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Landau pole in the Standard Model with weakly interacting Title scalar fields

Author(s) Hamada, Yuta; Kawana, Kiyoharu; Tsumura, Koji

Citation Physics Letters B (2015), 747: 238-244

Issue Date 2015-07-30

URL http://hdl.handle.net/2433/201624

Type Journal Article

Textversion publisher

Kyoto University Physics Letters B 747 (2015) 238–244

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Landau pole in the Standard Model with weakly interacting scalar ﬁelds ∗ Yuta Hamada , Kiyoharu Kawana, Koji Tsumura

Department of Physics, Kyoto University, Kyoto 606-8502, Japan a r t i c l e i n f o a b s t r a c t

Article history: We consider the Standard Model with a new scalar field X which is an nX representation of the SU(2)L Received 7 May 2015 with a hypercharge Y X . The renormalization group running effects on the new scalar quartic coupling Accepted 28 May 2015 constants are evaluated. Even if we set the scalar quartic coupling constants to be zero at the scale of the Available online 3 June 2015 new scalar field, the coupling constants are induced by the one-loop effect of the weak gauge bosons. Editor: J. Hisano 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 ≥ nX 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. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The discovery of the Higgs boson [1,2] and the determination sizable deviations in the observed Higgs boson couplings without of its mass open up the possibility that the Standard Model (SM) conflicting the stringent constraint from the electroweak ρ param- can be valid up to very high energy scale such as string/Planck eter. There are so many extended models motivated by various scale [3–16].1 In particular, the Higgs self-coupling constant and its motivations, so that it is important to constrain such possibilities beta function almost vanish at the same time around Planck scale in a generic way. if the top-quark mass Mt is 171 GeV. This fact may imply relations The triviality bound [54,55] is studied to give such a generic between physics at the weak scale and physics at Planck scale, e.g., constraint on the extended Higgs models [56,57]. Considering the Higgs inflation [26–33], a multiple point criticality principle [34] renormalization group equation (RGE) for the Higgs quartic cou- and a maximum entropy principle [35–39]. pling constant, the energy dependent coupling constant λ(Q 2) On the other hand, we know that the SM should be extended grows with an energy scale Q . At the end of the day, the run- to include dark matter, the origin of neutrino masses, and a mech- ning coupling constant blows up at a certain scale LP. The scale anism for generating the baryon number of the universe. Such an is called Landau pole (LP), where the coupling constant becomes extension often introduces a new scalar field X at around the elec- infinite 1/λ(LP) = 0. The absence of the LP at a certain scale, e.g., troweak/TeV or some intermediate scale which is charged under Plank scale, is often required to constrain the Higgs quartic cou- × the SU(2)L U (1)Y gauge group. If we add a sufficiently large pling (equivalently the Higgs boson mass in the SM) [54,55]. This SU(2)L isospin multiplet, the new field provides a good dark matter bound can be understood as a criterion for perturbativity of the candidate because direct interactions with SM particles are forbid- theory. den automatically by renomalizability [40–42]. It is known as the In this letter, we give the RGE analysis of the scalar quartic Type-II seesaw mechanism that Majorana masses for neutrinos can coupling constants in the SM with one more scalar multiplet X, be generated by introducing a vacuum expectation value of a scalar where the field X is an n representation under SU(2)L with a triplet with Y = 1 [43–46]. It is also known that additional scalar X hypercharge Y X . The model predicts the LP below Planck scale boson loops can increase the triple (SM-like) Higgs boson coupling for nX ≥ 4if X appears in the electroweak/TeV scale, where the [47,48], which helps to satisfy the sphaleron decoupling condi- scale of the LP is defined by the blowup of a coupling constant. tion for the electroweak baryogenesis [49–51]. A special choice of The LP we will derive from the scalar quartic coupling constants the Higgs multiplet, the Higgs septet with Y = 2 [52,53], can give 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 elec- * Corresponding author. E-mail address: [email protected] (Y. Hamada). troweak gauge couplings with large coefficients (due to the large 1 See Refs. [17–25] for the analysis in simple extensions of the SM. electroweak charges) in the beta functions, even if the initial values http://dx.doi.org/10.1016/j.physletb.2015.05.072 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Y. Hamada et al. / Physics Letters B 747 (2015) 238–244 239

Table 1 The independent coupling constants in the scalar potential are listed. We also show the dimension four extra couplings which contribute to the beta functions.

