Essay written for the Gravity Research Foundation 2012 Awards for Essays on Gravitation HIP-2012-08-T

Räsänen, Syksy

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arXiv:1203.6259v1  [gr-qc]  28 Mar 2012Preprint typeset in JHEP style - HYPER VERSION HIP-2012-08/THVacuum energy and dynamical symmetrybreaking in curved spacetimeSyksy Räsänen∗University of Helsinki, Department of Physicsand Helsinki Institute of PhysicsP.O. Box 64, FIN-00014 University of Helsinki, Finlandsyksy dot rasanen at iki dot fiAbstract: We argue that calculating vacuum energy requires quantum field theorywhose axioms are adapted to curved spacetime. In this context, we suggest that non-zero vacuum energy is connected to dynamical breaking of electroweak symmetry.The observed meV scale can be understood in terms of electroweak physics via anaive estimate. The scenario requires all particle masses to have a dynamical origin.Any Higgs particle has to be a composite, and the origin of vacuum energy might beprobed at the LHC.∗Essay written for the Gravity Research Foundation 2012 Awards for Essays on Gravitation.Submitted on March 28.The vacuum energy problem. Vacuum energy has a long history [1, 2], and ithas been brought to focus by cosmological observations which have in the past twodecades established that the universe expands faster than expected. The usual viewof the vacuum energy problem1 is that quantum field theory (QFT) predicts vacuumenergy, cosmological observations reveal its presence, and there is a huge discrepancybetween the predicted and measured values.One measure of the perceived severity of the problem is the popularity of an-thropic arguments. When no solution that would determine the value of a physicalquantity has been found, it may be tempting to take this as evidence that there isno solution, and that the measured value is due to environmental contingency. Thisroad was earlier taken with regard to spatial curvature [3], though anthropic argu-ments were eventually left by the wayside when a dynamical, testable, explanationwas found with the introduction of inflation. The appeal to anthropic argumentsindicates that the problem does not involve contradiction between theory and ob-servation, nor theoretical inconsistency, simply unmet expectations. In the vacuumenergy case, it is useful to consider these expectations carefully, because the simplesummary given above is somewhat misleading.No vacuum energy in flat spacetime QFT. First of all, the interpretation ofthe observations in terms of vacuum energy is not beyond reasonable doubt; apartfrom explanations involving exotic matter or modifications to gravity, it is possiblethat the change in the expansion rate is simply due to the known breakdown ofhomogeneity and isotropy at late times [4, 5]. Nevertheless, even if the cosmologicalobservations were explained by something else than vacuum energy, the questionof why vacuum energy is not large would remain. Let us assume that the correctexplanation for the observations is indeed vacuum energy, and look at the theoreticalside.It is often claimed that the Casimir effect shows that vacuum energy is real[1]. However, as discussed in [6], the Casimir effect can be calculated without anyreference to vacuum energy, and gives no more indication of the reality of vacuumenergy than any other loop contribution in QFT. Indeed, the Lamb shift was quotedto the same effect in the background material for the 2011 Nobel prize in physics,awarded for the discovery of accelerating expansion [7]. However, while loop effectsmay be related to vacuum fluctuations, these are distinct from vacuum energy2.1The terms vacuum energy and cosmological constant are often used interchangeably. Thecosmological constant is a geometrical term in the classical Einstein equation, and it changes theway spacetime curves in the absence of matter. Vacuum energy has its origin in quantum theory,and it is a form of matter which has the same effect as the cosmological constant. We only discussvacuum energy.2In particular, referring to vacuum energy as the energy of fluctuations of the vacuum is mis-leading, because the vacuum is an energy eigenstate, so there are no fluctuations in the energy.