Pion FF in QCD Sum Rules with NLCs
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1 Pion FF in QCD Sum Rules with NLCs A. Bakulev, A. Pimikov & N. Stefanis Bogoliubov Lab. Theor. Phys., JINR (Dubna, Russia) Pion FF in QCD Sum Rules with NLCs p.1
2 Content: Definition of pion form factor (FF) Pion FF in QCD Sum Rules with NLCs p.2
3 Content: Definition of pion form factor (FF) AAV correlator and corresponding OPE diagrams. Pion FF in QCD Sum Rules with NLCs p.2
4 Content: Definition of pion form factor (FF) AAV correlator and corresponding OPE diagrams. Non-local condensates in QCD calculations. Pion FF in QCD Sum Rules with NLCs p.2
5 Content: Definition of pion form factor (FF) AAV correlator and corresponding OPE diagrams. Non-local condensates in QCD calculations. QCD SR with non-local condensates. Pion FF in QCD Sum Rules with NLCs p.2
6 Content: Definition of pion form factor (FF) AAV correlator and corresponding OPE diagrams. Non-local condensates in QCD calculations. QCD SR with non-local condensates. QCD SR vs experimental data. Pion FF in QCD Sum Rules with NLCs p.2
7 Content: Definition of pion form factor (FF) AAV correlator and corresponding OPE diagrams. Non-local condensates in QCD calculations. QCD SR with non-local condensates. QCD SR vs experimental data. Local Duality approach and NLC SRs. Pion FF in QCD Sum Rules with NLCs p.2
8 Content: Definition of pion form factor (FF) AAV correlator and corresponding OPE diagrams. Non-local condensates in QCD calculations. QCD SR with non-local condensates. QCD SR vs experimental data. Local Duality approach and NLC SRs. Conclusions. Pion FF in QCD Sum Rules with NLCs p.2
9 Definition of pion FF Pion FF F π is defined by the matrix element q π + (p ) J μ (0) π + (p) = ( p + p ) μ F π(q 2 ), where J μ is the electromagnetic current, (p p) 2 = q 2 Q 2 is the photon virtuality, and pion FF is normalized to F π (0) = 1. p p Pion FF in QCD Sum Rules with NLCs p.3
10 Definition of pion FF Pion FF F π is defined by the matrix element q π + (p ) J μ (0) π + (p) = ( p + p ) μ F π(q 2 ), where J μ is the electromagnetic current, (p p) 2 = q 2 Q 2 is the photon virtuality, and p pion FF is normalized to F π (0) = 1. At asymptotically large Q 2 pqcd factorization gives pion FF p ϕ π ϕ π F π (Q 2 )= 8πα s(q 2 )f 2 π 9Q ϕ π (x, Q 2 ) x dx 2 in terms of twist-2 pion DA ϕ π (x, Q 2 ). Pion FF in QCD Sum Rules with NLCs p.3
11 AAV correlator Axial-Axial-Vector correlator can be used for studying pion FF by QCD SR technique: [ ] d 4 xd 4 ye i(qx p 2y) 0 T J + 5β (y)j μ (x)j 5α (0) 0 γ J μ J 5α J + 5β π + π + Pion FF in QCD Sum Rules with NLCs p.4
12 AAV correlator Axial-Axial-Vector correlator can be used for studying pion FF by QCD SR technique: [ ] d 4 xd 4 ye i(qx p 2y) 0 T J + 5β (y)j μ (x)j 5α (0) 0 γ J μ J 5α J + 5β π + π + where EM current J μ (x) =e u u(x)γ μ u(x)+e d d(x)γ μ d(x) Pion FF in QCD Sum Rules with NLCs p.4
13 AAV correlator Axial-Axial-Vector correlator can be used for studying pion FF by QCD SR technique: [ ] d 4 xd 4 ye i(qx p 2y) 0 T J + 5β (y)j μ (x)j 5α (0) 0 γ J μ J 5α J + 5β π + π + where EM current J μ (x) =e u u(x)γ μ u(x)+e d d(x)γ μ d(x) and axial-vector current: J 5α (x) =d(x)γ 5 γ α u(x). Pion FF in QCD Sum Rules with NLCs p.4
14 Diagramms for AAV-correlator 1 Perturbative LO term Nesterenko&Radyushkin Ioffe&Smilga [1982] Pion FF in QCD Sum Rules with NLCs p.5
15 Diagramms for AAV-correlator 1 Perturbative LO term Nesterenko&Radyushkin Ioffe&Smilga [1982] Perturbative NLO terms Braguta&Onishchenko [2004] Pion FF in QCD Sum Rules with NLCs p.5
16 Diagramms for AAV-correlator 1 Perturbative LO term Nesterenko&Radyushkin Ioffe&Smilga [1982] Perturbative NLO terms Braguta&Onishchenko [2004] Nonperturbative terms Nesterenko&Radyushkin Ioffe&Smilga [1982] local condensates Pion FF in QCD Sum Rules with NLCs p.5
17 Diagramms for AAV-correlator 1 Perturbative LO term Nesterenko&Radyushkin Ioffe&Smilga [1982] Perturbative NLO terms Braguta&Onishchenko [2004] + + Nonperturbative terms +... A. B.&Radyushkin [1991] nonlocal condensates Pion FF in QCD Sum Rules with NLCs p.5
18 Introducing NLC in QCD calculations T ( ψψ ) = ] ψψ + : ψψ :(Wick theorem) 0 T ( ψψ ) 0 = i 1 Ŝ 0 (x) +? QCD PT : ψψ : def =0 Pion FF in QCD Sum Rules with NLCs p.6
19 Introducing NLC in QCD calculations T ( ψψ ) = ] ψψ + : ψψ :(Wick theorem) 0 T ( ψψ ) 0 = i 1 Ŝ 0 (x) +? QCD PT : ψψ : def =0 QCD SR : ψ(0)ψ(0) : = qq CONST 0 [SVZ 79] Condensate Decay constants, masses of hadrons Pion FF in QCD Sum Rules with NLCs p.6
20 Introducing NLC in QCD calculations T ( ψψ ) = ] ψψ + : ψψ :(Wick theorem) 0 T ( ψψ ) 0 = i 1 Ŝ 0 (x) +? QCD PT : ψψ : def =0 QCD SR : ψ(0)ψ(0) : = qq CONST 0 NLC QCD SR : ψ(0)ψ(z): F S (z 2 )+ẑf V (z 2 ) [SVZ 79] Condensate M&R 86 Nonlocal condensate Decay constants, masses of hadrons Distribution Amplitudes, Form Factors Pion FF in QCD Sum Rules with NLCs p.6
21 Lattice data of Pisa group 1.4 ψ(0)ψ(z) ψ(0)ψ(0) /λ q z [Fm] Nonlocality of quark condensates from lattice data of Pisa group in comparison with local limit. Even at z 0.5 Fm nonlocality is quite important! Pion FF in QCD Sum Rules with NLCs p.7
22 Diagrams for T (J 1 (z)j 2 (0)) q μ q+k ν q k PT Pion FF in QCD Sum Rules with NLCs p.8
23 Diagrams for T (J 1 (z)j 2 (0)) q μ q+k ν q k q μ k=0 qq ν q PT q SVZ Pion FF in QCD Sum Rules with NLCs p.8
24 Diagrams for T (J 1 (z)j 2 (0)) q μ q+k ν q k q μ k=0 qq ν q PT q μ k=0 q(z)q(0) ν q q q+k SVZ NLC Quarks run through vacuum with nonzero momentum k 0: k 2 = ψd2 ψ ψψ = λ 2 q = GeV 2 Pion FF in QCD Sum Rules with NLCs p.8
25 Non-Local Condensates in QCD SR Illustration of NLC-model: q(0)q(z) = q(0)q(0) e z2 λ 2 q /8 Pion FF in QCD Sum Rules with NLCs p.9
26 Non-Local Condensates in QCD SR Illustration of NLC-model: q(0)q(z) = q(0)q(0) e z2 λ 2 q /8 A single scale parameter λ 2 q = k 2 characterizing the average momentum of quarks in QCD vacuum: λ 2 q = 0.4 ± 0.1 GeV 2 [ QCD SRs, 1987 ] 0.5 ± 0.05 GeV 2 [ QCD SRs, 1991 ] GeV 2 [ Lattice, ] Pion FF in QCD Sum Rules with NLCs p.9
27 Non-Local Condensates in QCD SR Illustration of NLC-model: q(0)q(z) = q(0)q(0) e z2 λ 2 q /8 A single scale parameter λ 2 q = k 2 characterizing the average momentum of quarks in QCD vacuum: λ 2 q = 0.4 ± 0.1 GeV 2 [ QCD SRs, 1987 ] 0.5 ± 0.05 GeV 2 [ QCD SRs, 1991 ] GeV 2 [ Lattice, ] Correlation length λ 1 q ρ-meson size Pion FF in QCD Sum Rules with NLCs p.9
28 Non-Local Condensates in QCD SR Illustration of NLC-model: q(0)q(z) = q(0)q(0) e z2 λ 2 q /8 A single scale parameter λ 2 q = k 2 characterizing the average momentum of quarks in QCD vacuum: 0.4 ± 0.1 GeV 2 [ QCD SRs, 1987 ] λ 2 q = 0.5 ± 0.05 GeV 2 [ QCD SRs, 1991 ] GeV 2 [ Lattice, ] Correlation length λ 1 q ρ-meson size Possible to include second (Λ 450 MeV) scale with q(0)q(z) qq e z Λ (not included here) z 1 Fm Pion FF in QCD Sum Rules with NLCs p.9
29 Non-Local Condensates in QCD SR Parameterization for scalar and vector condensates: ψ(0)ψ(x) = ψψ f S (α) e αx2 /4 dα ; 0 ψ(0)γ μ ψ(x) = ix μ A 0 f V (α) e αx2 /4 dα, where A 0 =2α s π ψψ 2 /81. 0 Pion FF in QCD Sum Rules with NLCs p.10
30 Non-Local Condensates in QCD SR Convenient to parameterize the 3-local condensate in fixed-point gauge by introduction of three scalar functions: ψ(0)γ μ ( gâν(x))ψ(y) = (x μ y ν g μν (xy))m 1 + (x μ x ν g μν x 2 )M 2 ; ψ(0)γ 5 γ μ ( gâν(x))ψ(y) = iε μνxy M 3, with M i (y 2,x 2, (x y) 2 )= A i dα 1 dα 2 dα 3 f i (α 1,α 2,α 3 ) e (α 1y 2 +α 2 x 2 +α 3 (x y) 2 )/4. 0 where A i = { 3 2, 2, 3 2 }A 0 [Mikhailov&Radyushkin 89]. Pion FF in QCD Sum Rules with NLCs p.10
31 Non-Local Condensates in QCD SR The minimal Gaussian ansatz: f S (α) =δ (α Λ) ; f V (α) =δ (α Λ) ; Λ λ 2 q/2; f i (α 1,α 2,α 3 )=δ (α 1 Λ) δ (α 2 Λ) δ (α 3 Λ). Only one parameter λ 2 q = GeV 2. Pion FF in QCD Sum Rules with NLCs p.10
32 Non-Local Condensates in QCD SR The minimal Gaussian ansatz: f S (α) =δ (α Λ) ; f V (α) =δ (α Λ) ; Λ λ 2 q/2; f i (α 1,α 2,α 3 )=δ (α 1 Λ) δ (α 2 Λ) δ (α 3 Λ). Only one parameter λ 2 q = GeV 2. Problems: QCD equations of motion are violated Vector current correlator is not transverse gauge invariance is broken Pion FF in QCD Sum Rules with NLCs p.10
33 Improved Gaussian model We modify functions f i : f imp i (α 1,α 2,α 3 )= (1 + X i x + Y i y + Z i z ) δ (α 1 xλ) δ (α 2 yλ) δ (α 3 zλ) Pion FF in QCD Sum Rules with NLCs p.11
34 Improved Gaussian model We modify functions f i : f imp i (α 1,α 2,α 3 )= (1 + X i x + Y i y + Z i z ) δ (α 1 xλ) δ (α 2 yλ) δ (α 3 zλ) What does it give us? Pion FF in QCD Sum Rules with NLCs p.11
35 Improved Gaussian model We modify functions f i : f imp i (α 1,α 2,α 3 )= (1 + X i x + Y i y + Z i z ) δ (α 1 xλ) δ (α 2 yλ) δ (α 3 zλ) What does it give us? If 12 (X 2 + Y 2 ) 9(X 1 + Y 1 )=1,x+ y =1, than QCD equations of motion are satisfied; Pion FF in QCD Sum Rules with NLCs p.11
36 Improved Gaussian model We modify functions f i : f imp i (α 1,α 2,α 3 )= (1 + X i x + Y i y + Z i z ) δ (α 1 xλ) δ (α 2 yλ) δ (α 3 zλ) What does it give us? If 12 (X 2 + Y 2 ) 9(X 1 + Y 1 )=1,x+ y =1, than QCD equations of motion are satisfied; We minimize nontransversity of polarization operator by special choice of model parameters: X 1 = ; Y 1 = Z 1 = ; x =0.788 ; X 2 = ; Y 2 = Z 2 = ; y =0.212 ; X 3 = ; Y 3 = Z 3 = ; z = Pion FF in QCD Sum Rules with NLCs p.11
37 NLC QCD SR The Borel SR for the pion FF based on three-point AAV correlator: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Approach Acc Condensates Q 2 -behavior of Φ OPE N&R, I&S 82 LO local const + Q 2 Pion FF in QCD Sum Rules with NLCs p.12
38 NLC QCD SR The Borel SR for the pion FF based on three-point AAV correlator: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Approach Acc Condensates Q 2 -behavior of Φ OPE N&R, I&S 82 LO local const + Q 2 B&R 91 LO local + nonlocal (const + Q 2 )(e Q2 λ 2 q + const) Pion FF in QCD Sum Rules with NLCs p.12
39 NLC QCD SR The Borel SR for the pion FF based on three-point AAV correlator: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Approach Acc Condensates Q 2 -behavior of Φ OPE N&R, I&S 82 LO local const + Q 2 B&R 91 LO local + nonlocal (const + Q 2 )(e Q2 λ 2 q + const) B&O 04 -LD NLO NO M 2 Φ OPE 0, s 0 =?(f π LD SR) Pion FF in QCD Sum Rules with NLCs p.12
40 NLC QCD SR The Borel SR for the pion FF based on three-point AAV correlator: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Approach Acc Condensates Q 2 -behavior of Φ OPE N&R, I&S 82 LO local const + Q 2 B&R 91 LO local + nonlocal (const + Q 2 )(e Q2 λ 2 q + const) B&O 04 -LD NLO NO M 2 Φ OPE 0, s 0 =?(f π LD SR) Here NLO nonlocal (const + Q 2 ) e Q2 λ 2 q Pion FF in QCD Sum Rules with NLCs p.12
41 NLC QCD SR The Borel SR for the pion FF based on three-point AAV correlator: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Approach Acc Condensates Q 2 -behavior of Φ OPE N&R, I&S 82 LO local const + Q 2 B&R 91 LO local + nonlocal (const + Q 2 )(e Q2 λ 2 q + const) B&O 04 -LD NLO NO M 2 Φ OPE 0, s 0 =?(f π LD SR) Here NLO nonlocal (const + Q 2 ) e Q2 λ 2 q Nonlocality improves Q 2 behavior of OPE widens region of applicability up to Q 2 10 GeV 2. Pion FF in QCD Sum Rules with NLCs p.12
