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Similarity Renormalization Group Evolution of Nucleon-Nucleon Interactions in the Subtracted Kernel Method Approach
Nuclear Physics B (Proc. Suppl.) 199 (2010) 215–218 www.elsevier.com/locate/npbps
Similarity Renormalization Group Evolution of Nucleon-Nucleon Interactions in the Subtracted Kernel Method Approach S. Szpigela , V. S. Tim´oteob , and F. de O. Dur˜aesa a
Centro de Ciˆencias e Humanidades, Universidade Presbiteriana Mackenzie, S˜ao Paulo, SP, Brazil
b
Faculdade de Tecnologia, Universidade Estadual de Campinas, Limeira, SP, Brazil
In this work we study the Similarity Renormalization Group (SRG) evolution of effective nucleon-nucleon (N N ) interactions derived using the Subtracted Kernel Method (SKM) approach. We present the results for the phaseshifts in the 1 S0 channel calculated using a SRG potential evolved from an initial effective potential obtained by implementing the SKM scheme for the leading-order N N interaction in chiral effective field theory (ChEFT).
1. INTRODUCTION The Similarity Renormalization Group (SRG) formalism, developed by Glazek and Wilson [1] and independently by Wegner [2], is an approach based on a series of continuous unitary transformations that evolve hamiltonians with a cutoff on energy differences. Such transformations are the group elements that give the method its name. Recently, the SRG approach has been applied to evolve phenomenological N N potentials, such as the Argonne V18 [3], and chiral effective field theory (ChEFT) N N potentials [4,5] to phaseshift equivalent softer forms, effectively decoupling the low-energy observables from the highenergy degrees of freedom. It has been shown that such a decoupling leads to more perturbative potentials, greatly simplifying calculations in nuclear few and many-body problems [6–8]. In this work we apply the SRG approach to evolve a N N potential derived within the framework of the Subtracted Kernel Method (SKM), a renormalization scheme based on a subtracted scattering equation [9–12]. We calculate the phase-shifts in the 1 S0 channel using a potential evolved through the SRG transformation from an initial effective N N potential derived by implementing the SKM scheme for the leading-order (LO) interaction in ChEFT, which consists of the one-pion exchange potential (OPEP) plus a Dirac-delta contact interaction. 0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2010.02.032
2. SIMILARITY RENORMALIZATION GROUP FORMALISM The general formulation of the SRG approach was developed by Glazek and Wilson [1] in the context of light-front hamiltonian field theory, aiming to obtain effective hamiltonians in which the couplings between high and low-energy states are eliminated, while avoiding artificial divergences due to small energy denominators. Consider an initial hamiltonian in the center of mass frame for a system of two nucleons, which can be written in the form H = Trel + V , where Trel is the relative kinetic energy and V is the N N potential. Here and in what follows we use units such that h ¯ = c = M = 1, where M is the nucleon mass. The similarity transformation is defined by a unitary operator designed to act on the hamiltonian and evolve it with a cutoff λ on free energy differences at the interaction vertices, Hλ ≡ U (λ) H U † (λ) ≡ Trel + fλ V λ ,
(1)
where fλ is a similarity function and V λ is called the reduced interaction. The similarity function fλ is a regularizing function which suppresses the matrix elements between states with free energy differences larger then the cutoff λ, such that the hamiltonian is driven towards a band-diagonal form as λ is lowered. Usually, fλ is chosen to be a smooth function of the similarity cutoff λ. A simpler choice is to use a step function.
