The conditions for the absence of gauge anomalies in effective field theories (EFT) are rivisited. General results from the cohomology of the BRST operator do not prevent potential anomalies arising from the non-renormalizable sector, when the gauge group is not semi-simple, like in the Standard Model EFT (SMEFT). By considering a simple explicit model that mimics the SMEFT properties, we compute the anomaly in the regularized theory, including a complete set of dimension six operators. We show that the dependence of the anomaly on the non-renormalizable part can be removed by adding a local counterterm to the theory. As a result the condition for gauge anomaly cancellation is completely controlled by the charge assignment of the fermion sector, as in the renormalizable theory.
It is often stated in the literature concerning D=4,6 compact Type IIB orientifolds that tadpole cancellation conditions i) uniquely fix the gauge group (up to Wilson lines and/or moving of branes) and ii) are equivalent to gauge anomaly cancellation. We study the relationship between tadpole and anomaly cancellation conditions and qualify both statements. In general the tadpole cancellation conditions imply gauge anomaly cancellation but are stronger than the latter conditions in D=4, N=1 orientifolds. We also find that tadpole cancellation conditions in Z_N D=4,6 compact orientifolds do not …
We prove that global anomaly cancellation requires more than one generation of quarks and leptons, provided that the standard model fields propagate in two universal extra dimensions. Furthermore, if the fermions of different generations have the same gauge charges and chiralities, then global anomaly cancellation implies that there must be three generations.
We extend our recently-proposed formalism for calculating anomalies of global and gauge symmetries using the Covariant Derivative Expansion to include a general class of operators that can appear in relativistic Effective Field Theories (EFTs). This allows us to prove that EFT operators involving general scalar, vector, and tensor couplings to fermion bilinears only give rise to irrelevant anomalies, which can be removed by an appropriate choice of counterterms, thereby confirming the absence of new constraints from anomaly cancellation on the Standard Model EFT.
Compactified string theories give rise to discrete symmetries which are essential if they are to provide a realistic low energy theory. We find that in a class of four dimensional string theories these symmetries are constrained by similar conditions to those discrete anomaly cancellation conditions found in the case the discrete symmetry is a residue of a spontaneously broken gauge symmetry. Such conditions strongly constrain the allowed form of the low energy effective theory.
We derive and generalize the RR twisted tadpole cancellation conditions necessary to obtain consistent D=4, Z_N orbifold compactifications of Type IIB string theory. At least two different types of branes (or antibranes with opposite RR charges) are introduced into the construction. The matter spectra and their contribution to the non-abelian gauge anomalies are computed. Their relation with the tadpole cancellation conditions is also reviewed. The presence of tachyons is a common feature for some of the non-supersymmetric systems of branes.
Many extensions of the Standard Model include an extra gauge boson, whose couplings to fermions are constrained by the requirement that anomalies cancel. We find a general solution to the resulting diophantine equations in the plausible case where the chiral fermion content is that of the Standard Model plus 3 right-handed neutrinos.
