A tale of twin Higgs: natural twin two Higgs doublet models

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

In original twin Higgs model, vacuum misalignment between electroweak and new physics scales is realized by adding explicit ...... breaking term. Introducing additional twin Higgs could accommodate spontaneous ...... breaking, which explains origin of this misalignment. We introduce a class of twin two Higgs doublet models with most general scalar potential, and discuss general conditions which trigger electroweak and ...... symmetry breaking. Various scenarios on realising the vacuum misalignment are systematically discussed in a natural composite two Higgs double model framework: explicit ...... breaking, radiative ...... breaking, tadpole-induced ...... breaking, and quartic-induced ...... breaking. We investigate the Higgs mass spectra and Higgs phenomenology in these scenarios.

Published for SISSA by Springer

Received: August 30, 2016 Accepted: December 20, 2016 Published: December 28, 2016

JHEP12(2016)143

A tale of twin Higgs: natural twin two Higgs doublet models

Jiang-Hao YuAmherst Center for Fundamental Interactions, Department of Physics, University of Massachusetts Amherst,710 North Pleasant St., Amherst, MA 01002, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected]

Abstract: In original twin Higgs model, vacuum misalignment between electroweak and new physics scales is realized by adding explicit Z2 breaking term. Introducing additional twin Higgs could accommodate spontaneous Z2 breaking, which explains origin of this misalignment. We introduce a class of twin two Higgs doublet models with most general scalar potential, and discuss general conditions which trigger electroweak and Z2 symmetry breaking. Various scenarios on realising the vacuum misalignment are systematically discussed in a natural composite two Higgs double model framework: explicit Z2 breaking, radiative Z2 breaking, tadpole-induced Z2 breaking, and quartic-induced Z2 breaking. We investigate the Higgs mass spectra and Higgs phenomenology in these scenarios.

Keywords: Beyond Standard Model, Higgs Physics, Technicolor and Composite Models

ArXiv ePrint: 1608.05713

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2016)143

Web End =10.1007/JHEP12(2016)143

Contents

1 Introduction 1

2 Original twin Higgs and vacuum misalignment 4

3 Twin two Higgs doublet models 73.1 General twin two Higgs potential 73.2 Fermion assignments 103.3 Radiative corrections 12

4 Symmetry breaking and vacuum misalignment 144.1 U(4)=U(3) breaking pattern 144.2 [U(4) [notdef] U(4)] = [U(3) [notdef] U(3)] breaking pattern 15

5 Spontaneous Z2 breaking in composite 2HDM 215.1 Radiative Z2 breaking 225.2 Tadpole induced Z2 breaking 235.3 Quartic induced Z2 breaking 24

6 Higgs phenomenology 246.1 Higgs mass spectra 246.2 Collider constraints 26

7 Conclusions 30

A Details in the U(4)/U(3) breaking pattern 31

B Details in the [U(4) U(4)] / [U(3) U(3)] breaking pattern 33

1 Introduction

The discovery of a 125 GeV Higgs boson at the LHC [1, 2] is a great triumph of the Standard Model (SM) of particle physics. Although it con rms the Higgs mechanism, it sharpens existing naturalness problem. Naturalness tells us that the weak scale should be insensitive to quantum e ects from physics at very higher scale. However, in SM, the large, quadratically divergent radiative corrections to the Higgs mass parameter destabilize the electroweak scale. From theoretical point of view, the SM should be well-behaved up to Planck scale. The existing hierarchy between the Planck and weak scales requires that the quantum corrections to the Higgs mass parameter should cancel against the Higgs bare mass to obtain the observed 125 GeV Higgs boson mass. The large cancellation indicates

{ 1 {

JHEP12(2016)143

existence of ne-tuning between the tree-level Higgs mass parameter and loop-level Higgs mass corrections. This is the well-known hierarchy problem [3].

The dynamical solution to the naturalness problem is to introduce a new symmetry which protects the Higgs mass against large radiative corrections. Under this direction are weak scale supersymmetry [4], and composite Higgs [5{7], etc. These new physics (NP) models introduce symmetry partners of the SM elds that cancel the quadratically divergent corrections to the Higgs boson mass. Because the dominant quantum correction to the Higgs mass involves in the SM top quark in the self-energy loop, the top quark partner is typically most relevant new particle to the quadratic cancellation. The new symmetry not only relates the top partner with the SM top quark, but also relates the Higgs coupling of the top partner to the one of the top quark. This enforces quadratic cancellation between the top quark and top partner contributions. Since the top partners typically carry SM color charge, the search limits of these top partners at the LHC have reached 700 800 GeV. This already leads to around 10% level of tuning between the weak

scale and NP scale. This is known as the little hierarchy problem.

One way to avoid the little hierarchy problem is the neutral naturalness [8{11], that symmetry partners are not charged under the SM gauge groups. This lowers the NP cuto scale, and thus softens the little hierarchy problem. The twin Higgs model [8, 9] [see also refs. [12{17] and [23{25]] introduces the mirror copy of the SM, the twin sector, which is neutral under the SM gauge group. The Higgs sector respects the approximate global U(4) symmetry, which is broken spontaneously to U(3) at NP scale f. The U(4) symmetry is broken at the loop level via radiative corrections from the gauge and Yukawa interactions. Thus the Higgs boson is the pseudo Goldstone Boson (PGB) of the symmetry breaking. Imposing a discrete Z2 symmetry between SM and twin sectors ensures that radiative corrections to the Higgs mass squared are still U(4) symmetric. Thus there is no quadratically divergent radiative corrections to the Higgs mass terms. At the same time, the Z2 symmetry needs to be broken at electroweak scale. Otherwise, the Z2 symmetry induces symmetric VEVs at NP scale. It is necessary to realize vacuum misalignment v f (and thus some level of little hierarchy) to separate the electroweak and NP scales.

This implies a moderate amount of tuning (approximately 2v2

f2 ).

If the Higgs boson is the PGB, the Higgs eld should respect the shift symmetry. The shift symmetry is approximately broken by radiative corrections. Considering radiative corrections only, the typical Higgs potential [18] could be parametrized as

V (h) [similarequal] af4 sin2

Here a and b denote radiative corrections, with the form

a [similarequal] b [similarequal]

JHEP12(2016)143

hf + bf4 sin4

hf : (1.1)

m2 f2 ; (1.2)

where g denotes the typical SM couplings, such as top Yukawa coupling, and m represents

the top partner mass. If there is no other contribution than the radiative corrections, the Higgs VEV can be obtained as

hh[angbracketright] =

pa=bf [similarequal] f: (1.3)

{ 2 {

g2 (4)2 log

To realize the vacuum misalignment, additional contributions need to be added to b or subtracted to a and have a=b [similarequal] v2=f2. In the littlest Higgs model [7], additional hard

quartic terms are added to b by hand to enhance the b. Instead, one could introduce soft term to a to reduce a. In the original twin Higgs model, the Z2 symmetry is broken explicitly by introducing soft or hard Z2 breaking terms in the scalar potential. The soft mass term is added only to visible or twin sector to reduce a. To soften the tuning between v and f, the Higgs sector is extended to incorporate two twin Higgses. Refs. [19, 20] introduce two twin Higgses, and several choices of the soft mass terms are introduced to breaking the Z2 symmetry and reduces level of ne tuning. In the supersymmetric realization of the twin

Higgs model [21, 22], two twin Higgses are also naturally introduced. In these literatures, the soft Z2 symmetry breaking term is introduced by hand, and its origin is unknown.

Actually two twin Higgs setup provides more variants of Z2 symmetry breaking.

The spontaneous Z2 breaking mechanism provides a complete description of the electroweak symmetry breaking and vacuum misalignment. The two twin Higgses are necessary to obtain such spontaneous breaking mechanism, without introducing the explicit Z2 breaking term. Refs. [26, 27] discussed the tadpole induced spontaneous Z2 breaking by introducing the bilinear term between two twin Higgses. Without bilinear term, the VEVs of the rst Higgs preserve Z2 while the other breaks it spontaneously. The bilinear Higgs mass term could transmit the Z2 breaking from the broken one to the unbroken one. It serves as the e ective tadpole induced symmetry breaking and induces the vacuum mis-alignment naturally. Ref. [28] realized that the spontaneous Z2 breaking could be realized even without tree-level bilinear term, the \radiative Z2 breaking". In this scenario, both the symmetry breaking and Z2 breaking are obtained by opposite but comparable radiative corrections from the gauge and Yukawa arrangements. It seems that it is very hard to realize such radiative Z2 breaking, because typically the gauge corrections is much smaller than the Yukawa corrections, and thus the cancellation in the Higgs mass squared term is not adequate. But the gauge corrections could be enhanced by adjusting the VEVs of the two twin Higgses to be hierarchical. Through this way, the purely radiative corrections could induce spontaneous Z2 breaking.

In this work, we consider the general conditions which trigger the electroweak symmetry and the Z2 breaking. Both the tadpole-induced and radiative Z2 breaking scenarios could be deduced from the general conditions. We nd that there is another novel spontaneous Z2 breaking mechanism. Instead of introducing the bilinear term in twin two Higgs potential, the quartic terms 4,5 could play the role of breaking Z2 symmetry spontaneously.

This is the \quartic induced Z2 breaking". Similar to radiative symmetry breaking, the tree-level quartic terms 4,5 contribute to cancellation of the Higgs mass squared term. At the same time, similar to the tadpole-induced scenario, turning on 4,5 gradually transits the VEV of one Higgs to another one of another Higgs. Thus it provides another natural way to realize vacuum misalignment.

To systematically classify various Z2 breaking scenarios, we investigate the most general scalar potential in the two twin Higgs doublet framework. Integrating out the twin particles, the visible Higgs sector contains the 2HDM scalar potential. But this 2HDM, (denoted as twin 2HDM) is di erent from the typical 2HDM, denoted as elementary 2HDM.

{ 3 {

JHEP12(2016)143

Depending on the breaking pattern, the scalars in twin 2HDM could be partially Gold-stone Bosons or completely Goldstone bosons. Through the twin 2HDM framework, physics behind the spontaneous Z2 breaking scenarios could be explained. The above radiative, tadpole induced, and quartic induced symmetry breaking mechanisms are also classi ed and considered in a uni ed framework with composite two twin Higgses. The collider phenomenology of the twin two Higgs models is quite similar to the one of elementary 2HDM, except that twin 2HDM also involves in the twin hadron phenomenology. Only when we identify the signatures of the twin hadrons from the twin Higgs decays, we will be able to distinguish the twin 2HDM from the elementary 2HDM.

The paper is organized as follows. In section 2 we brie y review the original twin Higgs and the vacuum misalignment in this model. In section 3 we introduce the most general scalar potential and its radiative corrections in the twin two Higgs model. Then we investigate the conditions for symmetry breaking and vacuum misalignment in section 4. Subsequently in section 5 we classify various Z2 symmetry breaking scenarios in a natural composite two Higgs doublet framework. Section 6 discuss Higgs phenomenology in each scenario. Finally we conclude this paper. In appendix A and B, we list the calculation details of the two twin Higgs models.

2 Original twin Higgs and vacuum misalignment

We rst brie y review the twin Higgs model [8, 9, 13{16] and how the vacuum misalignment is realized in this model. The original twin Higgs model consists of a mirror copy of the SM content, called the twin sector. We use the labels A and B to denote the SM and twin sector respectively. The twin sector is related to the SM sector by a Z2 exchange symmetry: A $ B. The Higgs sector consists of the SM Higgs doublet HA and the twin

Higgs doublet HB. Due to the Z2 symmetry, the Higgs potential preserves an approximate global symmetry U(4):

Vtree = 2(H2A + H2B) + (H2A + H2B)2 = 2H2 + H2; (2.1)

with the U(4) invariant eld H

JHEP12(2016)143

HA HB

!. If the 2 is positive, the global U(4) symme

try is spontaneously broken down to U(3) and there are seven Goldstone Bosons modes. Assuming the VEV [angbracketleft]H[angbracketright] = f lies along HB, three Goldstone bosons are eaten by the twin

gauge bosons, and the HA remains massless. Assuming the radial model is heavy, the eld

{ 4 {

H can be parametrized1 nonlinearly as

H

HA HB

!

