1. The Flavor of a Little Higgs with T-parity
PHENO 06
The Flavor of a Little Higgs with T-parity
Seung J. Lee
Cornell University
With Jay M. Hubisz and Gil Paz

2. Outline T-Parity and the Littlest Higgs model T-odd Flavor Mixing
Neutral Meson Mixing
Results and Constraints
Conclusions

3. Little Higgs Model (Arkani-Hamed, Cohen, Georgi)
Addressing “Little hierarchy problem”
Hierarchy problem: Radiative corrections to the Higgs mass become divergent at one-loop, but the Higgs mass should be light New physics should be introduced
at around ~1 TeV.
At energies below the cutoff scale of new physics, the effective theory contains a set of higher-dimensional non-renormalizable operators made up of SM particles, i.e or
But these operators have been constrained by experiments, which indicate that there should be no new physics up to L ~5 TeV
However, this constraint is in conflict with the naturalness requirement that new physics must enter ~1 TeV. “little hierarchy”

4. Little Higgs Model Idea: The Higgs as a pseudo-Goldstone boson.
-This idea has been pursued for ~30 years.
-Problem: To give a mass to the Higgs, explicit symmetry breaking terms are necessary, but quadratic divergences at one-loop are reintroduced.
-Solutions: “Collective symmetry breaking”
Written in the language of a non-linear sigma model:
-New particles appear around the global symmetry breaking scale f~1Tev, but the cutoff of the model can be as large as L ~4f~10TeV .
Two or more explicit symmetry breaking terms are needed to break all symmetries protecting the Higgs mass No quadratic divergences at one-loop.

5. The Littlest Higgs(Arkani-Hamed, Cohen, Katz, Nelson)
Based on an SU(5)/SO(5) non-linear sigma model Symmetry breaking by the VEV of a symmetric tensor :
describes GSB.
Structure of gauge symmetry breaking:
g1:
g2:
Higgs is an exact GSB under either SU(3)1 or SU(3)2. It gets mass only when both g1 and g2 are turned on (two copies of SU(2)  U(1) are gauged ) to break both SU(3)1 and SU(3)2.

6. T-Parity and Littlest Higgs
Cheng & Low hep-ph/
hep-ph/
Low hep-ph/
Little Higgs models have stringent EWP bounds.
(In most models, the symmetry breaking scale, f, needs to be ~4TeV to satisfy EWP constraints.)
Analogous to the role of R-parity in the MSSM, T-parity is a discrete symmetry added to solve the problems of the original Little Higgs model.
There are other models that solve these problems by extending gauge/global symmetry structures.
Introducing a discrete symmetry (T-parity) provides a weak scale dark matter candidate.
Except for the T-even top partner, all other BSM particles (i.e. WH, ZH, AH,…) introduced in the model are odd under T-parity.

7. T-Parity and the Littlest Higgs
T-parity exchanges the two gauge groups,
[SU(2)  U(1)]1 and [SU(2)  U(1)]2
WL ~ (W1+W2) (T-even), WH ~ (W1-W2) (T-odd)
T-parity enforces relations between couplings.
(i.e. g1 = g2)
However, to implement T-parity consistently in a phenomenologically viable way, we need to enlarge the fermion sector. (in order to cut off otherwise large four-fermion operators constrained by LEP and Drell-Yan processes)
A consistent and phenomenologically viable littlest Higgs with T-parity requires the introduction of “mirror fermions”
Opens up a new flavor structure in the model.

8. T-Parity and Littlest Higgs
T-parity Eigenstates
SU(2) SU(2) T-even T-odd
Royal SU(3) triplet
New fermion partners of top quark
Hubisz & Meade
Hubisz, Meade, Noble, Perelstein
T-odd “mirror fermion” mass has an upper-bound due to compositeness constraints. (in order to cut off dangerous four-fermion operators arising at one-loop.)

9. T-odd Flavor Mixing Interactions Gauge basis T-parity basis
Yukawa for T-odd fermions
Dirac masses
function of SU(5) Goldstone modes

10. T-odd Flavor Mixing: Mass eigenbasis
Diagonalization of new Yukawa terms introduces new CKM-like matrices: In analogy with the CKM transformation, the mass matrix is diagonalized by two U(3) matrices.
VH =UH, since SU(2)L symmetric
In the mass eigenbasis, interactions of SM fermions and T-odd fermions involve another T-odd gauge boson:
Recall:
New CKM type matrix:
These matrices are related through the SM CKM matrix:

11. Relevant Feynman rules
T-odd Flavor Mixing:
Relevant Feynman rules

12. T-odd Flavor Mixing: New Parameterization
Beyond SM, there are three new rotation angles, and one new CP violating phase:
- 2 unitary matrices, VHu and VHd: 3 rotations each, and 6 phases each.
- 6 quark fields which transform under SU(2)1, and 6 under SU(2)2. Each set of 6 quark absorb 5 phases.
- 2 unobservable overall phases (1 for each sector)
- 1 combination of VHu and VHd gives the SM CKM matrix, which has 3 rotations and 1 phase.

