Lecture 8

OverviewFinal lecture today! Can cover the following topics today: Sfermion, chargino and neutralino masses

Fine Tuning What this really means, how we may quantify it. How LHC squark, gluino and Higgs searches affect this

Changing universality assumptions Relaxing some constraints Using different breaking scheme inspired constraints

Non-minimal Supersymmetry Extend the chiral superfield contentExtend the gauge structure

Can give overview of all or focus on one or two?

MSSM Chiral Superfield Content

Left handed quark chiral 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!

Conjugate of right handed quark

superfields

MSSM Lagragngian densitySuperpotential

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

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:

Higgs Masses

Goldstone bosons

CP-even Higgs bosons

Charged Higgs boson

CP-odd Higgs boson

Sfermion masses

Softmass: (m2)ji Á

i Á¤j + 1

6ai j kÁi Áj Ák

m2~f L

~f L~f L + m2

~f R

~f R~f R + Amf

~f L~f R

(mF L )i j~f iL

~f ¤ jL + yf Af Hu=d

~f iL

~f ¤ jR

Flavour diagonal postulateHu=d ! vu=d

F-terms

FH u = ¹ Hd + yi ju

~ui ~Qj FH d = ¹ Hu + yi jd

~di~Qj + yi j

e~ei

~L j

jFH u j2 yu¹ Hd~u ~Q jFH d j2 yd¹ Hu~d~Q + ye¹ Hu~e~L

jFf R j2 + jFf L j2

D-terms

¢ ~f L= 1

4(T3 ~f Lg2 ¡ Y~f L

g02)(v2d ¡ v2

u)

¢ ~f R= ¡ 1

4Y~f Lg02(v2

d ¡ v2u)

Sfermion masses

L 3r dgen~f ¡ mass

= ¡ (et¤L

et¤R ) m2

et

µetLetR

¶¡

¡eb¤

Let¤R

¢m2

eb

µebLebR

¶¡ ( e¿¤

L e¿¤R ) m2

e¿

µe¿L

e¿R

¶

where

m2et =

µm2

Q3+ m2

t + ¢ ~uL mt(A¤t ¡ ¹ cot ¯)

mt(At ¡ ¹ ¤ cot ¯) m2u3

+ m2t + ¢ ~uR

¶:

m2eb =

µm2

Q3+ m2

b + ¢ ~dLmb(A¤

b ¡ ¹ tan¯)mb(Ab ¡ ¹ ¤ tan¯) m2

d3+ m2

b + ¢ ~dR

¶:

m2e¿ =

µm2

L 3+ m2

¿ + ¢ ~eL m¿ (A¤¿ ¡ ¹ tan¯)

m¿ (A¿ ¡ ¹ ¤ tan¯) m2e3

+ m2¿ + ¢ ~eR

¶:

Home exercise: find all the mistakes on the previous slide, then write in matrix form below and diagonalise.

Chargino and Neutralino masses

a) Find all the mass terms involving gauginos and Higgsinos in the MSSM,including soft SUSY breaking terms and any superpotential mass terms.

M1 ~B ~B + M2 ~Wa ~WaSoft masses:

Superpotential: ¹ ~Hu ~Hd

Kahler potential:

VEVs

(Home exercise)

Hints:

Chargino and Neutralino masses

b) Show that the mass terms for the charged states may be written as,

¡12[~g+ T X T ~g¡ + ~g¡ T X ~g+]+ h:c: = ¡

12

µ~g+

~g¡

¶T µ0 X T

X 0

¶ µ~g+

~g¡

¶+ h:c:

where the states ~g+ and ~g¡ are de ned as

~g+ =µ ~W+

~H +u

¶~g¡ =

µ ~W ¡

~H ¡d

¶

and

X =µ

M2p

2sin¯MWp2cos¯MW ¹

¶

with the MW = g2

4 v2 being themass of the W boson.Since X T 6= X we must perform a biunitary transformation to diagonalise

it.

