Maximally symmetric two Higgs doublet model with natural standard model alignment

Abstract

Abstract

We study the Higgs mass spectrum as predicted by a Maximally Symmetric Two Higgs Doublet Model (MS-2HDM) potential based on the SO(5) group, softly broken by bilinear Higgs mass terms. We show that the lightest Higgs sector resulting from this MS-2HDM becomes naturally aligned with that of the Standard Model (SM), independently of the charged Higgs boson mass and tan [beta]. In the context of Type-II 2HDM, SO(5) is the simplest of the three possible symmetry realizations of the scalar potential that can naturally lead to the SM alignment. Nevertheless, renormalization group effects due to the hypercharge gauge coupling g' and third-generation Yukawa couplings may break sizeably this alignment in the MS-2HDM, along with the custodial symmetry inherited by the SO(5) group. Using the current Higgs signal strength data from the LHC, which disfavour large deviations from the SM alignment limit, we derive lower mass bounds on the heavy Higgs sector as a function of tan [beta], which can be stronger than the existing limits for a wide range of parameters. In particular, we propose a new collider signal based on the observation of four top quarks to directly probe the heavy Higgs sector of the MS-2HDM during the run-II phase of the LHC.

Published for SISSA by Springer

Received: August 27, 2014

Revised: November 14, 2014 Accepted: November 18, 2014

Published: December 3, 2014

P.S. Bhupal Deva and Apostolos Pilaftsisa,b

aConsortium for Fundamental Physics, School of Physics and Astronomy, University of Manchester,Manchester, M13 9PL, United Kingdom

bCERN, Department of Physics, Theory Division,

CH-1211 Geneva 23, Switzerland

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study the Higgs mass spectrum as predicted by a Maximally Symmetric Two Higgs Doublet Model (MS-2HDM) potential based on the SO(5) group, softly broken by bilinear Higgs mass terms. We show that the lightest Higgs sector resulting from this MS-2HDM becomes naturally aligned with that of the Standard Model (SM), independently of the charged Higgs boson mass and tan . In the context of Type-II 2HDM, SO(5) is the simplest of the three possible symmetry realizations of the scalar potential that can naturally lead to the SM alignment. Nevertheless, renormalization group e ects due to the hypercharge gauge coupling g[prime] and third-generation Yukawa couplings may break sizeably this alignment in the MS-2HDM, along with the custodial symmetry inherited by the SO(5) group. Using the current Higgs signal strength data from the LHC, which disfavour large deviations from the SM alignment limit, we derive lower mass bounds on the heavy Higgs sector as a function of tan , which can be stronger than the existing limits for a wide range of parameters. In particular, we propose a new collider signal based on the observation of four top quarks to directly probe the heavy Higgs sector of the MS-2HDM during the run-II phase of the LHC.

Keywords: Higgs Physics, Beyond Standard Model

ArXiv ePrint: 1408.3405

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2014)024

Web End =10.1007/JHEP12(2014)024

Maximally symmetric two Higgs doublet model with natural standard model alignment

JHEP12(2014)024

Contents

1 Introduction 1

2 Maximally symmetric two Higgs doublet model potential 32.1 Custodial symmetries in the MS-2HDM 52.2 Scalar spectrum in the MS-2HDM 6

3 RG and soft breaking e ects 7

4 Misalignment predictions 10

5 Collider signals 145.1 Branching fractions 145.2 Charged Higgs signal 145.3 Heavy neutral Higgs signal 18

6 Conclusions 22

A Higgs spectrum and couplings in a general 2HDM 23

B Two-loop RGEs in a general 2HDM 25

1 Introduction

The discovery of a Higgs resonance with mass around 125 GeV at the LHC [1, 2] o ers an unprecedented opportunity for probing extended Higgs scenarios beyond the Standard Model (SM). Although the measured properties of the discovered Higgs boson show remarkable consistency with those predicted by the SM [3, 4], the current experimental data still leave open the possibility of new physics that results from an extended Higgs sector. In fact, several well-motivated new-physics scenarios require an enlarged Higgs sector, such as supersymmetry [5], in order to address a number of theoretical and cosmological issues, including the gauge hierarchy problem, the origin of the Dark Matter and the baryon asymmetry in our Universe. Here we follow a modest bottom-up approach and consider one of the simplest Higgs-sector extensions of the SM, namely the Two Higgs Doublet Model (2HDM) [6].

The 2HDM contains two complex scalar elds transforming as iso-doublets (2, 1) under the SM electroweak gauge group SU(2)L U(1)Y :

i = +i 0i

[parenrightBigg]

1

JHEP12(2014)024

, (1.1)

with i = 1, 2. In this doublet eld space 1,2, the general 2HDM potential reads

V = [notdef]21( 1 1) [notdef]22( 2 2) [bracketleftBig]

m212( 1 2) + H.c. [bracketrightBig]

+ 1( 1 1)2 + 2( 2 2)2 + 3( 1 1)( 2 2) + 4( 1 2)( 2 1)

+

12 5( 1 2)2 + 6( 1 1)( 1 2) + 7( 1 2)( 2 2) + H.c.[bracketrightbigg]

, (1.2)

which contains four real mass parameters [notdef]21,2, Re(m212), Im(m212), and ten real quartic couplings 1,2,3,4, Re( 5,6,7), and Im( 5,6,7). As a consequence, the vacuum structure of the general 2HDM can be quite rich [7], and in principle, can allow for a wide range of parameter space still compatible with the existing LHC constraints. However, additional requirements, such as the GlashowWeinberg condition [8, 9], must be imposed, so as to avoid Higgs interactions with unacceptably large avour changing neutral currents (FCNC) at the tree level. The GlashowWeinberg condition is satised by four discrete choices of tree-level Yukawa couplings between the Higgs doublets and SM fermions.1 By performing global ts to the current Higgs signals at the LHC and Tevatron in terms of the 2HDM parameter space, it has been shown [1118] that all four discrete 2HDM types are constrained to lie close to the so-called SM alignment limit, in which the mass eigenbasis of the CP-even scalar sector aligns with the SM gauge eigenbasis. Specically, in the Type-II (MSSM-type) 2HDM, the coupling of the SM-like Higgs to vector bosons is constrained to lie within 10% of the SM value at 95% CL [14, 1922].

In light of the present and upcoming LHC data, possible mechanisms that lead to the SM alignment limit within the 2HDM require further investigation and scrutiny. Naively, the SM alignment limit is often associated with the decoupling limit, in which all the nonstandard Higgs bosons are assumed to be much heavier than the electroweak scale so that the lightest CP-even scalar behaves like the SM Higgs boson. This SM alignment limit can also be achieved, without decoupling [2326].2 However, for small tan values, this is usually attributed to accidental cancellations in the 2HDM potential [26].

In this paper, we seek a symmetry of the 2HDM potential to naturally justify the alignment limit, without decoupling, independently of the kinematic parameters of the theory, such as the charged Higgs mass and tan . We show that a Maximally Symmetric 2HDM (MS-2HDM) potential based on the SO(5) group can naturally realize the alignment limit, where SO(5) acts on a bilinear eld space to be discussed in section 2. In section 3, we show that, in the context of Type-II 2HDM, the maximal symmetry group SO(5) is the simplest of the three possible symmetry realizations of the scalar potential having natural alignment. Nevertheless, as we analyze in section 3, renormalization group (RG) e ects due to the hypercharge gauge coupling g[prime] and third-generation Yukawa couplings, as well as soft-breaking mass parameters, violate explicitly the SO(5) symmetry, thereby inducing relevant deviations from the alignment limit. As we discuss in section 4, such deviations

1In general, the absence of tree-level avour-changing couplings of the neutral scalar elds can be guaranteed by requiring the Yukawa coupling matrices to be aligned in avour space [10].

2A similar situation was also discussed in an extension of the MSSM with a triplet scalar eld [27], where alignment without decoupling could be achieved in a parameter region at small tan [lessorsimilar] 10.

2

JHEP12(2014)024

lead to distinct predictions for the Higgs spectrum of the MS-2HDM. In section 5, we present a novel collider signature of the MS-2HDM with four top quarks as nal states. In section 6 we present our conclusions. Finally, several technical details related to our study have been relegated to appendices A and B.

2 Maximally symmetric two Higgs doublet model potential

In order to identify all accidental symmetries of the 2HDM potential, it is convenient to introduce the 8-dimensional complex multiplet [7, 28, 29]:

where

1

21 2 0. (2.3)

We must emphasize here that the bilinear eld space spanned by the 6-vector RA realizes an orthochronous SO(1, 5) symmetry group.

In terms of the 6-vector RA dened in (2.2), the 2HDM potential V given in (1.2) takes on a simple quadratic form:

V =

3

0

B

B

B

@

1

C

C

C

A

1 2

e 1

e 2

JHEP12(2014)024

, (2.1)

e i = i2 i (with i = 1, 2) and 2 is the second Pauli matrix. We should remark that the complex multiplet satises the Majorana property [7]: = C , where C = 2

0 2 is the charge-conjugation matrix, with 0 = 12[notdef]2 being the identity matrix. In terms

of the -multiplet, the following null 6-dimensional Lorentz vector can be dened [7, 29]:

RA A , (2.2)

where A = 0, 1, . . . , 5 and the six 8[notdef]8-dimensional matrices A may be expressed in terms

of the three Pauli matrices 1,2,3, as follows:

0,1,3 = 120 0,1,3 0, 2 =

1

23 2 0,

4 =

1

22 2 0, 5 =

1

2 MA RA +

14 LAB RARB , (2.4)

where MA and LAB are SO(1, 5) constant tensors that depend on the mass parameters and quartic couplings of the scalar potential V and their explicit forms may be found in [2932]. Requiring that the SU(2)L gauge-kinetic term of the multiplet remains canonical restricts the allowed set of rotations from SO(1,5) to SO(5),3 where only the spatial components RI (with I = 1, . . . , 5) transform, whereas the zeroth component R0 remains invariant. Consequently, in the absence of the hypercharge gauge coupling g[prime] and fermion Yukawa

3We note in passing that if the restriction of SU(2)L gauge invariance is lifted, the 2HDM is then equivalent to an ungauged theory with 8 real scalars and so the maximal symmetry group becomes the larger group O(8) [33].

couplings, the maximal symmetry group of the 2HDM is GR2HDM = SO(5). Given the group isomorphy SO(5) Sp(4)/Z2, the maximal symmetry group of the 2HDM in the original

-eld space is [29]4

G 2HDM = (Sp(4)/Z2) SU(2)L , (2.5) in the custodial symmetry limit of vanishing g[prime] and fermion Yukawa couplings. The quotient factor Z2 in (2.5) is needed to avoid double covering the group G 2HDM in the -space. One may note here that the 10 Lie generators of Sp(4) may be represented in the -space as

Ka = a 0 (with a = 0, 1, 2, . . . , 9), where

0 = 12 3 0 , 1 =

2 = 12 0 2 , 3 =

4 = 12 1 0 , 5 =

6 = 12 2 0 , 7 =

8 = 12 1 1 , 9 =

1

2 3 1 ,

1

2 3 3 ,

JHEP12(2014)024

1

2 1 3 , (2.6)

1

2 2 3 ,

1

2 2 1 ,

with the normalization: Tr( a b) = ab. Thus, the group G 2HDM includes the U(1)Y hypercharge group through the Sp(4) generator K0, whereas the 9 other Sp(4) generators listed in (2.6) are related to various Higgs Family and CP transformations [29]. On the other hand, the SU(2)L generators in the -space may be written as 0 0 (b/2) (with

b = 1, 2, 3), which manifestly commute with all Sp(4) generators Ka.

