Why the standard model

Joint work with Ali Chamseddine

(based on previous work with M. Marcolli)

1 Space-Time

Flat space (Poincar´e,Einstein, Minkowski)

ds2 = − dt2 + dx2 + dy2 + dz2

Curved space, gravitational potential gµν 2 µ ν ds = gµνdx dx Action principle Z 1 √ 4 SE[ gµν] = R g d x 16πG M

S = SE + SSM

Classical → Quantum (Feynman)

i S e ~

2 Standard Model

1 a a abc a b c 1 2 abc ade b c d e LSM = − ∂νg ∂νg − gsf ∂µg g g − g f f g g g g 2 µ µ ν µ ν 4 s µ ν µ ν

1 1 1 −∂ W +∂ W − − M 2W +W − − ∂ Z0∂ Z0 − M 2Z0Z0 − ∂ A ∂ A ν µ ν µ µ µ ν µ ν µ 2 µ µ µ ν µ ν 2 2cw 2 0 + − + − 0 + − − + −igcw(∂νZµ (Wµ Wν − Wν Wµ ) − Zν (Wµ ∂νWµ − Wµ ∂νWµ ) 0 + − − + + − + − +Zµ (Wν ∂νWµ − Wν ∂νWµ )) − igsw(∂νAµ(Wµ Wν − Wν Wµ ) + − − + + − − + −Aν(Wµ ∂νWµ − Wµ ∂νWµ ) + Aµ(Wν ∂νWµ − Wν ∂νWµ )) 1 1 − g2W +W −W +W − + g2W +W −W +W − 2 µ µ ν ν 2 µ ν µ ν 2 2 0 + 0 − 0 0 + − 2 2 + − + − +g cw(Zµ Wµ Zν Wν −Zµ Zµ Wν Wν )+g sw(AµWµ AνWν −AµAµWν Wν )

2 0 + − + − 0 + − 1 1 2 2 +g swcw(AµZ (W W −W W )−2AµZ W W )− ∂µH∂µH− m H ν µ ν ν µ µ ν ν 2 2 h

+ − 2 + − 1 0 0 1 2 0 0 −∂µφ ∂µφ − M φ φ − ∂µφ ∂µφ − M φ φ 2 2c2 µ w ¶ 2 4 2M 2M 1 2 0 0 + − 2M −βh + H + (H + φ φ + 2φ φ ) + αh g2 g 2 g2

3 ¡ ¢ 3 0 0 + − −gαhM H + Hφ φ + 2Hφ φ ¡ ¢ 1 2 4 0 4 + − 2 0 2 + − 2 + − 0 2 2 − g αh H + (φ ) + 4(φ φ ) + 4(φ ) φ φ + 4H φ φ + 2(φ ) H 8 1 M −gMW +W −H − g Z0Z0H µ µ 2 µ µ 2 cw ¡ ¢ 1 + 0 − − 0 − 0 + + 0 − ig W (φ ∂µφ − φ ∂µφ ) − W (φ ∂µφ − φ ∂µφ ) 2 µ µ ¡ ¢ 1 + − − − + + + g W (H∂µφ − φ ∂µH) + W (H∂µφ − φ ∂µH) 2 µ µ 2 1 1 0 0 0 sw 0 + − − + + g (Zµ (H∂µφ − φ ∂µH) − ig MZµ (Wµ φ − Wµ φ ) 2 cw cw 2 + − − + 1 − 2cw 0 + − − + +igswMAµ(Wµ φ − Wµ φ ) − ig Zµ (φ ∂µφ − φ ∂µφ ) 2cw ¡ ¢ + − − + 1 2 + − 2 0 2 + − +igswAµ(φ ∂µφ − φ ∂µφ ) − g W W H + (φ ) + 2φ φ 4 µ µ 1 1 ¡ ¢ − g2 Z0Z0 H2 + (φ0)2 + 2(2s2 − 1)2φ+φ− 2 µ µ w 8 cw 2 2 1 2 sw 0 0 + − − + 1 2 sw 0 + − − + − g Zµ φ (Wµ φ + Wµ φ ) − ig Zµ H(Wµ φ − Wµ φ ) 2 cw 2 cw

