A Novel Regularization Scheme for Nucleon-Nucleon Lattice Simulations with Effective Field Theory
.. A Novel Regularization Scheme for Nucleon-Nucleon Lattice Simulations withEffective Field Theory
M. Ahmadi ∗ Department of Physics, University of Tehran, PO Box 14395-547, Tehran, Iran
M. R. Hadizadeh † College of Engineering, Science, Technology and Agriculture,Central State University, Wilberforce, OH, 45384, USA andDepartment of Physics and Astronomy, Ohio University, Athens, OH, 45701, USA
M. Radin ‡ Department of Physics, K. N. Toosi University of Technology, PO Box 16315-1618 Tehran, Iran
S. Bayegan § Department of Physics, University of Tehran, PO Box 14395-547, Tehran, Iran (Dated: September 29, 2020)We propose a new regularization scheme to study the bound state of two-nucleon systems inLattice Effective Field Theory. Inspired by continuum EFT calculation, we study an exponentialregulator acting on the leading-order (LO) and next-to-leading order (NLO) interactions, consistingof local contact terms. By fitting the low-energy coefficients (LECs) to deuteron binding energyand the asymptotic normalization coefficient (ANC) on a lattice simulation, we extract the effectiverange expansion (ERE) parameters in the S channel to order p . We explore the impact ofdifferent powers of the regulator on the extracted ERE parameters for the lattice spacing a = 1 . . ≤ a ≤ . I. INTRODUCTION
Nuclear lattice effective field theory (NLEFT) is a model-independent and precision controlled approach for thecalculation of bound and scattering state properties in nuclear physics [1]. The novel combination of lattice methodswith an effective field theory approach has been pursued successfully for few- and many-body systems.The first attempts for an exact solution of infinite nuclear matter using Monte Carlo methods are performed inRef. [2], indicating that energy and saturation properties of symmetric nuclear matter can be reproduced from latticesimulations. The ab initio techniques combine the Monte Carlo methods with the low-energy EFT, known as chiraleffective field theory. Based on these approaches, our information for the scattering of light nuclei, and the ground-state properties of light-, medium-mass nuclei, as well as neutron matter has been compromised [3–8]. To improvethe efficiency of large-scale calculations of nucleus-nucleus scattering and reactions using Monte Carlo calculations,the adiabatic projection method is developed on lattice [9, 10]. The accuracy and efficiency of the method are testedon fermion-dimer scattering calculations in lattice EFT.The bound state of two nucleons on a lattice, mainly in the S − wave channel, is formulated in pionless EFT at theNLO [11]. The lattice spacing dependence of the RG flows is studied while the deuteron binding energy and the ANCare being fixed. L¨uscher has shown how one can connect the quantities obtained on a finite volume to the infinitevolume physical observables by connecting the box size dependence of energy eigenvalues on a lattice to the effectiverange parameter and the scattering length [12]. The exact solution of L¨uscher formula for the energy-levels of thetwo-nucleon system on a lattice with periodic boundary conditions for the extraction of scattering parameters has been ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]; (Corresponding author) a r X i v : . [ nu c l - t h ] S e p implemented by Beane et al. in a pionless EFT approach [13]. They have shown that lattice simulations with L ≥ L ≤
10 fm requires direct matching to p · cotδ in the spin-singlet channel andconsidering the mixing between the S − and D − wave remains challenging. The impact of the topological finite-volumecorrections in lattice calculations of three-nucleon bound state [14], the elastic scattering of fermion-dimer [15], andalso neutron-deuteron scattering at the very low energies [16] are studied in a pionless EFT approach.One of the main challenges of lattice calculations is the necessity to eliminate errors caused by the non-vanishinglattice spacing. One approach to eliminate the lattice artifacts is including the irrelevant higher-dimensional operatorsinto the lattice action, which leads to faster convergence to the continuum limit [17]. Since the lattice spacing serves asa natural UV regulator for the theory, another practical strategy is the application of a regulator to utilize the smearingof the contact interactions. Klein et al. have shown that the application of different regularization schemes leads tothe lattice spacing independence of observables for a wide range of the lattice spacing in the range 0 . ≤ a ≤ . LO), including two- and three-nucleon interactions, is performed in Ref. [19]. The binding energy correlation oftriton and helium-4 is studied for various lattice spacings a = 1 . , . , .
