One-loop considerations for coexisting vacua in the CP conserving 2HDM

Journal of High Energy Physics, Nov 2017

The Two-Higgs-Doublet model (2HDM) is a simple and viable extension of the Standard Model with a scalar potential complex enough that two minima may coexist. In this work we investigate if the procedure to identify our vacuum as the global minimum by tree-level formulas carries over to the one-loop corrected potential. In the CP conserving case, we identify two distinct types of coexisting minima — the regular ones (moderate tan β) and the non-regular ones (small or large tan β) — and conclude that the tree level expectation fails only for the non-regular type of coexisting minima. For the regular type, the sign of m 12 2 already precisely indicates which minima is the global one, even at one-loop.

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One-loop considerations for coexisting vacua in the CP conserving 2HDM

HJE One-loop considerations for coexisting vacua in the CP conserving 2HDM A.L. Cherchiglia 0 1 C.C. Nishi 0 1 Santo Andre 0 1 Brazil 0 1 0 Universidade Federal do ABC 1 Centro de Matematica, Computac~ao e Cognic~ao The Two-Higgs-Doublet model (2HDM) is a simple and viable extension of the Standard Model with a scalar potential complex enough that two minima may coexist. In this work we investigate if the procedure to identify our vacuum as the global minimum by tree-level formulas carries over to the one-loop corrected potential. In the CP conserving case, we identify two distinct types of coexisting minima | the regular ones (moderate tan ) and the non-regular ones (small or large tan ) | and conclude that the tree level expectation fails only for the non-regular type of coexisting minima. For the regular type, the sign of m212 already precisely indicates which minima is the global one, even at one-loop. Beyond Standard Model; Higgs Physics; Spontaneous Symmetry Breaking - 1 Introduction 2 Coexisting normal vacua at tree-level 3 E ective potential at one-loop 4 Parametrization and minimization at one-loop 5 Pole masses at one-loop 6 Numerical survey 7 Results 8 Conclusions A Deviation of a potential minimum B Finding more than one normal vacua C Matrix of second derivatives D Cubic derivatives E Cubic and quartic couplings F Self-energy for the scalars G Reparametrization symmetry After the discovery of the Higgs boson of 125 GeV mass in 2012 [1, 2], all the pieces of (2HDM) in which just another scalar doublet is added. This model features ve physical Higgs bosons, three neutral and two charged, instead of just one in the SM. It has been extensively studied in the literature (see e.g. ref. [4] for a review) partly because more fundamental theories, for instance the MSSM [5{7], require a similar extended scalar sector. The model also allows the possibility of other sources of CP violation [8, 9], a feature that gets even richer when more doublet copies are added [10]. Finally, a more complex scalar sector can generate a strong enough EW rst-order phase transition [11{27], a property that is lacking in the SM [28{33] but is necessary to explain the matter-antimatter asymmetry of our universe. Another feature that a more involved scalar potential encompasses is the possibility of di erent symmetry breaking patterns and the 2HDM is no exception [34{40]. With additional Higgs doublets, this complexity increases substantially [41{44]. In general, for a su ciently complex potential, there may even exist sets of parameters for which many local minima coexist. In this case, identifying which one is the global minimum might be a nontrivial task that involves solving a system of polynomial equations. Usually, when one is sure that only one minimum is present for a given choice of parameters, such a task is bypassed by trading some of the quadratic parameters in favor of the vevs and ensuring the extremum is a local minimum. In the 2HDM, that assumption does not hold in general and for some parameter ranges it is possible that up to two minima coexist for the same potential [35, 36]. So a worrisome possibility arises: we may be living in a metastable vacuum with the possibility to tunnel to the global minimum. That situation was described in ref. [45, 46] as our vacuum being a panic vacuum. One way of testing that situation is explicitly calculating the depth of the second minimum and comparing it to the depth of the rst. However, nding the location of the minima explicitly may be a di cult or, at least, computationally intensive task. Fortunately, in the same work, the authors developed a method capable of distinguishing if our vacuum is a panic vacuum by calculating a discriminant that depends only on the position of our vacuum (see ref. [47] for a more general test). Although many of the possible scenarios with coexisting minima are not favored by current LHC data [45, 46], we are interested here in studying if the simple use of this discriminant can be carried over to the one-loop corrected e ective potential. Already for the inert doublet model [34] it was found that the potential di erence of the coexisting minima can change sign when one-loop corrections are taken into account [48]. Therefore, the present work aims to verify the validity of such conclusions for a general CP conserving 2HDM with softly broken Z2. We focus on the case of two coexisting normal vacua and study the predictive power of tree-level formulas for the depth of the potential when one-loop corrections are considered. The outline of the paper is as follows: in section 2 we review the properties of the 2HDM with softly broken Z2 at tree-level focusing on the possibility of two coexisting minima. Some results can not be found in previous literature. In section 3 we review the form of the one-loop e ective potential for our case while section 4 explains our procedure for ensuring that our vacuum has the correct vacuum expectation value. The procedure to compute and x the pole mass of the SM Higgs boson to its experimental value is explained in section 5. The steps we performed to generate the numerical samples are { 2 { listed in section 6 and the resulting analysis is shown in section 7. Finally, the conclusions can be found in section 8. 