# Worldline Path-Integral Representations for Standard Model Propagators and Effective Actions

Authors: Somdatta Bhattacharya
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We develop the formalism to do worldline calculations relevant for the Standard Model. For that, we first figure out the worldline representations for the free propagators of massless chiral fermions of a single generation and gauge bosons of the Standard Model. Then we extend the formalism to the massive and dressed cases for the fermions and compute the QED vertex. We then go over fermionic one-loop effective actions and anomalies. To our knowledge, in the places where there has been an attempt at deriving the gauge boson propagator, the derivation is somewhat contrived, and we believe our derivation is more straightforward. Moreover, our incorporation of internal degrees of freedom is novel and sports some new features. The derivation of the QED vertex is also new. The treatment of the fermionic one-loop effective actions leads to a particularly economical derivation of chiral anomalies and the gauge anomaly freedom in the Standard Model, improving upon the state of the art in the literature. The appropriate worldline formalism developed thus sets the stage for Standard Model calculations beyond the tree and one-loop cases that incorporate Bern-Kosower type formulae for multiloop amplitudes, relevant for processes at the LHC.


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## Body

## 1. Introduction

That second quantization is equivalent to first quantization when formulated on singular 1D manifolds/graphs instead ofR1 has been known for quite some time, beginning with Feynman [1]. In fact this has been the main raison d’etre for the formulation of string theory in its present form, where one uses all possible super-Riemann surfaces in place of 1D manifolds. However, for field theories, it had been mainly existing as a curiosity and a testing ground for ideas to be applied to string theory, until Bern and Kosower ventured to show [2–4] that certain field theory computations were amenable to simplifications when looked upon as infinite tension/field theory limits of certain 4D heterotic string amplitudes. Strassler then proved [5] that the rules that Bern and Kosower had unearthed had a much simpler genesis in worldline formulations of the one-loop effective actions of field theory. That unleashed a whole industry of worldline computations, with generalizations to higher loops [6] and constant external backgrounds [7] among other things, culminating in a review article by Schubert [8], which contains an almost exhaustive list of references up to that point and which much of the subsequent literature also cites. A more recent and pedagogical review is [9].However most of the literature focussed on worldline path-integral representations of one-loop effective actions for spinning particles, which are formulated onS1, and their close cousins, the worldline representations for dressed propagators, that are formulated on segments of R1, did not receive as much attention, mainly because of their lack of utility in relevant computations but also because it was harder to formulate. In fact some of the latter’s avatars [10–16] existed even before the Bern-Kosower revolution, and some were formulated later on [17–23]. Particularly, [13, 19, 21–23] also handled the dressed propagators for the Abelian backgrounds. In any case it remains a relatively less researched subject to this day, compared to the effort devoted to the worldline path-integral representations of effective actions and its derivatives, like loop-level scattering amplitudes, and computations of higher loop beta-functions, whose original references are mostly found in [8].Moreover, most of the derivations for propagators were concerned only with pure spin-half particles, without any internal symmetries of any kind (with the exception of [12] which presciently anticipates some of what is done here). Also, the propagators were often derived not only from the worldline path-integrals per se, but from complicated derivatives thereof. Spin higher than 1/2 was treated in papers like [7, 24], but the results were not fully satisfactory in the sense that the connection to the path-integral was not immediate, and arrived at only after certain manipulations. We will see that the connection is more direct than what these papers would have us believe. And again, we will also see how internal symmetries can be incorporated more directly.The method for incorporating internal symmetries at the first quantized level had been well known for quite some time, starting with [25] and further elaborated upon in [26, 27]. However, their incorporation in worldline path-integrals for propagators [28, 29] and effective actions [30] did not somehow seem appealing until only recently. Moreover, these papers considered only scalar particles with internal symmetries and only included spin-half for the one-loop effective action [30].Here first of all we demonstrate how to incorporate spin as well as internal symmetries for both fermions and gauge bosons for generalSO(n) gauge groups (see [31, 32] for related work) and we specialize to the case of SO(10) for the chiral fermions of the SM, for the 16 fermions of the SM in a single generation transform as a spinorial representation of SO(10), upon including a right-handed antineutrino (see [33, 34] for related work). We derive the worldline path-integral representation of the propagators for the chiral fermions of the SM and the appropriate worldline Lagrangian includes worldline fermions pertaining to spin-half, as well as others pertaining to the SO(10) group.We also work out the path-integral representation for theSO(n) gauge boson propagator. The SU(3) gauge boson propagator can be derived from the SO(6) one, of which it is a subgroup (with certain caveats, cf. end of Section 5), and the SU(2) one can be obtained from the SO(3) one, of which it is a double cover. Or if one wishes to derive everything starting from SO(10), one can simply choose n=10. All these propagators are derived in the so-called coherent state bases, to be explained below, and it is straightforward to transform them to the index bases as we will see.The worldline actions involved for both spin-half and spin-one have local reparametrization symmetries and supersymmetries, which have to be fixed by choosing gauges, and that necessarily entails Faddeev-Popov ghosts. Here, in the initial derivations we simply gauge-fix and throw away the gauge volumes without any justifications, reserving the proper derivation for later, wherein we do a full Hamiltonian BRST analysis. In any case, as is well known and we find ourselves, the ghosts decouple trivially for the case of the open worldline.It is easy to see how to extend the analysis for the case of the graviton, but we do not do that here. Throughout, we make use of a technique outlined in [29], adapting the analysis there in terms of internal symmetry worldline fermions, to the spinning case here. We show that the propagators are related directly to the worldline path-integrals and not to any derivatives of it, albeit in the coherent state bases.After working out the free massless propagators, we concentrate on the dressed massive propagators and as an application derive the QED vertex by dressing the electron propagator by one photon. This simple result proves formidable to derive and it is gratifying that the various terms cancel subtly among each other to give the right result.Our next quest becomes derivation of fermionic one-loop effective actions and anomalies, both chiral and gauge, and we demonstrate gauge anomaly freedom in the SM in an especially economic fashion. Since the Atiyah-Singer index theorem is related to the chiral anomaly, the methods used here can be looked upon as yet another derivation of the theorem, when the background is nongravitational. These anomalies were first formulated in terms of worldline path-integrals in [35] and were put on more rigorous footing in [36] and the analysis finally culminated in a book [37]. One can compare the methods in these references with the methods here to appreciate the economy of arguments and calculational steps.One of the main aims of this paper is getting the quantum mechanical models whose wave-functions are the fields of the SM. Since the internal symmetry worldline fermions get transformed into fermionic creation and annihilation operators upon quantization (cf. Sections5 and 6) the different states in a spin-half or spin-one sector can simply be seen as excitations over ground states. Taking the cue from here, one might suppose that the other generations of the SM also have a similar genesis; namely, they too might be built upon the states of the first generation via the action of even more creation operators. We have been exploring this issue and have found that 6 more creation and annihilation operators are needed, which can be made to lead to 3 chiral and one vector-like generations upon imposition of GSO-like projection conditions. This necessarily enlarges the symmetry group from SO(10) to SO(16) and this then has ramifications for the scalar sector. The scalar sector in an SO(10) GUT is extremely rich, and there are certain advantages in making it even richer, like reducing the number of Yukawa couplings to just one with the SO(10) Yukawa couplings arising from spontaneous symmetry breaking from SO(16) to SO(10). A major problem for SO(16) is that its representations are known to be vector-like; however, that can be taken care of by the GSO-like projection operators on the spectrum of states, which enforces the condition that differently handed states that are exactly similar otherwise cannot both survive the projection.What we do here is mainly developing a formalism, to be applied to more involved processes than the ones considered here. So though there is no new physical result, we would like to stress that the formalism uncovered is novel, as is evident from the economy of arguments and calculational steps mentioned before, among other things. In particular, the thing that is primarily uncovered is the immediacy and simplicity of the coherent state formalism when interpreted in the way we have followed here.The rest of the paper is organized as follows. In Sections2 and 3, we derive the worldline representations of the massless propagators for bare spin-half and spin-one particles, respectively, which we believe are novel in themselves as they differ and improve upon the derivations present in the literature. We then do a full Hamiltonian BRST treatment for the spin-one case in Section 4. In Sections 5 and 6, we show how to incorporate internal symmetries into the mix, deriving the propagators for the SM gauge bosons and chiral fermions, respectively. In Section 7 we extend the formalism to massive dressed propagators for the spin-half cases and in Section 8, we use the formalism developed to compute the QED vertex. Then we launch into the topic of fermionic one-loop effective actions in Section 9, specializing to the case of QCD in Section 10, and derive the chiral anomaly and prove gauge anomaly freedom of the SM in Section 11. Finally we conclude in Section 12. There are three appendices that elaborate upon certain calculational steps.


