Gauge invariant Barr-Zee type contributions to fermionic EDMs in the two-Higgs doublet models
Tomohiro Abe, Junji Hisano, Teppei Kitahara, Kohsaku Tobioka
aa r X i v : . [ h e p - ph ] A p r KEK–TH–1682UT–13–38IPMU–13–0209November, 2013
Gauge invariant Barr-Zee type contributionsto fermionic EDMsin the two-Higgs doublet models
Tomohiro Abe (a) , Junji Hisano (b , c) , Teppei Kitahara (d) , and
Kohsaku Tobioka (c , d) (a) Theory Group, KEK, Tsukuba, 305-0801, Japan (b)
Department of Physics, Nagoya University, Nagoya 464-8602, Japan (c)
Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277–8583, Japan (d)
Department of Physics, University of Tokyo, Tokyo 113–0033, Japan
Abstract
We calculate all gauge invariant Barr-Zee type contributions to fermionic electric dipolemoments (EDMs) in the two-Higgs doublet models (2HDM) with softly broken Z symmetry.We start by studying the tensor structure of h → V V ′ part in the Barr-Zee diagrams, and wecalculate the effective couplings in a gauge invariant way by using the pinch technique. Then wecalculate all Barr-Zee diagrams relevant for electron and neutron EDMs. We make bounds onthe parameter space in type-I, type-II, type-X, and type-Y 2HDMs. The electron and neutronEDMs are complementary to each other in discrimination of the 2HDMs. Type-II and type-X2HDMs are strongly constrained by recent ACME experiment’s result, and future experimentsof electron and neutron EDMs may search O (10) TeV physics. Introduction
The standard model (SM) has been worked very well for a long time, and its last missing piece, theHiggs boson, was finally discovered by the Large Hadron Collider (LHC) experiment at CERN [1,2].This is a triumph of the SM and a great step to understand physics at the electroweak scale.However, there are many unsolved problems within the SM, for example, the observed dark matterparticles and baryon asymmetry in the Universe. From theoretical viewpoint, the gauge hierarchyproblem is still in question. Hence, there have been many attempts to solve such problems inframeworks beyond the SM.In a bottom-up approach towards new physics beyond the SM, an attractive option is to studythe two-Higgs doublet models (2HDMs). They are simple and may be low-energy effective theoriesof various new physics models. Since 2HDMs generally have dangerous flavor changing neutralcurrents (FCNCs), we particularly consider 2HDMs with softly broken Z symmetry which sup-presses the FCNCs. If two Higgs fields do not distinguish the generations of quarks and leptons,the models are classified, with respect to the Yukawa interactions, into four types: type-I, type-II,type-X, and type-Y. One of the important feature of 2HDMs is that there is a new CP violationsource in the Higgs potential.In general, the powerful tool to seek new physics including 2HDMs is of course the LHC whichmay directly probe physics up to a few TeV. Another possibility is provided by low energy precisionmeasurements, such as in flavor physics. The remarkable feature is that these measurements have apotential to investigate new physics beyond the LHC reach by orders of magnitude. In particular,the electric dipole moments (EDMs) are interesting because the EDMs are highly sensitive to CPviolation in physics beyond the SM. While the SM predictions of EDMs are much lower thanthe current experimental bounds, assuming the strong CP problem is solved by some mechanism,such as the Peccei-Quinn symmetry [3, 4], new physics around TeV scale would give large valueswithin the reach of the future EDM measurements [5]. In addition, the electroweak baryogenesis(EWBG) [6–9], which needs a new CP violation source, may lead to larger values of EDMs thanthe SM predictions.The EDM measurements, therefore, are concrete tests on 2HDMs containing a new CP phase.In the models, the one-loop contributions to the fermionic EDMs are too small to observed sincethose contributions are proportional to the third power of small Yukawa couplings. Some two-loopdiagrams, called the Barr-Zee diagrams [10], which we show in Fig. 1, may give sizable contributionsto the EDMs, since they are suppressed by only one power of small Yukawa couplings. Thesediagrams contain one-loop effective vertices, hγγ , hγZ , and H ∓ W ± γ . The Type-II case wasevaluated in Refs. [11, 12], but the results in the previous works are not gauge invariant. Weimprove this point by using the pinch technique [13–15] and make the Barr-Zee diagrams gaugeinvariant. We also study EDMs in the other three types as well as the type-II.We organize this paper as follows. In Sec. 2, we briefly review the 2HDMs with softly-broken Z symmetry. In Sec. 3, we study the tensor structure of the effective vertices which are neededto evaluate the Barr-Zee diagrams, and show the gauge invariant tensor structure. After that,we calculate the effective vertices explicitly and show that the diagrams which include the gaugebosons are not gauge invariant. This implies that we need some non-Barr-Zee diagrams to makethe effective vertices gauge invariant. We show it by using the pinch technique. The formulaeof the gauge invariant Barr-Zee diagrams are given in Sec. 4, and their numerical evaluation ispresented in Sec. 5. There we discuss the complementarity between the electron and neutron EDM1 a) (b) (c) (d) Figure 1: Barr-Zee diagrams, which contribute to fermionic EDMs at two-loop level.measurements in discrimination of 2HDMs, and also prospects of future experiments. Sec. 6 isdevoted to conclusions and discussion. Notations and details of the calculation are given in theAppendices.
We briefly review the models discussed in this paper. We have two Higgs doublets, H and H ,and they have the vacuum expectation values (VEVs). The Higgs doublets are parametrized asfollows, H i = π + i √ (cid:0) v i + σ i − iπ i (cid:1)! , ( i = 1 , . (2.1)In order to avoid the dangerous FCNC problems, we introduce the Z symmetry. The Z symmetryis assumed to be softly broken so that the domain-wall formation in the early universe is suppressed.Under this symmetry, the Higgs doublets are translated into H → + H and H → − H , and theHiggs potential is given as V = m H † H + m H † H − (cid:16)(cid:0) Re m + i Im m (cid:1) H † H + ( h.c. ) (cid:17) + 12 λ ( H † H ) + 12 λ ( H † H ) + λ ( H † H )( H † H ) + λ ( H † H )( H † H )+ (cid:16) λ e i φ ( H † H ) + ( h.c. ) (cid:17) . (2.2)The third and last terms in this potential contain complex parameters. While one of them canbe eliminated by redefinition of Higgs fields, another phase is physical so that CP symmetry isbroken. In this paper we take the Higgs VEVs, v and v , real using the gauge symmetry and alsoredefinition of a Higgs field. In this basis, two phases in the potential are related to each others bythe stationary condition of the potential, V ′ = 0. In this paper we choose φ as an input parameterfor CP violation.We also use the following variables for convenience in this paper,cos β = v v , sin β = v v , (2.3) M ≡ v + v v v Re m . (2.4)2able 1: Summary of the Higgs fields which couple to quarks and leptons in four types.Type I II X Y u H H H H d H H H H ℓ H H H H and where v = q v + v = ( √ G F ) − / ≃