2 Coexisting normal vacua at tree-level The general 2HDM potential at tree-level is We will be considering the real softly broken Z2 symmetric case where m212 and 5 are real real and we employ the parametrization We will also focus on CP conserving vacua where the vacuum expectation values are These vevs can be further parametrized by modulus and angle as h 1i = p 1 2 0 v1 ! ; h 2i = p 1 2 0 v2 ! : (v1; v2) = v (c ; s ) ; where v = 246 GeV for our vacuum and we use the shorthands c cos , s sin . We call this type of vacuum a normal vacuum and we denote our vacuum by NV [45, 46] with v1 > 0 and v2 > 0. By ensuring the existence of one normal vacuum (our vacuum), a scalar potential with xed parameters cannot simultaneously have another minimum of a di erent type, namely a charge breaking vacuum or a spontaneously CP breaking vacuum [37, 38]. Just another coexisting normal vacuum NV0 with vevs (v10; v20) may exist and this is the only case where two minima can coexist in the 2HDM potential at tree level [35, 36]: only two minima with the same residual symmetry may coexist. When the coexisting minima exist, we de ne the potential di erence as V VNV0 VNV ; so that V > 0 indicates that our vacuum is the global minimum. We use this convention for the one-loop potential as well. To describe the situation of two coexisting normal vacua in more detail, we can write the extremum equations for nonzero v1 and v2: 1 1 1 1 2 1v12 + We employ the usual shorthand 345 symmetric limit. The complete solutions of eq. (2.5) for m212 = 0 can be easily found: there are two degenerate extrema that spontaneously break Z2 | ZB+ and ZB | and two extrema that preserve Z2 | ZP1 and ZP2; the latter are often denoted as inert or inert-like vacuum (see ref. [48] and references therein). Only one of the pairs ZP1;2 or ZB may coexist as minima.1 They are characterized by (v1; v2) of the form = 1 1 2 As the m212v1v2 term is continuously turned on, the Z2 symmetry is soft but explicitly broken with a negative (positive) contribution in the rst (fourth) quadrant when m212 > 0. The opposite is true for negative m212. The e ect of adding the m212 term is di erent for the two types of coexisting minima which we denote by ZB and ZP1;2 from their m212 ! 0 limit. We also denote the ZB minima as regular and ZP1;2 as non-regular simply because it is much more probable to generate models with the former pair than the latter for generic values of tan and other parameters. The two degenerate spontaneously breaking minima ZB : (v1; v2) deviate to ZB+ : (v1; v2) and ZB : (v10; v20), respectively, and the degenerate potential depth, . We note that simultaneous sign ips of both v1; v2 is a gauge symmetry and do not count as a degeneracy. Hence we adopt the convention that v1 > 0 while v2 can attain both signs so that we only analyze the rst and fourth quadrant in the (v1; v2) plane. V0(v1; v2) = also deviates di erently lifting the degeneracy. In rst approximation in small m212 and in the deviation of the vevs, the potential depths change respectively by the amount V m212v1v2, so that the depth di erence of the two minima is VZB VZB VZB+ = V V+ 2m122v1v2 : 1Note that one of ZP1;2 may not be a minimum whereas ZB are always degenerate. 2As long as the solutions for vi2 give positive solutions. { 4 { ZB+ : where we adopt the convention that all v1; v2; v~1; v~2 are positive. The speci c values of the vevs are given by HJEP1(207)6 for the Z2 breaking extrema2 and (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) where by v2 1 2 with m2Hi = mi2i + 12 345v~j2, (ij) = (2; 1) or (1; 2), is the second derivative in the vi direction around ZPj . When m212 > 0, the deviations vi are positive and the two minima enter the rst quadrant. Otherwise they move to the fourth quadrant. The potential depth then changes from the Z2 limit m411 ; indeed it gets deeper as m212 increases from zero while the non-standard vacuum ZB respectively. The depth di erence of the coexisting ZP1;2 is VZP = V0jZP1 V0jZP2 m411 where we show the depth di erence calculated exactly against m212, both normalized by the appropriate power of v = 246 GeV (NV). Only potentials with two minima are selected and the free parameters are taken as ftan ; ; m2H+ ; m2A; m2H0 ; m2h; m122g ; with mh = 125 GeV, 1 tan 50, fmH+ ; mA; mH g ranging from 90 GeV to 1 TeV (mH > mh), 0:1 cos( 20;000 GeV2 m212 6000 GeV2 and is constrained near alignment, ) 0:1. Simple bounded from below and perturbativity constraints are also imposed [4]. The blue points end around m212=v2 0:07 because the nonstandard minimum gets pushed up until the point where it disappears. In contrast, the right panel shows the normalized depth di erence with respect to the ratio v0=v of the values for NV0 and NV. We can see for the blue points that the vacuum that lies deeper has a larger vacuum expectation value. The method we employed to calculate the location of NV0 is described in appendix B. { 5 { to the standard vev (NV). We con rm the approximation (2.11): for ZB the sign of m212 discriminates between our vacuum being the global minimum (m212 > 0) or just a local metastable vacuum (m212 < 0). This behavior, however, does not apply for the points (red) that deviate from the inert-like vacua, ZP1;2, where the nonstandard vacuum may lie deeper despite m212 being positive. For the generic values of t as used above the density of non-regular coexisting minima is very low so that only a handful of coexisting non-regular minima is obtained jointly with the regular points. To generate a su cient number of non-regular points we further produced another sample (most of the red points) by restricting 20 t 50 and positive m212. To accurately distinguish among the di erent cases, ref. [45, 46] constructed a very useful discriminant D that ensures that our vacuum is always the global minimum if D is positive.3 Since that discriminant was derived assuming that v1; v2 are both positive, we cannot apply it to NV0 when v20 < 0. So we rederived the discriminant allowing the vevs to be negative with the result D = 1 v4 m122(m121 k2m222)s2 (t2 k2) ; (2.18) where k ( 1= 2)1=4 and we have normalized to obtain a dimensionless quantity. This discriminant is useful because it can be obtained by using only the angle calculated in one vacuum and cases with only one minimum are automatically taken into account. The discriminating power of D is shown in gure 2 where the depth di erence is plotted against D calculated using NV. Obviously we could have calculated the discriminant for NV0, obtaining a D0 with sign opposite to D. That implies that the quantity that depends on the vevs, s2 (t2 k2), must have opposite signs when calculated for NV and NV0. Our main goal here is to analyze if the discriminant power of m212 and D carries over to the one-loop e ective potential. 3For D = 0 but m122 6= 0 the discriminant is inconclusive. { 6 { inant (2.18). 