Left-right symmetry, orbifold $S^1/Z_2$, and radiative breaking of $U(1)_{\rm R} \times U(1)_{\rm B-L}$

We study the origin of electroweak symmetry under the assumption that $SU(4)_{\rm C} \times SU(2)_{\rm L} \times SU(2)_{\rm R}$ is realized on a five-dimensional space-time. The Pati-Salam type gauge symmetry is reduced to $SU(3)_{\rm C} \times SU(2)_{\rm L} \times U(1)_{\rm R} \times U(1)_{\rm B-L}$ by orbifold breaking mechanism on the orbifold $S^1/Z_2$. The breakdown of residual gauge symmetries occurs radiatively via the Coleman-Weinberg mechanism, such that the $U(1)_{\rm R} \times U(1)_{\rm B-L}$ symmetry is broken down to $U(1)_{\rm Y}$ by the vacuum expectation value of an $SU(2)_{\rm L}$ singlet scalar field and the $SU(2)_{\rm L} \times U(1)_{\rm Y}$ symmetry is broken down to the electric one $U(1)_{\rm EM}$ by the vacuum expectation value of an $SU(2)_{\rm L}$ doublet scalar field regarded as the Higgs doublet. The negative Higgs squared mass term is originated from an interaction between the Higgs doublet and an $SU(2)_{\rm L}$ singlet scalar field as a Higgs portal. The vacuum stability is recovered due to the contributions from Kaluza-Klein modes of gauge bosons.


Introduction
bility in a four-dimensional (4D) model with G 3211 in Sect. 3, In the last section, we give conclusions and discussions.

Five-dimensional Pati-Salam model
The space-time is assumed to be factorized into a product of 4D Minkowski spacetime M 4 and the orbifold S 1 /Z 2 , whose coordinates are denoted by x µ (or x) (µ = 0, 1, 2, 3) and y, respectively. The 5D notation x M (M = 0, 1, 2, 3, 5) is also used with x 5 = y. The S 1 /Z 2 is obtained by dividing the circle S 1 (with the identification y ∼ y + 2πR) by the Z 2 transformation y → −y. Then, the point y is identified with −y on S 1 /Z 2 , and the space is regarded as an interval with length πR (R being the radius of S 1 ).
In the following, we formulate a Pati-Salam model on M 4 × S 1 /Z 2 . First we present particle contents in Table 1. In most cases, we pay attention to bosons under the assumption that matter fields (quarks and leptons) live on the 4D brane at y = 0. The gauge bosons possess several components such that Table 1: Gauge quantum numbers of bosons in 5D Pati-Salam model. bosons where T a C , T a L and T a R are generators of SU(4) C , SU(2) L and SU(2) R , respectively. We need a scalar field Φ B (x, y) that obeys the bi-fundamental representation under SU(2) L × SU(2) R , to construct Yukawa interactions on the brane. The Lagrangian density for bosons is given by where G a M N , W a LM N and W a RM N are field strengths of SU(4) C , SU(2) L and SU(2) R gauge bosons, respectively. The covariant derivative D M and the scalar potential V 5D are given by respectively. If we require the left-right symmetry that the theory should be invariant under the exchange (W a LM , Φ L ) into (W a RM , Φ R ), we obtain the conditions among couplings: We suppose that all scalar fields have no bulk masses. From the requirement that the Lagrangian density should be invariant under the translation T : y → y + 2πR and the Z 2 transformation P 0 : y → −y or it should be a single-valued function on the 5D space-time, non-trivial boundary conditions (BCs) of fields are allowed on S 1 /Z 2 .
For scalar fields, the following BCs are imposed on, Then, zero modes appear from the lower component of Φ R and the upper component of Φ B concerning SU(2) R , and they are denoted as φ R (x) and φ(x), respectively. Here, φ R (x) is the SU(2) L singlet scalar field and φ(x) is the SU(2) L doublet scalar field. The φ(x) is regarded as the Higgs doublet in the SM. We list gauge quantum numbers and mass spectra of bosons after compactification in Table 2. In Table 2, Q R is the U(1) R charge and Q B−L is the U(1) B−L charge defined by using the 15-th components of T a C . The fifth components of gauge bosons are would-be Nambu-Goldstone bosons and absorbed by the corresponding 4D gauge bosons.
After the dimensional reduction, we obtain the Lagrangian density: where G a µν , W a µν , R µν and N µν are field strengths of SU(3) C , SU(2) L , U(1) R and U(1) B−L gauge bosons, and L KK is the Lagrangian density containing Kaluza-Klein modes. Here, the covariant derivative D µ and the scalar potential V 4D are given by bosons Rµ (x, y) respectively. From the matching conditions between L 5D and L 4D at a scale M PS above the compactification scale M c (= 1/R), we obtain the relations: Note that fields from zero modes are massless at M PS and the value of λ r does not necessarily agree with that of λ there. (2.20). We refer to it as 3211 model. Particle contents of massless fields are listed in Table 3. In Table 3, the subscript A represents the generation of matter fields on the 4D brane and runs from 1 to 3. For a sake of reference, we denote values of the weak hypercharge defined by Y ≡ Q R + Q B−L and those of the

