We explore a common symmetrical origin for two long standing problems in particle physics: the strong CP and the fermion mass hierarchy problems. The Peccei--Quinn mechanism solves the former one with an anomalous global $U(1)_{\rm PQ}$ symmetry. Here we investigate how this $U(1)_{\rm PQ}$ could at the same time explain the fermion mass hierarchy. We work in the context of a four-Higgs-doublet model which explains all quark and charged fermion masses with natural, i.e.\ order $1$, Yukawa couplings. Moreover, the axion of the model constitutes a viable dark matter candidate and neutrino masses are incorporated via the standard type-I seesaw mechanism. A simple extension of the model allows for Dirac neutrinos.
I review the physics case for flavour violating axions. In particular, I argue that relaxing the assumption of the universality of the Peccei-Quinn current opens up new pathways, including: the relaxation of the Supernova bound on the axion mass, a possible connection with the Standard Model flavour puzzle and the experimental opportunity of discovering the axion via flavoured axion searches.
We present a comprehensive discussion of the phenomenology of flavourful axions, including both standard Peccei-Quinn (PQ) axions, associated with the solution to the strong $CP$ problem, and non-standard axion-like particles (ALPs). We give the flavourful axion-fermion and axion-photon couplings and calculate the branching ratios of heavy meson ($K$, $D$, $B$) decays involving a flavourful axion. We also calculate the mixing between axions and heavy mesons $ K^0 $, $ D^0 $, $ B^0 $ and $ B_s^0 $, which affects the meson oscillation probability and mass difference. Mixing also contributes to m…
The solution to the Strong CP problem is analysed within the Minimal Flavour Violation (MFV) context. An Abelian factor of the complete flavour symmetry of the fermionic kinetic terms may play the role of the Peccei-Quinn symmetry in traditional axion models. Its spontaneous breaking, due to the addition of a complex scalar field to the Standard Model scalar spectrum, generates the MFV axion, which may redefine away the QCD theta parameter. It differs from the traditional QCD axion for its couplings that are governed by the fermion charges under the axial Abelian symmetry. It is also distinct …
A local flavour symmetry acting on the quarks of the Standard Model can automatically give rise to an accidental global $U(1)$ which remains preserved from sources of explicit breaking up to a large operator dimension, while it gets spontaneously broken together with the flavour symmetry. Such non-fundamental symmetries are often endowed with a mixed QCD anomaly, so that the strong CP problem is automatically solved via the axion mechanism. We illustrate the general features required to realise this scenario, and we discuss a simple construction based on the flavour group $SU(3)\times SU(2) \t…
We show how an accidental $U(1)$ Peccei-Quinn (PQ) symmetry can arise from a discrete $A_4$ family symmetry combined with a discrete flavour symmetry $ \mathbb{Z}_3 \times \mathbb{Z}_5^2 $, in a realistic Pati-Salam unified theory of flavour. Imposing only these discrete flavour symmetries, the axion solution to the strong $ CP $ problem is protected from PQ-breaking operators to the required degree. A QCD axion arises from a linear combination of $ A_4 $ triplet flavons, which are also responsible for fermion flavour structures due to their vacuum alignments. We find that the requirement of a…
We propose a minimal model in which the flavour anomaly in the $b \to s μ^+ μ^-$ transition is connected to the breaking of Peccei-Quinn (PQ) symmetry. The flavour anomaly is explained from new physics contribution by introducing one generation of heavy quark and heavy lepton which are vector-like under the standard model (SM) gauge group but charged under a local $U(1)_X$ group. They mix with the SM quarks and leptons, inducing flavour-changing $Z^\prime$ couplings, which generates the $b \to s μ^+μ^-$ anomaly at tree level. On the other hand the new fermions are chiral under the global Pecce…
We consider a 5D extension of the DFSZ axion model that addresses both the axion quality and fermion mass hierarchy problems, and predicts flavour-dependent, off-diagonal axion-fermion couplings. The axion is part of a 5D complex scalar field charged under a U(1)$_{PQ}$ symmetry that is spontaneously broken in the bulk, and is insensitive to explicit PQ breaking on the UV boundary. Bulk Standard Model fermions interact with two Higgs doublets that can be localized on the UV boundary or propagate in the bulk to explain the fermion masses and mixings. When the Higgs doublets are localized on the…
