Correlation functions of conserved currents in four dimensional CFT. Yassen Stanev

Transcription

1 Correlation functions of conserved currents in four dimensional CFT Yassen Stanev INFN Sezione di Roma Tor Vergata Superfields, Pisa arxiv: [hep-th], (Nucl. Phys. B865 (2012) 200)

2 Plan Motivation Global Conformal Invariance. Bilocal fields, biharmonic bilocal fields. The scalar case. Complete classification of the 3-point functions of symmetric higher spin conserved currents. 4-point functions of U(1) currents (Aiming at the 4-point functions of the stress-energy tensor) Conclusions. Open problems

3 Motivation Conformal field theory : more constraints, still non-trivial. 3-point functions are (almost) fixed. Conserved currents : encode the symmetry. In particular the stress tensor should appear in the OPE of any field, since everything couples to gravity. 4-point functions are still difficult. Useful tool : the Conformal Partial Waves expansion. There are many CPW for fixed quantum numbers of the exchanged operator. Different 3-point functions different CPWs Coleman-Mandula theorem in CFT. Infinite number of conserved charges free theory? In 3D holds [Maldacena, Zhiboedov, 2011] In 4D? Personal [Ya.S. Bulg.J.Phys. 15 (1988)]

4 Conformal transformations Rescale the metric Poincaré Dilatations η µν Ω 2 (x)η µν Special conformal transformations x µ ρx µ x µ x µ + c µ x c.x + c 2 x 2 15 parameters : connected SO 0 (4, 2)/Z 2 cover SU(2, 2) Infinitesimal conformal transformations preserve the causal ordering. Finite conformal transformations can map spacelike into timelike separated pairs of points.

5 Representations Elementary induced (from the stability subgroup of x = 0) representations Operator O(x) : triple (j 1, j 2 ; ) (j 1, j 2 ) (half) integers : finite dimensional irreps of SL(2, C) : scale dimension (eigenvalue of the Dilation operator) Unitary representations [Mack, 1977] j 1 + j if j 1 j 2 = 0 j 1 + j if j 1 j 2 0 J (µ1...µ r)(x) symmetric traceless tensor of rank r representation (r/2, r/2; r ). Twist = r r 2

6 Global Conformal Invariance 1 Correlation functions are invariant under single valued action of SU(2, 2). Corollaries [Nikolov, Todorov; Bakalov, Ya.S., Rehren, ] All scale dimensions are integer. Huygens locality. Fields commute if (x 1 x 2 ) 2 0, for sufficiently large N. (x 2 12) N [O 1 (x 1 ), O 2 (x 2 )] = 0 Correlation functions are rational functions W (x 1,..., x n ) = µ ij C(µ ij ) i<j 1 ( x 2 ij ) µij x ij = x i x j, µ ij - integer Cluster decomposition : the 2-point function dominates. µ ij are bounded from above

7 Global Conformal Invariance - Bilocal fields Scalar Φ of scale dimension OPE 1 1 Φ (x 1 )Φ (x 2 ) = Φ (x 1 )Φ (x 2 ) + (x 2 V k(x 1, x 2 ) ) k k=1 V k (x 1, x 2 ) bilocal orthogonal conformal fields of weights (k, k), contain the contributions of the fields with twist=2k. V 1 (x 1, x 2 ) (the twist=2 fields) is biharmonic 1 V 1 (x 1, x 2 ) = 2 V 1 (x 1, x 2 ) = 0 Infinite number of conserved currents Biharmonicity constrains the correlation functions of V 1.

