Quantum backreaction in laser driven plasma

Quantum backreaction in laser-driven plasma

A. Conroy

_{, C. Fiedler}

_{, A. Noble}

_{, and D. A. Burton}

1

_{Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK}

2

_{Department of Physics, SUPA and University of Strathclyde, Glasgow, G4 0NG, UK}

June 22, 2019

Abstract

We present a new approach for investigating quantum effects in laser-driven plasma. Unlike the modelling strategies underpinning particle-in-cell codes that include the effects of quantum electrodynamics, our new field theory incorporates multi-particle effects from the outset. Our approach is based on the path-integral quantisation of a classical bi-scalar field theory describing the behaviour of a laser pulse propagating through an underdense plasma. Results established in the context of quantum field theory on curved spacetime are used to derive a non-linear, non-local, effective field theory that describes the evolution of the laser-driven plasma due to quantum fluctuations. As the first application of our new theory, we explore the behaviour of perturbations to fields describing a uniform, monochromatic, laser beam propagating through a uniform plasma. Our results suggest that quantum fluctuations could play a significant role in the evolution of an underdense plasma, with plasma frequency 10 THz, driven by a laser pulse with wavelength.150 nm, width 30µm and duration>100 fs. Such parameters should be realisable in an experiment using a facility such as the European XFEL.

1

Introduction

The new generation of high-power laser systems will drive the experimental study of high-intensity laser-matter interactions into novel territory. Forthcoming facilities [1] are expected to allow experimental investigations of uncharted parameter regimes using laser pulses with unprecedented peak intensities (greater than 1022_Wcm−2_).

In such regimes, the relativistic quantum-mechanical aspects of laser-matter interactions must be included [2,3], and this requirement has driven the development of particle-in-cell codes, such as EPOCH [4], that incorporate the effects of quantum electrodynamics. In such codes, the matter is represented by a large number of classical macro-particles whose electromagnetic fields, and the laser field, serve as a background in the perturbative calculation of single-particle matrix elements associated with each macro-particle. Each quantum process is assumed to be active only within a spacetime region whose size is negligible in comparison to the classical length and time scales of the laser and matter variables. Interactions between macro-particles in this model are implemented through their contributions to Maxwell equations as classical sources; as a consequence, the multi-particle effects calculated using this approach are classical, rather than quantum, in origin. Furthermore, tools such as EPOCH require high-performance computing facilities to be of greatest utility. Even without quantum effects, the 3-dimensional calculation of the propagation of micrometre-length laser pulses through many centimetres of underdense plasma is a considerable computational challenge. Such considerations have motivated the development of models of classical laser-driven plasma with reduced degrees of freedom, such as those underpinning INF&RNO [5], leading to a reduction of the computational burden by several orders of magnitude. The vigorous effort devoted to the development of such computational tools shows no sign of abating; however, the role of quantum theory in the context of reduced models is yet to be thoroughly investigated.

plasma. Hence, multi-particle considerations are included from the outset. The underpinning ingredients rely on standard approximations in classical laser-plasma theory; the slowly-varying envelope approximation for the laser pulse, and the ponderomotive approximation for the force exerted by the laser on the plasma electrons [6]. We exploit results established in the context of quantum field theory on curved spacetime to obtain the quantum corrections to the classical field equations. As the first application of our new theory, we show that the quantum fluctuations lead to the relationship

ω=

r

3 128π3/2

r

λe

w0

_αa2 0

(a2 0+ 1)3

14 _c2_κ2 √

ω0ωp

(1)

between the angular frequencyωand wavenumberκof a longitudinal perturbation to the laser-plasma variables in the rest frame of the plasma ions. The quantitya0is the dimensionless amplitude of the laser pulse,ω0is the frequency of the laser pulse,ωpis the plasma frequency in the unperturbed state,λeis the Compton wavelength

of the electron,α is the fine-structure constant, andw0 is the width of the laser pulse. Equation (1) is valid

for perturbations along the axis of the pulse whose wavelength is shorter than the length of the pulse, and the perturbation is static in the underlying classical theory (recovered in the limit_~→0).

Using (1), the characteristic time scale τ over which the length of a Gaussian disturbance approximately doubles due to quantum fluctations, in the rest frame of the plasma ions, satisfies

τ∼

r

128π3/2

3

r_w0

λe

_(a2 0+ 1)3

αa2 0

14√_ω0ωp c2 σ

2_{, (2)}

whereσ is the initial length of the disturbance. The implications of (2) are readily appreciated by expressing it in terms of the quantities ˇτ, ˇσ normalised with respect to the laser period 2π/ω0 and laser wavelength

λ0= 2πc/ω0, respectively. The approximate inequality

ˇ

τ_&534

s

w0λ0 λpλe

ˇ

σ2 (3)

results from noting that a0 = 1/ √

2 minimises the term within square parentheses in (2), where the length

λp = 2πc/ωp has been introduced. Moreover, the theory underpinning (1) requires the laser wavelengthλ0 to

be the shortest classical length scale in the analysis; thus, ˇσ >1 and

ˇ

τ_&534

s

w0λ0 λpλe

(4)

follows immediately.

A comparison of (4) with the number of oscillationsπw2

0/λ20 corresponding to the Rayleigh length of a laser

beam with waistw0 yields the upper bound

λ0_.0.128 (λeλpw03)

1/5 ₍₅₎

on the laser wavelength. If (5) is significantly violated then it is likely that classical diffraction will ensure the laser pulse disperses before the effects of quantum fluctuations become significant. For example, the represen-tative parameter choicew0=λp= 30µm describing a matched laser pulse yields 146 nm for the upper bound

onλ0. Thus, a comprehensive experimental investigation of our new results should be possible using an x-ray

laser (e.g. the European XFEL [8]) that produces pulses with duration greater than 100 fs.

2

Classical theory of laser-driven plasma

Our particular interest here is in the interaction of electrons with an intense laser pulse propagating through an underdense plasma, where the internal oscillations of the pulse determine the shortest significant classical length and time scales of the system. The large difference between the scales associated with the internal oscillations of the laser pulse and the behaviour of the wake behind the front of the pulse permits approximations to be introduced that greatly simplify the analysis. The precise details of the plasma electron motion due to the fast oscillations of the fields within the laser pulse are sacrificed to obtain an efficient model of the electron dynamics over distances that are much greater than the wavelength of the laser. Computationally efficient models in this context typically exploit the ponderomotive approximation for calculating the effect of the laser pulse on the plasma electrons in tandem with a slowly-varying envelope approximation for determining the influence of the matter on the laser pulse [5].