Running of gauge couplings
We study the running of gauge couplings. By solving the renormalization group equations (RGEs) of gauge couplings g i at the one-loop level, we obtain the solutions,  Table 4. In Table 4, we list b Y = 41/6 in the SM for a sake of completeness, and - Table 4: Gauge couplings and their coefficients of β functions.
From the matching conditions at M PS and M Z LR , we have the conditions: where M Z LR is the mass of gauge boson that becomes massive with the breakdown of U(1) R × U(1) B−L into U(1) Y . By combining with the solutions (3.2), we obtain the sum rule: where M Z is the Z boson mass given by M Z 91.19GeV. Using the values of ( the experimental values such that [25]

Scalar potential in 3211 model
We study the breakdown of U(1) R × U(1) B−L and the electroweak symmetry. The scalar potential at the tree level is given by V 4D in (2.22). The quartic couplings λ r , λ m , λ and the top Yukawa coupling y t obey the RGEs at the one-loop level, where the contributions from Kaluza-Klein modes are omitted. For a sake of completeness, we write down the RGEs of the Higgs quartic coupling λ and the top Yukawa coupling y t in the SM, The λ and y t run under the condition that the SM ones match those of 3211 model at M Z LR . We obtain an effective potential improved by the RGEs at the one-loop level, where ϕ 2 R = 2{(Reφ R ) 2 + (Imφ R ) 2 }, ϕ 2 = 2{(Reφ + ) 2 + (Imφ + ) 2 + (Reφ 0 ) 2 + (Imφ 0 ) 2 }, ϕ 4 R = (ϕ 2 R ) 2 , ϕ 4 = (ϕ 2 ) 2 , and B r , B m and B are given by,

