arXiv:1203.1779v1 [nucl-th] 8 Mar 2012 · 2012. 3. 9. · ily accomplished for spatial images...

Nuclear Physics B Proceedings Supplement 00 (2012) 1–37

Nuclear Physics BProceedingsSupplement

The Renormalization Group in Nuclear Physics

R. J. Furnstahl

Department of Physics, Ohio State University, Columbus, OH 43210, USA

Abstract

Modern techniques of the renormalization group (RG) combined with effective field theory (EFT) methods arerevolutionizing nuclear many-body physics. In these lectures we will explore the motivation for RG in low-energynuclear systems and its implementation in systems ranging from the deuteron to neutron stars, both formally and inpractice. Flow equation approaches applied to Hamiltonians both in free space and in the medium will be empha-sized. This is a conceptually simple technique to transform interactions to more perturbative and universal forms. Anunavoidable complication for nuclear systems from both the EFT and flow equation perspective is the need to treatmany-body forces and operators, so we will consider these aspects in some detail. We’ll finish with a survey of currentdevelopments and open problems in nuclear RG.

Keywords: Renormalization group, nuclear structure, three-body forces

1. Overview

The topic of these lectures is the use of renormal-ization group (RG) methods in low-energy nuclear sys-tems, which include the full range of atomic nuclei aswell as astrophysical systems such as neutron stars. Wewill examine why the RG has become an increasinglyuseful tool for nuclear physics theory over the last tenyears and consider how to apply RG technology bothformally and in practice. Of particular emphasis will beflow equation approaches applied to Hamiltonians bothin free space and in the medium, which are an accessiblebut powerful method to make nuclear physics computa-tionally more like quantum chemistry. We will see howinteractions are evolved to increasingly universal formand become more amenable to perturbative methods. Akey element in nuclear systems is the role of many-bodyforces and operators; dealing with their evolution is animportant on-going challenge.

The expected background for these lectures is a thor-ough knowledge of nonrelativistic quantum mechanics,

Email address: [email protected] ()

including scattering, the basics of quantum field the-ory, and linear algebra (it’s all matrices!). We will notassume a knowledge of nuclear structure or reactions,or even many-body physics beyond Hartree-Fock. Noadvanced computing experience is assumed (althoughMathematica or MATLAB knowledge will be very help-ful in exploring simple RG examples).

By necessity, we will only scratch the surface in theselectures. For a thorough treatment of flow equations formany-body systems not including nuclei, see the bookby Kehrein [1]. For more details on applications offlow-equation and similar renormalization group meth-ods to low-energy nuclear physics, the review article [2]and references therein are recommended.

2. Atomic nuclei at low resolution via RG

2.1. Goals and scope of low-energy nuclear physics

The playing field for low-energy nuclear physics isthe table of the nuclides, shown in Fig. 1. There are sev-eral hundred stable nuclei (black squares) but also sev-eral thousand unstable nuclei are known through exper-imental measurements. The total number of nuclides is

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v1 [

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R.J. Furnstahl / Nuclear Physics B Proceedings Supplement 00 (2012) 1–37 2

still unknown (see the region marked “terra incognita”),with theoretical estimates suggesting it could be as highas ten thousand! These unstable nuclei are the objectof scrutiny for new and planned experimental facilitiesaround the world. The challenge for low-energy nucleartheory is to describe their structure and reactions. We’llreturn in the final lecture to discuss the overlapping re-gions where the theoretical many-body methods listedin the figure can be applied.

Let’s start with some of the questions that drive low-energy nuclear physics research. These include generalquestions about the physics of nuclei [3]:

• How do protons and neutrons make stable nucleiand rare isotopes? Where are the limits?

• What is the equation of state of nucleonic matter?

• What is the origin of simple patterns observed incomplex nuclei?

• How do we describe fission, fusion, and other nu-clear reactions?

These topics inform and are in turn illuminated by ap-plications to other fields, such as astrophysics, whereone can ask:

• How did the elements from iron to uranium origi-nate?

• How do stars explode?

• What is the nature of neutron star matter?

or of fundamental symmetries:

• Why is there now more matter than antimatter inthe universe?

• What is the nature of the neutrinos, what are theirmasses, and how have they shaped the evolution ofthe universe?

Finally, there are applications, for which we are led toask: How can our knowledge of nuclei and our abil-ity to produce them benefit humankind? The impact isvery broad, encompassing the Life Sciences, MaterialSciences, Nuclear Energy, and National Security.

In Figure 2, the energy scales of nuclear physics areillustrated. There is an extended hierarchy, which is achallenge, but also an opportunity to make use of ef-fective field theory (EFT) and renormalization group(RG) techniques. The ratio of scales can become an ex-pansion parameter, leading to a systematic treatment atlower energies. The progression from top to bottom can

Figure 1: The nuclear landscape. A nuclide is specified by the numberof protons and neutrons [3].

Nuclear structure Nuclear reactions

Hot and dense quark-gluon matter

Hadron structure

Nuclear astrophysics New standard model

Applications of nuclear science

Hadron-Nuclear interface

Res

olut

ion

Effe

ctiv

e Fi

eld

Theo

ry

Figure 2: Hierarchy of nuclear degrees of freedom and associated en-ergy scales [3].


Figure 3: Some phenomenological potentials that accurately describeproton-neutron scattering up to laboratory energies of 300 MeV [4].

be viewed as a reduction in resolution. In these lectures,our focus is on the intermediate region only, where pro-tons and neutrons are the relevant degrees of freedom.But even within this limited scope, the concept of reduc-ing resolution by RG methods is extremely powerful.

2.2. Lowering the resolution with RG

What do we mean by resolution? Even the generalpublic these days is familiar with the concept of dig-ital resolution for computer screens, cellphones, tele-visions. High resolution is associated with more pix-els, which follows because as pixel size becomes smallcompared to characteristic scales in an image, greaterdetail is seen. In our discussion, we associate resolutionwith the Fourier transform space and the phenomenonof diffraction.

Recall the basic physics: if the wavelength of light iscomparable to or larger than an aperture, then diffractionis significant. If there are two sources, we say we canresolve them if the diffracted images don’t overlap toomuch. For a fixed angle between sources or details inthe object being observed, we find that the wavelengthdetermines whether or not we resolve the details. Beingunable to resolve details at long wavelength is generallyconsidered to be a disadvantage (e.g., for astronomicalobservations), but we turn it to an advantage.

A fundamental principle of any effective low-energydescription (not restricted to nuclear physics!) is thatif a system is probed at low energies, fine details are

Figure 4: Momentum space representation of the Argonne v18 (AV18)NN potential in the 1S0 channel [2].

not resolved, and one can instead use low-energy vari-ables for low-energy processes. Renormalization theorytells us that the short-distance structure can be replacedby something simpler without distorting low-energy ob-servables. The familiar analog from classical electro-dynamics is the replacement of a complicated chargeor current distribution with a truncated multipole ex-pansion. In the quantum case, the replacement can bedone by constructing a model, or in a systematic wayusing effective field theory. We emphasize that whileobservable quantities (such as cross sections) do notchange, the physics interpretation can (and generallydoes) change with resolution. What if there is no exter-nal probe? Then the particles still probe each other withresolution set by their de Broglie wavelengths. Low-density nuclear systems would seem to imply low reso-lution. But the picture is complicated by the nature oftraditional internucleon potentials.

Figure 3 shows several phenomenological potentialsthat reproduce nucleon-nucleon scattering phase shiftsup to about 300 MeV in lab energy. They are charac-terized by a long-range attractive tail from one-pion ex-change, intermediate attraction, and a strongly repulsiveshort-range “core”. For our purposes, the partial-wavemomentum-space representation, such as

〈k|VL=0|k′〉 ∝∫

r2 dr j0(kr) V(r) j0(k′r) (1)

for S-waves is more useful. Here k and k′ are the rel-ative momenta of the two nucleons. This is shown forthe AV18 potential in Fig. 4 in the 1S0 channel. (Thespin and isospin dependence of the nuclear interactions


Figure 5: Alternative momentum space representation of the AV18potential in the 1S0 channel [2].

is very important, unlike the situation in quantum chem-istry.) In the momentum basis, the potentials in Fig. 3are no longer diagonal, so we need three-dimensionalinformation, but we generally use the flat contour repre-sentation of the same information, as in Fig. 5. The RGevolution of potentials will be visualized as changes insuch pictures.

We work in units for which ~ = c = 1. Then thetypical relative momentum in the Fermi sea of any largenucleus is of order 1 fm−1 or 200 MeV. However, it isevident from Figs. 4 and 5 that there are large matrixelements connecting such momenta to much larger mo-menta. This is directly associated with the repulsivecore of the potential. For our discussion, we will adopt2 fm−1 as the (arbitrary but reasonable) dividing line be-tween low and high momentum for nuclei.

The consequences of the coupling to high momen-tum are readily seen in the probability density of theonly two-body bound state, the deuteron. Consider theArgonne v18 [5] curve in Fig. 6. The probability atsmall separations is significantly suppressed as a resultof high-momentum components in the wave function.This suppression, called “short-range correlations” inthis context, carries over to many-body wave functionsand greatly complicates basis expansions. For exam-ple, in a harmonic oscillator basis, which is frequentlythe choice for self-bound nuclei because it readily al-lows removal of center-of-mass contamination, conver-gence is greatly slowed by the need to accommodatehigh-momentum components. The factorial growth ofthe basis size with the number of nucleons then greatlylimits the reach of calculations.

0 2 4 6r [fm]

0

0.05

0.1

0.15

0.2

0.25

|ψ(r)

|2 [fm

-3]

Argonne v18

λ = 4.0 fm-1

λ = 3.0 fm-1

λ = 2.0 fm-1

3S1 deuteron probability density

Figure 6: Short-range correlations in the deuteron 3S1 probability den-sity from the AV18 potential (solid line). They are essentially elimi-nated by the RG evolution to lower flow parameter λ (see Section 3.1).

The underlying problem is that the resolution scaleinduced by the potential is mismatched with the basicscale of the low-energy nuclear states (given, for exam-ple, by the Fermi momentum). A solution is to elim-inate the coupling to high momentum. This is read-ily accomplished for spatial images (i.e., photographs)by Fourier transforming and then applying a low-passfilter—simply set the short wavelength parts to zero—and then transforming back. Let’s try that for ourHamiltonian by setting to zero all of the matrix elementsin Fig. 5 for k > 2 fm−1. We test the implications by see-ing the effect on scattering phase shifts in the region oflaboratory energies corresponding to k ≤ 2 fm−1.

The result is shown in Fig. 7. It would be unsurpris-ing that our filtered Hamiltonian fails close to the cut-off, but it is evident that there is a failure at all energies.What happened? The basic problem is that low mo-mentum and high momentum are coupled when solv-ing quantum mechanically for observables. For exam-ple, consider perturbation theory for the (tangent) of thephase shift (represented schematically here):

〈k|V |k〉 +∑

k′

〈k|V |k′〉〈k′|V |k〉(k2 − k′2)/m

+ · · · (2)

where a low momentum k is mixed with all other mo-menta at second order to a degree based on the size ofthe off-diagonal matrix elements. (As a computationalaside, although momentum is continuous in principle,in practice we work on a discrete grid. This means that


0 100 200 300E

lab (MeV)

−20

0

20

40

60

phas

e sh

ift

(deg

rees

)

1S

0

AV18 phase shifts

after low-pass filter

k = 2 fm−1

Figure 7: Effect of a low-pass filter on observables: the 1S0 phaseshifts. Note that the AV18 phase shifts reproduce experimentally ex-tracted phase shifts in this energy range.

Eq. (2) becomes a matrix equation. For two-body po-tentials, roughly 100 × 100 matrices are sufficient.) Thephase shift even at low energy or k will get large con-tributions from high k′ if the coupling matrix elements〈k|V |k′〉 are large.

How can we fix this? Our solution is to use a (short-distance) unitary transformation of the Hamiltonian ma-trix to decouple low and high energies. That is, insertthe operator U†U = 1 repeatedly:

En = 〈Ψn|H|Ψn〉 = 〈Ψn|U†)UHU†U |Ψn〉)≡ 〈Ψn|H|Ψn〉 . (3)

In doing so we have modified operators and wavefunc-tions but observables (measurable quantities) are un-changed. An appropriate choice of the unitary trans-formation can, in principle, achieve the desired decou-pling. This general approach has long been used in nu-clear structure physics and for other many-body applica-tions. The new feature here is the use of renormalizationgroup flow equations to create the net unitary transfor-mation via a series of infinitesimal transformations.

The renormalization group is well suited to this pur-pose. The common features of RG for critical phenom-ena and high-energy scattering are discussed by StevenWeinberg in an essay in Ref. [6]. He summarizes:

“The method in its most general form can Ithink be understood as a way to arrange invarious theories that the degrees of freedomthat you’re talking about are the relevant de-grees of freedom for the problem at hand.”

This is the essence of what is done with the low-momentum interaction approaches considered here: ar-range for the degrees of freedom for nuclear structureto be the relevant ones. This does not mean that otherdegrees of freedom cannot be used, but to again quoteWeinberg [6]: “You can use any degrees of freedom youwant, but if you use the wrong ones, you’ll be sorry.”

The consequences of using RG for high-energy (par-ticle) physics include improving perturbation theory,e.g., in QCD. A mismatch of energy scales can gener-ate large logarithms that ruins perturbative convergenceeven when couplings by themselves are small. The RGshifts strength between loop integrals and coupling con-stants to reduce these logs. For critical phenomena incondensed matter systems, the RG reveals the natureof observed universal behavior by filtering out short-distance degrees of freedom. We will see both theseaspects in our calculations of nuclear structure and re-actions. The end result can be said to make nuclearphysics look more like quantum chemistry, opening thedoor to a wider variety of techniques (such as many-body perturbation theory) and simplifying calculations(e.g., by improving convergence of basis expansions).

