PHYSICAL REVIEW D 97, 116012 (2018)
Decay rate of electroweak vacuum in the standard model and beyond
So Chigusa,1 Takeo Moroi,1 and Yutaro Shoji2 1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan 2Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa 277-8582, Japan
(Received 26 March 2018; published 14 June 2018)
We perform a precise calculation of the decay rate of the electroweak vacuum in the standard model as well as in models beyond the standard model. We use a recently developed technique to calculate the decay rate of a false vacuum, which provides a gauge invariant calculation of the decay rate at the one- loop level. We give a prescription to take into account the zero modes in association with translational, dilatational, and gauge symmetries. We calculate the decay rate per unit volume, γ, by using an analytic formula. The decay rate of the electroweak vacuum in the standard model is estimated to be 3 þ40þ184þ144þ2 log10γ × Gyr Gpc ¼ −582−45−329−218−1 , where the first, second, third, and fourth errors are due to the uncertainties of the Higgs mass, the top quark mass, the strong coupling constant and the choice of the renormalization scale, respectively. The analytic formula of the decay rate, as well as its fitting formula given in this paper, is also applicable to models that exhibit a classical scale invariance at a high energy scale. As an example, we consider extra fermions that couple to the standard model Higgs boson, and discuss their effects on the decay rate of the electroweak vacuum.
DOI: 10.1103/PhysRevD.97.116012
I. INTRODUCTION vacuum can be evaluated through a rate of bubble nucle- ation in unit volume and unit time as formulated in [26,27]. In the standard model (SM) of particle physics, it has The rate is expressed in the form of been known that the Higgs quartic coupling may become negative at a high scale through quantum corrections, so γ ¼ Ae−B; ð1:1Þ that the Higgs potential develops a deeper vacuum. The detailed shape of the Higgs potential depends on the Higgs where B is the action of a so-called bounce solution, and and the top masses; with the recently observed Higgs mass prefactor A is quantum corrections having mass dimension of ∼125 GeV, it has been known that the electroweak (EW) 4. The bounce solution is an Oð4Þ symmetric solution of vacuum is not absolutely stable if the SM is valid up to ∼1010 1 the Euclidean equations of motion, connecting the two GeV or higher. In such a case, the EW vacuum can vacua. Although the dominant suppression of the decay rate decay into the deeper vacuum through tunneling in quan- comes from B, the prefactor A is also important. This is tum field theory. The lifetime of the EW vacuum has been because of large quantum corrections from the top quarks one of the important topics in particle physics and and the gauge bosons. Thus, it is essential to calculate both cosmology. A and B to determine the decay rate precisely. In the SM, The decay rate of the EW vacuum has been discussed for there are infinite bounce solutions owing to (i) the classical a long time. The calculation of the decay rate at the one- scale invariance at a high energy scale, (ii) the global loop level first appeared in [15] and was also discussed in ð2Þ ð1Þ ð1Þ – symmetries corresponding to SU L × U Y=U EM,as other literature [16 25]. However, there are subtleties in the well as (iii) the translational invariance. For the calculation treatment of zero modes related to the gauge symmetry of the prefactor A, a proper procedure to take account of the breaking which make it difficult to perform a precise and effects of the zero modes related to (i) and (ii) were not well reliable calculation of the decay rate. The lifetime of a understood until recently. In addition, the previous calcu- lations of A were not performed in a gauge-invariant way, 1For the absolute stability of the EW vacuum in the SM, which made the gauge invariance of the result unclear. see [1–14]. Recently, a prescription for the treatments of the gauge zero modes was developed [28,29], based on which a Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. complete calculation of the decay rate of the EW vacuum Further distribution of this work must maintain attribution to became possible. The calculation has been performed by the author(s) and the published article’s title, journal citation, the present authors in a recent publication [25] and also by and DOI. Funded by SCOAP3. [24]. The purpose of this paper is to give a more complete
2470-0010=2018=97(11)=116012(34) 116012-1 Published by the American Physical Society SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) and detailed discussion about the calculation of the decay the quartic coupling constant negative at a high scale. In the rate. In [25], we have numerically evaluated the functional SM, the instability occurs when the Higgs amplitude determinants of fluctuation operators, which are necessary becomes much larger than the EW scale. Since the typical for the calculation of the decay rate. Here, we perform the field value for the bounce configuration is around that scale, calculation analytically; a part of the analytic results was we can neglect the quadratic term in the Higgs potential. first given in [24]. The effects of the zero modes and the In this section, we use a toy model with Uð1Þ gauge modes with the lowest angular momentum are carefully symmetry to derive relevant formulas. The calculation of taken into account, on which previous works had some the decay rate of the EW vacuum is almost parallel to that in confusion. We give fitting formulas of the functional the case with Uð1Þ gauge symmetry; the application to the determinants based on analytic results, which are useful SM case will be explained in the next section. for the numerical calculation of the decay rate. We also provide a C þþpackage to study the electroweak vacuum A. Setup stability (ELVAS), which is available at [30]. In this paper, we discuss the calculation of the decay rate Let us first summarize the setup of our analysis. We of the EW vacuum in detail. We first present a detailed study the decay rate of a false vacuum whose instability is formulation of the calculation of the decay rate at the one- due to an RG running of the quartic coupling constant of a loop level. We derive a complete set of analytic formulas scalar field, Φ. We assume that Φ is charged under the Uð1Þ that can be used for any models that exhibit classical scale gauge symmetry (with charge þ1); the kinetic term invariance at a high energy scale like the SM. Then, as one includes of the important applications, we calculate the decay rate of L ∋ ½ð∂ − ÞΦ †ð∂ − ÞΦ ð Þ the EW vacuum in the SM. We find that the lifetime of the kin μ igAμ μ igAμ ; 2:1 EW vacuum is much longer than the age of the Universe. There, we see that one-loop corrections from the top quark where Aμ is the gauge field and g is the gauge coupling and the gauge bosons are very large although there is an constant, while we consider the following scalar potential: accidental cancellation. It shows the importance of A for the evaluation of a decay rate. We also evaluate the decay VðΦÞ¼λðΦ†ΦÞ2: ð2:2Þ rates of the EW vacuum for models with extra fermions that couple to the Higgs field. In such models, the EW vacuum The quartic coupling, λ, depends on the renormalization tends to be destabilized compared with that of the SM since scale, μ, and is assumed to become negative at a high scale the quartic coupling of the Higgs field is strongly driven to due to the RG effect. As we have mentioned before, we a negative value. (For discussion about the stability of the neglect the quadratic term assuming that λ becomes negative EW vacuum in models with extra particles, see [22,31– at a much higher scale. In this setup, the scalar potential has 54].) We consider three models that contain, in addition to scale invariance at the classical level. In the application to the the SM particles, (i) vectorlike fermions having the same case of the SM, Φ corresponds to the Higgs doublet and λ SM charges as the left-handed quark and the right-handed corresponds to the Higgs quartic coupling constant. down quark, (ii) vectorlike fermions with the same SM Hereafter, we perform a detailed study of the effects of charges as left-handed lepton and right-handed electron, the fields coupled to Φ on the decay rate of the false and (iii) a right-handed neutrino. We give constraints on vacuum. We consider a Lagrangian that contains the their couplings and masses, requiring that the lifetime of the following interaction terms: EW vacuum be long enough. L ∋ κσ2jΦj2 þð Φψ¯ ψ þ Þþ ðΦÞ ð Þ This paper is organized as follows. In Sec. II,we int y L R H:c: V ; 2:3 summarize the formulation for the decay rate at the one- loop level, where we provide an analytic formula for each where σ is a real scalar field, and ψ L and ψ R are chiral field that couples to the Higgs boson. The detail of the fermions [with relevant Uð1Þ charges].2 We take y real and calculation is given in Appendixes A–D. In Sec. III,we κ > 0. We neglect dimensionful parameters that are evaluate the vacuum decay rate in the SM. Readers who are assumed to be much smaller than the typical scale of the interested only in the results can skip over the former bounce. In addition, gauge fixing is necessary to take into section to this section. In Sec. IV, we analyze decay rates in account the effects of gauge boson loops. Following models with extra fermions. Finally, we conclude in Sec. V. [29,55], we take the gauge fixing function of the following form: II. FORMULATION F ¼ ∂μAμ: ð2:4Þ We first discuss how we calculate the decay rate of the EW vacuum. In the SM, the EW vacuum becomes unstable due to the renormalization group (RG) running of the 2We assume that there exist other chiral fermions that cancel quartic coupling constant of the Higgs boson, which makes out the gauge anomaly.
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Then, the gauge fixing term and the Lagrangian of the where h and φ are the physical Higgs mode and the ghosts (denoted as c¯ and c) are given by Nambu-Goldstone (NG) mode, respectively. At the one- loop level, the prefactor can be decomposed as 1 L ¼ F 2 ð Þ ð Þ GF ; 2:5 A h 2ξ A −B ¼ AðσÞAðψÞAðAμ;φÞAðc;c¯Þ −B ð Þ e V e ; 2:13 4D L ¼ −¯∂ ∂ ð Þ FP c μ μc; 2:6 V AðXÞ where 4D is the volume of spacetime, and is the 3 contribution from particle X. Each of the factors has a where ξ is the gauge fixing parameter. form of The bounce solution is an Oð4Þ symmetric object [56,57], and transforms under a global Uð1Þ symmetry. ðXÞ wðXÞ ð Þ DetM Thus, choosing the center of the bounce at r ¼ 0 (with A X ¼ ; ð2:14Þ pffiffiffiffiffiffiffiffiffi McðXÞ r ≡ xμxμ), we can write the bounce solution as Det ð Þ where MðXÞ and Mc X are the fluctuation operators around 1 Φj ¼ pffiffiffi iθϕ¯ ð Þ ð Þ the bounce solution and around the false vacuum, respec- bounce e r ; 2:7 2 tively. Here, wðXÞ ¼ 1 for Dirac fermions and Faddeev- Popov ghosts, and wðXÞ ¼ −1=2 for the other bosonic with our choice of the gauge fixing function, without loss of fields. The fluctuation operator is defined as second θ ϕ¯ generality. Here, is a real parameter. The function, , derivatives of the action: obeys δ2SðXÞ 3 MðXÞδð4Þðx − yÞ¼ ; ð2:15Þ ∂2ϕ¯ ðrÞþ ∂ ϕ¯ ðrÞ − λϕ¯ 3ðrÞ¼0; ð2:8Þ δXðxÞδXðyÞ r r r where the brackets indicate the evaluation around the ∂ ϕ¯ ð0Þ¼0 ϕ¯ ð∞Þ¼0 cðXÞ with boundary conditions r and . For a bounce solution. In addition, M can be obtained from λ negative , we have a series of Fubini-Lipatov instanton MðXÞ with replacing ϕ¯ by the vacuum expectation value at solutions [58,59]: the false vacuum, i.e., ϕ¯ → 0. jλj −1 Since the Faddeev-Popov ghosts do not couple to the ϕ¯ ð Þ¼ϕ¯ 1 þ ϕ¯ 2 2 ð Þ r C Cr ; 2:9 Higgs boson with the present choice of the gauge fixing 8 ð ¯Þ function, Mðc;c¯Þ ¼ Mc c;c . Thus, we have which is parametrized by ϕ¯ (i.e., the field value at the C Aðc;c¯Þ ¼ 1: ð2:16Þ center of the bounce). We also define R, which gives the size of the bounce, as sffiffiffiffiffi B. Functional determinant 8 ≡ ϕ−1 ð Þ For the evaluation of the functional determinants, we first R jλj C : 2:10 decompose fluctuations into partial waves, making use of the Oð4Þ symmetry of the bounce [15]. The basis is The action of the bounce is given by ðΩÞ constructed from YJ;mA;mB , the hyperspherical function on S3 with Ω being a coordinate on S3. The decomposition 8π2 ¼ B ¼ : ð2:11Þ of each fluctuation is given in Appendix A. Here, J 3jλj 0; 1=2; 1; … is a non-negative half integer that labels the total angular momentum in four dimensions, and mA ϕ¯ Notice that the tree level action is independent of C owing and mB are the azimuthal quantum numbers for the A-spin ð4Þ ≃ ð2Þ ð2Þ to the classical scale invariance. and the B-spin of so su A × su B, respectively. Once the bounce solution is obtained, we may integrate The four-dimensional Laplacian operator acts on the hyper- over the fluctuation around it. We expand Φ as spherical function as
1 2 θ ¯ L Φ ¼ pffiffiffi ei ðϕ þ h þ iφÞ; ð2:12Þ −∂2Y ðΩÞ¼ Y ðΩÞ; ð2:17Þ 2 J;mA;mB r2 J;mA;mB with 3 ∂ In the non-Abelian case, one of μ in Eq. (2.6) is replaced by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the covariant derivative. The interaction of the ghosts with the ≡ 4 ð þ 1Þ ð Þ gauge field does not affect the following discussion. L J J : 2:18
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ðΩÞ In addition, YJ;mA;mB is normalized as Here, Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 κ dΩjY j2 ¼ 1: ð2:19Þ ¼ − 1 − 1 − 8 ð Þ J;mA;mB zκ 2 jλj ; 2:28 With the above expansion, the functional determinants sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 can be expressed as ¼ − 1 − 1 − 8 g ð Þ zg 2 jλj ; 2:29 ðXÞ ð Þ Y∞ ðXÞ n DetM X DetM J ¼ J ; ð2:20Þ y ðXÞ ðXÞ pffiffiffi Mc Mc zy ¼ i ; ð2:30Þ Det J¼0 Det J λ MðXÞ where J is the fluctuation operator for each partial where g is the gauge coupling constant, and ΓðzÞ is the ðψÞ ¼ð2 þ 1Þð2 þ 2Þ wave labeled by J. In addition, nJ J J gamma function. ðXÞ ¼ð2 þ 1Þ2 for fermions, and nJ J for the others. The explicit forms of the fluctuation operators are shown in C. Zero modes Appendix A. Using a theorem [29,60–63], the ratio of the In the calculation of the decay rate of the EW vacuum functional determinants can be calculated as with the present setup, there show up zero modes in association with dilatation, translation, and global trans- ð Þ MðXÞ Ψ Ψ −1 M h Det J Det Det formation of the bounce solution. Consequently, 0 , ¼ lim lim →0 ; ð2:21Þ ð Þ ðAμ;φÞ ðXÞ →∞ ˆ r ˆ h Mc r Ψ Ψ M1 2 and M0 have zero eigenvalues. Their determi- Det J Det Det = nants vanish as shown in Eqs. (2.24) and (2.27) [see also ˆ ˆ ˆ where Ψ ¼ðΨ1; Ψ2; …Þ and Ψ ¼ðΨ1; Ψ2; …Þ are sets of Eq. (A80)]; a naive inclusion of those results gives a independent solutions of divergent behavior of the decay rate, which requires a careful treatment of the zero modes. In the present case, we ðXÞ can consider the effect of each partial wave (labeled by J) M Ψi ¼ 0; ð2:22Þ J separately. Thus, in this subsection, we consider the case ðXÞ cðXÞ ˆ where the fluctuation operator M for a certain value of J M Ψi ¼ 0; ð2:23Þ J J has a zero eigenvalue and discuss how to take account of its ð Þ effect. Because only bosons have zero modes in the and are regular at r ¼ 0. Notice that, when M X is an J MðXÞ n × n object, there are n independent solutions that are calculation of vacuum decay rates, J is considered to be a fluctuation operator of a bosonic field. regular at r ¼ 0. Since Ψ and Ψˆ obey the same linear i i We first decompose the fluctuation of the bosonic field differential equation at r → ∞, the ratio of the two (denoted as X) into eigenstates of the angular momentum: determinants converges for each J. X With a Fubini-Lipatov instanton, we can calculate the X ∋ c G ðrÞY ðΩÞ; ð2:31Þ ratio analytically, as first pointed out in [24]. For the i i J;mA;mB i convenience of readers, we give the details of the calcu- lation in Appendix A. The ratios are given by where ci is the expansion coefficient while Gi obeys ð Þ DetM h 2Jð2J − 1Þ MðXÞG ¼ ω G ð Þ J ¼ ; ð2:24Þ J i i i; 2:32 McðhÞ ð2J þ 3Þð2J þ 2Þ Det J with ωi being the eigenvalue. We normalize them as MðσÞ Γð2 þ 1ÞΓð2 þ 2Þ Det J J J δ ¼ ; ð2:25Þ ij cðσÞ Γð2 þ 1 − ÞΓð2 þ 2 þ Þ hG jG i¼ ; ð : Þ DetM J zκ J zκ i j N 2 2 33 J i ðψÞ DetM ½Γð2J þ 2Þ 2 2 where the inner product is defined by J ¼ ; ð : Þ cðψÞ Γð2 þ 2 − ÞΓð2 þ 2 þ Þ 2 26 Z DetM J zy J zy J † hG jG i¼ drr3G ðrÞG ðrÞ: ð : Þ i j i j 2 34 ðAμ;φÞ DetM J Γð2J þ 1ÞΓð2J þ 2Þ 3 J ¼ : cðAμ;φÞ þ 1 Γð2 þ 1 − ÞΓð2 þ 2 þ Þ N ’ DetM J J zg J zg We leave i s unspecified since, as we see below, the final J result is independent of them. We denote the zero mode as ð : Þ 2 27 G0 (and hence ω0 ¼ 0).
