Rummukainen-Gottlieb's formula on two-particle system with different mass
aa r X i v : . [ h e p - l a t ] J un Rummukainen-Gottlieb’s formula on two-particle system with different mass
Ziwen Fu
Key Laboratory of Radiation Physics and Technology (Sichuan University) , Ministry of Education;Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, P. R. China.
L¨uscher established a non-perturbative formula to extract the elastic scattering phases from two-particle energy spectrum in a torus using lattice simulations. Rummukainen and Gottlieb furtherextend it to the moving frame, which is devoted to the system of two identical particles. In thiswork, we generalize Rummukainen-Gottlieb’s formula to the generic two-particle system where twoparticles are explicitly distinguishable, namely, the masses of the two particles are different. Thefinite size formula are achieved for both C v and C v symmetries. Our analytical results will bevery helpful for the study of some resonances, such as kappa, vector kaon, and so on. PACS numbers: 12.38.Gc, 11.15.Ha
I. INTRODUCTION
Many low energy hadrons, such as kappa, sigma, canobserved as resonances in the experiments. The energyeigenvalues of two-particle systems can be achieved bycalculating the propagators using lattice QCD. Hence, itis highly desired to connect these calculated energy eigen-states to the experimental scattering phases. L¨uscher [1–5] have established a non-perturbative formula to connecttwo-particle system’s energy in a torus with the elasticscattering phase. Rummukainen and Gottlieb further ex-tended L¨uscher’s formula to the moving frame (MF) [6].Moreover, Xu Feng et al generalized L¨uscher’s formulain an asymmetric box [7]. These formulae have been em-ployed in many different applications [8–19].For some cases, we have to use the generic two-particlesystem to extract the resonance parameters in the mov-ing frame. However, all of these aforementioned formu-lae in the moving frame can apply only to two identi-cal particle system. For example, to examine the be-havior of the κ resonance, it is highly desired for usto investigate πK scattering in the moving frame. Fora generic two-particle system in the moving frame, weshould change the Rummukainen-Gottlieb’s formulae,which is devoted to the system of two identical particles.To this purpose, we will strictly derive the equivalents ofthe Rummukainen-Gottlieb’s formulae for a generic two-particle system in the moving frame not only from the-oretical aspects, but also from practical considerations.This is very helpful for the lattice study since it providesa feasible method in the study of the κ decay, vector kaon K ∗ decay, and so on.The alterations of the Rummukainen-Gottlieb’s formu-lae are mainly relevant with the different symmetries oftwo-particle system in a torus . The representationsof the rotational group O ( h ) are decoupled into irre- It is Naruhito Ishizuka who first help us to find the correct sym-metry of the two-particle system with unequal mass. In there, weespecially thank him. Without his kind help, we can not finishthis work smoothly. ducible representations of the D h and D h cubic groupsfor the system of two identical particles with the non-zero total momentum in a torus [6]. In a generic two-particle system, the symmetry is reduced. For the caseof d = (0 , , C v instead of D h ; Asfor d = (0 , , C v . Hence,the final finite size formula for the generic two-particlesystem in the moving frame is certainly new.This paper is organized as follows. In Sec. II, wediscuss the basic properties of the generic two-particlestates in a torus. In Sec. III, we extend Rummukainen-Gottlieb’s formulae to the generic case and derive thefundamental relationship for the scattering phase inEq. (11), and in Sec. IV we present the symmetry consid-erations. Finally we give our brief conclusions in Sec. V. II. GENERIC TWO-PARTICLE SYSTEM ON ACUBIC BOX
