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FINE TUNING IN LITTLE HIGGS MODELS

VIEWS: 42 PAGES: 28

									THE FINE-TUNING PROBLEM
IN SUSY AND LITTLE HIGGS
                      Irene Hidalgo.
                        IFT, Madrid


Collaboration with:                    Pre-SUSY
A. Casas                               6 July 2005
J.R. Espinosa




                                       THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                   Outline
Hierarchy problem of SM.
Fine-tuning:
    SUSY
    Little Higgs
Conclusions.




                             THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
              Hierarchy problem of SM
SM as an effective theory valid up to a cut-off scale ΛSM → Radiative corrections
to the Higgs mass:
                                                                                          Veltman




No fine-tuning between tree-level and 1 loop contributions to mh →ΛSM≤ few
TeV ( “Big” hierarchy problem ) .
E.g. mh =130 GeV




                                                        THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
Tension between these bounds in ΛSM and the experimental bounds on the
effective scale of non-renormalizable operators (that parametrize new physics).


                                    1
                     L  LSM       2 O
                                    LH

Typically

                        LH > 10 TeV
                            ~

                 “Little” hierarchy problem


                                                       THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                                                        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.




                                                         THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                           FINE-TUNING
                                                                              Barbieri & Giudice

Standard definition of the fine-tuning parameters:
      v 2 (α1 , α 2 ,...)  v 2

                                            Δ                    Δ
                                                                                2
                                             ,
                                                 Total                          αi
                                                                     i


                     = 10  10% fine-tuning
                     = 100  1% fine-tuning



with αi the independent parameters of the model.


                                                         THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                                                                                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


                                                         THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
Top mass:   mt = 178 ± 4.3 GeV




with




But mh has not been measured:




                                 
                                 aver
                                  SM         < 2.5 TeV
                                     THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                                   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:
                                                                                     2
                   0 2             0 2                     1 2       2   0 2   0 2
   V m H   2
                         m H2
                                          (m H H  h.c.)  ( g  g ' ) H1  H 2 
                                             2   0   0
            1      1         2     2         3   1   2
                                                           8                      
     with       m12, 2   2  mH1, 2
                                2
                                           m3  B
                                            2




                                                               THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
Along the breaking direction in the H10, H20 space:
                           1 2 2 1 4
                        V  m v  v
                           2     4
where λ and m2 are functions of the soft masses and the μ-parameter at the initial
scale.
Minimization:                         Fine-tuning:
                 m2                             mi2
         v2                                    2
                                                v
MSSM
                         1 2                   1
                tree    ( g  g ' ) cos 2  cos2 2
                                   2     2

                         8                    15

      m  m c m s m s
         2      2 2         2 2      2                    ~2
                                             1.01  2.31m
                                                       2
                1          2       3 2


                                                           THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
Contourplot of the fine-tuning in the MSSM




                                 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
LOW SCALE SUSY
                                                                      2
                            F2                                   F 2 msoft
                    2
                   msoft   ~ 2 ,                   SUSY        ~ 4~ 2
                            M                                    M   M
with F the SUSY breaking scale and M the messenger scale.
- Gravity-mediated models: M~1019 GeV
- Low scale SUSY models: F and M of similar order ~ TeV

–   Concrete example:

                   W  2S T  H1  H 2 
                                                                  l
                                                                    H1H 2 2
                                                                 2M
                                                                t
                K  T  H1  H 2 
                           2                2          2                 4
                                                                     T
                                                                4M 2
               
                   1
                   M   2
                           T
                               2
                                   H   1
                                            2
                                                 H2
                                                       2
                                                                e
                                                                1 2 H1  H 2
                                                                2M
                                                                      4           4
                                                                                       
                                                                    ~
    where T is the singlet responsible for the breaking of SUSY and m = ΛS2 / M

                                                                                THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
 Integrating the singlet T out: 2HDM

                   ~
m12  m2   2  1m 2 , m3  0
       2                  2




         1 2                   ~2
                            2 m
1  2  ( g  g ' )  21 2
                   2

         4                    M
     1 2             2       ~
3  ( g  g ' )  2 (12 m 2  e1 2 )
              2

     4             M
       1 2           12   2
4   g  2 e1  2  2
              
       2              t  M
                          
                     l
5  0 , 6  7 
                     M
                                                       ~
                                          μ = 0.3 M , m =0.5 M , e1 = -2, αt = 1



                                                     THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                   Little Higgs Models
Stabilization of Higgs mass by making the Higgs a pseudo-Goldstone boson
resulting from a spontaneously broken approximate symmetry.
Spectrum:

       SM                    L. H.                H.E. cut-off


  mh ~ 200 GeV            m ~ f~ 1 TeV               ~ 4 f ~ 10 TeV

   New particles at 1 TeV than cancel quadratic divergences in mh.