As we will see below by an explicit construction [cf. (2.9) and section 2.1], it is not di cult to deduce that, in the custodial symmetry limit, the maximal symmetry group for an n Higgs Doublet Model (nHDM) will be

G nHDM = (Sp(2n)/Z2) SU(2)L , (2.7) in which case the multiplet becomes a Majorana 4n-dimensional complex vector.5 It is interesting to note that for the SM with n = 1 Higgs doublet, (2.7) yields the well-known result: G SM = (SU(2)C/Z2) SU(2)L, by virtue of the group isomorphy: Sp(2) SU(2)C,

where SU(2)C is the custodial symmetry group originally introduced in [36]. Hence, it is important to stress that (2.7) represents a general result that holds for any nHDM.

We may now identify all maximal symmetries of the 2HDM potential by classifying all proper, improper and semi-simple subgroups of SO(5) in the bilinear RI space. In this way, it was found [7, 29] that a 2HDM potential invariant under SU(2)L U(1)Y can

possess a maximum of 13 accidental symmetries. This symmetry classication extends the

4In [29], the symplectic group Sp(4) is denoted as SUM(4), where the 10 generators of the restricted U(4) group satisfying a Majorana (symplectic) condition were presented.

5Given an apparently deep connection between SO(2n + 1) and Sp(2n) groups [34, 35], both of which have n(2n+1) generators, one might be able to identify the necessary bilinears in the R-space for an nHDM. However, this is somewhat non-trivial for n 3, and therefore, we postpone this discussion to a future

dedicated study.

4

previous list of six symmetries reported in [31], where possible custodial symmetries of the theory were not included. Each of the 13 classied symmetries puts some restrictions on the kinematic parameters appearing in the 2HDM potential (1.2). In a specic diagonally reduced bilinear basis [37, 38], one has the general restrictions Im( 5) = 0 and 6 = 7, thus reducing the number of independent quartic couplings to seven. In the maximally symmetric SO(5) ( Sp(4)/Z2) limit, we have the following relations between the scalar

potential parameters [7, 29]:

[notdef]21 = [notdef]22 , m212 = 0 ,

2 = 1 , 3 = 2 1 , 4 = Re( 5) = 6 = 7 = 0 . (2.8)

Thus, in the SO(5) limit, the 2HDM potential (1.2) is parametrized by a single mass parameter [notdef]2 and a single quartic coupling :

V = [notdef]2 [parenleftBig][notdef]

1[notdef]2 + [notdef] 2[notdef]2[parenrightBig]

+

JHEP12(2014)024

[notdef] 1[notdef]2 + [notdef] 2[notdef]2[parenrightBig]2

2 . (2.9)

It is worth stressing that the MS-2HDM scalar potential in (2.9) is more minimal than the respective potential of the MSSM at the tree level. Even in the custodial symmetric limit g[prime] ! 0, the latter only possesses a smaller symmetry: O(2) O(3) SO(5), in the

5-dimensional bilinear RI space.

2.1 Custodial symmetries in the MS-2HDM

It is now interesting to discuss the implications of custodial symmetries for the Yukawa sector of the 2HDM. To this end, let us only consider the quark Yukawa sector of the theory, even though it is straightforward to extend our results to the lepton sector as well. The relevant part of the quark-Yukawa Lagrangian in the 2HDM can generally be written down as follows:

LqY =

QL(hu1 1 + hu2 2)uR + QL(hd1

[notdef]2

2 +

=

4

e 1 + hd2

e 2)dR

=L , dL

[parenrightbig] [parenleftBig]

, (2.10)

where QL (uL , dL)T is the SM quark iso-doublet and we have introduced a 12 [notdef] 6-

dimensional non-square Yukawa coupling matrix

H

1 , 2 ,

e 1 ,

e 2

[parenrightBig] H[parenleftBigg][parenleftBigg]

uR dR

[parenrightBigg]

0

B

B

B

@

hu1 03[notdef]3 hu2 03[notdef]3 03[notdef]3 hd1 03[notdef]3 hd2

1

C

C

C

A

. (2.11)

All the custodial symmetries of the 2HDM potential can be deduced by examining the Sp(4) generators Ka = a 0 in the -space, where a are explicitly given in (2.6).

Candidate Sp(4) generators of the custodial symmetry are those generators that do not

5

commute with the hypercharge generator K0, i.e. Ka with a = 4, 5, 6, 7, 8, 9. It is not di -cult to see that these six generators, together with K0, form three inequivalent realizations of the SU(2)C custodial symmetry [29]: (i) K0,4,6, (ii) K0,5,7 and (iii) K0,8,9.

In order to see the implications of the three custodial symmetries (i), (ii) and (iii) for the quark Yukawa sector, we impose a symmetry commutation relation on H after

generalizing it for non-square matrices:

a H H tb = 04[notdef]2 , (2.12) where H is expressed in the reduced 4[notdef]2-dimensional space, in which the 3[notdef]3 avour space

has been suppressed. In addition, we denote with tb = b/2 (with b = 1, 2, 3) the three 2 [notdef] 2 generators of the custodial SU(2)C group. One can immediately check that it holds

0 H H t3 = 04[notdef]2, which implies that the specic block structure of H in (2.11) respects

U(1)Y by construction, given the correspondence: 0 $ t3. In detail, imposing (2.12) for

the three SU(2)C symmetries, we obtain the following relations among the 3 [notdef] 3 up- and

down-type quark Yukawa coupling matrices:

(i) hu1 = ei hd1 and hu2 = ei hd2 ,(ii) hu1 = ei hd1 and hu2 = ei hd2 , (2.13)

(iii) hu1 = ei hd2 and hu2 = ei hd1 ,

where is an arbitrary angle unspecied by the symmetry constraint (2.12). We should stress again that only for a fully SO(5)-symmetric 2HDM, the three sets of solutions in (2.13) are equivalent. However, this is not in general true for scenarios that happen to realize only subgroups of SO(5), according to the symmetry classication given in [7, 29].

2.2 Scalar spectrum in the MS-2HDM

The masses and mixing in the Higgs sector of a general 2HDM are given in appendix A. After electroweak symmetry breaking in the MS-2HDM, we have the breaking pattern

SO(5) [angbracketleft]

1,2[angbracketright][negationslash]=0

! SO(4) , (2.14) which gives rise to a Higgs boson H with mass M2H = 2 2v2, whilst the remaining four scalar elds, denoted hereafter as h, a and h[notdef], are massless (pseudo)-Goldstone bosons. The latter is a consequence of the Goldstone theorem [39] and can be readily veried by means of (2.8) in (A.5). Thus, we identify H as the SM-like Higgs boson with the mixing angle = [cf. (A.7)]. We call this the SM alignment limit, which can be naturally attributed to the SO(5) symmetry of the theory.

In the exact SO(5)-symmetric limit, the scalar spectrum of the MS-2HDM is experimentally unacceptable, as the four massless pseudo-Goldstone particles, viz. h, a and h[notdef],

have sizeable couplings to the SM Z and W [notdef] bosons [cf. (A.9)]. These couplings induce additional decay channels, such as Z ! ha and W [notdef] ! h[notdef]h, which are experimentally

excluded [40]. Nevertheless, as we will see in the next section, the SO(5) symmetry of the original theory may be violated predominantly by RG e ects due to g[prime] and third-generation

Yukawa couplings, as well as by soft SO(5)-breaking mass parameters, thereby lifting the masses of these pseudo-Goldstone particles.

6

JHEP12(2014)024

3 RG and soft breaking e ects

As discussed in the previous section, the SO(5) symmetry that governs the MS-2HDM will be broken due to g[prime] and Yukawa coupling e ects, similar to the breaking of custodial symmetry in the SM. Therefore, an interesting question will be to explore whether these e ects are su cient to yield a viable Higgs spectrum at the weak scale. To address this question in a technically natural manner, we assume that the SO(5) symmetry is realized at some high scale [notdef]X. The physical mass spectrum at the electroweak scale is then obtained by the RG evolution of the 2HDM parameters given by (1.2). Using state-of-the-art two-loop RG equations given in appendix B, we examine the deviation of the Higgs spectrum from the SO(5)-symmetric limit due to g[prime] and Yukawa coupling e ects. This is illustrated in gure 1 for a typical choice of parameters in a Type-II realization of the 2HDM, even though the conclusions drawn from this gure have more general applicability. In particular, we obtain the following breaking pattern starting from a SU(2)L-gauged theory:

SO(5) SU(2)L

g[prime][negationslash]=0

! O(3) O(2) SU(2)L O(3) U(1)Y SU(2)L

Yukawa

! O(2) U(1)Y SU(2)L U(1)PQ U(1)Y SU(2)L

h 1,2[angbracketright][negationslash]=0

! U(1)em , (3.1)

where U(1)em is the electromagnetic group. In other words, RG-induced g[prime] e ects only lift the charged Higgs-boson mass Mh[notdef] , while the corresponding Yukawa coupling e ects also lift slightly the mass of the non-SM CP-even pseudo-Goldstone boson h. However, they still leave the CP-odd scalar a massless (see left panel of gure 1 for m212 = 0), which can be identied as a U(1)PQ axion [41]. The deviation of the scalar quartic couplings from the SO(5)-symmetric limit given in (2.8), thanks to g[prime] and Yukawa coupling e ects, is illustrated in gure 1 (right panel) for a simple choice of the single quartic coupling = 0 at the SO(5)-symmetry scale [notdef]X.