## 2. Propagator for the Spin-Half Particle

A spin-half particle is in reality a1∣1-dimensional object parametrized by a bosonic parameter τ and a fermionic parameter θ, and the embedding of these coordinates in spacetime is given, instead of xμ(τ), by Xμ(τ,θ), which embeds as (1)Xμτ,θ=xμτ+θψμτwhich in other words means that the electron sees a 4∣4-dimensional super-spacetime. The fermionic coordinates ψμ are nilpotent, so the fermionic submanifold is infinitesimal and compact; however, it is fibered over spacetime and, together, they form what is known as the Clifford-bundle, once the theory is quantized and ψμ become the Dirac matrices, γμ.The Lagrangian for a single massless electron is given by [38–41] (2)L=pμx˙μ+i2ψμψ˙μ-12epμpμ-iβψμpμ.We are working in the metric conventions (+---). This Lagrangian has local reparametrization symmetry, generated by (1/2)p2, and local worldline supersymmetry, generated by ψμpμ. It is convenient to transform the “Majorana” fermions ψμ in the kinetic term to “Dirac” fermions as follows: (3)L=pμx˙μ-12epμpμ-iβψμpμ+iψ-iψ˙i,where, by the doubling trick, (4)ψ0=12ψ-1+ψ1,ψ3=12ψ-1-ψ1,ψ2=i2ψ2+ψ-2,ψ1=12ψ-2-ψ2.The Majorana indices on the LHS are the ones in (2) and those on the right are the Dirac ones in (3). Upon Dirac quantization the barred objects in (3) become creation operators while the unbarred ones become annihilation operators, satisfying the commutation relations (5)ψ-i,ψj=δij.The creation operators act on a common vacuum to produce the fermionic subspace of the physical states that are precisely the spin-1/2 states. They are of the form (6)0,ψ-^10,ψ-^20,ψ-^1ψ-^20,where 0 collectively denotes the vacua of species 1 and 2. Now, if one assigns the following indices to the above states, (7)R1⟶0,L1⟶ψ-^10,L2⟶ψ-^20,R2⟶ψ-^1ψ-^20,where R and L denote right- and left-handed, the matrix elements of 2ψ^μ with respect to the above states yield just the matrix elements of the Dirac matrices γμ in the chiral representation; namely, (8)γμ=0σμσ-μ0.The above stems from converting 2ψ^μ to the Dirac basis and treating them as creation annihilation operators. The full wave-function is a direct product of elements of the bosonic Hilbert space and the fermionic Hilbert space above and together they form the components of a Dirac spinor.Now the theory has constraints, obtained from (3) by varying with respect to the auxiliary fields e and β and they are p2=0 and ψμpμ=0. The representation of a Dirac spinor is (9)α=αL,1αL,2αR,1αR,2=αR,10+αL,1ψ-^10+αL,2ψ-^20+αR,2ψ-^1ψ-^20.As a consequence of the above facts the constraint 2ψ^μpμ acting on the above state yields the Dirac equation (10)xγμpμα=-iγμ∂μαx=0.There is an alternate basis of states, formed by the fermionic coherent states, which satisfy (11)ψ^iψ1=ψ1iψ1,where ψ1i is a Grassmann number. This means the above state is of the form (12)ψ1=eψ-^iψ1i0because of the anticommutation relations between ψ^i and ψ-^i. Bra states ψ-2 are of the form (13)ψ-2=0eψ-2iψ^i,where they satisfy (14)ψ-2ψ-2i=ψ-2ψ-^i.This in particular means (15)ψ-2∣ψ1=eψ-2iψ1i.They also satisfy the properties (16)ψ-2ψ^i=∂∂ψ-2iψ-2,ψ-^iψ1=-∂∂ψ1iψ1.Let us now see whether the path-integral of the action (3) reproduces the Dirac propagator. One starts with the following path-integral form: (17)∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫DβDeVolSymDpei∫01dtpμx˙μ-1/2epμpμ-iβψμpμ+iψ-iψ˙i+ψ-iψi1,where the gauge volumes of the reparametrization symmetry and supersymmetry of the action have been divided and we have set certain boundary conditions for x’s and ψ’s and the choice of the boundary conditions will become clearer later. e and β are like gauge fields that transform under these symmetries and one has to gauge-fix them to avoid overcounting by restricting them to single gauge slices. The integral over the gauge-orbits would then cancel the volume factor. This would necessarily entail Faddeev-Popov ghosts, and we address the issue in Section 4, albeit in the context of the spin-1 particle. As we will see there, the dependence on the ghosts becomes trivial and this is related to the fact that, for an open line, the FP determinant becomes trivial on gauge fixing e to a constant and so does it for β. A similar analysis is applicable to the spin-half case as well. For the present we simply assume that, upon gauge fixation, one can simply throw away the gauge volumes. The boundary term is necessary in order for the field equation for ψi to be given by (d/dt)ψ-i=0 without incurring extra boundary terms1. First we simultaneously fix e to 2T and β to θ. Then upon integrating out pμ one gets (18)∫dTdθ∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-ei∫01dt1/4Tx˙2+iψ-iψ˙i-iθ/2Tψ-ix˙i+ψix-˙i+ψ-iψi1,where we have used (19)ψμx˙μ=ψ-ix˙i+ψix-˙i,where xi(x-i) have the same relations to xμ as ψi(ψ-i) have to ψμ (cf. (4)).Upon Wick rotatingt, one gets (20)∫dTdθ∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-e∫01dt-1/4Tx˙2-ψ-iψ˙i+θ/2Tψ-ix˙i+ψix-˙i+ψ-iψi1.Now, let us perform the integral over θ. We get (21)∫dT∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫dt12Tψ-ix˙i+ψix-˙ie∫01dt-1/4Tx˙2-ψ-iψ˙i+ψ-iψi1.Let us then concentrate on the fermionic integral. The free fermionic path-integral becomes [29] (22)∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-e∫01dt-ψ-iψ˙i+ψ-iψi1=eψ-2iψ1i.When one evaluates the correlators in (21), this part would be omnipresent, as is usual, as the normalization factor (cf. Appendix A). Now in evaluating the fermionic correlators in (21), we first split the fields into background and fluctuations (23)ψit′=ψ1i+κit′,κi0=0,ψ-it=ψ-2i+κ-it,κ-i1=0.The kinetic action for the fluctuations ∫01dTκ-iκ˙i can be inverted to give the propagator (24)κitκ-jt′=δijθt-t′.The fermionic part can thus be written as (25)12Teψ-2iψ1i∫dtψ-2i+κ-itx˙i+x-˙iψ1i+κit′,where the one-point functions are with respect to the action for the fluctuations. Now the one-point functions for κ-’s and κ’s vanish and using (26)∫κi0=0κ-i1=0DκDκ-e∫01dt-κ-iκ˙i=1one is left with (27)∫0∞dT12Teψ-2iψ1i∫dtψ-2ix˙i+x-˙iψ1i,where now the one-point functions are of x˙’s and are with respect to the action for x (cf. (18)). Again, x’s can be separated into background and fluctuations (28)xit=xi0+xi1-xi0t+yitwith a similar expansion for x-’s, where y(0)=y(1)=0, and one has (29)x˙it=xi1-xi0+y˙itand the x action becomes (30)-∫dt14T⁡x1-x02+2x1-x0iy˙i+y˙2=-∫dt14Tx1-x02+y˙2.Again, the y˙ one-point function vanishes and upon substituting the values for x(1) and x(0), and taking into account the contribution from the free integral over y’s, we get that the x˙ one-point function is (31)x˙it=14πT2xi′-xie-x′-x2/4T.Thus (27) becomes, upon incorporating similar effects for the x- part and performing the T integral, (32)eψ-2iψ1ix-i′-x-iψ1i+xi′-xiψ-2i2π2x′-x4,where one has used (33)12π2∫0∞dT2T12T2e-x′-x2/4T=12π2∫0∞dσσe-σx′-x2upon transforming 4T→σ-1, which finally becomes (34)=12π2x′-x4.Throughout, up to Section6, we are working with x′-x2≥0, since, otherwise, integrals such as in (33) blow up as σ→∞. In Sections 7 and 8 we will be working with x′-x2≤0, as will be clarified there. Beyond Section 8, where we are working with one-loop amplitudes, such considerations do not apply. But now notice that (32) is precisely (35)ψ-2x-i′-x-iψ^i+xi′-xiψ-^i2π2x′-x4ψ1,where ψ1 and ψ-2 are the coherent states defined above. Now expression (35) is equal to (36)ψ-2xμ′-xμψ^μ2π2x′-x4ψ1because of (19). This can be written as (37)-ψ-2ψ^μ∂μ′14π2x′-x2ψ1,where the prime on the derivative denotes differentiation with respect to x′. But as we saw above, the operator ψ^μ is precisely γμ/2 when acting on the basis (7), which is a complementary basis to the overcomplete coherent state one, and, transforming to this basis, one has (38)-12iγμ∂μ′14π2x′-x2j,where i and j stand for the four components of (7). But this is precisely equal to (39)i2x′,iγ·pp2x,j=i2x′,i1γ·px,j.Summarizing, (40)i2x′,ψ-21γ·px,ψ1=∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫DβDeVolSymDpei∫01dtpμx˙μ-1/2epμpμ-iβψμpμ+iψ-iψ˙i+ψ-iψi1,where γ means the usual gamma matrix when acting on the Dirac basis and an oscillation operator when acting on the coherent state basis.Thus we see that the massless Dirac propagator can be given a path-integral representation given by the above expression. Now we claim that this entire derivation essentially boils down to the above expression being equivalent to(41)∫0∞dT∫dθX′e-TH-iθQX,where H=p2 and Q=ψ^·p and where each of these is now an operator. X and X′ stand for (42)x⊗ψ1,x′⊗ψ-2,respectively. Upon integrating over the T and θ, one gets (43)iX′QHX=iX′ψ^·pp2X=i2x′,ψ-21γ·px,ψ1,where γ has the same meaning as that above, thus vindicating our claim. We will make use of similar equivalences later. In fact (41) has been the starting point for many of the earlier treatments [11–16, 19–23] and they have proceeded in the opposite route from this to the path-integral form. Here we have shown how to arrive at the above starting from the path-integral.We would like to remark that our treatment is closest in spirit to [12, 14], and we will see how to extend the formalism to the massive and dressed case in Section 7, with its vindication in Section 8, where we derive the QED vertex.


## 3. Propagator for the Spin-One Particle

Let us now come to the spin-1 particle. It is a1∣2-dimensional object with embedding in spacetime given by Xμ(τ,θ,θ-). Each xμ has two fermionic superpartners, ψ-μ and ψμ. The expansion is (44)Xμτ,θ,θ-=xμτ+θψ-μτ+θ-ψμτ+θθ-bμτ.bμ can be eliminated by its equation of motion as it appears in the action without a kinetic term. The locally supersymmetric N=2 worldline Lagrangian is given by [42, 43] (45)L=pμx˙μ+iψ-μψ˙μ-12epμpμ-iβψ-μpμ-iβ-ψμpμ-aψ-μψμ+sa,where there are additional terms in comparison to the N=1 Lagrangian in (2), whose significance is that they project the physical state space onto a subspace of specific Nψ-^ [42, 43], whose meaning will become clearer below.a is a worldline U(1) gauge field and ψ-μψμ generates a U(1) R-symmetry. In particular, choosing s = 0 projects to the Nψ-^=2 sector, the one relevant for two-forms, the antisymmetric tensor corresponding to Fμνψ-μψ-ν in this case. This is because once the fields are quantized, ψ’s become operators, and a term like ψ-^μψ^μ becomes ambiguous with respect to its ordering. To resolve this, one has to antisymmetrize it and define it as (1/2)(ψ-^μψ^μ-ψ^μψ-^μ), so that it picks up a normal ordering constant, upon being normal ordered, of value 2. So the operator that acts on physical states Φ(x,ψ-) is of the form ψ-^μψ^μ-2=ψ-μ(∂/∂ψ-μ)-2, and it projects onto 2-forms, when s=0. The sa term is evidently a Chern-Simmons term, appropriate for a 1D action.Now it is well known that quantization of the above Lagrangian gives rise to the free Maxwell equations forFμν [42, 43]. Thus the path-integral ought to give rise to the correlator for it. Upon Euclideanisation of the action, one gets the following path-integral form: (46)∫xμ0=xμxμ1=xμ′Dx∫ψμ0=ψμ1ψ-μ1=ψ-μ2DψDψ-×∫DaDβDβ-DeVolSymDpe∫01dtipμx˙μ-ψ-μψ˙μ-1/2epμpμ-iβψ-μpμ-iβ-ψμpμ-iaψ-μψμ+ψ-μψμ1.First we simultaneously fix e to 2T, β to θ, and β- to θ- and a to a constant which ranges from 0 to 2π, since it is a U(1) gauge field and hence compact (large gauge transformations lead us to identify ϕ with ϕ+2nπ, where ϕ=∫01dta(t) [43]). This constant we continue to call a and its associated Faddeev-Popov determinant is a constant independent of a which can be taken as (2π)-1. Then as before, we first integrate out pμ. It begets (47)∫dTdθdθ-da2π∫xμ0=xμxμ1=xμ′Dx∫ψμ0=ψμ1ψ-μ1=ψ-μ2DψDψ-e∫01dt-1/4Tx˙2-ψ-μψ˙μ+θ/2Tψ-μx˙μ+θ-/2Tψμx˙μ-ia-iθ-θ/2Tψ-μψμ+ψ-μψμ1.We will now evaluate this using the methods of [29]. First of all let us perform the integrals over θ and θ-. We get (48)∫dTda2π∫xμ0=xμxμ1=xμ′Dx∫ψμ0=ψμ1ψ-μ1=ψ-μ2DψDψ-∫dtdt′14T2⁡ψ-μx˙μtψνx˙νt′-∫dt12Tψ-μψμ×e∫01dt-1/4Tx˙2-ψ-μψ˙μ-iaψ-μψμ+ψ-μψμ1.Let us concentrate on the fermionic integral. First of all it is convenient to twist the fermions in the following manner: (49)ψt⟶ψteiat,ψ-t⟶ψ-te-iat,so that the free part of the fermionic path-integral becomes [29] (see Appendix B) (50)∫ψμ0=ψμ1ψ-μ1=e-iaψ-μ2DψDψ-e∫01dt-ψ-μψ˙μ+ψ-μψμ1=ee-iaψ-μ2ψμ1+2ia.As before, when one evaluates the correlators in (48), this part would be omnipresent, as the normalization factor. The extra factor of 2ia is generated because, in particular, the values of m and n/2 in (B.2) are both 2 in this case. Now in evaluating the fermionic part in (48), again we first split the fields into background and fluctuations (51)ψμt′=ψμ1+κμt′,κ0=0,ψ-μt=e-iaψ-μ2+κ-μt,κ-μ1=0.The kinetic action for the fluctuations ∫01dTκ-μκ˙μ can be inverted to give the propagator (52)κμtκ-νt′=ημνθt-t′.The fermionic part can thus be written as (53)∫02πda2πe2ia+e-iaψ-2·ψ1∫dtdt′e-iaψ-2μ+κ-μt14T2x˙μtx˙νt′ψ1ν+κνt′-∫dt12T⁡e-iaψ-2μ+κ-μtημνψ1ν+κνt,where the correlators are with respect to the action for the fluctuations. The a integral can be done as follows: (54)∫02πda2πeika+e-iaψ-2·ψ1=∫02πda2πeika∑n=041n!e-iaψ-2·ψ1n=1k!ψ-2·ψ1k,k≥00,k<0.Thus (53) becomes, upon substituting k=2 and using θ(0)=1/2, (55)∫dtdt′14T2ψ-2·ψ1ψ-2μx˙μtx˙νt′ψ1ν+12∫dtdt′ψ-2·ψ1214T2x˙μtx˙νt′θt-t′ημν-1T∫dtψ-2·ψ12since the one-point functions of κ’s vanish. Now, we must evaluate the rest of the integrals. Let us first do the x path-integral. Again, x’s can be separated into background and fluctuations (56)xμt=xμ0+xμ1-xμ0t+yμt,where y0=y1=0 so that (57)x˙μtx˙νt′=xμ1-xμ0xν1-xν0+xμ1-xμ0y˙νt′+xν1-xν0y˙μt+y˙μty˙νt′and the x action becomes (58)∫dt14Tx1-x02+2x1-x0·y˙+y˙2=∫dt14Tx1-x02+y˙2.Thus the x˙ correlator yields, upon substituting the values for x(1) and x(0) and again taking into account the contribution from the free integral over y’s, (59)x˙μtx˙νt′=14πT2xμ′-xμxν′-xν+ημν-2T1-δt-t′e-x′-x2/4T,where we have taken the y˙ correlator from [36] and used the fact that one-point functions vanish. Thus doing the t integral in (55), one gets, upon using ∫01dtdt′(1-δ(t-t′))=0, ∫01dtdt′θ(t-t′)=1/2, and ∫01dtdt′(1-δ(t-t′))θ(t-t′)=0, (60)14πT214T2xμ′-xμxν′-xνψ-2μψ1νψ-2·ψ1+14ψ-2·ψ12x′-x2-1Tψ-2·ψ12e-x′-x2/4Twhich, upon doing the T integral, reduces to (61)=ψ-2μψ1νψ-2·ψ1∂μ′∂ν′14π2x′-x2since (62)∫0∞dT14πT21Te-x′-x2/4T=1π2x′-x4,∫0∞dT14πT214T2e-x′-x2/4T=2π2x′-x6.We will now prove that (61) can be reexpressed as (63)
∫
0
2
π
d
a
2
π
ψ
-
2
ψ
-
^
μ
ψ
^
ν
e
-
i
a
ψ
-
^
μ
ψ
^
μ
-
2
ψ
1
∂
ν
′
∂
μ
′
1
4
π
2
x
′
-
x
2
.First of all the latter is equal, due to the property of coherent states, ψ-2ψ^ν=∂/∂ψ-2νψ-2, to (64)
∫
0
2
π
d
a
2
π
ψ
-
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∂
∂
ψ
-
2
ν
ψ
-
2
e
-
i
a
ψ
-
^
μ
ψ
^
μ
-
2
ψ
1
∂
ν
′
∂
μ
′
1
4
π
2
x
′
-
x
2which, from Appendix B, is (65)
=
∫
0
2
π
d
a
2
π
ψ
-
2
μ
∂
∂
ψ
-
2
ν
e
e
-
i
a
ψ
-
2
·
ψ
1
+
2
i
a
∂
ν
′
∂
μ
′
1
4
π
2
x
′
-
x
2
(66)
=
∫
0
2
π
da
2
π
⁡
ψ
-
2
μ
∂
∂
ψ
-
2
ν
∑
n
=
0
4
1
n
!
e
-
i
a
ψ
-
2
·
ψ
1
n
e
2
i
a
∂
ν
′
∂
μ
′
1
4
π
2
x
′
-
x
2from which only n=2 will survive the a integration, leading to (67)=ψ-2μ∂∂ψ-2ν12!ψ-2·ψ12∂ν′∂μ′14π2x′-x2=ψ-2μψ1νψ-2·ψ1∂μ′∂ν′14π2x′-x2which is the same as (61).But (63) can also be written as2(68)∫02πda2πx′,ψ-2ψ-^μpμψ^νpν1p2e-iaψ-^μψ^μ-2x,ψ1(69)=x′,ψ-2ψ-^μpμψ^νpν1p2δN^,2x,ψ1.Now again, this form could have been arrived at more directly by starting out with (70)∫0∞dT∫dθ∫dθ-∫daX′e-TH-iθQ--iθ-Q-iaN^-2X.And again, as in the previous section, the usual path that one traverses is from the above expression to the path-integral form in (46). The above equation yields (71)∫0∞dT∫dθ∫dθ-X′;ρ,σe-TH1-iθQ--iθ-Q-θQ-θ-QX;μ,ν(72)=-∫0∞dTX′;ρ,σe-THQ-QX;μ,ν=-X′;ρ,σQ-QHX;μ,ν,where (73)X;μ,ν=x⊗ψ-μψ-ν0and we have made a change of basis from that of the coherent states to the Maxwell one, which because of the delta function δN^,2 arising from (70) upon integrating over a is of the form (74)ψ-^μψ-^ν0=μ,ν;that is, the number of ψ-^ operators is restricted to 2. This gels well with the 2-form (gauge boson field strength) wave-function which is (75)Fμνxψ-^μψ-^ν0.In this basis, one has, from (72), (76)-x′⊗0ψ^κψ^φψ-^ρpρψ^σpσp2x⊗ψ-^μψ-^ν0.Notice that this is the same as (69) where the delta function δN^,2 acts on the ket there to reduce it to the above form, and then since the operator in the middle contains ψ-^ρ and ψ^σ the only bras which can have meaningful inner products are of the above form. We emphasize this because later we will see that, for the case of the left-handed spin-half particle in Section 5, the kets and bras will have different operator contents, since the operator in the middle will be of the form ψ^μpμ. Now, (77)ψ-^ρpρψ^σpσx⊗ψ-^μψ-^ν0=pρpσx⊗ημσψ-^ρψ-^ν-ηνσψ-^ρψ-^μ0.Thus taking the overlap (78)-x′;κ,φψ-^ρpρψ^σpσx;μ,νone gets (79)-x′⊗0ψ^κψ^φpρpσημσψ-^ρψ-^ν-ηνσψ-^ρψ-^μ0⊗x=-x′pρpσημσηφρηκν-ηφνηκρ-ηνσηφρηκμ-ηφμηκρx=-x′pφpμηκν-pκpμηφν-pφpνηκμ+pκpνηφμxand hence the propagator that is obtained from (76) in this basis is (80)-x′pφpμηκν-pκpμηφν-pφpνηκμ+pκpνηφμp2x.But this is precisely the propagator (81)∂φAκ∂μAν-∂κAφ∂μAν-∂φAκ∂νAμ+∂κAφ∂νAμor (82)Fφκx′Fμνxif the propagator Aμx′Aνx is (83)x′ημν-1-ξpμpν/p2p2xwhich is the gauge boson correlator in an arbitrary gauge; the freedom to choose ξ follows from the symmetry of (81) under interchange of the Lorentz indices3. Thus the appropriate path-integral representation of the gauge boson (field strength) propagator in the coherent state basis is (46), which in the light of all this is the same as (69), which in turn gives rise to the above propagator, when one transforms to the Maxwell basis, via (76). The derivation here can be compared with that in [24] to see the advantages of this method.