Here we derive the formalisms required for calculatingthe scattering phases in a cubic box, which are enoughfor studying lattice simulation data. However, in con-crete lattice calculation, we should address the latticeartifacts [20]. In this section we follow the Rummukainenand Gottlieb’s notations and conventions [6], and extendthem to the generic two-particle systems.Without loss of generality, we consider two particleswith masses m and m for particle 1 and particle 2,respectively. In this work we are specially interested ina system having a non-zero total momentum, namely,the moving frame [6]. Using a moving frame with totalmomentum P = (2 π/L ) d , d ∈ Z , the energy eigenvaluesfor two-particle system in the non-interacting case aregiven by [6] E MF = q m + p + q m + p , where p = | p | , p = | p | , and p , p denote the three-momenta of the particle 1 and particle 2, respectively,which obey the periodic boundary condition, p i = 2 πL n i , n i ∈ Z , and the relation p + p = P . (1)In the center-of-mass (CM) frame, the total CM mo-mentum disappears, namely, p ∗ = | p ∗ | , p ∗ = p ∗ = − p ∗ , where p ∗ = (2 π/L ) n , and n ∈ Z . Here and hereafter wedefine the CM momenta with an asterisk ( ∗ ). The possi-ble energy eigenstates of the generic two-particle systemare given by E CM = q m + p ∗ + q m + p ∗ . The relativistic four-momentum squared is invariant,and E CM is related to E MF in the moving frame throughthe Lorentz transformation E CM = E MF − P . In the moving frame, the center-of-mass is moving witha velocity of v = P /E MF . Using the standard Lorentztransformation with a boost factor γ = 1 / √ − v , the E CM can be obtained through E CM = γ − E MF , andmomenta p i and p ∗ are related by the standard Lorentztransformation, p = ~γ ( p ∗ + v E ∗ ) , p = − ~γ ( p ∗ − v E ∗ ) , (2)where E ∗ and E ∗ are energy eigenvalues of the particle 1and particle 2 in the center-of-mass frame, respectively, E ∗ = 12 E CM (cid:0) E CM + m − m (cid:1) ,E ∗ = 12 E CM (cid:0) E CM + m − m (cid:1) , (3)and the boost factor acts in the direction of v , here andhereafter we adopt the shorthand notation ~γ p = γ p k + p ⊥ , ~γ − p = γ − p k + p ⊥ , (4)where p k and p ⊥ are the p components which are par-allel and perpendicular to the CM velocity, respectively,namely, p k = p · v | v | v , p ⊥ = p − p k . (5)Thus, by inspecting Eqs. (1), (2) and (3), it can be seenthat the p ∗ are quantized to the values p ∗ = 2 πL r , r ∈ P d , where the set P d is P d = (cid:26) r (cid:12)(cid:12)(cid:12)(cid:12) r = ~γ − (cid:20) n + d · (cid:18) m − m E CM (cid:19)(cid:21) , n ∈ Z (cid:27) . (6) In the interacting case, the E CM is given by E CM = q m + k + q m + k , k = 2 πL q. (7)where q is no longer required to be an integer. Solvingthis equation for scattering momentum k we get k = 12 E CM q [ E CM − ( m − m ) ][ E CM − ( m + m ) ] . (8)We can rewrite Eq. (8) to an elegant form as k = E CM m − m ) E CM − m + m . (9)It is exactly this energy shift between the non-interacting situation and interacting case, namely, E CM − E CM (or equivalently | n | − q ), that we canevaluate two particle scattering phase.As it is done in Ref. [6], in the current study, we mainlyinvestigate two moving frames. One is d = (0 , , C v , only the irreducible representation A is rele-vant for two-particle scattering states in a cubic box withangular momentum l = 0. Another one is d = (0 , , C v , only the irreducible representation A is rel-evant. For the other cases, like d = (1 , , δ l with l = 1 , , , . . . are tiny in our concerned energy range, the s -wave phaseshift δ is connected to the scattering momentum k bytan δ ( k ) = γπ / q Z d (1; q ) , (10)where k = (2 π/L ) q , and the modified zeta function isdefined by Z d ( s ; q ) = X r ∈ P d | r | − q ) s , (11)and the set P d is P d = (cid:26) r (cid:12)(cid:12)(cid:12)(cid:12) r = ~γ − (cid:20) n + d · (cid:18) m − m E CM (cid:19)(cid:21) , n ∈ Z (cid:27) . (12)For Eq. (10), we note that the almost same result hasalready existed in Eq. (1) of Ref. [21], where the for-mula was just presented without any explanation. Wecan view our work as further confirming and strictlyproving this formula. The modified zeta function con-verges when Re 2 s > l + 3, and it can be analyticallycontinued to whole complex plane. The scattering mo-mentum k is defined from the invariant mass √ s as √ s = E CM = p k + m + p k + m . The calculationmethod of Z d (1; q ) is discussed in Appendix A and inRef. [20]. Using Eq. (10), we can obtain the phase shiftfrom the energy spectrum using lattice simulations. If wenow set m = m , all the results in Ref. [6] are elegantlyrecovered. III. DERIVATION OF THE PHASE SHIFTFORMULA