                                                       THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                         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  :



                                                      THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                            The Littlest Higgs

Tree-level Lagrangian:


                         (g1, g2 , g1´, g2´)   (1, 2)


constrained by

Radiative corrections:




                                                          THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                          The Littlest Higgs

The operators O 1 and O2 already at tree-level:




c and c’ unknown coefficients.
                                                     3
                    c  ctree  c1loop  ctree 
                                                     4
                    c'  c'tree  c'1loop    c'tree  24




                                                             THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                            The Littlest Higgs
Electroweak symmetry breaking.
At energies beneath m , integrating out the triplet:




with




          a  c( g 2  g '2 )  c' 1 , b  c( g12  g '1 )
                    2
                           2
                                     2                    2



                                                                THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                       Fine-tuning in the Littlest

Parametrization of the amount of fine-tuning:
   Rough estimate: heavy top contribution


            t mh2 =                                   2


   with
                                                t mh2  0.37 f 2
                          2  t2

     e.g. for f = 1 TeV, mh = 150 GeV


                         t mh2 / mh2  33
                                                     THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                            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

                              a  c( g 2  g '2 )  c' 1
                                        2
                                               2
                                                         2


                              b  c( g12  g '1 )
                                               2


                               M 2  (a  b ) f 2
                              1 /   1 / a  1 / b
                                                                           THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                       Fine-tuning in the Littlest
Case A. f = 1TeV , g’12= g’22= 2 g’2




    mh = 115 GeV                              mh = 250 GeV

                                                     THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                       Fine-tuning in the Littlest
Case A → c small → Implicit fine-tuning between ctree and c1-loop
    c tree instead of  c                                 cc               c1loop  ctree 
                                                                                                3
                                                                     tree
                                                                                                4
                                                               c'  c'tree  c'1loop  c'tree 24




       total with  c                                total with  c tree
                               mh = 250 GeV
                                                        THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                       Fine-tuning in the Littlest
Case B. f = 1TeV , g’12= g’22= 2 g’2   mh = 115 GeV
                                           Fine-tuning larger than case A.




    total with c tree                    Delicately tuned
                                                     THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                                   Littlest with T-parity
Extra symmetry: T-parity.                                                          Cheng & Low


    Coupling h2Φ is forbidden, and also direct couplings of SM fields to new gauge bosons .


    g1  g 2  2 g ,      g '1  g '2  2 g '
       1
      (a  b )
       4
    Parameters : c, c´ , λ1 , λ2

    Two cases:

         A)   λ1 < λ2
         B)   λ2 < λ1




                                                                     THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                               [SU(2)]2 x U(1)Y model
                                                                                          Peskin et al.
Differences from the Littlest:

    There is a quadratic divergence contribution to mh2 due to U(1)Y
                                                                              Case A
         mh = c
               2            g´2 2 /       162
    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.




                                                                       mh = 250 GeV
                                                                       THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                    Fine-tuning in the Simplest


Global [SU(3)×U(1)]2 / [SU(2)×U(1)]2                       f1 = f2 = 1 TeV

Two scales: f1 , f2 .
Radiatively induced δm2<0 :

Add tree-level mass μ2
Parameters: f1 , f2 , μ2, λ 1 , λ 2 .




                                              THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                           Conclusions
SM → hierarchy problem → Physics Beyond SM ~ few TeV.
SUSY
       MSSM
         Logarithmic and finite contributions from sparticles
         Bounds on sparticles masses MSSM ~5 % fine-tuned
                                  →
         λtree is small


       Low scale SUSY
         λtree is larger
                                     → Improvement in the fine-tuning problem
         No big effects of running




                                                         THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .
                           Conclusions

“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 3 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


                                               THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS   .

								
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