Hyperasymptotic approximation to the top, bottom and charm pole mass
HHyperasymptotic approximation to the top, bottom and charmpole mass
Cesar Ayala
Department of Physics, Universidad T´ecnica FedericoSanta Mar´ıa (UTFSM), Casilla 110-V, Valpara´ıso, Chile
Xabier Lobregat and Antonio Pineda
Grup de F´ısica Te`orica, Dept. F´ısica and IFAE-BIST,Universitat Aut`onoma de Barcelona,E-08193 Bellaterra (Barcelona), Spain (Dated: February 12, 2020)
Abstract
We construct hyperasymptotic expansions for the heavy quark pole mass regulated using theprincipal value (PV) prescription. We apply such hyperasymptotic expansions to the
B/D mesonmasses, and ¯Λ computed in the lattice. The issue of the uncertainty of the (top) pole mass iscritically reexamined. The present theoretical uncertainty in the relation between m t , the MS topmass, and m t, PV , the top pole mass regulated using the PV prescription, is numerically assessedto be δm t, PV = 28 MeV for m t = 163 GeV. a r X i v : . [ h e p - ph ] F e b ontents I. Introduction II. General formulas N large and µ ∼ Q (cid:29) Λ QCD . Eq. (3). Case 1) 4B. m PV , general formulas 7 III. m PV ( m ) in the large β approximation N large and µ ∼ m (cid:29) Λ QCD . Eq. (3). Case 1) 11B. (
N, µ ) → ∞ . Eq. (4). Case 2) 23 IV. Λ PV from lattice and B physics β -function coefficients in the Wilson action lattice scheme 27B. ¯Λ PV from lattice 281. Scheme dependence 31C. ¯Λ PV pot from lattice 32D. ¯Λ PV from B meson mass 34E. ( N, µ ) → ∞ . Eq. (4). Case 2B) 37 V. Top mass | u | = 1 renormalons 46 VI. Conclusions A. Evaluation of Ω d B. bottom and charm finite mass contributions to m t, PV References . INTRODUCTION
We construct hyperasymptotic expansions for the heavy quark pole mass regulated usingthe principal value (PV) prescription along the lines of [1]. We generalize the discussion ofthat reference by including possible ultraviolet renormalons. We then apply such expansionsto various observables. The name hyperasymptotic we borrow from [2]. For a treatise ofhyperasymptotic expansions in the context of ordinary differential equations see [3].In [1] we studied observables characterized by having a large scale Q (cid:29) Λ QCD , and forwhich the operator product expansion (OPE) is believed to be a good approximation. Wecomputed them within an hyperasymptotic expansion. More specifically, the perturbativepart of the OPE was summed up using the PV prescription: S PV . The difference between S PV and the full non-perturbative (NP) result is assumed to exactly scale as the intrinsicNP terms of the OPE. In general terms:Observable( Q Λ QCD ) = S PV ( α X ( Q )) + K (PV) X α γX ( Q ) Λ dX Q d (1 + O ( α X ( Q ))) + O ( Λ d (cid:48) X Q d (cid:48) ) , (1)where the last term refers to genuine higher order terms in the OPE ( d (cid:48) > d > S PV can not be computed exactly, we obtain it approximately along an hyperasymptoticexpansion (a combination of (truncated) perturbative sums and of NP corrections). Thisis possible if enough terms of the perturbative expansion are known, and if the divergentstructure of the leading renormalons of the observable is also known. This allows us to havea clear (parametric) control on the error of the computation. Two alternative methods wereconsidered in [1] depending on how the truncation of the leading perturbative sum S T ( α ) = N (cid:88) n =0 p ( X ) n α n +1 X ( µ ) (2)is made:1) N and µ ∼ Q large but finite: N = N P ≡ | d | πβ α X ( µ ) (cid:0) − c α X ( µ ) (cid:1) , (3)3) N → ∞ and µ → ∞ in a correlated way. We considered two options:A) N + 1 = N S ( α ) ≡ | d | πβ α X ( µ ) ; B) N = N A ( α ) ≡ | d | πβ α X ( µ ) (cid:0) − c (cid:48) α X ( Q ) (cid:1) (4)where c (cid:48) > c is arbitrary otherwise. Note that in case 1), c can partially simulatechanges on the scale or scheme of α X . d is the dimension associated to a given renormalon.Note that in this paper d can be positive (infrared renormalons) or negative (ultravioletrenormalons), unlike in [1], where only positive d ’s were considered. Note also that genuineNP corrections are only associated to positive d ’s.We will not study the modifications the inclusion of ultraviolet renormalons produce incase 2). In this paper we are mainly concerned in the scenario where the leading renormalonis of infrared nature and subleading renormalons can be ultraviolet and/or infrared. This isthe case of the pole mass. In such scenario the precision we can obtain in case 2) is limitedby the approximate knowledge of the leading infrared renormalon and we cannot add furtherto the discussion given in [1]. Different is the case 1), which we discuss in the next section.The structure of the paper is as follows. In Sec. II we review the general case. Comparedwith [1] we include the possible effect of ultraviolet renormalons. In Sec. III we study thepole mass of a heavy quark in the large β approximation. We use it as toy-model to testour methods. We then move to real QCD. In Sec. IV we study the B/D meson mass andlattice evaluations of ¯Λ. Finally, in Sec. V we do a dedicated study of the top mass.In general we will avoid to make explicit the scheme ( X ) and scale ( µ ) dependence unlessnecessary. II. GENERAL FORMULASA. N large and µ ∼ Q (cid:29) Λ QCD . Eq. (3) . Case 1)
This case was already discussed at length in [1]. We now give the general expression afterthe inclusion of ultraviolet renormalons (for a more detailed discussion see [4]). It can bewritten in the following way S PV ( Q ) = S P + (cid:88) {| d |} S | d | + (cid:88) { d> } Ω d + (cid:88) { d< } Ω d , (5)4here S P ≡ N P ( | d min | ) (cid:88) n =0 p n α n +1 ( µ ) ≡ S | d | =0 , (6)and ( | d | > S | d | ≡ N P ( | d (cid:48) | ) (cid:88) n = N P ( | d | )+1 ( p n − p ( as ) n ) α n +1 ( µ ) , (7)where the asymptotic behavior associated to renormalons with dimensions ≤ | d | is includedin p ( as ) n , and d (cid:48) is the dimension of the closest renormalon to the origin in the Borel planefulfilling that | d (cid:48) | > | d | . Ω d is a modification of the definition of terminant given in [5] that ismore suitable to our case. Whereas in [5] ( p N α N × the) terminant refers to the completationof the superasymptotic approximation to give the complete result, here Ω d is the completionof the part of the perturbative series associated to the singularity located at u ≡ β t π = d inthe Borel plane using the PV prescription. For the case of infrared renormalons ( d >
0) thegeneral analytic expression of Ω d can be found in [1]. For a generic ultraviolet renormalon( d <
0) that produces the asymptotic behavior p ( as ) n = Z XO d µ d Q d Γ( n + b (cid:48) + 1)Γ( b (cid:48) + 1) (cid:18) β πd (cid:19) n (cid:26) c b (cid:48) n + b (cid:48) + c b (cid:48) ( n + b (cid:48) )( n + b (cid:48) −
1) + . . . (cid:27) , (8)Ω d< reads Ω d< = ∆Ω UV ( db ) + c ∆Ω UV ( db −
1) + · · · , (9)where (we define η c ≡ − b (cid:48) + π | d | cβ − b (cid:48) = db − γ )∆Ω UV ( db ) = Z XO d µ d Q d ( − N P +1 b (cid:48) + 1) (cid:18) β π | d | (cid:19) N P +1 α N P +2 (cid:90) ∞ dx e − x x N P +1+ b (cid:48) xβ α π | d | (10)= Z XO d µ d Q d ( − N P +1 π Γ( b (cid:48) + 1) (cid:18) β | d | (cid:19) − b (cid:48) − / α ( µ ) / − b (cid:48) e − π | d | β α ( µ ) (cid:26) α ( µ ) π β | d | (cid:2) − η c (cid:3) + α ( µ ) π β | d | (cid:20) − η c − η c + 48 η c + 36 η c (cid:21) + O ( α ) (cid:27) . An sketch of how these computations are done is given in Appendix A. d< = (cid:112) α ( µ ) K ( P ) X Q | d | µ | d | e − π | d | β α ( µ ) (cid:18) β α ( µ )4 π (cid:19) − b (cid:48) (cid:26) K ( P ) X, α ( µ ) + ¯ K ( P ) X, α ( µ ) + O (cid:0) α ( µ ) (cid:1) (cid:27) , (11)where K ( P ) X ≡ Z X O d ( − N p +1 (cid:18) β π | d | (cid:19) − / b (cid:48) + 1) (cid:18) | d | (cid:19) − b (cid:48) , (12)¯ K ( P ) X, ≡ (cid:18) π (cid:19) / (cid:18) c β b (cid:48) √ π | d | + β | d |√ π ( − η c ) (cid:19) , (13)¯ K ( P ) X, ≡ (cid:18) π (cid:19) / (cid:18) c b (cid:48) β √ | d | π / + c b (cid:48) β ( − η c + 1) )24 √ | d | π / + β | d | / π / (cid:20) − η c − η c + 48 η c + 36 η c (cid:21)(cid:19) . (14)Note that in this case µ is in the denominator. If we set the anomalous dimensionto zero ( b (cid:48) = db ), Ω d< ∼ (cid:112) α ( µ ) Λ | d | QCD Q | d | µ | d | (unlike for infrared renormalons where Ω d> ∼ (cid:112) α ( µ ) Λ d QCD Q d ). If one takes µ very large this term will be quite small. In practice, if we take µ ∼ Q , we may need this term. S PV will be computed truncating the hyperasymptotic expansion in a systematic way.This means truncating Eq. (5) as follows (note that we always define D to be positive): S ( D,N )PV ( Q ) = (cid:88) {| d |} S | d | D, N ), we can state the parametric accuracy of S ( D,N )PV ( Q ). Forinstance for S (0 ,N P ) the error would be (up to a numerical and a √ α X factor) δS (0 ,N P ) ∼ O (cid:16) e −| d min | πβ αX ( Q ) (cid:17) . (16)This is what is commonly named the superasymptotic approximation. For S ( | d min | , the6arametric form of the error reads (up to a numerical and a possible α / X factor): δS ( | d min | , ∼ O (cid:16) e −| d min | πβ αX ( Q ) (1+ln( | d/d min | ) (cid:17) , (17)where d is the location of the next renormalon closest to the origin. This corresponds tothe first term in the hyperasymptotic approximation. The expression for the error in thegeneral case S ( D,N )PV ( Q ) reads ( N (cid:54) = N P but large) δS ( D,N ) ∼ O (cid:16) e − D πβ αX ( Q ) (1+ln( | d/D | ) α NX (cid:17) , (18)where d is the location of the next renormalon closest to the origin after D . B. m PV , general formulas For the case of the heavy quark mass, which we discuss at length in this paper, we have( m = m MS ( m MS )) m PV ( m ) = m P + m Ω m + N P (cid:88) n = N P +1 ( r n − r ( as ) n ) α n +1 ( µ ) + m Ω + m Ω − + O (cid:16) e − πβ α (1+ln(3 / (cid:17) , (19)where m P ≡ m + N P (cid:88) n =0 r n α n +1 ( µ ) ; (20)the coefficients r n for n ≤ m = (cid:112) α X ( µ ) K ( P ) X µm e − πβ αX ( µ ) (cid:18) β α X ( µ )4 π (cid:19) − b (cid:18) 1+ ¯ K ( P ) X, α X ( µ )+ ¯ K ( P ) X, α X ( µ )+ O (cid:0) α X ( µ ) (cid:1) (cid:19) , (21)7here now K ( P ) X and K ( P ) X,i read K ( P ) X = − Z Xm − b π Γ(1 + b ) β − / (cid:20) − η c + 13 (cid:21) , (22)¯ K ( P ) X, = β / ( π ) − η c + (cid:20) − b b (cid:18) η c + 13 (cid:19) − η c + 124 η c − (cid:21) , (23)¯ K ( P ) X, = β /π − η c + (cid:20) − w ( b − b (cid:18) η c + 512 (cid:19) + b b (cid:18) − η c − η c − η c − (cid:19) − η c − η c + 1144 η c + 196 η c − η c − (cid:21) , (24)where we have applied the general expression obtained in [1] to this case. In particular ( b and s n are defined in [1]), η c = − b + 2 πcβ − , b = s , w = (cid:18) s − s (cid:19) bb − . (25)Finally, r (as) n ( µ ) = Z Xm µ (cid:18) β π (cid:19) n ∞ (cid:88) k =0 c k Γ( n + 1 + b − k )Γ(1 + b − k ) . (26)The coefficients c k are pure functions of the β -function coefficients, as first shown in [10].They can be found in [11–13]. At low orders they read ( c = 1) c = s , c = 12 bb − s − s ) , c = 16 b ( b − b − 1) ( s − s s + 6 s ) . (27)Note that m ( N )OS = m + N (cid:88) n =0 r n α n +1 ( µ ) . (28)Therefore, m P is nothing but the pole mass truncated to order N = N P .Our knowledge of the other terminants, Ω and Ω − , is limited. We do not know therenormalization group structure of Ω − , except in the large β . On the other hand, therenormalization group structure of Ω is exactly known (provided the coefficients of thebeta function are known to all orders). The reason is that it is linked to the kinetic operatorof the HQET Lagrangian, the Wilson coefficient of which is protected by reparameterizationinvariance [14]. Therefore, it has no anomalous dimension and the Wilson coefficient is1 in dimensional regularization to all orders in perturbation theory. Still, in the large β Z X is equal to zero. If it is different from zero beyond thelarge β approximations has been a matter of debate [15]. We will retake this discussion inthe following sections.We also give the formulas that apply to Eq. (4), i.e. to case 2): the limit ( N, µ ) → ∞ .A general discussion can be found in [1]. It was argued that the limit 2A) was likely to belogarithmic divergent (see also [16]), and no formulas could be found that are valid beyondthe large β approximation. Therefore, we will not study this case further. For the limit2B) formulas with NP exponential accuracy were found in [1] generalizing results from [17].These formulas were valid beyond the large β approximation. For the specific case of thepole mass they read m PV = m A + K ( A ) X Λ X + O ( α Λ X ) , (29)where m A = m + lim µ →∞ ;2 B ) N A (cid:88) n =0 r n α n +1 ( µ ) , (30)and K ( A ) X = 2 πβ Z Xm (cid:18) β π (cid:19) b (cid:90) ∞− c (cid:48) , PV dx e − πβ x − x ) b . (31)As discussed in [1], there is more than one way to take the µ → ∞ ; 2B) limit. One is totake Eq. (67) of [1] for N A instead of limit B) of Eq. (4). Both methods are general butrequire the knowledge of r n and the beta function coefficients to all orders. This potentiallylimits their applicability in practice. Another option is to interpret the µ → ∞ limit as achange of scheme (where µ ∼ m ): α X (cid:48) ( µ ) = α X ( µ )1 + β π α X ( µ ) ln( µµ ) and N A ( α ) ≡ πβ α X (cid:48) ( µ ) (cid:0) − c (cid:48) α X ( µ ) (cid:1) . (32)This method still requires the knowledge of r n to all orders. On the other hand, there is noneed to know the β -function to all orders.Irrespectively of which of the above methods we use to take the µ → ∞ limit we havelim µ →∞ ;2 B ) N A (cid:88) n =0 r n α n +1 ( µ ) = (cid:90) πβ χ dte − t/α X ( µ ) B [ m PV − m ]( t ) , (33)where 2 /χ = 1 − c (cid:48) α ( µ ). The right-hand side of Eq. (33) can not be computed exactly.9n approximated determination can be obtained by approximating the Borel transform to( u ≡ β t π ) B [ m PV − m ]( t ) = N max (cid:88) n =0 ( r n − r ( as ) n ) n ! t n + Z m µ (1 − u ) b (cid:0) c (1 − u ) + + c (1 − u ) + · · · (cid:1) , (34)where N max is the number of perturbative coefficients that are known. III. m PV ( m ) IN THE LARGE β APPROXIMATION Here the discussion runs parallel to the discussion for the static potential in the large β approximation made in Section III of [1]. Nevertheless, we do not have the same analyticcontrol as for the static potential. Note also that now we have ultraviolet renormalons.Moreover, the pole mass has the extra complication that it is ultraviolet divergent andneeds renormalization. This makes the Borel transform more complicated and we do nothave the exact µ factorization one has in the static potential. We take the Borel transformfrom [18–20]: B [ m PV − m ]( u ) = m C F π (cid:20)(cid:18) m µ (cid:19) − u e − c MS u − u ) Γ( u )Γ(1 − u )Γ(3 − u ) − u + R ( u ) (cid:21) , (35)where c MS = − / u = β π t and R ( u ) = ∞ (cid:88) n =1 n !) d n dz n G ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z =0 u n − = − 52 + 3524 u + O ( u ) , (36) G ( u ) = − 13 (3 + 2 u ) Γ(4 + 2 u )Γ(1 − u )Γ (2 + u )Γ(3 + u ) . (37)This expression has been derived in the MS scheme. Whereas the scheme dependence of thefirst term in Eq. (35) can be reabsorbed in changes of µ and c MS (it would then be equivalentto a change of scale), controlling the scheme dependence of R ( u ) is more complicated. Wewill not care much, as R ( u ) has to do with the high energy behavior, and should only affect m P , the finite sum. Therefore, when we change from the MS to the lattice scheme, wewill just change c MS → c latt and leave R ( u ) unchanged. Strictly speaking then, the objectwe compute in the lattice scheme is not the pole mass, still it will have the same infrared10ehavior. The fact that we will obtain the same result after subtracting m P from m PV in both cases will be a nice confirmation that high-energy cancellation has effectively takenplace and what is left is low energy . The value of c latt that we use is the same to the one usedin [1]. To determine it we take the n f = 0 number for a Wilson action of d = 5 . c latt = − + πd β ). This is enough for our purposes, as we only use this scheme forchecking the consistency between the results obtained with different schemes. Note that thisyields two values of c latt if we introduce the n f dependence of β : c latt ( n f = 0) = − . c latt ( n f = 3) = − . A. N large and µ ∼ m (cid:29) Λ QCD . Eq. (3) . Case 1) We now take Eq. (19) in the large β approximation and truncate it at different orders inthe hyperasymptotic expansion. We then compare such truncations with the exact solution.We can study (even if in the large β approximation) up to which values of m the OPE is agood approximation of m PV . Remarkably enough we can actually check more than one termof the OPE (hyperasymptotic) expansion. Note that in the large β approximation Ω = 0,but not Ω − , which in the large β approximation reads ( η ( β ) c = πcβ − − = (cid:112) α ( µ ) K ( P ) X Λ X m µ (cid:26) K ( P ) X, α ( µ ) + ¯ K ( P ) X, α ( µ ) + O (cid:0) α ( µ ) (cid:1) (cid:27) , (38) K ( P ) X ≡ Z X − ( − N p +1 (cid:18) β π (cid:19) − / , Z X − = − C F e c X π , (39)¯ K ( P ) X, ≡ β π ( − η ( β )2 c ) , (40)¯ K ( P ) X, ≡ β π (cid:20) − η ( β ) c − η ( β )2 c + 48 η ( β )3 c + 36 η ( β )4 c (cid:21) . (41)We also explore the scheme dependence by performing the computation in the lattice andthe MS scheme. We will do these analyses for the cases with n f = 0 and n f = 3. The first in To make an analogy, the situation is similar to determinations of the infrared behavior of the energy ofan static source in lattice perturbation theory. In [21–23] two different discretizations were used for thestatic quark propagators. This affected the ultraviolet, but let the infrared behavior unchanged, as it wasnicely seen in those simulations. See also the discussion in [24]. β limit. In Figs. 1, 2, and 3 we plot m PV − m , m PV − m P , m PV − m P − m Ω m , m PV − m P − m Ω m − (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 , and m PV − m P − m Ω m − (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 − m Ω − with n f = 0 light flavours. In the counting of Eq. (15) this corresponds to (0,0), (0, N P ),(1,0), (1, N P ), (2,0) precision. We do such computation in the lattice (Fig. 1) and the MS(Fig. 2) scheme. In Fig. 3 we compare the results in the lattice and MS scheme. We observea very nice convergent pattern in all cases down to surprisingly small scales. To visualize thedependence on c , we show the band generated by the smallest positive and negative possiblevalues of c that yield integer values for N P . The size of the band generated by the differentvalues of c (the c dependence) decreases after introducing Ω m to its associated sum. On theother hand Ω − (an ultraviolet renormalon) gives a very small contribution, in particular inthe lattice scheme. This is consistent with interpreting the lattice scheme as the MS schemeusing a much higher renormalization scale µ for the scale of the strong coupling.Let us discuss the results in more detail. We first observe that the m dependence of m PV is basically eliminated in m PV − m P , as expected. This happens both in the lattice and MSscheme. The latter shows a stronger c dependence. This is to be expected, as in the MS, wetruncate at smaller orders in N . This makes the truncation error bigger. As we can see inthe upper panel of Fig. 3, both schemes yield consistent predictions for m PV − m P . We candraw some interesting observations out of this analysis. For m PV − m P is better to choose alarger factorization scale, if we have enough coefficients of the perturbative expansion. Thisis particularly so at large distances: We can still get good results at very large distances inthe lattice scheme.We now turn to m PV − m P − m Ω m . Adding the new correction brings much betteragreement with expectations (which we remind is to get zero). After the introduction of m Ω m , the MS scheme yields more accurate results than the lattice scheme. This can alreadybe seen in the upper panel of Fig. 3, and in greater detail in the lower panel of Fig. 3.12 PV - m ( a ) ( b ) ( c ) , ( d ) - - / m in r0 units r − ( c ) , ( d )( b ) - - - - / m in r0 units r − FIG. 1: Upper panel : We plot m PV − m (black line) and the differences: (a) m PV − m P (cyan),(b) m PV − m P − m Ω m (orange), (c) m PV − m P − m Ω m − (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 (green), and (d) m PV − m P − m Ω m − (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 − m Ω − (blue) in the large β approximation usingthe lattice scheme with n f = 0 light flavours. For each difference, the bands are generated by thedifference of the prediction produced by the smallest positive or negative possible values of c thatyields integer values for N P . The (c) and (d) bands are one on top of the other. Lower panel :As in the upper panel but in a smaller range. r − ≈ 400 MeV. The value of N P depends on thescale 1 /m we use. For instance for c positive, N P = 9 for 1 /m ∈ [0 . , . N P = 8 for 1 /m ∈ [0 . , . N P = 7 for 1 /m ∈ [0 . , . N P = 6 for 1 /m ∈ [0 . , . N P = 5 for1 /m ∈ [0 . , . N P = 4 for 1 /m ∈ [0 . , . N P = 3 for 1 /m ∈ [0 . , . PV - m ( a ) ( d ) ( c )( b ) - - / m in r0 units r − ( d )( b ) ( c ) - - - - / m in r0 units r − FIG. 2: As in Fig. 1 but in the MS scheme. The values of N P for c positive are, for instance, N P = 6for 1 /m = 0 . N P = 5 for 1 /m ∈ [0 . , . N P = 4 for 1 /m ∈ [0 . , . N P = 3 for1 /m ∈ [0 . , . N P = 2 for 1 /m ∈ [0 . , . N P = 1 for 1 /m ∈ [0 . , . N P = 0 for 1 /m ∈ [0 . , . m PV − m P − m Ω m shows some dependence on m , which is more pronounced in the latticethan in the MS scheme. As in the large β approximation the difference between bothschemes is somewhat equivalent to a change of scale, these results point to that µ = m inMS scheme