Figure 1 (left panel) also shows that g[prime] and Yukawa coupling e ects are not su cient to yield a viable Higgs spectrum at the weak scale, starting from a SO(5)-invariant boundary condition at some high scale [notdef]X. To minimally circumvent this problem, we need to include soft SO(5)-breaking e ects, by assuming a non-zero value for Re(m212) in the 2HDM potential (1.2). In the SO(5)-symmetric limit (2.8) for the scalar quartic couplings, but with Re(m212) [negationslash]= 0, we obtain the following mass spectrum [cf. (A.5)]:

M2H = 2 2v2 , M2h = M2a = M2h[notdef] =

JHEP12(2014)024

Re(m212)

s c , (3.2)

as well as an equality between the CP-even and CP-odd mixing angles: = , thus predicting an exact alignment for the SM-like Higgs boson H, simultaneously with an experimentally allowed heavy Higgs spectra (see left panel of gure 1 for m212 [negationslash]= 0). Note that in the alignment limit, the heavy Higgs sector is exactly degenerate [cf. (3.2)] at the

SO(5) symmetry-breaking scale, and at the low-energy scale, this degeneracy is mildly broken by the RG e ects. Thus, we obtain a quasi-degenerate heavy Higgs spectrum in the MS-2HDM, as illustrated in gure 1 (left panel). We emphasize that this is a

7

350

350

0.3

300

300

m122

0

h

h

a

a

h

h

H

H

0.2

ScalarMasses(GeV)

ScalarMasses(GeV)

ScalarQuarticCouplings

3

4

250

250

2

0.1

1

200

200

0.0

3-2

1

150

150

5,6,7

100

100

-0.1

50

50

-0.2

m122 = 0

0 2.0 2.5 3.0 3.5 4.0

0 2.0 2.5 3.0 3.5 4.0

- 2.0 2.5 3.0 3.5 4.0

0.3

Log10(

/GeV)

Log10(

/GeV)

Log10(

/GeV)

Figure 1. (left panel) The Higgs spectrum in the MS-2HDM without and with soft breaking e ects induced by m212. For m212 = 0, the pseudo-Goldstone boson a remains massless at tree-level, whereas h and h[notdef] receive small masses due to the g[prime] and Yukawa coupling e ects. For m212 [negationslash]= 0, one obtains

a quasi-degenerate heavy Higgs spectrum, cf. (3.2). (right panel) The RG evolution of the scalar quartic couplings under g[prime] and Yukawa coupling e ects. Here we have chosen [notdef]X = 2.5 [notdef] 104 GeV,

([notdef]X) = 0 and tan = 50 for illustration.

unique prediction of this model, valid even in the non-decoupling limit, and can be used to distinguish it from other 2HDM scenarios.

From (3.2), we notice that the alignment limit = is independent of the charged Higgs-boson mass Mh[notdef] and the value of tan . This is achieved without decoupling,i.e. without the need to consider the mass hierarchy Mh[notdef] v. Hence, in this softly

broken SO(5) 2HDM, we get natural SM alignment, without decoupling.6 It is instructive to analyze this last point in more detail. In the general CP-conserving 2HDM, the CP-even scalar mass matrix can be written down as [42, 43]

M2S = M2a s2 s c s c c2

[parenrightBigg]

JHEP12(2014)024

+ v2 2 1c2 + 5s2 + 2 6s c 34s c + 6c2 + 7s2 34s c + 6c2 + 7s2 2 2s2 + 5c2 + 2 7s c

[parenrightBigg]

c s s c

[parenrightBigg]

cM2S c s s c

[parenrightBigg]

, (3.3)

where M2a is given in (A.5), 34 3 + 4, and

cM2S =

bA

bC

[parenrightBigg]

, (3.4)

bC

bB

with

bA = 2v2

hc4 1 + s2 c2 345 + s4 2 + 2s c

c2 6 + s2 7[parenrightBig][bracketrightBig]

, (3.5)

bB = M2a + 5v2 + 2v2

hs2 c2

1 + 2 345[parenrightBig]

s c

c2 s2 [parenrightBig][parenleftBig]

6 7

[parenrightBig][bracketrightBig]

, (3.6)

bC = v2

hs3 c

i. (3.7)

6Strictly speaking, there will be one-loop threshold corrections to the e ective MS-2HDM potential, sourced from a non-zero Re(m212), which might lead to small misalignments. A simple estimate suggests that these corrections are of order 2/(162) and can therefore be safely neglected to a good approximation.

2 2 345[parenrightBig] c3 s

2 1 345[parenrightBig]+ c2

1 4s2 [parenrightBig]

6 + s2

4c2 1[parenrightBig] 7

8

cM2S in (3.3) is the respective 2 [notdef] 2 CP-even mass matrix written down in the so-called Higgs

eigenbasis [4447].Evidently, the SM alignment limit ! for the CP-even scalar mixing angle is

obtained, provided the o -diagonal elements of

cM2S in (3.4) vanish, i.e. for

Here we have used the short-hand notation: 345 3 + 4 + 5. Observe that

bC = 0 [24].

7t4 (2 2 345)t3 + 3( 6 7)t2 + (2 1 345)t 6 = 0 . (3.8)

In order to satisfy (3.8) for any value of tan , the coe cients of the polynomial in tan must identically vanish.7 Imposing this restriction, we conclude that all natural alignment solutions must satisfy the following condition:

1 = 2 = 345/2 , 6 = 7 = 0 . (3.9)

In particular, for 6 = 7 = 0, (3.8) has a solution

tan2 = 2 1 345

2 2 345

v4s2 c2

M2a + 5v2

From (3.7), this yields the quartic equation

> 0 , (3.10)

independent of Ma. After some algebra, the simple solution (3.10) to our general alignment condition (3.8) can be shown to be equivalent to that derived in [26, 48].

In the alignment limit, the two CP-even Higgs masses are given by the diagonal elements of

cM2S in (3.4):

M2H = 2v2( 1c4 + 345s2 c2 + 2s4 ) SMv2 , (3.11) M2h = M2a + 5v2 + 2v2s2 c2 ( 1 + 2 345) . (3.12)

On the other hand, in the limit Ma v, we can use a seesaw-like approximation

in (3.4) to obtain

M2H [similarequal] SMv2

JHEP12(2014)024

hs2

2 2 345[parenrightBig]

c2

2 1 345[parenrightBig][bracketrightBig]2, (3.13)

M2h [similarequal] M2a + 5v2 v2 . (3.14)

In (3.13) and (3.14), we have also included the possibility of decoupling via a large 5 coupling [25]. For large values of tan , e.g. tan > 10, we readily see that (3.13) reduces

to M2H [similarequal] 2 2v2, which again leads to a natural alignment.

As noted above, in the SO(5) symmetric limit of the conformal part of the 2HDM as given by (2.8) and (3.2), the SM alignment is achieved for any value of tan [cf. (3.10)]. In addition to SO(5), one may now wonder whether there are other classied symmetries of the 2HDM that lead to natural SM alignment, independently of tan and Ma. According to the classication given in table 1 of [29], we observe that, in the context of Type-II

7Notice that (3.8) is satised automatically in the SO(5) limit given in (2.8).

9

2HDM, there are only two other symmetries which lead to such natural SM alignment by satisfying (3.9), viz.8

(i) O(3) O(2) : 1 = 2 = 34/2, 5 = 6 = 7 = 0 , (3.15)(ii) Z2 [O(2)]2 : 1 = 2 = 345/2, 6 = 7 = 0 . (3.16) Both these symmetries also require [notdef]21 = [notdef]22 and m212 = 0. Note that in all the three naturally aligned scenarios, cf. (2.8), (3.15) and (3.16), tan as given in (3.10) consistently

gives an indenite answer 0/0. After spontaneous electroweak symmetry breaking, symmetry (i) predicts two pseudo-Goldstone bosons (h, a), whilst symmetry (ii) predicts only one pseudo-Goldstone boson, i.e. the CP-even Higgs boson h. However, a non-zero soft SO(5)-breaking mass parameter m212 can be introduced to render the pseudo-Goldstone bosons su ciently massive, in agreement with present experimental data, similar to the SO(5) case shown in gure 1. Even though the 2HDM scenarios based on the symmetries (i) and (ii) may be analyzed in a similar fashion, our focus here will be on the simplest realization of the SM alignment, namely, the MS-2HDM based on the SO(5) group. Nevertheless, the results that we will be deriving in the present study are quite generic and could apply to the less symmetric cases (i) and (ii) above as well.

Before concluding this section, we would like to comment that no CP violation is possible in the MS-2HDM, be it spontaneously or explicitly, at least up to one-loop level. This is due to the fact that the Higgs potential (2.9) remains CP-invariant after the RG and one-loop threshold e ects, even if a generic soft Z2-breaking term Im(m212ei) [negationslash]= 0

with an arbitrary CP-phase is present. In particular, as illustrated in gure 1 (right panel), a non-zero 5,6,7 cannot be induced via RG e ects, and this is true to any order in perturbation theory. On the other hand, one-loop threshold e ects could induce non-zero 5,6,7 of the following form:

5

23(m212)2 162[notdef]m212[notdef]2

JHEP12(2014)024

, 6

1 3m212 162[notdef]m212[notdef]

, 7

2 3m212 162[notdef]m212[notdef]

. (3.17)

However, the Higgs potential still remains CP-invariant, due to the fulllment of the following conditions [43]:

Im(m412 5) = Im(m212 6) = Im(m212 7) = 0 . (3.18)

Therefore, there is no CP-crisis arising from large contributions to electric dipole moments in the MS-2HDM, unlike in the case of MSSM.

4 Misalignment predictions

As discussed in section 3, a realistic Higgs spectrum can be obtained by softly breaking the maximal SO(5) symmetry of the 2HDM potential at some high scale [notdef]X by considering

Re(m212) [negationslash]= 0. As a consequence, there will be some deviation from the alignment limit in

8In Type-I 2HDM, there exists an additional possibility of realizing an exact Z2 symmetry [33] which leads to an exact alignment, i.e. in the context of the so-called inert 2HDM [49].

10

the low-energy Higgs spectrum. By requiring that the mass and couplings of the SM-like Higgs boson in our MS-2HDM are consistent with the latest Higgs data from the LHC [3, 4, 50], we can derive predictions for the remaining scalar spectrum and compare them with the existing (in)direct limits on the heavy Higgs sector. Our subsequent numerical results are derived for the Type-II 2HDM scenario, but the analysis could be easily extended to other 2HDM scenarios.

