Landau pole in the Standard Model with weakly interacting scalar fields

We consider the Standard Model with a new scalar field $X$ which is a $n_X^{}$ representation of the $SU(2)_L$ with a hypercharge $Y_X$. The renormalization group running effects on the new scalar quartic coupling constants are evaluated. Even if we set the scalar quartic coupling constants to be zero at the scale of the new scalar field, the coupling constants are induced by the one-loop effect of the weak gauge bosons. Once non-vanishing couplings are generated, the couplings rapidly increase by renormalization group effect of the quartic coupling constant itself. As a result, the Landau pole appears below Planck scale if $n_X^{}\geq 4$. We find that the scale of the obtained Landau pole is much lower than that evaluated by solving the one-loop beta function of the gauge coupling constants.

On the other hand, we know that the SM should be extended to include dark matter, the origin of neutrino masses, and a mechanism for generating the baryon number of the universe. Such an extension often introduces a new scalar field X at around the electroweak/TeV or some intermediate scale which is charged under the SU (2) L × U (1) Y gauge group. If we add a sufficiently large SU (2) L isospin multiplet, the new field provides a good dark matter candidate because direct interactions with SM particles are forbidden automatically by renomalizability [40][41][42]. It is known as the Type-II seesaw mechanism that Majorana masses for neutrinos can be generated by introducing a vacuum expectation value of a scalar triplet with Y = 1 [43][44][45][46]. It is also known that additional scalar boson loops can increase the triple (SMlike) Higgs boson coupling [47,48], which help to satisfy the sphaleron decoupling condition for the electroweak baryogenesis [49][50][51]. A special choice of the Higgs multiplet, the Higgs septet with Y = 2 [52,53], can give sizable deviations in the observed Higgs boson couplings without conflicting the stringent constraint from the electroweak ρ parameter. There are so many extended models motivated by various motivations, so that it is important to constrain such possibilities in a generic way.
The triviality bound [54,55] is studied to give such a generic constraint on the extended Higgs models [56,57]. Considering the renormalization group equation (RGE) for the Higgs quartic coupling constant, the energy de-1 See Refs. [17][18][19][20][21][22][23][24][25] for the analysis in simple extensions of the SM. pendent coupling constant λ(Q 2 ) grows with an energy scale Q. At the end of the day, the running coupling constant blows up at a certain scale Λ LP . The scale is called Landau pole (LP), where the coupling constant becomes infinite 1/λ(Λ LP ) = 0. The absence of the LP at a certain scale, e.g., Plank scale, is often required to constrain the Higgs quartic coupling (equivalently the Higgs boson mass in the SM) [54,55]. This bound can be understood as a criteria for perturbativity of the theory.
In this letter, we give the RGE analysis of the scalar quartic coupling constants in the SM with one more scalar multiplet X, where the field X is a n X representation under SU (2) L with a hypercharge Y X . The model predicts the LP below Planck scale for n X ≥ 4 if X appears in the electroweak/TeV scale, where the scale of the LP is defined by the blowup of a coupling constant. The LP we will derive from the scalar quartic coupling constants is much lower than that obtained by solving one-loop beta functions of the gauge coupling constants [58]. The point is that the quartic coupling constants of X are rapidly induced by the electroweak gauge couplings with large coefficients (due to the large electroweak charges) in the beta functions, even if the initial values of the quartic coupling constants are set to be zeros. Once a finite quartic coupling is injected, the RGE running of the quartic coupling constant leads the LP. In general, there are many degrees of freedom to choose the initial condition for the new coupling constants. Among them, we evaluate a conservative scale (the largest scale) of the LP.
The scalar potential of the model is given by where Φ is the SM Higgs doublet, X is a new scalar field, and the SU (2) L generator for X is T a X (a = 1, 2, 3). Among the scalar quartic coupling constants, some of them are related each other depending on the electroweak charges of X. For quadruplets, extra coupling constants (n X , YX) independent couplings dim-4 extra couplings are allowed, where we adopt the symmetric tensor notation, i.e., (X 111 , X 112 , X 122 , X 222 ) = (X 1 , X 2 / √ 3, X 3 / √ 3, X 4 ) for n X = 4. The former two coupling constants exist for Y X = 1/2, while the last does for Y X = 3/2. There is a similar coupling of λ Φ † 2 X 2 for a sextet. Note that there is an accidental global U (1) symmetry, if all the additional dimension four couplings are forbidden. We summarize the independent coupling constants and possible dimension four extra coupling constants of each model in Table I. Next, let us move to the RGE analysis. The scale below the mass of X, M X , we use the SM beta functions. Above M X , the runnings of the electroweak gauge couplings are modified to as where t = ln µ with µ being the renormalization scale, A = 1 (2) for a real (complex) scalar field, T (X) is the Dynkin index. We see that g 2 is non-asymptotic free if n X ≥ 7 (5) for a real (complex) field case. The runnings of the top-Yukawa coupling y t and the strong gauge coupling g 3 are unchanged at one-loop level. The one-loop RGEs of the scalar coupling constants are given in Appendix A.
