Current matrix element in HAL QCD's wave function equivalent potential method

We give a formula to calculate a matrix element of a conserved current in the effective quantum mechanics defined by the wave function equivalent potentials proposed by HAL QCD collaboration. As a first step, a non-relativistic field theory with two channel coupling is considered as the original theory, with which a wave function equivalent HAL QCD potential is obtained in a closed analytic form. The external field method is used to derive the formula by demanding that the result should agree with the original theory. With this formula, the matrix element is obtained by sandwiching the effective current operator between the left and the right eigen functions of the effective Hamiltonian associated with the HAL QCD potential. In addition to the naive one-body current, the effective current operator contains an additional two-body term emerging from the degrees of freedom which has been integrated out.


Introduction
The nuclear force plays a key role in understanding various properties of atomic nuclei. It is important not only for nuclear physics but also for astrophysics such as explosion of supernova and the structure of neutron stars. Enormous efforts have been devoted to investigations of the nuclear force, which includes phenomenological studies such as the meson exchange interactions [1], the chiral effective field theory [2] and the recently developed QCD based studies [3]. Since 90's, high-precision phase equivalent NN potentials have been available [4][5][6]. These high-precision potentials are so determined as to reproduce wide range of nucleonnucleon scattering data together with the deuteron properties.
A lattice QCD (LQCD) method to determine the nuclear force has been recently developed by HAL QCD collaboration [3,[7][8][9], which we will refer to as HAL QCD method. It has been applied to many systems [10,11]. With this method, LQCD is used to generate equal-time Nambu-Bethe-Salpeter (NBS) wave functions for NN system in the center of mass frame. By regarding these NBS wave functions as the NN wave functions, the NN potentials are defined by demanding that these NBS wave functions should be reproduced by Schrödinger equation below the inelastic threshold. We will refer to the potential thus defined as HAL QCD potential or wave-function equivalent potential. Note that, by using LSZ reduction formula, it is shown that these NBS wave functions have the asymptotic long distance behavior, which is parameterized by the scattering phase shift δ in exactly the same way as that of the non-relativistic quantum mechanics [7,12,13]. For instance, for s-wave This implies that HAL QCD potentials reproduce the scattering phase shifts together with the NBS wave functions. It is therefore a phase equivalent potential as well as a wave-function equivalent potential. These phase equivalent potentials can be used to obtain an effective NN quantum mechanics. However, although the scattering phase shifts are guaranteed to be reproduced by these effective NN quantum mechanics, it is not straightforward to calculate matrix elements. Note that phase equivalent potentials generate wave functions whose long distance behaviors are constrained by the scattering phase shift. However, there are no constraints imposed on their short and the medium distance behaviors. As a result, with the naive formula of matrix elements, the result depends on the choice of phase equivalent potentials, which suggests the existence of an additional contribution to absorb the difference. In fact, it is known that the electromagnetic current of the two-nucleon system has such an additional two-body contribution, i.e., µ (x) denotes the naive one-body nucleon current while J (2) µ (x) denotes the additional two-body current which is referred to as the exchange current. The exchange current is known to emerge from the charged mesons exchanged between the two nucleons. Because these charged mesons are "frozen" in the instantaneous potentials, their contribution must appear as an additional two-body operator in the effective NN quantum mechanics. It is known that the exchange current gives a dominant contribution in dγ → np reaction [14].
The current conservation imposes a constraint on the exchange current J (2) µ (x) as is assumed with r i and e i ≡ e 2 (1 + τ 3 ) i being the position and the charge operator acting on the isospin space of i-th nucleon, respectively [15,16]. In particular, this constraint implies (1) J (2) (x) does not vanish, if the potential V is either isospin dependent or is non-local. (2) An explicit form of J (2) (x) depends on particular choice of the potential V . Although it is rather a strong constraint, it is not strong enough to determine the complete form of J (2) µ (x). In fact, for the one pion exchange potential (OPEP) V = V OPEP , the constraint Eq. (2) is satisfied by two different currents J (2) (x) (i) Sachs current [17] and (ii) one pion exchange current (OPEC) [18]. OPEC is considered to be reasonable, because it is obtained by going back to the original theory (relativistic pionnucleon coupling model) with the demand that the Bremsstrahlung amplitude of the original theory should be reproduced in the effective quantum mechanics.
The same strategy does not work for the phenomenologically constructed phase equivalent potentials, because their connection to the original theory (QCD) is unclear. (For those potentials whose relation to QCD is unknown, a prescription to introduce a current operator which is conserved is proposed [15,16].) In contrast, since HAL QCD potentials are constructed in LQCD, it may be possible to derive an explicit form of the exchange current operator for the HAL QCD method. Note that once such formula is established for the HAL QCD method, it enables us to consider QCD matrix elements by the nuclear physics with the conventional nucleon degrees of freedom. There are many applications. In addition to the standard calculation of form factors, it can be applied to the nuclear EDM for for physics beyond the standard model [19], np → dγ in a big bang nucleosynthesis [20], etc.
In this paper, we consider a method to calculate a matrix element in an effective quantum mechanics assoicated with HAL QCD potentials. As a first step, we restrict ourselves to the matrix element of a conserved current. Instead of Lorentz covariant QCD, we employ a nonrelativistic Galilei covariant (field theoretical) coupled-channel model [21,22] as an original theory to present a formula to calculate the matrix element of a conserved current. This nonrelativistic model enables us to obtain a HAL QCD potential in a closed analytic form, which is used to define a non-relativistic effective quantum mechanics. We use a non-relativistic original theory, because, at this initial stage, we prefer to have the formula which involves as little approximation as possible. To obtain the formula to calculate a matrix element, we use the external field method, which was conveniently used in the Bethe-Salpeter framework [23][24][25]. The external field method enables us to obtain the formula in the effective quantum mechanics which agrees with the calculation in the original theory.
The paper is organized as follows. In Sec. 2, a second quantized non-relativistic coupledchannel model is introduced, which mimics np-np * coupling system. In Sec. 3, the nonrelativistic model is used as the original theory to obtain an effective quantum mechanics of np system below the np * threshold by integrating out the closed np * channel. This is done by using HAL QCD method. We will give a wave function equivalent HAL QCD potential in a closed analytic form which reproduces the equal-time Nambu-Bethe-Salpeter (NBS) wave functions of open np channel. In Sec. 4, we introduce an external gauge field and extend the HAL QCD potential in the external field. In Sec. 5.1, the external field method is used to derive a formula to calculate the current matrix element in the effective quantum mechanics so that the result agrees with the original theory. Readers may wonder why we employ the np-np * coupling model to consider the "exchange current" instead of those models where a potential is obtained by integrating out the exchanged mesons between two nucleons. This is because we determine to stick to the non-relativistic original theory to obtain an analytic expression of the formula of the matrix element. Note that, even with np-np * coupling model, the two-body current will appear after integrating out the closed np * channel to obtain the effective np potential.

