PHYSICAL REVIEW D 101, 055038 (2020)
Fate of the false vacuum in a singlet-doublet fermion extension model with RG-improved effective action
† Yu Cheng* and Wei Liao Institute of Modern Physics, School of Sciences, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, People’s Republic of China
(Received 30 September 2019; accepted 13 March 2020; published 30 March 2020)
We study the effective potential and the renormalization group (RG) improvement to the effective potential of a Higgs boson in a singlet-doublet fermion dark matter extension of the Standard Model (SM), and in general singlet-doublet fermion extension models with several copies of doublet fermions or singlet fermions. We study the stability of the electroweak vacuum with the RG-improved effective potential in these models beyond the SM. We study the decay of the electroweak vacuum using the RG-improved effective potential in these models beyond the SM. In this study we consider the quantum correction to the kinetic term in the effective action and consider the RG improvement of the kinetic term. Combining all these effects, we find that the decay rate of the false vacuum is slightly changed when calculated using the RG-improved effective action in the singlet-doublet fermion dark matter model. In general singlet- doublet fermion extension models, we find that the presence of several copies of doublet fermions can make the electroweak vacuum stable if the new Yukawa couplings are not large. If the new Yukawa couplings are large, the electroweak vacuum can be turned into metastable or unstable again by the presence of extra fermions.
DOI: 10.1103/PhysRevD.101.055038
I. INTRODUCTION Higgs potential with a negative Higgs quartic coupling [8– 10]. Calculation of vacuum decay using RG-improved Quantum contribution to the effective potential is effective potential in the SM does not give much difference, important to understand the properties of the scalar field, because the bounce solution is dominated by behavior at a e.g., the property of the ground state and the behavior at high energy scale and the RG-improved effective potential large energy scale. For example, radiative correction can in the SM at a high energy scale is accidentally close to the make a vacuum unstable and trigger spontaneous symmetry Higgs potential with running quartic coupling [12]. breaking [1]. The RG-improved effective potential, which In extension of the SM, the situation can be quite resums contributions of large logarithms, is important to different. The RG-improved effective potential is possible understand the behavior of effective potential at a large to be very different from the potential using a running energy scale. For example, the RG-improved effective Higgs quartic coupling. As an example, we consider a potential is crucial in reducing the dependence on the singlet-doublet fermion dark matter (SDFDM) extension of renormalization scale when calculating quantities related the SM. We show that the RG-improved effective potential with physical parameters [2–4]. can be quite different from the tree-level potential aided It is well known that a false vacuum can decay via with running Higgs quartic coupling. Then we study the tunneling [5–7] and become unstable. In the SM, the Higgs vacuum stability in this extension of SM. We study the false quartic self-coupling can become negative at an energy vacuum decay using RG-improved effective potential in scale around 1010 GeV. This makes the electroweak (EW) this SDFDM model. We also study the quantum contribu- vacuum unstable. The decay rate of the EW vacuum can be tion to the kinetic term in the effective action in this model calculated using an approximate bounce solution of the beyond the SM and consider the RG improvement of the kinetic term. After taking all these effects into account we *[email protected] find that the false vacuum decay rate is just slightly changed † [email protected] using the RG-improved effective action in the SDFDM model, although the RG effective potential is significantly Published by the American Physical Society under the terms of different from the tree-level form of the Higgs potential the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to with running quartic self-coupling. We also perform the author(s) and the published article’s title, journal citation, these analyses in general singlet-doublet fermion extension and DOI. Funded by SCOAP3. models in which several copies of singlet fermions or
2470-0010=2020=101(5)=055038(25) 055038-1 Published by the American Physical Society YU CHENG and WEI LIAO PHYS. REV. D 101, 055038 (2020) several copies of doublet fermions are considered. We find M χ0 0 d 1 † that the presence of several extra doublet fermions can M ¼ ¼ ULMUR ð3Þ 0 M χ0 make the EW vacuum stable if the new Yukawa couplings 2 are not large. with The article is organized as follows. In Sec. II, we first briefly review the SDFDM model. We study the threshold cos θL;R sin θL;R effect caused by the extra fermions in this model beyond U 4 L;R ¼ − θ θ ð Þ the SM and study the running of the Higgs quartic coupling sin L;R cos L;R in this model. Then we study the effective potential and the RG-improved effective potential in this model. In Sec. III and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we study the vacuum stability in the SDFDM model. We 1 2 − 2 − 4 2 calculate the renormalization of the kinetic term in the M χ0 ¼ TM TM DM ; ð5Þ 1 2 effective action in the SM and in the SDFDM model. We qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi study the RG improvement of the kinetic term and calculate 1 2 2 − 4 2 M χ0 ¼ TM þ TM DM ; ð6Þ the decay rate of the false vacuum. In Sec. IV, we do the 2 2 analysis in the general singlet-doublet fermion extension 2 1 2 2 2 1 2 2 1 2− model. Details of calculation are summarized in the where TM ¼ MS þ 2 y1v þ MD þ 2 y2v , DM ¼ 2 y1y2v Appendixes A–D. We summarize in the Conclusion. 0 0 MSMD. χ1 and χ2 are neutral fermion fermions in the diagonalized base with masses M χ0 and M χ0 respectively. 1 2 II. EFFECTIVE POTENTIAL AND The mixing angles θL;R can be solved as RG-IMPROVED EFFECTIVE POTENTIAL pffiffiffi IN THE SDFDM MODEL 2vðMSy1 þ MDy2Þ tan 2θ ¼ 2 ð7Þ L 2 − 2 v 2 − 2 A. The SDFDM model MD MS þ 2 ðy1 y2Þ In addition to the SM fields, the SDFDM model has pffiffiffi 0 − 2 SU(2) doublet fermions ψ ¼ðψ ; ψ ÞT with Y ¼ vðMSy2 þ MDy1Þ L;R L;R L;R tan 2θ ¼ 2 : ð8Þ −1 2 R 2 − 2 v 2 − 2 = and singlet fermions SL;R. Here, L (R) refers to the MD MS þ 2 ðy2 y1Þ left (right) chirality. As a singlet, S can be either a Dirac- pffiffiffi type or Majorana-type fermion [13,14]. In this model, the Writing H ¼ð0; ðv þ hÞ= 2ÞT, the interaction Lagrangian neutral fermion can be a dark matter candidate. The vacuum χ0 of dark matter fields 1;2 and the CP-even neutral Higgs properties of the SDFDM model with a Majorana-type field h is obtained from Eq. (1) as mass have been discussed in [15]. In this article we work on the Dirac-type mass [16]. 0 0 0 0 ΔL ¼ −y χ1 χ1h − y χ2 χ2h The relevant terms of ψ and S in the Lagrangian are A B − ½y χ0P χ0h þ y χ0P χ0h þ H:c: ; ð9Þ L ψ¯ ψ ¯ ∂ − ψ¯ ψ − ¯ C 1 R 2 D 1 L 2 SDFDM ¼ iD= þ Si=S MD L R MSSLSR − ψ¯ ˜ − ψ¯ ˜ where P ¼ð1 γ5Þ=2, y ¼ð−y2 cos θ sin θ − y1 × y1 LHSR y2 RHSL þ H:c:; ð1Þ R;L pffiffiffi A L R sin θ cos θ = 2, y y2 cos θ sin θ y1 sin θ × L pffiffiffiRÞ B ¼ð R L þ R where MD;S are the mass parameters, y1;2 the new Yukawa θ 2 θ θ − θ θ cosffiffiffi LÞ= , yC ¼ðy2 cos L cos R y1 sin L sin ffiffiffiRÞ= couplings, H is the SM Higgs doublet with Y ¼ 1=2, and p p 2 and y −y2 sin θ sin θ y1 cos θ cos θ = 2. ˜ D ¼ð L R þ L RÞ H ¼ iσ2H . We impose a Z2 symmetry in the Lagrangian with the new fermions ψ and S odd and SM fermions even B. Effective potential and RG-improved under the Z2 operation. This guarantees the lightest of these effective potential new fermions to be stable, and makes it a dark matter In the following, we will take ϕ to denote a neutral candidate if it is neutral. pffiffiffi After the EW symmetry breaking, the mass matrix of external field. ϕ= 2 corresponds to the CP-even neutral ψ ψ 0 SL;R and the neutral component of L;Rð L;RÞ is given as component of the Higgs doublet in the SM. The tree-level ! potential of ϕ is yp2ffiffiv MS 2 M ¼ ; ð2Þ 2 yp1ffiffiv mϕ λ 2 MD ϕ − ϕ2 ϕ4 V0ð Þ¼ 2 þ 4 ; ð10Þ where v ¼ 246 GeV. Mixings between these two neutral fermions are generated by this mass matrix. The mixing where λ is the Higgs quartic self-coupling in the SM and mϕ angles θL;R appear in the diagonalization of the mass matrix the mass term. A Coleman-Weinberg–type quantum cor- using two unitary mass matrices, that is rection to the potential can be calculated using a vacuum
055038-2 FATE OF THE FALSE VACUUM IN A SINGLET-DOUBLET … PHYS. REV. D 101, 055038 (2020) diagram by considering the quantum fluctuations around with the boundary condition λið0Þ¼λi. So the RG- the external field ϕ. The one-loop contribution of the extra improved effective potential can be written by simply fermion in the SDFDM model to the effective potential is substituting μ, λi, ϕ in the original effective potential with calculated as μðtÞ, λiðtÞ, ϕðtÞ. The RG-improved effective potential in the SDFDM 1 2 ϕ Ext 4 M χ1 ð Þ model is obtained by implementing the substitution men- V ¼ − M χ ðϕÞ ln − 3=2 1 64π2 1 μ2 tioned above into Eq. (12).Wehave 1 2 ϕ 4 M χ2 ð Þ ϕ SM ϕ SM ϕ Ext ϕ − M χ ðϕÞ ln − 3=2 ; ð11Þ Veffð ;tÞ¼V0 ð ;tÞþV1 ð ;tÞþV1 ð ;tÞ; ð18Þ 64π2 2 μ2 with 2 ϕ where M χ1; χ2 ð Þ are obtained from Eqs. (5) and (6) by replacing v with ϕ. μ is the renormalization scale chosen in m2 ðtÞ 1 SM ϕ − ϕ ϕ2 λ ϕ4 this calculation. In the limit ϕ ≫ v, we have approxi- V0 ð ;tÞ¼ 2 ðtÞþ4 ðtÞ ðtÞ; ð19Þ 2 2 2 2 2 mately M χ χ ϕ ≈ y ϕ =2;y ϕ =2. 1; 2 ð Þ 1 2 X 2 ð−1Þin M ðϕ;tÞ We arrive at a one-loop effective potential as follows: VSM ϕ;t i M4 ϕ;t i − c ; 1 ð Þ¼ 64π2 i ð Þ ln μ2 i ð20Þ i ðtÞ ϕ λ μ SM ϕ λ SM ϕ λ μ Ext ϕ λ μ Veffð ; i; Þ¼V0 ð ; ÞþV1 ð ; i; ÞþV1 ð ; i; Þ; X 2 −1 i M χ ϕ;t Ext ð Þ ni 4 i ð Þ ð12Þ V1 ðϕ;tÞ¼ M χ ðϕ;tÞ ln − 3=2 : 64π2 i μ2 i ðtÞ where λi denotes various parameters in the model. SM SM ð21Þ V0 ðϕ; λÞ is given in Eq. (10). V1 ðϕ; λi; μÞ is the one- loop contribution to the effective potential in the SM. The In Eq. (20) the index i ¼ H, G, f, W, Z runs over SM fields effective potential in the SM is known up to two loop in the loop, and cH;G;f ¼ 3=2;cW;Z ¼ 5=6. In Eq. (21) the [17,18]. VExt ϕ; λ ; μ is given in Eq. (11). 1 ð i Þ index i ¼ χ1; χ2 runs over extra neutral fermions in the In the vacuum stability analysis, we must consider the SDFDM model. ni is the number of degrees of freedom of behavior of the effective potential for the large external the fields. MiðϕÞ in Eq. (20) is given by field. That is to say, we must deal with potentially large logarithms of the type logðϕ=μÞ for a neutral external field 2 2 0 M ðϕ;tÞ¼κiðtÞϕ ðtÞ − κ ðtÞ: ð22Þ ϕ. The standard way to solve the problem is by means of i i the RG equation (RGE). Veff satisfies the RGE The values of n , κ and κ0 in the SM can be found in i i i ∂ ∂ ∂ Eq. (4) in Ref. [4] in the Landau gauge and in Ref. [19] both μ β − γϕ 0 þ i Veff ¼ ; ð13Þ in the Fermi gauge and in the Rξ gauge. For new ∂μ ∂λi ∂ϕ contributions in the SDFDM model, we have ni ¼ 1, β β λ γ 2 ϕ ≈ 2 ϕ2 2 2 ϕ2 2 where i is the function of parameter i, and the and M χ1; χ2 ð ;tÞ y1ðtÞ ðtÞ= ;y2ðtÞ ðtÞ= . In Eqs. (20) anomalous dimension of the scalar field. Straightforward and (21), ð−1Þi equals 1. For gauge and scalar bosons application of this method leads to a solution [2] ð−1Þi takes a positive sign, while for fermion fields it takes a negative sign. μ λ ϕ μ λ ϕ Veffð ; i; Þ¼Veffð ðtÞ; iðtÞ; ðtÞÞ; ð14Þ In the limit ϕ ≫ v, Eq. (18) can be written approximately as follows: where λ ϕ ϕ ≈ effð ;tÞ ϕ4 Veffð ;tÞ ; ð23Þ μðtÞ¼μet 4 λ ϕ ΓðtÞϕ where eff is an effective coupling. In vacuum stability ðtÞ¼e ð15Þ μ ϕ λ ϕ analysis, we generally take ðtÞ¼ ,so effð ;tÞ can be with written as [12] Z λ ðϕ;tÞ t eff ΓðtÞ¼− γðλðt0ÞÞdt0 ð16Þ 1 X 0 ≈e4ΓðtÞ λ t N κ2 t κ t e2ΓðtÞ − c : ð Þþ 4π 2 i i ð Þðlog ið Þ iÞ ð Þ i and λ ðtÞ the running coupling determined by the equation i ð24Þ λ d iðtÞ N κ c ¼ β ðλ ðtÞÞ; ð17Þ The values of coefficients i, i, and i appearing in dt i i Eq. (24) are listed in Table I.
