Approaching the Bottom Using Fine Lattices With Domain-Wall Fermions
KKEK-CP-351, OU-HET-921
Approaching the Bottom Using Fine Lattices WithDomain-Wall Fermions
Brendan Fahy ∗ a , Guido Cossu a , Shoji Hashimoto ab a High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan b School of High Energy Accelerator Science, SOKENDAI (The Graduate University forAdvanced Studies), Tsukuba 305-0801, Japan
We explore the heavy-quark mass region above the charm mass using Möbius domain-wallfermions on fine lattices at a = . . .
044 fm. We examine masses and decayconstants using a series of heavy quark masses up to 3 times the charm quark. We analyze thecutoff effects for heavy quarks above the charm and account for the leading order discretizationeffects using ideas from HQET. We extrapolate to the bottom quark mass and report preliminaryresults for f B and f B s . ∗ Speaker. 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 ] F e b pproaching the Bottom Using Fine Lattices With Domain-Wall Fermions Brendan Fahy
1. Introduction and Lattice Setup
Direct lattice simulation of the bottom quark is still a challenge for lattice QCD. One approachis to use effective actions specific for heavy quark physics such as non-relativistic QCD and matchback to QCD. Modern lattices are, on the other hand, much finer than in the past and cutoff effectsat the charm are small. Using very fine lattices, cutoff effects are manageable even above the charmmass [1], which allows us to produce results between the charm and the bottom with enough pointsto extrapolate to the bottom.The JLQCD collaboration has recently produced lattice ensembles with 2 + / a ≈ .
4, 3 .
6, and 4 . t with thephysical value from [4]. The parameters of each of the 15 lattices can be found in Table 1.Axial and pseudo-scalar two-point correlators were computed with the Iroiro++ software pack-age [5]. Correlators were improved using Z ( ± ) noise sources distributed over a single timeslice, and sources computed on many time slices on a single configuration are then averaged. Atotal of 400 −
600 measurements were carried out as detailed in Table 1. Each of these Z sourceswere computed both unsmeared and with Gaussian smearing. The source-sink combinations ofunsmeared-smeared and smeared-smeared were simultaneously fit to extract meson masses and de-cay constants. Due to good chiral symmetry of our domain-wall fermions the decay constants canbe computed directly from the psuedo-scalar currents utilizing the PCAC relation. These latticeshave also been used to compute semi-leptonic D decays in [6] and charm quark mass determinationfrom short-distance correlators in [7].
2. Charm Results
The first goal was to determine the decay constants f D and f D s . These were determined frompseudo-scalar correlators at the charm mass determined from a previous study [1] with an inputof the spin averaged c ¯ c mass. The values for f D ( s ) for all of our ensembles can be seen in Figure1. We preformed a global fit to all the ensembles assuming linear dependence on the light andstrange quark masses and on the lattice spacing squared. The plots show lines corresponding tothe continuum limit (black) as well as the line evaluated at a lattice spacing corresponding to ourcoarsest lattices. The difference between the continuum limit and our coarsest lattice is roughly2%. The plots for f D s (right panel) have fit lines which do not go through the cluster of pointsbecause the data are simulated at strange quark masses which sandwich the physical value.The results of the global fit evaluated at the physical point are f D = . ± . ± . , (2.1) f D s = . ± . ± . , (2.2)with the errors being the statistical error and the systematic error from the scale determination,respectively. 1 pproaching the Bottom Using Fine Lattices With Domain-Wall Fermions Brendan Fahy β L × T L am ud am s m π m π L β = .
17 32 ×
64 12 0.0035 0.040 230 3.0 8000.0070 0.030 310 4.0 8000.0070 0.040 310 4.0 8000.0120 0.030 400 5.2 8000.0120 0.040 400 5.2 8000.0190 0.030 500 6.5 8000.0190 0.040 500 6.5 80048 ×
96 12 0.0035 0.040 230 4.4 800 β = .
35 48 ×
96 8 0.0042 0.018 300 3.9 6000.0042 0.025 300 3.9 6000.0080 0.018 410 5.4 6000.0080 0.025 410 5.4 6000.0120 0.018 500 6.6 6000.0120 0.025 500 6.6 600 β = .
47 64 ×
128 8 0.0030 0.015 280 4.0 400
Table 1:
Parameters of the JLQCD gauge ensembles used in this work. Pion masses are rounded to thenearest 10 MeV. Inverse lattice spacings are a − = . ( ) GeV, 3 . ( ) GeV, and 4 . ( ) GeV for β = . , .
35 and 4 .
47 respectively. The L length in the domain-wall is 12 at β = .
