# Does a non-lagrangian field theory have a stress-energy tensor?

Most theories do not have a conserved energy-momentum tensor, regardless of whether they are Lagrangian or not. For example you need locality and Lorentz invariance. When you have those, you can define the energy-momentum tensor via the partition function (which always exists, it basically defines the QFT): $$\langle T_{\mu\nu}\rangle:=\frac{\delta}{\delta g^{\mu\nu}}Z[g]$$ You can define arbitrary correlation functions of $$T$$ by including insertions. And these functions define the operator $$T$$ itself. So $$T$$ is defined whenever $$Z$$ is a (differentiable) function of the metric, i.e., when you have a prescription to probe the dependence of the theory on the background metric. And this prescription is part of the definition of the theory: in order to define a QFT you must specify how the partition function is to be computed, for an arbitrary background. If you cannot or do not want to specify $$Z$$ for arbitrary $$g$$, then the derivative cannot be evaluated and $$T$$ is undefined. (This is not an unreasonable situation, e.g. I may be dealing with a theory that is anomalous and is only defined for a special set of metrics, e.g. Kähler. This theory does not make sense for arbitrary $$g$$, so I may not be able to evaluate $$Z[g]$$ for arbitrary $$g$$, and therefore $$T$$ may not exist).

If the theory admits an action, then the dependence of $$Z$$ on $$g$$ is straightforward: it is given by whatever the path-integral computes. If the theory does not admit an action, then you must give other prescriptions by which to compute $$Z$$. This prescription may or may not include a definition for arbitrary $$g$$; if it does not, then $$T$$ is in principle undefined.

But anyway, for fun consider the following very explicit example: $$\mathcal N=3$$ supersymmetry in $$d=4$$. This theory is known to be non-lagrangian. Indeed, if you write down the most general Lagrangian consistent with $$\mathcal N=3$$ SUSY, you can actually prove it preserves $$\mathcal N=4$$ SUSY as well. So any putative theory with strictly $$\mathcal N=3$$ symmetry cannot admit a Lagrangian. Such a theory was first constructed in arXiv:1512.06434, obtained almost simultaneously with arXiv:1512.03524. This latter paper analyses the consequences of the anomalous Ward identities for all symmetry currents, in particular the supercurrent and the energy-momentum tensor.

Not all non-Lagrangian theories have a stress-energy tensor, an example of this is the critical point of the long-range Ising model, which can be expressed as a "defect" field theory where the action consists of two pieces integrated over spaces of different dimensionality and hence has no single Lagrangian that would describe it.

See chapter 6 of "Conformal Invariance in the Long-Range Ising Model" by Paulos, Rychkov, van Rees, Zan for a discussion of this formulation and what the "missing" stress-energy tensor means for the Ward identities. That paper also refers us in a footnote to "Conformal symmetry in non-local field theories" by Rajabpour, where a "non-local stress tensor" is constructed for a class of theories where the usual kinetic term with a local Laplacian is replaced by the non-local fractional Laplacian, but this object does not behave like one would usually like a stress-energy tensor in a CFT to behave, in particular its operator product expansions are "wrong".