3.15 Renormalization of entropy due to fields of different spin

The effective action of a field of spin s can be written as
∫ (− )2s ∞ ds- − sΔ (s) W (s) = 2 šœ–2 s Tre . (131 )
The second-order covariant operators acting on the spin-s field can be represented in the following general form
Δ (s) = − ∇2 + X (s), (132 )
where the matrices X (s) depend on the chosen representation of the quantum field and are linear in the Riemann tensor. Here are some examples [43, 44]
(0) (1āˆ•2) 1- (1) X = ξR , X AB = 4R δAB , X μν = ±R μν , 1 1 X (3Aāˆ•B2,)μν = -R δABg μν − -R μναβ(γαγβ )AB , 4 2 X (2) = 1R (g g + g g ) − R g − R g − R − R , (133 ) μν,αβ 2 μα νβ μβ να αμ βν βν αμ μανβ ναμβ
where γα AB are gamma-matrices. The coefficient areg 1 in the small s expansion (67View Equation) – (69View Equation) of the heat kernel of operator (132View Equation) has the general form
∫ D (d) a(s1) = (--s---R − TrX (s)), (134 ) E 6
where Ds(d) is the dimension of the representation of spin s,
Ds=0 = 1 , Ds=1āˆ•2 = 2[dāˆ•2], Ds=1 = d , D = d2[dāˆ•2], D = (d-−-1)(d-+-2) . (135 ) s=3āˆ•2 s=2 2
D (d) s can be interpreted as the number of off-shell degrees of freedom.

Let us consider some particular cases.

Dirac fermions (s = 1āˆ•2).
The partition function for Dirac fermions is 1āˆ•2 (1āˆ•2) Z1 āˆ•2 = det Δ. In this case 1 [dāˆ•2] TrX (1āˆ•2) = 4 2 R and hence

š’Ÿ ∫ a(s1=1āˆ•2)= − -1āˆ•2- R , (136 ) 6 E
where š’Ÿ1āˆ•2 was introduced in Eq. (127View Equation). We note that the negative sign in Eq. (136View Equation) in combination with the negative sign for fermions in the effective action (131View Equation) gives the total positive contribution to Newton’s constant. The renormalization of Newton’s constant due to Dirac fermions is
1 1 1 š’Ÿ1āˆ•2 1 ------= --- + -----d−-2-------------d−-2 . (137 ) 4Gren 4G (4π ) 2 (d − 2 ) 6 šœ–
Comparison of this equation with the UV divergence of entropy (126View Equation) for spin-1/2 shows that the leading UV divergence in the entropy of spin-1/2 field is handled by the renormalization of Newton’s constant in the same manner as it was for a scalar field.

The Rarita–Schwinger field (s = 3 āˆ•2).
The partition function, including gauge fixing and the Faddeev–Popov ghost contribution, in this case, is

1āˆ•2 (3āˆ•2) −1 (1āˆ•2) Z3āˆ•2 = det Δ det Δ , (138 )
so that the appropriate heat kernel coefficient is
∫ (3āˆ•2) (1āˆ•2) š’Ÿ3āˆ•2- a1 = a1 − 2a = − 6 E R , (139 )
where š’Ÿ3āˆ•2 is introduced in Eq. (129View Equation). The renormalization of Newton’s constant
1 1 1 š’Ÿ3 āˆ•2 1 4G----= 4G-+ ----d−2---------6--šœ–d−2 (140 ) ren (4 π) 2 (d − 2)
then, similarly to the case of Dirac fermions, automatically renormalizes the entanglement entropy (126View Equation). However, this property, does not hold for all fields. The main role in the mismatch between the UV divergences in the entanglement entropy and in Newton’s constant is played by the non-minimal coupling terms (s) X, which appear in the field operators (132View Equation).
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