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˜ ˜ When there is a zero mode, the saddle point approxi- hG0jρjG0i¼2π: ð2:40Þ mation for the path integral breaks down and we need to go back to the original path integral. The path integral is Then, we have defined as the integration over the coefficient ci, and the Z ð Þ X Y ð Þ functional determinant of M originates from the follow- dci −1c c hG jðM X þνρÞjG i J pffiffiffiffiffiffi e 2 j k j J k ing integral: 2πN i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Z Y ð Þ hG˜ jG˜ i 1 Y 1 dci −1c c hG jM X jG i 0 0 pffiffiffiffiffiffi e 2 j k j J k ; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð2:41Þ 2πN 2π ν þ Oðν2Þ ω þ OðνÞ i i i≠0 i where the summations over the indices j and k are implicit. which gives Applying the saddle point approximation for the modes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi other than the zero mode, we get hG˜ jG˜ iY 1 0 0 pffiffiffiffiffi Z Y Z Y 2π ω 1 ðXÞ 1 i dci − c c hG jM jG i dc0 i≠0 pffiffiffiffiffiffi e 2 j k j J k ¼ pffiffiffiffiffiffi pffiffiffiffiffi : Z 2πN 2πN ω pffiffiffi Y ð Þ i i 0 i≠0 i dci −1c c hG jðM X þνρÞjG i ¼ lim ν pffiffiffiffiffiffi e 2 j k j J k : ð2:42Þ ν→0 2πN ð2:35Þ i i
ðXÞ The integration in the above expression is nothing but Notice that M , G and ω may depend on c0. J i i DetðMðXÞ þ νρÞ, and can be evaluated with the use of We are interested in the case where there exists a MðXÞ 1 1 symmetry (at least at the classical level) and the Eq. (2.21).If J is a × object, for example, we obtain Lagrangian is invariant under a transformation, which ð Þ we parametrize by z. The zero mode is in association with Det½M X þ νρ Ψˇ ðrÞ J ¼ ν lim þ Oðν2Þ; ð2:43Þ such a symmetry. Then, the transformation of the bounce ðXÞ →∞ ˆ DetMc r ΨðrÞ solution with z → z þ δz can be seen as a shift of X as J where → þ δ G˜ þ Oðδ 2Þ ð Þ X X z 0YJ;mA;mB z ; 2:36 MðXÞΨˇ ð Þ¼−ρð ÞΨð Þ ð Þ ˜ J r r r ; 2:44 where the function G0 is proportional to G0. The integration over c0 can thus be regarded as the integration over the while the functions Ψ and Ψˆ obey Eqs. (2.22) and (2.23), collective coordinate z: respectively. Then, we interpret the functional determinant sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the fluctuation operator with the zero eigenvalue as ˜ ˜ dc0 hG0jG0i pffiffiffiffiffiffi → ð Þ ð Þ−1 2 Z dz; 2:37 M X = Ψˇ ð Þ −1=2 2πN 0 2π Det J r → dz lim →∞ : ð2:45Þ McðXÞ r Ψˆ ð Þ Det J r and hence sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The above argument can be applied to the zero modes of Z Z our interest. In Appendix B, we obtain the following Y ð Þ hG˜ jG˜ iY 1 dci −1c c hG jM X jG i 0 0 pffiffiffiffiffiffi e 2 j k j J k → dz pffiffiffiffiffi: replacements to take care of the dilatational, translational, 2πN 2π ω i i i≠0 i and gauge zero modes: ð2:38Þ Z sffiffiffiffiffiffiffiffi MðhÞ −1=2 16π Det 0 → d ln R ; ð2:46Þ Next, we discuss how we evaluate the integrand of the cðhÞ jλj DetM0 right-hand side of Eq. (2.38). To omit the zero eigenvalue ! from the functional determinant, we introduce a regulator to ð Þ −2 M h 2 Det 1 2 32π V4 the fluctuation operator: = → D ; ð2:47Þ McðhÞ jλj R4 Det 1=2 MðXÞ → MðXÞ þ νρð Þ ð Þ J J r ; 2:39 ! sffiffiffiffiffiffiffiffi ðAμ;φÞ −1=2 Z DetM0 16π where ν is a small positive number, and ρðrÞ is an arbitrary → dθ : ð2:48Þ McðAμ;φÞ jλj function that satisfies Det 0
116012-5 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) κ 1 2 κ D. Renormalization ½ AðσÞ ¼ − þ þ ε ln εσ 2 ln σ After taking the product over J, we have a UV diver- jλj εσ εσ 3jλj gence and thus we need to renormalize the result. In this 1 − S ð ÞþOðε Þ ð Þ paper, we use the MS scheme, which is based on the 2 σ zκ σ ; 2:50 dimensional regularization. In this subsection, we explain 2 2 5 1 2 þ 2jλj how the divergences can be subtracted using counterterms ½ AðψÞ ¼ y þ þ y ε ln εψ 2 ln ψ in the MS scheme. jλj εψ εψ 3 jλj Since the dimensional regularization cannot be directly þ Sψ ðz ÞþOðεψ Þ; ð2:51Þ used in this evaluation, we first regularize the result by y using the angular momentum expansion as where the functions, Sσ and Sψ , are given in Eqs. (C13) and X∞ ðXÞ MðXÞ (C16), respectively. In addition, we define ðXÞ ðXÞ nJ Det J ½ln A ε ≡ w ln : ð2:49Þ Z X ð1 þ ε Þ2J McðXÞ 1 J¼0 X Det J ðhÞ 0ðhÞ A ¼ V4 d ln R A ; ð2:52Þ D R4 We call this regularization “angular momentum regulari- Z zation.” Here, ε is a positive number, which will be taken X AðAμ;φÞ ¼ dθA0ðAμ;φÞ: ð2:53Þ to be zero at the end of the calculation. Since the divergence is at most a power of J, the regularized sum converges. In Appendix C, we calculate the sum analytically, and obtain Then, the primed quantities are given by
1 jλj jλj X∞ ð2 þ 1Þ2 2 ð2 − 1Þ 0ðhÞ J J J ½ln A ε ¼ − ln þ 4 ln þ ln h 2 16π 32π 2J ð2 þ 3Þð2 þ 2Þ ð1 þ εhÞ J J J¼1 1 2 3 5 π 5 jλj ¼ 3 þ − ε − − 3γ − 6 A þ − þ Oðε Þ; ð : Þ ε2 ε ln h 4 E ln G 2 ln 3 2 ln 8 h 2 54 h h ∞ 1 jλj X ð2J þ 1Þ2 J Γð2J þ 1ÞΓð2J þ 2Þ 3 0ðAμ;φÞ ½ln A ε ¼ − ln þ ln A 2 16π ð1 þ ε Þ2J J þ 1 Γð2J þ 1 − z ÞΓð2J þ 2 þ z Þ J¼1=2 A g g 3g2 1 2 1 g4 3 1 1 jλj ¼ − − 1 þ − þ ε þ − γ − 2 A − jλj ε2 ε 3 jλj2 ln A 4 3 E ln G 2 ln 8π A A 3 3 − S ð Þ − Γð1 − ÞΓð2 þ ÞþOðε Þ ð Þ 2 σ zg 2 ln zg zg A ; 2:55
where AG ≃ 1.282 is the Glaisher number. where ð Þ Next, we relate the above results with those based on the δMðXÞ ¼ MðXÞ − Mc X ; ð2:58Þ dimensional regularization in D dimension, which contain ε¯ δMðXÞ ½ the regularization parameter, D, defined as and J is defined similarly. Here, OðδÞ indicates 1 2 hthe expansioni up to δ. The most important pointh is thati ð Þ ð Þ ¼ þ ln 4π − γE; ð2:56Þ DetM X DetM X ε¯ 4 − ln ð Þ has the same divergence as ln ð Þ D D b X b X DetM div DetM δMðXÞ with γE being the Euler’s constant. We convert the results does when does not have a derivative operator. based on two different regularizations by calculating the The first line of Eq. (2.57) can be calculated by directly following quantity: evaluating the traces in a momentum space with using ðXÞ −1 the dimensional regularization; the result is denoted as ½ AðXÞ ¼ ðXÞ ½ðMc Þ δMðXÞ ð Þ ln div w Tr ½ln A X . On the other hand, the second line of div;ε¯D ð Þ w X ð Þ −1 ð Þ −1 Eq. (2.57) can be evaluated as − ½ðMc X Þ δMðXÞðMc X Þ δMðXÞ 2 Tr ½McðXÞ þ δMðXÞ X∞ cðXÞ ðXÞ Det J J ˆ −1 ð1Þ ð Þ Det½M þ δM ln ¼ Tr½Ψ Ψ ¼ ðXÞ X J J McðXÞ w nJ ln ðXÞ ; Det J OðδM2Þ Mc J J¼0 Det J OðδM2Þ J 1 þ ½Ψˆ −1Ψð2Þ − ½Ψˆ −1Ψð1ÞΨˆ −1Ψð1Þ ð Þ ð2:57Þ Tr 2 Tr ; 2:59
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1 ε¯ where Subtracting = D, we obtain the renormalized prefactor in the MS scheme. McðXÞΨðpÞ ¼ −δMðXÞΨðp−1Þ ð Þ J J ; 2:60 E. Dilatational zero mode for p ¼ 1 and 2, and Ψð0Þ ¼ Ψˆ . Then, we calculate In the calculation of the decay rate of the EW vacuum, ðXÞ we have an integral over R in association with the classical ½ln A ε div; X scale invariance, as we saw in Eq. (2.46). So far, we have ð Þ ð Þ ð Þ X n X Det½Mc X þ δM X performed a one-loop calculation of the decay rate, based ≡ wðXÞ J ln J J : ð1 þ ε Þ2J cðXÞ on which the decay rate is found to behave as J X DetM OðδM2Þ J J Z ð1Þ ð2:61Þ 1 8π2 8π2β γðone-loopÞ ∝ d ln R exp − − λ ln μR ; R4 3jλðμÞj 3jλðμÞj2 We relate the results based on two regularizations by ð Þ replacing 2:67
ð1Þ ½ln AðXÞ → ½ln AðXÞ : ð2:62Þ where β is the one-loop β function of λ. (Here, we only div;εX div;ε¯D λ show the R and μ dependencies of the one-loop correc- The expressions of ½ln AðXÞ and ½ln AðXÞ for each tions.) Thus, using the purely one-loop result, the integral div;εX div;ε¯D field are given in Appendix D, where we further simplify does not converge. the relation so that the left-hand side only includes terms We expect, however, that the integration can converge that are divergent at the limit of εX → 0. We summarize the once higher order effects are properly included. To see the results below: detail of the path integral over the dilatational zero mode, (i) Scalar field: let us denote the decay rate as Z 1 2 κ −B ð Þ e eff þ þ ln εσ γ full ¼ ð Þ ε2 ε 3jλj d ln R 4 ; 2:68 σ σ R κ 1 μR → −1 − þ 1 þ γ þ ln ; ð2:63Þ B 3jλj 2ε¯ E 2 where eff fully takes account of all the effects of higher D order loops. B (ii) Higgs field: In order to discuss how eff should behave, it is instructive to rescale the coordinate variable as 1 2 1 μR rffiffiffiffiffi þ − ln ε → þ γ þ ln ; ð2:64Þ 2 ε h 2ε¯ E 2 jλj xμ ε h D ¯ h x˜μ ≡ ϕ xμ ¼ ; ð2:69Þ 8 C R (iii) Fermion: as well as the fields as 2 5 1 2 þ 2jλj þ þ y ε 2 ln ψ ˜ ¯ −1 εψ εψ 3 jλj Φ ≡ ϕ Φ; ð2:70Þ C y2 1 μR → − þ 1 þ γ þ ln σ˜ ≡ ϕ¯ −1σ ð Þ 3jλj 2ε¯ E 2 C ; 2:71 D 2 1 25 μ R ˜ ¯ −1 − þ þ γ þ ð Þ Aμ ≡ ϕ Aμ; ð : Þ 3 2ε¯ 4 E ln 2 ; 2:65 C 2 72 D ˜ ≡ ϕ¯ −1 ð Þ (iv) Gauge and NG fields: c C c; 2:73 3g2 1 2 1 g4 ¯˜ ≡ ϕ¯ −1 ¯ ð Þ − 1 þ þ þ ε c C c; 2:74 jλj ε2 ε 3 jλj2 ln A A A 2 4 jλj −1=4 1 2 1 μ ¯ −3=2 → − þ g þ g þ 1 þ γ þ R ψ˜ ≡ ϕ ψ: ð2:75Þ 3 jλj jλj2 2ε¯ E ln 2 8 C D 4 31 þ 1 þ g − π2 ð Þ 2 : 2:66 Using the rescaled fields, all the explicit scales disappear jλj 3 from the action as a result of scale invariance:
116012-7 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) Z Z 1 1 κ 4 4 ˜ pyffiffiffiffiffi pgffiffiffiffiffi If the effects of higher order loops are fully taken into d xL ¼ d x˜ L ; ; ; ð2:76Þ B μ ℏ ℏ˜ jλj jλj jλj account, eff should be independent of because the decay rate is a physical quantity; in such a case, we may choose where L is the total Lagrangian, any value of the renormalization scale μ. In the perturbative calculation, the μ dependence is expected to cancel out jλj order by order; as shown in Eq. (2.67), we can explicitly see ℏ˜ ¼ ℏ ð Þ 8 ; 2:77 the cancellation of the μ dependence at the one-loop level [64]. In our calculation so far, however, we only have the and L˜ includes canonically normalized rescaled fields one-loop result, in which μ dependence remains. As κ pyffiffiffiffi μ and depends only on the combinations jλj, and indicated in Eq. (2.81), the dependence shows up in jλj the form of lnpμR with p ¼ 1; 2; ….Ifj ln μRj ≫ 1, the pgffiffiffiffi. In addition, the rescaled bounce solution is given by jλj logarithmic terms from higher order loops may become comparable to the tree-level bounce action and the pertur- 1 bative calculation breaks down. In order to make our one- ð Þ 2 : 2:78 loop result reliable, we should take μ ∼ Oð1=RÞ, i.e., we 1 þ x˜ use R-dependent renormalization scale μ.4 With such a choice of μ (as well as with the use of coupling constants Based on L˜ and ℏ˜, we expect: κ y g evaluated at the renormalization scale μ), the integration (i) Only positive powers of , pffiffiffiffi and pffiffiffiffi appear in jλj jλj jλj over the size of the bounce is dominated only by the region the decay rate since there is no singularity when any where jλð1=RÞj becomes largest. In the case of the SM, the of these goes to zero. In particular, they cannot be in integration over the size of the bounce converges with this a logarithmic function. prescription as we show in the following section. (ii) When we renormalize divergences using dimen- sional regularization, we introduce a renormaliza- tion scale μ˜. It is always in a logarithmic func- F. Final result tion and is related to the original renormalization Here, we summarize the results obtained in the previous scale as subsections and Appendixes A–E. The decay rate with a resummation of important logarithmic terms is given by μ˜ ¼ μ ð Þ Z R: 2:79 1 γ ¼ dlnR ½A0ðhÞAðσÞAðψÞAðAμ;φÞe−B ; ð2:82Þ R4 MS;μ∼1=R (iii) In subtracting zero modes associated with transformations of Eq. (2.78), the result should be κ pyffiffiffiffi pgffiffiffiffi where again a polynomial of jλj, and . Notice that, jλpj ffiffiffiffiffiffiffiffi jλj ˜ 3 5 π 5 jλj μR for each zero mode, we have 1=ℏ since Eq. (2.37) ½ln A0ðhÞ ¼ − − 6 ln A þ ln − ln þ 3 ln ; MS 4 G 2 3 2 8 2 implies ð2:83Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ ˜ dc0 hG0jG0i 1 κ κ2 μ pffiffiffiffiffiffiffiffi ¼ dz : ð2:80Þ ½ AðσÞ ¼ − S ð Þþ þ 1 þ γ þ R ˜ ˜ ln σ zκ 2 E ln ; 2πℏN 0 2πℏ MS 2 jλj 3jλj 2 ð2:84Þ (iv) Quantum corrections have ℏ˜ l−1 at the lth loop ℏ˜ since the loop expansion is equivalent to the y4 μR ½lnAðψÞ ¼ − 1 þ γ þ ln expansion. MS 3jλj2 E 2 Based on the above arguments, B is expected to be eff 2 expressed as 2y 25 μR − þ γE þ ln þ Sψ ðzyÞ; ð2:85Þ 3jλj 4 2 8π2 jλðμÞj X∞ jλðμÞj l−1 B ¼ þ nzero þ eff ln ½ AðAμ;φÞ ¼ V þ½ A0ðAμ;φÞ ð Þ 3jλðμÞj 2 8 8 ln MS ln G ln MS; 2:86 l¼1 κðμÞ yðμÞ gðμÞ × Pl ; pffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffi ; ln μR ; ð2:81Þ 4 jλðμÞj jλðμÞj jλðμÞj This is equivalent to summing over large logarithmic terms appearing in higher loop corrections if we work with a fixed μ. Since we have calculated the decay rate at the one-loop level, it is P l where l is the contribution at the -loop level, and nzero is preferable to use, at least, the two-loop β-function to include the the number of zero modes. next-to-leading logarithmic corrections.