Chargino and Neutralino masses

c) Convinceyourself that onecan diagonalise this using unitary matrices U,V, so that

U¤X V ¡ 1 =

Ãm~§

10

0 m~§2

!

with ~Â+ = V~g+ and ~¡ = U~g¡

the check that

VX yX V ¡ 1 = U¤X X yUT =

Ãm2

~§1

0

0 m2~§

2

!

and use this to ¯nd expressions for the mass eigenstates m~§1

and m~§2

.

d) Convinceyourself that thesearethedoubly degenerateeigenvalues of the4x4 mass matrix, M y

~ÂM ~Â, where,

M ~Â =µ

0 X T

X 0

¶

Chargino and Neutralino masses

e) In a similar fashion write the mass terms for the neutral states as

Lneutralino mass = ¡12Ã0T M ~N Ã0 + c:c (1)

where Ã0T = ( ~B; ~W0; ~H 0d; ~H 0

u), giving the matrix M ~N .

M eN =

0

B@

M1 0 ¡ g0vd=2 g0vu=20 M2 gvd=2 ¡ gvu=2

¡ g0vd=2 gvd=2 0 ¡ ¹g0vu=2 ¡ gvu=2 ¡ ¹ 0

1

CA :

parameter space volume restricted by,

Parameter space point,

Tuning:

`` ``

Compare dimensionless variations in: ALL parameters vs ALL observables

Our ApproachPA & D.J.Millier PRD 76, 075010 (2007)

parameter space volume restricted by,

Parameter space point,

Tuning:

`` ``

Compare dimensionless variations in: parameters vs observables

Our ApproachPA & D.J.Millier PRD 76, 075010 (2007)

Probability of random point from lying in :

But remember any parameter space point is incredibly unlikely if all equally likely (flat prior)! Fine tuning is when a special qualitative feature ( ) is far less likely that other typical case ( )

mh = O(10¡ 17MP l)mh = O(MP lanck)

Any G << F

parameter space volume restricted by,

Parameter space point,

Tuning:

`` ``

Compare dimensionless variations in: parameters vs observables

Our ApproachPA & D.J.Millier PRD 76, 075010 (2007)

Probability of random point from lying in :

But what if :

) 4 large for all points (or all values of O)

Global sensitivity (Anderson & Castano 1995)

Any G << F

parameter space volume restricted by,

Parameter space point,

Tuning:

`` ``

Compare dimensionless variations in: parameters vs observables

Our ApproachPA & D.J.Millier PRD 76, 075010 (2007)

Probability of random point from lying in :

But what if :

) 4 large for all points (or all values of O)

Rescale to our expectation for

Regardless of measure details, fine tuning is increased when searches increase mass limits on squarks and gluinos:

M 2Z = 2(¡ j¹ j2

+0:076m20

+1:97M 21=2

+0:1A20

+0:38A0M1=2)

Search pushes up. M1=2

Larger cancellation required!

What about the Higgs?

Heavy stops ) Large soft masses

and large one loop corrections

M 2Z = 2(¡ j¹ j2 + 0:076m2

0 + 1:97M 21=2 + 0:1A2

0 + 0:38A0M1=2)

+@

@vd(¢ V) ¡ tan¯ @

@vu(¢ V)

vd(tan2 ¯ ¡ 1)

A relatively heavy Higgs requires heavy stops

Break cMSSM link between stop masses and light squarks and evade fine tuning

LEP bound ) tuning » 10¡ 100

mh = 126 GeV Tuning?Tentative LHC Higgs signal

Fine Tuning Summary

Most important consideration at the LHC (by far) is what do we seec Higgs? Beyond the standard model (BSM) signal?

If BSM signal is observed initially all efforts on understanding new physics. Eventually will know if new physics solves Hierachy Problem Residual tuning may also be a hint about highscale physics

If no SUSY signal? Where does that leave us? Subjective question, depends on tuning measure, but also prejudice Conventional wisdom: no observation ) SUSY is fine tuned! Motivation for low energy SUSY weakened (doesn’t remove fine tuning).

