Comparison of different source calculations in two-nucleon channel at large quark mass

We investigate a systematic error coming from higher excited state contributions in the energy shift of light nucleus in the two-nucleon channel by comparing two different source calculations with the exponential and wall sources. Since it is hard to obtain a clear signal of the wall source correlation function in a plateau region, we employ a large quark mass as the pion mass is 0.8 GeV in quenched QCD. We discuss the systematic error in the spin-triplet channel of the two-nucleon system, and the volume dependence of the energy shift.


Important condition of direct calculation
In the direct calculation [1][2][3][4][5][6][7][8][9] of the two-nucleon channel, the energy shift ∆E NN = 2m N − E NN with the nucleon mass m N and two-nucleon ground state energy E NN is determined from a plateau region of the ratio of the correlation functions R(t) = C NN (t)/C 2 N (t) with the two-nucleon correlation function C NN (t) in the spin-triplet channel and the single nucleon correlation function C N (t). An important condition of this determination is that ∆E NN should be determined in a region where both C NN (t) and C 2 N (t) have each plateau. It means that it is not enough to determine a plateau region from only R(t), but we need to investigate plateaus for C NN (t) and C 2 N (t). If one chooses a plateau from only R(t), it might cause an incorrect determination of ∆E NN , as discussed in the later sections. For example, when statistics is not enough, R(t) using the wall source has a plateau like behavior in early t region, where C NN (t) and C 2 N (t) do not have plateaus. In the following sections, we shall call the minimum t of the plateau region for C NN (t), C N (t), and R(t) as t S N , t S NN , and t S R , respectively, using the source S = E (exponential) or W (wall).

Simulation parameters
We calculate the two-nucleon correlation function C NN (t) in the spin-triplet channel as well as the single nucleon correlation function C N (t) in the quenched approximation. In this calculation, we employ Iwasaki gauge action at β = 2.416, corresponding to a = 0.128 fm [12]. The quark propagators are calculated with a tad-pole improved Wilson action with c SW = 1.378 at κ ud = 0.13482 corresponding to m π = 0.8 GeV and m N = 1.62 GeV. The actions and parameters are the same as in our previous works [1,2]. The temporal lattice size is fixed to 64, while the spatial size L is chosen to be 16, 20, and 32. In order to compare results with different source operators, we employ the exponential and wall sources. The exponential source at the time slice t is defined by where q(x, t) is the local quark field. The parameters A and B are chosen to obtain an early plateau of the effective nucleon mass on each volume. At the sink time slice, each nucleon operator is projected to zero momentum using the local quark field as in our previous calculations [2,5,7]. The number of the measurement of the correlation functions is tabulated in Table 1.

Results
In this section we present the results for twice the effective nucleon mass 2m eff N , the effective twonucleon energy E eff NN , and the effective energy shift ∆E eff NN evaluated from C N (t), C NN (t), and R(t), respectively, on each volume. The volume dependence of ∆E NN is also presented.   Figure 1. Effective twice nucleon mass 2m eff N (circle) and effective two-nucleon energy E eff NN (square) in the spin-triplet channel using the exponential (left panel) and wall (right panel) sources on L = 20. The vertical dot-dashed lines denote t S N and t S NN for S = E, W explained in the text. The horizontal dashed lines express the values of each plateau in the exponential source in both the panels.

L = 20
First we present the results of 2m eff N , E eff NN , and ∆E eff NN in both the exponential and wall sources on L = 20 as a typical result. Figure 1 shows the results for 2m eff N and E eff NN using the exponential (left panel) and wall (right panel) sources. The results of the exponential source have plateaus, which start from t = 12 as denoted by vertical dot-dashed line in the left panel. It means that t E N = t E NN = 12 in this case. The horizontal dashed lines in black and red represent the values of the plateaus for 2m eff N and E eff NN , respectively. The wall source results need longer t than the exponential source to have plateaus as shown in the right panel of Fig. 1. We determine t W N = 17 and t W NN = 16 from each plateau region, which are expressed by vertical dot-dashed lines. The same horizontal lines as in the left panel are shown in the right panel. Those lines are in good agreement with each plateau.
As discussed in Sec. 2, t S R , the minimum t of the plateau region of ∆E eff NN , should be larger or equal to t S N and t S NN . Thus, t E R = t E N = t E NN , and t W R = t W N , in this case. Figure 2 presents that ∆E eff NN of the exponential source has a reasonable plateau after t E R . On the other hand, the result of the wall source has a non-monotonic t dependence in t < 15.
One might choose t W R ∼ 14, if it is determined from only the wall source data in Fig. 2. Moreover, if the statistics is much smaller than the current calculation, the data around t = 5 would be also regarded as a plateau. However, the data in the small t region contain excited state contributions as shown in the right panel of Fig. 1. It suggests that it is easy to mistake the plateau region, when it is determined from only ∆E eff NN , especially in the case where ∆E eff NN has a non-monotonic t dependence, like the wall source data in the current study.
In the wall source, while the data in t ≥ t W R has the large error, it agrees with the plateau value of the exponential source within the error.

