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

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)

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

ConstructionChiral 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 integralTo 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 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

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)

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?

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?

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 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

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 potentialof the form

Corrections go as

For c2>0 turns a valleyinto 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 f

Not continuously connected to ordinary branch

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)

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!