Theory Foundations of the Two Higgs Doublet Model

Theory Foundations of the Two Higgs Doublet Model LIP, 11/06/2018 Pedro Ferreira ADF, ISEL e CFTC, UL

OUTLINE The Higgs mechanism in the Standard Model The Two-Higgs Doublet Model (2 HDM) Couplings between scalars and gauge bosons and fermions in the 2 HDM The vacuum structure of the 2 HDM Dark Matter in the 2 HDM – the Inert Model

Matter is composed of elementary particles - FERMIONS Interactions are mediated by particles – GAUGE BOSONS

One Higgs to Give to Them (almost) Them All. . . One Ring Mass to Rule All. . .

Before we go to TWO Higgs. . . WHAT IS THE HIGGS BOSON? HOW DOES IT WORK? WHY DO WE NEED THE HIGGS?

FUNDAMENTAL PRINCIPLE OF PARTICLE PHYSICS - When you have an unsolvable problem… … invent a new particle! WHY: MATTER + INTERACTIONS + SIMMETRY = ALL MASSES ZERO! HOW DO ELEMENTARY PARTICLES GAIN THEIR MASS? Due to their interactions with a misterious particle, which was only discovered in 2012, almost 50 years after it had been proposed THE HIGGS BOSON!

The beauty of the Higgs Mechanism is that allows us to preserve, in particle physics, all that’s GOOD about symmetries, but at the same time “breaks” them in such a way that elementary particles acquire mass – though not all of them! The photons and gluons remain massless! The way to accomplish this is, in a certain way, to leave the symmetry to its own devices: by itself, it will “break” in spontaneous manner…

BEFORE SPONTANEOUS SYMMETRY BREAKING: EMPTY SWIMMING POOL. AFTER SPONTANEOUS SYMMETRY BREAKING: : THE POOL IS FULL OF WATER!

Before we continue: • To give mass to elementary particles, a new field is introduced in theory, the Higgs field. • The state of minimal energy of this field – the vacuum of the model – breaks some symmetries. . . • BUT IT MUST LEAVE OTHER SYMMETRIES INTACT! • It is possible to show that if the vacuum of theory possesses some quantum numbers, then the symmetries behind those quantum numbers are BROKEN. • So, the Higgs field must be a SCALAR – spin 0, so that Angular Momentum Conservation still holds – and the vacuum must be NEUTRAL - so that electric charge is still conserved and Electromagnetism still works!

• It is the electroweak symmetry (with a gauge group given by SU(2)×U(1)) which must be broken, which implies the Higgs field must be a doublet of the gauge symmetry group, SU(2): • “h” is the Higgs Field, the one that LHC discovered. • “G+” and “G 0” are Goldstone bosons, non-physical fields which will vanish in theory. • “v” is the vacuum expectation value (VEV), the value that has at the vacuum – it is a constant, v = 246 Ge. V

How do you give a vev to a scalar field? Higgs Potential: μ 2 > 0 => minimum reached when |Φ| = 0 μ 2 < 0 => minimum reached when |Φ| 0

Mass and gauge symmetries

Mass and gauge symmetries

Mass and gauge symmetries

Mass and gauge symmetries

Mass and gauge symmetries

How does the Higgs work exactly. . . ? The kinetic + potential terms involving the potential are: CHOICE: Y = 1

We then find, reexpressing the lagrangian in terms of these physical fields,

What about fermions. . . ? DIRAC EQUATION DIRAC LAGRANGIAN Mass Term

To define an antiparticle, it is not sufficient to swap the SIGN of the electric charge – it is also necessary to swap parity. For neutrinos in particular this is very relevant… ALL FERMIONS ARE COMPOSED OF “LEFT” AND “RIGHT” COMPONENTS, AND THEIR WEAK INTERACTIONS ARE SENSITIVE TO THESE COMPONENTS. . .

• The Electromagnetic Force and the Strong Nuclear Force do not distinguish between the Left and Right components of the fermions. . . • . . . BUT THE WEAK INTERACTIONS DO! • The W bosons, for instance, ONLY INTERACT WITH THE LEFT PARTS OF THE FERMIONS! • Experimental results suggest that the LEFT component of the fermions “lives” inside a SU(2) DOUBLET, but the RIGHT component is described by an SU(2) SINGLET.

