerned. The rst radiative corrections to these Lagrangians, describing the eect of an additional photon exchange in the loop, were obtained in the seventies by Ritus [1618]. Using the exact propagators in a constant eld [13, 19], Ritus obtained the corresponding twoµ loop eective Lagrangians

k

(2)

Lscal,spin

in terms of certain two-parameter integrals.

Similar

x0

(2) two-parameter integral representations for Lscal,spin were obtained later by other authors, using either proper-time [20, 21] or dimensional regularisation [22, 23].

Although these

integrals are intractable analytically, closed-form expressions have been obtained for their kµ

weak-eld expansions for the purely electric or magnetic cases [34, 35]. In 1+1 dimensions, x0

x

0 0 the (spinor) EHL has also been calculated at the three-loop level, leading to four-parameter x0

x

integrals [2426]. In all these calculations it was assumed that the only diagram contributing to the EHL at the two-loop level is the one particle irreducible (`1PI') one shown in Fig. 1 (the double line as usual denotes the electron propagator in a constant eld). kµ

x0

x

x0

Figure 1. One-particle irreducible contribution to the two-loop EHL. kµ 2

At the same loop order, there is also the one-particle reducible (`1PR') diagram shown in Fig. 2.

x0

x

1 Figure 2. One-particle reducible contribution to the two-loop EHL.

However, since the one-photon amplitude in a constant eld formally vanishes on account of momentum conservation, this diagram was generally believed not to contribute. Recently, Gies and Karbstein [27] made the stunning discovery that actually this diagram

does

give a nite contribution, if one takes into account the divergence of the con-

necting photon propagator in the zero-momentum limit. A careful analysis of that limit led them to the simple formula

 2 

∂L(1) ∂L(1) . ∂F µν ∂Fµν

L(2)1PR =

(1.3)

Thus essentially all previous applications of the two-loop EHL now have to be reanalyzed to take the eects of this term into account. The purpose of the present paper is to point out that a similar 1PR addendum exists also for the QED scalar or electron propagators, here already at the one-loop level: usually in a constant eld one considers only the 1PI diagram shown in Fig. 3,

x0

x

Figure 3. One-loop propagator in a constant eld. but there is also a nite contribution from the 1PR one shown in Fig. 4 below. For the scalar QED case, we will explicitly obtain the contribution of this diagram, and obtain a generalization of (1.3). As a preparation for that calculation, we will rst rederive the Gies-Karbstein result in a way that allows us to combine the scalar and spinor QED cases, using the worldline formalism [21, 2833] (only the spinor QED case was considered in [27]).

2

Worldline rederivation of the Gies-Karbstein addendum to the EHL

2.1 Scalar QED The worldline path integral representation of the one-loop one-photon scalar QED amplitude in a constant eld

(1) Γscal [k, ε; F ]

Fµν

Z = −ie 0

∞

is [21, 32, 33]

dT −m2 T e T

Z

T

Z Dx

dτ ε · x˙ eik·x(τ ) e−

RT 0

x˙ 2 +iex·A ˙ 4

dτ




.

x(T )=x(0) 0 (2.1)

Here the momentum

k

2

is ingoing. Separating o the loop center of mass

x0 ,

x(τ ) = x0 + q(τ ) x0 ,

and choosing Fock-Schwinger gauge centered at

(1) Γscal [k, ε; F ]

Z = −ie

D

d x0 e

ik·x0

Z 0

∞

dT −m2 T e T

Z

(2.2)

one has [29]

q(T )=0

Z

Dq q(0)=0

T ik·q(τ )

dτ ε · q˙ e

e

−

RT 0

dτ

q˙ 2 + 21 ieq·F ·q˙ 4




.

0 (2.3)

 3 

The

x0

integration produces the delta function for momentum conservation.

Performing

the gaussian path integration along the lines of [21, 28, 32] gives

(1) Γscal [k, ε; F ]

∞

  1 sinZ dT 2T −D −m − = −e(2π) δ (k) (4πT ) 2 e det 2 T Z 0 Z T h1 i × dτ ε · G˙B · k exp k · GB · k 2 0 Z

D D

(2.4) where

Zµν ≡ eFµν T ,

and

GB , G˙B

are the coincidence limits of the constant eld worldline

Green's function and its rst derivative:

 T  Z · cot Z − 1 , 2Z 2 2Z i =i GB . ≡ G˙B (τ, τ ) = icotZ − Z T

GB ≡ GB (τ, τ ) = G˙B

(2.5) Since these coincidence limits are independent of

(1) Γscal [k, ε; F ]

