Bounds on 4D Conformal and Superconformal Field Theories

Transcription

1 Bounds on 4D Conformal and Superconformal Field Theories David Poland Harvard University November 30, 2010 (with David Simmons-Duffin [arxiv: ])

2 Motivation Near-conformal dynamics could play a role in BSM physics! Walking/Conformal Technicolor [Many people...] Conformal Sequestering [Luty, Sundrum 01; Schmaltz, Sundrum 06] Solution to µ/bµ problem [Roy, Schmaltz 07; Murayama, Nomura, DP 07] Flavor Hierarchies [Georgi, Nelson, Manohar 83; Nelson, Strassler 00]... Many ideas make assumptions about operator dimensions that are difficult to check Hard to calculate anything in non-susy theories! Lattice studies may be only hope. In N = 1 SCFTs, we actually know lots about chiral operators, but not much about non-chiral operators...

3 Example: Nelson-Strassler Flavor Models [ 00] Idea: Matter fields T i have large anomalous dimensions under some CFT, flavor hierarchies generated dynamically! W = T 1 O 1 +T 2 O 2 +y ij T i T j H +... Interactions of matter T i with CFT operators O i are marginal Yukawa couplings y ij flow to zero at rate controlled by dimt i Since T i are chiral, dimt i = 3 2 R T i (superconformal U(1) R ) Can write down lots of concrete models and then calculate dimensions using a-maximization! [DP, Simmons-Duffin 09]

4 Example: Nelson-Strassler Flavor Models [ 00] Soft-mass operators K 1 X XT Mpl 2 i T j also flow to zero Rate controlled by dimt i T j Maybe can solve SUSY flavor problem??? No way to calculate dimensions... Similar issue arises in Conformal Sequestering, µ/bµ solution Can we say anything about dimt T, given dimt? Recently, the papers: Rattazzi, Rychkov, Tonni, Vichi [arxiv: ] Rychkov, Vichi [arxiv: ] addressed a similar question in non-susy CFTs, deriving bounds on dimφ 2 as a function of dimφ...

5 Outline 1 CFT Review 2 Bounds from Crossing Relations 3 Superconformal Blocks 4 Bounds on CFTs and SCFTs 5 Outlook

6 CFT Review: Primary Operators In addition to Poincaré generators P a and M ab, CFTs have dilatations D and special conformal generators K a [K a,p b ] = 2η ab D 2M ab Primary operators O I (0) are defined by [K a,o I (0)] = 0 (descendants obtained by acting with P a )

7 CFT Review: Primary Operators Primary 2-pt and 3-pt functions fixed by conformal symmetry in terms of dimensions and spins, up to overall coefficients λ O O a 1...a l (x 1 )O b 1...b l (x 2 ) = Ia 1b 1...I a lb l φ(x 1 )φ(x 2 )O a 1...a l (x 3 ) = ( I ab = η ab 2 xa 12 xb 12 x 2 12 x 2 12 λ O x 2d +l 12 x l 23 x l 13, Z a = xa 31 x 2 xa x 2 32 ) Z a 1...Z a l Higher n-pt functions not fixed by conformal symmetry alone, but are determined once spectrum and λ O s are known...

8 CFT Review: Operator Product Expansion Let φ be a scalar primary of dimension d in a 4D CFT: φ(x)φ(0) = O φ φ λ O C I (x,p)o I (0) (OPE) Sum runs over primary O s C I (x,p) fixed by conformal symmetry [Dolan, Osborn 00] O I = O a 1...a l can be any spin-l Lorentz representation (traceless symmetric tensor) with l = 0,2,... Unitarity tells us that O l+2 δ l,0 and that λ O is real

9 CFT Review: Conformal Block Decomposition Use OPE to evaluate 4-point function φ(x 1 )φ(x 2 )φ(x 3 )φ(x 4 ) = λ 2 OC I (x 12, 2 )C J (x 34, 4 ) O I (x 2 )O J (x 4 ) O φ φ 1 x 2d 12 x2d 34 O φ φ λ 2 O g,l(u,v) u = x2 12 x2 34, v = x2 x 2 14 x2 23 conformally-invariant cross ratios. 13 x2 24 x 2 13 x2 24 g,l (u,v) conformal block ( = dimo and l = spin of O)

