Phenomenology of M-theory
compactifications on G2 manifolds
Bobby Acharya, KB, Gordon Kane, Piyush Kumar and Jing Shao,
hep-th/0701034,
B. Acharya, KB, G. Kane, P. Kumar and Diana Vaman
hep-th/0606262, Phys. Rev. Lett. 2006
and
B. Acharya, KB, P. Grajek, G. Kane, P. Kumar, and
Jing Shao - in progress
Konstantin Bobkov
MCTP, May 3, 2007
Outline
• Overview and summary of previous results
• Computation of soft SUSY breaking terms
• Electroweak symmetry breaking
• Precision gauge coupling unification
• LHC phenomenology
• Conclusions and future work
M-theory compactifications without flux
• All moduli are stabilized by the potential generated
by the strong gauge dynamics
• Supersymmetry is broken spontaneously in a unique
dS vacuum
• M Planck is the only dimensionful input parameter.
Generically ~30% of solutions give m3 2 ~ O(0.1  10)TeV
Hence – true solution to the hierarchy problem
• When the tree-level CC is set to zero for generic
compactifications with >100 moduli m3 2  O(100)TeV !
Overview of the model
• The full non-perturbative superpotential is
W  A1 e
a ib1 f
 A2e
ib2 f
where the gauge kinetic function
N
f   N i zi
i 1


~ 12
• Introduce an effective meson field   2QQ  0ei
2
bk 
ck
dual Coxeter number
SU(N): ck=N
SO(2N): ck=2N-2
E8:
ck=30
• For SU ( N c ) and SU (Q) hidden sector gauge
groups:
2
2
2
P  Nc 1
b1 
a
b2 
P ,
P , where
Q ,
• An N-parameter family of Kahler potentials
consistent with G2 holonomy and known to describe
accurately some explicit G2 moduli dynamics is given
by:
K  3 ln( 4 V7 )  
1/ 3
N
where the 7-dim volume
V7   s
i 1
ai
i
after we add
charged matter
N
7
and the positive rational parameters ai satisfy  ai 
3
i 1
Beasley-Witten: hep-th/0203061, Acharya, Denef, Valandro.
hep-th/0502060
• The N=1 supergravity scalar potential is given by
02
e
V
48V73
[(b A 
2
1

2 2 a  2b1 a
1 0
e

2  2b2 a
2
b A e
2
2

a b1 b1  a
2 0
 2b1b2 A1 A  e

 
cos b1  b2 N  t  a
 
 
 
 
2 2 a  2 b1 a
2  2 b2 a
a  b1  b2  a
  ai  3  a  b1 A1 0 e
 b2 A2 e
 b1  b2 A1 A20 e
N
2
i
i 1
(

)  3( A 

)
 
 cos b1  b2 N  t  a
 
 cos b1  b2 N  t  a
 
2 2 a  2 b1 a
1 0
e
 
2  2 b2 a
2
A e
2
 2 A1 A  e
 
  2b1a
3 2 2 2a  a
2  2 b2 a
 0 A1 0  2  1 e
 A2 e
4
 0

(

 
 a
 b1 b1 a
 2 A1 A   2  1e
cos b1  b2 N  t  a
 0

a
2 0
 
a  b1  b1  a
2 0
)]
)
Moduli Stabilization (dS)
• When Q  P there exists a dS minimum if the
following condition is satisfied, i.e.
V0  0 
28
0
 A2 P 

P ln 
 A1Q 
3 ai PQ  A1Q 

si 
ln 
14 Ni Q  P  A2 P 
with moduli vevs
with meson vev
 QP 2
8
3

QP
2
2
  1
 1

QP
QP
2
0
3
7
2 
  1


Q  P 
 A1Q   2

P ln 
 A2 P 
Moduli vevs and the SUGRA regime
3 ai PQ  A1Q 

si 
ln 
14 Ni Q  P  A2 P 
QP
Ak  P Ck
from threshold corrections
 C1 
Since ai~1/N we need to have large enough PQ ln  
in order to remain in the SUGRA regime si  1  C2 
•Friedmann-Witten: hep-th/0211269


