On-Shell neutral Higgs bosons in the NMSSM with complex parameters
DDESY–17–067IFT–UAM/CSIC–17–042
On-Shell neutral Higgs bosons inthe NMSSM with complex parameters
Florian Domingo , ∗ , Peter Drechsel † , and Sebastian Paßehr ‡ Instituto de F´ısica Te´orica (UAM/CSIC), Universidad Aut´onoma de Madrid, Cantoblanco,E-28049 Madrid, Spain Instituto de F´ısica de Cantabria (CSIC-UC), E-39005 Santander, Spain Deutsches Elektronensynchrotron DESY, Notkestraße 85, D–22607 Hamburg, Germany
The Next-to-Minimal Supersymmetric Standard model (NMSSM) appears as an interesting can-didate for the interpretation of the Higgs-measurement at the LHC and as a rich framework em-bedding physics beyond the Standard Model. We consider the renormalization of the Higgs sectorof this model in its CP -violating version, and propose a renormalization scheme for the calculationof on-shell Higgs masses. Moreover, the connection between the physical states and the tree-levelones is no longer trivial at the radiative level: a proper description of the corresponding transitionthus proves necessary in order to calculate Higgs production and decays at a consistent loop order.After discussing these formal aspects, we compare the results of our mass calculation to the outputof existing tools. We also study the relevance of the on-shell transition-matrix in the example of the h i → τ + τ − width. We find deviations between our full prescription and popular approximationsthat can exceed 10%. ∗ email: fl[email protected] † email: [email protected] ‡ email: [email protected] a r X i v : . [ h e p - ph ] J u l ontents CP -violating NMSSM 2 Z mix FeynHiggs in the MSSM-limit 113.2 Comparison in the CP -conserving limit 133.3 Comparison with NMSSMCALC Z mix and the h i → τ + τ − decays 18 Since the discovery of a Higgs-like particle with a mass around 125 GeV by the ATLAS and CMSexperiments [1, 2] at CERN, a lot of effort has been invested to reveal its nature as the particleresponsible for electroweak symmetry breaking. While within the present experimental uncertaintiesthe properties of the observed state are compatible with the predictions of the Standard Model(SM) [3] many other interpretations are possible as well, in particular as a Higgs boson of anextended Higgs sector.One of the prime candidates for physics beyond the SM is softly-broken supersymmetry (SUSY),which doubles the particle degrees of freedom by predicting two scalar partners for each SM fermion,as well as fermionic partners for all bosons—for reviews see [4, 5]. The Next-to-Minimal Super-symmetric Standard Model (NMSSM) [6, 7] is a well-motivated extension of the SM. In particularit provides a solution for the “ µ problem” [8] of the Minimal Supersymmetric Standard Model(MSSM), by naturally relating the µ parameter to a dynamical scale of the Higgs potential [9, 10].In contrast to the single Higgs doublet in the SM, the Higgs sector of the NMSSM containstwo Higgs doublets (like the MSSM) and one Higgs singlet. After electroweak symmetry breakingthe physical spectrum consists of five neutral Higgs bosons, h i ( i ∈ [1 , H ± . Ever since the Higgs discovery, the possibility to interpret this signal in terms ofan NMSSM (mostly) CP -even Higgs boson has been emphasized in Refs. [11–20]. In particular,it has been argued that such a solution came with improved naturalness compared to the MSSMinterpretation [21–29]. Moreover, several works have pointed out the possibility to accommodatedeviations from a strict SM behavior in the diphoton rate, in Higgs-pair production or in associatedproduction [30–47]. Admittedly, the viability of the extended NMSSM Higgs sector would becomforted by the detection of additional Higgs states. To this end, several search channels havebeen suggested, especially for states lighter than 125 GeV [48–65]. Another feature of the NMSSMphenomenology is the extended neutralino sector, due to the singlino.In contrast to the situation in the MSSM, CP -violation can already occur at the tree-level inthe NMSSM Higgs sector [10, 17, 66–82]. While low-energy observables place limits on such CP -violating scenarios [83], especially on MSSM-like phases [84], CP -violation beyond the SM appears asa well-motivated requirement for a successful baryogenesis [85]. Correspondingly, several computer1ools have been proposed in the past few years to promote the study of the CP -violating NMSSM: SPHENO [86–89] and
FlexibleSUSY [90, 91]—which employ
SARAH [92–95] in order to produce theirmodelfiles;
FlexibleSUSY contains components from
SoftSUSY [96, 97] and only the CP -conservingcase is explicitly mentioned for both—as well as NMSSMCALC [98, 99] and
NMSSMTools [100–103].In this work, we specialize in the Z -conserving version of the NMSSM, characterized by a scale-invariant superpotential. The main effort of our project consists in analyzing radiative corrections inthe Higgs sector of the CP -violating NMSSM. To serve this purpose, we elaborated a FeynArts [104,105] model file and a set of
Mathematica routines for the evaluation of the Higgs masses and wave-function normalization matrix at full one-loop order and beyond. These should serve as a basisfor a future inclusion of the CP -violating NMSSM in the FeynHiggs [106–112] package—originallydesigned for precise calculations of the masses, decays, and other properties of the Higgs bosons inthe CP -conserving or -violating MSSM. A first step in this direction is represented by Ref. [113],centering on the CP -conserving NMSSM. In the current paper, we expand this project further. Wefollow the general methodology of FeynHiggs , relying on a Feynman-diagrammatic calculation ofradiative corrections, which employs
FeynArts [104, 105],
FormCalc [114] and
LoopTools [114]. Ourchosen renormalization scheme differs somewhat from earlier proposals [88, 113, 115]. In particular,in our renormalization scheme, the electromagnetic coupling e —which is related to the fine-structureconstant α = e / (4 π )—is defined in terms of the Fermi constant G F measured in muon decays.In section 2, we shall introduce relevant notations and describe the renormalization procedureunderpinning our model file for the CP -violating NMSSM. In this section we also describe ourimplementation of higher-order corrections in the Higgs sector. A numerical evaluation of ourresults follows in section 3, where we will validate our calculation by a comparison with publiccodes. We will also insist on the relevance of the field-renormalization matrix for a consistentevaluation of the Higgs decays at the one-loop level, before a short conclusion in section 4. CP -violating NMSSM After a few general remarks concerning our notations and conventions, we present the renormaliza-tion conditions that we employ in our calculation. There, we focus on effects beyond the MSSMin the Higgs and higgsino sectors, since we otherwise align with the conventions of
FeynHiggs ,described in [109]. Then we discuss how to formally extract the loop-corrected Higgs masses andthe wave function normalization factors.