32 fm, and it is shown how the convergencetowards the Tjon line is reached for smaller lattice spacing. A systematic study of neutron-proton scattering, interms of the computationally efficient radial Hamiltonian method, is studied on a lattice EFT up to N LO [20]. Aregularization scheme is applied only to the LO contact interactions. The lattice spacing dependence of the scatteringobservables is explored for lattice spacings ranging from a = 1 .
97 fm down to a = 0 .
98 fm, and it is shown at a = 0 . et al. , the effect of the finite lattice sizeon the binding energies of light nuclei is explored by the construction of pionless EFT at the LO, where a gaussianregulator is applied on the contact terms [21].In this paper, we propose a regularization scheme, inspired by continuum EFT calculations, to study the two-nucleon systems on a lattice and extract the ERE parameters for a wide range of lattice spacing. In Sec. II, we brieflyreview the formalism of two-nucleon bound state on a lattice, projected in the S channel, using pionless EFT up toNLO. By introducing the Lagrangian and Hamiltonian of the two-nucleon system on a lattice, the explicit form of theLippmann-Schwinger equation is presented by considering the contact interactions between nucleons. In Sec. III, theprocedure of extraction of physical ERE parameters from finite volume energy eigenvalues is discussed. Our numericalresults for the lattice energy eigenvalues obtained for different lattice spacing parameters and different numbers oflattice nodes are presented in Sec. IV. Moreover, a new regularization scheme is introduced, and the impact of theregularization scheme on the ERE parameters is studied in detail. A conclusion is provided in Sec. V. All the energyeigenvalues obtained for different lattice spacing parameters are provided in the Appendix A. II. TWO-NUCLEON IN S CHANNEL IN PIONLESS LATTICE EFT UP TO NLO
At very low energies where the nucleon momentum is much smaller than the pion mass, i.e., Q (cid:28) m π , few-nucleonsystems are not sensitive to the details of the nucleon-nucleon interactions. So, an EFT is constructed by low energydegrees of freedom and the Lagrangian is formulated as all contact interactions between nucleons that are allowed bysymmetry. In this section, we consider the NLO Lagrangian of pionless EFT. The nucleon-nucleon interactions aredefined by an infinite number of local operators with an increasing number of derivatives acting on the nucleon fields.The isospin SU(2) symmetric and nonrelativistic Lagrangian in the continuum is given by L = N † (cid:20) i∂ t + ∇ M (cid:21) N − C (cid:0) N T P k N ) † ( N T P k N (cid:1) + C (cid:104) ( N T P k N ) † ( N T P k ←→∇ N ) + h.c. (cid:105) , (1)where N denotes the nonrelativistic nucleon field, M is nucleon mass, the low-energy constants (LECs) C and C are the zero-range interaction strengths, and ←→∇ = ←−∇ · ←−∇ − · ←−∇ · −→∇ + −→∇ · −→∇ . P k = √ σ σ k τ , with the vectorindices k = 1 , ,
3, is the projection operator for S channel, where σ and τ are the Pauli matrices acting on thespin and isospin spaces, respectively. The Hamiltonian corresponding to the Lagrangian of Eq. (1) is given by H = (cid:90) d x (cid:20) N † (cid:18) −∇ M (cid:19) N + C (cid:0) N T P k N ) † ( N T P k N (cid:1) − C (cid:16) ( N T P k N ) † ( N T P k ←→∇ N ) + h.c. (cid:17) (cid:21) . (2)To study the bound state of two-nucleon systems on a lattice, we utilize a cubic box of side length L with periodicboundary conditions. The lattice spacing between lattice nodes is a , so that L = N s a , where N s is the number ofnodes in each spatial direction. As it is shown in Ref. [11], in order to transform the Hamiltonian of Eq. (2) from thecontinuum to a dimensionless Hamiltonian on a lattice, one needs to apply the following substitutions N ( x ) → N n a − / , x → n a, (cid:90) d x → a (cid:88) n ,H → H L a − , M → M L a − , C → C L a , C → C L a , (3)where n = ( n , n , n ) is a three-dimensional vector with integer components and C L , C L and M L are dimensionlessparameters, corresponding to parameters C , C and M in continuum. The lattice Hamiltonian H L can be obtainedin terms of dimensionless quantities as H L = (cid:88) n (cid:20) − M L N † n ∇ L N n + C L (cid:0) N T n P k N n ) † ( N T n P k N n (cid:1) − C L (cid:110) ( N T n P k N