The strong coupling and its running to four loops in a minimal MOM scheme
We introduce the minimal momentum subtraction (MiniMOM) scheme for QCD. Its definition allows the strong coupling to be fixed solely through a determination of the gluon and ghost propagators. In Landau gauge this scheme has been implicit in the early studies of these propagators, especially in relation to their non-perturbative behaviour in the infrared and the associated infrared fixed-point. Here we concentrate on its perturbative use. We give the explicit perturbative definition of the scheme and the relation of its -function and running coupling to the scheme up to 4-loop order in general covariant gauges. We also demonstrate, by considering a selection of examples, that the apparent convergence of the relevant perturbative series can in some (though not all) cases be significantly improved by re-expanding the coupling version of this series in terms of the MiniMOM coupling, making the MiniMOM coupling also of potential interest in certain phenomenological applications.
keywords:strong coupling constant, 4-loop running, minimal subtraction, momentum subtraction
Pacs:12.38.Gc, 12.38.Aw, etc.
The coupling constant of the strong interaction, , is one of the fundamental parameters of the Standard Model. It is specified by its value in a particular renormalisation scheme at a chosen reference scale, , conventionally taken to be the scheme at the reference scale . A recent assessment of experimental results yields Bethke:2006ac (). This result, which is little changed if more recent experimental results are taken into account (see, e.g., Maltman:2008bx () and references therein), is in excellent agreement with two recent, slightly different, lattice determinations Maltman:2008bx (); Davies:2008sw () based on lattice perturbation theory analyses of short-distance-sensitive lattice observables computed using the MILC configurations.
Other schemes than the scheme are of course also possible. For example it has been proposed in vonSmekal:1997isvonSmekal:1997vx () that a particular product of dimensionless gluon and ghost dressing functions, and , in the Landau gauge can be used to define a non-perturbative running coupling via
where is the renormalised coupling at scale and the renormalisation condition,
is assumed. The definition (1) is particularly convenient since it allows the coupling and hence to be determined by measuring two-point functions on a lattice. Our first steps towards such a determination for were presented at the 2007 Lattice Conference Sternbeck:2007br (). Further updates of these preliminary results were reported last year vonSmekal:2008ma (); Sternbeck:2008au (). This project, which is ongoing, has the potential to provide an independent precision determination of from lattice simulations at purely perturbative scales, typically between and Sternbeck:2009la ().111The QCD scale parameter of the underlying scheme (the MiniMOM scheme, see below) is roughly 450 MeV for , or 430 MeV for . Non-perturbative contributions to the gluon and ghost dressing functions are at least of the order . They are negligible at scales above . As an important supplement we specify here the details of the renormalisation scheme underlying the coupling (1) for QCD in general covariant gauges. We also provide the explicit 4-loop function for in all such gauges. This information will be important to the previously mentioned lattice analysis and to quantifying the truncation error on the resulting determination.
The product in (1) is dimensionless and renormalisation group invariant, and it reduces to the running coupling of a perturbative momentum subtraction scheme (MOM) at large . This makes it a suitable candidate for a non-perturbative extension, though such extensions are, of course, not unique. The underlying renormalisation condition (2) respects infrared scaling (with vonSmekal:1997isvonSmekal:1997vx ()),
for , as predicted by a variety of functional continuum methods for Landau gauge QCD including studies of Dyson-Schwinger Equations (DSEs) Lerche:2002ep (), Stochastic Quantisation Zwanziger:2001kw (), and of the Functional Renormalisation Group Equations (FRGEs) Pawlowski:2003hq (). This conformal infrared behaviour of the purely gluonic correlations in Landau gauge QCD is consistent with the conditions for confinement in local quantum field theory Alkofer:2000wg (); Alkofer:2001iwAlkofer:2000mz (), but it is yet to be observed in lattice simulations.222In order to do that one needs a proper non-perturbative definition of BRST symmetry on the lattice which is possible in principle, but not realised in present lattice implementations of Landau gauge. For a recent discussion, see vonSmekal:2008ws ().
If the infrared scaling behaviour (3) is realised, the renormalisation condition (2) holds beyond perturbation theory and can be imposed at any (space-like) subtraction point . This is because the running coupling defined by (1) then approaches a finite infrared fixed-point, for , with obtained under a mild regularity assumption on the ghost-gluon vertex Lerche:2002ep (). For the purposes of this Letter it suffices, however, that the coupling (1), is well-defined perturbatively, independent of the infrared scaling behaviour in (3).
