Structure of conformal gravity in the presence of a scale breaking scalar field

Even though massive sources play a central role in theories of gravity, as long as the gravity theory is both classical and macroscopic, these masses can be treated as being purely mechanical or kinematical in nature. However, the actual generation of mass itself is intrinsically microscopic, being due to a quantum-field-theoretic, spontaneously broken Hilbert space vacuum in which a quantum field such as $\phi$ or a quantum fermion bilinear such as $\bar{\psi}\psi$ acquires a non-vanishing vacuum expectation value. Nonetheless, it is generally thought that macroscopic gravity is not sensitive to such microscopic effects, with gravity only needing to respond to the presence of a massive source irrespective of how that mass may or may not have arisen. And yet once one has an expectation value such as $S(x)=\langle\Omega|\phi|\Omega\rangle$ or $S(x)=\langle\Omega|\bar{\psi}\psi|\Omega\rangle$, one can ask whether this c-number $S(x)$ order parameter might itself extend to large distances and have macroscopic consequences. And if the vacuum is translation invariant the quantity $S(x)$ would be a constant and thus would indeed extend throughout all spacetime, and not only would it do so, it would then provide a potentially substantial contribution to the cosmological constant. Moreover, if evaluated in a coherent state $|C\rangle$, a quantity such as $S(x)=\langle C|\phi|C\rangle$ or $S(x)=\langle C|\bar{\psi}\psi|C\rangle$ would then be spacetime dependent and could thus have a long range spacetime dependence. With such a spacetime-dependent $S(x)$ carrying both energy and momentum it could contribute to the classical gravitational equations of motion in a possibly significant way. In addition, if Yukawa coupled to a fermion via a Lagrangian density term of the form $-hS(x)\bar{\psi}(x)\psi(x)$, such an $S(x)$ could modify the motion of a fermionic particle over large distances scales, so that its trajectory (or that of a star composed of fermions) in a gravitational field would then not be geodesic.

To see the nature of the issue in detail consider a minimally coupled scalar field with wave equation $\nabla_{\mu}\nabla^{\mu}S=0$ propagating in a static, spherically symmetric geometry with line element $ds^{2}=B(r)dt^{2}-A(r)dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta d\phi^{2}$ in an arbitrary metric theory of gravity. In such a geometry the wave equation would take the form $(ABr^{2})^{-1/2}\partial_{r}[r^{2}(B/A)^{1/2}\partial_{r}S]=0$, and the scalar field would acquire a radial dependence of the form $S(r)=c\int drA^{1/2}/r^{2}B^{1/2}$ where $c$ is an integration constant. If the metric coefficients $B(r)$ and $A(r)$ are fixed by some static source such as a galaxy with $N^{*}$ stars so that $A(r)$ and $B(r)$ are of order $N^{*}$, $S(r)$ would accordingly acquire a term of order $N^{*}$. But that would lead to a Yukawa coupling to a single star orbiting the galaxy that would also be of order $N^{*}$. Such a coupling could not be ascribed to the mass generation mechanism for a single star as it would be $N^{*}$ times too large, and not only that, it would give an effect that would vary from one galaxy to the next as $N^{*}$ itself would then vary.

Such a situation could for instance have been encountered in the scalar-tensor Brans-Dicke theory Brans1961 , in which a macroscopic scalar field is coupled to Einstein gravity. However, Brans and Dicke only allowed the scalar field to couple to the gravitational sector and not to the matter sector. And as emphasized in Weinberg1972 , any such coupling to the matter sector could impair the successes of the equivalence principle. The Brans-Dicke theory thus finesses the issue by not associating the macroscopic scalar field with mass generation, i.e., the Brans-Dicke scalar field does not Yukawa couple to propagating fermions. In the Brans-Dicke theory the lack of any such Yukawa coupling is essentially imposed by fiat. However, the typical Lagrangian density for a scalar field, viz. $-[\nabla_{\mu}S\nabla^{\mu}S/2-m^{2}S^{2}/2+\lambda S^{4}]$, has a discrete symmetry under $S\rightarrow-S$, and thus imposing this symmetry could actually forbid the presence of any $-hS\bar{\psi}\psi$ term as it would be odd under the discrete symmetry if $\bar{\psi}\psi$ is even. Alternatively, the scalar field could be associated with some very high mass scale such as a cosmological scale or a grandunified scale, scales that are not associated with low scale fermion masses.