For the SM-like Higgs boson mass, we will use the 3 allowed range from the recent CMS and ATLAS Higgs mass measurements [4, 50]:

MH 2

124.1, 126.6[bracketrightbig]

GeV . (4.1)

For the Higgs couplings to the SM vector bosons and fermions, we use the constraints in the (tan , ) plane derived from a recent global t for the Type-II 2HDM [21, 22].9

For a given set of SO(5) boundary conditions

[notdef]X, tan ([notdef]X), ([notdef]X)

, we thus require that the RG-evolved 2HDM parameters at the weak scale must satisfy the above constraints on the lightest CP-even Higgs boson sector. This requirement of alignment with the SM Higgs sector puts stringent constraints on the MS-2HDM parameter space, as shown in gure 2. Here the solid, dashed, and dotted blue shaded regions respectively show the 1, 2 and 3 excluded regions due to misalignment of the mixing angle from its allowed range derived from the global t. The shaded red region is theoretically inaccessible, as there is no viable solution to the RGEs in this area. We ensure that the remaining allowed (white) region satises the necessary theoretical constraints, i.e. positivity and vacuum stability of the Higgs potential, and perturbativity of the Higgs self-couplings [6]. From gure 2, we nd that there exists an upper limit of [notdef]X [lessorsimilar] 109 GeV on the SO(5)-breaking scale of the 2HDM potential, beyond which an ultraviolet completion of the theory must be invoked.

Moreover, for 105 GeV [lessorsimilar] [notdef]X [lessorsimilar] 109 GeV, only a narrow range of tan values are allowed.

For the allowed parameter space of our MS-2HDM as shown in gure 2, we obtain concrete predictions for the remaining Higgs spectrum. In particular, the alignment condition imposes a lower bound on the soft breaking parameter Re(m212), and hence, on the heavy

Higgs spectrum. We compare this limit with the existing experimental limits on the heavy Higgs sector of the 2HDM [40], and nd that the alignment limits obtained here are more stringent in a wide range of the parameter space. The most severe experimental constraint comes from the charged Higgs sector, which give signicant contributions to various avour observables, e.g. B ! Xs [5154]. For this, we use the global t results for the Type-II

2HDM from [21], which includes limits derived from electroweak precision data, as well as avour constraints from mBs and B ! Xs relevant for the low tan region. The

comparison of the existing limit on the charged Higgs-boson mass as a function of tan with our predicted limits from the alignment condition for a typical value of the boundary scale [notdef]X = 3 [notdef] 104 GeV is shown in gure 3. It is clear that the alignment limits are

stronger than the indirect limits, except in the very small and very large tan regimes. For tan [lessorsimilar] 1 region, the indirect limit obtained from the Z ! b

JHEP12(2014)024

b precision observable

9Note that in our convention, the couplings of the SM-like Higgs boson to vector bosons is proportional to cos( ) [cf. (A.8)]. Hence, the natural alignment limit is obtained for = , and not for = /2,

as conventionally used in literature.

11

JHEP12(2014)024

Figure 2. Alignment constraints in the (tan , [notdef]X)-plane of the maximally symmetric Type-II 2HDM. The blue shaded regions show the 1 (dotted), 2 (dashed) and 3 (solid) exclusion regions from the alignment condition. The red shaded region is theoretically excluded in this model.

becomes the strictest [21, 55]. Similarly, for the large tan [greaterorsimilar] 30 case, the alignment limit can be easily obtained [cf. (3.7)] without requiring a large soft-breaking parameter m212, and therefore, the lower limit on the charged Higgs mass derived from the misalignment condition becomes somewhat weaker in this regime.

From gure 2, it should be noted that for [notdef]X [greaterorsimilar] 105 GeV, phenomenologically acceptable alignment is not possible in the MS-2HDM for large tan and large m212, while keeping the lightest CP-even Higgs boson within the experimentally allowed range (4.1) and maintaining vacuum stability up to the scale [notdef]X. Therefore, [notdef]X [greaterorsimilar] 105 GeV also leads to an upper bound on the charged Higgs-boson mass Mh[notdef] from the misalignment condition, depending on tan . This is illustrated in gure 4 for [notdef]X = 105 GeV. Here the green shaded regions show the 1 (dotted), 2 (dashed) and 3 (solid) allowed regions, whereas the corresponding red shaded regions are the experimentally exclusion regions at 1 (dotted), 2 (dashed) and 3 (solid). On the other hand, for [notdef]X [lessorsimilar] 105 GeV, a phenomenologically acceptable aligned solution with an arbitrarily large m212 is allowed for any value of tan

[cf. gure 2], and hence in this case, there exists only a lower limit on Mh[notdef] , as shown by the blue shaded (exclusion) regions in gure 3.

Similar alignment constraints are obtained for the heavy neutral pseudo-Goldstone bosons h and a, which are predicted to be quasi-degenerate with the charged Higgs boson h[notdef] in the MS-2HDM [cf. (3.2)]. The current experimental lower limits on the heavy neutral

Higgs sector [40] are much weaker than the alignment constraints in this case. Thus, the MS-2HDM scenario provides a natural reason for the absence of a heavy Higgs signal below the top-quark threshold, and this has important consequences for the non-standard Higgs searches in the run-II phase of the LHC, as discussed in the following section.

12

Figure 3. The 1 (dotted), 2 (dashed) and 3 (solid) lower limits on the charged Higgs mass obtained from the alignment condition (blue lines) in the maximally symmetric Type-II 2HDM with [notdef]X = 3 [notdef] 104 GeV. For comparison, the corresponding lower limits from a global t are also shown

(red lines).

Figure 4. Alignment limits on the charged Higgs mass Mh in the maximally symmetric Type-II 2HDM with [notdef]X = 105 GeV. The dark green regions show the 1 (dotted), 2 (dashed) and 3 (solid) regions allowed by alignment constraints in the model. The existing lower limits from a global t are also shown (red lines) for comparison.

13

JHEP12(2014)024

5 Collider signals

In the alignment limit, the couplings of the lightest CP-even Higgs boson are exactly similar to the SM Higgs couplings, while the heavy CP-even Higgs boson preferentially couples to fermions (see appendix A). Therefore, two of the relevant Higgs production mechanisms at the LHC, namely, the vector boson fusion and Higgstrahlung processes are suppressed for the gaugephobic heavy neutral Higgs sector. As a consequence, the only relevant production channels to probe the neutral Higgs sector of the MS-2HDM are the gluon-gluon fusion and tth (bbh) associated production mechanisms at low (high) tan . For the charged Higgs sector of the MS-2HDM, the dominant production mode is the associated production process: gg ! tbh+ + t

bh, irrespective of tan .

5.1 Branching fractions

For our collider analysis, we calculate all the branching ratios of the heavy Higgs sector in the MS-2HDM as a function of their masses using the public C++ code 2HDMC [56, 57]. The results for tan = 2 and with SO(5)-symmetric boundary conditions at [notdef]X = 3 [notdef] 104 GeV are shown in gure 5 for illustration. It is clear that for the heavy neutral Higgs bosons, the tt decay mode is the dominant one over most of the MS-2HDM parameter space. However, this is true only for low tan [lessorsimilar] 5, since as we go to higher tan values, the bb decay mode becomes dominant, with a sub-dominant contribution from +, whereas the tt mode gets Yukawa suppressed. This is illustrated in gure 6, where we compare BR(h ! tt),

BR(h ! b

b) and BR(h ! +) for three representative values of tan = 2 (solid), 5

(dashed) and 10 (dotted). For the charged Higgs boson h+(), the tb(tb) mode is the dominant one over the entire parameter space, as shown in gure 5 for tan = 2, and this is true even for larger tan .

5.2 Charged Higgs signal

The detection of a charged Higgs boson will be an unequivocal evidence for a beyond SM Higgs sector, and in particular, a smoking gun signal for a 2HDM. For Mh[notdef] < Mt, i.e.

below the top-quark threshold, stringent collider limits have been set on its production directly through top quark decays t ! h+b, followed by h+ decays to + [58, 59], ++

jets [60] and cs [61, 62]. For charged Higgs boson masses above the top-quark threshold, the h+ ! t

b decay channel opens up, and quickly becomes the dominant channel. In fact, in the Type-II 2HDM, the h+tb coupling (A.14) implies that for Mh+ > Mt + Mb, the branching fraction of h+ ! t

b is almost 100% (cf. gure 5), independent of tan . This leads to mostly ttbb nal states at the LHC via

gg ! tbh+ + t

bh ! ttb

b . (5.1)

The experimental observation of this channel is challenging due to large QCD backgrounds and the non-trivial event topology, involving at least four b-jets [63]. Nevertheless, we should emphasize here that (5.1) is the most promising channel for the charged Higgs signal in the MS-2HDM, because other interesting possibilities, such as h[notdef] ! aW [notdef], hW [notdef] [64],

14

JHEP12(2014)024

1

0.1

HH WW ZZ bb

gg tt cc

BR

10-2

10-3

t

= 2

10-4

300 400 500 600 700 800 900 1000

Mh (GeV)

1

1

JHEP12(2014)024

0.1

ZH bb gg

tt cc

ss

tb WH ts

0.1

BR

10-2

BR

10-2

10-3

10-3

t

= 2

t

= 2

10-4

-4

300 400 500 600 700 800 900 1000

10 300 400 500 600 700 800 900 1000

Ma (GeV)

Mh (GeV)

Figure 5. The decay branching ratios of the heavy Higgs bosons in the maximally symmetric Type-II 2HDM for low tan .

1

tt (t

= 2) bb (t

= 2)

(t

= 2) tt (t

= 5) bb (t

= 5)

(t

= 5) tt (t

= 10) bb (t

= 10)

(t

= 10)

0.1

BR

10-2

10-3

-4

10 300 400 500 600 700 800 900 1000

Mh (GeV)

Figure 6. Comparison of BR(h ! tt), BR(h ! b

b) and BR(h ! +) for di erent values of

tan in the maximally symmetric Type-II 2HDM.

are not open in this scenario due to the kinematical constraints imposed by the quasi-degeneracy of the heavy Higgs sector [cf. (3.2) and gure 1 (left panel)].

A recent CMS study [65] has presented for the rst time a realistic analysis of the process (5.1), with the following decay chain:

gg ! h[notdef]tb ! ([lscript] [lscript]bb)([lscript][prime] [lscript][prime] b)b (5.2) ([lscript], [lscript][prime] beings electrons or muons). Using the ps = 8 TeV LHC data, they have derived

15

1000

100

10

tb)[fb]

BR(h

CMS 95% CL t

= 1 (14 TeV) t

= 2 (14 TeV) t

= 5 (14 TeV)

1

JHEP12(2014)024

0.1

200 400 600 800 1000

Mh (GeV)

Figure 7. Predictions for the cross section of the process (5.1) in the Type-II MS-2HDM at ps = 14 TeV LHC for various values of tan . For comparison, we have also shown the current 95% CL CMS upper limit from the ps = 8 TeV data [65].

95% CL upper limits on the production cross section (gg ! h[notdef]tb) times the branching

ratio BR(h[notdef] ! tb) as a function of the charged Higgs mass, as shown in gure 7. In the

same gure, we show the corresponding predictions at ps = 14 TeV LHC in the Type-II MS-2HDM for some representative values of tan . The cross section predictions were obtained at leading order (LO) by implementing the 2HDM in MadGraph5 [66, 67] and using the NNPDF2.3 PDF sets [68, 69].10 A comparison of these cross sections with the CMS limit suggests that the run-II phase of the LHC might be able to probe a portion of the MS-2HDM parameter space using the process (5.1).

