Isospin-0 ππ s-wave scattering length from twisted mass lattice QCD
L. Liu, S. Bacchio, P. Dimopoulos, J. Finkenrath, R. Frezzotti, C. Helmes, C. Jost, B. Knippschild, B. Kostrzewa, H. Liu, K. Ottnad, M. Petschlies, C. Urbach, M. Werner
IIsospin-0 ππ s-wave scattering length from twisted mass lattice QCD L. Liu, ∗ S. Bacchio,
2, 3
P. Dimopoulos,
4, 5
J. Finkenrath, R. Frezzotti, C. Helmes, C. Jost, B. Knippschild, B. Kostrzewa, H. Liu, K. Ottnad, M. Petschlies, C. Urbach, † and M. Werner (ETM Collaboration) Helmholz-Institut f¨ur Strahlen- und Kernphysik and Bethe Centerfor Theoretical Physics, Universit¨at Bonn, D-53115 Bonn, Germany Department of Physics, University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus Fakult¨at f¨ur Mathematik und Naturwissenschaften, BergischeUniversit¨at Wuppertal, 42119 Wuppertal, Germany Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche EnricoFermi, Compendio del Viminale, Piazza del Viminiale 1, I-00184, Rome, Italy Dipartimento di Fisica, Universit`a di Roma “Tor Vergata”,Via della Ricerca Scientifica 1, I-00133 Rome, Italy Computation-based Science and Technology Research Center,The Cyprus Institute, PO Box 27456, 1645 Nicosia, Cyprus Dipartimento di Fisica, Universit`a and INFN di Roma Tor Vergata, 00133 Roma, Italy Albert Einstein Center for Fundamental Physics, University of Bern, 3012 Bern, Switzerland
We present results for the isospin-0 ππ s-wave scattering length calculated withOsterwalder-Seiler valence quarks on Wilson twisted mass gauge configurations. We usethree N f = 2 ensembles with unitary (valence) pion mass at its physical value (250 MeV),at 240 MeV (320 MeV) and at 330 MeV (400 MeV), respectively. By using the stochasticLaplacian Heaviside quark smearing method, all quark propagation diagrams contributingto the isospin-0 ππ correlation function are computed with sufficient precision. The chi-ral extrapolation is performed to obtain the scattering length at the physical pion mass.Our result M π a I=00 = 0 . a r X i v : . [ h e p - l a t ] O c t ∗ Electronic address: [email protected] † Electronic address: [email protected]
I. INTRODUCTION
Quantum chromodynamics (QCD) is established as the fundamental theory of the strong in-teractions. QCD at low energies is largely determined by chiral symmetry, which is spontaneouslybroken. The effective theory of QCD at low energies is chiral perturbation theory ( χ PT) [1–3],representing a systematic expansion in the quark masses and momenta. Elastic ππ scattering pro-vides an ideal testing ground for the mechanism of chiral symmetry breaking. Since only the pions– the pseudo-Goldstone bosons of SU(2) chiral symmetry – are involved, the expansion is expectedto converge rapidly. In fact, the s-wave scattering length in the weakly repulsive isospin-2 channelcan be reproduced by leading order (LO) χ PT [4] with a deviation of only 0.5% when comparedto the results obtained from experiments combined with Roy equations [5].However, in the isospin-0 channel the situation is different: the interaction is attractive and muchstronger than in the isospin-2 channel. The agreement between LO χ PT and experiments for thes-wave scattering length in the isospin-0 channel is much less impressive: they deviate by around30% [4–6]. Moreover, this channel accommodates the lowest QCD resonance – the mysterious σ or f (500) scalar meson. Although it plays a crucial role in some fundamental features of QCD,its existence was disputed for a long time. Only recently it was established unambiguously withdispersive analyses and new experimental data, see Ref. [7] for a review.This makes a nonperturbative, ab initio computation of ππ interaction properties in the isospin-0channel directly from QCD highly desirable. Lattice QCD is the only available method to performsuch a computation with controlled systematic uncertainties. L¨uscher showed that the infinitevolume scattering parameters can be related to the discrete spectrum of the eigenstates in a finite-volume box [8, 9]. This allows one to compute scattering properties in lattice QCD, which isnecessarily implemented in a finite volume and Euclidean space-time.For the isospin-2 channel, many lattice results have become available. See Refs. [10–13] forthe most recent ones. For the isospin-0 channel the situation is more complicated mainly dueto the fermionic disconnected diagrams contributing here, which are challenging to compute inlattice QCD. To date there are only two full lattice QCD computations dedicated to this channel[11, 14]. In Ref. [11], the s-wave scattering length was computed for three unphysically large pionmasses. An extrapolation to the physical point was performed to obtain the scattering length atphysical pion mass. The authors of Ref. [14] on the other hand extracted many energy eigenstatesin the corresponding channel and obtained the scattering amplitudes for two values of pion mass– 236 MeV and 391 MeV. The information about the σ meson is deduced from the pole structure ensemble β c sw aµ (cid:96) ( L/a ) × T /a N conf cA . .
48 2.10 1.57551 0.009 48 ×
96 615 cA . .
48 2.10 1.57551 0.030 48 ×
96 352 cA . .