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L ∋ ¯ c þ ð Þ with Yukawa ytHqLtR H:c:; 3:2 2 4 1 2g g μR where qL is the left-handed third generation quark doublet, ½ A0ðAμ;φÞ ¼ þ þ 1 þ γ þ ln 2 E ln c MS 3 jλj jλj 2 tR is the right-handed antitop quark, and yt is the top Yukawa coupling constant. 4 31 1 1 − g − π2 − − γ − 2 Assuming that λ < 0, the bounce solution for the SM is 2 E ln AG jλj 3 4 3 given by 1 jλj 1 3 − ln þ ln π − SσðzgÞ 1 0 2 8 2 2 θaσa Hj ¼ pffiffiffi ei ; ð3:3Þ 3 bounce 2 ϕ¯ ðrÞ − Γð1 − ÞΓð2 þ Þ ð Þ 2 ln zg zg : 2:87 where σa is the Pauli matrices and function ϕ¯ is given by ϕ¯ Here, VG is the volume of the group space generated by the Eq. (2.9). In particular, remember that contains a free broken generators. The definitions of SσðzÞ, Sψ ðzÞ can be parameter, which we choose R, because of the classical found in Appendix C. Here, we note that the analytic results scale invariance. for the scalar, Higgs, and fermion contributions were first The results given in the previous section can be easily given in [24] with different expression. We emphasize that applied to the case of the SM. Taking account of the effects the final result does not depend on the gauge parameter, ξ, of the (physical) Higgs boson, top quark, and weak bosons 6 and hence our result is gauge invariant. The above result is (as well as NG bosons), the decay rate of the EW vacuum also applicable to the case where the Uð1Þ symmetry is not in the SM can be written in the following form: gauged as explained in Appendix E. Z 1 We have also derived fitting formulas of the functions γ ¼ ½A0ðhÞAðtÞAðW;Z;φÞ −B ð Þ d ln R 4 e : 3:4 necessary for the calculation of the decay rate; the result is R MS given in Appendix F. The fitting formulas are particularly useful for the numerical calculation of the decay rate. In As we have mentioned, the relevant renormalization scale μ ∼ ð1 Þ addition, a C þþ package to study the electroweak of the integrand is O =R ; in the following numerical μ ¼ 1 λðμÞ vacuum stability (ELVAS) is available at [30], which is analysis, we take =R unless otherwise stated. If also applicable to various models with (approximate) is positive, there is no bounce solution; the integrand is classical scale invariance. taken to be zero in such a case. In addition, since we neglect the mass term in the Higgs potential, 1=R should be much larger than the EW scale. This condition is automatically III. DECAY RATE OF THE EW VACUUM satisfied in the present analysis because λ becomes negative IN THE SM at the scale much higher than the electroweak scale. 0ðhÞ A. Decay rate The Higgs contribution A is given in Eq. (2.83), while the top-quark contribution is given by Now, we are in a position to discuss the decay rate of the EW vacuum in the SM. As we have discussed, the decay of ðtÞ ðψÞ ½ln A ¼ 3½ln A j → ; ð3:5Þ the EW vacuum is induced by the bounce configuration MS MS y yt whose energy scale is much higher than the EW scale. Thus, we approximate the Higgs potential as5 where the factor of 3 is the color factor. ð2Þ As for the gauge contributions, we have SU L × Uð1Þ =Uð1Þ broken symmetries, instead of Uð1Þ in ð Þ¼λð † Þ2 ð Þ Y EM V H H H ; 3:1 our previous example. Thus, we have a different volume of the group space, VG. To calculate VG, we first expand H where H is the Higgs doublet in the SM and λ is the Higgs around the bounce solution with θa ¼ 0 as λ quartic coupling constant. Notice that depends on the renormalization scale μ; in the SM, λ becomes negative 1 iðφ1 − iφ2Þ μ ∼1010 H ¼ pffiffiffi : ð3:6Þ when is above GeV with the best-fit values of the 2 ϕ¯ − iφ3 SM parameters. In addition, the relevant part of the Yukawa couplings are given by Here, φ1 and φ2 are NG bosons absorbed by charged W-bosons while φ3 is that by Z-boson. With the change of 5 We assume that the Higgs potential given in Eq. (3.1) is θa, the NG modes are transformed as applicable at a high scale. In particular, we assume that the effect of Planck suppressed operators, which may arise from the effect of quantum gravity, is negligible. For the discussion about the 6We checked that the effect of the bottom quark is numerically effect of Planck suppressed operators, see [65–73]. unimportant.
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a a a ˜ 2 φ → φ þ θ GGY0;0;0 þ Oðθ Þ; ð3:7Þ where pffiffiffiffiffiffiffi ˜ 2 ¯ GG ¼ 2π ϕ: ð3:8Þ The integration over the zero modes in association with the gauge transformation of the bounce solution can be θa replaced by the integrationZ over the parameter : 3 2 d θ ¼ 2π ≡ VSUð2Þ; ð3:9Þ with the above definition of θa. Then, following the argument given in Appendix B, the gauge contribution is evaluated as
½ AðW;Z;φÞ ¼ V þ 2½ A0ðAμ;φÞ ln MS ln SUð2Þ ln MS g→gW þ½ln A0ðAμ;φÞ j ; ð3:10Þ MS g→gZ
A0ðAμ;φÞ where is given in Eq. (2.87), and FIG. 1. Top: The RG evolution of the SM coupling constants as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ λ 0 2 þ 2 functions of (in units of GeV). The dashed line indicates < . g2 gY g2 The black dotted line shows the scale where ϕ¯ ¼ M , taking gW ¼ ;gZ ¼ ; ð3:11Þ C Pl 2 2 μ ¼ 1=R. The horizontal axis is common with the bottom panel. Bottom: The integrand of the decay rate with μ ¼ 1=R, taking the with g2 and g being the gauge coupling constants of Y central values for the SM parameters. In the shaded region, λ is SUð2Þ and Uð1Þ , respectively. L Y positive and the integrand is zero. B. Numerical results In order to understand the μ dependence of λ, let us show Now, let us evaluate the decay rate of the EW vacuum in one-loop RG equations of λ and y (although, in our the SM. The decay rate of the EW vacuum is very sensitive t numerical calculation, we use RG equations including to the coupling constants in the SM. In our numerical two- and three-loop effects and the contribution from the analysis, we use [74]: bottom and the tau Yukawa couplings):
mh ¼ 125.09 0.24; ð3:12Þ dλ g2 þ g2 g2 ¼ 173 1 0 6 ð Þ 16π2 ¼ 12λ 2λ þ 2 − Y 2 − 2 mt . . ; 3:13 μ yt 4 2 d ln one-loop α ð Þ¼0 1181 0 0011 ð Þ s mZ . . ; 3:14 2 þ 2 2 2 2 − 6 4 þ 6 gY g2 þ 12 g2 yt 4 4 ; where mh and mt are the Higgs mass and the top mass, α respectively, while s is the strong coupling constant. ð3:15Þ Following [75], the gauge couplings, the top Yukawa coupling, and the Higgs quartic coupling are determined μ ¼ dy 9 9 17 at mt; the calculations are done in the on-shell scheme 16π2 t ¼ 2 − 8 2 − 2 − 2 μ yt 2 yt g3 4 g2 12 gY : at next-to-next-to leading order precision. In addition, we d ln one-loop use three-loop QCD and one-loop QED β functions ð3:16Þ [76–78] together with values in [74] in order to determine the bottom and the tau Yukawa couplings at μ ¼ m . t 4 λ First, we show the RG evolution of the SM coupling At a low energy scale, the term proportional to yt drives to constants in Fig. 1. We use mainly three-loop β functions a negative value. As the scale increases, yt decreases while λ summarized in [75] and the central values for the SM gY increases, which brings back to a positive value. Notice ¯ that λ is bounded from below in the SM. parameters. The black dotted line indicates where ϕC ≃ 2 4 1018 We show the integrand of γ in the bottom panel of Fig. 1, reaches the Planck scale MPl . × GeV. We also show the running above the Planck scale, assuming together with that of there are no significant corrections from gravity. For Z 1 1010 GeV ≲ μ ≲ 1030 GeV, λ becomes negative; for such γ ¼ −B ð Þ tree d ln R 4 e : 3:17 a region, we use a dashed line to indicate λ < 0. R
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δBð1Þ for each X, where eff is the one-loop contribu- B ½ AðXÞ tion to eff, and ln MS is a contribution from particle X; the region that does not satisfy these conditions is excluded from the integration.7 By numerically integrating over R, we obtain8
½γ 3 ¼−582þ40þ184þ144þ2 ð Þ log10 Pl × Gyr Gpc −45−329−218−1 ; 3:18
3 þ40þ183þ145þ2 log10½γ∞ × Gyr Gpc ¼−580−44−328−218−1 ; ð3:19Þ
where the first, second, third, and fourth errors are due to the Higgs mass, the top mass, the strong coupling constant, and the renormalization scale, respectively. (In order to estimate the uncertainty due to the choice of the renorm- alization scale, we vary the renormalization scale from 1=2R to 2=R.) Currently, the largest error comes from the uncertainty of the top mass. With a better understanding of the top quark mass at the future LHC experiment [79–85], or even with at future eþe− colliders [86], a more accurate FIG. 2. Top: The integrand of the decay rate with the central determination of the decay rate will become possible. One values of SM parameters. The solid line corresponds to a result at can see that the predicted decay rate per unit volume is −4 the one-loop level and the dashed one corresponds to that at the extremely small, in particular, compared with H0 ≃ 3 3 tree level. The horizontal axis is common with the bottom figure. 10 Gyr Gpc (with H0 being the expansion rate of the ϕ¯ ¼ We show C MPl with the vertical dotted line. Bottom: The present Universe). Such a small decay rate is harmless for size of each quantum correction. The dashed line corresponds to realizing the present Universe observed.9 δBð1Þ eff . In Fig. 3, we show the decay rate in mh vs mt plane. In 4 the red region, γ becomes larger than H0, which we call the They are also shown in a linear scale in the top panel unstable region. In the yellow region, the EW vacuum is 4 of Fig. 2. metastable, meaning that 0 < γ
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vacuum may be affected if there exist extra particles. In particular, extra fermions coupled to the Higgs boson may destabilize the EW vacuum because the new Yukawa couplings tend to drive λ to a negative value through RG effects [32–43,46,48–52,54]. Consequently, the decay rate of the EW vacuum becomes larger than that in the SM. Potential candidates of such fermions include vectorlike fermions as well as right-handed neutrinos for the seesaw mechanism [109–111]. In this section, we consider several models with such extra fermions. We perform the RG analysis of the runnings of coupling constants with the effects of the extra fermions. We include two-loop effects of the extra fermions into the β functions, which can be calculated using the result in [112– 115]. We also take account of one-loop threshold correc- tions due to the extra fermions, which are summarized in Appendix G.11 For the integration over R, we follow the procedure in the SM case, as well as the following FIG. 3. The stability of the EW vacuum in the SM with a cutoff ϕ¯ ¼ treatments: of the integration at C MPl. The red region is unstable, the (i) We terminate the integration if any of the coupling yellow region is metastable, and the green region is absolutely stable. The dashed, solid, and dotted lines correspond to constants (in particular, Yukawapffiffiffiffiffiffi coupling constants of extra particles) exceeds 4π. αs ¼ 0.1192, 0.1181, and 0.1170, respectively. The black dot- 3 dashed lines indicate log10½γ × Gyr Gpc ¼0, −100, −300, and (ii) In order to maintain the classical scale invariance at a good accuracy, we require 1=R > 10M , where M −1000 with the central value of αs. The blue circles indicate 68, ex ex 95, and 99% C.L. constraints on the Higgs mass vs top mass is the mass scale of the new particles. plane assuming that their errors are independently Gaussian. A. Vectorlike fermions ¯ 18 of ϕC at the maximum of the integrand ranges from 10 1020 Here, we consider two examples of vectorlike fermions, to GeV. one is colored vectorlike fermions and the other is non- It is well known that, currently, our Universe is (almost) colored ones. We consider the case where the extra dominated by the dark energy. If it is a cosmological fermions have a Yukawa coupling with the SM Higgs constant, then our Universe will eventually become de boson. (We assume that the mixing between the extra Sitter space with the expansion rate of about fermions and the SM fermions is negligible.) 56.3 km= sec =Mpc [108]. Then, based on γ , the phase Pl We first consider colored vectorlike fermions, having transition rate within the Hubble volume of such a universe 10−580 −1 the same SM gauge quantum numbers as the left-handed is estimated to be Gyr , which may be regarded as quark doublet and the right-handed down quark, as well a decay rate of the EW vacuum in the SM. 1 ¯ as their vectorlike partners; we add Q ð3; 2; 6Þ, Q For comparison, we also perform a “tree-level” calculation 1 1 1 ð3¯; 2; − Þ, D ð3; 1; − Þ,andD¯ ð3¯; 1; Þ. [In the paren- of the decay rate using Eq. (3.17). The results 6 3 3 ð Þ ð Þ ð3Þ ð2Þ ½γ tree 3 ¼−575 ½γ tree theses, we show the quantum numbers of SU C, SU L, are log10 Pl ×GyrGpc and log10 ∞ × 3 and Uð1Þ .] The Yukawa terms in the Lagrangian are GyrGpc ¼−570. Thus, the difference between γ and Y given by γðtreeÞ turns out to be rather small. This is a consequence of an accidental cancellation among the contributions of ðSMÞ L ¼ L þ ¯ Φ ¯ þ Φ ¯ ð Þ several fields. In the bottom panel of Fig. 2, we show Yukawa Yukawa YD QD YD QD; 4:1 individual quantum corrections separately, as well as the total one-loop contribution. We can see that the large LðYukawaÞ where SM is the SM part. We also add the following quantum correction from the top quark is canceled by those mass terms: from the gauge bosons. We have also checked that the unstable region on the m vs m plane shifts upward by h t L ¼ ¯ þ ¯ ð Þ ðtreeÞ mass MQQQ MDDD: 4:2 Δmt ≃ 0.2 GeV if we use γ .
