The induced Newton constant depends on the regulator

Sakharov proposed that the Einstein–Hilbert term is generated by the vacuum fluctuations of the matter fields once a cutoff is imposed [Sakharov] [Visser]. If so, the matter content would fix the dimensionless product $G\Lambda^2$ of Newton’s constant and the squared cutoff. For free Lorentz-invariant fields with a proper-time regulator common to all species the leading term gives $G\Lambda^2=\pi/(\kappa_fc)$ , where $c$ is a rational number fixed by the spins and the curvature couplings and $\kappa_f$ depends on the shape of the regulator. Four admissible regulators differ by a factor $1.56$ (Proposition 1), so the product has content only relative to a prescription that defines $\Lambda$ . For the Standard Model content $c=\tfrac1{12}-2\xi$ , with $\xi$ the curvature coupling of the Higgs doublet, and the cutoff that reproduces the observed $G$ lies above the Planck mass unless $\xi$ is below about $-1$ (section 3).

If the couplings are chosen so that $c=0$ (for the Standard Model, at a suitable $\xi$ in ‘t Hooft–Feynman gauge, section 3.2), the quadratic divergence cancels, and $G$ then depends logarithmically on $\Lambda$ unless also $\sum_ic_im_i^2=0$ , which the Standard Model spectrum does not satisfy. The induced $1/G$ of that content, evaluated at one loop in this gauge, is negative and smaller in modulus than the observed value by a factor between about $4\times10^{-35}$ and $2\times10^{-33}$ . For the entropy route, in which $1/4G$ is read off from the area coefficient of a spectral-state entropy on a causal set, the question is whether that coefficient is fixed by the sprinkling. We prove a criterion for extracting it in the presence of logarithms (Proposition 2) and an identity for its dependence on the enclosing diamond (Propositions 4 and 5), derive the running of $G$ in two dimensions under an assumed form of the entropy, and compute the entropy in $2+1$ dimensions for two rules of truncation. In neither does the entropy of a fixed region become independent of the enclosing diamond.

The species counting is due to Larsen and Wilczek [LW] and Kabat [Kabat] for the entropy, and the combination $12c=N_S+2N_D-4N_V$ appears in the asymptotic-safety literature [DEP]. Frolov, Fursaev and Zelnikov [FFZ] showed that in a model with heavy constituents the induced constant is fixed by their masses, and related it to the Bekenstein–Hawking entropy. The relation between the entropy of a horizon and the renormalisation of $G$ goes back to Susskind and Uglum [SU] and Jacobson [Jac94]. Sorkin and Yazdi [SY] evaluated Sorkin’s spacetime entropy [Sor12] in the Sorkin–Johnston state [ASS12] on causal sets and found that truncating the Pauli–Jordan operator restores an area law in $1+1$ dimensions. Belenchia, Benincasa, Letizia and Liberati [BBLL] extended this to $2+1$ and $3+1$ dimensions. Proposition 1 adds to this the dependence of the induced coefficient on the shape of the regulator through two constants, $\kappa_f$ and $C_f$ ; section 3 evaluates them for the Standard Model spectrum, and section 4 tests the truncation rules of [SY] and [BBLL] against Propositions 4 and 5. Other approaches to the same question are mapped in the map of the quantum-gravity families examined here.

1. The induced Einstein–Hilbert term

Integrating out a free field on a curved background gives an effective action whose part linear in the scalar curvature $R$ has the form of the Einstein–Hilbert term. Its coefficient is a sum over species of a heat-kernel number times a proper-time integral that is cut off in the ultraviolet. In Euclidean signature a free field with a Laplace-type operator $\mathcal Q=-\Box-E+m^2$ contributes $\Gamma=-w\int_0^\infty\frac{ds}{s}f(\Lambda^2s)\,\mathrm{Tr}\,e^{-s\mathcal Q}$ to the effective action. Here $w=\tfrac12$ for a real boson and $w=-\tfrac12$ for a Dirac fermion (with $\mathcal Q=\mathcal D^2+m^2$ , the square of the Dirac operator $\mathcal D$ ), and $f$ is a regulator function common to all species with $f\to1$ for $u=\Lambda^2s\to\infty$ and $f(u)=O(u^{1+\eta})$ , $\eta>0$ , as $u\to0$ . The small- $s$ expansion of the heat kernel on a four-dimensional manifold is $\mathrm{Tr}\,e^{-s\mathcal Q}=(4\pi s)^{-2}e^{-sm^2}\int\sqrt g\,\big[\mathrm{tr}\,\mathcal H_0+s\,\mathrm{tr}\,\mathcal H_1+O(s^2)\big]$ with $\mathrm{tr}\,\mathcal H_0=\dim$ and $\mathrm{tr}\,\mathcal H_1=\mathrm{tr}(\tfrac R6+E)$ (the coefficients called $a_0$ and $a_1$ in [Vass]). Matching the term linear in $R$ with $\Gamma\supset-\frac1{16\pi G}\int\sqrt g\,R$ gives, for a set of species labelled by $i$ ,

\frac1{16\pi G}=\frac1{16\pi^2}\sum_ic_i\,\Lambda^2J\!\Big(\frac{m_i^2}{\Lambda^2}\Big),\qquad J(\epsilon)=\int_0^\infty\frac{du}{u^2}f(u)e^{-\epsilon u},\qquad c_i=w_i\,\frac{\mathrm{tr}\,a_1}{R}.\tag{1}

