# Third Generation Familons,

Factories, and Neutrino Cosmology

###### Abstract

We study the physics of spontaneously broken family symmetries acting on the third generation. Massless familons (or Majorons) associated with such broken symmetries are motivated especially by cosmological scenarios with decaying tau neutrinos. We first note that, in marked contrast with the case for the first two generations, constraints on third generation familon couplings are poor, and are, in fact, non-existent at present in the hadronic sector. We derive new bounds from – mixing, , , and astrophysics. The resulting constraints on familon decay constants are still much weaker than those for the first and second generation. We then discuss the promising prospects for significant improvements from searches for , , and with the current CLEO, ARGUS, and LEP data. Finally, we note that future constraints from CLEO III and the factories will probe decay constants beyond GeV, well within regions of parameter space favored by proposed scenarios in neutrino cosmology.

###### pacs:

^{†}

^{†}preprint: LBNL–40822 UCB–PTH–97/47 hep-ph/9709411 September 1997

## I Introduction

For over half a century, one of the major puzzles in particle physics
has been the question of why quark and lepton families replicate.
Although we have accumulated a wealth of data concerning the masses
and mixings of quarks and leptons, we still appear to be far from a
true understanding of family structure. In the absence of a concrete
model to consider, it is natural to postulate the existence of some
family symmetry [1, 2, 3] that plays a role in
determining the observed particle spectrum. Once we consider such a
family symmetry, we face a plethora of options. The symmetry may be
(1) discrete,^{1}^{1}1There is a subtle distinction between global
and gauged discrete symmetries [4]. For this phenomenological
analysis, however, they are equivalent. (2) continuous and local, or
(3) continuous and global. Within each of these categories, one may
choose any of a number of symmetry groups, and the overall family
symmetry may even be a combination of the three possibilities.

Of course, any exact family symmetry of the underlying theory must be spontaneously broken at some energy scale since we know that the quark and lepton masses are very different from one family to the next. For option (1), spontaneously broken discrete symmetries, domain walls are the only model-independent predictions, and these cannot be studied in particle physics laboratories. In case (2), the masses of the family gauge bosons of spontaneously broken local continuous symmetries can be constrained, e.g., from - mixing [5].

From a phenomenological point of view, however, possibility (3) is
particularly enticing, as it implies the existence of massless
Nambu–Goldstone bosons, called “familons,” from the spontaneously
broken family symmetry. This family symmetry may be either Abelian or
non-Abelian; Nambu--Goldstone bosons associated with the
spontaneous symmetry breaking of an Abelian lepton number symmetry are
often called ‘‘Majorons.’’^{2}^{2}2Majorons have been extensively
studied, and arise in a variety of models [6], including, for
example, supersymmetric theories with spontaneous -parity
breaking [7]. In this paper, we study a number of probes,
many of which are applicable to both Abelian and non-Abelian
symmetries. We use the generic name “familon” to denote the
associated Nambu–Goldstone bosons in either case. The existence of
new massless particles has many implications in particle physics,
astrophysics, and cosmology, and, as we will see, may be probed in a
wide variety of experiments. Moreover, the couplings of familons at
low energies are determined by the non-linear realization of the
family symmetry. These couplings are, e.g., of the form

(1) |

where is the family symmetry breaking scale, i.e., the familon decay constant, are the familons, are the generators of the broken symmetry, and the are fermion fields in terms of which the flavor symmetry is defined. The strength of the familon coupling is therefore inversely proportional to and can be constrained for a given family symmetry group in a model-independent manner.

Familon couplings between the first and second generations have been studied extensively and will be reviewed below. In contrast, however, couplings involving the third generation are largely unexplored, although they may have rather rich phenomenological and cosmological implications [8]. Current constraints in the lepton sector are relatively weak, with the best bounds coming from bounds [9], and there are at present no corresponding bounds reported in the hadronic sector (see, however, Ref. [10]). At the same time, it is a logical possibility that the familon couples preferentially to the third generation, and models have been proposed in which this is the case [11]. It is therefore interesting to explore the possibilities for improving (or setting) bounds on familon scales for the third generation, especially in light of the upcoming physics experiments.

