Hermitizing the HAL QCD potential in the derivative expansion
YYITP-19-83iTHEMS-Report-19
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 SciencesProgram (iTHEMS), RIKEN Saitama 351-0198, Japan (Dated: March 3, 2020)
Abstract
A formalism is given to hermitize the HAL QCD potential, which needs to be non-hermitianexcept the leading order (LO) local term in the derivative expansion as the Nambu-Bethe-Salpeter(NBS) wave functions for different energies are not orthogonal to each other. It is shown that thenon-hermitian potential can be hermitized order by order to all orders in the derivative expansion.In particular, the next-to-leading order (NLO) potential can be exactly hermitized without ap-proximation. The formalism is then applied to a simple case of ΞΞ( S ) scattering, for which theHAL QCD calculation is available to the NLO. The NLO term gives relatively small correctionsto the scattering phase shift and the LO analysis seems justified in this case. We also observethat the local part of the hermitized NLO potential works better than that of the non-hermitianNLO potential. The hermitian version of the HAL QCD potential is desirable for comparing itwith phenomenological interactions and also for using it as a two-body interaction in many bodysystems. PACS numbers: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ h e p - l a t ] M a r . INTRODUCTION Lattice quantum chromodynamics (QCD) is a successful non-perturbative method tostudy hadron physics from the underlying degrees of freedom, i.e. quarks and gluons. Massesof the single stable hadrons obtained from lattice QCD show good agreement with theexperimental results, and even hadron interactions have been recently explored in latticeQCD. Using the Nambu-Bethe-Salpeter (NBS) wave function, linked to the S-matrix inQCD [1–9], the hadron interactions have been investigated mainly by two methods: thefinite volume method [1] and the HAL QCD potential method [5–7]. Theoretically the twomethods 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 andtime dependences of the NBS wave function. Physical observables such as phase shifts andbinding energies are then calculated by solving the Schr¨odinger equation in infinite volumeusing the obtained potentials, since the asymptotic behavior of the NBS wave function isrelated to the T -matrix element and thus to the phase shifts [9]. In practice, the non-localpotential is given by the form of the derivative expansion, which is truncated by the firstfew orders [12].While the HAL QCD method has been successfully applied to a wide range of two (orthree) hadron systems at heavy pion masses [13–28] as well as at the nearly physical mass[29–35], there are some subtleties or issues in the method. One is the theoretical treatment ofthe 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. Weshow in Sec. II that non-hermitian potential defined in the derivative expansion can be madehermitian order by order in the derivative expansion. In particular, non-hermitian potentialwhich contains the second derivative at most can be made hermitian exactly, while non-hermitian potentials with higher order derivative than the second order can be shown to bemade hermitian potentials, using the mathematical induction for the order of the derivativeexpansion. In Sec. III, we apply our method to a non-hermitian HAL QCD potential for ΞΞin lattice QCD[12], which consists of local and second or first order derivative terms. We2how that the exactly hermitized potential gives the same scattering phase shifts with thosefrom the original non-hermitian potential but the contribution from its derivative term issmaller than the one from the original derivative term. The summary and conclusion of thispaper is given in Sec. IV. II. HERMITIZING THE NON-HERMITIAN POTENTIAL
