Low-energy effective action for pions and a dilatonic meson Maarten - PowerPoint PPT Presentation

Low-energy effective action for pions and a dilatonic meson Maarten Golterman San Francisco State University with Yigal Shamir , PRD94 (2016) 025020 arXiv:1610.01752, PRD95 (2017) 016003 Workshop APS Topical Group on Hadronic Physics, Feb. 3, 2017

A light, narrow flavor-singlet scalar — seen on the lattice(?) • SU (3) , N f = 8 fund. [LatKMI, LSD,..] Consistent low-energy theory must contain both pions and the flavor-singlet scalar LSD collaboration, PRD 93 (2016) 114514 • SU (3) , N f = 2 sextet [Fodor et al.] 1

Phases of SU ( N c ) with N f fundamental-rep Dirac fermions • running slows down when N f is increased ∂g 2 − b 1 b 2 16 π 2 g 4 − (16 π 2 ) 2 g 6 = ∂ log µ • two-loop IRFP g 2 ∗ develops when b 1 > 0 > b 2 • Gap equation ⇒ c = 4 π 2 ChSB when g 2 ( µ ) = g 2 3 C 2 c = π 2 ≃ 9 . 87 • SU (3) , fund. rep: g 2 • chirally broken if g c < g ∗ ( N f ) • emergent conformal symm. (IRFP) if g c > g ∗ ( N f ) • sill of conformal window: g ∗ ( N ∗ f ) = g c (note: N ∗ f not an integer) 2

Pseudo Nambu-Goldstone boson of approx dilatation symmetry? Φ i ( x ) → λ ∆ i Φ i ( λx ) , • dilatations: ∆ i scaling dimension of field Φ i ( x ) • dilatation current S µ = x ν T µν is classically conserved for m = 0 • Trace anomaly [Collins, Duncan, Joglekar, PRD 16 , 438 (1977)] ∂ µ S µ = T µµ = − T cl − T an T an = β ( g 2 ) 4 g 2 F 2 + γ m m ψψ T cl = mψψ β ( g 2 c ) ∝ N f − N ∗ • below conformal sill, expect at ChSB scale f hence, increasing N f towards N ∗ f ⇒ smaller β ( g c ) at ChSB scale ⇒ better scale invariance ⇒ “dilatonic” pNG boson, τ , gets lighter • use N f − N ∗ f as small parameter (issue: N f takes discrete values) 3

Low-energy EFT with dilatonic meson: power counting • standard ChPT: fermion mass m is a parameter of the microscopic theory that can be tuned continuously towards zero ⇒ Systematic expansion in m ∼ m 2 π and p 2 ; massless pions for m → 0 • issue: cannot turn off trace anomaly at fixed N c , N f • similar: cannot turn off U (1) A anomaly; but it vanishes for N c → ∞ ⇒ Systematic expansion in m , 1 /N c , and p 2 ; massless η ′ for m, 1 /N c → 0 • Here: Veneziano limit N f , N c → ∞ with n f = N f /N c fixed n f becomes a continuous parameter; theory depends only on g 2 N c and n f n ∗ f = lim N c →∞ N ∗ f ( N c ) /N c = sill of conformal window for N c → ∞ . f ) η at the ChSB scale T an ∼ ( n f − n ∗ • Assume: [ η = 1 in this talk] f , and p 2 ; ⇒ Systematic expansion in m , 1 /N c , n f − n ∗ massless dilatonic meson τ for m, 1 /N c , | n f − n ∗ f | → 0 4

Spurions in the microscopic theory (abelian symmetries) 4 F 2 + ψ / Dψ + θicg 2 F ˜ L MIC ( θ ) = 1 axial U (1) A symmetry: F • δ L MIC ( θ ) = 0 if U (1) A transformation of axionic spurion is θ → θ + α similar U (1) A transformation for singlet meson η ′ → η ′ + α explicit breaking: δ L MIC ( θ 0 ) = − icg 2 F ˜ • now set � θ � = θ 0 ⇒ F • � θ � = 0 not special! L MIC ( θ 0 = 0) not invariant! dilatations: d d x e σ ( d − 4) � 1 � � � 4 F 2 + · · · � � S MIC ( σ ) d 4 x L MIC (0) + σT an + · · · = = g 2 0 • S MIC ( σ ) invariant if σ transforms as dilatonic spurion e σ ( x ) → λe σ ( λx ) • again, explicit breaking: S MIC not invariant for any � σ � 5

EFT with pions Σ( x ) = e 2 iπ ( x ) /f π and dilatonic meson τ ( x ) • scale transformation: ( χ is fermion mass spurion, � χ � = m ) χ ( x ) → λ 1+ γ m χ ( λx ) source fields: σ ( x ) → σ ( λx ) + log λ , effective fields: τ ( x ) → τ ( λx ) + log λ , Σ( x ) → Σ( λx ) L EFT = ˜ L d → λ 4 ˜ ˜ L π + ˜ L τ + ˜ L m + ˜ L EFT • invariant low-energy theory: where ˜ V π ( τ − σ ) ( f 2 π / 4) e 2 τ tr ( ∂ µ Σ † ∂ µ Σ) L π = ˜ V τ ( τ − σ ) ( f 2 τ / 2) e 2 τ ( ∂ µ τ ) 2 L τ = � � ˜ − V M ( τ − σ ) ( f 2 π B π / 2) e (3 − γ m ) τ tr χ † Σ + Σ † χ L m = ˜ V d ( τ − σ ) f 2 τ B τ e 4 τ L d = with invariant potentials: V ( τ ( x ) − σ ( x )) → V ( τ ( λx ) − σ ( λx )) ⇒ No predictability without power counting! 6

