Symmetries of the Two-Higgs Doublet Model (2HDM)

Eddie SantosFinal Presentation, Physics 251 6/8/11

Outline

• Introduction to Higgs & Motivations for an Extended Higgs Sector

• Two-Higgs Doublet Model (2HDM)

• Basic Transformations and Higgs Family (HF) Symmetries of the 2HDM

• General CP (GCP) Symmetries of the 2HDM

• Multiple Symmetries

Higgs Boson: Motivations

• Standard model of particle physics gauge group: SU(3)cxSU(2)LxU(1)Y.

• Weak-force gauge bosons W+, W-, and Z bosons, and fermions naively predicted to be massless. Not true.

• Cannot acquire mass by radiative corrections, need Goldstone bosons to generate masses for Gauge Bosons.

• Introduce a complex scalar SU(2) doublet that acquires a vacuum expectation value (vev).

• Breaks electroweak symmetry and gives mass to W/Z bosons.

• Can couple to fermions via Yukawa terms, which generates fermion masses after the scalar acquires its vev.

Motivations for Multi-Higgs Doublet Models

• One Higgs Doublet is only the simplest scenario, still lots of freedom for an extended Higgs sector.

• Two-Higgs doublet model is simplest extension to SM Higgs sector.

• Peccei-Quinn solution to the Strong CP problem requires two-Higgs doublets.

• Minimal Supersymmetric Standard Model (MSSM) requires two-Higgs doublets to cancel potential gauge anomalies of the Higgsino superpartners.

• Allows an opportunity for CP violation.

Elements of the Two-Higgs Doublet Model (2HDM)

• The 2HDM consists of two hypercharge-one complex scalar SU(2) doublets:

Φa(x) =

�φ+a (x)

φ0a(x)

�where a=1,2

• Each Higgs doublet has 4 degrees of freedom, so between the two doublets, there are 8 total d.o.f.

• 3 of these d.o.f are “eaten” by the W/Z bosons, thus leaving 5 Higgs bosons: h0 (light CP-even), H0 (Heavier CP-even), A0 (CP-odd), and H+/H-.

• Each doublet acquires a vev:

�Φa� =�

0va√2

�where v2 = v21 + v22 = (246 GeV)2.

Tree-Level 2HDM Potential in a Generic Basis

VH = m211Φ

†1Φ1 +m2

22Φ†2Φ2 − [m2

12Φ†1Φ2 + (m2

12)∗Φ†

2Φ1] +1

2λ1(Φ

†1Φ1)

2 +1

2λ2(Φ

†2Φ2)

2

+λ3(Φ†1Φ1)(Φ

†2Φ2) + λ4(Φ

†1Φ2)(Φ

†2Φ1) +

1

2[λ5(Φ

†1Φ2)

2 + λ∗5(Φ

†2Φ1)

2]

+[λ6(Φ†1Φ2) + λ∗

6(Φ†2Φ1)](Φ

†1Φ1) + [λ7(Φ

†1Φ2) + λ∗

7(Φ†2Φ1)](Φ

†2Φ2)

• Six real parameters:

• Four potentially complex parameters:

• Constraints: bounded from below, stable, preserves U(1)EM.

m211,m

222,λ1,λ2,λ3,λ4.

m212,λ5,λ6,λ7.

Example of a 2HDM: MSSM

• CP-conserving (all parameters are real).

λ1 = λ2 =1

4(g22 + g21),

λ3 =1

4(g22 − g21),

λ4 = −1

2g22 ,

λ5 = λ6 = λ7 = 0.

Basis-Independent 2HDM Formalism

• Use Einstein summation convention, where a,b,c,d = 1,2.

Z1111 = λ1, Z2222 = λ2, Z1122 = Z2211 = λ3,

Z1221 = Z2112 = λ4, Z1212 = λ5, Z2121 = λ∗5,

Z1112 = Z1211 = λ6, Z1121 = Z2111 = λ∗6,

Z2212 = Z1222 = λ7, Z2221 = Z2122 = λ∗7.

Y11 = m211, Y12 = −m2

12,

Y21 = −(m212)

∗, Y22 = m222,

VH = Yabֆ

aΦb +1

2Zabcd

(Φ†aΦb)(Φ

†cΦd),

Basis-Independent 2HDM Formalism

• Wide array of possible basis choices for the 2HDM. Some bases carry parameters that are not “physical” in the generic 2HDM.

• Want to select basis-independent parameters that are uniquely defined, gauge invariant, and well-behaved under renormalization.

• Symmetry of quartic coupling: .

• Hermiticity of potential:

VH = Yabֆ

aΦb +1

2Zabcd

(Φ†aΦb)(Φ

†cΦd),

Zabcd = Zcdab

(Yab)∗ = Yba, (Zabcd)

∗ = Zdcba.