The original theory 2.1. Hamiltonian
We consider a second quantized non-relativistic Hamiltonian whereT ≡T 1 +T 2 +T 3 denotes the kinetic term witĥ which is used to mimic the np-np * coupling system [21,22], whereφ 0 (x),φ 1 (x) andφ 2 (x) correspond to the neutron (n), the proton (p) and an excited proton (p * ) with excitation energy ∆, respectively. (We use a bold font for three dimensional vectors. A variable with hat " * " is used to indicate that it is a field operator acting on the Fock space.) We employ the same non-relativistic mass m for all these three fields for Galilei covariance. Since we do not stick to the detail of the np-np * coupling system, we consider the scalar boson fields for simplicity which satisfy the equal-time commutation relation All the other combinations vanish. We consider eigenvalues and eigenvectors ofĤ aŝ where |n, P denotes an energy eigenstates with the normalization m, Q|n, P = δ mn · (2π) 3 δ 3 (Q − P ). P denotes the total spatial momentum, while n labels an "intrinsic excitation" in the center of mass frame. Due to the Galilei covariance, the energy eigenvalue E n (P 2 ) decomposes as where E n denotes the energy of the "intrinsic excitation" in the center of mass frame. We will refer to E n as "reduced" energy. (We use a variable with tilde " * " for "reduced" objects which have something to do with the center of mass frame such as the reduced NBS wave functions ψ(r) which will be introduced later.)

