$B-L$ as a Gauged Peccei-Quinn Symmetry

The gauged Peccei-Quinn (PQ) mechanism provides a simple prescription to embed the global PQ symmetry into a gauged $U(1)$ symmetry. As it originates from the gauged PQ symmetry, the global PQ symmetry can be protected from explicit breaking by quantum gravitational effects once appropriate charge assignment is given. In this paper, we identify the gauged PQ symmetry with the ${B-L}$ symmetry, which is obviously attractive as the ${B-L}$ gauge symmetry is the most authentic extension of the Standard Model. As we will show, a natural $B-L$ charge assignment can be found in a model motivated by the seesaw mechanism in the $SU(5)$ Grand Unified Theory. As a notable feature of this model, it does not require extra $SU(5)$ singlet matter fields other than the right-handed neutrinos to cancel the self and the gravitational anomalies.


I. INTRODUCTION
The strong CP problem is longstanding and probably one of the most puzzling issues in particle physics. Although the Peccei-Quinn (PQ) mechanism [1][2][3][4] provides a successful solution to the problem, it is not very satisfactory from a theoretical point of view, as it relies on a global Peccei-Quinn U (1) symmetry. The Peccei-Quinn symmetry is required to be almost exact but explicitly broken by the QCD anomaly. Even tiny explicit breaking terms of the PQ symmetry spoil the PQ mechanism. It is, on the other hand, conceived that any global symmetries are broken by quantum gravity effects [5][6][7][8][9][10]. Thus, the PQ mechanism brings up another question, the existence of such an almost but not exact global symmetry.
In [11], a general prescription to achieve a desirable PQ symmetry is proposed in which the PQ symmetry originates from a gauged U (1) symmetry, U (1) gP Q . The anomalies of U (1) gP Q are canceled between the contributions from two (or more) PQ charged sectors, while the inter-sector interactions between the PQ charged sectors are highly suppressed by appropriate U (1) gP Q charge assignment. As a result of the separation, a global PQ symmetry exist in addition to U (1) gP Q as an accidental symmetry. The accidental PQ symmetry is highly protected from explicit breaking by quantum gravitational effects as it originates from the gauge symmetry. The gauged PQ mechanism is a generalization of the mechanisms which achieve the PQ symmetry as an accidental symmetry resulting from (discrete) gauge symmetries [12][13][14][15][16][17][18][19][20].
In this paper, we discuss whether the B − L symmetry can play a role of the gauged PQ symmetry. The B − L gauge symmetry is the most authentic extension of the Standard Model (SM) which explains the tiny neutrino masses via the seesaw mechanism [21][22][23] (see also [24]). Therefore, the identification of the gauged PQ symmetry with B − L makes the gauged PQ mechanism more plausible. An intriguing coincidence between the righthanded neutrino mass scale appropriate for the thermal leptogenesis [25] (see [26][27][28], for review) and the PQ breaking scale which avoids astrophysical constraints also motivates this identification [29].
As will be shown, we find a natural B − L charge assignment motivated by the seesaw mechanism in the SU (5) Grand Unified Theory (GUT), with which the gauged PQ mech-anism is achieved. Notably, the charge assignment we find does not require extra SU (5) singlet matter fields other than the right-handed neutrinos to cancel the [U (1) gP Q ] 3 and the gravitational anomalies.
The organization of the paper is as follows. In section II, we discuss an appropriate B − L charge assignment so that it plays a role of U (1) gP Q . In section III, we discuss the properties of the axion and the global PQ symmetry. In section IV, we briefly discuss the domain wall problem. In section V, we discuss supersymetric (SUSY) extension of the model. The final section is devoted to our conclusions. Having the SU (5) GUT in mind, it is more convenient to consider "fiveness", 5(B − L) − 4Y , instead of B − L, as it commutes with the SU (5) gauge group. The fiveness charges of the matter fields are given by while the Higgs doublet, h, has a charge +2 (i.e. B − L = 0). 1 Here, we use the SU (5) GUT representations for the matter fields, i.e. 10 SM = (q L ,ū R ,ē R ) and5 SM = (d R , L ), whileN R denotes the right-handed neutrinos.
