Wilson flow and scale setting from lattice QCD
V.G. Bornyakov, R. Horsley, R. Hudspith, Y. Nakamura, H. Perlt, D. Pleiter, P.E.L. Rakow, G. Schierholz, A. Schiller, H. Stüben, J.M. Zanotti
aa r X i v : . [ h e p - l a t ] A ug ADP-15-25/T927DESY 15-154Edinburgh 2015/19Liverpool LTH 1054August 24, 2015
Wilson flow and scale setting from lattice QCD
V. G. Bornyakov a , R. Horsley b , R. Hudspith c , Y. Nakamura d ,H. Perlt e , D. Pleiter f , P. E. L. Rakow g , G. Schierholz h ,A. Schiller e , H. St¨uben i and J. M. Zanotti j – QCDSF-UKQCD Collaboration – a Institute for High Energy Physics, Protvino,142281 Protvino, Russia,Institute of Theoretical and Experimental Physics, Moscow,117259 Moscow, Russia,School of Biomedicine, Far Eastern Federal University,690950 Vladivostok, Russia b School of Physics and Astronomy, University of Edinburgh,Edinburgh EH9 3FD, UK c Department of Mathematics and Statistics, York University,Toronto, ON Canada M3J 1P3 d RIKEN Advanced Institute for Computational Science,Kobe, Hyogo 650-0047, Japan e Institut f¨ur Theoretische Physik, Universit¨at Leipzig,04109 Leipzig, Germany f J¨ulich Supercomputing Centre, Forschungszentrum J¨ulich,52425 J¨ulich, Germany,Institut f¨ur Theoretische Physik, Universit¨at Regensburg,93040 Regensburg, Germany g Theoretical Physics Division, Department of Mathematical Sciences,University of Liverpool, Liverpool L69 3BX, UK h Deutsches Elektronen-Synchrotron DESY,22603 Hamburg, Germany i Regionales Rechenzentrum, Universit¨at Hamburg,20146 Hamburg, Germany j CSSM, Department of Physics, University of Adelaide,Adelaide SA 5005, Australia bstract We give a determination of the phenomenological value of the Wilson(or gradient) flow scales t and w for 2 + 1 flavours of dynamical quarks.The simulations are performed keeping the average quark mass constant,which allows the approach to the physical point to be made in a controlledmanner. O ( a ) improved clover fermions are used and together with fourlattice spacings this allows the continuum extrapolation to be taken. Numerical lattice QCD simulations naturally determine dimensionless quantitiessuch as mass ratios and matrix element ratios, however determining a physicalvalue requires the introduction of a scale, usually taken from experiment. Ahadron mass, such as the proton mass, or decay constant, such as the pion decayconstant, are often used for this purpose. We discuss here setting the scaleusing flavour-singlet quantities, which in conjunction with simulations keepingthe average quark mass constant allow SU (3) flavour breaking expansions to beused. This is illustrated here using 2 + 1 clover fermions, and a determinationof the Wilson flow scales t and w is given. These are ‘secondary’ scales andare not experimentally accessible and thus they have to be matched to physicalquantities. These flow scales are cheap to compute from lattice simulations (forexample they do not require a knowledge of quark propagators) and accurate (forexample they do not require a determination of the potential which requires thelimit of a large distance). So once the phenomenological value of the flow scalesis known the determination of physical values becomes more tractable.Flow and flow variables were introduced by L¨uscher, [1]. We follow himhere, [2], in particular in our brief discussion of the t scale. Flow representsa smoothing of the gauge fields. We denote the flow time by t , and the linkvariables at this time by U µ ( x, t ) = exp( iT a θ aµ ( x, t )) which evolve according to dU µ ( x, t ) dt = iT a δS flow [ U ] δθ aµ ( x, t ) U µ ( x, t ) , with U µ ( x,
0) = U µ ( x ) , (1)with S flow [ U ] being the flow action, which does not have to be the same as theaction used to generate the gauge variable. ( x is just the normal 4-dimensionalEuclidean space-time.) Setting F ( t ) ≡ t h E ( t ) i , where E ( t ) = F a µν ( t ) , (2)then we define the t scale by F ( t ) | t = t ( c ) = c . (3)2lternatively, [3] define the w scale as t ddt F ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = w ( c ) = c , (4)where in both definitions c is a constant, conventionally taken as c = 0 .