IV. MODELS WITH EXTRA FERMIONS 11If we use the two-loop β functions instead of three-loop ones 3 in the SM calculation, the difference of log10½γ × Gyr Gpc is So far, we assumed that the SM is valid up to the Planck or around 40. Thus, the systematic error of neglecting three-loop some higher scale. However, the decay rate of the EW effects of the extra fermions is expected to be similar.
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FIG. 4. The decay rate of the EW vacuum for vectorlike quarks with (left) and without (right) the cutoff at the Planck scale. The red region is unstable, the yellow region is metastable, and the green region is absolutely stable. The solid lines show 3 log10½γ Gyr Gpc ¼−250; −500, and −1000. The dashed line corresponds to the constraint when we stop the integration at ϕ¯ ¼ 0 1 C . MPl.
¼ ϕ¯ ϕ¯ max For simplicity, we assume MQ MD.Wetakethenew C at the maximum of the integrand, C , can become Yukawa coupling constants and mass parameters real and much larger than the Planck scale in the case with extra ϕ¯ max positive. We also take fermions; we show C for the case with vectorlike colored fermions in the left panel of Fig. 6. (In the Y ðμ ¼ M Þ¼Y ¯ ðμ ¼ M Þ ≡ y : ð : Þ ϕ¯ max D D D D D 4 3 upper-left corner of the figure, the value of C becomes smaller; this is because, in such a region, the Yukawa As we have mentioned before, the scale dependence of the coupling constants become nonperturbative at a lower RG ¯ new Yukawa coupling constants is evaluated by using two- scale, which gives an upper bound on ϕC in the integration loop RG equations and one-loop threshold corrections (see over R.) To see the cutoff dependence of the decay rate, we Appendix G). show the constraint with terminating the integration at The calculation of the decay rate is parallel to the SM ϕ¯ ¼ 0 1 C . MPl in the left panel (dashed line). In addition, case, and the decay rate is given in the following form: when MD and yD are small, we have a region of absolute Z stability. This is because the addition of colored particles 1 ¯ ¯ γ ¼ ½A0ðhÞAðtÞAðAμ;φÞAðQ;DÞAðQ;DÞ −B makes the strong coupling constant larger than the SM dlnR 4 e MS;μ¼1=R; R case. It rapidly drives yt to a small value, which makes λ ð4:4Þ always positive. Requiring that the lifetime of the EW vacuum should be longer than the age of the Universe, we ðQ;D¯ Þ ðQ;D¯ Þ ≲ 0 35–0 5 103 ≲ ≲ 1015 where A and A are effects of the extra fermions obtain yD . . for GeV MD GeV. on the prefactor: The second example is noncolored extra fermions, having the same SM quantum numbers as leptons. We 1 ¯ 1 ¯ ðQ;D¯ Þ ðψÞ introduce L ð1; 2; − Þ, L ð1; 2; Þ, E ð1; 1; −1Þ, and E ½ln A ¼ 3½ln A j → ; ð4:5Þ 2 2 MS MS y YD¯ (1,1,1), and the Yukawa and mass terms in the Lagrangian ¯ are given by ½ln AðQ;DÞ ¼ 3½ln AðψÞ j : ð4:6Þ MS MS y→YD ðSMÞ L ¼ L þ ¯ Φ ¯ þ Φ ¯ ð Þ Yukawa Yukawa YE LE YE LE; 4:7 In Fig. 4, we show the contours of the constant decay L ¼ ¯ þ ¯ ð Þ rate on the MD vs yD plane. Here, we use the central values mass MLLL MEEE; 4:8 of the SM parameters. The meanings of the shading colors are the same as in the SM case. The left and the right panels respectively. For simplicity, we take ML ¼ ME and adopt show the results with and without imposing the condition the following renormalization conditions: ϕ¯ 116012-13 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) FIG. 5. The same as Fig. 4 but for vectorlike leptons. The decay rate of the EW vacuum is given by large in a wider parameter space, as indicated in the right Z panel of Fig. 6. 1 ¯ ¯ γ ¼ ½A0ðhÞAðtÞAðAμ;φÞAðL;EÞAðL;EÞ −B Requiring that the lifetime of the EW vacuum should be d ln R 4 e MS;μ¼1=R; R longer than the age of the Universe, we obtain yE ≲ 0.4–0.7 103 ≲ ≲ 1015 ð4:10Þ for GeV ME GeV. The constraint becomes significantly weaker for larger ME owing to the cutoff at the where Planck scale. ðL;E¯ Þ ðψÞ ½ln A ¼½ln A j → ; ð4:11Þ MS MS y YE¯ B. Right-handed neutrino ½ AðL;E¯ Þ ¼½ AðψÞ j ð Þ Next, we consider the case with right-handed neutrinos, ln MS ln MS y→Y : 4:12 E which is responsible for the active neutrino masses via the In Fig. 5, we show the contours of constant decay rate. seesaw mechanism [109–111]. For simplicity, we concen- Since the extra fermions are not colored, we do not have a trate on the case where only one mass eigenstate of the region of absolute stability. We observe a larger effect of the right-handed neutrinos, denoted as N, strongly couples to ϕ¯ max cutoff at the Planck scale. This is because C is typically the Higgs boson (as well as to the third generation lepton ϕ¯ γ ϕ¯ max FIG. 6. The value of C at the maximum of the integrand of , C . The left panel is for vectorlike quarks and the right is for vectorlike ½ϕ¯ max leptons. The numbers in the legends indicate log10 C =GeV . In the gray region, the EW vacuum is absolutely stable. 116012-14 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) FIG. 7. The same as Fig. 4 but for a right-handed neutrino. We also show lines indicating mν ¼ 0.05 eV and mν ¼ 0.08 eV with purple solid lines. Z 1 doublet). Then, the Yukawa and mass terms in the 0ð Þ ð Þ ð φÞ ð Þ −B γ ¼ dlnR ½A h A t A Aμ; A L;N e ; Lagrangian are R4 MS;μ¼1=R L ¼ LðSMÞ þ Φ ¯ ð Þ ð4:20Þ Yukawa Yukawa YN LN; 4:13 1 L ¼ ¯ ð Þ where mass 2 MNNN; 4:14 ½ AðL;NÞ ¼½ AðψÞ j ð Þ ln MS ln MS y→Y : 4:21 where, in this subsection, L denotes one of the lepton N doublets in the SM. We define In Fig. 7, we show the contour plots of the decay rate. Since it does not have any SM charges, the decay rate goes to the YNðμ ¼ MNÞ ≡ yN: ð4:15Þ SM value when yN goes to zero. The effect of the cutoff at the Planck scale is again large, which is because of a large Assuming that, for simplicity, the neutrino Yukawa ϕ¯ max C as shown in Fig. 8. The purple solid lines show the matrix is diagonal in the mass basis of right-handed left-handed neutrino mass. Requiring that the decay rate neutrinos, the following effective operator shows up by should be smaller than the age of the Universe, we obtain integrating out N: 12 15 yN ≲ 0.65–0.8 for 10 GeV ≲ MN ≲ 10 GeV. ΔL ¼ C ðΦ Þ2 ð Þ 4 L ; 4:16 with 2 yN CðMNÞ¼−2 : ð4:17Þ MN One of the active neutrino masses is related to the value of C at the EW scale as 2 ¼ ð Þ v ð Þ mν C mt 4 ; 4:18 with v ≃ 246 GeV being the vacuum expectation value of the Higgs boson. In our numerical calculation, we use the following one-loop RG equation to estimate the neutrino mass [116]: d 16π2 C ¼ð4λ þ 6y2 − 3g2ÞC: ð4:19Þ d ln μ t 2 In the SM with right-handed neutrinos, the decay rate of the EW vacuum is evaluated with FIG. 8. The same as Fig. 6 but for a right-handed neutrino. 116012-15 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) V. CONCLUSION also supported in part by the Program for Learning Graduate Schools, MEXT, Japan. In this paper, we have calculated the decay rate of the EW vacuum in the framework of the SM and also in various Note added.— models with extra fermions. We included the complete one- The analytical and numerical results given in loop corrections as well as large logarithmic terms in higher this paper are consistent with those given in the latest version of [24]; in the earlier version, (i) the calculation of loop corrections. We used a recently developed technique ¼ 0 to calculate functional determinants in the gauge sector, the J component of the gauge and NG contributions, (ii) the subtraction of the divergence, and (iii) the calculation which not only gives a prescription to perform a gauge ð2Þ invariant calculation of the decay rate but also allows us to of the volume of SU group, were not properly performed, calculate the functional determinants analytically. In addi- and have been corrected in the recent revision. The tion, in calculating the decay rate of the EW vacuum, zero differences between our numerical results and those in [24] come mainly from the difference of the threshold modes show up in association with the dilatational and gauge symmetries. We have properly taken into account corrections to the MS top Yukawa coupling constant, which their effects, which was not possible in previous calcu- is regarded as a theoretical uncertainty. In addition, there is a lations. We have given an analytic formula of the decay rate difference in the treatment of the integration over the bounce of the EW vacuum, which is also applicable to models that size, although it has little effect on the numerical results. exhibit classical scale invariance at a high energy scale. The decay rate of the EW vacuum is sensitive to the APPENDIX A: FUNCTIONAL DETERMINANT coupling constants in the SM and their RG behavior. We In this Appendix, we present analytic formulas for have used three-loop RG equations for the study of the RG various functional determinants. For simplicity, we con- behavior of the SM couplings. The result is used for the sider the case with Uð1Þ gauge interaction. The charge of precise calculation of the decay rate of the EW vacuum. the scalar field that is responsible for the instability, Φ,is The decay rate of the EW vacuum is estimated to be set to be þ1. Application of our