33-body quantization condition in a unitary formalism
Maxim Mai ∗† Institute for Nuclear Studies and Department of Physics, The George Washington University,Washington, DC 20052, USAE-mail: [email protected]
Unitarity identifies all power-law finite-volume effects and is, therefore, the crucial S-matrix prin-ciple for a mapping between experimental results and those of Lattice QCD calculations. In thiscontribution we review how 3-body unitarity constrains the form of the 3-body scattering ampli-tude parametrized by the tower of isobars. The result is discretized and projected to the irreduciblerepresentations of the cubic group, leading to a fully relativistic 3-body quantization condition.The latter is used to deduce the finite-volume excited level spectrum of the π + π + π + system,which agrees nicely with the available lattice results by the NPLQCD collaboration. The 36th Annual International Symposium on Lattice Field Theory - LATTICE201822-28 July, 2018Michigan State University, East Lansing, Michigan, USA. ∗ Speaker. † The speaker thanks the organizers of the conference for the opportunity to give this talk. This work is supportedby the German Research Association (MA 7156/1) and National Science Foundation (grant no. PHY-1452055). c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] O c t -body quantization condition Maxim Mai
Introduction
Many unsolved questions of QCD involve systems with 3 hadrons. One of the mostprominent examples is the so-called Roper-puzzle, which addresses the reversed mass-pattern ofthe N ( ) / − and the excited state of the nucleon N ( ) / + compared to the expectationsfrom the constituent quark model. The properties of the first can be accessed from the meson-baryon scattering amplitude, e.g., through a manifestly covariant and unitary approach [1]. Unfor-tunately, a similar treatment of the positive-parity state, the N ( ) / + , is more intricate due toits large branching ratio to the ππ N channels, see Ref. [2] for a recent review of several theoreticalapproaches. In the meson sector, the spin-exotics are states with quantum numbers which cannotbe formed by a quark-antiquark pair. Thus, the more complex structure of these states is believedto be an indicator for the gluonic degrees of freedom in QCD. Many of such hypothetical statescannot decay into two, but only into three pions. This also applies to ordinary mesons such as the a ( ) -meson. Thus, the identification of these novel states requires an in-depth understandingof the πππ system forming ordinary excited states.The only systematical, non-perturbative approach to the properties of strongly interacting sys-tems is Lattice QCD (LQCD). Such numerical calculations are performed on discretized Euclideanspace-time in finite volume and (depending on the technical intricacy) at unphysical quark masses.In the quest for time-independent quantities, such as resonance parameters, in systems with threehadrons the non-trivial issues are the finite-volume effects as well the subsequent chiral extrapo-lation. Therefore, while substantial computational and algorithmic advances have been made overthe last years in extracting finite-volume spectra from ab-initio LQCD calculations [3–8], theircomparison to phenomenology requires a so-called 3-body quantization condition, similar to thewell-established Lüscher’s method [9, 10] in the 2-body sector. Over the last years many explo-rations have been performed to this end [11–24] including alternative techniques [25, 26] proposedto obtain essential information on the system without the need of explicit parametrization of everyreaction channel.Over the last two years a convenient version of a fully relativistic 3-body quantization con-dition has been derived from the so-called isobar parametrization of the unitary 3-body scatteringamplitude [27] using discretization [28] and projection of the latter to the irreducible representa-tions of the cubic group [14]. In the present contribution we show the results derived in these worksas well as the recent application of this approach in the first-ever calculation of excited (i.e., above-threshold) energy eigenvalues [29] of a physical ( π + π + π + ) system. The ground state finite-volumeenergy for this system has been calculated by the NPLQCD collaboration [7, 8]. Three-body dynamics
A relativistic, infinite volume 3-body scattering amplitude in the isobarformalism [27] can be expressed in terms of on-shell, 2-body unitary 2 → → s -channel propagator with dissociation vertices attached to both ends.As further discussed in Ref. [27], the isobar can be associated with bound states, one or more res-onances, or a non-resonant 2-particle amplitude. The isobar formulation is not an approximationbut a re-parametrization of the full 2-body amplitude as shown in Ref. [30] and also discussed inRef. [11]. In the following we collect only the main results of the derivation and refer the readerfor details to Refs. [27, 31].The interaction of three spin-less particles of mass m and out- and in-going four-momenta1 -body quantization condition Maxim Mai T = +ˆ T is ˆ τ ˆ τ ˆ τ vv Figure 1:
