The MSR Mass
and the Renormalon Sum Rule
We provide a detailed description and analysis of a low-scale short-distance mass scheme, called the MSR mass, that is useful for high-precision top quark mass determinations, but can be applied for any heavy quark . In contrast to earlier low-scale short-distance mass schemes, the MSR scheme has a direct connection to the well known mass commonly used for high-energy applications, and is determined by heavy quark on-shell self-energy Feynman diagrams. Indeed, the MSR mass scheme can be viewed as the simplest extension of the mass concept to renormalization scales . The MSR mass depends on a scale that can be chosen freely, and its renormalization group evolution has a linear dependence on , which is known as R-evolution. Using R-evolution for the MSR mass we provide details of the derivation of an analytic expression for the normalization of the renormalon asymptotic behavior of the pole mass in perturbation theory. This is referred to as the renormalon sum rule, and can be applied to any perturbative series. The relations of the MSR mass scheme to other low-scale short-distance masses are analyzed as well.
Bhopal Bypass Road, Bhopal 462066, IndiaUniversity of Vienna, Faculty of Physics,
Boltzmanngasse 5, A-1090 Wien, AustriaDepartamento de Física Teórica II, Universidad Complutense de Madrid (UCM),
E-28040 Madrid, SpainCenter for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
Achieving higher precision in theoretical predictions in the framework of quantum chromo dynamics (QCD) is one of the main goals in high-energy physics and an essential ingredient in the indirect search for physics beyond the Standard Model. In this endeavor accurate determinations of the masses of the heavy charm, bottom and top quarks play an important role since they enter the description of many observables that are employed in consistency tests of the Standard Model and in the exploration of models of new physics. Because quark masses are formally-defined renormalized quantities and not physical observables, the quantities from which the heavy quark masses are extracted need to be computed in perturbative QCD to high order. Among the most precise recent high-order analyses to determine the heavy quark masses are QCD sum rules and the analysis of quarkonium energies for the charm and bottom quark masses Dehnadi:2011gc (); Bodenstein:2011ma (); Bodenstein:2011fv (); Hoang:2012us (); Chakraborty:2014aca (); Colquhoun:2014ica (); Beneke:2014pta (); Ayala:2014yxa (); Dehnadi:2015fra (); Erler:2016atg () and the top pair production threshold cross section at a future lepton collider for the top quark mass Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa (). Over time all of these analyses have been continuously updated and improved by computations of new QCD corrections, and more are being designed and studied currently to also allow for more precise determinations of the top quark mass from available LHC data Czakon:2013goa (); Khachatryan:2016mqs (); Aaboud:2016pbd (); Czakon:2016ckf (); Alioli:2013mxa (); Frixione:2014ala (); Chatrchyan:2013boa (); Kharchilava:1999yj ().
In all the analyses of Refs. Dehnadi:2011gc (); Bodenstein:2011ma (); Bodenstein:2011fv (); Hoang:2012us (); Chakraborty:2014aca (); Colquhoun:2014ica (); Beneke:2014pta (); Ayala:2014yxa (); Dehnadi:2015fra (); Erler:2016atg (); Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa () the use of short-distance mass schemes was essential to achieve a well-converging perturbative expansion and a precision in the mass determination well below the hadronization scale MeV. The heavy quark pole mass , which is the perturbation theory equivalent of the rest mass of an on-shell quark, on the other hand, leads to a substantially worse perturbative behavior due to its linear infrared-sensitivity, also known as the renormalon problem Bigi:1994em (); Beneke:1994sw (), and was therefore not adopted as a relevant mass scheme for analyses where a precision better than could be achieved. Nevertheless, the pole mass still served as an important intermediate mass scheme during computations because it determines the partonic (but unphysical) poles of heavy quark Green functions. Typical short-distance quark mass schemes which have been employed were the renormalization-scale dependent mass and so-called low-scale short-distance masses such as the kinetic mass Czarnecki:1997sz (), the potential-subtracted (PS) mass Beneke:1998rk (), the 1S mass Hoang:1998ng (); Hoang:1998hm (); Hoang:1999ye (), the renormalon-subtracted (RS) mass Pineda:2001zq () or the jet mass Jain:2008gb (); Fleming:2007qr (). The basic difference between the mass to the low-scale short-distance mass schemes is that the perturbative coefficients of its relation to the pole mass scale linearly with the heavy quark mass, , while for the low-scale short-distance mass schemes the corresponding series scales linearly with a scale . This feature enables the low-scale short-distance quark mass schemes to be used for predictions of quantities where the heavy quark dynamics is non-relativistic in nature and fluctuations at the scale of are integrated out. This is because radiative corrections to the mass in such quantities involve physical scales much smaller than . One very prominent example in the context of top quark physics is the non-relativistic heavy quarkonium dynamics inherent to the top-antitop pair production cross section at threshold at a future lepton collider Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa (), where the most important dynamical scale is the inverse Bohr radius GeV . On the other hand, the mass is a good scheme choice for quantities that involve energies much larger than , such as for high-energy total cross sections, or when the massive quark causes virtual and off-shell effects. This is because in such cases the heavy quark mass yields corrections that either scale with positive or negative powers of such that QCD corrections associated with the mass have a scaling that is linear in as well. The difference between the mass and the low-scale short-distance masses is most important for the case of the top quark because in this case the difference between and the dynamical low-energy scales can be very large numerically.
