Nuclear Force from Monte Carlo Simulations of Lattice Quantum Chromodynamics
aa r X i v : . [ h e p - ph ] O c t UTHEP-560, UTCCS-P-42TKYNT-08-06
Nuclear Force from Monte Carlo Simulations ofLattice Quantum Chromodynamics
S. Aoki
Graduate School of Pure and Applied Sciences, University of Tsukuba,1-1-1 Tennodai, Tsukuba 305-8571, JAPANRIKEN BNL Research Center, Brookhaven National Laboratory,Upton, New York 11973, USAE-mail: [email protected]
T. Hatsuda
Department of Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, JAPANE-mail: [email protected]
N. Ishii
Center for Computational Sciences, University of Tsukuba,1-1-1 Tennodai, Tsukuba 305-8571, JAPANE-mail: [email protected]
Abstract.
The nuclear force acting between protons and neutrons is studied in the MonteCarlo simulations of the fundamental theory of the strong interaction, the quantumchromodynamics defined on the hypercubic space-time lattice. After a brief summaryof the empirical nucleon-nucleon (NN) potentials which can fit the NN scatteringexperiments in high precision, we outline the basic formulation to derive the potentialbetween the extended objects such as the nucleons composed of quarks. The equal-time Bethe-Salpeter amplitude is a key ingredient for defining the NN potential on thelattice. We show the results of the numerical simulations on a 32 lattice with thelattice spacing a ≃ . ) in the quenched approximation.The calculation was carried out using the massively parallel computer Blue Gene/Lat KEK. We found that the calculated NN potential at low energy has basic featuresexpected from the empirical NN potentials; attraction at long and medium distancesand the repulsive core at short distance. Various future directions along this line ofresearch are also summarized.PACS numbers: 12.38.Gc, 13.75.Cs, 21.30.-x uclear Force from Lattice QCD
1. Introduction
One of the long standing problems in particle and nuclear physics is the origin of thestrong nuclear force which holds the nucleons (protons and neutrons) inside atomicnuclei. For the past half century, phenomenological fits of the proton-proton (pp)and neutron-proton (np) scattering data assuming empirical nucleon-nucleon (NN)potentials have been attempted [1, 2]: The potentials, which can fit more than 2000data points of the NN phase shift with χ / dof ≃ T lab < v potential [4] and Nijmegen potentials [5]. Alternativeapproach on the basis of the chiral perturbation theory has been also developed [6].Shown in Fig.1 are three examples of the empirical NN potentials in the S channel. ‡ From this figure, some characteristic features of the nuclear force can beseen:I. The long range part of the nuclear force ( r > < r < ρ , ω , and σ ). In particular, thespin-isospin independent attraction of about − ∼ −
100 MeV in this region playsan essential role for binding the atomic nuclei.III. The short range part ( r < r < S and S channels have been reported with the quenched lattice QCD simulations in ‡ A system of two nucleons with total spin s = (0 , L = ( S, P, D, · · · )and total angular momentum J = (0 , , , · · · ) is denoted as s +1 L J . uclear Force from Lattice QCD Figure 1.
Three examples of the high-precision NN potentials in the S channel.AV18 stands for the Argonne v potential [4], Reid93 stands for one of the Nijmegenpotentials [5] and Bonn stands for the Bonn potential [7]. I, II and III correspondto the long range part, medium range part and the short range part, respectively, asdiscussed in the text. [15, 16, 17]. Also the first results of the hyperon-nucleon potential have been reportedin [18, 19].In the following, we will outline the field theoretical derivation of the NN potentialfrom QCD [20] in Sec.2 and Sec.3. Then, we show how to define the potential in LQCDformalism in Sec.4. The basic setup and the method of our numerical simulations areshown in Sec.5. Some numerical results of the low energy NN potential taken from[15, 16] are shown in Sec.6. The last section is devoted to summary and concludingremarks with a discussion on the future directions.