Here we deduce the basic phase shift formula inEq. (10) for the generic two-particle system with spin-0. We utilize the Rummukainen and Gottlieb’s formulaein Ref. [6], and extend them to the generic two-particlesystem. To make the derivation simple, we are studyingthe system by the relativistic quantum mechanics.Throughout this section, we employ the metric ten-sor sign convention g µν = diag(1 , − , − , − p = p · p = p µ p µ ,etc, and express the quantities in natural units with ~ = c = 1. Here and hereafter we follow the originalnotations in Refs. [6]. A. Lorentz transformation of wave function
Let us consider the generic system of two spin-0 parti-cles with mass m , and m , respectively, in a cubic box.The two-particle system is described by the scalar wavefunction ψ ( x , x ), where x i = ( x i , x i ) , i = 1 , ψ ( x , x ) = ψ ′ ( x ′ , x ′ ) = ψ ′ (Λ x , Λ x ) , (13)where ( x ′ ) µ = Λ µν x ν defines the standard Lorentz trans-formation of the four-vector x .We can make the problem easier through the specialproperties of the CM frame. First let us study two non-interacting particles, and the wave functions of the sys-tem obey the Klein-Gordon (KG) equations (cid:0) ˆ p µ ˆ p µ − m (cid:1) ψ ( x , x ) = 0 , (cid:0) ˆ p µ ˆ p µ − m (cid:1) ψ ( x , x ) = 0 , (14)where ˆ p iµ , i = 1 , X = m x + m x m + m , (15) x = x − x , (16)where X is the position of the CM, and x is the relativecoordinate of two particles. Let us restrict ourselves tothe solutions which are the eigenstates of the CM mo-mentum operator. Then Eq. (14) can be changed into (cid:20) m M ˆ P µ ˆ P µ + ˆ p µ ˆ p µ − m M ˆ p µ · ˆ P µ + m (cid:21) ψ ( x, X )=0 , (17) (cid:20) m M ˆ P µ ˆ P µ + ˆ p µ ˆ p µ + 2 m M ˆ p µ · ˆ P µ + m (cid:21) ψ ( x, X )=0 , (18) where ˆ p = m ˆ p − m ˆ p m + m , (19)ˆ P = ˆ p + ˆ p , (20) M = m + m , (21)ˆ p is relative four-momentum operator, ˆ P is total four-momentum operator, and M is total mass of two parti-cles. Adding 1 /m × (17) to 1 /m × (18) and subtracting(17) from (18), respectively, yield (cid:20) M m m ˆ p µ ˆ p µ − M + ˆ P µ ˆ P µ (cid:21) ψ ( x, X ) = 0 , (22) (cid:20) ˆ p µ ˆ P µ − m − m M ˆ P µ ˆ P µ − m − m (cid:21) ψ ( x, X ) = 0 . (23)It is well-known that, without external potentials, thetotal momentum of the two-particle system is conserved,thus we can restraint ourselves to the eigenfunctions of P , namely, ψ ( x, X ) = e − iP µ X µ φ ( x ) , (24)where P µ is a constant time-like vector, and P is denotedthrough P = P µ P µ .In the present study, we are specially interested in theCM frame, which is denoted as the frame without thespatial components of the total momentum for the sys-tem, namely, P ∗ = 0. Thus, we can only take the positivekinetic energy solutions P ∗ = E ∗ CM > m + m into con-sideration. So, Eqs. (22) and (23) can be rewritten as, (cid:18) ˆ p ∗ µ ˆ p ∗ µ + E m m ( m + m ) − m m (cid:19) φ CM ( x ∗ ) = 0 , (25) (cid:18) ˆ p ∗ − E CM m − m m + m − m − m E CM (cid:19) φ CM ( x ∗ ) = 0 . (26)Eq. (26) indicates ˆ p ∗ φ CM ( x ∗ ) = 0 for m = m . By in-specting Eq. (25) and Eq. (26), we can reasonably assumethat the wave function φ CM ( x ∗ ) can be expressed as φ CM ( x ∗ ) ≡ e iβx ∗ φ CM ( x ∗ ) , (27)where x ∗ = x ∗ − x ∗ is the relative temporal separationof two particles, and β is a constant, namely, β = E CM m − m m + m + m − m E CM . (28)It is obvious that when m = m , β →
0. So, in the CMframe the wave function depends explicitly on the timevariable t ∗ ≡ X ∗ = ( m x ∗ + m x ∗ ) / ( m + m ), therelative spatial separation x ∗ = x ∗ − x ∗ , and the relativetemporal separation of the particles x ∗ , namely, ψ CM ( x ∗ , t ∗ ) = e − iE CM t ∗ e iβx ∗ φ CM ( x ∗ ) , (29)where the constant β is denoted in Eq. (28).Now let us discuss the case in the moving frame. Thetransformation from the MF to CM frame can be ex-pressed as r ∗ µ = Λ µν r ν , where r is any position four-vector and quantities without ∗ stand for these in themoving frame. Using the notation in (4), we have r ∗ = γ ( r + v · r ) , r ∗ = ~γ ( r + v r ) , (30)where γ is a boost factor, and v = P /P is the three-velocity of the CM in the moving frame. We can rewrite v to a form for