3. Werequire a value of c such that a ≪ √ t ≪ L , L being the lattice size and thisvalue was found to be a suitable choice, [2]. Alternative suggestions have beenmade, see e.g. [4, 5].In this article we shall give a determination of q t exp0 and w exp0 . There have beenseveral determinations for different numbers of flavours. These include quenched n f = 0 or quenched, [5, 6]; n f = 2, [7, 8]; n f = 2 + 1, [9, 3, 10]; n f = 2 + 1 + 1,[11, 12, 13]. We have also published preliminary results, [14].The plan of this article is as follows. In section 2 we describe our method ofapproaching the physical quark mass starting from a point on the SU (3) flavoursymmetric line, [15, 16]. We also discuss the general property of singlet quanti-ties, that they have a stationary point about this SU (3) flavour symmetric line.Section 3 gives examples of singlet quantities both hadronic and gluonic (i.e. inthis case t and w ) and also discusses their SU (3) flavour breaking expansions.Section 4 first gives our lattice conventions, ensembles used and numerical valuesof the singlet quantities. This is followed by section 5 in which the √ t and w scales are determined for several lattice spacings. In the next section, section 6 wetake the continuum result to give the final result. Finally in section 7 we compareour result with other results for n f = 2 + 1 flavours and give our conclusions. We consider extrapolations to the physical point from a point on the SU (3)flavour symmetric line keeping the average quark mass fixed, [15, 16], m = ( m u + m d + m s ) = const. . (5)This means that as the pion mass tends downwards to its physical value, thekaon mass increases upwards to its physical value. (In particular the kaon massis never larger than its physical value.) Possible scenarios are sketched in Fig. 1for a path from a point on the SU (3) flavour symmetric line to the physical pointfor the case of mass degenerate u and d quarks m l , see eq. (12). (For non-massdegenerate u and d quarks we would have instead a plane.) Shown in Fig. 1are the bare and renormalised quark masses for the case discussed here of cloverfermions. (Because the singlet and non-singlet pieces renormalise differently therenormalised quark axes are not orthogonal to each other, as further discussed in[16]. For chiral fermions this would not be the case.) In the left hand panel one3 m l* ,m s* ) m s =m l =m m=const m lR m sR m l m s (m l* ,m s* ) m s =m l X S =const m lR m sR m l m s Figure 1:
Possible scenarios for the path in the ( m u = m d ≡ ) m l – m s plane. In theLH panel m = const., while in the RH panel we hold other singlet quantities constant. common trajectory is sufficient, while in the right hand panel it depends on thesinglet quantity used, [16].With the condition of eq. (5) flavour singlet quantities turn out to have a sta-tionary point in the quark mass starting from a given point on the SU (3) flavoursymmetric line. As we shall see this potentially allows simpler extrapolations tothe physical point. This property may be shown by considering small changesabout a given point on the SU (3) flavour symmetric line. Let X S ( m u , m d , m s )be a flavour singlet object i.e. X S is invariant under the quark permutation sym-metry between u , d and s . Then Taylor expanding X S about a point m on the SU (3) flavour symmetric line m u = m d = m s = m ≡ m , m q = m + δm q , (6)gives X S ( m + δm u , m + δm d , m + δm s ) (7)= X S ( m, m, m ) + ∂X S ∂m u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δm u + ∂X S ∂m d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δm d + ∂X S ∂m s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δm s + O (( δm q ) ) . But on the symmetric line we have ∂X S ∂m u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∂X S ∂m d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∂X S ∂m s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (8)and on our chosen trajectory, eq. (5) δm u + δm d + δm s = 0 , (9)4hich together imply that X S ( m + δm u , m + δm d , m + δm s ) = X S ( m, m, m ) + O (( δm q ) ) . (10)In other words, the effect at first order