Non-Abelian basis tensor gauge theory
NNon-Abelian basis tensor gauge theory
Edward E. Basso ∗ and Daniel J. H. Chung † Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
Basis tensor gauge theory is a vierbein analog reformulation of ordinary gauge theoriesin which the difference of local field degrees of freedom has the interpretation of an objectsimilar to a Wilson line. Here we present a non-Abelian basis tensor gauge theory formal-ism. Unlike in the Abelian case, the map between the ordinary gauge field and the basistensor gauge field is nonlinear. To test the formalism, we compute the beta function and thetwo-point function at the one-loop level in non-Abelian basis tensor gauge theory and showthat it reproduces the well-known results from the usual formulation of non-Abelian gaugetheory.
1. INTRODUCTION
The Standard Model (SM) of particle physics [1–10] is usually formulated with gauge fieldsthat transform inhomogeneously under the gauge group: i.e. they are connections on principalbundles (see e.g. [11, 12]). This mechanism is used to construct covariant derivatives acting onmatter fields, which allows a simple recipe for constructing kinetic terms for local field theoriesliving on principal bundles. Gauge theories of this sort have a long history (see e.g. [6, 13–20])and are very economical in describing the physics locally at the cost of introducing redundanciesinto the system. Despite this long history, rewriting gauge theories in novel formalisms continueto offer insights into both computational techniques and ideas for physics beyond the SM (seee.g. [21–28]).The work of [29] gives a reformulation of U ( ) gauge theories in analogy with the vierbeinformalism of general relativity. In that paper, it was shown that the vierbein analog field G αβ transforms homogeneously under the U ( ) gauge group and satisfies certain constraints, in con-trast with the ordinary formulation in which the gauge field transforms inhomogeneously. The ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - t h ] J un nonlinear map between the ordinary A µ field and G αβ can be changed to a linear relationship us-ing a set of N unconstrained scalar fields θ a ( x ) in N dimensions. The field theory of θ a ( x ) is called basis tensor gauge theory (BTGT), which can be viewed as a theory of Wilson lines (e.g. [30–37]and references therein) modded by a particular symmetry that is required to allow only couplingsequivalent to ordinary gauge theories. In [38], the Ward identities of the theory were constructedand the theory was explicitly shown to be one-loop stable.In this work, we present a non-Abelian version of basis tensor gauge theory. Just as in theAbelian case, the interpretation of the basis tensor gauge field is similar to a Wilson line. Thismeans that the basis tensor field θ Aa ( x ) is more non-local when expressed in terms of the ordinarygauge potential A B µ . Unlike in the Abelian case, the map between θ Aa ( x ) and A B µ is nonlinear. Aperturbation theory can be defined in powers of θ Aa that allows us to have a finite power expan-sion map between θ Aa and A B µ . Just as in the Abelian case, we can impose a symmetry (BTGTsymmetry) to eliminate charge violating couplings and enforce positivity of the Hamiltonian.As the map between θ Aa and A B µ is nonlinear, unlike in the Abelian case, the choice of θ Aa variables to parameterize the gauge manifold target space is not motivated by simplicity. Onthe other hand, this motivation still exists since the number of functional degrees of freedombetween A B µ theories and θ Bm theories naturally match without imposing additional constraints onthe vierbein-like field that would make it difficult to quantize. The basis choice is also a naturalgeneralization of the Abelian construction (i.e. both are gauge group manifold target space fields),and it has the same relationship with the Wilson line as in the Abelian case. Furthermore, theBTGT symmetry representation that stabilizes the theory (e.g. enforces charge conservation andbounds the Hamiltonian from below) naturally generalizes the Abelian theory’s representation.To test the formalism we perturbatively compute the β -function and find that it matches theusual result non-Abelian gauge theory at one loop. We also compute the one-loop divergent con-tribution to the (cid:104) A µ ( x ) A ν ( y ) (cid:105) correlator, where A µ [ θ ] is now treated as a local composite operator.We find that before introducing the counter terms, the divergence that is obtained using the θ Aa formalism is the same as in the usual A A µ ( x ) formalism. This is an indication that the UV structureof ordinary gauge theories are faithfully reproduced by the non-Abelian BTGT theory.The order of presentation is as follows. In Section 2, we present the definition of non-Abelian In [29], we used upper indices to denote the components of θ a ( x ) field. In this work, the analogous index willappear as a lower index. basis tensor gauge theory. In Section 3, we present the path integral formulation of the BTGTtheory. This includes the perturbative expansion terms similar to what is done in nonlinear sigmamodels. To check that the quantum formulation of BTGT is stable and computable, in section4, we compute the β -function explicitly by renormalizing the two-point functions of the BTGTfield θ Aa , the ghost fields c ¯ c , and the θ c ¯ c vertex functions. In section 5, we compute the two-pointfunction (cid:104) A A µ ( x ) A B ν ( y ) (cid:105) at one-loop using the BTGT formalism. We check the transversality of thedivergent contribution consistent with gauge invariance and check that introducing the appropriatecomposite operator counter terms allow both (cid:104) θ Aa ( x ) θ Bb ( y ) (cid:105) and (cid:104) A A µ ( x ) A B ν ( y ) (cid:105) to be finite. Insection 6, we make a conjecture regarding what the relationship will be for the infinite number ofrenormalization constants based on the computations done in this paper. In section 7, we presentour conclusions. In Appendix A, we collect some of the less-standard notation and conventionsused in this paper. In Appendix B, we derive the relationship between the non-Abelian basis tensorfield and the ordinary gauge field. In Appendix C, we discuss the representations of gauge andBTGT symmetry transformations. In Appendix D, we list the Feynman rules for the theory.
2. NON-ABELIAN BTGT BASIS DEFINITION
In this section, we construct an explicit relationship between the vierbein analog field G andordinary non-Abelian gauge field A . We will work with 4 spacetime dimensions throughout thispaper to maintain simplicity and obvious physical relevance even though generalizations to dif-ferent spacetime dimensions are straightforward. All repeated indices will be summed unlessspecified otherwise. For example, whenever one side of an equation has indices specified, theother side of the equation may have repeated indices that are not summed.Given a field φ that is a complex scalar transforming under gauge transformations as φ k ( x ) → [ g ( x )] ks φ s ( x ) (1) [ g ( x )] ks ≡ (cid:16) e i Γ C ( x ) T C (cid:17) ks , (2)where ( T C ) ab are Hermitian generators of the gauge group in representation R , we define a Lorentztensor G α ( f ) β that exhibits the gauge group transformation property (cid:104) G α ( f ) β ( x ) (cid:105) i → (cid:104) G α ( f ) β ( x ) (cid:105) j (cid:2) g − ( x ) (cid:3) ji , (3)such that G α ( f ) β φ is gauge invariant, where f is a basis index that specifies a fixed direction in thegauge group representation space. The requirement of rank 2 comes from having enough func-tional degrees of freedom to match the gauge field functional degrees of freedom as explainedin [29]. More formally, G α ( f ) β ( x ) is a field that transforms as an ¯ R from the right under the non-Abelian gauge group representation and as a rank 2 Lorentz projection tensor. The index ( f ) in G α ( f ) β spans the dimension of the representation. Hence, G α ( f ) β contains 2 × D ( R ) real functionaldegrees of freedom (in 4-spacetime dimensions), where D ( R ) is the dimension of the representa-tion. The analogy with gravitational vierbeins ( e a ) µ can be identified as follows (similar to theAbelian case of [29]): the indices { f , α , β } are the analogs of the fictitious Minkowski space in-dex a of ( e a ) µ , and the representation of Eq. (3) is the analog of the diffeomorphism acting on the µ index of ( e a ) µ .To reproduce ordinary gauge theory with G α ( f ) β , we must be able to path integrate over un-constrained functions that match the number of degrees of freedom in A µ . This means that wemust eliminate the number of field degrees of freedom either by imposing a constraint through anintroduction of an auxiliary field or explicitly solving such a matching constraint. Since the gaugefield real functional degrees of freedom necessary for constructing covariant derivatives on funda-mental representation fields is 4 D ( A ) (where D ( A ) is the dimension of the adjoint representation),we need to eliminate 32 D ( R ) − D ( A ) degrees of freedom. We can accomplish this by choosingthe field degrees of freedom that represent G α ( f ) β to live on the target space of the gauge manifold,which will cause the D ( A ) dimension matching condition to be satisfied. We can then construct4 such sets with the help of a projection tensor (just as in the Abelian BTGT) to match 4 D ( A ) degrees of freedom in A µ : the gauge manifold target space fields are θ Ca where a ∈ { , , , } and C ∈ { , , ..., D ( A ) } .To find a map between G α ( f ) β and θ Ca , define an orthonormal set of spacetime-independentvectors ξ l ( f ) for f ∈ { , ..., dim R } that span the group representation vector space such that thefollowing completeness relationship is satisfied: δ kl = ∑ f ξ k ( f ) ξ ∗ l ( f ) . (4)The ξ ( f ) are defined to be invariant under gauge transformations.In the spirit of the Abelian case of [29], the vierbein analog in the non-Abelian gauge theorycan be defined as (cid:16)(cid:2) G ( f ) ( x ) (cid:3) γδ (cid:17) j = ξ ∗ l ( f ) (cid:104)(cid:0) exp (cid:2) − i θ Ma ( x ) H a T M (cid:3)(cid:1) γδ (cid:105) l j . (5)Here the objects H a with a ∈ { , ..., } are 4 × [ H a , H b ] =
0, which satisfy the completeness relationship ∑ a = ( H a ) µν = δ µν (6)and the orthonormality condition Tr (cid:16) H a H b (cid:17) = δ ab , (7)(just as in the Abelian case of [29]). These matrices can be chosen to have the following orthonor-mal projection property ( H a ) µν ( H b ) νβ = δ ab ( H a ) µβ no sum on a (8)and symmetry property ( H a ) µν = ( H a ) νµ . (9)The fields θ Ma ( x ) are real scalar fields which transform under gauge transformations as U a → e i Γ U a (10)where U a ≡ exp (cid:104) i θ Aa T A (cid:105) (11) Γ ≡ Γ B T B . (12)The reason why θ Aa is easier to work with than G ( f ) ( x ) is that it is unconstrained, similar to the π variable being easier to work with compared to U ( π ) in sigma models [6].There are several salient features to note regarding Eq. (5). Given the representation identity ψ C → ( g adj ) CS ψ S , (13)if ψ C T C → g [ ψ C T C ] g − , (14)where g adj is the adjoint representation group element (independent of the representation of g ), wemight naively expect that θ Ma has its M index transforming as an adjoint. However, this is not truebecause the transformation property of θ M is ξ ∗ ( f ) (cid:0) exp (cid:2) − i θ Ma ( x ) H a T M (cid:3)(cid:1) γδ → ξ ∗ ( f ) (cid:0) exp (cid:2) − i θ Ma ( x ) H a T M (cid:3)(cid:1) γδ g − ( x ) , (15)and not ξ ∗ ( f ) (cid:0) exp (cid:2) − i θ Ma ( x ) H a T M (cid:3)(cid:1) γδ → ξ ∗ ( f ) g ( x ) (cid:0) exp (cid:2) − i θ Ma ( x ) H a T M (cid:3)(cid:1) γδ g − ( x ) (16)in Eq. (5). Another aspect is that the index f in Eq. (5) runs from 1 to dim ( R ) components in G ( f ) ( x ) , but the number of independent scalar field degrees of freedom of G ( f ) ( x ) in terms of θ Am is the rank of the group times the spacetime dimension 4 (spanned by m ∈ { , ..., } ). This issimilar to the ordinary gauge field having dim ( R ) components of the f index in A M µ ( T M ) f k butcounting in terms of A M µ , the index M runs through the rank of the group.Another interesting relationship is the map between the ordinary non-Abelian gauge field and (cid:2) G ( f ) ( x ) (cid:3) γδ . As shown in Appendix B, the relationship is A µ = i (cid:104) G − αβ (cid:105) (cid:2) ∂ α G β µ (cid:3) (17)where G β µ are related to the basis tensor as (cid:2) G β µ (cid:3) qm = dim R ∑ f ξ q ( f ) (cid:2) G ( f ) β µ (cid:3) m . (18)We note that the relationship of U a and G αβ is G µλ = [ H b ] µλ U † b (19)according to Eq. (8). Owing to the projection property of Eq. (8) in a conveniently normalizedbasis, the ordinary non-Abelian gauge field can also be rewritten as A µ = iU a ˜ ∂ a µ U † a , (20)where ˜ ∂ a ν ≡ ( H a ) µν ∂ µ . (21)This can be seen simply by using Eq. (8) and Eq. (19); A µ = i ∑ a U a ( H a ) αβ ∑ b ∂ α ( H b ) β µ U † b (22) = i ∑ a ∑ b δ ab ( H a ) αµ U a ∂ α U † b (23) = i ∑ a U a ˜ ∂ a µ U † a . (24)As discussed in Appendix B, the relationship between the θ Aa field and the ordinary non-Abeliangauge fields can be written explicitly as A Q µ = ∑ c (cid:18)(cid:16)(cid:2) θ Jc f J (cid:3) − (cid:17) QR (cid:16) e θ Kc f K − (cid:17) RB ˜ ∂ c µ θ Bc (cid:19) , (25)where f J is a structure constant matrix having the components ( f J ) AB = f JAB . The non-AbelianEq. (25) reduces to the Abelian case of [29] in the limit that the structure constant matrix f → θ Bc and A differ by a minus sign compared to the original AbelianBTGT paper [29] because the sign convention for θ has been flipped (see Eq. (23) of that paperand Eq. (5) above). As we see in this expression, one key difference between the Abelian BTGTand the non-Abelian BTGT is that the map between the ordinary gauge field A and the θ fieldis linear in the Abelian case and nonlinear in the non-Abelian case. On the other hand, since θ Bc represents a solution to a first order differential equation, it still does have the interpretation of atype of object similar to a Wilson line.As noted in [29], because gauge invariance is insufficient to impose global charge conservation(unlike in the usual gauge theory formulation), we must impose a new symmetry introduced in [29]called a BTGT symmetry. The BTGT transformation in the non-Abelian case is U a → U a e iZ a (26) Z a ≡ Z Ba T B , (27)where Z Ba satisfies ( H a ) λ µ ∂ λ Z Ba = . (28)Because this transformation will not transform the gauge field variable when written in termsof the ordinary A M µ basis, this transformation is independent of the usual gauge transformations.Infinitesimally, Eqs. (3) and (26) can be rewritten as δ θ Aa = (cid:18) f · θ a exp [ f · θ a ] − (cid:19) AB Γ B + (cid:18) f · θ a − exp [ − f · θ a ] (cid:19) AB Z Ba (29)to linear order in Γ B and Z Ba , where ( f · θ a ) MN ≡ f CMN θ Ca . The derivation of this linearized trans-formation is presented in Appendix C. Finally, note that we can also write the combined gaugeand BTGT transformations acting on G αβ ( x ) as [ H f ] ψµ G µλ → e − iZ Bf ( x ) T B [ H f ] ψµ G µλ e − i Γ C ( x ) T C (30) Note that Ref. [29] uses the notation of having the basis tensor index c of θ c instead of θ Bc as in Eq. (5). and G µλ [ H f ] λ ν → e − iZ Bf ( x ) T B G µλ [ H f ] λ ν e − i Γ C ( x ) T C (31)This means that it is convenient to write gauge invariant and BTGT invariant fields in terms of ( H a ) β α G αβ ( x ) because of these simple transformation properties.
