A simple model of gauged lepton and baryon charges

We argue that simpler fermionic contents, responsible for the extension of the standard model with gauged lepton and baryon charges, can be constructed by assuming existence of so-called leptoquarks (j,k) with exotic electric charges q_j=1/2, q_k=-1/2. Some new features in our model are that (i) as the natural consequences of anomaly cancelation the right-handed neutrinos exist, and the number of the observed fermion families is equal to the number of the fundamental colors; (ii) although the lepton and baryon charges are conserved, the neutrinos can obtain small masses through the type I seesaw mechanism in similarity to the standard context, and the baryogenesis can be generated in several cases. They all are natural results due to the spontaneous breaking of these charges. Some constraints on the new physics via flavor changing and related phenomenologies such as the stable scalar with anomalous electric charge and interested processes at colliders are also discussed.


I. INTRODUCTION
It has been evidenced that the lepton charge (L) and the baryon charge (B) belong to exact symmetries up to very high scales much beyond TeV scale [1]. They may be only violated at the scales as of the grand unification theories such as that of the type I seesaw mechanism to explain the smallness of neutrino masses and that to describe the stability of proton in the proton decay question. It is therefore believed that these charges are of exact symmetries but spontaneously broken as the gauge symmetries. The possible phenomenologies of such theories at the TeV scale have been recently called for much attention [2][3][4][5][6][7].
If the lepton and baryon charges respectively behave as their own gauge symmetries, it is natural to extend the standard model (SM) gauge symmetry into a larger group G = SU(3) ⊗ SU(2) ⊗ U(1) Y ⊗ U(1) L ⊗ U(1) B , where the last two factors are the gauge groups of the lepton and baryon charges, respectively. The consistent condition of the model requires that all the anomalies associated with G must be canceled. However, in the literature the models of lepton and baryon charges gauged are actually complicated in the fermionic contents because they include many exotic particles as a result to cancel the anomalies. Also, the seesaw mechanism responsible for generating the neutrino small masses is quite complex due to the contributions of a large spectrum of neutral leptons [2][3][4][5][6][7]. Taking an example, a fourth family of leptons and a fourth family of quarks which are replications of the ordinary fermion families including the right-handed neutrinos were introduced into the SM, in which the fourth family exotic leptons have L = (+/−)3 and the exotic quarks having B = (+/−)1. The sign either plus or minus will appropriately take place depending on which chirality left or right assigned on the fermion multiplets [4,5]. Let us remind the reader that in these models the right-handed neutrinos of all the lepton families are in general not required on the background of the anomaly cancelation.
In the following we propose simpler fermionic contents which are free from all the anomalies. The leptonic and baryonic anomalies can be removed by introducing a single family of only a fermion kind of leptoquarks with appropriate quantum numbers. The leptonic content in our model is thus minimal since it only includes the ordinary right-handed neutrinos, as played in the usual type I seesaw mechanism, emerging as a natural requirement of the gravity-anomaly cancelation. From vanishing of another anomaly, the number of experimentally observed fermion families is found to be related to the number of the fundamental colors, which is equal to three. Advantages of the model such as simplicity in the seesaw mechanism responsible for the neutrino masses, the generation of baryon number asymmetry, phenomenologies associated with the leptoquarks, new gauge bosons, stable anomalously-charged scalar as well as some constraints on the model via flavor changing are also shown.
The rest of this article is organized as follows. In the next section, Sec. II, we construct the model by stressing on new fermionic contents. Section III is devoted to scalar particles, Yukawa interactions, fermion masses and some remarkable phenomenologies. In Sec. IV, we present constraints on the new physics such as flavor changing neutral currents (FCNCs) in the quark and lepton sectors, stable scalar, baryogenesis, and effects of the new particles as well as their possible processes and productions at the existing and future colliders. We summarize our results and make conclusions in the last section-Sec. V.