No BSM signal at all Hierarchy Problem motivated BSM models have tuning too. Nature is fine tuned? SM true up to Planck scale? Or we need some great new idea

Beyond the CMSSM(Relaxing high scale constraints)

Non-universal Higgs MSSM (NUHM)

Motivated since Higgs bosons do not fit into the same SU(5) or SO(10) GUT multiplets:

10

5*

+

1

+16

105

5*

Color triplets+

Beyond the CMSSM(Relaxing high scale constraints)

Non-universal Higgs MSSM (NUHM)

Impact: Higgs masses not linked to other scalar masses so strongly easier to fit EWSB constraints and other observables

Motivated since Higgs bosons do not fit into the same SU(5) or SO(10) GUT multiplets:

Very mild modification to the CMSSM

Beyond the CMSSM(Relaxing high scale constraints)

Non-universal Gaugino masses

For universal gauginos we have a (one loop) relation:

Testable predictions for gaugino universality!

Breaks ratio get different gaugino mass patterns:

One can also ignore the universality more parameters to consider the model with less prejudice, e.g. pMSSM

In gauge mediated symmetry breaking the SUSY breaking is transmitted from the hidden sector via SM gauge interactions of heavy messenger fields.

Chiral Messenger fields couple to Hidden sector SUSY breaking in messenger spectrum

SM Gauge interactions couple them to visible sectorLoops from gauge interactions with virtual messengers flavour diagonal soft masses.

Non-universal soft gaugino masses since they depend on gauge interactions!

Soft mass relations imposed at messenger scale

Gauge Mediation

More details and a more general definition given in Steve Abel’s lectures

Loop diagram:

Minimal Gauge Mediated SUSY Breaking (mGMSB)

Messenger fields form Complete SU(5) representations

From EWSB as in CMSSM

Number of SU(5) multiplets

Messenger scale

Beyond the MSSM

Non-minimal Supersymmetry

The fundamental motivations for Supersymmetry are:- The hierarchy problem (fine tuning)- Gauge Coupling Unification- Dark matter

None of these require Supersymmetry to be realised in a minimal form.

MSSM is not the only model we can consider!

The MSSM superpotential is written down before EWSB or SUSY breaking:

The problem

What mass should we use?

The natural choices would be 0 or MPlanck (or MGUT)

) it should know nothing about the EW scale.

Phenomenological Constraints ) ¹ ¼ 0.1 -1 TeV

(

¹-parameter has the dimension of mass!

The superpotential contains a mass scale!

)

Scale of origin

Forbidden by symmetry

)

Solve the -problem by introducing an extra singlet

[Another way is to use the Giudice-Masiero mechanism, which I won’t talk about here.]

Introduce a new iso-singlet neutral colorless chiral superfield , coupling together the usual two Higgs doublet superfields.

If S gains a vacuum expectation value we generate an effective -term, automatically of oder the electroweak scale

with

We must also modify the supersymmetry breaking terms to reflect the new structure

Yukawa terms effective -term

So our superpotential so far is

But this too has a problem – it has an extra U(1) Peccei-Quinn symmetry

Lagrangian invariant under the (global) transformation:

This extra U(1) is broken with electroweak symmetry breaking (by the effective -term)

massless axion!

Yukawa terms effective -termPQ breaking term

NMSSM Chiral Superfield Content

massless axion!

The superpotential of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) is

Yukawa terms effective -term

PQ breaking term

We also need new soft supersymmetry breaking terms in the Lagrangian:

[Dine, Fischler and Srednicki][Ellis, Gunion, Haber, Roszkowski, Zwirner]

Modified Higgs sector: 3 CP-even Higgs, 2 CP-odd Higgs (new real and imagnary scalar S)

“ Neutralino sector: 5 neutralinos (new fermion component of S)

Parameters:

The MSSM limit is ! 0, ! 0, keeping / and fixed.

and are forced to be reasonably small due to renormalisation group running.