2.3. Summary pointsLow-energy nuclear physics is made difficult by a

mismatch of energy scales inherent in conventional phe-nomenological potentials and those of nuclear structureand reactions. The renormalization group offers a wayout by lowering the resolution, which means decouplinglow- from high-momentum degrees of freedom.

3. Overview of flow equations

In Fig. 8, we show schematically two options for howthe RG can be used to decouple a Hamiltonian matrix.The more conventional approach on the left in Fig. 8lowers a cutoff Λ in momentum in small steps, withthe matrix elements adjusted by requiring some quan-tity such as the on-shell T-matrix to be invariant. (Inpractice this may be carried out by enforcing that thehalf-on-shell T-matrix is independent of Λ.) Matrix el-ements above Λ are zero and are therefore trivially de-coupled. When adapted to low-energy nuclear physics,this approach is typically referred to as “Vlow k” [7, 2].

The approach we will focus on is illustrated in Fig. 8on the right, in which the matrix is driven toward band-diagonal form, achieving decoupling again but withouttruncating the matrix. The corresponding RG was de-veloped in the early 1990’s by Wegner [8, 9, 10] for con-densed matter applications under the name “flow equa-tions” and independently by Glazek and Wilson [11] for


Λ0

Λ1

Λ2

k’

k

λ0

λ1

λ2

k’

k

Figure 8: Schematic Vlow k RG evolution (left) and flow equation RG evolution (right).

Figure 9: SRG flow of the AV18 NN potential in the 3S1 channel at selected values of the flow parameter λ [2].

solving quantum chromodynamics in light-front formal-ism under the name “similarity renormalization group”(SRG). Only in the last five years was it realized thatthe band-diagonal approach is particular well suited forlow-energy nuclear physics, where it is technically sim-pler and more versatile than other methods [12, 2]. Wewill apply formalism closer to the flow equation formu-lation, but generally use the simple abbreviation SRG.We have introduced a cutoff-like parameter λ that servesas a momentum decoupling scale. Elsewhere we willalso use the natural flow parameter s = 1/λ4.

3.1. Flow equation basics

The basic flow equation is a set of coupled differen-tial equations for the (discrete) matrix elements of theHamiltonian matrix (or the potential in practice, becausewe fix the kinetic energy matrix by construction). Let’ssee an example in action before stepping back and con-sidering details. For the two-body potential in a partial-

wave momentum basis, the flow equation takes the form

dVλ

dλ(k, k′) ∝ −(εk − εk′ )2Vλ(k, k′)

+∑

q

(εk + εk′ − 2εq)Vλ(k, q)Vλ(q, k′) , (4)

where k and k′ are the relative momenta and εk ≡

~2k2/M and we have omitted an inessential constant.The evolution is continuous, but snapshots at selectedλ values are shown in Fig. 9. (Note that the axes are thekinetic energy k2.) The evolution toward diagonaliza-tion is evident, with the width of the band in k2 roughlygiven by λ2.

It is evident that the off-diagonal matrix elements aredriven toward zero, so we expect that a low pass filterwill now be effective. Indeed it is, as shown in Fig. 10,where NN phase shifts for the AV18 potential in a va-riety of channels are compared with the results fromapplying a low-pass filter to the original (dotted) andevolved (dashed) potentials. The evolved result agreesup to the low-pass cutoff. Note that if this cutoff is notapplied, the phase shifts for the evolved Hamiltonian


0 500 1000-100

-50

0

50

AV18AV18 [kmax = 2.2 fm-1]

Vs [kmax = 2.2 fm-1]

0 500 1000

-50

0

50

100

0 500 1000

-30

-20

-10

0

0 500 1000

-40

-20

0

20

0 500 1000

Elab [MeV]

-10

-5

0

5

phas

e sh

ift [d

eg]

1S03S1

3P03F3

3D1

Figure 10: Nucleon-nucleon (NN) phase shifts showing the effect of a low-pass filter at kmax = 2.2 fm−1 in various partial-wave channels [13, 14].In each panel, the solid line is for the original AV18 potential while the dotted line is the result when the potential is set to zero for k, k′ > kmax.The dashed line is the results when the potential is evolved first and then cut at kmax.

agree precisely at all energies, because the transforma-tion is unitary in the two-body system (and this is pre-served to high numerical precision).

If we now revisit the consequence of a repulsive corefor the deuteron probability density, we see in Fig. 6that the “wound” in the wave function at small r is filledin as the core is transformed away. (Note: it may looklike the normalization is not conserved, but if we multi-plied by r2, we would see that the area under the curvesis the same, as are the large-r tails.) Thus the short-range correlations in the wave function are drasticallyreduced. This means that the physics interpretation ofvarious phenomena is altered, even though the observ-able quantities such as energies and cross sections areunchanged. (The long-range part of the wave functionis preserved; this is related to the asymptotic normal-ization constants, which can be extracted from experi-ment.) We cannot immediately visualize the changes inthe potential in coordinate space in a conventional plotlike Fig. 3, however, because it is now non-local, whichis to say it is not diagonal in coordinate representation:

V(r)ψ(r) −→∫

d3r′ V(r, r′)ψ(r′) . (5)

This is a technical problem for certain quantum many-body methods (such as Green’s function Monte Carlo)but not for methods using harmonic oscillator matrix el-

ements.We can visualize the evolution approximately by con-

sidering a local projection of the potential:

Vλ(r) =

∫d3r′ Vλ(r, r′) (6)

which leaves a local V(r) unchanged. For a non-localpotential, this roughly gives the action of the potentialon long-wavelength nucleons. This is shown in Figs. 11and 12, where in the former we see the central potentialdissolving and in the latter similar effects on the ten-sor part of the potential [15]. Also evident is the flow ofthe two potentials, initially quite different (potentials arenot observables!), toward a universal flow at the lowervalues of λ. Very recent work suggests that such a localprojection may capture most of the physics of the fullevolved potential, with the effects of the residual poten-tial calculable in perturbation theory. This may openthe door to using RG-evolved potential with quantumMonte Carlo methods.

The consequences of low-momentum potentials forharmonic-oscillator-basis calculations are illustrated inFigs. 13 and 14. (We’ll refer to the method used, whichis a direct diagonalization of the Hamiltonian matrix andtherefore is variational, as no-core full configuration, orNCFC.) The original potential in this case is alreadysoft (that is, there is much less coupling to high mo-


0 1 2 3 4r [fm]

−100

0

100

200

V(r

) [M

eV

]

λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

Figure 11: Local projection of AV18 and N3LO(500 MeV) potentials in 3S1 channel at different resolutions [15].

0 1 2 3 4r [fm]

−100

−50

0

50

V(r

) [M

eV

] λ = 20 fm−1

1 2 3 4r [fm]

λ = 4 fm−1

1 2 3 4r [fm]

λ = 3 fm−1

1 2 3 4r [fm]

λ = 2 fm−1

1 2 3 4r [fm]

λ = 1.5 fm−1

AV18

N3LO

Figure 12: Local projection of AV18 and N3LO(500 MeV) potentials in mixed 3S1–3D1 channel at different resolutions [15].

2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

Softenedwith SRG

VNNN

= N2LO

ground-state energy

Figure 13: Convergence of NCFC calculations of the 4He ground stateenergy with basis size at SRG different resolutions. The initial poten-tial includes both two- and three-body components [16, 17].

mentum than in the AV18 potential), but convergencein a harmonic oscillator basis with Nmax shells for ex-citation is slow (these are the “Original” curves). Notethat the matrix dimension grows rapidly with Nmax andthe number of nucleons A; for example Nmax = 8 hasdimension about 50,000 for 4He but over one million

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

und-S

tate

Ener

gy [

MeV

]

Lithium-6

expt.

Originalh- Ω = 20 MeV

Softened with SRG

ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

λ = 1.5 fm−1

λ = 2.0 fm−1

Figure 14: Convergence of NCFC calculations of the 6Li ground stateenergy with basis size at SRG different resolutions. The initial poten-tial includes both two- and three-body components [16, 17].

for 6Li (see Fig. 15 for other examples). But with SRGevolution, there is vastly improved convergence. Theconvergence is also smooth, which makes it possibleto reliably extrapolate partly converged results to theNmax → ∞ limit. Nevertheless, the rapid growth ofthe basis with A is still a major hindrance to calculat-


0 2 4 6 8 10 12 14

Nmax

101

102

103

104

105

106

107

108

109

Matr

ix d

imen

sio

n

4He

6Li

12C

16O

Figure 15: Dimension of the Hamiltonian matrix for NCFC/NCSMcalculations as a function of the harmonic oscillator basis size (Nmaxshells) for selected nuclei.

ing larger nuclei. One solution being explored by Rothand collaborators [18, 19] is to use importance samplingof matrix elements, in which only a fraction of the fullmatrix is used. This technique is enabled by the RGsoftening of the potential, which allows the importanceto be evaluated perturbatively.

Let’s now return to the basics of SRG flow equations.We wish to transform an initial hamiltonian, H = T + Vin a series of steps, labeled by the flow parameter s:

Hs = UsHU†s ≡ T + Vs (7)

withU†s Us = UsU†s = 1 . (8)

Note that the kinetic energy T is taken to be independentof s. Differentiating with respect to s:

dHs

ds=

dUs

dsU†s UsHU†s + UsHU†s Us

dU†sds

= [ηs,Hs] (9)

withηs ≡

dUs

dsU†s = −η†s . (10)

The anti-Hermitian generator ηs can be specified by acommutator of Hs with a Hermitian operator Gs:

ηs = [Gs,Hs] , (11)

which yields the flow equation (with T held fixed!),

dHs

ds=

dVs

ds= [[Gs,Hs],Hs] . (12)

The operator Gs determines the flow and there are manychoices one can consider.

Probably the simplest example we can consider is justa two-state system [20]. Let H = T + V , where

T |i〉 = εi|i〉 and Vi j ≡ 〈i|V | j〉 . (13)

Then we can choose Gs = T and

dHs

ds= [[T,Hs],Hs] (14)

becomes (with the s dependence implicit)

dds

Vi j = −(εi − ε j)2Vi j

+∑

k

(εi + ε j − 2εk)VikVk j . (15)

For a two-level system with i = a, b, we can expressthe flowing Hamiltonian in terms of Pauli matrices:

T =12

(εa + εb)I +12

(εa − εb)σz (16)

and

Vs =12

(Vaa + Vbb)I

+12

(Vaa − Vbb)σz + Vabσx . (17)

The solution to the SRG flow equation is easily found.It is convenient to parametrize the result in terms of θ(s)with constant ω:

dθds

= −2(εa − εb)ω sin θ(s) (18)

1 2 3 4 5 10

λ = 1/s1/4

0

2

4

6

8

En

erg

y

Vab

Vbb

Vaa

2-body matrix elements

εa = 3, ε

b = 9

Figure 16: SRG flow for a simple two-state system (see text).


Figure 17: SRG flow in stages. On the far left and far right are potentials in the 1S0 channel evolved to λ = 2.0 fm−1 and λ = 1.5 fm−1, respectively.The middle panels show the first (left) and second (right) terms on the right side of Eq. (23).

where

θ(s) = 2 tan−1[tan(θ(0)/2)e−2(εa−εb)ωs] , (19)

ω cos θ = (εa − εb + Vaa − Vbb)/2 , (20)

andω sin θ = Vab . (21)

The resulting flow is plotted for sample energies εa andεb in Fig. 16. We clearly see the off-diagonal matrix ele-ment Vab driven to zero. (Try reproducing this in Math-ematica!)

For a nuclear two-body (NN) potential in a partial-wave momentum basis with ηs = [T,Hs], we project onrelative momentum states |k〉 using 1 = 2

π

∫ ∞0 |q〉q

2 dq〈q|with ~2/M = 1. The flow equation reduces to:

dVs

ds= [[Trel,Vs],Hs] with Trel|k〉 = εk |k〉 (22)

and λ2 = 1/√

s. Trel is the relative kinetic energy of thenucleons. Then

dVλ

dλ(k, k′) ∝ −(εk − εk′ )2Vλ(k, k′)

+∑

q

(εk + εk′ − 2εq)Vλ(k, q)Vλ(q, k′) . (23)

This particular equation is for A = 2, but the results aregeneric if one lets k represent a set of Jacobi momenta.The first term in Eq. (23) drives Vλ toward the diagonal:

Vλ(k, k′) = Vλ=∞(k, k′) e−[(εk−εk′ )/λ2]2+ · · · , (24)

which can be visualized in Fig. 17. These panels repre-sent a sequence from λ = 2.0 to λ = 1.5. The potentialsat the beginning and the end are on the outside, whilethe two middle panels are the first and second term ofEq. (23). For off-diagonal matrix elements, the first termis numerically dominant and each element is driven tozero according to Eq. (24), with further off-diagonal ele-ments changing more rapidly. Note that the width of thediagonal is given roughly by λ2, in accord with Eq. (24).