The number $c_i$ , which we call the species character, is computed from the operator of the field:

field	operator	$\mathrm{tr}\,\mathcal H_1/R$	$w$	$c$
real scalar, coupling $\xi R\phi^2/2$	$-\Box+\xi R$	$\tfrac16-\xi$	$\tfrac12$	$\tfrac12(\tfrac16-\xi)$
Dirac fermion	$-\Box+\tfrac R4$ on 4 components	$4(\tfrac16-\tfrac14)=-\tfrac13$	$-\tfrac12$	$\tfrac16$
Weyl fermion	half of Dirac			$\tfrac1{12}$
Maxwell field, vector part	$-\Box\delta^a{}_b+R^a{}_b$ on vectors	$\tfrac46-1=-\tfrac13$	$\tfrac12$	$-\tfrac16$
Maxwell field, ghosts	$-\Box$ on two scalars	$\tfrac26=\tfrac13$	$-\tfrac12$	$-\tfrac16$
Maxwell field, total				$-\tfrac13$
Proca field	Maxwell plus one real scalar with $\xi=0$			$-\tfrac13+\tfrac1{12}=-\tfrac14$

For $\xi=0$ these give $12c=N_S+2N_D-4N_V$ for $N_S$ real scalars, $N_D$ Dirac fermions and $N_V$ gauge fields. The coefficient $\mathrm{tr}\,\mathcal H_1$ is quoted from [Vass]; it can be tested on the four-sphere of unit radius ( $R=12$ , volume $8\pi^2/3$ ), where the spectra are known. For the scalar Laplacian, with eigenvalues $l(l+3)$ and multiplicities $(2l+3)(l+1)(l+2)/6$ , the combination $[(4\pi s)^2\,\mathrm{Tr}\,e^{-s\mathcal Q}/\mathrm{Vol}-1]/s$ equals $2.0097$ at $s=0.005$ , against the limit $R/6=2$ . With the conformal coupling $\xi=\tfrac16$ (eigenvalues $l(l+3)+2$ ) it equals $-3.3\times10^{-4}$ , against $0$ . The Dirac operator has eigenvalues $\pm(n+2)$ with multiplicity $4(n+1)(n+2)(n+3)/6$ for each sign. For its square the combination per spinor component is $-0.9991$ at $s=0.005$ , against $\tfrac14\mathrm{tr}\,\mathcal H_1=-\tfrac1{12}R=-1$ .

Proposition 1 (Induced coefficient with a regulator). Let $f:(0,\infty)\to[0,1]$ be measurable, with $f(u)=O(u^{1+\eta})$ as $u\to0$ for some $\eta>0$ , $f(u)\to1$ as $u\to\infty$ , and $\int_1^\infty(1-f(u))\,du/u<\infty$ . Put $\kappa_f=\int_0^\infty u^{-2}f(u)\,du$ and $\alpha_f=\int_0^\infty u^{-1}\big(f(u)-\theta(u-1)\big)du$ , with $\theta$ the Heaviside function. Then as $\epsilon\to0$ ,

J(\epsilon)=\kappa_f+\epsilon\ln\epsilon+C_f\,\epsilon+o(\epsilon),\qquad C_f=\gamma_E-1-\alpha_f,\tag{2}

and consequently, with $c=\sum_ic_i$ and $\mu_1=\sum_ic_im_i^2$ , for fixed masses as $\Lambda\to\infty$ (terms with $m_i=0$ are omitted from the logarithmic sum),

\frac1{16\pi G}=\frac1{16\pi^2}\Big[\kappa_f\,c\,\Lambda^2-\sum_ic_im_i^2\ln\frac{\Lambda^2}{m_i^2}+C_f\mu_1\Big]+o(1).\tag{3}

(i) If $c\neq0$ , then $G\Lambda^2=\pi/(\kappa_fc)$ to leading order, which depends on $f$ through $\kappa_f$ . Only the sign and the zero set of $c$ are independent of $f$ for $\kappa_f>0$ .

(ii) If $c=0$ and $\mu_1=0$ , then $\frac1{16\pi G}=\frac1{16\pi^2}\sum_ic_im_i^2\ln m_i^2+o(1)$ , finite and independent of $f$ and of $\Lambda$ .

(iii) If $c=0$ and $\mu_1\neq0$ , then $1/16\pi G$ depends on $\ln\Lambda$ with coefficient $-\mu_1/8\pi^2$ and on $f$ through the constant $C_f\mu_1/16\pi^2$ .

The proof isolates the one non-analytic term of $J$ , $\epsilon\ln\epsilon$ , whose coefficient does not depend on $f$ . The constants $\kappa_f$ and $C_f$ carry all the dependence on the shape of the regulator.

Proof. The integral $\kappa_f$ converges at $u\to0$ by the assumption on $f$ and at infinity because $f\le1$ . Differentiating, $J'(\epsilon)=-\int_0^\infty u^{-1}f(u)e^{-\epsilon u}du$ . Write $f=\theta(u-1)+(f-\theta(u-1))$ . The first part gives $-E_1(\epsilon)=\ln\epsilon+\gamma_E+O(\epsilon)$ , with $E_1$ the exponential integral. The second part has an integrable integrand $(f-\theta(u-1))/u$ , by the assumptions at $u\to0$ and $u\to\infty$ , so it tends to $-\alpha_f$ by dominated convergence. Hence $J'(\epsilon)=\ln\epsilon+\gamma_E-\alpha_f+o(1)$ , and integrating from $0$ , where $J(0)=\kappa_f$ , gives $J(\epsilon)=\kappa_f+\epsilon\ln\epsilon+(\gamma_E-\alpha_f-1)\epsilon+o(\epsilon)$ . Inserting (2) in (1) with $\epsilon_i=m_i^2/\Lambda^2$ gives (3); the remainder is $\Lambda^2o(\epsilon_i)=o(1)$ at fixed $m_i$ . Part (i) is the leading term of (3), $1/16\pi G=\kappa_fc\Lambda^2/16\pi^2$ . For parts (ii) and (iii) the term $\kappa_fc\Lambda^2$ vanishes, and $-\sum_ic_im_i^2\ln(\Lambda^2/m_i^2)=\sum_ic_im_i^2\ln m_i^2-\mu_1\ln\Lambda^2$ . If $\mu_1=0$ this and $C_f\mu_1$ vanish as well. $\square$