For the top quark mass there are excellent prospects for very precise measurements in low-scale short-distance schemes such as the PS mass or the 1S mass from the top-antitop threshold inclusive cross section at a future lepton collider Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa (). Current studies indicate that a precision well below MeV can be achieved accounting for theoretical as well as experimental uncertainties Seidel:2013sqa (); Horiguchi:2013wra (); Vos:2016til (). Currently, the most precise measurements of the top quark mass come from reconstruction analyses at the LHC Khachatryan:2015hba (); Aaboud:2016igd () and the Tevatron Tevatron:2014cka () and have uncertainties at the level of MeV or larger. Moreover, the mass is obtained from multivariate fits involving multipurpose Monte Carlo (MC) event generators and thus represents a determination of the top quark mass parameter contained in the particular MC event generator. Recently, a first high-precision analysis on how the MC top quark mass parameter can be related to a field theoretically well-defined short-distance top quark mass was provided in Refs. Butenschoen:2016lpz (); Hoang:2017kmk () and general considerations on the relation were discussed in Ref. Hoang:2008xm (); Hoang:2014oea (). For the analysis, hadron level predictions for the 2-jettiness distribution Stewart:2010tn () for electron-positron collisions and QCD corrections together with the resummation of large logarithms at next-to-next-to leading order Fleming:2007qr (); Fleming:2007tv (); Hoang:2007vb () were employed. Since the 2-jettiness distribution is closely related to the invariant mass distribution of a single reconstructed top quark, the relevant dynamical scales inherent to the problem are governed by the width of the mass distribution which amounts to only about GeV in the peak region of the distribution where the sensitivity to the top mass is the highest. Interestingly, as was shown in Ref. Butenschoen:2016lpz (), the dynamical scales increase continuously considering the 2-jettiness distribution further away from the peak. In the analysis of Butenschoen:2016lpz () the MSR mass scheme was employed which depends on a scale and for which the dependence on is described by a renormalization group flow such that can be continuously adapted according to which part of the distribution is predicted. Other applications of the MSR mass using a flavor number dependent evolution in to account for the mass effects of lighter quarks were given in Ref. Hoang:2017btd (); Mateu:2017hlz (). In contrast to the -dependent mass , which evolves only logarithmically in , the MSR mass has logarithmic as well as linear dependence on .
The MSR mass scheme was succinctly introduced in Ref. Hoang:2008yj () and discussed conceptually in Ref. Hoang:2014oea (), but a detailed discussion has so far not been provided. A key purpose of this paper is to provide sufficient details such that phenomenological MSR mass analyses, such as the results of Ref. Butenschoen:2016lpz (), can be easily related to other common short-distance mass schemes that are being used in the literature.
The definition of the MSR mass given by the perturbative series for the MSR-pole mass difference is obtained directly from the -pole mass relation and is therefore the only low-scale short-distance mass suggested in the literature that is derived directly from on-shell heavy quark self-energy diagrams just like the mass.111The name ‘MSR mass’ arises from a combination of the letters ‘MS’ standing for the close relation to the mass and the letter ‘R’ standing for R-evolution. The MSR mass thus automatically inherits the clean and good infrared properties of the mass. Furthermore, by construction, the MSR mass matches to the mass for and is known to the same order as the series of without any further effort, which is currently from the results of Refs. Tarrach:1980up (); Gray:1990yh (); Melnikov:2000qh (); Chetyrkin:1999ys (); Chetyrkin:1999qi (); Marquard:2007uj (); Marquard:2015qpa (); Marquard:2016dcn (). As already argued in Refs. Hoang:2008yj (); Hoang:2008xm (), the MSR mass can therefore be considered as the natural modification of the “running” mass scheme concept for renormalization scales below , where the logarithmic evolution of the regular mass is known to be unphysical.