is close to the natural scale and minimize higher order corrections. Note that thelattice scheme computation is equivalent to the MS scheme choosing µ latt = µ MS e − clatt e c MS2 .14 m PV - m ) latt ( m PV - m ) MS ( alatt ) ( aMS ) ( bMS )( blatt ) - - / m in r0 units r − ( blatt ) ( bMS )( clatt ) , ( dlatt ) ( cMS ) ( dMS ) - - - - / m in r0 units r − FIG. 3: Comparison of lattice and MS scheme results for n f = 0 obtained in Figs. 1 and 2. Upper panel : We plot m PV − m and the differences: (a) m PV − m P , and (b) m PV − m P − m Ω m in the lattice and MS scheme with n f = 0 light flavours. Lower panel : Lower panel Figs. 1 and2 combined. This gives around a factor 30 (!). Once (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 is incorporated in theprediction most of the difference between schemes disappears. The effect of introducingΩ − is very small, in particular in the lattice scheme. This is to be expected, since thelattice scheme corresponds to a larger renormalization scale µ . In any case, the differencebetween schemes gets smaller and smaller as we go to higher orders in the hyperasymptotic15xpansion, in particular at short distances. We also want to stress that this analysis opensthe window to apply perturbation theory at rather large distances. Note that in the upperpanel plots in Figs. 1, 2, and 3, we have gone to very large distances.As some concluding remarks let us emphasize the following points. m PV − m P is moreor less constant with relatively large uncertainties. This is to be expected, as the nextcorrection in magnitude is m Ω m which is approximately constant (mildly modulated by (cid:112) α ( µ )). After introducing this term the error is much smaller and we can see more structure.In particular we are sensitive to (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 . Here we find (at the level ofprecision we have now) a sizable difference between lattice and MS. This can be expected: (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 is the object we expect to be more sensitive to the scale.Another interesting observation is that truncated sums behave better in the lattice schemethan in the MS scheme. Nevertheless, this could be misleading. The sums are truncated atthe minimal term. Therefore, one needs more terms in the lattice scheme. If the number ofterms is not an issue (which could be the case with dedicated numerical stochastic pertur-bation theory (NSPT) [26, 27] computations in the lattice scheme) then the lattice schemelooks better. But as soon as Ω m is introduced in the computation MS behaves better (atleast in the large β approximation).Overall, we observe a very nice convergence pattern up to (surprisingly) rather largescales in the lattice and MS scheme. The agreement with the theoretical prediction (whichis zero) is perfect at short distances. The estimated error is also expected to be small. Itwill be interesting to see if this also happens beyond the large β .We now turn to the n f = 3 case. To easy the comparison with [1], we use the samevalue: Λ MS ( n f = 3) = 174 MeV (which yields α ( M τ ) ≈ . MS (in the large β approximation) using the world average value of α . We note that Λ QCD for the physical case ( n f = 3) is smaller than for the n f = 0 case (ifone sets the physical scale according to r − ≈ 400 MeV). On top of that the running is lessimportant. All this points to that the convergence should be even better than in the n f = 0case (and it was quite good already there). We show our results in Figs. 4, 5 and 6 (these arethe analogous of Figs. 1, 2 and 3 but with n f = 3). These plots confirm our expectations.Down to scales as low as 667 MeV we see no sign of breakdown of the OPE. This is so inboth the lattice and the MS schemes. Note that the precision we get is extremely high aswe go to small scales: Using truncation (c): m P + m Ω m + (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 , one gets16 PV in the MS scheme with a precision below 1 MeV at scales of the order of the mass ofthe bottom, and in the lattice scheme with a precision below 2 MeV. Using truncation (d): m P + m Ω m + (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 + m Ω − , the precision does not significantly change,in particular in the lattice scheme. This reflects that ultraviolet renormalons play a minorrole. The rest of the discussion follows parallel the one for n f = 0.In the above numerics, we have used the exact expression for Ω m and Ω − . In fullQCD, we will not know the exact expression. Therefore, it makes sense to study howwell the exact result is reproduced by its semiclassical expansion. We observed in [1] thatΩ m is very well saturated by the first terms of such expansion. Truncating the expansionproduces differences much smaller than the typical precision of the different terms of thehyperasymptotic expansion. For Ω − , we compare in Table I and II the exact result andthe truncated semiclassical expansion for an illustrative set of values. We observe that theexact result is very well saturated by the first terms of the expansion computed in Eq. (38).Truncating the expansion produces differences much smaller than the typical precision ofthe different terms of the hyperasymptotic expansion. As expected n f = 3 is better than n f = 0. Note that in the large β approximation we exactly have Λ = µe − π/ ( β α ( µ )) .An alternative, very effective, presentation of the above results can be done by plotting therelative accuracy of the prediction at each order in α , and at each order of the superasymp-totic expansion. We note that we have one observable for each value of m . Therefore, forillustration, we take two extreme cases. We use m PV with m = 1 . 25 GeV and m = 163GeV. For the theoretical prediction we take the smallest positive value of c correspondingto lattice or MS scheme. We use the exact expressions for Ω m and Ω − . Nevertheless, theNNLO truncated expression for Ω m is precise enough to yield the same result. For Ω − wecould truncate earlier with no visible effect. We show the results in Fig. 7. We stress that Taking different values of c do not change the picture. The new points stand on top of the old ones wherethey overlap. PV - m ( a ) ( b ) ( c ) , ( d ) - - / m in GeV - G e V ( b ) ( c ) , ( d ) - - - - - - / m in GeV - G e V FIG. 4: As in Fig. 1 but with n f = 3 light flavours. The value of N P depends on the scale 1 /m weuse. For instance for c positive, N P = 1 for 1 /m = 0 . N P = 10 for 1 /m ∈ [0 . , . N P = 9for 1 /m ∈ [0 . , . N P = 8 for 1 /m ∈ [0 . , . N P = 7 for 1 /m ∈ [0 . , . N P = 6 for1 /m ∈ [0 . , . N P = 5 for 1 /m ∈ [0 . , . several terms of the hyperasymptotic expansion are included. First, we nicely see that, oncereached the minimum, N ∼ N P , both schemes yield similar precision, but in the latticescheme (bigger factorization scale µ ) more terms of the perturbative expansions are neededto reach the same precision. We can see a gap when Ω m is included, with significant better18 PV - m ( a )( b ) ( d ) ( c ) - - / m in GeV - G e V ( b )( c ) ( d ) - - - - - - / m in GeV - G e V FIG. 5: As in Fig. 1 but with n f = 3 light flavours and in the MS scheme. The value of N P depends on the scale 1 /m we use. For instance for c positive, N P = 7 for 1 /m = 0 . N P = 6for 1 /m = 0 . N P = 5 for 1 /m ∈ [0 . , . N P = 4 for 1 /m ∈ [0 . , . N P = 3 for1 /m ∈ [0 . , . N P = 2 for 1 /m ∈ [0 . , . N P = 1 for 1 /m ∈ [0 . , . precision in the MS scheme. One important lesson one may extrapolate from this exerciseis that, if the number of perturbative coefficients is fixed, the smaller the renormalizationscale µ , the better. One can obtain much better precision for an equal number of pertur-bative coefficients. Another observation is that the minimal term determined numerically19 m PV - m ) latt ( m PV - m ) MS ( aMS ) ( alatt ) ( bMS )( blatt ) - - / m in GeV - G e V ( blatt ) ( bMS )( dMS ) ( clatt ) , ( dlatt ) ( cMS ) - - - - - - / m in GeV - G e V FIG. 6: Comparison of lattice and MS scheme results for n f = 3. Upper panel : We plot m PV and the differences: (a) m PV − m P , and (b) m PV − m P − m Ω m in the lattice and MS scheme with n f = 3 light flavours. Lower panel : Fig. 4 and Fig. 5 combined. need not to coincide with the minimal term computed using N = N P (though it should notbe much different). The difference reflects how much the exact coefficient is saturated bythe asymptotic expression. The effect of Ω − is very small compared with the effect due toΩ m . For the case of the top ( m = 163 GeV) we can still see the sign alternating behavior20S-Scheme ( n f = 0) m in r − c m ΩExact (cid:12)(cid:12) ΩLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNNLOΩExact − (cid:12)(cid:12) × n f = 0) m in r − c m ΩExact × (cid:12)(cid:12) ΩLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNNLOΩExact − (cid:12)(cid:12) × TABLE I: m Ω − in the large β approximation for n f = 0 in r − units compared with Eq. (38)truncated at different powers of α . Upper panel computed in the MS scheme. Lower panel in thelattice scheme. Lattice seems to be better but both schemes yield very good results. of the perturbative series associated to the d = − d = 3. If one makes m small, m = 1 . 25 GeV, green and orangepoints mix in the MS scheme. This effect is more pronounced in the lattice scheme, whereone can continuously move from the orange to the green points. The effect of the ultravioletrenormalon is very small and the precision is set by the u = 3 / ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● m = 163 GeV - - - - - - α ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● m = - - α FIG. 7: | m PV − m HyperasymptoticPV | for m = 163 GeV (upper panel) and m = 1 . 25 GeV (lower panel).Blue points are | m PV − m N | . Orange points are | m PV − m P − m Ω m − (cid:80) Nn = N P +1 ( r n − r (as) n ) α n +1 | with c = 1 . / . 39 and c = 1 . / . 11 (the smallest positive values that yield integer N P ) in theMS and lattice scheme respectively for m = 163 / . 