On the ultimate uncertainty of the top quark pole mass
aa r X i v : . [ h e p - ph ] J un DESY 16-082TTP16-016TUM-HEP-1044/16arXiv:1605.03609 [hep-ph]June 02, 2017
On the ultimate uncertainty of the top quark pole mass
M. Beneke a , P. Marquard b , P. Nason c , M. Steinhauser da Physik Department T31,James-Franck-Straße, Technische Universität München,D–85748 Garching, Germany b Deutsches Elektronen Synchrotron DESY,Platanenallee 6, D–15738 Zeuthen, Germany c INFN, Sezione di Milano Bicocca, 20126 Milan, Italy d Institut für Theoretische Teilchenphysik,Karlsruhe Institute of Technology (KIT),D–76128 Karlsruhe, Germany
Abstract
We combine the known asymptotic behaviour of the QCD perturbation series expansion,which relates the pole mass of a heavy quark to the MS mass, with the exact series coeffi-cients up to the four-loop order to determine the ultimate uncertainty of the top-quark polemass due to the renormalon divergence. We perform extensive tests of our procedure byvarying the number of colours and flavours, as well as the scale of the strong coupling andthe MS mass. Including an estimate of the internal bottom and charm quark mass effect,we conclude that this uncertainty is around 110 MeV. We further estimate the additionalcontribution to the mass relation from the five-loop correction and beyond to be around300 MeV. Introduction
The top quark mass is a fundamental parameter of the Standard Model (SM). Due to itslarge size, it has non-negligible impact in the precision tests of the SM. After the discoveryof the Higgs boson and the measurement of its mass, the values of the W and top massare strongly correlated, such that a precise determination of both parameters would lead toa SM test of unprecedented precision [1]. Indeed, there is presently some tension betweenthe value of the top mass ± . GeV fitted from electroweak data and from its directmeasurement [1], for which the combination of the Tevatron and LHC data yields the 1.6 σ lower value of . ± . ± . GeV [2]. The value of the top mass is also crucial tothe issue of stability of the SM vacuum (see [3] for a recent analysis). The Higgs quarticcoupling decreases at high scales, eventually becoming negative. This evolution is verysensitive to the top mass value. For example, a top mass near 171 GeV would imply thatthe quartic coupling may vanish at the Planck scale, rather than turn negative.The standard direct determination of the top mass at hadron colliders, being based uponobservables that are related to the mass of the system comprising the top decay products,are quoted as measurements of the pole mass. On the other hand, it seems more natural touse the MS mass in both precision electroweak observables and in vacuum stability studies.In [4] the relation between the MS and pole mass for a heavy quark (the “mass conversionformula” from now on) has been computed to the fourth order in the strong coupling α s .Assuming the value of . GeV for the top-quark MS mass m t = m t ( m t ) , and assuming α (6) s ( m t ) = 0 . , we have [4] m P = 163 .
643 + 7 .
557 + 1 .
617 + 0 .
501 + (0 . ± . (1.1)for the series expansion of the mass conversion formula. The last term from the fourthorder correction is less than one half of the third order one.It is also known that the mass conversion formula is affected by infrared (IR) renor-malons [5–7]. This means that there are factorially growing terms of infrared origin in theperturbative expansion, such that the expansion starts to diverge at some order. If theseries is treated as an asymptotic expansion, the ambiguity in its resummation is of orderof a typical hadronic scale. Because of this, it is often stated that the ultimate accuracyof top pole mass cannot be below a few hundred MeV. One of the goals of this work is tomake this estimate more precise.It is remarkable that the perturbative relation between the pole and MS mass of aheavy quark appears to be dominated by the leading infrared renormalon already in loworders [8, 9]. This observation was used in previous work [10–12], and more recently in[14, 15] to estimate the unknown normalization of the leading IR renormalon, and mostlyapplied in the context of bottom physics. In the context of top physics, the importanceof this issue was raised recently in [16]. The purpose of this work is to combine the newlyavailable four-loop coefficient [4] in the mass conversion formula with the known structureof the first infrared renormalon singularity [7] to determine the normalization