Transcript TADPOLES AND SYMMETRIES IN HIGGS

Carla Biggio
Institut de Física d'Altes Energies
Universitat Autonoma de Barcelona
DIVERGENCIES AND SYMMETRIES
IN HIGGS-GAUGE
UNIFICATION THEORIES
based on CB & Quirós, Nucl.Phys.B703 (2004) 199 [hep-ph/0407348]
(see also hep-ph/0410226)
Outline:
1.
2.
3.
4.
5.
Introduction: motivations for Higgs-gauge unification theories
Gauge theories on orbifolds
Symmetries @ fixed points and localized terms
The residual Of symmetry
Conclusions and outlook
XL Rencontres de Moriond
ELECTROWEAK INTERACTIONS & UNIFIED THEORIES
La Thuile (IT), 5-12/03/04
A possible motivation:
Little Hierarchy Problem (LHP)
Barbieri & Strumia 00 Giudice 03
Standard Model (SM): effective theory with cutoff ΛSM
SM 
3GF

2
2
2
2
2
 mH 
2
m

m

m

4
m



200GeV
W
Z
H
t
SM

0.7TeV 
4 2 2



2
No fine-tuning → ΛSM ≤ 1 TeV
New physics ↔ non-renormalizable (dimension six) operators O
L  LSM 
1
O
2
 LH
Precision tests → ΛLH ≥ 5-10 TeV
One order of magnitude of discrepancy: LHP
2
A possible solution:
Supersymmetry (SUSY)
SUSY → no quadratic divergences → (grand) HP solved:
SUSY SM (MSSM) can be extended up to MPl
ΛSM ~ MSUSY
If R-parity is conserved → SUSY virtual loops are suppressed →
ΛLH ~ 4πΛSM
However:
→ LHP solved
• SUSY not yet been observed → fine-tuning
• SUSY breaking sector not well defined
•…
Worthwhile looking for alternative solutions
3
An alternative solution:
Higgs-gauge unification
Consider a gauge theory in a D-dimensional space-time

AMA  AA , AiA

4D Lorentz scalars → Higgs fields !
Randjbar-Daemi, Salam & Strathdee 83
4D Lorentz vector
they can acquire mass through the
Hosotani mechanism
Hosotani et al. 83-04
Higgs mass in the bulk is protected
by higher-dimensional gauge invariance
2 AiA ABj
4D theory
finite corrections ~ (1/R)2 allowed
D-dimensional theory
UV completion
E
ΛSM ~ 1/R ~ TeV
ΛD ≥ 10 TeV
→ LHP solved
4
Gauge theory in D dimensions
Spacetime: MD
Invariant under
gauge group G
(SO(1,D-1))
1
LD   FMAN F A M N  i  MD DM
4
coord.: xM = (x,yi)
Compactification on the orbifold
T  y  y u
u 
M4xTd/Gorb
5D: S1/Z2
d
y
Torus: y  y  u
k  Gorb
k  y  Rk y  u
Orbifold:
y
y = y + 2πR
Rk  SO(d )
y  Rk y  u
Fixed points: invariants under Gorb
k  yf  yf
y + 2πR
Circle S1
πR
0
-πR
y=-y
Orbifold
S1/Z2
0
bulk
πR
fixed points 5
Action of Gorb on the fields
k R ( y)  Rk  PkR (k 1  y)
acts on gauge and flavour indices
acts on Lorentz indices
unconstrained
fixed by requiring invariance of lagrangian

scalars: P0  1
k
vectors: P1  Rk
k
it can be used to break symmetries
S1 :
 x , y   n e n x


i
n
y
R


4D fields with mass
x , y   
 

 
n
R
n   

cos y n x
  y      y 
n 0
R 

n   


 x , y  n1 sin y n x
  y      y 
R 


• zero mode: only for 0 ( x )
   x  , k R   0
• @ yf

S1/Z2:
mn 



6
Gauge symmetry breaking @ yf
Why looking @ yf?
→ lagrangian terms localized @ the fixed points can be radiatively generated
(if compatibles with symmetries)
Georgi, Grant & Hailu 00
 
G  TA
 
Gorb
Hf  T
af
@ yf
G
y
T a f , Rk   0


y + 2πR
y
y = y + 2πR
G
πR
Non-zero fields @ yf:
Aa Aiaˆ
(with zero modes)
some derivatives of non-invariant fields
(without zero modes)
0
-πR
(for some i & â)
y=-y

