Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion

Scale Invariance and Conformal Invariance in Quantum Field Theory

Jean-Fran¸coisFortin

D´epartement de Physique, de G´eniePhysique et d’Optique Universit´eLaval, Qu´ebec, QC G1V 0A6, Canada

June 15-June 19, 2015 2015 CAP Congress

mostly based on arXiv:1208.3674 [hep-th]

with Benjam´ınGrinstein and Andreas Stergiou Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion The big picture

Theory space Renormalization group (RG) flow

Quantum field theory (QFT)

Scale field theory (SFT) fixed point limit cycle

Conformal field theory (CFT) Weyl consistency conditions fixed point c-theorem gradient flow

fixed point Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Why is it interesting ?

QFT phases Infrared (IR) free - With mass gap ⇒ Exponentially-decaying correlation functions (e.g. Higgs phase) - Without mass gap ⇒ Trivial power-law correlation functions (e.g. Abelian Coulomb phase) IR interacting - CFTs ⇒ Power-law correlation functions (e.g. non-Abelian Coulomb phase) - SFTs ⇒ ? Possible types of RG flows Strong coupling Weak coupling - Fixed points (e.g. Banks-Zaks fixed point Banks, Zaks (1982)) - Recurrent behaviors (e.g. limit cycles or ergodic behaviors) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Outline

1 Motivations

2 Scale and conformal invariance Preliminaries Scale invariance and recurrent behaviors

3 Weyl consistency conditions c-theorem Scale invariance implies conformal invariance

4 Discussion and conclusion Features and future work Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Preliminaries (d > 2)

Dilatation current Wess (1960) µ ν µ µ - D (x) = x Tν (x) − V (x) µ - Tν (x) any symmetric EM tensor following from spacetime nature of scale transformations - V µ(x) local operator (virial current) contributing to scale dimensions of fields µ - Freedom in choice of Tν (x) compensated by freedom in choice of V µ(x)

µ µ Scale invariance ⇒ Tµ (x) = ∂µV (x) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion

Conformal current Wess (1960)

µ λ µ λ 0µ λ ρµ - Cν (x) = vν (x)Tλ (x)−(∂λvν )(x)V (x)+(∂ρ∂λvν )(x)L (x) µ - Tλ (x) any symmetric EM tensor following from spacetime nature of conformal transformations - V 0µ(x) local operator corresponding to ambiguity in choice of dilatation current - Lρµ(x) local symmetric operator correcting position dependence of scale factor λ -( ∂λvν )(x) scale factor (general linear function of xν) µ - Freedom in choice of Tλ (x) compensated by freedom in choice of V 0µ(x) and Lρµ(x)

µ 0µ νµ Conformal invariance ⇒ Tµ (x) = ∂µV (x) = ∂µ∂νL (x) Conformal invariance ⇒ Existence of symmetric traceless energy-momentum tensor Polchinski (1988) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Scale without conformal invariance

Non-conformal scale-invariant QFTs Polchinski (1988)

µ µ Scale invariance ⇒ Tµ (x) = ∂µV (x)

µ νµ Conformal invariance ⇒ Tµ (x) = ∂µ∂νL (x) Scale without conformal invariance µ µ µ µ νµ ⇒ Tµ (x) = ∂µV (x) where V (x) 6= J (x) + ∂ν L (x) with µ ∂µJ (x) = 0

Constraints on possible virial current candidates - Gauge invariant (spatial integral) - Fixed d − 1 scale dimension in d spacetime dimensions

No suitable virial current ⇒ Scale invariance implies conformal invariance (examples: φp in d = n −  for (p, n) = (6, 3), (4, 4) and (3, 6)) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Virial current candidates (d = 4)

Most general classically scale-invariant renormalizable theory in d = 4 −  spacetime dimensions Jack, Osborn (1985)

1 1 − 1 A Aµν 1 2 2 µ L = − µ ZA 2 FµνF + 2 ZabZac DµφbD φc 4gA 1 ∗ 1 1 ∗ 1 1 2 2 ¯ µ 1 2 2 ¯ µ + 2 Zij Zik ψj iσ¯ Dµψk − 2 Zij Zik Dµψj iσ¯ ψk 1  λ − 4! µ (λZ )abcd φaφbφc φd   1 2 y 1 2 y ∗ ¯ ¯ − 2 µ (yZ )a|ij φaψi ψj − 2 µ (yZ )a|ij φaψi ψj

φa(x) real scalar fields

α ψi (x) Weyl fermions A Aµ (x) gauge fields Dimensional regularization with minimal subtraction Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Virial current candidates and new improved EM tensor

