Evolution equations for B -meson distribution amplitudes Yao Ji - PowerPoint PPT Presentation

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Evolution equations for B -meson distribution amplitudes Yao Ji University of Siegen Workshop LCDA 2020, MITP, Mainz Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 1 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Exclusive B –Decays • Heavy quark expansion methods ( m b ≫ Λ QCD ) • Soft-collinear factorization (final state particle energies ≫ Λ QCD ) Factorization Theorem : [M. Beneke, G. Buchalla, M. Neubert and Sachrajda (1999)] � 1 � M 1 M 2 | O i | B � = F B → M 1 (0) du T (1) ( u )Φ M 2 ( u ) 0 � ∞ � 1 du dv T (2) ( ω, u, v )Φ + ( ω )Φ M 1 ( u )Φ M 2 ( v ) + . . . + dω 0 0 u, v — momentum fractions � M 2 M ω — light quark energy 2 � M 2 I M T 2 ij in B-meson B I I � Φ M,B — distribution amplitudes + B T i B F j � M 1 M 1 M 1 B → γℓν ℓ provides the cleanest probe for unraveling the B -meson DAs Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 2 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Leading-twist distribution amplitude Definition [A. Grozin, M. Neubert (1997)] � � R | ¯ � 0 | q ( zn ) � n [ zn, 0] γ 5 h v (0) ¯ B ( v ) � = iF B ( µ ) Φ + ( z, µ ) • v µ is the heavy quark velocity • n µ is the light-like vector, n 2 = 0 , such that n · v = 1 • The twist-2 LCDA Φ + ( z − i 0 , µ ) is an analytic function of z in the lower half-plane Fourier transform � ∞ 1 dz e iωz Φ + ( z − i 0 , µ ) , φ + ( ω, µ ) = 2 π −∞ � ∞ dω e − iωz φ + ( ω, µ ) . Φ + ( z, µ ) = 0 • ω > 0 is the ( 2 × ) light quark energy in the b –quark rest frame Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 3 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Three-particle distribution amplitudes • Eight independent Lorentz structures [B. Geyer and O. Witzel (2005)] � � R | ¯ � 0 | q ( nz 1 ) gG µν ( nz 2 )Γ h v (0) ¯ B ( v ) � = � � � � = 1 2 F B ( µ ) Tr γ 5 Γ P + ( v µ γ ν − v ν γ µ ) Ψ A − Ψ V − iσ µν Ψ V − ( n µ v ν − n ν v µ ) X A � � − iǫ µναβ n α v β γ 5 � X A + iǫ µναβ n α γ β γ 5 � + ( n µ γ ν − n ν γ µ ) W + Y A Y A �� − ( n µ v ν − n ν v µ ) / n W + ( n µ γ ν − n ν γ µ ) / n Z ( z 1 , z 2 ; µ ) blue: [H. Kawamura, J. Kodaira, C. F. Qiao and K. Tanaka (2001)] red: [V. Braun, YJ and A. Manashov (2017)] � ∞ � ∞ dω 2 e − iω 1 z 1 − iω 2 z 2 ψ A ( ω 1 , ω 2 ) , Ψ A ( z 1 , z 2 ) = dω 1 0 0 � + ∞ 1 dz 1 dz 2 e iω 1 z 1 + iω 2 z 2 Ψ A ( z 1 − i 0 , z 2 − i 0) , ψ A ( ω 1 , ω 2 ) = etc. (2 π ) 2 −∞ Convenient for simple Lorentz strucutures. No definite collinear twist, not suitable for power counting in factorization. Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 4 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Collinear twist decomposition (1) • Twist-three q ( z 1 ) gG µν ( z 2 ) n ν / nγ µ ⊥ γ 5 h v (0) | ¯ 2 F B ( µ )Φ 3 ( z 1 , z 2 ; µ ) = � 0 | ¯ B ( v ) � Φ 3 = Ψ A − Ψ V , • Twist-four q ( z 1 ) gG µν ( z 2 ) n ν / nγ µ ⊥ γ 5 h v (0) | ¯ 2 F B ( µ )Φ 4 ( z 1 , z 2 ; µ ) = � 0 | ¯ ¯ B ( v ) � n µ n ν / nγ 5 h v (0) | ¯ 2 F B ( µ )Ψ 4 ( z 1 , z 2 ; µ ) = � 0 | ¯ q ( z 1 ) gG µν ( z 2 )¯ B ( v ) � n µ n ν / nh v (0) | ¯ 2 F B ( µ ) � q ( z 1 ) ig � Ψ 4 ( z 1 , z 2 ; µ ) = � 0 | ¯ G µν ( z 2 )¯ B ( v ) � Φ 4 = Ψ A + Ψ V , Ψ 4 = Ψ A + X A , Ψ 4 = Ψ V − � � X A , Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 5 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Collinear twist decomposition (2) • Twist-five n ν / 2 F B ( µ ) � nγ µ ⊥ γ 5 h v (0) | ¯ Φ 5 ( z 1 , z 2 ; µ ) = � 0 | ¯ q ( z 1 ) gG µν ( z 2 )¯ B ( v ) � n µ n ν / nγ 5 h v (0) | ¯ 2 F B ( µ )Ψ 5 ( z 1 , z 2 ; µ ) = � 0 | ¯ q ( z 1 ) gG µν ( z 2 )¯ ¯ B ( v ) � n µ n ν / nh v (0) | ¯ 2 F B ( µ ) � q ( z 1 ) ig � Ψ 5 ( z 1 , z 2 ; µ ) = � 0 | ¯ G µν ( z 2 )¯ ¯ B ( v ) � Φ 5 = Ψ A + Ψ V + 2 Y A − 2 � � Y A + 2 W , Ψ 5 = − Ψ A + X A − 2 Y A , Ψ 5 = − Ψ V − � � X A + 2 � Y A , • Twist-six n ν / 2 F B ( µ ) � nγ µ ⊥ γ 5 h v (0) | ¯ Φ 6 ( z 1 , z 2 ; µ ) = � 0 | ¯ q ( nz 1 ) gG µν ( nz 2 )¯ ¯ B ( v ) � Φ 6 = Ψ A − Ψ V + 2 Y A + 2 W + 2 � Y A − 4 Z Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 6 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Conformal spin and helicity assignment Ψ 4 + � Ψ 4 − � Ψ 5 + � Ψ 5 − � Φ 3 Φ 4 Ψ 4 Ψ 4 Φ 5 Ψ 5 Ψ 5 Φ 6 twist 3 4 4 4 5 5 5 6 j q 1 1 / 2 1 1 1 1 / 2 1 / 2 1 / 2 j g 3 / 2 3 / 2 1 1 1 / 2 1 1 1 / 2 chirality ↑ ↓ ( ↓ ↑ ) ↑ ↑ ( ↓ ↓ ) ↑ ↑ ( ↓ ↓ ) ↑ ↓ ( ↓ ↑ ) ↑ ↑ ( ↓ ↓ ) ↑ ↑ ( ↓ ↓ ) ↑ ↓ ( ↓ ↑ ) ↑ ↓ ( ↓ ↑ ) Table : The twist, conformal spins j q , j g of the constituent fields and chirality [same or opposite] of the three-particle B-meson DAs. • Asymptotic behavior at small momenta [V. Braun, I. Filyanov (1989)] f ( ω 1 , ω 2 ) ∼ ω 2 j 1 − 1 ω 2 j 2 − 1 f ∈ { φ 3 , φ 4 , ψ 4 , ˜ . ψ 4 . . . } 1 2 φ 3 ( ω 1 , ω 2 ) ∼ ω 1 ω 2 φ 4 ( ω 1 , ω 2 ) ∼ ω 2 ψ 4 ( ω 1 , ω 2 ) ∼ � 2 , 2 , ψ 4 ( ω 1 , ω 2 ) ∼ ω 1 ω 2 — agrees with [A. Khodjamirian, T. Mannel, N. Often (2006)] Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 7 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Relations between distribution amplitudes • Neglecting four-particle DAs qGGh v , q ¯ qqh v � � � � Ψ 4 ( z ) + � 2 ∂ 1 z 1 Φ 4 ( z ) = z 