later use as v = 2 πLE MF d = 2 πγLE CM d . (31)Considering the identity P µ X µ = P ∗ µ X ∗ µ , the Lorentztransformation in Eq. (13), and Eq. (24), the wave func-tion in the moving frame can be expressed as ψ MF ( x, X ) = e − iP µ X µ φ MF ( x ) , where φ MF ( x ) ≡ φ MF ( x , x ) = φ CM (cid:0) γ ( x + v · x ) , ~γ ( x + v x ) (cid:1) . (32)Thus, the wave function φ MF depends on time separa-tion x = x − x explicitly. However, in the movingframe we only consider two particles with the same timecoordinate, namely, x = 0. It corresponds to the tiltedplane ( x ∗ , x ∗ ) = ( γ v · x , ~γ x ) for the CM frame, since thewave function φ CM is dependent of the relative temporalseparation x ∗ , we can clearly observe the effect of thetilt to the wave function, and Eq. (32) take as φ MF (0 , x ) = φ CM ( γ v · x , ~γ x ) . (33)Using Eq. (27) and Eq. (31), we can rewrite Eq. (33) as φ MF (0 , x ) = e iβ ′ π d · x /L φ CM ( ~γ x ) . (34)where β ′ is a constant, namely, β ′ = m − m m + m + m − m E . (35)Eq. (34) has a simple physical meaning: the CM systemwatches the torus in the moving frame expanded by aboost factor γ in the direction of P , and the length scalesin perpendicular directions are hold. At last, Eq. (34)connects the MF wave function, ψ MF (0 , x , t, X ) = e − iE MF t + i P · X φ MF (0 , x ) , (36)to the CM frame wave function Eq. (29). By inspectingEqs. (25), (26) and (29), we, at last, achieve the wavefunction φ CM meeting the Helmholtz equation (HE)( ∇ x ∗ + k ∗ ) φ CM ( x ∗ ) = 0 , (37)where k ∗ = E CM m − m ) E CM − m + m . (38) This result is consistent with the solution in Ref. [22].The Eqs. (34) and (37) are very important when westudy the wave functions of our system. Thus Eq. (37) isa very important result, which represents one of the mainresults of the present work. In the following studies, wetake away the superscript ∗ from the quantities in the CMframe. We can easily check that if we take m = m , allthe corresponding results in Ref. [6] are restored. B. Modified singular d-periodic solutions of theHelmholtz equation
In our concrete problem, for the potential V µ ( x ) witha limit range [2], namely, V µ ( x ) = 0 for | x | > R, (39)we suppose that the KG equation (14) in the CM framestill has a square integral solution. In the CM frame theinteraction of the system is spherically symmetric. Thewave function of the system is usually given in sphericalharmonics φ CM ( x ) = ∞ X l =0 l X m = − l Y lm ( θ, ϕ ) φ lm ( x ) . (40)For x > R , φ CM is a solution of HE, and the radialfunctions φ lm meet the differential equation (cid:20) d d x + 2 x dd x − l ( l + 1) x + k (cid:21) φ lm ( x ) = 0 , (41)where k = E CM m − m ) E CM − m + m . (42)With the linear combinations of the spherical Bessel func-tions the solutions of Eq. (41) can be expressed as φ lm ( x ) = c lm [ a l ( k ) j l ( kx ) + b l ( k ) n l ( kx )] . (43)Although we do not know the radial equation in theregion x < R , by comparing the wave functions denotedin Eqs. (40,43), we can get the relation between the scat-tering phase and the coefficients a l and b l [2], namely, e i δ l ( k ) = a l ( k ) + ib l ( k ) a l ( k ) − ib l ( k ) . Because the radial equation a l and b l can be taken to bethe real number for k > δ l ( k ) is real. Thus, for a given l -sector, the phase shift δ l ( k ) can be expressed by energy E MF through k = E − P m − m ) E − P − m + m . In the moving frame, we now investigate two genericparticles confined in a torus of size L with periodicboundary conditions(PBC). The temporal direction ofthe torus is chosen to be infinite. The wave functions ψ MF in MF should be periodic with regard to the posi-tion of each particle, namely, ψ MF ( x , x ) = ψ MF ( x + l L, x + m L ) , l , m ∈ Z . (44)The wave function ψ MF is provided by (36), namely ψ MF ( x , x ) = exp (cid:18) i P · ( m x + m x ) m + m (cid:19) φ MF ( x − x ) . (45)Combining Eq. (44) and Eq. (45) gives P = 2 πL d , (46) φ MF ( x ) = e iπ m m m d · n φ MF ( x + n L ) , (47)where n = l − m , d , n ∈ Z and P is the total mo-mentum . The rule (47) separates the wave functionsinto the different sectors, which can be categorized bythe three-vector d . In the current study, we naturallyconcentrate on sectors d = (0 , ,
