Toward Baryon Distributions Amplitudes
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1 Toward Baryon Distributions Amplitudes Cédric Mezrag INFN Roma1 September 13 th, 2018 Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
2 Chapter 1: Dyson-Schwinger equations Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
3 Dyson-Schwinger Equations in a nutshell Dyson-Schwinger equations relate Green (Schwinger) functions among each other. One get in the case of QCD, an infinite system truncations are required. DSEs are usually solved in Euclidean space, yielding Schwinger functions instead of Green functions. (Although some works are performed to solve them directly in Minkowski space) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
4 Dyson-Schwinger Equations in a nutshell Dyson-Schwinger equations relate Green (Schwinger) functions among each other. One get in the case of QCD, an infinite system truncations are required. DSEs are usually solved in Euclidean space, yielding Schwinger functions instead of Green functions. (Although some works are performed to solve them directly in Minkowski space) Truncating the DSEs yields an non-perturbative approximation of QCD Schwinger functions. Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
5 The Gap Equation The gap equation for the quark propagator S(q): ( ) 1 = ( ) 1. Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
6 The Gap Equation The gap equation for the quark propagator S(q): ( ) 1 = ( ) 1. It has successfully described the quark mass behaviour: S(q) 1 = i /qa(q 2 ) + B(q 2 ) Non-perturbative description of the quark mass Dynamical mass generation Figure from Bashir et al. (2012) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
7 Baryon and Diquarks The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
8 Baryon and Diquarks The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. It predicts the existence of strong diquarks correlations inside the nucleon. p q p q p d a Ψ P = p d Γ a q Γ b b Ψ P Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
9 Baryon and Diquarks The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. It predicts the existence of strong diquarks correlations inside the nucleon. p q p q p d a Ψ P = p d Γ a q Γ b b Ψ Mostly two types of diquark are dynamically generated by the Faddeev equation: Scalar diquarks, Axial-Vector (AV) diquarks. P Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
10 Baryon and Diquarks The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. It predicts the existence of strong diquarks correlations inside the nucleon. p q p q p d a Ψ P = p d Γ a q Γ b b Ψ Mostly two types of diquark are dynamically generated by the Faddeev equation: Scalar diquarks, Axial-Vector (AV) diquarks. Can we understand the nucleon structure in terms of quark-diquarks correlations? Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28 P
11 Chapter 2: Baryon Distribution Amplitudes CM, J. Segovia, L. Chang, C.D. Roberts Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
12 Hadrons seen as Fock States Lightfront quantization allows to expand hadrons on a Fock basis: P, π β Ψ q q β q q + β Ψ q q,q q β q q, q q +... P, N β Ψ qqq β qqq + β Ψ qqq,q q β qqq, q q +... Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
13 Hadrons seen as Fock States Lightfront quantization allows to expand hadrons on a Fock basis: P, π β Ψ q q β q q + β Ψ q q,q q β q q, q q +... P, N β Ψ qqq β qqq + β Ψ qqq,q q β qqq, q q +... Non-perturbative physics is contained in the N-particles Lightfront-Wave Functions (LFWF) Ψ N Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
14 Hadrons seen as Fock States Lightfront quantization allows to expand hadrons on a Fock basis: P, π β Ψ q q β q q + β Ψ q q,q q β q q, q q +... P, N β Ψ qqq β qqq + β Ψ qqq,q q β qqq, q q +... Non-perturbative physics is contained in the N-particles Lightfront-Wave Functions (LFWF) Ψ N Schematically a distribution amplitude ϕ is related to the LFWF through: d 2 k ϕ(x) (2π) 2 Ψ(x, k ) S. Brodsky and G. Lepage, PRD 22, (1980) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
15 Nucleon Distribution Amplitudes 3 bodies matrix element: 0 ɛ ijk u i α(z 1 )u j β (z 2)d k γ (z 3 ) P Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
16 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
17 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] Usually, one defines ϕ = V A V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
18 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) Usually, one defines ϕ = V A 3 bodies Fock space interpretation (leading twist): [dx] P, = 8 uud [ϕ(x 1, x 2, x 3 ) 6x 1 x 2 x 3 +ϕ(x 2, x 1, x 3 ) 2T (x 1, x 2, x 3 ) ] Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
19 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) Usually, one defines ϕ = V A 3 bodies Fock space interpretation (leading twist): [dx] P, = 8 uud [ϕ(x 1, x 2, x 3 ) 6x 1 x 2 x 3 Isospin symmetry: +ϕ(x 2, x 1, x 3 ) 2T (x 1, x 2, x 3 ) ] 2T (x 1, x 2, x 3 ) = ϕ(x 1, x 3, x 2 ) + ϕ(x 2, x 3, x 1 ) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
20 Evolution and Asymptotic results Both ϕ and T are scale dependent objects: they obey evolution equations Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
21 Evolution and Asymptotic results Both ϕ and T are scale dependent objects: they obey evolution equations At large scale, they both yield the so-called asymptotic DA ϕ AS : Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
22 Evolution and Asymptotic results Both ϕ and T are scale dependent objects: they obey evolution equations At large scale, they both yield the so-called asymptotic DA ϕ AS : u x d x u x Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
23 Form Factors: Nucleon case Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
24 Form Factors: Nucleon case u x d x S. Brodsky and G. Lepage, PRD 22, (1980) u x η = 1 Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
25 Form Factors: Nucleon case u x d x S. Brodsky and G. Lepage, PRD 22, (1980) u x η = 0.5 Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
26 Form Factors: Nucleon case u x d x S. Brodsky and G. Lepage, PRD 22, (1980) u x η = 2 Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
27 Some previous studies of DA QCD Sum Rules V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246 (1984) Relativistic quark model Z. Dziembowski, PRD 37 (1988) Scalar diquark clustering Z. Dziembowski and J. Franklin, PRD 42 (1990) Phenomenological fit J. Bolz and P. Kroll, Z. Phys. A 356 (1996) Lightcone quark model B. Pasquini et al., PRD 80 (2009) Lightcone sum rules I. Anikin et al., PRD 88 (2013) Lattice Mellin moment computation G. Bali et al., JHEP Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
28 Faddeev WF Model Algebraic parametrisation inspired by the results obtained from DSEs and Faddeev equations. It is based on Nakanishi representation, which is completely general. We also assume the dynamical diquark correlations, both scalar and AV, and compare in the end with Lattice QCD one. This is a work in progress, an update of the previous baryon PDA work (which was suffering some issues) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
29 Nakanishi Representation γ n At all order of perturbation theory, one can write (Euclidean space): Γ(k, P) = N 0 1 dγ 1 We use a simpler version of the latter as follow: 1 Γ(q, P) = N 1 ρ n (γ, z) dz (γ + (k + z 2 P)2 ) n ρ n (z) dz (Λ 2 + (q + z 2 P)2 ) n Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
30 Nucleon Distribution Amplitude Operator point of view for every DA (and at every twist): 0 ɛ ijk ( u i (z 1)C /nu j (z2) ) /nd k (z 3) P, λ ϕ(x 1, x 2, x 3), Braun et al., Nucl.Phys. B589 (2000) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
31 Nucleon Distribution Amplitude Operator point of view for every DA (and at every twist): 0 ɛ ijk ( u i (z 1)C /nu j (z2) ) /nd k (z 3) P, λ ϕ(x 1, x 2, x 3), Braun et al., Nucl.Phys. B589 (2000) We can apply it on the wave function: p 1 p 2 O 21 ϕ p 3 O 3 ϕ Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
32 Nucleon Distribution Amplitude Operator point of view for every DA (and at every twist): 0 ɛ ijk ( u i (z 1)C /nu j (z2) ) /nd k (z 3) P, λ ϕ(x 1, x 2, x 3), We can apply it on the wave function: Braun et al., Nucl.Phys. B589 (2000) p1 p2 O 21 ϕ = p1 p2 O 21 ϕ + p2 p1 O 21 ϕ + p1 p2 O 21 ϕ p3 O 3 ϕ p3 O 3 ϕ p3 O 3 ϕ p3 O 3 ϕ Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
33 Nucleon Distribution Amplitude Operator point of view for every DA (and at every twist): 0 ɛ ijk ( u i (z 1)C /nu j (z2) ) /nd k (z 3) P, λ ϕ(x 1, x 2, x 3), We can apply it on the wave function: Braun et al., Nucl.Phys. B589 (2000) p1 p2 O 21 ϕ = p1 p2 O 21 ϕ + p2 p1 O 21 ϕ + p1 p2 O 21 ϕ p3 O 3 ϕ p3 O 3 ϕ p3 O 3 ϕ p3 O 3 ϕ The operator then selects the relevant component of the wave function. Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
34 Nucleon Distribution Amplitude Operator point of view for every DA (and at every twist): 0 ɛ ijk ( u i (z 1)C /nu j (z2) ) /nd k (z 3) P, λ ϕ(x 1, x 2, x 3), We can apply it on the wave function: Braun et al., Nucl.Phys. B589 (2000) p1 p2 O 21 ϕ = p1 p2 O 21 ϕ + p2 p1 O 21 ϕ + p1 p2 O 21 ϕ p3 O 3 ϕ p3 O 3 ϕ p3 O 3 ϕ p3 O 3 ϕ The operator then selects the relevant component of the wave function. Our ingredients are: Perturbative-like quark and diquark propagator Nakanishi based diquark Bethe-Salpeter-like amplitude (green disks) Nakanishi based quark-diquark amplitude (dark blue ellipses) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
35 Scalar Diquark BSA The model used: 1 = N γ dz n 1 (1 z 2 ) (Λ 2 + (q + z 2 P)2 ) Comparable to scalar diquark amplitude previously used: q,k.q q red curve from Segovia et al.,few Body Syst. 55 (2014) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
36 Diquark DA Scalar diquark [ ] φ(x) 1 M2 ln 1 + K 2 x(1 x) M 2 K 2 x(1 x) Asymptotic K^2 4M^2 K^2 16 M^2 Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
37 Diquark DA Scalar diquark [ ] φ(x) 1 M2 ln 1 + K 2 x(1 x) M 2 K 2 x(1 x) Asymptotic K^2 4M^2 K^2 16 M^2 Φ Π x Pion x Pion figure from L. Chang et al., PRL 110 (2013) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
38 Diquark DA Scalar diquark [ ] φ(x) 1 M2 ln 1 + K 2 x(1 x) M 2 K 2 x(1 x) Asymptotic K^2 4M^2 K^2 16 M^2 Φ Π x Pion x Pion figure from L. Chang et al., PRL 110 (2013) This results provide a broad and concave meson DA parametrisation The endpoint behaviour remains linear Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
39 Nucleon Quark-Diquark Amplitude 1 = N 1 (1 z 2 ) ρ(z) dz (Λ 2 + (q 1+3z 6 P) 2 ) 3 Preliminary estimations of the parameters through comparison to Chebychev moments: T0 q² q red curve from Segovia et al., Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
40 Nucleon Quark-Diquark Amplitude =N Z 1 dz 1 (1 z 2 )ρ (z) 2 3 (Λ2 + (q 1+3z 6 P) ) ÈqÈ T3Hq²L T2Hq²L T1Hq²L Preliminary estimations of the parameters through comparison to Chebychev moments: ÈqÈ ÈqÈ red curves from Segovia et al., Cédric Mezrag (INFN) Baryon DAs September 13th, / 28
41 Nucleon Quark-Diquark Amplitude =N Z 1 dz 1 (1 z 2 )ρ (z) 2 3 (Λ2 + (q 1+3z 6 P) ) ÈqÈ T3Hq²L T2Hq²L T1Hq²L Preliminary estimations of the parameters through comparison to Chebychev moments: ÈqÈ ÈqÈ red curves from Segovia et al., There are still some works necessary to improve the comparison of higher Chebychev moments Cédric Mezrag (INFN) Baryon DAs September 13th, / 28
42 Mellin Moments We do not compute the PDA directly but Mellin moments of it: x m 1 x n 2 = x1 dx 1 dx 2 x1 m x2 n ϕ(x 1, x 2, 1 x 1 x 2 ) 0 For a general moment x1 mx 2 n, we change the variable in such a way to right down our moments as: x m 1 x n 2 = 1 0 dα 1 α 0 dβ α m β n f (α, β) f is a complicated function involving the integration on 6 parameters Uniqueness of the Mellin moments of continuous functions allows us to identify f and ϕ Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
