Do que são feitas as coisasthibes.macsyma.org/dwlds/vst2019A.pdf · Do que são feitas as coisas ? Varese Salvador Timóteo Universidade Estadual de Campinas - UNICAMP Limeira -

Do que são feitas as coisas ?

Varese Salvador TimóteoUniversidade Estadual de Campinas - UNICAMP

Limeira - SP - Brazil

A força nuclear e o espalhamento NN

Varese Salvador TimóteoUniversidade Estadual de Campinas - UNICAMP

Limeira - SP - Brazil

Acknowledgements

CollaboratorsS. Szpigel (UPM) E. R. Arriola (UGR) E. Batista (UESB)

Effective theory principlePhysics at low energy (large distance) scales is insensitive to the

details of the physics at high energy (small distance) scales

S S0

nu

d

d

Nuclear ForcesPhenomenological forces (Argonne, Nijmegen, … )

High precision fits to scattering data, but too many parameters and no relation to QCD

Boson Exchange forces (Bonn, Paris, … )

Phenomenological short range + meson exchanges, hybrid approach

Chiral forces (LO, NLO, N2LO, N3LO, N4LO, N4LO+)

Chiral expansion, systematic improvement, QCD inspired, Quantum Field Theory

Effective Interactions Timeline

1991 1994 1998 2003 2005 2013

EFT formulationWeinberg

ED N2LOvan Kolck et al.

EI N2LOEpelbaum et al.

N3LOEntem

Machleidt

N3LO (SFR)Epelbaum et al.

optimized N2LOEkström et al.

Many other important works by: Kaiser, Robilotta, Ruiz Arriola, Frederico, Friar, ...

2015 2017

N4LOIdaho, Salamanca

Bochum, Bonn

N4LO+Idaho,

Salamanca

2018

N4LO+Bochum

Chiral Forces with pions & nucleons as fundamental d.o.f.

4

+... +... +...

+... +... +...

+... +... +... +...

2N Force 3N Force 4N Force 5N Force

LO(Q/⇤�)0

NLO(Q/⇤�)2

NNLO(Q/⇤�)3

N3LO(Q/⇤�)4

N4LO(Q/⇤�)5

N5LO(Q/⇤�)6

FIG. 1: Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons and dashed lines pions. Small dots, large soliddots, solid squares, triangles, diamonds, and stars denote vertexes of index �i = 0, 1, 2, 3, 4, and 6, respectively. Furtherexplanations are given in the text.

where the superscript denotes the order ⌫ of the expansion.Order by order, the long-range NN potential builds up as follows:

VLO ⌘ V (0) = V (0)

1⇡ (2.12)

VNLO ⌘ V (2) = VLO + V (2)

1⇡ + V (2)

2⇡ (2.13)

VNNLO ⌘ V (3) = VNLO + V (3)

1⇡ + V (3)

2⇡ (2.14)

VN3LO ⌘ V (4) = VNNLO + V (4)

1⇡ + V (4)

2⇡ + V (4)

3⇡ (2.15)

VN4LO ⌘ V (5) = VN3LO + V (5)

1⇡ + V (5)

2⇡ + V (5)

3⇡ (2.16)

where LO stands for leading order, NLO for next-to-leading order, etc..

10

0

50

100

δ [d

eg]

0.0

5.0

10.0

15.0

20.0

25.0

30.0

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

0

2

4

6

8

10

δ [d

eg]

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-0.5

-0.4

-0.3

-0.2

-0.1

3P03D2

3D3

3P2

FIG. 9: Cutoff dependence of phase shifts in attractive triplet channels at laboratory energies of 10 MeV (solid line), 50 MeV(dashed line), and 100 MeV (dotted line).

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-200-150-100

-500

50100150200

c 1 [fm

4 ]

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-10

-5

0

5

10

δ [d

eg]

3P0

3P0

FIG. 10: Fit result for the counterterm c1 as a function of the cutoff, and the resulting cutoff dependence of the 3P0 phaseshift at laboratory energies of 10 MeV (solid line), 50 MeV (dashed line), 100 MeV (dotted line), and 190 MeV (dash-dottedline).

situation in ordinary ChPT. We can describe this in thesame language used to discuss power counting in ChPT[5, 7, 8]: we represent typical nucleon momenta by Q andthe characteristic scale of QCD in the hadronic phase byMQCD. The effect of iterating an interaction in the ker-nel of the T matrix is twofold. First, one has an extrathree-dimensional momentum integral and an extra NN