A more general proof follows if we use the genera-tor advocated by Wegner, which includes the diagonalpart of the Hamiltonian, Hd. Call the diagonal elementsHii = ei, then (with ηs = [Hd,Hs]),

dHi j

ds= 〈i|[[Hd,Hs],Hs]| j〉

=∑

k

(ei + e j − 2ek)HikHk j

= 2∑

k

(ei − ek)|Hik |2 . (25)

But

dds

∑i

|Hii|2 = 2

∑i

HiidHii

ds

= 4∑i,k

ei(ei − ek)|Hik |2

= 2∑i,k

(ei − ek)2|Hik |2 ≥ 0 . (26)

Now use:

Tr H2s = const. =

∑ij

|Hij|2 =

∑i

|Hii|2 +

∑i,j

|Hij|2 ,

(27)to obtain

dds

∑i, j

|Hi j|2 = −

dds

∑i

|Hii|2

= −2∑i,k

(ei − ek)2|Hik |2 ≤ 0 . (28)

Thus, in the absence of degeneracies, the off-diagonalmatrix elements will decrease (or at least remain un-changed) [1].

This feature of the diagonal generator is desirable, butwe also note that for nuclear Hamiltonians in a momen-tum basis, the diagonal is completely dominated by thekinetic energy, so Hd ≈ Trel is a very good approxi-mation. Can this break down? Glazek and Perry [21]


showed that it can (see also Wendt et al. [22]). Re-consider the proof of diagonalization, but now withGs = Trel. Now we have Hii = ei and Trel|i〉 = εi|i〉,so that

dHii

ds= 2

∑k

(εi − εk)|Hik |2 . (29)

So consider

dds

∑i

|Hii|2 = 2

∑i

HiidHii

ds= 4

∑i,k

ei(εi − εk)|Hik |2 ,

(30)from which we conclude

dds

∑i, j

|Hi j|2 = −2

∑i,k

(ei − ek)(εi − εk)|Hik |2 . (31)

Thus the off-diagonal decrease depends on ei − ek ≈

εi − εk. But there is the possibility of this not being true,e.g., if there are spurious bound states as in large-cutoff

EFT [22].

3.2. Alternative generators

Other choices of generator are also possible. Recentwork by Shirley Li while an undergraduate physics ma-jor at Ohio State explored choices designed to accelerateevolution. In particular, one can choose Gs as

Gs = −Λ2

1 + Trel/Λ2 ≈ c + Trel + · · · (32)

or

Gs = −Λ2e−Trel/Λ2≈ c + Trel + · · · (33)

The expansions show that these reduce to the conven-tional Trel for momenta that are small compared to thecutoff parameter Λ (not to be confused with λ). ForΛ = 2 fm−1, the low energy part of the potential is stilldecoupled but there is much less evolution at high en-ergy, which makes it computationally much faster andallows evolution to low λ. See Ref. [23] for details.

The usual approach to Vlow k RG evolution (left dia-gram in Fig. 8) is based on the Lippmann-Schwingerequation for the half-on-shell T-matrix illustrated inFig. 18. A cutoff Λ is imposed on the integral in thesecond term and we demand that matrix elements of Tare invariant with an infinitesimal reduction of Λ. Thatis, we require dT (k, k′; Ek)/dΛ = 0, which establishesan RG equation for VΛ. In contrast to the SRG equation,which is second order in the running potential, the Vlow k

T

−k +k

−k′ +k′

= VΛ

−k +k

−k′ +k′

+ q ≤ Λ

VΛ

T

Figure 18: Schematic version of the Lippmann-Schwinger equationfor the T-matrix, with cutoff Λ on the intermediate states. The Vlow kpotential VΛ is determined by requiring half-on-shell matrix elementsof this equation to be invariant under changes in Λ[7, 2].

RG equation has the T-matrix on the right side. Thus

T (k′, k; k2) = VNN(k′, k)

+2πP

∫ Λ∞

0

VNN(k′, p) T (p, k; k2)k2 − p2 p2dp

= VΛlow k(k′, k)

+2πP

∫ Λ

0

VΛlow k(k′, p) T (p, k; k2)

k2 − p2 p2dp

(34)

for all k, k′ < Λ. (Note: we are using standing-waveboundary conditions for numerical reasons; this is oftencalled the K-matrix.) From dT/dΛ = 0, we get theVlow k RG equation:

ddΛ

VΛlow k(k′, k) =

2π

VΛlow k(k′,Λ) T Λ(Λ, k; Λ2)

1 − (k/Λ)2 . (35)

Note that the full T matrix appears on the right side,in contrast to the partial-wave SRG flow equation (withGs = Trel),

ddλ

Vλ(k, k′) ∝ −(εk − εk′ )2Vλ(k, k′)

+∑

q

(εk + εk′ − 2εq)Vλ(k, q)Vλ(q, k′) , (36)

which is only second-order in the potential. We can alsodefine smooth regulators for Vlow k as in Ref. [24].

The evolution of NN potentials using the Vlow k

method is illustrated for a sharp and smooth regulator inFig. 19. Comparing the Vlow k flow in these figures to theSRG flow in Fig. 9, we see the same decoupling of lowand high momentum. Other non-RG unitary transfor-mations (which perform the transformation all at once,rather than in steps) also decouple; an example is theUCOM method [25], which has close connections to theSRG (see Refs. [26, 27]).


Figure 19: Vlow k flow of AV18 in the 3S1 channel for a sharp (top) and smooth (bottom) regulator [24].

Figure 20: SRG sharp block-diagonal flow of AV18 in the 1P1 channel at λ = 4, 3, 2, and 1 fm−1 [28]. The initial N3LO potential is from Ref. [29]and Λ = 2 fm−1. The axes are in units of k2 from 0 to 11 fm−2 and the color scale is from −0.5 to +0.5 fm.

It is also possible to choose a generator that repro-duces the block diagonal (as opposed to band diagonal)form of the Vlow k RG shown schematically in Fig. 8,except that the transformation will be unitary. In partic-ular, we can use

dHs

ds= [[Gs,Hs],Hs] (37)

with

Gs =

(PHsP 0

0 QHsQ

), (38)

where projection operators P and Q = 1 − P are simplystep functions at a given Λ in partial-wave momentumrepresentation. An example of the subsequent flow isshown in Fig. 20 for the 1P1 channel [28]. To get thefully block-diagonal form, one would have to evolve toλ = ∞. But in practice, going to λ = 1 fm−1 is sufficientfor essentially complete decoupling at Λ = 2 fm−1.

The proof of block diagonalization (see Gubankova etal. [30, 31]) goes as follow. The generator ηs = [Gs,Hs]is non-zero only where Gs is zero, which means in theoff-diagonal blocks. This will then evolve the potentialin this same pattern (this is generically true, if one de-sires a different pattern [28]). A measure of off-diagonalcoupling of Hs is

Tr[(QHsP)†(QHsP)] = Tr[PHsQHsP] > 0 . (39)

Now we can calculate the derivative of this expressionby applying the SRG equation (37):

dds

Tr[PHsQHsP] =

Tr[PηsQ(QHsQHsP − QHsPHsP)]+ Tr[(PHsPHsQ − PHsQHsQ)QηsP]= −2 Tr[(QηsP)†(QηsP)] 6 0 . (40)


Thus the off-diagonal QHsP block will decrease as sincreases.

The low-momentum block of this SRG is found to beremarkably similar to the corresponding Vlow k RG po-tential. Examples are shown in Figs. 21 and 22. How-ever, proving that these different RG approaches shouldyield the same potential remains an open problem.

Figure 21: Comparison of momentum-space (a) Vlow k and (b) SRGblock-diagonal 3S1 potentials with Λ = 2 fm−1 [28].

Figure 22: Same as Fig. 21, only a surface plot..

3.3. Many-body forces

In Fig. 23, the convergence of the triton ground-stateenergy with harmonic oscillator basis size (Nmax~ω ex-citations) is shown for two chiral EFT NN potentialsand for the corresponding SRG-evolved potentials atλ’s from 4 fm−1 to 1 fm−1. In accord with our previ-ous discussion, we see increasingly rapid convergenceas λ decreases. (Note that there is softening alreadyat λ = 3 fm−1 for the N3LO EFT with Λ = 600 MeV,which corresponds to 3 fm−1. The moral is that it is notsufficient to simply compare the cutoff numerically tothe decoupling scale λ.) However, we also see that theconverged (or extrapolated) binding energies are differ-ent for each λ! How could this be, if the SRG is sup-posed to generate unitary transformations?

There are further signs of trouble, such as nuclearmatter failing to saturate after the two-body potential issoftened by RG evolution. Let’s review the facts aboutuniform nuclear matter. Figure 24 shows the binding

10 20 30N

max

10 20 30N

max

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

−5.5

−5.0

Et [

MeV

]

initialλ = 4 fm

−1

λ = 3 fm−1

λ = 2 fm−1

λ = 1 fm−1

550/600 MeV600 MeV

Figure 23: Convergence of three-body energies for two potentials andseveral sets of λ’s.

energy per particle for pure neutron matter (Z = 0) andsymmetric nuclear matter (N = Z = A/2) from cal-culations adjusted to be consistent with extrapolationsfrom nuclei. Neutron matter has positive pressure, whilesymmetric nuclear matter saturates; that is, there is aminimum at a density of about 0.16 fm3 with a bindingenergy of about −16 MeV/A. Reproducing this mini-mum with microscopic interactions fit only to few-bodydata is one of the holy grails of nuclear structure the-ory. But if we evolve an NN potential with either theVlow k RG or the SRG, we find that nuclear matter doesnot converge. This is shown for Vlow k by the “NN only”curves in Fig. 25 and similar behavior is found for SRG.

0 0.05 0.1 0.15 0.2 0.25-20

0

20

40

APRNRAPRRAPR

)-3n (fm

E/A

(MeV

)

Nuclear matter

Neutron matter

Figure 24: Binding energies of nuclear and neutron matter from Ak-mal et al. for several equations of state [32].


0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

−30

−25

−20

−15

−10

−5

0

Ener

gy/n

ucl

eon [

MeV

]

Λ = 1.8 fm−1

Λ = 2.8 fm−1

Λ = 1.8 fm−1

NN only

Λ = 2.8 fm−1

NN only

Vlow k

NN from N3LO (500 MeV)

3NF fit to E3H and r4He

Λ3NF

= 2.0 fm−1

3rd order pp+hh

NN + 3N

NN only

Figure 25: Nuclear matter at third order in perturbation theory in theparticle-particle channel (this appears sufficient for convergence) us-ing a Vlow k RG-evolved potential [33]. The “NN only” curves includeno three-body interactions while the “NN + 3N” curves include a 3NFfit to the triton binding energy and the alpha particle radius.

The failure of softened low-momentum potentials toreproduce nuclear matter saturation should sound famil-iar to anyone who knows the long history of low-energynuclear theory. There were active attempts to transformaway hard cores and soften the tensor interaction in thelate sixties and early seventies. But the requiem for softpotentials was given by Bethe in 1971 [34]: “Very softpotentials must be excluded because they do not givesaturation; they give too much binding and too high den-sity. In particular, a substantial tensor force is required.”The next thirty-five years were spent struggling to solveaccurately with such “hard” potentials. But the story isnot complete: the three-nucleon forces (3NF) were notproperly accounted for!

Three-body forces between protons and neutronshave a classical analog in tidal forces: the gravitationalforce on the Earth is not just the pairwise sum of Earth-Moon and Earth-Sun forces. Quantum mechanically, ananalog is with the three-body force between atoms andmolecules, which is called the Axilrod-Teller term anddates from 1943 [35]. The origin is from triple-dipolemutual polarization. It is a third-order perturbation cor-rection, so the weakness of the fine structure constantmeans that these forces are usually negligible in metalsand semiconductors. However, in solids bound by vander Waals potentials it can be significant; for example, itis ten percent of the binding energy in solid xenon [36].This is the same relative size as needed in the triton.

In general, three-body forces arise from eliminating

π, ρ, ω∆, N∗

π, ρ, ω

π, ρ, ω

π, ρ, ω

N

Figure 26: Sources of internucleon three-body forces. On the left, theintermediate state includes an excitation of the nucleon (∆ or N∗).On the right, the intermediate state includes a virtual anti-nucleon(N). When these degrees of freedom are integrated out, the result-ing nucleons-only vertex is a 3NF.

degrees of freedom. In the nuclear case, this can meaneliminating excited states of the nucleon (N∗ or ∆) orfrom relativistic effects; see Fig. 26 for diagrammaticrepresentations. If the intermediate states are not in-cluded in the low-energy degrees of freedom, we haveirreducible vertices with three nucleons in and three nu-cleons out. But 3NF’s also result from the decouplingof high-momentum intermediate states, whether theyare eliminated explicitly by a cutoff (as with the Vlow k

RG) or the degree of coupling modified (as with theSRG). Omitting three-body forces leads to model de-pendence: observables will depend on the decouplingscale, whether it is the Vlow k Λ or the SRG λ. This de-pendence also becomes a tool, because it is a diagnosticfor errors (more on this later). To eliminate this depen-dence, the 3NF at different Λ or λ must be either fit orevolved.

It is easy to see that RG flow equations lead to many-body operators. Consider the SRG operator flow equa-tion written with second-quantized a’s and a†’s:

dVs

ds=

[[∑a†a︸︷︷︸Gs

,∑

a†a†aa︸︷︷︸2-body

],∑

a†a†aa︸︷︷︸2-body

]= · · · +

∑a†a†a†aaa︸︷︷︸

3-body!

+ · · · (41)

where the second equality reflects that even if the initialHamiltonian is two-body, the commutators give rise tothree-body terms. (For future reference, recall that thecreation and destruction operators are always definedwith respect to a single-particle basis and a referencestate, which in this case is the vacuum.)