In the following, we consider the Z -conserving version of the NMSSM and neglect flavor-mixing.The superpotential of the NMSSM (showing only one generation of fermions/sfermions) reads W = Y u ˆ u (cid:16) ˆ H · ˆ Q (cid:17) − Y d ˆ d (cid:16) ˆ H · ˆ Q (cid:17) − Y e ˆ e (cid:16) ˆ H · ˆ L (cid:17) + λ ˆ S (cid:16) ˆ H · ˆ H (cid:17) + 13 κ ˆ S , (2.1)where ˆ Q , ˆ u , ˆ d , ˆ L , ˆ e , ˆ H , ˆ H , ˆ S denote the quark, lepton and Higgs superfields. The dot · standsfor the SU (2) L -invariant product. The Yukawa couplings in Eq. (2.1) can be complex in general.However, their phases can be absorbed in a redefinition of the quark and lepton superfields. Wemay write the scalar fields in ˆ H , ˆ H and ˆ S explicitly in terms of their (real and positive) vacuumexpectation values (vevs), v , v and v s , respectively, as well as their CP -even, CP -odd, and chargedcomponents, φ i , χ i , and φ ± i , H = e i ξ (cid:32) v + ( φ + iχ ) √ φ − (cid:33) , H = e i ξ (cid:32) φ +2 v + ( φ + iχ ) √ (cid:33) , S = e i ξ s (cid:104) v s + ( φ s + iχ s ) √ (cid:105) . (2.2)2ere ξ , ξ and ξ s are the phases of the two Higgs doublets and the Higgs singlet, respectively. It isconvenient to define the ratio tan β = v /v , the geometric mean of the doublet vevs v = (cid:112) v + v ,as well as the sum ξ = ξ + ξ of the doublet phases. Since ˆ S transforms as a singlet underthe SM-gauge transformations, the D -terms of the scalar potential are unchanged with respectto the MSSM. On the other hand, as compared to the MSSM, additional dimensionless, complexparameters λ = | λ | e i φ λ and κ = | κ | e i φ κ appear while the complex µ -term is absent. The latter isdynamically generated as an effective µ -term when the singlet field takes its vev, µ eff = | µ eff | e i φ µ = | λ | v s e i ( φ λ + ξ s ) . (2.3)In the NMSSM the phases ξ and ξ s only appear in the combinations φ λ + ξ s + ξ and φ κ +3 ξ s , so thatthey could be absorbed in a re-definition of φ λ and φ κ . Nevertheless, we will keep the dependenceon all phases of the Higgs sector explicitly, in order to allow for more flexibility on the choice ofinput.Soft SUSY-breaking in the NMSSM is parametrized by the complex trilinear soft-breaking param-eters A λ = | A λ | e i φ Aλ , A κ = | A κ | e i φ Aκ , A u = | A u | e i φ Au , A d = | A d | e i φ Ad , and A e = | A e | e i φ Ae ,as well as the real soft-breaking mass terms m , and m S for the Higgs fields, and m Q , m U , m D , m L and m E for the sfermions, L soft = − m |H | − m |H | − m S |S| − (cid:2) λ A λ S ( H · H ) + κ A κ S + h.c. (cid:3) − m Q | ˜ Q | − m U | ˜ u | − m D | ˜ d | − m L | ˜ L | − m E | ˜ e | − (cid:104) − Y u A u ˜ u (cid:16) H · ˜ Q (cid:17) + Y d A d ˜ d (cid:16) H · ˜ Q (cid:17) + Y e A e ˜ e (cid:16) H · ˜ L (cid:17) + h.c. (cid:105) . (2.4)Expanding the Higgs potential in terms of the charged Higgs fields ( φ +1 , φ +2 ) = ( φ − , φ − ) ∗ , andneutral Higgs fields ( φ, χ ) = ( φ , φ , φ s , χ , χ , χ s ) yields V H = − T ( φ, χ ) T + 12 ( φ, χ ) M ( φ, χ ) T + (cid:0) φ − , φ − (cid:1) M φ ± (cid:18) φ +1 φ +2 (cid:19) + · · · . (2.5)Here T = ( T φ , T φ , T φ s , T χ , T χ , T χ s ) denotes the tadpole coefficients of the neutral Higgs fields,and M and M φ ± denote the mass matrices of the neutral and charged Higgs bosons, respectively.Since M and M φ ± are symmetric and hermitian matrices, respectively, we diagonalize them byan orthogonal (6 ×
6) matrix U n and a unitary (2 ×
2) matrix U c , respectively, D hG = diag (cid:0) m h , m h , m h , m h , m h , (cid:1) = U n M U T n , (2.6a) D h ± G ± = diag (cid:0) M H ± , (cid:1) = U c M φ ± U † c . (2.6b)These transformations define the five neutral Higgs boson mass eigenstates, h i , ( i = 1 , . . . , G , as well as the charged Higgs and (would-be) Goldstone states, H ± and G ± , at the tree level,( h, G ) T ≡ ( h , h , h , h , h , G ) T = U n ( φ, χ ) T , (2.7a) (cid:0) H ± , G ± (cid:1) T = U c (cid:0) φ ± , φ ± (cid:1) T . (2.7b)It is convenient to decompose U n into two matrices U Gn and U n , where U Gn singularizes out theneutral Goldstone boson, U n ( φ, χ ) T = U n U Gn ( φ, χ ) T = U n ( φ , φ , φ s , A, χ s , G ) T = ( h, G ) T . (2.8)3n the CP -violating NMSSM the five fields h i are in general superpositions of the CP -even and -oddcomponents φ i and χ j . In the special case ofsin ( ξ − ξ s + φ λ − φ κ ) = 0 (2.9) CP -conservation is restored in the Higgs sector at the tree level, and the neutral mass matrix M becomes block-diagonal with two (3 ×
3) sub-matrices for the CP -even and -odd entries.The five linearly independent tadpole coefficients are related to soft-breaking terms and combi-nations of phases as m = −| µ eff | − M Z cos (2 β ) − ( | λ | v sin β ) − T φ √ v cos β + | µ eff | tan β (cid:18) | A λ | cos ζ + | κ | | µ eff || λ | cos ζ (cid:19) , (2.10a) m = −| µ eff | + 12 M Z cos (2 β ) − ( | λ | v cos β ) − T φ √ v sin β + | µ eff | tan β (cid:18) | A λ | cos ζ + | κ | | µ eff || λ | cos ζ (cid:19) , (2.10b) m s = −| λ | v − T φ s | λ |√ | µ eff | − | κ | | µ eff || λ | (cid:18) | A κ | cos ζ + 2 | κ | | µ eff || λ | (cid:19) + 12 | λ | v | µ eff | sin (2 β ) (cid:18) | A λ | cos ζ + 2 | κ | | µ eff || λ | cos ζ (cid:19) (2.10c)sin ζ = 1 | A λ | (cid:18) − | κ | | µ eff || λ | sin ζ − T χ √ | µ eff | v sin β (cid:19) , (2.10d)sin ζ = | λ | | κ | | µ eff | | A κ | (cid:18) | λ | v β ) (cid:18) | A λ | sin ζ − | κ | | µ eff || λ | sin ζ (cid:19) + T χ s √ (cid:19) , (2.10e)where the masses of the W and Z bosons are denoted by M W and M Z , respectively, and the phasescombine to ζ = ξ − ξ s + φ λ − φ κ , ζ = ξ + ξ s + φ A λ + φ λ and ζ = 3 ξ s + φ A κ + φ κ . The expressions ofEq. (2.10) make plain that the tadpole coefficients can substitute the five parameters m , m , m S , φ A λ and φ A κ , so that the latter will not be regarded as free parameters in the following. Finally, thetadpole coefficients in the (tree-level) mass basis, T h = ( T h , T h , T h , T h , T h , T h = U n T . Theminimization of V H at the chosen Higgs vevs is guaranteed through the condition that all tadpolecoefficients T vanish at the tree level.The trilinear parameter | A λ | can be expressed in terms of the charged Higgs mass M H ± as | A λ | cos ζ = − | κ | | µ eff || λ | cos ζ + (cid:0) M H ± − M W + | λ | v (cid:1) sin (2 β )2 | µ eff | . (2.11)Our renormalization scheme will also involve the fermionic superpartners of the Higgs bosons,known as the higgsinos. We thus introduce here the Dirac spinors ˜ H ± of the charged higgsino fields,as well as the Majorana spinor of the singlino ˜ S . In turn, these higgsino gauge eigenstates mix withthe gauginos to form the mass states known as the neutralinos and charginos—see e. g. Eq. (11)in Ref. [113], where the NMSSM parameters λ , κ , M , and µ eff should be promoted to complexvalues. Yet these mass states will play no role in the discussion below.4 .2 Renormalization of the Higgs potential In the past, radiative corrections to the Higgs masses of the CP -conserving NMSSM have beenconsidered in the effective potential approach, see e. g. Refs. [116–124]. This topic has also beenanalyzed from the perspective of a diagrammatic expansion, including radiative corrections frompart or the full set of the particle content of the NMSSM: see Refs. [113, 125, 126]. Both procedureshave also been employed for the CP -violating case: contributions to the effective potential havebeen discussed in Refs. [17, 67–82, 102], while contributions using the diagrammatic approach havebeen presented in Refs. [88, 115].In the present work, the radiative corrections to the Higgs sector are calculated in the diagram-matic approach. To this end, we first establish a list of the independent parameters appearing inthe linear and bilinear terms of the Higgs potential in Eq. (2.5): T h ,..., , M H ± , M W , M Z , e, tan β, | µ eff | , | λ | , φ λ , | κ | , φ κ , | A κ | , ξ, ξ s , (2.12)where the electromagnetic coupling e is related to the fine structure constant α by e = √ π α .Compared to Ref. [113], we use e as an independent parameter instead of the vev v . This choiceallows to renormalize e to its value derived from the Fermi constant G F and does not require thereparametrization procedure employed in Ref. [113]. The difference between the renormalizationemployed here, and the renormalization and reparametrization procedure described in Ref. [113] isa sub-leading effect of two-loop order, however. Other proposals in the literature consist