246 GeV . (2.5) G F is the Fermi constant. It is easy to find the charged Higgs boson mass, m H ± = M − v ( λ + λ cos(2 φ )) . (2.6)On the other hand, since CP symmetry is broken in the Higgs potential, we need to diagonalize a3 by 3 matrix to find the neutral Higgs masses.The Yukawa interaction in this model is given by L Yukawa = − q L e H y u u R − q L H i y d d R − ℓ L H j y e e R + h.c., (2.7)where e H = ǫH ∗ , and i, j = 1 or 2, depending on the type of 2HDMs. While up-type quarks coupleto only to H , leptons and down-type quarks couple to either H or H due to the Z symmetry.We summarize which Higgs fields couple to fermions in Table 1.The detail information of the models, such as mass eigenvalues, mixings, and interactions ofthe Higgs bosons, are given in Appendix A. In this section we calculate effective vertices relevant for the Barr-Zee diagrams in a gauge invariantway. To make our point clear, we start by exploring the relevant form of the effective vertices shownin Fig. 2. Then we calculate effective hγγ , hZγ and H ∓ W ± γ vertices . We also calculate the pinchterms to make the vertices gauge invariant. We study the tensor structure of the effective vertices shown in Fig. 2. This part has two Lorentzindices, and does not contain γ -matrices. Then it is generally written asΓ µν = A g µν + A p µ p ν + A p µ p ν + A p µ p ν + A p µ p ν + i Γ ǫ µνρσ p ρ p σ , (3.1)where p µ and p ν are the momenta of V and V , respectively, and their direction is outgoing. Weconsider the case that V is on-shell photon, and thus the terms proportional to p µ are dropped. In3igure 2: Effective Higgs boson-vector boson-vector boson vertices.addition, the gauge symmetry of photon requires Γ µν p µ = 0 . Then, the effective vertex for h - V - V in the case that V is on-shell photon is defined with only two form factors asΓ µν ( p , p ) =Γ( p , p ) ( − ( p p ) g µν + p µ p ν ) + i Γ ( p , p ) ǫ µνρσ p ρ p σ . (3.2)Note that this tensor structure is led from the gauge symmetry of on-shell photon. Then all theeffective vertices must be this form. We emphasize this point because sometimes this point seemsoverlooked, for example the tensor structure in Eq. (9) in Ref. [11] is different from Eq. (3.2).However, in the actual calculation, we would find terms proportional to p µ p ν and g µν , whichshould vanish and do not appear in Eq. (3.2), namely we would find the effective vertices become e Γ µν ( p , p ) =Γ µν ( p , p ) + Γ P ( p , p ) g µν + Γ D ( p , p ) p µ p ν , (3.3)where Γ µν ( p , p ) is defined in Eq. (3.2). These extra terms, Γ P and Γ D , are apparently againstthe gauge invariance, but, nevertheless, they would appear. See, for example, Eq. (9) in Ref. [11].As we will see the following sections, we find they disappear if we take on-shell conditions for allthe external legs. However, we should keep them off-shell except for a single photon because weuse the effective vertices to calculate the Barr-Zee diagrams. Hence we need to consider how todeal with these gauge variant terms.Fortunately, it is found that the p µ p ν term does not contribute to the EDMs at two-loop level. IfΓ µν ( p , p ) contains terms proportional to p µ p ν , the diagrams shown in Fig. 1 contain the followingstructures, u ( p + q ) ℓ/ p + q/ − ℓ/ − m f u ( p ) , (3.4) u ( p + q ) 1/ p + ℓ/ − m f ℓ/u ( p ) , (3.5)where Eq. (3.4) (Eq. (3.5)) comes from Figs. 1(a) and 1(c) (Figs. 1(b) and 1(d)). If we omit O ( y f ) terms, we can ignore the mass term in the fermion propagator and the mass of the externalfermions. Then, by using the equation of motion of the external fermions, u ( p + q )( ℓ/ − / p − q/ ) 1/ p + q/ − ℓ/ u ( p ) , (3.6) u ( p + q ) 1/ p + ℓ/ ( ℓ/ + / p ) u ( p ) . (3.7)4ow it is apparent that these terms do not contain σ µν γ structure because all the γ -matrices arecanceled out. Therefore the terms which are proportional to p µ p ν in the effective vertices do notcontribute to the EDMs. Then we can safely drop the Γ D term from Eq. (3.3).On the other hand, the Γ P term in Eq. (3.3) remains as long as we take off-shell conditions.This is nothing strange because the gauge invariance is promised for S -matrix, not for effectivecoupling. Then the gauge invariance will recover once we calculate non-Barr-Zee diagrams as wellas the Barr-Zee diagrams, namely a full two-loop order calculation manifestly gives the gaugeinvariant results. However, it is very tough work to accomplish it. Instead of the full two-looporder calculation, we make the effective vertex gauge invariant by borrowing some terms from non-Barr-Zee diagrams. This technique is known as the pinch technique, and the borrowed terms arecalled pinch terms [13–15]. As we will see in the fallowing section, we find that Γ P term in Eq. (3.3)is completely compensate with the pinch terms.Hereafter we calculate both − ( p p ) g µν + p µ p ν and g µν terms, and demonstrate the latter termcompletely vanishes thanks to the pinch terms. hγγ and hZγ vertices — W boson loop — Now we move on to calculate the effective vertices for hγγ and hZγ , which appear in Figs. 1(a) and1(b). In the following, p is the momentum of the external (on-shell) photon where p = 0, and p isthe momentum of the virtual gauge boson in the Barr-Zee diagram. Note that the diagrams whichcontain both W and H ± in the loop are absent in the 2HDM because g γW ± H ∓ = g ZW ± H ∓ = 0,where H ± is a physical charged scalar not a NG boson.In this subsection, we focus on W boson loops of the hγγ and hZγ effective vertices becausewe find these are not gauge invariant as long as we keep off-shell conditions. We work in ’t Hooft-Feynman gauge and find the hγγ and hZγ effective vertices are given byΓ µνhGγ ( p , p ) = + e (4 π ) m W g W W h g W W G × " Γ AhGγ ( p µ p ν − p p g µν ) + Γ PhGγ ( p − m G ) g µν + Γ BhGγ p µ p ν + Γ ChGγ [( p + p ) − m h ] g µν . (3.8)whereΓ AhGγ =4 (cid:18) − J ( m W ) + 6 J ( m W ) + m G m W ( J ( m W ) − J ( m W )) + (cid:18) − m G m W (cid:19) m h m W J ( m W ) (cid:19) , (3.9)Γ PhGγ = + 3 J ( m W ) , (3.10)Γ BhGγ = − J ( m W ) + m G m W ( J ( m W ) − J ( m W )) + 12 m G p (1 − J ( m W ))+ m G m W ( p + p ) p J ( m W ) , (3.11)5 ChGγ = − (cid:18) − m G m W (cid:19) J ( m W ) , (3.12)where G stands for Z or γ , and where J ( m ) = Z dx Z − x dy − p m x (1 − x ) − ( p + p ) − p m xy , (3.13) J ( m ) = Z dx Z − x dy xy − p m x (1 − x ) − ( p + p ) − p m xy . (3.14)The explicit forms of couplings, such as g W W h and g W W G , are given in Appendix A. This result isconsistent with previous works, for example in Eq. (9) in Ref. [11].Although the gauge invariance requires Γ P = Γ B = Γ C = 0 as we discussed in Sec. 3, it isnot satisfied in Eq. (3.8). So we should consider the gauge invariance for the EDM calculationcarefully. As discussed in Ref. [11], the Γ C term does not contribute to the EDMs. Because thisterm is proportional to inverse of neutral Higgs propagator, it can reduce neutral Higgs propagatorin Barr-Zee diagram. Then we can apply the vertex relation P h g Aℓℓh g W W h = 0, where g Aℓℓh is axial-scalar coupling of external fermion ℓ with neutral Higgs bosons h and P h is summationfor three neutral Higgs bosons. The Γ B terms do not contribute to the EDMs neither, becausethese terms do not keep σ µν γ structure as we discussed in Sec. 3.1. Then only the Γ P terms areproblematic. Actually the Γ P terms vanish once we consider the pinch contributions as will beshown.There are many two-loop diagrams which contribute to the EDMs, as well as the Barr-Zeediagrams. Once we calculate all the diagrams, the result must be gauge invariant. Therefore thegauge variant terms we discussed above should be canceled out by contributions from non-Barr-Zeediagrams. In order to see this cancellation, we do not need to calculate all the diagrams, but only the pinch contributions . The gauge invariance of Eq. (3.8) would be recovered by borrowing someterms from non-Barr-Zee diagrams.For this purpose, we calculate the diagrams shown in Fig. 3. These diagrams contain derivativecouplings which are contracted with the gamma matrices by the Lorentz index. Then, these termscancel out internal fermion propagators. We pick up the terms in which the fermion lines with redcolor in Fig. 3 are canceled out, and they are just the pinch contributions which make Barr-Zeecontributions gauge invariant. These terms are schematically shown in Fig. 4. In ’t Hooft-Feynmangauge , we findFig. 3 | pinch = X h Z ℓ i ˜Γ µνhGγ ( − q, ℓ ) i ( q − ℓ ) − m h − ig νρ ℓ − m G ( − iγ ρ g Gℓℓ ) i / p + q/ − ℓ/ − m f ( − ig ℓℓh ) , (3.15)where ˜Γ µνhGγ ( p , p ) = − g µν e (4 π ) g W W h m W g W W G ( p − m G ) J ( m W ) . (3.16) We show this vertex relation in Appendix A.3.4.
