QCD Evolution 2019 3-D STRUCTURE OF THE PION AND KAON FROM QCD'S DYSON-SCHWINGER EQUATIONS. Chao Shi Argonne National Laboratory 2019.05.14@ANL � 1
TMD PDFs The TMD PDFs are defined with correlation function with finite transverse separation Z d ξ − d 2 ξ ⊥ e i ( k + ξ − − k ⊥ · ξ ⊥ ) h P, S | ψ j (0) U n − � (0 , + ∞ ) U n − Φ ij ( x, k ⊥ , S ) = (+ ∞ , ξ ) ψ i ( ξ ) | P, S i ξ + =0 , � (2 π ) 3 � The TMD PDFs enter the general decomposition of the correlation function. ( ✏ ij ⊥ S j T k i 1 n + + ( k ⊥ · S ⊥ ) [/ S ⊥ , / n + ] +... n + − f ⊥ ⊥ Φ ( x, k ⊥ , S ) = f 1 / / n + + Λ g 1 L � 5 / g 1 T � 5 / n + + h 1 T � 5 1 T 2 M M 2 � [/ k ⊥ , / n + ] � 5 + ( k ⊥ · S ⊥ ) [/ k ⊥ , / n + ] [/ k ⊥ , / n + ] + Λ h ⊥ h ⊥ � 5 + ih ⊥ , 1 L 1 T 1 2 M 2 M 2 M M SIDIS Factorization TMDs Factorization DY � 2
<latexit sha1_base64="AJOo9fVXKcLQLEK0hqhbk6fdkHo=">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</latexit> <latexit sha1_base64="MKRGH2EaGKrqaTA8b+34JTQewxg=">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</latexit> <latexit sha1_base64="TFUfWUNOf/3acZMNCewjT+L8uk=">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</latexit> GPDs The generalized parton distribution introduces a finite momentum transfer Δ to the parent hadron, i.e, . ∆ 6 = 0 Z dz − � 2 π e ixP + z − h P � ∆ ψ i (0) γ + ψ j ( z ) | P + ∆ 2 | ¯ � 2 i � � z + = z ⊥ =0 ✓ ◆ ✓ ◆ 2 ) i σ + µ ∆ µ ✓ ◆� 1 P + ∆ P − ∆ u ( P + ∆ P − ∆ γ + u = H q ( x, ξ , t )¯ + E q ( x, ξ , t )¯ u u 2 2 2 M 2 P + GPDs show up in the factorization of DVCS et al. It encodes important information of hadrons, e.g., AM decomposition and spatial density distribution in the transverse plane. GPDs AM(Ji 1997) Factorization Deeply virtual Compton scattering IPD GPD (M. Burkardt 2000) � 3
Nonperturbative QCD (starting point of evolution) QCD Nonperturbative QCD methods 1. ADS/QCD 2. Dyson-Schwinger equations. 3. E ff ective theories and models, e.g., NJL model... 4. Light front QCD. 5. Lattice QCD. etc... Transverse momentum dependent distributions (TMD) 3-D tomography in the momentum space. Generalized parton distributions (GPD) 3-D picture of hadrons in the mixed spatial-momentum space. � 4
� � � � � QCD's DSEs Dyson-Schwinger equations: general relations between Green functions in quantum field theories. The connected 4-quark scattering amplitude satisfies the Dyson equation ✓ Quantum Field Theory ✓ Path Integral formulation 𝑈 𝐿 𝐿 𝑈 = + Non-perturbative 𝑄� ⟶ �𝑁� 𝑈 𝛺 𝛺 ⟹ 𝛺 𝐿 𝛺 Quark DSE: 𝑈 𝐿 𝐿 𝑈 Near mass pole, the quark scattering amplitude is dominated by hadron's Bethe-Salpeter WF . i Σ = i D 𝑄� ⟶ �𝑁� 𝑈 𝛺 𝛺 ⟹ 𝛺 𝐿 𝛺 γ i Γ 𝑈 𝐿 i S 𝐿 i 𝑈 = + b) Meson Bethe-Salpeter equation. i i S S Σ 0 0 i S i i S 𝑄� ⟶ �𝑁� 𝑈 𝛺 𝛺 ⟹ 𝛺 𝐿 𝛺 = � 5 on �
Chiral symmetry and AVWTI The hadron wave function can be solved by aligning the quark DSE and hadron BSE. D µ ν ( p − q ) q + p + p + P P Γ Γ K = − 1 − 1 + = Γ µ p − q − p − γ ν S 0 ( p ) S ( p ) S ( q ) To solve these equations, truncation is needed for the vertex and scattering kernel. A physically reasonable truncation scheme should preserve QCD's (nearly) chiral symmetry by respecting the Axial-Vector Ward-Takahashi Identity -1 -1 i γ 5 + i γ 5 − i ( m f + m g ) P µ Γ 5 µ = Γ 5 The AVWTI relates the vertex Γ and kernel K 𝑞� �𝑞 𝑞� �𝑞 � � � � � � 𝑞� �𝑞 𝑞� �𝑞 � � The simplest is the Rainbow-Ladder truncation γ µ λ a D ab γ µ λ a Γ a AND = = K µ ν µ γ ν λ b � 6
