Pion-pion scattering and the timelike pion form factor from N f =2+1 lattice QCD simulations using the stochastic LapH method
John Bulava, Ben Hörz, Brendan Fahy, K. J. Juge, Colin Morningstar, Chik Him Wong
PPion-pion scattering and the timelike pion formfactor from N f = + lattice QCD simulations usingthe stochastic LapH method John Bulava and Ben Hörz ∗ School of Mathematics, Trinity College DublinDublin 2, IrelandE-mail: [email protected] , [email protected] Brendan Fahy
High Energy Accelerator Research Organization (KEK)Ibaraki 305-0801, JapanE-mail: [email protected]
K. J. Juge
Dept. of Physics, University of the PacificStockton, CA 95211, USAE-mail: [email protected]
Colin Morningstar
Dept. of Physics, Carnegie Mellon UniversityPittsburgh, PA 15213, USAE-mail: [email protected]
Chik Him Wong
Dept. of Physics, University of WuppertalGaussstrasse 20, D-42119 GermanyE-mail: [email protected]
We report on progress applying the stochastic LapH method to estimate all-to-all propagators re-quired in correlation functions of multi-hadron operators relevant for pion-pion scattering. Large-volume results for I = I = N f = + m π = I = N f = + The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan ∗ Combined proceedings from the talks of J. Bulava and B. Hörz. 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). http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] N ov ion-pion scattering and the timelike pion form factor Lattice QCD simulations are inevitably carried out in finite volume and euclidean time, whichcomplicates scattering calculations [1]. Since most excited hadrons are unstable resonances whichappear experimentally as features in scattering cross sections, first-principles calculation of hadron-hadron scattering amplitudes is desirable. The relation between finite-volume two-hadron spectraand infinite-volume elastic scattering amplitudes was formulated by Lüscher [2, 3] more than twodecades ago and extended to moving frames in Ref. [4]. However, only recently have algorithmicadvances in lattice QCD spectroscopy enabled finite-volume spectra to be calculated efficiently inlarge volume with light pion masses.These advances concern the treatment of all-to-all propagators, which are required to givedefinite momenta to all hadrons and to treat valence-quark-line-disconnected Wick contractions.Laplacian-Heaviside (LapH) quark smearing projects the quark propagator onto the subspace span-ned by the lowest-lying N v eigenmodes of the three-dimensional covariant Laplace operator [5].Stochastic noise introduced only in the LapH subspace results in more efficient estimators for all-to-all quark propagators compared to noise on the entire lattice [6].The spatial profile of the LapH subspace projector is approximately Gaussian, as with otherquark smearing procedures. In order to maintain a constant physical smearing radius, N v mustincrease proportionally to the spatial volume. However, in Ref. [6] it was demonstrated that with amoderate amount of dilution in the LapH subspace, the number of quark matrix inversions can beheld constant as the spatial volume is increased without increasing the stochastic estimation errorrelative to the gauge noise. This enables all-to-all quark propagators to be estimated efficiently inlarge spatial volumes for a reasonable cost.The efficient treatment of all-to-all quark propagators in turn enables precision calculation ofcorrelation functions containing multi-hadron interpolating operators with definite momenta and/ordisconnected Wick contractions. From these correlation functions, finite-volume energies can beprecisely extracted, which then yield elastic scattering amplitudes. The application of these tech-niques to extract elastic pion-pion scattering amplitudes from large volume ensembles is the subjectof this work. Sec. 1 details a first application of the stochastic LapH method to large volume, inwhich the I = I = I = I = and I = phase shifts from a large-volume anisotropic lattice For a first large-volume application of the stochastic LapH method, we employ an anisotropiclattice regularization, in which the spatial lattice spacing ( a s ) is larger than the temporal one ( a t ).The renormalized anisotropy ξ R = a s a t is defined by demanding that pions satisfy the correct (con-tinuum) dispersion relation ( a t E π ) = ( a t m π ) + (cid:18) π a s ξ R L (cid:19) d , (1.1)where d ∈ Z is the quantized finite-volume momentum. Details on the anisotropic ensemble usedhere are found in Tab. 1. 2 ion-pion scattering and the timelike pion form factor ( L / a s ) × ( T / a t ) m π ( MeV ) a t ( fm ) ξ R m π L N cfg ×
256 240 0 .