1 2 0 + − − + 1 2 + − − + + g swAµφ (W φ + W φ ) + ig swAµH(W φ − W φ ) 2 µ µ 2 µ µ 2 sw 2 0 + − 2 2 + − −g (2cw − 1)Zµ Aµφ φ − g swAµAµφ φ cw

4 1 a σ µ σ a λ λ λ λ λ λ λ λ + igsλ (¯q γ q )g − ¯e (γ∂ + m )e − ν¯ γ∂ν − u¯ (γ∂ + m )u 2 ij i j µ e j u j ³ ´ λ λ λ λ µ λ 2 λ µ λ 1 λ µ λ −d¯ (γ∂ + m )d + igswAµ −(¯e γ e ) + (¯u γ u ) − (d¯ γ d ) j d j 3 j j 3 j j

ig 0 λ µ 5 λ λ µ 2 5 λ + Zµ {(¯ν γ (1 + γ )ν ) + (¯e γ (4sw − 1 − γ )e ) 4cw 4 8 +(d¯λγµ( s2 − 1 − γ5)dλ) + (¯uλγµ(1 − s2 + γ5)uλ)} j 3 w j j 3 w j ¡ ¢ ig + λ µ 5 λ λ µ 5 κ + √ Wµ (¯ν γ (1 + γ )e ) + (¯uj γ (1 + γ )Cλκdj ) 2 2 ig ¡ ¢ − λ µ 5 λ ¯κ † µ 5 λ + √ Wµ (¯e γ (1 + γ )ν ) + (dj C γ (1 + γ )uj ) 2 2 κλ ig mλ ¡ ¢ + √ e −φ+(¯νλ(1 − γ5)eλ) + φ−(¯eλ(1 + γ5)νλ) 2 2 M g mλ ¡ ¢ − e H(¯eλeλ) + iφ0(¯eλγ5eλ) 2 M ¡ ¢ ig + κ λ 5 κ λ λ 5 κ + √ φ −m (¯uj Cλκ(1 − γ )dj ) + mu(¯uj Cλκ(1 + γ )dj 2M 2 d ig ¡ ¢ − λ ¯λ † 5 κ κ ¯λ † 5 κ + √ φ m (dj C (1 + γ )uj ) − mu(dj C (1 − γ )uj 2M 2 d λκ λκ g mλ g mλ ig mλ ig mλ − u H(¯uλuλ) − d H(d¯λdλ) + u φ0(¯uλγ5uλ) − d φ0(d¯λγ5dλ) 2 M j j 2 M j j 2 M j j 2 M j j

5 a 2 a abc a b c + 2 2 + − 2 2 − + G¯ ∂ G + gsf ∂µG¯ G gµ + X¯ (∂ − M )X + X¯ (∂ − M )X

M 2 +X¯0(∂2 − )X0 + Y¯ ∂2Y + igc W +(∂ X¯0X− − ∂ X¯+X0) 2 w µ µ µ cw + − + − − 0 0 + +igswWµ (∂µYX¯ − ∂µX¯ Y ) + igcwWµ (∂µX¯ X − ∂µX¯ X ) − − + 0 + + − − +igswWµ (∂µX¯ Y − ∂µYX¯ ) + igcwZµ (∂µX¯ X − ∂µX¯ X ) + + − − +igswAµ(∂µX¯ X − ∂µX¯ X ) ³ ´ 1 1 − gM X¯+X+H + X¯−X−H + X¯0X0H 2 2 cw 1 − 2c2 ¡ ¢ 1 ¡ ¢ + w igM X¯+X0φ+ − X¯−X0φ− + igM X¯0X−φ+ − X¯0X+φ− 2cw 2cw ¡ ¢ ¡ ¢ 0 − + 0 + − 1 + + 0 − − 0 +igMsw X¯ X φ − X¯ X φ + igM X¯ X φ − X¯ X φ . 2

6 Notations

± 0 a – Gauge bosons : Aµ,Wµ ,Zµ, gµ κ κ σ – Quarks : uj , dj , collective : qj – Leptons : eλ, νλ – Higgs fields : H, φ0, φ+, φ− – Ghosts : Ga,X0,X+,X−,Y , λ λ λ – Masses : md, mu, me , mh,M (the latter is the mass of the W ) – Tadpole Constant βh – Cosine and sine of the weak mixing angle c , s w w √ – Coupling constants swg = 4πα (fine struc- m2 ture), g = strong, α = h s h 4M2 – Cabibbo–Kobayashi–Maskawa mixing matrix : Cλκ – Structure constants of SU(3) : f abc – The Gauge is the Feynman–t’Hooft gauge.