Mass term for the electron (any fermion. . . ): But. . . This is NOT gauge invariant (“neutral”) because e. L belongs to a doublet of SU(2), just like the Higgs field! (BLACKBOARD) It’s like trying to sum spins ½ and spin 0: we need another doublet to get a final term with “spin 0”: that is why we need to introduce the new doublet, the Higgs field.

And if. . . Interactions between the Higgs physical particle and the electrons Mass term for the electrons!!

And now for something completely different. . .

• LHC discovered a new particle (a scalar? ) with mass ~125 Ge. V. • Up to now, all is compatible with the Standard Model (SM) scalar particle. BORING! Two Higgs Dublet model, 2 HDM (Lee, 1973) : one of the easiest extensions of the SM, with a richer scalar sector. Can help explain the matter-antimatter asymmetry of the universe, provide dark matter candidates, … G. C. Branco, P. M. Ferreira, L. Lavoura, M. Rebelo, M. Sher, J. P Silva, Physics Reports 716, 1 (2012)

TWO HIGGS DOUBLET MODELS • They are the simplest Standard Model extension – instead of a single scalar doublet, we have two, Φ 1 and Φ 2. • They do not affect the most successful predictions of the Standard Model. • They have a richer scalar particle spectrum. • They are included in more general models, such as the Supersymmetric one. • They allow for the possibility of minima with spontaneous breaking of CP. . . (T. D. Lee, Phys. Rev. D 8 (1973) 1226)

The Two-Higgs Doublet potential Most general SU(2) × U(1) scalar potential (BLACKBOARD): m 212, λ 5, λ 6 and λ 7 complex - seemingly 14 independent real parameters Most frequently studied model: softly broken theory with a Z 2 symmetry, Φ 1 → - Φ 1 and Φ 2 → Φ 2, meaning λ 6, λ 7 = 0. It avoids potentially large flavour-changing neutral currents (FCNC – EXPLAIN!)

Softly broken Z 2 potential • EIGHT real independent parameters (all assumed real). Allows a decoupling limit. • The symmetry must be extended to the whole lagrangian, otherwise the model would not be renormalizable. Coupling to fermions MODEL I: Only Φ 2 couples to fermions. MODEL II: Φ 2 couples to up-quarks, Φ 1 to down quarks and leptons. . We’ll get back to this. . .

THEORETICAL BOUNDS ON 2 HDM SCALAR PARAMETERS Potential has to be bounded from below: Theory must respect unitarity:

Scalar sector of the 2 HDM is richer => more stuff to discover Two dublets => 4 neutral scalars (h, H, A) + 1 charged scalar (H±). h CP-even scalars H A - CP-odd scalar (pseudoscalar) h, H → γ γ h, H → ZZ, WW (real or off-shell) h, H → ff H → hh (if m. H>2 mh) … A→ γ γ A→ ZZ, WW A→ ff … Certain versions of the model provide a simple and natural candidate for Dark Matter – INERT MODEL, based on an unbroken discrete symmetry. Deshpande, Ma (1978); Ma (2006); Barbieri, Hall, Rychkov (2006); Honorez, Nezri, Oliver, Tytgat (2007)

Where do these scalars come from? Doublet field components: Both doublets may acquire vevs, v 1 and v 2, such that which are the solutions of the minimisation conditions,

Scalar mass eigenstates The masses of the scalar particles are the eigenvalues of the matrix of second derivatives at a minimum, This is an 8× 8 matrix, but which breaks into BLOCKS! (BLACKBOARD) Let us re-define the doublet components with the more used notation, The UPPER fields have electric charge, the LOWER fields are neutral. The mass matrices are

For the CHARGED SCALARS, This matrix has one zero eigenvalue – the charged goldstone boson, G+, which gives mass to the W gauge boson – and a non-zero eigenvalue, For the PSEUDOSCALARS, Also one zero-mass eigenstate – the neutral goldstone boson, G 0, which gives mass to the Z gauge boson – and a non-zero eigenvalue,