in fixing e from α ( M Z ) [88, 115].To all real and complex independent parameters, g r and g c , respectively, that are given inEq. (2.12), we apply the renormalization transformations g r → g r (1 + δZ r ) = g r + δg r , g c → g c (1 + δZ c ) = g c + δg c = g c + δ | g c | e i φ c + i g c δφ c . (2.13)The renormalization transformations for the Higgs, singlino and charged higgsino fields read H , → (cid:0) δZ H , (cid:1) H , , S → (cid:0) δZ S (cid:1) S , (2.14a)˜ H ± → (cid:0) δZ L˜ H ± P L + δZ R˜ H ± P R (cid:1) ˜ H ± , ˜ S → (cid:0) δZ L˜ S P L + δZ R˜ S P R (cid:1) ˜ S, (2.14b)with P L and P R denoting the left- and right-handed projectors, respectively. Since the singlino is aMajorana field, the corresponding wave-function counterterms δZ L˜ S and δZ R˜ S are complex conjugatesof one another. For the parameters M H ± , M W , M Z and tan β , which enter the one-loop calculation of the Higgsmasses in the MSSM as well, we follow the renormalization prescription outlined in Ref. [109]: theon-shell renormalization scheme is employed for the gauge boson masses, M Z and M W , and thecharged Higgs mass M H ± , while the parameter tan β is renormalized DR.We apply the minimization conditions in order to fix the tadpole counterterms: T (1) h i + δT h i = 0 , (2.15)where the T (1) h i correspond to the one-loop contributions to the tadpole parameters.The counterterm of the electromagnetic coupling e is fixed by δZ e = δZ Th e −
12 ∆ r NMSSM , (2.16a) δZ Th e = 12 Π γγ (0) + s w c w Σ γZT (0) M Z . (2.16b)5ere δZ Th e is the counterterm of the charge renormalization within the NMSSM according to thestatic (Thomson) limit. The quantities Π γγ (0) and Σ γZT (0) are respectively the derivative of thetransverse part of the photon self-energy and the transverse part of the photon– Z self-energy atzero momentum transfer. For the quantity ∆ r NMSSM , relating the elementary charge to the Fermiconstant G F measured in muon decays, we use the result of Ref. [127] (see also Ref. [128]). Thenumerical value for the electromagnetic coupling e in this parametrization is obtained from theFermi constant in the usual way as e = 2 M W s w (cid:112) √ G F . This choice differs from previous works,where either the charge renormalization condition was determined in terms of α ( M Z ) [115], orinstead v was renormalized DR and the result was subsequently reparametrized to use the valueof e derived from the Fermi constant [113].The remaining independent parameters and the field renormalization constants are renormalizedDR. We present a detailed description of the DR renormalization conditions that we apply. Theactual cancellation of UV-divergences, that we recover at the diagrammatic level, represents anon-trivial check for the validity of the FeynArts model-file employed for our calculation.The DR field renormalization constants for the Higgs fields are obtained as δZ H = −(cid:60) e (cid:34) d Σ (1) φ φ dp (cid:35) div , δZ H = −(cid:60) e (cid:34) d Σ (1) φ φ dp (cid:35) div , δZ S = −(cid:60) e (cid:34) d Σ (1) φ s φ s dp (cid:35) div , (2.17)where Σ (1) ii denotes the self-energy of field i at the one-loop order, and the subscript ’div’ denotesthe UV-divergent piece (along with the universal finite pieces that are associated in the DR scheme)of the quantity that it follows. The result does not depend on the momentum p .The field renormalization constants for the charged higgsino and the singlino fields are definedby the following conditions (the momentum p again does not matter) δZ L˜ H ± = − Σ vec L (1)˜ H ± ˜ H ± (cid:12)(cid:12)(cid:12) div , δZ R˜ H ± = − Σ vec R (1)˜ H ± ˜ H ± (cid:12)(cid:12)(cid:12) div , δZ L˜ S = − Σ vec L (1)˜ S ˜ S (cid:12)(cid:12)(cid:12) div , (2.18)where the self-energies of the fermion fields are decomposed into the left and right vector and thescalar contributions,Σ (1) ff (cid:0) p (cid:1) = Σ scal (1) ff (cid:0) p (cid:1) + p µ γ µ (cid:104) P L Σ vec L (1) ff (cid:0) p (cid:1) + P R Σ vec R (1) ff (cid:0) p (cid:1)(cid:105) . (2.19)The renormalization constants δ | λ | , δ | κ | , δφ λ , δφ κ , δξ , δξ s and δ | A κ | are fixed by DR conditionsimposed on trilinear vertices involving scalar, CP -even Higgs fields φ , ,s , the singlino ˜ S , and thecharged higgsino fields ˜ H ± , in the interaction basis in analogy to the procedure outlined in [129].The renormalization condition imposed on the renormalized three-point function ˆΓ ijk for threearbitrary fields i , j and k readsˆΓ ijk = Γ (0) ijk + Γ (1) ijk + δ Γ ijk ! = finite , ⇐ δ Γ ijk = − Γ (1) ijk (cid:12)(cid:12)(cid:12) div , (2.20)where Γ (0) ijk and Γ (1) ijk denote the vertex function at the tree-level and one-loop order, respectively,and δ Γ ijk denotes the counterterm. The counterterm δ Γ ijk is thus fixed by the divergent part ofthe vertex function. The renormalization constants of the independent parameters are subsequentlyfixed by linear relations to δ Γ ijk . 6 For δ | λ | and δφ λ we impose the DR renormalization condition of Eq. (2.20) on the verticesΓ (0)˜ S ˜ H − φ +1 = λ and Γ (0)˜ S ˜ H + φ − = λ ∗ , which yields δ | λ || λ | = − Γ (1)˜ S ˜ H − φ +1 Γ (0)˜ S ˜ H − φ +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) div + 12 (cid:0) δZ ˜ S + δZ R˜ H ± + δZ H (cid:1) + c. c. , (2.21a) δφ λ = − i Γ (1)˜ S ˜ H − φ +1 Γ (0)˜ S ˜ H − φ +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) div + 12 (cid:0) δZ ˜ S + δZ R˜ H ± + δZ H (cid:1) + c. c. (2.21b) • For δξ we impose the renormalization condition of Eq. (2.20) on the vertices Γ (0)˜ S ˜ H + φ − = λ e i ξ and Γ (0)˜ S ˜ H − φ +2 = λ ∗ e − i ξ . The counterterm reads δξ = − i Γ (1)˜ S ˜ H + φ − Γ (0)˜ S ˜ H + φ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) div + 12 (cid:0) δZ ˜ S + δZ L˜ H ± + δZ H (cid:1) + c. c. − δφ λ . (2.22) • We fix the renormalization constant δξ s for the phase ξ s by applying Eq. (2.20) on the verticesΓ (0)˜ H − , ˜ H + φ s = λ e i ξ s / √ (0)˜ H + , ˜ H − φ s = λ ∗ e − i ξ s / √
2, which yields δξ s = − i Γ (1)˜ H − , ˜ H + φ s Γ (0)˜ H − , ˜ H + φ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) div + 12 (cid:0) δZ L˜ H ± + δZ R˜ H ± + δZ φ s (cid:1) + c. c. − δφ λ . (2.23) • The absolute value and phase of κ are renormalized by δ | κ | and δφ κ . We fix both renor-malization constants by applying Eq. (2.20) on the vertex Γ (0)˜ S ˜ Sφ s = √ κ e i ξ s , which yields δ | κ || κ | = − Γ (1)˜ S ˜ Sφ s Γ (0)˜ S ˜ Sφ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) div + 12 (2 δZ ˜ S + δZ φ s ) + c. c. , (2.24a) δφ κ = − i Γ (1)˜ S ˜ Sφ s Γ (0)˜ S ˜ Sφ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) div + 12 (2 δZ ˜ S + δZ φ s ) + c. c. − δξ s . (2.24b) • The parameter | µ eff | could be renormalized in the on-shell scheme for one of the charginos orneutralinos [130–132]. However, such schemes cannot be stabilized over the whole parameterspace (due to mass-crossings). We thus prefer to apply the DR condition δ | µ eff | = | µ eff | (cid:18) δ | λ || λ | + 12 δZ S (cid:19) . (2.25) • In order to fix δ | A κ | , we impose the renormalization condition of Eq. (2.20) on the vertexΓ (0) φ s φ s φ s = −√ | κ | (cid:16) | κ | | µ eff || λ | + | A κ | cos ζ (cid:17) , ( ζ was defined after Eq. (2.10)). It reads δ | A κ | = −| A κ | (cid:18) δ | κ || κ | + δ cos ζ cos ζ (cid:19) − | κ | | µ eff || λ | cos ζ (cid:18) δ | µ eff || µ eff | − δ | λ || λ | + 2 δ | κ || κ | (cid:19) − δ Γ φ s φ s φ s √ | κ | cos ζ , (2.26a)7 ζ = (cid:18) δ | κ || κ | + 2 δ | λ || λ | − δ | µ eff | µ eff + δ sin (2 β )sin (2 β ) + δ sin ζ sin ζ + 2 δvv (cid:12)(cid:12)(cid:12)(cid:12) div (cid:19) cos ζ sin ζ + (cid:20) δ Γ φ s φ s φ s √ | κ | + 6 | κ | | µ eff || λ | (cid:18) δ | µ eff || µ eff | − δ | λ || λ | + 2 δ | κ || κ | (cid:19)(cid:21) sin ζ | A κ | + (cid:18) δT χ s | div − | λ | v cos β | µ eff | δT χ | div (cid:19) | λ | cos ζ √ | κ | | A κ | | µ eff | . (2.26b)where we used the one-loop relation δ cos ζ = − sin ζ δζ , and δv is not an independentcounterterm, but a quantity depending on the counterterms to the electroweak parameters δv = v (cid:18) δM W M W + δs w s w − δZ e (cid:19) , δs w = c s w (cid:18) δM Z M Z − δM W M W (cid:19) . (2.27)We performed various consistency checks of our model file at the one-loop order: • all the renormalized Higgs self-energies are UV-finite, for arbitrary values of the momentum, • all the vertex-diagram amplitudes of a Higgs state decaying to SM-particles or a pair ofcharginos/neutralinos are UV-finite, • the