Table Of ContentThe worldline approach to helicity flip in plane waves
Anton Ilderton1,∗ and Greger Torgrimsson1,†
1Department of Physics, Chalmers University, SE-41296 Gothenburg, Sweden
We apply worldline methods to the study of vacuum polarisation effects in plane wave back-
grounds, in both scalar and spinor QED. We calculate helicity-flip probabilities to one loop order
and treated exactly in the background field, and provide a toolkit of methods for use in investiga-
tions of higher-order processes. We also discuss the connections between the worldline, S-matrix,
and lightfront approaches to vacuum polarisation effects.
I. INTRODUCTION
6
1 Vacuum polarisation effects may be probed using strong electromagnetic fields, as may be generated by using
0 e.g. intense lasers. As such the measurement of “vacuum birefringence” [1] (a matterless analogue of birefringence
2 in optics) in the collision of x-ray and optical laser pulses, has been selected as a Flagship experiment at DESY [2],
n following [3]. For both the theoretical and experimental status of birefringence studies see the recent reviews [4, 5].
a Vacuum polarisation effects stem of course from photon-photon scattering [6] and are captured, to lowest order,
J by the one-loop polarisation tensor in a background field. It is usually enough, considering current experimental
9 abilities, toconsiderphoton-photoninteractionsatthelevelofthelow-energyEuler-Heisenbergeffectiveaction[7,8].
1 Asaccesstonewparameterregimesbecomesavailable[9], itwillthoughbecomeincreasinglyimportanttogobeyond
low-energyapproximations. Investigationsinthisdirectionaredrivennotonlybyphenomenologicalinterest,butalso
]
h by a desire to better understand higher-order, all-orders, and nonperturbative strong field effects in quantum field
t theory [10–14]. On this note, it is possible to calculate the polarisation tensor exactly for certain background fields.
-
p One example is a plane wave, as shown some years ago for a constant ‘crossed’ field in [15], and for arbitrary plane
e waveshapeandstrengthin[16](usingGreen’sfunctionmethods)and[17](usingoperatormethods). Thepolarisation
h
tensor in both plane waves and magnetic fields has recently been reconsidered by several groups [18–22]. This has
[
lead to many new insights, in particular with regard to realistic field geometries [23–25], which must be accounted for
1 in light-by-light scattering experiments [23, 26, 27] and vacuum birefringence experiments [2, 3, 5, 28].
v The original derivations of polarisation tensor results are rather involved, and it would be preferable to have
1 more transparent expressions in order to improve our understanding of strong-field vacuum polarisation effects: even
2
investigationsoftheplanewavecasecanrevealinsightswhichmaybeextendedtomoregeneralfieldconfigurations[25,
0
29]. Forthesereasonswewillherereconsidertheplanewavepolarisationtensorinanotherformalism, namelythatof
5
worldlinepathintegrals[30–36]–see[37]foranintroductionand[38]forareview. Theworldlineformalismhasproven
0
. powerfulforthestudyofmanytopics,onlyafewexamplesofwhicharepairproduction[39–51],photonsplitting[52],
1
QCD [53], string theories with contact interactions [54, 55], and two loop Euler-Heisenberg effective action [56, 57].
0
Worldline path integrals also lend themselves to numerical evaluation using Monte Carlo methods [58–61].
6
1 Our modest aim here is to use the worldline formalism to recover vacuum polarisation effects in inhomogeneous
: planewavebackgrounds,ofarbitrarystrengthandshape. (Thisformalismhaspreviouslybeenappliedtothestudyof
v
the polarisation tensor in constant fields [62] and in inhomogeneous magnetic fields [58]). As part of this calculation
i
X we will produce a new toolbox which can be used to apply worldline methods to more complicated higher-order
r strong-field processes in plane wave backgrounds.
a This paper is organised as follows. Directly below we introduce our notation and conventions, and then describe
the observables of interest, namely the helicity-flip probabilities in our chosen background field. In Section II we
perform the worldline calculation of the helicity flip probability. In Section III we extend this to the spinor case and
compare two different worldline methods for calculating the spin contribution. We discuss the results and conclude
in Section IV. The appendix contains normalisations for worldline path integrals.
A. Notation and conventions
Our background field depends only on the phase φ := n.x where n2 = 0, and is transverse. We can always choose
coordinates such that n.x=x0+x3 =x+, ‘lightfront time’, and then E and B point in the := x1,x2 directions.
⊥ { }
∗ [email protected]
† [email protected]
2
We define x− = x0 x3. We use lightfront gauge n.A = 0 for the background; residual gauge freedom can then
−
be used to remove a second component, such that the potential has only transverse components A A (x+) = 0.
⊥ ≡ ⊥ (cid:54)
The nonzero part of the field strength is F = ∂ A . This choice of potential has the advantage of making the
+⊥ + ⊥
physical (kinematic) particle momentum manifest in scattering calculations [63]. The fields here may take any shape
we choose. To emphasise that we treat the background exactly, i.e. without recourse to perturbation in the field
strength, we absorb the coupling into the field, writing a :=eA .
