1 Introduction

LPSC15154,

UCD-2015-001,

SCIPP-15/08

Scrutinizing the Alignment Limit

[2mm] in Two-Higgs-Doublet Models

Part 1:  GeV

Jérémy Bernon111Email: bernon@lpsc.in2p3.fr, John F. Gunion222Email: jfgunion@ucdavis.edu, Howard E. Haber333Email: haber@scipp.ucsc.edu, Yun Jiang444Email: yunjiang@ucdavis.edu, Sabine Kraml555Email: sabine.kraml@lpsc.in2p3.fr

Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53 Avenue des Martyrs, F-38026 Grenoble, France

[2mm] Department of Physics, University of California, Davis, CA 95616, USA

[2mm] Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA

In the alignment limit of a multi-doublet Higgs sector, one of the Higgs mass eigenstates aligns with the direction of the scalar field vacuum expectation values, and its couplings approach those of the Standard Model (SM) Higgs boson. We consider CP-conserving Two-Higgs-Doublet Models (2HDMs) of Type I and Type II near the alignment limit in which the lighter of the two CP-even Higgs bosons, , is the SM-like state observed at . In particular, we focus on the 2HDM parameter regime where the coupling of to gauge bosons approaches that of the SM. We review the theoretical structure and analyze the phenomenological implications of the regime of alignment limit without decoupling, in which the other Higgs scalar masses are not significantly larger than and thus do not decouple from the effective theory at the electroweak scale. For the numerical analysis, we perform scans of the 2HDM parameter space employing the software packages 2HDMC and Lilith, taking into account all relevant pre-LHC constraints, the latest constraints from the measurements of the Higgs signal at the LHC, as well as the most recent limits coming from searches for heavy Higgs-like states. We contrast these results with the alignment limit achieved via the decoupling of heavier scalar states, where is the only light Higgs scalar. Implications for Run 2 at the LHC, including expectations for observing the other scalar states, are also discussed.

## 1 Introduction

The minimal version of the Standard Model (SM) contains one complex Higgs doublet, resulting in one physical neutral CP-even Higgs boson after electroweak symmetry breaking. The discovery [1, 2] of a new particle with mass of about  [3] and properties that match very well those expected for a SM Higgs boson was a real triumph of Run 1 of the LHC. Fits of the Higgs couplings performed by ATLAS [4] and CMS [5] show no significant deviations from SM expectations. (A combined global fit of the Higgs couplings based on the Run 1 results was performed by some of us in [6].) However, one has to keep in mind that the present precisions on the Higgs couplings are, roughly, of the order of tens of percent, so substantial deviations are still possible. Indeed, the SM is not necessarily the ultimate theoretical structure responsible for electroweak symmetry breaking, and theories that go beyond the SM, such as supersymmetry, typically require an extended Higgs sector [7, 8, 9, 10]. Hence, the challenge for Run 2 of the LHC, and other future collider programs, is to determine whether the observed state is the SM Higgs boson, or whether it is part of a non-minimal Higgs sector of a more fundamental theory.

In this paper, we take Two-Higgs-Doublet Models (2HDMs) of Type I and Type II [11] as the prototypes for studying the effects of an extended Higgs sector. Our focus will be on a particularly interesting limit of these models, namely the case in which one of the neutral Higgs mass eigenstates is approximately aligned with the direction of the scalar field vacuum expectation values. In this case, the coupling to gauge bosons of the Higgs boson observed at the LHC tends towards the SM limit, .111We use the notation of coupling scale factors, or reduced couplings, employed in [6]: () for the coupling to gauge bosons, for the couplings to up-type and down-type fermions and for the loop-induced couplings to photons and gluons. This so-called alignment limit is most easily attained in the decoupling limit [12], where all the other non-SM-like Higgs scalars of the model are heavy. However, the alignment limit of the 2HDM can also be achieved in a parameter regime in which one or more of the non-SM-like Higgs scalars are light (and in some cases very light). This region of alignment without decoupling is a primary focus of this paper.

An extensive review of the status of 2HDMs of Type I and Type II was given in [13, 14]. Interpretations of the recently discovered Higgs boson at 125 GeV in the context of the 2HDMs were also studied in [15, 16, 17, 18, 19, 21, 20]. The possibility of alignment without decoupling was first noted in [12] and further clarified in [22, 23]. Previous studies of alignment without decoupling scenarios in the light of the LHC Higgs results were conducted in [24, 25, 26]. The specific case of additional light Higgs states in 2HDMs with mass below  GeV was studied in [27].