UV-divergences of the counterterms to gauge couplings, superpotential parameters or softterms are consistent with the corresponding one-loop beta functions (see e. g. Refs. [6, 133]), • in the CP -conserving limit, our parameters and couplings are identical to the findings of apreviously developed model file [113], • in the MSSM limit, we have found agreement of the values of all our couplings with theircounterparts in the complex MSSM, obtained with the model file of [134]. • we checked that φ λ + ξ + ξ s and φ κ + 3 ξ s were the only relevant combinations of the phases φ λ , φ κ , ξ and ξ s at the level of amplitudes, • finally, we checked explicitly, that the counterterms δφ λ , δφ κ , δξ and δξ s vanish when allNMSSM contributions are included, as pointed out in Ref. [115]. This can also be placed in theperspective of the β -functions [133, 135–139]: the phases from the superpotential parametershave no scale-dependence (at least up to two-loop order); since ξ and ξ s are spurious degreesof freedom, we could expect their counterterms to present the same vanishing behaviors as δφ λ and δφ κ . The Yukawa couplings of the top and bottom quarks, Y t and Y b , have a sizable impact on radiativecorrections to the Higgs masses. We present our prescriptions in this subsection.The top Yukawa coupling Y t = (cid:112) √ G F m t / sin β is defined by the on-shell top mass m t .For the bottom quark, we employ the running DR bottom-mass of the SM (containing one-loopQCD corrections), m b , at the scale m t [140]. Additionally, we subtract the possibly large tan β -enhanced one-loop contributions to m b —induced by gaugino–squark and higgsino–squark loops—from the numerical definition of Y b at the tree level: Y b = (cid:112) √ G F m b / [cos β | b | ], where ∆ b is discussed in e. g. Refs. [98, 140–146]. 8 .5 Higgs masses at higher orders
The masses of the Higgs bosons are obtained from the complex poles of the full propagator matrix.After rotating out the Goldstone mode the inverse propagator matrix for the five Higgs fields h i is a (5 ×
5) matrix that reads ˆ∆ − hh (cid:0) k (cid:1) = i (cid:104) k − D hh + ˆΣ hh (cid:0) k (cid:1)(cid:105) . (2.28)Here D hh = diag { m h , m h , m h , m h , m h } denotes the diagonalized mass matrix of the Higgsfields without the Goldstone at the tree level, and ˆΣ hh denotes the matrix of the renormalizedself-energy corrections of the neutral Higgs fields.The five complex poles of the propagator are given by the values of the squared external mo-mentum k for which the determinant of the inverse propagator matrix vanishes,det (cid:104) ˆ∆ − hh (cid:0) k (cid:1)(cid:105) k = M i ! = 0 , M i ! = M h i + i Γ h i M h i , i ∈ { , . . . , } , (2.29)where we have explicitly stated the connection between the pole M i , the Higgs mass M h i and thetotal width Γ h i for each Higgs field h i .In order to account for the imaginary parts of the poles of the propagator matrix, we perform anexpansion of the self-energies in terms of the imaginary part of the momentum, which is assumedto be small (also see section 4.3.5 of [147]), ˆΣ hh (cid:0) k (cid:1) ≈ ˆΣ hh (cid:0) (cid:60) e (cid:2) k (cid:3)(cid:1) + i (cid:61) m (cid:2) k (cid:3) dd k ˆΣ hh (cid:0) (cid:60) e (cid:2) k (cid:3)(cid:1) (2.30)In this work, the renormalized self-energy ˆΣ hh , ˆΣ hh (cid:0) k (cid:1) ≈ ˆΣ (1L) hh (cid:0) k (cid:1)(cid:12)(cid:12)(cid:12) NMSSM + ˆΣ (2L) hh (cid:0) k (cid:1)(cid:12)(cid:12)(cid:12) MSSM k = 0 . (2.31)is evaluated by taking into account the full contributions from the CP -violating NMSSM at one-looporder and, as an approximation, the MSSM-like contributions at two-loop order of O ( α t α s ) [148]and O (cid:0) α t (cid:1) [149, 150] at vanishing external momentum as implemented in FeynHiggs . We note that the two-loop O ( α b α s ) contributions to the Higgs self-energies are not includedin our calculation. Still, as we employ the running bottom mass in the definition of Y b entering ˆΣ (1L) hh (cid:0) k (cid:1)(cid:12)(cid:12)(cid:12) NMSSM , we expect that the missing two-loop piece is numerically subleading [151–153]. mix
In the Feynman-diagrammatic approach physical processes with external Higgs fields are definedin terms of the tree-level mass states h i . When higher-order contributions are considered, however,the tree-level mass states are not physical states. Indeed, radiative corrections induce additionalmass and kinematic mixing among the fields h i , and the poles of the tree-level propagators do notcoincide with M i . A relation between the amplitudes with an external tree-level Higgs mass stateand those with an external physical Higgs state is necessary (though this relation is trivial if thefields are renormalized on-shell). For example, for a Higgs decaying into two fermions f this relation Besides Higgs– G mixing, we neglect the kinetic Higgs– Z and Higgs–photon mixing, since they are sub-leading effectsof two- and three-loop order, respectively. Additional MSSM-like contributions, at two-loop order or beyond— e. g. resummation of large logarithms for heavysfermions [111]—could be incorporated as well (see [113]). However, we will confine our discussion in this paper tothe leading two-loop contributions.
9s given by the LSZ reduction formula, A [ h phys i → f ¯ f ] = Z mix ij A [ h j → f ¯ f ] . (2.32)Here the superscript ’phys’ denotes the amplitude with an external physical field. The coefficients Z mix ij can be expressed explicitly in terms of the full propagator matrix (see Refs. [154–157] and alsosection 5 . Z mix ij = (cid:20) i dd k (cid:16) ˆ∆ − hh ( k ) (cid:17) ii (cid:21) (cid:16) ˆ∆ hh ( k ) (cid:17) ij (cid:16) ˆ∆ hh ( k ) (cid:17) ii (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k →M i . (2.33)However, the analytical inversion becomes time-consuming in the case of a (5 ×
5) propagator matrix.Additionally, ˆ∆ hh needs to be evaluated at (or close to) its singular points M i , which can leadto numerical instabilities on the right-hand side of Eq. (2.33) (only the ratio of propagator matrixelements ( ˆ∆ hh ) ij is finite). In order to avoid these issues we employed an equivalent formulationof the coefficients Z mix ij , which is outlined below.We consider the following effective Lagrangian for the tree-level mass states h (tree) i , L eff = 12 i (cid:104) ˆ∆ − hh (cid:0) k (cid:1)(cid:105) ij h (tree) i h (tree) j (2.34)and set Z mix = ( Z mix ij ) as the transition matrix to the physical Higgs fields: h (loop) i = Z mix ij h (tree) j . Thestates h (loop) i are defined such that L eff = 12 (cid:2) k − M h i (cid:3) ( h (loop) i ) + O (cid:16)(cid:2) k − M h i (cid:3) (cid:17) . (2.35)In other terms, the h (loop) i should appear as on-shell fields with standard kinetic terms close to theirmass-pole. Thus the coefficients Z mix ij should satisfy the ‘eigenvalue’ conditions (cid:104) D hG − ˆΣ hG (cid:0) M h i (cid:1)(cid:105) kl Z mix il = M h i Z mix ik . (2.36)Once the roots of det (cid:104) ˆ∆ − hh (cid:0) k (cid:1)(cid:105) are known, the i -th line of Z mix is thus determined as the eigen-vector of D hG − ˆΣ hG for the eigenvalue M h i .Finally, the normalization of the i -th line of Z mix is specified by the following condition on thekinetic term, (cid:34) d ˆ∆ − hh d k (cid:0) M i (cid:1)(cid:35) kl Z mix ik Z mix il = (cid:34) I + d ˆΣ hG d k (cid:0) M i (cid:1)(cid:35) kl Z mix ik Z mix il = 1 , (2.37)such that the coefficients Z mix ij are uniquely specified (up to a sign without physical meaning) byEq. (2.35). For the study of effects from the normalization of Z mix , it is convenient to define the(squared) norm | Z i | of its rows, | Z i | = (cid:88) j = 1 (cid:12)(cid:12) Z mix ij (cid:12)(cid:12) , (2.38) Note that the system is non-linear due to the momentum dependence of ˆΣ hG . However, (cid:110) M h i , ( Z mix ) i (cid:111) is agenuine eigenstate of D hG − ˆΣ hG (cid:16) M h i (cid:17) . . e. | Z i | correspond to the norm of the eigenvectors, that are associated to the complex pole M h i ,see Eqs. (2.35) and (2.37).The determination of Z mix in terms of the eigenstates of ˆ∆ − hh (cid:0) k (cid:1) is numerically easier to handlethan its determination via Eq. (2.33). Applying the two defining conditions Eqs. (2.36) and (2.37)to the expression of Eq. (2.33), one can verify that both definitions of Z mix are identical.As we discussed, the components of the matrix Z mix establish the connection between the physicalfields and the tree-level external