VIEWS: 16 PAGES: 111 POSTED ON: 10/19/2011
Conformality Lost J.-W. Lee D. T. Son M. Stephanov D.B.K arXiv:0905.4752 Phys.Rev.D80:125005,2009 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Motivation: QCD at LARGE Nc and Nf lo rs vo rs Co Fla David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Motivation: QCD at LARGE Nc and Nf lo rs vo rs Co Fla Deﬁne x= Nf/Nc, treat as a continuous variable asymptotic freedom conformal trivial 0 ¯ ψψ = 0 xc x 11/2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Motivation: QCD at LARGE Nc and Nf lo rs vo rs Co Fla Deﬁne x= Nf/Nc, treat as a continuous variable asymptotic freedom conformal trivial 0 ¯ ψψ = 0 xc x 11/2 ks gauge coupling: α✱ ? -Za nt ks oi 2 n p Ba ed 9,1 98 :18 0 ﬁx .B196 .P hys l Nuc David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Motivation: QCD at LARGE Nc and Nf lo rs vo rs Co Fla Deﬁne x= Nf/Nc, treat as a continuous variable asymptotic freedom conformal trivial 0 ¯ ψψ = 0 xc x 11/2 ks gauge coupling: α✱ ? -Za nt ks oi 2 n p Ba ed 9,1 98 :18 0 ﬁx .B196 .P hys l Nuc What is the nature of this transition? How does the IR scale appear as conformality is lost? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Outline: I. A mechanism for vanishing conformal invariance II. The Berezinskii-Kosterlitz-Thouless (BKT) transition III. A quantum mechanics model: the 1/r2 potential IV. AdS/CFT V. Relativistic model: defect Yang-Mills VI. QCD with many ﬂavors? A partner theory QCD* with a nontrivial UV ﬁxed point? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A theory with an infrared conformal ﬁxed point at g=g has a zero in the beta function: β(g) ∂g ∂g β(g) = µ = g ∂µ ∂t g∗ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A theory with an infrared conformal ﬁxed point at g=g has a zero in the beta function: β(g) ∂g ∂g β(g) = µ = g ∂µ ∂t g∗ Suppose the theory has another parameter κ such that the ﬁxed point at g=g✱ vanishes for κ>κ✱ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A theory with an infrared conformal ﬁxed point at g=g has a zero in the beta function: β(g) ∂g ∂g β(g) = µ = g ∂µ ∂t g∗ Suppose the theory has another parameter κ such that the ﬁxed point at g=g✱ vanishes for κ>κ✱ Example: supersymmetric QCD is conformal for 3/2 ≤ Nf/Nc ≤ 3 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A theory with an infrared conformal ﬁxed point at g=g has a zero in the beta function: β(g) ∂g ∂g β(g) = µ = g ∂µ ∂t g∗ Suppose the theory has another parameter κ such that the ﬁxed point at g=g✱ vanishes for κ>κ✱ Example: supersymmetric QCD is conformal for 3/2 ≤ Nf/Nc ≤ 3 “κ” = Nf/Nc, “κ✱” = 3/2, 3 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A theory with an infrared conformal ﬁxed point at g=g has a zero in the beta function: β(g) ∂g ∂g β(g) = µ = g ∂µ ∂t g∗ Suppose the theory has another parameter κ such that the ﬁxed point at g=g✱ vanishes for κ>κ✱ Example: supersymmetric QCD is conformal for 3/2 ≤ Nf/Nc ≤ 3 “κ” = Nf/Nc, “κ✱” = 3/2, 3 How is conformality lost? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Three ways to lose an infrared ﬁxed point: #1: Fixed point runs to zero: β(g; κ) β(g; κ) ← κ <κ ∗ κ >κ ∗ g g David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Three ways to lose an infrared ﬁxed point: #1: Fixed point runs to zero: β(g; κ) β(g; κ) ← κ <κ ∗ κ >κ ∗ g g Example: Supersymmetric QCD at large Nc and Nf ➙ Increasing ﬂavors, leave conformal window. κ=Nf/Nc, κ✱=3 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Three ways to lose an infrared ﬁxed point: #1: Fixed point runs to zero: β(g; κ) β(g; κ) ← κ <κ ∗ κ >κ ∗ g g Example: Supersymmetric QCD at large Nc and Nf ➙ Increasing ﬂavors, leave conformal window. κ=Nf/Nc, κ✱=3 Nf/Nc~< 3 weak coupling Banks-Zaks conformal ﬁxed point Nf/Nc > 3 trivial QED-like “free electric” theory ~ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Three ways to lose an infrared ﬁxed point: #1: Fixed point runs to zero: β(g; κ) β(g; κ) ← κ <κ ∗ κ >κ ∗ g g Example: Supersymmetric QCD at large Nc and Nf ➙ Increasing ﬂavors, leave conformal window. κ=Nf/Nc, κ✱=3 Nf/Nc~< 3 weak coupling Banks-Zaks conformal ﬁxed point Nf/Nc > 3 trivial QED-like “free electric” theory ~ g2 FE ∼ 2 r ln (r ΛUV ) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #2: Fixed point