The total electric field and total magnetic field are each expressed a sum of two terms. The first term can be understood as the local average of the respective total field over the fast oscillations of the laser pulse, and the second term is the field of the laser itself. In particular, the total electric fieldEtotsatisfiesEtot=E−∂tA0

where, for notational convenience, the local average ofEtot is denotedE and the laser pulse is encoded by the

vector potentialA0. Likewise, the total magnetic fieldBtot isBtot=B+∇×A0.

The electrons in a cold fluid model of a laser-plasma satisfy

∂tp+ (v·∇)p=−

e2

2meγ

∇hA2₀i −e(E+v×B), γ=

s

1 + p

2

m2

ec2

+e

2_hA2 0i m2

ec2

, p=meγv (6)

in the relativistic ponderomotive approximation, wheremeis the rest mass of the electron,eis the elementary

charge andcis the speed of light in vacuo. The effect of the laser pulse on the electrons is determined byhA2 0i,

the square of the magnitude of the vector potentialA0 of the pulse averaged over its fast internal oscillations.

The vector fieldsv, p are understood as the averaged velocity and averaged momentum, respectively, of the plasma electrons. The contribution to γ2 _{(the square of the Lorentz factor γ) proportional to hA}2

0i is due to

the fast oscillatory motion of the electrons induced by the laser pulse.

The fieldsE, Bare produced from the averaged properties of the electrons; in particular,

∇·D=−en+ρi, ∇×H=∂tD−env+ji, ∇×E=−∂tB, ∇·B= 0, (7)

withD=ε0E,H=B/µ0, whereε0,µ0are the permittivity and permeability of the vacuum, respectively, and

nis the averaged electron number density. The fieldsρi,ji are the ion charge density and ion current density,

respectively, and are specified as data.

To proceed further, the properties of the laser potential must be specified. A popular strategy used in many studies of laser-plasma accelerators [6] is to solve

∂t2A0−c2∇2A0=−ωp2A0, ωp=

s

e2_n ε0meγ

(8)

for the vector potential A0, whereωp is the plasma frequency given by the local electron number density n.

For practical purposes, slowly-varying envelope, or eikonel, approximations are commonly used to remove the fast oscillations from (8) before further analysis. In addition to the separation of scales, a complete justification of the model (6), (7), (8) requires the dominant component of the electron momentum to be parallel to the direction of propagation of the laser pulse. We will exploit this facet of the classical model when developing the quantum theory in Section 3.

It can be shown [7] that applying the eikonel approximation to (8) yields the conservation of wave action

∂t(hA20iω0) +c2∇·(hA20ik0) = 0, (9)

where the local frequencyω0 and local wave vector k0 of the laser pulse satisfy ∂tk0 = −∇ω0 and the local

dispersion relation

ω₀2−c2k2₀=ω_p2. (10)

2.1

Reduction of the classical theory

There are numerous ways of developing effective quantum theories from the above system of classical field equations. A simple strategy is to focus on a regime in which the effects due to the interaction between the laser field and matter dominate over those directly connected to the averaged electromagnetic fields; thus,E,

Bare treated as negligible. To lowest order, the field equation forpis (6) withE,Bset to zero:

∂tp+ (v·∇)p=−

e2

2meγ

∇hA20i. (11)

Further simplification is achieved by focussing on potential flow. Equation (11) can be expressed as

∂tp−v×(∇×p) =−mec2∇γ, (12)

which is solved by p = ∇Ψ with the momentum potentiale Ψ satisfyinge ∂tΨ =e −mec2γ. The latter can be

rearranged to give

e2c2hA2₀i= (∂tΨ)e 2−c2(∇Ψ)e 2−m2_ec4. (13)

Although E, B feature in the lowest order Maxwell equations (7), their precise forms are not needed; the only necessary consequence of (7) is charge conservation. The electron number densitynis required to obtain the plasma frequencyωp, and charge conservation provides a suitable field equation forn. Since the ion charge

density is locally conserved, the electron number density must satisfy

∂tn+∇·(nv) = 0 (14)

which, using the expression forωp given in (8), yields

∂t(ωp2∂tΨ)e −c2∇·(ω_p2∇Ψ) = 0e . (15)

wheremev=∇Ψe/γ,∂tΨ =e −mec2γhave been employed.

The quantityε0ω2phA20i/2 has the physical dimensions of an energy density, and it can serve as the Lagrangian

density in an action principle for the field equations (9), (15). Expressing ε0ω2phA20i/2 in terms ofΨ and thee

phaseΦ, wheree ω0=−∂tΦ,e k=∇Φ, suggests the actione

S[Φe,Ψ] =e

ε0σ∗

e2_c2

Z

dtd3x1

2

(∂tΦ)e 2−c2(∇Φ)e 2 (∂tΨ)e 2−c2(∇Ψ)e 2−m2_ec4 (16)

where (10), (13) have been used to substituteωp,hA20i, respectively. Stationary variations of (16) with respect

toΦ,e Ψ yield (9), (15), respectively. The dimensionless constante σ∗has been introduced because (16) has been

obtained using dimensional reasoning and, althoughσ∗is inert in the classical theory, it will scale the quantum

corrections to the classical field equations.

Although substantial simplifications have been made to obtain (16), it is highly beneficial, from the perspec-tive of quantum theory, to replace (16) with its counterpart theory in one spatial dimension. This strategy is physically justified by noting that the laser-plasma variablesΦ,e Ψ change over a much shorter distance parallele

to the direction of propagation of the laser pulse than transverse to it. This property of the laser-plasma system is closely connected to the justification behind the introduction of (6), (8). Furthermore, within this approxi-mation, it is reasonable to chooseω2

phA20ito be expressible as a product of a function of (t, z) and a function of

(x, y). Thus, the quantityL∗ given by

L∗=

s

σ∗Rdxdy ω2phA20i ω2

phA20i|x=y=0

(17)

is constant, and we arrive at the estimate

S[Φe,Ψ]e ≈

ε0L2∗

e2_c2

Z

dtdz1

2

for the action (16). The linex=y= 0 has been chosen to lie along the centre of the laser pulse.