In other gauge, we would need other diagrams as well as shown in Fig. 3. a) (b) (c) (d) Figure 3: Diagrams containing the pinch terms for the effective hγγ and hZγ vertices. We pinchthe fermion lines shown with red color. The dashed lines attached to the fermion lines are thephysical scalars, and those not attached are would-be NG bosons. (a) (b)
Figure 4: Diagrams (a) and (b) are the diagrams after pinched away the red lines. Figs. 3(a) and3(b) and Figs. 3(c) and 3(d) become diagrams (a) and (b), respectively.Here, J is given in Eq. (3.13), g Gℓℓ and g ℓℓh are couplings of external fermion ℓ with gauge andHiggs bosons, respectively, and R ℓ = R d ℓ/ (2 π ) . Since Fig. 1(a) with effective hGγ vertices iscalculated asFig. 1(a) = X h Z ℓ i Γ µνhGγ ( − q, ℓ ) − ig νρ ℓ − m G i ( ℓ − q ) − m h ( − iγ ρ g Gℓℓ ) i / p + q/ − ℓ/ − m f ( − ig ℓℓh ) , (3.17)we find Eq. (3.16) is nothing but parts of the effective vertices by comparing Eq. (3.17) to Eq. (3.15),and cancels the second term in Eq. (3.8) (Γ P ) which is gauge variant term. In other words, thepinch term certainly cancels the gauge variant term and make the effective coupling gauge invariant.After adding the pinch terms, we finally find the gauge invariant W loop contributions to theeffective hγγ and hZγ vertices for the Barr-Zee diagrams,Γ µνhGγ ( p , p ) = e (4 π ) m W g W W h g W W G Γ AhGγ ( − ( p p ) g µν + p µ p ν ) , (3.18)where Γ AhGγ is given in Eq. (3.9). hγγ and hZγ vertices — fermion, H ± loop — For the Barr-Zee diagram calculation, we need other contributions to effective hγγ and hZγ vertices.We calculate the fermion loop contribution to the effective hγγ and hZγ vertices. We denote thefermion as f . Note that they are independent from the gauge fixing terms. Hence Γ P and Γ D in7q. (3.3) are zero. We find Γ and Γ defined in Eq. (3.2) areΓ hGγ ( p , p ) = + N c (4 π ) eQ f g Vffh ( g LGff + g RGff ) 2 m f (cid:0) J ( m f ) − J ( m f ) (cid:1) , (3.19)Γ hGγ ( p , p ) = + N c (4 π ) eQ f ( ig Affh )( g LGff + g RGff ) 2 m f J ( m f ) , (3.20)where N c is the color factor, for example N c = 3 for the top quark loop, Q f is the QED charge ofthe fermion in the loop, for example Q f = 2 / P and Γ D in Eq. (3.3) are zero. We find Γ and Γ defined in Eq. (3.2) areΓ hGγ ( p , p ) = − π ) eg H + H − h g GH + H − m H ± J ( m H ± ) , (3.21)Γ hGγ ( p , p ) =0 . (3.22) H ∓ W ± γ vertices — W , H ± loop — The effective vertices for H ∓ W ± γ , shown in Figs. 1(c) and 1(d), are also necessary to calculatethe all the Barr-Zee diagrams. Note that these Barr-Zee contributions have not been studied inthe literature yet, and we calculate for the first time them. To find a gauge invariant set for theBarr-Zee diagrams, we need to take into account for the pinch contributions. Calculations aretedious and long, so the details are given in Appendix C. After summing up all terms which arerelevant for the EDM calculations, we find the following gauge invariant effective vertex:Γ µνH − W + γ ( p , p ) = + 1(4 π ) ( p µ p ν − p p g µν ) × + X h eg W + H − h g W W h Z dz Z − z dy − yz − z + 4 − m H ± − m h m W yzm W (1 − z ) + m h z − p z (1 − z ) − p p yz − X h eg W + H − h g H + H − h Z dz Z − z dy yzm H ± (1 − z ) + m h z − p z (1 − z ) − p p yz ! , (3.23)Γ µνH + W − γ ( p , p ) = (cid:16) Γ µνH − W + γ ( p , p ) (cid:17) ∗ . (3.24)Here we have already omitted the terms which do not contribute to the EDM calculations. There might also be fermion loops in the effective H ∓ W ± γ vertices. It is found that thefermion loops in the effective H ∓ W ± γ vertices do not contribute to the EDMs if we consider onlythe CP phase in the Higgs potential in 2HDMs. While another CP phase is present in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, the contributions to the EDMs should be much suppressed dueto the GIM mechanism. Then, we do not calculate the fermion loop contributions to the effective H ∓ W ± γ vertices in this paper. These terms do not contribute to the on-shell H − → W − γ process neither. EDM from Barr-Zee diagram
In this section we calculate diagrams in Fig. 1. The EDM, d ℓ , for fermion ℓ is defined through H eff = i d ℓ ψ ℓ σ µν γ ψ ℓ F µν , (4.1)where σ µν = i γ µ , γ ν ] . (4.2)Once we get the gauge invariant effective vertex whose tensor structure is given in Eq. (3.2), wefind the neutral Higgs boson contributions to d ℓ as( d ℓ ) Fig. 1(a)+Fig. 1(b) = 12 X G = Z,γ X h (cid:0) g LGℓℓ + g RGℓℓ (cid:1) Z ℓ (cid:0) ig Aℓℓh Γ hGγ (0 , ℓ ) + g Vℓℓh Γ hGγ (0 , ℓ ) (cid:1) ℓ − m G ℓ − m h . (4.3)where g L ( R ) Gℓℓ is for couplings of left(right)-handed fermion ℓ with gauge boson G , and g V ( A ) ℓℓh is for(axial) scalar couplings with scalar boson h . Here we keep only the leading term for p and q , andignore mass term in the fermion propagator, and we have used a relation, ǫ µναβ γ α γ β = − iγ [ γ µ , γ ν ] . Note that we work in ’t Hooft-Feynman gauge in Eq. (4.3). If we work in other gauge, gaugeboson propagators contain the terms that proportional to ℓ ν and contract with the effective vertices.Since Γ µν ( − q, ℓ ) ℓ ν = 0, the terms proportional to ℓ ν in the gauge boson propagators always vanish.Therefore the Barr-Zee diagrams are gauge invariant as long as the effective vertices are gaugeinvariant.In the similar manner, we find the charged Higgs boson contribution to the leptonic EDMs as( d ℓ ) Fig. 1(c)+Fig. 1(d) = 12 √ es Z ℓ ℓ − m W ℓ − m H ± i Im (cid:0) g R ¯ νeH + Γ H − W + γ (0 , ℓ ) (cid:1) . (4.4)Here we have used the following relations, g L ¯ eνH − = (cid:0) g R ¯ νeH + (cid:1) ∗ , (4.5)Γ H + W − γ (0 , ℓ ) = (cid:0) Γ H − W + γ (0 , ℓ ) (cid:1) ∗ . (4.6)The charged Higgs contributions to the up-type and down-type quark EDMs are derived by replac-ing g R ¯ νeH + Γ H − W + γ in Eq. (4.4) by g R ¯ duH − Γ H + W − γ and g R ¯ udH + Γ H − W + γ , respectively. We denote s and c as sine and cosine of the Weinberg angle, respectively, in the following.The chromo-EDMs (cEDMs) also contribute to the neutron EDM. Its definition is similar toEq. (4.1), replace F µν by g s G µν , H eff = i d cq qg s σ µν γ G µν q, (4.7)where g s and G µν are the QCD coupling and the field strength of the gluon, respectively.The formulae of EDMs include complicated functions. Here, we show the approximated ex-pressions in the decoupling limit for qualitative discussion, while all plots are drawn by using the9xact formulae. The exact formula are given in Appendix B. In the decoupling limit all the non-SMparticles are degenerated, heavier than the electroweak scale, and decoupled from the SM sector.We can take such a limit by M → ∞ where M is defined in Eq. (2.4).Since the results depend on the Yukawa structure, we introduce the following notation tosimplify our expressions: G Ax = Type-I Type-II Type-X Type-Y u/c/t d/s/b − β − βe/µ/τ − β tan β − , (4.8) S x = u/c/t − d/s/b e/µ/τ , (4.9)where index A represents type of the model, and index x is for flavor.It is found that the EDMs for fermion ℓ in the decoupling limit are approximated to be (cid:18) d ℓ e (cid:19) W ≃ − X G Aℓ × e (cid:18)
15 + 2 ln (cid:18) M TeV (cid:19)(cid:19) (cid:0) g Lγℓℓ + g Rγℓℓ (cid:1) + g W W Z (cid:18) . .
71 ln (cid:18) M TeV (cid:19)(cid:19) (cid:0) g LZℓℓ + g RZℓℓ (cid:1)! , (4.10) (cid:18) d ℓ e (cid:19) top ≃ + X × e (cid:0) . G Aℓ + 7 . (cid:1) (cid:0) g Lγℓℓ + g Rγℓℓ (cid:1) + e (cid:0) . G Aℓ + 2 . (cid:1) (cid:0) g LZℓℓ + g RZℓℓ (cid:1)! , (4.11) (cid:18) d ℓ e (cid:19) bottom ≃ + X × e (cid:0) . G Aℓ + 0 . G Ab (cid:1) (cid:0) g Lγℓℓ + g Rγℓℓ (cid:1) + e (cid:0) . G Aℓ + 0 . G Ab (cid:1) (cid:0) g LZℓℓ + g RZℓℓ (cid:1)! , (4.12) (cid:18) d ℓ e (cid:19) tau ≃ + X × e (cid:0) . G Aℓ + 0 . G Aτ (cid:1) (cid:0) g Lγℓℓ + g Rγℓℓ (cid:1) + e (cid:0) . G Aℓ + 0 . G Aτ (cid:1) (cid:0) g LZℓℓ + g RZℓℓ (cid:1)! , (4.13) (cid:18) d ℓ e (cid:19) H ± ≃ + X G Aℓ × . e (cid:0) g Lγℓℓ + g Rγℓℓ (cid:1) + 0 . g ZH + H − (cid:0) g LZℓℓ + g RZℓℓ (cid:1)! , (4.14) (cid:18) d ℓ e (cid:19) HW γ ≃ − X G Aℓ S ℓ × (cid:18) .
23 + 0 .
20 ln (cid:18) M TeV (cid:19)(cid:19) , (4.15)( d cq ) top ≃ + X × g s (cid:0) . G Aq + 5 . (cid:1) , (4.16)( d cq ) bottom ≃ + X × g s (cid:0) . G Aq + 0 . G Ab (cid:1) , (4.17)10here X = 1(4 π ) m ℓ M cos βλ sin 2 φ, (4.18)and we use M S mass of M Z scale, m e = 0 .
511 MeV, m τ = 1 .
75 GeV, m u = 1 .
40 MeV, m t = 170 . m d = 2 .
92 MeV and m b = 2 .
94 GeV. Notice that the EDMs and cEDMs are proportionalto λ sin 2 φ = Im[ λ exp( i φ )], namely the imaginary part of the coupling which is needed for CPviolation.It is found that the W loop contributions are dominant in large parameter region. Among thecontributions from fermion loops, only the top quark contributions are relevant in the decouplinglimit as long as tan β .
