Supersymmetry and non-commutative geometry

NONCOMMUTATIVE GEOMETRY

Thijs van den BroekRadboud Univ. Nijmegen / NIKHEF May 22nd, 2011

SUPERSYMMETRY&Workshop Bayrischzell

Wednesday, May 30, 2012

INTRODUCTIONIntroWednesday, May 30, 2012

The project OutlookPreliminary resultsApproach ApplicaNon

Try to extend the Standard Model from NCG with supersymmetry

The research project

NoncommutaNve geometry & supersymmetryThijs van den Broek (RU Nijmegen)

(Everywhere: N=1 supersymmetry , i.e. MSSM)

Joint work with Walter van Suijlekom and Wim Beenakker






(So: no superfields or anything...)

How supersymmetric is the resulNng acNon?











Does it share the merits of ‘ordinary’ supersymmetry?

(E.g. hierarchy problem)












Can we predict anything from this?

(E.g. scalar masses, c.f Higgs mass)












Can we predict anything from this?

(E.g. scalar masses, c.f Higgs mass)

Why want this?






Why want this?








Why want this?

Promising BSM candidate.








Why want this?

To see what NCG might have in store for us.









Why want this?

To see what NCG might have in store for us.


UnificaNon of coupling constants:

vs






Take:

MoNvaNng example: super-‐QCD [1] (1/2)


1TvdB, W. D. van Suijlekom, Physics Letters B 699 (2011), 119–122

tensored with

where

parametrizing a 3-‐tuple and its conjugate.



Take:




tensored with

where

parametrizing a 3-‐tuple and its conjugate.

‘quark’ ‘anNquark’‘gluino’





Inner fluctuaNons

parametrize (anN)squark






Inner fluctuaNons


Gauge group : superpartners






Inner fluctuaNons



Inner product:

Spectral acNon , extra terms:






Inner fluctuaNons



Inner product:

Spectral acNon , extra terms:

SUSY automaNcally broken: (minus) mass terms for squarks.1TvdB, W. D. van Suijlekom, Physics Letters B 699 (2011), 119–122


APPROACHAPPRWednesday, May 30, 2012


More systemaNcal approach needed (cf. superfields)

The approach


More realisNc situaNons: calculaNons get out of handProblem:



More systemaNcal approach needed (cf. superfields)

The approach


More realisNc situaNons: calculaNons get out of hand

1) Define ‘supersymmetric spectral triple‘ Plan:

2) Prove ‘susy spectral triple’ supersymmetric acNon

spectral acNon

3) MSSM as a special case

Problem:



Finite spectral triple:

Intermezzo: Krajewski diagrams


Krajewski diagram:






Krajewski diagram:






Krajewski diagram:

...

......

...






Krajewski diagram:

...

......

...






Krajewski diagram:

...

......

...






Krajewski diagram:

Grading

...

......

...






Krajewski diagram:

Grading

...

......

...






Krajewski diagram:

Dirac operator

Grading

...

......

...






Krajewski diagram:

Dirac operator

Grading

...

......

...






Krajewski diagram:

Dirac operator

Grading

...

......

...






Krajewski diagram:

Dirac operator

Grading

...

......

...






Krajewski diagram:

Dirac operator

Grading

‘KO-‐dimension’

...

......

...



General scheme as in super-‐QCD:

Superpartners (1/2)


gauginos: Hilbert space (adjoint reps.)

ParIcle Superpartner

fermions: Hilbert space

sfermions: finite Dirac operator

Higgs: finite Dirac operator

Higgsinos: Hilbert space

gauge bosons: Dirac operator on



Superpartners (2/2)


Gauge group:



:






Superpartners (2/2)


Gauge group:



:






Superpartners (2/2)


Gauge group:



:






Superpartners (2/2)


Gauge group:



:






Superpartners (2/2)


Gauge group:



:






Problem

R-‐parity & KO-‐dimension (1/2)


the gaugino-‐sector (adjoint elements of ) incompaNble with



Problem




parts of finite spectral triple possibly of different KO-‐dimensions

In fact



Problem




an operator with:

SoluNon given:

two spectral triples

of KO-‐dimension (say)

Direct sum:

parts of finite spectral triple possibly of different KO-‐dimensions

In fact



Use to ‘even out’ the KO dimensions:



three new signs (‘super-‐KO-‐dimension’?)

Direct sum:



Use to ‘even out’ the KO dimensions:



three new signs (‘super-‐KO-‐dimension’?)

Direct sum:

Example

i.e.

Role

KO-‐dimensions 6 (SM) and 0 (gauginos) has:

‘R-‐parity’, where



DefiniNon

a spectral triple

A supersymmetric spectral triple


that is extended with a grading

saNsfying:

such that

where

We call an R-‐parity extended spectral triple:

with only

We write:



such that



DefiniNon An R-‐parity extended spectral triple is supersymmetric when:

(...)

each element that transforms under the gauge group

comes in both -‐values.

all allowed components of the -‐ part of the Dirac operator

are nonzero.