In this section, we propose a method to hermitize the non-hermitian Hamiltonian orderby order in terms of derivatives. We consider the non-hermitian Hamiltonian for the relativecoordinate of two identical particles without spin, which is given by H = H + ∞ (cid:88) n =0 V n , H = − m ∇ , (1)where V n is the potential with n -derivatives, and m is the mass of the particle, so that thereduced mass is given by m/
2. The explicit form of V n is denoted as V n := 1 n ! V i i ··· i n l ∇ i ∇ i · · · ∇ i n , (2)where the local function V i i ··· i n l is symmetric under exchanges of indices i i · · · i n andsummations over repeated indices are implicitly assumed. In this paper, we assume that theabove derivative expansion is convergent. See appendix A for some arguments. Except thelocal potential V , other V n> are non-hermitian. Note also that the (cid:126)r -dependence of V n isalso implicit.Since H + V is hermitian, we first consider V and V , which is the next-to-leading (1st)order, and more generally V n − and V n as the n -th order, for the hermitizing problem, andintroduce U n := V n + V n − . (3)The reason to treat V n and V n − together will be clear later. In terms of the derivativeexpansion for the potential, V is of leading order while V and V are of next-to-leading, sothat V is much larger in size than V or V at low energies.3 . n = 1 case At n = 1, the Hamiltonian is given by H (1) = H + V + U , (4)where the n = 1 potential U is rewritten as U = ˜ V + ˜ V , ˜ V := 12 ∇ i V ij ∇ j , ˜ V := ˜ V i ∇ i , ˜ V i := V i − (cid:0) ∇ j V ji (cid:1) . (5)Here ˜ V is hermitian, while ˜ V is not.The corresponding Schr¨odinger equation is given by H (1) ψ = Eψ, (6)which transforms to ˜ H (1) φ = Eφ, ˜ H (1) = R − H (1) R , (7)by the change of the wave function that ψ := R (1) φ with a local function R (1) = R , where˜ H (1) = H + ˜ V + ˜ V + (cid:26) ˜ V i − m R − ∇ i R + V ij R − ∇ j R (cid:27) ∇ i , (8)˜ V = V − m R − ∇ R + V i R − ∇ i R + 12 V ij (cid:0) R − ∇ i ∇ j R (cid:1) . (9)By demanding the condition that˜ V i − m R − ∇ i R + V ij R − ∇ j R = 0 , (10)˜ H (1) becomes hermitian as ˜ H (1) = H + ˜ V + ˜ V , where˜ V = V − (cid:16) ∇ i ˜ V i (cid:17) + m V i (cid:16) δ ij − m V ij (cid:17) − ˜ V j , (11) R − ∇ i R = m (cid:16) δ ij − m V ij (cid:17) − ˜ V j . (12)In the rotationally symmetric case such that˜ V i ( (cid:126)r ) := ˆ r i ˜ V ( r ) , V ij ( (cid:126)r ) := V a ( r )ˆ r i ˆ r j + V b ( r ) δ ij , R ( (cid:126)r ) := R ( r ) (13)with r := | (cid:126)r | and ˆ r i := r i /r , we have d R ( r ) d r = m V ( r )1 − m V ( r ) R ( r ) , V := V a + V b , (14)4hich can be solved as R ( r ) = exp m (cid:90) rr ∞ ˜ V ( s )1 − m V ( s ) d s , (15)where we assume V ( r ) = 0 and R ( r ) = 1 at sufficiently large r ≥ r ∞ . Thus the hermitianlocal potential ˜ V becomes˜ V = V − ˜ V r −
