Renormalon approach to Higher Twist Distribution Amplitudes

Transcription

1 Renormalon approach to Higher Twist Distribution Amplitudes Einan Gardi (Cambridge) plan conformal expansion: why, why not cancellation of ambiguities in the OPE example: quadratic UV divergence of twist 4 operators renormalon model: results convergence of the conformal expansion endpoint behavior Based on V.M. Braun, E. Gardi & S. Gottwald, Nucl. Phys. B (24) [hep-ph/41158] Distribution Amplitudes 26 p. 1/1

2 Conformal expansion any conformal expansion based ansatz, advantages consistent with EOM at each truncation order J max preserved by leading order QCD evolution high conformal spins are suppressed by J const ln µ2 F disadvantages practical only if converges fast convergence through evolution J const ln µ2 F holds only at high µ F Distribution Amplitudes 26 p. 2/1

3 Cancellation of ambiguities in the OPE Consider a physical amplitude near the lightcone, depending on x 1 x 2 where 1/Λ QCD. Factorization: short distance C (t) long distance DA φ (t) G(u, 2 ) = C (2) φ (2) }{{} twist C (4) φ (4) }{{} twist 4 + O( 4 ) Cancellation of µ F dependence: logarithmic: at each twist = evolution equations power: between different twists = renormalons: δ UV (twist 4) = 2 µ 2 F D(4 2) φ (2) = δ IR (twist 2) Distribution Amplitudes 26 p. 3/1

4 Three particle twist 4 DA and EOM relations d( z)[ z,vz]γν γ 5 gg µρ (vz)[vz,z]u(z) π + (p) = = f π + { Dα i e ipz(α 1 α 2 +α 3 v) pν [ ( p ρ g µν z µp ν pz pz (p µz ρ p ρ z µ )Φ (α i ) )] } Φ (α i ), ) p µ ( g ρν z ρp ν pz EOM relations between two and three particle DA: φ (4) 1 + φ (4) 2 (u) = u dα 1 1 u dα 2 ūα 1 uα 2 2α 2 3 [ 2Φ Φ ] (αi ), where α 3 = 1 α 1 α 2. Braun & Filyanov ( 9) Distribution Amplitudes 26 p. 4/1

5 Quadratic UV divergence of three particle twist 4 operators Consider a q qg twist 4 operator on the lightcone (z 2 = ): z vz z z vz z = Andersen ( 99) BGG ( 4) q2 d( z)γggαβ (vz)u(z) q1 = e i(q 1 +q 2 )z ( g λα g µ β g λβg µ α) 4π2 C F β dw e 5 3 w ( Λ 2 ) w B λ µ(w,q,z,v) B λ µ d q2 γ µ γ ρ Γu q1 I λρ (q 2, (1 + v)z) d q2 Γγ ρ γ µ u q1 I λρ (q 1, (1 v)z) where I λρ (q,z) d 4 k (2π) 4 k λ (q + k) ρ (k 2 ) (1+w) (q + k) 2 e ikz Distribution Amplitudes 26 p. 5/1

6 Twist 4 2 operator mixing relations B λ µ(w,q,z,v) has a pole at w = 1 owing to UV divergence: 1 I λρ (q,z) i w=1 = daae iqz(1 a) 32π 2 (1 w) [iq λ z ρ (1 a) iq ρ z λ a + g λρ + 12 ] a(1 a)q2 z λ z ρ. O ν µρ = d( z)γ ν γ 5 gg µρ (vz)u(z) ; ( ) y ± = z 1 (1±v)(1 a) { } 1 δ UV z µ gν ρ Oµρ ν = kλ 2 da(1 a)[ d( y + )z/γ 5 u(z) d( z)z/γ ] 5 u(y ) { } 1 δ UV z µ z ν Oµρ ν = kλ 2 z ρ daa[ d( y + )z/γ 5 u(z) d( z)z/γ ] 5 u(y ) Distribution Amplitudes 26 p. 6/1