3. PATH INTEGRAL FORMULATION
We define the quantized theory of G in this section using a path integral over the θ Aa variable inthis section. To this end, we begin by writing down the BTGT and gauge invariant action in termsof U a variable (defined in Eq. (11)). Next, we define a coupling constant expansion that allows usto match perturbative gauge theory computations. Afterwards, we construct the path integral over θ Aa . In this section, we construct the action for the basis tensor field θ Aa . Because of Eq. (25), anynon-Abelian gauge theory with finite powers of A µ will map to a field theory with an infinite powerseries in θ Kc . In this section, we construct the action of the usual Yang-Mills theory in terms of θ Aa .Recall that A µ is a BTGT transformation invariant (which we will refer to as a BTGT invariantfor short). Hence, we can construct BTGT invariant objects involving just θ Aa fields if we workwith our knowledge of the usual gauge kinetic terms. Using Eq. (20), we can write the action inthe usual way as L = − g T ( R ) Tr (cid:0) F µν F µν (cid:1) , (32)where the field strength is F µν = i [ D µ , D ν ] (33)and the covariant derivative in terms of U a is D µ = ∂ µ + ∑ a = U a ˜ ∂ a µ U † a . (34)More explicitly, we can expand the the field strength tensor as F µν = i (cid:32) ∂ µ ∑ a = U a ˜ ∂ a ν U † a − ∂ ν ∑ a = U a ˜ ∂ a µ U † a (cid:33) + i ∑ a , b = [ U a ˜ ∂ a µ U † a , U b ˜ ∂ b ν U † b ] . (35)When written in terms of components, we can identify L = − g (cid:16) ∂ µ A A ν ∂ µ A A ν − ∂ µ A A ν ∂ ν A A µ (cid:17) − g f ABC ∂ µ A A ν A B µ A C ν − g f ABC f AB C A B µ A C ν A B µ A C ν (36)with A A µ = iT ( R ) ∑ a = Tr (cid:16) T A U a ˜ ∂ a µ U † a (cid:17) . (37)Just as in the Abelian BTGT theory, we see that the theory has a 4-derivative kinetic term structure,which begs the question of whether the Hamiltonian is bounded from below [39–43]. Just as in theAbelian case [29], the Hamiltonian is indeed bounded from below because the BTGT symmetrygives rise to only field dependence on A A µ [ U a ] .The matter coupling can be written down by noting that under BTGT transformations, we have ∂ ψ (cid:20)(cid:16) H f (cid:17) ψα G αβ φ (cid:21) → e − iZ Bf ( x ) T B ∂ ψ (cid:20)(cid:16) H f (cid:17) ψα G αβ φ (cid:21) . (38)This means we can construct a gauge, Lorentz, and BTGT invariant combination ∑ f (cid:18) ∂ ψ (cid:20)(cid:16) H f (cid:17) ψ α G α β φ (cid:21)(cid:19) † g β β ∂ ψ (cid:20)(cid:16) H f (cid:17) ψα G αβ φ (cid:21) . (39)It is easy to check using Eqs. (19), (8), and (9) that this is equivalent to the usual gauge couplingto matter D µ φ † D µ φ : D µ φ † D µ φ = (cid:34) ∂ µ φ + ∑ a = ( H a ) λ µ U a ∂ λ U † a φ (cid:35) † (cid:34) ∂ µ φ + ∑ b = ( H b ) λ µ U b ∂ λ U † b φ (cid:35) . (40)We can of course write down a similar coupling for the fermions charged under the non-Abeliangauge group: L f K = Ψ (cid:34) i (cid:19)(cid:19) ∂ + i γ µ ∑ b = ( H b ) λ µ U b ∂ λ U † b (cid:35) Ψ . (41)We note that because of BTGT invariance, couplings of the form ∑ f (cid:104) G α ( f ) β φ (cid:105) (cid:104) G β ( f ) α φ (cid:105) (42)cannot be written down because they violate BTGT symmetry. There exists gauge and BTGTinvariant terms of the form ∑ a Tr (cid:16) U a U † a (cid:17) (43)that we might worry about. However, owing to their group representation structure, these areconstants and will not contribute nontrivially in flat spacetime.0 Written in terms of the θ Aa fields of Eq. (5), the Lagrangian is a power series in θ Aa . For pertur-bative computations, we only require a consistent truncation in the coupling constant. The usualperturbation theory proceeds through the identification A A µ → gA A µ . (44)Motivated by this and a need to truncate the power series of Eq. (5), we make the change ofvariables θ Aa → g θ Aa (45)and expand perturbatively about g →
0. However, given that Eqs. (44) and (45) match only to linearorder in g , the perturbative expansion of the A µ theory with g → θ Aa theory with g → A A µ → gA A µ perturbation theory to θ Aa → g θ Aa perturbationtheory to quadratic order in g , we must make the identification gA A µ = g ∑ a (cid:20) e g f · θ a − g f · θ a (cid:21) AB ˜ ∂ a µ θ Ba (46) ≈ g ˜ ∂ a µ θ Aa + g f ABC (cid:16) ˜ ∂ a µ θ Ba (cid:17) θ Ca + O ( g ) (47)at least to quadratic order in g . We explicitly then see a quadratic field identification with A µ . Inthis case, a two-point function in A µ becomes (cid:104) A A µ ( x ) A B ν ( y ) (cid:105) = ∑ a , b (cid:68) ( ˜ ∂ a µ θ Aa ( x ) + g f AC D θ D a ( x ) ˜ ∂ a µ θ C a ( x ) + . . . ) × ( ˜ ∂ b ν θ Bb ( y ) + g f BC D θ D b ( y ) ˜ ∂ b ν θ C b ( y ) + . . . ) (cid:69) . (48)Although this nonlinearity seems undesirable from the perspective of matching to ordinary non-Abelian field theory perturbative expansion in terms of A A µ , there may be an advantage since itallows us to map nontrivial composite non-local operator correlators in the language of A A µ field interms of correlators of the elementary θ Aa correlator. We will defer the exploration of this featureto a future work.1The power series can be explicitly written as A A µ = ∑ a (cid:20) e g f · θ a − g f · θ a (cid:21) AB ˜ ∂ a µ θ Ba (49) = ˜ ∂ a µ θ Aa + g f ABC (cid:16) ˜ ∂ a µ θ Ba (cid:17) θ Ca + g f ABE f CDE θ Ba θ Ca (cid:16) ˜ ∂ a µ θ Da (cid:17) + O (cid:0) g (cid:1) . (50)With the proper addition of the gauge fixing term, Eq. (32) takes the form L gauge = − F A , µν F A µν − ξ ∂ µ A A µ ∂ ν A A ν . (51)With Eq. (46) the gauge boson sector becomes L gauge = L θ + L θ + L θ + · · · = ∞ ∑ n = L θ n (52)where L θ = − (cid:16) ∂ µ ˜ ∂ ν a θ Aa (cid:17) δ AB (cid:16) ∂ µ ˜ ∂ b ν θ Bb − (cid:16) − ξ (cid:17) ∂ ν ˜ ∂ b µ θ Bb (cid:17) , (53) L θ = − g f ABC (cid:16) ∂ µ ˜ ∂ ν a θ Aa (cid:17) (cid:16) ˜ ∂ b µ θ Bb (cid:17) (cid:16) ˜ ∂ c ν θ Cc (cid:17) − g f ABC (cid:16) ∂ µ ˜ ∂ ν a θ Aa − (cid:16) − ξ (cid:17) ∂ ν ˜ ∂ µ a θ Aa (cid:17) (cid:16) ∂ µ (cid:16)(cid:16) ˜ ∂ b ν θ Bb (cid:17) θ Cb (cid:17)(cid:17) (54)and L θ = − g f EAB f ECD (cid:16) ˜ ∂ a µ θ Aa (cid:17) (cid:16) ˜ ∂ b ν θ Bb (cid:17) (cid:16) ˜ ∂ µ c θ Cc (cid:17) (cid:16) ˜ ∂ ν d θ Dd (cid:17) − g f EAB f ECD (cid:16) ˜ ∂ a µ θ Aa (cid:17) (cid:16) ˜ ∂ b ν θ Bb (cid:17) ∂ µ (cid:16)(cid:16) ˜ ∂ ν c θ Cc (cid:17) θ Dc (cid:17) − g f EAB f ECD (cid:16) ∂ µ ˜ ∂ ν a θ Aa − ∂ ν ˜ ∂ µ a θ Aa (cid:17) (cid:16) ˜ ∂ b µ θ Bb (cid:17) (cid:16) ˜ ∂ c ν θ Cc (cid:17) θ Dc − g f EAB f ECD (cid:16) ∂ µ (cid:16)(cid:16) ˜ ∂ a ν θ Aa (cid:17) θ Ba (cid:17) − (cid:16) − ξ (cid:17) ∂ ν (cid:16)(cid:16) ˜ ∂ a µ θ Aa (cid:17) θ Ba (cid:17)(cid:17) ∂ µ (cid:16)(cid:16) ˜ ∂ ν c θ Cc (cid:17) θ Dc (cid:17) − g f EAB f ECD (cid:16) ∂ µ ˜ ∂ ν a θ Aa − (cid:16) − ξ (cid:17) ∂ ν ˜ ∂ µ a θ Aa (cid:17) ∂ µ (cid:16) θ Bb θ Cb (cid:16) ˜ ∂ b ν θ Db (cid:17)(cid:17) . (55)If gauge fixing is accomplished using the Faddeev-Popov procedure, we can write down theghost Lagrangian coming from the delta-function involving the A A µ in the usual way: L gh1 = − ∂ µ ¯ c A D AB µ c B (56) = − ∂ µ ¯ c A δ AB ∂ µ c B + g f ABC ∂ µ ¯ c A c B A C µ (57)2where A C µ is given in terms of θ Aa explicitly in Eq. (46). To second order in g , the explicit expansionis L gh1 = − ∂ µ ¯ c A ∂ µ c A + g f ABC ˜ ∂ a µ θ Aa ∂ µ ¯ c B c C + g f ABE f CDE ( ˜ ∂ a µ θ Aa ) θ Ba ∂ µ ¯ c C c D + O (cid:0) g (cid:1) . (58)The ghost field couples to the gauge sector with quartic and higher power couplings unlike inthe usual vector potential formalism. If we formulate the path integral measure in terms of A µ andview the path integral in terms of θ Aa as a change of variables, then there will be additional ghostcontributions from D A = D θ nz det (cid:34) δ A A µ ( x ) δ θ B nz , b ( y ) (cid:35) , (59)where θ B nz , b stands for functions that are not annihilated by ( H b ) αβ ∂∂ x α . (60)The functional determinant can be written as usual as a Grassmannian integral yielding an addi-tional ghost Lagrangian: L gh2 = ¯ d Aa O ABab d Bb = ¯ d Aa ˜ ∂ µ a O AB µ b d Bb = − (cid:16) ˜ ∂ µ a ¯ d Aa (cid:17) O AB µ b d Bb (61)where we define the operator O AB µ b = (cid:20) ˆ dt e tg θ b · f (cid:21) AB (cid:16) H b (cid:17) λ µ (cid:126) ∂ λ + (cid:20) ˆ dt ˆ dse ( − s ) tg θ b · f tg f B e stg θ b · f (cid:21) AD (cid:16) H b (cid:17) λ µ (cid:0) ∂ λ θ Db (cid:1) (62) = (cid:20) δ AB + g f ABC θ Cb + g f AEC θ Cb f EBD θ Db (cid:21) ˜ ∂ b µ + (cid:20) ˆ dt ˆ dse ( − s ) tg θ b · f tg f B e stg θ b · f (cid:21) AD (cid:16) ˜ ∂ µ θ Db (cid:17) + O (cid:0) g (cid:1) . (63)We next work out the explicit Feynman rule factors. The inverse of the propagator in momentum space can be written as − iV ABab ( k ) = ∂ ( i L θ ) ∂ θ Aa ( k ) ∂ θ Bb ( − k ) (64) = − i (cid:0) k µ ˜ k ν a (cid:1) δ AB (cid:16) k µ ˜ k b ν − (cid:16) − ξ (cid:17) − k ν ˜ k b µ (cid:17) (65) = − i δ AB (cid:16) δ ab k k (cid:63) a k − (cid:16) − ξ (cid:17) ( k (cid:63) a k ) ( k (cid:63) b k ) (cid:17) , (66)3where we define the star product as k (cid:63) a k = ( H a ) µν k µ k ν . (67)The gauge propagator ∆ ABab ( k ) is given implicitly by ∑ c V ACac ( k ) ∆ CBcb ( k ) = δ AB δ ab , (68)the solution to which is − i ∆ ABab ( k ) = − i δ AB k k (cid:63) a k − i ε (cid:18) δ ab − ( − ξ ) k (cid:63) a kk (cid:19) , (69)where the i ε is the solution Feynman propagator pole prescription. If we assume a diagonal basisfor H a and a Wick rotation to Euclidean space, then this can be written as − i ∆ ABab ( k ) = − i δ AB k k a k b (cid:18) δ ab − ( − ξ ) k a k b k (cid:19) . (70)In position space the propagator can be written as ∆ ABab ( x − y ) = ˆ d k ( π ) e ik · ( x − y ) ∆ ABab ( k ) . (71) For Feynman rules with momenta satisfying k + k + k =