2. Bethe-Salpeter wave function and the NN potential
Let us start with the definition of the Bethe-Salpeter (BS) amplitude for the proton-neutron system,Ψ αβ ( x, y ) = h | T[ˆ n β ( y )ˆ p α ( x )] | p( ~q, s )n( ~q ′ , s ′ ); in i , (1)ˆ n β ( y ) = ε abc (cid:16) ˆ u a ( y ) Cγ ˆ d b ( y ) (cid:17) ˆ d cβ ( y ) , (2)ˆ p α ( x ) = ε abc (cid:16) ˆ u a ( x ) Cγ ˆ d b ( x ) (cid:17) ˆ u cα ( x ) , (3)where ( ~q, s ) and ( ~q ′ , s ′ ) denote the spatial momentum and the spin-state of the incomingproton and those of the neutron, respectively. The local composite operator for theneutron (proton) are denoted by ˆ n β ( y ) (ˆ p α ( x )) with the operators for the up-quark ˆ u ( x )and the down-quark ˆ d ( x ). Also, α and β denote the Dirac indices, a , b and c the color uclear Force from Lattice QCD C the charge conjugation matrix in the spinor space.One of the advantages to use local operator for the nucleon is that the Nishijimaand Zimmermann (NZ) reduction formula [21] for local composite fields can be utilized.In particular, in and out nucleon fields are defined through the Yang-Feldman equation, √ Z ˆ N in / out ( x ) = ˆ N ( x ) − Z S adv / ret ( x − x ′ ; m N ) ˆ J ( x ′ ) dx ′ , (4)where ˆ N ( x ) is the nucleon composite operator (ˆ n ( x ) or ˆ p ( x ) in Eqs.(2,3)), ˆ N in / out ( x ) isthe associated in/out field, m N is the physical nucleon mass, and ˆ J ( x ) ≡ ( i∂/ x − m N ) ˆ N ( x ).The advanced/retarded propagator for the free nucleon field with the mass m N is denotedby S adv / ret ( x − x ′ ; m N ). The normalization factor, √ Z , is the coupling strength of thecomposite operator ˆ N ( x ) to the physical nucleon state.Through the NZ reduction formula, the BS amplitude in Eq.(1) is related to thefour-point Green’s function G of the composite nucleons which is decomposed into thefree part and the scattering part, G = Z ( G (0)4 + G (sc)4 ), where G (0)4 is proportional to aproduct of the free nucleon propagators. After taking the equal time limit, x = y = t ,with ~r ≡ ~x − ~y , it is straight forward to rewrite Eq.(1) to the the following integralequation in the c.m. frame ( ~q ′ = − ~q ) [20],Ψ αβ ( ~r, t ) = ψ αβ ( ~r ; ~q, s, s ′ ) e − i √ ~q + m t , (5) ψ αβ ( ~r ; ~q, s, s ′ ) = Zu α ( ~q, s ) u β ( − ~q, s ′ ) e i~q · ~r + Z X γδ Z d k (2 π ) e i~k · ~r F αβ ; γδ ( ~k ; ~q ) u γ ( ~q, s ) u δ ( − ~q, s ′ ) . (6)Here u α ( ~q ; s ) is the positive-energy plain-wave solution of the Dirac equation and F αβ ; γδ ( ~k ; ~q ) is an integral kernel obtained from G (sc)4 after carrying out the k -integration.Hereafter, we call ψ αβ ( ~r ; ~q, s, s ′ ) as the Bethe-Salpeter wave function. The differentialform of the above equation is obtained by multiplying ( ~q + ∇ ) /m N to Eq.(6);1 m N ( ~q + ∇ ) ψ αβ ( ~r ; ~q, s, s ′ ) = K αβ ( ~r ; ~q, s, s ′ ) . (7)Note that we have not made any non-relativistic approximation to derive Eq.(7). Animportant observation is that the plain wave component of ψ αβ ( ~r ; ~q, s, s ′ ) is projected outby the operator ( ~q + ∇ ) so that the function K αβ ( ~r ; ~q, s, s ′ ) is localized in coordinatespace as long as | ~q | stays below the inelastic threshold as noted for pion-pion scatteringin [22]. This is equivalently said that the Fourier transform of K with respect to ~r ,which is proportional to the half off-shell T -matrix relating the the on-shell state withmomentum ~q and the off-shell state with momentum ~k , does not develop a real pole asa function of | ~k | , if | ~q | is below the inelastic threshold.