2 ∂ z 2 + 2 Ψ 4 ( z ) z = { z 1 , z 2 } as a consequence of Lorentz symmetry. [V. Braun, YJ and A. Manashov, (2017)] • Higher moments of φ + [A. G. Grozin and M. Neubert, (1997)] � ∞ � ∞ dω ω φ + ( ω ) = 4 Λ 2 + 2 E + 1 ¯ dω ω 2 φ + ( ω ) = 2¯ 3 λ 2 3 λ 2 Λ , H , 3 0 0 • Normalization conditions for higher-twist DAs: Φ 3 ( z = 0) = 1 Φ 4 ( z = 0) = 1 3( λ 2 E − λ 2 3( λ 2 E + λ 2 H ) , H ) , Ψ 4 ( z = 0) = 1 Ψ 4 ( z = 0) = 1 3 λ 2 � 3 λ 2 E , H , with λ 2 E /λ 2 H ∼ 0 . 5 by QCDSR. [A. Grozin and M. Neubert, (1997); T. Nishikawa and K. Tanaka, (2014)] Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 8 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary Two-particle distribution amplitudes of higher-twists • x 2 expansion of two-particle matrix element � ∞ � � dω e − iω ( vx ) � � B ( v ) � = − i q ( x )Γ[ x, 0] h v (0)] R | ¯ φ + ( ω, µ ) + x 2 g + ( ω, µ ) � 0 | [¯ 2 F B Tr γ 5 Γ P + 0 � ∞ � � 1 dω e − iω ( vx ) � � + i [ φ + − φ − ]( ω, µ ) + x 2 [ g + − g − ]( ω, µ ) 4 F B Tr γ 5 Γ P + / x + . . . vx 0 • Two-particle higher-twist DAs are related to three-particle through EOM: � 1 � � z d Φ − ( z ) = Φ + ( z ) + 2 z 2 dz + 1 udu Φ 3 ( z, uz ) 0 � 1 � � dz − 1 Φ + ( z ) − 1 z d 2 z 2 G + ( z ) = − 2 Φ − ( z ) − z 2 2 + iz ¯ Λ udu Ψ 4 ( z, uz ) ¯ 0 � 1 � � z d dz − 1 Φ − ( z ) − 1 2 z 2 G − ( z ) = − 2 + iz ¯ 2 Φ + ( z ) − z 2 Λ ¯ udu Ψ 5 ( z, uz ) 0 � � � 1 � � z d dz + 1 + 2 iz ¯ Φ + ( z ) + 2 z 2 Φ − ( z ) = Λ du u Φ 4 ( z, uz ) + Ψ 4 ( z, uz ) 0 [H. Kawamura, J. Kodaira, C. F. Qiao and K. Tanaka, (2001); V. M. Braun, YJ and A. N. Manashov, (2017)] Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 9 / 31

Background Classification Relations EOMs Renormalization Models Two-loop evolution Summary One-loop evolution of leading twist DA � � µ ∂ ∂µ + β ( a ) ∂ • RGE ∂a + H ( a ) Φ + ( z, µ ) = 0 , [B. Lange, M. Neubert (2003)] with H being the evolution kernel, usually presented as an integral operator. • One-loop evolution kernel � � � 1 du ¯ u H (1) Φ + ( z, µ ) = 4 C F [ln( i � µz ) + 1 / 2]Φ + ( z, µ ) + u [Φ + ( z, µ ) − Φ + (¯ uz, µ )] 0 µ = e γ E µ MS and ¯ where � u = 1 − u . [A. Grozin and M. Neubert, (1997); V. Braun, D. Ivanov and G. Korchemsky, (2004)] • Solution to one-loop RGE [G. Bell, T. Feldmann, Y.-M. Wang and M. W. Y. Yip, (2013); V. Braun and A. Manashov (2014)] � ∞ Φ + ( z, µ ) = − 1 ds s e is/z η + ( s, µ ) , z 2 0 2 CF αs ( µ ) ln η + ( s, µ ) = R ( s, µ, µ 0 ) η + ( s, µ 0 ) , R ( s, µ, µ 0 ) ∝ s β 0 αs ( µ 0) Come back later! Yao Ji (University of Siegen) Evolution of B -meson DAs Mainz, 14.01.2020 10 / 31