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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 (2HDM) Couplings between scalars and gauge bosons and fermions in the 2HDM

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Theory Foundations of the Two Higgs Doublet Model

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  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. 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!

  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. 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| 0 μ2> 0 => minimum reached when |Φ| = 0

  12. Mass and gauge symmetries

  13. Mass and gauge symmetries

  14. Mass and gauge symmetries

  15. Mass and gauge symmetries

  16. Mass and gauge symmetries

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

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

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

  20. To define an antiparticle, it is not sufficient to swap the SIGN of the electric charge – it is also necessary to swapparity. 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...

  21. 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.

  22. 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.

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

  24. And now for something completely different...

  25. 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)

  26. 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)

  27. The Two-Higgs Doublet potential Most general SU(2) × U(1) scalar potential (BLACKBOARD): m212, λ5, λ6andλ7complex - 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!)

  28. Softly broken Z2 potential • EIGHTreal 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...

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

  30. 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)

  31. 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,

  32. 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

  33. 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,

  34. 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

  35. 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,

  36. 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!

  37. Decoupling and Alignment

  38. 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: alwaystR → 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

  39. 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

  40. 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

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

  42. 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, πbreaks CP

  43. 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

  44. 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 IfNis a minimum, it is the deepest one, and stable against CBorCP!

  45. 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!

  46. 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)

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