3. The NN potential
In an abbreviated notation, ( ~r, α, β ) → x , and ( ~q, s, s ′ ) → q , Eq.(7) is written as( E q − H ) ψ ( x ; q ) = K ( x ; q ) , (8) uclear Force from Lattice QCD E q = ~q /m N and H = −∇ /m N . This equation defined in a finite box can be usedto extract various information on the NN scattering from the lattice QCD simulations:(i) Consider K ( x ; q ) as a measure to identify the length R beyond which the twonucleons do not interact. If we stay in such a region where K ( x > R ; q ) ≃ ψ ( x > R ; q ) can be expanded by the solution of the Helmholtzequation inside a finite box. Then one can extract the phase shift given the incomingenergy E q . This is the approach originally proposed in [23] and is later elaboratedto study hadron-hadron scatterings on the lattice [24, 22, 25, 26].(ii) Alternatively, one may extract the half off-shell T -matrix in momentum space bycalculating the left-hand-side of Eq.(8) in the coordinate space and making Fouriertransform with respect to x .(iii) One can go one-step further and define the non-local NN potential U ( x, x ′ ) from K ( x ; q ), so that Eq.(8) becomes the Schr¨odinger type equation.If we are interested only in the NN scattering phase shift in the free space, theprocedure (i) is certainly enough. On the other hand, if we are interested in applyingEq.(8) to the problems of bound states and the nuclear many-body system, (ii) and(iii) are useful since they give us the off-shell information in a well-defined manner inQCD. To see this explicitly for the case (iii), we introduce a set of functions labeled by q , { ˜ ψ ( x ; q ) } , which is dual to the set { ψ ( x ; q ) } in the following sense: Z dx ˜ ψ ( x ; q ) ψ ( x ; p ) = δ q,p . (9)As long as the dimensions of the x -space and p -space are the same and the elementsin { ψ ( x ; p ) } are linearly independent, such a dual basis exists and is unique. If thedimension of p -space is less than that of x -space, the dual basis exists but is not unique. § Assuming the existence (but not necessarily the uniqueness) of the dual basis, the non-local potential can be defined as U ( x, x ′ ) = Z dp K ( x, p ) ˜ ψ ( x ′ ; p ) . (10)The Eqs.(9,10) lead to the formula, K ( x ; q ) = R dx ′ U ( x, x ′ ) ψ ( x ′ ; q ), so that Eq.(8)becomes −∇ m N ψ αβ ( ~r ; ~q, s, s ′ ) + Z d r ′ U αβ ; γδ ( ~r, ~r ′ ) ψ γδ ( ~r ′ ; ~q, s, s ′ ) = E q ψ αβ ( ~r : ~q, s, s ′ ) . (11)Note that the non-local potential U can be rewritten in the form, U ( ~r, ~r ′ ) = V ( ~r, ∇ ) δ ( ~r − ~r ′ ) . (12) § This is easily seen as follows. Let us introduce a basis { e (1) , e (2) , · · · , e ( N ) } in the N -dimensionalvector space. The BS wave function in discretized coordinates ψ ( x ; p ) ≡ ψ ( i ; α ) corresponds to e i ( α ) with 1 ≤ i ≤ N and 1 ≤ α ≤ M ≤ N . If M = N , there exits a unique dual basis, { ˜ e (1) , ˜ e (2) , · · · , ˜ e ( N ) } satisfying ˜ e ( α ) · e ( β ) = δ αβ (see any textbook of linear algebra). If M < N , there is still a dual basis { ˜ e (1) , ˜ e (2) , · · · , ˜ e ( M ) } satisfying the above condition for α ≤ M and β ≤ M . However, it is not uniquebecause one always has a freedom to add linear combinations of ˜ e γ ( M + 1 ≤ γ ≤ N ) to the above dualbasis. uclear Force from Lattice QCD V ( ~r, ∇ ) under various symmetry constraints in the non-relativistic kinematics has been worked out by Okubo and Marshak [27]. If we furthermake the derivative expansion at low energies [28], we obtain the expression familiar inthe phenomenological potentials acting on the upper components of the wave function; V ( ~r, ∇ ) = V ( r ) + V σ ( r )( ~σ · ~σ ) + V τ ( r )( ~τ · ~τ ) + V στ ( r )( ~σ · ~σ )( ~τ · ~τ )+ V T ( r ) S + V T τ ( r ) S ( ~τ · ~τ ) + V LS ( r )( ~L · ~S ) + V LS τ ( r )( ~L · ~S )( ~τ · ~τ )+ O ( ∇ ) . (13)Here S = 3( ~σ · ~n )( ~σ · ~n ) − ~σ · ~σ is the tensor operator with ~n = ~r/ | ~r | , ~S = ( ~σ + ~σ ) / ~L = − i~r × ∇ the orbital angular momentum operator, and ~τ , are the isospin operators for the nucleons. Each component of the potential in Eq.(13)can be obtained by appropriate spin, isospin and angular momentum projection of theBS wave function. Also, the higher derivative terms of the potential in Eq.(13) can bededuced by combining the BS wave functions for different incident energies.It is in order here to remark that the structure of the non-local potential U ( x, x ′ ) isdirectly related to the nucleon interpolating operator adopted in defining the Bethe-Salpeter wave function. Different choices of the interpolating operator would givedifferent forms of the NN potential at short distance, although they give the samephase shift at asymptotic large distance. The advantage of working in QCD is that wecan unambiguously trace the connection between the NN potential and the interpolatingoperator.
