Supersymmetry breaking in STRING THEORY · 2013-12-19 · In these constructions, the closed (bulk)...
Transcript of Supersymmetry breaking in STRING THEORY · 2013-12-19 · In these constructions, the closed (bulk)...
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Supersymmetry breaking in STRING THEORY
Review based on old work in collaboration .with
I. Antoniadis, C. Angelantonj, N. Kitazawa, J. Mourad, S. Patil, G. Pradisi, A. Sagnotti et al.
(1998-2001, 2010,2012) + work in progress with J. Mourad, G. Pradisi and A. Sagnotti
Bucuresti, 19/12/2013
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Content • Orientifolds • Supersymmetry breaking by compactification • Tachyon-free non-BPS models (Brane
supersymmetry breaking ) • Internal magnetic fields intersecting
branes • Ground state and D-brane solutions in non-SUSY strings • Applications to Cosmology • Conclusions
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Orientifolds Prototype: type I strings = IIB/ , where is a left-right projection (involution). Orientifolds have closed and open strings. Cartoon picture of type II orientifolds (Sagnotti): open/closed strings, Dp-branes/O-planes
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D-branes/O-planes have Tension and Charges Crucial constraint: RR tadpole constraints UV finiteness Gauss law in internal space X
Dp
q(n)Dp +
X
Op
q(n)Op = 0 ; SUSY ! Tp = qp (1)
(Tp; qp)
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Supersymmetry breaking by compactification
The Scherk-Schwarz mechanism (ScherkSchwarz; Fayet; Rohm; Ferrara,Kounnas,Porrati,Zwirner) Main idea : use symmetries S of the higher-dimensional theory which do not commute with supersymmetry : R-symmetries or the fermion number (-1)^F.
After being transported around the compact space, bosonic and fermionic fields return to the initial value (at y=0) only up to a symmetry operation where the matrix is different for bosons and fermions.
©
©i(2¼R; x) = Uij(!)©j(0; x)
U 2 S
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This procedure is very similar to the breaking of supersymmetry at finite temperature The terms breaking supersymmetry generate UV finite effects, even at the field theory level.
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In models with D-branes there are two different ways in which SUSY can be broken by compactification (Antoniadis,E.D.,Sagnotti) - The D brane is parallel to the direction of breaking ; D brane spectrum experience tree-level SUSY breaking. This is the analog of the heterotic constructions: (tree-level) - the D brane is perpendicular to the direction of the breaking; massless D brane spectrum is SUSY at tree-level. SUSY breaking transmitted by radiative corrections from the brane massive states or from the gravitational sector.
m3=2 » M1=2 » 1R
m3=2 » 1R ; m0 » 1
R2MP; M1=2 » 1
R3M2P
(tree-level) (one-loop)
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Parallel and perpendicular Scherk-Schwarz breaking
Parallel dims TeV radii Perpendicular dims intermediate radii Problem : large cosmological constant
MSUSY » R¡2
MP
MSUSY » R¡1
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Tachyon-free non-BPS models (Brane supersymmetry breaking )
(Sugimoto; Antoniadis, E.D.,Sagnotti,1999)
In these constructions, the closed (bulk) sector is SUSY to lowest order, whereas SUSY is broken at the string scale on some stack of (anti)branes. - String consistency asks for the existence of exotic O9_{+} planes of positive RR charge. Then charge conservation /RR tadpoles ask for antibranes in the open sector.
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SUSY case (SO gauge group) : Bose-Fermi degeneracy
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Brane SUSY breaking case (USp gauge group): spectrum is "misaligned"
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system is non-BPS but tachyon-free. It breaks SUSY at string scale in the open sector, closed/gravitational sector SUSY at tree-level. - There is a NS-NS dilaton tadpole
Dp¡Op+
PDp q
(n)Dp +
POp q
(n)Op 6= 0
m » M2s =MP
- Singlet in the open string fermionic spectrum which can be correctly identified with the Goldstino realizing a nonlinear SUSY on antibranes (E.D.,Mourad; 2000).
- - No obvious candidate for decay to a SUSY vacuum - ( folklore : non-SUSY vacua decay into SUSY ones). - Suggestion : nonperturbative instabilites (Angelantonj,E.D.;2007)
- - low-string scale light moduli
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where
where implements a reparametrization and is a local Lorentz transformation.
SUSY acts in the open sector in the standard non-linear way: reparametrization + Lorentz
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So the effective action of this 10d USp(32) non-linear SUSY , tachyon-free string , seems consistent (other 6d and 4d similar examples known) . There are however several puzzles about this BSB (Brane Supersymmetry Breaking) theory and its lower dim. cousins : Are they really stable ? Not possible to have a super-Higgs mechanism in 10d. Degree of freedom of massless gravitino + goldstino 64 Degree of freedom of massive gravitino 128 However, in 9d this seems possible: Degree of freedom of massive 9d gravitino + = 64 What does this mean ???