The 3-particle scattering amplitude T , constructed from the particle-isobar scattering amplitudeˆ T is , isobar propagator ˆ τ and dissociation vertex v . The quantity in the parentheses on the right hand sideconsists of a fully connected and once disconnected parts in that order. q , q , q and p , p , p , respectively, is fully described by the S-matrix ( S ) related to the T-matrix ( T ) via S = : + i ( π ) δ (cid:0) ∑ i = ( q i − p i ) (cid:1) T . In the case of 3 → T is and isobar propagator ˆ τ as (cid:104) q , q , q | T | p , p , p (cid:105) = ∑ n = ∑ m = v ( q ¯ n , q ¯¯ n ) ˆ T ( q n , p m ; W ) v ( p ¯ m , p ¯¯ m ) , (1)ˆ T ( q n , p m ; W ) = (cid:32) ˆ τ ( σ qqq n ) ˆ T is ( q n , p m ; W ) ˆ τ ( σ ppp m ) − E qqq n ˆ τ ( σ qqq n )( π ) δ ( qqq n − ppp m ) (cid:33) , where P is the total four-momentum of the system, W = P and E ppp = (cid:112) ppp + m . All four-momenta p , q , ... are on-mass-shell, and the square of the invariant mass of the isobar reads σ qqq = W + m − W E qqq for the spectator momentum q . We work in the total center-of-mass framewhere PPP = v ( p , q ) of the isobar decaying in asymptotically stable par-ticles, e.g., ρ ( p + q ) → π ( p ) π ( q ) , is chosen to be cut-free in the physical energy region, whichis always possible. For the present study we choose the dissociation vertex to be of a particularlysimple form, v ( p , q ) : = λ f (( p − q ) ) with f such that it is 1 for ( p − q ) = T in Eq. (1) one obtainsˆ T is ( q , p ; W ) = B ( q , p ; W ) − (cid:90) d lll ( π ) B ( q , l ; W ) τ ( σ lll ) E lll ˆ T is ( l , p ; W ) , (2) B ( q , p ; W ) = − λ f (( P − p − q ) ) f (( P − p − q ) ) E qqq +++ ppp ( W − E qqq − E ppp − E qqq +++ ppp + i ε ) + C ( q , p ; W ) , where p and q denote the on-shell four-momenta of the in- and outgoing spectator, respectively.Additional terms C that are real functions of energy W and momenta in the physical region asdemanded by 3-body unitarity (3-body forces) can be added to B , see discussion in Ref. [27]. Wepostpone the introduction of multiple isobars and of spin and isospin for the isobars and the stableparticles to future work. As demonstrated in Ref. [27] the algebraic form of the isobar propagatoris fixed up to regular terms and can be written as1ˆ τ ( σ ) = σ − M − ∑ ± (cid:90) d kkk ( π ) ( λ f (( √ σ ± E kkk ) − kkk )) E kkk √ σ ( √ σ ± E kkk ) , (3)2 -body quantization condition Maxim Mai where M is a free parameter that can be used to fit (together with λ ) the 2-body amplitude corre-sponding to the considered isobar. We will refer to the integral term in Eq. (3) as self-energy in thefollowing. Note that the only principle used in the construction of the amplitude is 3-body unitarityfor the physical ( s -channel) region which is the only requirement needed to identify all power-lawfinite volume effects; like in the 2-body case, there are, of course, also left-hand singularities inthe infinite volume amplitude but they all contribute exponentially suppressed to the finite-volumeeffects. Three-body quantization condition
For the extraction of scattering information from latticecalculations, boundary conditions have to be imposed, and only discrete momenta are allowed. Inparticular, for a box of side length L and periodic boundary conditions the set of allowed momentareads 2 π / L · Z . For convenience, we order these momenta in “shells”, defined as sets of momentawhich are related to each other by cubic symmetry. The running index of these sets will be denotedin the following by s and its cardinality by ϑ ( s ) .For discretized momenta, the isobar-spectator amplitude becomes a matrix equation, whichin operator notation reads T = ( τ − + B ) − with τ and T being the isobar-propagator and the fullisobar-spectator scattering amplitude (combined terms in parenthesis in Fig. 1) in finite volume,respectively. Obviously, in this symbolic notation T can become singular for 3-body energies W fulfilling det ( τ − + B ) =