The above considerations suggest a relativistic bi-scalar field theory on 2-dimensional spacetime given by the action

S[Φ,Ψ] =~

Z

d2x√−η1

2η

µν_∂

µΦ∂νΦ (ηστ∂σΨ∂τΨ + 1) (19)

whereηµν is the Minkowski metric with signature (−,+), andµ, ν= 0,1. The fields Φ, Ψ,ηµν, and coordinates

x0_,x1_{are dimensionless. The action (19) can be obtained from (18) using the substitutions}

x0= ct

l∗

, x1= z

l∗

, Φe|x=y=0=

s

~e2

ε0m2ec3L2∗

Φ, Ψe|x=y=0=mecl∗Ψ. (20)

Note that the length scalesl∗, L∗ are unrelated; the former is inert (it has no direct physical meaning) and is

introduced solely for mathematical elegance, whereas the latter is proportional to the transverse size (width) of the laser pulse.

Unless otherwise indicated, for convenience we will henceforth adopt units in which the reduced Planck constant~is unity.

3

Quantum considerations

A perturbative exploration of some of the quantum implications of

S[Φ,Ψ] =

Z

d2x√−η1

2η

µν

∂µΦ∂νΦ (ηστ∂σΨ∂τΨ + 1) (21)

can be undertaken using the 1-loop effective action Γ given by

Γ[~Φ] =S[~Φ]−iln

Z

Df~exp(iΛ[f~])

(22)

where

Λ[f~] =1 2

Z

d2x

Z

d2y δ 2_S

δΦA(x)δΦB(y)

fA(x)fB(y) (23)

with the indicesA, B ranging over 1,2 and Φ1 = Φ, Φ2 = Ψ. For convenience, the notation ~Φ = (Φ1 Φ2)T, ~

f = (f1f2)T_{has been introduced, with T denoting matrix transposition. The functional Λ can be expressed as}

Λ[f~] =

Z

d2x√−η1

2

ηµν∂µΨ∂νΨ + 1

ηστ∂σf1∂τf1+ηµν∂µΦ∂νΦηστ∂σf2∂τf2

+ 4ηµν∂µΦ∂νf1ηστ∂σΨ∂τf2 (24)

or, equivalently,

Λ[f~] =

Z

d2x√−η1

2

~

f†Of~ (25)

using integration by parts, wheref~†is the Hermitian conjugate off~. In the above, and throughout the following, we adopt the minimal approach in whichf~is regarded as a map on an arbitrarily large torus; hence, boundary terms do not arise when integration by parts is used. The operatorOis given by

Of~=−∇(η)

µ

(ηστ∂σΨ∂τΨ + 1)ηµν 2ηµσ∂σΦηντ∂τΨ

2ηµσ_∂

σΨηντ∂τΦ ηστ∂σΦ∂τΦηµν

∂νf1

∂νf2

with∇(µη)the Levi-Civita covariant derivative induced fromηµν. SinceOis Hermitian with respect to the inner

product (~a,~b) =R

d2_x√_{−η ~a}†_{~b, Ohas real eigenvalues. By definition,}R

Df~exp(iΛ[f~]) = 1/pdet(−iO) where the functional determinant det(−iO) is formally equal to the product Πn(−iλn) of the non-zero eigenvalues

{−iλn}of−iO.

In general, it is not straightforward to analytically compute det(−iO). However, the calculation is trivial when Φ, Ψ are linear functions of Minkowski coordinates adapted toηµνbecause, in that case, the matrix within

the curly brackets in (26) is constant when expressed in those coordinates. Thus, the eigenfunctions ofOhave the form (a b)T_exp(il

µxµ), where the components of the wave 2-vectorlµ and the coefficientsa,bare constant,

and the pair of eigenvaluesλ_~+

l ,λ

−

~_l corresponding to eachlµ satisfies

λ_~+

l λ

−

~_l = (η στ_∂

σΨ∂τΨ + 1)ηγδ∂γΦ∂δΦ (ηµνlµlν)2−4(ηµσ∂σΦηντ∂τΨlµlν)2. (27)

For subsequent analysis, the fact that the right-hand side of (27) can be readily factorised is key. It follows that

λ_~+

l λ

−

~_l = (A µν

+ lµlν)(Aστ− lσlτ) (28)

where the symmetric tensorsAµν+ ,A

µν

− are

Aµν₊ =

q

ηστ_∂

σΨ∂τΨ + 1ηγδ∂γΦ∂δΦηµν+ηµσηντ∂σΦ∂τΨ +ηµσηντ∂σΨ∂τΦ, (29)

Aµν₋ =

q

ηστ_∂

σΨ∂τΨ + 1

ηγδ_∂

γΦ∂δΦηµν−ηµσηντ∂σΦ∂τΨ−ηµσηντ∂σΨ∂τΦ. (30)

Thus, det(−iO) = Π~l(iλ

+

~ l iλ

−

~_l ) = Π~l(−iλ

+

~_l ) Π~q(−iλ−~q) = det(−iO+) det(−iO−) where the second-order

differ-ential operatorsO+,O− are

O+f =−∇(η)

µ (A µν

+ ∂νf), O−f =−∇(µη)(A µν

− ∂νf). (31)

Hence, one can factoriseR

Df~exp(iΛ[f~]) as

Z

Df~exp(iΛ[f~]) =

Z

Dfexp(iΛ+[f])

Z

Dfexp(iΛ−[f])

(32)

where

Λ+[f] =

Z

d2x√−η1

2A

µν

+ ∂µf ∂νf, Λ−[f] =

Z

d2x√−η1

2A

µν

−∂µf ∂νf. (33)

The above considerations are strictly only applicable to the cases where Φ, Ψ are linear functions of Minkowski coordinates adapted toηµν. Fields Φ, Ψ of this type describe a non-evolving monochromatic laser

beam propagating through a uniform plasma. However, it is plausible that (29), (30), (32), (33) hold to a rea-sonable approximation when Φ, Ψ are more general. This assertion can be justified by appealing to the WKB approximation; seeking solutions toO~f=λ~fof the form~f=P∞

n=0ε

n_~a

nexp(iχ/ε), where~an, χare fields andε

is the WKB expansion parameter, leads to a pair of eigenvalues that satisfy (27) withlµ substituted by∂µχ/ε.

Those eigenvalues are identical to the eigenvalues ofO+,O− in the WKB approximation and, hence, (32), (33)

hold. This approach is analogous to using the Euler-Heisenberg action to describe QED vacuum polarisation even when the electromagnetic invariants are not constant. Although the derivation of the Euler-Heisenberg action requires the electromagnetic fields to be constant (the potentials are linear in Minkowski coordinates), it is not uncommon to use the result in more general circumstances.

