Identification of shallow two-body bound states in finite volume
aa r X i v : . [ h e p - l a t ] S e p Identification of shallow two-body bound states infinite volume
Shoichi Sasaki ∗ † Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, JAPANE-mail: [email protected]
Takeshi Yamazaki
Physics Department, University of Connecticut, Storrs, CT 06269-3046, USAE-mail: [email protected]
We discuss signatures of bound-state formation in finite volume via the L¨uscher finite size method.Assuming that the phase-shift formula in this method inherits all aspects of the quantum scatteringtheory, we may expect that the bound-state formation induces the sign of the scattering length tobe changed. If it were true, this fact provides us a distinctive identification of a shallow bound stateeven in finite volume through determination of whether the second lowest energy state appearsjust above the threshold. We also consider the bound-state pole condition in finite volume, basedon L¨uscher’s phase-shift formula and then find that the condition is fulfilled only in the infinitevolume limit, but its modification by finite size corrections is exponentially suppressed by thespatial lattice size L . These theoretical considerations are also numerically checked through latticesimulations to calculate the positronium spectrum in compact scalar QED, where the short-rangeinteraction between an electron and a positron is realized in the Higgs phase. The XXV International Symposium on Lattice Field TheoryJuly 30 - August 4 2007Regensburg, Germany ∗ Speaker. † The results of calculations were performed by using of RIKEN Super Combined Cluster (RSCC). S.S. is supportedby the JSPS for a Grant-in-Aid for Scientific Research (C) (No. 19540265). c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ dentification of shallow two-body bound states in finite volume
Shoichi Sasaki
1. Introduction
Signatures of bound-state formation in finite volume are of main interest in this paper. In theinfinite volume, the bound state is well defined since there is no continuum state below threshold.However, in a finite box on the lattice, all states have discrete energies. Even worse, the lowestenergy level of the elastic scattering state appears below threshold in the case if an interaction isattractive between two particles [1]. Therefore, there is an ambiguity to distinguish between theshallow (near-threshold) bound state and the lowest scattering state in finite volume in this sense.We may begin with a naive question: what is the legitimate definition of the shallow boundstate in the quantum mechanics? In the scattering theory, poles of the S -matrix or the scatteringamplitude correspond to bound states [2]. It is also known that the appearance of the S -wave boundstate is accompanied by an abrupt sign change of the S -wave scattering length [2]. It is interpretedthat formation of one bound-state raises the phase shift at threshold by p . This particular featureis generalized as Levinson’s theorem [2]. Thus, it is interesting to consider how the formationcondition of bound states is implemented in L ¨uscher’s finite size method, which is proposed as ageneral method for computing low-energy scattering phases of two particles in finite volume [1].In this paper, we discuss bound-state formation on the basis of the L ¨uscher’s phase-shift for-mula and then present our proposal for identifying the shallow bound state in finite volume. Toexhibit the validity and efficiency of our proposal, we perform numerical studies of the positron-ium spectroscopy in compact scalar QED model. In the Higgs phase of U ( ) gauge dynamics, thephoton is massive. Then, massive photons give rise to the short-ranged interparticle force betweenan electron and a positron, which is exponentially damped. In this model, we can control positro-nium formation in variation with the strength of the interparticle force and then explore distinctivesignatures of the bound-state formation in finite volume. The contents of this paper are based onour published work [3].
