Theoretical calculation of nuclear reactions of interest for Big - PowerPoint PPT Presentation

Theoretical calculation of nuclear reactions of interest for Big Bang Nucleosynthesis Candidate: Alex Gnech Advisors : Prof. Laura Elisa Marcucci (Univ. of Pisa) Prof. Michele Viviani (INFN Pisa) PhD thesis defense, April 23, 2020 1

Big Bang Nucleosynthesis • Predicts abundances of light elements • Network of reactions (cross-sections) ⇒ NO free parameters • Good agreement with Astrophysical Observations (A.O.) • A.O. more and more accurate ⇒ secondary products PDG, Phys. Rev. D 98 , 030001 (2018) 2

Is there a 6 Li problem? • The 6 Li abundance in the BBN 6 Li / 7 Li ∼ 10 − 5 BBN prediction 6 Li / 7 Li ∼ 5 × 10 − 3 measured in halo-stars [1] ⇒ results under debate • Possible solutions [2] • systematic errors in A.O. • new physics (BSM) appearing • incomplete knowledge of reaction cross-sections [1] Asplund et al. , Astrophys J. 664 , 229 (2006) [2] Fields, Ann. Rev. Nucl. Part. Phys. 61 , 47 (2011) 3

Motivations • Main uncertainties comes from α + d → 6 Li + γ [1] • Presence of non-thermal photons (BSM) ⇒ 7 Be + γ → p + 6 Li [2] ⇒ studied with p + 6 Li → 7 Be + γ • Both the reactions studied by the LUNA Collaboration[3] • Why theory? In the BBN energy range (50 < E < 400 keV) the measurements are very hard due to the Coulomb barrier The goal is the determination of the S-factor S ( E ) = E exp ( 2 πη ) σ ( E ) η = Z 1 Z 2 e 2 µ � 2 k [1] K.M. Nollett, et. al Phys. Rev. C 56 , 1144 (1997) [2] M. Kusukabe, et al. Phys. Rev. D 74 , 023526 (2006) [3] M. Anders, et al. Phys. Rev. Lett. 113 , 042501 (2014) 4

Theoretical approaches • Phenomenological approach ( p + 6 Li → 7 Be + γ ) • Nucleus = system of “pointlike” clusters ( 7 Be = p + 6 Li) • “Phenomenological” interactions between clusters • “Model dependent” prediction • numerically “Fast” • Ab-initio approach ( α + d → 6 Li + γ ) • Nucleus = system of A bodies interacting among themselves and with external probes • Realistic nucleon-nucleon and nucleon-probe interactions • Exact method to solve the quantum-mechanical problem • “True” predictions • numerically “Slow” ⇒ We limit the study to 6 Li 5

p + 6 Li → 7 Be + γ A.G. and L.E. Marcucci, Nucl. Phys. A 987 , 1 (2019) 6

Nuclear Physics Motivations • Is there a low-energy resonance? [1] • Photon angular distributions (for LUNA) [1] J.J. He et al. . Phys. Lett. B 725 , 287 (2013) 7

The p + 6 Li system in the cluster model • Clusters: p and 6 Li 185 • Intercluster potential (state dependent) 180 175 V ( r ) = − V 0 exp ( − a 0 r 2 ) δ [Deg] 4 S 3/2 170 165 2 S 1/2 Model Ref. [1] 160 • Elastic scattering data+bound 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 E p [MeV] state properties ⇒ cluster potential parameters J π Spin E (MeV) • Wave functions ⇒ prediction for 3 / 2 − GS 1/2 -5.6068 radiative capture 1 / 2 − FES 1/2 -5.1767 Dominated by E 1 transition σ ( E ) ∝ |� ψ 7 Be | E 1 | ψ p + 6 Li �| 2 8 [1] S.B. Dubovichenko et al. , Phys. Atom. Nucl. 74 , 1013 (2011)

The S-factor S ( E ) = E exp ( 2 πη )( σ 3 / 2 ( E ) + σ 1 / 2 ( E )) Branching ratio 140 Bare Ref. [1] σ 1 / 2 ( E ) � 120 Ref. [2] � th . ≃ 33 % � σ 1 / 2 ( E ) + σ 3 / 2 ( E ) 100 S(E) [eV b] 80 σ 1 / 2 ( E ) � 60 � exp . ≃ 39 % � σ 1 / 2 ( E ) + σ 3 / 2 ( E ) 40 20 Internal structure of 6 Li is 0 0 0.2 0.4 0.6 0.8 1 missing! E cm [MeV] [1] S.K. Switkowski, et al. Nucl. Phys. A 331 , 50 (1979) [2] J.J. He et al. , Phys. Lett. B 725 , 287 (2013) 9

The S-factor S ( E ) = E exp ( 2 πη )( S 2 3 / 2 σ 3 / 2 ( E ) + S 2 1 / 2 σ 1 / 2 ( E )) 140 Bare • S spectroscopic factor Final 120 Ref. [1] Ref. [2] 100 J π χ 2 χ 2 S J 0 / N S / N S(E) [eV b] 3 / 2 − 80 1.003 0.064 0.064 1 / 2 − 60 1.131 2.096 0.219 40 • Fitted on data of [1] 20 • Branching ratio well 0 0 0.2 0.4 0.6 0.8 1 reproduced E cm [MeV] [1] S.K. Switkowski, et al. Nucl. Phys. A 331 , 50 (1979) [2] J.J. He et al. , Phys. Lett. B 725 , 287 (2013) 10

The “He et al. ” resonance YES! • “He et al. ” suggested the presence of a resonance 140 Bare J π = ( 1 / 2 , 3 / 2 ) + , E r = 195 MeV 4 S 3/2 res. 120 Ref. [1] Γ p = 50 keV. Ref. [2] 100 S(E) [eV b] • Can we add the resonance in our 80 model? 60 40 20 0 0.2 0.4 0.6 0.8 1 E cm [MeV] • σ ( E ) = S 2 3 / 2 σ 3 / 2 ( E ) + S 2 1 / 2 σ 1 / 2 ( E ) + S 2 res σ res ( E ) • S 0 ≃ S 1 ∼ 1 • S res = 0 . 011 ⇒ small % of S=3/2 in 7 Be BUT... 11

The “He et al. ” resonance The 4 S 3 / 2 phase shift is NOT reproduced 180 160 140 120 δ [Deg] 100 4 S 3/2 80 60 40 no resonance 20 resonance Ref. [3] 0 0 0.2 0.4 0.6 0.8 1 E [MeV] We cannot add the resonance in our model [1] S.K. Switkowski, et al. Nucl. Phys. A 331 , 50 (1979) [2] J.J. He et al. , Phys. Lett. B 725 , 287 (2013) [3] S.B. Dubovichenko et al. , Phys. Atom. Nucl. 74 , 1013 (2011) 12

LUNA experimental setup Il nuovo cimento 42C , 116 (2019) -Courtesy of T. Chillery (LUNA Coll.) • Not a 4 π detector • The yield (= N γ / N p ) must be corrected by � W ( θ, E ) = a k ( E ) P k ( cos θ ) k ≥ 1 • Angle detector/beam θ 0 ≃ 55 ◦ ⇒ P 2 ( cos θ 0 ) ≃ 0 13

Photon angular distribution � W ( θ, E ) = a k ( E ) P k ( cos θ ) k ≥ 1 • a 1 and a 2 are the only significant coefficients • Dominated by the interference of E 1 (S-waves) and E 2 (P-waves) 1.5 1.5 This work This work Fit [1] Fit [1] 1.4 1.4 Ref. [1] Ref. [1] 1.3 1.3 1.2 1.2 a.u. a.u. 1.1 1.1 1 1 0.9 J π =3/2 - 0.9 J π =1/2 - 0.8 0.8 0.7 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 θ [deg] θ [deg] a 1 ∝ E 1 ( 2 S 3 / 2 ) × ( E 2 ( 2 P 1 / 2 ) − E 2 ( 2 P 3 / 2 )) ∼ 0 a 1 ∝ E 1 ( 2 S 3 / 2 ) × E 2 ( 2 P 3 / 2 ) [1] C.I. Tingwell, J. D. King and D.G. Sargood, Aust. J. Phys. 40 , 319 (1987) – E p = 0 . 5 MeV 14

