3: the mu term in the MSSM superpotential gives mass to the
higgsinos. The EWSB conditions require that mu be of the order of the
soft masses and of the order of the weak scale;

4: mu problem: if mu is a parameter of the SUSY theory valid up to
M_planck, why is it not of O(M_planck) itself? Giudice-Masiero
solution: mu is generated together by the soft terms by the mechanism
of SUSY breaking; NMSSM solution: mu arises at the weak scale as the
vev of an additional singlet.

5-8: Higgs sector of the NMSSM: the scalar, pseudoscalar and fermionic
components of the singlet superfield mix with the corresponding
components of the MSSM-like doublet superfields. In the pseudoscalar
sector, only P1 and P2 contribute to the Goldstone boson, but all
three pseudoscalar components contribute to the two physical
pseudoscalars A1 and A2.

9-10: in the limit in which the singlet vev is much larger than the
weak scale the singlet decouples from the MSSM-like Higgs sector;
however, the mass of the lightest MSSM-like Higgs retains an
additional, F-term induced contribution proportional to
lambda^2*sin(2Beta)^2.

11: through the effect of the additional quartic coupling, for large
lambda the lightest Higgs boson mass is maximized at low tanB (in
contrast with the case of the MSSM in which the mass is maximized at
large tanB).

12: another interesting feature of the NMSSM is the possibility that a
100-GeV Scalar is not excluded by the LEP. This can happen if a
sizeable singlet component suppresses the coupling to the Z, or if the
scalar decays mostly into two pseudoscalars which are in turn too
light to decay into b quarks. This scenario would also allow for
lighter squarks than is typical in the MSSM.

14-16: list of the calculations of the MSSM Higgs masses, from the
early calculations of 1991 to the three-loop results of last
year. Most public computer codes implement a full one-loop calculation
(a la PBMZ, or variations) and some linear combination of the
zero-momentum results of Slavich et al. and Heinemeyer et al. for the
two-loop quark/squark contributions.

17: the situation of the NMSSM is not quite as advanced. The one-loop
quark/squark contributions have been computed in the effective
potential approach; the one-loop gauge/gaugino and Higgs/higgsino
contributions, as well as the two-loop quark/squark contributions,
have been computed only at the leading-logarithmic level in the
renormalization group approach. All of these results are implemented
in the public computer code NMHDECAY.

18-20: the situation of the NMSSM is comparable to the situation of
the MSSM as it was in the mid nineties.

21: my paper with G. Degrassi, arXiv:0907.4682, provides a complete
one-loop calculation of the neutral Higgs masses (PBMZ-like) and an
effective potential calculation of the two-loop strong corrections.

22: summary of the one-loop calculation. We need to compute
self-energies and tadpole diagrams for the CP-even Higgses, plus the
self-energies of the CP-odd Higgses and of the gauge bosons.

23: summary of the two-loop calculation in the effective potential
approach: we need to compute the two-loop vacuum diagrams in terms of
field-dependent parameters and take the first and second derivatives
with respect to the fields.

24: the parameters in the top/stop sector depend on the Higgs fields
through the left-right mixing terms of the top and stop mass
matrices. Our set of field-dependent parameters includes the top and
stop squared masses, the stop mixing angle and a combination of the
phases of the left-right mixing terms.

25: compact formulae for the corrections to the CP-even mass
matrix. The first three lines correspond to the known MSSM results
while the last three are specific to the NMSSM.

26: definitions of the functions appearing in slide 25, in terms of
the derivatives of the effective potential w.r.t. the field-dependent
parameters. The derivatives are computed at the minimum of the
potential.

27-28: excerpt from arXiv:0907.4682, showing that the full two-loop
formulae for the derivatives fit in a four-page appendix.

30-32: effect of the one-loop corrections to the two lightest scalar
masses as a function of lambda, for a specific choice of the remaining
parameters. The comparison between the dotted and dashed lines shows
that the bulk of the correction to the lightest scalar mass comes from
the quark/squark contribution, as in the MSSM. However, the comparison
between the dashed and solid lines shows that the remaining
electroweak and higgs/higgsino contributions can amount to several GeV
for both masses.

33: effect of the one-loop corrections on the mixing between singlet
and doublets in the two lightest scalars. The plot shows the square of
the (i,3) element of the scalar mixing matrix a function of
lambda. For small lambda the lightest scalar is mostly doublet and the
other is mostly singlet, but when lambda increases they cross
over. The value of lambda for which they cross over depends crucially
on the accuracy of the calculation.

34: effect of the one-loop corrections to the lightest pseudocalar
mass as a function of lambda. In this scenario the lightest
pseudoscalar is mostly singlino thus it is not affected by the
quark/squark contributions. However, the higgs/higgsino contributions
can still amount to a few GeV.

35-39: effect of the two-loop corrections to the two lightest scalar
masses as a function of a common squark mass MS, for a specific choice
of the remaining parameters. The comparison between the dotted, dashed
and solid lines shows that the leading-logarithmic terms account only
for a fraction of the total two-loop corrections to the lightest scalar
mass, especially for small MS.

40: same plot as above with the opposite value of At. It shows that
the very good agreement in the previous slide between the
leading-logarithmic and "full" two-loop results for the mass of the
second-lightest scalar was due to an accidental cancellation.

41: effect of the two-loop corrections on the mixing between singlet
and doublets in the two lightest scalars. Smaller than the effect of
the one-loop corrections but still visible.

43: conclusions: the newly computed corrections are necessary for a
meaningful comparison between the MSSM and NMSSM predictions for the
Higgs sector.

45: to-do list: implement the new results into NMHDECAY; extend the
calculation to the charged Higgs; compute the two-loop corrections
controlled only by the Yukawa couplings.