7 Symmetries of S = SE + SSM

G = U(1) × SU(2) × SU(3)

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

Is there a space X whose group of diffeomor- phisms is directly of that form ?

Answer :

No : for ordinary “commutative spaces”

Yes : for noncommutative spaces

8 ∞ ∞ A = C (M,Mn(C)) = C (M) ⊗ Mn(C)

The group Inn(A) is locally isomorphic to the group G of smooth maps from M to the small gauge group G = PSU(n) (quotient of SU(n) by its center)

1 → Inn(A) → Aut(A) → Out(A) → 1

1 → Map(M,G) → G → Diff(M) → 1.

9 What is a metric in spectral geometry

– It contains the Riemannian paradigm (M, gµν) as a special case. – It does not require the commutativity of co- ordinates. – It contains spaces Xz of complex dimension z suitable for the Dim-Reg procedure. – It provides a way of expressing the full stan- dard model coupled to Einstein gravity as pure gravity on a modified space-time geo- metry. – It allows for quantum corrections to the geo- metry.

10 Infrared Ultraviolet

0 5 10 15 20 25 30 1014 Hz

Meter → Wave length (Krypton (1967) spec- trum of 86Kr then Caesium (1984) hyperfine levels of C133)

11 In fact, the actual definition of the unit of length m in the metric system is as a specific 9192631770 fraction 299792458 ∼ 30.6633... of the wave length of the radiation coming from the tran- sition between two hyperfine levels of the Ce- sium 133 atom. Indeed the speed of light is fixed once and for all at the value of

c = 299792458 m/s and the second s, which is the unit of time, is defined as the time taken by 9192631770 periods of the above radiation.

12 Spectral Triples

A spectral triple (A, H,D) is given by an involu- tive unital algebra A represented as operators in a Hilbert space H and a self-adjoint ope- rator D with compact resolvent such that all commutators [D, a] are bounded for a ∈ A.

A spectral triple is even if the Hilbert space H is endowed with a Z/2- grading γ which com- mutes with any a ∈ A and anticommutes with D.

13 Real Structure

A real structure of KO-dimension n ∈ Z/8 on a spectral triple (A, H,D) is an antilinear iso- metry J : H → H, with the property that

J2 = ε, JD = ε0DJ, and Jγ = ε00γJ (1) The numbers ε, ε0, ε00 ∈ {−1, 1} are a function of n mod 8 given by

n 0 1 2 3 4 5 6 7 ε 1 1 -1 -1 -1 -1 1 1 ε0 1 -1 1 1 1 -1 1 1 ε00 1 -1 1 -1

[a, b0] = 0 ∀ a, b ∈ A, b0 = Jb∗J−1 (2)

[[D, a], b0] = 0 ∀ a, b ∈ A . (3)

14 Our road to F is through the following steps 1. We classify the irreducible triplets (A, H,J). 2. We study the Z/2-gradings γ on H.

3. We classify the subalgebras AF ⊂ A which allow for an operator D that does not com- mute with the center of A but fulfills the “order one” condition (3)

0 [[D, a], b ] = 0 ∀ a, b ∈ AF .

15 Assume irreducibility, then one of the following cases holds

– The center Z(AC) is reduced to C. −1 – One has Z(AC) = C ⊕ C and Je1J = e2 where ej ∈ Z(AC) are the minimal projec- tions of Z(AC).

16 The case Z(AC) = C

Let H be a Hilbert space of dimension n. Then an irreducible solution with Z(AC) = C exists 2 iff n = k is a square. It is given by AC = Mk(C) acting by left multiplication on itself and antilinear involution

∗ J(x) = x , ∀x ∈ Mk(C) .

Three possibilities – A = Mk(C) (unitary case) – A = Mk(R) (orthogonal case) – A = Ma(H), for even k = 2a, (symplectic case)

17 The case Z(AC) = C ⊕ C

Let H be a Hilbert space of dimension n. Then an irreducible solution with Z(AC) = C ⊕ C exists iff n = 2k2 is twice a square. It is given by AC = Mk(C) ⊕ Mk(C) acting by left multi- plication on itself and antilinear involution

∗ ∗ J(x, y) = (y , x ) , ∀x, y ∈ Mk(C) .