16 2 16 2
2  5 
 
For SU(5): 2  2  10  ,where    ln  4 sin 
g GUT
gM
 q 

CSU ( 5)  e
 
1

2  5 

4 sin 
 q 


integers
 PQ ln  C1  can be made large
C 
 2  O(10-100)
dual Coxeter numbers
• When Q  P there exists a dS minimum with a tiny
CC if the following condition is satisfied, i.e.
V0  0 
moduli vevs
meson vev
8
3

QP
28
0
 A P
P ln  2 
 A1Q 
ai
6Q
si 
N i  3Q  P   8
1
8
02   
1
1
2
2
2

1

1
QP 4
QP QP
QP
• Recall that the gravitino mass is given by
3
m3 2
 g 
1


~ m pl
 m  8 V 3 2
7
 pl 
where
 g  m pl e
8 2

3Qg 2
 m pl e

2
Im f
3Q
Take the minimal possible value Q  P  3 and tune
V. 0  0 .Then
N
Im f   N i si 
i 1
14Q


 g  mpl e28 3  2.15 1014 GeV
• Scale of gaugino condensation is completely fixed!

m3 2  O(100)TeV
Computation of soft SUSY breaking terms
• Since we stabilized all the moduli explicitly, we can
compute all terms in the soft-breaking lagrangian
Nilles: Phys. Rept. 110 (1984) 1, Brignole et.al.: hep-th/9707209
• Tree-level gaugino masses. Assume SU(5) SUSY GUT
broken to MSSM.
M1 2  mp
e
Kˆ 2
K nm Fm  n f sm
2i Im f sm
where the SM gauge kinetic function
N
f sm   N ism zi
i 1
• Tree-level gaugino masses for dS vacua
M1 2




 i W


e
2
7

1


 m3 2


2
 A1Q   0 Q  P 
 AQ 
 
P ln 
02 P ln  1  
 A2 P  
 A2 P  
• The tree-level gaugino mass is always suppressed
for the entire class of dS vacua obtained in our model
V0  0 & Q  P  3 
 A1Q 
  84 - very robust
P ln 
 A2 P 
 M1 2  ei 0.024  m3 2
W
The suppression factor becomes completely fixed!
• Anomaly mediated gaugino masses
Gaillard et. al.: hep-th/09905122, Bagger et. al.: hep-th/9911029
M 
am
a
g a2

16 2
 

~ 
  Kˆ 2
*
  Kˆ 2 m
 Kˆ 2 m

3
C

C
e
W

C

C
e
F
K

2
C
e
F

ln
K








a
a
a
a
m
a
m










• Lift the Type IIA result to M-theory. Yields flavor
universal scalar masses
n
 (1   i ) 
~

K       

i 1   ( i ) 
1
2
where
 
tan i  ci si 
ci - constants
l - rational O(1)
Bertolini et. al.: hep-th/0512067
l
• Anomaly mediated gaugino masses. If we require
zero CC at tree-level and Q  P  3:

i  GUT  


M am


e

3
C

C

1
.
6556
C

C



 a
 a   0.048 Ca 

a
a
a
W
4  






i

1
l i sin 2i   m3 2
2




• Assume SU(5) SUSY GUT broken to MSSM
• Tree-level and anomaly contributions are almost the
same size but opposite sign. Hence, we get large
cancellations, especially when  GUT  1 25 - surprise!
Gaugino masses at the unification scale
• Recall that the m3 2 distribution peaked at O(100) TeV
• Hence, the gauginos are in the range O(0.1-1) TeV
• Gluinos are always relatively light – general prediction
of these G2 compactifications!
• Wino LSP
• Trilinear couplings. If we require zero CC at treelevel and Q  P  3 :
A  m3 2 e
 i W
C
(1.4876  0.024[10.45  2 ln Y

 14( P  3) 
 7 ln 

N


  1  (1   i )  1





 
   ln 
l i sin 2i          



i  2
 ( i )  2


])
• Hence, typically
A  m3 2
• Scalar masses. Universal because the lifted Type
IIA matter Kahler metric we used is diagonal. If we
require zero CC at tree-level and Q  P  3 :
 0.0013
2 
2