= exp

if

0

B

B

B

@

0 0 0 f

1

C

C

C

A

; =

0

B

B

B

@

0 0 0 h1 0 0 0 h2 0 0 0 h3 h 1 h 2 h 3 h0

1

C

C

C

A

; (2.3)

with hi = Rehi+iImhi

p2 to have correct eld normalization. Expanding out the exponential and taking the unitary gauge we obtain the explicit form

H =

0

B

B

@

f ih

phh sin

phhf

0

1

C

C

A

1

C

A

ih

0 f

1

2f hh

[similarequal]

0

B

@

; (2.4)

f cos

phhf

where the eld h denotes the SM Higgs doublet h = h+

h0

!.

The global symmetry U(4) is explicitly broken once the SM and its mirror gauge group SMA [notdef] SMB are gauged, and the Yukawa interactions are introduced. Both the gauge and

Yukawa interactions give rise to radiative corrections to the quadratic part of the scalar potential. The leading correction to the potential induced by gauging the SMA [notdef] SMB is

V

9 2 642

JHEP12(2016)143

g2AHAHA + g2BHBHB

HAHA + HBHB ; (2.5)

where gA and gB are the gauge couplings of the SMA [notdef] SMB gauge group. Here if the Z2

symmetry is imposed, the leading corrections to the quadratic part of the scalar potential accidentally respect the original U(4) symmetry. Thus corrections from the gauge sector cannot contribute to the masses of the Goldstone bosons. Similarly, consider the Yukawa sector by focusing on the top Yukawa couplings, which takes the form

L yAHAqAtA + yBHBqBtB + h:c:; (2.6)

where qA,B and tA,B are the left-handed SU(2)A,B doublet quark and right-handed SU(2)A,B singlet top quark in the SM and twin sectors. The leading corrections take the form

V

3 2 82

!

Z2: gA=gB

9g2 2

642

y2AHAHA + y2BHBHB

!

HAHA + HBHB : (2.7)

1Di erent notations on eld de nition and VEVs are used in literatures [8, 9]. Here we de ne the eld and take notation on eld VEVs [angbracketleft]HB[angbracketright] = f and [angbracketleft]HA[angbracketright] = v = 174 GeV. Using the same eld de nition,

another notation on eld VEVs [angbracketleft]HB[angbracketright] = f and [angbracketleft]HA[angbracketright] = v/p2 = 174 GeV are also used in literature [15].

Finally some literature [29] uses the following eld de nition

H exp

if

Z2: yA=yB

3y2 2

82

0

B

B

B

@

0 0 0

f p2

1

C

C

C

A

, =

0

B

B

B

@

0 0 0 h1 0 0 0 h2 0 0 0 h3 h 1 h 2 h 3 h0

1

C

C

C

A

, (2.2)

Note that the normalization of the hi is di erent, with Rehi + iImhi to have correct eld normalization. Under this notation, the VEVs are [angbracketleft]HB[angbracketright] = f/p2 and [angbracketleft]HA[angbracketright] = v/p2 = 174 GeV.

{ 5 {

Similarly the Z2 symmetry ensures that the leading corrections respect the U(4) symmetry. Therefore, there is no quadratically divergent contribution to the Higgs boson mass at one loop order.

Although the Z2 symmetry ensures the quadratically divergent corrections respect the U(4) symmetry, the gauge and Yukawa interactions still break the U(4) symmetry via the logarithmically divergent corrections. The leading logarithmically divergent corrections take the form

V

3y4 162

[notdef]HA[notdef]4 log 2y2[notdef]HA[notdef]2+ [notdef]HB[notdef]4 log 2 y2[notdef]HB[notdef]2

: (2.8)

The sub-leading corrections proportional to g4 take the similar form with opposite sign. However, since both the squared mass and quartic coupling come from the same loop-suppressed corrections, the VEV is obtained to be at the scale f as mentioned in introduction. This fact can be seen if we write the scalar potential including the radiative corrections, to good approximation, as

Vtot = 2([notdef]HA[notdef]2 + [notdef]HB[notdef]2) + ([notdef]HA[notdef]2 + [notdef]HB[notdef]2)2 + ([notdef]HA[notdef]4 + [notdef]HB[notdef]4); (2.9)

Here denotes the small U(4)-violating but Z2-preserving loop corrections on the quartic potential, with . According to eq. (2.8), the Yukawa interactions lead to [similarequal]

3y4 162 log

2y2f2 , while the gauge interactions give [similarequal]

JHEP12(2016)143

2g2f2 . It is interesting to note that the symmetry breaking structure is controlled by the sign of the :

if < 0 (such as, only including loop corrections from the gauge interactions), the

potential induces

hHA[angbracketright] = 0; [angbracketleft]HB[angbracketright] = f; (2.10) which breaks the Z2 symmetry spontaneously.

if > 0 (such as, adding loop corrections from the Yukawa interactions), the potential

induces

hHA[angbracketright] =

9g4 2562 log

fp2; [angbracketleft]HB[angbracketright] =

fp2; (2.11)

which preserves the Z2 symmetry.

The original twin Higgs belongs to the second case: the vacuum is equally aligned with the two sectors.

In order to realize the symmetry breaking at the electroweak scale, the VEV must be misaligned to be asymmetric with [angbracketleft]HA[angbracketright] = v f. This requires explicit Z2 symmetry

breaking by adding

either a soft Z2-breaking mass term

Vsoft m2A[notdef]HA[notdef]2; with m2A O

f2 162

2; (2.12)

or a hard Z2-breaking quartic termVhard A[notdef]HA[notdef]4; with A O(0:1) : (2.13)

{ 6 {

The approximate U(4) global symmetry is still valid since 2 m2A and A. The

Z2-breaking term pushes the VEVs: [angbracketleft]HA[angbracketright] ! v and [angbracketleft]HB[angbracketright] ! f, which gives the vacuum

misalignment. To obtain the correct VEV v, one needs to tune the Z2-breaking parameter.

In the case of the soft mass m2A, let us rewrite the scalar potential in terms of the Higgs doublet h. Taking expansion on the Higgs doublets

|HA[notdef]2 f2 sin2

phh f

! [similarequal]

hh; [notdef]HB[notdef]2 f2 cos2

phh f

! [similarequal]

f2 hh: (2.14)

we obtain the dominant Higgs potential

V (h)[similarequal]

m2A3y4f282 log 2 y2f2

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hh+ 3y4 162

log 2y2f2 +log 2 y2hh

(hh)2+O (hh)3 f2

:

(2.15)

If the m2A is smaller than f2 term, the mass term could be negative, which induces electroweak symmetry breaking, and the Higgs boson h obtains its mass. We minimize the potential and obtain the electroweak VEV from the tadpole condition

v2f2 = 1

m2A f2 ; (2.16)

3y4162 . To realize electroweak VEV, m2A should be comparable to the f2 term.

This implies a moderate tuning between f2 and m2A. We estimate the tuning using the following approximation:

m =

with [similarequal]

1

2 m2 m2h

2v2f2 ; (2.17)

where mh = 125 GeV. For a TeV scale f, this corresponds to around 15% tuning.

3 Twin two Higgs doublet models

3.1 General twin two Higgs potential

In this work, the visible sector is extended to the two Higgs doublet model, which is denoted as the 2HDM sector. The twin sector is exactly the mirror copy of the 2HDM sector and it is related to the 2HDM sector by the twin mirror parity Z2. It is convenient to label the 2HDM sector and its twin sector as A and B respectively. In the 2HDM, there are two Higgs doublets H1A and H2A. In the twin sector, two twin Higgs doublets H1B and

H2B are introduced and they are mapped into the 2HDM Higgses via the twin parity: H1B Z2

! H1A, H2B

Z2

[similarequal]

m2h 2 f2

! H2A. Similar to the original twin Higgs model, it is convenient to

de ne the U(4) invariant elds

H1

H1A

H1B

!; H2 H2A H2B

!: (3.1)

which respect the twin parity Z2.

{ 7 {

The scalar pontential of the elds H1 and H2 is similar to the two Higgs doublet model. In the generalized two Higgs doublet framework, we write the general twin Higgs potential

V (H1; H2) = 21[notdef]H1[notdef]2 22[notdef]H2[notdef]2 + 1([notdef]H1[notdef]2)2 + 2([notdef]H2[notdef]2)2 + 3[notdef]H1[notdef]2[notdef]H2[notdef]2

+m212

hH1H2 + h:c:i+ 4[notdef]H1H2[notdef]2 + 5 2

+ h( 6[notdef]H1[notdef]2 + 7[notdef]H2[notdef]2)H1H2 + h:c:i: (3.2)

Here all the parameters are taken to be real for simplicity. Note that refs. [19, 20] only contains 1,2,4 terms in the potential. The symmetries of the potential are recognised as follows:

First, of course, all the terms in the potential preserve the twin parity Z2 symmetry:

A $ B.

The rst line of the potential eq. (3.2) has the global U(4)1 [notdef] U(4)2 symmetry.

While the second and the third lines of the eq. (3.2) explicitly break the global

symmetry U(4)1 [notdef] U(4)2 ! U(4)V . If 5 is zero but 4 is non-zero, an additional

global U(1) symmetry exists.

To avoid tree-level Higgs mediated avor changing neutral current, similar to 2HDM, a softly-broken discrete symmetry Z[prime]2 : H1 ! H1 H2 ! H2 is imposed on the quartic

terms, which implies that 6 = 7 = 0, whereas m212 [negationslash]= 0 is still allowed.

The two Higgs sector is weakly gauged under the mirror SM gauge group. The gauge symmetry is applied2 under

SU(2)A [notdef] U(1)A 0

0 SU(2)B [notdef] U(1)B

The covariant kinetic terms of the Higgs elds are written as

Lkin = DH1DH1 + DH2DH2; (3.4)

where the covariant derivative is DHi = @Hi + igW Hi + ig[prime]BHi, with

W

0 0 0 f1

2To avoid massless twin photon, sometimes U(1)B gauge symmetry is not applied. Here we take the gauged U(1)B symmetry.

{ 8 {

h(H1H2)2 + h:c:

i

JHEP12(2016)143

!

U(4): (3.3)

W aAa 00 W aBa

!; B 12 BA 00 12 BB

!: (3.5)

The global symmetry is weakly broken by the loop e ects from the gauge interactions.If the mass terms 21,2 are positive, the elds H1 and H2 vacua take the form

hH1[angbracketright]

0

B

B

B

@

1

C

C

C

A

0

B

B

B

@

1

C

C

C

A

; [angbracketleft]H2[angbracketright]

0 0 0 f2

: (3.6)

Similar to the 2HDM model, let us de ne the mixing angle and scale f

tan

f2f1 ; f = q

f21 + f22: (3.7)

Depending on the global symmetry before the symmetry breaking, there could be seven or fourteen Goldstone bosons. In the following, we discuss the nonlinear parametrization of the elds Hi in U(4)=U(3) and [U(4) [notdef] U(4)]=[U(3) [notdef] U(3)] breaking patterns.

(1) U(4)/U(3) symmetry breaking. The most general scalar potential in eq. (3.2) exhibits the global U(4) symmetry. The VEVs will break the symmetries of the Lagrangian spontaneously:

global symmetry: U(4) ! U(3);

gauge symmetry: SU(2)A [notdef] U(1)A [notdef] SU(2)B [notdef] U(1)B ! SU(2)A [notdef] U(1)A: (3.8)

The SUSY twin Higgs model [21, 22] belongs to this breaking pattern.