## 4. The Hamiltonian BRST Treatment

In the above two derivations, we skirted the issue of ghosts that naturally arise when gauge-fixing a set of fields, and we will remedy the situation here by doing a full Hamiltonian BRST analysis. In particular, we will deal only with the spin-1 case and that too with a different choice of the Chern-Simmons parameters from the above section, which will lead us directly to the propagator in the Feynman-’t Hooft gauge. This is somewhat puzzling and we are yet to figure out a method that would yield (83) directly.The action that we will take is the same as before and is given by(84)∫01dTpμx˙μ+iψ-μψ˙μ-12epμpμ-iβψ-μpμ-iβ-ψμpμ-aψ-μψμ+sa,where s is an integer that takes value 1, instead of zero as in the above section.There are two bosonic gauge symmetries,pμpμ and ψ-μψμ, and two fermionic ones, ψ-μpμ and ψμpμ, and for each of them we need to define ghost fields CA = (C,C′,C-′,C′′) and ghost momenta PA = (P,P′,P-′,P′′) such that [PA,CB}=-iδAB. (C′,C-′) are bosonic ghosts and correspond to ψ-μpμ and ψμpμ, respectively, while the rest are fermionic ghosts and correspond to pμpμ and ψ-μψμ, respectively.We will now adapt the treatment of [44] to the case at hand. The treatment there deals with particles of any spin, corresponding to which there are many more ghosts Cij corresponding to the more generalized R-symmetry ψiμψμj instead of just ψ-μψμ and hence a more complicated analysis. The starting action is(85)∫dTpμx˙μ+iψiμψ˙μi-12epμpμ-iβiψiμpμ-i2aijψiμψjμ.Firstly, the quantum BRST operator can be written as a graded sum (86)Ω=∑p≥0Ωp.Then starting from (87)Ω0=CAGA=CH+CiQi+CijJij,where H=p2, Qi=ψi·p, and Jij=iψi·ψj, and imposing the nilpotency of the BRST charge, one can recursively obtain the higher operators. Then the quantum gauge-fixed Hamiltonian operator can be written as (88)Hqu=HBRST-iK,Ω,where the first term is the BRST-invariant Hamiltonian and K a gauge fixing fermion, the latter being BRST-invariant for any choice of K thanks to the nilpotency of Ω. Now since H itself enters as a constraint in (87) one can set HBRST=0 and thus have (89)Hqu=-iK,Ω.Then one can use the gauge fixing fermion (90)K=-E^APA,where E^A=(2T,0,θij) are the gauge fixations of (e,βi,aij) in the action in (85), where θij is a N×N skew diagonal matrix, dependent on N/2=n angular variables θk, with k=1,…,n.  N/2 is the spin of the particle. Note that there is no term akin to sa in (84) and it is only for N=2 that it manifests itself [42, 43]. In any case with these choices for E^A, the Hamiltonian operator becomes (91)Hqu=Tp2+12θijJij-θijCiPj-2θijCimPjmand, subsequently, the modified version of the above path-integral can be rewritten in terms of these variables.In the case at hand, sinceψ-μ and ψμ are related to ψμi in the following manner (92)ψ-μ=12ψ1μ+iψ2μ,ψμ=12ψ1μ-iψ2μone has (93)12θijJij=i2θ12ψ1·ψ2+θ21ψ2·ψ1=-iϑ2ψ1·ψ2-ψ2·ψ1=ϑψ-·ψ,where we have used the fact that, due to the skew diagonal nature of ϑ’s, θ12=-θ21=-ϑ. Also (C′,C-′) are related to Ci via (94)C′=12C1-iC2,C-′=12C1+iC2so that (95)θijCiPj=-ϑC1P2-C2P1=-i2ϑC′+C-′P′-P-′-C′-C-′P′+P-′=-iϑC-′P′-C′P-′=ϑC-′∂∂C-′-C′∂∂C′.Thus this is the difference of the bosonic ghost number operators, multiplied by ϑ (we have used ϑ instead of θ, in order to not confuse it with the gauge-fixed form of β of previous sections, and in fact it is rather a of the previous section). The last term 2θijCimPjm in (91) is equal to zero for N=2, because of the antisymmetry of the indices for each factor in the term. Hqu thus becomes(96)Hqu=Tpμpμ+ϑψ-μψμ+NC-′-NC′-sϑ,where we have incorporated the gauge-fixed form of sa that is present in our case. The path-integral then becomes (97)∫0∞dT∫02πdϑ∫DC∫DP∫Dp∫x0=xx1=x′Dx∫ψ0=ψ1ψ-1=ψ-2DψDψ-ei∫01dTpμx˙μ+iψ-μψ˙μ+C˙APA-Tpμpμ-ϑψ-μψμ+NC-′-NC′+sϑ+ψ-μψμ1+iCAPA1.Now, upon Wick rotation, where one also rotates PA→iPA and ϑ→iϑ one gets (98)∫0∞dT∫02πdϑ∫DC∫DP∫Dp∫x0=xx1=x′Dx∫ψ0=ψ1ψ-1=ψ-2DψDψ-e∫01dTipμx˙μ-ψ-μψ˙μ-C˙APA-Tpμpμ+iϑψ-μψμ+NC-′-NC′-isϑ+ψ-μψμ1+CAPA1.In general, one should also impose boundary conditions on the ghosts as well and a set of BRST-invariant boundary conditions is to set all the ghosts to zero at the boundaries [24]. Since the path-integrals for the ghosts all contribute unity (the contributions from the determinantal pieces are all powers of (T1-T0) [24], which is 1 here, and the boundary-value dependent pieces vanish in the exponent again leading to 1), doing the above integral, one again gets (Appendix B) (99)∫0∞dT∫02πdϑ2πee-iϑψ-2·ψ1-is-2ϑx′e-Tp2x,where the extra (−2) is again a normal ordering constant. The normal ordering constants from the other two number operators in (98) cancel each other out. Upon doing the ϑ integral one gets as before (100)x′,ψ-21p2δN^,1x,ψ1upon setting s=1, which upon reverting to the index basis yields (101)x′⊗0ψ^μ1p2ψ-^ν0⊗x=x′ημνp2x.Thus as noted above, one gets the propagator in the Feynman-’t Hooft gauge, and it would be nice to figure out a method that gives (83) directly.


## 5. Propagator for theSO(n) Gauge Boson

Let us now come to the case of the non-Abelian gauge boson for gauge groupSO(n). For it, on top of the ingredients necessary for spin, one has to incorporate those necessary for “color.” Thus the action (45) has to be supplemented in the following fashion: (102)L=pμx˙μ+iψ-μψ˙μ+iλ-iλ˙i-12epμpμ-iβψ-μpμ-iβ-ψμpμ-aψ-μψμ-bλ-iλi+m-n2b,where i runs over n values and m=2. We have again set s=0 in a possible sa term since we want to project to the two-form sector. Of course we could also set s to one, as in the previous section, in order to get the gauge boson propagator directly. Again, upon quantization, when λ’s become operators, the number operator λ-iλi becomes ambiguous and has to be antisymmetrized, so that upon normal ordering it picks up a factor of n/2. Thus choosing m-n/2 essentially projects to the m-form sector in the internal space. In the path-integral, the only extra ingredient would come from the λ and b integrals and the entire previous integral (47) would get supplemented by (Appendix B) (103)∫02πdb2π∫λ0=λ1λ-1=λ-2DλDλ-ei∫01dtiλ-iλ˙i-λ-ibλi+bm-n/2+λ-iλi1(104)=∫02πdb2πeimbee-ibλ-i2λi1=∫02πdb2πλ-i2e-ibλ-^iλ^i-mλi1=λ-i2δN^,2λi1for m=2. The usual route in the literature is to proceed from the second term in the above equation to the path-integral in (103) but as we have proven in Appendix B, with help from Appendix A, the first term in (104) follows directly from the path-integral. So for us Eq. (103) is the starting point, hence the inversion of the usual route. Again, since the delta function projects to states of the form λ-^aλ-^b0, so that the full wave-function is now (105)Fμνabxψ-^μψ-^νλ-^aλ-^b0,the entire expression for the gauge boson propagator gets supplemented by terms of the form, when one transforms from the coherent state basis to the gauge-index basis, (106)0λ^aλ^cλ-^bλ-^d0=-δabδcd-δadδcb.Hence the previous expression for the propagator simply gets augmented as follows: (107)-δabδcd-δadδcb∂μ∂νηρσ-∂μ∂σηρν-μ⟷ρ14π2x′-x2=Fμρacx′Fνσbdxso that the gauge boson propagator is simply (108)Aμacx′Aνbdx=-x′ημν-1-ξpμpν/p2p2xδabδcd-δadδcbwhich is the correct expression for the SO(n) gauge boson propagator.In order to get theSU(n) propagator, one can consider SO(2n) first and then restrict it to its SU(n) subgroup. Thus SU(3) can be obtained by starting with SO(6) and restricting it to its SU(3) subgroup. SU(2) on the other hand does not need this, as it is the double cover of SO(3), and there is no need to restrict it to any subgroup. One would however need to take appropriate linear combinations of SO(3)  λ’s, in order to get the SU(2) worldline fermions, with a concomitant change in the expression for the propagator. On the other hand if one wants to arrive at these gauge groups starting from some GUT gauge group, say SO(10), it can be implemented by choosing n=10. However, this is not entirely satisfactory, since when considering SU(3) one wants to have solely SU(3) gauge bosons propagating, and considering SO(6) would be superfluous. For this, one has to then take recourse to the analysis in [31], wherein how to get the first quantized theory for any arbitrary mixed symmetry tensor multiplet, and in particular the adjoint representations, has been shown.


