1. Non-perturbative particle production from SUSY flat directions – a spoiler of delayed thermalisation?
Anders Basbøll, University of Aarhus
COSMO 2008: Madison, WI
based on: AB, D Maybury, F Riva and SM West:
Nonperturbative flat direction decay, Phys. Rev. D 76, (2007) and
AB: SUSY Flat Direction Decay – the prospect of particle production and preheating investigated in the unitary gauge, Phys. Rev. D (2008)

2. (Super)potential and flatness
Superpotential (Yukawa couplings same as in SM)
Scalar potential: F- and D-term
T: symmetry generators
Flatness: V=0
All F- and D-terms vanish
V=0 only for exact SUSY (unbroken) and no non-renormalisable terms
All ~300 flat directions of SUSY are broken by non-renormalisable terms of order n=4 to n=9

3. Flat direction evolution – from Dine-Randall-Thomas ArXiv:hep-ph/9507453
Flat direction (QLD, example)
Flatness easy: Colour, weak charge, hypercharge sum to zero
m: dimension
with canonical field
Equation of motion
- with Hubble friction

4. Evolution during and at end of inflation
Potential (n=k*m – k integer in Xk)
Last term: non-renormalisable SUSY-breaking
c of order unity. Positive sign of c necessary (50%chance? - depends on Kähler potential)
With H dominating, induced mass terms and self-couplings dominate over SUSY-breaking terms
Minimum at ^16 GeV?!
End of inflation: SUSY mass terms dominate
Oscillation around minimum
Potential has phase dependent part:
Baryon number if B-L not conserved by FD

5. Consequences of Flat Directions - Allahverdi, Mazumdar hep-ph/0603244v2 and others
FD induce masses to inflation decay products:
No preheating
Energy is stored in the flat direction.
Only when the FD decays – Reheating.
Reheating temperature 10^3-10^7 GEV
This avoids the (lack of) gravitino problem
FD itself a candidate for the Inflaton (LLE,UDD) Allahverdi et al. hep-ph/ v2
Olive+Peloso hep-ph/ :
Valid only if decay is nonperturbative SPOILED if FD decays rapidly - likely.

6. The framework Exitations of fields Lagrangian
Exitations must be defined, such that mixed kinetic terms are avoided!
Orthogonal transformation
to make U-term disappear:
”New” Lagrangian
Diagonalisation

7. Non-perturbative particle production
Conformal fields
Equation of motion
Mass matrix
Changing eigenvectors

8. Multi-field Bogolyubov - Nilles, Peloso and Sorbo ArXiv:0103202
Differential equations
generalised Bogolyubov coefficients (in GR: particle production from rapidly changing vacua)
Non-diagonal part -
not present in single
field case
Particle number
In our framework
Rapidly changing eigenSTATES give particles

9. LLE – in detail VEV's: generic feature: Flatness independent of phase
Covariant deriv.
Lagrangian
F, W: Hypercharge, Weak field strength tensors
Mixed kinetic terms – unphysical Goldstones
only time derivative of phase - therefore also only 0'th component of Gauge fields matter

10. Gauging Goldstones away - go to unitary gauge
U(1) (hypercharge) transformation
with
SU(2) (weak charge) transformation
P³: 3rd Pauli matrix
new VEV's (phase differences gone)

11. Diagonal Goldstones removed
still gives

12. All Goldstones removed

13. U-matrix U from Lagrangian
Partial integration (Ignoring surface terms)

14. Mass matrix
Notice: block diagonal – split in 2 sectors corresponding to off- [sector 2] and diagonal [sector 1] connection to VEV – generic feature

15. results – LLE and UDD (seperately)
Make transformation eliminating U.
J-matrix=0
No preheating!
UDD exactly the same.
3 phases, 2 gauged away.
J=0, no preheating

16. QLQLQLE – Fields Squarks with identical SU2-charge chosen
...and 12 fields with no VEV

17. Mass matrix eigenvalues
States 1,2 from diagonal, 3,4 from off-diagonal parts of SU(2) X U(1)
States 5-12 diagonal+off-diagonal SU(3)

18. J – Matrix: Particle production - from 1 flat direction!!!
Higgses mix with light states in both diagonal part of SU(2) X U(1)
and off-diagonal part of SU(2)

19. 2 Flat directions: UDD+LLE
Give VEV's to same fields as before!
Now 6 phases – only 4 diagonal generators.
However, it is 1 phase for LLE and 1 for UDD that survives.
U-matrix block diagonal: Fields and phase of LLE in one block. Fields and phase of UDD in another block.
J=0 – no preheating.
Very encouraging for the cosmological role of SUSY flat directions!

20. QLD+LLE – overlapping directions
QLD and LLE can co-exist. They can have VEV in the same field. A: relation between VEV's.
”Overlapping field” must have ”pythagoras” size
... and 6 fields with no VEV

21. Mass matrix Seperated sectors Sector 2 diagonal
Sector 1 phases (fields 6,7) have time dependent mass elements

22. Particle production! 2 terms giving 2 contributions to J NZ= nonzero
Notice: J_2 cannot balance this: lower part zero

23. To be done – and recent papers
How much different can VEV's be to get the effects of more directions -how much different from 0 is allowed for A [or log(A)]
Investigate which directions will get large VEV's - or make some kind of statistical analysis of probability.
Polarisation and numerics:
Gümrükçüoğlu et al. arxiv: v1
Conserved global charges (NEXT TALK?):
Allahverdi et al. arxiv: v2

24. Conclusion
SUSY Flat directions can have crucial influence on (p)reheating and offer a very nice explanation of the gravitino problem, baryogenesis and even offer a ”known” particle as a candidate for inflation.
Preheating serious threat to this.
This deserves further investigation
....and I'm looking for a job!