0, the vertex function iV ABCabc ( k , k , k ) can be written as iV ABCabc ( k , k , k ) = ∂ ( i L θ ) ∂ θ Aa ( k ) ∂ θ Bb ( k ) ∂ θ Cc ( k ) (72) = ig f ABC { δ bc ( k (cid:63) b k ) k (cid:63) a ( k − k ) + δ ac ( k (cid:63) c k ) k (cid:63) b ( k − k )+ δ ab ( k (cid:63) b k ) k (cid:63) c ( k − k ) + δ abc (cid:2) k k (cid:63) a ( k − k ) + k k (cid:63) a ( k − k )+ k k (cid:63) a ( k − k ) (cid:3) − (cid:16) − ξ (cid:17) [ δ bc ( k (cid:63) a k ) k (cid:63) b ( k − k )+ δ ac ( k (cid:63) b k ) k (cid:63) c ( k − k ) + δ ab ( k (cid:63) c k ) k (cid:63) a ( k − k )] } . (73)If we assume a diagonal basis for H a , then we get iV ABCabc ( k , k , k ) = ig f ABC (cid:32) ∑ i = V ( i ) abc ( k , k , k ) + (cid:16) − ξ (cid:17) V ( ) abc ( k , k , k ) (cid:33) (74)4with V ( ) abc ( k , k , k ) = + k a k b k c ( δ bc ( k a − k a ) + δ ac ( k b − k b ) + δ ab ( k c − k c )) (75) V ( ) abc ( k , k , k ) = + δ abc (cid:0) k k a ( k a − k a ) + k k a ( k a − k a ) + k k a ( k a − k a ) (cid:1) (76) V ( ) abc ( k , k , k ) = − (cid:0) δ bc k a k b ( k b − k b ) + δ ac k b k a ( k a − k a ) + δ ab k c k a ( k a − k a ) (cid:1) . (77)Setting ξ = V ( ) abc can be ignored. Treelevel ξ -dependent vertex terms are an interesting distinction from the usual vector potential gaugetheory. The numbering here is organized according to powers of A µ that contribute to these θ a vertices in the following way: f ABC ∂ µ A A ν A B µ A C ν → V ( ) (78) ∂ µ A A ν ∂ µ A A ν → V ( ) (79) (cid:16) − ξ (cid:17) ∂ µ A A µ ∂ ν A A ν → (cid:16) − ξ (cid:17) V ( ) . (80) The quartic vertex (or four θ vertex) can be written as iV ABCDabcd ( k , k , k , k ) = ∂ ( i L θ ) ∂ θ Aa ( k ) ∂ θ Bb ( k ) ∂ θ Cc ( k ) ∂ θ Dd ( k ) (81) = ig (cid:32) ∑ i = V ABCD ( i ) abcd + (cid:16) − ξ (cid:17) ∑ i = V ABCD ( i ) abcd (cid:33) (82)5where we define 8 terms as V ABCD ( ) abcd = − f ABE f CDE δ ac δ bd ( k (cid:63) a k ) ( k (cid:63) b k ) + perms . (83) V ABCD ( ) abcd = − f ABE f CDE δ bcd ( k (cid:63) a ( k + k )) ( k (cid:63) b k ) + perms . (84) V ABCD ( ) abcd = − f ABE f CDE δ acd ( k (cid:63) b k ) ( k (cid:63) c k ) + perms . (85) V ABCD ( ) abcd = + f ABE f CDE δ ab δ cd ( k (cid:63) b k ) ( k (cid:63) c k ) + perms . (86) V ABCD ( ) abcd = + f ABE f CDE δ abcd ( k + k ) ( k (cid:63) a k ) + perms . (87) V ABCD ( ) abcd = + f ABE f CDE δ abcd k ( k (cid:63) a k ) + perms . (88) V ABCD ( ) abcd = − f ABE f CDE δ ab δ cd ( k (cid:63) a ( k + k )) ( k (cid:63) c ( k + k )) + perms . (89) V ABCD ( ) abcd = − f ABE f CDE δ bcd ( k (cid:63) a k ) ( k (cid:63) b k ) + perms . (90)Here we are using the notation f ABC = f CAB = f ABC for convenience. The numbering here isorganized according to powers of A µ that contribute to these θ a vertices in the following way: f ABE f CDE A µ A A ν B ν A C µ A D ν → V ( ) (91) f ABC ∂ µ A A ν A B µ A C ν → V ( ) + V ( ) + V ( ) (92) ∂ µ A A ν ∂ µ A A ν → V ( ) + V ( ) (93) (cid:16) − ξ (cid:17) ∂ µ A A µ ∂ ν A A ν → (cid:16) − ξ (cid:17) (cid:0) V ( ) + V ( ) (cid:1) . (94)Let’s consider the evaluation of the permutations in each of these terms.Consider first V ( ) . Note that since ABCD = BADC = CDAB = DCBA , we get a symmetryfactor of 4. This means we can write V ABCD ( ) abcd = − f ABE f CDE ( δ ac δ bd ( k (cid:63) a k ) ( k (cid:63) b k ) − δ ad δ bc ( k (cid:63) a k ) ( k (cid:63) b k )) − f ACE f BDE ( δ ab δ cd ( k (cid:63) a k ) ( k (cid:63) c k ) − δ ad δ bc ( k (cid:63) a k ) ( k (cid:63) b k )) − f ADE f BCE ( δ ab δ cd ( k (cid:63) a k ) ( k (cid:63) c k ) − δ ac δ bd ( k (cid:63) a k ) ( k (cid:63) b k )) . (95)If we assume a diagonal basis for H a , this simplifies further to V ABCD ( ) abcd = − k a k b k c k d (cid:16) f ABE f CDE ( δ ac δ bd − δ ad δ bc ) + f ACE f BDE ( δ ab δ cd − δ ad δ bc )+ f ADE f BCE ( δ ab δ cd − δ ac δ bd ) (cid:17) , (96)6which takes on a form proportional to the quartic A µ vertex in the usual formalism. Similarly, weobtain other seven terms of the quartic BTGT vertex by writing the rest of the permutations. Thefull results can be found in Appendix D. The generating function for A µ correlators in the usual formalism is given by the path integral Z [ J ] = ˆ D A D ¯ c D c exp (cid:18) iS [ A , ¯ c , c ] + i ˆ d xJ · A (cid:19) , (97)where S [ A , ¯ c , c ] = ˆ d x (cid:18) − F A µν F A µν − ξ ( ∂ · A ) − ∂ µ ¯ c A D AB µ c B (cid:19) (98)is the Yang-Mills action with gauge fixing and ghosts.Now make A A µ ( x ) = A A µ [ θ ( x )] a composite operator as specified by Eq. (25). This changeaffects both the action and the path measure. The generating function is now Z [ J ] = ˆ D θ D ¯ c D c D ¯ d D d e iS [ A [ θ ] , ¯ c , c ]+ iS gh2 [ θ , ¯ d , d ]+ i ´ d xJ · A [ θ ] , (99)where ¯ d , d are the additional ghosts defined in Eq. (61) and the additional ghost action is S gh2 = ´ d x L gh2 .We will now construct a generating function for correlators of A µ and θ a . We define K Aa as asource for θ Aa and define the new generating function as¯ Z [ J , K ] = ˆ D θ D ¯ c D c D ¯ d D de iS [ A [ θ ] , ¯ c , c ]+ iS gh2 [ θ , ¯ d , d ]+ i ´ d x ( J · A [ θ ]+ K Aa θ Aa ) . (100)In this paper, Eq. (100) will be our definition of the quantized theory and this will be used tocalculate both the θ a and A µ correlators. The difference from the generating function of the A µ formalism shown in Eq. (99) is that A µ is now a composite operator in terms of θ a fields and thepath integral is now over θ a instead of A µ . We will find through explicit computations below that S gh2 [ θ , ¯ d , d ] (the action describing the ghosts coming from the transformation from A B µ to θ Aa ) doesnot contribute to the divergent structure (in dimensional regularization) in the processes that wecompute in this paper. It would be interesting to elucidate this decoupling in a future work.For perturbative computations, we split apart the Yang-Mills action Eq. (98) in the followingway S [ A [ θ ] , ¯ c , c ] = S int [ A [ θ ] , ¯ c , c ] + ˆ d x L θ , (101)7where L θ is defined in Eq. (53). Then we can rewrite all powers of θ a higher than quadratic asfunctional derivatives with respect to iK a . The generating function Eq. (100) can then be writtenas ¯ Z [ J , K ] = ˆ D θ D ¯ c D c D ¯ d D de iS int [ A [ θ ] , ¯ c , c ]+ iS gh2 [ θ , ¯ d , d ]+ i ´ d xJ · A [ θ ] e i ´ d x ( L θ + K Aa θ Aa ) (102) = ˆ D ¯ c D c D ¯ d D d e iS int [ A [ δ i δ K ] , ¯ c , c ] + iS gh2 [ δ i δ K , ¯ d , d ]+ i ´ d xJ · A [ δ i δ K ] × ˆ D θ e i ´ d x ( L θ + K Aa θ Aa ) (103) = N e i ´ d xJ · A [ δ i δ K ] ˆ D ¯ c D c D ¯ d D d e iS int [ A [ δ i δ K ] , ¯ c , c ] + iS gh2 [ δ i δ K , ¯ d , d ] × e i ´ d xd yK Aa ( x ) ∆ ABab ( x − y ) K Bb ( y ) (104)where N is a normalization constant. Eq. (104) is what was used to derive the Feynman rules ofnon-Abelian BTGT, which are presented in Appendix D.
4. BETA FUNCTION COMPUTATION
In this section, we show that the beta function at one loop for non-Abelian BTGT is β ( g ) = − C ( A ) g π (105)which is the same result as the usual A µ formalism of Yang Mills theory. This lends support tothe quantum consistency of the formalism and its faithful representation of the usual non-Abeliangauge theory perturbative content. This result is achieved by computing the renormalization con-stants of the counter-terms of the θ a and ghost quadratic terms and the θ a ¯ cc ghost-gauge vertex.The relevant terms in the Lagrangian are L (cid:51) − Z θ (cid:16) ∂ µ ˜ ∂ a ν θ Aa − ∂ ν ˜ ∂ a µ θ Aa (cid:17) ∂ µ ˜ ∂ ν b θ Ab − ξ Z ξ θ ∂ ν ˜ ∂ a µ θ Aa ∂ µ ˜ ∂ ν b θ Ab − Z ¯ cc ∂ µ ¯ c ∂ µ c + Z g ¯ cc θ g f ABC ∂ µ ¯ c A c B ˜ ∂ µ a θ Ca . (106)These renormalization constants are computed in MS with d = − ε dimensional regularizationto be Z θ = + C ( A ) g π ε + O (cid:0) g (cid:1) (107) Z ¯ cc = + C ( A ) g π ε + O (cid:0) g (cid:1) (108) Z g θ ¯ cc = + C ( A ) g π ε + O (cid:0) g (cid:1) , (109)8which implies Eq. (105) since Z g = Z g θ ¯ cc Z / θ Z ¯ cc = − C ( A ) g π ε + O (cid:0) g (cid:1) (110)In the following subsections, we compute Eqs. (107), (108) and (109). We display a largeamount of details since this BTGT formalism is new and how the formalism works is one of themain results of this paper. For convenience we choose the Feynman gauge ξ = ( H a ) µν : ( H a ) µν = g µ a g ν a g aa (no sum over a ) . We will be using the minimalsubtraction scheme and dimensional regularization with d = − ε to determine the renormalizationconstants. We will also be using the shorthand ˆ (cid:96) ≡ ˆ d d (cid:96) ( π ) d (111)In the computation below, many zeros appear for the following reasons. In dimensional regu-larization, we utilize the identity ˆ d n (cid:96) ( π ) n (cid:96) n + k ∝ δ k , (112)where n > , k are integers and where as is customary, we do not distinguish raised or loweredindices on Kronecker delta functions whenever contextually the Lorentzian metric information isirrelevant. Other diagrams are zero due to the anti-symmetric nature of f ABC . Yet other diagramsare zero due to the identity δ ab ( − δ ab ) = δ ab − δ ab = . (113) Z θ and Z ξ θ The relevant diagrams are defined in Fig. 1. It is understood that when we write symbols suchas D without indices, the implicit indices are understood be of the form ( D ) ABab ( k ) . The θ a selfenergy can be written as i Π ABab ( k ) = ∑ i = ( D i ) ABab ( k ) + ( D c . t . ) ABab ( k ) (114)9 Figure 1: Self energy diagrams for θ a θ self energy diagram 1 Diagram 1 in Fig. 1 is given by ( D ) ABab = ˆ (cid:96) ∑ cde f (cid:0) igV ACDacd ( k , (cid:96) ) (cid:1) (cid:0) − i δ CE δ ce (cid:1) (cid:0) − i δ DF δ ce (cid:1) (cid:16) igV BEFbe f ( − k , − (cid:96) ) (cid:17) (cid:96) (cid:96) c ( (cid:96) + k ) ( (cid:96) d + k d ) (115) = g f ACD f BCD ∑ i = ∑ j = ˆ d (cid:96) ( π ) ∑ cd V ( i ) acd ( k , (cid:96) ) V ( j ) bcd ( − k , − (cid:96) ) (cid:96) ( (cid:96) + k ) (cid:96) c ( (cid:96) d + k d ) (116) = g C ( A ) δ AB ∑ i = ∑ j = (cid:16) D ( i , j ) (cid:17) ab (117)where in the last line we define the sub-diagrams (cid:16) D ( i , j ) (cid:17) ab = ˆ d (cid:96) ( π ) ∑ cd V ( i ) acd ( k , (cid:96) ) V ( j ) bcd ( − k , − (cid:96) ) (cid:96) ( (cid:96) + k ) (cid:96) c ( (cid:96) d + k d ) (118)The sums over i and j in Eq. (117) only go from 1 to 2 because (cid:16) − ξ (cid:17) V ( ) abc = R ξ gauge, the sums in Eq. (117) would go from 1 to 3. Due to the symmetryof the diagram, we also know that (cid:16) D ( j , i ) (cid:17) ABab ( k ) = (cid:16) D ( i , j ) (cid:17) BAba ( − k ) (119)which means there are only three independent terms to compute in Eq. (117).0We start with (cid:16) D ( , ) (cid:17) ab = ˆ d (cid:96) ( π ) ∑ cd V ( ) acd ( k , (cid:96) ) V ( ) bcd ( − k , − (cid:96) ) (cid:96) ( (cid:96) + k ) (cid:96) c ( (cid:96) d + k d ) (120) = ˆ d (cid:96) ( π ) ∑ cd k a k b (cid:96) c ( (cid:96) d + k d ) N abcd (cid:96) ( (cid:96) + k ) (cid:96) c ( (cid:96) d + k d ) (121) = k a k b ˆ d (cid:96) ( π ) ∑ cd N abcd (cid:96) ( (cid:96) + k ) (122)where the numerator is N abcd = ( − δ cd ( (cid:96) a + k a ) + δ ad ( (cid:96) c + k c ) + δ ac ( (cid:96) d − k d )) × ( − δ cd ( (cid:96) b + k b ) + δ bd ( (cid:96) c + k c ) + δ bc ( (cid:96) d − k d )) . (123)Summing over c and d yields ∑ cd N abcd = (cid:96) a (cid:96) b + (cid:96) a k b + k a (cid:96) b − k a k b + (cid:16) ( (cid:96) + k ) + ( (cid:96) − k ) (cid:17) δ ab (124)and applying this to Eq. (122) gives (cid:16) D ( , ) (cid:17) ab = k a k b ˆ (cid:96) (cid:96) a (cid:96) b + (cid:96) a k b + k a (cid:96) b − k a k b + (cid:16) ( (cid:96) + k ) + ( (cid:96) − k ) (cid:17) δ ab (cid:96) ( (cid:96) + k ) (125) =