runs off to inﬁnity: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #2: Fixed point runs off to inﬁnity: β(g; α) κ >κ ∗ β(g; α) κ ≤ κ∗ ← David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #2: Fixed point runs off to inﬁnity: β(g; α) κ >κ ∗ β(g; α) κ ≤ κ∗ ← Possible example? SQCD again ➙ κ=Nf/Nc, κ✱=3/2 For κ≤κ✱ get “free magnetic phase” [Seiberg] David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #2: Fixed point runs off to inﬁnity: β(g; α) κ >κ ∗ β(g; α) κ ≤ κ∗ ← Possible example? SQCD again ➙ κ=Nf/Nc, κ✱=3/2 For κ≤κ✱ get “free magnetic phase” [Seiberg] ➤ electric theory dual to a QED-like magnetic theory: g ln (r ΛUV ) 2 2 gM FE ∼ FM ∼ gM ∼ 1/g r2 r2 ln (r ΛUV ) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #3: UV and IR ﬁxed points annihilate: A toy model: β(g; κ) = (κ − κ∗ ) − (g − g∗ )2 √ κ ≥ κ∗ : g± = g∗ ± κ − κ∗ UV, IR ﬁxed points κ = κ∗ ﬁxed points merge κ <κ ∗ conformality lost David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #3: UV and IR ﬁxed points annihilate: β(g; κ) κ >κ ∗ g A toy model: g− g+ β(g; κ) = (κ − κ∗ ) − (g − g∗ )2 √ κ ≥ κ∗ : g± = g∗ ± κ − κ∗ UV, IR ﬁxed points κ = κ∗ ﬁxed points merge κ <κ ∗ conformality lost David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #3: UV and IR ﬁxed points annihilate: β(g; κ) κ >κ ∗ g A toy model: g− g+ β(g; κ) = (κ − κ∗ ) − (g − g∗ )2 β(g; κ) κ = κ∗ √ κ ≥ κ∗ : g± = g∗ ± κ − κ∗ g∗ g UV, IR ﬁxed points κ = κ∗ ﬁxed points merge κ <κ ∗ conformality lost David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 #3: UV and IR ﬁxed points annihilate: β(g; κ) κ >κ ∗ g A toy model: g− g+ β(g; κ) = (κ − κ∗ ) − (g − g∗ )2 β(g; κ) κ = κ∗ √ κ ≥ κ∗ : g± = g∗ ± κ − κ∗ g∗ g UV, IR ﬁxed points κ = κ∗ ﬁxed points merge κ <κ ∗ β(g; κ) κ <κ ∗ conformality lost g David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 What happens close to the transition on the nonconformal side? β(g; κ) UV g∗ IR κ κ∗ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 What happens close to the transition on the nonconformal side? β(g; κ) UV g∗ IR κ κ∗ i. Start: g = gUV < g✱ in the UV ii. g grows, stalling near g✱ iii. g strong at scale ΛIR David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 What happens close to the transition on the nonconformal side? β(g; κ) UV g∗ IR κ κ∗ = ln µ i. Start: g = gUV < g✱ in the UV R dg − ii. g grows, stalling near g✱ ΛIR ΛUV e β(g) iii. g strong at scale ΛIR −√ π = ΛUV e |κ−κ∗ | David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 What happens close to the transition on the nonconformal side? β(g; κ) UV g∗ IR κ κ∗ = ln µ i. Start: g = gUV < g✱ in the UV R dg − ii. g grows, stalling near g✱ ΛIR ΛUV e β(g) iii. g strong at scale ΛIR −√ π = ΛUV e |κ−κ∗ | (Not like 2nd order phase transition: ΛIR ΛUV |κ − κ∗ | ) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 π −√ ΛIR ΛUV e |κ−κ∗ | Scaling behavior of toy model is reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition (an “inﬁnite order” phase transition) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 π −√ ΛIR ΛUV e |κ−κ∗ | Scaling behavior of toy model is reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition (an “inﬁnite order” phase transition) BKT: a classical phase transition in the 2-d XY-model David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 π −√ ΛIR ΛUV e |κ−κ∗ | Scaling behavior of toy model is reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition (an “inﬁnite order” phase transition) BKT: a classical phase transition in the 2-d XY-model Vortices in XY model box size R, vortex core size a: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 π −√ ΛIR ΛUV e |κ−κ∗ | Scaling behavior of toy model is reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition (an “inﬁnite order” phase transition) BKT: a classical phase transition in the 2-d XY-model Vortices in XY model box size R, vortex core size a: E = E0 ln R/a , S = 2 ln R/a F = E − T S = (E0 − 2T ) ln R/a David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 π −√ ΛIR ΛUV e |κ−κ∗ | Scaling behavior of toy model is reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition (an “inﬁnite order” phase transition) BKT: a classical phase transition in the 2-d XY-model Vortices in XY model box size R, vortex core size a: E = E0 ln R/a , S = 2 ln R/a F = E − T S = (E0 − 2T ) ln R/a Vortices condense for T>Tc = E0/2 ; √ b/ T −Tc ξ ae can show correlation length forms: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 π −√ ΛIR ΛUV e |κ−κ∗ | Scaling behavior of toy model is reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition (an “inﬁnite order” phase transition) BKT: a classical phase transition in the 2-d XY-model Vortices in XY model box size R, vortex core size a: E = E0 ln R/a , S = 2 ln R/a F = E − T S = (E0 − 2T ) ln R/a Vortices condense for T>Tc = E0/2 ; √ b/ T −Tc ξ ae can show correlation length forms: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 RG analysis of the BKT transition XY model = Coulomb gas (vortices = point-like charges with ln(r) Coulomb interaction): David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 RG analysis of the BKT transition XY model = Coulomb gas (vortices = point-like charges with ln(r) Coulomb interaction): Sum over vortex fugacity positions/numbers N+ N− N+ N− z z R P − d2 x T ( φ)2 +i Z=N d2 xi d2 yj Dφ e 2 i,j (φ(xi )−φ(yj )) N+ !N− ! i=1 j=1 N+ ,N− Coulomb ﬁeld vortices anti-vortices David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 RG analysis of the BKT transition XY model = Coulomb gas (vortices = point-like charges with ln(r) Coulomb interaction): Sum over vortex fugacity positions/numbers N+ N− N+ N− z z R P − d2 x T ( φ)2 +i Z=N d2 xi d2 yj Dφ e 2 i,j (φ(xi )−φ(yj )) N+ !N− ! i=1 j=1 N+ ,N− Coulomb ﬁeld vortices anti-vortices R − d2 x [ T ( φ)2 −2z cos φ] =N Dφ e 2 temp. fugacity David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 RG analysis of the BKT transition XY model = Coulomb gas (vortices = point-like charges with ln(r) Coulomb interaction): Sum over vortex fugacity positions/numbers N+ N− N+ N− z z R P − d2 x T ( φ)2 +i Z=N d2 xi d2 yj Dφ e 2 i,j (φ(xi )−φ(yj )) N+ !N− ! i=1 j=1 N+ ,N− Coulomb ﬁeld vortices anti-vortices R − d2 x [ T ( φ)2 −2z cos φ] =N Dφ e 2 temp. fugacity The XY model is equivalent to the Sine-Gordon model David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Classical XY model BKT transition = zero temperature quantum transition in Sine-Gordon model: T L= ( φ) − 2z cos φ 2 2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Classical XY model BKT transition = zero temperature quantum transition in Sine-Gordon model: T L= ( φ) − 2z cos φ 2 2 1 2z New variables: u=1− , v= 8πT T Λ2 Perturbative β-functions: βu = −2v 2 , βv = −2uv David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Classical XY model BKT transition = zero temperature quantum transition in Sine-Gordon model: T L= ( φ) − 2z cos φ 2 2 1 2z New variables: u=1− , v= 8πT T Λ2 Perturbative β-functions: βu = −2v 2 , βv = −2uv Λ = UV cutoff at vortex core Dimensionful quantities in units of XY model interaction strength David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Classical XY model BKT transition = zero temperature quantum transition in Sine-Gordon model: T L= ( φ) − 2z cos φ 2 2 1 2z New variables: u=1− , v= 8πT T Λ2 Perturbative β-functions: βu = −2v 2 , βv = −2uv Λ = UV cutoff at vortex core v Dimensionful quantities in units of XY model interaction strength •T<Tc u •bound vortices •trivially conformal David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Classical XY model BKT transition = zero temperature quantum transition in Sine-Gordon model: T L= ( φ) − 2z cos φ 2 2 1 2z New variables: u=1− , v= 8πT T Λ2 Perturbative β-functions: βu = −2v 2 , βv = −2uv Λ = UV cutoff at vortex core v Dimensionful quantities in units of XY model interaction strength •T<Tc u •bound vortices •T>Tc •trivially conformal •Coulomb gas •screening length David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 1 2z u=1− v= v 8πT , T Λ2 κ/τ τ βu = −2v 2 , βv = −2uv Newer variables: u τ = (u + v) , κ = (u2 − v 2 ) βτ = κ − τ 2 , βκ = 0 Nonperturbative region κ >κ ∗ (T<Tc) κ = κ∗ κ∗ = 0 κ <κ ∗ (T>Tc) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Correlation length in BKT transition: κ>0: Conformal (bound vortices) For small negative κ, assume τ small & positive in UV T=Tc κ<0 ﬁnite ξ τ blows up in RG time (unbound vortices) dτ π t= =− √ β(τ ) 2 −κ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Correlation length in BKT transition: κ>0: Conformal (bound vortices) For