arXiv:hep-lat/0509093v1 22 Sep 2005

izaeA oo2 -08 oe Italy E-mail: Rome, I-00185 INFN-Se 2, and Moro Sapienza” A. “La Piazzale Roma di Università Fisica, di Dip. Villadoro Giovanni E-mail: Kingdom United 1BJ, SO17 Southampton Southampton, of Highfield, University Astronomy, and Physics of Sachrajda School Christopher Jüttner, Andreas Flynn, Jonathan Simulations Lattice in Conditions Boundary Twisted Partially rnt olg,Dbi,Ireland Dublin, College, Trinity 2005 July Theory 25-30 Field Lattice on Symposium International XXIIIrd c oyih we yteato()udrtetrso h Creat the of terms the under author(s) the by owned Copyright † ∗ arxeeet)i iuainwt ata wsigo s a on twisting partial with simulation a interactions in elements) state matrix final without quantities for suppressed vdnefrtersl httefiievlm fet fteb the of effects volume finite the determ that result be the cannot for amplitudes evidence the that implies conditions ary Speaker. SHEP-0530 euecia etrainter oivsiaetitdan twisted investigate to theory perturbation chiral use We hc lo cest oet te hnitgrmlilso multiples integer than other momenta to access allow which rvdWlo . Wilson proved ume [email protected] {jflynn,juettner,cts}@phys.soton.ac.uk L 3 For . K → ππ ∗ easw hwta h raigo ssi ymtyb h tw the by symmetry isospin of breaking the that show we decays v omn trbto-oCmeca-hrAieLicen Attribution-NonCommercial-ShareAlike Commons ive , atal wse onayconditions boundary twisted partially d msnmse n -to-vacuum and masses (meson in iRoma di zione udr odtosaeexponentially are conditions oundary aof ea ndi eea.W n numerical find We general. in ined 2 f π / N L f † naltiewt pta vol- spatial with lattice a on = ce. o-etraieyim- non-perturbatively 2

http://pos.sissa.it/ se bound- isted PoS(LAT2005)352 PoS(LAT2005)352 i U (1.1) (1.2) , where i n s, but some × L / π partially twisted decays), it is not 2 Christopher Sachrajda = boundary conditions ππ i ices the gaps between p , where we take the → ) i x K ( q twisted e-volume corrections, which ) the finite-volume corrections t is not necessary to generate r a single , the flavour a making the method much more T ions, with exponential precision a i o, limiting the phenomenological using twisted boundary conditions such as θ i ) with periodic boundary conditions angian corresponding to the twisted ( , one can also use 3 , such as masses or matrix elements bosons can be evaluated using Chiral h that these corrections are negligibly L eoretically and numerically and in this , exp ) = i . x i ( V L θ q )= i i θ eas. x i + ( chiral symmetry and impose the twisted bound- e q π R i L 2 ) U i )= N n 352 / 2 , are quantized according to ( L p ~ = )= ’s, arbitrary momenta can be reached. In refs. [3, 4] we + i i SU i L p θ x Bedaque proposed the use of × ( + q L 1 i ) x ( N q ( SU in which the sea quarks satisfy periodic boundary condition ), e.g. ) x ( are integers. This implies that on currently available latt q i n 3 and the , of local operators between states consisting of the vacuum o symmetry breaking induced bynevertheless the remain twist exponentially small. only affects the finit possible in general to extract the physical matrix elements . (at least without introducing new ideas) boundary conditions or all of the valencein quarks satisfy the twisted volume. boundary This condit new implies that configurations in for every unquenched choice simulationspracticable. of i twisting angle, 2 , The choice of boundary conditions for the fields only affects In lattice simulations of QCD on a cubic volume ( See also the references cited in [1, 2] for earlier related id 1 1 1. For physical quantities without final state interactions 2. For physical quantities without final state interactions 3. For amplitudes which do involve final state interactions ( = so that the momentum spectrum is have investigated the use of twistednote boundary we conditions briefly th report the conclusions of these studies. 2. Finite-Volume Effects with Twisted Boundary Conditions The three main results from ref. [3] are: i imposed on the fields, hadronic momenta, neighbouring momenta are large, typicallyreach about of the 500 MeV simulations. or In s ref. [1, 2] and when extracting physical quantitiessmall. we need Infrared to effects establis dominated byPerturbation the Theory pseudo-Goldstone and in ref.boundary [3] conditions. we Consider derive the chiral Lagr We now briefly discuss each of these results in turn. 2.1 Result 1. Twisted Boundary Conditions in Lattice Simulations 1. Introduction on the fields ( ary conditions on the quark fields Thus by varying the twisting angles, the PoS(LAT2005)352 . (2.1) (2.2) (2.3) mply ! quarks. i s, except θ d cos and 1 quark with (a) 3 u = ∑ i u 2 3

