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Ufuk Aydemir Department of Physics and Astronomy, Uppsala University

Pati-Salam unification from noncommutative geometry and the TeV-scale WR boson

Collaborators: Djordje Minic, Tatsu Takeuchi, Chen Sun (Virginia Tech)

“The Higgs mass and the emergence of new Physics” UA, Djordje Minic, Tatsu Takeuchi, Physics Letters B 724 (2013) 301, [arXiv:1304.6092]

edd“The Higgs Mass, Superconnections and the TeV-scale Left-Right Symmetric Model” UA, Djordje Minic, Tatsu Takeuchi, Chen Sun, Phys.Rev. D91 (2015) 045020, [arXiv:1409.7574]

“Pati-Salam unification from noncommutative geometry and the TeV-scale WR boson” UA, Djordje Minic, Tatsu Takeuchi, Chen Sun, accepted for International Journal of Modern Physics A [arXiv:1512.00568]

The Discovery of the Higgs and BSM

• The SM works great in its realm but no answer for questions beyond it. - many issues remains unresolved; need for new physics.

• No signs of new physics at LHC yet; SUSY, technicolor, extra dims. 
- A big disappointment for BSM theories.

• The last building block of the SM, the Higgs, detected at 126 GeV. 
- a great success for the high energy physics community.

• Maybe the picture is completely different than what we expect - maybe time for a paradigm change?

• Why is the SM so robust? • Could it be a sign for a deeper reason than what we see? Like a geometry? • Is it possible to get more out of the SM beyond the EFT approach? - A possible UV/IR mixing (due to a geometry)?

Lie Supergroup (Graded Lie group), SU(N/M)Z2 grading

• H (N) – NxN

• H (M)– MxM

• θ –NxM matrices

(N+M)

Bosonic

Fermionic (Grassmanian numbers)

Here, unlike in SU(N), the supertrace is invariant—not the trace.

Therefore, the supertrace is the vanishing quantity.

Traceless part of H (N) and H (M) generates SU(N) and SU(M) Nonvanishing trace part defines U(1)

SU(N/M) ⊇ SU(N)xSU(M)xU(1)

SU(2/1), superconnections and supercurvaturesWe write the anti-commutative superconnection in the form

3x3

M is 2x2 and N is 1x1 supermatrices-- SU(2/1) g-even (valued ever one-form).

φ and φ are 1x2 and 2x1 supermatrices-– SU(2/1) g-odd (valued over zero-form).

Supercurvature:

Ne’eman-Sternberg rule for supermatrices.

SU(2/1) embedding

Compare it to the exp. value sin2θW=0.2312

▪ ▪

▪ ▪

Interpreting the sin2θW=0.25 is the value at Λs,

Λs≅4 TeV!

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Superconnection approach to the SM

▪ Embed SU(2)L x U(1)Y gauge and Higgs fields into a single SU(2/1) superconnection.

▪ Gauge-Higgs unification. More aesthetic. Gives Higgs field a geometric meaning.

Ne’eman and Sternberg (1990) (uses Quillen’s superconnections)

▪ The formalism gives the EW part of the SM with some extra constraints: 
 - Modulo the Higgs mass. It should either be added ad-hoc which breaks the
 SU(2/1) structure explicitly or Coleman-Weinberg type of introducing. 
 - But the kinetic and the potential terms for the Higgs sector come out automatically.

▪ SU(2/1) is not a symmetry of the theory. Gauging it would also be problematic. 
 - Rather, it can be interpreted as an emergent geometric structure.

▪ The formalism selects 4 TeV as the energy scale of its emergence.

▪ Wrong Higgs mass. Can be corrected by SU(2/2) extension— superconnection formalism for the left-right symmetric model. The energy scale here is also 4 TeV!

LHC

▪A possible UV/IR connection which manifests itself as the non-decoupling in the Higgs sector of the left-right symmetric model.

Superconnections Non-commutative geometry?

Generalized exterior derivative in superconnection——Matrix derivative

Non commutative geometryChang-Yeong Lee (1997),

Coquereaux and collaborators.

Superconnections make more sense if the underlying theory is NCG.

Brane 1 Brane 2

fL fR

L R

Jeven Jeven

Jodd= Higgs• Requires another new scale Λ! • Offers a new approach for Hierarchy problem. • Modification of RG running due to UV\IR mixing, 
-not only ingrate from UV down but IR up to get 
the self dual fixed point. SM with LRSM could provide
such a fixed point.

• Similar examples in non-commutative field theory.

Extra discrete dimension—Higgs bridges the gap!

▪ The formalism is more meaningful if the underlying theory is Connes’ non-commutative geometry. Discrete extra dimension consisting of two points. The even part of superconnection (gauge fields) connects the fermions with same chirality in each brane, while its odd part (Higgs field) bridges the gab between two branes and hence connects the fermions with different chirality.