4. Effective central potential on the lattice
The BS wave function in the S -wave on the lattice with the lattice spacing a and thespatial lattice volume L is obtained by ψ ( r ) = 124 X R ∈ O L X ~x P σαβ h | ˆ n β ( R [ ~r ] + ~x )ˆ p α ( ~x ) | pn; q i . (14)The summation over R ∈ O is taken for the cubic transformation group to project outthe S -wave. k The summation over ~x is to select the state with zero total momentum. Wetake the upper components of the Dirac indices to construct the spin singlet (triplet)channel by P σ =singlet αβ = ( σ ) αβ ( P σ =triplet αβ = ( σ ) αβ ). The BS wave function ψ ( ~r ) isunderstood as the probability amplitude to find “neutron-like” three-quarks located atpoint ~x + ~r and “proton-like” three-quarks located at point ~x .In the actual simulations, the BS wave function is obtained from the four-pointcorrelator, G ( ~x, ~y, t ; t ) = D (cid:12)(cid:12)(cid:12) ˆ n β ( ~y, t )ˆ p α ( ~x, t ) J pn ( t ) (cid:12)(cid:12)(cid:12) E = X n A n h | ˆ n β ( ~y )ˆ p α ( ~x ) | n i e − E n ( t − t ) . (15)Here J pn ( t ) is a wall source located at t = t , which is defined by J pn ( t ) = P σαβ P ~x,~y ˆ p α ( ~x, t )ˆ n β ( ~y, t ). The eigen-energy and the eigen-state of the six quark system k In principle, this projection cannot remove possible contamination from the higher orbital waves with L ≥
4, although these contributions are expected to be negligible. uclear Force from Lattice QCD E n and | n i , respectively, with the matrix element A n ( t ) = h n |J pn ( t ) | i .For ( t − t ) /a ≫
1, the G and hence the wave function ψ are dominated by the lowestenergy state.The lowest energy state created by the wall source J pn ( t ) contains not only the S -wave component but also the D -wave component induced by the tensor force. Inprinciple, they can be disentangled by preparing appropriate operator sets for the sink.Study along this line to extract the mixing between the S -wave and the D -wave atlow energies has been put forward recently in [29]. In the present paper, instead ofmaking such decomposition, we define an “effective” central potential V C ( r ) accordingto Refs.[15, 16]: V C ( r ) = E q + 1 m N ∇ ψ ( r ) ψ ( r ) . (16)Note that one can test the non-locality of the potential U ( x, x ′ ) by evaluating theeffective central potential for different energies. If there arises appreciable energydependence in V C ( r ), it is a signature of the necessity of high derivative terms in Eq.(13).
5. Setup of the lattice simulations
In lattice QCD simulations, the vacuum expectation value of an operator O ( q, ¯ q, U ) isdefined as hOi = Z − Z Y ℓ dU ( ℓ ) Y x dq ( x ) d ¯ q ( x ) O ( q, ¯ q, U ) e − S f ( q, ¯ q,U ) − S g ( U ) (17)= Z − Z Y ℓ dU ( ℓ ) Q ( U ) det M ( U ) e − S g ( U ) . (18)where Z = R Q ℓ dU ( ℓ ) Q x dq ( x ) d ¯ q ( x ) e − S f ( q, ¯ q,U ) − S g ( U ) is the QCD partition function, S f = P x,x ′ ¯ q ( x ) M xx ′ ( U ) q ( x ′ ) is the quark part of the action, and S g is the gluon part ofthe action. The quark field q ( x ) is defined on each site x of the hypercubic space-timelattice, while the gluon field U ( ℓ ) denoted by 3 × ℓ . In Eq.(18), the integration of the quark fields is carried out analytically. Inthe quenched approximation adopted in our simulation, the virtual fermion loop denotedby det M ( U ) is set to be 1 and the integration over link variables U is performed usingthe importance sampling method [30].We employ the standard plaquette gauge action on a 32 lattice with the bare QCDcoupling constant β = 6 /g = 5 .
