Impersonating the Standard Model Higgs Boson: Alignment without Decoupling

# Impersonating the Standard Model Higgs Boson: Alignment without Decoupling

Marcela Carena, Ian Low, Nausheen R. Shah, and Carlos E. M. Wagner Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510
Enrico Fermi Institute, University of Chicago, Chicago, IL 60637
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106
High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439
Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
Michigan Center for Theoretical Physics, Department of Physics,
University of Michigan, Ann Arbor, MI 48109
###### Abstract

In models with an extended Higgs sector there exists an alignment limit, in which the lightest CP-even Higgs boson mimics the Standard Model Higgs. The alignment limit is commonly associated with the decoupling limit, where all non-standard scalars are significantly heavier than the boson. However, alignment can occur irrespective of the mass scale of the rest of the Higgs sector. In this work we discuss the general conditions that lead to “alignment without decoupling”, therefore allowing for the existence of additional non-standard Higgs bosons at the weak scale. The values of for which this happens are derived in terms of the effective Higgs quartic couplings in general two-Higgs-doublet models as well as in supersymmetric theories, including the MSSM and the NMSSM. Moreover, we study the information encoded in the variations of the SM Higgs-fermion couplings to explore regions in the parameter space.

## I Introduction

The Standard Model (SM) with one Higgs doublet is the simplest realization of electroweak symmetry breaking and provides a very good description of all data collected so far at hadron and lepton colliders. This includes measurements associated with the recently discovered 125 GeV Higgs boson at the CERN LHC :2012gk (); :2012gu (). In this model, the Higgs field receives a vacuum expectation value (VEV), GeV, which breaks the electroweak gauge symmetry and gives masses to the fundamental fermions and gauge bosons. The couplings of these particles to the Higgs boson are fixed by their masses and . On the other hand, Higgs self interactions are controlled by the quartic coupling in the Higgs potential, which in turn is given by the Higgs mass and . Therefore, interactions of the SM Higgs boson with fermions, gauge bosons and with itself are completely determined.

Extensions of the SM commonly lead to modifications of the Higgs couplings, especially if there exist new particles interacting with the Higgs or if there is an extended Higgs sector. The size of these modifications may be naively estimated in the decoupling limit to be:

 O(v2m2new) ≈ O(5%) ×(1 TeVmnew)2 , (1)

where is the scale of new particles. Therefore, for new particles below the TeV scale, changes in the Higgs couplings from the SM expectations are quite small. Such an estimate supports the fact that due to the large uncertainties in present measurements, no significant deviations from the SM Higgs properties should be identifiable in present data, if all new particles are at or above the TeV scale. At the same time, it stresses the need for precision Higgs measurements to uncover possible signs of new physics.

Conversely, Eq. (1) implies that, if in the future, refined measurements of the properties of the 125 GeV Higgs boson continue to be consistent with those of a SM Higgs boson, no new light particles interacting with the SM-like Higgs are to be expected. However, this estimate is only valid in the so called decoupling limit, where all non-standard Higgs bosons are significantly heavier than the gauge boson. On the other hand, current searches for these particles do not exclude the possibility of additional Higgs bosons in the hundred to several hundred GeV mass range. Given that initial data appears to disfavor large deviations with respect to the SM Higgs description Low:2012rj (), it is of special interest to consider models of extended Higgs sectors containing a CP-even Higgs that has properties mimicking quite precisely the SM ones, even if the non-standard Higgs bosons are light.

A well-known example is that of general two-Higgs-doublet models (2HDMs) Branco:2011iw (); Craig:2012vn (), in which the heavy CP-even Higgs could be the SM-like Higgs boson. However, in this case the 2HDM parameter space becomes very restrictive, with masses of the non-standard scalars of the order of the and boson masses, and is severely constrained by data Christensen:2012ei (). On the other hand, the possibility of the lightest CP-even Higgs mimicking the SM Higgs, referred to as “alignment” in Ref. Craig:2013hca (), is much less constrained and usually associated with the decoupling limit. The less known and more interesting case of alignment without recourse to decoupling deserves further study.