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¯ ¯ TABLE I. The coefficients in Eq. (24) for the background Rξ gauge [19]. ξW and ξZ are the gauge-fixing parameters in the background þ 0 Rξ gauge, G and G the Goldstone bosons, C and CZ the ghost fields, χ1 and χ2 are the dark matter particles in the SDFDM model. ¯ ¯ ¯ ¯ For ξW ¼ ξZ ¼ 0, Eq. (24) reproduces the one-loop result in the Landau gauge, and for ξW ¼ ξZ ¼ 1, we get the result in the ’t Hooft- Feynman gauge.
þ ptWZhG G0 C CZ χ1 χ2
Ni −12 631 2 1 −2 −1 −1 −1 3 5 5 3 3 3 3 3 3 3 ci 2 6 6 2 2 2 2 2 2 2 y2 g2 g2 g02 ξ¯ g2 ξ¯ g2 g0 ξ¯ g2 ξ¯ g2 g0 y2 y2 κi t þ 3λ λ W λ Zð þ Þ W W ð þ Þ 1 2 2 4 4 þ 4 þ 4 4 4 2 2
We note that the two-loop contributions of strong does not give rise to change of the initial parameters as coupling and the top Yukawa to the effective potential significant as that of changing Yukawa couplings. λ can be written in the eff as Therefore, we will always choose mass parameters as given in Table II and concentrate on the impact of different 4 2− y Yukawa couplings y1 2 in the remaining part of the article. λ loopðϕ;tÞ ≈ e4ΓðtÞ t 8g2ð3r2 − 8r þ 9Þ ; eff ð4πÞ4 s t t With these initial values in Table II, we then run the 3 π2 parameters all the way up to MPl scale. For RGE running, − 2 3 2 − 16 23 we use three-loop SM β functions [17]. We also include 2 yt rt rt þ þ 3 ; ð25Þ one-loop contributions of new particles in the SDFDM
2 model to the β functions of these SM parameters. For new yt 2Γ where rt ¼ ln 2 þ . We can see in Eq. (25) that the two- parameters in the SDFDM model, we use one-loop β loop contributions from top loops are of the order of functions which can be extracted using PyR@TE2 [21]. The 6 4 yt =ð4πÞ , while the one-loop terms are of the order of results are shown in Appendix B. 4 2 yt =ð4πÞ as can be seen in (24). The two-loop contributions from the new fermions in the SDFDM model are similar. TABLE II. All the parameters are renormalized at the top pole We expect that these new two-loop contributions would be mass (M ) scale in the MS scheme. BMP1: y1 ¼ y2 ¼ 0.25, much smaller than the one-loop contribution if the new t MS ¼ 1000 GeV, MD ¼ 1000 GeV; BMP2: y1 ¼ y2 ¼ 0.35, 1 2 Yukawa couplings y and y are not much larger than the MS ¼ 1000 GeV, MD ¼ 1000 GeV; BMP3: y1 ¼ y2 ¼ 0.4, top Yukawa. So in this work, we do not take into account MS ¼ 1000 GeV, MD ¼ 1000 GeV; BMP4: y1 ¼ y2 ¼ 0.6, these two-loop contributions from the new particles in the MS ¼ 1000 GeV, MD ¼ 1000 GeV; The superscript indicates SDFDM model. For similar reasons, we do not consider that the NNNLO pure QCD effects are also included. BMPs two-loop contributions of new fermions to the β function. means benchmark points. More detailed analysis on this aspect, in particular for the Initial values in the MS scheme for RGE running case with very large Yukawa coupling, is outside the scope μ λ of the present article. ¼ Mt yt g2 gY
SMLO 0.12917 0.99561 0.65294 0.34972 MS * C. Running parameters in the scheme SMNNLO 0.12604 0.93690 0.64779 0.35830 BMP1 * To study the vacuum stability of a model at high energy SDFDMNLO 0.12549 0.93526 0.64573 0.35752 scale, we need to know the value of coupling constants at BMP2 * SDFDMNLO 0.12554 0.93368 0.64574 0.35630 low energy scale and then run them to the Plank scale BMP3 * SDFDMNLO 0.12586 0.93269 0.64573 0.35553 according to RGEs. To determine these parameters at low 4 * SDFDMBMP 0.13126 0.92744 0.64573 0.35144 energy scale, the threshold corrections must be taken into NLO account. In this article we work with the modified minimal subtraction (MS) scheme and use the strategy in [17,20] to TABLE III. y1 ¼ y2 ¼ 0.35 for all three cases with different evaluate one-loop threshold corrections and determine the * initial values for RGE. The details of the corrections are masses of the new particles. The superscript indicates that the NNNLO pure QCD effects are also included. summarized in Appendix A. Using these results, we find coupling constants in the MS scheme at μ ¼ Mt scale Effects of different masses on initial values which are different for the SM and for the SDFDM model. μ ¼ M λ y g2 g We list some of the results in Table II. Both the change of t t Y M M 800 GeV 0.12564 0.93402* 0.64599 0.35650 the Yukawa couplings y1 2 and the change of the mass term S ¼ D ¼ ; 1000 * have an effect on the corrections. We can see in Tables II MS ¼ MD ¼ GeV 0.12554 0.93368 0.64574 0.35630 M ¼ M ¼ 1200 GeV 0.12546 0.93340* 0.64552 0.35613 and III that changing the mass scale of dark matter particles S D
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λ λ λ FIG. 1. (a) ðtÞ up to MPl for the SM and for various Yukawa couplings in the SDFDM model. (b) Running ðtÞ and eff ðtÞ up to MPl scale for the SDFDM model.