Since the MSR mass is designed to be employed for scales , it can be useful – for applications where a clean treatment of virtual massive-flavor effects is important – to integrate out the virtual effects of the massive quark from the MSR mass definition. We therefore introduce two types of MSR masses, one where the virtual effects of the massive quark are integrated out, called the natural MSR mass, and one where these effects are not integrated out, called the practical MSR mass. The difference between these two versions of the MSR mass is quite small and very well behaved for all values in the perturbative region, and the practical definition should be perfectly fine for most phenomenological applications. But the natural definition has conceptual advantages as its evolution for scales does not include the virtual effects of the massive quark , which is conceptually cleaner since these belong physically to the scale .
We note that the R-evolution concept of a running heavy quark mass scheme for scales elaborated in Ref. Hoang:2008yj () has already been suggested a long time ago in Refs. Voloshin:1992wg (); Bigi:1997fj (). The R-evolution equation we discuss for the MSR mass was already quoted explicitly for the renormalization group evolution of the kinetic mass Czarnecki:1997sz () at in these references, but the conceptual implications of R-evolution and its connection to the renormalon problem in the perturbative relations between short-distance masses and the pole mass were first studied systematically in Ref. Hoang:2008yj (). The second main purpose of this paper is to give further details on R-evolution and also to discuss its relation to the Borel transformation focusing mainly on the case of the MSR mass. We note that the concept of R-evolution is quite general and can in principle be applied to any short-distance mass which depends on a variable infrared cutoff scale (such as the PS and the RS masses) or to cutoff-dependent QCD matrix elements with arbitrary dimensions. In fact, R-evolution has already been examined and applied in a number of other applications which include the factorization-scale dependence in the context of the operator product expansion Hoang:2009yr (), the scale dependence of the non-perturbative soft radiation matrix element in high-precision determinations of the strong coupling from event-shape distributions Abbate:2010xh (); Abbate:2012jh (); Hoang:2014wka (); Hoang:2015hka (), even accounting for the finite mass effects of light quarks Gritschacher:2013tza (); Pietrulewicz:2014qza () and hadrons Mateu:2012nk (); Hoang:2014wka ().
The basic feature of the R-evolution concept is that for the difference of MSR masses at two scales, , its linear dependence on the renormalization scale provides, completely within perturbation theory, a resummation of the terms in the asymptotic series associated to the pole-mass renormalon ambiguity to all orders. The R-evolution then resums the factorially growing terms in a systematic way that is -renormalon free and, at the same time also sums all large logarithms that arise if and are widely separated. This cannot be achieved by more common purely logarithmic renormalization group equations, but is fully compatible with a Wilsonian renormalization group setup. We note that the summations carried out by the R-evolution was achieved prior to Ref. Hoang:2008yj () for the RS mass in Bali:2003jq () (see also Ref. Campanario:2005np ()). Their method (and the RS mass) is based on using an approximate expression for the Borel transform function. The summation for a difference of RS masses (for scales and ) is obtained by computing the inverse Borel integral over the difference of the two respective Borel functions. This method and R-evolution lead to consistent results, but the R-evolution does not rely on the knowledge of the Borel functions.
The essential and probably most interesting conceptual feature of the perturbative series of the R-evolution equations is that it provides a systematic reordering of the terms in the asymptotic series associated to the renormalon ambiguity in leading, subleading, subsubleading, etc. contributions. So using the analytic solution of the R-evolution equations allows one to derive analytically (i.e. without any numerical procedure or modeling) the Borel-transform of a given perturbative series from the perspective that it carries an renormalon ambiguity. As a result one can rigorously derive an analytic expression for the normalization of the non-analytic terms in the Borel transform that are characteristic for the renormalon. The analytic result for this normalization factor was already given and discussed in Ref. Hoang:2008yj (), but no details on the derivation were provided. We take the opportunity to show the details of the derivation here. We call the analytic result for the normalization of the renormalon ambiguity the sum rule, because it can be quickly applied to any given perturbative series. To demonstrate the use and the high sensitivity of the renormalon sum rule we apply it also to a number of other cases, pointing out subtleties in its application to avoid inconsistencies and misinterpretations of the results.