In order to make a rough estimate of the ps = 14 TeV LHC sensitivity to the charged Higgs signal (5.1) in the MS-2HDM, we perform a parton level simulation of the signal and background events using MadGraph5 aMC@NLO [66, 67]. For the event reconstruction, we use some basic selection cuts on the transverse momentum, pseudo-rapidity and dilepton invariant mass, following the recent CMS analysis [65]:

p[lscript]T > 20 GeV, [notdef] [lscript][notdef] < 2.5, R[lscript][lscript] > 0.4,

M[lscript][lscript] > 12 GeV, [notdef]M[lscript][lscript] MZ[notdef] > 10 GeV,

pjT > 30 GeV, [notdef] j[notdef] < 2.4, /

ET > 40 GeV. (5.3)

Jets are reconstructed using the anti-kT clustering algorithm [71] with a distance parameter of 0.5. Since four b-jets are expected in the nal state, at least two b-tagged jets are required in the signal events, and we assume the b-tagging e ciency for each of them to be 70%.

The inclusive SM cross section for pp ! ttb

b+X is 18 pb at NLO, with roughly 30%

uncertainty due to higher order QCD corrections [72]. Most of the QCD background for

10For an updated and improved next-to-leading order (NLO) calculation, see [70].

16

the 4b + 2[lscript] + /

ET nal state given by (5.2) can be reduced signicantly by reconstructing at least one top-quark. As we will show below, the remaining irreducible background due to SM ttbb production can be suppressed with respect to the signal by reconstructing the charged Higgs boson mass, once a valid signal region is dened, e.g. in terms of an observed excess of events at the LHC in future. For the semi-leptonic decay mode of top-quarks as in (5.2), one cannot directly use an invariant mass observable to infer Mh[notdef] , as both the neutrinos in the nal state give rise to missing momentum. A useful quantity in this case is the MT2 variable, also known as the stransverse mass [73], dened as

MT2 = min

n/ pTa+/pTb=/pT [bracerightBig][bracketleftBig]

, (5.4)

where [notdef]a[notdef], [notdef]b[notdef] stand for the two sets of particles in the nal state, each containing a

neutrino with part of the missing transverse momentum (/pTa,b ). Minimization over all

possible sums of these two momenta gives the observed missing transverse momentum /

pT ,

whose magnitude is the same as /

ET in our specic case. In (5.4), mTi (with i =a,b) is the usual transverse mass variable for the system [notdef]i[notdef], dened as

m2Ti =

Xvisible

ETi + /ETi

!2

[parenleftBigg][summationdisplay]

visible

For the correct combination of the nal state particles in (5.2), i.e. for [notdef]a[notdef] = ([lscript] [lscript]bb) and {b[notdef] = ([lscript][prime] [lscript][prime] bb) in (5.4), the maximum value of MT2 represents the charged Higgs boson

mass, with the MT2 distribution smoothly dropping to zero at this point. This is illustrated in gure 8 (left panel) for a typical choice of Mh[notdef] = 300 GeV. For comparison, we also show the MT2 distribution for the SM background, which obviously does not have a sharp endpoint. Thus, for a given hypothesized signal region dened in terms of an excess due to Mh[notdef] , we may impose an additional cut on MT2 Mh[notdef] to enhance the signal (5.2) over

the irreducible SM background.

Apart from the decay chain (5.2) as considered in the CMS analysis [65], we also examine another decay chain involving hadronic decay modes of the secondary top-quark from the charged Higgs decay, i.e.

gg ! h[notdef]tb ! (jjbb)([lscript] [lscript]b)b . (5.6) In this case, the charged Higgs boson mass can be reconstructed using the invariant mass

Mjjbb for the correct combination of the b-quark jets. This is illustrated in gure 8 (right panel) for Mh[notdef] = 300 GeV, along with the expected SM background. Thus, for the decay chain (5.6), one can use an invariant mass cut of Mjjbb around Mh[notdef] to observe the signal over the irreducible SM background. Note that the hadronic mode (5.6) has a larger branching ratio, although from the experimental point of view, one has to deal with the uncertainties in the jet energy measurements, combinatorics and the resulting uncertainties in the invariant mass reconstruction of multiparticle nal states.

Thus, in principle, we can obtain an observable charged Higgs signal in the MS-2HDM above the irreducible SM background by using one of the methods shown in gure 8 to

17

max [notdef]mTa, mTb[notdef]

[bracketrightBig]

pTi + /

pTi

!2. (5.5)

JHEP12(2014)024

Figure 8. An illustration of the charged Higgs boson mass reconstruction using the MT2 (left panel) and invariant mass (right panel) observables. The irreducible SM background distribution is also shown for comparison.

reconstruct e ciently the charged Higgs boson mass. Assuming this, we present an estimate of the signal to background ratio for the charged Higgs signal given by (5.1) at ps = 14 TeV LHC with 300 fb1 for some typical values of tan in gure 9. Since the mass of the charged Higgs boson is a priori unknown, we vary the charged Higgs mass, and for each value of Mh[notdef] , we assume that it can be reconstructed around its actual value within 30 GeV uncertainty. We believe such a mass resolution is possible experimentally, given the fact that the width of the charged Higgs boson h[notdef] in the mass range shown in gure 9 is always smaller than the chosen mass uncertainty.11 From gure 9, we see that the ttbb channel (5.1) is e ective for charged Higgs searches at the LHC for low tan values. Note that the production cross section (gg ! tbh+) decreases rapidly with increasing tan

due to the Yukawa suppression [cf. (A.14)], even though the branching fraction of h+ ! tb remains close to 100%.

5.3 Heavy neutral Higgs signal

Since the heavy CP-even Higgs boson in the MS-2HDM is gaugephobic, most of the existing collider limits derived using the decay modes h ! W W [74, 75] and h ! ZZ [76] do not

apply in this case. The only existing searches relevant to the heavy CP-even sector of the MS-2HDM scenario are those based on gg ! h ! + and gg ! b

bh ! b

b+ [77, 78].

However, due to the relatively small branching ratio of h ! +, the model-independent

upper limits derived in [77, 78] are easily satised for the heavy Higgs spectrum presented here. Similarly, the h ! branching ratio in the MS-2HDM is 102 103 times smaller

than that for the SM Higgs boson; therefore, the cross section limits derived from the channel [79, 80] are also easily satised. So far there have been no direct searches for heavy neutral Higgs bosons involving ttand/or bb nal states, mainly due to the challenges associated with uncertainties in the jet energy scales and the combinatorics arising from complicated multiparticle nal states in a busy QCD environment. Nevertheless, these channels become pronounced in the MS-2HDM scenario, and hence, we will make here a preliminary attempt to study them.

11For instance, for tan = 2, h = 1.9 GeV at Mh = 300 GeV and h = 22 GeV at Mh = 2 TeV.

18

JHEP12(2014)024

Figure 9. Predicted number of events for the ttbb signal from the charged pseudo-Goldstone boson in the MS-2HDM at ps = 14 TeV LHC with 300 fb1 integrated luminosity. The results are shown for three di erent values of tan =1 (green solid), 2 (blue dashed) and 5 (orange solid). The irreducible SM background (red dotted) is controlled by assuming an e cient mass reconstruction technique, as described in the text.

It is worth mentioning here that the Higgs pair production process pp ! h ! HH

(see e.g. [81]) is another interesting possibility. However, as shown in (A.12), the h ! HH

decay mode should also vanish in the exact alignment limit ! , just like the h ! V V

decay modes. Therefore, the LHC limits derived using the h ! HH channel [82, 83]

are applicable only below the top threshold Mh 2Mt in the MS-2HDM (cf. gure 5).

On the other hand, the lower limits on the heavy Higgs sector, as derived in section 4 (e.g. gures 3 and 4) strongly suggest a mass spectrum above the tt threshold, where the h ! HH branching fraction drops orders of magnitude below that of h ! tt (b

b) at low

(high) tan .

In light of the above discussion, we propose a new search channel for the heavy neutral Higgs boson in the MS-2HDM via the tttt nal state:

gg ! tth ! tttt. (5.7)

Such four top nal states have been proposed before in the context of other exotic searches at the LHC, e.g. composite top [8486], low-scale extra-dimensions [87, 88] and SUSY with light stops and gluinos [89]. However, their relevance for heavy Higgs searches have not been explored so far. We note here that the existing 95% CL experimental upper limit on the four top production cross section is 59 fb from ATLAS [90] and 32 fb from CMS [91], whereas the SM prediction for the inclusive cross section of the process pp ! tttt+ X is

about 10-15 fb [92].

To get a rough estimate of the signal to background ratio for our new four-top signal, we perform a parton-level simulation of the signal and background events at LO in QCD using MadGraph5 aMC@NLO [66, 67] with NNPDF2.3 PDF sets [68, 69]. For the inclusive

19

JHEP12(2014)024

10

t

= 1 (14 TeV) t

= 2 (14 TeV) t

= 5 (14 TeV)

- ) [ fb ]

t

1

BR(h

0.1

10-2

500 1000 1500 2000

Mh (GeV)

Figure 10. Predictions for the cross section of the process (5.7) in the Type-II MS-2HDM at ps = 14 TeV LHC for various values of tan .

SM cross section for the four-top nal state at ps = 14 TeV LHC, we obtain 11.85 fb, whereas our proposed four-top signal cross sections are found to be comparable or smaller depending on Mh and tan , as shown in gure 10. However, since we expect one of the tt pairs coming from an on-shell h decay to have an invariant mass around Mh, we can

use this information to signicantly boost the signal over the irreducible SM background. Note that all the predicted cross sections shown in gure 10 are well below the current experimental upper bound [91].

Depending on the W decay mode from t ! W b, there are 35 nal states for four top

decays. According to a recent ATLAS analysis [93], the experimentally favoured channel is the semi-leptonic/hadronic nal state with two same-sign isolated leptons. Although the branching fraction for this topology (4.19%) is smaller than most of the other channels, the presence of two same-sign leptons in the nal state allows us to reduce the large QCD background substantially, including that due to the SM production of ttbb+jets.12

Therefore, we will only consider the following decay chain in our preliminary analysis:

gg ! tth ! (tt)(tt) ! [parenleftBig]

([lscript][notdef] [lscript]b)(jjb)

JHEP12(2014)024

[parenrightBig][parenleftBig]

([lscript][prime][notdef] [lscript][prime] b)(jjb)

[parenrightBig]

. (5.8)

For event reconstruction, we will use the same selection cuts as in (5.3), and in addition, following [93], we require the scalar sum of the pT of all leptons and jets (dened as HT )

to exceed 350 GeV.