(nX , Y X ) independent couplings dim-4 extra couplings †2 2 †2 (4, 1/2)λ,κ, κ ,λX ,λX X , X †3 (4, 3/2)λ,κ, κ ,λX ,λX X (5, real)λ,κ,λX × × (5, 1)λ,κ, κ ,λX ,λX ,λX × (5, 2)λ,κ, κ ,λX ,λX ,λX †2 2 (6, 1/2)λ,κ, κ ,λX ,λX ,λX X × (6, 3/2)λ,κ, κ ,λX ,λX ,λX × (6, 5/2)λ,κ, κ ,λX ,λX ,λX × (7, real)λ,κ,λX ,λX × (7, 1)λ,κ, κ ,λX ,λX ,λX ,λX × (7, 2)λ,κ, κ ,λX ,λX ,λX ,λX × (7, 3)λ,κ, κ ,λX ,λX ,λX ,λX = Fig. 1. A contour plot of the scale of the LP as functions of λX , λX at μ M X = + = with M X 100 GeV. The red and blue lines represent to λX 1/4λX 0and + = of the quartic coupling constants are set to be zeros. Once a finite λX 9/4λX 0, respectively. The left-bottom side of the red or blue line corre- quartic coupling is injected, the RGE running of the quartic cou- sponds to the unbounded scalar potential. pling constant leads the LP. In general, there are many degrees of freedom to choose the initial condition for the new coupling con- As a set of initial conditions for the RGEs, we take stants. Among them, we evaluate a conservative scale (the largest = =···= = = scale) of the LP. λX (M X ) λX (M X ) κ(M X ) κ (M X ) 0, (7) The scalar potential of the model is given by and evaluate the scale of the LP. This set gives a conservative eval- 2 † † 2 2 † † 2 uation for the scale of the LP under the condition that the scalar V =−μ + λ( ) + M X X + λX |X X| X potential is bounded from below. We use the initial values of SM + | † || † |+ † a † a κ X X κ (X T X X)( T) parameters in Ref. [12] and take M H = 125.7GeV, Mt = 173 GeV, + † a 2 + † a b 2 +··· which are not sensitive to our analysis. By calculating the RGE, the λX (X T X X) λX (X T X T X X) , (1) existence of the LP is found below Planck scale for nX ≥ 4. The ··· where is the SM Higgs doublet, X is a new scalar field, and blowup of the quartic coupling constants λX , λX , of X occurs a = the SU(2)L generator for X is T X (a 1, 2, 3). Among the scalar because the quantum corrections to these couplings are propor- quartic coupling constants, some of them are related to each other tional to the fourth power of the electroweak charges of X. This depending on the electroweak charges of X. For quadruplets, extra gives a rapid injection of non-zero value to the quartic coupling coupling constants are allowed, constants, and eventually results in the LP. Let us discuss the initial condition for the new quartic couplings † † i † † i jk λ 2 X = λ 2 X ii , (2) = † X † X j k constants. For an