3 E ective potential at one-loop We can now consider the e ective potential with the one-loop contribution V = V0 + V1l ; V1l = 1 The masses Mk2('i) correspond to the scalar- eld-dependent eigenvalues of the tree-level mass matrices of all particles of the theory while is the renormalization scale. We are already assuming a renormalization scheme with minimal subtraction (MS) and, for the gauge sector, the Landau gauge and dimensional reduction (DRED), following the scheme of ref. [49, 50]. The parameters contained in V0 are thus the renormalized parameters. The integer coe cients jckj count all the degrees of freedom for each particle k including color, charge and spin, while the sign of ck is determined by its boson/fermion character: positive for bosons and negative for fermions. For example, for the top quark we have ct = 3 2 2 corresponding to its 3 colors, 2 particle/antiparticle and 2 spin degrees of freedom. We should note that the e ective potential is generically a gauge dependent quantity but its value at an extremum is not [51]. As we will focus on normal vacua, we can consider that the e ective potential depends only on the two real values '1; '2 in the real neutral directions:4 1 = p 1 2 0 ! '1 ; 2 = p 1 2 0 ! '2 : (3.3) 4For a generic eld dependence modulo gauge freedom, we would need two more real directions. { 7 { We reserve the symbols v1; v2 in eq. (2.2) to values at a minimum. So the eld-dependent gauge boson masses retain the same functional form as in the SM with v2 = v12 + v2: 2 M W2 ('i) = 4 1 g2('21 + '22) ; 1 4 MZ2 ('i) = where yt;b are the Yukawa couplings of the third family quarks normalized to the SM values and the enhancement factor 1=hs i should be considered as the xed value at the NV minimum at one-loop. We emphasize that information with the bracket h i. For the type II, we should replace the Mb dependence on '2 and hs i by '1 and hc i respectively. We will see that, as usual, the top correction dominates the fermion loops and the di erence between type I or type II is negligible for the one-loop corrections except for excessively large tan which we do not consider. It is also justi ed that we only consider the e ects of the top and bottom quarks; see gure 3 and comments in the text. For the scalar contribution we need to calculate the eigenvalues of the matrix of second derivatives of V0 for generic values of 'i. These mass matrices are shown in appendix C and their eigenvalues correspond to MS2('i) of the 8 scalars S 2 fG ; H ; G0; A0; H0; h0g. Due to charge and CP conservation, the mass matrices are still separated into three sectors: two charged scalars and its antiparticles, two CP odd scalars and two CP even scalars. We emphasize that e.g. MG2 0 ('i) is nonvanishing at 'i away from any tree-level minimum. It is the second derivative of the whole e ective potential at one-loop that will vanish in the directions of the Goldstone modes. 4 Parametrization and minimization at one-loop We are interested in surveying the cases where the e ective potential at one-loop (3.1) continues to have two local minima, one of which should be our vacuum with v = pv12 + v22 = 246 GeV. The vevs v1; v2 no longer satisfy the tree-level minimization relations in (2.5) but should now minimize the whole e ective potential V0 + V1l. We need a convenient parametrization to ensure that one minimum has the appropriate value of v. To parametrize V0, we will use as input the usual 8 quantities fv1; v2; ; m2H+ ; m2A; m2H0 ; m2h; m122g ; (4.1) where vi satisfy the minimization equations (2.5). It is clear that these quantities de ne V0 unambiguously by xing the 8 parameters fm211; m222; m212; 1 ; 2 ; 3 ; 4; 5g; see e.g. ref. [45]. When we add the one-loop contribution, it is clear that the true minimum will { 8 { We can see that mi2i ! 0 and V1l ! 0 in the limit where we turn o all couplings of the scalars to other particles including self-couplings. For small couplings, it is also expected that the physical masses are close to the masses fm2H+ ; m2A; m2H0 ; m2hg used as input. It is possible to use a di erent renormalization scheme where all the masses and mixing angles at tree level are maintained at one-loop [52]. Our scheme, however, avoids the need to deal with infrared divergences coming from the vanishing Goldstone masses [22{27, 53, 54]. This problem is more severe at higher loop orders [55]. Now the minimization equations at one-loop can be separated into a tree-level part which leads to the tree-level equations written in (2.5), and a one-loop part that de nes mi2i by = 0 ; i = 1; 2 ; mi2i = i = 1; 2 : (4.2) (4.3) (4.4) (4.5) (4.6) be shifted by a small amount from the position (v1; v2) at tree-level. Instead of correcting for that shift, we add the nite counterterms to the potential and adjust the values of mi2i so that v1; v2 continue to be a minimum at one-loop.5 This means that the one-loop e ective potential (3.1) is now rewritten as V = We can separate the derivative of V1l into its contribution from scalars (S), vector bosons (V ) and fermions (F ): 1 h3syti22 mt2 ln mt2 2 1 + g2m2W ln 1 + h3sybi22 mb2 ln m2 b 2 m2W 2 1 ; 1 ; for each i = 1; 2. The dimensionless coe cients iS are given in appendix D and we use lowercase letters for mW;Z;t;b because they correspond to the actual values in the SM when we use our vacuum NV. For charged particles the contribution of the antiparticle can be taken into account by doubling the contribution of the particle. The fermion part corresponds to the type I model. For the type II model, we must replace hs i ! h c i in the couplings to the b quark. We can see in gure 3 that the contributions from scalars are 5This procedure is equivalent to trading m121; m222 in favor of v1; v2 [48]. { 9 { large for m211 and m222 while the top contribution is also large and negative for m222. The contribution from the bottom quark is negligible for tan 50 and there is no appreciable di erence between the type I or type II model. Thus for de niteness we consider the type I model. We also note that the scalar masses and coe cients depend on m211; m222 and (4.5) must be solved self-consistently. The only remaining task is to write MS2(vi) in terms of the input parameters (4.1). We note that MS2(vi) should be computed from the second derivatives of V0 + V . But the part coming from V0 at 'i = vi corresponds to the usual masses at tree-level because (v1; v2) still corresponds to a minimum of V0. Therefore, these matrices will have the generic form HJEP1(207)6 Speci cally, the mass matrices for the di erent sectors read M(vi) = M tree + M : s 2 s c s 2 s c s c ! c 2 + s c ! c 2 + m211 m211 m222 ! m222 ! ; ; Mchar(vi) = m2H+ Modd(vi) = m2A Meven(vi) = (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) 2 5 are the masses squared for the charged Higgs and the pseudoscalar at tree-level; see e.g. [4]. Clearly the rst and second matrices contain each a vanishing eigenvalue corresponding to the charged (G ) and neutral (G0) Goldstone bosons in the limit where (p2 Im 0 p 1 ; 2 Im 02) and (p2 Re 01; p from the parametrization i = ( i+; i0)T. 2 Re 02) in eqs. (4.8), (4.9) and (4.10), respectively, mi2i ! 