1. 6D Supergravity and the Cosmological Constant
A 6D Search for Self-Tuning
C.P. Burgess
with Y. Aghababaie, S. Parameswaran and F. Quevedo
hep-th/

2. Outline The Cosmological Constant Problem 6D Brane Worlds
Naturalness Issues
Self-tuning Proposals
6D Brane Worlds
Large Extra Dimensions in Salam-Sezgin Supergravity
The 4D Vacuum Energy
Brane-Bulk Classical Self-Tuning
UV Sensitive Bulk Quantum Effects
Outlook
08/04/2019
SUSY in the Desert

3. Outline The Cosmological Constant Problem 6D Brane Worlds
Naturalness Issues
Self-tuning Proposals
6D Brane Worlds
Large Extra Dimensions in Salam-Sezgin Supergravity
The 4D Vacuum Energy
Brane-Bulk Classical Self-Tuning
UV Sensitive Bulk Quantum Effects
Outlook
08/04/2019
SUSY in the Desert

4. Outline The Cosmological Constant Problem 6D Brane Worlds
Naturalness Issues
Self-tuning Proposals
6D Brane Worlds
Large Extra Dimensions in Salam-Sezgin Supergravity
The 4D Vacuum Energy
Brane-Bulk Classical Self-Tuning
UV Sensitive Bulk Quantum Effects
Outlook
08/04/2019
SUSY in the Desert

5. Outline The Cosmological Constant Problem 6D Brane Worlds
Naturalness Issues
Self-tuning Proposals
6D Brane Worlds
Large Extra Dimensions in Salam-Sezgin Supergravity
The 4D Vacuum Energy
Brane-Bulk Classical Self-Tuning
UV Sensitive Bulk Quantum Effects
Outlook
08/04/2019
SUSY in the Desert

6. The Cosmological Constant
Universal energy density appears to be presently dominated by matter with p/r < -0.7
Why Now?
Naturally understand why now if:
CC is parameterically r ~ (Mw2/Mp )4
Dark Matter has weak-interaction cross sections. (eg WIMPS)
Arkani-Hamed, Hall, Kolda & Murayama
08/04/2019
SUSY in the Desert

7. The Naturalness Problem
‘Relevant’ effective interactions are dominantly sensitive to the highest-energy scales whose integration can produce them.
Cosmological Constant: M = 10-3 eV
The CC Problem
Newton’s Constant: M = 1018 GeV
A success!
Higgs Mass: M = 102 GeV
The Heirarchy Problem
(in 4 dimensions)
08/04/2019
SUSY in the Desert

8. Two Parts to the Problem
Q1: At high energies, M:
Why is the relevant parameter small in the underlying microscopic theory at scale M?
Q2: At low energies, m:
Why does the parameter stay small as scales between M and m are integrated out?
In 4D supersymmetry or scale invariance can help, but only down to the scale at which they break.
Why is r1/4 ~ 10-3 eV if Dm > 100 GeV?
08/04/2019
SUSY in the Desert

9. Self-Tuning and the No-Go Theorem
Self-Tuning models try to arrange that the field equations for other fields, f, ensure that flat solutions exist to Einstein’s equations.
A generic problem (in 4D):
Either this equation fails or solutions to the f equation do not exist:
L = c (det g)1/2 exp[4 F(f)]
Weinberg
08/04/2019
SUSY in the Desert

10. Self-Tuning: One mo’ time…..
The effective theory above 10-3 eV need not be 4 dimensional:
In ADD† 6D model, 2 internal dimensions can be quite large:
Radius a can be as large 0.1 mm if 6D gravity scale near TeV scale: Mp ~ a Mw2
Tests of gravity require a no larger than ~ 0.1 mm.*
Corresponding KK energy is of order the cc: /a ~ Mw2/Mp ~ 10-3 eV
Try self-tuning in 6D Sugra
†Arkani-Hamad, Dvali & Dimopoulos
* Cullen & Perelstein; Hanhart, Philips, Reddy & Savage; Atwood et al
* Astrophysical bounds can be stronger.
08/04/2019
SUSY in the Desert

11. Outline The Cosmological Constant Problem 6D Brane Worlds
Naturalness Issues
Self-tuning Proposals
6D Brane Worlds
Large Extra Dimensions in Salam-Sezgin Supergravity
The 4D Vacuum Energy
Brane-Bulk Classical Self-Tuning
UV Sensitive Bulk Quantum Effects
Outlook
08/04/2019
SUSY in the Desert

12. N=2 6D Chiral Supergravity
Nishino & Sezgin
Salam & Sezgin
Randjbar-Daemi, Salam, Sezgin & Strathdee
6D Gravity + Tensor + Gauge + Hyper-multiplets.
Anomaly cancellation:
G = E6 × E7 × U1 Q = Sp(456,1)/[Sp(456) × Sp(1)]
‘Gauged’ supergravity gives dilaton potential.
v(0) = 2 g12
08/04/2019
SUSY in the Desert

13. Compactification Admits solution with constant dilaton:
Metric: Minkowski4 x S2, Unit magnetic flux: Fmn = f emn
4D N=1 Supersymmetry if flux is in U1 gauge group factor
08/04/2019
SUSY in the Desert