12 ˜ V (cid:48) + m V − m V , ˜ V := V − V a r − V (cid:48) a + V (cid:48) b , (16)where the prime (cid:48) means the derivative with respect to r . B. n = 2 case In the previous subsection, we show that the non-hermitian potential at n = 1 can bemade hermitian without any approximations. In this subsection, we proceed to the nextorder, the n = 2 case, where some truncations are required for a number of derivatives, aswe will see.The n = 2 Hamiltonian is given by H (2) = H (1) + U , (17)where the n = 2 potential U can be written as U = 14! ∇ i ∇ j U ijkl , ∇ k ∇ l + 13! U ijk , ∇ i ∇ j ∇ k + 12! ∇ i U ij , ∇ j + U i , ∇ i (18):= U , + U , + U , + U , , (19)and U , and U , are hermitian, while U , and U , are not. In terms of the original V and V , we have U ijkl , := V ijkl , U ijk , := V ijk − ∇ l V ijkl , U ij , := −
24! ( ∇ k ∇ l V ijkl ) , U i , := 14! ( ∇ j ∇ k ∇ l V ijkl ) . (20)
1. General case
The change of the wave function ψ = R (2) φ at n = 2 is given by R (2) = R (1 + R ) , R := R , + R , , (21)5here the n = 1 term R is already determined in the previous subsection, while the n = 2term R contains the local function R , without derivatives and R , with second derivativesas R , := 12! R ij , ∇ i ∇ j . (22)As will be seen later, we can make H (2) hermitian without the first derivative term, R , .The transformed Hamiltonian ˜ H is given by˜ H (2) := ( R (2) ) − H (2) R (2) (cid:39) (1 − R ) R − ( H (1) + U ) R (1 + R ) (cid:39) ˜ H (1) + R − U R + (cid:104) ˜ H (1) , R (cid:105) , ˜ H (1) = H + ˜ V + ˜ V := ˜ V + H , , (23)where ˜ H (1) is already made hermitian by R and we neglect higher order terms such as O ( R ) and O ( R H ).We first consider ˜ U := R − U R , which is evaluated as˜ U = U , + ˜ U , + ˜ U , + ˜ U . + ˜ U , , (24)where ˜ U ,n consists of n -th derivative terms, and ˜ U , can be taken to be hermitian while˜ U , is always hermitian. (Note that U , is defined to be hermitian.) Explicit forms of ˜ U ,n in terms of U ,l are too complicated but unnecessary for our argument hereafter.Similarly we can write X := (cid:104) ˜ H (1) , R , (cid:105) = X , + X , , (25) X := (cid:104) ˜ H (1) , R , (cid:105) = X , + X , + X ., + X , , (26)where n in X k,n represents the number of derivatives, and X k, n is taken to be hermitian.Explicitly, we have X , := 13! X ijk , ∇ i ∇ j ∇ k , X , := 12! ∇ i X ij , ∇ j , X , := X i , ∇ i , (27) X ijk , = (cid:104) H il , ( ∇ l R jk , ) − R il , ( ∇ l H jk , ) (cid:105) + 2 permutations , (28) X ij , = − R il , ( ∇ k ∇ l H kj , ) + 12 (cid:8) H kl , ( ∇ k ∇ l R ij , ) + ( ∇ k H kl , )( ∇ l R ij , ) − R kl , ( ∇ k ∇ l H ij , ) (cid:9) , (29) X i , = − R ij , ( ∇ j ˜ V ) − R kl , ( ∇ j ∇ k ∇ l H ij , ) −
12 ( ∇ j X ij , ) , (30) X , = − R ij , ( ∇ i ∇ j ˜ V ) , (31)6nd X , = H ij , ( ∇ j R , ) ∇ i , X , = 12 H ij , ( ∇ i ∇ j R , ) + 12 ( ∇ i H ij , )( ∇ j R , ) , (32)where H , is defined in eq. (23).The transformed Hamiltonian becomes˜ H (2) = ˜ H (1) + U , + ˜ U , + ˜ U , + ˜ U , + ˜ U , + X , + X , + X , + X , + X , + X , . (33)To remove non-hermitian 3rd derivative terms, R , must satisfy˜ U ijk , + X ijk , = 0 , (34)which becomes a linear 1st order partial differential equation for R , . Once R , is deter-mined from this equation, X , , X , and X , are completely fixed. To remove non-hermitian1st derivative terms, R , must satisfy˜ U i , + X i , + H ij , ( ∇ j R , ) = 0 , (35)which again becomes a linear 1st order partial differential equation for R , , so that we caneasily solve it to fix X , .We finally obtain˜ H (2) = ˜ H (2) + U , + ˜ U , + ˜ U , + X , + X , + X , , (36)which is manifestly hermitian at n = 2. R . and R . for the rotationally symmetric case In order to demonstrate that equations for R , and R , can be