0. We use the basis ( 1+ ; 2+), Diagonalization of the tree-level part of Meven(vi) de nes the angle through where Using the same notation we can nd the eigenvalues shifted by mi2i as U y MterveeenU = diag(m2H0 ; m2h0 ) ; U = c s s ! c : U y+ Mchar(vi)U + = diag(MG2 + ; M H2+ ) ; U y0 Modd(vi)U 0 = diag(MG2 0 ; MA20 ) ; U y0 Meven(vi)U 0 = diag(M H20 ; Mh20 ) ; where the angles +; 0; 0 are shifted from ; ; by a small amount due to mi2i: + = + + ; 0 = + 0 ; 0 = + : The explicit forms for +;0 and can be seen in appendix D. The previous section showed a way of ensuring that one of the minima of the e ective potential at one-loop corresponded to our vacuum with v = 246 GeV and that tan = v2=v1 could be used as input at one-loop. The following task to describe a realistic 2HDM at one-loop is to ensure that the SM higgs boson mass corresponds to the experimentally measured value [56]: It is clear that at one-loop we cannot use this value for the tree-level parameter mh in (4.1). Instead, we must check that the pole mass corresponds to (5.1).6 We follow ref. [50] to calculate the pole masses m^S of all the scalars S including the SM higgs boson by computing the self-energies of the theory at one-loop. We focus in this section on the CP even sector which will give rise to the pole masses of h and H restricted to the case m^H > m^ h. The self-energy for the other sectors are given in appendix F. The scalar self-couplings can be extracted from V0. Given a set of real scalars Si that interact through the quartic vertex igijkl and cubic vertex igijk, the self-energy ij for Si{Sj coming from scalars in the loop with incoming momentum p2 = s is given by [49, 50] 16 2 iSj (s) = 1 X gijkkA(mk) + 1 X gijklB(mk; ml; s) ; 2 assuming we are in the basis with diagonalized masses (quadratic part of V0 + V ). The A and B functions are the Passarino-Veltman functions [59] which, in the notation of ref. [48], read 6Another possibility is to use a more physical renormalization condition to ensure this [52]. The renormalization of the entire theory is discussed in ref. [57, 58]. 2 k Z 1 0 A(m) m2 log B(m1; m2; s) dt log 2 kl 1 ; m2 2 t m21 + (1 t) m22 2 t(1 t)s : (5.1) (5.2) (5.3) (5.4) (5.5) The A function represents the one-loop graph with one quartic vertex (tadpole) and the B function represents the one-loop graph with two vertices and two internal lines. Hence the factors 1=2 are the symmetry factors that appear in front of the Feynman diagrams for identical elds. These functions appear after renormalizing these diagrams using the MS prescription. Simpli cations for vanishing s can be found in ref. [48]. The contributions coming from gauge bosons and fermions in the loop are also shown in appendix F and all the necessary cubic and quartic couplings of the theory are explicitly shown in appendix E. For example, the SM higgs self-energy is given by where gh2h2 = gh4 and ghh2 = gh3 are the quartic and cubic self-couplings for h and all MS refer to MS(vi). However, the mixing to the other CP even scalar H in Hh cannot be neglected. Due to charge and CP conservation, the self-energy for the di erent scalars decouple into separate pieces for the CP even, CP odd and charged sectors. The self-energy for the CP even sector is given by the matrix even(s) = HH (s) hH (s) Hh(s)! hh(s) : The one-loop pole squared masses m^2H ; m^ 2h for the CP even scalars H; h are the solutions where we take only the real part of the self-energy because we are not interested in the decay widths. The m^2h corresponds to the solution that continually approaches Mh2 in the limit where we turn o the interactions. A similar consideration applies to all pole squared masses. 6 Numerical survey We describe here the procedure we used to survey the models. For de niteness, we adopted the relevant xed parameters of the SM to be g = 0:6483 ; g0 = 0:3587 ; v = 246:954 GeV ; yt = 0:93697 ; yb = 0:023937 : (5.6) (5.7) (6.1) The rst four values were taken from refs. [55, 60] as the running parameters of the SM at the top pole mass = 173:34 GeV and the bottom Yukawa was adapted from its mass value mb = 4:18 GeV [56]. Among these parameters, only the top Yukawa appreciably a ects the one-loop e ective potential together with the quartic scalar self-couplings. We use a xed renormalization scale = 300 GeV for all calculations and note that the running of yt from the top mass scale only amounts to a small di erence. The running for the rest of the parameters are even less relevant. We remark that any choice of the renormalization scale is allowed, since the di erence in depth of the potential at two extrema is a renormalization scale-independent quantity [61]. However, from a practical point of view, it is desirable that the logarithms in the e ective potential do not become too \large" so as to lead to numerical instabilities [62]. We have checked our calculation for some di erent values of the renormalization scale and the value of = 300 GeV proved to be a stable choice. Among the input parameters in (4.1), we xed the standard vev v as above and took the rest of the parameters randomly in the range shown in the rst row of table 1, restricted to mH > mh. After checking for simple perturbativity and bounded from below conditions at tree-level,7 we picked only the points where the shifted masses squared MS2(vi) were 7Since we work with a xed renormalization scale where the one-loop corrections are not large, we expect the tree level relations to be valid to a good approximation [64{66]. For bounded from below conditions, this is a conservative choice as one-loop corrections may enlarge the possible parameter space [67]. Sample t cos( ) mH+ (GeV) mA (GeV) mH (GeV) mh (GeV) G NR 1 20 50 50 0:1 0:1 90 1000 90 1000 90 1000 0 150 6000 6000 dash denotes the same range as in the previous row. positive and the solutions for the shift mi2i in (4.5) were real. Then we further selected only the points where the pole mass m^h calculated as in section 5 fell in the experimental range of (5.1).8 In this way, we generated the sample G with 294437 points among which 4525 had two minima at one-loop, 17 of which were non-regular. To nd the second minimum, we explicitly minimized the real part of the e ective potential (3.1) starting from the non-standard minimum at tree-level and then retained only the points where the value of the potential at that minimum was real [63]. Given the small number of nonregular points, we generated another sample denoted as NR focusing only on non-regular points by imposing large t as in the second row of table 1; other ranges were kept the same, except for m212 which was chosen positive. Sample NR thus contained 185905 points among which 1563 had two minima at one-loop. Hereafter, unless explicitly speci ed, we will consider only the joint sample of G and NR. However, we would like to emphasize that, even though samples G and NR have a similar number of points, the parameter space scanned by sample NR represents only a small portion of the parameter space probed by sample G. This justi es our de nition of non-regular points. After the selections described above, the input parameters mH+ ; mA; mH get roughly con ned to the range [90; 500] and the non-standard higgsses acquire pole masses in a similar range with slightly smaller maximal value for m^H+ . In contrast, the distribution for t is homogeneous in the range of table 1 for the whole sample but, as we select only the points with two minima at one-loop, it gets separated into two ranges, 1 t . 