## 6. Propagators for the Chiral Fermions of the Standard Model

The strange hypercharge quantum numbers of the fermions of the Standard Model have natural geneses if one assumes that they stem from anSU(5) or SO(10) GUT. Thus the sixteen chiral fermions of a single generation can be broken up into sets that transform as totally antisymmetric tensors of rank n of the SU(5) subgroup of the SO(10) gauge group. For the 16 of SO(10), n would take values 0, 2, and 4, corresponding to the 1, 10, and 5¯ of SU(5). We believe nature is telling us something important here and it might prove to be productive if one tries to explore the possible microscopic genesis of such a pattern. Let us first consider the fermions to be Dirac spinors. Choosing the above values of n essentially restricts the states to (109)Ψi,i1i2⋯inxλ-^i1λ-^i2⋯λ-^in0,Ψ…(x) being the position space wave-function, which for n=0 forms the 1, for n=2, forms the 10, and for n=4 forms the 5¯ of SU(5) [33, 34, 45]. The beauty of it is that all the delicate hypercharge assignments come about precisely (see end of section), upon assuming (110)Y=12λ-^4λ^4+λ-^5λ^5-13λ-^1λ^1+λ-^2λ^2+λ-^3λ^3,where λ^i, for i= 1 to 3, encode color while the rest encode flavor. In fact, this would precisely be the oscillator basis representation of a diagonal generator of the SU(5) group, another tantalizing hint that this group might have to do something with nature.The first indexi in (109) is the Dirac index and is associated with the states in (7), which we have suppressed for simplicity. The relevant action would have to be supplemented as follows: (111)L=pμx˙μ+iψ-jψ˙j+iλ-iλ˙i-12epμpμ+iβψμpμ-aλ-iλi+n-52a,where λ’s are worldline fermions for internal symmetries and i runs over 1 to 5. a is an auxiliary field that imposes the restriction that the rank of the tensor be n. The relevant path-integral would be (112)∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫λk0=λk1λ-k1=λ-k2DλDλ-∫Da×∫DβDeVolSymDpe∫01dtipμx˙μ-ψ-jψ˙j-λ-iλ˙i-1/2epμpμ-iβψμpμ-iaλ-iλi+in-5/2a+λ-iλi1+ψ-iψi1.As before, the result of the λ and a path-integrals is (113)∫da2πeinaee-iaλ-k2λk1=λ-i2δN^,nλi1which, upon reverting from the coherent state basis to that in (109) (since the delta function restricts it to states of that form only), yields (114)0λ^i1λ^i2⋯λ^inλ-^j1λ-^j2⋯λ-^jn0=δj1j2⋯jni1i2⋯inwhich in turn yields the following form of the propagator: (115)i2x′,i1γ·px,jδj1j2⋯jni1i2⋯in,where [i1i2⋯in] denotes complete antisymmetrization with respect to the indices, which is the correct expression for (116)Ψii1i2⋯inx′Ψ-j,j1j2⋯jnx.Now if one considers that the SM fermions consist only of left-handed ones, then one has to factor in a term that effectively projects to the left-handed states. From (7), it is evident that if one imposes that the number of ψ-^i operators be one, one is done. Thus the action has to be modified as follows: (117)pμx˙μ+iψ-jψ˙j+iλ-iλ˙i-12epμpμ-iβψμpμ-aλ-iλi+n-52a-bψ-jψj,where b is the auxiliary field as before that imposes the said condition. In this case the normal ordering constant is 1 since j runs over 2 values, and hence, in the absence of any Chern-Simmons-like pure b term, the projection is to the N^ψ-=1 sector. Using the same strategy as in Sections 2 and 3, it can easily be seen that, upon integrating over ψ-’s and x’s, the path-integral corresponding to the above Lagrangian would finally boil down to (sans the λ part) (118)ix′ψ-2ψ^μpμ1p2δN^,1x,ψ1and as before, transforming to the Dirac basis, one essentially gets either (119)ix′⊗0ψ^2ψ^1ψ^μpμp2x⊗ψ-^j0or (120)ix′⊗0ψ^μpμp2x⊗ψ-^j0as the surviving inner products, since the delta function essentially projects the ket to the N=1 sector, which is the sector of the left-handed states, and the operator in the middle has only a single ψ^μ which mandates that the bras are either twofold in ψ^i operator content or zero, implying that they are right-handed states (cf. (7)). These can be combined and generated in terms of left-handed states by inserting ψ^0, taking the form (121)ix′⊗0ψ^iψ^0ψ^μpμp2x⊗ψ-^j0which can easily be verified by using (7). Since this is the same as (122)i2x′,iγ0γμpμp2x,j,where both the states are left-handed, one can write this as (123)i2x′,iσμpμp2x,jwhich is the propagator for Weyl fermions. Upon augmenting with the internal symmetry part as in (115), it has the form (124)i2x′,iσμpμp2x,jδj1j2⋯jni1i2⋯in.This thus has the following path-integral representation in the coherent state basis: (125)∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫λk0=λk1λ-k1=λ-k2DλDλ-∫Da∫Db×∫DβDeVolSymDpe∫01dtipμx˙μ-ψ-jψ˙j-λ-iλ˙i-1/2epμpμ+iβψμpμ-iaλ-iλi-ibψ-iψi+in-5/2a+λ-iλi1+ψ-iψi1.which differs from (112) in the -ibψ-iψi term.For completeness we give the representations of the SM fermions in Table1, where χ1 and χ2 stand for λ4 and λ5 above.Table 1
Fermionic states of the 1st chiral generation of the SM. In our indication of theSU(5) multiplet to which each state belongs we use the notation nY where n is the dimension of the SU(5) representation in which the state transforms and Y is the (SU(5)) U(1)Y hypercharge.
0ψ^1⊗νL,1c⊂10 0ψ^1χ^1λ^1λ^2λ^3⊗νL,1⊂5--1/2 0ψ^1χ^1χ^2⊗eL,1c⊂101 0ψ^1χ^2λ^1λ^2λ^3⊗eL,1⊂5--1/2 0ψ^1ϵmijλ^iλ^j⊗um,L,1c⊂10-2/3 0ψ^1χ^1λ^m⊗um,L,1⊂101/6 0ψ^1χ^1χ^2ϵmijλ^iλ^j⊗dm,L,1c⊂5-1/3 0ψ^1χ^2λ^m⊗dm,L,1⊂101/6We have written them in terms of bras to avoid putting bars over all the creation operators. We have also made explicit only the first component of the Weyl fermions, hence the index 1 on all of them, and the second components can be obtained simply by replacingψ^1 with ψ^2. Factors of 1/2 associated with each ϵ tensor are assumed to be implicit. The states are written in terms of direct products of the bosonic and fermionic parts, with the bosonic parts forming the position space wave-functions upon conjugating them with x’s. Thus, it is evident that all these states are the result of a set of fermionic creation operators acting on the vacuum and are superpartners of each other in the sense of ordinary spacetime supersymmetry. Akin to the fields in the latter, which can be expanded in terms of the Grassmann parameters θi belonging to a superfield Ψ(xμ,θi), the fermionic multiplets here too form a supermultiplet that results from expanding Φ(xμ,ψ-j,λ-i) and retaining terms even in λ- and odd in ψ-. In a different way of looking at it, Φ(xμ,ψ-j,λ-i) can be understood to arise as follows: (126)Φxμ,ψ-j,λ-i=x,ψ-,λ-∑nΨi,i1i2⋯inxλ-^i1λ-^i2⋯λ-^inψ-^i0,where the different Ψ’s, which are the wave-functions of fermions of the SM, are seen to belong in a single supermultiplet Φ.Let us now come to the issue of the various charges of the SM. The hypercharge operator is(127)Y=-13λ-^iλ^i+12χ-^iχ^iand the isospin operator is (128)T3L=12χ-^1χ^1-χ-^2χ^2.Hence the charge operator is (129)Q=Y+T3L=-13λ-^iλ^i+χ-^1χ^1.Also, (130)B-L=1-23λ-^iλ^i.It is straightforward to check that all the charge and hypercharge assignments for the chiral generations are correct and, as we said, can be regarded as an explanation of the strange hypercharge assignments and the fractional charge assignments in the Standard Model. The way these computations are done is by noting that each of the bilinears above is actually a number operator for the different species of worldline fermions. So, for instance, (131)Y=-13∑iNλ-i+12∑iNχ-i,and hence the action on say λ-^mχ-^1ψ-^10⊗um,L,1 would give -1/3(1)+1/2(1)=1/6. In a similar spirit, the rest of them can be easily computed to be (132)B-LνeLc=1,B-LνeL=-1,B-LeL+=1,B-LeL-=-1,B-LuLαc=-13,B-LuLα=13,B-LdLαc=-13,B-LdLα=13,T3LνeLc=0,T3LνeL=12,T3LeL+=0,T3LeL-=-12,T3LuLαc=0,T3LuLα=12,T3LdLαc=0,T3LdLα=-12,YνeLc=0,YνeL=-12,YeL+=1,YeL-=-12,YuLαc=-23,YuLα=16,YdLαc=13,YdLα=16,QνeLc=0,QνeL=0,QeL+=1,QeL-=-1,QuLαc=-23,QuLα=23,QdLαc=13,QdLα=-13.


## 7. The Massive Fermion Propagator in a Gauge Field Background

Let us now extend our methods to the case of the dressed and massive fermion propagator. This is given by simply incorporating masses and backgrounds in (43): (133)i2x′,ψ-2,λ-21γ·p-gAijλ-^iλ^j-mx,ψ1,λ1,where γ has the same meaning as before, when acting on coherent states and Aμij=(AμaTa)ij. Again, this can be written as (134)i2x′,ψ-2,λ-2γ·p-gAijλ-^iλ^j+mp-gAijλ-^iλ^j2-12gσμνFμνijλ-^iλ^j-m2x,ψ1,λ1,where σμν=i/2[γμ,γν] and Fμν is the full non-Abelian gauge boson. First of all there is no need to introduce any ψ5 in the worldline representation as is usual for the case of the massive fermion, since the identity that multiplies the mass in the above expression is identity in the Dirac index basis as well and would automatically place the mass in the right place in the numerator when the transformation to that basis is done. One can then call the numerator Q and the denominator H and as before exponentiate both (cf. (41)) (135)∫dT∫dθX′,λ-2e-TH-iθQX,λ1.This can be cast in the (Wick-rotated) path-integral form (for a fermion that is an SU(n) tensor of rank p, cf. (125)) (136)∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫λk0=λk1λ-k1=λ-k2DλDλ-∫Da∫DβDeVolSymDpexp∫01dtipμx˙μ-ψ-jψ˙j-λ-iλ˙i-12eπμπμ-m2-iβ2ψμπμ+m+i2egψμFμνijλ-iλjψν-iaλ-iλi+ip-n2a+λ-iλi1+ψ-jψj1,where πμ=pμ-gAμijλ-iλj. See [13, 19, 21–23] for the pioneering treatment of the dressed propagator for the Abelian case and [12] for the non-Abelian one, the details here being somewhat akin to the latter reference, modulo the projection to the SU(n) tensor and the explicit use of the formalism made in the next section.The point might be raised that theβ term breaks supersymmetry since there is no ψ5, but as long as no use is made of supersymmetry, it isok to work without a ψ5. Also the entire approach’s vindication lies in the analysis in the next section, which gives the right form for the QED vertex, when restricted to the Abelian case and considering dressing by a single photon. Upon absorbing the factor of 2 into β and calling m/2m′ (only in the term multiplying β) and integrating out the p, one ends up with (137)∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫λk0=λk1λ-k1=λ-k2DλDλ-∫Da∫DβDeVolSymexp∫01dt-12ex˙μ-βψμ2-iβm′+12em2-ψ-jψ˙j-λ-iλ˙i+igx˙μAμijλ-iλj+i2egψμFμνijλ-iλjψν-iaλ-iλi+ip-n2a+λ-iλi1+ψ-jψj1as the worldline representation of the massive dressed fermionic propagator.