Similar to the original twin Higgs, there are seven Goldstone bosons. To isolate the Goldstone bosons in the elds, similar to 2HDM, it is convenient to work in the Higgs basis by rotating the elds

H = H1 cos + H2 sin ; H[prime] = H1 sin + H2 cos ; (3.9)

After rotation, only the eld H obtain VEV. Similar to the original twin Higgs, the eld H can be parametrized non-linearly. After rotation, the two elds becomes

H = exp

2

64

JHEP12(2016)143

i f

0

B

@

02[notdef]2 01[notdef]2 h 02[notdef]1 0 C h C N

1

C

A

3

75

0

B

@

01[notdef]2 0f

1

C

A

; H[prime] =

0

B

B

B

@

H+ H0 + iA0

H[prime]+

H[prime]0 + iA[prime]0

1

C

C

C

A

: (3.10)

where the eld h denotes the SM Higgs doublet h = h+

h0

!, and C[notdef] and N are Goldstone

bosons in the B sector, which are absorbed by the twin gauge bosons. Therefore, similar to the original twin Higgs, taking the expansion, the eld H takes the form

H =

0

B

B

@

f ih

phh sin

phhf

0

1

C

C

A

[similarequal]

0

B

@

1

C

A

ih

0 f

1

2f hh

; (3.11)

Here the eld H plays the role of the twin Higgs as the original twin Higgs model. Another eld H[prime] does not obtain VEV, and thus it is just another scalar quadruplet in this model.

(2) [U(4)U(4)]/[U(3)U(3)] symmetry breaking. Now let us consider the special

scalar potential with larger global symmetry. If only the rst line exists, The potential exhibits the exact global U(4) [notdef] U(4) symmetry. Here the soft mass term m212 and the

{ 9 {

f cos

phhf

quartic 4 and 5 terms are taken to be small, and thus the U(4)[notdef]U(4) symmetry becomes

approximate. The VEVs will break the symmetries of the Lagrangian spontaneously:

global symmetry: U(4) [notdef] U(4) ! U(3) [notdef] U(3);

gauge symmetry: SU(2)A [notdef] U(1)A [notdef] SU(2)B [notdef] U(1)B ! SU(2)A [notdef] U(1)A: (3.12) In this case, the approximate global symmetry breaking is U(4)1 [notdef] U(4)2 ! U(3)1 [notdef]

U(3)2. Let us parametrize the elds H1 and H2 nonlinearly in terms of the nonlinear sigma elds. Assuming the radial models 1 and 2 in H1 and H2 are heavy, the elds H1 and H2 are parametrized nonlinearly as

H1 =exp

2

64

JHEP12(2016)143

i f1

0

B

@

02[notdef]2 01[notdef]2 h1 02[notdef]1 0 C1 h 1 C 1 N1

1

C

A

3

75

0

B

@

01[notdef]2 0 f1

1

C

A

; H2 =exp

2

64

i f2

0

B

@

02[notdef]2 01[notdef]2 h2 02[notdef]1 0 C2 h 2 C 2 N2

1

C

A

3

75

0

B

@

01[notdef]2 0 f2

1

C

A

:

(3.13)

Expanding out the exponential we obtain the explicit form

Hi =

0

B

B

B

@

fi ihiHi sin

Hifi

fi iCiHi sin

Hifi

Hifi

1

C

C

C

A

[similarequal]

0

B

@

ihi iCi fi

1

2f hihi + iNi

1

C

A

; (3.14)

fi cos

Hifi + f iNiHi sin

where Hi =

qhihi + C iCi + N2i. Here the doublets hi = h+i

h0i

!

are Goldstone bosons in

the sector A, and Ci; Ni are Goldstone bosons in the sector B. When U(4) [notdef] U(4)-breaking

terms exist, one combination of the h[notdef]i, and one combination of the C[notdef]i becomes pseudo Goldstone bosons.

3.2 Fermion assignments

In the twin Higgs model, the SM fermions are extended to include mirror fermions:

qA(3; 2; 1=6; 1; 1; 0) Z2

! qB(1; 1; 0; 3; 2; 1=6); uA(3; 1; 2=3; 1; 1; 0) Z2

! uB(1; 1; 0; 3; 1; 2=3); dA(3; 1; 1=3; 1; 1; 0)

! dB(1; 1; 0; 3; 1; 1=3); (3.15) where the quantum number assignments are (SU(3)A; SU(2)A; U(1)A; SU(3)B; SU(2)B;

U(1)B). If there are two twin Higgses, the general Yukawa interactions could be written as

LYuk = y1 (qAH1AtA + qBH1BtB) + (1 $ 2) + h:c: (3.16) Similar to 2HDM, it is possible to induce Higgs mediated FCNC processes in visible sector.

To avoid such problem, the discrete Z[prime]2 symmetry H1 ! H1; H2 ! H2 can also be applied

to the fermion contents, which are identi ed as the Type-I, II, X, Y 2HDMs [30]. Here for simplicity, we adopt the type-I Yukawa structure: all fermions only couple with H1.

Similar to 2HDM, it is straightforward to extend type-I Yukawa structure to other Yukawa structures.

{ 10 {

Z2

(1) Fermion assignment: mirror fermions. In this setup, similar to the original twin Higgs model, the 2HDM top Yukawa interactions are

LYuk = y (qAH1AtA + qBH1BtB) + h:c: (3.17)

In the above Lagrangian the U(4) symmetry is explicitly broken by the Yukawa terms. Similar to the SM fermions, the mirror fermions are treated as the chiral fermions. The fermion masses are

m2tA = y2H21A; m2tB = y2H21B [similarequal] y2f2 y2H21A; (3.18)

where the relation m2tA + m2tB [similarequal] y2f2 indicates the quadratically divergent cancellation.

Of course, it is also possible to treat the mirror fermions vector-like [31] with

Lmass = M(qBqB + tBtB) + h:c: (3.19)

Here additional fermion degree of freedoms are introduced to make the mirror fermions vector-like. This will lift the mirror fermion masses but not a ect the quadratically divergent cancellation in the Higgs potential. Here we only take chiral fermion case.

(2) Fermion assignment: SU(6)SU(4) fermions. To keep the U(4) invariant form,

the following fermions [8, 9] are introduced:

Q = qA (3; 2; 1=6; 1; 1; 0) + ~qA (3; 1; 2=3; 1; 2; 1=2)

+qB (1; 1; 0; 3; 2; 1=6) + ~qB(1; 2; 1=2; 3; 1; 2=3);

U = tA (3; 1; 2=3; 1; 1; 0) + tB (1; 1; 0; 3; 1; 2=3);

D = bA (3; 1; 1=3; 1; 1; 0) + bB (1; 1; 0; 3; 1; 1=3): (3.20)

In the SU(6) [notdef] SU(4) invariant form, the fermions are assembled as

Q = qA ~qA ~qB qB

JHEP12(2016)143

!; U =tA tB

!: (3.21)

Similar for the leptons. The U(4) [notdef] U(4) invariant top Yukawa interactions are written as

LYuk = yH1QU + h:c: =

H1A H1B

qA ~qA

~qB qB

!

tA tB

!

+ h:c: (3.22)

To lift the non-SM fermions masses, additional vector-like fermion mass terms are introduced as

Lmass =

~M(~qA~qA + ~qB ~qB) + h:c: (3.23)

The vector-like mass terms exhibit U(4) [notdef] U(4) breaking e ects in the Yukawa sector.

Expanding the Yukawa interactions, we obtain

LYuk = y (qAH1AtA + qBH1BtB + H1A~qBtB + H1B ~qAtA) + h:c: (3.24)

Thus the mass matrices are

Lmass =

qA ~qA

H1A 0

H1B ~M

!

tA

~qA

!

+ qB ~qB

H1B 0

H1A ~M

!

tB

~qB

!

+ h:c: (3.25)

{ 11 {

3.3 Radiative corrections

The gauge and Yukawa interactions break the global symmetry explicitly, which generate the scalar potential for the pseudo-Goldstone bosons. The one-loop Coleman-Weinberg potential in Landau gauge is

VCW(H1; H2) = 1642 STr

m4(H1,2) log m2(H1,2) 2 32 ; (3.26)

where the super-trace STr is taken among all the dynamical elds that have the Higgs dependent masses. The Higgs dependent gauge boson masses are

m2WA =

JHEP12(2016)143

g2

g2

2 [notdef]H1A[notdef]2 + [notdef]H2A[notdef]2

; (3.27)

for the SU(2) [notdef]SU(2) gauge bosons, and similarly for the U(1) [notdef]U(1) gauge boson masses.

The Higgs dependent top sector masses in the fermion assignment I are

m2tA = y2[notdef]H1A[notdef]2; m2tB = y2[notdef]H1B[notdef]2: (3.28)

The eld dependent top sector masses in the fermion assignment II are

m2tA,t[prime]A =

; m2WB =

2 [notdef]H1B[notdef]2 + [notdef]H2B[notdef]2

y2[notdef]H1A[notdef]2+y2[notdef]H1B[notdef]2+M22

1

2

p(y2[notdef]H1A[notdef]2+y2[notdef]H1B[notdef]2+M2)24y2[notdef]H1A[notdef]2M2;

m2tB,t[prime]B =

y2[notdef]H1A[notdef]2+y2[notdef]H1B[notdef]2+M22

1

2

p(y2[notdef]H1A[notdef]2+y2[notdef]H1B[notdef]2+M2)24y2[notdef]H1B[notdef]2M2:(3.29)

Let us examine how the quadratic divergence cancels at the one-loop again due to the Z2 symmetry. The leading corrections to the quadratic part of the scalar potential are

V

9g2 2

642

H1AH1A + H1BH1B + H2AH2A + H2BH2B ; (3.30)

from the gauge sector, and

H1AH1A + H1BH1B ; (3.31)

due to the Yukawa interactions in the top sector. Note that both quadratic contributions respect the original U(4) symmetry, and thus there is no quadratically divergent contribution to the Higgs boson masses. Therefore the leading corrections are the quartic terms in the e ective potential. The radiative corrections to the gauge sector is

V

V

3y2 2

82

3g4 162

[notdef]H

1A

[notdef]2 + [notdef]H2A[notdef]2

2 log 2 g2([notdef]H1A[notdef]2 + [notdef]H2A[notdef]2)

+ [notdef]H1B[notdef]2 + [notdef]H2B[notdef]2

2 log 2 g2([notdef]H1B[notdef]2 + [notdef]H2B[notdef]2)

: (3.32)

Similarly for the U(1) sector. The radiative corrections to the top sector in the mirror fermion model are

V

3y4 162

|H1A[notdef]2

2 log 2 g2[notdef]H1A[notdef]2+

[notdef]H1B[notdef]2

2 log 2 g2[notdef]H1B[notdef]2

: (3.33)

{ 12 {

In most general case, the dominant contributions of the radiative corrections could be parametrized as

Vrad.cor. = 1 [notdef]H1A[notdef]4+[notdef]H1B[notdef]4

+ 2

[notdef]H2A[notdef]4+[notdef]H2B[notdef]4

+ 3

[notdef]H1A[notdef]2[notdef]H2A[notdef]2+[notdef]H1B[notdef]2[notdef]H2B[notdef]2

h(H1AH2A)2+(H1BH2B)2+h:c:i: (3.34)

Note that there could have 6,7 terms in the scalar potential (just like the 6,7 terms in 2HDM). However, since we have taken the 6,7 terms to be zero, and we adopt the Type-I Yukawa structure, the radiative corrections could not generate 6,7 terms. We list the coe cients in eq. (3.34):

1 [similarequal]

1 162

+ 4

[notdef]H1AH2A[notdef]2+[notdef]H1BH2B[notdef]2 + 5 2

JHEP12(2016)143

94g4 +32g2g[prime]2 +34g[prime]4 log 2 f2 ;

2 [similarequal]

1 162

94g4 +32g2g[prime]2 +34g[prime]4 log 2 f2 ;

3 [similarequal]

1 162

92g4 3g2g[prime]2 +32g[prime]4 log 2 f2 ;

4 [similarequal]

1162 6g2g[prime]2

log 2

f2 ;

5 = 0: (3.35)

from gauge interactions [19]. In the Type-I Yukawa structure, the Yukawa interactions induce

1 [similarequal] +

1162 (3y4) log

2f2 ; 2,3,4,5 = 0; (3.36)

for the fermion assignment I and

1 [similarequal]

3 162

y2M2=f2

M2 y2f2

M2 log M2 + y2f2M2 y2f2 logM2 + y2f2 y2f2

; 2,3,4,5 = 0;

(3.37)

for the fermion assignment II [8, 9]. In other Yukawa structures, the Yukawa radiative corrections could be di erent. Here other non-logarithm contributions and small radiative contributions from scalar self-interactions are neglected.