We note that also other methods to determine the normalization factor have been used. In Ref. Pineda:2001zq () it was determined from a computation of the residue of the Borel transform of the series following a proposal in Ref. Lee:1996yk (). This approach, which we call Borel method can also be carried out analytically and provides the correct result, but has been observed to converge very slowly. We can identify the reason for this analytically from the solutions for the R-evolution equations, and we also discuss the connection of this method to our sum rule based on explicit analytic expressions. In Ref. Bali:2013pla () the normalization factor was computed taking the ratio of the -th term of the series to the asymptotic behavior. This ratio method converges very fast and provides results very similar to the sum rule. Recently, the ratio method was applied in Ref. Beneke:2016cbu (), accounting for the corrections to the pole- mass relation Marquard:2015qpa (); Marquard:2016dcn (). We show that our sum rule provides results that are in full agreement with the ones obtained in Ref. Beneke:2016cbu () and also leads to very similar uncertainties.
The paper is organized as follows: In Sec. 2 we provide the definition of the natural and practical MSR masses, and , based on the perturbative series of the -pole mass relation , and we also analyze the difference between these two MSR masses. This section provides the conventions we use for the coefficients of perturbative series, but it can otherwise be skipped by the reader not interested in the MSR masses. In Sec. 3 we present the R-evolution equations which describe the scale dependence of the MSR masses and we also show explicitly how the solutions of the R-evolution equations sum large logarithms together with the high-order asymptotic series terms related to the renormalon. We in particular show for the top quark mass under which conditions the use of the R-evolution equations and its resummation is essential and superior to renormalon-free fixed-order perturbation theory, which does not sum any large logarithms. To our knowledge, such an analysis has not been provided in the literature before. We also point out that the solution of the R-evolution equations is intrinsically related to carrying out an inverse Borel transform over differences of functions in the Borel plane such that the singularities related to the renormalon cancel. In Sec. 4 we present the analytic derivation of the renormalon sum rule and demonstrate its utility by a detailed analysis concerning the normalization of the renormalon ambiguity in the series for the difference of the pole mass and the MSR masses. The derivation of the sum rule allows to derive a new alternative expression for the high-order asymptotic behavior of a series that contains an renormalon which we discuss as well. To demonstrate the high sensitivity of the sum rule and to explain its consistent (and inconsistent) application we discuss its strong flavor number dependence and apply it to the massive quark vacuum polarization function, the series for the PS mass-pole mass difference, the QCD -function, and the hadronic R-ratio. This section can be bypassed by the reader not interested in applications of the sum rule, but we note that Sec. 4.5.3 discusses implications for the PS mass that are relevant for Sec. 5 and may be important for high-precision top quark mass determinations. Some subtle issues in the relation of the MSR masses to the PS, 1S and masses are discussed in Sec. 5. Finally, we conclude in Sec. 6. The paper also contains two appendices. In App. A we specify our convention for the QCD -function coefficients and present a number of expressions and formulae for coefficients, quantities and matching relations that arise in the discussion of R-evolution, the renormalon and on various mass definitions throughout this paper. In App. B we provide details on the relation of the Borel method and our sum rule method to determine the normalization of the renormalon ambiguity of the pole mass. Finally, in App. C we quote the coefficients that define the PS and the 1S masses for the convenience of the reader and also show how the MSR masses can be obtained from a given value of the 1S mass in the non-relativistic and -expansion counting scheme Hoang:1998hm (); Hoang:1998ng ().
2 MSR Mass Setup
2.1 Basic Idea of the MSR Mass
The mass serves as the standard short-distance mass scheme for many high-energy applications with physical scales of the order or larger than the mass of the quark . It relies on the subtraction of the divergences in the common scheme in the on-shell self-energy corrections calculated in dimensional regularization. Despite the fact that it is an unphysical (i.e. theoretically designed) mass definition, it is infrared-safe and gauge invariant to all orders Tarrach:1980up (); Kronfeld:1998di () and its series relation to the pole mass thus serves as the cleanest way to precisely quantify the renormalon ambiguity of the pole mass. The relation of to the pole mass in the approximation that the masses of all quarks lighter than are zero reads
where stands for the strong coupling that renormalization-group (RG) evolves with active flavors, see Eq. (67). The coefficients at are known analytically from Refs. Tarrach:1980up (); Gray:1990yh (); Chetyrkin:1999ys (); Chetyrkin:1999qi (); Melnikov:2000qh (); Marquard:2007uj (). The coefficient was determined numerically in Refs. Marquard:2015qpa (); Marquard:2016dcn (), and the quoted numerical uncertainties have been taken from Ref. Marquard:2016dcn (). Using the method of Ref. Kataev:2015gvt () the uncertainties of the -dependent terms may be further reduced. Using renormalon calculus Bigi:1994em (); Beneke:1994sw (); Beneke:1998ui () one can show that the high-order asymptotic behavior series of Eq. (1) has an ambiguity of order , which depends on the number of massless quarks (indicated by the superscript) but is independent of the actual value of .