legs. In the literature this matrix Z mix is often replaced by simplifiedversions neglecting the momentum dependence of the self-energies. With the aim of performingnumerical comparisons in the following section, we introduce two such approximate definitions ofthe relation between loop-corrected and tree-level fields: • The first approach consists in freezing the momentum to k = 0 in the self-energy of Eq. (2.28).This assumption is known as the effective potential approximation. In this approach theinverse propagator matrix ∆ − hh ( k = 0), as given by Eq. (2.28), is diagonalized by a simpleorthogonal matrix U , which approximates Z mix . • Another choice consists in replacing the momentum dependence of the self-energy in Eq. (2.28)by (cid:104) ˆΣ hh (cid:0) k (cid:1)(cid:105) ij → (cid:104) ˆΣ hh (cid:16) ( m h i + m h j ) / (cid:17)(cid:105) ij (given in the basis of the tree-level mass states).This procedure aims at more closely mimicking the actual values of the self-energies involvedin the mass calculation. In this approach the inverse propagator is also diagonalized by anorthogonal matrix U m .While these procedures capture the mixing effects induced by radiative corrections, at least partially,it is nevertheless obvious that they miss the normalization of the fields outlined in Eq. (2.37). Thismeans in particular that the norm as defined in Eq. (2.38) will always be identical to 1 if the matrix Z mix is approximated by either U or U m . We will discuss the impact of these approximations inthe following section. In this section we present the results of our Higgs mass calculation and compare them with theoutput of public tools for several CP -violating scenarios. We also investigate the relevance of thematrix Z mix for transition amplitudes in the example of the one-loop corrected decays of one Higgsfield into a tau/anti-tau pair, h i → τ + τ − .The choice for the top-quark mass is m t = 173 . m t , and all stop-parameters are on-shell parameters.From the point of view of the Higgs phenomenology we test the scenarios presented in thissection with the full set of experimental constraints and signals implemented in the public tools HiggsBounds-4.3.1 [159–164] and
HiggsSignals-1.3.1 [164, 165].
FeynHiggs in the MSSM-limit
In the limit of vanishing λ and κ , the singlet superfield decouples from the MSSM sector and oneis left with an effective MSSM—the µ eff term persists as long as κ ∼ λ . We may then compare ourresults for the Higgs masses and the matrix Z mix in this limit to those of FeynHiggs-2.12.0 . Forthis to be meaningful, we adjust the settings of
FeynHiggs , so that they match the higher-ordercontributions and renormalization scheme of our NMSSM calculation. In particular, we imposethat the one-loop field-renormalization constants and tan β are DR-renormalized and select full one-loop and leading two-loop MSSM contributions of O ( α t α s + α t ). We also require that FeynHiggs takes the tan β -enhanced contributions to the down-type Yukawa couplings into account. Thecorresponding FeynHiggs input flags read
FHSetFlags[4,0,0,3,0,2,0,0,1,1] .11 π - π - π - π π π π π ϕ A t M h [ G e V ] M h M h - π - π - π - π π π π π ϕ A t M h i [ G e V ] our two loop FeynHiggs two loop
Figure 1 . Masses of the light, SM-like state h (left plot) and the two heavy states h and h (rightplot) as a function of φ A t . The solid lines denote the masses obtained with our calculation, while thesquares denote the masses obtained with FeynHiggs with the options indicated in the text. The scenario isrepresentative of the MSSM-limit of the NMSSM and we employ the following parameters: λ = κ = 10 − ,tan β = 10, m H ± = 500 GeV, µ eff = 250 GeV, A κ = −
100 GeV, m ˜ F = 1 . | A t | = A b = 2 . M = M = M / . Z - π - π - π - π π π π π Z - π - π - π - π π π π π Z - π - π - π - π π π π π Z - π - π - π - π π π π π Z - π - π - π - π π π π π Z - π - π - π - π π π π π Z - π - π - π - π π π π π Z - π - π - π - π π π π π Z - π - π - π - π π π π πϕ A t ϕ A t ϕ A t our two loop FeynHiggs two loop
Figure 2 . Modules for the nine elements of the matrix Z mix , obtained with our calculation (solid line)and FeynHiggs (squares). The colors follow the convention of Fig. 1: red for the coefficients defining thewave-function normalization of the light, SM-like state h , blue and green for the coefficients correspondingto the heavy states h and h , respectively. The parameters are chosen as in Fig. 1.
12e consider a region in the parameter space of the NMSSM with the following characteristics: λ = κ = 10 − , tan β = 10, m H ± = 500 GeV, µ eff = 250 GeV, A κ = −
100 GeV; the sfermion softmasses are set to the universal value of 1 . . | A t | = A b = 2 . M = M = M / . φ A t . Avariation of φ A t (or of any MSSM-like phase) in such a naive direction is of limited phenomenologicalinterest, since in this case limits from Electric Dipole Moments (EDM) are violated almost as soonas CP , see e.g. Ref. [84]. In the following we dismiss this issue, however, and allow φ A t to vary overits full range. Indeed, we are only interested in comparing our results with those of FeynHiggs . Dueto the largely SM-like properties of the state with a mass close to 125 GeV, our scenario appears toretain characteristics that are compatible with the experimental data implemented in
HiggsBounds and
HiggsSignals , over the whole range of φ A t . The results for the Higgs masses are displayedin Fig. 1. We observe a near perfect agreement between our results (solid curves) and those of FeynHiggs (squares) with differences of order MeV. This agreement is expected, since we closelyfollow the procedure for the renormalization and processing of the MSSM-like input of
FeynHiggs .Moreover, due to the small values for λ and κ , deviations induced by genuine NMSSM effectsremain negligible. The results for the elements of the matrix Z mix are displayed in Fig. 2. Again,we find a very good agreement between our results and FeynHiggs with differences below 1 (cid:104) forthe modules. CP -conserving limit We now turn away from the MSSM limit. Our mass calculation can be confronted to the routinespresented in [113] in the CP -conserving case. Both approaches employ an identical renormalizationscheme in this limit, with the exception of the electroweak vev, which receives a DR renormalizationin [113] while we parametrize v in terms of M W , M Z and e (see Eq. (2.27)). However, in [113] theinput for v is obtained via a reparametrization from our scheme (the scheme using α ( M Z ) as input isalso considered), as explained in section 2.3 of that reference. Therefore, both mass predictions aredirectly comparable and the mismatch between them should be understood as an effect of two-loopelectroweak order, due to the approximations in the reparametrization used by [113].We consider the following region in the parameter space of the NMSSM: κ = λ/
2, tan β = 10, M H ± = 1 TeV, µ eff = 125 GeV, A κ = −
70 GeV; the soft masses are taken as in the previoussubsection while the trilinear soft sfermion couplings are all set to 0 . A t = 1 . λ from ∼ .
5. The masses of the three lightestHiggs states are displayed in the plot on the left-hand side of Fig. 3. The dominantly SM-likestate is the heaviest of the three (green curve). The two lighter states are dominantly singlet, CP -even (red curve) or CP -odd (blue curve). In the MSSM-limit, these three states are significantlylighter than 125 GeV: this results in an unsatisfactory phenomenological situation in view of theLHC measurements. With increasing λ , the CP -even singlet-doublet mixing uplifts the mass ofthe dominantly SM-like state, leading to phenomenologically viable characteristics—as tested with HiggsSignals —for λ ∼ .
2. There, we observe that h possesses a sizable doublet component anda mass M h (cid:39)
100 GeV, so that this state could offer an interpretation of the local excess observedat LEP in e + e − → Z + ( H → b ¯ b ) searches [166].We then focus on the comparison with the masses predicted by [113]. The plot on the left-handside of Fig. 3 illustrates a general agreement between our calculation (solid curves) and the resultsof [113] (squares). On the right-hand side of Fig. 3, we display the mass differences between the twoprocedures, which are due to differences of two-loop order induced by the reparametrization usedby [113]. We observe vanishing effects in the MSSM-limit while the mass differences eventuallyreach O (40 MeV) for λ (cid:39) .