In this paper, we will study what we believe to be the most sensitive probes of couplings of familons to the third generation, primarily to leptons and quarks. We show that dedicated analyses of existent data from CLEO, ARGUS, and LEP could probe family symmetry breaking scales up to GeV and may be significantly improved at future factories. Simply because this is largely unexplored physics, there is a high discovery potential for familons at these facilities.

Familon couplings to the third generation are also of interest from a cosmological point of view. The mass of the neutrino is still allowed to be as large as 18.2 MeV experimentally [12]. A heavy neutrino has interesting consequences for both big-bang nucleosynthesis (BBN) [13, 14, 15, 16] and large scale structure formation [17, 18, 19, 20], as will be discussed in Sec. VI. Since a heavy neutrino ( eV, where is the expansion rate of the universe in units of 100 km/sec/Mpc) must decay in order not to overclose the universe, an invisible decay into a lighter neutrino and a massless boson, such as a familon (or Majoron), is typically required. (The three neutrino mode is strongly disfavored and therefore the familon mode is most preferred [13].) There is therefore an interesting interplay between experimental searches for familons and scenarios requiring heavy neutrinos, and, as we will see, future collider experiments and analyses may severely constrain a number of such cosmologically motivated scenarios.

This paper is organized as follows. We begin in Sec. II with a discussion of familon interactions. In particular, we emphasize that the familon interactions of particles in the same gauge multiplet are expected to be comparable. In Sec. III we consider constraints on familon interactions that may be inferred from current experimental data, concentrating on familon couplings to the third generation. Current bounds on third generation couplings from astrophysical considerations are presented in Sec. IV. We then describe some promising prospects for detecting familons in physics at future experiments in Sec. V. Finally, we note some of the interesting cosmological implications in Sec. VI and give our conclusions in Sec. VII.

## Ii Familon Interactions

The standard model contains 15 particle states in each of the 3 generations. These states are distinguished by the SU(3)SU(2)U(1) gauge interactions, which divide each generation into 5 multiplets: , , , , and . The gauge interactions therefore break the flavor symmetry group from U(45) to U(3). In the standard model, the flavor group U(3) is broken explicitly to by Yukawa couplings. However, in extensions of the standard model in which one hopes to gain some understanding of the pattern of fermion masses and mixings, some subgroup of the flavor group may be an exact symmetry of the Lagrangian that is broken spontaneously by the vacuum, and it is this possibility we consider here.

The massless Nambu-Goldstone bosons of the spontaneously broken flavor symmetry, familons [1, 2, 3], have interactions given by the couplings

(2) |

where are the familon fields, and are flavor
currents.^{3}^{3}3Throughout this study, we will assume that no
additional light degrees of freedom are introduced by other new
physics. The interactions are suppressed by , the scale at which
the flavor symmetry is spontaneously broken. Note that familons are
derivatively coupled,^{4}^{4}4If the flavor symmetry is anomalous,
familons may also have non-derivative, flavor-diagonal couplings. We
will not consider such couplings here. and so do not mediate
long-range () forces. The most general current composed of two fermion fields takes either the form

(3) |

or

(4) |

where are the projection operators , and are generational indices, and are the spontaneously broken generators of the family symmetry. The fields and are fermion mass eigenstates, which we assume here to be also flavor eigenstates. (The more general case is described below.) Using the form of the current given in Eq. (3), the familon interaction may be written as

(5) | |||||

where in the last step we have integrated by parts and then substituted the equations of motion. The second line of Eq. (5) is of course only valid for on-shell fermions such as external leptons, whereas in hadronic matrix elements and processes including off-shell fermions, the derivative coupling of the first line must be used.

We see that familons may mediate or be produced in family-changing processes. They may also couple to identical fermions , but only through axial couplings. What processes are mediated by familons depends on the particular family symmetry group that is broken. For example, for O() groups, the generators are anti-symmetric, and so do not generate flavor-diagonal interactions. However, they do generate interactions like , where we have considered axial vector current interactions as an example. Familons from O() groups may therefore mediate neutral meson mixing, which we will consider in Sec. III.2. The situation is reversed for SU() groups. Here, flavor-diagonal couplings exist. However, if we consider any SU(2) subgroup and form the complex familon , the off-diagonal interactions are given by , and we see that exchange cannot induce neutral meson mixing.