Conserved Currents and Conserved Charges
The HamiltonianĤ has two U (1) symmetries U n (1) and U p (1). They are generated by the conserved chargesQ respectively.Q n corresponds to the conservation of n-number, whereasQ p corresponds to the conservation of p-number which also counts p * . These charges are associated with the non-relativistic conserved currentsĵ denotes the Heisenberg operators for α = 0, 1, 2. By using the Heisenberg's equation i∂ 0φα (x) = φ α (x),Ĥ , it is straightforward to see that these two

The two-particle subspaces
We restrict ourselves to the two-particle subspace with (Q n , Q p ) = (1, 1), which we will refer to as F . The subspace F is spanned by all the state vectors of the form where |0 denotes the non-relativistic vacuum defined by the relation for α = 0, 1, 2 and all x ∈ R 3 with 0|0 = 1. We introduce a cutoff by using a projection operator Note that this cutoff is Galilei covariant. We use it to define a truncated subspace F ∆ by The truncated subspace F ∆ consists of all states in F which exist below the np * threshold. From Sec. 3, we will use HAL QCD method to construct an effective quantum mechanics for states in F ∆ by integrating out all the states |n, P with E n > ∆.
3. The HALQCD potential and the effective quantum mechanics

The HAL QCD potentials
We use HAL QCD method to obtain an effective np potential (wave function equivalent HAL QCD potential). This is carried out in two steps. We first construct the (reduced) HAL QCD potential in the center of mass frame. To do this, we require the Schrödinger equation to reproduce the reduced NBS wave functions of np channel below the np * threshold. We then use Galilei covariance to generalize the reduced potential for a general Galilei frame, with which (full) NBS wave functions satisfy the Schrödinger equation.
To obtain the reduced HAL QCD potential V(r, r ′ ), we demand that, for any states |n, P = 0 ∈ F ∆ , the reduced NBS wave function of the np channel ψ 1 (r|n) ≡ 0|φ 0 (r/2)φ 1 (−r/2)|n, P = 0 satisfies Schrödinger equation where E n denotes the energy eigenvalue of |n, P = 0 . The demand is satisfied by the following energy-independent non-local potential V(r, r ′ ) as where ψ 2 (r|m) ≡ 0|φ 0 (r/2)φ 2 (−r/2)|m, P = 0 . ψ ∨ 1 (r|m) denotes a dual basis associated with a linearly independent set of reduced NBS wave functions ψ 1 (r|m) |m, P = 0 ∈ F ∆ . The dual basis satisfies the orthogonality relation for m, n with E m , E n < ∆. An explicit form of dual basis is given, for instance, by where N nm ≡ d 3 r ψ * 1 (r|n) ψ 1 (r|m) denotes the norm kernel. It is straightforward to see Eq.
where the orthogonality relation Eq. (24) is used. Thus the Schrödinger equation (22) reduces to the coupled channel equation (21). We use Galilei covariance to generalize the reduced HAL QCD potential V(r, r ′ ) to the (full) HAL QCD potential V(x, y; x ′ , y ′ ) by Note that, for any states |n, P ∈ F ∆ , (full) NBS wave functions of the np channel ψ 1 (x, y|n, P ) ≡ 0|φ 0 (x)φ 1 (y)|n, P satisfy Schrödinger equation with this potential as where the last line is obtained by using Eq. (26). 7

The effective quantum mechanics
The HAL QCD potentials Eq. (23) and Eq. (27) are used to define an effective quantum mechanics. For later convenience, we give a summary of the eigenvalue property of the effective quantum mechanics. We begin with the reduced system in the center of mass frame. The eigenvalue relations are given as where H denotes the effective Hamiltonian, whose actions on the left and the right wave functions are defined as χ L n (r) and χ R n (r) denote the left and the right eigen functions associated with the energy eigenvalue E n . They satisfy the orthogonality relations Note that, in the upper relation, the summation is not restricted to E n < ∆. Since V(r, r ′ ) is not Hermitian in general, χ L n (r) is not a complex conjugate of χ R n (r). We note that, since V is defined to reproduce the NBS wave functions in the elastic region, we have for the states |n, P = 0 ∈ F ∆ . Due to the upper relation, the scattering phase shift of the original theory is reproduced by the effective quantum mechanics through the right eigen function χ R n (r) for the energy region E n < ∆. We then use Galilei covariance to generalize these relations to arbitrary Galilei frames. By defining the eigenvalue relations are given as H χ R n,P (x, y) = E n (P 2 )χ R n,P (x, y) where H denotes the effective Hamiltonian, whose actions on the left and right wave functions are defined as χ L n (x, y) and χ R n (x, y) are the left and right eigen functions, respectively, associated with the energy eigenvalue E n (P 2 ) ≡ E n + 1 4m P 2 . They satisfy the orthogonality relations and Needless to say, for the states |n, P ∈ F ∆ , the right eigen functions agree with the NBS wave functions as E n (P 2 ) = E n (P 2 ).

