Noncommutative models for Particle and Cosmology, Lecture II

Matilde Marcolli

Villa de Leyva school, July 2011

Matilde Marcolli NCG models for particles and cosmology, II This lecture based on Ali Chamseddine, , Matilde Marcolli, and the Standard Model with mixing, ATMP 11 (2007) 991–1090, arXiv:hep-th/0610241 Symmetries and NCG Symmetries of gravity coupled to matter: G = U(1) × SU(2) × SU(3)

G = Map(M, G) o Diff(M)

Is it G = Diff(X )? Not for a , yes for an NC space ∞ Example: A = C (M, Mn(C)) G = PSU(n)

1 → Inn(A) → Aut(A) → Out(A) → 1

1 → Map(M, G) → G → Diff(M) → 1.

Matilde Marcolli NCG models for particles and cosmology, II Symmetries viewpoint: can think of X = M × F noncommutative with G = Diff(X ) pure gravity symmetries for X combining gravity and gauge symmetries together (no a priori distinction between “base” and “fiber” directions)

Want same with action functional for pure gravity on NC space X = M × F giving gravity coupled to matter on M

Matilde Marcolli NCG models for particles and cosmology, II Product geometry M × F Two spectral triples (Ai , Hi , Di , γi , Ji ) of KO-dim 4 and 6:

A = A1 ⊗ A2 H = H1 ⊗ H2

D = D1 ⊗ 1 + γ1 ⊗ D2

γ = γ1 ⊗ γ2 J = J1 ⊗ J2 Case of 4-dimensional spin manifold M and finite NC geometry F :

∞ ∞ A = C (M) ⊗ AF = C (M, AF )

2 2 H = L (M, S) ⊗ HF = L (M, S ⊗ HF )

D = /∂M ⊗ 1 + γ5 ⊗ DF

DF chosen in the moduli space described last time

Matilde Marcolli NCG models for particles and cosmology, II Dimension of NC spaces: different notions of dimension for a spectral triple (A, H, D) Metric dimension: growth of eigenvalues of Dirac operator KO-dimension (mod 8): sign commutation relations of J, γ, D Dimension spectrum: poles of zeta functions −s ζa,D (s) = Tr(a|D| ) For first two agree and third contains usual dim; for NC spaces not same: DimSp ⊂ C can have non-integer and non-real points, KO not always metric dim mod 8, see F case X = M × F metrically four dim 4 = 4 + 0; KO-dim is 10 = 4 + 6 (equal 2 mod 8); DimSp k ∈ Z≥0 with k ≤ 4

Matilde Marcolli NCG models for particles and cosmology, II Variant: almost commutative

∞ 2 (C (M, E), L (M, E ⊗ S), DE )

M smooth manifold, E algebra bundle: fiber Ex finite dimensional algebra AF C ∞(M, E) smooth sections of a algebra bundle E E S Dirac operator DE = c ◦ (∇ ⊗ 1 + 1 ⊗ ∇ ) with spin connection ∇S and hermitian connection on bundle Compatible grading and real structure An equivalent intrinsic (abstract) characterization in: Branimir Ca´ci´c,´ A reconstruction theorem for almost-commutative spectral triples, arXiv:1101.5908 Here on assume for simplicity product M × F

Matilde Marcolli NCG models for particles and cosmology, II Inner fluctuations and gauge fields Setup: Right A-module structure on H

ξ b = b0 ξ, ξ ∈ H , b ∈ A

Unitary group, adjoint representation:

ξ ∈ H → Ad(u) ξ = u ξ u∗ ξ ∈ H

Inner fluctuations:

0 −1 D → DA = D + A + ε JAJ with A = A∗ self-adjoint operator of the form X A = aj [D, bj ] , aj , bj ∈ A

Note: not an equivalence relation (finite geometry, can fluctuate D to zero) but like “self Morita equivalences”