16. This can be understood in the following fashion: the leadingeffect originates in the Higgs mass matrix at the tree level, where an explicit dependence on v renormalized e reparametrized mass di ff erence from scheme of eM h M h M h λ M h i [ G e V ] Δ M h Δ M h Δ M h - - - - λ Δ M h i [ G e V ] Figure 3 . Masses of the three lightest Higgs states h (red), h (blue), h (green) in the CP -conserving limitfor varying λ = 2 κ and the following input: tan β = 10, M H ± = 1 TeV, µ eff = 125 GeV, A κ = −
70 GeV, m ˜ F = 1 . A t = 2 TeV, A f (cid:54) = t = 0 . M = M = M / . appears only through terms of the form λ v and κ v (quadratically for the doublet and singlet massentries, and linearly for the doublet–singlet mixing). These terms are processed differently in bothapproaches: in [113] v is regarded as an independent DR parameter, while in our calculation v is adependent quantity that is expressed in terms of the independent parameters M W , M Z and e . Whilethe reparametrization of [113] should restore the agreement between the two procedures, neglectedeffects of two-loop electroweak order in this reparametrization result in a small mismatch. Since theterms that convey this mismatch come with prefactors λ or κ , the difference vanishes in the MSSMlimit ( λ, κ → β (cid:29)
1, it is possible tounderstand why the mass of the CP -odd singlet (blue curve) is largely insensitive to the mismatch:terms ∝ ( λ v ) in the CP -odd singlet mass entry are suppressed as 1 / tan β . Additionally, leadingone-loop radiative corrections of O ( α t ) induce further dependence on the processing of v . However,these corrections are suppressed for the points of Fig. 3, as the stops are relatively light.On the whole, the numerical mismatch with the procedure of [113] is very minor, which placesour current code in the direct continuity of this earlier work. NMSSMCALCNMSSMCALC is particularly suitable for a comparison with our calculation, since its mixed DR/on-shell renormalization scheme is relatively close to the one that we use. Yet, we note severaldifferences between the prescriptions implemented by
NMSSMCALC and the procedure that we haveoutlined in section 2 (defining our “default” calculation). First,
NMSSMCALC applies a renormalizationscheme for the electric charge employing α ( M Z ) as input—whereas we decided to define α via itsrelation to G F . Then the input parameters in the stop sector are defined in the DR schemein NMSSMCALC —while we employ on-shell definitions. Additionally, we resum large tan β effectsfrom our definition of the bottom Yukawa, contrarily to the Higgs-mass calculation of NMSSMCALC .Finally,
NMSSMCALC includes only O ( α t α s ) corrections at the two-loop order—where we consider We remind the reader that both in [113] and in our calculation M W and M Z are chosen as independent, on-shellparameters. Therefore, the corresponding terms in the Higgs mass-matrix are not affected by the differences in therenormalization/reparametrization discussed here. Note that it is somewhat more involved to compare our results quantitatively with RGE-based tools, as the inputrequires a conversion to the appropriate scheme (usually DR) and a running to the correct input scale [167]. Forthis reason, we shall confine our discussion to comparisons with
NMSSMCALC , which shares closer characteristics withour approach. A similar comparison for real parameters has been presented in [168]. (cid:0) α t (cid:1) effects as well. However, the two-loop O ( α t α s ) contributions of NMSSMCALC are exhaustivein the NMSSM (including corrections for the self-energies with at least one external singlet field)—whereas ours are obtained in the MSSM approximation.These observations mean that our mass-calculation is not directly (at least, not quantitatively)comparable to the predictions of
NMSSMCALC , since, of the items listed above, the first few producea deviation relative to the scheme, while the later ones generate a mismatch of higher orders.Consequently, several adjustments need to be performed in order to make a comparison meaningfuland control the sources of deviations. Thus,
NMSSMCALC has been adjusted in view of acceptingon-shell input in the stop sector. Moreover, we also establish a “modified” version of our routinesthat attempts to mimic the choices of
NMSSMCALC — i. e. employing α ( M Z ), discarding large-tan β effects for Y b and subtracting O (cid:0) α t (cid:1) corrections—although we cannot currently include O ( α t α s )corrections beyond the MSSM, so that this effect should control the difference of our modifiedversion with NMSSMCALC . Beyond this comparison with
NMSSMCALC , we will also try to quantify themagnitude of the other higher-order effects that distinguish our “default” result from
NMSSMCALC .First, we consider the regime of the NMSSM with low tan β and large λ . This region in parameterspace is well-known for maximizing the specific NMSSM tree-level contributions to the mass of theSM-like Higgs state as well as for stimulating singlet–doublet mixing effects and other genuineaspects of the NMSSM phenomenology. We employ the following parameters: λ = 0 . | κ | = 0 . β = 2, M H ± = 1170 GeV, µ eff = 500 GeV, A κ = −
70 GeV; the soft masses are taken as in Fig. 1with the exception of the squarks of the third generation, for which the soft masses and trilinearcouplings are set to 500 GeV and 100 GeV, respectively. In the regime under consideration genuineNMSSM effects are indeed sufficient to produce a SM-like state in the observed mass-range withoutrequiring large top/stop corrections. We vary the phase φ κ (we restrict to a range where the tree-level squared Higgs masses remain positive). We note that, contrarily to MSSM-like phases, thephases from the singlet sector are allowed a wide range of variation without conflicting with themeasured EDM [83, 169].The results for the mass prediction are presented in Fig. 4. At vanishing φ κ the mass of theSM-like state (in blue) is somewhat low, m h ∼
120 GeV, so that this point in parameter space has avery marginal agreement with the observed characteristics of the Higgs state. For non-vanishing φ κ ,however, a CP -violating mixing with the lighter pseudoscalar singlet (in red) develops: this effectincreases the mass of the light mostly CP -even state h but affects its otherwise SM-like propertiesonly in a subleading way. Consequently, we recover an excellent agreement with the LHC results—as tested by HiggsSignals and
HiggsBounds —for e. g. φ κ (cid:39) − .
11. Additionally, the dominantly CP -odd singlet h then has a mass close to 100 GeV. As it acquires a doublet CP -even componentvia mixing, it could explain the LEP local excess in b ¯ b final states [166]. The mostly CP -evensinglet h (in green), with mass at ∼
210 GeV plays no significant role. The masses of the heavierdoublet-like fields h and h are approximately constant and close to M H ± .In Fig. 4, we observe a good agreement between our results (solid lines), computed as describedin section 2, and the predictions of NMSSMCALC (squares), although the corresponding masses aredefined in different schemes and at different orders. For a more quantitative comparison, we turn toour “modified” scheme for the mass calculation. On the left-hand side of Fig. 5, we plot the deviationbetween the corresponding results and the predictions of
NMSSMCALC for the three lightest Higgsstates. We checked that the one-loop results are virtually identical, so that the differences between
NMSSMCALC and our calculation are entirely controlled by two-loop effects. We observe typicaldeviations of order 0 . O ( α t α s ) correctionsbeyond the MSSM-approximation. As could be expected, the masses of the mostly singlet states(red and green lines) tend to exhibit the largest effect, though the mass-predictions for the SM- We thank K. Walz for providing a modified version of
NMSSMCALC for this feature. ur default two loop NMSSMCalc two loop M h M h M h - π - π - π - π π π
16 3 π π ϕ κ M h i [ G e V ] Figure 4 . Masses of the three lighter Higgs fields as a function of φ κ for the scenario λ = 0 . | κ | = 0 . β = 2, M H ± = 1170 GeV, µ eff = 500 GeV, A κ = −
70 GeV, m ˜ Q , ˜ T, ˜ B = 0 . A t = A b = 0 . M = M = M / . CP -odd, singlet-like state h ;blue is associated to the essentially SM-like state h and green corresponds to the mostly CP -even singlet-like state h . We display our “default” result for the masses (solid curves) as well as the predictions of NMSSMCALC (squares). Δ M h Δ M h Δ M h - π - π - π - π π π
16 3 π π - ϕ κ Δ M h i [ G e V ] Δ M h Δ M h Δ M h - π - π - π - π π π
16 3 π π - ϕ κ Δ M h i [ G e V ] our modi fi ed two loop minus NMSSMCalc two loop two - loop shift by ( α t ) Figure 5 . Impact of two-loop contributions in the scenario of Fig. 4. On the left-hand side,∆ M h i = M h i − M NC h i correspond to the mass-differences (for each of the three lightest Higgs states) be-tween the predictions of our “modified” scheme and NMSSMCALC : O ( α t α s ) should dominate these deviations.On the right-hand side, ∆ M h i corresponds to the mass-shifts associated to O (cid:0) α t (cid:1) contributions, which arecalculated in our “default” scheme. The color of the curves match the convention of Fig. 4. like state may still differ by ∼ . φ κ (cid:39) O (cid:0) α t (cid:1) effects, which is quantified in our “default” scheme. Here again,the typical impact on the masses is of order 1 GeV. Expectedly, the masses of the almost puresinglet states (red curve at φ κ (cid:39) CP -odd singlet (red curve) is only affectedwhen the corresponding state acquires a non-vanishing doublet component ( φ κ (cid:54) = 0).In Fig. 6, we compare the U matrix elements that are delivered by NMSSMCALC (squares) withours (solid line;
NMSSMCALC does not provide Z mix ). The results show a satisfactory agreement alsoat this level. 16 ur default two loop NMSSMCalc two loop U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π U - π - π - π - π π π
16 3 π π ϕ κ ϕ κ ϕ κ Figure 6 . The matrix-elements | U ij | in our calculation (full curves) and in NMSSMCALC (squares) for thescenario of Fig. 4.