14. Supersymmetry Breaking
Supersymmetry is broken on brane at the brane scale.
Dmb ~ Mw
Bulk contains gravity plus superpartners, all of whom couple to branes with gravitational strength.
Supersymmetry breaking filters into the bulk, with 4D bulk multiplets split by
DmB ~ Mw2/Mp ~ 1/a
~ 10-3 eV
08/04/2019
SUSY in the Desert

15. Outline The Cosmological Constant Problem 6D Brane Worlds
Naturalness Issues
Self-tuning Proposals
6D Brane Worlds
Large Extra Dimensions in Salam-Sezgin Supergravity
The 4D Vacuum Energy
Brane-Bulk Classical Self-Tuning
UV Sensitive Bulk Quantum Effects
Outlook
08/04/2019
SUSY in the Desert

16. Potential Contributions to the CC
Step 0:
No bare 6D cc allowed by supersymmetry.
Step 1: Brane Modes
Integrate out brane modes exactly, generating T ~ Mw4
Step 2: Bulk Modes (Classical)
Classically integrate out all massive KK modes.
Corresponds to eliminating KK modes from the action using their classical field equations – Self – tuning solution can cancel the contribution of Step 1.
Step 3: Bulk Modes (Quantum)
Integrate out the quantum fluctuations of the bulk fields.
UV sensitive contributions are crucial!
08/04/2019
SUSY in the Desert

17. Self-Tuning I: The metric
Brane tensions cancel the bulk curvature once classical integration over the metric KK modes is performed.
Chen, Luty & Ponton
08/04/2019
SUSY in the Desert

18. Self-Tuning II: The dilaton
‡Arkani-Hamed, Dimopoulos, Kaloper & Sundrum;
Kachru, Schulz and Silverstein
Bulk contributions also cancel up to a total derivative once the classical integration over the dilaton KK modes is performed.
Result vanishes if dilaton is nonsingular everywhere and l = 0.
Does this share the singularities of 5D self-tuning models?‡
08/04/2019
SUSY in the Desert

19. Do Solutions Exist?
* Deser; Carroll & Guica; Navarro
Solutions to 6D equations with branes exist for which reff is zero.
Eg: solutions with two identical flat branes and constant dilaton.
Back-reaction removes wedge of angular width proportional to T.*
BUT: Requires magnetic flux not in U1, and couplings tuned to: g1/g = 1 – e.
08/04/2019
SUSY in the Desert

20. Bulk Quantum Corrections
Low-energy contributions just right: (Dm)4
What of ultraviolet-dangerous contributions?
eg: (Dm)2 Mp2 ~ Mw4 or (Dm)2 Mw2 ~ Mw2/a2
These arise from modes with E » 1/a, and so integrating them out gives local 6D interactions.
eg: (det g)1/2 Mw4 R ~ Mw (det g)1/2 Mw2 R2 ~ Mw2/a2
Mw4 R term renormalizes 6D Newton Constant.
In absence of branes, nonrenormalization theorems preclude higher-curvature terms giving corrections to reff for a supersymmetric compactification.
What happens with branes?
08/04/2019
SUSY in the Desert

21. A Toy Model
Sc – S0 = (divergent brane term) because Sc – S0 = 0 if J = 0.
08/04/2019
SUSY in the Desert

22. Outline The Cosmological Constant Problem 6D Brane Worlds
Naturalness Issues
Self-tuning Proposals
6D Brane Worlds
Large Extra Dimensions in Salam-Sezgin Supergravity
The 4D Vacuum Energy
Brane-Bulk Classical Self-Tuning
UV Sensitive Bulk Quantum Effects
Outlook
08/04/2019
SUSY in the Desert

23. Cosmic Six-Dimensional Virtues
6D supergravity with large extra dimensions helps with parts of the cc problem.
Supersymmetry forbids a bare 6D cc.
Particle superpartners are not observed because supersymmetry is badly broken on the branes, BUT integrating out brane particles only results in a large brane tension, localized in 6D.
Classically, brane tensions can cancel the bulk fields they produce in the extra dimensions. Untuned & singularity free?
The bulk SUSY-breaking scale is the right size for the 4D cc.
Can supersymmetry adequately protect the UV part of the integration over bulk modes?
08/04/2019
SUSY in the Desert

24. Observational Implications
If so, relating the cosmological constant scale to the heirarchy problem using large extra dimensions would imply many observational consequences:
Having extra dimensional radii at the cc scale implies deviations from Newton’s Law at submillimetre scales.
Bulk gravitational modes, and their superpartners, interact at LHC energies with weak-interaction cross sections.
Astrophysical and cosmological bounds which can already constrain large extra dimensions must be evaded, possibly with interesting and observable consequences.
08/04/2019
SUSY in the Desert

25. Many Open Issues
Do 6D self-tuned solutions exist which are everywhere nonsingular and for which no couplings are tuned?
Rugby ball requires special choice: g1/g = 1 – e.
Can the UV bulk quantum corrections be small enough?
Explicit calculations are possible with the rugby ball.
How are the extra dimensions stabilized?
Both electroweak and cc hierarchies rely on a » ls
Is the Dark Energy presently evolving?
Is contact with string theory possible?
Is Salam-Sezgin supergravity a low-energy string limit?
Other dimensions and connections to stringy scale invariance?
08/04/2019
SUSY in the Desert