A few examples of “alignment without decoupling” have been considered in the literature. The first one was presented over a decade ago by Gunion and Haber in Ref. Gunion:2002zf (). Their main focus was to emphasize the SM-like behavior of the lightest CP-even Higgs of a 2HDM in the decoupling limit. However, they also demonstrated that it can behave like a SM Higgs without decoupling the non-SM-like scalars. Much more recently, a similar situation was discussed in Ref. Delgado:2013zfa (), where an extension of the Minimal Supersymmetric Standard Model (MSSM) with a triplet scalar was studied. It was found that after integrating out the triplet scalar, a SM-like Higgs boson and additional light scalars are left in the spectrum for low values of . Another recent study, Ref. Craig:2013hca (), presented a scanning over the parameter space of general 2HDMs. Solutions were found fulfilling alignment without decoupling and the phenomenological implications were investigated.

It is obvious that the possibility of alignment without decoupling would have far-reaching implications for physics beyond the SM searches. However, its existence has remained obscure and has sometimes been attributed to accidental cancellations in the scalar potential. A simple way to understand how one of the CP-even Higgs bosons in a 2HDM mimics the SM Higgs is to realize that the alignment limit occurs whenever the mass eigenbasis in the CP-even sector aligns with the basis in which the electroweak gauge bosons receive all of their masses from only one of the Higgs doublets 111This would imply that the other, non-standard CP-even Higgs has no tree-level couplings to the gauge bosons. However, there are still couplings to SM fermions in general. Therefore the non-standard Higgs boson is not inert.. From this perspective, it is clear that the alignment limit does not require the non-standard Higgs bosons to be heavy. After presenting the general conditions for the alignment limit in 2HDMs, we analyze in detail the possible implications for well motivated models containing two Higgs doublets. In particular, we consider the MSSM as well as its generalization to the next-to-minimal supersymmetric standard model (NMSSM), where an extra singlet is added. Along the way, we analyze the extent to which precision measurements of Higgs-fermion couplings could be useful in probing regions of parameters that are difficult to access through direct non-standard Higgs boson searches.

This article is organized as follows. In the next section we define the notation and briefly review the scalar potential and the Higgs couplings in general, renormalizable 2HDMs. In Section III we derive the alignment condition in the decoupling regime in terms of the eigenvectors of the CP-even Higgs mass matrix, which provides a simple analytical understanding of alignment. We then write down the general conditions for alignment without decoupling. In Section IV we study the alignment limit in general 2HDMs and provide new perspectives on previous works. Detailed studies on the parameter space of the MSSM and beyond are presented in Section V, which is followed by the conclusion in Section VI.

## Ii Overview of 2HDM

### ii.1 Scalar Potential

We follow the notation in Ref. Haber:1993an () for the scalar potential of the most general two-Higgs-doublet extension of the SM:

 V = m211Φ†1Φ1+m222Φ†2Φ2−m212(Φ†1Φ2+h.c.)+12λ1(Φ†1Φ1)2+12λ2(Φ†2Φ2)2 (2) +λ3(Φ†1Φ1)(Φ†2Φ2)+λ4(Φ†1Φ2)(Φ†2Φ1) +{12λ5(Φ†1Φ2)2+[λ6(Φ†1Φ1)+λ7(Φ†2Φ2)]Φ†1Φ2+h.c.} ,

where

 Φi=⎛⎝ϕ+i1√2(ϕ0i+ia0i)⎞⎠ . (3)

We will assume CP conservation and that the minimum of the potential is at

 ⟨Φi⟩=1√2(0vi) , (4)

where

 v≡√v21+v22≈246 GeV ,tβ≡tanβ=v2v1 . (5)