Matilde Marcolli NCG models for particles and cosmology, II Properties of inner fluctuations( A, H, D, J) 1 ∗ Gauge potential A ∈ ΩD , A = A Unitary u ∈ A, then

Ad(u)(D + A + ε0 JAJ−1)Ad(u∗) =

0 −1 D + γu(A) + ε J γu(A) J ∗ ∗ where γu(A) = u [D, u ] + u A u 0 1 ∗ D = D + A (with A ∈ ΩD , A = A ) then

0 0 0 1 D + B = D + A , A = A + B ∈ ΩD

1 ∗ ∀B ∈ ΩD0 B = B D0 = D + A + ε0 JAJ−1 then

0 0 −1 0 0 0 −1 0 1 D + B + ε JBJ = D + A + ε JA J A = A + B ∈ ΩD

1 ∗ ∀B ∈ ΩD0 B = B

Matilde Marcolli NCG models for particles and cosmology, II Gauge bosons and Unitary U(A) = {u ∈ A | uu∗ = u∗u = 1} Special unitary

SU(AF ) = {u ∈ U(AF ) | det(u) = 1}

det of action of u on HF Up to a finite abelian group

SU(AF ) ∼ U(1) × SU(2) × SU(3)

Unimod subgr of U(A) adjoint rep Ad(u) on H is gauge group of SM Unimodular inner fluctuations (in M directions) ⇒ gauge bosons of SM: U(1), SU(2) and SU(3) gauge bosons Inner fluctuations in F direction ⇒ Higgs field

Matilde Marcolli NCG models for particles and cosmology, II More on Gauge bosons (1,0) P 0 Inner fluctuations A = i ai [/∂M ⊗ 1, ai ] with with 0 0 0 0 ∞ ai = (λi , qi , mi ), ai = (λi , qi , mi ) in A = C (M, AF ) P 0 P 0 U(1) gauge field Λ = i λi dλi = i λi [/∂M ⊗ 1, λi ] P 0 P α SU(2) gauge field Q = i qi dqi , with q = f0 + α ifασ and P 0 α Q = α fα[/∂M ⊗ 1, ifασ ] 0 P 0 P 0 U(3) gauge field V = i mi dmi = i mi [/∂M ⊗ 1, mi ] reduce the gauge field V 0 to SU(3) passing to unimodular subgroup SU(AF ) and unimodular gauge potential Tr(A) = 0

 Λ 0 0  1 1 V 0 = −V − 0 Λ 0 = −V − Λ1 3   3 3 0 0 Λ

Matilde Marcolli NCG models for particles and cosmology, II Gauge bosons and hypercharges The (1, 0) part of A + JAJ−1 acts on quarks and leptons by

 4  3 Λ + V 0 0 0 2  0 − 3 Λ + V 0 0   1   0 0 Q11 + 3 Λ + VQ12  1 0 0 Q21 Q22 + 3 Λ + V

 0 0 0 0   0 −2Λ 0 0     0 0 Q11 − Λ Q12  0 0 Q21 Q22 − Λ ⇒ correct hypercharges!

Matilde Marcolli NCG models for particles and cosmology, II More on Higgs boson Inner fluctuations A(0,1) in the F -space direction

X 0 (0,1) (0,1) ai [γ5 ⊗ DF , ai ](x)|Hf = γ5 ⊗ (Aq + A` ) i

 0 X   0 Y  A(0,1) = ⊗ 1 A(0,1) = q X 0 0 3 1 Y 0 0

 ∗ ∗   0 0  Υuϕ1 Υuϕ2 0 Υuϕ1 Υd ϕ2 X = ∗ ∗ and X = 0 0 −Υd ϕ¯2 Υd ϕ¯1 −Υuϕ¯2 Υd ϕ¯1  ∗ ∗   0 0  Υνϕ1 Υνϕ2 0 Υνϕ1 Υe ϕ2 Y = ∗ ∗ and Y = 0 0 −Υe ϕ¯2 Υe ϕ¯1 −Υνϕ¯2 Υe ϕ¯1 P 0 0 P 0 0 P 0 0 ¯0 ϕ1 = λi (αi − λi ), ϕ2 = λi βi ϕ1 = αi (λi − αi ) + βi βi and 0 P 0 ¯0 0 ϕ2 = (−αi βi + βi (λi − α¯i )), for ai (x) = (λi , qi , mi ) and  α β  a0(x) = (λ0, q0, m0) and q = i i i i −β¯ α¯

