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Please note: Images will appear in color online but will be printed in black and white.ArticleTitle Axial Anomaly and the Triality Symmetry of Leptons and Hadrons

Article Sub-Title

Article CopyRight Springer-Verlag Wien(This will be the copyright line in the final PDF)

Journal Name Few-Body Systems

Corresponding Author Family Name FuruiParticle

Given Name SadatakaSuffix

Division Faculty of Science and Engineering

Organization Teikyo University

Address 1-1 Toyosatodai, 320-8551, Utsunomiya, Japan

Email furui@umb.teikyo-u.ac.jp

Schedule

Received 4 May 2013

Revised

Accepted 25 April 2014

Abstract We apply the supersymmetric model of É. Cartan to the pseudoscalar meson decay into two photons, , and . In the book of É. Cartan published in 1966, Dirac spinors t (A, B) and t

(C, D) and vector fields E and E′ were introduced and five supersymmetric transformations G 23, G 12, G 13,G 123 and G 132 were considered. The Pauli spinor is treated as a quaternion and the Dirac spinor is treated asan octonion. In the pseudoscalar meson decay, when the two final vector fields belong to the same group(EE or E′E′), we call the diagram rescattering diagram. When they belong to different groups (EE′), thediagram is called twisted diagram. Assuming the triality selection rules of octonions, dark matter is interpretedas matter emitting photons in a different triality sector than that of electromagnetic probes in our world.

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Few-Body SystDOI 10.1007/s00601-014-0897-0

Sadataka Furui1

Axial Anomaly and the Triality Symmetry of Leptons2

and Hadrons3

4

Received: 4 May 2013 / Accepted: 25 April 20145

© Springer-Verlag Wien 20146

Abstract We apply the supersymmetric model of É. Cartan to the pseudoscalar meson decay into two pho-7

tons, π0 → γ γ , η → γ γ and η′ → γ γ . In the book of É. Cartan published in 1966, Dirac spinors8

t (A, B) and t (C, D) and vector fields E and E ′ were introduced and five supersymmetric transformations9

G23, G12, G13, G123 and G132 were considered. The Pauli spinor is treated as a quaternion and the Dirac10

spinor is treated as an octonion. In the pseudoscalar meson decay, when the two final vector fields belong to11

the same group (E E or E ′E ′), we call the diagram rescattering diagram. When they belong to different groups12

(E E ′), the diagram is called twisted diagram. Assuming the triality selection rules of octonions, dark matter13

is interpreted as matter emitting photons in a different triality sector than that of electromagnetic probes in our14

world.15

1 Introduction16

According to Hawking and Mlodinow, physical theories are assembly of mathematical models or assembly of17

rules that connect elements of models and observables [1]. In QED, complex numbers and quaternion are used18

and in QCD also the same number system is used. A quaternion operates on a two-component spinor. Pauli19

spinor is a two-component spinor, but Dirac spinor is a four component spinor, and there is an octonion that20

operates on a four component spinor, which has the triality symmetry.21

In the low-energy world there are two kinds of Dirac spinors, leptons e, μ and τ , and quarks u, d and s.22

Quarks have three-color degrees of freedom. When color degrees of freedom is large, the quark loop degrees23

of freedom U (1)-anomaly due to difference of masses of e, μ and τ is suppressed, but in the infrared there24

appears quark condensates 〈0|qq|〉 �= 0, and U (1) symmetry is broken and there appears η′ meson in addition25

to π±, π0, K ±, K 0, K0 and η, that plays essential roles [2].26

The anomaly was accounted for in the effective theory by the Wess–Zumino–Witten term [3,4] and the27

effective Lagrangian in the presence of external source28

LQC D = L0QC D + qγμ(vμ + γ5aμ)q − q(s − iγ5 p)q29

− g2

16π2θ ǫαβλμtrc(GαβGμν) (1)30

with the vacuum angle θ was introduced in [5], and studied for a study of the mixing of η and η′ [6–10].31

The singlet axial current is normalized by introducing a number of flavors N f as32

A0μ = 1

√

N f

qγμγ5q (2)33

S. Furui (B)Faculty of Science and Engineering, Teikyo University, 1-1 Toyosatodai, Utsunomiya 320-8551, JapanE-mail: furui@umb.teikyo-u.ac.jp

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S. Furui

and the divergence34

∂μ A0μ =

√

2N f

1

16π2trcǫ

αβμνGαβGμν =√

2N f ω (3)35

The η′ field defined in [5] is36

⟨

0|A0μ|η′⟩ = i pμF0,

⟨

0|ω|η′⟩ =√

2N f

τ

F037

where the mass of η′ and the parameter τ are related by38

Mη′ 2 = 2N f τ

F20

(4)39

The triangle diagram yields the electromagnetic anomaly of the axial quark charge current40

∂μ j5λμ = − e2

16π2 ǫαβμν Fαβ Fμν · tr [τλQ2]41

where Fμν is the electromagnetic field strength,42

Q =( 2

3 00 − 1

3

)

43

is the quark electric charge and tr is over colors and flavors [11].44

The Nambu–Goldstone boson π’s decay into two gamma rays is described by divergence of the axial45

current. The Adler–Bardeen [12]’s theorem says that higher-order effects in the triangular diagram of pion46

decay into two photons can be incorporated in the renormalization47

∂μ j5μ = ǫαβμν Fαβ Fμν

e20

8π2

(

1 − 3e40

64π4 log�2

m2

)

48

The axial anomaly is not exhausted by the single loop, and radiative corrections were evaluated by several49

authors [13–15]. Ioffe calculated the divergence of the axial current from the triangle diagram using the gluonic50

field strength as51

∂μ j5λμ = − αs

4πǫαβμνGαβGμν · Nc52

where Nc = 3 is the number of colors.53

The decay process of Nambu–Goldstone boson π0 can be derived from current algebra and constrained54

by the symmetry of the vector current, whose triality sector is fixed. The gluons exchanged between the quark55

triangle, and the quark rectangle would be constrained to be in a triality sector, and the twisted diagram does56

not contribute. The standard chiral effective theory predicts [14]57

Ŵ(π0 → γ γ ) = α2

32π3

mπ3

fπ2 = 7.7eV.58

Experimentally, Ŵ(π0 → γ γ ) = 7.82 ± 0.14 ± 0.17eV [16].59

Divergence of an isoscalar axial current could play a role in η → γ γ and/or η′ → γ γ decay processes, in60

which instanton contribute and the twisted diagram appears. A simple theoretical extension of π0 → γ γ to61

η → γ γ predicts [14]62

Ŵ(η → γ γ ) = α2

32π3

1

3

mη3

fη2 = 0.13keV63

Experimentally, Ŵ(η → γ γ ) = 0.510 ± 0.026keV [16] is about 4 times larger.64

A simple multiplication of (mη′/mη)3 predicts65

Ŵ(η′ → γ γ ) = α2

32π3

1

3

mη′ 3

fη′ 2= 0.69keV66

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Triality Symmetry of Leptons and Hadrons

Experimental data of Ŵ(η′ → γ γ ) = 4.34 ± 0.14keV [16] is about six times larger, and the average decay67

width of η and η′ is 5 times larger.68

In [17], one loop correction to the Wess-Zumino Lagrangean governing π0, η → γ γ decay widths were69

studied. They used the eighth member of SU(3) octet, η8 and the SU(3) singlet η0 as70

|η〉 = cos θ |η8〉 − sin θ |η0〉71

∣

∣η′⟩ = sin θ |η8〉 + cos θ |η0〉 (5)72

and obtained θ ≃ −20◦. Ref. [5,8] also obtained θ = −20◦, but the studies of vacuum using quark-flavor73

basis rather than the octet-singlet basis turned out to fit data of η, η′ better [18,19].74

The one-mixing angle scheme turned out to be valid only in large Nc limit [5,7,8]. Bhagwat et al. [10]75

defined the kernel of Bethe–Salpeter equation of pseudoscalar mesons as76

K = KL + K A77

where KL is the reading order term and K A is the term associated with the anomaly, which is parametrized as78

(K A)turs(q, p, P) = −ξ((q − p)2)

{

cos2 θξ [ζ γ5]rs[ζ γ5]tu + sin2 θξ [ζ γ · Pγ5]rs[ζ γ · P γ5]tu

}

79

ξ(k2) = (2π)4ξ δ4(k) (6)80

with a dimensionless coupling constant ξ as a parameter, and the quark mass function81

ζ = diag[ 1

M Du

,1

M Dd

,1

M Ds

, · · · ], M Df = M f (s = 0) (7)82

where f is the quark flavor.83

When the basis of non-strange η and strange η,84

|ηN S〉 = 1√2(|uu〉 + |dd〉) = 1√

3|η8〉 +

√

2

3|η0〉85

|ηS〉 = |ss〉 = −√

2

3|η8〉 + 1√

3|η0〉 (8)86

are used,87

|η〉 = cos φ |ηN S〉 − sin φ |ηS〉88

∣

∣η′⟩ = sin φ |ηN S〉 + cos φ |ηS〉 . (9)89

The experimental value φ = 41.88◦ corresponds to |ηN S〉 = 0.427, |ηS〉 = 0.0860, and θ = φ−arctan√

2 =90

φ − 54.74◦ = −12.9◦ [9]. When the parameter ξ = 0.076 was taken, θη = −15.4◦ and θ ′η = −15.7◦ were91

obtained [10].92

In a phenomenological model using the quark-flavor basis and assuming mixing angle φq ≃ φs ∼ 40◦,93

the decay width of η and η′ could be fitted [19], and a lattice simulation using the similar bases of (9), and94

including the loop of c quarks, obtained also φ = 46◦ [20].95

2 Triality Symmetry of É. Cartan96

É. Cartan [21] studied algebra of system of spinors and vectors, which have the triality symmetry. He con-97

sidered in the euclidean space E2ν , the semispinors φ which are specified by an even number of indices:98

ξ0, ξ23, ξ31, ξ12, ξ1234, ξ14, ξ24 and ξ34 and semispinors ψ which are specified by an odd number of indices:99

The spinor bases A, B, C, D are expressed as100

A = ξ14σx + ξ24σy + ξ34σz + ξ0I101

B = ξ23σx + ξ31σy + ξ12σz + ξ1234I102

C = ξ1σx + ξ2σy + ξ3σz + ξ4I103

D = ξ234σx + ξ314σy + ξ124σz + ξ123I (10)104

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S. Furui

and the vector fields are expressed as105

E = x1 i + x2 j + x3k + x4I106

E ′ = x ′1 i + x ′

2 j + x ′3k + x ′

4I. (11)107

The coupling of spinors to vector particles xi and x ′j are expressed as tφC Xψ . É. Cartan considered 5 superspace108

transformations G23, G12, G13, G123 and G132. By the transformation G12, the spinor t (A, B) is transformed109

to E, E ′ in which the 4th component of the vector field x4 and x ′4 are interchanged, and E, E ′ are transformed110

to t (A, B) in which ξ1234 and ξ0 are interchanged.111

The operator G23 does not transform vectors to spinors or spinors to vectors, but its operation on Dirac112

fermions is a charge conjugation,113

G23

(

A

B

)

=(

C

D

)

, G23

(

C

D

)

= −(

A

B

)

.114

and its operation on vector field E and E ′ is an interchange of the 4th component.115

G23

(

E

E ′

)

= G23

(

x1i + x2j + x3k + x4Ix ′

1i + x ′2j + x ′

3k + x ′4I

)

116

=(

x1i + x2j + x3k − x ′4I

x ′1i + x ′

2j + x ′3k − x4I

)

.117

When the vector fields are selfdual, and the energy is null, a singular behavior can be expected from G23 E118

and G23 E ′.119

When G13 operates on left-handed fermion the 4th component of A and B, ξ0 and ξ1234, respectively are120

interchanged:121

G13

(

A

B

)

= G13

(

ξ14σx + ξ24σy + ξ34σz + ξ0Iξ23σx + ξ31σy + ξ12σz + ξ1234I

)

,122

=(

ξ14σx + ξ24σy + ξ34σz + ξ1234Iξ23σx + ξ31σy + ξ12σz + ξ0I

)

.123

G13

(

E

E ′

)

=(

D

C

)

=(

ξ234σx + ξ314σy + ξ124σz + ξ123Iξ1σx + ξ2σy + ξ3σz + ξ4I

)

124

In quantum electrodynamics, the success of Dirac spinors which are described by quaternions is established,125

but in QCD, it is not so evident. I study in this paper, consequences of the interchange of the 4th component126

in the octonion by extending the QCD using octonions. The QCD in quaternion basis was studied in [22–26],127

and recently, I discussed the axial anomaly using octonion bases and considered rescattering diagrams in [27].128

The vector particles that propagate between the triangle diagram and the square diagram can be photons as129

well as gluons. In low energy, the effect of gluons is expected to be important, but transition from two gluons130

to two photons via square diagram or the gluon and photon scattering at low energy is not well understood131

[28].132

In the rescattering diagram, I considered the change of vector particles xi and x j to x ′i and x ′

j without133

changing the polarization direction.134

In chiral gauge theory, left-handed quarks and left-handed leptons are defined as135

QL =(

u

d

)

L L =(

ν

l

)

136

and right-handed leptons �R are defined as137

� ′L i

= σ 2�∗Ri

�′†L i

= �TRi

σ 2138

The left-handed fermion t (A, B) is defined as a fermion and its right-handed anti-particle (C, D) is defined139

as a new right-handed fermion.140

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Triality Symmetry of Leptons and Hadrons

The fermionic Lagrangian in the standard model is141

L = �iγ μ

(

∂μ − ig Aaμta

r

(

1 − γ 5

2

))

�142

In É. Cartan’s convention, the projection operator 1−γ 5

2 is contained in φ or ψ and143

� = (φ, ψ)

(

0 11 0

)

= (ψ, φ)144

Quarks and leptons belong to their triality sectors, and I assume that the electromagnetic interaction has the145

triality selection rules, or electron e, muon μ and tauon τ react to quarks in their same triality sector. Quarks146

may emit photons in an arbitrary triality sector, but if photons from quarks in the triality sector different from147

that of leptons are not detected, and if μ → eγ is not observed, ( its actual experimental probability is less148

than 10−11), the triality selection rule can be established.149

The vector field E and E ′ of É. Cartan is a kind of generalization of the electro magnetic field Aμ. When150

a pair of vector particles E, E or E ′E ′ transform into a quark-antiquark pair and return to two same type of151

vectors E, E or E ′, E ′, we call the diagram a rescattering diagram. The quark-antiquark system may have152

isospin 1, like π . When different types E, E ′ transform into a quark-antiquark pair and return to E, E ′, we153

call the diagram a twisted diagram. The corresponding quark-antiquark systems have isospin 0, like η and η′.154

3 Rescattering Diagrams155

Due to the γ5 operator in the axial vector vertex, the triangle diagram introduces an operator containing 4156

components ∂i , x j , ∂k and xm , combined by the anti-commutation factor ǫi jkm . Emitted two vector particles157

can be absorbed by a fermion, and the fermion can emit two vector particles. When the polarizations of the158

two vector particles do not change in the two emissions, I call the diagram as a rescattering diagram.159

In QED, a e60 term appears from rescattering of two photons of two different polarizations, as shown in160

Figs. 1, 2, 3, 4, 5, 6, 7 and 8. They contain square diagrams contoured by four quark lines. In Figs. 1, 2, 3161

and 4, axial vector current couples to fermions with an even number of indices, and In Figs. 5, 6, 7, and 8, it162

couples to fermions with an odd number of indices.163

I choose the propagator between two vector particles emission points in the A1x2x3 or A1x ′2x ′

3 diagram and164

between the final x ′2x ′

3 or x2x3 emission points to be spinless. In this model, the A1(A′1)σx vertex is produced165

from the product:166

ξ12σz × ξ31σy → −A1σx , ξ34σz × ξ24σy → −A1σx , ξ3σz × ξ2σy → −A′1σx167

and ξ124σz × ξ314σy → −A′1σx .168

Similarly, A2(A′2)σy vertex is produced from the product:169

ξ23σx × ξ12σz → −A′2σy, ξ14σx × ξ34σz → −A′

2σy, ξ1σx × ξ3σz → −A2σy170

and ξ124σz × ξ234σx → A2σy171

Similarly, A3(A′3)σz vertex is produced from the product:172

ξ31σy × ξ23σx → −A′3σz, ξ24σy × ξ14σx → −A′

3σz, ξ2σy × ξ1σx → −A3σz173

and ξ314σy × ξ234σx → −A3σz174

The amplitude that includes the half circle of Figs. 1 and 2 is175

T ′321(k, q) = −e2

∫

d4 p

(2π)4 T r [γ31

p − (k − q) − mγ1γ5

1

p − q − mγ2

1

p − k − m]176

∝ G ′3σ (k)G ′

2ρ(q)ǫ3σ2ρ (12)177

where G ′λμ(k) = kλx ′

μ − kμx ′λ, and ǫ3124 = 1 and ǫ3421 = −1.178

The amplitude that includes the half circle of Figs. 3 and 4 is179

T321(k, q) ∝ G3σ (k)G2ρ(q)ǫ3σ2ρ180

where Gλμ(k) = kλxμ − kμxλ.181

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S. Furui

Fig. 1 The half circle diagram of an axial anomaly. A1x ′3x ′

2 type and its rescattering

Fig. 2 The half circle diagram of an axial anomaly. A1x ′3x ′

2 type and its rescattering

Fig. 3 A1x3x2 type and its rescattering

Fig. 4 A1x2x3 type and its rescattering

4 Twisted Diagrams182

When the vector particles are self-dual, there is no reason to restrict an emission of two vector particles in183

the triangle diagram to be of type xi x j or x ′i x

′j . When the emitted vector particles are x4x ′

i or x ′4xi , where184

i = 1, 2, 3, the fermion square diagram can emit two vector particles x j x ′k or x ′

j xk where j �= k and they are185

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Triality Symmetry of Leptons and Hadrons

Fig. 5 A′1x3x2 type and its rescattering

Fig. 6 A′1x2x3 type and its rescattering

Fig. 7 A′1x ′

3x ′2 type and its rescattering

Fig. 8 A′1x ′

2x ′3 type and its rescattering

not equal to ′i ′ and ’4’. When the emitted vector particles are x ′2x3 or x2x ′

3, the fermion square can emit x1x ′4186

or x ′1x4, respectively.187

When the vector fields can be treated as self-dual, i.e. xi and x ′j interact with ξ1234 and ξ0, one could188

consider topologically complicated processes in which after twice rotations, (the first via ξ1234 and the second189

via ξ0) the initial configuration reappears.190

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S. Furui

Fig. 9 The axial anomaly diagram via A1x1x ′4 to A1x ′

3x2

Fig. 10 The axial anomaly diagram via A1x1x ′4 to A1x3x ′

2

G13 transforms (A1, A2, A3, A4) → (A1, A2, A3, B4) and (B1, B2, B3, B4) → (B1, B2, B3, A4). In other191

words, ξ1234 is replaced by ξ0. Since the intermediate quark propagator that absorbs and emits x1x ′4 or x4x ′

1 is192

not expected to be in the eigenstate of G13, the quark ξ123 that absorbs x ′4 is transformed to ξ0 but through the193

mixed component of ξ1234, it will emit x1 and changes to ξ1 as shown in Fig. 9.194

When vector particles are selfdual and xi and x ′i are indistinguishable, the square in the left hand side of195

Fig. 9 and Fig. 10 give an amplitude, in which a spinor ξa emits a vector particle xb and changes to a spinor196

ξc is represented as ξa, xb, ξc, of the type197

(ξ1234, x ′4), ξ1, x ′

3, ξ24, x2, ξ123, x ′4, ξ0, x1,198

(ξ0, x1), ξ123, x3, ξ34, x ′2, ξ1, x1, ξ1234, x ′

4.199

They emit x2, x ′2 and x3, x ′

3 but do not contain x4 and x ′1.200

They are included in the amplitudes201

T3241(K , Q, q, k) ∝ G ′3ρ(K )G2τ (Q)G4υ(q)′G1σ (k)ǫ3ρ2τ ǫ4υ1σ

202

T3214(K , Q, k, q) ∝ G3ρ(K )G ′2τ (Q)G1σ (k)G ′

4υ(q)ǫ3ρ2τ ǫ1σ4υ .203

The amplitude that includes the right hand loop of Figs. 9 and 10 are204

T ′′41(k, q) ∝ G ′

4υ(q)G1σ (k)ǫ4υ1σ205

T ′′′14(k, q) ∝ G ′

1σ (k)G4υ(q)ǫ1σ4υ . (13)206

They are included in the amplitudes207

T324141μ(K , Q, q, k)) = −e2g6∫

d4 p

(2π)4

∫

d4 P

(2π)4T r [γ3

1

(P + K ) − mγ2

1

(P + K + Q) − m208

×γ11

P − mγ4

1

(P + k) − m] 1

k2 + ǫ

1

q2 + ǫδ(P + K + Q − p − k − q)209

×T r [γ41

(p − k + q) − mγμγ5

1

p − mγ1

1

(p − k) − m]210

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Fig. 11 The diagram via A1x1x ′4 to A1x ′

3x2

Fig. 12 The diagram via A1x1x ′4 to A1x3x ′

2

∝ G ′3ρ(K )G2τ (Q)G ′

4υ(q)G1σ (k)ǫ3ρ2τ ǫ4υ1σ G ′4υ(q)G1σ (k)ǫ4υ1σ

211

∝ G ′3ρ(K )G2τ (Q)ǫ3ρ2τ (G ′

43(q)G12(k) − G ′42(q)G13(k))2

212

T321414μ(K , Q, k, q) = −e2g6∫

d4 p

(2π)4

∫

d4 P

(2π)4 T r [γ31

(P + K ) − mγ2

1

(P + K + Q) − m213

×γ11

P − mγ4

1

(P + q) − m] 1

k2 + ǫ

1

q2 + ǫδ(P + K + Q − p − q − k)214

×T r [γ41

(p − q + k) − mγμγ5

1

p − mγ1

1

(p − q) − m]215

∝ G3ρ(K )G ′2τ (Q)G1σ (k)G ′

4υ(q)ǫ3ρ2τ ǫ1σ4υG1σ (k)G ′4υ(q)ǫ1σ4υ

216

∝ .G3ρ(K )G ′2τ (Q)ǫ3ρ2τ (G13(k)G ′

42(q) − G12(k)G ′43(q))2

217

In Figs. 9, 10, 11, 12, 13, 14, 15 and 16, the quark spinor ξ1234 I absorbs vectors xi , the spinor ξ0 I absorbs x ′i .218

Gluons polarized in the four directions x ′1, x ′

2, x3, x4, x1, x2, x ′3, x ′

4 or x ′1, x2, x ′

3, x4 ( and xi , x ′i interchanged)219

appear on the twisted diagrams.220

The vector x4 and x ′1 have common coupling to spinor ξ123 and ξ1. In Fig. 9, the product ξ1σx ×ξ24σy → x ′

3k221

and ξ24σy × ξ123 I → x2j the final two vectors x ′3k × x2j makes a vector product -x1i. After emission of x2,222

ξ123 emits a x ′4 and becomes ξ0, and as shown in Fig. 10, absorbs x ′

4 and becomes ξ123. It emits x3 and x ′2 and223

becomes ξ1. It emits x1 and goes to ξ1234. ξ1234 absorbs x1 as shown in Fig. 9 and emits x ′3 and x2 and returns224

to ξ123. In Figs. 11 and 12, the spinor ξ123 is replaced by ξ1.225

The relative sign of x ′3k × x2j from Fig. 9 and x ′

2j × x3k from Fig. 10 is cancelled by the relative sign of226

the product of 6 vertices ξ∗x jξ∗′ , where j runs from 1 to 4, and ξ∗, ξ∗′ run ξα, ξαβ , ξαβγ or ξ1234 that appear227

on the figures.228

The vector x ′4 and x1 have the common coupling to spinor ξ234 and ξ4. In Figs. 13, 14, 15 and 16, the same229

mechanism occurs as in Figs. 9, 10, 11 and 12, in which ξ123 is replaced by ξ234 and ξ1 is replaced by ξ4.230

Diagrams for vertices via A2x2x ′4 to A2x ′

1x3, etc. and via A3x3x ′4 to A3x1x ′

2, etc. are similar.231

In Figs. 17, 18, 19, 20, 21, 22 23 and 24, I show the cases of the vector particles x1/x ′1 and x ′

4/x4 in the232

final state. In Figs. 17 and 18, the quark propagator between x ′2 and x3 is ξ314 on the side of A1 vertex, but233

ξ1234 or ξ0 on the side of x1x4′. ξ1234 and ξ0 are the 4th component of spinor B and A, respectively. When G13234

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Fig. 13 The diagram via A1x4x ′1 to A1x ′

2x3

Fig. 14 The diagram via A1x4x ′1 to A1x2x ′

3

Fig. 15 The diagram via A1x4x ′1 to A1x ′

2x3

Fig. 16 The diagram via A1x4x ′1 to A1x2x ′

3

operates on t (A, B), ξ1234 and ξ0 are interchanged. Therefore, a mixing of ξ1234 and ξ0 inside the quark loop235

occur.236

The Figs. 17 and 18 are contained in the amplitude237

T1432(K , Q, k, q) ∝ G1ρ(K )G ′4τ (Q)G3υ(k)G ′

2σ (q)ǫ1ρ4τ ǫ3υ2σ238

T1423(K , Q, q, k) ∝ G ′1ρ(K )G4τ (Q)G ′

2σ (q)G3υ(k)ǫ1ρ4τ ǫ2σ3υ .239

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Triality Symmetry of Leptons and Hadrons

Fig. 17 The diagram via A1x ′2x3 to A1x1x ′

4

Fig. 18 The diagram via A1x ′2x3 to A1x ′

1x4

Fig. 19 The diagram via A1x ′2x3 to A1x1x ′

4

Fig. 20 The diagram via A1x ′2x3 to A1x ′

1x4

Mixing of ξ1234 and ξ0 is important. Experimentally, x1′x4 or x1x4

′ state will not be detected, but via240

rescattering it changes to x2′x3 or x2x3

′ state.241

The diagrams for A2x ′1x3, A2x1x ′

3 and A3x ′2x1, A3x2x ′

1 are similar.242

In the twisted diagrams, a mixing of ξ1234 and ξ0 is assumed, which appears after transformations of243

G12, G13, G123 and G132. The two vector particles which appear on the triangle diagram belong to different244

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Fig. 21 The diagram via A1x ′3x2 to A1x1x ′

4

Fig. 22 The diagram via A1x ′3x2 to A1x ′

1x4

Fig. 23 The diagram via A1x ′3x2 to A1x1x ′

4

Fig. 24 The diagram via A1x ′3x2 to A1x ′

1x4

groups E and E ′, respectively. Such processes would contribute in the instanton and do not enter in decays of245

a Nambu-Goldstone boson.246

A possible origin of large φ is a contribution of the twisted diagram, in which the 4th component of the quark247

that runs on the loop becomes a mixed state due to the operation of G13 in the transition to the intermediate248

state.249

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5 Discussion and Conclusion250

Whether É. Cartan’s spinor matches the dynamics of QCD is not trivial. However, if physical theories are251

assembly of mathematical models or assembly of rules that connect elements of models and observables [1],252

there is a possibility that the triality symmetry plays a role in QCD. The possible role of the triality property253

of SO(8) group in the quark system was discussed in [29]. In my model, the triality selection rule in the254

electromagnetic interaction of leptons was incorporated, and no selection rule was assumed in the interaction255

of quarks.256

Photons emitted from matter made of quarks that belong to a triality sector different from that of electro-257

magnetic probes, will not be detected, and the matter will be assigned as a dark matter. Dark matter search is258

done by the Xenon100 detector [30], and a direct detection of composite dark matter through lattice calculation259

of electromagnetic form fators and comparison to the data of Xenon100 was proposed [31]. The dark matter260

with masses less than 10TeV was excluded in this analysis. However, the detection through electromagnetic261

probes in our world would be impossible due to the triality selection rule.262

The dark matter search was done also in AMS-02 by measuring positron fraction in cosmic rays [32].263

The signal that is detected in our electromagnetic probe would not originate from dark matter, but rather from264

pulsars, as a recent analysis suggests [33].265

A comparison of the decay width of η → γ γ and η′ → γ γ could be a place to study the effect of triality266

symmetry, including η or η′ decay into intermediadte two vector particles in twisted diagrams. An investigation267

of the photon-gluon scattering contribution to the structure functions of deep inelastic scattering for unpolarized268

as well as polarized photons and gluons [28] could provide helpful information. The Primakoff production269

of η and η′ in the Coulomb field of a nucleon also shows enhancement as compared to the production of π270

mesons [34–37].271

In [26], I considered three massless neutrinos in different triality sectors interacting with each other and272

produced one heavy and two degenerate light neutrinos. νe, νμ and ντ have their lepton partners. I expect e, μ273

and τ are sensitive to flavors, but blind to the triality of neutrinos, quarks and gluons, and that they are sensitive274

to the triality of electromagnetic waves. If electromagnetic waves from different triality sectors cannot be275

detected by electromagnetic probes in our world, we can understand the presence of dark matter.276

In the rescattering or twisted diagrams, qq state that decay into two vector particles appears. Brodsky and277

Shrock [38,39] discuss problems in the expectation value of qq in QCD, which gives too large cosmological278

constant ��, and claimed that �� has the spacial support within hadrons. The recent review [40] explains the279

region of matter distribution reachable with terrestrial facilities. Whether the spacial support and the region280

where the trialty sector agrees with that of our electromagnetic probes could match is under investigation.281

Acknowledgments The author thanks Craig Roberts for sending helpful references and Stan Brodsky for helpful information282

and comments.283

Appendix: Conjecture on the Structure of the Vacuum of Universe284

In order to obtain physical quantities like decay width of π, η and η′ mesons, it is necessary to regularize285

Feynmann integral. Lüscher [41] started from the space-time lattice286

� = {x ǫ Z4| − L/2 < xμ ≤ L/2, μ = 1, 2, 3, 4}287

and link variables U (x, μ) defined, when x ǫ � and x + μ ǫ �. The gauge transformation g(x) of the gauge288

group G acts on the gauge field U , and the compact Lie group G acts in a differential manner on the field289

manifold F ,290

g ǫ G, U ǫ F → g · U = U g ǫ F291

The orbit manifold M = F/G is a differential manifold and with some measure on F defined as dμ(U ) and292

some integrable function defined as f (U ), one introduces notations as follows.293

We define basis of the Lie algebra LG of G as T a, a = 1, · · · , dG , general element of LG as X = XaT a and294

for any differential function F(U ) on F295

δX F(U ) = Xa

{

∂

∂Ya

F(e−Y · U )

}

Y=0,296

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is defined.297

For any subset of F , defined as N , a set298

[N ] = {U ǫ F |g · U ǫ N for some g ǫ G}299

is defined as a union of all gauge orbits passing through N .300

It was shown in [41] that, by using invariant measure dg on G,301

χ(U ) ={

1 if Uǫ N ,

0 otherwise302

and the linear operator303

L(U ) · X = δX F(U ) for all X ǫLG,304

one can consider for a set h ǫ G and g = e−X h, a gauge fixing function305

F(g · U ) = L(h · U ) · X + O(X2).306

For any function f (U ) supported in N , the integral307

∫

F

dμ(U ) f (U ) ∼∫

F

dμ(U ) f (U )

∫

G

dgχ(g · U )det L(g · U )δ(F(g · U ))308

reduces in the neighbourhood of h to309

K

∫

F

dμ(U ) f (U )det L(U )δ(F(U ))310

where K is a constant independent of f .311

We consider the vacuum near N of our universe, and the universe transformed by G23, G12, G13, G123312

and G132.313

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