The seesaw mechanism is implemented by assuming that the right-handed neutrinos obtain Majorana masses from spontaneous breaking of fiveness. In this paper, we assume that the Majorana masses are provided by the vacuum expectation value (VEV) of a gauge singlet scalar field with fiveness, −10, i.e., which couples to the right handed neutrinos, Here, y N denotes a coupling constant, with which the Majorana mass is given by M N = y N φ . By integrating out the right-handed neutrinos, the tiny neutrino masses are obtained, where y also denotes a coupling constant.
Now, let us identify the gauged PQ symmetry with B − L, i.e., fiveness. Following the general prescription of the gauged PQ mechanism in [11], let us introduce extra matter multiplets which obtain a mass from the VEV of φ; with y K being a coupling constant. 2 Here, the extra multiplets (5 K ,5 K ) are assumed to form the 5 and5 representations of the SU (5) gauge group, respectively. As in the KSVZ axion model [30,31], the Ward identity of the fiveness current, j 5 , obtains an anomalous contribution from the extra multiplets, Here, F a (a = 1, 2, 3) are the gauge field strengths of the Standard Model and g a the corresponding SM gauge coupling constants. The Lorentz indices and the gauge group representation indices are suppressed. The factor −10 corresponds to the charge of the bi-linear, 5 K5K (see Eq. (5)).
In the gauged PQ mechanism, the U (1) gP Q gauge anomalies are canceled by a contribution from another set of the PQ charged sector. For that purpose, let us also introduce 10-flavors of extra matter multiplets (5 K ,5 K ). We assume that they obtain masses from a VEV of a 2 The reason why the extra multiplets couple not to φ but φ * will become clear shortly.
complex scalar field φ whose fiveness charge is +1; where the charge of the bi-linear, 5 K5 K , is set to be +1. With this choice, the anomalous contributions of the Ward identity in (6) are canceled by the one from (5 K ,5 K ), i.e., The fiveness charges of the respective extra multiplets are chosen as follows. To avoid stable extra matter fields, we assume that5 K and5 K can mix with5 SM , so that respectively. As a notable feature of this charge assignment, it cancels the [U (1) gP Q ] 3 and the gravitational anomalies automatically without introducing additional SM singlet fields.
The anomaly cancellation without singlet fields other than the right-handed neutrinos is by far advantageous compared with the previous models [11,12,32]. The singlet fields required for the anomaly cancellation tend to be rather light and longlived, which make the thermal history of the universe complicated [32]. The anomaly cancellation of the present model is, therefore, a very important success as it is partly motivated by thermal leptogenesis which requires a high reheating temperature after inflation, i.e., T R 10 9 GeV [26][27][28].
Under the fiveness symmetry, the interactions are restricted to Here,5 collectively denotes (5 SM ,5 K ,5 K ), and V (φ, φ , h) is the scalar potential. The coupling coefficients are omitted for notational simplicity. At the renormalizable level, the above Lagrangian possesses a global U (1) symmetry, which is identified with the global PQ symmetry. The global PQ symmetry corresponds to a phase rotation of a gauge invariant combination, φφ 10 , while the other fields are rotated appropriately. The global PQ charges of the individual fields are generically given by for {SM,N R , 5 K ,5} and {5 K }, respectively. Here, q 5 denotes the fivness charge of each field, and Q φ,φ are the global PQ charges of φ and φ with Q φ /Q φ = −10, respectively.
The global PQ symmetry is broken by the QCD anomaly. In fact, under the global PQ rotation with a rotation angle α P Q , the Lagrangian shifts by, It should be noted that the normalization factor of Eq. (14) is independent of the choice of the global PQ charge assignment for the individual fields.
Since the global PQ symmetry is just an accidental one, it is also broken by the Planck suppressed operators explicitly. However, due to the gauged fiveness symmetry, no PQsymmetry breaking operators such as φ n or φ n (n > 0) are allowed. As a result, the explicit breaking terms of the global PQ symmetry are highly suppressed, and the lowest dimensional ones are given by, where M P L 2.44 × 10 18 is the reduced Planck scale. As we will see in the next section, the breaking terms are acceptably small not to spoil the PQ mechanism in a certain parameter space.

III. AXION AND GLOBAL PQ SYMMETRY
To see the properties of the accidental global PQ symmetry, let us decompose the axion from the would-be Goldstone boson of fiveness. Both of them originate from the phase components of φ and φ ; where f a,b are the decay constants and we keep only the Goldstone modes, a and b. The domains of the phase components are given respectively.