solved, we explicitlydetermine R , and R , for the rotationally symmetric case.For this case, we have˜ U ijk , := V a ( r )ˆ r i ˆ r j ˆ r k + V b ( r ) (cid:8) ˆ r i δ jk + ˆ r j δ ki + ˆ r k δ ij (cid:9) , ˜ U i , := V ( r )ˆ r i , (37) H ij , := H a ( r )ˆ r i ˆ r j + H b ( r ) δ ij , R ij , := R a ( r )ˆ r i ˆ r j + R b ( r ) δ ij , R , := R , ( r ) , (38)which lead to X ijk , = 3 X a ( r )ˆ r i ˆ r j ˆ r k + X b ( r ) (cid:8) ˆ r i δ jk + ˆ r j δ ki + ˆ r k δ ij (cid:9) , (39) X a := ( H a + H b ) R (cid:48) a − ( R a + R b ) H (cid:48) a − r ( R a H b − H a R b ) , (40) X b := ( H a + H b ) R (cid:48) b − ( R a + R b ) H (cid:48) b + 2 r ( R a H b − H a R b ) . (41)7hus eq. (34) gives H R (cid:48) a − H (cid:48) a R − r ( R a H b − H a R b ) = V a , (42) H R (cid:48) b − H (cid:48) b R + 2 r ( R a H b − H a R b ) = V b (43)with R := R a + R b and H := H a + H b , which is simplified as H R (cid:48) − H (cid:48) R = V , V := V a + 3 V b . (44)This equation can be easily solved as R = C ( r ) H ( r ) , C ( r ) := (cid:90) rr ∞ ds V ( s )3 H ( s ) , (45)where we assume the s -integral to be finite. In other words, singularities of the integrandbetween 0 < s < r ∞ are all integrable. From the original equations, individual terms aregiven by R a ( r ) = C ( r ) H a ( r ) + r (cid:90) rr ∞ ds H b ( s ) V a ( s ) − H a ( s ) V b ( s ) s H ( s ) , (46) R b ( r ) = C ( r ) H b ( r ) + r (cid:90) rr ∞ ds H a ( s ) V b ( s ) − H b ( s ) V a ( s ) s H ( s ) . (47)Once R , ( (cid:126)r ) is determined, eq. (30) fixes X i , and eq. (35) becomes V ( r ) + X ( r ) + H ( r ) R (cid:48) , ( r ) = 0 , (48)which can be solved as R , ( r ) = − (cid:90) rr ∞ ds V ( s ) + X ( s ) H ( s ) , (49)where X i , := X ( r )ˆ r i , (50)and X ( r ) is expressed in terms of ˜ V , H a , H b , V a and V b . C. All orders
We now argue that we can make the total Hamiltonian hermitian order by order in thederivative expansion. The total Hamiltonian is given in the derivative expansion as H = H + V + ∞ (cid:88) l =1 U l , U l := V l + V l − , (51)8hile the n -th order Hamiltonian is denoted as H ( n ) = H + V + n (cid:88) l =1 U l . (52)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 derivativeterms, so that U n = n (cid:88) k =1 U n,k , U n, k : hermitian , (53)where k of U n,k represents the number of derivatives in this terms, while n corresponds to theorder of this term. Throughout this subsection, we use the similar notations also for otherquantities such as F n,k , which is the n -th order term with k derivatives, and is hermitian foreven k .The transformation operator R is expanded as R = R (cid:32) ∞ (cid:88) l =2 R l (cid:33) , (54)where R n is expanded in terms of even numbers of derivatives as R n := n − (cid:88) k =0 R n, k . (55)In order to prove that H can be made hermitian order by order, we use the mathematicalinduction. We have already seen that H and H can be made hermitian by R and R (1 + R ), respectively.We next assume that H n can be made hermitian by R ( n ) = R (1 + (cid:80) nl =2 R l ) at the n -thorder as ˜ H ( n ) := ( R ( n ) ) − H ( n ) R ( n ) (cid:39) n (cid:88) k =0 ˜ H ( n )2 k + ∆ ˜ H n +1 , (56)where ˜ H ( n )2 k are all hermitian with 2 k derivatives and contain terms whose orders are lessthan or equal to n , while ∆ ˜ H n +1 is non-hermitian at ( n + 1)-th order and consists of termssuch as (cid:32) s (cid:89) i =1 R k i (cid:33) × R − U l R × (1 or R m ) , s = 0 , , , · · · , < k i , l, m ≤ n with the constraint s (cid:88) i =1 ( k i −