We choose so that and write and . The five mass eigenstates are: two CP-even scalars, and , with , one CP-odd scalar, , and a charged pair, . The mass parameters, and , can be eliminated by imposing the minimization condition Haber:1993an ():

 m211−tβm212+12v2c2β(λ1+3λ6tβ+~λ3t2β+λ7t3β)=0 , (6) m222−t−1βm212+12v2s2β(λ2+3λ7t−1β+~λ3t−2β+λ6t−3β)=0 , (7)

where . It then follows that Haber:1993an ()

 m2A=2m212s2β−12v2(2λ5+λ6t−1β+λ7tβ) , (8)

and the mass-squared matrix for the CP-even scalars can be expressed as

 M2=(M211M212M212M222)≡m2A(s2β−sβcβ−sβcβc2β)+v2(ccL11L12L12L22) , (9)

where

 L11 = λ1c2β+2λ6sβcβ+λ5s2β , (10) L12 = (λ3+λ4)sβcβ+λ6c2β+λ7s2β , (11) L22 = λ2s2β+2λ7sβcβ+λ5c2β . (12)

The mixing angle, , in the CP-even sector is defined as

 (Hh)=(cαsα−sαcα)(ϕ01ϕ02)≡R(α)(ϕ01ϕ02) , (13)

where and . This leads to

 RT(α)(m2H00m2h)R(α)=(M211M212M212M222) . (14)

From the component in the above equation we see

 (m2H−m2h)sαcα=M212 , (15)

which implies has the same sign as . There are two possible sign choices:

 (I) −π2≤α≤π2:cα≥0  and  Sign(sα)=Sign(M212) , (16) (II) − 0≤α≤π:sα≥0  and  Sign(cα)=Sign(M212) . (17)

We will discuss the implications of these two sign choices in some detail below.

The eigenvector associated with the eigenvalue corresponds to the second row in , Eq. (13), and satisfies

 (M211M212M212M222) (−sαcα)=m2h (−sαcα) , (18)

giving rise to two equivalent representations for :

 tα=M212M211−m2h=M222−m2hM212 . (19)

The equivalence of the two representations is guaranteed by the characteristic equation, , where is the identity matrix. Moreover, since

 m2h≤M2ii≤m2H ,for   i=1,2 , (20)

due to the “level repulsion” of eigenvalues of symmetric matrices, in both representations , consistent with the sign choices specified above.

Eq. (19) allows us to solve for the mixing angle, , in terms of or , depending on one’s preference. For example, in the sign choice (I) we have the following two representations:

 sα = M212√(M212)2+(M211−m2h)2 ,m2H=M211(M211−m2h)+(M212)2M211−m2h, (21) sα = Sign(M212)M222−m2h√(M212)2+(M222−m2h)2 ,m2H=M222(M222−m2h)+(M212)2M222−m2h, (22)

where the expression for follows from solving for the corresponding eigenvalue equation for .

One can verify that Eqs. (21) and (22) lead to the expected limiting behavior when . For example, for Eq. (21), if , the smaller mass eigenvalue, , is given by . Then in Eq. (21) we have and . As expected the lightest CP-even Higgs is mostly in this case. On the other hand, if then is mostly and , since

 (M211−m2h)=(M212)2|M211−M222|+O((M212)4) , (23)

which also implies in this case. The behavior of Eq. (22) can be verified in a similar fashion.

### ii.2 Higgs Couplings

The Higgs boson couplings to gauge bosons in 2HDMs follow from gauge invariance and have the same parametric dependence on the CP-even mixing angle, , and the angle in any 2HDM, namely,

 ghVV=sβ−αgV ,gHVV=cβ−αgV , (24)

where is the SM value for bosons.

