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Hierarchy, Naturalness evjas-2013 Hierarchy, Naturalness... seeking help from symmetry yet again ? Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 0 evjas-2013 The S.M. – complete ? • In the last ∼50 years most aspects of the S.M have been increasingly well tested – this includes the “extreme regimes” such as top physics; Bs os- + − cillations; Rare decays [Bs µ µ : Phys.Rev.Lett.110, 021801 (2013)] → • EWSB was the exception – but now we seem to have a Higgs boson (is it the only one? is it the S.M. one?) • This minimal option [the Higgs + nothing else] comes with poten- tially serious internal consistency problems. • The “Hierarchy (or Naturalness) Problem” is discussed here. Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 1 evjas-2013 Back to the Big SM Picture • Reminder: processes like W boson pair production or WW scattering violate unitarity at ∼2 TeV • To restore unitarity “minimally” (i.e. one extra particle) the only possibility is – a SM-like Higgs Boson: spin zero, and – Yukawa couplings proportional to particle masses • But... a new problem creeps in... – such a fundamental scalar has quadratically divergent mass corrections • This is known as the Naturalness or Fine Tuning or Hierarchy problem. The required energy to probe this “conflict” seems to be the TeV region. =⇒ LHC Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 2 evjas-2013 Boundary Physics Scales • One example: ΛQCD ≈ 300 MeV – represents a broad boundary between the perturbative and non-perturbative regimes of QCD • For the EW interactions we can set a ΛEW ≈ 1 TeV as determined by vacuum stability requirements, perturbativeness alternative: we did observe the Higgs, and of the higgs self coupling, etc ΛEW ≈ v ≈ 246 GeV • Above ΛEW the next “new” physics scale that we know of would be necessarily associated with gravitation: 2 hc¯ 5 18 – The Planck Mass: MP c = ≈ 2.4 × 10 GeV 8πGN where presumably quantum gravity effects must become relevant, 2 −33 with associated (de Broglie) length scale λP =¯h/(MP c ) ≈ 10 m • Or else you consider the possibility of the RGE meeting of the SM coupling 16 constants at the GUT scale ΛGUT = 10 GeV Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 3 evjas-2013 Grand-Unification Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 4 evjas-2013 The Hierarchy of Scales • The distance between ΛEW and the next higher scale poses a potential conflict for a fundamental scalar field in the SM • Consider the one-loop order corrections for the mass of a particle, using the ΛUV cut-off regularization plus counter-term renormalization scheme • Example, the QED case: for the electron mass the 1st order radiative corrections have a logarithmic divergence given by phys bare 3α ΛUV me = me (1 + ln( )+ · · ·) 2π me • You can check that for all meaningful values of ΛUV the magnitudes of phys bare 80 ⇒ phys ∼ bare me and me will be the same. example: ΛUV = 10 GeV me 2 me • Next consider more general cases... Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 5 evjas-2013 Mass corrections at 1-loop Consider a fermion coupled to a higgs-like scalar, both massive and • m = λ v/√2 (i.e. the Yukawa f h interaction) f f − µ 2 ∗ = iψγ¯ ∂µψ (λ ψ¯LφψR + h.c.) m φ φ L − f − h The mass renormalization for f will be • 3λ2 m 2 f f ΛUV δmf = log + − 32π2 m2 ··· f which is of the same -magnitude as m O f Now consider the limit m 0 ( δm = 0) • f → ⇒ f – the fermion and the higgs decouple iα iβ – ψL and ψR are independently gauge-symmetric (ψL e ψL ; ψR e ψR) → → – So, m 0 increases the symmetry of the theory as expected because the f → Yukawa breaking is proportional to mass as a consequence, δmf mf ∗ ≈ • Fermions – logarithmic sensitivity to high mass scales • Scalars – quadratic sensitivity to high mass scales Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 6 evjas-2013 Scalars behave quite differently Masses of fundamental scalars are not stable against radiative quantum corrections. λf = f.H.f Yukawa coupling f S λS = Higgs self coupling H H (a) (b) 2 2 |λf | − 2 2 (a) ∆MH = 16π2 2ΛUV + 6mf ln(ΛUV /mf ) + ... 2 λS 2 − 2 (b) ∆MH = 16π2 +2ΛUV 2mS ln(ΛUV /mS ) + ... where, ΛUV is a momentum cutoff used to regulate the loop integrals. a fundamental scalar has quadratically divergent mass corrections Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 7 evjas-2013 Numbers, within the S.M. 2 • The LHC has seen a Higgs candidate at MH =125 GeV/c • Radiative EW(SM) mass corrections integrated up to Λ (LO) will shift the bare value of the higgs mass upwards by (see C.Quigg’s lectures) 2 2 2 6GF Λ 2 1 2 1 2 1 2 2 Λ ∆MH = mt mW mZ mH (125GeV ) √2π2 − 2 − 4 − 4 ≈ × 400GeV • In particular, if the “desert” is true, 2 2 −1 18 2 Λ ∼ MPlanck = (8πGNewton) ∼ (2.4 × 10 GeV) • However, for SM consistency w/ EW, Λ must be below ∼ 1 TeV... • The way out is the existence of larger