Ward Identity and Basis Tensor Gauge Theory at One Loop

Basis tensor gauge theory (BTGT) is a reformulation of ordinary gauge theory that is an analog of the vierbein formulation of gravity and is related to the Wilson line formulation. To match ordinary gauge theories coupled to matter, the BTGT formalism requires a continuous symmetry that we call the BTGT symmetry in addition to the ordinary gauge symmetry. After classically interpreting the BTGT symmetry, we construct using the BTGT formalism the Ward identities associated with the BTGT symmetry and the ordinary gauge symmetry. As a way of testing the quantum stability and the consistency of the Ward identities with a known regularization method, we explicitly renormalize the scalar QED at one-loop using dimensional regularization using the BTGT formalism.

To match the BTGT action to that of the usual gauge theory coupled to scalars, we had to impose a new local symmetry in [20] which we will call the BTGT symmetry. In this paper, we investigate the Ward identities associated with the BTGT symmetry and the usual U(1) gauge symmetry within the BTGT formalism as a step in building practical computational tools and checking the theory's quantum stability. First, we find that the BTGT symmetry current itself can be classically interpreted as a decomposition of A µ equations of motion in basis tensor components. We will also find a relationship of a particular combination of BTGT current conservation and the residual gauge symmetry current conservation in ξ -fixed ordinary A µ field theory formalism. Next, we use the effective action formalism to derive the BTGT and the U(1) Ward identities in both configuration space and momentum space within the BTGT formalism. These identities are then explicitly applied to 1-loop renormalization of scalar QED. We find that dimensional regularization preserves the BTGT symmetry (in addition to the U(1) as expected). The explicit computations also highlight the utility of the basis tensor star product J c K ≡ J µ (H c ) µν K ν in computing the Feynman graphs within the BTGT formalism. Through this explicit renormalization exercise, we confirm that BTGT is stable at one loop.
The order of presentation is as follows. In the next section, we give a quick overview of the basis tensor gauge theory formalism. In Sec. 3, we construct the BTGT Noether current and give a classical interpretation. In Sec. 4, we briefly review the effective action method of generating Ward identities. In Sec. 5, we compute the Ward identities for both the BTGT symmetry and U(1) symmetry in the context of the BTGT formalism. In Sec. 6, we use the BTGT formalism Feynman rules to explicitly renormalize the scalar QED at 1-loop, checking the quantum stability of the theory as well as the consistency of the dimensional regularization with the Ward identities.
We conclude in Sec. 7 by speculating on future research directions. The Appendix presents basic identities of (H a ) µ ν useful for Feynman diagram computations. There, we also point out a minor typo in equation 36 of [20] 2. REVIEW OF BTGT FORMALISM In this section we give a very brief review of [20]. For more details, we refer the reader to the original article.
A vierbein formulation of Einstein gravity relies on finding a basis of spacetime vector fields that transforms as an (1, 1) under SO(3, 1) ⊗ diffeomorphism. By contracting with vierbeins, a non-singlet diffeomorphism tensor turns into a set of diffeomorphism scalar fields that transform as a non-singlet SO(3, 1) tensor components. The vierbein's relationship with the Christoffel symbols (i.e. the gravitational gauge fields) can be viewed as the vierbeins being solutions to a set of nonlinear partial differential equations involving the Christoffel symbols. The vierbein analog in the context in which ordinary compact Lie groups replace diffeomorphism group is the focus of BTGT.
In the U(1) BTGT of [20], the field G α β is the analog of the gravitational vierbein and it transforms as a (2, 1) under SO(3, 1) ⊗U(1). The constraint of matching to the usual U(1) connection (i.e. the analog of the gravitational vierbein relationship to the Christoffel symbols Γ α µν ) is The number of degrees of freedom in G α β is reduced by introducing BTGT fields θ a (i.e. 4 scalar fields in 4 spacetime dimensions) through of θ a to A µ is then which says that θ a is similar to the Wilson line.
To construct a field theory of θ a (i.e. BTGT theory) that matches the ordinary gauge theory, one must impose the following independent continuous symmetries: 1. Ordinary gauge symmetry: if φ has charge e under U(1), then the U(1) gauge symmetry is 2. A BTGT symmetry: vary just the BTGT field by a restricted class of functions δ θ a = Z a (x) where (H a ) α µ ∂ α Z a (x) = 0 and there is no sum over the repeated a index in this equation.
At the renormalizable level, this set of symmetries reproduces the ordinary U(1) gauge theory action. For example, in the case of scalar QED, the partition function is where the Lagrangian terms for the ξ -gauge fixed action are given by Eq. (14)- (17).
Correlators of certain differences of θ a map to correlators of integrals over A µ .