Although one can replace (31) with more complicated second-order linear operators and, via the WKB approximation, motivate more complicated expressions for Λ+, Λ−, the choice (33) is perhaps the most natural.

This conclusion is supported by the special case in whichησλ_∂

σΨ∂λΨ −1 and∂µΦ≈∂µΨ. In this particular

situation

Λ[f~]≈

Z

d2x√−η1

2

follows from (24). Introducing the new variables ˇf1= (f1+f2)/ √

2, ˇf2= (−f1+f2)/ √

2 immediately yields

Λ≈

Z

d2x√−η1

2 A

µν

+ ∂µfˇ1∂νfˇ1+A

µν

− ∂µfˇ2∂νfˇ2

(35)

where Aµν₊, Aµν₋ are given by (29), (30) with the substitutions ∂µΨ →∂µΦ, ηστ∂σΨ∂τΨ + 1→ ηστ∂σΦ∂τΦ.

The functional measure satisfiesDf~=Df~ˇbecause the transformation betweenf~andf~ˇis a constant rotation, and we obtain

Z

Df~ˇexp(iΛ[f~])≈

Z

Df1ˇ exp(iΛ+[ ˇf1])

Z

Df2ˇ exp(iΛ−[ ˇf2])

(36)

as required. Unfortunately, when considered as a metric,Aµν₊ does not have a Lorentzian signature in this case and the classical states in this regime are not perturbatively stable. Hence, the validity of the effective action Γ is questionable in this regime. Nevertheless, the above considerations suggest the use of (29), (30), (32), (33) in cases whereAµν+ ,A

µν

− both have Lorentzian signatures.

The metric signatures of A+

µν, A−µν can be deduced from the signs of the eigenvalues of the tensors M µ

+ν,

M₋µν given by

M₊µν=δµν+B η µω_(X

ωYν+YωXν), M−µν =δνµ−B η µω_(X

ωYν+YωXν), (37)

where the timelike unit normalised covector fieldsXµ,Yν and scalar fieldB (satisfyingB >1) are

Xµ =

∂µΦ

√ −ηστ_∂

σΦ∂τΦ

, Yµ=

∂µΨ

√ −ηστ_∂

σΨ∂τΨ

, B=

s

ηστ_∂ σΨ∂τΨ

ηγδ_∂

γΨ∂δΨ + 1

. (38)

The eigenvalues ofM₊µν,M−µν are{1−B(coshχ+1), 1−B(coshχ−1)}and{1+B(cosχ+1), 1+B(coshχ−1)},

respectively, where coshχ=−ηµν_X

µYν. By inspection, the eigenvalues ofM−µν are always positive and, since

M₋µν =ηνσAσµ− /CwhereCis a positive scalar field, it follows thatA

µν

− has the same signature asηµν. However,

one of the eigenvalues ofM₊µν =ηνσAσµ+ /Cis always negative and the sign of the remaining eigenvalue depends

on the properties of the fields. The remaining eigenvalue must be negative forAµν₊ to be Lorentzian, albeit of opposite signature toηµν. Thus, coshχ >(1 +B)/B, which can be expressed as the condition

−ηµν∂µΦ∂νΨ>

q

ηστ_∂

σΦ∂τΦ (ηµν∂µΨ∂νΨ + 1) +

p

ηστ_∂

σΦ∂τΦηµν∂µΨ∂νΨ. (39)

Henceforth, we will only consider the regime in which (39) is satisfied.

An explicit expression for the effective action Γ is readily obtained by appealing to studies of the behaviour of the quantum vacuum in curved spacetimes. The required results emerge when (32) is expressed in terms of a massless field theory on a dilatonic curved background. The pair of metricsg+

µν,g−µν and the pair of dilatons

ϕ+_{, ϕ}− _are

g₊µν = A

µν

+

p

A+

, ϕ+=−

1

4ln(A+), g

µν

− =

Aµν₋

p

A−

, ϕ−=−

1

4ln(A−) (40)

whereA+,A− are the determinants of the tensorsA+νµ=ηµσAσν+ ,A

ν

−µ =ηµσAσν− , respectively. It follows that

(33) can be expressed as

Λ+[f] =

Z

d2xp−g+1

2exp(−2ϕ+)g

µν

+ ∂µf ∂νf, Λ−[f] =

Z

d2xp−g−1

2exp(−2ϕ−)g

µν

− ∂µf ∂νf (41)

withg+,g− the determinants ofgµν+,gµν−, respectively, and the effective action (22) decomposes as

Γ =S−iln

Z

Df exp(iΛ+[f])

−iln

Z

Df exp(iΛ−[f])

The effective actionW given by

exp(iW[gµν, ϕ]) =

Z

Df exp

_i

2

Z

d2x√−g exp(−2ϕ)(∇f)2

(43)

describes the coupling of a dilatonϕ to a Lorentzian metricgµν, and their self-couplings, due to the vacuum

fluctuations of a massless scalar fieldf. Its exact renormalised form is [9, 10]

W[gµν, ϕ] =

1 24π

Z

d2x√−g

1 4R

−1_{R+R(ψ+ 2ϕ)−3(∇ϕ)}2

−1R−(∇ϕ)2−4∇ϕ· ∇ψ−(∇ψ)2

(44)

− 1

4πlnµ

Z

d2x√−g(∇ϕ)2

where (∇f)2_{=∇f· ∇f, ∇ϕ· ∇ψ=g}µν_∇

µϕ∇νψ, =gµν∇µ∇ν, ∇µ is the Levi-Civita covariant derivative

given bygµν, andRis the scalar curvature ofgµν. The conventions used for the Riemann tensor and Ricci tensor

underpinningR are given in Ref. [11]. The scalar fieldψthat appears in (44) captures some of the freedom in the choice of the measureDf. In particular, the quantity R

Dfexp(ihf, fi) is chosen to be a field-independent constant, where the inner product h·,·iis given by ha, bi =R

d2x√−gexp(−2ψ)a∗b. The constant µ in (44) emerges from the zeta-function regularisation technique used to derive (44) and, in general, must be fixed using additional information.

