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 − 2 ∗ = iψγ¯ ∂ψ (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).

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– 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. Jord˜ao,SP (Jan. 2013) 15 evjas-2013 The Goldstone Boson Equivalence Theorem

• In the high energy limit, the pseudo-scalar Goldstone boson that is ab- sorbed by the gauge vector boson controls the behavior of the longitudinal polarization state of that gauge boson. (more+refs: P. & S. sec. 21.2)

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at high energies, the W absorption/emission

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Ï = · Ç ´ µ amplitudes become equal to those of the GB that

E was “eaten” to form this longitudinal state

• Experiments at the LHC can search for new strong-like interactions (using

the Higgs sector) through Wℓ Wℓ boson reactions such as VB fusion.

• These data analyses can be performed in a fairly model independent way.

– Experimentally it is possible to isolate the Wℓ Wℓ component in W W final states by means of kinematic cuts.

Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 16 evjas-2013 Consequences of the GBET

• N-G Bosons as helpers in the search for effects that are unique to some hypothetical new strong (QCD-like) regime

• + − Through the GBET, Wℓ Wℓ scattering in the new strong interactions − regime becomes related to the ordinary π+ π scattering in low energy QCD.



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– access to J = 0 (Higgs-type) and J =1(ρ-type) resonances.

Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 17 evjas-2013 Simplified Calculations

+ + Wℓ Wℓ + + + ··· − − = Wℓ Wℓ

“Goldstone Boson Scattering”

+ + + ···O 2 = + (MW /s)

− • VBF−→ W +W is actively pursued by the LHC experiments

• Needs significantly more integrated luminosity (2015?)

− • A repeat of the π+π -physics of the 1950’s ?

Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 18