0. The latter is commonly referred to as the quantization condition anddetermines the finite-volume spectrum of the 3-body system in question. Note that intermediatestates with more than three particles are explicitly excluded from the formalism. This limits therange of validity of the present approach to the next-higher multi-particle channel (being, e.g., 5 π in the case of the πππ system).Similarly to the infinite volume case, the technical obstacle solving the above (symbolicallydefined) quantization condition is its high dimensionality. Projection to the irreducible representa-tions (irreps) of the cubic group Γ ∈ { A , A , E , T , T } reduces the quantization condition greatlyand has the advantage that the obtained spectra can be compared directly with the LQCD results. Aparticularly convenient projection procedure has been introduced in Ref. [14]. Having large simi-larities to the partial wave projection techniques in infinite volume, it defines an orthonormal basisof functions on each shell. Projection of the above quantization condition to such basis functionsis described in detail in Sec. IV of Ref. [14], and leads to a diagonal condition in the irrep-index Γ det (cid:18) B Γ ss (cid:48) uu (cid:48) ( W ) + E s L ϑ ( s ) τ − s ( W ) δ ss (cid:48) δ uu (cid:48) (cid:19) = . (4)Here, the determinant is taken with respect to the shell-index s ( (cid:48) ) and basis-index u ( (cid:48) ) , while E s : = E ppp and τ s = τ ppp with ppp being a momentum on the shell s . An important consequence ofthe breakdown of the spherical and therefore Lorentz symmetry is that the isobar-propagator has tobe boosted into the isobar rest frame before discretizing the momenta. Denoting the boost of themomentum xxx by the momentum ppp (spectator momentum) as kkk ∗ xxx ,,, ppp and the corresponding Jacobianby J ppp , the isobar propagator in finite volume reads1 τ ppp = σ ppp − M − J ppp L ∑ xxx ∈ π L Z ∑ ± (cid:16) λ f (cid:16)(cid:0) P ∗ ppp ± k ∗ xxx , ppp (cid:1) (cid:17)(cid:17) √ σ ppp E kkk ∗ xxx , ppp (cid:16) √ σ ppp ± E kkk ∗ xxx , ppp (cid:17) . (5)3 -body quantization condition Maxim Mai πππ + πππ + πππ + πππ + πππ + m π [ m π phys ] E [ m π ] m π [ m π phys ] E [ m π ] Figure 2: Left : Prediction of 2-body energy levels (full) as a function of m π with dashed lines denotingnon-interacting levels. Right : Prediction of excited energy levels for the π + π + π + -system as a functionof pion mass with non-interacting levels represented by dashed lines. The insets show the zoom-in on theground level, where the lattice data [7, 8] are shown in red. Here P ∗ qqq : = ( √ σ qqq , ) is the four-momentum of the isobar (2-pion system) boosted to its referenceframe. For a given absolute value of the spectator momentum ppp , the range of validity of the boostformula and, therefore, of the discretized propagator τ is limited to σ ppp >
0. However, alreadybelow the 2-particle threshold σ ppp < ( M ) the regular summation theorem applies and the sum canbe replaced by the integral up to exponentially suppressed terms.In conclusion, we note that both the isobar-spectator kernel B as well as τ − in Eq. (4) canbecome singular separately. However, these singularities cancel each other exactly as shown ex-plicitly in Ref. [28], leaving one with the singularities from genuine three-body dynamics only. Finite-volume spectrum of the π + π + π + system The quantization condition derived in Eq. (4)has a particularly simple form and has been tested on a hypothetical scenario of three spin-less parti-cles, two of each interacting via a Breit-Wigner like resonance, see Ref. [28]. However, no assump-tions have been made in the derivation of the quantization condition about the form of the 2-particleinteraction in the sub-channels. In the following we demonstrate the application of the quantiza-tion condition (4) to the physical system of π + π + π + . The 2-body sub-channel interaction of thissystem is repulsive and serves, therefore, as an ideal test bed for the applicability of the proposed3-body quantization condition. Fortunately, LQCD data in this (repulsive) channel are availablefrom the NPLQCD Collaboration [7, 8] for L = . m π ∈ { , , , } MeV. Ourprogram consists of prediction of the full (up to the 4 π threshold) finite-volume 2-body spectrumusing experimentally available data. Subsequently, we will fix the remaining parameter (genuine3-body coupling) to the ground-state energy level of the π + π + π + system [7, 8], predicting higherlevels up to W = m π .In the following we specify the parameters of the quantization condition (4) following the find-ings of Ref. [29]. First, the system in question is in relative S -wave such that for the finite-volumeanalysis we fix Γ = A + . The form-factor f ( Q ) ( Q being the difference of the four-momenta ofthe dissociation products) yields a smooth cutoff of an otherwise log-divergent self-energy partof the isobar propagator (third term in τ − of Eq. (5)). Note, that this cutoff-dependence can-cels in the full quantization condition (4) by the functions C and M . Specifically, we chose here4 -body quantization condition Maxim Mai f ( Q ) = / ( + e − ( Λ / − ) + Q / ) with Λ =
42 in units of m π . Second, we have tested various formsof the coupling λ and found that taking λ = ( M − σ ) (cid:18) d π + T LO − ¯ T NLO T (cid:19) − , (6)where T LO and ¯ T NLO are the leading and next-to-leading (without the s-channel loop) order chiralamplitudes [32], respectively, yields the Inverse Amplitude Method (IAM) for T : = v ˆ τ v . Here d = .