2. Bound-state formation in L ¨uscher’s formula
In quantum scattering theory, the formation condition of bound states is implemented as apole in the S -matrix or scattering amplitude. Here, an important question naturally arises as to howbound-state formation is studied through L ¨uscher’s phase-shift formula [1]. Intuitively, the polecondition of the S -matrix: S = e i d ( p ) = cot d ( p )+ i cot d ( p ) − i is expressed ascot d ( p ) = i , (2.1)which is satisfied at p = − g where positive real g represents the binding momentum. In fact, aswe will discuss in the following, such a condition is fulfilled only in the infinite volume. Howeverthe finite-volume corrections on this pole condition are exponentially suppressed by the size ofspatial extent L .It was shown by L ¨uscher that the S -wave phase shift d can be calculated by measuring therelative momentum of two particles p in a finite box L with a spatial size L through the relationtan d ( p ) = p / p q Z ( , q ) at q = Lp n / p , (2.2)2 dentification of shallow two-body bound states in finite volume Shoichi Sasaki where the generalized zeta function Z ( s , q ) ≡ √ p (cid:229) n ∈ Z ( n − q ) − s is defined through analyticcontinuation in s from the region s > / s = negative q , an exponentially convergentexpression of the zeta function Z ( s , q ) has been derived in Ref. [4]. For s =
1, it is given by Z ( , q ) = − p / p − q + (cid:229) n ∈ Z ′ p / √ n e − p √ − q n , (2.3)where (cid:229) ′ n ∈ Z means the summation without n = ( , , ) . We now insert Eq. (2.3) into Eq. (2.2) andthen obtain the following formula, which is mathematically equivalent to Eq. (2.2) for negative q :cot d ( p ) = i + p i (cid:229) n ∈ Z ′ p − q n e − p √ − q n . (2.4)The second term in the r.h.s. of Eq. (2.4) vanishes in the limit of q → − ¥ . It clearly indicatesthat negative infinite q is responsible for the bound-state formation. Therefore, in this limit, therelative momentum squared p approaches − g , which must be non-zero. Meanwhile, the negativeinfinite q turns out to be the infinite volume limit.According to the original paper [1], for negative q , we introduce the phase s ( k ) , which isdefined by an analytic continuation of d into the complex p plane through the relation tan s ( k ) = − i tan d ( p ) , where k = − ip . As a result, the bound-state pole condition in the infinite volumereads cot s ( g ) = − g [1]. Then, Eq. (2.4) can be rewritten in terms ofthe phase s aslim k → g cot s ( k ) = − + ¥ (cid:229) n = N n √ n L g e −√ n L g = − + L g h e − L g + O ( e −√ L g ) i , (2.5)where the factor N n is the number of integer vectors n ∈ Z with n = n . Therefore, it is found thatalthough a bound-state pole condition is fulfilled only in the infinite volume limit, its modificationby finite size corrections is exponentially suppressed by the spatial extent L in a finite box L . Wecan learn from Eq. (2.5) that “shallow bound states” are supposed to receive larger finite volumecorrections than those of “tightly bound states” since the expansion parameter is scaled by thebinding momentum g .
3. Novel view from Levinson’s theorem
If the S -wave scattering length a , which is defined through a = lim p → tan d ( p ) / p , is suffi-ciently smaller than the spatial size L , one can make a Taylor expansion of the phase-shift formula(2.2) around q =
0, and then obtain the asymptotic solution of Eq. (2.2). Under the condition p ≪ m where m represents the reduced mass of two particles, the solution is given by D E ≈ − p a m L (cid:20) + c a L + c (cid:16) a L (cid:17) (cid:21) + O ( L − ) , (3.1)which corresponds to the energy shift of the lowest scattering state from the threshold energy. Thecoefficients are c = − . c = . † Although it was pointed out how the bound-state pole condition could be implemented in his phase-shift formulain the original paper [1], this important fact has been firstly reported in Ref. [3]. dentification of shallow two-body bound states in finite volume Shoichi Sasaki
Eq. (3.1). The lowest energy level of the elastic scattering state appears below threshold ( D E < a >
0) between two particles. This point makesit difficult to distinguish between near-threshold bound states and scattering states on the lattice.Here, a crucial question arises: once the S -wave bound states are formed, what is the fate ofthe lowest S -wave scattering state? The answer to this question might provide a hint to resolveour main issue of how to distinguish between “shallow bound states” and scattering states. Anaive expectation from Levinson’s theorem in quantum mechanics is that the energy shift relativeto a threshold turns out to be opposite in comparison to the case where there is no bound state.Levinson’s theorem relates the elastic scattering phase shift d l for the l -th partial wave at zerorelative momentum to the total number of bound states ( N l ) in a beautiful relation d l ( ) = N l p † .Therefore, if an S -wave bound state is formed in a given channel, the S -wave scattering phaseshift should always be positive at low energies. This positiveness of the scattering phase shiftis consistent with a consequence of the attractive interaction. Conversely, the S -wave scatteringlength may become negative ( a < above threshold . Therefore, the spectra of the scatteringstates quite resembles the one in the case of the repulsive interaction . If it were true, we can observea significant difference in spectra above the threshold between the two systems: one has at leastone bound state (bound system) and the other has no bound state (unbound system).