In not Yukawa coupling the scalar field to the matter sector the Brans-Dicke theory avoids another problem. Specifically, while the c-number $S(x)$ might describe the vacuum or a coherent state, quantizing fluctuations around either such configuration would give rise to propagating quantum scalar fields, which if Yukawa coupled to fermions would mediate interactions between fermions, in direct analog to photon or graviton exchange. These interactions would then be long range if the quantized scalar field is massless. There appears to be no sign of any such long range interaction, and concomitantly, there actually appears to be no known scalar (or pseudoscalar) particle that is in fact massless. Thus for the $SU(2)\times U(1)$ electroweak theory for instance, while the vacuum expectation value of the Higgs field gives rise to fermion masses, the associated fluctuating scalar Higgs field that is excited out of the Higgs vacuum is actually massive, to thus lead to acceptably short range rather than long range interactions. In addition, the electroweak Higgs field is accompanied by a set of massless Goldstone bosons, and initially the exchange of these particles could give rise to long range interactions. However, via the Englert-Brout-Higgs mechanism these Goldstone bosons are incorporated into a matching set of gauge bosons that then become massive, with no massless Goldstone bosons being left in the spectrum. The electroweak theory thus nicely protects itself from the presence of any long range interaction that could be mediated by spin zero scalar or pseudoscalar particles, with this all being done at the microscopic, not macroscopic, level.

While these remarks would apply to any metric theory of gravity, for the conformal gravity theory of interest to us in this paper there are additional considerations. Specifically, in a theory that has a conformal symmetry there are no mass scales at the level of the Lagrangian. Consequently, all mass must be generated in the vacuum, and thus both the stars in a galaxy and a test particle that is orbiting around the galaxy must all acquire their masses by one and the same mechanism. Thus unlike in the Brans-Dicke case, one cannot decouple the mass generation mechanism for the stars in a gravitational source (the stars that fix the above $A(r)$ and $B(r)$ coefficients) from the mass generation mechanism for the test star that orbits around the galaxy. Thus if the scalar field does couple to the stars in the source, it must couple to the test particle too, and one is not free to ignore this coupling. Moreover, if one were to introduce a discrete $S\rightarrow-S$ symmetry, while it might then prevent the scalar field from coupling to the test particle, it would equally prevent the scalar field from coupling to and giving mass to the particles in the source as well.

An additional, and quite unusual aspect of the conformal gravity theory is that even if there is a radially dependent macroscopic scalar field, according to Brihaye2009 it can actually hide itself by not contributing to the energy-momentum tensor in the region exterior to a static, spherically symmetric source. Its presence thus does not modify the exterior region solution for the metric that would have been obtained had it not been present at all. However, such a scalar field is not completely without consequence since, as noted in Horne2016 and Hobson2021 , its Yukawa coupling to a test particle (as realized via a test particle action of the form $I_{T}=-h\int S(x)ds$) would lead to a trajectory for the test particle that is not geodesic. Such non-geodesic behavior would represent a challenge to the dark-matter-free conformal gravity fits to 138 galactic rotation curves given in Mannheim2011 ; Mannheim2012 ; O'Brien2012 , fits that were based on the use of geodesic orbits. Thus in contrast to the Brans-Dicke study where the scalar field couples to the geometry but not to the test particle, in the conformal case the scalar field couples to the test particle but does not affect the geometry in the exterior region.

It is the purpose of this paper to revisit the analyses of Brihaye2009 , Horne2016 and Hobson2021 , and show that they actually reinforce the point we raised above, namely that one cannot associate a radially varying, macroscopic scalar field with mass generation in the first place. Now initially this might not appear to be the case since the very fact that the scalar field causes the exterior region $T_{\mu\nu}$ to vanish means that the strength of the scalar field is not fixed by the number $N^{*}$ of stars in the source. However, it turns out that we need to reexamine the study made in Brihaye2009 . Specifically, for a general static, spherically symmetric energy-momentum tensor there are only three independent components: $T_{00}$, $T_{rr}$ and $T_{\theta\theta}$. However, for the traceless $T_{\mu\nu}$ required of a conformal theory there are only two, and as will become apparent below, the specific structure of the conformal gravitational equations actually make $T^{0}_{\phantom{0}0}-T^{r}_{\phantom{r}r}$ and $T^{r}_{\phantom{r}r}$ the optimal choice. And as noted in Brihaye2009 , the vanishing of $T^{0}_{\phantom{0}0}-T^{r}_{\phantom{r}r}$ in the region exterior to the matter source would force a classical, macroscopic scalar field to have the form $S(r)=1/(b+ar)$; with the very vanishing of $T^{0}_{\phantom{0}0}-T^{r}_{\phantom{r}r}$ allowing the parameters $a$ and $b$ to not be fixed by any matter fields that are located in the interior region.