8 Global Conformal Invariance - The scalar case If in the theory there is a scalar Φ of scale dimension if = 2, then all the functions are equal to the ones in the theory of a finite set of free massless scalar fields with global symmetry algebra O(N), U(N) or Sp(2N) If = 3 or = 4, the possible 4-point functions are classified, there are unsolved questions for the positivity of higher point functions Some preliminary results for N = 1 SUSY theories Beyond the scalar case? Higher spin fields not much studied, since Many possible conformal structures Operator mixing problem The necessary bilocal fields are more complicated

9 Conservation Conserved currents µ1 J (µ1...µ r)(x) = 0 compatible with conformal invariance if r = r + 2. Twist=2. Useful parametrization J r (a, x) = a µ1... a µr J µ1...µ r (x), a 2 = 0 Simple special conformal transformations [C λ, J r (a, x)] = (2x λ (x. x + r ) x 2 x λ + 2 a λ x. a 2 a.x a λ)j r (a, x) Complicated conservation law [Bargmann, Todorov, 1977] ((a. a + 1) a. x 12 a. x a ) J r (a, x) = 0

10 2-point functions Conformal invariant 2-point function 0 J r1 (a, x 1 ) J r2 (b, x 2 ) 0 = δ r1r 2 c(r 1 ) (R ab) r1 x 4 12, x ij = x i x j, c(r) normalization constant, R ab - 2-point conformal covariant of weights (1, 1) in (x 1,x 2 ) R ab = R ab (x 12 ) = 1 ( x 2 a.b 2 a.x ) 12 b.x x No trace subtractions (a 2 = b 2 = 0) 2-point function conserved + OPE ALL n-point functions conserved

11 3-point functions Find ALL 3-point functions G(r 1, r 2, r 3 ) = 0 J r1 (a, x 1 ) J r2 (b, x 2 ) J r3 (c, x 3 ) 0 =? J ri (x i ) conserved a 2 = b 2 = c 2 = 0 Primitive covariants? Parity (even / odd)? Symmetry under permutations? How many functions (for fixed r 1 r 2 r 3 )? Explicit form? Generating function?

12 Primitive covariants 1 6 independent parity even covariants R ab = R ab (x 12 ) of weights (1, 1, 0) R ac = R ac (x 13 ) of weights (1, 0, 1) R bc = R bc (x 23 ) of weights (0, 1, 1) L a = L a (123) = a.x 12 x 2 12 L b = L b (231) = b.x 23 x 2 23 L c = L c (312) = c.x 31 x 2 13 a.x 13 x 2 13 b.x 21 x 2 12 c.x 32 x 2 23 of weights (1, 0, 0) of weights (0, 1, 0) of weights (0, 0, 1)

13 Primitive covariants 2 Unique parity odd covariant of weights (1, 1, 1) in (x 1, x 2, x 3 ) O abc = 1 [ 2 (a.x12 + a.x 13 ) ɛ b,c,x12,x (b.x 12 b.x 23 ) ɛ a,c,x12,x23 x 2 12 x2 13 x2 23 2(c.x 13 + c.x 23 ) ɛ a,b,x13,x23 + x 2 12ɛ a,b,c,x12 x 2 13ɛ a,b,c,x13 + x 2 23ɛ a,b,c,x23], where ɛ a,b,c,z = a µ b ν c ρ z τ ɛ µνρτ. In D = 4, R and L are independent, any even power of O abc can be expressed in terms of R and L. In D 5, R and L are independent, no odd 3-point covariant. In D = 3, R and L are not independent [Osborn, Petkou, 1994] 2R ab R ac R bc + (R ab L c + R ac L b + R bc L a + 2L a L b L c ) 2 = 0, 3 different odd 3-point covariants. [Giombi, Prakash, Yin, 2011]

14 Primitive covariants 3 Send one of the points x to infinity. The prescription is Conformal inversion x x /x 2 Translation x 0 J (µ1...µ r)( ) = lim x x2 r r i=1 ( δµ νi i 2 x ) µ i x νi x 2 J (ν1...ν r)(x). Send x 1, x 3 0, the 3-point function depends only on a, b, c and z = x 2, six even structures, a.b, a.c, b.c, a.z, b.z and c.z (since a 2 = b 2 = c 2 = 0, and the powers of z 2 are determined by the scaling properties). unique odd structure, a µ b ν c ρ z τ ɛ µνρτ. Invert the argument: Any Lorentz and scale invariant function of a, b, c, z can be uplifted to conformal invariant 3-point function.