Four regulators satisfy the assumptions. They are the sharp cutoff $f=\theta(u-1)$ , with $\kappa_f=1$ ; $f=e^{-1/u}$ , with $\kappa_f=1$ ; $f=e^{-1/u^2}$ , with $\kappa_f=\sqrt\pi/2=0.8862$ ; and $f=(1-e^{-u})^2$ , with $\kappa_f=2\ln2=1.3863$ . The last is the regulator of two Pauli–Villars fields, of squared masses $\Lambda^2$ and $2\Lambda^2$ and weights $-2$ and $+1$ . Their constants are $C_f=-0.4228,\,0.1544,\,-0.1342,\,-0.3069$ . The ratio of the largest to the smallest $\kappa_f$ is $1.564$ , which is the factor by which $G\Lambda^2$ varies among these four.

Rescaling $\Lambda^2\to\kappa_f\Lambda^2$ turns every such regulator into one with $\kappa_f=1$ , so the $f$ -dependence in (i) concerns the definition of $\Lambda$ . A value of $G\Lambda^2$ has content only once $\Lambda$ is defined by a prescription that does not refer to $G$ , such as the mass of a Pauli–Villars field or a lattice spacing, and different prescriptions give different values. Proposition 1 does not show that no such prescription exists. In case (ii) there is no cutoff to define. The product $G\,m_*^2$ is a function of the ratios of the masses, and $G$ is set by the overall mass scale $m_*$ of the species in the sum.

2. Counting species

The character $c$ of a matter content is the sum of the rows of the table and does not depend on masses. For the Standard Model with 45 Weyl fermions, twelve gauge fields (with ghosts) and a Higgs doublet of four real components with coupling $\xi R\,H^\dagger H$ , each real component contributes $\tfrac12(\tfrac16-\xi)$ , and $c=2(\tfrac16-\xi)+\tfrac{45}{12}-4=\tfrac1{12}-2\xi .$ This equals $\tfrac1{12}$ at $\xi=0$ , $-\tfrac14$ at $\xi=\tfrac16$ , and vanishes at $\xi=\tfrac1{24}$ . With three right-handed neutrinos there are 48 Weyl fermions and $c=\tfrac13-2\xi$ , which is $\tfrac13$ at $\xi=0$ and $0$ at $\xi=\tfrac16$ . At $\xi=\tfrac16$ the four conformally coupled scalars contribute $0$ , the fermions $4$ and the gauge fields $-4$ . The vanishing is arithmetic, and no mechanism that keeps it at higher order is claimed. In the cutoff case the sign of $c$ is the sign of $1/G$ , so a matter content with $c<0$ has a negative induced coefficient.

3. The Standard Model content as the source of $G$

3.1 A cutoff with $c>0$

In the cutoff case the leading term of (3) equals the observed value $1/16\pi G=M_{\rm P}^2/16\pi=2.97\times10^{36}$ GeV $^2$ , with $M_{\rm P}=G^{-1/2}=1.221\times10^{19}$ GeV, when $\Lambda=\sqrt{\pi/(\kappa_fc)}\,M_{\rm P}$ . This requires $c>0$ , that is $\xi<\tfrac1{24}$ for the Standard Model alone and $\xi<\tfrac16$ with three right-handed neutrinos. At $\xi=0$ the four regulators give:

regulator	$\kappa_f$	$\Lambda/M_{\rm P}$ for $c=\tfrac13$	$\Lambda/M_{\rm P}$ for $c=\tfrac1{12}$
sharp, $e^{-1/u}$	$1$	$3.07$	$6.14$
$e^{-1/u^2}$	$0.886$	$3.26$	$6.52$
$(1-e^{-u})^2$	$1.386$	$2.61$	$5.21$

The required cutoff is above the Planck mass for every regulator in the table, and its value is set by the regulator. A cutoff $\Lambda\le M_{\rm P}$ needs $c\ge\pi/\kappa_f$ . For the Standard Model alone this means $\xi\le(\tfrac1{12}-\pi/\kappa_f)/2=-1.53,\,-1.73,\,-1.09$ for $\kappa_f=1,\,0.886,\,1.386$ . Three right-handed neutrinos raise each threshold by $\tfrac18$ , to $-1.40,\,-1.61,\,-0.97$ .

3.2 The quadratic divergence cancelled

In the case $c=0$ the cutoff term of (3) is absent. We take the Standard Model with three right-handed neutrinos at $\xi=\tfrac16$ , where the unbroken-phase count of section 2 gives $c=0$ . Then (3) reads $\frac1{16\pi G}=-\frac1{16\pi^2}\big[\mu_1\ln\Lambda^2-\sum_ic_im_i^2\ln m_i^2-C_f\mu_1\big]$ , and Proposition 1(iii) applies whenever $\mu_1\neq0$ .

