1. No category

2015-04-07
Hand-in Exercise:
Higgs Physics and Beyond Standard Model
1. Higgs mass: quadratic divergence
Calculate the one-loop corrections to the Higgs (φ) self-energy (propagator) from a scalar
(S) and a fermion (f ) with couplings:
yf
mf
= −i √
φφSS : −iλS , φf f¯ : −i
v
2
where v ≈ 246 GeV is the vacuum expectation value of the Higgs field.
Since we are only interested in the ultraviolet properties of the diagrams you can neglect
the momentum of the Higgs particle compared to the loop momentum and use that
Z
Λ
d4 k
1
Λ2
Λ2
2
=
+ O m log 2
(2π)4 k 2 − m2
16π 2
m
m2φ
a) Do this calculation in the SM with S being the Higgs field itself so that λS = 2
v
and f is the t-quark with mass mt . (Remember to take into account the symmetry
factor in the scalar case and the number of colours for the t-quark)
b) Show that if instead λS = yt2 , then the quadratic divergence is canceled provided
that the scalar S comes with the same number of colours as the t-quark. (A proper
calculation shows that in a supersymmetric theory the logarithmic divergence also
cancels if mf = mS )
A proper calculation in the SM1 gives the corrections to the Higgs mass
3Λ2
m2φ = m2φ,0 + 2 2 4m2t − 2m2W − m2Z − m2φ
8π v
2. Gauge coupling unification
At one-loop order the renormalization group equations for the gauge couplings (α3 = αs ,
α
5α
, α1 =
) have the following form
α2 =
2
3 cos2 θW
sin θW
d
bi 2
Q αi =
α
dQ
2π i
33
41 19
SSM
= ( , 1, −3)
with bSM
= ( , − , −7) and bM
i
i
10
6
5
Make a plot with the inverse couplings in the range 102 GeV < Q < 1019 GeV for the
following cases (be careful to choose appropriate starting values):
a) Only SM
b) SM up to 103 GeV and from then on MSSM
c) SM up to 105 GeV and from then on MSSM
P.T.O.
1
M. J. G. Veltman, “The Infrared - Ultraviolet Connection,” Acta Phys. Polon. B 12 (1981) 437.
1
3. Signal strength
Consider the 2HDM in the Higgs basis:
√ +
1
2G
Φ1 = √
0
2 v + H1 + iG
,
1
Φ2 = √
2
√
2H +
H2 + iA
where the physical fields (mH > mh ):
H = H1 cos(β − α) − H2 sin(β − α)
h = H1 sin(β − α) + H2 cos(β − α)
In an arbitrary basis, characterized by tan β = vˆ2 /ˆ
v1 , the Higgs doublets are given by
ˆ 1 = Φ1 cos β − Φ2 sin β
Φ
ˆ 2 = Φ1 sin β + Φ2 cos β
Φ
The Yukawa sector (for quark fields) in the Higgs basis can be written as
¯ L κD Φ1 DR + (U¯ , D)
¯ L ρD Φ2 DR + (U¯ , D)
¯ L κU Φ
˜ 1 DR + (U¯ , D)
¯ L ρU Φ
˜ 2 DR + h.c.
−LY = (U¯ , D)


 
u
d



c
s  are vectors in flavour space and κF , ρF (where
, D =
where U =
t
b
F = D or√U ) are 3 × 3 matrices with κF being related to the diagonal mass matrix M F
by κF = 2M F /v whereas ρF in the most general case is arbitrary.
ˆ1 → Φ
ˆ 1, Φ
ˆ 2 → −Φ
ˆ 2 and FR → ±FR
a) Show that if one imposes a Z2 -symmetry under Φ
F
F
F
in some arbitrary basis then it follows that ρ = cot βκ or ρ = − tan βκF in the
two respective cases.
b) Use the result in (a) to calculate the ratio of the Yukawa couplings to the h, H
and A Higgses in a type II 2HDM (which has a Z2 -symmetry under DR → DR ,
UR → −UR ) to the ones in the SM.
c) Use the result in (b) to calculate the signal strength for h → ZZ in a type II 2HDM
in the limit that the production of the Higgs is completely dominated by gluon-gluon
fusion via a t-loop and that its width is dominated by the decay to b¯b. The signal
strength µZZ is in the general case defined as
P
σk (pp → h + Xk ) × BR(h → ZZ)
.
µZZ = P k
k σk (pp → HSM + Xk ) × BR(HSM → ZZ)
You should find that in the limit above µZZ =
sin2 (β − α)
.
tan2 β tan2 α
d) Make a plot of the region in the (tan β, sin(β − α))-plane, which is compatible with
the latest data on the Higgs signal strength in the ZZ (or 4l) channel2 . You can
limit yourself to the ranges 1 < tan β < 50, 0 < sin(β − α) < 1.
2
G. Aad et al. [ATLAS
and CMS Collaborations], “Combined Measurement of the Higgs Boson
√
Mass in pp Collisions at s = 7 and 8 TeV with the ATLAS and CMS Experiments,” arXiv:1503.07589
[hep-ex].
2