Up to this point, we have ignored possible mass mixing effects. In general, if the flavor eigenstates are related to the mass eigenstates by

(6) |

where is a unitary mixing matrix, the familon interactions are given by

(7) | |||||

where . Mass mixings may therefore generate flavor-diagonal interactions from flavor off-diagonal interactions, and vice versa. For example, in the case of an Abelian U(1) symmetry, mass mixing effects may generate flavor-changing interactions. They may also extend non-maximal family symmetries to couplings involving all three generations; for example, a U(2) symmetry between the first and second families, may, after rotation to mass eigenstates, result in familon interactions involving the third generation.

While the phenomenology of familons varies from group to group, it is important to note that gauge symmetry relates the familon interactions of particles in the same gauge multiplet. As an example, let us consider a spontaneously broken lepton flavor symmetry. The familon interaction is then given by

(8) |

where the SU(2) lepton doublets are in the flavor eigenstate basis. This interaction therefore generates familon interactions for both the charged leptons and neutrinos. In the presence of neutrino masses, the flavor eigenstates may not correspond to mass eigenstates. The familon interactions in the mass basis are then

(9) |

where , and we have defined and . and are therefore related by a similarity transformation, and in the presence of mass mixing, the couplings of the interactions of and are not necessarily identical. However, in the absence of fine-tuning, we expect these couplings to be of the same magnitude. Bounds on one familon interaction may thus be considered to imply comparable bounds on the other interactions linked by gauge symmetry.

Because the familon interactions of particles in the same gauge multiplet are comparable in the absence of fine-tuning, there are many more relations in theories with enlarged gauge groups. For example, for SU(5) grand unified theories (GUTs), the particles are expected to have comparable familon interactions, as are the particles . A particularly relevant example for our study below is that, in the GUT framework, bounds on familon decays of mesons imply bounds on familon decays of tau neutrinos in the absence of fine-tuning.

Flavor mixing effects also induce familon couplings of fields with different generational indices. In the quark sector, for example, substituting the quark doublet for in the discussion above, Eq. (9) becomes

(10) |

where and are related by the CKM matrix through

(11) |

We see that in general, couplings to all generations are induced by flavor mixings. For example, a familon with flavor-diagonal coupling to in the up sector couples not only to , but also to, for example, and . The induced couplings to first and second generation quarks in this case are CKM-suppressed, but may still lead to significant bounds when, as is often the case, these induced couplings are much more strongly constrained. We will consider the constraints on mixing-induced couplings from decays in Sec. III.1 and from supernova cooling in Sec. IV.3.

Finally, note in Eq. (3) that the strength of the interaction depends not only on , but also on and the couplings . In the following sections, we will present a variety of bounds on combinations of these couplings, and it is important that we define our conventions and normalizations. We will always define our interaction as

(12) |

and similarly for and ; the superscripts of the couplings will often be omitted when they are obvious from the context. In presenting our bounds, it will be convenient to define

(13) |

where . In addition, as many of our bounds are to a good approximation independent of the chirality of the interaction and so only dependent on the combination , we define

(14) |

## Iii Bounds from accelerator data

As described in the previous section, familons may take part in flavor-changing processes, and bounds on such processes lead to lower bounds on the familon energy scale. For familons mediating transitions between the first and second generation, such bounds are rather stringent. In contrast, similar bounds involving the third generation are much weaker, with the previously reported constraints limited only to bounds from rare decays. We are thus motivated to focus on the third generation. In Sec. III.1, we begin by reviewing and contrasting such bounds, and then discuss the implications of flavor eigenstate mixings. We then go on to derive new bounds from a variety of processes. In Sec. III.2 we consider familon-mediated processes such as neutral meson mixing and rare leptonic decays of mesons. Finally, in Sec. III.3 we consider possible analyses at LEP and extrapolate a preliminary ALEPH bound on to a bound on .

