Hermitizing the HAL QCD potential in the derivative expansion

Hermitizing the HAL QCD potential in the derivative expansion Sinya Aoki, ∗ Takumi Iritani, † and Koichi Yazaki ‡ Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Theoretical Research Division, Nishina Center, RIKEN, Saitama 351-0198, Japan Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN Saitama 351-0198, Japan (Dated: September 4, 2019)


I. INTRODUCTION
Lattice quantum chromodynamics (QCD) is a successful non-perturbative method to study hadron physics from the underlying degrees of freedom, i.e. quarks and gluons. Masses of the single stable hadrons obtained from lattice QCD show good agreement with the experimental results, and even hadron interactions have been recently explored in lattice QCD. Using the Nambu-Bethe-Salpeter (NBS) wave function, linked to the S-matrix in QCD [1][2][3][4][5][6][7][8][9], the hadron interactions have been investigated mainly by two methods: the finite volume method [1] and the HAL QCD potential method [5][6][7]. Theoretically the two methods in principle give same results of the scattering phase shifts between two hadrons, while in practice they sometimes show different numerical results for two baryon systems, whose origin has been clarified recently in Refs. [10,11].
The HAL QCD method utilizes the NBS wave function in non-asymptotic (interacting) region, and extract the non-local but energy-independent potentials from the space and time dependences of the NBS wave function. Physical observables such as phase shifts and binding energies are then calculated by solving the Schrödinger equation in infinite volume using the obtained potentials, since the asymptotic behavior of the NBS wave function is related to the T -matrix element and thus to the phase shifts [9]. In practice, the non-local potential is given by the form of the derivative expansion, which is truncated by the first few orders [12].
While the HAL QCD method has been successfully applied to a wide range of two (or three) hadron systems at heavy pion masses [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] as well as at the nearly physical mass [29][30][31][32][33][34][35], there are some subtleties or issues in the method. One is the theoretical treatment of the bound states in this method, which has been recently clarified in Ref. [36]. In this paper, we consider the other issue, non-hermiticity of the potential in the HAL QCD method. We show in Sec. II that non-hermitian potential defined in the derivative expansion can be made hermitian order by order in the derivative expansion. In particular, non-hermitian potential which contains the second derivative at most can be made hermitian exactly, while nonhermitian potentials with higher order derivative than the second order can be shown to be made hermitian potentials, using the mathematical induction for the order of the derivative expansion. In Sec. III, we apply our method to a non-hermitian HAL QCD potential for ΞΞ in lattice QCD [12], which consists of local and second or first order derivative terms. We show that the exactly hermitized potential gives the same scattering phase shifts with those from the original non-hermitian potential but the contribution from its derivative term is smaller than the one from the original derivative term. The summary and conclusion of this paper is given in Sec. IV.

II. HERMITIZING THE NON-HERMITIAN POTENTIAL
In this section, we propose a method to hermitize the non-hermitian Hamiltonian order by order in terms of derivatives. We consider the non-hermitian Hamiltonian for the relative coordinate of two identical particles without spin, which is given by where V n is the potential with n-derivatives, and m is the mass of the particle, so that the reduced mass is given by m/2. The explicit form of V n is denoted as where the local function V i 1 i 2 ···in l is symmetric under exchanges of indices i 1 i 2 · · · i n and summations over repeated indices are implicitly assumed. Except the local potential V 0 , other V n>0 are non-hermitian. Note also that the r-dependence of V n is also implicit.
Since H 0 + V 0 is hermitian, we first consider V 1 and V 2 , which is the next-to-leading (1st) order, and more generally V 2n−1 and V 2n as the n-th order, for the hermitizing problem, and introduce The reason to treat V 2n and V 2n−1 together will be clear later. In terms of the derivative expansion for the potential, V 0 is of leading order while V 1 and V 2 are of next-to-leading, so that V 0 is much larger in size than V 1 or V 2 at low energies.
A. n = 1 case At n = 1, the Hamiltonian is given by In the rotationally symmetric case such that with r := | r| andr i := r i /r, we have which can be solved as where we assume V 1 (r) = 0 and R 1 (r) = 1 at sufficiently large r ≥ r ∞ . Thus the hermitian local potentialṼ 0 becomes where the prime means the derivative with respect to r.
In the previous subsection, we show that the non-hermitian potential at n = 1 can be made hermitian without any approximations. In this subsection, we proceed to the next order, the n = 2 case, where some truncations are required for a number of derivatives, as we will see.
The n = 2 Hamiltonian is given by where the n = 2 potential U 2 can be written as and U 2,4 and U 2,2 are hermitian, while U 2,3 and U 2,1 are not. In terms of the original V 4 and ). (20)

