Transcript Document

Lecture 7
Tuesday…
Superfield content of the MSSM
Strong
Weak
hypercharge
Gauge group is that of SM:
Vector superfields of the MSSM
MSSM Chiral Superfield Content
Left handed quark
chiral superfields
Conjugate of right
handed quark
superfields
Note: left handed fermions are described by chiral superfields,
right handed fermions by anti-chiral superfields.
Superpotential is a function of chiral superfields only so we include right handed fermions
by taking the conjugate, which transforms as a left handed superfield!
MSSM Lagragngian density
Superpotential
With the gauge structure, superfield content and Superpotential now specified
we can construct the MSSM Lagrangian.
A SUSY signature at the LHC
Superfield
strength
Kahler
potential
Lightest
supersymmetric
particle (LSP)
R-parity
conservation
signal
Contributes to:
- MSSM is phenomenologically viable model currently searched for at the LHC
-Predicts many new physical states:
- Very large number of parameters (105)!
- These parameters arise due to our ignorance of how SUSY is broken.
Electroweak Symmetry Breaking (EWSB)
Recall in the SM the Higgs potential is:
Vacuum Expectation Value (vev)
Underlying SU(2) invariance ) the direction of the vev in SU(2) space is arbitrary.
Any choice breaks SU(2) £ U(1)Y in the vacuum, choosing
All SU(2) £ U(1)Y genererators broken:
But for this choice
Showing the components’
charge under unbroken
generator Q
EWSB
Recall in the SM the Higgs potential is:
In the MSSM the full scalar potential is given by:
Extract Higgs terms:
EWSB
And after a lot of algebra…
The Higgs Potential
VH
=
(m 2H d + j¹ j 2 )(jH d0 j 2 + jH d¡ j 2 ) + (m 2H u + j¹ j 2 )(jH u+ j 2 + jH u0 j 2 )
¡ 02
¢
1 2
¡
¡ 2
+
0 0
02
+ 2
0 2 2
+ B ¹ (H u H d ¡ H u H d + h.c.) + (g + g ) jH d j + jH d j ¡ jH u j ¡ jH u j
8
1
+ g2 (H u+ ¤ H d0 + H u0 ¤ H d¡ )(H u+ H d0 ¤ + H u0 H d¡ ¤ )
2
EWSB conditions
VH
=
(m 2H d + j¹ j 2 )(jH d0 j 2 + jH d¡ j 2 ) + (m 2H u + j¹ j 2 )(jH u+ j 2 + jH u0 j 2 )
¡
¢2
1
+ B ¹ (H u+ H d¡ ¡ H u0 H d0 + h.c.) + (g2 + g02 ) jH d0 j 2 + jH d¡ j 2 ¡ jH u+ j 2 ¡ jH u0 j 2
8
1
+ g2 (H u+ ¤ H d0 + H u0 ¤ H d¡ )(H u+ H d0 ¤ + H u0 H d¡ ¤ )
2
As in the SM, underlying SU(2)W invariance means we can choose one component
of one doublet to have no vev:
Choose:
B¹ term unfavorable for stable
EWSB minima
EWSB conditions
VH
=
(m 2H d + j¹ j 2 )jH d0 j 2 + (m2H u + j¹ j 2 )jH u0 j 2 ¡ B ¹ (H u0 H d0 + h.c.)
¡ 02
¢
1 2
02
0 2 2
+ (g + g ) jH d j ¡ jH u j
8
Only phase in
potential
First consider:
To ensure potential is bounded from below:
(m2H d + m2H u + 2j¹ j 2 ) ¸ 2B ¹ cosÁ
Choosing phase to maximise contribution of B¹ reduces potential:
For the origin in field space,
we have a Hessian of,
EWSB conditions
VH
=
(m 2H d + j¹ j 2 )jH d0 j 2 + (m2H u + j¹ j 2 )jH u0 j 2 ¡ B ¹ (H u0 H d0 + h.c.)
¡ 02
¢
1 2
02
0 2 2
+ (g + g ) jH d j ¡ jH u j
8
For successful EWSB:
With:
(m2H d + m2H u + 2j¹ j 2 )
¸
2B ¹
(m2H d + j¹ j 2 )(m2H u + j¹ j 2 )
·
(B ¹ ) 2
Recall from SUSY breaking section, gravity mediation implies:
Take minimal set of couplings:
Universal soft scalar mass:
(warning: minimal flavour
Universal soft gaugino mass:
diagonal couplings not
motivated here, just postulated)
Universal soft trilinear mass:
Universal soft bilinear mass:
Fits into a SUSY Grand unified Theory where chiral superfields all transform together:
Idea: Single scale for universalities, determined from gauge coupling unification!
Constrained MSSM:
Radiative EWSB
Renormalisation group equations (RGEs) connect soft masses at MX to the EW scale.
RGEs naturally trigger
EWSB:
(m2H d + j¹ j 2 )(m2H u + j¹ j 2 ) ·
Runs negative
(B ¹ ) 2
Constrained MSSM (cMSSM)
(Slope 1 from Snowmass points and slopes)
Higgs Bosons in the MSSM
8 scalar Higgs degrees of freedom
Note:
3 longitudinal modes for
5 Physical Higgs bosons
no mass mixing term between neutral and charged components,
nor between real and imaginary components.
CP-even Higgs
bosons
CP-odd Higgs
boson
Goldstone
bosons
Charged Higgs
boson
CP-odd mass matrix
VH
2
(m 2H d + j¹ j 2 )(I mH d0 ) 2 + (m2H u + j¹ j 2 )(I mH u0 ) 2 + B ¹ I m(H u0 )I m(H d0 )
¡
¢
1 2
02
0 2
0 2
0 2
0 2 2
+ (g + g ) (ReH d ) + (I mH d ) ¡ (ReH u ) ¡ (I mH u )
8
Included for vevs
Eigenvalue equation
Massless Goldstone boson
CP-odd Higgs
Charged Higgs mass matrix
VH
=
(m 2H d + j¹ j 2 )jH d¡ j 2 + (m 2H u + j¹ j 2 )jH u+ j 2 + B ¹ (H u+ H d¡ + h.c.)
¡ 02
¢
1 2
¡ 2
02
+ 2
0 2 2
+ (g + g ) H d j + jH d j ¡ jH u j ¡ jH u j
8
1
+ g2 (H u+ ¤ H d0 + H u0 ¤ H d¡ )(H u+ H d0 ¤ + H u0 H d¡ ¤ )
2
Massless Goldstone boson
Charged Higgs
CP-Even neutral Higgs mass matrix
Taylor expand:
Upper bound:
Consequence of quartic coupling fixed in terms of gauge couplings
( compare with free ¸ parameter in SM)
Upper bound:
Consequence of quartic coupling fixed in terms of gauge couplings
(compare with free ¸ parameter in SM)
Radiative corrections significantly raise this
Including radiative
corrections