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S. Szpigel et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 215–218
The similarity transformation can be defined in terms of an anti-hermitian operator ηλ which generates infinitesimal changes of the cutoff λ, dU † (λ) = −ηλ† . (2) dλ Using this definition and the unitarity of U (λ) we can derive a first-order differential equation for the evolution of the hamiltonian,
ηλ = U (λ)
dHλ = [Hλ , ηλ ] , (3) dλ with the boundary condition Hλ |λ→∞ ≡ H. In the application of the SRG described in this work, we employ the formulation developed by Wegner [2], based on a flow equation that governs the unitary evolution of the hamiltonian dHs = [ηs , Hs ] , (4) ds with the boundary condition Hs |s→0 ≡ H. The flow parameter s has dimensions of (energy)−2 and ranges from 0 to ∞. In terms of a similarity cutoff λ, here with dimensions of momentum, the flow parameter is given by the relation s = λ−4 . Wegner’s flow equation is analogous to Eq.(3), but the specific form ηs = [Gs , Hs ] is chosen for the anti-hermitian operator that generates the unitary transformation, which gives dHs = [[Gs , Hs ], Hs ] . (5) ds Such a choice for the generator ηs corresponds in the Glazek-Wilson formulation to the choice of a gaussian similarity function fλ with uniform width λ . The operator Gs defines the generator ηs and so specifies the flow of the hamiltonian. Wegner’s choice in the original formulation is the full diagonal part of the hamiltonian in a given basis, Gs = diag(Hs ). A simpler choice is to use the free hamiltonian, Gs = Trel . Using the generator ηs = [Trel , Hs ], Wegner’s flow equation for the SRG evolution of the N N potential is given by dVs (p, p ) ds
= + ×
−(p2 − p2 )2 Vs (p, p ) 2 ∞ dq q 2 (p2 + p2 − 2q 2 ) π 0 (6) Vs (p, q) Vs (q, p ).
For simplicity, we are using Vs (p, p ) as a short notation for the projected N N potential matrix (JLL S;I) elements Vs (p, p ) in a partial-wave relative momentum space basis, | q(LS)J; I , with normalization such that 2 ∞ 1= dq q 2 | q(LS)J; I q(LS)J; I |, (7) π 0 where the indexes J, L(L ), S and I respectively denote the total angular momentum, the orbital angular momentum, the spin and the isospin quantum numbers of the N N state. 3. SUBTRACTED KERNEL METHOD APPROACH In this section we present the main aspects of the Subtracted Kernel Method (SKM) [9–12]. We begin by considering the formal LippmannSchwinger (LS) equation for the T -matrix of a two-nucleon system, written in operator form as T (E) = V + V G+ 0 (E) T (E) ,
(8) G+ 0 (E)
where V is the interaction potential and is the free Green’s function for the two-nucleon system with outgoing-wave boundary conditions, given in terms of the free hamiltonian H0 by −1 G+ . 0 (E) = [E − H0 + i]
(9)
For singular N N potentials, Eq. (8) becomes illdefined due to the ultraviolet divergencies which appear in the momentum integration. In the SKM approach, a regularized and renormalized LS equation is derived by performing subtractions in the propagator at a certain energy scale. Consider the N N potential in LO ChEFT, which consists of the one-pion exchange potential (OPEP) plus a Dirac-delta contact interaction. In momentum space, the projection of such a potential in the 1 S0 channel is given by V = VOPEP + C0 , where C0 is the strength of the contact interaction. Using Eq. (8), the potential V can be formally written in terms of the T -matrix at a given energy scale −μ2 , 2 2 −1 . (10) V = T (−μ2 ) 1 + G+ 0 (−μ ) T (−μ ) For convenience we choose a negative energy for the subtraction scale, such that the free Green’s 2 function G+ 0 (−μ ) is real.
S. Szpigel et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 215–218
From Eqs. (8) and (10), we obtain the subtracted kernel LS equation for the T -matrix T (E)
=
T (−μ2 )
+
2 T (−μ2 ) G+ R (E; −μ )T (E),
(11)
+ + 2 2 where G+ R (E; −μ ) ≡ G0 (E) − G0 (−μ ). For the LO ChEFT potential, which contains only Dirac-delta singular interactions, Eq. (11) provides a finite solution for the T -matrix at any given energy E, once its value at the subtraction scale −μ2 is known. A simple ansatz for the input matrix T (−μ2 ), which is called “driving term”, consists in considering that
T (−μ2 ) ≡ V (−μ2 ) = VOPEP + C0 (−μ2 ) ,
(12)
where the renormalized strength of the contact interaction C0 (−μ2 ) at the subtraction scale −μ2 is fixed by fitting data for scattering observables. Once C0 (−μ2 ) is fixed, and so T (−μ2 ) is known, a fixed-point renormalized potential VR (invariant with respect to −μ2 ) can be formally defined from Eq. (10) [10], which satisfies the integral equation (in operator form) 2 VR = T (−μ2 ) − T (−μ2 ) G+ 0 (−μ ) VR .