2 


 
m2  m32 2 1 
l

sin
2


l

sin
4


2
l

sin
2

i ii
i
i
i
i
i

4






• Universal heavy scalars m  m3 2



in superpotential from Kahler potential. (Guidice-Masiero)
•  - problem
physical

 W * Kˆ 2
 ~ ~
Kˆ 2 m
  
e    m3 2 Z  e F  m Z  K H u K H d
W

~ ~
B  K H u K H d


1 2




1 2
 
 W * Kˆ 2

Kˆ 2 m ˆ
2

 e F K m   m ln    m3 2  2m3 2  V0 Z 
e

W



• Witten argued for his G2 embeddings that   parameter can vanish if there is a discrete symmetry
• If the Higgs bilinear coefficient Z ~ O (1) then
typically expect  ~ O ( m3 2 )
• Phase of  - interesting, we can study it

Electroweak Symmetry Breaking
• In most models REWSB is accommodated but not
predicted, i.e. one picks tan  and then finds  , B
which give the experimental value of M Z
• We can do better with almost no experimental
constraints:
• tan  ~ O (1) since  , B ~ O ( m3 2 )
• Generate REWSB robustly for “natural” values of 
, B ~ O ( m3 2 ) from theory
• Prediction of M Z alone depends on precise values of
 and B
M3/2=35TeV
1 < Zeff < 1.65
Z eff   m3 2
• Generic value M Z ~ O(m3 2 )
• Fine tuning – Little Hierarchy Problem
• Since tan  ~ O(1) , the Higgs cannot be too heavy
PRECISION GAUGE UNIFICATION
M 11 
●
●
●
●
 M Pl
12
7
V
M11
M compact 
2VQ1 3
M unif

 M compact  qe

2

5
13




Threshold corrections to gauge couplings from KK
modes (these are constants) and heavy Higgs triplets are
computable.
Can compute Munif at which couplings unify, in terms
of Mcompact and thresholds, which in turn depend on
microscopic parameters.
Phenomenologically allowed values – put constraints
on microscopic parameters.
The SU(5) Model – checked that it is consistent with
precision gauge unification.
Details:
– Here, big cancellation between the tree-level and anomaly
contributions to gaugino masses, so get large sensitivity on  GUT
– Gaugino masses depend on  GUT , BUT  GUT in turn depends on
corrections to gauge couplings from low scale superpartner
thresholds, so feedback.
– Squarks and sleptons in complete multiplets so do not affect
unification, but higgs, higgsinos, and gauginos do – μ, large so
unification depends mostly on M3/M2 (not like split susy)
– For SU(5) if higgs triplets lighter than Munif their threshold
contributions make unification harder, so assume triplets as heavy
as unification scale.
– Scan parameter space of  GUT and threshold corrections, find
good region for  GUT ~ 1 26.5 in full two-loop analysis, for
reasonable range of threshold corrections.
α1-1
α2-1
α3-1
t = log10 (Q/1GeV)
Two loop precision gauge unification for the SU(5) model
M3
M2
M1
After RG evolution, can plot M1, M2, M3 at low scale as a
1
function of  GUT
for 27TeV  m3 2  50TeV ( here   0 )
M3
M2
M1
Can also plot M1, M2, M3 at low scale as a function of m3 2
In both plots 119GeV  mh  123GeV as 1.58  Z eff  1.67
• Moduli masses:
one is heavy M   600  m3 2
N-1 are light m  1.96  m3 2
• Meson is mixed with the heavy modulus
m  2.82  m3 2
•Since m3 2 ~ O(100)TeV, probably no moduli or
gravitino problem
• Scalars are heavy, hence FCNC are suppressed
LHC phenomenology
• Relatively light gluino and very heavy squarks and
sleptons
• Significant gluino pair production– easily see them
at LHC.
• Gluino decays are charge symmetric, hence we
predict a very small charge asymmetry in the
number of events with one or two leptons and # of
jets  2
• In well understood mechanisms of moduli
stabilization in Type IIB such as KKLT and “Large
Volume” the squarks are lighter and the up-type
squark pair production and the squark-gluino
production are dominant. Hence the large charge
asymmetry is preserved all the way down
Example
For m3 2  35TeV,   45.7TeV get tan   1.45
Compute physical masses:
mg  733GeV
mN~2  228.7GeV
mC~  111.76GeV mN~1  111.6GeV
1
Dominant production modes:
mh  121GeV
1
 GUT
 26.4
almost degenerate!
 ( gg  g~g~ )  1.33 pb
(s-channel gluon exchange)
 (qq  g~g~)  0.46 pb
~~
0
 (qq  C1C1 )  6.2 pb (s-channel Z exchange)
~ ~