32 2.10 1.57551 0.060 32 ×
64 337TABLE I: The gauge ensembles used in this study. The labeling of the ensembles follows the notations inRef. [18, 19]. In addition to the relevant input parameters we give the lattice volume (
L/a ) × T /a and thenumber of evaluated configurations N conf . in the scattering amplitudes at the two unphysical pion masses, respectively.In this work we compute the scattering length of the isospin-0 ππ channel in twisted masslattice QCD [15] and L¨uscher’s finite volume method [8, 9]. As discussed in Ref. [16], the explicitisospin breaking of the twisted mass quark action makes it prohibitively complicated to studythis channel with this action. To circumvent this complication we use a mixed action approachwith Osterwalder-Seiler quarks [17] in the valence sector, which preserves isospin symmetry. Thisapproach introduces additional lattice artefacts due to unitarity breaking. These lattice artefactsare of O ( a ) and will vanish only in the continuum limit. In particular, due to isospin breaking inthe sea there is possibly residual mixing with I = 2 , I z = 0. Since we use only one value of latticespacing, systematic uncertainties in our results are not fully controlled. Further calculations areneeded to explicitly address these uncertainties. However, they are beyond the scope of this work.This paper is organized as follows. The lattice setup is discussed in Sec. II. L¨uscher’s finitevolume method is introduced in Sec. III. In Sec. IV we present the computation of the finite volumespectrum of the isospin-0 ππ system. The result for the scattering length is given in Sec. V. Thelast section is devoted to a brief summary and discussions. II. LATTICE ACTION
The results presented in this paper are based on the gauge configurations generated by theEuropean Twisted Mass Collaboration (ETMC) with Wilson clover twisted mass quark action atmaximal twist [15]. The gauge action is the Iwasaki gauge action [20]. We use three N f = 2ensembles with pion mass at the physical value, at 240 MeV and at 330 MeV, respectively. Thelattice spacing is a = 0 . O ( a )lattice artefacts. In Table I we list the three ensembles with the relevant input parameters, thelattice volume and the number of configurations. More details about the ensembles are presentedin Ref. [18].The sea quarks are described by the Wilson clover twisted mass action. The Dirac operator forthe light quark doublet consists of the Wilson twisted mass Dirac operator [15] combined with theclover term (in the so-called physical basis) D (cid:96) = (cid:101) ∇ − iγ τ (cid:20) W cr + i c sw σ µν F µν (cid:21) + µ (cid:96) , (1)with (cid:101) ∇ = γ µ ( ∇ ∗ µ + ∇ µ ) / ∇ µ and ∇ ∗ µ the forward and backward lattice covariant derivatives.Here c sw is the so-called Sheikoleslami-Wohlert improvement coefficient [21] multiplying the cloverterm and W cr = − ra ∇ ∗ µ ∇ µ + m cr , with m cr the critical mass. µ (cid:96) is the average up/down (twisted)quark mass. a is the lattice spacing and r = 1 the Wilson parameter. D (cid:96) acts on a flavor doubletspinor ψ (cid:96) = ( u, d ) T . In our case the clover term is not used for O ( a ) improvement but serves tosignificantly reduce the effects of isospin breaking [18].The critical mass has been determined as described in Refs. [19, 22]. This guarantees automatic O ( a ) improvement [23], which is one of the main advantages of the Wilson twisted mass formulationof lattice QCD.In the valence sector we introduce quarks in the so-called Osterwalder-Seiler (OS) discretiza-tion [17]. The corresponding up and down single flavor lattice Dirac operators read D ± (cid:96) = (cid:101) ∇ ± iγ (cid:20) W cr + i c sw σ µν F µν (cid:21) + µ OS (cid:96) . (2)From this definition it is apparent that OS up and down quarks have explicit SU(2) isospin symme-try if for both e.g. D + (cid:96) was used. The matching of OS to unitary actions is performed by matchingthe quark mass values, i.e. µ OS (cid:96) = µ (cid:96) . The value of m cr in the OS action can be shown to beidentical to the unitary one and O ( a ) improvement stays valid [17]. Moreover, we have shown inRef. [24] that in such a mixed action approach disconnected contributions to η and η (cid:48) mesons canbe computed and the results agree with the unitary ones [25] in the continuum limit. Therefore,this mixed action approach should works also in the case of ππ scattering, where disconnectedcontributions can be expected to be less important, since OZI suppression is in place. However,there is a potential complication arising from the double poles in flavor-neutral meson propagatorspresent in a quenched or partially quenched theory [26]. The scalar correlators with disconnecteddiagrams suffer from unphysical contributions due to the double poles. The unphysical contribu-tions to the a and ππ correlators have been studied in Refs. [27–30]. In this work, we are notgoing to consider this problem since the formula of these unphysical contributions for our partiallyquenched approach is not available. Also, as will be presented in Sec. IV, our numerical resultsdo not indicate large unphysical contributions. All the correlators we computed numerically arepositive within the obtained statistics and are well described by a single exponential function of t/a in a reasonably large time range. This would not be the case if there were large unphysicalcontributions as shown in Refs. [27–30]. Nevertheless, one should keep in mind that the effects ofthe double poles may cause uncertainties that are not considered in our results.Masses computed with OS valence quarks differ from those computed with twisted mass valencequarks by lattice artefact of O ( a ), in particular( M OS π ) − ( M π ) = O ( a ) . For twisted clover fermions this difference is much reduced as compared to twisted massfermions [18], however, the effect is still sizable. We use the OS pion mass in this paper, withthe consequence that the pion mass value of the cA2.09.48 ensemble takes a value larger (around250 MeV) than the physical one.As a smearing scheme we use the stochastic Laplacian Heavyside (sLapH) method [31, 32] forour computation. The details of the sLapH parameter choices for a set of N f = 2 + 1 + 1 Wilsontwisted mass ensembles are given in Ref. [13]. The parameters for the ensembles used in this workare the same as those for N f = 2 + 1 + 1 ensembles with the corresponding lattice volume. III. L ¨USCHER’S FINITE VOLUME METHOD
