The and other short-comings of the (SM)

Thorsten Feldmann [email protected]

Seminar WiSe 20/21 – Extensions of the Standard Model 4. January 2021

TRR 257 particle phenomenology after the Higgs discovery

Th. Feldmann Hierarchy Problem 1 / 25 Current Understanding of Fundamental Physics

4 classical forces:

electromagnetic, weak and strong nuclear forces + | {z } → SM symmetry: SU(3)C × SU(2)L × U(1)Y →

elementary particles ():

Quarks + Leptons (3 families + anti-particles) electric charges (2/3,-1/3) electric charges (0,-1)

additional observations: - -antibaryon Asymmetry in the (BAU) . . . - Neutrinos have tiny (but finite) mass . . . - A scalar Higgs with mass of ∼ 125 GeV . . . - Abundance of (DM) in the Universe . . .[ 7→ Ch. Papior] ...

Th. Feldmann Hierarchy Problem 2 / 25 Current Understanding of Fundamental Physics

4 classical forces:

electromagnetic, weak and strong nuclear forces + gravity | {z } → SM symmetry: SU(3)C × SU(2)L × U(1)Y → General Relativity

elementary matter particles (fermions):

Quarks + Leptons (3 families + anti-particles) electric charges (2/3,-1/3) electric charges (0,-1)

additional observations: - Baryon-antibaryon Asymmetry in the Universe (BAU)... - Neutrinos have tiny (but finite) mass . . . - A scalar with mass of ∼ 125 GeV . . . - Abundance of Dark Matter (DM) in the Universe . . .[ 7→ Ch. Papior] ...

Th. Feldmann Hierarchy Problem 2 / 25 Table of Contents

1 The Standard Model (recap. of subtle issues) 2 The Hierarchy Problem 3 (some) Popular NP Ideas 4 Conclusion/Outlook → SMEFT

Th. Feldmann Hierarchy Problem 3 / 25 The Standard Model

Th. Feldmann Hierarchy Problem 4 / 25 Theoretical features of the SM

Formulated as a : (special relativity + QM) → restricted to renormalizable operators of dim≤ 4 in Lagrange density Interactions from Gauge Principle: → fundamental forces mediated by massless Vector Quarks and leptons distinguished by Chirality: → different quantum numbers for left- and right-handed fermions → gauge symmetry forbids explicit masses → right-handed neutrinos decouple (all SM quantum numbers vanish)

3 1 X X X L = − X X µν + F¯ (i) (iD/) F (i) gauge 4 µν X=G,W ,Y i=1 F=QL,UR ,DR ,LL,ER

µν ˜ theoretically possible additional term ∝ G Gµν that violates CP

experimentally constrained to ' 0 −→ strong CP problem [7→ J. Steinberg]

Th. Feldmann Hierarchy Problem 5 / 25 → the running couplings in the SM

3 independent gauge couplings g1,2,3 depend on energy resolution µ 2 gi (αi = 4π )

∂α i = β (µ) = −β(0) α2 + ... ∂ ln µ2 i i i 2 1 1 (0) µ ⇒ ' + βi ln 2 αi (µ) αi (µ0) µ0

coefficients in the ”β-functions” depend on particle content

WARNING: U(1)Y coupling in the SM has arbitrary normalization ! 15 α3 and α2 tend to meet above 10 GeV

Unification of EW and strong forces? ← modified particle content between EW scale and 1015 GeV ? ← embed SU(3) × SU(2) × U(1) in larger (non-abelian) group ? [7→ A. Boushmelev]

Th. Feldmann Hierarchy Problem 6 / 25 The Higgs and EWSB

Add a Complex Scalar Higgs Doublet ! φ+(x) φ(x) = φ0(x) Postulate a non-trivial scalar potential

2 2 †  †  Lpot = −V (φ)= µ φ φ − λ φ φ

q 2 → non-vanishing VEV hφ0i = √v = µ ' 246 GeV 2 2λ

→ spontaneous symmetry breaking: SU(2)L × U(1)Y → U(1)Q → 3 massless Goldstone modes g±, g0

