Overview from Nuclear Lattice Effective Field Theory

Serdar Elhatisari Nuclear Lattice EFT Collaboration HISKP, Universität Bonn

Workshop on Polarized light ion physics with EIC Ghent University, Belgium February 5-9, 2018 Nuclear Lattice Effective Field Theory collaboration

Serdar Elhatisari (Bonn) Ning Li (MSU) Evgeny Epelbaum (Bochum) Bing-Nan Lu (MSU) Nico Klein (Bonn) Thomas Luu (Jülich) Hermann Krebs (Bochum) Ulf-G. Meißner (Bonn/Jülich) Timo Lähde (Jülich) Gautam Rupak (MSU) Dean Lee (MSU) Gianluca Stellin (Bonn) Outline

Introduction Lattice effective ﬁeld theory Adiabatic projection method : scattering and reactions on the lattice Degree of locality of nuclear forces Nuclear clusters : probing for alpha clusters Pinhole Algorithm : density proﬁles for nuclei Summary 4/4/2017 nuchart1.gif (1054×560)

Ab initio nuclear structure and nuclear scattering

2 nuclear structure :

Source: rarfaxp.riken.go.jp/ gibelin/Nuchart/

2 nuclear scattering : ... processes relevant for stellar astrophysics £ scattering of alpha particles : 4He+4He→4He+4He £ triple-alpha reaction : 4He + 4He + 4He → 12C + γ £ alpha capture on carbon : 4He + 12C → 16O + γ . .

http://rarfaxp.riken.go.jp/~gibelin/Nuchart/nuchart1.gif 1/1 Progress in ab initio nuclear structure and nuclear scattering

Unexpectedly large charge radii of neutron-rich calcium isotopes. Garcia Ruiz et al., Nature Phys. 12, 594 (2016).

Structure of 78Ni from ﬁrst principles computations. Hagen, Jansen, & Papenbrock, PRL 117, 172501 (2016).

A nucleus-dependent valence-space approach to nuclear structure. Stroberg et al., PRL 118, 032502 (2017).

Ab initio many-body calculations of the 3H(d, n)4He and 3He(d, p)4He fusion. Navratil & Quaglioni, PRL 108, 042503 (2012).

3He(α,γ)7Be and 3H(α,γ)7Li astrophysical S factors from the no-core shell model with continuum Dohet-Eraly, J. et al. PLB B 757 (2016) 430-436.

Elastic proton scattering of medium mass nuclei from coupled-cluster theory. Hagen & Michel PRC 86, 021602 (2012).

Coupling the Lorentz Integral Transform (LIT) and the Coupled Cluster (CC) Methods. Orlandini, G. et al. , Few Body Syst. 55, 907-911 (2014). Nuclear LEFT: ab initio nuclear structure and scattering theory

2 Lattice EFT calculations for A = 3, 4, 6, 12 nuclei, PRL 104 (2010) 142501 2 Ab initio calculation of the Hoyle state, PRL 106 (2011) 192501

2 Structure and rotations of the Hoyle state, PRL 109 (2012) 252501 2 Viability of Carbon-Based Life as a Function of the Light Quark Mass, PRL 110 (2013) 112502 2 Radiative capture reactions in lattice effective ﬁeld theory, PRL 111 (2013) 032502 2 Ab initio calculation of the Spectrum and Structure of 16O, PRL 112 (2014) 102501

2 Ab initio alpha-alpha scattering, Nature 528, 111-114 (2015).

2 Nuclear Binding Near a Quantum Phase Transition, PRL 117, 132501 (2016). 2 Ab initio calculations of the isotopic dependence of nuclear clustering. �� PRL 119, 222505 (2017).

�� E� – Eα�A/4 � �� ∞ αα αα ��

� λ�� λ�� λ�� λ� �� λ�� λ�� λ ��

� ��! �� ! �� ! � �� !

�� 25

�� 11 Lattice effective ﬁeld theory

Lattice effective ﬁeld theory is a powerful numerical method formulated in the framework of chiral effective ﬁeld theory.

Accessible by Lattice QCD

100 early quark-gluon universe plasma

heavy-ion

collisions Accessible by

10 gas of light

[MeV] Nucleons T nuclei nuclear Lattice EFT liquid excited nuclei superfluid neutron star crust neutron star core 푎 1 퐿 10-3 10-2 10-1 1 -3 r [fm ] rN

Fig. courtesy of D. Lee 5 Chiral EFT for nucleons: nuclear forces

Chiral effective ﬁeld theory organizes the nuclear interactions as an expansion in powers of momenta and other low energy scales such as the pion mass (Q/Λχ). Rep. Prog. Phys. 75 (2012) 016301 N Kalantar-Nayestanaki et al

2N force 3N force 4N force

NLO

N 2LO

N3 LO

Fig. courtesy of E.Epelbaum Figure 4. Hierarchy of nuclear forces in chiral EFT based on Weinberg’s power counting. Solid and dashed lines denote nucleons and pions, respectively.Ordonez etSolid al. dots,’94; Friar ﬁlled & circles Coon and ’94; ﬁlled Kaiser squares et al. refer, ’97; respectively,Epelbaum et to al. the ’98,’03,’05,’15; leading, subleading Kaiser and ’99-’01; sub-subleading vertices in the effective Lagrangian. The crossed square denotes 2N contact interactions with 4 derivatives. Higa et al. ’03; ...

1 GeV, respectively. A recent review on the methodology and with the superscripts referring to the power of the expansion applications of chiral perturbation theory in the Goldstone– parameter Q/χ . The long-range part of the nuclear boson and single-nucleon sectors can be found in [66]. force is dominated by 1π-exchange with the 2π-exchange Chiral EFT, however, cannot be directly applied to contributions starting at next-to-leading order (NLO). The low-energy few-nucleon scattering. The strong nature of expressions for the potential up to next-to-next-to-leading the nuclear force that manifests itself in the appearance of order (N2LO) in the heavy-baryon formulation [70, 71] are self-bound atomic nuclei invalidates a naive application of rather compact and have been independently derived by several perturbation theory. Weinberg pointed out that the breakdown authors using a variety of different methods [72–75]. 2π- and of perturbation theory can be traced back to the infrared 3π-exchange contributions at next-to-next-to-next-to-leading 3 enhancement of reducible diagrams that involve few-nucleon order (N LO) have been worked out by Kaiser [76–79] and cuts [67, 68]. The irreducible part of the amplitude that gives are considerably more involved, see also [80]. While the 2 rise to the nuclear force is, however, not affected by the infrared 2π-exchange at NLO and, especially, at N LO generates a enhancement and is thus accessible within chiral EFT. These rather strong potential at distances of the order of the inverse important observations have triggered intense research activity pion mass [74], the leading 3π-exchange contributions turn out 3 towards the systematic derivation of the nuclear forces in chiral to be negligible. The N LO contributions to the 2π-exchange EFT and their applications to the nuclear few- and many-body potential were also derived in the covariant formulation of problem. chiral EFT [81] by Higa et al [82, 83]. The LECs entering the pion-exchange contributions up to N3LO are known from It should also be emphasized that an EFT can be pion–nucleon scattering and related processes [84–86]. We formulated that is only valid at typical momenta well below note that some of them, especially the ones from L(3) , are the pion mass. This framework allows one to take into πN presently not very accurately determined. In the isospin account the unnaturally large NN scattering lengths but loses limit, the short-range part of the potential involves 2, 9, 24 the connection with the chiral symmetry of QCD. It has independent terms at LO, NLO/N2LO, N3LO, respectively. been used with great success not only for nuclear systems The corresponding LECs were fixed from the two-nucleon [69, 14]. This so-called pion-less EFT requires a 3NF in data leading to the accurate N3LO potentials of Entem and leading order. The corresponding hard scale is given by Machleidt (EM) [87] and Epelbaum, Glockle¨ and Meißner the pion mass and, therefore, intermediate-energy observables (EGM) [88]. These two potentials differ in the treatment of cannot be predicted within this framework. In the following, relativistic corrections (including the form of the employed we restrict ourselves to chiral EFT. dynamical equation), IB terms and the form of the regulator The EFT expansion of the nuclear force based on functions. There are also differences in the adopted values the standard chiral power counting (i.e. assuming that all of certain pion–nucleon LECs. Finally, EGM provide an operators in the effective Lagrangian scale according to a naive estimation of the theoretical uncertainty by means of the dimensional analysis) is visualized in figure 4. variation of the cutoffs in some natural ranges. This important It provides a natural qualitative explanation of the issue is not addressed in [87], where a single excellent observed hierarchy of two-, three- and more-nucleon forces fit to neutron–proton (proton–proton) scattering data with 2 2 with V2NV3NV4N .... The expansion of the 2NF χ /datum = 1.10 (χ /datum = 1.50) in the energy range has the form from 0 to 290 MeV is given. We refer the reader to [87, 88] for more details. In figure 5, we compare NN phase shifts obtained = (0) (2) (3) (4) ··· V2N V2N + V2N + V2N + V2N + , (1) with these models with predictions from phenomenological

5 Chiral EFT for nucleons: NN scattering phase shifts

80 180 10 10

70 160 8 8 60 6 6 140 50 4 4 120 40 2 2

) [degrees] ) [degrees] 100 0 1 30 [degrees] 0 [degrees] 0 S S 1 2 1 3 80 ε ε δ ( 20 NPWA δ ( -2 -2 LO 10 NLO,N2LO 60 -4 -4 N3LO 0 40 -6 -6 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 pCM [MeV] pCM [MeV] pCM [MeV] pCM [MeV]

10 20 5 14

12 5 15 0 10

0 8 10 -5 6 ) [degrees] ) [degrees] ) [degrees] ) [degrees] 1 -5 0 1 2 4 P P P P

1 3 5 3 3

δ ( δ ( δ ( -10 δ ( 2 -10 0 0 -15 -15 -2 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 pCM [MeV] pCM [MeV] pCM [MeV] pCM [MeV]

10 2 16 5 14 8 0 4 12 -2 3 6 10 -4 8 2 4

) [degrees] 6 ) [degrees] -6 ) [degrees] ) [degrees] 1 2 1 2 3 D D D D

1 2 4 3 -8 3 3 0 δ ( δ ( δ ( 2 δ ( 0 -10 -1 0 -2 -12 -2 -2 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 pCM [MeV] pCM [MeV] pCM [MeV] pCM [MeV]

Figure 1: Neutron-proton scattering phase shifts versus center-of-mass momenta. Coarse 1 lattice spaces,a=a t = (100MeV)− = 1.97 fm, are used in thefits. The two-pion-exchange potentials are not included specifically since they can be absorbed into the contact forces through a redefinition of the low-energy constants.

4 Lattice Monte Carlo calculations

Transfer matrix operator formalism M = : exp(−H at) :

Microscopic Hamiltonian H = Hfree + V

( ) R ∗ ∗ Z Lt = Tr(MLt ) = Dc Dc exp[−S(c, c )] Creutz, Found. Phys. 30 (2000) 487.

The exact equivalence of several different lattice formulations. Lee, PRC 78:024001, (2008); Prog.Part.Nucl.Phys., 63:117-154 (2009)

e−E0 at = lim Z(Lt+1)/Z(Lt) Lt→∞

These amplitudes are computed with the Hybrid Monte Carlo methods. Phys. Lett. B195, 216-222 (1987), Phys. Rev. D35, 2531-2542 (1987). Lattice Monte Carlo calculations

Nuclear forces posses approximate SU(4) symmetry. HSU(4) acts as an approximate and inexpensive low energy filter at few first/last time steps. Significant suppression of sign oscillations. Chen, Lee, Schäfer, PRL 93 (2004) 242302

0 0 Lt −at HSU(4) 0 0 |ψI (τ )i = [MSU(4)] |ψI i MSU(4) = : e : τ = Lt at

For time steps in midsection, the full HLO Hamiltonian is used.

Lt 0 −at HLO |ψI (τ)i = [MLO] |ψI (τ )i MLO = : e : τ = Lt at

The ground state energy at LO can be extracted from

(Lt+1) Z hψ (τ/2)|MLO|ψ (τ/2)i e−ELO at = lim LO = lim I I (L ) Lt→∞ t Lt→∞ hψI (τ/2)|ψI (τ/2)i ZLO Lattice Monte Carlo calculations

Higher order calculations (perturbative) ho = NLO, NNLO, ···

−at(HLO+Vho) Mho = : e : where the potential Vho is treated perturbatively.

The higher order correction to the ground state energy can be extracted from

(Lt+1) Z hψ (τ/2)|Mho|ψ (τ/2)i e−∆Eho at = ho = I I (L +1) t hψI (τ/2)|MLO|ψI (τ/2)i ZLO Lattice Monte Carlo calculations

Compute observable O

The observable O at LO

hψ |[M ]Lt O [M ]Lt |ψ i hOi = lim I LO LO I 0,LO 2Lt+1 Lt→∞ hψI |[MLO] |ψI i

The observable O at (NLO, NNLO, ··· )

hψ |[M ]Lt−1 M O [M ]Lt |ψ i hOi = lim I LO ho LO I 0,ho Lt Lt Lt→∞ hψI |[MLO] Mho [MLO] |ψI i Lattice EFT: (Euclidean time) projection Monte Carlo

e−Hτ τ = Ltat

훑 훑 Euclidean time Euclidean

퐿 2 evolve nucleons forward in Euclidean time 푎 2 allow them to interact Auxiliary ﬁeld Monte Carlo

Use a Gaussian integral identity

r C 2 1 Z s2 √ exp − N† N = ds exp − + C s N† N 2 2π 2 s is an auxiliary field coupled to particle density. Each nucleon evolves as if a single particle in a fluctuating background of pion fields and auxiliary fields.

풔푰

훑 훑 풔흅

풔 Euclidean time Euclidean Adiabatic projection method

Scattering and reactions on the lattice

The ﬁrst part use Euclidean time projection to construct an ab initio low-energy cluster Hamiltonian, called the adiabatic Hamiltonian.

The second part compute the two-cluster scattering phase shifts or reaction amplitudes using the adiabatic Hamiltonian. −/2 /2

1 Rupak, Lee., PRL 111 (2013) 032502. interacting free Pine, Lee, Rupak, EPJA 49 (2013) 151.

SE, Lee, PRC 90, 064001 (2014). 0.5 Rw ) r ) ( p

Rokash, Pine, SE, Lee, Epelbaum, Krebs, PRC 92,054612 (2015) ( 0 R

SE, Lee, Meißner, Rupak, EPJA 52: 174 (2016) 0

-0.5 0 10 20 30 40 50 60 r (fm) Adiabatic projection method

The method constructs a low energy effective theory for the clusters.

Use initial states parameterized by the relative spatial separation between |~ i = |~ + ~ i ⊗ |~i clusters, and project them in Euclidean R ∑ r R 1 r 2 time. ~r

푛′ , 푛′ 푥 푦 ~ −H τ ~ |Riτ = e |Ri dressed cluster state

푅 The adiabatic projection in Euclidean time gives a systematically improvable description of the low- lying scattering cluster states.

푛푥, 푛푦 In the limit of large Euclidean projection time the description becomes exact. Adiabatic projection method

~ −H τ ~ |Riτ = e |Ri dressed cluster state (not orthogonal)

Hamiltonian matrix Norm matrix ~ ~ 0 ~ ~ 0 [Hτ]~R,~R0 = τ hR|H|R iτ [Nτ]~R,~R0 = τ hR|R iτ

a h −1/2i h −1/2i [Hτ]~ ~ 0 = Nτ [Hτ]~ 00~ 000 Nτ R,R ∑ ~R~R00 R R ~R000~R0 ~R00,~R000

The structure of the adiabatic Hamiltonian,[ a] , is similar to the Hamiltonian Hτ ~R,~R0 matrix used in calculations of ab initio no-core shell model/resonating group method (NCSM/RGM) for nuclear scattering and reactions.

Navratil, Quaglioni, PRC 83, 044609 (2011). Navratil, Roth, Quaglioni, PLB 704, 379 (2011). Navratil, Quaglioni, PRL 108, 042503 (2012). Adiabatic projection method

The radiative capture process p(n, γ)d. N − d scattering. 4

) 10 20 50 70 100 2 -2 at MSU. Part of this work was completed at the INT.

− 0 10 Partial support provided by the U.S. Department of En- n−d ergy grant DE-FG02- 03ER41260 for D.L. and the U.S. -3 n−d (EFT) 10 −15 δ = 0.6 NSF Grant No. PHY-0969378 for G.R. p−d δ = 0.4 p−d (EFT) |M| (MeV -4 δ = 0 10 δ = 0.6 −30 Breakup δ = 0.4 δ = 0 -5

10 (degrees) −45 ∗ [email protected] 0

80 δ † dean [email protected] 60 [1] P.−60 Navratil,Quartet− R. Roth,S and S. Quaglioni, Phys. Lett. B δ = 0.6 704, 379 (2011).