For the (CP-EVEN) SCALARS, This matrix has two non-zero eigenvalues – the lightest is usually called “h” and the heavier, “H”. Definition of β angle: Definition of α angle (h, H: CP-even scalars): (without loss of generality: -π/2 ≤ α ≤ + π/2) The α angle is the diagonalization angle of the 2× 2 mass matrix of the CP-even scalars

Gauge boson masses in 2 HDM Obtained, like in the SM, from the kinetic terms of the scalar terms, excpet now there are contributions from TWO doublets: with identical definitions for the covariant derivatives of the doublets, from which we obtain similar mass terms for the gauge fields, and therefore the masses for the gauge bosons are very similar to the SM expressions,

Notice that since we are only including doublets one of th most stringent Electroweak precision constraints is automatically verified (at tree-level), DO IN BLACKBOARD, SM vs 2 HDM!

Decoupling and Alignment

Coupling to Fermions Each type of fermion only couples to ONE of the doublets. Four possibilities, with the convention that the up-quarks always couple to Φ 2 (neutrinos not considered!): Up quarks Down quarks Leptons How does this work? It corresponds to the way each field is chosen to transform under the Z 2 symmetry. SCALAR DOUBLETS: Φ 1 → - Φ 1 , Φ 2 → Φ 2 FERMION (LEFT) DOUBLETS: QL → QL, LL → LL FERMION (RIGHT) SINGLETS: always t. R → t. R (for all up quarks) but four possible choices for b. R (and al down quarks) and τR (and all charged leptons).

It’s actually very simple. . . Most general quark and lepton yukawa lagrangian in the 2 HDM (3 rd generation only): Φ 1 → - Φ 1 Φ 2 → Φ 2 QL → QL LL → LL MODEL I: b. R and τR do not change. ALL FERMIONS COUPLE ONLY TO 2

MODEL II: b. R → - b. R , τR → - τR Φ 1 → - Φ 1 Φ 2 → Φ 2 QL → QL LL → LL UP QUARKS COUPLE TO 2 DOWN QUARKS AND CHARGED LEPTON COUPLE TO 1 LEPTON-SPECIFIC MODEL: b. R → b. R , τR → - τR FLIPPED MODEL: b. R → - b. R , τR → τR

Constraints from b-physics and others Most important and reliable constraints from b → sγ and Γ(Z → bb) observables. Significant theory uncertainties!

The vacuum structure of the 2 HDM Vaccuum structure more rich => different types of stationary points/minima possible! The NORMAL minimum, The CHARGE BREAKING (CB) minimum, with c 2 has electric charge => breaks U(1)em The CP BREAKING minimum, with θ ≠ 0, π breaks CP

Would there be any problem if the potential had two of these minima simultaneously? Answer: there might be, if the CB minimum, for instance, were “deeper” than the normal one (metastable). Local minimum NORMAL Global minimum – CHARGE BREAKING

THEOREM: if a Normal Minimum exists, the Global minimum of theory is Normal - the photon is guaranteed to be massless. (not so in SUSY, for instance) Barroso, Ferreira, Santos If N is a minimum, it is the deepest one, and stable against CB or CP!

But there is another possibility – AT MOST TWO NORMAL MINIMA COEXISTING… Ivanov The relationship between the depths of the potential at both minima is given by Now there isn’t an obvious minimum, each of them can be the deepest!

So, though our vacuum cannot tunnel to a deper CB or CP minimum, there is another scary prospect… Local minimum NORMAL Global minimum – ALSO NORMAL PANIC VACUUM!! Cannot occur for SUSY, models with exact U(1) or Z 2 symmetries (exception – INERT MODEL!) Can occur if there is soft symmetry breaking! (or for potentials with only CP symmetry, or no symmetry at all)

Inert vacua – preserve Z 2 symmetry • The INERT minimum, • Since only Φ 1 has Yukawa couplings, fermions are massive – “OUR” minimum. WHY BOTHER? In the inert minimum, the second doublet originates perfect Dark Matter candidates! Inert neutral scalars do not couple to fermions or have triple vertices with gauge bosons. Only possible with EXACT Z 2 symmetry – m 12 term must be zero (no decoupling limit, extra scalar masses cannot be too heavy).

