Lecture 7 (Theory Part 6 )
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Transcript of Lecture 7 (Theory Part 6 )
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Lecture 7(Theory Part 6 )
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SUSY BREAKINGGravity Mediation example
(Recap of Part 6)
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Soft SUSY breaking Lagrangian[Shown to be soft to all orders, L. Girardello, M. Grisaru]
All dimension 3 or less,) all coefficients have mass dimension!
) relationships between dimensionless couplings maintained!
(Recap of Part 6)
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Minimal Supersymmetric Standard Model (MSSM)
Gauge group is that of SM:
Strong Weak hypercharge
Vector superfields of the MSSM
(Recap of Part 6)
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MSSM Chiral Superfield Content
Left handed quark chiral superfields
Conjugate of right handed quark
superfields
(Recap of Part 6)
Two Higgs doublets
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MSSM
Lepton number violating
Baryon number violating
R-parity All SM particles + Higgs bosons:
All SUSY particles:
) SUSY particles appear in even numbers ) SUSY pair production
) Lightest Supersymmetric Particle (LSP) is stable!
Gives rise to a Dark Matter candidate.
Evade proton decay:
Superpotential
(Recap of Part 6)
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Part 6
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4.2 MSSM Lagragngian densitySuperpotential
With the gauge structure, superfield content and Superpotential now specified we can construct the MSSM Lagrangian.
SM-like Yukawa coupling H-f-f
Higgs-squark-quark couplings with same Yukawa coupling!
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4.2 MSSM Lagragngian densitySuperpotential
With the gauge structure, superfield content and Superpotential now specified we can construct the MSSM Lagrangian.
Quartic scalar couplings again from the same Yukawa coupling
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4.2 MSSM Lagragngian densitySuperpotential
With the gauge structure, superfield content and Superpotential now specified we can construct the MSSM Lagrangian.
Non-abelian self interactions from gauge-kinetic term
Gauge-gaugino-gaugino SUSY version of this
[See page 86 of Drees, Godbole, Roy]
Auxialliary D-term
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4.2 MSSM Lagragngian densitySuperpotential
With the gauge structure, superfield content and Superpotential now specified we can construct the MSSM Lagrangian.
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Scalar covariant derivative Usual gauge-fermion-
fermion vertex
Gaugino interactions from Kahler potential
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- 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.
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4.3 Electroweak Symmetry Breaking (EWSB)
Recall in the SM the Higgs potential is:
Underlying SU(2) invariance ) the direction of the vev in SU(2) space is arbitrary.
Vacuum Expectation Value (vev)
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
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EWSB
Recall in the SM the Higgs potential is:
In the MSSM the full scalar potential is given by:
Extract Higgs terms:
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EWSB
And after a lot of algebra…
VH = (m2H d
+ j¹ j2)(jH 0dj2 + jH ¡
d j2) + (m2H u
+ j¹ j2)(jH +u j2 + jH 0
u j2)
+B¹ (H +u H ¡
d ¡ H 0uH 0
d + h.c.) +18(g2 + g02)
¡jH 0
d j2 + jH ¡d j2 ¡ jH +
u j2 ¡ jH 0u j2
¢2
+12g2(H + ¤
u H 0d + H 0¤
u H ¡d )(H +
u H 0¤d + H 0
uH ¡ ¤d )
The Higgs Potential
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EWSB conditions
VH = (m2H d
+ j¹ j2)(jH 0dj2 + jH ¡
d j2) + (m2H u
+ j¹ j2)(jH +u j2 + jH 0
u j2)
+B¹ (H +u H ¡
d ¡ H 0uH 0
d + h.c.) +18(g2 + g02)
¡jH 0
d j2 + jH ¡d j2 ¡ jH +
u j2 ¡ jH 0u j2
¢2
+12g2(H + ¤
u H 0d + H 0¤
u H ¡d )(H +
u H 0¤d + H 0
uH ¡ ¤d )
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
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EWSB conditions
VH = (m2H d
+ j¹ j2)jH 0d j2 + (m2
H u+ j¹ j2)jH 0
u j2 ¡ B¹ (H 0uH 0
d + h.c.)
+18(g2 + g02)
¡jH 0
d j2 ¡ jH 0u j2
¢2
First consider:
(m2H d
+ m2H u
+ 2j¹ j2) ¸ 2B¹ cosÁ
To ensure potential is bounded from below:
Only phase in potential
Choosing phase to maximise contribution of B¹ reduces potential:
For the origin in field space, we have a Hessian of,
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EWSB conditions
VH = (m2H d
+ j¹ j2)jH 0d j2 + (m2
H u+ j¹ j2)jH 0
u j2 ¡ B¹ (H 0uH 0
d + h.c.)
+18(g2 + g02)
¡jH 0
d j2 ¡ jH 0u j2
¢2
For successful EWSB:(m2
H d+ m2
H u+ 2j¹ j2) ¸ 2B¹
(m2H d
+ j¹ j2)(m2H u
+ j¹ j2) · (B¹ )2
With:
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Recall from SUSY breaking section, gravity mediation implies:
Take minimal set of couplings:(warning: minimal flavour diagonal couplings not motivated here, just postulated)
Universal soft scalar mass:
Universal soft gaugino mass:
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:
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Radiative EWSBRenormalisation group equations (RGEs) connect soft masses at MX to the EW scale.
RGEs naturally trigger EWSB:
(m2H d
+ j¹ j2)(m2H u
+ j¹ j2) · (B¹ )2
Runs negative
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4.3 Higgs Bosons in the MSSM
8 scalar Higgs degrees of freedom 3 longitudinal modes for 5 Physical Higgs bosons
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4.3 Higgs Bosons in the MSSM
8 scalar Higgs degrees of freedom 3 longitudinal modes for 5 Physical Higgs bosons
Note: no mass mixing term between neutral and charged components, nor between real and imaginary components.
Goldstone bosons
CP-even Higgs bosons
Charged Higgs boson
CP-odd Higgs boson
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CP-odd mass matrix
VH 2 (m2H d
+ j¹ j2)(I mH 0d)2 + (m2
H u+ j¹ j2)(I mH 0
u)2 + B¹ I m(H 0u)I m(H 0
d)
+18(g2 + g02)
¡(ReH 0
d)2 + (I mH 0d)2 ¡ (ReH 0
u)2 ¡ (I mH 0u)2¢2
Included for vevs
Eigenvalue equation
Massless Goldstone boson
CP-odd Higgs
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Charged Higgs mass matrix
VH = (m2H d
+ j¹ j2)jH ¡d j2 + (m2
H u+ j¹ j2)jH +
u j2 + B¹ (H +u H ¡
d + h.c.)
+18(g2 + g02)
¡H 0
d j2 + jH ¡d j2 ¡ jH +
u j2 ¡ jH 0u j2
¢2
+12g2(H + ¤
u H 0d + H 0¤
u H ¡d )(H +
u H 0¤d + H 0
uH ¡ ¤d )
Massless Goldstone boson
Charged Higgs
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CP-Even neutral Higgs mass matrix
VH 2 (m2H d
+ j¹ j2)(ReH 0d)2 + (m2
H u+ j¹ j2)(ReH 0
u)2 + B¹ Re(H 0u)Re(H 0
d)
+18(g2 + g02)
¡(ReH 0
d)2 ¡ (ReH 0u)2¢2
Taylor expand:
Upper bound:
Consequence of quartic coupling fixed in terms of gauge couplings ( compare with free ¸ parameter in SM)
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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