µ µ
B. Helicity flip
Ratherthancalculatethepolarisationtensorwecalculatedirectlytheobservablesofinterest,namelythescattering
amplitudes for photons of momentum l l(cid:48) and helicity state (cid:15) (cid:15)(cid:48) . (These could also be obtained by first
µ → µ µ → µ
calculating the polarisation tensor and then contracting with appropriate asymptotic states, see e.g. [17, 20, 21] and
references therein.) However, on-shell scattering of single photons is automatically forward in a plane wave. This is
a consequence of the integrability of the equations of motion; classically, the transverse canonical momenta and the
physical longitudinal momentum are conserved in the Lorentz force equation, and quantum mechanically the fermion
propagator depends nontrivially only on x+, which leads to overall conservation of three momentum components in
scattering amplitudes. While these conservation laws force an on-shell photon to scatter forward, internal degrees of
freedom can still change. In particular, the photon helicity can flip, which is the microscopic description of vacuum
birefringence [20]. We therefore consider the total probability of helicity-flip P , which is (with appropriately
flip
normalised wave packets taken into account, see below), just the probability for forward-scattering plus helicity-flip.
Inordertoguideuswebrieflymentiontwopropertiesoftheflipprobabilitybeforebeginningthecalculation. First,
an S-matrix calculation, or a calculation in lightfront perturbation theory, expresses the flip probability as a double
integral [16, 20, 21] over two lightfront times φ and θ which are related to the spacetime vertex positions φ θ/2 in
±
the relevant Feynman diagram, see Fig. 1. Second, the integrand in the probability depends on the ‘effective mass’
M2 =m2 a2 + a 2 , (1)
−(cid:104) (cid:105) (cid:104) (cid:105)
defined by the moving average [64]
φ+θ/2
1 (cid:90)
f = dϕf(ϕ). (2)
(cid:104) (cid:105) θ
φ−θ/2
The effective mass appears classically after averaging a particle’s kinetic momentum π over propagation time,
µ
π πµ = M2. It appears in scattering probabilities after integrating over final states [66], and in scattering
µ
(cid:104) (cid:105)(cid:104) (cid:105)
amplitudes after performing loop integrals [20, 64]; see e.g. [65] for a discussion of the phenomenology of the effective
mass. We now proceed with our worldline calculation.
II. SCALAR QED
In the worldline formalism the helicity-flip amplitude in scalar QED (“sQED”) is [62]
∞ 1 1
(cid:90) dT (cid:73) (cid:90) (cid:90)
T=(ie)2 xµeiS dσ(cid:48) eil(cid:48).x(σ(cid:48))(cid:15)(cid:48).x˙(σ(cid:48)) dσ e−il.x(σ)(cid:15).x˙(σ), (3)
T D
0 0 0
where the worldline action S describes a relativistic particle coupled to the background field a ,
µ
1
T (cid:90) x˙2
S = m2 dτ +a(x).x˙ , (4)
− 2 − 2T
0
andadotisaderivativewithrespecttoτ,whichparameterisestheworldline. Asusual(seee.g. [62])weexponentiate
the vertex operators in (3) and write their contribution as a source term in the action, remembering to keep, at the
end, only terms which are linear in all polarisation vectors:
∞ 1 1
(cid:90) dT (cid:90) (cid:73) (cid:20) (cid:90) (cid:21)
T=(ie)2 dσ(cid:48)dσ xµexp iS i dτ J xµ(τ) , (5)
T D − µ
0 0 0
3
φ+θ/2
p = sl
φ θ/2 − −
−
γ γ l l l l
− − − −
θ p l = (1 s)p
−− − − − −
FIG. 1. Left: The Furry-Feynman diagram for photon-photon scattering in a background field, to one loop. The double
line represents fermions dressed (to all orders) by the background. Middle: The two vertices have lightfront-time coordinates
x+ =φ±θ/2. Right: In a plane wave the ‘minus’ momentum component is conserved, as in vacuum. The momentum flowing
around the loop is parameterised in terms of the momentum fraction s:=n.p/n.l=p /l .
− −
where the source is
J =l δ(τ σ) l(cid:48)δ(τ σ(cid:48))+(cid:15) δ˙(τ σ) (cid:15)(cid:48) δ˙(τ σ(cid:48)). (6)
µ µ − − µ − µ − − µ −
We divide each coordinate x into a centre of mass x and an oscillating, non-constant piece y, so xµ =xµ+yµ, where
c c
(see the appendix)
1
(cid:90)
dτyµ(τ)=0. (7)
0
The x⊥ and x− integrals can be performed immediately to yield the momentum conservation laws described above:
c c
∞ 1 1
(ie)2 (cid:90) (cid:90) dT (cid:90) (cid:73) (cid:20) (cid:90) (cid:21)
T= (2π)3δ3 (l(cid:48) l) dx+ dσ(cid:48)dσT2 yµexp i(l(cid:48) l )x++iS i dτ J yµ(τ) , (8)
2 ⊥,− − c T D +− + c − µ
0 0 0
where the leading 1/2 comes from the Jacobian for the change to lightfront variables in d4x . We will perform the
c
integrals above from right to left. Since our photon is on-shell, momentum conservation sets l(cid:48) =l , i.e. scattering is
+ +
forward, and the first term in the exponent of (8) drops out.