of changing the strange quark mass iscancelled by the change in the light quark mass, so we know that X S must havea stationary point on the SU (3) flavour symmetric line.Thus provided the quadratic terms in eq. (10) are small it will not matterwhich singlet quantity we use, all are equivalent to keeping m = const., thescenario of the left hand panel of Fig. 1. We shall consider here hadronic singlet quantities, such as independent aver-ages with respect to quark permutations of the pseudoscalar, vector and nucleonoctets, [15, 16], and also gluonic quantities derived from the Wilson flow, [2, 3].In particular we shall consider X π = ( M K + + M K + M π + + M π − + M K − + M K ) = (2 M K + M π ) X ρ = ( M K ∗ + + M K ∗ + M ρ + + M ρ − + M K ∗ + M K ∗− ) = (2 M K ∗ + M ρ ) X N = ( M p + M n + M + + M − + M + M − ) = ( M N + M + M ) X t = 1 /t X w = 1 /w . (11)(The charge conjugate mesons have the same masses, at least for pure QCD. Forexample M π + = M π − , M K + = M K − and M K = M K , obviously no such similarresult holds for the baryons.) The second expressions are the masses for massdegenerate u and d quarks, m u = m d ≡ m l . (12)Except for X π , which is naturally an average over quadratic masses, it does notmatter whether we consider linear or quadratic averages – quadratic averageswere found to give slightly better fits for heavy partially quenched masses (upto the charm quark); in the small quark mass range considered here this is lessimportant. Note also that for the Wilson flow singlets we consider the inverse ofthe flow variable as then all singlet quantities have the same dimensions.Other possibilities, as discussed in [16] include the further nucleon octet singlet X = ( M + M ) = ( M + M ) , (13)and baryon decuplet singlets X = ( M ++ + M − + M − ) = (2 M + M ) X ∗ = ( M + + M + M ∗ + + M ∗− + M ∗ + M ∗− ) = ( M + M + M ) X ∗ = M ∗ = M ∗ . (14)5inglet quantity GeV X exp π X exp ρ X exp N X expΛ X exp∆ X expΞ ∗ X expΣ ∗ Table 1:
Experimental values for various singlet quantities, averaging over (masses) (to 4 decimal places). Further possibilities can be constructed from ‘fictitious’ particles, such as a ‘nu-cleon’, N s , with three mass degenerate quarks at the strange quark mass. At the SU (3) flavour symmetric points, all the baryon octet hadrons are mass degenerate(at least from a QCD perspective), so we expect that away from this point thereis no (or very little) difference between X N and X , i.e. X N = X Λ . The same ar-gument holds for the various baryon decuplet possibilities, i.e. X ∆ = X Ξ ∗ = X Σ ∗ .How far does this extend? Let us consider the experimental (or phenomeno-logical) singlet hadron mass results as given in Table 1 . It is seen that evenafter the extrapolation from the SU (3) flavour line to the experimental point,then X exp N ≈ X expΛ and X exp∆ ≈ X expΞ ∗ ≈ X expΣ ∗ the worst discrepancy (between X expΞ ∗ and X expΣ ∗ ) is only a fraction of a percent. This indicates that quite likely the X S are constant over a large interval.However as the baryon decuplet possibilities are numerically substantiallynoisier, we shall only use the baryon octet hadrons here.An equivalent statement (for singlet quantities built from hadron masses) isfound by considering the SU (3) flavour breaking expansion. As discussed in[15, 16] we have for the octet mesons, π + ( ud ), π − ( du ), K + ( us ), K − ( su ), K ( ds )and K ( sd ) not lying at the centre of the octet M ( ab ) = M + α π ( δm a + δm b ) + O (( δm q ) ) , (15)where a and b are u , d or s quarks. Similar results hold for the vector mesons. Forthe p ( uud ), n ( ddu ), Σ + ( uus ), Σ − ( dds ), Ξ ( ssu ), Ξ − ( ssd ) baryons on the outer Only when necessary and for clarity do we distinguish between experimental and latticemasses – X exp S and X lat S respectively. As we are not considering mass differences, then the effect of electromagnetic effects is small,and so we can disregard them here. For example for the lightest particles – the pseudoscalaroctet, the value given in Table 1 for X π is to be compared with the value upon using Dashen’stheorem which gives 0 . ∼ < .