### iii.1 Decays to familons

We begin by considering bounds from decays of mesons and leptons to familons. Normalizing the relevant familon scale according to Eq. (12), we find

(15) |

where . In the limit of exact flavor SU(3) symmetry, the form factor at zero momentum transfer has a fixed normalization, . For leptonic decays , the exact tree-level partial decay width in the limit of massless is given by

(16) |

where here .

The strongest bound on any flavor scale is derived from the constraint on exotic decay. Using the above expressions, the experimental result [21] leads to the bound

(17) |

Note that the limit on bounds only the vectorial familon coupling; the axial coupling is unconstrained. For the leptonic sector, Jodidio et al. report the constraint [22], which they obtain under the assumption of a vector-like familon coupling. This can be converted into the bound

(18) |

For familon interactions of arbitrary chirality, the slightly weaker constraint

(19) |

may be obtained from the bound (90% CL) [23].

We now compare these bounds to those available in the third generation. The ARGUS collaboration [9] has bounded the branching fractions of decays into light bosons and found the limits and . These imply the following constraints on the flavor scale:

(20) | |||||

(21) |

We see that the bounds on flavor scales in the leptonic sector are significantly less stringent for third generation couplings than for those involving only the first two. The discrepancy is even more pronounced in the hadronic sector, where there are as yet no reported bounds on flavor scales from decays.

It is also worth noting, however, that strong bounds on a particular flavor scale, such as the one on , may imply significant bounds on other flavor scales as well. These bounds are induced by the flavor-mixing effects discussed in Sec. II and are thus model-dependent. As an example let us now assume that flavor and mass eigenstates coincide for up-type quarks. A given familon coupling in the up sector requires, by gauge invariance, a corresponding coupling in the down sector. For example, from Eqs. (10) and (11) we see that the coupling induces the coupling , which mediates the rare decay . Assuming complex familons, the Hermitian conjugate coupling gives a similar contribution to the decay into the complex conjugate familon. Summing both decay widths and comparing to the bound on in Eq. (17), one can derive the mixing induced bound

(22) |

Under similar assumptions, we find . Note, however, that such bounds do not apply if the mass and flavor bases are aligned in the down sector [24] or if the couplings are purely axial.

### iii.2 Familon-mediated processes

In this section we derive new constraints on the scale of spontaneous flavor symmetry breaking by considering non-standard familon contributions to neutral meson mixing and existing bounds on rare leptonic decays such as .

A familon contribution to neutral meson mixing requires a real flavor group to be spontaneously broken in the corresponding sector, such that the same real familon scalar field couples to the quark current and its Hermitian conjugate. For concreteness, let us consider the system; similar formulae hold (at least approximately) for other neutral meson systems. Assuming the general coupling structure

(23) |

we find a familon contribution to the mass splitting of

(24) |

Eq. (24) may be derived by taking the matrix element of the non-local operator

(25) |

between and states and using the definition of the pseudoscalar decay constant, . The subscripts , , , and in Eq. (25) are color indices. Between two color singlet states, there are two contributions. The first one arises from and with a familon in the -channel. In this case, the momentum transfer through the familon propagator is , and after a vacuum insertion, it is easy to verify that this contribution is as in Eq. (24), but without the factor of . However, there is also a -channel contribution from and , which may be evaluated by a Fierz transformation and then a vacuum insertion as before. For a heavy–light system like the meson, one may assume the free-quark picture, in which the momentum transfer is governed by the energy of the “static” quark , and, in the numerator, the derivative acting on the quark current gives again a factor of . Using and including the relative color factor of , one can estimate the -channel contribution to be times the -channel contribution, which leads to Eq. (24).

Our result should be fairly reliable for the meson. For and mesons, the evaluation of the -channel momentum transfer is more ambiguous. However, because this contribution is suppressed relative to the -channel part, we expect the result of Eq. (24) to be reasonably accurate in these cases as well. We also note that a vector-like familon interaction does not contribute to the mass splitting, at least in the heavy quark approximation . Although one might expect a vector contribution to appear in the -channel contribution after the Fierz rearrangement, one finds that the term proportional to contains axial vector and pseudoscalar contributions of equal magnitude but opposite sign.

