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Parametric Co-Linear Axion Photon Instability

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Parametric Co-Linear Axion Photon Instability Parametric co-linear axion photon instability K. A. Beyer,1, ∗ G. Marocco,1, y C. Danson,2, 3 R. Bingham,4, 5 and G. Gregori1 1Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK 2AWE, Aldermaston, Reading RG7 4PR, UK 3OxCHEDS, Clarendon Laboratory, Department of Physics, University of Oxford, UK 4Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK 5Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK Axions and axion-like particles generically couple to QED via the axion-photon-photon inter- action. This leads to a modification of Maxwell's equations known in the literature as axion- electrodynamics. The new form of Maxwell's equations gives rise to a new parametric instability in which a strong pump decays into a scattered light wave and an axion. This axion mode grows exponentially in time and leads to a change in the polarisation of the initial laser beam, therefore providing a signal for detection. Currently operating laser systems can put bounds on the axion parameter space, however longer pulselengths are necessary to reach the current best laboratory bounds of light-shining through wall experiments. I. INTRODUCTION gated by the PVLAS collaboration, axions source bire- fringence [17] of light propagating through a strong con- Despite the many successes of the Standard Model, it is stant magnetic field. This is a consequence of the altered thought to be incomplete. The strong sector of the Stan- dispersion relation [15]. In this work we extend previous dard Model allows CP violation, which is constrained to analyses to look at systems in plane-wave backgrounds. cancel the electroweak contribution by measurements of We will demonstrate the existence of a new parametric the electromagnetic dipole moment of the neutron [1,2]. decay instability of one photon in the seed pulse into an This is known as the strong CP problem, which can be axion and a secondary photon, satisfying solved in an elegant fashion by introducing a new chiral, anomalous U(1)PQ symmetry, as pointed out by Peccei and Quinn [3,4]. Such global symmetry, after spon- !0 = !γ + !a and k0 = kγ + ka: (2) taneously breaking at a high scale, dynamically drives the CP violating parameter θ¯ to 0. The pseudo Nambu- Goldstone boson originating from the symmetry breaking Here, subscript γ denotes the scattered photon and a process is called the axion [5,6]. It was pointed out that the axion. Physically, small fluctuations in the axion axion models can account for the dark matter content field source electromagnetic fields. Such fields then in- of the universe [7{9]. Axion-like particles (ALPs) may teract with the pump pulse and source the axion field. abundantly arise in extensions of the Standard Model, Hence, a positive feedback is generated which leads to such as the low energy spectrum of String Theory [10, 11]. the growth of the axion fluctuations. A similar idea was In the following we will use the term axion to refer to investigated in [18] for the case of parametric excitation both, the CP restoring QCD axion and ALPs. of axions, where a different ansatz is used for the time- Axions generically couple to Standard Model photons dependence of the fields. Additionally, the stimulated via a dimension 5 operator decay of photons (or more specifically, plasmons) into axions have been considered in [19]. L = g aE · B: (1) aγγ aγγ The polarisation-dependent coupling (1) forces the with a coupling strength parametrized by g . The in- scattered mode to have a polarisation orthogonal to the arXiv:2108.01489v1 [hep-ph] 3 Aug 2021 aγγ clusion of equation1 modifies Maxwell's equations [12]. seed pulse. Hence, the laser polarisation depletes. This Axion electrodynamics in vacuum has been widely ap- rate of polarisation change scales as gaγγ , as opposed to 2 plied to axion detection experiments (for a review see, the gaγγ scaling of conventional birefringence or dichro- e.g., [13]). A range of phenomena like the axion-photon ism experiments [16]. mass mixing in magnetic field backgrounds [14, 15] or This paper is organised as follows: In section II we axion sourced birefringence [16] are extensively studied derive the set of equations coupling the axion to elec- in the literature. trodynamics in a strong pump background and combine The propagation of an intense laser beam receives cor- them to find the dispersion relation of the system. In rections due to the presence of axions. As was investi- section III we solve the dispersion relation. For simplic- ity we limit our analysis to the co-linear limit and leave the general treatment for a later paper. In sectionIV we ∗ Authors to whom correspondence should be addressed: kon- comment on the significance of this axion-photon insta- [email protected] bility. Within the paper we employ natural units with y [email protected] ~ = c = kB = 4π"0 = 1. 