To cancel the anomalies associated with the gauge symmetry G = SU(3) ⊗ SU(2) ⊗ U(1) Y ⊗ U(1) L ⊗ U(1) B , we introduce into the SM particle content a single kind of only colored-fermions j and k with L = −1, B = −1 and electric charges q j = 1/2, q k = −1/2, called leptoquarks. In addition, the right-handed neutrinos corresponding to the SM lepton families will be included in order to cancel the gravity anomaly. The particle content in our model under G transform as where a = 1, 2, 3 is family index, and the values in the parentheses denote quantum numbers based [SU(2)] 2 U(1) L = (3 ∼ number of SM lepton families) × (1) [Gravity] 2 U(1) L = 3 × (1 ∼ lepton-charge of left-handed neutrino) −3 × (1 ∼ lepton-charge of right-handed neutrino) = 0.
The last equation shows that the right-handed neutrinos are required. If one supposes that the leptoquarks have no exotic lepton charges as we took L = −1, then the number of ordinary lepton families is equal to the number of the fundamental color (as we can see above from the [SU(2)] 2 U(1) L anomaly cancelation). Otherwise, the family number will be a multiple of the color number. The similar one with the baryonic anomaly implies that the number of the observed quark families is also related to the color number. The leptoquarks possessing B = −1 as we put are in order to cancel the baryonic anomalies. It is noteworthy that the [SU(2)] 2 U(1) Y anomaly was removed because of Y (F L ) = 0, and this is why we assumed q j = −q k = 1/2. In summary, the baryonic and leptonic anomalies as stored in the SM particle content are all canceled due to the presence of just leptoquarks (j, k).
It is also noted that we can have another fermion content if one reverses the chirality of leptoquarks, simultaneously changes their sign of baryon and lepton charges (L = +1, B = +1): In the following, we consider the case with the first fermionic content as given in the table above. The scalar sector will be introduced as usual to generate the mass for the particles and to make the model viable.
One can check that all the ordinary fermions, charged leptons and quarks, gain masses in the same with the SM. The neutrinos ν L and ν R at this step obtain Dirac mass as follows where H 0 = 174 GeV is the vacuum expectation value (VEV) of H 0 , and M D is a 3 × 3 matrix in a and b family indices. It is easily checked that the leptoquarks do not mix with the ordinary quarks, and have masses proportional to H 0 . They should be uncharacteristically heavy like top quark, but different from the top quark because these particles possess the unusual charges such as lepton, baryon and electricity.
Next, let us introduce a scalar singlet S L ∼ (1, 1, 0, −2, 0) which couples to the right-handed neutrinos ν R . The Yukawa interaction is then When S L develops a VEV, it provides not only Majorana mass for ν R : which is also a 3 × 3 matrix in family indices, but also a necessary mass for the U(1) L gauge boson Let us note that the lepton charge is spontaneously broken by the S L scalar. To be consistent with the effective theory, we An interesting result from our proposal is that three active neutrinos (∼ ν L ) gain masses via a type I seesaw mechanism similar to the simplest seesaw extension of the SM: which is quite different from the previous proposals [4,6]. This is also a simple one like the fermionic content in our model.
To avoid having stable colored particles we will introduce the following scalar doublets charged under U(1) L,B that couples the leptoquarks to the ordinary quarks: The Yukawa interactions are then The leptoquarks can now decay into a scalar and SM ordinary quark. It is noteworthy that in our model the φ and χ scalars could not develop a VEV because of electric charge conservation, which is also a new feature and in contradiction to [4]. Consequently, the leptoquarks do not mix in mass with the ordinary quarks (notice also that they are different in electric charges), and the FCNCs at the tree level never appear. ordinary quarks. With the electric charge conservation, this implies that at least one of the scalars is absolutely stable. Unfortunately, it maynot be a dark matter since it has a electric charge (see also [12,13] for similar matters).
Finally, the baryon charge should be spontaneously broken too. This may be achieved by another scalar singlet S B charged under U(1) B . Then, the U(1) B gauge boson Z ′ B will gain a mass proportional to S B . The consistent condition with the effective theory also implies S B ≫ H 0 .