12 ˜ k µ a ˜ k ν b ˆ (cid:96) (cid:96) µ (cid:96) ν + (cid:96) µ k ν + k µ (cid:96) ν − k µ k ν + (cid:16) ( (cid:96) + k ) + ( (cid:96) − k ) (cid:17) g µν (cid:96) ( (cid:96) + k ) . (126)The momentum integral of Eq. (126) is identical to the one that appears the usual non-Abelian A µ formalism. We can evaluate it using the usual Feynman parameterization technique to obtain (cid:16) D ( , ) (cid:17) ab =
12 ˜ k µ a ˜ k ν b ˆ dx ˆ d d q ( π ) d (cid:0) q + (cid:0) − x + x (cid:1) k (cid:1) g µν − (cid:0) + x − x (cid:1) k µ k ν [ q + x ( − x ) k ] (127)We are only interested in the divergent part, which in dimensional regularization with d = − ε isdiv (cid:16)(cid:16) D ( , ) (cid:17) ab (cid:17) = (cid:18) k k a δ ab − k a k b (cid:19) i π ε (128)which has the same form numerically as the usual non-Abelian A µ formalism.We now compute (cid:16) D ( , ) (cid:17) ab = ˆ d d (cid:96) ( π ) d ∑ cd V ( ) acd ( k , (cid:96) ) V ( ) bcd ( − k , − (cid:96) ) (cid:96) ( (cid:96) + k ) (cid:96) c ( (cid:96) d + k d ) (129) = ˆ d d (cid:96) ( π ) d N ab (cid:96) ( (cid:96) + k ) (cid:96) a ( (cid:96) a + k a ) (130)1where the numerator is N ab = ( δ ab − ) k b (cid:96) a ( (cid:96) a + k a ) ( (cid:96) b + k b ) × (cid:16) k k a ( (cid:96) a + k a ) − (cid:96) (cid:96) a ( (cid:96) a + k a ) + ( (cid:96) + k ) (cid:0) (cid:96) a − k a (cid:1)(cid:17) . (131)The divergent part of Eq. (130) isdiv (cid:16)(cid:16) D ( , ) (cid:17) ab (cid:17) = ( δ ab − ) k b (cid:18) k k a δ ab i π ε (cid:19) = . (132)This is identically zero because of Eq. (113). Due to the symmetry of the diagram we also knowthat div (cid:16)(cid:16) D ( , ) (cid:17) ab (cid:17) = . (133)Finally, we compute (cid:16) D ( , ) (cid:17) ab = ˆ d d (cid:96) ( π ) d ∑ cd V ( ) acd ( k , (cid:96) ) V ( ) bcd ( k , (cid:96) ) (cid:96) ( (cid:96) + k ) (cid:96) c ( (cid:96) d + k d ) (134) = ∑ cd δ acd δ bcd ˆ d d (cid:96) ( π ) d n a ( k , (cid:96) ) n b ( k , (cid:96) ) (cid:96) ( (cid:96) + k ) (cid:96) c ( (cid:96) d + k d ) (135) = δ ab ˆ d d (cid:96) ( π ) d n a ( k , (cid:96) ) (cid:96) ( (cid:96) + k ) (cid:96) a ( (cid:96) a + k a ) (136)where n a ( k , (cid:96) ) = k k a ( (cid:96) a + k a ) − (cid:96) (cid:96) a ( (cid:96) a + k a ) + ( (cid:96) + k ) (cid:0) (cid:96) a − k a (cid:1) . (137)The divergent part of Eq. (136) isdiv (cid:16)(cid:16) D ( , ) (cid:17) ab (cid:17) = i π ε (cid:18) k k a δ ab (cid:19) . (138)After summing the contributions from the sub-diagrams given by Eqs. (128), (132), (133), and(138), we find that divergent part of the first diagram isdiv (cid:16) ( D ) ABab (cid:17) = C ( A ) g π ε (cid:18) i δ AB k k a δ ab − i δ AB k a k b (cid:19) . (139) θ self energy diagram 2 The second diagram is given by ( D ) ABab = ˆ d (cid:96) ( π ) ∑ cd (cid:18) − i δ cd δ CD (cid:96) (cid:96) c (cid:19) iV ABCDabcd ( k , − k , (cid:96), − (cid:96) ) (140) = g ˆ d (cid:96) ( π ) ∑ c ∑ i = V ABCC ( i ) abcc ( k , − k , (cid:96), − (cid:96) ) (cid:96) (cid:96) c ; (141)2the seventh and eighth terms of Eq. (141) don’t contribute because ξ =
1. The following identityis useful in evaluating the divergent part of Eq. (141):div (cid:32) ˆ d (cid:96) ( π ) (cid:96) N a a (cid:96) N b b (cid:96) (cid:96) a (cid:33) = δ N a δ N b div (cid:32) ˆ d (cid:96) ( π ) (cid:96) (cid:96) a (cid:33) (142) = δ N a δ N b i Γ (cid:0) ε (cid:1) Γ (cid:0) − (cid:1) ( π ) Γ (cid:0) (cid:1) (143) = δ N a δ N b (cid:18) − i π ε (cid:19) . (144)Since Eq. (144) is zero in dimensional regularization unless N a = N b =
0, we ignore any term inthe numerator of Eq. (141) that has any positive power of (cid:96) to find the divergence. We need toignore any term that has k = + (cid:96) or k = − (cid:96) since they proportional to (cid:96) .The divergent part of the first four terms of Eq. (141) vanishes due to either Lorentz invarianceor Eq. (144). The only non zero divergent contributions come from the fifth term, which is givenby V ABCC ( ) abcc = + f ACE f BCE δ abc (cid:104) ( k + k ) ( k − k ) (cid:63) a ( k − k )+ ( k + k ) ( k − k ) (cid:63) a ( k − k ) (cid:105) (145) → f ACE f BCE δ abc (cid:16) ( k ) k (cid:63) a k + ( k ) ( k ) (cid:63) a ( k ) (cid:17) (146) → − f ACE f BCE δ ab k k a , (147)and the sixth term, given by V ABCC ( ) abcc = f ACE f BCE δ abc (cid:2) k k (cid:63) a ( k − k ) + k k (cid:63) a ( k − k ) + k k (cid:63) a ( k − k ) + k k (cid:63) a ( k − k )+ k k (cid:63) a ( k − k ) + k k (cid:63) a ( k − k ) + k k (cid:63) a ( k − k ) + k k (cid:63) a ( k − k ) (cid:3) → f ACE f BCE δ ab (cid:0) k k (cid:63) a ( − k ) + k k (cid:63) a ( − k ) + k k (cid:63) a ( − k ) + k k (cid:63) a ( − k ) (cid:1) (148) = + f ACE f BCE δ ab k k a . (149)Applying the results from Eq. (147) and Eq. (149) to Eq. (141) yields the following divergentcontribution div (cid:16) ( D ) ABab (cid:17) = − C ( A ) g π ε (cid:16) i δ AB k k a δ ab (cid:17) . (150)3 θ self energy diagram 3 The ghost-loop diagram 3 of Fig. 1 receives contributions from the ghosts of Eq. (57), whichwe label as D ( gh1 ) and the ghosts of Eq. (61), which we label as D ( gh2 ) : ( D ) ABab = (cid:16) D ( gh1 ) (cid:17) ABab + (cid:16) D ( gh2 ) (cid:17) ABab (151)where (cid:16) D ( gh1 ) (cid:17) ABab = ( − ) ˆ p igV A , CDa ( k , p + k , p ) i ∆ CF ( p + k ) i ∆ DE ( p ) igV B , EFb ( − k , p , p + k ) (152) = ( − ) g ˆ d p ( π ) (cid:0) f ACD k (cid:63) a ( p + k ) (cid:1) (cid:0) f BDC ( − k ) (cid:63) b p (cid:1) p ( p + k ) (153) = g f ACD f BCD (cid:0) − ˜ k µ a ˜ k ν b (cid:1) ˆ d p ( π ) ( p + k ) µ p ν p ( p + k ) (154)and (cid:16) D ( gh2 ) (cid:17) ABab = ( − ) g ∑ c , d ˆ p f ACD δ acd ( k − p ) (cid:63) a ( p + k ) f BDC δ bdc ( − k − p − k ) (cid:63) a p ( p c + k c ) p d (155) = − g C ( A ) δ AB δ ab ˆ p ( p a − k a ) ( p a + k a ) ( p a + k a ) p a ( p a + k a ) p a . (156)Using the usual Feynman parameterization, the integral of Eq. (154) becomes ˆ d d p ( π ) d ( p + k ) µ p ν p ( p + k ) = i ( π ) ε ˆ dx (cid:18) − g µν x ( − x ) k − x ( − x ) k µ k ν (cid:19) + finite (157) = i π ε (cid:18) − k g µν − k µ k ν (cid:19) + finite (158)and therefore div (cid:18)(cid:16) D ( gh1 ) (cid:17) ABab (cid:19) = i δ AB C ( A ) g π ε (cid:18) k k a δ ab + k a k b (cid:19) . (159)The divergent part of D ( gh2 ) in dimensional regularization is zero because of Eq. (112) for n = (cid:18)(cid:16) D ( gh2 ) (cid:17) ABab (cid:19) = . (160)As noted before, it is interesting that the ghosts arising from transforming A B µ to θ Ac do not con-tribute to the divergent structure here. Combining these results, we conclude thatdiv (cid:16) ( D ) ABab (cid:17) = i δ AB C ( A ) g π ε (cid:18) k k a δ ab + k a k b (cid:19) . (161)This ghost contribution will be important for restoring the transverse structure of the gauge bosonpropagator.4 θ self energy diagram 4 Similar to diagram 3, diagram 4 of Fig. 1 describes ghost contributions to the propagator. Thesehowever do not have any external momenta flowing through the ghost-lines. Just as in diagram3, this has a contribution coming from the usual gauge-fixing ghost and the ghost associated withtransforming the field coordinates from A B µ to θ Ac : ( D ) ABab = (cid:16) D ( gh1 ) (cid:17) ABab + (cid:16) D ( gh2 ) (cid:17) ABab (162)We find the first ghost contribution to be (cid:16) D ( gh1 ) (cid:17) ABab = ( − ) ˆ d p ( π ) ig V AB , CDab ( k , − k , p , p ) i ∆ CD ( p ) (163) = ( − ) g ˆ d p ( π ) f ABE f CCE δ ab k a p a p (164) = (cid:16) D ( gh2 ) (cid:17) ABab = ( − ) ∑ c ˆ d p ( π ) ig V AB , CDab , cd ( k , − k , p , p ) i ∆ CDcd ( p ) (166) = − ig δ ab ∑ c ˆ d p ( π ) f ACE f BCE δ abc (( p + k ) (cid:63) a p + ( p − k ) (cid:63) a p ) p a (167) = − ig f ACE f BCE δ ab ˆ d p ( π ) p a p a . (168)Using the identity Eq. (112), this also vanishes:div (cid:18)(cid:16) D ( gh2 ) (cid:17) ABab (cid:19) = . (169)Therefore, we conclude div (cid:16) ( D ) ABab (cid:17) = θ propagator in dimensional regularization.5 θ self energy counter-term The counter-term diagram yields ( D c . t . ) ABab = − i δ AB (cid:18) ( Z θ − ) δ AB (cid:0) k k a δ ab − k a k b (cid:1) + ξ ( Z ξ θ − ) k a k b (cid:19) (171) = − i δ AB (cid:18) ( Z θ − ) k k a δ ab + ( Z ξ θ − Z θ ) k a k b (cid:19) . (172)To have a finite self energy, we require the divergent parts of these diagrams to cancel out. Thesum of Eqs. (139), (150), (161), and (170) isdiv (cid:32) ∑ i = ( D i ) ABab (cid:33) = i δ AB C ( A ) g π ε (cid:18) k k a δ ab − k a k b (cid:19) (173)and therefore the renormalization constants are Z θ = + C ( A ) g π ε (174)and Z ξ θ = + C ( A ) g π ε . (175)It is interesting that despite the nontransversality of the divergent part of the θ propagator seenhere, the divergent part of the usual gauge field propagator when computed in the BTGT formalismwill maintain transversality, as we will demonstrate below. ξ θ Note that Z ξ = Z θ Z ξ θ = + C ( A ) g π ε = Z A = Z A Z ξ A (176)where Z A is gauge kinetic renormalization constant in the usual gauge theory formalism. This is anontrivial check of the theory. It shows that ξ B = Z ξ ξ R has the same scaling behavior in BTGT as inthe usual formalism. It is interesting that while Z ξ A = g , Z ξ θ − (cid:54) =
0. This doesnot indicate a violation of gauge invariance because the gauge fixing parameter ξ (parameterizingthe coefficient of the gauge fixing chosen to be of the same form as in ordinary gauge theories with A a µ → A a µ [ θ ] ) is still renormalized by the same renormalization constant of Z ξ as in the ordinarygauge theory formalism and Z θ (cid:54) = Z A .6 Figure 2: Ghost self energy diagrams