Now in their study Brihaye and Verbin and then subsequently Horne and then Hobson and Lasenby took $T^{r}_{\phantom{r}r}$ to vanish as well. However, we shall show below that this form for $S(r)$ does not in and of itself actually cause $T^{r}_{\phantom{r}r}$ to vanish. That this could in principle be the case is due to the fact that for the general line element $ds^{2}=B(r)dt^{2}-A(r)dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta d\phi^{2}$ the covariant conservation of a traceless $T_{\mu\nu}$ leads to

Consequently, at best the vanishing of $T^{0}_{\phantom{0}0}-T^{r}_{\phantom{r}r}$ could only constrain $T^{r}_{\phantom{r}r}$ to be of the form $A/r^{4}$ where $A$ is a constant. In our study below we will use the full set of equations of motion of the theory to show that $T^{r}_{\phantom{r}r}$ does not in fact vanish if $b=0$, but does vanish if $b\neq 0$. With $b\neq 0$ we now can set all the components of $T_{\mu\nu}$ to zero and thus establish the result that had been assumed in Brihaye2009 ; Horne2016 ; Hobson2021 where the coefficient $b$ had been taken to be non-zero. In order to actually get $T^{r}_{\phantom{r}r}$ to vanish we have to use the equations of motion, and this now does lead us to a relation between $a$ and $b$ and the metric coefficients produced by the interior region matter source, to thus force $a$ and $b$ to be of order $N^{*}$ in the galactic case. (As noted in Horne2016 , one can obtain a relation between the $a$ and $b$ parameters and the metric coefficients through the scalar field equation of motion, but as we show below, this does not force $T^{r}_{\phantom{r}r}$ to vanish.)

While mass generation is microscopic, nonetheless it is of interest to examine what would happen in the conformal theory if mass generation were to lead to a scalar field $S(x)$ that did vary macroscopically and couple to a test particle with an action of the form $I_{T}=-h\int S(x)ds$. In this paper we shall examine this option in detail in the explicitly solvable model introduced in Brihaye2009 , and explicitly demonstrate that such a test particle action would give rise to masses for test stars that were not of order one solar mass but of order the mass of the galaxy around which they orbit. Since it is thus not valid to ascribe mass generation to such a macroscopic field, the concerns raised in Horne2016 and Hobson2021 (studies that used this $I_{T}$ test particle action) are thus invalidated.

If mass generation is not to be associated with an $S(x)$ that varies macroscopically then it must be associated with an $S(x)$ that only varies on microscopic scales while being constant on large scales. Structures in which the order parameter $S(x)$ only varies within the particles that it gives mass to are well established in flat spacetime dynamical theories where they are known as extended models or bag models of hadrons, i.e., models of hadrons with non-zero radii. In particular, they can arise in flat spacetime theories that have no intrinsic dimensional parameters. And it was this particular aspect of mass generation that first led Mannheim1994 ; Mannheim1994a the present author to generalize these flat space concepts to curved space in a gravity theory, conformal gravity, in which there again would be no intrinsic dimensional parameters. Moreover, such extended structures are actually necessary in the conformal theory since the mechanism that generates mass dynamically generates extended structures at the same time, to thereby enable particles to localize into the stars and galaxies that are of relevance to astrophysics and cosmology. Moreover, in these dynamical models coherent states such as $|C\rangle$ are in and of themselves not actually stable. (They would be stable if they had a non-trivial topology, though that is not the case here.) The coherent states only become stable in the presence of a positive energy fermion excited out of the filled negative energy Dirac sea. And in this stabilization mechanism the fermion becomes localized and the scalar field only varies from a constant value within the localization region. Thus the very nature of dynamical mass generation is such that there can be no macroscopic order parameters that vary radially on large distance scales. This is completely consistent with our showing in this paper that one could not ascribe mass generation to macroscopic scalar fields that do vary on large distance scales.