Subsequently, we present our results in another region of the parameter space: λ = 0 . | κ | = 0 . β = 25, m H ± = 1 TeV, µ eff = 200 GeV, A κ = −
750 GeV, the gaugino soft masses as well asthe soft masses for the sfermions of first and second generations are chosen as before; for the thirdgeneration, the soft sfermion mass is set to 1 . − CP -even state and the heavy CP -even and CP -odddoublet-like states receive comparable masses of the order of 1 TeV. This results in a sizable mixingfor the corresponding fields h , h and h , which includes both singlet–doublet admixture as wellas CP -violation (for non-vanishing φ κ ). The SM-like Higgs state has a mass close to ∼
124 GeV onthe whole range of φ κ , which leads to a good agreement with the Higgs properties measured at theLHC (as tested with HiggsSignals ). The heaviest state h has a mass of ∼ . NMSSMCALC is represented by the squares at about 118 GeV,which is substantially smaller than ours (by ≈ NMSSMCALC (our“modified result”, dotted curve), this discrepancy is considerably reduced. In fact, the differencebetween our full result and
NMSSMCALC ’s is largely driven by the O (cid:0) α t (cid:1) two-loop contributions,missing in NMSSMCALC . Again, both results are virtually identical at the one-loop order.On the right-hand side of Fig. 7, we turn to the heavier states h , h and h of this scenario.Our default results (full curves) are compatible with the predictions of NMSSMCALC (squares). Thediscrepancies are of order 1–3 GeV only, which should be considered from both the perspective of the17 π - π - π - π π π π π ϕ κ M h [ G e V ] M h M h M h - π - π - π - π π π π π ϕ κ M h i [ G e V ] our default two loop our modi fi ed two loop NMSSMCalc two loop
Figure 7 . Mass predictions as a function of φ κ for the lightest, mostly SM-like Higgs state h on the left-hand side, and the states h (blue), h (green) and h (orange) on the right-hand side. The latter involveessentially the CP -even singlet and the two CP -even and CP -odd heavy-doublet degrees of freedom that mixsubstantially. The full curves correspond to our default result; the squares are obtained with NMSSMCALC ;the dotted line represents our modified result. The parameters are chosen as follows: λ = 0 . | κ | = 0 . β = 25, m H ± = 1 TeV, µ eff = 200 GeV, A κ = −
750 GeV, m ˜ Q , ˜ T, ˜ B = 1 . A t = A b = − M = M = M / . different renormalization scheme of the electric coupling e and the different two-loop contributions.Actually, the mass predictions match almost exactly when comparing NMSSMCALC with our modifiedscheme (dotted curves). The corresponding deviations are shown on the left-hand side of Fig. 8 andfall in the range of 100 MeV. In this precise case, the difference between our results and
NMSSMCALC is essentially driven by the resummation of large-tan β effects in the b -quark Yukawa coupling. Onthe right-hand side of Fig. 8, we quantify the associated mass-shift and find an impact of a few GeV.Finally, we turn to the U matrix elements for h , h and h in Fig. 9. There, we observe sizabledeviations between our default result (solid curves) and NMSSMCALC (squares), which, however,have no deep-reason to agree in view of the diverging options. If we keep in mind that the maindifference between our full scheme and
NMSSMCALC is controlled by the large-tan β corrections to Y b in this precise scenario, it is not surprising to observe large shifts, as mixing angles are indeedvery sensitive to small deviations in the mass-matrix for states that are very close in mass. Thesedifferences largely vanish when we identify the output of NMSSMCALC with our modified results(dotted lines), which is better equipped for comparisons with this code.In summary, the results of our mass calculation are largely compatible with the predictions of
NMSSMCALC . Deviations with a magnitude of O (GeV) are indeed within the expected range if we allowfor the different renormalization scheme of e and higher-order contributions. Such discrepancies tendto be reduced sizable when we modify our routines to adopt the assumptions of NMSSMCALC . Theimpact of two-loop O ( α t α s ) contributions beyond the MSSM, two-loop O (cid:0) α t (cid:1) and the resummationof large-tan β effects in the Yukawa couplings can be clearly identified from this comparison. As afinal remark, let us emphasize that the uncertainty on the Higgs-mass calculation from unknownhigher-order contributions and parametric errors may reach several GeV [108, 167, 168]. mix and the h i → τ + τ − decays The matrix Z mix is not an observable quantity in itself. It is a renormalization-scheme dependentobject relating the tree-level mass states of the Higgs sector to the physical Higgs fields. For on-shellrenormalized fields Z mix is trivial. In any other renormalization scheme, however, it is mandatoryto include this transition to the physical fields for a proper description of external legs in Feynmandiagrams at higher orders. 18 M h Δ M h Δ M h - π - π - π - π π π π π - - ϕ κ Δ M h i [ G e V ] Δ M h Δ M h Δ M h - π - π - π - π π π π π - - - ϕ κ Δ M h i [ G e V ] our modi fi ed two loop minus NMSSMCalc two loop two - loop shift by resummed m b Figure 8 . Impact of the higher-order effects in the mass-predictions for h , h and h in the scenarioof Fig. 7. On the left-hand side, we show the deviation in mass between NMSSMCALC and our modifiedscheme (controlled by O ( α t α s ) corrections beyond the MSSM-approximation). The plot on the right-handside illustrates the impact of large-tan β effects in Y b : the considered mass-difference is that induced in ourdefault scheme by the ∆ b term. The colors follow the conventions of Fig. 7. The discontinuities at φ κ = ± π originate from the mass calculation in NMSSMCALC . our default two loop our modi fi ed two loop NMSSMCalc two loop U - π - π - π - π π π π π U - π - π - π - π π π π π U - π - π - π - π π π π π U - π - π - π - π π π π π U - π - π - π - π π π π π U - π - π - π - π π π π π U - π - π - π - π π π π π U - π - π - π - π π π π π U - π - π - π - π π π π π ϕ κ ϕ κ ϕ κ Figure 9 . The matrix-elements | U ij | in our calculation (solid curves), in NMSSMCALC (squares), and in ourmodified calculation closer to the options of
NMSSMCALC (dotted) for the scenario of Fig. 7. h M h M h - π - π - π - π π π π π ϕ κ M h i [ G e V ] Z Z Z Z Z - π - π - π - π π π π π ϕ κ Z i Figure 10 . On the left-hand side, the masses of the three lighter Higgs states are depicted as a functionof φ κ , including all available two-loop contributions. On the right-hand side, the squared norms | Z i | (asdefined in Eq. (2.38)) of the five eigenvectors defining Z mix are shown. The scenario is characterized by λ = 0 . | κ | = 0 .
