bla Conformal Symmetry and the Weak Scale Hermann Nicolai MPI f¨ ur Gravitationsphysik, Potsdam (Albert Einstein Institut) Based on joint work with Krzysztof A. Meissner [hep-th/0612165, arXiv:0710.2840, arXiv:0803.2814, arXiv:0809.1338[hep-th]] NB: Conformal symmetry is an old subject! [see e.g. H.Kastrup, arXiv:0808.2730 for an historical survey and references]
Mass Generation and Hierarchy • Fact: Standard Model (= SM) of elementary par- ticle physics is conformally invariant at tree level except for explicit mass term m 2 Φ † Φ in potential → masses for vector bosons, quarks and leptons. • Why m 2 < 0 rather than m 2 > 0 ? • Quantum corrections δm 2 ∼ Λ 2 ⇒ why m H ≪ M Pl ? (with UV cutoff Λ = scale of ‘new physics’) – stabilization/explanation of hierarchy? • Most popular proposal: SM − → MSSM or NMSSM: use supersymmetry to control quantum corrections via cancellation of quadratic divergences ⇒ δm 2 ∼ Λ 2 SUSY ln(Λ 2 / Λ 2 SUSY )
Landau Poles Large scalar self-coupling ↔ Landau pole ( A > 0) µdy y 0 dµ = Ay 2 = ⇒ y ( µ ) = 1 − Ay 0 ln( µ/µ 0 ) Thus we are left with two possibilities: • Theory strongly coupled for ln( µ/µ 0 ) ∼ ( Ay 0 ) − 1 • Or: theory does not exist (rigorously as a QFT) General features of RG evolution of couplings in SM: • Coupled RG equations (linking α s to other cou- plings) also give rise to infrared (IR) Landau poles • With SM-like bosonic and fermionic matter, UV and IR Landau poles are (generically) unavoidable .
The demise of relativistic quantum field theory Or: Why we need quantum gravity! • Breakdown of any extension of the standard model (supersymmetric or not) that stays within the frame- work of relativistic quantum field theory is probably unavoidable [as it appears to be for λφ 4 4 ]. • Therefore the main challenge is to delay breakdown until M Pl where a proper theory of quantum gravity is expected to replace quantum field theory. • How the MSSM achieves this: scalar self-couplings tied to gauge coupling λ ∝ g 2 by supersymmetry, and thus controlled by gauge coupling evolution. √ ⇒ m H ≤ 2 m Z in (non-exotic variants of) MSSM.
Conformal invariance and the Standard Model Can classically unbroken conformal symmetry stabilize the weak scale w.r.t. the Planck scale? Claim: Yes, if • there are no intermediate mass scales between m W and M Pl (‘grand desert scenario’); and • the RG evolved couplings exhibit neither Landau poles nor instabilities (of the effective potential) over this whole range of energies. Thus: is it possible to explain all mass scales from a single scale v via the quantum mechanical breaking of conformal invariance (i.e. via conformal anomaly) → Hierarchy ‘natural’ in the sense of ’t Hooft? [See also: W. Bardeen, FERMILAB-CONF-95-391-T, FERMILAB-CONF-95-377-T]
Evidence for large scales other than M Pl ? m X ≥ O (10 15 GeV) ? • (SUSY?) Grand Unification: – But: proton refuses to decay (so far, at least!) – SUSY GUTs: unification of gauge couplings at ≥ O (10 16 GeV) • Light neutrinos ( m ν ≤ O (1 eV) ) and heavy neutrinos → most popular (and most plausible) explanation of observed mass patterns via seesaw mechanism: ∼ m 2 ∼ M ≥ O (10 12 GeV)? m (1) M , m D = O ( m W ) ⇒ m (2) D ν ν • Resolution of strong CP problem ⇒ need axion a ( x ) . Limits e.g. from axion cooling in stars ⇒ L = 1 aF µν ˜ with f a ≥ O (10 10 GeV) F µν 4 f a NB: axion is (still) an attractive CDM candidate.