Wehavethatl.(cid:15)(cid:48) =l.(cid:15)=0. Wecansimplifymattersbyobservingthatwecanalwaysmakeagaugetransformation
such that (cid:15)+ =(cid:15)(cid:48)+ =0. It is not necessary to make this transformation, but it reduces the number of terms we have
to consider1. The next step is to perform the yµ integrals. To do so we shift y+ and y⊥ by a solution to the equations
of motion. The classical y+ is found by solving
1
1 1 (cid:90)
y¨+(τ)=J+(τ) = y+(τ)= dτ(cid:48) G(τ,τ(cid:48))J+(τ(cid:48)), (9)
T cl ⇒ T cl
0
in which G is the inverse of d2/dτ2 on the space of periodic functions with zero average [36], hence
d2
G(τ,τ(cid:48))=δ(τ τ(cid:48)) 1. (10)
dτ2 − −
(Forthecaseofconstantbackgroundfieldstheactionremainsquadraticinthecoordinates, anditiscommontothen
perform calculations using the background-modified worldline propagator. Here though it suffices to use the free G,
in both scalar and spinor QED. The reason why will become apparent below.) We will see that the two terms in (10)
correspond precisely to the two field-dependent terms in the effective mass (1). Having determined the classical y+
we can find the classical y⊥:
1
1 1 (cid:90)
y¨⊥(τ)=J⊥(τ) a˙⊥(x++y+ +y+) = y⊥(τ)= dτ(cid:48) G(τ,τ(cid:48))J˜⊥(τ(cid:48)), (11)
T cl − c cl ⇒ T cl
0
1 This is easily shown provided l (cid:54)=0. If l =0 then l =0 and hence (cid:15) =0 automatically in order to fulfil (cid:15).l=0; however in this
− − ⊥ −
casetheprobephotontravelsparalleltothelaserandthereisnohelicityflip,sowecanignorethiscase.
4
where (recalling that only the perp components of a are nonzero)
J˜ =J (τ) a˙ (x++y+(τ)+y+(τ)). (12)
µ µ − µ cl
Note that we have done nothing with y−. Shifting y by the classical solution, the amplitude T becomes
∞ 1 1 1
(ie)2 (cid:90) (cid:90) dT (cid:90) (cid:73) (cid:20) m2T T (cid:90) (cid:90) y˙2 (cid:21)
T= (2π)3δ3 (l(cid:48) l) dx+ dσ(cid:48)dσT2 yµexp i i J˜ GJ˜µ i +y+J , (13)
2 ⊥,− − c T D − 2 − 2 µ − 2T +
0 0 0 0
which still looks formidable due to the appearance of y+, which is to be integrated over, inside the background field
and therefore inside Jµ and J˜µ. However, we observe that the y− integral can be evaluated exactly, as in the free
theory, to give a delta functional which kills the oscillatory y+ in the rest of the integral. (A similar trick is applied
to pair production in [46], see also [32, 33].) Hence all the y–integrals may be evaluated precisely, just as in the free
theory,eventhoughtheyarenotGaussian. Thex andy integralscontributetogetherthefree-theoryfactor(2πT)−2,
c
see (A2). What remains is
∞ 1 1
e2 (cid:90) (cid:90) dT (cid:90) (cid:20) m2T T (cid:90) (cid:21)
T= (2π)3δ3 (l(cid:48) l) dx+ (2πT)−2 dσ(cid:48)dσexp i i J˜ GJ˜µ , (14)
− 2 ⊥,− − c T − 2 − 2 µ
0 0 0
where now J˜ = J (τ) a˙ (x+ +y+(τ)). The expression (14) is almost identical to that which would be obtained
µ µ − µ cl
in the free theory, except that the source here contains the gauge field. In fact the exponent in (14) is nothing but
S[x ], that is the action evaluated on the classical trajectory obeying
cl
1
x¨µ =Jµ+eFµ x˙ν . (15)
T cl ν cl
Inotherwordsthesemiclassicalapproximationistheexactresulthere. ThisisatypicalpropertyofGaussianintegrals,
i.e. of free theories or theories with constant background fields [39, 67]. Here our fields are inhomogeneous but the
semiclassical result is still exact; we comment further on this in Sect. IV. That the semiclassical result is exact is also
awellknownpropertyoftheVolkovwavefunctionswhichsolvetheDiracandKlein-Gordonequationsinaplanewave
background.
Wenowevaluatetheexponentin(14)andexpandthepolarisationtermsbackout. Aswearelookingatascattering
amplitude there are only a few terms, and one quickly arrives at
∞
(ie)2 (cid:90) (cid:90) dT
T= (2π)3δ3(l(cid:48) l) dx+ , (16)
2(2π)2 − c T Z
0
in which we have defined
1
:=(cid:90) dσ(cid:48)dσ (cid:15)(cid:48).(cid:0) a a (cid:1)(cid:15).(cid:0) a a (cid:1)exp(cid:20) iT(cid:18)m2 a2 + a 2(cid:19)(cid:21), (17)
Z (cid:104) (cid:105)− σ(cid:48) (cid:104) (cid:105)− σ − 2 −(cid:104)(cid:104) (cid:105)(cid:105) (cid:104)(cid:104) (cid:105)(cid:105)
0
with a worldline average which arises from contractions aG¨a;
(cid:104)(cid:104)···(cid:105)(cid:105)
1
(cid:90)
(cid:0) (cid:1)
a = dτa x++y+(τ) . (18)
(cid:104)(cid:104) (cid:105)(cid:105) cl
0
The structure in the exponent of is similar to that in the effective mass (1), but the average is taken over the
Z
worldline rather than a spacetime variable as in (2). ‘Projecting’ the average into target space is in fact the key to
performing the σ, σ(cid:48)–integrals. One can show using the explicit form of y+ that, writing s:= σ σ(cid:48) ,
cl | − |
Tl+s(1−s)/2
1 (cid:90)
a = dya(x++y). (19)
(cid:104)(cid:104) (cid:105)(cid:105) Tl+s(1 s) c
−
−Tl+s(1−s)/2
5
Now change variable in (16) from T to
θ :=Tl+s(1 s), (20)
−
and change notation to φ:=n.x =x+. The worldline average then becomes exactly equal to (2), = , and
hence the exponent in recovercs theceffective mass (1). In the contraction aG¨a the two terms in(cid:104)(cid:104)·(1··0(cid:105))(cid:105)gen(cid:104)·e·ra·(cid:105)te the
averages over a2 and ovZer a squared, respectively. Without the “background charge density” smeared over the loop,
µ
represented by the ‘1’ in (10) [36], the effective mass would not be correct. The averages over lightfront time appear
in the Feynman diagram approach after performing the transverse loop momentum integrals; these integrals do not
appear explicitly in the worldline approach.