I show that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for Calabi-Yau compactification to six dimensions. This reflects the fact that K3 is the only non-trivial CY manifold in two complex dimensions. I explicitly construct the Green-Schwarz counterterms and derive sum rules for charges of additional enhanced U(1) factors and compare the results with all possible Abelian orbifold constructions of K3. This includes asymmetric orbifolds as well, showing that it is possible to regain a geometrical interpretation for this class of…
We investigate some aspects of anomaly cancellation realized by the subtraction of an anomaly pole, stressing on some of its properties in superspace. In a local formulation these subtractions can be described in terms of a physical scalar, an axion and related ghosts. They appear to be necessary for the unitarization of the theory in the ultraviolet, but they may generate an infrared instability of the corresponding effective action, signalled by ghost condensation. In particular the subtraction of the superanomaly multiplet by a pole in superspace is of dubious significance, due to the diffe…
We construct a seven-dimensional brane world in a slice of AdS_7, where the boundary matter content is fixed by the cancellation of anomalies. The seven-dimensional minimal N=2 gauged supergravity is compactified on the orbifold S^1/Z_2, and the supersymmetric bulk-boundary Lagrangian is consistently derived for boundary vector and hypermultipets up to fermionic bilinear terms. Anomaly cancellation then fixes the boundary gauge coupling in terms of the seven-dimensional Planck mass, and a topological mass parameter of the Chern-Simons term. In addition for gauge groups containing the standard …
The current flow from the bulk is due to the anomaly on the brane-but the absence of current flow is not, necessarily, due to anomaly cancellation, but to the absence of the chiral zero modes themselves, due to the existence of the layered phase. This can be understood in terms of the difference between the Chern-Simons terms in three and five dimensions. Thus the anomaly cancellation in four dimensions, which is essential for shielding the boundary from quantum effects within the bulk, makes sense only along the transition line between the layered and the Coulomb phase, which, in turn, requir…
We consider six-dimensional supergravity compactified on orbifolds with Wilson lines and bulk flux. Torus Wilson lines are decomposed into Wilson lines around the orbifold fixed points, and twisted boundary conditions of matter fields are related to fractional localized flux. Both, orbifold singularities and flux lead to chiral fermions in four dimensions. We show that in addition to the standard bulk and fixed point anomalies the Green-Schwarz term also cancels the four-dimensional anomaly induced by the flux background. The two axions contained in the antisymmetric tensor field both contribu…
In this note we review the role of homotopy groups in determining non-perturbative (henceforth `global') gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of $π_d(G)$ is neither a necessary nor a sufficient condition for there being a possible global anomaly in a $d$-dimensional chiral gauge theory with gauge group $G$. To showcase the failure of sufficiency, we revisit `global anomalies' that have been previously studied in 6d gauge theories with $G=SU(2)$, $SU(3)$, or $G_2$. Even though $π_6(G) \neq 0$, the bordism groups …
It has been recently shown that the requirement of anomaly cancellation in a (non-supersymmetric) six-dimensional version of the standard model fixes the field content to the known three generations. We discuss the phenomenological consequences of the cancellation of the local anomalies: the strong CP problem is solved and the fundamental scale of the theory is bounded by the physics of the axion. Neutrinos acquire a mass in the range suggested by atmospheric experiments.
A perturbatively renormalized Abelian Higgs-Kibble model with a chirally coupled fermion is considered. The Slavnov identity is fulfilled to all orders of perturbation theory, which is crucial for renormalizability in models with vector bosons. BRS invariance, i.e. the validity of the identity, forces the chiral anomaly to be cancelled by Wess-Zumino counterterms. This procedure preserves the renormalizability in the one-loop approximation but it violates the Froissart bounds for partial wave amplitudes above some energy and destroys renormalizability from the second order in h bar onwards due…
We consider the breaking of the global conservation of gauge field charges which are commonly thought to survive the spontaneous breakdown of gauge symmetry brought about by Kalb-Ramond fields. Depending on the dilaton field and also the size of the compactifying space, the global charge breaking may take place due to world sheet instantons. In going to 3+1 dimensions one could have a serious problem in order to produce the hierarchies between the quark and the charged lepton masses using the mass protecting charges with the Green-Schwartz anomaly cancellation. Various unnatural features of th…
A possibility of 5D gauge unification of $SU(2)_L \times U(1)_Y$ in $SU(3)_W$ is examined. The orbifold compactification allows fixed points where $SU(2)_L\times U(1)_Y$ representations can be assigned. We present a few possibilities which give long proton lifetime, top-bottom mass hierarchy from geometry, and reasonable neutrino masses. In general, these {\it chiral models} can lead to fixed point anomalies. We can show easily, due to the simplicity of the model, that these anomalies are cancelled by the relevant Chern-Simons terms for all the models we consider. It is also shown that the fix…
To investigate $Z'$ phenomenology model-independent, we combine chiral effective theory with anomaly cancellation conditions without any other model input. We focus on $Z'$ mixings with $γ-Z$ in both mass and kinetic parts and calculate contributions to oblique $S,T,U$. The three sets of anomaly-free fermion $U(1)'$ charges parameterize the $Z'$ interactions with fermions. The cancellation of the $[U(1)']^3$ anomaly and mixing gravitational-gauge anomaly determines the number of right-handed neutrinos. We also find a novel relation between the charge assignments and Stueckelberg coupling in te…