Power counting hierarchy from matching correlation functions • recall microscopic theory β ( g 2 ) � � ∂ ∂σ ( x ) L MIC 4 g 2 [ F 2 ( x )] ∼ n f − n ∗ � � = T an ( x ) = � � f � � σ = χ =0 χ =0 • effective theory � n � � ∂ τ B τ e 4 τ ( x ) + · · · V ( n ) ˜ L EFT f 2 � � � − = τ ( x ) � d ∂σ ( x ) � σ = χ =0 ∞ � c n ( τ − σ ) n ⇒ f ) n ) c n = O (( n f − n ∗ V ( τ − σ ) = where n =0 ⇒ Only a finite number of LECs at each order! 7

Leading order lagrangian: L = L π + L τ + L m + L d ( f 2 π / 4) e 2 τ tr ( ∂ µ Σ † ∂ µ Σ) L π = ( f 2 τ / 2) e 2 τ ( ∂ µ τ ) 2 L τ = π B π / 2) e (3 − γ ∗ � Σ + Σ † � − ( mf 2 m ) τ tr L m = c 11 τ )] f 2 τ B τ e 4 τ c 00 + ( n f − n ∗ L d = [˜ f )(˜ c 01 + ˜ f )( τ − 1 / 4) ˆ τ ˆ c 11 ( n f − n ∗ f 2 B τ e 4 τ • use τ shift and redefine LECs to get L d = ˜ • m = renormalized mass at ChSB scale ⇒ choose γ m ( g ) = γ m ( g ∗ ) = γ ∗ m , where g ∗ is IRFP at the sill of the conformal window • corrections are accounted for by expansion in n f − n ∗ f 8

Classical vacuum in the chiral limit • Dilatonic meson’s potential: V cl ( τ ) ∝ V d ( τ ) e 4 τ = ˜ c 11 ( n f − n ∗ f )( τ − 1 / 4) e 4 τ f ) ⇒ V cl ( τ ) bounded from below c 11 < 0 (recall n f < n ∗ • Self-consistency: ˜ • Effective theory at leading order seems “almost” scale invariant ◦ But: linear term in V d ( τ ) crucial; reflects breaking of scale invariance by running in microscopic theory ◦ Going to n f > n ∗ f , EFT classical potential becomes unbounded from below ⇒ EFT “knows” it cannot be used inside conformal window (where EFT has the wrong degrees of freedom)! 9

Tree-level masses • m = 0 : shifted classical vacuum: v = � τ � = 0 B τ = e 2 v [pre-shift] B τ f ) ˆ ˆ ◦ dilatonic meson mass: m 2 c 11 ( n f − n ∗ τ = 4˜ B τ ⇒ m τ vanishes for n f → n ∗ f V cl ( τ ) = V d ( τ ) e 4 τ − m M e (3 − γ ∗ m ) τ • m > 0 : ⇒ v ( m ) increases monotonically with m (from v (0) = 0 ) f ) ˆ ◦ dilatonic meson mass: m 2 B τ e 2 v ( m ) � c 11 ( n f − n ∗ 1 + (1 + γ ∗ � τ = 4˜ m ) v ( m ) ⇒ m τ increases monotonically with m B π me (1 − γ ∗ π = 2 ˆ ◦ pion mass: m 2 m ) v ( m ) ⇒ m π increases with m faster than ordinary ChPT for γ ∗ m < 1 10

Varying n f towards n ∗ f • what happens at the conformal sill? Λ 2 ≪ Λ 2 ∼ m 2 m 2 O ( n f − n ∗ � � π,τ = f ) + O ( m/ Λ) non − NGB ⇒ even though chiral symm. breaking scale Λ → 0 when n f → n ∗ f , m π,τ vanish faster, consistent with EFT framework • condensate enhancement (needs ˜ c 00 > 0 ) � � ψψ = − B π v = − 1 ˜ c 00 e − γ ∗ m v 4 − f ) (“gauge” choice ˜ c 01 = 0 ) ˆ c 11 ( n f − n ∗ f π ˜ f 3 π 11

Summary & further comments • light scalar found in chirally broken “walking” theories β ( g 2 c ) ∝ n f − n ∗ • crude dynamical model (2-loop + gap equation): f = n f − 4 T an ∼ ( n f − n ∗ • main assumption: f ) at the onset of ChSB • EFT allows for systematic treatment of both pions and the dilatonic meson ◦ for two-index (and higher) irreps, asymptotic freedom forbids N f → ∞ can try the EFT anyway, for fixed N c , N f assuming N f − N ∗ f small with a non-integer N ∗ f close to (and above) integer N f ◦ can add SM fermions as in other composite Higgs models fermion-dilaton coupling ∝ fermion mass 12

Matching the trace anomaly • dilatation current: S µ = x ν Θ µν = x ν ( T µν + K µν / 3) ˆ f τ 3 ( − δ µν p 2 + p µ p ν ) e ipx � 0 | Θ µν ( x ) | τ � = ip µ ˆ f τ e ipx � 0 | S µ ( x ) | τ � = • anomalous divergence shows up at leading order in EFT: m ) m f 2 π B π τ B τ e 4 τ + (1 + γ ∗ e (3 − γ ∗ c 11 ( n f − n ∗ f ) f 2 m ) τ tr (Σ + Σ † ) ∂ µ S µ = ˜ 2 − β ( g 2 ) 4 g 2 F 2 (EFT) − (1 + γ ∗ = m ) m ψψ (EFT) ˆ f 2 π m 2 � � ◦ GMOR relation for m → 0 : − (2 m/N f ) ψψ = π ˆ − ( β ( g 2 ) /g 2 ) F 2 � f 2 τ m 2 � ◦ GMOR-like relation for dilatonic meson: = τ 13