Electroproduction of Nucleon Resonances in QCD I.V. Anikin JINR, - PowerPoint PPT Presentation

Electroproduction of Nucleon Resonances in QCD I.V. Anikin JINR, Dubna / University of Regensburg based on Phys. Rev. D 88 , 114021 (2013), Phys. Rev. D 92 , 014018 (2015) in collaboration with V.M. Braun and N. Offen October 4, 2015 I.V. Anikin Electroproduction of Nucleon Resonances

Main results • We derive light-cone sum rules for the electromagnetic nucleon and N ∗ (1535) resonance form factors including the next-to-leading-order corrections for the contribution of twist-three and twist-four operators and a consistent treatment of the nucleon mass corrections. • The soft contributions have been calculated in terms of small transverse distance quantities using dispersion relations and quark-hadron duality. I.V. Anikin Electroproduction of Nucleon Resonances

• The form factors have been expressed in terms of nucleon wave functions at small transverse separations (DAs), without any additional parameters. • For N ∗ (1535), we find that the form factors are dominated by the twist-four distribution amplitudes that are related to the P -wave three-quark wave functions, i.e. to contributions of orbital angular momentum. • The distribution amplitudes have been extracted from the comparison with the experimental data on form factors and compared to the results of lattice QCD simulations. A self-consistent picture emerges, with the three valence quarks carrying 40% : 30% : 30% of the proton momentum. I.V. Anikin Electroproduction of Nucleon Resonances

Essential stages The work consists of three parts: • Calculations within LCSR; • Factorization of amplitude at LO up to twist-6 and at NLO up to twist-4. We calculate 22 coefficient functions at NLO and 20 of them are new ones. To avoid the mixture with the so-called evanescent operators, we use the renormalization procedure for operators with open Dirac indices; • Derivation of distribution amplitudes. In particular, light-cone expansion to the twist-4 accuracy of the three-quark matrix elements with generic quark positions I.V. Anikin Electroproduction of Nucleon Resonances

LCSRs for form factors: General structure The LCSR approach allows one to calculate the form factors in terms of the nucleon (proton) DAs. To this end we consider the correlation function � d 4 x e iqx � 0 | T [ η (0) j em T ν ( P , q ) = i ν ( x )] | P � where T denotes time-ordering and η (0) is the Ioffe interpolating current η ( x ) = ǫ ijk � � u i ( x ) C γ µ u j ( x ) γ 5 γ µ d k ( x ) , � 0 | η (0) | P � = λ 1 m N N ( P ) . I.V. Anikin Electroproduction of Nucleon Resonances

The matrix element of the electromagnetic current u ( x ) γ µ u ( x ) + e d ¯ j em µ ( x ) = e u ¯ d ( x ) γ µ d ( x ) taken between nucleon states is conventionally written in terms of the Dirac and Pauli form factors F 1 ( Q 2 ) and F 2 ( Q 2 ): � � γ µ F 1 ( Q 2 ) − i σ µν q ν µ (0) | P � = ¯ F 2 ( Q 2 ) � P ′ | j em N ( P ′ ) N ( P ) . 2 m N In terms of the electric G E ( Q 2 ) and magnetic G M ( Q 2 ) Sachs form factors, we have G M ( Q 2 ) F 1 ( Q 2 ) + F 2 ( Q 2 ) , = F 1 ( Q 2 ) − Q 2 G E ( Q 2 ) F 2 ( Q 2 ) . = 4 m 2 N I.V. Anikin Electroproduction of Nucleon Resonances

The matrix element of the electromagnetic current j em between ν spin-1/2 states of opposite parity can be parametrized in terms of two independent form factors, which can be chosen as � N ∗ ( P ′ ) | j em u N ∗ ( P ′ ) γ 5 Γ ν u N ( P ) , ν | N ( P ) � = ¯ Γ ν = G 1 ( q 2 ) qq ν − q 2 γ ν ) − i G 2 ( q 2 ) σ νρ q ρ , (ˆ m 2 m N N where q = P ′ − P is the momentum transfer and Q 2 = − q 2 . I.V. Anikin Electroproduction of Nucleon Resonances

The helicity amplitudes A 1 / 2 ( Q 2 ) and S 1 / 2 ( Q 2 ) for the electroproduction of N ∗ (1535) can be expressed in terms of the form factors [Aznauryan ’08] : � � Q 2 G 1 ( Q 2 ) + m N ( m N ∗ − m N ) G 2 ( Q 2 ) = e B , A 1 / 2 � � = eBC ( m N − m N ∗ ) G 1 ( Q 2 ) + m N G 2 ( Q 2 ) S 1 / 2 . √ 2 √ Here e = 4 πα is the elementary charge and B , C are kinematic factors defined as � Q 2 + ( m N ∗ + m N ) 2 B = N ) , 2 m 5 N ( m 2 N ∗ − m 2 � 1 + ( Q 2 − m 2 N ∗ + m 2 N ) 2 = . C 4 Q 2 m 2 N ∗ I.V. Anikin Electroproduction of Nucleon Resonances

Light-Cone Basis We define a light-like vector n µ by the condition q µ = P . q n 2 = 0 , q 2 = q 2 ⊥ = − Q 2 . P . nn µ + q ⊥ q · n = 0 , µ , and introduce the second light-like vector as m 2 p µ = P µ − 1 p 2 = 0 , N P · n , 2 n µ and µν = g µν − 1 g ⊥ pn ( p µ n ν + p ν n µ ) I.V. Anikin Electroproduction of Nucleon Resonances

Light-Cone Sum Rules We consider the “plus” spinor projection of the correlation function involving the “plus” component of the electromagnetic current, which can be parametrized in terms of two invariant functions � � m N A ( Q 2 , P ′ 2 ) + ˆ q ⊥ B ( Q 2 , P ′ 2 ) N + ( P ) , Λ + T + = p + where Q 2 = − q 2 and P ′ 2 = ( P − q ) 2 and N ± ( P ) = Λ ± N ( P ) , Λ + = ˆ p ˆ n Λ − = ˆ n ˆ p 2 pn , 2 pn I.V. Anikin Electroproduction of Nucleon Resonances

Making the Borel transformation 1 → e − s / M 2 , s − P ′ 2 − we derive the sum rules (for nucleon) � s 0 1 ds e ( m 2 N − s ) / M 2 Im A QCD ( Q 2 , s ) , 2 λ 1 F 1 ( Q 2 ) = π 0 � s 0 1 ds e ( m 2 N − s ) / M 2 Im B QCD ( Q 2 , s ) . λ 1 F 2 ( Q 2 ) = π 0 and (for N ∗ (1535)) � s 0 1 Q 2 G 1 ( Q 2 ) 2 λ N 1 ds e ( m 2 N − s ) / M 2 Im A QCD ( Q 2 , s ) , = m N m N ∗ π 0 � s 0 1 ds e ( m 2 N − s ) / M 2 Im B QCD ( Q 2 , s ) . − 2 λ N 1 G 2 ( Q 2 ) = π 0 I.V. Anikin Electroproduction of Nucleon Resonances

The correlation functions A ( Q 2 , P ′ 2 ) and B ( Q 2 , P ′ 2 ) can be written as a sum: A = e d A d + e u A u , B = e d B d + e u B u . Each of the functions has a perturbative expansion which we write as A = A LO + α s ( µ ) A NLO + . . . 3 π and similar for B ; µ is the renormalization scale. I.V. Anikin Electroproduction of Nucleon Resonances

For consistency with our NLO calculation we rewrite LO results in a different form, expanding all kinematic factors in powers of m 2 N / Q 2 : We keep all corrections O ( m 2 N / Q 2 ) but neglect terms O ( m 4 N / Q 4 ) etc. which is consistent with taking into account contributions of twist-three, -four, -five (and, partially, twist-six) in the OPE. I.V. Anikin Electroproduction of Nucleon Resonances

The following Feynman diagrams contribute to the NLO amplitude. Figure: NLO corrections to the light-cone sum rule for baryon form factors. I.V. Anikin Electroproduction of Nucleon Resonances

The NLO corrections read [IVA, V.Braun, N.Offen’13] . Q 2 A NLO = q � � � � � V k ( x i ) C V k q ( x i , W ) + A k ( x i ) C A k = [ dx i ] q ( x i , W ) k =1 , 3 �� � � ( x i ) C V ( m ) ( x i ) C A ( m ) V ( m ) ( x i , W ) + A ( m ) + 2 2 ( x i , W ) q q 2 2 m =1 , 2 , 3 + O (twist-5) , where W = Q 2 + P ′ 2 . Q 2 I.V. Anikin Electroproduction of Nucleon Resonances