Lower-dimensional (hyper)surfaces that can carry gauge or gauge/gravitational anomalies occur in many areas of physics: one-plus-one-dimensional boundaries or two-dimensional defect surfaces in condensed matter systems, four-dimensional brane-worlds in higher-dimensional cosmologies or various branes and orbifold planes in string or M-theory. In all cases we may have (quantum) anomalies localized on these hypersurfaces that are only cancelled by ``anomaly inflow'' from certain topological interactions in the bulk. Proper cancellation between these anomaly contributions of different origin requ…
Generalized matter couplings to four-dimensional supersymmetric sigma models on general Kaehler manifolds are presented, preserving all holomorphic symmetries. Our generalization allows assignment of arbitrary U(1) charges to additional matter fermions, in all representations of (the holomorphic part of) the isometry group. This can be used to eliminate unwanted gamma_5 anomalies, in particular for the U(1) symmetry arising from the complex structure of the target space. A consistent gauging of this isometry group, or any of its subgroups, then becomes possible. When gauged in the presence of …
We present an orbifold compactification of the minimal seven dimensional supergravity. The vacuum is a slice of AdS_7 where six-branes of opposite tension are located at the orbifold fixed points. The cancellation of gauge and gravitational anomalies restricts the gauge group and matter content on the boundaries. In addition anomaly cancellation fixes the boundary gauge couplings in terms of the gravitational constant, and the mass parameter of the Chern-Simons term.
We establish the non-perturbative validity of the gauge anomaly cancellation condition in an effective electroweak theory of massless fermions with finite momentum cut-off and Fermi interaction. The requirement that the current is conserved up to terms smaller than the energy divided by the cut-off scale, which is the natural condition as gauge invariance is only emerging, produces the same constraint on charges as in the Standard Model. The result holds at a non-perturbative level as the functional integrals are expressed by convergent power series expansions and are analytic in a finite doma…
The anomaly found by Callan and Harvey is shown to be cancelled in a three-dimensional noncommutative gauge theory coupled to a fermion with a mass function depending on one spatial coordinate (domain wall mass). This evaluation has been done for the fermion in the fundamental and adjoint representations of the gauge group in the limit of small noncommutativity $θ$ parameter.
$Z \to γγ$ decay amplitude is considered and proven to be zero due to properties of polarization vectors and Bose statistics. Triangular diagrams for a pseudoscalar $\to γγ$ and $Z \to γγ$ processes with massive fermions in the loop are explicitely calculated. In the Standard Model axial anomaly vanishes in the sum of these diagrams as Z boson is mixed with one of the Goldstone bosons.
An interesting family gauge boson (FGB) model (Model A) has been proposed by Sumino. The model can give FGBs with a considerably low energy scale in spite of the sever constraints form the observed $K^0$-$\bar{K}^0$ mixing and so on. An essential idea in Model A is in the so-called Sumino cancellation mechanism between QED and FGB diagrams. However, Model A is not anomaly free and, besides, it causes effective interactions with $ΔN_{\rm family}=2$. In order to avoid these problems, a revised Sumino model with an inverted mass hierarchy (Model B) has proposed, but, in this time, it cannot satis…
We reconsider the Adler-Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin-Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approac…
It is pointed out that there are infinite classes of cases based on gauge groups of the form SU(p)xSU(q)xU(1) in which gauge anomalies cancel non-trivially for small sets of fermion multiplets that include symmetric tensor representations. These cancellations are non-trivial in the sense that no group-theoretic explanation in terms of embedding in a larger simple group is apparent. The cases presented here could be useful for model building and lead to models with extra leptons and an extra U(1) gauge interaction under which the Standard Model fermions have distinctive charges. In many cases t…
Whilst the prospect of new $Z'$ gauge bosons with only axial couplings to the Standard Model (SM) fermions is widely discussed, examples of anomaly-free renormalisable models are lacking in the literature. We look to remedy this by constructing several motivated examples. Specifically, we consider axial vectors which couple universally to all SM fermions, as well as those which are generation-specific, leptophilic, and leptophobic. Anomaly cancellation typically requires the presence of new coloured and charged chiral fermions, and we argue that in a large class of models masses of these new s…
In a minimal extension of the Standard Model, in which new neutral fermions have been introduced, we show that the requirement of vanishing anomalies fixes the hypercharges of all fermions uniquely. This naturally leads to electric charge quantization in this minimal scenario which has features similar to the Standard Model: invariance under the gauge group $SU(2)_L\otimes U(1)_Y$, conservation of the total lepton number and masslessness for the ordinary neutrinos. Such minimal models might arise as low-energy realizations of some heterotic superstring models or $SO(10)$ grand unified theories…
We discuss some implications of anomaly cancellation in the standard model with (i) the color group extended to $SU(N_c)$, and (ii) the leptonic sector extended to allow right-handed components for neutrinos.