10 CFT Review: Conformal Blocks Explicit formula [Dolan, Osborn 00] g,l (u,v) = ( 1)l 2 l zz z z [k +l(z)k l 2 (z) z z] k β (x) = x β/2 2F 1 (β/2,β/2,β;x), where u = zz and v = (1 z)(1 z). Similar expressions in other even dimensions, recursion relations known in odd dimensions Alternatively can be viewed as eigenfunctions of the quadratic casimir of the conformal group [Dolan, Osborn 03]

11 CFT Review: Crossing Relations Four-point function φ(x 1 )φ(x 2 )φ(x 3 )φ(x 4 ) is clearly symmetric under permutations of x i After OPE, symmetry is non-manifest! Switching x 1 x 3 gives the crossing relation : O φ φ λ 2 Og,l (u,v) = ( u v) d O φ φ λ 2 Og,l (v,u) 1 O 4 = 1 O Other permutations give no new information

12 Outline 1 CFT Review 2 Bounds from Crossing Relations 3 Superconformal Blocks 4 Bounds on CFTs and SCFTs 5 Outlook

13 Review: Method of Rattazzi et. al. [arxiv: ] Let s study the OPE coefficient of a particular O 0 φ φ We can rewrite crossing relation as where λ 2 O 0 F 0,l 0 (u,v) } {{ } O 0 = 1 }{{} unit op. λ 2 OF,l (u,v), O O 0 } {{ } everything else F,l (u,v) vd g,l (u,v) u d g,l (v,u) u d v d.

14 Review: Method of Rattazzi et. al. [arxiv: ] Idea: Find a linear functional α such that α(f 0,l 0 ) = 1, and α(f,l ) 0, for all other O φ φ. Applying to both sides: α ( λ 2 ) O 0 F 0,l 0 = α(1 λ 2 O F,l) O O 0 λ 2 O 0 since λ 2 O 0 by unitarity. = α(1) λ 2 O α(f,l) α(1) O O 0

15 Review: Method of Rattazzi et. al. [arxiv: ] To make the bound λ 2 O 0 α(1) as strong as possible: Minimize α(1) subject to the constraints α(f 0,l 0 ) = 1 and α(f,l ) 0 (O O 0 ). This is an infinite dimensional linear programming problem... to use known algorithms (e.g., simplex) we must make it finite Can take α to be linear combinations of derivatives at some point in z,z space α : F(z,z) a mn z m zf(1/2,1/2) n m+n 2k Discretize constraints to α(f i,l i ) 0 for D = {( i,l i )} Take k,d to recover optimal bound

16 Review: Method of Rattazzi et. al. [arxiv: ] Can make any assumptions about the spectrum that we want! E.g., can assume that all scalars appearing in the OPE φ φ have dimension larger than some min = dimo 0 If λ 2 O 0 α(1) < 0, there is a contradiction with unitarity and the assumed spectrum can be ruled out By scanning over different min, one can obtain bounds on dimφ 2 as a function of d = dimφ

17 Bounds on dimφ 2 (taken from arxiv: )

18 Limitations Only looked at single real φ, can t distinguish between O s in different global symmetry reps E.g., chiral Φ in an N = 1 SCFT: Re[Φ] Re[Φ] contains O s from both Φ Φ and Φ Φ Φ Φ Φ , with dimφ 2 = 2dimΦ: Φ 2 satisfies bound and we learn nothing about Φ Φ... Supersymmetry also relates different conformal primaries, so we should additionally take this information into account Let s try to generalize the method to deal with this case! (see Rattazzi, Rychkov, Vichi [arxiv: ] for more on global symmetries)

19 Outline 1 CFT Review 2 Bounds from Crossing Relations 3 Superconformal Blocks 4 Bounds on CFTs and SCFTs 5 Outlook

20 N = 1 Superconformal Algebra dim +1 P a +1/2 Q α Q α 0 M αβ D,R M α β 1/2 S α S α 1 K a, {Q,Q} = P {S,S} = K Superconformal primary means [S,O(0)] = [S,O(0)] = 0 Descendants obtained by acting with P,Q,Q Chiral means [Q,φ(0)] = 0

21 Superconformal Block Decomposition φ: scalar chiral superconformal primary of dimension d in an SCFT (lowest component of chiral superfield Φ) φ(x 1 )φ (x 2 )φ(x 3 )φ (x 4 ) = 1 x 2d 12 x2d 34 O Φ Φ λ O 2 ( 1) l G,l (u,v) Sum over superconformal primaries O I with zero R-charge λ O real for even spin O I, imaginary for odd spin O I x 1 x 3 gives crossing relation only involving O I Φ Φ Must organize superconformal descendants into reps of the conformal subalgebra...