2% difference. For the baryonsfor X N it is ∼ < . we have M ( aab ) = M N + A (2 δm a + δm b ) + A ( δm b − δm a ) + O (( δm q ) ) . (16)All the expansion coefficients are functions of m . It is easy to check that thismeans that X S = M S + O (( δm l ) ), S = π , ρ and N in agreement with eq. (10).‘Fan’ plots from the symmetric point down to the physical point are welldescribed by the linear behaviour of eqs. (15) and (16) as shown in [15, 16, 19, 20]which further supports the earlier statement that X S is constant over a largequark mass range.Although in our extrapolations, we shall not be using chiral perturbationtheory, χ PT, it is natural to ask about its relationship to the SU (3) flavourbreaking expansion. This also allows a check on eq. (10), assuming we are ina region where χ PT is valid. This was investigated in [15, 16] for hadron masssinglets and we now extend the argument to t (and w ), using the result of [21].Using the notation of this paper and for mass degenerate u and d quark masseswe find t = T ( χ ) " πf ) ( k + k ′′ )( χ s − χ l ) + · · · , (17)where T ( χ ) = t , ch " k (4 πf ) χ + 8 k (4 πf ) χ ln χ Λ + 9 k ′ (4 πf ) χ (18)being the value of t on the symmetric line ( t , ch is the value in the chiral limit). k i are constants, f the pion decay constant again in the chiral limit and χ l = B m l , χ s = B m s , χ = (2 χ l + χ s ).As expected, there is no linear term, and the first term we see is quadratic inthe SU (3) breaking. Further details are given in Appendix A. We consider 2+1 non-perturbatively O ( a ) improved clover fermions, as describedin [17]. The relation between the bare quark masses m q in lattice units and thelattice mass parameters κ q is given by [15, 16] m q = 12 κ q − κ c ! with q ∈ { l, s, } , (19) For non-degenerate u and d quark mass, the Lambda and Sigma particles mix. This mixingis however very small. This was investigated in [20], where using the notation there, it wasshown that ( M + M ) = P A = M N + O (( δm q ) ), i.e. again we have no linear term inthe quark mass. Note also that when the u and d quark masses are degenerate, then no mixingoccurs. V κ κ l κ s M π L L [fm]5.80 48 ×
96 0.122760 0.122760 0.122760 6.95 2.825.80 48 ×
96 0.122810 0.122810 0.122810 6.11 2.825.80 48 ×
96 0.122810 0.122880 0.122670 5.11 2.825.80 48 ×
96 0.122810 0.122940 0.122551 4.01 2.825.80 48 ×
96 0.122870 0.122870 0.122870 4.96 2.82
Table 2:
Parameters for β = 5 .
80. Each block has the same κ , i.e. constant m . β V κ κ l κ s M π L L [fm]5.65 32 ×
64 0.121975 0.121975 0.121975 4.99 2.195.65 32 ×
64 0.122005 0.122005 0.122005 4.67 2.195.65 32 ×
64 0.122005 0.122078 0.121859 4.00 2.19 × ×
64 0.122030 0.122030 0.122030 4.32 2.195.65 32 ×
64 0.122050 0.122050 0.122050 4.03 2.19
Table 3:
Parameters for β = 5 .
65. The entries in italics have M π L < where in simulations we have mass degenerate u and d quarks, i.e. m u = m d ≡ m l and the s quark has mass m s . Along the SU (3) flavour mass degenerate line, thecommon quark mass is denoted by m (or equivalently by κ ) and where vanishingof the quark mass along this line determines κ c . Along the m = m = constantline gives from eqs. (6) and (19) the SU (3) flavour breaking mass parameter as δm q = 12 κ q − κ ! . (20)We see that κ c has dropped out of this equation, so we do not need its explicitvalue here. Along this trajectory the choice of quark masses is restricted and wehave κ s = 1 κ − κ l , (21)so once we have decided on a κ , then a given κ l determines κ s .We consider four beta values β = 5 .
8, 5 .
65, 5 .
50, 5 .
40 (where β = 10 /g withour conventions). In Tables 2, 3, 4 and 5 we give parameters of the runs. Theentries in italics have M π L < β = 5 .