illustration, we focus on the case (nX , Y X ) †2 2 † † ik jk (4, 3/2) with an additional Z2 symmetry, which forbids the term λ 2 X = λ 2 X X , (3) 3 † X2 † X2 i j kk † X. In this case, by imposing the boundedness of the scalar †3 = † † † ijk potential for |X1|4 and |X2|4 directions, we obtain the following λ †3 X λ †3 X , (4) X X i j k conditions2 111 112 where we adopt the symmetric√ √ tensor notation, i.e., (X , X , 1 9 122 222 1 2 3 4 X , X ) = (X , X / 3, X / 3, X ) for n = 4. The former two λX + λ ≥ 0,λX + λ ≥ 0. (8) X 4 X 4 X coupling constants exist for Y X = 1/2, while the last does for 1 4 2 4 Y = 3/2. There is a similar coupling of λ 2 for a sextet. Note This requirements comes from the |X | and |X | terms in the X † X2 that there is an accidental global U (1) symmetry, if all the addi- scalar potential of X, tional dimension four couplings are forbidden. We summarize the λ |X† X|2 + λ (X† T a X)2 independent coupling constants and possible dimension four extra X X coupling constants of each model in Table 1. 1 4 2 4 9 1 4 1 2 4 Next, let us move to the RGE analysis. The scale below the mass = λX (|X | +|X | ) + λ |X | + |X | +···, (9) X 4 4 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 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 condi- dg 1 41 1 Y = + AY2 n g3 , (5) tions are also come from |X3|4 and |X4|4 directions. The contour 2 X X Y dt 16π 6 6 plot for the scale of the LPs together with the above two condi- dg2 1 19 1 tions is shown in Fig. 1 as functions of λX (M X ) and λ (M X ). One = − + AT(X) g3, (6) X 2 2 can see that λX = λ = 0actually gives the conservative scale (the dt 16π 6 6 X largest scale) of the LP. Even if we introduce nonzero κ, κ , λ 3 , = = † X where t ln μ with μ being the renormalization scale, A 1 (2) these effects decrease the scale of the LP since these contributions for a real (complex) scalar field, T (X) is the Dynkin index. We to the beta functions of λX and λX are always positive. For the see that g2 is non-asymptotic free if n ≥ 7 (5) for a real (com- = X (nX , Y X ) (4, 1/2) case, the similar arguments are applicable for plex) field case. The runnings of the top-Yukawa coupling yt and the strong gauge coupling g3 are unchanged at one-loop level. The one-loop RGEs of the scalar coupling constants are given in Ap- 2 Note that this is not the sufficient condition for the boundedness but the nec- pendix A. essary condition. 240 Y. Hamada et al. / Physics Letters B 747 (2015) 238–244