As the evolution continues, there inevitably will beA-body forces (and operators) generated. Is this a prob-lem? Not if these “induced” many-body forces are thesame size as those that naturally occur. Indeed, nuclearthree-body forces are already needed in almost all po-tentials in common use to get even the triton binding


π π π

c1, c3, c4 cD cE

Figure 27: Leading three-body forces from chiral EFT. These contri-butions represent three different ranges: long-range 2-pion exchange,short-range contact with one-pion exchange, and pure contact interac-tion.

energy correct. In fact, low-energy effective theory tellus generalized diagrams such as those in Fig. 26 withfour or more legs imply that there are A-body forces(and operators) initially!

However, there is a natural hierarchy predicted fromchiral EFT, whose leading contributions are given inFig. 27 (we’ll return to this in Section 4.1 and supplyadditional details). If we stop the flow equations be-fore induced A-body forces are unnaturally large or ifwe can tailor the SRG Gs to suppress their growth, wewill be ok. (Another option is to choose a non-vacuumreference state, which is what is done with in-mediumSRG, to be discussed later.) Note that analytic boundson A-body growth have not been derived, so we need toexplicitly monitor the contribution in different systems.But the bottom line that makes the SRG attractive asa method to soften nuclear Hamlitonians is that it is atractable method to evolve many-body operators.

To include the 3NF using SRG with normal-orderingin the vacuum, we start with the SRG flow equationdHs/ds = [[Gs,Hs],Hs] (e.g., with Gs = Trel). Theright side is evaluated without solving bound-state orscattering equations, unlike the situation with Vlow k, sothe SRG can be applied directly in the three-particlespace. The key observation is that for normal-orderingin the vacuum, A-body operators are completely fixed inthe A-particle subspace. Thus we can first solve for theevolution of the two-body potential in the A = 2 space,with no mention of the 3NF (either initial or induced),and then use this NN potential in the equations appliedto A = 3.

What about spectator nucleons? There is a decou-pling of the 3NF part. We can see this from the first-quantized version of the SRG flow equation,

dVs

ds=

dV12

ds+

dV13

ds+

dV23

ds+

dV123

ds= [[Trel,Vs],Hs] , (42)

where we isolate the contributions from each pair andthe 3NF. Using each SRG equation for the two-bodyderivatives, we can cancel them against terms on the

1 2 3 4 5 6 7 10 20λ [fm

−1]

−8.6

−8.4

−8.2

−8.0

−7.8

−7.6

−7.4

Gro

und-

Stat

e E

nerg

y [M

eV]

NN-onlyNN + NNN-inducedNN + NNN

3H

Expt.

Figure 28: Triton binding energy during SRG evolution. The threecurves are for an initial potential with only NN components wherethe induced 3NF is not kept (“NN-only”), for the same initial NNpotential but keeping the induced 3NF (“NN + NNN-induced”), andwith an initial NNN included as well (“NN + NNN”).

right side. The result is [12]:

dV123

ds= [[T12,V12], (T3 + V13 + V23 + V123)]

+ 123→ 132 + 123→ 231+[[Trel,V123],Hs] . (43)

The key is that there are no “multi-valued” two-bodyinteractions remaining (i.e., dependence on the excita-tion energy of unlinked spectators); all the terms areconnected. An implementation of these equations in amomentum basis would be very useful and has very re-cently been achieved by Hebeler [37]. But an alternativeapproach has also succeeded: a direct solution in a dis-crete basis [38, 16, 17].

The idea is that the SRG flow equation is an opera-tor equation, and thus we can choose to evolve in anybasis. If one chooses a discrete basis, than a separateevolution of the three-body part is not needed. This wasfirst done for nuclei by Jurgenson and collaborators in2009 using an anti-symmetrized Jacobi harmonic oscil-lator (HO) basis [16]. The technology for working withsuch a basis had already been well established for appli-cations to the no-core shell model (NCSM) [39]. Thisapproach leads to SRG-evolved matrix elements of thepotential directly in the HO basis, which is just whatis needed for many-body applications such as NCFC orcoupled cluster.

In Fig. 28, the comparison of two-body-only to fulltwo-plus-three-body evolution is shown for the triton(3H). The NN-only curve uses the evolved two-body


1 2 3 4 5 10 20

λ [fm−1

]

−29

−28

−27

−26

−25

−24

Gro

und

-Sta

te E

ner

gy [

MeV

]

NN-only

NN+NNN-induced+NNN-initial

4He N

3LO (500 MeV)

Expt.

Figure 29: Alpha particle binding energy during SRG evolution. Thecurves correspond to those in Fig. 28.

7.6 7.8 8 8.2 8.4 8.6 8.8

Eb(3H) [MeV]

24

25

26

27

28

29

30

Eb(4

He)

[M

eV]

Tjon line for NN-only potentials

SRG NN-only

SRG NN+NNN (λ >1.7 fm−1

)

8.45 8.528.2

28.3

28.4

N3LO

λ=3.0λ=1.2

λ=2.5

λ=1.5

λ=2.0

λ=1.8

Expt.

(500 MeV)

Figure 30: Correlation plot of the binding energies of the alpha parti-cle and the triton. The dotted line connects (approximately) the locusof points found for phenomenological potentials, and is known as theTjon line [40].

potential. The change in energy with λ reflects the vi-olation of unitarity by omission of the induced three-body force. When this induced 3NF is included (“NN+ NNN–induced”), the energy is independent of λ forA = 3. If we now turn to the alpha particle (4He) inFig. 29, we see similar behavior, except now the inclu-sion of the induced 3NF does not lead to a completelyflat curve at the lowest λ values. If there is sufficientconvergence, this is a signal of missing induced 4NF.

In both cases, it is evident that starting with an initialNN-only interaction (in this case, an N3LO(500 MeV)

1 2 3 4 5 10

λ

−40

−30

−20

−10

0

10

20

30

40

g.s

. E

xp

ecta

tio

n V

alu

e

<Trel><V

NN>

<V3N

>

1 2 3 4 5 10

λ

−1

−0.5

0

<Trel><V

NN>

<V3N

>

3H

h- ω = 28

Nmax

= 18

Nmax

-A3 = 32 NN+NNN

Figure 31: Contributions to the triton binding energy during SRG evo-lution. Plotted are the expectation values of the kinetic energy, thetwo-body potential, and the three-body potential [17].

interaction [29]), does not reproduce experiment. Thethird line in each plot of Figs. 28 and 29 shows that aninitial 3NF (labeled NNN) contribution leads to a goodreproduction of experiment. The triton energy is part ofthe fit of this initial force, but the alpha particle energyis a prediction. Note that the magnitude of the NN-onlyvariation is comparable to the initial 3NF needed. Thisis an example of the natural size of the 3NF being mani-fested by the running of the potential (which is, in effect,the beta function).

The nature of the evolution is illustrated in Fig. 30,which is a correlation plot of the binding energies ineach nucleus. The dotted line is known as the Tjonline for NN-only phenomenological potentials. It wasfound that different potentials that fit NN scattering datagave different binding energies, but that they clusteredaround this line. With the SRG evolution starting withjust an NN potential, the path follows the line, passingfairly close to the experimental point. With an initialNNN force and keeping the induced 3-body part, thetrajectory is greatly reduced (see inset), at least until λis small.

Figures 31 and 32 show individual contributions tothe energy in the form of ground-state matrix elementsof the kinetic energy, two-body, three-body, and (im-plied) four-body potentials. The hierarchy of contri-butions is quite clear but the graphs also manifest thestrong cancellations between the NN and kinetic energycontributions. These cancellations magnify the impactof higher-body forces. Even so, it appears that a trunca-tion including the NNN but omitting higher-body forces


1 2 3 4 5 10

λ

−100

−80

−60

−40

−20

0

20

40

60

80

g.s

. E

xpec

tati

on V

alu

e

<Trel

>

<VNN

>

<V3N

>

<V4N

>

1 2 3 4 5 10

λ

−4

−2

0

<Trel

>

<VNN

>

<V3N

>

<V4N

>

4He

h- ω = 28

Nmax

= 18

Nmax

-A3 = 32NN+NNN

Figure 32: Contributions to the alpha particle binding energy duringSRG evolution. Plotted are the expectation values of the kinetic en-ergy, the two-body potential, the three-body potential, and the four-body potential [17].

is workable, particularly with λ > 1.5 fm−1. But whatabout the A dependence of the 4NF (and beyond)?

This A dependence is the topic of current research. InFig. 33, results for 6Li are shown [17]. Assessing theseresults is made difficult because of insufficient conver-gence of comparison calculations (the shaded areas) andof calculations at the large λ values. Note, however, thatthe variations are with the 1 MeV level; nevertheless weexpect to do better.

Roth and collaborators have since used importancetruncated NCSM (IT-NCSM) to extend these calcula-tions to much higher Nmax and all the way to 16O [19].They find that the SRG with initial NN-only but includ-ing the induced NNN shows only small running withλ. On the other hand, with increasing A they find sig-nificant deviations. This has been traced to the influ-ence of the initial long-range NNN interaction. If theylower the cutoff of this part of the interaction, then ap-proximate SRG unitarity is restored and with coupled-cluster methods they find reasonable results even formedium-size nuclei (although not yet with fully consis-tent Hamiltonians) [41].

3.4. Summary pointsRenormalization group flow equations dramatically

reduce correlation in many-body wave functions, lead-ing to faster convergence of many-body calculations.Flow equations (SRG) achieve this lower resolutionby decoupling via a series of unitary transformations,which leave observables invariant (if no approximationsare made) but alter the physics interpretation. Few-body

1 1.2 1.5 1.8 2.0 2.2 2.5

λ [fm−1

]

−34

−33

−32

−31

−30

−29

−28

−27

Eg

s [M

eV] NN-only

NN+NNN-inducedNN+NNN

6Li N

3LO (500 MeV)

extrapolated Nmax

= 4-8(10)

using best hΩ

expt.

from othercalculations

Figure 33: SRG running of the 6Li ground-state energy. The error barsare estimates of the errors from extrapolating the results to Nmax = ∞.The gray shaded region shows the uncertainty from NSCM calcula-tions using a Lee-Suzuki effective interaction [17].

forces are inevitable, but the flow-equation approach al-lows the evolution of vacuum interactions.

4. Features of SRG applied to nuclear problems

4.1. Chiral EFT, many-body forces, and the SRG

Before continuing with the SRG for nuclear systems,let’s say a bit more about chiral effective field theory(EFT). In the SRG flow equations, the input interactionis merely an initial condition; the equations are the samewhether we start with an EFT potential or a more phe-nomenological potential such as Argonne v18. However,increasingly nuclear theorists are moving toward usingEFT interactions because they promise a more system-atic construction of many-body forces and consistentoperators.

There are three fundamental ingredients of an effec-tive field theory (e.g., see Ref. [42]). The first is to usethe most general lagrangian with low-energy degrees offreedom consistent with the global and local symme-tries of the underlying theory. For nuclei, the underly-ing theory is quantum chromodynamics (QCD). We canidentify a hierarchy of nuclear QCD scales:

• MQCD ∼ 1 GeV [Mhadrons, 4π fπ]

• Mnuc ∼ 100 MeV [kF, fπ, mπ, δ∆N]

• M2nuc/MQCD ∼ 10 MeV [nuclear binding energy

per nucleon]


For all but the lightest nuclei, the EFT of choice atpresent draws a line between the first two levels to de-fine the high-energy and low-energy scales. This is thechiral EFT, with degrees of freedom consisting of thenucleon (proton and neutron) and the pion [43]. In thenear future, the ∆ resonance will be included as wellbecause of the small mass difference with the nucleon,δ∆N . Besides the usual space-time symmetries, termsin the chiral EFT lagrangian are constrained by the re-quirements of spontaneously broken (as well as explic-itly broken) chiral symmetry [43].

The other two ingredients are the declaration of a reg-ularization and renormalization scheme, and the identi-fication of a well-defined power counting based on well-defined expansion parameters. The separation of scalesprovides the expansion parameter as ratios of Q/MQCD,where Q is one of the quantities lumped together aboveas Mnuc. The chiral EFT potentials used here are derivedusing a momentum cutoff and what is called “Wein-berg counting”, in which the counting is done at thelevel of an irreducible potential that is summed non-perturbatively with the Lippmann-Schwinger equation.This scheme has been criticized because it does not al-low systematic renormalization (meaning, in this con-text, the removal of cutoff dependence at each order),which limits the range of cutoffs used. This in turn hin-ders the validation of the EFT, because the sensitivityto the cutoff is used as a measure of uncertainties, SeeRef. [43] for a thorough overview of chiral EFT for nu-clei and the status of the power counting and renormal-ization controversies.

With any scheme, however, the power counting im-plies a hierarchy of many-body contributions. Wein-berg counting associates a power ν of Q/MQCD with di-agrams for the potential, where

ν = −4 + 2N + 2L +∑

i

(di + ni − 2) . (44)

The definitions and details of its implementation canbe found in Ref. [43]. For our purposes, the relevantterm is “2N”, which says that adding a nucleon to gofrom an A–body potential to an A + 1–body potentialgenerally suppresses the contribution by Q2/MQCD. Inthe theory without ∆’s, the suppression of the leading3NF compared to the leading NN interaction is actually(Q/MQCD)3, and a four-body force first appears at or-der (Q/MQCD)4 [43]. It is this hierarchy that we want topreserve as we run our SRG flow equations.

As noted, the flow equation technology discussedhere does not rely on a particular implementation ofchiral EFT, except that the SRG is inherently non-perturbative. This apparently excludes alternative renor-

malization and power counting schemes that require aperturbative treatment beyond leading order [44, 45,46]. The current belief is that the two approaches shouldgive comparable results as long as the EFT cutoff istaken to be of order MQCD, but the issue is far from set-tled [43].