(13)
Replacing V by VR in Eq. (8), we obtain the LS equation for the renormalized T -matrix: TR (E) = VR + VR G+ 0 (E) TR .
(14)
The renormalized potential VR is not welldefined for singular interactions. Nevertheless, for the LO ChEFT potential, Eq. (14) gives a finite solution for the T -matrix, which is equivalent to the solution obtained from the subtracted kernel LS equation, i.e. TR (E) = T (E). 4. RESULTS AND DISCUSSION When considering the application of the SRG approach to evolve the effective N N potential in LO ChEFT, we use the fixed-point potential VR derived through the SKM scheme as the initial potential. For convenience, we implement the SKM procedure using the K-matrix instead of the T -matrix. In the LS equation for the K-matrix the i prescription is replaced by the principal value, such that the K-matrix is real.
217
We also introduce a momentum cutoff Λ, which is convenient for the numerical calculations when we further consider the evolution of VR through the SRG transformation, allowing us to work with a finite-range gaussian grid of relative momentum integration points. It is important to emphasize that the cutoff Λ just plays the role of an instrumental regulator for the numerical integrations, whose effects on the calculated quantities should vanish in the limit Λ → ∞. The subtracted kernel LS equation for the 1 S0 channel K-matrix, with the cutoff Λ included through a step function θ(Λ − q), is given by K(p, p ; k 2 )
K(p, p ; −μ2 ) ∞ 2 P dq q 2 θ(Λ − q) + π 0 2 μ + k2 × K(p, q; −μ2 ) μ2 + q 2 K(q, p ; k 2 ) , (15) × k2 − q2 √ where k = E is the on-shell momentum and the driving term K(p, p ; −μ2 ) is given by V (p, p ; −μ2 ) = VOPEP (p, p ) + C0 (−μ2 ). The renormalized strength C0 (−μ2 ) is fixed at the subtraction scale −μ2 by fitting the experimental value for the scattering length in the 1 S0 channel, as = −23.7 fm, with an accuracy corresponding to an absolute error of 10−5 . Once the renormalized strength C0s (−μ2 ) is fixed, we obtain the renormalized potential VR (p, p ) from the driving term K(p, p ; −μ2 ) by solving the integral equation derived from Eq. (13). By construction, VR (p, p ) should be independent of the subtraction scale −μ2 . However, a residual dependence on the scale μ is generated by the fitting procedure used to fix C0 (−μ2 ). We solve Eq. (6) numerically, obtaining an exact (non-perturbative) solution for the SRG evolved potential. The boundary condition is set at s = 0 (λ → ∞), such that the initial potential Vs=0 (p, p ) is given by VR (p, p ). The relative momentum space is discretized on a grid of N (= 200) gaussian integration points, leading to a system of N 2 first-order non-linear coupled differential equations which is solved by using an adaptative fifth-order Runge-Kutta algorithm. =
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S. Szpigel et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 215–218
In Fig. (1) we show the result obtained for the SRG evolution of the SKM-LO ChEFT potential in the 1 S0 channel. The initial potential VR (p, p ) (λ → ∞) was calculated by fixing C0 (−μ2 ) at μ = 25 fm−1 and using a momentum cutoff Λ = 25 fm−1 . As one can observe, the initial potential has non-zero off-diagonal matrix elements which extend up to high momenta. As the similarity cutoff λ is lowered, the off-diagonal matrix elements are systematically suppressed while the low-momentum components are enhanced. This result clearly shows that the SKM-LO ChEFT potential evolved through the SRG transformation is driven towards a band-diagonal form. Figure 2. Phase-shifts in the 1 S0 channel as a function of ELAB for the initial SKM-LO ChEFT potential and the SRG evolved potentials.