Fig. 2. The solid and dashed lines correspond to the numerical results and the approximated results in Eq. (14).

λ , λ and λ 2 . Although the effect of λ 2 is unclear, we Table 2 X X † X † X2 numerically confirmed that this coupling constant also leads to the The fitted results of the scale of LPs with the initial condition in Eqs (7). The po- lower scale of the LP. Thus, we conclude that Eqs. (7) are the ini- sitions of LP derived from one-loop beta function of the gauge couplings taking M = 100 GeV are also given. tial condition for the conservative evaluation of the LP. The same X (n , Y X ) LP [ GeV] g [GeV] g [GeV] discussions would be applied to higher isospin multiplets. We also X Y 2 × 16 1.57 × 41 confirmed these points numerically. The detailed and generalized (4, 1/2) 6.4 10 (M X /100 GeV) 1.2 10 – × 15 1.21 × 30 analysis will be presented in our future publications [59]. (4, 3/2) 1.3 10 (M X /100 GeV) 3.1 10 – 11 1.38 42 (5, real) 9.5 × 10 (M X /100 GeV) 9.6 × 10 – In Fig. 2, we present the conservative scale of the LP as a 9 1.28 34 488 (5, 1) 4.9 × 10 (M X /100 GeV) 9.0 × 10 9.0 × 10 function of M X in the SM with one more scalar multiplet (n = 8 1.13 22 488 X (5, 2) 7.9 × 10 (M X /100 GeV) 5.7 × 10 9.0 × 10 3 6 1.17 40 32 4, 5, 6, 7). We consider all the possible assignments of the hyper- (6, 1/2) 4.6 × 10 (M X /100 GeV) 1.6 × 10 2.7 × 10 × 6 1.16 × 26 × 32 charges, where fractional electric charges are not allowed for new (6, 3/2) 4.2 10 (M X /100 GeV) 5.2 10 2.7 10 × 6 1.08 × 16 × 32 particles. For a Y = 0field, a real scalar condition is assumed. (6, 5/2) 1.6 10 (M X /100 GeV) 3.2 10 2.7 10 X 6 1.13 42 56 (7, real) 1.0 × 10 (M X /100 GeV) 9.6 × 10 1.3 × 10 The solid lines in each plot represent the numerical results, while 5 1.11 32 15 (7, 1) 1.2 × 10 (M X /100 GeV) 3.6 × 10 1.4 × 10 the dashed lines show the approximated results obtained from the 5 1.10 19 15 (7, 2) 1.1 × 10 (M X /100 GeV) 2.2 × 10 1.4 × 10 4 1.06 12 15 analytic study as we will discussed later in this letter. We fit the (7, 3) 6.6 × 10 (M X /100 GeV) 1.2 × 10 1.4 × 10 solid lines in Fig. 2 by the scale of the LP and M X with an exponent. The results are listed in Table 2. For comparisons, we also of the LP from the scalar quartic coupling constant is much lower present the positions of the LP which are calculated by solving gi than . the one-loop beta function of gauge couplings. gi In order to understand the results analytically, we focus on the beta function of the quartic coupling of X in which the gauge cou- 1 1 pling g(μ) is approximated by the constant g(M ). Then, we rede- g = M X exp X i 2B 2 ··· 4 i gi (M X ) fine the quartic coupling constants λX , such that g terms only appear in one beta function. By this redefinition, we correctly take Bi,SM/Bi Mt 1 1 into account the induced quartic coupling constant from the gauge = M X exp , (10) M 2B 2 couplings at one-loop level, since we begin with a conservative as- X i gi (Mt ) sumption given in Eqs. (7). We call the new coupling constants = ˜ ··· where i Y , 2, and Bi,(SM) is the coefficient of the beta function λX , . Consequently, the RGE is simplified as 2 of gi including a factor 1/16π . We see that the conservative scale dλX 1 2 ˜ ˜ 2 = (c0 − c1λX + c2λ + c3λX λ + c4λ ), (11) dt 16π 2 X X X 3 ˜ Requiring the tree-level unitarity of SU(2)L gauge interactions, n ≤ 8(9) is ob- dλ 1 X X = −˜ + ˜ 2 + ˜ ˜ + ˜ ˜ 2 tained for a complex (real) scalar multiplet [60,61]. ( c λX c λ c λX λ c λ ). (12) dt 16π 2 1 2 X 3 X 4 X Y. Hamada et al. / Physics Letters B 747 (2015) 238–244 241