43 Preliminary Results 0.4 u x d x u x d x u x 1 Nucleon DA (Scalar Diquark only) u x 1 Asymptotic DA Nucleon DA is skewed compared to the asymptotic one These properties are consequences of our quark-diquark picture Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
44 Comparison with lattice < x i > ϕ= Dx x i ϕ(x 1, x 2, x 3) 0.40 xi Lattice 2016 Lattice 2014 Scalar Only Asymptotic Value Lattice data from V.Braun et al, PRD 89 (2014) G. Bali et al., JHEP Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
45 Summary Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
46 Summary Baryon PDA DSE compatible framework for Baryon PDAs. Simple Nakanishi representation works for the nucleon PDA. Preliminary results assuming a scalar diquarks For the Future Priority : calculation of AV diquark part Improvement of our various components (ie propagators and amplitudes) Calculation of the Dirac form factor. Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
47 Addendum: Meson Form Factors and beyond Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
48 n = 1 Mellin Moment x 1 = 1 0 dx ϕ(x) ln [1 + κx(1 x)] 1 x φ ln (x) 1 κx(1 x) x(1 x) φ ln (x) (x(1 x)) ν x(1 x) x x 1 x 1 As Φ x Asymptotic Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
49 Form Factors Q 2 F (Q 2 ) = N [dx i ][dy i ]ϕ(x, ζ 2 x )T (x, y, Q 2, ζ 2 x, ζ 2 y )ϕ(y, ζ 2 y ) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
50 Form Factors Q 2 F (Q 2 ) = N LO Kernel and NLO kernels are known T 0 α S(µ 2 R ) (1 x)(1 y) [dx i ][dy i ]ϕ(x, ζ 2 x )T (x, y, Q 2, ζ 2 x, ζ 2 y )ϕ(y, ζ 2 y ) T 1 α2 s (µ 2 R ) (1 x)(1 y) (f UV (µ 2 R ) + f IR(ζ 2 ) + f finite ) R Field et al., NPB (1981) F. Dittes and A. Radyushkin, YF (1981) B. Melic et al., PRD (1999) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
51 Pion FF The UV scale dependent term behaves like: ( ( µ f UV (µ 2 2 )) R ) β 0 5/3 ln((1 x)(1 y)) + ln R Q 2 Here I take two examples: the standard choice of ζx 2 = ζy 2 = µ 2 = Q 2 /4 the regularised BLM-PMC scale ζ 2 x = ζy 2 = µ 2 = e 5/3 Q 2 /4 S. Brodsky et al., PRD (1983) S. Brodsky and L. Di Giustino, PRD (2011) Take the PDA model coming from the scalar diquark: φ(x) 1 ln [1 + κx(1 x)] κx(1 x) κ is fitted to the lattice Mellin Moment Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
52 Pion FF F1 Q² F0 Q² Standard Scale BLM Scale Q²F Q² LO Standard NLO Standard LO Reg. BLM NLO Reg. BLM Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
53 Pion FF F1 Q² F0 Q² Standard Scale BLM Scale Q²F Q² LO Standard NLO Standard LO Reg. BLM NLO Reg. BLM LO Effective Coupling Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
54 Pion FF The UV scale dependent term behaves like: ( ( µ f UV (µ 2 2 )) R ) β 0 5/3 ln((1 x)(1 y)) + ln R Q 2 Here I take two examples: the standard choice of ζx 2 = ζy 2 = µ 2 = Q 2 /4 the regularised BLM-PMC scale ζ 2 x = ζy 2 = µ 2 = e 5/3 Q 2 /4 S. Brodsky et al., PRD (1983) S. Brodsky and L. Di Giustino, PRD (2011) BLM scale reduces significantly the impact of the NLO corrections and increase dramatically the LO one. Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
55 DVMP LO Transition Form Factor x 1 ( )) µ 2 At NLO : g UV β 0 (5/3 ln((1 u)(1 v)) + ln R Q 2 Shape effects are also magnified D. Müller et al., Nucl.Phys. B884 (2014) Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
56 DVMP LO Transition Form Factor x 1 ( )) µ 2 At NLO : g UV β 0 (5/3 ln((1 u)(1 v)) + ln R Q 2 Shape effects are also magnified Bottom Line A good knowledge of the PDA is a key point to perform reliable extraction of GPDs though DVMP Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
57 DVMP LO Transition Form Factor x 1 ( )) µ 2 At NLO : g UV β 0 (5/3 ln((1 u)(1 v)) + ln R Q 2 Shape effects are also magnified Optimism This workshop shows that we are on good tracks to achieve it Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
58 Thank you for your attention Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
59 Back up slides Cédric Mezrag (INFN) Baryon DAs September 13 th, / 28
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