REFERENCES
Figure 1. SRG evolution of the SKM-LO ChEFT potential in the 1 S0 channel (in units of fm). In Fig. (2) we show the phase-shifts in the 1 S0 channel obtained for the initial SKM-LO ChEFT potential VR (p, p ) and for the potentials evolved through the SRG transformation up to several values of the similarity cutoff λ. As expected for a unitary transformation, the results are the same (apart from relative differences smaller than 10−9 due to numerical errors). Acknowledgments: This work was supported by CNPq, FAPESP and Instituto Presbiteriano Mackenzie through Fundo Mackenzie de Pesquisa.
1. S. D. Glazek and K. G. Wilson, Phys. Rev. D48, 5863 (1993); Phys. Rev. D49, 4214 (1994). 2. F. Wegner, Ann. Physik 3, 77 (1994). 3. R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C51, 38 (1995). 4. D. R. Entem and R. Machleidt, Phys. Rev. C68, 041001 (2003). 5. E. Epelbaum, W. Gl¨ockle and U. G. Meissner, Nucl. Phys. A747, 362 (2005). 6. S. K. Bogner, R. J. Furnstahl and R. J. Perry, Phys. Rev. C75, 061001 (2007) 7. S. K. Bogner, R. J. Furnstahl, R. J. Perry and A. Schwenk, Phys. Lett. B649, 488 (2007). 8. E. D. Jurgenson, S. K. Bogner, R. J. Furnstahl and R. J. Perry, Phys. Rev. C78, 014003 (2008). 9. T. Frederico, V. S. Tim´ oteo and L. Tomio, Nucl. Phys. A653, 209 (1999). 10. T. Frederico, A. Delfino, L. Tomio and V. S. Tim´oteo, [arXiv: hep-ph/0101065]. 11. V. S. Tim´ oteo, T. Frederico, L. Tomio and A. Delfino, Phys. Lett. B621, 109 (2005). 12. V. S. Tim´ oteo, T. Frederico, L. Tomio, and A. Delfino, Nucl. Phys. A790, 406c (2007).
Similarity Renormalization Group Evolution of Nucleon-Nucleon Interactions in the Subtracted Kernel Method Approach S. Szpigela , V. S. Tim´oteob , and F. de O. Dur˜aesa a
Centro de Ciˆencias e Humanidades, Universidade Presbiteriana Mackenzie, S˜ao Paulo, SP, Brazil
b
Faculdade de Tecnologia, Universidade Estadual de Campinas, Limeira, SP, Brazil
In this work we study the Similarity Renormalization Group (SRG) evolution of effective nucleon-nucleon (N N ) interactions derived using the Subtracted Kernel Method (SKM) approach. We present the results for the phaseshifts in the 1 S0 channel calculated using a SRG potential evolved from an initial effective potential obtained by implementing the SKM scheme for the leading-order N N interaction in chiral effective field theory (ChEFT).
