The XXVIII International Symposium on Lattice Field Theory, June 14-19, 2010, Villasimius, Sardinia ItalyInternational audienceWhile the masses of light hadrons have been extensively studied in lattice QCD simulations, there exist only a few exploratory calculations of the strong decay widths of hadronic resonances. We will present preliminary results of a computation of the rho meson width obtained using $N_f=2+1$ flavor simulations. The work is based on L\"uscher's formalism and its extension to moving frames

Durr, S

Fodor, Z

Frison, J

Hoelbling, C

Katz, SD

Krieg, S

Kurth, T

Lellouch, L

Lippert, T

Portelli, A

Ramos, A

Szabo, KK

English

arXiv

ELTE Digital Institutional Repository (EDIT)

Rho decay width from the latticeJ. Frison∗a, S. Durrb,c, Z. Fodorb,c,d , C. Hoelblingb, S.D. Katzb,d , S. Kriegb,c, T. Kurthb,L. Lelloucha, T. Lippertb,c, A. Portellia, A. Ramosa and K.K. Szabob(Budapest-Marseille-Wuppertal Collaboration)aCentre de Physique Théorique †, Case 907, CNRS Luminy, F-13288 Marseille Cedex 9, France.bBergische Universität Wuppertal, Gaussstr. 20, D-42119 Wuppertal, Germany.cJülich Supercomputing Centre, Forschungszentrum Jülich, D-52425 Jülich, Germany.dInstitute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary.E-mail: frison@cpt.univ-mrs.frWhile the masses of light hadrons have been extensively studied in lattice QCD simulations, thereexist only a few exploratory calculations of the strong decay widths of hadronic resonances. Wewill present preliminary results of a computation of the rho meson width obtained using N f = 2+1flavor simulations. The work is based on Lüscher’s formalism and its extension to moving frames.The XXVIII International Symposium on Lattice Field TheoryJune 14-19,2010Villasimius, Sardinia Italy∗Speaker.†CPT is research unit UMR 6207 of the CNRS and of the universities Aix-Marseille II, Aix-Marseille I and SudToulon-Var, and is affiliated with the FRUMAM.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/arXiv:1011.3413v1  [hep-lat]  15 Nov 2010Rho decay width from the lattice J. FrisonThe light hadron masses have been extensively and quite successfully studied in lattice QCDsimulations [5, 16]. However, the study of strong decays remains a challenge that only a fewexploratory calculations have addressed so far [14, 15].Until recently this issue could be ignored : firstly because the sea quarks required for a multi-body decay are not present in the quenched approximation, and then because energy conservationleaves little or no phase space in those decays for unphysically large light quark masses. But as weget closer to the physical point and improve our precision, this issue has to be addressed for us tobe able to reliably determine masses of resonant states. Additionally, it is exciting way to test thecharacteristic dynamical effects of sea quarks.A finite-volume formalism has been developped by Lüscher [7–11], and extended to movingframes [12], which describes the modification of the quantization condition of scattering statesmomenta under interactions. We use this framework to compute the rho decay width on a subsetof the Budapest-Marseille-Wuppertal collaboration configurations [1, 2], which features improvedWilson fermions with N f = 2+1 flavors of sea quarks.1. Introducing the two-pions levels1.1 Basics of avoided crossingsIf ρ were not coupled to ππ , the spectrum as a function of box size L would only consist of :• a ρ state, whose energy would be a constant mρ up to exponentially suppressed finite-volumecorrections [7], and• free two-pions states, whose momenta~k = (2π/L)~n, and energies E =√m2π +~k2 are quan-tized.For some particular box sizes the ρ states crosses one of the free two-pions states. At these pointswe would have