Why isn't the renormalization of composite operators determined by the renormalization of elementary fields?

Question

In typical expositions of renormalization of composite operators, one needs such renormalization to tame the divergences of some matrix elements such as (say in an interacting scalar theory)

\[\langle \alpha|\phi^2(x)|\beta\rangle\]

One renormalizes $\phi^2$ by introducing the renormalized composite operator

\[(\phi^2)_R=Z_{(\phi^2)}\phi^2.\]

My question is, doesn't the above relation imply $Z_{(\phi^2)}=Z^2_{\phi}$, where $Z_{\phi}$ is the field strength renormalization of $\phi$? That is, once the elementary field is renormalized, how come one can still have the freedom to renormalize the composite operator?

Useful scholarpedia introduction on the subject(still scratching my head reading it): http://www.scholarpedia.org/article/Local_operator#Definition_of_normal_products_and_operator-mixing

Comments

In the book ''The theory of quark and gluon interactions'' by F.J. Yndurain (4th edition), there is on p.76 beginning with (3.4.2) an explicit example for the 1-loop renormalization of a composite operator in QCD. 

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＠ArnoldNeumaier, thanks for the new reference!

Answer 1

One has no extra freedom in renormalizing the composite fields (i.e., no new parameters in the renormalization prescription).

However, the normalization factors are not multiplicative since the operator product expansion is singular, and the limits do not commute as it would be needed to show multiplicativity. Therefore the renormalization of $\phi^2$ is quite different from the square of the renormalization of $\phi$ - which is not even defined, since all local fields are distributions only that cannot be multiplied. (In conformal field theories, not even the scaling dimensions add, as one would expect from a naive argument.)

Comments

 I must admit there must be something very elementary that I don't understand: In the renormalized perturbation theory, when we write down the renormalized Lagrangian, the $\phi^2$ terms simply become $Z^2\phi^2$, where $Z$ is the field strength renormalization factor. However in composite operator renormalization this does not look so. What's the difference in the two cases?

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The thing is you don't have such arbitrariness in the renormalization of field strength, mass and other coupling constants. Due to LSZ, S-matrix becomes a very solid anchoring point for us to write down the renormalization conditions, and seemingly different renormalization schemes either result from different regularization schemes, or different schemes of relating correlation function to S-matrix, so the freedom of such renormalizations is very limited. What would be the "anchoring point" for the renormalization of local products of operators?

 I gradually realize it is really a question about the meaning of local products, maybe I'll start a new thread if necessary.

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Stefan Hollands has several papers on perturbative OPE construction (for scalar fields), which ''proves'' the existence of all local products (i.e., demonstrates their existence at the level of rigor of theoretical physics).

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@JiaYiyang: As the whole procedure and how to talk about it is ill-defined, it is difficult for me to identify in full detail how to make it well-defined. My way of processing the literature on ill-defined objects is simply to take for granted the properties and tricks that proved to work, and trust an authour unless another one contradicts. So on that level I can say little more than Weinberg, apart from providing my own ill-defined intuition.

Independent of that I am looking for a coherent way to organize things in a more consistent way. In the case of operator products, the rigorous interpretation (though nowhere rigorously completed) is along the lines of Hollands: For free fields, the well-defined variations of the products are the normally ordered products with respect to a vacuum state (or more generally a coherent state - these typically provide inequivalent representations of the CCR). Thus $\phi^2$ should be understood as $:\phi^2:$, and $\phi^4\sim(\phi^2)^2$ as $:\phi^4:=:(:\phi^2:)^2:$. In the case with interactions, everything is deformed according to finite rules that were derived by Hollands in perturbation theory. This deformation defines the renormalized interaction, and $\phi$ and $\phi^2$ are defomed differently. I believe that there lies the future, and some version of this might become fully rigorous one day.

How these two pictures (the traditional one and the near-rigorous one) relate to each other is again given only by some ill-defined intuition that I tried to convey. Don't attach too much meaning to it.

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@ArnoldNeumaier, thanks, but I'm hardly convinced, let me think about it.

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I guess all I'm asking is a fixed recipe of relating all kinds of local products to S-matrix, you are probably right that OPE is the right way since I know OPE can be used to study cross-sections. But I've never studied OPE in detail, and it's far from clear to me such local products can be systematically defined without inconsistencies. I shall read more on the subject. Thanks for your help.

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@JiaYiyang: Only observable asymptotic (local or string-localized) fields [which are very few compared to the mass of possible local products] can be related to the S-matrix, as the latter is a transformation between the asymptotic states of the theory. This means one can find there only fields corresponding to bound states.

Due to confinement, there are no asymptotic  observable fields. The asymtoptic gauge fields in a formal S-matrix expansion are not observable. This means on the algebraic level that they must have imaginary mass (tachyonic) contributions in their propagator, and hence cannot occur as operators on a well-defined asymptotic Hilbert space. This is the deeper reason for the infrared problems in all gauge theories.

Answer 2

This answer relies pretty heavily on http://www.scholarpedia.org/article/Local_operator. Some material can also be found in Peskin and Schröder and Weinberg, but the above article is quite self contained and gives a good overview.

I think there is a misconception regarding different types of divergences in QFT's. Consider for example a renormalizable QFT of a scalar field \(\phi_0\), where this denotes the bare field. The standard procedure then is that you rescale/shift the fields, masses and coupling constants so that your Lagrangian is expressed in terms of renormalized (physical) fields, masses, coupling constants and counterterms. The counterterms are fixed by appropriate renormalization conditions. Then all Green's-functions of the renormalized fields \(\phi_R= Z^{-\frac{1}{2}} \phi_0\) at different spacetime positions are well defined. However that does not mean that for example \(\langle \phi_R(x) \phi_R(x) \rangle_0\) is finite (0 denotes the interacting vacuum here). As an example consider  a non interacting theory of a scalar field (where no field renormalization occurs at all). \( \phi(x) \phi(x) \) is ill defined since for example the vacuum expectation value (VEV) diverges. A regular operator mimicking the \(\phi^2\) operator in the free case is the normal ordered version \(: \phi^2 :\) which can be written (schematically) as \(: \phi^2 : = \phi^2 -\mathbb{1} \Delta(0) \), where the propagator and the identity operator appear. 

This procedure of normal ordering inspires the proper definition of local regular operators in any interacting QFT. For any operator \(O(x)\) that is a product of elementary (and already renormalized) field operators and their derivatives at the same spacetime point we can define an operator \(N_{\delta}[O(x)]\), which can be expressed as a finite sum \(\Sigma_l Z_l O_l(x)\) where \(Z_l\) are (usually) infinite​ constants, and \(O_l(x)\) are other local field operators built from elementary fields and their derivatives. In some special cases \(N_{\delta}[O(x)]= Z O(x)\), but in the general case other operators appear on the righthandside. This is then called operator mixing. \(\delta\) is a parameter related to the mass dimension of the operator. 

The prescription on how to construct these renormalized versions of local operators requires additional conditions, called the BPHZ normalization conditions. All of the above is a highly abbreviated and simplified version of http://www.scholarpedia.org/article/Local_operator. This article also mentions why operators like the electromagnetic current \(J(x)\) which consists of 2 fermion fields does not require this procedure. Due to it being conserved/ gauge invariant we have \(N_{\delta}[J(x)]=J(x)\). I hope this helps a bit