n ) † ( N T n P k ←→∇ L N n ) + h.c. (cid:111) (cid:21) , (4)where ∇ L and ←→∇ L represent the discretization of the dimensionless Laplacian. By considering the nucleon operator N n in momentum space as N n = 1 N / s (cid:88) p L e i p L · n a p L , (5)the lattice Hamiltonian of Eq. (4) leads to H L = (cid:88) p L P L M L a † p L a p L + 1 N s (cid:88) p L , p (cid:48) L (cid:0) C L + 4 C L ( P L + P (cid:48) L ) (cid:1) (cid:16) a † p L P k a †− p L (cid:17) (cid:0) a p (cid:48) L P k a − p (cid:48) L (cid:1) . (6)The momentum argument P L obtained from the free nucleon lattice action, improvement up to O ( a ), defined as [6] P L ≡ (cid:88) i =1 (cid:18) ω − ω cos( p i ) + ω cos(2 p i ) − ω cos(3 p i ) (cid:19) . (7)where the components of the lattice momentum p L ≡ ( p , p , p ) under the periodic boundary condition takes thevalues p i = 2 πN s ˆ p i , − N s < ˆ p i ≤ N s , i = 1 , , . (8)As it is shown in Ref. [6], the hopping coefficients ω i in the improved free nucleon action eliminate lattice artifacts inthe Taylor expansion of single-nucleon dispersion relation around p L = 0 up to the indicated order. The coefficients ω i for different level of improvement up to O ( a ), are listed in Table. I. It should be noticed that the O ( a n )-improvedlattice action corresponds to a lattice derivative which contains 2 n +2 nearest neighbors, or a total of 2 n +3 lattice sites.It means unimproved, O ( a )-improved, and O ( a )-improved actions are corresponding to three-, five-, and seven-pointformula, respectively. By considering the lattice Hamiltonian of Eq. (6), the lattice form of Lippmann-Schwingerequation for two-nucleon bound state can be obtained as [11] ψ ( p L ) = 1 E L − P L M L · N s (cid:88) p (cid:48) L (cid:0) C L + 4 C L ( P L + P (cid:48) L ) (cid:1) ψ ( p (cid:48) L ) , (9)where E L = Ea is the dimensionless two-nucleon binding energy and ψ ( p L ) is the discretized two-nucleon wavefunction.TABLE I: Hopping coefficients ω i for different levels of improvement up to O ( a ) in the free nucleon lattice action[6]. unimproved O ( a )-improved O ( a )-improved ω / / ω / / ω /
12 3 / ω / III. EXTRACTION OF EFFECTIVE RANGE EXPANSION PARAMETERS IN LATTICE
By solving the discretized form of the LippmannSchwinger equation of (9), one can obtain the two-nucleon energyeigenvalues on the lattice. In the following, we briefly show how the L¨uscher formula can be used to extract the EREparameters in S channel by having the deuteron binding energy spectrum on the lattice. L¨uscher has shown howone can connect the physical quantities in a finite volume to the real physics by connecting the box size dependenceof the energy eigenvalues in a finite volume to the infinite volume scattering matrix. As it is shown in Ref. [13],the low-momentum behavior of the S − wave phase shift δ , for two-nucleons with a relative momentum p , can bedescribed by the following ERE p · cot δ ( p ) = − a ( S ) + 12 r ( S ) p + . . . = 1 πL S ( η ) , (10)where a ( S ) and r ( S ) refer to the scattering length and the effective range, respectively. S ( η ) is the three-dimensionalzeta function with the dimensionless argument η = (cid:18) Lp π (cid:19) . For | η | < S ( η ) can be expanded in powers of η as S ( η ) = − η + S + S η + S η + S η + . . . (11)where the first few coefficients S i are given as S = − . , S = 16 . , S = 8 . , S = 6 . ,S = 6 . , S = 6 . , S = 6 . , S = 6 . . (12)By considering the connection between the two-nucleon energy levels E = E L /a = p M and the argument η , i.e. , E L = ηaM ( 2 πL ) , one can obtain a set of η for a set of energy eigenvalues E L obtained for a given lattice parameter a and different values of N s or the box side length L . By using Eq. (11), the function 1 πL S ( η ) can be obtained fordifferent values of η dictated by energy eigenvalues E L . Finally by using a linear fitting to Eq. (10), one can extractthe ERE parameters a ( S ) and r ( S ) . IV. NUMERICAL RESULTSA. LECs and different levels of improvement in the lattice momentum argument P L In this section, we study the effect of different levels of improvement, up to O ( a ), in the lattice momentum definedin Eq. (7) to solve the lattice form of Lippmann-Schwinger Eq. (9). To this aim, we solve the discretized Lippmann-Schwinger equation for the lattice spacing a = 1 .