We investigate to what extent a generic, generation-dependent $U(1)$ symmetry acting on the quark Yukawa operators can reduce the number of free parameters by forcing some entries in the Yukawa matrices to vanish. The maximal reduction compatible with CP violation yields nine real parameters and one phase, which matches the number of physical observables, implying that such models have no free parameters. We derive a set of results: (i) the only possible structures have the form $M_4 \oplus M_5$, where the subscripts indicate the number of real parameters in the Yukawa matrices, (ii) there are…
Motivated by the recent Xenon1T result, we study a leptophilic flavour-dependent anomaly-free axion-like particle (ALP) and its effects on charged-lepton flavour violation (CLFV). We present two representative models. The first one considers that the ALP origins from the flavon that generates the charged-lepton masses. The second model assumes a larger flavour symmetry such that more general mixings in the charged-lepton are possible, while maintaining flavour-dependent ALP couplings. We find that a keV ALP explaining the Xenon1T result is still viable for lepton flavour violation and stellar …
We study UV-complete Froggatt-Nielsen-like models for the generation of mass and mixing hierarchies, assuming that the integrated heavy fields are chiral with respect to an abelian Froggatt-Nielsen symmetry. It modifies the mixed anomalies with respect to the Standard Model gauge group, which opens up the possibility to gauge the Froggatt-Nielsen symmetry without the need to introduce additional spectator fermions, while keeping mass matrices usually associated to anomalous flavour symmetries. We give specific examples where this happens, and we study the flavourful axion which arises from an …
We study the consequences of the existence and breaking of a Peccei--Quinn symmetry within the context of a dynamical model of electroweak symmetry breaking based on broken gauged flavour symmetries. We perform an estimate of the axion mass by including flavour instanton effects and show that, for low cut--offs, the axion is sufficiently massive to prevent it from being phenomenologically unacceptable. We conclude with an examination of the strong CP problem and show that our axion cannot solve the problem, though we indicate ways in which the model can be extended so that the strong CP proble…
An accidental U(1) Peccei-Quinn (PQ) symmetry automatically arises in a class of SO(10) unified theories upon gauging the SU(3)$_{f}$ flavour group. The PQ symmetry is protected by the $\mathbb{Z}_4 \times \mathbb{Z}_3$ center of $\text{SO(10)} \times \text{SU(3)}_f$ up to effective operators of canonical dimension six. However, high-scale contributions to the axion potential posing a PQ quality problem arise only at $d=9$. In the pre-inflationary PQ breaking scenario the axion mass window is predicted to be $m_a \in [7 \times 10^{-8}, 10^{-3}]$ eV, where the lower end is bounded by the seesaw…
We propose a unified model of scalar particles that addresses the flavour hierarchies, solves the strong CP problem, delivers a dark matter candidate, and provides the trigger for electroweak symmetry breaking. Besides furnishing a unification of the recently proposed axiflavon with a Goldstone-Higgs sector, the scenario can also be seen as adding a model of flavour (and strong CP conservation along with axion dark matter) to elementary Goldstone-Higgs setups. In particular, we derive bounds on the axion decay constant from the need to generate a SM-like Higgs potential at low energies, which …
In this talk, a unified model of scalar particles that addresses the flavour hierarchies, solves the strong CP problem, delivers a dark matter candidate, and radiatively triggers electroweak symmetry breaking is discussed. The recently proposed axiflavon is embedded together with an (elementary) Goldstone Higgs-sector in a single multiplet (and thereby also a model of flavour and strong CP conservation for the latter is provided). Bounds on the axion decay constant follow from requiring a SM-like Higgs potential at low energies and are confronted with constraints from flavour physics and astro…
We propose a gauged two-Higgs-doublet model (2HDM) featuring an anomalous Peccei-Quinn symmetry, $U(1)_{PQ}$. Dangerous tree-level flavour-changing neutral currents, common in 2HDMs, are forbidden by the extra gauge symmetry, $U(1)_X$. In our construction, the solutions to the important issues of neutrino masses, dark matter and the strong CP problem are interrelated. Neutrino masses are generated via a Dirac seesaw mechanism and are suppressed by the ratio of the $U(1)_X$ and the $U(1)_{PQ}$ breaking scales. Naturally small neutrino masses suggest that the breaking of $U(1)_X$ occurs at a rel…
We show how gauged flavour breaking and N = 1 supersymmetry breaking can be dynamically aligned to produce natural models. Supersymmetry is broken in a metastable vacuum, while a weakly gauged flavour symmetry is identified with an SU(3)_F subgroup of the global symmetry of the SQCD model. We find that alignment can easily occur either through strongly coupled flavour models, with SQCD bound states playing the role of flavons, or entirely weakly coupled and renormalizable models with elementary SU(3)_F adjoint flavons. In both cases it is essential that the SU(3)_F breaking, and hence all the …
