Transcript Slide 1

Lecture 8
Overview
Final 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 content
Extend the gauge structure
Can give overview of all or focus on one or two?
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.
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
Higgs Masses
CP-even Higgs
bosons
CP-odd Higgs
boson
Goldstone
bosons
Charged Higgs
boson
Sfermion masses
Softmass: (m2 ) j Ái Á¤ + 1 ai j k Ái Áj Ák
j
i
6
Flavour diagonal postulate
H u=d ! vu=d
(mF L ) i j f~Li f~L¤ j + yf A f H u=d f~Li f~R¤ j
m2f~ f~L f~L + m2f~ f~R f~R + Amf f~L f~R
L
R
F-terms
~j
~i Q
FH u = ¹ H d + yui j u
jFH u j 2
~
~Q
yu ¹ H d u
~Q
ij ~ ~
~
FH d = ¹ H u + ydi j d
i j + ye ei L j
~Q
~ + ye¹ H u ~
yd ¹ H u d
eL~
jFH d j 2
jFf R j 2 + jFf L j 2
D-terms
¢ f~L =
1
4 (T3f~L
¢ f~R = ¡
g2 ¡ Yf~L g02 )(vd2 ¡ vu2 )
1
4 Yf~L
g02 (vd2 ¡ vu2 )
Sfermion masses
Home exercise: find all the mistakes on the previous slide, then write in matrix
form below and diagonalise.
µ
dgen
¤
e
L 3r
=
¡
(
t
L
~
f ¡ m ass
e
t ¤R ) me2t
e
tL
e
tR
¶
¡
¡
eb¤
L
¢
2
e
t ¤R m e
b
µ
ebL
ebR
¶
µ
¡ (¿
eL¤
where
µ
me2t =
µ
2
me
=
b
µ
2
me
=
¿
m 2Q 3
¶
m 2t
+
+ ¢ u~L
m t (A t ¡ ¹ ¤ cot ¯)
m t (A ¤t ¡ ¹ cot ¯)
m 2u 3 + m 2t + ¢ u~R
m 2Q 3 + m 2b + ¢ d~L
m b(A b ¡ ¹ ¤ t an ¯)
m b(A ¤b ¡ ¹ t an ¯)
m 2d + m 2b + ¢ d~R
m 2L 3
m ¿ (A ¤¿ ¡ ¹ t an ¯)
m 2e3 + m 2¿ + ¢ e~R
m 2¿
+
+ ¢ e~L
m ¿ (A ¿ ¡ ¹ ¤ t an ¯)
:
¶
:
3
¶
:
2
¿
eR¤ ) m e
¿
¿
eL
¿
eR
¶
Chargino and Neutralino masses
(Home exercise)
a) Find all t he mass t erms involving gauginos and Higgsinos in t he MSSM,
including soft SUSY breaking t erms and any superpot ent ial mass t erms.
Hints:
Soft masses:
Superpotential:
~ aW
~a
M 1 B~B~ + M 2 W
¹ H~u H~d
VEVs
Kahler potential:
Chargino and Neutralino masses
b) Show t hat t he mass t erms for t he charged st at es may be writ t en as,
1 +T T ¡
¡ [~
g X g
~ + g
~¡
2
1
T
+
Xg
~ ] + h:c: = ¡
2
µ
+
g
~
g
~¡
¶T µ
0
X
T
X
0
¶µ
+
g
~
g
~¡
¶
+ h:c:
where t he st at es g
~+ and g
~¡ are de¯ned as
µ
g
~+ =
~+
W
H~u+
¶
µ
g
~¡ =
~¡
W
H~ ¡
¶
d
and
µ
X =
g2
4
p
M2
2 cos¯M W
p
2 sin ¯M W
¹
¶
wit h t he M W =
v2 being t he mass of t he W boson.
Since X T 6
= X we must perform a biunit ary t ransformat ion t o diagonalise
it .
Chargino and Neutralino masses
c) Convince yourself t hat one can diagonalise t his using unit ary mat rices U,
V , so t hat
Ã
¤
U XV
¡ 1
=
m Â~§
1
0
0
m Â~§
!
~¡
~¡ = U g
~+ and Â
~+ = V g
wit h Â
2
t he check t hat
Ã
V X yX V ¡
1
= U¤X X yUT =
!
m 2Â~§
0
0
m 2Â~§
1
2
and use t his t o ¯nd expressions for t he mass eigenst at es m Â~§ and m Â~§ .