constant anddiscuss its impact on top physics. We also perform an analysis of the dependence on thenumber of colours and flavours, which is by itself of interest, and stability tests with respect1o variations of the scale of the strong coupling and MS mass. This leads to an expressionfor the mass conversion factor including an estimate of the contributions beyond four loops,and an estimate of the irreducible error. The renormalon divergence is a manifestation of the fact that the mass conversion formula,while infrared finite is sensitive to small loop momentum. In the case of the pole massthis sensitivity is particularly strong, namely linear, resulting in rapid divergence of theperturbative expansion, and an infrared sensitivity of order Λ QCD [5, 6]. The ambiguity indefining the pole mass is therefore of similar size. This is not surprising as the pole mass ofa quark is not an observable due to confinement and the difference with the physical heavymeson masses is also of order Λ QCD . Unlike other heavy quarks, the top quark decays onhadronic time scales, and thus the propagator pole position acquires an imaginary part.The renormalon divergence is not altered [17] by the fact that the top quark is unstablewith a width larger than Λ QCD and hence does not form bound states. The finite widthsimplifies the perturbative treatment of top quarks, since it provides a natural IR cut-off,and there exists no quantity for which the pole mass would ever be relevant. But theinfrared sensitivity of the QCD corrections to the mass conversion factor, which causes thedivergence, remains unaffected by the width.Slightly more technically, the divergence arises from logarithmic enhancements of theloop integrand. Heuristically, this can be understood by noticing that the running couplingevaluated at the scale l of the loop momentum has the expansion α s ( l ) = 1 b ln l / Λ = α s ( m )1 − α s ( m ) b ln m /l = ∞ X α ns ( m ) b n ln n m l . (2.1)The IR contribution to the last loop integration in the ( n + 1) -loop order then takes theform δm ( n +1) ∝ α n +1 s ( m ) Z m d l b n ln n m l = m (2 b ) n α n +1 s ( m ) n ! . (2.2)With this behaviour the series of mass corrections reaches a minimal term of order m (2 b ) n α n +1 s n ! ≈ m α s n − n ( √ πn n +1 / e − n ) ≈ m r πα s b exp (cid:18) − b α s (cid:19) ≈ r πα s b Λ QCD , (2.3)when n ≈ / (2 b α s ) and then diverges. Asymptotic expansions can sometimes be summedusing the Borel transform. Given a power series f ( α s ) = ∞ X n =1 c n α ns , (2.4)the corresponding Borel transform is defined by B [ f ]( t ) = ∞ X n =0 c n +1 t n n ! . (2.5)2he Borel integral Z ∞ dt e − t/α s B [ f ]( t ) (2.6)has the same series expansion as f ( α s ) and provides the exact result under suitable condi-tions. However, for the case of (2.2), where c n +1 = (2 b ) n n ! , the Borel integral Z ∞ dt e − t/α s − b t (2.7)cannot be performed because of the pole at t = 1 / (2 b ) . We can introduce some prescriptionfor handling the pole in the integral, as, for example, the principal value prescription.Whether or not this reconstructs the exact result, an ambiguity remains, quantified by theimaginary part of the integral when going above or below the singular point. A commonlyused procedure is to define this ambiguity to be equal to the imaginary part of the integraldivided by Pi (see, e.g., [18], section 5.2). For (2.7), this yields Λ QCD / (2 b ) . (2.8)In the range of α s values considered in this paper, the ambiguity is close to the size of thesmallest term in (2.3). It can be shown [7] that while the precise asymptotic behaviour of the mass conversionformula differs from the simple ansatz employed in this section for illustration, as discussedbelow, the ambiguity is exactly proportional to Λ QCD , which evaluates to about
MeVin the MS scheme. In the remainder of this work, we aim to quantify the proportionalityfactor. We write the perturbative expansion of the mass conversion formula as m P = m ( µ m ) (cid:18) ∞ X n =1 c n ( µ, µ m , m ( µ m )) α ns ( µ ) (cid:19) . (3.1)Here α s ( µ ) is the MS coupling in the n l light flavours theory, and m ( µ m ) stands for the MS mass evaluated at the scale µ m . (In the following we will consider different scale choices forthe heavy quark mass and the strong coupling constant). We also use m to denote the MS mass evaluated self-consistently at a scale equal to the mass itself, i.e. m = m ( m ) . (3.2) Note, however, the different parametric dependence on α