aa r X i v : . [ h e p - l a t ] J u l Myth of scattering in finite volume
Peng Guo
1, 2, ∗ Department of Physics and Engineering, California State University, Bakersfield, CA 93311, USA Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: July 10, 2020)In this notes, we illustrate why the infinite volume scattering amplitude is in fact dispensablewhen it comes to formulating few-body quantization condition in finite volume. Only subprocessinteractions or interactions associated subprocess amplitudes are essential and fundamental ingre-dients of quantization conditions. After these ingredients are determined, infinite volume scatteringamplitude can be computed separately. The underlying reasons are rooted in facts that (1) the finalphysical process is generated by all subprocess or interactions among particles; (2) the ultimate goalof quantization condition in finite volume is to find stationary solutions of few-body system. Thatis to say, in the end, it all comes down to the solving of eigenvalue problem, ˆ H | n i = E n | n i . I. INTRODUCTION
In past few years, much efforts [1–29] have been putinto the study of few-hadron scattering in finite volume,aiming to mapping out few-hadron scattering informa-tion from lattice QCD results. Such a program is moti-vated by the fact that lattice QCD computation is nor-mally performed in a periodic box in Euclidean space,the physical amplitude is usually not directly accessedand discrete energy spectrum is the primary observablesfrom lattice computation. All the dynamical informationis encoded in the discrete energy spectrum. Since latticeQCD is considered as ab-initio calculation of QCD, map-ping out few-hadron dynamical information from latticeenergy spectrum may be useful for number of reasons,such as, helping to understand the nature of some res-onances and determine the fundamental parameters ofQCD or QCD inspired effective theory.Many good progresses have been made in past fewyears toward this direction, a couple of things have be-come quite clear:(1) Such an effort can be accomplished, however, it isnormally done not in a direct way. In another word, ex-cept in two-body case, few-body scattering amplitude isin fact not directly computed from lattice results. Onlyingredients are extracted directly, such as interaction po-tentials, two-body scattering amplitudes, or K -matrixcomponents, depending on a specific approach. The scat-tering or decay amplitudes have to be computed in a sep-arate step by assembling all these ingredients together.Metaphorically speaking, lattice QCD is like IKEA storein modern days, only components of a furniture are of-fered, and assembling has to be carried out separatelyin order to see the whole picture. One of many rea-sons for this situation is that unlike two-body scatteringamplitude that may be parameterized by a set of pa-rameters, such as phase shifts, a simple analytic formof parametrization of few-body scattering amplitudes isusually not available. Hence, scattering amplitudes or ∗ [email protected] decay amplitudes conventionally have to be computedthough a set of coupled integral equations with few fun-damental ingredients of interactions as kernels. One typ-ical example is Faddeev equations approach [30–32] byusing two-body scattering amplitudes as central ingredi-ents which may be parameterized in a relative easier way.This can also be understood by the fact that all complexphysical processes are generated by subprocesses, in thecase of few-body scattering, few-body scattering ampli-tudes are connected to subprocess by integral equations.(2) Essentially, all different groups are more or less do-ing the same thing with different twist. Two steps pro-cedures are adopted. Step one, formulating quantizationcondition and extracting central ingredients by fitting lat-tice data. Step two, computing infinite volume scatter-ing amplitude. The choice of central ingredients, suchas, interaction parameters of effective theory, parame-ters of K -matrix, or interaction potentials, etc, and howquantization conditions are formulated differ from groupto group. Ultimately, these central ingredients must bemodeled one way or another in order to fit lattice data.