15 Primitive covariants 4 R, L and O : basis for functions of tensors of well defined parity; Not sufficient for spinors, spin-tensors, more complicated representations which are not eigenstates of parity; Generalization [Todorov, 2012] Complex auxiliary commuting spinors λ i, λ j. New primitive 2-point invariants P ij P ji, type (1, 1) in (λ, λ) P 12 = λ 1ˆx 12 λ2 x 2 12, ˆx = x µ σ µ R ab is not primitive, bilinear in P R ab = 2 P 12 P 21, a µ = λ 1 σ µ λ1, b ν = λ 2 σ ν λ2 Invariants of type (3, 1) or (1, 3) in (λ, λ). What is the complete basis of such invariants?

16 3-point functions - Ansatz G(r 1, r 2, r 3 ) = 0 J r1 (a, x 1 ) J r2 (b, x 2 ) J r3 (c, x 3 ) 0 Even G(r 1, r 2, r 3 ) even = 1 x 2 C 12 x2 13 x2 {k,n} e Rk1 ab Rk2 acr k3 bc Ln1 23 {k,n} k 1 + k 2 + n 1 = r 1, k 1 + k 3 + n 2 = r 2, k 2 + k 3 + n 3 = r 3. a L n2 b Ln3 c, Odd G(r 1, r 2, r 3 ) odd = O abc x 2 C 12 x2 13 x2 {k,n} o Rk1 ab Rk2 acr k3 bc Ln1 23 {k,n} a L n2 b Ln3 c, k 1 + k 2 + n 1 = r 1 1, k 1 + k 3 + n 2 = r 2 1, k 2 + k 3 + n 3 = r 3 1.

17 3-point functions - Permutation symmetry Under the simultaneous exchange of (a, x 1 ) with (b, x 2 ), L a L b, L b L a, L c L c, R ab R ab, R ac R bc, R bc R ac, O abc O abc, and similar under the other permutations (L change sign, while R and O do not). By comparing the parity of n 1 + n 2 + n 3 and r 1 + r 2 + r 3 : If two of the r i are equal the even functions are symmetric/ antisymmetric under permutation if r 1 + r 2 + r 3 is even/ odd. the odd functions are symmetric/ antisymmetric under permutation if r 1 + r 2 + r 3 is odd/ even. the 3-point functions of three equal currents of rank r are necessarily of the even/odd type for r even/odd.

18 3-point functions - How many? Known only for small rank (r 1, r 2 2) r 1 = r 2 = r 3 = 1 : 2 even, 1 odd [Schreier, 1971] r 1 = r 2 = 1, r 3 = 2 : 2 even r 1 = r 2 = r 3 = 2 : 3 even [Ya.S, 1988] generalized for D > 4 [Osborn, Petkou, 1994] r 1 = 1, r 2 = r 3 = 2 : 1 odd [Erdmenger, 1999] r 1 = r 2 = 2, r 3 > 2, D 4 : 3 even; Conjecture for the general case [Costa, Penedones, Poland, Rychkov, 2011] Imposing the conservation conditions Even functions : min(r 1, r 2, r 3 ) + 1 Odd functions : min(r 1, r 2, r 3 ) Free massless fields transforming in the representations (s, 0) and (0, s) of the Lorentz group.