2 II. AXION-PHOTON DISPERSION RELATION In this colinear limit, we may use the residual gauge free- dom to set φ = 0. The remaining equation of motion for The axion-photon vertex (1) introduces a source and the axion is then current to the vacuum Maxwell's equations [12, 14] 2 2 2 @t − r + ma a = gaγγ (@tA) · (r × A): (7) r · E = gaγγ (ra) · B; r · B = 0; (3) r × E = −@tB; We perform a linear stability analysis by setting r × B = @tE + gaγγ (E × ra − B@ta) : A = A cos (! t − k · x) + A~ ; a =a; ~ (8) The electric and magnetic fields in the axion photon cou- 0 0 0 pling term (1) are defined as where A~ ; a~ A0. Note that this assumption is only E = −@tA − rφ, (4) valid for early times before the instability turns strongly B = r × A: non-linear. To solve this system of coupled partial differ- ential equations, we first perform a spatial Fourier trans- We employ Coulomb gauge r · A = 0, and work in the form of all fields colinear limit where all momenta, are parallel to find the wave equations for the gauge potentials Z d3x f (k; t) ≡ f (x; t) e−ik·x (9) 2 2 3=2 @t − r A = gaγγ r × (a@tA) (5) (2π) and to find a coupled set of linearised ordinary differential r2φ = 0: (6) equations !0 −@2 − k2 A~ (k; t) = −g (k × A ) a~ e−i!0t − a~ ei!0t ; (10) t aγγ 2 0 − + gaγγ h i −@2 − k2 − m2 a~ (k; t) = A · ei!0t (! k − ik @ ) × A~ − e−i!0t (! k − ik @ ) × A~ : (11) t a 2 0 0 + 0 t + 0 − 0 t − with subscript ± denoting the dependency on k ± k0. III. GROWTH RATE To analyse the coupled system's behaviour we make an Ansatz for the temporal dependency of the fields We would now like to solve equation (13) for real k, f (k; t) = f (k) e−i!(k)t (12) while allowing complex !. The dispersion relation is a 6th-order polynomial in !. Immediately the two real which reduces the differential equations (10) and (11) to roots !(k) = k can be discarded in the following as we are algebraic equations in the fields. They are solved for interested in unstable modes. Such unstable modes are frequencies satisfying the dispersion relation characterised by a complex frequency with non-vanishing imaginary part Γ ≡ = (!) which defines the growth rate. 2 2 th a gaγγ A0 k− k+ We proceed by working with the remaining 4 -order D (!; k) = !0 (!0k − !(k)k0) γ + γ ; polynomial and splitting ! into its real and imaginary 4 D− D+ (13) parts !(k) = !a(k) + iΓ(k) for !a(k); Γ 2 R. The result- a 2 2 2 γ ing real and imaginary equations are linearly independent where k = jkj, D (!; k) ≡ ! − k − ma, D± ≡ (! ± 2 2 and must be satisfied individually. !0) − (k ± k0) , and functions of (! ± 2!0; k ± 2k0) are neglected as off-resonant and hence sub-dominant. The imaginary equation is a cubic polynomial in the The rest of the paper is concerned with an investigation growth rate Γ with one trivial solution Γ = 0 and two of this dispersion relation. non-trivial ones s 2 3 2 3 2 2 2 2 3k! + 2! − 4!a! − k − m (!a + k) + g A k! Γ = ± a a 0 a aγγ 0 0 : (14) k − 2!a 3 From our ansatz (12) we know that a positive Γ corre- sponds to growth, hence we focus only on the positive root. After substitution of Γ into the real part of the equation we are then left with a single algebraic equation for !a(k). Figure1 shows that in our case, the frequency !a(k) is very small and thus motivating an expansion in !a, which we find to be 2 2! (gaγγ A0) 2 ma !a(k) = k (gaγγ A0) − : (15) 16 !0 Here we dropped terms of order 6 in (gaγγ A0), (ma=!0) and k as being small. We may then express the growth rate as s 2 2 2 ma k Γ(k) = !0 (gaγγ A0) − − ; (16) FIG. 1. The plot shows the growing solution to the dispersion !0 k0 relation (13). The blue solid curve shows the real part !a(k) where we have again dropped higher order terms in corresponding to the modes frequency while the dashed yellow curve depicts the growth rate Γ. Note that here we included (g A ), (m =! ) and (k=k ). aγγ 0 a 0 0 the O(k2) contributions to show the cutoff. As a reference we Upon a closer look at (16) the qualitative behaviour of include the vacuum dispersion relation for an axion with the the numerical solution depicted in fig1 is recovered.
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