L¨uscher’s finite volume method provides a direct relation between the energy eigenvalues of atwo-particle system in a finite box and the scattering phase shift of the two particles in the infinitevolume. Considering two particles with mass m and m in a cubic box of size L , the energy ofthis system in the center-of-mass frame reads E = (cid:113) m + (cid:126)k + (cid:113) m + (cid:126)k , (3)where (cid:126)k is the scattering momentum. For the following discussion, it is convenient to define adimensionless variable q via q = (cid:126)k L (2 π ) , (4)which differs from an integer due to the interaction of the two particles.The general form of L¨uscher’s formula reads [9]:det (cid:34) e iδ l δ ll (cid:48) δ nn (cid:48) − M Γ ln,l (cid:48) n (cid:48) + i M Γ ln,l (cid:48) n (cid:48) − i (cid:35) = 0 , (5)where δ l is the phase shift of the partial wave with angular momentum { l } , Γ denotes an irreduciblerepresentation (irrep) of the cubic group. The matrix in the determinant is labeled by the pair( l, n ), in which { l } are the angular momenta subduced into the irrep Γ and n is an index indicatingthe n th occurrence of that l in the irrep. The matrix element M Γ ln,l (cid:48) n (cid:48) is a complicated function of q but can be computed numerically.In this work we are interested in the s-wave low energy scattering in the isospin-0 ππ channel.Therefore, we will compute only the lowest energy level in the center-of-mass frame. In this caseone should consider the irrep A +1 . Assuming that the effects of the partial waves with l ≥ q cot δ ( k ) = Z (1; q ) π / , (6)where Z (1; q ) is the L¨uscher zeta-function which can be evaluated numerically for given valueof q . Using the effective range expansion of s-wave elastic scattering near threshold, we have k cot δ ( k ) = 1 a + 12 r k + O ( k ) , (7)where a is the scattering length and r is the effective range parameter. Once the finite volumeenergy E is determined from lattice QCD simulations, the scattering length a can be calculatedfrom the following relation 2 πL Z (1; q ) π / = 1 a + 12 r k + O ( k ) . (8)It should be pointed out that L¨uscher’s formulas presented here, i.e. Eqs. 5 and 6, are for thescattering processes with k >
0. The phase shift δ ( k ) in the continuum is only defined forpositive k . In the case of negative k , one can introduce a phase σ ( k ) which is related to δ ( k )by analytic continuation of tan σ ( k ) = − i tan δ ( k ) [9]. Only when there is a bound state at thecorresponding energy, the phase σ ( k ) has physical interpretation and its value equals to − π/ π ) in the continuum and infinite volume limit. For the purpose of this paper – calculatingthe scattering length, we will only consider the lowest energy level in the center-of-mass frame. Sincethe interaction in the isospin-0 ππ channel is attractive, this energy level is below the threshold,i.e. k <
0. For convenience, in the following we will always use the notation k cot δ ( k ), which isunderstood as the analytically continued form for k <
0. Please note that Eq. 8 holds for bothpositive and negative k as long as the modulus of k is close to zero. IV. FINITE VOLUME SPECTRUM
In lattice QCD, the discrete spectra of hadronic states are extracted from the correlation func-tions of the interpolating operators that resemble the states. Due to the isospin symmetry breakingof the twisted mass quark action, it is difficult to investigate the isospin-0 ππ channel directly inunitary twisted mass lattice QCD [16]. For this reason we use a mixed action approach with the OSaction [17] in the valence sector and choose D + (cid:96) in Eq. 2 for both up and down quarks, so that theisospin symmetry is guaranteed in the valence sector. In this section we describe our methodologyto calculate the energies of the isospin-0 ππ system. A. Computation of the correlation functions
We define the interpolating operator that represents the isospin-0 ππ state in terms of OSvalence quarks O I=0 ππ ( t ) = 1 √ π + π − ( t ) + π − π + ( t ) + π π ( t )) , (9)with single pion operators summed over spatial coordinates x to project to zero momentum π + ( t ) = (cid:88) x ¯ dγ u ( x , t ) , π − ( t ) = (cid:88) x ¯ uγ d ( x , t ) ,π ( t ) = (cid:88) x √ uγ u − ¯ dγ d )( x , t ) . (10)Here u and d represent the OS up and down quarks, respectively. With OS valence quarks all threepions are mass degenerate and will be denoted as M OS π .The energy of the isospin-0 ππ state can be computed from the exponential decay in time ofthe correlation function C ππ ( t ) = 1 T T − (cid:88) t src =0 (cid:104)O I=0 ππ ( t + t src ) ( O I=0 ππ ) † ( t src ) (cid:105) , (11)where T is the temporal lattice extend. The four diagrams contributing to this correlation function,namely the direct connected diagram D ( t ), the cross diagram X ( t ), the box diagram B ( t ) and thevacuum diagram V ( t ), are depicted in Fig. 1 (a)-(d). The correlation function can be expressed interms of all relevant diagrams as C ππ ( t ) = 2 D ( t ) + X ( t ) − B ( t ) + 3 V ( t ) . (12) C ππ and the contributions from individual diagrams D, X, B and V are plotted in Fig. 2 for thethree ensembles. H a L D H t L H b L X H t L H c L B H t L H d L V H t L H e L H f L H g L FIG. 1: Diagrams contributing to the correlation functions. (a)–(d) represent the usual contributions to C ππ , while (e)–(f) need to be taken into account due to unitarity breaking effects. Even though we have full SU(2) isospin symmetry in the valence sector when using OS valencequarks as described above, we have to consider effects of unitarity breaking. This may in particularhappen due to the vacuum diagram V ( t ). There is no symmetry available to prevent this diagramto couple for instance to intermediate states of n ≥ n = 1 (and maybe n = 2) will dominate the large Euclidean time behavior of the correlationfunction C ππ , if the overlap of the used interpolating operators with these states is nonzero. Inorder to resolve this mixing, we build a 2 × C ij ( t ) = 1 T T − (cid:88) t src =0 (cid:104)O i ( t + t src ) O † j ( t src ) (cid:105) (13)with i, j labeling the operator O I=0 ππ and the unitary neutral pion operator π ,uni ( t ) = (cid:88) x √ uγ u − ¯ d (cid:48) γ d (cid:48) )( x , t ) , (14)where u and d (cid:48) are the (unitary) Wilson clover twisted mass up and down quarks. We use d (cid:48) todistinguish it from OS down quark in Eq. 10. The twisted mass up quark coincides with the OS up quark with our matching scheme of the OS to the unitary action.The diagrams contributing to the correlation function of the unitary neutral pion operator aredepicted in Fig. 1 (e) - (f). The two operators couple solely via the vacuum diagram, see Fig. 1 (g).0 t/a -1 l og [ C ( t ) ] cA2.09.48 D ( t ) − B ( t ) X ( t )3 V ( t )2 D ( t ) − B ( t )+ X ( t )+3 V ( t ) t/a -1 l og [ C ( t ) ] cA2.30.48 D ( t ) − B ( t ) X ( t )3 V ( t )2 D ( t ) − B ( t )+ X ( t )+3 V ( t ) t/a -1 l og [ C ( t ) ] cA2.60.32 D ( t ) − B ( t ) X ( t )3 V ( t )2 D ( t ) − B ( t )+ X ( t )+3 V ( t ) FIG. 2: Correlation functions of the operator O I=0 ππ and the single diagrams D, X, B, V for the three ensembleslisted in Table I.
