Scale without conformal invariance

Scale without conformal invariance Andy Stergiou Department of Physics, UCSD based on arxiv:1106.2540, 1107.3840, 1110.1634, 1202.4757 with Jean-François Fortin and Benjamín Grinstein

Outline The physics: Background and motivation New improved SE tensor and scale invariance How generators of dilatations generate dilatations Associated RG flows On the -theorem Stability and scheme changes The examples: Method of search Scalars; scalars and spinors in =4 General QFT in Future work and conclusion

The physics

Scale and conformal invariance The symmetry group of QFTs usually consists of the Poincaré group and an internal symmetry group. The Poincaré group can be extended to include supersymmetry, and/or conformal transformations. But, in principle, only scale transformations could be allowed, without special conformal ones. In a lot of examples, if the theory is unitary and scale-invariant, then it is automatically conformally invariant.

Scale? conformal invariance In =2 the answer is yes! (Polchinski, 1988; Zamolodchikov, 1986) Operating assumptions: Unitarity Finiteness of correlators of SE tensor Relaxing these assumptions, leads to examples. (Riva & Cardy, 2006; Hull & Townsend, 1986) Many examples in there has been no proof. have suggested the same answer, but Counterexamples are classical: Roman s field theory for a scalar(jackiw & Pi, 2011) Free Maxwell theory in = 4 Holography: Kerr-AdS black holes in =57 Extension of Liouville theory (Jackiw & Pi, 2011; El-Showk, Nakayama & Rychkov, 2011) (Awad & Johnson, 1999) (Iorio, O Raifeartaigh, Sachs & Wiesendanger, 1997)

Is there room for scale without conformal invariance? Maybe there is a proof à la Polchinski, since so many examples show that scale implies conformal invariance. But it would be much more interesting if such a proof did not exist: Conformal theories are the most tractable nontrivial QFTs, and we can hope that scale-invariant theories are also tractable, while almost completely unexplored Since the symmetry is weaker, they must be richer than CFTs Possibly, there s a whole new class of QFTs!

Conformal invariance Object of interest: Stress-energy tensor, A theory has conformal invariance iff T µ µ = µ ν L µν T µν (symmetric) Equivalently, there is a new improved SE tensor, Θ µ µ =0 Θ µν, with (Callan, Coleman & Jackiw, 1970) Conformal implies scale, but algebra doesn t imply the converse: [M µν P ρ ]=(η µρ P ν η νρ P µ ) [M µν K ρ ]=(η µρ K ν η νρ K µ ) [M µν M ρσ ]=(η µρ M νσ η νρ M µσ + η νσ M µρ η µσ M νρ ) [D P µ ]= P µ [D K µ ]=K µ [K µ P ν ]=2(η µν D M µν )

Scale invariance If we ask for just scale invariance, then the dilatation current µ = ν Θ µν V µ Virial current is conserved if Θ µ µ = µ V µ with V µ = J µ + ν L µν where µ J µ =0. (Polchinski, 1988) In a scale-invariant theory the trace of the SE tensor is nonzero, but just a total derivative.

Scale without conformal invariance The question of scale without conformal invariance can thus be asked as follows: (Polchinski, 1988) Are there nontrivial candidates for V µ? Constraints on virial: Gauge-invariant Scaling dimension 1 in spacetime dimensions Attempt: φ 4 theory in =4 Answer: No nontrivial candidate for V µ No possibility for scale without conformal invariance Same conclusion for φ 6 in =3 and φ 3 in =6.

Most general QFT in =4 But in more complicated theories there are nontrivial candidates for V µ. Consider the most general renormalizable QFT: L = µ Z A 1 4 2 A F A µνf Aµν + 1 2 Z 1 2 Z 1 2 D µ φ D µ φ + 1 2 Z 1 2 Z 1 2 ψ σ µ D µ ψ 1 2 Z 1 2 Z 1 2 D µ ψ σ µ ψ 1 4! µ (λz λ ) φ φ φ φ 1 2 µ 2 (Z ) φ ψ ψ 1 2 µ 2 (Z ) φ ψ ψ

Nontrivial virial The virial is V µ = Q φ D µ φ P ψ σ µ ψ with Q = Q P = P Q and P also satisfy conditions for gauge invariance of V µ. The virial generates an internal transformation of the fields. It cannot be improved away from µ = ν Θ µν V µ.

Scale vs conformal Assume that we have a theory in 4 dimensions with scale but without conformal invariance. Then the virial Has no anomalous dimension (scaling dimension exactly 3) Is gauge invariant Is not conserved This is impossible in a unitary CFT (Mack, 1977 (also Intriligator, Grinstein & Rothstein, 2008)) It is however within bounds on operator dimensions in nonconformal scale-invariant unitary theories.(intriligator, Grinstein & Rothstein, 2008) Differences between scale and conformal invariance start showing up immediately!