The overall radiative corrections are the sum over gauge boson and fermion contributions. Note that the above radiative corrections are independent of the breaking patterns. It is valid for both U(4)=U(3) and [U(4) [notdef] U(4)] = [U(3) [notdef] U(3)] patterns. Given the gauge

and fermion assignments, the radiative corrections is completely determined by gauge and Yukawa couplings. In the following, we take general form of 15. In the numerical calcu

lation, we take the values from the fermion assignments I:

1 = 0:09; 2 = 0:004; 3 = 0:005; 4 = 0:002; 5 = 0; (benchmark point):

(3.38)

This serves as our benchmark point in the following discussions.

{ 13 {

4 Symmetry breaking and vacuum misalignment

The radiative corrections calculated in the above section trigger spontaneous symmetry breaking, and induce VEVs for the h01 and h02 components in H1A,2A de ned in eq. (3.13).

We could determine the VEVs of h01,2 using general tadpole conditions in the loop-induced scalar potential. Depending on global symmetry breaking patterns, the symmetry breaking and vacuum misalignment are quite distinct. We discuss the symmetry breaking and vacuum misalignment in U(4)=U(3) and [U(4) [notdef] U(4)] = [U(3) [notdef] U(3)] breaking patterns,

respectively.

4.1 U(4)/U(3) breaking pattern

In this breaking pattern, due to existence of the m212 term and 45 terms in the potential,

the global symmetry breaking pattern is U(4) ! U(3), with seven Goldstone bosons gener

ated. The 15 terms further trigger spontaneous symmetry breaking, and some Goldstone

bosons become PGBs.

The radiative corrections from the gauge and Yukawa interactions trigger symmetry breaking on A sector. According to eq. (3.10), only one combination of the two twin Higgses H1,2 obtains VEV. Denoting the VEV [angbracketleft]h

JHEP12(2016)143

0

[angbracketright]

f we obtain the eld VEVs in the Higgs basis,

or the H1,2 basis:

hH[angbracketright] =

0

B

B

B

@

0f sin 0f cos

1

C

C

C

A

; [angbracketleft]H[prime][angbracketright] 0; or [angbracketleft]H1[angbracketright]

0

B

B

B

@

0f1 sin

0f1 cos

1

C

C

C

A

; [angbracketleft]H2[angbracketright]

0

B

B

B

@

0f2 sin

0f2 cos

1

C

C

C

A

:

(4.1)

Let us calculate the VEV = [angbracketleft]h0[angbracketright]=f using the tadpole conditions. The tadpole

conditions determine only only the mass-squared parameters 21,2, but also the VEV . The full tadpole conditions are listed in the appendix A. Here we only list the tadpole conditions which determine the VEV:

f21 1 + f22 345

cos 2 = 0: (4.2)

If f1 [negationslash]= f2 and 1 [negationslash]= 2, the two conditions give rise to cos(2 ) = 0 ) =

cos 2 = 0; f22 2 + f21 345

4 ) [angbracketleft]HA[angbracketright] = [angbracketleft]HB[angbracketright] = f=p2: (4.3)

Note that the VEVs are equally aligned because of the Z2 symmetry. Similar to the original twin Higgs model, adding soft or hard breaking terms will realize vacuum misalignment.

Here we add the soft mass breaking terms in the scalar potential

Vsoft = m21A[notdef]H1A[notdef]2 m22A[notdef]H2A[notdef]2 m212A h

H1AH2A + h:c:

i

: (4.4)

Taking the soft breaking terms into account, we obtain new relevant tadpole conditions

F1A + 2 1 + t2 345

cos 2 = 0;

F2A + 2t2 2 + 345

cos 2 = 0: (4.5)

{ 14 {

2.

0.2

0.10

F

F

0.08

0.1

1.5

t

=3

3

t

=0.1

0.5

1

1.5

)

]

0.06

F 2 A

1

TeV

(

0

F 2 A

t

=0.1

[

F 1 A

1.

f 1

0.04

3

-0.1

0.02

t

=1

1

0.5

t

=0.1

0.00

t

=3

t

=0.1

-0.2

0.2

0

16 8

3 16 4

0

16 8

3 16 4

-0.20

-0.15

-0.10

-0.05 0.00 0.05

F1A

JHEP12(2016)143

Figure 1. On the left panel, the correlations between F1A and F2A for di erent t are shown. On the middle, the F1A (solid lines) and F2A (dashed lines) as function of for di erent t . On the right, the scale f1 as function of for di erent t . For all the gures, here the benchmark parameters 15 in eq. (3.38) are used.

where F1A = m

21A+m212At2

f21 and F2A = m

22A+t1 m212Af21 . From the above relations we see that

could be less than =4 only if there are the following relations

F1A 2t2 2 + 345

= F2A 2 1 + t2 345

; F1A < 2 1 + t2 345

: (4.6)

Since both F1A and F2A are free parameters, These relations could be easily satis ed.

The left panel of the gure 1 shows the correlations between F1A and F2A given the t

and benchmark parameters in eq. (3.38). Given the soft mass term F1A or F2A, we could determine the VEV using tadpole conditions in eq. (4.5). Figure 1 (middle) shows the F1A or F2A vesus the value for di erent tan . It shows that the solution < =4 does exist, and thus the vacuum misalignment is realized.

Although the tadpole conditions in eq. (4.5) determine , we need to know the scale f to further determine v. To obtain the VEV v at electroweak scale, the following condition should be imposed

f sin = f1

q1 + t2 sin = v = 174 GeV: (4.7)

Given tan , there are relations between and f1, which are shown in gure 1 (right). In summary, given appropriate values of m21A (or m22A) and t , and f1 are totally determined, and the vacuum misalignment could be relized.

4.2 [U(4) U(4)] / [U(3) U(3)] breaking patternIf the tree-level breaking terms are small, the potential exhibits approximate U(4) [notdef] U(4)

global symmetry and exact Z2 symmetry. In the global symmetry breaking pattern [U(4) [notdef] U(4)] = [U(3) [notdef] U(3)], 14 Goldstone bosons are generated after symmetry break

ing. The 15 terms further trigger spontaneous symmetry breaking, and some Goldstone

bosons become PGBs.

{ 15 {

The gauge and Yukawa interactions radiatively generate the symmetry breaking for the Goldstone bosons h1,2. Denoting

1 [angbracketleft]

h1[angbracketright]f1 ; 2 [angbracketleft]

h2[angbracketright]f2 ; (4.8)

we parametrize the VEVs of the elds H1,2 as

hH1[angbracketright]

0

B

B

B

@

0f1 sin 1 0f1 cos 1

1

C

C

C

A

; [angbracketleft]H2[angbracketright]

0

B

B

B

@

0f2 sin 2 0f2 cos 2

1

C

C

C

A

: (4.9)

The tadpole conditions determine not only the mass-squared parameters 21,2, but also VEVs 1,2. The full tadpole conditions are presented in appendix B. Here we only list the two tadpole conditions3 which determine the VEVs:

sin 4 1 + 1 sin 4 2 + 2 sin 2( 1 + 2) = 0;sin 4 1 1 sin 4 2 + 2 sin 2( 1 2) + 2F sin 2( 1 2) 4Fm sin( 1 2) = 0; (4.10)

where we denote

1 t4

345

45

2 1 ; 2 t2

: (4.11)

Since the parameters ( 1; 2) only depend on t and radiative corrections 15, we denotes

the parameters ( 1; 2) as \radiative breaking parameters". Given the gauge and fermion assignments, ( 1; 2) are uniquely determined. In Type-I fermion assignment, 1 > 0 and 25 < 0 indicates 1 < 0 and 2 < 0. In the following, we will focus on the region 1 < 0

and 2 < 0.4 On the other hand, (F ; Fm) depend on both radiative parameters and tree-level U(4) [notdef]U(4) breaking terms 45 and m212, denoted as \tree breaking parameters".

Given radiative and tree-level breaking parameters, ( 1; 2) are uniquely determined by the tadpole conditions.

The rst tadpole condition in eq. (4.10) describes the relation between 1 and 2. In gure 2 we plot the correlation contours between ( 1; 2) for di erent ( 1; 2) imposed by the rst tadpole condition in eq. (4.10). Several features are in order. First, depending on the size of [notdef] 1 + 2[notdef], the contours live in regions: 2 < 1 if [notdef] 1 + 2[notdef] > 1, and 2 > 1

if [notdef] 1 + 2[notdef] < 1. Second, 2 determines intersection point 1 between the contour curve

and the x-axis, or 2 between the curve and y-axis. If 2 is zero, the intersection point is either 1 = =4 or 2 = =4. From gure 2 (right), the smaller 2, the smaller (lager) 1 if [notdef] 1 + 2[notdef] > 1 ([notdef] 1 + 2[notdef] < 1). Third, 1 only controls the convex behaviour of these

contours. From the left and middle panels of gure 2, the smaller 1, the larger convex behaviour for the contours if [notdef] 1 + 2[notdef] > 1, while vice versa for [notdef] 1 + 2[notdef] < 1.

3In ref. [26], only 1 and m212 terms are included in the tadpole conditions. Thus the tadpole conditions in ref. [26] could be treated as a special case of these general conditions.

4If 1 < 0, 2 > 0 or 1 > 0, 2 < 0, the Z2 symmetry breaking could also be realized. For example, if 1 < 0, 2 > 0, the Z2 symmetry breaking happens when 1 < 2 < /4. This could happen in di erent fermion assignments.

{ 16 {

1 ; Fm t

m212 1f21

JHEP12(2016)143

1 ; F t2

4

4

4

3

16

3

16

3

16

8

8

8

16

16

16

2

2

2

(-

0.6, -2 )

0

0

0

-

16

-

16

-

16

JHEP12(2016)143

- 8

0

16 8

3 16 4

- 8

0

16 8

3 16 4

- 8

0

16 8

3 16 4

1

1

1

Figure 2. The contour lines exhibit the relation between ( 1; 2) imposed by the rst tadpole condition in eq. (4.10). Each contour is labeled by the radiative parameters ( 1; 2). The left

(middle) panel shows contours for di erent 1 with 2 = 0(0:6). The right panel shows contours

for di erent 2 with 1 = 0:6.

The second tadpole condition provides us another relation on ( 1; 2), which is shown as another contour in the ( 1; 2) plane. Together with the contour from rst tadpole condition, the two contours uniquely determine value of ( 1; 2) which is the intersection point between two contours. Similar to gure 2, we plot the ( 1; 2) contours imposed by the second tadpole condition. To clearly present e ects of each parameters, we rst turn o tree-level breaking parameters (F ; Fm). In this case, the two conditions reduce to

sin 4 1 + 1 sin 4 2 + 2 sin 2( 1 + 2) = 0;sin 4 1 1 sin 4 2 + 2 sin 2( 1 2) = 0: (4.12)

Figure 3 shows the ( 1; 2) contours imposed by two conditions for di erent ( 1; 2). We note that the two conditions are symmetric under 1 $ 1 if [notdef] 1 + 2[notdef] > 1, while they

are symmetric under 2 $ 2 if [notdef] 1 + 2[notdef] < 1. This symmetric behaviour can be seen

from the upper left and middle panels of gure 3. Therefore we can determine the solution for ( 1; 2):

( 2 = 0; 1 =4; for [notdef] 1 + 2[notdef] > 1 1 = 0; 2 =4; for [notdef] 1 + 2[notdef] < 1:

(4.13)

This indicates only one Higgs Hi obtains VEV. From the left panel of gure 3, if 2 = 0, we have either 1 = =4 (if [notdef] 1 + 2[notdef] > 1) or 2 = =4 (if [notdef] 1 + 2[notdef] < 1). According to

the middle panel, when 2 < 0, we have either 1 < =4 (if [notdef] 1 + 2[notdef] > 1) or 2 > =4

(if [notdef] 1 + 2[notdef] < 1). On the right panel, it shows as 2 decreases, the value of 1 decreases.