25 GeV. Green points are | m PV − m P − m Ω m − (cid:80) N P n = N P +1 ( r n − r (as) n ) α n +1 − m Ω − − (cid:80) Nn =2 N P +1 ( r n − r (as) n ) α n +1 | , where in the last sum the twofirst renormalons are subtracted. Change of color correspond to the inclusion of Ω m and Ω − . Fullpoints have been computed in the MS scheme and empty points in the lattice scheme. We workwith n f = 3. n f = 3) m in GeV c m ΩExact (cid:12)(cid:12) ΩLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNNLOΩExact − (cid:12)(cid:12) × n f = 3) m in GeV c m ΩExact × (cid:12)(cid:12) ΩLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNLOΩExact − (cid:12)(cid:12) × (cid:12)(cid:12) ΩNNLOΩExact − (cid:12)(cid:12) × TABLE II: m Ω − in the large β approximation for n f = 3 in GeV units compared with Eq. (38)truncated at different powers of α . Upper panel computed in the MS scheme. Lower panel in thelattice scheme. Lattice seems to be better but both schemes yield very good results. B. ( N, µ ) → ∞ . Eq. (4) . Case 2) We take Eq. (29) in the large β limit by setting b = 0. As before, we have no analyticexpressions to compare with (unlike the case of the static potential). Therefore, we directly23ocus on taking the limit 2B) and numerically check its convergence and how it compareswith method 1).Method 2B) has the pleasant feature that the generated O (Λ QCD ) correction complieswith the OPE. It also yields results that do not depend on N (and µ ) anymore. Still, ithas some errors and does not reach the precision of method 1). There is a residual schemedependence associated to uncomputed terms of O ( α Λ QCD ). Part of it can be estimated bythe residual dependence in c (cid:48) . In order to estimate it, we compute m A for different valuesof c (cid:48) . On the one hand c (cid:48) cannot be very large, as c (cid:48) α ( m ) should be relatively close to zero.On the other hand we cannot make c (cid:48) α ( m ) to get arbitrary close to zero, as the O (Λ QCD )correction diverges logarithmically in c (cid:48) . We also note that there is a value of c (cid:48) = c (cid:48) min thatmakes that K ( A ) X = 0 so that the O (Λ QCD ) correction vanishes. Therefore, we compute m A for different values of c (cid:48) . For illustration we show some results in Fig. 8. We draw linesfor m PV − m A − K ( A ) X Λ X at c (cid:48) = 1 and c (cid:48) = c min generating a band. We also explore thedependence on the scheme by comparing the results in the lattice and MS scheme. We stressagain that, in the large β approximation, lattice and MS schemes basically correspond to aredefinition of µ , but quite large indeed. On the other hand the final result is µ independent.Nevertheless, the way the µ → ∞ limit is taken is fixed by N A , as defined in Eq. (4), whichis dependent on µ . This explains why different results are obtained.In Fig. 8, we also compare with results obtained using method 1), more specifically wecompare with m PV − m P − m Ω m , as they both have analogous power accuracy (thoughmethod 1) is parametrically more precise). For Ω m we take the exact expression but usingits approximated expression does not change the discussion, as the difference is very small.What we see is that the MS scheme yields more precise predictions than the lattice scheme,and that method 1) yields considerable better results than method 2B).Another issue specific of method 2B) is to determine how large we need to take N (andconsequently µ ) of the truncated sum such that it approximates well m A . For illustrativepurposes we show the convergence in Fig. 9 for n f = 3 in the lattice and MS scheme. Wefind that we have to go to relatively large values of µ (and N ) to get it precise. This can24 alatt ) ( aMS ) ( blatt )( bMS ) - - - - - - - - / m r0 units ( alatt ) ( bMS )( blatt )( aMS ) - - - - - - - - / m GeV - FIG. 8: Upper panel : We plot (a) m PV − m A − K ( A ) X Λ X for n f = 0 in the lattice and MS scheme.For each case, we generate bands by computing m A with c (cid:48) = 1 and c (cid:48) = c (cid:48) min = 0 . m P V − m P − m Ω m obtained with method 1) with the bands generated for Fig.3. Lower panel : As the upper panel with n f = 3, c (cid:48) min = 0 . 534 and taking the the bands obtainedwith method 1) for Fig. 6 for (b) m PV − m P − m Ω m . be a problem if one wants to go beyond the large β . This problem would be less severe ifone can use the asymptotic expression for the coefficients beyond certain n . Nicely enough,we find that the use of asymptotic expression for the coefficients for n > N ∗ ( ∼ ∼ m A by the truncated sum is25 PV - m A - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X - - - - - - - - - - - - / m in GeV - latt - Scheme m PV - m A - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X m PV - m + - K X ( A ) Λ X - - - - - - - - / m in GeV - MS - Scheme FIG. 9: Upper panel : We plot m PV − m A − K ( A ) X Λ X for n f = 3 in the lattice scheme with c = 1versus the truncated sums m PV − (cid:80) N A n =0 r n α n +1 ( µ ) − K ( A ) X Λ X , where µ is fixed using N A definedin Eq. (4). Lower panel : As in the upper panel but in the MS scheme. more costly for small values of c (cid:48) . IV. Λ PV FROM LATTICE AND B PHYSICS We now abandon the large- β approximation. Our aim is to determine ¯Λ PV . We willdetermine it first in gluedynamics ( n f = 0) in Sec. IV B. To study the scheme dependenceof the result it will be useful to estimate the higher order coefficients of the β function in26he Wilson action lattice scheme. We do so in the next section. A. β -function coefficients in the Wilson action lattice scheme β β β β − . × − . × − . × − . × TABLE III: Estimates of the coefficients of the beta function for the bare coupling in the latticescheme using renormalon dominance and Z MS m = 0 . 62 [23]. The error quoted in the table givesthe difference with the values of the beta coefficients obtained if one uses instead Z MS m = 0 . Z MS m . β r Λ l a tt - L3 - L4 - L5 - L6 - L7 - L FIG. 10: Same caption as in Fig. 1 of [29] including more terms in the perturbative expansionusing the β -function coefficients listed in Table III. In [22, 23] it was shown that renormalon dominance allowed to give an accurate value for β latt assuming that c (see Eq. (44)) is already saturated by the renormalon in the MSscheme. We can estimate higher order terms of the β function in the lattice scheme (usingthe Wilson action) by also assuming that for n > c n in the MS scheme aresaturated by the renormalon. We show such estimates in Table III. These coefficients of the27 function improve the agreement with the phenomenological parameterization of α latt (1 /a )obtained in [28] in the range β ∈ (6 , . 8) (see Fig. 10). It is also worth mentioning that weobserve a geometrical growth of the coefficients of the β function. Elucubrative, this wouldindicate that the beta function in this scheme has a finite radius of convergence, and onecan take the ansatz β latt ( α ) = ν ddν α (cid:39) − α (cid:40) (cid:88) n =0 β n (cid:16) α π (cid:17) n +1 − . × (cid:16) α π (cid:17) − α π (cid:41) , (42)which would have a pole at around β = 6 /g (cid:39) . B. ¯Λ PV from lattice We determine ¯Λ PV in gluedynamics ( n f = 0) from the energy of a meson made of a staticquark and a light valence quark: E MC ( a ) = δm PVlatt + ¯Λ PV + O ( a Λ ) . (43) δm PVlatt has the following asymptotic series in powers of α = α latt (1 /a ): δm PVlatt ∼ ∞ (cid:88) n =0 a c n α n +1 , (44)where r (as) n ( ν ) ν = c (as) n , since m PV and δm PVlatt have the same leading infared renormalon (locatedat d = 1). The coefficients c n are known from n = 0 ÷ 19 in the lattice scheme for a Wilsonaction [21–23]. We then adapt Eq. (19) to δm PVlatt to determine ¯Λ PV :¯Λ PV ( n f = 0) = E MC ( a ) − δm P latt − a Ω m − N (cid:48) =2 N P (cid:88) N P +1 a [ c n − c (as) n ] α n +1 + O ( a Λ ) . (45)where δm P latt = (cid:80) N P n =0 1 a c n α n +1 . In the counting of Eq. (15) this corresponds to (1, N P )precision. The expression we use for Ω m is Eq. (21) truncated to O ( α ) here and in the restof the paper. The error committed by this truncation is smaller than the error associated28 ●●●●●●●●● E MC ( a ) ( c )( b )( d ) ( a ) ●●●●●●●●●● ( a )( d ) ( b ) ( c ) FIG. 11: Upper panel : E MC is the Montecarlo lattice data [30–32]. The continuous lines aredrawn to guide the eye. The other lines correspond to Eq. (45) truncated at different orders inthe hyperasymptotic expansion. (a) E MC ( a ) − δm P (1 /a ), (b) E MC ( a ) − δm P (1 /a ) − a Ω m , (c) E MC ( a ) − δm P (1 /a ) − a Ω m − (cid:80) N (cid:48) =2 N P N P +1 1 a [ c n − c (as) n ] α n +1 (in this last case we include the errorof the lattice points in the middle of the band), (d) is the fit of the right-hand-side of Eq. (45)to ¯Λ PV ( n f = 0) − Ka . For each difference and for the final fit, the bands are generated by thedifference of the prediction produced by the smallest positive or negative possible values of c thatyield integer values for N P . Lower panel : As in the upper panel but in a smaller range. r − ≈ Z m . Therefore, we will neglect it in the following. The renormalon behavior associatedto subleading renormalons of E MC ( a ) is not well known, except that the next singularityin the Borel plane is expected to be at | u | = 1 (d=2). Therefore, we stop the secondperturbative expansion at N (cid:48) = 2 N P such that the reminder should be of O ( a Λ ). For thecoefficients c (as) n we use Z lattm ( n f = 0) = 17 . . 0) [23]. We also truncate the 1 /n expansionin Eq. (26) to O (1 /n ). This means using the estimates for β and β listed in TableIII. We take E MC (a) from [30–32]. These points expand over the following energy range:1 /a ∼ . r − ÷ . r − . We show our results in Fig. 11. They follow the same logic thanFigs. 1-6 in Sec. III. We observe that the subtraction of the perturbative expansion accountsfor most of the 1 /a dependence. Still we have enough precision to be sensitive to O ( a Λ )effects. A strict fit setting the O ( a Λ ) correction to zero gives a large χ ∼ − 7. Theinclusion of a pure Ka term to Eq. (45) gives a good fit . The statistical error is small andthe χ = 1 . / . 06 (for the smallest | c | with positive/negative c value) is good. Overall,we obtain (using the smallest c positive, which means N P = 7 except for β = 5 . N P = 6) ¯Λ PV = 1 . r − (stat . ) − . . ( c ) +0 . − . ( Z m ) +0 . − . . (46)This number is not very different from the number obtained in [29] using a superasymptoticapproximation truncated at the minimal term determined numerically (typically this alwaysgives slightly better results than truncating at the minimal term predicted by theory).Let us now discuss the error budget in Eq. (46). The first error is the statistical errorof the fit. The remaining errors are different ways to estimate the error produced by theapproximate knowledge of the hyperasymptotic expansion. One possibility is to take themodulus of the difference with the evaluation using the c negative with the smallest possiblemodulus. This is the second error we quote in Eq. (46). The last error we include is due tothe variation of Z lattm ( n f = 0) = 17 . . 