˜ = ˜ However, even if we set λX 0at M X , nonzero λX is induced finite quartic coupling constants are originally generated by the ˜ by the c2 term, which affects the running of λX through the c3 gauge couplings, and then the RGE of the quartic coupling constant 4 term. Therefore, we redefine λX in order to cancel the c3 term. leads the LP. Roughly speaking, the LP is induced by two-loop ra- As a result, we have diative corrections to the RGEs. We can also evaluate the exponent + ˜ of M X by putting gY (2)(M X ) gY (2)(Mt ) BY (2),SM ln(M X /Mt ). Al- dλX 1 ˜ ˜ 2 though the analytic expression is too complicate to write, we ob- = (c˜0 − c˜1λX + c˜2λ ), (13) dt 16π 2 X tain ∼ 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 taking λ˜ = ··· = 0. This simplified differential equation can be X order. Firstly, the position of the LP does not depend on the nor- solved analytically. The position of the LP is calculated as malization of λ˜ . We can understand the reason from Eqs. (13) X 2 and (14) as follows; let us define the new scalar coupling constant LP 16π π − 1 c˜1 = exp − tan 1 λ˜ (M ) − , (14) having a different normalization as λ˜ N := a λ˜ where a is a real ˜ X X ˜ X X M X c2d 2 d 2c2 ˜ N ˜ constant. Although the RGE of λX takes the same form as λX , the ˜ ˜2 coefficients are different. They are a c0, c1 and c2/a. By using these c0 c1 ˜ N ˜ d = − . (15) coefficients and λ (M X ) = a λX (M X ), one can easily show that the c˜ ˜2 X 2 4c2 ˜ N LP for λX calculated by Eq. (14) gives the same result as that of ˜ As an example, we show the concrete procedure for the case with λX . Therefore, we do not need to care about the normalizations of nX = 4. Concentrating on the quartic coupling constants of X, we scalar couplings when we study the LP. have Secondly, we discuss the nonzero value of the initial condition. ˜ Let us start with Eq. (14). Expanding Eq. (14) around 1/λX (M X ) = dλX 1 2 99 2 = + 32λ + 30λX λ + λ 0, we get dt 16π 2 X X 2 X 2 297 LP 16π 1 ˜ 2 + 4 4 + 4 − 2 2 − 2 = exp + O 1/λX (M X ) , (24) 6Y X gY g2 12Y X λX gY 45λX g2 , (16) ˜ ˜ 8 M X c2 λX (M X ) dλ 1 X = + + 2 + 2 2 2 which means that the scale of the LP is determined by only the 24λX λX 42λX 12Y X gY g2 ˜ dt 16π 2 initial value and c2. The conditions to neglect higher order terms ˜ are λX (M X ) c˜1/(2c˜2), d. Numerically, c˜1/(2c˜2) ≈ 0.28, d ≈ 0.31 − 2 2 − 2 for (n , Y ) = (4, 3/2). 12Y X λX gY 45λX g2 . (17) X X Though we have focused on a single scalar extension of the SM = = = = in this letter, more matter fields may be added. Since the increase Here we put gY (μ) gY (M X ), g2(μ) g2(M X ) and κ κ = = = = of scalar fields leads to the large coefficient in the beta functions λ †2 λ †2 2 λ †3 λ †3 0. At first, we redefine λX in X X X X in general, we expect that the scale of the LP becomes lower. If we order to cancel the constant term (12Y 2 g2 g2 term). Hence, we X Y 2 add fermions in addition to scalars, the Yukawa couplings among take 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 λ = λ + η λX , (18) X X the beta function at the one-loop level. ≡− 2 2 2 4 4 + 297 4 Finally, we comment on the implications of the new physics where η 12Y X gY g2/(6Y X gY 8 g2). The requirement of the 2 2 ˜ beyond the SM. In the Minimal dark matter models [40–42], in- vanishing gY g2 term fixes λX up to the normalization, while we teractions of new scalar multiplets are also bounded by the relic still have the degree of freedom to redefine λX . Using this, we ˜ abundance of dark matter and the data for direct/indirect searches. define λX as It is very interesting to combine our study with such constraints ˜ = + ˜ [59]. In some class of the seesaw models for the neutrino mass λX λX ξ λX , (19) generation, Majorana fermions are introduced together with scalar ˜ ˜ and choose ξ in such a way that the λX λX term vanishes in the multiplets, which may or may not accommodate a dark matter ˜ beta function of λX . Then, we obtain candidate. These models are constrained not only by the relic abundance but also by the neutrino oscillation data. In these mod- −20 + 3η − 99η2 5(80 + 72η + 621η2) ξ = ± . (20) els, the effects of Majorana fermions on the RGEs are expected to 16η + 24η2 + 99η3 16η + 24η2 + 99η3 be small, because the required Yukawa coupling is generally small if we add these new fermions in the electroweak/TeV scale. In The coefficients c0, c1, c2 are given by addition, models with a flavor symmetry are also very intriguing 297 4 4 4 candidates for this kind of analyses. Because the scalar fields are c˜0 = g + 6Y g , (21) 8 2 X Y embedded in the multiplet under the flavor symmetry, which pro- 2 2 2 vides a large number of SU(2)L multiplets. Increase of the number c˜1 = 12Y g + 45g , (22) X Y 2 of multiplet also gives large contribution to the RGE, which would 99 2 2 99 3 make the LP smaller [59]. c˜2 = 32 − 30η + η + 8ηξ + 12η ξ + η ξ. (23) 2 2 In conclusion, we have investigated the scale of the LP us-