035 3 . ( ) . Table 1:
Details for the anisotropic N f = + The anisotropy is crucial in defining the center-of-mass momentum (discussed later), and thuscare must be taken in its determination. We employ two different methods which are consistentwithin statistical errors. In the first we calculate a t E π for all pions with d ≤ N b = χ fits. Wethen perform correlated χ fits on each bootstrap sample to Eq. 1.1 to obtain ξ R . Our seconddetermination of ξ R simultaneously fits all of these correlation functions to obtain ξ R directly.In order to calculate elastic scattering phase shifts, we require correlation functions containingtwo-pion interpolating operators which transform irreducibly according to the lattice symmetries.In particular, we require such correlation functions at various total momenta, and construct ap-propriate operators which transform irreducibly under the corresponding little groups according toRef. [8]. While this operator construction procedure can be used to construct spatially-extendedoperators with gauge-covariantly displaced quark fields (which are ideal for high-lying resonancestates), we employ only single-site hadron operators in this work.In order to extract the finite-volume energies in each irrep we evaluate a matrix of correlationfunctions C i j ( t ) = (cid:104) O i ( t ) ¯ O j ( ) (cid:105) and solve the generalized eigenvalue problem (GEVP) C ( t d ) v ( t , t d ) = λ ( t , t d ) C ( t ) v ( t , t d ) (1.2)for a single ( t , t d ) . The eigenvectors { v n ( t , t d ) } from this diagonalization define ‘optimal’ inter-polating operators [9] whose correlation matrix ˆ C i j ( t ) = (cid:104) O i ( t ) ¯ O j ( ) (cid:105) = ( v i ( t , t d ) , C ( t ) v j ( t , t d )) ismostly diagonal. In order to get a preliminary idea of the spectrum, we perform single exponentialfits to the diagonal elements of this rotated correlation matrix to extract the energies, taking careto vary ( t , t d ) to ensure that any systematic error associated with the off-diagonal elements of therotated correlation matrix is smaller than the statistical one. We find that generally the variationof the GEVP parameters has little effect on the extracted energies. The fitting range [ t min , t max ] ischosen by fixing t max = a t and varying t min until a suitable χ is achieved and a plateau is evident.Examples of such t min plots are shown in Fig. 1 which illustrate both the quality of the plateaux andinsensitivity to the GEVP parameters.In addition to finite-volume energies, we can use GEVP eigenvectors to estimate the overlaps Z in = |(cid:104) | ˆ O i | n (cid:105)| between our operators and the finite-volume Hamiltonian eigenstates. We estimatethese overlaps by constructing the ratio Z in ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ j C i j ( t ) v n j ( t , t d ) e − En t (cid:113) ˆ C nn ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1.3)where E n is the fitted energy, and taking t = a t .The I = J PG = − + state appearsare shown in Fig. 2. There we show center-of-mass energies E cm = √ E − P , with P = π L d , in3 ion-pion scattering and the timelike pion form factor
10 20 300.110.1150.120.1250.130.1350.140.145 t /a min t f i t E t a , Level 0 +1u = 0, T d ) t , 20a t ) = (10a d , t (t ) t , 14a t ) = (7a d , t (t
10 20 300.160.1650.170.1750.180.1850.19 t /a min t f i t E t a , Level 2 +1 = 2, A d = 3 op N = 4 op N t /a min t f i t E t a , Level 1 + = 3, E d = 2 op ), N t , 14a t ) = (7a d ,t (t = 3 op ), N t , 20a t ) = (10a d , t (t Figure 1:
Representative t min plots from exponential fits to diagonal elements of the rotated correlationmatrices. Each plot shows results from two different choices for the GEVP parameters. p m cm E +1u = 0, T d +1 = 1, A d + = 1, E d +1 = 2, A d +1 = 2, B d +2 = 2, B d +1 = 3, A d + = 3, E d +1 = 4, A d + = 4, E d ( ) r ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) r ( ) p ( ) p ( ) r ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p Figure 2: I = units of m π as well as overlaps onto our interpolators. As the ρ meson is present in these irreps,we employ local single-meson operators (denoted ρ ( d ) ) as well as two-pion operators, which aredenoted π ( d ) π ( d ) where d = ( d + d ) . Correlation functions with equivalent total momentaare averaged for each d .A clear picture emerges from these energies and overlaps. It is only two-pion interpolatingoperators which have significant overlap with states far below E cm m π ≈ .