General case
The change of the wave function ψ = R (2) φ at n = 2 is given by where the n = 1 term R 1 is already determined in the previous subsection, while the n = 2 term R 2 contains the local function R 2,0 without derivatives and R 2,2 with second derivatives as As will be seen later, we can make H (2) hermitian without the first derivative term, R 2,1 .
The transformed HamiltonianH 2 is given bỹ whereH (1) is already made hermitian by R 1 and we neglect higher order terms such as whereŨ 2,n consists of n-th derivative terms, andŨ 2,2 can be taken to be hermitian whilẽ U 2,0 is always hermitian. (Note that U 2,4 is defined to be hermitian.) Explicit forms ofŨ 2,n in terms of U 2,l are too complicated but unnecessary for our argument hereafter.
Similarly we can write where n in X k,n represents the number of derivatives, and X k,2n is taken to be hermitian.
Explicitly, we have and where H 1,2 is defined in eq. (23).

The transformed Hamiltonian becomes
To remove non-hermitian 3rd derivative terms, R 2,2 must satisfỹ which becomes a linear 1st order partial differential equation for R 2,2 . Once R 2,2 is determined from this equation, X 2,2 , X 2,1 and X 2,0 are completely fixed. To remove non-hermitian 1st derivative terms, R 2,0 must satisfỹ which again becomes a linear 1st order partial differential equation for R 2,0 , so that we can easily solve it to fix X 0,0 .
We finally obtaiñ which is manifestly hermitian at n = 2.
2. R 2.2 and R 2.0 for the rotationally symmetric case In order to demonstrate that equations for R 2,2 and R 2,0 can be solved, we explicitly determine R 2,2 and R 2,0 for the rotationally symmetric case.
For this case, we havẽ which lead to Thus eq. (34) gives with R 2+ := R 2a + R 2b and H 2+ := H 2a + H 2b , which is simplified as This equation can be easily solved as where we assume the s-integral to be finite. In other words, singularities of the integrand between 0 < s < r ∞ are all integrable. From the original equations, individual terms are given by Once R 2,2 ( r) is determined, eq. (30) fixes X i 2,1 and eq. (35) becomes which can be solved as where and X 1 (r) is expressed in terms ofṼ 0 , H 2a , H 2b , V 3a and V 3b .

C. All orders
We now argue that we can make the total Hamiltonian hermitian order by order in the derivative expansion. The total Hamiltonian is given in the derivative expansion as while the n-th order Hamiltonian is denoted as In previous subsections, we have already shown that H (1) and H (2) can be made hermitian.
As before, we make the even-derivative terms hermitian by introducing lower derivative terms, so that where k of U n,k represents the number of derivatives in this terms, while n corresponds to the order of this term. Throughout this subsection, we use the similar notations also for other quantities such as F n,k , which is the n-th order term with k derivatives, and is hermitian for even k.
The transformation operator R is expanded as where R n is expanded in terms of even numbers of derivatives as In order to prove that H can be made hermitian order by order, we use the mathematical induction. We have already seen that H 1 and H 2 can be made hermitian by R 1 and R 1 (1 + R 2 ), respectively.
We next assume that H n can be made hermitian by R (n) = R 1 (1 + n l=2 R l ) at the n-th order asH 2k are all hermitian with 2k derivatives and contain terms whose orders are less than or equal to n, while ∆H n+1 is non-hermitian at (n + 1)-th order and consists of terms such as (k i − 1) + 2l + 2(m − 1) = 2(n + 1), so that we can write where k denotes the number of derivatives in ∆H n+1,k , which is hermitian for even k.
We now consider the transformed Hamiltonian at the (n + 1)-th order as where the second term is evaluated as while the 3rd term becomes Using the assumption of the mathematical induction in eq. (56), we havẽ Using n + 1 unknown R n+1,2k with k = 0, 1, · · · , n, we can remove non-hermitian contributions inH (n+1) (the second line in eq. (61) ), as shown below.
We first remove (2n + 1)th order derivative terms in the second line by requiring which fixes R n+1,2n .
Repeating this procedure, we can remove all non-hermitian contributions as follows. For general k = 0, 1, 2, · · · , n, we have where n l=k+1 X n+1,2k+1 [l] has already been determined from R n+1,2l with l = n, n − 1, n − 2, · · · , k + 1. Thus the above condition fixes R n+1,2k in X n+1,2k−1 [k]. Therefore it is shown that H (n+1) can be made hermitian as The proof that non-hermitian H can be made hermitian order by order is thus completed by the mathematical induction.
We note here that the n-th order Hamiltonian, given by Eqs. (2) and (3), contains 2n + 1 new unknown functions to be extracted from the NBS wave functions generated by lattice QCD calculations. Thus, in order to perform the n-th order analysis, we totally need (n + 1) 2 NBS wave functions, which must be independent beyond numerical uncertainties.
This may give rise to severe limitations in applying the present formalism to the cases where the higher order terms are important. In the next section we will see that, in the case of ΞΞ( 1 S 0 ) scattering, the LO potential already gives a good approximation for the scattering phase shift, while the NLO corrections to the phase shift gradually appear as the energy increases.

