Charm physics with Moebius Domain Wall Fermions
Andreas Jüttner, Francesco Sanfilippo, Justus Tobias Tsang, Peter Boyle, Marina Marinkovic, Shoji Hashimoto, Takashi Kaneko, Yong-Gwi Cho
aa r X i v : . [ h e p - l a t ] J a n Charm physics with Moebius Domain Wall Fermions
Andreas Jüttner, Francesco Sanfilippo, Justus Tobias Tsang ∗ School of Physics and Astronomy, University of SouthamptonSO17 1BJ Southampton, United KingdomE-mail: [email protected]
Peter Boyle,
School of Physics and Astronomy, University of EdinburghEH9 3JZ, Edinburgh, United Kingdom
Marina Marinkovi ´c,
School of Physics and Astronomy, University of SouthamptonSO17 1BJ Southampton, United Kingdom andCERN, Physics Department, 1211 Geneva 23, Switzerland
Shoji Hashimoto, Takashi Kaneko
High Energy Accelerator Research Organization (KEK) andSchool of High Energy Accelerator Science, The Graduate University for Advanced Studies,Tsukuba 305-0801.
Yong-Gwi Cho
Graduate School of Pure and Applied Sciences, University of Tsukuba,Tsukuba, Ibaraki 305-8571.
We present results showing that Domain Wall fermions are a suitable discretisation for the simu-lation of heavy quarks. This is done by a continuum scaling study of charm quarks in a MöbiusDomain Wall formalism using a quenched set-up. We find that discretisation effects remain wellcontrolled by the choice of Domain Wall parameters preparing the ground work for the ongoingdynamical 2 + f charm program of RBC/UKQCD. The 32nd International Symposium on Lattice Field Theory23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ harm physics with Moebius Domain Wall Fermions
Justus Tobias Tsang b L / a N HB N sep t int ( Q top ) t ( Q ) a − [ GeV ] L [ FD]4.41 16 10000 100 15(3) 10.5(1.6) 2.00(07) 1.579(55)4.66 24 20000 200 159(60) 74(22) 2.81(09) 1.686(52)4.89 32 600000 500 215(100) 167(80) 3.80(12) 1.664(51)5.20 48 1400000 40000 139 ( ) ×
200 58 ( ) ×
200 5.64(22) 1.683(64)
Table 1:
Run parameters and autocorrelation times for the topological charge for the simulated tree-level-improved Symanzik gauge ensembles. The lattice volume is kept fixed to L ≈ . ∼ × t int ( Q ) allows for the autocorrelation between the saved config-urations to be neglected.
1. Introduction
To determine potential channels for New Physics (NP) it is necessary to test and over-constrainthe Standard Model (SM) by performing model independent theoretical predictions with high pre-cision. For this all systematic errors need to be controlled. Currently the heavy quark sector (charmand bottom) is rather unexplored compared to the light quark sector (see [1]) which makes it in-teresting for numerical simulations in Lattice Quantum Chromodynamics (LQCD). However, thesimultaneous simulation of light and heavy (charm) quarks poses a difficulty. Namely, in order tocontrol finite volume effects for the light quarks large lattice volumes are required, whilst to resolvethe heavy quarks it is necessary to have fine lattice spacings a , which is very costly to achieve atthe same time. The work presented in this talk investigates the feasibility of simulating charmedmesonic quantities within a Domain Wall formalism to lay the ground work for RBC/UKQCD’songoing 2 + f charm program at the physical point whilst maintaining chiral symmetry and hence O ( a ) improvement [2].To this end we produced 4 tree-level improved Symanzik gauge ensembles with inverse latticespacings a − between 2GeV and 5 . h s -like D s -like and h c -like quantities.Using the quenched approximation allows the exploration of very fine lattice spacings which are nototherwise affordable. Furthermore we developed the expertise and understanding of the parameterspace which is needed for the dynamical simulations and are now set-up for the aforementionedongoing 2 + f measurements presented at this conference [2]. A similar study comparing differentvalence discretisations for the heavy quarks was carried out on the same gauge ensembles and alsopresented at this conference [3].