2, while single- ρ interpola-tors have overlap for states with energies at or above this value. Due to G -parity, the lowest-lyinginelastic threshold is at E cm = m π . While we are able to precisely extract energies near and abovethis threshold, they cannot be used to extract infinite-volume scattering information. Although ex-tensions of the Lüscher formula above three-hadron thresholds has been developed [10, 11, 12, 13],a rigorous treatment of four-hadron thresholds is still lacking.4 ion-pion scattering and the timelike pion form factor In order to obtain elastic scattering phase shifts, we first define the kinematic quantities E cm = (cid:112) E − P , γ = EE cm , q = E − m π , u = L q ( π ) , (1.4)where E is the fitted two-hadron energy. Generally, the relation between these quantities and theinfinite-volume scattering matrix takes the form det { + F ( d , γ ) ( u )[ S ( E cm ) − ] } =
0, where F is aknown kinematic function and S is the infinite-volume scattering matrix. This relation holds up tocorrections which are exponential in the volume and the determinant is taken over the usual ( (cid:96), m ) indices of partial waves, in which F is non-diagonal.For this work we ignore higher partial waves. Precisely, this means neglecting (cid:96) ≥ I = p -wave scattering and (cid:96) ≥ I = s -wave scattering. After applyingthis approximation and block-diagonalizing in lattice irreps, the above relation takes the particularlysimple form q (cid:96) + cot δ (cid:96) = − q (cid:96) + φ ( d , γ , Λ ) ( u ) (1.5)for each finite-volume irrep Λ . The functions φ ( d , γ , Λ ) ( u ) involve Rummukainen-Gottlieb-Lüschershifted zeta functions ( Z ( d , γ ) lm ( u ) ) and are given in e.g. Ref. [14] for I = (cid:96) =
1. For I =
2, Eq. 1.5takes the even simpler form q cm cot δ ( q ) = γ L √ π Z ( d , γ ) ( u ) (1.6)for both the A + g and A + irreps used in this work. It should be noted that near the two-pion threshold q (cid:96) + cot δ (cid:96) is analytic and thus Eq. 1.5 is valid for also for negative q . In order to efficientlyevaluate these zeta functions we employ a method described in Ref. [15] which agrees with ourimplementation of Appendix A in Ref. [14].While the single-exponential fits discussed above provide preliminary spectra, the center-of-mass momentum and thus the scattering phase shifts are very sensitive to these energies. For two-hadron dominated levels, the energies will be very close to their non-interacting values. Therefore,for these levels it is beneficial to perform fits to the ratio of two and single-hadron correlators (asin Ref. [16] but here generalized to arbitrary momenta) R ( t ) = (cid:104) O d , d ( t ) ¯ O d , d ( ) (cid:105)(cid:104) O d ( t ) ¯ O d ( ) (cid:105)(cid:104) O d ( t ) ¯ O d ( ) (cid:105) , (1.7)where ˆ O d , d is a optimized operator for an eigenstate dominated by individual pions with momentaof magnitude d and d , respectively. The O d are single-pion operators with momentum d . Singleexponential fits to these ratios directly yield this energy difference, and typically have less excitedstate contamination than ordinary single exponential fits. However, the excited state contamina-tion in such fits may not be monotonically decreasing, which can complicate the identification ofthe plateau region in some cases. Another method to better resolve differences between the two-and single-hadron energies is to perform simultaneous fits to the two hadron correlators includedin the ratio of Eq. 1.7. While both methods yield consistent results, we quote values from thesimultaneous fits, as they result in more conservative statistical errors.5 ion-pion scattering and the timelike pion form factor p m cm E d c o t p m c m p +1u = 0, T p +1 = 1, A p + = 1, E p +1 = 2, A p +1 = 2, B p +2 = 2, B p +1 = 3, A p + = 3, E p +1 = 4, A p + = 4, E p p m cm q d c o t p m c m q +1g = 0, A d +1 = 1, A d +1 = 2, A d +1 = 3, A d Figure 3:
The I = p -wave scattering phase shift (left) and I = s -wave phase shift (right) together withfits to q (cid:96) + cot δ (cid:96) described in the text. The resultant scattering phase shifts are shown in Fig. 3 for both I = I =
2, together withfits to q (cid:96) + cot δ (cid:96) . Care must be taken to treat the correlation between x − and y -errors as well asamong the data points in these fits. Upon every call to the correlated χ function, we estimate thenecessary covariance matrix using the N b =
800 bootstrap samples of these secondary observables.The minimization is performed on each bootstrap sample, where bootstrap replica of the meansare used together with the covariance matrix to construct the correlated χ . For I = p -wavescattering, we fit to a relativistic Breit-Wigner form (cid:18) q cm m π (cid:19) cot δ = (cid:32) m ρ m π − E cm m π (cid:33) π E cm g ρππ m π (1.8)and obtain g ρππ = . ( ) , m ρ m π = . ( ) , χ d . o . f . = .
