Evolution equations beyond one loop from conformal symmetry. V. M. Braun

Transcription

1 Evolution equations beyond one loop from conformal symmetry V. M. Braun University of Regensburg based on V. Braun, A. Manashov, Eur. Phys. J. C [arxiv: [hep-th V. Braun, A. Manashov, arxiv: [hep-ph Santa Fe, 05/16/2014 V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

2 Conformal symmetry of QCD is broken by quantum corrections is it lost completely or still implies smth useful? Usual vague conjecture: Q = Q conformal + βg g Q Generalized Crewther relation Broadhurst, Kataev,... NLO GPD evolution equations... Müller, Belitsky& Müller off-diagonal elements of the NLO mixing matrix are determined by one-loop special conformal anomaly V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

3 Our motivation: Make precise the separation of conformal part and corrections Emphasize conformal symmetry friendly representation of the results Explore the road to NNLO three-loop evolution equations for GPDs Difference to D.Müller: Instead of considering consequences of broken conformal symmetry in QCD we make use of exact conformal symmetry of a modified theory: QCD in 4 2ǫ dimensions at critical coupling β QCD a s = 2a s [ ǫ β 0a s +... a s = 4πǫ/β β QCD a s = 0 Premium: Exact symmetry algebraic group-theory methods Answer obtained directly in MS scheme Light-ray operator basis, do not need to restore evolution eqs. from local operators. V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

4 Collinear Subgroup P = P z Special conformal transformation x x = x 1 + 2ax translations x x = x + c dilatations x x = λx form the so-called collinear subgroup SL2, R p + = p = px p + x 1 2 p 0 + p z 1 2 p 0 p z 0 α α = a α + b, ad bc = 1 c α + d aα + b Φα Φ α = c α + d 2j Φ cα + d where Φx Φx = Φαn is the quantum field with scaling dimension l and spin projection s living on the light-ray Conformal spin: j = l + s/2 V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

5 Light-ray operators There are two possibilities to specify leading twist operators 1 by an operator function of field coordinates on the light cone n 2 = 0 z1 m z2 k [Oz 1, z 2 [ qz 1n/nqz 2n m!k! [Dm n q0/nd nq0 k m,k 2 by a polynomial that details the number of derivatives in local operators [D m n q0/nd k nq0 = P 1, 2[Oz 1, z 2 z1 =z 2 =0, Pu 1, u 2 = u m 1 u k 2 Generators of conformal transformations act differently in these representations S 0 + = z2 1 z1 + z 2 2 z2 + 2z 1 + z 2 S 0 0 = z 1 z1 + z 2 z2 + 2 S 0 = z 1 z2 Conformal light-ray operators S 0 f = 0 f = z1 z2n S 0 + = u1 u2 S 0 0 = u 1 u1 + u 2 u2 + 2 S 0 = u1 2 u 1 + u 2 2 u u1 + u2 Conformal local operators S 0 f = 0 f = C 3/2 n u 1 u 2 Light-ray operator representation is natural for conformal symmetry V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

6 Light-ray operators satisfy the RG equation Balitsky, Braun 89 M M + βg g + À [Oz 1, z 2 = 0 where À is an integral operator acting on the light-cone coordinates of the fields: 1 1 À [Oz 1, z 2 = dα dβ hα, β [Oz12, α z β z α 12 z 1ᾱ + z 2α ᾱ = 1 α One can show that the powers [Oz 1, z 2 z 1 z 2 N are eigenfunctions of À, and the corresponding eigenvalues are the anomalous dimensions of local operators of spin N with N 1 derivatives γ N = dαdβ hα, β1 α β N 1 V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

7 One-loop evolution equations Expect [À 1, S 0 α = 0 = h 1 α, β = hτ, τ = αβ ᾱ β i.e. function of two variables reduces to a function of one variable can be restored from anomalous dimensions Confirmed by a direct calculation h 1 α, β = 4C F [ δ +τ + θ1 τ 1 2 δαδβ where 1 1 [ dαdβ δ + τf z12 α, zβ 21 dα dβ δτ f z12 α, zβ 21 f z 1, z = dα ᾱ [2f z 1, z 2 f z α12 α, z 2 f z 1, z α21 0 Combines DGLAP, ERBL and GPD evolution equations in the most compact form V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