0) [23] (correlated with the error of c n ). The error itproduces in Ω m is small. Comparatively, most of the error associated to Z m comes from thedifferences in (cid:80) N (cid:48) =2 N P N P +1 1 a [ c n − c (as) n ] α n +1 evaluated at different Z m . Whereas (cid:80) N (cid:48) =2 N P N P +1 1 a [ c n − Unlike for the pole mass, it is not clear what is the operator of the OPE that would produce the NPcorrection and the associated u = 1 renormalon. Therefore, if for the pole mass we can be certain that theNP correction has the form Ka , without any anomalous dimensions nor any nontrivial ln( a ) dependence,we can not exclude the possibility that this O ( a ) correction may have a non trivial anomalous dimensionand/or ln( a ) dependence. (as) n ] α n +1 is quite small for the central value of Z m , it significantly changes after variationof Z m . This variation is only partially compensated by the variation of the coefficients c n ,which have smaller errors, producing a significant change in (cid:80) N (cid:48) =2 N P N P +1 1 a [ c n − c (as) n ] α n +1 . Wehave also determined the central value in Eq. (46) not including the O (1 /n ) corrections inthe asymptotic expressions for c (as) n . The difference we obtain is -0.08. This is significant,showing that the 1 /n corrections are sizable in the lattice scheme. On the other hand, thedifference is well inside the error associated to Z m . Actually, the difference with evaluationsincluding the O (1 /n ) corrections in the asymptotic expressions for c (as) n is -0.03. Thisshows a convergent pattern, which we illustrate in Table IV. Overall, the largest sourceof uncertainty comes from the incomplete knowledge of (cid:80) N (cid:48) =2 N P N P +1 1 a [ c n − c (as) n ] α n +1 , whichis closely linked to the incomplete knowledge of Z m . This discussion points to that moreaccurate determinations of Z m can be possible and, then, that the error of ¯Λ PV associatedto Z m could be made smaller. We believe these issues deserve further study that we leavefor future work.latt O (cid:0) n (cid:1) O ( n ) O ( n ) MS N tr = 7 N tr = 6 N tr = 5 N tr = 4¯Λ PV . 33 1 . 42 1 . 45 ¯Λ PV . 48 1 . 52 1 . 59 1 . TABLE IV: Determinations of ¯Λ PV in the lattice and MS scheme from fits of ¯Λ PV − Ka to theright hand side of Eq. (45). The first three numbers show the impact in the fit of including the O (1 /n m ) corrections for m = 2, 3, 4 in the asymptotic expressions for c ( as ) n in the lattice scheme(in the MS this effect is negligible). The other numbers are the fit of ¯Λ PV in the MS scheme, using α MS = α latt (1 + (cid:80) N tr n =0 d n α n latt ) truncanted at N tr = 4 , , , One error that we do not include here is the error associated to α . From the lattice pointof view, we are talking of the relation between α (1 /a ) and r . We use the phenomenologicalformula deduced in [28]. The error of this formula is claimed to be around 0.5-1% in therange β ∈ (5 . , . 92) (having a look to Fig. 10 a more conservative range could be (6,6.8)). 1. Scheme dependence It is interesting to consider the scheme dependence of Eq. (46). In [29] relative largedifferences were found for fits to ¯Λ after (approximated) scheme conversion to the MS scheme.The real problem is not transforming the coefficients c n from the lattice to the MS scheme,but transforming α latt to α MS with enough precision (in a way we need the relation between31 latt and α MS with NP, exponential, accuracy). This needs the coefficients of the β functionin the lattice scheme to high orders. We show estimates in Table III. The inclusion of thesecoefficients of the β function makes that the determinations of ¯Λ PV in the MS and latticescheme approach each other as we include more terms in the perturbative expansion of therelation between α MS and α latt . We show the comparison in Table IV. C. ¯Λ PV pot from lattice As an extra check of the method, we now consider the ground state energy of two staticsources in the fundamental representation at a fixed distance r computed in the lattice: E Σ + g ( r ; a ). This object has the same renormalon as twice the pole mass. Following [33] wedefine the quantity ¯Λ pot ( a ) ≡ E Σ + g ( r ; a )2 + ∆ , (47)where ∆ is just a constant to fix the normalization at r = r . For ¯Λ pot ( a ) we perform anOPE assuming r (cid:29) a , and compute it using the PV prescription. We then have¯Λ PV pot ≡ E Σ + g ( r ; a )2 + ∆ − δm P latt − a Ω m − N (cid:48) =2 N P (cid:88) n = N P +1 a [ c n − c (as) n ] α n +1 + O ( a Λ ) . (48)We show the results in Fig. 12. The lattice data is taken from [34, 35], as analyzed in[33]. A nicely flat curve appears. This object does not show O ( a Λ ) artifacts. This isconsistent with the discussion in [28], though there the discussion was only made for energydifferences. This has the potentially important consequence that potentials computed withdifferent β ’s can be related with perturbation theory with good accuracy. There is no needto subtract independent constants for each β ≡ /g .32 ●●●●●●●●● Λ pot ( a ) ( b )( a ) ( c ) ●●●●●●●●●● ( c ) ( b )( a ) FIG. 12: Upper panel : ¯Λ pot ( a ) is the Montecarlo lattice data [34, 35], as analyzed in [33]. Thecontinuous lines are drawn to guide the eye. The other lines correspond to Eq. (48) truncated atdifferent orders in the hyperasymptotic expansion: (a) ¯Λ pot ( a ) − δm P (1 /a ), (b) ¯Λ pot ( a ) − δm P (1 /a ) − a Ω m , (c) ¯Λ pot ( a ) − δm P (1 /a ) − a Ω m − (cid:80) N (cid:48) =2 N P N P +1 1 a [ c n − c (as) n ] α n +1 (in this last case we include theerror of the lattice points in the middle of the band). For each difference the bands are generatedby the difference of the prediction produced by the smallest positive or negative possible values of c that yield integer values for N P . Lower panel : As in the upper panel but in a smaller range. r − ≈ 400 MeV. . ¯Λ PV from B meson mass We now move to the physical case with n f = 3 light quarks. Using HQET we approximatethe B/D meson mass by (we use spin averaged masses) m B ( D ) = m PV + ¯Λ PV + O (cid:18) m PV (cid:19) . (49)It is not the aim of this paper to determine m b (nor m c ). We are rather interested to knowthe error associated to determinations of m PV if m is known, and vice versa. We will thenlater use this analysis for the top quark mass determination. For this purpose we use itshyperasymptotic approximation m PV ( m b/c ) = m P ( m b/c ) + m b/c Ω m + N (cid:48) =2 N P (cid:88) n = N P +1 [ r n − r (as) n ] α n +1 + · · · . (50)To make the error analysis we use the bottom case, and take m b = 4 . 186 GeV from [36].We obtain (we have added a -2 MeV to the relation between the MS bottom mass and thepole mass due to the charm quark [13]) m b, PV = 4836( µ ) +8 − ( Z m ) − ( α ) +8 − MeV . (51)For the variation of µ we take the range µ ∈ ( m b / , m b ). For Z m we take Z MS m ( n f = 3) =0 . α we take Λ ( n f =3)MS = 332 ± 17 MeV from [37].The central value has been obtained with N P = 3 ( c = 0 . (cid:80) N (cid:48) =2 N P n = N P +1 [ r n − r (as) n ] α n +1 term should roughly scale as (assuming the next renormalon islocated at | u | = 1) ∼ e − πβ αX ( µ ) (1+ln(2)) . (52)This is the expected scaling if µ ∼ m . Nevertheless, the dependence on µ will be quitedifferent depending on whether the next renormalon is ultraviolet ( ∼ µ − ) or infrared ( ∼ µ ). Actually, the magnitude is also expected to be different, being more important foran eventual infrared renormalon. As the situation is somewhat uncertain we do not dwell34urther in this issue. To roughly estimate the size of subleading terms we could compute with N P = 2 ( c = 1 . , N max − N P ).The difference is below 1 MeV (after including [ r − r (as)3 ] α , otherwise the difference is 7.5MeV). Even computing with N P = 1, which formally allows us to reach the next renormalonlocated at 2 N P = 2 (i.e. (1, N P ) precision in the counting of Eq. (15)), the difference is ∼ µ scale dependence of m P ( m b/c ) − m b/c Ω m also gives ameasure of the uncomputed (cid:80) N (cid:48) =2 N P N P +1 [ r n − r (as) n ] α n +1 term, as such scale dependence shouldcancel in the total sum. We will then take it as the associated error. This is the first errorquoted in Eq. (54). Actually, the error associated to Z m is also a measure of the lack ofknowledge of higher order terms in perturbation theory. Therefore, there is some degree ofdouble counting by considering these two errors separately.It is interesting to analyze the error of the superasymptotic approximation of m PV (onlycomputing m P ). If we vary µ in the range µ ∈ ( m b / , m b ), we obtain ( N P = 3, c = 0 . m b, P = 5077( µ ) +134 − MeV . (53)Nicely enough, it agrees with Eq. (51) within one sigma. This is also so if we take N P = 2 ( c =1 . m b, P = 4922 +107 − MeV. We find that the scale dependence of the superasymptoticapproximation is large. Therefore the inclusion of m Ω m is crucial to make the result muchmore scale independent. On the other hand note that there is no error associated to Z m atthis order (which is, in any case, comparatively small).Using Eq. (51) and Eq. (49) we can determine ¯Λ PV . We work at leading order in 1 /m .We obtain ¯Λ PV = 477( µ ) − ( Z m ) +11 − ( α ) − ( O (1 /m )) +46 − MeV , (54)where we have included an extra error source compared with Eq. (51). This extra error isassociated to the O (1 /m ) corrections. The existence or not of genuine NP 1 /m correctionsmay introduce a significant error. In case they exist, if we take the hyperfine energy splittingas a measure of 1 /m corrections, we find shifts from the central values of order ∼ 46 MeVand ∼ 140 MeV for B and D mesons respectively. As Eq. (54) has been obtained from the B meson spin-average mass we conservatively estimate the error associated to genuine NP1 /m corrections to be of order ∼ 46 MeV, as it is the most we can do from phenomenologyand perturbation theory. Let us recall however that recent lattice simulations point to much35maller genuine NP 1 /m corrections for the spin-independent average [38].Earlier direct determinations of ¯Λ PV or m PV can be found in [39, 40]. The formulas areequivalent to those used here to one order less (using N P = N max = 2). They also includeless terms in the sum in Eq. (26). More recently, a determination of ¯Λ has been obtainedin [38] using lattice data. In this case the formulas are equivalent to those used here since N P = 3 (see Eq. (57) of Ref. [1]) except for the fact that the scale µ was always fixed equalto the heavy quark mass and that the mass was obtained in the MRS scheme [41]. In thisreference is also given the relation between the PV and the MRS mass. Using it we obtain(where we combine quadratically the error of Z MS m and Λ MS )¯Λ PV − ¯Λ MRS = cos( πb ) 4 π Γ( − b )2 b β Z Xm Λ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n f =3 = − . (55)The prediction of [38] translates then to ¯Λ PV = 435(31), where we only include the errorquoted in [38]. In particular, we do not include the error in Eq. (55). Note that Eq. (55)scales like O (Λ QCD ), whereas m Ω m scales like O ( √ α Λ QCD ). There is a 40 MeV differencewith the number given in Eq. (54). 