0 2 2 → 1 massive scalar mode h with mh = 2λ v

Th. Feldmann Hierarchy Problem 7 / 25 and Gauge-Boson Masses

Masses for gauge bosons from kinetic Higgs terms

† µ LHiggs−kin. =( Dµφ )(D φ) → Goldstone modes become longitudinal modes for W ± and Z 0 → gauge-boson masses proportional to VEV p g g2 + (g0)2 M = v , M2 = v , W 2 Z 2

→ Higgs VEV sets the interaction range of weak force: g2 GF = √ 2 4 2 MW

→ weak mixing angle between photon and Z 0

e = g sin θW , MW = MZ cos θW (at tree level)

dimensionful parameter µ2 → v 2 introduced ”by hand” sets the energy scale for EW dynamics (range of weak force)

Th. Feldmann Hierarchy Problem 8 / 25 → the Custodial Symmetry of the Higgs Potential

The Higgs potential has a larger symmetry than SU(2)L × U(1)Y This can be seen by introducing the notation ! ! φ0∗ φ+ Φ = ≡ φ˜ , φ −φ− φ0

such that the potential can also be written as

2 2 h † i  h † i V = −Lpot = −µ tr Φ Φ + λ tr Φ Φ

This is obviously symmetric under SU(2)L × SU(2)R transformations,

† Φ → UL Φ UR

protects SM relations against quantum corrections → ”Custodial Symmetry” → confirmed by ElectroWeak Precision Tests (LEP etc.) the custodial symmetry is broken by

(a) the gauge couplings, which only respect SU(2)L × U(1)Y (b) the Yukawa couplings to fermions (see below)

Th. Feldmann Hierarchy Problem 9 / 25 Higgs Yukawa Couplings and Fermion Masses

Higgs doublet interacts with chiral fermions via Yukawa Couplings:

ij ¯ i ˜ j ij ¯ i j ij ¯i j Lyuk = −YU QL φ UR − YD QL φ DR − YE LL φ ER + h.c.

→ fermion mass matrices from Higgs Yukawa couplings v v Mij = √ Y ij , Mij = √ Y ij U 2 U D 2 D diagonalized with different trafos for left-handed up- and down-quarks: → CKM matrix for quark transitions: 3 mixing angles + 1 weak CP-violating phase → hierarchical pattern for fermion masses: v mu ∼ md  ms  mc < mb ≪ mt ∼ √ 2 me  mµ  mτ → massless (left-handed) neutrinos, no lepton-flavour violation → accidental symmetries: baryon number B and lepton number L

Th. Feldmann Hierarchy Problem 10 / 25 → Cancellation of chiral gauge anomalies

gauge symmetries in QFT can be broken by quantum corrections (”triangle anomalies”) proportional to trace of the representation matrices for the involved gauge bosons, with opposite sign for left- and right-handed fermions

Example ”U(1)3” (for one family): 8 1 2 right-handed : tr[Y 3] = 3 · y 3 + 3 · y 3 + y 3 = 3 · − 3 · − 1 = − UR DR ER 27 27 9 1 1 2 left-handed : −tr[Y 3] = −3 · 2 · y 3 − 2 · y 3 = −6 · + 2 · = + QL LL 216 8 9 → fermions must come in complete families! → can/must not be accidental! → more symmetry?

Th. Feldmann Hierarchy Problem 11 / 25 Standard Model: Goods and Bads

+ predictive power and efficiency (more experimental observables than free parameters) + unification of electromagnetic and weak forces (tested to high accuracy with EWPTs) + CKM mechanism confirmed as dominant source of flavour decays (BaBar, Belle I+II, LHCb, . . . ) + Higgs boson observed with ”reasonable” mass value (LHC)

- SM does not include gravity (not to mention ) - enters as ad-hoc parameter in Higgs potential - SM does not explain fermion charges and origin of families - SM does not explain hierarchies in Yukawa couplings - CP violation in CKM matrix not sufficient to explain BAU - Strong CP parameter fine-tuned to zero ? - SM does not account for non-vanishing neutrino masses - SM does not provide a viable DM candidate

Th. Feldmann Hierarchy Problem 12 / 25 Hierarchy Problem

Th. Feldmann Hierarchy Problem 13 / 25 Quantum corrections to mass parameter in potential

2-point function:

2 2 assume that SM is valid for momenta p < ΛUV 1-loop contribution to dimensionful parameter µ2 in Higgs potential:

2  2 02  2 ΛUV 9 g + 3 g 2 δµ = 3 λ + − 3 yt 1−loop 16π2 8

15 for ΛUV & 10 GeV one has δµ2  µ2 + δµ2 = µ2 ∼ m2, v 2 quantum UV quantum physical h

⇒ THE ”Hierarchy Problem in the SM”

Th. Feldmann Hierarchy Problem 14 / 25 How could the scale ΛUV come into play? The Landau pole, associated with the scale µLandau where the U(1)Y charge α1 diverges 19 the Planck scale MPlanck ∼ 10 GeV, where quantum gravitational effects become relevant

15 the Grand Unification Scale MGUT ∼ 10 GeV, where strong and electroweak forces unify any mass parameters of New (super-)heavy Particles which have not yet been seen in particle experiment (e.g. dark matter) ...

Th. Feldmann Hierarchy Problem 15 / 25 Possible Explanations / Solutions

2 2 1 the values of µ |UV and the coupling constants entering δµ |quantum are fine-tuned to yield the observed physical value of µ2 at the electroweak scale: 2 2 ⇒ one out of many possible where, by accident, µ  ΛUV

2 the SM is only valid up to energies of the order of ΛNP ∼ a few TeV: ⇒ new particles/interactions, possibly to be found at the LHC

⇒ still, the ”new physics model” has to explain why ΛNP  ΛUV

Conclusions for Research Directions: exp.: search for ”Physics beyond the SM” theo.: investigate ”NP Models” that can address the hierarchy problem (and possibly some of the other problems/puzzles in the SM) pheno.: precise predictions for observables in the SM or with NP

Th. Feldmann Hierarchy Problem 16 / 25 Popular NP Ideas

Th. Feldmann Hierarchy Problem 17 / 25 Example 1: (low-scale) (SUSY)

Idea: bosons and fermions contribute with different sign to δµ2 implement ”Supersymmetry" on the level of the Lagrangian ⇒ bosonic and fermionic contributions cancel to arbitrary loop-order

→ new partners for SM particles with opposite spin/statistics → in particular, any elementary scalar comes with a fermionic partner

complex Higgs boson fields φ ↔ chiral fermions: ”” φ˜

→ masses of higgsinos are ”protected” by chiral symmetry, i.e.

δmφ˜ ∝ mφ˜  ΛUV

→ masses of Higgs and higgsinos related by SUSY, i.e.

δmφ = δmφ˜  ΛUV #

Th. Feldmann Hierarchy Problem 18 / 25 Goods and Bads of (low-energy) SUSY

+ theoretically appealing as only non-trivial extension of Poincare symmetry + SUSY can account for a spin-2 −→ (SUGRA) + allows unification of strong and electroweak couplings in SU(5) GUT + naturally favours a light Higgs boson as observed at the LHC + SUSY can be supplemented with discrete R-parity ⇒ candidates for stable weakly-interacting dark matter + SUSY provides additional mechanisms for neutrino-mass generation + /- new sources of CP violation (→ BAU / electric dipole moments)

- SUSY must be broken at low energies: ⇒ need to postulate different options for ”soft” SUSY breaking ⇒O (100) new parameters to be fitted to experimental data - new sources for flavour transitions ⇒ ”NP flavour problem”

- SUSY still does not explain origin of fermion families - strong CP problem remains

Th. Feldmann Hierarchy Problem 19 / 25 Example 2: Higgs as a Pseudo-Goldstone Boson

Idea: the physical Higgs boson might be a (pseudo-) Goldstone boson of an approximate global symmetry that is spontaneously broken by some new strong dynamics

→ Higgs scalars are not fundamental degrees of freedom: ”Composite Higgs” → UV effects cut-off by ”form factors” from strong dynamics → Generic models with composite Higgs scalars have a ”” 2 2 → mh  Λstrong can be naturally realized from Goldstone’s theorem 2 2 (analogous to mπ  mproton) Various symmetry groups with ”Collective Symmetry Breaking” [7→ Ch. Schneider]

Quadratic divergences cancel between contributions of same spin/statistics !

