Shamit Kachru- A comment on gravitational waves and the scale of supersymmetry breaking

8/3/2019 Shamit Kachru- A comment on gravitational waves and the scale of supersymmetry breaking

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A comment on gravitational waves and

the scale of supersymmetry breaking

Based on work with T. He and A. Westphal

Shamit Kachru (KITP & Stanford)

(arXiv : 1003.4265)

Friday, May 21, 2010


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I. Introduction

There are many reasons to be particularly excited about

our prospects for learning new facts about fundamentalphysics in the next few years. Two of the biggest (and most

expensive!) such reasons are:



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One will teach us about physics at the weak scale, and theother about the inflationary cosmology which preceeded

the big bang (reheating).

It is of course quite clear that lessons coming fromcosmological distance scales have significant impact on our

thinking about short distance physics, and vice versa.Friday, May 21, 2010


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So: what are the possible new lessons, or interactions,between cosmology and particle theory that could

intertwine the LHC and Planck results?

* The leading candidate for stabilizing the physics of the

weak scale, in many peoples minds, is supersymmetry. Andwhere there is supersymmetry, there is (probably!)supergravity.

The SUGRA potential takes the form:

V= eK

i

|DiW|2 3

|W|2

M2

P



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And in a nearly flat-space vacuum, of the sort which weinhabit, this implies that the dominant F-component (the

scale of SUSY-breaking) is related to the expectation valueof the superpotential in vacuum.

Probably the single most important observable in a realisticsupersymmetric theory is the scale of SUSY breaking.

Equivalently, it is the gravitino mass:

m2

3/2 eK |W|

2

M4P



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It is often more natural to talk in terms of the Hubbleparameter

V = 3M2PH2

Near term observation of primordial gravitational waveswould pin this at 1014 GeV.

So to be maximally reductive, the next few years mayteach us two fundamental numbers:

the scale of supersymmetry breaking, and the scale ofinflation.

In common models, we would think that the scale of

supersymmetry breaking is going to be:Friday, May 21, 2010


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m3/2 TeV gravity mediation

m3/2


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If this so-called KL bound is correct, then detection ofprimordial gravitational waves would rule out SUSY at the

LHC, and vice versa.

My purpose in this talk is to explain the rough logic that ledKallosh and Linde to propose this bound, and to explain

why we think that despite their arguments it is very clearlyNOT a universal (or even generic) bound on the behavior

of inflation models in string theory.

Although string theory is an intricate subject, in fact thebasic point that leads to the new constraint is just the fact

that there are extra dimensions of space-time.



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II. String Theory, Extra Dimensions, and all that...

The supersymmetric string theories famously live in tendimensions. Realistic models are made by curling up six of

these on, say, a Calabi-Yau manifold:

By Yaus theorem, there is a moduli space of Ricci-flatmetrics on the Calabi-Yau space. At the very least, this is aone parameter family of metrics, controlled by the volume

of the Calabi-Yau.Friday, May 21, 2010


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In terms of the 4D effective field theory, this means that inaddition to whatever fields one has to represent the

Standard Model particles, dark matter, and the inflaton, one

has a (complexified) volume modulus:

The complexification is expected because 4D chiralsuperfields contain complex scalars, and in string theory,

the imaginary component is a period of a Ramond-Ramondgauge field.

In the simplest Calabi-Yau compactifications, such fieldsenjoy shift symmetries and give rise to axions in the low-

energy theory. The partner of the volume modulus is then

such an axion.

T =J J+ i

C4 = + ia



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What new physics can we expect of a theory with extradimensions and a volume modulus?

* In dimensional reduction, to go to 4D Einstein frame, onemust re-scale the metric by a Weyl transformation that

depends on the volume. This rescales energies in the 4D

effective potential by a factor:V

1

3

Of course this must arise naturally in the 4D effectivesupergravity, and it does:

K = 3ln(T + T)



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Now, you might think other dependence on the volume inspecific terms in V in 10D Einstein frame could complicate

matters, but locality in the extra dimensions implies that nosource of energy can grow faster than the volume itself as

one expands the extra dimensions.

The Weyl re-scaling, on the other hand, falls off as inversevolume squared.

Therefore, all terms in the potential in 4D Einstein framevanish if one allows the extra dimensions to decompactify.

