Pion photo- and electroproduction and the chiral MAID interface Stefan Scherer Institute for Nuclear Physics, Johannes Gutenberg University Mainz Uppsala University, 20 May 2016 work in collaboration with 1 Marius Hilt, Bj¨ orn C. Lehnhart, Lothar Tiator 1 Phys. Rev. C 87 , 045204 (2013), Phys. Rev. C 88 , 055207 (2013)
1. Introduction 2. Renormalization and power counting 3. Application to pion photo- and electroproduction 4. Summary and outlook
1. Introduction Effective field theory ... if one writes down the most general possible La- grangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S–matrix consistent with analyticity, perturbative unitar- ity, cluster decomposition and the assumed symmetry principles. ... 2 2 S. Weinberg, Physica A 96 , 327 (1979)
... if we include in the Lagrangian all of the infinite num- ber of interactions allowed by symmetries, then there will be a counterterm available to cancel every ultraviolet di- vergence. ... 3 3 S. Weinberg, The Quantum Theory of Fields , Vol. I, 1995, Chap. 12
Fundamental theory Effective field theory QCD ChPT dof quarks & gluons Goldstone bosons (+ other hadrons) parameters g 3 + quark masses ( ∞ # of ) LECs + quark masses
Simplified analogies between multipole expansion and EFT Multipole expansion EFT | � x | ≫ R q ≪ Λ χ x ) = � L eff = � Y lm ( θ,φ ) 1 φ ( � q lm c lm L lm lm lm 2 l +1 r l +1 multipole moment q lm LEC c lm Y lm ( θ,φ ) 1 Structures L lm 2 l +1 r l +1 • In principle, infinite number of terms. Actual calculation: Truncation at finite order. • Systematic improvement possible.
Perturbative calculations in effective field theory require two main ingredients 1. Knowledge of the most general effective Lagrangian (a) Mesonic ChPT [SU(3) × SU(3)] 4 ( π, K, η ) 2 + 10 + 2 + 90 + 4 + 23 + . . . ���� � �� � � �� � O ( q 2 ) O ( q 4 ) O ( q 6 ) – q : Small quantity such as a pion mass – Even powers – Two-loop level 4 Gasser, Leutwyler (1985), Fearing, Scherer (1996), Bijnens, Colangelo, Ecker (1999), Ebertsh¨ auser, Fearing, Scherer (2002) Bijnens, Girlanda, Talavera (2002)
(b) Baryonic ChPT [SU(2) × SU(2) × U(1)] 5 ( π, N ) 2 + 7 + 23 + 118 + . . . ���� ���� ���� ���� O ( q 2 ) O ( q 3 ) O ( q 4 ) O ( q ) – Odd and even powers (spin) – One-loop level Each term comes with an independent low-energy constant (LEC) Lowest-order Lagrangians: F , M 2 = 2 B ˆ m , m , g A Higher-order Lagrangians: l i , c i , d i , e i , . . . 5 Gasser, Sainio, ˇ Svarc (1988), Bernard, Kaiser, Meißner (1995), Ecker, Mojˇ ziˇ s (1996), Fettes, Meißner, Mojˇ ziˇ s, Steininger (2000)
2. Consistent expansion scheme for observables (a) Tree-level diagrams, loop diagrams � ultraviolet diver- gences, regularization (of infinities) (b) Renormalization condition (c) Power counting scheme for renormalized diagrams (d) Remove regularization ChPT: Momentum and quark mass expansion at fixed ratio m quark /q 2 6 6 J. Gasser and H. Leutwyler, Annals Phys. 158 , 142 (1984)
2. Renormalization and power counting • Most general Lagrangian L eff = L π + L πN = L (2) + L (4) + . . . + L (1) πN + L (2) πN + . . . π π Basic Lagrangian � � g A Ψ − 1 L (1) iγ µ ∂ µ − m Ψ γ µ γ 5 τ a ∂ µ π a Ψ + · · · πN = ¯ ¯ Ψ 2 F m , g A , and F denote the chiral limit of the physical nucleon mass, the axial-vector coupling constant, and the pion-decay constant, respectively
• Power counting: Associate chiral order D with a diagram – Square of the lowest-order pion mass: M 2 = B ( m u + m d ) ∼ O ( q 2 ) – Nucleon mass in the chiral limit m ∼ O ( q 0 ) – Loop integration in n dimensions ∼ O ( q n ) – Vertex from L (2 k ) ∼ O ( q 2 k ) π – Vertex from L ( k ) πN ∼ O ( q k ) – Nucleon propagator ∼ O ( q − 1 ) – Pion propagator ∼ O ( q − 2 )
• Renormalization – Regularize (typically dimensional regularization) � d n k i I ( M 2 , µ 2 , n ) = µ 4 − n k 2 − M 2 + i 0 + (2 π ) n � � �� = M 2 M 2 R + ln + O ( n − 4) , 16 π 2 µ 2 2 n − 4 − [ ln (4 π ) + Γ ′ (1)] − 1 → = ∞ R – Adjust counterterms such that they absorb all the diver- gences occurring in the calculation of loop diagrams – Renormalization prescription: Adjust finite pieces such that renormalized diagrams satisfy a given power counting