97 fm with the number of nodes N s = 20. The equation can besolved with both direct and Lanczos methods. Our numerical analysis shows that the runtime of the calculationswith the direct approach increases exponentially with the number of nodes N s . For instance, a direct diagonalizationof the kernel of Eq. (9) for N s = 20 takes about 90 minutes, while an iterative solution with the Lanczos technique(see Appendix C2 of Ref. [22]) takes about 1 second, both performed on a single-node CPU desktop. While we areconvinced that both methods yield the same results for lattice deuteron binding energy and wave function, we performall the calculations with the Lanczos technique to save runtime. The Eq. (9) is an eigenvalue equation in the form of λ ψ = K ( E L ) · ψ with the eigenvalue λ = 1. Since the kernel of the equation K ( E L ) is energy dependent, the solutionof the eigenvalue Eq. (9) can be started by an initial guess for the energy E L and the search in the binding energy isstopped when | − λ | ≤ − .The LEC C L at LO is fitted to deuteron binding energy E d = − . C L and C L are determined simultaneously by fitting to deuteron binding energy as well as the asymptotic normalization coefficientANC = 0 . − . . The value of ANC is extracted from the expression for the S − wave asymptotic normalizationcoefficient ANC = √ π (cid:113) k − r (3 S k [23], with k = (cid:112) M | E d | and the experimental value of r ( S ) = 1 . ψ ( p L ) to the analytical wave function ψ ( p L ) = A + BM L | E L | + P L , with ANC = B π .To extract the physical values of LECs C L and C L , Eq. (9) is solved for a wide range of coefficients C L and C L .In Table II, we have listed the obtained LECs at LO and NLO for different levels of improvement. As we can seeat LO, the improvements up to O ( a ) and O ( a ) lead to about 17% and 23% increasing in the absolute value of C L , respectively. While at NLO, the improvements up to O ( a ) and O ( a ) lead to about 4% (14%) and 6% (18%)increasing (decreasing) in the absolute value of C L ( C L ), respectively. In order to minimize the lattice artifacts inour numerical study, for the rest of the paper we use O ( a )-improvement in the lattice momentum P L .TABLE II: The LECs C L and C L obtained at LO and NLO for different levels of improvement in the latticemomentum P L , defined in Eq. (7), to reproduce deuteron binding energy E d = − . . − . for the lattice parameter a = 1 .
97 fm and N s = 20. Improvement Level C L C L · − E (MeV) ANC (fm − . ) LO unimproved − . − . . O ( a )-improved − . − . . O ( a )-improved − . − . . NLO unimproved − . . − . . O ( a )-improved − . . − . . O ( a )-improved − . . − . . B. A New Regularization Scheme in Lattice
In this section, we introduce a new regularization scheme and study its impact on the ERE parameters a ( S ) and r ( S ) obtained from the lattice energy eigenvalues E L for different values of lattice spacing. Inspired by continuumEFT calculations [24], we consider the exponential regulators in the lattice nucleon-nucleon interactions V LNN ( P L , P (cid:48) L )as V LNN ( P L , P (cid:48) L ) → V LNN ( P L , P (cid:48) L ) · f ( P L ) · f ( P (cid:48) L ) , (13)where the regulators are defined as f ( P L ) = 1 f exp( − b · P n/ L /n ); f = 1 N s (cid:88) p L exp( − b · P n/ L /n ) . (14)It should be noticed that the P n/ L is calculated from the lattice momentum argument P L , defined in Eq. (7). Theregulator parameter b is dependent on the lattice spacing parameter a and is defined as b · a = A . A typical valueof the regularization parameter in our calculations for the lattice spacing a = 1 .