In the broken phase the list of species with masses, and with it the evaluation of $\mu_1$ , depends on the gauge. In the ‘t Hooft–Feynman gauge each massive vector together with its ghosts has $c=-\tfrac13$ (vector $-\tfrac16$ , ghosts $-\tfrac16$ ). The three Goldstone modes carry the coupling $\xi=\tfrac16$ of the doublet and have $c=0$ , and so does the Higgs boson, so that the total is $c=0$ as in the unbroken phase. In unitary gauge the $W^\pm$ and $Z$ are Proca fields with $c=-\tfrac14$ each and the Goldstone modes are absent. The total is then $4-3-\tfrac34=\tfrac14$ , because a Proca field has a minimally coupled longitudinal mode with $c=\tfrac1{12}$ instead of $0$ , and three of them add $\tfrac14$ . The quadratic divergence cancels in the first gauge and not in the second, so the off-shell induced action is gauge dependent. We use the first, because $c=0$ is the condition that defines the case.

A tree-level contribution $-\xi v^2/2$ to $1/16\pi G$ from the Higgs vacuum value $v=246$ GeV is not part of Proposition 1. It equals $-5.0\times10^3$ GeV $^2$ at $\xi=\tfrac16$ , has the same sign as the loop term, and is also far below the observed value.

The inputs are the masses (GeV) top $172.6$ , bottom $4.18$ , charm $1.27$ , strange $0.093$ , down $0.0047$ , up $0.0022$ , tau $1.777$ , muon $0.1057$ , electron $0.000511$ , $W$ $80.38$ , $Z$ $91.19$ , with neutrinos massless and ghost masses equal to the vector masses. The characters are $c=\tfrac12$ for each coloured quark (three colours, $3\times\tfrac16$ ), $\tfrac16$ for each charged lepton, $-\tfrac23$ for the $W^\pm$ pair and $-\tfrac13$ for the $Z$ . Then

\mu_1=7.8\times10^3\ \mathrm{GeV}^2,\tag{4}

with the fermions contributing $+14.9\times10^3$ GeV $^2$ , almost all of it from the top quark, and the vectors $-7.1\times10^3$ GeV $^2$ . The logarithmic term of (3) has the value $1/16\pi G=-110,\,-339,\,-1.7\times10^3,\,-3.8\times10^3$ GeV $^2$ for $\Lambda=10^3,\,10^4,\,10^{10},\,1.2\times10^{19}$ GeV. It is negative, and smaller in modulus than the observed value by $3.7\times10^{-35}$ , $1.1\times10^{-34}$ , $5.8\times10^{-34}$ and $1.3\times10^{-33}$ . The regulator-dependent constant $C_f\mu_1/16\pi^2$ is at most $21$ GeV $^2$ in modulus for the four regulators of section 1, which is $19\%$ of the logarithmic term at $\Lambda=1$ TeV and negligible at the larger cutoffs.

For the Standard Model alone at $\xi=\tfrac1{24}$ , $c=0$ with the Higgs boson and the three Goldstone modes each contributing $\tfrac1{16}$ (Higgs mass $125.25$ GeV, Goldstone masses equal to the vector masses in this gauge). Then $\mu_1=10.1\times10^3$ GeV $^2$ , and the logarithmic term runs from $-178$ to $-4.9\times10^3$ GeV $^2$ over the same range of $\Lambda$ , which is smaller in modulus than the observed value by $6\times10^{-35}$ to $1.7\times10^{-33}$ . These are one-loop evaluations with the structure of Proposition 1. They assume free fields with the masses of the broken phase, the species characters of section 1 and the gauge stated above.

Consequently the Standard Model content fixes $1/G$ only through a cutoff above the Planck mass, for the Standard Model alone with $\xi>-1.09$ or with three right-handed neutrinos and $\xi>-0.97$ , and for each of the four regulators. With the quadratic divergence cancelled, in ‘t Hooft–Feynman gauge, it gives a value roughly 33 to 35 orders of magnitude below the observed one. A sector that supplies the observed $1/G$ with $c=0$ needs $\sum_ic_im_i^2\ln(\ldots)\sim16\pi^2M_{\rm P}^2/16\pi$ , so masses of order $10^{18}$ to $10^{19}$ GeV, and $G$ is then set by whatever is heaviest.

4. The area coefficient of a causal-set entropy

In the entanglement picture the coefficient of $A/4G$ in the entropy of a horizon is renormalised by the same fluctuations that renormalise $1/G$ [SU] [Jac94] [LW], and for a minimally coupled scalar the divergent area term of the entanglement entropy equals $A/4G$ with the induced $G$ [SU] [LW]. On a causal set the density $\rho$ supplies the cutoff. If the area coefficient of the entropy of a region is a fixed number times $\rho^{(d-2)/d}A$ , this defines $G$ in units of the discreteness length $\ell=\rho^{-1/d}$ . We examine whether that coefficient is fixed.

For a free scalar on a sprinkled causal set let $K$ be the retarded Green function, a matrix supported on related pairs, and $i\Delta=i(K-K^T)$ the Pauli–Jordan operator, Hermitian and imaginary. The Sorkin–Johnston state has two-point function $\mathsf W$ equal to the positive part of $i\Delta$ . For a region $\mathcal R$ with restrictions $\mathsf W_{\mathcal R}$ and $(i\Delta)_{\mathcal R}$ , the spacetime entanglement entropy of Sorkin [Sor12] is $S=\sum_\lambda\lambda\ln\lvert\lambda\rvert$ over the generalised eigenvalues $\lambda$ of $\mathsf W_{\mathcal R}\psi=\lambda\,(i\Delta)_{\mathcal R}\psi$ , restricted to the range of $(i\Delta)_{\mathcal R}$ . For a single mode with symplectic eigenvalue $\nu\ge\tfrac12$ the pair $\lambda=\tfrac12\pm\nu$ gives $(\nu+\tfrac12)\ln(\nu+\tfrac12)-(\nu-\tfrac12)\ln(\nu-\tfrac12)$ , the Bose–Einstein entropy, which is why the formula is the entropy of the Gaussian state.

