Non-p erturbative lo w energy amplitudes in non-lo al hiral qua rk mo del Piotr K otk o Jagiellonian Universit y , Krak � w 49 Cra o w S ho ol of Theo reti al Physi s, Zak opane 2009
OUTLINE Non-p erturbative input to amplitudes fo r ex lusive p ro esses is analyzed within full non-lo al hiral qua rk mo del T w o examples: Photon Distribution Amplitudes Pion-photon T ransition Distribution Amplitudes Sp e ial attention is paid to the question of inheriting QCD p rop erties b y obje ts al ulated in the e�e tive mo del • • • • •
T ABLE OF CONTENTS 1 Intro du tion 2 Chiral Qua rk Mo del and its p roblems 3 Photon Distribution Amplitudes 4 T ransition Distribution Amplitudes 5 Con lusions
INTRODUCTION F a to rization of the amplitudes fo r ex lusive p ro esses in the p resen e of the 1 , 2 , 3 ha rd s ale M = ( HARD ) ⊗ ( SOFT ) HARD pa rt an b e al ulated in p erturbation theo ry SOFT pa rt is a subje t to the non-p erturbative treatment Examples: � Distribution Amplitudes (D A) Generalized P a rton Distributions (GPD) T ransition Distribution Amplitudes (TD A) Ho w to a ess the SOFT pa rt ? extra tion from the exp eriment latti e al ulations lo w energy e�e tive mo dels ⇒ ⇒ 1 2 3 Efremov, Radyushkin; Bro dsky , Lepage; Collins, F rankfurt, Strikman � � ⇒ ⇒ ⇒
INTRODUCTION ( ontinued...) SOFT pa rt pa rametrizes matrix elements of ertain non-lo al qua rk (gluon) op erato rs on the light- one, e.g. H ′ ˛ ψ ( y ) O ψ ( x ) H they should p ossess p rop erties o riginating from QCD symmetries (e.g. Lo rentz inva rian e, W a rd identities, axial anomaly) it is not obvious that e�e tive mo dels do inherit all QCD symmetries p ossible p roblems with o rre t p rop erties of SOFT pa rt in the e�e tive mo dels ˛ ¯ ˛ ˙ ¸ ˛ ⇒ = ⇒ = ⇒
CHIRAL QUARK MODEL ( χ QM) F o r simpli it y w e onsider pions only . In o rder to obtain onsidered matrix elements w e need the mo del of qua rk-pion intera tions. A t lo w energy s ales sp ontaneous hiral symmetry b reaking ( χ SB) pla ys very imp o rtant role the mo del should in o rp o rate χ SB there app ea r onstituent qua rk mass M ∼ 350 Me V 1 The simplest mo del is the semi-b osonized Nambu-Jona-Lasinio mo del ( hiral limit) 4 5 ) ψ ( x ) S d x ¯ ψ ( x ) ( i � D − MU γ lo = i a π a ( x ) γ 5 ( x ) = where U γ exp , with F π = 93 Me V . 5 F π τ In o rder to get �nite qua rk lo ops w e need to imp ose some kind of regula rization (but w e annot remove the uto� pa rameter at the end) ⇒ Ho w ever to get o rre t results fo r anomalous p ro esses w e have to ⇒ remove regula rization P a rti ula r regula rization s heme la ks motivation in terms of QCD... ˆ 1 see e.g. S.P . Klevansky n o • • •
NON-LOCAL χ QM The most natural w a y of regula rizing qua rk lo ops momentum dep endent onstituent qua rk mass M ≡ M ( k ) 4 4 d k d l 5 ( k − S ψ ( k ) M ( k ) U γ l ) M ( l ) ψ ( l ) Int = 8 ( 2 π ) 2 ( k ) where usually one de�nes M ( k ) = M F , and F ( 0 ) = 1 , F ( k → ∞ ) → 0 . This a tion w as �derived� from QCD instanton va uum theo ry , with Eu lidean 1 analyti al exp ression fo r M ( k ) . Problem: momentum dep endent mass naive ve to r urrent ¯ is not onserved = ⇒ lo al vertex γ µ has to b e repla ed b y the non-lo al one Γ µ 2 , 3 , 4 The p re ise fo rm of the vertex is un onstrained and has to b e mo deled . ˆ One of the simplest solution is ¯ p p k µ + p µ Γ µ ( k , p ) = γ µ − 2 ( M ( k ) − M ( p )) 2 − k p The on rete mo del is sp e i�ed b y giving M ( k ) and the fo rm of the verti es. 1 2 3 4 Diak onov, P etrov; Bo wler, Birse; B. Holdom, R. Lewis; A. Bzdak, M. Praszalo wi z ψγ µ ψ ⇒ ⇒