A coherent treatment of the mass effects of lighter quarks is beyond the scope of this paper, and we therefore use the approximation that all flavors lighter than are massless. These mass corrections come from the insertion of massive virtual quark loops in the self-energy Feynman diagrams and start at . At this order and at the mass corrections from the virtual massive quark loops have been calculated analytically for all mass values in Ref. Gray:1990yh () and Bekavac:2007tk (), respectively. The dominant linear mass corrections at were determined in Ref. Hoang:2000fm (). At and the mass corrections are not yet known, but the corrections in the limit of large virtual quark masses are encoded in the ultraheavy flavor threshold matching relations of the RG-evolution at scales above Chetyrkin:1997un ().
The idea of the MSR mass is based on the fact that the ambiguity of the perturbative series on the RHS of Eq. (1) does not depend on the value , as already mentioned above. This is an exact mathematical statement within the context of the calculus for asymptotic series and means that we can replace the term by the arbitrary scale on the RHS of Eq. (1) and use the resulting perturbative series as the definition of the -dependent MSR mass scheme. It was pointed out in Ref. Hoang:2014oea () that, for a given value of , one can also interpret the MSR mass field theoretically as having a mass renormalization constant that contains the on-shell self-energy corrections of the pole mass only for scales larger than . In other words, the pole mass and the MSR mass at the scale differ by self-energy corrections from scales below : while the pole mass absorbs all self-energy corrections for quantum fluctuations up to scales , the MSR mass at the scale absorbs only self-energy corrections between and . Since the pole mass renormalon problem is related to the self-energy corrections from the scale , this explains why the MSR mass is a short-distance mass. In this illustrative context the mass absorbs no self-energy corrections up to the scale . Since the scale is variable, the MSR mass can serve as a short-distance mass definition for applications governed by different physical scales and thus can also interpolate between them. Since the MSR mass is expected to have applications primarily for , it is further suitable to change the scheme from dynamical flavors, which includes the UV effects of the quark , to a scheme with dynamical flavors. This can be achieved in two ways, either by simply rewriting in terms of , or by integrating out the virtual loop corrections of the quark . This results in two different ways to define the MSR mass, where we call the former the practical MSR mass and the latter the natural MSR mass, either one having advantages depending on the application.
We note that the notion of a scale-dependent short-distance mass which was first suggested in Refs. Voloshin:1992wg (); Bigi:1997fj () has also been adopted for the kinetic Czarnecki:1997sz (), the PS Beneke:1998rk (), RS Pineda:2001zq () and jet masses Jain:2008gb (); Fleming:2007tv (). However, none of these short-distance masses is defined directly from the on-shell self-energy diagrams of the massive quark such as the MSR mass. This has a number of advantages, for example when discussing heavy flavor symmetry properties in the pole- mass relation of different heavy quarks.
2.2 Natural MSR Mass
The natural MSR mass definition is obtained by integrating out the corrections from the heavy quark virtual loops in the self-energy diagrams of the massive quark , such that its relation to the pole mass reads
where the coefficients are given in Eq. (2). The natural MSR mass only accounts for gluonic and massless quark corrections, and has a non-trivial matching relation to the mass. The matching between the natural MSR mass and the mass can be derived from the relation [ ]
and will be discussed in more detail in Sec. 5.3.
We note that, formally, the natural MSR mass (as well as the practical MSR mass discussed in the next subsection) agrees with the pole mass in the limit . However, taking this limit is ambiguous as it involves evolving through the Landau pole of the strong coupling and dealing with its non-perturbative definition for . This issue is a manifestation of the renormalon problem of the pole mass.