10. In the similar manner, we can make approximation of cEDMs. Thediagrams with charged Higgs boson in hγγ , hZγ , and H ∓ W ± γ couplings are smaller than the othercontributions. Note that the contributions from Z boson exchange diagrams are proportional to (cid:0) g LZℓℓ + g RZℓℓ (cid:1) . Although this factor is numerically small at electron EDM case, one must not ignoreat quark EDM case. Actually, Z boson exchange diagrams occupy 30–50% of all contribution atdown quark EDM case.In the decoupling limit the bottom quark and tau lepton contributions are small because oftheir small Yukawa couplings. In the non-decoupling region, however, these are not necessarilyvalid. Their leading contributions are given by diagrams in which heavy Higgs propagate, andtheir values are approximately O ( X G Aℓ G Ab/τ ( m h − m h ) /M ), where h and h is the heaviest andthe next heaviest Higgs bosons, respectively. These contributions are enhanced by tan β whentan β ≫
1. Thus, when tan β is large, the contribution may be sizable in the non decouplingregion. Now we evaluate the EDMs numerically. At first, in Fig. 5, we show the numerical improvementby the pinch contributions. Here we consider the electron EDM in the type-II 2HDM. The verticalaxis in the Fig. 5 is difference of the gauge invariant EDM contribution and non-invariant one, ∆,defined as ∆ = ( d e ) gauge non-inv. − ( d e ) gauge inv. ( d e ) gauge inv. , (5.1)where the gauge non-invariant EDM contribution ( d e ) gauge non-inv. is gotten by calculating onlyBarr-Zee diagrams [11, 12]. The horizontal axis is the mass of charged Higgs boson. We taketan β = 10, λ = λ = λ = λ sin 2 φ = 0 . λ is uniquely determined. We find that the pinch contributions are 5–8%. This isnot big improvement from the numerical point of view. However, we would like to emphasize thatour result is now gauge invariant, which must be satisfied when we discuss observables.Next, we discuss dependence of the electron EDM on the types of 2HDMs. The contributionsfrom each types of diagrams to the electron EDM for type-I and II cases in Figs. 6 and 7, respectively.Here we take tan β = 3 or 50, and λ = λ = λ = λ sin 2 φ = 0 . W boson contribution to h → γγ is dominant and that allcontributions to the electron EDM are proportional to 1 / tan β for tan β &
1. On the other hand,11
00 400 600 800 10000246810 M H + @ GeV D D @ % D Figure 5: Numerical improvement of electron EDM by the pinch contributions in the type-II 2HDM.We take tan β = 10, λ = λ = λ = λ sin 2 φ = 0 . β is large, the W boson and top quark contributions are not suppressed and the bottom quarkand tau lepton contributions also become dominant due to the non-decoupling effect. Since thesigns of the bottom quark and tau lepton contributions are opposite to that of the W boson, theaccidental cancellation occurs in some parameter region. Thus, the tan β dependence is non-trivialin the type-II case.In Figs. 8, the electron EDM is shown in four types of 2HDMs as functions of tan β and chargedHiggs boson mass. We take λ = λ = λ = λ sin 2 φ = 0 . λ = 0 .
25. The regions filled withred color in the figures show the excluded regions by the latest upper bound on electron EDM,which is derived by the ACME experiment, | d e | < . × − e cm (90% CL [3]) . (5.2)The blue dashed lines are the future prospects given in Table 2.experiments sensitivities on d e Fr [16] 1 × − e cmYbF molecule [17] 1 × − e cmWN ion [18] 1 × − e cmTable 2: Future prospects on electron EDM.The electron EDM in the type-X and Y models has similar behavior to the type-II and I ones,respectively, because leptons couple to H in type-I and Y models, and to H in type-II and Xmodels. We find that type-II and type-X 2HDMs are strongly constrained by the recent ACMEexperimental result, except for regions where the cancellation among diagrams occurs, as shownin Fig. 8. Furthermore, the future experiments could cover wide parameter regions with chargedHiggs mass smaller than 1 TeV even in type-I and Y cases.Next let us consider the neutron EDM. Even when the Peccei-Quinn mechanism [19] is operative,the neutron EDM is generated by higher-dimensional CP-violating operators in QCD, such as quark12
00 400 600 800 1000 - ´ - - ´ - - ´ - - ´ - M H + @ GeV D d e (cid:144) e @ c m D Type I tan
Β = total W H ± + H ¡ W ± Γ tb Τ
200 400 600 800 1000 - ´ - - ´ - - ´ - - ´ - - ´ - - ´ - - ´ - ´ - M H + @ GeV D d e (cid:144) e @ c m D Type I tan
Β = Figure 6: Anatomy of the type-I electron EDM. Various Barr-Zee contributions to the electronEDM are shown as functions of charged Higgs mass M + H . We take tan β = 3 or 50, and λ = λ = λ = λ sin 2 φ = 0 .
5. The mass of lightest neutral Higgs is 126 GeV. We see that W loop is thedominant contribution. The qualitative feature are independent from tan β .
200 400 600 800 1000 - ´ - - ´ - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d e (cid:144) e @ c m D Type II tan
Β = total W H ± + H ¡ W ± Γ tb Τ
200 400 600 800 1000 - ´ - - ´ - ´ - M H + @ GeV D d e (cid:144) e @ c m D Type II tan
Β = Figure 7: Anatomy of the type-II electron EDM. The input parameters are the same as in Fig. 6.In contrast of the type-I case, the qualitative feature depends on tan β . For large tan β , bottomquark and tau lepton contributions are sizable due to the tan β enhancement of their Yukawacouplings.EDMs and also cEDMs with mass dimension up to 5. The neutron EDM is evaluated from theup and down quarks EDM and cEDM with the QCD sum rules [20–22]. The evaluation still O (1)uncertainties from the excited state contribution to the correlation function [20], and also frominput parameters [21]. In this paper we use the result in Ref. [22] since it gives more conservativeprediction for the neutron EDM, d n =0 . d d − . d u + e (0 . d cd + 0 . d cu ) . (5.3)Here, the Peccei-Quinn mechanism is assumed.Before going to evaluate the neutron EDM, we discuss behaviors of the quark EDMs and cEDMsin the 2HDMs. We plot the contributions from each types of diagrams to the down and up quarkEDMs and cEDMs in the type-I case in Figs. 9 and 10. The input parameters are the same asin Fig 6. We see that the W boson and top quark contributions give the dominant contributionsto the EDMs and cEDMs, respectively, and the tan β dependence is 1 / tan β , as expected fromEq. (4.18). It is found that the sizes of cEDMs and EDMs are comparable to each others so that13
00 400 600 800 10001020304050 eEDM Type I t a n β M [GeV] H + -1×10 -30 -1×10 -29 -5×10 -30 -5×10 -29 -5×10 -31 -1×10 -28 (a)
200 400 600 800 10001020304050 eEDM Type II t a n β M [GeV] H + -28 -27 -28 -1×10 -28 -28 -1×10 -27 (b)
200 400 600 800 10001020304050 eEDM Type X t a n β M [GeV] H + -27 -28 -28 -28 -5×10 -28 -1×10 -27 (c)
200 400 600 800 10001020304050 eEDM Type Y t a n β M [GeV] H + -1×10 -29 -5×10 -30 -5×10 -29 -1×10 -28 (d) Figure 8: Electron EDM on charged Higgs boson mass and tan β plane in four types of 2HDMs.We take λ = λ = λ = λ sin 2 φ = 0 . λ = 0 .
25. The regions filled with red color show thecurrent bound [3]. The blue dashed lines are the future prospects given in Table 2.14oth contributions have to be included in evaluation of the neutron EDM.In Figs. 11 and 12, the contributions from each types of diagrams to the down and up quarkEDMs and cEDMs in the type-II case are also shown. The EDMs and cEDMs have qualitativelydifferent behaviors from the type-I case. We find that the largest contribution to the neutron EDMcomes from down quark cEDM. The top quark loop dominates in the down quark cEDM (andalso the up quark cEDM) for small tan β , while the bottom quark one quickly dominates it whentan β is large. The later comes from the non-decoupling effect. Thus, the neutron EDM wouldbe enhanced when tan β is large. It is also found that the down quark EDM has similar behaviorto the electron EDM in the type-II case, though it is smaller than the down quark cEDM in theneutron EDM.Here, we ignore the QCD corrections to the quark EDMs and cEDMs. The QCD correctionsmay change them up to O (10)% [23, 24], while the neutron EDM evaluation from the quark EDMsand cEDMs may have larger uncertainties. See Ref. [24] for evaluation for the QCD corrections tothe Barr-Zee diagrams.Now we show the neutron EDM in four types of 2HDMs in Fig. 13. The regions filled with redcolor in Fig. 13 show the excluded region by the current neutron EDM data, | d n | < . × − e cm (90% CL [4]) . (5.4)The blue dashed lines are the future prospects given in Table 3.Table 3: Future prospects for neutron EDMexperiments sensitivities on | d n | cyro EDM [25] 1 . × − e cmPSI (Phase II) [26] 5 × − e cmIt is found that the neutron EDM in the type-X case has similar behavior to the type-I inlow tan β region because the down quark Yukawa couplings in these two types are the same. Thedifference in high tan β region between Figs. 13(a) and 13(c) is due to the large tan β enhancementof the tau lepton Yukawa coupling. The behavior of the neutron EDM in the type-Y case is quitesimilar to the type-II case. This is because the cEDM contribution is dominant in both cases.It is found in comparison of Fig. 8 with Fig. 13 that both measurements of the electron andneutron EDMs are complementary to each others in order to discriminate the 2HDMs. We maychoose one from the four models in future.Before closing this section, we would like to give a comment on the constraints on the parameterspace. We have shown that some parameter regions are constrained by EDMs in Figs. 8 and 13.The constrained regions have an overlap with other constraints, such as flavor physics [27, 28] ordirect search of heavy Higgs bosons [29]. Note that it is known that the custodial SU (2) symmetryis broken in the Higgs potential in 2HDMs with the CP violation, and ρ parameter might deviatefrom one at the one loop level [30]. However, if heavy Higgs boson mass scale M is large or ifcoupling λ − λ are not large, this contribution is small. We checked that this contribution doesnot conflict with the current bound in all figure of this paper.15
00 400 600 800 1000 - ´ - - ´ - - ´ - - ´ - - ´ - ´ - M H + @ GeV D d d (cid:144) e @ c m D Type I tan
Β = total W H ± + H ¡ W ± Γ tb Τ
200 400 600 800 1000 - ´ - - ´ - - ´ - - ´ - ´ - M H + @ GeV D d d (cid:144) e @ c m D Type I tan
Β =
200 400 600 800 1000 - ´ - ´ - ´ - ´ - M H + @ GeV D d C d @ c m D Type I tan
Β =
200 400 600 800 1000 - ´ - ´ - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d C d @ c m D Type I tan
Β = Figure 9: Anatomy of the type-I down quark EDM and cEDM. Various Barr-Zee contributionsto the EDM and cEDM are shown as functions of charged Higgs mass M + H . We take tan β = 3and 50. Other input parameters are the same as in Fig 6. We see that W and top give dominantcontributions to EDM and cEDM, respectively.