Schrodinger propagator. Second, one has an extra factorof the potential. After the cutoff dependence is removedby renormalization, the contribution to the NN T ma-trix from an NN intermediate state is expected to beO(mNQ/4π). This is a factor mN/Q ≫ 1 larger thanin analogous states in ordinary ChPT, and it is due tothe small energy of intermediate states containing nucle-

10

0

50

100

δ [d

eg]

0.0

5.0

10.0

15.0

20.0

25.0

30.0

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

0

2

4

6

8

10

δ [d

eg]

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-0.5

-0.4

-0.3

-0.2

-0.1

3P03D2

3D3

3P2


2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-200-150-100

-500

50100150200

c 1 [fm

4 ]

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-10

-5

0

5

10

δ [d

eg]

3P0

3P0




10

0

50

100

δ [d

eg]

0.0

5.0

10.0

15.0

20.0

25.0

30.0

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

0

2

4

6

8

10

δ [d

eg]

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-0.5

-0.4

-0.3

-0.2

-0.1

3P03D2

3D3

3P2


2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-200-150-100

-500

50100150200

c 1 [fm

4 ]

2 4 6 8 10 12 14 16 18 20

Λ [fm-1]

-10

-5

0

5

10

δ [d

eg]

3P0

3P0




Nogga, Timmermans, van Kolck, Phys Rev C 72 (2005) 054006

10-2 10-1 100 101 10210-6

10-5

10-4

10-3

10-2

10-1

100

101

SKM 1S0 µ=1.030 fm-1

LO NLO NNLO 2c

Δδ

/ δ

ELAB (MeV)10-2 10-1 100 101 102

10-6

10-5

10-4

10-3

10-2

10-1

100

101

SKM 1S0 µ=1.030 fm-1

LO NLO NNLO 3c

Δδ

/ δ

ELAB (MeV)

Szpigel & VST, J Phys G 39 (2012) 105102

LO

N2LO

see works from Machleidt et al. and Epelbaum et al.

Two nucleons

See works byvan Kolck, Machleidt, Entem, Epelbaum, Meissner, …

Kaplan, Savage, Weise, Beane, Bedaque, …

Renormalization strategiesModify the potential

V (p, p0)! V (p, p0) f(p, p0)

V (p, p0)! 0 when p,p0 !1

T (p, p0) = V (p, p0) +Z

dq q2 V (p, q) G(q) T (q, p0)

Modify the LS equation

V (p, p0)!1 when p,p0 !1

V (p, p0)! V (p, p0)

Tn(p, p0) = V (n)(p, p0) +Z

dq q2 V (n)(p, q) G(n)(q) Tn(q, p0)

SUBTRACTIVE RENORMALIZATION

T (E) = V + V G(E) T (E)

T (E) = T (�µ2) + T (�µ2)G(E)T (E)� T (�µ2)G(�µ2)T (E)

T (E) = T (�µ2) + T (�µ2)[G(E)�G(�µ2)⇤ ⇥� ⌅Kernel Subtraction

]T (E)

T. Frederico, V.S. Timóteo and L. Tomio, Nucl. Phys. A 653 (1999) 209

T (�µ2) = V + V G(�µ2) T (�µ2) ⇥ T (�µ2) = V�1 + G0(�µ2) T (�µ2)

⇥

V =�1 + T (�µ2) G(�µ2)

⇥�1T (�µ2)

EXAMPLE: PURE CONTACT INTERACTIONT (E) = T (�µ2) + T (�µ2)[G(E)�G(�µ2)⇤ ⇥� ⌅

Kernel Subtraction

]T (E)

Same result as pionless EFT at LO with DR or CR !

T (�µ2) = V (�µ2) = C0(�µ2)

T (p, p�; k2) = C0(�µ2) +2⇤

⇤dq q2 C0(�µ2)

�1

k2 � q2 + i�� 1�µ2 � q2

⇥T (q, p�; k2)

T (p, p⇥; k2) =�

1C0(�µ2)

+ (µ + ik)⇥�1

a = T (k2 = 0) C0(�µ2) =a

1� µ a

T (p, p⇥; k2) =�1a

+ ik

⇥�1

EXAMPLE 2: OPE + CONTACT INTERACTION

T. Frederico, VST and L. Tomio, Nucl. Phys. A 653 (1999) 209

⇤ ! 1

n = 1

RECURSIVENESS

T (k2) = V (n)(�µ2; k2) + V (n)(�µ2; k2)G(n)(k2;�µ2)T (k2)

V (n) = [1 � (�µ2 � k2)n�1V (n�1)G(n)0 (�µ2)]�1 V (n�1)