17 and 8 on the finerlattices. The spatial extent satisfies m π L (cid:38) ≥ . β = .
17 ensembles and 50 in the others.
3. Above Charm
Since the cutoff effects at the charm mass were small we expect that the results at slightlyheavier quark masses are also under control. We produced heavy-light and heavy-strange correla-tors using a set of heavy masses above the charm mass. These were chosen to be above the charmmass by factors of 1 .
25 producing a sequence m n = ( . ) n m c . The bare charm mass is determinedin a separate study [1]. The values for m n used on different β values are shown in Table 2. For thefiner lattices we produced five bare quark masses above the charm mass, while we limited to onlythree masses above charm for our coarsest lattice as the bare quark mass ( . ) m c exceeds 1 . f hx √ m hx where f is the pseudo-scalar decay constant, m is the pseudo-scalar meson mass,and x is either a strange ( s ) quark or light ( (cid:96) ) quark . This was chosen because in the heavy quarklimit, m h → ∞ , the combination f hx √ m hx is known to scale as a constant up to the anomalousdimension contribution (see below). Plots of this verses the inverse meson mass, for heavy-light(left) or heavy-strange (right), are shown in Figure 2. The results on all ensembles are plottedtogether with the cluster of points on the right side being the values at the charm mass and moving2 pproaching the Bottom Using Fine Lattices With Domain-Wall Fermions Brendan Fahy m π [GeV^2] f D [ M e V ] Linear continuum fitfit at β =4.17 m π [GeV^2] f D s [ M e V ] β = 4 . , L = 32 , m s = 0 . β = 4 . , L = 32 , m s = 0 . β = 4 . , L = 48 , m s = 0 . β = 4 . , L = 48 , m s = 0 . β = 4 . , L = 48 , m s = 0 . β = 4 . , L = 64 , m s = 0 . Figure 1:
Plots of f D (left) and f D s (right) vs m π . The fit lines correspond to the continuum limit (black)and the coarsest lattice (blue). They are linear in m π and are interpolated to the physical strange point using2 m K − m π . Beta m = m c m m m m m Table 2:
Bare heavy quark masses where m is the charm quark mass m n = λ n m . to the left each cluster being the next choice for the heavy quark mass. It is clear that on thecoarsest lattice (blue) discretization effects become significant already at ( . ) m c and heavier. Inthe next section we attempt to account for the leading order a dependence. Here as a first attemptwe simply perform a global fit to all of the data including a term to account for a as well as a m effects.The global fit is performed using an ansatz f hx √ m hx = (cid:0) Φ phys (cid:1) (cid:18) + C m h + C m h (cid:19) Φ phys = (cid:18) + γ S (cid:0) M K − M π (cid:1) + γ P (cid:18) M π − (cid:16) M phys π (cid:17) (cid:19) + γ A a + γ MA ( ma ) (cid:19) . (3.1)The basic assumption is the dependence on the inverse meson mass is a polynomial preserving aconstant in the limit of m → ∞ . The Φ phys accounts for the extrapolation to the physical pion mass,interpolation to the physical strange quark mass using 2 M K − M π , and extrapolation to the cotinuumlimit with both m a and a terms. Here the fit excludes the heaviest points on the β = .
17 and β = .
35 lattices which have a bare quark mass above 0 . ( ma ) and higher are expected to be more significant. The fit curves corresponding to each β value3 pproaching the Bottom Using Fine Lattices With Domain-Wall Fermions Brendan Fahy /m hl [1/GeV] f h l p m h l [ G e V ^ ( / )] Continuum fitfit at β =4.17fit at β =4.35fit at β =4.47 /m hs [1/GeV] f h s p m h s [ G e V ^ ( / )] Continuum fitfit at β =4.17fit at β =4.35fit at β =4.47 Figure 2:
Values for f √ m for heavy-light (left) and heavy-strange (right) on all of our ensembles for eachof the different heavy-quark masses from Table 2. Data points are those of β = .
17 (blue), 4 .
35 (red) and4 .
47 (magenta). Each horizontal cluster of points shows values for a particular choice of heavy quark masswith the value at the charm mass on the far right and increasing heavy quark masses going to the left. Theblack line indicates a global fit in physical limit as discussed in the text. The colored lines indicate the samefit parameters evaluated at the finite lattice spacings corresponding to our three choices of β . The emptypoints with error bars show the statistical uncertainly of the fit at the physical values for m B and m B s . and physical light/strange masses are shown in Figure 2, We observe that the data points drift awayfrom the continuum curve (black) as ma gets large. In particular the heaviest points at β = . .
35 suffer from strong discretization effects.