The scalar potential that consists of doublets H, φ, χ and singlets S L , S B can be easily written.
Here we notice that all the charged scalars as contained in the doublets do not mix among them and with the others, and by themselves become mass eigenstates.

A. Flavor changing
First we consider the flavor violation in the quark sector. As mentioned above, there is no FCNC at the tree level because of the gauge symmetry. We also know that the ordinary quarks have couplings to the leptoquarks. This will lead to the FCNC processes in the ordinary quark sector at one loop level via exchange of the leptoquarks. In this work, we consider the decay b → sγ and processes associated with K 0 −K 0 mixing (the mixings D 0 −D 0 , B 0 −B 0 and decays s → dγ, b → dγ, t → cγ and so on can be similarly calculated).
The contributions to the K 0 −K 0 mixing come from the box diagrams as shown in Fig. 1.
After integrating out the heavy particles with a characteristic mass scale M , the amplitude is proportional 4 is some product of h ′d , h ′k , or h ′j and appropriate quark mixing matrix elements. For M of order 100 GeV, it is negligible provided h ′ < 10 −2 , which is similar to [4].
The contributions to b → sγ are given in Fig. 2. In order to arrive at the branching ratio Br(b → sγ), we divide as usual the decay width Γ(b → sγ) by the theoretical expression for the semileptonic decay width Γ(b → Xeν) and multiply this ratio with the measured semileptonic branching ratio Br(b → Xeν) ≃ 0.1 [1]: The semileptonic decay width (see, for example, [9]) is where The decay width in the model can be evaluated as follows Hence eq. (15) becomes: Now The calculation in the SM for this branching is in good agreement with the experiments Br(b → sγ) ≃ 3.55 × 10 −4 [1]. In our case, the new physics does not give contribution (i.e. does not contradict the SM result) if M is of order 100 GeV and h ′d < 0.1.
The flavor changing in the lepton sector is the same as in the simplest seesaw extension of the SM. As such, it is to be noted that the charged lepton flavor violations such as µ → eγ and µ → 3e are highly suppressed [8].

B. Stable anomalously-charged scalars
As mentioned our model contains the scalars χ and φ with the anomalous electric charges under which at least one of them may be very long-lived. Indeed, due to the electric charge conservation the lightest anomalously-charged scalar (assuming χ ±1/6 ) never decays, even if included quantum corrections. This is similar to the supersymmetric models with R-parity conservation there the lightest superparticle (LSP) can be charged-scalars such as stau, stop, sbottom or the lightest messenger in some scenario of the gauge-mediated supersymmetry breaking [14] (see also [12,13]).
In addition, our proposal is not similar to the supersymmetric models there the long-lived charged particles are the next-to-LSP (NLSP) [15].
If our stable scalar χ is produced at a collider, it can easily escape the detector. There is no simple way to measure its lifetime (however, for searches of long-lived charged NLSPs, see [16]).
It is important to determine whether the lifetime is indeed finite or if the particle is stable on cosmological timescales. Its cosmological evolution can be obtained from the Boltzmann equation: where n χ is the number density, n eq χ is that of equilibrium, H is the Hubble parameter, and σv is a thermal-averaged annihilation cross section times their relative velocity.
The dominant contributions to the annihilation cross section would be model-dependent. In our case, it comes from the annihilation processes into γγ and Zγ. Moreover, due to an anomalous electric charge 1/6 for χ that its electromagnetic coupling is weak, such annihilation processes are very rare. It follows that the bounds of χ mass in our model may be much lower than the other cases [13]. The typical lower bound on the mass of such stable particles in various models is able to as low as 1 TeV [13]. Hence, the lower bound for the χ mass is equal to (1/6) 2 × 1 TeV ≃ 277 GeV (see, for example, Chuzhoy and Kolb in [13]). This is in agreement with a search by Byrne, Kolda and Regan in [13] for the lower mass bound of the squarks and gluinos about 230 GeV. It is also in agreement with searches by Smith et al. in [13] and Yamagata et al. in [13].