The constraint on the flavor scale results in principle from the requirement that the combined standard model and familon contributions do not exceed the measured value. However, when considering nonstandard contributions, it is also uncertain what one should take as the standard model contribution. For example, the reported value [25] for is derived from mixing under the assumption that the standard model gives the only contribution. As a conservative bound, we simply compare the familon contributions directly to the corresponding measured values. The results are summarized in Table 1. We take the decay constants to be MeV and MeV from recent lattice results [26], and MeV [25]. Since we use the measured mass splitting (not its error), the bounds from and will only improve when the size of the standard model contribution can be quantified independently. For the , where only the upper bound on the mass splitting is known, future experiments will improve the bound.

Bound | ||
---|---|---|

[25] | GeV | |

[27] | GeV | |

[25] | GeV |

We next consider rare leptonic decays of neutral mesons, mediated by familon exchange. Such decays are possible if the same familon couples to both quarks and leptons. This is guaranteed in grand unified scenarios, where quarks and leptons are in the same gauge multiplet. In general the relevant interaction can be written in terms of effective vector and axial vector couplings that parametrize the familon couplings and mixing angles of a particular model. For example, the process can be mediated by the interaction Lagrangian

(26) |

Note that the constants and may be different in the hadronic and leptonic sectors. Also, even if familon couplings always include third generation flavor eigenstates, mixing effects may induce transitions like .

With the interaction defined in Eq. (26) one obtains a width of

(27) |

where , and we have displayed the leading piece. In the limit where the lighter lepton is massless, the result is independent of the chirality of the interaction and depends only on the combination of lepton couplings . Expressions for other similar processes are obtained by replacing the coupling constants and the meson and lepton masses accordingly. Limits on the flavor scales from current experimental bounds on rare leptonic decays are given in Table 2.

The bounds of Tables 1 and 2 are significantly weaker than those presented in Sec. III.1. This is especially true in Table 2, as rare leptonic meson decays are dependent on the flavor scale to the fourth power. However, such processes set bounds on third generation hadronic familon couplings, which were previously unconstrained. It is also important to note that the bounds on familon couplings to the first two generations are also interesting, as they constrain axial couplings, whereas the bound from decay reviewed in the previous section bounds only vector-like couplings.

Branching Ratio Upper Bound | Bound | |
---|---|---|

[28] | GeV | |

[28] | GeV | |

[28] | GeV | |

[29] | GeV | |

[30] | GeV |

### iii.3 Constraints from LEP

Currently, there are no reported experimental bounds on decays . One can, however, infer a constraint from ALEPH’s preliminary bound on [31]. By searching for events with large missing energy, they placed the constraint (90% CL). One can rescale this constraint to obtain an upper bound on .

The analysis for relies on the distribution [32], where is defined by

(28) |

in each hemisphere of -tagged events. Here, is half of the center-of-momentum energy, , where and are the visible invariant masses in the same and opposite hemispheres, respectively, and is the total visible energy in the hemisphere. improves the estimate of the actual missing energy in the hemisphere by correcting for the fact that the hemisphere with larger invariant mass typically has higher energy.

The backgrounds from and are suppressed by rejecting events with identified or in the relevant hemisphere. Up to this point, we do not expect significant differences in efficiencies between the mode and the mode. They then required . The efficiencies for this requirement obviously differ between the two decay modes, since the mode has two missing neutrinos, resulting in a harder spectrum than that of the mode. The spectrum of both modes may be calculated by convoluting the theoretical missing energy distribution in three-body () and two-body () decays with the measured fragmentation function [33]. We find that the ratio of efficiencies is 0.43 with little dependence on the details of the fragmentation function. By scaling the reported upper bound by this factor, we find

(29) |

Using the expression of Eq. (16) with the substitution of for , this corresponds to a limit on the flavor scale of

(30) |

Note that this analysis does not require an energetic strange particle, and so the constraint of Eq. (29) is actually on the sum . Thus, for , the bound on the flavor scale given in Eq. (30) improves by a factor . The bound of Eq. (30) is enhanced by the fact that the SM decay width is greatly suppressed by , which increases the sensitivity of decays to small exotic decay widths.