2. Ensembles
We perform all our measurements on four sets of O ( ) independent, tree-level-improvedSymanzik gauge configurations [4, 5], where the inverse lattice spacing is varied in the range 2 − harm physics with Moebius Domain Wall Fermions Justus Tobias Tsang a /t t / r β=4.41β=4.66β=4.89 β -2 -1 a ( f m ) Figure 1:
Left: Determination of the lattice spacing with the Sommer scale parameter r and flow parameter t as a cross check of the determination of the lattice spacing. Right: log a as a function of b to determinefinite size effects. The larger symbols correspond to the 4 ensembles used for measurements, the smaller tothe auxiliary ensembles with volumes of 1 . − . . ∼ . Q = p a (cid:229) x Tr { F mn ( x ) ˜ F mn ( x ) } , (2.1)where F mn represents a clover definition of the field strength tensor. The topological charge wasclosely monitored to ensure ergodic sampling. The square topological charge Q is consideredto be a slowest mode relevant for the dynamics of the simulated system [7] and we use its auto-correlation time (found as described in [8]) to estimate the minimal separation between the savedconfigurations, so that they can be considered independent. We use the Wilson flow [9] to set the scale of our simulations, in particular, we use the w -scale which was proposed in [10]. From this we also take the physical scale w phys0 = . ( )( ) fm,computed for 2 + f simulations. The program package GLU used for the Wilson flow measure-ments was kindly provided by Jamie Hudspith. We additionally perform a cross check of the scalesetting procedure by measuring the Sommer scale parameter r and an additional flow parameter t [11] on the 3 coarsest ensembles. It is found that these agree well (cf. l.h.s of figure 1). Theone-loop beta function was used to estimate the parameters of the simulations. From the r.h.s. offigure 1 it can be seen that all lattices with volumes larger than ∼ . . https://github.com/RJhudspith/GLU harm physics with Moebius Domain Wall Fermions Justus Tobias Tsang PS (GeV) f P S (cid:1) m P S / f n o r m P S (cid:0) m n o r m P S M5 = 1.0M5 = 1.2M5 = 1.3M5 = 1.4M5 = 1.5M5 = 1.6M5 = 1.8M5 = 1.9 PS (GeV) f P S (cid:1) m P S / f n o r m P S (cid:0) m n o r m P S M5 = 1.8M5 = 1.6M5 = 1.3
Figure 2:
Left: Behaviour of f PS √ m PS for heavy-heavy pseudo scalar mesons as a function of the inversepseudo scalar mass m PS for different values of M on the coarsest ensemble using Shamir DWFs with L s =
16. The data is normalised at m normPS = . h c mass. Right: Overlay of data fromthe two coarsest ensembles for 3 choices of M . The dotted (dashed) curves depict data from the (second)coarsest ensemble. effects of the particular definition of the Wilson flow parameter are negligible in the volume rangewe are interested in.
3. Parameter Choices
In this study we aim to determine the range of parameter space in which Domain Wall fermions(DWF) can be used to reliably simulate heavy quarks with O(a) improvement. The use of domainwall fermions introduces two new parameters which can be chosen freely: The Domain Wall height M and the extent of the fifth dimension L s . In this study we use two different versions of theDomain Wall formalism both of which are used in the RBC/UKQCD 2 + f simulations, namelythe ’standard’ Shamir
Domain Wall fermions [12, 13] and the more recently developed
Möbius
Domain Wall fermions [14 – 16]. Möbius Domain Wall fermions are a rescaled version of theShamir case, allowing to simulate the same physics with half the extent in the 5th dimension andtherefore decreasing the computational cost. In particular choosing L Möbius s = L Shamir s / am bare q & . f PS √ m PS as heavier pseudo scalar massesare approached (compare left hand panel of figure 2). The right hand panel of figure 2 suggests thatthis behaviour is very sensitive to the choice of M . Additional to the pseudo scalar mass for whichthis bending sets in, the size of the cut-off effects also depends strongly on the choice of M . Wefound that the optimal choice of the Domain Wall height is given by M = .