ªM
µ =
=
ª9
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Internal magnetic fields intersecting branes
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Type II B/ with internal magnetic fields
0 = £ I
T-dual
Type II A/ with intersecting branes
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Ground state and D-brane solutions in non-SUSY strings
SUSY breaking generates dilaton potential / NS-NS tadpoles. String frame :
V » e¡©(BSB) , V = const (Scherk¡ Schwarz)
In both cases, in the Einstein frame they are of the type V » e°©
For 10d Sugimoto model ° = 3=2
The vacuum/ground state is not 10d. The maximal symmetric Solutions have - SO(1,8) (space dependent) or - S0(9) (time-dependent) symmetry.
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The SO(1,8), space-dependent case
In the gauge the solution can be found explicitly
e2Be°© = const
It has naked singularities at y=0 and y = ; the internal coordinate y becomes compact. However, Planck mass, YM constant are finite.
1
In the SO(9), time-dependent case we get
t=0 is a big-bang singularity
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Applications to Cosmology
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S =1
2·2
ZdDx
p¡g
·R ¡ 1
2(@Á)2 ¡ V (Á) + : : :
¸• Consider the action for gravity and a scalar φ :
• Look for cosmological solutions of the type
ds2 = ¡ e 2B(t) dt2 + e 2A(t) dx ¢ dx
• Make the convenient gauge choice V (Á) e2B = M2
(Halliwell, 1987) ……………… (E.D,Mourad, 2000) (Russo, 2004) ………………..
¯ =
rd¡ 1
d¡ 2; ¿ = M ¯ t ; ' =
¯ Áp2
; A = (d¡ 1) A
• In expanding phase : Ä' + _'p
1 + _' 2 +¡1 + _' 2
¢ 1
2V
@V
@'= 0
• Let :
• OUR CASE : V = exp( 2 ° ') ¡! 1
2 V
@V
@ '= °
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A climbing scalar in d dim’s
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• γ < 1 ? Both signs of speed a. “Climbing” solution (ϕ climbs, then descends):
b. “Descending” solution (ϕ only descends ):
_' =1
2
·r1¡ °
1 + °coth
³¿
2
p1¡ °2
´¡r
1 + °
1¡ °tanh
³¿
2
p1¡ °2
´¸
_' =1
2
·r1¡ °
1 + °tanh
³¿
2
p1¡ °2
´¡r
1 + °
1¡ °coth
³¿
2
p1¡ °2
´¸
vl = ¡ °p1¡ ° 2Limiting τ- speed (LM attractor):
γ 1 : LM attractor & descending solution disappear
_' =1
2 ¿¡ ¿
2• γ ≥ 1 ? Climbing ! E.g. for γ=1 :
CLIMBING : in ALL asymptotically exponential potentials with γ≥ 1 !
NOTE : only ϕo . Early speed singularity time !
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Climbing and Inflation
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a. “Hard” exponential of Brane SUSY Breaking b. “Soft” exponential (γ < 1/√3):
Non-BPS D3 brane gives γ = 1/2 [+ stabilization of Φs]
V (Á) = M4 ¡
e 2' + e 2 ° '¢
(Sen , 1998) (E.D .J.Mourad, A.Sagnotti 2001)
BSB “Hard exponential“ makes initial climbing phase inevitable
“Soft exponential” drives inflation during subsequent descent
Would need :
° ¼ 1
12
ϕo : “hardness” of kick !
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Numerical Power Spectra
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²Á ´ ¡_H
H2; ´Á ´
VÁÁV
nS ¡ 1 = 2( ´Á ¡ 3 ²Á) ;
nT ¡ 1 = ¡ 2 ²ÁPS;T »Z
dk
kknS;T¡1
Key features: 1. Harder “kicks” make ϕ reach later the attractor 2. Even with mild kicks the time scale is 103- 104 in t M ! 3. η re-equilibrates slowly
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31
WMAP9/Planck power spectrum :
NOTE : ¯̄¯̄¢C`
C`
¯̄¯̄ =
r2
2` + 1
Qualitatively the low-k tail
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Conclusions
There are several known ways to break SUSY in string theory: - by compactification: parallel and perpendicular to branes - by non-BPS configurations (BSB) - by internal magnetic fields/brane rotations - by closed-string fluxes
Few 4d « realistic » models with broken SUSY constructed along
these lines .
Conceptual problems in defining the ground state of the theory. To date, all static solutions I know of have naked singularities.
Time-depedent solutions seem to have interesting early-time cosmological interpretations. A realistic setting should combine SUSY breaking with moduli stabilization.