text: 1 Introduction Previously we have speculated [1–12] that 6 top + 6 anti-top quarks should be so strongly bound that the bound states, having masses small compared to the total mass of 12 top quarks, would eﬀectively function as elementary particles to ﬁrst approximation. In this case they can be added into loop calculations as “new” particles in the theory and seen, if produced at LHC, as resonances. Now the Higgs boson decay into two photons and its production via the gluon fusion mechanism are loop induced processes in the Standard Model1 (SM) and therefore relatively suppressed. So both these processes should be sensitive to loop contributions from our proposed new bound states. These new bound states are supposedly held together by the exchange of Higgs bosons and gluons between all the constituents. This is because the Higgs ﬁeld leads to attraction between tt, tt¯ and t¯t¯ quarks and the Yukawa coupling constant g in the t Lagrangian, describing the interaction of top-quarks with the Higgs boson: 1 √g L = D Φ DµΦ + t ψ ψ Φ + h.c. (1) 2 µ H H 2 tL tR H ∼ is large enough: g 1. If these bound states are indeed very light compared to the sum of t the constituent masses, we expect eﬀects arising from them which are not included in the usual type of essentially perturbative calculations using only the fundamental particles of the SM as elementary particles. In the present article we consider these eﬀects in the amplitudes for Higgs diphoton decay and for Higgs production via gluon fusion. In fact we estimate the scale factors (κ γ and κ ) by which the usual SM amplitudes for these processes have to be scaled when the g eﬀects of our proposed strongly bound states are included. The ATLAS and CMS collaborations have extracted values for these scale factors from their LHC data: κ = 0.97 (2σ interval 0.59 to 1.30), κ = 0.83 (2σ interval 0.63 to 1.03) (2) γ g for CMS [13], and ± ± κ = 1.20 0.15, κ = 1.04 0.14 (3) γ g for ATLAS [14, 15]. These experimental values are consistent with the usual SM values κ = 1, κ = 1, but are also consistent with signiﬁcant corrections to the usual SM values. γ g We confront our estimates of the corrections from our new bound states with these data and discuss the implications for our model. In estimating these corrections from our strongly bound states, we have to make a number of severe approximations: 1. We use non-relativistic calculations to evaluate the interaction vertices involving the bound states, in order to avoid Bethe Salpeter equations. This also means that we ignore the ﬁnite speed of Higgs exchange and use an instantaneous Higgs exchange potential. Consequently the spin of the quarks eﬀectively becomes an internal decoupled degree of freedom for our calculation. 1Here of course we refer to the usual Standard Model one loop diagrams involving just the fundamental SM particles. In this paper we are claiming that the true Standard Model calculation should include the loop contributions from our new bound states. 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Introduction

text: Previously

text: we

text: have

text: speculated

text: that

text: top

text: anti-top

text: quarks

text: should

text: be

text: so

text: strongly

text: bound

text: that

text: the

text: bound

text: states,

text: having

text: masses

text: small

text: compared

text: to

text: the

text: total

text: mass

text: of

text: top

text: quarks,

text: would

text: eﬀectively

text: function

text: as

text: elementary

text: particles

text: to

text: ﬁrst

text: approximation.

text: In

text: this

text: case

text: they

text: can

text: be

text: added

text: into

text: loop

text: calculations

text: as

text: “new”

text: particles

text: in

text: the

text: theory

text: and

text: seen,

text: if

text: produced

text: at

text: LHC,

text: as

text: resonances.

text: Now

text: the

text: Higgs

text: boson

text: decay

text: into

text: two

text: photons

text: and

text: its

text: production

text: via

text: the

text: gluon

text: fusion

text: mechanism

text: are

text: loop

text: induced

text: processes

text: in

text: the

text: Standard

text: (SM)

text: and

text: therefore

text: relatively

text: suppressed.

text: So

text: both

text: these

text: processes

text: should

text: be

text: sensitive

text: to

text: loop

text: contributions

text: from

text: our

text: proposed

text: new

text: bound

text: states.

text: These

text: new

text: bound

text: states

text: are

text: supposedly

text: held

text: together

text: by

text: the

text: exchange

text: of

text: Higgs

text: bosons

text: and

text: gluons

text: between

text: all

text: the

text: constituents.

text: This

text: is

text: because

text: the

text: Higgs

text: ﬁeld

text: leads

text: to

text: attraction

text: between

text: tt,

text: tt¯

text: and

text: t¯t¯

text: quarks

text: and

text: the

text: Yukawa

text: coupling

text: constant

text: g

text: in

text: the

text: t

text: Lagrangian,

text: describing

text: the

text: interaction

text: of

text: top-quarks

text: with

text: the

text: Higgs

text: boson:

text: √g

text: L

text: D

text: Φ

text: DµΦ

text: t

text: ψ

text: ψ

text: Φ

text: h.c.

text: µ

text: H

text: H

text: tL

text: tR

text: H

text: is

text: large

text: enough:

text: g

text: If

text: these

text: bound

text: states

text: are

text: indeed

text: very

text: light

text: compared

text: to

text: the

text: sum

text: of

text: t

text: the

text: constituent

text: masses,

text: we

text: expect

text: eﬀects

text: arising

text: from

text: them

text: which

text: are

text: not

text: included

text: in

text: the

text: usual

text: type

text: of

text: essentially

text: perturbative

text: calculations

text: using

text: only

text: the

text: fundamental

text: particles

text: of

text: the

text: SM

text: as

text: elementary

text: particles.

text: In

text: the

text: present

text: article

text: we

text: consider

text: these

text: eﬀects

text: in

text: the

text: amplitudes

text: for

text: Higgs

text: diphoton

text: decay

text: and

text: for

text: Higgs

text: production

text: via

text: gluon

text: fusion.

text: In

text: fact

text: we

text: estimate

text: the

text: scale

text: factors

text: (κ

text: γ

text: and

text: κ

text: by

text: which

text: the

text: usual

text: SM

text: amplitudes

text: for

text: these

text: processes

text: have

text: to

text: be

text: scaled

text: when

text: the

text: g

text: eﬀects

text: of

text: our

text: proposed

text: strongly

text: bound

text: states

text: are

text: included.

text: The

text: ATLAS

text: and

text: CMS

text: collaborations

text: have

text: extracted

text: values

text: for

text: these

text: scale

text: factors

text: from

text: their

text: LHC

text: data:

text: κ

text: interval

text: to

text: κ

text: interval

text: to

text: γ

text: g

text: for

text: CMS

text: and

text: κ

text: κ

text: γ

text: g

text: for

text: ATLAS

text: These

text: experimental

text: values

text: are

text: consistent

text: with

text: the

text: usual

text: SM

text: values

text: κ

text: κ

text: but

text: are

text: also

text: consistent

text: with

text: signiﬁcant

text: corrections

text: to

text: the

text: usual

text: SM

text: values.

text: γ

text: g

text: We

text: confront

text: our

text: estimates

text: of

text: the

text: corrections

text: from

text: our

text: new

text: bound

text: states

text: with

text: these

text: data

text: and

text: discuss

text: the

text: implications

text: for

text: our

text: model.

text: In

text: estimating

text: these

text: corrections

text: from

text: our

text: strongly

text: bound

text: states,

text: we

text: have

text: to

text: make

text: a

text: number

text: of

text: severe

text: approximations:

text: We

text: use

text: non-relativistic

text: calculations

text: to

text: evaluate

text: the

text: interaction

text: vertices

text: involving

text: the

text: bound

text: states,

text: in

text: order

text: to

text: avoid

text: Bethe

text: Salpeter

text: equations.

text: This

text: also

text: means

text: that

text: we

text: ignore

text: the

text: ﬁnite

text: speed

text: of

text: Higgs

text: exchange

text: and

text: use

text: an

text: instantaneous

text: Higgs

text: exchange

text: potential.

text: Consequently

text: the

text: spin

text: of

text: the

text: quarks

text: eﬀectively

text: becomes

text: an

text: internal

text: decoupled

text: degree

text: of

text: freedom

text: for

text: our

text: calculation.

text: of

text: course

text: we

text: refer

text: to

text: the

text: usual

text: Standard

text: Model

text: one

text: loop

text: diagrams

text: involving

text: just

text: the

text: fundamental

text: SM

text: particles.

text: In

text: this

text: paper

text: we

text: are

text: claiming

text: that

text: the

text: true

text: Standard

text: Model

text: calculation

text: should

text: include

text: the

text: loop

text: contributions

text: from

text: our

text: new

text: bound

text: states.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Fig. 3: CMS ( 68%, 95% and 99.7% CL) and ATLAS (68% and 95% CL) Collaboration likelihood contours for the scale factors κ and κ . The bent line represents the correlation | | γ g between κ and κ predicted by our model for positive values of the amplitude K. γ g 13 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Fig.

label: paragraph

text: CMS

label: caption

text: and

text: CL)

text: and

text: ATLAS

text: and

text: CL)

text: Collaboration

text: likelihood

text: contours

text: for

text: the

text: scale

text: factors

text: κ

text: and

text: κ

text: The

text: bent

text: line

text: represents

text: the

text: correlation

text: γ

text: g

text: between

text: κ

text: and

text: κ

text: predicted

text: by

text: our

text: model

text: for

text: positive

text: values

text: of

text: the

text: amplitude

text: K.

text: γ

text: g

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: (cid:112) correct relativistic normalization. We also introduce a normalization factor 2m ∗ for ∗ S1 the antisymmetric tensor ﬁeld V associated with the spin-1 bound state S . µν 1 Appendix B: On describing the polarizability In this appendix we discuss the interaction of γ-quanta with the bound states S and ∗ S given by the diagram of Fig. 2b and the eﬀective interaction obtained by integrating 1 ∗ out S and given by the diagram of Fig. 4. This eﬀective interaction, estimated in the 1 non-relativistic limit of the bound state S being essentially at rest, is described by the following polarization term in the Hamiltonian: −1 H = α E(cid:126) 2, (40) pol. pol. 2 where the constant α is called the polarizability. A model of polarizability is given in pol. Refs. [46, 47]. (cid:126) When a neutral bound state is placed in an electric ﬁeld E, the positively charged part of it gets pulled in the direction of the ﬁeld, and the negative cloud gets pulled in the direction opposite to the ﬁeld. This results in the centres of positive and negative charge no longer being in the same place, so the bound state obtains a small dipole moment (cid:126) d. Experimentally, it is found that for small ﬁelds this dipole moment is approximately proportional to the applied ﬁeld: (cid:126) (cid:126) d = α E. (41) pol. Let us now consider the dipole moment transition amplitude d = qR for the bound | | 1 1 ∗ state S, where q = +2/3 e and R is the radius of transition from S to the S -particle, 1 1 which is a 2p excited state of the spin-0 6t + 6t¯ bound state S. We make an estimate of the value of d based on the atomic physics calculation of the dipole moment matrix 1 elements of the hydrogen atom. According to the atomic physics terminology, we have: √ | | 27 ≈ R =< 2, 1, 0 z 1, 0, 0 >= 2 a 0.744a , (42) 1 35 B B where a is the radius of the Bohr Hydrogen-atom-like bound state S, in which we B approximate the interaction of each top quark by a central 1/r potential (neglecting the Higgs mass) while only half of the other 11 particles5 are present in the “nucleus” [10]. The variable z denotes the position coordinate in an axial coordinate system. Therefore we have: 4π ≈ 5.2 a = . (43) B 11(g2/2)m m 2 t ≈ t t for a running top quark Yukawa coupling g 0.935. t ∗ In the philosophy that the excited state S of the ground state S is a 2p state with 1 an excitation energy equal to 3/4 of the ground state binding energy E , we obtain [47] Ry the contribution of each of the 12 quark constituents to the polarizability of S to be: | | | | 2d2 < 1s z 2p > 2 α = 1 = 2q2 (44) pol. ∆E ∆E 5We note that our estimate of the loop amplitude A in appendix C is insensitive to the number Z of S quarks we take to be in the “nucleus” and therefore to the radius a . B 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: correct

text: relativistic

text: normalization.

text: We

text: also

text: introduce

text: a

text: normalization

text: factor

text: for

text: the

text: antisymmetric

text: tensor

text: ﬁeld

text: V

text: associated

text: with

text: the

text: bound

text: state

text: S

text: µν

text: Appendix

label: section

text: B:

text: On

text: describing

text: the

text: polarizability

text: In

text: this

text: appendix

text: we

text: discuss

text: the

text: interaction

text: of

text: γ-quanta

text: with

text: the

text: bound

text: states

text: S

text: and

text: S

text: given

text: by

text: the

text: diagram

text: of

text: Fig.

text: and

text: the

text: eﬀective

text: interaction

text: obtained

text: by

text: integrating

text: out

text: S

text: and

text: given

text: by

text: the

text: diagram

text: of

text: Fig.

text: This

text: eﬀective

text: interaction,

text: estimated

text: in

text: the

text: non-relativistic

text: limit

text: of

text: the

text: bound

text: state

text: S

text: being

text: essentially

text: at

text: rest,

text: is

text: described

text: by

text: the

text: following

text: polarization

text: term

text: in

text: the

text: Hamiltonian:

text: −1

text: H

text: α

text: pol.

text: pol.

text: where

text: the

text: constant

text: α

text: is

text: called

text: the

text: polarizability.

text: A

text: model

text: of

text: polarizability

text: is

text: given

text: in

text: pol.

text: Refs.

text: When

text: a

text: neutral

text: bound

text: state

text: is

text: placed

text: in

text: an

text: electric

text: ﬁeld

text: E,

text: the

text: positively

text: charged

text: part

text: of

text: it

text: gets

text: pulled

text: in

text: the

text: direction

text: of

text: the

text: ﬁeld,

text: and

text: the

text: negative

text: cloud

text: gets

text: pulled

text: in

text: the

text: direction

text: opposite

text: to

text: the

text: ﬁeld.

text: This

text: results

text: in

text: the

text: centres

text: of

text: positive

text: and

text: negative

text: charge

text: no

text: longer

text: being

text: in

text: the

text: same

text: place,

text: so

text: the

text: bound

text: state

text: obtains

text: a

text: small

text: dipole

text: moment

text: d.

text: Experimentally,

text: it

text: is

text: found

text: that

text: for

text: small

text: ﬁelds

text: this

text: dipole

text: moment

text: is

text: approximately

text: proportional

text: to

text: the

text: applied

text: ﬁeld:

text: d

text: α

text: E.

text: pol.

text: Let

text: us

text: now

text: consider

text: the

text: dipole

text: moment

text: transition

text: amplitude

text: d

text: qR

text: for

text: the

text: bound

text: state

text: S,

text: where

text: q

text: e

text: and

text: R

text: is

text: the

text: radius

text: of

text: transition

text: from

text: S

text: to

text: the

text: S

text: -particle,

text: which

text: is

text: a

text: excited

text: state

text: of

text: the

text: bound

text: state

text: S.

text: We

text: make

text: an

text: estimate

text: of

text: the

text: value

text: of

text: d

text: based

text: on

text: the

text: atomic

text: physics

text: calculation

text: of

text: the

text: dipole

text: moment

text: matrix

text: elements

text: of

text: the

text: hydrogen

text: atom.

text: According

text: to

text: the

text: atomic

text: physics

text: terminology,

text: we

text: have:

text: R

text: z

text: a

text: B

text: B

text: where

text: a

text: is

text: the

text: radius

text: of

text: the

text: Bohr

text: Hydrogen-atom-like

text: bound

text: state

text: S,

text: in

text: which

text: we

text: B

text: approximate

text: the

text: interaction

text: of

text: each

text: top

text: quark

text: by

text: a

text: central

text: potential

text: (neglecting

text: the

text: Higgs

text: mass)

text: while

text: only

text: half

text: of

text: the

text: other

text: are

text: present

text: in

text: the

text: “nucleus”

text: The

text: variable

text: z

text: denotes

text: the

text: position

text: coordinate

text: in

text: an

text: axial

text: coordinate

text: system.

text: Therefore

text: we

text: have:

text: 4π

text: a

text: B

text: m

text: t

text: t

text: t

text: for

text: a

text: running

text: top

text: quark

text: Yukawa

text: coupling

text: g

text: t

text: In

text: the

text: philosophy

text: that

text: the

text: excited

text: state

text: S

text: of

text: the

text: ground

text: state

text: S

text: is

text: a

text: state

text: with

text: an

text: excitation

text: energy

text: equal

text: to

text: of

text: the

text: ground

text: state

text: binding

text: energy

text: E

text: we

text: obtain

text: Ry

text: the

text: contribution

text: of

text: each

text: of

text: the

text: quark

text: constituents

text: to

text: the

text: polarizability

text: of

text: S

text: to

text: be:

text: z

text: α

text: pol.

text: ∆E

text: ∆E

text: note

text: that

text: our

text: estimate

text: of

text: the

text: loop

text: amplitude

text: A

text: in

text: appendix

text: C

text: is

text: insensitive

text: to

text: the

text: number

text: Z

text: of

text: S

text: quarks

text: we

text: take

text: to

text: be

text: in

text: the

text: “nucleus”

text: and

text: therefore

text: to

text: the

text: radius

text: a

text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ (cid:29) We now assume the bound state S is very heavy: m ∗ m , m /2, so we can 1 ∗ S1 S H replace the propagator of the bound state S using the approximation: 1 Π (cid:48) (cid:48) ≈ −g (cid:48)g (cid:48) −µνµ ν µµ νν . (59) q2 m2∗ m2∗ S S 1 1 If we further make the identiﬁcation − ≈ ∆E = m ∗ m m ∗ (60) S S S 1 1 in the expression (44) for the polarizability, we obtain the following result: 1 d2 = α m ∗. (61) (cid:90) 1 2 pol. S1 Then, using the approximation (59) and the relations (61) and (15), we obtain: 144α d4q 1 · 1 × I = i pol. m m2 F (q2) − − − S v t S (2π)4 0 (q + k )2 m2 (q k )2 m2 1 S 2 S · · − · · [(2k k )((cid:15) (cid:15) ) 2(k (cid:15) )(k (cid:15) )]. (62) 1 2 1 2 1 2 2 1 We note that we get the same expression for I from Fig. 4, using the eﬀective interaction S (51) supplemented with the form factor F (q2). This provides a useful consistency check 0 on our calculation and also explains why the mass m ∗ cancels out from our expression S 1 (62) for I . S We now make t(cid:90)he approximation of neglecting the photon momenta k as small 1,2 compared to m in the propagators of the expression (62) and obtain: S 144α d4q 1 · · − · · I = i pol. m m2 F (q2) − [(2k k )((cid:15) (cid:15) ) 2(k (cid:15) )(k (cid:15) )]. (63) S v t S (2π)4 0 (q2 m2 )2 1 2 1 2 1 2 2 1 (cid:90) S − After the Wick rotation q0 = iq4 , q2 = q2 we have: E E −144α d4q 1 · · − · · I = pol. m m2 E F (q2 ) [(2k k )((cid:15) (cid:15) ) 2(k (cid:15) )(k (cid:15) )]. (64) S v t S (2π)4 0 E (q2 + m2 )2 1 2 1 2 1 2 2 1 E S The symbol E occurring as an index in (64) means that we are now using a Euclidean metric. Taking into account that d4q q2 dq2 E = E E , (cid:90) (2π)4 16π2 we obtain: −144α q2 dq2 1 · · − · · I = pol. m m2 E E F (q2 ) [(2k k )((cid:15) (cid:15) ) 2(k (cid:15) )(k (cid:15) )]. (65) S v t S 16π2 0 E (q2 + m2 )2 1 2 1 2 1 2 2 1 E S We can also choose the polarizations (cid:15) for the photons to be orthogonal to both i · photon momenta k and k . From the kinematics(cid:90)it is easy to see that 2k k = m2 on 1 2 1 2 H shell, and so we have: ∞ − m2 · 144α q2 dq2 I = H ((cid:15) (cid:15) ) pol. m m2 F (q2 ) E E . (66) S 16π2 1 2 v t S 0 E (q2 + m2 )2 0 E S 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: now

text: assume

text: the

text: bound

text: state

text: S

text: is

text: very

text: heavy:

text: m

text: m

text: m

text: so

text: we

text: can

text: S

text: H

text: replace

text: the

text: propagator

text: of

text: the

text: bound

text: state

text: S

text: using

text: the

text: approximation:

text: Π

text: −g

text: −µνµ

text: ν

text: µµ

text: νν

text: S

text: S

text: If

text: we

text: further

text: make

text: the

text: identiﬁcation

text: ∆E

text: m

text: m

text: m

text: S

text: S

text: S

text: in

text: the

text: expression

text: for

text: the

text: polarizability,

text: we

text: obtain

text: the

text: following

text: result:

text: α

text: m

text: pol.

text: Then,

text: using

text: the

text: approximation

text: and

text: the

text: relations

text: and

text: we

text: obtain:

text: I

text: i

text: pol.

text: m

text: F

text: S

text: v

text: t

text: S

text: (q

text: k

text: (q

text: k

text: S

text: S

text: k

text: )(k

text: We

text: note

text: that

text: we

text: get

text: the

text: same

text: expression

text: for

text: I

text: from

text: Fig.

text: using

text: the

text: eﬀective

text: interaction

text: S

text: supplemented

text: with

text: the

text: form

text: factor

text: F

text: This

text: provides

text: a

text: useful

text: consistency

text: check

text: on

text: our

text: calculation

text: and

text: also

text: explains

text: why

text: the

text: mass

text: m

text: cancels

text: out

text: from

text: our

text: expression

text: S

text: for

text: I

text: S

text: We

text: now

text: make

text: approximation

text: of

text: neglecting

text: the

text: photon

text: momenta

text: k

text: as

text: small

text: compared

text: to

text: m

text: in

text: the

text: propagators

text: of

text: the

text: expression

text: and

text: obtain:

text: S

text: I

text: i

text: pol.

text: m

text: F

text: k

text: )(k

text: S

text: v

text: t

text: S

text: S

text: After

text: the

text: Wick

text: rotation

text: we

text: have:

text: E

text: E

text: I

text: pol.

text: m

text: E

text: F

text: k

text: )(k

text: S

text: v

text: t

text: S

text: E

text: E

text: S

text: The

text: symbol

text: E

text: occurring

text: as

text: an

text: index

text: in

text: means

text: that

text: we

text: are

text: now

text: using

text: a

text: Euclidean

text: metric.

text: Taking

text: into

text: account

text: that

text: E

text: E

text: E

text: we

text: obtain:

text: I

text: pol.

text: m

text: E

text: E

text: F

text: k

text: )(k

text: S

text: v

text: t

text: S

text: E

text: E

text: S

text: We

text: can

text: also

text: choose

text: the

text: polarizations

text: for

text: the

text: photons

text: to

text: be

text: orthogonal

text: to

text: both

text: i

text: photon

text: momenta

text: k

text: and

text: k

text: From

text: the

text: is

text: easy

text: to

text: see

text: that

text: k

text: on

text: H

text: shell,

text: and

text: so

text: we

text: have:

text: I

text: H

text: pol.

text: m

text: F

text: E

text: E

text: S

text: v

text: t

text: S

text: E

text: E

text: S

text: ##LTLine##

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text: (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) It then follows from (85) that 44 m ≈ 352m m 2 m 3 m 3 t t t t t A = 0.018 = 0.034 . (87) 3 m F 27m2 m3 a4 m m m F H F B H F F where we have used the expression (43) for a from appendix B. B References [1] C.D. Froggatt and H.B. Nielsen, Surveys High Energy Phys. 18, 55-75 (2003) [arXiv: hep-ph/0308144]. [2] C.D.Froggatt, H.B.Nielsen, in Proc. to the Euroconference on Symmetries Beyond the Standard Model, (DMFA, Zaloznistvo, 2003) pp. 73-89 [ArXiv: hep-ph/0312218]. [3] C.D. Froggatt, H.B. Nielsen and L.V. Laperashvili, Int. J. Mod. Phys. A 20, 1268 (2005) [arXiv: hep-ph/0406110]. [4] C.D. Froggatt, L.V. Laperashvili and H.B. Nielsen, Phys. Atom. Nucl. 69, 67 (2006) [Yad. Fiz. 69, 3 (2006)]; [arXiv: hep-ph/0407102]. [5] C.D. Froggatt, L.V. Laperashvili and H.B. Nielsen, in: Proceedings of 13th Interna- tional Seminar on High-Energy Physics: QUARKS-2004, arXiv: hep-ph/0410243. [6] C.D. Froggatt, PASCOS 2004 Particles, Strings and Cosmology, edited by G. Alver- son, E. Barberis, P. Nath and M.T. Vaughn (World Scientiﬁc, Singapore, 2005) pp. 325-334 [arXiv: hep-ph/0412337]. [7] C.D. Froggatt, L.V. Laperashvili, R.B. Nevzorov, H.B. Nielsen and C.R. Das, Report CHEP-PKU-1-04-2008, CERN-PH-/2008-051TH, arXiv:0804.4506. [8] C.D. Froggatt and H.B. Nielsen, in Proceedings of the 34th International Con- ference on High Energy Physics (ICHEP08), Philadelphia, 2008, eConf C080730, [arXiv:0810.0475]. [9] C.D. Froggatt and H.B. Nielsen, Phys. Rev. D 80, 034033 (2009) [arXiv:0811.2089]. [10] C.R. Das, C.D. Froggatt, L.V. Laperashvili and H.B. Nielsen, Int. J. Mod. Phys. A 26, 2503 (2011) [arXiv:0812.0828]. [11] C.R. Das, C.D. Froggatt, L.V. Laperashvili and H.B. Nielsen, in Proceedings of the 14th Lomonosov Conference on Elementary Particle Physics, edited by Alexander I. Studenikin (World Scientiﬁc, Singapore, 2010) pp. 379-381 [arXiv:0908.4514]. [12] C.D. Froggatt, C.R. Das, L.V. Laperashvili and H.B. Nielsen, Yad. Fiz., 76, 172 (2013) [arXiv:1212.2168]. [13] CMS Collaboration, CERN report CMS-PAS-HIG-13-005. [14] ATLAS Collaboration, ATLAS-CONF-2013-079. 25 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: It

text: then

text: follows

text: from

text: that

text: m

text: m

text: m

text: m

text: t

text: t

text: t

text: t

text: t

text: A

text: m

text: F

text: m

text: m

text: m

text: F

text: H

text: F

text: B

text: H

text: F

text: F

text: where

text: we

text: have

text: used

text: the

text: expression

text: for

text: a

text: from

text: appendix

text: B.

text: B

text: References

label: reference

text: C.D.

text: Froggatt

text: and

text: H.B.

text: Nielsen,

text: Surveys

text: High

text: Energy

text: Phys.

text: [arXiv:

text: C.D.Froggatt,

text: H.B.Nielsen,

text: in

text: Proc.

text: to

text: the

text: Euroconference

text: on

text: Symmetries

text: Beyond

text: the

text: Standard

text: Model,

text: (DMFA,

text: Zaloznistvo,

text: pp.

text: [ArXiv:

text: C.D.

text: Froggatt,

text: H.B.

text: Nielsen

text: and

text: L.V.

text: Laperashvili,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: [arXiv:

text: C.D.

text: Froggatt,

text: L.V.

text: Laperashvili

text: and

text: H.B.

text: Nielsen,

text: Phys.

text: Atom.

text: Nucl.

text: [Yad.

text: Fiz.

text: [arXiv:

text: C.D.

text: Froggatt,

text: L.V.

text: Laperashvili

text: and

text: H.B.

text: Nielsen,

text: in:

text: Proceedings

text: of

text: Interna-

text: tional

text: Seminar

text: on

text: High-Energy

text: Physics:

text: arXiv:

text: C.D.

text: Froggatt,

text: PASCOS

text: Particles,

text: Strings

text: and

text: Cosmology,

text: edited

text: by

text: G.

text: Alver-

text: son,

text: E.

text: Barberis,

text: P.

text: Nath

text: and

text: M.T.

text: Vaughn

text: (World

text: Scientiﬁc,

text: Singapore,

text: pp.

text: [arXiv:

text: C.D.

text: Froggatt,

text: L.V.

text: Laperashvili,

text: R.B.

text: Nevzorov,

text: H.B.

text: Nielsen

text: and

text: C.R.

text: Das,

text: Report

text: C.D.

text: Froggatt

text: and

text: H.B.

text: Nielsen,

text: in

text: Proceedings

text: of

text: the

text: International

text: Con-

text: ference

text: on

text: High

text: Energy

text: Physics

text: Philadelphia,

text: eConf

text: C.D.

text: Froggatt

text: and

text: H.B.

text: Nielsen,

text: Phys.

text: Rev.

text: D

text: C.R.

text: Das,

text: C.D.

text: Froggatt,

text: L.V.

text: Laperashvili

text: and

text: H.B.

text: Nielsen,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: C.R.

text: Das,

text: C.D.

text: Froggatt,

text: L.V.

text: Laperashvili

text: and

text: H.B.

text: Nielsen,

text: in

text: Proceedings

text: of

text: the

text: Lomonosov

text: Conference

text: on

text: Elementary

text: Particle

text: Physics,

text: edited

text: by

text: Alexander

text: I.

text: Studenikin

text: (World

text: Scientiﬁc,

text: Singapore,

text: pp.

text: C.D.

text: Froggatt,

text: C.R.

text: Das,

text: L.V.

text: Laperashvili

text: and

text: H.B.

text: Nielsen,

text: Yad.

text: Fiz.,

text: CMS

text: Collaboration,

text: CERN

text: report

text: ATLAS

text: Collaboration,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [15] ATLAS Collaboration (Georges Aad et al.), Phys. Lett. B 726, 88 (2013) [arXiv:1307.1427]. [16] C. D. Froggatt and H. B. Nielsen, arXiv:1403.7177. [17] D.L. Bennett, H.B. Nielsen, Int. J. Mod. Phys. A 9, 5155 (1994). [18] D.L. Bennett, C.D. Froggatt, H.B. Nielsen, Report NBI-HE-94-44, in Proceedings of the 27th International Conference on High Energy Physics, Glasgow, 1994, edited by P.Bussey and I.Knowles (IOP Publishing Ltd, 1995) pp. 557-559. [19] D.L. Bennett, C.D. Froggatt, H.B. Nielsen, Perspectives in Particle Physics ’94, edited by D.Klabucar, I.Picek and D.Tadic (World Scientiﬁc, Singapore 1995) pp. 255-279 [arXiv: hep-ph/9504294]. [20] C.D. Froggatt and H.B. Nielsen, Talk at 5th Hellenic School and Workshop on Ele- mentary Particles, Corfu, 1995, Conference C95-09-03.1 [arXiv: hep-ph/9607375]. [21] D.L. Bennett, C.D. Froggatt, H.B. Nielsen, in Proceedings of Seoul 1997, Recent de- velopments in nonperturbative quantum ﬁeld theory, Conference C97-05-26, pp. 362- 393 [arXiv: hep-ph/9710407]. [22] D.L. Bennett, H.B. Nielsen, in Proc. to the Euroconference on Symmetries Beyond the Standard Model, (DMFA, Zaloznistvo, 2003) pp. 235-246 [arXiv: hep-ph/0312218]. [23] H.B. Nielsen and M. Ninomiya, arXiv: hep-th/0701018. [24] D.L. Bennett, in Proceedings to the 12th Workshop on ’What Comes Beyond the Stan- dard Models? Bled, 2009 (DMFA, Zaloznistvo, Ljubljana, 2009) [arXiv:0912.4532]. [25] C. D. Froggatt, H. B. Nielsen, Phys. Lett. B 368, 96 (1996) [arXiv: hep-ph/9511371]. [26] N. Cabbibo, L. Maiani, G. Parisiand R. Petronzio, Nucl. Phys. B 158 (1979) 295. [27] P. Q. Hung, Phys. Rev. Lett. 42 (1979) 873. [28] M. Lindner, Z. Phys. C 31 (1986) 295. [29] M. Sher, Phys. Lett. B 317 (1993) 159: Addendum Phys. Lett. B 331 (1994) 448. [30] G. Altarelli and G. Isidori, Phys. Lett. B 337 (1994) 141. [31] J. A. Casas, J. R. Espinosa and M. Quiros Phys. Lett. B 342 (1995) 171. [32] J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, A. Riotto and A. Strumia, Phys. Lett. B709 (2012) 222 [arXiv:1112.3022]. [33] F. Bezrukov, M. Y. Kalmykov, B. A. Kniehl and M. Shaposhnikov, JHEP 1210 (2012) 140 [arXiv:1205.2893]. [34] G. Degrassi, S. Di Vita, J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori, A. Strumia, JHEP 1208, 098 (2012) [arXiv:1205.6497]. 26

text: ATLAS

text: Collaboration

text: (Georges

text: Aad

text: et

text: al.),

text: Phys.

text: Lett.

text: B

text: C.

text: D.

text: Froggatt

text: and

text: H.

text: B.

text: Nielsen,

text: D.L.

text: Bennett,

text: H.B.

text: Nielsen,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: D.L.

text: Bennett,

text: C.D.

text: Froggatt,

text: H.B.

text: Nielsen,

text: Report

text: in

text: Proceedings

text: of

text: the

text: International

text: Conference

text: on

text: High

text: Energy

text: Physics,

text: Glasgow,

text: edited

text: by

text: P.Bussey

text: and

text: I.Knowles

text: (IOP

text: Publishing

text: Ltd,

text: pp.

text: D.L.

text: Bennett,

text: C.D.

text: Froggatt,

text: H.B.

text: Nielsen,

text: Perspectives

text: in

text: Particle

text: Physics

text: edited

text: by

text: D.Klabucar,

text: I.Picek

text: and

text: D.Tadic

text: (World

text: Scientiﬁc,

text: Singapore

text: pp.

text: [arXiv:

text: C.D.

text: Froggatt

text: and

text: H.B.

text: Nielsen,

text: Talk

text: at

text: Hellenic

text: School

text: and

text: Workshop

text: on

text: Ele-

text: mentary

text: Particles,

text: Corfu,

text: Conference

text: [arXiv:

text: D.L.

text: Bennett,

text: C.D.

text: Froggatt,

text: H.B.

text: Nielsen,

text: in

text: Proceedings

text: of

text: Seoul

text: Recent

text: de-

text: velopments

text: in

text: nonperturbative

text: quantum

text: ﬁeld

text: theory,

text: Conference

text: pp.

text: [arXiv:

text: D.L.

text: Bennett,

text: H.B.

text: Nielsen,

text: in

text: Proc.

text: to

text: the

text: Euroconference

text: on

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text: Beyond

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text: Standard

text: Model,

text: (DMFA,

text: Zaloznistvo,

text: pp.

text: [arXiv:

text: H.B.

text: Nielsen

text: and

text: M.

text: Ninomiya,

text: arXiv:

text: D.L.

text: Bennett,

text: in

text: Proceedings

text: to

text: the

text: Workshop

text: on

text: ’What

text: Comes

text: Beyond

text: the

text: Stan-

text: dard

text: Models?

text: Bled,

text: (DMFA,

text: Zaloznistvo,

text: Ljubljana,

text: C.

text: D.

text: Froggatt,

text: H.

text: B.

text: Nielsen,

text: Phys.

text: Lett.

text: B

text: [arXiv:

text: N.

text: Cabbibo,

text: L.

text: Maiani,

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text: Parisiand

text: R.

text: Petronzio,

text: Nucl.

text: Phys.

text: B

text: P.

text: Q.

text: Hung,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Lindner,

text: Z.

text: Phys.

text: C

text: M.

text: Sher,

text: Phys.

text: Lett.

text: B

text: Addendum

text: Phys.

text: Lett.

text: B

text: G.

text: Altarelli

text: and

text: G.

text: Isidori,

text: Phys.

text: Lett.

text: B

text: J.

text: A.

text: Casas,

text: J.

text: R.

text: Espinosa

text: and

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text: Quiros

text: Phys.

text: Lett.

text: B

text: J.

text: Elias-Miro,

text: J.

text: R.

text: Espinosa,

text: G.

text: F.

text: Giudice,

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text: Isidori,

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text: Riotto

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text: Strumia,

text: Phys.

text: Lett.

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text: Bezrukov,

text: M.

text: Y.

text: Kalmykov,

text: B.

text: A.

text: Kniehl

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text: Shaposhnikov,

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text: Degrassi,

text: S.

text: Di

text: Vita,

text: J.

text: Elias-Miro,

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text: G G | +2α HFF¯ N Q2A + α HSS A 2. (11) m c t F m m S F t S In the above equation, the notations F and S refer to the spin-1/2 and spin-0 new bound states, respectively. If we could treat the bound state F as an elementary particle, the ≈ amplitude A would be given by A = A (τ ) 1/3. The amplitude A corresponds to F F 1/2 F S the loop diagram of Fig. 2b and arises from the polarizability of the self-conjugate scalar ∗ ∗ bound state S, due to its dipole interaction SS γ with the spin-1 excited state S . If we ∗ 1 1 could treat the bound states S and S as elementary particles, the amplitude A would 1 S diverge. Our evaluation of the polarizability and the amplitude A is discussed in Section S 7 with more details given in the appendices. | | The electric charge of the fermion F in units of e is Q = Q and N = 3 is the F t c number of colors. So, in Eqs. (4) and (11) we have: 4 N Q2 = . (12) c t 3 Using the well-known SM relations √ 1 √g v2 = , m = t v, t 2G 2 F the W boson and top quark couplings to the Higgs are given by G G 2 HWW = 2 Htt¯ = . (13) m m v W t We use an “impulse approximation” to calculate the couplings of the Higgs ﬁeld H to the 11 quark bound state F and the 12 quark bound state S, in which we add the interactions of H with each of the t and t quarks individually as if they were free particles. The coupling to F is then given by √g m t t G = 11 = 11 . (14) HFF¯ 2 v As described in appendix A, we now obtain the coupling constant between the three scalar particles SSH by comparing with a non-relativistic interaction of the 6t + 6t¯ bound state S with the Higgs ﬁeld H: √g · m m t t S G = 12 2m = 24 . (15) HSS S 2 v Finally we have: → α2m3 | 4 44 m | Γ(H γγ) = H A (τ ) + A (τ ) + t A + 12A 2. (16) 16π3v2 W W 3 t t 3 m F S F which can be rewritten in the form → α2m3 | 4 − | Γ(H γγ) = H A (τ ) + A (τ ) K 2. (17) 16π3v2 W W 3 t t 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: G

text: G

text: HFF¯

text: N

text: α

text: HSS

text: A

text: m

text: c

text: t

text: F

text: m

text: m

text: S

text: F

text: t

text: S

text: In

text: the

text: above

text: equation,

text: the

text: notations

text: F

text: and

text: S

text: refer

text: to

text: the

text: and

text: new

text: bound

text: states,

text: respectively.

text: If

text: we

text: could

text: treat

text: the

text: bound

text: state

text: F

text: as

text: an

text: elementary

text: particle,

text: the

text: amplitude

text: A

text: would

text: be

text: given

text: by

text: A

text: A

text: (τ

text: The

text: amplitude

text: A

text: corresponds

text: to

text: F

text: F

text: F

text: S

text: the

text: loop

text: diagram

text: of

text: Fig.

text: and

text: arises

text: from

text: the

text: polarizability

text: of

text: the

text: self-conjugate

text: scalar

text: bound

text: state

text: S,

text: due

text: to

text: its

text: dipole

text: interaction

text: SS

text: γ

text: with

text: the

text: excited

text: state

text: S

text: If

text: we

text: could

text: treat

text: the

text: bound

text: states

text: S

text: and

text: S

text: as

text: elementary

text: particles,

text: the

text: amplitude

text: A

text: would

text: S

text: diverge.

text: Our

text: evaluation

text: of

text: the

text: polarizability

text: and

text: the

text: amplitude

text: A

text: is

text: discussed

text: in

text: Section

text: S

text: with

text: more

text: details

text: given

text: in

text: the

text: appendices.

text: The

text: electric

text: charge

text: of

text: the

text: fermion

text: F

text: in

text: units

text: of

text: e

text: is

text: Q

text: Q

text: and

text: N

text: is

text: the

text: F

text: t

text: c

text: number

text: of

text: colors.

text: So,

text: in

text: Eqs.

text: and

text: we

text: have:

text: N

text: c

text: t

text: Using

text: the

text: well-known

text: SM

text: relations

text: √g

text: m

text: t

text: v,

text: t

text: F

text: the

text: W

text: boson

text: and

text: top

text: quark

text: couplings

text: to

text: the

text: Higgs

text: are

text: given

text: by

text: G

text: G

text: HWW

text: Htt¯

text: m

text: m

text: v

text: W

text: t

text: We

text: use

text: an

text: “impulse

text: approximation”

text: to

text: calculate

text: the

text: couplings

text: of

text: the

text: Higgs

text: ﬁeld

text: H

text: to

text: the

text: quark

text: bound

text: state

text: F

text: and

text: the

text: quark

text: bound

text: state

text: S,

text: in

text: which

text: we

text: add

text: the

text: interactions

text: of

text: H

text: with

text: each

text: of

text: the

text: t

text: and

text: t

text: quarks

text: individually

text: as

text: if

text: they

text: were

text: free

text: particles.

text: The

text: coupling

text: to

text: F

text: is

text: then

text: given

text: by

text: √g

text: m

text: t

text: t

text: G

text: HFF¯

text: v

text: As

text: described

text: in

text: appendix

text: A,

text: we

text: now

text: obtain

text: the

text: coupling

text: constant

text: between

text: the

text: three

text: scalar

text: particles

text: SSH

text: by

text: comparing

text: with

text: a

text: non-relativistic

text: interaction

text: of

text: the

text: bound

text: state

text: S

text: with

text: the

text: Higgs

text: ﬁeld

text: H:

text: √g

text: m

text: m

text: t

text: t

text: S

text: G

text: HSS

text: S

text: v

text: Finally

text: we

text: have:

text: m

text: Γ(H

text: γγ)

text: H

text: A

text: (τ

text: A

text: (τ

text: t

text: A

text: W

text: W

text: t

text: t

text: m

text: F

text: S

text: F

text: which

text: can

text: be

text: rewritten

text: in

text: the

text: form

text: Γ(H

text: γγ)

text: H

text: A

text: (τ

text: A

text: (τ

text: K

text: W

text: W

text: t

text: t

text: ##LTLine##

text: ##LTLine##

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text: The Standard Model in Noncommutative Geometry and Morita equivalence Francesco D’Andrea1,2 and Ludwik Dabrowski3 (cid:44) 1 Universit`a degli Studi di Napoli Federico II, P.le Tecchio 80, I-80125 Napoli 2 I.N.F.N. Sezione di Napoli, Complesso MSA, Via Cintia, I-80126 Napoli 3 Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, I-34136 Trieste Abstract We discuss some properties of the spectral triple (A , H , D , J , γ ) describing the F F F F F internal space in the noncommutative geometry approach to the Standard Model, with C ⊕ H ⊕ C A = M ( ). We show that, if we want H to be a Morita equivalence F 3 F bimodule between A and the associated Cliﬀord algebra, two terms must be added F to the Dirac operator; we then study its relation with the orientability condition for a spectral triple. We also illustrate what changes if one considers a spectral triple with a C ⊕ C ⊕ C degenerate representation, based on the complex algebra B = M ( ) M ( ). F 2 3 1 Introduction In the spectral action approach to (quantum) ﬁeld theory, the space of the theory is the product of an ordinary spin manifold M with a ﬁnite noncommutative space (cf. [11, 23] and ⊗ references therein). States of the system are represented by unit vectors in L2(M, S) H, → where L2(M, S) are square integrable sections of the spinor bundle S M and H is a ﬁnite- dimensional Hilbert space representing the internal degrees of freedom of a particle. The ∞ algebra containing the observables is the tensor product of smooth functions C (M) on M with certain ﬁnite dimensional algebra A. More precisely, one has an “almost commutative” geometry described by a product of spectral triples, with Dirac operator constructed from the Dirac operator of M and certain selfadjoint operator (a Hermitian matrix) D on H. A deep algebraic characterization of the space of Dirac spinor ﬁelds L2(M, S) on a spin C manifold is as the Morita equivalence bimodule between C(M) and the algebra (cid:96)(M) of sections of the Cliﬀord bundle of M. It is natural to investigate if also the ﬁnite- dimensional spectral triple of the Standard Model (A, H, D) describes a (noncommutative) spin manifold, and in particular if the elements of H are in some sense “spinors”. This condition – which we name “property (M)” in Def. 4 – can be precisely formulated again in C terms of Morita equivalence involving A and certain noncommutative analogue of (cid:96)(M), and is satisﬁed in some basic examples like e.g. Einstein-Yang-Mills systems. MSC-class 2010: Primary: 58B34; Secondary: 46L87, 81T13. Keywords: noncommutative geometry; spectral action; Standard Model. 1 ##LTLine##

text: The

label: title

text: Standard

text: Model

text: in

text: Noncommutative

text: Geometry

text: and

text: Morita

text: equivalence

text: Francesco

label: author

text: and

text: Ludwik

text: Universit`a

text: degli

text: Studi

text: di

text: Napoli

text: Federico

text: II,

text: P.le

text: Tecchio

text: Napoli

text: I.N.F.N.

text: Sezione

text: di

text: Napoli,

text: Complesso

text: MSA,

text: Via

text: Cintia,

text: Napoli

text: Scuola

text: Internazionale

text: Superiore

text: di

text: Studi

text: Avanzati

text: (SISSA),

text: via

text: Bonomea

text: Trieste

text: Abstract

text: We

label: abstract

text: discuss

text: some

text: properties

text: of

text: the

text: spectral

text: triple

text: (A

text: H

text: D

text: J

text: γ

text: describing

text: the

text: F

text: F

text: F

text: F

text: F

text: internal

text: space

text: in

text: the

text: noncommutative

text: geometry

text: approach

text: to

text: the

text: Standard

text: Model,

text: with

text: C

text: H

text: C

text: A

text: M

text: We

text: show

text: that,

text: if

text: we

text: want

text: H

text: to

text: be

text: a

text: Morita

text: equivalence

text: F

text: F

text: bimodule

text: between

text: A

text: and

text: the

text: associated

text: Cliﬀord

text: algebra,

text: two

text: terms

text: must

text: be

text: added

text: F

text: to

text: the

text: Dirac

text: operator;

text: we

text: then

text: study

text: its

text: relation

text: with

text: the

text: orientability

text: condition

text: for

text: a

text: spectral

text: triple.

text: We

text: also

text: illustrate

text: what

text: changes

text: if

text: one

text: considers

text: a

text: spectral

text: triple

text: with

text: a

text: C

text: C

text: C

text: degenerate

text: representation,

text: based

text: on

text: the

text: complex

text: algebra

text: B

text: M

text: M

text: F

text: Introduction

text: In

text: the

text: spectral

text: action

text: approach

text: to

text: (quantum)

text: ﬁeld

text: theory,

text: the

text: space

text: of

text: the

text: theory

text: is

text: the

text: product

text: of

text: an

text: ordinary

text: spin

text: manifold

text: M

text: with

text: a

text: ﬁnite

text: noncommutative

text: space

text: (cf.

text: and

text: references

text: therein).

text: States

text: of

text: the

text: system

text: are

text: represented

text: by

text: unit

text: vectors

text: in

text: S)

text: H,

text: where

text: S)

text: are

text: square

text: integrable

text: sections

text: of

text: the

text: spinor

text: bundle

text: S

text: M

text: and

text: H

text: is

text: a

text: ﬁnite-

text: dimensional

text: Hilbert

text: space

text: representing

text: the

text: internal

text: degrees

text: of

text: freedom

text: of

text: a

text: particle.

text: The

text: algebra

text: containing

text: the

text: observables

text: is

text: the

text: tensor

text: product

text: of

text: smooth

text: functions

text: C

text: (M)

text: on

text: M

text: with

text: certain

text: ﬁnite

text: dimensional

text: algebra

text: A.

text: More

text: precisely,

text: one

text: has

text: an

text: “almost

text: commutative”

text: geometry

text: described

text: by

text: a

text: product

text: of

text: spectral

text: triples,

text: with

text: Dirac

text: operator

text: constructed

text: from

text: the

text: Dirac

text: operator

text: of

text: M

text: and

text: certain

text: selfadjoint

text: operator

text: (a

text: Hermitian

text: matrix)

text: D

text: on

text: H.

text: A

text: deep

text: algebraic

text: characterization

text: of

text: the

text: space

text: of

text: Dirac

text: spinor

text: ﬁelds

text: S)

text: on

text: a

text: spin

text: C

text: manifold

text: is

text: as

text: the

text: Morita

text: equivalence

text: bimodule

text: between

text: C(M)

text: and

text: the

text: algebra

text: of

text: sections

text: of

text: the

text: Cliﬀord

text: bundle

text: of

text: M.

text: It

text: is

text: natural

text: to

text: investigate

text: if

text: also

text: the

text: ﬁnite-

text: dimensional

text: spectral

text: triple

text: of

text: the

text: Standard

text: Model

text: (A,

text: H,

text: D)

text: describes

text: a

text: (noncommutative)

text: spin

text: manifold,

text: and

text: in

text: particular

text: if

text: the

text: elements

text: of

text: H

text: are

text: in

text: some

text: sense

text: “spinors”.

text: This

text: condition

text: which

text: we

text: name

text: “property

text: (M)”

text: in

text: Def.

text: can

text: be

text: precisely

text: formulated

text: again

text: in

text: C

text: terms

text: of

text: Morita

text: equivalence

text: involving

text: A

text: and

text: certain

text: noncommutative

text: analogue

text: of

text: and

text: is

text: satisﬁed

text: in

text: some

text: basic

text: examples

text: like

text: e.g.

text: Einstein-Yang-Mills

text: systems.

text: MSC-class

text: Primary:

text: Secondary:

text: Keywords:

text: noncommutative

text: geometry;

text: spectral

text: action;

text: Standard

text: Model.

text: ##LTLine##


text: § we’ll see that in the Standard Model example, the spectral triple is not orientable (cf. ) (cid:48) ∩ ◦ (cid:48) and A (A ) is not zero. F F We now come back to the Standard Model. In the rest of the paper, we employ ◦ § A , A , H , J as deﬁned in , but the same results are valid for the algebra B and F F F F § F the representation discussed in . ∗ Proposition 11. An operator D = D as in Prop. 9 commutes with J if and only if it F F F is of the form: D = D + J D J F 0 F 0 F ∗ ∈ ◦ (cid:48) with D = D (A ) . 0 0 F (cid:55)→ (cid:48) → ◦ (cid:48) Proof. x J x¯J gives a bijection A (A ) . The condition J D J = D gives F F F F F F F F − − (J D J D ) + (J D J D ) = 0 . F 0 F 1 F 1 F 0 (cid:48) ◦ (cid:48) Since the ﬁrst term is in A and the second in (A ) , the sum is zero if and only if both − F− (cid:48) ∩ F ◦ (cid:48) (cid:48) − J D J D and J D J D belong to A (A ) . Called D = D J D J , one F 0 F 1 F 1 F 0 F F 1 F 0 F has the decomposition (cid:48) D = D + J D J + D . F 0 F 0 F − (cid:48) − (cid:48) (cid:48) From J D J D = J D J D one deduces that J and D must commute. So F F F F F F F (cid:48) (cid:48) D = (D + D /2) + J (D + D /2)J and we get the decomposition (16), after renaming F 0 F 0 F (cid:48) → D + D /2 D . 0 0 ∗ Decompose D = S + iT with S and T selfadjoint. Since J is antilinear and J = J : 0 F F F − ∗ − D D = 2i(T J T J ) F F F F which must be zero. But this implies − D = S + J SJ + i(T J T J ) = S + J SJ . F F F F F F F (cid:32) (cid:4) Renaming S D (which now is selfadjoint) we conclude the proof. 0 C Remark 12. Note that the D term does not contribute to (cid:96)(A ) (it commutes with A ). 1 F F Then, for any Dirac operator as in Prop. 9, we can ﬁnd one commuting with J (replacing F C D by J D J ) without changing the Cliﬀord algebra (cid:96)(A ). In particular, the property 1 F 0 F F (M) puts constrains only on D . 0 It is useful to reformulate Prop. 9 and Prop. 11 as follows. Let ⊗ D := (Υ e + Υ¯ e ) e , (15) R R 51 R 15 11 ∈ C ∈ (cid:48) ∩ ◦ (cid:48) with Υ . Note that D A (A ) and J D = D J . R R F F F R R F ∗ Proposition 13. The most general D = D satisfying the 1st order condition is F F D = D + D + D (16) F 0 1 R ∗ ∈ ◦ (cid:48) ∗ ∈ (cid:48) ⊗ where D = D (A ) and D = D A have null entry in direction of e e ⊗ 0 0 F 1 1 F 15 11 and e e , and D = J D J if D and J commute. 51 11 1 F 0 F F F 13

text: we’ll

text: see

text: that

text: in

text: the

text: Standard

text: Model

text: example,

text: the

text: spectral

text: triple

text: is

text: not

text: orientable

text: (cf.

text: and

text: A

text: (A

text: is

text: not

text: zero.

text: F

text: F

text: We

text: now

text: come

text: back

text: to

text: the

text: Standard

text: Model.

text: In

text: the

text: rest

text: of

text: the

text: paper,

text: we

text: employ

text: A

text: A

text: H

text: J

text: as

text: deﬁned

text: in

text: but

text: the

text: same

text: results

text: are

text: valid

text: for

text: the

text: algebra

text: B

text: and

text: F

text: F

text: F

text: F

text: F

text: the

text: representation

text: discussed

text: in

text: Proposition

text: An

text: operator

text: D

text: D

text: as

text: in

text: Prop.

text: commutes

text: with

text: J

text: if

text: and

text: only

text: if

text: it

text: F

text: F

text: F

text: is

text: of

text: the

text: form:

text: D

text: D

text: J

text: D

text: J

text: F

text: F

text: F

text: with

text: D

text: D

text: (A

text: F

text: Proof.

text: x

text: J

text: x¯J

text: gives

text: a

text: bijection

text: A

text: (A

text: The

text: condition

text: J

text: D

text: J

text: D

text: gives

text: F

text: F

text: F

text: F

text: F

text: F

text: F

text: F

text: (J

text: D

text: J

text: D

text: (J

text: D

text: J

text: D

text: F

text: F

text: F

text: F

text: Since

text: the

text: ﬁrst

text: term

text: is

text: in

text: A

text: and

text: the

text: second

text: in

text: (A

text: the

text: sum

text: is

text: zero

text: if

text: and

text: only

text: if

text: both

text: F−

text: F

text: J

text: D

text: J

text: D

text: and

text: J

text: D

text: J

text: D

text: belong

text: to

text: A

text: (A

text: Called

text: D

text: D

text: J

text: D

text: J

text: one

text: F

text: F

text: F

text: F

text: F

text: F

text: F

text: F

text: has

text: the

text: decomposition

text: D

text: D

text: J

text: D

text: J

text: D

text: F

text: F

text: F

text: From

text: J

text: D

text: J

text: D

text: J

text: D

text: J

text: D

text: one

text: deduces

text: that

text: J

text: and

text: D

text: must

text: commute.

text: So

text: F

text: F

text: F

text: F

text: F

text: F

text: F

text: D

text: (D

text: D

text: J

text: (D

text: D

text: and

text: we

text: get

text: the

text: decomposition

text: after

text: renaming

text: F

text: F

text: F

text: D

text: D

text: D

text: Decompose

text: D

text: S

text: iT

text: with

text: S

text: and

text: T

text: selfadjoint.

text: Since

text: J

text: is

text: antilinear

text: and

text: J

text: J

text: F

text: F

text: F

text: D

text: D

text: J

text: T

text: J

text: F

text: F

text: F

text: F

text: which

text: must

text: be

text: zero.

text: But

text: this

text: implies

text: D

text: S

text: J

text: SJ

text: i(T

text: J

text: T

text: J

text: S

text: J

text: SJ

text: F

text: F

text: F

text: F

text: F

text: F

text: F

text: Renaming

text: S

text: D

text: (which

text: now

text: is

text: selfadjoint)

text: we

text: conclude

text: the

text: proof.

text: C

text: Remark

text: Note

text: that

text: the

text: D

text: term

text: does

text: not

text: contribute

text: to

text: (it

text: commutes

text: with

text: A

text: F

text: F

text: Then,

text: for

text: any

text: Dirac

text: operator

text: as

text: in

text: Prop.

text: we

text: can

text: ﬁnd

text: one

text: commuting

text: with

text: J

text: (replacing

text: F

text: C

text: D

text: by

text: J

text: D

text: J

text: without

text: changing

text: the

text: Cliﬀord

text: algebra

text: In

text: particular,

text: the

text: property

text: F

text: F

text: F

text: (M)

text: puts

text: constrains

text: only

text: on

text: D

text: It

text: is

text: useful

text: to

text: reformulate

text: Prop.

text: and

text: Prop.

text: as

text: follows.

text: Let

text: D

text: (Υ

text: e

text: Υ¯

text: e

text: e

text: R

text: R

text: R

text: C

text: with

text: Υ

text: Note

text: that

text: D

text: A

text: (A

text: and

text: J

text: D

text: D

text: J

text: R

text: R

text: F

text: F

text: F

text: R

text: R

text: F

text: Proposition

text: The

text: most

text: general

text: D

text: D

text: satisfying

text: the

text: order

text: condition

text: is

text: F

text: F

text: D

text: D

text: D

text: D

text: F

text: R

text: where

text: D

text: D

text: (A

text: and

text: D

text: D

text: A

text: have

text: entry

text: in

text: direction

text: of

text: e

text: e

text: F

text: F

text: and

text: e

text: e

text: and

text: D

text: J

text: D

text: J

text: if

text: D

text: and

text: J

text: commute.

text: F

text: F

text: F

text: F


text: § ii) Let D and γ be as in . If the property (M), with or without grading, holds then F F • (cid:126) each summand in D must have at least one circled coeﬃcient ( ) diﬀerent from zero; 0 • in each of the ﬁrst two summands, in both the 1st and 2nd row there must be at least one non-zero element; • in the ﬁrst summand: at least one element in the 5th row must be non-zero and at least one element in the upper-right block must be non-zero. Proof. It is enough to give the proof for the property (M) with grading (the weaker one). C ◦ We apply Lemma 16 to A = (cid:96)(A ) , B = (A )C and H = H . Let D and γ be as § § F e F F 0 F in or . Note that A is generated by A , [D , A ] and γ . Moreover, due to the 1st order F 0 F F condition, A and B are mutually commuting. ∈ (cid:48) ∩ ◦ (cid:48) (cid:48) ∩ (cid:48) Any operator X A (A ) commuting with D and γ belongs to A B (since it F F 0 F ∈ also commute with [D , A ]). If we can exhibit such an X and prove that X / Z(B), then 0 F    the property (M) with grading is not satisﬁed. Note that Z(B) has elements: (cid:34) (cid:35)   (cid:34) (cid:35)            λ 0 0 0   λ 0 0 0     (cid:48)  ◦ 1 0 ⊗ 0 0 0 ⊗ 0 λ 0 0 a = + , 0 0 0 α1 0 1 0 0 3 β1 2 0 0 0 (cid:48) ∈ C § ⊗ − with λ, λ , α, β . For D , γ as in , the operator X = e (1 e ) does the job: 0 F 55 11 1) it commutes with D and γ , 0 F (cid:48) ⊗ C ∈ 2) it belongs to A = C M ( ) (e C : take q = 1 and all other coeﬃcients F F 4 55 F 22 zero in (11)), ◦ (cid:48) 3) it belongs to (A ) (cf. Lemma 8), F 4) and it does not belong to Z(B). § Let now D and γ be as in . Concerning the ﬁrst summand: 0 F • (cid:126) ⊗ if all the circled terms ( ) are zero, then X = (e + e + e ) e satisﬁes the 66 77 88 11 conditions (1-4) above; • if all the elements in the 1st resp. 2nd row are zero (and then also in 1st resp. 2nd ⊗ ⊗ column, by hermiticity), then X = e e resp. X = e e satisﬁes (1-4); 11 11 22 11 • ⊗ if all the elements in the 5th row are zero, similarly by hermiticity X = e e 55 11 satisﬁes the conditions (1-4) above; • if all the elements in the upper-right blocks are zero, then X = (e + e + e + 11 22 33 ⊗ e ) e satisﬁes the conditions (1-4) above. 44 11 Concerning the second summand: • if all the elements in the 1st resp. 2nd row are zero (and then also in 1st resp. 2nd ⊗ − ⊗ − column, by hermiticity), then X = e (1 e ) resp. X = e (1 e ) satisﬁes 11 11 22 11 the conditions (1-4) above; 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ii)

text: Let

text: D

text: and

text: γ

text: be

text: as

text: in

text: If

text: the

text: property

text: (M),

text: with

text: or

text: without

text: grading,

text: holds

text: then

text: F

text: F

text: each

text: summand

text: in

text: D

text: must

text: have

text: at

text: least

text: one

text: circled

text: coeﬃcient

text: diﬀerent

text: from

text: zero;

text: in

text: each

text: of

text: the

text: ﬁrst

text: two

text: summands,

text: in

text: both

text: the

text: and

text: row

text: there

text: must

text: be

text: at

text: least

text: one

text: non-zero

text: element;

text: in

text: the

text: ﬁrst

text: summand:

text: at

text: least

text: one

text: element

text: in

text: the

text: row

text: must

text: be

text: non-zero

text: and

text: at

text: least

text: one

text: element

text: in

text: the

text: upper-right

text: block

text: must

text: be

text: non-zero.

text: Proof.

text: It

text: is

text: enough

text: to

text: give

text: the

text: proof

text: for

text: the

text: property

text: (M)

text: with

text: grading

text: (the

text: weaker

text: one).

text: C

text: We

text: apply

text: Lemma

text: to

text: A

text: B

text: (A

text: )C

text: and

text: H

text: H

text: Let

text: D

text: and

text: γ

text: be

text: as

text: F

text: e

text: F

text: F

text: F

text: in

text: or

text: Note

text: that

text: A

text: is

text: generated

text: by

text: A

text: [D

text: A

text: and

text: γ

text: Moreover,

text: due

text: to

text: the

text: order

text: F

text: F

text: F

text: condition,

text: A

text: and

text: B

text: are

text: mutually

text: commuting.

text: Any

text: operator

text: X

text: A

text: (A

text: commuting

text: with

text: D

text: and

text: γ

text: belongs

text: to

text: A

text: B

text: (since

text: it

text: F

text: F

text: F

text: also

text: commute

text: with

text: [D

text: A

text: If

text: we

text: can

text: exhibit

text: such

text: an

text: X

text: and

text: prove

text: that

text: X

text: Z(B),

text: then

text: F

text: the

text: property

text: (M)

text: with

text: grading

text: is

text: not

text: satisﬁed.

text: Note

text: that

text: Z(B)

text: has

text: elements:

text: λ

text: λ

text: λ

text: a

text: α1

text: β1

text: C

text: with

text: λ,

text: λ

text: α,

text: β

text: For

text: D

text: γ

text: as

text: in

text: the

text: operator

text: X

text: e

text: e

text: does

text: the

text: job:

text: F

text: it

text: commutes

text: with

text: D

text: and

text: γ

text: F

text: C

text: it

text: belongs

text: to

text: A

text: C

text: M

text: (e

text: C

text: take

text: q

text: and

text: all

text: other

text: coeﬃcients

text: F

text: F

text: F

text: zero

text: in

text: it

text: belongs

text: to

text: (A

text: (cf.

text: Lemma

text: F

text: and

text: it

text: does

text: not

text: belong

text: to

text: Z(B).

text: Let

text: now

text: D

text: and

text: γ

text: be

text: as

text: in

text: Concerning

text: the

text: ﬁrst

text: summand:

text: F

text: if

text: all

text: the

text: circled

text: terms

text: are

text: zero,

text: then

text: X

text: (e

text: e

text: e

text: e

text: satisﬁes

text: the

text: conditions

text: above;

text: if

text: all

text: the

text: elements

text: in

text: the

text: resp.

text: row

text: are

text: zero

text: (and

text: then

text: also

text: in

text: resp.

text: column,

text: by

text: hermiticity),

text: then

text: X

text: e

text: e

text: resp.

text: X

text: e

text: e

text: satisﬁes

text: if

text: all

text: the

text: elements

text: in

text: the

text: row

text: are

text: zero,

text: similarly

text: by

text: hermiticity

text: X

text: e

text: e

text: satisﬁes

text: the

text: conditions

text: above;

text: if

text: all

text: the

text: elements

text: in

text: the

text: upper-right

text: blocks

text: are

text: zero,

text: then

text: X

text: (e

text: e

text: e

text: e

text: e

text: satisﬁes

text: the

text: conditions

text: above.

text: Concerning

text: the

text: second

text: summand:

text: if

text: all

text: the

text: elements

text: in

text: the

text: resp.

text: row

text: are

text: zero

text: (and

text: then

text: also

text: in

text: resp.

text: column,

text: by

text: hermiticity),

text: then

text: X

text: e

text: e

text: resp.

text: X

text: e

text: e

text: satisﬁes

text: the

text: conditions

text: above;

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Concerning the third resp. fourth summands: • (cid:126) ⊗ ⊗ if the circled terms ( ) are zero, X = e e resp. X = e (e + e ) satisﬁes 55 22 55 33 44 (cid:4) the conditions (1-4) above. § Corollary 18. Let D be as in and γ one of the two gradings (17) or (18). If ∆ = 0 F F or Ω = 0, the property (M), with or without grading, is not satisﬁed. § (cid:54) (cid:54) The operator D in , with Ω = 0 and ∆ = 0, represents a minimal modiﬁcation of the F Dirac operator of [3, 9, 5, 11] which satisﬁes all the conditions in Prop. 17. We will now show that for such an operator, the Morita condition is satisﬁed. 6.1 Morita with a grading This section is devoted to prove the following theorem. § § Theorem 19. Let γ be as in , D as in with all coeﬃcients diﬀerent from zero, and F F assume that at least one of the following conditions holds: (cid:54) ± (cid:54) ± 1. Υ = Υ , 2. Υ = Υ . ν u e d Then, the spectral triple: i) does not satisfy the property (M); ii) it satisﬁes the property (M) with grading. C (cid:54) C ∈ C As a corollary, (cid:96)(A ) = (cid:96)(A ) , so γ / (cid:96)(A ) . F o F e F F o We need a preliminary lemma. From now on, we assume that the hypothesis of Thm. 19 are satisﬁed. (cid:0) (cid:1) Lemma 20. The A -bimodule Ω1 is generated by the four elements: F (cid:0) (cid:1) ⊗ − ⊗ ω = e Υ e + Υ (1 e ) , ξ = e e , ν 31 ν 11 u 11 52 11 ⊗ − ⊗ ω = e Υ e + Υ (1 e ) , η = e 1 , e 42 e 11 d 11 56 and their adjoints. Proof. A linear basis of (A )C is given by the elements: F ⊗ ⊗ X := e 1 with i, j = 3, 4, Y := e 1 , ij ij 22 ⊗ ⊗ Z := e 1 with k, l = 6, 7, 8, T := (e + e ) 1 . kl kl 11 55 ∗ For any projection p2 = p = p , the commutator [D , p] = [D , p2] = p[D , p] + [D , p]p F F F F − is a linear combination of p[D , p] and its adjoint [D , p]p. Hence X [D , X ] and F F 33 F 33 X [D , X ] can be taken as generators, instead of [D , X ] and [D , X ]. An explicit 44 F 44 F 33 F 44 computation gives: − − X [D , X ] = ω , X [D , X ] = ω . 33 F 33 ν 44 F 44 e Note that [D , X ] is also the adjoint of [D , X ], and F 34 F 43 ∗ − [D , X ] = (X ω ) X ω F 43 34 e 43 ν 18

text: Concerning

text: the

text: third

text: resp.

text: fourth

text: summands:

label: list

text: if

text: the

text: circled

text: terms

text: are

text: zero,

text: X

text: e

text: e

text: resp.

text: X

text: e

text: (e

text: e

text: satisﬁes

text: the

text: conditions

text: above.

text: Corollary

text: Let

text: D

text: be

text: as

text: in

text: and

text: γ

text: one

text: of

text: the

text: two

text: gradings

text: or

text: If

text: F

text: F

text: or

text: Ω

text: the

text: property

text: (M),

text: with

text: or

text: without

text: grading,

text: is

text: not

text: satisﬁed.

text: The

text: operator

text: D

text: in

text: with

text: Ω

text: and

text: represents

text: a

text: minimal

text: modiﬁcation

text: of

text: the

text: F

text: Dirac

text: operator

text: of

text: which

text: satisﬁes

text: all

text: the

text: conditions

text: in

text: Prop.

text: We

text: will

text: now

text: show

text: that

text: for

text: such

text: an

text: operator,

text: the

text: Morita

text: condition

text: is

text: satisﬁed.

text: Morita

text: with

text: a

text: grading

text: This

text: section

text: is

text: devoted

text: to

text: prove

text: the

text: following

text: theorem.

text: Theorem

text: Let

text: γ

text: be

text: as

text: in

text: D

text: as

text: in

text: with

text: all

text: coeﬃcients

text: diﬀerent

text: from

text: zero,

text: and

text: F

text: F

text: assume

text: that

text: at

text: least

text: one

text: of

text: the

text: following

text: conditions

text: holds:

text: Υ

text: Υ

text: Υ

text: Υ

text: ν

text: u

text: e

text: d

text: Then,

text: the

text: spectral

text: triple:

text: i)

text: does

text: not

text: satisfy

text: the

text: property

text: (M);

text: ii)

text: it

text: satisﬁes

text: the

text: property

text: (M)

text: with

text: grading.

text: C

text: C

text: C

text: As

text: a

text: corollary,

text: so

text: γ

text: F

text: o

text: F

text: e

text: F

text: F

text: o

text: We

text: need

text: a

text: preliminary

text: lemma.

text: From

text: now

text: on,

text: we

text: assume

text: that

text: the

text: hypothesis

text: of

text: Thm.

text: are

text: satisﬁed.

text: Lemma

text: The

text: A

text: -bimodule

text: Ω1

text: is

text: generated

text: by

text: the

text: four

text: elements:

text: F

text: ω

text: e

text: Υ

text: e

text: Υ

text: e

text: ξ

text: e

text: e

text: ν

text: ν

text: u

text: ω

text: e

text: Υ

text: e

text: Υ

text: e

text: η

text: e

text: e

text: e

text: d

text: and

text: their

text: adjoints.

text: Proof.

text: A

text: linear

text: basis

text: of

text: (A

text: )C

text: is

text: given

text: by

text: the

text: elements:

text: F

text: X

text: e

text: with

text: i,

text: j

text: Y

text: e

text: ij

text: ij

text: Z

text: e

text: with

text: k,

text: l

text: T

text: (e

text: e

text: kl

text: kl

text: For

text: any

text: projection

text: p

text: p

text: the

text: commutator

text: [D

text: p]

text: [D

text: p[D

text: p]

text: [D

text: p]p

text: F

text: F

text: F

text: F

text: is

text: a

text: linear

text: combination

text: of

text: p[D

text: p]

text: and

text: its

text: adjoint

text: [D

text: p]p.

text: Hence

text: X

text: [D

text: X

text: and

text: F

text: F

text: F

text: X

text: [D

text: X

text: can

text: be

text: taken

text: as

text: generators,

text: instead

text: of

text: [D

text: X

text: and

text: [D

text: X

text: An

text: explicit

text: F

text: F

text: F

text: computation

text: gives:

text: X

text: [D

text: X

text: ω

text: X

text: [D

text: X

text: ω

text: F

text: ν

text: F

text: e

text: Note

text: that

text: [D

text: X

text: is

text: also

text: the

text: adjoint

text: of

text: [D

text: X

text: and

text: F

text: F

text: [D

text: X

text: (X

text: ω

text: X

text: ω

text: F

text: e

text: ν


text: experimental observation threshold. Their ﬁne-tuning in order to get the correct value of the Higgs mass could be studied in future work. Of the two terms, the Ω-term is compatible with both the original and the modiﬁed § § grading of and . The ∆-term on the other hand is compatible (anticommutes) only with the modiﬁed grading. Such a term may potentially (see [21]) mix quarks and leptons, and although it may seem exotic, it is also necessary for the irreducibility of the spectral § triple (cf. ): without this term, the leptonic and quark sectors of H carry each one a F sub-spectral triple. A third additional term (with coeﬃcient Γ) in D is instead necessary F (though non suﬃcient) if one wants the spectral triple to be also orientable, cf. Prop. 26 for the precise statement. Concerning the grading, the one in (18) is minimal a modiﬁcation the one in (17): they agree on leptons and have opposite sign on baryons (quarks). A study of the physical consequences of this modiﬁcation are beyond the scope of this paper. The internal grading contributes to the grading of the full spectral triple, product of the ﬁnite-dimensional one with the canonical spectral triple of a 4-dimensional spin manifold. The full grading is used to project out from the Hilbert space unphysical degrees of freedom (and partially solve the quadrupling of degrees of freedom, cf. [11] for the details). Thus, changing the grading in principle could aﬀect this part of the theory, which should be studied in the future. We close by stressing again that the aim of this paper was a mathematical study of the property (M), and few related issues, but a detailed analysis of the physical aspects of our model (for example, understanding what happens to the Higgs mass) goes beyond the scope of the paper and is postponed to future works. Acknowledgments F.D. was partially supported by UniNA and Compagnia di San Paolo in the framework of the program STAR 2013. L.D. was partially supported by the Italian grant PRIN 2010 “Operator Algebras, Noncommutative Geometry and Applications” and by the Polish grant “Harmonia”. References [1] J. Bhowmick, F. D’Andrea and L. Dabrowski, Quantum Isometries of the ﬁnite noncommutative (cid:44) geometry of the Standard Model, Commun. Math. Phys. 307 (2011), 101–131. [2] J. Bhowmick, F. D’Andrea, B. Das and L. Dabrowski, Quantum gauge symmetries in Noncom- (cid:44) mutative Geometry, J. Noncomm. Geom. 8 (2014), 433–471. [3] A.H. Chamseddine and A. Connes, The Spectral Action Principle, Commun. Math. Phys. 186 (1997), 731–750. [4] A.H. Chamseddine and A. Connes, Resilience of the Spectral Standard Model, JHEP 1209 (2012) 104. [5] A.H. Chamseddine, A. Connes and M. Marcolli, Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007), 991–1090. [6] A.H. Chamseddine, A. Connes and W.D. van Suijlekom, Inner Fluctuations in Noncommutative Geometry without the ﬁrst order condition, J. Geom. Phys. 73 (2013), 222–234. [7] A.H. Chamseddine, A. Connes and W.D. van Suijlekom, Beyond the Spectral Standard Model: Emergence of Pati-Salam Uniﬁcation, JHEP 1311 (2013), 132. [8] A. Connes, Noncommutative Geometry, Academic Press, 1994. 24

text: experimental

text: observation

text: threshold.

text: Their

text: ﬁne-tuning

text: in

text: order

text: to

text: get

text: the

text: correct

text: value

text: of

text: the

text: Higgs

text: mass

text: could

text: be

text: studied

text: in

text: future

text: work.

text: Of

text: the

text: two

text: terms,

text: the

text: Ω-term

text: is

text: compatible

text: with

text: both

text: the

text: original

text: and

text: the

text: modiﬁed

text: grading

text: of

text: and

text: The

text: ∆-term

text: on

text: the

text: other

text: hand

text: is

text: compatible

text: (anticommutes)

text: only

text: with

text: the

text: modiﬁed

text: grading.

text: Such

text: a

text: term

text: may

text: potentially

text: (see

text: mix

text: quarks

text: and

text: leptons,

text: and

text: although

text: it

text: may

text: seem

text: exotic,

text: it

text: is

text: also

text: necessary

text: for

text: the

text: irreducibility

text: of

text: the

text: spectral

text: triple

text: (cf.

text: without

text: this

text: term,

text: the

text: leptonic

text: and

text: quark

text: sectors

text: of

text: H

text: carry

text: each

text: one

text: a

text: F

text: sub-spectral

text: triple.

text: A

text: third

text: additional

text: term

text: (with

text: coeﬃcient

text: Γ)

text: in

text: D

text: is

text: instead

text: necessary

text: F

text: (though

text: non

text: suﬃcient)

text: if

text: one

text: wants

text: the

text: spectral

text: triple

text: to

text: be

text: also

text: orientable,

text: cf.

text: Prop.

text: for

text: the

text: precise

text: statement.

text: Concerning

text: the

text: grading,

text: the

text: one

text: in

text: is

text: minimal

text: a

text: modiﬁcation

text: the

text: one

text: in

text: they

text: agree

text: on

text: leptons

text: and

text: have

text: opposite

text: sign

text: on

text: baryons

text: (quarks).

text: A

text: study

text: of

text: the

text: physical

text: consequences

text: of

text: this

text: modiﬁcation

text: are

text: beyond

text: the

text: scope

text: of

text: this

text: paper.

text: The

text: internal

text: grading

text: contributes

text: to

text: the

text: grading

text: of

text: the

text: full

text: spectral

text: triple,

text: product

text: of

text: the

text: ﬁnite-dimensional

text: one

text: with

text: the

text: canonical

text: spectral

text: triple

text: of

text: a

text: spin

text: manifold.

text: The

text: full

text: grading

text: is

text: used

text: to

text: project

text: out

text: from

text: the

text: Hilbert

text: space

text: unphysical

text: degrees

text: of

text: freedom

text: (and

text: partially

text: solve

text: the

text: quadrupling

text: of

text: degrees

text: of

text: freedom,

text: cf.

text: for

text: the

text: details).

text: Thus,

text: changing

text: the

text: grading

text: in

text: principle

text: could

text: aﬀect

text: this

text: part

text: of

text: the

text: theory,

text: which

text: should

text: be

text: studied

text: in

text: the

text: future.

text: We

text: close

text: by

text: stressing

text: again

text: that

text: the

text: aim

text: of

text: this

text: paper

text: was

text: a

text: mathematical

text: study

text: of

text: the

text: property

text: (M),

text: and

text: few

text: related

text: issues,

text: but

text: a

text: detailed

text: analysis

text: of

text: the

text: physical

text: aspects

text: of

text: our

text: model

text: (for

text: example,

text: understanding

text: what

text: happens

text: to

text: the

text: Higgs

text: mass)

text: goes

text: beyond

text: the

text: scope

text: of

text: the

text: paper

text: and

text: is

text: postponed

text: to

text: future

text: works.

text: Acknowledgments

text: F.D.

text: was

text: partially

text: supported

text: by

text: UniNA

text: and

text: Compagnia

text: di

text: San

text: Paolo

text: in

text: the

text: framework

text: of

text: the

text: program

text: STAR

text: L.D.

text: was

text: partially

text: supported

text: by

text: the

text: Italian

text: grant

text: PRIN

text: “Operator

text: Algebras,

text: Noncommutative

text: Geometry

text: and

text: Applications”

text: and

text: by

text: the

text: Polish

text: grant

text: “Harmonia”.

text: References

text: J.

text: Bhowmick,

text: F.

text: D’Andrea

text: and

text: L.

text: Dabrowski,

text: Quantum

text: Isometries

text: of

text: the

text: ﬁnite

text: noncommutative

text: geometry

text: of

text: the

text: Standard

text: Model,

text: Commun.

text: Math.

text: Phys.

text: J.

text: Bhowmick,

text: F.

text: D’Andrea,

text: B.

text: Das

text: and

text: L.

text: Dabrowski,

text: Quantum

text: gauge

text: symmetries

text: in

text: Noncom-

text: mutative

text: Geometry,

text: J.

text: Noncomm.

text: Geom.

text: A.H.

text: Chamseddine

text: and

text: A.

text: Connes,

text: The

text: Spectral

text: Action

text: Principle,

text: Commun.

text: Math.

text: Phys.

text: A.H.

text: Chamseddine

text: and

text: A.

text: Connes,

text: Resilience

text: of

text: the

text: Spectral

text: Standard

text: Model,

text: JHEP

text: A.H.

text: Chamseddine,

text: A.

text: Connes

text: and

text: M.

text: Marcolli,

text: Gravity

text: and

text: the

text: standard

text: model

text: with

text: neutrino

text: mixing,

text: Adv.

text: Theor.

text: Math.

text: Phys.

text: A.H.

text: Chamseddine,

text: A.

text: Connes

text: and

text: W.D.

text: van

text: Suijlekom,

text: Inner

text: Fluctuations

text: in

text: Noncommutative

text: Geometry

text: without

text: the

text: ﬁrst

text: order

text: condition,

text: J.

text: Geom.

text: Phys.

text: A.H.

text: Chamseddine,

text: A.

text: Connes

text: and

text: W.D.

text: van

text: Suijlekom,

text: Beyond

text: the

text: Spectral

text: Standard

text: Model:

text: Emergence

text: of

text: Pati-Salam

text: Uniﬁcation,

text: JHEP

text: A.

text: Connes,

text: Noncommutative

text: Geometry,

text: Academic

text: Press,


text: (cid:55)→ Since π is degenerate, the map u π(u)Jπ(u)J is not a representation of U(A) in Aut(H) (cid:55)→ (it doesn’t map 1 1, and u into an invertible operator). A unitary representation ρ of U(A) on H is given by ρ(u) := π¯ (u) + Jπ¯ (u)J + π (u)Jπ (u)J . (6) 0 0 1 1 Indeed π¯ (1) = id , π (1) = id and Jπ¯ (1)J = id . So ρ(1) = 1. Moreover, π¯ , 0 H¯0 1 H1 0 H0 0 Jπ¯ ( . )J, π and Jπ ( . )J are mutually commuting, hence ρ is multiplicative, and from 0 1 1 ∗ ∗ ρ(u)ρ(u ) = ρ(uu ) = 1 we deduce that the representation is also unitary. Basically, we are considering the direct sum of three representations of U(A): the fun- damental associated to π¯ and its dual, and the adjoint representation of π . 0 1 § In we exhibit a possible choice of the above data, such that U(A) contains (strictly) | the gauge group G of the Standard Model (modulo a ﬁnite subgroup), and ρ gives SM GSM the correct representation. 3 From particles to algebras In this section we give a review of the derivation of the data (A , H , J ) from physical F F F considerations, and collect at the end few results about the algebra and its commutant that will be useful in the following sections. In some sense, these data reﬂect the “topology” of the ﬁnite noncommutative manifold describing the internal space of the Standard Model, § while the Dirac operator encodes the metric properties. In we explain how to get the same gauge group from a spectral triple based on the complex algebra (2) with a degenerate representation. For simplicity, we work with only one generation of leptons/quarks. 3.1 The gauge group of the Standard Model (cid:101) Let × × G := U(1) SU(2) SU(3) SM be the usual gauge group of the Standard Model, let H be the ﬁnite-dimensional Hilbert (cid:101) space representing the internal degrees of freedom of elementary fermions. Let us recall ⊕ ∗ ∗ what is the representation of G . We have a decomposition H = F F , with F the SM (cid:18) (cid:19) (cid:18) (cid:19) dual space of F . The vector space F (for fermions) has basis { } ν uc ν uc L L R { R}c=1,2,3 eL dcL c=1,2,3 eR dcR c=1,2,3 where ν stands for neutrino, e for electron, uc for up-quark and dc for down-quark with ↑ color c = 1, 2, 3, L, R stands for left-handed resp. right-handed. We will use the label for ↓ the ﬁrst particle in each column (neutrino or quark up) and for the second one (electron or quark down). Left-handed doublets carry the fundamental representation of SU(2), while ↑ right handed particles are SU(2)-invariant; in particular, the particle in each doublet has ↓ − weak isospin I = 1/2 and the has weak isospin I = 1/2. The SU(2)-singlets have 3,w 3,w weak isospin I = 0. Each one of the color triplets carry the fundamental representation 3,w of SU(3), the other particles being SU(3)-invariant. Each particle carries a 1-dimensional → ∈ Z representation λ λ3Yw of U(1), where Yw 13 is the weak hypercharge; it is computed 6 ##LTLine##

text: Since

text: π

text: is

text: degenerate,

text: the

text: map

text: u

text: π(u)Jπ(u)J

text: is

text: not

text: a

text: representation

text: of

text: U(A)

text: in

text: Aut(H)

text: (it

text: doesn’t

text: map

text: and

text: u

text: into

text: an

text: invertible

text: operator).

text: A

text: unitary

text: representation

text: ρ

text: of

text: U(A)

text: on

text: H

text: is

text: given

text: by

text: ρ(u)

label: equation

text: π¯

text: (u)

text: Jπ¯

text: (u)J

text: π

text: (u)Jπ

text: (u)J

text: Indeed

text: π¯

text: id

text: π

text: id

text: and

text: Jπ¯

text: id

text: So

text: Moreover,

text: π¯

text: Jπ¯

text: )J,

text: π

text: and

text: Jπ

text: )J

text: are

text: mutually

text: commuting,

text: hence

text: ρ

text: is

text: multiplicative,

text: and

text: from

text: ρ(u)ρ(u

text: ρ(uu

text: we

text: deduce

text: that

text: the

text: representation

text: is

text: also

text: unitary.

text: Basically,

text: we

text: are

text: considering

text: the

text: direct

text: sum

text: of

text: three

text: representations

text: of

text: U(A):

text: the

text: fun-

text: damental

text: associated

text: to

text: π¯

text: and

text: its

text: dual,

text: and

text: the

text: adjoint

text: representation

text: of

text: π

text: In

text: we

text: exhibit

text: a

text: possible

text: choice

text: of

text: the

text: above

text: data,

text: such

text: that

text: U(A)

text: contains

text: (strictly)

text: the

text: gauge

text: group

text: G

text: of

text: the

text: Standard

text: Model

text: (modulo

text: a

text: ﬁnite

text: subgroup),

text: and

text: ρ

text: gives

text: SM

text: GSM

text: the

text: correct

text: representation.

text: From

text: particles

text: to

text: algebras

text: In

text: this

text: section

text: we

text: give

text: a

text: review

text: of

text: the

text: derivation

text: of

text: the

text: data

text: (A

text: H

text: J

text: from

text: physical

text: F

text: F

text: F

text: considerations,

text: and

text: collect

text: at

text: the

text: end

text: few

text: results

text: about

text: the

text: algebra

text: and

text: its

text: commutant

text: that

text: will

text: be

text: useful

text: in

text: the

text: following

text: sections.

text: In

text: some

text: sense,

text: these

text: data

text: reﬂect

text: the

text: “topology”

text: of

text: the

text: ﬁnite

text: noncommutative

text: manifold

text: describing

text: the

text: internal

text: space

text: of

text: the

text: Standard

text: Model,

text: while

text: the

text: Dirac

text: operator

text: encodes

text: the

text: metric

text: properties.

text: In

text: we

text: explain

text: how

text: to

text: get

text: the

text: same

text: gauge

text: group

text: from

text: a

text: spectral

text: triple

text: based

text: on

text: the

text: complex

text: algebra

text: with

text: a

text: degenerate

text: representation.

text: For

text: simplicity,

text: we

text: work

text: with

text: only

text: one

text: generation

text: of

text: leptons/quarks.

text: The

text: gauge

text: group

text: of

text: the

text: Standard

text: Model

text: Let

text: G

text: SM

text: be

text: the

text: usual

text: gauge

text: group

text: of

text: the

text: Standard

text: Model,

text: let

text: H

text: be

text: the

text: ﬁnite-dimensional

text: Hilbert

text: space

text: representing

text: the

text: internal

text: degrees

text: of

text: freedom

text: of

text: elementary

text: fermions.

text: Let

text: us

text: recall

text: what

text: is

text: the

text: representation

text: of

text: G

text: We

text: have

text: a

text: decomposition

text: H

text: F

text: F

text: with

text: F

text: the

text: SM

text: dual

text: space

text: of

text: F

text: The

text: vector

text: space

text: F

text: (for

text: fermions)

text: has

text: basis

text: ν

text: uc

text: ν

text: uc

text: L

text: L

text: R

text: eL

text: dcL

text: eR

text: dcR

text: where

text: ν

text: stands

text: for

text: neutrino,

text: e

text: for

text: electron,

text: uc

text: for

text: up-quark

text: and

text: dc

text: for

text: down-quark

text: with

text: color

text: c

text: L,

text: R

text: stands

text: for

text: left-handed

text: resp.

text: right-handed.

text: We

text: will

text: use

text: the

text: label

text: for

text: the

text: ﬁrst

text: particle

text: in

text: each

text: column

text: (neutrino

text: or

text: quark

text: up)

text: and

text: for

text: the

text: second

text: one

text: (electron

text: or

text: quark

text: down).

text: Left-handed

text: doublets

text: carry

text: the

text: fundamental

text: representation

text: of

text: while

text: right

text: handed

text: particles

text: are

text: in

text: particular,

text: the

text: particle

text: in

text: each

text: doublet

text: has

text: weak

text: isospin

text: I

text: and

text: the

text: has

text: weak

text: isospin

text: I

text: The

text: have

text: weak

text: isospin

text: I

text: Each

text: one

text: of

text: the

text: color

text: triplets

text: carry

text: the

text: fundamental

text: representation

text: of

text: the

text: other

text: particles

text: being

text: Each

text: particle

text: carries

text: a

text: Z

text: representation

text: λ

text: of

text: where

text: Yw

text: is

text: the

text: weak

text: hypercharge;

text: it

text: is

text: computed

text: ##LTLine##


text: z x + z = 0 1 2 sin z5 J C z x + z = −z 1 2 3 D G(D) sL−TJT JT z x + z = z 1 2 3 Jx + K = −z 4 Jx + K = 0 sin z5 J Figure 4: Pictorial illustration of the phase space of (18) when Assumption 3 is satisﬁed. The second and third bullet of this assumption imply that the intersection between C and the domains depicted in dark gray and light gray, respectively, is empty. In order to present a constructive Lyapunov function design, we ﬁrst introduce the function G¯ : R R n → n as G¯(x) := Lx+H+M(Jx+K)+sLJT max(0, z x+z ), (19) 1 2 R where the parameter M ∈ n is to be designed. Note that if x ∈ D, then G¯(x) = G(x) = Lx+H. Since G(D) ∩ D = ∅, Deﬁnition 1 implies that d(x, y) = 0 if and only if x = y, or x = G(y) or y = G(x). To design a Lyapunov function V , we note that (13) requires that V (x, y) = 0 if and only if d(x, y) = 0. Hence, we propose the following piecewise quadratic Lyapunov function V (x, y) = min((cid:107)x − y(cid:107)2 , (cid:107)x − G¯(y)(cid:107)2 , (cid:107)G¯(x) − y(cid:107)2 ), (20) P0 Ps Ps R where the positive deﬁnite matrices P , P ∈ n×n are to be designed. While this function is 0 s not smooth, we restrict our attention to a suﬃciently small sub-level set where, as we will show in Lemma 4, the function is smooth. Design of Lyapunov function parameters To design the parameters P , P and M of the Lyapunov function V in (20), we distinguish 0 s the domains S := {(x, y) ∈ (C∪D)2| V (x, y) = (cid:107)x − y(cid:107)2 }, (21a) 0 P0 S := {(x, y) ∈ (C∪D)2| V (x, y) = (cid:107)x − G¯(y)(cid:107)2 }, (21b) 1 Ps S := {(x, y) ∈ (C∪D)2| V (x, y) = (cid:107)G¯(x) − y(cid:107)2 }. (21c) 2 Ps The following lemma characterises the possibility of jumps from these sets, as illustrated in Fig. 5. R Lemma 3. Consider the hybrid system (18) and V in (20). Let M ∈ n satisfy (JL−1M+1)s < 0, let P , P (cid:31) 0 and let Assumption 3 hold. There exists a v > 0 such that for the sub-level set 0 s L V := V −1([0, v ]) the following statements hold: L L 1) V consists of the union of the mutually disconnected sets S ∩ V , S ∩ V and S ∩ V . L 0 L 1 L 2 L (cid:0) (cid:1) (cid:0) (cid:1) 2) S ∩ V ∩ (D × (C ∪ D)) = ∅ and S ∩ V ∩ (C ∪ D) × D) = ∅. 1 L 2 L 3) it holds that z y +z < 0 for all xT yT T ∈ S ∩V and z x+z < 0 for all xT yT T ∈ 1 2 1 L 1 2 S ∩ V . 2 L 13 ##LTFigure## ##LTFigure## ##LTFigure##

text: z

text: x

text: z

text: sin

text: J

text: C

text: z

text: x

text: z

text: −z

text: D

text: G(D)

text: sL−TJT

text: JT

text: z

text: x

text: z

text: z

text: Jx

text: K

text: −z

text: Jx

text: K

text: sin

text: J

text: Figure

text: Pictorial

text: illustration

text: of

text: the

text: phase

text: space

text: of

text: when

text: Assumption

text: is

text: satisﬁed.

text: The

text: second

text: and

text: third

text: bullet

text: of

text: this

text: assumption

text: imply

text: that

text: the

text: intersection

text: between

text: C

text: and

text: the

text: domains

text: depicted

text: in

text: dark

text: gray

text: and

text: light

text: gray,

text: respectively,

text: is

text: empty.

text: In

text: order

text: to

text: present

text: a

text: constructive

text: Lyapunov

text: function

text: design,

text: we

text: ﬁrst

text: introduce

text: the

text: function

text: G¯

text: R

text: R

text: n

text: n

text: as

text: G¯(x)

text: Lx+H+M(Jx+K)+sLJT

text: z

text: x+z

text: R

text: where

text: the

text: parameter

text: M

text: n

text: is

text: to

text: be

text: designed.

text: Note

text: that

text: if

text: x

text: D,

text: then

text: G¯(x)

text: G(x)

text: Lx+H.

text: Since

text: G(D)

text: D

text: Deﬁnition

text: implies

text: that

text: d(x,

text: y)

text: if

text: and

text: only

text: if

text: x

text: y,

text: or

text: x

text: G(y)

text: or

text: y

text: G(x).

text: To

text: design

text: a

text: Lyapunov

text: function

text: V

text: we

text: note

text: that

text: requires

text: that

text: V

text: (x,

text: y)

text: if

text: and

text: only

text: if

text: d(x,

text: y)

text: Hence,

text: we

text: propose

text: the

text: following

text: piecewise

text: quadratic

text: Lyapunov

text: function

text: V

text: (x,

text: y)

text: Ps

text: Ps

text: R

text: where

text: the

text: positive

text: deﬁnite

text: matrices

text: P

text: P

text: n×n

text: are

text: to

text: be

text: designed.

text: While

text: this

text: function

text: is

text: s

text: not

text: smooth,

text: we

text: restrict

text: our

text: attention

text: to

text: a

text: suﬃciently

text: small

text: sub-level

text: set

text: where,

text: as

text: we

text: will

text: show

text: in

text: Lemma

text: the

text: function

text: is

text: smooth.

text: Design

text: of

text: Lyapunov

text: function

text: parameters

text: To

text: design

text: the

text: parameters

text: P

text: P

text: and

text: M

text: of

text: the

text: Lyapunov

text: function

text: V

text: in

text: we

text: distinguish

text: s

text: the

text: domains

text: S

text: {(x,

text: y)

text: V

text: (x,

text: y)

text: S

text: {(x,

text: y)

text: V

text: (x,

text: y)

text: Ps

text: S

text: {(x,

text: y)

text: V

text: (x,

text: y)

text: Ps

text: The

text: following

text: lemma

text: characterises

text: the

text: possibility

text: of

text: jumps

text: from

text: these

text: sets,

text: as

text: illustrated

text: in

text: Fig.

text: R

text: Lemma

text: Consider

text: the

text: hybrid

text: system

text: and

text: V

text: in

text: Let

text: M

text: n

text: satisfy

text: let

text: P

text: P

text: and

text: let

text: Assumption

text: hold.

text: There

text: exists

text: a

text: v

text: such

text: that

text: for

text: the

text: sub-level

text: set

text: s

text: L

text: V

text: V

text: v

text: the

text: following

text: statements

text: hold:

text: L

text: L

text: V

text: consists

text: of

text: the

text: union

text: of

text: the

text: mutually

text: disconnected

text: sets

text: S

text: V

text: S

text: V

text: and

text: S

text: V

text: L

text: L

text: L

text: L

text: S

text: V

text: (D

text: (C

text: D))

text: and

text: S

text: V

text: (C

text: D)

text: D)

text: L

text: L

text: it

text: holds

text: that

text: z

text: y

text: +z

text: for

text: all

text: xT

text: yT

text: T

text: S

text: ∩V

text: and

text: z

text: x+z

text: for

text: all

text: xT

text: yT

text: T

text: L

text: S

text: V

text: L

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##