(3) When it comes to the quantization condition infinite volume, in spite of how one choose to formulatequantization condition, in the end, it is all about findingstationary solutions of few-body system. Stationary solu-tions can be obtained equivalently by either finding polepositions of scattering amplitude or solving homogeneousFaddeev type equations directly. To put it simply, it isabout solving eigenvalue problem, ˆ H | n i = E n | n i withperiodic boundary condition constraint. This is in factreflected by how the energy spectrum are extracted fromEuclidean space correlation function in lattice computa-tion, hO ( t ) O † (0) i ∝ X n hO (0) e − ˆ Ht | n ih n |O † (0) i = X n |hO (0) | n i| e − E n t . (1)From this angle of view, we can also see why the scatter-ing amplitude is not a mandatory ingredient if one’s aimis only to find out discrete energy spectrum of few-hadronsystem.In this notes, we aim to illustrate above mentionedthree claims in a way as simple as possible without dis-tractions of all the fancy and complicated technical dress-up. In Sec.II, we will start with a simple example to showhow quantization condition is formulated in two-bodysector in general and it is relation to L¨uscher formula [33].In Sec.III, we will illustrate how the situation is compli-cated by subprocess pair-wise interactions in three-bodyproblems, and show how the quantization condition canbe formulated properly and discuss why infinite volumescattering amplitude is in fact dispensable in formulatingquantization condition. A summary is given in Sec.IV. II. TWO-BODY INTERACTION ANDL ¨USCHER FORMULA
From this point on, all the discussions will be carriedout in terms of quantum mechanical operators, so thatthe scattering process regardless boundary conditions canbe formulated on an equal footing symbolically. As far asonly short-range interactions are considered, i.e. inter-action range is much shorter than size of box, boundarycondition will only have significant impact on analyticproperties of Green’s functions. Most importantly, con-clusion can be drawn based on the simple discussion evenwithout digging into all the distracting technical details.Let’s first introduce a T -matrix operator, scatteringprocess is conventionally described by inhomogeneousLippmann-Schwinger equation [32],ˆ t ( E ) = ˆ V + ˆ V ˆ G ( E )ˆ t ( E ) , (2)where ˆ V stands for potential operator. Free Green’s func-tion operator, ˆ G , is given byˆ G ( E ) = 1 E − ˆ H + iǫ = X k | k ih k | E − E k + iǫ , (3)where ˆ H is free particles Hamiltonian, andˆ H | k i = E k | k i . In infinite volume, eigenstates of ˆ H , | k i , are contin-uous. Hence, infinite volume Green’s function ˆ G ( ∞ )0 ( E )develop a branch cut on real E axis that ultimately splitˆ t ( ∞ ) ( E ) into physical and unphysical amplitudes definedin first and second Riemann sheets respectively. Thephysical amplitude is defined on real E axis with ǫ → G ( ∞ )0 ( E ) has both contin-uous principle part and imaginary part, therefore, witha real ˆ V , the solution of equation1ˆ t ( ∞ ) ( E ) = 1ˆ V − ˆ G ( ∞ )0 ( E ) = 0 (4)does not exist not only on real E axis but also on com-plex plane in physical sheet. Hence, in infinite volume, stationary bound state solutions can be found only belowtwo-particle threshold, where imaginary part of ˆ G ( ∞ )0 ( E )vanishes.On the contrary, in finite volume, eigenstates | k i be-come discrete, the branch cut of ˆ G ( ∞ )0 ( E ) is replaced bydiscrete δ ( E − E k ) poles in finite volume Green’s func-tion ˆ G ( L )0 ( E ). The principle part of ˆ G ( L )0 ( E ) becomesperiodic, and imaginary part of ˆ G ( L )0 ( E ) vanishes as faras E = E k . Therefore, the solution of1ˆ t ( L ) ( E ) = 1ˆ V − ˆ G ( L )0 ( E ) = 0 (5)can be found on entire real E axis in finite volume.Equivalently, the pole position of ˆ t ( L ) ( E ) are associ-ated with stationary state solutions of homogeneousLippmann-Schwinger equation,ˆ t ( E ) = ˆ V ˆ G ( E ) ˆ T ( E ) . (6)Using relation ˆ t = ˆ V Ψ, Eq.(6) is thus can be convertedinto Ψ( E ) = ˆ V ˆ G ( E )Ψ( E ) , (7)which describes stationary bound states of system.In summary, the stationary solutions can be found byeither looking for pole positions of solution of inhomoge-neous Lippmann-Schwinger equation,ˆ t ( E ) = 1 V − ˆ G ( E ) , (8)or equivalently solving homogeneous Lippmann-Schwinger equation, Eq.