Algebraic condition for SI The new improved SE tensor is (Robertson, 1991) Θ µ µ () = β A F 2 µνf A Aµν + γ 3 D 2 φ φ γ ψ σ µ D µ ψ + γ D µ ψ σ µ ψ A 1 4! (β γ λ γ λ γ λ γ λ )φ φ φ φ 1 2 (β γ γ γ )φ ψ ψ + h.c. The divergence of the dilatation current is µ µ () = β A F 2 µνf A Aµν +(γ 3 + Q )D 2 φ φ (γ + P ) ψ σ µ D µ ψ +(γ + P )D µ ψ σ µ ψ A 1 4! (β γ λ γ λ γ λ γ λ )φ φ φ φ 1 2 (β γ γ γ )φ ψ ψ + h.c.

Algebraic condition for SI We can now use the equations of motion to find that µ µ () =0 when β A =0 β = Q λ Q λ Q λ Q λ β = Q P P A solution to these equations with nonzero Q and/or P defines a theory with scale but without conformal invariance. These equations are true to all orders in perturbation theory.

How do generators of dilatations generate dilatations? The naive Ward identity for scale invariance becomes the Callan Symanzik equation in the quantum theory: µ µ + β + γ 4 δ () Γ[() µ] =0 δ () The dilatation generator can be redefined to account for anomalous dimensions, but not for beta functions. (Coleman & Jackiw, 1971)

How do generators of dilatations generate dilatations? µ µ + β + γ 4 () δ δ () Γ[() µ] =0

How do generators of dilatations generate dilatations? µ µ + β + γ 4 () δ δ () Γ[() µ] =0 µ µ +(γ + Q ) 4 () δ δ () Γ[() µ] =0 The dilatation generator can be redefined to account for these very special beta functions too! (Fortin, Grinstein & AS, 2011)

How do generators of dilatations generate dilatations? D = 3 0 = 3 ( µ Θ 0µ V 0 ) E.g. in a theory with scalars and fermions [Dφ ()] = ( + 1)φ () Q φ () [Dψ ()] = ( + 3 2 )ψ () P ψ () There are contributions to scaling dimensions from the beta functions: = δ + Q + γ = 3 2 δ + P + γ

RG trajectories in SIFTs If we find a point where the theory is scale-invariant, then there must be an RG trajectory through that point. What is the nature of such a trajectory? Schematically: = Q The system can be solved, and if is antisymmetric or anti- Hermitian we ll get a periodic or quasi-periodic solution. Q

Running of the couplings If we have a scale-invariant point, then all points where and are scale-invariant. and are constant and so is orthogonal and unitary.

Is this physical? It seems that these special beta functions can be shifted into the anomalous dimension by a field redefinition: λ ()φ φ φ φ λ (0)φφ φ φ φ = Q φ True, but this is a scale-dependent field redefinition. Such redefinitions change what we mean by a beta function. Any RG flow can be generated by an appropriate scale-dependent field redefinition anyway. (Latorre & Morris, 2001)

Scale invariance Recurrence Trajectories that go through scale-invariant points are periodic or quasi-periodic! We can get both limit cycles and ergodic behavior. This is behavior first speculated to exist in QFTs by Wilson in 1971! Such behavior in a field theory is in sharp contrast with the strongest version of the -theorem. The usual intuition is that massless degrees of freedom are lost as one coarse-grains. This intuition is violated on scale-invariant trajectories. If scale-invariant trajectories exist, then RG flows are not gradient flows.

Aside: -theorem Strongest version: Positive-definite = G β Strong version: 0 Weak version: UV > IR

Known results In the most general 4d theory there is a proof of the strong version of the -theorem at two loops at weak coupling. (Jack & Osborn, 1990) The proof relies on using Weyl consistency conditions in the curved-space theory. To all orders checked (two loops) there s actually a proof of the strongest version.(jack & Osborn, 1990) A claim for the proof of the weak version of the -theorem was recently made. (Komargodski & Schwimmer, 2011)

SIFTs hidden in Osborn s results When spin-1 operators of dimension 3, cf. virial, are included, then Osborn shows that (Osborn, 1991) = G B + [ W ] B B = β (Q) Because of this, = (G [ W ] )β B There s a -function that becomes zero both at fixed points and on scale-invariant trajectories! (Fortin, Grinstein & AS, 2012)

Scale invariance Recurrence

Stability in CFTs To study stability we linearize around the fixed point: β() =[() ] β + = Very easy to solve: () = +( 0 ) exp β = The eigenvalues of β/ = approach to the fixed point. determine the nature of the

Stability in SIFTs To study stability we linearize around the trajectory: β() =β = () +[() ()] β + = () Non-trivial to solve because everything is RG-time-dependent. But there must be a variable that takes the RG-time-dependence out of β/. = () Using the comoving variable δ() =[() ()] Q δ() = δ()s + S = β + Q = (0)

Stability in SIFTs The eigenvalues of the stability matrix β S = + Q = (0) tell us if a deformation is attractive or repulsive. S has scheme-independent eigenvalues. S always has a zero eigenvalue with corresponding eigenvector the beta function on the trajectory.