3:8 for the regular minima, and 24 . t 50 for the non-regular ones. To check our numerically implemented formulas, we performed the following consistency checks: 1. Vanishing of the pole masses for the Goldstone bosons G ; G0 for zero external momentum for the three cases where we successively add the one-loop scalar, vector boson and fermion contributions. 2. Equality of the pole masses for the CP even higgsses H; h for zero external momentum and the eigenvalues of the explicitly calculated second derivative matrix of the e ective potential in the real neutral directions (3.3) in all three cases of successive addition of the one-loop scalar, vector boson and fermion contributions. 8The exact adopted procedure di ers slightly with respect to the range of mh: instead of post-selecting only the values of mh for which the pole mass coincided with the experimental range, we randomly selected the input mh in the range [0; 200] GeV and then later varied only this value searching for the correct pole mass m^h. If a solution were found, we kept that point. This procedure resulted in the approximately homogeneous distribution of mh in the range shown in table 1. This modi cation speeded up the generation of points. HJEP1(207)6 m222 (green points) while the fermions contribute negatively. The fermion contribution to while the contribution to m222 (red curve) is practically the same for the type I and II models m211 (orange curve) applies only to the type II case but vanishes for the type I case. The contributions from the gauge bosons are shown in the purple dashed line. only whereas the red points consider all the contribution in the type I or II model. 7 Results Let us rst quantify the shifts mi2i in (4.5) for the di erent contributions. The contribution coming from the gauge bosons only depends on the value of v and, for fermions, it depends on v and . The scalar contribution depends on many parameters coming from the scalar potential. The dependence of the di erent contributions on tan are shown in gure 3. We can see that the dominant contribution for m211 comes from the scalars whereas m222 also has large positive contributions from scalars but they are partly canceled by the negative contribution from fermions (top). Such a partial cancellation in m222 can be clearly seen in gure 4 where we show the contribution only from scalars (green points) and all the blue (red) points represent (non-)regular points. contributions (red points). The orange curve in gure 3, which quanti es the bottom contribution to m211 in the type II model, shows that it is negligible in the range of tan we are interested in and our calculation that uses the type I model applies equally well to the type II case. All the di erent contributions are calculated using eq. (4.6) and the scalar contribution in particular depends on the mi2i themselves and these are taken as the total contributions. The remaining contributions of gauge bosons (purple dashed curve in gure 3) are much smaller. The contribution from scalars are calculated using the whole sample (G + NR) described in section 6 which also includes the points with only one vacuum. The deviation of the location of our vacuum when we add V do V0 is illustrated in gure 5 where we show the ratio of t for V0 + V (ttree+ ) to that of V0 (t ) against the ratio of the vev for V0+ V (vtree+ ) to that of V0 (v). We only show the points with two coexisting minima and divide the points between the regular ones (blue) and the non-regular ones (red). We can see that as mi2i get shifted by mi2i all points with two minima have their vevs decreased while t mostly increases for the regular points and mostly decreases for the non-regular points. Note that the non-regular points only consider large t whereas the regular points only include moderate t roughly up to 3:8. As the location of (v1; v2) for our vacuum is the same for V0 and the one-loop corrected potential (4.3) we can also interpret this plot as the modi cation of the vev location of V0 + V compared to V0 + V + V1l. If we had considered all the points including the points with only our vacuum, the majority of points would follow the behavior of the regular points in blue. For the case of coexisting vacua, we can see a clear di erence in behavior between the regular points and the non-regular ones in gure 6 where we plot the potential di erence with respect to m212. The left panel shows these quantities using the tree-level potential V0 while the right panel is the same plot for the one-loop potential (4.3). We can clearly see that the potential di erence goes continuously to zero as m212 ! 0 for the blue points representing the regular minima. Moreover, for positive m212 our vacuum is guaranteed to be the global minimum in both tree level or one-loop potential. This behavior is opposite potential (2.1) is considered. This plot should be compared with gure 1. Right: the full 1-loop e ective potential (4.3) is considered. The green points represent a change in sign of the tree level prediction for the potential di erence. (red), scalars (purple), and vectors bosons (green) to the one-loop potential di erence of the right panel of gure 6. In the left plot only regular points are considered while the right plot shows the behavior of the non-regular ones. for negative m212. The reason is that only m212 breaks explicitly the Z2 symmetry of the theory and it controls the degeneracy breaking of the spontaneously breaking minima. This behavior is not followed by the non-regular minima that do not have degenerate minima in the m212 ! 