G(n)(k2;�µ2) = (�µ2 � k2)n [G0(�µ2)]n G0(k2)

T. Frederico, L. Tomio, A. Delfino, Phys. Lett. B 481 (2000) 143

G0 =1

k2 � q2

V. S. Timóteo, T. Frederico, A. Delfino, L. Tomio, Phys. Lett. B 621 (2005) 109-118

Renormalization Group Invariance

⇥V (n)

⇥µ2= �V (n) ⇥G(n)(k2;�µ2)

⇥µ2V (n)

⌅

⌅µ2V (1)(p, p�) =

2⇥

�dq q2 V (1)(p, q)

1(µ2 + q2)2

V (1)(q, p�)

⌅

⌅µ2V (n)(p, p⇥) =

2⇥

�dq q2 V (n)(p, q)

n(µ2 + k2)n�1

(µ2 + q2)n+1V (n)(q, p⇥)

�T

�µ2= 0

EXAMPLE 3: MULTIPLE SUBTRACTIONS (N2LO)

��

�

�

��

��

��

�

��

��

��

��

��

��

��

� �

��

��

� �!"��#� �!"��#Pha

se s

hift

(deg

rees

)

Cutoff (fm )-1��

�

�

��

��

��

�

��

��

��

��

�

��

��

��

�

��

��

� �!"��#� �!"��# Pha

se s

hift

(deg

rees

)Cutoff (fm )-1

pwa pwa

LECs can be set to reproduce the partial wave analysis at infinite cutoff

V.S.T., T. Frederico, A. Delfino, L. Tomio, Phys. Rev. C 84, 064005 (2011)

n = 4

Scale dependence at N3LO

4 Advances in High Energy Physics

Nijmegen! = 0.1 fm−1

! = 0.2 fm−1

! = 0.3 fm−1

! = 0.4 fm−1

! = 0.5 fm−1

−5−4−3−2−1

0

#(d

eg)

50 100 150 2000Elab (MeV)


! = 0.7 fm−1

! = 0.8 fm−1

! = 0.9 fm−1

50 100 150 2000Elab (MeV)

−5−4−3−2−1

0

#(d

eg)


! = 2.0 fm−1

! = 3.0 fm−1

! = 4.0 fm−1

! = 5.0 fm−1

−5−4−3−2−1

0

#(d

eg)

50 100 150 2000Elab (MeV)


! = 7.0 fm−1

! = 8.0 fm−1

! = 9.0 fm−1

50 100 150 2000Elab (MeV)

−5−4−3−2−1

0

#(d

eg)


! = 0.2 fm−1

! = 0.3 fm−1

! = 0.4 fm−1

! = 0.5 fm−1

50 100 150 2000Elab (MeV)

−3.0−2.5−2.0−1.5−1.0−0.5

0.0

#(d

eg)


! = 0.7 fm−1

! = 0.8 fm−1

! = 0.9 fm−1

−3.0−2.5−2.0−1.5−1.0−0.5

0.0

#(d

eg)

50 100 150 2000Elab (MeV)


! = 2.0 fm−1

! = 3.0 fm−1

! = 4.0 fm−1

! = 5.0 fm−1

50 100 150 2000Elab (MeV)

−3.0−2.5−2.0−1.5−1.0−0.5

0.0

#(d

eg)


! = 7.0 fm−1

! = 8.0 fm−1

! = 9.0 fm−1

−3.0−2.5−2.0−1.5−1.0−0.5

0.0#

(deg

)

50 100 150 2000Elab (MeV)

1F33F3

1F33F3

1F33F3

1F3

3F3

Figure 1: (Color on-line) phase shifts in 1!3 and 3!3 uncoupled channels calculated from the solution of the subtracted LS equation for the"-matrix with five subtractions for the N3LO-EM potential for several values of the renormalization scale compared to the Nijmegen partialwave analysis.

4 Advances in High Energy Physics


! = 0.2 fm−1

! = 0.3 fm−1

! = 0.4 fm−1

! = 0.5 fm−1

−5−4−3−2−1

0

#(d

eg)

50 100 150 2000Elab (MeV)


! = 0.7 fm−1

! = 0.8 fm−1

! = 0.9 fm−1

50 100 150 2000Elab (MeV)

−5−4−3−2−1

0

#(d

eg)


! = 2.0 fm−1

! = 3.0 fm−1

! = 4.0 fm−1

! = 5.0 fm−1

−5−4−3−2−1

0

#(d

eg)

50 100 150 2000Elab (MeV)


! = 7.0 fm−1

! = 8.0 fm−1

! = 9.0 fm−1

50 100 150 2000Elab (MeV)