This paper is organized as follows. After introducing the conformal theory in the rest of this section, in Sec. II we present the exact solution to the conformal theory in the static, spherically symmetric case in which there are no matter fields in the exterior region. In this analysis we present the linear potentials that are key to the conformal gravity theory fitting to galactic rotation curves, noting that there are actually two linear potentials of relevance, one due to the stars inside a local inhomogeneous matter source such as a galaxy and the other being due to the homogeneous global background cosmology. In Sec. III we follow Brihaye2009 and augment the theory with a scalar field that is to be present in this exterior region, and investigate its implications for the behavior of the scalar field $T_{\mu\nu}$. This analysis enables us to establish that the linear potential remains in the solution and that the $a$ and $b$ parameters in the $S(r)=1/(b+ar)$ solution are of galactic and even cosmological scale. In Sec. IV we discuss the implications of coupling the same scalar field to a test particle. As noted in Horne2016 the linear potential contribution is removed from the test particle orbit in the $S=1/(b+ar)$ case, while it is not present at all in the $S=1/b$ case, with the effect of the linear potential thus being lost in both of the cases. However, in either case we establish that at the same time this would lead to a test particle that would acquire an unacceptable galactic or cosmological sized mass. In addition, we identify some inconsistencies in the calculation given in Horne2016 . Moreover, while, as noted in Horne2016 ; Hobson2021 , one can actually remove the key linear potential by a conformal transformation when there is just a single source and radial symmetry, in Sec. V we show that this is not the case for a multiparticle source such as a spiral galaxy with axial not radial symmetry. Finally, in Sec. VI we show that through use of the microscopic scalar field $\langle C|\bar{\psi}\psi|C\rangle$ that is appropriate to dynamical symmetry breaking test particles then move on standard geodesics. That this is the case is because in the self-consistent vacuum the conformal symmetry is broken in both the vacuum and the action, with the fermion that is excited out of the vacuum not being constrained to propagate according to a conformal invariant equation of motion, and with the linear potential term then not being removed even though the scalar field is macroscopically constant. In an appendix we follow Mannheim2006 and discuss the cosmological origin of the global linear potential, with there being a conformal and coordinate equivalence between the static, spherically symmetric geometries of relevance to galactic rotation curves and the comoving Robertson-Walker geometry of relevance to cosmology.

Conformal gravity has been advanced as a candidate alternate theory of gravity (see e.g. the reviews Mannheim2006 ; Mannheim2017 and references therein). The theory is a pure metric theory of gravity that possesses all of the general coordinate invariance and equivalence principle structure of standard gravity while augmenting it with an additional symmetry, local conformal invariance, in which the action is left invariant under local conformal transformations on the metric of the form $g_{\mu\nu}(x)\rightarrow e^{2\alpha(x)}g_{\mu\nu}(x)$ with arbitrary local phase $\alpha(x)$. Under such a symmetry a gravitational sector action that is to be a polynomial function of the Riemann tensor is uniquely prescribed, and with use of the Gauss-Bonnet theorem is given by (see e.g. Mannheim2006 )

Here $\alpha_{g}$ is a dimensionless gravitational coupling constant, and

is the conformal Weyl tensor. (Here and throughout unless indicated to the contrary we follow the notation and conventions of Weinberg1972 .)

With the Weyl action $I_{\rm W}$ given in (1.2) being a fourth-order derivative function of the metric, functional variation with respect to the metric $g_{\mu\nu}(x)$ generates fourth-order derivative gravitational equations of motion of the form Mannheim2006

where the functions $W^{\mu\nu}_{(1)}$ and $W^{\mu\nu}_{(2)}$ (respectively associated with the $(R^{\alpha}_{\phantom{\alpha}\alpha})^{2}$ and $R_{\mu\kappa}R^{\mu\kappa}$ terms in (1.2)) are given by

and where $T^{\mu\nu}$ is the conformal invariant and thus traceless energy-momentum tensor associated with a conformal matter source. With the action being both general coordinate invariant and conformal invariant $W^{\mu\nu}$ is automatically covariantly conserved and covariantly traceless (i.e., it obeys $\nabla_{\nu}W^{\mu\nu}=0$, $g_{\mu\nu}W^{\mu\nu}=0$) without the need to impose any gravitational equation of motion. And with $I_{\rm W}$ being conformal invariant $W_{\mu\nu}$ transforms as $W_{\mu\nu}\rightarrow e^{-2\alpha(x)}W_{\mu\nu}$ under a local conformal transformation. In addition the Weyl tensor vanishes in geometries that are conformal to flat, so that in the homogeneous and isotropic conformal to flat background Robertson-Walker and de Sitter geometries of interest to cosmology the background $W^{\mu\nu}$ is zero. Despite this, fluctuations (i.e., cosmological inhomogeneities such as the galaxies and stars of interest to us in this paper) around that background are not conformal to flat, with the fluctuating $\delta W^{\mu\nu}$ then not being zero.