We can see the evolution of λðtÞ both in the SM and in where Scl is the Euclidean action of bounce solution and At λ ϕ the SDFDM model in Fig. 1(a). We see that the min in the is the quantum correction. For fluctuation of the field, At SDFDM model, the minimum of λðtÞ in the RGE running, is given as is negative and is more negative than in the SM. This 2 0 −∂2 00 ϕ −1=2 indicates that in the SDFDM model the EW vacuum is Scl det ½ þ U ð BÞ At ¼ 2 2 00 ; ð27Þ unstable and the lifetime of the EW vacuum could be much 4π det½−∂ þ U ðϕ0Þ shorter owing to new physics effects. The greater the Yukawa couplings y1 and y2, the greater the destabilization where ∂2 is operated on Euclidean space and det0 the effects of the SDFDM model. determinant omitting the zero mode contribution. ϕ0 is the λ λ As shown in Eq. (24), eff differs from . In the SM, the field value in the false vacuum which can be taken as zero difference λ − λ is always positive and negligible near eff as an approximation. ϕB refers to the spherical symmetric the Planck scale as shown in Ref. [12]. The situation is bounce solution to the Euclidean field equation. ϕB satisfies different in the SDFDM model. As we can see in Fig. 1(b), λ − λ is not negligible in the SDFDM model. In fact, λ 2 eff eff 2 d ϕB 3 dϕB 4ΓðtÞ −∂ ϕ þ U0ðϕ Þ¼− − þ U0ðϕ Þ¼0; ð28Þ is suppressed by the e factor in Eq. (24) which comes B B dr2 r dr B from the contribution of the anomalous dimension. As Λ 0 we can see, the instability scale I, the energy scale at where U means derivative of U withffiffiffi respect to the field. In λ λ p which effðtÞ or ðtÞ becomes zero, is larger when deter- the case under consideration, ϕ= 2 is the CP-even neutral λ mined by effðtÞ. This is the case both in the SM and in the component of the Higgs doublet in the SM. If there are SDFDM model. other particles coupled to the bounce field, their contribu- tions to the determinants should also be taken into account, III. VACUUM STABILITY AND LIFETIME as happens in the SM and in the singlet-doublet fermion OF THE VACUUM extension models considered in this article. λ 4 For a potential UðϕÞ¼4 ϕ with a negative λ, the As we have seen in the last section, RG improvement to calculation leads to [8] the effective potential can be quite significant in the SDFDM model. We need to consider the effects of RG- 8π2 improved effective potential in the calculation of the Scl ¼ 3 λ : ð29Þ vacuum decay rate. The decay rate of the false vacuum j j can be computed by finding a bounce solution to the field In the SM, there is a mass of the Higgs field. The Higgs equations in Euclidean space [5–7]. For a potential UðϕÞ, mass can be safely neglected in this calculation because the the decay rate per unit time per unit volume, Γ , can be t bounce solution is dominated by the behavior at large field expressed as values, so that the potential can be written approximately as ϕ4 λ − a form. In quantum theory, is a quantity running with Γ Scl t ¼ Ate ; ð26Þ energy scale. To simplify calculation, λ can be taken at a
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λ 2 2 sufficiently large energy scale M so that ðMÞ is negative 2Γ 8π 2 8π 8π2 λ Scl ¼ e Z2 × ˜ 2Γ ¼ðZ2Þ ˜ : ð35Þ and varies slowly with energy scale. So Scl ¼ =j ðMÞj 3jλje =Z2 3jλj in this case. It has been shown that this scale dependence of S in the false vacuum decay rate is canceled ΓðtÞ cl Scl depends on Z2 but is independent of the e factor. when taking into account one-loop correction from the Similar to the case of obtaining Eq. (29), running param- determinant [8,9]. eters in Eqs. (34) and (35) are understood to be at an To fully take into account quantum corrections, we need arbitrary large energy scale M. The leading dependence on to consider the effective action. As long as the field varies M in the decay rate would be canceled by including slowly with space and time, we can compute the effective quantum correction from the determinant, similar to analy- action using derivative expansion [1]. Neglecting terms sis in Refs. [8,9]. 2 with a higher derivative, we can write the effective action in Similarly, one can find that the ðSclÞ factor in Eq. (27), Euclidean space for external field ϕ as which comes from the zero mode contribution, becomes 2 ˜ 2Γ 2 Z ½8π =ð3jλje =Z2Þ . The ratio of determinants in Eq. (27) 1 ϕ 4 ∂ ϕ 2 ϕ ϕ 0 − 2Γ ∂2 3λ˜ 4Γϕ˜2 − 2Γ ∂2 −1=2 Seff½ ¼ d x ð μ Þ Z2ð ÞþVeffð Þ : ð30Þ becomes jdet ½ e Z2 þ e B = det½ e Z2 j 2 0 −∂2 3 λ˜ 2Γϕ˜ 2 −∂2 −1=2 which equals to jdet ½ þ ð =Z2Þe B =det½ j × 2Γ 2 2 ðe Z2Þ when including effects omitting four zero modes. Z2 can be obtained from the p terms in the Feynman Γ ˜ It is easy to see that if taking ϕB ¼ e ϕB the nonzero diagrams as shown in Appendix C. It is renormalized to 2 ˜ 2Γ ˜2 ˜ eigenvalues of operator −∂ 3 λ=Z2 e ϕ for ϕ sat- 2 ϕ 0 1 þ ð Þ B B make Z ð ¼ Þ¼ which makes the kinetic term going 2 back to the standard form when there is no external field. isfying Eq. (34) would be the same of the operator −∂ þ 3 λ˜ ϕ2 ϕ The results in the ’t Hooft-Feynman gauge are summarized ð =Z2Þ B for B satisfying in Table IX for the SM, and in Table X for new − ∂2ϕ λϕ˜ 3 0 contributions in the SDFDM model. In the large ϕ limit, Z2 B þ B ¼ : ð36Þ we can simplify the result. We obtain Z2 for the SM in Eq. (C14), and Z2 for the SDFDM model in Eq. (C16).As So eventually we find that the decay rate is again expressed we can see, the explicit dependences on ϕ are canceled in by Eq. (26) but with Scl expressed by Eq. (35) and with these results. 2 0 −∂2 3 λ˜ ϕ2 −1=2 RG improvement of the kinetic term can be studied S det ½ þ ð =Z2Þ B A ¼ cl ; ð37Þ similar to the effective potential. The kinetic term in the t 4π2 det½−∂2 effective action is the one-particle irreducible self-energy Γ2. It satisfies the RG equation in which ϕB satisfies Eq. (36). We see that the final result ΓðtÞ ΓðtÞ depends on Z2 but does not depend on e . The factor e ∂ ∂ μ β − 2γ Γ ϕ 0 comes from the wave function renormalization but can be ∂μ þ i ∂λ 2ð Þ¼ : ð31Þ i associated with an arbitrariness in relating ϕ with a renormalization scale. So it is not surprising to see that The equation can be solved in a way similar to solving the physical result does not depend on it. One can actually V ðϕÞ. Solving this equation gives rise to Z2ðϕ;tÞ with all eff redefine, from the very beginning, the external field ϕ of the parameters λi in Z2ðϕÞ substituted by λiðtÞ and with 2ΓðtÞ Euclidean action in Eq. (32) in the path integral and arrive an e factor in the kinetic term. So we arrive at an at this conclusion. Euclidean action, Note that the idea that physical quantities should not Z depend on eΓ has been expressed in [11]. In this article, 1 λ˜ 4 2ΓðtÞ 2 4ΓðtÞ ðtÞ 4 S ¼ d x e Z2ðϕ;tÞ ð∂μϕÞ þ e ϕ ; ð32Þ the authors redefine the field and introduce canonically 2 4 normalized field in effective action. In our convention, the ϕ ϕ canonically normalized fieldffiffiffiffiffi can is related to with the where λ˜ is only different from Eq. (24) by a factor e4ΓðtÞ, ϕ ϕ Γp ϕ equation d can=d ¼ e Z2. The solutionffiffiffiffiffi of canffiffiffiffiffican be Γp Γp that is found approximately as ϕ e Z2ϕ γe Z2ϕ in can ¼ pffiffiffiffiffiþ 1 X which the derivative with respect to Z2, which gives λ˜ λ κ2 κ 2ΓðtÞ − ðtÞ¼ ðtÞþ 2 Ni i ðtÞðlog iðtÞe ciÞ: ð33Þ the contribution of higher order, has been neglected. As can ð4πÞ i be found numerically, the anomalous dimension γ is at most The Euclidean equation of bounce solution becomes a few of a thousand in high energy scale if the new Yukawa couplings areffiffiffiffiffi not too large. Consequently, we can take 2 3 2Γ Γp − ∂ ϕ˜ λ˜ϕ˜ ðtÞ 0 ϕ ¼ e Z2ϕ approximately in high energy scale, which Z2 B þ Be ¼ : ð34Þ can pffiffiffiffiffi agrees with the result in [11] up to the factor Z2. The λ From the bounce action in Eq. (29), one can immediately coupling can, introduced in [11] using the canonically ˜ 2 deduce that the bounce action becomes normalized field, can be found approximately as λ=ðZ2Þ in
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FIG. 2. (a) The behavior of Z2 at large energy scale in the SM; (b) The behavior of Z2 at large energy scale with y1 ¼ y2 ¼ 0.35 and MS ¼ MD ¼ 1000 GeV in the SDFDM model. our case. This agrees with the result in (35) which also In vacuum stability analysis, we call the vacuum stable if ˜ 2 ϕ λ˜ 0 depends on λ=ðZ2Þ . So the real quartic coupling which the potential at large keeps positive. This requires > λ λ λ˜ λ˜ 0 controls physical quantities is can, not eff, or just if Z2 is for an energy scale up to the Planck scale. If < at an close to one. As will be shown in detail later, Z2 is indeed energy scale but with P0 < 1, it means that the lifetime of close to one and the difference between λ˜ and λ in the SM is the false vacuum is greater than the age of the Universe. In also very small at high energy scale. In the SDFDM model, this case we call the vacuum metastable. Other scenarios λ˜ − λ λ − λ can be similarly defined. In summary, we list them as could be smaller than eff shown in Fig. 1(b). λ˜ − λ follows: However, can still be significant in some cases, as will λ˜ 0 μ (i) stable: > for
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˜ ˜ FIG. 3. (a) Comparison between λ and λ in the SM. (b) Comparison between λ and λ with y1 ¼ y2 ¼ 0.25 and MS ¼ MD ¼ 1000 GeV in the SDFDM model.
Table IV. This means that the one-loop corrections to comparison, we also list the results just using λðtÞ. We can ˜ effective potential have little effects on the tunneling prob- see that using λðtÞ and Z2 in the effective action leads to ability in the SM. In the SDFDM model, the situation can be some differences in the probability of false vacuum decay. different. As can be seen in Fig. 3(b), λ˜ðtÞ and λðtÞ at high For the case of the SM, we can see that the lifetime of the EW energy scale are not as close as in the SM. In this plot, λ˜ t and vacuum computed using effective action is slightly longer ð Þ ˜ λðtÞ all approach their minima before the Planck scale. But than that computed just using λðtÞ although λðtÞ and λðtÞ are their values at the minima and the energy scales of the very close at high energy scale. This is caused mainly by the 2 minima are not as close as in the SM, as can be seen in presence of Z2 in the effective action. In the SM, the ðZ2Þ Table IV.InFig4, we give more plots with larger y1 and y2.In term in Eq. (35) is about 1.02 which makes Scl slightly larger these cases, the difference between λ˜ðtÞ and λðtÞ is more and leads to a smaller decay rate. In the SDFDM model, the λ˜ λ significant. The larger the Yukawa coupling y1 and y2,the difference between and is significant, and the Z2 factor larger the difference. We can see that the difference between λ˜ increases with the increase of the Yukawa couplings y1 and y2. Therefore, both the Z2 factor and the increasing value of and λ in Fig. 4(b) is not as significant as the difference λ˜ − λ makes the lifetime calculated using effective action between λ and λ in Fig. 1(b). However, λ˜ − λ is still eff longer than that computed just using λ. significant in this case. In these cases in Fig. 4,bothλ˜ðtÞ and λ In Fig. 5, we compare the two ways of obtaining the ðtÞ have no minimum for energy scale below the Planck tunneling probability. The green (blue) region indicates that Λ scale. The energy scale of bounce, B, is chosen as the Planck the EW vacuum is metastable (unstable), and the red region scale for these two cases. We note that the positive sign of ˜ means that the EW vacuum is nonperturbative. We find that λ − λ shown in Fig. 4 means that the lifetime calculated using the one-loop effect on effective action slightly enlarges the ˜ λ in these plots is longer than that computed by using λ. parameter space for the vacuum to be metastable. In Table IV, we list more numerical results for the SM and The parameter space of the singlet-doublet fermion for some benchmark points in the SDFDM model. As a dark matter model is constrained by phenomenological
TABLE IV. The results computed by using λðtÞ and λ˜ðtÞ are presented. Three benchmark models are BPM1 (y1 ¼ y2 ¼ 0.25;MS ¼ MD ¼ 1000 GeV), BPM2(y1 ¼ y2 ¼ 0.35;MS ¼ MD ¼ 1000 GeV), BPM3(y1 ¼ y2 ¼ 0.4;MS ¼ MD ¼ 1000 λ λ λ˜ λ˜ μ GeV). min is the minimal value of the running . min is the minimal value of the running . min is the energy scale when ˜ minimal value of λ or λ is achieved. P0 represents the EW vacuum decay probability.