Basis Transformations

• The most general symmetry group between the two doublets is SL(2,C), but SL(2,C) changes the form of the kinetic term. U(2) is the most general subgroup of SL(2,C) that leaves the kinetic term unchanged. Thus,

Φa → Φ�a = UabΦb,

Φ†a → Φ�†

a = Φ†bU †ba.

where the unitary 2 x 2 matrices U∈U(2) satisfy the condition

U †aeUeb = δab

Basis Transformations

• Consequently, the coefficients must transform as

and the vevs transform as

va → v�a = Uabvb.

Yab → Y �ab = UaαYαβU

†βb,

Zabcd → Z �abcd = UaαUcγZαβγδU

†βbU †δd.

2HDM Higgs Family (HF) Symmetries

• Let us assume the scalar potential has some explicit internal symmetry such that the coefficients of the potential stay exactly the same:

• S is a unitary matrix, thus the kinetic terms are unchanged. As a result of the symmetry,

Φa → ΦSa = SabΦb.

Yab = Y Sab = SaαYαβS

∗βb,

Zabcd = ZSabcd = SaαScγZαβγδS

∗βbS

∗δd

2HDM Higgs Family (HF) Symmetries

• The HF symmetry group must be a subgroup of the full U(2) transformation group. However, there is always a U(1) subgroup of U(2) in which the 2HDM scalar potential is invariant, which is the U(1)Y symmetry group:

• Invariance of U(1)Y is trivially guaranteed by the invariance SU(2)LxU(1)Y.

• HF symmetries are defined to be those Higgs family symmetries that are orthogonal to U(1)Y⊂U(2).

U(1)Y : Φa = eiθΦa.

Examples of HF symmetries

• Discrete Z2 preclude flavor-changing neutral currents (Glashow & Weinberg, Paschos):

• Actually the same as under a change of basis, though they have different impacts on the parameters of the potential.

• Peccei-Quinn continuous U(1): but a potential invariant under U(1) is invariant under Z2.

• Existence of any of these symmetries is enough to guarantee the existence of a basis in which all the potential parameters are real (CP-invariance).

Z2 : Φ1 → Φ1, Φ2 → −Φ2.

Π2 : Φ1 ↔ Φ2,

U(1) : Φ1 → e−iθΦ1, Φ2 → eiθΦ2

Simple Higgs Family Symmetry

• Consider a symmetry group G and require the following property: Scalar potential is invariant under a particular group element g (not equal to e) of G, sufficient to guarantee the scalar potential is invariant under the entire group G.

• Such symmetries are called simple Higgs Family symmetries.

• Examples: Discrete cyclic group Zn, Peccei-Quinn (PQ) U(1) symmetry.

• In the 2HDM, there are only two independent classes of simple HF symmetries: discrete Z2 flavor symmetry and a continuous PQ U(1) flavor symmetry.

Maximal HF Symmetry group

• Examine the largest possible HF symmetry group, U(2). We determine the maximal HF symmetry group to be the HF symmetry that is orthogonal to U(1)Y. We begin by nothing that U(2)≅SU(2)⊗U(1)Y/Z2. In our class, we’ve become all too familiar with the isomorphism:

• Therefore, U(2)≅SO(3)⊗U(1)Y, and we identify SO(3) as the maximal HF symmetry group. Imposes the following constraints:

• All parameters are real, therefore the Higgs sector obeying maximal HF symmetry is CP-conserving.

SO(3) ∼= SU(2)/Z2.

Generalized CP (GCP) Symmetries

• CP Transformation on scalar fields:

• GCP combines basis transformation (with unitary matrix X) with CP transformation:

• Leaves kinetic terms invariant.

• Potential invariant under GCP if and only if

Y ∗ab = X∗

aαYαβXβb = (X†Y X)ab,

Z∗abcd = X∗

aαX∗cγZαβγδXβbXδd.

GCP and Basis Transformations

• If the scalar potential is invariant under a GCP transformation with a unitary matrix X, then after an additional basis transformation, the potential is invariant under a new GCP transformation with a new unitary matrix X’ = UXUT.

• For every matrix X, there exists a unitary matrix U such that X’ can be reduced to the form (Ecker, Grimus, & Neufeld 1987):

• These leads to three classes of GCP symmetries:

X � = UXUT =

�cos θ sin θ− sin θ cos θ

�, where 0 ≤ θ ≤ π/2.

CP1 : θ = 0, CP2 : θ = π/2, CP3 : 0 < θ < π/2.

CP1: θ=0

• With θ=0, X’ is the unit matrix, and we get a standard CP transformation:

• The couplings are subject to the constraint

• This forces all couplings to be real, which means are all real, just what we’d expect for a CP symmetry.

CP1 : Φa → Φ∗a

Yab = Y ∗ab, Zabcd = Z∗

abcd.

m212,λ5,λ6,λ7.

CP2: θ=π/2

• With θ=π/2, X’ takes the form

• If this symmetry holds in one basis, it holds in all basis choices (Davidson & Haber, 2005).