a degeneracy but nothing particular would happen. But if now we turn on interac-tions, the situation becomes very different: we know from quantum mechanics that two eigenstatescannot cross. The interaction mixes ρ and ππ into new eingenstates which exhibit an avoidedlevel crossing phenomenon (Fig. 1). These avoided crossings therefore contain information on thecoupling between the states, and hence on the ρ width.1.2 Lüscher’s FormulaIn a finite box we expect the pions to interact then propagate over small distance, then interactagain, and so on. This can be represented as an expansion in a series of two-particles-irreduciblekernels as shown in Fig. 2. When this expansion is resummed, the interactions move F poles(free two-pions states) to 1−MF zeros [13]. The zeros of 1−MF correspond to the solutions ofLüscher’s formula, which was first derived in [8].Lüscher’s formula is only valid below the 4π inelastic threshold. Expressed in a form exhibit-ing its role of quantization condition, it reads (up to exponentially small terms in mπL) :Φ(q) = nπ−δ (q), (1.1)using the reduced momentum q = kL/2π and the δ phase shift in the I = J = 1 channel. Φ is aknown kinematical function of q, expressing the breaking of Lorentz invariance by the cubic box.2Rho decay width from the lattice J. Frison2 3 4 5 6 7 8mπL22,533,54E/mπ2 3 4 5 6 7 8mπL22,533,54E/mπFigure 1: The left side shows the spectrum in the free case as a function of mπ L, for mρ/mπ = 3 and ignoringexponential corrections. This free spectrum is reproduced in dash lines on the right side and then we showits deformation with different couplings. The model is the one used in section 1.3, with the coupling gettingstronger from green to red.A M M M A’FF F...= AF 11−MF A′Figure 2: Here we represent the finite-volume correction of an arbitrary correlator, computed under theinelastic threshold up to exponentially small corrections. M is the infinite-volume amputated four-pointfunction, while F is the difference between finite-volume loops and infinite-volume ones. M is obtained asa resummation of a volume-independant series of 2-particle-irreducible kernels.1.3 The g-coupling modelLüscher’s equation gives us the phase shift for a few discrete values of the momentum, deter-mined by the parameters of the lattice. It would be difficult and very costly numerically to signifi-cantly increase the number of such momenta. Therefore we need a model to reconstruct the phaseshift from a few points. Moreover, this model will be useful to make the necessary extrapolations(mπ → mphysπ , a→ 0, L→ ∞, ... ). Following [11] we use both an effective range approximationand an effective lagrangian Le f f = gεabcρaµπb∂ µπc.The effective range approximation parametrizes the phase shift in terms of a Taylor expansionaround kρ =√E2/4−m2π , where kρ is the momentum of ρ ′s decay products in infinite-volume:k3Wcotδ = b(k2− k2ρ). (1.2)The second ingredient allows us to parametrize the coupling of the ρ to ππ states in terms ofa constant coupling g that should have a rather small dependance on the quark mass. On the otherhand the width Γ is strongly dependant on kinematics :Γρ =g26π·k3ρm2ρ. (1.3)Using experimental values for the masses and the width it yields g' 6.0.3Rho decay width from the lattice J. Frison2. Results with a single interpolating operatorThe formulae 1.2 and 1.3 can be combined into a very simple form, giving the correction toget the ρ mass from the energy E of an arbitrary eigenstate in the I = J = 1 channel :m2ρ = E2− g26πk3EcotΦ(kL/2π) where k =√E24−m2π (2.1)In this expression, the unknowns are mπ and g. Instead of using two energy levels at a givenquark mass, one can extract the ground state energy level at different quark masses. But the kine-matics required