1
2
d) Convince yourself t hat t hese are t he doubly degenerat e eigenvalues of t he
4x4 mass mat rix, M Â~y M Â~ , where,
µ
M Â~ =
0
X
T
X
0
¶
Chargino and Neutralino masses
e) In a similar fashion writ e t he mass t erms for t he neut ral st at es as
L neut ralino mass = ¡
1 0T
à M N~ Ã0 + c:c
2
~ 0 ; H~ 0 ; H~ 0 ), giving t he mat rix M ~ .
where Ã0 T = ( B~; W
u
d
N
0
1
M1
0
¡ g0vd =2 g0vu =2
0
M2
gvd =2 ¡ gvu =2 C
B
M Ne = @ 0
A:
¡ g vd =2 gvd =2
0
¡ ¹
g0vu =2 ¡ gvu =2
¡ ¹
0
(1)
Our Approach
PA & D.J.Millier PRD 76, 075010 (2007)
Compare dimensionless variations in:
ALL parameters vs ALL observables
Parameter space point,
parameter space volume restricted by,
``
Tuning:
``
Our Approach
PA & D.J.Millier PRD 76, 075010 (2007)
Compare dimensionless variations in:
parameters vs observables
Parameter space point,
parameter space volume restricted by,
``
``
Tuning:
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 ( mh = O(10¡ 17 M P l )) is
far less likely that other typical case (mh = O(M P l an ck ))
Our Approach
PA & D.J.Millier PRD 76, 075010 (2007)
Compare dimensionless variations in:
parameters vs observables
Parameter space point,
parameter space volume restricted by,
``
``
Tuning:
Probability of random point from
lying in
:
But what if :
Any G << F
) 4
large for all points (or all values of O)
Global sensitivity (Anderson & Castano 1995)
Our Approach
PA & D.J.Millier PRD 76, 075010 (2007)
Compare dimensionless variations in:
parameters vs observables
Parameter space point,
parameter space volume restricted by,
``
``
Tuning:
Probability of random point from
lying in
:
But what if :
Any G << F
) 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 Z2
=
2(¡ j¹ j 2
+ 0:076m 20
2
+ 1:97M 1=2
+ 0:1A 20
+ 0:38A 0 M 1=2 )
Search pushes M 1=2 up.
Larger cancellation required!
Heavy stops
)
Large soft masses
2
M Z2 = 2(¡ j¹ j 2 + 0:076m 20 + 1:97M 1=2
+ 0:1A 20 + 0:38A 0 M 1=2 )
+
@
@v d
(¢ V ) ¡ t an ¯ @@
v u (¢ V )
vd (t an2 ¯ ¡ 1)
and large one loop corrections
Break cMSSM link between stop masses and light squarks and evade fine tuning
What about the Higgs?
A relatively heavy Higgs requires heavy stops
Tentative LHC Higgs signal
mh = 126 GeV
LEP bound
Tuning?
)
tuning » 10 ¡ 100
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
+
16
5*
+
1
5
10
+
5*
Color
triplets
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:
Very mild modification to the CMSSM
Impact: Higgs masses not linked to other scalar masses so strongly
easier to fit EWSB constraints and other observables
Beyond the CMSSM
(Relaxing high scale constraints)
For universal gauginos we have a (one loop) relation:
Testable predictions for gaugino universality!
Non-universal Gaugino masses
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
Gauge Mediation
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 sector
Loops from gauge interactions with virtual messengers
flavour diagonal soft masses.
Loop diagram:
Soft mass relations imposed at messenger scale
Non-universal soft gaugino masses since they depend on gauge interactions!
More details and a more general definition given in Steve Abel’s lectures
Minimal Gauge Mediated SUSY Breaking (mGMSB)
Messenger fields form Complete SU(5) representations
Number of SU(5)
multiplets
Messenger scale
From EWSB as
in CMSSM
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  problem
 The MSSM superpotential is written down before EWSB or SUSY
breaking:
) it should know nothing about the EW scale.
(
¹-parameter has the
dimension of mass!
 The superpotential contains a mass scale!
 What mass should we use?
)
Forbidden
by
symmetry
The natural choices would be 0 or MPlanck (or MGUT)
 Phenomenological Constraints ) ¹ ¼ 0.1 -1 TeV
)
Scale of origin
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
the usual two Higgs doublet superfields.