200 400 600 800 1000 - ´ - - ´ - ´ - ´ - M H + @ GeV D d u (cid:144) e @ c m D Type I tan
Β = total W H ± + H ¡ W ± Γ tb Τ
200 400 600 800 1000 - ´ - - ´ - - ´ - ´ - ´ - ´ - M H + @ GeV D d u (cid:144) e @ c m D Type I tan
Β =
200 400 600 800 1000 - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d C u @ c m D Type I tan
Β =
200 400 600 800 100002. ´ - ´ - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d C u @ c m D Type I tan
Β = Figure 10: Anatomy of the type-I up quark EDM and cEDM. We taketan β = 3 and 50. Otherinput parameters are the same as in Fig 6. We see that W and top give dominant contributions.16
00 400 600 800 1000 - ´ - - ´ - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d d (cid:144) e @ c m D Type II tan
Β = total W H ± + H ¡ W ± Γ tb Τ
200 400 600 800 1000 - ´ - ´ - M H + @ GeV D d d (cid:144) e @ c m D Type II tan
Β =
200 400 600 800 100001. ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d C d @ c m D Type II tan
Β =
200 400 600 800 100001. ´ - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d C d @ c m D Type II tan
Β = Figure 11: Anatomy of the type-II down quark EDM and cEDM. We take tan β = 3 and 50. Otherinput parameters are the same as in Fig 6. In contrast of the type-I case, the qualitative featuredepends on tan β . For large tan β , the bottom quark and tau lepton contributions are sizable dueto the tan β enhancement of their Yukawa couplings.
200 400 600 800 1000 - ´ - - ´ - ´ - ´ - M H + @ GeV D d u (cid:144) e @ c m D Type II tan
Β = total W H ± + H ¡ W ± Γ tb Τ
200 400 600 800 1000 - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d u (cid:144) e @ c m D Type II tan
Β =
200 400 600 800 1000 - ´ - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d C u @ c m D Type II tan
Β =
200 400 600 800 100001. ´ - ´ - ´ - ´ - ´ - ´ - M H + @ GeV D d C u @ c m D Type II tan
Β = Figure 12: Anatomy of the type-II up quark EDM and cEDM. We take tan β = 3 and 50. Otherinput parameters are the same as in Fig 6. In contrast of the type-I case, the qualitative featuredepends on tan β . For large tan β , bottom and tau contributions are sizable due to the tan β enhancement of their Yukawa couplings. 17
00 400 600 800 10001020304050 nEDM Type I M [GeV] H + t a n β -28 -29 -30 (a)
200 400 600 800 10001020304050 nEDM Type II M [GeV] H + t a n β -27 -26 -26 (b)
200 400 600 800 10001020304050 nEDM Type X t a n β M [GeV] H + -28 -29 -30 -1×10 -29 -1×10 -30 (c)
200 400 600 800 10001020304050 nEDM Type Y t a n β M [GeV] H + -27 -26 -26 (d) Figure 13: Neutron EDM on charged Higgs boson mass and tan β plane. The input parametersare the same as in Fig. fig:eEDM. The region filled with red color show the current bound [4]. Theblue dashed lines are the future prospects given in Table 3.18
00 1000 5000 1 ´ ´ - - - - - M H + @ GeV D - d e (cid:144) e @ c m D eEDM Type I
500 1000 5000 1 ´ ´ - - - - - M H + @ GeV D d e (cid:144) e @ c m D eEDM Type II
500 1000 5000 1 ´ ´ - - - - - - M H + @ GeV D d n (cid:144) e @ c m D nEDM Type II Figure 14: Electron and Neutron EDMs at large m H ± region in the type-II case. We take tan β = 10, λ = λ = λ = λ sin 2 φ = 0 . In this paper, we evaluated fermionic EDMs in 2HDMs with softly broken Z symmetry. We startedby calculating the Barr-Zee diagrams in a gauge invariant way by using the pinch technique. Themodification by the gauge invariant calculation is 5–8% numerically. This does not change theprevious result drastically, but important because physical quantities must be calculated in a gaugeinvariant way. We evaluated the electron and neutron EDMs in all four types in the 2HDMs. Wefind that type-II and type-X 2HDMs are strongly constrained by the latest ACME experimentbound on the electron EDM. The electron and neutron EDM measurements will improve in thefuture experiments. They are possible to seek physics at O (10) TeV scale (Fig. 6). The electronand neutron EDMs have different sensitivities on the 2HDMs, and they are complementary to eachother in discrimination of the type of 2HDMs.We have not addressed that the contributions from non-Barr-Zee type diagrams in this pa-per. Although they are naively expected to be smaller than the contributions from the Barr-Zeediagrams, they would become important once experiments find the EDMs and start precise mea-surements. To evaluate them, we need to calculate all diagrams at two-loop level. This issue may19e discussed elsewhere.It is worth referring to relation between EWBG and EDMs. In the 2HDMs, it is known thatEWBG may occur through a strongly first order electroweak phase transition [31–36]. For example,Ref. [35] numerically showed that the 2HDMs with softly-broken Z symmetry may accommodatea strongly first order phase transition when the lightest neutral Higgs boson is around 125 GeV.In order to achieve the EWBG, one needs some CP violation phases in Higgs potential. The EDMsearches could indirectly constrain parameter space which achieve the EWBG. In this paper, wefind that low tan β regions in 2HDMs are disfavored by electron EDM. On the other hand, in fact,a strongly first order phase transition, which is needed for EWBG, prefers low tan β region [35].Therefore there is a tension between EWBG and current bound on the EDM. Acknowledgments
The authors thank the Yukawa Institute for Theoretical Physics at Kyoto University, where thiswork was initiated during the YITP workshop on “LHC vs. Beyond the Standard Model (YITP-W-12-21)”, March 19-25, 2013, and also acknowledge the participants of the workshop for veryactive discussions. They would also like to thank Koji Tsumura, Eibun Senaha, Ryosuke Sato,Yasuhiro Yamamoto, and Motoi Endo for useful discussions and comments. The work of J.H. issupported by Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports,and Culture (MEXT), Japan, No. 24340047, No. 23104011 and No. 22244021, and also by WorldPremier International Research Center Initiative (WPI Initiative), MEXT, Japan. The figures forthis paper were drawn using Feynmf [37].
A 2HDMs
In this appendix, we present mass spectrum and also interactions in 2HDMs, which are used intext.
A.1 Relations between mass and gauge eigenstates
While eight scalar fields are present in 2HDMs, σ , , π ± , , π , , as in Eq. (2.1), those states are not mass eigenstates, namely their mass matrices are not diagonal-ized. We call them the gauge eigenstates. Corresponding to them, there are eight mass eigenstates,which we denote them as h , , (neutral Higgs bosons) ,H ± (charged Higgs bosons) ,π Z,W ± (would-be NG bosons) . (cid:18) π Z π A (cid:19) = (cid:18) cos β sin β − sin β cos β (cid:19) (cid:18) π π (cid:19) , (cid:18) π W ± H ± (cid:19) = (cid:18) cos β sin β − sin β cos β (cid:19) (cid:18) π ± π ± (cid:19) . (A.1)The matrix U for physical neutral Higgs bosons is given by a 3 by 3 matrix as h h h = U T σ σ π A = ω σ h ω σ h ω π A h ω σ h ω σ h ω π A h ω σ h ω σ h ω π A h σ σ π A (A.2)where X X ω Xi ω Xj = δ ij , X i ω Xi ω Yi = δ XY . (A.3)These relations are useful to find relations among some couplings. A.2 Higgs masses in 2HDMs
The mass terms for the neutral physical Higgs bosons are given by
L ⊃ − (cid:0) σ σ π A (cid:1) f M σ σ π A , (A.4)where (cid:16) f M (cid:17) = v λ + M sin β, (cid:16) f M (cid:17) = v λ + M cos β, (cid:16) f M (cid:17) = M − v λ cos(2 φ ) , (cid:16) f M (cid:17) = (cid:16) f M (cid:17) = (cid:0) v λ − M (cid:1) sin β cos β, (cid:16) f M (cid:17) = (cid:16) f M (cid:17) = 12 v λ sin(2 φ ) sin β, (cid:16) f M (cid:17) = (cid:16) f M (cid:17) = 12 v λ sin(2 φ ) cos β, (A.5)where λ = λ + λ + λ cos(2 φ ) , (A.6)and M is defined in Eq. (2.4). This mass matrix satisfies f M = U m h m h m h U T . (A.7)21n large M limit, we find the following expressions for mass and mixing angles. m h = v λ + v λ + 2 v v λ v + O ( M − ) ,m h = M (cid:0) O ( M − ) (cid:1) ,m h = M (cid:0) O ( M − ) (cid:1) . ω σ h ω σ h ω π A h = cos β (cid:0) − X sin β (cid:1) sin β (cid:0) X cos β (cid:1) − v v λ sin(2 φ ) M + O ( M − ) , ω σ h ω σ h ω π A h = − sin β sin θ cos β sin θ cos θ + O ( M − ) , ω σ h ω σ h ω π A h = − sin β cos θ cos β cos θ − sin θ + O ( M − ) , (A.8)where tan(2 θ ) = − (cos β − sin β )sin β cos β ( λ + λ − λ ) + λ cos 2 φ λ sin(2 φ ) ,X = v λ − v λ − ( v − v ) λ M . (A.9) A.3 Interactions in 2HDMs
Couplings which are relevant to calculation for the gauge invariant Barr-Zee contributions arewritten in this subsection. Our convention of the sign in covariant derivative is D µ = ∂ µ + igV µ . (A.10) A.3.1 V - ¯ f - f couplings These couplings are the same as the SM case, but we show them here to establish our conventions.For neutral gauge bosons,
L ⊃ − X G = γ,Z f γ µ g Gff f G µ , (A.11)where g Gff contains chirality structure, g Gff = g LGff P L + g RGff P R , (A.12)22here g Lγff = eQ,g Rγff = eQ,g LZff = esc (cid:0) T − s Q (cid:1) ,g RZff = esc (cid:0) − s Q (cid:1) . (A.13)For W boson, L ⊃ − √ uγ µ g W ud dW + µ + h.c., (A.14)where g W ud = V CKM es P L , (A.15)where V CKM is for the CKM matrix.