2.3 Practical MSR Mass
The practical MSR mass definition is directly related to the -pole perturbative series of Eq. (1). To obtain its defining series one rewrites as a series in in Eq. (1) using the matching relation given in Eq. (73) and then replaces by , obtaining
The practical MSR mass still accounts for the virtual corrections from the massive quark Q with an evolving mass and has the convenient feature that it agrees with the mass at the scale of the mass to all orders in perturbation theory [ ]:
The formula for the difference of the natural and practical MSR masses at the same scale up to reads
In Fig. 1 the difference between the natural and the practical MSR top quark masses is shown for between and GeV (here ).222Throughout this article we use and GeV. The numerical difference between these two masses is quite small. The natural MSR mass is larger than the practical MSR mass and the difference increases with reaching about MeV at GeV. The error bands reflect variations of the renormalization scale in between and , showing very good convergence, exhibiting a perturbative error of MeV for GeV and below MeV for GeV due to missing terms of and higher. This indicates that the different way how the natural and practical MSR masses treat the virtual massive quark effects does not reintroduce any infrared sensitivity, as is expected since the mass of the virtual quark provides an infrared cutoff. The numerical uncertainties in the correction are below the level of MeV and negligible. Note that the difference between the natural and the practical MSR masses at the common scale starts at and that the uncertainty band from scale variation is an underestimate at this lowest order. However, the series results and error bands at show good behavior and convergence. In Ref. Butenschoen:2016lpz () the practical MSR mass was employed, but the numerical difference to the natural MSR mass is subdominant to the uncertainties obtained in the analysis there.
In the rest of the paper we will simply use the notation of the MSR mass with the definition when the difference between the natural and practical definitions and the value of are insignificant but we will specify explicitly our use of the practical or the natural MSR masses (or any other mass scheme) and the massless flavor number for any numerical analysis.
The dependence of the MSR mass on the scale is described by the R-evolution equation Hoang:2008yj (), which is derived from the logarithmic derivative of the defining equations (3) and (5) and using that the pole mass is independent:
The overall minus sign on the RHS of Eq. (9) indicates that the MSR mass always decreases with . Note that this equation applies to all MSR schemes and we have therefore suppressed the superscript on the ’s. The crucial feature of the R-evolution equation is that it is free from the ambiguity contained in the series that relates the MSR mass to the pole mass because the ambiguity is -independent. This is directly related to the fact that for determining the R-evolution equation also the overall linear factor of on the RHS of Eqs. (3) and (5) has to be accounted for. Therefore the R-evolution equation does not only have a logarithmic dependence on , as common to usual renormalization group equations (RGEs), but also a linear one. Both of these issues are actually tied together conceptually. The numerical expressions for the coefficients for the natural and practical MSR masses are given explicitly in Eqs. (77) and (78). We implement renormalization scale variation in the R-evolution equation by simply expanding in Eq. (9) as a series in and by varying , typically in the range . In principle one may also consider varying the boundaries of integration, as it is common for usual RGEs, but only the former way of implementing scale variations in the R-evolution leads to variations of the scale solely in logarithms, which is the standard used for the usual logarithmic RGEs.
By solving the R-evolution equation one sums, at the same time and systematically, the asymptotic renormalon series as well as the large logarithmic terms in to all orders in a manner free from the renormalon:
It is straightforward to solve the R-evolution equation numerically and it shows very good perturbative stability even for low values of very close to the Landau pole Hoang:2009yr () in the perturbative strong coupling. Details of how to solve the R-evolution equations analytically have already been given in Hoang:2008yj () and shall not be repeated here.
It is instructive to briefly discuss what the solution of the R-evolution achieves by considering the difference of the MSR mass, , in the context of fixed-order perturbation theory (FOPT), where it is well-known that the renormalon ambiguity contained in the series for and the series for only cancel if one expands in with a common renormalization scale . This is nicely illustrated in the /LL (leading log) approximation where the pole-MSR mass relation has the all order form
The series by itself is divergent and not summable, but
is easily seen to be convergent. In the context of FOPT, when the sum over is truncated, the unavoidable appearance of large logarithms for let’s say may degrade the convergence and cause sizable perturbative uncertainties. Due to the additional linear dependence on and , as shown in Eq. (13), these logarithms cannot be summed by common logarithmic renormalization group (RG) equations. The same type of logarithms also appear for example in the relation of any other low-scale short-distance mass to the mass and their effects can be significant particularly for the top quark. By solving the R-evolution equation one sums, at the same time and systematically, the asymptotic terms in the renormalon series as well as the large logarithmic terms in to all orders in a manner free from the renormalon. It is again instructive to see how this is achieved in the /LL approximation of Eq. (12), which explicitly shows the factorial growth of the perturbative series. When calculating the derivative to get the R-evolution equation, the whole series collapses exactly (i.e. without any truncation!) to
which is the one-loop version of Eq. (9). Moreover, the exact solution of the R-evolution equation at this order
can be easily seen to be exactly equal to the RHS of Eq. (13) which sums the renormalon series and the large logarithms at the same time into a convergent series.