D D

τ,

we can simplify (2.4) to

∞

Z

= −e(2π) δ (k)

dT (4πT ) 0

×ε · G˙B · k exp

−m2 T −D 2

h1 2

e

det

− 12



sinZ Z



i k · GB · k . (2.6)

From (2.6), we can now simply construct the reducible diagram 2 by taking two copies of it, and connecting them with a photon propagator in Feynman gauge. This gives

(2)1PR Γscal

dD k (1) (1) 0 0 Γ [k, ε; F ]Γ [k , ε ; F ] 0 scal scal D 2 (2π) k k →−k,εµ ε0ν →η µν   Z Z Z ∞ dD k 2 D ik·x0 −D −m2 T − 21 sinZ 2 det d x0 e dT (4πT ) e =e (2π)D k 2 Z 0   Z Z ∞ 0 D 1 sinZ 0 2 0 × dD x00 e−ik·x0 dT 0 (4πT 0 )− 2 e−m T det− 2 Z0 0 h1 i 0 0 ×k · G˙B · G˙B · k exp k · (GB + GB )·k . 2 Z

=

Here we have used that

G˙B (τ, τ )

is an odd function of the antisymmetric matrix

(2.7)

F,

and

therefore antisymmetric too. Now we change variables from of mass

x+

x0 , x00

to

x+ ≡ 21 (x0 + x00 ), x− ≡ x0 − x00 .

The center

remains xed, and will be the argument of the two-loop eective Lagrangian.

Integrating out

x− ,

we get

 4 

(2)1PR Lscal (x

∞



 sinZ dT (4πT ) e ≡ x+ ) = e det Z 0   Z ∞ 0 0 0 −D −m2 T 0 − 21 sinZ 2 dT (4πT ) e × det Z0 0 Z h1 i 0 ·k k · G˙B · G˙B 0 × dD k δ D (k) exp k · (G + G ) · k . B B k2 2 2

Z

−D −m2 T 2

− 12

(2.8)

The crucial observation of [27] was that, although the presence of the delta function would seem to kill the one-photon amplitude (2.6), after the sewing this is not the case anymore; terms with only two factors of momentum in the numerator will survive the momentum integration, and give a nite result. Thus, in the presence of the delta function, the exponential factor in the last line can be replaced by unity. Lorentz invariance can then be invoked to set

Z

dD k δ D (k)

η µν kµ kν = . k2 D

(2.9)

Thus one has nally

(2)1PR

Lscal

∞

 sinZ Z   Z 0∞ 0 1 sinZ 2T 0 0 0 −D −m − × dT (4πT ) 2 e det 2 Z0 0
1 0 × tr G˙B · G˙B . D

= e2

Z

D

2

1

dT (4πT )− 2 e−m T det− 2



(2.10)

Now, we use the fact that the scalar EHL (1.2) can also be rewritten more compactly as [29]

Z

(1)

Lscal (F ) =

0

∞

  D 1 sinZ dT 2 (4πT )− 2 e−m T det− 2 . T Z (2.11)

Using the identity

ln det = tr ln

as well as

−iG˙B = cot Z − and setting

D = 4,

1 d sin Z = ln Z dZ Z

it is then easy to verify the Gies-Karbstein equation (1.3).

 5 

(2.12)

2.2 Spinor QED The generalization of the one-photon amplitude (2.6) to the spinor-loop case in the worldline formalism involves, apart from the global normalization, only a change of the determinant factor, and an application of the Bern-Kosower rules [36, 37] to replace where

G˙B

by

G˙B − GF ,

GF = −itanZ : (1) Γspin [k, ε; F ]

∞

  dT −D −m2 T − 12 tanZ 2 = 2e d x0 e (4πT ) e det T Z 0 Z T h1 i dτ ε · (G˙B − GF ) · k exp k · GB · k . × 2 0 Z

D

ik·x0

Z

(2.13) The procedure is the same as in the scalar case mutatis mutandis: the spinor QED EHL should be rewritten as [29]

(1) Lspin (F )

Z = −2 0

∞

  1 tanZ dT 2T −m − −D det 2 (4πT ) 2 e T Z (2.14)

x0

and instead of the identity (2.12) one has to use

x

d tanZ 1 + tanZ = ln . Z dZ Z

−i(G˙B − GF ) = cot Z −

(2.15)

One then nds the same identity as in the scalar case, eq. (1.3).

3

An analogous addendum for the scalar propagator

We will now generalize the Gies-Karbstein addendum to the case where, instead of two closed loops, we have one closed loop and an open line, i.e.

to the self-energy diagram

shown in Fig. 4. Here we will restrict ourselves to the scalar QED case.

x0

kµ

x0

x

Figure 4. One-particle reducible contribution to the one-loop scalar propagator in a 2 constant eld.