µ µ µ Virial current V (x) = QabφaD φb − Pij ψ¯i iσ¯ ψj

- Qba = −Qab ∗ - Pji = −Pij

µ New improved energy-momentum tensor [Θν (x)] Callan, Coleman, Jackiw (1970) - Finite and not renormalized (vanishing anomalous dimension) - Anomalous trace Osborn (1989,1991) & Jack, Osborn (1990)

µ BA A Aµν 1 [Θµ (x)] = 3 [FµνF ] − 4! Babcd [φaφbφc φd ] 2gA δ − 1 B [φ ψ ψ ] + h.c.
− ((δ + Γ)f ) · A 2 a|ij a i j δf Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion

Anomalous trace δ [Θ µ(x)] = BI [O (x)] − ((δ + Γ)f ) · A µ I δf

µ Conserved dilatation current ∂µD (x) = 0 (up to EOMs)

BI = QI ≡ −(gQ)I

µ Conserved conformal current ∂µCν (x) = 0 (up to EOMs)

BI = 0

⇒ Both SFT (Q 6= 0) and CFT (Q = 0) can be treated simultaneously Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Virial current and unitarity bounds

New improved energy-momentum tensor ⇒ Finite and not renormalized Callan, Coleman, Jackiw (1970)

Operators related to EOMs ⇒ Finite and not renormalized Politzer (1980) & Robertson (1991) Virial current ⇒ Finite and not renormalized - Unconserved current with scale dimension exactly 3

Unitarity bounds for conformal versus scale-invariant QFTs Grinstein, Intriligator, Rothstein (2008)

Non-trivial virial current ⇒ Non-conformal scale-invariant QFTs Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion RG flows along scale-invariant trajectories

Scale-invariant solution (λabcd , ya|ij , gA) ⇒ RG trajectory

¯ λabcd (t) = Zba0a(t)Zbb0b(t)Zbc0c (t)Zbd0d (t)λa0b0c0d0

y¯a|ij (t) = Zba0a(t)Zbi 0i (t)Zbj0j (t)ya0|i 0j0

g¯A(t) = gA

Qt
) Zba0a(t) = e 0 a a t = ln(µ /µ) (RG time) Pt
0 0 Zbi i (t) = e i 0i ¯ (λabcd (t, g, λ, y), y¯a|ij (t, g, λ, y), g¯A(t, g, λ, y)) also scale-invariant solution

Qab and Pij constant along RG trajectory

Zbab(t) orthogonal and Zbij (t) unitary ⇒ Always non-vanishing beta-functions along scale-invariant trajectory Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Scale invariance and recurrent behaviors

RG flows along scale-invariant trajectories ⇒ Recurrent behaviors ! Lorenz (1963,1964), Wilson (1971) & Kogut, Wilson (1974) Virial current ⇒ Transformation in symmetry group of kinetic terms (SO(NS ) × U(NF ))

- Zbab(t) and Zbij (t) in SO(NS ) × U(NF ) - Qab antisymmetric and Pij antihermitian ⇒ Purely imaginary eigenvalues

⇒ Periodic (limit cycle) or quasi-periodic (ergodicity) scale-invariant trajectories Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Recurrent behaviors

µ ν µ µ Intuition from D (x) = x Tν (x) − V (x)

ν µ RG flow ⇒ Generated by scale transformation (x Tν (x)) RG flow ⇒ Related to virial current through conservation of dilatation current Virial current ⇒ Generates internal transformation of the fields

- Internal transformation in compact group SO(NS ) × U(NF ) ⇒ Rotate back to or close to identity

RG flow return back to or close to identity ⇒ Recurrent behavior Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Why dilatation generators generate dilatations

Dilatation generators do not generate dilatations in non-scale-invariant QFTs Coleman, Jackiw (1971) Quantum anomalies at low orders - Anomalous dimensions ⇒ Possible to absorb into redefinition of scale dimensions of fields " Preserve scale invariance

Quantum anomalies at high orders - Beta-functions ⇒ Not possible to absorb % Break scale invariance Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion

Why dilatation generators generate dilatations in scale-invariant QFTs ? Beta-functions on scale-invariant trajectories - Both vertex correction and wavefunction renormalization contributions - Very specific form for vertex correction contribution - Equivalent in form to wavefunction renormalization contribution (redundant operators) ⇒ Also possible to absorb into redefinition of scale dimensions of fields " Preserve scale invariance ! Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion

Beta-functions from vertex corrections and wavefunction renormalizations (d = 4 spacetime dimensions)

dλ B = − abcd abcd dt λ = −(λγ )abcd + λa0bcd Γa0a + λab0cd Γb0b + λabc0d Γc0c + λabcd 0 Γd 0d

dya|ij y B = − = −(yγ ) + y 0 Γ 0 + y 0 Γ 0 + y 0 Γ 0 a|ij dt a|ij a |ij a a a|i j i i a|ij j j dg B = − A = γ g (no sum) A dt A A Beta-functions on scale-invariant trajectories

Babcd = −λa0bcd Qa0a − λab0cd Qb0b − λabc0d Qc0c − λabcd 0 Qd 0d

Ba|ij = −ya0|ij Qa0a − ya|i 0j Pi 0i − ya|ij0 Pj0j

BA = 0 Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Ward identity for scale invariance

Callan-Symanzik equation for effective action Callan (1970) & Symanzik (1970)  Z  ∂ I ∂ I 4 δ M +B I +Γ J d x fI (x) Γ[f (x), g, M] = 0 ∂M ∂g δfJ (x)

In non-scale-invariant QFTs " Anomalous dimensions % Beta-functions  Z  ∂ I 4 δ M + (Γ + Q)J d x fI (x) Γ[f (x), g, M] = 0 ∂M δfJ (x) In scale-invariant QFTs " Anomalous dimensions " Beta-functions (redundant operators) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Poincar´ealgebra augmented with dilatation charge

Beta-functions on scale-invariant trajectories - Quantum-mechanical generation of scale dimensions - Appropriate scale dimensions required by virial current ⇒ Conserved dilatation current Dµ(x) Poincar´ealgebra with dilatation charge D = R d3x D0(x)

[Mµν, Mρσ] = −i(ηµρMνσ − ηνρMµσ + ηνσMµρ − ηµσMνρ)

[Mµν, Pρ] = −i(ηµρPν − ηνρPµ)

[D, Pµ] = −iPµ

Algebra action on fields OI (x)

[Mµν, OI (x)] = −i(xµ∂ν − xν∂µ + Σµν)OI (x)

[Pµ, OI (x)] = −i∂µOI (x)

[D, OI (x)] = −i(x · ∂ + ∆)OI (x) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion

New classical scale dimensions of fields due to virial current

[D, φa(x)] = −i(x · ∂ + 1)φa(x) − iQabφb(x) 3 [D, ψi (x)] = −i(x · ∂ + 2 )ψi (x) − iPij ψj (x)

- How do non-conformal scale-invariant QFTs know about new scale dimensions ? ⇒ Generated by beta-functions !

Quantum-mechanical scale dimensions of fields

∆ab = δab + Γab + Qab 3 ∆ij = 2 δij + Γij + Pij Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion c-theorem

c-theorem Barnes, Intriligator, Wecht, Wright (2004) RG flow ⇒ Irreversible process (integrating out DOFs)

c(g) ∼ measure of number of massless DOFs c-theorem and implications for SFT

- weak( cIR < cUV ) Komargodski, Schwimmer (2011) & Luty, Polchinski, Rattazzi (2012) dc - stronger( dt ≤ 0) Osborn (1989,1991) & Jack, Osborn (1990) - ////////strongest (RG flows as gradient flows) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Gradient flow

Gradient flow dg I ∂c(g) BI (g) = − = G IJ (g) dt ∂g J - G IJ positive-definite metric - Potential c(g) function of couplings Potential c(g) monotonically decreasing along RG trajectory dc(g(t)) = −G (g)BI BJ ≤ 0 dt IJ - Recurrent behaviors (scale-invariant trajectories)⇔ / Gradient flows (scale implies conformal invariance) Wallace, Zia (1975) ⇒ Another way to prove scale implies conformal invariance - Different than proof for d = 2 unitarity interacting QFTs with well-defined correlation functions Zamolodchikov (1986) & Polchinski (1988) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion c-theorem and gradient flow at weak coupling

Weyl consistency conditions Osborn (1989,1991) & Jack, Osborn (1990)

∂c(g) dc(g(t)) = (G + A )βJ ⇒ = −βI G (g)βJ ∂g I IJ IJ dt IJ

- Curved spacetime ⇒ Background metric with spacetime-dependent couplings ⇒ (Weak-coupling) RG flow recurrent behaviors forbidden at all loops Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Local and global renormalized operators

I Global renormalized operator OI (x) = ∂L(x)/∂g Finite global insertion in Green functions ⇒ I R d −i∂h...i/∂g = h d x OI (x) ...i