The fermion couplings, on the other hand, take different forms in different 2HDMs. However, it is common to require the absence of tree-level flavor-changing neutral currents (FCNC) by imposing the Glashow-Weinberg condition Glashow:1976nt (). This condition requires fermions with the same quantum numbers to couple to a single Higgs doublet and leads to four different types of 2HDMs Branco:2011iw ().222Typically the Glashow-Weinberg condition requires a discrete symmetry: , which demands in the general scalar potential given in Eq. (2). Amongst them the most popular ones are: the type I model, where all SM fermions couple to one doublet, and the type II model, where the up-type fermions couple to one doublet and down-type fermions couple to the other. In one of the other two models, up-type quarks and leptons couple to the same doublet, while down-type quarks couple to the other. The remaining one has all the quarks coupled to one Higgs doublet while the leptons couple to the other one. In what follows we base the discussion on the type II model, although our analysis can be easily adapted to all four types of 2HDMs.

In type II models, where at tree-level and only couple to down-type and up-type fermions, respectively, the tree-level Higgs couplings to fermions are

 ghdd = −sαcβgf=(sβ−α−tβcβ−α)gf ,ghuu=cαsβgf=(sβ−α+t−1βcβ−α)gf , (25) gHdd = −cαcβgf=(cβ−α+tβsβ−α)gf ,gHuu=sαsβgf=(cβ−α−t−1βsβ−α)gf , (26)

where is the coupling of the Higgs to the corresponding fermions in the SM.

We are interested in the alignment limit, where the lightest CP-even Higgs mimics the SM one. We will begin by solving for the conditions for which the Higgs couplings to fermions have the same magnitude as in the SM: . There are four possibilities, which can be divided in two cases:

 i) ghdd=ghuu=±gf, ii) ghdd=−ghuu=±gf.

 sα=∓cβ ,cα=±sβ , (27)

which then implies

 cβ−α=0andsβ−α=±1 . (28)

Couplings of the CP-even Higgs bosons now become

 ghVV→±gV , ghff→±gf , gHVV→0 , gHdd→±tβgf , gHuu→∓t−1βgf , (29)

where the upper and lower signs correspond to and , respectively. This is the alignment limit. The heavy CP-even Higgs couplings to SM gauge bosons vanish in this limit since it does not acquire a VEV. In other words, the alignment limit is the limit where the mass eigenbasis in the CP-even sector coincides with the basis where the gauge bosons receive all of their masses from one of the doublets. As such, the non-SM-like CP-even Higgs does not couple to the gauge bosons at the tree-level. However, in this basis still has non-vanishing couplings to SM fermions. This feature remains true in all four types of 2HDMs, as can be seen, for example, by inspecting Table 2 in Ref. Craig:2013hca (). It is important to observe that results in an overall sign difference in the couplings of the SM-like Higgs and, hence, has no physical consequences.

On the other hand, fulfillment of case ii) requires

 sα=∓cβ ,cα=∓sβ , (30)

which gives

 cβ−α=∓s2β ,sβ−α=±c2β . (31)

We see that the coupling does not tend to the SM value in this case and alignment is not reached. However, in the limit ,

 s2β=2tβ1+t2β≈2tβ ,c2β=1−t2β1+t2β≈−1 , (32)

we observe that the CP-even Higgs couplings become, to linear order in ,

 ghVV = ∓gV ,ghdd=±gf ,ghuu=∓gf , (33) gHVV = ∓2t−1βgV ,gHdd=∓tβgf ,gHuu=∓t−1βgf . (34)

Hence, if Eq. (30) is required, one obtains that the lightest CP-even Higgs couplings to down-type fermions have the opposite sign as compared to its couplings to both the vector bosons and up-type fermions, although all couplings have the same strength as in the SM. If, instead of the large limit, one takes , then it is straightforward to check that has the opposite sign to and . It is worth noting that, in type II 2HDMs, leads to an unacceptably large top Yukawa coupling and should be avoided. However, the scenario of “wrong-sign” down-type fermion couplings of the SM-like Higgs in the large limit is clearly of phenomenological importance. A detailed study of this scenario is beyond the scope of the present work.

Similar arguments can be made in the case in which it is the heavy Higgs that behaves as the SM Higgs. For this to occur,

 sβ−α=0 (35)

and therefore . In the following, we shall concentrate on the most likely case in which the lightest CP-even Higgs satisfies the alignment condition. The heavy Higgs case can be treated in an analogous way.

