Thermal production of gravitinos

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1 Thermal production of gravitinos Slava Rychkov Scuola Normale Superiore & INFN, Pisa Università di Padova April hep-ph/ with Alessandro Strumia (to appear in PhysRevD)

2 Outline Gravitino in Cosmology Physics of Gravitino Production at High T Applications

3 Gravitino basics SUSY predicts new particles a) superpartners of SM particles (sparticles) b) gravitino. Gravitino ψ µ couples to particles/sparticles via supercurrent S µ with gravitational strength, cannot be seen at colliders: L int = 1 2M Pl ψ µ S µ However: will be produced at high T in the Early Universe (more precisely, during/after reheating)

4 Reheating Reminder Slow-roll inflation ends when inflaton φ reaches minimum of its potential V (φ) and starts coherent oscillations Oscillation amplitude decreases while inflaton φ decays into SM particles and thermalizes them Reheating temperature T RH temperature when radiation density of SM starts dominating the expansion of the Universe (transition to radiation dominated FRW). T RH is the maximal observable temperature of the Universe: prior history has been washed out by entropy released during the inflaton decay.

5 Gravitino Cosmology In SUSY world, a SUSY version of SM (like MSSM) gets thermalized, and gravitinos start being produced. Gravitino overproduction has to be watched: If LSP, has to satisfy Ω 3/2 < Ω observed DM 20% If not LSP, decay during or after BBN, with damaging consequences Gravitino production rate estimate: γ = dn 3/2 dt = γ 0 T 6 M 2 Pl Efficient production during/immediately after reheating, when T maximal The energy density of gravitinos after reheating: Ω 3/2 Ω observed DM 1.5γ 0 ( m3/2 100 GeV Today s problem: compute γ 0. ) ( T RH GeV )

6 Gravitino production I. Scatterings Interaction Lagrangian L int = 1 2M Pl ψ µ S µ Super-QCD supercurrent S µ SQCD = (gluon) (gluino)+(quark) (squark). There are 10 SQCD 2 2 processes giving gravitino production: IR-divergent (with gluon t-channel exchange). E.g.: gluon+gluino gluon+gravitino IR-convergent. E.g.: gluon+gluon gluino+gravitino

7 Best result up to now was [Bolz,Brandenburg,Buchmuller 2000] γ scattering = γ 0 T 6 M 2 Pl γ 0 = 0.5g 2 3 log 1.2 g M 2 gluino 3m 2 3/2 + O(g 4 3 ) Here g 3 is SU(3) gauge coupling, g at T GeV. Comments M 2 gluino /m2 3/2 term: related to goldstino ( longitudinal massive gravitino). Dominates for m 3/2 M gluino log g 3 dependence: related to how the IR divergence is regulated by gluon thermal mass m gluon g 3 T. (Debye screening of electric fields in plasma)

8 Gravitino production II. Decays We identify an important subclass of O(g3 4 ) corrections. Let s compare typical production rate due to scattering and decay, keeping track of phase space factors of π. γ scat 1 π 4σ T 6 g2 3 π 5 T 6 M 2 Pl γ decay 1 π 2Γ T 3 g4 3 T 6 π 3 M 2 Pl = O(g 2 3 π2 ) γ scat where we used Γ m T Γ rest, Γ rest m3 πm Pl, m g 3 T γ decay, although formally higher order, may be sizeable. Needs to be included.

9 Our result NB: Once γ decay is added, further contributions are expected to be suppressed by powers of g/π (usual expansion parameter in field theory at finite T ). Perturbative expansion does not fail, simply some new effect appears only at higher order.

10 Calculation ingredients Gravitino and goldstino couplings - equivalence theorem Dispersion relations at finite temperature - thermal spectral densities Avoiding double counting - subtracted scattering rates

11 Goldstino and Super-Higgs mechanism Phenomenology assumes existence of heavy hidden sector where SUSY is broken spontaneously. At E M mess L low energy = L visible + L soft-susy-breaking + i χ/ χ + 1 2F χ µ S µ visible (where F is a SUSY-breaking vev in hidden sector, m soft F /M mess ) After coupling to SUGRA, gravitino eats goldstino χ and becomes massive (super-higgs mechanism [Deser,Zumino 1977]): Ψ µ = ψ µ 2 µ χ m 3m 3/2 = F 3/2 3MPl = in gravity-mediation models m 3/2 m soft = in gauge-mediation m 3/2 m soft possible.

12 Equivalence theorem Although goldstino is the longitudinal component of massive gravitino, at energies E m 3/2 gravitino and goldstino do not mix = can study their production independently, and in the massless approximation: γ(ψ) γ(ψ) + γ(χ), L int = 1 2M Pl ψ µ S µ vis + 1 2F χ µ S µ vis Goldstino has dim-6 coupling. Naively could expect production rate γ(χ) T 8 (wrong) F 2 This is wrong because Goldstino must decouple in the limit when SUSY breaking goes to zero. In fact, µ S µ visible m soft and thus [Leigh, Rattazzi 1995], [Ellis et al 1995] γ(χ) m2 soft T 6 F 2 m2 soft m 2 γ(ψ) 3/2

13 Thermal effects Particles in thermal plasma get masses m thermal gt (g any coupling). Reminder : Debye screening of electric field in QED. Electric potential φ creates unequal chemical potentials µ = ±eφ for e ± = induced charge density ( ) µ ρ ind = e ( ) 1 n e + n e = 3 e T 3 = 1 T 3 e2 T 2 φ This induced density gives mass term in the Poisson equation: φ + m 2 Debye φ = Qδ(r), m Debye = 1 3 et. = Static electric field decays away exponentially on scale m 1 Debye.