A.3.2 Yukawa couplings
The Yukawa interaction terms are described as − (cid:0) u d (cid:1) m diag. u + P s g uus s P s g uds + s + P s g dus − s − m diag. d + P s g dds s ! (cid:18) ud (cid:19) , (A.16)where s = h , h , h , π Z , and s ± = H ± , π W ± . We define g V and g A as g = g V + iγ g A . (A.17)Finally we find explicit expressions of the couplings. For the neutral Higgs bosons, g Vuuh = m diag. u v β ω σ h ,g Auuh = m diag. u v β ω π A h ,g Vddh = m diag. d v β ω σ h ( i = 1) m diag. d v β ω σ h ( i = 2) ,g Addh = m diag. d v tan βω π A h ( i = 1) − m diag. d v β ω π A h ( i = 2) . (A.18)Here, i corresponds to the same suffix of H i which couples to down-type quarks.23or the physical charged Higgs boson, g VudH + = 1 √ V CKM m diag. d v i ( − δ i sin β + δ i cos β ) − m diag. u v V CKM cos β ! ,g AudH + = − i √ V CKM m diag. d v i ( − δ i sin β + δ i cos β ) + m diag. u v V CKM cos β ! ,g VduH − = − √ V † CKM m diag. u v cos β − m diag. d v i V † CKM ( − δ i sin β + δ i cos β ) ! ,g AduH − = i √ V † CKM m diag. u v cos β + m diag. d v i V † CKM ( − δ i sin β + δ i cos β ) ! , (A.19)where i in the suffix is again the same suffix of H i which couples to down-type quarks. Sometimethe followings are useful: g udH + = g LudH + P L + g RudH + P R = + √ " − m diag. u v V CKM β ! P L + V CKM m diag. d v (cid:18) − δ i tan β + δ i β (cid:19)! P R ,g duH − = g LduH − P L + g RduH − P R = −√ " − m diag. d v V † CKM (cid:18) − δ i tan β + δ i β (cid:19)! P L + V † CKM m diag. u v β ! P R . (A.20) A.3.3 L WWW
These couplings are the same as the SM case, but we show them here to establish our conventions.
L ⊃ − X G = γ,Z ig W W G (cid:26) ( ∂ α W + β ) W − µ G ν ( g αµ g βν − g αν g βµ )+ W + β ( ∂ α W − µ ) G ν ( g αν g βµ − g αβ g µν )+ W + β W − µ ( ∂ α G ν )( g αβ g µν − g αµ g βν ) (cid:27) , (A.21)where g W W γ = e,g W W Z = es c. (A.22) A.3.4 W - W - h couplings L ⊃ X h g W W h W + µ W − µ h + 12 g ZZh Z µ Z µ h, (A.23)24here g W W h =2 m W v (cid:2) cos βω σ h + sin βω σ h (cid:3) ,g ZZh =2 m Z v (cid:2) cos βω σ h + sin βω σ h (cid:3) . (A.24)By using Eq. (A.3), we find that X h g Aℓℓh g W W h = 0 . (A.25) A.3.5 V - H + - H − couplings L ⊃ + i (cid:0) H + ∂ µ H − − H − ∂ µ H + (cid:1) (cid:0) g γH + H − A µ + g ZH + H − Z µ (cid:1) , (A.26)where g γH + H − = e,g ZH + H − = 12 esc ( c − s ) . (A.27) A.3.6 W ± - H ∓ - h couplings L ⊃ + ig W − H + h (cid:0) h∂ µ H + − H + ∂ µ h (cid:1) W − µ + ig W + H − h (cid:0) h∂ µ H − − H − ∂ µ h (cid:1) W + µ , (A.28)where g W ± H ∓ h = ± es (cid:0) − sin βω σ h + cos βω σ h ∓ iω π A h (cid:1) . (A.29)By using Eq. (A.3), we find that X h g W + H − h g W W h = 0 . (A.30) A.3.7 s + - s − - h couplings L ⊃ + g H + H − h H + H − h + g π W + π W − h π W + π W − h + g π W + H − h π W + H − h + g H + π W − h H + π W − h + 12 g π Z π Z h π Z π Z h, (A.31)25here g H + H − h = + v v (cid:0) − v λ + v ( − λ + λ + λ cos(2 φ )) (cid:1) ω σ h + v v (cid:0) − v λ + v ( − λ + λ + λ cos(2 φ )) (cid:1) ω σ h + v v v λ sin(2 φ ) ω π A h ,g π W + π W − h = − m h m W g W W h ,g π W + H − h = − m H ± − m h m W g W + H − h ,g H + π W − h = + m H ± − m h m W g W − H + h ,g π Z π Z h = − m h m Z g ZZh . (A.32) A.3.8 W ± - π ∓ - h couplings L ⊃ + ig W − π + h (cid:0) h∂ µ π + − π + ∂ µ h (cid:1) W − µ + ig W + π − h (cid:0) h∂ µ π − − π − ∂ µ h (cid:1) W + µ , (A.33)where g W ± π ∓ h = ± m W g W W h . (A.34) A.3.9 V - W ± - π ∓ couplings L ⊃ + X G = γ,Z (cid:0) g GW − π + G µ W − µ π + + g GW + π − G µ W + µ π − (cid:1) , (A.35)where g γW ∓ π ± = + em W ,g ZW ∓ π ± = − esm Z . (A.36) A.3.10 Some four-point couplings
L ⊃ + g H − π W + π W − π W + H − π W + π W − π W + + 12 g H − π W + π Z π Z H − π W + π Z π Z + g H − π W + H − H + H − π W + H − H + , (A.37)26here g H − π W + π W − π W + = X h m W g W + H − h g π W + π W − h ,g H − π W + π Z π Z = X h m W g W + H − h g π Z π Z h ,g H − π W + H − H + = X h m W g W + H − h g H + H − h . (A.38) B EDM formula details
In this section we present formulae for the Barr-Zee contributions to fermionic EDMs and cEDMs.
