Ab-initio methods for light nuclei from low to high resolution James P. Vary Iowa State University, Ames, Iowa, USA Polarized light ion physics with EIC Ghent, Belgium Feb. 5 – 9, 2018 Meeting Topics include: * Neutron spin structure from polarized deep-inelastic scattering on light nuclei (d, 3He) * Nuclear fragmentation and final-state interactions in high-energy processes * Spin-orbit effects and azimuthal asymmetries in scattering on proton and light nuclei * Tensor-polarized deuteron in low- and high-energy processes * Theoretical methods for light nuclear structure: Few-body, Lattice, Light-front * Nuclear structure at variable scales: Effective degrees of freedom, EFT methods * Quarks and gluons in light nuclei: EMC effect, non-nucleonic degrees of freedom * Diffraction and nuclear shadowing in DIS on light nuclei * Polarized light ion beams: Sources, acceleration, polarimetry * Forward detection of spectators and nuclear fragments at EIC
Hot and/or dense quark-gluon matter Quark-gluon percolation QCD Resolution Hadron structure quark models Hadron-Nuclear interface ab initio Effective Field Theory CI Nuclear structure Nuclear reactions DFT Third Law of Progress in Theoretical collective Nuclear astrophysics Physics by Weinberg: and “ You may use any degrees of Applications of nuclear science freedom you like to describe a algebraic physical system, but if you use the models wrong ones, you’ll be sorry! ” Adapted from W. Nazarewicz
Effective Nucleon Interaction (Chiral Perturbation Theory) Chiral perturbation theory ( χ PT) allows for controlled power series expansion υ ⎛ ⎞ Expansion parameter : Q ⎜ ⎟ , Q − momentum transfer, ⎜ ⎟ Λ χ ⎝ ⎠ Λ χ ≈ 1 GeV , χ - symmetry breaking scale Within χ PT 2 π -NNN Low Energy Constants (LEC) are related to the NN-interaction LECs {c i }. C D C E Terms suggested within the Chiral Perturbation Theory Regularization is essential, which is also implicit within the Harmonic Oscillator (HO) wave function basis (see below) R. Machleidt and D.R. Entem, Phys. Rep. 503, 1 (2011); R. Machleidt, D. R. Entem, nucl-th/0503025 E. Epelbaum, H. Krebs, U.-G Meissner, Eur. Phys. J. A51, 53 (2015); Phys. Rev. Lett. 115, 122301 (2015)
No-Core Configuration Interaction calculations Barrett, Navrátil, Vary, Ab initio no-core shell model , PPNP69, 131 (2013) Given a Hamiltonian operator p j ) 2 ( ⃗ p i − ⃗ ˆ � � � = + V ij + V ijk + . . . H 2 m A i<j i<j i<j<k solve the eigenvalue problem for wavefunction of A nucleons ˆ H Ψ ( r 1 , . . . , r A ) = λ Ψ ( r 1 , . . . , r A ) Expand wavefunction in basis states | Ψ ⟩ = � a i | Φ i ⟩ Expand eigenstates in basis states Diagonalize Hamiltonian matrix H ij = ⟨ Φ j | ˆ H | Φ i ⟩ No Core Full Configuration (NCFC) – All A nucleons treated equally No-Core CI: all A nucleons are treated the same Complete basis − → exact result In practice truncate basis study behavior of observables as function of truncation Progress in Ab Initio Techniques in Nuclear Physics, Feb. 2015, TRIUMF , Vancouver – p. 2/50
Basis expansion Ψ ( r 1 , . . . , r A ) = � a i Φ i ( r 1 , . . . , r A ) Many-Body basis states Φ i ( r 1 , . . . , r A ) Slater Determinants ( ) with α = ( n , l , s , j , m j ) φ α r Single-Particle basis states φ ik ( r k ) quantum numbers n , l , s , j , m j k Harmonic Oscillator (HO), natural orbitals, Radial wavefunctions: Harmonic Oscillator, Woods-Saxon, Coulomb-Sturmian, Complex Scaled HO, Berggren,. . . Wood–Saxon, Coulomb–Sturmian, Berggren (for resonant states) M -scheme: Many-Body basis states eigenstates of ˆ J z A � ˆ J z | Φ i ⟩ = M | Φ i ⟩ = m ik | Φ i ⟩ k =1 N max truncation: Many-Body basis states satisfy A N max runs from zero ∑ ( ) α ≤ N 0 + N max 2 n + l � � � 2 n ik + l ik ≤ N 0 + N max to computational limit. ! Ω ( N max , ) fix HO basis α occ. k =1 Alternatives: Full Configuration Interaction (single-particle basis truncation) Importance Truncation Roth, PRC79, 064324 (2009) No-Core Monte-Carlo Shell Model Abe et al , PRC86, 054301 (2012) SU(3) Truncation Dytrych et al , PRL111, 252501 (2013) Progress in Ab Initio Techniques in Nuclear Physics, Feb. 2015, TRIUMF , Vancouver – p. 3/50