43 (1.9)where the coupling is in good agreement with the experimental value. For I = s -wave scattering,we employ the NLO effective range expansion to the points at low momenta q cm ≤ m π (cid:18) q cm m π (cid:19) cot δ = m π a + ( m π r ) (cid:18) q cm m π (cid:19) (1.10)and obtain m π a I = = − . ( ) , m π r = . ( . ) , χ / d . o . f . = . . (1.11)The JLab group [17] has recently calculated the I = p -wave phase shift on this same ensem-ble using the distillation method of Ref. [5]. While their results are more precise than those quotedhere, significantly more Dirac matrix inversions are required. Using the notation of Ref. [6], weemploy (TF, SF, LI8) dilution for fixed quark lines and (TI16, SF, LI8) for relative ones. Our useof five fixed and two relative lines (with eight source times) results in N D = N D = ion-pion scattering and the timelike pion form factor ( L / a ) × ( T / a ) m π ( MeV ) a ( fm ) m π L N cfg ×
128 280 0 .
065 4 . Table 2:
Details for the isotropic N f = + I = phase shifts and the timelike pion form factor on a CLS ensemble Motivated by the efficacy of the stochastic LapH method in the large-volume calculation de-scribed in Sec. 1, we now employ it to isotropic N f = + τ exp asthe continuum limit is approached. This means source and sink times must be chosen carefully inorder to avoid boundary effects. Based on the observables considered in Refs. [18, 20] and on thebehavior of the LapH eigenvalues, we use t = T / = a .In the gauge-covariant 3-D Laplace operator used to define the LapH subspace we stoutsmear [21] the gauge links with parameters ( ρ , n ρ ) = ( . , ) . We choose a LapH cutoff σ s ≈ ( a σ s ) ≈ .
11 and N v = N v isexpected. We employ the same (TF, SF, LI8) dilution scheme for fixed lines, but for relative linesuse (TI8, SF, LI8) as the temporal lattice spacing is larger. We use four independent fixed quarklines, a single relative line, and one source time t = a . The N D =
384 Dirac matrix inversionsper configuration are performed using the DFL-SAP-GCR solver [22] implemented in openQCD ,which we have integrated into the stochastic LapH codebase. The results from these inversions areprojected onto the LapH subspace and stored for later use in other calculations, such as Ref. [23].Finally, these ensembles employ RHMC and twisted mass re-weighting, and we multiply all pri-mary observables by the corresponding re-weighting factors.The renormalization and O ( a ) improvement of composite operators is simplified on an isotropiclattice. Furthermore, the regularization employed by these CLS ensembles is well studied and manyrenormalization and improvement coefficients have been previously determined. Therefore, in ad-dition to applying the methods of Sec. 1 to calculate the I = p -wave scattering phase shift, wealso calculate a matrix element of the vector current with two-pion states. The simplest such matrixelement of phenomenological relevance is the timelike pion form factor. As before we are restrictedto the elastic region which, for this ensemble is 2 m π ≤ E cm ≤ m K . In this region the timelike pionform factor | F π ( E cm ) | can be defined as [24] R ( s ) ≡ σ ( e + e − → hadrons ) πα ( s ) / ( s ) = (cid:18) − m π s (cid:19) | F π ( √ s ) | , (2.1) http://luscher.web.cern.ch/luscher/openQCD/ ion-pion scattering and the timelike pion form factor where the denominator in R ( s ) is the tree-level cross section σ ( e + e − → µ + µ − ) for s = E (cid:29) m µ .Effectively, this form factor describes QCD corrections to the coupling of a (virtual) photon to twopions. It is phenomenologically relevant because of (e.g.) its relation to the low-energy contributionto the hadronic vacuum polarization (HVP) Π ( Q ) . For spacelike four-momentum