Massive Dual Spinless Fields Revisited

Massive dual spin zero fields are reconsidered in four spacetime dimensions. A closed-form Lagrangian is presented that describes a field coupled to the gradient of its own energy-momentum tensor.


Introduction
As indicated in the Abstract, the point of this paper is to find an explicit Lagrangian for the dual form of a massive scalar field self-coupled in a particular way to its own energy-momentum tensor. This boils down to a well-defined mathematical problem whose solution is given here, thereby completing some research initiated and published long ago in this journal [1].
After first presenting a concise mathematical statement of the problem, and then giving a closed-form solution in terms of elementary functions, the field theory that led to the problem is re-examined from a fresh perspective. The net result is a very direct approach that leads to both the problem and its solution.

Some History
Here I reconsider research first pursued in collaboration with Peter Freund, in an effort to tie up some loose ends. In the spring of 1980, when I was a post-doctoral fellow in Yoichiro Nambu's theory group at The Enrico Fermi Institute, Peter and I were confronted by a pair of partial differential equations (see [1] p 417).
where m and g are constants. We noticed in passing that these PDEs imply the secondary condition [3] and we then looked for a solution to (1-3) as a series in g beginning with To simplify the equations to follow, I will rescale g = mκ so that the constant m always appears in (1)(2)(3) only in the combination v/m. Thus I may as well set m = 1, and hence κ = g. I can then restore the parameter m in any subsequent solution for L by the substitution L (u, v) → m 2 L (u, v/m). Clearly, there is a two-parameter family of exact solutions to these PDEs which depends only on v, namely, where a and b are constants. However, for the model field theory that gave rise to the partial differential equations (1,2), this linear function of v amounts to a topological term in the action and therefore gives no contribution to the bulk equations of motion. Moreover, L 0 (v) contributes only a (cosmological) constant term to the canonical energy-momentum tensor. So, in the context of our 1980 paper [1] where solutions of (1,2) were sought which gave more interesting contributions, this L 0 (v) was not worth noting. Nevertheless, it reappeared in another context, somewhat later [2].

Completing Some Unfinished Business
It so happened in 1980 that Peter and I did not find an exact L (u, v) to solve the PDEs (1,2). In fact, we reported then only the terms given in (4). Here I wish to present an exact, closed-form solution to all orders in g.
The crucial feature leading to this particular solution is that the dependence on v is only through the linear combination v − gu. The result is where as a series Fortunately, the 3 F 2 hypergeometric function in (7) reduces to elementary functions. For real w, Nevertheless, the solution (6) was first obtained in its series form (7) and only afterwards was it expressed as a special case of the hypergeometric 3 F 2 , with its subsequent simplification to elementary functions. More generally, it is not so difficult to establish that solutions to (1-3) necessarily have the form where the function G is differentiable, and H is integrable, but otherwise not yet determined, as befits the general solution of a more easily solvable 1st-order PDE, albeit nonlinear: Note in (9) the return of an explicit term linear in v. This term arises as the particular solution of the inhomogeneous 1st-order PDE that results from integrating (10) and exponentiating, namely, The functions G and H are now constrained by additional conditions that lie hidden within (1) and (2). I will leave it to the reader to flesh out those additional conditions. I will not go through that analysis here. Instead, I will reconsider the model field theory that led to the partial differential equations (1,2) in light of the exact solution (6). That solution provides a good vantage point to view and analyze the model.

The Model Revisited
Consider a Lagrangian density L (u, v) depending on a vector field V µ through the two scalar variables, This vector field is to be understood in terms of an antisymmetric, rank 3, tensor gauge field, V αβγ , i.e. the four-dimensional spacetime dual of a massive scalar [1], with its corresponding gauge invariant field strength, The bulk field equations that follow from the action of L (u, v) by varying V µ are simply where the partial derivatives of L are designated by L u ≡ ∂L (u, v) /∂u and L v ≡ ∂L (u, v) /∂v. An obvious inference from these field equations is that the on-shell vector V µ is a gradient of a scalar Φ, if and only if L u is a function of L v . For example, if L u has a linear relation to L v with L u = a + bL v for constants a and b, the field equations give But in any case, on-shell the combination U µ = V µ L u is a spacetime gradient. An additional gradient of the field equations then gives Thus the vector V µ is a gradient of a scalar, as in (15), such that if and only if for some scalar function Ω, Simplification Now for simplicity, demand that L u = a + bL v for constants a and b, in accordance with V µ being a gradient, as in (15) and (20). This linear condition is immediately integrated to obtain where L (v + bu) is a differentiable function of the linear combination v + bu. The field equations (14) are now That is to say, the scalar in (21) is Ω = 2ab + 2b 2 L ′ .

