arxiv:2111.01980v1 [hep-ph] 3 nov 2021

November 8, 2021 18:47 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in schubert page 1

1

Tadpole contribution to magnetic photon-graviton conversion

N. Ahmadiniaz

Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, GermanyE-mail: [email protected]

F. Bastianelli

Dipartimento di Fisica ed Astronomia, Universita di Bologna, Via Irnerio 46, I-40126 Bologna,

Italy and INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy

E-mail: [email protected]

F. Karbstein

Helmholtz-Institut Jena, Frobelstieg 3, 07743 Jena, Germany and Theoretisch-Physikalisches

Institut, Friedrich-Schiller-Universitat Jena, Max-Wien-Platz 1, 07743 Jena, GermanyE-mail: [email protected]

C. Schubert

Instituto de Fısica y Matematicas Universidad Michoacana de San Nicolas de Hidalgo Edificio

C-3, Apdo. Postal 2-82 C.P. 58040, Morelia, Michoacan, Mexico

E-mail: [email protected]

Photon-graviton conversion in a magnetic field is a process that is usually studied at treelevel, but the one-loop corrections due to scalars and spinors have also been calculated.

Differently from the tree-level process, at one-loop one finds the amplitude to depend on

the photon polarization, leading to dichroism. However, previous calculations overlookeda tadpole contribution of the type that was considered to be vanishing in QED for decades

but erroneously so, as shown by H. Gies and one of the authors in 2016. Here we compute

this missing diagram in closed form, and show that it does not contribute to dichroism.

Keywords: photon-graviton; tadpole; Einstein-Maxwell

1. Introduction: photon-graviton conversion

Einstein-Maxwell theory contains a tree-level vertex for photon-graviton conversion

in a constant electromagnetic field:

1

2κhµν

(Fµαfνα + fµα F

να)− 1

4κhµµF

αβfαβ . (1)

Here hµν denotes the graviton, fµν the photon, Fµν the external field, and κ the

gravitational coupling constant.

This interaction leads to photon-graviton oscillations similar to the better-known

neutrino or photon-axion oscillations1–11 (see12 for a recent application to gravita-

tional waves).

In momentum space, this vertex becomes

Γ(tree)(k, ε;F ) = εµνεαΠµν,α(tree)(k;F ), Πµν,α

(tree)(k;F ) = − iκ2Cµν,α (2)

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1


2

with

Cµν,α = Fµαkν + F ναkµ −(F · k

)µδνα −

(F · k

)νδµα +

(F · k

)αδµν . (3)

Since in a constant field the four-momentum is preserved, kµ here is the four-

momentum of the photon as well as of the graviton.

For the photon polarizations, as is customary we will use the Lorentz frame

where E and B are collinear, and choose the polarization basis εα⊥, εα‖ , where εα‖ lies

in the plane spanned by k and B or E, and εα⊥ is perpendicular to it. For convenience

we construct also the graviton polarization tensor using the same vectors:

ε⊕µν = ε⊥µε⊥ν − ε‖µε‖ν , ε⊗µν = ε⊥µε‖ν + ε‖µε⊥ν . (4)

From the CP-invariance of Einstein-Maxwell theory one can then derive the follow-

ing selection rules:

• For a purely magnetic field ε⊕ couples only to ε⊥ and ε⊗ only to ε‖.• For a purely electric field ε⊕ couples only to ε‖ and ε⊗ only to ε⊥.

2. One-loop photon-graviton vacuum polarization

In13,14 the worldline formalism was used along the lines of15–17 to study the one-loop

corrections to this amplitude due to a scalar or spinor loop, see Fig. 1.

Fig. 1. One-loop correction to the photon-graviton amplitude in a constant field.

Here we employ the usual double-line notation for the full propagator in the

external electromagnetic field, Fig. 2., whereΓ

1-loopHE [A] =

= + + + · · ·

FIG. 1: Diagrammatic representation of the one-loop Heisenberg-Euler effective action. The dou-

ble line denotes the dressed fermion propagator accounting for arbitrarily many couplings to the

external field A, represented by the wiggly lines ending at crosses.

with Γl-loopHE ∼ (α

π)l−1, where α = e2

4π" 1

137is the fine-structure constant; we use the Heaviside-

Lorentz System with c = ! = 1. At each loop order l, Γl-loopHE =

∫xLl-loop

HE accounts for an

infinite number of couplings to the external field, and thus is fully nonperturbative in the

parameter eA. For completeness, we sketch the expansion to two-loop order in the following.