Many extensions of the Standard Model (SM) predict new neutral vector bosons at energies accessible by the Large Hadron Collider (LHC). We study an extension of the SM with new chiral fermions subject to non-trivial anomaly cancellations. If the new fermions have SM charges, but are too heavy to be created at LHC, and the SM fermions are not charged under the extra gauge field, one would expect that this new sector remains completely invisible at LHC. We show, however, that a non-trivial anomaly cancellation between the new heavy fermions may give rise to observable effects in the gauge boson …
Quantum fluctuations lead to an anomalous violation of parity symmetry in quantum electrodynamics for an even number of spatial dimensions. While the leading parity-odd electric current vanishes in vacuum, we uncover a non-cancellation of the anomaly for strong electric fields with distinct macroscopic signatures. We perform real-time lattice simulations with fully dynamical gauge fields and Wilson fermions in $2+1$ space-time dimensions. In the static field limit, relevant at early times, we solve the problem analytically. Our results point out the fundamental role of quantum anomalies for st…
We extend the approach of Dolgov and Zakharov to the axial anomaly to study polarized radiative decays. We analyse the pattern of mass singularity cancellation in the corresponding decay rates. We compare polarized and unpolarized cases. The cancellation of the infrared and collinear singularities is verified to all powers of the lepton mass for the $π^+$ and $Z^0$ polarized radiative decays.
A Fermion in 2+1 dimensions, with a mass function which depends on one spatial coordinate and passes through a zero ( a domain wall mass), is considered. In this model, originally proposed by Callan and Harvey, the gauge variation of the effective gauge action mainly consists of two terms. One comes from the induced Chern-Simons term and the other from the chiral fermions, bound to the 1+1 dimensional wall, and they are expected to cancel each other. Though there exist arguments in favour of this, based on the possible forms of the effective action valid far from the wall and some facts about …
I remark that the weak mixing angle in the standard model may be computed even in the absence of a grand unification symmetry. In particular, if there is an additional gauged $U(1)$ symmetry at some large scale which can be made anomaly-free only by a Green-Schwarz (GS) mechanism, this typically results in a prediction for the weak angle. In the case of the standard model one can see that the standard Peccei-Quinn symmetry may be gauged and the anomalies cancelled through a GS mechanism. Remarkably enough, cancelation of anomalies works only for the `canonical' value $sin^2θ_W=3/8$. In the cas…
We comment on aspects of discrete anomaly conditions focussing particularly on $R$ symmetries. We review the Green-Schwarz cancellation of discrete anomalies, providing a heuristic explanation why, in the heterotic string, only the "model-independent dilaton" transforms non-linearly under discrete symmetries; this argument suggests that, in other theories, multiple fields might play a role in anomaly cancellations, further weakening any anomaly constraints at low energies. We provide examples in open string theories of non-universal discrete anomalies at low energies. We then consider the fact…
The standard model has for massless quarks and leptons "miraculously" no triangle anomalies due to the fact that the sum of all possible traces $Tr [τ^{Ai}$$ τ^{Bj}$$ τ^{Ck}]$ --- where $τ^{Ai}, τ^{Bi}$ and $τ^{Ck}$ are the generators of one, of two or of three of the groups $SU(3), SU(2)$ and $U(1)$ --- over the representations of one family of the left handed fermions and anti-fermions (and separately of the right handed fermions and anti-fermions), contributing to the triangle currents, is equal to zero. It is demonstrated in this paper that this cancellation of the standard model triangle …
We study a gauged $B-L$ extension of the standard model where the new fermions with fractional $B-L$ charges that play the role of keeping the model anomaly free can also explain the origin of neutrino mass at one loop level as well as dark matter. We discuss two different versions of the model to realise fermion and scalar dark matter, both of which guarantee the dark matter stability by a remnant discrete symmetry to which $U(1)_{B-L}$ gauge symmetry gets spontaneously broken down to. Apart from giving rise to the observed neutrino mass and dark matter abundance, the model also has tantalisi…
In a variant of chiral color with the electroweak gauge group generalized to $SU(3)_L \times U(1)$ anomaly cancellation occurs more readily than in the $SU(2)_L \times U(1)$ case. Three families are required by anomaly cancellation and the top family appears non-sequentially.