10 MeV can be understood because the value of m b usedin [38] is around 10 MeV bigger. Another 10 MeV can be understood by the inclusion of1 /m nonperturbative effects. The remaining 20 MeV difference are more difficult to identify,though they are well inside uncertainties. Leaving aside the different α ’s used, anothersource of difference is the value of Z m . The value used in [38] comes from [42] (where theeffect of scale variation was not included in the error analysis). This determination useda sum rule that is free of the leading pole mass renormalon. The possibility of using sumrules to determine the normalization of renormalons was first considered in [43]. For thedetermination of Z m , sum rules were first used in [12]. Later sum rule analyses can be foundin [44]. Alternatively one can use the ratio of the exact and asymptotic expression of thecoefficients r n to determine Z m as in [13, 21–23, 45]. For an extra discussion on this issuesee [46]. Finally, it is worth mentioning that Z m can be determined either from the staticpotential or from the pole mass (and its relatives). The only value of Z m that uses the staticpotential is from [13]. A preference for determinations of Z m from the static potential canbe theoretically motivated, as they are less affected by subleading renormalons. There areno ultraviolet renormalons and the next infrared renormalon is located at u = 3 / 2. On the36ther hand, the pole mass is expected to have renormalons at | u | = 1. Only in the event thatthere is no u = 1 renormalon and the effect of the u = − E. ( N, µ ) → ∞ . Eq. (4) . Case 2B) All previous determinations of ¯Λ PV have been obtained using limit 1). For completenesswe have also explored how limit 2B) performs for ¯Λ PV , even though it is, in principle, lessprecise. We have considered the different methods to take the limit 2B) discussed at theend of Sec. II B, and compared with the numbers obtained above. We first consider theevaluation of m PV using the right hand side of Eq. (33) with Eq. (34). The central value isdetermined using c (cid:48) min = 1 . K ( A )MS = 0, and µ = m b = 4 . PV = 453 MeV. The difference with Eq. (54) is 24 MeV, which is quitereasonable. We can also explore the µ dependence. Taking the variation µ ∈ ( m b / , m b ),we obtain ¯Λ PV = 453 − ( µ ) MeV. Comparatively with Eq. (54) the µ scale dependence ismuch larger. We next consider the limit as taken in Eq. (32). This requires the knowledgeof the coefficients r n to all orders. For n > N A = 3000 (though it already converges at smaller values of N A ).Remarkably enough, we obtain the same result than before: 453 MeV. There is a residualdependence on c (cid:48) . For illustration, if we take instead c (cid:48) = 2, we obtain ¯Λ PV = 438 MeV(the result using the right-hand side of Eq. (33) with Eq. (34) yields the same number), andthe scale dependence is larger: ¯Λ PV = 438 − ( µ ). The value of c (cid:48) we have used to make theanalysis can be an issue. As discussed in [1, 17], taking χ − N A are needed. This problem aminorates by taking largervalues of c (cid:48) . Since for the limit as taken in Eq. (32) we can go to very large N A this is nota problem. We have also performed a similar analysis with n f = 0 and r units, relevantfor the analyses performed in Sec. IV B. The discussion follows parallel to the one we justhad with the difference that we now know 20 coefficients of the perturbative expansion. Thevalue we obtain: ¯Λ PV = 1 . r − (using a quadratic fit) is indeed quite close to the valueobtained in Sec. IV B, though less precise. 37e have more problems with the other ways to take the µ → ∞ limit discussed at theend of Sec. II B. The direct use of N A in Eq. (4) or of N A in Eq. (67) in [1] requires, besidesthe coefficients r n to all orders, the β -function coefficients to all orders as well. We do nothave them. Instead we use truncated version of the β function. This makes the numericalcalculation much more challenging, since the running in µ is more complicated. Therefore,we had problems to go to very large N A . For N A ≥ 200 we find instabilities is some cases. Asmentioned before, the value of c (cid:48) we use to make the analysis can be an issue. Taking χ − c (cid:48) . In the lattice scheme determination of quenched ¯Λ PV we indeed observe convergenceto the value obtained before using c (cid:48) = 2. Using c (cid:48) min = 1 . 076 the convergence is less good.Determinations in the MS scheme do not show convergence if we stop at N A ≤ c (cid:48) = 2 rather than c (cid:48) min . Overall, as the precision we getwith method 2B) is worse than with method 1), we will not study this limit in more detail. V. TOP MASSA. About the pole mass ambiguity The top quark mass is one of the key parameters of the standard model. A lot ofexperimental work has been devoted to its determination (see for instance [47–49]). Whereasthis is a matter of debate, it is typically assumed that the masses obtained from experimentcorrespond to the pole mass. Thus, there has been an ongoing discussion on the intrinsicuncertainty of these determinations (see for instance [45, 50], and [51] for a more recentdiscussion). We believe that, without further qualifications, the question is ill posed, or maylead to confusion. It is well known that the pole mass is well defined (infrared finite andgauge independent) at finite (albeit arbitrary) order in perturbation theory [52]. It is alsowell known that such series is divergent . Therefore, no numerical value can be assigned tothe infinite sum of the perturbative series of the pole mass. Truncated sums are well definedbut depend on the order of truncation (a detailed discussion relevant for the analysis made inthe present paper can be found in [1]). These truncated sums can be related with observables Actually this is only proven in the large β approximation [19, 20], and it is also supported by numericalanalyses [21–23], but there is no analytic proof. 38r with intermediate definitions of the heavy quark mass, like the PV mass (which regulatesvia Borel resummation the infinite sum), in a well-defined way.In this context, the shortest answer to the above posed question is that the ambiguity(of a well-defined mass) is zero . As a matter of principle, m PV (or m P ) can be defined witharbitrary accuracy (this also applies to any threshold mass), if one computes high enoughorders of the perturbative series, and if m is given. One can discuss (actually one cancompute) the scheme/scale dependence (if they have) of them. In this respect, there is nomuch conceptual difference with respect to asking about the scheme/scale dependence ofminimal subtraction schemes for the heavy quark masses.A quite a different question is to determine the typical difference (that not ambiguity)between (reasonable) different definitions of the pole mass. The short answer to this questionis that the differences are (at most) of order Λ QCD for (reasonable) different definitions ofthe pole mass. We emphasize that one can not be more precise unless stating the specificdefinition used for the pole mass. For instance, the difference between m PV and m P is of O ( √ α Λ QCD ) with a known prefactor. Truncating the perturbative series at order N near N ∗ are also legitimate definitions of the pole mass. The typical difference between truncatingat different N is of order Λ QCD : see for instance Eq. (62) of [22]. One could even use M B as a definition for the pole mass. Its difference with m PV is of order Λ QCD . If one defines animaginary mass by doing the Borel integral just above the positive real axis, the differencewith m PV is of O ( i Λ QCD ). The authors of [45] choose to divide this number by π and takethe modulus as their definition of the ambiguity. These examples illustrate that, even if theambiguity is of O (Λ QCD ), the coefficient multiplying Λ QCD is arbitrary. Overall, it should beclear that no much more can be said, and we are indeed against of dwelling too much on thisissue. Instead, we strongly advocate to avoid generic discussions about the pole mass, whichis not well defined beyond perturbation theory, and restrict the discussion to the precisionand errors of specific, NP well-defined, heavy quark masses the perturbative expansion ofwhich can be related with the perturbative expansion of the pole mass.Once working with NP well-defined heavy quark masses like m t, PV or m t,P , we can addressthe more relevant question of determining the precision with which m t can be determined if m t, PV or m t,P is known (and vice versa, if m t is known what is the uncertainty of m PV ) withnowadays knowledge of the perturbative expansion. In other words, with which precisionthe theoretical expression is known. For reference we will take the value m t = 163 GeV in39he following. We will see in the next section that indeed the precision is quite good andthat the error is significantly smaller than typical numbers assigned for the ambiguity of thepole mass. We will not dwell in this paper on the precision with which m t, PV or m t,P canbe determined from experiment as such discussion is observable dependent. B. Decoupling and running We now turn to an issue specific to the top quark (as compared with the bottom andcharm quark). The top quark mass is much larger than Λ QCD . The latter is the scale thatcharacterizes renormalon associated effects and it is the precision we want to achieve. Thisobviously generates ratios of quite disparate scales. In the context of threshold masses withan explicit infrared cutoff ν f , this calls for resummation of the large logarithms: ln ν f /m t .This is possible, and first done in [33] in the RS scheme (see also [46] for an extra discussionon this issue). Here, we approach the problem in a different way. We want to work withexpansions for the perturbative series of the pole mass truncated at the minimal term: m P , and to improve upon it using hyperasymptotic expansions. Nevertheless, at the scaleof the top mass, we do not have enough terms to reach the asymptotic behavior of theperturbative expansion. We use instead that the top quark pole mass and the pole mass ofa fictitious top quark with mass m (cid:48) t share the same leading infrared renormalon. Therefore,the leading infrared renormalon cancels in the difference. We can then decrease the topmass in a renormalon free way until we reach a top mass low enough that we can use thehyperasymptotic expansion. Such renormalon free running is determined by the followingfunction (not compulsory to take µ = m but it simplifies the computation) F ( m, n f ) ≡ ddm ( m PV ( m ) − m ) (cid:39) ddm (cid:88) n =0 r ( n f ) n ( m ; µ = m ) α n +1( n f ) ( m ) ≡ N +1 (cid:88) n =1 f n ( m ) (cid:18) α ( n f ) ( m ) π (cid:19) n . (56)This formula is correct up to N ∼ N P , since m Ω m and r ( as ) n are independent of m (seeEq. (19)), so that their derivative with respect to m vanishes. The coefficients r ( n f ) n areevaluated for n f massless particles. In the context of the MSR threshold mass the runningis implemented in a similar way (see, for instance, [44]). Eq. (56) makes explicit that suchrunning is just a natural consequence of the relation between