1) and d = (0 , , d , we have φ CM ( x ) = e iπ d · n (cid:18) m − m E (cid:19) φ CM ( x + ~γ n L ) , n ∈ Z . (48)For simpleness, we refer to the functions obeying the pe-riodicity rule (48) as modified d -periodic rule. As shownlater, the modified d -periodic rule (48) is a milestone inthis work.Now we deduce the general solutions of the HE satisfy-ing the modified d -periodic rule (48). Except the modi-fied d -periodicity, our derivation follows the works in sec-tion 4 . φ as a singular modified d -periodic solution of the HE, ifit is a smooth function denoted for all x = ~γ n L , n ∈ Z ,and it meets the HE, namely,( ∇ + k ) φ ( x ) = 0 , for k >
0, and meets the modified d -periodic rule,i.e., φ ( x ) = e iαπ d · n φ ( x + ~γ n L ) , n ∈ Z , (49)here and hereafter, for compactness, we defined a factor α , namely, α ≡ m − m E CM . (50) Eq. (47) can also be written as φ MF ( x ) = e − iπ m m m d · n φ MF ( x + n L ) . Following the almost same procedures and addressing the cor-responding formulae, we can arrive at the same final numericalresults. For our case, d · n is an integer. When m = m , α = 1, this is the case Rummukainenand Gottlieb studied in Ref. [6]. Moreover, the wavefunction is required to be bounded by a power of 1 / | x | around the origin, namelylim x → | x Λ+1 φ ( x ) | < ∞ for the positive integer Λ, the degree of φ . For our aim,it is enough to study the regular values of k , namely k = 2 πL (cid:12)(cid:12)(cid:12) ~γ − (cid:16) n + α d (cid:17)(cid:12)(cid:12)(cid:12) , n ∈ Z . We denote the Green function G d ( x ; k ) = γ − L − X p ∈ Γ e i p · x p − k , (51)whereΓ = (cid:26) p ∈ R (cid:12)(cid:12)(cid:12) p = 2 πL ~γ − (cid:16) n + α d (cid:17) , n ∈ Z (cid:27) . Eq. (51) is well-defined due to the non-singular k . If weselect k = (2 π/L ) ~γ − (cid:0) m + α d (cid:1) , m ∈ Z , then k · ( x + ~γ n L ) = k · x + απ d · n + 2 π m · n , where n ∈ Z , and the Green function G d ( x ; k ) meetsclearly the modified d -periodic rule, as we expected. Fur-thermore, it obeys( ∇ + k ) G d ( x ; k ) = − X n ∈ Z e iαπ d · n δ ( x + ~γ n L ) . We can easily verify that the function G d ( x ; k ) is a sin-gular modified d -periodic solution of Helmholtz equationwith degree 1. Other singular periodic solutions can beeasily achieved by differentiating G d with x . And G d lm ( x ; k ) = Y lm ( ∇ ) G d ( x ; k ) , (52)where the harmonic polynomials Y lm ( x ) = x l Y lm ( θ, ϕ ).We can easily verify that the functions G d lm are singu-lar modified d -periodic solutions of the HE, and theymake up a complete set of solutions so that any singularmodified d -periodic solution of degree Λ is just a linearcombination of G d lm ( x ; p ) for l ≤ Λ [2]. In the region0 < x < L/ G d lm can be expanded in usualspherical harmonics, namely G d lm ( x ; k ) = ( − l k l +1 π [ n l ( kx ) Y lm ( θ, ϕ ) + ∞ X l ′ =0 l X m ′ = − l M d lm,l ′ m ′ ( k ) j l ′ ( kx ) Y l ′ m ′ ( θ, ϕ ) . (53)The regular part has the coefficients M d lm,l ′ m ′ ( k ). In theconcrete calculation, only a few of them is needed, andwe also provide its general expression, namely, M d lm,l ′ m ′ ( k ) = ( − l γπ / l + l ′ X j = | l − l ′ | j X s = − j i j q j +1 Z d js (1; q ) C lm,js,l ′ m ′ , (54)where q = kL/ (2 π ). Using Wigner 3 j -symbols [23] thetensor C lm,js,l ′ m ′ can be expressed as C lm,js,l ′ m ′ = ( − m ′ i l − j + l ′ p (2 l + 1)(2 j + 1)(2 l ′ + 1) × (cid:18) l j l ′ m s − m ′ (cid:19) (cid:18) l j l ′ (cid:19) . (55)The modified zeta function is formally denoted by Z d lm ( s ; q ) = X r ∈ P d Y lm ( r )( r − q ) s , (56)where P d = n r ∈ R (cid:12)(cid:12)(cid:12) r = ~γ − (cid:16) n + α d (cid:17) , n ∈ Z o . In Table I we summarized the expressions of M d lm,l ′ m ′ for l, l ′ ≤
3. For compactness, we denoted w lm = 1 π / √ l + 1 γ − q − l − Z d lm (1; q ) . (57)The Wigner 3 j -symbol values can be obtained inRef. [23]. We can easily verify that, if we set m = m ,all of the above definitions and formulae nicely reduce tothe those obtained in Ref. [6], as we expected. Of course,if we further select d = 0, the moving frame and the CMframe coincide, γ → P d → Z , and they furtherneatly reduce to that in Ref. [2]. The Table I can becompared with the Table 3 in Ref. [6], which summariesthe matrix elements for m = m . We should stress thatthe functions w , w and w appear in Table I. If weset m = m , then w → w →