Thus we could obtain appropriate asymmetric vacuum 1 when we vary 2. When we take

| 1 + 2[notdef] > 1, 1 could be smaller than =4 as we vary 2. Thus even without tree-level

breaking parameters, the vacuum misalignment could still happen. This is the scenario of radiative Z2 symmetry breaking [28].

{ 17 {

4

4

4

3

16

3

16

3

16

(-

0.5,0,0

(0,

-0.6,0

(-

0.6,0,0

8

8

8

(-

0.9,0,0 )

(-

0.3,

-0.6,0 )

(-

0.6,

-0.3,0 )

16

16

2

2

16

2

0

0

0

(- - )

(-

2,

-0.6,0 )

-

-

(-

1,

-0.6,0 )

-

(-

0.6,

-1,0 )

16

16

16

(-

1.1,0,0 )

(-

0.5,

-0.6,0 )

(-

0.6,

-0.5,0 )

- 8

- 8 -

16

0

16 8

3 16 4

- 8

- 8 -

16

0

16 8

3 16 4

- 8

- 8 -

16

0

16 8

3 16 4

JHEP12(2016)143

1

1

1

4

4

4

3

16

3

16

3

16

8

8

8

16

16

2

2

2

16

0

(-

3,0,0.3

)

0

0

(- - )

(-

0.5,

-0.6,0.3 )

(-

0.6,

-0.5,0.3 )

-

(-

1.1,0,0.3 )

16

-

16

-

16

- 8

0

16 8

3 16 4

- 8

0

16 8

3 16 4

- 8

0

16 8

3 16 4

1

1

1

Figure 3. The contour lines exhibit the relation between ( 1; 2) imposed by the rst (dashed lines) and second (solid lines) tadpole conditions. Each contour is labeled by the radiative parameters ( 1; 2; Fm). The left (middle) panel shows contours for di erent 1 with 2 = 0(0:6). The right

panel shows contours for di erent 2 with 1 = 0:6. The upper and lower panels correspond to

Fm = 0 and 0:3 respectively.

Turning on tree-level breaking terms (F ; Fm) will change the contours between 1 and 2 imposed by the second tadpole condition. For simplicity, let us turn on single tree-level breaking term: F or Fm. Figure 3 (lower panel) shows the ( 1; 2) contours imposed by two conditions for di erent ( 1; 2; Fm). For comparison, we use the same values of the ( 1; 2) in both the upper and lower panels of gure 3. We nd that turning on Fm shifts the intersection point between the contour and the x-axis to lower 1, and also change the convex behavior of the contour. Thus Fm plays a similar role as 2. Figure 3 (left) show that even 2 is zero, turning on Fm will obtain the following solution:

2 < 1 =4; for [notdef] 1 + 2[notdef] > 1: (4.14)

The vacuum misalignment could be realized via the bilinear term m212. This is the scenario of tadpole induced Z2 symmetry breaking [26, 27]. The middle and right panels of gure 3 show that turning on 2 will also obtain viable solutions. And the larger 2, the smaller 1. Our discussion on the tree-level breaking term Fm could also be applied to the case

{ 18 {

8

4

4

t

=0.5

f

=0.3TeV

t

=0.5

3

16

3

16

f

=0.42TeV

t

=1

t

=1

f

=0.41TeV

3

32

t

=1

t

=0.5

8

8

t

=1.5

t

=1.5

16

16

f

=0.39TeV

t

=1.5

(-

1,0.6,0.9

)

16

2

2

0

(-

1,0.6,0 )

2

0

t

=0.5

-

16

-

16

32

- 8

- 8

t

=1.5

-

3

16

-

3

16

=0.5TeV f =0.8TeV f =1.5TeV f =5TeV

f

0 0

32

16

3 32 8

- 4

0

16 8

3 16 4

- 4

0

16 8

3 16 4

JHEP12(2016)143

1

1

1

Figure 4. The solid contours on the ( 1; 2) plane show the VEV relations for xed t = 1:5 and di erent f1 = 0:5; 0:75; 1; 2:5 TeV. The dashed lines show the VEV relations for di erent t = 0:5; 1; 1:5; 2 with xed f1 = 0:5 TeV (the left panel) and f1 = 1 TeV (the right panel).

with only F . The results are quite similar to the one in gure 3. In this case, the 45 plays the role to obtain vacuum misalignment. This is a new scenario: quartic induced Z2 symmetry breaking.

The set of parameters ( 1; 2; F ; Fm) could only determine ( 1; 2), but not the VEVs (v1; v2). To obtain the electroweak VEVs, additional condition (the VEV condition) needs to be imposed:

f21 sin2 1 + f22 sin2 2 = v2; (4.15)

where v = 174 GeV. Given t and 1,2, we could determine f1 and f2. Figure 4 (left) shows the VEV contour curves on the ( 1, 2) plane for di erent f1 and t . f1 determines the intersection point between the curve and the x-axis, while t determines the curvature behaviour of the curve. The middle and right panels of gure 4 show that once t is xed, f1 could be determined and thus the VEV contour is xed. There is one special case. In gure 4 (middle panel), when the tree-level breaking term is o , f1 keeps the same for di erent t . So in radiative Z2 case, f1 can be determined by two parameters ( 1; 2).

Given the vacuum misalignment condition 1 < =4, we could estimate the parameter region for the tree-level breaking parameters Fm; F and global symmetry breaking scales f1,2. Figure 5 (left) shows the values of Fm or F as functions of 1, which determines 1 for di erent t . Interestingly, even when Fm or F is absent, we could still obtain 1 < =4, which corresponds to the radiative breaking scenario. Figure 5 (right) shows that once 1 (and t ) is known, f1 is totally determined. And the larger t , the larger f1. This relation is quite general and does not depend on scenarios. Thus in tadpole or quartic induced symmetry breaking, only two independent parameters are needed, which are typically taken to be Fm(F ) and t . If there is no tree-level breaking term, only one parameter t could determine the VEVs.

Finally let us summarize what we have obtained so far from the tadpole conditions. The tadpole conditions determine ( 1; 2), which depends on ( 1; 2) and/or (Fm; F ).

Given 1,2 0, the vacuum misalignment requires 2 < 1 < =4 with [notdef] 1 + 2[notdef] > 1.

{ 19 {

10

2.

t

=1.9

F

F

5

1.5

t

=2.5

t

]

F morF

0

=3

TeV

[

1.

f 1

-5

0.5

JHEP12(2016)143

0.2

10 0

16 8

3 16 4

0

16 8

3 16 4

1

1

Figure 5. On the left panel, the relations between 1, and Fm (solid lines) or F (dashed lines) are shown. On the right panel, the contours show the relations between 1 and f1 imposed by the VEV condition for di erent t .

Three scenarios are discussed to obtain this misalignment. We classify these scenarios according to parameters ( 1; 2) and (Fm; F ):

Radiative Z2 breaking [28], when 1 [negationslash]= 0; 2 [negationslash]= 0. Since there is no tree-level breaking

term, the tree-level potential is U(4) [notdef] U(4) invariant:

VU(4)[notdef]U(4) = 21[notdef]H1[notdef]2 22[notdef]H2[notdef]2 + 1([notdef]H1[notdef]2)2 + 2([notdef]H2[notdef]2)2 + 3[notdef]H1[notdef]2[notdef]H2[notdef]2: (4.16)

The radiative corrections to the scalar potential are shown in eq. (3.34). The 1 determines whether the electroweak symmetry breaking could happen, while the 2 determines whether vacuum misalignment could happen. Since 1 < 0; 2 < 0, solu

tion of the asymmetric vacua has 1 < =4 and 2 0. The two tadpole conditions

reduce to one

sin 4 1 + 2 sin 2( 1) = 0: (4.17)

Thus 1 only depends on 2: the larger 2 the smaller 1. Although gauge corrections are much smaller than Yukawa corrections, 2 could be large if t 1. When 2

approaches one, 1 approaches to zero.

m212-induced Z2 breaking [26, 27], when 1 [negationslash]= 0; m212 [negationslash]= 0. The tree-level potential is

VTadpole = VU(4)[notdef]U(4) + m212

hH1H2 + h:c:i: (4.18)

The dominant radiative corrections to the scalar potential are the same as the radiative Z2 breaking case. Similarly 1 determines whether the electroweak symmetry breaking could happen. However, in the parameter region that t is small, 2 alone can not obtain small enough 1. And in certain case 2 is also small. In those cases, the parameter m212 could play the role to obtain appropriate 1. It is the m212 which determines whether vacuum misalignment could happen. As explained in ref. [26]

{ 20 {

and next section, the m212 plays the role of the tadpole terms, which transits from 1 to 2, and obtains 1 < =4.

45-induced Z2 breaking, when 1 [negationslash]= 0; 45 [negationslash]= 0. The tree-level potential is VQuartic = VU(4)[notdef]U(4) + 4[notdef]H1H2[notdef]2 + 5 h

(H1H2)2 + h:c:

i

: (4.19)

Similarly 1 determines whether the electroweak symmetry breaking could happen. Even when 2 exists, it is the 4,5 controls the vacuum misalignment. Furthermore, negative 4,5 is favored to obtain appropriate 1.

It is also possible that both m212 and 4,5 terms exist in the potential. In this case, it is the m212 and 4,5 which determine whether vacuum misalignment could happen. This scenario is mixture of the tadpole and quartic induced Z2 breaking scenarios.

5 Spontaneous Z2 breaking in composite 2HDM

In above section, we discussed how the tadpole conditions determine the electroweak vacua ( 1; 2). Three di erent mechanisms could lead to the vacuum misalignment 2 < 1 < =4, and realize the spontaneous Z2 breaking. Let us understand the physics behind these Z2 breaking scenarios.

Since the electroweak symmetry breaking only involves in the PGBs in visible A sector, we will simplify the original scalar potential in eq. (3.2) and eq. (3.34) by setting

H1B [similarequal]

0 f1 cos

H1 f1

JHEP12(2016)143

!; H2B [similarequal]0 f2 cos

H2 f2

!: (5.1)

Approximately, [notdef]HiB[notdef]2 [similarequal] f2i [notdef]HiA[notdef]2 are obtained. Expanding the potential to the quartic

order, we obtain the approximated potential of the visible sector in the 2HDM framework:

V (H1A; H2A) = 21A[notdef]H1A[notdef]2 22A[notdef]H2A[notdef]2 + 1A[notdef]H1A[notdef]4 + 2A[notdef]H2A[notdef]4 + 3A[notdef]H1A[notdef]2[notdef]H2A[notdef]2

+m2A12

hH1AH2A + h:c:i+ 4A[notdef]H1AH2A[notdef]2 + 5A 2

h(H1AH2A)2 + h:c:

i

+ h( 6A[notdef]H1A[notdef]2 + 7A[notdef]H2A[notdef]2)H1AH2A + h:c:i: (5.2)

Unlike the elementary 2HDM potential, in this composite 2HDM potential, all the coe -cients in the potential are proportional to the tree-level and loop-induced breaking terms:

21A = 2 1f21 + ( 345 + 45)f22 + m212t ;

22A = 2 2f22 + ( 345 + 45)f21 + m212t1 ;

1A = 8 13 +

3 t2 +

m212 12f21

345 + 45

t ;

2A = 8 2

345 + 45

3 +

3 t2 +

m212 12f22

t1 ;

3A = 345 + 45 + m212

2f1f2 ;

{ 21 {

m2A12 = m212 + 45f1f2; 4A = 4 + 4; 5A = 5 + 5;

6A = 2 453 t +

m212

6f21

; 7A = 2 45

3 t1 +

m212

6f22

: (5.3)

Note that there is no dependence on the tree-level parameters 13. Since the H1A and H2A

are pseudo-Goldstone bosons, we identify this scenario as composite 2HDM, to distinguish it from the elementary 2HDM.

