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Then the susceptibility χ ′′ $_{ρρ}$( q , ω ) can be obtained from the real part of Eq. (80), 43,50 

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where I$_{1}$ ( q ), I$_{2}$ ( q ), I$_{3}$ ( q ), and I$_{4}$ ( q ) are given in Ap- pendix C. 

Equation (87) is one of the central results of this paper, it’s worth exploring some of its features and limits. First, one can show that Eq. (87) satisfies both the thermody- namic sum rule, Eq. (60), and f-sum rule, Eq. (61). Sec- ond, the first and second terms in Eq. (87) produce two Brillouin doublets centered at ω = ± c$_{L}$q and ω = ± c$_{2}$q with widths D$_{L}$q 2 and D$_{2}$q $^{2}$, respectively. The third and fourth terms in Eq. (87) are negligible near the Brillouin doublets, and in fact these terms vanish in the limit of zero dissipation. Therefore the non-dissipative density- density correlation function, Eq. (58), is obtained from the first two terms in the limit D$_{L}$, D$_{2}$ → 0. Finally, for normal solids ( ρ$_{s}$ = 0), the second term in Eq. (87) van- ishes, and there is only one Brillouin doublet due to the longitudinal first sound modes. In this case the fourth term in Eq. (87) becomes the Rayleigh peak of the de- fect diffusion mode centered at ω = 0. Therefore, we see that in analogy with a superfluid, 48 the defect diffusion peak that exists in a normal solid will split into a Bril- louin doublet of second sound modes upon entering the supersolid phase . 

To get a sense of the size of this effect, let’s substi- tute some physically realistic numbers into the correla- tion function. Assuming ρ$_{s}$ ≪ ρ$_{0}$ and γ = Λ = Σ = 0, we have 

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where the longitudinal sound speed of normal solid c$_{NS}$ is given in Eq. (50), the defect diffusion constant D$_{Δ}$ ≡ Γ /χ , and α ≡ ρ$_{0}$χc 2 $_{NS}$. We show in Fig. 1 the normalized density-density correlation functions of a normal solid and a supersolid. We have used the first sound speed c$_{NS}$ = 550 m/s, the density ρ = 0 . 19048 g/cm $^{3}$, the isothermal compressibility χ = 0 . 29615 × 10 − 8 cm s $^{2}$/g for $^{4}$He solid51,52 at the molar volume 21 cm $^{3}$/mole, the viscosity of $^{4}$He fluid of 2 × 10 − 5 g/cm s, the typical wave number involved in light scattering q − 1 = 100 nm, and Γ = 8 × 10 − 11 cm $^{3}$s/g. In Fig. 2 we show the splitting of the Rayleigh peak due to defect diffusion in a normal solid into a Brillouin doublet of second sound modes, for two values of the supersolid fraction. 

Starting from general conservation laws and sym- metry principles we derived the effective action for a supersolid—a state of matter with simultaneous broken translational symmetry and Bose condensation. The re- sulting effective action in Eq. (21) is one of the two im- portant results of this work, and will be further developed in subsequent work on vortex and dislocation dynamics in supersolids. In this work, however, we used the lin- 

## V. CONCLUSION 