The result of each functional integral in (42) follows immediately from (44) using the respective substitutions

gµν =gµν+, ϕ= ϕ+, µ = µ+ and gµν = g−µν, ϕ = ϕ−, µ = µ−. The natural inner product in both cases is

R

d2_x√_{−η a}∗_{b, and sinceg}+_=g−_{=ηfollows from (40), we setψ= 0 inh·,·i. In summary, an effective theory}

describing the self-interaction of a laser-driven plasma due to quantum vacuum fluctuations is

Γ[Φ,Ψ] =S[Φ,Ψ] +w[g+_µν, ϕ+, µ+] +w[g_µν−, ϕ−, µ−] (45)

where

w[gµν, ϕ, µ] =

1 24π

Z

d2x√−g

₁

4R

−1_{R+ 2Rϕ−3(∇ϕ)}2

−1R−(∇ϕ)2

− 1

4πlnµ

Z

d2x√−g(∇ϕ)2 (46)

andg+

µν, gµν−, ϕ+,ϕ− depend on Φ, Ψ according to (29), (30), (40). The fields must satisfyηµν∂µΦ∂νΦ<0,

ηµν_∂

µΨ∂νΨ<−1, and the condition (39).

4

Field equations for

Φ

and

Ψ

Stationary variations of the action (45) with respect to Φ, Ψ lead to field equations describing a laser-driven plasma that include the backreaction of the quantum fluctuations. The field equations arising from the Φ variation and Ψ variation can be expressed as

∇(η)

σ (εβ σ_+Bσ

++B

σ

−) = 0, ∇(ση)(αζ σ_+Cσ

++C

σ

−) = 0, (47)

respectively, where

α=ηστ∂σΦ∂τΦ, βµ=ηµν∂νΦ, ε=ηστ∂σΨ∂τΨ + 1, ζµ=ηµν∂νΨ, (48)

with

Bσ+=

1

√ −η

δw+ δAµν+

ηµν

r

ε αβ

σ_{+ 2ζ}µ_ηνσ

, B₋σ =√1 −η

δw−

δAµν₋

ηµν

r

ε αβ

σ

−2ζµηνσ

, (49)

Cσ

+=

1

√ −η

δw+ δAµν₊

ηµν

r

α εζ

σ_{+ 2β}µ_ηνσ

, Cσ

− =

1

√ −η

δw−

δAµν₋

ηµν

r

α εζ

σ_−2βµ_ηνσ

wherew+=w[g+µν, ϕ+, µ+], w− =w[g−µν, ϕ−, µ−].

The structure of (49), (50) follows because the effective metricsg+

µν,g−µνand dilatonsϕ+,ϕ−can be expressed

solely in terms ofAµν₊ ,Aµν₋, respectively. The details follow using

δw+=

Z

d2xδw+ δAµν+

δAµν₊ , δw−=

Z

d2xδw− δAµν₋ δA

µν

− , (51)

with

δAµν₊ =

ηµν

r

ε αβ

σ_{+ 2ζ}(µ_ην)σ

∂σδΦ +

ηµν

r

α εζ

σ_{+ 2β}(µ_ην)σ

∂σδΨ, (52)

δAµν₋ =

ηµν

r

ε αβ

σ_−2ζ(µ_ην)σ

∂σδΦ +

ηµν

r

α εζ

σ_−2β(µ_ην)σ

∂σδΨ, (53)

which emerge from (29), (30). The parentheses enclosing indices denote symmetrisation with the standard weighting; e.g. 2β(µην)σ = βµηνσ+βνηµσ. As usual, the variations δΦ, δΨ are chosen to have compact support; thus, no boundary terms arise during the derivation of the field equations.

The remainder of this section is focussed on determining the functional derivatives ofw+,w− with respect

toAµν+ ,A

µν

− , respectively. To achieve this goal, it is fruitful to briefly return to the most natural variables for

expressingw+, w−; the effective metrics and dilatons.

4.1

Variations of

w

with respect to

g

and

ϕ

Sincew+ (or w−) is simplywevaluated at particular values of its arguments, we can capture the variations of

w+ with respect tog

µν

+ ,ϕ+ (org

µν

− , ϕ−) by appealing solely to the variations of wwith respect togµν, ϕ.

Taking care of the inverse D’Alembertian operator−1 _{using the techniques given in Ref. [12], we find that}

the functional derivatives ofwwith respect togµν_,ϕare

48π √

−g δw

δgµν =−∇µ∇ν

−1_R+g

µνR−

1 4gµν∇

σ

−1R∇σ−1R+

1 2∇ν

−1_R∇

µ−1R

+ 3gµν(∇ϕ)2−1R+ 3gµν∇σ−1(∇ϕ)2∇σ−1R−6∇(ν−1(∇ϕ)2∇µ)−1R

−6(∇µϕ∇νϕ)−1R+ 6∇µ∇ν−1(∇ϕ)2−4∇µ∇νϕ+ 4gµνϕ

+gµν(−5 + 6 lnµ) (∇ϕ)2−2(∇µϕ∇νϕ)(1 + 6 lnµ), (54)

12π √

−g δw

δϕ = 3∇µ

−1_R∇µ_{ϕ+ 3}

−1R+ 1 + 6 lnµϕ+R. (55)

For convenience, indices have been lowered (or raised) using the metric tensorgµν (or its inverse gµν) in (54),

(55).

4.2

Variation of

w

with respect to

A

The appropriate combination of (54), (55) that appears in the field equations (47) emerges upon introducing the variableAµν _=e−2ϕ_gµν_{, where 4ϕ=−ln(A) withA the determinant ofA}ν

µ =ηµσAσν. Hence, δgµν =

e2ϕδAµν_{+ 2g}µν_{δϕand 4δϕ=−e}2ϕ_g

µνδAµν. It follows

δw=

Z

d2x

_δw

δgµνδg

µν₊δw

δϕδϕ

=

Z

d2x

_δw

δgµν −

1 4

2 δw

δgστg

στ₊δw

δϕ

gµν

e2ϕδAµν ₍₅₆₎

and we obtain

1

√ −η

δw δAµν =

1

√ −g

δw δgµν −

1 4

2 δw

δgστg

στ₊δw

δϕ

gµν

e2ϕ. (57)

Hence, the field equations (47) for Φ, Ψ are specified by substituting the functional derivatives found in (49), (50) with

1

√ −η

δw+ δAµν+

= _p1

−g+

_δw

+ δgµν+

−1

4

2δw+

δgστ+

g₊στ+δw+

δϕ+

g_µν+

e2ϕ+_{, (58)}

1

√ −η

δw−

δAµν₋ =

1

p

−g−

_δw

−

δg₋µν −

1 4

2δw−

δgστ

−

g₋στ+δw−

δϕ−

g_µν−

e2ϕ−_{. (59)}

The right-hand side of (58) (or (59)) is given by substitutingg+

µν,ϕ+,µ+ (org−µν,ϕ−,µ−) into (54), (55).