Result with λðtÞ Result with λ˜ðtÞ in effective potential λ μ P λ˜ Λ μ P min log10ð min=GeVÞ log10ð 0Þ min Z2ð BÞ log10ð min=GeVÞ log10ð 0Þ SM −0.0148 17.46 −535.34 −0.0150 1.0116 18.07 −543.35 BMP1 −0.0176 17.60 −413.72 −0.0165 1.0152 18.23 −474.68 BMP2 −0.0406 MPl −38.98 −0.0346 1.0182 MPl −99.73 BMP3 −0.0661 MPl Unstable −0.0539 1.0231 MPl Unstable
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˜ FIG. 4. Comparison between λðtÞ and λðtÞ in the SDFDM model. (a) y1 ¼ y2 ¼ 0.35 and MS ¼ MD ¼ 1000 GeV. (b) y1 ¼ y2 ¼ 0.4 and MS ¼ MD ¼ 1000 GeV.
4 FIG. 5. Status of the EW vacuum in the y1 − y2 plane with MS ¼ MD ¼ 1000 GeV. The left panel is given by using the ϕ potential and the running λðtÞ, and the right panel is computed by using effective action and λ˜ðtÞ. The green, blue, and red regions indicate that the EW vacuum is metastable, unstable and nonperturbative. considerations of dark matter, such as the direct detection XN XQ XN L ψ¯ = ψ ¯ ∂ − ψ¯ ˜ ψ and the constraint from the dark matter relic density. It is general ¼ niD n þ Sqi=Sq LnðMDÞnn Rn found in [16] that the dark matter Yukawa couplings must n¼1 q¼1 n¼1 −3 be very small, i.e., jy1j; jy2j ≲ 4 × 10 , and that the masses XQ ≲733 − ¯ ˜ of dark matter particle are constrained to be M χ1 GeV SLqðMSÞqqSRq − ≲ 0 1 q¼1 and jM χ2 M χ1 j=M χ1 . . The vacuum stability analy- sis of this model does not give a constraint on the parameter XN XQ − ψ¯ ˜ ψ¯ ˜ space stronger than these phenomenological constraints. ðYRnq LnHSRq þ YLnq RnHSLqÞþH:c:; n¼1 q¼1 IV. GENERAL SINGLET-DOUBLET FERMION ð39Þ EXTENSION MODEL A more general singlet-doublet fermion extension of the where we have chosen to work in the base that the mass ˜ ˜ SM can be considered. In general, we can add N copies of matrices MD and MS are diagonal and real. SU(2) doublet fermions ψ Ln;Rnðn ¼ 1; …;NÞ and Q copies After the EW symmetry breaking, the mass matrix of the 1 … ψ ψ − of singlet fermions SLq;Rqðq ¼ ; ;QÞ. The relevant charged components of Ln;Rnð Ln;RnÞ are not changed. ψ − χ− Lagrangian can be written as We simply denote Ln;Rn as Ln;Rn. The N copies of neutral
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0 χ0 component of ψ Ln;Rnðψ Þ and Q copies of SLq;Rq using two unitary matrices UL and UR with ¼ Ln;Rn 0 0 0 T χ ; χ ; …; χ T M get a mass term. Introducing SL;R ¼ðSL1;R1; …;SLQ;RQÞ ð 1 2 NþQÞ , the mass matrix can be diagonal- T and ψ L;R ¼ðψ L1;R1; …; ψ LN;RNÞ , we can write the mass ized and becomes term as d † M ¼ diagfM χ0 ;Mχ0 ; …;Mχ0 g¼U MUR; ð43Þ 1 2 NþQ L SR ð S¯ ψ¯ 0 ÞM þ H:c: ð40Þ L L ψ 0 0 R where M χ0 is the mass of the χ field. The interaction i i Lagrangian of χ0 and the CP-even neutral Higgs field h is with the mass matrix M given as i obtained as ! M˜ pvffiffi Yþ S 2 L ΔL −χ0 χ0 − χ0 † χ0 M ¼ : ð41Þ h ¼ YPR h Y PL h; ð44Þ pvffiffi ˜ 2 YR MD where Y is the matrix of Yukawa coupling Here Y and Y are the N × Q matrices of Yukawa L R coupling given in (39). 1 0 Yþ ffiffiffi þ L Performing a field transformation Y ¼ p U UR: ð45Þ 2 L Y 0 R SL;R U χ0 42 The interaction Lagrangians of χ0, χ− and the gauge ψ 0 ¼ L;R L;R ð Þ i n L;R bosons become
XN g2 þ 0 μ − þ g2 − μ 0 − ΔL ¼ pffiffiffi ðU Þ χ γ P χ Wμ þ pffiffiffi ðU Þ χ γ P χ Wμ W;Z 2 L i;nþQ i L n 2 L nþQ;i n L i n¼1 XN 2θ g2 þ 0 μ 0 − g2 cos w − U U χ γ P χ Zμ L → R χ −g2 θ =Aμ − =Zμ χ : þ 2 θ ð L Þi;nþQð LÞnþQ;j i L j þð Þ þ n sin w 2 θ n ð46Þ cos w n¼1 cos w
Here θw is the Weinberg angle. B. Model II In numerical analysis we consider three typical models. In this model we add N copies of SU(2) doublet fermions ψ ψ 0 ψ − T L;R ¼ð L;R; L;RÞ and only one copy of singlet fermion A. Model I SL;R. The doublet fermions are all coupled with the only ˜ This model includes N copies of SU(2) doublet fermions singlet. We assume that MD are proportional to the unit and N copies of singlet fermions. We assume the mass matrix and all generations of doublet fermions couple with ˜ ˜ matrix matrices MS and MD are proportional to the unit the singlet fermion with the same strength: matrix. We also assume the Yukawa couplings YL and YR are diagonal and are proportional to the unit matrix. The XN L ψ¯ = ψ ¯ ∂ − ψ¯ ψ − ¯ relevant Lagrangian is model II ¼ ð niD n þ Si=S MD n n MSSSÞ n¼1 XN XN ˜ ˜ − ðy1ψ¯ HS þ y2ψ¯ HS ÞþH:c: ð48Þ L ¼ ðψ¯ iD= ψ þ S¯ i=∂S − M ψ¯ ψ − M S¯ S Þ Ln R Rn L model I n n n n D n n S n n n¼1 n¼1 XN − ψ¯ ˜ ψ¯ ˜ After EW symmetry breaking, the mass matrix M is ðy1 LnHSRn þ y2 RnHSLnÞþH:c: ð47Þ obtained as n¼1 0 1 y2ffiffiv y2ffiffiv MS p p This model basically introduces N generations of singlet- B 2 2 C B y1ffiffiv 0 C doublet fermions and there are no couplings between B p MD C B 2 C generations. So the mass matrix can be diagonalized in M ¼ B . . . . C: ð49Þ B . . . . C the same way as in Eq. (2) for each generation and there are @ . . . . A N copies of neutral fermions χ1 and χ2 in the diagonalized y1v pffiffi 0 M base with masses given in (5) and (6). 2 D
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In a suitable base, the singlet S can be considered TABLE V. Threshold corrections to couplings when y1 ¼ y2 ¼ 0 25 1000 * coupled only to one of the linear combinations of ψ , . for model I, MD ¼ MS ¼ GeV. The superscript P pffiffiffiffi n 1ffiffiffi N indicates that the NNNLO QCD effects are included. i.e., Ψ ¼ p 1 ψ n, with effective couplings Ny1 and pffiffiffiffi N n¼ Ny2 in the new base. Other orthogonal linear combina- Threshold effects for different N in model I tions of ψ do not couple to the singlet fermion. So the n μ ¼ Mt λ yt g2 gY mass matrix can be diagonalized to a form N ¼ 2 0.12495 0.93361* 0.64367 0.35859 5 * d N ¼ 0.12330 0.92868 0.63750 0.35902 M ¼ diagfM χ0 ;Mχ0 ;M ; …;M g; ð50Þ * 1 2 D D N ¼ 7 0.12221 0.92539 0.63338 0.35932 where the M 0 and M 0 are the masses of the two neutral χ1 χ2 0 0 fields, χ1 and χ2, which couple to the neutral Higgs field. D. Vacuum stability in general singlet-doublet We can obtain fermion extension models qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 In this section, we study the vacuum stability in the three 2 − 2 − 4 2 M χ0 ¼ TN TN DN ; ð51Þ models just introduced using RG-improved effective 1 2 action. For model I, we can write down immediately the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 contribution to the RG-improved effective potential follow- 2 2 − 4 2 M χ0 ¼ TN þ TN DN ; ð52Þ ing Eq. (21). We get 2 2 X i 2 2 2 2 2 2 2 2 −1 M χ ϕ;t N N N − Ext ð Þ ni 4 i ð Þ where TN ¼ MS þ 2 y1v þ MD þ 2 y2v , DN ¼ 2 y1y2v V1 ðϕ;tÞ¼N M χ ðϕ;tÞ ln − 3=2 : 64π2 i μ2ðtÞ MSMD. i ð55Þ C. Model III For models II and III, the RG-improved effective potential In the third model, we consider extending the SM by can be obtained by simply substituting the neutral fermions adding N copies of singlet fermions SL;R with only one masses (51) and (52) into Eq. (21). copy of SU(2) doublet fermion. The singlet fermions are ˜ Threshold corrections to couplings in the general singlet- all coupled with the only doublet. We assume that MS are doublet fermion extension model are given in Appendix D. proportional to the unit matrix and all generations of singlet In Tables V–VII, we show some numerical results of fermions couple with the doublet fermion with the same couplings in the MS scheme for the three models shown strength. above. New contributions to the Z2 factor in three models The mass matrix is given as are shown Appendix D. The one-loop β functions in the 0 1 three models are given in Appendix D1, D2, D3. M 0 yp2ffiffiv B S 2 C B C B . . . C . . . y2ffiffiv TABLE VI. Threshold corrections to couplings when y1 ¼ B . . . p C * B 2 C y2 ¼ 0.2 for modelII, M ¼ M ¼ 1000 GeV. The superscript M ¼ B C: ð53Þ D S B . C indicates that the NNNLO QCD effects are included. B 0 . C @ MS A Threshold effects for different N in model II yp1ffiffiv yp1ffiffiv 2 2 MD μ ¼ Mt λ yt g2 gY * Similar to the case in model II, the doublet ψ can be N ¼ 2 0.12545 0.93480 0.64367 0.35686 5 * considered coupled only to one of the linear combinations N ¼ 0.12644 0.93164 0.63750 0.35340 P pffiffiffiffi 7 * 1ffiffiffi N N ¼ 0.12835 0.92954 0.63714 0.34576 of Sq, i.e., p 1 Sq, with effective couplings Ny1 and pffiffiffiffi N q¼ Ny2 in a suitable base. Other orthogonal linear combi- nations of Sn do not couple to the doublet fermion. So the TABLE VII. Threshold corrections to couplings when y1 ¼ 0 2 1000 * mass matrix can be diagonalized to a form y2 ¼ . for model III, MD ¼MS ¼ GeV. The superscript indicates that the NNNLO QCD effects are included. d M ¼ diagfM χ0 ;Mχ0 ;M ; …;M g: ð54Þ 1 2 S S Threshold effects for different N in model III
μ ¼ Mt λ yt g2 gY The M χ0 and M χ0 are the masses of the two neutral fields, 1 2 0 0 N ¼ 2 0.12545 0.93479* 0.64573 0.35809 χ1 and χ2, which couple to the neutral Higgs field. The N ¼ 5 0.12644 0.93164* 0.64573 0.35563 expressions of M χ0 and M χ0 are the same as in (51) * 1 2 N ¼ 7 0.12835 0.92954 0.64573 0.35400 and (52).
