PDF Non-commutative Geometry & the Standard Model

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NON-COMMUTATIVE GEOMETRY &

THE STANDARD MODEL

Tatsu Takeuchi Virginia Tech

March 26, 2014 @ TWCU

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Collaborators:

•  Djordje Minic (Virginia Tech)

•  Ufuk Aydemir (Virginia Tech → Uppsala)

•  Chen Sun (Virginia Tech)

•  U. Aydemir, D. Minic, and T. Takeuchi

"The Higgs Mass and the Emergence of New Physics"

Physics Letters B 724 (2013) 301 [arXiv:1304.6092]

•  Work in Progress (no conclusions yet)

•  Collaborators welcome

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The Discovery of the Higgs @ 126 GeV

•  The Standard Model may be more robust against quantum corrections than previously believed

•  Is there a deeper reason for the Standard Model?

•  Many proposed models of physics Beyond the Standard Model are ugly and unconvincing

•  Do prettier models exist?

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The Superconnection Model

•  Y. Ne’eman, “Irreducible Gauge Theory of a Consolidated Salam-Weinberg Model,” Phys. Lett. B 81 (1979) 190.

•  D. Fairlie, “Higgs Fields and the determination of the Weinberg Angle,” Phys. Lett. B 82 (1979) 97.

•  D. Quillen, “Superconnections and the Chern Character,”

Topology 24 (1985) 89.

•  Y. Ne’eman, S. Sternberg, D. Fairlie, “Superconnections for electroweak su(2/1) and extensions, and the mass of the Higgs,” Physics Reports 406 (2005) 303.

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su(2/1) superalgebra:

λ₁^S =

0 1 0 1 0 0 0 0 0

!

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, λ₂^S =

0 −i 0

i 0 0

0 0 0

!

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, λ₃^S =

1 0 0 0 −1 0 0 0 0

!

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, λ₄^S =

0 0 1 0 0 0 1 0 0

!

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,

λ₅^S =

0 0 −i 0 0 0

i 0 0

!

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, λ₆^S =

0 0 0 0 0 1 0 1 0

!

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, λ₇^S =

0 0 0 0 0 −i 0 i 0

!

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, λ₈^S = 1 3

−1 0 1

0 −1 0 1 0 −2

!

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λ_i^S^,λ_j^S

!" #$= 2if_ijkλ_k^S^, ^i, ^j,^k =1, 2, 3

λ_i^S,λ_m^S

!" #$= 2if_imkλ_k^S, m,n = 4, 5, 6, 7

λ_i^S^,λ₈^S

!" #$= 0, _!"λ₈^S^,λ_j^S_#$= 2if_8mkλ_k^S^,

{

λ_m^S^,λ_n^S

}

⁼ ^2d^mni^λⁱ^S ⁻ ³^δ^mn^λ⁸^S

Only λ₈ is different from su(3).

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su(2/1) superconnection:

J_i = W_i =W_i^µdx_µ (i =1, 2, 3), J₈ = B = B^µdx_µ,

J₄ ∓iJ₅ = 2φ^±^, ^J₆ −iJ₇ = 2φ⁰^, ^J₆ +iJ₇ = 2φ^0*

J = g

2 λ_a^S

a=1 8

∑

^J^a ⁼ ^g₂

W_iτ_i − 1

3 B 2φ 2φ − 2

3 B

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%%

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&

' ((

(( (

F = dJ +J ∧J, S = − 1

g²

∫

Tr[F⋅F^*]^,

→ L =−1

4Wⁱ_µνWⁱ^µν − 1

4 B_µνB^µν +(D_µφ⁾^*^(D^µφ⁾− g²

2 (φ^*φ⁾² with g'

g = 1

3 → sin²θ_W = 1 4

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Pros & Cons:

•  Electroweak quantum numbers come out naturally by assigning the fermions to su(2/1) representations

•  Full SU(2/1) supersymmetry cannot be imposed or gauged.

(The Higgs is bosonic!)

•  The values of sin²θ_W and the Higgs quartic coupling are predicted

→ two less free parameters

→ is it any different from arbitrarily choosing these numbers?

→ the condition sin²θ_W=¼ is only true at ~4 TeV

→ emergent effective model at 4 TeV?

→ underlying noncommutative geometry?

→ Higgs mass is predicted but comes out to be ~170 GeV

→ Higgs mass prediction can be fixed by introducing more scalar degrees of freedom

→ Extension to su(2/2)?