5

Linearised field equations

The simplest exact solutions to (47) describe a uniform, monochromatic, laser beam propagating through a uniform plasma. In this case,βµ_{, ζ}ν _{are covariantly constant with respect to the Levi-Civita connection ∇}(η)

of the flat spacetime metric ηµν. Using (48), it is clear that the classical terms in (47) immediately vanish.

The dilatons ϕ+, ϕ− and effective metrics gµν+, g−µν are constructed solely from tensors that are covariantly

constant with respect to ∇(η)_{; thus, they are also covariantly constant with respect to ∇}(η)_{. It follows that}

the components of the effective metrics are constant in a Minkowski coordinate system adapted toηµν; thus,

their Christoffel symbols are zero. In addition to the fact that the dilatons are constant, we conclude that the curvatures of the effective metrics are zero. Inspection of (54), (55) shows that the quantum corrections to the classical field equations are zero as required.

We will now uncover the impact of the quantum backreaction on perturbations to the exact solutions describing a uniform, monochromatic, laser beam propagating through a uniform plasma. Throughout the following, we will use a ‘bar’ to denote fields and operators associated with the unperturbed exact solutions. For simplicity, we will use Minkowski coordinates adapted toηµν; thus, for the reasons given above, all components

of ‘bar’ tensors are constant. Introducing the substitutions

gµν = ¯gµν+g₍₁₎µν, ϕ= ¯ϕ+ϕ(1) (60)

in (54), (55) gives

48π √

−g δw

δgµν =−∂µ∂ν¯

−1_R

(1)+ ¯gµνR(1)−4∂µ∂νϕ(1)+ 4¯gµν¯ϕ(1), (61)

12π √

−g δw

δϕ =R(1)+ (1 + 6 lnµ) ¯ϕ(1) (62)

to first order in the perturbationsgµν₍₁₎,ϕ(1). The scalar curvature perturbationR(1) is

R(1) =−∂µ∂νg µν

(1)+ ¯gµν¯g

µν

(1), (63)

and ¯= ¯gµν_∂ µ∂ν.

Hence, (47), (49), (50), (58), (59), (61), (62) together give

0 = ¯ε₍η)Φ(1)+ 2 ¯βµζ¯ν∂µ∂νΨ(1)

+ 1 96π

r

¯

ε

¯

α

¯

βσ∂σ

e2 ¯ϕ+₋₂₍₍

η)/¯+)R₍₁₎+ −c+R+₍₁₎−8(η)ϕ+₍₁₎+ 2c+(1−6 lnµ+) ¯+ϕ+₍₁₎

+e2 ¯ϕ−₋₂₍

(η)/¯−)R−₍₁₎−c−R−₍₁₎−8(η)ϕ−₍₁₎+ 2c−(1−6 lnµ−) ¯−ϕ−₍₁₎

+ 1 48π

¯

ζµ

e2 ¯ϕ+

−2(₍η)/¯+)∂µR+₍₁₎−¯g+µν∂ ν_R+

(1)−8∂µ(η)ϕ+(1)+ 2(1−6 lnµ+)¯g +

µν∂ ν_ϕ+

(1)

−e2 ¯ϕ−

−2(₍η)/¯−)∂µR−₍₁₎−¯g−µν∂ ν_R−

(1)−8∂µ(η)ϕ−(1)+ 2(1−6 lnµ−)¯g

−

µν∂ ν_ϕ−

and

0 = ¯α₍η)Ψ(1)+ 2 ¯βµζ¯ν∂µ∂νΦ(1)

+ 1 96π r ¯ α ¯ ε ¯

ζσ∂σ

e2 ¯ϕ+₋₂₍

(η)/¯+)R+₍₁₎−c+R+₍₁₎−8(η)ϕ+₍₁₎+ 2c+(1−6 lnµ+) ¯+ϕ+₍₁₎

+e2 ¯ϕ−

−2(₍η)/¯−)R−(1)−c

−_R−

(1)−8(η)ϕ−(1)+ 2c

−_{(1−6 lnµ}

−) ¯−ϕ−(1)

+ 1 48π

¯

βµ

e2 ¯ϕ+₋₂₍₍

η)/¯+)∂µR+₍₁₎−g¯µν+∂ ν_R+

(1)−8∂µ(η)ϕ +

(1)+ 2(1−6 lnµ+)¯g +

µν∂ ν_ϕ+

(1)

−e2 ¯ϕ−₋₂₍

(η)/¯−)∂µR−₍₁₎−¯g−µν∂ ν_R−

(1)−8∂µ(η)ϕ

−

(1)+ 2(1−6 lnµ−)¯g

−

µν∂ ν_ϕ−

(1) , (65)

where∂µ=ηµν∂ν,(η)=ηµν∂µ∂ν,c+= ¯g µν

+ ηµν,c− = ¯g µν

− ηµν, and

R₍₁₎+ =−∂µ∂νg µν

+(1)+ ¯g +

µν¯

+_gµν

+(1), R

−

(1)=−∂µ∂νg µν

−(1)+ ¯g

−

µν¯−g µν

−(1). (66)

The perturbations to the effective metrics and dilatons are given in terms of Φ(1), Ψ(1) by

g₊₍₁₎µν = 2ϕ+₍₁₎¯gµν₊ +e2 ¯ϕ+_Aµν

+(1), g

µν

−(1) = 2ϕ

−

(1)¯g

µν

− +e

2 ¯ϕ−_Aµν

−(1), (67)

4ϕ+₍₁₎ =−e2 ¯ϕ+_¯g+

µνA µν

+(1), 4ϕ

−

(1)=−e 2 ¯ϕ−_¯g−

µνA µν

−(1), (68)

with

Aµν₊₍₁₎=

ηµν r ¯ ε ¯ α ¯

βσ+ 2 ¯ζ(µην)σ

∂σΦ(1)+

ηµν r ¯ α ¯ ε ¯

ζσ+ 2 ¯β(µην)σ

∂σΨ(1), (69)

Aµν₋₍₁₎ =

ηµν r ¯ ε ¯ α ¯

βσ−2 ¯ζ(µην)σ

∂σΦ(1)+

ηµν r ¯ α ¯ ε ¯

ζσ−2 ¯β(µην)σ

∂σΨ(1). (70)