Considering experimental as well as theoretical uncertainties, the expected precision for coupling measurements at the LHC after collecting 300 fb of data is about 4–6% for the coupling to gauge bosons, and of the level of 6–13% for the couplings to fermions [28]. The precision improves by roughly a factor of 2 for at the high-luminosity run of the LHC with 3000 fb. At a future international linear collider (ILC) with  GeV to  TeV, one may measure the couplings to fermions at the percent level, and the coupling to gauge bosons at the sub-percent level. A detailed discussion of the prospects of various future colliders can be found in [28].

We take this envisaged accuracy on as the starting point for the numerical analysis of the alignment case. Concretely, we investigate the parameter spaces of the 2HDMs of Type I and Type II assuming that the observed state is the , the lighter of the two CP-even Higgs bosons in these models, and imposing that (note that in any model whose Higgs sector consists of only doublets and/or singlets). The case of the heavier CP-even being the state at is discussed in a separate paper [29].

Taking into account all relevant theoretical and phenomenological constraints, including the signal strengths of the observed Higgs boson, as well as the most recent limits from the non-observation of any other Higgs-like states, we then analyse the phenomenological consequences of this scenario. In particular, we study the variations in the couplings to fermions and in the triple-Higgs couplings that are possible as a function of the amount of alignment when the other Higgs states are light, and contrast this to what happens in the decoupling regime. Moreover, we study the prospects to discover the additional Higgs states when they are light.

The public tools used in this study include 2HDMC [30] for computing couplings and decay widths and for testing theoretical constraints within the 2HDM context, Lilith 1.1.2 [31] for evaluating the Higgs signal strength constraints, and SusHi-1.3.0 [32] and VBFNLO-2.6.3 [33] for computing production cross sections at the LHC.

The paper is organised as follows. In Section 2 we first review the theoretical structure of the 2HDM. A softly-broken discrete -symmetric scalar potential is introduced using a basis of scalar doublet fields (called the -basis) in which a the symmetry is manifest. The Higgs basis is then introduced, which provides an elegant framework for exhibiting the alignment limit. We then provide a comprehensive discussion of the Higgs couplings in the alignment regime. In Section 3, we explain the setup of the numerical analysis and the tools used. The results are presented in Section 4. Section 5 contains our conclusions. In Appendix A, detailed formulae relating the quartic coefficients of the Higgs potential in the -basis to those of the Higgs basis are given. Some useful analytical expressions regarding the trilinear Higgs self-couplings in terms of physical Higgs masses are collected in Appendix B.

NB: This version of the paper has been updated to include exactly the same constraints as Part 2 with  GeV [29].

## 2 CP-conserving 2HDM of Types I and II

In this section, we review the theoretical structure of the two-Higgs doublet model. Comprehensive reviews of the model can also be found in, e.g.[34, 23, 12, 35]. In order to avoid tree-level Higgs-mediated flavor changing neutral currents (FCNCs), we shall impose a Type-I or II structure on the Higgs-fermion interactions. This structure can be naturally implemented [36, 37] by imposing a discrete symmetry on the dimension-four terms of the Higgs Lagrangian. This discrete symmetry is softly-broken by mass terms that appear in the Higgs scalar potential. Nevertheless, the absence of tree-level Higgs-mediated FCNCs is maintained, and FCNC effects generated at one loop are all small enough to be consistent with phenomenological constraints over a significant fraction of the 2HDM parameter space [38, 39, 40, 41].

Even with the imposition of the softly-broken discrete symmetry mentioned above, new CP-violating phenomena in the Higgs sector are still possible, either explicitly due to a physical complex phase that cannot be removed from the scalar potential parameters or spontaneously due to a CP-violating vacuum state. To simplify the analysis in this paper, we shall assume that these CP-violating effects are absent, in which case one can choose a basis of scalar doublet Higgs fields such that all scalar potential parameters and the two neutral Higgs field vacuum expectation values are simultaneously real. Moreover, we assume that only the neutral Higgs fields acquire non-zero vacuum expectation values, i.e. the scalar potential does not admit the possibility of stable charge-breaking minima [43, 44].