6, minimising cut-offeffects at the same time as postponing the unphysical behaviour to very close to charm even on thecoarsest ensemble. In the limit L s → ¥ , Domain Wall fermions maintain exact chiral symmetry onthe lattice. However for finite L s there is a residual chiral symmetry breaking that can be quantifiedby the residual mass m res which can be defined from the axial Ward Identity for the Domain Wall4 harm physics with Moebius Domain Wall Fermions Justus Tobias Tsang at -4 -3 -2 -1 a m e ff r e s am=0.1am=0.2am=0.3am=0.4am=0.5am=0.6am=0.7 am q -4 -3 -2 -1 a m r e s Figure 3:
Left: Behaviour of the effective residual mass m effres as a function of time. m effres is found by evaluating(3.2) on every time slice. Right: The residual mass taken at the centre of the lattice as a function of the barequark mass. Both panels are for the coarsest ensemble with Shamir Domain Wall fermions and L s = fermions (cid:10) ¶ m A m ( x ) P ( ) (cid:11) = m h P ( x ) P ( ) i + (cid:10) J q ( x ) P ( ) (cid:11) , (3.1)so that am res = (cid:229) x J q ( x ) P ( ) (cid:229) x P ( x ) P ( ) . (3.2)Here A m is the conserved axial current, P is the pseudo scalar density and J q is a pseudo scalarcurrent that mediates between the boundaries of the 5th dimensions and the centre of the 5th di-mension.To investigate the above mentioned unphysical behaviour the residual mass was closely moni-tored (compare figure 3) and from this it can be seen that for am q & . Möbius
Domain Wall fermionsimulations to am q ≤ . L s by choosing it as L Möbius s =
12 (correspondingto L Shamir s =
24) as opposed to L Shamir s =
16 as in figures 2 and 3.
4. Analysis and Results
The physical quantities of which we investigate the continuum scaling are the decay constantsof strange-strange ( h s ), heavy-strange ( D s -like) and heavy-heavy ( h c -like) pseudo scalar mesons.Our strategy is as follows:On each ensemble we simulate two closely spaced strange quark masses and various ’heavy’quark masses with am q ≤ . D s and h s masses are interpolated to the physical strange quark mass, by matching the h s mass to thepublished value [17]. To be able to take a continuum limit at the same physical point the decayconstants for h c and D phys s are now interpolated to common reference masses m refPS on each ensem-ble. Since we are not interested in physical predictions in the quenched approximation we do notcalculate renormalisation constants for these quantities, but instead take ratios at a reference massto cancel the renormalisation constant. This reference mass is chosen as m normPS = . harm physics with Moebius Domain Wall Fermions Justus Tobias Tsang a (GeV −1 ) f P S (cid:1) m P S / f n o r m P S (cid:0) m n o r m P S m refPS =2.5 GeVm refPS =2.2 GeVm refPS =1.9 GeVm refPS =1.6 GeVm refPS =1.3 GeV a (GeV −1 ) f P S (cid:1) m P S / f n o r m P S (cid:0) m n o r m P S m refPS =3.5 GeVm refPS =3.0 GeVm refPS =2.5 GeVm refPS =2.0 GeVm refPS =1.5 GeV Figure 4:
Continuum limit of the ratio of decay constants at different pseudo scalar masses for D s (left) and h c (right) using Möbius Domain Wall Fermions. The data is normalised at m normPS = . a . For the heaviest two reference masses the coarsest ensemble doesnot allow a simulation with am q ≤ .
4, so the continuum limit is taken from the three finer ensembles only. of further investigations towards the feasibility of B -physics and contact to HQET we will considerthe expression f PS √ m PS which approaches a constant in the static limit. The resulting continuumlimit is presented in figure 4. From the two panels in figure 4 it is clear that the continuum approachwith the chosen parameters is very flat and well described by O ( a ) scaling. This is further encour-aged by the fact that the residual mass remained well behaved and under control for the simulateddata points, confirming the O ( a ) improvement.
5. Conclusion and Outlook
From these results we conclude that Möbius Domain Wall fermions are a suitable discretisationfor charm physics. We expect the qualitative behaviour of our findings to remain unchanged whengoing beyond the quenched approximation allowing us to dynamically simulate QCD with 2 + f whilst keeping discretisation errors under control. This is very encouraging for our ongoingdynamical efforts (first results were presented in Jüttner’s talk at this conference [2]).Finally this quenched data will be used to test the feasibility of doing B -physics with DomainWall fermions, by combining it with results in the static limit as well as using the ratio method [18]. Acknowledgments
The research leading to these results has received funding from the European Research Councilunder the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agree-ment 279757. The authors gratefully acknowledge computing time granted through the STFCfunded DiRAC facility (grants ST/K005790/1, ST/K005804/1, ST/K000411/1, ST/H008845/1).PB acknowledges support from STFC grants ST/L000458/1 and ST/J000329/1.6 harm physics with Moebius Domain Wall Fermions
Justus Tobias Tsang
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