s of (2.3) and (2.8). The correct dependenceis that of (2.8), for the following reason: The typical width of the region where the minimal term isattained grows parametrically as p / (2 b α s ) . The accuracy of an asymptotic series is better estimatedby the minimal term times the factor accounting for the number of terms in this region, which makes(2.3) parametrically consistent with (2.8). Numerically, this factor turns out to be of order one for theapplications considered in this paper, as will be confirmed in section 4 below. In case of doubt, the estimatefrom the ambiguity of the Borel integral should be the preferred choice. n behaviour of theperturbative coefficients [7] (and [18] , eq. (5.90)) c n ( µ, µ m , m ( µ m )) −→ n →∞ N c (as) n ( µ, m ( µ m )) ≡ N µm ( µ m ) ˜ c (as) n , (3.3)where ˜ c (as) n +1 = (2 b ) n Γ( n + 1 + b )Γ(1 + b ) (cid:18) s n + b + s ( n + b )( n + b −
1) + · · · (cid:19) . (3.4)It is remarkable that b = b / (2 b ) and the s i coefficients of the sub-leading O (1 /n i ) be-haviour can all be given in terms of the coefficients of the beta-function [7]. The relevantexpressions are collected in appendix A. We also note that the scale µ m at which m isevaluated does not appear explicitly on the right-hand side of (3.3) and hence is irrelevantin (3.1) as far as the large- n behaviour is concerned. The dependence on the scale µ of thestrong coupling is compensated by the factor µ in front of ˜ c (as) n +1 in (3.3). With these defi-nitions the normalization N is independent of µ and µ m . It cannot however be computedrigorously with present perturbative techniques in general, but in the limit of large negativeor positive n l it assumes the value [5] lim | n l |→∞ N = C F π × e , (3.5)which equals . for n c = 3 ( C F = 4 / ).In the following we compare the exactly known low-order coefficients of the perturbativeexpansion in the mass conversion relation with their expected asymptotic behaviour. Bydefinition (see (3.3)) the normalization N is given by N = lim n →∞ c n ( µ, µ m , m ( µ m )) c (as) n ( µ, m ( µ m )) . (3.6)We now determine N by evaluating the above expression for n = 1 , , , , for which c n ( µ, µ m , m ( µ m )) is known. To this end the result of [4] for the four-loop coefficient hasbeen expressed in terms of the strong coupling constant with n l flavours rather than n l + 1 ,since the asymptotic expression refers to the n l massless flavour theory. We also use un-published results [19] for the n l , n c , µ and µ m dependence of the four-loop coefficient. Inaddition to the ratio c n /c (as) n for n from 1 to 4 we consider the relative difference betweenthe N estimates performed using the third and the fourth order coefficients, defined as ∆ = 2 | c /c (as)3 − c /c (as)4 || c /c (as)3 + c /c (as)4 | . (3.7)The value of ∆ can be considered to be an estimate of how close is the third ordercoefficient to the asymptotic value. It is likely to be an overestimate of the deviation of the The perturbative coefficients r n in this reference are related to those employed here by r n = c n +1 .With this notation the number of loops contributing to c n is n . N .We report our results in table 1 for µ m = m and the three values µ = m , µ = m/ and µ = 2 m of the coupling renormalization scale. The number of colours has been fixed to n c = 3 in this table, and the number of light flavours was varied from a very large negativevalue (equivalent to the large- n l limit) up to n l = 10 . In columns 2 to 5 we show the ratios c n /c (as) n , that correspond to an estimate of N according to (3.6) for finite n . In the lastcolumn we give ∆ . The ± numbers account for the change in N due to the numericaluncertainty in the calculation of the exact four-loop conversion coefficient, which is about . on the n l independent term for µ = µ m = m ( µ m ) .We first discuss the result for µ = m . For n l very large and negative the value of N isclose to the one predicted by (3.5). The value of ∆ corresponds to a deviation of thethird order coefficient from the asymptotic result, which is indeed the case, and the fourth-order value is already much closer. As n l increases, the value of N decreases, reaching . and . for n l = 4 and 5, respectively, with a 9 and 13% variation when goingfrom the third to the fourth order coefficient. As n l increases, ∆ also increases, so thatfor n l above 7 the N values obtained from the third and fourth order coefficients differ byfactors of order 1. This behaviour is not unexpected: by increasing the number of lightflavours the first coefficient of the β function, b , decreases (it vanishes for n l = 33 / ),hence the renormalon dominance is delayed to higher orders. We shall comment further onthe n l dependence below.When considering different choices of the renormalization scale, we see that the µ = m/ case leads to larger variations than µ = 2 m . The large n l limit yields a value that isabout 10% higher than the exact result but the associated value ∆ ≈ is also large,indicating that the series is not as close to the asymptotic regime as for µ = m . For theinteresting cases n l = 4 and n l = 5 , ∆ is also more than a factor of two larger thanfor µ = m . Again, this behaviour is not unexpected. The coefficients c n depend onlyon logarithms of µ/m up to the ( n − th power. Eq. (3.3) shows that these logarithmsmust asymptotically exponentiate to µ/m , which clearly happens less efficiently at finiteorder when ln( µ/m ) is larger. Hence we expect the best approximation to the asymptoticbehaviour to occur when µ ≈ m . Fig. 1 shows that this is indeed the case for large − n l . Itfurther shows a plateau around µ ≈ m and a more rapid departure from the exact resultfor µ smaller then m than for larger µ , as also seen in table 1.We also determine the normalization N for different values of n c and show the resultfor ∆ in fig. 2. We generically find ∆ < . except in regions where b is small, where wedo not expect our method to work. Fig. 2 therefore demonstrates that the exact four-loopcoefficient indeed matches the asymptotic formula (3.1) in the expected range of n c and n l values, comprising those of physical interest.For the following a reliable determination of N and an estimate of its error is par-ticularly important for n c = 3 , n l = 5 , corresponding to the case of the top quark. We We may note that the contribution from sub-leading renormalon poles to c n is of order / n relative tothe leading one, but there is a further suppression for the case at hand due to a small numerical coefficient,at least in the large- n l limit, see [18]. /m = 1 n l c /c (as)1 c /c (as)2 c /c (as)3 c /c (as)4 ∆ − . . . . . −
10 0 . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . − . − . − . ± . . ± . µ/m = 0 . − . . . . . −
10 0 . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . − . ± . . ± . . − . − . − . ± . . ± . µ/m = 2 − . . . . . −
10 0 . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . . . . ± . . ± . . − . − . − . ± . . ± . Table 1 . The values of N obtained from the coefficients of the perturbative expansion up to thefourth order for several values of n l . Three values of the renormalization scale are considered. determine the error by varying the two renormalization scales independently, that is wevary µ/m ( µ m ) and µ m /m ( µ m ) independently between 0.5 and 2, compute N from c /c (as)4 as above, and determine the error on N from the maximal variation. The dependence of N on the two scale ratios is shown in fig. 3. With this definition our error estimate on N neither depends on the value of the heavy quark mass nor the one of the strong coupling.We find N = 0 . +0 . − . ( µ and µ m ) ± .
002 ( c ) . (3.8)As a further check we note that when the subleading term s ( s and s ) is removed in6 .2 0.5 1.0 2.0 5.00.60.81.01.21.41.61.82.0 Μ (cid:144) m N n l = - Figure 1 . The normalization N as a function of µ/m varied by a factor five around the centralscale. The dashed line shows the exact value e / / (3 π ) = 0 . ... . −10 −5 0 5 10 n l n c ⊗ . . . . . . . . . . Figure 2 . ∆ as a function of n c and n l , for µ = µ m = m . The cross corresponds to the caserelevant for top, i.e. n c = 3 and n l = 5 . (3.4), the central value changes very little to 0.4573 (0.4584). A similar method to determine the normalization of the leading pole mass renormalon,albeit without variations of µ m and n c , has already been used in [14]. More precisely, insteadof the four-loop pole mass considered here the three-loop static potential was employed toarrive at the best estimate, based on the fact that the pole mass and static potentialleading renormalon normalizations are rigorously related by a factor of − / . Their valuesare indeed in good agreement with ours, though deteriorating with increasing n l . Theapproach to the exact value for large negative n l was also observed in [14].The authors of [14] also determined the normalization N as a function of n l and noted Using the five-loop beta-function