1. THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS
Irene Hidalgo.
IFT, Madrid
Collaboration with: Pre-SUSY
A. Casas July 2005
J.R. Espinosa

2. Outline Hierarchy problem of SM. Fine-tuning: Conclusions. SUSY
Little Higgs
Conclusions.

3. Hierarchy problem of SM
SM as an effective theory valid up to a cut-off scale ΛSM → Radiative corrections to the Higgs mass:
No fine-tuning between tree-level and 1 loop contributions to mh → ΛSM≤ few TeV ( “Big” hierarchy problem ) .
E.g. mh =130 GeV
Veltman

4. Tension between these bounds in ΛSM and the experimental bounds on the effective scale of non-renormalizable operators (that parametrize new physics).
Typically
“Little” hierarchy problem  LH
10 TeV ~
>

5. Veltman´s condition (1-loop):
Kolda & Murayama
Veltman´s condition (1-loop):
ΛSM could be larger than expected if Veltman´s condition is fulfilled.
At higher order this condition becomes cut-off dependent.

6. FINE-TUNING  = 100  1% fine-tuning
Barbieri & Giudice
Standard definition of the fine-tuning parameters: ,
with αi the independent parameters of the model.
 =  10% fine-tuning
 = 100  1% fine-tuning

7. SM: ΛSM as an indepedent parameter Veltman´s throat Contourplot of ΔΛ
Kolda & Murayama
SM: ΛSM as an indepedent parameter Veltman´s throat
Contourplot of ΔΛ
Other relevant parameters in the SM for the fine-tuninng: λ t and λ

8.  < 2.5 TeV Top mass: mt = 178 ± 4.3 GeV with
But mh has not been measured:
 < 2.5 TeV SM
aver

9. SUSY models
SUSY:
There are the same number of bosonic and fermionic degrees of freedom.
The hierarchy problem is solved due to the cancellation of quadratic divergences of the Higgs mass.
The Minimal Supersymmetric extension of the SM: the MSSM
Higgs sector: 2 doublets, H1 and H2 .
Tree-level scalar potential:
with

10. Along the breaking direction in the H10, H20 space:
where λ and m2 are functions of the soft masses and the μ-parameter at the initial scale.
Minimization: Fine-tuning:
MSSM

11. Contourplot of the fine-tuning in the MSSM

12. LOW SCALE SUSY
with the SUSY breaking scale and M the messenger scale.
- Gravity-mediated models: M~1019 GeV
- Low scale SUSY models: and M of similar order ~ TeV
Concrete example:
where T is the singlet responsible for the breaking of SUSY and m = ΛS2 / M ~

13. ~ Integrating the singlet T out: 2HDM
μ = 0.3 M , m =0.5 M , e1 = -2, αt = 1 ~

14. Little Higgs Models mh ~ 200 GeV SM L. H. H.E. cut-off m ~ f~ 1 TeV
Stabilization of Higgs mass by making the Higgs a pseudo-Goldstone boson resulting from a spontaneously broken approximate symmetry.
Spectrum:
New particles at 1 TeV than cancel quadratic divergences in mh. SM
L. H.
H.E. cut-off
mh ~ 200 GeV
m ~ f~ 1 TeV
 ~ 4 f ~ 10 TeV

15. The Littlest Higgs
Arkani-Hamed et al.
The Littlest Higgs is a non-linear σ model based on a global SU(5) symmetry which is spontaneously broken to SO(5) at the scale f ~ 1 TeV.
An [SU(2)×U(1)]2 subgroup of SU(5) is gauged, and is spontaneously broken to the diagonal SU(2)×U(1) subgroup.
New states that cancel the quadratic divergences:
Heavy top T :
Extra gauge bosons W’ , B’ : ,
- Triplet  :

16. The Littlest Higgs Tree-level Lagrangian: Radiative corrections:
(g1, g2 , g1´, g2´)
(1, 2)
constrained by

17. The Littlest Higgs The operators O 1 and O2 already at tree-level:
c and c’ unknown coefficients.

18. The Littlest Higgs Electroweak symmetry breaking.
At energies beneath m , integrating out the triplet:
with

19. Fine-tuning in the Littlest
Parametrization of the amount of fine-tuning:
Rough estimate: heavy top contribution
t mh2 =
 2
with
 2 t2
t mh2  f 2
e.g. for f = 1 TeV, mh = 150 GeV
t mh2 / mh2  33

20. Fine-tuning in the Littlest
But heavy top contribution is not all.
Using the standard definition of fine-tuning parameters.
Parameters in Littlest: c, c´ , λ1 , λ2 , g1 , g2 , g´1 , g´2 . (Constraints between them)
Two regions:
A) λ ≈ λb « λa ≈ M2Φ/f2
B) λ ≈ λa « λb ≈ M2Φ/f2

21. Fine-tuning in the Littlest
Case A. f = 1TeV , g’12= g’22= 2 g’2
mh = 250 GeV
mh = 115 GeV

22. Fine-tuning in the Littlest
Case A → c small → Implicit fine-tuning between ctree and c1-loop
 c instead of  c
tree
 total with  c
 total with  c
mh = 250 GeV
tree

23. Fine-tuning in the Littlest
mh = 115 GeV
Case B. f = 1TeV , g’12= g’22= 2 g’2
Fine-tuning larger than case A.
 total with c
Delicately tuned
tree

24. Littlest with T-parity
Cheng & Low
Extra symmetry: T-parity.
Coupling h2Φ is forbidden, and also direct couplings of SM fields to new gauge bosons.
Parameters : c, c´ , λ1 , λ2
Two cases:
A) λ1 < λ2
B) λ2 < λ1

25.  mh2 = c g´2 2 / 162 [SU(2)]2 x U(1)Y model mh = 250 GeV Case A
Peskin et al.
Differences from the Littlest:
There is a quadratic divergence contribution to mh2 due to U(1)Y
Absence of the heavy B’ boson.
Two regions (A and B heirs of the Littlest):
Case A similar fine-tuning as Littlest.
Case B is worse in terms of fine-tuning.
Case A
 mh2 = c g´2 2 / 162
mh = 250 GeV

26. Fine-tuning in the Simplest
f1 = f2 = 1 TeV
Global [SU(3)×U(1)]2 / [SU(2)×U(1)]2
Two scales: f1 , f2 .
Radiatively induced δm2<0 :
Add tree-level mass μ2
Parameters: f1 , f2 , μ2, λ 1 , λ 2 .

27. Conclusions → MSSM ~5 % fine-tuned
SM → hierarchy problem → Physics Beyond SM ~ few TeV.
SUSY
MSSM
Logarithmic and finite contributions from sparticles
Bounds on sparticles masses
λtree is small
Low scale SUSY
λtree is larger
No big effects of running
→ MSSM ~5 % fine-tuned
→ Improvement in the fine-tuning problem

28. Conclusions Minimum value of Δ accessible by varying the parameters
“Little Higgs” models.
Rough estimate with the heavy top contribution : few % fine-tuned.
Taking into account the standard definition of fine-tuning and all the parameters in the studied models:
More fine-tuned than the rough estimate due to implicit tunings between the parameters of the models to work properly and have the correct EW scale.
Minimum value of Δ accessible
by varying the parameters