97 fm is b = 0 .
01, which leads to theconstant parameter A = 7 . · − fm . In Fig. 1, we have shown the regulator f ( P L ) as a function of thelattice momentum P . L for three exponential powers n = 1 , , b = 0 .
01. The lattice The factor √ π comes from the normalization of the spherical harmonics. P L0.5 f( P L ) n = 1n = 2n = 3 FIG. 1: The functional form of the regulator f ( P L ), defined in Eq. (14), for n = 1 , , b = 0 . P L is obtained for N s = 20. To study the effect of the regulators on the prediction of the EREparameters a ( S ) and r ( S ) , we solve Eq. (9) with different regulator powers for the lattice spacing a = 1 .
97 fm and N s = 20. For each power of the regulator, we refit the LECs in such a way that C L and C L reproduce the deuteronbinding energy and the ANC. Then by having the LECs, we resolve Eq. (9) to calculate the energy eigenvalues E L for smaller values of N s , in the domain 4 ≤ N s ≤
20. Finally, by applying the L¨uscher formula, as discussed in Sec.III, we extract the ERE parameters from the energy eigenvalues. We implement the same steps at the LO, where theonly LEC parameter C L reproduces the deuteron binding energy, and we have no control over the ANC. In Table III,we have presented our numerical results for the prediction of the ERE parameters a ( S ) and r ( S ) , with differentpowers of the regulator. At the NLO, deuteron binding energy and ANC are both used as inputs to extract the LECs C L and C L , while at the LO, the only input to extract C L is deuteron binding energy. As we can see, applying theregulator leads to a correction in the ERE parameters, and it seems the power n = 1 leads to more corrections than n = 2 and n = 3.TABLE III: Deuteron binding energy, ANC, and the ERE parameters a ( S ) and r ( S ) calculated for the latticespacing parameter a = 1 .
97 fm. n, b indicates the parameters of the regulator, defined in Eq. (14). The numbers inparentheses are the uncertainties in the last digits.
Order n, b C L C L · − E (MeV) ANC (fm − . ) a ( S ) (fm) r ( S ) (fm)LO 1 , − . − . . . . , . − . − . . . . , . − . − . . . . , . − . − . . . . , − . . − . . . . , . − . . − . . . . , . − . . − . . . . , . − . . − . . . . − − − − . . . . In Fig. 2, we have shown the effective range function, in the S neutron-proton channel, calculated for latticespacing a = 1 .
97 fm as a function of the square of relative momentum. The results are shown at the LO and NLO.As we have discussed earlier, by using a linear fit to our data and matching to Eq. (10), one can extract the infinitevolume ERE parameters from the finite volume energy eigenvalues. The impact of different power of regulators (for n = 1 , ,
3) on our data for the effective range function is shown. As we can see, all regulators, independent oftheir power, are increasing the slope and decreasing the absolute value of the vertical intercept of the effective rangefunction, indicating an increase in the scattering length and effective range parameter. In the following, we discuss -p (MeV ) -50-49-48-47-46-45-44-43-42 p . c o t ( d ) ( M e V ) (n = 1, b = 0.0)(n = 1, b = 0.01)(n = 2, b = 0.01)(n = 3, b = 0.01) (n = 1, b = 0.0)(n = 1, b = 0.01)(n = 2, b = 0.01)(n = 3, b = 0.01) LO -p (MeV ) -60-55-50-45-40-35 p . c o t ( d ) ( M e V ) (n = 1, b = 0.0)(n = 1, b = 0.01)(n = 2, b = 0.01)(n = 3, b = 0.01) (n = 1, b = 0.0)(n = 1, b = 0.01)(n = 2, b = 0.01)(n = 3, b = 0.01) NLO
FIG. 2: Effective range function in the S neutron-proton channel for the lattice spacing a = 1 .