0 limit. Some points (green) in which our vacuum is not the global one at tree-level even get inverted and become the global minimum as the one-loop corrections are added. The same conclusion is reached if we had compared the one-loop potential to V0 + V instead: some cases where our minimum is not the deepest at tree-level becomes the global minimum at one-loop. We can have an idea of the di erent one-loop contributions in gure 7 where we separate the potential di erence in gure 6 into its di erent contributions. Regarding the regular points (left plot), it can be seen that the one-loop potential di erence is almost entirely due ence. The color coding follows gure 6. to the tree-level contribution (blue) since the contributions from V (yellow), fermions (red) and scalars (purple) approximately cancel each other while the contribution from gauge bosons (green) is negligible compared to the others. This behavior justi es our choice for the renormalization scale. For the non-regular points (right plot), no clear pattern emerges. We can also see in gure 8 that there are points where the potential di erence is raised as well as points where it is lowered by the one-loop corrections for both regular and non-regular points. Considering that the discriminant (2.18) test is applicable for all cases where at least one normal vacuum is known, we can investigate if it is still a good predictor for the oneloop potential with two coexisting minima. We can adapt the tree-level discriminant to one-loop in the following three ways: D1 = eq. (2.18) ; D2 = eq. (2.18) mi2i!mi2i+ mi2i; v!v(0); ! (0) ; D3 = eq. (2.18) mi2i!mi2i+ mi2i : (7.1) The quantity D1 is the discriminant calculated with V0 as the whole potential while D2 is calculated by using V0 + V in (4.3). In the latter case, the quadratic parameters mi2i get shifted and the location of the minimum, denoted above as (v(0); (0)) or as tree + (superscript/subscript) in gure 5, do not coincide with the ones used as input, denoted as (v; ). The last adaptation D3 considers the shifts in the quadratic parameters but keeps the vevs as (v; ). We will test here if any of these discriminants are capable of distinguishing if our vacuum is the global one at one-loop only using parameters at treelevel.9 9Strictly speaking, some calculation at one-loop is required for some of these quantities depending on how the calculation is set up. Using the splitting of the potential in the form (4.3), D1 is the most natural quantity to use and there is no one-loop calculation required. If V0 + V is considered as the potential at tree-level, D2 or D3 are the natural quantities depending on which minimum is taken. points. The green points mark the cases where D1 predicts the opposite depth di erence. For the regular coexisting minima, gure 6 shows that m212 is already a good predictor of the global minimum, but we can test the discriminants in (7.1). The result of this test is shown in gure 9 where the potential di erence at one-loop is plotted against the three discriminants. We can see in the left and middle plots that D1 and D2 correctly predict the global minimum of the one-loop e ective potential while the right plot shows that D3 fails for more than 10% of the points. In constrast, the failure of all the discriminants (7.1) for the non-regular points (of samples G and NR) can be seen in gure 10 which shows the potential depth di erence as a function of the discriminants, similarly to the regular points in gure 9; note that the horizontal scale is very di erent. The green points in the second and fourth quadrants mark the cases where the discriminant D1 predicts the opposite behavior at one-loop. We can clearly see that all discriminants fail for a signi cant portion of points, not necessarily for the same ones. From the property of D1, we could have seen its wrong prediction in gure 6 as well. At last, to make sure that the points for which the discriminant test fails include phenomenologically realistic points, we have checked the viability of the red/green points in gure 6 by considering phenomenological constraints implemented in the 2HDMC code [68, 69]. We found that in the case of the type I model around half of the green points are allowed by experimental constraints such as the S, T, U precision electroweak parameters and data from colliders implemented in the HiggsBounds and HiggsSignals packages [70{ 73], while in the type II model they are all excluded. For the type II model we have also included constraints coming from Rb measurements [74] as well as B meson decays [75{78] which prove to be very strong by setting mH+ > 480 GeV independently of the value of tan . For type I, since the red/green points have tan > 10, these constraints do not impose any further restrictions [79, 80]. We also checked that all points still respect simple bounded from below and perturbativity constraints [4]. 8 Conclusions We have studied the one-loop properties of the real Two-Higgs-doublet model with softly broken Z2 with respect to the possibility of two coexisting normal vacua. The softly broken nature must remain at one-loop and the case of two coexisting normal minima can be classi ed into two very distinct types depending on their nature in the vanishing m212 limit: regular minima that spontaneously break the symmetry and the non-regular minima (or minimum) that preserve the symmetry, i.e., they are inert-like in the symmetric limit. Since in the rst case the two minima are degenerate in the symmetry limit, even at oneloop, they are connected by the Z2 symmetry and then they should di er only by the sign of v2. After the inclusion of the m212 term, the sign of v2 continue to be opposite for our vacuum and the non-standard one so that the two regular minima are found in the rst and fourth quadrants in the (v1; v2) plane. In contrast, the non-regular minima deviate from the inert-like minima and both deviate to the rst quadrant when m212 is positive. The vacua that spontaneously break Z2 in the m212 ! 