B.1 Fermion loops ( hγγ and hZγ ) After substituting Eqs. (3.19) and (3.20) for Eq. (4.3), we find the fermion loop contributions tothe EDMs for fermion ℓ are (cid:18) d ℓ e (cid:19) fermion = − m ℓ (4 π ) √ G F X f X h X G = γ,Z N c Q f (cid:0) g LGℓℓ + g RGℓℓ (cid:1) × " g Aℓℓh m ℓ /v g Vffh m f /v I G ( m f , m h ) + g Vℓℓh m ℓ /v g Affh m f /v I G ( m f , m h ) , (B.1)where I G ( m f , m h ) = (cid:0) g LGff + g RGff (cid:1) m f m h − m G ( I ( m f , m G ) − I ( m f , m h )) , I G ( m f , m h ) = (cid:0) g LGff + g RGff (cid:1) m f m h − m G ( I ( m f , m G ) − I ( m f , m h )) , (B.2)and where I ( m , m ) = Z dz (1 − z (1 − z )) m m − m z (1 − z ) ln m z (1 − z ) m ,I ( m , m ) = Z dz m m − m z (1 − z ) ln m z (1 − z ) m . (B.4) The functions f ( z ) and g ( z ) in Refs. [10, 11] are related to I and I as follows: I ( m , m ) = − m m f (cid:18) m m (cid:19) , I ( m , m ) = − m m g (cid:18) m m (cid:19) . (B.3) .2 Charged Higgs loops ( hγγ and hZγ ) By substituting the result in Eq. (3.21) into Eq. (4.3), we find the charged Higgs contribution tothe EDMs, (cid:18) d ℓ e (cid:19) scalar = + m ℓ (4 π ) √ G F X h X G = γ,Z (cid:0) g LGℓℓ + g RGℓℓ (cid:1) g Aℓℓh m ℓ /v g H + H − h v I G ( m H ± , m h ) , (B.5)where I G ( m H ± , m h ) = − g GH + H − v m h − m G × " ( I ( m H ± , m G ) − I ( m H ± , m h )) − ( I ( m H ± , m G ) − I ( m H ± , m h )) . (B.6) B.3 W loops ( hγγ and hZγ ) The EDM contributions from W boson loops are (cid:18) d ℓ e (cid:19) W = + m ℓ (4 π ) √ G F X h X G = γ,Z (cid:0) g LGℓℓ + g RGℓℓ (cid:1) g Aℓℓh m ℓ /v g W W h m W /v I GW ( m h ) , (B.7)where I GW ( m h ) = g W W G m W m h − m G × " − (cid:26)(cid:18) − m G m W (cid:19) + (cid:18) − m G m W (cid:19) m h m W (cid:27) (cid:2) I ( m W , m h ) − I ( m W , m G ) (cid:3) + (cid:26)(cid:18) − m G m W (cid:19) + 14 (cid:18) − m G m W + (cid:18) − m G m W (cid:19) m h m W (cid:19)(cid:27) (cid:2) I ( m W , m h ) − I ( m W , m G ) (cid:3) . (B.8)We note that when one chooses G = γ in Eq. (B.7) and drops m h /m W terms in Eq. (B.8), theEDM contribution from W boson loops becomes consistent with original result of Barr and Zee [10],where they ignored diagrams which contain only NG boson in the loop in Fig. 1 for simplicity andHiggs-NG bosons interaction is proportional to m h /m W (see Eq. (A.32)). B.4 H ∓ W ± γ In this paper we compute for the first time the EDM contributions from H ∓ W ± γ vertices whichare generated by W and charged Higgs boson loops. The detail of this derivation is given inAppendix C. The contributions to the EDMs are d ℓ e = − m ℓ (4 π ) √ G F S ℓ X h (cid:18) g Aℓℓh m ℓ /v g W W h m W /v e s I ( m h , m H ± ) + g Aℓℓh m ℓ /v g H + H − h v I ( m h , m H ± ) (cid:19) , (B.9)28here I ( m h , m H ± ) = m W m H ± − m W ( I ( m W , m h ) − I ( m H ± , m h )) , I ( m h , m H ± ) = m W m H ± − m W ( I ( m W , m h ) − I ( m H ± , m h )) , (B.10)and where I ( m , m h ) = Z dz (cid:18) z (1 − z ) − − z ) + m H ± − m h m W z (1 − z ) (cid:19) × m m W (1 − z ) + m h z − m z (1 − z ) ln (cid:18) m W (1 − z ) + m h zm z (1 − z ) (cid:19) ,I ( m , m h ) =2 Z dz m z (1 − z ) m H ± (1 − z ) + m h z − m z (1 − z ) ln (cid:18) m H ± (1 − z ) + m h zm z (1 − z ) (cid:19) . (B.11)Here we have used the following relations among the coupling,Im g R ¯ νeH + √ m e /v g W + H − h e/ (2 s ) ! = g Aeeh m e /v , Im g R ¯ udH + √ m d /v g W + H − h e/ (2 s ) ! = g Addh m d /v , Im g R ¯ duH − √ m u /v g W − H + h e/ (2 s ) ! = g Auuh m u /v . (B.12) B.5 CEDMs
The effective Hamiltonian for the cEDM is defined as Eq. (4.7). We find d cq = + m q (4 π ) √ G F X f X h g s m f m h " g Aqqh m q /v g Vffh m f /v I ( m f , m h ) + g Vqqh m q /v g Affh m f /v I ( m f , m h ) . (B.13) C Derivation for effective H − W + γ vertex In this appendix, we present explicit derivation of the effective H − W + γ vertex, which is generatedfrom bosonic loop diagrams, in 2HDMs.There are two types of loop diagrams; vertex corrections (Fig. 15) and wave function corrections(Fig. 16). The diagrams in Fig. 16(d) give nothing because of C -invariance. The contributions fromFig. 16(c) is always proportional to p ν . Thus they do not contribute to the on-shell amplitude of H ∓ → W ∓ γ nor the EDM at two-loop level by the same discussion in Sec. 3. Hence what we needto calculate are only the diagrams in Figs. 15, 16(a), and 16(b). In this section, we calculate thesediagrams in ’t Hooft-Feynman gauge. 29 a) (b) (c) (d)(e) (f) (g) (h) Figure 15: Diagrams for the vertex corrections to H − W + γ . Figs. 15(a)–15(f) depend on the gaugefixing parameter ξ , while Figs. 15(g) and 15(h) are independent of ξ . (a) (b) (c) (d) Figure 16: Diagrams of wave function type corrections.First, let us consider the diagrams in Figs. 15(a)–15(f). These diagrams depend on the gauge30xing parameter of W boson. We find X Figs. 15(a)-15(f) = + i (4 π ) D/ Γ(3 − D/
2) ( p µ p ν − p p g µν ) X h eg W + H − h g W W h × Z x + y + z =1 − yz − z + 4 − m H ± − m h m W yz (cid:2) m W (1 − z ) + m h z − p z (1 − z ) − p p yz (cid:3) − D/ + i (4 π ) D/ g µν X h eg W + H − h g W W h × " Γ(2 − D/ Z dz − (1 + z ) (cid:2) m W z + m h (1 − z ) − p H z (1 − z ) (cid:3) − D/ − m H ± − m h m W Γ(2 − D/ Z dz ( − z ) (cid:2) m W z + m h (1 − z ) − p H z (1 − z ) (cid:3) − D/ + (cid:0) p H − m H ± (cid:1) Γ(3 − D/ × Z x + y + z =1 (cid:2) m W (1 − z ) + m h z − p z (1 − z ) − p p yz (cid:3) − D/ , (C.1)where p H = ( p + p ) . We find g µν terms, which are not gauge invariant. We will show theseterms are canceled with other diagrams, that is, the pinch contributions.The diagrams in Figs. 15(g) and 15(h) are independent from the gauge fixing parameter.Fig. 15(g) + Fig. 15(h) = − i (4 π ) D/ Γ(3 − D/
2) ( p µ p ν − p p g µν ) X h eg W + H − h g H + H − h × Z x + y + z =1 yz (cid:2) m H ± (1 − z ) + m h z − p z (1 − z ) − p p yz (cid:3) − D/ − i (4 π ) D/ Γ(2 − D/ g µν X h eg W + H − h g H + H − h × Z dz − z (cid:2) m H ± z + m h (1 − z ) − p H z (1 − z ) (cid:3) − D/ . (C.2)Next we calculate the diagrams in Figs. 16(a) and 16(b). First we define the following notationfor self-energies: = i Π µH − W + ( p ) = ip µ Π H − W + ( p ) , (C.3)= i Π H − π W + ( p ) . (C.4)The direction of the momentum of Π µH − W + is shown in the figure. Using this notation, we findFig. 16(a) + Fig. 16(b) = − ig µν p H − m W (cid:16) − em W i Π H − π W + ( p H ) − em W Π H − W + ( p H ) (cid:17) + (cid:0) p − m W (cid:1) − ig µν p H − m W (cid:0) − e Π H − W + ( p H ) (cid:1) . (C.5)31 a) (b) (c) Figure 17: Diagrams for Π H − W + . (a) (b) (c) (d)(e) (f) (g) (h) Figure 18: Diagrams for Π H − π W + . The last one is for the counter term.Here we ignored p µ p ν terms because they do not contribute to the EDMs as we discussed in Sec. 3.Note that the (cid:0) p − m W (cid:1) term does not also contribute to the on-shell amplitudes nor the EDMsat two-loop level. If we calculate the EDMs with this term, we immediately see that q dependencecompletely canceled out. Thus, we only need the first term in Eq. (C.5).Fig. 17 shows the diagrams for Π H − W + ( p ). We find i Π H − W + ( p ) = + i (4 π ) D/ Γ(2 − D/ g W W h g W + H − h Z dx − (2 − x ) + m H ± − m h m W ( x − ) (cid:2) m W (1 − x ) + m h x − p x (1 − x ) (cid:3) − D/ + i (4 π ) D/ Γ(2 − D/ g H + H − h g W + H − h Z dx − x (cid:2) m H ± x + m h (1 − x ) − p x (1 − x ) (cid:3) − D/ . (C.6)Fig. 18 shows the diagrams for Π H − π W + ( p ). We find Figs. 18(d)–18(g) are canceled byFig. 