## 8. Applications to QED

Here we derive the QED vertex using the worldline formalism developed in the previous section, adapted to the case of an Abelian gauge boson. It has to be formulated, like the propagator, on the open line, since one has an incoming and an outgoing propagator, with one photon insertion. One has to take the Fourier transform of the resultant expression with respect to the propagator endpoints to arrive at the momentum space expression. Since the path-integral that we start with, (156), corresponds to the propagator in position space with photon insertions (once Aμ are converted to plane waves), the momentum space expression in general would involve external leg factors which have to be amputated in order to arrive at the expression for the vertex.Let us first work with the worldline expression for the scalar case since it would be present as a part of the spinor case. It has its genesis in the Fourier transform of the expression below(138)∫xμ0=xμ′xμ1=xμDx∫DeVolSymexp⁡∫01dt-12ex˙μ2+12em2+igx˙μAμ.The external A legs are to be represented by the plane-wave expansion of the A, leading to vertex operators. Explicitly, one has (139)igx˙μAμ=igε·x˙teik·xt.This can be evaluated by exponentiating the ε part and the resultant expression for an n-photon amplitude can be written as e∫dtJt·xt with (140)Jμt=∑i=1nikiμ+εiμ∂tiδt-ti.It is given by (upon fixing e to 2T) (141)∫0∞dT∫xμ0=xμ′xμ1=xμDxe∫01dt-1/4Tx˙2+m2T+J·x.We first separate x into background and quantum parts as follows: (142)xμt=xμ′+xμ-xμ′t+yμtso that the above expression takes the form (143)∫0∞dTem2T∫yμ0=0yμ1=0Dye∫01dt-1/4Ty˙2+x-x′2+Jt·x+x-x′t+yt.Integrating out y, one gets (144)∫0∞dTem2T4πT2e∫01dt-1/4Tx-x′2+Jt·x+x-x′t-T∫01dtdt′JμtημνΔt,t′Jνt′,In the above equation we see that since we have em2T, the integrand is divergent as T→∞. This has its genesis in implementing (cf. (134))(145)1p2-m2=∫0∞dTe-Tp2-m2and can be remedied by taking instead(146)1p2-m2=-∫0∞dTeTp2-m2which is equivalent to replacing T→-T everywhere. Then, one is then necessarily working in the p2≤0 regime, in order to make the above integral convergent. So now, one has(147)-∫0∞dTe-m2T4πT2e∫01dt1/4Tx-x′2+Jt·x+x-x′t+T∫01dtdt′JμtημνΔt,t′Jνt′.As a result, it is implicit in our assumptions thatx-x′2≤0; that is, the spacetime interval can only be spacelike or null, which is compatible with p2≤0. Thus one is dealing with the offshell propagator, where [29, 46] (148)Δt,t′=tt′+12t-t′-12t+t′.The n-photon amplitude then becomes (149)Ax′,x;ε1,k1,…,εn,kn=-∫0∞dTe-m2T4πT2exp⁡∫01dt14Tx-x′2×∏i=1n∫dtiexp∫01dt∑i=1nikiμ+εiμ∂tiδt-tixμ+x-x′μt∫01dtdt′T∑i=1nikiμ+εiμ∂tiδt-tiημνΔt,t′∑j=1nikjν+εjν∂tjδt′-tj=-∫0∞dTe-m2T+1/4Tx-x′24πT2∏i=1n∫dtiexp⁡∑i=1niki·x′+x-x′ti+εi·x-x′exp⁡-T∑i,i′=1nki·ki′Δii′-i2εi·ki′Δii′•-εi·εi′Δii′••m.l.,where Δii′=Δ(ti,ti′) and where the dot on the LHS implies derivative with respect to the first argument and that on the right implies with respect to the second one (150)Δ•t,t′=t′-θt′-t,Δ••t,t′=1-δt-t′.Now, as argued in [46], upon taking Fourier transforms (151)Ap′,p;ε1,k1,…,εn,kn=∫dDx∫dDx′eip·x+p′·x′Ax′,x;ε1,k1,…,εn,kn=∫dDx+∫dDx-eip+p′·x++i/2p-p′·x-Ax+-x-2,x++x-2;ε1,k1,…,εn,kn,with x-≔x-x′ and x+≔x+x′/2, the integral over the “center of mass” x+ yields the energy-momentum conservation delta function (2π)DδD(p+p′+∑ki), whereas the integral over the “distance” x- is Gaussian. Hence, after some simple manipulations, the amplitude reduces to (152)
A
p
′
,
p
;
ε
1
,
k
1
,
…
,
ε
n
,
k
n
=
-
i
g
2
2
π
D
δ
D
p
+
p
′
+
∑
k
i
∫
0
∞
d
T
e
T
p
2
-
m
2
∏
i
=
1
2
∫
0
1
d
t
i
×
exp
⁡
-
T
2
p
·
∑
i
=
1
2
-
k
i
t
i
+
i
ε
i
-
T
∑
i
,
j
=
1
n
k
i
·
k
j
t
i
-
t
j
2
-
t
i
+
t
j
2
-
i
ε
i
·
k
j
s
i
g
n
t
i
-
t
j
-
1
+
ε
i
·
ε
j
δ
t
i
-
t
j
m.l.
,since we are working in the spacelike/null region, p2≤0, and hence eT(p2-m2) is not divergent as T→∞.Let us now come to the case of the derivation of the vertex in spinor QED. Let us first state what we expect. Since the vertex is justγμ, when a plane-wave photon and two external electron propagators couple to it, we in general expect the momentum space expression (153)ig2πDδDp+p′+k1p-mε1p′-m(154)=-ig2πDδDp+p′+kp+mεp+k-mp2-m2p+k2-m2(155)=ig-p·ε2p+k+εp2+p·k-m2+iγσγ5ϵσμνρpμενkρ+2mε·p-imσμνεμkν×∫01dt1p2-m2+2p·kt2,where in the last line we have used Feynman reparametrization to represent the denominator, with k2=0, assuming that the photon is on-shell. Also we have used ε·k=0 in the numerator, implying that we are working in the Lorentz gauge and suppressed the momentum conserving delta function. As we can see, this expression is formidable, and we claim that the following is its worldline path-integral representation in the coherent state basis: (156)∫xμ0=xμxμ1=xμ′Dx∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-∫DβDeVolSymexp⁡∫01dt-12ex˙μ-βψμ2-iβm′+12em2-ψ-jψ˙j-igx˙μAμ+i2egψμFμνψν+ψ-jψj1lin⁡,where, as in the previous section, m′=m/2. The overall subscript lin implies that only the linear order term in the expansion of the exponent of the interaction term -igx˙μAμ+i/2egψμFμνψν is to be retained, as the dressing is only by a single photon. It has all the right ingredients for reproducing the above expression, as we will see, and it is essentially (137), adapted to the case of the Abelian background. The new vertex operators are (upon fixing e to -2T) (157)igx˙μAμ+i2egψμFμνψν=igε·x˙t-2iTε·ψtk·ψteik·xt.The relevant amplitude involving just one photon is (upon fixing β to θ and integrating over it) (158)-∫0∞dTe-m2T∫01dt1dt2-12Tψt1·x˙t1-im′igε·x˙t2-2iTε·ψt2k·ψt2eik·xt2. Let us first work out the x parts.This would be given by(159)-ig∫0∞dTe-m2T∫dθ∫01dt2e∫dtJt·xt-θ/2T∫dt1ψt1·x˙t1lin.with (160)Jμt=ikμ+εμ∂t2δt-t2.The general amplitude involving multiple photons and the susy charge operator ψ(t1)·x˙(t1), upon Fourier transforming to momentum space, is (161)
A
p
′
,
p
;
ε
1
,
k
1
,
…
,
ε
n
,
k
n
=
-
i
g
2
2
π
D
δ
D
p
+
p
′
+
∑
k
i
∫
0
∞
d
T
e
T
p
2
-
m
2
∏
i
=
1
n
∫
0
1
d
t
i
×
exp
⁡
-
T
2
p
·
∑
i
=
1
n
-
k
i
t
i
+
i
ε
i
-
T
∑
i
,
j
=
1
n
k
i
·
k
j
t
i
-
t
j
2
-
t
i
+
t
j
2
-
i
ε
i
·
k
j
s
i
g
n
t
i
-
t
j
-
1
+
ε
i
·
ε
j
δ
t
i
-
t
j
m.l.
×
i
∫
0
1
d
t
1
ψ
t
1
·
p
+
i
∑
i
∫
0
t
i
d
t
1
ψ
μ
t
1
k
i
μ
+
∑
i
ψ
μ
t
i
ε
i
μ
⋯
,where we have done the θ integral and where the ellipses denote other ψ insertions that might be present. Here we have simply adapted the analysis above for scalars to include the effects of the additional source in (159). The expectation value in the last line is with respect to the ψ path-integral. So at this point we need to evaluate the ψ correlators. As is evident from (158), we would need 1, 2, and 3 point ψ correlators. The point made is that we now need the correlators for the Majorana fermions ψμ, which we have to derive from the Dirac form of the action, as it is the only form amenable for the coherent state formalism. The relevant path-integral is given by (162)∫ψi0=ψ1iψ-i1=ψ-2iDψDψ-e∫01dt-ψ-iψ˙i+χ-iψi+χiψ-i+ψ-iψi1.Again, ψ’s can be split into background and quantum parts as follows: (163)ψ-i=ψ-2i+ψ-qi,ψi=ψ1i+ψqiand the resultant path-integral looks like (164)eψ-2iψ1i∫ψqi0=0ψ-qi1=0DψDψ-e∫01dt-ψ-qiψ˙qi+χ-iψ1i+ψqi+ψ-2i+ψ-qiχi,where the prefactor is the usual normalization factor. Upon integration, the above becomes (cf. Appendix A (see (A.18))) (165)=eψ-2iψ1ie-∫01dtdt′χ-itθt-t′χit′-∫01dtχ-itψ1i+ψ-2iχitand this can be written in the Majorana basis as (166)eψ-2iψ1iexp⁡∫01dtχt·ψb-14∫01dtdt′χμtημνθt′-t-θt-t′+Kμνχνt′.Here χμ are precisely the sources for the Majorana fermions ψμ, where χμ is related to χ-i and χi as(167)χ0=12χ-1-χ1,χ3=12χ-1+χ1,χ2=i2χ-2-χ2,χ1=12χ-2+χ2and ψbμ is given by (168)ψb0=12ψ-21+ψ11,ψb3=12ψ-21-ψ11,ψb2=i2ψ12+ψ-22,ψb1=12ψ-22-ψ12.Also, (169)K=1-ii-1.Let us first tackle 3-point correlator. It is given by (170)ψαt1ψβt2ψγt2=∂∂χαt1∂∂χβt2∂∂χγt2eψ-2iψ1iexp⁡∫01dtχt·ψb+14∫01dtdt′χμtημνsignt-t′-Kμνχνt′χ=0=eψ-2iψ1i12Kβγψbα+12ηαγsignt1-t2-Kαγψbβ-12ψbγηαβsignt1-t2-Kαβ-ψbγψbβψbα.It can be easily checked that, upon transforming to the Dirac basis, it assumes the form (171)122ηαγsignt1-t2γβ-γγηαβsignt1-t2-γγγβγαwith γ,β,α all different in the last term, in which case it can be written as (172)122ηαγsignt1-t2γβ-γγηαβsignt1-t2+iϵσγβαγσγ5.In order to derive this, one has to use (168), (11) and (14), and the following:(173)ψ^0=12ψ-^1+ψ^1,ψ^3=12ψ-^1-ψ^1,ψ^2=i2ψ^2+ψ-^2,ψ^1=12ψ-^2-ψ^2.We also have to use the facts that, for instance, (174)ψ-2ψ^iψ-^jψ1=ψ-2δij-ψ-^jψ^iψ1=eψ-2iψ1iδij-ψ-2jψ1i=eψ-2iψ1iδij+ψ1iψ-2jand hence when three different ψ^μ’s are involved, (175)ψ-2ψ^γψ^βψ^αψ1=eψ-2iψ1i-12Kβγψbα+12Kαγψbβ-12ψbγKαβ+ψbγψbβψbαwith the expression vanishing if any of ψ^’s are the same, thus automatically antisymmetrizing the expressions, leading to 1/22γγγβγα in the Dirac basis, since ψ^μ in the Dirac basis becomes 1/2γμ. Next, let us come to the 2-point correlator. It is given by (176)
ψ
β
t
2
ψ
γ
t
2
=
∂
∂
χ
β
t
2
∂
∂
χ
γ
t
2
e
ψ
-
2
i
ψ
1
i
exp
⁡
∫
0
1
d
t
χ
t
·
ψ
b
+
1
4
∫
0
1
d
t
d
t
′
χ
μ
t
η
μ
ν
s
i
g
n
t
-
t
′
-
K
μ
ν
χ
ν
t
′
χ
=
0
(177)
=
e
ψ
-
2
i
ψ
1
i
1
2
K
β
γ
-
ψ
b
γ
ψ
b
β
≃
i
2
σ
γ
β
.and the one-point function is simply (178)ψμt≃12γμ.So we are now ready to tackle (161) for one photon, which corresponds to the following part of (158): (179)∫0∞dT∫01dt1dt212Tψt1·x˙t1igε·x˙t2eik·xt2and it would be (180)-ig2πDδDp+p′+k∫0∞dTeTp2-m2∫01dt2×exp⁡2Tp·kt2-i2Tp·εi∫01dt1ψt1·p+i∫0t2dt1ψμt1kμ+ψμt2εμlin.,where lin means linear in εμ. Using the one-point function above, (178), we would get (upon extracting the part linear in ε) (181)Ap′,p;ε,k=-12ig2πDδDp+p′+k∫0∞dTeTp2-m2∫01dt2×exp⁡2Tp·kt22Tp·εp+kt2+ε,(182)=ig2∫01dt21p2-m2+2p·kt22-p·εp+kt2+12εp2-m2+2p·kt2,where we have suppressed the delta function. Let us then come to the first mass term in (158), namely, (183)-gm′∫0∞dT∫01dt1dt2ε·x˙t2eik·xt2.Using the master formula (161), we have (184)
-
g
m
′
∫
0
∞
d
T
e
T
p
2
-
m
2
∫
0
1
d
t
2
exp
⁡
2
T
p
·
k
t
2
+
i
2
T
p
·
ε
lin.
e
ψ
-
2
i
ψ
1
i
(185)
=
i
g
m
2
∫
0
1
d
t
2
1
p
2
-
m
2
+
2
p
·
k
t
2
2
p
·
ε
e
ψ
-
2
i
ψ
1
i
,where eψ-2iψ1i is the ubiquitous part coming from the fermion path-integral, which means that when one transforms to the Dirac basis, it would yield identity for this term. Next let us consider the following term in (158): (186)-ig∫0∞dT∫01dt2dt112Tψt1·x˙t12iTε·ψt2k·ψt2eik·xt2.Again, using the master formula (161) for the x-part, we get (187)-2g∫0∞dTeTp2-m2T∫01dt2exp⁡2Tp·kt2i∫dt1ψt1·p+i∫0t2dt1ψμt1kμε·ψt2k·ψt2.Now, upon using (172) and (188)∫01dt1dt2signt1-t2ft2=∫01dt22t2-1ft2we get (189)-2ig∫01dt21p2-m2+2p·kt22122εp·k2t2-1-p·εk2t2-1+iϵσγβαγσγ5pαεβkγ,where we have used k·ε=k2=ϵσγβαγσγ5kαεβkγ=0. This when combined with (182) yields (190)
i
g
1
2
∫
0
1
d
t
2
1
p
2
-
m
2
+
2
p
·
k
t
2
2
-
p
·
ε
2
p
+
k
+
ε
p
2
-
m
2
+
p
·
k
-
i
ϵ
σ
γ
β
α
γ
σ
γ
5
p
α
ε
β
k
γ
.Now we come to the remaining term in (158): (191)gm′∫0∞dT∫01dt2dt12iTε·ψt2k·ψt2eik·xt2which upon using the master formula for the x-part and using (177) yields (192)-gm′∫0∞dTeTp2-m2T∫01dt2exp2Tp·kt2σγβεβkγ(193)=ig2m∫01dt21p2-m2+2p·kt22i2σγβεβkγ.Thus collecting terms from (185), (190), and (193), we finally end up with (194)ig12∫01dt21p2-m2+2p·kt22imσγβεβkγ-p·ε2p+k+εp2-m2+p·k-iϵσγβαγσγ5pαεβkγ+2mp·εwhich is precisely the same as (155) up to a factor of 1/2. Now that we essentially have (153), we can amputate the external legs and get εμγμ from which the vertex can be obtained by taking derivative with respect to εμ. We have thus obtained the QED Feynman rule for the vertex starting from the worldline formalism.The obvious next step would be computing the Compton scattering amplitude, which is a repeat of the above calculation with two photon vertex operators instead of one in (158). But since the single photon case is formidable enough, we are currently investigating methods that can vastly simplify the calculations, allowing us to tackle even more involved cases like the Lamb-shift calculation. One can then also venture into the applications to the non-Abelian case.


