This paper suggests a solution to the problem of some apparently excessive contributions to the cosmological constant from Standard-Model condensates

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Brodsky, Stanley J.

Shrock, Robert

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Work supported in part by US Department of Energy contract DE-AC02-76SF00515Standard-Model Condensates and the Cosmological ConstantStanley J. Brodskya,b and Robert Shrockb(a) Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 and(b) C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794We suggest a solution to the problem of some apparently excessive contributions to the cosmo-logical constant from Standard-Model condensates.PACS numbers: 95.36.+x, 98.80.-k, 98.80.QcOne of the most challenging problems in physics is thatof the cosmological constant Λ (recent reviews include [1]-[3]. This enters in the Einstein gravitational field equa-tions as [4, 5]Rµν − 12gµνR− Λgµν = (8piGN )Tµν , (1)where Rµν , R, gµν , Tµν , and GN are the Ricci curvaturetensor, the scalar curvature, the metric tensor, the stress-energy tensor, and Newton’s constant. One definesρΛ =Λ8piGN(2)andΩΛ =Λ3H20=ρΛρc, (3)whereρc =3H208piGN, (4)and H0 = (a˙/a)0 is the Hubble constant in the presentera, with a(t) being the Friedmann-Robertson-Walkerscale parameter [4, 6]. Long before the current period ofprecision cosmology, it was known that ΩΛ could not belarger than O(1). In the context of quantum field theory,this was very difficult to understand, because estimatesof the contributions to ρΛ from (i) vacuum condensatesof quark and gluon fields in quantum chromodynamics(QCD) and the vacuum expectation value of the Higgsfield hypothesized in the Standard Model (SM) to be re-sponsible for electroweak symmetry breaking, and from(ii) zero-point energies of quantum fields appear to betoo large by many orders of magnitude. Observations ofsupernovae showed the accelerated expansion of the uni-verse and are consistent with the hypothesis that this isdue to a cosmological constant, ΩΛ ≃ 0.76 [7, 8, 9].Here we shall propose a solution to the problem ofQCD condensate contributions to ρΛ. We also com-ment on other contributions of type (i) and (ii). Twoimportant condensates in QCD are the quark conden-sates 〈q¯q〉 ≡ 〈∑Nca=1 q¯aqa〉, where q is a quark whosecurrent-quark mass is small compared with the confine-ment scale ΛQCD ≃ 250 MeV, and the gluon condensate,〈GµνGµν〉 ≡ 〈∑N2c−1a=1 GaµνGaµν〉, where Gaµν = ∂µAaν −∂νAaµ + gscabcAbµAcν , a, b, c denote the color indices, gs isthe color SU(3)c gauge coupling, Nc = 3, and cabc arethe structure constants for SU(3)c. These condensatesform at times of order 10−5 sec. in the early universeas the temperature T decreases below the confinement-deconfinent temperature Tdec ≃ 200 MeV. For T <<Tdec, in the conventional quantum field theory view, thesecondensates are considered to be constants throughoutspace. If this were true, then they would contribute(δρΛ)QCD ∼ Λ4QCD, so that (δΩΛ)QCD ≃ 1045. How-ever, we have argued in Ref. [13] that, contrary to thisconventional view, these condensates (and also higher-order ones such as 〈(q¯q)2〉 and 〈(q¯q)GµνGµν〉) have spa-tial support within hadrons, not extending throughoutall of space. The reason for this is that the condensatesarise because of quark and gluon interactions, and theseparticles are confined within hadrons [14]. We have ar-gued that, consequently, these QCD condensates shouldreally be considered as comprising part of the masses ofhadrons. Hence, we conclude that their effect on gravityis already included in the baryon term Ωb in Ωm and, assuch, they do not contribute to ΩΛ.Another excessive type-(i) contribution to ρΛ is con-ventionally viewed as arising from the vacuum expecta-tion value of the Standard-Model Higgs field, vEW =2−1/4G−1/2F = 246 GeV, giving (δρΛ)EW ∼ v4EW andhence (δΩΛ)EW ∼ 1056. Similar numbers are obtainedfrom Higgs vacuum expectation values in supersymmet-ric extensions of the Standard Model (recalling that thesupersymmetry breaking scale is expected to be the TeVscale). However, it is possible that electroweak symme-try breaking is dynamical; for example, it may resultfrom the formation of a bilinear condensate of fermionsF (called technifermions) subject to an asymptoticallyfree, vectorial, confining gauge interaction, commonlycalled technicolor (TC), that gets strong on the TeV scale[15]. In such theories there is no fundamental Higgs field.Technicolor theories are challenged by, but may be ableto survive, constraints from precision electroweak data.By our arguments in [13], in a technicolor theory, thetechnifermion and technigluon condensates would havespatial support in the technihadrons and techniglueballsand would contribute to the masses of these states. Westress that, just as was true for the QCD condensates,these technifermion and technigluon condensates wouldnot contribute to ρΛ. Hence, if a technicolor-type mech-anism should turn out to be responsible for