Noncommutative Geometry in the LHC-Erapeople.na.infn.it/~semgr4/doc/131205.pdf · Noncommutative...

Noncommutative Geometry in the LHC-Era


Christoph A. StephanInstitut für MathematikUniversität Potsdam

Napoli2013


Overview

1 Basic Ideas

2 Geometry

3 Physics

4 Beyond the Standard Model

5 Conclusions

Noncommutative Geometry in the LHC-EraBasic Ideas

Overview

1 Basic Ideas

2 Geometry

3 Physics


5 Conclusions


The Aim of Noncommutative Geometry:

Aim (what everybody tries to do):To unify general relativity (GR) and the standard model ofparticle physics (SM) on the same geometrical level. Thismeans to describe gravity and the electro-weak and strongforces as gravitational forces of a generalised “space-time”.

Idea (what many people have done before):Find generalised “space-time” with symmetries of GeneralRelativity and the Standard Model (and perhaps more).

Possible Solution (A. Connes):Try a special type of noncommutative geometry


Analogy: Almost-comm. geometry! Kaluza-Klein space

. . . . .

M

F

. . . . .

M

F

. . . . .

M

F

. . . . .

M

F

. . . . .

M

F

Idea:M " C!(M), F " some ”finite space”,differential geometry" spectral triple


Almost-commutative Geometry

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

C!(M)C!(M)C!(M)C!(M)C!(M)

AfAfAfAfAf

Replacing manifolds by algebrasextra dimension: F " Af = M1(K) #M2(K) # . . .

Kaluza-Klein space: M $ F " A = C!(M) %Af

Noncommutative Geometry in the LHC-EraBasic IdeasGeneral Relativity

Schematic Structure of GR:Gravity emerges as a pseudo-force associated to thespace-time symmetries, i.e. the diffeomorphisms of themanifold M.

Space-Time(S-T)

4-dim Manifold M

S-T SymmetriesDiffeomorphisms

of M

GravityEinstein-Hilbert

Action

act on

leave invariantDynamics

Noncommutative Geometry in the LHC-EraBasic IdeasGeneral Relativity & Standard Model: The spectral point of view

Euclidean space(-time)!

Space(-Time)(S-T)

4-dim Manifold M

S-T SymmetriesDiffeomorphisms

of Mequiv. equiv.

act on

Spectral Triple(C!(M), /!,H)

AutomorphismsAut(C!(M))

GravitySpectral Action =

E-H Act.+Cosm.Const.

act on



Almost-Commutative Standard Model (A.Chamseddine,A.Connes):

Almost-CommutativeSpectral Triple

A = C!(M) %Af

Int.+Ext. SymmetriesAut(A) & Diff(M)!

U(1) $ SU(2) $ SU(3)

Spectral Action =E-H Act.+Cosm.Const.+ Stand. Model Action

act on



The almost-commutative standard model automaticallyproduces:

• The combined Einstein-Hilbert and standard model action• A cosmological constant• The Higgs boson with the correct quartic Higgs potential

The Dirac operator plays a multiple role:

D = /! % 1f + /" + #5%!

Higgs & GaugeBosons

Metric of M,Internal Metric

Particle Dynamics,Ferm. Mass Matrix

Noncommutative Geometry in the LHC-EraGeometry

Overview

1 Basic Ideas

2 Geometry

3 Physics


5 Conclusions

Noncommutative Geometry in the LHC-EraGeometrySpectral Triples: the Input

An even, real spectral triple (A,H,D)

The ingredients (A. Connes):A real, associative, unital pre-C"-algebra AA Hilbert space H on which the algebra A is faithfullyrepresented via a representation $

A self-adjoint operator D with compact resolvent,the Dirac operatorAn anti-unitary operator J on H, the real structureor charge conjugationA unitary operator # on H, the chiralityor volume element

Noncommutative Geometry in the LHC-EraGeometryThe Classical Conditions

The axioms of noncommutative geometry (A. Connes):Axiom 1: Classical Dimension n (we assume n even)Axiom 2: RegularityAxiom 3: FinitenessAxiom 4: First Order of the Dirac OperatorAxiom 5: RealityAxiom 6: OrientabilityAxiom 7: Poincaré Duality

Noncommutative Geometry in the LHC-EraGeometryThe Reconstruction Theorem

Connes’ Reconstruction Theorem (sloppy version):Compact Riemannian spin manifolds are equivalent to realspectral triples with A commutative.