C. Baryogenesis
In this model the proton decay is discarded because the effective operator QQQL could not be generated with provided the model particle content. Also, the neutrinoless double beta decay with the effective operator QQQQLL is explicitly suppressed. All these are the natural consequences due to the local lepton-number and baryon-number conservations. In the literature, the above processes are generally known to be prevented up to the very high scale as of the grand unification theories, where the lepton and baryon numbers may be violated. In our model the status is different. Although these processes cannot happen due to the gauge symmetries, there are still spontaneous breaking of baryon-number and lepton-number due to the VEVs of baryon-charged S B and lepton-charged S L , respectively. And, the scales for these breakings in principle may be arbitrary but should be greater than the electroweak scale. These VEVs are just the sources for associated phenomena to be happened, independent of the explicit gauge symmetries.
As an example, the conservation of the lepton number and baryon number is the reason why we cannot generate the baryon number asymmetry as in the standard technics through explicit lepton/baryon violation interactions as such baryogenesis via the grand unification or baryogenesis via leptogenesis [18]. The baryogenesis in this model may appear either one of the following cases: 1. We may realize a baryon asymmetry via the spontaneous symmetry breaking of the baryon number at the TeV scale due to S B as well as the ordinary quark products resulting from the decay of the unstable leptoquarks. The procedure for achieving the excess of baryon number can closely follow Harvey and Turner in Ref. [17]. The calculations in Ref. [5] in the models like ours have shown that this is possible.
2. Baryogenesis via leptogenesis: Let us recall that the neutrinos in our model gain the masses via the type I seesaw mechanism. The difference here is that the Majorana masses for the right-handed neutrinos are generated as a result from the spontaneous breaking of the lepton number due to S L . The leptogenesis can be obtained via this source due to a nontrivial vacuum of the lepton number. The VEV for S L must be very high and the procedure for deriving an excess of lepton number, thus the baryon number, is similar to [18].
A detailed calculation for these cases to be included in the current work is out of the scope of this letter, and we will devote it to a further publication.

D. Other aspects
Our model has two new gauge bosons as mentioned Z ′ L and Z ′ B , respectively gaining the masses via the lepton-and baryon-number breaking VEVs. If they are much heavier than the electroweak scale (particularly, this may exist in the second case of the baryogenesis as mentioned) their contributions to the collider phenomenology and affectations to the Higgs potential could be negligible.
However, as shown above, in the first case the breaking of the baryon number at the TeV scale may be responsible for the baryogenesis. Then the Z ′ B gets a mass in this scale. We can also have a Z ′ L light as of Z ′ B (this may only happen in the first case since the second case needs a very high scale of lepton number breaking). The Z ′ L or Z ′ B can then contribute to the known processes, e.g. e + e − → Z ′ L → τ + τ − and pp → Z ′ B → tt, or new processes to observe the lepton number violation pp → νν provided that Z ′ L and Z ′ B mixing, which can be used to search for (for a detailed evaluation, see [4] and references therein).
At the LHC and ILC, we may have interesting processes due to the decays of the leptoquarks.
Here we focus on the case where the leptoquarks decay into a stable scalar and a top quark. The channels are (i) LHC: pp →jj → χ −1/6 χ +1/6t t, where the first process is possibly mediated by γ, Z, gluon, Z ′ B (if Z ′ B lies in TeV scale with its large enough gauge coupling, this contribution is also important) and (ii) ILC: e + e − →jj → χ −1/6 χ +1/6t t, where the mediations of the first process may be γ, Z, Z ′ L (if Z ′ L lies in TeV scale and its gauge coupling is large enough). There is a possible portion of χ −1/6 χ +1/6 fusion into γγ, γZ, or a light fermion pairf f via photon exchange but all these are very rare due to a charge 1/6 for χ. The final state of those processes at each stage may have a missing energy and att pair, which is worth to looking for. Finally, we can have typical processes with the stable scalar χ ±1/6 such as a direct channel e + e − → χ −1/6 χ +1/6 at the ILC and γγ → χ −1/6 χ +1/6 at the photon-photon collider. All the above processes are devoted to the forthcoming experimental considerations.