Another nontrivial check of the formalism would be to calculate Z g θ and Z ξ g θ and check thatthey satisfy Z g θ Z ξ g θ = + C ( A ) g π ε + O (cid:0) g (cid:1) = Z ξ , (177)but we defer this to a future work. Z ¯ cc The renormalization constant Z ¯ cc is determined by the ghost self energy. The one loop diagramsthat contribute to the ghost self energy are given in Fig. 2.The first diagram in Fig. 2 is ( D ) AB = g f CAD f CDB ∑ c ˆ (cid:96) ( − (cid:96) c p c ) (cid:96) c ( (cid:96) c + p c ) (cid:96) c (cid:96) ( (cid:96) + p ) (178) = g C ( A ) δ AB ˆ (cid:96) p + p · (cid:96)(cid:96) ( (cid:96) + p ) (179) = g C ( A ) δ AB p ˆ dx ( − x ) ˆ q [ q + x ( − x ) p ] . (180)The divergent part of this isdiv (cid:16) ( D ) AB (cid:17) = − C ( A ) g π ε (cid:16) − i δ AB p (cid:17) . (181)The second diagram in Fig. 2 vanishes identically because of the anti-symmetric property of f CDE : ( D ) AB = g ∑ c , d ˆ (cid:96) V CD , ABcd ( (cid:96), − (cid:96), p ) ∆ CDcd ( (cid:96) ) (182) = g ∑ c , d δ cd (cid:18) f CDE f ABE ( (cid:96) c + (cid:96) c ) p c (cid:19) δ cd δ CD (cid:96) (cid:96) c (183) = . (184)7 Figure 3: Ghost- θ vertex one loop diagrams The counter-term diagram is given by ( D c . t . ) AB = − i ( Z ¯ cc − ) δ AB p . (185)In order to make the ghost self energy finite, we find that Z ¯ cc = + C ( A ) g π ε + O (cid:0) g (cid:1) . (186)Note that Eq. (186) is the same result that is obtained in the usual computation with A a µ fields.This is most likely part of a general result discussed in more detail in 4.4. Z g θ ¯ cc Let’s now compute the θ a -ghost interaction in our continuing efforts to derive Eq. (105). Therelevant diagrams are defined in Fig. 3.One of the surprises in the computation below will be that the first diagram D of Fig. 3 van-ishes. This is in contrast with the case in which θ Aa is replaced by A A µ . Another interesting aspect8of the computation will be that diagrams D and D each violate the BTGT symmetry in thedivergence, but their sum has a cancellation that thereby preserves the BTGT symmetry. θ vertex diagrams 1 and 2 Diagram 1 in Fig. 3 is given by ( D ) ABCa = g f EBD f FDC f AEF ∑ e , f ˆ (cid:96) ( − (cid:96) e q e ) (cid:0) (cid:96) f + k f (cid:1) (cid:0) (cid:96) f + q f (cid:1) (cid:16) V ( ) ae f ( k , (cid:96) ) + V ( ) ae f ( k , (cid:96) ) (cid:17) (cid:96) ( (cid:96) + k ) ( (cid:96) + q ) (cid:96) e (cid:0) (cid:96) f + k f (cid:1) (187) = (cid:16) D ( ) (cid:17) ABCa + (cid:16) D ( ) (cid:17) ABCa (188)where we have denoted the V ( n ) ae f contributions as D ( n ) which we will evaluate separately. Throughthe identity f AEF f EBD f FDC = − f FEA f EDB f DFC = − f ABC C ( A ) , (189)the first contribution can be written as (cid:16) D ( ) (cid:17) ABCa = − g f ABC C ( A ) ∑ e , f ˆ (cid:96) ( − (cid:96) e q e ) (cid:0) (cid:96) f + k f (cid:1) (cid:0) (cid:96) f + q f (cid:1) V ( ) ae f ( k , (cid:96) ) (cid:96) ( (cid:96) + k ) ( (cid:96) + q ) (cid:96) e (cid:0) (cid:96) f + k f (cid:1) (190) = − g f ABC C ( A ) k a × ˆ (cid:96) q a ( (cid:96) + q ) · ( k − (cid:96) ) − ( (cid:96) a + q a ) q · ( (cid:96) + k ) + ( (cid:96) a + k a ) q · ( (cid:96) + q ) (cid:96) ( (cid:96) + k ) ( (cid:96) + q ) . (191)A divergence only occurs when the numerator is at (cid:96) or higher powers in (cid:96) . There are no termshigher than (cid:96) and therefore the maximum degree of divergence is zero. This means that we canignore the dependence on the external momenta in the denominator:div (cid:18)(cid:16) D ( ) (cid:17) ABCa (cid:19) = − g f ABC C ( A ) k a div (cid:32) ˆ (cid:96) ( (cid:96) · q ) (cid:96) a − (cid:96) q a (cid:96) ( (cid:96) + k ) ( (cid:96) + q ) (cid:33) (192) = − g f ABC C ( A ) k a (cid:18) − q a i π ε (cid:19) (193) = + C ( A ) g π ε (cid:16) ig f ABC k a q a (cid:17) . (194)9The second contribution to this diagram is (cid:16) D ( ) (cid:17) ABCa = − g f ABC C ( A ) ∑ e , f ˆ (cid:96) ( − (cid:96) e q e ) (cid:0) (cid:96) f + k f (cid:1) (cid:0) (cid:96) f + q f (cid:1) V ( ) ae f ( k , (cid:96) ) (cid:96) ( (cid:96) + k ) ( (cid:96) + q ) (cid:96) e (cid:0) (cid:96) f + k f (cid:1) (195) = g f ABC C ( A ) q a ˆ (cid:96) (cid:96) a ( (cid:96) a + k a ) ( (cid:96) a + q a ) × (cid:16) k k a ( (cid:96) a + k a ) − (cid:96) (cid:96) a ( (cid:96) a + k a ) + ( (cid:96) + k ) (cid:0) (cid:96) a − k a (cid:1)(cid:17) (cid:96) ( (cid:96) + k ) ( (cid:96) + q ) (cid:96) a ( (cid:96) a + k a ) . (196)The divergent part evaluates todiv (cid:18)(cid:16) D ( ) (cid:17) ABCa (cid:19) = − C ( A ) g π ε (cid:16) ig f ABC k a q a (cid:17) . (197)Summing these contributions together givesdiv (cid:16) ( D ) ABCa (cid:17) = (cid:18) + − (cid:19) C ( A ) g π ε (cid:16) ig f ABC k a q a (cid:17) = . (198)The result of the diagram D calculation with θ Aa replaced with A A µ is equivalent to Eq. (194)(see e.g. [44]). The difference between this result and Eq. (198) is a manifestation of how θ Aa isdifferent from A A µ .Diagram 2 is given by ( D ) ABCa = g ∑ f ˆ (cid:96) V F , BDf ( − (cid:96) + p , q ) V A , DEa ( k , (cid:96) + k ) V F , ECf ( (cid:96) − p , (cid:96) )( (cid:96) + k ) (cid:96) ( (cid:96) − p ) (cid:0) (cid:96) f − p f (cid:1) (199) = g f FBD f ADE f FEC ∑ f ˆ (cid:96) (cid:0) − (cid:96) f + p f (cid:1) q f k a ( (cid:96) a + k a ) (cid:0) (cid:96) f − p f (cid:1) (cid:96) f ( (cid:96) + k ) (cid:96) ( (cid:96) − p ) (cid:0) (cid:96) f − p f (cid:1) (200) = g C ( A ) f ABC k a ˆ (cid:96) ( (cid:96) a + k a ) ∑ f q f (cid:96) f ( (cid:96) + k ) (cid:96) ( (cid:96) − p ) , (201)and the divergent part of this diagram is thereforediv (cid:16) ( D ) ABCa (cid:17) = + C ( A ) g π ε (cid:16) i f ABC k a q a (cid:17) . (202)The 1 / θ Aa with A A µ in the usual gauge theory.0 θ vertex diagram 3 and 4 Diagram 3 evaluates to ( D ) ABCa = g ˆ (cid:96) ∑ d V AD , BEad ( k , − (cid:96) ; q , (cid:96) + p ) V D , ECd ( (cid:96) ; (cid:96) + p , p ) (cid:96) (cid:96) d ( (cid:96) + p ) (203) = g f ADF f BEF f DEC ∑ d ˆ (cid:96) δ ad ( k d + (cid:96) d ) q d (cid:96) d ( (cid:96) d + p d ) (cid:96) (cid:96) d ( (cid:96) + p ) (204) = g C ( A ) f ABC q a ˆ (cid:96) ( (cid:96) a + k a ) (cid:96) a ( (cid:96) a + p a ) (cid:96) ( (cid:96) + p ) (cid:96) a (205)and after integrating, we find the divergent part isdiv (cid:16) ( D ) ABCa (cid:17) = C ( A ) g π ε (cid:16) ig f ABC (cid:17) (cid:18) q a k a + q a (cid:19) . (206)Diagram 4 evaluates to ( D ) ABCa = g ˆ (cid:96) ∑ d V D , BEd ( − (cid:96) ; q , (cid:96) + p ) V AD , ECad ( k , (cid:96) ; (cid:96) + q , p ) (cid:96) (cid:96) d ( (cid:96) + q ) (207) = g f DBE f ADF f ECF ˆ (cid:96) ∑ d ( − (cid:96) d q d ) δ ad ( k a − (cid:96) a ) ( (cid:96) a + q a ) (cid:96) ( (cid:96) + q ) (cid:96) d (208) = − g C ( A ) f ABC q a ˆ (cid:96) (cid:96) a ( (cid:96) a − k a ) ( (cid:96) a + q a ) (cid:96) ( (cid:96) + q ) (cid:96) a , (209)and the divergent part isdiv (cid:16) ( D ) ABCa (cid:17) = C ( A ) g π ε (cid:16) ig f ABC (cid:17) (cid:18) q a k a − q a (cid:19) . (210)Even though the divergent parts of D and D separately lead to new counter terms that wouldviolate BTGT and gauge invariance, their sum does not. The BTGT violating term proportional to q a cancels and we are left withdiv (cid:16) ( D ) ABCa + ( D ) ABCa (cid:17) = C ( A ) g π ε (cid:16) ig f ABC q a k a (cid:17) . (211)This contribution does not have an analog in the ordinary gauge theory formalism in which thereis no quartic coupling of the gauge sector to the ghosts.1 θ vertex diagram 5 Diagram 5 in Fig. 3 is given by ( D ) ABCa = g ∑ d , e ˆ (cid:96) V ADEade ( k , (cid:96) ) V DE , BCde ( − (cid:96), (cid:96) + k ; q , p ) (cid:96) ( (cid:96) + k ) (cid:96) d ( (cid:96) e + (cid:96) e ) (212) = g ∑ d , e ˆ (cid:96) f ADE V ade ( k , (cid:96) ) δ de f DEF f BCF q d ( − (cid:96) d − k d ) (cid:96) ( (cid:96) + k ) (cid:96) d ( (cid:96) e + (cid:96) e ) (213) = − g C ( A ) f ABC ∑ d ˆ (cid:96) V add ( k , (cid:96) ) q d ( (cid:96) d + k d ) (cid:96) ( (cid:96) + k ) (cid:96) d ( (cid:96) d + k d ) (214) = (cid:16) D ( ) (cid:17) ABCa + (cid:16) D ( ) (cid:17) ABCa . (215)We find that div (cid:18)(cid:16) D ( ) (cid:17) ABCa (cid:19) = , (216)div (cid:16) D ( ) (cid:17) = − C ( A ) g π ε (cid:16) ig f ABC q a k a (cid:17) , (217)and D ( ) = (cid:16) ( D ) ABCa (cid:17) = − C ( A ) g π ε (cid:16) ig f ABC q a k a (cid:17) . (218)Diagram 6 in Fig. 3 is ( D ) ABCa = g ∑ d ˆ (cid:96) V ADD , BCadd ( k , (cid:96), − (cid:96) ; p , q ) (cid:96) (cid:96) d (219) = g ∑ d ˆ (cid:96) δ add q a f BCF (cid:0) + f FDG f DAG ( k a + (cid:96) a ) + f FDG f ADG ( (cid:96) a − k a ) (cid:1) (cid:96) (cid:96) d (220) = g f BCF (cid:16) − C ( A ) δ AF (cid:17) q a ˆ (cid:96) k a (cid:96) (cid:96) a . (221)The divergent part turns out to bediv (cid:16) ( D ) ABCa (cid:17) = + C ( A ) g π ε (cid:16) ig f ABC q a k a (cid:17) . (222) The counter term is ( D c . t . ) ABCa = (cid:0) Z g θ ¯ cc − (cid:1) (cid:16) ig f ABC q a k a (cid:17) . (223)2After summing the contributions from Eqs. (198), (202), (211), (218), and (222), we immediatelyfind the renormalization constant Z g θ ¯ cc = + C ( A ) g π ε + O (cid:0) g (cid:1) . (224)Hence, we have finally accomplished our computation of the Z g given by Eq. (110) using thenon-Abelian BTGT formalism. Thus, as mentioned at the beginning of this section where we em-barked on an explicit computation of the beta function, it is gratifying to see that the θ Aa formalismcan be used to reproduce the perturbative results of the A A µ formalism. The true physics advantageof using the non-Abelian BTGT formalism has yet to be discovered, but its existence is expectedsince simple correlators in θ Aa will map to nonlinear and nonlocal A B µ correlators. Here we give another perspective on the beta function computation which we have explicitlycarried out in the previous subsections. We expect the correlator (cid:10) ΨΨ (cid:11) to be independent of thegauge formalism chosen for any matter or ghost field Ψ because the change from the A µ formalismto θ a formalism does not depend on Ψ . In other words, assuming (cid:10) ΨΨ (cid:11) ( A ) = (cid:10) ΨΨ (cid:11) ( θ ) , (225)and using the Callan–Symanzik equation (cid:20) ∂∂ ln µ + β ( g ) ∂∂ g − ξ ∂ ln Z ξ ∂ ln µ ∂∂ ξ + ∂ ln Z ΨΨ ∂ ln µ (cid:21) (cid:10) ΨΨ (cid:11) = , (226)we infer that β ( A ) ( g ) = β ( θ ) ( g ) , (227) Z ( A ) ξ = Z ( θ ) ξ , (228)and Z ( A ) ΨΨ = Z ( θ ) ΨΨ . (229)Even more generally, the anomalous dimension of any matter or ghost field should be independentof the gauge formalism.3
5. COMPOSITE OPERATOR CORRELATOR