This has some obvious implications:



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* Any positive energy configuration has a possibility ofrelaxing away its energy by rolling to infinity, unless it is

meta-stabilized.

100 150 200 250

1

2

3

4

V

* While it has been argued that metastable supersymmetrybreaking configurations with positive vacuum energy existin this context, any additional energy on top of the meta-

stable vacuum (e.g. associated with inflation) creates a

danger of destabilizing the vacuum.Friday, May 21, 2010


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Let us put some more detail into this discussion, and seewhat kind of bounds we get.

III. Bounds on inflation in stabilized IIB orientifolds?

A popular class of models work in the following way.One compactifies IIB string theory on a Calabi-Yau(orientifold), and stabilizes almost all of the moduli

guaranteed by Yaus theorem using magnetic flux in the

extra dimensions. Below the mass scale set by thisflux-induced potential, the effective theory has a single

modulus, T. The shift symmetry of the axion in T,prevents it from receiving a flux-induced mass.



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The axion shift symmetry is broken (as in the case of theQCD axion) by strong dynamics, or by D-brane instanton

effects. The resulting effective theory has:

W = W0 + AeaT

.

There is a supersymmetric AdS vacuum, whose depth is:

|VAdS| = 3eK|W0|

2.

Supersymmetry-breaking dynamics, occurring atexponentially low scales, cancels this negative vacuum

energy and yields a metastable false vacuum.Friday, May 21, 2010


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100 150 200 250

-2

-1

1

2

V

The supersymmetry breaking dynamics, in terms of this

potential, can be thought of as adding an exponentiallysmall term which depends on some inverse power of the

volume (like all energies do, in 4D Einstein frame).

The important point is that when the energy of themetastable SUSY-breaking can be treated as a smallperturbation of the AdS vacuum, the barrier heightpreventing decay of the new de Sitter vacuum is:



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VB |VAdS|.

But since the SUSY-breaking F-term had to precisely(almost) cancel the AdS vacuum energy, we get a relation

between the barrier height and the gravitino mass!

Finally, let us add our inflaton. The energy in the inflatonalso depends inversely on the volume, and to prevent adistortion of the potential that creates an instability to

decompactify, we need:

100 150 200 250

1

2

3

4

V

eK|DW|2 O(10)VB.



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But that is precisely the energy density contained in theinflaton field! Therefore,

H

2

=

V

3 eK

|DW|2

O(10)VB O(10)|VAdS|,

which is a relation between the Hubble scale of inflation,and the gravitino mass. This led Kallosh & Linde to

postulate a bound on generic string theory inflation:

This bound is not related to inflation itself, but rather to adesire to avoid decompactification during inflation intheories with extra dimensions. Stringy and other extra-

dimensional UV completions can introduce extra

constraints on inflation.

H m3/2



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IV. Evading the KL bound

While the logic that has been described is clearly correct

given the assumptions, there is an obvious assumption thatone can relax to evade this bound. Our goal will be to

exhibit models that have all of the same generic SUGRAstructure that was assumed in deriving the bound, but

which give

H >> m3/2



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Issues I will sweep under the rug:

* The problem is one which is evident in a wide class ofstring-inspired supergravities. We will work in that

frameworkwithout deriving the model from string theory.

* The problem as stated has nothing to do withobservational facts about the precise model of inflationwhich governed our world. We will try to exhibit easy

violations of the bound without requiring that the inflationmodel be realistic in its observables.

I believe that because the essence of the problem is as I described it, omitting realism of the inflationary perturbationspectrum & a string derivation is actually fine. Our mechanism could presumably be generalized to realistic models; we

worked instead in the most obvious setting that could display gross violations of the proposed bound, with models of large-field chaotic inflation.



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The clear strategy to pursue is the following:

* The depth of the AdS vacuum that is being lifted by

supersymmetry breaking & inflationary energy density,determines the barrier height.

* For models where SUSY is relevant to the hierarchy

problem, this gives very low barrier height, because thedepth of the AdS vacuum (the expectation value of W) is

fixed by the gravitino mass.

This is easily fixed. Intuitively, the flux-induced parameter

0 is setting the scales. But who says one cannotgeneralize this structure, so that one has:



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W = W0() + AeaT

varies adiabatically as inflaton rolls

For instance, one can design a model where:

1 2 5 10 20 501015

1012

109

106

0.001

1

W

Figure 2: The vev of the superpotential |W| = |W(, X(), T())| plotted as a functionof the inflaton with X and T adiabatically tracking their instantaneous minima.