−5−4−3−2−1

0

#(d

eg)


! = 0.2 fm−1

! = 0.3 fm−1

! = 0.4 fm−1

! = 0.5 fm−1

50 100 150 2000Elab (MeV)

−3.0−2.5−2.0−1.5−1.0−0.5

0.0

#(d

eg)


! = 0.7 fm−1

! = 0.8 fm−1

! = 0.9 fm−1

−3.0−2.5−2.0−1.5−1.0−0.5

0.0

#(d

eg)

50 100 150 2000Elab (MeV)


! = 2.0 fm−1

! = 3.0 fm−1

! = 4.0 fm−1

! = 5.0 fm−1

50 100 150 2000Elab (MeV)

−3.0−2.5−2.0−1.5−1.0−0.5

0.0

#(d

eg)


! = 7.0 fm−1

! = 8.0 fm−1

! = 9.0 fm−1

−3.0−2.5−2.0−1.5−1.0−0.5

0.0

#(d

eg)

50 100 150 2000Elab (MeV)

1F33F3

1F33F3

1F33F3

1F3

3F3

Figure 1: (Color on-line) phase shifts in 1!3 and 3!3 uncoupled channels calculated from the solution of the subtracted LS equation for the"-matrix with five subtractions for the N3LO-EM potential for several values of the renormalization scale compared to the Nijmegen partialwave analysis.

no contacts

E. F. Batista, S. Szpigel and VST, AHEP (2017) 2316247

400 MeV < Λ < 1000 MeVsharp cutoff

20 MeV < μ < 2000 MeVmultiple subtractions

smooth cutoff Λ = 500 MeV

0 50 100 150 200-5

-4

-3

-2

-1

0

Elab (MeV)

phaseshift

(degrees

)1F3

no contacts

Cutoff vs Subtractions at N4LO

Subtractions now provide scale independence

IMPLICIT RENORMALIZATION

unitary evolution of a hamiltonian H = Trel + V with a flowparameter s that ranges from 0 to1,

dHs

ds= [⌘s,Hs] , (7)

where ⌘s = [Gs,Hs] is an anti-hermitian operator that generatesthe unitary transformations. We take the Block-diagonal SRGgenerator [17] given by

Gs = HBDs ⌘

0BBBBBBBB@

PHsP 0

0 QHsQ

1CCCCCCCCA . (8)

where P and Q = 1 � P are projection operators. The flow pa-rameter s has dimensions of [energy]�2 and in terms of a sim-ilarity cuto↵ � with dimension of momentum is given by therelation s = ��4. The flow equation is to be solved with theboundary condition Hs|s!s0

⌘ Hs0 . Using that Trel is indepen-dent of s, we obtain

dVs

ds= [⌘s,Hs] . (9)

In a partial-wave relative momentum space basis, the projectionoperators are determined in terms of a momentum cuto↵ scale⇤ that divides the momentum space into a low-momentum P-space (p < ⇤) and a high-momentum Q-space (p > ⇤),

P ⌘ ✓(⇤ � p); Q ⌘ ✓(p � ⇤) . (10)

The potential Vs can be written as,

Vs ⌘

0BBBBBBBB@

PVsP PVsQ

QVsP QVsQ

1CCCCCCCCA . (11)

By choosing the block-diagonal generator, the matrix-elementsinside the o↵-diagonal blocks PVsQ and QVsP are suppressedas the flow parameter s increases (or as the similarity cuto↵� decreases), such that the hamiltonian is driven to a block-diagonal form,

lim�!0

V� = PVlowkP + QVhighkQ =

0BBBBBBBB@

Vlow k 0

0 Vhigh k

1CCCCCCCCA (12)

Thus, in the limit � ! 0 the P-space and the Q-space becomecompletely decoupled. Thus, while unitarity implies ��(p) =�(p) for any � one has

lim�!0��(p) = �lowk(p) + �highk(p) (13)

where �lowk(p) = �(p)✓(⇤� p) and �highk(p) = �(p)✓(p�⇤) arethe phase shifts of the Vlow k and Vhigh k potentials respectively(see Eq. (4)).