In terms of θ a,b , fiveness is realized by, Here, α(x) denotes a gauge parameter field with q a = −10 and q b = +1, Y µ the gauge field, and g the coupling constant, respectively. The gauge invariant effective Lagrangian of the Goldstone modes is given by, where the covariant derivatives are defined by The gauge invariant axion, A (∝ q b θ a − q a θ b ), and the would-be Goldstone mode, B, are given by By using A and B, the effective Lagrangian is reduced to, The second term is the Stückelberg mass term of the gauge boson with m Y being the gauge boson mass, Through the mass term, the would-be Goldstone mode B is absorbed into Y µ by the Higgs mechanism. The effective decay constant of the axion A is given by, Given F A , the domain of the gauge invariant axion is given by when |q a | and |q b | are relatively prime integers [11].
The global PQ symmetry defined in the previous section is realized by a shift of where α P Q ranges from 0 to 2π. In fact, the phase of the gauge invariant combination φφ 10 rotates by as in Eq. (13).
After integrating out the extra multiplets, the axion obtains anomalous couplings to the SM gauge fields, The constraint on the VEVs of φ and φ . The gray shaded region is excluded by ∆θ < 10 −10 for the non-SUSY model (see Eq. (31)). The orange lines are the contours of the effective decay constant F A . In the blue shaded region, φ > φ .
Here, we have used the fact that the numbers of extra multiplets coupling to φ and φ are giving by N a = q b = 1 and N b = −q a = 10. By substituting Eq. (22), the anomalous coupling is reduce to, which reproduces the axial anomaly of Eq. (14) by the shift of the axion in Eq. (27). Through this term, the axion obtains a mass from the anomalous coupling below the QCD scale, with which the QCD vacuum angle is erased.
In the presence of the explicit breaking terms in Eq. (15), the QCD vacuum angle is slightly shifted by 3 where m a denotes the axion mass. Such a small shift should be consistent with the current experimental upper limit on the θ angle, θ 10 −10 [33].
In It should also be noted that the "inter-sector" interactions via5 do not lead to explicit breaking of the global PQ symmetry. To see this, it is most convenient to choose Q φ = 0 and Q φ = 1 (see Eq. (12)), which leads to the global PQ charges, The non-vanishing couplings to the neutrinos can also be understood from the fact that the axion in the present model also plays a role of the Majoron [35] which is obvious in the limit of φ φ . However, it seems very difficult to test the direct couplings between the axion and the neutrinos in laboratory experiments. 4 Note that φφ 10 is the lowest dimensional operators among all the global PQ breaking operators. In this case, no larger explicit breaking terms are generated by radiative corrections other than the anomalous breaking terms given in Eq. (30).

IV. DOMAIN WALL PROBLEM
Here, let us briefly discuss the domain wall problem and axion dark matter. As discussed in [32], the model suffers from the domain wall problem for φ φ when global PQ symmetry breaking takes place after inflation. To avoid the domain wall problem, we assume either one of the following possibilities; (i) Both phase transitions of φ = 0 and φ = 0 take place before inflation.
The latter possibility is available as the fiveness charges of φ and φ are relatively prime and |q a | : |q b | = 10 : 1. 5 For the first possibility, the cosmic axion abundance is given by, for the initial misalignment angle θ a = O(1) [36]. Thus, in the allowed parameter region in Fig. 1, i.e., F A 10 10 GeV, relic axion abundance is a subdominant component of dark matter. It should be also noted that the Hubble constant during inflation is required to satisfy, to avoid the axion isocurvature problem (see Refs. [37,38]). 6 For the second possibility, the cosmic axion abundance is dominated by the one from the decay of the string-domain wall networks [39], Ω a h 2 0.035 ± 0.012 F A 10 10 GeV Thus, the relic axion from the string-domain wall network can be the dominant component of dark matter at the corner of the parameter space in Fig. 1. To avoid symmetry restoration 5 The domain wall problem might also be solved for φ ∼ φ even if both the phase transitions take place after inflation. To confirm this possibility, detailed numerical simulations are required. 6 Here, we do not assume that the axion is the dominant component of dark matter but use the axion relic abundance in Eq. (33) to derive the constraint.
after inflation, we also require that the maximum temperature during reheating [40], does not exceed φ , which leads to Here, we use the effective massless degrees of freedom g * 200, though the condition does not depend on g * significantly.