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λ λ FIG. 6. (a) ðtÞ up to MPl for different values of N in model I. The value of Yukawa couplings y1 and y2 are 0.25. (b) Running ðtÞ and λ˜ 2 ðtÞ up to MPl scale for N ¼ , 5, 7 in model I.
We can see the evolution of λðtÞ in three models in lines, i.e., and for N ¼ 5, 7, the dashed line for the stable Figs. 6(a), 7(a) and 8(a). Here we choose y1 ¼ y2 ¼ 0.25 in bound, the solid line in the middle for the metastable bound model I and y1 ¼ y2 ¼ 0.2 in models II and III. We can and the dotted line for the unstable bound. In Fig. 10(a), see that the minimum of λðtÞ decreases as N is increased there are two types of lines for all cases of N ¼ 2, 5 and 7, for these parameters in all these cases. In Figs. 6(b), 7(b) i.e., the solid line for the metastable bound and the dotted and 8(b), we compare the evolution of λðtÞ and λ˜ in line for the unstable bound. all models. The λ˜ is bigger than λ due to the one-loop We can see that in both models I and II the vacuum Coleman-Weinberg–type corrections. The difference becomes stable when y1 and y2 are small and N is large. between λ˜ and λ increases with the increase of N. This is quite different from the result when N is small. In comparison, we can also see that in model III, there are no We study the status of EW vacuum in the y1-y2 plane for different N in the three models. Note that in Figs. 9(a) such regions in the parameter space for which the EW and 9(b), there are two types of lines for N ¼ 2, i.e., the vacuum becomes stable when N is large. This happens solid line for the metastable bound, the dotted line for the because the extra copies of fermion doublets in models I β unstable bound. For N ¼ 5 and 7, there are three types of and II give positive contributions to the functions of g1
λ λ FIG. 7. (a) ðtÞ up to MPl for different values of N in model II. The value of Yukawa couplings y1 and y2 are 0.2. (b) Running ðtÞ and λ˜ 2 ðtÞ up to MPl scale for N ¼ , 5, 7 in model II.
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FIG. 8. (a) λðtÞ up to MPl for different values of N in model III. The value of Yukawa couplings y1 and y2 are 0.2. (b) Running λðtÞ and ˜ λðtÞ up to MPl scale for N ¼ 2, 5, 7 in model III.
FIG. 9. (a) Status of the EW vacuum in the y1-y2 plane for N ¼ 2, 5, 7 in model I. (b) Status of the EW vacuum in the y1-y2 plane for N ¼ 2, 5, 7 in model II. In both cases, the dashed line is the stable bound, the solid line the metastable bound, and the dotted line is the unstable bound.
and g2, as can be seen in (D5), (D6), (D14) and (D15).For that the EW vacuum becomes stable for these parameters λ 0 larger N, this effect drives g1 and g2 running to larger values in model II. On the other hand, min < when y1 ¼ y2 ¼ with faster rate. When y1 and y2 are small, the contribution 0.2 in model II. The results here are consistent with the of larger g1 and g2 can even make the β function of λ results indicated in Fig. 9. When Yukawa couplings λ 0 turning into a positive value, as can be seen in Fig. 10(b).If become bigger, min < and the vacuum become unstable increasing the value of Yukawa coupling, the β function can in both models. be turned into negative again, as can be seen in Fig. 10(b). We note that in Figs. 9(a), 9(b) and 10(a), we can also The impact of these running effects can be seen clearly in find that the metastable region and unstable region become λ Figs. 11(a) and 11(b). In Fig. 11(a), we can see that min > smaller with the increase of N. This occurs because when 0 for y1 ¼ y2 ¼ 0.1 and N ¼ 5, 7. This means that the EW y1 and y2 are bigger, the new physics effects of extra vacuum becomes stable for these parameters in model I. On fermions would dominate the running of λ. More copies of λ 0 0 25 λ the contrary, min < when y1 ¼ y2 ¼ . even for N ¼ extra fermions would make running faster to a negative 5 λ 0 and 7. We can also see in Fig. 11(b) that min > when value. We can similarly compare two ways of obtaining the y1 ¼ y2 ¼ 0.05 and for both N ¼ 5 and 7. This means tunneling probability, as done for the SDFDM model in
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FIG. 10. (a) Status of the EW vacuum in the y1-y2 plane for N ¼ 2, 5, 7 in model III. The solid line is the metastable bound, the dotted line the unstable bound. (b) β function of λ in model I. The solid line is for y1 ¼ y2 ¼ 0.1, the dashed line for y1 ¼ y2 ¼ 0.25.
FIG. 11. (a) λðtÞ up to MPl for different values of Yukawa couplings in model I, The solid lines are for y1 ¼ y2 ¼ 0.1 and the dashed lines for y1 ¼ y2 ¼ 0.25. (b) λðtÞ up to MPl for different values of Yukawa couplings in model II. The solid lines are for y1 ¼ y2 ¼ 0.05 and the dashed lines are for y1 ¼ y2 ¼ 0.2.
Fig. 5. The one-loop effective action again slightly modifies Using the method of derivative expansion, we have the parameter space presented in Figs. 9(a), 9(b) and 10(a). studied the quantum correction to the effective action. We have calculated the renormalization on the kinetic term in the effective action in the case with external field. We have V. CONCLUSION studied the RG improvement of the kinetic term. Using the In summary, we have studied the one-loop Coleman- RG-improved kinetic term and the RG-improved effective Weinberg–type effective potential of the Higgs boson in a potential, we calculate the decay rate of the false vacuum. single-doublet fermion dark matter extension of the SM. We We find that the factor arising from the anomalous have calculated the threshold effect of these fermions in this dimension which appears in the kinetic term and the model beyond the SM and have studied the RG running of effective potential cancels in the decay rate. Taking all parameters in the MS scheme. We have studied the RG these considerations into account, we find that the decay improvement to the effective potential. We have studied the rate of the false vacuum is slightly changed by the effective vacuum stability using the RG-improved effective potential. action.
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We have also done all these studies in general singlet- TABLE VIII. Input values of physical observables used to fix doublet fermion extension models. We perform a numerical the SM fundamental parameters λ, m, yt, g2, and gY . MW, MZ, analysis in three typical extension models with N copies of Mh, and Mt are the pole masses of the W boson, of the Z boson, single-doublet fermions, N copies of doublet fermions and of the Higgs boson, and of the top quark, respectively. Gμ is the μ α 3 N copies of singlet fermions separately. We find that several Fermi constant for decay, and 3 is the SUð Þc gauge coupling μ copies of fermion doublet can make the β function of λ at the scale ¼ MZ in the MS scheme. become positive in some regions of parameter space when Input values of SM observables Yukawa couplings of these extra fermions are small. Consequently, in models with a small value of Yukawa Observables Values couplings and large number of copies of fermion doublet, MW 80.384 0.014 GeV 91 1876 0 0021 the EW vacuum can become stable. For a large value of the MZ . . GeV 125 15 0 24 Yukawa coupling, the EW vacuum can again be turned Mh . . GeV M 173.34 0.76 GeV into metastable or unstable. We also find that the difference t pffiffiffi 2 −1=2 246.21971 0.00006 GeV between Higgs self-coupling λ and λ˜, the effective self- v ¼ð GμÞ α 0 1184 0 0007 coupling after including Coleman-Weinberg type correc- 3ðMZÞ . . tion, becomes larger when the number of copies of singlet fermions or doublet fermions is increased. In the θ μ¯ θ − δθ general singlet-doublet fermion extension models, the ð Þ¼ jfin; ðA2Þ decay rate of the false vacuum is also slightly changed by the effective action. where the subscript fin denotes the finite part of the quantity δθ, obtained after subtracting the terms propor- tional to 1=ϵ þ γ − lnð4πÞ. Difference at two-loop order has ACKNOWLEDGMENTS been neglected in this expression. This work is supported by National Science Foundation The physical parameters which