0, and w →
0, andRummukainen-Gottlieb’s results is elegantly restored.So far, we can arrive at Λ X l =0 l X m = − l v lm G d lm ( x , k ) = Λ X l =0 l X m = − l c lm [ a l ( k ) j l ( kx ) + b l ( k ) n l ( kx )] Y lm ( θ, ϕ ) (58)for the constants c lm and v lm . And we obtain c lm a l ( k ) = Λ X l ′ =0 l ′ X m ′ = − l ′ c l ′ m ′ b l ′ ( k ) M d l ′ m ′ ,lm ( k ) . (59) M l ′ m ′ ,lm can be viewed as the matrix element of the op-erator M . We can express Eq. (59) as a matrix equa-tion, C ( A − BM ) = 0 , where matrix A ( lm ) , ( l ′ m ′ ) = a l ( p ) δ l,l ′ δ m,m ′ (similar for B ). We can denote the phaseshift matrix [2, 6], e iδ = A + iBA − iB . The determinant condition arrives at [6]det (cid:2) e iδ ( M − i ) − ( M + i ) (cid:3) = 0 . (60) IV. SYMMETRY DISCUSSIONS
When the moving and CM frames coincide, the two-particle system has a cubic symmetry and the wave func-tions transforms under the representations of the cubicgroup O h . On the other hand, according to Eq. (34),if two frames are not equivalent, the Lorentz translationboost from the moving frame to the CM frame and onlysome subgroups of O h group survive [6].In this work, we are mainly interested in a boost alongone of the coordinate axes, namely d = (0 , , , , → (1 , , γ ), andthe corresponding symmetry group is tetragonal pointgroup C v , which has 8 elements: 4 rotations throughan angle ( nπ/ n = 0 , , ,
3, around the x -axis;and all four of the above multiplied by the reflection withrespect to the (1 , O h , C v , and C v . The tetrago-nal group C v has four 1-dimensional representations A , A , B , B , and one 2-dimensional representation E [24].The representations of the rotational group are reducedinto irreducible representations of C v as:Γ (0) = A , Γ (1) = A ⊕ E , (61)Γ (2) = A ⊕ B ⊕ B ⊕ E .
The representations can be obtained through the char-acter tables [24] or by enumerating harmonic polynomialsof degree l which transform under the representations of C v . The basis polynomials for the corresponding repre-sentations are summarized in Table III for l ≤ C v has four 1-dimensional rep-resentations A , A , B , B [24]. The representations ofthe rotational group are further reduced into irreduciblerepresentations of C v as:Γ (0) = A , Γ (1) = A ⊕ B ⊕ B , (62)Γ (2) = A ⊕ A ⊕ B ⊕ B . Similarly, we can obtain the basis polynomials for C v representation, which are listed in Table IV for l ≤ A sector. We will therefore concentrate on thissector. As is seen, up to l ≤ s -wave, p -wave and d -wavecontribute to this sector. The other symmetry sectors canbe easily worked out in the same way.First, let us consider the angular momentum cutoffΛ = 0. From the reduction relations (61,62) and Ta-bles III, IV, we note that only M d , belongs to thissector, and Eq. (60) can be written astan δ ( k ) = 1 M d , = γqπ / Z d (1; q ) , q = L π k. (63)This is our essential result. l m l ′ m ′ M d lm,l ′ m ′ w iw w +2 w w − w −√ w i q w i q w i q w i q w w + w + w w + w − w − √ w w − w + w − iw − √ w − √ w − √ w + √ w − √ w − i q w − i q w − i q w − i q w − i q w − i q w i q w i w i q w w + w + w + w w + w + w − w − √ w + √ w w − w + w − √ w − √ w w − w + w − w TABLE I: The matrix elements M d lm,l ′ m ′ for d = (0 , , , m = m and l, l ′ ≤ d point group classification N elements (0 , , O h cubic 48(0 , , a ) C v tetragonal 8(0 , a, a ) C v orthorhombic 4TABLE II: The categorizations of the boosts on a cubic boxand its reduction of the cubic group. The first column showsthe boost direction and a is a non-zero real number. Here weadopted the Schonflies notation [24]. If the angular momentum cutoff Λ = 1, we shouldtake the sector l = 1 into consideration, and the matrixin Eq. (60) is two-dimensional. Hence, the determinantcondition contains both phase shifts δ and δ , (cid:2) e iδ ( m − i ) − ( m + i ) (cid:3) (cid:2) e iδ ( m − i ) − ( m + i ) (cid:3) = m (cid:0) e iδ − (cid:1) (cid:0) e iδ − (cid:1) , (64)where we denote m ab ≡ M d a ,b . If δ = 0 (mod π ), aswhat we expected, Eq. (64) reduces nicely to Eq. (63).If δ = 0. Normally we can reasonably suppose that thelowest l -channel dominate the scattering phase. This