(6) or Eq.(7) directly. Thisstatement in fact is true regardless boundary conditions.In infinite volume, only stationary solutions can be foundare bound states below two-particle threshold. However,in finite volume, infinite discrete solutions can be foundeven above two-particle threshold. Therefore, findingstationary solutions is nothing but solving bound stateproblems with certain boundary condition constraint onwave function, ˆ H Ψ = E Ψ . (9)The finite volume quantization condition given inEq.(5) can be recasted in terms of ˆ T ( ∞ ) by assumingthat short-range interaction potential ˆ V remains same inboth finite and infinite volume, thus we obtain1ˆ V = 1ˆ t ( ∞ ) ( E ) + ˆ G ( ∞ )0 ( E ) . (10)The finite volume quantization condition now is given by1ˆ t ( ∞ ) ( E ) + ˆ G ( ∞ )0 ( E ) − ˆ G ( L )0 ( E ) = 0 , (11)which is nothing but L¨uscher formula [33]. In two-bodycase, due to the fact that the interaction potential and ˆ t has a relatively simple relation, so the quantization con-dition can be formulated in terms of infinite volume scat-tering amplitude ˆ t ( ∞ ) directly. III. FEW-BODY INTERACTIONA. T -matrix formalism It is very tempting to extend and generalize the previ-ous described procedure into few-body sectors, unfortu-nately the situation in few-body case is complicated byphysical processes where some particles are disconnectedfrom and not interacting with rest of particles. The con-clusion and argument we are going to draw and makein fact can be made in general, however, for the sake ofsimplicity, we will only use three-body interaction as asimple example in follows. It is sufficient to just make apoint.Let’s now consider three-particle interacting with pair-wise interactions, three-particle scattering process againis described by inhomogeneous Lippmann-Schwingerequation [30–32],ˆ T ( E ) = ˆ V + ˆ V ˆ G ( E ) ˆ T ( E ) , (12)where ˆ V = X i =1 ˆ V i and ˆ V i stands for interaction potential operator betweenj-th and k-th particles. It has been a well-known factthat pair-wise potentials yield the appearance of a δ -function in the kernel of Eq.(12) because of momen-tum conservation of third spectator particle. This δ -function persist during all orders of iterations, hence, thekernel ˆ V ˆ G ( E ) in three-body case is not compact, andLippmann-Schwinger equation in three-body case can-not be solved by Fredholm method [30–32]. To overcomethis difficulty, Faddeev approach normally is introducedby splitting Eq.(12) into coupled equations,ˆ T i ( E ) = ˆ V i + ˆ V i ˆ G ( E ) ˆ T ( E ) , i = 1 , , , (13)and ˆ T ( E ) = X i =1 ˆ T i ( E ) . Rearranging above equations, one obtain inhomogeneousFaddeev equations ˆ T ( E )ˆ T ( E )ˆ T ( E ) = ˆ t ( E )ˆ t ( E )ˆ t ( E ) + ˆ K ( E ) ˆ G ( E ) ˆ T ( E )ˆ T ( E )ˆ T ( E ) , (14)now with a compact kernel matrixˆ K ( E ) = t ( E ) ˆ t ( E )ˆ t ( E ) 0 ˆ t ( E )ˆ t ( E ) ˆ t ( E ) 0 , (15) where ˆ t i ( E ) is subprocess amplitude and given byˆ t i ( E ) = 1 V i − ˆ G ( E ) . The solutions of ˆ T -matrix is hence given by ˆ T ( E )ˆ T ( E )ˆ T ( E ) = h I − ˆ K ( E ) ˆ G ( E ) i − ˆ t ( E )ˆ t ( E )ˆ t ( E ) . (16)The stationary state solutions are associated with thepole positions of Faddeev equations solutions, hence aredetermined by finding solutions ofdet h I − ˆ K ( E ) ˆ G ( E ) i = 0 . (17)Similar to tow-body case, in infinite volume, station-ary state solutions can only be found below three-bodythreshold as bound state solutions. In finite volume,stationary state solutions exist even above three-bodythreshold due to periodic boundary condition. Equiv-alently, stationary state solutions can also be found byusing homogeneous Faddeev equations directly, h I − ˆ K ( E ) ˆ G ( E ) i ˆ T ( E )ˆ T ( E )ˆ T ( E ) = 0 . (18)Although three-body quantization conditiondet " K ( E ) − ˆ G ( E ) = 0 , (19)resemble two-body quantization condition given inEq.(5), unlike two-body case, ˆ K now depend on ˆ t i , seeEq.