coefficient from [13], which appeared after this analysis was finished,allows us to compute the next sub-asymptotic term s in (3.4) (see appendix). We find that N changes bya negligible amount to 0.4606. igure 3 . The normalization N as a function of µ/m ( µ m ) and µ m /m ( µ m ) . that it tends to zero in the range n l = 12 . . . close to the conformal window. We confirmthis behaviour in our analysis, see Figure 4. To understand why the normalization of theleading renormalon is forced to be small in this n l region, we look at the explicit expressionof for c (as) n from (3.3) for n = 4 , c (as)4 = (2 b ) (1 + b )(2 + b )(3 + b ) (cid:18) s b + s (3 + b )(2 + b ) + · · · (cid:19) . (3.9)The region n l = 12 . . . is approximately centred around the value of n l , where b vanishes,hence b = b / (2 b ) becomes large. As soon as b ≫ n , where n is the order from which N is determined (here n = 4 ), the individual terms in the above expression behave as c (as)4 = (2 b ) n (cid:18) b b (cid:19) n (cid:16) s b + s b + · · · (cid:17) ∼ b ) n (cid:18) b + b + · · · (cid:19) , (3.10) - - - n l N Figure 4 . The normalization N (for n c = 3 , µ = µ m = m ) as a function of n l (black). The bluedots show /b . c (as)4 ∼ / (2 b ) n becomes very large, hence N must becomesmall to fit the given value of the exact four-loop coefficient c , and b) the series of sub-leading asymptotic terms s , s , etc. breaks down, hence the extracted value of N iscompletely unreliable. The smallness of N is therefore a technical artifact of the method,which ceases to be valid when b becomes large compared to n , and the question whether N is small in the conformal window cannot be answered. In fact, while small b makesrenormalon behaviour less relevant to low orders due to the diminished (2 b ) n factor, thereseems to be no reason why the normalization N should vanish when the theory becomesconformal non-perturbatively. m P – m conversion factor to all orders and the ultimate top polemass uncertainty In the following we use two methods to estimate the remainder of the mass conversion rela-tion beyond the exactly known four-loop accuracy and to estimate the intrinsic ambiguityof summing the assumed asymptotic expansion. The first relies on truncation of the expan-sion and an estimate of the minimal term. The second on Borel summation. We restrictourselves to the case of the top quark mass ( n c = 3 , n l = 5 ) and choose µ = µ m = m .We begin by writing m P ( n ) = m n X k =1 c k α ks ! , (4.1)where the coefficients are the exact ones up to the fourth order in α s , and determined fromthe asymptotic formula (3.4) (with normalization fitted to the fourth order term) for theterms of order 5 and higher. We would like to define the best value of m P as the value atwhich its increment with n is minimal. More precisely, we define ∆( n + 1 /
2) = m P ( n + 1) − m P ( n ) , (4.2)which is a decreasing function of n up to a certain value n beyond which it begins toincrease due to the renormalon divergence of the series expansion. By interpolating ∆ witha quadratic form in the three points n − / , n + 1 / , n + 3 / , we find its minimum at(generally non-integer) n min = n + 1 / − ∆( n + 3 / − ∆( n − / n + 3 /
2) + ∆( n − / − n + 1 / . (4.3)By interpolating linearly the value of m P ( n min ) between n and n + 1 we get m c P = m P ( n )(∆( n + 3 / − ∆( n + 1 / m P ( n + 1)(∆( n − / − ∆( n + 1 / n + 3 /
2) + ∆( n − / − n + 1 / (4.4)as the best value of the pole mass. We note that with this prescription, if ∆( n − /
2) =∆( n + 3 / , then m c P corresponds to ( m P ( n ) + m P ( n + 1)) / , as one would intuitivelyexpect, while for ∆( n − / ≫ ∆( n + 3 / ( ∆( n − / ≪ ∆( n + 3 / ), we obtain m P ( n + 1) ( m P ( n ) ). 9 ˜ c (as) j ˜ c (as) j α js . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . Table 2 . The coefficients ˜ c (as) j above the fourth order. Their value multiplied by the correspondingpower of α s = 0 . is also reported. We now estimate the correction to the top pole mass due to terms of order higher thanfour by δ (5+) m P = N µ X k =5 ˜ c (as) k α ks ( µ ) , (4.5)where ˜ c (as) j is defined in (3.4), and the barred sum represents the procedure we have justoutlined for the evaluation of the (divergent) sum. We report in table 2 the values of ˜ c (as) j beyond the fourth order term. Eq. (4.5) can be easily computed for any value of α s and µ and is well approximated by the second-order Taylor series around the reference value: δ (5+) m P = N µ × − .
