The Impact of the Higgs on Einstein's Gravity

TL;DR

This paper reviews how the discovery of the Higgs mechanism influences Einstein's gravity, showing that gravitation can be consistent with the Standard Model and is renormalizable.

Contribution

It provides an updated review of deriving Einstein's equations from quantum field theory post-Higgs discovery, emphasizing the compatibility of gravity with the Standard Model.

Findings

Gravitation is renormalizable within the Standard Model framework.

Einstein's gravity remains consistent with quantum field theory after Higgs discovery.

The derivation of Einstein's equations from quantum theory is revisited with new insights.

Abstract

We present an updated review of Kraichnan's derivation of Einstein's equations from quantum field theory, including the period after the discovery of the Higgs mechanism. Gravitation in the Einstein sense is seen to be renormalizable and consistent with the Standard Model of Fundamental Interactions.

Equations

$$\begin{array}[]{ll}{\cal L}_{Fierz-Pauli}\!=\!\frac{1}{4}[h_{,\mu}h^{,\mu}\!-\!h_{\nu\rho,\mu}h^{\nu\rho,\mu}-2h_{\mu\nu}{}^{,\mu}h^{,\nu}+\vspace{1mm}\\ 2h_{\nu\rho,\mu}h^{\mu\rho,\nu}-m^{2}(h_{\mu\nu}h^{\mu\nu}-h^{2})],\end{array}$$

$$\frac{\partial U}{\partial h_{\mu \nu }}=h^{\mu \nu }-h{\eta }^{\mu \nu }=0.$$

$${\square }^2{h}_{\mu \nu }=0.$$

$$R_{\mu \nu \rho \sigma }{X}^{\rho \sigma }=\lambda X_{\mu \nu },\text{mbox}nosumon\rho \text{mbox}and\sigma ,$$

$${\square }^2{\psi }_{\mu \nu }=0,{\psi }_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}h{\eta }_{\mu \nu }.$$

$$g_{\mu \nu }={\eta }_{\mu \nu }+\delta h_{\mu \nu }+\delta h_{\mu \nu }^2+\cdots .$$

$$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=\kappa T_{\mu \nu },$$

$$\frac{\delta }{\delta g_{\mu \nu }}\int (R-\kappa {\mathcal{L}}_{source})\sqrt{-g}dv=0.$$

$$D^{\mu }{F}_{\mu \nu }=4\pi J_{\nu },D^{\mu }{F}_{\mu \nu }^{\ast }=0,$$

$$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }.$$

$$\frac{1}{\kappa }\frac{\delta }{\delta g_{\mu \nu }}\int R\sqrt{-g}dv=\frac{\delta }{\delta g_{\mu \nu }}{\mathcal{L}}_{source}\sqrt{-g}dv,$$

$$r_{Sch}=\frac{2\kappa \hslash }{c\lambda }.$$

$${\mathcal{P}}_4\times U(1)\times SU(2)\times SU(3).$$

$$\begin{array}[]{ll}{\cal L}=-\frac{1}{4}tr\left[F^{\mu\nu}F_{\mu\nu}+(D^{\mu}\phi)^{\dagger}(D_{\mu}\phi)\right.-\vspace{2mm}\\ \left.\mu^{2}(\phi^{\dagger}\phi)+\lambda(\phi^{\dagger}\phi)^{2}\right],\;\lambda>0,\end{array}$$

$$\phi=\left(\begin{array}[]{cc}\phi_{1}\\ \phi_{2}\end{array}\right),\;\;\phi^{\dagger}=\left(\phi^{*}_{1},\phi^{*}_{2}\right).$$

$$\frac{\partial U}{\partial \varphi }={\mu }^2{\varphi }^{†}+2\lambda ({\varphi }^{†}\varphi ){\varphi }^{†}=0,$$

$${(\Delta m}_{Higgs}{)}^2=(m_{Higgsa}{)}^2-(m_{Higgsb}{)}^2=0.$$

Full text

I The Gravitating Quantum Fields

The derivation of Einstein’s equations from the spin-2 quantum field theory was first obtained by Robert KraichnanKraichnan1955 ; Kraichnan1956 , an assistant of Einstein at Princeton during the period 1949-1950. According to the storyFeynman1955 , Einstein suggested to Kraichnan an investigation on the emergence of singularities from the non-linearity of the gravitational equations.