would be used in of China (NSFC), Grant No. 11875130. Eq. (A2),suchasμ2 and λ, the quadratic and quartic couplings in the Higgs potential, the vacuum expect- ation value v, the top Yukawa coupling y , the gauge APPENDIX A: THRESHOLD EFFECT AND t couplings g2 and g of SUð2Þ ×Uð1Þ group, can be PARAMETERS IN THE MS SCHEME Y L Y determined from physical observables, such as the pole 1. General strategy for one-loop matching mass of the Higgs boson ðMhÞ, the pole mass of the top To study the vacuum stability of a model at high energy quark ðMtÞ, the pole mass of the Z boson ðMZÞ,the scale, we need to know the value of coupling constants at pole mass of the W boson ðMWÞ, and the Fermi low energy scale and then run them to the Plank scale constant ðGμÞ. These physical observables are listed according to RGEs. To determine these parameters at low in Table VIII. If knowing the corresponding counter- energy scale, the threshold corrections must be taken into terms in the physical scheme, the MS couplings are account. In this article we use the strategy in [17,20] to then obtained using (A2). For example, if knowing δλ, evaluate one-loop corrections. All the loop calculations are we then obtain λðμ¯Þ in the MS scheme. More details are performed in the MS scheme in which all the parameters explained as follows. are gauge invariant and have gauge-invariant renormaliza- We follow Ref. [20] to fix the notation. We write the tion group equations [24]. classical Higgs potential in bare quantities as A parameter in the MS scheme, e.g., θðμ¯Þ, can be 2 † † 2 obtained from renormalized parameter θ in the physical V ¼ −μ0Φ Φ þ λ0ðΦ ΦÞ ðA3Þ scheme which is directly related to physical observables. The connection between θ and θðμ¯Þ to one-loop order can with θ be found by noting that the unrenormalized 0 is related to 0 1 the renormalized couplings by ϕþ B C Φ @ qffiffi A θ θ − δθ θ μ¯ − δθ ¼ 1 : ðA4Þ 0 ¼ ¼ ð Þ MS; ðA1Þ 2ðϕ1 þ iϕ2 þ v0Þ where δθ and δθ are the corresponding counterterms. By MS λ λ − δλ − δ μ2 μ2 − δμ2 definition δθ subtracts only the divergent part propor- Setting 0 ¼ ;v0 ¼ v v; 0 ¼ , where MS λ μ tional to 1=ϵ þ γ − lnð4πÞ in dimensional regularization , v and are regarded as renormalized quantities, with d ¼ 4 − 2ϵ being the space-time dimension. Since the we write divergent parts in the δθ and δθ counterterms are of the MS − δ same form, θðμ¯Þ can be rewritten as V ¼ VðrÞ V ðA5Þ
055038-15 YU CHENG and WEI LIAO PHYS. REV. D 101, 055038 (2020) with δv is obtained using this expression with δg2 known from other conditions which can be found in Eq. (28a) of [25]. 1 Putting δv, Eqs. (A12) and (A14) into (A8) and (A9), one V ¼ λ ϕþϕ−ðϕþϕ− þ ϕ2 þ ϕ2Þþ ðϕ2 þ ϕ2Þ2 ðrÞ 1 2 4 1 2 can then obtain δλ and δμ2. They are as follows: 1 2 2 þ − 2 2 þ λvϕ1½ϕ þ ϕ þ 2ϕ ϕ þ2λv ϕ ðA6Þ 1 1 2 2 1 δμ2 Π M2 3T=v ; ¼ 2 ½Re hhð hÞþ ðA15Þ and δλ=λ ReΠ M2 T=v =M2 − ReΠ M2 =M2 ¼½ hhð hÞþ h wwð WÞ W 1 2δ δ δλ ϕþϕ− ϕþϕ− ϕ2 ϕ2 ϕ2 ϕ2 2 þ g2=g2; ðA16Þ V ¼ ð Þð þ 1 þ 2Þþ4 ð 1 þ 2Þ 2 2 2 2 þ − δ Π 2 − δ þ½λδv þ vδλ ϕ1½ϕ þ ϕ þ 2ϕ ϕ v=v ¼ Re wwðMWÞ=ð MWÞ g2=g2: ðA17Þ 1 2 1 1 δτ ϕþϕ− ϕ2 δ 2 ϕ2 δτϕ We can get the expressions of the counterterms of the other þ þ 2 þ Mh 1 þ v 1; ðA7Þ 2 2 parameters in a similar way. Ignoring the contribution of higher order, we list the one- where loop results of counterterms as follows: δ 2 3 2δλ 6λ δ − δμ2 1 Mh ¼ v þ v v ; ðA8Þ Gμ 1 1 Tð Þ δð1Þλ ¼ pffiffiffiM2 Δrð Þ þ þReΠ ðM2Þ ; ðA18Þ 2 h 0 M2 v hh h δτ ¼ v2δλ þ 2λvδv − δμ2: ðA9Þ h 1 2 2 ð1Þ Gμ = ReΠ M Δr ð1Þ ffiffiffi 2 ttð t Þ 0 v is determined at tree level by Gμ as shown in Table VIII. δ yt ¼ 2 p Mt þ ; ðA19Þ 2 2 2 In order to determine δλ, δv and δμ we need three Mt δτ δτϕ constraints. The strategy is to adjust so that the v 1 ffiffiffi 2 p 1 2 ReΠ M 1 term in Eq. (A7) cancels the tadpole diagrams. Calling iT δð1Þ 2 = wwð WÞ Δ ð Þ g2 ¼ð GμÞ MW 2 þ r0 ; ðA20Þ the sum of the tadpole diagrams with the external legs MW extracted, we have the condition qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi δð1Þ 2 1=2 2 − 2 gY ¼ð GμÞ MZ MW δτ ¼ −T=v: ðA10Þ Π 2 − Π 2 Re zzðMZÞ Re wwðMWÞ Δ ð1Þ × 2 − 2 þ r0 ; A second constraint is conveniently obtained by demanding MZ MW 1 ϕ2 that the coefficient of the term proportional to 2 1 in VðrÞ be ðA21Þ the physical mass of the Higgs boson. So we have
2 2 where superscripts 1 in these equations indicate that they 2λ 1 Mh ¼ v ðA11Þ ð Þ are results at one-loop order. Δr0 in the above equations can be written as a sum of several terms [17], and δM2 is fixed by condition of on-shell renormaliza- h ffiffiffi tion, i.e., ð1Þ p 1 1 A 2 1 Δ ð Þ ð Þ − WW Bð Þ Eð1Þ r0 ¼ VW 2 þ W þ ; ðA22Þ δ 2 Π 2 M Gμ Mh ¼ Re hhðMhÞ; ðA12Þ W where Π ðM2Þ is the Higgs boson self-energy evaluated where AWW is the W boson self-energy at zero momentum, hh h B on shell. A third constraint is provided by Eq. (9b) of VW the vertex contribution in the muon decay process, W E Ref. [25]: the box contribution, and a term due to the renormaliza- tion of external legs. They are all computed at zero external 2 2 momentum. Thus we eventually get the MS parameter to δM ¼ ReΠwwðM Þ; ðA13Þ W W one-loop order as follows [15,17]: Π 2 where wwðMWÞ is the W boson self-energy evaluated Gμ λ μ¯ ffiffiffi 2 − δð1Þλ on shell. Recalling that the W-mass counterterm is ð Þ¼p Mh ; ðA23Þ given by [25] 2 fin 1 2 1 Gμ = 2 2 2 μ¯ 2 ffiffiffi 2 − δð1Þ δM ¼ ðv g2δg2 þ g vδvÞ; ðA14Þ ytð Þ¼ p Mt yt ; ðA24Þ W 2 2 2 fin
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FIG. 12. Contributions of extra fermions to the self-energy for (a) the Higgs boson, (b) Z boson, and (d) and (e) W bosons, as well as to χ (c) the tadpole of the Higgs boson. ð1;2Þ are dark sector fermions.
pffiffiffi As singlet-doublet fermions in the SDFDM model 1=2 ð1Þ g2ðμ¯Þ¼2ð 2GμÞ MW − δ g2 ; ðA25Þ fin do not couple to SM leptons, Eq. (A22) can be further qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi simplified as pffiffiffi μ¯ 2 2 1=2 2 − 2 − δð1Þ gYð Þ¼ ð GμÞ MZ MW gY : ðA26Þ fin Δ − AWW r0 ¼ 2 : ðA27Þ MW
2. MS parameters in the SDFDM model Summing over all the loop contributions and using the To determine the initial values of running couplings, we matching conditions, we get coupling constants in the MS 173 use the equations given in the last section. Since the scheme at Mt ¼ GeV energy scale and for the SDFDM threshold corrections have been done to NNLO in the model respectively. SM, we only need to calculate the contribution of extra We summarize here the one-loop corrections to λ from fermions in the SDFDM model. All the relevant Feynman new particles in the SDFDM model by using Eq. (A18).We δð1Þλ δð1Þλ diagrams for computing jfin with extra fermions are write SDFDM in terms of finite parts of the Passarino- listed in Fig. 12. Veltman functions,
Z 2 1 2 1 − 2 − 1 − 2 2 1 − M − xM1 þð xÞM2 xð xÞp A0ðMÞ¼M ln 2 ;B0ðM1;M2;pÞ¼ ln 2 dx: ðA28Þ μ¯ 0 μ¯
The one-loop result is
1 Gμ 2 2 2 δð Þλ ffiffiffi 4 0 − 2 − 4 0 0 jfin ¼ p 2 fyA½ A0ðM χ Þ ðMh M χ0 ÞB0ðM χ ;Mχ ;MhÞ 2ð4πÞ 1 1 1 1 2 2 2 4 0 − 2 − 4 0 0 þ y ½ A0ðM χ Þ ðM M χ0 ÞB0ðM χ ;Mχ ;MhÞ B 2 h 2 2 2 2 2 2 2 þ 2y ½A0ðM χ0 ÞþA0ðM χ0 Þ − ðM − M 0 − M 0 ÞB0ðM χ0 ;Mχ0 ;MhÞ C 1 2 h χ1 χ2 1 2 2 2 2 2 þ 2y ½A0ðM χ0 ÞþA0ðM χ0 Þ − ðM − M 0 − M 0 ÞB0ðM χ0 ;Mχ0 ;MhÞ þ8yCyDM χ0 M χ0 B0ðM χ0 ;Mχ0 ;MhÞg D 1 2 h χ1 χ2 1 2 1 2 1 2 ffiffiffiGμ Gffiffiffiμ 2 ð1Þ þ p ½−4yAM χ0 A0ðM χ0 Þ − 4yBM χ0 A0ðM χ0 Þ þ p M Δr0 j ; ðA29Þ 2ð4πÞ2v 1 1 2 2 2 h fin
055038-17 YU CHENG and WEI LIAO PHYS. REV. D 101, 055038 (2020) where yA;B;C;D has been given in the text following Eq. (9) and 2 2 2 1 M χ0 2 − ð1Þ 2 2 1 M χ 2 2 Δ θ θ 0 − − − r0 ¼ 2 ðsin L þ sin RÞ 2 2 A0ðM χ Þ 2 2 A0ðM χ ÞþM χ þ M χ0 4π − − 1 − − 1 fin ð vÞ M χ M χ0 M χ M χ0 1 1 M 0 M χ− χ1 8 θ θ − − 0 þ sin L sin R 2 2 ðA0ðM χ Þ A0ðM χ ÞÞ − − 1 M χ M χ0 1 2 2 2 M χ0 2 − 2 2 2 M χ 2 2 θ θ 0 − − − þðcos L þ cos RÞ 2 2 A0ðM χ Þ 2 2 A0ðM χ ÞþM χ þ M χ0 − − 2 − − 2 M χ M χ0 M χ M χ0 2 2 M χ0 M χ− 2 8 cos θ cos θ A0 M χ− − A0 M χ0 : A30 þ L R 2 2 ð ð Þ ð 2 ÞÞ ð Þ M χ− − M 0 χ2