18 Intrinsic description

AC = EndC(W ) ⊕ EndC(V ) .

∗ E = HomC(V,W ) , E = HomC(W, V )

H = E ⊕ E∗ ,J(ξ, η) = (η∗, ξ∗)

19 F of KO-dimension 6 (mod 8)

In the case Z(AC) = C, let γ be a Z/2-grading of H such that γAγ−1 = A and Jγ = ²00γJ for ²00 = ±1. Then ²00 = 1. ⇓ case Z(AC) = C is excluded

20 Z/2-grading

We make the hypothesis that both the grading and the real form come by assuming that the vector space W is a right vector space over H and is non-trivially Z/2-graded. ⇓ Simplest case : W = H2, V = C4

(The space E = HomC(V,W ) is related to the classification of instantons)

21 Up to an automorphism of Aev, there exists ev a unique involutive subalgebra AF ⊂ A of maximal dimension admitting off-diagonal Di- rac operators. It is given by

AF = {(λ⊕q, λ⊕m) | λ ∈ C , q ∈ H , m ∈ M3(C)}

⊂ H ⊕ H ⊕ M4(C) , using a field morphism C → H. The involutive algebra AF is isomorphic to C ⊕ H ⊕ M3(C) and together with its representation in (H, J, γ) it gives the noncommutative geometry F .

22 The product geometry

A = A1 ⊗ A2 , H = H1 ⊗ H2 ,

D = D1 ⊗1+γ1 ⊗D2 , γ = γ1 ⊗γ2 ,J = J1 ⊗J2

∞ ∞ A = C (M) ⊗ AF = C (M, AF )

2 2 H = L (M,S) ⊗ HF = L (M,S ⊗ HF )

D = ∂/M ⊗ 1 + γ5 ⊗ DF where ∂/M is the Dirac operator on M.

23 Spectral Model

Let M be a Riemannian spin 4-manifold and F the finite noncommutative geometry of KO- dimension 6 described above. Let M × F be endowed with the product metric. 1. The unimodular subgroup of the unitary group acting by the adjoint representation Ad(u) in H is the group of gauge transfor- mations of SM. 2. The unimodular inner fluctuations of the metric give the gauge bosons of SM. 3. The full standard model (with neutrino mixing and seesaw mechanism) minimally coupled to Einstein gravity is given in Euclidean form by the action functional 1 S = Tr(f(D /Λ))+ h J ξ,˜ D ξ˜i , ξ˜∈ H+, A 2 A cl where DA is the Dirac operator with the unimodular inner fluctuations. 24 Predictions

– Unification of couplings g2 f 1 5 3 0 = , g2 = g2 = g2 . 2π2 4 3 2 3 1 – See-saw mechanism for neutrino masses with large MR ∼ Λ. – The mass matrices satisfy the constraint at unification X σ 2 σ 2 σ 2 σ 2 2 (mν ) +(me ) +3 (mu) +3 (md) = 8 M σ X σ 2 σ 2 σ 2 σ 2 Y2 = (yν ) + (ye ) + 3 (yu) + 3 (yd ) σ

2 Y2(S) = 4 g . This yields a value of the top mass which is 1.04 times the observed value when ne- glecting the yukawa couplings of the bottom quarks etc...and is hence compatible with ex- periment.

25 – The Higgs scattering parameter b ˜λ(Λ) = g2 . 3 a2 The numerical solution to the RG equations with the boundary value λ0 = 0.356 at Λ = 17 10 GeV gives λ(MZ) ∼ 0.241 and a Higgs mass of the order of 170 GeV.