5.1 Radiative Z2 breakingIn this scenario, the tree-level breaking terms m212 and 4,5 do not exist. From the above potential, the Higgs mass squared terms reduce to

2H1A = f21 2 1 + 345t2

JHEP12(2016)143

; 2H2A = f22

2 2 + 345t2 ; (5.4)

and the quartic terms reduce to

1A = 8 1

3 +

3 t2 ; 2A =

3 t2 : (5.5)

Since H1A has negative mass-squared 2H1A < 0, H1A gets VEV. While H2A has pos

itive mass-squared 2H2A > 0, there is no VEV for H2A. The asymmetric vacua hH2B[angbracketright] = f2 [angbracketleft]H2A[angbracketright] = 0 indicate the spontaneously broken Z2 symmetry between

H2A $ H2B. Furthermore, the term [notdef]H1B[notdef]2[notdef]H2B[notdef]2 generates additional mass term

3453 t2

345

8 2

345

3 +

for H1A radiatively.5 This additional mass term triggers the Z2 symmetry breaking between H1A $ H1B spontaneously.

The composite 2HDM potential reduces to

Vinert [notdef] 21A[notdef][notdef]H1A[notdef]2 + 1A[notdef]H1A[notdef]4 + [notdef] 22A[notdef][notdef]H2A[notdef]2 + 2A[notdef]H2A[notdef]4 + 345[notdef]H1A[notdef]2[notdef]H2A[notdef]2 ;(5.6)

Note that the 4,5 terms are absent. The form of the potential is the same as the inert Higgs doublet potential [32], although all the terms are generated radiatively. The rst two terms in the potential, determine the electroweak vacuum:

v2 = 21A= 1A [similarequal]

38 2 + 345t2 = 1

f21: (5.7)

To obtain the electroweak VEV v = 174 GeV, the two terms in the equation should cancel with each other. We know although contributions 1 (from Yukawa corrections) and 345 (from gauge corrections) have opposite sign, the adequate cancellation will not happen if 1 345. To have adequate cancellation, the second term in the mass-squared 2H2A

needs to be enhanced by assigning large t . On the other hand, because 1A keeps large compared to the mass-squared 2H2A, the electroweak VEV is obtained.

Finally, let us write the masses of the PGBs. The Higgs boson mass reads

m2h = 2 21A = 4 1 + 2 345t2

f21; (5.8)

5If the Z2 symmetry between H2A $ H2B is exact, the terms [notdef]H1B[notdef]2[notdef]H2B[notdef]2 and [notdef]H1A[notdef]2[notdef]H2A[notdef]2 generate

opposite but equal mass terms for H1A. Thus the Z2 symmetry between H1A $ H1B is still unbroken.

{ 22 {

and the masses of the charged and neutral scalars in the inert doublet are

m2H[notdef] = 22A + 345v2; m2A0 = 22A + 345v2: (5.9)

In elementary 2HDM model, the masses of the charged and CP-odd neutral scalar are only proportional to 4,5, which are very small. In this scenario, the inert scalar masses also have 2,3 dependences, which induce large masses for the inert scalars. Therefore, the radiative Z2 breaking scenario can be viewed as a natural UV completion of the inert Higgs doublet model.

5.2 Tadpole induced Z2 breaking

The radiative Z2 breaking scenario can only realize the electroweak symmetry breaking when 345 is non-zero, and if the enhancement from t 1 exists. Otherwise, vacuum

misalignment cannot be obtained by purely radiative Z2 breaking. Thus when 345 is zero, or t 1, the tadpole induced Z2 breaking scenario should play the role of electroweak

symmetry breaking. However, the price to pay is introducing additional m212 term.

Let us turn on m212 gradually to see how the VEVs 1,2 vary. When m212 term is o , from the radiative breaking scenario, the VEVs have

hH1A[angbracketright] [similarequal] f1; [angbracketleft]H2A[angbracketright] [similarequal] 0: (5.10)

If t 1, we obtain 21A 22A due to 1 345. Thus mh1 is much heavier than mh2.

When gradually turning on m212 term, h2 starts to obtain small VEV. This can be seen from the potential by assuming h1 is too heavy and decoupled. After integrating out h1, the potential generates an e ective tadpole term. The h2 potential is dominated by the tadpole and quadratic terms

V (H2A) 22Ah22 + m212f1h2 (5.11)

Thus h2 obtain VEV

hH2A[angbracketright] m212f1= 22A; (5.12)

which gradually becomes large as we increase m212. At the same time, the VEV of h1 decreases. This can be seen from the H1A potential. Assuming the VEV [angbracketleft]h2[angbracketright] is small, the

relevant H1A potential is

V (H1A) [similarequal] 2 1f21 + 345f22 + m212t

[notdef]H1A[notdef]4: (5.13)

Here the tadpole contribution is negligible due to [angbracketleft]h1[angbracketright] > [angbracketleft]h2[angbracketright]. From the potential, we

see that as the m212 becomes larger, there is large cancellation in the quadratic term, and

thus the VEV [angbracketleft]h1[angbracketright] becomes smaller. Therefore, the bilinear term m212 plays the role of the

e ective tadpole. As the e ective tadpole term increases, the VEV 1 decreases from =4, while the VEV 2 increases from 0. The vacuum misalignment 2 < 1 < =4 could be realized when an appropriate m212 term is taken.

{ 23 {

JHEP12(2016)143

|H1A[notdef]2 +

8 1

345

3 +

3 t2 +

m212 12f21

t

5.3 Quartic induced Z2 breaking

In this scenario, only quartic breaking terms 4,5 are kept in the tree level potential. Unlike the m212 case, the quartic breaking scenario works for both small t and large t regions.

The 45 terms appear in both quadratic term 21A and the bilinear term in the potential. The quadratic term has

21A = f21 2 1 + ( 345 + 45) t2

: (5.14)

We see that both 345 and 45 contribute to the quadratic term to have the opposite 1 corrections. Furthermore, it also generates the e ective tadpole term, which transits 1 to

2. Therefore, the quartic induced breaking scenario has the ingredients of the radiative and tadpole scenarios to break the Z2 symmetry. In this scenario, there should be much larger viable parameter regions which generate the appropriate Z2 breaking.

6 Higgs phenomenology

6.1 Higgs mass spectra

In this natural composite 2HDM framework, the Higgs sector contains two Higgs doublets H1A; H2A in A sector, and another two Higgs doublets H1B; H2B (with two neutral radial mode decoupled) in B sector. Six exact GBs: three (z[notdef],0) from HiA and three (C[notdef]; N0) from HiB are generated. All of them are eaten by gauge bosons in A and B sectors. Depending on the breaking pattern, other particles than the exact GBs in the scalar multiplets could be PGBs or just scalar particles.

We present details of the mass spectra in two breaking pattern in appendix A and B. Here we summarize main features of the mass spectra based on results in appendix A and B.

Explicit soft Z2 breaking In the Higgs basis, the eld H plays the role of twin

Higgs, while another eld H[prime] is just additional scalar U(4) multiplet. Thus among seven GBs, six are eaten by gauge bosons, and one PGB is the Higgs boson. For the additional scalars in H[prime], the masses are

m2H[notdef] = m2A =

m212s c 5f2: (6.1)

which only depends on U(4) breaking parameters m212; m212A and 4,5 in the potential. If the tree-level terms 4,5 do not exist, then all the new scalars have degenerate masses. In SUSY extension of the twin Higgs model [22], the mass spectra are much simpli ed due to the global symmetries.

Radiative Z2 breaking

The global symmetry breaking is [U(4) [notdef] U(4)] ! [U(3) [notdef] U(3)]. All the scalar

components in visible sector are PGBs. Furthermore, only H1A obtains VEV, and

{ 24 {

JHEP12(2016)143

m212 + m212A s c

1

2( 4 + 5)f2;

m2H[prime][notdef] = m2A[prime] =

2000

2000

t=

0.4

m12

=500 GeV

m12A

1000

1000

=200 GeV

H+,A

]

]

GeV

GeV

[

500

[

500

MassSpectra

MassSpectra

H'+,A'

Neutral scalar H

200

200

mh

=

125 GeV

100

The Higgs h

100

JHEP12(2016)143

50 0

16 8

3 16 4

50 0

16 8

3 16 4

1

1

2000

2000

Tadpole Z2

with t=

3

Quartic Z2

=

2.5

1000

1000

]

]

GeV

H'+,A'

GeV

[

500

[

500

MassSpectra

MassSpectra

200

H+,A

200

mh

=

125 GeV

scalar H

100

The Higgs h

Neutral scalar H

100

50 0

16 8

3 16 4

50 0

16 8

3 16 4

1

1

Figure 6. The masses spectra as the function of 1 in four Z2 breaking scenarios. The particles in the mass spectra are charged and neutral CP odd Higgses (H[notdef]; A0) in visible sector, charged and neutral CP odd Higgses (H[prime][notdef]; A[prime]0) in twin sector, and two CP even Higgses (h; H) in visible sector.

H2A is an inert Higgs doublet. In the twin sector, since both H1B and H2B have VEVs, the PGBs in twin sector mix together. The PGBs mass eigenstates are

m2H[notdef] =2 2f22 345f21 cos 2 1 45f21 sin2 1; m2H[prime][notdef] = 45(f21 cos2 1+f22);

m2A0 =2 2f22 345f21 cos 2 12 5f21 sin2 1; m2A[prime]0 =2 5(f21 cos2 1+f22): (6.2)

Tadpole-induced Z2 breakingSimilar to the radiative breaking scenario, all the scalar components except the radial modes in twin two Higgs doublets are PGBs. The di erence between two scenarios is that in this tadpole scenario there are mixing between two Higgs doublet in A sector, with mixing angle A, de ned in appendix B. All the masses of the

{ 25 {

charged Higgses and CP odd Higgses depend on m212 and are nearly degenerate when 1,2 are much smaller than =4. The mass spectra read

m2H[notdef] = m2A01 =

: (6.3)

Quartic-induced Z2 breakingSimilar to the tadpole induced breaking scenario, all the masses of the charged Higgses and CP odd Higgses depend on 4,5. The di erence between quartic and tadpole scenarios is that in this scenario there are mass splittings between charged and neutral CP odd Higgses unless 4 = 5. The charged scalar masses are

m2H[notdef] = 45(1 + cot 1 cot 2) f21 sin2 1 + f22 sin2 2

m2H[prime][notdef] = 45(1 + tan 1 tan 2) f21 cos2 1 + f22 cos2 2

and the CP-odd scalar masses are presented in appendix B.

In all scenarios, the SM Higgs boson originates from the mixing between h1 and h2 in visible sector.6 We present the mass matrices of the Higgs boson in appendix A and B. Figure 6 shows the mass spectra in above four scenarios. The independent parameters in four scenarios are taken to be ( 1; t ; m12 = 500; m12A = 200) (explicit Z2 breaking), 1 (radiative breaking), 1; t = 3 (tadpole breaking), and 1; t = 2:8 (quartic breaking). Figure 6 shows that typically charge and neutral CP odd Higgses (H[notdef]; A0) in visible

sector have degenerate masses, and similarly for charged and neutral CP odd Higgses (H[prime][notdef]; A[prime]0) in twin sector. In the spontaneous Z2 breaking scenarios, there are only two free parameters (t ; 1). Imposing the condition of the 125 GeV Higgs boson mass provides additional constraint on the model parameters. In gure 6, if we identify the Higgs boson mass to be 125 GeV (dashed line), the 1 totally xed once we x t .