observables and their OPEs (forillustration, it follows from the fact that M B − M D , the B minus D meson mass difference40s free of the leading infrared renormalon), and not linked to an specific threshold massdefinition.There is still another issue specific to the top quark: there are two heavy quarks (thebottom and charm), with masses much larger than Λ QCD , that generate extra corrections tothe pole-MS mass relation due to the finite mass of the bottom and charm quark. Therefore,we have for m ∼ m t m PV ( m ) = m + N max (cid:88) n =0 r ( n f ) n ( m ; µ = m ) α n +1( n f ) ( m ) + δm ( n f ) b ( m ) + δm ( n f ) c ( m ) + δm ( n f ) bc ( m ) , (57)where it is implicit that N max (the number of known terms of the perturbative expansion) isnot large enough to see the decoupling of the bottom nor charm and certainly N max < N P . n f stands for the number of active flavours. At the top mass scale we take n f = 5. The O ( α ) term of δm ( n f ) Q was computed in [53] and the O ( α ) term in [54]. Note as well thatat O ( α ) there is a new contribution including a vacuum polarization of the bottom andcharm at the same time. We name it δm ( n f ) bc and it has been computed in [50].As we decrease the value of m t the bottom and charm quark will decouple. This decou-pling will be absorbed in δm ( n f ) b/c/bc , which are polynomials in powers of α ( n f ) . In general thisis not just changing n f in the original expressions from n f = 5 to n f = 4 or 3. The explicitexpressions can be found in the Appendix B.The renormalon is associated to scales smaller than the bottom and charm quark masses.Therefore, such scales should be decoupled before we talk about the hyperasymptotic ex-pansion. As we have mentioned above we do such decoupling by varying the mass of thetop till reaching a fictitious top with a mass small enough such that, first the bottom, andlater the charm, decouple. Overall, our final formula is the following: m PV ( m t ) = m t + (cid:90) m t µ b dm (cid:18) F ( m, 5) + ddm ( δm (5) b ( m ) + δm (5) c ( m ) + δm (5)( bc ) ( m )) (cid:19) + (cid:90) µ b µ c dm (cid:18) F ( m, 4) + ddm ( δm (4) b ( m ) + δm (4) c ( m ) + δm (4)( bc ) ( m )) (cid:19) + m PV ( µ c ) − µ c . (58)We emphasize that F ( m, n f ) is expanded in powers of α before integration. We take µ b small enough such that the bottom decouples and µ c small enough such that the bottom41nd charm decouple, and also such that we reach the asymptotic limit of the pole-MS massperturbative expansion with the existing known coefficients. Therefore, m PV ( µ c ) = m P ( µ c )+ µ c Ω m + δm (3) b ( µ c )+ δm (3) c ( µ c )+ δm (3)( bc ) ( µ c )+ O ( µ c e − πβ α ( µc ) (1+ln(2)) ) . (59)The O ( µ c e − πβ α ( µc ) (1+ln(2)) ) term stands for subleading corrections in the hyperasymptoticexpansions, which are not known.Let us now discuss in more detail the dependence on the bottom and charm quark, inparticular the effects associated to the fact that they have masses much bigger than Λ QCD (for the analysis we take m b = 4 . 186 GeV and m c = 1 . 223 GeV [36] but the sensitivity to thespecific values we use is very tiny). As already discussed in [19], the natural scale of a n -loopintegral is not m t but m t e − n . For the case of the bottom versus charm quark it was observedin [13] that the charm quark effectively decouples at order α /α for the case of the charmquark effects in the bottom pole mass-MS mass relation. If we lower the mass of the top wecan also observe at which scales it is more convenient to decouple the bottom and charmquark in the top pole mass-MS mass relation. This can be illustrated in Fig. 13, where weplot the corrections associated to the bottom and charm with and without decoupling interms of the fictitious top mass (assuming a single heavy quark). Obviously for very largetop masses it is not convenient to do the decoupling. Nevertheless, as we decrease the massof the top it becomes much more effective to decouple, first the bottom, and afterwards thecharm quark. Once this is done the corrections due to the bottom and charm masses toEq. (59) are very small. Comparatively to other errors, the uncertainty associated to the O ( α ) corrections is negligible. Also the correction associated to the bottom and charmquark masses to Eq. (58) is, comparatively to the total running, very small. From thisanalysis we will take as central values µ b = 20 GeV and µ c = 5 GeV. For these values we In that reference MeV should read GeV instead from Eq. (8) to Eq. (12). B10 ) ,O ( α )( B7 ) ,O ( α )( B10 ) ,O ( α )( B7 ) ,O ( α ) 50 100 150020406080100120 m t [ GeV ] δ m b [ M e V ] ( B10 ) ,O ( α )( B7 ) ,O ( α )( B10 ) ,O ( α )( B7 ) ,O ( α ) 50 100 150050100150 m t [ GeV ] δ m c [ M e V ] FIG. 13: Upper panel : Plot of the correction to the PV mass of a top mass with varying m t massdue to a heavy quark with MS mass equal to 4.185 GeV (bottom) with and without decoupling(assuming a single heavy quark). Lower panel : As in the upper panel with a heavy quark withMS equal to 1.223 GeV (charm). We use Eqs. (B7) and (B10). (cid:90) m t µ b dm ddm ( δm (5) b ( m ) + δm (5) c ( m ) + δm (5)( bc ) ( m ))+ (cid:90) µ b µ c dm ddm ( δm (4) b ( m ) + δm (4) c ( m ) + δm (4)( bc ) ( m ))+ δm (3) b ( µ c ) + δm (3) c ( µ c ) + δm (3)( bc ) ( µ c ) = − . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O ( α ) + 0 . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O ( α ) = − . . (60)The specific value depends on µ b and µ c but the good convergence and smallness of thiscorrection holds true for other values of µ b and µ c . The implementation of the decouplingof the bottom and charm in [50] produces a much larger correction. An even larger effectis observed in the implementation performed in [45], where the perturbative expansion isalways performed at the scale of the top mass (using renormalon based estimates for thehigher order coefficients), decoupling the bottom, and later the charm, depending on theorder of perturbation theory. Therefore, we take our numbers as optima, and the errornegligible compared with other uncertainties.We next explore the convergence pattern of the perturbative expansion. We first considerthe perturbative expansion associated to F . We find (cid:90) m t µ b dm F ( m, 5) + (cid:90) µ b µ c dm F ( m, 4) = 8445 + 837 + 53 − 43 = 9291(22) MeV . (61)We observe a convergent pattern. For the last two terms the convergence deteriorates. Onthe other hand the perturbative expansion becomes sign alternating. This may indicatesensitivity to the u = − ∼ − / × (the last computedterm) (see [5]). Therefore, we take it as the error of the truncation of the perturbativeexpansion, which is the error we quote in Eq. (61). We also explore the dependence ofEq. (58) on µ b and µ c . The dependence is very small, as we can see in Fig. 14. For µ c thevariation is negligible, and for µ b one gets variations of ∼ µ b oraround 20 GeV. Therefore, we will neglect it for the total error budget. We emphasize that these arguments do not apply to IR renormalons (and in particular to the u = 1 / NNLONNLONLOLO μ c [ GeV ] m PV ( m t ) [ G e V ] μ b = 20 GeV NNNLONNLONLOLO 10 20 30 40 50172.2172.4172.6172.8173.0173.2 μ b [ GeV ] m PV ( m t ) [ G e V ] μ c = FIG. 14: Plots of Eq. (58) in terms of µ b (upper panel) and µ c (lower panel) truncating theperturbative expansion of F ( m, n f ) at different orders in α in Eq. (61). In the upper figure we set µ c = 5 GeV. In the lower figure we set µ b = 20 GeV. The other source of error is associated to the approximate determination of Eq. (59)(except for the δm q terms, which have already been taken into account in Eq. (60)). Theerror analysis is equal to the one in Eq. (54) adapted by changing m b = 4 . 186 GeV → µ c = 545eV (the error associated to α is only computed for the full Eq. (58))( m P ( µ c ) + µ c Ω m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ c =5 GeV = 5744( µ ) +7 − ( Z m ) +9 − MeV . (62)Finally, we also include the error associated to α . Combining all errors we obtain m t, PV (163MeV) = 173033(h . o . ) +22 − ( µ ) +7 − ( Z m ) +9 − ( α ) +119 − MeV . (63)By far the largest uncertainty is associated to α . For the purely theoretical error budget,the error is associated to higher order corrections in perturbation theory. They show up indifferent ways. One is the approximate knowledge of Z m , which shows up in Ω m . The otheris the error in µ , which is a measure of the O ( e − πβ αX ( µ ) (1+ln(2)) ) corrections to Eq. (59). h.o.stands for the error associated to higher order terms in perturbation theory of Eq. (61).All these errors would profit from higher order perturbative computations. We have alsoexplored other sources of uncertainty, and find them to be comparatively very small: theerror (and the effect) associated to the finite mass of the bottom and charm quark is foundto be very small, and similarly for variations in the values of µ b and µ c .It is also useful to make the error estimate of the ratio of the PV and MS top mass. Weobtain ( m t = 163 GeV)[ m t, PV m t − × = 6155(h . o . ) +13 − ( µ ) +4 − ( Z m ) +6 − ( α ) +73 − . (64)Note that there is no ambiguity error associated to this number. Except for α all errorsare associated to the lack of knowledge of higher order terms of the perturbative expansion.In comparison with [45] we find that our result is less sensitive to Z m and to its associatederror. C. | u | = 1 renormalons The perturbative expansion of F ( m, n f ) is free of the u = 1 / u = 1 and u = − 1. The existence of an infrared renormalon46 ( m, n f ) f f f f f n f = 0 4 / . 11 25 . 52 18 . n f = 3 4 / . 32 12 . − . n f = 6 4 / . − . − . β /exact) n f = 10 / − . × − . × − . × − . × (Large β ) n f = 0 4 / . − . 31 114 . 33 -377.22(Large β ) n f = 3 4 / . − . 57 62 . 62 -169.04(Large β ) n f = 6 4 / . − . 58 29 . 