Electromagnetic Interactions in a Chiral Effective Lagrangian for Nuclei Brian D. Serot Indiana University ² Outline ¤ Motivation for Lorentz-Covariant EFT ¤ EFT Lagrangian ¤ Chiral Symmetry Transformations ¤ DFT for Nuclei: Strategy and Justification ¤ Isovector Vector and Axial-Vector Currents ¤ Electromagnetic Interactions Supported in part by the NSF and DOE Collaborators 1993–2005 Sergei Ananyan Dick Furnstahl Carrie Halkyard Ying Hu Jeff Mcintire Horst Muller¨ Jorge Piekarewicz John Rusnak Huabin Tang Dirk Walecka Why Use Hadrons? ² We focus on low-energy, long-range physics, and all observables are colorless. ² Hadrons (baryons and mesons) are the actual particles observed in experiments. ² Colored quarks and gluons participate only in intermediate states, and such “off-shell” behavior is unobservable. ² So pick the most efficient degrees of freedom! We have to parametrize the hamiltonian anyway, since we don’t know its true form. ² We are interested primarily in the nuclear many-body problem, so include “collective” degrees of freedom like scalar and vector fields. ² Only nucleons and pions are “real” (stable) particles; other fields are always virtual and just parameterize the NN interaction (or EM form factors). Why Impose Lorentz Covariance? ² The scalar and vector mean fields in nuclei are large (several hundred MeV). This is a new energy scale. The scalar and vector fields cancel to produce a small binding energy. ¤ Consistent with QCD sum-rule results (size and density dependence). ¤ Consistent with chiral power counting (two-body energy/nucleon 2 is of order ½0=f¼). ² Large mean fields produce important relativistic interaction effects. ¤ Velocity-dependent NN interaction provides a new saturation mechanism. ¤ Scalar and vector mean fields add to produce correct spin-orbit force. (Compare “fine” structure in atoms and nuclei.) ¤ Successful prediction of nucleon–nucleus spin observables in the RIA and energy dependence of the optical potential. ¤ Explains pseudospin symmetry in nuclei. There really is relativity in nuclei! Why Use Effective Field Theory? Lorentz-covariant hadronic field theories ´ Quantum HadroDynamics ² Interpret QHD lagrangians as nonrenormalizable LEFT’s ¤ known long-range interactions constrained by symmetries; ¤ a complete set of generic short-range interactions; ¤ the borderline is characterized by breakdown scale ¤ of EFT. For QHD, ¤ ¼ 600 MeV (empirically). ² When based on a local, Lorentz-invariant lagrangian density, EFT is the most general way to parameterize observables consistent with the principles of quantum mechanics, special relativity, unitarity, cluster decomposition, microscopic causality, and the desired internal symmetries. ² It’s not necessary to derive L from QCD ¤ Use a general L that respects the symmetries. ¤ By construction, this provides a general parametrization for < energies » ¤ (remove redundancies). ² The freedom to redefine and transform the fields =) infinitely many representations of low-energy QCD physics Advantages for Electroweak Interactions in Nuclei ² We use the same degrees of freedom to describe the nuclear interactions and nuclear currents. ² The QHD/EFT lagrangian exhibits the symmetries of QCD. ¤ Chiral SU(2)L £ SU(2)R symmetry is nonlinear. ¤ Unbroken isovector subgroup SU(2)V symmetry is linear. ¤ These are global symmetries. ¤ Electromagnetic interactions [local U(1)Q gauge symmetry] is straightforward to include. ² At momenta small compared to ¤, short-distance physics (like nucleon substructure) is only partially resolved and can be described with a derivative expansion. ² The lagrangian parameters can be calibrated to ¤ properties of isolated hadrons, ¤ ¼N scattering, ¤ bulk and single-particle properties of nuclei, ¤ NN scattering. Strategy ² Assign an index to each term in the lagrangian: º = d + n=2 + b. ¤ d = number of derivatives (except on nucleons). ¤ n = number of nucleon fields. ¤ b = number of non-Goldstone bosons. ² Organize L in powers of º and truncate; this gives a reliable expansion in inverse powers of a “heavy” mass scale ¤ ¼ M . For heavy nuclei, essentially an expansion in kF=M . Fields ² Nucleon (N ), Lorentz scalar (Á = “sigma”) [chiral scalar] ² Lorentz vector (V¹ = “omega” ; V¹º ´ @¹Vº ¡ @º V¹) [ " ] ² Pion: U ´ exp(i¿ ¢¼=f¼) ;» ´ exp(i¿ ¢¼ =2f¼) ; ¡ ¢ i y y together with a¹ ´ ¡ 2 » @¹» ¡ »@¹» ; ¡ ¢ i y y v¹ ´ ¡ 2 » @¹» + »@¹» ; v¹º ´ ¡i[a¹; aº ] : 1 ² Rho: ½¹ ´ 2 ¿ ¢½¹ ;D¹½º ´ @¹½º + i[v¹; ½º ] ; ½¹º = D¹½º ¡ Dº ½¹ + i g½[½¹; ½º ] : EFT Lagrangian (4) LQHD = LN + L¼N + LM ¹ ¹ = N (iγ [D¹ + ig½½¹ + igvV¹] + gA γ γ5a¹ ¡ M + gsÁ) N f g f g · ¡ ½ ½ N ½ σ¹º N ¡ v v NV σ¹º N ¡ ¼ N v σ¹º N 4M ¹º 4M ¹º M ¹º 4¯ + ¼ NN Tr (a a¹) + L(4) M ¹ ¼N ¡ ¢ 1 ¹ 1 2 ¹ y + 2 @¹Á@ Á + 4 f¼ Tr @¹U@ U 1 ¹º 1 ¹º ¡ 2 Tr (½¹º ½ ) ¡ 4 V¹º V 2 µ 2 2 ¶ 2f¼ ¹º 1 gsÁ ´2 gs Á 2 ¹ ¡g½¼¼ 2 Tr (½¹º v )+ 2 1 + ´1 + 2 mvV¹V m½ M 2 M µ ¶ g Á + 1 ³ g2(V V ¹)2 + 1 + ´ s m2 Tr (½ ½¹) 4! 0 v ¹ ½ M ½ ¹ µ ¶ 1 · g Á · g2Á2 ¡m2Á2 + 3 s + 4 s : s 2 3! M 4! M 2 ² LQHD contains all nonredundant terms through order º = 4. ² We see standard noninteracting hadron terms © Yukawa nucleon–meson couplings © anomalous-moment interactions © pion–nucleon and meson nonlinearities: nontrivial dynamics. Transformation Laws In our EFT (QHD) lagrangian: ² Chiral SU(2)L £ SU(2)R symmetry is nonlinear. ² Isovector subgroup SU(2)V symmetry is linear. ² These are global symmetries. ² Vector transformations: L = exp(i¯ ¢ ¿ =2) = R ² Axial-vector transformations: L = exp(i® ¢ ¿ =2) ;R = exp(¡i® ¢ ¿ =2) ² Field transformations: (all objects are matrices) U(x) ! LU(x)Ry ; »(x) ! L»(x)hy(x) = h(x)»(x)Ry [defines h(x)] N(x) ! h(x)N(x) [generally; h(x) is local] y ½¹(x) ! h(x)½¹(x)h (x) : ² Chirally covariant derivatives: D¹N ´ (@¹ + iv¹)N : D¹N ! h(x)(D¹N) ; y D¹½º ´ @¹½º + i[v¹; ½º ]: D¹½º ! h(x)(D¹½º )h (x) Discussion ² To realize the nonlinear SU(2)L £ SU(2)R symmetry, the lagrangian must include pions explicitly. ² Note that U , », and ½¹ are 2 £ 2 matrices. ² For isospin transformations, L = R = h (constants); the transformations are linear. ² For general transformations: L 6= R. [Axial transformations have L = Ry.] ¤ Now h(x) is nontrivial and contains pion fields. ¤ So h(x)N(x) mixes nucleons with any number of pions: the transformation is nonlinear. ² The only field or tensor that transforms inhomogeneously is y y v¹ ! hv¹h ¡ ih@¹h . This allows for the construction of chirally covariant derivatives. ² This is NOT the linear sigma model; the