We compare the effectiveness of two hypotheses, Natural Flavour Conservation (NFC) and Minimal Flavour Violation (MFV), in suppressing the strength of flavour-changing neutral-currents (FCNCs) in models with more than one Higgs doublet. We show that the MFV hypothesis, in its general formulation, is more stable in suppressing FCNCs than the hypothesis of NFC alone when quantum corrections are taken into account. The phenomenological implications of the two scenarios are discussed analysing meson-antimeson mixing observables and the rare decays B -> mu+ mu-. We demonstrate that, introducing fla…
The axion solution to the strong CP problem makes use of a global Peccei-Quinn (PQ) U(1) symmetry which is susceptible to violations from quantum gravitational effects. We show explicitly how discrete gauge symmetries can protect the axion from such violations. PQ symmetry emerges as an accidental global symmetry from discrete gauge symmetries which are subgroups of the anomalous U(1) of string origin. We also show how the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) axion model provides a natural solution to mu problem of supersymmetry as mu ~ M_{SUSY} ~ M^2_{PQ}/M_{Pl}.
Technically Natural Baryogenesis with the QCD Axion
University of California Santa Cruz（加州大学圣克鲁兹分校）；University of Washington（华盛顿大学）；International Centre for Theoretical Physics（国际理论物理中心）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We revisit the supersymmetric CP problem and find that it can be naturally resolved if the origin of CP violation is closely related to the origin of flavour structures. In this case, the supersymmetry breaking dynamics do not bring in any new CP-violating phases. This mechanism requires hermitian Yukawa matrices which naturally arise in models with a U(3) flavour symmetry. The neutron electric dipole moment (NEDM) is predicted to be within one-two orders of magnitude below the current experimental limit. The model also predicts a strong correlation between A_{CP}(b --> s gamma)and the NEDM. T…
Some implications of flavour unification in supersymmetric Grand Unified Theories are briefly reviewed. We argue that the gauge hierarchy phenomenon properly interpreted in terms of the natural missing VEV solution for the adjoint scalar in general SU(N) SUSY GUT could give insight into both of the basic features of low-energy physics accommodated in MSSM just as some strict colour-flavour (or colour-family) interrelation, so the SU(2) symmetry structure of weak interactions. There, apart from the ordinary triple matter of MSSM, just three families of the pseudo-Goldstone states appear in the …
An inverted mass hierarchy in the squark sector, as in so-called "natural supersymmetry", requires non-universal boundary conditions at the mediation scale of supersymmetry breaking. We propose a formalism to define such boundary conditions in a basis-independent manner and apply it to generic scenarios where the third-generation squarks are light, while the first two generation squarks are heavy and near-degenerate. We show that not only is our formalism particularly well-suited to study such hierarchical squark mass patterns, but in addition the resulting soft terms at the TeV scale are mani…
We propose a minimal realization of the Peccei Quinn mechanism in a realistic SU(5) model, where the axion mass is directly connected to the grand-unification scale. By taking into account constraints from proton decay, collider searches and gauge coupling unification, we predict the axion mass: $m_a \in [4.8, 6.6]$ neV. The upper bound can be relaxed up to $m_a < 330$ neV, at the cost of tuning the flavour structure of the proton decay operators. The predicted mass window will be complementarily probed by the axion dark matter experiments ABRACADABRA and CASPER-Electric, which could provide a…
We revisit flavour effects in soft leptogenesis relaxing the assumption of universality for the soft supersymmetry breaking terms. We find that with respect to the case in which the heavy sneutrinos decay with equal rates and equal CP asymmetries for all lepton flavours, hierarchical flavour configurations can enhance the efficiency by more than two orders of magnitude. This translates in more than three order of magnitude with respect to the one-flavour approximation. We verify that lepton flavour equilibration effects related to off-diagonal soft slepton masses are ineffective for damping th…
Using the particle content of the fundamental {27} supermultiplet of E_6, naturally small Dirac neutrino masses are obtained in the context of SU(3)_C times SU(2)_L times U(1)_Y times U(1)_{chi}, where U(1)_{chi} comes from the decomposition E_6 to SO(10) times U(1)_{psi}, then SO(10) to SU(5) times U(1)_{chi}. New observable consequences are predicted at the TeV scale. An axionic solution of the strong CP problem may be included at no extra cost.