## 9. One-Loop Effective Actions

Let us now consider the analogous representations for one-loop effective actions and anomalies. The usual one-loop effective action for a fermionic field in a gauge field background is given by(195)12ln⁡Det-γ·D2=12ln⁡Det-D2-12gσμνFμνaTa=12Trln⁡-D2-12gσμνFμνaTa,where Dμ=∂μ+igAμaTa and can be expressed as (196)=-12Tr∫0∞dTTexp⁡-T-D2-12gσμνFμνaTawhich has the following path-integral representation [5]: (197)-12∫dTT∫PDx∫APDψDψ-e∫01dt-1/4Tx˙2-ψ-jψ˙jTrPei∫dtgx˙μAμaTa+TgψμFμνaTaψν,where the trace is over a super-Wilson loop that is the worldline supersymmetric generalization of the normal Wilson loop. P denotes periodic, while AP denotes antiperiodic.Now it can be shown that the effective action of a Dirac fermion that transforms as anSU(5) tensor of rank n would be given by the following path-integral (see [29–34] for related approaches): (198)∫PDx∫APDψDψ-∫APDλDλ-∫Da×∫DeVolSymDpe∫01dtipμx˙μ-ψ-jψ˙j-λ-iλ˙i-1/2eπμπμ+i/2egψμFμνijλ-iλjψν-iaλ-iλi+in-5/2a,where πμ=pμ-gAμijλ-iλj with Aμij=(AμaTa)ij and Fμνij is the full self-interacting field strength with Ta belonging to the Standard Model gauge group. λ’s are 5 in number, as in the above section, in order to accommodate the full SM gauge group. We have dropped the prefactor of -1/2, which is understood to be implicit in what follows. β, the auxiliary field multiplying the susy generator, has been fixed to zero, since it is antiperiodic and an antiperiodic field has no zero modes. Integrating out pμ at first yields(199)∫dTT∫PDx∫APDψDψ-∫APDλDλ-∫Da×e∫01dt-1/4Tx˙2+igx˙μAμijλ-iλj+iTgψμFμνijλ-iλjψν-ψ-jψ˙j-λ-iλ˙i-iaλ-iλi+in-5/2a,where we have fixed e to 2T and for a circle the measure for the modular integral is 1/T, which then tallies with (197).Let us first show that, for a fermion transforming in and only in the fundamental representation of the gauge group (n=1), the integral over λ and λ- gives rise precisely to the trace of the super-Wilson loop in (197). We will compute the λ and λ- integrals for open intervals first and then identify the endpoints with a change in sign. There is a subtlety involved here in that, for instance, the integral over T for a closed loop for the bosonic part involves a measure factor of 1/T that would be missing if one arrived at the closed loop by simply sewing open lines which do not have such a term, and so one must watch out for such factors here as well. However, in the fermionic case at hand the measures involved are trivial for both closed and open loops and there are no problems4.Let us call-(gx˙μAμij+TgψμFμνijψν), Mij(t). Let us again twist λ’s as in (49). Then the open ended path-integral becomes (200)∫λi0=λi1λ-i1=λ-i2e-iaDλDλ-e-∫dtλ-itd/dt+iMijtλjt,where the understanding is that, at the end, we will identify λi(0)=-λi(1) and integrate over it. Since this is quadratic in λ, it can be evaluated. We first make a transformation Mij(t)→Uik-1(t)Mkl(t)Ulj(t)=Mij′(t) and λj(t)→Ujm-1(t)λm(t)=λj′(t), and λ-i(t)→λ-n(t)Uni(t)=λ-i′(t). Since M’s are traceless and Hermitian, U’s are to be chosen to be special unitary, to maintain these properties. Then the Lagrangian transforms as (201)λ-itddt+iMijtλjt=λ-n′tddt+Uni-1tddtUimt+iUnk-1tMklUlmtλm′t.We choose U(t) such that (202)Uni-1tddtUimt+iUnk-1tMklUlmt=iMn′tδnm,where ∑nMn′=0, owing to the tracelessness property of M and the specialness of U’s. Thus we have (203)ddtUU-1+iM=iUM′U-1or (204)iM=UddtU-1+iUM′U-1.Then we have (205)∫λi′0=λi1′λ′-i1=λ-i2′e-iaDλ′Dλ′-Det-1UtU-1te-∫dtλ′-itd/dt+iMi′tλi′t.The Jacobian determinant is 1. The path-integral becomes (Appendix A) (cf. (A.26)) (206)Tei/2∫dt∑iMi′t∏ieλ-i2′e-iaTe-i∫dtMi′tλi1′-λi2′-λi2′,where T implies time-ordering. Now this is equivalent to (Appendix C) (207)ee-iaλ-m2Te-i∫dtMmpλp1-λ-m2λm2.Finally, let us identify the endpoint λ’s after incorporating a minus sign, so that the above expression gets converted to, upon reinstating the integrals over the boundary values, (208)∫dλ-dλda2πeiae-e-iaλ-mTe-i∫dtMmpλp+λ-mλm,where one has fixed a to a constant modulus that one continues to call a and used n=1. Then doing the a integral begets (since only the e-iaλ-mTe-i∫dtMmpλp term would survive the integration with all higher and lower powers dropping out) (209)-∫dλ-dλλ-mTe-i∫dtMmpλpe-λ-mλmwhich is nothing but(210)TrPe-i∫dtM,where time-ordering has been replaced by path-ordering. Hence, we have proved that, for fundamental fermions, the integral over λ’s gives rise to the super-Wilson loop: (211)TrPei∫dtgx˙μAμij+TgψμFμνijψνthus establishing the connection between (198) and (197) for n=1 that we set out to prove, and this has been considered as a starting point for introducing the λ fermions in some places.However, as is evident, this will no longer work for fermions of higher rank representations requiring modifications such as those reported in [31]. But there is an economical way of considering the one-loop effective actions for SM fermions. And that is to treat them holistically, namely, considering the loop-running of all the fermions together. The way to consider this is by imposing the condition (212)-1λ-iλi=1on the spectrum of states. Notice that this condition automatically selects out the representations with even Nλ, namely, the 16 in Section 6. Thus imposing this condition essentially allows all the fermions to run in the loops, except that the right-handed antineutrino just goes along for a free ride as it is a singlet of the SM and hence does not couple to the gauge bosons. Now, inserting 1/2(1+(-1)λ-iλi) in a trace over the full state spectrum projects out the states satisfying (-1)λ-iλi=-1, leaving the states satisfying the above condition only. In the path-integral representation of the trace, this insertion is equivalent to the taking of a sum of the unrestricted effective actions with antiperiodic and periodic boundary conditions for λ’s [8]. The expression to be considered is thus (213)14∫dTT∫PDx∫AP-∫PDψDψ-×∫AP+∫PDλDλ-e∫01dt-1/4Tx˙2+igx˙μAμijλ-iλj+iTgψμFμνijλ-iλjψν-ψ-jψ˙j-λ-iλ˙i,where we have in addition incorporated the fact that the fermions are left-handed leading one to insert 1/2(1-γ5) in the trace over states, which projects out the right-handed states. This is equivalent to the taking of a difference of terms with antiperiodic and periodic boundary conditions for ψ’s in the path-integral, since γ5=(-1)ψ-iψi, as can be easily seen by the action of this operator on the states in (7). γ5 in this basis turns out to be (214)γ5=-1001.Notice that there are no additional terms that impose restrictions to specific sectors, as in Section 6. We will again resort to evaluating the integral for an open line and identifying the endpoints at the end, with and without minus signs built in. Thus, for an open line, with λ-m(1)=λ-m2 and λm(0)=λm1, one has (215)eλ-m2Pe-i∫dtMmpλp1-λ-m2λm2,where M has the same meaning as before, and hence adding the boundary contribution and identifying the endpoints as λm(1)=-λm(0) for antiperiodic boundary conditions and λm(1)=λm(0) for periodic, one has (216)∫dλ-dλe-λ-mPe-i∫dtMmpλp+λ-mλm+∫dλ-dλe--λ-mPe-i∫dtMmpλp+λ-mλm=det⁡Pei∫dtgx˙μAμaTa+TgψμFμνaTaψν+1+det⁡1-Pei∫dtgx˙μAμaTa+TgψμFμνaTaψν.This expression is valid for the full SU(3)c×SU(2)W×U(1)Y background of the Standard Model. In particular, for SU(3)c the electrons and neutrinos go for free rides in the loops, as they are color singlets.