The change of variable (20) maps the intrinsic length of the worldline path, T, onto θ, the spacetime separation
between Feynman diagram vertices in the n.x direction. (This could have been guessed at the beginning: both θ
and T are integrals over the positive real line.) When rewritten in terms of the new variables the terms outside the
exponent in (17) become
a(x++y+(σ(cid:48)))=a(φ+θ/2), a(x++y+(σ))=a(φ θ/2), (21)
c cl c cl −
independent of σ, σ(cid:48), the integrals over which can now be performed. Because the integrand depends on σ,σ(cid:48) only
in the combination s(1 s), one can show directly that there remains no dependence on σ(cid:48) after evaluati{on of}the σ
integral. This is just as−in the theory without background [36]. Hence we drop the σ(cid:48) integral (which contributes a
factorofunity)andintegrateoverσ s. Writingx M2/n.l thisintegralis, withK themodifiedBesselfunctions,
j
→ ≡
1
(cid:90) ds exp(cid:18) i x (cid:19)=ixe−ix(cid:0)K (ix) K (ix)(cid:1)=: (x). (22)
− 2s(1 s) 1 − 0 I1
0 −
This integral over the relative vertex position σ σ(cid:48) appears in the Feynman diagram and lightfront approaches as
−
an integral over the lightfront momentum fraction s = n.p/n.l = p /l , where p is the momentum of the fermion in
− −
the loop, see Fig. 1, right hand panel. In those approaches the integral limits arise from momentum conservation,
but here they were in place from the beginning. The appearance of the “s(1 s)” factors is also typical of lightfront
−
wavefunctions [68–70].
Collecting factors, we find the near-final result
∞
T= (2π)3δ (l(cid:48) l) α (cid:90) dφ(cid:90) dθ (cid:0)θM2(cid:1)(cid:15)(cid:48).(cid:0)a(φ+θ/2) a (cid:1)(cid:15).(cid:0)a(φ θ/2) a (cid:1). (23)
− ⊥,− − 2π θ I1 n.l −(cid:104) (cid:105) − −(cid:104) (cid:105)
0
To compactify notation we define
A:=(cid:15). a , A¯ :=(cid:15)(cid:48). a , A :=∂ A; A :=∂ A, and so on. (24)
θ θ φ φ
(cid:104) (cid:105) (cid:104) (cid:105)
With this and the results
(cid:15).(cid:0)a(φ θ/2) a (cid:1)=θ(cid:0)1A A (cid:1), (25)
± ± −(cid:104) (cid:105) 2 φ± θ
we can write our amplitude in the form
T= (2π)3δ (l l(cid:48))n.l . (26)
− ⊥,− − Mscalar
To obtain the probability we take the modulus squared of (26) and integrate over final states. This integral converts
one factor of (2π)3δ3 into a 1/n.l, while the second delta function is regularised by including a photon wavepacket
from the beginning, and gives a second factor of 1/n.l, see [71]. Hence we obtain P = 2 or, explicitly,
flip flip
|M |
Pflip,scalar =(cid:12)(cid:12)(cid:12)(cid:12)2απn1.l (cid:90)∞dφ(cid:90)∞dθθ I1(cid:0)θnM.l2(cid:1)(A¯θ+ 21A¯φ)(Aθ− 21Aφ)(cid:12)(cid:12)(cid:12)(cid:12)2 , (27)
−∞ 0
which is rather compact, and of the same form as the spinor QED result found in [20] using lightfront methods. The
tworemainingintegralsoverthelightfronttimesφandθcanbeperformedanalyticallyonlyforspecialcases(butthen
one must give up performing the s-integral exactly [72]), and in general must be tackled numerically. For examples
and details of the numerical method of integration see [20].
6
A. Analyticity
In evaluating the contraction J˜µGJ˜ in (14) to obtain (17) there arises a term
µ
1
(cid:90)
dτ y˙+a(x++y+). (28)
cl c cl
0
Thisintegralisexactand,bytheboundaryconditions,vanishes,providedthatthepotentialaisanintegrablefunction
without singularities. This is of course natural, and we remark only that similar terms appear in worldline instanton
calculations of pair production, see e.g. [50] and the discussion of the argument principle in [49].