Energy-momentum tensors
In [1] Peter and I say that, given (1-3), the field equations for V µ amount to (20) along with the "simple, indeed elegant" statement where g has units of length, and θ is the trace of the conformally improved energy-momentum tensor. Be that as it may, there is a less oracular method to reach this form for the field equations in light of the simplification (22). As is well-known, there may be two distinct expressions for energy-momentum tensors that result from any Lagrangian. From (22) the canonical results for Θ µν , and its trace Θ = Θ µ µ , are immediately seen to be Although not manifestly symmetric, it is nonetheless true that Θ Surprisingly different results follow from covariantizing (22) with respect to an arbitrary background metric g µν , varying the action for − det g αβ L with respect to that metric, and then taking the flat-space limit. This procedure gives the "gravitational" energy-momentum tensor and its trace: The unusual structure exhibited in this tensor follows because in curved spacetime V µ as defined by (13) is a relative contravariant vector of weight +1 with no dependence on the metric, so ∂ µ V µ is a relative scalar of weight +1 also with no dependence on g µν , and V µ V µ = g µν V µ V ν is a relative scalar of weight +2 where all dependence on the metric is shown explicitly. Hence the absolute scalar version of L (u, v) is given by where again all the metric dependence is shown explicitly.
It is straightforward to check on-shell conservation of either (25) or (26), separately. However, it turns out the flat-space equations of motion can now be written in the form (24) provided a linear combination of Θ µν .

(28)
The trace is then

Field equation redux
Since various scales have been previously chosen to set m = 1, the field equations (20) and (23) give for the left-hand side of (24) On the other hand, from (29) for any constant c, The choice 2ac = b reconciles the spurious ∂ µ u term to give the desired form provided the function L satisfies the second-order nonlinear equation But note, the constant c can be set to a convenient nonzero value by further rescalings.
For example, if (a, L) → ab 2c , aL 2bc , along with the previous choice 2ac = b → a = 1, the equation for L becomes 1 + 1 2b Finally, rescaling z → w/b gives 1 + 1 2 The solution of this equation for L ′ with initial condition L ′ (0) = 0 is Imposing the additional initial condition L (0) = 0, this integrates immediately to Comparison with (8) shows that Given the previous rescalings, namely, L (u, v) = au + L (v + bu) → ab 2c u + 1 2bc L (w = bz) a=1 , the Lagrangian density for the model becomes As before, v = ∂ µ V µ , u = V µ V µ , and z = v + bu. Note that the term linear in z in (39) cancels out upon power series expansion, so the result agrees with (4) up to and including all terms of O V 3 .
To comport to the conventions in [1], choose b = −g and c = g, so that z = v − gu, to find Now restore m via the coordinate rescaling x µ → mx µ , hence v → v/m and L (u, v) → m 2 L (u, v/m), thereby converting (32) into the form (24), with θ = m 2 Θ.

Discussion
The conventional integral equation form of (24), including a free-field term with + m 2 V (0) µ = 0, is given by where Θ (y) depends implicitly on the field V ν (y) and G is the usual isotropic, homogeneous, Dirichlet boundary condition Green function that solves + m 2 G (x − y) = δ 4 (x − y). The free-field term must be a gradient, is also a gradient. Integration by parts followed by an overall integration then gives where now Θ (y) depends implicitly on Φ (y). That is to say, On the one hand, this is not surprising, since there is a long-known construction of an explicit local Lagrangian that leads directly to this form for the scalar field equations [6]. (It amounts to the Goldstone model after scalar field redefinition.) Taking a gradient to reverse the steps above then leads back to (43). On the other hand, it is far from obvious that Θ [Φ (x)] can be re-expressed as a local function of V µ = ∂ µ Φ, and that Θ [V µ (x)] follows in turn from a local, closed-form Lagrangian for V µ . The main point of this paper was to show that, indeed, there is an L such that all this is true.
Were Θ due to anything other than V µ , field equations of the form (24) would easily follow from i.e. a simple direct coupling of the vector to the gradient of any other traced energy-momentum tensor. With a pinch of plausibility, this calls to mind the axion coupling, albeit without the group theoretical and topological underpinnings, not to mention the phenomenology. In any case, Peter and I certainly did not have axions in mind in 1980 when we wrote [1]. As best I can recall, we had only some embryonic thoughts about massive gravity. In that context we speculated (see [1] p 418) that g/m ∼ L Hubble L Planck = 4.7 × 10 −5 m 2 = 1/ 4.2 × 10 −3 eV 2 . In retrospect, we were both struck by the fact that this guess is approximately the same as phenomenological lower limits for 1/m 2 axion . There is one more noteworthy piece of unfinished business in [1], namely, a closed-form Lagrangian for a massive spin 2 field coupled to the four-dimensional curl of its own energy-momentum tensor, where the spin 2 field is not the usual symmetric tensor, but rather the rank three tensor T [λµ]ν [7]. For progress on this additional unfinished business, please see [8]. With enough effort, perhaps a complete formulation of this spin 2 model will also be available soon, along with a few other variations on the theme of fields coupled to Θ µν .
In closing, so far as I can tell, Peter had little if any interest in totally antisymmetric tensor gauge fields prior to our paper [1]. But he quickly pursued the subject in stellar fashion with his subsequent work on dimensional compactification [2]. While all this work is still conjectural, at the very least it provided and continues to provide fundamental research problems in theoretical physics, especially for doctoral students.