As a set of initial conditions for the RGEs, we take and evaluate the scale of the LP. This set gives a conservative evaluation for the scale of the LP under the condition that the scalar potential is bounded from below. We use the initial values of SM parameters in Ref. [12] and take M H = 125.7 GeV, M t = 173 GeV, which are not sensitive to our analysis. By calculating the RGE, the existence of the LP is found below Planck scale for n X ≥ 4. The blowup of the quartic coupling constants λ X , λ ′ X , · · · of X occurs because the quantum corrections to these couplings are proportional to the fourth power of the electroweak charges of X. This gives a rapid injection of non-zero value to the quartic coupling constants, and eventually results in the LP.
Let us discuss the initial condition for the new quartic couplings constants. For an illustration, we focus on the case (n X , Y X ) = (4, 3/2) with an additional Z 2 symmetry, which forbids the term Φ † 3 X. In this case, by imposing the boundedness of the scalar potential for |X 1 | 4 and |X 2 | 4 directions, we obtain the following conditions 2 This requirements comes from the |X 1 | 4 and |X 2 | 4 terms in the scalar potential of X, where the second term in the r.h.s. of the equation results from (X † T 3 X) 2 with T 3 = diag(3/2, 1/2, −1/2, −3/2). The same conditions are also come from |X 3 | 4 and |X 4 | 4 directions. The contour plot for the scale of the LPs together with the above two conditions is shown in Fig. 1 as functions of λ X (M X ) and λ ′ X (M X ). One can see that λ X = λ ′ X = 0 actually gives the conservative scale (the largest scale) of the LP. Even if we introduce nonzero κ, κ ′ , λ Φ † 3 X , these effects decrease the scale of the LP since these contributions to the beta functions of λ X and λ ′ X are always positive. For the (n X , Y X ) = (4, 1/2) case, the similar arguments are applicable for λ X , λ ′ X and λ ΦΦ † 2 X . Although the effect of λ Φ † 2 X 2 is unclear, we numerically confirmed that this coupling constant also leads to the lower scale of the LP. Thus, we conclude that Eqs. applied to higher isospin multiplets. We also confirmed these points numerically. The detailed and generalized analysis will be presented in our future publications [59].
In Fig. 2, we present the conservative scale of the LP as a function of M X in the SM with one more scalar multiplet (n X = 4, 5, 6, 7). 3 We consider all the possible assignments of the hypercharges, where fractional electric charges are not allowed for new particles. For a Y X = 0 field, a real scalar condition is assumed. The solid lines in each plot represent the numerical results, while the dashed lines show the approximated results obtained from the analytic study as we will discussed later in this letter. We fit the solid lines in Fig. 2 by the scale of the LP and M X with an exponent. The results are listed in Table II. For comparisons, we also present the positions of the LP Λ gi which are calculated by solving the oneloop beta function of gauge couplings.
where i = Y, 2, and B i,(SM) is the coefficient of the beta function of g i including a factor 1/16π 2 . 3 Requiring the tree-level unitarity of SU (2) L gauge interactions, n X ≤ 8(9) is obtained for a complex (real) scalar multiplet [60,61].   (7). The positions of LP derived from one-loop beta function of the gauge couplings taking MX = 100 GeV are also given.
We see that the conservative scale of the LP from the scalar quartic coupling constant is much lower than Λ gi .