1, tan β = 2, m H ± = 1 . µ eff = 500 GeV, A κ = −
100 GeV, m ˜ Q , ˜ T, ˜ B = 0 . A t = A b = 0 . M = M = M / . A remarkable aspect of Z mix is that the eigenvectors that it contains do not preserve unitarity withrespect to the tree-level fields. Instead, they satisfy the normalization condition given in Eq. (2.37).This is a feature that the approximations U and U m are unable to capture (by construction). Ina first step, we will show that the norms in Z mix can differ from 1 by a few percent in the schemethat we have described in section 2. Beyond the normalization of the fields, U and U m also differfrom Z mix in that they diagonalize the mass-matrix away from the poles of the propagator.However, as we wrote above, Z mix is a scheme-dependent object and we should not pay excessiveattention to its actual structure. In order to characterize its role in an observable quantity, we willconsider the h i → τ + τ − decays at the one-loop level. We have chosen this particular channel as it isone of the main fermionic Higgs decays and proves technically simple to implement in a predictiveway. Moreover, one-loop corrections are of purely electroweak nature—QCD contributions occuronly at three-loop order and beyond—so that radiative corrections are expected to be moderate.This allows for a clean appreciation—free of large higher-order uncertainties—of the impact of thewave-function normalization matrix Z mix . Radiative corrections are computed with our model file,except for the QED contributions, which are included according to the prescriptions of Refs. [170,171]. There, Z mix intervenes in the decay amplitudes of the physical fields according to Eq. (2.32)(we dismiss the superscript ’phys’ throughout this section). We will show that the substitution of Z mix by the approximations U and U m may lead to sizable deviations in certain regions of the theNMSSM parameter space. This result confirms the outcome of similar studies in the MSSM [109].We turn to the following NMSSM input: the parameters are chosen as in Fig. 4, except for M H ± = 1 . A κ = −
100 GeV. We have plotted the masses of the three lighter Higgs fieldsas a function of φ κ in the plot on the left-hand side of Fig. 10. For vanishing φ κ the lightestHiggs state h is SM-like and we checked with HiggsBounds and
HiggsSignals that this point isconsistent with the experimental data. The dominantly CP -odd, singlet-like state h is only slightlyheavier than the state h in this case. For increasing values of φ κ the mixing of the states h and h tends to lower the mass of the SM-like state h , which eventually becomes too light to accommodatethe experimental data. The dominantly CP -even, singlet-like state h has a near constant mass of ∼
210 GeV for all depicted values of φ κ . The two heavier, CP -even and CP -odd doublet-like stateshave masses close to ∼ . Details on the calculation of the decays at the one-loop level will be presented in a future publication. | Z i | of the eigenvectors—see Eq. (2.38)—in this scenario areshown in the plot on the right-hand side of Fig. 10. We observe a departure from the value 1—whichwould correspond to a unitary transition, as modeled by the approximations U and U m —by a fewpercent. The local extrema at φ κ (cid:39) | Z | (red curve) and | Z | (blue curve) are associated tothe sudden disappearance of the mixing between the light CP -odd singlet and the SM-like statesat φ κ = 0 ( CP -conserving limit). The discontinuities of | Z | and | Z | (green curve) at φ κ (cid:39) ± . ± . h → W + W − , h → Z , h → h h ).These “spikes” are associated to the singularities of the first derivatives of the loop functions involvedin the determination of Z mix —the apparent singularities actually come with a finite height due tothe imaginary parts of the poles. A proper description of these threshold regions would requirethat the interactions among the daughter particles (of the decays at threshold) are properly takeninto account, which would result in e. g. interactions between the Higgs state and bound-states or s -waves of the daughter particles. This, however goes beyond the scope of the present work.We now turn to the decay widths Γ( h i → τ + τ − ) in the scenario of Fig. 10. The widths aredisplayed in the left column of Fig. 11 in the exhaustive description of the Higgs external leg ( i. e. employing Z mix ; solid curves), in the U m approximation (dashed lines) and in the U approximation(dotted lines), for the five Higgs mass-eigenstates. We observe a sharp variation close to φ κ = 0 forthe decays of h and h , both in the full and approximate descriptions. It is associated to the mixingthat develops between the SM-like state h and the dominantly CP -odd, singlet-like state h : thiseffect transfers part of the doublet component of h to h , so that the second state acquires a non-vanishing coupling to SM fermions at the expense of the first. The sum of the decay widths for boththese states remains approximately constant in the vicinity of φ κ = 0. The width Γ( h → τ + τ − )appears to be an order of magnitude smaller than the corresponding widths for h and h , an effectthat is associated to the dominantly CP -even, singlet-like nature of h . Still, Γ( h → τ + τ − ) nearlydoubles in the considered interval of φ κ , while the mass of h is fairly stable: we can understand thisfact in terms of the acquisition of a larger doublet component, which is channeled by the increasedproximity of the masses of h and h . The widths of h and h are essentially constant with onlysmall relative changes. The general φ κ -dependency of the approximated and the full results arevery similar. Yet, a systematic shift can be observed, especially in the case of U . This is consistentwith the findings of similar studies in the context of the MSSM [109].On the right-hand side of Fig. 11, we show the difference between the full and the approximateresults, ∆Γ = Γ − Γ appr , normalized to the more accurate one obtained with Z mix . When Z mix isapproximated by U (dotted lines), the typical discrepancy averages 4%, although the deviationreaches beyond 10% in the case of the decays of the two lightest Higgs states in the vicinity of, butnot at, φ κ (cid:39)
0. We stress that this interval close to φ κ = 0 corresponds to the phenomenologicallyrelevant region from the perspective of the measured Higgs properties. The approximation of Z mix by U m tends to provide better estimates of the full result, though deviations reach up to ∼ Z mix areintimately related to the proximity in mass of the SM-like and dominantly CP -odd, singlet-likestates: as the approximations capture the dependence on the external momentum either partially( U m ) or not at all ( U ), the gap between the diagonal elements of the Higgs mass-matrix, hencethe mixing between the two states, is not quantified properly. While this precise configurationmight appear somewhat anecdotal, we wish to point out the popularity of NMSSM scenarios witha sizable singlet–doublet mixing. Dismissing this extreme case, the approximations of Z mix by U ,and to a lesser extent by U m , still generate errors of the order of a few percent at the level of thedecay widths. In view of the precision of the measurements achievable at the LHC [3, 172–175],such discrepancies may appear of secondary importance. In the long run, however, if the Higgscouplings are studied more closely, e. g. at a linear collider [176–179], one would have to try andkeep such sources of error to a minimum. 21 π - π - π - π π π π π Γ ( h → τ + τ - ) [ M e V ] - π - π - π - π π π π π Δ ΓΓ ( h → τ + τ - ) - π - π - π - π π π π π Γ ( h → τ + τ - ) [ M e V ] - π - π - π - π π π π π - - - Δ ΓΓ ( h → τ + τ - ) - π - π - π - π π π π π Γ ( h → τ + τ - ) [ M e V ] - π - π - π - π π π π π - - - - - Δ ΓΓ ( h → τ + τ - ) - π - π - π - π π π π π Γ ( h → τ + τ - ) [ M e V ] - π - π - π - π π π π π - - - - - - Δ ΓΓ ( h → τ + τ - ) - π - π - π - π π π π π Γ ( h → τ + τ - ) [ M e V ] - π - π - π - π π π π π - - - - Δ ΓΓ ( h → τ + τ - ) ϕ κ ϕ κ mixing via Z mix mixing via U m mixing via U Figure 11 . In the left column, we show the decay widths Γ( h i → τ + τ − ) in the scenario of Fig. 10 for thefive neutral Higgs states. The widths are computed at the one-loop level, and the mixing of the external,physical Higgs fields is expressed in terms of the matrix Z mix (solid), or approximated by the matrices U m (dashed) or U (dotted). In the right column, the differences ∆Γ = Γ − Γ appr between the widths obtainedwith Z mix and its approximate treatments U m (dashed) and U (dotted) are depicted, normalized to thewidth Γ obtained with Z mix . Conclusions
In this paper, we have discussed the renormalization of the NMSSM Higgs sector, including complexparameters. Radiative contributions to the Higgs self-energies have been included up to the leadingtwo-loop MSSM-like effects of O (cid:0) α t α s + α t (cid:1) . Beyond the calculation of on-shell Higgs masses inthis scheme, we were interested in determining the transition matrix Z mix between the mass- andtree-level states. The latter plays an essential role in the proper description of external Higgs legsin physical processes at the radiative level.Our predictions for the Higgs masses have been compared to the calculations of existing toolsin several NMSSM scenarios. In the MSSM-limit of the model, we have recovered an excellentagreement with FeynHiggs . For non-vanishing λ and κ , we first compared our Higgs-mass predictionwith the findings of a previous extension of FeynHiggs to the NMSSM in the case of real parameters,and found nearly identical results. Second, we compared our calculation in the case of complexparameters with
NMSSMCalc and found values of the Higgs masses which are compatible, althoughsmall differences emerge as a result of different processing of the two-loop pieces, both for low andhigh tan β .Finally, we investigated the impact of the transition matrix Z mix on the h i → τ + τ − width ina scenario with low tan β and large λ , where the SM-like Higgs state may have a sizable mixingwith the CP -odd singlet. We compared the full one-loop calculation of the width— i. e. including Z mix —with the popular approximations U and U m —which are determined for fixed, unphysicalmomenta. We found typical deviations at the percent level, although larger effects can develop inthe presence of almost-degenerate states, especially in the U approximation. Such precision effectswill matter when the measurement of fermionic Higgs couplings reaches comparable accuracy.In its current form, our mass-computing tool is contained within a Mathematica package. Intime, our routines should be incorporated in an extension of
FeynHiggs to the NMSSM.
Acknowledgments
We thank S. Heinemeyer for helpful comments on the manuscript. The work of F. Domingo issupported in part by CICYT (Grant FPA 2013-40715-P), in part by the MEINCOP Spain undercontract FPA2016-78022-P, and in part by the Spanish “Agencia Estatal de Investigaci´on” (AEI)and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA2016-78645-P. The work of P. Drechsel and S. Paßehr is supported by the Collaborative Research CenterSFB676 of the DFG, “Particles, Strings and the early Universe.”
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