The unbroken Z 2 symmetry acts as a conserved quantum number – the “darkness” of each particle. All the scalars of the first doublet (only h!) have “darkness” equal to +1, all the scalars of the second doublet (H, A and H !) have “darkness” equal to -1. Fermions and gauge bosons are not “dark”, thus their “charge” under this symetry is +1. Since this quantum number is preserved, many interactions/vertices are no longer possible:

The “dark” scalars do not couple directly to fermions, which will make them xtremely difficult to detect – perfect dark matter candidates! Sin the “darkness” quantum number must be conserved, “dark” scalars must be produced in pairs! Which means that the lightest of the “dark” scalars will be STABLE (cannot decay to another “dark” scalar). “Dark” scalars wold manifest as missing energy in detectors. . . The Inert Model is a very simple Dark Matter Model, but it is heavily constrained from cosmological/astrophysical dark matter search data.

CP VIOLATION IN 2 HDM • In the SM, the only source of CP violation is the CKM matrix. • The CKM matrix arises because of a “mismatch” between the rotations needed to diagonalize the up-type and down-type. quark mass matrices: CKM matrix: • The CKM matrix has a complex phase in it – it would not be there if there were only ONE or TWO generations!

• This is an EXPLICIT breaking of the CP symmetry – because the Yukawa couplings are complex to begin with, the SM is CP-violating “by hand”. • Any effects of CP-violation in the SM are all proportional to the quantity called the Jarlskog invariant (combination of quark masses and CKM matrix elements) and do not affect scalar interactions until at least the three -loop level. • In the 2 HDM there are other possibilities of getting CP violation. . .

Spontaneous CP violation (SCPV) in 2 HDM • Impossible in the SM. • The actual reason why the 2 HDM was created. • To happen, model needs to: (a) be CP-invariant to begin with (“real”); (b) have a vacuum which breaks CP (“complex”) • Simplest form of the CP transformation of scalar doublets: (forget about the parity transformation, it essentially is complex conjugation) • Simplest 2 HDM where SCPV can occur: softly broken Z 2, all parameters real – the usual model!

• But the BIG difference is that, for some regions of parameter space, the potential may have a different vaccum. Instead of REAL VEVS, there is another possibility, The CP BREAKING minimum, with θ ≠ 0, π breaks CP It is easy to see that we need BOTH the soft breaking term m 12 and the quartic coupling λ 5 to obtain anon-trivial phase θ – those are the only two terms which give a dependence on θ, so that with the vevs above the potential becomes Which means that the minimisation conditions will require

• What are the consequences of spontaneous CP violation? • THERE CEASES TO BE A DISTINCTION BETWEEN SCALARS AND PSEUDOSCALARS! THE NEUTRAL MASS MATRIX MIXES REAL AND IMAGINARY PARTS! • Instead of a 4 × 4 mass matrix breaking into two 2 × 2 submatrices corresponding to the CP-even and CP-odd components, now the matrix yields the neutral Goldstone boson and leaves a 3 × 3 mass matrix corresponding to a mix of CP even and odd states (BLACKBOARD). • All three scalar mass eigenstates, h 1, h 2 and h 3, can now decay to pairs of Z bosons, for instance! • Instead of diagonalization with the angles β and α, we now need FOUR: still β and three angles, α 1, α 2, and α 3. • The couplings to gauge bosons and fermions now are more complicated functions of these new angles.

Explicit Soft CP violation in 2 HDM • Another possibility is to emulate what happens in the SM: the CP symmetry is broke ab initio, explicitly, in the Yukawa sector and ALSO in the scalar sector. • But to avoid FCNC, the Z 2 symmetry is still used. . . • . . . and we still consider the softly broken Z 2, but now parameters in the potential can be complex! • Minimisation conditions become

• The phases of the two parameters are NOT independent, the minimisation conditions yield • This model is known as the Complex Two-Higgs Doublet Model (C 2 HDM) and it is one of the simplest models to describe the possibility of mixed-CP scalars at the LHC. • Scalar mass matrix again 3 × 3 , diagonalized by rotation matrix R, written in terms of three angles, α 1, α 2, and α 3 like the SCPV case.