Another consideration is whether one could bypassthe SRG by simply applying chiral EFT with a lowercutoff. Indeed, there exists low-cutoff implementationson the market that display similar characteristics tolow-momentum RG-evolved interactions. However, thelower cutoff also means the effective expansion param-eter is smaller, and therefore the truncation error is re-duced. With the RG, one preserves the truncation errorfrom the cutoff of order MQCD. An additional advantageof the RG is the controlled variation of the decouplingscale, which provides a tool for assessing errors fromthe Hamiltonian truncation and many-body approxima-tions. But this is also not a settled issue and furtherstudy would be welcome.

4.2. More perturbative nuclear systems flow equationsEarlier we mentioned the role of RG in high-energy

physics in improving perturbation theory. Much of low-energy nuclear physics is intrinsically non-perturbativebecause of large scattering lengths and bound states, sohow do we quantify “perturbativeness”? We study theconvergence of the Born series for scattering to see howthis can be done.

Consider whether the Born series for the T-matrix op-erator at a given (complex) z,

T (z) = V + V1

z − H0V + V

1z − H0

V1

z − H0V + · · · (45)

converges. This is something like a geometric series,for which we know clearly the convergence criterion:1 + w + w2 + · · · diverges if |w| ≥ 1. We get a clue forhow to use this if we consider a bound state |b〉 and thespecial value z = Eb, which is the bound-state energy.Then we can rearrange the Schrodinger equation

(H0 + V)|b〉 = Eb|b〉 (46)

to the form1

Eb − H0V |b〉 = |b〉 (47)

and look at T (Eb)|b〉. Using Eq. (47) repeatedly, thedivergence is manifest, i.e., we get V(1 + 1 + 1 + · · · )|b〉.

Now we see that we can generalize Eq. (47) for fixedE by looking for eigenstates of (E −H0)−1V with eigen-value ην(E),

1E − H0

V |Γν〉 = ην(E)|Γν〉 . (48)


Figure 34: The potentials of Fig. 3 inverted as part of the Weinbergeigenvalue analysis (see text).

Then with T applied to these eigenstates, there is mani-festly a divergence for |ην(E)| ≥ 1:

T (E)|Γν〉 = V |Γν〉(1 + ην + η2ν + · · · ) . (49)

So we characterize the perturbativeness of a potentialat energy E by the ην(E) with the largest magnitude,which dictates convergence of the T matrix. This anal-ysis follows work in the early 1960’s by Weinberg [48],so we call ην a “Weinberg eigenvalue”, although othershave made similar treatments of the convergence of theLippmann-Schwinger series.

If we compare Eq. (48) to Eq. (47), we can expressthe convergence criterion for E < 0 as saying that T (E)

2 3 4 5 6 7Λ [fm-1]

0

0.5

1

1.5

2

2.5

| ην |

free space, η > 0free space, η < 0

kf = 1.35 fm-1, η > 0

kf = 1.35 fm-1, η < 0

3S1 (Ecm = -2.223 MeV)

Figure 35: Vlow k RG evolution of the largest positive and negativeWeinberg eigenvalue at fixed energy (corresponding to the deuteronbinding energy) for AV18 in the 3S1 channel [47].

AV18CD-BonnN3LO (500 MeV)

−3 −2 −1 1

Re η

−3

−2

−1

1

Im η1S0

−3 −2 −1 1

Re η

−3

−2

−1

1

Im η

AV18CD-BonnN3LO

3S1−3D1

Figure 36: Trajectories of Weinberg eigenvalues in the complex planefor several realistic NN potentials in two channels. The symbols gofrom 0 MeV on the axis to 25, 66, 100, and 150 MeV [47].

diverges if there exists a bound state at E for V/ην with|ην| ≥ 1. Or, in other words, what is the largest ην forwhich V/ην supports a bound state at E? We have con-vergence only for ην < 1. This means we’ll have twotypes of eigenvalues, because ην could be negative (“re-pulsive”) as well as positive (“attractive”). The negativeeigenvalue corresponds to looking for bound states witha scaled “flipped” potential, as in Fig. 34. We see thatthe repulsive core becomes a deep attractive well, im-plying that we will have a large negative eigenvalue inthese cases. But then we expect that RG evolution to asoftened form, which eliminates the core, should resultin decreased eigenvalues.

These expectations hold in practice, as illustrated inFig. 35, which shows the evolution of the largest posi-tive and negative Weinberg eigenvalues as a function ofthe Vlow k cutoff Λ. (Very similar behavior is observedfor the SRG with λ replacing Λ.) In free space, thelargest attractive eigenvalue is unity at all Λ’s, corre-


−3 −2 −1 1

Re η

−3

−2

−1

1

Im η

Λ = 10 fm-1

Λ = 7 fm-1

Λ = 5 fm-1

Λ = 4 fm-1

Λ = 3 fm-1

Λ = 2 fm-1

N3LO

1S0

−3 −2 −1 1

Re η

−3

−2

−1

1

Im η

Λ = 10 fm-1

Λ = 7 fm-1

Λ = 5 fm-1

Λ = 4 fm-1

Λ = 3 fm-1

Λ = 2 fm-1

N3LO

3S1−3D1

Figure 37: Trajectories of Weinberg eigenvalues in the complex plane for the AV18 NN potential in two channels at various states in a Vlow k RGevolution (labeled by Λ). The symbols go from 0 MeV on the axis to 25, 66, 100, and 150 MeV [47].

1.5 2 2.5 3 3.5 4Λ (fm-1)

−1

−0.8

−0.6

−0.4

−0.2

0

η ν(E=0

)

Argonne v18

N2LO-550/600 [19]

N3LO-550/600 [14]

N3LO [13]

1S0

1.5 2 2.5 3 3.5 4Λ (fm-1)

−2

−1.5

−1

−0.5

0η ν(E

=0)

Argonne v18

N2LO-550/600

N3LO-550/600

N3LO [Entem]

3S1−3D1

Figure 38: Repulsive (negative) Weinberg eigenvalues at E = 0 for several N3LO chiral EFT potentials as a function of Vlow k Λ [47].

sponding to the deuteron bound state. But the negativeeigenvalue starts very large (because of the repulsivecore) and drops dramatically as the potential is evolved,ending up well less than unity, indicating it is perturba-tive (but the positive eigenvalue still makes this channelnonperturbative). The situation is even more dramaticin the medium. The other curves in the figure showthe result from considering the T matrix in the nuclearmedium, where Pauli blocking effects are included inthe intermediate states. We see a further reduction ofthe negative eigenvalue and now the positive eigenvalueis perturbative as well: the deuteron has dissolved!

Starting from negative energies, we can follow theevolution of the largest eigenvalue into the complexplane as we increase E to positive values. Figures 36and 37 show paths in the complex plane for Weinberg

eigenvalues, with the symbols indicating energies of 0,25, 66, 100, and 150 MeV. The shaded region is the unitcircle; the potential is perturbative for energies whereboth eigenvalues lie inside. The softening effect of theRG evolution is manifest in Fig. 37 and the eigenvaluesprovide a quantitative measure of the perturbativeness.It’s also clear from Fig. 36 that at least the particularchiral potential used there is already quite soft. How-ever, Fig. 38 shows that significant additional soften-ing is possible with RG evolution. In all of these plots,the differences between the 1S0 channel and the 3S0–3D0 coupled channel stem from the latter having an ad-ditional source of nonperturbative behavior: the short-range tensor force.

The increasing “perturbativeness” at finite density isdocumented again in Fig. 39. At typical nuclear densi-


0 0.2 0.4 0.6 0.8 1 1.2 1.4kF [fm-1]

-1

-0.5

0

0.5

1

η ν(Bd)

Λ = 4.0 fm-1

Λ = 3.0 fm-1

Λ = 2.0 fm-1

3S1 with Pauli blocking

Figure 39: Dependence of the largest Weinberg eigenvalues on densityfor several evolved Vlow k potentials [47].

ties with 1 fm−1 ≤ kF ≤ 1.3 fm−1, both positive and neg-ative eigenvalues are small at the lowest Λ’s. This im-plies that nuclear matter may actually be perturbative!(Note that at the Fermi surface, pairing as a nonpertur-bative phenomenon is revealed by |ην| > 1 [49].) Wecan understand how this happens from Fig. 40, whichshows the phase space available to two nucleons thatscatter in the medium. Pauli blocking means they mustgo outside the two Fermi spheres, but the volume is in-creasingly restricted with decreasing Λ. In addition, themagnitudes of the matrix elements that scatter such par-ticles decrease as well [50].

F: |P/2 ± k| < kF

Λ: |P/2 ± k| > kF

P/2

k

Λ

kF

|k| < Λand

Figure 40: Overlapping Fermi spheres showing available phase spacefor two nucleons excited above the Fermi surface [50].

Perturbation theory in the particle-particle channel isshown in Fig. 41 for the high-resolution Argonne v18potential initially and after evolution by the Vlow k RGto low resolution (Λ = 1.9 fm−1). Whereas many-bodyperturbation theory (MBPT) manifestly diverges for the

0.8 1 1.2 1.4 1.6kf [fm

-1]

-100

-50

0

50

100

150

E/A

[MeV

]

1st order2nd order pp ladder3rd order pp ladder

Vlow k ( Λ=1.9 fm-1)

Argonne v18

Figure 41: Many-body perturbation theory for symmetric nuclearmatter up to third order in the particle-particle channel [33].

original potential, it converges for the low-momentuminteraction at second order (at least in this channel; morecomplete many-body approximations must be studied tobe more definite). It is also evident that there is no sat-uration. But adding a 3NF fit only to few-body prop-erties, as in Fig. 42, shows that the empirical saturationpoint can be reproduced with an uncertainty of about 2–3 MeV/particle. On-going work to improve this resultincludes the development of SRG evolution for the 3NFin momentum space [37] and of coupled cluster meth-ods for infinite matter with 3NF’s to provide a high-order resummation of perturbation theory to test con-vergence.

4.3. Universality from flow equationsAnother general aspect of RG flows known from the

study of critical phenomena is the appearance of univer-sal behavior. In the application of RG to nuclear inter-actions, the universality we observe is that distinct ini-tial NN potentials that reproduce the experimental low-energy scattering phase shifts, are found to collapse to asingle universal potential. We’ve already seen an indi-cation of this, but here we document it in more detail.

We focus on the SRG, but very similar conclusionsare found for Vlow k evolution. In Figs. 43 and 44, S-wave N3LO chiral EFT potentials from Refs. [29] and[51] are evolved with the SRG. Although the level oftruncation is the same and the cutoffs approximatelyequal, the methods of regulating the potential differ(particularly for the two-pion exchange part). The resultis very different looking initial interactions. This is not


0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

−20

−15

−10

−5

0

5E

ner

gy

/nu

cleo

n [

MeV

] Λ = 1.8 fm−1

Λ = 2.0 fm−1

Λ = 2.2 fm−1

Λ = 2.8 fm−1

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

Hartree-Fock

Empiricalsaturationpoint 2nd order

Vlow k

NN from N3LO (500 MeV)

3NF fit to E3H and r4He

3rd order pp+hh

2.0 < Λ3NF

< 2.5 fm−1

Figure 42: Many-body perturbation theory calculations of symmetric nuclear matter with Vlow k NN potentials at four different cutoffs and theleading-order 3NF from chiral EFT [33]. The two free parameters in the 3NF are fit at each density to the triton binding energy and the alphaparticle radius.

a concern, because the NN potential is not an observ-able. We also observe there is significant off-diagonalstrength coupling low and medium momenta in the ini-tial potentials.

As the potentials are evolved, we see the characteris-tic driving toward the diagonal, with the diagonal widthin k2 given roughly by λ2. At the end of the evolutionshown, the interactions still look quite different at firstglance. However, if we focus on the low-momentumregion, where k2 < 2 fm−2, they appear much more sim-ilar. We can quantify this by taking a slice along theedge (i.e., V(k, 0) and along the diagonal (i.e., V(k, k))and plot these quantities for these potentials and two ad-ditional ones. This is done in Fig. 45. We see a dra-matic collapse of the interaction between λ = 5 fm−1

and λ = 2 fm−1 for the region of k below λ (or maybe3/4 λ). An open question under active investigation iswhether evolved 3NF interactions will be universal.

4.4. Operator evolution via flow equationsWe have focused almost exclusively on the evolution

of the Hamiltonian, but an RG transformation will alsochange the operators associated with measurable quan-tities. It is essential to be able to start with operatorsconsistent with the Hamiltonian and then to evolve themmaintaining this consistency. The first step is a primemotivation for using EFT; we will assume that we haveconsistent initial operators in hand. The second stepcan be technically difficult, especially since we will in-evitably induce many-body operators as we evolve (forthe same arguments as we made for the Hamiltonian).This is where the SRG is particularly advantageous, be-cause it is technically feasible to evolve operators alongwith the Hamiltonian.

The SRG evolution with s (recall s = 1/λ4) of anyoperator O is given by:

Os = UsOU†s , (50)

so Os evolves via

dOs

ds= [[Gs,Hs],Os] , (51)

where we use the same Gs to evolve the Hamiltonianand all other operators. While we can directly evolveany operator like this in parallel to the evolution of theHamiltonian, in practice it is more efficient and numer-ically robust to either evolve the unitary transformationUs itself:

dUs

ds= ηsUs = [Gs,Hs]Us , (52)

with initial value Us=0 = 1, or calculate it directly fromthe eigenvectors of Hs=0 and Hs:

Us =∑

i

|ψi(s)〉〈ψi(0)| . (53)

Then any operator is directly evolved to the desired sby applying Eq. (50) as a matrix multiplication. Thesecond method works well in practice.