The external field
In order to consider the matrix element of the conserved U p (1) current in the effective quantum mechanics, we employ the external field method. We introduce an external U p (1) gauge field A µ (x, t), (µ = 0, 1, 2, 3) and demand that the response of the effective quantum mechanics to the external field should be the same as the original theory. To do this, we construct a HAL QCD potential in the external field. To avoid unnecessary complexity, we restrict ourselves to those external field A µ (x, t) which are non-zero only for the time region t > 0.

Hamiltonian in the external field
We consider the coupling of the original theory to the external field. In order for the coupling to be consistent with the conserved U p (1) current of Eq. (11), the Hamiltonian should bê where the kinetic termT ≡T 0 +T 1 [A t ] +T 2 [A t ] couples to the external fields aŝ whereas the interaction termV ≡ α,β=1,2Vαβ does not couple. (IfV were to couple to the external field, the conserved U p (1) current would need an additional term.) The subscript "t"

Truncated Hamiltonian and Truncated time-evolution in the external field
Time-dependence of the external field causes an unwanted transition to np * above the inelastic threshold, which is harmful in constructing a low-energy effective quantum mechanics below the np * threshold. In order to suppress such an unwanted transition, we insert the projection operatorP ∆ at every step of the time evolution. This is done by replacing the Hamiltonian by the truncated HamiltonianĤ generates a time-evolution, which will be referred to as the truncated time-evolution. The truncated time-evolution is denoted byÛ ∆ (t, s; A), which is explicitly expressed as a time-ordered product aŝ Note that U ∆ (t, s; A) is a solution of the initial value problems withÛ ∆ (t = s, s, A) =Î.

The truncated NBS wave functions in the external field
The equal-time NBS wave function in the external field with the truncated time-evolution is defined as for α = 1, 2, whereφ α (x, t; A) denotes the Heisenberg operator with the truncated timeevolution asφ We will refer to Eq. (46) as the truncated NBS wave function. Note that, unlike Eq. (19), ψ α (x, y, t; A|n, P ) does not factorize any more, since the external field breaks the Galilei covariance.
As will be shown in Appendix. A, the truncated NBS wave functions satisfy the coupled channel Schrödinger equations with We give several comments. (1) Sinceφ 0 (x, t) does not have Q p charge, the derivative for x remains to be an ordinary one, i.e., ∂ x in Eq. (48). (2) The additional term ∆V αβ (· · · ) originates from the cutoffP ∆ inĤ ∆ [A t ].

The HAL QCD potential in the external field
We define HAL QCD potential in the presence of the external field by demanding that, for any states |n, P ∈ F ∆ , the time-dependent Schrödinger equation should reproduce the truncated NBS wave functions for np channel as We note that, due to the time-dependence of the external field, we need to use the timedependent Schrödinger equation to define HAL QCD potential. The demand is satisfied by where ψ α (x ′′ − y ′′ |m) and ψ ∨ 1 (x ′ − y ′ |m) denote the reduced NBS wave function and the dual basis defined at Eq. (20) and Eq. (25), respectively, in the absence of external field. The proof is straightforward, which is given in Appendix. B.

The current matrix element in the effective quantum mechanics
In Eq. (51) in Sec. 4, we derived Schrödinger equation in the external field which is satisfied by the truncated NBS wave functions. In this section, we use this Schrödinger equation to derive a formula to calculate a current matrix element in the effective quantum mechanics associated with HAL QCD potentials. In Sec. 5.1, we will just give the formula together with several remarks. The derivation of the formula will be given in Sec. 5.2.