CLASSICAL BTGT SYMMETRY CURRENT
To write scalar QED in the BTGT formalism, a new local symmetry (in addition to the usual U(1) gauge symmetry) was introduced in [20] which we will refer to as the BTGT symmetry in this paper. To have a physical interpretation of the continuous BTGT symmetry, we construct the Noether current associated with this symmetry in this section. More explicitly, we seek the Noether currents associated with the BTGT symmetry defined as satisfying the constraint equation where there is no sum over the repeated a index in this equation. Note that even though Z a (x) is reminiscent of a pure U(1) gauge field function θ (x) appearing in δ A µ = ∂ µ θ , there is no gauge charged tensor transformation here since all the gauge charged matter fields are held fixed and only the θ a transforms as δ θ a = Z a (x). Furthermore, each Z a (x) for different a indices are independent.
To gain some intuition of the mathematical procedure for constructing the Noether current of a constrained symmetry representation, it is useful to review a mathematically analogous more familiar symmetry: the residual gauge transformation of a ξ -gauge fixed scalar QED theory. With the gauge fixing term there still exists a continuous residual gauge transformation where Eq. (9) is the analog of the constraint Eq. (6). Using the standard Noether construction with the non-symmetry deformation parameterized through an arbitrary continuous function ε(x) as 2 it is straightforward to show Since the classical equations of motion in ξ -gauge satisfies where J ν U(1) is a Noether current associated with the global subgroup of U(1), we see that Eq. (11) indeed is satisfied owing to the antisymmetric property of the field strength tensor and the global U(1) current conservation.
We can carry out the same exercise with the BTGT symmetry of Eq. (5). We consider the ξ -gauge fixed scalar QED in the BTGT basis [20]: In these expressions, the summation over repeated indices is implied, and we will assume below that repeated indices are summed unless stated otherwise explicitly or if it is clear from the context. This action is invariant under the BTGT symmetry Eq. (5). The gauge fixing term L GF breaks the usual U(1) gauge transformations which when written in the BTGT formalism are where θ is an arbitrary smooth function, but it does preserve the global U(1) subgroup.
We find the conservation of Noether current associated with the BTGT symmetry to be where we have introduced the notation This current is ordinary From this, one also sees that the current is invariant under residual gauge transformations of Eq. The BTGT current Eq. (20) can be interpreted in terms of A µ by noting that the classical (see [20] for details) allows us to identify where Hence, the sum of the BTGT currents itself is the equation of motion in terms of the usual gauge field A µ . As we will see below, the equation of motion for θ a differs from the equation of motion for A µ by a derivative. Hence, BTGT current can be interpreted as the decomposition of A µ equation of motion in basis tensor components (to be distinguished from the decomposition of θ a equation of motion in basis tensor components).
To check B µ a conservation using the classical equations of motion, note Eq. (19) can be rewritten as for each a choice. Since this is precisely the equation of motion of θ a , the BTGT classical current is manifestly conserved for classical θ b fields satisfying the equation of motion. On the other hand, the equation of motion for A µ is the sum of the currents shown in Eq. (26) and not the equation of motion for θ b itself: i.e. the difference is a particular type of derivative.
with the Noether current conservation coming from the global subgroup of the U(1) gauge symmetry (which obviously is not broken even for finite ξ ): This means that the ordinary U(1) current conservation and the sum of the BTGT current conservation together enforces 1 which is the same as Eq. (11) enforced by the residual gauge symmetry.