97 fm, with andwithout regulators. The solid red line indicates the results obtained by bare contact interactions, while the blue,green, and orange dashed lines are corresponding to the results obtained with regularized interactions with powers n = 1, 2, and 3, respectively.the impact of the regulator function on the ERE parameters extracted from different lattice spacing. In the first step,we have calculated the lattice energy eigenvalues with and without the regularized interactions for different latticespacing values. To this aim, we have considered a regulator with a power one. Starting with N s = 20, we extractthe LECs C L and C L for different lattice spacing parameters a = 1 . , . , . , . , . C L and positive C L for all considered latticespacing parameters. Then by having the physical LECs, we have obtained a spectrum of the energy eigenvaluesby lowering the number of nodes to N s = 4. Finally, by using L¨uscher formula in Eq. (10), we extract the EREparameters. In Fig. 3, our numerical results for deuteron binding energies obtained from the solution of Eq. (9), areshown as a function of the number of lattice nodes N s , with and without using the regularized interactions. All thecalculated energy eigenvalues used in Fig. 3 are given in the Appendix A. The obtained effective range functions with N s -5-4.5-4-3.5-3-2.5-2 E ( M e V ) a = 2.60 fma = 2.30 fma = 1.97 fma = 1.70 fma = 1.40 fmExp. No regulator N s -5-4.5-4-3.5-3-2.5-2 E ( M e V ) a = 2.60 fma = 2.30 fma = 1.97 fma = 1.70 fma = 1.40 fmExp. Regulator (n=1)
FIG. 3: Deuteron binding energy as a function of N s for different lattice spacing parameter a . In the left panel, theresults are obtained with no regulator, whereas in the right panel, a regulator with power n = 1 and the regulatorparameter b = 0 .
01 is applied.different lattice spacing a = 1 . , . , . , . , . C L and C L are fitted to the experimental values of deuteron binding energy and ANC with N s = 20.As we can see, the regularization scheme for lattice spacing greater than 2 fm, brings the scattering length parameters a ( S ) very close to the experimental value. Similarly, the regularization scheme increases the effective ranges r ( S ) tovalues closer to the corresponding experimental value. So, we are confident that the introduced regularization schemeimproves the extracted ERE parameters for different lattice spacing at NLO pionless EFT. It should be mentionedthat we have not manipulated the regularization parameter b to reach the same ERE parameters for different latticespacing. As it is shown earlier, the regulator parameter b is dependent on the lattice spacing a as b = A /a , whilethe value of A is considered to be constant for all lattice spacing. While the regularization scheme for smaller latticespacing doesn’t match the ERE parameters precisely to the corresponding experimental data, it brings them closerto the experimental data. -p (MeV ) -60-55-50-45-40-35 p . c o t ( d ) ( M e V ) a = 1.40 fma = 1.70 fma = 1.97 fma = 2.30 fma = 2.60 fm No regulator -p (MeV ) -55-50-45-40-35 p . c o t ( d ) ( M e V ) a = 1.40 fma = 1.70 fma = 1.97 fma = 2.30 fma = 2.60 fm a = 1.40 fma = 1.70 fma = 1.97 fma = 2.30 fma = 2.60 fm Regulator (n=1)
FIG. 4: Effective range function in the S neutron-proton channel for different values of lattice spacing parameter a .TABLE IV: Deuteron binding energy, ANC and the ERE parameters a ( S ) and r ( S ) calculated for different latticespacing parameter a with and without implementing the regularization scheme, suggested in Eqs. (13) and (14).The numbers in parentheses are the uncertainties in the last digits. a (fm) C L C L · − E (MeV) ANC (fm − . ) a ( S ) (fm) r ( S ) (fm) No Regulator . − . . − . . . . . − . . − . . . . . − . . − . . . . . − . . − . . . . . − . . − . . . . With Regulator ( n = 1 , b = A /a ; A = 0 . )1 . − .
064 +5 . − . . . . . − . . − . . . . . − . . − . . . . . − . . − . . . . . − . . − . . . . − − − . . . . In Table V, we have compared our ERE parameters extracted for lattice spacing a = 1 .