The formula to calculate a matrix element
Suppose that we have the potential V(x, y; x ′ , y ′ ; A t ) with which Schrödinger equation in the external field is satisfied by the truncated NBS wave function of np channel for any states |n, P ∈ F ∆ in the form Eq. (51). Then the matrix element of the currentĵ µ p (z) is calculated for any states |m, Q , |n, P ∈ F ∆ in the effective np quantum mechanics by the formula where χ L m,Q (x, y) and χ R n,P (x, y) denote the left and the right eigen functions of the effective Hamiltonian H in the absence of the external field (See Eq. (35)). K µ (x, y; x ′ , y ′ ; z) denotes the effective current operator, which is defined by The derivation of Eq. (53) is given in Sec. 5.2. We give several remarks.
(1) In the conventional quantum mechanics, matrix elements are obtained by sandwiching an operator with a state vector and its Hermitian conjugate. In contrast, in our effective quantum mechanics associated with the HAL QCD potential, an operator is sandwiched by the left and right eigen functions of the effective Hamiltonian H. Since HAL QCD potentials are not Hermitian in general, this could be a natural generalization. (2) The first terms of the effective current operator K µ (x, y; x ′ , y ′ ; z) correspond to the naive one-body current carried by a single proton as In contrast, the second terms are the two-body current, which correspond to "exchange current". The two-body currents originate from the states above the np * threshold which have been integrated out during the construction of the HAL QCD potential. (3) In a realistic situation, HAL QCD potentials is constructed not by the method which was employed in the previous sections but by the derivative expansion using the NBS wave functions as inputs [7,22]. However, as we shall see in Sec. 5.2, the derivation of the formula Eq. (53) given in Sec. 5.2 does not depend on how HAL QCD potential in the external field is constructed. All we need to use the formula is that there is a potential with which Schrödinger equation in the external field is satisfied by the truncated NBS wave functions. (4) It would be interesting to argue the gauge covariance property as was done in Ref. [16].
However, the cutoff which we have introduced in the Hamiltonian makes the situation complicated. Since the same matrix elements as the original theory can be reproduced, we do not stick too much to this point in this paper. (5) Application of two functional derivative δ/δA µ to Eq. (56) in Sec. 5.2 does not lead to m, Q|j µ p (z 1 )j ν p (z 2 )|n, P but to m, Q|j µ p (z 1 )P ∆ j ν p (z 2 )|n, P . To calculate the former matrix element, an additional consideration is needed.

The derivation
For notational convenience, we arrange Eq. (51) as where the effective Hamiltonian H[A t ] acts on the truncated NBS wave function as 1 (x, y, t; A|n, P ) 1 (x ′ , y ′ , t; A|n, P ). (57) We apply the functional derivative δ/δA µ (z, z 0 ) to both sides of Eq. (56) and then set the external field A µ ≡ 0 to have , which reduces to the explicit expression of K µ given in Eq. (54).

13
In the original theory, δψ (∆) /δA µ (z) is expressed as where we used Eq. (C1) to derive the second line. The existence of To express δψ By comparing Eq. (60) and Eq. (62), we arrive at the formula Eq. (53).

Summary and conclusion
We have considered how to deal with a matrix element in HAL QCD's potential method. HAL QCD method is a lattice QCD (LQCD) method to obtain a potential (HAL QCD potential) which is faithful to the scattering phase shift. This is supported by the fact that the HAL QCD potential is defined by demanding Schrödinger equation should reproduce the equaltime Nambu-Bethe-Salpeter (NBS) wave functions, and the NBS wave functions contain the scattering phase shift in its long distance part in exactly the same way as that of the scattering wave functions of the non-relativistic quantum mechanics. Therefore the effective NN quantum mechanics associated HAL QCD potential is supported to reproduce the scattering phase shift. However, an additional consideration is needed to calculate the matrix elements. In fact, there have been no arguments concerning the relation between the matrix elements calculated from the effective NN quantum mechanics associated with the HAL QCD potential and QCD, the original theory. As a first step to considering a matrix element in HAL QCD method, we have considered a simplified non-relativistic field theoretical model instead of Lorentz covariant QCD. We have employed a two-channel coupling model as the original theory where np-np * coupling is mimicked (np-np * coupling model). By integrating out the closed np * channel with HAL QCD method, we have obtained an effective np potential (HAL QCD potential) which is used to define the effective np quantum mechanics. Due to the simplicity of our np-np * coupling model, we have obtained the HAL QCD potential in a closed analytic form.
We have used the external field method and obtained a formula to calculate a matrix element of a conserved current in the effective np quantum mechanics by demanding that the response of the effective quantum mechanics to the external field is the same as that of the original theory. With our formula, the matrix element is calculated by sandwiching the effective current operator between the left and the right eigen functions of the effective np Hamiltonian. The effective current operator consists of two parts (1) naive one-body current and (2) the twobody current which corresponds to the exchange current. In our np-np * coupling model, the two body current emerges from the states above the np * threshold which have been integrated out when obtaining the effective np potential.
To extend the formula for QCD, several generalizations are needed. To use the formula in relativistic original theories, it is necessary to generalize the HAL QCD potential for a boosted Lorentz frame. Note that it is only in the center of mass frame where the asymptotic forms of NBS wave functions Eq. (1) agree with those of the scattering wave functions of the quantum mechanics. To consider a system of composite particles, we have to take into account the form factors. To use the formula in LQCD, we have to deal with the time-evolution in an external field with a cutoff. To do these things, we may have to introduce several approximations.