We begin by noting that the fermionic integral in Eq. (7) can be written as a functional

determinant,

iSψ[A + q] = ln det(−i /D[A + q] + m

). (21)

If evaluated at q = 0, this quantity amounts to the one-loop Heisenberg-Euler effective action

in the external field A, i.e., Γ1-loopHE [A] = Sψ[A]; for a graphical representation, cf. Fig. 1.

Since Sψ is a one-loop expression, the two-loop order of the Schwinger functional is already

obtained by performing the photonic fluctuation integral ∼ Dq to Gaußian order. For this,

we expand Sψ about the external field A,

Sψ[A + q] = Sψ[A] +

∫ (S

(1)ψ [A]

)µqµ +

1

2

∫∫qµ

(S

(2)ψ [A]

)µνqν + O(q3), (22)

where we employed the shorthand notation

(S

(n)ψ

)σ1...σn[A] :=

δnSψ[A]

δAσ1 . . . δAσn

∣∣∣∣A=A

. (23)

The first-order term corresponds to a one-loop photon current induced by the field A, and

the Hessian is related to the one-loop photon polarization tensor Πµν [A] :=(S

(2)ψ

)µν[A]

evaluated in the external field A; for completeness note that this definition of the photon

polarization tensor differs from that of [57] by an overall minus sign. To Gaußian order, we

ignore the terms of O(q3) in the exponent, resulting in

eiW [A] " eiSψ [A]

∫Dq ei

∫ (S

(1)ψ [A]

)µ

qµ− i2

∫∫qµ

(D−1−Π[A]

)µν

qν . (24)

10

Fig. 2. Full scalar or spinor propagator in a constant field.

These calculations lead to the same type of two-parameter integrals as for

the well-studied case of the one-loop photon vacuum polarisation in a constant

field18–29. Both the scalar and the spinor-loop amplitudes are UV divergent, but

multiplicatively renormalizable. For example, the scalar-loop contribution in di-

mensional regularization displays the pole

Πµν,αscal,div(k) =

ie2κ

3(4π)21

D − 4Cµν,α


3

where Cµν,α is the tree level vertex (3).

For studying the relative importance of the one-loop amplitudes it is useful to

normalize them by the tree-level amplitude ( the ‘bar’ on Π denotes renormalization)

ΠAascal,spin(ω, B, E) ≡

ΠAascal,spin(ω, B, E)

− i2κC

Aa(5)

where A = ⊕,⊗ and a =⊥, ‖ , ω = ωm , B = eB

m2 , E = eEm2 .

It then becomes obvious that the one-loop amplitudes become quantitatively

relevant only for field strengths close to the critical ones B, E ≈ 1. Macroscopic

fields of this magnitude are known to exist only for the magnetic case, so that we

will set E = 0 in the following. In14 it was shown that for this purely magnetic case

the parameter integrals allow for a straightforward numerical evaluation as long as

the photon energy ω is below the pair-creation threshold ωcr,

ω⊕⊥cr,scal = ω⊗‖cr,scal = 2

√1 + B ,

ω⊕⊥cr,spin = 1 +

√1 + 2B ,

ω⊗‖cr,spin = 2 .

(6)

There also a number of special cases were studied that allow for more explicit

representations:

(1) The case of small B and arbitrary ω leads to single - parameter integrals over

Airy functions.

(2) In the large B limit one finds a logarithmic growth in the field strength,

ΠAascal(ω, B)

B→∞∼ − α

12πln(B) , (7)

ΠAaspin(ω, B)

B→∞∼ − α

3πln(B) . (8)

(3) In the limit of vanishing photon energy the amplitudes can be related to the

corresponding one-loop effective Lagrangians:

Π⊕⊥scal,spin(ω = 0, B) = −2πα

m4

( 1

B

∂

∂B+

∂2

∂B2

)LEHscal,spin(B) , (9)

Π⊗‖scal,spin(ω = 0, B) = −4πα

m4

1

B

∂

∂BLEHscal,spin(B) . (10)

Here LEHscal,spin(B) denotes the one-loop effective Lagrangian in a constant mag-

netic field, obtained for the spinor QED case by Heisenberg and Euler30 and


4

for scalar QED by Weisskopf31:

LEHscal(B) = − m4

16π2

∫ ∞

0

ds

s3e−is

[Bs

sin(Bs)− (Bs)2

6− 1

], (11)

LEHspin(B) =

m4

8π2

∫ ∞

0

ds

s3e−is

[Bs

tan(Bs)+

(Bs)2

3− 1

]. (12)

3. Dichroism

For realistic parameters, the one-loop corrections turn out to be small compared

to the tree-level amplitudes. However, there is also a qualitative difference to the

tree-level amplitude. As has been emphasized in9, the tree level photon-graviton

conversion does, contrary to the better known photon-axion case, not lead to a

dichroism effect for photon beams. This is because both photon polarization com-

ponents have equal conversion rates. This symmetry gets broken by the loop cor-

rections. Although this effect is, of course, tiny and hardly measurable in the near

future, an exhaustive analysis by Ahlers et al.32 has shown that it is still the lead-

ing contribution to magnetic dichroism in the standard model (including standard

gravity)!

4. Quantum electrodynamics in external fields

In vacuum QED, there is, of course, no one-photon amplitude because of Furry’s

theorem. In the presence of an external field, however, one-photon tadpole diagrams

such as the one in Fig. 3 will in general be nonzero.

Eq. (2) follows straightforwardly from Eq. (4).

Prior to the recent analysis performed in Ref. [1], this contribution was believed to vanish

in constant fields [7, 9], based on the argument that (2π)dδ(d)(k) kα = 0 in Eq. (5), thereby

missing the subtlety that, upon convolution with the infrared divergent photon propagator,

the current (5) might induce finite 1PR tadpole contributions in Feynman diagrams. Finally,

note that L2-loopHE may alternatively be viewed as a one-loop calculation performed with the

charged particle propagator in the constant field already accounting for quantum corrections

at one-loop level. The latter is depicted in Fig. 2. Identifying the in- and outgoing charged

+

FIG. 2: One-loop corrections to the charged particle propagator in an external electromagnetic

field.

particle lines in Fig. 2 to form loops, the diagrams shown in Fig. 1 are recovered.

B. Charged particle propagators in constant electromagnetic fields

Meanwhile, Refs. [5, 6] have explicitly determined the one-loop tadpole corrections for

the propagators of both charged scalar and spinor particles in constant fields within the

worldline formalism. The respective Feynman diagram is depicted in Fig. 2 (right). Their

finding is that in momentum space this 1PR contribution to the charged particle propagator

can be expressed similarly to Eq. (2) as

G1-loop(k|F )∣∣1PR

=∂G0-loop(k|F )

∂F µν

∂L1-loopHE

∂Fµν, (7)

where G0-loop(k|F ) denotes the charged scalar/spinor propagator dressed to all orders in the

constant external field F [4, 15], depicted as double solid line in Figs. 1 and 2.

III. FROM 1PI TO 1PR TADPOLE DIAGRAMS IN CONSTANT FIELDS

Here, we argue that the structural similarity of Eqs. (2) and (7) is not a coincidence,

but rather a direct consequence of the fact that no momentum can be transferred from the

induced photon current (5) in constant fields. We find that this implies that all 1PR tadpole

contributions to a given quantity follow by differentiations of lower loop diagrams for the

5

Fig. 3. Full scalar or spinor propagator in a constant field.

If the external field is constant, then the one-photon amplitude Fig. 4 can still

be shown to vanish.

kµ

Fig. 4. One photon amplitude in a constant field.

The argument goes as follows:


5

(1) A constant field emits only photons with zero energy-momentum, thus there is

a factor of δ(k).

(2) Because of gauge invariance, this diagram in a momentum expansion starts with

the term linear in momentum.

(3) δ(k)kµ = 0.

Since the tadpole vanishes, it has been assumed for decades that also any dia-

gram containing it can be discarded. For example, in the book “Quantum Electro-

dynamics with Unstable Vacuum” by E.S. Fradkin, D.M. Gitman and S.M. Shvarts-

man34 it is stated (on page 225) even more generally that, “Thus, in the constant

and homogeneous external field combined with that of a plane-wave, all the diagrams

containing the causal current J µ(x) (i.e., those containing tadpoles having a causal

propagator Sc(x, y)), are equal to zero.”