19 Generating function 1 Even 3-point functions G even = 1 x 2 12 x2 13 x2 23 w v w 1 2 ζ X abc w 2 ζ X abc w 1 2 ζ2 R ab R ac R bc, Odd 3-point functions G odd = 1 x 2 12 x2 13 x2 23 w v ζ O abc w 2 ζ X abc w 1 2 ζ2 R ab R ac R bc, where X abc = 2 L a L b L c + R ab L c + R ac L b + R bc L a, u = 2 (1 L a )(1 L b )(1 L c )+R ab (1 L c )+R ac (1 L b )+R bc (1 L a ), v = u R ab R ac R bc, w = u + v

20 Generating function 2 Rescale L a ρ 1 L a, L b ρ 2 L b, L c ρ 3 L c, R ab ρ 1 ρ 2 R ab, R ac ρ 1 ρ 3 R ac, R bc ρ 2 ρ 3 R bc, O abc ρ 1 ρ 2 ρ 3 O abc, X abc ρ 1 ρ 2 ρ 3 X abc, Expand in power series in ρ i and ζ. The coefficients of ρ r1 1 ρr2 2 ρr3 3 ζn even / odd 3-point functions of conserved currents of ranks r 1, r 2, r 3, for any value of n from 0 / 1 up to min(r 1, r 2, r 3 ).

21 Generating function - Factorization 1 Auxiliary conformal covariants I r of weights (r, r, r) in (x 1, x 2, x 3 ) z X abc 1 z X abc 1 2 z2 R ab R ac R bc = r z r I r, I 0 = 1, I 1 = 1/2 X abc, I 2 = 1/2 (R ab R ac R bc + X 2 abc ), etc. G(r, r, r) = I r x 2 12 x2 13 x2 23 even 3-point function for three conserved currents of equal rank r. The (even) generating function can be written as G even = r ζ r G r, G r = G(r, r, r) v w r. For fixed power of ζ, common factor G(r, r, r)

22 Generating function - Factorization 2 The conserved 3-point functions can be organized in families. Each family has a common factor G(r, r, r), conserved 3-point function. Due to the free field construction in terms of (s, 0) and (0, s) fields [Gelfond, Skvortsov, Vasiliev, 2006] G 0 : free complex scalar; G 1 : free (Dirac) fermion; G 2 : free Maxwell field;... G 2s : free (s, 0) and (0, s) fields. The trace subtraction terms do not respect factorization.

23 Generating function - Derivation 1 Acting with D ab = R ab La Lb, D ac and D bc on L r1 a L r2 b Lr3 c G(r 1, r 2, r 3 ) even = 1 x 2 12 x2 13 x2 23 Solving the conservation conditions G(r 1, r 2, r 3 ; r) = G(r, r, r) f (r) ab f (r) P (D ab, D ac, D bc ) L r1 a L r2 b Lr3 c. ac f (r) bc (Lr1 r a L r2 r b Lc r3 r ) is conserved for any integer 0 r min(r 1, r 2, r 3 ), if f (r) ab = k=0 ( 1 R ) k ab k k!(k + r)! 2 L k a k L k b. factorization of G(r, r, r); factorization of f ab, f ac, f bc.

24 Generating function - Derivation 2 Generating function for fixed r G r = r 1,r 2,r 3=r N (r) (r 1,r 2,r 3) G(r 1, r 2, r 3 ; r), with arbitrary non-vanishing coefficients. Our choice C r r 1 C r r 2 C r r 3 gives N (r) (r 1,r 2,r 3) Lr1 r a r 1,r 2,r 3=r L r2 r b L r3 r c = 1 [(1 L a )(1 L b )(1 L c )] r+1. Generalization for D 4. [Zhiboedov, 2012] Acting on e La e L b e Lc Both factorization properties remain true also for D > 4

25 4-point function of conserved abelian currents Conformal invariant 4-point function 0 J α1 (x 1 )J α2 (x 2 )J α3 (x 3 )J α4 (x 4 ) 0 =? conserved U(1) currents of scale dimension J = 3 Building blocks out of 4 points 6 are 2-point covariants R(ij) α J α (x) = 0 12 are 3-point covariants L(ijk) only 8 are independent : L(123) + L(134) = L(124) etc. 2 conformal invariant cross-ratios s = x2 12x 2 34 x 2 13 x2 24, t = x2 14x 2 23 x 2 13 x2 24