scalar field Á is a chiral scalar. It is NOT the chiral partner of the pion. Important Things to Remember ² Off-shell behavior is not observable. Choose the dynamical variables that are most efficient (still unknown). ² Vacuum dynamics involves short-range physics. Don’t calculate it, but parametrize it in a few fitted constants. (Computation of hadronic loops =) unnatural coefficients.) Use valence nucleons only. ² Although fields and couplings are local, nucleon substructure is also included: ¤ Example: NNσ ! g(σ)NNσ ¤ But define: Á ´ g(σ)σ ; [g(0) = 1]; then invert for σ(Á). ¤ Then: g(σ)NNσ + p(σ) = NNÁ + aÁ2 + bÁ3 + cÁ4 + ¢ ¢ ¢ ² Nucleon EM structure included in a derivative expansion: e e L = ¡ NA¹γ (1 + ¿ )N ¡ F ¹º Nf¸(0) +¸(1)¿ gσ N EM 2 ¹ 3 4M 3 ¹º e + ¢ ¢ ¢ ¡ @ F ¹º N(f¯(0) +¯(1)¿ gγ )N 2M 2 º 3 ¹ e ¡ @2@ F ¹º N(f±(0) + ±(1)¿ gγ )N + ¢ ¢ ¢ + VMD ; M 4 º 3 ¹ s;v which generates e, ¸, rrms, ::: This works at long distances (low momenta). ² This Electromagnetic lagrangian is valid to lowest order in the electric charge and the pion fields. When combined with the VMD terms, we find the following results for the nucleon EM form factors (Q ´ four-momentum transfer): (0) 2 2 (0) 2 1 ¯ Q gv Q F1 (Q ) = ¡ 2 ¡ 2 2 + ¢ ¢ ¢ 2 2 M 3gγ Q + mv (1) 2 2 (1) 2 1 ¯ Q g½ Q F1 (Q ) = ¡ 2 ¡ 2 2 + ¢ ¢ ¢ 2 2 M 2gγ Q + m½ 2 (0) 2 ¸p + ¸n fvgv Q F2 (Q ) = ¡ 2 2 + ¢ ¢ ¢ 2 3gγ Q + mv 2 (1) 2 ¸p ¡ ¸n f½g½ Q F2 (Q ) = ¡ 2 2 + ¢ ¢ ¢ 2 2gγ Q + m½ ² Using the single-nucleon charge and anomalous rms radii allows (t) one to determine the ¯ and fi parameters. (The constants gv and g½ are determined from nuclear properties, and gγ is fit to the leptonic ½ ! e+e¡ decay.) ² These expressions allow one to describe the contributions of nucleon EM structure to nuclear charge form factors without introducing ad hoc form factors at the NNγ vertex. (Naive) Dimensional Analysis: NDA [Georgi & Manohar, 1984] ² Low-energy QCD is expected to contain two mass scales: f¼ ¼ 93 MeV ; ¤ ¼ 500 to 800 MeV: The first is related to chiral symmetry and resulting Goldstone bosons (pions); the second signals non-Goldstone-boson (“heavy”) physics. ² NDA rules for a generic term in the energy functional: " ! ! ! ! # ` m n p 2 2 NN 1 © 1 W @ C [f¼ ¤ ] 2 f¼ ¤ m! ¤ n! ¤ ¤ ² “Naturalness” =) dimensionless C is of order unity. ² Provides expansion parameters at finite density: © W ½s ½B 0 ¼ ¼ 1=2 ; 2 ¼ 2 ¼ 1=5 at ½B ¤ ¤ f¼¤ f¼¤ ² Allows truncation and calibration with quantitatively accurate fits to bulk and single-particle nuclear observables (binding energies, charge densities, single-particle spectra near the Fermi surface). Density Functional Theory ² Construct the ground-state energy functional from the lagrangian using a mean-field (”factorized”) approximation: ¤ A functional of scalar (½s) and baryon (½B) densities. ¤ Lorentz scalar and vector fields are interpreted as Kohn–Sham single-particle potentials. Dirac (quasi)nucleons move in these local potentials. ² Kohn–Sham theorem [1965]: The exact ground-state scalar and vector densities, energy, and chemical potential for the fully interacting many-fermion system can be reproduced by a collection of (quasi)fermions moving in appropriately defined, self-consistent, local, classical fields.