We also comment on the coupling since it may have a significant impact on strategies in direct searches Craig:2013hca (). The coupling of the heavy Higgs to the lightest Higgs is given by

 gHhh = v4[−12 λ1cβcαs2α−12 λ2sβsαc2α+~λ3(−4cα−β+6s2αsα+β) (36) v4+3λ6(−4s2αsα+β+8sαc2αcβ)+3λ7(8s2αcαsβ−4c2αsα+β)] .

One can rewrite Eq. (36) as

 gHhh = −3vsβc3β{[sαβcαβ(λ1sαβ+~λ3t2βcαβ+λ6tβ(2cαβ+sαβ))+λ7t3βc3αβ] (37) − [sαβcαβ(λ2t2βcαβ+~λ3sαβ+λ7tβ(2sαβ+cαβ))+λ6t−1βs3αβ]}−~λ3cα−β,

where and tend to 1 in the alignment limit. We shall demonstrate in the next section that the alignment conditions in general 2HDMs imply that the coupling vanishes.

## Iii Alignment without decoupling

### iii.1 Derivation of the Conditions for Alignment

One of the main results of this work is to find the generic conditions for obtaining alignment without decoupling. The decoupling limit, where the low-energy spectrum contains only the SM and no new light scalars, is only a subset of the more general alignment limit in Eq. (28). In particular, quite generically, there exist regions of parameter space where one attains the alignment limit with new light scalars not far above GeV.

It is instructive to first derive the alignment limit in the usual decoupling regime but in a slightly different manner. Consider the eigenvalue equation of the CP-even Higgs mass matrix, Eq. (18), which, using Eq. (9), becomes

 (s2β−sβcβ−sβcβc2β) (−sαcα)=−v2m2A(L11L12L12L22)  (−sαcα)+m2hm2A (−sαcα) . (38)

Decoupling is defined by taking all non-SM-like scalar masses to be much heavier than the SM-like Higgs mass, . Then we see that at leading order in and , the right-hand side of Eq. (38) can be ignored, and the eigenvalue equation reduces to

 (s2β−sβcβ−sβcβc2β) (−sαcα)≈0 , (39)

leading to the well-known decoupling limit Gunion:2002zf (): . This is also exactly the alignment limit.

Here we make the key observation that while decoupling achieves alignment by neglecting the right-hand side of Eq. (38), alignment can also be obtained if the right-hand side of Eq. (38) vanishes identically, independent of :

 v2(L11L12L12L22) (−sαcα)=m2h (−sαcα) . (40)

More explicitly, since in the alignment limit, we can re-write the above matrix equation as two algebraic equations: 333The same conditions can also be derived using results presented in Ref. Gunion:2002zf ().

 (C1) : m2h=v2L11+tβv2L12=v2(λ1c2β+3λ6sβcβ+~λ3s2β+λ7tβs2β) , (41) (C2) : m2h=v2L22+1tβv2L12=v2(λ2s2β+3λ7sβcβ+~λ3c2β+λ6t−1βc2β) . (42)

Recall that . In the above is the SM-like Higgs mass, measured to be about 125 GeV, and is known once a model is specified. Notice that (C1) depends on all the quartic couplings in the scalar potential except , while (C2) depends on all the quartics but . If there exists a satisfying the above equations, then the alignment limit would occur for arbitrary values of and does not require non-SM-like scalars to be heavy!

Henceforth we will consider the coupled equations given in Eqs. (41) and (42) as required conditions for alignment. When the model parameters satisfy them, the lightest CP-even Higgs boson behaves exactly like a SM Higgs boson even if the non-SM-like scalars are light. A detailed analysis of the physical solutions will be presented in the next Section.

### iii.2 Departure from Alignment

Phenomenologically it seems likely that alignment will only be realized approximately, rather than exactly. Therefore it is important to consider small departures from the alignment limit, which we do in this subsection.