Final Yields 1x10 -12 1x10 -12 Yield [1/part] Yield [1/part] 1x10 -13 1x10 -13 DC->0, Wout DC->429, Wout DC->0, Win DC->429, Win 100 150 200 250 300 350 400 100 150 200 250 300 350 400 E p [keV] E p [keV] J π = 3 / 2 − J π = 1 / 2 − • Correction to the ground state negligible ( a 1 ∼ 0) • Correction to the first excited state ∼ 6 − 9 % [1] [1] Courtesy of R. Depalo (LUNA Coll.) 15

Conclusions I • Calculation of the S-factor ⇒ nice agreement with the data • A resonance? ⇒ not possible in the cluster model • Photon angular distribution ⇒ Important correction to first excited state yields 16

α + d → 6 Li + γ A.G., M. Viviani and L.E. Marcucci, arXiv:2004.05814 (2020) 17

Nuclear Physics Motivations • 6 Li has an exotic structure • Weakly bound nucleus • Strong clusterization • Study of electromagnetic moments • Small and negative electric quadrupole moment • Asymptotic Normalization Coefficients ( ⇒ S-factor) • Dark matter search ⇒ CRESST Coll. 18

Ab-initio approach • Which nuclear potential? • Which method to solve the Schrödinger Equation? 19

Chiral interaction ( χ EFT) chiral symmetry QCD → χ EFT − − − − − − − − • Low Energy Theory ( Λ χ ∼ 1 GeV) • N , π as d.o.f. • high energy d.o.f. integrated out → Low Energy Constants • Perturbative expansion ( ∝ ( Q / Λ χ ) ν ) • Various phenomena in a consistent framework ( A.G. and M. Viviani, Time-reversal violation in light nuclei , PRC 101 , 024004 (2020). ) D.R. Entem, et al. Phys. Rev. C 96 , 024004 (2017) E. Epelbaum, et al. Phys. Rev. Lett. 115 , 122301 (2015) 20

Nuclear chiral potential • Non-relativistic expansion • Regularization with a cutoff ( Λ C = 400 − 600 MeV) • LECs fitted to the NN experimental scattering data • The chiral convergence must be checked a posteriori D.R. Entem, et al. , Phys. Rev. C 96 , 024004 (2017) • Controlled theoretical uncertainties 21

The Hyperspherical Harmonic method p 2 � � � i H = 2 M + V ( i , j ) + W ( i , j , k ) + . . . i i < j i < j < k Search for accurate solution of H Ψ = E Ψ • Variational approach • Expansion of Ψ on the basis of Hyperspherical Harmonic (HH) functions • [L.E. Marcucci, J. Dohet-Eraly, L. Girlanda, A.G., A. Kievsky, and M. Viviani, Front. Phys. 8 , 69 (2020)] • Applied for A = 3 , 4 bound and scattering states For A = 6 implemented from scratch 22

The HH wave function • Jacobi vectors � ξ 1 , . . . , � ξ N ⇒ CoM completely decoupled • Hyperangular variables ρ = � 5 k = 1 ( ξ k ) 2 , Ω = { ˆ ξ k √ ξ i , φ i } , cos φ k = ξ 2 1 + ... + ξ 2 k � ∂ 2 � T = − � 2 ∂ρ − L 2 (Ω) ∂ρ 2 + D − 1 ∂ ρ 2 m ρ • Expansion on a base ⇒ Hyperspherical Harmonics (HH) L 2 (Ω) Y [ K ] (Ω) = K ( K + 13 ) Y [ K ] (Ω) • The variational wave function � � � ψ A = a l , [ K ] f l ( ρ ) Y [ K ] (Ω A − 1 ) χ S ⊗ χ T , l , [ K ] • Check convergence on K 23

The HH wave function • Sum over the permutations ⇒ antisymmetrization • Transformation Coefficients (TC) Y [ K ] (Ω ′ ) = � K = K ′ a [ K ] , [ K ′ ] Y [ K ′ ] (Ω) [ K ′ ] • Sum over the permutations rewritten in terms of the transformation coefficients � (Ω perm ) = � perm Y KLSTJ [ α ′ ] a KLSTJ [ α ] , [ α ′ ] Y KLSTJ (Ω) [ α ′ ] [ α ] • Basis states are linearly dependent ⇒ orthogonalization procedure 24