Residual global symmetry K
Gersdorff, Irges & Quirós 02
0
πR
H K
H K
7
Effective 4D lagrangian


d
L   d y  LD   L f  ( y  y f ) 
f


eff
4
d
The symmetries @ yf are:
Lf → most general 4D lagrangian
compatible with symmetries @ yf
Gorb SO(3,1) Hf K
Forbidden terms:
If
aˆ
i
bˆ
j
A is Gorb-invariant  a “shift” symmetry forbids a direct mass term: 1 A A
Allowed terms:
2
Fa F a
→ localized kinetic term for
Fa F a
→ localized anomaly
→ If
aˆ
i
Aa
Aiaˆ and Fija are orbifold invariant
Fija F aij
→ localized quartic coupling for
Fiaˆ F aiˆ 
→ localized kinetic term for
Aiaˆ
(D≥6)
Aiaˆ
All these are dimension FOUR operators → renormalize logarithmically
8
… another (worse) allowed term…
If
Hf = U(1)a x …
and
Aiaˆ and Fija are orbifold invariant
ˆ
ˆ
Fija   i Aaj   j Aia  gf abcˆ Aib Acjˆ
is invariant under Gorb SO(3,1) Hf K →
i Aaj
ˆ
• mass term for Aib A cjˆ
• tadpole for
can be radiatively generated @ yf
This is a dimension TWO operator → quadratic divergencies
D≥6 it seems it always exists
• D=6 (QFT)
Gersdorff, Irges & Quirós 02 Csaki, Grojean & Murayama 02
Scrucca, Serone, Silvestrini & Wulzer 03 (SSSW03)
• D=10 (strings) Groot-Nibbelink et al. 03
 How can we avoid this?
1. global cancellation of tadpoles SSSW03
2.
9
But…
another symmetry must be considered
CB & Quirós, Nucl.Phys.B703 (2004) 199
SO(1,D-1)
d-dimensional smooth manifold:
at each point can be defined a
TANGENT SPACE →
SO(d)
y
as G
so SO(d)
Gorb
y
y = y + 2πR
when orbifolding:
Gorb
y + 2πR
SO(1,3)x
SO(d)
πR
Hf
k
such that  R , H f   0
Of
such that
0
-πR
 Pk , O f   0
y=-y
πR
0
SO(1,3)xOf
The symmetries @ yf are:
Gorb SO(3,1) Hf
 Can this Of forbid the tadpole?
K Of
10
The tadpole Fij and the symmetry Of
If Of=SO(2) x … then the Levi-Civita tensor ij exists
→
 ij Fija
is Of invariant
If Of=SO(p1) x SO(p2) x … (pi>2)
→ TADPOLES ARE ALLOWED
then the Levi-Civita tensor is

i1i2 ...i p
→ only invariants constructed with pi-forms are allowed

i1i2 . . .i p
Bi1i2 . . .i p
→ NO TADPOLES
Sufficient condition for the absence of localized tadpoles
Of = SO(p)
pi > 2
i
BQ’04
Of is orbifold-dependent: we studied the Td/ZN case
11
Orbifolds Td/ZN
(d even)
Of depends on RNf:
on Td/ZN → RNf ~ diag(r1…ri…rd/2) with ri rotation in the i-plane
d /2
If Nf>2

O f   SO(2)i
SO(2)  SO(d)
i 1
d /2
 in every subspace (y2i-1,y2i) IJ exists →
If Nf=2
which acts on (y2i-1,y2i)
→
RNf= -1

Of=SO(d)

2i
c 
i 1
i
I , J  2i 1
 IJ FIJa  d / 2 ( y  y f )
[-1,SO(d)] = 0
the Levi-Civita tensor is
 i i ...i
12
d
→ only invariants constructed with d-forms are allowed
→

TADPOLES ONLY FOR d=2 (D=6)
Td/Z2 → explicitely checked @ 1- and 2-loop for any D
valid also
for odd D
BQ’04
12
Conclusions
In Higgs-gauge unification theories (Higgs = Ai)
• bulk gauge symmetry G prevents the Higgs from adquiring
a quadratically divergent mass in the bulk
• “shift” symmetry K forbids a direct mass @ yf
a
If Hf = U(1)a x … Fij can be radiatively generated @ yf
giving rise to a quadratically divergent mass for the Higgs
Fija can be generated ↔ it is Of-invariant
Of  SO(d)
such that
Of , Pk   0
If Of=SO(p1) x SO(p2) x … (pi>2) → NO TADPOLES
If Of=SO(2) x … → TADPOLES
 IJ FIJa
Td/ZN (d even, N>2): if Nf>2 → Of=SO(2) x … x SO(2) → TADPOLES
Td/Z2 (any d): Of=SO(d) → TADPOLES ONLY FOR d=2 (D=6)
13
Outlook
The absence of tadpoles is a necessary but not sufficient condition
for a realistic theory of EWSB without SUSY
Other issues:
• REALISTIC HIGGS MASS
D>6
(D=5 no quartic coupling, D=6 tadpoles)
Td/Z2 → d Higgs fields  non-minimal models
→ we have to obtain only one SM Higgs
even if this is achieved
→ Higgs mass must be in agreement with LEP bounds
• FLAVOUR PROBLEM
- matter fermions in the bulk coupled to a background
which localizes them at different locations
Burdman & Nomura 02
- matter fermions localized and mixed with extra heavy
bulk fermions
Csaki, Grojean & Murayama 02
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