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With this obvious idea in mind, lets try to build a simplemodel. Well see that simple models exist, but there are

generic supergravity issues that arise with large-fieldinflation which make them less simple than you might have

thought.

For instance, ignoring the additional volume modulus, wenote that a large-field model in the spirit of Lindes chaoticinflation, involving just the inflaton chiral multiplet, is very

hard to get in N=1 supergravity.

This is because with a polynomial superpotential (and ashift-symmetric K), at large fields

>> MP|DW

|

2


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So, the supergravity potential slopes downward at largefields, and goes negative.

If we want to avoid futzing around with the field range(which is against the spirit of large-field inflation), weshould probably include a second field. Kawasaki, Yamaguchi,

Yanagida

A simple class of large-field models with a 2-fieldinflationary sector, which circumvents this problem and also

allows large violations of the KL bound, is:

K =

1

2(

+

)2

+ XX (X

X)

2

3log(T+T)

W = W0 g(X) + f(X)n + eaT

with : g(X) = 1 +O(X) and f(X) = b + X+O(X2)

O(1), sign important

generic functions, we just wroteout the terms that matter

Any function of real partis fine here



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Let us explain how and why anyone would write downsuch an ugly model.

1. In the final vacuum when X and the inflaton vanish, thisreduces to the low-energy effective theory that governs

many of the string models of moduli stabilization.

2. In the Kahler potential, we have assumed that there is ashift symmetry for :

= + i

It is broken by alpha, which can naturally be very small.Friday, May 21, 2010


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3. The superpotential can be natural if there is anapproximate R-symmetry under which X is neutral and theinflaton carries charge 2/n. The unknown functions of X

can be arbitrary, only the terms we explicitly wrote outever matter in the dynamics.

Why do the single correction in K and the few terms inf(X) and g(X) that we wrote out, matter?

K =

1

2 (+ )2

+ XX (XX)2

3 log(T+ T)

W = W0 g(X) + f(X)n + eaT

with : g(X) = 1 +O(X) and f(X) = b + X+O(X2)



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1. The constant in g is necessary to get the correctstructure after inflation.

2. The linear term in f guarantees that the X F-termdominates the energy during inflation. This avoids the

problem that we noted with large-field inflation in

supergravity (negativity of the potential at large fields!).

3. The constant term in f guarantees that at large fields, thevalue of the effective seen by the T-modulus does

indeed vary adiabatically with the inflaton.

0

4. Given the large F-term for X during inflation, the quarticcorrection in K keeps X stabilized at the origin.



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V. Dynamics of these modelsA. Dynamics of the volume modulus T during inflation

So, with reasonable choices for all of the parameters,this class of models gives chaotic inflation with

V() 2nThese models are close to beingruled out for n>2...

The dominant energy source is

|FX |2 = eK|DXW|

2 1/3

This is one of the canonical forms assumed also for theterms induced by SUSY-breaking starting in the final AdS

vacuum. So the dynamics of T is indeed to adiabaticallyFriday, May 21, 2010


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track an instantaneous vacuum:

Tmin() T0 : DTW()|T0

= 0

We can check validity of SUGRA and the single instantonapproximation:

DTW() 0 eaTmin()

1

1 + 23aTmin()

W0,eff.()

where : W0,eff.() W0 + (b + X)n .

So demanding

|W0,eff.()| |W0 + (b+X)n| 1 || = |Im| < 60

does guarantee that the instantaneous minimum will



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occur where the volume is large and the instanton action ishealthy.

Then, as long as one satisfies an avoidance ofoveruplifting with the given instantaneous effective value

of , there will be a nice minimum for T. This gives

the constraint:

0

|F2| + |F2X|

3eK/2|

W

| O(1)

which is indeed just the KL bound for the instantaneousvalue of W.



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B. Decompactification constraints

i. At large inflaton vevs

When the inflaton vev is large, we can approximate:

|F2

| + |F2X |3eK/2|W|

|FX |3eK/2|W|

eK/2n3eK/2bn

13b

.

b was just some parameter of O(1) in the theory. Thisshows that there is no problem at large fields.