By this procedure, we can evaluate the LP in the analytical calcula- ing the one-loop RGE in the SM with one more scalar nX -plet = tions. The approximated lines are also plotted in Fig. 2. Comparing X (nX 4, 5, 6, 7with all possible hypercharge assignments). The ≥ the numerical results and our analytic results in these figures, the LP is found below Planck scale for nX 4, if we introduce a new above expressions fit the numerical results within an order of mag- scalar multiplet at electroweak/TeV scale. This means that any sin- ≥ nitude error. From this analytical analysis, we confirm that the gle scalar field extension with nX 4of 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 4 We checked that the effect of the c4 term is smaller than that of the c3 term. values for the quartic coupling constant of new scalar fields are set 242 Y. Hamada et al. / Physics Letters B 747 (2015) 238–244 dλ †2 2 1 to be zeros at the scale of new particles. Nevertheless, the quartic X = − 2 + 2 4λ 2 6y λ 2 coupling constants are induced from the electroweak gauge inter- dt 16π 2 † X t † X2 actions. The induced coupling constants are rapidly enhanced, and 3 2 2 2 2 finally hit the LP. We have calculated the LP as a function of the − λ 2 g − 6Y λ 2 g − 27λ 2 g † X2 Y X † X2 Y † X2 2 mass of X, and listed a fitting formula of the LP with an exponent. 2 + 8κλ 2 + 4κ λ 2 The results are consistent with our approximated formula within † X2 † X2 an order of magnitude error, which is derived from the simplified RGE with the conservative assumption. The obtained LP in each + 4λλ 2 + 4λ λ 2 − 11λ λ 2 , † X2 X † X2 X † X2 model is much smaller than that previously calculated by solving the beta functions of the gauge couplings [58]. These new results dλ 2 † X 1 2 are very useful and generic constraints on the beyond the SM in- = + 12λλ 2 + 9y λ 2 dt 16π 2 † X t † X cluding new scalar multiplets. 9 2 2 2 2 − g λ 2 − 3Y g λ 2 − 18g λ 2 Acknowledgements 4 Y † X X Y † X 2 † X + + 5 This work is supported by the Grant-in-Aid for Japan Society 6κλ †2 κ λ †2 X 2 X for the Promotion of Science (JSPS) Fellows No. 25·1107 (Y.H.) and No. 27·1771 (K.K.). K.T.’s work is supported in part by the − 40 ∗ λ †2 2 λ 2 , MEXT Grant-in-Aid for Scientific Research on Innovative Areas 3 X † X No. 26104704. dλ †3 λ †3 9 X = X + 12λ + 9y2 − 18g2 − g2 − 3Y 2 g2 dt 16π 2 t 2 4 Y X Y Appendix A. Renormalization group equations 15 + 6κ + κ . (A.1) We calculate the RGEs by using the general formula in Ref. [62]. 2