One of the key differences of non-Abelian BTGT from Abelian BTGT is the appearance of thenonlinearity in the map between the θ Aa variable and the ordinary gauge field A A µ variable. Hence,any A A µ [ θ ] correlator computation in ordinary field theory turns into a composite operator corre-lation computation beyond the leading order in the coupling constant expansion. To demonstrateexplicitly that we can recover the gauge dynamics of A A µ at the quantum level using the non-AbelianBTGT formalism, we give in this section an example of the requisite composite operator renormal-ization. We will find that the transverse divergent structure of the two-point function is recoveredonly after including the composite operator renormalization, indicating the self-consistency of theformalism and that ordinary gauge invariance is not spoiled by the nonlinear field redefinition andthe BTGT symmetry. We will also show in this section that there is a sufficient number of counterterm coefficients to preserve finiteness of both θ Aa and A B µ correlators without spoiling the gaugeand BTGT symmetries, lending further evidence that the θ Aa theory is a consistent rewriting of the A A µ theory.More explicitly, define the two-point momentum space Green’s function by G AB µν ( k ) = ˆ d x e − ik · x (cid:68) A A µ ( x ) A B ν ( ) (cid:69) (230) = ˆ d x e − ik · x δ i δ J µ A ( x ) δ i δ J ν B ( ) ¯ Z [ J , K ] (cid:12)(cid:12)(cid:12)(cid:12) J = , K = (231)where ¯ Z [ J , K ] is the generating function defined in Eq. (100). The difference from the usual gen-erating function Eq. (99) is that A µ is now a composite operator in terms of θ a fields and thepath integral is now over θ a instead of A µ . Using dimensional regularization with d = − ε , wewill demonstrate below that the divergent part of the momentum space Green’s function for A µ istransverse and exactly the same as the typical formulation before introducing counter terms.div (cid:16) G AB µν ( k ) (cid:17) = C ( A ) g π ε i δ AB (cid:18) g µν k − k µ k ν k (cid:19) + div (cid:16) ( D c . t . ) AB µν + ( D c . t . ) AB µν + ( D c . t . ) AB µν (cid:17) (232)Furthermore, after introducing counter terms, we will find that both (cid:10) θ Aa θ Bb (cid:11) and (cid:68) A A µ A B ν (cid:69) can bemade finite without changing the symmetries of the theory. The details of the (cid:68) A A µ A B ν (cid:69) computa-tion are presented below.This calculation simplifies significantly when using the Feynman gauge. This is due to thegauge propagator becoming diagonal in the BTGT indices, which greatly simplifies the sums.4 Figure 4: Diagram to compute the A µ [ θ ] two-point correlator. The blob in D refers to all 1PI sub-diagramsand is proportional to the θ a self energy. A further breakdown is shown in Fig. 5. The tree level diagram for the two-point A µ correlator in the Feynman gauge is = ∑ a , b (cid:16) − i ˜ k a µ (cid:17) i ∆ ABab ( k ) (cid:16) i ˜ k b ν (cid:17) (233) = − i δ AB k ∑ a ˜ k a µ ˜ k a ν k a (234) = − i δ AB k ∑ a ( H a ) µν (235) = − i δ AB k g µν (236)as expected. The structure is essentially identical to Abelian BTGT at this level of approximation. Next we consider the one-loop diagrams determining the composite operator counter-terms.The diagrams involved in evaluating (cid:68) A A µ A B ν (cid:69) at one loop are shown in Fig. 4.The first diagram in Fig. 4 is given by ( D ) AB µν = ∑ i = (cid:16) D ( i ) (cid:17) AB µν (237)5where i runs through the two possible terms of the θ vertex and (cid:16) D ( i ) (cid:17) AB µν = g C ( A ) δ AB ∑ a , b ˆ d d (cid:96) ( π ) d (cid:16) (cid:96) a µ + ˜ k a µ (cid:17) V ( i ) baa ( k , (cid:96) ) (cid:96) a (cid:96) ( (cid:96) a + k a ) ( (cid:96) + k ) ˜ k b ν k b k . (238)The θ vertex in Eq. (238) can be written as V ( ) baa ( k , (cid:96) ) = k b (cid:96) a ( (cid:96) a + k a ) ( δ ab − ) ( (cid:96) b + k b ) (239) V ( ) baa ( k , (cid:96) ) = δ ab (cid:16) k k a ( (cid:96) a + k a ) − (cid:96) (cid:96) a ( (cid:96) a + k a ) + ( (cid:96) + k ) (cid:0) (cid:96) a − k a (cid:1)(cid:17) (240)where there is no sum over a or b . Using Eq. (239), we find (cid:16) D ( ) (cid:17) AB µν = g C ( A ) δ AB ∑ a , b k b ˜ k b ν k b k ( δ ab − ) ˆ d d (cid:96) ( π ) d (cid:16) (cid:96) a µ + ˜ k a µ (cid:17) (cid:96) a ( (cid:96) b + k b ) ( (cid:96) a − k a ) (cid:96) a (cid:96) ( (cid:96) a + k a ) ( (cid:96) + k ) (241) = g C ( A ) δ AB ∑ a , b k b ˜ k ν b k b k ( δ ab − ) (cid:18) δ a µ δ ab i π ε + finite (cid:19) (242) = + finite . (243)From Eq. (240), we find (cid:16) D ( ) (cid:17) AB µν = g C ( A ) δ AB ∑ a ˜ k ν a k a k (cid:18)(cid:0) k µ a (cid:1) i π ε + finite (cid:19) (244) = − C ( A ) g π ε (cid:18) i δ AB g µν k (cid:19) + finite. (245)Adding up the contributions givesdiv (cid:16) ( D ) AB µν (cid:17) = − C ( A ) g π ε (cid:18) i δ AB g µν k (cid:19) . (246)The symmetry between diagrams 1 and 2 of Fig. 4 is given { A , k } ↔ { B , − k } , and we can thereforeconclude without computationdiv (cid:16) ( D ) AB µν (cid:17) = − C ( A ) g π ε (cid:18) i δ AB g µν k (cid:19) . (247)The third diagram in Fig. 4 is given by ( D ) AB µν = ∑ b , b (cid:48) , c , d ˆ (cid:96) ig V A , B (cid:48) CD µ , b (cid:48) cd ( k ; − k , (cid:96), − (cid:96) ) i ∆ B (cid:48) Bb (cid:48) b ( k ) i ∆ CDcd ( (cid:96) ) i ˜ k b ν (248) = g ∑ b , c ˜ k b ν k k b ˆ (cid:96) δ bcc + f ACE f BCE (cid:16) − ˜ (cid:96) b µ + ˜ k b µ (cid:17) + f ACE f BCE (cid:16) ˜ (cid:96) b µ + ˜ k b µ (cid:17) (cid:96) (cid:96) c (249) = + C ( A ) g π ε (cid:18) i δ AB g µν k (cid:19) + finite . (250)6 Figure 5: Breakdown of ( D ) AB µν from Fig. 4; they are equivalent to the θ a self energy diagrams of Fig. 1. Since the fourth diagram in Fig. 4 must be the same as D up to { A , k } ↔ { B , − k } , we can imme-diately write div (cid:16) ( D ) AB µν (cid:17) = + C ( A ) g π ε (cid:18) i δ AB g µν k (cid:19) . (251)Diagram 5 in Fig. 4 is ( D ) AB µν = ∑ a , b (cid:16) g (cid:17) ˆ d d (cid:96) ( π ) d f ACD (cid:16) − (cid:96) a µ − ˜ k a µ (cid:17) f BCD (cid:0) (cid:96) b ν + ˜ k b ν (cid:1) (cid:96) (cid:96) a ( (cid:96) + k ) ( (cid:96) b + k b ) . (252)This momentum integral does not UV diverge for d =
4: i. e.div (cid:16) ( D ) AB µν (cid:17) = . (253) θ self energy diagrams Diagram 6 in Fig. 4 is the sum of all 1PI sub-diagrams as shown in Fig. 5. Using the results ofSection 4.1, we have div (cid:16) Π ABab ( k ) (cid:17) = C ( A ) g π ε δ AB (cid:18) k k a δ ab − k a k b (cid:19) (254)where Π ABab ( k ) is the θ a self energy. The divergent part of diagram 6 is given bydiv (cid:16) ( D ) AB µν (cid:17) = ∑ a , b , a (cid:48) , b (cid:48) (cid:16) − i ˜ k a µ (cid:17) i ∆ AA (cid:48) aa (cid:48) ( k ) div (cid:16) i Π A (cid:48) B (cid:48) a (cid:48) b (cid:48) ( k ) (cid:17) i ∆ B (cid:48) Bb (cid:48) b ( k ) (cid:16) i ˜ k b ν (cid:17) (255) = − ∑ a , b ˜ k a µ ˜ k b ν k k a k b i div (cid:16) Π ABab ( k ) (cid:17) (256) = C ( A ) g π ε i δ AB (cid:18) g µν k − k µ k ν k (cid:19) . (257)As expected, the divergences of Fig. 5 are completely canceled out by the renormalization con-stants Z θ and Z ξ θ that arise from D c . t . in Fig. 4.7 Adding up the contributions from the six diagrams of Fig. 4, given by Eqs. (246), (247), (250),(251), (253), and (257) gives the divergent part of the two-point A correlator before renormaliza-tion: ∑ i = div (cid:16) ( D i ) AB µν (cid:17) = C ( A ) g π ε i δ AB (cid:18)(cid:18) − + + (cid:19) g µν k − k µ k ν k (cid:19) (258) = C ( A ) g π ε i δ AB (cid:18) g µν k − k µ k ν k (cid:19) . (259)It has the expected transverse property and the same numerical value as in the usual A µ formula-tion. While the k µ k ν term receives a contribution from only diagram D , the g µν term receivescontributions from six diagrams D through D .Now we need to renormalize both θ a and the composite operator A µ [ θ ] and show that bothcorrelators are finite without introducing any counter terms that spoil gauge invariance, BTGTinvariance, or Lorentz invariance. The composite operator counter terms in the Lagrangian are ofthe form L c . t . (cid:51) ( Z J θ − ) J A µ ˜ ∂ a µ θ Aa + (cid:16) Z gJ θ − (cid:17) g f ABC J A µ ˜ ∂ a µ θ Ba θ C + . . . (260)and to preserve BTGT invariance the counter terms have to obey certain relations given by Z J = Z J θ Z / θ = Z Jg θ Z g Z θ = Z Jg θ Z g Z / θ = . . . (261)where we have defined Z J to be the ratio of the bare source J to the renormalized source J : J ≡ Z J J .The Z J θ counter-term occurs in diagrams D c . t . and D c . t . of Fig. 4, which evaluate to ( D c . t . ) AB µν = ∑ a , b (cid:16) − i ( Z J θ − ) ˜ k a µ (cid:17) i ∆ ABab ( k ) (cid:16) i ˜ k b ν (cid:17) (262) = ( Z J θ − ) i δ AB k ∑ a ˜ k a µ ˜ k a ν k a (263) = ( Z J θ − ) (cid:18) − i δ AB k g µν (cid:19) , (264)and ( D c . t . ) AB µν = ∑ a , b (cid:16) − i ˜ k a µ (cid:17) i ∆ ABab ( k ) (cid:16) i ( Z J θ − ) ˜ k b ν (cid:17) (265) = ( Z J θ − ) (cid:18) − i δ AB k g µν (cid:19) . (266)8Using the results of Section 4.1, we find ( D c . t . ) AB µν = ∑ a , b , a (cid:48) , b (cid:48) − i ˜ k a µ i ∆ AA (cid:48) aa (cid:48) ( k ) i δ AB (cid:18) ( Z θ − ) k k a δ ab + (cid:18) Z ξ θ − Z θ (cid:19) k a k b (cid:19) (267) × i ∆ B (cid:48) Bb (cid:48) b ( k ) i ˜ k b ν = ∑ a , b ˜ k a µ ˜ k b ν k k a k b i δ AB (cid:18) C ( A ) g π ε k k a δ ab − C ( A ) g π ε k a k b (cid:19) + O (cid:0) g (cid:1) (268) = C ( A ) g π ε δ AB i (cid:18) − g µν k + k µ k ν k (cid:19) + O (cid:0) g (cid:1) . (269)The divergence of all these diagrams cancel to make the two-point A µ correlator finite:div (cid:16) G AB µν ( k ) (cid:17) = div (cid:32) ∑ i = ( D i ) AB µν + ( D c . t . ) AB µν + ( D c . t . ) AB µν + ( D c . t . ) AB µν (cid:33) (270) = (cid:18) C ( A ) g π ε + ( Z J θ ) (cid:19) (cid:18) − i δ AB k g µν (cid:19) (271) = . (272)The renormalization constant Z J θ is therefore Z J θ = + C ( A ) g π ε + O (cid:0) g (cid:1) . (273)Using this, Eq. (107), and Eq. (261), we see that Z − J = Z / θ Z J θ = + C ( A ) g π ε + O (cid:0) g (cid:1) = Z / A . (274)The self-consistency of the renormalization, Z / θ / Z J θ equals Z / A , is as expected from the externalsource coupling in the usual A A µ theory being of the form L J (cid:51) J µ A A A µ (275)where A A µ are renormalized fields, while in the BTGT formulation the source coupling is definedwith a composite operator renormalization constant Z J θ as seen in Eq. (260).