86 for the chosen Λ , see Ref. [29] for more details. Such an amplitude has a substantiallylarger range of validity in the 2-body energy and resembles the chiral expansion up to next-to-leading order exactly as argued in Ref. [33]. Indeed, taking the low-energy constants from Ref. [32]we have found that this ansatz perfectly reproduced the phase shifts from experiment. The predictedfinite-volume spectrum of the π + π + system, extracted from the corresponding equation when re-placing ˆ τ (cid:55)→ τ lies on top of the LQCD data [7, 8] as depicted in Fig. 2. This approach agrees withthe lattice data even at pion masses as large as ≈
600 MeV, see Ref. [29] for further discussions.With the 2-body input fixed, the only remaining unknown of the 3-body quantization conditionremains the genuine 3-body force term C ( q , p ; W ) . The functional form of this term is not known.We found, however, that the simplest choice C ( q , p , W ) = c δ ( ) ( ppp − qqq ) leads to a good fit to theLQCD data [7, 8] ( χ = .
05 for c = . ± . · − ). The value of constant c turns out to beof the same order of magnitude as the η L term introduced on the level of Hamiltonian in the large-volume expansion formula [34]. The comparison with the data as well as prediction of the excitedlevels for the π + π + π + system is depicted in the right panel of Fig. 2. While no uncertainty bands(from the 2-body and 3-body input) are depicted there, they are discussed in Ref. [29].In conclusion, we have analyzed the finite-volume spectrum for the π + π + and π + π + π + sys-tems using experimental data and a non-perturbative ansatz for the 2-body amplitude. The π + π + energy levels in finite volume have been predicted and agree nicely with the available lattice data.Using this input and fitting the genuine 3-body contact term to the threshold level determined by theNPLQCD collaboration we have predicted the finite volume spectrum of the π + π + π + system up to W = m π . This is the first prediction of excited levels in a physical 3-body system. The extensionsof this approach to multi-channel systems and systems with higher spin is work in progress. References [1] P. C. Bruns, M. Mai and U. G. Meißner, Phys. Lett. B , 254 (2011) [arXiv:1012.2233 [nucl-th]].[2] L. Alvarez-Ruso, (Bled Workshops in Physics. Vol. 11 No. 1) [arXiv:1011.0609 [nucl-th]].[3] C. B. Lang, L. Leskovec, D. Mohler and S. Prelovsek, JHEP , 162 (2014) [arXiv:1401.2088[hep-lat]].[4] C. B. Lang, L. Leskovec, M. Padmanath and S. Prelovsek, Phys. Rev. D , no. 1, 014510 (2017)[arXiv:1610.01422 [hep-lat]].[5] A. L. Kiratidis, W. Kamleh, D. B. Leinweber, Z. W. Liu, F. M. Stokes and A. W. Thomas, Phys. Rev.D , no. 7, 074507 (2017) [arXiv:1608.03051 [hep-lat]].