• SM with a quadruplet scalar field • SM with a real quintet scalar field dλ 1 3 9 3 = + 2 − 4 + 4 + 4 + 2 2 dλ 1 2 4 3 4 9 4 3 2 2 24λ 6yt gY g2 gY g2 = + 24λ − 6y + g + g + g g dt 16π 2 8 8 4 dt 16π 2 t 8 Y 8 2 4 Y 2 5 + 2 − 2 − 2 + 2 + 2 5 12λyt 3λgY 9λg2 4κ κ + κ2 + 12λy2 − 3λg2 − 9λg2 , 4 2 t Y 2 40 2 2 2 + |λ 2 | + 8|λ 2 | + 24λ , dλ 1 † X2 † X †3 X 2 4 2 2 9 X = + 26λ + 108g − 72λX g + 2κ , dt 16π 2 X 2 2 dλX 1 99 = + 32λ2 + 30λ λ + λ 2 2 X X X X dκ 1 4 2 dt 16π 2 = + 18g + 12κλ + 14κλX + 6y κ dt 16π 2 2 t 297 + 4 4 + 4 − 2 2 − 2 6Y X gY g2 12Y X λX gY 45λX g2 3 81 8 − κ g2 − κ g2 + 4κ2 . (A.2) 2 Y 2 2 + 2 + | |2 2κ 2 λ †2 2 , X • SM with a complex quintet scalar field dλ 1 X = + + 2 + 2 2 2 24λX λX 42λX 12Y X gY g2 dλ 1 3 9 3 dt 16π 2 = + 24λ2 − 6y4 + g4 + g4 + g2 g2 dt 16π 2 t 8 Y 8 2 4 Y 2 1 8 − 2 2 − 2 + 2 − | |2 12Y X λX gY 45λX g2 κ λ †2 2 , 5 2 9 X + 12λy2 − 3λg2 − 9λg2 + 5κ2 + κ 2 , t Y 2 2 dκ 1 45 15 = + 3Y 2 g4 + g4 + 4κ2 + κ 2 2 X Y 2 dλX 1 2 dt 16π 4 4 = + 36λ + 48λX λ dt 16 2 X X 3 π + 2 − 2 2 − 2 − 2 6yt κ 6Y X κ gY κ gY 27κ g2 + + + 2 2 720λX λX 1152λX λX 3168λX + + + 4 4 2 2 2 2 12κλ 20κλX 15κλX + 6Y g − 12Y λX g − 72λX g + 2κ , XY X Y 2 40 + 2 + | |2 + | |2 dλ 1 18λ 3 λ †2 2 6 λ †2 , X = + + 2 + − 2 † X X X 24λX λ 84λ 408λ λ 84λ 3 dt 16π 2 X X X X X d 1 κ = + 2 2 + 4 + 2 2 2 − 2 2 12Y X g g 3g2 12Y X gY g2 12Y X λX gY dt 16π 2 Y 2 2 1 2 2 3 2 2 2 2 − 72λ g + κ , + 6y κ − κ g − 6Y κ g − 27κ g X 2 2 t 2 Y X Y 2 + 8κκ + 4κ λ + 4κ λ + 31κ λ dλX 1 2 2 X X = + 8λ + 24λ λ − 32λ λ + 368λ 2 X X X X X X 64 8 dt 16π + 2 + | |2 + | |2 24λ 3 λ †2 2 λ †2 , + 4 − 2 2 − 2 † X 9 X 3 X 6g2 12Y X λX gY 72λX g2 , Y. Hamada et al. / Physics Letters B 747 (2015) 238–244 243 dκ 1 dλ 2 λ 2 = + 2 4 + 4 † X2 † X2 961 3Y g 18g = + 4λ + 4λX − 31λ + λ dt 16π 2 X Y 2 dt 16π 2 X 4 X + + + + + 2 3 12κλ 24κλX 24κλX 360κλX 6yt κ + 2 − 2 − 2 2 − 2 + + 6yt gY 6Y X gY 57g2 4κ 8κ . 2 3 81 − κ g2 − 6Y 2 κ g2 − κ g2 + 4κ2 + 6κ 2 , 2 Y X Y 2 2 (A.4) dκ 1 • SM with a real septet scalar field = + 12Y g2 g2 2 X Y 2 dt 16π dλ 1 9 3 = + 24λ2 − 6y4 + g4 + g4 2 2 t 2 Y + 4κ λ + 4κ λX + 60κ λ + 60κ λ + 6y κ dt 16π 8 8 X X t 3 7 3 2 2 2 81 2 + 2 − 2 − 2 + 2 2 + 2 − κ g − 6Y κ g − κ g + 8κκ . (A.3) 12λyt 3λgY 9λg2 gY g2 κ , 2 Y X