We show that a large mixing between the right-handed charm and top squarks (i) is allowed by low-energy flavour constraints; (ii) reduces the experimental bound on the stop mass; (iii) has a mild, but beneficial, effect on fine-tuning; (iv) leads to interesting signatures at the LHC not presently investigated by experiments. We estimate the current bound on the stop mass, in presence of flavour mixing, and discuss the new collider signatures. The signal in the t anti-c (c anti-t) + missing E_T channel is large enough that it can be immediately searched for experimentally, while the signature w…
In the Type I see-saw model, the naturalness requirement that corrections to the electroweak $μ$ parameter not exceed 1 TeV results in a rough bound on the lightest right-handed neutrino mass, $M_{N_1}\lesssim 3\times 10^7$ GeV. In this letter we derive generic bounds applicable in any three-flavour Type I see-saw model. We find $M_{N_1}\lesssim 4\times 10^7$ GeV and $M_{N_2}\lesssim 7\times 10^7$ GeV. In the limit of one massless neutrino, there is no naturalness bound on $M_{N_3}$ in the Poincare protected decoupling limit. Our results confirm that no Type I see-saw model can explain the obs…
We consider a minimal left-right model with a Peccei-Quinn symmetry, where generalised charge conjugation plays the role of the left-right symmetry. We show how the spontaneous breaking of the Peccei-Quinn symmetry by a scalar singlet can provide us with solutions not only to the strong CP and dark matter problems but can also help to generate naturally suppressed off-diagonal CKM elements and small neutrino masses via the inverse seesaw mechanism. For this, we make use of an economical scalar sector composed of a bi-doublet, two doublets and a singlet only. As a result of the new gauge bosons…
In the absence of any additional assumption it is natural to conjecture that sizeable flavour-mixing mass entries, $Δm^2$, may appear in the mass matrices of the scalars of the MSSM, i.e. $Δm^2\sim O(m^2)$. This flavour violation can still be reconciled with the experiment if the gaugino mass, $M_{1/2}$, is large enough to yield (through the renormalization group running) a sufficiently small $Δm^2 / m^2$ at low energy. This leads to a gaugino dominance framework (i.e. $M_{1/2}^2\gg m^2$), which permits a remarkably model--independent analysis. We study this possibility focussing our attention…
We formulate a consistent model predicting, in the leading order approximation, maximal atmospheric mixing angle, vanishing reactor angle and tan θ_12 = 1/ϕ where ϕ is the Golden Ratio. The model is based on the flavour symmetry A5 \times Z5 \times Z3, spontaneously broken by a set of flavon fields. By minimizing the scalar potential of the theory up to the next-to-leading order in the symmetry breaking parameter, we demonstrate that this mixing pattern is naturally achieved in a finite portion of the parameter space, through the vacuum alignment of the flavon fields. The leading order approxi…
We investigate the deviation in the couplings of the standard model (SM) like Higgs boson ($h$) with a mass of 125 GeV from the prediction of the SM in multi-doublet models within the framework where flavour changing neutral currents at the tree level are naturally forbidden. After we present the general expressions for the modified gauge and Yukawa couplings for $h$, we show the correlation between the deviation in the Yukawa coupling for the tau lepton $hτ^+τ^-$ and that for the bottom quark $hb\bar{b}$ under the assumption of a non-zero deviation in the $hVV$ $(V=W,Z)$ couplings in two Higg…
In the context of supersymmetric theories, a weakly broken gauged SO(3) flavour symmetry is used to produce two highly degenerate right handed (RH) neutrinos. It is then shown that this SO(3) flavour symmetry is compatible with all fermion masses and mixings if it is supplemented with a further SU(3) flavour symmetry. A specific Susy breaking model is used to generate the light neutrino masses as well as a natural model of TeV scale resonant leptogenesis.