The computation of the disconnected diagrams, e.g. Fig. 1 (c), (d), (f), and (g), requires thequark propagator from a time slice t to the same time for every t . This has been a challenge inlattice QCD for decades. By using the stochastic LapH quark smearing method [31, 32], we haveall-to-all propagators available. The disconnected diagrams can be computed directly from thosepropagators.We can reduce lattice artefacts in the vacuum diagrams following the ideas worked out inRef. [24]. In the continuum limit the difference between u + ( d + ) quarks corresponding to D + (cid:96) and1 u − ( d − ) quarks corresponding to D − (cid:96) vanishes [17]. Therefore, we can write O ( a ) = (cid:104) ¯ u + d + ( x ) ¯ d + u + ( y ) − ¯ u − d − ( x ) ¯ d − u − ( y ) (cid:105) = Tr { S + ( x, y ) S + ( y, x ) } − Tr { S − ( x, y ) S − ( y, x ) } = Tr { S + ( x, y ) S + ( y, x ) } − Tr { ( S + ( x, y ) S + ( y, x )) † } = 2 i Im Tr { S + ( x, y ) S + ( y, x ) } , where S ± ≡ ( D ± (cid:96) ) − are the OS quark propagators and we have used ( S + ) † = γ S − γ . This showsthat the imaginary part of the loop needed in the contraction of the vacuum diagram V is a purelattice artefact and we will drop it in the computation. The same argument holds for the vacuumdiagrams shown in Fig. 1 (f) and (g). B. Determination of the energies
Due to the finite temporal extend T of the lattice, the correlation functions of multiparticleoperators are polluted by so-called thermal states [33]. In the case of interest here, there is aconstant contribution to C ππ ( t ) of the form ∝ |(cid:104) π ± , | O I =0 ππ | π ± , (cid:105)| · e − M OS π T , which vanishes in the infinite volume limit T → ∞ . However, at finite T -values it will dominate thecorrelation function at large Euclidean time. To remove this artefact we define a shifted correlationmatrix ˜ C ( t ) = C ( t ) − C ( t + δt ) . (15)The new matrix ˜ C is then free of any constant pollution from the thermal states. The value of δt can be adjusted for optimal results. We take δt = 4 in our analysis. Note that the shiftingprocedure also subtracts any constants stemming from vacuum expectation values.The energy levels can then be obtained by solving the generalized eigenvalue problem(GEVP) [34] ˜ C ( t ) v n ( t, t ) = λ n ( t, t ) ˜ C ( t ) v n ( t, t ) . (16)The eigenvalues λ n ( t, t ) are expected to have the following time dependence λ n ( t, t ) = A n sinh (cid:20)(cid:18) T − t − δt (cid:19) E n (cid:21) . (17)2 t/a a E t/a a E FIG. 3: Effective energies for the ensemble cA2.09.48. The grey bars indicate the fitted values of theenergies and the fit ranges. The left and right panels correspond to before and after removing the excitedstate contaminations, respectively.
The sinh-like behavior comes from the shifting of the correlation functions in Eq. 15. The energies E n are then obtained by fitting the above functional form to the eigenvalues λ n ( t, t ) in the rangewhere the effective energy shows a plateau. The value of t should be chosen such that thecorrelation function at t is dominated by the states we are interested in. We tried various t values in the range of 1 to 7 and found negligible differences in the energies. In the following weset t = 1. With the two operators used here, we obtain two energy levels, one of which is farbelow the other one. The lower one agrees with the unitary neutral pion mass, while the higherone is close to 2 M OS π . Hence, we identify the higher one to be the isospin-0 ππ state. In principle,multi neutral and charged unitary pion states could also appear in the spectrum. To resolve thesestates, more operators need to be included. We have tried so and merely found increased statisticalerrors of the I = 0 ππ state. Therefore, we cannot finally exclude possible contaminations fromsuch states.As an example, the effective energies of the two eigenvalues for ensemble cA2.09.48 are shownin Fig. 3 (a). The fitted energies and fit ranges are indicated by the grey bands in the plot.To further improve our results we adopt a method to remove the excited state contaminations[35], which we have recently used successfully to study η and η (cid:48) mesons [25, 36]. It is based on theassumption that vacuum diagrams are only sizeable for low lying states, but negligible for higher3excited states. Of course, the validity of this assumption must be checked in the Monte-Carlo data.In our case we know there is a very sizable disconnected contribution to the unitary neutral pion,which represents a pure lattice artefact [37]. A similar contribution has not been found to anyother unitary correlation function. For the ππ correlation function there are indications that thedisconnected contribution is already small by itself [11].The connected contractions contributing to ˜ C are computed with sufficient precision, so we canreliably determine the ground states in the connected correlators and subtract the excited statecontributions. We then build a correlation matrix ˜ C