A special feature of Landau gauge which underlies the definition (1) is the non-renormalisation of the ghost-gluon vertex Taylor:1971ff (). The possibility of taking advantage of this non-renormalisation has been criticised in the past Boucaud:2005ce (); Boucaud:2008ji () on the grounds that the Landau-gauge ghost-gluon vertex acquires a momentum dependence in all common MOM schemes, already at one-loop level Celmaster:1979km (), despite the absence of ultraviolet divergences in this vertex in Landau gauge. There is, however, no contradiction here at all, as will be explained in the next section. The basic reason is that the non-renormalisation of the ghost-gluon vertex ensures the existence of a scheme for which the notion of (1) as a running coupling makes sense and which this running coupling implicitly defines, without the need to use an asymmetric momentum scheme Boucaud:2008gn (). We call it the MiniMOM scheme for reasons that will become clear below. A useful feature of the MiniMOM scheme is that it can be related to the scheme at four-loops vonSmekal:2008ma (); Sternbeck:2008au () without the need to compute vertices to three loops in perturbation theory.
After providing more details on the MiniMOM coupling and its relation to the coupling below, we consider, in Sec. 4, the and MiniMOM versions of the perturbative series entering a selection of phenomenological applications, demonstrating that, in some cases, the apparent convergence of the series is much improved by the use of the MiniMOM coupling. Our conclusions and outlook are given in Sec. 5.
2 The minimal MOM scheme
Some confusion in the literature concerning the running coupling of Eq. (1) arose in relation to the misconception that this definition rests on the non-renormalisation of the ghost-gluon vertex in Landau gauge Taylor:1971ff (). This definition is not, however, based on requiring that the ghost-gluon vertex reduce to the tree-level one at a symmetric subtraction point . In particular, the scheme underlying (1) is not the MOMh scheme of Ref. Chetyrkin:2000fd (), but is, instead, a minimal MOM (MiniMOM) scheme, which is defined as follows:
As in every MOM scheme, the gluon and ghost renormalisation constants and are defined by requiring
the perturbative realisation of (2) valid for where is the scale parameter of the scheme. and are the dressing functions in the renormalised gluon and ghost propagators of Landau-gauge QCD, which are in (Euclidean) momentum space of the form
respectively. Instead of imposing additional analogous renormalisation conditions on vertex functions, requiring certain vertex structures to equal their tree-level counter parts at some symmetric or asymmetric subtraction point, we here supplement (4) by the further condition
where is the renormalisation constant of the ghost-gluon vertex, whose momentum dependence is thus the same as in the minimal subtraction schemes. The renormalisation constants for the three and four gluon vertex functions are then determined by the Slavnov-Taylor identities as usual,
The extension to quarks is straightforward, and this defines the MiniMOM scheme: momentum subtraction for the gluon, ghost and quark propagators, and minimal () subtraction for the ghost-gluon vertex (7) together with the Slavnov-Taylor identities to fix the remaining vertices, including the quark-gluon vertex. Note that the renormalisation constants for these remaining vertices then differ from those in the scheme by ratios of the propagator (field) renormalisation constants in the MiniMOM () and schemes, for example,
Likewise, from the general definition of the renormalised coupling constant,
the MiniMOM and scheme couplings, with the definition in (7), are related by
Note that for this conversion we only need to know the perturbative expansions of the ghost and gluon propagators but not of any vertex structures. Rather, with the MOM renormalisation conditions for these propagators (4), or their non-perturbative extension (2) for that matter, we simply obtain,
from the gluon and ghost dressing functions evaluated at in the scheme. Alternatively, we can relate the MiniMOM coupling to the MOMh scheme just as easily via
All these conversion identities are valid for arbitrary linear-covariant gauges and not restricted to Landau gauge. The special feature of Landau gauge is that there . Because the Landau gauge ghost-gluon vertex trivially reduces to its tree-level form when one of the ghost momenta is set to zero, the MiniMOM scheme then agrees with the asymmetric scheme of Ref. Chetyrkin:2000dq () (called the scheme in Ref. Boucaud:2008gn ()) which is defined from renormalising precisely this vertex structure. The MiniMOM scheme is defined, however, so as to not require knowledge of any vertex structure beyond the scheme contributions as determined by their ultra-violet divergences. This makes it particularly useful for a lattice determination of from the perturbative behaviour of QCD Green’s functions as described in Refs. Sternbeck:2007br (); vonSmekal:2008ma (); Sternbeck:2008au ().
For early 2 and 3-loop calculations in Feynman gauge, see Tarasov:1976efTarasov:1980kxTarasov:1980au (). The complete 2-loop results for general gauge parameters are given in Davydychev:1997vhDavydychev:1998kj (). To obtain the 4-loop version of the conversion between and from Eq. (12), we use the 3-loop expressions for the gluon and ghost self-energies found in Appendix C of Chetyrkin:2000dq () which, with , for massless quarks yields
where, with , , and ,
This conversion depends on the gauge parameter as in every other momentum subtraction scheme, though in the MiniMOM scheme, this dependence is comparatively weak. At one-loop level, for example, the coefficient of the leading dependence around in the above conversion is 3 times smaller than that of the asymmetric scheme (which coincides with the MiniMOM scheme at ).