Q 2 B NLO = q � � � V 1 ( x i ) D V 1 q ( x i , W ) + A 1 ( x i ) D A 1 = [ dx i ] q ( x i , W ) + O (twist-5) . It turns out that C V (1) d ( x i , W ) = C A (1) 2 d ( x i , W ) = 0. 2 I.V. Anikin Electroproduction of Nucleon Resonances

x 2 C V 1 d ( x i ) = � � 2 x 2 x 3 3( L − 2) g 1 ( x 3 ) + 2( L − 1) g 11 ( x 3 , x 3 ) + g 21 ( x 3 , x 3 ) � � + 2 x 2 + (4 L − 3) x 3 h 11 ( x 3 ) + (3 − 4 L )¯ x 1 h 11 (¯ x 1 ) � � + 2 x 3 h 21 ( x 3 ) − 2¯ x 1 h 21 (¯ x 1 ) − 2 3( x 2 / x 3 )(2 L − 3) + 5 L − 7 h 12 ( x 3 ) � � +2(5 L − 7) h 12 (¯ x 1 ) − 6( x 2 / x 3 ) + 5 h 22 ( x 3 ) + 5 h 22 (¯ x 1 ) + (6 / x 3 )( L − 2) h 13 ( x 3 ) − (6 / ¯ x 1 )( L − 2) h 13 (¯ x 1 ) +(3 / x 3 ) h 23 ( x 3 ) − (3 / ¯ x 1 ) h 23 (¯ x 1 ) , I.V. Anikin Electroproduction of Nucleon Resonances

x 2 C A 1 d ( x i ) = 3¯ x 1 h 11 (¯ x 1 ) − 3 x 3 h 11 ( x 3 ) + 2(3 L − 10) h 12 (¯ x 1 ) − 2(3 L − 10) h 12 ( x 3 ) + 3 h 22 (¯ x 1 ) − 3 h 22 ( x 3 ) − (6 / ¯ x 1 )( L − 3) h 13 (¯ x 1 ) + (6 / x 3 )( L − 3) h 13 ( x 3 ) − (3 / ¯ x 1 ) h 23 (¯ x 1 ) + (3 / x 3 ) h 23 ( x 3 ) , where g nk ( y , x ; W ) = ln n [1 − yW − i η ] ( − 1 + xW + i η ) k , h nk ( x ; W ) = ln n [1 − xW − i η ] ( W + i η ) k with n = 0 , 1 , 2 and k = 1 , 2 , 3. For n = 0 the first argument becomes dummy,i.e g k ( x ; W ) ≡ g 0 k ( ∗ , x ; W ) , I.V. Anikin Electroproduction of Nucleon Resonances

Results Discussion of parameters Schematically, the general structure of form factors has the following form: 2 2 � � � + f N + f N F = F tw-4 F tw-3 η 1 i F tw-4 ϕ ij F tw-3 + . 0 η 1 i ϕ ij f N λ 1 λ 1 i =0 , 1 i =1 j =0; j ≤ i Or, in other words, we have � � • tw-3: ϕ 10 , ϕ 11 , ϕ 20 , ϕ 21 , ϕ 22 , f N ; � � • tw-4: η 10 , η 11 , λ 1 ; I.V. Anikin Electroproduction of Nucleon Resonances

The other parameters that enter LCSRs are • the interval of duality (continuum threshold) s 0 ( s 0 = 2 . 25 GeV 2 ); • Borel parameter M 2 ( M 2 = 1 . 5 GeV 2 and M 2 = 2 GeV 2 and M 2 ≃ s 0 ) ; • factorization/renormalization scale µ 2 ( µ 2 = 2 GeV 2 and µ 2 ∼ (1 − x ) Q 2 − xP ′ 2 or µ 2 ≤ (1 − x 0 ) Q 2 + x 0 M 2 ≤ 2 s 0 Q 2 s 0 + Q 2 < 2 s 0 ). • We use a two-loop expression for the QCD coupling with Λ (4) QCD = 326 MeV resulting in the value α s (2 GeV 2 ) = 0 . 374. I.V. Anikin Electroproduction of Nucleon Resonances