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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]
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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)?
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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)
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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.
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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.
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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.
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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
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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
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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
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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