6.2 Collider constraints

Let us rst consider the visible sector. The visible sector contains the same particle contents as the ones in the 2HDM. The phenomenology in visible sector should be very similar to the one in 2HDM, except that there could be additional decay channels to the twin particles. For simplicity, we take the Type-I Yukawa structure in this work, although other Yukawa structure, such as Type-II, X, Y, are possible. Let us setup the notation similar to 2HDM.7 According to appendix A and B, the 2HDM mixing angles and electroweak VEV

6The exception is that in radiative breaking scenario, there is no mixing between h1 and h2. The Higgs mass is proportional to the breaking parameters 15 and/or m212( 4,5).

7Note that de nition of the mixing angle is opposite from the typical notation, such as ref. [30].

{ 26 {

m212f1 sin 1f2 sin 2 f21 sin2 1 + f22 sin2 2

;

m2H[prime][notdef] = m2A02 =

m212f1 cos 1f2 cos 2 f21 cos2 1 + f22 cos2 2

JHEP12(2016)143

;

: (6.4)

are di erent in two breaking patterns, de ned as

U(4)=U(3)

8

>

>

<

>

>

:

= f2f1 ; mixing angle between charged/CP-odd scalars; v = f sin ; electroweak vacuum;

; mixing angle between CP even scalars;

(6.5)

U(4)2=U(3)2 8

>

>

<

>

>

:

A = f2 sin 2f1sin 1 ; mixing angle between charged/CP-odd scalars; v =pf1 sin 1+f2 sin 2; electroweak vacuum;

; mixing angle between CP even scalars:

(6.6)

The normalized Higgs couplings to the SM gauge bosons and fermions are

JHEP12(2016)143

hV V

ghV V

gSMhVV

=

8

>

>

<

>

>

:

cos c ; Explicit Z2 Breaking; cos 1; Radiative Z2 Breaking;

cos 1c c A + cos 2s s A; Tadpole and Quartic Z2 Breaking;

(6.7)

hff

yhff

ySMhff

8

>

>

<

>

>

:

=

cos c c ; Explicit Z2 Breaking;cos 1; Radiative Z2 Breaking;cos 1 c c A ; Tadpole and Quartic Z2 Breaking:

(6.8)

v and gSMhff = mfv . These Higgs couplings are constrained by the Higgs coupling measurements at the LHC [37, 38]. The charged and

CP-odd neutral scalars in visible sector have the same constraints as the one in 2HDM. On the other hand, the CP-even neutral scalars have additional decay channels to the twin particles, and thus need additional care.

The twin sector includes another two Higgs doublet H1B,2B, the mirror gauge bosons, and mirror fermions. The mirror gauge bosons are mirror photon, and mirror WB; ZB,

which absorb three GBs in two Higgs doublets. For simplicity, two radial modes in H1B,2B

are assumed to be decoupled. The physical scalars in twin sector are charged and neutral CP odd scalars H[prime][notdef]; A[prime]0. The mirror fermions have very rich twin hadron phenomenology [33] because they are charged under the mirror QCD. Since the twin fermions are mirror copy of the SM particles, the mirror fermion phenomenology should be similar to the original twin Higgs. For simplicity, we take the fermion setup in the fraternal twin Higgs model [33], and leave more general discussion for future. The fermionic ingredients of the fraternal twin Higgs setup are summarized as follows:

To avoid the twin SU(3) and twin SU(2) anomalies, the whole third generation twin

fermions are introduced: twin top, bottom, tau, and twin tau neutrino, but not the rst two generations;

The fermion Yukawa interactions are taken to be the fermion assignment I in our

discussion;

The twin SU(3) has con nement, which indicates the existence of the twin glue-balls,

and twin bottomonium and hadrons below con nement scalar [prime]3 O(10) QCD.

{ 27 {

Here the SM couplings are taken to be gSMhVV = 2m

2V

To be speci c, we take the twin bottom Yukawa coupling the same as the bottom Yukawa coupling, which indicates mbB [similarequal] mb fv . Thus the Higgs boson could decay into bB: h !

bB bB. Because twin fermions are SM charge neutral, some of them could be dark matter

candidate. This has been discussed in refs. [33{36].

The Higgs boson and the heavier CP even neutral scalar provide connection between visible and twin sector. The Higgs boson also couples to the twin particles due to its PGB feature. Here we denote the VEV in twin sector v[prime] f cos (v[prime]

pf21 cos 21 + f22 cos 22), and mixing angle ( B) in explicit (spontaneous) breaking pattern. The normalized Higgs couplings to the twin gauge bosons and fermions are

[prime]hV V

JHEP12(2016)143

ghVBVB gSMhVBVB

8

>

>

<

>

>

:

=

sin c ; Explicit Z2 Breaking; sin 1; Radiative Z2 Breaking;

sin 1c c B + sin 2s s B; Tadpole and Quartic Z2 Breaking;

(6.9)

[prime]hff

yhfBfB

ySMhfBfB

8

>

>

<

>

>

:

=

sin c c ; Explicit Z2 Breaking;sin 1; Radiative Z2 Breaking;sin 1 c c B ; Tadpole and Quartic Z2 Breaking:

(6.10)

Here the SM-like couplings are taken to be gSMhVBVB =

2m2VBv[prime] and gSMhfBfB = mfBv[prime] . The Higgs

fB; VBVB; ABAB. Since in general the normalized couplings of the Higgs boson to the twin gauge bosons and fermions are di erent, the calculation of the signal strength is not just a simple scaling. We calculate the Higgs invisible decay widths based on the above couplings.

We take the latest LHC results on the Higgs coupling measurements [37, 38] and Higgs invisible decays [39], and perform a global t on the model parameters. In the tadpole and quartic Z2 breaking scenarios, if we x the parameter t , there is only one free parameter.

Thus in all the spontaneous Z2 breaking scenarios, we will vary 1 and x t . Furthermore, the explicit breaking scenario is not considered here, since it should be less constrained than the other three scenarios. In the following, we perform a global tting on the Higgs signal strength. In the case where the Higgs coupling measurements are well within the Gaussian statistical regime, the likelihood function is de ned

2 log L( ) = ~2( ) =

n

invisible decay channels are h ! fB

Xi=1[ i ^ i( 1)]2 i : (6.11)

Based on Higgs signal strengths at the 8 TeV LHC with 20.7 fb1 data [37], a statistical analysis is performed by the Lilith package [40]. Figure 7 (left panel) shows the log-likelihood pro le (2 log L) as the function of 1, in three scenarios. Here in tadpole

and quartic scenarios we x the parameter t = 3 and t = 2:5. Up to the 2 level, the exclusion limits in three scenarios are that 1 should typically be less than 0:2. This put very strong constraints on the model parameter. Looking back to gure 6, we note that both this Higgs coupling constraints and the requirement on 125 GeV Higgs mass should be satis ed. The tadpole and quartic breaking scenarios are viable, but there is tension

{ 28 {

10

1

9

3

R

Br

8

Tadpole Z t

=3

0.8

SignalStrength

)

2LogL

6

Radiative Z

0.6

Tadpole Z

2

1

(-

4

0.4

Radiative Z

Invis. Decay Bounds

2

0.2

1

Quartic Z

0 0

16 8

3 16 4

0 0

16 8

3 16 4

JHEP12(2016)143

1

1

0.4

0.3

0.2

0.1

T

0.0

-0.1

-0.2

-0.3

-0.3

-0.2

-0.1 0.0 0.1 0.2 0.3 0.4

S

Figure 7. On the left, it shows the log-likelihood pro le (2 log L) as the function of 1 in three

scenarios. Here 1; 2; 3 errors are also shown. On the middle, it shows the signal strength in gluon fusion production and subsequent V V decays, and Higgs invisible branching ratio as function of 1 in three scenarios. The bound on the invisible decay branching ratio is Brinv < 0:23. On the right, the S T oblique parameter contours at the 1; 2 levels are shown. The dotted points

are the parameter points in three scenarios: radiative (orange color), tadpole (blue), and quartic (green) Z2 breaking scenarios.

between the Higgs coupling constraints and the 125 GeV Higgs boson mass requirement in the radiative breaking scenario. If the U(4) fermion assignment is taken in the radiative breaking scenario, such tension does not exist, and there are viable 1 parameter regions which could satisfy both conditions. This viable fermion assignment has been discussed in ref. [28]. Although the invisible decay width has been taken into account indirectly in the above global tting, we would like to consider constraints from the direct searches on the Higgs invisible decays. The updated upper limits on the invisible decay branching ratio is Brinv < 0:23 [39]. Figure 7 (middle panel) shows the invisible decay branching ratio as the function of 1. As a comparison, we also plot the signal strength in the gluon fusion process gg ! h1 ! V V . From gure 7, we see that the direct searches on the invisible decays put

much weaker constraints than the Higgs coupling measurements. The high luminosity LHC will improve sensitivity of signal strengths to around 5% assuming current uncertainty with 3 ab1 luminosity [41]. Thus we should be able to explore more parameter regions at the high luminosity LHC.

{ 29 {

According to the updated results on oblique parameters via G tter package [42], the S, T parameters have S = 0:05 [notdef] 0:09; T = 0:11 [notdef] 0:13, with correlation coe cients of +0:90

between S and T . In this model, the S and T parameters contains two contributions: corrections from possible radial modes, and corrections from 2HDM scalars. The corrections from radial modes takes the form

S [similarequal]

16 sin2 log

mmh ; T [similarequal]

3 8c2W

sin2 log m

mh ; (6.12)

with radial modes , while the 2HDM corrections [43] are roughly

S [similarequal]

m2H + m2A 2m2H[notdef] 24m2A

JHEP12(2016)143

; T [similarequal]

(m2H[notdef] m2A)(m2H[notdef] m2H

48s2W m2W m2A

: (6.13)

In our numerical study, the complete form of the S; T parameters [30, 43] are used. From the above, we see that if the radial modes decouple, or if the heavy scalars are degenerate, the oblique corrections are negligible. Figure 7 (right panel) plots the predicted S; T values in three scenarios, which we vary the parameter 1 while xing t = 3 in tadpole scenario, and t = 2:5 in quartic scenario. According to the S T oblique parameter contours

at the 1; 2 levels, we note that most of S; T parameter points are within the 2 level contour. Thus the precision electroweak test usually provide weaker constraints on the model parameters than the one from the Higgs coupling measurements.

Let us brie y discuss the distinct signatures of this model. First, like the original twin Higgs model, the twin hadron phenomenology [33] provides us very distinct signatures from other models. Furthermore, the additional charged and neutral scalars provide us a way to distinguish this model from the original twin Higgs. This has been explored in the 2HDM contents for the general case [30] and the inert case [44]. Finally, to distinguish it from the typical 2HDM, the signatures from the twin H[prime][notdef] and A[prime]0 need to be explored. Furthermore, if the radial modes are not so heavy (thus not decoupled), exploring the radial mode decay channels could provide us di erent signatures from the typical 2HDM model. The detailed collider phenomenology would require studies of their own. Furthermore, the ne-tuning argument provides additional theoretical constraints on the models. We leave the detailed studies of these [45] in future.

7 Conclusions

In this work, we investigated a class of twin two Higgs models, in which the Higgs sector is extended to incorporate two twin Higgses and the global symmetry breaking pattern could be either U(4) ! U(3) or [U(4) [notdef] U(4)] ! [U(3) [notdef] U(3)]. The SM Higgs boson is

identi ed as one of the pseudo Goldstone Bosons after symmetry breaking. The discrete Z2 symmetry protects the Higgs mass term against the quadratically divergent radiative corrections. However, the Z2 symmetry needs to be broken to generate electroweak scale, which should be separated from the new physics scale. Typically the soft or hard explicit Z2 breaking terms were introduced to do so. We found that in the twin two Higgs setup, it is possible to realize spontaneous Z2 breaking, without the need of explicit Z2 breaking terms.