4. Numerical results
To explore signatures of bound-state formation on the lattice, we consider a bound state(positronium) between an electron and a positron in the compact QED with scalar matter [3]: S SQED [ U , F , Y ] = b (cid:229) plaq . (cid:2) − ´ { U x , mn } (cid:3) − h (cid:229) link ´ { F ∗ x U x , m F x + m } + (cid:229) sites Y x D W [ U ] x , y Y y , (4.1)where b = / e and the constraint | F x | = U ( ) gauge theory coupled to both scalar matter (Higgs) fields F and fermion (electron) fields Y .In this study, we treat the fermion fields in the quenched approximation.Our purpose is to study the S -wave bound state and scattering states through L ¨uscher’s finitesize method, which is only applied to the short-ranged interaction case. Thus, we fix b = . h = . U ( ) -Higgs action to simulate the Higgs phase of U ( ) gauge dynamics,where massive photons give rise to the short-ranged interparticle force between an electron anda positron. We generate U ( ) gauge configurations with a parameter set, ( b , h ) = ( . , . ) , on L ×
32 lattices with several spatial sizes, L = , , , ,
28 and 32. Details of our simulationsare found in Ref. [3].Once the parameters of the compact U ( ) -Higgs part, ( b , h ) , are fixed, the strength of aninterparticle force between electrons should be frozen on given gauge configurations. However,if we consider the fictitious Q -charged electron, the interparticle force can be controlled by this † Strictly speaking, this form is only valid unless zero-energy resonances exist. dentification of shallow two-body bound states in finite volume Shoichi Sasaki t ground1st excited2nd excited Q=3 L=28 t ground1st excited2nd excited Q=4 L=28
Figure 1:
The effective mass plots for each eigenvalue of the transfer matrix in the S channel on the latticewith L =
28. Full circles, squares and diamonds represent the ground state, the first excited state and thesecond excited state. The left (right) panel is for Q = Q = charge Q since the interparticle force is proportional to (charge Q ) . Within the quenched ap-proximation, this trick of the Q -charged electron is easily implemented by replacing U ( ) linkfields as U x , m −→ U Qx , m = P Qi = U x , m into the Wilson-Dirac matrix [3]. According to our previouspilot study [5], numerical simulations are performed with two parameter sets for fermion (electron)fields, ( Q , k ) =(3, 0.1639) and (4, 0.2222). As we will see later, the former case ( Q =
3) corre-sponds to the unbound system , while the latter case ( Q =
4) corresponds to the bound system wherethe positronium state can be formed. Here k , which is the hopping parameter of the Wilson-Diracmatrix, is adjusted to yield almost the same electron masses M e ≈ . S and S states of the e − e + system, where the electron-positron bound state (positronium) could be formed even in the Higgs phase. S and S positro-nium are described by the bilinear pseudo-scalar operator Y x g Y x and vector operator Y x g m Y x respectively. Therefore, we may construct the four-point functions of electron-positron statesbased on the above operators. We are interested in not only the lowest level of two-particlespectra, but also the 2nd and 3rd lowest levels. In order to extract a few low-lying energy lev-els of two-particle system, we utilize the diagonalization method [6]. We consider three typesof operators for this purpose: W P ( t ) = L − (cid:229) x Y ( x , t ) GY ( x , t ) , W W ( t ) = L − (cid:229) x , y Y ( y , t ) GY ( x , t ) and W M ( t ) = L − (cid:229) x , y Y ( y , t ) GY ( x , t ) e i p · ( x − y ) where p = p L ( , , ) and G = g ( g m ) for the S ( S ) e − e + state. We construct the 3 × G i j ( t ) = h | W i ( t ) W † j ( ) | i and then employ a diagonalization of a transfer matrix. As shown in Fig. 1, thediagonalization method with our chosen three operators successfully separates the first excited stateand the second excited state from the ground state [3]. In Figs. 2, we show energies of the ground state and also excited states in the e − e + system as afunction of spatial lattice size L . The dashed lines and curves represent the threshold energies 2 M e and 2 E e ( p ) , which are evaluated by measured energies of the single electron with zero momentum p = p L ( , , ) and nonzero lowest momenta p = p L ( , , ) respectively. Two left panels are forthe S channels, while the right panel is for the S channel.5 dentification of shallow two-body bound states in finite volume Shoichi Sasaki L Q=3, S L Q=4, S L Q=4, S Figure 2:
Energies of the ground state and excited states in the S (the left and middle panels) and S (theright panel) channels of the e − e + system as functions of spatial lattice size. The left figure is for Q = Q = Let us focus on results in the S channel. The energy level of the ground state for Q = L -dependence toward thethreshold energy is observed as spatial size L increases. This is consistent with a behavior of thelowest scattering state predicted by Eq. (3.1) for the weakly attractive interaction without boundstates. On the other hand, we clearly see the presence of a bound state for Q =
4, which certainlyremains finite energy gap from the threshold even in the infinite-volume limit. The most strikingfeature is our observed L -dependence of the energy level of the second lowest state for ( Q = from above . The energy shift vanishes asthe spatial size L increases. Therefore, the second lowest energy state must be the lowest scatteringstate with the repulsivelike scattering length ( a < E e ( p ) for Q = Q = S channel for Q =
4, the binding energy B is rather large as B ≈ M e /
2. The observedbound state should be a “tightly bound state” rather than a “shallow bound state”. On the otherhand, we observe that the bound state in the S channel (the right panel) is much near the thresholdenergy. Although the S ground state lies too close to the threshold energy to be assured of bound-state formation, the distinctive signature of bound state is given by an information of the excitedstate spectra. The second lowest state appears just above the first threshold 2 M e , but far from thesecond threshold 2 E e ( p ) . Therefore, we can conclude: the S ground state should be the shallowbound state, of which formation clearly induces the sign of the scattering length to change [3]. A rigorous way to test for bound-state formation would be to use an asymptotic formula forfinite volume correction to the pole condition as Eq. (2.5). In Figs.3, we plot the value of cot s versus the spatial lattice extent L for either S (left) and S (right) channels. Full circles aremeasured value at five different lattice volumes. At first glance, we observe that the phase cot s dentification of shallow two-body bound states in finite volume Shoichi Sasaki
10 15 20 25 30 35 40 L −1−0.998−0.996−0.994 c o t s ( k ) S Q=4
10 15 20 25 30 35 40 L −1−0.9−0.8−0.7−0.6−0.5−0.4 c o t s ( k ) S Q=4
Figure 3: cot s in the S (left) and S (right) channel for Q = L . gradually approaches − L increases for either channels.We next examine the L -dependence of cot s by reference to Eq. (2.5), where the finite volumecorrections on the bound-state pole condition are theoretically predicted. The solid and dashedcurves represent fit results with a single leading exponential term and three (six) exponential termsin the S ( S ) channel. All five data points are used for those fits in the S channel, while thefour data points in the region 20 ≤ L ≤
32 are used in the S channel. The fitting with the three(six) exponential terms yields a convergent result of g in the S ( S ) channel. Either fit curves inFigs. 3 reproduce all data points except for data at the smallest L in the S channel. Therefore, weconfirm that the ground state in the S channel at least for L ≥
20 can be identified as a shallowbound state without ambiguity.
5. Summary and conclusion
In this paper, we have discussed formation of an S-wave bound-state in finite volume on thebasis of L ¨uscher’s phase-shift formula. We have first showed that although a bound-state polecondition is fulfilled only in the infinite volume limit, its modification by the finite size correctionsis exponentially suppressed by the spatial extent L in a finite box L . We have also confirmed thatthe appearance of the S-wave bound state is accompanied by an abrupt sign change of the S-wavescattering length even in finite volume through numerical simulations. This distinctive behaviormay help us to discriminate the shallow bound state from the lowest energy level of the scatteringstate in finite volume simulations. References [1] M. L¨uscher, Commun. Math. Phys. , 177 (1986), Nucl. Phys. B , 531 (1991).[2] R. G. Newton, “Scattering Theory of Waves and Particles”, 2nd ed. (Springer, New York, 1982).[3] S. Sasaki and T. Yamazaki, Phys. Rev. D , 114507 (2006).[4] E. Elizalde, Commun. Math. Phys. , 83 (1998).[5] S. Sasaki and T. Yamazaki, PoS LAT2005 , 061 (2006).[6] M. L¨uscher and U. Wolff, Nucl. Phys. B , 222 (1990)., 222 (1990).