Coleman-Weinberg Mechanism (1973) • Idea: spontaneous symmetry breaking by radiative corrections = ⇒ can small mass scales be explained via conformal anomaly and effective potential ? 4 ϕ 4 + 9 λ 2 ϕ 4 � ϕ 2 V ( ϕ ) = λ 4 ϕ 4 → V eff ( ϕ ) = λ � � � ln + C 0 64 π 2 µ 2 • But: when can we trust one-loop approximation? – Radiative breaking spurious for pure ϕ 4 theory – Scalar electrodynamics: consistent for λ ∼ e 4 [See e.g.: Sher, Phys.Rep.179(1989)273; Ford,Jones,Stephenson,Einhorn, Nucl.Phys.B395(1993)17; Chishtie,Elias,Mann,McKeon,Steele, NPB743(2006)104] • And: can this be made to work for real world (=SM)? – m H > 115 GeV and m top = 174 GeV
Regularization and Renormalization • Conformal invariance must be broken explicitly for computation of quantum corrections via regulator mass scale with any regularization. • Most convenient: dimensional regularization d 4 k d 4 − 2 ǫ k � � v 2 ǫ → (2 π ) 4 (2 π ) 4 − 2 ǫ • Renormalize by requiring exact conformal invari- ance of the local part of the effective action ⇒ pre- serve anomalous Ward identity T µµ ( φ ) = β ( λ ) O 4 ( φ ) • (Renormalized) effective action to any order: – no mass terms ( ∝ v 2 ) in divergent or finite parts – conformal symmetry broken only by logarithmic terms containing L ≡ ln( φ 2 /v 2 ) (to any order!)
RG improved effective potential One (real) scalar field ϕ coupled to non-scalar fields W eff ≡ W eff ( ϕ, g, v ) = ϕ 4 f ( L, g ) L ≡ ln( ϕ 2 /v 2 ) for Improved effective potential must obey RG equation v ∂ β j ( g ) ∂ + γ ( g ) ϕ ∂ � W eff ( ϕ, g, v ) = 0 ∂v + ∂g j ∂ϕ j Therefore [see also: Curtright,Ghandour, Ann.Phys.112(1978)237] − 2 ∂ β j ( g ) ∂ ˜ � f ( L, g ) = 0 γ ( g ) ∂L + + 4˜ ∂g j j with ˜ β ( g ) ≡ β ( g ) / (1 − γ ( g )) and ˜ γ ( g ) ≡ γ ( g ) / (1 − γ ( g )) ⇒ g j ( L ) /dL ) = ˜ Running couplings ˆ g j ( L ) from 2(dˆ β j (ˆ g ) .
• General solution (with arbitrary function F ) � L � � f ( L, g ) ≡ F (ˆ g 1 ( L ) , ˆ g 2 ( L ) , . . . ) exp 2 γ (ˆ ˜ g ( t ))d t 0 The choice F ( L, g ) = ˆ g 1 ( L ) ( g 1 = scalar self-coupling) yields correct � → 0 limit. • The textbook example: pure (massless) φ 4 theory ϕ 4 W eff ( ϕ ) = 1 λ ( L ) ϕ 4 = λ ˆ 1 − (9 λ/ 16 π 2 ) L = V eff ( ϕ ) + O ( λ 3 L 2 ) 4 · 4 captures leading log contributions to all orders. • Explains spuriousness of symmetry breaking for V eff via restoration of convexity by RG improvement ⇒ W eff ( ϕ ) has only trivial minimum at � ϕ � = 0 !
An almost realistic example QCD coupled to colorless real scalar field φ L = − 1 qγ µ D µ q + 1 qq − g 4Tr F µν F µν + i ¯ 2 ∂ µ φ∂ µ φ + g Y φ ¯ 4 φ 4 Cancellations in β -functions 2dˆ y 2dˆ x 2dˆ z x 2 , y 2 + a 2 ˆ x 2 − b 2 ˆ z 2 d L = a 1 ˆ x ˆ y − a 3 ˆ d L = b 1 ˆ x ˆ z , d L = − 2 c ˆ with x ≡ g 2 z ≡ g 2 y ≡ g 4 π 2 ≡ α s s Y 4 π 2 , 4 π 2 , π Explicit closed form solutions of one-loop β -function equations available for general coefficients a i , b i , c [Faivre,Branchina, PR D72 (2005) 065017; Chishtie et al., hep-ph/0701148; MN, arXiv:0809.1338] Our general formula for W eff allows more detailed study of range of validity of one-loop CW potential.