Christopher Sachrajda (a) (c) (b) 2 / 3 ) L π L π m − m e π 2 ( ng Feynman rules with Partially satisfies the boundary conditions ons for processes with final state is that expected of a particle com- π 2 , the leptonic decay constant of the 2 V , the leptonic decay constant of the π , n terms of its twist by eq. (1.2). The f ) K π 2 itions as expected, they nevertheless m  f f N no longer states with definite isospin. 3 3 4 ion relation, ! es). The difference of the correction (  3 4 or partially quenched QCD. As an illus- θ ~ volume corrections are different for the + pin by the boundary conditions. i SU − y conditions: − ns to θ →− / L i n R ~ 0 θ ) 0 π π cos f π 2 N ’s are the generators of the Cartan subalgebra f 1 ( a cos ∆

= , the difference of the twists of the 3 i 1 T i 0 channel. The twisted boundary conditions break , SU = + ∑ , d 3 i = 352 / 3 2 1 θ ρ × ∑ , ! I 2 L π 2 1 − / 2 ) 2 i 3 / , / m ) L 3 3 + u N ) L L π . The results in eq.(2.2) illustrate the point that whereas ) L i ( L θ π π ) m L π = π θ m − π m m ∞ e − ρ m − = π m ( e , SU e π 2 i π 2 π ( 2 cos 2 X θ ( ( E π 2 2 1 / π f 2 π 2 3 2 π π 2 m π = f ∑ f i )) m m 4 9 decays in the group. In the chiral Lagrangian these boundary conditions i 3 1 ∞ − − ( −