L = 16
The results for 2m eff N and E eff NN using the exponential and wall sources are plotted in the left and right panels of Fig. 3, respectively. The results are similar to the ones on L = 20 in the previous subsection.   Figure 3. The same figures as Fig. 1, but in the L = 16 case.
The data of the exponential source have plateaus, which start from t = 12, and the ones of the wall source need longer t to have plateaus. It is noted that comparing with the results on L = 20 and 16 we observe 0.02% finite volume effect in m N on this volume of the spatial extent 2.0 fm.
The results of ∆E eff NN are shown in Fig. 4. The exponential source has a reasonable plateau after t E R = t E N = t E NN as in the L = 20 case. The t dependence of the wall source in the smaller t region becomes larger as the volume decreases comparing with the result in Fig. 2. While the error of the wall source is large after t W R = t W N , the data is consistent with the plateau value of the exponential source.
It is noted that on L = 16 the consistent results with the exponential and wall sources are not obtained even in 2m eff N , when the number of the measurement of the wall source is half of the current calculation. This suggests that a huge statistics is necessary to obtain statistically stable result from the wall source even in the single nucleon mass.

L = 32
The left panel of Fig. 5 presents that the results for 2m eff N and E eff NN with the exponential source are similar to the ones in the L = 16 and 20 cases.
On the other hand, the wall source results look different from the ones on the other volumes. The result of E eff NN with the wall source in t ≤ 20 is larger than the one with the exponential source represented by the red dashed line. One of the reasons is that the contribution of the two-nucleon scattering state with almost zero relative momentum, which corresponds to the first excited state in this system, becomes relatively larger than the one of the ground state in C NN (t) with the wall source as the volume increases. Another reason is that the energy of the first excited state is larger than 2m N in this system, where one bound state exists [2]. Thus, it is harder to obtain the same plateau as the one of the exponential source, expressed by the red dashed line in the right panel of Fig. 5, from the wall source as the volume increases. From the data, we cannot determine t NN of the wall source. In the following analysis, it is assumed that t W N = t W NN in the wall source result. From the above reasons, it is expected that on much larger volumes than the current calculation E eff NN of the wall source would become larger than 2m eff N in a large t region. Then, it would go down to agree with the plateau value of the exponential source in much larger t region. Figure 6 shows the results of ∆E eff NN with both the sources. It is surprising that the wall source result has a mild t dependence in the small t region, although the data for 2m eff N and E eff NN largely depend on t in the same region as shown in the right panel of Fig. 5. If a plateau of ∆E eff NN is determined from only the wall source data in smaller statistics than the present calculation, one might choose much smaller t region as a plateau than t W R . While it is not as good as the smaller volumes, we observe a plateau after t E R in the exponential source result on this volume. Although the wall source result has large error after t W R , it is not inconsistent with the plateau of the exponential source. We expect that the plateau of E eff NN with the wall source is obtained in a region of the much larger t than the smaller volumes. In order to confirm this expectation, it is an important future work to observe clear signal of the wall source after t W R . From the comparisons including the ones in the smaller volumes, we conclude that the results using the exponential and wall source are consistent with each other in each plateau region. Thus, contaminations of excited states in ∆E eff NN obtained from the plateau region are negligible in our calculation.   Figure 5. The same figures as Fig. 1, but in the L = 32 case.

Volume dependence
The result of ∆E NN on the three volumes with the exponential source are plotted in Fig. 7 together with our previous result [2]. We neglect the wall source data in the following due to the much larger error.
The result of the current calculation denoted by the filled circle has much smaller statistical error, and is reasonably consistent with the fit curve using the previous data, so that the result indicates that the existence of a bound state in this system. Recently HALQCD Collaboration suggested that the volume dependence of ∆E NN obtained from the direct calculation is too small comparing to the one expected from the effective range expansion [13]. However, this argument is assumed that the effective range expansion is valid in p 2 < 0 region in the continuum theory, and there is no finite volume effect in the two-nucleon interaction. In the comparison between the expectation in the ideal case and the lattice data, there could be several sources of systematic errors, such as finite lattice spacing and finite volume effects, which may deform the two-nucleon interaction. In order to understand the current situation, it is an important future work to investigate such systematic errors in the ∆E NN calculation.
It is noted that even if there is a finite volume effect in ∆E NN , which cannot be treated by the finite volume method [14,15], we consider that the signal of the existence of the bound state is meaningful in our calculation, because we discuss the existence in the infinite volume limit, so that our result does not contain the finite volume effect.

Summary
We have carried out the high precision calculation of the spin-triplet two-nucleon channel at the large quark mass, corresponding to m π = 0.8 GeV in the quenched approximation to investigate a systematic error of ∆E NN coming from excited states by comparing the results with the two different source calculations using the exponential and wall sources on the three volumes. Though it might be a trivial, we discuss the important condition to calculate ∆E NN . When the condition is satisfied, the two sources give the consistent results of ∆E eff NN in each plateau region, while the wall source data has the large error due to the late plateau. From this comparison, we have concluded that the systematic error from higher excited states is negligible in our calculation.
There are several important future works, such as comparing the current result with the one obtained from the generalized eigenvalue problem [16], and investigations of systematic errors in ∆E NN . It is also an important future work to clarify the qualitative difference between the direct calculation and HALQCD method in the point of view of the definitions of the scattering amplitude in quantum field theory and quantum mechanics [17].