results to the case of general gauge groups is straightforward. We are interested ½γ 3 ¼−582þ40þ184þ144þ2 ð Þ log10 Pl × Gyr Gpc −45−329−218−1 ; 5:1 in the case where the Lagrangian has (classical) scale invariance; the potential of Φ is given in Eq. (2.2), and the where the errors come from the Higgs mass, the top mass, bounce solution is obtained as Eq. (2.7). the strong coupling constant, and the renormalization scale, respectively. Here, only the bounce configurations with 1. Scalar contribution amplitude smaller than the Planck scale are taken into account; for the decay rate of the EW vacuum, such a cutoff We first consider a real scalar field, σ, which couples to of the bounce amplitude does not significantly affect the Φ as result as far as it is not so far from the Planck scale. Since 2 † −4 3 3 V ¼ κσ Φ Φ; ðA1Þ H0 ≃ 10 Gyr Gpc , the lifetime is long enough compared with the age of the Universe. κ We have also considered models with extra fermions. where is a positive coupling constant. The contribution to Since they typically make the EW vacuum more unstable, the prefactor is given by the constraints on their masses and couplings are phenom- 1 ½−∂2 þ κϕ¯ 2 AðσÞ ¼ − Det ð Þ enologically important. We have analyzed the decay rate ln ln 2 : A2 for the extensions of the SM with (i) vectorlike quarks, 2 Det½−∂ (ii) vectorlike leptons, and (iii) a right-handed neutrino. We σ have obtained a constraint on the parameter space for each First, we expand into partial waves: model, requiring that the lifetime be long enough. The ðσÞ σðr; ΩÞ¼α ðrÞY ðΩÞ: ðA3Þ results constrain the Yukawa couplings that are larger than J;mA;mB J;mA;mB about 0.3–0.5 if we do not consider the cutoff at the Planck ð Þ scale. The effect of the cutoff was found to be rather large Here and hereafter, α X denotes the radial mode J;mA;mB and the constraints on the Yukawa couplings become function of X. For notational simplicity, the summations weaker, at most, by 0.3 after including the cutoff. over J, mA, and mB are implicit. Since the fluctuation operator for partial waves does not 2 ACKNOWLEDGMENTS depend on mA and mB, we have ð2J þ 1Þ degeneracy for each J. Summing up all the contributions, The authors are grateful to A. J. Andreassen for helpful ∞ correspondence. This work was supported by the Grant- 1 X Det½−Δ þ κϕ¯ 2 in-Aid for Scientific Research C (No. 26400239), and ln AðσÞ ¼ − ð2J þ 1Þ2 ln J ; ðA4Þ 2 Det½−Δ Innovative Areas (No. 16H06490). The work of S. C. was J¼0 J 116012-16 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) where where the function fðσÞ satisfies 3 2 Δ ¼ ∂2 þ ∂ − L ð Þ J r r 2 : A5 r r ¯ 2 ðσÞ ð−ΔJ þ κϕ Þf ¼ 0; ðA7Þ Then, using Eq. (2.21),wehave ½−Δ þ κϕ¯ 2 ðσÞð Þ Det J f r ðσÞð Þ 2J ¼ 1 ¼ lim ; ðA6Þ and limr→0f r =r . The solution of the above ½−Δ →∞ 2J Det J r r differential equation is given by jλj 1þzκ jλj ðσÞ ¼ 2J 1 þ ϕ¯ 2 2 1 þ 2ð þ 1Þþ ;2ð þ 1Þ; − ϕ¯ 2 2 ð Þ f r 8 Cr 2F1 zκ; J zκ J 8 Cr ; A8 ð ; ; Þ ðYukawaÞ where 2F1 a; b c z is the hypergeometric function and L ¼ yΦψ¯ Lψ R þ H:c: ðA14Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 κ ≡ − 1 − 1 − 8 ð Þ The contribution to the prefactor is given by zκ 2 jλj : A9 ½∂ þ pyffiffi ϕ¯ → ∞ Det = 2 Taking the limit of r , we have ln AðψÞ ¼ ln Det½=∂ ¯ 2 Det½−ΔJ þ κϕ Γð2J þ 1ÞΓð2J þ 2Þ ½ð∂ þ pyffiffi ϕ¯ Þð−∂ þ pyffiffi ϕ¯ Þ ¼ : ðA10Þ 1 Det = 2 = 2 Det½−Δ Γð2J þ 1 − zκÞΓð2J þ 2 þ zκÞ ¼ ln : ðA15Þ J 2 Det½−=∂=∂ Taking the basis given in [117], we expand it as 2. Higgs contribution ∞ ðψÞ X DetM Using the bounce solution given in Eq. (2.9), the Higgs ln AðψÞ ¼ ð2J þ 1Þð2J þ 2Þ ln J ; ðA16Þ McðψÞ mode fluctuation is parametrized as J¼0 Det J 1 pffiffiffi ¯ ðhÞ where Φ ¼ ½ϕðrÞþα ðrÞYJ;m ;m ðΩÞ : ðA11Þ 2 J;mA;mB A B 0 1 2 y ¯ 2 pyffiffi ¯ 0 −ΔJ þ 2 ϕ ϕ Then, the Higgs contribution to the prefactor is given by MðψÞ ¼ @ 2 A ð Þ J 2 ; A17 pyffiffi ¯ 0 y ¯ 2 ∞ ϕ −Δ þ1 2 þ ϕ 1 X Det½−Δ − 3jλjϕ¯ 2 2 J = 2 ln AðhÞ ¼ − ð2J þ 1Þ2 ln J : ðA12Þ 2 Det½−Δ ðψÞ ðψÞ J¼0 J ϕ¯ 0 ¼ ϕ¯ Mc M with d =dr, and J is obtained from J by replacing ϕ¯ → 0 and ϕ¯ 0 → 0. The functional determinant of the Higgs mode can be Using Eq. (2.21),wehave obtained with the same procedure as the case of the scalar contribution, taking κ → −3jλj: MðψÞ ½Ψ Ψ ½Ψ Ψ −1 Det J ¼ Det 1 2 Det 1 2 ½−Δ − 3jλjϕ¯ 2 2 ð2 − 1Þ ðψÞ lim 4 þ1 limr→0 4 þ1 ; Det J J J Mc r→∞ r J r J ¼ : ðA13Þ Det J Det½−Δ ð2J þ 3Þð2J þ 2Þ J ðA18Þ As we can see, the above ratio vanishes for J ¼ 0 and J ¼ 1=2, which are due to the scale invariance and the where translational invariance, respectively. MðψÞΨ ð Þ¼0 ð Þ J i r : A19 3. Fermion contribution Let us consider chiral fermions ψ L and ψ R with the Solutions of Eq. (A19) can be expressed by using two following Yukawa term: functions χðψÞ and ηðψÞ as 116012-17 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) 0 pffiffi 1 0 1 2 1 ðψÞ ∂ 2Jþ3χ 2 1 2Jþ3 ðψÞ For the first solution, we take ϕ¯ 2Jþ3 rr i − 2 ¯ 2 2 þ3 ∂ r η Ψ ¼ @ y r Aþ @ y ϕ r J r i A; i ðψÞ χ 0 ðψÞ ðψÞ ðψÞ i χ1 ðrÞ¼f ðrÞ; η1 ðrÞ¼0; ðA23Þ ðA20Þ where the function fðψÞ satisfies where i ¼ 1 and 2 are for two independent solutions of χðψÞ ηðψÞ Eq. (A19), and and obey ϕ¯ 0 1 2 2Jþ3 y ¯ 2 ðψÞ −Δ þ1 2 þ ∂ r þ ϕ f ¼ 0; ðA24Þ ϕ¯ 0 1 2 J = ϕ¯ 2Jþ3 r 2 2Jþ3 y ¯ 2 ðψÞ r −Δ þ1 2 þ ∂ r þ ϕ χ J = ϕ¯ r2Jþ3 r 2 i pffiffiffi ¯ 0 and 2ϕ 1 ðψÞ ¼ ∂ r2Jþ3η ; ðA21Þ ϕ¯ 2 2Jþ3 r i y r ðψÞð Þ f r 0 2 lim ¼ 1: ðA25Þ ϕ¯ 1 →0 2Jþ1 2Jþ3 y ¯ 2 ðψÞ r r −Δ þ1 2 þ ∂ r þ ϕ η ¼ 0: ðA22Þ J = ϕ¯ r2Jþ3 r 2 i The analytic formula of fðψÞ is given by −ipyffiffiffi jλj jλj y y jλj fðψÞðrÞ¼r2Jþ1 1 þ ϕ¯ 2 r2 F 2J þ 2 − i pffiffiffiffiffi ; 1 − i pffiffiffiffiffi ;2J þ 3; − ϕ¯ 2 r2 ; ðA26Þ 8 C 2 1 jλj jλj 8 C ðψÞ and fðψÞ behaves as DetM ½Γð2J þ 2Þ 2 2 J ¼ : ðA33Þ McðψÞ Γð2J þ 2 − i pyffiffiÞΓð2J þ 2 þ i pyffiffiÞ ðψÞð Þ 8 Γð2 þ 1ÞΓð2 þ 3Þ Det J λ λ f r ¼ J J lim 2 −1 2 y y : r→∞ J jλjϕ¯ Γð2 þ 2 − pffiffiÞΓð2 þ 2 þ pffiffiÞ r C J i λ J i λ 4. Gauge contribution ðA27Þ We consider the contributions from gauge bosons, NG For the second solution, we take bosons, and Faddeev-Popov ghosts. The Lagrangian is given in the following form: ðψÞ η ðrÞ¼fðψÞðrÞ: ðA28Þ 1 2 L ¼ þ½ð∂ − ÞΦ †ð∂ − ÞΦ 4 FμνFμν μ igAμ μ igAμ χðψÞ Then, 2 behaves as þ ðΦÞþL þ L ð Þ V GF FP; A34 pffiffiffi ðψÞ ¯ 00 χ ð Þ 2ϕ ð0Þ þ 1 where Fμν is the field strength tensor, and 2 r ¼ − J ð Þ lim 2 þ3 ¯ 2 ; A29 r→0 r J yϕ 2J þ 3 1 C L ¼ F 2 ð Þ GF 2ξ ; A35 ðψÞ pffiffiffi χ ðrÞ 2 ½Γð2J þ 2Þ 2 2 ¼ ð Þ L ¼ ¯ð−∂ ∂ Þ ð Þ lim 2 þ1 y y : A30 FP c μ μ c; A36 r→∞ J ϕ¯ Γð2 þ 2 − pffiffiÞΓð2 þ 2 þ pffiffiÞ r y C J i λ J i λ with Using these solutions, we get F ¼ ∂μAμ: ðA37Þ ½Ψ Ψ 8ð þ 1Þ Det 1 2 ¼ J ð Þ lim 4 þ1 2 2 ; A31 Since Faddeev-Popov ghosts do not directly couple to r→0 J ϕ¯ r y C the Higgs field with our choice of the gauge fixing function, we have Det½Ψ1Ψ2 lim Aðc;c¯Þ ¼ 0 ð Þ →∞ 4Jþ1 ln : A38 r r 2 2 8ðJ þ 1Þ ½Γð2J þ 2Þ At the one-loop level, ln AðAμ;φÞ is given by ¼ ; ðA32Þ 2ϕ¯ 2 Γð2 þ 2 − pyffiffiÞΓð2 þ 2 þ pyffiffiÞ y J i λ J i λ 1 DetMðAμ;φÞ ln AðAμ;φÞ ¼ − ln ; ðA39Þ 2 cðAμ;φÞ and hence DetM 116012-18 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) where 0 1 1 2 2 ¯ 2 ¯ ¯ B −∂ δμν þ 1 − ∂μ∂ν þ g ϕ gð∂νϕÞ − gϕ∂ν C MðAμ;φÞ ¼ @ ξ A; ðA40Þ ¯ ¯ 2 2gð∂μϕÞþgϕ∂μ −∂ þ Vφφ 2 2 with Vφφ ¼ d V=dφ . For the partial wave expansions, we use the following basis: ðSÞ xμ ðLÞ r ðT1Þ ð1Þ Aμðr; ΩÞ¼α ðrÞ YJ;m ;m ðΩÞþα ðrÞ ∂μYJ;m ;m ðΩÞþα ðrÞiεμνρσVν LρσYJ;m ;m ðΩÞ J;mA;mB r A B J;mA;mB L A B J;mA;mB A B ðT2Þ ð2Þ þ α ðrÞiεμνρσVν LρσY ðΩÞ; ðA41Þ J;mA;mB J;mA;mB ðφÞ φðr; ΩÞ¼α ðrÞY ðΩÞ; ðA42Þ J;mA;mB J;mA;mB ðiÞ where Vν ’s are arbitrary independent vectors and εμνρσ is a fully antisymmetric tensor. Then, we have ðS;φÞ X∞ ðS;L;φÞ ðTÞ 1 DetM0 1 DetM DetM ln AðAμ;φÞ ¼ − ln − ð2J þ 1Þ2 ln J þ 2 ln J ; ðA43Þ 2 McðS;φÞ 2 McðS;L;φÞ McðTÞ Det 0 J¼1=2 Det J Det J where ! 1 ð−Δ þ ξ 2ϕ¯ 2Þ ϕ¯ 0 − ϕ¯ ∂ ð φÞ ξ 1=2 g g g r M S; ¼ ; ðA44Þ 0 2 ϕ¯ 0 þ ϕ¯ 1 ∂ 3 −Δ þ g g r3 rr 0 Vφφ 0 1 0 1 3 2 ¯ 2 2L ¯ 0 ¯ 1 −Δ þ 2 þ g ϕ − 2 gϕ − gϕ∂ Δ − ∂ 0 J r r r 1=2 L r r ð φÞ B C 1 B C M S;L; ¼ B − 2L −Δ − 1 þ 2ϕ¯ 2 − L ϕ¯ C þ 1 − B L 3 L2 C ð Þ 2 J 2 g g 4 ∂ − 2 0 ; A45 J @ r r r A ξ @ r rr r A 2 ϕ¯ 0 þ ϕ¯ 1 ∂ 3 − L ϕ¯ −Δ þ 000 g g r3 rr r g J Vφφ and L 2ϕ¯ 0 1 2ϕ¯ 0 Δ χ ¼ η þ ∂ r3 ζ − ξζ ; ðA48Þ J i r g2ϕ¯ 3 i r3 r g2ϕ¯ 3 i i ð Þ M T ¼ −Δ þ g2ϕ¯ 2: ðA46Þ J J ϕ¯ 0 1 ϕ¯ 0 Δ − 2 ∂ 2 − 2ϕ¯ 2 η ¼ −2 L ζ ð Þ J ¯ 2 rr g i ¯ i; A49 MðS;φÞΨ ¼ 0 ϕ r r ϕ Three independent solutions of J i (with i ¼ 1–3) can be constructed from the functions, χi, ηi, 12 Δ ζ ¼ 0: ðA50Þ and ζi,as[28,29] J i 0 1 0 1 0 1 We also note useful relations: ΨðtopÞ L 1 η i ∂rχi 2ϕ¯ 2 i B C B C B r g C 1 Ψ ≡ B ðmidÞ C ¼ B L χ C þ B 1 1 2 C 3 ðtopÞ L ðmidÞ i @ Ψ A @ i A @ 2 ¯ 2 2 ∂ r η A ∂ r Ψ ¼ Ψ − ξζ ; ðA51Þ i r g ϕ r r i r3 r i r i i ðbotÞ ¯ Ψ gϕχi 0 i 0 1 1 ð Þ L ð Þ 2ϕ¯ 0 ∂ Ψ mid ¼ Ψ top þ η ð Þ − ζ rr i i i; A52 B g2ϕ¯ 3 i C r r þ B 0 C ð Þ @ A; A47 ¯ ð Þ rgϕ ð Þ 1 1 1 1 ζ Ψ bot ¼ Ψ mid − ∂ r2η þ ζ : ðA53Þ gϕ¯ i i L i gϕ¯ L r r i gϕ¯ i The first solution is obtained by setting ζ1 ¼ 0 and where χi, ηi, and ζi obey η1 ¼ 0 as 12 η η The function in the present analysis corresponds to =L in χ ¼ 2J ð Þ [28,29]; such a rescaling makes the formulas simpler. 