Ward identity for off shell photon?

[CAF123]

Consider an amplitude for some subprocess involving an off shell external state photon with polarisation $\epsilon_{\mu}$ and momentum $q_{\mu}$, stripped of the polarisation vectors so that e.g $T = \epsilon_{\mu} \epsilon_{\nu}^* T^{\mu \nu}$ ($\epsilon_{\nu}^*$ is polarisation vector for some other lorentz index carrying object).

Does the Ward identity $q_{\mu} T^{\mu \nu} = 0$ hold for off shell external states? Wikipedia mentions particularly on shell but if this is true why? I have reason to believe it holds for off shell states as well but I am just looking to understand why so.

Thanks!

 

[vanhees71]

Do you mean the photon polarization tensor (or self-energy of the photon)? Then there's a Ward-Takahashi identity,
$$q_{\mu} \Pi^{\mu \nu}(q)=0.$$

 

[CAF123]

I just mean the part of the amplitude with the lorentz indices stripped off - I'm not sure if this is called the 'photon polarisation tensor' but it is the quantity $T^{\mu \nu}$ in $T = \epsilon_{\mu} \epsilon_{\nu}^* T^{\mu \nu}$. The Ward identity says that if I replace $\epsilon_{\mu} \rightarrow q_{\mu}$ then I get $q_{\mu} \epsilon_{\nu}^* T^{\mu \nu} = 0$. But I am wondering if this also holds for off shell $q_{\mu}$? A quick search on the net has some literature with fragments of text saying it also holds for off shell states but there is no reason given.
Thanks!

 

[vanhees71]

You have to define your quantities, so that we can help you. What's $T^{\mu \nu}$?

 

[CAF123]

$T_{\mu \mu}$ is defined as the amplitude for an off shell photon initiated subprocess. The photon carries lorentz index $\mu$ and the object produced in such a subprocess carries index $\nu$. So basically it encodes all the feynman rules for the process but with the explicit polarisation vectors stripped off, ie $T = \epsilon_{\mu} \epsilon_{\nu}^* T^{\mu \nu}$.. . Do you require further information?

 

[CAF123]

$T_{\mu \nu}$ is defined as the amplitude for an off shell photon initiated subprocess. The photon carries lorentz index $\mu$ and the object produced in such a subprocess carries index $\nu$. So basically it encodes all the feynman rules for the process but with the explicit polarisation vectors stripped off, ie $T = \epsilon_{\mu} \epsilon_{\nu}^* T^{\mu \nu}$.. . Do you require further information?

 

[CAF123]

In many books one finds that $q_{\mu} T^{\mu \nu} = 0$ expresses the fact that there can be no longitudinal degrees of freedom carried by a particle whose momentum is $q_{\mu}$ with polarisation vector $\epsilon_{\mu}$. (see, e.g here https://en.wikipedia.org/wiki/Ward–Takahashi_identity#Ward_identity - the $\mathcal M^{\mu}$ there is analogous to my $T^{\mu \nu}$)

Yet the equation holds for off shell particles too, which I believe has support mathematically based on the fact the $j^{\mu}$ current is still conserved following an effective mass term in the lagrangian for the photon so this would not break a global U(1) symmetry. So for off shell photons this 'physical interpretation' described above in terms of no propagation of longitudinal degrees of freedom must break down?

 

[vanhees71]

The Ward-Takahashi identities are for general $n$-point functions or (equivalently) proper vertex functions (i.e., 1PI amputated diagrams which are needed to define renormalization conditions and corresponding counterterms for renormalization). The Ward identity is for on-shell quantities figuring into $S$-matrix elements.

In the Abelian case there is no problem with mass terms of the gauge bosons whatsoever. In the Stueckelberg formalism you can, by introducing an additional scalar field, which in a certain class of gauges that are similar to the $R_{\xi}$ gauges in the Higgsed non-Abelian, stay non-interacting. See, e.g., Collins, Renormalization, Cambridge University Press. This trick, however, doesn't work in the non-Abelian case, for which the only way to get a consistent theory with massive gauge bosons is the Anderson-Higgs-Kibble-Englert-Brout-et-al formalism.

 

[CAF123]

I see, thanks! So does the identity $q \cdot \mathcal M = 0$ that follows for off-shell $q_{\mu}$ not strictly a Ward identity but a Ward-like identity? Ward like in the sense that the equation content is the same but $q_{\mu}$ is not on-shell? So perhaps this identity would belong to the class of Ward-Takahashi identities?

 

[vanhees71]

For the Ward-Takahashi identity, see here

https://en.wikipedia.org/wiki/Ward–Takahashi_identity#Ward.E2.80.93Takahashi_identity

It connects an $n$-point function (or proper vertex function) to an $(n-1)$-point function, while the Ward identity applies to on-shell quantities, i.e., all the external legs representing the $n$-point function put on-shell (including the photon line in question (with four-momentum $k^{\mu}$ in the Wikipedia article).