1) + ( l −
1) + ( m −
1) = n . Therefore, themaximum number of derivatives in ∆ ˜ H ( n +1) is 2 s (cid:88) i =1 ( k i −
1) + 2 l + 2( m −
1) = 2( n + 1), sothat we can write ∆ ˜ H n +1 = n +1) (cid:88) k =0 ∆ ˜ H n +1 ,k , (57)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˜ H ( n +1) := (cid:0) R ( n +1) (cid:1) − H ( n +1) R ( n +1) (cid:39) (cid:0) R ( n ) (cid:1) − H ( n ) R ( n ) + ˜ U n +1 + (cid:104) ˜ H (1) , R n +1 (cid:105) , (58)where the second term is evaluated as˜ U n +1 := R − U n +1 R = n +1) (cid:88) k =0 ˜ U n +1 ,k , (59)while the 3rd term becomes (cid:104) ˜ H (1) , R n +1 (cid:105) = n (cid:88) k =0 X n +1 [ k ] , X n +1 [ k ] := (cid:104) ˜ H (1) , R n +1 , k (cid:105) = k +1 (cid:88) l =1 X n +1 ,l [ k ] . (60)Using the assumption of the mathematical induction in eq. (56), we have˜ H ( n +1) = n (cid:88) k =0 (cid:32) ˜ H ( n )2 k + k (cid:88) l =1 X n +1 , l [ k ] (cid:33) + n +1 (cid:88) k =0 (cid:16) ∆ ˜ H n +1 , k + ˜ U n +1 , k (cid:17) + n (cid:88) k =0 (cid:32) ∆ ˜ H n +1 , k +1 + ˜ U n +1 , k +1 + k (cid:88) l =0 X n +1 , l +1 [ k ] (cid:33) . (61)Using n + 1 unknown R n +1 , k with k = 0 , , · · · , n , we can remove non-hermitian contribu-tions in ˜ H ( n +1) (the second line in eq. (61) ), as shown below.We first remove (2 n + 1)th order derivative terms in the second line by requiring∆ ˜ H n +1 , n +1 + ˜ U n +1 , n +1 + X n +1 , n +1 [ n ] = 0 , (62)which fixes R n +1 , n .The next condition becomes∆ ˜ H n +1 , n − + ˜ U n +1 , n − + X n +1 , n − [ n ] + X n +1 , n − [ n −
1] = 0 , (63)10here X n +1 , n − [ n ] is already determined completely from R n +1 , n . Therefore, the aboveequation determines R n +1 , n − in X n +1 , n − [ n − k = 0 , , , · · · , n , we have∆ ˜ H n +1 , k +1 + ˜ U n +1 , k +1 + n (cid:88) l = k +1 X n +1 , k +1 [ l ] + X n +1 , k − [ k ] = 0 , (64)where n (cid:88) l = k +1 X n +1 , k +1 [ l ] has already been determined from R n +1 , l with l = n, n − , n − , · · · , k + 1. Thus the above condition fixes R n +1 , k in X n +1 , k − [ k ]. Therefore it is shownthat H ( n +1) can be made hermitian as˜ H ( n +1) = n (cid:88) k =0 (cid:32) ˜ H ( n )2 k + k (cid:88) l =1 X n +1 , l [ k ] (cid:33) + n +1 (cid:88) k =0 (cid:16) ∆ ˜ H n +1 , k + ˜ U n +1 , k (cid:17) . (65)The proof that non-hermitian H can be made hermitian order by order is thus completedby the mathematical induction.We note here that the n-th order Hamiltonian, given by Eqs. (2) and (3), contains 2 n + 1new unknown functions to be extracted from the NBS wave functions generated by latticeQCD calculations. Thus, in order to perform the n -th order analysis, we totally need( n + 1) 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 wherethe higher order terms are important. In the next section we will see that, in the case ofΞΞ( S ) scattering, the LO potential already gives a good approximation for the scatteringphase shift, while the NLO corrections to the phase shift gradually appear as the energyincreases. III. HERMITIZING THE NLO POTENTIAL FOR THE
ΞΞ( S ) SYSTEM
In this section, we actually apply the method in Sec. II to lattice QCD data. We considerthe ΞΞ( S ) system, whose potential is much more precise than those for N N or N Λ.thanks to more strange quarks in the system, in order to make the NLO analysis numericallypossible. The potential for the ΞΞ( S ) was calculated on 2+1 flavor QCD ensembles[37]at a = 0 .