A BRIEF REVIEW OF EFFECTIVE ACTION GENERATING WARD IDENTITIES
We would now like to study the quantum current conservation associated with the BTGT symmetry: i.e. generate Ward identities. For this goal, we briefly review here the effective action formalism for generating Ward identities [29]. This allows us to then reduce the generation of Ward identities to a set of functional derivatives, which we will use to obtain explicit Ward identities for the BTGT and gauge symmetry.
Let ϕ be a vector of fields (e.g. ϕ = (θ a , φ , φ * , ...) and J = (J a , J φ , J φ * , ...)). The generating functional W for the connected Green's functions G and the effective action is its Legendre transform: The effective action Γ can be interpreted as a collection of amplitudes where Γ (n) (x 1 , ..., x n ) are external momenta truncated (with full propagators) 1PI graphs.
It is well known that one can choose counter terms perturbatively such that only gauge invariant counter terms renormalize the theory as long as the regulators do not spoil gauge invariance. This leads to the gauge invariant effective action if we subtract out the tree-level gauge-fixing action from the total effective action:Γ At this point, one simply applies the functional derivative representation of the symmetry transformation and pick out functionally independent coefficients to generate the Ward identities.
As a simple illustration, consider scalar QED in the usual A µ gauge field representation. The gauge transformation mixes the 3-point function involving (A µ , φ * , φ ) fields and 2-point functions obtained from removing A µ due to the inhomogeneous nature of δ A µ . Hence, the relevant components of the effective action areΓ To apply the symmetry transformations on this object, one needs a functional representation of Eq. (38): .
Ward identities are generated through Explicitly, one finds the well known Ward identity:

BTGT FORMALISM WARD IDENTITIES
In this section, we use the method of the previous section to derive the Ward identities in the BTGT formalism both for the U(1) gauge symmetry and the BTGT symmetry.

U(1) in BTGT formalism
Here we consider the U(1) gauge symmetry transformations of Eq. (18). The functional derivative representation analog of Eq. (41) for this symmetry is First, applying this operator to the 2-point function we find Let's see if this is satisfied by the tree-level propagator. Note that in Fourier space, the 2-point coefficient is given by the inverse of the propagator: where ∆ bc (k) is the propagator defined by To obtainΓ ab , we subtract out the gauge fixing term: We can rewrite this as and Fourier transform as to obtain The gauge fixing term subtraction therefore giveŝ and the Ward identity is to all orders in perturbation theory.
At tree level, we have after subtraction the expression We can sum over f and find confirming the Ward identity for U(1) at tree-level.
Next, let's consider the 3-point function.
we find for the 3-point function Ward identity In Fourier space, with Feynman diagrams drawn with outgoing scalar momentum p 2 = p and incoming scalar momentum p 1 = −q, and incoming θ c field momentum p 3 = −k, we find whereĜ c is the Fourier space representation ofΓ c : e.g.
At tree level, Eq. (62) is = −e(p + q) · k which indeed is an identity.

BTGT symmetry
Next, we investigate the Ward identities associated with the BTGT symmetry Eq. (5). The functional derivative representation is Apply this to the two-point function which is symmetric in both a ↔ b and x ↔ y. We find We cannot take a simple functional derivative with respect to Z c (x) because that variable is constrained. One can solve the constraint trivially in Fourier space: where F a (k) is unconstrained. Eq. (69) implies that the 2-point contribution in momentum space is the momentum vector with the (H c ) α β component projected out. This can be trivially checked with the tree level action: where there is no sum over the repeated indices here. Since Eq. (77) becomesĜ we see that Eq. (79) is manifestly satisfied at tree level owing to the orthogonality property manifest in Eq. (A3).

1-LOOP RENORMALIZATION
One way to check the quantum stability of the BTGT formalism is to explicitly renormalize

Action and Feynman rules
Here, we will write down the Feynman rules for scalar QED in the BTGT formulation specified by Eqs. (14)- (17). We will use the summation convention that we sum over all repeated Latin indices unless specified otherwise. For our objective of testing the BTGT coupling at one loop, we have set the scalar quartic self-coupling to zero at tree level. The Feynman rules for this scalar QED theory is shown in Fig. 1. In the figure, the photon propagator has a factor written in terms of In the rest of this section, we use these Feynman rules in the Feynman gauge (ξ = 1) for an explicit 1-loop renormalization of scalar QED.

Vacuum Polarization
In this subsection, we compute the two vacuum polarization graphs shown in Fig. 2. All diagrams in this subsection and the subsequent subsections will be computed in the Feynman gauge.
The first graph we consider is the one of Fig. 2a). The Feynman rules give The structure of this diagram is similar to that of ordinary scalar QED other than the appearance of star products in the numerator: The quartic vertex graph of Fig. 2b) gives where We can add Eqs. (83) and (86) to obtain Note that the scalar mass dependent longitudinal mode has decoupled from the photon propagator, restoring gauge invariance (i.e. leaving the term proportional to the transverse projection operator P T µν ). Eq. (88) generates the counter term where we used Eq. (A1). That is why we were able to absorb the divergence using only gauge invariant counter terms. Next, the BTGT Ward identity Eq. (71) is satisfied by Eq. (88) since where we used Eq. (A3). Hence, dimensional regularization also preserves the 2-point function BTGT Ward identity.