We consider a two Higgs-doublet model with $S_3$ symmetry, which implies a $π\over 2$ rather than 0 relative phase between the vacuum expectation values $<Φ_1>$ and $<Φ_2>$. The corresponding Feynman rules are derived accordingly and the transformation of the Higgs fields from the weak to the mass eigenstates includes not only an angle rotation but also a phase transformation. In this model, both doublets couple to the same type of fermions and the flavour-changing neutral currents are naturally suppressed. We also demonstrate that the Type III natural flavour-conserving model is valid at tree…
We present predictions for $μ\to e γ$ and $τ\to μγ$ in the CMSSM in which there is a natural hierarchy of neutrino masses resulting from the sequential dominance of three right-handed neutrinos. We perform a global analysis of this model in the $(m_0,M_{1/2})$ plane, assuming radiative electroweak symmetry breaking, and including all observed laboratory data. We confirm that a large (small) $τ\to μγ$ rate results from the dominant right-handed neutrino being the heaviest (lightest) one. We show that the $μ\to e γ$ rate may determine the order of the sub-dominant neutrino Yukawa couplings in th…
We consider an enlarged color sector which solves the strong CP problem via new massless fermions. The spontaneous breaking of a unified color group into QCD and another confining group provides a source of naturally large axion mass $m_a$ due to small size instantons. This extra source of axion mass respects automatically the alignment of the vacuum, ensuring a low-energy CP-conserving vacuum. The mechanism does not appeal to a $Z_2$ 'mirror' copy of the SM, nor does it require any fine-tuning of the axion-related couplings at the unification scale. There is no light axion and uncharacteristi…
We argue that coherent oscillations of the axion field excited by the misalignment mechanism and non-thermal leptogenesis by the saxion decay can naturally explain the observed abundance of dark matter and baryon asymmetry, thus providing a solution to the baryon-dark matter coincidence problem. The successful axionic co-genesis requires a supersymmetry breaking scale of O(10^{6-7}) GeV, which is consistent with the recently discovered standard-model like Higgs boson of mass about 126 GeV. Although the saxion generically decays into a pair of axions, their abundance sensitively depends on the …
After a brief introduction to the so-called flavour problem, we discuss the role of rare K decays in probing the mechanism of quark-flavour mixing. Particular attention is devoted to the formulation of the Minimal Flavour Violation hypothesis, as a general and natural solution to the flavour problem, and to the fundamental role of K -> pi nu nu-bar decays in testing this scenario.
Starting from the general cosine form for the axion effective potential, we quantize the axion and show that the result is described by a naturally flat potential, if interactions with other particles are not considered. This feature therefore restores the would-be Goldstone-boson nature of the axion, and we calculate the corresponding vacuum energy density, which does not need to be too large by orders of magnitude compared to Dark Energy.