## Iv Bounds from Astrophysics

In this section, we discuss constraints on third generation familon couplings from astrophysics. We begin in Sec. IV.1 with constraints on direct (tree-level) couplings. Second and third generation particles are absent in almost all astrophysical objects. The exception is supernovae, where all three neutrino species are thermalized in the core. We therefore consider what bounds on familon couplings to neutrinos may be obtained by supernova observations. Couplings of familons to the third generation may also radiatively induce couplings to first generation particles. Although such induced couplings are suppressed by loop factors, they are so stringently bounded by constraints from supernovae, white dwarfs, and red giants that interesting bounds also result. These are studied in Sec. IV.2. Finally, mixings of flavor eigenstates may also induce couplings of familons to the first generation; such effects are discussed in Sec. IV.3. It is important to note that, while the bounds derived in this section are rather strong in certain cases, they are also typically more model-dependent than, for example, the accelerator bounds of the previous section. We therefore specify the necessary conditions for each bound in detail in each case.

### iv.1 Bounds from direct couplings

In 1987, the Kamiokande group and the IMB group independently detected neutrinos emitted from supernova SN 1987A. They observed that the neutrino pulse lasted for a few seconds. Furthermore, their results indicate that neutrinos carried off about erg from the supernova. The observed duration time and neutrino flux can be well explained by the generally accepted theory of core collapse, and the observations confirmed the idea that most of the released energy in the cooling process is carried off by neutrinos. Exotic light particles, such as familons, may affect the agreement of theory and observation, since they can also carry off a significant energy fraction. The core of the supernova is hot () and dense, and so neutrinos are thermalized in the core and can be a source of familon emission. If the energy fraction carried away by familons is substantial, the duration time of the neutrino pulse becomes much shorter than the observed value. In order not to affect the standard cooling process, the familon luminosity must be smaller than the neutrino luminosity, i.e., less than erg/sec.

This constraint can be satisfied in two different regimes of the familon coupling strength. For sufficiently high flavor scales , the familon interaction is weak enough that familons are rarely produced and the familon luminosity is suppressed. On the other hand, for sufficiently low flavor scales, although familons are readily produced, they interact so strongly that they become thermalized and trapped in the core as well, thus decreasing the familon luminosity. Therefore, there are two parameter regions consistent with observations, high and low , with an excluded region in the middle.

These types of constraints have been discussed by Choi and
Santamaria [34] in the context of a Majoron model. We
modify their discussions slightly for the familon case. To simplify
the analysis, we will look at two extreme scenarios. First we
consider a diagonal familon coupling to , as in the case of
an Abelian family symmetry, and second we analyze a purely
off-diagonal coupling, as in the case of an O(2) family
symmetry.^{5}^{5}5Throughout our discussions, we assume that
neutrinos are Majorana particles. Observations of supernova SN1987A
imply that Dirac neutrinos must be lighter than
3 keV [36] or heavier than 31 MeV [37]. As
we will discuss in Sec. VI, most of the interesting
mass range from a cosmological point of view is therefore excluded.
For a general family symmetry, one expects familons with both diagonal
and off-diagonal couplings; a generalization to such cases is
straightforward. In this subsection, we also neglect possible
mismatches between the flavor and mass eigenstates, and assume that
the relative angles relating the two are small.
Such mismatches will be discussed in
Sec. IV.3. Finally, we assume that ,
are negligible compared to , as suggested
from laboratory constraints as well as the corresponding masses of the
charged leptons.

#### Familon with diagonal coupling

Here we consider a purely diagonal familon coupling to , such as in models with a U(1) family symmetry acting on the third-generation lepton doublet . The relevant interaction is given by

(31) |

Let us first consider the case where the familon can freely escape the core of the supernova. Based on the interaction given in Eq. (31), potentially significant processes of familon production are the neutrino scatterings and , the latter process being allowed due to background matter effects. The familon luminosities due to these processes are given in Ref. [34]:

(32) | |||||

(33) |

If either one of the above luminosities is larger than erg/sec, the cooling process of the supernova may be dominated by familon emission, and the duration time of the neutrino pulse becomes shorter than . Imposing the constraint that the familon luminosities given in Eqs. (32) and (33) are smaller than erg/sec, we obtain the following constraints:

(34) | |||||

(35) |

Familon volume emission is sufficiently small when both Eqs. (34) and (35) are satisfied.