φ (deg) 40 δ = 0.4 [2] P. Navratil and S. Quaglioni, Phys. Rev. Lett. 108, δ = 0 −75 δ = 0.6 042503 (2012). 20 δ = 0.4 δ = 0 [3] T. Neff, Phys. Rev. Lett. 106, 042502 (2011). 0 [4] O. Jensen, G. Hagen, T. Papenbrock, D. Dean, and 10 20 50 70 100 −90 p (MeV) J. Vaagen, 0 20 Phys. Rev. 40 C 82 60 , 014310 80 (2010). 100 120 140 [5] K. M. Nollett and R. Wiringa,p (MeV) Phys. Rev. C 83, 041001 Rupak, Lee., PRL 111 (2013) 032502. (2011). [6] M. Lage,SE, U.-G. Lee, Meißner,Rupak. Meißner, and A. EPJ Rusetsky, 52:174 (2016). Phys. Lett. FIG. 4. Comparison of lattice results with continuum results B 681, 439 (2009). from Eq. (5) forδ=0.6,0.4 and results extrapolated toδ = 0. [7] V. Bernard, M. Lage, U.-G. Meißner, and A. Rusetsky, In the top panel we show and in the bottom panel we JHEP 1101, 019 (2011). show the phase angleφ in degrees. |M| [8] H. B. Meyer, (2012), arXiv:1202.6675 [hep-lat]. [9] R. A. Briceno and Z. Davoudi, (2012), arXiv:1204.1110 [hep-lat]. will be discussed in forthcoming publications. In this [10] K. Polejaeva and A. Rusetsky, Eur. Phys. J. A 48, 67 study we have demonstrated the second part of the (2012). method using the example of radiative neutron capture [11] C. Chin and P. S. Julienne, Phys. Rev. A 71, 012713 reactionp(n,γ)d at leading order in pionless EFT. We (2005). have shown that extrapolations in the parameterδ are [12] T. M. Hanna, T. Köhler, and K. Burnett, Phys. Rev. A an effective way to removefinite volume errors in the lat- 75, 013606 (2007). [13] C. Weber, G. Barontini, J. Catani, G. Thalhammer, tice calculations. We were able to reproduce the known M. Inguscio, and F. Minardi, Phys. Rev. A 78, 061601 continuum result for the capture amplitude at infinite (2008). volume with an error of within 2% for the magnitude [14] O. Machtey, Z. Shotan, N. Gross, and L. Khaykovich, and an error of within 4% for the phase angle. These Phys. Rev. Lett. 108, 210406 (2012). remaining errors are consistent with the size of the small [15] B. Bazak, E. Liverts, and N. Barnea, Phys. Rev. A 86, discretization errors due to lattice spacing. 043611 (2012). This method should be applicable for calculating pho- [16] D. Lee, Prog. Part. Nucl. Phys. 63, 117 (2009). [17] B. Borasoy, H. Krebs, D. Lee, and U.-G. Meißner, Nucl. tonuclear reaction rates involving many nuclear systems. Phys. A 768, 179 (2006). This includes halo nuclei with a single valence nucleon. [18] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, 14 15 For example, in C(n,γ) C the carbon-15 nucleus is Phys. Rev. Lett. 104, 142501 (2010). a halo nucleus with a neutron separation energy of [19] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Eur. only 1.22 MeV. At low energy the incoming state can Phys. J. A 45, 335 (2010). be treated as a point-like neutron and a tightly-bound [20] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, carbon-14 core in halo EFT [27]. But ab initio lattice Phys. Rev. Lett. 106, 192501 (2011). [21] E. Epelbaum, H. Krebs, T. A. Lahde, D. Lee, and U.-G. calculations should be able to go well beyond halo sys- Meißner, Phys. Rev. Lett. 109, 252501 (2012). tems. This method could also be applied to more com- [22] J.-W. Chen and M. J. Savage, Phys. Rev. C 60, 065205 plicated systems such as alpha capture on carbon-12 or (1999). oxygen-16 as occurs in helium burning. We are hopeful [23] G. Rupak, Nucl.Phys. A 678, 405 (2000). that such systems will studied in the near future using [24] M. Luscher, Commun. Math. Phys. 105, 153 (1986). methods outlined here. [25] V. Efros, W. Leidemann, G. Orlandini, and N. Barnea, Acknowledgments.— The authors thank E. Epelbaum, J. Phys. G 34, R459 (2007). [26] H. Kamada, Y. Koike, and W. Gloeckle, Prog. Theor. R. Higa, H. Krebs, T. Lähde,U.-G. Meißner for use- Phys. 109, 869 (2003). ful discussions. G.R. thanks R. Higa for valuable dis- [27] G. Rupak, L. Fernando, and A. Vaghani, Phys. Rev. C cussions on the formulation of the Green’s function ap- 86, 044608 (2012). proach. Computing support was provided by the HPCC Alpha-alpha scattering

S-wave D-wave 200 200 180 Higa et al. Afzal et al. LO (no Coulomb) NLO 160 160 150 NNLO (degrees) 0 δ

120 120 120 0 1 2 3 E Lab (MeV) (degrees) (degrees)

0 80 2 80 δ δ

40 40 Afzal et al. LO (no Coulomb) NLO NNLO 0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 E E Lab (MeV) Lab (MeV)

Afzal, Ahmad, Ali, Rev. Mod. Phys. 41, 247 (1969) Higa, Hammer, van Kolck, Nucl.Phys. A809, 171 (2008) SE, Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, Nature 528, 111-114 (2015) Degree of locality of nuclear forces

Interaction A Interaction B

Local Introducing the nonlocal short-range interactions reduce the Monte Carlo sign oscillation signiﬁcantly. r′ r

Nonlocal Degree of locality of nuclear forces

Interaction A Interaction B

0 0 0 Nonlocal short-range interactions V(r, r ) Local short-range interactions V(r, r ) = U(r)δ(r − r ) 0 One-pion exchange interaction Nonlocal short-range interactions V(r, r ) (+ Coulomb interaction) One-pion exchange interaction (+ Coulomb interaction) a b c d 80 180 15 Nijmegen PWA 5 Continuum LO 60 120 10 Lattice LO-A 0 Lattice LO-B 40 -5 5 60 20 -10 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150

e f g h 3 0 6 0 2 4 -2 -5 1 2 -4

(deg) ε or δ 0 -10 0 -6 -1 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150

i j k l 10 5

1 0

5 0 0 -1

0 -1 -5 -2 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150

pcms (MeV)

22 Nucleus A (LO) B (LO) A (LO + Coulomb) B (LO + Coulomb) Experiment 3H -7.82(5) -7.78(12) -7.82(5) -7.78(12) -8.482 3He -7.82(5) -7.78(12) -7.08(5) -7.09(12) -7.718 4He -29.36(4) -29.19(6) -28.62(4) -28.45(6) -28.296 Degree of locality of nuclear forces

4He – 4He S-wave scattering 180 A (LO) A (LO + Coulomb) B (LO) B (LO + Coulomb) 135 Afzal et al.

90 (degrees)

0 45 δ

-45 0 2 4 6 8 10 E Lab (MeV)

Nucleus A (LO) B (LO) A (LO + Coulomb) B (LO + Coulomb) Experiment 8Be -58.61(14) -59.73(6) -56.51(14) -57.29(7) -56.591 12C -88.2(3) -95.0(5) -84.0(3) -89.9(5) -92.162 16O -117.5(6) -135.4(7) -110.5(6) -126.0(7) -127.619 20Ne -148(1) -178(1) -137(1) -164(1) -160.645

SE, Li, Rokash, Alarcon, Du, Klein, Lu, Meißner, Epelbaum, Krebs, Lähde, Lee, Rupak, PRL 117, 132501 (2016) Degree of locality of nuclear forces

Interaction A Interaction B

0 0 0 Nonlocal short-range interactions V(r, r ) Local short-range interactions V(r, r ) = U(r)δ(r − r ) 0 One-pion exchange interaction Nonlocal short-range interactions V(r, r ) �� (+ Coulomb interaction) One-pion exchange interaction (+ Coulomb interaction) �� 4 Tight-binding approximation

40 A (LO) B (LO) � 20 C (LO) r �� 0

-20 Local

(MeV) -40 TB V -60 � � r′ -80 �� r -100 Nonlocal 2 2.5 3 3.5 4 4.5 5 5.5 6 r (fm) 32

FIG.The 2: Tight-binding alpha-alpha potential. interaction We plot isthe sensitive tight-binding to potential the degree versus radial of locality distance of for the the LO interaction. interactions for A, B, and C, where C is the interaction used in several previous lattice calculations [11, 18]. Interaction A is shown with blue squares and solid line, B is drawn with red crosses and dashed line, and C is presented with orange circles and short-dashed line.

16 12 8 λ16 = 0.2(1) for O, λ12 = 0.3(1) for C, and λ8 = 0.7(1) for Be. One finds a sudden change in the nucleon-nucleon density correlations at long distances as λ crosses the critical point, going from a continuum state to a self-bound system. As λ increases further beyond this critical value, the nucleus becomes more tightly bound, gradually losing its alpha cluster substructure and becoming more like a nuclear liquid droplet. The quantum phase transition at λ∞ = 0.0(1) is the corresponding phenomenon in the many-body system, a first-order phase transition occurring for infinite matter.

FIG. 3: Zero-temperature phase diagram. We show the zero-temperature phase diagram as a function of the parameter λ in the interaction Vλ = (1 − λ)VA + λVB without Coulomb included. The blue filled circles indicate neutrons, the red filled circles indicate protons, and the small arrows attached to the circles indicate spin direction. We show a first-order quantum phase transition from a Bose gas to nuclear liquid at the point where the scattering length aαα crosses zero. We have also plotted the alpha-like nuclear ground state energies EA for A nucleons up to A = 20 relative to the corresponding multi-alpha threshold 8 EαA/4. The last alpha-like nucleus to be bound is Be at the unitarity point where |aαα| = ∞.

EA – Eα A/4 aαα = 0 |aαα| = ∞

λ20 λ16 λ12 λ8

λ = 0 λ = 1 λ

A = 8

A = 12

A = 16

A = 20

Alpha gas Nuclear liquid

By adjusting λ in ab initio calculations, we have a new tool for studying alpha cluster states such as the Hoyle state 12 of C and possible rotational excitations of the Hoyle state [1–6]. By tuning λ to the unitarity point |aαα| = ∞, we can continuously connect the Hoyle state wave function without Coulomb interactions to a universal Efimov trimer [7,8, 27]. An Efimov trimer is one of an infinite tower of three-body states for bosons in the large scattering-length limit, with intriguing mathematical properties such as fractal-like discrete scale invariance. Another interesting system is the second 0+ state of 16O[28], which should be continuously connected to a universal Efimov tetramer [27, 29, 30]. This connection to Efimov states is now being investigated in follow-up work. The ability to tune the energies of alpha cluster states relative to alpha-separation thresholds provides a new window on wave functions and rotational excitations of alpha cluster states. By studying the λ-dependence of nuclear energy levels one can also identify the Nuclear binding near a quantum phase transition

Nucleus A (LO) B (LO) A (LO + Coulomb) B (LO + Coulomb) Experiment 4He -29.36(4) -29.19(6) -28.62(4) -28.45(6) -28.296 8Be -58.61(14) -59.73(6) -56.51(14) -57.29(7) -56.591 12C -88.2(3) -95.0(5) -84.0(3) -89.9(5) -92.162 16O -117.5(6) -135.4(7) -110.5(6) -126.0(7) -127.619 20Ne -148(1) -178(1) -137(1) -164(1) -160.645

E8 Be = 1.997(6) E4He E12 �� C = 3.00(1) E4He E16 O = 4.00(2) E4He E20 Ne = 5.03(3) E4He

SE, Li, Rokash, Alarcon, Du, Klein, Lu, Meißner, Epelbaum, Krebs, Lähde, Lee, Rupak, PRL 117, 132501 (2016)

29 Nuclear binding near a quantum phase transition

V = (1 − λ)VA + λVB

E� – Eα�A/4 �αα�� αα��∞

λ�� λ�� λ�� λ�

λ�� λ�� λ

��!