Matilde Marcolli NCG models for particles and cosmology, II More on Higgs boson Discrete part of inner fluctuations: quaternion valued function H = ϕ1 + ϕ2j or ϕ = (ϕ1, ϕ2)

2 1,0 2 0,1 2 1,0 0,1 DA = (D ) + 14 ⊗ (D ) − γ5 [D , 14 ⊗ D ] √ 1,0 0,1 µ s 0,1 [D , 14 ⊗ D ] = −1 γ [(∇µ + Aµ), 14 ⊗ D ] 2 ∗ ∗ s This gives DA = ∇ ∇ − E where ∇ ∇ Laplacian of ∇ = ∇ + A 1 X −E = s ⊗ id + γµγν ⊗ − i γ γµ ⊗ (D ϕ) + 1 ⊗ (D0,1)2 4 Fµν 5 M µ 4 µ<ν with s = −R scalar curvature and Fµν curvature of A i i D ϕ = ∂ ϕ + g W αϕ σα − g B ϕ µ µ 2 2 µ 2 1 µ SU(2) and U(1) gauge potentials

Matilde Marcolli NCG models for particles and cosmology, II The spectral action functional Ali Chamseddine, Alain Connes, The spectral action principle, Comm. Math. Phys. 186 (1997), no. 3, 731–750.

A good action functional for noncommutative geometries

Tr(f (D/Λ))

D Dirac, Λ mass scale, f > 0 even smooth function (cutoff approx) Simple dimension spectrum ⇒ expansion for Λ → ∞ Z X k −k Tr(f (D/Λ)) ∼ fk Λ − |D| + f (0) ζD (0) + o(1), k

R ∞ k−1 with fk = 0 f (v) v dv momenta of f −s where DimSp(A, H, D) = poles of ζb,D (s) = Tr(b|D| )

Matilde Marcolli NCG models for particles and cosmology, II Asymptotic expansion of the spectral action

−t∆ X α Tr(e ) ∼ aα t (t → 0) and the ζ function −s/2 ζD (s) = Tr(∆ )

Non-zero term aα with α < 0 ⇒ pole of ζD at −2α with 2 a Res ζ (s) = α s=−2α D Γ(−α)

No log t terms ⇒ regularity at 0 for ζD with ζD (0) = a0

Matilde Marcolli NCG models for particles and cosmology, II Get first statement from Z ∞ −s −s/2 1 −t∆ s/2−1 |D| = ∆ = s  e t dt Γ 2 0

R 1 α+s/2−1 −1 with 0 t dt = (α + s/2) . Second statement from 1 s s  ∼ as s → 0 Γ 2 2 contrib to ζD (0) from pole part at s = 0 of Z ∞ Tr(e−t∆) ts/2−1 dt 0

R 1 s/2−1 2 given by a0 0 t dt = a0 s

Matilde Marcolli NCG models for particles and cosmology, II Spectral action with fermionic terms 1 S = Tr(f (D /Λ)) + h J ξ,˜ D ξ˜i , ξ˜ ∈ H+, A 2 A cl DA = Dirac with unimodular inner fluctuations, J = real structure, + Hcl = classical spinors, Grassmann variables Fermionic terms 1 hJξ,˜ D ξ˜i 2 A antisymmetric bilinear form A(ξ˜) on + Hcl = {ξ ∈ Hcl | γξ = ξ} ⇒ nonzero on Grassmann variables Euclidean functional integral ⇒ Pfaffian Z − 1 A(ξ˜) Pf (A) = e 2 D[ξ˜] avoids doubling problem of previous models based on symmetric hξ, DAξi for NC space with KO-dim=0 Matilde Marcolli NCG models for particles and cosmology, II Grassmann variables Anticommuting variables with basic integration rule Z ξ dξ = 1