09 fm on a 64 lattice with heavy up/down quark masses and the physical strange11 IG. 1: NLO potentials without V (NLO A ): (Left) V NLO A ( r ) (black open diamonds), togetherwth V LO0 ( r ) from the wall source (red open circles) and the smeared source (blue open squares).(Right) V NLO A ( r ) multiplied by m π (green open crosses) . quark mass, m π = 0 .
51 GeV, m K = 0 .
62 GeV, m N = 1 .
32 GeV and m Ξ = 1 .
46 GeV. SeeRef. [12] for more details.The rotationally symmetric potential in the previous section for the n = 1 case can berewritten as V ( (cid:126)r, ∇ ) = V ( r ) + V ( r ) ˆ r i ∇ i + V ( r ) ∇ + V ( r ) (cid:126)L , (66)where the expression in the previous section is recovered by replacing V ( r ) → V ( r ) − V a ( r ) r , V ( r ) →
12 ( V a ( r ) + V b ( r )) , V ( r ) → V a ( r )2 r . (67)Since the V ( r ) term does not contribute to the S wave scattering, we ignor this term inthe present analysis. Having only two NBS wave functions, one from the wall source, theother from the smeared source, available from the previous calculation [12], we consider twodifferent extractions of potentials, one with V ( r ) = 0 (NLO A ), the other with V ( r ) = 0(NLO B ), in the present analysis. A. NLO A : NLO analysis without V We first consider the NLO A potential for ΞΞ( S ), V NLO A ( (cid:126)r, ∇ ) = V NLO A ( r ) + V NLO A ( r ) ∇ , (68)12 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r [fm] V ( r ) [ M e V ] V V ( k / m ) ( k ) [ deg .] V V V + i V i V + V FIG. 2: (Left) ˜ V NLO A ( r ) (blue dashed line) for ΞΞ( S ) together with V NLO A ( r ) (red solid line).(Right) Scattering phase shifts for ΞΞ( S ) from V NLO A (red solid line), ˜ V NLO A (blue dashed line), V NLO A ( (cid:126)r, ∇ ) (green dotted line), and ˜ V NLO A ( (cid:126)r, ∇ ) (yellow solid line). where V NLO A ( r ), together with V LO(wall)0 ( r ) from the wall source and V LO(smeared)0 ( r ) fromthe smeared source, are plotted in Fig. 1 (Left), and m π V NLO A ( r ) is given in Fig. 1 (Right).Small differences between V NLO A ( r ) and V LO(wall)0 ( r ) are observed at short distance, due tocontributions from V NLO A ( r ), which is non-zero only at r < V NLO A ( (cid:126)r, ∇ ) = ˜ V NLO A ( r ) + ∇ i V NLO A ( r ) ∇ i , (69)where ˜ V NLO A ( r ) = V NLO A ( r ) + ( V NLO A ) (cid:48) ( r ) r + ( V NLO A ) (cid:48)(cid:48) ( r )2 + m Ξ { ( V NLO A ) (cid:48) ( r ) } − m Ξ V NLO A ( r ) . (70)Here ˜ V is calculated from the fitted functions for V NLO A ( r ) and V NLO A ( r ) in Ref. [12],together with the 1st and second derivatives of V analytically calculated from the fittedfunction. We compare ˜ V NLO A ( r ) with V NLO A ( r ) in Fig. 2 (Left), while we plot the phaseshifts of the ΞΞ( S ) obtained with ˜ V NLO A ( r ), V NLO A ( r ), V NLO A ( (cid:126)r, ∇ ) and ˜ V NLO A ( (cid:126)r, ∇ )in Fig. 2 (Right). For visibility, only central values are given here. Errors of ˜ V NLO A arecomparable to those of V NLO A , which can be seen in Fig. 1 (Left). We notice that the leadingorder term V NLO A of the original non-hermitian NLO A potential describes the behavior ofthe scattering phase shift rather well at low energy but show some deviation from the13 IG. 3: (Left) V NLO B ( r ) (yellow open triangles) as a function of r at short distances, togetherwith V LO(wall)0 ( r ) (red open circles) and V NLO A ( r ) (black open triangles). (Right) V NLO B ( r ) (purpleopen triangles). NLO A result at high energies, while the local term ˜ V NLO A of the hermitized NLO A potentialdescribes the scattering phase shift in a wider energy ranges, 0 ≤ ( k/m π ) ≤ . k/m π ) (cid:39) .
