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  Effective field theories and gauge anomalies cancellation

+ 4 like - 0 dislike

Let us assume some theory which consists of sets of generations of fermions (let us call them $A$ and $B$). Fermions $A$ have some gauge group $G_{A}$ (for example, SM), while fermions $B$ are charged under another group $G_{B}$ as well as under some subgroup of $G_{A}$ (for example, $SU(2)\times U_{Y}(1)$). Group $G_{B}$ (for example, $U_{B}(1)$) is spontaneously broken by vacuum averaged values $v_{B}$ due to a Higgs-like mechanism (for case of $G_{B}$ isomorphism to $U_{B}(1)$ - with Higgs-like field $h_{B} = v_{B}e^{i\theta_{B}}$).

Theories of $A$ and $B$ fermions are anomaly-free separately, while the mixed anomalies are cancelled by fixing charges of $B$ fermions under $G_{B}$ and $G_{A}$.

Let us then assume that all $B$ fermions are very massive; we thus can integrate them out for getting an effective action $\Gamma$ that contains an interaction between $G_{A}$ and $G_{B}$ gauge fields. I have a few questions about anomalies in $\Gamma$.

  1. It seems to me that by this naive integration (without introduction of some non-invariant terms) we make the theory anomalous; but if the mother theory is gauge invariant, gauge invariance must hold in the effective theory too. Is this true?
  2. An effective action, which is derived after integrating out massive fermions, consists of Wess-Zumino terms $$ \tag 1 \Gamma_{WZ} = c_{1} \int d^{4}x \theta_{B}F_{A}^{\mu\nu}\tilde{F}^{A}_{\mu \nu} + c_{2}\int d^{4}x \theta_{B}F_{A}^{\mu\nu}\tilde{F}^{B}_{\mu \nu}, \quad \tilde {F}_{\mu \nu} = \epsilon_{\mu \nu \alpha \beta}F^{\alpha \beta}, $$ which are connected with mixed anomalies and arise from triangular diagrams in the correlator $\langle \bar{\psi}_{B}\gamma_{5}\psi_{B} \rangle$. To contract the gauge-variant part of $(1)$ we have to introduce a generalized Chern-Simons counterterm $$ \tag 2 \Delta \Gamma_{CS} = c_{3}\int d^{4}x \epsilon^{\mu \nu \alpha \beta}A_{\mu}\left(B_{\nu}\partial_{\alpha}A_{\beta} + \frac{1}{3}e_{A}\epsilon_{abc}A^{a}_{\nu}A^{b}_{\alpha}A^{c}_{\beta}\right), $$ which arises from process $B_{\mu} \to A^{a}_{\mu}A^{b}_{\nu}$ in one-loop approximation. Is this true?
  3. The coefficients in front of Wess-Zumino terms are determined uniquely (are regularization independent), which is connected with the fact that they collect all anomomaly effects in theory. Is this true?
  4. Finally, are Wess-Zumino terms the only terms that break unitarity in an effective action $\Gamma$ before introducing the counterterm? Or is this a completely wrong statement?

Some prehistory

The questions have arisen after reading of article, in which there is an explanation how nontrivial anomaly cancellation in a fundamental theory provides effects of non-decoupling of massive fermions. As example there is the toy-model with two sets of chiral fermions whose Lagrangian has $U_{X}(1), U_{Y}(1)$ symmetries; they then are integrated out; after that effective operators unsuppressed by fermions masses arise; look at Eq.(7). I want to know about the nature of these terms; the first two terms I identify as Wess-Zumino terms; they are regularization independent, as is claimed in an article. The last term arises, if I understand correctly, as counterterm which arises for the process $A_{1} \to A_{2},A_{2}$.

But in some articles (for example, Preskill's Gauge anomalies) Wess-Zumino terms $(1)$ as well as counterterm $(2)$ are interpreted as counterterms which initially aren't included in an effective action (look at page 25 for a detailed discussion of contraction of gauge-variant terms in an effective action of $SU(2)\times U(1)$ theory). So there is a bad mix in my head about anomalies contraction in an effective field theory.

An edit

It seems that the following is the case. Let's temporarily turn off $G_{B}$ interactions. This provides that both $A, B$ fermions interact only with $G_{A}$ fields. Then there is the fact that the lepton fermions $A$ anomaly is cancelled by the fermions $B$ anomaly. Let's then integrate $B$ fermions out. The resulting effective field theory must be anomaly-free, so that it must contain some term which is changed as well as $B$ fermions part of mother action under gauge transformation. This term is called the Wess-Zumino term, $\Gamma_{WZ}[U, A_{L},A_{R}]$, where $A_{L/R}$ denotes a gauge field that interacts with left or right $B$ fermions $\frac{1 \mp \gamma_{5}}{2}\psi_{B}$ (for example, $A_{L} = \gamma + Z$, $A_{R} = \gamma$). By denoting the action which consists of $A$ fermions as $S_{A}$ which has anomaly, $$ \delta S_{A} = \Gamma_{anomaly}, $$ we have $$ \delta_{anomalous} \left( \Gamma_{WZ} + S_{A}\right) = -\Gamma_{anomaly} + \Gamma_{anomaly} = 0 $$ Let's then introduce $G_{B}$ interactions. It seems that only mixed anomalies cancellation is interesting. So maybe it is convenient to introduce $\Gamma_{WZ}[U, A_{L} + b, A_{R} + b]$ term, where $b$ corresponds to a set of background vector fields corresponding to the adjoint representation of $G_{B}$. Then $$ \delta_{anomalous} \left( \Gamma_{WZ} + S_{A}\right) = -\Gamma_{anomaly} + \Gamma_{anomaly} + \Gamma_{anomaly}[\varphi , b, A_{L}, A_{R}] \neq 0 $$ We need to introduce a counterterm $\Gamma_{ct}[b, A_{L}, A_{R}, \varphi]$ (if it exists) which has variation equal to $\Gamma_{anomaly}[b, A_{L}, A_{R}]$. It is possible, and the sum of $\Gamma_{WZ} + \Gamma_{ct}$ contains new interactions of type $$ b \wedge Z \wedge \partial Z,\quad b \wedge Z \wedge F^{\gamma},\quad b \wedge Z \wedge \partial b $$ (this is important for the first linked article).

This post imported from StackExchange Physics at 2015-05-11 19:55 (UTC), posted by SE-user Name YYY

asked May 9, 2015 in Theoretical Physics by NAME_XXX (1,060 points) [ revision history ]
edited May 12, 2015 by Arnold Neumaier
Comment to the question (v9): Consider adding references to make the question more accessible to the reader and focus the answers.

This post imported from StackExchange Physics at 2015-05-11 19:55 (UTC), posted by SE-user Qmechanic

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