To simplify our study of operator evolution, we con-sider the simplest possible operator: the momentumnumber operator a†qaq. In Fig. 46, we show the momen-tum distribution in the deuteron, i.e., < ψd |a

†qaq|ψd > for

two different realistic potentials, AV18 and CD-Bonn.As implied by the y-axis label, the momentum distri-bution is just the square of the deuteron wave functionin momentum space. The results for the two potentials


Figure 43: SRG evolution of two chiral EFT potentials in the 1S0 channel [2].

Figure 44: SRG evolution of two chiral EFT potentials in the 3S1 channel [2].

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [fm−1

]

−2.0

−1.0

0.0

1.0

Vλ(k

,0)

[fm

]

550/600 [E/G/M]

600/700 [E/G/M]

500 [E/M]

600 [E/M]

−2.0

−1.0

0.0

1.0

Vλ(k

,0)

[fm

]

λ = 2.0 fm−1

λ = 5.0 fm−1

1S

0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [fm−1

]

−2.0

−1.0

0.0

1.0

Vλ(k

,k)

[fm

]

550/600 [E/G/M]

600/700 [E/G/M]

500 [E/M]

600 [E/M]

−2.0

−1.0

0.0

1.0

Vλ(k

,k)

[fm

]

λ = 2.0 fm−1

λ = 5.0 fm−1

1S

0

Figure 45: SRG flow toward universality for several chiral EFT potentials [2].


0 1 2 3 4 5

q [fm−1

]

10−5

10−4

10−3

10−2

10−1

100

101

102

4π [

u(q)

2 + w

(q)2 ]

[fm

3 ]

AV18Vs at λ = 2 fm

−1

Vs at λ = 1.5 fm−1

CD-Bonn

Figure 46: Momentum distribution of the deuteron for the AV18 po-tential and CD-Bonn potentials, compared to results from SRG poten-tials evolved to λ = 2.0 fm−1 and 1.5 fm−1.

0 1 2 3 4

q [fm−1]

10-5

10-4

10-3

10-2

10-1

100

101

102

4π [u

(q)2 +

w(q

)2 ] [fm

3 ]

N3LO unevolvedλ = 2.0 fm−1

λ = 1.5 fm−1

(a

qaq) deuteron

Figure 47: Momentum distribution for an unevolved chiral N3LOpotential compared to results from SRG potentials evolved to λ =

2.0 fm−1 and 1.5 fm−1 [52].

agree up to about 2 fm−1 and then are different. If weevolve the AV18 potential and the momentum operator,then the matrix element < ψλd |Oλ|ψ

λd > will be the same

for any λ and the curve will exactly reproduce the AV18deuteron momentum distribution. However, if we cal-culate < ψλd |a

†qaq|ψ

λd > for λ = 2 fm−1 and 1.5 fm−1, that

is, we use the evolved wave function but the unevolvedoperator, then we get the other curves. Besides directlyillustrating that the high-q part of the momentum distri-bution is not an observable, since we can change it atwill by unitary transformations, this manifests that thehigh-momentum part of the wave function is removed.

The latter is potentially disturbing, because if theevolved operator is supposed to reproduce a high mo-mentum result when the evolved wave function has avanishingly small component at that momentum, thismay be because the operator is becoming pathological.To explore this further, we consider the momentum dis-tribution in Fig. 47 at low (q = 0.34 fm−1) and moder-ately high (q = 3.0 fm−1) values of the momentum [52].In Fig. 48 we plot the integrand of⟨

ψλd∣∣∣ (Ua†qaqU†)

∣∣∣ψλd⟩ , (54)

at each of these two q values. The full integral is themomentum distribution at those q’s, so the plots tellus where the strength of the operator lies. For thelow-momentum operator, there is little renormalization,but the nature of the high-momentum operator changescompletely. Originally, the integral comes entirely fromthe region of q = 3.0 fm−1, but the evolution of the oper-ator shifts its strength entirely to low momentum. Thisresult is similar for other operators, such as electromag-netic form factors [52].

As we move to A ≥ 3, the operator evolution and ex-traction process becomes more involved. A flowchartfor the procedure is given in Fig. 49. Imagine we ini-tially have a one-body operator and we want to evolveand then evaluate it in an A-particle nucleus. The dif-ficulty is that n-body components are induced as weevolve and these must be separated out so we can cor-rectly embed them in the nucleus. In particular, toembed an n-body operator in an anti-symmetrized A-particle nucleus, we need an embedding factor of

(An

), so

we need to isolate the components first.With the usual SRG generators, there is no evolu-

tion of one-body operators. This is easiest to see insecond quantization, where the commutators on theright side yield operators that have at least four cre-ation/destruction operators, meaning it is two- andhigher-body. To isolate the two-body part, we firstevolve the Hamiltonian in the two-particle basis to findthe unitary operator U(2)

s and use it to evolve our opera-tor. Then if we subtract the embedded one-body opera-tor, we will have our two-body part. The one- and two-body parts are then embedded in the 3-particle basis andsubtracted from the evolved operator in that basis. Andso on until we reach the desired level of truncation, atwhich point we embed in the A-particle basis and per-form the A-particle calculation. More details and exam-ples can be found about operator evolution in Ref. [52].

4.5. Computational aspectsBefore concluding this lecture, we’d like to make

some brief comments on the computational aspects of


Figure 48: Integrand of momentum distribution operator [52].

Figure 49: Schematic of the SRG operator evolution and embedding process [53].


the calculations behind the figures we’ve seen. As notedearlier, the continuous momentum is discretized into afinite number of momentum points. The subsequent dis-cretization of integrals leads directly to matrices, andmost of the manipulations are efficiently cast in this lan-guage. For example, momentum-space flow equationshave integrals like:

I(p, q) ≡∫

dk k2 V(p, k)V(k, q) . (55)

The usual choice of discretization is to use gaussianquadrature to accurately evaluate integrals with a mini-mum of points. (The size is not an issue for two-bodyoperators, but becomes critical for higher-body opera-tors.) We introduce gaussian nodes and weights kn,wn

with (n = 1,N) to reduce integrals to finite sums:∫dk f (k) ≈

∑n

wn f (kn) . (56)

(Note: these sets of nodes and weights are generally acombination of separate smaller rules over adjacent in-tervals with a total of N points.) Then I(p, q) → Ii j,where p = ki and q = k j, and

Ii j =∑

n

k2nwn VinVn j →

∑n

VinVn j (57)

whereVi j ≡

√wiki Vi j k j

√w j . (58)

This allows us to solve SRG equations and integralequations for phase shifts, Schrodinger equation in mo-mentum representation. In practice, N ≈ 100 gausspoints is adequate for accurate nucleon-nucleon partialwaves.

A computer code that carries out SRG evolution canbe remarkable simple. Here is a possible pseudocodethat is suitable:

1. Set up basis (e.g., momentum grid with gaussianquadrature or HO wave functions with Nmax)

2. Calculate (or input) the initial Hamiltonian and Gs

matrix elements (including any weight factors)3. Reshape the right side [[Gs,Hs],Hs] to a vector

and pass it to a coupled differential equation solver4. Integrate Vs to desired s (or λ = s−1/4)5. Diagonalize Hs with standard symmetric eigen-

solver =⇒ energies and eigenvectors6. Form U =

∑i |ψ

(i)s 〉〈ψ

(i)s=0| from the eigenvectors

7. Output or plot or calculate observables

Such a code has been implemented in MATLAB, Math-ematica, Python, C++, and Fortran-90. While any basiscan be used, so far only discretized momentum and har-monic oscillator bases have been implemented. Notethat the same procedure (and even the same code insome cases) is relevant for a many-particle basis, butthe number of differential equations will grow rapidly.For accurate results in the two-body evolution, 1002 =

10, 000 matrix elements are needed, so there are thatmany differential equations. For an accurate 3NF evo-lution in a harmonic oscillator basis, at least 20 mil-lion coupled differential equations need to be solved.This sounds intimidating, but is well within reach ofMATLAB (for example) on a machine with a moderateamount of memory.

4.6. Summary points

Chiral EFT establishes a hierarchy of many-bodyforces. Using flow equations to run to low resolutionmakes many-body calculations more perturbative andinteractions flow to universal form (at least for NN;this is not yet established for the 3NF). Operators canbe evolved consistently with interaction. Long-distanceoperators change very little, while short-distance oper-ators renormalize significantly, accompanied in somecases by a change in physics interpretation. The basicSRG flow equations in a partial wave momentum basiscan be cast in a form that involves just matrix manipu-lations and the solution of ordinary first-order, coupleddifferential equations.

5. Nuclear applications and open questions

In this final lecture, we take a look at the in-mediumsimilarity renormalization group (IM-SRG) and make abroad survey of nuclear applications. We conclude witha summary of open problems and new challenges.

5.1. In-medium similarity renormalization group

We start with a review of Hartree-Fock. The Hartree-Fock wave function is the best single Slater determinant

|ΨHF〉 = detφi(x), i = 1 · · · A , x = (r, σ, τ) (59)

in the variational sense. The φi(x) single-particle wavefunctions satisfy non-local Schrodinger equations:

−∇2

2Mφi(x) + VH(x)φi(x) +

∫dy VE(x, y)φi(y) = εiφi(x)

(60)


with direct

VH(x) =

∫dy

A∑j=1

|φ j(y)|2v(x, y) (61)

and exchange

VE(x, y) = −v(x, y)A∑

j=1

φ j(x)φ∗j(y) (62)

potentials [54, 55]. The direct and exchange potentialsare shown diagrammatically in Fig. 50, with the right-most diagram an abbreviated form called a Hugenholtzdiagram [56]. We must solve self-consistently using oc-cupied orbitals for VH and VE . Then Slater determinantsfrom all orbitals form an A-body basis.

+ =⇒

Figure 50: Feynman and Hugenholtz diagrams for Hartree-Fock.

The in-medium SRG (IM-SRG) for nuclei, developedrecently by Tsukiyama, Bogner, and Schwenk [57], ap-plies the flow equations in an A–body system using adifferent reference state than the vacuum. For exam-ple, we can choose the Hartree-Fock ground state as areference state. The appealing consequence is that, un-like the free-space SRG evolution, the in-medium SRGcan approximately evolve 3, ..., A-body operators usingonly two-body machinery. However, also in contrast tothe free-space SRG, the in-medium evolution must berepeated for each nucleus or density.

The key to the IM-SRG simplification is the use ofnormal-ordering with respect to a finite-density refer-ence state. That is, starting from the second-quantizedHamiltonian with two- and three-body interactions,

H =∑12

T12a†1a2 +1

(2!)2

∑1234

〈12|V |34〉a†1a†2a4a3

+1

(3!)2

∑123456

〈123|V (3)|456〉a†1a†2a†3a6a5a4 , (63)

all operators are normal-ordered with respect to a finite-density Fermi vacuum |Φ〉 (for example, the Hartree-Fock ground state or the non-interacting Fermi sea innuclear matter), as opposed to the zero-particle vacuum.

Wick’s theorem can then be used to exactly write H as

H = E0 +∑12

f12a†

1a2

+1

(2!)2

∑1234

〈12|Γ|34〉a†1a†2a4a3

+1

(3!)2

∑123456

〈123|Γ(3)|456〉a†1a†2a†3a6a5a4 , (64)

where the zero-, one-, and two-body normal-orderedterms are given by

E0 = 〈Φ|H|Φ〉 =∑

1

T11n1

+12

∑12

〈12|V |12〉n1n2

+13!

∑123

〈123|V (3)|123〉n1n2n3 , (65)

f12 = T12 +∑

i

〈1i|V |2i〉ni

+12

∑i j

〈1i j|V (3)|2i j〉nin j , (66)

〈12|Γ|34〉 = 〈12|V |34〉 +∑

i

〈12i|V (3)|34i〉ni , (67)

and ni = θ(εF−εi) denotes the sharp occupation numbersin the reference state, with Fermi level or Fermi energyεF.

By construction, the normal-ordered strings of cre-ation and annihilation operators obey 〈Φ|a†1 · · · an|Φ〉 =

0. It is evident from Eqs. (65)–(67) that the coeffi-cients of the normal-ordered zero-, one-, and two-bodyterms include contributions from the three-body interac-tion V (3) through sums over the occupied single-particlestates in the reference state |Φ〉. Therefore, truncat-ing the in-medium SRG equations to two-body normal-ordered operators will (approximately) evolve inducedthree- and higher-body interactions through the density-dependent coefficients of the zero-, one-, and two-bodyoperators in Eq. (64).

The in-medium SRG flow equations at the normal-ordered two-body level are obtained by evaluatingdH/ds = [η,H] with the normal-ordered HamiltonianH = E0+ f +Γ and the SRG generator η = η1b+η2b (withone- and two-body terms) and neglecting three- andhigher-body normal-ordered terms. For infinite mat-ter, a natural generator choice is η = [ f ,Γ] in analogywith the free-space SRG. In this case, the explicit formof the SRG equations simplifies because η1b = 0 and


fi j = fi δi j. This leads to

dE0

ds=

12

∑1234

( f12 − f34)|Γ1234|2 n1n2n3n4 , (68)

d f1ds

=∑234

( f41 − f23)|Γ4123|2

× (n2n3n4 + n2n3n4) , (69)

dΓ1234

ds= −( f12 − f34)2 Γ1234

+12

∑ab

( f12 + f34 − 2 fab)Γ12abΓab34

× (1 − na − nb)

+∑ab

(na − nb)

×Γa1b3Γb2a4

[( fa1 − fb3) − ( fb2 − fa4)

]− Γa2b3Γb1a4

[( fa2 − fb3) − ( fb1 − fa4)

], (70)

where the single-particle indices refer to momentumstates and include spin and isospin labels.