in that case are contradictory: on the one hand we would get no signal for the widthif the state is far below the first free two-pion state; on the other hand, near the first crossing onecannot disentangle the contributions of the ρ mass and width to the measured energy level.In [5] we considered only situations on which the ground state obtained with a single operatoris far from the first crossing. Thus, in combined fits of our 6-stout data, we obtained a precisedetermination of mρ (mρ ∼ 5%), but a much less precise determination of g : g = 9.5±4.6.3. Using two interpolating operators3.1 Generalized Eigenvalue ProblemNow we are going to quickly present the principle of extraction of several energies by thevariationnal method. Let us first assume that we can compute observables only affected by Neigenstates. We will show that energies can be computed from a set of N×N cross-correlatorsCi j(t) =〈0 | Oi(t)O j(0) | 0〉, with i, j = 1 . . .N. (3.1)We first decompose them on the energy eigenstates and express them in a compact matrix form :Ci j(t) = ∑n〈0 | Oi | n〉e−Ent〈n | O j | 0〉(3.2)= ∑m,n(V †)i,m ·Dm,n(t) ·Vn, j, (3.3)so Vn, j is the nth-state content of the Oi operator and D is a diagonal matrix containing the expo-nentials of the energies. Now we immediatly see that C(t)C−1(t0) is diagonal in the eigenstatesbasis :C(t)C−1(t0) =V †D(t)V(V †D(t0)V)−1= (V †)D(t− t0)(V †)−1, (3.4)and its eigenvectors give the energies through :λi = e−Ei(t−t0). (3.5)This is exact if and only if the operators are linearly independent (so we can invert the V s). Notethat in the case N = 1 this is simply the effective mass method.In practice we can only compute the matrix of operators obtained from a few N operators.Thus only the N low-lying levels are asymptotically known, and higher-levels are treated as acontamination at short times. The energies are computed with errors of order exp[(EN −EN+1)t]and exp[(EN−EN+1)t0], so t and t0 must be large (t−t0 is not very important, since it only enters inpolynomial prefactors). A more precise study of high-levels corrections on some derived quantitiesis made in [9].4Rho decay width from the lattice J. Frison~p ~q−~p −~q-~p ~q−~p −~q+~p ~q−~p −~q+~p ~q−~p −~q-~p ~q−~p −~q-~p ~q−~p −~q~p ~q−~P-~p ~q−~P~P−~p −~q-~P−~p −~qFigure 3: The contractions of ππ → ππ (top), ππ → ρ (bottom-left) and ρ → ππ (bottom-right), theρ → ρ being trivial. Time flows upward from 0 to t. Black dots represent an explicit summation whereasshaded dots represent a noise-noise contact. Between those dots we can have Q propagators (one line) or Wpropagator (two lines).3.2 Operators and contractionsTo implement the generalized eigenvalue approch described above, we consider two operators.The first is the point ρ meson operatorρi = ūγiu− d̄γid. (3.6)The second operator must be non-local to have a sufficiently independent coupling to scatteringstates. We construct it from local pion operators :ππ i(~p,~q) = (pi−qi)[π+(~p)π−(~q)−π−(~p)π+(~q)]. (3.7)Then, the cross-correlator is computed by contracting “stochastic propagators”. Following [14] weuse two kinds of “stochastic propagators” :Q(~x, t |~q, ts,ξ j) = ∑~yD−1(~x, t;~y, ts) · [ei~p·~yξ j(~y)] (3.8)W (~x, t |~k, t1 |~q, ts) = ∑~zD−1(~x, t;~z, t1) · [ei~k·~zγ5Q(~z, t1 |~q, ts)] (3.9)The contraction are described in Fig. 3, and here we make explicit the first one :G1stππ→ρ = ∑j,~xe−i~P~x〈Q(~x, t |~0, ts,ξ j)W †(~x, t | −~p, ts | −~q, ts,ξ j)γ5γ3〉. (3.10)4. Preliminary resultsWe use the simulation setup of the Budapest-Marseille-Wuppertal collaboration [5, 6] with 2levels of HEX smearing [1–3], featuring N f = 2+1 flavors of tree-level improved Clover fermions[6] and the tree-level improved Lüscher-Weisz gauge action [17]. We choose two simulations, forwhich