3.16 The puzzle of non-minimal coupling

The simplest case to consider is that of a

Non-minimally coupled scalar field.
In this case, one has TrX (0) = ξR, where ξ is the parameter of non-minimal coupling. The renormalization of Newton’s constant

( ) --1--- 1-- ------1-------- 1- --1- 4G = 4G + d−22 6 − ξ šœ–d−2 (141 ) ren (4π) (d − 2)
is modified due to the presence of the non-minimal coupling ξ in the scalar field operator. At the same time, the entropy calculation on a Ricci flat background (¯R = 0) is not affected by the non-minimal coupling since the field operator for this background is identical to the minimal one. This simple reasoning shows that the area law in the case of a non-minimally coupled scalar field is the same as in the case of the minimal scalar field (81View Equation). Clearly, there is a mismatch between the renormalization of Newton’s constant and the renormalization of the entanglement entropy. One concludes that, in the presence of non-minimal coupling, when the Riemann tensor appears explicitly in the action of the quantum field, the UV divergence of the entanglement entropy cannot be handled by the standard renormalization of Newton’s constant. The mismatch in the entropy is
non-min (− ξ ) A(Σ ) S ξ = -----------d−2-šœ–d−2-. (142 ) (d − 2)(4π) 2
It is an important fact that there is no known way to give a statistical meaning to this entropy. Moreover, (142View Equation) does not have a definite sign and may become negative if ξ is positive. In some respects, this term is similar to the classical Bekenstein–Hawking entropy: both entropies, at least in the framework of the conventional field theory, do not have a well-defined statistical meaning. There is a hope that in string theory the terms similar to Eq. (142View Equation) may acquire a better meaning. However, this question is still open. We should note that on a space with a conical singularity one can consider the Ricci scalar in the non-minimal scalar operator as the complete curvature including the δ-like singular term as in Eq. (55View Equation). Then, the differential operator − (∇2 + ξR ) contains a delta-like potential concentrated on the horizon surface Σ. The presence of this potential modifies the surface terms in the heat kernel in such a way that [198Jump To The Next Citation Point]
∫ a Σ(ξ) = aΣ(ξ = 0) − 4πξ(1 − α ) 1 + O (1 − α)2 , (143 ) 1 1 Σ
where aΣ(ξ = 0) 1 is the surface term aΣ 1 (70View Equation) without the non-minimal coupling and the term 2 O (1 − α) is ill defined (something like 2 δ (0)). However, it does not affect the entropy calculation. If we now apply the replica trick and calculate the entropy corresponding to the theory with the heat kernel with the surface term (143View Equation) we get that [198Jump To The Next Citation Point, 163, 16]
( ) --------1---------- 1- Sdiv = d−22 d−2 6 − ξ A (Σ) . (144 ) (d − 2)(4π) šœ–
This divergence takes a form consistent with the UV divergence of Newton’constant (141View Equation). However, we cannot interpret this entropy as a contribution to the entanglement entropy since the presence of the delta-like potential in the Euclidean field operator is not motivated from the point of view of the original Lorentzian theory, for which the entanglement entropy is calculated. Moreover, Eq. (144View Equation) is not positive if ξ > 1āˆ•6, while the entanglement entropy is supposed to be a positive quantity.

Similar features are shared by other non–minimally-coupled fields.

Abelian vector field.
After gauge fixing, the partition function of an abelian gauge field is

Z = det −1āˆ•2 Δ (1+)⋅ det Δ (0), (145 )
where (1) Δ + is operator defined in Eq. (132View Equation) with sign + in the matrix (1) X μν (133View Equation). For the effective action We ff = − lnZ we find,
∫ ( ∫ ) 1- ∞ ds----1---- We ff = − 2 šœ–2 s (4πs)dāˆ•2 E 1 + a1 s + .. , (146 )
( ) (s=1) (s=0) š’Ÿ1 (d) ∫ a1 = a1 − 2a 1 = ------ − 1 R , (147 ) 6 E
and š’Ÿ1(d) = d − 2 is the number of on-shell degrees of freedom of the abelian vector field. The renormalization of Newton’s constant is
( ) 1 1 1 š’Ÿ1 (d) 1 ------= ---+ ----d−2-------- ------ − 1 -d−2 . (148 ) 4Gren 4G (4π) 2 (d − 2) 6 šœ–
Comparison with Eq. (126View Equation) shows that there is again a mismatch between the UV divergences in the entropy and in Newton’s constant. This mismatch stems from the non-minimal term (1) X μν in the Laplace type field operator for the vector field.

Massless graviton.
The partition function of a massless graviton in d dimensions, after gauge fixing and adding the Faddeev–Popov ghost contribution, is

Z = det− 1āˆ•2Δ (2) ⋅ detΔ (1)⋅ det−1āˆ•2Δ (0), (149 ) −
where Δ (1−) is operator defined in Eq. (132View Equation) with sign − in the matrix X (μ1ν) (133View Equation). The operator Δ (2) governs the dynamics of the tensor perturbations, which satisfy the condition μ 1 ∇ (hμν − 2gμνh ) = 0. The operator (0) Δ is due to the contribution of the conformal mode, while the determinant of operator (1) Δ − is due to the Faddeev–Popov ghosts. Hence, one has in this case that
( ) s=2 (s=1) (s=0) š’Ÿ2 (d) ∫ d2 − d + 4 a1 = a1 − 2a1 + a 1 = ------ − c(d) R , c(d) = ---------- , (150 ) 6 E 2
where š’Ÿ2(d) = d(d−3) 2 is the number of on-shell degrees of freedom of a massless spin-2 particle. The renormalization of Newton’s constant is
1 1 1 ( š’Ÿ (d ) ) 1 ------= --- + -----d−2------- --2---− c(d) -d−2 . (151 ) 4Gren 4G (4π ) 2 (d − 2) 6 šœ–
Again, we observe the mismatch between the UV divergent terms in the entropy (126View Equation) and in Newton’s constant, this time due to the graviton. To summarize, the UV divergences in the entanglement entropy of minimally-coupled scalars and fermions are properly renormalized by the redefinition of Newton’s constant. It happens that each minimally-coupled field (no matter bosonic or fermionic) contributes positively to Newton’s constant and positively to the entropy of the black hole. The contributions to both quantities come proportionally, which allows the simultaneous renormalization of both quantities. The mismatch between the UV divergences in the entropy and in Newton’s constant appears for gauge bosons: the abelian (and non-abelian) vector fields and gravitons. The source of the mismatch are those non-minimal terms (s) X in the field operator, which contribute negatively to Newton’s constant and do not make any contribution to the entanglement entropy of the black hole. At the moment of writing of this review the appropriate treatment of the entropy of non–minimally-coupled fields is not yet available.
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