transfer Q , theonce-subtracted dispersion relation Π ( ) − Π ( Q ) = Q (cid:90) ∞ ds ρ ( s ) s ( s + Q ) , ρ ( s ) = R ( s ) π (2.2)expresses the HVP in terms of R ( s ) . Typically, this dispersion relation is not used in lattice QCDcalculations of the HVP which is instead calculated directly from current-current correlation func-tions. Furthermore, the relation between R ( s ) and | F π ( E cm ) | given in Eq. 2.1 is valid only in theelastic region, while the integral in Eq. 2.2 is unbounded from above. However, direct lattice calcu-lations of the HVP require fully disconnected Wick contractions (which are typically ignored), andsuffer from large statistical errors and finite-volume effects in the low- Q region [25]. Therefore,a ‘hybrid’ determination which combines lattice data from both Π ( Q ) and | F π ( E cm ) | may be thebest approach .In analogy with earlier work by Lellouch and Luscher [26], a relation between the infinite-volume | F π ( E cm ) | and finite-volume matrix elements (up to exponential finite-volume corrections)was derived by Meyer in Ref. [27] | F π ( E cm ) | = g ( d , Λ ) ( γ ) (cid:32) u d φ ( d , Λ ) ( u ) d u + q cm ∂ δ ( q cm ) ∂ q cm (cid:33) π E q | A ( d , Λ ) | , (2.3) g ( d , Λ ) ( γ ) = (cid:40) / γ if Λ = A γ else , where φ ( d , Λ ) ( u ) is given in Eq. 1.5 and A ( d , Λ ) = (cid:104) | V ( d , Λ ) | d Λ n (cid:105) (2.4)is a finite-volume matrix element involving an I = d inirrep Λ below inelastic threshold. While this relation was derived only for total zero momentumin Ref. [27], it may be straight-forwardly extended to non-zero total momentum using the argu-mentation of Ref. [28]. A proof-of-principle application of this relation was performed recently inRef. [29] while a similar matrix element is calculated in Ref. [30].Since we work in the isospin limit, the electromagnetic current ˆ J em i = ¯ u γ i u − ¯ d γ i d − . . . isreplaced by the isospin current V ai = ¯ ψγ i τ a ψ , ψ = (cid:16) ud (cid:17) , (2.5)where τ a is an SU ( ) generator. Furthermore, we project this current onto finite-volume irreps bydefining V ( d , Λ ) = b ( d , Λ ) i V i . The vectors b ( d , Λ ) are given in Tab. 3 for all irreps used in this work.In order to obtain the derivative of the p -wave phase shift required in Eq. 2.3, we first perform the We thank Harvey Meyer for clarifying this point. ion-pion scattering and the timelike pion form factor Reference momentum d Irrep Λ b [000] T u (1,0,0)[00n] A (0,0,1) E (0,1,0)[0nn] A √ (0,1,1) B (1,0,0) B √ (0,-1,1)[nnn] A √ (1,1,1) E √ (1,-1,0) Table 3:
Linear combinations of components of the vector current such that V ( d , Λ ) = b ( d , Λ ) i V i transformsirreducibly according to the irrep Λ . complete phase shift analysis and parametrize the energy dependence of δ using the Breit-Wignerform of Eq. 1.8. The derivative is then evaluated on each bootstrap sample and taken together withbootstrap samples of the current matrix elements of Eq. 2.4 to obtain samples of | F π ( E cm ) | .In order to obtain the current matrix elements in Eq. 2.4, we must evaluate correlation functionsof the form D i ( t ) = (cid:104) V ( d , Λ ) ( t + t ) ¯ O ( d , Λ ) i ( t ) (cid:105) , (2.6)where the { O ( d , Λ ) i } are interpolators used in the GEVP of Eq. 1.2. In order to ensure an O ( a ) approach to the continuum limit, the currents V ( d , Λ ) are linear combinations of the renormalized, O ( a ) -improved local vector current ( V R ) µ defined as [31] ( V R ) a µ = Z V ( + b V am q ) { V a µ + ac V ˜ ∂ ν T a µν } , (2.7)where Z V , b V , and ac V are renormalization and improvement coefficients, am q the bare quarkmass in lattice units, T a µν = i ¯ ψσ µν τ a ψ , and ˜ ∂ ν the symmetrized lattice derivative. A