7. The corresponding lattice spacing is determined to be1 /a = 1 . a ≃ .
137 fm) from the ρ meson mass in the chiral limit [31]. Then,the physical size of our lattice becomes L ≃ . κ = 0 . , . . uclear Force from Lattice QCD κ N conf m π [MeV] m N [MeV] ( t − t ) /a E q ( S ) [MeV] E q ( S ) [MeV]0.1640 1000 732.1(4) 1558.4(63) 7 − . − . − . − . − . − . Table 1.
The number of gauge configurations N conf , the pion mass m π , the nucleonmass m N , time-slice t − t on which BS wave functions are measured, and the non-relativistic energies E q ≡ q /m N for S and S channels. For our numerical simulations, we use IBM Blue Gene/L at KEK, which consists of10,240 computation nodes with total theoretical performance of 57.3 TFlops. A modifiedversion of the CPS++ (the Columbia Physics System) [33] is used to generate quenchedgauge configurations and propagators of quarks. Most of the computational time isdevoted to the calculation of the four-point function of nucleons, for which our codeachieves 34–48 % of peak performance. Totally about 4000 hours are used by queueswith 512 nodes for the calculations of the effective central potentials corresponding tothe three values of hopping parameters. The number of sampled gauge configurations N conf , the pion mass m π , and the nucleon mass m N are summarized in Table 1. (For κ = 0 . t/a = t /a = 5. The BS wave functionsare measured on the time-slice ( t − t ) /a = 7 , , κ = 0 . , . , . t -dependence of the NNpotential. We employ the nearest neighbor representation of the discretized Laplacianas ∇ f ( ~x ) ≡ P i =1 { f ( ~x + a~n i ) + f ( ~x − a~n i ) } − f ( ~x ), where ~n i denotes the unit vectoralong the i -th coordinate axis. BS wave functions are fully measured for r < . r > . q is obtained by fitting the BS wave function with the Green’s function ina finite and periodic box [23]: G ( ~r ; q ) = 1 L X ~n ∈ Z e i (2 π/L ) ~n · ~r (2 π/L ) ~n − q , (19)which satisfies ( ∇ + q ) G ( ~r ; q ) = − δ L ( ~r ) with δ L ( ~r ) being the periodic delta-function.In the actual calculation, Eq.(19) is rewritten in terms of the heat kernel H satisfying theheat equation, ∂ t H ( t, ~r ) = ∇ H ( t, ~r ) with the initial condition, H ( t → , ~r ) = δ L ( ~r ).The fits are performed outside the range of NN interaction determined by ∇ ψ ( ~r ) /ψ ( ~r )[25].
6. Numerical results
Fig.2 (upper panel) shows the BS wave functions in S and S channels for κ = 0 . r < . uclear Force from Lattice QCD NN w a v e f un c t i on φ (r) r [fm] S S -2 -1 0 1 2 -2 -1 0 1 20.51.01.5 φ (x,y,z=0; S ) x[fm] y[fm] φ (x,y,z=0; S ) V C (r) [ M e V ] r [fm] -50 0 501000.0 0.5 1.0 1.5 2.0 S S Figure 2. (Upper panel) The NN wave functions in S and S channels. The insetis a 3D plot of the wave function φ ( x, y, z = 0; S ). (Lower panel) The NN effectivecentral potential in the S and S channels for m π = 529 MeV ( κ = 0 . repulsion at short distance. Fig.2 (lower panel) shows the reconstructed NN potentialsfor κ = 0 . S and S channels. See Table 1,for the values of the non-relativistic energies E q ≡ q /m N in Eq.(16).We show the NN potentials for three different quark masses in S channel in Fig.3. As the quark mass decreases, the repulsive core at short distance is enhanced rapidly,whereas the attraction at medium distance is modestly enhanced. This indicates that itis important to perform the lattice QCD calculations for the lighter quark mass regionin order to make quantitative comparison of our results with the observables in the realworld.Although there exist both attraction and repulsion, the net effect of our potentialis attractive at low energies as shown by the scattering lengths calculated from uclear Force from Lattice QCD V C (r) [ M e V ] r [fm] -50 0 50 100 1500.0 0.5 1.0 1.5 2.0m π =380MeVm π =529MeVm π =731MeV Figure 3.