Th. Feldmann Hierarchy Problem 20 / 25 Composite Higgs Models: Goods and Bads

+ much fewer new parameters than in SUSY + custodial symmetry can be implemented to obey EWPTs + Higgs boson mass protected by symmetry + Variants with discrete T-parity provide DM candidates + third fermion family can be distinguished by different quantum numbers + /- additional sources of CP violation

- still does not address gravity - still does not explain fundamental origin of fermion charges - still does not address strong CP problem - (usually) does not address neutrino masses

Th. Feldmann Hierarchy Problem 21 / 25 Example 3: Extra Space-Time Dimensions

Consider field operators on an extended geometry: SM dynamics that we observe is restricted to a 4-dim. subspace

Key concept: Kaluza-Klein decomposition consider a real scalar field on a (4 + 1) dimensional torus of radius R φ = φ(x0, x1, x2, x3; y) ≡ φ(x0, x1, x2, x3; y + 2πR) the action for a massless 5D scalar then reads Z Z 4 1 h 2 2i  2 2 S = d x dy (∂µφ) − (∂y φ) ⇒ ∂ − ∂ φ = 0 2 µ y the solutions with the periodic b.c. thus read ∞ 1 X φ(x, y) = √ φ(n)(x) ei n y/R , φ(n)∗ = φ(−n) 2πR n=−∞ and the effective 4-dim Lagrangian follows as

2 2 2 X (n) 2 (n) 2 n Leff(x) = ∂µφ (x) − m φ (x) , m = n n R2 n≥0

Th. Feldmann Hierarchy Problem 22 / 25 Some aspects of extra-dimensional models: Kaluza-Klein towers of massive ”resonances” on 4-dim subspace NP mass scale set by geometry (radius) of the non-trivial metric, including gravitational excitations, can be naturally implemented for flat extra dimensions, all couplings in the higher-dimensional theory are rescaled in the effective 4-dimensional theory, in particular 2 2+d MPlanck = M(4+d) Vol(d) → geometries with singular sub-manifolds → ”” scenarios (”large Extra Dim’s”) circumvent hierarchy problem – fundamental mass scale in (4+d) dim. in TeV range. ”warped extra dimensions” [Randall/Sundrum models]

2 2 R  µ ν 2 ds = ηµν dx dx − z z2 can simultaneously explain hierarchies in Higgs potential and fermion couplings Dualities between weakly coupled gravitational theories in extra dimensions and strongly coupled dynamics in 4 dimensions . . . in particular, global symmetries in 4D can be related to gauge symmetries in 5D ! → connection between Composite Higgs Models and Extra-dimensional Models

Th. Feldmann Hierarchy Problem 23 / 25 Conclusions and Outlook

Important open Questions in the SM why is the EW scale so much smaller than the Planck scale? where does the SM gauge group come from? why do quarks have fractional charges? why do the fermion multiplets come in 3 copies (families)? what is the reason for the hierarchies in the Yukawa matrices? solution for the strong CP problem?

Direct or indirect Hints for New Physics? Dark Matter (only astrophysical/cosmological obs.)

anomalous magnetic moment (g − 2)µ of the muon (ca. 3σ deviation)

neutrino oscillations (mν(i) − mν(j) 6= 0) baryon–antibaryon asymmetry in the universe (needs more CP violation) recently: Hints on violation of lepton-flavour universality in b-quark decays

No direct signals for new particles at the LHC !

Th. Feldmann Hierarchy Problem 24 / 25 The Standard Model Effective Field Theory (SMEFT)

In absence of direct signals for new particles at the LHC:

→ maybe the answer to the hierarchy problem has to be postponed?

For the time being . . .

Parameterize NP effects in terms of an effective field theory, that has the same symmetries as the SM, but includes operators with higher mass dimension, whose effects at low energies are suppressed by inverse powers of the NP scale,

(i) C5 X C6 i LSMEFT = LSM + O5 + 2 O6 + ... ΛNP Λ i NP

one unique operator of dim-5 (”Weinberg operator”) gives rise to neutrino masses of the Majorana type (|∆L| = 2); flavour matrix C5 can be fitted to neutrino-oscillation data many possible dim-6 operators with many new flavour structures → 2499 new parameters, to be constrained by precision tests of experimental observables + theory insight

Th. Feldmann Hierarchy Problem 25 / 25