Non-commutative geometry of Connes• Reformulate the notions of geometry in operator algebraic terms

Connes-Lott (1990)

• Basic objects: — a (possibly NC) algebra A
 — a Hilbert space H 
 — a self-adjoint operator D on M

generalization of manifold

metric structure

• Riemannian geometry as a special case:

• The action functional gives the pure YM theory in the ordinary Riemannian geometry

• Appropriately generalized Riemannian space: W = W + H SYM= SYM+ SH

generalized Dirac operator A is commutative

Regular gauge + Higgs fields

Regular YM kinetic terms + Higgs fields

• —- The basic fields are spinor fields
—- The bosonic fields come from diff. forms on H 
—- Input: Enter the fermion representations and the symm. breaking

algebra A generalized metric

Similar to superconnection case

• Spectral Triple: {A, H, D}

f ∈ A, df̂ = [D, f̂ ], d2 = 0

f̂ : representation of f

M: Compact spin-4 manifold S: vector bundle of spinors on M

L2 (M,S): Hilbert space of square integrable section of S

(C1(M), L2(M,S),r)

M ⇥ F

The Spectral Standard Model of NCG

• Spacetime is extended to a product of a continuous four dimensional manifold by a finite discrete space with non-commutative geometry.

Two points: Fermions on sheets at each point. Chirality introduced. Gauge transformations: unitary inner automorphisms of the algebra.

M ⇥ F two point space with NCG

S=

Cut-off function

Spectrum of the Dirac operator

• Gravity and the symmetries of the SM emerge. • SO(10) boundary conditions from the action—RG running from GUT scale. • Higgs mass of 170 GeV— same as in superconnection approach! • Same cure here as well: Taking into account extra dofs (singlet and neutrino) 
 to accommodate the right Higgs mass.

• Claimed that SU(2)L x SU(2)R x U(1)B-L emerges from SSM! Another sign for a possible SU(2/2) connection.

SM + GR+CC+ Weyl terms

The Spectral Pati-Salam

inner automorphisms

A = C1(M)⌦ (HR �HL �M4(C))

3 versions of Pati-Salam

Left-right symmetric one

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Recent LHC signals

• ATLAS and CMS recently reported an excess in various search channels in

the invariant mass region of 1.8 - 2.0 TeV.

• The largest deviation from the background occurs in heavy bosons hadronically

decaying to the W Z channel at around 2 TeV with a local significance of 3.4σ

and a global of 2.5σ.

• Potential signals in the HW and dijet final states.

• It was recently discussed in Brehmer et.al. (arXiv:1507.00013[hep-ph]) that these signals can be explained by a heavy gauge boson WR of the TeV-scale left-right model, with a single coupling gR =0.4.

• We analyze the compatibility of the unified left-right symmetric Pati-Salam models motivated by non-commutative geometry and the TeV scale right-handed W boson suggested by recent LHC data.

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Emerges

Smaller chains

Boundary/matching conditions

Ordering of scales

the most general chain for G2213 at MR not relevantSU(2)R ! U(1)R

{

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ai’s depend on the particle content in the relevant interval

Talking to the low energy world

MR=5 TeV and gR(MR)=0.4 ??

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• Depends on the particle content in each interval. • Need to know the scalar content— how to determine?

In the literature Extended Survival Hypothesis (ESH) used:

ESH states that at every step of the symmetry breaking chain, the only scalars which survive below the corresponding symmetry breaking scale are the ones which acquire vacuum expectation values (VEV’s) at the subsequent levels of the symmetry breaking.

Left-Right Symmetric Pati-Salam:

Minimal fine tuning

ESH

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Way off..

Model C�

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Relax the ESH

Light color scalars

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Outlook:

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▪ The superconnection approach to the SM based on the supergroup SU(2/1) and its extension to the LRM from the SU(2/2) fits in surprisingly well.

▪ We discussed the possibility that non-commutative geometry is the underlying theory of the emergent low energy SU(2/1)—SU(2/2) superconnection structure.

▪ Requirement of an extra scale associated with the discrete extra dimension of NCG, with the possibility of UV/IR mixing, can bring in a new insight to hierarchy problem.

▪ This (geometrical approach) may offer a road towards reading the possible implications/signals of Plank scale physics at the low energies via decoupling (possibly in the Higgs sector); there may be more to the EFT approach.

▪ NCG geometrizes the SM and places it on a similar footing to gravity.

▪ The framework is restrictive and therefore testable. Pati-Salam type models of NCG 
 will be challenged if the recent LHC signals turn out to be statistically significant.

▪ On the other hand, the current treatments available may not capture the true nature and predictions of NCG formalism.

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Thank you..

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Extra slides

Schucker, hep-th/0409077

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A real spectral triple is given by 5 items:

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where the Weyl tensor:

Schucker, hep-th/0409077