6. COUNTER TERM PREDICTIONS AND SLAVNOV-TAYLOR IDENTITIES
The Slavnov-Taylor identities have yet to be formally derived or shown to exist for the BTGTformalism. This is an interesting area for future study. The one loop calculations done thus far9show that g and ξ scale as expected, and A scales as expected when written as a composite operatorof θ . Assuming that the symmetries in BTGT are preserved in a way similar to the explicitlycomputed processes in this paper, we state in this section a set of concrete generalizations for theone loop counter term factors for the θ n -vertex.We expect the BTGT formulation of the Slavnov-Taylor identities to show that the followingholds Z g n − θ n = Z n − g Z n / θ ( n ≥ ) (276) Z ξ − g n − θ n = Z − ξ Z n − g Z n / θ ( n ≥ ) (277) Z g n θ n ¯ cc = Z ¯ cc Z ng Z n / θ ( n ≥ ) . (278)Based on calculated value in Eq. (174), the predictions are Z g n − θ n = + + n C ( A ) g π ε + O (cid:0) g (cid:1) ( n ≥ ) (279) Z ξ − g n − θ n = + + n C ( A ) g π ε + O (cid:0) g (cid:1) ( n ≥ ) (280) Z g n θ n ¯ cc = + + n C ( A ) g π ε + O (cid:0) g (cid:1) ( n ≥ ) . (281)We have explicitly computed the n = n = n = n = θ vertex diagrams. Also ofinterest is the n = θ ¯ cc vertex.The factors Z g , Z ξ and Z ¯ cc are unchanged by the choice of using either the BTGT field θ a orthe vector potential A µ to describe the gauge boson sector. We could have started by assuming thatthe following relations would hold: Z ( θ ) g = Z ( A ) g = − C ( A ) g π ε + O (cid:0) g (cid:1) (282) Z ( θ ) ξ = Z ( A ) ξ = + C ( A ) g π ε + O (cid:0) g (cid:1) (283) Z ( θ ) ¯ cc = Z ( A ) ¯ cc = + C ( A ) g π ε + O (cid:0) g (cid:1) , (284)where Z ( θ ) is calculated in θ a formalism and Z ( A ) in the A µ formalism. Therefore, Z θ is the onlya priori undetermined parameter in Eqs. (276), (277), and (281) . Since we have done four compu-tations and there was only one a priori undetermined parameter, we have done three independentnontrivial checks of the gauge invariance of this theory at one loop level. This result gives usconfidence that gauge invariance in the BTGT formalism is preserved in perturbation theory.0
7. CONCLUSIONS
We have constructed a non-Abelian basis tensor gauge theory (BTGT) which gives an alternateformulation of usual non-Abelian gauge theory in terms of the vierbein analog for ordinary gaugebundles. For example, the basis tensor that couples to matter transforming as N of SU ( N ) has therepresentation ¯ N and has the Lorentz transformation properties of a rank 2 projection tensor. Tomatch the usual gauge theory formalism, the basis tensor must satisfy Eq. (17) and the couplingsmust be symmetric under a non-gauge symmetry called BTGT symmetry that is identical to theBTGT transformation of the Abelian case. To have a simple match in the number of degrees offreedom between the ordinary gauge theory formalism and the BTGT formalism, we have decidedto choose the scalar fields θ Aa that parameterize the basis tensor to be in the target space of thegauge manifold just as in Abelian BTGT. As in the Abelian BTGT case, the map between θ Fc is a nonlocal functional of A B µ . More explicitly, θ c is a type of path-ordered line integral of A µ ,and hence is related to Wilson lines. However, unlike in the Abelian case, the map between A B µ and θ Fc is nonlinear, where the nonlinearities form a power series of the structure constants. Thismeans that any A B µ correlator computation is a composite operator correlator with respect to the θ Fc elementary field theory requiring composite operator counter terms.The Feynman rules for the 1-loop order and O ( g ) computations were explicitly presented.We have tested non-Abelian BTGT to one-loop and O ( g ) (where g is the usual gauge coupling),using θ Fc are the elementary field degrees of freedom, by computing the beta function of the gaugecoupling and finding it to be identical to the usual formulation. We have also computed the gaugefield 2-point function to the same one-loop accuracy and found identical results as in the usualgauge theory formulation. In particular, we found that the UV divergent part of the correlatoris transverse just as in the usual gauge theory formulation. Furthermore, the composite operatorcounter terms are sufficient to make both the A B µ correlator and θ Fc correlators finite.Through these explicit computations, we have also given several nontrivial checks that therenormalization constants in the minimal subtraction scheme are identical to those of the usualgauge theory formalism. Although we defer a formal BRST construction for this theory to afuture work, the nontrivial checks indicate that there will be no insurmountable obstacles to itsformulation.Although the nonlinearities in the map between A B µ and θ Ca might make this choice of formalismseem unnecessarily complicated, it is a natural choice from several considerations. First, it leads1to a natural match in the number of functional degrees of freedom of a gauge theory. Second, itis a continuous deformation (as a function of group structure constants) of a simple linear map inthe case of Abelian theories. Third, its semblance with nonlinear sigma-model parameterizationsmay allow several extensions of this work using the techniques that have been developed for sigmamodels. Fourth, the BTGT symmetry which stabilizes the Hamiltonian and the gauge symmetryhave elegant representations given by Eqs. (10) and (26). Note also that from the perspectiveof having a nontrivial transformation that may lead to new insights into the usual gauge theoryformulations, such nonlinear maps are more promising. On the other hand, it is important to keepin mind, just as in the usual sigma model parameterizations, this choice of using θ Ca is far fromunique even though there is uniqueness of the map between the vierbein-like field (cid:2) G ( f ) ( x ) (cid:3) γδ (which θ Ca parameterizes) and the gauge field A µ if we stipulate that the gauged matter kineticterm be locally gauge equivalent to that without a gauge field.Many extensions of this work on BTGT theory beyond explicit constructions of BRST formal-ism are self-evident. To complete the tests of this formalism’s equivalence with the usual StandardModel formulation, BTGT should also be tested in the contexts of spontaneous symmetry breakingand curved spacetime. Since this is a formalism most naturally suited for exploring Wilson lines, itwould be interesting to reformulate the Eikonal phase re-summing soft gluonic effects [45–49] inthis formalism and investigate whether any new insights or simplifications can arise. The enhancedlocal nature of BTGT for dealing with nonlocal quantities such as Wilson lines also suggests ex-ploring its applications in lattice gauge theory [50, 51]. The gauge field representation iU a ˜ ∂ a µ U † a also is reminiscent of the sigma model representation used in [52] to explore topological aspectsof the theories with spontaneously broken global symmetries. This suggests there may be a way tomore conveniently explore the topological aspects of gauge theories using the BTGT formalism.The precise connection between the generalized global symmetries of [53] and the symmetries ofBTGT remains to be clarified. For physics beyond the standard model, it would be interesting tosee if the gauge fields can be interpreted as Nambu-Goldstone bosons of a spontaneously brokentheory since A µ = iU a ˜ ∂ a µ U † a are suggestive of a sigma model. Acknowledgments
DJHC was supported in part by the DOE through grant DE-SC0017647. DJHC would like tothank Lisa Everett for comments on the manuscript. All of our Feynman diagrams in this paper2were made using the help of TikZ-Feynman [54].
Appendix A: Relevant Notation
This section lists the various notations and conventions used throughout this paper. The metricsignature chosen was g µν = diag ( − , + , + , +) . (A1)If ψ µ ( a ) for a ∈ { , , , } are 4 orthonormal Lorentz 4-vectors, we can write an explicit represen-tation of the projection tensors ( H a ) µν as ( H a ) µν = ψ µ ( a ) ψ ( a ) ν g aa . (A2)The H a matrices are commutative.Using these projection tensors ( H a ) µν , we define the following notation related to them. Wedefine the tilde notation as ˜ A µ a ≡ ( H a ) µν A ν (A3)to denote the contraction between H a and any 4-vector A µ . Note that ˜ A a µ = ˜ A µ a because there is nocovariant/contravariant distinction for the BTGT index unlike a Lorentz index µ . Also, we definethe star product as A (cid:63) a B ≡ ( H a ) µν A µ B ν (A4)for any two 4-vectors A µ and B µ . Using the tilde notation defined above, we have the followingidentities A (cid:63) a B = ˜ A µ a B µ = A µ ˜ B µ a = g µν ˜ A µ a ˜ B ν a (A5)We define the product of two Kronecker deltas as δ abc ≡ δ ab δ bc = δ ac δ bc = δ ab δ ac no sum over a , b , c . (A6)moving to Euclideanized space via Wick rotation , we can unambiguously define for any fourvector p µ p a ≡ ψ µ ( a ) p µ (A7)that satisfies p (cid:63) a p = p a and ∑ a = p a = p . (A8)3The group structure constant f ABC is defined by the Lie bracket (cid:104) T A , T B (cid:105) = i f ABC T C (A9)where T A are basis elements of the Lie algebra such that e iT A Γ A are group elements for somefunction Γ A ( x ) . We take the basis of generators such that f ABC is completely anti-symmetric.Given this anti-symmetry, we can define without ambiguity the following f ABC = f CAB = f ABC . (A10)Note that f ABC = f CAB = f BCA .Note that Ref. [29] uses the notation of having the basis tensor index c of θ c (with c ∈{ , , , } ) instead of θ Bc (with c ∈ { , , , } ) as in Eq. (5). Also, the sign convention for θ has been flipped between Eq. (23) of Ref. [29] and Eq. (5).In the Feynman diagrams, all momenta that flow into a vertex are assigned a positive value. Appendix B: The relationship between non-Abelian basis tensor and ordinary gauge fields A µ Here we follow the equivalence-principle-like procedure of [29] to construct the relationship ofnon-Abelian basis tensor and the ordinary non-Abelian gauge field A µ ( x ) .Start with a gauge frame such that the Lagrangian at spacetime point x looks like there is nogauge field (i.e. trivial Chern-Simons number vacuum): L φ ( x ) = ∂ µ ˜ φ a ∂ µ ˜ φ ∗ a ( x ) . (B1)We demand in this special gauge frame that the vierbein-like tensor field has the following valueat point x : ˜ G αβ ( x ) = S αβ ( x ) . (B2)Upon making a gauge transformation to move to the general frame, we have φ ( x ) = e i θ C ( x ) T C ˜ φ ( x ) , (B3)The gauge field in the new frame is ˜ D µ ˜ φ = ˜ g − D µ ˜ g ˜ g − φ (B4)4where ˜ g = e i θ C ( x ) T C . (B5)Hence, we find ∂ µ ˜ φ = ˜ g − ( ∂ µ − iA µ ) φ (B6)where the right hand side can be also be written in terms of ˜ φ as ∂ µ ˜ φ = (cid:2) ∂ µ + ˜ g − ∂ µ ˜ g − i ˜ g − A µ ˜ g (cid:3) ˜ φ . (B7)This implies 0 = (cid:2) ˜ g − ∂ µ ˜ g − i ˜ g − A µ ˜ g (cid:3) ˜ φ (B8)or equivalently A µ ( x ) = − i (cid:2) ∂ µ ˜ g ( x ) (cid:3) ˜ g − ( x ) (B9)which is pure gauge only at a single point x and not for all spacetime (just as in the Abelianconstruction).We can use Eq. (B9) to find the map between G αβ and A µ . Since G αβ is defined to obey thetransformation rule of Eq. (3): G α ( f ) β ( x ) φ ( x ) = ˜ G α ( f ) β ( x ) ˜ φ ( x ) (B10)where φ ( x ) = ˜ g ( x ) ˜ φ ( x ) . (B11)This means G α ( f ) β ( x ) = ˜ G α ( f ) β ( x ) ˜ g − ( x ) . (B12)Similarly as in [29], choose ∂ α ˜ G α ( f ) β ( x ) =
0. To solve for the right hand side of Eq. (B9), we takethe derivative (cid:2) G ( f ) β µ (cid:3) m [ ∂ α ˜ g ] ml + (cid:2) ∂ α G ( f ) β µ (cid:3) m [ ˜ g ] ml = δ ks = dim R ∑ f ξ k ( f ) ξ ∗ s ( f ) (B14)where the ξ ( f ) are constant vectors in the group representation space. This allows us to rewriteEq. (B13) as ξ ∗ s ( f ) (cid:2) G β µ (cid:3) sm [ ∂ α ˜ g ] ml + ξ ∗ s ( f ) (cid:2) ∂ α G β µ (cid:3) sm [ ˜ g ] ml = ξ ∗ s ( f ) (cid:2) G β µ (cid:3) sm ≡ (cid:2) G ( f ) β µ (cid:3) m . (B16)Multiplying both sides by ξ q ( f ) and summing, we find ∑ f ξ q ( f ) ξ ∗ s ( f ) (cid:2) G β µ (cid:3) sm [ ∂ α ˜ g ] ml = − ∑ f ξ q ( f ) ξ ∗ s ( f ) (cid:2) ∂ α G β µ (cid:3) sm [ ˜ g ] ml (B17)to arrive at (cid:2) G β µ (cid:3) qm [ ∂ α ˜ g ] ml = − (cid:2) ∂ α G β µ (cid:3) qm [ ˜ g ] ml (B18) Require that the inverse of (cid:2) G β µ (cid:3) qm exists such that (cid:104) G − λ β (cid:105) bq (cid:2) G β µ (cid:3) qm = δ λµ δ bm (B19)Eq. (B18) then becomes δ λµ [ ∂ α ˜ g ] bl (cid:2) ˜ g − (cid:3) ls = − (cid:104) G − λ β (cid:105) bq (cid:2) ∂ α G β µ (cid:3) qs (B20)After setting λ = α , we sum over α to obtain A µ = i (cid:104) G − αβ (cid:105) (cid:2) ∂ α G β µ (cid:3) (B21)where Eq. (B16) gives the explicit relationship to the basis tensor as (cid:2) G β µ (cid:3) qm = dim R ∑ f ξ q ( f ) (cid:2) G ( f ) β µ (cid:3) m . (B22)Eq. (B21) can also be expressed in terms of derivative of the basis tensor G ( f ) β µ as A µ = i dim R ∑ f (cid:104) G − αβ (cid:105) bq ξ q ( f ) (cid:2) ∂ α G ( f ) β µ (cid:3) s (B23)where one notes (cid:104) G − αβ (cid:105) bq ξ q ( f ) is an object that satisfies the identity dim R ∑ f (cid:104) G − αβ (cid:105) bq ξ q ( f ) (cid:2) G ( f ) β µ (cid:3) s = δ bs δ αµ . (B24)One can check that the non-Abelian basis tensor of Eq. (5) satisfies Eq. (B19). Using the iden-tity ddx exp [ O ( x )] = ˆ dy exp [( − y ) O ( x )] dO ( x ) dx exp [ yO ( x )] (B25)for a matrix O , we can evaluate Eq. (B21) as A Q µ ( x ) = ∑ c (cid:18)(cid:16)(cid:2) θ Jc f J (cid:3) − (cid:17) QR (cid:16) e θ Kc f K − (cid:17) RB ˜ ∂ c µ θ Bc (cid:19) (B26)where f J is a structure constant matrix having the components ( f J ) AB = f JAB .6 Appendix C: Gauge and BTGT transforms