the two lowest-lying scattering states are near the crossing.5Rho decay width from the lattice J. Frison0 5 10 15t00,20,40,60,81|Re vin|, |Im vin|ρ<->1ρ<->2ππ<->1ππ<->2GEVP Eigenvectors at mπ=200MeVFigure 4: Components of generalized eigenvectors, normalized and obtained from operators normalizedso that 〈0 | O(1)O(0) | 0〉 = 1. The eigenvectors are very sensitive to both higher-level contaminationand statistical errors. For t < 6 higher-state contamination is obvious, but for 6 ≤ t ≤ 9 or even more theeigenvectors are constant as should be.0 5 10 15t00,20,40,60,81aEE1E2Energies at mπ=340MeVFigure 5: The two energies obtained from solving generalized eigenvalue problem show a clean plateau,and their energy difference is significant. The same goes for mπ = 200 MeV.The first point is for mπ ' 200 MeV, with β = 3.31 and a = 0.116 fm with a lattice size of323× 48. Contractions were computed in the center-of-mass frame ~P = (0,0,0), in which the ρmass nearly crosses the π(0,0,2π/L)π(0,0,−2π/L) free two-pion state.The other point has mπ ' 340 MeV, with β = 3.31 and a = 0.116 fm. Here, the lattice sizeis 243× 48. The analysis is performed in the moving frame ~P = (0,0,2π/L), in which the ρ cannearly "decay" into π(0,0,2π/L)π(0,0,0).Using Eq. 2.1 on the two energies resulting from the variational method we get (see Fig. 4 and6Rho decay width from the lattice J. Frison5) :g = 5.5±2.9 for the mπ ' 200MeV point, (4.1)g = 6.6±3.4 for the mπ ' 340MeV point, (4.2)where the error is purely statistical. Combining the two results we getg = 6.0±2.2, (4.3)which is in good agreement with experimental data.AknowledgmentsComputations were performed using HPC resources from GENCI-[CCRT/IDRIS] (grant 52275),from FZ Jülich and from the DECI-5 project HADWIDTH, as well as clusters at Wuppertal andCPT. This work is supported in part by EU grants FP7/2007-2013/ERC no208740, MRTN-CT-2006-035482 (FLAVIAnet), OTKA grant AT049652 DFG grant FO 502/2, SFB-TR 55, U.S. De-partment of Energy Grant No. DE-FG02-05ER25681, by CNRS grants GDR no2921 and PICSno4707.References[1] S. Durr et al., “Lattice QCD at the physical point: light quark masses”, arXiv:1011.2403[2] S. Durr et al., “Lattice QCD at the physical point: simulation and analysis detail”, arXiv:1011.2711[3] T. Kurth et al., “Scaling study for 2 HEX smeared fermions: hadron and quark masses.“,arXiv:1011.1780[4] S. Durr et al. Phys. Rev. D81:054507, 2010, arXiv:1001.4692[5] S. Durr et al. Science, 322:1224–1227, 2008[6] S. Durr et al. Phys. Rev. D79:014501, 2009, arXiv:0802.2706[7] M. Luscher. Commun. Math. Phys., 104:177, 1986.[8] M. Luscher. Commun. Math. Phys., 105:153–188, 1986.[9] Martin Luscher and Ulli Wolff. Nucl. Phys., B339:222–252, 1990.[10] Martin Luscher. Nucl. Phys., B354:531–578, 1991.[11] Martin Luscher. Nucl. Phys., B364:237–254, 1991.[12] Steven A. Gottlieb and K. Rummukainen. Nucl. Phys. Proc. Suppl., 47:819–822, 1996.[13] C.H. Kim et al. Nucl. Phys., B727:218–243, 2005[14] S. Aoki et al. Phys. Rev. D, 76:094506, 2007[15] Xu Feng XXVII International Symposium on Lattice Field Theory, July 26-31, 2009, Beijing[16] C. Hoelbling XXVIII International Symposium on Lattice Field Theory, July 14-19, 2010, Villasimius[17] M. Lücher and P. Weisz, Commun. Math. Phys. 97:59, 19857

Rho decay width from the lattice

Frison, J.

Durr, S.

Ramos, A.

Szabo, K.K.

Fodor, Z.

Hoelbling, C.

Katz, S.D.

Krieg, S.

Kurth, T.

Lellouch, L.

Lippert, T.

Portelli, A.

Juelich Shared Electronic Resources

Katz, S. D.

Lellouch, Laurent

Szabo, K. K.

HAL AMU

https://edit.elte.hu/xmlui/bitstream/handle/10831/48069/Rho%20decay%20width%20from%20the%20lattice.pdf?sequence=1&amp;isAllowed=y

Rho decay width from the lattice

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