Let us note that the similar processes can happen in Tevatron and LEP, if the new particles are assumed to be light. The Tevatron and LEP then provide constraints on the model. Otherwise, those processes would been evaded if the new particles are much heavier than the electroweak scale.
In this case the model is explicitly consistent with the effective theory, and we have a natural seesaw mechanism for the neutrinos as well as the baryogenesis appearing in the mentioned second case, that all are quite similar to the standard context. Anyway, a constraint on the new physics at the TeV scale using the existing colliders are worth, and a more detailed analysis on the model's consequences as briefly mentioned are needed. All these are large subjects out of scope of this letter. We will study of these issues to be published elsewhere in a near future.
In our model, with such a heavy leptoquark generation it is well-established that the gluon gluon fusion cross section for the SM Higgs is larger by a factor 9 [10]. However, the new results from CDF and D0 do not rule out our model when the Higgs mass is 114 GeV < M H < 120 GeV, or when M H > 200 GeV [11]. Notice that for a large mixing between H and the singlets S L and S B (if these scalars are assumed to be light enough) one can relax those constraints. These are similarities to the former proposal [4].

V. CONCLUSIONS
We have proposed a simple, predictive model of the gauged lepton and baryon charges. All the anomalies were removed by the presence of only the leptoquarks j +1/2 and k −1/2 . The numbers of the observed lepton and quark families have been proved to be equal to the number of the fundamental colors, which is just three. Let us note that in the standard model the number of fermion families is left arbitrary, and thus fail to answer this question. The right-handed neutrinos have been naturally existed as a requirement of the gravity-anomaly cancelation. This is an interesting feature in comparison with such particles as required for the SO(10) grand unification.
In contradiction to the previous proposals [4], the lepton sector in our model is minimal since it only assumes the mentioned right-handed neutrinos.
The conservation of the lepton and baryon numbers will explicitly prevent the proton decay and neutrinoless double beta decay, but the spontaneous breaking of these charges will explain for associated low-energy phenomena such as neutrino mass and baryogenesis. Indeed, the small masses of the neutrinos have been explained by the type I seesaw mechanism in similarity to the simplest seesaw extension of the standard model. But, in our case the Majorana masses for the right-handed neutrinos get naturally generated as a result of spontaneous lepton-number breaking due to the VEV of lepton-charged S L scalar, although this symmetry is exact. A standard technic for the baryogenesis via leptogenesis is therefore followed. In other case, the spontaneous breaking of baryon number at TeV scale as well as the decay of the leptoquarks may also generate the baryogenesis. We particularly stress that the neutrino masses and leptogenesis as obtained in our model provide an insight into the standard contexts.
The flavor changing neutral current processes such as K 0 −K 0 mixing and b → sγ have been considered and evaluated. Here the new physics constrained is in natural consistency with the standard model. The model contains an interesting stable scalar as associated with an anomalous electric charge that can be very long-lived. This is similar to the supersymmetric models where the LSP is electrically charged such as stau, stop or sbottom. However, our scalar is different from the ones mentioned because it has very weak electromagnetic coupling. A naive evaluation has shown that the low bound on its mass may be as low as 277 GeV.
If the scales of lepton-and baryon-number breaking are much larger than the electroweak scale, the S L and S B would not affect the Higgs potential, and the gauge bosons associated with these charges Z ′ L and Z ′ B do not contribute to the collider phenomenology. However, if these scales are as low as required in a case for the baryogenesis, they will take place. Also, existing as an answer of consistency of the model the new leptoquarks j, k and the new scalars χ and φ with the anomalous electric charges (where almost the new particles fast decay, only one is the stable scalar as mentioned) have the interesting phenomenologies in the colliders such as the LHC, ILC and photon-photon collisions, which can be worth to search for. In addition, such similar processes can also happen in the existing accelerators such as the Tevatron and LEP, if the new particles are light enough, which provide constraints on the model. A detailed study on these issues are necessary to be published elsewhere.