Since the light cone is left invariant under a local conformal transformation, in a conformal invariant theory there are no mass scales at the level of the Lagrangian, with all particles being massless. The generation of fermion mass scales in a conformal invariant theory has to be done in the vacuum, and is characterized by an order parameter $S(x)$ (a vacuum expectation value). For the fermion and scalar field sector we take the matter action to be of the form

where $h$ and $\lambda$ are dimensionless coupling constants. Here $V^{\mu}_{c}(x)$ is a vierbein and $\Gamma_{\mu}=-(1/8)[\gamma_{a},\gamma_{b}](V^{b}_{\nu}\partial_{\mu}V^{a\nu}+V^{b}_{\lambda}\Gamma^{\lambda}_{\phantom{\lambda}\nu\mu}V^{a\nu})$ is the spin connection, with the $\gamma_{a}$ being a set of fixed axis Dirac gamma matrices and $\Gamma^{\lambda}_{\phantom{\lambda}\nu\mu}$ being the Levi-Civita connection $(1/2)g^{\lambda\sigma}[\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}]$. As such, this action is locally conformal invariant under $g_{\mu\nu}(x)\rightarrow e^{2\alpha(x)}g_{\mu\nu}(x)$, $V^{a}_{\mu}(x)\rightarrow e^{\alpha(x)}V^{a}_{\mu}(x)$, $\psi(x)\rightarrow e^{-3\alpha(x)/2}\psi(x)$, $S(x)\rightarrow e^{-\alpha(x)}S(x)$. Variation of this action with respect to $\psi(x)$ and $S(x)$ yields the matter field equations of motion

while variation with respect to the metric yields an energy-momentum tensor of the form (first without use of any matter field equation of motion and then with)

Use of the matter field equations of motion then confirms that the matter sector energy-momentum tensor is indeed traceless. Moreover, from the conformal invariance of $I_{\rm M}$ it follows that $T_{\mu\nu}$ transforms as $T_{\mu\nu}\rightarrow e^{-2\alpha(x)}T_{\mu\nu}$ under a local conformal transformation. With $W_{\mu\nu}$ transforming the same way, the $4\alpha_{g}W_{\mu\nu}=T_{\mu\nu}$ equation of motion is conformal invariant. We now proceed to look for static, spherically symmetric solutions to these equations. As such, these selfsame equations could be considered to be microscopic or macroscopic, and so we first explore them as macroscopic equations, and defer a discussion of them as microscopic ones to Sec. VI.

To obtain an exact solution to the equations of motion in the exterior $R_{0}<r<R_{1}$ region in the case when there are non-vanishing fields in that region, we restrict the source to just a macroscopic scalar field $S(x)$. Following Brihaye2009 we look to see if we can have a non-trivial scalar field and yet nonetheless still get $T_{\mu\nu}$ to vanish in the exterior $R_{0}<r<R_{1}$ region. Thus in the static, spherically symmetric case we take $S(x)$ to depend on the radial coordinate $r$. Then, given the metric in (2.6) and a scalar sector of the form that (1.8) and (1.9) would take if the fermion sector is excluded, in the exterior $R_{0}<r<R_{1}$ region we obtain

while the pure scalar sector of (1.8) becomes

To find an exterior solution it was suggested in Brihaye2009 to set $T^{0}_{{\phantom{0}}0}-T^{r}_{{\phantom{r}}r}=0$ footnoteC . This then requires that $SS^{\prime\prime}=2S^{\prime 2}$, with integrals

where $a$ and $b$ are integration constants. If we insert (3.3) into (3.2) we obtain

With (3.4) we can rewrite (3.5) as

Now the intent of Brihaye2009 , Horne2016 and Hobson2021 was to have the vanishing of $T^{0}_{{\phantom{0}}0}-T^{r}_{{\phantom{r}}r}$ cause $T^{r}_{{\phantom{r}}r}$ to vanish as well. As we see from (3.6), for an as yet unspecified form for $B(r)$ this is not the case (and not even if we were to set $a=0$). To see why $T^{r}_{{\phantom{r}}r}$ could not vanish even in principle absent further information of some sort, we recall that we had noted above that because of its tracelessness and covariant conservation $T_{\mu\nu}$ and thus equally $W_{\mu\nu}$ obey