Supersymmetric Dark Energy Neven Bilić Ruđer Bošković Institute Zagreb, Croatia BW2011, Donji Milanovac, 31 Aug 2011
Outline1. Introductory remarks a) Cosmological considerations b) Vacuum energy2. Motivation for SUSY3. Summary4. The model5. Calculations of the vacuum energy density and pressure
1.Introductory remarks a) Cosmological considerationsIn a homogeneous, isotropic and spatially flat spacetime(FRW), i.e., with metric ds dt a(t ) dx 2 2 2 2Einstein’s equations take the form a 8 G 0 2 H 2 T0 a 3 a 4 G 0 (T0 Ti ) i a 3 iwhere Tµν is the energy momentum tensorThe Hubble “constant” H describes the rate of the expansion.
Owing to the isotropy we can set T00 ; T11 T22 T33 pIn a hydrodynamical description in which Tµνrepresents a perfect fluid. i.e., T ( p )u u pgp and ρ may be identified with pressure and density.This identification is correct only in a comoving frame,i.e., when the cosmic fluid velocity takes the form 1 u ( g00 ,0,0,0); u ( ,0,0,0) g00
We obtain the Friedmann equations a 8 G 2 H 2 a 3 a 4 G ( 3 p) a 3
b) vacuum energy If we assume T g then p and we reproduce Einsten’s equations with a cosmologicalconstant equal to 8 G
In this case the metric takes the form 2 ds dt e 2 2 2 Ht dx We have a universe with an accelerating expansion.This metric describes the so called flat patch of de Sitter(dS) spacetime with the de Sitter symmetry group.Thus, a nonzero cosmological constant implies the dSsymmetry group of space-time rather than the Poincar´egroup which is the space time symmetry group of Minkowskispace.
It is generally accepted that the cosmological constantterm which was introduced ad-hoc in the Einstein-Hilbertaction is related to the vacuum energy density ofmatter fields. It is often stated that the vacuum energydensity estimated in a quantum field theory is by about 120orders of magnitude larger than the value required byastrophysical and cosmological observations. e.g., S.Weinberg, Rev. Mod. Phys., 61 (2000)
Consider a real scalar field. Assuming the so calledminimal interaction, the Lagrangian is 1 L g V ( ) 2with the correspoding energy-momentum tensor T g L
We define vac = T H > 0 0 1 pvac Ti i 3 iWhere < A> denotes the vacuum expectation value ofan operator A. In FRW spacetime 1 1 vac 2 ( )2 V ( ) 2 2 2a 1 1 pvac 2 ( )2 V ( ) 2 2 6a
For a free massive field in flat spacetime one finds 1 d 3k vac k 2 m2 2 (2 )3 3 2 1 d k k pvac 6 (2 )3 k 2 m2and with a 3-dim momentum cutoff K we obtain K4 m2 K 2 1 K2 vac ln 2 .... 16 2 16 2 64 2 m 1 K 4 1 m2 K 2 1 K2pvac ln 2 .... 3 16 2 3 16 2 64 2 m
Assuming that the ordinary field theory is valid up tothe scale of quantum gravity, i.e. the Planck scale, we find K4 4 mPl vac ... 1073 GeV 4 16 2 16 2 compared with the observed value vac 1047 GeV4
Fine tuning problemIn addition to the vacuum fluctuations of the field there mayexist an independent cosmological term Λ equivalent to 8 Gso that one would find an effective vacuum energy eff vac In order to reproduce the observed value one needsa cancelation of these two terms up to 120 decimal places!The problem is actually much more severe as we have manyother contributions to the vacuum energy from different fields with different interactions and all these contributions mustsomehow cancel to give the observed vacuum energy.
Adding gravityIn fact, there would be no problem if there were no gravity!In flat space one can renormalize the vacuum energyby subtracting the divergent contributions since the energyis defined up to an arbitrary additive constant.However , in curved space this cannot be easily done becausethe energy is a source of the gravitational field andadding (even constant) energy changes the spacetimegeometry .