��

25 SE, Li, Rokash, Alarcon, Du, Klein, Lu, Meißner, Epelbaum, Krebs, Lähde, Lee, Rupak, PRL 117, 132501 (2016) �� Ground state energies at LO

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SE, Epelbaum, Krebs, Lähde, Lee, Li, Lu, Meißner, Rupak, PRL 119, 222505 (2017)

�� 11 Nuclear clusters: probing for alpha clusters

ρ(~n) : the total nucleon density operator on the lattice site ~n . 4 ρ4 = ∑ : ρ (~n)/4! : is deﬁned to construct a probe for alpha clusters . ~n

Similarly ρ3 is to construct a second probe for alpha clusters only in nuclei with even Z and even N where 3H and 3He clusters are not energetically favorable. 3 ρ3 = ∑ : ρ (~n)/3! : ~n

ρ3 and ρ4 depend on the short-distance regulator, the lattice spacing a. However the regularization-scale dependence of ρ3 and ρ4 does not depend on the nucleus being considered.

Therefore, by deﬁning ρ3,α and ρ4,α as the corresponding values for the alpha particle, we consider the ratios ρ3/ρ3,α and ρ4/ρ4,α that are free from short-distance divergences and are model-independent quantities up to contributions from higher-dimensional operators in an operator product expansion. Nuclear clusters: probing for alpha clusters

ρ3 ∆α /Nα the ρ3-entanglement of the alpha clusters , ρ3 where ∆α = ρ3/ρ3,α − Nα .

푁훼 = 4

푁훼 = 3

푁훼 = 2

푁훼 = 1

Nucleus 4,6,8He 8,10,12,14Be 12,14,16,18,20,22C 16,18,20,22,24,26,28O ρ3 ∆α /Nα 0.0 − 0.03 0.20 − 0.35 0.25 − 0.50 0.50 − 0.75 Nuclear clusters: probing for alpha clusters

Our results show that the transition from cluster-like states in light systems to nuclear liquid-like states in heavier systems should not be viewed as a simple suppression of multi-nucleon short-distance correlations, but rather an increasing entanglement of the nucleons involved in the multi-nucleon correlations.

SE, Epelbaum, Krebs, Lähde, Lee, Li, Lu, Meißner, Rupak, PRL 119, 222505 (2017) Density proﬁles for nuclei: pinhole algoritm

The simulations with auxiliary-ﬁeld Monte Carlo methods involve quantum states that are super- position of many different center-of-mass positions. Therefore, the density distrubitions of the nucleons cannot be computed direclty.

Consider a screen placed at the middle time step having pinholes with spin and isospin labels that allow nucleons with the corresponding spin and isospin to pass.

푖5, 푗5 푖7, 푗7 푖3, 푗3

휏푖 = 휏 푖2, 푗2

푖4, 푗4 푖6, 푗6

푖8, 푗8 휏/2 푖1, 푗1

Euclidean time Euclidean 휏푖 = 0 Density proﬁles for nuclei: pinhole algoritm

This opaque screen corresponds to the insertion of the normal-ordered A-body density operator,

(~ ~ ) = (~ ) (~ ) ρi1,j1,...,iA,jA n1,..., nA : ρi1,j1 n1 ... ρiA,jA nA : where † is the density operator for nucleon with spin and isospin . ρi,j(~n) = ai,j(~n)ai,j(~n) i j

We performe Monte Carlo sampling of the amplitude,

(~ ~ ) = h ( )| (~ ~ )| ( )i Ai1,j1,...,iA,jA n1,..., nA, Lt ψI τ ρi1,j1,...,iA,jA n1,..., nA ψI τ

푖5, 푗5 푖7, 푗7 푖3, 푗3

휏푖 = 휏 푖2, 푗2

푖4, 푗4 푖6, 푗6

푖8, 푗8 휏/2 푖1, 푗1

Euclidean time Euclidean 휏푖 = 0 Pinhole algoritm: proton and neutron densities

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��SE, Epelbaum, Krebs, Lähde,�� Lee, Li, Lu, Meißner, Rupak, PRL 119, 222505 (2017) �� 18 16

FIG. S11. The magnitude of the 12C electric form factor, |F (q)|, versus momentum transfer q in units of fm−1. The error bars indicate one standard deviation errors from the stochastic noise of the Monte Carlo simulations. For comparison we also show experimental results [55].