22 Superconformal Block Derivation Multiplet built from O (generically) contains four conformal primaries with vanishing R-charge and definite spin: name operator dim spin O O l J,N QQO+#PO +1 l+1,l 1 D Q 2 Q 2 O+#PQQO+#PPO +2 l Superconformal symmetry fixes coefficients of φφ J, φφ N, φφ D in terms of φφ O Must also normalize J,N,D to have canonical 2-pt functions Superconformal block is then a sum of g,l s for O,J,N,D

23 Superconformal Block Derivation We find, 1 G,l = g,l ( +l) 2( +l +1) g +1,l+1 ( l 2) 8( l 1) g +1,l 1 + ( +l)( l 2) 16( +l+1)( l 1) g +2,l. When unitarity bound l+2 is saturated, multiplet is shortened. G,l can also be determined from consistency with N = 2 superconformal blocks computed by Dolan and Osborn [ 01]. 1 after plenty of algebra

24 Outline 1 CFT Review 2 Bounds from Crossing Relations 3 Superconformal Blocks 4 Bounds on CFTs and SCFTs 5 Outlook

25 Bounds on Dimension of Φ Φ Isolating the lowest dimension scalar Φ Φ Φ Φ, we have λ Φ Φ 2 F min,0 = 1 λ O 2 F,l, O Φ Φ where min = dimφ Φ, and F,l is F,l with g,l ( 1) l G,l. Now minimize α(1) subject to α(f,0 ) 0 for all min, α(f,l ) 0 for all l +2 and l 1, α(f min,0) = 1 If α(1) < 0, we get λ Φ Φ 2 < 0 = Φ Φ can t have dim min

26 Upper Bound on Dimension of Φ Φ max( Φ Φ) Ruled Out = 2d d Scanning over min, minimizing α(1) over 21 dimensional space of derivatives

27 Flavor Currents If φ transforms under flavor symmetry with charges T I, conserved currents J I appear in the φ φ OPE: φφ J I i 2π 2TI (Ward id.) Flavor current conformal blocks are then determined by current 2-pt functions J I J J 3 4π 4τIJ φφ φφ 1 3 τ IJT I T J g 3,1 φφ φφ τ IJ T I T J G 2,0 (general CFTs), (SCFTs), where τ IJ = (τ IJ ) 1 (in SCFTs, τ IJ = 3Tr(RT I T J )).

28 Upper Bounds on τ IJ T I T J τ IJT I T J charged scalar (CFT) Ruled Out 0 d τ IJT I T J chiral primary (SCFT) Ruled Out 2 SQCD {}}{ 0 d Example: SUSY QCD with 3 2 N c < N f < 3N c, consider MM MM : d = 3 3Nc N f and τ IJ T I T J = 2 N f 1 3 Nc 2 In dual AdS 5, (8π 2 L)τ IJ = gij 2. Gauge coupling can t be too strong in presence of charged scalar.

29 The Stress Tensor Ward identity ensures T ab φ φ TT is proportional to the central charge c (trace anomaly 16π 2 T a a = c(weyl) 2 a(euler)) In an SCFT, T lives in the supercurrent multiplet J a = J a R +θσ bθt ab +..., and c is determined in terms of U(1) R anomalies Conformal block contributions are φφφφ d2 90c g 4,2 φφ φφ d2 36c G 3,1 (general CFTs) (SCFTs)

30 Lower Bound on c in General CFT c Ruled Out general CFT free scalar d c SCFT Ruled Out free chiral superfield d Scalar with d 1 contributes 1 degree of freedom In dual AdS 5, c π 2 L 3 MP 3. Gravity can t be too strong in presence of bulk scalar. See also Rattazzi, Rychkov, Vichi [arxiv: ]

31 Outline 1 CFT Review 2 Bounds from Crossing Relations 3 Superconformal Blocks 4 Bounds on CFTs and SCFTs 5 Outlook

32 Outlook We calculated: Superconformal blocks Bound dimφ Φ < f Φ Φ(d) Bound τ IJ T I T J f τ (d) in CFT, SCFT Bound c f c (d) in CFT, SCFT In the future, we d like: Stronger bounds to make contact with BSM motivation! Improved numerics and better algorithms. SUSY theories that come close to saturating bounds on τ,c. Bounds in other numbers of dimensions. Understand bounds from bulk dual perspective.