g 0.25 0.2 0.15 0.1 0.05 0 50 100 150 200 L L The scalar self-coupling ˆ λ ( L ) • ˆ λ ( L ) remains small over large range of values for L in spite of large logarithms (for ˆ λ (0) L ) • Landau pole at L > 200 and IR barrier Λ IR > 0 • Approximation can be trusted for ˆ λ ( L ) small
W 10 8 6 4 2 –1 1 2 L 3 L The RG improved effective potential W eff ( ϕ ) . • Convex function, unlike unimproved potential V eff . • Λ IR > 0 ⇒ enforces symmetry breaking � ϕ � � = 0 • Minimum safely within perturbative range • Cancellations in β -functions are crucial
A Minimalistic Proposal • Minimal extension of SM with classical conformal symmetry (i.e. no tree level mass terms) and : – right-chiral neutrinos – enlarged scalar sector: Φ and φ • No large intermediate scales (‘grand desert’) ⇒ no grand unification, no low energy SUSY [also: M. Shaposhnikov, arXiv:0708.3550[hep-th]; R. Foot et al., arXiv:0709.2750[hep-ph]] • All mass scales from effective (CW) potential: – no new scales required to explain m ν < 1 eV if Yukawa couplings vary over Y ∼ O (1) – O (10 − 5 ) – no new scales required to explain f a ≥ O (10 12 GeV)
Minimally Extended Standard Model • Start from conformally invariant (and therefore renor- malizable) Lagrangian L = L kin + L ′ with: L ′ := ij E j + ¯ ij D j + ¯ ij U j + � ¯ L i Φ Y E Q i ǫ Φ ∗ Y D Q i ǫ Φ ∗ Y U +¯ ij ν j ij ν j L i ǫ Φ ∗ Y ν R + φν iT R C Y M � R + h . c . − − λ 1 4 (Φ † Φ) 2 − λ 2 2 ( φ † φ )(Φ † Φ) − λ 3 4 ( φ † φ ) 2 the ‘ νMSM ’] [See also Shaposhnikov, Tkachev, PLB639(2006)104: • Besides usual SU (2) doublet Φ : new scalar field φ ( x ) � ia ( x ) � √ φ ( x ) = ϕ ( x ) exp 2 µ • No mass terms, all coupling constants dimensionless • Y U ij , Y E ij , Y M real and diagonal ij Y D ij , Y ν ij complex → parametrize family mixing (CKM)
Effective potential at one loop V eff ( H, ϕ ) = λ 1 H 4 + λ 2 H 2 ϕ 2 + λ 3 ϕ 4 � H 2 9 � w H 4 ln 16 π 2 α 2 + v 2 4 2 4 � λ 1 H 2 + λ 2 ϕ 2 3 � 256 π 2 ( λ 1 H 2 + λ 2 ϕ 2 ) 2 ln + v 2 � λ 2 H 2 + λ 3 ϕ 2 � 1 256 π 2 ( λ 2 H 2 + λ 3 ϕ 2 ) 2 ln + v 2 1 � F + � 1 � F − � 64 π 2 F 2 64 π 2 F 2 + + ln + − ln v 2 v 2 � H 2 � ϕ 2 � � 6 1 t H 4 ln M ϕ 4 ln 32 π 2 g 4 32 π 2 Y 4 − − v 2 v 2 H 2 ≡ Φ † Φ and ϕ 2 ≡ φ † φ with �� 3 λ 1 − λ 2 � 2 F ± ( H, ϕ ) ≡ 3 λ 1 + λ 2 H 2 + 3 λ 3 + λ 2 H 2 − 3 λ 3 − λ 2 ϕ 2 ± + λ 2 ϕ 2 2 ϕ 2 H 2 4 4 4 4
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