26 4-point function - Assumptions 1 0 J α1 (x 1 )J α2 (x 2 )J α3 (x 3 )J α4 (x 4 ) 0 = 1 x 4 13 x4 24 Even, conformal invariant function Full permutation symmetry P i LLLL, LLR, RR i P i({l, R}) f i (s, t) x 1 x 2 : s s t, t 1 t x 1 x 3 : s t L(ijk) are not independent x 1 x 4 : s 1 s, t t s Conservation Gives conditions both on P i and on f i

27 4-point function - Assumptions 2 0 J α1 (x 1 )J α2 (x 2 )J α3 (x 3 )J α4 (x 4 ) 0 = 1 x 4 13 x4 24 i P i({l, R}) f i (s, t) Weak positivity Only unitary representations appear in the OPE of two currents. j 1 + j if j 1j 2 = 0 j 1 + j if j 1j 2 0 constrains the singularities of the functions f i for small x kl. Weak positivity is a necessary, but by far not sufficient condition to ensure positivity all the coefficients of the Conformal Partial Wave Expansion are positive Rational functions f i. Very strong assumption. Excludes all theories with anomalous dimensions. Natural in GCI theories or theories with infinite conserved currents f i are polynomials in s, 1/s, t and 1/t.

28 4-point function - Solution Only 3 solutions Free complex massless scalar Free massless fermion New J α1 (x 1 )J α2 (x 2 )J α3 (x 3 )J α4 (x 4 ) = ɛ α1 µ1 ν1 ρ1 ɛ α2 µ2 ν2 ρ2 ɛ α3 µ3 ν3 ρ3 α4 µ4 ν4 ρ4 ɛ R µ1µ 2 (x 12 )R ν1ν 3 (x 13 )R ρ1ρ 4 (x 14 )R ρ2ρ 3 (x 23 )R ν2ν 4 (x 24 )R µ3µ 4 (x 34 ) Permutation symmetry and conservation are manifest Denominators like x 2 12x 2 13x 2 14, no free field realization Leading short distance singularity 1 x 2 12 Leading light-cone singularity. and 1 x 4 12 x µ 12 xν 12 x 4 12

29 4-point function - OPE 1 The leading terms : dimension 4 fields z 4 : J µ (x 1 ) J ν (x 2 ) : = z ρ z τ A ([µ,ρ],[ν,τ]) (x 2 ) where z = x 12. Θ ρτ is the stress-energy tensor Φ 4 is a scalar field of dimension four + (z µ z ρ δτ ν + z ν z ρ δ τ µ η µν z ρ z τ ) Θ ρτ (x 2 ) ) + (z µ z ν + ηµν 2 z2 Φ 4 (x 2 ) + O(z 3 ), A ([µ,ρ],[ν,τ]) is a tensor of scale dimension four Belongs to the (reducible) representation (2, 0) (0, 2): antisymmetric in µ, ρ and ν, τ, symmetric under the exchange of the two pairs, traceless in any pair of indices, satisfies ɛ µνρτ A ([µ,ρ],[ν,τ]) = 0 (symmetric in ρ and τ).

30 The subleading O(z 3 ) terms 4-point function - OPE 2 z 4 : J µ (x 1 ) J ν (x 2 ) : = z ρ z τ A ([µ,ρ],[ν,τ]) (x 2 ) + (z µ z ρ δτ ν + z ν z ρ δ τ µ η µν z ρ z τ ) Θ ρτ (x 2 ) ) + (z µ z ν + ηµν 2 z2 Φ 4 (x 2 ) + O(z 3 ), Θ ρτ even rank twist two symmetric tensors Φ 4 higher (larger than two) twist operators A ([µ,ρ],[ν,τ]) family of tensors of scale dimension 4 + n belonging to (2 + n/2, n/2) (n/2, 2 + n/2) Only unitary representations in the OPE Weak positivity OK.