On the other hand, if familon interactions are strong enough, familons effectively scatter off the neutrinos in the background and get thermalized and trapped in the supernova. Once this happens, a thermal sphere of familons is formed, just like the thermal neutrino sphere, and familons can only be emitted from the surface. The familon luminosity essentially obeys the formula of blackbody emission with the surface temperature of the familon sphere. The important point is that, once the familon is trapped, the familon luminosity decreases as the familon interaction becomes stronger. This can be understood in the following way: as the familon interaction becomes stronger, familons can be thermalized with a lower temperature. (Notice that the scattering rate increases for higher temperature.) The surface temperature of the familon sphere then decreases, and hence the luminosity is suppressed. Therefore, the familon luminosity can be small enough when the scale is sufficiently low. Following Ref. [34] we find that the cooling through familon emission is sufficiently suppressed (i.e., is less than ) when either one of the following constraints is satisfied:

(36) | |||||

(37) |

Of course, if the familon has strong interactions with other light particles (the photon, electron, or neutron), familons may be trapped by other processes as well. This gives additional regimes where the earlier constraints of Eqs. (34) and (35) can be evaded.

In Fig. 1, we show the upper and lower bounds on the diagonal coupling as a function of the neutrino mass . The region above the upper line is allowed because familon emission is sufficiently suppressed by the flavor scale . This line is basically determined by Eq. (34); the slight bump is due to Eq. (35). As we can see, the lower bound on is at most 1 TeV for the maximum allowed value of the tau neutrino mass (18.2 MeV), and it becomes less stringent as the mass becomes smaller. The lower boundary is determined by Eq. (36), which supercedes Eq. (37). The region below this line is also allowed because the familon is trapped in the core and the contribution to the cooling is again sufficiently small.

#### Familon with off-diagonal coupling

Here we discuss SN 1987A constraints on a familon which has only an off-diagonal coupling, such as in the case of an O(2) family symmetry. The low-energy Lagrangian of the model can be written as

(38) |

The inclusion of in the discussion is straightforward. The familons are produced by via -channel exchange, via -channel exchange, or the decays . Following Ref. [34] again,

(39) | |||||

(40) | |||||

(41) |

We require that all of these familon luminosities are smaller than erg/sec, and obtain the following constraints:

(42) | |||||

(43) |

where the lifetime of is given by

(44) |

Increasing the interaction strength further into the excluded region, familons eventually become trapped and rendered harmless again. This occurs when any one of the following constraints are satisfied:

(45) | |||||

(46) |

As mentioned before, if the familon has strong interactions with other light particles, these interactions may lead to thermalization of familons as well, resulting in additional allowed regions for low flavor scales.

The resulting excluded region is fairly similar to that of the diagonal coupling case. The constraint from the decay process Eq. (43) is important only for smaller or larger masses , values that are outside our range of interest. In addition, the small bump in Fig. 1 now disappears due to the absence of the process. The dominant constraints are therefore from Eqs. (42) and (45) in the off-diagonal case, which differ from the dominant constraints of Eqs. (34) and (36) in the diagonal coupling case only by a small constant factor. The boundary of the excluded region is therefore given by the lines of Fig. 1 shifted downwards by a factor of 1.2 in .

### iv.2 Bounds from loop-induced couplings

In the previous subsection, we considered astrophysical constraints on tree-level familon couplings to . In addition, however, astrophysical bounds may also be used to constrain familon couplings to all other particles, as these couplings may induce couplings of familons to electrons and nucleons at the loop level. While these induced couplings are suppressed by the usual loop factors, the bounds on familon couplings to first generation particles are so stringent that these constraints may be strong in certain cases. In fact, we will see below that the contributions to induced couplings are proportional to fermion masses, and so these constraints are particularly relevant for couplings of familons to third generation fermions. In this subsection, we will estimate the induced couplings for various choices of the family symmetry group and determine what lower bounds on flavor scales result from current astrophysical constraints. For simplicity, we will limit our discussion here to familons with flavor-diagonal couplings to the third generation, and ignore possible rotations relating the flavor and mass eigenstates. Extensions of this analysis to more general cases are straightforward.