The untruncated Sorkin–Johnston entropy obeys a volume law in $1+1$ dimensions [SY]. An area law is recovered if the modes of $i\Delta$ with small eigenvalues are projected out of $\mathsf W$ and $i\Delta$ , once for the whole causal set and once for the region (the double truncation of [SY] [BBLL]). We call a rule for the two truncations a scheme. Throughout, $N$ is the number of points of the causal set, $n$ the number in the region, $d$ the dimension, and a knee factor $k$ sets the number of retained modes. Three schemes are used. The untruncated state is scheme U. The mode-count scheme (M) retains globally the $k_N=\mathrm{round}(kN^{(d-1)/d})$ largest positive modes of $i\Delta$ and their negative partners, and then $\mathrm{round}(kn^{(d-1)/d})$ positive modes (and partners) of the restricted operator. The threshold scheme (T) retains the same $k_N$ modes globally and applies to the restricted operator the absolute threshold equal to the smallest retained global eigenvalue.

4.1 Logarithms and the extraction of the area coefficient

Let $F(\rho)$ be the mean over sprinklings of density $\rho$ of an observable of a fixed region. A thinning of a sprinkling is a sprinkling of lower density, so $\rho\,\partial_\rho$ acts on $F$ as the generator of density dilations, and the powers $\rho^\beta$ are its eigenvectors. A power times a logarithm, $\rho^\beta\ln\rho$ , is a generalised eigenvector: $(\rho\,\partial_\rho-\beta)$ maps it to $\rho^\beta$ . The coefficient of the area term of the entropy of a boundary of dimension $d-1$ is the component of $F$ at the exponent $\beta_A=(d-2)/d$ . It is extracted by a function $P(\rho\,\partial_\rho)$ with $P$ vanishing on the exponents of the other terms and equal to $1$ at $\beta_A$ . For horizon counts, the extraction of one power of the density from counts at several thinnings is treated on the page on horizon counts and the area law. Here we ask what such an extraction returns when some terms carry a logarithm.

Proposition 2 (Extraction of a coefficient in the presence of logarithms). Let $\mathcal B$ be a finite set of exponents with $\beta_A\notin\mathcal B$ , and let

F(\rho)=\sum_{\beta\in\mathcal B}\rho^\beta(a_\beta+b_\beta\ln\rho)+\rho^{\beta_A}(b_A\ln\rho+a_A).

Let $P$ be an entire function with $P(\beta)=0$ for $\beta\in\mathcal B$ and $P(\beta_A)=1$ . Then

P(\rho\,\partial_\rho)F=\sum_{\beta\in\mathcal B}P'(\beta)\,b_\beta\,\rho^\beta+\rho^{\beta_A}\big(b_A\ln\rho+a_A+P'(\beta_A)\,b_A\big).\tag{5}

Consequently: (a) $P(\rho\,\partial_\rho)$ removes the whole term $\rho^\beta(a_\beta+b_\beta\ln\rho)$ , $\beta\in\mathcal B$ , if and only if $P'(\beta)=0$ as well, that is $\beta$ is a double zero of $P$ (or $b_\beta=0$ ); (b) the coefficient $b_A$ of $\rho^{\beta_A}\ln\rho$ in $P(\rho\,\partial_\rho)F$ is the same for every admissible $P$ ; (c) the constant $a_A$ of the area term shifts by $P'(\beta_A)\,b_A$ and is therefore determined only up to the choice of $P$ when $b_A\neq0$ .

Proof. Since $\rho\,\partial_\rho\rho^\beta=\beta\rho^\beta$ , we have $P(\rho\,\partial_\rho)\rho^\beta=P(\beta)\rho^\beta$ . Because $\rho^\beta\ln\rho=\partial_\beta\rho^\beta$ and $\rho\,\partial_\rho$ commutes with $\partial_\beta$ , $P(\rho\,\partial_\rho)\rho^\beta\ln\rho=\partial_\beta\big[P(\beta)\rho^\beta\big]=P'(\beta)\rho^\beta+P(\beta)\rho^\beta\ln\rho$ . Insert $P(\beta)=0$ on $\mathcal B$ and $P(\beta_A)=1$ to obtain (5). Statements (a) to (c) read off the coefficients. For (c), replacing $P(\beta)$ by $P(\beta)(1+\delta(\beta-\beta_A))$ keeps the admissibility conditions and changes $P'(\beta_A)$ by $\delta$ . $\square$

In two dimensions the area exponent is $\beta_A=0$ , because the boundary of a region is a set of points, and a logarithm of the density is expected on general grounds. The entropy of an interval of length $\ell_{\rm IR}$ in a conformal field theory with central charge $c_{\rm cft}$ is $\tfrac{c_{\rm cft}}3\ln(\ell_{\rm IR}/\ell_{\rm UV})$ for a short-distance cutoff $\ell_{\rm UV}$ . With two endpoints and $\ell_{\rm UV}\propto\rho^{-1/2}$ this is $\tfrac{c_{\rm cft}}{12}\ln(\rho\,\ell_{\rm IR}^2)$ per boundary point plus terms independent of $\rho$ . Proposition 2 then gives a derivation under an assumed form of the entropy of the sprinkled theory.

