2.10 Entropy of massive fields

The heat kernel of a massive field described by the wave operator 𝒟 = − ∇2 + m2 is expressed in terms of the heat kernel of a massless field,
′ ′ − m2s K (m⁄=0 )(x,x ,s) = K (m=0)(x,x ,s) ⋅ e .
Thus, one finds
(m ⁄=0) (m=0 ) −m2s TrK α (s) = TrK α (s) ⋅ e , (29 )
where the trace of the heat kernel for vanishing mass is given by Eq. (27View Equation). Therefore, the entanglement entropy of a massive field is
A (Σ) ∫ ∞ ds − m2s Sm ⁄=0 = 12-(4π)(d−-2)∕2 2 sd∕2e . (30 ) 𝜖
In particular, if d = 4, one finds that
S = A-(Σ-)( 1-+ 2m2 ln 𝜖 + m2 lnm2 + m2 (γ − 1) + O (𝜖m )). (31 ) m⁄=0 12(4π) 𝜖2
The logarithmic term in the entropy that is due to the mass of the field appears in any even dimension d. The presence of a UV finite term proportional to the (d − 2)-th power of mass is the other general feature of (30View Equation), (31View Equation).
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