Lecture 7(Theory Part 6 )

SUSY BREAKINGGravity Mediation example

(Recap of Part 6)

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)

Minimal Supersymmetric Standard Model (MSSM)

Gauge group is that of SM:

Strong Weak hypercharge

Vector superfields of the MSSM

(Recap of Part 6)

MSSM Chiral Superfield Content

Left handed quark chiral superfields

Conjugate of right handed quark

superfields

(Recap of Part 6)

Two Higgs doublets

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)

Part 6

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!

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

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

4.2 MSSM Lagragngian densitySuperpotential

With the gauge structure, superfield content and Superpotential now specified we can construct the MSSM Lagrangian.

Scalar covariant derivative Usual gauge-fermion-

fermion vertex

Gaugino interactions from Kahler potential

- 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.

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

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…

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

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

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,

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:

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:

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

4.3 Higgs Bosons in the MSSM

8 scalar Higgs degrees of freedom 3 longitudinal modes for 5 Physical Higgs bosons

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

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

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

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)

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