sub from the subtracted connected and theoriginal disconnected correlators. If disconnected contributions were relevant only for the groundstates, one should find – after diagonalizing ˜ C sub – a plateau for both states from small values of t on. Note that with this procedure only the small t behavior of the correlation functions is altered.To be more specific, the connected contributions to the correlation function C ππ ( t ) is given by C con ππ ( t ) = 2 D ( t ) + X ( t ) − B ( t ) . (18)We fit the functional form Eq. 17 to the shifted correlator ˜ C con ππ ( t ) = C con ππ ( t ) − C con ππ ( t − δt ). Afterobtaining the parameters A and E con ππ from the fit, the connected correlator is reconstructed as˜ C con , sub ππ ( t ) = A sinh (cid:20)(cid:18) T − t − δt (cid:19) E con ππ (cid:21) , (19)in which the excited state contaminations are subtracted. We repeat the fit to the data of ˜ C con ππ ( t ) formany different fit ranges. The expectation values of the fit parameters are computed as the weightedmedian over these many fits [13]. By doing this, the systematics arising from different fit ranges isexpected to be taken into account. The full correlator is then given by ˜ C sub ππ ( t ) = ˜ C con , sub ππ ( t )+3 ˜ V ( t ),where ˜ V ( t ) is the shifted vacuum correlator ˜ V ( t ) = V ( t ) − V ( t + δt ). The same procedure isperformed for the unitary π correlation function.In Fig. 3 (b), we present the effective energies of the two eigenvalues of the subtracted correlatormatrix ˜ C sub for the same ensemble as in Fig. 3 (a). Clearly a plateau appears at much earlier t -valuecompared to Fig. 3 (a), while the fitted energies agree very well. Therefore, we use this procedure– which allows us to determine in particular the interacting energy E ππ with much higher accuracy– for the results presented here.The effective energies after removing the exited states for the ensembles cA . .
48 and cA . . E ππ and the unitary π mass M π obtainedfrom the procedure described above. The unitary charged pion mass M π ± and the OS pion mass M OS π are added for convenience.4 t/a a E cA2.30.48 t/a a E cA2.60.32 FIG. 4: Effective energies after removing the excited states contaminations for the ensembles cA . .
48 and cA . . aM π ± aM π aM OS π aE ππ a ˆ E ππ cA . .
48 0.06212(6) 0.058(2) 0.11985(15) 0.2356(4) 0.2350(4) cA . .
48 0.11197(7) 0.108(2) 0.15214(11) 0.3010(3) 0.3009(3) cA . .
32 0.15781(15) 0.149(2) 0.18844(24) 0.3647(5) 0.3645(5)TABLE II: Pion masses and the ππ interacting energies in lattice units for the three ensembles. In order to estimate possible artefacts from the mixing with the unitary π , we also determinedthe energy ˆ E ππ by fitting to only the single correlator ˜ C sub ππ ( t ), without including the operator forthe unitary neutral pion. The values are given in the last column of Table II. One can see that themean value of ˆ E ππ is slightly lower than E ππ for all three ensembles, but they agree very well witheach other considering the statistical error. Keeping in mind that π mixing is purely a latticeartefact, the agreement between E ππ and ˆ E ππ indicates that we are not likely to suffer severe latticeartefact here. This can be also seen in the small mass splitting between the unitary charged andneutral pions.5 Ensemble ( ak ) ak cot δ ( k ) ar k M OS π a I=00 M OS π /f OS π cA . .
48 -0.00049(4) 0.168(19) 0.0037(3)(2) 0.730(83)(1) 1.86(2) cA . .
48 -0.00050(4) 0.167(19) 0.0042(3)(2) 0.94(11)(0) 2.21(1) cA . .
32 -0.00224(9) 0.074(7) 0.0224(9)(22) – 2.63(1)TABLE III: The values of squared scattering momentum k , k cot δ ( k ), r k (see appendix), M OS π a I=00 and M OS π /f OS π for the three considered ensembles. Values for k , k cot δ and r k are in lattice units. The firsterror is the statistical error, the second error comes from the uncertainty of r , see Appendix A. V. RESULTSA. Scattering length
The scattering momentum k is calculated from Eq. 3 with the energies E ππ and the OS pionmasses listed in Table II. Then the scattering length can be obtained from Eq. 8. Considering therelatively strong interaction in the isospin-0 ππ channel, one has to investigate the contribution of O ( k ) and higher order terms in the effective range expansion. Since we only have one energy levelfor each pion mass, we are not able to determine the effective range r with our lattice simulations.We rely on the r values determined from χ PT [2]. See Appendix A for the details of the r valuesused in our analysis.The values of k , k cot δ ( k ) and r k in lattice units for all three ensembles are given inTable III. For the ensembles cA . .
48 and cA . .
48 the scattering momentum is small due tothe large volume. Therefore, the contribution of r k is expected to be small. As visible fromTable III, the value of r k is indeed less than 3% of k cot δ ( k ) for these two ensembles. Wecompute the scattering length from Eq. 8 by ignoring the O ( k ) term, which is well justified forthe ensembles cA . .
48 and cA . .