B. Zero modes
Asourresultsparallelthosefoundusinglightfrontquantisationweshouldcommentontheappearanceoflightfront
zero modes in this formalism. Zero modes are states for which the momentum component p =0, and there is a vast
−
literature on the role they play and the problems they can cause through the appearance of 1/p factors, see [69, 70]
−
and references therein. For this discussion it is enough to check that the zero modes do not introduce divergences
into our calculations [20, 21].
The change of variable (20) may look suspect when s 0,1 , that is when σ = σ(cid:48), σ = 1,σ(cid:48) = 0 , or σ(cid:48) =
∈ { } { } {
1,σ = 0 . For these values the vertex operators lie at the same position on the worldline. Now, we have already
}
identified that s corresponds to the momentum fraction p /l flowing around the loop. From the right-hand diagram
− −
in Fig. 1 this means that s=0,1 corresponds to a zero mode flowing through the upper or lower portion of the loop,
respectively. We observe that y+ 0 at these points, which implies that the background field vanishes from J˜ in
cl →
(14), and we return to the free theory, where the helicity-flip amplitude must vanish. This suggests that we should
not expect problems from such points. Indeed it is seen directly that there are no singularities in (17) as s 0,1 ,
→{ }
sothezeromodesdonotcauseproblems. Alsoasexpected,usingthaty+ 0andthat a a (x+)from(19),the
cl → (cid:104) µ(cid:105)→ µ c
vertex operator insertions vanish in (17) so that there is no contribution to the flip probability from the zero modes.
III. SPINOR QED
We now turn to the helicity-flip amplitude in spinor QED. Spin contributions are included by inserting a “spin
factor” into the scalar worldline integral, of form [31, 37, 38]
1
1 (cid:20) iT (cid:90) (cid:21)
Spin= tr P exp dτσµν , (29)
4 − 4 Fµν
0
where the fieldstrength contains both the background and scattered photons (definition below), σµν = i[γµ,γν],
Fµν 2
the trace is over the Dirac matrices, and P stands for path-ordering. (We define Spin with an extra factor of
1/4 compared to [37], and we work in Minkowski rather than Euclidean space.) It is sometimes more convenient
to rewrite Spin as a path integral over Grassmann variables on the worldline; doing so makes explicit a worldline
supersymmetry [73], the use of which in worldline calculations is reviewed in [38]. Another method for evaluating the
Grassmann integral, for arbitrary background fields, is presented in [74]. There the Grassmann fields are integrated
out without approximation to obtain a worldline path integral for spinor QED which depends only on the worldline
coordinates, as for sQED. Here we will first consider the form (29), since only a few terms in the expansion of the
exponent are nonzero, but we will also use the Grassmann approach, below. (The form (29) was also used in [40] to
obtain the spinor contribution to pair production.)
A. Feynman’s Dirac-trace approach
The helicity flip amplitude in QED is
∞ 1
(cid:90) dT (cid:73) (cid:20) T (cid:90) x˙2 (cid:21)
T =2e2 xexp im2 i dτ + .x˙ Spin, (30)
flip T D − 2 − 2T A
0 0
7
where =a +(cid:15) e−il.x+(cid:15)(cid:48) eil(cid:48).x, and the fieldstrength in the spin factor is =∂ ∂ . As before we
retain Aonµly thoµse teµrms whichµare linear in both (cid:15) and (cid:15)(cid:48).FNote that (30) is obtaiFnµeνd fromµAtνhe−sQνEADµ expression (3)
by inserting the spin-factor (29) and multiplying by two.
We begin by expanding out the exponential in the spin factor, using
i
σµν =n/a/(cid:48) i/l/(cid:15)e−il.x+i/l(cid:48)/(cid:15)(cid:48)eil(cid:48).x . (31)
− 2 Fµν −
Onlyafewofthetermsarenonzerobecauseofourchosenbackgroundandprocess. Ananalogoustruncationofterms
occurs when one solves the Lorentz force equation in a plane wave: a path-ordered exponential truncates at second
order because the field strength F is nilpotent of order three.