k 1 1 p p p � k � • Example: Contribution to nucleon mass Goal: D = n · 1 − 2 · 1 − 1 · 1 + 2 · 1 = n − 1
� � Σ = − 3 g 2 p + m ) I N + M 2 ( / A 0 ( / p + m ) I Nπ ( − p, 0) + · · · 4 F 2 0 Apply � MS renormalization scheme Σ r = − 3 g 2 � = � M 2 ( / I r Ar O ( q 2 ) p + m ) Nπ ( − p, 0) + . . . 4 F 2 � �� � r 1 = − 16 π 2 + . . . GSS 7 : It turns out that loops have a much more complicated low-energy structure if baryons are included. Because the nu- cleon mass m N does not vanish in the chiral limit, the mass scale m (nucleon mass in the chiral limit) occurs in the effec- tive Lagrangian L (1) πN ... . This complicates life a lot. 7 J. Gasser, M. E. Sainio, A. ˇ Svarc, Nucl. Phys. B307 , 779 (1988)
One possible solution: Extended on-mass-shell (EOMS) scheme 8 Main idea: Perform additional subtractions such that renormal- ized diagrams satisfy the power counting Motivation for this approach 9 Terms violating the power counting are analytic in small quan- tities (and can thus be absorbed in a renormalization of coun- terterms) • Example (chiral limit) � d n k i H ( p 2 , m 2 ; n ) = − [( k − p ) 2 − m 2 + i 0 + ][ k 2 + i 0 + ] (2 π ) n 8 T. Fuchs, J. Gegelia, G. Japaridze, S. Scherer, Phys. Rev. D 68 , 056005 (2003) 9 J. Gegelia and G. Japaridze, Phys. Rev. D 60 , 114038 (1999)
Small quantity ∆ = p 2 − m 2 = O ( q ) m 2 We want the renormalized integral to be of order D = n − 1 − 2 = n − 3 Result of integration H ∼ F ( n, ∆) + ∆ n − 3 G ( n, ∆) • F and G are hypergeometric functions • analytic in ∆ for arbitrary n
Observation 10 F corresponds to first expanding the integrand in small quanti- ties and then performing the integration ⇒ Algorithm: Expand integrand in small quantities and subtract those (integrated) terms whose order is smaller than suggested by the power counting 10 J. Gegelia, G. Japaridze, K. S. Turashvili, Theor. Math. Phys. 101 , 1313 (1994)
Here: � � d n k � i H subtr = − � � ( k 2 − 2 k · p + i 0 + )( k 2 + i 0 + ) (2 π ) n p 2 = m 2 1 = − 2¯ λ + 16 π 2 + O ( n − 4) where � �� λ = m n − 4 n − 4 − 1 1 � ln(4 π ) + Γ ′ (1) + 1 ¯ (4 π ) 2 2 H R = H − H subtr = O ( q n − 3 )
Chiral versus loop expansion ππ : MS πN : MS πN : EOMS D D D ✻ ✻ ✻ 6 6 6 ✈ ✈ ✈ ✈ ✈ ✈ ✈ 5 5 ✈ ✈ ✈ ✈ ✈ ✈ 4 4 4 ✈ ✈ ✈ ✈ ✈ ✈ 3 3 ✈ ✈ ✈ ✈ ✈ 2 2 2 ✈ ✈ ✈ ✈ ✈ 1 1 ✈ ✈ ✲ ✲ ✲ 0 1 2 N L 0 1 2 N L 0 1 2 N L
3. Application to pion photo- and electroproduction e ( k i ) + N ( p i ) → e ( k f ) + N ( p f ) + π ( q ) One-photon-exchange approximation e ( k f ) e ( k i ) γ ∗ ( k ) π ( q ) N ( p f ) N ( p i )
Invariant amplitude M = leptonic vertex × i propagator × hadronic vertex = ǫ µ M µ ǫ µ = e ¯ u ( k f ) γ µ u ( k i ) M µ = − ie � N ( p f ) , π ( q ) | J µ (0) | N ( p i ) � , k 2 Current conservation k µ M µ = 0 Parameterization in terms of six invariant amplitudes � � 6 � M µ = ¯ A i ( s, t, u ) M µ u ( p f , s f ) u ( p i , s i ) , u ( p, s ) : Dirac spinor i i =1 Mandelstam variables s = ( p i + k ) 2 , t = ( p i − q ) 2 , u = ( p i − p f ) 2 , Q 2 = − k 2 s + t + u = 2 m 2 N + M 2 π − Q 2 ,
Lorentz structures 1 = − i M µ 2 γ 5 ( γ µ / kγ µ ) , k − / � � � � � � q − 1 q µ − 1 M µ P µ k · 2 k µ 2 = 2 iγ 5 2 k − k · P , M µ 3 = − iγ 5 ( γ µ k · q − / kq µ ) , M µ kP µ ) − 2 m N M µ 4 = − 2 iγ 5 ( γ µ k · P − / 1 , � k µ k · q − q µ k 2 � M µ 5 = iγ 5 , � kk µ − γ µ k 2 � M µ / 6 = − iγ 5 , where P = 1 2( p i + p f ) Current conservation k µ M µ i = 0 , i = 1 , . . . , 6 .
cm frame: � k = − � p i , � q = − � p f ✒ π ( � q ) � � � � � γ ∗ ( � � k ) Θ π � ✲ � ✛ � � N ( � p i ) � � � N ( � p f ) � � � ✠ 6 � u ( p i , s i ) = 4 πW A i M µ χ † M = ǫ µ ¯ u ( p f , s f ) f F χ i i m N i =1 • χ : Pauli spinor, W = √ s • Gauge transformation ( � longitudinal multipoles) a µ = ǫ µ − k µ ǫ 0 k 0
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