48. The values of M OS π a I=00 for these two ensembles are alsogiven in Table III. The first error is the statistical error and the second error comes from theuncertainty of the effective range r .As for the ensemble cA2.60.32, the value of r k is rather large compared to k cot δ ( k ). Thisindicates that the effective range expansion up to O ( k ) might not be a good approximation andthe O ( k ) term might not be negligible. Since the contribution of O ( k ) is unclear, we refrain fromgiving the scattering length for this ensemble. There are two possible reasons for the invalidity ofthe effective range expansion. First, due to the relatively small volume of the ensemble cA2.60.32,the value of k is not small enough to make the expansion converge rapidly. Second, at the pionmass around 400 MeV, which is the OS pion mass of the ensemble cA2.60.32, there might be virtual6 M OSπ /f OSπ M O S π a I = FIG. 5: Chiral extrapolation using only the data point with lower pion mass. The grey band represents theuncertainty. The red point indicates the extrapolated value at physical pion mass. or bound state poles appearing in the isospin-0 ππ scattering amplitude [38–43]. The appearance ofsuch states will give a very large scattering length – positively (negatively) large if it was a virtual(bound) state. Hence, the leading order in the effective range expansion, i.e. a , becomes verysmall compared to the higher orders. In this case the NLO term r k can contribute significantlycompared to the LO term even when k is small. Assuming that the contribution of O ( k ) termis not bigger than the O ( k ) term, we can qualitatively estimate the scattering length for thisensemble to be a very large positive number, which features a virtual state. However, we do notexclude the possibility of a bound state because we do not include single meson operators when wecompute the matrix of correlation functions. Including these operators might change the resultingspectrum and thus the scattering length. B. Chiral extrapolation
In order to obtain the scattering length at the physical pion mass, one needs to perform achiral extrapolation. ππ scattering has been studied extensively in χ PT in the literature [2, 4, 44–46]. Since we only have two data points, we fit the NLO χ PT formula, which contains one freeparameter, to our data. When expressed in terms of M π and f π computed from lattice simulations,the formula reads [11] M π a I=00 = 7 M π πf π (cid:20) − M π π f π (cid:18) M π f π − − (cid:96) I=0 ππ (cid:19)(cid:21) , (20)7where (cid:96) I=0 ππ is a combination of the low energy coefficients ¯ l i ’s : (cid:96) I=0 ππ = 4021 ¯ l + 8021 ¯ l −
57 ¯ l + 4¯ l + 9 ln M π f π,phy . (21)In this expression, the renormalization scale is set to be the physical pion decay constant f π,phy .Note that we work in the normalization with f π,phy = 130 . χ PTto perform the chiral extrapolation. The χ PT for the mixed action with twisted mass sea quarksand OS valence quarks is presented in Ref. [47]. However, using the two data points at one valueof lattice spacing, we are not able to implement the mixed action χ PT formula. With this caveatin mind, we proceed with our analysis.The OS pion decay constant f OS π has not been determined by ETMC yet. We compute f OS π forthe three ensembles used in this paper. The values of M OS π /f OS π are collected in the last columnof Table III. The details of their computation are presented in Appendix B. We recall that the OSpion mass values are larger than their unitary counterpart, see Table II, such that our lowest OSpion mass value is at around 250 MeV.The method we are applying here is valid only in the elastic region. Therefore, the pion massvalues must be small enough to be below threshold where the σ meson becomes stable. Thisthreshold is not known exactly, but results obtained with the 1-loop inverse amplitude method [40](see also Refs. [38, 39, 48]) suggest that M π <
400 MeV should be safe [41]. Our two data points areobtained at pion mass around 250 MeV and 320 MeV respectively, both are below this threshold.Furthermore, the pion mass value should also be small enough to make the chiral expansion valid.To be safe, we perform the chiral extrapolation using only the data point with the lower pion mass( 250 MeV). The results of this extrapolation are given in Table IV as fit-1 and illustrated in Fig. 5.The results of the fit with the two data points, which are given in Table IV as fit-2, agree withfit-1 within errors. We take the difference as an estimate of the systematics arising from chiralextrapolation. This leads to our final result for the scattering length: M π a I=00 = 0 . stat (6) sys . (22)We remark here that the extrapolation is strongly constrained since M OS π a I =00 must vanish in thechiral limit. This explains the small statistical uncertainty on the value extrapolated to the physicalpoint.We compare our result in Table V to other results available in the literature. Our result for M π a I =00 is lower, but within errors still compatible with most recent experimental, lattice and Roy8 fit-1 fit-2 M π a I=00 (Phy.) 0.198(9) 0.192(5) (cid:96)
I=0 ππ χ / dof – 0.75/1TABLE IV: Results of the NLO chiral fit. fit-1 includes only one data point, namely cA2.09.48, while fit-2includes both, cA2.09.48 and cA2.30.48. M π a I=00 (cid:96)
I=0 ππ This work 0 . . . . . . − CGL [46] 0 . . . . . . . . M π a I=00 and (cid:96)
I=0 ππ . equations results. Due to our comparably low value for M π a I =00 , the value for (cid:96) I =0 ππ is also relativelylow. This is a direct consequence of the NLO χ PT description we are using.
VI. DISCUSSION AND SUMMARY
In this paper, the isospin-0 ππ scattering is studied with L¨uscher’s finite volume formalism intwisted mass lattice QCD. We use a mixed action approach with the OS action in the valencesector to circumvent the complications arising from isospin symmetry breaking in the twisted massquark action. The stochastic LapH quark smearing method is used to compute all-to-all quarkpropagators, which are required to compute the quark disconnected diagrams contributing to theisospin-0 ππ correlation function. The lowest energy level in the rest frame is extracted for three N f = 2 ensembles with three different pion mass values. The scattering length is computed withL¨uscher’s formula for the two ensembles with the lowest pion mass values. For the third ensemblewith the largest pion mass value the scattering length cannot be determined reliably. In thecomputation of the scattering length we include the O ( k ) term in the effective range expansionusing values for the effective range, which we compute using χ PT. The chiral extrapolation of M π a I =00 is performed using NLO χ PT. Extrapolated to the physical value of M π /f π , our resultis M π a I =00 = 0 . χ PT is rather limited. We cannot exclude thatsuch contributions are sizable.For these reasons a future study should include several lattice spacing values and ideally en-sembles at the physical point. In order to avoid isospin breaking and unitarity breaking effects, wewill repeat this computation with an action without isospin breaking.