1. INTRODUCTION The Similarity Renormalization Group (SRG) formalism, developed by Glazek and Wilson [1] and independently by Wegner [2], is an approach based on a series of continuous unitary transformations that evolve hamiltonians with a cutoff on energy differences. Such transformations are the group elements that give the method its name. Recently, the SRG approach has been applied to evolve phenomenological N N potentials, such as the Argonne V18 [3], and chiral effective field theory (ChEFT) N N potentials [4,5] to phaseshift equivalent softer forms, effectively decoupling the low-energy observables from the highenergy degrees of freedom. It has been shown that such a decoupling leads to more perturbative potentials, greatly simplifying calculations in nuclear few and many-body problems [6–8]. In this work we apply the SRG approach to evolve a N N potential derived within the framework of the Subtracted Kernel Method (SKM), a renormalization scheme based on a subtracted scattering equation [9–12]. We calculate the phase-shifts in the 1 S0 channel using a potential evolved through the SRG transformation from an initial effective N N potential derived by implementing the SKM scheme for the leading-order (LO) interaction in ChEFT, which consists of the one-pion exchange potential (OPEP) plus a Dirac-delta contact interaction. 0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2010.02.032
2. SIMILARITY RENORMALIZATION GROUP FORMALISM The general formulation of the SRG approach was developed by Glazek and Wilson [1] in the context of light-front hamiltonian field theory, aiming to obtain effective hamiltonians in which the couplings between high and low-energy states are eliminated, while avoiding artificial divergences due to small energy denominators. Consider an initial hamiltonian in the center of mass frame for a system of two nucleons, which can be written in the form H = Trel + V , where Trel is the relative kinetic energy and V is the N N potential. Here and in what follows we use units such that h ¯ = c = M = 1, where M is the nucleon mass. The similarity transformation is defined by a unitary operator designed to act on the hamiltonian and evolve it with a cutoff λ on free energy differences at the interaction vertices, Hλ ≡ U (λ) H U † (λ) ≡ Trel + fλ V λ ,
(1)
where fλ is a similarity function and V λ is called the reduced interaction. The similarity function fλ is a regularizing function which suppresses the matrix elements between states with free energy differences larger then the cutoff λ, such that the hamiltonian is driven towards a band-diagonal form as λ is lowered. Usually, fλ is chosen to be a smooth function of the similarity cutoff λ. A simpler choice is to use a step function.
216
S. Szpigel et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 215–218
The similarity transformation can be defined in terms of an anti-hermitian operator ηλ which generates infinitesimal changes of the cutoff λ, dU † (λ) = −ηλ† . (2) dλ Using this definition and the unitarity of U (λ) we can derive a first-order differential equation for the evolution of the hamiltonian,
ηλ = U (λ)
dHλ = [Hλ , ηλ ] , (3) dλ with the boundary condition Hλ |λ→∞ ≡ H. In the application of the SRG described in this work, we employ the formulation developed by Wegner [2], based on a flow equation that governs the unitary evolution of the hamiltonian dHs = [ηs , Hs ] , (4) ds with the boundary condition Hs |s→0 ≡ H. The flow parameter s has dimensions of (energy)−2 and ranges from 0 to ∞. In terms of a similarity cutoff λ, here with dimensions of momentum, the flow parameter is given by the relation s = λ−4 . Wegner’s flow equation is analogous to Eq.(3), but the specific form ηs = [Gs , Hs ] is chosen for the anti-hermitian operator that generates the unitary transformation, which gives dHs = [[Gs , Hs ], Hs ] . (5) ds Such a choice for the generator ηs corresponds in the Glazek-Wilson formulation to the choice of a gaussian similarity function fλ with uniform width λ . The operator Gs defines the generator ηs and so specifies the flow of the hamiltonian. Wegner’s choice in the original formulation is the full diagonal part of the hamiltonian in a given basis, Gs = diag(Hs ). A simpler choice is to use the free hamiltonian, Gs = Trel . Using the generator ηs = [Trel , Hs ], Wegner’s flow equation for the SRG evolution of the N N potential is given by dVs (p, p ) ds
= + ×
−(p2 − p2 )2 Vs (p, p ) 2 ∞ dq q 2 (p2 + p2 − 2q 2 ) π 0 (6) Vs (p, q) Vs (q, p ).