is representation l = 0 l = 1 l = 2 indices A x x − x A B x − x B x x E x i x i x i = 1 , C v . quite right in the low-energy elastic scattering [2, 6].If we expand δ = δ + ∆ , where δ meets Eq. (63)and ∆ is a perturbative term, we can give the first ordercorrection, namely,∆ ( k ) = σ ( k ) δ ( k ) . The function σ ( k ) stands for the sensitivity of the higherscattering phases. For C v symmetry, it is given by σ ( k ) = − m m + 1 , representation l = 0 l = 1 l = 2 A x x − x A x x B x x x B x x x TABLE IV: The basis polynomials for the irreducible repre-sentations of C v . which is not usually small. To make sure that theEq. (63) is a well approximation, the phase shift δ ( k )(mod π ) should be small. Fortunately, the physicalcase is often like this way: the elastic scattering is oftendominated by the lowest l - channel.The sensitivity function σ ( q ) can be calculated usingthe matrix elements provided in Eq. (57) discussed in Ap-pendices A and B. The sensitivity function σ ( q ) for C v symmetry with α = 1 .
15 and γ = 1 .
177 is illustrated inFig. 1, here γ is a boost factor, and α -factor is defined inEq. (50), which are typical values we used in Ref. [20]. InFig. 1 the lower panel is the same sensitivity function asthe upper panel just to display some detailed variations. FIG. 1: The sensitivity σ ( q ) for C v symmetry with param-eters α = 1 .
15 and γ = 1 . In the work, we also calculate the sensitivity function σ ( q ) using the some typical α and γ values, we found that it often varies in the range 0 −
20, in Figs. 2, 3,and 4, we plotted just three of them. We can note thatthe sensitivity function σ ( q ) is finite for all q >
0. Forsome special values of q , however, the sensitivity func-tion σ ( q ) has a sharp peak. For other values of q awayfrom these values, σ ( p ) is always not too large.We also note that, when q →
0, the sensitivity function σ ( q ) is often large. However, it does not usually causeany problem because it is nicely canceled out by δ whichis of q order at small q [2, 6]. Thus, for the range 0 05 and γ = 1 . σ ( q ) for C v symmetry with param-eters α = 1 . γ = 1 . We should bear in mind that if δ ( k ) is not small, itis very difficult to extract the phase shift from energyspectrum [6]. In principle, we can still extract the s -wave scattering phase from Eq. (64) through dividing the p -wave phase shift by lattice simulations at various en-ergy [6], because the corrections due to higher scattering FIG. 4: The sensitivity σ ( q ) for C v symmetry with param-eters α = 1 . 15 and γ = 1 . phases can also be estimated from lattice calculations.For example, from Table III, it is easily noted that, forlattices with C v symmetry, by inspecting energy eigen-values of E symmetry, we can get an approximate esti-mate for the p -wave scattering phase δ which dominatesthis symmetry sector. It seems to be too difficult, how-ever naturally, it is still possible to calculate the energyspectrum, this is our future tasks.If we choose the sector d = 0, the moving andthe CM frames coincide, γ → P d → Z ,and Eq. (64) nicely reduces to the form given inRef. [2]. Of course, if we select m = m and P d → (cid:8) r ∈ R (cid:12)(cid:12) r = ~γ − ( n + d / (cid:9) , n ∈ Z , and Eq. (64)neatly reduces to the form presented in Ref. [6]. Theseare what we expected.As for Λ = 2 or higher, it is quite complicated. Seethe relevant discussions in Ref. [7]. Bearing in mind thatthis work is an exploratory study for some systems likethe πK system, the main purpose is to present some con-ceptual and theoretical issues. V. CONCLUSION In the current work we have strictly investigatedthe scattering states of two-particle with unequal mass,and the best-efforts are paid to derive the modified d -periodic rule which is crucial to the alteration of theRummukainen-Gottlieb’s formula. The finite size expres-sions, which can be regarded as a generalization of theRummukainen-Gottlieb’s formulae to the generic two-particle system in the moving frame, are developed. Wealso checked that all the Rummukainen-Gottlieb’s resultsin Ref. [6] is nicely restored if we set m = m .Since the so-called κ meson is a low-lying scalar mesonwith strangeness, a study of κ meson decay is an explicitexploration of