604 + 14 . (cid:18) α s ( µ )0 . − (cid:19) + 9 . (cid:18) α s ( µ )0 . − (cid:19) ! . (4.6)For typical values of N ≈ . and µ ≈ GeV the formula is accurate at the sub-MeVlevel for a ± variation of the strong coupling constant.We now adopt the PDG value α s ( M Z ) = 0 . ± . , and take µ = m =163 . GeV for definiteness. With this input we find α s ( µ ) = 0 . for the (fiveflavour) strong coupling constant and . GeV for the top pole mass using the four-loopconversion formula. From the values reported in the table and the value of N given in (3.8)we obtain for the series remainder δ (5+) m P = 0 . +0 . − . ( N ) ± .
001 ( c ) ± .
011 ( α s ) ± . (ambiguity) GeV , (4.7)where we show the error due to the uncertainty in the normalization N , the four-loop coef-ficient c , and α s ( M Z ) . For the irreducible renormalon ambiguity we tentatively estimatethe size of the first omitted term by the value of ∆( n − / . For the top mass conversionfactor we find m c P /m = 1 . +0 . − . ( N ) ± . c ) ± . α s ) . (ambiguity) . (4.8)We also computed the change of the conversion factor under variations of µ/m and µ m /m ,simultaneously in the exact four-loop part and the remainder, accounting for the dependenceof N on µ and µ m (fig. 3). This leads to +0 . − . , which we do not include above, since itis strongly correlated with the uncertainty of the same order from N alone.In the second method we first compute the Borel transform of the asymptotic seriescoefficients ˜ c (as) in (3.3), which gives B [˜ c (as) ]( t ) = 1(1 − b t ) b + s b − b t ) b + s b ( b −
1) 1(1 − b t ) − b + . . . , (4.9)and then the Borel sum BS [˜ c (as) ]( α s ) = Z ∞ dt e − t/α s B [˜ c (as) ]( t ) . (4.10)Since the series is not Borel-summable due to the IR renormalon singularity at t = 1 / (2 b ) ,we define the sum as the principal value and estimate the ambiguity as the imaginary partof the integral when the contour is deformed into the upper complex plane, divided byPi. This procedure is known to usually give a reliable estimate [18], close to the sum tothe minimal term and the estimate of the summation ambiguity by the smallest term inthe series. The Borel sum can easily be computed analytically, since (with the contourdeformed into the upper complex plane) Z ∞ dt e − t/α s − b t ) γ = α s ( − b α s ) γ e − / (2 b α s ) Γ(1 − γ, − / (2 b α s )) , (4.11)where Γ( a, z ) denotes the incomplete Gamma function. The remainder of the mass conver-sion formula is obtained by subtracting the first four coefficients, resulting in δ (5+) m P = N µ BS [˜ c (as) ]( α s ( µ )) − X k =1 ˜ c (as) k α s ( µ ) k ! . (4.12)With parameter input as above, we find δ (5+) m P = 0 . +0 . − . ( N ) ± .
001 ( c ) ± .
010 ( α s ) ± . (ambiguity) GeV , (4.13)which is close to the result (4.7) from the previous method. For any value of α s and µ the result can again be determined accurately in the phenomenologically relevant regionaccording to the fit formula δ (5+) m P = N µ × − .