4. Implicit Renormalization: Low cut-o↵ evolution

At low cut-o↵s ⇤ we may approximate the hermitian e↵ec-tive interaction by a polymomial,

V⇤(p0, p) = C0 +C2(p2 + p02)+ C4(p4 + p04) +C04 p2 p02 + . . . . (14)

where C0,C2,C4,C04, . . . are real coe�cients depending on ⇤to be determined. This corresponds to a theory with contactinteractions only. Using the potential of Eq. (14) the LS Eq. (3)reduces to a system of algebraic equations which solution iswell known (see e.g. Ref. [20]). At lowest leading order (LO)we just keep the leading term C0 and get

C0(⇤) =↵0

1 � 2⇤↵0⇡

, (15)

showing that lim⇤!0 V⇤(0, 0) = ↵0. Going to Next-to-leafingorder (NLO) we obtain

�1↵0⇤

=4⇣�2c2

2 + 90⇡4 + 15(3c0 + 2c2)⇡2⌘

9⇡⇣c2

2 � 10c0⇡2⌘ , (16)

r0⇤ =16⇣c2

2 + 12⇡2c2 + 9⇡4⌘

⇡�c2 + 6⇡2�2 �

12c2⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵0⇤

+3c2⇡

⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵2

0⇤2,

where c0 = 4⇡⇤C0, c2 = 4⇡⇤3C2. In the second equation wehave eliminated C0 in terms of ↵0. This leads for any cut-o↵⇤ to the mapping (↵0, r0) ! (C0,C2). At this level of approxi-mation there are two branches and we choose the one consistentwith the LO one for⇤! 0, see Eq. (15) and Fig. 2. We will de-note LO by C(0)

0 and NLO by C(2)0 and C(2)

0 . One should note thatin the case of the 3S 1 channel C(0)

0 is singular and the derivativesof C(0)

2 and C(2)2 are discontinuous at ⇤ = ⇡/2↵0 ⇠ 0.3 fm�1,

which is the momentum scale where the deuteron bound-stateappears. The strong resemblance of both 1S 0 and 3S 1 at thescales around ⇤ ⇠ 1fm�1 is just a reminiscent of the SU(4)Wigner symmetry for the two-nucleon system [26, 7, 27, 28].

One can in principle improve by including more terms be-yond second order in Eq. (14). The problem is that there aretwo such terms C4 and C04 [20] but there is only one low en-ergy parameter in the ERE, v2 in Eq. (6). This is so becausescattering does not depend just on the on-shell potential. Thus,the implicit renormalization is not unique beyond NLO. Thisis just a manifestation of the ambiguities of the inverse scatter-ing problem which can only be fixed after three or higher bodyproperties are taken into account 1. Clearly, and even for the C0and C2 coe�cients, increasing ⇤ values one starts seeing morehigh energy details of the theory.

Even at NLO the question is how small must be the cut-o↵scale so that Eq. (14) works. There is a maximum value ⇤WBfor the cuto↵ scale ⇤ above which one cannot fix the strengthsof the contact interactions C(0)

2 and C(2)2 by fitting the experi-

mental values of both the scattering length ↵0 and the e↵ec-tive range r0 while keeping the renormalized potential hermi-tian. This limit corresponds to the Wigner causality bound re-alized as an o↵-shell unitarity condition [20, 30]. Indeed, for

1Actually from a dimensional point of view the two-body operators withfour derivatives are suppressed as compared to contact three body operators.The o↵-shellness of the two body problem can be equivalently be translatedinto some three-body properties [29].

3


dHs

ds= [⌘s,Hs] , (7)


Gs = HBDs ⌘

0BBBBBBBB@

PHsP 0

0 QHsQ

1CCCCCCCCA . (8)



dVs

ds= [⌘s,Hs] . (9)


P ⌘ ✓(⇤ � p); Q ⌘ ✓(p � ⇤) . (10)


Vs ⌘

0BBBBBBBB@

PVsP PVsQ

QVsP QVsQ

1CCCCCCCCA . (11)


lim�!0


0BBBBBBBB@

Vlow k 0

0 Vhigh k

1CCCCCCCCA (12)






V⇤(p0, p) = C0 +C2(p2 + p02)+ C4(p4 + p04) +C04 p2 p02 + . . . . (14)


C0(⇤) =↵0

1 � 2⇤↵0⇡

, (15)


�1↵0⇤

=4⇣�2c2

2 + 90⇡4 + 15(3c0 + 2c2)⇡2⌘

9⇡⇣c2

2 � 10c0⇡2⌘ , (16)

r0⇤ =16⇣c2

2 + 12⇡2c2 + 9⇡4⌘

⇡�c2 + 6⇡2�2 �

12c2⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵0⇤

+3c2⇡

⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵2

0⇤2,












3


dHs

ds= [⌘s,Hs] , (7)


Gs = HBDs ⌘

0BBBBBBBB@

PHsP 0

0 QHsQ

1CCCCCCCCA . (8)



dVs

ds= [⌘s,Hs] . (9)