counterterms that must be inter- preted as manifestations of new physics at some higher energy scale Λ¯ Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 8 evjas-2013 The Hierarchy Problem QED − like finite 2 1 − 2 · 2 δMH = 2 (λS λf ) ΛUV + + 8π log. diverg. terms If we consider that the natural cut-off can only be the GUT or Planck scales • then in order to have MH within the bounds of ΛEW a counter-term adjustment • of 1 part in 1015 is required at each order in perturbation theory... This is NOT a renormalizability problem. The quadratic divergences can be • renormalized away in the same way as is done for the log divergence in QED. — the problem is the fine tuning of the counter-term CRITICISM The problem is a consequence of attributing a physical • significance to the cut-off ΛUV — is it really connected to physical scales ? And if the SM is not contained in a theory of gravitation ? • — why think of ΛUV MP ? −→ Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 9 evjas-2013 Comments • If the δMH calculation is performed in the DR scheme (D =4+ ǫ) – one obtains only 1/ǫ singularities which are absorbed into the defini- tions of the counter-terms as usual – the scales (or tuning) conflict does not show up • A divergent MH means a divergent VEV, and so the fine tuning applies not just to the Higgs but to all particles (?) • It is generally thought that if there is legitimate need to stabilize the EW sector within the observed scale – then the stabilizing mechanism is associated with new physics beyond the current (minimal) scheme • Another well known hierarchy problem is the spectrum of fermion masses 6 (e.g. why mt ≈ 10 me ?) — are these two hierarchy puzzles related ? Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 10 evjas-2013 The SUSY cure 2 1 2 2 QED like finite δMH = 2 (λS λf ) ΛUV + − + 8π − · log. diverg. terms Boson and fermion loops have opposite signs. Exact cancellations are possible • 2 (say: λf = λS ) but require a deep relationship between fermions and bosons. – If each SM fermion is accompanied by two complex scalars with identical coupling strengths then the cancellation is guaranteed to all orders. This kind of perfect conspiracy can only arise from • a fermion-boson symmetry This constitutes one of the foremost arguments in favor of • SUPERSYMMETRY Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 11 evjas-2013 A Famous Hierarchy Problem/Cure 17 • The elementary (point-like: r < 10− cm) electron self-energy crisis in mobsc2 e2 m c2 e = + ∆ e classical electrodynamics 4πr 4 which 0.5MeV = 10 MeV + ?? ?? physics – a one percent difference in the bare mass changes the observed mass mobs r< 13cm 200-fold !! plus: corrections surpass e for 10− • The hierarchy problem in the electron self-mass reaches stability through QED vacuum quantum fluctuations and a symmetry principle (positrons). · e e · ¨ e ¨ ¡ ¡ ¡ e e e ÔÓÐÖÞØÓÒ – The CP symmetry doubles the number of existing particles. Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 12 evjas-2013 Another hint of a Chiral Anomalies deep fermion-boson connection ? • Weak gauge bosons couple differently to RH and LH fermions, with pro- found implications in most aspects of HEP theory and phenomenology. • Theory wise, one of the most significant (and problematic) consequences is the introduction of chiral anomalies, a nasty type of current non- conservation. • Theories (such as QED or QCD) in which the gauge bosons couple equally to the RH and LH species do not exibit chiral anomalies. • In the EW theory, the cancellation of chiral anomalies stems from – the tight structure displayed by the fermion generations – the quarks-only three color degrees of freedom – the assigned fermion hypercharges • This suggests the existence of an underlying relationship among fermions and bosons, and some deeper structure to which the generations pattern is a consequence. Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 13 evjas-2013 Discussing Nambu- Goldstone Bosons are they physical degrees of freedom ? Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 14 evjas-2013 About SSB, and the two known sources of mass • QCD contains the -Higgsless- breaking of a global symmetry (the chiral symmetry of quarks) – as a result, 3 N-G bosons appear in the spectrum • Hadron masses are generated by the QCD energy densities as created by color confinement (see Lattice QCD) – this is by far the (quantitatively) dominant mass in the “non-dark” Universe • EWSB is the breaking of a local gauge symmetry, and needs a Higgs boson – the (Higgs-given) “intrinsic” masses compete with QCD only in so far as the 3rd generation is concerned – the 3 N-G bosons are “eaten” and become the longitudinal polarization modes of the 3 massive gauge bosons – In this case, are the 3 N-G bosons also physical ? Arthur Maciel, C.
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