One can therefore replace a compact 4-dim. Riemannianspace-timeM by the spectral triple (C!(M),H, /!).

Noncommutative Geometry in the LHC-EraGeometryFinite Spectral Triples

Finite spectral triples:Af = M1(K) #M2(K) # . . .K = R, C or H

Aute(Mn(C)) = Unc(Mn(C)) = U(n),Hf ' CN

N is the total number of particlesleft-/right-handed particles/antiparticles counted separatelyDf ( MN(C), Df is the fermionic mass matrix.

Axioms" Restrictions for Df and Hf

Noncommutative Geometry in the LHC-EraGeometryAlmost-Commutative Geometry

Almost-commutative geometry:An almost-commutative spectral triple (A,H,D) is a tensorproduct of a spectral triple of a manifold M(AM = C!(M),HM ,DM = /!) with dimensions nM > 0and a finite spectral triple(Af ,Hf ,Df ) with metric dimension nf = 0(i.e. Af matrix algebra).

A = C!(M) %Af , H = HM %Hf ,

J = JM % Jf , # = #5 % #f ,

D = /! % 1f + #5 % Df

Aut(C!(M) %Af ) 'Diff(M)! Unc(Af )

Noncommutative Geometry in the LHC-EraGeometryAlmost-Commutative Geometry

The geometric setup imposes constraints:mathematical axioms" Restrictions on particle contentsymmetries of finite space" determines gauge grouprepresentation of matrix algebra" representation of gauge group(only fundamental and adjoint representations)Dirac operator" allowed mass terms / Higgs fields

Noncommutative Geometry in the LHC-EraPhysics

Overview

1 Basic Ideas

2 Geometry

3 Physics


5 Conclusions

Noncommutative Geometry in the LHC-EraPhysicsThe Action

The fluctuated Dirac operatorThe Dirac operator /! % 1f + #5 % Df is fluctuated with innerunitaries Unc(Af ) and becomes

D = /! % 1f + /" + #5 % !

The Spectral Action (A. Connes, A. Chamseddine 1996)(",D") + SD(#) with " ( H

(",D") = fermionic actionincludes Yukawa couplings& fermion–gauge boson interactions

SD(#) = % eigenvalues of D up to cut-off #= Einstein-Hilbert action + Cosm. Const.+ full bosonic SM action + constraints at #

constraints => less free parameters than classical SM


The bosonic spectral action:Spectral action (bosonic part) of Dirac operator D is the numberof eigenvalues of D in the interval [)#,#] (with # ( R+):

SD(#) = Tr f!

D2

!2

"

,

where Tr is the L2-trace over the space of spinor fields,and f is cut-off function with support in [0,+1] which is constantnear 0.


Asymptotic expansion of the Spectral ActionFrom the heat trace asymptotics for # " *

Tr#

e#D2

!2

$

+%

n$0#2#na2n(D2)

(with Seeley-deWitt coefficients a2n(D2))one gets an asymptotics for the spectral action

SD(#) = Tr f!

D2

!2

"

+ #4 f4 a0(D2) + #2 f2 a2(D2) + #0 f0 a4(D2)

as # " *.Here f4, f2, f0 are moments of the cut-off function f .


Spectral action for Connes-Chamseddine Dirac operatorFor the Connes-Chamseddine Dirac operator(or fluctuated Dirac operator)

D := /! + /" + #5 % !

we find the Seeley-deWitt coefficients

a2(D2) = )dim(Hf )96!2

&

MR dvol ) 1

48!2

&

Mtr(!2)dvol

a4(D2) = 11 dim(Hf )720 &(M) ) dim(Hf )

320!2

&

M,W,2 dvol

+ 18!2

&

Mtr'

[-Hf ,!](

+ tr(!4)dvol

+ 596!2

&

Mtr'

$2f(

dvol + 148!2

&

MR tr(!2)dvol

Noncommutative Geometry in the LHC-EraPhysicsThe Standard Model

The standard modelAf = C # H #M3(C)(#C)

Unc(Af ) = SU(2) $U(3)Hf = HSM Hilbert space of minimal standard modelfermion multipletsDf : Fermionic mass matrix with CKM matrix and PMNSmatrixDf " ! Higgs field(s) by inner fluctuationsMajorana masses and SeeSaw mechanism forright-handed neutrinos (J. Barrett & A. Connes ’06)


Constraints on the SM parameters at the cut-off #:

5 g21 = NSM g22 = NSM g23 = 3 Y 22H

"24 = 3

4 Y2

g1, g2, g3: U(1)Y , SUw(2), SUc(3) gauge couplings': quartic Higgs couplingY2: trace of the Yukawa matrix squaredH: trace of the Yukawa matrix to the fourth powerNSM : number of standard model generations


Consequences from the SM constraints:

Input:Big Desertg1(mZ ) = 0.3575, g2(mZ ) = 0.6514, g3(mZ ) = 1.221renormalisation group equations(mtop = 171.2± 2.1 GeV)

Output:g22(#) = g23(#) at # = 1.1$ 1017 GeVmtop < 190 GeVno 4th SM generation

Excluded by Tevatron & LHC since:mSMS .= 168.3± 2.5 GeV53 g1(#)2 .= g2(#)2

Noncommutative Geometry in the LHC-EraPhysicsClassifying Almost-Commutative Spectral Triples

How unique is the Standard Model?The aim: Classifying the internal spaces

Af = M1(K) #M2(K) # . . .

with respect to the number of summands in the algebrawith respect to physical criteria

Little ReminderFor the Standard Model we have

Af = C # H #M3(C)(#C)

or alternatively

Af = C #M2(C) #M3(C) # C

Noncommutative Geometry in the LHC-EraPhysicsThe Requirements

Physicist’s ”shopping list” (B. Iochum, T. Schücker, C.S. ’03):

The physical models emerging from the spectral action arerequired to

be irreducible i.e. to have the smallest possible internalHilbert space (minimal approach)allow a non-degenerate Fermionic mass spectrumbe free of harmful anomalieshave unbroken colour groupspossess no uncharged massless Fermions

Noncommutative Geometry in the LHC-EraPhysicsThe Results

Classification Results(B.Iochum, J.-H. Jureit, T.Schücker, C.S. 2003-2008):# sum. in Af KO 0 KO 6

1 no model no model2 no model no model3 SM2 no model4 SM2, SM2,

el.-str.1 el.-str.16 SM2 + el.-str.1,

2 $ el.-str.1

1 Electro-Strong Model: ”electron+proton”, no Higgs,Af = C # C # C #Mn(C),Ggauge = U(1) $ SU(n)/SO(n)/Sp(n)

2 first family, colour group = SU(n)/SO(n)/Sp(n)

Noncommutative Geometry in the LHC-EraBeyond the Standard Model

Overview

1 Basic Ideas

2 Geometry

3 Physics


5 Conclusions

Noncommutative Geometry in the LHC-EraBeyond the Standard Model

Beyond SM: the general strategy (bottom-up approach)find finite geometry that has SM as sub-model (tricky)=> particle content, gauge group & representationmake sure everything is anomaly freecompute the spectral action => constraints on parametersdetermine the cut-off scale # with suitable sub-setof the constraintsuse renorm. group equations to obtain low energy valuesof (hopefully) interesting parameters(Higgs couplings, Yukawa couplings)check with experiment! (and here we usually fail)

Noncommutative Geometry in the LHC-EraBeyond the Standard ModelThe BSM failures

Excluded models:Extensions of the Standard Model:

AC-model (C.S. ’05):new particles (A and C) with opposite hyperchargedark matter as bound AC-states (Fargion,Khlopov,C.S. ’05)(-model (C.S. ’07): new particles with SUc(D)-colourVector-Doublet Model (Squellari, C.S. ’07):new SUw(2)-vector doublets

Problem for models: • mSMS / 170 GeV• constraints on g1, g2, g3 at #.

Saving these models?Some of these models, e.g. the AC-model may perhaps beextended to comply with experimental data!

Noncommutative Geometry in the LHC-EraBeyond the Standard ModelNew Scalars

SM + U(1)X scaler field + U(1)X fermion singlet (C.S. 2009):Internal space: C #M2(C) #M3(C) # C # C # C

Gauge group: U(1)Y $ SU(2)w $ SU(3)c $ U(1)XNew fermions: U(1)X -vector singlets (X -particles)

neutral w.r.t SM gauge group , MX + #

New scalar: U(1)X singlet ), neutral w.r.t SM gauge group

Lscalar = )µ21|H|2 + "16 |H|4 ) µ22|)|

2 + "26 |)|4 + "3

3 |H|2|)|2

U(1)Y $ SU(2)w $ SU(3)c $ U(1)X " U(1)el . $ SU(3)c

Lferm+gauge = X̄LMXXR+g#,X *̄R)XL+ h.c. +1/g24Fµ#X FX ,µ#


The constraints at #:

only top-quark & *$

valid at g2 = g3=> # = 1.1$ 1017 GeVg22 = "1

24(3g2t +g2!)2

3g4t +g4!