– Newton constant (f2 ∼ 5f0) – No proton decay 1 a a abc a b c 1 2 abc ade b c d e + − LSM = − 2 ∂νgµ∂νgµ − gsf ∂µgν gµgν − 4 gs f f gµgνgµgν − ∂νWµ ∂νWµ − 2 + − 1 0 0 1 2 0 0 1 0 + − M W W − ∂νZ ∂νZ − 2 M Z Z − ∂µAν∂µAν − igcw(∂νZ (W W − µ µ 2 µ µ 2cw µ µ 2 µ µ ν + − 0 + − − + 0 + − − + Wν Wµ ) − Zν (Wµ ∂νWµ − Wµ ∂νWµ ) + Zµ(Wν ∂νWµ − Wν ∂νWµ )) − + − + − + − − + + − igsw(∂νAµ(Wµ Wν − Wν Wµ ) − Aν(Wµ ∂νWµ − Wµ ∂νWµ ) + Aµ(Wν ∂νWµ − − + 1 2 + − + − 1 2 + − + − 2 2 0 + 0 − Wν ∂νWµ )) − 2 g Wµ Wµ Wν Wν + 2 g Wµ Wν Wµ Wν + g cw(ZµWµ Zν Wν − 0 0 + − 2 2 + − + − 2 0 + − ZµZµWν Wν ) + g sw(AµWµ AνWν − AµAµWν Wν ) + g swcw(AµZν (Wµ Wν − W +W −) − 2A Z0W +W −) − 1 ∂ H∂ H − 2M 2α H2 − ∂ φ+∂ φ− − 1 ∂ φ0∂ φ0 − ν µ µ ³µ ν ν 2 µ µ h ´ µ µ 2 µ µ 2M 2 2M 1 2 0 0 + − 2M 4 βh g2 + g H + 2 (H + φ φ + 2φ φ ) + g2 αh − 3 0 0 + − gαhM (H + Hφ φ + 2Hφ φ ) − 1 2 4 0 4 + − 2 0 2 + − 2 + − 0 2 2 8 g αh (H + (φ ) + 4(φ φ ) + 4(φ ) φ φ + 4H φ φ + 2(φ ) H ) − + − 1 M 0 0 gMWµ Wµ H − g 2 ZµZµH − ¡ 2 cw ¢ 1 ig W +(φ0∂ φ− − φ−∂ φ0) − W −(φ0∂ φ+ − φ+∂ φ0) + ¡ 2 µ µ µ µ ¢µ µ 1 g W +(H∂ φ− − φ−∂ H) + W −(H∂ φ+ − φ+∂ H) + 1 g 1 (Z0(H∂ φ0 − φ0∂ H) + 2 µ µ µ µ µ µ 2 cw µ µ µ 2 M ( 1 Z0∂ φ0 +W +∂ φ− +W −∂ φ+)−ig sw MZ0(W +φ− −W −φ+)+igs MA (W +φ− − cw µ µ µ µ µ µ cw µ µ µ w µ µ 2 W −φ+) − ig 1−2cw Z0(φ+∂ φ− − φ−∂ φ+) + igs A (φ+∂ φ− − φ−∂ φ+) − µ 2cw µ µ µ w µ µ µ 1 2 + − 2 0 2 + − 1 2 1 0 0 2 0 2 2 2 + − g W W (H + (φ ) + 2φ φ ) − g 2 Z Z (H + (φ ) + 2(2s − 1) φ φ ) − 4 µ µ 8 cw µ µ w 2 2 1 g2 sw Z0φ0(W +φ− + W −φ+) − 1 ig2 sw Z0H(W +φ− − W −φ+) + 1 g2s A φ0(W +φ− + 2 cw µ µ µ 2 cw µ µ µ 2 w µ µ W −φ+) + 1 ig2s A H(W +φ− − W −φ+) − g2 sw (2c2 − 1)Z0A φ+φ− − µ 2 w µ µ µ cw w µ µ g2s2 A A φ+φ− + 1 ig λa (¯qσγµqσ)ga − e¯λ(γ∂ + mλ)eλ − ν¯λ(γ∂ + mλ)νλ − u¯λ(γ∂ + w µ µ 2 s ij i j µ ¡ e ν ¢j λ λ ¯λ λ λ λ µ λ 2 λ µ λ 1 ¯λ µ λ mu)uj − dj (γ∂ + md )dj + igswAµ −(¯e γ e ) + 3 (¯uj γ uj ) − 3 (dj γ dj ) + ig Z0{(¯νλγµ(1 + γ5)νλ) + (¯eλγµ(4s2 − 1 − γ5)eλ) + (d¯λγµ( 4 s2 − 1 − γ5)dλ) + 4cw µ ¡ w j 3 w j ¢ (¯uλγµ(1 − 8 s2 + γ5)uλ)} + ig√ W + (¯νλγµ(1 + γ5)U lep eκ) + (¯uλγµ(1 + γ5)C dκ) + j 3 w ³j 2 2 µ λκ j ´ λκ j † † ig√ W − (¯eκU lep γµ(1 + γ5)νλ) + (d¯κC γµ(1 + γ5)uλ) + 2 2 µ¡ κλ j κλ j ¢ ig√ φ+ −mκ(¯νλU lep (1 − γ5)eκ) + mλ(¯νλU lep (1 + γ5)eκ + 2M³ 2 e λκ ν λκ ´ † † mλ ig√ φ− mλ(¯eλU lep (1 + γ5)νκ) − mκ(¯eλU lep (1 − γ5)νκ − g ν H(¯νλνλ) − 2M 2 e λκ ν λκ 2 M λ λ λ g me H(¯eλeλ) + ig mν φ0(¯νλγ5νλ) − ig me φ0(¯eλγ5eλ) − 1 ν¯ M R (1 − γ )ˆν − 2 M 2 M ¡ 2 M 4 λ λκ 5 κ ¢ 1 ν¯ M R (1 − γ )ˆν + ig√ φ+ −mκ(¯uλC (1 − γ5)dκ) + mλ(¯uλC (1 + γ5)dκ + 4 λ λκ ³ 5 κ 2M 2 d j λκ j ´u j λκ j † † mλ ig√ φ− mλ(d¯λC (1 + γ5)uκ) − mκ(d¯λC (1 − γ5)uκ − g u H(¯uλuλ) − 2M 2 d j λκ j u j λκ j 2 M j j mλ λ mλ g d ¯λ λ ig mu 0 λ 5 λ ig d 0 ¯λ 5 λ ¯a 2 a abc ¯a b c 2 M H(dj dj ) + 2 M φ (¯uj γ uj ) − 2 M φ (dj γ dj ) + G ∂ G + gsf ∂µG G gµ + ¯ + 2 2 + ¯ − 2 2 − ¯ 0 2 M 2 0 ¯ 2 + ¯ 0 − X (∂ − M )X + X (∂ − M )X +X (∂ − 2 )X + Y ∂ Y + igcwW (∂µX X − cw µ ¯ + 0 + ¯ − ¯ + − ¯ − 0 ∂µX X )+igswWµ (∂µYX − ∂µX Y ) + igcwWµ (∂µX X − ¯ 0 + − ¯ − ¯ + 0 ¯ + + ∂µX X )+igswWµ (∂µX Y − ∂µYX ) + igcwZµ(∂µX X − ¯ − − ¯ + + ³ ∂µX X )+igswAµ(∂µX ´X − 2 ¡ ¢ ¯ − − 1 ¯ + + ¯ − − 1 ¯ 0 0 1−2cw ¯ + 0 + ¯ − 0 − ∂µX X )− gM X X H + X X H + 2 X X H + igM X X φ − X X φ + 2 ¡ ¢ cw ¡ 2cw ¢ 1 ¯ 0 − + ¯ 0 + − ¯ 0 − + ¯ 0 + − igM X X φ − X X φ +igMsw X X φ − X X φ + 2cw ¡ ¢ 1 ¯ + + 0 ¯ − − 0 2 igM X X φ − X X φ . 1 Inner fluctuations of metric