Calculation of three-body forces at N 3 LO Goal Low Energy Calculate matrix elements of 3NF in a partial- Nuclear wave decomposed form which is suitable for Physics different few- and many-body frameworks International Collaboration Challenge J. Golak, R. Skibinski, K. Tolponicki, H. Witala Due to the large number of matrix elements, E. Epelbaum, H. Krebs the calculation is extremely expensive. A. Nogga R. Furnstahl Strategy S. Binder, A. Calci, K. Hebeler, Develop an efficient code which allows to J. Langhammer, R. Roth treat arbitrary local 3N interactions. P . Maris, J. Vary (Krebs and Hebeler) H. Kamada
Initial LENPIC Collaboration results: Chiral NN results for 6 Li by Chiral order Orange: Chiral order uncertainties; Blue/Green: Many-body method uncertainties S. Binder, et al, Phys. Rev. C 93, 044002 (2016); arXiv:1505.07218
0 Experimental data -10 + , 1/2) π , T) = (1/2 2 LO chiral NN potential (J LO, NLO, and N Chiral truncation uncertainty estimate -20 + , 0) + , 1) (0 (0 + , 0) (1 Ground state energy (MeV) -30 + , 2) (0 -40 - , 1/2) + , 1) (3/2 (2 - , 3/2) -50 (3/2 -60 + , 0) - , 1/2) (0 (3/2 -70 + , 0) (3 -80 -90 3 H 4 He 7 Li 8 He 8 Li 8 Be 9 Li 9 Be 10 B 6 He 6 Li S. Binder, et al., LENPIC Collaboration, in preparation
Preliminary LENPIC results with Chiral NN only and R = 1.0 fm, IA for operator S. Binder, et al., LENPIC Collaboration, in preparation Good chiral convergence and all are close to expt, where available 4 3 2 magnetic moment 1 0 -1 -2 3 H 3 He 6 Li 7 Li 7 Be 8 Li 8 B 9 Li 9 Be 9 B 9 C 10 B
Dirac’s forms of relativistic dynamics [Dirac, Rev. Mod. Phys. 21 , 392 1949] Dirac’s Forms of Relativistic Dynamics [Dirac, Rev.Mod.Phys. ’49] Instant form is the well-known form of dynamics starting with x 0 = t = 0 K i = M 0 i , J i = 12 ε ijk M jk , ε ijk = (+1,-1,0) for (cyclic, anti-cyclic, repeated) indeces Front form defines QCD on the light front (LF) x + , t + z = 0 . Front form defines relativistic dynamics on the light front (LF): x + = x 0 +x 3 = t+z = 0 P ± , P 0 ± P 3 , ~ P ⊥ , ( P 1 , P 2 ) , x ± , x 0 ± x 3 , ~ x ⊥ , ( x 1 , x 2 ) , E i = M + i , E + = M + − , F i = M − i , K i = M 0 i , J i = 1 2 ✏ ijk M jk . instant form front form point form x + , x 0 + x 3 √ t = x 0 ⌧ , time variable t 2 − ~ x 2 − a 2 quantization surface P − , P 0 − P 3 P µ H = P 0 Hamiltonian P, ~ ~ P ⊥ , P + , ~ ~ J, ~ ~ J − E ⊥ , E + , J z J K kinematical ~ ~ ~ F ⊥ , P − K, P 0 P, P 0 dynamical p 0 = p − = ( ~ p µ = mv µ ( v 2 = 1 ) dispersion p 2 + m 2 p p 2 ⊥ + m 2 ) /p + ~ relation Adapted from talk by Yang Li
Discretized Light Cone Quantization Pauli & Brodsky c1985 Basis Light Front Quantization* φ * + + f α [ ] ∑ ( ) = ( ) a α ( ) a α Operator-valued x f α x x distribution function α { } satisfy usual (anti-) commutation rules. where a α ( ) are arbitrary except for conditions: Furthermore, f α x * ∫ ( ) f α ' ( ) d 3 x = δ αα ' Orthonormal: f α x x * = δ 3 x − ∑ ( ) f α ( ) ( ) Complete: f α x x ' x ' α ( ) => Wide range of choices for and our initial choice is f a x ik + x − Ψ n , m ( ρ , ϕ ) = Ne ik + x − f n , m ( ρ ) χ m ( ϕ ) ( ) = Ne f α x *J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath, G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, PRC 81, 035205 (2010). ArXiv:0905:1411
Set of transverse 2D HO modes for n=4 m=0 m=1 m=2 m=3 m=4 J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath, G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, PRC 81, 035205 (2010). ArXiv:0905:1411
Symmetries & Constraints ∑ = B b i i ∑ = Q e i i ∑ + s i ) = J z ( m i i ∑ = K k i Finite basis regulators i [ ] ≤ N max ∑ 2 n i + | m i | + 1 i Global Color Singlets (QCD) Light Front Gauge Optional - Fock space cutoffs H → H + λ H CM
Efffec%ve Yukawa Model in BLFQ Q a%on Wenyang Qi We Qian, , et t al. al. in in pr prepar para%o Inspira+on: Chiral Perturba+on Theory * Four-nucleon-leg contact term + one pion exchange LO LF treatment: Approximate the contact term by heavy scalar boson exchange + effec[ve one pion exchange Basis Light-Front Quan+za+on(BLFQ) Approach: Hamiltonian formalism Rela[vis[c theory Light-front wave func[ons provides direct access to all physical observables * R. Machleidt, D.R. Entem, Phys.Rept.503:1-75 (2011)
2.04 2.03 2.02 Mass / m f 2.01 2.00 � = 0.395 N max = K max = 15 Yukawa 1.98909 1.99 ( scalar boson ) m b = 0.1 m f 1.98907 b = 0.35 m f 1.98 - 3 - 2 - 1 0 1 2 3 M J
Light Front (LF) Hamiltonian Defined by its Elementary Vertices in LF Gauge QED & QCD QCD
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