λ 3 2 9 2 9 2 2 Plugging Eq. (A29) into Eq. (A23) we obtain at one-loop SM yb yt yb 2 2 g2 g1 β ðy Þ¼ þ þ yτ − 8g3 − − ; order in the SDFDM model. Contributions of extra fer- b ð4πÞ2 2 2 4 4 ð1Þ ð1Þ ð1Þ mions to δ y j , δ g2j and δ g j can be similarly t fin fin Y fin B6 obtained. Plugging them into Eqs. (A24), (A25) and (A26) ð Þ we obtain relevant parameters at one-loop order in the 5 2 9 2 9 2 SM yτ 2 2 yτ g2 g1 SDFDM model. Using these parameters in the MS scheme, β yτ 3y 3y − − ; ð Þ¼ 4π 2 t þ b þ 2 4 4 ðB7Þ we then carry out the calculation of the effective action in ð Þ the MS scheme. 1 9 2 9 2 SM 2 2 2 g2 g1 β ðλÞ¼ 2λ 12λ þ 6y þ 6y þ 2yτ − − ð4πÞ2 t b 2 10 β γ 4 4 2 2 APPENDIX B: ONE-LOOP AND FUNCTION 4 4 4 9g2 27g1 9g2g1 − 6y − 6y − 2yτ : IN THE SDFDM MODEL t b þ 8 þ 200 þ 20 β The function and the anomalous dimension can be ðB8Þ decomposed into two parts: In this article we focus on the SDFDM model with βtotal ¼ βSM þ βSDFDM; γtotal ¼ γSM þ γSDFDM; ðB1Þ Dirac-type mass. Here we show one-loop contributions of new particles in the SDFDM model to the β functions where βSM and γSM are the β function and the anoma- of the SM parameters and the one-loop β functions of new lous dimension in the SM, while βSDFDM and γSDFDM parameters in the SDFDM model. They can be extracted are the contributions from new particles in the using the Python tool PyR@TE2 [21]. They are as follows. β SDFDM model. The functions of the SM parameters receive one-loop The β functions in the SM are known to three-loop [17]. contributions of new particles in the SDFDM model as We list the one-loop results as follows: follows: 1 2 1 41 SDFDM 3 βSM 3 β ðg1Þ¼ g1; ðB9Þ ðg1Þ¼ 2 g1; ðB2Þ 4π 2 5 ð4πÞ 10 ð Þ 1 2 1 19 βSDFDM 3 SM 3 ðg2Þ¼ 2 g2; ðB10Þ β ðg2Þ¼ − g2; ðB3Þ ð4πÞ 3 ð4πÞ2 6 1 βSDFDM 2 2 1 ðyτÞ¼ 2 ðy1 þ y2Þyτ; ðB11Þ SM 3 ð4πÞ β ðg3Þ¼ ð−7Þg3; ðB4Þ ð4πÞ2 1 βSDFDM y y2 y2 y ; ð bÞ¼ 4π 2 ð 1 þ 2Þ b ðB12Þ y 9y2 3y2 9g2 17g2 ð Þ βSM t t b 2 − 8 2 − 2 − 1 ðytÞ¼ 2 þ þ yτ g3 ; ð4πÞ 2 2 4 20 1 SDFDM 2 2 β ðy Þ¼ ðy1 þ y2Þy ; ðB13Þ ðB5Þ t ð4πÞ2 t
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1 SDFDM 4 4 2 2 β ðλÞ¼ ½−2y1 − 2y2 þ 4λðy1 þ y2Þ : ðB14Þ ð4πÞ2
The one-loop β functions of new parameters in the SDFDM model are as follows: 1 5 9 9 SDFDM 3 2 2 2 2 2 2 β ðy1Þ¼ y1 þ y1y2 − g1y1 − g2y1 þ 3y y1 þ 3y y1 þ yτ y1 ; ðB15Þ ð4πÞ2 2 20 4 t b 1 5 9 9 SDFDM 3 2 2 2 2 2 2 β ðy2Þ¼ y2 þ y1y2 − g1y2 − g2y2 þ 3y y2 þ 3y y2 þ yτ y2 : ðB16Þ ð4πÞ2 2 20 4 t b
2 5 2 λ Note here that g1ðg1 ¼ 3 gYÞ, g2, g3 are the gauge couplings, yt, yb, yτ, y1, and y2 are the Yukawa couplings, and is the Higgs quartic coupling. The one-loop anomalous dimension of the Higgs field is 1 9 9 1 total SM SDFDM 2 2 2 2 2 2 2 γ ¼ γ þ γ ¼ g2 þ g1 − 3y − 3y − yτ þ ð−y1 − y2Þ: ðB17Þ ð4πÞ2 4 20 t b ð4πÞ2
¯ 2 ξ¯ ¯ 2 APPENDIX C: RENORMALIZATION OF mC ¼ WmW; ðC3Þ KINETIC TERM IN EFFECTIVE ACTION m¯ 2 ¼ ξ¯ m¯ 2 ; ðC4Þ CZ Z Z We compute effective action of an external field ¯ 2 ¯ 2 ξ¯ ¯ 2 using derivative expansion. As long as the field varies mGþ ¼ mG þ WmW; ðC5Þ slowly with respect to space and time, this is a valid ¯ 2 ¯ 2 ξ¯ ¯ 2 approximation. Keeping derivatives up to second order, mG0 ¼ mG þ ZmZ: ðC6Þ the Euclidean effective action for a neutral scalar ϕ is written as Z 1 ϕ 4 ϕ ∂ ϕ 2 ϕ Seff½ ¼ d x Veffð Þþ2 ð μ Þ Z2ð Þ ; ðC1Þ where Veff is the effective potential. The one-loop result of Veff in the SM in the background Rξ gauge 2 is given in [19]. Z2 can be obtained from the p terms in self-energy Feynman diagrams. We renormalize Z2 to make Z2ðϕ¼0Þ¼1 which means that the kinetic term goes back to the standard form when there is no external field.
1. Feynman rules in background Rξ gauge The Feynman rules with external field ϕ in the SM and in the SDFDM model are given in Figs. 13 and 14. Here, we only list the vertices that we need in the Z2 calculation. We have introduced
¯ 2 − 2 λϕ2 ¯ 2 − 2 3λϕ2 FIG. 13. Propagators for SM fields with external field ϕ in mG ¼ mϕ þ ; mH ¼ mϕ þ ; ðC2Þ ¯ 2 ¯ 2 ¯ ¯ ¯ ϕ background Rξ gauge. mh, mG, mW, mZ, mf are the -dependent 2 masses, which can be defined as m¯ 2 ¼−m2 þλϕ2, m¯ 2 ¼ −m2 þ where mϕ is the mass term in the Higgs potential given in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG ϕ H ϕ 2 1 1 2 2 y 2 3λϕ ¯ ϕ ¯ 0 ϕ ¯ pfffiffi ϕ (10). The other ϕ-dependent masses can be obtained by , mW ¼ 2 g , mZ ¼ 2 g þ g , mf ¼ 2 . mϕ is the substituting the vacuum expectation value v with ϕ. mass term in the Higgs potential, yf is the Yukawa coupling for We define the field-dependent masses of Goldstone the alternative SM fermion. Note here that G and G0 are the bosons and ghost particles as Goldstone bosons, CZ and C are the ghost fields.
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FIG. 14. Vertices with external field ϕ for the SM in background Rξ gauge. m¯ W and m¯ Z are the ϕ-dependent masses as given in Fig. 13. ¯ ¯ ξW and ξZ are the gauge fixing parameters in background Rξ gauge.
2. Z2 factor in the SM Higgs self-energy diagram in Fig. 15. Notations in [26] are used for the integrals calculated in the modified minimal For simplicity, we calculate Z2 in the ’t Hooft-Feynman ¯ ¯ 2 subtraction scheme: gauge with ξW ¼ ξZ ¼ 1. Z2 comes from the p term in the
FIG. 15. Self-energy diagrams contributing to the Z2 factor in the SM. G and G0 are the Goldstone bosons, CZ and C are the ghost fields.
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2 i 2 TABLE IX. p terms from the self-energy diagram in the SM 2 B0ðm1;m2;p Þ 16π which contribute to Z2. Note that in these results we only list the Z d 1 fermion loop contribution from the top quark. μϵ d k ¼ 0 2 2 2 2 ; ðC7Þ ð2πÞd ðk − m Þððk þ pÞ − m Þ 1 18λ2ϕ2 1 ¯ 1 2 (a) 16π2 ð B0ðmHÞÞ 1 2 2 1 (b) 2 ð4λ ϕ B0ðm¯ G ÞÞ i 2 16π 2 B0ðm1;m2;p Þ 1 2λ2ϕ2 1 ¯ 16π (c) 16π2 ð B0ðmG0 ÞÞ Z 1 2 2 1 d (d) 2 4g m¯ B m¯ ϵ d k 1 16π ð W 0ð W ÞÞ ¼ μ ; ðC8Þ 2 0 2π d 2 − 2 2 − 2 (e) 1 4g ¯ 2 1 ¯ ð Þ ðk m1Þððk þ pÞ m2Þ 2 ð 2 m B0ðmZÞÞ 16π cos ðθwÞ Z (f) 1 g2 ¯ 2 1 ¯ 2 ð− 2 m B0ðmC ÞÞ 16π 4 cos ðθwÞ CZ Z (g) 1 − g2 ¯ 2 1 ¯ 16π2 ð 4 m B0ðmC ÞÞ 2 0 1 2 C B0ðm1;m2;p Þ¼B0ðm1;m2ÞþB0ðm1;m2Þ · p 1 g2 2 2 1 0 (h) − −2 ¯ ¯ ¯ − ¯ −2 ¯ − ¯ 16π2 ð 4 Þ½ð mG− þm þ ÞB0ðmG ;mW Þ B0ðmG ;mW Þ 4 W þ þ þ Oðp Þþ ; ðC9Þ 1 g2 2 2 1 0 (i) − −2 ¯ ¯ − ¯ þ ¯ −2 ¯ þ ¯ 16π2 ð 4 Þ½ð mGþ þmW ÞB0ðmG ;mW− Þ B0ðmG ;mW− Þ 2 2 0 1 1 g þg0 2 2 1 0 (j) 2 −2m¯ m¯ B m¯ ; m¯ − 2B m¯ ; m¯ where B0ðm1;m2Þ and B0ðm1;m2Þ can be expressed as 16π ð 4 Þ½ð G0 þ ZÞ 0ð G0 ZÞ 0ð G0 ZÞ 1 2 0 2 1 (k) 2 ð−gt Þ½−B0ðm¯ tÞþ4m¯ t B0ðm¯ tÞ 2 2 16π 2 m1 2 m2 m ln 2 − m ln 2 0 1 1 μ 2 μ B0ðm1;m2Þ¼ þ 2 2 ; ðC10Þ m2 − m1 1 B1 m : 0ð Þ¼6 2 ðC13Þ m2 m 2 2 2 2 1 1 m1m2 ln 2 1 m1 þ m2 m2 2 B0ðm1;m2Þ¼ − : ðC11Þ We list the p terms of each self-diagram in Table IX. 2 2 − 2 2 2 − 2 3 2 ðm1 m2Þ ðm1 m2Þ Summing over all the p term contributions, we obtain the Z2 factor in the SM. Since the RG equation for the kinetic 0 1 term in the effective action can be solved in a way similar to When m1 ¼ m2 ¼ m, B0ðm1;m2Þ and B0ðm1;m2Þ can ϕ 0 1 the solution to Veffð Þ, we can obtain the RG-improved be written as B0ðmÞ and B0ðmÞ. They are expressed as kinetic term by replacing ϕ, μ, λi with ϕðtÞ, μðtÞ and λðtÞ. Their expressions or equations are shown in Eqs. (15) and m2 B0 m −ln ; (17). Taking μðtÞ¼ϕ as mentioned above, we get the RG- 0ð Þ¼ μ2 ðC12Þ improved Z2 factor in the SM for large ϕ field:
1 8λ2 4λ2 2 2 2 02 SM 2 02 ð g þ g Þ Z2 ¼ 1 þ λ þ þ þ ð2g þ g Þ − 16π2 12λ þ 3g2 12λ þ 3ðg2 þ g02Þ 3 24 2 − 4λ 2 4 4λþg 2 2 4 2 4 2Γ 1 2 2Γ 4 1 2 2Γ 2 1 ð g þ g Þlnð 2 Þ 2λ 8λ 4λg g ln λe g e − g ln g e g þ g g þ g − ð þ Þ ð þ 4 Þ ð4 Þ g þ 8π2 16λ3 þ 4λ2 16 8λ þ 2
2 02 2 02 2 02 2 4λþg þg 2 2 2 2 2 2 2 1 −ð4λðg þ g Þþðg þ g Þ Þlnð 2 2 Þ 2λ 0 8λ 0 0 − g þg0 þ g þ g ðg þ g Þþðg þ g Þ 16π2 16λ3 þ 4λ2 16 4λ g2 g02 g2 g02 2 ln λe2Γ 1 g2 g02 e2Γ − g2 g02 2ln 1 g2 g02 e2Γ 2 02 − ð ð þ Þþð þ Þ Þ ð þ 4 ð þ Þ Þ ð þ Þ ð4 ð þ Þ Þ g þ g 8λ þ 2 3 2 2 − yt 2Γ 2 2 16π2 ln 2 e yt þ 3 yt ðC14Þ with Z t ΓðtÞ¼− γðλðt0ÞÞdt0: ðC15Þ 0
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0 0 FIG. 16. Self-energy diagrams contributing to Z2 factor by extra fermions in the SDFDM model. χ1 and χ2 are the new extra fermions in the SDFDM model.