ONE LOOP CALCULA TIONS As the mass dep enden e on momentum w e tak e n 2 F ( k ) = 2 − Λ 2 + k i ǫ al ulations in Eu lidean as w ell as Mink o wski spa e �analyti al� solutions The ansatz ab ove leads to set of p oles in the omplex plane. Using some tri ks w e an exp ress the lo op integral with N p ropagato rs as 1.0 4 n + 1 D κ n � 1 d 0.8 M M 1 N f f g ( κ, η i i i N ) n � 3 1 . . . „ N η i « i 1 , . . . , η − Λ 1 D N ( 2 π ) 0.6 i 1 ,..., i F inst � k � N F � k � 0.4 where g is a fun tion ontaining only N p oles, η a re solutions of i • 2 = 4 n + 1 + 4 n − ( M / Λ) z z 0 and f a re some numb ers omp osed from η . i i 0.2 Higher t wist light one amplitudes delta t yp e singula rities in the 0.0 0 1 2 3 4 5 • b ounda ries of physi al supp o rt � k � � GeV � Praszalo wi z, Rost w o ro wski, Bzdak, P .K. ˆ X . . . η ⇒
APPLICA TION I Simplest SOFT obje ts = Distribution Amplitudes (D A) Example: radiative ve to r meson de a y V → S γ and V ( q ) S ( q + P ) Photon D A. Relevant o o rdinates a re de�ned b y t w o light-lik e ve to rs n = ( 1 , 0 , 0 , − 1 ) , ˜ n = ( 1 , 0 , 0 , 1 ) . Then w e PSfrag repla ements an de omp ose any ve to r v as v + v − v µ = n µ + n µ + v µ T 2 ˜ 2 γ ( P ) General de�nition: ⇒ 1 2 ´ ˆ i ( 2 u − 1 ) λ P + φ O 2 ” 0 ˛ ψ ( λ n ) O ψ ( − λ n ) ˛ γ ( P ) F O P du e u , P 0 where O = { σ µν , γ µ , γ µ γ and F O a re relevant fo rm fa to rs. 5 } P . Ball, Bro wn; Arriola, Bronio wski, Do rokhov; “ ˛ ˛ ˙ ¸ ` ∼
2 ´ As an example onsider ve to r photon D A φ V u , P urrent onservation in QCD: F V ( 0 ) = 0. nonlo al N χ QM al ulations: urrent PSfrag repla ements Using full non-lo al photon-qua rk γ ( P ) vertex and leaving pure QCD ve to r urrent op erato r w e re over F 0 ) = 0. V ( Diagram in the right has t w o ontributions: hadroni and in�nite ` p erturbative pa rt o rresp onding to ⇒ ele tromagneti ingredient of the photon. Amplitudes up to t wist-4 in all • ¯ hannels have b een al ulated. ψγ µ ψ M = 350 Me V , n = 1 • F V � P 2 � local vertex 0.4 non � local vertex 0.3 0.2 0.1 • 1.0 � P 2 � GeV 2 � 0.2 0.4 0.6 0.8 � 0.1 � 0.2
APPLICA TION I I Mo re demanding obje ts to study in Chiral Qua rk Mo dels: PSfrag repla ements T ransition Distribution Amplitudes app ea ring fo r example in π + π − → γ ∗ γ in 1 the fo rw a rd region . e − γ ∗ ( q Kinemati s: 2 ) π − ( q 1 ) 2 high virtualit y Q of the upp er photon e − lo w momentum transfer to the lo w er blob u d 2 = ( 2 = 2 π + ( P 1 ) γ ( P P P t ≪ Q 2 ) 2 − 1 ) Example: V e to r TD A (VTD A) TD A d λ i λ Xp + < γ ( P e d ( − λ n ) γ µ u ( λ n ) | π + ( P P P V ( X , ξ, t ) 2 ) | 1 ) > ∼ ε µναβ ε ∗ 1 α 2 β 2 π 2 2 1 where ξ = − 2 ∆ + / p + with p = 2 ( P P is so alled sk ew edness. 1 + 2 ) • • ¯ ∆ 1 Pire, Szymano wski ˆ ν
Prop erties of VTD A o riginating from QCD: n n + n − 1 + . . . + p olynomialit y dX X V ( X , ξ, t ) = a a a n ξ n − 1 ξ 0 2 no rmalization is �xed b y axial anomaly dX V ( X , ξ, t = 0 ) = 1 / 2 π PSfrag repla ements nonlo al N χ QM al ulations: urrents P olynomialit y is satis�ed. γ ( P 2 ) π + ( P 1 ) W e obtain o rre t no rmalization only when b oth ve to r urrents a re non-lo al. VTD A is related to pion-photon transition fo rm • ´ fa to r • ´ dX V ( X , ξ, t ) ∼ F πγ ( t ) ¯ ψγ µ ψ 0 ontrolling γ ∗ γ → π rea tion. • New BaBa r data a re available (29 Ma y) whi h ast some new light on pion Distribution Ampli- • tudes... this is urrently under investigation... M = 350 Me V , n = 1 , ξ = 0 . 5 V � X � full model ˆ 0.10 local model 0.05 1.0 X � 1.0 � 0.5 0.5 � 0.05 ⇒
SUMMARY Non-lo al hiral qua rk mo del allo ws fo r analyzing lo w energy matrix elements Ho w ever, b efo re using in real p ro esses they have to b e evolved (s ale of e�e tive mo dels is lo w) - not dis ussed In o rder to mak e al ulations onsistent w e have to use mo di�ed urrents The fo rm of the full urrents is not restri ted and has to b e mo delled Ho w ever, in general it is not lea r y et whi h urrents w e should mo dify and when • Case of full axial urrent is mo re di� ult - not dis ussed First analysis of pion-photon T ransition Distribution Amplitudes in non-lo al mo del • • • • • •
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