Y 2 2 4 2 dλX 1 • SM with a sextet scalar field = + 30λ2 + 2448λ λ + 51840λ 2 2 X X X X dt 16π dλ 1 3 9 − 144λ g2 + 2κ2 , = + 24λ2 − 6y4 + g4 + g4 X 2 dt 16 2 t 8 Y 8 2 π dλ 1 X = + 6g4 + 1530λ 2 + 24λ λ − 144λ g2 , 3 2 2 X X X X 2 + 2 − 2 − 2 + 2 2 dt 16π 12λyt 3λgY 9λg2 gY g2 4 dκ 1 = + 36g4 + 12κλ + 18κλ + 6y2κ 35 28 2 2 X t + 2 + 2 + | |2 dt 16π 6κ κ λ †2 2 , 8 5 X 3 153 − 2 − 2 + 2 κ gY κ g2 4κ . (A.5) dλX 1 3535 2 2 = + 40λ2 + 70λ λ + λ λ 2 X X X X X dt 16π 2 • SM with a complex septet scalar field 8525 271975 + λ λ + λ 2 + 6Y 4 g4 dλ 1 3 9 4 X X 16 X X Y = + 24λ2 − 6y4 + g4 + g4 dt 16π 2 t 8 Y 8 2 2 2 2 2 11 2 − 12Y λX g − 105λX g + 2κ − |λ 2 | , 3 X Y 2 8 † X2 + 2 − 2 − 2 + 2 2 + 2 + 2 12yt λ 9λg2 3λgY g2 gY 7κ 7κ , 4 dλ 1 2115 X = + + 2 + 24λX λX 136λX λX λX dλX 1 2 dt 16π 2 2 = + 44λ + 96λX λ + 3744λX λ dt 16π 2 X X X 265 2 4 2 2 2 + 2 + + − λ + 3g + 12Y g g 77184λX 9216λX λX 31104λX λX 2 X 2 X Y 2 + + 2 + 4 4 90432λX λX 2859264λX 6Y X gY 2 2 2 1 2 7 2 − 12Y λ g − 105λ g + κ − |λ 2 | , 2 2 2 2 X X Y X 2 † X2 − 12Y λ g − 144λ g + 2κ , 2 25 X X Y X 2 dλ 1 dλX 1 2 1715 2 X = + + 2 + = + 8λ + 24λX λ + 2λ λ + λ 24λX λX 204λX 768λX λX dt 16π 2 X X X X 2 X dt 16π 2 2 2 + 7008λ + 44616λ λ − 73824λ λ + 4 − 2 − 2 2 + | |2 X X X X X 6g2 105λX g2 12Y X λX gY λ †2 2 , 25 X + 2408676λ 2 + 3g4 − 12Y 2 λ g2 − 144λ g2 X 2 X X Y X 2 dκ 1 105 1 = + 3Y 2 g4 + g4 + 12Y 2 g2 g2 + κ 2 , dt 16π 2 X Y 4 2 X 2 Y 2 3535 dλ 1 + + + + X 2 12κλ 28κλX 35κλ κλ = + 8λ + 24λX λ + 96λ λ X 4 X dt 16π 2 X X X X 3 + 2 + + + 2 + 2 − 2 − 2 − 2 2 1588λX 912λX λX 5312λX λX 22696λX 6yt κ 57κ g2 κ gY 6Y X κ gY 2 + 4 − 2 − 2 2 6g2 144λX g2 12Y X λX gY , 35 56 + 2 + 2 + | |2 4κ κ λ †2 2 , dλ 1 4 5 X X = + 16λ λ − 80λ 2 + 24λ λ 2 X X X X X dt 16π dκ 1 2 2 − + − 2 = + 12Y X g g 128λX λX 3056λX λX 12200λX dt 16π 2 Y 2 − 2 2 − 2 12Y X λX gY 144λX g2 , 697 2 + 4κ λ + 4κ λX + 101κ λ + κ λ + 6y κ X 4 X t dκ 1 2 4 4 = + 3Y g + 36g + 12κλ + 32κλX dt 16π 2 X Y 2 3 2 2 2 2 − κ g − 6Y κ g − 57κ g 2 Y X Y 2 + 48κλ + 1872κλ + 4608κλ + 6y2κ X X X t 64 + + | |2 3 2 2 2 153 2 2 2 8κκ λ †2 2 , − κ g − 6Y κ g − κ g + 4κ + 12κ , 25 X 2 Y X Y 2 2 244 Y. Hamada et al. / Physics Letters B 747 (2015) 238–244

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