50 on 32 ×
64 lattices with degenerate quarkmasses of κ = 0 . . . . SU (3) flavour symmetric point on the constant m trajectory to the physical point. As can be seen for some of the κ values we have8 V κ κ l κ s M π L L [fm]5.50 32 ×
64 0.120900 0.120900 0.120900 5.59 2.375.50 32 ×
64 0.120900 0.121040 0.120620 4.32 2.37 × × ×
96 0.120900 0.121166 0.120371 4.10 3.555.50 32 ×
64 0.120920 0.120920 0.120920 5.27 2.375.50 32 ×
64 0.120920 0.121050 0.120661 4.10 2.375.50 32 ×
64 0.120950 0.120950 0.120950 4.83 2.375.50 32 ×
64 0.120950 0.121040 0.120770 3.97 2.37 × ×
64 0.120990 0.120990 0.120990 4.11 2.37
Table 4:
Parameters for β = 5 . β V κ κ l κ s M π L L [fm]5.40 24 ×
48 0.119860 0.119860 0.119860 4.98 1.965.40 24 ×
48 0.119895 0.119895 0.119895 4.54 1.965.40 24 ×
48 0.119930 0.119930 0.119930 4.11 1.96 × × Table 5:
Parameters for β = 5 . extended the constant m = m trajectories down in the direction of the physicalpoint.While changing the β value gives the greatest change to the singlet terms,smaller effects occur on the SU (3) flavour symmetric line as we change m . Thiswill help us to locate the initial κ , the starting point for the trajectory in the m s – m l plane leading to the physical point. Note that if m is held constant, then afurther advantage of this condition is that for clover fermions the O ( a ) improvedcoupling constant remains unchanged as˜ g = g (1 + b g ( g ) am ) , (22)although this is unlikely to lead to any large effect.For orientation the SU (3) symmetric point has a pion mass of about ∼
450 MeV and we reach down to about ∼
260 MeV.The specific components used in the flow discretisation here are (f[low] , g[auge action] , o[bservable]) The flow Wilson action here means S flow [ U ] = P Re Tr[1 − U plaq ]. , S[ymanzik [tree level]] , C[lover]) , (23)and the Runge-Kutta discretisation is used for the flow equation, [2].One can improve the scaling behaviour, which is expected to have O ( a )corrections. For example for √ t following [11] we can write F ( t )1 + C a t + . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = t ( c ) = c ⇒ t = t C F F ′ a t + . . . ! . (24)Inverting this and relabelling t → t gives t = t − C F F ′ a t + . . . ! , (25)where F x = F ( t x ), F ′ x = tdF ( t ) /dt | t x , with x ≡ f go ) = ( W SC ), C = − /
72 [18] so we expect the gradient to be +ve.
We first investigate the constancy of the singlet quantities X S , as discussed insection 2. In Fig. 2 and 3 we plot X S for S = t , N , w ρ and π against M π /X π π /X π ( X S l a t ) S = t S = NS = w S = ρ S = π ( β , κ ) = (5,80,0.122810) Figure 2:
Top to bottom ( X lat S ) for S = t (circles), N (right triangles), w (squares), ρ (left triangles) and π (up triangles) for ( β, κ ) = (5 . , . .0 0.2 0.4 0.6 0.8 1.0M π /X π ( X S l a t ) S = t S = NS = w S = ρ S = π ( β , κ )=(5.50,0.120900) Figure 3: ( X lat S ) for S = t , N , w , ρ and π , (top to bottom) for ( β, κ ) =(5 . , . M π L < (which is equivalent to 1 /κ l ). The value at M π /X π = 1 corresponds to the SU (3)symmetric point and the vertical dashed lines correspond to the physical point.No structure or trend is seen in the results and they are compatible with( X lat S ) (and hence X lat S ) a constant down to the vicinity of the physical point.The constant fits exclude points with M π L <
4. However these additional pointsall have M π L > X S as discussed in sections 2 and 3 is supported by the numericalresults.Furthermore any (significant) quadratic term would mean that each quan-tity S starts at a slightly different point on the SU (3) flavour symmetric line(but all with the same gradient) and the trajectories would then all focus atthe experimental point. We have considered the partially quenched expansion(up to cubic terms in the quark mass) and have determined that along the uni-tary line considered here these higher order terms are negligible – only whenthe quark mass is in the vicinity of the charm quark mass do these non-linearterms become appreciable. Thus practically we have a unique starting point onthe SU (3) flavour symmetric line for the trajectory to the physical point, i.e.we have the situation for at least S = t , N , w , ρ and π of the left panel ofFig. 