preliminarydetermination of the renormalization coefficient Z V for the CLS lattice regularization has beenprovided in Ref. [32], while b V , and ac V are calculated to 1-loop in Ref. [33]. For this preliminarywork, we subsitute the unrenormalized PCAC mass am PCAC for am q which holds at tree level,so that O ( a ) improvement is formally implemented only at this order. However, this effect issuppressed by the small quark mass.In the stochastic LapH framework interpolators are built from quark fields projected onto theLapH subspace, while quark fields appearing in the current are local. However, this can be easilyaccommodated by forming the meson functions in Eq. 32 of Ref. [6] with quark sinks which are notprojected into the LapH subspace. The unprojected ‘current’ functions are otherwise completelyanalogous to the meson functions in the construction of correlators but are calculated immediatelyafter the Dirac matrix inversions but before the sinks are projected and written out to disk. Inpractice, correlation functions required for the matrix elements A ( ) n = (cid:104) | V ( d , Λ ) | d Λ n (cid:105) , A ( ) n = (cid:104) | b ( d , Λ ) i ˜ ∂ ν T ai ν | d Λ n (cid:105) (2.8)9 ion-pion scattering and the timelike pion form factor are calculated and analyzed separately.We turn now to extraction of the matrix elements given Eq. 2.8. In analogy with our procedurefor the energies, we measure the correlation functions of Eq. 2.6 and with the GEVP eigenvectorsform their ‘optimized’ counterparts ˆ D i ( t ) = ( D ( t ) , v i ( t , t d )) , where the inner product is taken overthe GEVP indices. Up to GEVP corrections (treated as a systematic error as described in theprevious section) these optimized correlation functions have the large-time behaviorlim t → ∞ ˆ D i ( t ) = (cid:104) | V ( d , Λ ) | d Λ i (cid:105) (cid:104) d Λ i | ¯ O ( d , Λ ) i | (cid:105) × e − E ( d , Λ ) i ( t − t ) . (2.9)This suggests three different ratios which tend to the matrix elements of interest: R ( i ) ( t ) = ˆ D i ( t ) ˆ C ii ( t ) e − E i ( t − t ) , R ( i ) ( t ) = ˆ D i ( t ) (cid:104) d Λ n | ¯ O ( d , Λ ) i | (cid:105) ˆ C ii ( t ) , R ( i ) ( t ) = ˆ D i ( t ) (cid:104) d Λ n | ¯ O ( d , Λ ) i | (cid:105) e − E i ( t − t ) , (2.10)where the overlaps and energies appearing in these expressions are obtained from fits to the di-agonal elements of the rotated correlation matrix. The value of the matrix element is taken to bea plateau average of the ratio over a suitable region. Alternatively, simultaneous fits to ˆ C ii ( t ) andˆ D i ( t ) may also be used to extract the desired matrix elements. Generally we find that all fourof these different determinations yield results which are consistent within statistical errors for fitranges in the plateau region. p m cm E d c o t p m c m q +1u = 0, T d +1 = 1, A d + = 1, E d +1 = 2, A d +1 = 2, B d +2 = 2, B d +1 = 3, A d + = 3, E d +1 = 4, A d + = 4, E d p m cm E) (cid:176) ( d Figure 4:
The p -wave scattering phase shift on the N200 CLS lattice. Shown are q cot δ (left) and δ (right) together with a Breit-Wigner fit to q cot δ . The resultant fit parameters are given in the text. Thelowest inelastic threshold is at 2 m K ≈ . m π . Points above this threshold are shown on the graphs but notincluded in the fit. Our results from the phase shift determination on this lattice are shown in Fig. 4. As mentionedabove, the lowest inelastic threshold in this channel is due to two kaons at 2 m K ≈ . m π . Althoughwe have a number of energy levels above this threshold, they are not included in the final analysisbut shown for illustration. As before, we fit q cot δ to the Breit-Wigner form of Eq. 1.8 whichgives g ρππ = . ( ) , m ρ m π = . ( ) , χ d . o . f . = .