As in the charged Higgs boson case [cf. (5.2)], the heavy Higgs mass can be reconstructed from the signal given by (5.8) using the MT2 endpoint technique. The correct combination of visible nal states in (5.8) will lead to a smooth drop at Mh in the MT2

distribution, as illustrated in gure 11 for a typical choice of Mh = 450 GeV. As shown in the same gure, the SM background does not exhibit such a feature, and hence, an additional selection cut on MT2 Mh can be used to enhance the signal to background

ratio in the signal region.

Our simulation results for the predicted number of signal and background events for the process (5.8) at ps = 14 TeV LHC with 300 fb1 luminosity are shown in gure 12.

12For a detailed analysis of the reducible and irreducible four top background, see [93].

20

Figure 11. An illustration of the heavy CP-even Higgs boson mass reconstruction using MT2

observable. The irreducible SM background distribution is also shown for comparison.

Figure 12. Predicted number of events for the ttttsignal from the neutral pseudo-Goldstone boson in the MS-2HDM at ps = 14 TeV LHC with 300 fb1 integrated luminosity. The results are shown for three di erent values of tan =1 (green solid), 2 (blue dashed) and 5 (orange solid). The SM background (red dotted) is controlled by assuming an e cient mass reconstruction technique, as outlined in the text.

The signal events are shown for three representative values of tan . Here we vary the a priori unknown heavy Higgs mass, and for each value of Mh, we assume that it can be reconstructed around its actual value within 30 GeV uncertainty, which should be feasible experimentally, given the fact that the decay width of h is always smaller the chosen mass uncertainty window for the entire mass range shown in gure 12. From this preliminary analysis, we nd that the tttt channel provides the most promising collider signal to probe the heavy Higgs sector in the MS-2HDM at low values of tan [lessorsimilar] 5.

The above analysis is also applicable for the CP-odd Higgs boson a, which has similar production cross sections and tt branching fractions as the CP-even Higgs h. However, the tth(a) production cross section as well as the h(a) ! tt branching ratio decreases

with increasing tan . This is due to the fact that the htt coupling in the alignment limit is cos / sin cot , which is same as the att coupling [cf. (A.13)]. Thus, the high

21

JHEP12(2014)024

tan region of the MS-2HDM cannot be searched via the tttt channel proposed above, and one needs to consider the channels involving down-sector Yukawa couplings, e.g. bbbb and bb+, which are very challenging in the LHC environment [63]. For instance, the

SM bbbb cross section at ps = 14 TeV LHC is about 140 pb at NLO [94], whereas the pp ! b

bh ! b

bbb signal cross section for Mh = 300 GeV and tan = 10 is only about 0.3 pb at NLO, as estimated using the public FORTRAN code SusHi [95, 96]. In practice, one would require a sophisticated jet substructure technique [97, 98] to disentangle such a tiny signal from the huge QCD background. It is also worth commenting here that the simpler process pp ! h ! tt (b

b) at low (high) tan similarly su ers from a huge SM tt (bb) QCD background, even after imposing an Mtt (bb) cut.

Before concluding this section, we should clarify that although we were able to obtain a sizeable signal-to-background ratio in the low tan regime for the signals (5.2), (5.6) and (5.8) using e cient mass reconstruction techniques in the signal region, as described in the subsections 5.2 and 5.3 above, these results are valid only at the parton level. In a realistic detector environment, the sharp features of the signal shown in gures 8 and 11 may not survive, and therefore, the signal-to-background ratio might get somewhat reduced than that shown here. However, a detailed realistic detector-level analysis of these signals in our MS-2HDM scenario, including realistic top reconstruction e ciencies and smearing e ects, is beyond the scope and main focus of this article, and is being pursued in a separate dedicated study.

6 Conclusions

We have analyzed the symmetries of the 2HDM scalar potential to naturally justify the so-called SM alignment limit, independently of the heavy Higgs spectrum and the value of tan . We show that in the Type-II 2HDM, there exist only three di erent symmetry realizations, cf. (2.8), (3.15) and (3.16), which could lead to a natural alignment by satisfying (3.8) for any value of tan . In the context of the Maximally Symmetric Two Higgs Doublet Model based on the SO(5) group, we demonstrate how small deviations from this alignment limit are naturally induced by RG e ects due to the hypercharge gauge coupling g[prime] and third generation Yukawa couplings, which also break the custodial symmetry of the theory. In addition, a non-zero soft SO(5)-breaking mass parameter is required to yield a viable Higgs spectrum consistent with the existing experimental constraints. Employing the current Higgs signal strength data from the LHC, which disfavour large deviations from the alignment limit, we derive important constraints on the 2HDM parameter space. In particular, we predict lower limits on the heavy Higgs spectrum, which prevail the present limits in a wide range of parameter space. Depending on the scale where the maximal symmetry could be realized in nature, we also obtain an upper limit on the heavy Higgs masses in certain cases, which could be completely probed during the run-II phase of the LHC. Finally, we propose a new collider signal with four top quarks in the nal state, which can become a valuable observational tool to directly probe the heavy Higgs sector of the 2HDM in the SM alignment limit for low values of tan . It would be interesting to investigate how this tool could be applied to supersymmetric theories in the alignment limit.

22

JHEP12(2014)024

Acknowledgments

A.P. thanks Celso Nishi for a clarifying remark that led us to include the fourth footnote in this article. P.S.B.D. thanks Otto Eberhardt and Martin Wiebusch for helpful discussions regarding their global t results in [21, 22]. We also thank Rohini Godbole, Romain Madar, Yvonne Peters and Jos Santiago for valuable comments on the four top analysis. The work of P.S.B.D. and A.P. is supported by the Lancaster-Manchester-She eld Consortium for Fundamental Physics under STFC grant ST/J000418/1.

A Higgs spectrum and couplings in a general 2HDM

Here we will restrict our discussion to 2HDM potentials realizing CP-conserving vacua. In this case, the minimization of a CP-conserving 2HDM potential (1.2) yields the following real non-negative vacuum expectation values (VEVs):

h 1[angbracketright] =

1 p2[parenleftBigg][parenleftBigg]

JHEP12(2014)024

0 v1

!, [angbracketleft] 2[angbracketright] =1p2[parenleftBigg]0 v2

!, (A.1)

where v =

pv21 + v22 = 246.2 GeV is the SM electroweak VEV and for later convenience, we dene tan v2/v1. The two scalar doublets can be expanded in terms of eight real

scalar elds as follows:

j = +j

1p2 (vj + j + iaj)

[parenrightBigg]

, (A.2)

with j = 1, 2. After spontaneous symmetry breaking, there are three Goldstone modes (G[notdef], G0), which become the longitudinal components of the SM W [notdef] and Z bosons. Thus, there are ve remaining physical scalar mass eigenstates: two CP-even (h, H), one CP-odd (a) and two charged (h[notdef]) scalars. The mixing in the CP-odd and charged sectors is governed by the angle dened above:

G[notdef] h[notdef]

[parenrightBigg]

= c s

s c

[parenrightBigg][parenleftBigg][parenleftBigg]

[notdef]1 [notdef]2

[parenrightBigg]

,

G0 a

[parenrightBigg]

= c s

s c

[parenrightBigg][parenleftBigg][parenleftBigg]

a1 a2

[parenrightBigg]

, (A.3)

where c cos , s sin . On the other hand, in the CP-even sector, we have a new

mixing angle :

H h

[parenrightBigg]

= c s

s c

[parenrightBigg][parenleftBigg][parenleftBigg]

1 2

[parenrightBigg]

. (A.4)

23

The corresponding physical mass eigenvalues are given by [42, 43]

M2h[notdef] =

m212 s c

v22s c 6c2 + 7s2

v2

2 ( 4 + 5) +

[parenrightbig]

,

M2a = M2h[notdef] +

v2

2 ( 4 5) ,

M2H = 12

h(A + B)

p(A B)2 + 4C2

[bracketrightBig]

,

M2h = 12

h(A + B) +

p(A B)2 + 4C2

[bracketrightBig]

, (A.5)

JHEP12(2014)024

where we have dened tan 2 = 2C/(A B), andA = M2as2 + v2 2 1c2 + 5s2 + 2 6s c

[parenrightbig]

,

B = M2ac2 + v2 2 2s2 + 5c2 + 2 7s c

, (A.6)

C = M2as c + v2 34s c + 6c2 + 7s2

[parenrightbig]

.

[parenrightbig]

with 34 = 3 + 4. The SM Higgs eld is given by

HSM = 1 cos + 2 sin = H cos( ) + h sin( ) . (A.7) From (A.7), the couplings of h and H to the gauge bosons (V = W [notdef], Z) with respect to the SM Higgs couplings gHSMV V are given by

ghV V = s , gHV V = c . (A.8)

Similarly, unitarity constraints uniquely x the other Higgs-Higgs-V couplings [5]:

ghaZ = g2 cos w c , gHaZ =

g2 cos w s ,

gh+hW =

g2c , gh

+HW =

g2s , gah[notdef]W =

g2 , (A.9)

(where w is the weak mixing angle) in order to satisfy the sum rules [99]

g2haZ + g2HaZ = 1

cos2 w (g2h+hW + g2h+HW ) =

1 4M2Z

g2HSMZZ , (A.10)

g2HSMW+W . (A.11)

For our subsequent discussion, we also write down the h-H-H coupling [24]:

ghHH = s

4vs c

h2(M2h + 2M2H)s2 2(M2a + v2 5)(s2 + 3s2 )

v2( 6 7)(c2 + 3c2 ) v2( 6 + 7)(1 + 3c2( ))[bracketrightBig]

g2h+hW + g2h+HW = g2ah[notdef]W =

1 4M2W

. (A.12)

Note that the coupling ghHH is proportional to s and so vanishes identically in the

alignment limit ! .

To obtain a phenomenologically acceptable theory, we need to forbid Higgs interactions with tree-level FCNCs. This can be accomplished minimally by imposing appropriate

24

discrete Z2 symmetries, which will explicitly break in general the custodial symmetries of the theory. By convention, we may take uR to couple to 2, i.e. hu1 = 0, and then 1 ( 2)

to couple to dR, with hd2 = 0 (hd1 = 0), in a Type-II (Type-I) realization of the 2HDM. As our interest is in the Type-II 2HDM, we only list the Yukawa couplings of the neutral scalars with respect to those of HSM for this class of models [5]:

ghtt = cos / sin , ghbb = sin / cos ,

gHtt = sin / sin , gHbb = cos / cos ,

gatt = cot , gabb = tan . (A.13)

Finally, we also write down the coupling of the charged scalar to the third-generation quarks [5]:

ghtb =

g2p2MW [mt cot (1 + 5) + mb tan (1 5)] . (A.14)

B Two-loop RGEs in a general 2HDM

In this section, we present the two-loop renormalization group equations (RGEs) for the general 2HDM given by (1.2). These results were obtained using the general prescription given in [100102], as implemented in the public Mathematica package SARAH 4 [103, 104].