1. To check this an alternative description is provided by a re-arrangement11f X π /X S = (2 M K + M π ) /X S to give2 M K − M π X S = X π X S − M π X S , (26)for S = N, ρ, t , w . So plotting (2 M K − M π ) /X S against M π /X S with constantgradient − S . Hence in Fig. 1 left panel thegradient is − − M K − π /X S2 ( M K − M π ) / X S S = NS = ρ S = t S = w ( β , κ )=(5.80,0.122810) Figure 4: (2 M K − M π ) /X S against M π /X S , together with the fit from eq. (26) for( β, κ ) = (5 . , . S = N (right triangles), ρ (left triangles), t (circles)and w (squares). The stars correspond to the experimental values for S = ρ , N upperand lower respectively. M π ) /X S versus M π /X S with S = N , ρ , t , w for the same data sets as in Fig. 2and 3. Again straight lines (from the fit function) describe the data very well.However as can be seen the lines for the S = N and ρ cases do not quite gothrough their physical points (denoted by stars). This is because the κ usedwhile close is not quite the value required for the correct path. We shall infuture denote this point by κ ∗ . For example, it can be seen that the β = 5 . κ = 0 . κ = 0 . κ ∗ than for the β = 5 . κ = 0 . κ ∗ giving the beginning of the path from the SU (3) flavour symmetric line to the physical point.12 .00 0.05 0.10 0.15 0.20 0.25 0.30M π /X S2 ( M K − M π ) / X S S = NS = ρ S = t S = w ( β , κ )=(5.50,0.120900) Figure 5: (2 M K − M π ) /X S against M π /X S , together with the fit from eq. (26) for( β, κ ) = (5 . , . S = N , ρ , t , w . The notation is as for Fig. 4. Opaquepoints are not included in the fit as they have M π L < Using these results we now take X S = const. to define the scale, i.e. we set X lat S = const. = a S X exp S . (27)We take the experimental hadron mass results as given in Table 1. As X t , X w are secondary quantities, i.e. X exp t , X exp w are not experimentally known, they haveto be determined.If we now normalise a S ( κ ) = ( X lat S ( κ )) ( X exp S ) , (28)this provides an estimate for the lattice spacing using the singlet quantity S ,which is also a function of the point on the SU (3) flavour symmetric line, i.e. κ (we have indicated this by writing a S ( κ ) for the lattice spacing).We now vary κ , searching for the location where the various a S ( κ ) cross,providing a value for the common lattice spacing a (and κ ∗ ). While ideally wewould wish the crossing of all the lines to occur at a single point leading to acommon lattice spacing, this, of course, does not quite happen. So we considerpairs of singlet quantities and determine the crossing points, together with theassociated (bootstrap) error. In particular we apply this to the pairs( π, N ) , ( π, ρ ) . (29)13e now use these crossings to adjust X t and X w so they also go throughthese points. This determines q t exp0 , w exp0 . For example we have( w exp0 ) ≡ X exp w ) = a ( X lat w ) . (30)For example in Figs. 6 and 7 we plot a S ( κ ) from eq. (28) (in fm ) against κ a S = ( X s l a t / X s e x p ) [f m ] S = NS = π S = t S = w β =5.80 ( π ,N) Figure 6: a S against 1 /κ for S = π (up triangles), N (right triangles) and t (circles), w (squares) together with linear fits for β = 5 . κ for β = 5 .
80 and 5 .
50 for the singlet quantities S = π and N together with S = t and w . Where we have three κ values a linear fit in 1 /κ is made, while ifthere are four κ values available then a quadratic fit is made. (However it madevery little difference to the later results whether the results from the linear orquadratic fit is used, as mainly interpolations between the X lat S data is sufficient.)Also plotted is S = t and w , again together with appropriate fits. The latticevalues have been adjusted with a common factor so these singlet quantities alsocross at the same value as ( π, N ), which is equivalent to a determination of √ t and w exp0 as indicated in eq. (30). This procedure is then repeated for the pair( π, ρ ).For completeness we also take a weighted average of both the ( π, N ) and ( π, ρ )crossings to determine the best (1 /κ ∗ , a ). These values are given in Table 6.14 .264 8.266 8.268 8.270 8.2721/ κ a S = ( X S l a t ) / X S e x p2 [f m ] S = π S = NS = t S = w β =5.50 ( π ,N) Figure 7: a S against 1 /κ for S = π , N and t , w together with quadratic fits for β = 5 .
50. Notation as for Fig. 6. β /κ ∗ a [fm ] κ ∗ a [fm]5 .
80 8.14197(12) 0.00346(04) 0.122820(2) 0.0588(03)5 .
65 8.19602(15) 0.00468(06) 0.122010(2) 0.0684(04)5 .
50 8.26844(13) 0.00547(06) 0.120942(2) 0.0740(04)5 .