Top left entry of CP-odd mass matrix. Becomes MSSM MA in MSSM limit.

minimisation conditions

Finally:

Supersymmetric Models

Minimal Supersymmetric Standard Model (MSSM)

Next to Minimal Supersymmetric Standard Model (NMSSM)[Dine, Fischler and Srednicki] [Ellis, Gunion, Haber, Roszkowski, Zwirner]

Decouple the axion PQSNMSSM

Alternative solution to Peccei–Quinn symmetry :

Linear S term nMSSM

Eat the axion Z0 models (e.g. USSM, E6SSM)

In the latter we extend the gauge group of the SM with an extra gauged U(1)0!

When U(1)0 is broken as S gets a vev, Z0 eats the masless axion to become massive vector boson!

Supersymmetric Models

Minimal Supersymmetric Standard Model (MSSM)

Next to Minimal Supersymmetric Standard Model (NMSSM)

Other variants: nmMSSM, PQSNMSSM.

U(1) extended Supersymmetric Standard Model (USSM)

Exceptional Supersymmetric Standard Model (E6SSM)

[Dine, Fischler and Srednicki][Ellis, Gunion, Haber, Roszkowski, Zwirner]

[S.F. King, S. Moretti, R. Nevzrov, Phys.Rev. D73 (2006) 035009]

Yukawa terms effective -term

USSM Chiral Superfield Content

Problem: to avoid gauge anomalies

Yukawa terms effective -term

USSM Chiral Superfield Content

Problem: to avoid gauge anomalies

Charges not specified in the definition of the USSM

U(1) extended Supersymmetric Standard Model (USSM)

Yukawa terms effective -term

Modified Higgs sector: 3 CP-even Higgs, 2 CP-odd Higgs (new real and imagnary scalar S) Modified Neutralino sector: 6 neutralinos: (new singlino + Zprimino )

Modified Gauge sector, new Z0

Disclaimer: I work on the E6SSM

Final part included for vanity

For anomaly cancelation, one can use complete E6 matter multiplets

New U(1)0 from E6

E6 inspired models

Matter from 3 complete generations of E6

) automatic cancellation of gauge anomalies!

In the E6SSM ) right-handed neutrino is a gauge singlet

All the SM matter fields are contained in one 27-plet of E6 per generation.

27

10, 1

5*, 2

5*, - 3

5, - 2

1, 0

+

+

+

+

U(1)N chargeSU(5) reps.

1, 5+ singlets

right handed neutrino

3 generations of “Higgs”

exotic quarks

Exceptional Supersymmetric Standard Model(E6SSM)

[Phys.Rev. D73 (2006) 035009 , Phys.Lett. B634 (2006) 278-284 S.F.King, S.Moretti & R. Nevzorov]

E6SSM Chiral Superfield Content

Note: In it’s usual form there are also two extra SU(2) doublets included for single step gauge coupling unification, but these are negleected here for simplicity.

SUSY Theory space

Gauge group(vector

superfields)

Chiral superfieldsMinimal

superfields

Complete E6 multiplets

E6SSM

MSSMNMSSM

USSM

Thank you for listening

End of Supersymmetry Lecture course

Sfermion masses

m2~f L

~f L~f ¤L + m2

~f R

~f ¤R

~f R + Amf~f L

~f ¤R

¹ mf u cot ¯ ~f uR

~f uL ¹ mf d tan¯ ~f d

R~f d

L

(where f u and f d are up and down type fermions respectively)

¢ ~f L~f L

~f ¤L ¢ ~f R

~f R~f ¤R

L 3rdgen~f ¡ mass

= ¡ (et¤L

et¤R ) m2

et

µetLetR

¶¡

¡eb¤

Let¤R

¢m2

eb

µebLebR

¶¡ ( e¿¤

L e¿¤R ) m2

e¿

µe¿L

e¿R

¶

where

m2et =

µm2

Q3+ m2

t + ¢ ~uL v(A¤t sin¯ ¡ ¹ yt cos¯)

v(At sin¯ ¡ ¹ ¤yt cos¯) m2u3

+ m2t + ¢ ~uR

¶:

m2et =

µm2

Q3+ m2

b + ¢ ~dLv(A¤

t sin¯ ¡ ¹ yt cos¯)v(At sin¯ ¡ ¹ ¤yt cos¯) m2

d3+ m2

t + ¢ ~dR

¶: