SU(2/1) superchiral self-duality: a new quantum, algebraic and geometric paradigm to describe the electroweak interactions

We propose an extension of the Yang-Mills paradigm from Lie algebras to internal chiral superalgebras. We replace the Lie algebra-valued connection one-form A, by a superalgebra-valued polyform A˜ mixing exterior-forms of all degrees and satisfying the chiral self-duality condition A˜=*A˜χ, where χ denotes the superalgebra grading operator. This superconnection contains Yang-Mills vectors valued in the even Lie subalgebra, together with scalars and self-dual tensors valued in the odd module, all coupling only to the charge parity CP-positive Fermions. The Fermion quantum loops then induce the usual Yang-Mills-scalar Lagrangian, the self-dual Avdeev-Chizhov propagator of the tensors, plus a new vector-scalar-tensor vertex and several quartic terms which match the geometric definition of the supercurvature. Applied to the SU(2/1) Lie-Kac simple superalgebra, which naturally classifies all the elementary particles, the resulting quantum field theory is anomaly-free and the interactions are governed by the super-Killing metric and by the structure constants of the superalgebra.

From a geometrical perspective, our new complementary treatment of the exterior bundle (the polyform superconnection) with the spinor bundle (the chiral Fermions) reflects an element of bona fide internal supersymmetry. The signs generated in the quantum loops by the tensorial structures of the propagators and by the orientations of the chiral Fermion propagators match the signs generated by the grading of the superalgebra in the Clebsch-Gordan calculations and their interplay with the super-Jacobi identity. The theory is superalgebraic despite the fact that all the gauge fields are Bosons and all the matter fields are Fermions, as requested by the spin-statistics relation.
The theory respects Einstein's distinction between the force fields, which are geometrized as superalgebra-valued polyforms, and matter fields, represented by pointlike chiral Fermions. The odd couplings play with the orientation of space, which is represented by the opposite helicities of the massless left/right Fermions, and with the orientations of the p-form matter fields and their Hodge duals. This is very different from the Wess-Zumino supersymmetry which couples Bosons to Fermions, i.e. force fields to matter fields. Although the latter is much more developed, one should remember that not a single known particle is the supersymmetric partner of another known particle, for example the neutrino is not the partner of the photon, whereas on the contrary, all the known elementary particles, the leptons and the quarks, naturally fall in superchiral SU(2/1) multiplets. Finally, the superconnection offers a way geometrize the Higgs fields.
In section 2, we introduce in details the new paradigm of a self superchiral superconnection. In sections 3 and 4, we show how the knowledge of the couplings of the Fermions to the scalar, vector and 2-tensor components of the superconnection induce their propagators and interactions. In section 5 to 7, we analyze the quantum anomalies. The notations are given explicitly in appendix A to E.

The new superchiral superconnection A
The de Rham complex over a 4-dimensional differentiable manifold is the space of all differential exterior-forms of all degrees from 0 to 4: A = ϕ + a + b + c + e. In Yang-Mills theory, the scalars ϕ are considered as zero-forms, i.e. ordinary functions, and the Yang-Mills vector a μ can be identified with the components of a Lie algebra-valued Cartan connection one-form a = a μ a λ a dx μ where the λ a are the generators of the Lie algebra. The connection a defines the parallel transport on the manifold and specifies how to rotate the fields in internal space under an infinitesimal displacement in the base space by replacing the Cartan exterior differential d by the covariant exterior differential D = d + a. Since exterior-forms of even degree (ϕ, b, e) commute, and exterior-forms of odd degree (a, c) anticommute, it is natural (see Ne'eman-Thierry-Mieg [10], Quillen [11,12]) to associate the ℤ 2 grading of the exterior-forms to the ℤ 2 grading of a superalgebra (appendix A) and to try to define a superconnection as a globally odd form, that is to keep only the odd exterior-forms of degree 1 and 3, a + c = a μ a dx μ + c μνρ a dx μ dx ν dx ρ /6 λ a which are valued in the even Lie subalgebra, together with the even forms of degree 0, 2 and 4 ϕ i + b μν i dx μ dx ν /2 + e μνρσ even forms commute ϕ i a a = a a ϕ i , whereas we need (A.3) to generate the antisymmetric commutator of the even and odd matrices. The paradox is resolved by invoking the superalgebra charge chirality matrix χ (see the details in appendix A), which defines the supertrace of the superalgebra, commutes with the even matrices and anticommutes with the odd matrices: ST r(M) = T r(χ M), χ, λ a = 0, χ, λ i = 0. (2.1) Our final definition of the superconnection is The presence of the superalgebra-grading-matrix χ ensures that the signs arising in the construction of the curvature polyform F , and in the action of D on all fields, are always consistent with the brackets and structure relations of the superalgebra [13]. As a result, the curvature F defined as the square of the covariant differential F = DD is valued in the adjoint representation of the superalgebra, defines a linear map, and satisfies the Bianchi identity DF = 0, which in turn implies that the covariant differential is associative (DD)D = D(DD). This geometric construction is satisfactory, but it does not yet explain the structure of the electroweak interactions.
The new concept presented here is, firstly, to consider, in Minkowski 4-dimensional spacetime with signature (− + ++), a self-dual superconnection A =* A, where * denotes the Hodge duality which maps p-forms onto (4 − p)-forms (appendix C). In Yang-Mills theory, the connection a is a 1-form, its dual *a is a 3-form, so a Yang-Mills connection cannot be self-dual and we are only familiar with the self-dual topological theories satisfying F =* F. But because a superconnection is composed of exterior-forms of all degrees, its 1-form component a can be the dual a =* c of its 3-form component c and the concept of a self-dual superconnection makes sense. This constraint has a remarkable consequence when we consider the action of the superconnection on chiral spinors. To construct this action, we saturate the Lorentz indices of the component p-forms with Dirac γ matrices, effectively defining a map in spinor space using the Dirac-Feynman slash operator. The classic Dirac mapping a = a μ dx μ a = a μ σ μ + σ μ is generalized to antisymmetric tensors of any rank, for example b = 1 2 b μν dx μ dx ν b = 1 2 b μν σ μ σ ν + σ μ σ ν . As all our spinors are chiral, we use the γ 5 diagonal notation γ μ 1 + γ 5 /2 + γ μ 1 − γ 5 /2 σ μ + σ μ as explained in appendix B. However, the anti-symmetric product of p Pauli matrices can be rewritten as a product of 4 − p Pauli matrices contracted with the antisymmetric Levi-Civita ϵ symbol (B.7). Therefore the Dirac operator associated to a p-form ω can be rewritten as ± the Dirac operator associated to its c 1 − γ 5 2 = 1 6 c μνρ σ μ σ ν σ ρ = i 6 c μνρ ϵ μν ρσ σ σ = *c μ σ μ (2.3) Applying this transformation to the 2, 3 and 4 forms (b, c, e), the Dirac operator associated to the superconnection A acting on the left Fermions can be rewritten as A 1 − γ 5 2 = ϕ +* e 1 − γ 5 2 + a +* c μ σ μ + 1 2 b +* b μν σ μ σ ν , (2.4) whereas the Dirac operator associated to the superconnection acting on the right Fermions can be rewritten as Each parenthesized term pairs a p-form to the dual of the matching (4−p)-form. As a result, see the details in appendix C, a self-dual superconnection annihilates the right Fermions and mutatis mutandis an anti-self-dual superconnection annihilates the left Fermions To describe the electroweak interactions, we need to act both on left and on right Fermions, but with different kinds of forces. In a superalgebra framework, the charge chirality operator χ (2.1) that we have already introduced in the definition (2.2) of the superconnection provides this distinction and we postulate that our superconnection should, in addition, be superchiral A =* Aχ .  For the scalar fields, we have Φ = ϕ +* e and Φ = ϕ −* e which act as  absorb left Fermions and emit right Fermions, as illustrated below in the Feynman diagrams presented in sections 3 and 4. Our point is that the superchiral constraint allows us to derive from first principles the same interactions that had to be imposed in the previous SU(2/1) literature to force the gauge superalgebra to look like the standard model. The price we pay is the appearance of a new scalar sector represented by the BB fields.
The reader should notice that the 2-form component of the curvature polyform F (2.2) reads in these new notations (2.11) generating inside the Lagrangian F 2 a new scalar-vector-tensor interaction F ( B, Φ + Φ, B ). As shown below, this term plays a crucial role in the self consistency of the theory.
Given these algebraic and geometrical definitions, let us now study how the Dirac action of the superconnection on the chiral Fermions gets promoted in the quantum field theory into the definition of the propagators and interactions of its components, the complex scalar field Φ, the vector A, and the complex self-dual anti-self-dual antisymmetric tensor BB all correctly satisfying the spin-statistics relation.

The Avdeev-Chizhov propagator is induced by the Fermion loop
In their seminal study [14], Avdeev and Chizhov have introduced a new type of quantum field: a self-dual and anti-self-dual antisymmetric tensor B and B satisfying in Minkowski space the conditions B =* B, B = −* B, (3.1) where * denotes the Hodge dual (C.1) in 4-dimensional Minkowski space-time with signature (− + ++): and ϵ is the fully antisymmetric symbol with ϵ 0123 = 1. These fields coincide with the antisymmetric tensor fields identified in (2.10) as part of the superchiral superconnection A: compare (3.2) with (C.1), (C.5) and the definition of the Hodge dual of the field components in (C.6) and (C.8).
Until their discovery, the existence of a Lagrangian compatible with the self-duality condition seemed unlikely and its structure appeared at first complicated. Some efforts were needed to demonstrate that the Avdeev-Chizhov tensors describe a complex scalar field with one real degree of freedom for B and one for B and to delineate their possible interactions [15,16]. With hindsight, we can reconstruct the model just from the rules of quantum field theory. The possible couplings of a 2-tensor to a chiral Fermion are strongly constrained by Lorentz invariance. The μν indices must act on the Fermions via the antisymmetrized product of two Pauli matrices (see appendix B for our precise notations) and this product is by itself self-dual: σσ = P + σσ, σσ = P − σσ, (3.3) where P ± are the self-duality projectors P μνρσ ± = 1 4 g μν g ρσ − g μρ g νσ ∓ iϵ μνρσ , P + P + = P + , P − P − = P − , P + P − = P − P + = 0 . (3.4) Therefore, the only antisymmetric tensors which can couple to chiral Fermions are self or anti-self-dual. The anti-self-dual field B absorbs right states and emits left states, and the self-dual field B absorbs left states and emits right states according to the Feynman diagrams: Assuming the standard propagator for the chiral Fermions defined by the Lagrangian (3.5) the knowledge of these 2 vertices is sufficient to compute the pole part of the propagator of the BB field by closing the Fermion loop: Carefully computing this Feynman diagram (appendix E), we recover the tensorial structure of Avdeev-Chizhov propagator [14] ℒ B = − κ ij g μν ∂ α B αμ i ∂ β B βν j , (3.6) however [6], an unexpected consequence of the chiral couplings of the BB fields (2.10) is that the κ ij metric is calculated as a chiral trace: The theory hesitates between a Lie algebra like metric: Tr(λ i λ j ), and a Lie-Kac superalgebra supermetric: STr(λ i λ j ). The resolution of this dilemma depends on the number and types of chiral Fermions described by the model and is discussed below in section 6.

The Bosonic interaction terms are induced by the Fermion loops
Following our above discussion of the Avdeev-Chizhov fields, we now extend the method to determine the propagators and self interactions of the remaining components of the superchiral superconnection. We postulate the generalized Dirac Lagrangian Where D = χd + A, and A is our new superchiral superconnection (2.7). The renormalization of the wave function upon inclusion of a Fermion loop as above gives the well known propagator of the scalars and the vectors, as well as the Avdeev-Chizhov propagator [14] as derived in (3.6): where the same κij metrics controls the scalar (4.2) and tensor propagator (3.7). The vector metric κ ab = g ab = 1 2 T r λ a λ b is the only term that is purely algebraic and does not hesitate.
The interaction terms are given by the pole part of the Fermion loops with 3 external fields. The Feynman diagrams induce the expected covariant derivative minimal coupling with a caveat [6]: since the orientation of the loop is correlated with the chirality of the looping Fermions, the interaction term hidden in the definition of the covariant derivative is given by the chiral trace t aij = T r (1 + χ)λ a λ i λ j − (1 − χ)λ a λ j λ j = T r λ a λ i , λ j + ST r λ a λ i , λ j . (4.6) As found for the tensor propagators (3.7) and (4.2), the t aij interaction terms (4.6) are neither fish nor meat. They hesitate between a Lie algebra trace and a Lie-Kac supertrace. They are not universal. They depend on the Fermion content of the model.
Another novelty is the apparition of a new mixed ABΦ coupling, which must be considered as a genuine component of the superchiral minimal coupling, and is induced by the Feynman diagrams: The tensorial structure of these counterterms is unusual because the propagator (3.6) of the BB field has a rather complex structure P μναβ + k α g βγ k δ P γδρσ − / k 2 2 . (4.7) When we perform the calculation, we get with the same strength as in (4.3) the interaction: This is the only term which is Lorentz invariant and invariant under the Lie subalgebra. The coupling matrix t aij is the same mixture (4.6) of trace and supertrace which appeared above in DΦ and DB, and is common to AΦΦ, ABB and ABΦ because the Φ and the B fields have the same chiral interactions to the Fermions (2.9), (2.10). Regrouping all terms we get The interesting point is that the F coupling cannot be freely adjusted. It comes as a consequence of the D coupling of all the connection fields to the Fermion and should be considered as an indispensable part of the minimal coupling of the Avdeev Chizhov fields.
The same coupling appears in (2.11) as part of the classic Lagrangian F 2

The Adler-Bell-Jackiw vector anomaly viewed as superalgebraic
The main surprise of the previous calculations is that the theory seems to hesitate between a Lie algebra and a Lie superalgebra structure. The scalar propagator κ ij (3.7) and the vector-scalar or vector-tensor vertex t aij (4.6) contain a Lie algebra and a Lie superalgebra tensor, which cannot both be well defined at the same time. But a posteriori, this is not so surprising; this situation is actually very well known in physics. If we compute just as before the chiral Fermion loop contributions to the triple vector interaction: we also obtain two types of terms: Z f = T r λ a λ b , λ c A aμ A bν ∂ μ A ν c , d abc = ST r λ a λ b , λ c . To conclude, the triple-vector vertex (5.1) also hesitates between a Lie algebra and a Lie superalgebra structure. The Adler-Bell-Jackiw anomalous term (5.1) is superalgebraic in nature and cancels out if the supertrace of the Casimir of rank 3 (A.7) of the Lie subalgebra vanishes.
The Fermion loop counterterms to the quartic vertices A 4 , A 2 ΦΦ, A 2 BB, A 2 ΦB, A 2 BΦ also contain anomalies, but they automatically follow the structure of the cubic terms because of the Lie algebra Ward identities. For example the classic A 4 vertices are the complements of the A 3 vector terms in the classic Yang-Mills Lagrangian Tr(F 2 ). The A 2 BΦ counterterm is the complement of the ABΦ term in the (F B, Φ ) Lagrangian. The quartic potentials Φ 2 Φ 2 , ΦΦBB and B 2 B 2 remain to be studied.

Classification of the anomaly-free superchiral supeconnections
We have identified three obstructions to the construction of the quantum field theory: (3.7), (4.6) and (5.1). We wish to show here that these hesitations between trace and supertrace are resolved in many superchiral models.
Consider first the scalar anomalies. Since the trace operator is invariant under circular permutation, we can use the closure relation (A.3) of the superalgebra to rewrite the trace term in (3.7) as T r λ i λ j = T r λ j λ i = 1 2 T r λ i , λ j = 1 2 d ij a T r λ a . (6.1) In the same way, we can rewrite (4.6) as T r λ a λ i , λ j = T r λ a λ i λ j − λ a λ j λ i = T r λ i λ j λ a − λ i λ a λ j = − T r λ i λ a , λ j = − f aj k T r λ i λ k = − 1 2 d ik b f aj k T r λ b . (6.2) Hence if all the even generators satisfy the constraint T r λ a = 0, Indeed, the complete rank 3 super-Casimir tensor cancels for each family, removing any potential measure anomaly STr(λ a [λ i , λ j ]) in the ABB and ABΦ triangle diagrams.

Discussion
The concept of a superconnection defined as an odd polyform, a linear combination of exterior-forms of all degrees valued in a Lie superalgebra (2.2), was first introduced by Thierry-Mieg and Ne'eman in 1982 [10] in terms of the primitive forms (ϕ, a, b, c, e) as A = (ϕ + b + e) i λ i + (a + c) a λ a , (8.1) and then by Quillen and Mathai in their seminal papers [11,12] and recently modified in [13]. In Quillen [11,12], the covariant differential is defined as D = d + A + L, where L = L i λ i is as for us a mixed exterior-form of even degree valued in the odd module of the superalgebra. But because L must be odd relative to the differential calculus to ensure that the curvature F = DD defines a linear map, Quillen assumes that the components L i of L are valued in another graded algebra which anticommute with the exterior-forms. The difficulty is that these partially anticommuting L i cannot be represented in quantum field theory by commuting scalar fields. This is probably why the works of Ne'eman and Sternberg [17] or of the Marseille-Mainz group (see for example [18,19]) who have all adopted the Quillen formalism, stop short of the quantum theory. In our construction [13], the components ϕ i of ϕ = ϕ i λ i are just ordinary commuting functions. Nevertheless ϕ is odd with respect to our differential calculus as requested by Quillen [11], because the λ i matrices anticommute with the chirality χ (2.1) which decorates our exterior differential d = χd (2.2). As a result, the commuting ϕ i can be represented by Bose scalars and we can develop a quantum field theory formalism as here. This modifies the calculation of the superconnection cohomology [12] which should be reexamined and we conjecture that the Adler-Bell-Jackiw quantum anomalies play a role as obstructions in this purely geometrical context. However, an inconvenience of the superconnection formalism is the excessive number of component fields. In a similar way, the counterterm to the scalar and tensor interactions (4.6) involve algebraic antisymmetric structure constants f aij =Tr(λ a [λ i , λ j ]) although (ij) are odd indices. Our result is that all these unwanted terms cancel out if all the even generators are traceless (6.3). This is true in particular in the standard model of the fundamental interactions when we apply the BIM [9] mechanism whereby each lepton family is balanced by its pair of quarks (section 7). The complete rank-3 super-Casimir tensor (A.7) vanishes and only the representation independent universal couplings f abc =Tr(λ a [λ b , λ c ]) and d aij =STr(λ a {λ i , λ j }) survive. The resulting scalar-vector-tensor theory is therefore, at one-loop, superalgebraic and anomaly-free.

The new self-dual superchiral constraints
Another very interesting consequence of our superchiral structure is the induction by the Fermion loops of a new scalar-vector-tensor triple interaction (4.8) which reproduces, if and only is we apply the BIM mechanism, the structure of the square of the geometric supercurvature (2.11). Once again, Differential Geometry and Quantum Field Theory agree, conditional on the elimination of the Adler-Bell-Jackiw anomaly.
These results are tantalizing from a theoretical point of view, yet very surprising: the coupling (2.8)-(2.10) of the vectors χλ a and of the scalars λ i (1 ± χ) are outside the naive superalgebra generated by the (λ a , λ i ) matrices (appendix A); the couplings to the Fermions are all even (they transform Fermions into Fermions, not Fermions into Bosons as in Wess-Zumino supersymmetry); yet the signs induced by the helicity of the Fermion propagators restore the superalgebraic structure (4.6), if and only if the model is anomaly-free. The expected minimal coupling of the vectors to the scalars and the tensors, via the covariant derivatives, is also necessarily completed by a new scalar-vector-tensor vertex (4.8) which modifies the asymptotic behavior of the coupling of the scalars and tensors to the Fermions. A deeper understanding of these equations must be possible.
These results are also curious from a phenomenological point of view, even if the superalgebraic structure is a direct consequence of the experimentally verified BIM mechanism whereby the chiral quantum anomalies are canceled by the balances of the leptons against the quarks. However, the model is highly constrained and offers no choice. The field content is defined by the differential geometry, the dynamics are induced by the Fermion loops, and there are no free parameters, except that the Cabbibo-Kobayashi-Maskawa angles can be understood as specifying the details of the 3-generations indecomposable representations of SU(2/1) (see [18,19] and [6], appendix H).
Many problems remain. Having established the self interactions of the Boson fields (4.9), one has to examine if the theory is renormalizable and in particular if the counterterms involving Boson loops have the correct Lorentz structure, which seems likely, and the correct algebraic structure, which is non-trivial as we only have Lie-algebra Ward identities.
The scalar potential has to be evaluated. The symmetry breaking pattern of the model must be studied. Finally, the crucial open question is the eventual existence of a symmetry associated to the odd-generators of the superalgebra. and satisfy the super-Jacobi relation with 3 cyclic permuted terms: The quadratic Casimir tensor (g ab , g ij ), also called the super-Killing metric, is defined as The even part g ab of the metric is as usual symmetric, but because the odd generators anticommute (A.2) with the chirality hidden in the supertrace (A.1), its odd part g ij is antisymmetric. The structure constants can be recovered from the supertrace of a product of 3 matrices f abc = g ae f bc e = 1 2 ST r λ a λ b , λ c − , d aij = g ae d ij e = 1 2 ST r λ a λ i , λ j + .
(A. 6) The cubic Casimir tensor is defined as The Casimirs use the 'wrong' type of commutator, otherwise, using equation (A.3), they could be simplified. We have g ai = C abi = C ijk = 0 since the diagonal elements of the product of an odd number of odd matrices necessarily vanish. Using these tensors, we can construct the super-Casimir operators K 2 = g AB λ A λ B , K 3 = C ABC λ A λ B λ C , (A.8) where the upper index metric g AB is the inverse of the lower metric g AB , summation over the repeated indices is implied and ranges over even and odd values A, B = a, b, …, i, j …, and the indices of C ABC are raised using g AB . The Casimir operators K 2 and K 3 commute with all the generators of the superalgebra. In an irreducible representation, they are represented by a multiple of the identity matrix. In SU(2/1), which has rank 2, they form a basis of its enveloping superalgebra.

Appendix B Pauli matrices
Because SO(4) is isomorphic to SU(2)+SU (2) and we have P L + P R = 1, P L P R = P R P L = 0, P L P L = P L , P R P R = P R . (B. 2) The spin-one Pauli matrices σ map the right spinors on the left spinors and the σ matrices map the left spinors on the right spinors. σ = P L σ P R , σ = P R σ P L .
(C. 9) Mutatis mutandis, if A is anti-self-dual, A annihilates the left Fermions. Hence, if d + p ≠ 0, the integral of a power of k vanishes. In negative dimension d = −1, we recover the Berezin integral ∫ d −1 θ θ = I 1 .
(D. 6) In the renormalization procedure, we treat I 1 as a non standard number. In any equation, all terms linear in I 1 must be isolated and treated separately from the other terms just like we treat the imaginary part of a complex equation, or the term proportional to 5 in 5ℚ arithmetic. If in the Lagrangian, the global sum of all the counterterms proportional to I 1 cancels out, the theory is called renormalizable.
Let us now consider the wave function renormalization of a Fermion. Factorizing out the Dirac matrix and the sign conventions which play no role in the present discussion we get an integral of the form Z p μ = ∫ d 4 k k μ k 2 (k + p) 2 .
(D. 7) To evaluate this integral, we complete the square in the numerator 0 = ∫ d 4 k 1 k 2 = ∫ d 4 k (k + p) 2 k 2 (k + p) 2 = = ∫ d 4 k k 2 k 2 (k + p) 2 + 2p μ∫ d 4 k k μ k 2 (k + p) 2 + p 2 ∫ d 4 k 1 k 2 (k + p) 2 = = 0 + 2p μ Z p μ + p 2 I 1 (D. 8) Hence, we conclude that Z p μ = ∫ d 4 k k μ k 2 (k + p) 2 = − 1 2 p μ I 1 . proportional to g μν and to g ρσ in the trace of the six Pauli matrices, a generalization of (B.6), can be dropped and the terms proportional to the epsilon symbol can be dropped if they contain αβ or can be absorbed by the BB fields if they contain μν or ρσ. We are left with 6 equivalent contractions, 4 coming from the ggg trace, and 2 coming from gϵ, compensating the factor 1/6 coming from the loop integral (E.3). Finally Z μνρσ ij = − g ij P μναβ − p α p γ g βδ P γδρσ + , (E.4) our equation (4.7), which is equivalent to (3.6), as the P ± projectors (C.2) can be absorbed by the (B, B) fields.
The calculation of the Feynman diagrams leading to (4.6) and (4.8) are analogous, with traces involving up to 8 Pauli matrices and a flurry of ϵ symbols which can be all reabsorbed in the self-duality of the BB fields. The loop integral generates several terms like in (E.3), summing up to the only Lorentz invariant contraction F μν B μν which is equivalent to ϵ μνρσ F μν B ρσ since B is self-dual.