In this appendix, we derive an explicit expression for the finite and linearized gauge and BTGTtransforms of the θ Aa field. The key simplification occurs from the fact the θ Aa parameterizes thegroup manifold. As a result U a ≡ e i θ a has a relatively simple transformation law governed by afirst order differential equation. The result is U a → e i Γ U a e iZ a (C1)The BTGT symmetry can then be seen as a result of the constant of integration. The BTGTsymmetry in Eq. (C1) can also be viewed as the symmetry inherent in the covariant derivative asdefined by D µ ( · ) = ∑ a U a ˜ ∂ a µ ( U † a · ) . (C2)Let’s start with the vector potential given by (cid:0) A µ (cid:1) i j = i ∑ a (cid:16) e i θ a ˜ ∂ a µ e − i θ a (cid:17) i j = i ∑ a (cid:16) U a ˜ ∂ a µ U † a (cid:17) i j (C3)where θ a ≡ θ Aa T A and U a ≡ e i θ a . From now on the group indexes i , j will be dropped and impliedby matrix multiplication. In this appendix, repeated lower-case Latin indices will not be implicitlysummed. Under an infinitesimal gauge transformation parameterized by Γ A , we have δ ˜ A a µ = ( H a ) νµ δ A ν = (cid:104) ˜ D a µ , Γ (cid:105) = ˜ ∂ a µ Γ − i (cid:104) ˜ A a µ , Γ (cid:105) (C4)where Γ ≡ T A Γ A and D µ = ∂ µ − iA µ . In terms of U a this is δ (cid:16) iU a ˜ ∂ a µ U † a (cid:17) = ˜ ∂ a µ Γ + (cid:104) U a ˜ ∂ a µ U † a , Γ (cid:105) (C5) = U a ˜ ∂ a µ (cid:16) U † a Γ U a (cid:17) U † a . (C6)To first order in variations, unitarity implies δ U † a = − U † a δ U a U † a (which is equivalent to keepingall θ Aa real). This can be used to reexpress the left hand side of Eq. (C6) as δ (cid:16) U a ˜ ∂ a µ U † a (cid:17) = δ U a ˜ ∂ a µ U † a + U a ˜ ∂ a µ (cid:16) δ U † a (cid:17) (C7) = − U a ˜ ∂ a µ (cid:16) U † a δ U a (cid:17) U † a . (C8)Combining Eqs. (C6) and (C8), we arrive at the following first order differential equation:˜ ∂ a µ (cid:16) − iU † a δ U a (cid:17) = ˜ ∂ a µ (cid:16) U † a Γ U a (cid:17) . (C9)7The general solution to Eq. (C9) is − iU † a δ U a = U † a Γ U a + Z a (no sum over a ) , (C10)where Z a is an infinitesimal zero mode that satisfies˜ ∂ a µ Z a = . (no sum over a ) (C11)Inhomogeneously transforming θ a by this zero mode is the BTGT symmetry of Eq. (26).Since − iU † a δ U a is an element of the Lie algebra spanned by T A and U a is unitary, we choosethe boundary conditions of Eq. (C9) such that Z a ≡ Z Aa T A for some real components Z Aa that eachsatisfy the zero mode equation. Thus we have the result − i δ U a U † a = Γ + U a Z a U † a . (C12)To solve for the components δ θ Aa − i δ U a U † a = − i δ (cid:16) e i θ a (cid:17) e − i θ a (C13) = ˆ dt e it θ a δ θ a e i ( − t ) θ a e − i θ a (C14) = ˆ dt e it θ a δ θ a e − it θ a (C15) = ˆ dt ∞ ∑ n = ( − it ) n n ! [[ . . . [[ δ θ a , θ a ] , θ a ] . . . ] , θ a ] (C16) = ˆ dt ∞ ∑ n = ( − it ) n n ! (cid:104)(cid:104) . . . (cid:104)(cid:104) T B , T C (cid:105) , T C (cid:105) . . . (cid:105) , T C n (cid:105) θ C a · · · θ C n a δ θ Ba (C17)where we made use of e − C Be C = + [ B , C ] + [[ B , C ] , C ] + [[[ B , C ] , C ] , C ] + . . . . (C18)Note that (cid:104) T B , T C (cid:105) = iT D f DBC = iT A (cid:16) f C (cid:17) AB (C19) (cid:104)(cid:104) T B , T C (cid:105) , T C (cid:105) = i (cid:104) T D , T C (cid:105) f DBC = i T A f ADC f DBC = i T A (cid:16) f C f C (cid:17) AB . (C20)Using iteration it is straight forward to show that (cid:104)(cid:104) . . . (cid:104) T B , T C (cid:105) . . . (cid:105) , T C n (cid:105) = i n T A (cid:16) f C n · · · f C (cid:17) AB (C21)8such that the Eq. (C17) becomes − i δ U a U † a = T A ˆ dt ∞ ∑ n = t n n ! (cid:16) f C n · · · f C (cid:17) AB θ C a · · · θ C n a δ θ Ba = T A (cid:18) e f · θ a − f · θ a (cid:19) AB δ θ Ba . (C22)Another useful identity in solving for δ θ Aa is U a Z a U † a = e i θ a Z a e − i θ a (C23) = ∞ ∑ n = n ! [[ . . . [[ Z a , θ a ] , θ a ] . . . ] , θ a ] (C24) = T A ∞ ∑ n = n ! (cid:16) f C n · · · f C (cid:17) AB θ C a · · · θ C n a Z Ba (C25) = T A (cid:16) e f · θ a (cid:17) AB Z Ba . (C26)We can eliminate T A from both Eq. (C22) and Eq. (C26) to obtain (cid:18) e f · θ a − f · θ a (cid:19) AB δ θ Ba = Γ A + (cid:16) e f · θ a (cid:17) AB Z Ba . (C27)From here, we can immediately solve for δ θ Aa as δ θ Aa = (cid:18) f · θ a e f · θ a − (cid:19) AB Γ B + (cid:18) f · θ a − e − f · θ a (cid:19) AB Z Ba . (C28)Again, both Γ A and Z Aa are infinitesimal parameters in Eq. (C28).Next, we will express the finite gauge and BTGT transformations as a left and right multiplica-tion of a group element representation. Start by writing the condition for δ U a as δ U a = i ( ε Γ U a + U a ε Z a ) = ε ( i Γ U a + iU a Z a ) (C29)where we added ε to Γ and Z to emphasize that the transformation is infinitesimal. We can thenrewrite the infinitesimal transformation using the exponential map as U a → U (cid:48) a = U a + i ε Γ U a + iU a ε Z a (C30) = ( + i ε Γ ) U a ( + i ε Z a ) + O (cid:0) ε (cid:1) (C31) = e i ε Γ U a e i ε Z a + O (cid:0) ε (cid:1) . (C32)Next, if we apply the infinitesimal transformation twice, we see U (cid:48)(cid:48) a = e i ε Γ U (cid:48) a e i ε Z a = e i ε Γ e i ε Γ U a e i ε Z a e i ε Z a = e i ε Γ U a e i ε Z a . (C33)9 Figure 6: Propagators
Thus, we can then iterate this for N = ε times to obtain the finite gauge transformation U a → e i Γ U a e iZ a (C34)which gives an elegant finite gauge and BTGT transformation expression. This can also be ex-pressed as e i θ a → e i Γ e iU a Z a U † a e i θ a . (C35) Appendix D: Feynman Rules
The Feynman rules for non-Abelian BTGT are given in the following figures. Fig. 6 showsthe propagators for the gauge field θ Aa and ghost fields c A and d Aa . Fig. 7 shows the first three θ n vertices that exist for all integer n ≥
3. There are an infinite number of such vertices, but they aresuppressed by higher powers of the gauge coupling g . The explicit form of the θ vertex is notgiven in this paper because it was lengthy to show and was not necessary for the computationsshown in this paper. It can be derived by expanding the Yang-Mills actions written in terms of A [ θ ] and keeping the θ terms.Fig. 8 shows the first three ghost gauge interaction terms. Qualitatively, they are of the form V θ n ¯ cc ∼ g n θ n ¯ cc for all n ≥
1. Like in the case of V θ n , there are an infinite number of such verticesbut are suppressed by higher power of g .The composite operator A A µ [ θ ] defined in Eq. (49) can be computed using the vertices of Fig. 10.0 Figure 7: Gauge interaction vertices up to quartic order in θ .Figure 8: Ghost gauge vertices up to third order in θ
1. Explicit Vertex Expressions
This section contains vertex expressions that were defined in the Feynman rules figures. The θ ¯ cc vertex V ABC , DEabc ( k , k , k ; q ) defined in Figure 8 is iV ABC , DEabc = i δ abc ˜ q a µ f DEF (cid:16) f FAG f BCG (cid:0) k µ − k µ (cid:1) + f FBG f CAG (cid:0) k µ − k µ (cid:1) + f FCG f ABG (cid:0) k µ − k µ (cid:1)(cid:17) , (D1)1 Figure 9: Additional ghost gauge vertices up to second order in θ Figure 10: Composite operator vertices up to third order in θ Figure 11: Quadratic counter-terms Figure 12: Interaction vertex counter-terms where the momenta are constrained to satisfy q = k + k + k + p . The θ ¯ dd vertex V AB , CDab , cd ( k , k ; q , p ) defined in Fig. 9 with q = k + k + p is iV AB , CDab , cd = i δ abcd ˜ q a µ (cid:16) f ABE f CDE ( k µ − k µ ) + f ACE f BDE ( p µ − k µ ) + f ADE f BCE ( p µ − k µ ) (cid:17) . (D2)The J θ vertex V A , BCD µ , bcd ( k , k , k ) defined in Fig. 10 is iV A , BCD µ , bcd = i δ bcd (cid:16) f ABE f CDE ( ˜ k b µ − ˜ k b µ ) + f ACE f BDE ( ˜ k b µ − ˜ k b µ ) + f ADE f BCE ( ˜ k b µ − ˜ k b µ ) (cid:17) , (D3)where the composite operator momentum is k = − k − k − k .
2. Quartic vertex terms
Here, we are using the notation δ bcd = δ bc δ cd such as to avoid confusion regarding summation.The quartic BTGT gauge vertex in Fig. 7 is given by iV ABCDabcd = ig (cid:32) ∑ i = V ABCD ( i ) abcd ( k , k , k , k ) + (cid:16) − ξ (cid:17) ∑ i = V ABCD ( i ) abcd ( k , k , k , k ) (cid:33) (D4)3where the momenta k i must sum to zero. In a diagonal basis for H a , the eight terms are given by V ABCD ( ) abcd = − k a k b k c k d (cid:16) f ABE f CDE ( δ ac δ bd − δ ad δ bc ) + f ACE f BDE ( δ ab δ cd − δ ad δ bc )+ f ADE f BCE ( δ ab δ cd − δ ac δ bd ) (cid:17) (D5) V ABCD ( ) abcd = − (cid:110) f ABE f CDE [ δ bcd k a ( k a + k a ) k b ( k b − k b ) + δ acd k b ( k b + k b ) k a ( k a − k a )+ δ abd k c ( k c + k c ) k a ( k a − k a ) + δ abc k d ( k d + k d ) k a ( k a − k a )]+ f ACE f BDE [ δ bcd k a ( k a + k a ) k b ( k b − k b ) + δ abd k c ( k c + k c ) k a ( k a − k a )+ δ acd k b ( k b + k b ) k d ( k d − k d ) + δ abc k (cid:63) d ( k d + k d ) k b ( k b − k b )]+ f ADE f BCE [ δ bcd k a ( k a + k a ) k d ( k d − k d ) + δ abc k c ( k c + k c ) k a ( k a − k a )+ δ acd k b ( k b + k b ) k c ( k c − k c ) + δ abd k c ( k c + k c ) k b ( k b − k b )] (cid:111) (D6) V ABCD ( ) abcd = − (cid:110) f ABE f CDE [ δ acd k b k b k a ( k a − k a ) + δ bcd k a k a k b ( k b − k b )+ δ abc k d k d k a ( k a − k a ) + δ abd k c k c k a ( k a − k a )]+ f ACE f BDE [ δ abd k c k c k a ( k a − k a ) + δ bcd k a k a k b ( k b − k b )+ δ abc k d k d k a ( k a − k a ) + δ acd k b k b k a ( k a − k a )]+ f ADE f BCE [ δ abc k d k d k a ( k a − k a ) + δ bcd k a k a k b ( k b − k b )+ δ abd k c k c k a ( k a − k a ) + δ acd k b k b k a ( k a − k a )] (cid:111) (D7) V ABCD ( ) abcd = (cid:110) f ABE f CDE δ ab δ cd [ k a k a ( k c − k c )( k c − k c )+ k c k c ( k a − k a )( k a − k a )]+ f ACE f BDE δ ac δ bd [ k a k a ( k b − k b ) ( k b − k b )+ k b k b ( k a − k a ) ( k a − k a )]+ f ADE f CDE δ ad δ bc [ k a k a ( k b − k b )( k b − k b )+ k b k b ( k a − k a )( k a − k a )] (cid:111) (D8) V ABCD ( ) abcd = δ abcd (cid:110) f ABE f CDE ( k + k ) ( k a − k a ) ( k a − k a )+ f ACE f BDE ( k + k ) ( k a − k a ) ( k a − k a )+ f ADE f BCE ( k + k ) ( k a − k a ) ( k a − k a ) (cid:111) (D9)4 V ABCD ( ) abcd = δ abcd (cid:110) f ABE f CDE (cid:2)(cid:0) k k a − k k a (cid:1) ( k a − k a ) + (cid:0) k k a − k k a (cid:1) ( k a − k a ) (cid:3) + f ACE f BDE (cid:2)(cid:0) k k a − k k a (cid:1) ( k a − k a ) + (cid:0) k k a − k k a (cid:1) ( k a − k a ) (cid:3) + f ADE f BCE (cid:2)(cid:0) k k a − k k a (cid:1) ( k a − k a ) + (cid:0) k k a − k k a (cid:1) ( k a − k a ) (cid:3)(cid:111) (D10) V ABCD ( ) abcd = − (cid:110) f ABE f CDE δ ab δ cd ( k a − k a ) ( k a + k a ) ( k c − k c ) ( k c + k c )+ f ACE f BDE δ ac δ bd ( k a − k a ) ( k a + k a ) ( k b − k b ) ( k b + k b )+ f ADE f BCE δ ad δ bc ( k a − k a ) ( k a + k a ) ( k b − k b ) ( k b + k b ) (cid:111) (D11) V ABCD ( ) abcd = − (cid:110) f ABE f CDE (cid:2) δ bcd k a k b ( k b − k b ) + δ acd k b k a ( k a − k a )+ δ abd k c k a ( k a − k a ) + δ abc k d k a ( k a − k a ) (cid:3) + f ACE f BDE (cid:2) δ bcd k a k b ( k b − k b ) + δ abd k c k a ( k a − k a )+ δ acd k b k a ( k a − k a ) + δ abc k d k a ( k a − k a ) (cid:3) + f ADE f BCE (cid:2) δ bcd k a k b ( k b − k b ) + δ abc k d k a ( k a − k a )+ δ acd k b k a ( k a − k a ) + δ abd k c k a ( k a − k a ) (cid:3)(cid:111) . (D12) [1] S. L. Glashow, Partial Symmetries of Weak Interactions , Nucl. Phys. (1961) 579–588. 1[2] S. Weinberg, A Model of Leptons , Phys. Rev. Lett. (1967) 1264–1266.[3] A. Salam, Weak and Electromagnetic Interactions , Conf. Proc.
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