For simplicity, we are using Vs (p, p ) as a short notation for the projected N N potential matrix (JLL S;I) elements Vs (p, p ) in a partial-wave relative momentum space basis, | q(LS)J; I , with normalization such that 2 ∞ 1= dq q 2 | q(LS)J; I q(LS)J; I |, (7) π 0 where the indexes J, L(L ), S and I respectively denote the total angular momentum, the orbital angular momentum, the spin and the isospin quantum numbers of the N N state. 3. SUBTRACTED KERNEL METHOD APPROACH In this section we present the main aspects of the Subtracted Kernel Method (SKM) [9–12]. We begin by considering the formal LippmannSchwinger (LS) equation for the T -matrix of a two-nucleon system, written in operator form as T (E) = V + V G+ 0 (E) T (E) ,
(8) G+ 0 (E)
where V is the interaction potential and is the free Green’s function for the two-nucleon system with outgoing-wave boundary conditions, given in terms of the free hamiltonian H0 by −1 G+ . 0 (E) = [E − H0 + i]
(9)
For singular N N potentials, Eq. (8) becomes illdefined due to the ultraviolet divergencies which appear in the momentum integration. In the SKM approach, a regularized and renormalized LS equation is derived by performing subtractions in the propagator at a certain energy scale. Consider the N N potential in LO ChEFT, which consists of the one-pion exchange potential (OPEP) plus a Dirac-delta contact interaction. In momentum space, the projection of such a potential in the 1 S0 channel is given by V = VOPEP + C0 , where C0 is the strength of the contact interaction. Using Eq. (8), the potential V can be formally written in terms of the T -matrix at a given energy scale −μ2 , 2 2 −1 . (10) V = T (−μ2 ) 1 + G+ 0 (−μ ) T (−μ ) For convenience we choose a negative energy for the subtraction scale, such that the free Green’s 2 function G+ 0 (−μ ) is real.
S. Szpigel et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 215–218
From Eqs. (8) and (10), we obtain the subtracted kernel LS equation for the T -matrix T (E)
=
T (−μ2 )
+
2 T (−μ2 ) G+ R (E; −μ )T (E),
(11)
+ + 2 2 where G+ R (E; −μ ) ≡ G0 (E) − G0 (−μ ). For the LO ChEFT potential, which contains only Dirac-delta singular interactions, Eq. (11) provides a finite solution for the T -matrix at any given energy E, once its value at the subtraction scale −μ2 is known. A simple ansatz for the input matrix T (−μ2 ), which is called “driving term”, consists in considering that
T (−μ2 ) ≡ V (−μ2 ) = VOPEP + C0 (−μ2 ) ,
(12)
where the renormalized strength of the contact interaction C0 (−μ2 ) at the subtraction scale −μ2 is fixed by fitting data for scattering observables. Once C0 (−μ2 ) is fixed, and so T (−μ2 ) is known, a fixed-point renormalized potential VR (invariant with respect to −μ2 ) can be formally defined from Eq. (10) [10], which satisfies the integral equation (in operator form) 2 VR = T (−μ2 ) − T (−μ2 ) G+ 0 (−μ ) VR .
(13)
Replacing V by VR in Eq. (8), we obtain the LS equation for the renormalized T -matrix: TR (E) = VR + VR G+ 0 (E) TR .
(14)
The renormalized potential VR is not welldefined for singular interactions. Nevertheless, for the LO ChEFT potential, Eq. (14) gives a finite solution for the T -matrix, which is equivalent to the solution obtained from the subtracted kernel LS equation, i.e. TR (E) = T (E). 4. RESULTS AND DISCUSSION When considering the application of the SRG approach to evolve the effective N N potential in LO ChEFT, we use the fixed-point potential VR derived through the SKM scheme as the initial potential. For convenience, we implement the SKM procedure using the K-matrix instead of the T -matrix. In the LS equation for the K-matrix the i prescription is replaced by the principal value, such that the K-matrix is real.