V X ) quarks satisfying periodic boundary conditions and the ππ                  2 / s The Feynman rules for the Chiral Lagrangian are the usual one N − 3 ( → . ) ) → L unitary matrices and the † L π L the leading corrections are found to be: U i K ( ± and π m ± U N K ∞ X − ) f K d m i ( f e × π x ∆ → ( 2 ≡ N ( Σ L i 2 π X 2 π U f is the meson mass and / m ) ρ 3 / X , the coset representative of ( π )= ) ∆ L x m ( →− As an example consider the finite-volume corrections in The derivation of the Chiral Lagrangian and the correspondi To illustrate the problem in using twisted boundary conditi + Σ i ± x ± π -meson, with the -. As ( f π f isospin symmetry which implies that energy eigenstates are π Twisted Boundary Conditions in Lattice Simulations to be diagonal that tration of the results consider the finite-volume correctio Twisted boundary conditions follows similar steps to that f 2.2 Result 2. the finite-volume corrections do depend onremain the boundary exponentially cond small (seeterms ref. [3] in eq.(2.2) for is further a manifestation exampl of the violation of isos of the (light) flavour Σ where where This example illustrates again that,three cases in but general, they the are always finite- exponentially small. 2.3 Result 3. interactions consider K untwisted, (b) fully twisted (c) partially twisted boundar that the momentum spectrum forposed the of Pseudo-Goldstone a bosons quark-antiquark pair, each witheffect a of momentum the given twist i is a change in the charged meson’s dispers ∆ PoS(LAT2005)352 2 L / 2 (2.4) . The θ L / + pectively θ ~ π 2 ) there are π ± m m . as powers of = p 2 2 ± > π = p ~ E − Christopher Sachrajda π s that the terms with + π E 0 and ~ decays above the two-pion = s and 0 . π π ππ ) p ~ ven if we were able to overcome p decays, the sea quarks contribute m ~ for both the valence and the sea e ity boundary conditions proposed ( ially twisted boundary conditions 2 → f s momentum follows eq.(2.1) and cal study of partially twisted bound- ete momenta in a finite volume by pion threshold ( ππ p s is analogous to the inconsistencies = ~ K er the two pions to be in the centre-of- . Without new ideas, it not therefore ustrated by the diagram: · l 2 ~ 0 e Poisson summation formula: me energy eigenstates to the infinite- π = iL → re different. 0 I e reducible in the s-channel. In evaluating i π K by the following diagram: 3 E ) p 0 ππ p | 3 π transitions which complicate the analysis very = amplitudes using twisted boundary conditions 2 d 0 ( 0 = I I ) π and 0 Z ππ ππ 0 3 π ( 352 / 4 Z = I → ∈ ∑ l i ~ ↔ K − ππ 0 | )= π p ~ + ( = ··· ··· π f I and so the benefit of using partial twisting are not so clear. p ~ ∑ π 3 quark satisfy twisted and periodic boundary conditions res 1 L d and ). The finite-volume effects which decrease only slowly, i.e 2 u E (corresponding to the cut in infinite volume) and this implie ) p ~ ( f The interacting theory contains The twist angle is now fixed to be This section describes the results of an exploratory numeri There is a further difficulty which we would like to exhibit. E 0 in eq. (2.4) are not exponentially small. For 2 6= quarks, there is aand breakdown the of finite-volume corrections unitarity are inducedencountered not by in under quenched the control. and part Thi partially quenched QCD for Therefore, unless the same boundary conditions are imposed finite-volume corrections we replace the sums over the discr The circles represent insertions which are two-particle ir integrals over continuous infinite-volume momenta using th the volume, come from the propagation of two-pion states ill mass frame and let the Twisted Boundary Conditions in Lattice Simulations This is manifest even in the free theory. As an example consid making it explicit that the energy and isospin eigenstates a to the final-state interactions as illustrated for example, ary conditions, in which we investigate, whether the meson’ with finite-volume corrections under control. or circumvent the problem aboveby (for Christ example, and by Kim using [5] G-par poles in threshold in refs. [6, 7] . 3. Numerical study of partially twisted boundary condition significantly making it impossible to relate the finite-volu volume energy and isospin eigenstates not possible to determine physical corresponding energies of the two-pion states are then l For two-particles propagating with an energy above the two- so that the lowest momenta of the neutral and charged pions ar PoS(LAT2005)352 ) L 0, = / 32, | 16 π = ( lat 0 and 2 × | ~ 2 Wil- p are in- ~ 3 L | = × 566 eq. (2.1) P = . lat d vertical 3 16 Z p lat f ~ | p √ ~ = N respectively, of the pseudo 0 but ) a ~ and 0 0 i / ) and , = π T | L L 11 d / P / ( × θ ~ | Christopher Sachrajda π π 3 0 2 ) 2 = , 697 a . However, when com- decays with isospin 0. . 0 u / × ≡ h d ( 0 , θ ~ L u 2 P ( − and refer the reader to ref. = θ ππ ~ ntum. Deviations for large Z y conditions or only Fourier √ ) ρ = 2, re lattice artifacts are small. . → = m 11 5 lat rbatively improved rix elements increases smoothly | / ( K p conditions with similar momenta ~ π noise in the data. The combined lat . The (red) diamonds and (green) = m p d at the Fourier momenta ~ lements and hadronic structure. We 697 | θ β . [4]) for all possible pairs of valence ~ . um resolution in lattice phenomenol- } and of the matrix element and 0 ) ) e demonstrate that the use of partially ρ 3 s in which the correlation function was oundary conditions cannot be applied in f ng correlation functions focused on the 0 , = and , e lighter data set. The plots in figure 1 3 L ρ s such as , u ut final-state interactions. We now look for- / 3 θ ~ m ( and π / 2 π π , f m 0 and . The four points with ) d and the matrix element 0 = ( , 352 / 5 θ ~ ρ π f lat , p ~ 0 and ( , and 13550 corresponding to u . The positions of the discrete Fourier momenta ) . are denoted by (black) circles. The three main results are: ), the errors are found to be comparable. 2 θ and we also monitored the un-improved bare decay constants ~ π 0 ) ) f L , ρ L / 0 E / pL , ~ π under the variation of the momentum. ( 2 π 2 ( P 2 ( and Z -mesons (with masses below the two-pion threshold) satisfy × 0, 2 ~ π ρ 3 which can be reached without twisting are indicated by dashe √ E 13500 and 0 √ ∈ { . π or d 0 and , 2 u L π and 0, but with all possible pairs of = θ ~ ~ × / L 3 π = sea / √ κ π lat are independent of the twisting angle. 2 p = ~ P . In addition we also evaluated all quantities at × and ) val 2 0 π κ , and matrix element 2 √ L ρ , / × f L π statistical and systematic error on the meson masses and mat when increasing the meson’s momentum by varying the angles paring results obtained with twisted and(i.e. periodic boundary around 2 dependent of the twistingmomenta angles are for of the small same values magnitude when oftransformation using to the twisted induce boundar mome momentum. very well, particularly for small values of the momentum whe 2 1fm, / 2 . The study has been carried out on 200 independent, non-pertu Here we discuss only the data for the set at We evaluated mesonic correlation functions (details in ref √ π 0 ± 1. The energies of 3. Twisted boundary conditions do not introduce additional 2. The values of the leptonic decay constants and 2 , , ≈ , π π 0 f son gauge configurations with the following specifications: scalar density 0 squares represent the results obtained with Twisted Boundary Conditions in Lattice Simulations whether the meson decay constants ( for vanishing twist. The jack-knife analysis of the resulti lines. In each plot theevaluated with (blue) triangles correspond to point momentum dependence of 2 [4] for details andshow qualitatively our equivalent results results as a for function th of ward to applying these techniques to studies of weak matrix e twisted boundary conditions does indeed improve theogy moment at relatively little cost for physical quantities witho have also demonstrated that twisted andgeneral partially to twisted physical b quantities with final state interaction 4. Conclusions The theoretical and numerical results presented in this not combined with all possible pairs of (see also [8, 9]). quark twisting angles a PoS(LAT2005)352 . P Z 120 120 ispersion (2003) 229 (2004) 207 0. 100 100 = 553 581 d θ ~ 80 − 70 80 70 . ror in the case of the u . 0 θ 0 ~ Christopher Sachrajda decay constant and 2 2 ) ) = = ρ 60 60 ρ ρ pL ~ 0. Note also in all the four pL ~ ( m ( m and / = / π 2 π π 40 40 θ ~ m m = (2004) 408 [arXiv:nucl-th/0405002]. 1 θ ~ 20 20 ration], arXiv:hep-lat/0506016. = ts for the PSfrag replacements PSfrag replacements 595 lat (2003) 365 [arXiv:hep-lat/0210003]. p . B ~ 0 0 nd G. Villadoro, Phys. Lett. B nd G. Villadoro, Phys. Lett. B 119