Sometimes also additional arguments have been given; in the book “Effective

Lagrangians in Quantum Electrodynamics” by Dittrich and Reuter33 it is argued

that the “handcuff” diagram of Fig. 5 vanishes because of Lorenz invariance.

+ + + . . .

FIG. 3: One-particle reducible diagrams constituting the last term in Eq. (26). For the definition

of the double line, cf. Fig. 1.

Γ2-loopHE [A] = +

FIG. 4: Diagrams constituting the two-loop Heisenberg-Euler effective action. Obviously, we have

Γ2-loopHE = Γ2-loop

HE

∣∣1PI

+Γ2-loopHE

∣∣1PR

. Note that the first diagram amounts to the leftmost one in Fig. 2,

where it is drawn in a slightly different way; for the definition of the double line, cf. Fig. 1.

corresponding to the Dyson series of the full one-loop resummed photon propagator. In the

last term of Eq. (26), this resummed propagator interconnects two one-loop photon currents

∼(S

(1)ψ [A]

). All the diagrams arising when adopting the expansion (29) in the last term in

Eq. (26) are one-particle reducible; see Fig. 3.

In turn, the two-loop Heisenberg-Euler effective action consists of a 1PI and a 1PR

diagram and is given by

Γ2-loopHE [A] =

1

2Tr(DΠ[A])

︸︷︷︸=:Γ2-loop

HE

∣∣1PI

+1

2

∫∫ (S

(1)ψ [A]

)µDµν

(S

(1)ψ [A]

)ν

︸︷︷︸=:Γ2-loop

HE

∣∣1PR

. (30)

The existence as a matter of principle of the 1PR term in Eq. (30) has been known for a

long time. It has, however, been argued that this term vanishes for constant external fields

[4, 54]. Let us reproduce this argument for reasons of completeness: a crucial building block

of the 1PR term is(S

(1)ψ [A]

)ν, which corresponds to the one-loop photon current which will

be called jµ1-loop[A] below. For a constant external field, jµ

1-loop[A] does not depend on any

spacetime point x either. On the other hand, jµ1-loop[A] is a Lorentz 4-vector. The vector

index of the current can only be generated from the building blocks F , ∂ and x. However,

for constant fields ∂µF νκ = 0 and for an x independent current, all conceivable combinations

with one vector index vanish and so does the current (an explicit verification of this fact in

momentum space is given below).

While this part of the argument holds true in the full analysis, it does not necessarily

imply that the 1PR diagram in Fig. 4 vanishes. In fact, the two currents in the 1PR diagram

are convoluted with a photon propagator, describing a long-range force with an IR singularity

∼ 1/p2 in the propagator. Hence, it is a quantitative question as to whether the currents

approaching zero are outbalanced by the IR singularity of the photon propagator. In the

12

Fig. 5. Handcuff diagram in a constant field.

However, in 2016 H. Gies and one of the authors35 noted that such diagrams

can give finite values because of the infrared divergence of the connecting photon

propagator. In dimensional regularization, the key integral is∫dDk δD(k)

kµkν

k2=ηµν

D. (13)

Applying this integral to the handcuff diagram one finds a non-vanishing result,

which can be expressed in the following simple way in terms of the one-loop Euler-

Heisenberg Lagrangian36

L1PRspin =

1

2

∂L(EH)spin

∂Fµν∂L(EH)

spin

∂Fµν

(the superscript ‘1PR’ stands for “one-particle reducible”). This adds on to the

standard diagram for the two-loop Euler-Heisenberg Lagrangian, studied by V.I.

Ritus39 half a century ago:

L(EH)2−loopspin = (14)

Along the same lines, it was found in37,38 that the one-loop tadpole contribution

Fig. 3 to the scalar or spinor propagator in a constant field is also non-vanishing,


6

and given by

S1PR(p) =∂S(p)

∂Fµν

∂L(EH)

∂Fµν

where S(p) denotes the tree-level propagator in the field.

5. Tadpole contribution to the photon-graviton amplitude

Returning to the photon-graviton amplitude in a constant field, the point of the

present talk is that this amplitude, too, has a previously overlooked tadpole contri-

bution, shown in Fig. 6.

458 The coupling to gravity and electromagnetism

4)

p p0

k0 = k + k0

k

k0

=========

p p0

k0

k0

k

=========

6)

k k = �k0

7)

k k

+

k k

+

k k

3

Figure 17.6 Tree-level vertex for photon-graviton conversion in a constant field.