{ 30 {

We performed a systematical study on the general Z2 breaking conditions in a natural composite two Higgs doublet framework, and discussed various possible scenarios which could realize the vacuum misalignment. In the radiative Z2 breaking scenario, given the appropriate fermion assignments, the Z2 symmetry could be spontaneously broken purely due to the radiative corrections to the Higgs potential. In this scenario, only one Higgs obtains the electroweak vacuum, and the other is just an inert Higgs. The tadpole-induced

Z2 breaking scenario can also be classi ed in this twin two Higgs doublet framework. In this scenario, the bilinear term in the scalar potential triggers the spontaneous Z2 breaking. We also proposed a novel scenario: the quartic-induced Z2 breaking scenario. In this scenario, the 4,5 terms instead of the bilinear term in the scalar potential trigger the spontaneous

Z2 breaking.

In the twin two Higgs models, we discussed phenomenology of the Higgs sector in the composite two Higgs doublet framework. Although the particle contents in the scalar sector are the same in each scenario, the Higgs mass spectra are quite distinct for each Z2 breaking scenarios. The radiative Z2 breaking scenario includes an inert Higgs doublet with degenerated masses. Both the tadpole-induced and quartic-induced Z2 breaking scenarios contain additional scalars in two Higgs doublet model with not so degenerated masses. We calculated various Higgs couplings and utilized the the Higgs coupling measurements at the current LHC to constrain the model parameters. The additional scalars from the Higgs sector should be able to be probed at the Run-2 LHC and future colliders.

Acknowledgments

The author would like to thank Can Kilic and Nathaniel Craig for valuable discussions. This work was supported by DOE Grant DE-SC0011095.

A Details in the U(4)/U(3) breaking pattern

In this breaking pattern, the scalar potential takes the form

VU(4) = 21[notdef]H1[notdef]2 22[notdef]H2[notdef]2 + 1([notdef]H1[notdef]2)2 + 2([notdef]H2[notdef]2)2 + 3[notdef]H1[notdef]2[notdef]H2[notdef]2+m212

hH1H2 + h:c:i+ 4[notdef]H1H2[notdef]2 + 5 h(H1H2)2 + h:c:i;

Vrad.cor. = + 1 [notdef]H1A[notdef]4+[notdef]H1B[notdef]4

: (A.1)

As a special case, the supersymmetric extension of the twin Higgs model is one speci c realization of this breaking pattern. We identify the speci c terms in the scalar potential in SUSY twin Higgs model [22] as

21 = (m2Hu + 2); 22 = (m2Hd + 2); m212 = b;

1,2,3,5 = 0; 4 = 2; 1 = 2 = 3

2 =

{ 31 {

JHEP12(2016)143

+ 2

+ 3

[notdef]H2A[notdef]4+[notdef]H2B[notdef]4

+ 4

[notdef]H1AH2A[notdef]2 + [notdef]H1BH2B[notdef]2 + 5 2

h(H1AH2A)2 + (H1BH2B)2 + h:c:i;

[notdef]H1A[notdef]2[notdef]H2A[notdef]2+[notdef]H1B[notdef]2[notdef]H2B[notdef]2

Vsoft = m21A[notdef]H1A[notdef]2 m22A[notdef]H2A[notdef]2 m212A h

H1AH2A + h:c:

i

g2 + g[prime]2

8 ; 4,5 = 0: (A.2)

Thus all our discussion about U(4)=U(3) breaking pattern can be applied to the SUSY twin Higgs model.

The tadpole conditions are

21 = f2

f1 m212 + ( 1 + 2 1)f21 + ( 345 + 345=2) f22 + f21 1 + f22 345=2

cos 2 ;

22 = f1

f2 m212 + ( 2 + 2 2)f22 + ( 345 + 345=2) f21 + f22 2 + f21 345=2

cos 2 ;

m21A =

f2f1 m212A + f21 1 + f22 345

cos 2 ;

m22A =

f1f2 m212A + f22 2 + f21 345

cos 2 : (A.3)

Similar to the 2HDM, rotating to the Higgs basis

H = H1 cos + H2 sin ; H[prime] = H1 sin + H2 cos : (A.4)

In the Higgs basis, the masses of the charged gauge bosons are

m2G[notdef] = m2C[notdef] = 0; (Goldstone Bosons);

m2H[notdef] =

m212A + m212f1f2 45 sin2 45 (f21 + f22);

m2H[prime][notdef] =

JHEP12(2016)143

m212f1f2 45 cos2 45 (f21 + f22): (A.5)

The mass matrices of the neutral CP-odd gauge bosons are

m2G0 = m2N0 = 0; (Goldstone Bosons);

m2A0A0 =

m212A + m212f1f2 2 5 sin2 ( 4 5) cos2 2 5 (f21 + f22);

m2A[prime]0A[prime]0 =

m212f1f2 2 5 cos2 ( 4 5) sin2 2 5 (f21 + f22);

m2A0A[prime]0 =

2 sin 2 (f21 + f22); (A.6)

Performing a further rotation on the elds (A0; A[prime]0), we obtain the mass eigenvalues

m2A,A[prime] = m2A0A0 + m2A[prime]0A[prime]0 [notdef] q

4 5

(m2A0A0 m2A[prime]0A[prime]0)2 4m4A0A[prime]0: (A.7)

Assuming radial mode H[prime]0 is heavy, the mass matrices of the CP-even gauge bosons are

m2h0h0 = 4 1f21 sin2 +

f2(m212 + m212A)f1 45f22 cos2 ;

m2H0H0 = 4 2f22 sin2 +

f1(m212 + m212A)f2 45f21 cos2 ;

m2h0H0 = m212 m212A + 45f1f2 + (2 345 + 45)f1f2 sin2 : (A.8)

{ 32 {

Similar to 2HDM, we could further rotate the eld with a rotation angle : h

H

= c s

s c

!

h0 H0

!; (A.9)

we obtain the mass eigenstates

m2h,H = m2h0h0 + m2H0H0 q

(m2h0h0 m2H0H0)2 4m4h0H0: (A.10)

Here we identify h as the SM Higgs boson.

B Details in the [U(4) U(4)] / [U(3) U(3)] breaking pattern The general scalar potential reads

VU(4)[notdef]U(4) = 21[notdef]H1[notdef]2 22[notdef]H2[notdef]2 + 1([notdef]H1[notdef]2)2 + 2([notdef]H2[notdef]2)2 + 3[notdef]H1[notdef]2[notdef]H2[notdef]2;

Vbreaking = m212

hH1H2 + h:c:

i

JHEP12(2016)143

+ 4[notdef]H1H2[notdef]2 +

5

h(H1H2)2 + h:c:

i

Vrad.cor. = 1 [notdef]H1A[notdef]4+[notdef]H1B[notdef]4

+ 2

[notdef]H2A[notdef]4+[notdef]H2B[notdef]4

2 + 3

[notdef]H1A[notdef]2[notdef]H2A[notdef]2+[notdef]H1B[notdef]2[notdef]H2B[notdef]2

h(H1AH2A)2+(H1BH2B)2+h:c:i: (B.1)

Here due to existence of the small tree-level breaking terms, the U(4) [notdef] U(4) symmetry is

approximate.The tadpole conditions are

21 = ( 1+2 1)f21+ 12( 345+2 3+ 45)f22+

+ 4

[notdef]H1AH2A[notdef]2+[notdef]H1BH2B[notdef]2 + 5 2

sin 2 22 sin 2 1 45f22

f2 sin( 1+ 2)f1 sin 2 1 m212; (B.2)

22 = ( 2+2 2)f22+ 12( 345+2 3+ 45)f21+

sin 2 12 sin 2 2 45f21

f1 sin( 1+ 2)f2 sin 2 2 m212; (B.3)

and

f41 1 sin 4 1+f42 2 sin 4 2 = f21f22 345 sin 2( 1 + 2); (B.4)

f41 1 sin 4 1f42 2 sin 4 2 = 4f1f2m212 sin( 1 2)f21f22( 345+2 45) sin 2( 1 2): (B.5)

Similar to the 2HDM, let us de ne the mixing angles of the VEVs in the A and B sectors

f2 cos 2f1 cos 1 : (B.6)Then we will perform a rotation from the H1; H2 basis

h[notdef]

H[notdef]

tan A = f2 sin 2

f1 sin 1 ; tan B =

= cos A sin A sin A cos A

!

h[notdef]1 h[notdef]2

!;

C[notdef]

H[prime][notdef]!=cos B sin B

!

C[notdef]1 C[notdef]2

!;

sin B cos B

z0 A01

= cos A sin A sin A cos A

!

Imh01 Imh02

!;

N0A02!=cos B sin B

!

N01 N02

sin B cos B

!:

(B.7)

{ 33 {

The charged mass spectra have

m2C[notdef] = m2h[notdef] = 0; (exact Goldstone bosons);

m2H[notdef] = f21 sin2 1 + f22 sin2 2

45 + 45(1 + cot 1 cot 2)

m212f1 sin 1f2 sin 2

;

m2H[prime][notdef] = f21 cos2 1+f22 cos2 2

45+ 45(1+tan 1 tan 2)

m212f1 cos 1f2 cos 2

: (B.8)

Similarly, the CP-odd neutral masses have

m2N0 = m2z0 = 0; (exact Goldstone bosons);

m2A01 = f21 sin2 1 + f22 sin2 2

JHEP12(2016)143

2 5 + 2 5 + 45 cot 1 cot 2

m212f1 sin 1f2 sin 2

;

m2A02 = f21 cos2 1 + f22 cos2 2

2 5 + 2 5 + 45 tan 1 tan 2

m212f1 cos 1f2 cos 2

;

m2A01A02 = ( 4 5)q

: (B.9)

Note that there are mixings between (A01; A02), a further rotation from (A01; A02) to the mass eigenstates (A0; A[prime]0) are needed with the mass eigenstates

m2A0,A[prime]0 = m2A01 + m2A02 [notdef]

q(m2A01 m2A02 )2 4m4A01A02: (B.10)

Finally, we obtain the masses for the SM-like Higgs boson and heavier Higgs boson. Assuming the radial modes are heavy, we obtain the the 2[notdef]2 mass matrices in the Re h01; Re h02

basis

M2Higgs =

f21 sin2 1 + f22 sin2 2

f

21 cos2 1 + f22 cos2 2

m2h1h1 m2h1h2 m2h1h2 m2h2h2

!: (B.11)

The mass matrices read

m2h1h1 =

1f21

345f22

2 (1 3 cos 4 1)

2 cos 2 1 cos 2 2

45f22 2

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

m212f2 f1

sin( 1 + 2)

sin 2 1 ;

m2h2h2 =

2f22

345f21

2 (1 3 cos 4 2)

2 cos 2 1 cos 2 2

45f21 2

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

m212f1 f2

sin( 1 + 2)

sin 2 2 ;

345f1f2

( 345+ 45)f1f2

2 cos 2( 1 2)m212 cos( 1 2): (B.12)

Similar to 2HDM, let us rotate the (h1; h2) to the mass eigenstates (h; H) with rotation angle :

m2h1h2 =

2 cos 2( 1+ 2)+

h

= cos sin sin cos

!

h1 h2

!: (B.13)

H

{ 34 {

Here the rotation angle is de ned as

tan 2 = 2m2h1h2 m2h1h1 m2h2h2

; (B.14)

and the mass eigenvalues are

m2h,H = m2h1 + m2h2 [notdef] q

(m2h1 m2h2)2 4m4h1h2: (B.15)

We identify h as the SM Higgs boson.

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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