We first exhibit the Higgs scalar potential, the corresponding Higgs scalar spectrum and the Higgs-fermion interactions subject to the restrictions discussed above. Motivated by the Higgs data, we then examine the conditions that yield an approximately SM-like Higgs boson.

### 2.1 Higgs scalar potential

Let and denote two complex , SU(2) doublet scalar fields. The most general gauge invariant renormalizable scalar potential is given by

 V = m211Φ†1Φ1+m222Φ†2Φ2−[m212Φ†1Φ2+h.c.]+12λ1(Φ†1Φ1)2+12λ2(Φ†2Φ2)2+λ3(Φ†1Φ1)(Φ†2Φ2) (1) +λ4(Φ†1Φ2)(Φ†2Φ1)+{12λ5(Φ†1Φ2)2+[λ6(Φ†1Φ1)+λ7(Φ†2Φ2)]Φ†1Φ2+h.c.}.

In general, , , and can be complex. As noted above, to avoid tree-level Higgs-mediated FCNCs, we impose a softly-broken discrete symmetry, and on the quartic terms of Eq. (1), which implies that , whereas is allowed. In this basis of scalar doublet fields (denoted as the -basis), the discrete symmetry of the quartic terms of Eq. (1) is manifest. Furthermore, we assume that the scalar fields can be rephased such that and are both real. The resulting scalar potential is then explicitly CP-conserving.

The scalar fields will develop non-zero vacuum expectation values if the Higgs mass matrix has at least one negative eigenvalue. We assume that the parameters of the scalar potential are chosen such that the minimum of the scalar potential respects the U(1) gauge symmetry. Then, the scalar field vacuum expectations values are of the form

 ⟨Φ1⟩=1√2(0v1),⟨Φ2⟩=1√2(0v2). (2)

As noted in Appendix B of Ref. [12], if , then the vacuum is CP-conserving and the vacuum expectation values and can be chosen to be non-negative without loss of generality. In this case, the corresponding potential minimum conditions are:222Here and in the following, we use the shorthand notation , , , , , , , , etc.

 m211 = m212tβ−12v2(λ1c2β+λ345s2β), (3) m222 = m212t−1β−12v2(λ2s2β+λ345c2β), (4)

where we have defined:

 λ345≡λ3+λ4+λ5,tβ≡tanβ≡v2v1, (5)

where , and

 v2≡v21+v22=4m2Wg2=(246 GeV)2. (6)

Of the original eight scalar degrees of freedom, three Goldstone bosons ( and ) are absorbed (“eaten”) by the and . The remaining five physical Higgs particles are: two CP-even scalars ( and , with ), one CP-odd scalar () and a charged Higgs pair (). The resulting squared-masses for the CP-odd and charged Higgs states are

 m2A = ¯¯¯¯¯m2−λ5v2, (7) m2H± = m2A+12v2(λ5−λ4), (8)

where

 ¯¯¯¯¯m2≡2m212s2β. (9)

The two neutral CP-even Higgs states mix according to the following squared-mass matrix:

 M2≡(λ1v2c2β+(m2A+λ5v2)s2β[λ345v2−(m2A+λ5v2)]sβcβ[λ345v2−(m2A+λ5v2)]sβcβλ2v2s2β+(m2A+λ5v2)c2β). (10)

Defining the physical mass eigenstates

 H = (√2ReΦ01−v1)cα+(√2ReΦ02−v2)sα, (11) h = −(√2ReΦ01−v1)sα+(√2ReΦ02−v2)cα, (12)

the masses and mixing angle are found from the diagonalization process

 (m2H00m2h)=(−cαsα−sαcα)(M211M212M212M222)([]cccα−sαsα−cα) =(M211c2α+2M212cαsα+M222s2αM212(c2α−s2α)+(M222−M211)sαcαM212(c2α−s2α)+(M222−M211)sαcαM211s2α−2M212cαsα+M222c2α). (13)

Note that the two equations, and , yield the following result:

 |M212|=√(m2H−M211)(M211−m2h)=√(M222−m2h)(M211−m2h). (14)

Explicitly, the squared-masses of the neutral CP-even Higgs bosons are given by

 m2H,h=12[M211+M222±Δ], (15)

where and the non-negative quantity is defined by

 Δ≡√(M211−M222)2+4(M212)2. (16)