III. HERMITIZING THE NLO POTENTIAL FOR THE ΞΞ( 1 S 0 ) SYSTEM
In this section, we actually make non-hermitian potential for the ΞΞ( 1 S 0 ) system obtained in lattice QCD hermitian, using the method in Sec. II.
The rotationally symmetric potential in the previous section for the n = 1 case can be rewritten as where the expression in the previous section is recovered by replacing Since the V 3 (r) term does not contribute to the S wave scattering, we ignor this term in the present analysis. Having only two NBS wave functions, one from the wall source, the other from the smeared source, available from the previous calculation [12], we consider two different extractions of potentials, one with V 1 (r) = 0 (NLO A ), the other with V 2 (r) = 0 (NLO B ), in the present analysis.
A. NLO A : NLO analysis without V 1 We first consider the NLO A potential for ΞΞ( 1 S 0 ), where V NLO A 0 (r), together with V LO(wall) 0 (r) from the wall source and V LO(smeared) 0 (r) from the smeared source, are plotted in Fig. 1 (Left), and m 2 π V NLO A 2 (r) is given in Fig. 1 (Right).
Small differences between V NLO A 0 (r) and V LO(wall) 0 (r) are observed at short distance, due to contributions from V NLO A 2 (r), which is non-zero only at r < 1 fm.
According to the procedure in Sec. II, we can make this non-hermitian potential hermitian  whereṼ . We next consider the NLO analysis without V 2 (NLO B ), whose potential is given by (r) at short distances.
According to the procedure in Sec. II, we convert this non-hermitian NLO B potential to a local hermitian potentialṼ NLO B 0 (r), wherẽ which is shown in Fig. 4

IV. SUMMARY AND CONCLUDING REMARKS
The HAL QCD potential expressed as an energy independent non-local potential is known to be non-hermitian due to the nature of the Nambu Bethe Salpeter (NBS) wave function used to extract it: While the leading order (LO) term in the derivative expansion of the potential is local and hermitian, the higher order terms are in general non-hermitian. In this paper, we have formulated a way of hermitizing it in the derivative expansion. Since the hermitized potential can be expressed to contain only even number of derivatives, we classify the first and second order derivative terms as the next-to-leading order (NLO) and in general (2n − 1) and 2n derivative terms as the n-th order. Starting from the NLO terms, which can be made hermitian exactly, we have shown that the higher order terms can be hermitized order by order to all orders using the mathematical induction in the derivative expansion.
In order to see the feasibility of our formalism, we applied it to the case of ΞΞ( 1 S 0 ) scattering for which two independent NBS wave functions were available from the lattice QCD calculations [12]. Since two NBS wave functions are insufficient for the full NLO analysis which requires three unknown functions, V 0 (r), V 1 (r) and V 2 (r), we carried out two NLO analyses, one without V 1 (NLO A ) and the other without V 2 (NLO B ). Although the two hermitized potentials,V NLO A andV NLO B , look very different, the former containing a second order derivative term while the latter being purely local, they give essentially the same phase shifts within the uncertainties of the calculations. This agreement indicates that the obtained NLO phase shift can be regarded approximately as the yet unknown exact one. By comparing it to the LO phase shifts obtained in ref. [12], we find that the LO phase shift from the NBS wave function with the wall source is very similar to the NLO phase shift at low energies while it is slightly larger at higher energies. The LO analysis with the wall source is thus well justified for the ΞΞ( 1 S 0 ) scattering.
While the non-hermitian potential is fine as long as we are interested in the two-body observables such as scattering amplitudes and binding energies, the hermitian version is more convenient for a comparison with phenomenological interactions and also for using it as a two-body interaction in many-body systems.