V. SUPERSYMMETRIC EXTENSION
The SUSY extension of the present model is straightforward. The SM matter fields, the right-handed neutrinos, and the extra multiplets are simply extended to corresponding supermultiplets with the same fiveness charges given in Eqs. (1) and (9) Under the fiveness symmetry, the superpotential is restricted to 7 Here, X and Y are introduced to make φ and φ obtain non-vanishing VEVs, which are neutral under fiveness. 8 The coupling coefficients are again omitted for notational simplicity.
The SUSY extension again possesses the global PQ symmetry as in the case of the non-SUSY model.
In addition to fiveness, we also assume that a discrete subgroup of U (1) R , Z N R (N > 2), 7 More generally, the Higgs bi-linear, H u H d , also couples to X and Y . We assume that the soft masses of the Higgs doublets are positive and larger than those of φ's and φ 's, so that the Higgs doublets do not obtain VEVs from the couplings to X and Y . We may also restrict those couplings by some symmetry. 8 See [32] for details of the SUSY extension of the gauged PQ mechanism. TABLE I. The charge assignment of the fiveness symmetry and the gauged Z 4R symmetry. Here, we fix the Z 4R charges of the Higgs doublets to 0 which is motivated by pure gravity mediation model [41]. An extra multiplet (5 E ,5 E ) is introduced to cancel the Z 4R -SU (5) 2 anomaly [42].
is an exact discrete gauge symmetry. This assumption is crucial to allow the VEV of the superpotential, and hence, the supersymmetry breaking scale much smaller than the Planck scale. 9 In the following, we take the simplest possibility, Z 4R with the charge assignment given in Tab. I, which is free from Z 4R -SU (5) 2 anomaly and the gravitational anomaly. 10 It should be noted that the mixed anomalies of Z 4R and fiveness do not put constraints on charges since they depend on the normalization of the heavy spectrum [45][46][47][48][49][50][51][52][53][54]. 11 Under fiveness and the gauged Z 4R symmetry, the lowest dimensional operators which break the global PQ symmetry are given by, where m 3/2 denotes the gravitino mass. Compared with Eq. (15), the explicit breaking is suppressed by a factor of (m 3/2 /M PL ) 2 . Accordingly, the shift of the QCD vacuum angle is given by, 9 R-symmetry is also relevant for SUSY breaking vacua to be stable [43,44]. 10 It should be noticed that there is no need to add extra SU (5) singlet fields to cancel the anomalies. 11 GUT models consistent with the Z 4R symmetry are discussed in, e.g., [55,56].  (41)). The orange lines are the contours of the effective decay constant F A . In the blue shaded region, φ > φ . The gray shaded lower regions are excluded as the gauge coupling constants become non-perturbative below the GUT scale. The thin green region is excluded by the Axion Dark Matter eXperiment (ADMX) [57] where the dark matter density is assumed to be dominated by the relic axion.
where we assume φ = φ and φ = φ for simplicity. 12 In Fig. 2, we show the constraints on the VEVs of φ and φ from the experimental upper limit on ∆θ. Here, we take the gravitino mass, m 3/2 100 TeV, which is favored to avoid the cosmological gravitino problem for T R 10 9 GeV [58][59][60]. For m 3/2 100 TeV, the scalar partner and the fermionic partner of the axion also do not cause cosmological problems as they obtain the masses of the order of the gravitino mass and decay rather fast [61].
In the figure, the gray shaded region is excluded by the constraint on ∆θ 10 −10 . Due to the suppression of the breaking term in Eq. (40), the higher value of φ is allowed compared with the non-SUSY model. The higher φ is advantageous to avoid symmetry restoration after inflation (see Eq. (37)), with which the domain wall problem is avoided in the possibility (ii) (see section III). Accordingly, the decay constant can also be as high as about 10 11−12 GeV, which also allows the axion to be the dominant dark matter component (see Eq. (35)). Therefore, we find that the SUSY extension of the model is more successful. 13 It should be noted that the 11-flavors of extra multiplets at the intermediate scale make 12 The following argument can be easily extended to the cases with φ = φ and φ = φ . 13 As in [32], we will discuss a possibility where SUSY and B − L are broken simultaneously elsewhere.
the renormalization group running of the MSSM gauge coupling constants asymptotic nonfree. Thus, the masses of them are bounded from below so that perturbative unification is achieved. In the figure, the gray shaded lower region shows the contour of the renormalization scale M * at which at least one of g 1,2,3 becomes 4π. Here, we use the one-loop renormalization group equations assuming that the extra quarks obtain masses of φ and φ , respectively. 14 The result shows that the perturbative unification can be easily achieved for φ 10 9-10 GeV even in the presence of 11-flavors of the extra multiplets.