1. Of course, non-hermitian NLO A potential V NLO A ( (cid:126)r, ∇ ) and the hermitized NLO A potential ˜ V NLO A ( (cid:126)r, ∇ ) give identical results at allenergies by construction. B. NLO B : NLO analysis without V We next consider the NLO analysis without V (NLO B ), whose potential is given by V NLO B ( (cid:126)r, ∇ ) = V NLO B ( r ) + V NLO B ( r ) ˆ r i ∇ i . (71)Fig. 3 (Left) shows V NLO B ( r ) (yellow open triangles), together with V LO(wall)0 ( r ) (red opencircles) and V NLO A ( r ) (black open triangles), at 0.45 fm ≤ r ≤ V NLO B ( r ) (purple open triangles) at r ≤ . V NLO A ( r ), V NLO B ( r ) deviateslargely from V LO(wall)0 ( r ) at short distances.According to the procedure in Sec. II, we convert this non-hermitian NLO B potential toa local hermitian potential ˜ V NLO B ( r ), where˜ V NLO B ( r ) = V NLO B ( r ) − V NLO B ( r ) r −
12 ( V NLO B ) (cid:48) ( r ) + m Ξ V NLO B ( r )) , (72)14 IG. 4: (Left) ˜ V LO B ( r ) (blue dotted line) for ΞΞ( S ) together with V LO(wall)0 ( r ) (red solidline) and V NLO B ( r ) (yellow dashed line). (Only central values are plotted for visibility.) (Right)Scattering phase shifts δ ( k ) for ΞΞ( S ) from V LO(wall)0 ( r ) (red solid circles), V NLO B ( r ) (yellowsolid down-triangles), V NLO A ( r ) (black solid diamonds), V NLO B ( (cid:126)r, ∇ ) (purple solid up-triangles), V NLO A ( (cid:126)r, ∇ ) (green crosses) and ˜ V NLO B ( r ) (blue open squares). which is shown in Fig. 4 (Left) by blue dotted line, together with V LO(wall)0 ( r ) (red solidline) and V NLO B ( r ) (yellow dashed line). The attractive pocket of ˜ V NLO B ( r ) around r (cid:39) . V LO(wall)0 ( r ) or V NLO B ( r ). Fig. 4 (Right) compares thescattering phase shifts δ ( k ) among V LO(wall)0 ( r ) (red solid circles), V NLO B ( r ) (yellow soliddown-triangles), V NLO A ( r ) (black solid diamonds), V NLO B ( (cid:126)r, ∇ ) (purple solid up-triangles), V NLO A ( (cid:126)r, ∇ ) (green crosses) and ˜ V NLO B ( r ) (blue open squares). By construction, ˜ V NLO B ( r )and V NLO B ( (cid:126)r, ∇ ) give identical results, which also agree well with δ ( k ) from V NLO A ( r ).As mentioned before, V LO(wall)0 ( r ) and V NLO A ( r ) give good approximations at low energybut show small deviations as energies increase. However δ ( k ) from V NLO B ( r ) deviatesfrom others even at low energies. Finally it is noted that two different NLO potentials, V NLO A ( (cid:126)r, ∇ ) (green solid crosses) and V NLO B ( (cid:126)r, ∇ ) (purple solid up-triangles), agree welleven at high energies, though the first derivative term ( V NLO B ( r )) has larger effects thanthe second derivative term ( V NLO A ( r )) on the scattering phase shift.15 V. SUMMARY AND CONCLUDING REMARKS
The HAL QCD potential expressed as an energy independent non-local potential is knownto be non-hermitian due to the nature of the Nambu Bethe Salpeter (NBS) wave functionused to extract it: While the leading order (LO) term in the derivative expansion of thepotential is local and hermitian, the higher order terms are in general non-hermitian. Inthis paper, we have formulated a way of hermitizing it in the derivative expansion. Sincethe hermitized potential can be expressed to contain only even number of derivatives, weclassify the first and second order derivative terms as the next-to-leading order (NLO) andin general (2 n −