P ⌘ ✓(⇤ � p); Q ⌘ ✓(p � ⇤) . (10)


Vs ⌘

0BBBBBBBB@

PVsP PVsQ

QVsP QVsQ

1CCCCCCCCA . (11)


lim�!0


0BBBBBBBB@

Vlow k 0

0 Vhigh k

1CCCCCCCCA (12)






V⇤(p0, p) = C0 +C2(p2 + p02)+ C4(p4 + p04) +C04 p2 p02 + . . . . (14)


C0(⇤) =↵0

1 � 2⇤↵0⇡

, (15)


�1↵0⇤

=4⇣�2c2

2 + 90⇡4 + 15(3c0 + 2c2)⇡2⌘

9⇡⇣c2

2 � 10c0⇡2⌘ , (16)

r0⇤ =16⇣c2

2 + 12⇡2c2 + 9⇡4⌘

⇡�c2 + 6⇡2�2 �

12c2⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵0⇤

+3c2⇡

⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵2

0⇤2,












3


dHs

ds= [⌘s,Hs] , (7)


Gs = HBDs ⌘

0BBBBBBBB@

PHsP 0

0 QHsQ

1CCCCCCCCA . (8)



dVs

ds= [⌘s,Hs] . (9)


P ⌘ ✓(⇤ � p); Q ⌘ ✓(p � ⇤) . (10)


Vs ⌘

0BBBBBBBB@

PVsP PVsQ

QVsP QVsQ

1CCCCCCCCA . (11)


lim�!0


0BBBBBBBB@

Vlow k 0

0 Vhigh k

1CCCCCCCCA (12)






V⇤(p0, p) = C0 +C2(p2 + p02)+ C4(p4 + p04) +C04 p2 p02 + . . . . (14)


C0(⇤) =↵0

1 � 2⇤↵0⇡

, (15)


�1↵0⇤

=4⇣�2c2

2 + 90⇡4 + 15(3c0 + 2c2)⇡2⌘

9⇡⇣c2

2 � 10c0⇡2⌘ , (16)

r0⇤ =16⇣c2

2 + 12⇡2c2 + 9⇡4⌘

⇡�c2 + 6⇡2�2 �

12c2⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵0⇤

+3c2⇡

⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵2

0⇤2,












3


dHs

ds= [⌘s,Hs] , (7)


Gs = HBDs ⌘

0BBBBBBBB@

PHsP 0

0 QHsQ

1CCCCCCCCA . (8)



dVs

ds= [⌘s,Hs] . (9)


P ⌘ ✓(⇤ � p); Q ⌘ ✓(p � ⇤) . (10)


Vs ⌘

0BBBBBBBB@

PVsP PVsQ

QVsP QVsQ

1CCCCCCCCA . (11)


lim�!0


0BBBBBBBB@

Vlow k 0

0 Vhigh k

1CCCCCCCCA (12)






V⇤(p0, p) = C0 +C2(p2 + p02)+ C4(p4 + p04) +C04 p2 p02 + . . . . (14)


C0(⇤) =↵0

1 � 2⇤↵0⇡

, (15)


�1↵0⇤

=4⇣�2c2

2 + 90⇡4 + 15(3c0 + 2c2)⇡2⌘

9⇡⇣c2

2 � 10c0⇡2⌘ , (16)

r0⇤ =16⇣c2

2 + 12⇡2c2 + 9⇡4⌘

⇡�c2 + 6⇡2�2 �

12c2⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵0⇤

+3c2⇡

⇣c2 + 12⇡2

⌘

�c2 + 6⇡2�2

1↵2

0⇤2,












3

0,0 0,5 1,0 1,5 2,0 2,5-20

-15

-10

-5

0

5

10

contact theory - sharp cutoff

Λ2 C

(2)

2 (f

m)

Λ (fm-1)

1S0

3S1

0,0 0,5 1,0 1,5 2,0 2,5-25

-20

-15

-10

-5

0

5

10

15

20


C(2

)0

(fm

)

Λ (fm-1)

1S0

3S1

0,0 0,5 1,0 1,5 2,0 2,5-25

-20

-15

-10

-5

0

5

10

15

20

C(0

)0

(fm

)

Λ (fm-1)

1S0

3S1


Contact theory in the continuum, regulated by a sharp cutoff

(C0, C2) ! (↵0, r0)

C0 ! ↵0LO:

NLO:

LO NLO

NNLO

EXPLICIT RENORMALIZATION - 1S0 & 3S1

The software elements required for solving the Faddeev equation are displayed in Fig. 11 and the main steps maybe summarized as follows:

• set the momentum grids, the splines and the NN potential on the grid points,

• compute the NN potential at K ! V(pi,K), V(K, p0i) and V(K,K),

• solve the Lippman-Schwinger equation to obtain the o↵-shell T -matrix T (p, p0; K),

• perform the interpolations with the modified cubic splines for having T (p, ⇡1) and S (⇡2),

• carry out the angular and momentum integrations to get Kpi q j ,

• determine the eigenvalues of the Faddeev kernel matrix ⇣l,

• find the energy at which the largest eigenvalue is one, which means f (Kt) = 0 and Et = �K2t ,

• iterate the Faddeev equation to obtain the wave function (p, q).

Figure 11: Software design for the three-nucleon system bound state and scattering.

5.2. SRG evolved chiral forcesThe basic problem of quantum systems is to diagonalize the hamiltonian which, in general, are very complicated

due to the number of particles in the system. Since the kinetic energy is diagonal, the potential is the piece which spoilsthe diagonalness. If the interation part of the hamiltonian is pre-diagonalized, diagonalization of the hamiltonian isfaster. That is the underlying reason for the succsess of the SRG in producing interacions which provide fasterconvergence in many-body calculations.

SRG evolved interations are obtained from a renormalization group flow equation controled by a flow parameterwhich essentially determines the o↵-shellness of the interaction or in other words how close to the diagonal form itis. In momentum space, the partial-wave projected SRG flow equation is given by a set of Np

2 non-linear coupledintegro-diferential equations

dd�

V�(p, p0) = � (p2 � p02)2 V�(p, p0) +2⇡

Zdp00 p002

⇣p2 + p002

⌘V�(p, p00) V�(p00, p0) , (55)

where � = 1/�4 and � is the SRG cuto↵. So � = 0 ( � = 1 ) corresponds to no evolution (original o↵-shellness) and� = 1 ( ⇢ = 0 ) corresponds to a complete evolution towards a diagonal form (no o↵-shellness).

In order to solve the three-nucleon bound state with an SRG evolved interaction the SRG evolution needs to beperformed before the potential goes into the Lippman-Schwinger equation to generate de o↵-shell T -matrix.

17

Physics Letters B 735 (2014) 149

Annals of Physics 371 (2016) 398

Implicit x Explicit - 1S0 & 3S1

0,0 0,5 1,0 1,5 2,0-20

-15

-10

-5

0

5

101S0 - 50 pts - Pmax

= 5 fm-1

λ = 0.1 fm-1C

(2)

0 (f

m)

Λ (fm-1)

contact - smooth cutoff - n=16 SRG-BD

0,0 0,5 1,0 1,5 2,0-4

-2

0

2

4

6

8

101S0 - 50 pts - Pmax

= 5 fm-1

λ = 0.1 fm-1

Λ2 C

(2)

2 (f

m2 )

Λ (fm-1)

0,0 0,5 1,0 1,5 2,0-20

-15

-10

-5

0

5

10

15

203S1 - 50 pts - Pmax

= 5 fm-1- λ = 0.1 fm-1

C(2

)0

(fm

)

Λ (fm-1)

contact - smooth cutoff - n=16 SRG-BD

0,0 0,5 1,0 1,5 2,0-20

-15

-10

-5

0

5

10

15

20

253S1 - 50 pts - Pmax

= 5 fm-1- λ = 0.1 fm-1

Λ2 C

(2)

2 (f

m)

Λ (fm-1)

0,0 0,5 1,0 1,5 2,0 2,5-25

-20

-15

-10

-5

0

5

10

15

20

C(0

)0

(fm

)

Λ (fm-1)

1S0

3S1


0,0 0,5 1,0 1,5 2,0 2,5-25

-20

-15

-10

-5

0

5

10

15

20


C(2

)0

(fm

)

Λ (fm-1)

1S0

3S1

0,0 0,5 1,0 1,5 2,0 2,5-20

-15

-10

-5

0

5

10


Λ2 C

(2)

2 (f

m)

Λ (fm-1)

1S0

3S1

Figure 2: C(0)0 , C(0)

2 and C(2)2 for the contact theory in the continuum regulated by a sharp momentum cuto↵ for the 1S 0 channel and the 3S 1 channel. The parameters

are determined from the solution of the LS equation for the on-shell K-matrix by fitting the ERE parameters

⇤ > ⇤WB ⇠ 1.9 fm�1 in the case of the 1S 0 channel and⇤ > ⇤WB ⇠ 2.4 fm�1 in the case of the 3S 1 channel, the pa-rameters C(0)

2 and C(2)2 diverge before taking complex values

and hence violating the hermiticity of the e↵ective potential inEq. (14).