An antisymmetric bilinear form A(ξ1, ξ2): if ordinary commuting variables A(ξ, ξ) = 0 but not on Grassmann variables 0 0 0 Example: 2-dim case A(ξ , ξ) = a(ξ1ξ2 − ξ2ξ1), if ξ1 and ξ2 anticommute, with integration rule as above

Z 1 Z − A(ξ,ξ) −aξ1ξ2 e 2 D[ξ] = e dξ1dξ2 = a

Pfaffian as functional integral: antisymmetric quadratic form Z − 1 A(ξ,ξ) Pf (A) = e 2 D[ξ]

Method to treat Majorana in the Euclidean setting

Matilde Marcolli NCG models for particles and cosmology, II Fermionic part of SM Lagrangian Explicit computation of 1 hJξ,˜ D ξ˜i 2 A gives part of SM Larangian with •LHf = coupling of Higgs to fermions •Lgf = coupling of gauge bosons to fermions •Lf = fermion terms

Matilde Marcolli NCG models for particles and cosmology, II Bosonic part of the spectral action

1 f Z √ S = (48 f Λ4 − f Λ2 c + 0 d) g d 4x π2 4 2 4 96 f Λ2 − f c Z √ + 2 0 R g d 4x 24π2 f Z 11 √ + 0 ( R∗R∗ − 3 C C µνρσ) g d 4x 10 π2 6 µνρσ (−2 a f Λ2 + e f ) Z √ + 2 0 |ϕ|2 g d 4x π2 f a Z √ + 0 |D ϕ|2 g d 4x 2 π2 µ f a Z √ − 0 R |ϕ|2 g d 4x 12 π2 f b Z √ + 0 |ϕ|4 g d 4x 2 π2 f Z 5 √ + 0 (g 2 G i G µνi + g 2 F α F µνα + g 2 B Bµν ) g d 4x, 2 π2 3 µν 2 µν 3 1 µν

Matilde Marcolli NCG models for particles and cosmology, II Parameters:

f0, f2, f4 free parameters, f0 = f (0) and, for k > 0, Z ∞ k−1 fk = f (v)v dv. 0 a, b, c, d, e functions of Yukawa parameters of νMSM

† † † † a = Tr(Yν Yν + Ye Ye + 3(Yu Yu + Yd Yd )) † 2 † 2 † 2 † 2 b = Tr((Yν Yν) + (Ye Ye ) + 3(Yu Yu) + 3(Yd Yd ) ) c = Tr(MM†) d = Tr((MM†)2)

† † e = Tr(MM Yν Yν).

Matilde Marcolli NCG models for particles and cosmology, II 2 ∗ Gilkey’s theorem using DA = ∇ ∇ − E µν µ Differential operator P = − (g I ∂µ∂ν + A ∂µ + B) with A, B bundle endomorphisms, m = dim M Z −tP X n−m Tr e ∼ t 2 an(x, P) dv(x) n≥0 M ∗ µ µ P = ∇ ∇ − E and E;µ := ∇µ∇ E 1 ∇ = ∂ + ω0 , ω0 = g (Aν + Γν · id) µ µ µ µ 2 µν µν 0 0 0 ρ 0 E = B − g (∂µ ων + ωµ ων − Γµν ωρ) 0 0 0 0 Ωµν = ∂µ ων − ∂ν ωµ + [ωµ, ων] Seeley-DeWitt coefficients −m/2 a0(x, P) = (4π) Tr(id) −m/2 R  a2(x, P) = (4π) Tr − 6 id + E −m/2 1 µ 2 µν a4(x, P) = (4π) 360 Tr(−12R;µ + 5R − 2Rµν R µνρσ 2 µ + 2Rµνρσ R − 60 RE + 180 E + 60 E;µ µν + 30 Ωµν Ω )