The mixing angle , which is defined modulo , is evaluated by setting the off-diagonal elements of the CP-even scalar squared-mass matrix given in Eq. (13) to zero. It is often convenient to restrict the range of the mixing angle to . In this case, is non-negative and is given by

 cα=√Δ+M211−M2222Δ= ⎷M211−m2hm2H−m2h, (17)

and the sign of is given by the sign of . Explicitly, we have

 sα=√2M212√Δ(Δ+M211−M222)=sgn(M212)√m2H−M211m2H−m2h. (18)

In deriving Eqs. (17) and (18), we have assumed that . The case of is singular; in this case, the angle is undefined since any two linearly independent combinations of and can serve as the physical states. In the rest of this paper, we shall not consider this mass-degenerate case further.

### 2.2 SM-limit in the Higgs basis

The scalar potential given in Eq. (1) is expressed in the -basis of scalar doublet fields in which the discrete symmetry of the quartic terms is manifest. It will prove convenient to re-express the scalar doublet fields in the Higgs basis [45, 46], defined by

 H1=(H+1H01)≡Φ1cβ+Φ2sβ,H2=(H+2H02)≡−Φ1sβ+Φ2cβ, (19)

so that and . The scalar doublet possesses SM tree-level couplings to all the SM particles. Therefore, if one of the CP-even neutral Higgs mass eigenstates is SM-like, then it must be approximately aligned with the real part of the neutral field .

The scalar potential, when expressed in terms of the doublet fields, and , has the same form as Eq. (1),

 V = Y1H†1H1+Y2H†2H2+Y3[H†1H2+h.c.]+12Z1(H†1H1)2+12Z2(H†2H2)2+Z3(H†1H1)(H†2H2) (20) +Z4(H†1H2)(H†2H1)+{12Z5(H†1H2)2+[Z6(H†1H1)+Z7(H†2H2)]H†1H2+h.c.},

where the are real linear combinations of the and the are real linear combinations of the . In particular, since , we have [46, 47]333To make contact with the notation of Ref. [12], , , , , , and .

 Z1 ≡ λ1c4β+λ2s4β+12λ345s22β, (21) Z2 ≡ λ1s4β+λ2c4β+12λ345s22β, (22) Zi ≡ 14s22β[λ1+λ2−2λ345]+λi,(for i=3,4 or 5), (23) Z6 ≡ −12s2β[λ1c2β−λ2s2β−λ345c2β], (24) Z7 ≡ −12s2β[λ1s2β−λ2c2β+λ345c2β]. (25)

Since there are five nonzero and seven nonzero , there must be two relations. The following two identities are satisfied if , ,  [47]:444For , , the -basis and the Higgs basis coincide, in which case and , , are independent quantities. For , the two relations are and , and is an independent quantity.

 Z2 = Z1+2(Z6+Z7)cot2β, (26) Z345 = Z1+2Z6cot2β−(Z6−Z7)tan2β, (27)

where . One can invert the expressions given in Eqs. (21)–(25), subject to the relations given by Eqs. (26) and (27). The results are given in Appendix A.

The squared mass parameters are given by

 Y1 = m211c2β+m222s2β−m212s2β, (28) Y2 = m211s2β+m222c2β+m212s2β, (29) Y3 = 12(m222−m211)s2β−m212c2β. (30)

and are fixed by the scalar potential minimum conditions,

 Y1=−12Z1v2,Y3=−12Z6v2. (31)

Using Eqs. (9) and (31), we can express in terms of , and ,

 ¯¯¯¯¯m2=Y2+12Z1v2+Z6v2cot2β. (32)

The masses of and are given by

 m2H± = Y2+12Z3v2, (33) m2A = Y2+12(Z3+Z4−Z5)v2. (34)

It is straightforward to compute the CP-even Higgs squared-mass matrix in the Higgs basis [48, 45],

 M2H=(Z1v2Z6v2Z6v2m2A+Z5v2). (35)

From Eq. (35), one can immediately derive the conditions that yield a SM-like Higgs boson. Since and , the couplings of are precisely those of the Standard Model. Thus a SM-like Higgs boson exists if is an approximate mass eigenstate. That is, the mixing of and is subdominant, which implies that either and/or , . Moreover, if in addition , then is SM-like, whereas if , then is SM-like. In both cases, the squared-mass of the SM-like Higgs boson is approximately equal to .