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(2005) 73 [arXiv:hep-lat/0411033]. 0.1

P 0.2 0.1 0.3

0.12 0.08 0.06 0.04 0.02 Z

0.15 0.25 E π δ , which correspond to the cases 609 L / 352 / 6 π 2 , 120 120 0 = 70. The plots in the first line illustrate the results for the d | . (2002) 054502 [arXiv:hep-lat/0107021]. (2004) 014501 [arXiv:hep-lat/0403007]. 0 100 lat 100 p (2004) 82 [arXiv:nucl-th/0402051]. ~ = | ρ D 65 D 70 m 593 80 70 / 80 70 . . π 0 0 m 2 2 ) ) = = (empty and full symbols respectively) and the associated er 60 60 ρ ρ pL ρ pL ~ ~ ( m ( m , Phys. Rev. , Phys. Rev. / / π π 40 40 m m et al. et al. and the π 20 PSfrag replacements PSfrag replacements 20 Results for the case 0 0 [arXiv:hep-lat/0211043]. [arXiv:hep-lat/0308014]. 0 1

0

0.2 0.8 0.6 0.4

/ as a function of the momentum. The second line shows the resul

0.3 0.2 0.1 0.4 ) ( aE ρ π

/ [1] P. F. Bedaque, Phys. Lett. B [2] G. M. de Divitiis, R. Petronzio[3] and N. C. Tantalo, Phys. T. Sachrajda Lett and G. Villadoro, Phys.[4] Lett. B J. M. Flynn, A. Jüttner and[5] C. T. Sachrajda C. [UKQCD h. Collabo Kim and N. H. Christ, Nucl. Phys. Proc. Suppl. [6] C. J. D. Lin, G. Martinelli, E. Pallante, C. T. Sachrajda a [7] C. J. D. Lin, G. Martinelli, E. Pallante, C. T. Sachrajda a [8] C.R. Allton, [9] C.R. Allton, f a ρ π 2 plots the superimposed data points at Figure 1: π Twisted Boundary Conditions in Lattice Simulations relation for the References In each plot the horizontal lines represent the central valu