Matilde Marcolli NCG models for particles and cosmology, II Normalization and coefficients √ a f0 Rescale Higgs field H = π ϕ to normalize kinetic term R 1 2 √ 4 2 |DµH| g d x Normalize Yang-Mills terms 1 i µνi 1 α µνα 1 µν 4 GµνG + 4 FµνF + 4 BµνB Normalized form: Z Z 1 √ 4 √ 4 S = 2 R g d x + γ0 g d x 2κ0 Z Z µνρσ√ 4 ∗ ∗√ 4 + α0 Cµνρσ C g d x + τ0 R R g d x 1 Z √ Z √ + |DH|2 g d 4x − µ2 |H|2 g d 4x 2 0 Z Z 2 √ 4 4 √ 4 − ξ0 R |H| g d x + λ0 |H| g d x 1 Z √ + (G i G µνi + F α F µνα + B Bµν ) g d 4x 4 µν µν µν

∗ ∗ 1 µνρσ αβ γδ where R R = 4  αβγδR µν R ρσ integrates to the Euler characteristic χ(M) and C µνρσ Weyl curvature

Matilde Marcolli NCG models for particles and cosmology, II Coefficients

2 1 96f2Λ − f0c 1 4 2 f0 2κ2 = 2 γ0 = 2 (48f4Λ − f2Λ c + d) 0 24π π 4 3f 11f α = − 0 τ = 0 0 10π2 0 60π2 2 2 f2Λ e 1 µ0 = 2 − ξ0 = 12 f0 a π2b λ0 = 2 2f0a

Energy scale: Unification (1015 – 1017 GeV)

g 2f 1 0 = 2π2 4 Preferred energy scale, unification of coupling constants

Matilde Marcolli NCG models for particles and cosmology, II Renormalization group flow The coefficients a, b, c, d, e (depend on Yukawa parameters) run with the RGE flow Initial conditions at unification energy: compatibility with physics at low energies RGE in the MSM case 2 Running of coupling constants at one loop: αi = gi /(4π) 41 19 β = (4π)−2 b g 3, with b = ( , − , −7), gi i i i 6 6

−1 −1 41 Λ α1 (Λ) = α1 (MZ ) − log 12π MZ −1 −1 19 Λ α2 (Λ) = α2 (MZ ) + log 12π MZ −1 −1 42 Λ α3 (Λ) = α3 (MZ ) + log 12π MZ 0 MZ ∼ 91.188 GeV mass of Z boson Matilde Marcolli NCG models for particles and cosmology, II At one loop RGE for coupling constants decouples from Yukawa parameters (not at 2 loops!)

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Well known triangle problem: with known low energy values 2 2 2 constants don’t meet at unification g3 = g2 = 5g1 /3

Matilde Marcolli NCG models for particles and cosmology, II Geometry point of view At one loop coupling constants decouple from Yukawa parameters Solving for coupling constants, RGE flow defines a vector field on moduli space C3 × C1 of Dirac operators on the finite NC space F Subvarieties invariant under flow are relations between the SM parameters that hold at all energies At two loops or higher, RGE flow on a rank three vector bundle (fiber = coupling constants) over the moduli space C3 × C1 Geometric problem: studying the flow and the geometry of invariant subvarieties on the moduli space

Matilde Marcolli NCG models for particles and cosmology, II Constraints at unification The geometry of the model imposes conditions at unification energy: specific to this NCG model λ parameter constraint 2 π b(Λunif ) λ(Λunif ) = 2 2f0 a(Λunif ) Higgs vacuum constraint √ af 2M 0 = W π g See-saw mechanism and c constraint 2 2 2f2Λunif 6f2Λunif ≤ c(Λunif ) ≤ f0 f0 Mass relation at unification X 2 2 2 2 2 (mν + me + 3mu + 3md )|Λ=Λunif = 8MW |Λ=Λunif generations Need to have compatibility with low energy behavior