14 Analogous kinetic-theory calculation can be done applying timedependent EM fields to the plasma. = modified dispersion relation for EM wave propagation. NB: longitudinal waves ( E k) can propagate at finite T.

15 Equivalently : can find modified dispersion by computing Compton scattering corrections to photon propagator in plasma: In thermal field theory formalism, these are computed as 1-loop corrections to photon self-energy where we have to use finite-t propagators S F (p) = (/p + m) ( 1 p 2 m 2 + iε + i2πn F (p 0 )δ(p 2 m 2 ) with corrected imaginary parts to account for presence of particles in the thermal bath. )

16 Resumming these 1-loop corrections, we get photon propagator at finite temperature: D µν = i=t,l P i µν p 2 Π i (ω, k) + iε where P T,L µν are transverse and longitudinal projectors. At T = 0 Lorentz invariance fixes Π T = Π L, however at finite-t Lorentz invariance is broken by thermal bath = more general structure. Propagator poles give dispersion relations. Self-energies Π T,L also have imaginary parts for ω/k < 1 related to Landau damping (energy transfer between EM waves and electrons in the plasma) = continuum spectral densities below the light cone. ρ T,L (ω, k) = Im 1 p 2 Π T,L + iε = Z δ(ω ω T,L(k)) + ρ cont (ω, k)

17 Spectral densities parton densities, i.e. relevant photon distribution becomes d 3 k 2ω n B(ω)dω d3 k 2ω ρ(ω, k)n B(ω)dω Spectral densities satisfy sum rules meaning essentially that total number of photons remain the same.

18 Summary of thermal effects Here we discussed photon in QED plasma, but similar effects exist for electrons and other particles in other plasmas. In particular in QCD and in Super-QCD (with different values of thermal masses). Modified dispersion relation - screening of IR divergences Modified quasiparticle distribution a là parton distribution functions in QCD Spectral densities (imaginary parts of 1-loop resummed propagators) accurately encode both effects.

19 Gravitino production Come back to our problem of gravitino production from Super-QCD plasma. If we don t take finite-t effects on particle propagation into account, the leading effect is due to 2 2 scatterings like gluon+gluino gluon+gravitino: However, these give IR divergent production rate γ scat g 3 2 T 6 M 2 Pl log T µ where µ is the IR regulator. Can guess µ m Debye g 3 T. If we want to do better than this estimate, have to use the full thermal gluon propagator.

20 γ Im Π This useful piece of formalism relates gravitino production rate to imaginary part of its propagator: γ = ṅ 3/2 = d 3 k (2π) 3 E n F (E)Im Π(p) Π = 1 4MPl 2 Tr[Πµν 3/2 S ν (p)s µ ( p) T ] (Π 3/2 µν = 1 2 γ µ/pγ ν /pη µν is the tensor used to sum over gravitino polarizations) γ Im NB:True to all orders in the plasma couplings; to leading order in 1 M Pl.

21 Cutting rules Thermal field theory cutting rules allow to connect γ ImΠ formalism with direct calculation of production rate via scatterings. = 0 (no phase space) Various cuts of 2-loop diagrams correspond to terms in scattering amplitude squared. self-energy corrections describe squares of individual diagrams: = 2 While 1PI irreducible diagrams correspond to cross terms: =

22 Resummation: Decay diagram Decay diagram, has to be computed from finite temperature propagators: = Decay diagram overcounts some contributions which are part of γ scattering. These are all individual 2 2 diagrams squared. E.g. gluon+gluino gluon+gravitino is the sum of 4 diagrams: λ B s λ Ψ λ B t g Ψ λ λ B u B x λ Ψ g g g g g g g γ scattering B s + B t + B u + B x 2

23 Subtracted scattering rate It turns out that terms B s,t,u 2 are all contained in Decay diagram. Thus we define subtracted scattering rate γ subtracted B s + B t + B u + B x 2 B s 2 B t 2 B u 2 So defined γ sub is IR-finite! term B t 2 is subtracted away. This is because the only IR-divergent = Removing double counting also removes IR divergence. Full scattering rate is defined as γ = γ Decay + γ subtracted

24 Result and applications Gravitino production rate coefficient γ 0 as a function of SQCD coupling: γ = dn 3/2 dt = γ 0 T 6 M 2 Pl 1 + M 2 gluino 3m 2 3/2 γ 0

25 Bounds for gravitino Dark Matter and Reheating temperature Reheating temperature in GeV m 1/2 = 150 GeV m 1/2 = 1 TeV Problematic within thermal leptogenesis Gravitino mass in GeV Contours show regions of (m 3/2, T RH ) plane where gravitino abundance equals observed DM abundance (3σ levels). Shaded region is problematic for standard thermal leptogenesis.

26 Conclusions Most accurate to date computation of gravitino abundance in post-inflationary universe Main contributions: scatterings and decays in Super-QCD plasma Also included: O(g 2 2 ), O(g2 Y ), O(λ2 top ) effects. Stronger constraints on coexistence of Gravitino DM + Thermal Leptogenesis