µν
Whenselectingthetermsthatarelinearin(cid:15)and(cid:15)(cid:48) itisconvenienttoseparatethetotalamplitudeintothreeparts,
writing T =T +T +T . The term T is identical to the amplitude in sQED up to the factor of 2, and receives
flip 0 1 2 0
both (cid:15) and (cid:15)(cid:48) from the .x˙ term in (30). The term T receives one of the polarisation vectors from .x˙ and the other
1
from Spin. The term TA receives both (cid:15) and (cid:15)(cid:48) from Spin. Writing arguments as subscripts to comApactify notation,
2
the spinor contribution to T is obtained from
1
1
1 (cid:90) T2
tr dτ dτ θ(τ τ ) (n/a/(cid:48) i/l/(cid:15)e−il.x+i/l(cid:48)/(cid:15)(cid:48)eil(cid:48).x) (n/a/(cid:48) i/l/(cid:15)e−il.x+i/l(cid:48)/(cid:15)(cid:48)eil(cid:48).x) , (32)
4 2 1 2− 1 4 − τ2 − τ1
0
and the spinor contribution to T is obtained from
2
1
1 (cid:90) T4
tr dτ θ (n/a/(cid:48) i/l/(cid:15)e−il.x+i/l(cid:48)/(cid:15)(cid:48)eil(cid:48).x) (...) (...) (...) , (33)
4 4321 432116 − τ4 τ3 τ2 τ1
0
where θ = θ(τ τ )θ(τ τ )θ(τ τ ). After performing the traces the three ‘partial amplitudes’ T can be
4321 4 3 3 2 2 1 i
− − −
written in terms of the same current J as in (6) as
∞ 1
(cid:90) dT (cid:73) (cid:20) (cid:90) x˙2 (cid:21)
T =2e2 e−iTm2/2 xI exp i +a.x˙ +J.x , (34)
i T D i − 2T
0 0
where the three integrands I are, writing “lin ” for the instruction to select the term linear in (cid:15),
i (cid:15)
1
(cid:90)
I = dσdσ(cid:48) lin lin , (35)
0 (cid:15) (cid:15)(cid:48)
−
0
1 1 1
iT2 (cid:90) (cid:16) (cid:90) (cid:90) (cid:17)
I = dσdσ(cid:48) n.l dτ (cid:15).a(cid:48) lin +n.l(cid:48) dτ (cid:15)(cid:48).a(cid:48) lin , (36)
1 4 (cid:15)(cid:48) (cid:15)
0 0 0
1
T4 (cid:90) (cid:12)
I =n.ln.l(cid:48) dτ (cid:12) (θ +θ +θ +θ )((cid:15).a(cid:48)(cid:15)(cid:48).a(cid:48)+(cid:15).a(cid:48)(cid:15)(cid:48).a(cid:48)). (37)
2 8 4321(cid:12)τ3=σ,τ1=σ(cid:48) 4321 2143 3214 1432 4 2 2 4
0
Allexplicitreferencetospinorshasnowbeeneliminated,andweareleftwithaworldlineintegraloverthecoordinates,
as in [74]. The integral can be performed with the same method as for the scalar case, above. T is the same as in
0
sQED up to a factor of 2:
∞ 1
T0 =−(2π)3δ−,⊥(l(cid:48)−l)απ (cid:90) dφ(cid:90) dθ θ(cid:90) ds(cid:0)A¯θ+ 21A¯φ(cid:1)(cid:0)Aθ− 21Aφ(cid:1)e−2n.ipθsM(12−s) . (38)
0 0
For T we use
1
1 φ+θ/2
(cid:90) 1 (cid:90)
dτ f(cid:48)(x++y+(τ;σ,σ(cid:48)))= dϕf(cid:48)(ϕ)= f , (39)
cl θ (cid:104) (cid:105)φ
0 φ−θ/2
8
and that the integrand again depends only on the combination s(1 s) with s= σ σ(cid:48) . This gives
− | − |
∞ 1
T1 =−(2π)3δ−,⊥(l(cid:48)−l)4απ (cid:90) dφ(cid:90) dθ θ(cid:90) s(1ds s) (cid:16)−A¯[φAθ]+A¯φAφ(cid:17)e−2n.ipθsM(12−s) . (40)
0 0 −
The term T is more complicated. By using the symmetry τ τ and τ τ (σ σ(cid:48)) we can write
2 4 2 3 1
↔ ↔ ↔
1
I =n.l2T4 (cid:90) dτ (θ +θ )(cid:0)(cid:15).a(cid:48)(cid:15)(cid:48).a(cid:48)+((cid:15) (cid:15)(cid:48))(cid:1). (41)
2 4 4321 4321 3214 4 2 ↔
0
Next we use the explicit solution y+(τ;σ,σ(cid:48)) (recall τ =σ and τ =σ(cid:48)) to perform the integrals over τ and τ ,
cl 3 1 2 4
(cid:90)τ3 (cid:90)τ1 (cid:90)1
dτ a(cid:48)(x++y+)=s a , dτ a(cid:48)(x++y+)+ dτ a(cid:48)(x++y+)=(1 s) a . (42)
2 cl (cid:104) (cid:105)φ 4 cl 4 cl − (cid:104) (cid:105)φ
τ1 0 τ3
The integrand for the remaining integrals over σ and σ(cid:48) again only depends on s(1 s), except for a factor of θ
31
which simply gives a factor of 1/2. The final result for T is −
2
∞ 1
T2 =(2π)3δ−,⊥(l(cid:48)−l)4απ (cid:90) dφ(cid:90) dθ θ(cid:90) s(1ds s) A¯φAφe−2n.ipθsM(12−s) . (43)
0 0 −
Note that (43) is cancelled entirely by a similar term in (40). The integrals over s give either introduced in (22) or
1
I
1
(cid:90) ds (cid:18) x (cid:19)
(x):= exp i =e−ixK (ix). (44)
I2 2s(1 s) − 2s(1 s) 0
0 − −
Adding the three contributions (38), (40) and (43) together we find
∞
α(cid:90) (cid:90) (cid:20) 1 (cid:21)
T = (2π)3δ (l(cid:48) l) dφ dθ θ (A¯ + 1A¯ )(A 1A ) A¯ A
flip − −,⊥ − π θ 2 φ θ− 2 φ I1− 2 [φ θ]I2 (45)
0
= (2π)3δ (l l(cid:48))n.l ,
− ⊥,− − Mspinor
inexactagreementwiththeresultsin[20]foundusinglightfrontquantisationandstandardFeynmandiagrammethods.