5. Numerical results

The Block-Diagonal-SRG equations, Eq.(9), have to besolved numerically on a momentum grid with N-points yield-ing 4 ⇥ N2 non-linear first order coupled di↵erential equations.Furthermore, an auxiliary numerical cut-o↵ Pmax = N�p mustalso be introduced. It is interesting to test the space dimen-sions needed to solve the contact theory close to the contin-uum. This is shown in Fig. 3 where one sees that large N isneeded to reproduce the continuum limit. We will set N = 50and Pmax = 5fm�1 to our SRG calculations, solve the systemof 4 ⇥ N2 non-linear first-order coupled di↵erential equationsby using an adaptative fifth-order Runge-Kutta algorithm as inRef. [30] and compare the results to the contact interaction withthe same N and Pmax. We check unitarity by comparing phase-shifts along the � evolution, ��(p) = �(p). The sharp momen-tum projectors in Eq. (10) may be regularized as smooth pro-jectors [31] (Q ⌘ 1 � P)

P = ⇥(⇤ � p) = limn!1

e�(p/⇤)n, (17)

and we will take the values n = 2, 4, 8, 16 to check convergence.We want to compare the running of the coe�cients C0 and C2with the cut-o↵ ⇤ in the contact theory potential at NLO to therunning of the corresponding coe�cients C0 and C2 with theVlowk cuto↵ (⌘ ⇤) extracted from a polynomial fit of the BD-SRG-evolved gaussian potential,

V�,⇤(p, p0) = C0 + C2 (p2 + p02) + · · · . (18)

The parameters C and L in the initial gaussian potential (�,⇤!1), defined by Eq. (5), and the coe�cients C0 and C2 in thecontact theory potential at NLO are determined from the so-lution of the LS equation for the K-matrix on the finite mo-mentum grid by fitting the experimental values of the scatter-ing length ↵0 and the e↵ective range r0. The coe�cients C0

and C2 are determined by fitting the diagonal matrix-elementsof the BD-SRG-evolved potential for the lowest momenta withthe polynomial form and the finite momentum grid.

In Fig. 4 we show the results for C0 and ⇤2C2 extracted fromthe 1S 0 channel and the 3S 1 channel BD-SRG-evolved gaus-sian potentials on a grid (with N = 50 gauss points and Pmax =5 fm�1) and down to the lowest SRG cuto↵ � = 0.1fm�1, com-pared to C0 and ⇤2C2 obtained for the contact theory potentialat NLO (on the same grid) regulated by a smooth exponentialmomentum cuto↵ with sharpness parameter n = 16. As wesee, there is a remarkably good agreement between the coe�-cients extracted from the BD-SRG-evolved potential and thoseobtained for the contact theory in the limit �! 0.

It is important to point out that the agreement between therunning of the coe�cients C0 and C2 in the contact theory po-tential and the running of the coe�cients C0 and C2 extractedfrom the BD-SRG-evolved gaussian potential as the similaritycuto↵ � decreases below ⇤ can be traced to the decoupling be-tween the P-space and the Q-space, which follows a similarpattern. Thus, in the limit � ! 0 we expect to achieve a highdegree of agreement for cuto↵s ⇤ up to ⇤WB determined by theWigner bound for the contact theory.

The overlapp between the discretized explicit and implicitnumerical solutions is verified in a wide range of cut-o↵s ⇤.If continuum accuracy was to be judged from the slow conver-gence pattern of Fig. 3, the equivalent BD-SRG calculationswould be out of question. Thus, the continuum limit �p! 0 isbetter and more simply represented by the implicit approach.

For the 1S 0 and 3S 1 neutron-proton scattering states thisrange is within 0.5fm�1

⇤ 1.5fm�1. This is a welcomefeature, since it suggests that the bulk of the e↵ective interac-tion and its scale dependence can directly be extracted from lowenergy NN data. This was the purpose of a previous analysis [7]where the Skyrme force parameters just deducible from the NNinteraction in S- and P-waves were determined. Of course, thepresumably small corrections due to long distance e↵ects likeOne-Pion-Exchange (OPE) in the determination of the e↵ec-tive interaction at these low cut-o↵s remains to be seen (see e.g.Ref. [32]).

4

Szpigel, Ruiz Arriola, VST, Physics Letters B 728 (2014) 596–601

Szpigel, Ruiz Arriola, VST, Annals of Physics 353 (2015) 129–149

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