g22 = "224

g22 = "324

3g2t +g2!g2!

g22 = 14 (3g2t + g2# )

free parameters: |0)1|, g4mSMS + 120) 130 GeVProblem:

)

5/3g1 .= g2 = g3

1000800600400200

200

600

300

400

500

100

v2/ GeV

m/GeV

Mass EVs of scalar fields forv2 =

22 |0)1|,2

2|0H1| = 246 GeV, g4(mZ ) = 0.3


SM + U(1)X scalar field + new fermions (C.S. ’13):SM as a sub-model: comme il faut!Internal space: C #M2(C) #M3(C) # C #6

i=1 Ci

gauge group: U(1)Y $ SU(2)w $ SU(3)c $ U(1)X

new fermions in each SM-generation:X 1l # X 2

l # X 3l : (0,1,1,+1) # (0,1,1,+1) # (0,1,1,0)

X 1r # X 2

r # X 3r : (0,1,1,+1) # (0,1,1,0) # (0,1,1,+1)

Vw% , Vw

r : (0, 2̄,1,0)Vc

% , Vcr : ()1/6,1, 3̄,0)

new scalar: ) : (0,1,1,+1)


The Lagrangian (scalar potential & new terms):

Lscalar = )µ21|H|2 ) µ22|)|2 + "1

6 |H|4 + "26 |)|4 + "3

3 |H|2|)|2

Lferm = g#,X1 *̄r)X 1% + X̄ 1

% mXX 1r + gX2 X̄ 2

% )X 2r

+gX3 X̄ 3% )X 3

r + V̄ c% mcVc

r + V̄ w% mwVw

r + h.c.

Lgauge = 1g24Fµ#X FX ,µ#

Symmetry breaking:U(1)Y $SU(2)w$SU(3)c$U(1)X " U(1)e%.$SU(3)c$Z2


The constraints at #:

g2(#) = g3(#) =*

76 g1(#) =

*

43 g4(#)

'1(#) = 36 HY2 g2(#)2, '2(#) = 36

tr(g4!,X1

)

tr(g2!,X1

)2g2(#)2

'3(#) = 36 tr(g2! )Y2 g2(#)2

Y2(#) = tr(g2#,X1)(#) + tr(g2X1)(#) + tr(g2X2)(#) = 6 g2(#)2

Some simplifications:Y2 3 3gtop + g#"

tr(g2X1)(#) 3 tr(g2X2)(#) 3 0tr(g2

#,X1)(#) 3 g#,X (#)2 = 6 g2(#)2

(mw )ij 3 #, (mc)ij 3 1015 GeV


Results for 1-loop renormalisation groups:

Constraints=> # 3 2$ 1018 GeVmtop 3 172.9± 1.5 GeVm&1,SMS 3 125± 1.1 GeVm&2 3 445± 139 GeVmZX 3 254± 87 GeVg4(mZ ) 3 0.36mX2,X3 ! 50 GeVfree parameter: |0)1|

1000800600400200

200

600

300

400

500

100

22 |0)1| GeV

m/GeV

Mass EVs of scalar fields


Running of the gauge couplings with normalisation factors

2520 351550

1,2

1

0,4

3010

0,6

0,8

t[log(GeV)]

)

7/6g1(t)

g2(t)

g3(t)

)

4/3g4(t)

Noncommutative Geometry in the LHC-EraConclusions

Overview

1 Basic Ideas

2 Geometry

3 Physics


5 Conclusions

Noncommutative Geometry in the LHC-EraConclusions

Questions & to-do-listIs the SM + scalar model compatible with LHC-data?Does the SM + scalar model contain viable dark mattercandidates?Explore parameter space (g#,X1 , g2X2 , g2X3 , mX1 , mVw , mVc )Extend renormalisation group analysis to n-loop, n / 2Is the geometry a “sub-geometry” of a Connes-Chamseddine-type geometry?Classify Models beyond the Standard ModelSpectral triples with Lorentzian signature(A. Rennie, M. Paschke, R. Verch, K. van den Dungen...)

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