20 (2.11)10 ion-pion scattering and the timelike pion form factor which is somewhat lower than the experimental value of g ρππ . The Breit-Wigner parametriza-tion of the phase shift in Eq. 2.3 together with the matrix elements A ( ) , A ( ) and renormaliza-tion/improvement coefficients of Eq. 2.7 enables the extraction | F π ( E cm ) | , which is shown inFig. 5. Also shown in that figure is the ratio of the matrix element appearing in the O ( a ) termover the leading order one. We see that this O ( a ) matrix element grows from ∼
10% of the leadingone at low momenta to ∼
30% at our largest momentum. Due to the small 1-loop value of ac V ,this term therefore has no significant effect on our final results. However, non-perturbative deter-minations of these improvement coefficients can be considerably larger than the 1-loop value [34]possibly increasing the influence of this term to the few-percent level. In future work , an alterna-tive calculation which employs the point-split vector current will give an additional handle on themagnitude of these O ( a ) effects. p m cm E ) | c m ( E p | F +1u = 0, T d +1 = 1, A d + = 1, E d +1 = 2, A d +1 = 2, B d +2 = 2, B d +1 = 3, A d + = 3, E d +1 = 4, A d + = 4, E d p m cm E (0) A (1) A Figure 5:
The timelike pion form factor together with the expected Gounaris-Sakurai parametrization usingthe previously-calculated m ρ and g ρππ (left). The ratio of the matrix element which contributes at O ( a ) overthe leading order one is shown on the right for each of the form factor data points. Also shown in Fig. 5 is the Gounaris-Sakurai parametrization of | F π ( E cm ) | which (using thenotation of Ref. [35]) is F GS π ( √ s ) = f q h ( √ s ) − q ρ h ( m ρ ) + b ( q − q ρ ) − q √ s i , (2.12) b = − h ( m ρ ) − π g ρππ − q ρ m ρ h (cid:48) ( m ρ ) , f = − m π π − q ρ h ( m ρ ) − b m ρ , h ( √ s ) π q cm √ s ln (cid:18) √ s + q cm m π (cid:19) , where q ρ is the center-of-mass momentum at the resonance energy. The curve shown in Fig. 5 isnot a fit but a ‘prediction’ using the values of m ρ and g ρππ obtained from the phase shift analysis.We see that this GS model fits our data rather well.
3. Conclusion
A first large-volume application of the stochastic LapH method to calculate pion-pion scatter-ing on an anisotropic lattice has been performed. This results in a good precision for both the I = I = I = p -wave irreps, stochastic11 ion-pion scattering and the timelike pion form factor LapH is sufficiently precise to resolve the differences for I = s -wave scattering scattering phaseshift as well. The fewer points below inelastic threshold here are due to the reduced number ofirreps in which the s -wave contributes. A first look at scattering in the final isospin combination( I =
0) is underway, but complicated by both ‘annihilation’ Wick contractions and the need for avev subtraction when d = I = p -wave scatteringphase shift on a single 48 ×
128 ensemble with m π = a = . m ρ and g ρππ with comparable statistical precision. Apart from the scattering phase shift,we also calculate the timelike pion form-factor, which is related to the hadronic vacuum polariza-tion at low four-momentum transfer. Given the moderate number of inversions required, we plan toincrease our current level of statistics for this form factor by using an additional source time. Fur-thermore, while we average over equivalent orientations of the total momentum d in calculation ofthe correlation functions used for δ , we have not done so for those used for | F π ( Q ) | . Preliminarytests indicate that this averaging has a significant impact on the precision of the form factor and itwill be performed on the additional source time, possibly resulting in a significant reduction in thestatistical errors. Finally, calculations of the scattering phase shift and timelike pion form factor onadditional CLS ensembles are underway. Acknowledgments
BH is supported by Science Foundation Ireland under Grant No. 11/RFP/PHY3218. CJM ac-knowledges support from the U.S. NSF under award PHY-1306805 and through TeraGrid/XSEDEresources provided by TACC, SDSC, and NICS under grant number TG-MCA07S017. The authorswish to acknowledge the DJEI/DES/SFI/HEA Irish Centre for High-End Computing (ICHEC) forthe provision of computational facilities and support. The CLS consortium also acknowledgesPRACE for awarding access to resource FERMI based in Italy at CINECA, Bologna and to re-source SuperMUC based in Germany at LRZ, Munich. Furthermore, this work was supported bya grant from the Swiss National Supercomputing Centre (CSCS) under project ID s384. We aregrateful for the support received by the computer centers.
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