If you do not trust me, here is a plot:

0 10 20 30 40 50

1.5

2.0

2.5

3.0

FX2

W2



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ii. At small inflaton vevs

In this region, instead, the F-term of the inflaton itself

dominates. The constraint to avoid decompactificationbecomes:

|F|3e

K/2

|W| O(1).

On the other hand, evaluating the functional forms gives:

|F|3eK/2|W| n

bn1

3(bn +W0)

1

So there is indeed a danger of losing the minimum during

the exit from large-field inflation.Friday, May 21, 2010


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To avoid disaster, the function of the inflaton that appearsthere needs to satisfy:

max

|F|

3eK/2|W|

=

n 13

b

W0(n 1)

1/n

O(1).

This requires that alpha should not be much larger than

W0.

This is theoretically fine, but will lead to unrealistic models.

Since this is a constraint arising from the behavior at smallfields, while our inflation occurs at large fields, we wouldguess that by slightly complicating the exit (as in hybridinflation), one could make completely realistic theories.



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VI. Horse-trading : achievable hierarchies vs realisticinflation

So, finally, let us see how these models fare by computingthe density perturbations and the spectral index, and seeing

whether we can match data while getting parametrically

large hierarchies. At large fields:

V |FX |2 22n

and the density fluctuations are given by:

=

1

1502V

=60



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In reality this is measured, but lets pretend its a freeparameter and derive relations between observables.

Computing the vev of the inflaton at the 60th e-folding andplugging in, we can find the relation:

=

103n

n+160 ,

Plugging this into the decompactification constraint atsmall fields:

max

|F|

3eK/2|W|

=

n 13

b

W0(n 1)

1/n

O(1).

one finds:(n1) 10

3bn

n+160 W0(n 1)1/n

23,


23

nn+1W ( 1)


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Finally, putting in the value of that is consistent withintermediate scale SUSY-breaking, we get:

W0

23

n1

n+160 W0(n 1)103bn

.

2, 103

3, 105

5, 107

8, 1010

10, 1013

1.0 10.05.02.0 3.01.5 7.0

1011

108

105

0.01

Minn

Figure 3: Minimum value of n in the inflaton potential V() 2n necessary toachieve a given /, at fixed W0 = 1015, satisfying the no-decompactification-constraint eq. (3.20). Points are labelled in the format (n,O(|Wi|/|W0|)), where Wi =

W(60, X(60), T(60)) is the initial superpotential (and O(|x|) here denotes the order ofmagnitude of |x|).

i.e.


If instead we fix the density perturbations to come out


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If instead we fix the density perturbations to come outcorrectly, we obtain a relation between and n:W0

W0 103nb

23

n1n

n+160 (n 1)

.

2,103

3,105

5,107

7,1010

10,1013

1.0 10.05.02.0 3.01.5 7.0

1017

1014

1011

108

Minn

W0

Figure 4: Minimum value ofn in the inflaton potential V() 2n necessary to achieve agiven post-inflationary vacuum VEV of the superpotential W0, at fixed / = 2105,satisfying the no-decompactification-constraint eq. (3.20). Points are labelled in the format

(n,O(|Wi|/|W0|)), where Wi =

W(60, X(60), T(60))

is the initial superpotential (and

O(|x|) here denotes the order of magnitude of |x|).



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Since observation requires n < 2, this model cannot be fullyrealistic. Even with n < 2, it does have

H >> m3/2

but would certainly not have low enough supersymmetry

breaking scale to explain the gauge hierarchy.

Of course, our goal here was NOT to write a fully realistictheory, but to illustrate that the tension between high-scale

inflation and low-energy supersymmetry in string theory isnot at all a generic feature. Our analysis was confined to

supergravities containing the basic ingredients used inproving the KL bound.



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The obvious things to do, to make the case moreconvincing, would be:

* Write down models which are fully realistic, at the levelof supergravity with a T-modulus. This can presumably bedone by slightly complicating the exit from inflation as in

hybrid models, since our main constraints (that led toconflict with data) actually came from the small-fieldregion.

* Embed high-scale inflation with low-energysupersymmetry (with or without a realistic set of

inflationary observables) in full string compactifications.

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Shamit Kachru- A comment on gravitational waves and the scale of supersymmetry breaking