Conceptually, the solution of the R-evolution equation is directly related to the Borel space integral over the Borel transform for the series for . Since this has not been shown in Hoang:2008yj () we briefly outline this calculation here at the /LL level. Starting from Eq. (15) one can shuffle the integration over into an integral over by using the QCD -function and the relation . Using the variable one can then rewrite the integral as [ ]
where the two integrals in the last line are just the difference of the MSR masses at to the pole mass, and the pole mass ambiguity is encoded in the singularity at , which arises because ,
Upon changing variables to the Borel plane parameter and writing in terms of and in both integrals, this gives
is the well-known Borel transform with respect to of the /LL series in Eq. (12). In Eq. (18) the singular and non-analytic contributions contained in the individual Borel functions cancel and the integral becomes ambiguity-free.
To illustrate the impact of using R-evolution compared to using FOPT we show in Fig. 2 the difference of natural MSR masses for in fixed-order perturbation theory (FOPT) and with R-evolution. The curves in Fig. 2 show for GeV in FOPT for the common renormalization scale between and at 1 loop (cyan), 2 loop (green), 3 loop (blue) and 4 loops (red). We see a good convergence for around , but a deterioration of the series when gets closer to either or . For the series even gets out of bounds and breaks down completely. If one uses scale variation as an estimate of the remaining perturbative error, one therefore obtains a significant dependence on the choice of the lower bound of the variation, and one has no other choice than to abandon in an ad hoc manner scales closer to to estimate the scale variation error. The curves in Fig. 2 show for GeV from numerically solving the R-evolution equation as a function of the renormalization scale parameter between and . The color coding for the order of the R-evolution equation used for the evaluation is the same as for Fig. 2. As explained below Eq. (9), the parameter is the renormalization scaling parameter in the R-evolution equation which determines by how much the scale in differs from the scale . Thus a variation between and means that in the solution of the R-evolution equations scales between and are covered at each value of along the evolution, which in this case includes scales between and GeV. Comparing the curves in Fig. 2 and 2 we see that the renormalization scale variation in the R-evolved results is much smaller than the one of FOPT. For the FOPT result with scale variation between – which we pick by hand – and we obtain GeV at (1, 2, 3, 4) loops. Using R-evolution with variation between and we obtain GeV which is fully compatible with the FOPT result, but shows more stability and smaller errors. It is also quite instructive to see that using R-evolution the 3-loop result is significantly closer to the 4-loop result than the corresponding 3-loop FOPT result. The results show that for employing R-evolution to calculate MSR mass differences is clearly superior to FO perturbation theory.
To compare to a situation where the scales and are of similar size we have also shown in Figs. 2 and 2 the results for in FOPT and from R-evolution for GeV. Here the results from both approaches are completely equivalent showing that the logarithm is not large and the summation of the renormalon contributions from higher orders only constitutes very small effects. Furthermore using renormalization scales close to or in FOPT is not problematic. Numerically, using FOPT with scale variations between and we obtain GeV at loops, while using R-evolution with variations between and we obtain GeV. We find that FOPT and R-evolution give equivalent results even for GeV, and that the use of R-evolution is essential for . Overall we see that, if and are of similar size, FO perturbation theory and R-evolution lead to equivalent results, but that it is in general safer to use R-evolution. So the situation is very similar to the one we encounter when considering the relation of the strong coupling for two different renormalization scales.