Since the alignment limit is characterized by , it is customary to parametrize the departure from alignment by considering a Taylor-expansions in Gunion:2002zf (); Craig:2013hca (), which defines the deviation of the couplings from the SM values. However, this parametrization has the drawback that deviations in the Higgs coupling to down-type fermions are really controlled by , which could be when is large. Therefore, we choose to parametrize the departure from the alignment limit by a parameter which is related to by

 cβ−α=t−1βη ,sβ−α=√1−t−2βη2 . (43)

Then at leading order in , the Higgs couplings become

 ghVV ≈ (1−12t−2βη2)gV ,gHVV≈t−1βη gV , (44) ghdd ≈ (1−η)gf ,gHdd≈tβ(1+t−2βη)gf , (45) ghuu ≈ (1+t−2βη)gf ,gHuu≈−t−1β(1−η)gf . (46)

We see characterizes the departure from the alignment limit of not only but also . On the other hand, the deviation in the and are given by , which is doubly suppressed in the large regime. Moreover, terms neglected above are of order and are never multiplied by positive powers of , which could invalidate the expansion in when is large.

There are some interesting features regarding the pattern of deviations. First, whether the coupling to fermions is suppressed or enhanced relative to the SM values, is determined by the sign of : and are suppressed (enhanced) for positive (negative) , while the trend in and is the opposite. In addition, as 0, the approach to the SM values is the fastest in and the slowest in . This is especially true in the large regime, which motivates focusing on precise measurements of in type II 2HDMs.

Our parametrization of can also be obtained by modifying Eq. (39), which defines the alignment limit, as follows:

 (s2β−sβcβ−sβcβc2β) (−sαcα)=t−1βη(−sβcβ) . (47)

The eignevalue equation for in Eq. (40) is modified accordingly,

 (48)

From the above, taking and expanding to first order in , we obtain the “near-alignment conditions”,

 (C1′) : m2h=v2L11+tβv2L12+η[tβ(1+t−2β)v2L12−m2A] , (49) (C2′) : m2h=v2L22+tβ−1v2L12−η[t−1β(1+t−2β)v2L12−m2A] . (50)

We will return to study these two conditions in the next section, after first analyzing solutions for alignment without decoupling in general 2HDMs.

## Iv Alignment in General 2HDM

In what follows we solve for the alignment conditions (C1) and (C2), assuming all the scalar couplings are independent of . This is not true in general, as radiative corrections to the scalar potential often introduce a dependence in the quartic couplings that are not present at the tree-level. However, this assumption allows us to analyze the solutions analytically and obtain the necessary intuition to understand more complicated situations.

When all the quartics are independent of , the conditions (C1) and (C2) may be re-written as cubic equations in , with coefficients that depend on and the quartic couplings in the scalar potential,

 (C1) : (m2h−λ1v2)+(m2h−~λ3v2)t2β=v2(3λ6tβ+λ7t3β) , (51) (C2) : (m2h−λ2v2)+(m2h−~λ3v2)t−2β=v2(3λ7t−1β+λ6t−3β) . (52)

Alignment without decoupling occurs only if there is (at least) a common physical solution for between the two cubic equations.444Since in our convention, a physical solution means a real positive root of the cubic equation. From this perspective it may appear that alignment without decoupling is a rare and fine-tuned phenomenon. However, as we will show below, there are situations where a common physical solution would exist between (C1) and (C2) without fine-tuning.

Regarding the coupling of the heaviest CP-even Higgs to the lightest one, it is now easy to see from Eqs. (51) and (52) that each term inside the square brackets in Eq. (37) tends to in the alignment limit, and hence, as stated in Ref. Craig:2013hca (), vanishes.