## 10. QCD

Let us concentrate on QCD. For it, one can consider thatλi’s range over 3 values instead of five, and one has the representations (217)λ-i1⋯λ-ip0,where p ranges from 0 to 3, with 0 and 3 corresponding to color singlets and the other 2 values corresponding to the fundamental and bifundamental/antifundamental quarks. Now, in general there are other λ-’s also present that correspond to flavor and we are suppressing those as they do not couple to the color gauge bosons (cf. Table 1). The bifundamentals have the fermionic operator content λ-iλ-j0 and hence correspond to (3⊗3)as=3¯. Since one is considering all the representations, there is no need for a projection operator, and hence the fermionic boundary conditions for λ’s are only antiperiodic. However the fermions are still left-handed, so the other projection operator applies. QCD does not distinguish between left-handed and right-handed, but on the other hand considering a left-handed antiquark is equivalent to considering a right-handed quark, which is why one can work with only left-handed fermions, even in the case of QCD. Thus the effective path-integral over λ’s is just (218)∫dλ-dλe-λ-mPe-i∫dtMmpλp+λ-mλm.This can be easily evaluated by expanding the first term and realizing that the expansion terminates after the third order term. Thus one has [33] (219)∫dλ-dλ1-λ-mPe-i∫dtMmpλp+12λ-mPe-i∫dtMmpλpλ-m′Pe-i∫dtMm′p′λp′-16λ-mPe-i∫dtMmpλpλ-m′Pe-i∫dtMm′p′λp′λ-m′′Pe-i∫dtMm′′p′′λp′′e-λ-mλm.Let us call Pe-i∫dtMmp, Bmp. Let us first consider the third term. For it, the contribution from the exponential is just 1 (since the terms apart from the exponent completely saturate the measure as for SU(3) there are a total of 3λ’s and 3λ-’s, and hence only the first term in the expansion of the exponent, that is, 1, would contribute in this case) and hence, it is given by (220)-16ϵmm′m′′ϵpp′p′′BmpBm′p′Bm′′p′′=-det⁡B.Then the above expression becomes, upon using, (221)∫dλ-dλ⋯λiλ-j⋯e-λ-mλm=⋯δij⋯and the expression (220) can be written5(222)1+TrB+12TrB2-12TrB2-det⁡B.Note that the naive assumption of quarks transforming in the fundamental representation only would have led to just the second term, and the actual expression is richer. Now since M are traceless and Hermitian, B belongs to the special unitary group and hence det⁡B=1, and hence the full expression just reduces to (223)TrB+12TrB2-12TrB2.To check whether we have the right expression it might be instructive to work out the trace terms for fundamentals and bifundamentals separately. For fundamentals, one already knows the answer; it is given byTrB (cf. (210)). For bifundamentals, the path-integral would be (224)12∫dTT∫PDx∫AP-∫PDψDψ-∫APDλDλ-∫Da×e∫01dt-1/4Tx˙2+igx˙μAμijλ-iλj+iTgψμFμνijλ-iλjψν-ψ-jψ˙j-λ-iλ˙i-iaλ-iλi+i2-3/2a.Again, doing the λ path-integral, one has (225)∫dλ-dλda2πe2iae-e-iaλ-mTe-i∫dtMmpλp+λ-mλm=12!∫dλ-dλλ-mTe-i∫dtMmpλpλ-m′Te-i∫dtMm′p′λp′e-λ-mλm=12TrB2-12TrB2and hence we have essentially reproduced (223).The contributions from the singlets are trivial and cancel each other out, sincedet⁡B=1. The full effective action is thus (226)12∫dTT∫PDx∫AP-∫PDψDψ-e∫01dt-1/4Tx˙2-ψ-jψ˙jTrPei∫dtgx˙μAμaTa+TgψμFμνaTaψν+12TrPei∫dtgx˙μAμaTa+TgψμFμνaTaψν2-12TrPei∫dtgx˙μAμaTa+TgψμFμνaTaψν2which can also be written, because of (218), as (227)
1
2
∫
d
T
T
∫
P
D
x
∫
A
P
-
∫
P
D
ψ
D
ψ
-
e
∫
0
1
d
t
-
1
/
4
T
x
˙
2
-
ψ
-
j
ψ
˙
j
det
⁡
P
e
i
∫
d
t
g
x
˙
μ
A
μ
a
T
a
+
T
g
ψ
μ
F
μ
ν
a
T
a
ψ
ν
+
1
,where Ta corresponds to the SU(3) group.


## 11. Anomalies and Anomaly Cancellations in the Standard Model

The chiral anomaly is well known to be given byTrγ5, the trace being inclusive of the position and other internal coordinates as well. Since the trace over the position coordinates formally gives rise to infinity while that over γ5 gives rise to zero, this object needs to be regularized. It is well known [47] that when the regularized trace is performed, one gets(228)Trγ5=-132π2g2∫d4xϵμνρσTrFμνFρσ.Another avatar of the same thing is the anomaly in ∂μJμ5, with Jμ5 being given by Ψ-iγμγ5Ψi, where i is the internal index of the gauge group in question, and it is well known that the anomaly is given by (229)∂μJμ5=-116π2g2∫d4xϵμνρσTrFμνFρσ,where Fμν is FaμνTa and Ta’s are the gauge group generators. Here the fermions transform in the fundamental of the gauge group. Trγ5 is also well known as the Atiyah-Singer index (for the nongravitational case considered here).On the other hand when considering the chiral non-Abelian or gauge anomaly in the SM, instead of computing piece by piece, one might try to look for a way of summing over the entire SM spectrum of left-handed states (i.e., the16), and indeed one can do that, as we saw in the preceding sections, by simply considering [35] (230)iηa14Trγ51-γ5λ-^iTaijλ^j1+-1λ-^iλ^i=iηa14TrCSγ51-γ5λ-^iTaijλ^j1+-1λ-^iλ^i,where the usual projection operators (1-γ5) and (1+(-1)λ-^iλ^i) have been introduced that project to a single generation SM spectrum. CS denotes coherent states. Now to start with, on the LHS the “internal” trace is over states of the form 0λ^iλ^j⋯ψ^i in the spin and internal space (cf. Section 6) and the various charges for the various generators Ta arise out of the action on these states by λ-^iTijaλ^j for SU(3)×SU(2)×U(1). So when, on the RHS, we pass over to the trace over the coherent states λ,ψ, it is λ-^iTijaλ^j that we should continue considering as the generators. Since we are summing over the entire SM spectrum, this must turn out to be identically zero if the SM is to be nonanomalous. In [35], the expression for the gauge anomaly was taken to be computed piece by piece by considering for a fermion transforming as an n(231)iηaTrγ5Λ-^iT~aijΛ^j,T~’s being the n-dimensional representations of the generators of the SM, with the understanding that one must take the trace over states of the form Λ-^i0 only (with i running over 1 to n), namely, the n-dimensional fundamental representation. Then the total anomaly was supposed to be obtained essentially by summing over the contributions for the different n’s that form the states of the SM (cf. Section 6). But here we demonstrate that there is no need to adopt such a piecemeal approach and incorporate the different representations of the SM gauge group at once and sum over the entire spectrum, by making use of the GSO-like projectors above, and the result turns out to be zero. In doing this, we are of course committing to the fact that the representations of the SM are given by (109), and this gives further vindication that that is correct.As is well known [47] the above expressions can be regularized by introducing regulators in the following manner (ignoring the prefactor of 1/4 in the second expression for the time being): (232)Trγ5⟶limT→0Trγ5e-T-∂+igA2-1/2gσμνFμν,Trγ51-γ5λ-^iTaijλ^j1+-1λ-^iλ^i⟶limT→0TrCSλ-^iTaijλ^j-1ψ-^iψ^i-11+-1λ-^iλ^ie-T-∂+igA2-i/2gσμνFμν,where in the first expression the gauge group is SU(N) for some N, while, in the second, it is the full SM gauge group. The trace in x’s means one has to again take path-integrals with periodic boundary conditions in them. Analogous to the previous expression (213), the path-integral representations of the above expressions would be(233)limT→0∫PDx∫Da∫PDψDψ-∫APDλDλ-e∫01dt-1/4Tx˙2+igx˙μAμijλ-iλj+iTgψμFμνijλ-iλjψν-ψ-jψ˙j-λ-iλ˙i-iaλ-iλi+i1-n/2a,(234)limT→0∫PDx∫AP-∫PDψDψ-∫AP+∫PDλDλ-×λ-iTaijλje∫01dt-1/4Tx˙2+igx˙μAμijλ-iλj+iTgψμFμνijλ-iλjψν-ψ-jψ˙j-λ-iλ˙i,respectively.The difference between the above two expressions is to be noted. In the first expression, one is considering Dirac fermionsΨi in the fundamental representation of the gauge group and hence the extra a-dependent terms in it (as per the above sections). In the second expression, one is summing over even-dimensional SU(5) representations, which together form the 16 of SO(10), and that is taken care of by the sum over periodic and antiperiodic boundary conditions, and there is no necessity of introducing a dependent terms.Now the above expressions can be evaluated by taking advantage of the fact thatT→0 and scaling some of the variables with respect to T in a manner that makes most of the 0 + 1-dimensional field theory amplitudes redundant. The details are as in [36, 37]. In particular, one scales ψ’s by setting ψ=ψ′/2T and then dropping the prime but leaves λ’s and x’s invariant.In this case, thexμ(t) field theory can be made redundant by expanding it as xμ+qμ(t), where xμ is the “background” and qμ(t) is the quantum fluctuation. qμ(t) Green’s functions are proportional to T whereas the vertices are T independent after the scalings and hence all the q-dependent contributions coming from q˙(t)·Aμij(x+q(t))λ-iλj and ψμFμνij(x+q(t))λ-iλjψν to the path-integral would simply drop out when the limit T→0 is taken. All that the xμ(t) part then contributes to is the measure ∫d4x coming from the integral over the background xμ, the 1/2ψ·Fij(x)·ψλ-iλj factor, and the free path-integral over qμ(t) which contributes a factor (4πT)-2, the (2T)-2 part of which is absorbed by the ψ scalings.Thus the relevant expressions boil down to, ignoring the factor of1/(2π)2, and gauge-fixing a, (235)limT→0∫d4x∫da2π∫APDλDλ-∫PDψDψ-e-∫01dt1/2Tψ-jψ˙j+λ-iλ˙i-igλ-l1/2ψ·Flmx·ψλm-iaλ-iλi+i1-n/2aand the corresponding expression for (234). Now the ψ correlators also become proportional to T upon scaling and the vertices become T independent, thereby making its field theory redundant as well, and in what follows we will make heavy use of that. First of all the two different boundary conditions in (234) present two separate ways of going about the problem. In the case of periodic boundary conditions for ψi’s one can separate them into background and quantum parts, as periodic boundary conditions allow for zero modes, so that even if the quantum part is rendered redundant by the scalings, there is a finite nontrivial contribution coming from the zero modes or the background part. This is essentially because in the measure there are integrals over the zero modes and these need to be saturated, and that comes from the zero modes in the integrand.However for the antiperiodic boundary conditions there are no zero modes and the wholeψi(t) field is quantum and hence rendered redundant by the scalings. And since in expression (234) there are no terms independent of the ψi(t) in the interaction term after the x dependent terms have been eliminated, the entire path-integral vanishes. Hence we need to bother only with the periodic case.


### 11.1. The Chiral Anomaly

Here we will assume that the quantum part ofψi(t) has been rendered redundant and all that remains in the expression is the zero mode which we will again call ψi. The λ and a path-integrals can be written as (236)∫da2π∫APDλDλ-e-∫01dtλ-iλ˙i-igλ-l1/2ψ·Flmx·ψλm-iaλ-iλi+i1-n/2a.Going by (208) and (210), this would simply beget (237)TrPe-i∫dtM,where in this case ∫01dtM=g/2ψ·F·ψ. This then reproduces precisely, upon restoring the factor of 1/(2π)2 above (235) and the integrals over the x and ψ zero modes, the standard expression [35, 36] for the chiral anomaly for fermions, namely, (238)12π2∫d4x∫d4ψTreig/2ψ·F·ψwhich simply yields (239)-132π2g2∫d4xϵμνρσTrRFμνFρσthus tallying with (228). This can be compared with the derivations in [36, 37], to appreciate the economy of arguments and mathematical steps. Note that we hence have also derived the formula for the Atiyah-Singer index for the case at hand.