The physical mass eigenstates are identified from Eq. (11), (12) and (19) as

 H = (√2ReH01−v)cβ−α−√2ReH02sβ−α, (36) h = (√2ReH01−v)sβ−α+√2ReH02cβ−α. (37)

Then, Eqs. (15) and (16) yield

 (38)

where

 ΔH≡√[m2A+(Z5−Z1)v2]2+4Z26v4. (39)

 |Z6|v2=√(m2H−Z1v2)(Z1v2−m2h). (40)

Comparing Eqs. (11) and (12) with Eqs. (36) and (37), we identify the corresponding mixing angle by , which is defined modulo . Diagonalizing the squared mass matrix, Eq. (35), it is straightforward to derive the following expressions:

 Z1v2 = m2hs2β−α+m2Hc2β−α, (41) Z6v2 = (m2h−m2H)sβ−αcβ−α, (42) m2A+Z5v2 = m2Hs2β−α+m2hc2β−α. (43)

It follows that

 m2h = (Z1+Z6cβ−αsβ−α)v2, (44) m2H = m2A+(Z5−Z6cβ−αsβ−α)v2. (45)

Note that Eq. (42) implies that555Having established a convention where , we are no longer free to redefine the Higgs basis field . Consequently, the sign of is meaningful in this convention.

 Z6sβ−αcβ−α≤0. (46)

One can also derive expressions for and either directly from Eqs. (41) and (42) or by using Eqs. (17) and (18) with replaced by . Using Eq. (46), the sign of the product is fixed by the sign of . However, since is defined only modulo , we are free to choose a convention where either or is always non-negative.666Such a convention, if adopted, would replace the convention employed in Eq. (17) in which is taken to be non-negative. In a convention where is non-negative (this is a convenient choice when the is SM-like),

 cβ−α=−sgn(Z6) ⎷Z1v2−m2hm2H−m2h=−Z6v2√(m2H−m2h)(m2H−Z1v2), (47)

where we have used Eq. (40) to obtain the second form for in Eq. (47).

Finally, we record the following useful formula that is easily obtained from Eqs. (7) and (A.10),777In Eq. (48), the term in the expression for that is proportional to is never greater than for all values of , since Eqs. (24) and (25) imply that .

 ¯¯¯¯¯m2=m2A+Z5v2+12(Z6−Z7)v2tan2β. (48)

Combining Eq. (48) with Eqs. (42) and (43) yields

 Z7v2=(m2h−m2H)sβ−αcβ−α+2cot2β[m2Hs2β−α+m2hc2β−α−¯¯¯¯¯m2]. (49)

Using Eqs. (26) and (27), one can likewise obtain expressions for and in terms of , , and . However, these expressions are not particularly illuminating, so we do not write them out explicitly here.

### 2.3 Higgs couplings and the alignment limit

As noted in the previous subsection, the Higgs basis field behaves precisely as the Standard Model Higgs boson. Thus, if one of the neutral CP-even Higgs mass eigenstates is approximately aligned with , then its properties will approximately coincide with those of the SM Higgs boson. Thus, we shall define the alignment limit as the limit in which the one of the two neutral CP-even Higgs mass eigenstates aligns with the direction of the scalar field vacuum expectation values. Defined in this way, it is clear that the alignment limit is independent of the choice of basis for the two Higgs doublet fields. Nevertheless, the alignment limit is most clearly exhibited in the Higgs basis. In light of Eqs. (36) and (37), the alignment limit corresponds either to the limit of if is identified as the SM-like Higgs boson, or to the limit of if is identified as the SM-like Higgs boson.

Consider first the case of a SM-like , with  GeV. In this case, , , and . It follows from Eq. (47) that the alignment limit can be achieved in two ways: (i) or (ii) . The case of (or equivalently ) is called the decoupling limit in the literature.888More precisely, we are assuming that . Since is a dimensionless coefficient in the Higgs basis scalar potential, we are implicitly assuming that cannot get too large without spoiling perturbativity and/or unitarity. One might roughly expect , in which case provides a reasonable indication of the domain of the decoupling limit. In this case, one finds that , so one can integrate out the heavy scalar states below the scale of . The effective Higgs theory below the scale is a theory with one Higgs doublet and corresponds to the Higgs sector of the Standard Model. Thus not surprisingly, is a SM-like Higgs boson. However, it is possible to achieve the alignment limit even if the masses of all scalar states are similar in magnitude in the limit of . This is the case of alignment without decoupling and the main focus of this study. Finally, if both and are satisfied, the alignment is even more pronounced; when relevant we shall denote this case as the double decoupling limit.