Matilde Marcolli NCG models for particles and cosmology, II 2 Mass relation at unification Y2(S) = 4g

X σ 2 σ 2 σ 2 σ 2 Y2 = (yν ) + (ye ) + 3 (yu ) + 3 (yd ) σ

g σ κ (k(↑3))σκ = 2M mu δσ g µ ρ † (k(↓3))σκ = 2M md CσµδµCρκ g σ κ (k(↑1))σκ = 2M mν δσ g µ lep ρ lep† (k(↓1))σκ = 2M me U σµδµU ρκ j δi = Kronecker delta, then constraint:

∗ ∗ ∗ ∗ 2 Tr(k(↑1)k(↑1) + k(↓1)k(↓1) + 3(k(↑3)k(↑3) + k(↓3)k(↓3))) = 2 g

⇒ mass matrices satisfy

X σ 2 σ 2 σ 2 σ 2 2 (mν ) + (me ) + 3 (mu ) + 3 (md ) = 8 M σ

Matilde Marcolli NCG models for particles and cosmology, II See-saw mechanism: D = D(Y ) Dirac  ∗ ∗  0 Mν MR 0  Mν 0 0 0   ¯ ∗   MR 0 0 Mν  0 0 M¯ ν 0 on subspace (νR , νL, ν¯R , ν¯L): largest eigenvalue of MR ∼ Λ unification scale. Take MR = x kR in flat space, Higgs vacuum v 2 small (w/resp to unif scale) ∂uTr(f (DA/Λ)) = 0 u = x 2 ∗ 2 2 f2 Λ Tr(kR kR ) x = ∗ 2 f0 Tr((kR kR ) ) Dirac mass Mν ∼ Fermi energy v 1 q (±m ± m2 + 4 v 2) 2 R R v 2 two eigenvalues ∼ ±mR and two ∼ ± mR Compare with estimates 7 12 16 (mR )1 ≥ 10 GeV , (mR )2 ≥ 10 GeV , (mR )3 ≥ 10 GeV

Matilde Marcolli NCG models for particles and cosmology, II Low energy limit: compatibilities and predictions Running of top Yukawa coupling (dominant term): v √ (y σ) = (mσ), 2 · · dy 1 9  t = y 3 − a g 2 + b g 2 + c g 2 y , dt 16π2 2 t 1 2 3 t 17 9 (a, b, c) = ( , , 8) 12 4 ⇒ value of top quark mass agrees with known (1.04 times if neglect other Yukawa couplings)

Matilde Marcolli NCG models for particles and cosmology, II Top quark running using mass relation at unification

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σ correction to MSM flow by yν for τ neutrino (allowed to be comparably large by see-saw) lowers value

Matilde Marcolli NCG models for particles and cosmology, II Higgs mass prediction using RGE for MSM Higgs scattering parameter:

Z 2 Z f0 4 √ 4 π b 4 √ 4 2 b |ϕ| g d x = 2 |H| g d x 2 π 2 f0 a

⇒ relation at unification (λ˜ is |H|4 coupling)

b λ˜(Λ) = g 2 3 a2 Running of Higgs scattering parameter: dλ 1 = λγ + (12λ2 + B) dt 8π2 1 3 γ = (12y 2 − 9g 2 − 3g 2) B = (3g 4 + 2g 2g 2 + g 4) − 3y 4 16π2 t 2 1 16 2 1 2 1 t

Matilde Marcolli NCG models for particles and cosmology, II Higgs estimate (in MSM approximation for RGE flow)

M2 √ 2M m2 = 8λ , m = 2λ H g 2 H g

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Matilde Marcolli NCG models for particles and cosmology, II Next time RGE flow for νMSM and see-saw scales Sensitive dependence on initial condition and constraints RGE scales and the cosmology timeline Gravitational terms and RGE running Models of the Very Early Universe

Matilde Marcolli NCG models for particles and cosmology, II