 6 

3.1 The propagator in conguration space Let us start in

x-space.

In addition to the one-photon amplitude in the eld, we now

also need the scalar propagator

0

Dxx [k, ε; F ]

between points

x

and

x0 ,

in the constant eld

background and with one photon attached. Similarly to (2.7), we can then construct the 1PR self-energy diagram

xx0 (1)1P R

Dscal

0

Dxx (1)1P R

Z =

by sewing,

dD k (1) 0 0 xx0 Γ [k , ε ; F ]D [k, ε; F ] . 0 scal (2π)D k 2 scal k →−k, εµ ε0ν →η µν (3.1)

In the worldline formalism, the amplitude

0

Dxx [k, ε; F ]

can be obtained from the open-line

master formula of [38]. Specializing eq. (3.7) there to the one-photon case, one gets

  Z ∞ 1 D sin Z − 1 x− Z·cot Z·x− 0 0 2 Dxx [k, ε; F ] = (−ie) eik·x dT e−m T (4πT )− 2 det− 2 e 4T Z 0  µ τ Z T x− 2ie i k·x− +k·∆(τ,τ )·k+ 2e x ·F ·◦∆(τ )·k • ◦ • T − ^ ^ ×εµ dτ − 2i ∆(τ, τ ) · k − x− · F · ∆ (τ ) e T . ^ ^ T T 0 (3.2) Here

x− ≡ x − x0 ,

and instead of the worldline Green's function

the string-inspired condition (`SI') ing Green's function

∆(τ, 0) = 0. ^

∆(τ, τ 0 ) ^

RT 0

obeying

dτ (dτ 0 )G

GB (τ, τ 0 ),

which fullls

(τ, τ 0 )

= 0, we encounter the correspondB 0 Dirichlet boundary conditions (`DBC') ∆(0, τ ) = ^

Since those boundary conditions break the translation invariance in

τ,

this

Green's function is not a function of the dierence of the arguments, so that one has to distinguish between its two partial derivatives. A convenient notation [38, 39] is to denote dierentiation with respect to

τ (τ 0 )

by a

•,

and the denite integral

RT 0

to the left (right) respectively. For the same reason, also the coincidence and its derivatives are not constants any more, so that the

τ -integral

RT

dτ 0 ) by a ◦ 0 limits of ∆(τ, τ )

dτ

(

0

^

will have to be dealt

with. From the sewing relation (3.1) together with (2.6) and (2.9) it is clear that only the terms linear in independent of

k in the integrand of (3.2) will k , but it can be omitted, since

survive the sewing (there is also a term it leads to a

k -integral

that vanishes by

antisymmetry). Collecting those gives

  Z ∞ D 1 sin Z − 1 x− Z·cot Z·x− 0 2 Dxx [k, ε; F ] = (−ie) dT e−m T (4πT )− 2 det− 2 e 4T Z k 0     Z T 2e x− 2ie τ × dτ k · ix0 + i x− − ◦∆T (τ ) · F · x− − x− · F · ◦∆• (τ ) − 2i•∆T (τ, τ ) · ε . ^ ^ T T ^ T T 0 (3.3) Combining this equation with (3.1), (2.6) and (2.9) we get

 7 

  Z 0 1 sinZ e2 ∞ 0 2T 0 0 −D −m − dT (4πT ) 2 e = −i det 2 D 0 Z0   Z ∞ D 1 sin Z − 1 x− Z·cot Z·x− 2 × dT e−m T (4πT )− 2 det− 2 e 4T Z 0      τ 2e x− 2ie 0 ×tr G˙B · ix0 + i x− − ◦∆T (τ ) · F · x− − x− · F · ◦∆• (τ ) − 2i•∆T (τ, τ ) . ^ ^ T T ^ T T

xx0 (1)1P R Dscal

(3.4) For the calculation of the function

∆ ^

τ

integral, it will be convenient to rewrite the DBC Green's

in terms of the SI one

GB ,

using the relation [38]

2∆(τ, τ 0 ) = GB (τ, τ 0 ) − GB (τ, 0) − GB (0, τ 0 ) + GB (0, 0) .