1) and 2 n 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 behermitized order by order to all orders using the mathematical induction in the derivativeexpansion.In order to see the feasibility of our formalism, we applied it to the case of ΞΞ( S ) scat-tering for which two independent NBS wave functions were available from the lattice QCDcalculations[12]. Since two NBS wave functions are insufficient for the full NLO analysiswhich requires three unknown functions, V ( r ), V ( r ) and V ( r ), we carried out two NLOanalyses, one without V (NLO A ) and the other without V (NLO B ). Although the twohermitized potentials, ¯ V NLO A and ¯ V NLO B , look very different, the former containing a sec-ond order derivative term while the latter being purely local, they give essentially the samephase shifts within the uncertainties of the calculations. This agreement indicates that theobtained NLO phase shift can be regarded approximately as the yet unknown exact one. Bycomparing it to the LO phase shifts obtained in ref. [12], we find that the LO phase shiftfrom the NBS wave function with the wall source is very similar to the NLO phase shiftat low energies while it is slightly larger at higher energies. The LO analysis with the wallsource is thus well justified for the ΞΞ( S ) scattering.While the non-hermitian potential is fine as long as we are interested in the two-bodyobservables such as scattering amplitudes and binding energies, the hermitian version ismore convenient for a comparison with phenomenological interactions and also for using itas a two-body interaction in many-body systems.16 ppendix A: Convergence of the derivative expansion for non-local potentials In this appendix, we briefly discuss an issue on the convergence of the derivative expansionfor non-local potentials.Let us consider a non-local potential V ( x , y ), which can be expressed in terms of thederivative expansion as V ( x , y ) = ∞ (cid:88) n =0 n ! V i i ··· i n l ( x ) ∇ i ∇ i · · · ∇ i n δ (3) ( x − y ) , (A1)where no symmetry is assumed for generality.Applying this potential to a plane wave e i k · x , we obtain¯ V ( x , k ) e i k · x ≡ (cid:90) d y V ( x , y ) e i k · y = ∞ (cid:88) n =0 i n n ! V i i ··· i n l ( x ) k i k i · · · k i n e i k · x (A2)= e i k · x (cid:90) d r V ( x , x + r ) e i k · r = e i k · x ∞ (cid:88) n =0 (cid:90) d r V ( x , x + r ) i n n ! ( k · r ) n . (A3)By equating (A2) and (A3), we have V i i ··· i n l ( x ) = (cid:90) d r V ( x , x + r ) r i r i · · · r i n ≡ V l ( x ) (cid:104) r i r i · · · r i n (cid:105) x , (A4)where we define the n -th moment of non-locality, (cid:104) r i r i · · · r i n (cid:105) x , satisfying (cid:104) (cid:105) x = 1, and V l ( x ) is the term at n = 0 (the local term) in the derivative expansion eq. (A1). Then¯ V ( x , k ) can be expanded as ¯ V ( x , k ) = V l ( x ) ∞ (cid:88) n =0 i n n ! (cid:104) ( k · r ) n (cid:105) x . (A5)The convergence of the sum in eq. (A5) is guaranteed if the absolute magnitude of the n -thmoment of non-locality grows slower than n αn with α < n → ∞ , though the convergencerate depends on x and k , reflecting the detail for the non-locality of V ( x , x + r ). It is clearthat the sum converges as k →
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