18(h), so we do not calculate them. Fig. 18(h) is the counter term for H – π W mixing, and itis also related with the counter terms for the Higgs tadpoles (Fig. 19(i)), δ H − π W + = X h m W g W + H − h δ h , (C.7)32 a) (b) (c) (d)(e) (f) (g) (h) (i) Figure 19: Tadpoles diagrams.where δ ’s are defined through L ⊃ − δ H − π W + H − π W + + X h δ h h. (C.8)It is easy to find this relation by analyzing the Higgs potential. We take renormalization conditionsin which all tadpole diagrams are completely canceled by their counter terms. Then δ H − π W + is notarbitrary but should be calculated from the tadpole diagrams and Eq. (C.8). We show the tadpolediagrams in Fig. 19. After calculating tadpole diagrams, using Eq. (C.7), we findFig. 18(h) = − (Fig. 18(d) + Fig. 18(e) + Fig. 18(f) + Fig. 18(g))+ i X h (cid:18) m H ± − m h m W g W W h g W + H − h + g H + H − h g W + H − h m W (cid:19) Z ℓ ℓ − m h − i X h g H + H − h g W + H − h m W Z ℓ ℓ − m H ± + i X h m h m W g H + H − h g W + H − h Z ℓ ℓ − m W . (C.9)Now we have calculated all the diagrams shown in Fig. 18, and we find i Π H − π W + = + Γ(2 − D/ π ) D/ m W g W W h g W + H − h × Z dx " p (1 + 2 x ) + m h (cid:2) m W x + m h (1 − x ) − p x (1 − x ) (cid:3) − D/ + m H ± − m h m W m W − p (1 − x ) (cid:2) m W x + m h (1 − x ) − p x (1 − x ) (cid:3) − D/ − Γ(2 − D/ π ) D/ m W g H + H − h g W + H − h p Z dx − x (cid:2) m H ± x + m h (1 − x ) − p x (1 − x ) (cid:3) − D/ . (C.10)33e have finished preparing to calculate Fig. 16(a) + Fig. 16(b). Substituting Eqs. (C.6) and(C.10) into the first term in Eq. (C.5), then we findFig. 16(a) + Fig. 16(b) = − ig µν π ) D/ Γ(2 − D/ eg W + H − h × " g W W h Z dx − (1 + x ) + m H ± − m h m W (1 − x ) (cid:2) m W x + m h (1 − x ) − p H x (1 − x ) (cid:3) − D/ + g W W h p H − m H ± p H − m W ) Z dx (cid:2) m W x + m h (1 − x ) − p H x (1 − x ) (cid:3) − D/ + g H + H − h Z dx − x (cid:2) m H ± x + m h (1 − x ) − p H x (1 − x ) (cid:3) − D/ . (C.11)Here we dropped the ( p − m W ) g µν term because it does not contribute to what we are interestedin. Note that the first term in the bracket in Eq. (C.11) is canceled with Eq. (C.1), and the secondterm is canceled with Eq. (C.2).So far we have calculated many diagrams, vertex corrections and wave function corrections.The corrections are not so simple and some of them canceled out, so we give a short summary sofar here. After summing up all the correction we have calculated so far, we find+ i (4 π ) D/ Γ(3 − D/
2) ( p µ p ν − p p g µν ) × + X h eg W + H − h g W W h Z x + y + z =1 − yz − z + 4 − m H ± − m h m W yz (cid:2) m W (1 − z ) + m h z − p z (1 − z ) − p p yz (cid:3) − D/ − X h eg W + H − h g H + H − h Z x + y + z =1 yz (cid:2) m H ± (1 − z ) + m h z − p z (1 − z ) − p p yz (cid:3) − D/ ! + i (4 π ) D/ g µν (cid:0) p H − m H ± (cid:1) X h eg W + H − h g W W h × " +Γ(3 − D/ Z x + y + z =1 (cid:2) m W (1 − z ) + m h z − p z (1 − z ) − p p yz (cid:3) − D/ − Γ(2 − D/
2) 12( p H − m W ) Z dx (cid:2) m W x + m h (1 − x ) − p H x (1 − x ) (cid:3) − D/ . (C.12)Note that the last two terms are not gauge invariant in the sense that we discussed in Sec. 3. Sincethey are proportional to p H − m H ± , if we take the charged Higgs boson on-shell, they are droppedand the result becomes gauge invariant. However, now we need to take the charged Higgs bosonoff-shell, so we still need some other terms to cancel them.To find a gauge invariant set for the Barr-Zee diagrams, we need to take into account for thepinch contributions shown in Fig. 20. After pinching the fermion propagators with red color in34 a) (b) −→ (c) (d) Figure 20: Pinch contributions.Figs. 20(a) and 20(b), the pinch contributions for H − W + γ effective vertex for the Barr-Zee dia-grams arise. They are schematically shown in Figs. 20(c) and 20(d). We denote their contributionsas Γ µν P and i Π P , respectively. Then we find i Γ µν P ( p , p ) = − i Γ(3 − D/ π ) D/ eg W + H − h g W W h g µν × (cid:0) p H − m H ± (cid:1) Z x + y + z =1 (cid:2) m W (1 − z ) + m h z − p z (1 − z ) − p p yz ) (cid:3) − D/ , (C.13) i Π P ( p H ) = + Γ(2 − D/ π ) − D/ m W g W + H − h g W W h (cid:0) p H − m H ± (cid:1) × Z dx (cid:2) m W x + m h (1 − x ) − p H x (1 − x ) (cid:3) − D/ . (C.14)Using Eq. (C.5), we find that Γ µν P and i Π P completely cancel the second term in Eq. (C.12), namelythese pinch contributions really make the effective vertex correction gauge invariant. References [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B , 1 (2012) [arXiv:1207.7214 [hep-ex]].[2] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 30 (2012) [arXiv:1207.7235[hep-ex]].[3] J. Baron et al. [ ACME Collaboration], arXiv:1310.7534 [physics.atom-ph].[4] C. A. Baker, D. D. Doyle, P. Geltenbort, K. Green, M. G. D. van der Grinten, P. G. Harris,P. Iaydjiev and S. N. Ivanov et al. , Phys. Rev. Lett. , 131801 (2006) [hep-ex/0602020].[5] For reviews, M. Pospelov and A. Ritz, Annals Phys. , 119 (2005) [hep-ph/0504231];M. Raidal, A. van der Schaaf, I. Bigi, M. L. Mangano, Y. K. Semertzidis, S. Abel, S. Albinoand S. Antusch et al. , Eur. Phys. J. C , 13 (2008) [arXiv:0801.1826 [hep-ph]];T. Fukuyama, Int. J. Mod. Phys. A , 1230015 (2012) [arXiv:1201.4252 [hep-ph]].[6] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B , 36 (1985).357] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. , 27 (1993)[hep-ph/9302210].[8] V. A. Rubakov and M. E. Shaposhnikov, Usp. Fiz. Nauk , 493 (1996) [Phys. Usp. , 461(1996)] [hep-ph/9603208].[9] M. Trodden, Rev. Mod. Phys. , 1463 (1999) [hep-ph/9803479].[10] S. M. Barr and A. Zee, Phys. Rev. Lett. , 21 (1990) [Erratum-ibid. , 2920 (1990)].[11] R. G. Leigh, S. Paban and R. M. Xu, Nucl. Phys. B , 45 (1991).[12] D. Chang, W. -Y. Keung and T. C. Yuan, Phys. Rev. D , 14 (1991).[13] G. Degrassi and A. Sirlin, Phys. Rev. D , 3104 (1992).[14] G. Degrassi and A. Sirlin, Nucl. Phys. B , 73 (1992).[15] A. Denner, Fortsch. Phys. , 307 (1993) [arXiv:0709.1075 [hep-ph]].[16] Y. Sakemi, K. Harada, T. Hayamizu, M. Itoh, H. Kawamura, S. Liu, H. S. Nataraj andA. Oikawa et al. , J. Phys. Conf. Ser. , 012051 (2011).[17] D. M. Kara, I. J. Smallman, J. J. Hudson, B. E. Sauer, M. R. Tarbutt and E. A. Hinds, NewJ. Phys. , 103051 (2012) [arXiv:1208.4507 [physics.atom-ph]].[18] D. Kawall, J. Phys. Conf. Ser. , 012031 (2011).[19] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440 (1977).[20] M. Pospelov and A. Ritz, Phys. Rev. D , 073015 (2001) [hep-ph/0010037].[21] J. Hisano, J. Y. Lee, N. Nagata and Y. Shimizu, Phys. Rev. D , 114044 (2012)[arXiv:1204.2653 [hep-ph]].[22] K. Fuyuto, J. Hisano, N. Nagata and K. Tsumura, arXiv:1308.6493 [hep-ph].[23] G. Degrassi, E. Franco, S. Marchetti and L. Silvestrini, JHEP , 044 (2005)[hep-ph/0510137].[24] J. Hisano, K. Tsumura and M. J. S. Yang, Phys. Lett. B , 473 (2012) [arXiv:1205.2212[hep-ph]].[25] S. N. Balashov, K. Green, M. G. D. van der Grinten, P. G. Harris, H. Kraus,J. M. Pendlebury, D. B. Shiers and M. A. H. Tucker et al. , arXiv:0709.2428 [hep-ex].[26] K. Kirch,http://vmsstreamer1.fnal.gov/Lectures/Colloquium/presentations/130213Kirch.pdf[27] U. Haisch, arXiv:0805.2141 [hep-ph].[28] F. Mahmoudi and O. Stal, Phys. Rev. D , 035016 (2010) [arXiv:0907.1791 [hep-ph]].3629] [CMS Collaboration], CMS-PAS-HIG-12-050.[30] A. Pomarol and R. Vega, Nucl. Phys. B , 3 (1994) [hep-ph/9305272].[31] N. Turok and J. Zadrozny, Nucl. Phys. B , 471 (1991).[32] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B , 86 (1991).[33] J. M. Cline and P. -A. Lemieux, Phys. Rev. D , 3873 (1997) [hep-ph/9609240].[34] L. Fromme, S. J. Huber and M. Seniuch, JHEP , 038 (2006) [hep-ph/0605242].[35] G. C. Dorsch, S. J. Huber and J. M. No, JHEP , 029 (2013) [arXiv:1305.6610 [hep-ph]].[36] J. M. Cline, K. Kainulainen and M. Trott, JHEP , 089 (2011) [arXiv:1107.3559[hep-ph]].[37] T. Ohl, Comput. Phys. Commun.90