While the in-medium SRG equations are of secondorder in the interactions, the flow equations build upnon-perturbative physics via the successive interferencebetween the particle-particle and the two particle-holechannels in the SRG equation for Γ, Eq. (70), andbetween the two-particle–one-hole and two-hole–one-particle channels for f , Eq. (69). In terms of diagrams,one can imagine iterating the SRG equations in incre-ments of δs. At each additional increment δs, the in-teractions from the previous step are inserted back intothe right side of the SRG equations. Iterating this pro-cedure, one sees that the SRG accumulates complicatedparticle-particle and particle-hole correlations to all or-ders.

With the choice of generator η = [ f ,Γ], the Hamil-tonian is driven towards the diagonal. This meansthat Hartree-Fock becomes increasingly dominant withthe off-diagonal Γ matrix elements being driven tozero. As with the free-space SRG, it is convenient formomentum-space evolution to switch to the flow vari-able λ ≡ s−1/4, which is a measure of the resulting band-diagonal width of Γ. In the limit λ → 0, Hartree-Fockbecomes exact for the evolved Hamiltonian; the zero-body term, E0, approaches the interacting ground-stateenergy, f approaches fully dressed single-particle ener-gies, and the remaining diagonal matrix elements of Γ

approach a generalization of the quasiparticle interac-tion in Landau’s theory of Fermi liquids [1].

An approximate solution of the E0 flow equation for

0 2 4 6 8 10 12

λ [fm-1

]

-30

-25

-20

-15

-10

-5

0

E/A

[M

eV] E

0

EBHF

(in-medium SRG)

EBHF

(free-space SRG)

kF = 1.4 fm

-1

N3LO(500)

Figure 51: Symmetric nuclear matter energy/particle at kF = 1.4 fm−1

as evolved in the IM-SRG [57].

0 2 4 6 8 10 12

λ [fm-1

]

8

10

12

14

16E

/A [

MeV

]

E0

EBHF

(in-medium SRG)

EBHF

(free-space SRG)

kF = 1.35 fm

-1

N3LO(500)

Figure 52: Neutron matter energy/particle at kF = 1.35 fm−1 asevolved in the IM-SRG [57].

symmetric nuclear matter and neutron matter as a func-tion of λ is shown in Figs. 51 and 52 for two differ-ent Fermi momenta kF (corresponding to different den-sities). As expected, the in-medium SRG drives theHamiltonian to a form where Hartree-Fock becomesexact in the limit λ → 0. In contrast to the lad-der approximation based on NN-only SRG interactionsevolved in free space, the same many-body calcula-tion using interactions evolved with the in-medium SRGat the two-body level gives energies that are approxi-mately independent of λ. This indicates that truncationsbased on normal-ordering efficiently include the dom-inant induced many-body interactions via the density-


Figure 53: “Off-diagonal” terms (e.g., 2p2h sectors) driven to zero as s increases, decoupling them from the Hartree-Fock reference state, whichbecomes exact as s→ ∞ [57].

H. Hergert - NSCL, Michigan State University - Ohio State University Theory Seminar, 22/02/11

In-Medium SRG: Finite Nuclei

rapidly approaches perturbation theory & quasi-exact results (CC, NSCM): weak correlationsE0

[K. Tsukiyama, S. K. Bogner & A. Schwenk, arXiv: 1006.3639 [nucl-th]]

4He

emax = 3

Tuesday, February 22, 2011

Figure 54: IM-SRG flow for the energy of the alpha particle [57].

H. Hergert - NSCL, Michigan State University - Ohio State University Theory Seminar, 22/02/11

In-Medium SRG: Finite Nuclei

In-Medium SRG resultscomparable to CCSD(T)

In-Medium SRG: Groundstate Energies

H. Hergert – NSCL, Michigan State University — TU Darmstadt, 11/06/2010

16 20 24 28 32 36h_ ! [MeV]

-28.5

-28

-27.5

-27

-26.5

Gro

und-

stat

e en

ergy

[MeV

]

emax = 4emax = 5emax = 6emax = 7emax = 8

CCSD

CCSD(T)

16 20 24 28 32 36h_ ! [MeV]

-640

-620

-600

-580

-560

Gro

und-

stat

e en

ergy

[MeV

]

emax = 4emax = 5emax = 6emax = 7emax = 8

CCSD

CCSD(T)

4He 40Ca

In-Medium SRG resultscomparable to CCSD(T)

35

[K. Tsukiyama, S. K. Bogner & A. Schwenk, arXiv: 1006.3639 [nucl-th]]

Tuesday, February 22, 2011

Figure 55: IM-SRG convergence in finite nuclei compared to coupled cluster CCSD and CCSD(T) calculations [57].


dependent zero-, one-, and two-body normal-orderedterms.

In a similar manner, the in-medium SRG can be usedas an ab initio method for finite nuclei. Figure 53 showsthe off-diagonal normal-ordered two-body matrix ele-ments Γ being driven to zero with the IM-SRG evolu-tion. Figure 54 shows the evolution of the ground-stateenergy of 4He [57]. As the flow parameter s increases,the E0 flow and second-order (in Γ) many-body per-turbation theory contributions approach each other, aswas the case for the infinite matter results in Fig. 51.In addition, the convergence behavior with increasingharmonic-oscillator spaces in Fig. 55 for 4He and 40 Cais very promising. Based on these calculations, the in-medium SRG truncated at the normal-ordered two-bodylevel appears to give accuracies comparable to coupled-cluster calculations truncated at the singles and doubles(CCSD) level. Finally, we note that the in-medium SRGis a promising method for non-perturbative calculationsof valence shell-model effective interactions and opera-tors.

5.2. Implications of RG for nuclear calculations

There are many on-going and potential applicationsof low resolution methods for calculations of nuclearstructure and reactions. In many cases RG techniquesare used explicitly but there are also examples of lowresolution being achieved by other means. Here we’llsurvey some of both types. This will be far from a com-prehensive list because it focuses largely on develop-ments associated entirely or in part with a project calledUNEDF, which stands for Universal Nuclear EnergyDensity Functional.

UNEDF is a collaboration of more than fifty physi-cists, applied mathematicians, and computer scientistsin the United States plus many international collabora-tors, funded through the U.S. Department of Energy’sSciDAC program. The long-term vision of the projectis to arrive at a comprehensive and quantitative descrip-tion of nuclei and their reactions. The focused mis-sion is to construct, optimize, validate, and apply en-ergy density functionals for structure and reactions, butto carry out this mission the team has developed manycrosscutting physics collaborations where none existedpreviously between the main physics areas: ab initiostructure, ab initio functionals, DFT applications, DFTextensions, reactions. These interconnections are indi-cated schematically in the UNEDF strategy diagram inFig. 56. This type of large-scale collaboration repre-sents a transformation in how low-energy nuclear the-ory is done. UNEDF has been very productive, with

over 200 publications to date, including 11 Physical Re-view Letters and a Science article in the 2011 calendaryear alone. Further background, references, and scien-tific highlights can be found at unedf.org, the projectwebsite.

One of the important tools for nuclear structure en-abled by low-resolution interactions is the diagonaliza-tion of enormous but very sparse Hamiltonian matrices,usually in a harmonic oscillator basis to permit center-of-mass effects to be excluded. This is referred to as no-core full configuration (NCFC) or no-core shell model(NCSM), depending on the context. Two recent ex-amples of what is enabled are represented in Fig. 57.On the left are Gamov-Teller matrix elements from alarge-scale calculation of Carbon-14 using a soft chiralEFT potential [58]. They highlight the critical role ofthe 3NF in suppressing the beta decay rate, which ex-plains the anomalously long lifetime of 14C (which isused to great advantage for dating artifacts!). On theright is the low-lying spectrum of Fluorine-14, whichis unstable to proton decay. This spectrum was pre-dicted in advance of the experimental measurements(not a common occurrence until now!), which meantsolving a Hamiltonian matrix of dimension 2 billion us-ing 30,000 cores with a soft interaction (derived frominverse scattering rather than RG, but with similar char-acteristics) [59, 60]. The predictions and measurementagree within the combined experimental and theoreticaluncertainties. These calculations would not be possiblewith the potentials of Fig. 3. RG-softened interactionswill allow many more of these confrontations of exper-iment with theory in the future.

One of the principal aims of the UNEDF project isto calculate reliable reaction cross sections for astro-physics, nuclear energy, and national security, for whichextensions of standard phenomenology is insufficient.The interplay of structure and reactions is essential fora successful description of exotic nuclei as well. Suchinterplay is characteristic of the ab initio no-core shellmodel/resonating-group method (NCSM/RGM), whichtreats bound and scattering states within a unified frame-work using fundamental interactions between all nucle-ons. A quantitative proof-of-principle calculation of thisapproach is shown in Fig. 58 [61, 62]. A wide rangeof applications is now possible including 3H(d, n)4Hefusion and the 7Be(p, γ)8B reaction important for solarneutrino physics, and many more to come. In Fig. 59,the first-ever ab-initio calculation of the 7Be(p, γ)8B as-trophysical S-factor is shown [63]. This calculation usesNCSM/RGM with an N3LO NN interaction evolved bythe SRG to a fine-tuned value of λ = 1.86 fm−1. Theab initio theory predicts both the normalization and


!"#$%&'()*%'+,(-%*./0,(

1"*#23*&$(

45.%'6&5$%.(

!"#$%&'(-17(819(:.%$;<#3*./.0%*#,=(>,??%0',(5'%&@/*+(

Nuclear Density Functional Theory

and Extensions

A&*,(53B,(

?%0C3B(

45.%'6&5$%.(

-/&+3*&$/D&23*(7'"*#&23*EB/&+3*&$/D&23*((

A3*0%(F&'$3(

0'"*#&0%B(.G&#%(

45.%'6&5$%.(

;"$$(.G&#%(

>,??%0',('%.03'&23*(A"$2<'%;%'%*#%(-17(:HFA=(

7/?%(B%G%*B%*0(-17(:7-819=(IJKL(

45.%'6&5$%.(

Ab Initio Configuration Interaction

!"#$%

!"#$%

!"#$%

!"#$%

Compound Nucleus and Direct Reaction

Theory F3"G$%B(FC&**%$.(-M9L(

+$35&$(G'3G%'2%.(

45.%'6&5$%.(

!"#$% #'3..(.%#23*.(

%N#/0%B(.0&0%.(B%#&,.(O../3*(

.G%#0'3.#3G/#(/*;3'?&23*(

+$35&$(G'3G%'2%.(.G%#0'3.#3G,(.#&P%'/*+(

4G2#&$(G30%*2&$.(

Figure 56: Strategy diagram for UNEDF.

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

GT

mat

rix

ele

men

t

no 3NF forceswith 3NF forces (c

D= -0.2)

with 3NF forces (cD

= -2.0)

s p sd pf sdg pfh sdgi pfhj sdgik pfhjl

shell

-0.1

0

0.1

0.2

0.3

0.2924

Figure 57: Left: Matrix elements from Carbon-14 lifetime calculation [58]. Right: Fluorine-14 spectrum predicted by NCSM and measuredexperimentally[59, 60].


Figure 58: NCSM/RGM calculation of neutron scattering from thealpha particle using SRG-evolved interactions, compared with exper-imental measurements [61, 62].

6 !!"!#$%&'#((((((!

Lawrence Livermore National Laboratory

7Be(p,!)8B astrophysical S-factor!!! The 7Be(p,!)8B is the final step in the

nucleosynthetic chain leading to 8B

!! ~10% error in latest S17(0): dominated by uncertainty in theoretical models

!! NCSM/RGM results with largest realistic model space

•! SRG-N3LO NN potential (" = 1.86 fm-1)

•! Nmax = 10

•! p+7Be(g.s., 1/2-, 7/2-, 5/21-, 5/22

-)

!! Sep. energy: 136 keV (Expt. 137 keV)"

!! S17(0)=19.4 eV b on the lower side of, but consistent with latest evaluation"

!! Run time: ~150,000 CPU hrs

Reactions important for solar astrophysics P. Navrátil, R. Roth and S. Quaglioni, submitted to Phys. Rev. Lett., arXiv:1105.5977

0 0.5 1 1.5 2 2.5E

kin [MeV]

0

5

10

15

20

25

30

35

S1

7 [

eV b

]

Filippone

StriederHammacheHassBaby

Junghans

SchuemannDavidsNCSM/RGM E1

7Be(p,!)

8B

(a)

Ab initio theory predicts simultaneously

both normalization and shape of S17.

Figure 59: First ever ab initio calculations of 7Be(p, γ)8B astrophysi-cal S-factor [63]. Uses SRG-N3LO with λ = 1.86 fm−1.

the shape of S 17. This very promising techniques hasmany additional applications. The inclusion of an SRG-evolved 3NF is planned for the near future.

UNEDF members have demonstrated that the pow-erful coupled-cluster (CC) ab initio method, which is aworkhorse in quantum chemistry, can be used to accu-rately calculate closed-shell medium-mass nuclei suchas 40C, 48C, 56Ni with chiral EFT two-body interactionsor the RG-softened versions [64, 65], as well as protonhalo nuclei like 19F [66]. A proof-of-principle conver-gence curve (ground-state energy versus the size of theorbital space) for 56Ni with an SRG-evolved potential isshown in Fig. 60. The CC formalism has been extendedto include NNN forces and their inclusion in calcula-tions of the heavier nuclei will break new barriers.