Proposition 3 (Two dimensions, conditional on the form of the entropy). Suppose the mean entropy per boundary point of a region of a sprinkled theory of $N_{\rm sp}$ massless scalars ( $c_{\rm cft}=1$ each) in $1+1$ dimensions has the form of $F$ in Proposition 2 with $\beta_A=0$ and $b_A=N_{\rm sp}c_{\rm cft}/12$ . That is, its exponent- $0$ part is $\tfrac{N_{\rm sp}c_{\rm cft}}{12}\ln(\rho\,\ell_{\rm IR}^2)+a$ for an infrared scale $\ell_{\rm IR}$ and a constant $a$ , and the remaining terms are powers $\rho^\beta$ , $\beta\in\mathcal B$ , possibly with logarithms. Define $1/4G_{\rm eff}$ as the exponent- $0$ component of the entropy per boundary point, extracted by an admissible $P$ . Then $1/4G_{\rm eff}=a_P+\tfrac{N_{\rm sp}c_{\rm cft}}{12}\ln(\rho\,\ell_{\rm IR}^2)$ , where the constant $a_P=a+P'(0)\,N_{\rm sp}c_{\rm cft}/12$ depends on $P$ and the slope $N_{\rm sp}c_{\rm cft}/12$ does not. In particular $1/4G_{\rm eff}$ depends on the density, and there is no density-independent induced $G$ unless $N_{\rm sp}c_{\rm cft}=0$ .

Proof. Write the exponent- $0$ part as $b_A\ln\rho+a_A$ with $a_A=a+b_A\ln\ell_{\rm IR}^2$ . By (5) with $\beta_A=0$ the exponent- $0$ component of $P(\rho\,\partial_\rho)F$ is $b_A\ln\rho+a_A+P'(0)b_A=b_A\ln(\rho\,\ell_{\rm IR}^2)+a+P'(0)b_A$ . By Proposition 2(b) and (c), a term in $\rho^0\ln\rho$ sits at the area exponent and no admissible $P$ removes it, while the constant shifts with $P$ . $\square$

The assumption is tested numerically in $1+1$ dimensions. We computed the spacetime entropy of the causal diamond $\lvert t\rvert+\lvert x\rvert<\tfrac12$ inside the diamond $\lvert t\rvert+\lvert x\rvert<1$ , with the massless retarded function $K=\tfrac12$ on related pairs, for nested sprinklings with $N=250,354,500,707,1000,1414,2000$ points (the same points thinned), in schemes M and T. The slope $dS/d\ln N$ expected from $c_{\rm cft}=1$ and two endpoints is $\tfrac16=0.167$ .

knee factor $k$	slope, scheme M	slope, scheme T	$\langle S\rangle$ at $N=2000$ , M	$\langle S\rangle$ at $N=2000$ , T
$0.5$	$0.158\pm0.009$	$0.160\pm0.017$	$0.92$	$0.66$
$1$	$0.156\pm0.012$	$0.193\pm0.017$	$1.31$	$1.00$
$2$	$0.057\pm0.042$	$0.182\pm0.023$	$2.18$	$1.68$

The slopes are fits of the entropy against $\ln N$ for each of six independent sprinklings, with mean and standard error. For $k=0.5$ and $k=1$ both schemes give slopes within about $1.5$ standard errors of $\tfrac16$ . With $k=2$ scheme M retains so many modes that the entropy no longer grows with $N$ , while T stays near $\tfrac16$ . The constant term depends on the scheme and on $k$ : the mean entropy at $N=2000$ changes by more than a factor $2$ between $k=0.5$ and $k=2$ . The logarithmic coefficient assumed in Proposition 3 is therefore reproduced where the truncation is not too weak, and the constant is not a property of the sprinkling alone.

4.2 Dilation identities in $2+1$ dimensions

Consider a sprinkling in a causal diamond $D(L)=\{\lvert t\rvert+r<L\}$ in $2+1$ dimensions and the concentric sub-diamond $D(h)$ , whose edge is the circle of length $2\pi h$ . An area law would read $\langle S\rangle=2\pi s_1\,h\,\rho^{1/3}+S_0$ with $s_1$ a property of the substrate and $S_0$ independent of $h$ . The statements below need one hypothesis on the scheme, which holds for U, M and T: the scheme depends on the causal order and on the numbers $N$ and $n$ , and on nothing else. The retarded function used below depends on the order alone.

Proposition 4 (Dilation invariance and the outer-diamond identity). If the scheme depends only on the causal order and on $N$ and $n$ , then $\langle S\rangle=\Phi(p,q)$ with $p=h\rho^{1/3}$ and $q=h/L$ . Define $s_1^h=(2\pi\rho^{1/3})^{-1}\partial_h\langle S\rangle\vert_{\rho,L}$ and $s_1^\rho=(2\pi h)^{-1}\partial_{\rho^{1/3}}\langle S\rangle\vert_{h,L}$ . Then

s_1^h-s_1^\rho=-\frac1{2\pi\,h\,\rho^{1/3}}\;\frac{\partial\langle S\rangle}{\partial\ln L}\bigg\vert_{h,\rho}.\tag{6}

Proof. The dilation by a factor $\sigma$ maps a sprinkling of density $\rho$ in $D(L)$ to one of density $\sigma^{-3}\rho$ in $D(\sigma L)$ , the sub-diamond $D(h)$ to $D(\sigma h)$ , and preserves the order, hence the retarded function up to a constant factor and the numbers $N,n$ . The matrices $\mathsf W$ and $i\Delta$ scale together, the generalised eigenvalues do not change, and $\langle S\rangle$ is invariant under $(h,L,\rho)\to(\sigma h,\sigma L,\sigma^{-3}\rho)$ . The invariants are $p$ and $q$ . Now $\partial_h\langle S\rangle=\rho^{1/3}\Phi_p+\Phi_q/L$ , $\partial_{\rho^{1/3}}\langle S\rangle=h\Phi_p$ and $\partial\langle S\rangle/\partial\ln L=-q\Phi_q$ . Hence $2\pi(s_1^h-s_1^\rho)=\Phi_q/(L\rho^{1/3})=-\partial_{\ln L}\langle S\rangle/(h\rho^{1/3})$ . $\square$

So the area coefficient read from the dependence on the size of the region ( $s_1^h$ ) equals the one read from the dependence on the density ( $s_1^\rho$ ) exactly when the entropy of a fixed region does not depend on the diamond that contains it. A stronger statement holds for a decomposition of the entropy into an area term and a term that does not depend on the region.