5.1

Plane-wave perturbations and their dispersion relations

Inspection of (64), (65) shows that the classical behaviour of the perturbations Φ(1), Ψ(1) is determined by the

linear equations

¯

ε(η)Φ(1)+ 2 ¯βµζ¯ν∂µ∂νΨ(1)= 0, α¯(η)Ψ(1)+ 2 ¯βµζ¯ν∂µ∂νΦ(1)= 0. (71)

Using the plane-wave ans¨atze Φ(1) ∝exp(ikx), Ψ(1) ∝exp(ikx) in (71), wherekx≡kµxµ, leads to the dispersion

relation ¯Aµν+kµkνA¯σω− kσkω = 0 where ¯A µν

+ = √

¯

αε η¯ µν _{+ 2 ¯β}(µ_ζ¯ν)_{, ¯A}µν

− =

√

¯

αε η¯ µν_{−2 ¯β}(µ_ζ¯ν)_{. Furthermore, the}

presence of 1/¯+ _{within the quantum corrections in (64), (65) suggests that the terms denoted “. . .” inside the}

equations

0 = ¯ε₍η)Φ(1)+ 2 ¯βµζ¯ν∂µ∂νΨ(1)

+ 1 96π r ¯ ε ¯ α ¯

βσ∂σ

e2 ¯ϕ+₋₂₍

(η)/¯+)R+₍₁₎+. . . +

1 48π

¯

ζµe2 ¯ϕ+₋₂₍

(η)/¯+)∂µR+₍₁₎+. . . , (72)

0 = ¯α(η)Ψ(1)+ 2 ¯βµζ¯ν∂µ∂νΦ(1)

+ 1 96π r ¯ α ¯ ε ¯

ζσ∂σe2 ¯ϕ+−2((η)/¯+)R+(1)+. . . +

1 48π

¯

βµ

e2 ¯ϕ+

are negligible close to the classical solution satisfying ¯Aµν+ kµkν = 0. Likewise, the presence of 1/¯− within the

quantum corrections in (64), (65) suggests that the terms denoted “. . .” inside

0 = ¯ε(η)Φ(1)+ 2 ¯βµζ¯ν∂µ∂νΨ(1)

+ 1 96π

r

¯

ε

¯

α

¯

βσ∂σe2 ¯ϕ−−2((η)/¯−)R−(1)+. . . +

1 48π

¯

ζµ

−e2 ¯ϕ−

−2(₍η)/¯−)∂µR−₍₁₎+. . . ,

(74)

0 = ¯α₍η)Ψ(1)+ 2 ¯βµζ¯ν∂µ∂νΦ(1)

+ 1 96π

r

¯

α

¯

ε

¯

ζσ∂σe2 ¯ϕ−−2((η)/¯−)R−(1)+. . . +

1 48π

¯

βµ

−e2 ¯ϕ−

−2(₍η)/¯−)∂µR−₍₁₎+. . .

(75)

are negligible close to the classical solution satisfying ¯Aµν₋ kµkν = 0. Note that the unknown constantsµ+,µ−

in (64), (65) do not contribute in this regime.

Focussing on (72), (73), the above considerations suggest a perturbative analysis of

0 = ¯gµν₊ kµkν(¯ε k·kΦ(1)+ 2 ¯βkζk¯ Ψ(1)) +2 e2 ¯ϕ+

48π

r_ε¯

¯

αi

¯

βk+ 2iζk¯

k·k R+₍₁₎+O(3), (76)

0 = ¯gµν₊ kµkν( ¯α k·kΨ(1)+ 2 ¯βkζk¯ Φ(1)) +2 e2 ¯ϕ+

48π

r

¯

α

¯

εi

¯

ζk+ 2iβk¯

k·k R+₍₁₎+O(3), (77)

where ¯gµν₊ kµkν =O(), k·k ≡ηµνkµkν, ¯βk ≡β¯µkµ, and ¯ζk≡ζ¯µkµ. The perturbation parameter has been

introduced for clarity of exposition, and the-orders of the terms have been allocateda posterioriso that the working is self-consistent. The parameteris merely a device for capturing perturbative orders, and can be set to unity at the end of the calculation. Thus,R+₍₁₎=kµkνgµν₊₍₁₎+O() follows from (66), and so

¯

g₊µνkµkν(¯εk·kΦ(1)+ 2 ¯βkζk¯ Ψ(1))−2

1

48πa(aΦ(1)+bΨ(1)) =O(

3_{), (78)}

¯

g₊µνkµkν( ¯αk·kΨ(1)+ 2 ¯βkζk¯ Φ(1))−2

1

48πb(aΦ(1)+bΨ(1)) =O(

3_{) (79)}

emerge from (76), (77) using (69), where the quantitiesa,bare

a=

r_ε¯

¯

α

¯

βk+ 2 ¯ζk

k·k e2 ¯ϕ+_{, b=}

r_α¯

¯

ε

¯

ζk+ 2 ¯βk

k·k e2 ¯ϕ+_{. (80)}

Equations (78), (79) together form a homogeneous linear system for Φ(1), Ψ(1). For a non-zero solution to

exist, the condition

¯

g₊µνkµkν

¯

ε k·k 2 ¯βkζk¯

2 ¯βkζk¯ α k¯ ·k

−2 1

48π

a2 _ab ab b2

=O(6) (81)

on the matrix determinant of the coefficients of the linear system must be satisfied. Equation (81) is the dispersion relation

e2 ¯ϕ+_{( ¯A}µν

+kµkν)2A¯σω− kσkω−2

_k·k

48π(a

2_α¯+b2_ε¯)− ab

12π

¯

βkζk¯

=O(6) (82)

where ¯Aµν₊ kµkν =

√

¯

αε k¯ ·k+ 2 ¯βkζk¯ and ¯Aµν₋ kµkν =

√

¯

αε k¯ ·k−2 ¯βkζk¯ have been used. Introducing the substitution 2 ¯βkζk¯ = ¯Aµν+ kµkν−

√

¯

αε k¯ ·kin the final term of (82), and introducing ¯Aµν₋kµkν = 2

√

¯

αε k¯ ·k−

¯

Aµν+kµkν in the first term, yields

e2 ¯ϕ+_{( ¯A}µν

+kµkν)2−2

1 96π

a

_α¯

¯

ε

14

+b

_ε¯

¯

α

142

since ¯Aµν+ kµkν=O(). The latter follows from ¯g µν

+ kµkν =O(). Hence, substitutinga,b using (80) gives

√

¯

αε k¯ ·k+ 2 ¯βkζk¯ =

3 4

1

√

6π

|k·k|

(√α¯ε¯+ ¯β·ζ¯)2_−β¯_·β¯_ζ¯_·ζ¯1/4 _α¯ ¯ ε 14 ¯ ζk+ _ε¯ ¯ α 14 ¯ βk

+O(2) (84)

wheree−4 ¯ϕ+_{= (}√_{α¯ε¯+ ¯β·ζ}¯₎2_−β¯_·β¯_ζ¯_·ζ¯_{has been used to eliminate ¯ϕ}

+, which follows becausee−4 ¯ϕ+ is equal

to the determinant of the tensor ¯A+νµ=ηµσA¯σν+ .