For completeness we note that in the case of a SM-like we have , and . Here, it is more convenient to employ a convention where is non-negative. One can then use Eqs. (40), (46) and (47) to obtain an expression for . In a convention where is non-negative,

 sβ−α=−sgn(Z6) ⎷m2H−Z1v2m2H−m2h=−Z6v2√(m2H−m2h)(Z1v2−m2h). (50)

Taking  GeV, there is no decoupling limit as in the case of a SM-like . However, the alignment limit without decoupling can be achieved in the limit of . This case will be discussed in detail in [29].

We now turn to the tree-level Higgs couplings. Denoting the SM Higgs boson by , the coupling of the CP-even Higgs bosons to (where or ) normalized to the coupling is given by

 ChV=sβ−α,CHV=cβ−α. (51)

As expected, if is a SM-like Higgs boson then in the alignment limit, whereas if is a SM-like Higgs boson then in the alignment limit.

Next, we consider the Higgs boson couplings to fermions. The most general renormalizable Yukawa couplings of the two Higgs doublets to a single generation of up and down-type quarks and leptons (using third generation notation) is given by

 −LYuk=Y1b¯¯bRΦi∗1QiL+Y2b¯¯bRΦi∗2QiL+Y1τ¯¯¯τRΦi∗1LiL+Y2τ¯¯¯τRΦi∗2LiL+ϵij[Y1t¯tRQiLΦj1+Y2t¯tRQiLΦj2]+h.c., (52)

where , , and are the doublet left handed quark and lepton fields and , and are the singlet right-handed quark and lepton fields. However, if all terms in Eq. (52) are present, then tree-level Higgs-mediated FCNCs would be present, in conflict with experimental constraints. To avoid tree-level Higgs-mediated FCNCs, we extend the discrete symmetry to the Higgs-fermion Lagrangian. There are four possible choices for the transformation properties of the fermions with respect to , which we exhibit in Table 1.

For simplicity, we shall assume in this paper that the pattern of the Higgs couplings to down-type quarks and leptons is the same. This leaves two possible choices for the Higgs-fermion couplings [11]:

 Type I:IY1t=Y1b=Y1τ=0, (53) Type II:Y1t=Y2b=Y2τ=0. (54)

In particular, the pattern of fermion couplings to the neutral Higgs bosons in the Type I and Type II models is exhibited in Table 2.

In the strict alignment limit, the fermion couplings to the SM-like Higgs boson should approach their Standard Model values. To see this explicitly, we note the identities,

 cosαsinβ = sβ−α+cotβcβ−α, (55) −sinαcosβ = sβ−α−tanβcβ−α, (56) sinαsinβ = cβ−α−cotβsβ−α, (57) cosαcosβ = cβ−α+tanβsβ−α. (58)

If is the SM-like Higgs boson, then in the limit of , the fermion couplings of approach their Standard Model values. However, if , then the alignment limit is realized in the Type-II Yukawa couplings to down-type fermions only if . That is, if but , then the couplings and the couplings are SM-like whereas the and couplings deviate from their Standard Model values. Thus the approach to the alignment limit is delayed when . We denote this phenomenon as the delayed alignment limit. Similar considerations apply if ; however, this region of parameter space is disfavored as the corresponding coupling quickly becomes non-perturbative if is too large.

Finally, we examine the trilinear Higgs self-couplings. Using the results of Ref. [12] (see also Ref. [48]), the three-Higgs vertex Feynman rules (including the corresponding symmetry factor for identical particles but excluding an overall factor of ) are given by:

 ghAA = −v[(Z3+Z4−Z5)sβ−α+Z7cβ−α], (59) gHAA = −v[(Z3+Z4−Z5)cβ−α−Z7sβ−α], (60) ghHH = −3v[Z1sβ−αc2β−α+Z345sβ−α(13−c2β−α)+Z6cβ−α(1−3s2β−α)+Z7s2β−αcβ−α], (61) gHhh = −3v[Z1cβ−αs2β−α+Z345cβ−α(13−s2β−α)−Z6s