42 NLC QCD SR The Borel SR for the pion FF based on three-point AAV correlator: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Approach Acc Condensates Q 2 -behavior of Φ OPE N&R, I&S 82 LO local const + Q 2 B&R 91 LO local + nonlocal (const + Q 2 )(e Q2 λ 2 q + const) B&O 04 -LD NLO NO M 2 Φ OPE 0, s 0 =?(f π LD SR) Here NLO nonlocal (const + Q 2 ) e Q2 λ 2 q Nonlocality improves Q 2 behavior of OPE widens region of applicability up to Q 2 10 GeV 2. We use model-independent expression for Φ OPE -term obtained by A. B.&Radyushkin, but significantly different model of condensate s nonlocality. Pion FF in QCD Sum Rules with NLCs p.12
43 NLC QCD SR Result for Q 2 F π (Q 2 ) 0.8 Q 2 F π (Q 2 ) curve approach NLC QCD SR Q 2 [GeV 2 ] Pion FF from: SRs with NLC (blue solid line), Pion FF in QCD Sum Rules with NLCs p.13
44 NLC QCD SR Result for Q 2 F π (Q 2 ) 0.8 Q 2 F π (Q 2 ) curve approach NLC QCD SR QCD SR LD SR 0.2 Q 2 [GeV 2 ] Pion FF from: SRs with NLC (blue solid line), standard QCD SRs (red dashed line) [N&R I&S 82], O(α s ) Local Duality (black dashed line) [B&O 04], Pion FF in QCD Sum Rules with NLCs p.13
45 NLC QCD SR Result for Q 2 F π (Q 2 ) 0.8 Q 2 F π (Q 2 ) curve approach NLC QCD SR QCD SR LD SR AdS/QCD (BT) AdS/QCD (GR) Q 2 [GeV 2 ] Pion FF from: SRs with NLC (blue solid line), standard QCD SRs (red dashed line) [N&R I&S 82], O(α s ) Local Duality (black dashed line) [B&O 04], AdS/QCD (green dashed line) [B&T 07] and (green dot-dashed line) [G&R 07] Pion FF in QCD Sum Rules with NLCs p.13
46 NLC QCD SR Result for Q 2 F π (Q 2 ) 0.8 Q 2 F π (Q 2 ) curve approach NLC QCD SR QCD SR LD SR AdS/QCD (BT) 0.2 Q 2 [GeV 2 ] AdS/QCD (GR) [JLab 08] Pion FF from: SRs with NLC (blue solid line), standard QCD SRs (red dashed line) [N&R I&S 82], O(α s ) Local Duality (black dashed line) [B&O 04], AdS/QCD (green dashed line) [B&T 07] and (green dot-dashed line) [G&R 07] in comparison with [JLab 08] ( ) experimental data. Pion FF in QCD Sum Rules with NLCs p.13
47 NLC QCD SR Result for Q 2 F π (Q 2 ) 0.8 Q 2 F π (Q 2 ) curve approach NLC QCD SR QCD SR LD SR AdS/QCD (BT) 0.2 Q 2 [GeV 2 ] AdS/QCD (GR) [JLab 08] [Cornell 78] Pion FF from: SRs with NLC (blue solid line), standard QCD SRs (red dashed line) [N&R I&S 82], O(α s ) Local Duality (black dashed line) [B&O 04], AdS/QCD (green dashed line) [B&T 07] and (green dot-dashed line) [G&R 07] in comparison with [JLab 08] ( ) and [Cornell 78] ( ) experimental data. Pion FF in QCD Sum Rules with NLCs p.13
48 NLC QCD SR vs. Lattice QCD results 0.8 Q 2 F π (Q 2 ) curve approach NLC QCD SR QCD SR LD SR AdS/QCD (BT) 0.2 Q 2 [GeV 2 ] AdS/QCD (GR) [JLab 08] [Cornell 78] Pion FF from: SRs with NLC (blue solid line), in comparison with recent lattice results by D. Brommel et al. [Eur. Phys. J., C51 (2007) 335]. Pion FF in QCD Sum Rules with NLCs p.14
49 Local Duality vs Sum Rules for pion FF Borel SR: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). 1 s 0 (Q 2 )[GeV 2 ] Q 2 [GeV 2 ] Pion FF in QCD Sum Rules with NLCs p.15