The equivalent calculation using (74), (75) instead of (72), (73) yields

√

¯

αε k¯ ·k−2 ¯βkζk¯ =

3 4

1

√

6π

|k·k|

(√α¯ε¯−β¯·ζ¯)2_−β¯_·β¯_ζ¯_·ζ¯1/4 ¯ α ¯ ε 1₄ ¯ ζk− ¯ ε ¯ α 1₄ ¯ βk

+O(2) (85)

since ¯Aµν₋ kµkµ =O(), and e−4 ¯ϕ− = (

√

¯

αε¯−β¯·ζ¯)2_−β¯_·β¯_ζ¯_·ζ¯_{has been used to eliminate ¯ϕ}

−. The latter

is obtained becausee−4 ¯ϕ− _{is equal to the determinant of the tensor ¯A}

−νµ =ηµσA¯σν− . Finally, (39) yields the

condition

−β¯·ζ >¯ √α¯ε¯+

q

¯

β·β¯ζ¯·ζ¯ (86)

on the unperturbed fields.

Henceforth, for notational convenience, we will dropO(2_{) from (84) and (85), set = 1 in the remaining}

terms, and treat the results as equalities. The physical consequences of those results readily emerge by intro-ducing an ultrarelativistic approximation for ¯βµ _{via the asymptotic series ¯β}µ_{= ¯β}µ

[−1]/χ+χβ¯

µ

[1]+O(χ

2_{). Like}

, the sole reason for introducing the parameter χ is to track the perturbative orders of terms, and it will be set to unity after use. Note that the condition ¯β[−1]·β¯[−1] = 0 is a key ingredient in our approach, and so |β¯·β|¯ =O(1),|β¯·ζ|¯ =O(1/χ). We find that the same result

|ζk|¯ =√χ3

8 1

√

6π

_ζ¯_·ζ¯_{+ 1}

¯

α[0]

14 _|k·k|

q

|ζ¯·β¯[−1]|

+O(χ3/2), (87)

where ¯α = ¯α[0]+O(χ2), emerges from both (84) and (85). Equating the coefficients of equal powers of χ

throughout the condition (86) yields the requirement|ζ¯·β¯[−1]|>0, which is trivially satisfied because ¯ζ·ζ <¯ 0

and ¯β[−1]·β¯[−1]= 0.

Equation (87) can be reduced further by orthogonally decomposing the spacetime metric ηµν _{with respect}

to the timelike, unit normalised, vectornµ_{= ¯ζ}µ_/p

−ζ¯·ζ¯. It followsηµν _=−nµ_nν_+nµ

⊥nν⊥ where the spacelike

vectornµ_⊥ is orthogonal tonµ_{, i.e. n}

⊥·n= 0. Thus, (87) gives

|nk|=√χ3

8 1

√

6π

_ζ¯_·ζ¯_{+ 1}

|ζ¯·ζ|¯3_α¯ [0]

14 _(n

⊥k)2

q

|n·β¯[−1]|

+O(χ3/2). (88)

Equation (88) has been simplified usingnk = O(χ1/2_{). The quantities |nk| and |n}

⊥k| are the dimensionless

frequency and dimensionless wavenumber, respectively, of the perturbations to the laser-plasma variables in the rest frame of the plasma electrons. Furthermore, the introduction of nµ _{makes the physical content of}

the ultrarelativistic approximation used in passing from (84), (85) to (87) straightforward to appreciate. The quantity|n·β¯| is the dimensionless frequency of the laser in the rest frame of the plasma electrons, and the ultrarelativistic approximation is admissible because the frequency of the laser is much larger than the plasma frequency.

Finally, we will now express (88) in terms of the dimensionful variables that were introduced in the context of the underlying 3-dimensional classical theory in Section 2. DroppingO(χ3/2) from (88), and setting χ to unity, yields the dispersion relation

ω= 3 8

1

√

6π

~e2

ε0m2

ec3L2∗

14 _a2 0

(a2 0+ 1)3

14 _c2_κ2 √

ω0ωp

_x=y=0

The angular frequencyω and wavenumber κof the perturbation in the rest frame of the plasma electrons are given byω=c|nk|/l∗andκ=|n⊥k|/l∗, respectively, wherel∗is the inert length scale used in the construction

of (19). The laser frequencyω0, plasma frequencyωp, and dimensionless laser amplitude

a0=

ephA2 0i mec

(90)

emerge using the substitutions

|n·β[¯−1]|=

l∗ω0

c

_x=y=0

, α[0]¯ =−ε0m 2

ec3L2∗

~e2

l2

∗ωp2

c2

_x=y=0

, ζ¯·ζ¯=−(a20+ 1)|x=y=0, (91)

which hold to lowest order in perturbation theory. The details of (91) follow from (20),βµ=∂µΦ,ζµ=∂µΨ,

(∂tΦ)e 2−c2(∇Φ)e 2=ω2p, (∂tΨ)e 2−c2(∇Ψ)e 2−m2ec4=e2c2hA20i, (92)

and∂xΦe|x=y=0 =∂yΦe|x=y=0 = 0, ∂xΨe|x=y=0 =∂yΨe|x=y=0 = 0. Equation (1) follows immediately from (89),

upon replacingL∗ with the width of the laser pulse.

Acknowledgements

We thank Robin W Tucker for useful discussions. This work was supported by the UK Engineering and Physical Sciences Research Council grant EP/N028694/1 (D.A.B., A.C., A.N.), the Cockcroft Institute (A.C.), and the Lancaster University Faculty of Science and Technology (C.F.). All of the results can be fully reproduced using the methods described in the paper.

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