### 11.2. The Gauge Anomaly

Let us now deal with the chiral non-Abelian anomaly or gauge anomaly. The expression corresponding to (234) is given by(240)
i
η
a
l
i
m
T
→
0
∫
P
D
x
∫
A
P
-
∫
P
D
ψ
D
ψ
-
∫
A
P
+
∫
P
D
λ
D
λ
-
×
λ
-
i
T
a
i
j
λ
j
e
∫
0
1
d
t
-
1
/
4
T
x
˙
2
+
i
g
x
˙
μ
A
μ
i
j
λ
-
i
λ
j
+
i
T
g
ψ
μ
F
μ
ν
i
j
λ
-
i
λ
j
ψ
ν
-
ψ
-
j
ψ
˙
j
-
λ
-
i
λ
˙
i
,where Ta belongs to the SM gauge group and i ranges over 5 values. The iηaλ-iTaijλj term can be exponentiated as follows: (241)
l
i
m
T
→
0
∫
P
D
x
∫
A
P
-
∫
P
D
ψ
D
ψ
-
∫
A
P
+
∫
P
D
λ
D
λ
-
×
e
∫
0
1
d
t
-
1
/
4
T
x
˙
2
+
i
g
x
˙
μ
A
μ
i
j
λ
-
i
λ
j
+
i
T
g
ψ
μ
F
μ
ν
i
j
λ
-
i
λ
j
ψ
ν
+
i
η
a
λ
-
i
T
a
i
j
λ
j
-
ψ
-
j
ψ
˙
j
-
λ
-
i
λ
˙
i
,where the understanding is that the anomaly would be given by the term linear in ηa. The antiperiodic boundary conditions for ψ would not contribute and we need to deal only with the periodic case. And as before, it would boil down to first evaluating (242)∫AP+∫PDλDλ-e∫01dti/2gψμFμνijλ-iλjψν+iηaλ-iTaijλj-λ-iλ˙i.We would have as in previous sections (243)∫dλ-dλe-λ-mei/2gψ·Fx·ψ+2ηaTampλp+λ-mλm+∫dλ-dλe--λ-mei/2gψ·Fx·ψ+2ηaTampλp+λ-mλm(244)=2∫dλ-dλ1+12λ-mei/2gψ·Fx·ψ+2ηaTampλp2+14!λ-mei/2gψ·Fx·ψ+2ηaTampλp4e-λ-mλm.So let us again call the matrix ei/2[gψ·F(x)·ψ+2ηaTa], B. As before, one has ((TrB)2-Tr(B2)) from the second term, and from the fourth (245)112∫dλdλ-λ-mBmpλpλ-m′Bm′p′λp′λ-m′′Bm′′p′′λp′′λ-m′′′Bm′′′p′′′λp′′′e-λ-mλm=112TrB4-6TrB2TrB2+3TrB22+8TrB3TrB-6TrB4.And, since the trace of [gψ·F(x)·ψ+2ηaTa] would vanish, the second term, for instance, in the above expression would look like (246)-12Tr1-i3!32ηaTagψ·Fx·ψ2Tr1-i3!382ηaTagψ·Fx·ψ2Tr1-i3!382ηaTagψ·Fx·ψ2,where one has kept terms that are linear in ηa and quartic in ψ’s in each trace, and the overall coefficient for this term looks like (247)i58Tr12TrηaTagψ·Fx·ψ2=i1258TrηaTagψ·Fx·ψ2since Tr(1)=5, as the indices range over 5 values for SU(5) (which is inherent in the assumptions, since λ’s and λ-’s run over five values in order to generate the full SM fermion spectrum).Now the sum over the coefficients of all such terms in (244) simply vanishes. The 1 would not survive the ψ integration. Doing the remaining integration over ψ’s and factoring out ηa, one simply gets that the anomaly is, ignoring prefactors involving π and other numerical coefficients, (248)0g2∫d4x∫d4ψTrTaψ·Fx·ψ2=0g2∫d4xϵμνρσTrTaFμνFρσwhich is the correct expression for the gauge anomaly, with a vanishing coefficient which thus demonstrates the gauge anomaly freedom of the SM, since Ta can be any of its generators and one has summed over the full fermionic spectrum. Note that Ta and F’s in the above expression are in the fundamental representation. Note also that this gives just the leading order term of the anomaly—in that Fμν=∂μAν-∂νAμ only—and not the consistent one [35], which can however be computed via the Wess-Zumino consistency conditions.Notice that the above expressions could also have been obtained by considering eachn separately by simply incorporating terms of the form -iaλ-iλi+i(m-5/2)a in the action, where n=5Cm, and it is evident that the above automatically represents the results of taking m=0,2 and 4 together, for n=1, 10, and 5-, respectively.Thus we see that taking a holistic approach and treating all the 16 states of the SM together by simply taking a sum over the periodic and antiperiodic boundary conditions forλ’s are a rather economical way of deriving the gauge anomaly freedom of the SM.


## 11.1. The Chiral Anomaly

Here we will assume that the quantum part ofψi(t) has been rendered redundant and all that remains in the expression is the zero mode which we will again call ψi. The λ and a path-integrals can be written as (236)∫da2π∫APDλDλ-e-∫01dtλ-iλ˙i-igλ-l1/2ψ·Flmx·ψλm-iaλ-iλi+i1-n/2a.Going by (208) and (210), this would simply beget (237)TrPe-i∫dtM,where in this case ∫01dtM=g/2ψ·F·ψ. This then reproduces precisely, upon restoring the factor of 1/(2π)2 above (235) and the integrals over the x and ψ zero modes, the standard expression [35, 36] for the chiral anomaly for fermions, namely, (238)12π2∫d4x∫d4ψTreig/2ψ·F·ψwhich simply yields (239)-132π2g2∫d4xϵμνρσTrRFμνFρσthus tallying with (228). This can be compared with the derivations in [36, 37], to appreciate the economy of arguments and mathematical steps. Note that we hence have also derived the formula for the Atiyah-Singer index for the case at hand.


## 11.2. The Gauge Anomaly

Let us now deal with the chiral non-Abelian anomaly or gauge anomaly. The expression corresponding to (234) is given by(240)
i
η
a
l
i
m
T
→
0
∫
P
D
x
∫
A
P
-
∫
P
D
ψ
D
ψ
-
∫
A
P
+
∫
P
D
λ
D
λ
-
×
λ
-
i
T
a
i
j
λ
j
e
∫
0
1
d
t
-
1
/
4
T
x
˙
2
+
i
g
x
˙
μ
A
μ
i
j
λ
-
i
λ
j
+
i
T
g
ψ
μ
F
μ
ν
i
j
λ
-
i
λ
j
ψ
ν
-
ψ
-
j
ψ
˙
j
-
λ
-
i
λ
˙
i
,where Ta belongs to the SM gauge group and i ranges over 5 values. The iηaλ-iTaijλj term can be exponentiated as follows: (241)
l
i
m
T
→
0
∫
P
D
x
∫
A
P
-
∫
P
D
ψ
D
ψ
-
∫
A
P
+
∫
P
D
λ
D
λ
-
×
e
∫
0
1
d
t
-
1
/
4
T
x
˙
2
+
i
g
x
˙
μ
A
μ
i
j
λ
-
i
λ
j
+
i
T
g
ψ
μ
F
μ
ν
i
j
λ
-
i
λ
j
ψ
ν
+
i
η
a
λ
-
i
T
a
i
j
λ
j
-
ψ
-
j
ψ
˙
j
-
λ
-
i
λ
˙
i
,where the understanding is that the anomaly would be given by the term linear in ηa. The antiperiodic boundary conditions for ψ would not contribute and we need to deal only with the periodic case. And as before, it would boil down to first evaluating (242)∫AP+∫PDλDλ-e∫01dti/2gψμFμνijλ-iλjψν+iηaλ-iTaijλj-λ-iλ˙i.We would have as in previous sections (243)∫dλ-dλe-λ-mei/2gψ·Fx·ψ+2ηaTampλp+λ-mλm+∫dλ-dλe--λ-mei/2gψ·Fx·ψ+2ηaTampλp+λ-mλm(244)=2∫dλ-dλ1+12λ-mei/2gψ·Fx·ψ+2ηaTampλp2+14!λ-mei/2gψ·Fx·ψ+2ηaTampλp4e-λ-mλm.So let us again call the matrix ei/2[gψ·F(x)·ψ+2ηaTa], B. As before, one has ((TrB)2-Tr(B2)) from the second term, and from the fourth (245)112∫dλdλ-λ-mBmpλpλ-m′Bm′p′λp′λ-m′′Bm′′p′′λp′′λ-m′′′Bm′′′p′′′λp′′′e-λ-mλm=112TrB4-6TrB2TrB2+3TrB22+8TrB3TrB-6TrB4.And, since the trace of [gψ·F(x)·ψ+2ηaTa] would vanish, the second term, for instance, in the above expression would look like (246)-12Tr1-i3!32ηaTagψ·Fx·ψ2Tr1-i3!382ηaTagψ·Fx·ψ2Tr1-i3!382ηaTagψ·Fx·ψ2,where one has kept terms that are linear in ηa and quartic in ψ’s in each trace, and the overall coefficient for this term looks like (247)i58Tr12TrηaTagψ·Fx·ψ2=i1258TrηaTagψ·Fx·ψ2since Tr(1)=5, as the indices range over 5 values for SU(5) (which is inherent in the assumptions, since λ’s and λ-’s run over five values in order to generate the full SM fermion spectrum).Now the sum over the coefficients of all such terms in (244) simply vanishes. The 1 would not survive the ψ integration. Doing the remaining integration over ψ’s and factoring out ηa, one simply gets that the anomaly is, ignoring prefactors involving π and other numerical coefficients, (248)0g2∫d4x∫d4ψTrTaψ·Fx·ψ2=0g2∫d4xϵμνρσTrTaFμνFρσwhich is the correct expression for the gauge anomaly, with a vanishing coefficient which thus demonstrates the gauge anomaly freedom of the SM, since Ta can be any of its generators and one has summed over the full fermionic spectrum. Note that Ta and F’s in the above expression are in the fundamental representation. Note also that this gives just the leading order term of the anomaly—in that Fμν=∂μAν-∂νAμ only—and not the consistent one [35], which can however be computed via the Wess-Zumino consistency conditions.Notice that the above expressions could also have been obtained by considering eachn separately by simply incorporating terms of the form -iaλ-iλi+i(m-5/2)a in the action, where n=5Cm, and it is evident that the above automatically represents the results of taking m=0,2 and 4 together, for n=1, 10, and 5-, respectively.Thus we see that taking a holistic approach and treating all the 16 states of the SM together by simply taking a sum over the periodic and antiperiodic boundary conditions forλ’s are a rather economical way of deriving the gauge anomaly freedom of the SM.


## 12. Conclusions

We have thus given representations of propagators for the gauge bosons and chiral fermions of a single generation of the SM (both free and dressed, massless and massive for the fermions) and its fermionic one-loop effective actions and anomalies, based upon the worldline formalism of quantum field theory. Our approach relied upon the use of fermionic worldline fields to represent the additional spin and color information of the particle multiplets, and associated worldline gauge fields to project onto appropriate sectors of the Hilbert space. Such an approach had been initiated in [13, 19, 21–23] in the context of dressed propagators in the Abelian case and we have extended the formalism to the non-Abelian case (Section 7) (see [12] for related approaches) and made concrete use of the Abelian version in deriving the QED vertex (Section 8). We also applied BRST quantization to verify that the correct gauge fixing procedures had been adopted (Section 4). Since the various SM fields arise as excitations of the quantized worldline theories (Sections 5 and 6), this then gives one the quantum mechanical models, whose wave-functions are the various fields of the SM. String models that try to achieve the same thing have the problem that the wave-functions that arise out of its quantization encompass too many other fields apart from the SM. However, unlike string theory, it is not out of a unique single model that the various fields of the SM arise (we have considered only gauge bosons and fermions), but two different sets of models for the fermions and gauge bosons, and that is a disadvantage.Our approach also proved, as we said, amenable to the determination of chiral anomalies and demonstrations of part of the cancellations of the gauge ones (Section11).With the details discussed in this paper, it would seem appropriate to consider application to specific physical problems; an example, building upon [48], would be to calculate propagators in a constant (non-)Abelian background. One might also contemplate extending the derivation of the QED vertex to the non-Abelian case, as well as to higher point functions, leading to, for instance, the derivation of the Compton scattering amplitude in QED. And then, extending from that point onward, one might think of applying the formalism to compute amplitudes for processes at the LHC.Mostly, the main purport of the paper is setting up a formalism for doing more advanced calculations relevant for high energy processes. To date, people have mostly dealt with one and higher loop amplitudes in the worldline formalism, skirting the issue of tree amplitudes as it proved to be more formidable. We have taken the first steps here towards the latter program, uncovering the utility of coherent states in possible future applications. However, as we said before, we are currently engaged in building even more powerful methods stemming from all this that give rise to these things much more economically.The fact that the 3 generations of SM fermions organize themselves as 3 distinct spinors ofSO(10) as well as the fact that the explanation of the hypercharge quantum numbers is a natural explanation if the corresponding generator is a diagonal one of SU(5) seem in our opinion, to be tantalizing hints that the structure of nature does indeed make use of these groups, especially of SO(10) (SU(5) being a subgroup). From the structure of the excitations, arrived at through the action of worldline fermionic creation operators, it is tempting to propose that even the higher generations arise via the action of more creation operators on the fields of the first generation. We are also working on such a model, through which we find that there are ramifications on the scalar sector as well, in the sense that the scalar sector is populated with a panoply of scalar fields that couple to the fermions in such a way that it leads to a particularly economical set of couplings (4 or 5) from which everything in the SM could potentially arise. That is, one can essentially trade the arbitrariness in the number of parameters in the SM with a proliferation of the number of scalar fields in an extension of it. However we have to admit that such a model is already hugely constrained by nonoccurrences of such scalars at the LHC, and our hope is that there might be some natural mechanism that automatically renders these scalars much more massive than the Higgs. We believe that the reduction in the arbitrariness of the parameters in the SM is a feature interesting enough to merit attention.



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*Source: 2165731-2017-11-19.xml*