One defines a right A-module structure on H by ξ b = b0 ξ , ∀ ξ ∈ H , b ∈ A The unitary group of the algebra A then acts by the “adjoint representation” in H in the form

ξ ∈ H → Ad(u) ξ = u ξ u∗ , ∀ ξ ∈ H The inner fluctuations of the metric are given by 0 −1 D → DA = D + A + ε JAJ where A is a self-adjoint operator of the form X A = aj[D, bj] , aj, bj ∈ A.

7 The bosonic part of the spectral action is Z 1 f √ S = (48 f Λ4 − f Λ2 c + 0 d) g d4x(1) π2 4 2 4 Z 96 f Λ2 − f c √ + 2 0 R g d4x 2 24Z π f0 11 ∗ ∗ µνρσ √ 4 + ( R R − 3 Cµνρσ C ) g d x 10 π2 6 Z (−2 a f Λ2 + e f ) √ + 2 0 |ϕ|2 g d4x 2 Z π f √ + 0 a |D ϕ|2 g d4x 2 µ 2 π Z f √ − 0 a R |ϕ|2 g d4x 12 π2 f Z + 0 (g2 Gi Gµνi + g2 F α F µνα 2 π2 3 µν 2 µν 5 2 µν √ 4 + g1 Bµν B ) g d x 3 Z f √ + 0 b |ϕ|4 g d4x 2 π2 where 1 R∗R∗ = ²µνρσ² RαβRγδ 4 αβγδ µν ρσ is the Euler characteristic. 10 We first perform a trivial rescaling of the Higgs field ϕ so that kinetic terms are normalized. To normalize the Higgs fields kinetic energy we have to rescale ϕ to : √ a f H = 0 ϕ , π so that the kinetic term becomes Z 1 2 √ 4 |DµH| g d x 2 The normalization of the kinetic terms imposes a relation between the coupling constants g1, g2, g3 and the coefficient f0, of the form g2 f 1 5 3 0 = , g2 = g2 = g2 . 2π2 4 3 2 3 1

11 The bosonic action then takes the form à R 1 µνρσ ∗ ∗ S = 2 R + α0 Cµνρσ C + γ0 + τ0 R R 2κ0 1 i µνi 1 α µνα 1 µν + G G + F F + Bµν B 4 µν 4 µν 4 ! 1 2 2 2 2 4 √ 4 + 2|Dµ H| − µ0|H| − ξ0 R |H| + λ0|H| g d x, where 2 1 96 f2 Λ −f0 c 2 = 2 κ0 12 π 2 µ2 = 2 f2 Λ − e 0 f0 a α = − 3 f0 0 10 π2 τ = 11 f0 0 60 π2 γ = 1 (48 f Λ4 − f Λ2 c + f0 d) 0 π2 4 2 4 2 λ = π b 0 2 f0 a2 1 ξ0 = 12

12 Standard Model notation notation Spectral Action √ √ Higgs Boson ϕ = ( 2M + H − iφ0, −i 2φ+) H = √1 a (1 + ψ) Inner metric(0,1) g 2 g

0 ± a (1,0) Gauge bosons Aµ,Zµ,Wµ , gµ (B, W, V ) Inner metric

(0,1) Fermion masses mu, mν Y(↑3) = δ(↑3),Y(↑1) = δ(↑1) Dirac in ↑ u, ν

κ † (0,1) CKM matrix Cλ , md Y(↓3) = C δ3, ↓ C Dirac in (↓ 3) Masses down

lep lep lep† (0,1) Lepton mixing U λκ, me Y(↓1) = U δ(↓1) U Dirac in (↓ 1) Masses leptons e

(0,1) Majorana MR YR Dirac on mass matrix ER ⊕ JF ER

2 2 5 2 Gauge couplings g1 = g tan(θw), g2 = g, g3 = gs g3 = g2 = 3 g1 Fixed at unification

2 1 2 mh 2 b Higgs scattering 8 g αh, αh = 4M 2 λ0 = g a2 Fixed at parameter unification

2 Tadpole constant β , (−α M 2 + βh ) |ϕ|2 µ2 = 2 f2Λ − e −µ2 |H|2 h h 2 0 f0 a 0

(1,0) Graviton gµν ∂/M Dirac

Table 1. Conversion from Spectral Action to Standard Model

1 The mass relation

The relation between the mass matrices comes from the equality of the Yukawa coupling terms LHf .

After Wick rotation to Euclidean and the chiral iπγ transformation U = e 4 5 ⊗1 they are the same provided the following equalities hold

g σ κ (k )σκ = m δ (↑3) 2M u σ g µ ρ † (k )σκ = m Cσµδ C (↓3) 2M d µ ρκ g σ κ (k )σκ = m δ (↑1) 2M ν σ g µ lep ρ lep† (k )σκ = m U σµδ U (↓1) 2M e µ ρκ j Here the symbol δi is the Kronecker delta (not to be confused with the previous notation δ↑↓).