2 TABLE X. p terms from the self-energy diagram contributed 3. Z factor in the SDFDM model by extra fermions in the SDFDM model. Here we define 2 − θ θ − θ θ θ θ The Feynman diagrams in the SDFDM model contrib- A ¼ð y2 cos L sin R y1 sin L cos RÞ, B ¼ y2 cos R sin Lþ 2 y1 sinθR cosθL, C ¼ y2 cos θL cos θR − y1 sin θL sin θR, D ¼ uting to the Higgs self-energy are shown in Fig. 16. p term −y2 sin θ sin θ þ y1 cos θ cos θ . m¯ χ0 , m¯ χ0 are the masses of contributions to Z2 in these diagrams are summarized in L R L R 1 2 2 new particles under the external field ϕ. They are obtained by Table X. Summing over all the p term contributions in substituting v in Eqs. (5) and (6) with ϕ. Tables IX and X, we obtain the Z2 factor in the SDFDM model. In the large ϕ limit, the Z2 factor in the SDFDM 1 2 2 1 0 − 4 ¯ ¯ 0 − ¯ 0 (a) 16π2 ð A Þð m χ0 B0ðm χ Þ B0ðm χ Þ model can be expressed as 1 1 1 1 2 2 1 0 − 4 ¯ ¯ 0 − ¯ 0 (b) 16π2 ð B Þð m χ0 B0ðm χ Þ B0ðm χ Þ 2 2 2 2 2 1 1 y2 y2 2Γ 2 − CD 2 ¯ 0 ¯ 0 ¯ 0 ¯ 0 SDFDM SM (c) 16π2 ð 2 Þð ðm χ m χ B0ðm χ ; m χ Þ Z ¼ Z − ln e þ 1 2 1 2 2 2 16π2 2 3 1 CD 1 (d) 2 ð− Þð2ðm¯ χ0 m¯ χ0 B ðm¯ χ0 ; m¯ χ0 Þ 16π 2 1 2 0 1 2 2 2 y1 y1 2Γ 2 1 2 2 2 1 0 − (e) − D ¯ ¯ ¯ 0 ¯ 0 − ¯ 0 ¯ 0 2 ln e þ ; ðC16Þ 16π2 ð 2 Þ½ðm χ0 þ m χ0 ÞB0ðm χ ; m χ Þ B0ðm χ ; m χ Þ 16π 2 3 1 2 1 2 1 2 1 2 2 2 1 0 (f) − C ¯ ¯ ¯ 0 ¯ 0 − ¯ 0 ¯ 0 16π2 ð 2 Þ½ðm χ0 þ m χ0 ÞB0ðm χ ; m χ Þ B0ðm χ ; m χ Þ 1 2 1 2 1 2 SM where Z2 is given in Eq. (C14).
APPENDIX D: THRESHOLD EFFECT, Z2 AND β FUNCTION IN THE GENERAL SINGLET-DOUBLET FERMION EXTENSION MODEL We summarize here the one-loop corrections to λ from new particles in the general singlet-doublet extension model using Eq. (A18). The one-loop result is
QXþN QXþN 1 Gμ 2 2 2 δð Þλ ffiffiffi 2 þ 0 0 − − − 0 0 jfin ¼ p 2 f YijYji½A0ðM χ ÞþA0ðM χ Þ ðMh M χ0 M χ0 ÞB0ðM χ ;Mχ ;MhÞ 2 4π i j i j i j ð Þ i¼1 j¼1 NXþQ Gμ 2 þ þ 0 0 0 0 ffiffiffi −2 − 2 þ 0 0 þ ðYijYji þ YijYjiÞM χ M χ B0ðM χ ;Mχ ;MhÞg þ p ½ð Yii Yii ÞM χ A0ðM χ Þ i j i j 2 4π 2 i i ð Þ v i¼1 Gμ 1 þ pffiffiffi M2Δrð Þj ; ðD1Þ 2 h 0 fin
Δ ð1Þ where Y has been given in Eq. (45) and the new contribution to r0 jfin is
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XN 1 M χ0 M χ− ð1Þ † † i k Δ 4 − − 0 r0 ¼ 2 ððULÞi;k QðURÞkþQ;i þðURÞi;k QðULÞkþQ;iÞ 2 2 ðA0ðM χ Þ A0ðM χ ÞÞ 4π þ þ − − k i fin ð vÞ 1 M χ M χ0 k¼ k i XN NXþQ † † þ ððULÞi;kþQðULÞkþQ;i þðURÞi;kþQðURÞkþQ;iÞ k¼1 i¼1 2 2 2 M χ0 2M χ− i k 2 2 0 − − − × 2 2 A0ðM χ Þ 2 2 A0ðM χ ÞþM χ þ M χ0 : ðD2Þ − − i − − k k M χ M χ0 M χ M χ0 i k i k i
λ 1 Plugging Eq. (D1) into Eq. (A23) we obtain at one- model I 4 4 2 2 β ðλÞ¼ ½−Nð2y1 þ 2y2Þþ4Nλðy1 þ y2Þ : loop order in the general singlet-doublet extension model. ð4πÞ2 ð1Þ ð1Þ Contributions of extra fermions to δ y j , δ g2j and t fin fin ðD10Þ δð1Þg j can be similarly obtained. Plugging them into Y fin 1 3 9 Eqs. (A24)–(A26) we obtain relevant parameters at one- model I 3 3 2 2 β ðy1Þ¼ y1 þ Ny1 þ Ny1y2 − g1y1 loop order in the general model. Using these parameters in ð4πÞ2 2 20 the MS scheme, we then carry out the calculation of the 9 2 2 2 2 − g 1 3 1 3 1 τ 1 ; effective action in the MS scheme. 4 2y þ yt y þ yby þ y y ðD11Þ In the large ϕ limit, the Z2 factor in model I can be 1 3 9 expressed as model I 3 3 2 2 β ðy2Þ¼ y2 þ Ny2 þ Ny1y2 − g1y2 ð4πÞ2 2 20 Ny2 y2 2 9 model I SM − 2 2 2Γ 2 2 2 2 Z2 ¼ Z2 2 ln e þ − y2 3y y2 3y y2 yτ y2 : 16π 2 3 4 g2 þ t þ b þ ðD12Þ Ny2 y2 2 − 1 1 e2Γ : The one-loop contribution of new particles to the anoma- 16π2 ln 2 þ 3 ðD3Þ lous dimension is 1 The Z2 factor in models II and III can be expressed as model I 2 2 γ ¼ Nð−y1 − y2Þ: ðD13Þ ð4πÞ2 Ny2 Ny2 2 Zmodel II;III ZSM − 2 2 e2Γ 2 ¼ 2 16π2 ln 2 þ 3 2. Model II 2 2 2 − Ny1 Ny1 2Γ 2 ln e þ : ðD4Þ 1 2 16π 2 3 model II 3 β ðg1Þ¼ N g1; ðD14Þ ð4πÞ2 5 β The one-loop functions and the anomalous dimension 1 2 model II 3 of the three singlet-doublet extension models are as β g2 Ng ; ð Þ¼ 4π 2 3 2 ðD15Þ follows. ð Þ 1 βmodel II 2 2 ðyτÞ¼ 2 Nðy1 þ y2Þyτ; ðD16Þ 1. Model I ð4πÞ 1 βmodel II y N y2 y2 y ; 1 2 ð bÞ¼ 4π 2 ð 1 þ 2Þ b ðD17Þ βmodel I 3 ð Þ ðg1Þ¼ 2 N g1; ðD5Þ ð4πÞ 5 1 βmodel II 2 2 1 2 ðytÞ¼ 2 Nðy1 þ y2Þyt; ðD18Þ βmodel I 3 ð4πÞ ðg2Þ¼ 2 Ng2; ðD6Þ ð4πÞ 3 1 model II 2 4 4 2 2 β ðλÞ¼ ½−N ð2y1 þ 2y2Þþ4Nλðy1 þ y2Þ : 1 ð4πÞ2 βmodel I 2 2 ðyτÞ¼ 2 Nðy1 þ y2Þyτ; ðD7Þ ð4πÞ ðD19Þ 1 1 5 9 9 model I 2 2 model II 3 2 2 2 β ðy Þ¼ Nðy1 þ y2Þy ; ðD8Þ β ðy1Þ¼ Ny1 þ Ny1y2 − g1y1 − g2y1 b ð4πÞ2 b ð4πÞ2 2 20 4 1 model I 2 2 2 2 2 β ðy Þ¼ Nðy1 þ y2Þy ; ðD9Þ þ 3y y1 þ 3y y1 þ yτ y1 ; ðD20Þ t ð4πÞ2 t t b
055038-23 YU CHENG and WEI LIAO PHYS. REV. D 101, 055038 (2020) 1 5 9 9 1 model II 3 2 2 2 model III 2 2 β ðy2Þ¼ Ny2 þ Ny1y2 − g1y2 − g2y2 β ðy Þ¼ Nðy1 þ y2Þy ; ðD27Þ ð4πÞ2 2 20 4 t ð4πÞ2 t 3 2 3 2 2 1 þ yt y2 þ yby2 þ yτ y2 : ðD21Þ modelIII 2 4 4 2 2 β ðλÞ¼ ½−N ð2y1 þ 2y2Þþ4Nλðy1 þ y2Þ : ð4πÞ2 The one-loop contribution of new particles to the anoma- ðD28Þ lous dimension is 1 5 9 9 1 βmodel III 3 2 − 2 − 2 γmodel II − 2 − 2 ðy1Þ¼ 2 Ny1 þ Ny1y2 g1y1 g2y1 ¼ 2 Nð y1 y2Þ: ðD22Þ ð4πÞ 2 20 4 ð4πÞ 3 2 3 2 2 þ yt y1 þ yby1 þ yτ y1 ; ðD29Þ 3. Model III 1 5 9 9 model III 3 2 2 2 β ðy2Þ¼ Ny2 þ Ny1y2 − g1y2 − g2y2 1 2 ð4πÞ2 2 2 4 βmodel III 3 ðg1Þ¼ 2 5 g1; ðD23Þ ð4πÞ 2 2 2 þ 3y y2 þ 3y y2 þ yτ y2 : ðD30Þ 1 2 t b model III 3 β g2 g ; ð Þ¼ 4π 2 3 2 ðD24Þ ð Þ The one-loop contribution of new particles to the anoma- 1 model III 2 2 lous dimension is β yτ N y y yτ; ð Þ¼ 4π 2 ð 1 þ 2Þ ðD25Þ ð Þ 1 1 γmodel III ¼ Nð−y2 − y2Þ: ðD31Þ model III 2 2 4π 2 1 2 β ðy Þ¼ Nðy1 þ y2Þy ; ðD26Þ ð Þ b ð4πÞ2 b
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