Matthias Strohmaier. given at QCD Evolution , JLAB. University of Regensburg. Introduction Method Analysis Conclusion

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1 Three-loop evolution equation for non-singlet leading-twist operator based on: V. M. Braun, A. N. Manashov, S. Moch, M. S., not yet published, arxiv: Matthias Strohmaier University of Regensburg given at QCD Evolution , JLAB

2 Motivation & Remarkable archievements in experimental accuracy intended to increase even more in near future (e.g. JLAB 12 GeV update or EIC) needs to be matched by theoretical accuracy NNLO is becoming the state-of-the-art in most fields NNLO evolution for off-forwards distributions needs to be investigated

3 Operators under consideration Non-local light-ray twist-two operator n 2 = 0 D + := nµd µ [O (n) (x;z 1,z 2)] = Z q(x+nz 1)/nq(x+nz 2) Generating object for local operators [O (n) (x;z 1,z 2)] = l,m=0 We tacitely assume different flavors. z l 1z m 2 l!m! [ZO]lm(x), Olm(x) = q(x)( D +) l /n( D +) m q(x). Renormalization factor in MS - scheme Z = 1+ k=0 1 ǫ k l=k a l Z (l) k, a = αs 4π. Relation to evolution kernel H = d dlnµ ln(z), m γl lm = d m dlnµ ln(zl lm )

4 Evolution equation RGE [µ µ +β(a) a +H]O(x;z 1,z 2) =0, Balitsky, Braun 89 [µ µ +β(a) a + ˆγ]O lm (x) =0. Radyushkin, etc. 84 QCD β-function β(a) = µ a µ = 2a(ǫ+aβ0 +a2 β ). }{{} β QCD (a) General form [HO](z 1,z 2) = 1 dα dβh(α,β)o(z α 12,z β 21), z α 12 = (1 α)z 1 +αz 2.

5 Conformal symmetry at the classical level Three symmetry generators on the light-cone Translations: S (0) = z1 z2 Dilatations: S (0) 0 = z 1 z1 +z 1 z1 +2(j 1 +j 2) Spec. conf. transf.: S (0) + = z 2 1 z1 +z 2 2 z2 +2j 1z 1 +2j 2z 2 All three commute with the LO evolution kernel [S (0) α,h (1) ] = 0

6 Evolution equations at the LO Shape of the evolution kernel fixed by [S (0) +,H(1) ] = 0 [H (1) O](z 1,z 2) = 1 0 dα 1 0 dβh (1) inv (τ)o(zα 12,z β 21), τ = αβ ᾱ β. Anomalous dimension γ N of local operators with spin N are eigenvalues 1 H(z 1 z 2) N = (z 1 z 2) N dα dβh(α,β)(1 α β) N = (z 1 z 2) N γ N Knowledge of anomalous dimension γ N of local operators with spin N allows to reconstruct kernel h(α,β), if h(α,β) = h inv(τ) h inv(τ) = +i i P N(x) : Legendre polynomial ( ) 1+τ dn(2n +1)γ NP N 1 τ (1)

7 Conformal symmetry beyond LO Conformal symmetry breaks down! canonical conformal symmetry broken Find critical point β(a ) = 0 a = a (ǫ). exact symmetry can be reconstructed order by order (in modified theory in d = 4 2ǫ) done by conformal Ward identities δ C [O(x;z 1,z 2)][O(y;w 1,w 2)] = 0 Corrections added to canonical generators S =S (0) S 0 =S (0) 0 + S 0(a ) S + =S (0) + + S +(a ) O(a ): V.M. Braun, A.N. Manashov, Eur.Phys.J. C73 (2013) 2544 O(a 2 ): V.M. Braun, A.N. Manashov, S. Moch, M. S., JHEP 03 (2016) 142

8 Constraints on evolution equations from conformal symmetry The three generators must commute with evolution kernel [S α,h] = 0 At fixed perturbative order l 1 +,H(l) ] = [S (0) j=1 [ S (j) +,H(l j) ] (2) lhs: differential equation for evolution kernel at O(α l s) rhs: only operators to at most O(αs l 1 ) enter.

9 Solution for evolution kernel Eq. (2) fixes H (l) only up to solutions of homogeneous equation [S (0) +,H(l) inv ] = 0 Split H (l) = H (l) inv + H(l) Determine H (l) as solution of l 1 +, H (l) ] = [S (0) j=1 [ S (j) +,H (l j) ] Fix H (l) inv, i.e. canonically invariant part, with the knowledge of l-loop anomalous dimensions γ (l) N by h inv(τ) = +i i dn(2n +1) [ γ N γ N ] PN ( 1+τ 1 τ )

10 Non-invariant part Result for evolution kernel in MS-scheme H (3) = H (3) inv +T (1)( β ) 1 2 H(2) inv + 2 T(1) T (β (1) ) H(1) (β ) 2 H(1) +T (β (2) ) H(1) +T (1) 2 +[H (2) inv,x(1) ]+ 1 [ T (1) H (1),X (1)] + 1 H 2 2[ (1),X (1) ] 2 H (1) ( [T +[H (1),X (2) ]+β (1) 0,X (1)] + [ H (1),X (1) [[ H (1),X (1)],X (1)] 1 H 2 2[ (1),X (2) ] 2 All of the new operators are defined by eq s a la [S (0) +,T] = F( S +,H), [S (0) +,X] = F( S +,H). ] )

11 Non-invariant part: Side remark Finite scheme-transformation U(a) = e X(a) [ UHU 1 ](3) = H (3) inv +T (1)( β ) 1 2 H(2) inv + 2 T(1) T (β (1) ) H(1) (β ) 2 H(1) +T (β (2) ) H(1) +T (1) 2 Satisfies new evolution equation [µ µ +β(a) a ( 1+lnU ) +UHU 1 ][UO(z 1,z 2)] = 0.