(3.5)

^

Using the collection of formulas given in [38], it is then easy to compute (note that all matrices appearing in the integrand are built from the same

Fµν ,

and thus all commute

with each other):

T

Z

dτ (◦∆)T (τ ) = ^

0

Z

0

T

(3.6)

dτ ◦∆• (τ ) = 0 ,

(3.7)

^

T2 GB , ^ 2 0 Z T T dτ (•∆)T (τ, τ ) = − G˙B , ^ 2 0
i iT 4 h ˙ 2 dτ ◦∆• (τ ) · (◦∆)T (τ ) = · 2i G + cot(Z) − Z · csc Z . B ^ ^ 8Z 2 Z

Z

0 T

T

T2 GB , 2

dτ ◦∆• (τ )τ = −

Here it is understood that, on the right-hand side,

GB

and

G˙B

(3.9)

(3.10)

are taken at their coincidence

limits, given in (2.5). Putting things together, and now also setting

xx0 (1)1P R Dscal

(3.8)

D = 4,

we have

  Z 0 1 sinZ e2 ∞ 0 2T 0 0 −D −m − = −i dT (4πT ) 2 e det 2 4 0 Z0   Z ∞ D 1 sin Z − 1 x− Z·cot Z·x− 2 × dT e−m T (4πT )− 2 det− 2 e 4T Z 0 h i
1 0 × iT tr(G˙B · G˙B ) − x− · cot Z − Z · csc2 Z · G˙ 0 B · x− + ix · G˙ 0 B · x0 . 2 (3.11)

One would expect this to fulll an equation generalizing the Gies-Karbstein equation (1.3). And indeed, comparing (3.11) with (1.2) and the corresponding representation of the photonless scalar propagator in the eld [38, 40]

 8 

xx0 Dscal [F ]

Z =

∞

−m2 T

dT e

−D 2

(4πT )

− 12



det

0

 sin Z − 1 x− Z·cot Z·x− e 4T Z (3.12)

it is easy to see that

(1)

0

(1)

xx ∂L ∂Dscal ie xx0 µ ∂Lscal 0ν scal = + Dscal x x . ∂Fµν ∂F µν 2 ∂F µν

xx0 (1)1P R Dscal

(3.13)

We remark that the form of the second term is specic for our choice of Fock-Schwinger gauge centered at

x0 .

However, it is well-known how to convert our expression for the scalar

propagator into an arbitrary gauge, as outlined in [2].

3.2 The propagator in momentum space Fourier transforming (3.13) we obtain the corresponding equation for the momentum space propagator:

(1)

(1)1P R Dscal Here

Dscal (p)

∂Dscal (p) ∂Lscal = . ∂Fµν ∂F µν

(3.14)

denotes the constant eld propagator in momentum space [20, 38]

∞

Z Dscal (p) =

dT e−m

2T

e−T p·

tanZ ·p −1 Z det 2

[cos Z] .

(3.15)

0 Note that the second term on the right-hand side of (3.13) does not survive the Fourier transform, since the momentum space propagator depends only on a single momentum, so (1)

that the antisymmetric matrix

(1)1P R Dscal (p)

∂Lscal ∂F µν cannot be saturated. Explicitly, one nds

 Z ∞ Z 1 1 tanZ ie2 ∞ 0 sin Z 0 2T 0 2 0 −D −m − dT (4πT ) 2 e det 2 dT T e−m T det− 2 [cos Z] e−T p· Z ·p = 0 2 0 Z 0 h i sin Z · cos Z − Z ˙ 0 1 ˙B0 ) . × Tp · · G · p + tr(tanZ · G (3.16) B Z 2 · cos2 Z 2

We have also veried (3.16) by a direct momentum space calculation, based on the momentum space worldline representation of the one-photon dressed scalar propagator given in [38].

4

Summary and outlook

We have shown that a nite 1PR diagram, similar to the one found in [27] for the twoloop EHL, exists for the scalar or electron propagator in a constant eld already at the

 9 

one-loop level. For the scalar propagator, we have explicitly calculated this term in FockSchwinger gauge using the worldline formalism, leading to equations generalizing the GiesKarbstein equation [27] in both

x-space

and momentum space.

The calculation for the

spinor propagator is more involved and will be presented elsewhere. It will be interesting to see how this 1PR contribution aects quantities associated to the QED propagator in a constant eld such as the leading asymptotic

ln2

eB term in a m2

strong magnetic eld (see, e.g. [41] and refs. therein) or the famous Ritus mass shift [42]. Clearly, the Gies-Karbstein mechanism will lead to a plethora of such terms at higher loop orders in constant-eld QED. Finally, we deem it important to stress the fact that the non-vanishing of the GiesKarbstein addendum relies on the photon being exactly massless.

Therefore the experi-

mental verication of terms of this type might be promising as a way to improve on the existing lower bounds on the photon mass.

Acknowledgments We would like to thank Idrish Huet for helpful discussions. Both authors thank CONACYT for nancial support through grant Ciencias Basicas 2014 No. 242461.

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