In order to understand the results analytically, we focus on the beta function of the quartic coupling of X in which the gauge coupling g(µ) is approximated by the constant g(M X ). Then, we redefine the quartic coupling constants λ ′ X , · · · such that g 4 terms only appear in one beta function. By this redefinition, we correctly take into account the induced quartic coupling constant from the gauge couplings at one-loop level, since we begin with a conservative assumption given in Eqs. (7). We call the new coupling constantsλ ′ X , · · · . Consequently, the RGE is simplified as However, even if we setλ ′ X = 0 at M X , nonzeroλ ′ X is induced by thec ′ 2 term, which affects the running of λ X through the c 3 term. 4 Therefore, we redefine λ X in order to cancel the c 3 term. As a result, we have takingλ ′ X = · · · = 0. This simplified differential equation can be solved analytically. The position of the LP is calculated as As an example, we show the concrete procedure for the case with n X = 4. Concentrating on the quartic coupling constants of X, we have Here we put g Y (µ) = g Y (M X ), g 2 (µ) = g 2 (M X ) and κ = At first, we redefine λ ′ X in order to cancel the constant term (12Y 2 X g 2 Y g 2 2 term). Hence, we takẽ where η ≡ −12Y 2 X g 2 Y g 2 2 /(6Y 4 X g 4 Y + 297 8 g 4 2 ). The requirement of the vanishing g 2 Y g 2 2 term fixesλ ′ X up to the nor-malization, while we still have the degree of freedom to redefine λ X . Using this, we defineλ X as and choose ξ in such a way that theλ Xλ ′ X term vanishes in the beta function ofλ X . Then, we obtain The coefficients c 0 , c 1 , c 2 are given bỹ By this procedure, we can evaluate the LP in the analytical calculations. The approximated lines are also plotted in Fig. 2. Comparing the numerical results and our analytic results in these figures, the above expressions fit the numerical results within an order of magnitude error. From this analytical analysis, we confirm that the finite quartic coupling constants are originally generated by the gauge couplings, and then the RGE of the quartic coupling constant leads the LP. Roughly speaking, the LP is induced by two-loop radiative corrections to the RGEs. We can also evaluate the exponent of M X by putting Although the analytic expression is too complicate to write, we obtain ∼ 1.52 and ∼ 1.28 for (4, 1/2) and (4, 3/2), respectively.
Before concluding this letter, a few more remarks are given in order. Firstly, the position of the LP does not depend on the normalization ofλ X . We can understand the reason from Eqs. (13) and (14) as follows; let us define the new scalar coupling constant having a different normalization asλ N X := aλ X where a is a real constant. Although the RGE ofλ N X takes the same form asλ X , the coefficients are different. They are a c 0 , c 1 and c 2 /a. By using these coefficients andλ N X (M X ) = aλ X (M X ), one can easily show that the LP forλ N X calculated by Eq. (14) gives the same result as that ofλ X . Therefore, we do not need to care about the normalizations of scalar couplings when we study the LP.
Though we have focused on a single scalar extension of the SM in this letter, more matter fields may be added. Since the increase of scalar fields leads to the large coefficient in the beta functions in general, we expect that the scale of the LP becomes lower. If we add fermions in addition to scalars, the Yukawa couplings among them may be allowed. In such a case, the scale of the LP can be large since the Yukawa coupling gives the negative contribution to the beta function at the one-loop level.
Finally, we comment on the implications of the new physics beyond the SM. In the Minimal dark matter models [40][41][42], interactions of new scalar multiplets are also bounded by the relic abundance of dark matter and the data for direct/indirect searches. It is very interesting to combine our study with such constraints [59]. In some class of the seesaw models for the neutrino mass generation, Majorana fermions are introduced together with scalar multiplets, which may or may not accommodate a dark matter candidate. These models are constrained not only by the relic abundance but also by the neutrino oscillation data. In these models, the effects of Majorana fermions on the RGEs are expected to be small, because the required Yukawa coupling is generally small if we add these new fermions in the electroweak/TeV scale. In addition, models with a flavor symmetry are also very intriguing candidates for this kind of analyses. Because the scalar fields are embedded in the multiplet under the flavor symmetry, which provides a large number of SU (2) L multiplets. Increase of the number of multiplet also gives large contribution to the RGE, which would make the LP smaller [59].
In conclusion, we have investigated the scale of the LP using the one-loop RGE in the SM with one more scalar n X -plet X (n X = 4, 5, 6, 7 with all possible hypercharge assignments). The LP is found below Planck scale for n X ≥ 4, if we introduce a new scalar multiplet at electroweak/TeV scale. This means that any single scalar field extension with n X ≥ 4 of the SM is not allowed, if we impose the absence of the LP up to Planck scale. The scale is evaluated with the conservative initial condition, where the initial values for the quartic coupling constant of new scalar fields are set to be zeros at the scale of new particles. Nevertheless, the quartic coupling constants are induced from the electroweak gauge interactions. The induced coupling constants are rapidly enhanced, and finally hit the LP. We have calculated the LP as a function of the mass of X, and listed a fitting formula of the LP with an exponent. The results are consistent with our approximated formula within an order of magnitude error, which is derived from the simplified RGE with the conservative assumption. The obtained LP in each model is much smaller than that previously calculated by solving the beta functions of the gauge couplings [58]. These new results are very useful and generic constraints on the beyond the SM including new scalar multiplets.