12 Invariant part I Large-N asymptotics γ N = f(j N +β QCD (a)+ 1 γn) jn = N +2, 2 f(j N) = f(1 j N) f(j N) = f(j 2 ) J 2 = j N(j N 1), Large-spin expansion: f(j N) = f (0) lnj 2 +f (const) + n f (n) (lnj 2 ) J 2n In all known cases the function f(n) turns out to be simpler than γ N. Our solution for the non-invariant part turns out to be a lucky one: γ inv(n) = f(j N).

13 Invariant part II Reminder: We need to solve Eq. (1): h inv(τ) = +i i ( ) 1+τ dn(2n +1)f(j N)P N 1 τ

14 Invariant part II Reminder: We need to solve Eq. (1): h inv(τ) = +i i ( ) 1+τ dn(2n +1)f(j N)P N 1 τ Did not find a smart way to solve this problem analytically!

15 Invariant part III Schedule fix leading asymptotic terms to some order parametrize the remainder by some simple function with only few fit parameters H (3) invo(z1,z2) = Γ(3) cusp +χ (3) 1 0 ( ) dαᾱ 2O(z 1,z 2) O(z12,z α 2) O(z 1,z21) α α ᾱ ( dα dβ χ (3) inv (τ)+χp(3) inv )O(z (τ)p12 12,z α 21). β 0 O(z1,z2)+ 1 Leading two terms (Γ cusp,χ 0) can simple be taken from literature: S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B 688 (2004)

16 Invariant part IV {( χ inv(τ) = C2 F 176ζ π2 N c ϕ3(τ) 64ϕ4(τ) ( 16π ) ( ) 3632 ϕ 1(τ)+ 16π2 ϕ 2(τ) 9 3 ) ( ζ } ) ln(τ/ τ)+δχ inv(τ) 3200π π five more color structures with ϕ i(τ) harmonic polylogarithm with at most weight i. Convers the asymptotic expansion dαdβχ inv(τ)(1 α β) N 1/J 2,...,1/J 10,lnJ 2 /J 2, whatever remains

17 Invariant part V Restrictions on δχ(τ): As simple as possible (concerning number of parameters) Reciprocity respecting: moments are functions of J 2 = j N(j N 1). We make the Ansatz: δχ (3) inv (τ) = H 0 (1+4aτ/ τ) 5/2 [ 1+a τ τ (4 6b) ] H 0, with e.g. for structure C2 F N c : H 0 = 368ζ π2 9 a = b = π4 9

18 Accuracy of approximation (for n f = 4) Compare χ inv(τ) in our parametrization with what one gets by numerical solution of χ exact inv (τ) = ( ) dn(2n C +1) γinv(n)pn 1+τ with 1 τ γ inv(n) = f(j N) 2Γ cusp(s 1(N) 1) χ 0. 1 χinv χ exact inv 1 χp inv χ P,exact inv τ τ 1 χeven inv χ even,exact inv 1 χodd inv χ odd,exact inv τ τ

19 Numerical impact of NNLO How big is the NNLO correction? α s/π = 0.1, n f = 4. χinv(τ) χ P inv(τ) τ τ Seems to be not too small, however: the leading contributions are Γ cusp = aγ (1) ( a+80.53a ) = aγ (1) ( ), χ 0 = aχ (1) 0 ( a 141.3a2 +...) = aχ (1) 0 ( ).

20 OPE of light-ray operator In general O(z 1,z 2) = n,mϕ nm(z 1,z 2)O nm, O nm(x) = P nm( z1, z2 ) q(z 1)q(z 2) Particular choice: ϕ nm(z 1,z 2) = ω nm(s (0) + )m n (z 1 z 2) n, ( ) O nm = ( z1 + z2 ) m Cn 3/2 z1 z2 O(z 1,z 2) z1 + z2 z1 =z 2 =0 There exists a scalar product: P n m ( z 1, z2 )ϕ nm(z 1,z 2) = δ n nδ m m z1 =z 2 =0 z1 =z 2 =0., m n.

21 Relation between mixing matrix and evolution kernel Compare evolution equation for non-local and local operators: ( µ ) [O(z 1,z 2)] = H[O(z 1,z 2)], µ +β(a) a ( µ µ +β(a) a ) [O nk ] = n γ nn [O n k]. n =0 Local matrix is given by matrix elements of scalar product: nm H n m P nm( 1, 2)Hϕ n m(z 1,z 2) z=0 = γ nn

22 Three-loop results for mixing matrix We have found for the first few elements (taking n f = 4) ( ζ 0 3 ) γ (3) ζ = Comparing different orders in perturbative expansion γ = a γ (1)[ Id+a γ ] with a γ = 1+16a a a a a a a

23 and Outlook What can be done next? Exact solution for H most likely irrelevant Evolution kernel in momentum space Sample application for physical process, e.g. pion DA in γ γ π What other sets of local operators might be of interest for physical applications? Big goal: extend analysis for singlet operators!!!

24 Thank you for your attention Any questions?