14 It might seem at first sight that one can simply use the above equation to define the matrices kx but this overlooks the fact that one has the constraint

∗ ∗ Tr(k(↑1)k(↑1) + k(↓1)k(↓1)

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

The mass matrices thus satisfy the constraint X σ 2 σ 2 σ 2 σ 2 2 (mν ) + (me ) + 3 (mu) + 3 (md) = 8 M σ

15 Running of coupling constants

At one loop :

−2 3 41 19 βg = (4π) b g , with b = ( , − , −7), 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 where MZ is the mass of the Z vector boson.

16 0.09 Couplings

0.08 53 Α1

Α2 0.07

Α3 0.06

0.05

0.04

0.03

0.02

log10H̐GeVL 2.5 5 7.5 10 12.5 15 17.5

17 Constraint on Higgs scattering parameter

Z Z f √ π2 b √ 0 b |ϕ|4 g d4x = |H|4 g d4x 2 2 2 π 2 f0 a gives a further relation in our theory between the ˜λ|H|4 coupling and the gauge couplings to be imposed at the scale Λ. This is of the form b ˜λ(Λ) = g2 . 3 a2

18 Running of Higgs scattering parameter

dλ 1 = λγ + (12λ2 + B) dt 8π2 where 1 γ = (12 y2 − 9 g2 − 3 g2) 16π2 t 2 1 3 B = (3 g4 + 2g2 g2 + g4) − 3 y4 . 16 2 1 2 1 t

19 Higgs mass

The Higgs mass is then given by M2 √ 2M m2 = 8λ , m = 2λ H g2 H g

The numerical solution to these equations with 17 the boundary value λ0 = 0.356 at Λ = 10 GeV gives λ(MZ) ∼ 0.241 and a Higgs mass of the order of 170 GeV.

20 x=log10H̐GeVL

ΛHlogH̐MZLL 0.28

0.26

0.24

x 0 2.5 5 7.5 10 12.5 15 0.22

0.2

21 The fermion–boson mass relation

σ In terms of the Yukawa couplings (y· ) the mass constraint reads as 2 X v σ 2 σ 2 σ 2 σ 2 2 2 (yν ) +(ye ) +3 (yu) +3 (yd ) = 2 g v , 2 σ 2M with v = g the vacuum expectation value of the Higgs, as above.

In the traditional notation for the Standard Model the combination X σ 2 σ 2 σ 2 σ 2 Y2 = (yν ) + (ye ) + 3 (yu) + 3 (yd ) σ is denoted by Y2 = Y2(S). Thus, the mass constraint is of the form

2 Y2(S) = 4 g .

22 Running of top Yukawa coupling

v √ (yσ) = (mσ), 2 · · · ³ ´ ¸ dyt 1 9 3 2 2 2 = y − a g + b g + c g yt , dt 16π2 2 t 1 2 3 17 9 (a, b, c) = ( , , 8) 12 4

23 yt=0.596

yt=0.516 1.1

log10H̐GeVL 0 2.5 5 7.5 10 12.5 15 17.5

0.9

0.8

0.7

0.6

0.5

24 Gravitational terms

Z Ã 1 µνρσ 2 R + α0 Cµνρσ C + γ0 2κ0 ! ∗ ∗ 2 √ 4 +τ0 R R − ξ0 R |H| g d x,

Curvature square terms : Z Ã ! 1 µνρσ ω 2 θ √ 4 Cµνρσ C − R + E g d x 2η 3η η Z Z 1 √ 1 √ χ(M) = E g d4x = R∗R∗ g d4x 32π2 32π2

25 Running of gravitational terms

It is gauge independent and known (cf. Avra- midi)

1 133 2 βη = − η (4π)2 10 1 25 + 1098 ω + 200 ω2 βω = − η (4π)2 60 1 7 (56 − 171 θ) βθ = η (4π)2 90

26 log10H̐GeVL 2.5 5 7.5 10 12.5 15 17.5

-0.4

-0.6

-0.8

-1 ΗHlogH̐MZLL

-1.2

27 log10H̐GeVL 0 2.5 5 7.5 10 12.5 15

-0.0025

-0.005

-0.0075 ΩHlogH̐MZLL

-0.01

-0.0125

-0.015

-0.0175

28 log10H̐GeVL 2.5 5 7.5 10 12.5 15 17.5

-0.2

-0.4 ΘHlogH̐MZLL

-0.6

29 34