15)
(3.17) The effective potential V eff (µ) satisfies the renormalization conditions such that and does not depend on µ, that is, The first derivative of V eff by fields are given by From the stationary conditions we obtain the relations: and, by combining them, the relation: whereλ r ,λ m andλ are defined bỹ and | ϕ R means the value at µ = ϕ R . We find that the breakdown of residual gauge symmetries occurs radiatively via the Coleman-Weinberg mechanism, such that the U(1) R × U(1) B−L symmetry is broken down to U(1) Y at the scale v R ≡ ϕ R that satisfies (3.24) and the SU(2) L × U(1) Y symmetry is broken down to U(1) EM by ϕ . The hierarchy between ϕ R and ϕ comes from the difference of magnitude among couplingsλ r ,λ m andλ, as seen from (3.23).
After the breakdown of U(1) R × U(1) B−L , a gauge boson Z LRµ (x) acquires the mass The Z LRµ (x) and B µ (x) (a gauge boson relating to U(1) Y ) are given as linear combinations such that where the mixing angle θ R is defined by tan θ R ≡ g B−L /g R . Using the stationary conditions, we obtain the following formula for mass matrix elements, Here we choose ξ = 1.3, i.e., M Z LR 1819 GeV, and M c = 1 × 10 12 GeV, i.e., η 1.7, as a bench mark. In this case, v R is estimated as GeV (3.34) and the mass matrix elements of scalar fields are estimated as After diagonalizing the mass matrix, the mass of ϕ R -dominated component is evaluated as The third term in the right hand side of (3.14) or (2.22) and its radiative corrections (4-th term in the right hand side of (3.14)) are Higgs portal. By replacing ϕ R into its VEV, we obtain the following squared mass of Higgs boson approximately as From a numerical analysis, we obtain the negative squared mass because of λ m < 3B m . It can be interpreted that the Higgs mechanism occurs effectively. The runnings of various couplings including λ r , λ m and λ are depicted in Figure  3. The values of λ r and λ m at v R are estimated using stationary conditions (3.22) and λ(v R ) with ϕ ≈ 246 GeV. Here, contributions from Kaluza-Klein modes of gauge bosons are added, but those from Kaluza-Klein modes of scalar fields are not considered because they are negligible small when λ r , λ m and λ take tiny values. The running of λ is almost same as that in the SM because contributions from gluon and top quark are dominant. From Figure 3, we find that the vacuum stability is recovered by the rapid increase of λ due to contributions from Kaluza-Klein modes of gauge bosons. We suppose that the vacuum stability problem could be solved by changing the running of λ if M c is less than 10 7 GeV. But, in this case, λ can generally blow up infinity much less than M PS due to the threshold corrections of various Kaluza-Klein modes.

Conclusions and discussions
We have studied the origin of electroweak symmetry under the assumption that SU(4) C × SU(2) L × SU(2) R is realized on the 5D space-time M 4 × S 1 /Z 2 . The Pati-Salam type gauge symmetry is reduced to SU(3) C × SU(2) L × U(1) R × U(1) B−L at a high-energy scale M PS above the compactification scale M c by orbifold breaking mechanism on S 1 /Z 2 . The breakdown of residual gauge symmetries occurs radiatively via the Coleman-Weinberg mechanism, such that the U(1) R × U(1) B−L symmetry is broken down to U(1) Y by the VEV of an SU(2) L singlet scalar field and the SU(2) L × U(1) Y symmetry is broken down to the electric one U(1) EM by the VEV of the Higgs doublet, using the negative squared mass originated from an interaction between the Higgs doublet and an SU(2) L singlet scalar field as a Higgs portal. The vacuum stability can be recovered by the contributions from Kaluza-Klein modes appearing at M c and above there. Figure 3: The running of various couplings. The red, green, blue, violet, black, aqua, purple, dark brown and orange lines stand for the evolution of g Y , g 2 , g 3 , g R , g B−L , y t , λ r , λ m and λ, respectively.
Our 3211 model has an excellent feature that M PS is almost determined as M PS = O(10 13 ) GeV from the gauge coupling unification of SU(3) C and U(1) B−L into SU(4) C and the left-right symmetry between SU(2) L and SU(2) R . On the contrary, the breaking scale v R of U(1) R × U(1) B−L is not fixed from the information of gauge couplings alone. The criterion of naturalness can favor v R close to the weak scale.
Our 3211 model has almost same particle contents as those in a minimal B − L extension of the SM proposed in [26][27][28][29]. Main differences of our model and the B − L extended SM are U B−L charge assignment of SU(2) L singlet scalar field φ R and the interactions between U(1) gauge bosons and matter fields. In our model, the ν RA and φ R have U(1) B−L charge of −1/2 and 1/2, respectively. Then, allowed interaction terms between them are not renormalizable ones but non-renormalizable ones, e.g., (f AB /Λ)φ 2 R ν c RA ν RA , where Λ is a high-energy scale such as M PS . Hence small Majorana masses appear after the breakdown of U(1) R × U(1) B−L and the seesaw mechanism does not work at the TeV scale. In this paper, we focus on physics of gauge symmetry breaking sector. It would be meaningful to investigate flavor physics relating to quarks and leptons in our model. It would be also important to clarify the relationship between our model and the B − L extended SM through the study of gauge kinetic mixing and so on.