315 + 12 . (cid:18) α s ( µ )0 . − (cid:19) + 4 . (cid:18) α s ( µ )0 . − (cid:19) ! . (4.14)For the top mass conversion factor itself, we find m c P /m = 1 . +0 . − . ( N ) ± . c ) ± . α s ) ± . (ambiguity) . (4.15)11n this case, the scale variation is +0 . − . .The ultimate uncertainty on the top quark pole mass, which we identify with theambiguity of about 70 MeV, is smaller than estimates from the large- n l limit, because thenormalization N is smaller. We also note that dividing the imaginary part of the Borelintegral by Pi to obtain the ambiguity is a convention that has proven reliable in contextswhere the quantity in question is amenable of a non-perturbative definition [18]. This is notthe case for the pole mass, so that we cannot ask how well the divergent series approximatesthe exact, non-perturbative result. The point is rather that the pole mass can in principlebe used as a reasonable perturbative reference parameter, as long as computing additionalorders does not require increasingly larger shifts in the reference value. The dividing-by-Piconvention therefore appears reasonable, since, if the imaginary part of the Borel transformwas instead used to estimate the ambiguity, it would be almost as large as the known four-loop term, where the series is clearly still in the regime of decreasing terms. We observethat, in any case, even if the ambiguity were taken to be the imaginary part of the Borelintegral itself, the resulting estimate of would still be significantly below the uncertaintythat can conceivably be achieved at hadron colliders. The analysis assumed up to now that the five lighter quarks are massless. Since the typicalloop momentum at order α n +1 s is of order m t e − n in the regime where the series is dom-inated by the leading renormalon divergence, we expect internal quark mass effects fromthe bottom and charm quark to become more important in higher orders. Furthermore,the minimal term is attained when the typical loop momentum is of order Λ QCD , hence theambiguity should be determined by Λ -parameter Λ (3) QCD in the three-flavour scheme, exclud-ing the bottom and charm quark. In this section we estimate the effect of the finite bottomand charm quark mass on the top mass conversion factor and the ultimate uncertainty.The decoupling of internal quark loops from quarks with masses m q ≫ Λ QCD in therenormalon asymptotic behaviour was studied analytically and numerically in the large- n l limit [9]. The analysis showed that the asymptotic behaviour of the series in a theory with n l quarks of which n m are massive, approaches the series of the theory with n l − n m masslessquarks when both are expressed in terms of the MS coupling α ( n l − n m ) s ( m t ) in the n l − n m flavour scheme. Based on this observation it has been argued [14] that the bottom massconversion factor should be expressed in terms of α (3) s ( m b ) rather than the four-flavourcoupling α (4) s ( m b ) . For the two- and three-loop coefficients, for which the mass dependence Note that (4.14) in [9] does not apply term by term, but only as a transformation of the entire series.Term by term the approximation holds, if the right-hand side of (4.14) is multiplied by the factor exp πβ (3)0 ln m b m c ! / − α (3) s π ln m b m c ! n +1 , which follows from (4.16) in [9]. Here we put β (3)0 into the exponent rather than β (4)0 as in (4.16), since inthe presence of a massive quark, the leading singularity is slightly shifted to u = 1 / × β (4)0 /β (3)0 when u isdefined as − β (4)0 t .
12s known [20, 21], it was shown that this substitution indeed renders the charm mass effectalmost negligible.This procedure does not work for top, however, since the masses of the bottom andcharm quark are too small in relation to m t to express the entire series in terms of the four-or three-flavour coupling. Instead, we switch from the five- to the four-flavour scheme at theorder, where the typical internal loop momentum is of order m b , which is O ( α s ) , and fromthe four- to the three-flavour scheme at O ( α s ) . Since the mass effect is not known for c atthe four-loop order, and since c n beyond the four-loop order can only be estimated assumingdominance of the first renormalon (as done above), this implies the following procedure: (a)at two- and three-loops we include the known mass dependence, but c is approximated bythe massless value. For given top MS mass, this increases the top pole mass by 11 (2-loop)+ 16 (3-loop) MeV, adopting m b = 4 . GeV and m c = 1 . GeV. Since the c n increaseas n l decreases, the mass effect is also expected to be positive in higher orders. Henceapproximating c by its massless value underestimates the mass effect. (b) At five-loop, weuse c (as)5 [ α (4) s ( m t )] with c (as)5 determined as described in sect. 3, but with the normalization N m = 0 . and beta-function coefficients for the four-flavour theory, n l = 4 . (c) Beyondfive loops, the remainder and the ambiguity is calculated according to (4.12) (with obviousmodification, since we sum the terms from six rather than five loops), but with the three-flavour scheme coupling α (3) s ( m t ) and normalization N m = 0 . . Since the bottom andcharm quarks are not yet completely decoupled at the five- to seven-loop order, and since anextra quark flavour decreases the c n , we expect that (b) and (c) overestimate the mass effect,since the approximation assumes that bottom and charm are already decoupled completely.The sum of (b) and (c) adds another 53 MeV to the top pole mass, such that the total masseffect is estimated to be 80 MeV. Since the bottom is neither heavy enough to be decoupledin low orders, nor light enough to be ignored, where in both cases a massless approximationcan be justified, there is an inherent uncertainty in the above estimate. However, as arguedabove, the errors in the approximations are expected to go in opposite directions, hence weconsider (80 ± MeV a conservative estimate of the internal bottom and charm quarkmass effect on the top pole mass. The 30 MeV error estimate arises from an estimate of theneglected mass effect on c by extrapolation from the known lower orders. We have alsochecked that the approximation described here works well in models for the series inspiredby the large- n l limit.Including the internal mass effect into the massless results (4.13) and (4.15), we obtainfor the series remainder from the five-loop order δ (5+) m P = 0 . +0 . − . ( N ) ± .