8 Beyond one loop in d = 4 2ǫ at critical coupling Exact conformal symmetry, but the generators are modified by quantum corrections S =S 0, as = α s 4π S 0 =S 0 0 ǫ À a s, À as = as À S + =S z1 + z2 ǫ a s À 1 + as z 1 z Oǫ 2, where arxiv: ᾱ [ +[Oz 1, z 2 = 2C F dα α + ln α [Oz12, α z 2 [Oz 1, z21 α 0 Modified generators satisfy usual SL2 commutation relations [S 0, S ± = ±S ±, [S +, S = 2S 0 V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

9 Beyond one loop in d = 4 2ǫ at critical coupling II Expanding the commutation relations in powers of a s [S 0 +, À 1 = 0, [S 0 +, À 2 = [À 1, S 1 +, [S 0 +, À 3 = [À 1, S [À 2, S 1 +, etc. A nested set of inhomogenious first order differential equations for À 1 Their solution determines À k up to an SL2-invariant term The r.h.s. involves À k and S m + at one oder less compared to the l.h.s. D.Müller Back to the d = 4 world: In MS scheme À does not depend on ǫ À a s = a s À 1 + a s 2 À À a s = a s À 1 + a 2 s À µ µ + À as [Oz 1, z 2 = 0 µ µ + βg g + À a s [Oz 1, z 2 = 0 Reexpanding a s = a s ǫ ǫ = ǫa s the kernel in d = 4 for arbitrary coupling can be restored from the kernel in d = 4 2ǫ at the critical coupling V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

10 Procedure tested Scalar ON φ 4 theory in d = 4 in three loops arxiv: Scalar SUN φ 3 theory in d = 6 in two loops arxiv: QCD flavor-nonsinglet in two loops arxiv: For a NLO calculation one needs Calculate S + = S a s S 1 + one loop from the conformal Ward identity Find a particular solution to the differential equation [S 0 +, À 2 = [À 1, S 1 + non-invariant part of the evolution equation Restore invariant part solution of [S 0 +, À 2 = 0 from known anomalous dimensions In QCD h 2 α, β = 8CFh α, β + 4C FC A h 2 2 τ + 4b0C Fh 2 3 α, β V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

11 Special conformal Ward identity Conformal symmetry at the critical coupling implies S z x2 n x [Onz 1, z 2, [O nx, w 1, w 2 = 0 nx = nx = 0 To find explicit expression for S +, consider Ward identity δ+ [O n z [O n x, w + [O n z δ + [O n x, w δ + S R [O n z [O n x, w = 0 Then δ + [O n x; w 1, w 2 = x 2 n x[o n x; w 1, w 2 δ + [O n 0; z 1, z 2 = 2n n S 0 + ǫz 1 + z 2 as 2 [À 1, z 1 + z 2 [O n 0; z 1, z 2 and for the last term δ + S QCD = 4ǫ d d xx nl QCD + 2d 2 n µ d d x δ BRST c a A a µ. Ò Ò V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

12 Two-loop evolution kernels [ h 2 1 α, β = δ +τ φα + φβ + ϕα, β + θ τ 2 Li 2 τ + ln 2 τ + ln τ 1 + τ τ [ln 2 τ/τ 2τ ln τ/τ + 6ζ π θ τ h 2 2 α, β = 1 π 2 4 δ + τ 2θ τ[ Li 2 τ Li ln2 τ 1 τ ln τ θ τ h 2 3 α, β = δ +τ [ln 2 τ/τ 2τ ln τ/τ + [ ln ᾱ + ln β [ θ τ 6ζ3 2 3 π ln1 α β δαδβ, δαδβ, δαδβ. ln τ where [ 3 φα = ln ᾱ 2 ln ᾱ ᾱ ᾱ ln α, [ 1 ϕα, β = θ1 τ 2 ln2 1 α β ln2 ᾱ ln2 β ln α ln ᾱ ln β ln β 1 2 ln α 1 2 ln β + ᾱ β ln ᾱ + ln β α β, V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13

13 Conclusions QCD evolution equations possess a "hidden" conformal symmetry n-loop evolution kernels for twist-two operators in MS scheme can be restored from the n 1-loop calculation of the special conformal anomaly and n-loop anomalous dimensions SL2 symmetry properties manifest in the light-ray operator representation V. M. Braun Regensburg Evolution equations from conformal symmetry Santa Fe, 05/16/ / 13