small negative κ, assume τ small & positive in UV T=Tc κ<0 ﬁnite ξ τ blows up in RG time (unbound vortices) dτ π t= =− √ β(τ ) 2 −κ ...giving rise to an IR scale (like ΛQCD) which sets the scale for the ﬁnite correlation length for α<0: 1 2√π ξBKT ∼ e −α Λ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 So far: • BKT transition = loss of conformality via ﬁxed point merger • Mechanism of ﬁxed point merger in general gives rise to “BKT scaling”: π −√ ΛIR ΛUV e |κ−κ∗ | David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 So far: • BKT transition = loss of conformality via ﬁxed point merger • Mechanism of ﬁxed point merger in general gives rise to “BKT scaling”: π −√ ΛIR ΛUV e |κ−κ∗ | Next: other examples: • QM with 1/r2 potential • AdS/CFT • Defect Yang-Mills • QCD with many ﬂavors David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Example: QM in d-dimensions with 1/r2 potential Vr κ − 2 + V (r) − k 2 ψ=0, V (r) = 2 r David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Example: QM in d-dimensions with 1/r2 potential Vr κ − 2 + V (r) − k 2 ψ=0, V (r) = 2 r k=0 solutions: ψ = c− r ν− + c+ r ν+ 2 d−2 √ d−2 ν± − ± κ − κ∗ κ∗ = − 2 2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Example: QM in d-dimensions with 1/r2 potential Vr κ − 2 + V (r) − k 2 ψ=0, V (r) = 2 r k=0 solutions: ψ = c− r ν− + c+ r ν+ 2 d−2 √ d−2 ν± − ± κ − κ∗ κ∗ = − 2 2 • valid for κ✱ < κ < (κ✱+1) • κ < κ✱: ν± complex, no ground state • κ = κ✱: ν+ = ν- • κ > (κ✱+1): rν- too singular to normalize David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 κ − 2 + V (r) − k 2 ψ=0, V (r) = 2 r k=0 solutions: ψ = c− r ν− + c+ r ν+ 2 d−2 √ d−2 ν± = ± κ − κ∗ , κ∗ = − 2 2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 κ − 2 + V (r) − k 2 ψ=0, V (r) = 2 r k=0 solutions: ψ = c− r ν− + c+ r ν+ 2 d−2 √ d−2 ν± = ± κ − κ∗ , κ∗ = − 2 2 • c+ =0 or c-=0 are scale invariant solutions • If c+≠0, ψ → c+rν+ for large r (ν+ > ν-) • to make sense of BC at r=0, introduce δ-function: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 κ − 2 + V (r) − k 2 ψ=0, V (r) = 2 r k=0 solutions: ψ = c− r ν− + c+ r ν+ 2 d−2 √ d−2 ν± = ± κ − κ∗ , κ∗ = − 2 2 • c+ =0 or c-=0 are scale invariant solutions • If c+≠0, ψ → c+rν+ for large r (ν+ > ν-) • to make sense of BC at r=0, introduce δ-function: κ V (r) = 2 − gδ (d) (r) r David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 κ − 2 + V (r) − k 2 ψ=0, V (r) = 2 r k=0 solutions: ψ = c− r ν− + c+ r ν+ 2 d−2 √ d−2 ν± = ± κ − κ∗ , κ∗ = − 2 2 • c+ =0 or c-=0 are scale invariant solutions • If c+≠0, ψ → c+rν+ for large r (ν+ > ν-) • to make sense of BC at r=0, introduce δ-function: κ V (r) = 2 − gδ (d) (r) r • rν+ dominates at large r -- corresponds to IR ﬁxed point of g • rν- dominates at small r -- corresponds to UV ﬁxed point of g David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 I. Non-perturbative RG treatment of 1/r2 potential: la tor regulate with square well: egu Vr U r0 κ/r2 r > r0 V (r) = 2 2 −g/r0 −g/r0 r > r0 κ/r2 E=0 solution for r>r0: ψ = c− r ν− + c+ r ν+ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 I. Non-perturbative RG treatment of 1/r2 potential: la tor regulate with square well: egu Vr U r0 κ/r2 r > r0 V (r) = 2 2 −g/r0 −g/r0 r > r0 κ/r2 E=0 solution for r>r0: ψ = c− r ν− + c+ r ν+ Solve for c+/c- (a physical dimensionful quantity) and require invariance: d(c+/c-)/dr0 = 0: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 I. Non-perturbative RG treatment of 1/r2 potential: la tor regulate with square well: egu Vr U r0 κ/r2 r > r0 V (r) = 2 2 −g/r0 −g/r0 r > r0 κ/r2 E=0 solution for r>r0: ψ = c− r ν− + c+ r ν+ Solve for c+/c- (a physical dimensionful quantity) and require invariance: d(c+/c-)/dr0 = 0: Find exact β-function for g. Eg, for d=3: Β g, Α √ √ √ 2√ 2 g κ + g cot g − g cot g g β = √ √ 2 √g 1 g 2 3 − cot g + g csc Α 0 κ=0 Α 14 κ=-1/4 κ✱ = -¼ , g✱ ≈ 1.36 Α 12 κ=-1/2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Aside: Even better to deﬁne a modiﬁed coupling constant √ √ g Jd/2 ( g) γ= √ Jd/2−1 ( g) Condition d(c+/c-)/dr0 yields exact β-function in d-dimensions: ∂γ d−2 βγ = = (κ − κ∗ ) − (γ − γ∗ )2 , γ∗ = ∂t 2 • Toy model is exact! • γ is a periodic function of g, γ=±∞ equivalent • Aside: Limit cycle behavior for κ<κ✱: describes “Eﬁmov