Scheme changes Q and P are scheme-independent. CFTs SIFTs Existence of fixed point Existence of SI trajectory Eigenvalues of γ. Eigenvalues of γ + Q. β β + Q Eigenvalues of. Eigenvalues of. β First coefficient in. First coefficient in γ. (Same) (Same)

The examples

Scale? conformal invariance In multi-flavor becomes φ 4 (Scalars in =4 ) theory the condition for scale invariance from T µ µ where antisymmetric. permutations, with from virial V µ = Q φ µ φ Q In, a solution to is automatically a solution that sets both sides to zero. This can be shown at one and two loops. (Fortin, Grinstein & AS, 2011) (Polchinski, 1988) No possibility for scale without conformal invariance!

Scale? conformal invariance (Scalars and spinors in =4 ) The condition for scale invariance is and where anti-hermitian. with and are solved at fixed points at one loop:(dorigoni & Rychkov, 2009) = β (1-loop) =0 =0

Scale? conformal invariance (Scalars and spinors in =4 ) At two loops, however,(fortin, Grinstein & AS, 2011) and there s a chance that = Re( β(2-loop) ) = 0 and have solutions that are not fixed points. Necessary (but not sufficient) condition for the existence of scale without conformal invariance is satisfied.

Specific models With one scalar and any number of Weyl spinors, scale implies conformal invariance to all orders in perturbation theory. (Fortin, Grinstein & AS, 2011) We need at least two scalars and at least one Weyl spinor. In we want to find well-defined examples: Unitary With bounded scalar potential

Scale? conformal invariance (Scalars and spinors in =4 ) Is there really a solution to and with Q and/or nonzero? P Search à la Wilson Fisher: λ = 1 λ () = 1 () 1 2 Q = 2 Q () P = 2 P () Plug into and and solve order by order in ( 3/2, 2,...).

Scale? conformal invariance (Scalars and spinors in =4 ) First order: System of coupled nonlinear equations Many solutions Throw away bad ones Beyond first order: Use solutions of first order System of linear equations Unique solution

Second order in Specific model: 2 scalars and 2 Weyl spinors (17 couplings) Q 12 = and Do we get a nonzero? Im P 12 = 7 independent couplings undetermined (2) 1 + 24 2

Second order in Specific model: 2 scalars and 2 Weyl spinors (17 couplings) Q 12 = and Im P 12 = undetermined 7 independent couplings Not at two loops... 1 = 2 2 = 1 12 (2) 1 + 24 2 =0 (but (3) =0) Chance for scale without conformal invariance is not utilized???

Second order in Specific model: 2 scalars and 2 Weyl spinors (17 couplings) Q 12 = and Im P 12 = undetermined 7 independent couplings Not at two loops... FRUSTRATING 1 = 2 2 = 1 12 (2) 1 + 24 2 =0 (but (3) =0) Chance for scale without conformal invariance is not utilized???

More loops in CFTs β

More loops in SIFTs β

Third order in New diagrams contribute to, e.g....

12 diagrams in total contribute to : We computed these diagrams and (3) =0 We thus have the first example of scale without conformal invariance!!!!!! Third order in (3) 71 + 3( 1 +2 2 +2 3 + 4 +2 5 +4 6 +8 7 )+ This theory is unitary with bounded-from-below scalar potential. (3) We have a limit cycle with frequency. The eigenvalues of the stability matrix tell us that there are 5 attractive and 1 repulsive direction. 4( 8 +2 9 +3 10 +4 11 + 58 12 )

Oscillating*couplings 0.6 λ 1 λ 2 0.4 0.2 λ 5 λ 4 λ 3 2 4 6 8 t 10 9 0.2 0.4 y 3 y 5 y 1 y 2

Scale conformal invariance Limit cycle is established! For ergodic behavior we need at least 4 scalars and 2 spinors (at most 59 couplings). The solutions have been found in an expansion in, and they disappear when (like, e.g., the Wilson Fisher fixed point). The limit leads to strong coupling, and so we cannot claim a result in.

Scale conformal invariance =4 has always been useful in the study of properties of the renormalization group. But we want to study theories in integer spacetime dimensions. In we can add gauge fields and go to a Banks Zaks fixed point for the gauge coupling. We have found an example an SU(2) gauge theory with two singlet scalars, two pairs of fundamental+anti-fundamental Weyl spinors, and pairs of sterile Weyl spinors. Problems: The scalar potential is unbounded from below. The three-loop beta function might spoil the result.

Conclusion Scale conformal invariance (established in ). Scale-invariant theories are less constrained than conformal theories with novel unexplored features. Scale invariance recurrent behaviors. Phenomenological applications: Cyclic unparticle physics. (Fortin, Grinstein & AS, 2011) Future work: Examples in integer dimensions BSM phenomenology Supersymmetry Holographic description