1 r ; A54 116012-19 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) resulting in The last solution can be obtained with 0 1 2 −1 2 J 2J Jr ζ3 ¼ r : ðA64Þ B C 2J−1 Ψ1 ¼ @ Lr A: ðA55Þ The asymptotic form of η3 is given by gϕ¯ r2J η ð Þ jλjϕ¯ 2 For the second solution, we set ζ2 ¼ 0 and 3 r ¼ L C ð Þ lim 2 þ2 ; A65 r→0 r J 16ðJ þ 1Þ ðηÞ η2 ¼ f ; ðA56Þ η3ðrÞ L ðηÞ ¼ ð Þ lim 2 : A66 where the function f satisfies r→∞ r J 2ðJ þ 1Þ ϕ¯ 0 1 2 2 2 ðηÞ Using Eqs. (A51)–(A53),wehave Δ − 2 ∂ r − g ϕ¯ f ¼ 0; ðA57Þ J ϕ¯ r2 r 0 1 − ξ r2Jþ1 and B 4 C B Jξ 2Jþ1 C limΨ3 ¼ B − 2 r C; ðA67Þ ðηÞð Þ r→0 @ L A f r ¼ 1 ð Þ 1 2J lim 2 : A58 ϕ¯ r r→0 r J g C 0 −ξð þ1Þ 1 We can find an analytic formula of fðηÞ as J J r2Jþ1 B 4ðJþ1Þ C B J½ðJþ2Þ−ξðJþ1Þ 2Jþ1 C jλj zg lim Ψ3 ¼ B 2 ð þ1Þ r C: ðA68Þ ðηÞ ¼ 2J 1 þ ϕ¯ 2 2 r→∞ @ L J A f r Cr 8 g ðJþ2Þ−ξðJþ1Þ 2J jλjϕ¯ ð þ1Þ2 r jλj C J × F 1 þ z ; 2ðJ þ 1Þþz ;2ðJ þ 1Þ; − ϕ¯ 2 r2 ; 2 1 g g 8 C We also need three independent solutions around the ðA59Þ false vacuum. We take 0 1 2J−1 ðJþ1Þξ−J 2Jþ1 2Jr 2 r 0 with B 2L C 0 1 ˆ ˆ ˆ B 2 −1 ðJþ1Þξ−ðJþ2Þ 2 þ1 C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðΨ1Ψ2Ψ3Þ¼B J J 0 C: ðA69Þ @ Lr 4LðJþ1Þ r A 1 B g2 C ¼ − @1 − 1 − 8 A ð Þ 002J zg 2 jλj : A60 r Then, using Eq. (2.21) with combining the above Then, we get expressions, we obtain the ratio of the functional determi- nants for S, L, and NG modes as fðηÞðrÞ 8 Γð2J þ 1ÞΓð2J þ 2Þ ¼ ð Þ ðS;L;φÞ lim 2 −2 2 : A61 M Γð2 þ 1ÞΓð2 þ 2Þ r→∞ J jλjϕ¯ Γð2 þ 1 − ÞΓð2 þ 2 þ Þ Det J J J J r C J zg J zg ¼ : McðS;L;φÞ J þ 1 Γð2J þ 1 − z ÞΓð2J þ 2 þ z Þ Det J g g – Using Eqs. (A51) (A53), we obtain ðA70Þ 0 1 2 þ1 Lr J The functional determinant for T modes can be obtained by B 8ðJþ1Þ C B Jþ2 2Jþ1 C using the result for the scalar contribution. With the limΨ2 ¼ B 4ð þ1Þ r C; ðA62Þ 2 r→0 @ J A replacement κ → g in Eq. (A10): − 2 Jþ1 2J ϕ¯ r g C L ð Þ DetM T Γð2J þ 1ÞΓð2J þ 2Þ J ¼ : ðA71Þ Γð2 þ 1ÞΓð2 þ 2Þ ðTÞ J J Mc Γð2J þ 1 − zgÞΓð2J þ 2 þ zgÞ lim Ψ2 ¼ Det J →∞ r Γð2J þ 1 − zgÞΓð2J þ 2 þ zgÞ 0 1 ¼ 0 MðS;φÞΨ ¼ 0 4L 2J−1 For J , the solutions of 0 can be decom- ¯ 2 r ln r B ð2Jþ1Þjλjϕ C posed as [28,29] B 8ð þ1Þ C B J 2J−1 C ð Þ × ð2 þ1Þjλjϕ¯ 2 r ln r : A63 ! ¯ 0 ! @ J C A ðtopÞ 2ϕ Ψ ∂ χ − 2 ¯ 3 ζi − 2J 2J Ψ ≡ i ¼ r i þ g ϕ ð Þ ϕ¯ r i ; A72 gL C ðbotÞ ¯ 1 Ψ gϕχi ζ i gϕ¯ i 116012-20 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) pffiffiffiffiffiffiffi with i ¼ 1 and 2 for two independent solutions, where the ϕ¯ G˜ ð Þ¼ 2π2 d ð Þ functions χ and ζ satisfy D r ¯ : B2 i i dϕC 1 2ϕ¯ 0 3 The second term in the right-hand side of Eq. (B1) is Δ0χ ¼ ∂ r ζ − ξζ ; ðA73Þ i r3 r g2ϕ¯ 3 i i nothing but the change of the amplitude of the dilatational ð Þ zero mode; one can easily check M h G˜ ¼ 0. Δ ζ ¼ 0 ð Þ 0 D 0 i : A74 In order to translate the path integral over the dilatational zero mode to the integration over ϕ¯ , we calculate Notice that there are useful relations: C ¯ 2 Det½−Δ0 − 3jλjϕ þ νρ ðrÞ 1 ð Þ D ð Þ ∂ 3Ψ top ¼ −ξζ ð Þ ; B3 3 rr i; A75 Det½−Δ0 r i 13 1 ð Þ ð Þ 1 with ∂ Ψ bot ¼ Ψ top þ ∂ ζ : ðA76Þ r ϕ¯ i i 2ϕ¯ 2 r i g g 15 2 ¯ 2 ¯ 2 ρDðrÞ¼ jλj ϕ ϕðrÞ : ðB4Þ The first solution is given by 16π C 0 Notice that the condition (2.40) is satisfied with the above Ψ1 ¼ ϕ¯ : ðA77Þ choice of ρD, i.e., ϕ¯ C Z 3G˜ 2 ð Þρ ð Þ¼2π ð Þ The second solution can be obtained with drr D r D r : B5 ζ2 ¼ 1; ðA78Þ The ratio of the determinant can be obtained from Eq. (A10) κ→−3jλjþ 15 jλj2ϕ¯ 2 ν by replacing 16π C . Expanding the result giving rise to with respect to ν, we get −ξ r 4 ½−Δ − 3jλjϕ¯ 2 þ νρ ð Þ jλj Ψ ¼ ð Þ Det 0 D r 2 2 2 2 : A79 ¼ −ν ϕ¯ þ Oðν Þ ¯ r C : −ξgϕ 8 Det½−Δ0 16π ðB6Þ Then, using Eq. (2.21), we obtain ð φÞ Thus, we obtain DetM S; 0 ¼ 0: ðA80Þ cðS;φÞ ½−Δ − 3jλjϕ¯ 2 −1=2 DetM0 Det 0 Det½−Δ0 The vanishing of the above ratio is due to the existence of Z sffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffi 16π 1 16π zero mode. The treatment of the zero mode is discussed in → ϕ¯ ¼ ð Þ d C jλj ¯ d ln R jλj : B7 Appendix B. ϕC Here, we take an absolute value since there is a negative APPENDIX B: ZERO MODES mode [26,27]. In this Appendix, we discuss the zero modes associated with the dilatation, translation, and global transformation. 2. Translational zero mode In J ¼ 1=2 mode of the Higgs fluctuation, translational 1. Dilatational zero mode zero modes exist. The translation is parametrized by the In J ¼ 0 mode of the Higgs fluctuation, there exists a center of the bounce. The shift of the center of the bounce to dilatational zero mode. The dilatational transformation is aμ can be absorbed by the transformation of the Higgs ¯ ¯ ¯ ¯ parametrized by ϕC; with ϕC → ϕC þ δϕC, such a change mode as can be absorbed by ˜ h → h þ aμGTY1=2;μ þ ; ðB8Þ ¯ ˜ h → h þ δϕCGDY0;0;0 þ ; ðB1Þ 13A prescription used in [25] is consistent with our argument, δϕ¯ ρ where we neglect higher order terms in C, and where D is a constant. 116012-21 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) where 3. Gauge zero mode ¯ In J ¼ 0 mode of the gauge field, we have a gauge zero ˜ π dϕ G ðrÞ¼− pffiffiffi ; ðB9Þ mode. For the case of the Uð1Þ gauge symmetry, the T 2 dr bounce solution is parametrized as Eq. (2.7) with the and parameter θ. The path integral over the gauge zero mode pffiffiffi can be understood as the integration over the variable θ,as 2 we see below. Y1 2 μðxˆÞ¼ xˆμ; ðB10Þ = ; π The effect of the shift θ → θ þ δθ can be absorbed by the transformation of the NG mode as with xˆμ ¼ xμ=jxj. Notice that Y1=2;μ is given by a linear combination of Y1=2;mA;mB with the same normalization as ˜ φ → φ þ δθGGY0;0;0 þ ; ðB19Þ Y1=2;mA;mB . Thus, as noted in [27], the path integral over the translational zero modes can be converted to the integration where over the spacetime volume. pffiffiffiffiffiffiffi ˜ 2 ¯ In order to take care of the translational zero mode, we GG ¼ 2π ϕ: ðB20Þ calculate ¯ 2 Using the equation of motion of the bounce solution, the Det½−Δ1 2 − 3jλjϕ þ ρ ðrÞν = T ; ðB11Þ following relation holds: Det½−Δ1 2 = 0 MðS;φÞ ¼ 0 ð Þ with 0 ˜ : B21 GG 3jλj ρ ð Þ¼ ð Þ T r 4π : B12 In order to deal with gauge zero mode, we calculate ð φÞ One can see that Det½M S; þ νρ ðrÞ Z 0 G ; ðB22Þ cðS;φÞ 3 ˜ 2 DetM0 drr GTðrÞρTðrÞ¼2π; ðB13Þ with which is consistent with Eq. (2.40). jλj2 Following the argument in Sec. II C, we have ¯ ¯ ρGðrÞ¼ ϕCϕ: ðB23Þ 1 16π −2 ¯ 2 −2 Z ˇ ðhÞ Det½−Δ1 2 −3jλjϕ f1 2 Notice that the following relation holds: = → d4a lim = A ; ðB14Þ →∞ Z Det½−Δ1=2 r r 3 ˜ 2 drr GGðrÞρGðrÞ¼2π: ðB24Þ where For the evaluation of the ratio (B22) at the leading order ½−Δ − 3jλjϕ¯ 2 ˇ ðhÞ ¼ −ρ ðhÞ ð Þ ν Ψˇ 1=2 f1=2 Tf1=2; B15 in , we introduce the function 1 satisfying with ðS;φÞ ˇ M0 Ψ1ðrÞ¼−ρGðrÞΨ1ðrÞ; ðB25Þ ¯ ð Þ 4 dϕ f h ¼ − : ðB16Þ with which 1=2 jλjϕ¯ 3 C dr Z ˇ ðhÞ MðS;φÞ −1=2 ½Ψˇ ð ÞΨ ð Þ −1=2 The function f1=2 behaves as Det 0 Det 1 r 2 r → dθ lim →∞ cðS;φÞ r ˇ ðhÞ DetM0 r f1=2 1 lim ¼ : ðB17Þ ½Ψ ð ÞΨ ð Þ 1=2 →∞ ¯ 2 Det 1 r 2 r r r 4πϕ × lim →0 : C r r Thus, we obtain ð Þ B26 ¯ 2 −2 Det½−Δ1 2 − 3jλjϕ = ˇ We can decompose Ψ1 as Det½−Δ1=2 0 1 32π 2 jλj 2 32π 2 V ¯ 0 ¯ 2 4D 2ϕ ˇ → ϕ V ¼ ð Þ ∂ χˇ − 2 3 ζ jλj 8 C 4D jλj 4 ; B18 ˇ r 1 @ g ϕ¯ 1 A R Ψ1 ¼ þ ; ðB27Þ ¯ 1 gϕχˇ1 ζˇ V gϕ¯ 1 where 4D is the spacetime volume. 116012-22 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) ˇ where χˇ1 and ζ1 satisfies APPENDIX C: INFINITE SUM OVER ANGULAR MOMENTUM 1 2ϕ¯ 0 3 ˇ ˇ Δ0χˇ1 ¼ ∂ r ζ1 − ξζ1; ðB28Þ In this Appendix, we perform various infinite sums 3 r 2ϕ¯ 3 r g appearing in the calculation of functional determinants. ¯ 2 ˇ ϕ We first evaluate the following sum: Δ0ζ1 ¼ gρ : ðB29Þ G ϕ¯ C ∞ X ð2J þ 1Þ2 Γð2J þ 1ÞΓð2J þ 2Þ ζˇ IσðzÞ¼ ln ; We can solve 1 as ð1 þ ε Þ2J Γð2 þ 1 − ÞΓð2 þ 2 þ Þ J¼0 σ J z J z jλjg ζˇ ð Þ¼− ϕ¯ ð Þ ð Þ ðC1Þ 1 r 16π r ; B30 ˇ based on which Ψ1 should behave as which can be used for the calculation of the scalar 0 1 contribution to the prefactor. Notice that J ¼ 0; 1=2; 1; … ξg 4πϕ¯ is half integer. In addition, here, z is a complex number ˇ @ Cr A lim Ψ1 ¼ : ðB31Þ satisfying r→∞ jλj 16π −2 < ℜðzÞ < 1: ðC2Þ Thus, we obtain ε 0 sffiffiffiffiffiffiffiffi For σ > , the sum converges thanks to the fac- −1 2 Z 2J ðS;φÞ = tor 1=ð1 þ εσÞ . DetM0 16π → dθ : ðB32Þ First, we rewrite the log-gamma functions with integrals McðS;φÞ jλj Det 0 of digamma functions as ∞ Z X ð2 þ 1Þ2 2Jþ1 ð Þ¼ J ½ψ ð Þ − ψ ð − Þþψ ð þ 1Þ − ψ ð þ 1 þ Þ ð Þ Iσ z 2J dx Γ x Γ x z Γ x Γ x z ; C3 ð1 þ ε Þ ∞ J¼0 σ where ψ ΓðzÞ is the digamma function. Then, we use the following relation: Z 1 uy−1 − ux−1 ψ ΓðxÞ − ψ ΓðyÞ¼ du; ðC4Þ 0 1 − u which is valid for ℜðxÞ > 0 and ℜðyÞ > 0, and we obtain ∞ Z Z X ð2 þ 1Þ2 2Jþ1 1 x−z−1ð1 − zÞð1 − zþ1Þ ð Þ¼ J u u u ð Þ Iσ z 2J dx du : C5 ð1 þ ε Þ ∞ 0 1 − J¼0 σ u Notice that this is verified only in region (C2). Then, we interchange the two integrals,14 and integrate over x first. Consequently, it becomes Z X∞ ð2 þ 1Þ2 1 2Jð1 − zÞð1 − 1þzÞ ð Þ¼ J u u u ð Þ Iσ z 2J du z : C6 ð1 þ ε Þ 0 ð1 − Þ J¼0 σ u u ln u Notice that the integration over u is convergent. Since we have regularized the sum, the result should be finite. Thus, we can take the sum first: Z 1 ð1 − zÞð1 − 1þzÞð1 þ þ ε Þ ð Þ¼ð1 þ ε Þ2 u u u σ ð Þ Iσ z σ du z 3 : C7 0 u ð1 − uÞð1 þ εσ − uÞ ln u 14This is justified when Z Z 2 þ1 1 x−z−1 z zþ1 J u ð1 − u Þð1 − u Þ dx du < ∞: ∞ 0 1 − u 116012-23 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) We can see that new poles appear at u ¼ 1 þ εσ, but the integral is still convergent for positive εσ. Then, using Z 1 − uz 1 ¼ −z dtuzt; ðC8Þ ln u 0 we obtain Z Z 1 1 1þz 2 ð1 − u Þð1 þ u þ εσÞ σð Þ¼− ð1 þ εσÞ ð Þ I z z du dt zð1−tÞ 3 : C9 0 0 u ð1 − uÞð1 þ εσ − uÞ Interchanging the integrals and integrating over u, we get Z 1 ð1 þ ε Þð − 1 þ ε − ε Þ ð1 þ ε Þ2ð2 þ ε Þ½ψ ð1 þ − Þ − ψ ð1 þ Þ ð Þ¼ σ z tz σ σ þ σ σ Γ tz z Γ tz Iσ z z dt 2 3 0 ε ε σ σ 2 þð3 þ 2 − 2 Þε þð1 þ − Þ2ε2 1 þ z tz σ z tz σ 1 1 þ − ;2þ − ; 3 2F1 ; tz z tz z ð1 þ tz − zÞεσ 1 þ εσ ð1 þ ε Þð2 þ ε − 2 ε þ 2 2ε2Þ 1 − σ σ tz σ t z σ 1 1 þ ;2þ ; ð Þ 3 2F1 ; tz tz ; C10 ð1 þ tzÞεσ 1 þ εσ where 2F1ða; b; c; zÞ is the hypergeometric function. Since we do not need higher order terms in εσ, we expand Iσ as Z 1 1 1 ε ð Þ¼ − ð þ 1Þ − ð þ 1Þð4 þ − 2 Þþln σ 2ð1 þ Þð1 þ 2 − 6 þ 6 2 Þ Iσ z dt 2 z z z z z tz z z z tz t z 0 ε 2ε 6 σ σ 53 1 1 3 − þ 2 − þ þ ð1 − Þ ð − Þþ ð Þ þ z ½−7 þ 14 þ 6ð1 − Þ2 ð − Þþ6 2 ð Þ z z 36 t 6 t H tz z 6 tH tz 12 t t H tz z t H tz 4 þ z ½−1 þ 3ð1 − Þ þ 3ð1 − Þ3 ð − Þþ3 3 ð Þ þ Oðε Þ ð Þ 9 t t t H tz z t H tz σ ; C11 where HðzÞ is the harmonic number. After the final integral, we obtain 1 2 1 ð Þ¼− ð þ 1Þ − ð þ 1Þþ 2ð þ 1Þ2 ε þ S ð ÞþOðϵ Þ ð Þ Iσ z 2 z z z z z z ln σ σ z σ ; C12 εσ εσ 6 where 1 1 S ð Þ¼ ð1 þ Þð1 þ 2 Þ½ Γð1 þ Þ − Γð1 − Þ − þ 2 þ ½ψ ð−2Þð1 þ Þþψ ð−2Þð1 − Þ σ z 6 z z z ln z ln z z z 6 Γ z Γ z 1 35 4 þð1 þ 2 Þ½ψ ð−3Þð1 þ Þ − ψ ð−3Þð1 − Þ − 2½ψ ð−4Þð1 þ Þþψ ð−4Þð1 − Þ þ γ 2ð þ 1Þ2 − − 2 − z z Γ z Γ z Γ z Γ z 6 Ez z z 36 z 18 1 ζð3Þ þ 2π þ 2 A þ : ð Þ 2 ln ln G 2π2 C13 We can repeat a similar calculation for ∞ X ð2J þ 1Þð2J þ 2Þ ½Γð2J þ 2Þ 2 2 Iψ ðzÞ¼ ln ; ðC14Þ ð1 þ ε Þ2J Γð2 þ 2 þ ÞΓð2 þ 2 − Þ J¼0 ψ J z J z which can be used for the calculation of the fermionic contribution to the prefactor. Here, −2 < ℜðzÞ < 2. The result is 2 5 1 ð Þ¼− 2 − 2 þ 2ð 2 − 2Þ ε þ S ð ÞþOðϵ Þ ð Þ Iψ z 2 z z z z ln ψ ψ z ψ ; C15 εψ εψ 3 116012-24 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) where 2 1 S ð Þ¼ ð 2 − 1Þ½ Γð1 þ Þ − Γð1 − Þ − 2 2 − ½ψ ð−2Þð1 þ Þþψ ð−2Þð1 − Þ þ 4 ½ψ ð−3Þð1 þ Þ − ψ ð−3Þð1 − Þ ψ z 3 z z ln z ln z z 3 Γ z Γ z z Γ z Γ z 1 2 ζð3Þ − 4½ψ ð−4Þð1 þ Þþψ ð−4Þð1 − Þ þ 2ð 2 − 2Þγ − z ð 2 þ 31Þþ4 þ ð Þ Γ z Γ z 3 z z E 9 z ln AG π2 : C16 Next, we consider ½ AðσÞ First, we calculate ln div with angular momentum ∞ regularization with using εσ as a regularization parameter. X ð2J þ 1Þ2 2Jð2J − 1Þ I ¼ ln ; ðC17Þ The expansion of Eq. (2.59) is exactly the same as that with h ð1 þ ε Þ2J ð2 þ 3Þð2 þ 2Þ J¼1 h J J respect to κ. Thus, we get which is for the calculation of the Higgs contribution. Using ½−Δ þκϕ¯ 2 a similar technique as in the previous cases, we obtain Det J ln ½−Δ Z Det J Oðκ2Þ X∞ 2 ð2J þ 1Þ 3 1 1 2κ 1 2κ 2 1 1 ¼ − þ ð Þ ð1Þ Ih 2J dx : C18 ¼ þ þ −ψ ð2 þ1Þ ð1 þ ε Þ 0 þ 2 þ 2 − 1 2 J ; J¼1 h x J x J jλj2Jþ1 jλj 2ð2Jþ1Þ 2Jþ1 ðD1Þ Then, performing the sum first, Ih is obtained as 6 12 3 ψ ðnÞðzÞ J ¼ − − þ 6 ε þ þ 6γ þ 12 where is the polygamma function. Summing over , Ih ε2 ε ln h 2 E ln AG we get h h þ 9 2 þ 5 3 þ Oðε Þ ð Þ κ 1 2 κ ln ln h : C19 ½ AðσÞ ¼ − þ þ ε ln div;εσ 2 ln σ jλj εσ εσ 3jλj Finally, for the gauge and NG contributions, let us κ κ consider − 1 þ þ Oðε Þ ð Þ jλj 18jλj σ : D2 ∞ X ð2J þ 1Þ2 2J IA ¼ ln : ðC20Þ As is expected, it has the same divergence as ð1 þ ε Þ2J 2J þ 2 J¼1=2 A ½ AðσÞ ln εσ does. Next, we directly calculate ½ln AðσÞ by using the Similar to Ih, it is expressed as div Z dimensional regularization. Using the ordinary Feynman X∞ ð2 þ 1Þ2 2 1 ¼ − J ð Þ rules, we obtain IA 2J dx ; C21 ð1 þ ε Þ 0 x þ 2J J¼1=2 A ðσÞ ½lnA ε¯ div;ZD which results in 1 dDk 1 ¼ − κ F½ϕ¯ 2 ð0Þ 2 4 2 3 2 2 ð2πÞD k2 I ¼ − − þ ln ε − þ γ þ ln 2 Z A ε2 ε 3 A 2 3 E κ2 dDk dDk0 1 1 A A − F½ϕ¯ 2 ðk − k0Þ F½ϕ¯ 2 ðk0 − kÞ ; 2 ð2πÞD ð2πÞD k2 k02 þ 4 ln AG þ OðϵAÞ: ðC22Þ ðD3Þ APPENDIX D: RENORMALIZATION where F½ is the Fourier transform of the argument. With WITH MS SCHEME performing the integration, we obtain In this Appendix, we relate the regularization based on κ2 1 5 μ the angular momentum expansion, which we call angular ðσÞ R ½ln A ε¯ ¼ þ þ γ þ ln : ðD4Þ div; D 2 2ε¯ 6 E 2 momentum regularization, and the dimensional regulariza- 3jλj D tion. In particular, we derive the relation between εX, which ε¯ Comparing Eqs. (D2) and (D4), we obtain the relation shows up in the angular momentum regularization, and D, which is for dimensional regularization. between the two regularizations as 1 2 κ κ 1 μ þ þ ε → −1 − þ 1 þ γ þ R 1. Scalar field 2 ln σ E ln : εσ εσ 3jλj 3jλj 2ε¯ 2 Let us start with the scalar contribution. For this purpose, D ½ AðσÞ ðD5Þ let us consider ln div, defined in (2.57). 116012-25 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) 2. Higgs Notice that the zero modes have nothing to do with the UV ε ε¯ divergence. For the Higgs case, the relation between h and D can be obtained from Eq. (D5) with the replacement of κ → −3jλj: 3. Fermion Next, we derive the relation for the fermionic contribu- 1 2 1 μR þ − ε → þ γ þ ð Þ tion. As discussed in [24], it is convenient to expand with ε2 ε ln h 2ε¯ E ln 2 : D6 h h D respect to y. The expansion in Eq. (2.57) is equivalent to 8" # " # " # 9 ∞ < ðψÞ ðψÞ ðψÞ = X ð2J þ 1Þð2J þ 2Þ DetM DetM DetM ½ln AðψÞ ¼ ln J þ ln J;diag − ln J;diag ; ðD7Þ div;εψ ð1 þ ε Þ2J : cðψÞ cðψÞ cðψÞ ; ¼0 ψ M 2 M M J Det J Oðy Þ Det J;diag Oðy4Þ Det J;diag Oðy2Þ where 0 1 y2 ¯ 2 ðψÞ B −ΔJ þ 2 ϕ 0 C M ¼ @ A; ðD8Þ J;diag y2 ¯ 2 0 −ΔJþ1=2 þ 2 ϕ McðψÞ MðψÞ ϕ¯ → 0 and J is obtained from J by taking . From Eq. (A10),wehave ðψÞ DetM Γð2J þ 1ÞΓð2J þ 2ÞΓð2J þ 2ÞΓð2J þ 3Þ J;diag ¼ ; ðD9Þ McðψÞ Γð2J þ 1 − z˜ ÞΓð2J þ 2 þ z˜ ÞΓð2J þ 2 − z˜ ÞΓð2J þ 3 þ z˜ Þ Det J;diag y y y y where 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 B y2 C ˜ ¼ − @1 − 1 − 4 A ð Þ zy 2 jλj : D10 Then, together with Eq. (A33), we have 2 2 5 1 2 þ 2jλj 31 2 1 4 ½ AðψÞ ¼ y þ þ y ε þ y þ y þ Oðε Þ ð Þ ln div;εψ 2 ln ψ 2 ψ : D11 jλj εψ εψ 3 jλj 9 jλj 18 jλj ½ AðψÞ We can also calculate ln div with dimensional regularization: 4 2 ðψÞ y 1 5 μR 2y 1 13 μR ½ln A ε¯ ¼ − þ þ γ þ ln − þ þ γ þ ln : ðD12Þ div; D 2 2ε¯ 6 E 2 3jλj 2ε¯ 12 E 2 3jλj D D Thus, we obtain 2 5 1 2 þ 2jλj 2 1 μ 2 1 25 μ þ þ y ε → − y þ 1 þ γ þ R − þ þ γ þ R ð Þ 2 ε 3 jλj ln ψ 3jλj 2ε¯ E ln 2 3 2ε¯ 4 E ln 2 : D13 εψ ψ D D F ¼ ∂ − ξ ϕφ¯ ð Þ 4. Gauge and NG fields BG μAμ g ; D14 Finally, we consider the gauge and NG contributions. which we call the background gauge. We may perform a Although we may use Eq. (2.57) to subtract the divergence, calculation of the prefactor A with this choice of the gauge it is more convenient to use the expression of the prefactor fixing function. We have checked that, irrespective of the with a different choice of the gauge fixing function from choice of ξ, the J>0 contribution from the gauge and NG that in Eq. (A37). sectors agrees with the one that we have obtained using the Here, we use the result with the following choice of the gauge fixing function in (A37). We also comment here that, gauge fixing function: in the background gauge, the treatment of the gauge zero 116012-26 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) mode is highly nontrivial. However, such an issue is where the fluctuation operator in the background gauge is unimportant for the following discussion because the zero given by mode does not affect the behavior of the divergence, which we will discuss below. ξ ¼ 1 2 2 ¯ 2 ¯ For simplicity, we take in the following. Then, −∂ δμν þ g ϕ 2gð∂νϕÞ MðAμ;φÞ ¼ ð Þ using the same basis as Eq. (A42), we define BG ¯ 2 2 ¯ 2 : D16 2gð∂μϕÞ −∂ þ Vφφ þ g ϕ ðAμ;φÞ ðAμ;φÞ 1 DetM ln A ¼ − ln BG ; ðD15Þ BG 2 McðAμ;φÞ Det BG With the angular momentum decomposition, we have MðS;φÞ X∞ MðS;L;φÞ MðTÞ ðAμ;φÞ 1 Det 0 1 Det Det ln A ¼ − ln ;BG − ð2J þ 1Þ2 ln J;BG þ 2 ln J;BG ; ðD17Þ BG 2 McðS;φÞ 2 McðS;L;φÞ McðTÞ Det 0;BG J¼1=2 Det J;BG Det J;BG where 0 1 3 2 ¯ 2 2L ¯ 0 −Δ þ 2 þ g ϕ − 2 2gϕ B J r r C ðS;L;φÞ B 2L 1 2 ¯ 2 C M ¼ B − 2 −Δ − 2 þ g ϕ 0 C; ðD18Þ J;BG @ r J r A ¯ 0 2 ¯ 2 2gϕ 0 −ΔJ þ Vφφ þ g ϕ for the partial waves with J>0, and 0 1 −Δ þ g2ϕ¯ 2 2gϕ¯ 0 MðS;φÞ ¼ @ 1=2 A ð Þ 0;BG ¯ 0 2 ¯ 2 ; D19 2gϕ −Δ0 þ Vφφ þ g ϕ for J ¼ 0. In addition, MðTÞ ¼ −Δ þ 2ϕ¯ 2 ð Þ J;BG J g ; D20 for the T mode. Remember that the T modes exist only for J>0. In the background gauge, Faddeev-Popov ghosts also couple to the bounce; the fluctuation operator of the ghosts is given by Mðc;c¯Þ ¼ −∂2 þ 2ϕ¯ 2 ð Þ BG g ; D21 and, after the angular momentum decomposition, we have Mðc;c¯Þ ¼ −Δ þ 2ϕ¯ 2 ð Þ J;BG J g : D22 Then, we define X∞ ðc;c¯Þ ð ¯Þ DetM ln A c;c ¼ ð2J þ 1Þ2 ln J;BG ; ðD23Þ BG Mcðc;c¯Þ J¼0 Det J;BG where the hatted fluctuation operators are defined through the replacements of ϕ¯ → 0 and ϕ¯ 0 → 0. MðS;L;φÞ One feature of the above gauge fixing is that, by a rotation using an orthogonal transformation, J;BG becomes 0 qffiffiffiffiffiffiffiffi 1 −Δ þ 