50 Local Duality vs Sum Rules for pion FF Borel SR: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Local Duality approximation: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ). Pion FF in QCD Sum Rules with NLCs p.15
51 Local Duality vs Sum Rules for pion FF Borel SR: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ) e (s 1+s 2 )/M 2 +Φ OPE (Q 2,M 2 ). Local Duality approximation: f 2 π F π (Q 2 )= s 0 0 ds 1 ds 2 ρ 3 (s 1,s 2,Q 2 ). In general s 0 = s LD 0 (Q 2 ) s 0 (Q 2 ). Pion FF in QCD Sum Rules with NLCs p.15
52 Approximation of LD result In general s 0 = s LD 0 (Q 2 ) s 0 (Q 2 ): s LD 0 (Q 2 )[GeV 2 ] Q 2 [GeV 2 ] Pion FF in QCD Sum Rules with NLCs p.16
53 Approximation of LD result This is the reason for pion FF underestimation in Braguta Lucha Melikhov (2008) approach: Pion FF in QCD Sum Rules with NLCs p.16
54 Conclusion Taking into account nonlocality of condensates makes QCD SR stable and widens region of applicability up to Q 2 10 GeV 2. Pion FF in QCD Sum Rules with NLCs p.17
55 Conclusion Taking into account nonlocality of condensates makes QCD SR stable and widens region of applicability up to Q 2 10 GeV 2. We use the model-independent expression for Φ OPE -term obtained by A. B.&Radyushkin [1991], but significantly different model of NLCs with nonlocality parameter λ 2 q =0.4GeV 2. Pion FF in QCD Sum Rules with NLCs p.17
56 Conclusion Taking into account nonlocality of condensates makes QCD SR stable and widens region of applicability up to Q 2 10 GeV 2. We use the model-independent expression for Φ OPE -term obtained by A. B.&Radyushkin [1991], but significantly different model of NLCs with nonlocality parameter λ 2 q =0.4GeV 2. NLO corrections to double spectral density, obtained by Braguta&Onishchenko [2004], are large. Taking them into account in Local Duality approach suffers from underestimation of s LD 0 (Q 2 ): our results show that s LD 0 (Q 2 ) grows with Q 2. Pion FF in QCD Sum Rules with NLCs p.17
57 Conclusion Taking into account nonlocality of condensates makes QCD SR stable and widens region of applicability up to Q 2 10 GeV 2. We use the model-independent expression for Φ OPE -term obtained by A. B.&Radyushkin [1991], but significantly different model of NLCs with nonlocality parameter λ 2 q =0.4GeV 2. NLO corrections to double spectral density, obtained by Braguta&Onishchenko [2004], are large. Taking them into account in Local Duality approach suffers from underestimation of s LD 0 (Q 2 ): our results show that s LD 0 (Q 2 ) grows with Q 2. QCD SR method with NLCs for the pion FF gives us a strip of predictions. This strip appears to be in a good agreement with existing experimental data of JLab and Cornell, as well as with lattice data. Pion FF in QCD Sum Rules with NLCs p.17
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