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

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

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

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

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

6. WHY: THE HIGGS BOSON! 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! 6

7. 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…

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

9. BUT IT MUST LEAVE OTHER SYMMETRIES INTACT!
Before we continue:
To give mass to elementary particles, a new field is introduced in the 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 the 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!

10. 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 “G0” are Goldstone bosons, non-physical fields which will vanish in the theory.
“v” is the vacuum expectation value (VEV), the value that  has at the vacuum – it is a constant,
v = 246 GeV

11. How do you give a vev to a scalar field?
Higgs Potential:
μ2 < 0 => minimum reached when |Φ|
μ2 > 0 => minimum reached when |Φ| = 0

12. Mass and gauge symmetries

13. Mass and gauge symmetries

15. Mass and gauge symmetries

16. Mass and gauge symmetries

17. Mass and gauge symmetries

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

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

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

22. For neutrinos in particular this is very relevant…
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...

23. 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.

24. Mass term for the electron (any fermion...):
But...
This is NOT gauge invariant (“neutral”) because eL 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.

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

27. And now for something completely different...

28. G.C. Branco, P.M. Ferreira, L. Lavoura, M. Rebelo, M. Sher, J.P Silva,
LHC discovered a new particle (a scalar?) with mass ~125 GeV.
Up to now, all is compatible with the Standard Model (SM) scalar particle.
BORING!
Two Higgs Dublet model, 2HDM (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)

29. TWO HIGGS DOUBLET MODELS
They are the simplest Standard Model extension – instead of a single scalar doublet, we have two, Φ1 and Φ2.
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. D8 (1973) 1226)

30. The Two-Higgs Doublet potential
Most general SU(2) × U(1) scalar potential (BLACKBOARD):
m212, λ5, λ6 and λ7 complex - seemingly 14 independent real parameters
Most frequently studied model: softly broken theory with a Z2 symmetry,
Φ1 → - Φ1 and Φ2 → Φ2, meaning λ6, λ7 = 0.
It avoids potentially large flavour-changing neutral currents (FCNC – EXPLAIN!)

31. Softly broken Z2 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 Φ2 couples to fermions.
MODEL II: Φ2 couples to up-quarks, Φ1 to down quarks and leptons.
We’ll get back to this...

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

33. Scalar sector of the 2HDM is richer => more stuff to discover
Two dublets => 4 neutral scalars (h, H, A) + 1 charged scalar (H±).
h, H → γ γ
h, H → ZZ, WW (real or off-shell)
h, H → ff
H → hh (if mH>2mh) … h H
A CP-odd scalar
(pseudoscalar)
CP-even scalars
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)

34. Where do these scalars come from?
Doublet field components:
Both doublets may acquire vevs, v1 and v2, such that
which are the solutions of the minimisation conditions,

35. 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

36. 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, G0, which gives mass to the Z gauge boson – and a non-zero eigenvalue,

37. (h, H: CP-even scalars):
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

38. Gauge boson masses in 2HDM
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,

39. 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 2HDM!

40. Decoupling and Alignment

41. 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 Φ2 (neutrinos not considered!):
How does this work? It corresponds to the way each field is chosen to transform under the Z2 symmetry.
SCALAR DOUBLETS: Φ1 → - Φ1 , Φ2 → Φ2
FERMION (LEFT) DOUBLETS: QL → QL, LL → LL
FERMION (RIGHT) SINGLETS: always tR → tR (for all up quarks) but four possible choices for bR (and al down quarks) and τR (and all charged leptons).
Up quarks
Down quarks
Leptons

42. It’s actually very simple...
Most general quark and lepton yukawa lagrangian in the 2HDM (3rd generation only):
MODEL I: bR and τR do not change.
Φ1 → - Φ1
Φ2 → Φ2
QL → QL
LL → LL
ALL FERMIONS COUPLE ONLY TO 2

43. DOWN QUARKS AND CHARGED LEPTON COUPLE TO 1
Φ1 → - Φ1
Φ2 → Φ2
QL → QL
LL → LL
MODEL II: bR → - bR , τR → - τR
LEPTON-SPECIFIC MODEL: bR → bR , τR → - τR
FLIPPED MODEL: bR → - bR , τR → τR
UP QUARKS COUPLE TO 2
DOWN QUARKS AND CHARGED LEPTON COUPLE TO 1

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

46. The vacuum structure of the 2HDM
Vaccuum structure more rich => different types of stationary points/minima possible!
The NORMAL minimum,
The CHARGE BREAKING (CB) minimum, with
c2 has electric charge => breaks U(1)em
The CP BREAKING minimum, with
θ ≠ 0, π breaks CP

47. Global minimum – CHARGE BREAKING
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

48. If N is a minimum, it is the deepest one, and stable against CB or CP!
THEOREM: if a Normal Minimum exists, the Global minimum of the 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!

49. Now there isn’t an obvious minimum, each of them can be the deepest!
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!

50. 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 Z2 symmetries
(exception – INERT MODEL!)
Can occur if there is soft symmetry breaking!
(or for potentials with only CP symmetry, or
no symmetry at all)

51. Inert vacua – preserve Z2 symmetry
The INERT minimum,
Since only Φ1 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 Z2 symmetry – m12 term must be zero (no decoupling limit, extra scalar masses cannot be too heavy).

52. The unbroken Z2 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:

53. 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.

54. CP VIOLATION IN 2HDM
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!

55. 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 2HDM there are other possibilities of getting CP violation...

56. Spontaneous CP violation (SCPV) in 2HDM
Impossible in the SM.
The actual reason why the 2HDM 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 2HDM where SCPV can occur: softly broken Z2, all parameters real – the usual model!

57. Which means that the minimisation conditions will require
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,
It is easy to see that we need BOTH the soft breaking term m12 and the quartic coupling λ5 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
The CP BREAKING minimum, with
θ ≠ 0, π breaks CP

58. 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, h1, h2 and h3, 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, α1, α2, and α3.
The couplings to gauge bosons and fermions now are more complicated functions of these new angles.

59. Explicit Soft CP violation in 2HDM
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 Z2 symmetry is still used...
... and we still consider the softly broken Z2, but now parameters in the potential can be complex!
Minimisation conditions become

60. The phases of the two parameters are NOT independent, the minimisation conditions yield
This model is known as the Complex Two-Higgs Doublet Model (C2HDM) 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, α1, α2, and α3 like the SCPV case.