### iv.1 Alignment for Vanishing Values of λ6,7

It is useful to consider solutions to the alignment conditions (C1) and (C2) when and , which can be enforced by the symmetries and . Then (C1) and (C2) collapse into quadratic equations:

 (C1) → (m2h−λ1v2)+(m2h−~λ3v2)t2β=0 , (53) (C2) → (m2h−~λ3v2)+(m2h−λ1v2)t2β=0 . (54)

We see that a solution exists for whenever

 λSM=λ1+~λ32, (55)

where we have expressed the SM-like Higgs mass as

 m2h=λSMv2 . (56)

From Eq. (55) we see that the above solution, , is obviously special, since it demands to be the average of and .

We next relax the condition while still keeping . Recall that the Glashow-Weinberg condition Glashow:1976nt () on the absence of tree-level FCNC requires a discrete symmetry, , which enforces at the tree-level. The two quadratic equations have a common root if and only if the determinant of the Coefficient Matrix of the two quadratic equations vanishes,

 (57)

Then the positive root can be expressed in terms of ,

 t(0)β=√λ1−λSMλSM−~λ3 . (58)

We see from Eqs. (57) and (58), that a real value of can exist only if the set of parameters has one of the two orderings

 λ1,λ2 ≥ λSM ≥ ~λ3 , (59)

or

 λ1,λ2 ≤ λSM ≤ ~λ3 . (60)

A solution for can be found using the following procedure: once one of the conditions in Eqs. (59) or (60) is satisfied, Eq. (58) leads to the alignment solution for a given . However, Eq. (57) must also be satisfied, which is then used to solve for the desired so that is a root of (C2) as well. More specifically, the relations

 λ2−λSM = λSM−~λ3(t(0)β)2 = λ1−λSM(t(0)β)4 (61)

must be fulfilled. Therefore, the alignment solution demands a specific relationship between the quartic couplings of the 2HDM. In addition, it is clear from Eqs. (58) and (61) that if all the quartic couplings are , as well, unless and are very close to , or is taken to be much larger than . For examples, could be achieved for , or for . Our discussion so far applies to alignment limit scenarios studied, for instance, in Refs. Delgado:2013zfa (); Craig:2013hca (), both of which set .

### iv.2 Alignment for Non-Zero λ6,7

The symmetry leading to is broken softly by . Thus a phenomenologically interesting scenario is to consider small but non-zero . Therefore, in this subsection we study solutions to the alignment conditions (C1) and (C2) under the assumptions

 λ6,7 ≪ 1 . (62)

Although general solutions of cubic algebraic equations exist, much insight can be gained by first solving for the cubic roots of (C1) as a perturbation to the quadratic solution ,

 t(±)β = t(0)β±32λ6λSM−~λ3±λ7(λ1−λSM)(λSM−~λ3)2+O(λ26,λ27) . (63)

The solutions lie in the same branch as , to which they reduce in the limit . In addition, both solutions are again given our assumptions. More importantly, similar to , specific fine-tuned relations between the quartic couplings are required to ensure are also cubic roots of (C2).

However, a new solution also appears,

 t(1)β = λSM−~λ3λ7−3λ6λSM−~λ3−λ7(λ1−λSM)(λSM−~λ3)2+O(λ26,λ27) . (64)

The solution belongs to a new branch that disappears when and exists provided the condition

 Sign(λSM−~λ3)=Sign(λ7) (65)

is satisfied. For , as is natural due to the assumption , we are led to . As an example, for , one obtains by solving for the cubic root of (C1) exactly. Lower values of may be obtained for somewhat larger values of and/or larger values of .

The solution is an example of alignment without decoupling that does not require fine-tuning. This is because the condition (C2), in the limits and , becomes insensitive to all quartic couplings but :

 m2h−λ2v2=O⎛⎝1t2β,λ7tβ,λ6t3β⎞⎠ . (66)

Unlike the fine-tuned relation in Eq. (57), in this case is determined by the input parameter , or equivalently , and is insensitive to other quartic couplings in the scalar potential. Therefore, provided the condition given in Eq. (65) is fulfilled, the value of the quartic couplings, and , are still free parameters and thus can be varied, leading to different values of for which alignment occurs.

For the purpose of demonstration, let us again use the example below Eq. (65), . The condition that