31 4-point function - Positivity 1 What about positivity? Two general strategies 1 Expand the function. One needs the expressions for the CPW Known only for Scalar external operators (very explicit) [Dolan, Osborn, 2001, 2004] Symmetric tensor exchange, external operators with spin (not very explicit) [Costa, Penedones, Poland, Rychkov, 2011] General case (very implicit) [Simmons-Duffin, 2012] 2 Reduction of n-point functions to n 1 point ones. Applies only for fields of integer scale dimension (GCI) (explicit only for symmetric tensors). [Neumann, Rehren, Wallenhorst, 2011] Use the special properties of the function

32 4-point function - Positivity 2 All the twist two symmetric tensor contributions respect positivity Two different 3-point functions of two currents and a rank r tensor. Scalar : leading light-cone singularity 1/x 6 12 G 0 r1=r2=1 1 ( x 2 12 x2 13 x2 23 (1 L c) 3 2 L a L b R ab 4 L a L b L c + 2 R ab L c R bc L a R ac L b + ) 2 L a L b L 2 c R ab L 2 c + L a L c R bc + L b L c R ac + R ac R bc Fermion : leading light-cone singularity 1/x 4 12 G(1, 1, r) ψ X abc L r 1 c x 2 12 x2 13 x2 23 All the contributions are of the fermion type

33 4-point function - Positivity 3 Scalar conformal bi-field of weights (2, 2) in (x 1, x 2 ) [ ] V (x 1, x 2 ) = (x 12 ) µ (x 12 ) ν : J µ (x 1 ) J ν (x 2 ) : x : J µ (x 1 ) J µ (x 2 ) : V : projection on the even rank symmetric tensors F ρτ = V (x 1, x 2 ) J ρ (x 3 ) J τ (x 4 ) 1/x 2 12 F ρτ : contributions from the twist two even rank symmetric tensors F ρτ new = c F ρτ ψ, c > 0 The contributions of all the twist two even rank symmetric traceless tensors are the same positive.

34 4-point function - Unique stress tensor Several different 3-point functions, different OPE structures, different light cone singularities : 1/x 6 12 and 1/x 4 12 The tress-energy tensor contribution JJJJ ϕ + JJJJ new Θ = OPE of the form J(x 1 ) J(x 2 ) = ( cϕ x 6 12 ( c 2 ϕ x 6 12 x c2 n x 4 12 x4 34 ) ΘΘ. + c ) ( n cϕ x 4 T x 6 c ) n 12 x 4 T , 12 T 1 and T 2 are two distinct (orthogonal) symmetric traceless tensors However T 1 T 2 = 0, T 1 T 1 = T 2 T 2 = 1 2 ΘΘ. JJJJ ψ + JJJJ new Θ = c2 ψ + c2 n x 4 12 x4 34 ΘΘ, consistent with an OPE involving only one conserved tensor of each rank.

35 4-point function - Speculations In the Maxwell theory naturally appears the operator F µν F αβ + 1 ( η µα F ν ρf ρβ η να F µ ρf ρβ η µβ F ν ρf ρα + η νβ F µ ρf ρα) 2 1 ( η µα η νβ η µβ η να) F τρ F ρτ ɛµναβ ɛ κλρτ F κλ F ρτ, all the properties of A ([µ,ν],[α,β]) Non-abelian construction J µ = F a [µν] V ν a N ɛ µ ν ρ τ f abc V ν a V ρ b V τ c, where µ F a [µν] = 0 = [ µ, V ν a ].

36 Conclusions Complete classification of the 3-point functions of conserved symmetric traceless higher spin currents in 4D CFT Compact generating functions (both for the parity even and parity odd case) Surprising factorization properties Correspondence with free (s, 0) and (0, s) fields The rational 4-point functions of conserved abelian currents Unique non-free function OPE analysis Only unitary representations Candidate for interaction term of fermion and gauge fields

37 Open problems 3-point functions Understand better the factorization Implications on the Conformal Partial Waves More general conserved currents 4-point function of U(1) currents Positivity? Higher point functions? Non-abelian case? Relax the rationality assumption 4-point functions of the stress-tensor? The structure of the conserved currents and charges? Are there non-free theories with higher spin conserved currents?