The purpose of this note is to show that Kraichnan’s derivation of Einstein’s equations implies that Einstein’s gravitational theory is renormalizable, offering a definitive and unique solution to the long standing search for quantum gravity, including its implications to the Standard Model of Fundamental Interactions (The Standard Model for short). We start below with a brief review and update of Kraichnan’s derivation. In the next section we show why that result implies in perturbative quantum gravity. In section 3 we suggest a way to include Einstein’s gravitation in the Standard Model, using the Higgs mechanism.

We begin when Kraichnan’s noted the similarity between the linear gravitational wave equations and the wave equation for a massless spin-2 quantum field in the Minkowski space-time. The Lagrangian of a spin-2 field $h_{\mu\nu}$ was originally derived by Fierz and PauliFierzPauli as (all Greek indices $\mu,\nu...$ run from [math] to $3$. Unless explicitly indicated all small case Latin indices run from 1 to 3.)

[TABLE]

where we have denoted $h=\eta_{\mu\nu}h^{\mu\nu}$ and the field potential energy is $U=(h_{\mu\nu}h^{\mu\nu}-h^{2})$. The observables of such field were defined as being the non-zero spatial components $h_{ij}$, satisfying the minimal energy condition

[TABLE]

With this condition, it follows that $h=0$ and one degree of freedom is eliminated. Consequently the Fierz-Pauli field has only five non-zero independent degrees of freedom (dof). From the spin-statistics theorem we infer also that $h_{\mu\nu}$ is a spin-2 field. The Fierz-Pauli field equations are $(\Box^{2}-m^{2})h_{\mu\nu}=0$ and in the massless case they become simply

[TABLE]

On the other hand, the observables of Einstein’s gravitational field are given by the real non-zero solutions of the eigenvalue equations of the curvature tensorPirani .

[TABLE]

where $X^{\mu\nu}$ are the corresponding eigenvectors. Since these eigenvectors are anti-symmetric 2-forms, there are at most six independent solutions, corresponding to six real eigenvalues, including the trivial solution $\lambda=0$, which corresponds to the vacuum state of the gravitational field. Thus, we are left also with only five non-trivial eigenvalues $\lambda$ to be measured by the observers, so that Einstein’s gravitation has $dof=5$ or equivalently, it is a spin-2 field.

The linear gravitational wave equations written in the de Donder coordinate gauge are

[TABLE]

Comparing with Eq. (2), we find that if the above $h_{\mu\nu}$ coincide with the spin-2 field defined by Eq. (2), then the above wave equations becomes identical to the Fierz-Pauli wave equations Eq. (2) in an arbitrary reference frame. Such independence of a specific coordinate gauge suggests that gravitation could be obtained from sequence of small perturbations of the Minkowski metric by the Fierz-Pauli field $h_{\mu\nu}$ as

[TABLE]

After a laborious sequence of calculations, the comparison of each term of such perturbation with an arbitrary conserved energy-momentum tensor $T_{\mu\nu}$, Kraichnan obtained Einstein’s equations, with the only difference that the coupling constant with the source is an arbitrary proportionality factor $\kappa$, not related to the Newtonian gravitational constant $G$:

[TABLE]

Such equation is defined in the Minkowski space-time, but from this point on, Kraichnan just followed Einstein’s geometrization of the gravitational field: Define a new space-time whose metric $g_{\mu\nu}$ is a solution of Eq. (5) written in an arbitrary reference frame; introduce the principles of general covariance and of equivalence; and finally define the observables of the gravitational field (the space-time curvature tensor).

Notice that we could also have derived directly the Ricci scalar using Kraichnan’s perturbative construction, subsequently obtaining the Einstein-Hilbert Lagrangian $R\sqrt{-g}$ and finally applying the variational principle

[TABLE]

Therefore we may infer that Kraichnan’s result expressed by Eq. (5) is unique up to the addition of a divergence-free term. It also suggests that the Einstein-Hilbert Lagrangian can be regarded as a non-linear version of the Fierz-Pauli variational principle for massless spin-2 fields.

Similar derivations of Einstein’s gravitation from quantum field theory were also obtained by several authors in the period from 1950-1970, like S. Gupta, showing that the Einstein tensor could be obtained from a perturbation of the Minkowski metric by a quantum fieldGupta1954 . Richard Feynman in his Lectures on Gravitation at Caltech also attempted to derive Einstein’s gravitation from a spin-2 quantum fields, starting from Feynman’s diagrams. A comprehensive analysis of these results can be found in the foreword by John Preskill and Kip Thorne to Feynman’s Lecture Notes on GravitationFeynman1955 .