Squaring the amplitude yields, as above, P= 2, with
|M|
Pflip,spinor =(cid:12)(cid:12)(cid:12)(cid:12)απn1.l (cid:90)∞dφ(cid:90)∞dθθ (cid:18)I1(cid:0)θnM.l2(cid:1)(A¯θ+ 21A¯φ)(Aθ− 12Aφ)− 12I2(cid:0)θnM.l2(cid:1)A¯[φAθ])(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)2 . (46)
−∞ 0
B. Grassmann aproach
The spin factor (29) can be expressed as a path integral over anticommuting Grassmann variables ψµ [37] (replace
T in that paper by iT/2 to go to our conventions),
1
1(cid:90) (cid:20) 1(cid:90) (cid:21)
Spin= ψexp ψ ψ˙µ+Tψµ ψν , (47)
4 D − 2 µ Fµν
0
and the integral is calculated with anti-periodic boundary conditions ψ(1) = ψ(0). We begin by expanding the
−
exponential in the polarisation vectors. This gives three terms, as above. The next step is to remove the gauge field
from the action via the change of variable
1
(cid:90) 1
ψµ(τ) ψµ(τ) Tnµ dτ(cid:48) G (τ τ(cid:48))a(cid:48).ψ , G (τ τ(cid:48))= sign(τ τ(cid:48)). (48)
→ − F − F − 2 −
0
9
Note that only ψ− is changed, and that only ψ⊥ appears under the integral in (48). After this change of variable the
exponent appearing in our amplitude becomes that of the free theory:
1
1(cid:90) (cid:26) 1 (cid:27) (cid:20) 1(cid:90) (cid:21)
Spin= ψ 1 iT T2 2 exp ψ ψ˙µ ,
4 D − J − 2 J − 2 µ
0
(49)
1
:=(cid:90) dτdτ(cid:48)(cid:0)ψµ(τ) TnµG (τ τ(cid:48))a(cid:48).ψ(τ(cid:48))(cid:1)(cid:16)l(cid:48)(cid:15)(cid:48)eil(cid:48).x(τ) l (cid:15) e−il.x(τ)(cid:17)ψν(τ).
J − F − µ ν − µ ν
0
(The same change of variables in the effective action would remove all dependence on the field, turning it into the
effective action of the free theory; this is because there is no Schwinger pair production in a plane wave.) The
Grassmann integral can now be performed using the same Wick contractions as in the free theory [37]. This is trivial
for the first term in (49), and a straightforward calculation shows that the second term agrees with that obtained by
performing the trace in (32). The third term in (49) requires a longer calculation. At an intermediate step, one finds
1
T4n.l2 (cid:90)
dτ e−il.x(τ3)eil(cid:48).x(τ1)(cid:15).a(cid:48)(cid:15)(cid:48).a(cid:48)(1 16GF GF GF GF ). (50)
16 4321 2 4 − 43 32 21 14
0
By writing GF =θ θ and using that θ +θ =1 one can show that (50) is equal to (37), and one recovers (34).
ij ij− ji ij ji
The rest of the calculation is identical to that above.
IV. DISCUSSION AND CONCLUSIONS
We have derived the photon helicity-flip probability in inhomogeneous plane wave backgrounds using the worldline
formalism. The calculation was performed to one loop, but exactly in all other parameters. The calculation in scalar
QED is direct and the final result compact. Both this and the QED calculation recover the expressions previously
found using lightfront methods, thus our results also serve to confirm the advantages of this approach to strong field
QED [14].
ThesQEDcalculationdoesseemsimplerintheworldlineformalismthaninapproachesbasedontheVolkovsolution,
and so offers a promising tool for the calculation of other processes, including for example multi-photon emission: by
combining the methods presented here with those in [53, 75] is should be possible to obtain useful expressions for
the n-photon emission amplitudes, for arbitrarily high n. It is fair to say that the calculation of spin contributions
is involved. Nevertheless, we have seen that two different methods can be applied, and one or the other may offer
simplifications depending on the process in question. For the problem here, using the Dirac trace is simpler than
using a Grassmann integral: there are fewer steps with the former, even though the calculations as a whole are very
similar.
We have seen that the semiclassical approximation to our worldline integrals remains exact, even though the
background is inhomogeneous. This provides a simple recipe for the calculation of other processes: 1) exponentiate
thevertexoperatorstoobtainthesource,2)evaluatetheclassicalactionontheclassicalpath,3)re-expandthevertex
operators. Finally, 4) compute as many integrals as possible. The calculation of the classical path will go through as
above with the appropriate source and propagator: one first solves for x+, and then for x⊥. This method should hold
for all processes in plane wave backgrounds.
ACKNOWLEDGMENTS
A. I. thanks N. Ahmadiniaz, S. Meuren and C. Schubert for many useful discussions. A. I. is supported by the Olle
Engkvist Foundation, grant 2014/744.
Appendix A: Conventions
The reparameterisation-invariant measure over the worldline coordinates is [76, 77]
(cid:73) (cid:90) (cid:73)
x=√T dx y , (A1)
c
D D
10
in which x is the centre-of-mass of the loop, y is the nonconstant and oscillatory part of the closed path, and the
c
factor of √T arises from changing variable from the Fourier zero mode of the loop to x . In four dimensions the
c
measure obeys
1
(cid:73) (cid:20) i (cid:90) (cid:21) (cid:90)
xexp dτ x˙2 =(2πT)−2 d4x , (A2)
D − 2T c
0
with the final integral giving the spacetime volume. The fermionic path integral measure is, in four dimensions,
normalised to [37]
1
(cid:90) (cid:20) 1(cid:90) (cid:21)
ψexp dτ ψ ψ˙µ =4. (A3)
D − 2 µ
0
and is calculated with anti-periodic boundary conditions ψ(1)= ψ(0).