Under the expectation that nature is natural, we extend the Standard Model to include SUSY to stabilize the electroweak sector and PQ symmetry to stabilize the QCD sector. Then natural SUSY arises from a Kim-Nilles solution to the SUSY mu problem which allows for a little hierarchy where mu ~ f_a^2/M_P ~ 100-300 GeV while the SUSY particle mass scale m(SUSY)~ 1-10 TeV >> mu. Dark matter then consists of two particles: a higgsino-like WIMP and a SUSY DFSZ axion. The range of allowed axion mass values m(axion) depends on the mixed axion-higgsino relic density. The range of m(axion) is actually r…
We show that a recently proposed framework that provides a simple connection between Majorana neutrinos and an invisible axion in minimal scalar extensions of the standard electroweak model can be naturally embedded in a classically scale-invariant setup. The explicit breaking of the scale invariance à la Coleman-Weinberg generates the Peccei-Quinn and electroweak scales. The spontaneous breaking of the chiral $U(1)_{PQ}$ triggers the generation of neutrino masses via Type-II seesaw and, at the same time, provides a dynamical solution to the strong CP problem as well as the axion as a dark mat…
We propose a landscape of many axions, where the axion potential receives various contributions from shift symmetry breaking effects. We show that the existence of the axion with a super-Planckian decay constant is very common in the axion landscape for a wide range of numbers of axions and shift symmetry breaking terms, because of the accidental alignment of axions. The effective inflation model is either natural or multi-natural inflation in the axion landscape, depending on the number of axions and the shift symmetry breaking terms. The tension between BICEP2 and Planck could be due to smal…
We study the Dine-Fischler-Srednicki (DFS) model in the light of the recent Higgs LHC results and electroweak precision data. The DFS model is a natural extension of the two-Higgs doublet model endowed with a Peccei-Quinn symmetry and leading to a physically acceptable axion. For generic couplings, the model reproduces the minimal Standard Model showing only tiny deviations (extreme decoupling scenario) whereas all additional degrees of freedom (with the exception of the axion) are very heavy. Recently, new corners of this model have been highlighted where it may exhibit enlarged global symmet…
We show that Peccei-Quinn and lepton number symmetries can be a natural outcome in a 3-3-1 model with right-handed neutrinos after imposing a Z_11 x Z_2 symmetry. This symmetry is suitably accommodated in this model when we augmented its spectrum by including merely one singlet scalar field. We work out the breaking of the Peccei-Quinn symmetry, yielding the axion, and study the phenomenological consequences. The main result of this work is that the solution to the strong CP problem can be implemented in a natural way, implying an invisible axion phenomenologically unconstrained, free of domai…
The recent tension between local and early measurements of the Hubble constant can be explained in a particle physics context. A mechanism is presented where this tension is alleviated due to the presence of a Majoron, arising from the spontaneous breaking of Lepton Number. The lightness of the active neutrinos is consistently explained. Moreover, this mechanism is shown to be embeddable in the Minimal (Lepton) Flavour Violating context, providing a correct description of fermion masses and mixings, and protecting the flavour sector from large deviations from the Standard Model predictions. A …
By eschewing finetuning from the electroweak and QCD sectors of supersymmetry (natural supersymmetry or SUSY), and by invoking the Kim-Nilles solution to the SUSY mu problem, one is lead to models wherein the dark matter is comprised of a mixture of axions and higgsino-like WIMPs. Over a large range of Peccei-Quinn breaking scale f_a~ 10^9-10^{12} GeV, one then expects about 90-95% axion dark matter. In such a scenario, both axion and WIMP direct detection may be expected.
If electroweak dynamical symmetry breaking is due to a chiral condensate of color sextet quarks, dynamics analagous to ``walking technicolor'' will enhance the condensate by orders of magnitude compared to the electroweak chiral scale. This enhancement compensates for the exponential suppression of electroweak scale color instanton interactions. As a result the $η_6$ axion can naturally aquire an electroweak scale mass.
The complex field $Φ$ containing the QCD axion has a natural portal coupling to the Higgs doublet of the form $λ_{h ϕ}|Φ|^2 |H|^2$. Here we consider the possibility that $λ_{h ϕ}$ has a natural magnitude for a dimensionless coupling, $λ_{h ϕ} \sim 0.1-1$. This is possible if the total mass squared parameter of the Higgs in the vacuum, including quadratic divergent and the $Φ$ vacuum expectation value contributions, is renormalized to reproduce the observed Higgs boson mass. It is then possible for the axion sector to stabilize the electroweak vacuum. We show the requirement of electroweak vacu…
The discovery of the Higgs boson along with the vigorous confirmation of the SM coupled with non-appearance of new (SUSY) particles has led to an apparent naturalness crisis. We argue the crisis stems from overestimates of finetuning. A proper evaluation of naturalness in SUSY leads to light higgsinos with mass mu~ 100-300 GeV while other sparticles may inhabit the multi-TeV range at little cost to naturalness. Special signatures like dilepton mass edges in gluino pair events and same-sign dibosons from wino pairs might occur at LHC13. An ILC with \sqrt{s}>2 mu would make the definitive test o…
We show that the hierarchically small mu term in supersymmetric theories is a consequence of two identical pairs of Higgs doublets taking a democratic form for their mass matrix. We briefly discuss the discrete symmetry S2xS2 toward the democratic mass matrix. Then, we show that there results an approximate Peccei-Quinn symmetry and hence the value mu is related to the axion decay constant.