Along a similar line, it was also suggested that a short range form of gravitation could result from the perturbations of the Minkowski metric by a massive spin-2 fieldvDam ; Zakharov , leading to what is known today as massive gravityClaudia . Accordingly, Einstein’s massless gravitational field would result from the zero mass limit of such massive gravity. However, it has been argued that such massive gravity would be plagued with ghosts, which would remain even after the zero mass limitDeserBoulware . A short range gravitational field was also suggested as a spin-2 field independent of the space-time metric, similar to the electromagnetic field produced by a local currentSalam . In spite of the ongoing debate on the existence short range gravity, here we concentrate in the most relevant implication of Eq. (5), perturbative quantum gravity.

II Quantum Gravity

The renormalization of gauge fields was proved by ’tHooft & VeltmanVeltman , but the same procedure did not work for the gravitational field of General Relativity. This can be understood from a fundamental difference between the two theories: In gauge theory, the coupling constant g appears in the covariant derivative $D_{\mu}=\partial_{\mu}+\textbf{g}\;A_{\mu}$, where $A_{\mu}$ is the gauge field, solution of the Yang-Mills field equations

[TABLE]

where $F_{\mu\nu}=[D_{\mu},D_{\nu}]$ and where $F^{*}$ is the dual of $F$. As we see, the values of g can be used as a renormalization parameter “before” it is committed to the dynamical equations.

On the other hand, in General Relativity the coupling constant has a specific value $\frac{8\pi G}{c^{4}}$ with physical dimensions, which was fixed at level of the dynamical equations

[TABLE]

It follows tat the Einstein gravitational field in GR cannot be perturbatively quantized because the coupling constant is fixed by the dynamical equations, a conclusion reached by Goroff-SagnottiGoroff ,

However, in the case of the Kraichnan derivation, the source $T_{\mu\nu}$ and the corresponding coupling constant $\kappa$ were not specified and they depend of the scale of observations, from the range of quantum field theory, to the classical Newtonian level where the sources are predominantly composed of ordinary matter, up to the cosmological scale where gravitation may also also couple with dark matter in the universe.

Following a reasoning similar to that of Goroff-Sagnotti, the gravitational field defined by Eq. (5) can be written in a form that simulates a gauge field where the covariant derivative is modified to $\nabla_{\mu}=\partial_{\mu}+\sqrt{\kappa}\Gamma_{\mu}$, for $\kappa\neq 0$, and simultaneously writing the variational principle as

[TABLE]

Consequently, the gravitational field resulting from Eq. (5) is renormalizable in the sense of Goroff-Sagnotti.

Notice that in the vacuum case, ($\kappa=0$) the equations Eq. (5) correspond to the non-linear version of the Fierz-Pauli field without any source, whose interpretation may be that of a non-linear quantum spin-2 field. One interesting example of such quantum field is given by the spherically symmetric vacuum solution of Eq. (5), describing a quantum black-hole, with Schwarzschild radius $r_{Sch}=\frac{2GM}{c^{2}}$, where $G$ is replaced by $\kappa$ and $M$ cannot be identified with a Newtonian mass. However at the level of quantum field theory $M$ can be identified with a quantum of energy $M=E_{0}=\hbar c/\lambda$, so that the “ quantum Schwarzschild black hole” has the Schwarzschild radius

[TABLE]

In principle, such quantum black hole may be experimentally detected by adapting the experimental setup previously proposed to detect black holes from extra dimensions at the tev energy scale. In that extra dimensional case, the existence of tev black holes was justified precisely because in higher dimensions the Newtonian gravitational constant does not depend on $G$ and it must be replaced by a higher dimensional coupling constantMyersPerry ; Dimopolous ; ADD ; Seong ; Chamblin . The novelty here is in the fact that such quantum black hole is defined in the four dimensional space-time.

III Gravitation in the Standard Model

The derivation of Eq. (5) from quantum field theory implies that the gravitational hierarchy ceases to exist, and quantum gravity exists at tev. Thus, at least in principle gravitation has the right to be explicitly present in the Standard Model.