(15). Thus, ˆ K is affected by boundary condition aswell, infinite volume ˆ K ( ∞ ) and finite volume ˆ K ( L ) havequite different analytical properties. Therefore, a simpleL¨uscher formula type quantization condition is no longeravailable in few-body case. The few-body quantizationcondition cannot be easily formulated in terms of infinitevolume scattering amplitude ˆ T ( ∞ ) directly. However, theconnection between finite and infinite volume dynamicscan be made again based on the assumption that poten-tials remain short-range and same in both finite volumeand infinite volume. Hence, subprocess amplitudes ˆ t ( L ) i can be related to ˆ t ( ∞ ) i by1ˆ t ( L ) i ( E ) = 1ˆ t ( ∞ ) i ( E ) + ˆ G ( ∞ )0 ( E ) − ˆ G ( L )0 ( E ) . (20)The ˆ K ( L ) now can be parameterized by using either ˆ V i orˆ t ( ∞ ) i as basic ingredients, and the finite volume quanti-zation condition can be formulated in terms of either ˆ V i or ˆ t ( ∞ ) i as well. After the determination of basic ingredi-ents, ˆ V i or ˆ t ( ∞ ) i by fitting lattice results, if one is inter-ested in obtaining the infinite volume three-body scatter-ing amplitude as well, then ˆ T ( ∞ ) has to be computed ina separate step by using infinite volume inhomogeneousFaddeev equations, Eq.(16).In summary, infinite volume few-body scattering am-plitude is in fact not a necessary component when itcomes to the formulating few-body quantization condi-tion. The finite volume few-body quantization conditiondet " K ( L ) ( E ) − ˆ G ( L )0 ( E ) = 0can be obtained in terms of some basic ingredients of in-finite volume few-body scattering amplitudes alone. Theinfinite volume few-body scattering amplitude can becomputed separately by Faddeev equations after basicingredients are determined. These basic ingredients canbe chosen either as interaction potential ˆ V i or their asso-ciated subprocess amplitudesˆ t ( ∞ ) i ( E ) = h ˆ V − i − ˆ G ( ∞ )0 ( E ) i − . B. K -matrix formalism The three-body scattering process may also be de-scribed by using K -matrix formalism [34, 35],ˆ K ( E ) = ˆ V + ˆ V ˆ G P ( E ) ˆ K ( E ) , (21)where ˆ G P ( E ) stands for the principle part of ˆ G ( E ),and ˆ T -matrix and ˆ K -matrix are related byˆ T ( E ) = ˆ K ( E ) − iπδ ( E − ˆ H ) ˆ K ( E ) ˆ T ( E ) . (22)After carrying out Faddeev’s procedure, ˆ K = P i =1 ˆ K i ,the solution of ˆ K -matrix is given by ˆ K ( E )ˆ K ( E )ˆ K ( E ) = h I − ˆ K P ( E ) ˆ G P ( E ) i − ˆ t P ( E )ˆ t P ( E )ˆ t P ( E ) , (23) where ˆ t i P ( E ) = 1 V i − ˆ G P ( E )and ˆ K P ( E ) = t P ( E ) ˆ t P ( E )ˆ t P ( E ) 0 ˆ t P ( E )ˆ t P ( E ) ˆ t P ( E ) 0 . (24)Hence, we can conclude that K -matrix formalism isequivalent to T -matrix formalism and doesn’t change thepicture fundamentally. In the end, the quantization con-dition can be formulated in terms of either ˆ V i or ˆ t ( ∞ ) i P aswell, and infinite volume scattering amplitude still needto be computed in a separate step. IV. SUMMARY
In summary, we have illustrated that except in two-body case, infinite volume few-body scattering ampli-tudes are in fact not a mandatory component for formu-lating finite volume quantization condition. The essen-tial ingredients of formulating finite volume quantizationcondition can be chosen as either interactions potentialˆ V i or corresponding subprocess amplitudesˆ t ( ∞ ) i = h ˆ V − i − ˆ G i − . The total few-body amplitude ˆ T ( ∞ ) can be computed ina separate step. In fact, if only obtaining quantizationcondition is one’s aim, few-body amplitude ˆ T ( ∞ ) is notneeded at all. It comes as no surprise due to the factthat the final few-body physical process is in fact gen-erated by all subprocess or interactions. In the end, itall comes down to the problem of finding stationary so-lutions of system with periodic boundary condition con-staint, which can be accomplished by either looking forthe pole of solutions of inhomogeneous Faddeev equationsor solving homogeneous equations directly. [1] S. Kreuzer and H. W. Ham-mer, Phys. Lett. B673 , 260 (2009),arXiv:0811.0159 [nucl-th].[2] S. Kreuzer and H. W. Ham-mer, Eur. Phys. J.
A43 , 229 (2010),arXiv:0910.2191 [nucl-th].[3] S. Kreuzer and H. W. Grießhammer,Eur. Phys. J.
A48 , 93 (2012), arXiv:1205.0277 [nucl-th].[4] K. Polejaeva and A. Rusetsky,Eur. Phys. J.
A48 , 67 (2012), arXiv:1203.1241 [hep-lat].[5] R. A. Brice˜no and Z. Davoudi,Phys. Rev.