<latexit sha1_base64="s50cy/oHEd/8KrqyzZ+rmlhS/g=">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</latexit> DSEs highlights and status Dynamical chiral symmetry breaking Z ( p 2 ) S ( p ) = i/ p + M ( p 2 ) m=0 M>>0 (M.S. Bhagwat et al, PRC2003) Hadron spectrum Pieter Maris and Craig D. Roberts, PRC 1997 Gernot Eichmann, PRL 2010 Jorge Segovia, et al, PRL 2015 G Eichmann, C S. Fischer, W Heupel , PLB 2016 tetraquark hybrid Shu-Sheng Xu, eta al 2018 Form factors and parton distribution Pieter Maris and Peter Tandy, PRC 2000, 2002, Lei Chang, et al PRL 2013, G Eichmann PRD2011 Elastic and transition form factor Parton distribution amplitude Lei Chang, et al PRL 2013, Ian Cloet, et al, PRL 2013, Parton distribution function Chao Shi et al PLB 2014, Cedric Mezrag, et al PLB 2019 GPD & TMD Trang Nguyen, et al PRD 2011, Kyle Bednar, et al PRL (in review) 2019 (Cedric Mezrag et al PLB 2015, Chao Shi, et al PRL 2019) � 7
<latexit sha1_base64="/XgVOICXRIzoRDGSpZhy51cqSM=">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</latexit> TMDs & GPDs: Covariant approach Why pion and kaon? Pion (and kaon) has the dual roles of being both a QCD bound state and also the Goldstone boson of DCSB. DCSB contributes 99% mass in visible universe. The massness of proton and masslessness of pion are closely related and both deserve studying. Pion (and kaon) is among the few hadrons whose parton structure can be experimentally measured, through, e.g., Drell-Yan and Sullivan process (o ff -shell pion). ` 0 ✓ F π ( x, Q 2 ) ` Pion also enters the description of nucleon by meson cloud. For a � P X q quark-core nucleon, pion cloud reduces its mass by ~20%, X modifies nucleon's EM radius, and provides the sea quark content ⇡ + proton u p ( x ) 6 = ¯ hence its asymmetry. ¯ d p ( x ) neutron Sullivan process + Theoretically, pion and kaon have been well studied in DSEs, there is no free parameter, the TMDs and GPDs pose new challenge. � 8 � 8
TMDs & GPDs: Covariant approach p + = P DSEs: & b) Γ i S p − Covariant approach: Compute the triangle diagrams in terms of fully covariant propagators/vertices with appropriate truncations. Light-front approach: Extract from pion's Bethe-Salpeter wave functions the LFWFs and calculate TMDs and GPDs using overlap representation. Impulse Approximation: TMD & GPD Γ Γ � 9
TMDs & GPDs: Covariant approach TMDs & GPDs: Light-front approach p + = P DSEs: & b) Γ i S p − Covariant approach: Compute the triangle diagrams in terms of fully covariant propagators/vertices with appropriate truncations. Light-front approach: Extract from pion's Bethe-Salpeter wave functions the LFWFs and calculate TMDs and GPDs using overlap representation. TMD & GPD � 10 � 10
Recommend
More recommend
