Pairing in high-T_{C} superconductors is mediated by Compton scattering of virtual photons between two charge reservoirs; type I and type II [1]. Typically, the type I reservoir hosts the primary superconducting condensate while the type II reservoir provides mediating charges. However, in the case of H_{3}S, for example, the two reservoirs are identical and perform both roles. As presented elsewhere, the optimal transition temperature for high-T_{C} superconductors obey the algebraic equation,

T_{C0} =* k*_{B}^{–1} β*/*ℓζ =* k*_{B}^{–1} (Λ/ℓ) *e*^{2}/ζ .

Part of the above equation, the Coulomb potential *e*^{2}/ζ factor, was previously derived (see section 3 of [2]) by drawing an analogy with Allen-Dynes theory [3], which considers BCS superconductivity in the strong-coupling limit, obtaining T_{C} ∝ (〈ω^{2}〉λ)^{1/2}, where 〈ω^{2}〉 is the average square phonon frequency and λ the electron-phonon coupling parameter, with Coulomb repulsion being ignored [4] This derivation evaluates 〈ω^{2}〉λ by an integral over the Fermi surface of the square of electron-ion interaction forces. Evaluated in real space and determined by the inter-reservoir electronic Coulomb interaction (assumed to be unscreened within the unit cell [29]), the force is given as *F*(*r*) = ±*e*^{2}*r* (*r*^{2} + ζ^{2})^{–3/2}, which is constrained to the in-plane or longitudinal component for electronic charges in planes separated by a transverse distance ζ and projected in-plane radial distance *r*. Integrating 2π*r**F*^{2}(*r*) over *r* from 0 to ∞ and taking the square-root yields (π/2)^{1/2}*e*^{2}/ζ, whence the factor *e*^{2}/ζ in T_{C0} is obtained. In practice, this interaction is local, corresponding to a limited range of integration, *r* ≤ ζ; this results in the same potential factor, but with a smaller numerical coefficient.

The origin of the factor Λ/ℓ in equation (1) may be understood by considering the Compton scattering probability. Owing to thermal smearing of both the Fermi surface and superconducting gap, one expects ample phase space for quasiparticle creation by Compton scattering events. The essentially instantaneous nature of quasiparticle creation, compared to the significantly slower relaxation process, allows for unscreened Coulomb interactions between electronic charges in adjacent reservoirs. Equation (1) thus derives fundamentally from quantum fluctuations within the virtual photon field between the two charge reservoirs inducing Compton scattering of quasiparticles confined essentially within their respective reservoirs.

Scattering of a quasiparticle from an initial equilibrium state ** k** on the Fermi surface to a final state

**via a Compton scattered virtual photon precipitates a corresponding shift from**

*k’***to**

*q***in the photon wavevector. The final state wavevector**

*q’**is expected to contain a small transverse component, such that*

**k’****and**

*k***remain essentially transverse to the direction of the separation distance ζ. Such electron-photon scattering events can be likened to the problem of electron-impurity scattering, where transport electrons confined to near the Fermi surface with Fermi wavevector**

*k’**k*

_{F}are scattered by real space localized potentials with a mean free path spacing of ℓ

_{mfp}; the probability for a single such scattering event scales as (

*k*

_{F}ℓ

_{mfp})

^{–1}. In the Compton scattering picture, virtual photons of wave vector

**confined within the real space region between the two interacting reservoirs defined by ζ act in analogy to transport electrons and scatter off of the essentially localized electrons/holes of spacing ℓ (acting in analogy to impurities). The analogy entails a tenable difference between electron and photon time scales, given Fermi velocity**

*q**v*

_{F}<<

*c*. Thus, the probability of scattering a virtual photon from initial state

**to final state**

*q***is given by the differential (**

*q’**q’*ℓ)

^{–1}– (

*q*ℓ)

^{–1}= Δƛℓ

^{–1}= ƛ

_{C}(1 – cos θ)ℓ

^{–1}, where ƛ

_{C}=

*ћ*/

*m*

_{e}

*c*is the reduced Compton wavelength and θ is the angle between the incident and scattered photons (note that setting θ = π yields Δƛ = 2ƛ

_{C}). The momentum perturbation associated with electron recoil is very small, since |

**–**

*q***| << k**

*q’*_{F}for virtual photons of energy

*e*

^{2}/ζ. Identifying Λ as proportional to ƛ

_{C}and integrating over the relevant angles of the incident and scattered photons one obtains,

TC0 ∝ (ƛC/ℓ) *e*2/ζ ,

a result which can be (holistically) interpreted as the product of the scattering probability and the energy of the scattering interaction. Thus, the mediation of the high-*T*_{C} superconducting state is facilitated through Compton scattering between virtual photons and unscreened quasiparticles.

1. D. R. Harshman and A. T. Fiory, Journal of Physics: Condensed Matter **29**, 445702 (2017)

2. D. R. Harshman, A. T. Fiory and J. D. Dow, J. Phys.: Condens. Matter **23**, 295701 (2011); **23** 349501 (2011).

3. P. B. Allen and R. C. Dynes, Phys. Rev. B **12**, 905 1975.

4. See Eq. (25) in [2]