2. Motivation for SUSYQuestion No 1Can supersymmetry cure the mentioned problems?At least we know that in a field theory with exact SUSY thevacuum energy, and hence the cosmological constant, isequal to zero as the contributions of fermions and bosons tothe vacuum energy precisely cancel!Unfortunately, in the real world SUSY is broken at smallenergy scales. The scale of SUSY breaking required byparticle physics phenomenology must be of the order of 1 TeVor larger implying Λ still by about 60 orders of magnitude toolarge.
Question No 2How does the SUSY vacuum behave in curved spacetime,e.g., in de Sitter spacetime?Our aim is to investigate the fate of vacuum energy whenan unbroken supersymmetric model is embedded inspatially flat, homogeneous and isotropic spacetime.In addition, we assume the presence of a dark energytype of substance obeying the equation of statepDE =wρDE, with w<0.
The space time symmetry group of an exact SUSY is the Poincar´e group.. The lack of Poincar´e symmetry will liftthe Fermi-Bose degeneracy and the energy density ofvacuum fluctuations will be nonzero. This type of “soft” supersymmetry breaking is known in supersymmetric field theory at finite temperature where the Fermi-Bose degeneracy is lifted by statistics. Das and Kaku, Phys. Rev. D 18 (1978) Girardello, Grisaru and Salomonson, Nucl. Phys. B 178 (1981)
3. SummaryThe final expressions for the vacuum energy density andpressure are free from all divergent and finite flat-spacetimeterms.The dominant contributions come from the leading termswhich diverge quadratically. N cut a 2 2 2 1 ( cut ln cut ) 2 8 a2 N cut a 2 2 a p 2 2 2 1 ( cut ln cut ) 2 24 a a 3 2 NB, Phys Rev D 2011 2 cut mPl N λ an arbitrary positive parameter 0 1 N number of chiral species
Combining effects of dark energy with the equation of statepDE = wρDE and vacuum fluctuations of the supersymmetricfield we find the effective equation of state peff weff eff DE 2 eff weff w 1 3 1 Friedman equations take the standard form a 2 8 2 G eff a 3 a 4 G(1 3weff ) eff a 3
1. Imposing a short distance cutoff of the order mPl we havefound that the leading term in the energy density of vacuumfluctuations is of the same order as dark energy (H2 mPl2)and no fine tuning is needed2. The contribution of the vacuum fluctuations to the effectiveequation of state is always positive and, hence, it goes againstacceleration!A similar conclusion was drawn by M. Maggiore, PRD (2011)who considers massless scalar fields only and removes theflat-spacetime contribution by hand.3. If we require accelerating expansion, i.e., that the effectiveequation of state satisfies weff < −1/3, the range −1 < w < −1/3is compatible with 0 < λ < 1/2, whereas w < −1 (phantom)would imply λ> 1/2.
4. The modelWe consider the Wess-Zumino model with N species andcalculate the energy momentum tensorof vacuum fluctuationsin a general FRW space time. The supersymmetric Lagrangianfor N chiral superfields Φi has the form W(Φ) denotes the superpotential for which we takeBailin and Love, Supersymmetric Gauge Field Theory and String Theory( 1999)
From now on, for simplicity, we suppress the dependenceon the species index i. Eliminating auxiliary fields byequations of motion the Lagrangian may be recast in theformwhere ϕi are the complex scalar and Ψi the Majoranaspinor fields. are the curved space time gamma matrices The symbol ea denotes inverse of the vierbein.
In the chiral (m→0) limit, this Lagrangian becomes invariant under the chiral U(1) transformation: This symmetry reflects the R-invariance of the cubic superpotential
The action may be written aswhere LB and LF are the boson and fermion LagrangiansThe Lagrangian for a complex scalar field ϕ may beexpressed as a Lagrangians for two real fields, σ and π
The potential for the scalar fields then readsVariation of the action with respect to Ψ yields the Diracequation of the form
Effective actionWe introduce the background fields and andredefine the fields ; The effective action at one loop order is is given byS0 is the classical part of the action and S(2) is the part ofthe action which is quadratic in quantum fields.