0 limit behave rather regularly and we can distinguish which coexisting minimum is the global one by just examining the sign of m212: when it is negative our vacuum is only a metastable local one and the opposite is true if it is positive. For this type of coexisting vacua, the discriminant at tree level (D1;2 in eq. (7.1)) is still a good predictor of the nature of the minimum it is calculated with. For the non-regular coexisting vacua, m212 is positive in the convention that our vacuum has both vevs positive and it cannot be used as an indicator. At tree level, the discriminant of ref. [45, 46] is a very convenient way of testing if our vacuum is the global one because only the location of our minimum is required. However, at one-loop, this discriminant is not a precise indicator of which minima is the global one. We have found realistic cases where our vacuum is not the global minimum at tree-level but it becomes the global one after the addition of the one-loop corrections. Few cases for the opposite behavior were also found. As the discriminant e ectively distinguishes the sign of the potential di erence between the coexisting minima at tree-level, the latter itself is also a not good indicator for the non-regular minima. This is reminiscent of the exact Z2 symmetric case (inert model) investigated in ref. [48]. On the other hand, we were unable to nd a discriminant that works for both regular and non-regular minima at one-loop. Finding a simple and precise criterion for global minimum does not seem to be an easy task as that was not achieved even in the simpler exact Z2 limit. We also emphasize that for our parametrization which enforces our vacuum to be a minimum from the start, and for the chosen generic ranges of parameters (sample G), the occurrence of non-regular minima, as suggested by its name, is much rarer: only 38% correspond to the non-regular cases and, among then, only 0:3% are coexisting minima. In summary, the soft-breaking term m212 controls the lifting of the degeneracy of the regular coexisting minima (moderate tan ) even at one-loop and can be used as the sole indicator of which minimum is the global one. That is not true when two coexisting nonregular vacua (large or small tan ) exist: the discriminant that is a precise indicator at tree-level is not reliable at one-loop and explicit calculation of the potential depths must be carried out. Acknowledgments HJEP1(207)6 A.L.C. acknowledges nancial support by Brazilian CAPES (Coordenac~ao de Aperfeicoamento de Pessoal de N vel Superior) and C.C.N. acknowledges partial support by Brazilian Fapesp grant 2014/19164-6 and 2013/22079-8, and CNPq 308578/2016-3. The authors thank Pedro Ferreira for useful discussions while this work was developed. A Deviation of a potential minimum Take a potential V0(') depending on real scalar elds 'i for which we know a minimum (extremum) 'i = 'i satisfying We want to quantify how the location of the minimum and the value of the potential deviate when we add a small perturbation U on the potential as Since the rst term vanishes due to (A.1), we get the deviation to rst order where Ui The deviation in the value of the minimum can be equally expanded around ' as We assume V0 has no at direction around '. We rst quantify the deviation of the location of the minimum as The derivative of the perturbed potential gives 'j + O( 2) : V = V0 + U : 'i = 'i + 'i : 'j = Mji1Ui ; V (' + ') = V0(') + U (') 1 2 UiMij1Uj + O( 3) ; (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) after using (A.5). Generically the third term contributes to deepen the potential. That is the dominant contribution when the perturbation vanishes on the unperturbed minimum: U (') = 0. The latter happens for the m212 term when perturbing the inert-like minima while for the spontaneously broken Z2 minima the dominant term is the second, linear in the perturbation. (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) Finding more than one normal vacua Let us rewrite the minimization equations in eq. (2.5) as where A X = M ; A = 345 1 345 2 ! ; v2! 2m212=v1v2 depends on the vevs. end, we can formally write (B.1) as X = A 1M and equate v12v22 = 4m412= 2. We obtain To nd solutions (v1; v2) of (B.1) for m212 6= 0, we rst need an equation for . For that 2 2 = [m121 2] ; where 1 = ! 0 and m413245!m12v112v22=4. When m212 6= 0, we can nd the possible values for 1m222 and 2 = 345m222 2 2m121. The m212 ! 0 limit is taken as (extrema) from the quartic equation 2 = 2[m121 2] : trema in this case [35, 36]. We also can see that the root 345 +p We know there are at most four real solutions coinciding with the maximal number of ex1 2 from the lefthandside is always positive due to bounded below conditions. Possible solutions for (B.1) depends on the sign of and we allow both signs for v1v2. We can see that for the same , ipping the sign of m212 is equivalent to ipping the sign of v1v2. Once some solution = 0 is found, we can nd the vevs from the relation X = A 1M After ensuring these expression are positive, we can extract v1; v2 and tan with the sign convention where v1 > 0 and v12 = v22 = 2(m222 2(m211 ( 345 ( 345 )2 )2 2) 1) 1 2 1 2 > 0 ; > 0 : sign(v2) = sign( m122) : The derivatives are with respect to the elds around the values in eq. (2.2). D Cubic derivatives The coe cients iS in (4.6) are de ned as (quadratic) = 0 1 + a 2 p m211 m212 m212! : m222 a = 1; 2 iS = SS where S = fG+; H+; G0; A; H; hg, sec = fchar, odd, eveng refers to the di erent scalar sectors and Usec diagonalizes Msec. The last subindex SS refers to one of the diagonal entries following the ordering in (4.13). Explicit calculation leads to Matrix of second derivatives The mass matrices at tree-level for the charged, CP-odd and CP-even scalar sectors prior to imposing the minimization conditions are given respectively by Mchar('i) = h Modd('i) = h Meven('i) = h i = i = The term (quadratic) refers to the quadratic contribution given by + (quadratic) ; (C.1) + (quadratic) ; (C.2) + (quadratic) : (C.3) 1A0 = 1s20 + 345c20 1H0 = 3 1c2 0 + 345s 0 s 0 + 2c 0 1h0 = 3 1s2 0 + 345c 0 c 0 2s 0 v2 v2 v2 v2 v2 v1 v2 v1 ; ; ; ; ; ; (C.4) (C.5) (D.1) (D.2) where the mixing angles +; 0; 0 were de ned in (4.14) with tan 2 + = tan 2 0 = tan 2 = m2H+ m2A0 The coe cients 2S can be obtained from 1S for each scalar S above by using the replace Cubic and quartic couplings 0;+ ! angles Given that the cubic and quartic couplings do not depend on the quadratic parameters, they correspond to the tree-level ones listed, e.g., in ref. [81] if we are in the limit 0 ! , . If we want the couplings with these corrections, we can adapt the rotation ! 0 or ! + or ! 