, coupling together
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
So our superpotential so far is
Yukawa terms
effective -term
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!
massless axion!
NMSSM Chiral Superfield Content
Yukawa terms
effective -term
PQ breaking term
The superpotential of the Next-to-Minimal Supersymmetric Standard Model
(NMSSM) is
[Dine, Fischler and Srednicki]
[Ellis, Gunion, Haber, Roszkowski, Zwirner]
Yukawa terms
effective -term
PQ breaking term
We also need new soft supersymmetry breaking terms in the Lagrangian:
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:
Top left entry of CP-odd
mass matrix. Becomes MSSM
MA in MSSM limit.
minimisation
conditions
Finally:
 The MSSM limit is  ! 0,  ! 0, keeping / and  fixed.
  and  are forced to be reasonably small due to renormalisation group running.
Supersymmetric Models
 Minimal Supersymmetric Standard Model (MSSM)
 Next to Minimal Supersymmetric Standard Model (NMSSM)
[Dine, Fischler and Srednicki] [Ellis, Gunion, Haber, Roszkowski, Zwirner]
Alternative solution to Peccei–Quinn symmetry :
Decouple the axion
PQSNMSSM
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)
[Dine, Fischler and Srednicki]
[Ellis, Gunion, Haber, Roszkowski, Zwirner]
 Other variants: nmMSSM, PQSNMSSM.
 U(1) extended Supersymmetric Standard Model (USSM)
 Exceptional Supersymmetric Standard Model (E6SSM)
[S.F. King, S. Moretti, R. Nevzrov, Phys.Rev. D73 (2006) 035009]
USSM Chiral Superfield Content
Yukawa terms
Problem: to avoid gauge anomalies
effective -term
USSM Chiral Superfield Content
Yukawa terms
Problem: to avoid gauge anomalies
Charges not specified in the definition of the USSM
effective -term
U(1) extended Supersymmetric Standard Model (USSM)
Yukawa terms
effective -term
Modified Gauge sector, new Z0
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 )
Disclaimer:
I work on the E6SSM
Final part included for vanity
E6 inspired models
For anomaly cancelation, one can use complete E6 matter multiplets
New U(1)0 from E6
 Matter from 3 complete generations of E6
) automatic cancellation of gauge anomalies!
 In the E6SSM
) right-handed neutrino is a gauge singlet
Exceptional Supersymmetric Standard Model
(E6SSM)
[Phys.Rev. D73 (2006) 035009 , Phys.Lett. B634 (2006) 278-284
S.F.King, S.Moretti & R. Nevzorov]
All the SM matter fields are contained in one 27-plet of E6 per generation.
10, 1
3 generations
of “Higgs”
+
5*, 2
+
5*,
27
exotic
quarks
-3
+
5, - 2
+
1, 5
singlets
+
1, 0
SU(5) reps.
right handed neutrino
U(1)N charge
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)
USSM
E6SSM
MSSM
Minimal
superfields
NMSSM
Complete E6
multiplets
Chiral
superfields
End of Supersymmetry Lecture course
Thank you for listening
Sfermion masses
m2f~ f~L f~L¤ + m2f~ f~R¤ f~R + Amf f~L f~R¤
L
R
~
¢ f~L f~L f~L¤ ¢ f~R f~R f~R¤
¹ mf u cot ¯ f~u R f~u L ¹ mf d tan ¯ f d R f~d L
(where f u and f d are up and down type fermions respectively)
µ
¶
µ
¶
µ
¶
¡
¢
e
e
tL
¿
eL
bL
dgen
2
2
2
¤
¤
¤
¤
¤
¤
e
e
e
e
L 3r
=
¡
(
t
t
)
m
¡
m
¡
(
¿
e
¿
e
)
m
b
t
L
R
L
R
L
R
ebR
e
¿
f~¡ m ass
et e
e
b
tR
¿
eR
where
µ
me2t =
µ
me2t =
m 2Q 3 + m 2t + ¢ u~L
v(A t sin ¯ ¡ ¹ ¤ yt cos¯)
v(A ¤t sin ¯ ¡ ¹ yt cos¯)
m 2u 3 + m 2t + ¢ u~R
m 2Q 3 + m 2b + ¢ d~L
v(A t sin ¯ ¡ ¹ ¤ yt cos¯)
v(A ¤t sin ¯ ¡ ¹ yt cos¯)
m 2d + m 2t + ¢ d~R
3
¶
:
¶
: