Similarity Renormalization Group Evolution of Three-Nucleon Forces in a Hyperspherical Momentum Representation
SSimilarity Renormalization Group Evolution of Three-Nucleon Forcesin a Hyperspherical Momentum Representation
K. A. Wendt ∗ Department of Physics, The Ohio State University, Columbus, OH 43210 (Dated: November 1, 2018)A new framework for computing the Similarity Renormalization Group (SRG) evolution of three-nucleon forces (3NF) in momentum representation is presented. The use of antisymmetric three-particle hyperspherical momentum states ensures unitary evolutions within certain basis truncations,much like antisymmetric harmonic oscillator SRG evolutions. Additionally, in each partial wavethe T rel-SRG regulator is exactly represented, similar to recent 3NF momentum representationevolutions. Unitary equivalence is demonstrated for the triton using several chiral two- plus three-nucleon interactions. This method allows for a clean visualization of the evolution of the three-nucleon forces, which manifests the SRG decoupling pattern and low-momentum universality. PACS numbers: 21.30.-x,05.10.Cc,13.75.Cs
The Similarity Renormalization Group (SRG), as for-mulated for nuclei in Refs. [1–3], renormalizes andthereby softens inter-nucleon interactions. One solves aflow equation to generate a unitary flow of the Hamilto-nian H = T rel + V , dds H s = [ η s , H s ] , (1)where T rel is the relative kinetic energy and V consistsof all inter-nucleon interactions. The details of the floware controlled by choosing an anti-hermitian generator η s . Most of the studies in nuclear physics to date use η s = (cid:2) T rel , H s (cid:3) . (2)The T rel-SRG flow of two-nucleon forces (2NF) has beenstudied in detail using both momentum representation [1,4–7] and in a discrete harmonic oscillator (HO) basis [8].In contrast, three-nucleon forces (3NF) have primarilybeen studied in the HO basis [8–12] with only recent workin momentum representation [13, 14].We have developed an alternative momentum represen-tation SRG evolution that exploits hyperspherical har-monics (HH) to build a hybrid method combining therelative strengths of the previous HO SRG implemen-tations and the recent momentum representation SRGimplementations. This is achieved by representing thethree-body Hamiltonian in antisymmetric HH momen-tum states, which leads to a permutationally closed trun-cated representation of the interaction. Our hybrid mo-mentum representation method provides evolved interac-tion matrix elements that are applicable directly in infi-nite systems and are easily projected to a HO basis witharbitrary frequency for use in finite systems. Extensionto alternative SRG generators and four-body interactionsare straightforward. Finally, many features of the SRG ∗ Electronic address: [email protected] flow for the 3NF become easy to visualize in a HH mo-mentum representation.The partial-wave momentum representation flow equa-tion for the two-nucleon system can be written as dds V α,βs ( k, k (cid:48) ) = − ( k − k (cid:48) ) V α,βs ( k, k (cid:48) )+ (cid:88) γ (cid:90) ∞ q dq ( k + k (cid:48) − q ) V α,γs ( k, q ) V γ,βs ( q, k (cid:48) ) , (3)where Greek letters abbreviate the collection ( l, s, j, t, t z )of orbital angular momentum, spin, total angular mo-mentum, isospin, and isospin projection quantum num-bers. The first term on the right side of Eq. (3) drivesthe decoupling and also suggests a different parameter touse for studying the T rel-SRG: λ = (cid:114) µ (cid:126) s − / , (4)where µ is the reduced mass for the relative Hamil-tonian and λ has units of fm − . When solving the T rel-SRG flow equation for the 2NF, the partial wavesdecouple into 1 × × V s = 3 V s N + V s N and the three-body T rel (see [8]). Because the antisymmetrizer is block di-agonal in this basis, and the initial truncated Hamilto-nian is completely contained in some finite set of blocks,the SRG flow equation will not induce non-zero matrixelements into blocks outside of the original truncation. a r X i v : . [ nu c l - t h ] A p r The evolved 2NF matrix elements can be embeddedand subtracted from V s to isolate the evolved three-bodyforce, V Ns = V s − V Ns . While being formally exact, fi-nite numerical precision means the subtraction leaves theevolved 3NF with some small residual from the evolved2NF matrix elements. This may have an effect when us-ing the evolved 3NF in a larger nucleus [11, 12]), howeverthis effect has not been studied in detail. An additionalissue with using a discrete basis is that the T rel-SRG flowequation induces a regulator that is naturally describedin a continuum representation. Evolving in momentumrepresentation and then embedding in the HO basis canyield different evolved matrix elements than evolving di-rectly in the HO basis due to truncated contractions inthe flow equation. Additionally, for each oscillator pa-rameter used in an HO basis calculation, the SRG evo-lution needs to be recomputed. For these reasons, it ismore natural to solve for the three-body flow equationsforces in momentum representation and then project intothe HO basis.In a recent implementation of a momentum represen-tation SRG 3NF [13], the 3NF is represented in a partial-wave basis, (cid:12)(cid:12) k k ; [( l s ) j ( l ) j ] J ; ( t ) T T z (cid:11) ≡ | k k ; α (cid:105) , (5) (cid:104) k (cid:48) k (cid:48) ; α (cid:48) | k k ; α (cid:105) = δ ( k − k (cid:48) ) k δ ( k − k (cid:48) ) k δ α , α (cid:48) , (6)where subscript 1 refers to the relative coordinates forparticle 1 and 2, subscript 2 refers to relative coordinateof the third particle in the center of mass frame of par-ticles 1 and 2, and J, T, T z are total coordinates for thethree-body systems. Solving the three-body SRG flowequation in this momentum representation comes withtwo related difficulties. Expressing the 2NF in this ba-sis, there is a spectator delta function that is numeri-cally difficult to represent. To overcome this, Ref. [13]applied a separated form of the the flow equation fromRef. [1]. This form expresses the flow equation as coupledflow equations for the 2NF and 3NF, where in the 3NFflow equation, the 2NF spectator delta function can inte-grated out analytically. Upon solving the flow equation,isolated solutions for the 2NF and 3NF are generatedautomatically, and the resulting forces can be projectedinto a HO basis with any oscillator parameter for usein many-body calculations. Since the 3NF flow equa-tion is explicitly separated from the 2NF flow equation,the possible 2NF contamination in the 3NF matrix ele-ments, which can occur in the HO SRG evolutions, is notpresent. A second difficulty is that anti-symmetrizationof the 2NF is not block diagonal within any partial-wavetruncation of this basis. This enters through terms inthe flow equation such as A V N A × T rel × V N , where A is the antisymmetrizer, will induce non-zero matrixelements in partial waves outside the initial truncation.Ignoring these matrix elements generates a unitary vi-olating error. In current evolutions based on [13], thiserror is at the ∼ . | k k ; α (cid:105) → | QG ; α (cid:105) , (7) (cid:104) QG ; α | Q (cid:48) G (cid:48) ; α (cid:48) (cid:105) = δ ( Q − Q (cid:48) ) Q δ G,G (cid:48) δ α , α (cid:48) , (8)where Q = (cid:112) k + k is the hyper-momentum and G is the grand angular momentum. Using results fromRefs. [17, 18], we embed the 2NF between particles 1and 2 into the three-body HH momentum basis.For identical particles (equal mass), all dependence ofany orthogonal transformation operator ( O ), such as per-mutation operators, is completely encoded between thepartial waves and is block diagonal in G [19], thereforethey are independent of the remaining continuous coor-dinate Q , (cid:104) Q (cid:48) G (cid:48) ; α (cid:48) | O | QG ; α (cid:105) = (cid:104) G ; α (cid:48) | O | G ; α (cid:105) δ G (cid:48) ,G δ ( Q − Q (cid:48) ) Q . (9)We build the antisymmetrizer A in blocks of fixed G ,diagonalize it, and construct a set of linearly independentantisymmetric states in a manner similar to Refs. [20, 21], A | Gi (cid:105) = | Gi (cid:105) , (10) | Gi (cid:105) = (cid:88) α c Gi, α | G ; α (cid:105) . (11)We project from our initial basis into a complete set ofnon-degenerate antisymmetric states that have definite Q and G , (cid:104) Q (cid:48) G (cid:48) i (cid:48) | V | QGi (cid:105) = (cid:88) n, α ,n (cid:48) , α (cid:48) c G (cid:48) i (cid:48) , α (cid:48) c Gi, α (cid:104) Q (cid:48) G (cid:48) α | ˆ3 V N + ˆ V N | QG α (cid:48) (cid:105) , (12)This will also build in interactions between particles 1 , ,
3. Now the Hamiltonian is translationally invari-ant and antisymmetric, computed in the manner of [21],but using a momentum representation instead of a dis-crete HO basis.This antisymmetric representation can be inserted intothe SRG flow equation. In this form, the flow equationfor the total 2+3NF is nearly identical in form to Eq. (3), dds V a,bs ( Q, Q (cid:48) ) = − ( Q − Q (cid:48) ) V a,bs ( Q, Q (cid:48) )+ (cid:88) c (cid:90) ∞ P dP ( Q + Q (cid:48) − P ) V a,cs ( Q, P ) V c,bs ( P, Q (cid:48) ) , (13) ∞ λ [fm − ]8.58.48.38.28.1 E B [ M e V ] Expt.550/600 MeVNN-onlyNN+3N-inducedNN+3N initial 450/500 MeV600/500 MeV450/700 MeV600/700 MeV − − − s [fm ] FIG. 1: (color online) HH SRG evolution of the triton bind-ing energy using a N LO 500 /
600 MeV interaction, includingonly the 2NF, the 2NF + induced 3NF, and the 2NF + full3NF (see Ref. [8]. The dashed lines are for the other N LO2NFs+3NFs( See [22, 23]). with a, b, c each specifying a hyperspherical channel | Gi ; JT T z (cid:105) . A critical difference between this flow equa-tion and the flow equation solved in [13] is that ourequation cannot induce non-zero matrix elements aboveour partial-wave truncation, provided all antisymmetricstates up to some G max are included. This enforcesunitary observables even when using a finite momen-tum partial-wave basis. Finally, we isolate the 3NF from V s ( Q, Q (cid:48) ) in a manner identical to what is done for HOevolutions. Full details on our embedding and flow equa-tion will be presented in a following work [27].Using the Sundials CVODE solver [24], we evolved sev-eral triton Hamiltonians, setting the proton and neutronmass to twice the proton-neutron reduced mass. This isdone to simplify the evolution and embedding of different2NF isospin channels for this initial work. We truncatethe hyperspherical harmonic basis at G max = 20. Weuse the chiral N LO 2N+3N Hamiltonians from [22, 23]to explore unitary renormalization of the chiral effectivetheory with inital 3NFs while allowing a direct compari-son with the 3NF SRG evolutions presented in Ref [13].These approximations give a result for the triton bind-ing energy within ∼ − .
482 MeV [23]), using 60 momentum states in eachpartial wave. Analyzing our convergence with respect to G max indicates that the remaining error is dominated byour approximate handling of the different nucleon masses,and finite quadrature effects. Q [fm]10 -3 -2 -1 Q | Ψ ( Q ) | [ f m ] λ [fm − ] ∞ . . FIG. 2: (color online) SRG evolution of the triton probabilitydistribution as a function of hypermomentum Q for severaldifferent SRG λ s for the interactions in Fig. 1. These areplotted as bands that span the range of the wave functionsfrom different initial 2N+3N N LO interactions.
