In a gauge theory, the tree-level Lagrangian is gauge invariant. Is it possible that a gauge non-invariant term induced in the Lagrangian via loop-effects (may be by integrating out certain fields)?
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Gauge variant terms in the action arise in two different cases:
1) When we want to quantize the gauge theory, we always need to fix the gauge. In order to fix the gauge correctly (which involves the reduction of the number of configurations of the gauge fields over which we integrate) we have to modify the action by introducing ghosts - the fields with indefinite norm in the Hilbert space. In the case of non-abelian theories (and in the case of the abelian ones, when we impose the non-linear in the gauge field gauge fixing condition) the resulting action isn't gauge invariant - at least under "usual" gauge transformations. However, instead of this "usual" symmetry there arises another symmetry realizing the gauge invariance - the so-called BRST symmetry. The resulting action is BRST-invariant, which is realized by the so-called Slavnov-Taylor identities.
2) In certain theories (chiral gauge theories) these identities are broken by the so-called gauge anomaly. The anomaly is coded in the 3-point effective action, which contains the information about the triangle diagram with fermion chiral currents running into it; in fact it is one-loop exact. It generates the anomalous gauge current $J^{\mu}$ conservation, $$ \partial_{\mu}J^{\mu} = A(x), \quad \text{where } A(x) \ \text{is the anomaly functional} $$ Although the anomaly functional is local one (i.e., it is the integral of the polynomial of derivarives in momentum space), the effective action $\Gamma$ generating the anomaly, $$ \tag 1 \delta_{\epsilon}\Gamma = -\int A(x)\cdot \epsilon , $$ is non-local, so You can't add a counter-term coinciding the anomaly without breaking the locality of the theory.
3) If the initial gauge theory is anomaly-free (i.e., gauge-invariant), but there is non-trivial gauge anomaly cancellation between different fermions, then, after integrating out some of these fermions, corresponding effective action must contain the fixed gauge-variant terms in order to preserve the gauge invariance. In a literature these terms are called the Wess-Zumino terms. In consistent gauge-invariant theories they are local! This is because typically these theories include the scalar sector which is interpreted as Goldstone fields sector. It could serve as the Higgs-like sector associated with Higgs mechanism, or be the physical particles like the pseudo-scalar meson octet. It turns out that the the pole from the non-local effective action is "absorbed" by the Goldstones $\varphi$, and instead the non-local action we have terms like $$ \tag 2 \Gamma_{WZ} \simeq \int d^{4}x \varphi (x) A(x) $$ Under the gauge transformation we have $\varphi(x) \to \varphi(x) +\epsilon$, so the gauge variation of $(2)$ reproduces $(1)$.
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Naively speaking, a gauge invariant Lagrangian in a quantum field theory (QFT) can not have any non gauge invariant because the presence of any such term would make the theory non renormalizable. That's why one of our best known theory - quantum electrodynamics (QED) has only gauge invariant terms.
That was the thinking of most quantum field theorists until the late 1960s. The discovery of renormalizable group (RG) idea in QFT has lead to a change in attitudes. RG idea also helped us understand why renormalization works in a QFT and why it is perfectly correct to subtract 2 infinite quantities to obtain a finite result.
An idea closely associated to RG is the effective field theory. Effective field theory is a way to reformulate a given QFT so that a cut off is applied to the momentum or energy scale of the theory, keeping only renormalizable terms. In spite of the presence of a cut-off, it is possible to recover the continuum limit of the theory. All non renormalizable terms in the theory are suppressed by increasing powers of the ratio of cut-off energy to the Planck energy. This idea grew out of the RG work by Wilson in 1974 and was developed by Weinberg in a series of papers from 1975 to 1990.
Effective field theory idea helps to explain why QED has only 3 terms at low energy and all 3 terms are gauge invariant and the theory is completely renormalizable. Why would nature be so kind? The answer lies in the fact that actually this theory contains an infinite number of non renormalizable terms - including an infinite number of non gauge invariant terms. All of these terms are suppressed at low energy as described in the previous paragraph.
So to finally answer your question, yes a Lagrangian for a given QFT can have a non gauge invariant and hence a non renormalizable term, but an effective theory formulation of this theory would suppress such a term and it would make virtually zero contribution to the final quantity calculated by such a theory. But such terms are waiting for us as we probe energy scales close to Planck scale!