We note that the possibility to sum the renormalon-type logarithms displayed in Eq. (13) by considering the Borel integral over the difference of Borel transforms as shown in Eq. (18) was pointed out already in Ref. Bali:2003jq () prior to Ref. Hoang:2008yj (). However, this exact equivalence [ via a transformation of variables as given below Eq. (17) ] of R-evolution and the method using the integration over Borel transform differences can only be analytically shown at the /LL approximation. Beyond that, both approaches sum up the same type of logarithms but differ in subleading terms. Numerically, both approaches converge to the same result and have comparable order-by-order convergence. From a practical point of view, however, the concept of R-evolution may be considered more general. This is because R-evolution can be applied directly to any series having the form of (3) or (5) while using the Borel integration method requires that the corresponding Borel transforms are known or constructed beforehand. For general series, such as for the difference of MSR masses as discussed above, this is not possible without making additional approximations. In practice, the approach of Ref. Bali:2003jq () to sum the renormalon-type logarithms has therefore only been applied for series (referred to as RS-schemes) which were explicitly derived from a given expression for the Borel transform.
4 Analytic Borel Transform and Renormalon Sum Rule
Using the solution of the R-evolution equation it is possible to derive, analytically and rigorously, an expression for the Borel transform of the MSR-pole mass relation. This Borel transform is designed to focus on the singular contributions that quantify the renormalon of the pole mass. This result was already quoted in the letter Hoang:2008yj () where, however, no details on the derivation could be given due to lack of space. In the following we provide these details on how to obtain the analytic result for the normalization of the singular terms. The analytic results for the normalization can be applied to other perturbative series as a probe of renormalon ambiguities, and we therefore call it the renormalon sum rule. This sum rule was first given in Ref. Hoang:2008yj (), and is very sensitive to even subtle effects if corrections are known. We apply the sum rule to obtain an updated determination of the size of the pole mass ambiguity, accounting for the results of Refs. Marquard:2015qpa (); Marquard:2016dcn () which became available recently but were unknown when Ref. Hoang:2008yj () appeared. To demonstrate the sum rule’s capabilities to probe renormalon ambiguities in perturbative series and to clarify subtleties in how to use it properly, we also apply it to a few other cases. Interestingly, the analytic manipulations arising in the derivation of the sum rule lead to an alternative expression for the high-order asymptotic behavior of a series that contains an renormalon. This expression differs from the well known asymptotic formula which is known since a long time from Beneke:1994rs (), and we therefore discuss it as well.
The analytic derivation for the Borel transform of the MSR-pole mass relation starts from its expression related to the solution of the R-evolution equation given in Eq. (9) which was already derived in Ref. Hoang:2008yj ().
where in the second line we changed variable to and used the identity (72) to scale out , and in the third line we employed the coefficients given in Eq. (81). The expression in Eq. (20) gives an all-order representation of the original series that is more useful for analyzing renormalon issues than Eqs. (3) and (5). This is because using the R-evolution equation of Eq. (9) (which is linear in ) and its solution, provides, through the sum in , a reordering of the original series in leading and subleading series of terms from the perspective of their numerical importance in the asymptotic high order behavior related to the renormalon. This allows to derive rigorously a representation of the Borel transform [ given in Eq. (4.1) ] reflecting efficiently the hierarchy of leading and subleading terms with respect to the renormalon, which is the information that is not contained in the original series. That such a separation is possible in a systematic way may not be obvious, but it is achieved by the R-evolution equation. We stress that the result of Eq. (4.1) should not be considered as the exact expression for the Borel transform because it does not encode information on possible poles (or non-analytic cuts) other than at . We note that these poles and the associated renormalons can be studied by considering solutions of R-evolution equations involving powers of different from the linear dependence shown in Eq. (9), see Ambar:thesis ().
Here the cut in the gamma functions for cancels in the difference for each in the sum, and the result on the RHS is real. We mention that the first term () in the sum over provides the summation of the leading terms in the approximation shown in Eqs. (13) and (15). In Eq. (20) the cut still remains and arises from the integration of the Landau pole in the strong coupling located at in the integral in the next-to-last line. The resulting imaginary part in the numerical expression corresponds to the imaginary part that arises in the inverse Borel integral for , see Eq. (18), and simply reflects the ambiguity of the pole mass. From the point of view of the analytic solution of Eq. (20) based on a perturbative expansion, the imaginary part is well-defined and analytically unique.
To proceed we asymptotically expand the incomplete gamma function in inverse powers of (i.e. powers of )
where the coefficients are given in Eq. (79), and coincide with the coefficients defined in Ref. Beneke:1994rs (). We stress that the equality in Eq. (4.1) is the asymptotic expansion and is not an identity, so that the imaginary part due to the cut in the incomplete gamma function does not arise on the RHS. Inserting Eq. (4.1) in Eq. (20) gives
We then perform the Borel transform with respect to powers of according to the rule giving
Using identities for the hypergeometric function we can rewrite