−
[1] J.S. Toll, PhD thesis, Princeton, 1952 (unpublished).
[2] See the HIBEF website at http://www.hzdr.de/db/Cms?pNid=427&pOid=35325
[3] T. Heinzl, B. Liesfeld, K. U. Amthor, H. Schwoerer, R. Sauerbrey and A. Wipf, Opt. Commun. 267 (2006) 318 [hep-
ph/0601076].
[4] B. King and T. Heinzl, arXiv:1510.08456 [hep-ph].
[5] H.-P. Schlenvoigt, T. E. Cowan, T. Heinzl, R. Sauerbrey, U. Schramm, to appear.
[6] O. Halpern, Phys. Rev. 44, 855 (1934).
[7] H. Euler and B. Kockel, Naturwiss. 23, 246 (1935).
[8] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936) [physics/0605038].
[9] Extreme Light Infrastructure Nuclear Physics pillar, Bucharest-Magurele, Romania. http://www.eli-np.ro/
[10] G. M. Shore, Nucl. Phys. B 778 (2007) 219 [hep-th/0701185].
[11] G. V. Dunne, Eur. Phys. J. D 55 (2009) 327 [arXiv:0812.3163 [hep-th]].
[12] I. Huet, D. G. C. McKeon and C. Schubert, JHEP 1012 (2010) 036 [arXiv:1010.5315 [hep-th]].
[13] I.Huet,M.RauschdeTraubenbergandC.Schubert,Int.J.Mod.Phys.Conf.Ser.14(2012)383[arXiv:1112.1049[hep-th]].
[14] B. L. G. Bakker et al., Nucl. Phys. Proc. Suppl. 251-252 (2014) 165 [arXiv:1309.6333 [hep-ph]].
[15] N.B. Narozhnyi, Zh. Eksp. Teor. Fiz. 55, 714 (1968) [Sov. Phys. JETP 28, (1969)].
[16] W. Becker and H. Mitter, J. Phys. A 8, 1638 (1975).
[17] V. N. Baier, A. I. Milshtein and V. M. Strakhovenko, Zh. Eksp. Teor. Fiz. 69, 1893 (1975) [Sov. Phys. JETP 42, 961
(1976)].
[18] S. Meuren, C. H. Keitel and A. Di Piazza, Phys. Rev. D 88, 013007 (2013) [arXiv:1304.7672 [hep-ph]].
[19] F. Karbstein, Phys. Rev. D 88 (2013) 8, 085033 [arXiv:1308.6184 [hep-th]].
[20] V. Dinu, T. Heinzl, A. Ilderton, M. Marklund and G. Torgrimsson, Phys. Rev. D 89 (2014) 12, 125003 [arXiv:1312.6419
[hep-ph]].
[21] S. Meuren, K. Z. Hatsagortsyan, C. H. Keitel and A. Di Piazza, Phys. Rev. D 91 (2015) 1, 013009 [arXiv:1406.7235
[hep-ph]].
[22] F. Karbstein and R. Shaisultanov, Phys. Rev. D 91 (2015) 8, 085027 [arXiv:1503.00532 [hep-ph]].
[23] H. Gies, F. Karbstein and N. Seegert, New J. Phys. 15, 083002 (2013) [arXiv:1305.2320 [hep-ph]].
[24] H. Gies, F. Karbstein and N. Seegert, New J. Phys. 17 (2015) 4, 043060 [arXiv:1412.0951 [hep-ph]].
[25] V. Dinu, T. Heinzl, A. Ilderton, M. Marklund and G. Torgrimsson, Phys. Rev. D 90 (2014) 4, 045025 [arXiv:1405.7291
[hep-ph]].
[26] E. Lundstrom, G. Brodin, J. Lundin, M. Marklund, R. Bingham, J. Collier, J. T. Mendonca and P. Norreys, Phys. Rev.
Lett. 96, 083602 (2006) [hep-ph/0510076].
[27] B. King and C. H. Keitel, New J. Phys. 14 (2012) 103002 [arXiv:1202.3339 [hep-ph]].
[28] F. Karbstein, H. Gies, M. Reuter and M. Zepf, Phys. Rev. D 92 (2015) 7, 071301 [arXiv:1507.01084 [hep-ph]].
[29] G. Torgrimsson, Few Body Syst. 56 (2015) 6-9, 615 [arXiv:1409.8069 [hep-ph]].
[30] R. P. Feynman, Phys. Rev. 80 (1950) 440.
[31] R. P. Feynman, Phys. Rev. 84 (1951) 108.
[32] M. B. Halpern and P. Senjanovic, Phys. Rev. D 15 (1977) 1655.
[33] M. B. Halpern, A. Jevicki and P. Senjanovic, Phys. Rev. D 16 (1977) 2476.
[34] M. B. Halpern and W. Siegel, Phys. Rev. D 16 (1977) 2486.
[35] Z. Bern and D. A. Kosower, Nucl. Phys. B 379 (1992) 451.
[36] M. J. Strassler, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205].
[37] C. Schubert, “Lectures on the worldline formalism", available at https://indico.cern.ch/event/206621/mater ial/5/0.pdf