40 8.33823(25) 0.00669(16) 0.119930(4) 0.0818(09)
Table 6:
Determined values of 1 /κ ∗ and a [fm ]. For completeness in the third andfourth columns we also give κ ∗ and a [fm] directly. We are now in a position to perform the last, continuum, extrapolation. InFigs. 8, 9 we show these extrapolations from the pairs ( π, N ) and ( π, ρ ). Asanticipated the gradients in a , while small, are positive (c.f. eq. (25)) with the( π, ρ ) results being slightly larger than the ( π, N ) results.Finally a weighted average from these continuum results (i.e. for a = 0) givesour final results q t exp0 = 0 . , w exp0 = 0 . . (31)The first error is statistical, while the second (finite volume), the third ( SU (3)flavour breaking expansion) and fourth (scale) are systematic errors as discussedin Appendix B. 15 .0000 0.0025 0.0050 0.0075a [fm ]0.120.130.140.150.160.170.180.190.20 t / , w [f m ] t w ( π ,N) Figure 8: √ t and w (in fm) against a (in fm ) from the ( π, N ) crossing togetherwith a linear fit. [fm ]0.120.130.140.150.160.170.180.190.20 t / , w [f m ] t w ( π , ρ) Figure 9: √ t and w (in fm) against a (in fm ) from the ( π, ρ ) crossing togetherwith a linear fit. Conclusions
In this article we have described a method for determining the trajectory toapproach the physical point, and demonstrated (theoretically and numerically)that singlet quantities remain constant as we approach this point. This enablesus by considering pairs of singlet quantities to determine ‘best’ lattice spacingsand starting values for the path. By matching these results to the flow variables t and w this enables a determination of their physical values, see eq. (31).In Fig. 10 we compare these results with other determinations for n f = 2 + 1 t [fm] BMW 12
HotQCD 14QCDSF−UKQCD 15
RBC−UKQCD 14 w [fm] Figure 10: q t exp0 , left panel and w exp0 , right panel in fm for BMW 12 [3], HotQCD14 [9], RBC-UKQCD 14 [10], together with the present results. flavours, namely BMW 12 [3], HotQCD 14 [9] and RBC-UKQCD 14 [10]. (Thegiven errors are taken in quadrature.) Reasonable consistency is found betweenthe different determinations.In conclusion we have determined in this article the flow scales for t and w .These are ‘secondary’ scales and while having the advantage of being cheap andaccurate to determine from lattice simulations are not directly experimentallyaccessible and thus have to be matched to physical quantities. Acknowledgements
The numerical configuration generation (using the BQCD lattice QCD program[23]) and data analysis (using the Chroma software library [24]) was carried out17n the IBM BlueGene/Q using DIRAC 2 resources (EPCC, Edinburgh, UK), theBlueGene/P and Q at NIC (J¨ulich, Germany), the Lomonosov at MSU (Moscow,Russia) and the SGI ICE 8200 and Cray XC30 at HLRN (The North-German Su-percomputer Alliance) and on the NCI National Facility in Canberra, Australia(supported by the Australian Commonwealth Government). HP was supportedby DFG Grant No. SCHI 422/10-1. PELR was supported in part by the STFCunder contract ST/G00062X/1 and JMZ was supported by the Australian Re-search Council Grant No. FT100100005 and DP140103067. We thank all fundingagencies.
AppendixA Singlet chiral perturbation theory: Wilsonflow
We want to check that the chiral perturbation theory results for the Wilson flow,as given in [21] are consistent with the SU (3) flavour symmetry expansion [16]. A.1 Pseudoscalar Meson masses