13. T-odd Flavor Mixing: Summary
T-odd fermion masses have an upper limit from four fermion operators- could be light. (MTeV <4.8 TeV)
New flavor physics.
Interesting phenomenology.
Many new parameters.
BSM, there are 3 new rotation angles, and one new CP violating phase.
Model dependent.

14. Neutral Meson Mixing Two different mass scales:
-Electro weak scale v: MWL=gv/2, v=246 GeV
-LH global symmetry breaking scale f: MWH=gf, f=TeV
New interactions involving T-odd fermions lead to effective Hamiltonians for processes such as neutral meson mixing and rare decays.
We first focused on neutral meson mixing. (Taking account of rare decays and EDM is under preparation now.) s d
Integrate out heavy particles
Effective Hamiltonian d s

15. Neutral Meson Mixing
SM contribution to the Heff that governs neutral meson mixing:
Inami-Lim function-gauge dependent
where
and the mi are the masses of the quarks in the loop
When summing over the different flavors, the gauge dependence cancels due to the unitarity of the CKM matrix.
QCD correction
gauge-independent
In the limit of quark mass degeneracy, Heff=0 (GIM).
Expect GIM for T-odd fermions from flavor constraints.
-Take T-odd Yukawa eigenstates to be nearly degenerate.

16. Neutral Meson Mixing
Box diagrams involving T-odd gauge bosons and scalars that contribute to the neutral meson mixing.

17. Neutral Meson Mixing
T-odd sector contribution to the Heff that governs neutral meson mixing (using unitarity):
Gauge-independent
function
QCD correction
Since off-diagonal CKM elements are extremely small, this can be large compared to the SM contribution
Naively suppressed
Heff for the neutral meson mixing is basically the same for the D, K, and B meson mixing.
From Heff , can be calculated, so that comparison with experimental data can constrain the T-odd fermion masses.

18. Results and Constraints
Too many unknown parameters compared to the number of observables we have from neutral meson mixing.
Instead of VHd or VHu, Vd and Vu are physical in the T-parity model, while in the SM their combination is physical.
For the case that Vd and Vu are nearly diagonal, one universal feature is that GIM mechanism is working for the first two generations:
Their masses are nearly degenerate, but neutral meson mixing is nearly independent of the mass of the third generation T-odd fermion due to the smallness of off-diagonal elements.
For the case of having large off-diagonal elements, the entire T-odd vector-like quarks must be degenerate within a few percent.
In the absence of any guiding mechanism, no particular texture is preferred.

19. Results and Constraints
m12 is the average mass of the first two generations of T-odd fermions
m12 is the mass splitting of the first two generations
In order of the darkest contours to lightest, m12 =500, 1000, 2000, 3000 GeV
Vu=1, Vd=VCKM , f=1 TeV
All constraints arise from neutral K and B mixing, while D system is unaffected.
Experimental constraint: for B and K system, contribution from T-odd fermions not exceed 30% of the SM contribution to the mass splittings and K.
m3 is not constrained. (due to smallness of Vti.

20. Results and Constraints
Vd=1, Vu=VCKM
f=1 TeV
The only constraints arise from D system mass splitting.
Experimental constraint:
m12 < 4.6 10-14 GeV

21. Results and Constraints
Going away from the diagonal: One large angle
d13= 0, put CP phase entirely into up-type relax the constraint.
d13= SM13 , degeneracy for all three generations.

22. Results and Constraints
Going away from the diagonal: all angles large
d13= 0, put CP phase entirely into up-type relax the constraint.
d13= SM13 , sharp degeneracy for all three generations.

23. Results and Constraints
f=1 TeV, after taking BS mixing bound into account: become more constraining
f=1 TeV, d13= 0, before BS mixing bound was imposed

24. Conclusions
The Littlest Higgs with T-parity necessarily introduces new fermions. (It becomes more like SUSY)
These fermions add a new flavor structure to the model.
CP violation is a major source of constraints
For order one mixing parameters, we found that the T-odd fermion spectrum must be degenerate to within a few percent.
In certain cases, this constraint can be alleviated for the third generation.
However, a more detailed study taking into account flavor changing processes such as rare decays would be necessary to confirm that these regions of parameter space are indeed allowed. (under preparation now)