Figure 1 shows the SRG evolution of the triton bindingenergy for the N LO 550 /
600 MeV interaction using only2N and 2N+3N induced forces, as well as the binding en-ergy for various the N LO 2NFs + 3NFs. Using onlythe evolved 2N force, we see a non-unitary flow. This isfrom a missing induced 3NF, which when included yieldsa unitary flow. This behavior is identical to what hasbeen documented in Refs. [8, 13]. Since we maintain ex-act antisymmetry in our finite truncated basis, the errorin our ground-state eigenvalue is determined by the pre-cision of our ODE solver. For the evolutions presented,we find this error to be about 1 eV.In Fig. 2, we plot the evolution of the triton momen-tum distribution, Q (cid:80) Gi (cid:104) QGi | Ψ (cid:105) . Each color band isthe span of the hyper-momentum probability for a setof different initial 2N+3N N LO interactions. As theSRG renormalizes the interactions, we see two criticalfeatures. The first and most drastic is the suppressionof the high-momentum tail. This feature has been wellstudied for the deuteron [3, 25] and is expected for all T rel-SRG evolved wave functions. The second criticalfeature is that the range spanned by wave functions fordifferent interactions collapses to a universal form, signal-ing low-momentum universality, again in a manner verysimilar to what has been seen in the deuteron [25]. Thiscollapse is also seen in individual partial waves.The interaction is plotted in Figs. 3 and 4 for the low-est fully antisymmetric partial wave along the top rowas a function of the SRG flow parameter λ . Along thebottom row is a two-body channel, a mixture of S and FIG. 3: (color online) Color Contour plot of the 3NF (upper row) and 2NF (lower row) as a function of the λ for the 550/600MeV potential. The lowest antisymmetric H.H. partial wave is plotted as well as the two-body partial waves that in embeddedkernel for for this three-body partial wave.FIG. 4: (color online) Same as Figure 3 but using the 600/700 MeV potential.FIG. 5: (color online) Contour plot of the integrand of triton 3NF expectation value, I ( Q, Q (cid:48) ) = M (cid:126) Ψ i ( Q )Ψ † j ( Q (cid:48) )( QQ (cid:48) ) V NFi,j ( Q, Q (cid:48) ). The upper row is the integrand for several initial N LO chiral potentials, whilethe lower row is the integrand for the SRG evolved interactions. S , that is embedded in this three-body partial wave. Asthe interaction is softened via the SRG, we see the banddiagonalization/decoupling that has been described forthe 2NF in Ref. [1], which is as expected from the flowequation Eq. (13). Also similar to the 2NF SRG results,we find that the 3NF interactions are collapsing to a uni-versal low-momentum form, though the degree of univer-sality is not as extreme as seen for the 2NF evolutions(see [3, 4]). We see a similar pattern in the other lowlying partial waves.In Fig. 5, we plot the integrand for triton expectationvalue of the 3NF, I ( Q, Q (cid:48) ) = ( Q (cid:48) Q ) (cid:88) G,i ; G (cid:48) i (cid:48) (cid:104) Ψ | QGi (cid:105)(cid:104) Q (cid:48) G (cid:48) i (cid:48) | Ψ (cid:105)× (cid:104) QGi | V N | Q (cid:48) G (cid:48) i (cid:48) (cid:105) . (14)The SRG moves critical strength from the high-momentum region of the integrand to low momentum.In the process, it develops universal integrands for the3NF’s expectation value, rendering the low-momentumuniversality obvious.A feature in our interactions is an apparent non-smoothness at small momenta ( ∼ − ) and near thediagonal. These can seen in Fig. 3 near the diagonalaround 1 fm − for λ = 1 .