97 fm, by different powersof the regulator, with the results of other studies.TABLE V: Comparison of our ERE parameters in the S channel, obtained with and without the application of theregularization scheme, with the results of other groups. The parameters ( n, b ) indicate the regulator parameters,introduced in Eq. (14). The numbers in parentheses are the uncertainties in the last digits. Method a a ( S ) (fm) r ( S ) (fm)Present ( n, b )LO (1 ,
0) 1 .
97 fm 4 . . , .
01) 1 .
97 fm 4 . . , .
01) 1 .
97 fm 4 . . , .
01) 1 .
97 fm 4 . . ,
0) 1 .
97 fm 5 . . , .
01) 1 .
97 fm 5 . . , .
01) 1 .
97 fm 5 . . , .
01) 1 .
97 fm 5 . . et al. (LO Pionless EFT) [4] 1 .
97 fm 4 . . .
97 fm 4 . . et al. (LO Pionless EFT) [16] 2 fm 4 .
50 0 . et al. (LO Pionless EFT) [18] 1 .
97 fm 5 . . et al. (LO pionfull EFT) [18] 1 .
97 fm 5 . . et al. (LO pionfull EFT) [20] 1 .
97 fm 5 . . et al. (NLO pionfull EFT) [20] 1 .
97 fm 5 . . et al. (N LO pionfull EFT) [20] 1 .
97 fm 5 . . . . V. CONCLUSION
In this paper, we have studied the impact of a new regularization scheme on the extraction of the ERE parametersof S channel for different lattice spacing in a pionless effective field theory up to NLO. We first use the deuteronbinding energy and the ANC to fix the LECs of the contact interactions by solving the lattice form of the Lippmann-Schwinger equation with Lanczos technique. Then we employ L¨uscher’s finite-volume relation to extract the S − waveERE parameters r ( S ) and a ( S ) from the lattice energy eigenvalues corresponding to the different lattice size. Thelattice spacing dependence of the ERE parameters is studied in the range 1 . ≤ a ≤ . O ( a )-improved lattice action is considered. The impact of different powers of the exponential regulator isstudied for the lattice spacing a = 1 .
97 fm, and it is shown that they have an almost similar influence on the extractedERE parameters. The introduced regulator is applied to different lattice spacing, leading to an improvement on theextraction of the ERE parameters, and brings them close to the experimental data for a ≥ Acknowledgments
We thank Koji Harada for sharing their results, which allowed us to validate our codes for the solution of theLippmann-Schwinger equation for two-nucleon-bound states on a lattice. The work of M. R. Hadizadeh was supportedby the National Science Foundation under grant NSF-PHY-2000029 with Central State University.
Appendix A: Two-Nucleon Energy Eigenvalues
In Tables VII-XI, we provide our numerical results for the solution of the Lippmann-Schwinger equation, given inEq. (9), with the LECs given in Table VI, for different values of lattice spacing parameter a and different number oflattice nodes N s . [1] T. A. L¨ahde and U.-G. Meißner, Nuclear Lattice Effective Field Theory: An Introduction , Vol. 957 (Springer, 2019).[2] H.-M. M¨uller, S. E. Koonin, R. Seki, and U. van Kolck, Nuclear matter on a lattice, Phys. Rev. C , 044320 (2000). C L and C L fitted to deuteron binding energy and ANC for different lattice spacingparameter a with and without implementing the regularization scheme, introduced in Eqs. (13) and (14). n indicates the power of the exponential regulator. a (fm) C L C L · − No Regulator . − . . . − . . . − . . . − . . . − . . With Regulator ( n = 1)1 . − .
064 +5 . . − . . . − . . . − . . . − . . With Regulator ( n = 2)1 . − . . With Regulator ( n = 3)1 . − . . TABLE VII: Deuteron binding energy calculated for different values of N s with the lattice spacing a = 1 . n, b ) indicate the regulator parameters, introduced in Eq. (14). N s NLO ( n = 1 , b = 0) NLO ( n = 1 , b = 2 . · − )20 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . TABLE VIII: The same as Table VII, but for a = 1 . N s NLO ( n = 1 , b = 0) NLO ( n = 1 , b = 1 . · − )20 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . a = 1 .