2ϕ¯ 2 02J ϕ¯ 0 B J−1=2 g 2Jþ1g C B qffiffiffiffiffiffiffiffi C ð φÞ B C BtM S;L; B ¼ B 0 −Δ þ 2ϕ¯ 2 −2 Jþ1 ϕ¯ 0 C; ðD24Þ J;BG B Jþ1=2 g 2Jþ1g C @ qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi A 2 J ϕ¯ 0 −2 Jþ1 ϕ¯ 0 −Δ þ 2 þ 2ϕ¯ 2 2Jþ1g 2Jþ1g J ma g 116012-27 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) ðAμ;φÞ ðc;c¯Þ with ½ A0ðAμ;φÞ − ½ A − ½ A ¼ð Þ 0 1 ln ln BG div ln BG div finite ; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ð Þ B J − Jþ1 0 C D27 B 2Jþ1 2Jþ1 C B qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi C ¼ B C ð Þ where ½ is defined accordingly to Eq. (2.57). B B Jþ1 J 0 C: D25 div @ 2Jþ1 2Jþ1 A The relation between the angular momentum regulariza- 001 tion and dimensional regularization for the gauge and NG contributions can be understood by evaluating ðAμ;φÞ ð ¯Þ Notice that, in the new basis, the fluctuation operator ½ln A þ½ln A c;c . McðS;L;φÞ BG div BG div around the false vacuum, J;BG , becomes diagonal. This Let us evaluate the divergent part with εA regularization. makes the calculation of the functional determinant easier. Before proceeding, we define the following quantities: Following [28], we can show that ½−Δ þ ϕ¯ 2 ð1Þð Þ¼ Det J C ð Þ ðS;L;φÞ ðTÞ KJ C ln ; D28 M M Det½−Δ 2 Det J þ 2 Det J J OðC Þ ln ð φÞ ln ð Þ 2 3 Mc S;L; Mc T ¯ 0 Det J Det J −Δ Cϕ 6 J 7 ðS;L;φÞ ðTÞ ðc;c¯Þ Det ¯ 0 DetM DetM DetM ð2Þ 6 Cϕ −Δ þ1 2 7 ¼ ln J;BG þ 2ln J;BG − 2ln J;BG : K ðCÞ¼6ln J = 7 : ðD29Þ cðS;L;φÞ cðTÞ cðc;c¯Þ J 6 −Δ 0 7 DetM DetM DetM 4 J 5 J;BG J;BG J;BG Det 0 −ΔJþ1=2 ðD26Þ OðC2Þ ð2Þ A0ðAμ;φÞ ð Þ Thus, a divergent part can be subtracted from ln as Notice that KJ C can be also expressed as 2 3 −Δ þC2ϕ¯ 2 Cϕ¯ 0 6 J 7 6 Det ¯ 0 2 ¯ 2 7 2 ¯ 2 2 ¯ 2 ð2Þ 6 Cϕ −Δ þ1 2 þC ϕ 7 Det½−Δ þC ϕ Det½−Δ þ1 2 þC ϕ ð Þ¼ J = − J − J = KJ C 6ln 7 ln ln : 6 −ΔJ 0 7 Det½−Δ Oð 2Þ Det½−Δ þ1 2 2 4 Det 5 J C J = OðC Þ 0 −ΔJþ1=2 OðC2Þ ðD30Þ ð1Þð Þ ð2Þð Þ In addition, KJ C and KJ C can be analytically calculated as 2 ð1Þ 2C 1 2C 1 1 K ðCÞ¼ þ þ − ψ ð1Þð2J þ 1Þ ; ðD31Þ J jλj 2J þ 1 jλj 2ð2J þ 1Þ2 2J þ 1 and 2 ð2Þ C ð1Þ 2 1 K ðCÞ¼ 4ψ ð2J þ 2Þ − − : ðD32Þ J jλj Γ 2J þ 1 J þ 1 Then, 1 MðS;φÞ Mðc;c¯Þ ðAμ;φÞ ðc;c¯Þ Det 0;BG Det 0;BG ½ln A ε þ½ln A ε ¼ − ln þ ln BG div; A BG div; A 2 McðS;φÞ Mcðc;c¯Þ Det 0;BG OðδM2Þ Det 0;BG OðδM2Þ ∞ ðS;L;φÞ 1 X ð2J þ 1Þ2 DetM − ln J;BG ; ðD33Þ 2 ð1 þ ε Þ2J McðS;L;φÞ J¼1=2 A Det J;BG OðδM2Þ where MðS;φÞ Det 0;BG ð1Þ 2 ð1Þ 2 ð2Þ ln ¼ K ðg ÞþK0 ðg − jλjÞ þ K0 ð2gÞ; ðD34Þ McðS;φÞ 1=2 Det 0;BG OðδM2Þ 116012-28 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) 2 3 0 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi ðS;L;φÞ 6 DetM 7 ð1Þ ð1Þ ð1Þ ð2ÞB J þ 1 C ð2Þ J 4ln J;BG 5 ¼ K ðg2ÞþK ðg2ÞþK ðg2 − jλjÞþ K @−2 gAþ K 2 g ; McðS;L;φÞ Jþ1=2 J−1=2 J J 2J þ 1 J−1=2 2J þ 1 Det J;BG OðδM2Þ ðD35Þ Mðc;c¯Þ Det 0;BG ð1Þ 2 ln ¼ K0 ðg Þ: ðD36Þ Mcðc;c¯Þ Det 0;BG OðδM2Þ ¯ 2 Notice that Vφφ ¼ −jλjϕ , and that the Faddeev-Popov and T-mode contributions cancel out for J>0. Summing over J,we obtain 2 4 2 4 ð φÞ ð ¯Þ 3g 1 2 1 g 17 g g ½ A Aμ; þ½ A c;c ¼ − − 1 þ − þ ε þ þ þ ð59 − 6π2ÞþOðε Þ ln ε ln ε 2 2 ln A 2 A : BG div; A BG div; A jλj ε ε 3 jλj 18 3jλj 6jλj A A ðD37Þ With the dimensional regularization, it becomes 2 4 2 4 ð φÞ ð ¯Þ 1 2g g 1 μR 5 7g g ½ A Aμ; þ½ A c;c ¼ þ þ þ γ þ þ þ þ ð Þ ln ε¯ ln ε¯ 2 E ln 2 : D38 BG div; D BG div; D 3 jλj 2ε¯ 2 18 3jλj jλj D 2jλj Thus, we obtain 3 2 1 2 1 4 1 2 2 4 1 μ 4 31 g −1 þ þ þ g ε → − þ g þ g þ 1 þ γ þ R þ 1 þ g − π2 ð Þ jλj ε2 ε 3 jλj2 ln A 3 jλj jλj2 2ε¯ E ln 2 jλj2 3 : D39 A A D APPENDIX E: VACUUM DECAY WITH APPENDIX F: NUMERICAL RECIPE GLOBAL SYMMETRY In this Appendix, we give fitting formulas of the For completeness, we discuss the case where the prefactors at the one-loop level. Contrary to the analytic field that is responsible for the decay transforms under a formulas including various special functions with complex global symmetry, although it is not the case of the SM arguments, which may be inconvenient for numerical Higgs field. In such a case, we need to take into account calculations, the fitting formulas give a simple procedure quantum corrections from the associated NG bosons. to perform a numerical calculation of the decay rate with Similarly to the gauge contributions, the NG fluctuation saving computational time. Compared to the analytic operator has zero modes in association with the breaking of expressions, the errors of the fitting formulas are 0.05% the global symmetry. or smaller. A C þþpackage based on our fitting formulas Let us consider a Uð1Þ symmetry, for simplicity. The for the study of the electroweak vacuum stability (ELVAS) contribution from the NG boson, φ,isgivenby can be found at [30]. (i) Higgs 1 Det½−∂2 − jλjϕ¯ 2 ln AðφÞ ¼ − 2 Det½−∂2 X∞ −½ A0ðhÞ ¼ −0 99192944327027 1 J ln MS . ¼ − ð2J þ 1Þ2 ln : ðE1Þ 2 þ 1 þ 2 5 jλj − 3 μ ð Þ J¼0 J . ln ln R: F1 As we can see, there is a zero mode for J ¼ 0. Since it is → 0 ½ AðφÞ obtained in the limit of g in Eq. (2.27), ln MS is (ii) Scalar given by Eqs. (2.86) and (2.87) with taking g ¼ zg ¼ 0. Let x ¼ κ=jλj.Forx<0.7, 116012-29 SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) −½ AðσÞ ¼ −0 239133939224974 2 þ 0 222222222222222 3 − 0 134704602106396 4 ln MS . x . x . x þ 0.102278606592866x5 − 0.0839329261179402x6 þ 0.0715956882048009x7 − 0.0625481711576628x8 þ 0.0555697470602515x9 − 0.0500042455037409x10 − 0.333333333333333x2 ln μR: ðF2Þ For x>0.7, −½ AðσÞ ¼ −0 0261559272783723 þ 0 0000886704923163256 4 þ 0 0000962000962000962 3 ln MS . . =x . =x þ 0.000198412698412698=x2 þ 0.00105820105820106=x þ 0.111111111111111x − 0.181204187497805x2 þð−0.0055555555555556 þ 0.166666666666667x2Þ ln x − 0.333333333333333x2 ln μR: ðF3Þ (iii) Fermion Let x ¼ y2=jλj.Forx<1.3, −½ AðψÞ ¼ 0 64493454511661 þ0 005114971505109 2 −0 0366953662258276 3 þ0 00476307962690785 4 ln MS . x . x . x . x −0.000845451274112082x5 þ0.000168244913551417x6 −0.0000353785958610453x7 þ7.67709260595572×10−6x8 þð0.66666666666667xþ0.333333333333333x2ÞlnμR: ðF4Þ For x>1.3, −½ AðψÞ ¼ −0 227732960077634 þ 0 00260942760942761 3 þ 0 00271164021164021 2 ln MS . . =x . =x þ 0.00820105820105820=x þ 0.53790187962670x þ 0.296728717591129x2 þð−0.06111111111111111 − 0.3333333333333333x − 0.1666666666666666x2Þ ln x þð0.66666666666667x þ 0.333333333333333x2Þ ln μR: ðF5Þ (iv) Gauge Let x ¼ g2=jλj.Forx<1.4, −½ A0ðAμ;φÞ ¼ −0 96686103284373 − 1 76813696868318 þ 0 61593151565841 2 þ 0 145084271024101 3 ln MS . . x . x . x − 0.0241469799983579x4 þ 0.00555917805602827x5 − 0.00145020891759152x6 þ 0.000402580447036276x7 − 0.000115821925959136x8 þ 0.5 ln jλj þð−0.333333333333333 − 2x − x2Þ ln μR: ðF6Þ For x>1.4, −½ A0ðAμ;φÞ ¼−27 0091748854198þ0 000266011476948977 4 þ0 000288600288600289 3 ln MS . . =x . =x þ0.000595238095238095=x2 þ0.00317460317460317=xþ1.56519636465016x −0.07988363024944x2 þð−3.54033527491510×10−6=x5 −0.0000404609745704583=x4 −0.00051790047450187=x3 −0.0082864075920299=x2 −0.265165042944955=x pffiffiffi s þ4.24264068711929Þ xarcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þð−6.01666666666667þ0.5x2Þlnx s2 þ79164837199872x9 þ1.5ln½3.14159265358979ð−98796.7402597403þ136316.571428571x−136594.285714286x2 þ92160x3 þ7372800x4 þ6553600x5Þ þ0.5lnjλjþð−0.333333333333333−2x−x2ÞlnμR; ðF7Þ where s ¼ 7 þ 80x þ 1024x2 þ 16384x3 þ 524288x4 − 8388608x5: ðF8Þ 116012-30 DECAY RATE OF ELECTROWEAK VACUUM IN THE … PHYS. REV. D 97, 116012 (2018) APPENDIX G: THRESHOLD CORRECTIONS 1 μ2 Δ ¼ − g2 log ; ðG10Þ g2 3 2 2 In this Appendix, we summarize the one-loop threshold ME corrections for the coupling constants in the models with Δ ¼ 0; ðG11Þ extra fermions discussed in Sec. IV. We parametrize the g3 threshold corrections as 1 μ2 1 Δ ¼ −2y y2 log − ; ðG12Þ yt t E 2 2 3 1 ME cðbelowÞ ¼ c þ Δ ; ðG1Þ 16π2 c 1 μ2 1 Δ ¼ −2y y2 log − ; ðG13Þ yb b E 2 2 3 ð Þ ME where c below is a coupling constant below the matching scale, while c is that above the scale. For the notational 2 2 1 μ 1 Δ Δ ¼ −2yτy log − ; ðG14Þ simplicity, we only show the quantity c for each coupling yτ E 2 2 3 ME constant. Notice that Δc depends on the extra fermion mass ¼ MX (X D, E, N). In our analysis, we take the matching 1 μ2 1 1 μ2 4 Δ ¼ −8λ 2 − þ 4 4 − scale to be equal to MX. λ yE 2log 2 3 yE 2log 2 3 : (i) Model with vectorlike quarks Q, Q¯ , D, and D¯ ME ME ð Þ 2 G15 1 2 μ Δ ¼ − g1 log ; ðG2Þ g1 5 2 15 MD (iii) Model with right-handed neutrino N μ2 Δ ¼ − 2 ð Þ g2 g2 log 2 ; G3 Δg ¼ 0 ði ¼ 1; 2; 3Þ; ðG16Þ MD i μ2 2 2 Δ ¼ −g3 log ; ðG4Þ 1 μ 1 g3 M2 Δ ¼ −y y2 log þ ; ðG17Þ D yt t N 2 M2 4 1 μ2 1 N Δ ¼ −6 2 − ð Þ y ytyD log 2 ; G5 t 2 M 3 1 μ2 1 D 2 2 Δ ¼ −y y log þ ; ðG18Þ 1 μ 1 yb b N 2 M2 4 Δ ¼ −6y y2 log − ; ðG6Þ N yb b D 2 M2 3 D 2 1 μ2 3 2 1 μ 1 Δ ¼ 2 þ ð Þ Δ ¼ −6yτy log − ; ðG7Þ yτ yτyN log 2 ; G19 yτ D 2 2 3 4 M 8 MD N 2 2 1 μ 1 1 μ 4 2 2 Δ ¼ −24λ 2 − þ 12 4 − 1 μ 1 1 μ 1 λ y log 2 y log 2 : Δ ¼ −4λ 2 þ þ 2 4 − D 2 3 D 2 3 λ y log 2 y log 2 : MD MD N 2 4 N 2 2 MN MN ð Þ G8 ðG20Þ (ii) Model with vectorlike leptons L, L¯ , E, and E¯ 2 15 3 2 μ For simplicity, we assume that N couples only to third Δ ¼ − g log ; ðG9Þ g1 5 1 2 generation leptons. ME [1] N. 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