– 1 –QFT formulated in flat spacetime is sensitive only to differences in energy: anarbitrary constant can be added to the Lagrangian without changing the physics,so QFTs which differ only by a constant energy are equivalent. As the concept ofabsolute energy is not part of the physical content of flat spacetime QFT, it is evenin principle impossible to obtain any prediction about the value of vacuum energy,any more than it is possible to predict gauge-fixing parameters.Arguments about vacuum energy in flat spacetime are therefore at most sug-gestive, and suggestions are easy to misconstrue. For example, it has been arguedthat calculating vacuum energy by integrating over momentum space modes with acut-off Λ much larger than the masses present in the theory shows that it is of theorder Λ4. However, the resulting energy-momentum tensor is traceless, so it behaveslike radiation and does not lead to accelerating expansion, as is well known [1, 8].When the integration range is extended to infinity, the equation of state becomesindefinite as the result diverges.In any case, vacuum expectation values in QFT do not reduce to simple momen-tum mode decomposition [8]. From the field theory point of view, vacuum energy isa renormalisable term (supersymmetric theories excepted), and its value is arbitrary,like particle masses. However, whereas masses are fixed by observation, vacuum en-ergy is not observable in flat spacetime QFT. If spacetime is curved, the situation isdifferent, since gravity responds to absolute and not relative amounts of energy. Thenew piece of information which has to be supplied is how quantum fields couple togravity.Symmetry breaking in curved spacetime QFT. The effect of quantum fieldson spacetime curvature can be studied in semiclassical quantum gravity, where space-time is treated classically and its dynamics is sourced by the expectation value ofthe energy-momentum tensor. Vacuum energy then becomes a physical quantitywith arbitrary magnitude. However, the vacuum state can evolve as the universeexpands. A well-known problem is that changes in the vacuum energy associatedwith the Higgs field are of the order of the electroweak scale. While vacuum energymay be simply set to the value that would explain the late-time observations, thisdoes not offer any insight into why it has this value.In the above discussion it has been assumed that we can use the rules of flatspacetime QFT to calculate vacuum expectation values. However, the structure ofQFT in flat spacetime is rooted in special properties of Minkowski space which curvedspacetime does not share. The crux of the vacuum energy issue is the relation betweenthe energy-momentum tensor and spacetime curvature, so we should consider howthe situation changes when QFT is formulated consistently in curved spacetime.In QFT based on axioms suited to curved spacetime, the vacuum energy of amassive free scalar field is arbitrary for non-zero mass, but necessarily vanishes forzero mass, in contrast to the flat spacetime theory, in which the vacuum energy is– 2 –always arbitrary [9, 10]. Vacuum expectation values are expected to depend non-analytically on the parameters of the theory, and it has been suggested that thiscould lead to vacuum energy much smaller than the natural scale of the theory, andthat this might be relevant for present-day cosmic acceleration [10].To elaborate further, we can propose the following scenario. Since the StandardModel Higgs field is massive, it might seem that the above result is not relevant.However, if there is no elementary Higgs field, the situation changes [4]. If it is alsotrue for fermions and vector fields that vacuum energy vanishes for zero mass andthere are no elementary scalars, so that all masses are dynamically generated, thenthis would explain why the vacuum energy is zero. (This is sometimes known asthe “old” vacuum energy problem.) Two questions immediately arise. What hap-pens when interactions are included, in particular those which dynamically generatemasses and composite particles? And second, is it possible to understand the valueof vacuum energy which would explain the observations? These issues turn out tobe related.It may be that as the symmetry which keeps particles at zero mass is dynamicallybroken, it can no longer enforce vanishing vacuum energy. Vacuum energy shouldthus vanish as coupling constants go to zero, the theory becomes free and symmetryis restored. Vacuum energy should also depend non-analytically on the couplings.For a coupling constant g, perhaps the simplest possibility is e−1/g2. In electroweakphysics, the scale of particle masses is related to the Higgs vacuum expectation valuev ≈ 246 GeV. Taking g2 = α ≈ 1/137, the vacuum energy density could then naivelybe expected to beρvac ∼ e−1/g2v4 ≈ (0.33 meV)4 . (1)This is close to the value that would explain the observations, ρvac = 3H2M2PlΩΛ ≈(h/0.7)2(ΩΛ/0.7) (2.3 meV)4. The estimate is exponentially sensitive to insertinge2 = 4πα instead of α and to the presence of other factors such as weak mixingangles, and the functional form is mere conjecture. Nevertheless, this does showthat the scale which would explain late-time