The two-loop RGEs for the SU(3)c, SU(2)L and U(1)Y gauge couplings are respectively given by

Dg3 =

7g33 162 +

JHEP12(2014)024

g33 2564

26g23 +92g22 +116 g[prime]2 2y2b 2y2t[parenrightbigg]

, (B.1)

Dg2 =

3g32 162 +

g32 2564

12g23 + 8g22 + 2g[prime]2 32y2b 32y2t 12y2[parenrightbigg]

, (B.2)

Dg[prime] = 7g[prime]3 162 +

443 g23 + 6g22 +1049 g[prime]2 56y2b 176 y2t 52y2[parenrightbigg]

, (B.3)

where D d/d ln [notdef] and [notdef] is the usual t-Hooft mass employed in the regularization of

ultraviolet divergences in loop integrals. Note that at the one-loop level, the gauge coupling RGEs do not depend on the Yukawa and scalar couplings, whilst at two loops, they do depend on the quark and lepton Yukawa couplings, for which we have only kept the dominant third-generation contributions.

Similarly for the Yukawa RGEs, we will only consider the third-generation Yukawa couplings, such that for Type-II 2HDM, we have

mt = v2 p2hu2,33

v2p2yt , mb() =

g[prime]3

2564

v1p2yb() . (B.4)

With this approximation, the third-generation Yukawa coupling RGEs are given by

Dyt = yt 162

8g23 94g22 1712g[prime]2 +92y2t +1 2y2b[parenrightbigg]

+ yt

2564

108g43 214 g42 +1267216 g[prime]4 + 9g22g23 34g22g[prime]2 +199 g23g[prime]2 + 6 22 + 23

25

v1 p2hd(e)1,33

+ 3 4 + 24 + 32( 25 + 26 + 3 27) + [parenleftbigg]

16

3 g23 +

33 16g22

41144g[prime]2 2 3 + 2 4[parenrightbigg]

y2b

+ 36g23 + 22516 g22 +13116 g[prime]2 12 2[parenrightbigg]

y2t 52y4b 12y4t 52y2by2t 34y2by2[bracketrightbigg]

, (B.5)

8g23 94g22 512g[prime]2 +92y2b +12y2t + y2[parenrightbigg]

+ yb

2564

108g43 214 g42 113216g[prime]4 + 9g22g23 94g22g[prime]2 +319 g23g[prime]2 + 6 21 + 23

+ 3 4 + 24 + 32( 25 + 3 26 + 27) + [parenleftbigg]

16

3 g23 +

Dyb = yb 162

33 16g22

53144g[prime]2 2 3 + 2 4[parenrightbigg]

y2t

JHEP12(2014)024

+

158 g22 +258 g[prime]2[parenrightbigg]

y2 +

36g23 + 22516 g22 +7916g[prime]2 12 1[parenrightbigg] y2b

12y4b

94y4

52y4t

52y2by2t

9 4y2by2[bracketrightbigg]

, (B.6)

Dy = y 162

94g22 154 g[prime]2 + 3y2b +5 2y2[parenrightbigg]

214 g42 +1618 g[prime]4 +94g22g[prime]2 + 6 21 + 23 + 3 4 + 24

+32( 25 + 3 26 + 27) + [parenleftbigg]

20g23 + 45

8 g22 +

+ y

2564

25 24g[prime]2[parenrightbigg]

y2b

16516 g22 +17916 g[prime]2 12 1[parenrightbigg]

y2 274 y4b 3y4 274 y2by2 94y2by2t[bracketrightbigg]

. (B.7)

Similarly, the two-loop RGEs for the VEVs are given by

Dv1 = v1 162

34(3g22 + g[prime]2) 3y2b y2[bracketrightbigg]

+ v1

2564

43532 g42 14932 g[prime]4 316g22g[prime]2 6 21 23 3 4 24

+

32( 25 + 3 26 + 27) [parenleftbigg]

20g23 + 45

8 g22 +

25 24g[prime]2[parenrightbigg]

y2b [parenleftbigg]

15

8 g22 +

25

8 g[prime]2[parenrightbigg]

y2

+27

4 y4b +

94y4 +

9 4y2by2t[bracketrightbigg]

3v2 5124

(2 1 + 345) 6 + (2 2 + 345) 7[bracketrightbigg]

, (B.8)

Dv2 = v2 162

34(3g22 + g[prime]2) 3y2t[bracketrightbigg]

+ v2

2564

43532 g42 14932 g[prime]4 316g22g[prime]2 6 22 23 3 4 24

32( 25 + 26 + 3 27) + [parenleftbigg]

20g23 + 45

8 g22 +

85 24g[prime]2[parenrightbigg]

y2t + 94y2by2t +

27

4 y4t[bracketrightbigg]

(2 1 + 345) 6 + (2 2 + 345) 7[bracketrightbigg]

. (B.9)

The two-loop RGEs for all the scalar quartic couplings appearing in (1.2) in the Type-II

26

3v1 5124

2HDM are given by

D 1 = 1 162

"38(3g42 + g[prime]4 + 2g22g[prime]2) 3 1(3g22 + g[prime]2) + 24 21 + 2 23 + 2 3 4 + 24

+ 25 + 12 26 + 4 1(3y2b + y2) 6y4b 2y4[bracketrightBigg]

+ 1

2564

[bracketleftBigg]

116 291g62 101g42g[prime]2 191g22g[prime]4 131g[prime]6

[parenrightbig]

18(51g42 78g22g[prime]2 217g[prime]4) 1 +

52(3g42 + g[prime]4) 3 +

54(3g42 + 2g22g[prime]2 + g[prime]4) 4

+ (3g22 + g[prime]2)(36 21 + 4 23 + 4 3 4 + 24 + 18 26) + g[prime]2( 24 25) 312 31 20 1 23 8 33 6 34 20 1 3 4 4(5 1 + 3 4) 23 4(3 1 + 4 3) 24 2(7 1 + 10 3 + 11 4) 25 2(159 1 + 33 3 + 35 4 + 37 5) 26 4(9 3 + 7 4 + 5 5) 6 7 + 2(3 1 9 3 7 4 5 5) 27

94g42 92g22g[prime]2 54g[prime]4 [parenleftbigg]452 g22 + 80g23 +256 g[prime]2[parenrightbigg]

1 + 36(4 21 + 26)

y2b

32g23 43g[prime]2 + 3 1

y4b 6(2 23 + 2 3 4 + 24 + 25 + 6 26)y2t

JHEP12(2014)024

34g42 112 g22g[prime]2 +254 g[prime]4 52(3g22 + 5g[prime]2) 1 + 12(4 21 + 26) y2

(4g[prime]2 + 1)y4 9 1y2by2t 6y2ty4b + 30y6b + 10y6[bracketrightBigg]

, (B.10)

D 2 = 1 162

"38(3g42 + g[prime]4 + 2g22g[prime]2) 3 2(3g22 + g[prime]2) + 24 22 + 2 23 + 2 3 4

+ 24 + 25 + 12 27 + 12 2y2t 6y4t[bracketrightBigg]

+ 1

2564

[bracketleftBigg]

116 291g62 101g42g[prime]2 191g22g[prime]4 131g[prime]6

[parenrightbig]

18 51g42 78g22g[prime]2 217g[prime]4

[parenrightbig]

2 + 52(3g42 + g[prime]4) 3 +

54(3g42 + 2g22g[prime]2 + g[prime]4) 4

+ (3g22 + g[prime]2)(36 22 + 4 23 + 4 3 4 + 24 + 18 27) + g[prime]2( 24 25) 312 32 20 2 23 8 33 6 34 20 2 3 4 4(5 2 + 3 3) 23 4(3 2 + 4 3) 24 2(7 2 + 10 3 + 11 4) 25 + 2(3 2 9 3 7 4 5 5) 26 4(9 3 + 7 4 + 5 5) 6 7 2(159 2 + 33 3 + 35 4 + 37 5) 27

6(2 23 + 2 3 4 + 24 + 25 + 6 27)y2b

94g42

21

2 g22g[prime]2 +

19

4 g[prime]4

452 g22 + 80g23 +856 g[prime]2[parenrightbigg]

2 + 36(4 22 + 27)

y2t 9 2y2by2t 6y2by4t

32g23 + 83g[prime]2 + 3 2[parenrightbigg]

y4t + 30y6t 2(2 23 + 2 3 4 + 24 + 25 + 6 27)y2[bracketrightBigg]

, (B.11)

27

D 3 = 1 162

"34(3g42 + g[prime]4 2g22g[prime]2) 3 3(3g22 + g[prime]2) + 4( 1 + 2)(3 3 + 4) + 4 23

+ 2( 24 + 25) + 4( 26 + 27 + 4 6 7) + 2 3(3y2b + y2 + 3y2t) 12y2by2t[bracketrightBigg]

+ 1

2564

"18 291g62 + 11g42g[prime]2 + 101g22g[prime]4 131g[prime]6

[parenrightbig]

18(111g42 22g22g[prime]2 197g[prime]4) 3 + 2(3g22 + g[prime]2)[12( 1 + 2) 3 + 23 + 24] 4( 21 + 22)(15 3 + 4 4)

4( 1 + 2)(18 23 + 7 24 + 8 3 4 + 9 25) 12( 33 + 34 + g22 3 4) +

+ 52 9g42 2g22g[prime]2 + 3g[prime]4

[parenrightbig]

( 1 + 2)

152 g42 3g22g[prime]2 +52g[prime]4[parenrightbigg]

4 + 4(9g22 + 2g[prime]2)( 1 + 2) 4

4 3 4( 3 + 4 4) 2(9 3 + 22 4) 25 4g[prime]2( 24 25) + 2g[prime]2( 26 + 27)

4(31 1 + 11 2) 26 4(15 3 + 17 4 + 17 5)( 26 + 27) + 4(27g22 + 8g[prime]2) 6 7 8[11( 1 + 2 + 2 3 + 4) + 9 5] 6 7 4(11 1 + 31 2) 27

14(9g42 5g[prime]4 + 18g22g[prime]2) 40g23 +

25 12g[prime]2

3

JHEP12(2014)024

45

4 g22 +

y2b

1 4

n3g42 + 25g[prime]4 + 22g22g[prime]2 + 5(3g22 + 5g[prime]2) 3

oy2

(2 23 + 24 + 4 1(3 3 + 4) + 25 + 4 26 + 8 6 7)

