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Higher Derivative Scalars in Supergravity
Jean-Luc LehnersMax Planck Institute for Gravitational Physics
Albert Einstein Institute
Based on work with Michael Köhn and Burt Ovrut
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MotivationAssume N =1 supersymmetry is a good symmetry at an early phaseAim to construct a corresponding effective theory for scalar fieldsCan be applied to inflation, ekpyrosis, ...
Extension of 1012.3748,1103.0003 (Khoury, JLL, Ovrut) 1109.0293 (Baumann, Green)
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General FeaturesMultiple scalars, as a chiral multiplet contains two real scalarsNatural setting for some curvaton models of inflation and entropic mechanism in ekpyrosisSusy constrains scalar field actions
e.g. consequences for non-gaussianity
New effects from eliminating auxiliary fields
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ConstructionChiral multiplet
Spin ½ Auxiliary field
Superspace
Complex scalar
Kähler potential
e.g.
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First concentrate on where
Rewrite
Strategy: construct first - everything else will follow easily! For need two more fields and two more derivatives/four superspace derivatives since
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Only two “clean” possibilities (want not )
chiral integralTo go to supergravity integrate over curved superspace and use curved chiral projector
contains Ricci scalar
and
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Includes
Second scalar not of P(X) form
Interesting – modifies gravity sector too!
More worrying – Auxiliary field not auxiliary anymore!
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Focus on
which equals
- Scalar action- No new coupling to Ricci scalar- No kinetic term for auxiliary field F- All terms involving auxiliary fields of supergravity multiplet also involve fermions
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P(X) in supergravityAll lower components of contain fermions!
Hence now easy to construct sugra extension of any term that contains as a factor:To get use
but now with
In this way one can build up P(X,f) as a Taylor series
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Ghost CondensateWhen the kinetic function P(X) has a minimum, develop a time-dependent vev for : f
Typical action:
Minimum corresponds to dS spacePerturbations around minimum allow stable violations of NEC for short periods of time
Can be used to model dark energy or non-singular bounces
X
P(X)
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Ghost condensate in supergravity
Omitting the second real scalar, up to quadratic order in fermions action becomes:
Vacuum breaks Lorentz invariance, manifested by wrong sign spatial gradient term for goldstinoMixed mass term for gravitino-goldstino super-Higgs?
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Super-HiggsSusy transformation
Usual F-term breaking: DW≠0, A=0 Gravitino eats goldstino and becomes massive
Here W=0, but √2A = f = t, hence goldstino also shifts by a constant:
However, there is no superpotential and hence no mass term for the gravitino - so what happens?
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Redefine gravitino to get rid of mixed mass term:
Action
- Gravitino remains massless!- Goldstino remains present, otherwise degrees of
freedom would be lost- Goldstino kinetic term has a different
normalization This is the indication that susy is really broken
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Eliminating the auxiliary field F
Add only X - equation of motion for F is
Equation for F is cubicraises interesting question as to how one
defines the quantum theorythere are now new solutions that correspond
to new branches of the theory
2
coefficient of X2
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Perturbing around usual solution
X term contributes
For small c2, solve
Hence a new, higher-derivative kinetic term modifies the potential
2
Corrections to kinetic term Corrections to
the potential
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Example: W=A
Leads to a potentialof the form
Corrections go as
For c2>0 turns a valleyinto a mexican hat!
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New Branch of Supergravity
Turn superpotential off: W=0
Then eq for F reads
Solved not only by F=0, but also by
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Without fermions, whole action becomes
- Ordinary kinetic term has vanished- A potential (depending on the Kähler
potential) has appeared
Scale of potential: Mass of f
Not continuously connected to ordinary branch
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Dynamics: for
the action becomes
In a θ~ x background, need c2>0 so thatρisn’t a ghostThen the potential is positive, which is unusual for supergravity(the size of the potential is limited by the vev of θ)
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Summary• Break susy with ghost condensate
Unusual way of breaking supersymmetry: the gravitino remains massless, and a kinetic term for the “goldstino” remains present
• Auxiliary field F leads to new effectso Solutions that are close to the standard solution for F
imply that the new higher-derivative kinetic terms correct both the kinetic terms and the potential
o New solutions for F lead to entirely new branches of the theory. Their physical significance is not clear yet!