To evaluate the strength of the induced coupling, we will begin by considering the low energy effective theory below the flavor scale . In this approach, the theory is specified by the flavor symmetry, that is, the low energy derivative couplings of the familon, and no further knowledge of the mechanisms of flavor symmetry breaking is required. With the assumptions given above, the dominant contribution to the induced couplings is from the - mixing graph shown in Fig. 2. Here is any one of the third generation particles directly coupled to the familon, and , , or . (There are also additional contributions from penguin-like diagrams, but these are suppressed by mixing angles, e.g., in the case of .) Let us define the - coupling as

(47) |

and the - coupling as

(48) |

where , , and . The induced - mixing from the fermion loop is divergent, and the logarithmically-enhanced contribution is

(49) | |||||

where is the number of colors of the fermion , and
is the effective ultraviolet cutoff of the order of the
flavor scale . Note that and the
fermion charge drops out: the Ward–Takahashi identity
guarantees that the piece in the
vertex gives a completely transverse vacuum polarization amplitude
proportional to , which vanishes
when contracted with . The leading contributions to
the induced couplings are determined by the amount of current
non-conservation, i.e., the masses of the particles in the loop,
and the third generation couplings therefore give the most important
contributions.^{6}^{6}6Note that the radiatively-induced mixing
operator can be written in the manifestly gauge invariant form
. This operator may be present at tree-level if the familon couples
to “Higgs number,” but we will not consider this case.

The mixing of Eq. (49) is logarithmically-enhanced and so typically gives the leading contribution if present. However, there are cases in which this term is not present or is highly suppressed. First, it may be that the amount of current non-conservation is itself suppressed by inverse powers of the flavor scale. For instance, in the singlet Majoron model [38], lepton number conservation in the low-energy theory is violated only by neutrino masses which are of order due to the seesaw mechanism. The neutrino loop contribution to the -Majoron mixing is then and highly suppressed. Second, if the familon coupling is vector-like so that , the logarithmically-enhanced term is absent. For example, a familon coupled to the (possibly generation-dependent) baryon number current has this property. Finally, this mixing is also absent if the familon has only flavor off-diagonal couplings. In all of these cases, the contribution of Eq. (49) is absent or suppressed, and the leading contributions to - mixing come from non-logarithmically-enhanced threshold corrections which may be of order . Such corrections are sensitive to physics at the flavor symmetry breaking scale, and are therefore model-dependent.

With these caveats in mind, we now assume that the leading contribution given in Eq. (49) is present, and determine bounds on for various flavor symmetries. The logarithmically-enhanced mixing induces an effective coupling

(50) | |||||

where in the last step we have integrated by parts and substituted the equations of motion for . For example, for a familon coupled to , , , and . For a familon coupled to , the contributions from both and must be summed.

The effective coupling of Eq. (50) is constrained from various sources. For the case , a stringent constraint is provided by red giants. Familon-electron couplings lead to additional sources of red giant cooling, which, if too large, would destroy the agreement between the observed population of red giants in globular clusters and stellar evolution theory. Such constraints have been studied extensively in the literature [39, 40]. The current best upper limit on the coupling is [40]

(51) |

for .^{7}^{7}7For a larger coupling, the familon may be trapped in red
giants and not contribute to their cooling. Still, they can be
emitted from the Sun and change its dynamics significantly. For yet
larger couplings, familons may be trapped in the Sun as well, but then
they contribute to the thermal transport. Combination of these
constraints exclude all couplings to electrons larger than this one.
See Ref. [35] for further details. The strongest bound
on the family symmetry breaking scale is for familon couplings that
are dominantly proportional to , as, for example, when a
familon is coupled only to . Such a case results in the bound

(52) |

where we have taken . Weaker, but still significant constraints are obtained if the familon coupling is dominated by , as when the familon couples only to . The bound in this case is

(53) |

Notice that, in the case where the familon couples to and with the same charge, the bound of Eq. (53) also holds for the corresponding flavor scales. However, possibly stronger bounds may also be possible if model-dependent non-logarithmically enhanced terms proprtional to are present. If the familon contribution to the induced coupling is dominantly proportional to , we find the constraint