Proposition 5 (Area-plus-background decomposition). Under the hypothesis of Proposition 4, the following are equivalent: (a) $\langle S\rangle=2\pi s_1\,h\rho^{1/3}+S_0(L,\rho)$ for all $(h,L,\rho)$ with $s_1$ constant; (b) $\Phi(p,q)=2\pi s_1p+\zeta(p/q)$ for a function $\zeta$ . In that case $\partial\langle S\rangle/\partial\ln L\vert_{h,\rho}=L\rho^{1/3}\zeta'(L\rho^{1/3})$ does not depend on $h$ .

Proof. At fixed $(L,\rho)$ , varying $h$ moves $(p,q)$ along the ray $p/q=L\rho^{1/3}$ . Statement (a) says that $\Phi-2\pi s_1p$ is independent of $h$ at fixed $(L,\rho)$ , that is constant on each ray, which is (b), with $S_0(L,\rho)=\zeta(L\rho^{1/3})$ . Differentiating $S_0$ with respect to $\ln L$ at fixed $\rho$ gives the last statement. $\square$

4.3 Numerical results in $2+1$ dimensions

We took sprinklings of $N=2000$ points in $D(1)$ and the same sprinklings thinned to $1414$ points, so that $\rho=3N/2\pi$ (the volume of $D(1)$ is $2\pi/3$ ) and $\rho^{1/3}=9.85$ and $8.77$ . The entropy $\langle S\rangle$ was computed for the concentric sub-diamonds $h=0.30,0.375,0.45,0.525,0.60$ , whose boundary lengths $2\pi h\rho^{1/3}$ range from $16$ to $37$ in units of $\ell$ . The kernel is $K_{ij}=(I_{ij}+1)^{-1/3}$ if $i$ precedes $j$ and $0$ otherwise, where $I_{ij}$ is the number of elements between them (the discretised retarded function of [BBLL]; the overall constant drops out of $S$ ). Schemes M and T use $k=1$ . The coefficient $s_1^h$ is the slope of the least-squares line through the five values of $\langle S\rangle$ against $2\pi h\rho^{1/3}$ , computed for each sprinkling. The table gives the mean and standard error over eight sprinklings.

scheme	$s_1^h$ , $N=1414$	$s_1^h$ , $N=2000$
U	$7.73\pm0.13$	$9.42\pm0.11$
M	$0.046\pm0.013$	$0.072\pm0.012$
T	$0.179\pm0.005$	$0.187\pm0.007$

The untruncated entropy is of volume type. Its mean at $N=1414$ is $27,56,90,117,156$ for the five values of $h$ , which is $0.5$ to $0.75$ per point of the region, and the fitted $s_1^h$ is not an area coefficient. In scheme M the mean entropy at $N=2000$ , $9.7,11.0,11.7,11.6,11.1$ , rises and saturates with $h$ , so a straight line in $h$ does not describe it. In scheme T the means $0.69,1.24,2.16,3.06,4.13$ are close to linear in $h$ .

The coefficient read from the density dependence at fixed $h$ is $s_1^\rho=0.424\pm0.024$ for M and $0.137\pm0.014$ for T. It comes from a least-squares fit through the origin of $\Delta S$ between $N=2000$ and $N=1414$ against $2\pi h\,\Delta\rho^{1/3}$ over the five values of $h$ , so it is an average over $h$ of the pointwise $s_1^\rho$ of Proposition 4, which may depend on $h$ . The difference $s_1^h-s_1^\rho$ , which by (6) measures the dependence of the entropy on the enclosing diamond, is $-0.35\pm0.03$ for M and $+0.05\pm0.02$ for T (errors treated as independent, using $N=2000$ for $s_1^h$ ).

To measure that dependence directly we sprinkled $D(1.26)$ at the density of $1414$ points in $D(1)$ , so that $D(1.26)$ holds about twice as many points as $D(1)$ . We computed the entropy of the same sub-diamond as a region of $D(1)$ (the points of the sprinkling inside $D(1)$ ) and of $D(1.26)$ . The finite difference $\Delta\langle S\rangle/\ln1.26$ estimates $\partial\langle S\rangle/\partial\ln L$ . We used $40$ independent sprinklings, with $1326$ to $1497$ points in $D(1)$ and $2730$ to $2959$ in $D(1.26)$ . The table gives the mean and standard error of the estimate. The values at different $h$ come from the same sprinklings and are correlated, so differences between them are formed sprinkling by sprinkling.

scheme	$h=0.30$	$h=0.45$	$h=0.60$
M	$2.0\pm0.2$	$8.2\pm0.2$	$16.4\pm0.2$
T	$-2.2\pm0.3$	$-1.8\pm0.2$	$-1.80\pm0.14$

For M the entropy of a fixed region changes by $0.5$ , $1.9$ and $3.8$ nats when the enclosing diamond is enlarged by $1.26$ , and the derivative grows with $h$ . By Proposition 5 the decomposition into an area term and a term independent of $h$ fails. For T the change is $-0.5$ , $-0.4$ and $-0.4$ nats. The difference between the derivatives at $h=0.60$ and $h=0.30$ is $0.4\pm0.4$ , and a constant fitted to the three derivatives, with their covariance estimated from the sample, is $-1.90\pm0.10$ ( $\chi^2=1.5$ for two degrees of freedom). No dependence on $h$ is resolved at $40$ sprinklings, so the data for T are consistent with the decomposition of Proposition 5, with $s_1=s_1^h\approx0.19$ and a background term that depends on $L$ and $\rho$ . The density-direction value $s_1^\rho$ differs from $s_1^h$ because of that term.