0 for the couplings that do not mix the charged sector with the CP-odd sector because in the latter there is an ambiguity in distinguishing + from 0. Another ambiguity arises if the couplings are written in terms of (v; ) instead of (v1; v2) because then we need to distinguish coming from the vevs and the 0;+ coming from the diagonalization of the shifted mass matrices. We adopt the convention that igABCD is the Feynman rule associated to the vertex ABCD. This is opposite to e.g. ref. [81]. We also abbreviate e.g. gG0G0G0G0 as gG04. The essential set of quartic couplings is 2c20 + 345c2 0 ; 2c2+ + 3c2 +) 1 1 1 gG04 = 3 1c40 + 2s40 + 2 345s20c20 ; gh2G02 = 1s2 0c20 + 2c2 0s20 + 345c20 0 gHhG02 = 2 s2 0( gG02A2 = 43 s22 0( 1 + 2) + 1 2 345c22 0 ; = 1s2+c20 + 2s20c2+ + 3c2+ 0 gG02H+H gG02G+H = 2 s2 +( gh2G0A = 2 s2 0( gG0A3 = gG0AH+H = 3 1 1 2 v1( 1 i 2 gG02G+G Coupling = = More couplings can be obtained from simple replacements; see table 2. The essential set of cubic couplings is ghG02 = = v2 2s2+ c 0 v1 1c2+ s 0 + 3(v2c2+ c 0 v1s2+ s 0 )+ 1 2 45s2 + (v1c 0 v2s 0 ) ; 3)s2 + s 0 + 2 v2( 2 1 3)s2 + c 0 + 1 2 45c2 + (v1c 0 v2s 0 ) ; ( 4 5)(v1s 0 v2c 0 ) : The remaining couplings can be obtained through reparametrization symmetries as shown Replacement Coupling Obtainable from Finally we note that, although the reparametrization symmetry already allows a huge simpli cation in the computation of the cubic and quartic couplings needed for our calculation, their use can be error-prone. In our routines we have adopted a di erent approach, namely we expanded the tree-level scalar potential in terms of physical elds S = fh; H; A; G0; G+; H+; G ; H g and performed derivatives to obtain the desired couplings. For instance, gh3G0 = ghH2 = Si=0 (E.3) HJEP1(207)6 Some couplings are absent because CP is conserved and A; G0 are CP odd. (F.1c) F Self-energy for the scalars We show here the di erent contributions to the self-energy of scalars due to scalars (S), fermions (F ) and gauge bosons (V ) in the loop. We rst list the contributions from scalars in the loop for the di erent sectors. For the CP odd sector we have gG0SS0 B(MS; MS0 ; s) + 2jgG0G+H j2B(MG+ ; MH ; s) ; 2 (F.1a) gA2SS0 B(MS; MS0 ; s) + 2jgAG+H j2B(MG+ ; MH ; s) ; (F.1b) 16 2 SAA(s) = gA2S2 A(MS) + gA2SSA(MS) 16 2 SAG0 (s) = 16 2 S G0A(s) = gAG0S2 A(MS) + gAG0SSA(MS) 1 2 X S=G0;A;H;h X S=G+;H+ gASS0 gG0SS0 B(MS; MS0 ; s) + 2gAG+H gG0G+H B(MG+ ; MH ; s) : 1 2 + 1 2 + + X X S=G0;A;H;h S = G0; A S0=H;h X S=G0;A;H;h X S = G0; A S0=H;h X S = G0; A S0=H;h X S=G+;H+ X S=G+;H+ X S=G+;H+ X The self-energy for the charged sector is G+G+ (s) = gG+G S2 A(MS) + gG+G SSA(MS) H+H+ (s) = gH+H S2 A(MS) + gH+H SSA(MS) In the CP even sector we have 16 2 SHH (s) = G+H+ (s) = 16 2 S H+G+ (s) = gG+H S2 A(MS) + gG+H SSA(MS) X S=G+;H+ gG+SS0 gH+SS0 B(MS; MS0 ; s) jgG+SS0 j2B(MS; MS0 ; s) X S=G+;H+ X S=G+;H+ jgH+SS0 j2B(MS; MS0 ; s) 1 2 X S=G0;A;H;h 1 2 + 1 2 + + X X X X S=G0;A;H;h S = G+; H+ S0=G0;A;H;h S=G0;A;H;h S = G+; H+ S0=G0;A;H;h X S = G+; H+ S0=H;h (F.2a) (F.2b) (F.2c) (F.2d) (F.2e) (F.2f) (F.3a) (F.3b) (F.3c) (F.3d) (F.3e) (F.3f) ) (F.4) (F.5) the general formula [49, 50]: SFS0 (s) = Nc 16 2 ( tr YfSf0 (YfS0f0 )y 2 where Nc is the number of colors of the fermion in the loop, the B function was given in (5.4) while BF F (mf ; mf0 ) is de ned as BF F (mf ; mf0 ; s) mf mf0 tr[YfSf0 YfS0f0 ]B(mf ; mf0 ; s) ; Note that couplings such as gG+G A0 = 0 due to CP conservation. The fermionic corrections to the propagator of a scalar S to a scalar S0 are given by BF F (mf ; mf0 ; s) [(s mf mf0 )B(mf ; mf0 ; s) A(mf ) A(mf0 )] : VS;So0dd(s) = SS0 V; char(s) = g 2 g 2 g 2 H A G0 H G 1 1 pyt2 cs 0 1 pyt2 ss 0 1 i pyt2 cs 0 5 bb pyb2 Cbh1 pyb CH 1 2 b i pyb2 CbA 5 i pyt2 ss 0 5 i pyb2 CbG 5 ybCbH PR ybCbG PR yt cs + PL yt ss + PL We denote the vertex of the scalar Sk to the fermions f and f 0 by contain the 5 matrix while Y refers to the transformation These Yukawa couplings are listed in table 4 where the coe cients CfS depend on the model used as in table 5. Finally, the contributions coming from the gauge bosons in the loop are given by iYfkf0 and they may SS0 16 2 4c2w CSeoS00 CSeo0S00 BSV (MS00 ; mZ ; s) + e+ 3 3 e+ + SS0 4c2w A(mZ ) + A(mW ) 16 2 4c2w CSeoS00 CSeo0S00 BSV (MS00 ; mZ ; s) + + SS0 4c2w A(mZ ) + A(mW ) 1 Ce+ ; ; 1 Co+ o+ 3 2 3 2 16 2 4 SS00 CS0S00 BSV (MS00 ; mW ; s) + 4 SS00 CS0S00 BSV (MS00 ; mW ; s) + SS0 s2w BSV (MS; 0; s) + cot22w BSV (MS; mZ ; s) + s2wCS+CS+0 s2wm2Z BV V (mZ ; mW ; s) + m2W BV V (0; mW ; s) o+ 1 Co+ 2 SS00 CS0S00 BSV (MS00 ; mW ; s) + SS0 3s2w cot22w A(mZ ) + A(mW ) ; 3 2 (F.6) (F.7) (F.8) A G c 0 0 H G c + 0 s + 0 c + 0 H G s + 0 c + 0 H s 0 c 0 C+ S s H c G + + where there is an implicit summation on S00. The loop functions BSV (mV ; mS; s) and BV V (mV ; mV 0 ; s) are de ned as BSV (mS; mV ; s) = 4m2V m2V 0 p 4 4m2V m2V 0 B(0; 0; s) + B(mS; 0; s) + B(mV ; 0; s) (m2V + m2V 0 4m2V m2V 0 m2S m2V m2V A(mV ) 4m2V A(mV 0 ) : 4 m2V 0 B(mV ; mV 0 ; s) B(0; mV 0 ; s) s A(mV ) ; (F.9) BV V (mV ; mV 0 ; s) = 2B(mV ; mV 0 ; s) + The couplings CSxS0 and CSx can be read from the following tables. G Reparametrization symmetry Here we list some useful reparametrization symmetries that allow us to relate di erent quartic and cubic scalar couplings that are numerous. These relations help us to check di erent couplings or deduce new ones from a smaller set. Moreover, given simple relations, they minimize errors and speed up numerical implementation. The simplest reparametrization is to exchange elds of the same type, one in 1 and the other in 2, together with a 90 shift in the respective mixing angle: 8H ! h h ! >: 0 ! H 0 + =2 8G0 > < 0 A > : ! A ! 0 G 0 0 ! 0 + =2 Qeven : Qodd : Qchar : ghG02 hG02 ! ghG02 j 0! 0 =2 This is what allowed us to relate gHG02 to ghG02 in (E.2a) after 0 ! 0 HG02 by the inverse transformation of Qeven. Another example is gHhAG0 in (E.1k): it is odd by the replacement 0 ! 0 + =2 or 0 ! 0 + =2. The reparametrization symmetry above arises by noting that the original elds in 1;2 are left invariant by the transformations and consequently the potential is also invariant. (F.10) 8 G+ H+ > < > : ! + ! ! H+ G+ + + =2 : (G.1) =2 since Q12 : Qe!o : 1 $ 2 ; m121 $ m222 ; 1 $ 8m121 $ m222 ; 1 $ : v1 $ v2 2 ; > < 8 H $ h ; G 0 $ A ; < 8 0 ! 0 ! > : + ! =2 =2 =2 H! G0! A0 ; G0! A0 ! H! 8 0 $ v1 ! > :v2 ! 0 ; 0 ; + : 0 iv1 iv2 The last type of symmetry we can explore for reparametrization is the original gauge invariance. Discrete subgroups are the most useful. For example, the reparametrization arises because of invariance by the gauge transformation a ! 1 i ! a ; a = 1; 2 : For special eld con gurations, we can also de ne an exchange reparametrization between charged elds and CP even neutral elds as below If we take the case of CP even elds, 1 2 ! c 0 H s 0 h s 0 H + c 0 h ! ; we see 1;2 are the same after the transformation Qeven. The same is true for the other pair of scalars and diagonalization angles. The other reparametrization symmetry we can use is the exchange symmetry of the original potential (2.1), restricted10 to the case of the CP conserving softly broken Z2, For the elds with de nite masses at tree level this transformation reads (G.2) (G.3) (G.4) ; (G.5) (G.6) (G.7) (G.9) and G+ = G1=p2; H+ = H1=p2 can be chosen by electromagnetic gauge invariance. Similarly, Qc$e : Qc$o : H! G1! H1 G0! A0 G2! H2 0 $ + ; for G0 = A0 = 0 ; 0 $ + ; for H = h = 0 ; (G.8) and G+ = iG2=p2; H+ = iH2=p2 can be chosen. 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A. L. Cherchiglia, C. C. Nishi. One-loop considerations for coexisting vacua in the CP conserving 2HDM, Journal of High Energy Physics, 2017, 106, DOI: 10.1007/JHEP11(2017)106