states” David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 II. Perturbative RG treatment of κ/r2 potential: κ✱ ≡ -(d-2)2/4 so work in d=2+ε David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 II. Perturbative RG treatment of κ/r2 potential: κ✱ ≡ -(d-2)2/4 so work in d=2+ε 2 δ-function | ψ| gπ † † S = d † dt d x iψ ∂t ψ − + ψ ψ ψψ 2m 4 κ − dt d x d y ψ (t, x)ψ (t, y) d d † † ψ(t, y)ψ(t, x) |x − y|2 i propagator: ω − p2 /2m contact vertex: iπgµ− 2πiκ 1 “meson exchange”: |q| David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 II. Perturbative RG treatment of κ/r2 potential: κ✱ ≡ -(d-2)2/4 so work in d=2+ε 2 δ-function | ψ| gπ † † S = d † dt d x iψ ∂t ψ − + ψ ψ ψψ 2m 4 κ − dt d x d y ψ (t, x)ψ (t, y) d d † † ψ(t, y)ψ(t, x) |x − y|2 i propagator: ω − p2 /2m Find g runs: + 2 ∂g contact vertex: iπgµ− β(g; κ) = µ = κ+ − (g − )2 ∂µ 4 2πiκ 1 “meson exchange”: |q| David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 II. Perturbative RG treatment of κ/r2 potential: κ✱ ≡ -(d-2)2/4 so work in d=2+ε 2 δ-function | ψ| gπ † † S = d † dt d x iψ ∂t ψ − + ψ ψ ψψ 2m 4 κ − dt d x d y ψ (t, x)ψ (t, y) d d † † ψ(t, y)ψ(t, x) |x − y|2 i propagator: ω − p2 /2m Find g runs: + 2 ∂g contact vertex: iπgµ− β(g; κ) = µ = κ+ − (g − )2 ∂µ 4 2πiκ 1 Same as toy model! κ✱ = -ε2/4, g✱ =ε “meson exchange”: |q| Exact, ε=1: κ✱ = -1/4, g✱ =1.36 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 II. Perturbative RG treatment of κ/r2 potential: κ✱ ≡ -(d-2)2/4 so work in d=2+ε 2 δ-function | ψ| gπ † † S = d † dt d x iψ ∂t ψ − + ψ ψ ψψ 2m 4 κ − dt d x d y ψ (t, x)ψ (t, y) d d † † ψ(t, y)ψ(t, x) |x − y|2 i propagator: ω − p2 /2m Find g runs: + 2 ∂g contact vertex: iπgµ− β(g; κ) = µ = κ+ − (g − )2 ∂µ 4 2πiκ 1 Same as toy model! κ✱ = -ε2/4, g✱ =ε “meson exchange”: |q| Exact, ε=1: κ✱ = -1/4, g✱ =1.36 κ>κ✱: conformal Λ2 Λ2 √ IR UV −2π/ κ∗ −κ κ=κ✱: critical B∼ ∼ e m m κ<κ✱: g blows up in IR BKT scaling bound state energy David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conformal phases: measure correlations, not β-functions! Look at operator scaling dimensions: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conformal phases: measure correlations, not β-functions! Look at operator scaling dimensions: From Nishida & Son, 2007: • Replace V(r1-r2) ➞ V(r1-r2) + ½ ω2|r12+r22| • Compute 2-particle ground state energy E0 • Operator dimension of ψψ is Δψψ =E0/ω David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conformal phases: measure correlations, not β-functions! Look at operator scaling dimensions: From Nishida & Son, 2007: • Replace V(r1-r2) ➞ V(r1-r2) + ½ ω2|r12+r22| • Compute 2-particle ground state energy E0 • Operator dimension of ψψ is Δψψ =E0/ω 2-particle wave- function at |r1-r2|=0 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conformal phases: measure correlations, not β-functions! Look at operator scaling dimensions: From Nishida & Son, 2007: • Replace V(r1-r2) ➞ V(r1-r2) + ½ ω2|r12+r22| • Compute 2-particle ground state energy E0 • Operator dimension of ψψ is Δψψ =E0/ω 2-particle wave- function at |r1-r2|=0 As the two conformal theories merge when κ➝ κ✱ , operator dimensions in the two CFTs merge David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conformal phases: measure correlations, not β-functions! Look at operator scaling dimensions: From Nishida & Son, 2007: • Replace V(r1-r2) ➞ V(r1-r2) + ½ ω2|r12+r22| • Compute 2-particle ground state energy E0 • Operator dimension of ψψ is Δψψ =E0/ω 2-particle wave- function at |r1-r2|=0 As the two conformal theories merge when κ➝ κ✱ , operator dimensions in the two CFTs merge For 1/r2 potential -- ﬁnd for the two conformal theories: d+2 √ “+” = UV ﬁxed point [ψψ]: ∆± = (d + ν± ) = ± κ − κ∗ 2 “-” = IR ﬁxed point Note: (Δ++Δ-) = (d+2): scaling dimension of nonrelativistic spacetime. David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Analog in AdS/CFT: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Analog in AdS/CFT: d 1 AdS: ds = 2 2 dz + 2 2 dxi z i=1 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Analog in AdS/CFT: d 1 AdS: ds = 2 2 dz + 2 2 dxi z i=1 Massive scalar in the bulk two solutions to eq. of motion, corresponding to two different CFT’s: ϕ = c+ z ∆+ + c− z ∆− r dim op erato ∆± = d ± m2 + d 2 ≡ d ± m2 − m2 Δ± = 2 2 2 ∗ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Analog in AdS/CFT: d 1 AdS: ds = 2 2 dz + 2 2 dxi z i=1 Massive scalar in the bulk two solutions to eq. of motion, corresponding to two different CFT’s: ϕ = c+ z ∆+ + c− z ∆− r dim op erato ∆± = d ± m2 + d 2 ≡ d ± m2 − m2 Δ± = 2 2 2 ∗ AdS QM • (Δ++Δ-)=d= spacetime dim of CFT • (Δ+ψψ+Δ-ψψ)=(d+2)= conformal wt. of nonrelativistic d-space+time David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Analog in AdS/CFT: d 1 AdS: ds = 2 2 dz + 2 2 dxi z i=1 Massive scalar in the bulk two solutions to eq. of motion, corresponding to two different CFT’s: ϕ = c+ z ∆+ + c− z ∆− r dim op erato ∆± = d ± m2 + d 2 ≡ d ± m2 − m2 Δ± = 2 2 2 ∗ AdS QM • (Δ++Δ-)=d= spacetime dim of CFT • (Δ+ψψ+Δ-ψψ)=(d+2)= conformal wt. of nonrelativistic d-space+time • when m2 = m✱2 = -d2/4 , Δ±=d/2 • κ = κ✱ = -(d-2)2/4 Δ±=(d+2)/2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Analog in AdS/CFT: d 1 AdS: ds = 2 2 dz + 2 2 dxi z i=1 Massive scalar in the bulk two solutions to eq. of motion, corresponding to two different CFT’s: ϕ = c+ z ∆+ + c− z ∆− r dim op erato ∆± = d ± m2 + d 2 ≡ d ± m2 − m2 Δ± = 2 2 2 ∗ AdS QM • (Δ++Δ-)=d= spacetime dim of CFT • (Δ+ψψ+Δ-ψψ)=(d+2)= conformal wt. of nonrelativistic d-space+time • when m2 = m✱2 = -d2/4 , Δ±=d/2 • κ = κ✱ = -(d-2)2/4 Δ±=(d+2)/2 • Instability (no AdS or CFT) for • Conformality lost for κ < κ✱ m2 < m✱2 (B-F bound) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 AdS/CFT cont’d: As with QM example, 2 different solutions 2 different CFTs David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 AdS/CFT cont’d: As with QM example, 2 different solutions 2 different CFTs ϕ = ϕ0 z ∆ + : Zgrav. = ZCFT [ϕ0 ] ϕ − − ϕ 0 z ∆+ −→ z→0 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 AdS/CFT cont’d: As with QM example, 2 different solutions 2 different CFTs ϕ = ϕ0 z ∆ + : Zgrav. = ZCFT [ϕ0 ] ϕ − − ϕ 0 z ∆+ −→ z→0 S = SCFT + d d x φ0 O David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 AdS/CFT cont’d: As with QM example, 2 different solutions 2 different CFTs ϕ = ϕ0 z ∆ + : Zgrav. = ZCFT [ϕ0 ] ϕ − − ϕ 0 z ∆+ −→ z→0 S = SCFT + d d x φ0 O ϕ = J z ∆− : Zgrav. = ZCFT [J] ϕ− − Jz ∆− −→ z→0 R dd xJϕ = Dϕ ZCFT [ϕ]e i David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 AdS/CFT cont’d: As with QM example, 2 different solutions 2 different CFTs ϕ = ϕ0 z ∆ + : Zgrav. = ZCFT [ϕ0 ] ϕ − − ϕ 0 z ∆+ −→ z→0 S = SCFT + d d x φ0 O ϕ = J z ∆− : Zgrav. = ZCFT [J] ϕ− − Jz ∆− −→ z→0 R dd xJϕ = Dϕ ZCFT [ϕ]e i UV ﬁne-tuning: m2φ2...adds OO operator. Eg: O=ψψ, OO =ψψψψ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 AdS/CFT cont’d: As with QM example, 2 different solutions 2 different CFTs ϕ = ϕ0 z ∆ + : Zgrav. = ZCFT [ϕ0 ] ϕ − − ϕ 0 z ∆+ −→ z→0 S = SCFT + d d x φ0 O ϕ = J z ∆− : Zgrav. = ZCFT [J] ϕ− − Jz ∆− −→ z→0 R dd xJϕ = Dϕ ZCFT [ϕ]e i UV ﬁne-tuning: m2φ2...adds OO operator. Eg: O=ψψ, OO =ψψψψ analog of δ(r) in QM example tuned to unstable UV ﬁxed pt. David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A relativistic example: defect Yang-Mills theory d spatial dimensions David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A relativistic example: defect Yang-Mills theory d spatial dimensions Charged relativistic fermions on a d-dimensional defect + 4D conformal gauge theory (eg, N=4 SYM) S= d d+1 ¯ µ Dµ ψ − 1 x iψγ a d4 x Fµν F a,µν 4g 2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 A relativistic example: defect Yang-Mills theory d spatial dimensions Charged relativistic fermions on a d-dimensional defect + 4D conformal gauge theory (eg, N=4 SYM) S= d d+1 ¯ µ Dµ ψ − 1 x iψγ a d4 x Fµν F a,µν 4g 2 g doesn’t run David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 g doesn’t run by construction Expect a phase transition as a function of g: ¯ 0 g < g∗ ψψ = ΛdIR g > g∗ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 g doesn’t run by construction Expect a phase transition as a function of g: ¯ 0 g < g∗ ψψ = ΛdIR g > g∗ Add a contact interaction to the theory (as in QM & AdS/CFT examples!) and study its running: c ¯ ∆S = d d+1 x − (ψγµ Ta ψ)2 2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 g doesn’t run by construction Expect a phase transition as a function of g: ¯ 0 g < g∗ ψψ = ΛdIR g > g∗ Add a contact interaction to the