Acknowledgments
We thank the members of ETMC for the most enjoyable collaboration. The computer time forthis project was made available to us by the John von Neumann-Institute for Computing (NIC)on the Jureca and Juqueen systems in J¨ulich. We thank A. Rusetsky and Zhi-Hui Guo for veryuseful discussions and R. Brice˜no for valuable comments. We are grateful to Ulf-G. Meißner forcarefully reading this manuscript and helpful comments. This project was funded by the DFGas a project in the Sino-German CRC110. S. B. has received funding from the Horizon 2020research and innovation program of the European Commission under the Marie Sklodowska-Curieprogramme GrantNo. 642069. This work was granted access to the HPC resources IDRIS under theallocation 52271 made by GENCI. The open source software packages tmLQCD [49], Lemon [50],DD α AMG [51] and R [52] have been used.
Appendix A: Effective range from χ PT In order to investigate the contribution of the O ( k ) term in the effective range expansion, weneed to know the value of the effective range r . As explained in Section V A, we estimate r from χ PT.In Ref. [2], the chiral expansion of the threshold parameter b to NLO is given as b = 12 πf π (cid:26) M π f π (cid:20) − π ln M π µ + 32 l r + 24 l r + 4 l r − π (cid:21)(cid:27) , (A1)where µ is the renormalization scale, l r , l r and l r are the renormalized, quark mass independentcouplings. In this expression, we have replaced the low energy parameters M and F with their(lattice) physical values M π and f π using the NLO chiral expansions of M π and f π , which are given0 Ensemble cA2.09.48 cA2.30.48 cA2.60.32 r /a -14.9(0.8) -17(1) -20(2)TABLE VI: The effective range in lattice unit. in the same reference. Please also note that our convention of f π ( ∼
130 MeV) is different from the F π ( ∼ r is related to b as r = − b M π . In order to avoid the uncertainty inlattice scale setting, we write r in lattice units as a function of the dimensionless parameters aM π and M π /f π : r a = − b M π aM π = − aM π M π πf π (cid:26) M π f π (cid:20) − π ln M π f π + 32 l r + 24 l r + 4 l r − π (cid:21)(cid:27) . Here the renormalization scale µ is set to be the physical pion decay constant f phy π . To writethe formula as a function of M π /f π , we have replaced f phy π with f π . The corrections due to thisreplacement appear at next-to-next-to-leading order.We take the values of the scale independent couplings ¯ l , ¯ l and ¯ l determined in Ref. [53]:¯ l = − . ± . , ¯ l = 4 . ± . , ¯ l = 4 . ± . . (A2)From the relations between l ri and ¯ l i l ri = γ i π (cid:18) ¯ l i + ln M π µ (cid:19) (A3)with γ = 1 / γ = 2 / γ = 2, we calculate the values of l ri at µ = f phy π : l r = − . , l r = 0 . , l r = 0 . . (A4)The effective range is calculated with the l ri ’s in Eq. A4 and the values of aM OS π and M OS π /f OS π in Table II and Table III. The results of r /a for the three ensembles are presented in Table VI. Theerrors are estimated from the errors of l ri ’s and the statistical uncertainties of aM OS π and M OS π /f OS π . Appendix B: Determination of the OS f π values The chiral extrapolation of the I = 0 scattering length is most conveniently performed in M π /f π .For this reason we need to compute also the OS pion decay constant. It is given by the following1relation [54] f OS π = Z A (cid:104) | A | π (cid:105) M OS π , (B1)with the (OS valence quark) axial vector component A = ¯ uγ γ d . The corresponding renormal-ization constant Z A has been determined in Ref. [55] for the action and β -value used here. Itreads Z A = 0 . . The matrix element (cid:104) | A | π (cid:105) can be determined from suitable correlation functions. We used theoperator O A = (cid:88) x ¯ uγ γ d ( x , t )together with π + from Eq. 10 to build a 2 × [1] S. Weinberg, Physica A96 , 327 (1979).[2] J. Gasser and H. Leutwyler, Ann. Phys. , 142 (1984).[3] J. Gasser and H. Leutwyler, Phys. Lett.
B188 , 477 (1987).[4] S. Weinberg, Phys. Rev. Lett. , 616 (1966).[5] NA48-2
Collaboration, J. R. Batley et al. , Eur. Phys. J.
C70 , 635 (2010).[6] S. Pislak et al. , Phys. Rev.
D67 , 072004 (2003), arXiv:hep-ex/0301040 [hep-ex] , [Erratum: Phys.Rev.D81,119903(2010)].[7] J. R. Pelaez, Phys. Rept. , 1 (2016), arXiv:1510.00653 [hep-ph] .[8] M. L¨uscher, Commun.Math.Phys. , 153 (1986).[9] M. L¨uscher, Nucl.Phys.
B354 , 531 (1991).[10] T. Yagi, S. Hashimoto, O. Morimatsu and M. Ohtani, arXiv:1108.2970 [hep-lat] .[11] Z. Fu, Phys. Rev.
D87 , 074501 (2013), arXiv:1303.0517 [hep-lat] .[12]
PACS-CS
Collaboration, K. Sasaki, N. Ishizuka, M. Oka and T. Yamazaki, Phys.Rev.
D89 , 054502(2014), arXiv:1311.7226 [hep-lat] .[13]
ETM
Collaboration, C. Helmes et al. , JHEP , 109 (2015), arXiv:1506.00408 [hep-lat] . [14] R. A. Briceno, J. J. Dudek, R. G. Edwards and D. J. Wilson, Phys. Rev. Lett. , 022002 (2017), arXiv:1607.05900 [hep-ph] .[15] ALPHA
Collaboration, R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz, JHEP , 058 (2001), hep-lat/0101001 .[16] M. I. Buchoff, J.-W. Chen and A. Walker-Loud, Phys.Rev. D79 , 074503 (2009), arXiv:0810.2464[hep-lat] .[17] R. Frezzotti and G. C. Rossi, JHEP , 070 (2004), arXiv:hep-lat/0407002 .[18] ETM
Collaboration, A. Abdel-Rehim et al. , arXiv:1507.05068 [hep-lat] .[19] ETM
Collaboration, R. Baron et al. , JHEP , 111 (2010), arXiv:1004.5284 [hep-lat] .[20] Y. Iwasaki, Nucl. Phys. B258 , 141 (1985).[21] B. Sheikholeslami and R. Wohlert, Nucl.Phys.