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Assessment:

•  The Superconnection formalism is interesting only if it can be motivated by an underlying noncommutative geometry.

•  What is noncommutative geometry?

→ many reviews exist but none are particularly useful for model builders

•  Some of the more accessible ones:

•  J. C. Varilly, “An Introduction to Noncommutative Geometry,”

arXiv:physics/9709045

•  J. Madore, “Noncommutative Geometry for Pedestrians,” arXiv:gr- qc/9906059

•  P. Bongaarts, “A short introduction to noncommutative geometry,”

2004

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Algebraic Geometry:

•  In mathematics, geometry is discussed in terms of algebras and never in terms of sets of “points”

→ Mathematical papers difficult to decipher for physicists

•  A well known example of a noncommutative geometry is that of quantum mechanical phase space

→ The notion of “points” becomes fuzzy

→ Some type of “discreteness” is implied by the non- commutativity.

Geometry of a manifold M Noncommutative Geometry?

! ⇑

Algebra of functions defined on M ⇒ Noncommutative Algebras

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Noncommutative Geometry of Connes:

•  Alain Connes, 1982 Fields Medalist.

•  Spectral Triple: {A,H,D}

A: C* algebra (commutative or non-commutative)

H: Hilbert space on which A is represented by operators D: Dirac operator which defines the differential operator d

•  Using d, one can construct differential forms → differential geometry

f ∈ A, dfˆ =[D, ˆf ], d² = 0 fˆ : representation of f

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Spectral Standard Model 1

•  A. Connes and J. Lott, “Particle Models and

Noncommutative Geometry,” Nucl. Phys. B (Proc. Suppl.) 18B (1990) 29.

A = C^∞(Z) ⊗ (C ⊕ H ⊕ M₃(C)), Z : 4d spin manifold H = L²(S)⊗ (E + E), S : vector bundle of spinors on Z E = finite dimensional Hilbert space (flavor space)

D = γ^µ∂_µ ⊗ I +γ₅ ^Y ⁰

0 Y

%

&

' (

)*, Y = 0 M

M 0

%

&

' (

)*

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Spectral Standard Model 2

•  A. H. Chamseddine and A. Connes, “The Spectral Action Principle,” Commun. Math. Phys. 186 (1997) 731.

•  It is claimed that the action is independent of the choice of cutoff function.

Action = Tr[χ⁽^D² ^/ Λ²)]+ ψ^, ^Dψ

= Standard Model Action satisfying SO(10) GUT condition + fixed Higgs quartic coupling

χ : smooth cutoff function

α₃^(Λ) =α₂^(Λ⁾= 5

3α₁^(Λ), λ^(Λ) = 16π

3 α₃^(Λ⁾

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Pros:

•  4d spacetime is extended by a discrete dimension

consisting of 3 (?) points with a particular noncommutative geometry

•  The gauge group is connected to the geometry of the extended spacetime → no freedom to choose the gauge group once the geometry is fixed.

•  The su(2/1) connection comes out naturally,

→ SU(2) lives on one brane and U(1) on the other

→ The Higgs is the connection in the discrete direction

•  Provides justification and constrains the superconnection approach (though the parameters are constrained

differently at a different scale).

•  In principle, gravity can be treated on the same footing as the other interactions.

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Cons:

•  Matter context must be put in by hand by selecting the Hilbert space properly.

•  Quark and lepton mixing matrices must be put into the Dirac operator by hand.

•  How is the model different from the Standard Model with a particular boundary condition at the GUT scale? Isn’t it just a rewriting of the Standard Model? Is anything really gained?

•  Higgs mass prediction comes out to be 170 GeV

→ need to include more scalar degrees of freedom.

A. H. Chamseddine and A. Connes, “Resilience of the Spectral Standard Model,” JHEP 1209 (2012) 104

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Problems & Work in Progress:

•  Can a noncommutative geometry be used to motivate the emergence of the su(2/1) (or su(2/2)) superconnection

model at ~4 TeV? If yes, what would the observable consequences be?

•  Does the scale of the extra discrete dimension introduce a new scale which stabilizes the Higgs mass?

•  How much freedom does the Spectral Standard Model actually have in the choices of A, H and D?

•  What is the relation of Connes-Lott-Chamseddine to other approaches in the literature? Do better approaches exist?

•  Can the flavor structure of the Standard Model also be encoded into a noncommutative geometry?

•  Etc.