217
We also introduce a momentum cutoff Λ, which is convenient for the numerical calculations when we further consider the evolution of VR through the SRG transformation, allowing us to work with a finite-range gaussian grid of relative momentum integration points. It is important to emphasize that the cutoff Λ just plays the role of an instrumental regulator for the numerical integrations, whose effects on the calculated quantities should vanish in the limit Λ → ∞. The subtracted kernel LS equation for the 1 S0 channel K-matrix, with the cutoff Λ included through a step function θ(Λ − q), is given by K(p, p ; k 2 )
K(p, p ; −μ2 ) ∞ 2 P dq q 2 θ(Λ − q) + π 0 2 μ + k2 × K(p, q; −μ2 ) μ2 + q 2 K(q, p ; k 2 ) , (15) × k2 − q2 √ where k = E is the on-shell momentum and the driving term K(p, p ; −μ2 ) is given by V (p, p ; −μ2 ) = VOPEP (p, p ) + C0 (−μ2 ). The renormalized strength C0 (−μ2 ) is fixed at the subtraction scale −μ2 by fitting the experimental value for the scattering length in the 1 S0 channel, as = −23.7 fm, with an accuracy corresponding to an absolute error of 10−5 . Once the renormalized strength C0s (−μ2 ) is fixed, we obtain the renormalized potential VR (p, p ) from the driving term K(p, p ; −μ2 ) by solving the integral equation derived from Eq. (13). By construction, VR (p, p ) should be independent of the subtraction scale −μ2 . However, a residual dependence on the scale μ is generated by the fitting procedure used to fix C0 (−μ2 ). We solve Eq. (6) numerically, obtaining an exact (non-perturbative) solution for the SRG evolved potential. The boundary condition is set at s = 0 (λ → ∞), such that the initial potential Vs=0 (p, p ) is given by VR (p, p ). The relative momentum space is discretized on a grid of N (= 200) gaussian integration points, leading to a system of N 2 first-order non-linear coupled differential equations which is solved by using an adaptative fifth-order Runge-Kutta algorithm. =
218
S. Szpigel et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 215–218
In Fig. (1) we show the result obtained for the SRG evolution of the SKM-LO ChEFT potential in the 1 S0 channel. The initial potential VR (p, p ) (λ → ∞) was calculated by fixing C0 (−μ2 ) at μ = 25 fm−1 and using a momentum cutoff Λ = 25 fm−1 . As one can observe, the initial potential has non-zero off-diagonal matrix elements which extend up to high momenta. As the similarity cutoff λ is lowered, the off-diagonal matrix elements are systematically suppressed while the low-momentum components are enhanced. This result clearly shows that the SKM-LO ChEFT potential evolved through the SRG transformation is driven towards a band-diagonal form. Figure 2. Phase-shifts in the 1 S0 channel as a function of ELAB for the initial SKM-LO ChEFT potential and the SRG evolved potentials.
REFERENCES
Figure 1. SRG evolution of the SKM-LO ChEFT potential in the 1 S0 channel (in units of fm). In Fig. (2) we show the phase-shifts in the 1 S0 channel obtained for the initial SKM-LO ChEFT potential VR (p, p ) and for the potentials evolved through the SRG transformation up to several values of the similarity cutoff λ. As expected for a unitary transformation, the results are the same (apart from relative differences smaller than 10−9 due to numerical errors). Acknowledgments: This work was supported by CNPq, FAPESP and Instituto Presbiteriano Mackenzie through Fundo Mackenzie de Pesquisa.
1. S. D. Glazek and K. G. Wilson, Phys. Rev. D48, 5863 (1993); Phys. Rev. D49, 4214 (1994). 2. F. Wegner, Ann. Physik 3, 77 (1994). 3. R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C51, 38 (1995). 4. D. R. Entem and R. Machleidt, Phys. Rev. C68, 041001 (2003). 5. E. Epelbaum, W. Gl¨ockle and U. G. Meissner, Nucl. Phys. A747, 362 (2005). 6. S. K. Bogner, R. J. Furnstahl and R. J. Perry, Phys. Rev. C75, 061001 (2007) 7. S. K. Bogner, R. J. Furnstahl, R. J. Perry and A. Schwenk, Phys. Lett. B649, 488 (2007). 8. E. D. Jurgenson, S. K. Bogner, R. J. Furnstahl and R. J. Perry, Phys. Rev. C78, 014003 (2008). 9. T. Frederico, V. S. Tim´ oteo and L. Tomio, Nucl. Phys. A653, 209 (1999). 10. T. Frederico, A. Delfino, L. Tomio and V. S. Tim´oteo, [arXiv: hep-ph/0101065]. 11. V. S. Tim´ oteo, T. Frederico, L. Tomio and A. Delfino, Phys. Lett. B621, 109 (2005). 12. V. S. Tim´ oteo, T. Frederico, L. Tomio, and A. Delfino, Nucl. Phys. A790, 406c (2007).