NOTES). This tree-level vertex we can read off from (17.14),

�(tree)(k0, ✏0; k, "; F ) = ✏0µ⌫"↵⇧µ⌫,↵(tree)(k; F ), ⇧µ⌫,↵

(tree)(k; F ) = � i

2Cµ⌫,↵ (17.68)

where the tensor Cµ⌫,↵ was given in (17.15). CHECK THE SIGNAt the one-loop level, this on-shell process will have the three contributions shown

in Fig. 17.7, which we call �(irr), �(tadpole) and �(ext).

4)

p p0

k0 = k + k0

k

k0

=========

p p0

k0

k0

k

=========

6)

k k

7)

k k

+

k k

+

k k

2

�(irr) �(tadpole) �(ext)

Figure 17.7 Three one-loop contributions to photon-graviton conversion.

The rest of this chapter will be devoted to the calculation of these three contribu-tions to the photon-graviton conversion amplitude, and to see how they are relatedby the gravitational Ward identity. Since this is a relatively lengthy calculation,here we will restrict ourselves to the more important fermionic case (for the scalarcase see the NOTES).

17.8.2 Irreducible contribution to photon-graviton conversion

Let us start with the irreducible contribution �(irr)spin . Using the machinery developed

above, its worldline representation can be written as

�(irr)spin(k0, ✏0; k, "; F ) = (�ie)(�

4)(�1

2)2

D2

Z 1

0

dT

Te�m2T (4⇡T )

�D2 det�

12

tan(Z)

Z

�

⇥DV G

spin[k0, ✏0]V�spin[k, "]

E(17.69)

Fig. 6. Tadpole contribution to photon-graviton conversion.

Using the integral (13) it is easy to show that its contribution to the one-loop

amplitudes with a scalar or spinor loop can be written as

Γ(tadpole)scal (kα, εβ ; εµν ;Fκλ) = −i α

8πκ(ε · F · ε · k + ε · ε · F · k

)

×∫ ∞

0

dz

ze−

m2

eB zcoth(z)− 1/z

sinh(z), (15)

Γ(tadpole)spin (kα, εβ ; εµν ;Fκλ) = i

α

4πκ(ε · F · ε · k + ε · ε · F · k

)

×∫ ∞

0

dz

ze−

m2

eB zcoth(z)− tanh(z)− 1/z

tanh(z). (16)

Expanding out the tadpole in powers of the external field we see that the leading

term, which is linear in the field (Fig. 7), is removed by the photon wave function

renormalization.

k k

k k

k k

1

Fig. 7. Expanding the tadpole to lowest order in the field.

This gives the renormalized amplitudes


7

Γ(tadpole)scal,ren (kα, εβ ; εµν ;Fκλ) = −i α

8πκ(ε · F · ε · k + ε · ε · F · k

)

×∫ ∞

0

dz

ze−

m2

eB z

[coth(z)− 1/z

sinh(z)− 1

3

],

Γ(tadpole)spin,ren (kα, εβ ; εµν ;Fκλ) = i

α

4πκ(ε · F · ε · k + ε · ε · F · k

)

×∫ ∞

0

dz

ze−

m2

eB z

[coth(z)− tanh(z)− 1/z

tanh(z)+

2

3

].

6. Comparison with the main diagram

These amplitudes are of a structure similar to what we got from the main diagram

Fig. 1 in the limit of low photon energy ω, eqs. (9), (10).

However, they do not contribute to dichroism since the polarizations are still

bound up in the tree-level vertex (ε ·F · ε ·k+ε · ε ·F ·k). Thus the above-mentioned

analysis of Ahlers et al.32 remains unaffected by the presence of the tadpole diagram.

7. Summary and Outlook

• We have presented the first example of a non-vanishing diagram in Einstein-

Maxwell theory involving a tadpole in a constant field.

• This diagram does not contribute to magnetic dichroism.

• A more quantitative analysis is in progress.

• In the ultra strong-field limit, the tadpoles have been shown even to dominate

the (multi-loop) effective action in QED40. It would be interesting to extend

this analysis to the Einstein-Maxwell case.

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http://arxiv.org/abs/hep-th/9610191

http://arxiv.org/abs/hep-th/9610191









arxiv:2111.01980v1 [hep-ph] 3 nov 2021

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