• CP2 invariance implies:

• Under these conditions, we can always find a basis where all the parameters are real (Gunion & Haber, 2005), implying that if a potential is invariant under CP2, then there is a basis where CP2 still holds and the potential is also invariant under CP1.

X � =

�0 1−1 0

�=⇒ Φa → �abΦ

∗b .

m211 = m2

22,m212 = 0,λ1 = λ2,λ6 = −λ7.

CP3: 0<θ<π/2

• Analysis is a little more complicated (see Ferreira, Haber, Silva, 2009) but the constraints imposed on the potential are:

• Results are independent of the angle so long as it is between the bounds!

Impact of Symmetries on 2HDM Potential

Ferreira, Haber, & Silva (2009)

Multiple Applications of GCP Symmetries

• If we apply a GCP transformation twice, we get

• Thus, We can apply this to our three classes of GCP symmetries previously discussed:

• CP1: In this case, we have X=1, so (CP1)2 = 1. Thus a CP1 invariant potential is invariant under the symmetry group Z2 = {1,CP1}.

• CP2: Here which implies (CP2)2 = -1. Thus a CP2 invariant potential is invariant under the symmetry group Z4 = {1,CP2,-1,-CP2}.

(ΦGCPa )GCP = Xaα(Φ

GCPα )∗ = XaαX

∗αbΦb.

(GCP)2 = XX∗.

X =

�0 1−1 0

�,

Multiple Applications of GCP Symmetries

• CP2(cont): If we denote (Z2)Y = {1,-1} as the two-element discrete subgroup of the global hypercharge U(1)Y, then the Z4/(Z2)Y≅Z2, and we conclude that the CP2-invariant potential exhibits a Z2 symmetry orthogonal to the Higgs flavor symmetries of the potential.

• CP3: The matrix X is given by which can not be reduced to the identity. A CP3 invariant potential does however, exhibit a Z2 symmetry that is orthogonal to the Higgs flavor symmetries of the potential.

X =

�cos 2θ sin 2θ− sin 2θ cos 2θ

�,

Multiple Symmetries and GCP

• Consider possibility of imposing more than one symmetry requirement on the Higgs potential.

• Case 1: Impose Z2 and Π2 symmetry in the same basis. This leads to:

• This is simply CP2 in the basis λ6=0. Using Z2 ⊕ Π2 to denote the simultaneous symmetries of Z2 and Π2, we conclude Z2 ⊕ Π2 = CP2 in the basis λ6=0.

• Note that we cannot do this for CP1. CP1 reduces the potential to 9 independent real parameters. The smallest HF symmetry Z2 can only reduce the potential to 8 real parameters.

Multiple Symmetries and GCP

• Case 2: Impose U(1) and Π2 symmetry in the same basis. This leads to:

• This isn’t the CP3 explicitly shown, but we can choose a basis transformation U that takes Re λ5= λ1 - λ3 - λ4 , where λ5 = 0. Thus we conclude U(1) ⊕ Π2 = CP3 in some basis.

• Note that we cannot do this for CP1. CP1 reduces the potential to 9 independent real parameters. The smallest HF symmetry Z2 can only reduce the potential to 8 real parameters.

• Case 3: Impose U(1) (PQ) and CP3 in the same basis. We obtain precisely the constraints resulting from the SO(3) HF symmetry. Thus U(1) ⊕ CP3 = SO(3).

Maximal Symmetry Group of the Potential Orthogonal to U(1)Y

• The standard CP symmetry CP1 is a discrete Z2 symmetry is not a subgroup of the U(2) basis transformation group.

• We have noted that 2HDM scalar potentials that exhibit any nontrivial HF symmetry G is automatically CP conserving. Thus the maximal symmetry group is thus the semi-direct product of G and Z2. The maximal symmetry groups are listed below (ex: U(1)⋊ Z2 ≅ SO(2)⋊Z2≅O(2), and SO(3)⊗Z2≅O(3)).

Conclusions

• Higgs mechanism provides masses for W/Z bosons and fermions. Two-Higgs doublets is the simplest extension to the SM and contains 5 Higgs bosons.

• Can impose discrete symmetries and continuous symmetries that lead to constraints on the parameters of the 2HDM.

• Three classes of Generalized CP (GCP) symmetries, which are related to multiple applications of the aforementioned discrete and continuous symmetries in specific bases.

Bibliography

J. F. Gunion, H. E. Haber, G. Kane and S. Dawson, The Higgs Hunter’s Guide (Perseus Publishing, Cambridge, MA, 1990).

S. Davidson and H. E. Haber, Phys. Rev. D 72, 035004 (2005)

P. M. Ferreira, H. E. Haber, and J. P. Silva Phys. Rev. D 79, 116004 (2009)

P. M. Ferreira and J. P. Silva, Phys. Rev. D 78, 116007 (2008)