100

10-1

10-2

|F(q)| experiment L = 7 t L = 9 t -3 L = 11 10 t L = 13 12 14 t Pinhole algoritm: C and C electricL = 15 form factor t 10-4 0 0.5 1 1.5 2 2.5 3 q (fm-1) 16

12 FIG. S12. The magnitude−1 of the 14C electric form factor, |F (q)|, versus momentum transfer q in units of fm−1. The error bars indicate one FIG. S11. The magnitude of the C electric form factor, |F (q)|, versus12 momentum transfer q in units of fm . The error bars indicate one 14 standard deviation errors from the stochasticThe magnitude noise of the Monte of the Carlo simulations.C electric Forstandard comparison form deviationfactor we also errors show from experimental the stochasticThe results magnitude noise [55]. of the Monte of the Carlo simulations.C electric For comparison form factor we also show experimental results [32].

100 100

10-1 10-1

10-2 10-2

|F(q)| experiment

|F(q)| experiment L = 7 L = 7 t t L = 9 L = 9 t t -3 L = 11 -3 L = 11 10 t 10 t L = 13 L = 13 t t L = 15 L = 15 t t 10-4 10-4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 q (fm-1) q (fm-1)

FIG. S12. The magnitude of the 14C electric form factor, |F (q)|, versus momentum transfer q in units ofTABLE fm−1. I.The Observed error bars charge indicate radii one from electron scattering and proton and neutron radii at leading order. SE, Epelbaum, Krebs, Lähde, Lee, Li, Lu, Meißner, Rupak, PRL 119, 222505 (2017) standard deviation errors from the stochastic noise of the Monte Carlo simulations. For comparison we also show experimentalnucleus resultsobserved [32]. charge radius proton radius (LO) neutron radius (LO) 12C 2.472(16) fm [56], 2.481(6) fm [57] 2.40(7) fm 2.39(5) fm 100 14C 2.497(17) fm [56] 2.43(7) fm 2.56(7) fm 16C — 2.46(10) fm 2.65(8) fm

10-1 [2] C. Beck, ed., Clusters in Nuclei, vol. 3 of Lecture Notes in Physics (Springer-Verlag, 2014). [3] Y. Funaki, H. Horiuchi, and A. Tohsaki, Prog. Part. Nucl. Phys. 82, 78 (2015). 10-2 [4] M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, and U.-G. Meißner (2017), 1705.06192. |F(q)| experiment [5] F. Hoyle, Astrophys. J. Suppl. 1, 121 (1954). L = 7 t [6] C. Cook, W. A. Fowler, C. C. Lauritsen, and T. Lauritsen, Phys. Rev. 107, 508 (1957). L = 9 t -3 L = 11 10 t L = 13 t L = 15 t 10-4 0 0.5 1 1.5 2 2.5 3 q (fm-1)

TABLE I. Observed charge radii from electron scattering and proton and neutron radii at leading order. nucleus observed charge radius proton radius (LO) neutron radius (LO) 12C 2.472(16) fm [56], 2.481(6) fm [57] 2.40(7) fm 2.39(5) fm 14C 2.497(17) fm [56] 2.43(7) fm 2.56(7) fm 16C — 2.46(10) fm 2.65(8) fm

[2] C. Beck, ed., Clusters in Nuclei, vol. 3 of Lecture Notes in Physics (Springer-Verlag, 2014). [3] Y. Funaki, H. Horiuchi, and A. Tohsaki, Prog. Part. Nucl. Phys. 82, 78 (2015). [4] M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, and U.-G. Meißner (2017), 1705.06192. [5] F. Hoyle, Astrophys. J. Suppl. 1, 121 (1954). [6] C. Cook, W. A. Fowler, C. C. Lauritsen, and T. Lauritsen, Phys. Rev. 107, 508 (1957). ��

��Pinhole algoritm: measure of alpha cluster geometry �� Consider the triangular shape formed by the three spin-up protons.

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20 20 SE, Epelbaum, Krebs, Lähde, Lee, Li, Lu, Meißner, Rupak, PRL 119, 222505 (2017)

20 Summary

Scattering and reaction processes involving alpha particle are in reach of ab initio methods and this has opened the door towards using experimental data from collisions of heavier nuclei as input to improve ab initio nuclear structure theory.

Understanding of the connection between the degree of locality of nuclear forces and nuclear structure has led to a more efﬁcient set of lattice chiral EFT interactions (to be proven by the higher order corrections).

The pinhole algorithm has been developed for the auxiliary-ﬁeld Monte Carlo methods for the calculation of arbitrary density correlations with respect to the center of mass.

Thanks! Extras