These results are for one kernel, the schemes defined above, up to about $3000$ points, and regions of boundary length $16$ to $37$ in units of $\ell$ . They show that in the mode-count scheme the entropy has no area-plus-background form in this range, and that in the threshold scheme the value of $s_1$ depends on how it is read: $0.19$ from the size of the region, $0.14$ from the density. They do not show that no scheme exists for which $s_1$ is a property of the sprinkling alone. For M and T the coefficients at $1414$ and $2000$ points agree within errors, so no trend with $N$ is resolved; the untruncated coefficient does not (from $7.73$ to $9.42$ ), as befits a volume-type entropy.

5. Gauge dependence, higher loops and truncation dependence

A spectrum of positive norm with $c=\mu_1=0$ and masses of order $10^{18}$ to $10^{19}$ GeV would give a finite, regulator-independent $G$ , a function of the mass ratios and of one overall mass (Proposition 1(ii)). No such spectrum is exhibited here.

The vanishing of $c$ at $\xi=\tfrac16$ with three right-handed neutrinos holds at one loop. A two-loop computation of the term linear in $R$ , where $\xi=\tfrac16$ is not protected, would show whether it survives. The broken-phase count depends on the gauge, as stated in section 3.2, and a computation in a family of gauges would show which part of $\mu_1$ is gauge independent.

A regulator built from Pauli–Villars fields has an indefinite spectral weight. The entropy of the Lorentz-invariant positive-energy vacuum of such a regulated theory is treated on the page on entropy in indefinite-metric theories. Constructions in which the ultraviolet entanglement per unit area is identified with $1/4G$ as an input [Jac15] [BM] take Newton’s constant as a parameter, so the question of its value does not arise there.

For the causal-set entropy, Propositions 4 and 5 give the criterion that any truncation rule or state must meet: $\partial\langle S\rangle/\partial\ln L$ at fixed $h$ and $\rho$ must vanish. A state defined by local Hadamard conditions, or by a different regularisation of the Pauli–Jordan operator, can be tested against it with the same computation. Whether a scheme exists for which the derivative tends to $0$ as $L\rho^{1/3}\to\infty$ for a fixed region is open. If the volume term of the untruncated entropy is an eigenvector of $\rho\,\partial_\rho$ at the exponent $1$ with no accompanying logarithm, a suitable $P$ in Proposition 2 removes it in the mean; testing this requires the variance of the untruncated entropy, which the mean does not control.

References

Sakharov, “Vacuum quantum fluctuations in curved space and the theory of gravitation,” Dokl. Akad. Nauk SSSR 177 (1967) 70; reprinted in Gen. Relativ. Gravit. 32 (2000) 365.
Visser, “Sakharov’s induced gravity: a modern perspective,” Mod. Phys. Lett. A 17 (2002) 977, arXiv:gr-qc/0204062.
Susskind, Uglum, “Black hole entropy in canonical quantum gravity and superstring theory,” Phys. Rev. D 50 (1994) 2700, arXiv:hep-th/9401070.
Jacobson, “Black hole entropy and induced gravity,” arXiv:gr-qc/9404039.
Larsen, Wilczek, “Renormalization of black hole entropy and of the gravitational coupling constant,” Nucl. Phys. B 458 (1996) 249, arXiv:hep-th/9506066.
Kabat, “Black hole entropy and entropy of entanglement,” Nucl. Phys. B 453 (1995) 281, arXiv:hep-th/9503016.
Frolov, Fursaev, Zelnikov, “Statistical origin of black hole entropy in induced gravity,” Nucl. Phys. B 486 (1997) 339, arXiv:hep-th/9607104.
Donà, Eichhorn, Percacci, “Matter matters in asymptotically safe quantum gravity,” Phys. Rev. D 89 (2014) 084035, arXiv:1311.2898.
Vassilevich, “Heat kernel expansion: user’s manual,” Phys. Rep. 388 (2003) 279, arXiv:hep-th/0306138.
Jacobson, “Entanglement equilibrium and the Einstein equation,” Phys. Rev. Lett. 116 (2016) 201101, arXiv:1505.04753.
Bianchi, Myers, “On the architecture of spacetime geometry,” arXiv:1212.5183.
Sorkin, “Expressing entropy globally in terms of (4D) field-correlations,” arXiv:1205.2953.
Afshordi, Aslanbeigi, Sorkin, “A distinguished vacuum state for a quantum field in a curved spacetime: formalism, features, and cosmology,” arXiv:1205.1296.
Sorkin, Yazdi, “Entanglement entropy in causal set theory,” Class. Quantum Grav. 35 (2018) 074004, arXiv:1611.10281.
Belenchia, Benincasa, Letizia, Liberati, “On the entanglement entropy of quantum fields in causal sets,” Class. Quantum Grav. 35 (2018) 074002, arXiv:1712.04227.