030 ( m b,c ) ± .
009 ( α s ) ± . (ambiguity) GeV , (5.1)where we now dropped the negligible uncertainty from the massless four-loop coefficient c . Apart from the shift of the value of δ (5+) m P the ambiguity has increased to 108 MeV,which is mainly due to the fact that Λ (3) QCD is larger than Λ (5) QCD . Note that the ambiguity isindependent of the precise value of the bottom and charm mass, as long as m b , m c ≫ Λ QCD .This also implies that it is the same for any heavy quark, including the bottom quark,since it depends only on the infrared properties of the theory, which is QCD with threeapproximately massless flavours. 13or the top mass conversion factor itself, we find m c P /m = 1 . +0 . − . ( N ) ± . m b,c ) ± . α s ) ± . (ambiguity) . (5.2)The scale variation remains as for (4.15). We adopt (5.1) and (5.2) as our final results. Giventhe MS mass, the top quark pole mass is determined by this relation with an accuracy of1.1 per mil, half of which is due to the irreducible uncertainty of the relation itself. We employed the four-loop coefficient in the pole- MS quark mass relation, which has re-cently become available [4], and knowledge of the leading asymptotic behaviour of theseries expansion of the mass conversion factor [7] to estimate the remainder of the seriesfrom terms above the four-loop order and the intrinsic ambiguity due to the asymptoticnature of the series. For the case of the top quark we find about MeV for the for-mer, including an estimate of the effect of the internal bottom and charm quark mass,and
MeV for the ambiguity, which also represents the ultimate precision that can beobtained for the pole mass. The ambiguity of
MeV is far below the accuracy that canconceivably be achieved at the Large Hadron Collider, but larger than the one foreseen intheoretical and experimental studies [22, 23] of a scan of the top pair production thresholdat a high-energy e + e − collider. In this case the pole mass ceases to be a useful concept andother mass definitions must be employed. Acknowledgements
This work is supported by the BMBF grants 05H15WOCAA (MB) and 05H15VKCCA(MS). PM was supported in part by the EU Network HIGGSTOOLS PITN-GA-2012-316704. MB thanks the Kavli Institute for Theoretical Physics, Santa Barbara, for hospi-tality while this work was completed.
A Summary of formulae
In this Appendix, in order to make contact with the notation of [7, 18], we define the QCDbeta-function as β ( α s ) = µ ∂α s ( µ ) ∂µ = β α s + β α s + . . . , (A.1)With this convention β = − (11 n c / − n l / / (4 π ) , while in the main text we used b i = − β i > (for small n l ).We adopt the MS scheme with n l massless quark flavours. (Theheavy quark whose mass is considered here is decoupled.) The constants that appear in(3.4) are given by [7, 18] b = − β / (2 β ) and s = (cid:18) − β (cid:19) (cid:18) − β β + β β (cid:19) , (A.2)14 = (cid:18) − β (cid:19) (cid:18) β β + β β − β β β − β β β + β β + β β (cid:19) , (A.3) s = (cid:18) − β (cid:19) (cid:18) − β β − β β − β β + β β β + 3 β β β + β β β − β β β − β β β − β β β − β β β + β β − β β + β β β + β β (cid:19) . (A.4)Note that we have corrected some misprints in the expression for b and s given in [18](eqs. (5.91) and (5.92)) as already noted in [10]. The result for s was not given explicitlyin [18]. References [1]
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