Infinite local insertion in Green functions ⇒ hOI (x) ...i

I Local renormalized operator [OI (x)] = δA/δg (x) Finite local insertion in Green functions ⇒ µ h[OI (x)] ...i = h(OI (x) − ∂µJI (x)) ...i µ µ ¯ µ Infinite current JI (x) = −(NI )abφaD φb + (MI )ij ψi iσ¯ ψj ∗ -( NI )ba = −(NI )ab and (MI )ji = −(MI )ij (i) (i) P NI P MI - NI = i≥1 i and MI = i≥1 i Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Computations of new divergences

p p p p

p p p p

1 1 (N ) = − (y ∗ δ − y ∗ δ ) + h.c. + finite c|ij ab 16π2 2 a|ij bc b|ij ac Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Finite contributions to EM tensor

Anomalous trace Osborn (1989,1991) & Jack, Osborn (1990)

δ [Θ µ(x)] = βI [O ] − D [S φ Dµφ − R ψ¯ iσ¯µψ ] − ((δ + γ)f ) · A µ I µ ab a b ij i j δf

1 1 ( 2 −δ) 2 I kI  I I f0 = µ Z (g)f g0 = µ (g + L (g)) 1 1 I 2 (1) ˆI I I (1) J I (1) γˆ = ( 2 − δ) − kI g ∂I Z β = −kI g  − kI L + kJ g ∂J L I (1) I (1) S = −kI g NI R = −kI g MI Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Ambiguities in RG functions

Relevant quantities Osborn (1989,1991) & Jack, Osborn (1990)

1 Square root of wavefunction renormalization Z 2 1 1 1 1 T 1 1 T T 1 - Freedom Z 2 → Z˜ 2 = OZ 2 with Z = Z 2 Z 2 → Z 2 O OZ 2 T P O(i) - O O = 1 and O = 1 + i≥1 i

I (1) Extra freedom with ω = kI g ∂I O

1 1 (1) (1) 2 (1) 2 (1) (1) I (1) I (1) (1) I (1) Z → Z + O L → L − (gO ) NI → NI − ∂I O γˆ → γˆ − ω βˆI → βˆI − (gω)I S → S + ω

Invariant anomalous trace δ [Θ µ(x)] = (βI + (gS)I )[O ] − ((δ + γ + S)f ) · A µ I δf δ = BI [O ] − ((δ + Γ)f ) · A I δf Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Scale and conformal invariance

“Correct” RG flow ⇒ BI = βI + (gS)I = −(gQ)I - SFTs (Q 6= 0) ⇒ limit cycles (BI = −(gQ)I 6= 0) - CFTs (Q = 0) ⇒ fixed points (BI = 0)

“Old” RG flow ⇒ βI = −(g(S + Q))I - SFTs (Q 6= 0) ⇒ fixed points (S = −Q) and limit cycles (S 6= −Q) - CFTs (Q = 0) ⇒ fixed points (S = 0) and limit cycles (S 6= 0)

⇒ Systematic understanding of SFTs and CFTs through “correct” RG flow (unless S vanishes identically) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Computation of S

(one-loop) (two-loop) S = S = 0 due to symmetry of contributions to NI

2 3 5 ∗ ∗ 3 ∗ ∗ ∗ (16π ) Sab = 8 tr(ya yc yd ye )λbcde + 8 tr(ya yc yd yd yb yc ) − {a ↔ b} + h.c.

S 6= 0 ⇒ Examples of CFTs with S 6= 0 exist JFF, Grinstein, Stergiou (2012) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Generalized c-theorem

Weyl consistency conditions and local current conservation Osborn (1989,1991) & Jack, Osborn (1990)

∂c(g) dc(g(t)) = (G + A )BJ ⇒ = − BI G BJ ∂g I IJ IJ dt IJ - Curved spacetime ⇒ Background metric with spacetime-dependent couplings - Spin-one operator of dimension 3 ⇒ Background gauge fields with gauge-dependent couplings ⇒ (Weak-coupling) RG flow recurrent behaviors allowed at all loops

Scale invariance implies conformal invariance JFF, Grinstein, Stergiou (2012) & Luty, Polchinski, Rattazzi (2012) Motivations Scale and conformal invariance Weyl consistency conditions Discussion and conclusion Features and future work

Features of SFTs and CFTs Correct RG flow - SFTs ⇒ Recurrent behaviors - CFTs ⇒ Fixed points

Generalized c-theorem ⇒ Only CFTs allowed ⇒ Scale invariance implies conformal invariance - Unexpected CFTs with expected behaviors

Future work

Proof at strong coupling Farnsworth, Luty, Prelipina (2013) 6d analysis Grinstein, Stergiou, Stone, Zhong (2014,2015)

Thank you !