electroweakSLAC-PUB-13166arXiv:0803.2554v1[hep-th]March 2008Submitted to Physical Review Letters2symmetry breaking, then there would not be any prob-lem with a supposedly excessive contribution to ρΛ fora Higgs vacuum expectation value. Indeed, stable tech-nihadrons in certain technicolor theories may be viabledark-matter candidates [18].We next comment briefly on type-(ii) contributions.The formal expression for the energy density E/V dueto zero-point energies of a quantum field correspondingto a particle of mass m isE/V =∫d3k(2pi)3ω(k)2, (5)where the energy is ω(k) =√k2 +m2. However, first,this expression is unsatisfactory, because it is (quarti-cally) divergent. In modern particle physics one wouldtend to regard this divergence as indicating that one isusing an low-energy effective field theory, and one wouldimpose an ultraviolet cutoff MUV on the momentum in-tegration, reflecting the upper range of validity of thislow-energy theory. Since neither the left- nor right-handside of eq. (5) is Lorentz-invariant, this cutoff proce-dure is more dubious than the analogous procedure forFeynman integrals of the form∫d4k I(k, p) in quantumfield theory, where I(k, p1, ..., pn) is a Lorentz-invariantintegrand function depending on some set of 4-momentap1, ..., pn. If, nevertheless, one proceeds to use such acutoff, then, since a mass scale characterizing quantumgravity (QG) is MPl = G−1/2N = 1.2 × 1019 GeV, onewould infer that (δρΛ)QG ∼ M4Pl/(16pi2), and hence(δΩΛ)QG ∼ 10120. With the various mass scales char-acterizing the electroweak symmetry breaking and par-ticle masses in the Standard Model, one similarly wouldobtain (δΩΛ)SM ∼ 1056. Given the fact that eq. (5) isnot Lorentz-invariant, one may well question the logic ofconsidering it as a contribution to the Lorentz-invariantquantity ρΛ [19, 20]. Indeed, one could plausibly arguethat, as an energy density, it should instead be part ofT00 in the energy-momentum tensor Tµν . Phrased in adifferent way, if one argues that it should be associatedwith the Λgµν term, then there must be a negative corre-sponding zero-point pressure satisfying p = −ρ, but thesource for such a negative pressure is not evident in eq.(5).The light-front (LF) quantization of the StandardModel provides another perspective. In this case, theHiggs field has the form [24] φ = ω+ϕ where ω is a clas-sical zero mode determined by minimizing the Yukawapotential V (φ) of the SM Lagrangian, and ϕ is the quan-tized field which creates the physical Higgs particle. Thecoupling of the leptons, quarks, and vector bosons to thezero mode ω give these particles their masses. The elec-troweak phenomenology of the LF-quantized StandardModel is in fact identical to the usual formulation [24].In contrast to conventional instant-form Standard Modelis trivial in the light-front formulation [22, 23], and thereis no zero-point fluctuation in the light-front theory sinceω is a classical quantity. Although this eliminates anywould-be type-(ii) contributions of zero-point fluctua-tions to the cosmological constant, the contribution tothe electroweak action from the Standard Model Yukawapotential V (ω) evaluated at its minimum would, as in theconventional analysis, yield an excessively large type-(i)contribution (δΩΛ)EW ∼ 1056. Thus the light-front for-mulation of the Standard Model based on a fundamentalelementary Higgs field evidently does not solve the prob-lem with type-(i) electroweak contributions to ΩΛ. How-ever, as we have noted above, theories with dynamicalelectroweak symmetry breaking, such as technicolor, areable to solve the problem with type-(i) contributions.In summary, we have suggested a solution to what hashitherto commonly been regarded as an excessively largecontribution to the cosmological constant by QCD con-densates. We have argued that these condensates do not,in fact, contribute to ΩΛ; instead, they have spatial sup-port within hadrons and, as such, should really be consid-ered as contributing to the masses of these hadrons andhence to Ωb. We have also suggested a possible solutionto what would be an excessive contribution to ΩΛ from ahypothetical Higgs vacuum expectation value; the solu-tion would be applicable if electroweak symmetry break-ing occurs via a technicolor-type mechanism.This research was partially supported by grants DE-AC02-76SF00515 (SJB) and NSF-PHY-06-53342 (RS).Preprint SLAC-PUB-13166, YITP-SB-08-09.[1] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559(2003).[2] S. Carroll, Liv. Rev. Rel. 4, 1 (2001); T. Padmanabhan,Phys. Rept. 380, 235 (2003); Gen. Rel. Grav. 40, 529(2008); E. J. Copeland, M. Sami, and S. Tsujikawa, Int.J. Mod. Phys. D 15, 1753 (2006); V. Sahni and A. A.Starobinsky, Int. J. Mod. Phys. D 15, 2105 (2006); S.Bludman, astro-ph/0702085; J. Frieman, M. S. 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Standard Model Condensates and the Cosmological Constant

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Standard Model Condensates and the Cosmological Constant

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