D87 , 094507 (2013),arXiv:1212.3398 [hep-lat].[6] M. T. Hansen and S. R. Sharpe,Phys. Rev.
D90 , 116003 (2014), arXiv:1408.5933 [hep-lat].[7] M. T. Hansen and S. R. Sharpe,Phys. Rev.
D92 , 114509 (2015),arXiv:1504.04248 [hep-lat].[8] M. T. Hansen and S. R. Sharpe,Phys. Rev.
D93 , 096006 (2016), [Erratum: Phys.Rev.D96,no.3,039901(2017)], arXiv:1602.00324 [hep-lat].[9] R. A. Brice˜no, M. T. Hansen, and S. R.Sharpe, Phys. Rev.
D95 , 074510 (2017),arXiv:1701.07465 [hep-lat].[10] H.-W. Hammer, J.-Y. Pang, and A. Rusetsky,JHEP , 109 (2017), arXiv:1706.07700 [hep-lat].[11] H. W. Hammer, J. Y. Pang, and A. Rusetsky,JHEP , 115 (2017), arXiv:1707.02176 [hep-lat]. [12] U.-G. Meißner, G. R´ıos, and A. Ruset-sky, Phys. Rev. Lett. , 091602 (2015), [Erra-tum: Phys. Rev. Lett.117,no.6,069902(2016)],arXiv:1412.4969 [hep-lat].[13] M. Mai and M. D¨oring, Eur. Phys. J. A53 , 240 (2017),arXiv:1709.08222 [hep-lat].[14] M. Mai and M. D¨oring,Phys. Rev. Lett. , 062503 (2019),arXiv:1807.04746 [hep-lat].[15] M. D¨oring, H. W. Hammer, M. Mai, J. Y. Pang, § . A.Rusetsky, and J. Wu, Phys. Rev. D97 , 114508 (2018),arXiv:1802.03362 [hep-lat].[16] F. Romero-L´opez, A. Rusetsky, andC. Urbach, Eur. Phys. J.
C78 , 846 (2018),arXiv:1806.02367 [hep-lat].[17] P. Guo, Phys. Rev.
D95 , 054508 (2017),arXiv:1607.03184 [hep-lat].[18] P. Guo and V. Gasparian, Phys. Lett.
B774 , 441 (2017),arXiv:1701.00438 [hep-lat].[19] P. Guo and V. Gasparian,Phys. Rev.
D97 , 014504 (2018),arXiv:1709.08255 [hep-lat].[20] P. Guo and T. Morris, Phys. Rev.
D99 , 014501 (2019),arXiv:1808.07397 [hep-lat].[21] T. D. Blanton, F. Romero-L´opez, and S. R. Sharpe,JHEP , 106 (2019), arXiv:1901.07095 [hep-lat].[22] F. Romero-L´opez, S. R. Sharpe, T. D. Blanton,R. A. Brice˜no, and M. T. Hansen, (2019), arXiv:1908.02411 [hep-lat].[23] T. D. Blanton, F. Romero-L´opez, and S. R. Sharpe,(2019), arXiv:1909.02973 [hep-lat].[24] M. Mai, M. D¨oring, C. Culver, and A. Alexandru,(2019), arXiv:1909.05749 [hep-lat].[25] P. Guo, M. D¨oring, and A. P. Szczepa-niak, Phys. Rev. D98 , 094502 (2018),arXiv:1810.01261 [hep-lat].[26] P. Guo, Phys. Lett.
B804 , 135370 (2020),arXiv:1908.08081 [hep-lat].[27] P. Guo and M. D¨oring, Phys. Rev.
D101 , 034501 (2020),arXiv:1910.08624 [hep-lat].[28] P. Guo, Phys. Rev.
D101 , 054512 (2020),arXiv:2002.04111 [hep-lat].[29] P. Guo and B. Long, Phys. Rev. D , 094510 (2020),arXiv:2002.09266 [hep-lat].[30] L. D. Faddeev,
Mathematical Aspects of the Three-body Problem in the Quantum Scattering Theory (IPST, 1963).[31] L. D. Faddeev, Sov. Phys. JETP , 1014 (1961), [Zh.Eksp. Teor. Fiz.39,1459(1960)].[32] R. Newton, Scattering Theory of Waves and Particles (New York, USA: Springer, 1982).[33] M. L¨uscher, Nucl. Phys.
B354 , 531 (1991).[34] K. Kowalski, Phys. Rev. D , 395 (1972).[35] N. Mishima and M. Yamazaki,Progress of Theoretical Physics34