46 -61.86 TABLE V: The coefficients f n of F ( m, n f ). Note that f ( n f = 0) has a 9% error from thedetermination in [9]. The n f = 10 case is used as a test for comparison with the large β . Thelast three (four) rows are the coefficients f n in the large β approximation. at u = 1 has been a matter of debate [15]. The existence of an ultraviolet renormalon at u = − β approximation [19, 20] but not beyond. Withrespect to this discussion some interesting observations can be drawn out of our analysis.The coefficients f n show an interesting dependence in n f (with changes of sign of differentpowers of n f ). In Table V we give the numbers of f n for different values of n f and also inthe large β approximation. We observe that for n f = 3 the O ( α ) flips sign. For n f = 6,the O ( α ) and O ( α ) flip sign. The situation is somewhat puzzling. Let us first note thatthe sign of the coefficients would be interchanged compared with the large β predictions(for n f = 3). This could still be understood from a u = − Z X − flips signfrom the large β prediction to real QCD. This would indicate a large dependence of Z X − on n f compared with what has been seen for Z Xm , where the large β approximation gave theright sign and order of magnitude. For n f → ∞ , the results agree with QED expectations( β becomes negative and the perturbative series is non sign-alternating). For n f = 6 weobserve that the last two terms are negative. One may then wonder if what we are seeingfor n f = 6 (and maybe also for n f = 3) is that the u = − f n to clarify this issue. It is usual lore that infrared renormalons dominate over ultraviolet ones (this is somewhatbased on large β analyses where ultraviolet renormalons are typically suppressed by the The coefficients of the perturbative expansion of the pole mass itself are also a polynomial in powers of n f . The sign dependence of the different powers of n f has been studied in [55, 56]. ∼ e d cX whereas infrared renormalons are enhanced by the factor ∼ e − d cX ). If wetake this seriously, and also the numbers we obtain for f n as an indication of the existenceof the u = − u = 1 renormalon is indeed zero. Inthis respect, it is worth mentioning the analysis of [38] where the NP correction associatedto the u = 1 renormalon was found to be zero within errors. This is consistent with thisdiscussion.On the theoretical side it is also interesting to see where the u = 1 renormalon wouldshow up in a perturbative computation of the heavy quarkonium mass. For the purposes ofthis discussion, the heavy quarkonium mass would read M nl = 2 m Q + (cid:104) p m Q (cid:105) nl + (cid:104) V (cid:105) nl + (cid:104) V m Q (cid:105) nl + O (cid:32) m Q (cid:33) , (65)where V is the static potential, and V is the 1 /m Q potential. OPE analyses in the staticlimit show that V does not have renormalon at u = 1. The virial theorem: (cid:104) p m Q (cid:105) nl = (cid:104) rV (cid:48) (cid:105) nl ,also guaranties that the kinetic term does not have such u = 1 renormalon. Therefore, anypossible u = 1 infrared renormalon of the pole mass should cancel with the analogousinfrared renormalon of the V /m Q potential. The fact that the latter can be written in aclosed way in terms of Wilson loops [57] may open a venue on which to study this issue infurther detail. This is postponed to future work. VI. CONCLUSIONS In this paper we have constructed hyperasymptotic expansions for the heavy quark polemass (and for associated quantities) regulated using the PV prescription along the lines of [1].We generalize the discussion of that reference by including possible ultraviolet renormalons.Such organization of the computation allows us to have a parametric control of the errorcommitted when truncating the hyperasymptotic expansion.In Sec. III the hyperasymptotic expansion of the pole mass of a heavy quark in thelarge β is computed. We use it as a toy-model observable to test our methods. It worksas expected. We can see the u = 1 / u = − u = 3 / 2. Compared with the staticpotential case studied in [1] in the large β approximation, infrared renormalons are located48t the same points in the Borel plane. On the other hand, the pole mass has ultravioletrenormalons, whereas the static potential does not. In practice the main difference comesfrom the relevance of the u = − u = − N ∼ × πβ α ,otherwise the perturbative series would start to diverge, as we can observe in Fig. 7 inthe MS scheme. Nevertheless, the importance of this renormalon heavily depends on thefactorization scale µ one uses. If one takes µ high enough, one could indeed do perturbationtheory until N ∼ × πβ α , where the u = 3 / u = − µ , in Fig. 7. One should keep in mind, though, that one needs perturbationtheory to a much higher order in the lattice scheme to reach the same precision than in theMS scheme. We expect this qualitative behavior of ultraviolet renormalons to also hold truebeyond the large β approximation.We next move to real QCD. We have performed determinations of ¯Λ PV using quenchedlattice QCD. For these observables perturbative expansions to high orders are available[21–23]. This allows us to test the method and go beyond the superasymptotic and theleading term in the hyperasymptotic approximation. We observe O ( a Λ ) corrections forthe B meson mass in the static approximation, but not for an analogous observable fromthe static potential. Nevertheless, we do not have enough precision to quantitatively studythese effects. The limiting factor is the error of the normalization of the leading renormalon,and, related, the lack of knowledge of the higher order beta function coefficients. Thelatter affects the O (1 /n ) corrections to the asymptotic formula of the perturbative seriescoefficients. These effects are sizable in the lattice scheme. On the other hand they are quitesmall in the MS scheme. On top of that the higher order coefficients of the perturbativeexpansion of δm latt are not known with enough precision to disentangle the subleadingrenormalon (their error is strongly correlated with the error of Z m ). All these considerationsforbid quantitative analyses beyond the leading term in the hyperasymptotic approximation.Further investigations are needed to improve on these issues, particularly on the error of Z m ,which also affects the discussion below.We also determine ¯Λ PV from the physical B meson mass assuming that the MS heavyquark mass is known. The result can be found in Eq. (54). In this analysis, we determine theerror associated to the incomplete knowledge of the perturbative expansion in determinations49f the heavy quark mass. We translate this result to the case of the top mass, which westudy in detail in Sec. V. In this section the issue of the uncertainty of the (top) pole massis critically reexamined. In particular, the bottom and charm quark finite mass effects arecarefully incorporated. In our implementation we find these to be very small. We find thepresent uncertainty in the relation between m t and m PV to be (for m t = 163 GeV) m t, PV (163MeV) = 173033(th) +25 − ( α ) +119 − MeV , (66)[ m t, PV m t − × = 6155 (th) +15 − ( α ) +73 − , (67)where we have combined the theoretical errors quoted in Eqs. (63) and (64) in quadrature.There is no ambiguity associated to the renormalon in this number. The precision is sys-tematically improvable the more terms of the perturbative expansion get to be known inthe future. Interestingly enough, it seems we have found some evidence for the existence ofthe next renormalon at u = − u = 1. We believe thismakes very timely a quantitative determination of the renormalization group structure ofthe u = − Acknowledgments We thank M. Steinhauser for comments on the manuscript. C.A. thanks the IFAE groupat Universitat Aut`onoma de Barcelona for warm hospitality during part of this work. Thiswork was supported in part by the Spanish FPA2017-86989-P and SEV-2016-0588 grantsfrom the ministerio de Ciencia, Innovaci´on y Universidades, and the 2017SGR1069 grantfrom the Generalitat de Catalunya; and by the Chilean FONDECYT Postdoctoral GrantNo. 3170116, and by FONDECYT Regular Grant No. 1180344. Appendix A: Evaluation of Ω d We here briefly sketch how we compute the integrals that appear in Ω d . For d < d > 0, we can use also such formulas (see for instance Eq. (47) of Chapter XXI). In thiscase, we can also alternatively perform the integration in the following way. For simplicity,50e take the case b = 0 and d = 1, as the method is similar for the more general case. I = (cid:90) ∞ , PV due − πβ α u (2 u ) N +1 − u = − e − πβ α (cid:90) ∞− , PV dyy e − πβ α y e (cid:16) πβ α (1 − cα )+1 (cid:17) ln(1+2 y ) , (A1)where in the second equality we set N = N P according to Eq. (3). We also do the changeof variables (where K = c − β π ) y = x √ − Kα (cid:114) β απ . (A2)We can then expand the exponent in powers of α and x : I = − e − πβ α (cid:90) ∞− √ πβ α (1 − Kα ) , PV dxx e − x e − πβ α K x (cid:113) β απ √ − Kα + x (cid:113) β απ √ − Kα + ··· . (A3)The inferior limit of the integral is then extended to −∞ . We then have I (cid:39) − e − πβ α (cid:90) ∞−∞ , PV dxx e − x e − πβ α K x (cid:113) β απ √ − Kα + x (cid:113) β απ √ − Kα + ··· (cid:39) e − πβ α (cid:112) β α (cid:18) − πβ K (cid:19) + · · · . (A4)Irrespectively of considering d > d > 0, it is not clear to us what is the asymptoticstructure of this expansion. This is something that we are investigating. In any case, atpresent, we have not seen evidence of asymptotic behavior of this expansion for all cases wehave considered. This does not preclude however that if we go to higher orders we will findan asymptotic behavior for this perturbative expansion.51 ppendix B: bottom and charm finite mass contributions to m t, PV We define δm (1) q ≡ m (cid:18)(cid:18) − m q m (cid:19) (cid:18) − m q m (cid:19) (cid:18) Li (cid:18) m q m (cid:19) − 12 ln (cid:18) m q m (cid:19) + ln (cid:18) − m q m (cid:19) × ln (cid:18) m q m (cid:19) − π (cid:19) + (cid:18) m q m (cid:19) (cid:18) m q m (cid:19) (cid:18) Li (cid:18) − m q m (cid:19) − 12 ln (cid:18) m q m (cid:19) +ln (cid:18) m q m (cid:19) ln (cid:18) m q m (cid:19) + π (cid:19) − m q m (cid:18) ln (cid:18) m q m (cid:19) + 32 (cid:19) + ln (cid:18) m q m (cid:19) + π (cid:19) (B1)(note that this coefficient is n f -independent), δm (2 ,n f ) q = m (cid:20) h (cid:18) m q m (cid:19) + w (cid:18) , m q m (cid:19) + n f p (cid:18) m q m (cid:19)(cid:21) , (B2)where n f = 5 for q = b and n f = 4 for q = c , and we use the representation for the functions h ( x ), w ( x, y ) and p ( x ) given in Ref. [50], and δm (2) bc = m w (cid:18) m b m , m c m (cid:19) . (B3)52e then have δm (5) b/c = δm (1) b/c α ( m ) π + δm (2 , / b/c α ( m ) π (B4) δm (5) bc = δm (2) bc α ( m ) π (B5) δm (4) b = (cid:104) δm (1) b + δm (1) b,dec (cid:105) α ( m ) π + (cid:104) δm (2 , b + δm (2) b,dec (cid:105) α ( m ) π (B6) δm (4) c = δm (1) c α ( m ) π + δm (2 , c α ( m ) π (B7) δm (4) bc = (cid:104) δm (2) bc + δm (2) bc,dec (cid:105) α ( m ) π (B8) δm (3) b = (cid:104) δm (1) b + δm (1) b,dec (cid:105) α ( m ) π + (cid:104) δm (2 , b + δm (2) b,dec (cid:105) α ( m ) π (B9) δm (3) c = (cid:104) δm (1) c + δm (1) c,dec (cid:105) α ( m ) π + (cid:104) δm (2 , c + δm (2) c,dec (cid:105) α ( m ) π (B10) δm (3) bc = (cid:104) δm (2) bc + δm (2) bc,dec + δm (2) cb,dec (cid:105) α ( m ) π , (B11)where δm ( i )( q, dec) are generated by the decoupling and read δm (1)( q, dec) = − m (cid:18) (cid:18) m q m (cid:19) + π (cid:19) , (B12) δm (2 ,n f )( q,dec ) = m (cid:40)(cid:20) ζ (3) + 13 π − (cid:18) π 54 + 71432 (cid:19) ln (cid:18) m m q (cid:19)(cid:21) n f + 8Li (cid:0) (cid:1) − ζ (3) + 61 π − π − (2)81 + 281 π ln (2) − π ln(2)+ (cid:18) − ζ (3) + π π ln(2) (cid:19) ln (cid:18) m m q (cid:19) + 127 ln (cid:18) m m q (cid:19)(cid:27) + 13 ln (cid:18) m m q (cid:19) δm (1) q . 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