the three-flavor structure of the low-energyhadronic interactions, which is not directly probed in ππ scattering, therefore, it is a significant step for us under- standing the dynamical aspect of hadron reactions withQCD. Moreover, BES collaboration recently carried outsome experimental measurements [25, 26] to investigate κ resonance mass and its decay width. With the modi-fied formula in Eq. (63) and our strict discussion of thisformula from theoretical aspects, now it will be possibleto compute the resonance masses and perhaps its de-cay widths of some resonances including possible exotichadrons as well as traditional hadrons like κ and vectorkaon K ∗ , etc., directly from lattice simulation in a correctmanner. We have already used these formulae to prelimi-narily analyze our πK scattering at I = 1 / Acknowledgments The author thanks Naruhito Ishizuka for kindly help-ing us about group symmetry, and we would also liketo thank Sasa Prelovsek, Carleton DeTar, and Martin J.Savage for their encouraging and enlightening comments. Appendix A: The calculation of zeta function The method for evaluating the zeta function for d = 0has been discussed by L¨uscher in Ref. [2]. Rummukainenand Gottlieb extended this discussion in the MF for d = 0 , α = 1 [6]. The formalism used here is furtheradapted to the case of d = 0 , α = 1, and we just presentthe essential formulae.We first denotes the heat kernel on a modified d -periodic torus in Eq. (48), namely, K d ( t, x ) = 1(2 π ) X r ∈ P d e i r · x − t r (A1)where the summation for r is carried out over the set P d = n r (cid:12)(cid:12)(cid:12) r = ~γ − (cid:16) n + α d (cid:17) , n ∈ Z o , (A2)here the factor α is denoted in Eq. (50), and the operation ~γ − is is defined in Eq. (4). Following from Poisson’sidentity, we can rewrite the heat kernel as K d ( t, x ) = γ πt ) e i / α d · x × X n ∈ Z e − iαπ d · n exp (cid:20) − t ( x − π~γ n ) (cid:21) . (A3)The expression in Eq. (A1) is fast convergent for large t ,and the expression in Eq. (A3) is useful for small t . Wecan denote the truncated heat kernel K λ d ( t, x ) by K λ d ( t, x ) = K d ( t, x ) − X r ∈ P d , | r | <λ exp( i r · x − t r ) . Y lm ( − i ∇ x ) to heat kernels, K λ d ,lm ( t, x ) = Y lm ( − i ∇ x ) K λ d ( t, x ) . (A4)We can easily show that the zeta function has a rapidlyconvergent integral expression Z d lm (1; q )= X r ∈ P d , | r | <λ Y lm ( r ) r − q +(2 π ) Z ∞ dt (cid:18) e tq K λ d ,lm ( t, ) − γδ l, δ m, π t / (cid:19) . (A5)To calculate the integrand, we use the Eq. (A1) when t ≥ 1, and the Eq. (A3) in the case of t < 1. The cutoff λ is chosen such that λ > Re q . We can easily ver-ify that, when m = m (or equivalently α = 1), theRummukainen-Gottlieb’s result in Ref. [6] is restored. Appendix B: The evaluation of the zeta function Z ( s ; q ) In this appendix we briefly discuss one useful methodfor numerical evaluation of zeta function Z ( s ; q ). Here we follow the methods and notations in Ref. [11].The definition of the zeta function Z d ( s ; q ) inEq. (56) is r π · Z d ( s ; q ) = X r ∈ P d r ( r − q ) s , (B1)where the summation for r is carried out over the set P d = n r (cid:12)(cid:12)(cid:12) r = ~γ − (cid:16) n + α d (cid:17) , n ∈ Z o , (B2)here the factor α is denoted in Eq. (50). The operation ~γ − is is defined in Eq. (4). We consider that the value q can be a positive or negative.First we consider the case of q > 0, and we can sepa-rate the summation in Z d ( s ; q ) into two parts as X r ∈ P d r ( r − q ) s = X r 0, it is not necessary for us toseparate the summation in Z ( s ; q ), and it can be alsowritten in an integral form. Following the same pro-cedures, we arrive at the same expression in Eq. (B6). Hence, Eq. (B6) can be applied for both cases.Substituting d = (0 , , 1) into Eq. (B6) we obtain thezeta function in Eq. (56) r π · Z d (1; q ) = X r ∈ P d r e − ( r − q ) r − q + √ π Z d t e tq (cid:16) πt (cid:17) X n ∈ Z n sin( απn ) e − ( π~γ n ) /t . 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q r ( r − q ) s . (B3)The second term can be written in an integral form, X r >q r ( r − q ) s = 1Γ( s ) X r >q r (cid:20)Z d t t s − e − t ( r − q ) + Z ∞ d t t s − e − t ( r − q ) (cid:21) = 1Γ( s ) Z d tt s − e q t X r r e − r t − X r