Higher Derivative Scalars in Supergravity Jean-Luc Lehners Max Planck Institute for Gravitational Physics Albert Einstein Institute Based on work with Michael Köhn and Burt Ovrut

Motivation Assume N =1 supersymmetry is a good symmetry at an early phase Aim to construct a corresponding effective theory for scalar fields Can be applied to inflation, ekpyrosis,... Extension of , (Khoury, JLL, Ovrut) (Baumann, Green)

General Features Multiple scalars, as a chiral multiplet contains two real scalars Natural setting for some curvaton models of inflation and entropic mechanism in ekpyrosis Susy constrains scalar field actions e.g. consequences for non-gaussianity New effects from eliminating auxiliary fields

Construction Chiral multiplet Spin ½ Auxiliary field Superspace Complex scalar Kähler potential e.g.

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

Only two “clean” possibilities (want not ) chiral integral To go to supergravity integrate over curved superspace and use curved chiral projector contains Ricci scalar and

Includes Second scalar not of P(X) form Interesting – modifies gravity sector too! More worrying – Auxiliary field not auxiliary anymore!

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

P(X) in supergravity All 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,  ) as a Taylor series

Ghost Condensate When the kinetic function P(X) has a minimum, develop a time- dependent vev for  Typical action: Minimum corresponds to dS space Perturbations 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)

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 goldstino Mixed mass term for gravitino-goldstino super-Higgs?

Super-Higgs Susy transformation Usual F-term breaking: DW≠0, A=0 Gravitino eats goldstino and becomes massive Here W=0, but √2A =  = t, hence goldstino also shifts by a constant: However, there is no superpotential and hence no mass term for the gravitino - so what happens?

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

Eliminating the auxiliary field F Add only X - equation of motion for F is Equation for F is cubic raises interesting question as to how one defines the quantum theory there are now new solutions that correspond to new branches of the theory 2 coefficient of X 2

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

Example: W=A Leads to a potential of the form Corrections go as For c2>0 turns a valley into a mexican hat!

New Branch of Supergravity Turn superpotential off: W=0 Then eq for F reads Solved not only by F=0, but also by

Without fermions, whole action becomes - Ordinary kinetic term has vanished -A potential (depending on the Kähler potential) has appeared Scale of potential: Mass of  Not continuously connected to ordinary branch

Dynamics: for the action becomes In a θ~ x background, need c2>0 so thatρisn’t a ghost Then the potential is positive, which is unusual for supergravity (the size of the potential is limited by the vev of θ)

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 effects o 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!