7 Renormalon model: twist 4 DA in terms of twist 2 DA Using the twist 4 2 operator mixing with the definitions of 3-particle twist 4 DA (l.h.s) and twist 2 DA (r.h.s): δ UV {Φ (α 1,α 2,α 3 )} = 1 [ φπ (α 1 ) 2 ikλ2 φ ] π(α 2 ), 1 α 1 1 α 2 { δ UV Φ (α 1,α 2,α 3 ) } [ = ikλ 2 α2 φ π (α 1 ) (1 α 1 ) α ] 1φ π (α 2 ) 2 (1 α 2 ) 2 Upon fixing the overall normalization (one parameter!) using dγ ν ig G µρ u π + (p) = δ2 3 f π[p ρ g µν p µ g ρν ]; δ 2 QCD SR.2 GeV 2, UV ren. ambiguities translate into a model: ikλ 2 δ 2 /3. Distribution Amplitudes 26 p. 7/1

8 Renormalon model vs. conformal expansion based model 3 particle twist 4 pion DA α 3(g) = 1 α 1(q) α 2( q) Renormalon model assuming asymptotic leading twist DA: Φ (α i ) = δ 2 [α 1 α 2 ], [ Φ (α i ) = 2δ 2 1 α 1 α 2 1 ] 1 α 1 1 α 2 First two orders in the conformal expansion (J = 3, 4): Φ BF (α i ) = 1δ 2 (α 1 α 2 )α 3 2 [1 + 6ǫ(1 2α 3 )], Φ BF (α i ) = 12ǫδ 2 α 1 α 2 α 3 (α 1 α 2 ) Braun&Filyanov ( 9) Qualitatively different for vanishing gluon momentum α 3. Distribution Amplitudes 26 p. 8/1

9 Renormalon model for two particle twist 4 DA Using the 3-particle twist 4 DA with the EOM relations: 1 { φ (4) 1 (u) = δ2 1 dv φ π (v) 6 v 2 φ (4) 2 (u) = δ2 6 1 [ ( u + (v u) ln 1 u v) ] θ(v > u) + 1 v ( [ū + (u v) ln 1 ū ) ] θ(v < u)}, 2 v { (u ) 2 (ū ) 2 } dv φ π (v) θ(v > u) + θ(v < u) v v In physical amplitudes δ UV (twist 4) + δ IR (twist 2) =. Indeed, the same expressions are obtained considering IR renormalons in twist 2 coefficient functions. Distribution Amplitudes 26 p. 9/1

10 Convergence of the conformal expansion for 2-particle twist-4 DA We can examine the convergence of the conformal expansion by expanding the renormalon model in this basis: φ (4) 2 (u) = 4δ 2 u 2 (1 u) 2 J=3,5,7,... 2J 1 J(J 1) 2 (J 2) P (2,2) J 3 (2u 1) the good φ (4) 2 (u) the bad φ (4) 1 (u) the ugly φ (4) 1 + φ (4) u u u Distribution Amplitudes 26 p. 1/1

11 Endpoint behavior of 2 particle twist 4 DA Renormalon model (assuming asymptotic leading twist DA): [ ] [ ] } φ (4) 1 (u) = δ {ū 2 ln(ū) Li 2 (ū) + u ln(u) Li 2 (u) uū + π2 6 [( ) ] π δ u + O(u 2 ). Its conformal expansion (at any truncation order): { ] } φ (4) 5 1 (u) = δ [u 2 2 ū 2 + δ 2 O(u 2 ) 2 J=3 The expansion does not converge uniformly at the endpoints! Distribution Amplitudes 26 p. 11/1

12 Conclusions Renormalons: model for higher twist DA in terms of the leading twist DA. consistent with all EOM relations! single parameter(!) for the entire set of twist 4 DA Already available for π and ρ (BGG) and for K (BBL). Convergence of the conformal expansion 3 particle DA: no convergence for fixed α 3. 2 particle DA: converges away from endpoints, but qualitatively different slower endpoint behavior High J contributions: renormalon model is an upper bound, since evolution J const ln µ2 F Distribution Amplitudes 26 p. 12/1