4. HQET Corrections
To understand the leading order cutoff effects we use ideas from Heavy Quark Effective The-ory (HQET). We closely follow the discussion of [8, 9], which was mainly applied to the Wilson-type fermions.Expanding the energy of a free quark for low momentum we obtain E ≈ m + p m + . . . , whereon the lattice the “rest mass”, m , may not be equal to the “kinetic mass”, m . These correctionswere computed years ago for Wilson fermions [8] which give simple corrections that have beenused to design actions suited for heavy quarks [9]. In the case of domain wall fermions the expres-sions for these corrections are not as simple.Starting with the propagator for domain-wall fermions [10] and expanding in low momen-tum we obtain m and m at tree level as well as the wave-function renormalization, A DWKLM . Theexpressions for these factors are, m = log (cid:18) − W + (cid:113) ( − W ) − (cid:19) , (4.1) m = (cid:113) W − W (cid:16) Q + − W ( Q + )+( Q − )( W + W ) (cid:17) , (4.2) A DWKLM = ( − m ) (cid:104) + (cid:113) Q + W (cid:105) , (4.3)4 pproaching the Bottom Using Fine Lattices With Domain-Wall Fermions Brendan Fahy / ( m hl + m − m ) [1/GeV] ˆ f h l q ˆ m h l / C ( µ ) [ G e V ^ ( / )] Continuum fitfit at β =4.17fit at β =4.35fit at β =4.47 / ( m hs + m − m ) [1/GeV] ˆ f h s q ˆ m h s / C ( µ ) [ G e V ^ ( / )] Continuum fitfit at β =4.17fit at β =4.35fit at β =4.47 Figure 3:
Same as Figure 2 with the corrections described in Section 4. The anomalous dimension factorsis also included. The fit now accounts for leading order a m effects so the correction used is γ MA α s ( ma ) .The black error bars on the fit give the statistical uncertainty at the B and B s meson mass respectively. where Q = (cid:18) + m − m (cid:19) and W = + Q − (cid:112) Q + Q . (4.4)According to [8, 9] we can then re-scale the heavy quark mass in favor of m from m by adding m − m , since the kinetic mass controls the motion of the heavy quark inside the meson. Similarly,we divide the amplitude by A DWKLM to eliminate the leading discretization effect for the heavy quark.These corrections, however, turned out to be insufficient to account for the heavy quarks prop-agator at short distances. If we numerically integrate the free propagator and divided by A KLM , theyagree at large time separations but for small separation they disagree. This deviation from a simpleexponential behavior may be due to the non-locality of domain-wall fermions, which becomes rela-tively more significant at large quark masses. To fully capture such lattice artifacts we numericallyintegrate the free propagator for each bare heavy quark mass, and divide our correlators by theseand multiply back by the continuum result, so that the heavy quark propagator coincides that of thecontinuum at least at tree level.Applying these corrections we match the decay constants from QCD to HQET with a factoraccounting for the anomalous dimension C ( µ ) . Perturbatve calculation of C ( µ ) is available upto three loop, α s [11]. These results and fit are shown in Figure 3. These results exhibit lessdivergence from the continuum limit of the values for the heaviest quark masses indicating thatwe have successfully account for the bulk of the leading a m cutoff effects. The global fit to thecorrected data is performed in the same manner as before, see (3.1), but replacing the m a termwith γ MA α s ( ma ) as we have accounted for the tree level corrections. The continuum limit (blackline) as well as the fit for the individual finite lattice spacings (colored lines) are plotted. The fit5 pproaching the Bottom Using Fine Lattices With Domain-Wall Fermions Brendan Fahy function evaluated at the B and B s mass yields, f B = . ± . ± . f B s = . ± . ± . . (4.6)These results are within 2 σ of the current FLAG average [12].
5. Summary
Using fine lattices we are able to obtain results for pseudo-scalar decay constants above thecharm mass at nearly the bottom mass. Accounting for the leading discretization effects we are ableto extrapolate to the bottom quark mass and predict f B and f B s . This is all done without requiringa specialized action for heavy quarks as they were treated with the same domain wall action usedfor light quarks.Numerical simulations are performed on Hitachi SR16000 and IBM System Blue Gene Solu-tion at KEK under the support of its Large Scale Simulation Program (No. 16/17-14). This researchis supported in part by the Grant-in-Aid of the MEXT (No. 26247043, 25800147) and by MEXTas “Priority Issue on Post-K computer” (Elucidation of the Fundamental Laws and Evolution of theUniverse) and JICFuS. References [1] JLQCD collaboration, B. Fahy, G. Cossu, S. Hashimoto, T. Kaneko, J. Noaki and M. Tomii,
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