cosmological observations may emergenaturally from electroweak physics in curved spacetime.We can outline steps to take to see whether the above conjecture holds any truth.First we should consider the vacuum energy of free fermions and vector bosons incurved spacetime QFT, then introduce a model of symmetry breaking and trace whathappens to vacuum energy. Eventually, a full model of dynamical mass generationshould be considered: realistic models tend to have a new scale in addition to theelectroweak scale, and the dynamics can be rather involved [11].It may not be necessary to deal with spacetime curvature. The key ingredient isthe new set of axioms: this is not flat spacetime QFT, but the flat spacetime limitof curved spacetime QFT. The experimental success of QFT based on flat spacetimeaxioms shows that the curved spacetime theory must reduce extremely accurately to– 3 –the flat spacetime version as far as non-gravitational physics is concerned. However,when it comes to effects which are absent or ambiguous in the flat spacetime theoryand to which particle physics experiments are not sensitive, the results can be verydifferent.Conclusions. If vacuum energy is the cause of the increased expansion rate at latetimes, its value is tiny compared to particle physics scales. A natural explanationwould be that there is a symmetry which sets vacuum energy exactly to zero, andwhich is slightly broken. In the present scenario, the symmetry is related to particleshaving zero mass, and it is broken by the emergence of new effective degrees offreedom in dynamical breaking of electroweak symmetry. Vacuum energy is thus,like massive particles, an emergent phenomenon. Dynamical mass generation is nota small effect, but its impact on vacuum energy is diluted exponentially by the non-analytical dependence of vacuum expectation values on coupling constants.The scenario requires that there are no massive elementary scalar fields, in par-ticular that any Higgs particle is a composite, which is a qualitative prediction forcollider physics. As vacuum energy is related to the spectrum of masses of particlesat the electroweak scale, the vacuum energy problem is connected to the Higgs masshierarchy problem and it may be possible to probe it at the LHC. It is also necessarythat there are no particles with masses larger than the electroweak scale, or that thecontribution of such particles to vacuum energy is correspondingly more suppressed.On the theoretical side, the dynamical generation of a mass scale and the choice ofthe axioms of QFT relates vacuum energy to the Millennium problem of definingYang-Mills theory and understanding the mass gap [12].Even if the idea turns out to be wrong, it shows that addressing the vacuumenergy problem does not necessarily require delving into speculative physics distantfrom observations, and that there is little reason to abandon hope that a solutionexists.References[1] P.J.E. Peebles and B. Ratra, The Cosmological constant and dark energy, Rev. Mod. Phys.75 (2003) 559 [arXiv:astro-ph/0207347][2] H. Kragh, Preludes to dark energy: Zero-point energy and vacuum speculations[arXiv:1111.4623 [physics.hist-ph]][3] C.B. Collins and S.W. Hawking, Why is the Universe isotropic?, Astrophys. J. 180 (1973)317[4] S. Räsänen, Backreaction: directions of progress [arXiv:1102.0408 [astro-ph.CO]][5] T. Buchert and S. Räsänen, Backreaction in late-time cosmology [arXiv:1112.5335[astro-ph.CO]][6] R.L. Jaffe, The Casimir effect and the quantum vacuum, Phys. Rev. D72 (2005) 021301[arXiv:hep-th/0503158][7] http://www.nobelprize.org/nobel prizes/physics/laureates/2011/advanced.html– 4 –[8] S. Hollands and R.M. Wald, Quantum field theory is not merely quantum mechanics appliedto low energy effective degrees of freedom Gen. Rel. Grav. 36 (2004) 2595[arXiv:gr-qc/0405082][9] S. Hollands and R.M. Wald, Axiomatic quantum field theory in curved spacetime, Commun.Math. Phys. 293 (2010) 85 [arXiv:0803.2003 [gr-qc]][10] S. Hollands and R.M. Wald, Quantum field theory in curved spacetime, the operator productexpansion and dark energy, Gen. Rel. Grav. 40 (2008) 2051, Int. J. Mod. Phys. D17 (2009)2607 [arXiv:0805.3419 [gr-qc]][11] K. Lane, Two lectures on technicolor [arXiv:hep-ph/0202255]C.T. Hill and E.H. Simmons, Strong dynamics and electroweak symmetry breaking, Phys.Rept. 381 (2003) 235, Erratum: ibid. 390 (2004) 553 [arXiv:hep-ph/0203079].[12] http://www.claymath.org/millennium/Yang-Mills Theory/– 5 –

Vacuum energy and dynamical symmetry breaking in curved spacetime

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Vacuum energy and dynamical symmetry breaking in curved spacetime

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