97 fm. N s LO NLO
No Reg. n = 1 n = 2 n = 3 No Reg. n = 1 n = 2 n = 320 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . TABLE X: The same as Table VII, but for a = 2 . N s NLO ( n = 1 , b = 0) NLO ( n = 1 , b = 6 . · − )20 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . TABLE XI: The same as Table VII, but for a = 2 . N s NLO ( n = 1 , b = 0) NLO ( n = 1 , b = 4 . · − )20 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . , 134863 (2019).[4] B. Borasoy, E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Lattice simulations for light nuclei: Chiral effective fieldtheory at leading order, The European Physical Journal A , 105 (2007).[5] B. Borasoy, E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Chiral effective field theory on the lattice at next-to-leadingorder, The European Physical Journal A , 343 (2008).[6] D. Lee, Lattice simulations for few-and many-body systems, Progress in Particle and Nuclear Physics , 117 (2009).[7] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Modern theory of nuclear forces, Reviews of Modern Physics , 1773(2009). [8] R. Machleidt and D. R. Entem, Chiral effective field theory and nuclear forces, Physics Reports , 1 (2011).[9] M. Pine, D. Lee, and G. Rupak, Adiabatic projection method for scattering and reactions on the lattice, The EuropeanPhysical Journal A , 151 (2013).[10] S. Elhatisari, D. Lee, U.-G. Meißner, and G. Rupak, Nucleon-deuteron scattering using the adiabatic projection method,The European Physical Journal A , 174 (2016).[11] K. Harada, S. Sasabe, and M. Yahiro, Numerical study of renormalization group flows of nuclear effective field theorywithout pions on a lattice, Physical Review C , 024004 (2016).[12] M. Luscher, Volume dependence of the energy spectrum in massive quantum field theories. 2. scattering states, Commun.Math. Phys. , 153 (1986).[13] S. R. Beane, P. F. Bedaque, A. Parreno, and M. J. Savage, Two nucleons on a lattice, Physics Letters B , 106 (2004).[14] S. Bour, S. K¨onig, D. Lee, H.-W. Hammer, and U.-G. Meißner, Topological phases for bound states moving in a finitevolume, Physical Review D , 091503 (2011).[15] S. Bour, H.-W. Hammer, D. Lee, and U.-G. Meißner, Benchmark calculations for elastic fermion-dimer scattering, PhysicalReview C , 034003 (2012).[16] A. Rokash, E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Finite volume effects in low-energy neutron–deuteronscattering, Journal of Physics G: Nuclear and Particle Physics , 015105 (2013).[17] N. Klein, D. Lee, and U.-G. Meißner, Lattice improvement in lattice effective field theory, The European Physical JournalA , 1 (2018).[18] N. Klein, D. Lee, W. Liu, and U.-G. Meißner, Regularization methods for nuclear lattice effective field theory, PhysicsLetters B , 511 (2015).[19] N. Klein, S. Elhatisari, T. A. L¨ahde, D. Lee, and U.-G. Meißner, The tjon band in nuclear lattice effective field theory,The European Physical Journal A , 121 (2018).[20] J. M. Alarc´on, D. Du, N. Klein, T. A. L¨ahde, D. Lee, N. Li, B.-N. Lu, T. Luu, and U.-G. Meißner, Neutron-protonscattering at next-to-next-to-leading order in nuclear lattice effective field theory, The European Physical Journal A ,83 (2017).[21] M. Eliyahu, B. Bazak, and a. N. Barnea, Extrapolating lattice qcd results using effective field theory, arXiv preprintarXiv:1912.07017 (2019).[22] M. Hadizadeh, M. T. Yamashita, L. Tomio, A. Delfino, and T. Frederico, Binding and structure of tetramers in the scalinglimit, Physical Review A , 023610 (2012).[23] D. R. Phillips, G. Rupak, and M. J. Savage, Improving the convergence of nn effective field theory, Physics Letters B ,209 (2000).[24] E. Epelbaum, W. Gl¨ockle, and U.-G. Meißner, Nuclear forces from chiral lagrangians using the method of unitary trans-formation ii: The two-nucleon system, Nuclear Physics A671