B259 , 572 (1985).[22] T. Chiarappa et al. , Eur.Phys.J.
C50 , 373 (2007), arXiv:hep-lat/0606011 [hep-lat] .[23] R. Frezzotti and G. C. Rossi, JHEP , 007 (2004), hep-lat/0306014 .[24] ETM
Collaboration, K. Ottnad, C. Urbach and F. Zimmermann, Nucl. Phys.
B896 , 470 (2015), arXiv:1501.02645 [hep-lat] .[25]
ETM
Collaboration, C. Michael, K. Ottnad and C. Urbach, Phys.Rev.Lett. , 181602 (2013), arXiv:1310.1207 [hep-lat] .[26] C. W. Bernard, M. Golterman, J. Labrenz, S. R. Sharpe and A. Ukawa, Nucl. Phys. Proc. Suppl. ,334 (1994).[27] C. Aubin, J. Laiho and R. S. Van de Water, Phys. Rev. D77 , 114501 (2008), arXiv:0803.0129[hep-lat] .[28] W. A. Bardeen, A. Duncan, E. Eichten, N. Isgur and H. Thacker, Phys.Rev.
D65 , 014509 (2001), arXiv:hep-lat/0106008 [hep-lat] .[29] M. Golterman, T. Izubuchi and Y. Shamir, Phys. Rev.
D71 , 114508 (2005), arXiv:hep-lat/0504013[hep-lat] .[30] C. W. Bernard and M. F. L. Golterman, Phys. Rev.
D53 , 476 (1996), arXiv:hep-lat/9507004[hep-lat] .[31]
Hadron Spectrum
Collaboration, M. Peardon et al. , Phys. Rev.
D80 , 054506 (2009), arXiv:0905.2160 [hep-lat] .[32] C. Morningstar et al. , Phys. Rev.
D83 , 114505 (2011), arXiv:1104.3870 [hep-lat] .[33] X. Feng, K. Jansen and D. B. Renner, Phys.Lett.
B684 , 268 (2010), arXiv:0909.3255 [hep-lat] .[34] C. Michael and I. Teasdale, Nucl.Phys.
B215 , 433 (1983).[35] H. Neff, N. Eicker, T. Lippert, J. W. Negele and K. Schilling, Phys.Rev.
D64 , 114509 (2001), arXiv:hep-lat/0106016 [hep-lat] .[36]
ETM
Collaboration, K. Jansen, C. Michael and C. Urbach, Eur.Phys.J.
C58 , 261 (2008), arXiv:0804.3871 [hep-lat] .[37] P. Dimopoulos, R. Frezzotti, C. Michael, G. C. Rossi and C. Urbach, Phys. Rev.
D81 , 034509 (2010), arXiv:0908.0451 [hep-lat] .[38] M. Albaladejo and J. A. Oller, Phys. Rev. D86 , 034003 (2012), arXiv:1205.6606 [hep-ph] .[39] J. R. Pelaez and G. Rios, Phys. Rev.
D82 , 114002 (2010), arXiv:1010.6008 [hep-ph] .[40] C. Hanhart, J. R. Pelaez and G. Rios, Phys. Rev. Lett. , 152001 (2008), arXiv:0801.2871[hep-ph] .[41] C.Hanhart, Remarks on the pion mass dependence of the sigma meson, private communication.[42] V. Bernard, M. Lage, U. G. Meissner and A. Rusetsky, JHEP , 019 (2011), arXiv:1010.6018[hep-lat] .[43] M. Doring, U.-G. Meissner, E. Oset and A. Rusetsky, Eur. Phys. J. A47 , 139 (2011), arXiv:1107.3988[hep-lat] .[44] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. Sainio, Nucl.Phys.
B508 , 263 (1997), arXiv:hep-ph/9707291 [hep-ph] .[45] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. E. Sainio, Phys. Lett.
B374 , 210 (1996), arXiv:hep-ph/9511397 [hep-ph] .[46] G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys.
B603 , 125 (2001), arXiv:hep-ph/0103088[hep-ph] .[47] A. Walker-Loud, arXiv:0904.2404 [hep-lat] .[48] C. Hanhart, J. R. Pelaez and G. Rios, Phys. Lett.
B739 , 375 (2014), arXiv:1407.7452 [hep-ph] .[49] K. Jansen and C. Urbach, Comput.Phys.Commun. , 2717 (2009), arXiv:0905.3331 [hep-lat] .[50]
ETM
Collaboration, A. Deuzeman, S. Reker and C. Urbach, arXiv:1106.4177 [hep-lat] .[51] C. Alexandrou et al. , accepted by PRD (2016), arXiv:1610.02370 [hep-lat] .[52] R Development Core Team,
R: A language and environment for statistical computing , R Foundationfor Statistical Computing, Vienna, Austria, 2005, ISBN 3-900051-07-0.[53] J. Bijnens and G. Ecker, Ann. Rev. Nucl. Part. Sci. , 149 (2014), arXiv:1405.6488 [hep-ph] .[54] ETM
Collaboration, M. Constantinou et al. , JHEP , 068 (2010), arXiv:1004.1115 [hep-lat] .[55] ETM
Collaboration, C. Alexandrou, M. Constantinou and H. Panagopoulos, arXiv:1509.00213[hep-lat] .[56]
ETM
Collaboration, P. Boucaud et al. , Comput. Phys. Commun. , 695 (2008), arXiv:0803.0224[hep-lat]arXiv:0803.0224[hep-lat]