theory (as in QM & AdS/CFT examples!) and study its running: c ¯ ∆S = d d+1 x − (ψγµ Ta ψ)2 2 Phase transition is in perturbative regime for d=1+ε (spatial dimensions of “defect”): compute β-function David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 = cc (¯ µ ta ψ)2 ¯ L = ······− (ψγ µta ψ)2 − ψγ L 2 1 c ¯ µ a 22 1 µ ψAµ − c (¯ µ t ψ) ) − µ ψγ a 2 d44 x Faa Faa + · · · (55) ψAµ − 2(ψγ t ψ) ) − 4 d x Fµν Fµν + · · · µν µν (55) 2 4 = β(c)β(c): (c) = a 1/ factor from the gluon propagator (40) and contains a 1/ factor from the gluon propagator (40) and ontains c: r c: 1/ε pole for d=(1+ε) Nc 2 g 22 g 1/! pole from photon 1/! pole from photon (c) = cc− Nc cc − c) = − 2 − propagator in (1+!)+1 D propagator in (1+!)+1 D (56) (56) 2π 2π 2π 2π g where β(c) has a double zero, g∗∗ where β(c) has a double zero, π22 g∗ = π = π22 Same as gap eq, except 2CAA"N Same as gap eq, except 2C "N Nc= π ∗ N g∗ c= g∗ (57) (57) N Ncc is substitution also works for mass gap at g>g his substitution also works for mass gap at g>g** RG equation, RG equation, ∂c ∂c = β(c) (58) ∂ln µ = β(c) ∂ ln µ (58) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 = cc (¯ µ ta ψ)2 ¯ L = ······− (ψγ µta ψ)2 − ψγ L 2 1 c ¯ µ a 22 1 µ ψAµ − c (¯ µ t ψ) ) − µ ψγ a 2 d44 x Faa Faa + · · · (55) ψAµ − 2(ψγ t ψ) ) − 4 d x Fµν Fµν + · · ·µν µν (55) 2 4 = β(c)β(c): (c) = a 1/ factor from the gluon propagator (40) and contains a 1/ factor from the gluon propagator (40) and ontains c: r c: 1/ε pole for d=(1+ε) Nc 2g 2 g 22 g 1/! pole from photon 1/! pole from photon Nc − − − c − Nc c2 propagator in (1+!)+1 D (c) = cc−= cc − = − β(c) 2 propagator in (1+!)+1 D (56) c) (56) 2π 2π 2π 2π 2π 2π 2 1 π 2 2 N 2 zero, c π g∗∗ where = 2πhas a double zero, β(c) g where β(c) has a double − 2 Nc −g 2π c− Nc π2 π = g∗ = π22 Same as gap eq, except 2CAA"N Same as gap eq, except 2C "N Nc= π ∗ N g∗ c= g∗ (57) (57) N Ncc is substitution also works for mass gap at g>g his substitution also works for mass gap at g>g** RG equation, RG equation, ∂c ∂c = β(c) (58) ∂ln µ = β(c) ∂ ln µ (58) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 = cc (¯ µ ta ψ)2 ¯ L = ······− (ψγ µta ψ)2 − ψγ L 2 1 c ¯ µ a 22 1 µ ψAµ − c (¯ µ t ψ) ) − µ ψγ a 2 d44 x Faa Faa + · · · (55) ψAµ − 2(ψγ t ψ) ) − 4 d x Fµν Fµν + · · ·µν µν (55) 2 4 = β(c)β(c): (c) = a 1/ factor from the gluon propagator (40) and contains a 1/ factor from the gluon propagator (40) and ontains c: r c: 1/ε pole for d=(1+ε) Nc 2g 2 g 22 g 1/! pole from photon 1/! pole from photon Nc − − − c − Nc c2 propagator in (1+!)+1 D (c) = cc−= cc − = − β(c) 2 propagator in (1+!)+1 D (56) c) (56) 2π 2π 2π 2π 2π 2π 2 1 π 2 2 N 2 zero, c π g∗∗ where = 2πhas a double zero, β(c) g where β(c) has a double − 2 Nc −g 2π c− Nc π2 π = g∗ = π22 Same as gap eq, except 2CAA"N Same as gap eq, except 2C "N Nc= π ∗ N g∗ c= g∗ (57) (57) • Find BKT transition at g2 = g✱2 = (επ)2/Nc N Ncc is substitutionΛ exp[-π/√(g2-g gap at g>g also works for mass 2 his substitution also works for mass gap at g>g** ΛIR ~ RG equation, UV equation, ✱ )] RG • Schwinger-Dyson ∂c gap eq (rainbow approx) gives ∂c = β(c) (58) qualitatively same results ∂ ln µ = β(c) (58) ∂ ln µ David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Back to QCD at LARGE Nc and Nf: asymptotic freedom conformal trivial 0 ¯ ψψ = 0 xc x 11/2 ks Za t gauge coupling: α✱ ? ks- oin an d p B e ﬁx 0 Transition at x=xc? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Back to QCD at LARGE Nc and Nf: asymptotic freedom conformal trivial 0 ¯ ψψ = 0 xc x 11/2 ks Za t gauge coupling: α✱ ? ks- oin an d p B e ﬁx 0 Transition at x=xc? Schwinger-Dyson (rainbow approximation): David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Back to QCD at LARGE Nc and Nf: asymptotic freedom conformal trivial 0 ¯ ψψ = 0 xc x 11/2 ks Za t gauge coupling: α✱ ? ks- oin an d p B e ﬁx 0 Transition at x=xc? Schwinger-Dyson (rainbow approximation): Found: BKT scaling for <ψψ>...not rigorous, but qualitatively correct? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. 3 Δ+ QCD Δψψ 2 Δ+ + Δ- = 4? - QCD* Δ 1 xBZ x x✱ =11/2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. 3 Δ+ Free fermions QCD Δψψ 2 Δ+ + Δ- = 4? - QCD* Δ 1 xBZ x x✱ =11/2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. 