After the experimental evidence of the Higgs, the Higgs mechanism explains how the Higgs boson and the spin-1 bosons $Z^{0}$ and $W^{\pm}$ acquire their masses. All these masses belong to the spectrum of eigenvalues of the second order Casimir operator $C^{2}=P_{\mu}P^{\mu}$, where $P_{\mu}$ denote the translation operators of the Poincaré group. If so, then the consistency with the Lie algebra structure of the Standard Model, require that the Poincaré Lie algebra ${\cal P}_{4}$ should explicitly combine with the gauge field symmetries, something like

[TABLE]

However, such symmetry combination stumbles in the classic problem of high energy physics, the no-go theorem for the mixing of the Poincaré and gauge symmetriesORaifeartaigh . In short, that theorem tells that the action of ${\cal P}_{4}$ over any operator of $SU(n)$ is idempotent, in the sense that the (squared) mass difference of two particles $a$ and $b$ belonging to the same multiplet of an $SU(n)$ internal symmetry vanishes: $\Delta m^{2}=m^{2}_{a}-m^{2}_{b}=0$, which is a blatant disagreement with the observations. Such symmetry mixing problem has never been solved. At the best it was averted by replacing the Poincaré Lie algebra by a graded Lie algebraColeman . In the following, we will see why the solution of such problem is implicit in the structure of the Higgs potential.

Consider the well known interaction of the Higgs with the electroweak field, described by the Lagrangian (here for simplicity we take $\textbf{g}=1$)

[TABLE]

where $D_{\mu}=\partial_{\mu}+iA_{\mu}$ and $A_{\mu}$ is the electroweak field potential. The Higgs is a complex column

[TABLE]

The Lagrangian Eq. (8) is invariant under the Poincaré groups and under the electroweak symmetry $U(1)\times SU(2)$. The Higgs potential is $U(\phi_{1},\phi_{2})$, so that the minimum energy condition for the Higgs field is

[TABLE]

whose solution $\phi=0$ is naturally excluded, as it represents an unstable vacuum. Since $\lambda>0$, we obtain also an infinite number of non-trivial real and stable vacuum solutions by choosing $-\mu^{2}=m^{2}$, where $m^{2}$ is the Higgs mass. Those stable solutions are represented by the points of a circumference $|\phi|^{2}=\phi_{1}^{2}+\phi_{2}^{2}=a^{2}=m^{2}/2\lambda$ located at the bottom of the Higgs potential.

The exclusion of the false vacuum $\phi=0$ has the mathematical implication that the Higgs potential is a genuine revolution surface, generated by the global rotation group $SO(2)_{Higgs\,global}$, through the axis $U(\phi_{1},\phi_{2})$, which does not touch the surface. As a revolution surface, the Higgs potential can be represented as “cassino roulette”, where the bottom circumference is the Higgs vacuum. In this case there are two spontaneously broken symmetries: The first one breaks the global symmetry $SO(2)_{Higgs\,global}$, associating the Higgs mass $m^{2}=-\mu^{2}$, which has the same value for all points of that circumference. This is a consequence of the the nilpotent action of the Poincaré translations over $SO(2)_{Higgs\,global}$, implying in a zero mass splitting. That is, for any two points $a$ and $b$ of the vacuum circumference we obtain the same Higgs mass:

[TABLE]

Secondly, while the global symmetry $SO(2)_{Higgs\,global}$ remains broken, the local gauge symmetry $U(1)\times SU(2)$ is also broken, without the interference of the nilpotent action of the Poincaré translations.

The current issue on the possible instabilities of the Higgs vacuum refers to the possible quantum fluctuation of the Higgs potential caused by quantum gravity, producing lower energy stable solutionsGregory ; Bednyakov . Such quantum gravity interference on the Higgs potential must be consistent with the Einstein-Kraichnan quantum gravity and with the standard model, where the Higgs potential may have lower energy stable vacuum states as surfaces of revolution, looking like a rotating wavelet.

In conclusion, Kraichnan’s derivation of Einstein’s equations is correct and unique, and it has some interesting consequences to the understanding of Einstein’s relativistic gravitation. Not only because gravitation can be generated by a massless spin-2 quantum field, but also because the emergence of quantum black holes, ultimately answers Einstein’s question concerning the origin of singularities: These quantum black holes are evidences that the gravitational singularities emerge from quantum field theory as shown by Kraichnan.

Finally, General Relativity is still with us as a low energy limit of the Einstein-Kraichnan Relativistic gravitation. In such limit, the sources of the gravitational field are essentially composed of ordinary matter, with $\kappa=\frac{8\pi G}{c^{4}}$.

Acknowledgments

VBB thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for partial financial support of the research project no 305835/2016-5. MDM thanks prof. S. Deser for a valuable comment on a previous version of this report. Both authors wish to add a posthumous thanks to prof. Leite Lopes.