For the quadratic part we findEffective masses
4.Calculations of the vacuum energy density and pressureWe need the vacuum expectation value of the energy-momentum tensor. The energy-momentutensor is derivedfrom S(2) as
It is convenient to work in the conformal frame withmetric 2 ds a( ) (d dx ) 2 2 2where the proper time t of the isotropic observers isrelated to the conformal time η as dt a( )dIn particular, we will be interested in de Sitter space-time with 1 ae Ht H
• Scalar fieldsAs in the flat space time, each real scalar field operatoris decomposed as The function χk(η) satisfies the field equation Where ’ denotes a derivative with respect to the conformal time η . [N.D. Birell, P.C.W. Davies, Quantum Fields in Curved Space]
If m ≠ 0, the solutions may be constructed by making use of the WKB ansatzwhere the function Wk (η)may be found by solving the fieldequation iteratively up to an arbitrary order in adiabaticexpansion. L.E. Parker and D.J. Toms, Quantum Field Theory in Curved Spacetime To second adiabatic order we find where
The vacuum expectation value of the components ofthe energy-momentum tensor for each scalar is thencalculated from
• Spinor fieldsRescaling the Majorana field aswe obtain the usual flat space-time Dirac eq. with timedependent effective mass am.The quantization of is now straightforward .The Majorana field may be decomposed as
The spinor uks is given bywith the helicity eigenstates vks is related to uks by charge conjugation
The mode functions ςk satisfy the equationIn massless case the solutions to (a) are plane waves.For m≠ 0 two methods have been used to solve (a) fora general spatially flat FRW space-time: 1) expandingin negative powers of m 2 k 2 and solving a recursive setof differential equations Baacke and Patzold, Phys. Rev. D 62 (2000)b) using a WKB ansatz similar to the boson case and theadiabatic expansion Cherkas and Kalashnikov, JCAP 0701(2007)
The divergent contributions to these expressions werecalculated for a general spatially flat FRW metric.Baacke and Patzold, Phys. Rev. D 62 (2000)
From T00 we find the boson and fermion contributions tothe vacuum energy density
To make the results finite we need to regularize the integrals. We use a simple 3-dim momentum cutoff regularization for the following reasons1. It is the only regularization scheme with a clear physical meaning: one discards the part of the momentum integral over those momenta where a different, yet unknown physics should appear.2. We apply this in a cosmological context where we have a preferred reference frame: the frame fixed by the CMB background or large scale matter distribution.3. As we have an unbroken SUSY, the cancelation of the flat-spacetime contributions takes place irrespective what regularization method we use.
We change the integration variable to the physical momentum p = k/a and introduce a cutoff of the order Λcut ~mPl. The leading terms yield N a 2 2 2 cut 1 ( cut ln cut ) 2 8 a 2 N cut a 2 2 a p 2 2 2 1 ( cut ln cut ) 2 24 a aClearly, we do not reproduce the usual vacuum equation of state.E.g., in the de Sitter background 1 pvac vac 3
It is convenient to introduce a free dimensionless cutoffparameter of order such that 3 2 2 cut mPl N Then, the vacuum energy 3 a2 8 G a 2is of the order H2 mPl2The factor 1/N is introduced to make the result independentof the number of species. A similar natural cutoff has beenrecently proposed in order to resolve the so called speciesproblem of black-hole entropy. Dvali and Solodukhin, arXiv:0806.3976 Dvali and Gomez PLB (2009)
Concluding remarkWe do not reproduce the vacuum energy-momentum tensor in the form Tvac vac g required by Lorentz invariance.One may argue that our result is an artifact of the 3-dimregularization which is not Lorentz covariant. However,even a Lorentz covariant approach (e.g., Schwinger -de Witt expansion) would give something like Tvac Rg R Where ∙∙∙ denote higher order terms in Riemann tensor,involving its contractions and covariant derivatives.