First we need to set out some notation. The quark masses for the 2 + 1 case arebest denoted by χ q , defined through χ l ≡ B m l χ s ≡ B m s . (32)To simplify expressions, it is useful to define some additional χ variables: χ ≡ (2 χ l + χ s ) χ π ≡ χ l χ K ≡ ( χ s + χ l ) χ η ≡ (2 χ s + χ l ) , (33)and a logarithmic function µ P ≡ χ P (4 πf ) ln χ P Λ ≈ M P (4 πf ) ln M P Λ , P ∈ π, K, η . (34)In this notation the NLO pseudoscalar meson masses are [22] M π = χ π (cid:26) q χ + q χ π + µ π − µ η (cid:27) K = χ K (cid:26) q χ + q χ K + 23 µ η (cid:27) M η = χ η (cid:26) q χ + q χ η + 2 µ K − µ η (cid:27) + χ π (cid:26) − µ π + 23 µ K + 13 µ η (cid:27) + q ( χ s − χ l ) (35)where q = 48 f (2 L − L ) q = 16 f (2 L − L ) q = 1289 f (3 L + L ) . (36) A.2 Wilson Flow scale, t In [21], eq.(4.10), B¨ar and Golterman give the form expected for the quantity t at NNLO in chiral perturbation theory. t = t , ch " k (4 πf ) (2 M K + M π )+ 1(4 πf ) (3 k − k ) M π µ π + 4 k M K µ K + k M π − M K ) µ η + k M η µ η ! + k (4 πf ) (2 M K + M π ) + k (4 πf ) ( M K − M π ) (37)The free parameters in this expression are k , k , k , k . Most terms in the ex-pression are obviously symmetric, but the k term has been written in a way thatobscures its symmetry.Using eqs. (35) to translate eq. (37) into χ variables t = t , ch " k (4 πf ) χ + k (4 πf ) (3 χ π µ π + 4 χ K µ K + χ η µ η )+ 9 k ′ (4 πf ) χ + k ′ πf ) ( χ s − χ l ) (38)with k ′ = k + (4 πf ) k ( q + q ) k ′ = k + (4 πf ) k q . (39) In this Appendix we work to order χ q , dropping terms of order χ q . χ , t = T " k (4 πf ) χ π ln χ π χ + 4 χ K ln χ K χ + χ η ln χ η χ ! + k ′′ πf ) ( χ s − χ l ) (40)with k ′′ = k ′ + 209 k ln χ Λ (41)and T = t , ch " k (4 πf ) χ + 8 k (4 πf ) χ ln χ Λ + 9 k ′ (4 πf ) χ (42)being the value of t on the symmetric line. We can Taylor expand eq. (40) aboutthe symmetric point, the result is t = T " πf ) ( k + k ′′ )( χ s − χ l ) + · · · . (43)As expected, there is no linear term, and the first term we see is quadratic in the SU (3) breaking. B Systematic errors
We follow here the more general discussion given in Appendix A of [19].
B.1 Finite lattice volume
Clearly the argument given in section 2 that X S is flat along the SU (3) flavoursymmetric point holds for any volume. As discussed in [16] for an estimate offinite volume effects, a suitable expression is given by X S ( L ) = X S (cid:16) c S [ f L ( M π ) + 2 f L ( M K )] (cid:17) . (44)Lowest order χ PT, [25, 26] indicates that reasonable functional forms for f L ( M )are f L ( M ) = ( aM ) e − ML ( M L ) / , meson ,f L ( M ) = ( aM ) e − ML ( X N L ) , baryon . (45)20 .00000 0.00002 0.00004 0.00006(f L (M π )+2f L (M K ))/30.000.050.100.150.200.250.30 ( X S l a t ) S = NS = ρ S = π κ =0.120900 Figure 11: ( X lat S ) versus ( f L ( M π ) + 2 f L ( M K )) / β, κ ) = (5 . , . N (squares), ρ (diamonds) and π (upper triangles). The left-most clusters of pointsare from the 32 ×
64 and 48 ×
96 lattices, while the right cluster is from 24 × In Fig. 11 we plot ( f L ( M π ) + 2 f L ( M K )) / X lat S ) for S = N , ρ and π on48 ×
96, 32 ×
64 and additionally 24 ×
48 lattices for β = 5 .
50 and κ = 0 . X π , X N , X ρ ) thechanges are about (1 . , . , . B.2 SU (3) flavour breaking expansion We first note that in Figs. 2, 3 from the SU (3) flavour symmetric line down tothe physical point lies in the range | δm l | ∼ < .
01 (and | δm s | ∼ < . SU (3) flavour breaking expansion, eq. (15) or (16). The nextorder in the expansion is multiplied by a further δm q . So we expect that everyincrease in the order leads to a decrease by an order of magnitude or more (oftenby a factor ∼
20) in the series. So we believe that convergence is very good forhyperons. (Such an expansion is good compared to most approaches availableto QCD.) Nevetheless we have, however, made tests with a linear or quadratic21t for example for the nucleon in Fig. 3 and followed this through the analysis.The final change in central value for √ t and w was not large, we include it asa (second) systematic error. B.3 Physical scale
As mentioned in footnote 2 physical values of hadron masses have a small elec-tromagetic component. Although we disregard this in our analysis, we make asmall allowance here, and take q t exp0 , w exp0 to also have a similar error as X π , i.e.a systematic error of ∼ .
2% due to electromagnetic effects.
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