[6] A. Woss, C. E. Thomas, J. J. Dudek, R. G. Edwards and D. J. Wilson, JHEP , 043 (2018)[arXiv:1802.05580 [hep-lat]]. -body quantization condition Maxim Mai[7] S. R. Beane, W. Detmold, T. C. Luu, K. Orginos, M. J. Savage and A. Torok, Phys. Rev. Lett. ,082004 (2008) [arXiv:0710.1827 [hep-lat]].[8] W. Detmold, M. J. Savage, A. Torok, S. R. Beane, T. C. Luu, K. Orginos and A. Parreno, Phys. Rev. D , 014507 (2008) [arXiv:0803.2728 [hep-lat]].[9] M. Luscher, Commun. Math. Phys. , 153 (1986).[10] M. Luscher, Nucl. Phys. B , 531 (1991).[11] H.-W. Hammer, J.-Y. Pang and A. Rusetsky, JHEP , 115 (2017) [arXiv:1707.02176 [hep-lat]].[12] H. W. Hammer, J. Y. Pang and A. Rusetsky, JHEP , 109 (2017) [arXiv:1706.07700 [hep-lat]].[13] R. A. Briceño, M. T. Hansen and S. R. Sharpe, Phys. Rev. D , no. 7, 074510 (2017)[arXiv:1701.07465 [hep-lat]].[14] M. Döring, H. W. Hammer, M. Mai, J.-Y. Pang, A. Rusetsky and J. Wu, Phys. Rev. D , no. 11,114508 (2018) [arXiv:1802.03362 [hep-lat]].[15] S. R. Sharpe, Phys. Rev. D , no. 5, 054515 (2017) [arXiv:1707.04279 [hep-lat]].[16] P. Guo, Phys. Rev. D , no. 5, 054508 (2017) [arXiv:1607.03184 [hep-lat]].[17] M. T. Hansen and S. R. Sharpe, Phys. Rev. D , no. 3, 034501 (2017) [arXiv:1609.04317 [hep-lat]].[18] M. T. Hansen and S. R. Sharpe, Phys. Rev. D , no. 9, 096006 (2016) Erratum: [Phys. Rev. D , no.3, 039901 (2017)] [arXiv:1602.00324 [hep-lat]].[19] M. T. Hansen and S. R. Sharpe, Phys. Rev. D , no. 11, 114509 (2015) [arXiv:1504.04248 [hep-lat]].[20] U. G. Meißner, G. Ríos and A. Rusetsky, Phys. Rev. Lett. , no. 9, 091602 (2015) Erratum: [Phys.Rev. Lett. , no. 6, 069902 (2016)] [arXiv:1412.4969 [hep-lat]].[21] R. A. Briceno and Z. Davoudi, Phys. Rev. D , no. 9, 094507 (2013) [arXiv:1212.3398 [hep-lat]].[22] S. Bour, H.-W. Hammer, D. Lee and U. G. Meißner, Phys. Rev. C , 034003 (2012)[arXiv:1206.1765 [nucl-th]].[23] S. Kreuzer and H. W. Grießhammer, Eur. Phys. J. A , 93 (2012) [arXiv:1205.0277 [nucl-th]].[24] K. Polejaeva and A. Rusetsky, Eur. Phys. J. A , 67 (2012) [arXiv:1203.1241 [hep-lat]].[25] D. Agadjanov, M. Doring, M. Mai, U. G. Meißner and A. Rusetsky, JHEP , 043 (2016)[arXiv:1603.07205 [hep-lat]].[26] M. T. Hansen, H. B. Meyer and D. Robaina, Phys. Rev. D , no. 9, 094513 (2017)[arXiv:1704.08993 [hep-lat]].[27] M. Mai, B. Hu, M. Doring, A. Pilloni and A. Szczepaniak, Eur. Phys. J. A , no. 9, 177 (2017)[arXiv:1706.06118 [nucl-th]].[28] M. Mai and M. Döring, Eur. Phys. J. A , no. 12, 240 (2017) [arXiv:1709.08222 [hep-lat]].[29] M. Mai and M. Doring, arXiv:1807.04746 [hep-lat].[30] P. F. Bedaque and H. W. Griesshammer, Nucl. Phys. A , 357 (2000) [nucl-th/9907077].[31] M. Mai, B. Hu, M. Doring, A. Pilloni and A. Szczepaniak, PoS Hadron , 140 (2018).[32] J. Gasser and H. Leutwyler, Annals Phys. , 142 (1984).[33] T. N. Truong, Phys. Rev. Lett. , 2526 (1988).[34] W. Detmold and M. J. Savage, Phys. Rev. D , 057502 (2008) [arXiv:0801.0763 [hep-lat]]., 057502 (2008) [arXiv:0801.0763 [hep-lat]].