Figure 60: Convergence of CCSD for Ni-56 evolved N3LO to λ =

2.5 fm−1.

The in-medium SRG diagonalization of closed-shellnuclei such as 40C [57], discussed in Section 5.1 is acomplementary approach to CC and is one of severaladvances in our understanding of the phenomenologicalnuclear shell model enabled by softened potentials. An-other is the direct use of MBPT to examine the effectof the 3NF on the location of the neutron dripline: thelimits of nuclear existence where an added neutron is nolonger bound—it “drips” away. The new physics is indi-cated schematically in Fig. 61. It’s been established ex-perimentally that as you add neutrons to stable oxygen-16, the neutrons stay bound until 24O. But adding onemore proton to get fluorine extends the dripline all theway to 31F. This result is not predicted by previous mi-croscopic calculations using NN interactions, becausethe single-particle neutron energy levels that get filledare predicted to be bound (leftmost panel in Fig. 62),leading to 28O as the calculated dripline. The phe-nomenological shell model, in which matrix elements ofthe Hamiltonian are fit to nearby nuclei, has a very dif-ferent pattern (e.g., compare the d3/2 single-particle en-

The oxygen anomaly - impact of 3N forces include “normal-ordered” 2-body part of 3N forces (enhanced by core A)

leads to repulsive interactions between can understand partly based on Pauli

d3/2 orbital remains unbound from 16O to 28O

first microscopic explanation of the oxygen anomaly Otsuka, Suzuki, Holt, AS, Akaishi (2010)

Figure 61: Influence of a three-body force on valence neutrons in oxy-gen isotopes.


The oxygen anomaly - impact of 3N forces include “normal-ordered” 2-body part of 3N forces (enhanced by core A)

leads to repulsive interactions between can understand partly based on Pauli

d3/2 orbital remains unbound from 16O to 28O

first microscopic explanation of the oxygen anomaly Otsuka, Suzuki, Holt, AS, Akaishi (2010)

Figure 62: Three-body force impact on oxygen single-particle levels. Left: NN-only, middle: phenomenological forces, right: NN + 3NF.

ergies in the middle and left panels of Fig. 62) and pre-dicts the correct dripline. However, recent calculationswith a Vlow k RG force and fitted 3NF yield the rightpanel of Fig. 62. When the 3NF effect is added [67],the interaction of valence neutrons with a core neutron,as in Fig. 61, is repulsive, pushing up the d3/2 level sothat the dripline is at 24O.

The apparent success of MBPT with low-momentumpotentials has been tested by Roth and collabora-tors [68], who have done calculations of closed shellnuclei across the mass table in second-order perturba-tion theory (first-order is Hartree-Fock). Results forSRG evolved interactions from an initial NN chiral EFTand including the induced 3NF show excellent indepen-dence of the flow parameter λ (see Fig. 63) and both theenergies and radii are in good agreement with coupled

Systematics: E/A and Rch

-25

-20

-15

-10

-5

0

.

E/A[M

eV]

4He16O

24O34Si

40Ca48Ca

48Ni56Ni

60Ni78Ni

88Sr90Zr

100Sn114Sn

132Sn146Gd

208Pb

2

3

4

5

.

Rch[f

m]

HF+PT

HF

NN + 3N-induced

!α = 0.02 fm4

"α = 0.04 fm4

!

α = 0.08 fm4

experiment

emax = 14E3,max = 12

ℏΩ = 28MeV

28

Figure 63: Second-order MBPT applied to closed shell nuclei withSRG-evolved NN interactions, including the induced 3NF [68]. Sev-eral different values of the flow parameter are shown (note that α hereis the same as s = 1/λ4).

cluster results [64, 65]. However, adding a 3NF leads tolarge λ dependence. Very recent results from Roth et al.that use importance truncated NCSM as well as coupledcluster calculations show that the long-range 3NF is thesource to apparent large 4NF contributions for oxygenand heavier nuclei, causing a strong dependence on theflow parameter. However, by using a lower cutoff forthe initial 3NF, remarkable agreement with experimen-tal binding energies is achieved with fits only to few-body properties [19, 41]. Work is in progress to identifySRG generators to better control the RG evolution of theinitial 3NF.

We have already seen the convergence of MBPT forsymmetric infinite nuclear matter. Perturbation theoryis even more controlled for pure neutron matter, as il-lustrated in Figs. 64 and 65, where Vlow k RG-evolvedinteractions and fitted 3NF’s are used in calculations ofthe neutron matter energy per nucleon as a function ofthe density [69]. The cutoff dependence of the result isused to estimate the many-body uncertainty. Figure 64shows that the 3NF contribution is important (compareto the NN-only curves) but that the dominant theoret-ical uncertainty is the value of the coupling constantsfor the long-range part of the N2LO chiral EFT 3NF. InFig. 65, comparisons with non-perturbative calculationsdemonstrate the consistency of the much easier MBPTcalculations. The results for the neutron matter equationof state have been used by Hebeler et al. to provide tightconstraints on neutron star masses and radii [70].

The MBPT results for infinite matter are also valu-able input for work to develop a microscopic nuclearenergy density functional (EDF). A key tool to incorpo-rate microscopic input into nuclear EDF’s is the den-sity matrix expansion (DME) originally proposed byNegele and Vautherin, which has been revived and im-


Figure 64: Neutron matter energy/particle versus density calculatedusing a Vlow k RG-evolved NN interaction plus fit 3NF. The arrowshows the importance of the 3NF and the width of the band indicatesthe uncertainties due to 3NF couplings [69].

Figure 65: Neutron matter energy/particle versus density as in Fig. 64compared to other calculations [69].

proved in the UNEDF project. The DME provides aroute to an EDF based on microscopic nuclear interac-tions through a quasi-local expansion of the energy interms of various local densities and currents, includingresummations that can treat long-range one- and two-pion exchange interactions given by chiral EFT. Withsufficiently soft microscopic interactions, many-bodyperturbation theory (MBPT) for nuclei is a quantitativeframework for implementing the DME. The formal de-velopment of DME with MBPT is on-going, but thereare already hybrid formulations between purely ab ini-

Vtrap

Figure 66: Schematic representation of interacting neutrons in a (the-oretical) trapping potential.

tio and phenomenological functionals [71], which allowimprovements to be made while more systematic func-tionals are developed.

Several DME implementations strategies have beendeveloped, with the first tests recently made againstab-initio calculations using a semi-realistic interaction(Minnesota) in trapped neutron drops [72]. Neutrondrops are a powerful theoretical laboratory for improv-ing existing nuclear energy functionals, with particu-lar value in providing microscopic input needed forneutron-rich nuclei, where there are fewer constraintsfrom experiment. The necessity of an external potential(because the untrapped system is unbound, with pos-itive pressure) is turned into a virtue by allowing ex-ternal control over the environment (see Fig. 66 for a

2.0 2.5 3.0

radius [fm]

0.85

0.9

0.95

Eto

t/(N

4/3

hΩ

) NCFCHFPSABHFfit

2.0 2.5 3.0

radius [fm]

0.8

0.85

0.9

Eto

t/(N

4/3

hΩ

) Minnesota potential

N = 8

N = 20

Figure 67: Calculations of the ground-state energies and radii oftrapped systems of twenty (top) and eight (bottom) neutron drops [72].The different clusters of points are for different harmonic oscillatortrap frequencies. The NCFC results are exact within the error bars.They are compared to various approximate microscopic functions,with the BHF and “fit” calculations expected to be the best approx-imations.


schematic of the trapped neutrons). Density functionaltheory, which provides the theoretical underpinning forthe microscopic EDF’s, dictates that the same functionalapplies for any external potential, which can thereforebe varied to probe and isolate different aspects of theEDF. Results summarized in Fig. 67 show promisingagreement with the DME functionals and the (essen-tially exact) ab initio results using NCFC. Many moredevelopments in this line will be forthcoming.

5.3. Summary and survey of open problems

In these lectures, we’ve made a whirlwind tool ofatomic nuclei at low resolution. With the renormaliza-tion group (RG), the strategy has been to lower the reso-lution and track dependence on it. We’ve seen how highresolution leads to coupling of low momenta to highmomenta, which hinders solutions of the many-bodyproblem for low-energy properties. With RG evolution,correlations in wave functions are reduced dramatically,leading to faster convergence of many-body methods. Aconsequence is that non-local potentials and many-bodyoperators are induced, so these must be accommodated.

Flow equations (SRG) achieve low resolution by de-coupling. This can be in the form of band or block diag-onalization of the Hamiltonian matrix. The flow equa-tions implement a series of unitary transformations, inwhich observables (measurable quantities) are not al-tered but the physics interpretation can change! In thenuclear case, the usual plan is to evolve until few-bodyforces start to grow rapidly, or to use an in-medium ver-sion of the SRG.

With the RG, cutoff dependence becomes a new toolin low-energy nuclear physics. The basic idea is that,in principle, observables should be unchanged with RGevolution. In practice, there are approximations in theRG implementation and in calculating nuclear observ-ables. These come from truncation or approximation of“induced” many-body forces/operators and from many-body approximations. For nuclei there can be dramaticchanges even with apparently small changes in the reso-lution scale. We can use these changes as diagnostics ofapproximations and to estimate theoretical errors. Somespecific applications of cutoff dependence include:

• using cutoff dependence at different orders in anEFT expansion, which carries over to the corre-sponding RG-evolved interactions;

• using the running of ground-state energies withcutoff in few-body systems to estimate errors andidentify correlations (e.g., the Tjon line);

• in nuclear matter calculations, validating MBPTconvergence and setting lower bounds on the er-rors from uncertainties in many-body interactions;

• in calculations of finite nuclei, diagnosing missingmany-body forces;

• identifying and characterizing scheme-dependentobservables, such as spectroscopic factors.

The possibilities have really only begun to be explored.We have seen glimpses of the many promising ap-

plications of RG methods to nuclei. Configuration in-teraction and coupled cluster approaches using softenedinteractions converge faster, opening up new possibil-ities and allowing the limits of computational feasibil-ity to be extended. Ground-breaking ab-initio reactioncalculations are now possible. Applications of low-momentum interactions to microscopic shell model cal-culations bring new understanding to phenomenologicalresults, highlighting the role of three-body forces. Be-cause many-body perturbation theory (MBPT) is feasi-ble with the evolved interactions, the door is opened toconstructive nuclear density functional theory.

There are also many open questions and difficultproblems in applying RG to low-energy nuclear physics.Here is a subset:

• Power counting for evolved many-body operators.That is, how do we anticipate the size of contri-butions from induced many-body interactions andother operators? This is essential if we are tohave reliable estimates of theoretical errors, be-cause truncations are unavoidable. We need bothanalytic estimates to guide us as well as more ex-tensive numerical tests. Many of the same issuesapply to chiral EFT; can the additional informationavailable from SRG flow parameter dependencehelp with analyzing or even constructing EFT’s?

• Only a few possibilities for SRG generators havebeen considered for nuclear systems. Can otherchoices for the SRG Gs operator help to control thegrowth of many-body forces? Can convergence beimproved in the harmonic oscillator basis, whichis limited by an infrared cutoff as well as an ultra-violet cutoff? Can a generator be found to drivenon-local potentials to local form, so they can beused with quantum Monte Carlo methods? Or canthe SRG equations be formulated to directly pro-duce a local projection and a perturbative residualinteraction?

• An apparent close connection between the block-diagonal generator SRG and the “standard” Vlow k


RG has been established empirically, but a formaldemonstration of the connection and its limits hasnot been made.

• What other bases for SRG evolution would be ad-vantageous? The need for a momentum-space im-plementation for evolution in the A = 3 space andbeyond is foremost. Beyond providing necessarychecks of evolution in the harmonic oscillator ba-sis, the evolved interactions in this form could bedirectly applied to test MBPT in infinite matter andto test nuclear scaling. Another possibility is to usehyperspherical coordinates, which combine the ad-vantage of a discrete basis with better asymptoticbehavior (and which would be useful for visualiza-tion of many-body forces).

• There are many open questions and problems in-volving operators. These include formal issuessuch as the scaling of many-body operators andtechnical issues such as how to handle boosts ofoperators that are not galilean invariant. And thereare simply many applications that are yet to bemade (e.g., electroweak processes).

• The flow to universal form exhibited by two-bodyinteractions has been clear from the beginning ofRG applications to nuclei, but whether this samebehavior is expected for many-body interactions orfor other operators is still open.

• How can we use more of the power of the RG?

There is no shortage of opportunities and challenges!

References

[1] S. Kehrein, The Flow Equation Approach to Many-Particle Sys-tems, Springer, Berlin, 2006.

[2] S. K. Bogner, R. J. Furnstahl, A. Schwenk, From low-momentum interactions to nuclear structure, Prog. Part. Nucl.Phys. 65 (2010) 94–147.

[3] DOE/NSF Nuclear Science Advisory Committe, The Frontiersof Nuclear Science: A Long-Range Plan, 2007.

[4] S. Aoki, T. Hatsuda, N. Ishii, Nuclear Force from Monte CarloSimulations of Lattice Quantum Chromodynamics, Comput.Sci. Dis. 1 (2008) 015009.

[5] R. B. Wiringa, V. G. J. Stoks, R. Schiavilla, An Accuratenucleon-nucleon potential with charge independence breaking,Phys. Rev. C 51 (1995) 38–51.

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