40 fm − and in Fig. 5 in the lowerpanels. This error is not seen when projecting matrix el-ements from [13] into our representation. Therefore, itseems to be a result of computing the 3NF by subtractingthe evolved 2NF from the evolved 2NF+3NF matrix el-ements, similar to the HO evolution. This suggests thatthis small error may exist in the oscillator matrix ele-ments that have been used in recent calculations (suchas [9, 12, 26]), which could become significant in largersystems and should be studied further.Another advantage of our approach is that our mo-mentum representation Hamiltonian can be completely diagonalized on a given momentum quadrature. Exploit-ing this, is it possible to construct the unitary operatordescribes the SRG, permitting the evolution of variousoperators within momentum space.We have developed a fully unitary momentum repre-sentation SRG 3NF evolution and applied it to chiralN LO inter-nucleon interactions in the triton channel.This was accomplished by expressing the Hamiltonian infully antisymmetric hyperspherical plane waves. Exam-ining the evolution in this representation, we find that allfeatures observed in 2NF momentum representation evo-lutions are manifest in the triton system: suppression ofhigh-momentum tail, collapse of wave functions and in-teractions from different initial Hamiltonians to univer-sal forms, decoupling between high- and low-momentumstates, unitarity of the eigenvalues to the precision ofODE solver used. In the process, we identified featuresof evolved 3NF interactions that are a result of the finiteprecision cancellation of the 2NF, for which the natureand effect in heavier systems needs to be investigated.Using the HH momentum representation, the momentumrepresentation evolution of the four-nucleon force shouldbe straightforward.We gratefully acknowledge R. J. Furnstahl, K. Hebeler,and H. Hergert for many helpful discussions, and R.J. Perry, S. More and B. S. Dainton for useful com-ments, and E. Epelbaum and A. Nogga for provid-ing the initial N LO 3NF matrix elements to the OSUgroup and N. Barnea for notes on HH calculations.I thank the Ohio Supercomputing Center for the useof their computational resources. This work was sup-ported in part by the National Science Foundation un-der Grants No. PHY–0758125 and No. PHY–1002478,the UNEDF SciDAC Collaboration under DOE GrantDE-FC02-07ER41457, and by the DOE Office of Sci-ence Graduate Fellowship (SCGF) program under DOEcontract number DE-AC05-06OR23100. [1] S. K. Bogner, R. J. Furnstahl, and R. J. Perry, Phys.Rev. C (2007).[2] S. K. Bogner, R. J. Furnstahl, R. J. Perry, andA. Schwenk, Phys. Lett. B , 488 (2007).[3] E. D. Jurgenson, S. K. Bogner, R. J. Furnstahl, and R. J.Perry, Phys. Rev. C , 014003 (2008).[4] K. A. Wendt, R. J. Furnstahl, and R. J. Perry, Phys.Rev. C (2011).[5] K. A. Wendt, R. J. Furnstahl, and S. Ramanan, Phys.Rev. C , 014003 (2012).[6] W. Li, E. R. Anderson, and R. J. Furnstahl, Phys. Rev.C , 054002 (2011).[7] E. R. Anderson, S. K. Bogner, R. J. Furnstahl, E. D.Jurgenson, R. J. Perry, and A. Schwenk, Phys. Rev. C , 037001 (2008).[8] E. D. Jurgenson, P. Navr´atil, and R. J. Furnstahl, Phys.Rev. Lett. , 082501 (2009).[9] E. Jurgenson, P. Maris, R. Furnstahl, P. Navratil, W. Or- mand, et al. (2013), 1302.5473.[10] E. D. Jurgenson, P. Navr´atil, and R. J. Furnstahl, Phys.Rev. C , 034301 (2011).[11] S. Binder, J. Langhammer, A. Calci, P. Navr´atil, andR. Roth, Phys. Rev. C , 021303 (2013), ISSN 0556-2813.[12] R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer,and P. Navr´atil, Phys. Rev. Lett. , 052501 (2012).[13] K. Hebeler, Phys. Rev. C , 021002 (2012).[14] O. Akerlund, E. J. Lindgren, J. Bergsten, B. Grevholm,P. Lerner, R. Linscott, C. Forss´en, and L. Platter, Eur.Phys. J. A , 122 (2011).[15] K. Hebeler, private communicaton.[16] N. Barnea, W. Leidemann, and G. Orlandini, Nucl. Phys.A , 565 (2001).[17] M. Viviani, L. E. Marcucci, S. Rosati, A. Kievsky, andL. Girlanda, Few-Body Syst. , 159 (2006).[18] A. Novoselsky and N. Barnea, Phys. Rev. A , 2777 (1995).[19] J. Raynal and J. Revai, Nuovo Cimento A , 612 (1970).[20] N. Barnea, Phys. Rev. A , 1135 (1999).[21] P. Navr´atil, G. Kamuntaviˇcius, and B. Barrett, Phys.Rev. C (2000).[22] E. Epelbaum, Progress in Particle and Nuclear Physics , 654 (2006), ISSN 01466410.[23] E. Epelbaum, A. Nogga, W. Gl¨ockle, H. Kamada, U.-G.Meiß ner, and H. Witaa, Phys. Rev. C , 064001 (2002).[24] A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward, ACMTransactions on Mathematical Software , 363 (2005).[25] E. R. Anderson, S. K. Bogner, R. J. Furnstahl, and R. J.Perry, Phys. Rev. C , 054001 (2010).[26] H. Hergert, S. K. Bogner, S. Binder, a. Calci, J. Lang-hammer, R. Roth, and a. Schwenk, Phys. Rev. C87