Central potentials in the S channel for three different quark masses inthe quenched QCD simulations on a (4 . lattice. κ m π [MeV] a ( S ) [fm] a ( S ) [fm]0.1640 732.1(4) 0 . . . . . . Table 2.
Scattering lengths obtained from the L¨uscher’s formula [23] in the spin-singlet and spin-triplet channels for different quark masses. the L¨uscher’s formula in Table 2 [23]. The attractive nature of our potential isqualitatively understood by the Born approximation formula for the scattering length a ≃ − m N R V C ( r ) r dr. Owing to the volume factor r dr , the attraction at mediumdistance can overcome the repulsive core at short distance. However, there is aconsiderable discrepancy between the scattering lengths in Table 2 and the empiricalvalues, a (exp)0 ( S ) ∼
20 fm and a (exp)0 ( S ) ∼ − uclear Force from Lattice QCD
7. Summary and concluding remarks
In this article, we have outlined the basic notion of the nucleon-nucleon potential andits field-theoretical derivation from the equal-time Bethe-Salpeter amplitude. Such aformulation allows us to extract the potential between extended objects by using thelattice QCD simulations. The central part of the NN potential at low energies wasobtained in lattice QCD simulations with quenched approximation. It was found thatthe NN potential calculated on the lattice at low energy shows all the basic featuresexpected from the empirical NN potentials determined from the NN scattering data;attraction at long and medium distances and the repulsion at short distance. This isthe first step toward the understanding of atomic nuclei from the fundamental law ofthe strong interaction, the quantum chromodynamics.There are a number of directions to be explored on the basis of our approach:1. Energy dependence of the NN potential in Eq.(16) should be studied to test thenon-locality of the potential U ( x, x ′ ) and the validity of its derivative expansion.This is currently under investigation by changing the spatial boundary conditionof the fermion field [34].2. The tensor force, which mixes the states with different orbital angular momentumby two units, is a unique feature of the nuclear force and plays an essential rolefor the deuteron binding. This is also under investigation by projecting out the S component and D component separately from the exact two-nucleon wavefunction with J = 1 on the lattice [29, 35]. The spin-orbit force, which is known tobe strong at short distances in empirical NN force, should be also studied.3. Three nucleon force is thought to play important roles in nuclear structure andalso in the equation of state of high density matter. Since the experimentalinformation is scarce, simulations of the three nucleons on the lattice combinedwith appropriate generalizations of the formulas in Section 2 may lead to thefirst principle determination of the three nucleon potential in the future. Withthese generalizations of the present approach, one may eventually make a firm linkbetween QCD and the physics of nuclear structure [36].4. The hyperon-nucleon (YN) and hyperon-hyperon (YY) potentials are essential forunderstanding the properties of hyper nuclei and the hyperonic matter inside theneutron stars. However, the experimental data are very limited due to the shortlife-time of hyperons. On the lattice, NN, YN and YY interactions can be treatedin the same footing since the difference is only the mass of the strange quark.Recently, the ΞN potential [18, 19] and the Λ N potential [37] are examined as afirst step toward systematic derivation of the hyperon interactions.5. To compare the NN potential on the lattice with experimental observables, itis necessary to carry out full QCD simulations which take into account thedynamical quark loops. Study of the nuclear force with the use of the full-QCD configurations generated by PACS-CS Collaboration [38] is currently under uclear Force from Lattice QCD Acknowledgments
This research was partly supported by the Ministry of Education, Culture, Sports,Science and Technology, Grant-in-Aid Nos. 18540253, 19540261 and 20340047.Numerical simulations were supported by the Large Scale Simulation Program No.07-07(FY2007) of High Energy Accelerator Research Organization (KEK). We are gratefulfor authors and maintainers of
CPS++ [33], of which a modified version is used formeasurement done in this work.
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