[bracerightbig](3y2b + y2)

n94g42 +212 g22g[prime]2 +194 g[prime]4 [parenleftBig]40g23 + 454 g22 +85 12g[prime]2

3

+ 6(12 2 3 + 2 23 + 4 2 4 + 24 + 25 + 8 6 7 + 4 27)

oy2t

64g23 +

43g[prime]2 15 3

y2by2t + 36y4by2t 92 3[3(y4b + y4t) + y4] + 36y2by4t[bracketrightBigg]

, (B.12)

D 4 = 1 162

"3g22g[prime]2 3 4(3g22 + g[prime]2) + 4( 1 + 2 + 2 3) 4 + 4 24 + 8 25

+ 10( 26 + 27) + 4 6 7 + 2 4

3(y2b + y2t) + y2

[bracerightbig]+ 12y2by2t

[bracketrightBigg]

+ 1

2564

[bracketleftBigg]

73

2 g22g[prime]4 + 2g22g[prime]2

14g42g[prime]2

5( 1 + 2) + 3

[bracerightbig]

18 231g42 102g22g[prime]2 157g[prime]4

4 + 4

2g[prime]2( 1 + 2) 7( 21 + 22 + 23)

4

+ 4(9g22 + g[prime]2) 3 4 40( 1 + 2)(2 3 4 + 24) + 2(9g22 + 4g[prime]2) 24 28 3 24

+ 2(27g22 + 8g[prime]2) 25 48( 1 + 2 + 3) 25 26 4 25 + 8g[prime]2 6 7

+ 2

27g22 + 7g[prime]2 2(18 3 + 17 4 + 20 5) ( 26 + 27) 4(37 1 + 5 2) 26 8[notdef]5( 1 + 2 + 2 3 + 4 4) + 12 5[notdef] 6 7 4(5 1 + 37 2) 27+ n9g22g[prime]2 + 40g23 + 45

4 g22 +

25 12g[prime]2

4 12[2( 1 + 3) 4 + 24 + 2 25

28

+ 5 26 + 6 7]

oy2b +

n21g22g[prime]2 + 40g23 + 45

4 g22 +

85 12g[prime]2

4 12[2( 2 + 3) 4

+ 24 + 2 25 + 6 7 + 5 27]

oy2t +

64g23 + 43g[prime]2 24 3 33 4[parenrightBig] y2by2t

+ n11g22g[prime]2 + 54 3g22 + 5g[prime]2

[parenrightbig]

4 8( 1 + 3) 4 4 24 8 25

20 26 4 6 7[bracerightBig]

y2

9 2 4

3(y4b + y4t) + y4 24y2by2t(y2b + y2t)

[bracerightbig]

[bracketrightBigg]

, (B.13)

D 5 = 1 162

[bracketleftBigg]

3 5(3g22 + g[prime]2) + 4( 1 + 2 + 2 3 + 3 4) 5 + 10( 26 + 27)

+ 4 6 7 + 2 5

3(y2b + y2t) + y2

[bracerightbig][bracketrightBigg]

+ 1

2564

JHEP12(2014)024

[bracketleftBigg]

1 8

231g42 38g22g[prime]2 157g[prime]4[parenrightBig]

5 + 4(9g22 + 4g[prime]2) 3 5

4

ng[prime]2( 1 + 2) + 7( 21 + 22 + 23) o 5 8( 1 + 2)(10 3 + 11 4) 5+ 4[notdef]6(3g22 + g[prime]2) 19 3 8 4[notdef] 4 5 + 6 35 4(37 1 + 5 2) 26 4(5 1 + 37 2) 27 + 2[notdef]27g22 + 10g[prime]2 2(18 3 + 19 4 + 18 5)[notdef]( 26 + 27) 4[notdef]g[prime]2 + 10( 1 + 2 + 2 3) + 22 4 + 37 5[notdef] 6 7+[braceleftBig][parenleftBigg]

40g23 + 45

4 g22 +

25 12g[prime]2

5 12[2( 1 + 3) 5 + 3 4 5 + 5 26 + 6 7][bracerightBig] y2b

[braceleftBig][parenleftBigg]

+ 40g23 + 45

2 g22 +

85 12g[prime]2

5 12[2( 2 + 3) 5 + 3 4 5 + 6 7 + 5 27[bracerightBig] y2t

+

15

4 g22 +

[braceleftBig][parenleftBigg]

4 g[prime]2

25 5 8( 1 + 3) 5 12 4 5 20 26 4 6 7[bracerightBig] y2

1

2 5[notdef]3(y4b + y4t) + y4[notdef] 33 5y2by2t[bracketrightBigg]

, (B.14)

D 6 = 1 162

[bracketleftBigg]

3 6(3g22 + g[prime]2) + 2(12 1 + 3 3 + 4 4) 6 + 2(3 3 + 2 4) 7

+ 10 5 6 + 2 5 7 + 3 6(3y2b + y2t + y2)

[bracketrightBigg]

+ 1

2564

[bracketleftBigg]

18(141g42 58g22g[prime]2 187g[prime]4) 6 + 6(3g22 + g[prime]2)(6 1 + 3) 6

6(53 21 + 22) 6 4(33 1 + 9 2 + 8 3) 3 6 + 2(18g22 + 5g[prime]2) 4 6 2(70 1 + 14 2 + 34 3 + 17 4) 4 6 + 2(27g22 + 10g[prime]2) 5 6 4(37 1 + 5 2 + 18 3 + 19 4 + 9 5) 5 6 111 36 42 37+ 54(9g42 + 2g22g[prime]2 + 3g[prime]4) 7 + 12(3g22 + g[prime]2) 3 7 36( 1 + 2 + 3) 3 7 + 2(9g22 4 + 4g[prime]2) 4 7 2[notdef]14( 1 + 2 + 2 3) 4 + 17 4[notdef] 4 7 2[notdef]g[prime]2 + 10( 1 + 2 + 2 3) + 22 4 + 21 5[notdef] 5 7 3(42 6 + 11 7) 6 7

29

+ n60g23 + 1358 g22 +258 g[prime]2 6(24 1 + 3 3 + 4 4 + 5 5)[bracerightBig] 6y2b

+ n20g23 + 458 g22 +8524g[prime]2 6(3 3 + 4 4 + 5 5)[bracerightBig] 6y2t

n158 (3g22 + 5g[prime]2) 2( 1 + 3 3 + 4 4 + 5 5)[bracerightBig] 6y2

12(3 3 + 2 4 + 5) 7y2t

14(27y4t + 33y4b + 11y4) 6 21 6y2by2t[bracketrightBigg]

3 7(3g22 + g[prime]2) + 2(12 2 + 3 3 + 4 4) 7 + 2(3 3 + 2 4) 6

+ 10 5 7 + 2 5 6 + 7(3y2b + 9y2t + y2)

[bracketrightBigg]

"54(9g42 + 2g22g[prime]2 + 3g[prime]4) 6 + 12(3g22 + g[prime]2) 3 6

36( 1 + 2 + 3) 3 + 2(9g22 + 4g[prime]2) 4 6 28( 1 + 2 + 2 3) 4 6 34 24 6 2g[prime]2 5 6 4[notdef]5( 1 + 2 + 2 3) + 11 4[notdef] 5 6 42( 25 + 26) 6

18(141g42 58g22g[prime]2 187g[prime]4) 7 + 6 21 7 + 36(3g22 + g[prime]2) 2 7 318 22 7 + 6(3g22 + g[prime]2) 3 7 12(3 1 + 11 2) 3 7 32 23 7 + 2(18g22 + 5g[prime]2) 4 7 4(7 1 + 35 2 + 17 3) 4 7 34 24 7 + 2(27g22 + 10g[prime]2) 5 7 4(5 1 + 37 2 + 18 3 + 19 4) 5 7 36 25 7 33 26 7 126 6 27 111 37 12(3 3 + 2 4 + 5) 6y2b+ n20g23 + 458 g22 +2524g[prime]2 6(3 3 + 4 4 + 5 5)[bracerightBig] 7y2b

+ n60g23 + 1358 g22 +858 g[prime]2 6(24 2 + 3 3 + 4 4 + 5 5)[bracerightBig] 7y2t

4(3 3 + 2 4 + 5) 6y2 + [braceleftBig]

58(3g22 + 5g[prime]2) 6 3 8 4 10 5[bracerightBig]

. (B.16)

Finally, the two-loop RGE for the soft mass parameter is given by

D(m212) = 1 162

[bracketleftBigg]

116(243g42 30g22g[prime]2 153g[prime]4)m212 + 3

m212 + 4(3g22 + g[prime]2)( 3 + 2 4 + 3 5)m212

30

+

, (B.15)

D 7 = 1 162

[bracketleftBigg]

JHEP12(2014)024

+ 1

2564

7y2

14(33y4t + 27y4b + 9y4) 7 21 7y2by2t[bracketrightBigg]

32(3g22 + g[prime]2)m212 + 2( 3 + 2 4 + 3 5)m212

+ 2(3y2b + 3y2t + y2)m212 + 12( 6[notdef]21 + 7[notdef]22)

[bracketrightBigg]

+ 1

2564

+ 4( 26 + 27)

[bracketleftBigg]

2 21 + 22) + 25

12( 1 + 2) 345m212 6( 3 4 + 2 3 5 + 2 4 5 + 6 6 7)m212 + n20g23 + 458 g22 +2524g[prime]2 6( 3 + 2 4 + 3 5)[bracerightBig] y2bm212

+ n20g23 + 458 g22 +8524g[prime]2 6( 3 + 2 4 + 3 5)[bracerightBig] y2tm212

+

n58(3g22 + 5g[prime]2) 2( 3 + 2 4 + 3 5)[bracerightBig] y2m212

+ 24(3g22 + g[prime]2)( 6[notdef]21 + 7[notdef]22) 72( 1 6[notdef]21 + 2 7[notdef]22) 12 345[notdef](2 6 + 7)[notdef]21 + ( 6 + 2 7)[notdef]22[notdef]

24[notdef](3y2b + y2) 6[notdef]21 + 3y2t 7[notdef]22[notdef]

94(3y4b + 3y4t + y4)m212[bracketrightBigg]

Note that the mass parameters [notdef]21,2 are removed by the tadpole conditions:

@V

@v1 = 0 = [notdef]21v1 m212v2 +

@V

@v2 = 0 = [notdef]22v2 m212v1 +

JHEP12(2014)024

(B.17)

1

2

2 1v31 + 345v1v22 + 3 6v21v2 + 7v32[bracketrightbig]

, (B.18)

2 2v32 + 345v2v21 + 6v31 + 3 7v2v21[bracketrightbig]

, (B.19)

and hence, it is not necessary to write down their RGEs explicitly.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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