Vertex correction
In this subsection, we compute the vertex corrections shown in Fig. 3. Let's first consider the diagram Fig. 3a). We find with a θ c insertion Figure 3: One loop cubic vertex corrections when the scalar quartic self-coupling is zero at tree level. Note that we have set the incoming scalar momentum to zero to simplify the computation as is sometimes done [30].
Next, the diagram Fig. 3b) evaluates to which vanishes because of Lorentz symmetry and the fact that we have set the incoming scalar momentum to zero.
The most interesting diagram is Fig. 3c) iΓ where and which a priori looks different from the usual computation particularly because of the tensor structure in the denominator. By boosting to the diagonal frame of H c , we can simplify this. After-wards, we boost back to find Combining this with Eq. (101), we find the third diagram contributes Combining the three vertex correction diagrams, we thus arrive at This leads to (for example in the minimal subtraction renormalization scheme) With Eq. (108), we can also check that the 3-point BTGT Ward identity Eq. (79) is preserved by the dimensional regularization. Before adding the counter term, we have the regularized con-  where the novel tensor structure in the denominator can be handled by boosting to to the (H b ) µ ν diagonal frame as in the vertex corrections. This results in which matches the usual computation results. The diagram of Fig. 4b) gives which has no novel tensor structure (as the numerator and the denominator cancel) and does not contribute to the Z 2 counter term in dimensional regularization as usual.
Using Eq. (115) and i(Z 2 − 1)k 2 counter term, we thus find We explicitly see that Z 1 = Z 2 in the minimal subtraction scheme which is the prediction of the while we find the right hand side from Eq. (115) to be Hence, the U(1) Ward identity is preserved by the dimensional regularization within the BTGT formalism as expected.

4-point function
In this subsection we would like to check the U(1) Ward identity prediction Z 1 = Z 4 for scalar QED. If we turn off the scalar quartic self-interaction, we only need to evaluate the diagram shown in Fig. 5: where k 1 is the incoming left external photon momentum and k 2 is the incoming right external photon momentum. Note that unlike in the case of A µ field theory, we cannot set k 1 = 0 to obtain the desired counterterm. To handle the BTGT specific tensor structure in the denominator, we can go to the (H f ) µ ν diagonal basis as done before to evaluate this integral. We find where there is no sum over the repeated indices here. Adding this to the counter-term, the renormalization constant Z 4 can be extracted in the minimal subtraction scheme as matching the expected result of the U(1) Ward identity.
Given that the Z i at one loop has the same result as in the ordinary formulation of scalar QED, we know that the β -function for this theory will be the same: i.e.
where α ≡ e 2 (4π) −1 . Furthermore, this explicit computation of Z 4 was a nontrivial test of the (two θ )-(two scalar) coupling loop computation within the BTGT formalism. To check whether or not dimensional regularization is consistent with the BTGT symmetry Ward identity and the BTGT formalism in general, explicit one loop renormalization of scalar QED was carried out. All two and three point Ward identities associated with BTGT symmetry and U(1) are shown to be consistent with dimensional regularization. Novel dot products in the form of A c B appear both in the numerator and the denominator, but the renormalization constants in the minimal subtraction scheme are identical to the results from the standard computational formalism. It is clear from these explicit computations that the BTGT formalism is stable at one loop.
There are many future research directions for BTGT. It would be interesting to see if the non-Abelian gauge theories can be expressed in the BTGT formalism. This involves constructing a solution to the nonlinear constraint equation in a fashion similar to what was done for the Abelian theory. It would also be interesting to find practical applications for this theory in computing nonlocal correlators or in lattice gauge theory. This formalism should also be tested in the contexts of spontaneous symmetry breaking and curved spacetime. Since θ a field is very close to a Wilson line and since it is charged under a BTGT symmetry, the Abelian BTGT symmetry may be related to generalized global symmetries [31]. It would be worth investigating the precise connection. For beyond the standard model physics, it would be interesting to see if the basis tensor fields is the Lorentz boost matrix that boosts away from the diagonal basis.