VI. CONCLUSIONS AND DISCUSSIONS
In this paper, we consider the gauged PQ mechanism where the gauged PQ symmetry is identified with the B − L symmetry (fiveness). As the B − L gauge symmetry is the most plausible extension of the SM, the identification of the gauged PQ symmetry with B − L is very attractive. An intriguing coincidence between the B − L breaking scale appropriate for the thermal leptogenesis and the favored PQ breaking scale from the astrophysical constraints also motivates this identification.
We found a natural B − L charge assignment motivated by the seesaw mechanism in the SU (5) GUT, with which the gauged PQ mechanism is achieved. There, the global PQ symmetry breaking effects are suppressed by the gauged fiveness symmetry so that the successful PQ mechanism is realized. As a notable feature, the fiveness charge assignment does not require extra SU (5) singlet matter fields other than the right-handed neutrinos to cancel the [U (1) gP Q ] 3 anomaly and the gravitational anomaly. This feature is advantageous since the singlet fields required for anomaly cancellation tend to be rather light and longlived, and hence, often cause cosmological problems. As a result, we find that the gauged PQ mechanism based on the B−L symmetry is successfully consistent with thermal leptogenesis.
We also discussed the SUSY extension where the Z 4R symmetry is also assumed. As has been shown, a larger effective decay constant is allowed in the SUSY model, as explicit breaking of the global PQ symmetry is more suppressed. Resultantly, the upper limit on 14 The masses of the sfermions, the heavy charged/neutral Higgs boson, the Higgsinos, and (5 E ,5 E ) are at the gravitino mass scale, m 3/2 100-1000 TeV. The gaugino masses are, on the other hand, assumed to be in the TeV scale as expected by anomaly mediation [62,63]. This is motivated by the pure gravity mediation model in [41] (see also Refs. [64][65][66][67] for similar models), where the Higgsino mass is generated from the R-symmetry breaking [68].
the effective decay constant is extended to which corresponds to the axion mass, m a 1.9 µeV The dark matter axion in this mass range can be detected by the ongoing ADMX-G2 experiment [69] and future ADMX-HF experiment [70].
In the SUSY model, it should be also noted that Z 4R is spontaneously broken down to the Z 2R symmetry. 15 Thus, the lightest supersymmetric particle in the MSSM sector also contributes to the dark matter density. Therefore, the model predicts a wide range of dark matter scenario from axion dominated dark matter to the LSP dominated dark matter, which can be tested by future extensive dark matter searches.
As emphasized above, the fiveness anomalies are canceled without introducing singlet fields other than the right-handed neutrinos. Although this feature is advantageous from The anomaly free charge assignment of 5's is fixed in the following way. For all the nflavors of (5,5) to have masses in the intermediate scale, they need to couple to the order parameters of fiveness. As we assume the seesaw mechanism, we have a natural candidate of such an order parameter, a complex scalar field, φ, with a fiveness charge −10. In order to make all the n-flavors of (5,5) massive while achieving anomaly free fiveness, however, it is required to introduce one more complex scalar, φ , with the fiveness charge q φ .
In the presence of φ and φ , the mass terms of (5,5) are generated from L = φ 55 + φ * 5 5 + φ 5 5 + φ * 5 5 , where the coupling coefficients are again omitted. Here, 5's are devided into {5,5 ,5 ,5 } 16 The choice of −3 just defines the normalization of fiveness. whose fiveness charges are given by, 5(+13) , 5 (−7) , 5 (−q φ + 3) , 5 (q φ + 3) , respectively. We allocate N 5 , N 5 , N 5 and N 5 flavors to {5,5 ,5 ,5 } with N 5 + N 5 + N 5 + N 5 = n. The anomaly free conditions of fiveness are given by, By solving the anomaly free conditions, we find only two sets of solutions, or N 5 = 7 , N 5 = 1 , N 5 = 3 , N 5 = 0 , q φ = 20 , both of which corresponds to n = 11. 17 Here, we restrict ourselves to n < 22. The first charge assignment is nothing but the fiveness charges assumed in this paper, while the later is another possibility. In this sense, we find that the number of the flavors, n = 11, is a unique choice within n < 22, and the fiveness charge assignment in this paper is one of the only two possibilities, where the second possibility is not suitable for the gauged PQ mechanism.