(s-channel
exchange)
W
 (qq  C1 N1 )  12.1 pb
Decay modes:
~
~
g  tt N 2
~
~
tb C1  t b C1
~
jets  C1
~
bb N1
~
qq N1
~
qq N 2
~37% ;
~20.7% ;
~19% ;
~8.3% ;
~
~

N 2  C1  W
~
C1  W 
~ 50% ;
~
~

C1    N1
~12% ;
~3% ;
very soft!
mC~  m N~1  160MeV
~
10
τ C~ ~ 10 sec  C1 is quasi-stable!
1
1
~ 50% ;
Signatures
• Lots of tops and bottoms.
Estimated fraction of events (inclusive):
4 tops
14%
same sign tops
23%
same sign bottoms 29%
• Observable # of events with the same sign dileptons
and trileptons. Simulated with 5fb-1 using Pythia/PGS
with L2 trigger (tried 100,198 events; 8,448 passed the
trigger; L2 trigger is used to reduce the SM background)
Same sign dileptons
172
Trileptons
112
2mg~  1466 GeV
After L2 cuts
Before L2 cuts
L2 cut
~
~
C1     N1
Before L2 cuts
After L2 cuts
Dark Matter
• LSP is Wino-like when the CC is tuned
• LSPs annihilate very efficiently so can’t generate
enough thermal relic density
• Moduli and gravitino are heavy enough not to spoil the
BBN. They can potentially be used to generate enough
non-thermal relic density.
• Moduli and gravitinos primarily decay into gauginos
and gauge bosons
• Have computed the couplings and decay widths
• For naïve estimates the relic density is too large
Phases
• In the superpotential:
i1
a ia
0
W  A1 e  e e
e
 
i ( 2  b2 t  N )
 
 
ib1t  N b1s  N
A  e
1
 
 
i 2 ib2 t  N b2 s  N
 A2 e e
 
 
a b1s  N i[( b1 b2 ) t  N 1  2  a ]
0
e
 A2 e
 
b2 s  N

• Minimizing with respect to the axions ti and 
 
fixes cos[(b1  b2 )t  N  1   2  a ]  1
• Gaugino masses as well as normalized trilinears
 
have the same phase given by   W  (2  b2t  N )
• Another possible phase   comes from the Higgs

bilinear, generating the - term
 
• Each Yukawa has a phase   2 l  t
Conclusions
• All moduli are stabilized by the potential generated by
the strong gauge dynamics
• Supersymmetry is broken spontaneously in a unique
dS vacuum
• Derive m3 2  O(100)TeV from CC=0
• Gauge coupling unification and REWSB are generic
• Obtain tan  ~ O (1) => the Higgs cannot be heavy
• Distinct spectrum: light gauginos and heavy scalars
• Wino LSP for CC=0, DM is non-thermal
• Relatively light gluino – easily seen at the LHC
• Quasi-stable lightest chargino – hard track, probably
won’t reach the muon detector
Our Future Work
• Understand better the Kahler potential and the
assumptions we made about its form
• Compute the threshold corrections explicitly and
demonstrate that the CC can be discretely tuned
• Our axions are massless, must be fixed by the
instanton corrections. Axions in this class of vacua
may be candidates for quintessence
• Weak and strong CP violation
• Dark matter, Baryogenesis, Inflation
• Flavor, Yukawa couplings and neutrino masses