DefiniNon

a spectral triple that is extended with a grading

saNsfying:




such that



DefiniNon An R-‐parity extended spectral triple is supersymmetric when:

(...)

each element that transforms under the gauge group

comes in both -‐values.

Hope (sNll) The acNon resulNng from such a spectral triple (via the spectral acNon principle) is automaNcally supersymmetric.

all allowed components of the -‐ part of the Dirac operator

are nonzero.

DefiniNon

a spectral triple that is extended with a grading

saNsfying:



APPLICATIONAPPRWednesday, May 30, 2012

The project OutlookPreliminary resultsApproach ApplicaIon

A nice way to look at things is provided by Chamseddine & Connes [2]:

Why the SM?


Chamseddine & Connes, Why the Standard Model, 0706.3688v1 [hep-‐th]

Look for irreducible soluNons of a pair :




Why the SM?




Either: acNng on

with




Why the SM?




Either: acNng on

with

Or: acNng on

with




Why the SM?




IncompaNble with

Either: acNng on

with

Or: acNng on

with




Why the SM?




IncompaNble with

Either: acNng on

with

Or: acNng on

with



ObservaNon:

Why the SM


Why the MSSM

with

Given the soluNon for the algebra we we can take not only but in addiNon to that also the soluNon for each of the two components of the algebra:



ObservaNon:

Why the SM


Why the MSSM

with


(From )

There is an R-‐parity operator:



ObservaNon:

Why the SM


Why the MSSM

SM parNcles

with


(From )




ObservaNon:

Why the SM


Why the MSSM

“Gaugino’s”SM parNcles

with


(From )




ObservaNon:

Why the SM


Why the MSSM

“Gaugino’s”SM parNcles

with


(Krajewski diagrams: representaNons have a solid fill.)

(From )




The supersymmetric spectral triple for the MSSM’


IniNal situaNon:

Three steps to the (MS)SM

1.





IniNal situaNon:


1.





IniNal situaNon:


1.




2.

Three steps to the (MS)SM:


As the result of a grading:

A vs A^C

1.




2.




A vs A^C

1.




2.




A vs A^C

1.




2.




A vs A^C

1.




2.




A vs A^C

1.




2.




A vs A^C

1.



NoncommutaNve geometry & supersymmetryThijs van den Broek (RU Nijmegen) NoncommutaNve geometry & supersymmetryThijs van den Broek (RU Nijmegen)

3.



By adding a Majorana massfor the right handed neutrino

2.

1.




3.




2.

1.




3.



By adding a Majorana massfor the right handed neutrino Bino

2.

1.




3.




Gluino

Bino

2.

1.




3.




Gluino

Bino

Wino/Zino

2.

1.




3.




Gluino

Bino

Higgsinos Wino/Zino

2.

1.




3.




Gluino

Bino

Higgsinos Wino/Zino

+ new parNcles

2.

1.


PRELIMINARY RESULTSPrelWednesday, May 30, 2012


Gauge group | UnificaNon


The gauge group:

is sNll





We sNll have coupling constant unificaNon:

The gauge group:

is sNll





We sNll have coupling constant unificaNon:

This happens only because we have more parNcles than the MSSM itself

provides!

The gauge group:

is sNll



Fermion doubling | Chiral anomalies


Copies of fermions exceed those of gaugino’s by a factor of four.

Change inner product in:





Hypercharges:







All come in pairs of opposite charges: chiral anomalies cancel

Hypercharges:





Comments on supersymmetry


NCG treats bosons & fermions differently

No auxiliary fields (on-‐shell descripNon)

AutomaNcally broken by sfermion masses








Nonetheless: definitely susy-‐like properIes









Try to prove susy modulo sfermion potenNal terms:

1. prove susy for both soluNons given by C&C:





2. prove that susy stays intact upon breaking





Try to prove susy modulo sfermion potenNal terms:

1. prove susy for both soluNons given by C&C:


OUTLOOKOUTWednesday, May 30, 2012


Summary & Outlook


‘Supersymmetric spectral triple’

Supersymmetric acNon / explicit susy transformaNons

✓?



Summary & Outlook




✓?

Coupling constant unificaNon

Applied to SM-‐algebra gives MSSM’

Gauge group intact, anomaly free theory

✓✓✓

Role & effects extra parNcles??



Summary & Outlook




✓?

PredicNons??




✓✓✓




Summary & Outlook




✓?

PredicNons??




✓✓✓


For more (conclusive) results: stay tuned!


Technology

Supersymmetry and non-commutative geometry