We note that the QCD phases at large finite density respect 't Hooft anomaly matching conditions. Specifically the spectrum of the light excitations possesses the correct quantum numbers required to obey global anomaly constraints. We argue that 't Hooft constraints can be used at finite density along with non perturbative methods to help selecting the correct phase.
We consider scenarios with a heavy $Z^\prime$ gauge boson with flavour non-universal quark and lepton couplings with the goal to illustrate how the cancellation of gauge anomalies generated by the additional U(1)$^\prime$ gauge symmetry would imply correlations between FCNC processes in the quark sector, in the lepton sector and most interestingly between quark flavour and lepton flavour violating processes. We present scenarios with only left-handed flavour-violating $Z^\prime$ couplings and those with also right-handed flavour-violating couplings. These are characterized by a small number of…
In the framework of the chiral quark-soliton model of the nucleon we investigate the properties of the polarized quark distribution. In particular we analyse the so called anomalous difference between the representations of the quark distribution functions in terms of occupied and non-occupied quark states. By an explicit analytical calculation it is shown that this anomaly is absent in the polarized isoscalar distribution Δu + Δd, which is ultaviolet finite. In the case of the polarized isovector quark distribution Δu - Δd the anomaly can be cancelled by a Pauli-Villars subtraction which is a…
Using the Fujikawa and the heat-kernel methods we make a complete and detailed computation of the global, gauge and gravitational anomalies present in the Standard Model defined on a curved space time with torsion. We find new contributions coming from curvature and torsion terms to the leptonic number anomaly (so that $B-L$ is not conserved any more), to the $U(1)_Y$ gauge and to the mixed $U(1)_Y$-gravitational anomalies, but the gauge anomaly cancellation conditions on the hypercharges remain the same. We also find that the condition, usually related to the cancellation of the mixed $U(1)_Y…
Nonstandard Solution for Anomaly Cancellation as Seesaw Neutrino Origin in the SM
Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University（上海交大李政道研究所与物理与天文学院）；Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院）
This thesis work analyzes basic field theoretical aspects of a class of models motivated by orientifold vacua of string theory and some of their phenomenological applications at the Large Hadron Collider. They extend the gauge structure of the Standard Model by anomalous extra U(1) symmetries, which involve Stuckelberg axions for anomaly cancellation and are accompanied by Chern-Simons interactions. In particular, these effective actions are characterized by a physical pseudoscalar (the axi-Higgs) in the CP-odd spectrum, which has the properties of a generalized Peccei-Quinn axion, with indepe…
The anomaly cancellation equations for the $U(1)$ gauge group can be written as a cubic equation in $n-1$ integer variables, where $n$ is the number of Weyl fermions carrying the $U(1)$ charge. We solve this Diophantine cubic equation by providing a parametrization of the charges in terms of $n-2$ integers, and prove that this is the most general solution.
A Lorentz and gauge symmetry preserving regularization method has been proposed recently in 4 dimension based on Euclidean momentum cutoff. It is shown that the triangle anomaly can be calculated unambiguously with this new improved cutoff. The anticommutator of γ^{5} and γ^μ multiplied by five gamma is proportional to terms that do not vanish under a divergent loop-momentum integral, but cancel otherwise.