3 Δ+ Free fermions QCD Δψψ 2 Δ+ + Δ- = 4? - QCD* Δ Free boson 1 xBZ x x✱ =11/2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. 3 Δ+ Free fermions QCD Δψψ 2 Δ+ + Δ- = 4? - QCD* Δ Free boson 1 xBZ x x✱ =11/2 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. 3 Δ+ Free fermions QCD Δψψ 2 Δ+ + Δ- = 4? - QCD* Δ Free boson 1 xBZ x x✱ =11/2 Near Banks-Zaks (IR) ﬁxed point: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. 3 Δ+ Free fermions QCD Δψψ 2 Δ+ + Δ- = 4? - QCD* Δ Free boson 1 xBZ x x✱ =11/2 Near Banks-Zaks (IR) ﬁxed point: QCD: + Δ ψψ = 3 - # g2Nc (almost free quarks) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conjecture: loss of conformality for QCD at xc is of BKT type, due to ﬁxed point merger. 3 Δ+ Free fermions QCD Δψψ 2 Δ+ + Δ- = 4? - QCD* Δ Free boson 1 xBZ x x✱ =11/2 Near Banks-Zaks (IR) ﬁxed point: QCD: Partner theory QCD*: + - + Δψψ = d-Δψψ = 1+ # g2Nc Δ ψψ =3-# g2Nc (almost free quarks) (almost free scalar?) David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 WANTED ☛ Conformal theory defined at nontrivial UV fixed point to merge with QCD at x=xc LAST SEEN WITH WEAKLY COUPLED SCALAR David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Haven’t found a Lorentz invariant WANTED perturbative example with: ☛ Conformal theory (i) weakly coupled scalar; defined at nontrivial UV fixed point (ii) full SU(Nf)xSU(Nf) chiral symmetry to merge with QCD (iii) Matching anomalies at x=xc LAST SEEN WITH WEAKLY COUPLED SCALAR David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Haven’t found a Lorentz invariant WANTED perturbative example with: ☛ Conformal theory (i) weakly coupled scalar; defined at nontrivial UV fixed point (ii) full SU(Nf)xSU(Nf) chiral symmetry to merge with QCD (iii) Matching anomalies at x=xc Look for nonperturbative QCD* on the lattice? LAST SEEN WITH WEAKLY One place to start: strong/weak transition for QCD COUPLED SCALAR with Nf in conformal window? (A. Hasenfratz) conformal phase strong coupling phase ∗ ∗ g 0 g− g+ QCD* possibly at g+* ? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conclusions: David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conclusions: I. Fixed point annihilation appears to be a generic mechanism for the loss of conformality David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conclusions: I. Fixed point annihilation appears to be a generic mechanism for the loss of conformality II. Leads to similar scaling as in the BKT transition: ΛIR ~ ΛUV e[-π/√(-κ-κ✱)] David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conclusions: I. Fixed point annihilation appears to be a generic mechanism for the loss of conformality II. Leads to similar scaling as in the BKT transition: ΛIR ~ ΛUV e[-π/√(-κ-κ✱)] III. Both relativistic & non-relativistic examples David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conclusions: I. Fixed point annihilation appears to be a generic mechanism for the loss of conformality II. Leads to similar scaling as in the BKT transition: ΛIR ~ ΛUV e[-π/√(-κ-κ✱)] III. Both relativistic & non-relativistic examples IV. Analog in AdS/CFT; implications for AdS below the Breitenlohner-Freedman bound? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conclusions: I. Fixed point annihilation appears to be a generic mechanism for the loss of conformality II. Leads to similar scaling as in the BKT transition: ΛIR ~ ΛUV e[-π/√(-κ-κ✱)] III. Both relativistic & non-relativistic examples IV. Analog in AdS/CFT; implications for AdS below the Breitenlohner-Freedman bound? V. Implications for QCD with many ﬂavors? Is there a pair of conformal QCD theories? What is QCD*? David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 Conclusions: I. Fixed point annihilation appears to be a generic mechanism for the loss of conformality II. Leads to similar scaling as in the BKT transition: ΛIR ~ ΛUV e[-π/√(-κ-κ✱)] III. Both relativistic & non-relativistic examples IV. Analog in AdS/CFT; implications for AdS below the Breitenlohner-Freedman bound? V. Implications for QCD with many ﬂavors? Is there a pair of conformal QCD theories? What is QCD*? VI. Finding QCD* should be on ﬁeld theory / lattice QCD “to-do” list. David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010 David B. Kaplan INT Feb. 22 , 2010 Monday, February 22, 2010