For the same conversion can be obtained from the product of the scheme-invariant propagators given for Landau gauge in Sec. 4 of Chetyrkin:2004mf (). We verified the general result (15) for and and from the explicit expressions given there. Another check of our conversion, up to and including , can be obtained using Eq. (13) in Landau gauge, where , with the 2-loop expression for in the symmetric scheme from Chetyrkin:2000fd (), together with the 3-loop conversion from to as given there.333There appears to be a typo in Table 2 of Chetyrkin:2000fd (), the entry for the 2-loop coefficient of proportional to should probably read instead of .
The numerical values of the in Landau gauge are given explicitly, for and and , in Tab. 1. As an illustration of the weakness of the dependence on , the corresponding , results are
with similar, slightly less -dependent, results for .
In order to compare actual values of in the MiniMOM scheme (at ) to the corresponding ones in the scheme we give two important examples. First, at the mass of the boson, GeV with Bethke:2006ac (), from Eq. (14) we obtain
for , while at the mass of the lepton, GeV,
where, to be specific, we have used , the value obtained by running down to the scale using the standard 4-loop running Chetyrkin:1997sg ().
We conclude this section with a few comments on quark mass effects. As in every off-shell subtraction scheme, the running coupling and beta function of the MiniMOM scheme in principle depend on the quark masses. This is evident from its relation to the (mass independent) scheme, Eq. (12), in which the scheme gluon and ghost dressing functions will depend on the masses in the quark loops. These have not been included and our conversion formulas are therefore strictly speaking valid only for massless quarks. To fully account for finite quark masses at this level one would need the corresponding 3-loop expressions for the gluon and ghost dressing functions with massive quark loops which have not been calculated to our knowledge as yet.
We can estimate the leading quark mass effects, however, which will affect the conversion formula at the 2-loop level. These are obtained from the 1-loop vacuum polarisation with massive fermions in the gluon self-energy, and they lead to an increase of (via ) as compared to the massless case. For each quark flavour with mass one then separately obtains a transcendental function of which approaches for , i.e., for . Using the current upper limits from the Particle Data Group for the average up/down mass of MeV and the strange mass of MeV as commonly given at GeV, we observe a maximum increase in by 0.15% at GeV as compared to the massless value given in Tab. 1. At GeV, with correspondingly larger light quark masses, the same upper bound for the increase in reaches 1%. At larger scales the effect rapidly decreases. In particular, the explicit comparisons in Eqs. (17) and (18) remain unaffected by the corresponding changes in (note that even with the charm and bottom quark masses included, the increase in will be less than 0.1% at as compared to the massless flavour value there).
The most noticeable quark mass effects will of course occur right at the decoupling scales . At the charm threshold, for example, with GeV, the quark mass contributions to the vacuum polarisation lead to an increase in by around 13% as compared to the massless value (at GeV this increase is reduced by a factor of 2 already). Considerable charm-quark mass effects should therefore be expected when converting the 4 flavour MiniMOM coupling to the coupling in the phenomenoligically interesting range between 1 and 4 GeV. Without a more detailed knowledge of these effects an conversion would therefore not be advisable. Here we use the MiniMOM to conversion up to at most for which the quark mass effects are very small. For now, matching to the and regimes should always be done after the conversion, for the coupling in the usual way.
There are no charm and bottom quarks in the lattice determinations of the MiniMOM coupling which are presently restricted to and and which will be extended to light flavours in due course. At the relevant high momentum scales quark mass effects should then be completely negligible in the conversion to . In addition, it is always a possibility to remove any small residual light-quark mass effects by extrapolation, if necessary.
3 Beta-function coefficients of the MiniMOM coupling
The running of the coupling constant as the scale, , changes is controlled by the (scheme-dependent) function which, at small couplings, is defined by
where . The function in the scheme is known to 4-loop order, the expressions for the corresponding coefficients , , for general and , being given in Refs. vanRitbergen:1997vaCzakon:2004bu (). These results, together with the relation between the and MiniMOM coupling given in Eqs. (14) and (15), and the 3-loop version of the expression , for the running of the renormalised gauge parameter, , with the gluon anomalous dimension, the 3-loop expression for which can be found in Appendix D of Ref. Chetyrkin:2000dq (), allow us to obtain the function coefficients in the MiniMOM scheme to 4-loop order. For general and , we find
where the all vanish in Landau gauge.444The Landau gauge version of these results were first presented in Refs. vonSmekal:2008ma (); Sternbeck:2008au () and subsequently confirmed in Ref. Boucaud:2008gn (). The general expressions for the are rather long and unilluminating, and hence not included here. The results for the phenomenologically most interesting case, , however, are given in Tab. 2.
The numerical values of the ’s in Landau gauge are given, for and and , in Tab. 3. For the reader’s convenience, we give also the numerical results for general and ,
which results serve to illustrate the weakness of the dependence in the vicinity of Landau gauge. Note that, while the first coefficient, is gauge independent, and universal, the coefficients beginning with are gauge dependent, as is typical of momentum subtraction schemes (as usual, the universal value of is obtained only in Landau gauge).
4 Comparing perturbative expansions
The definition of the running coupling in (1) has been widely used, and continues to be widely used, in phenomenological applications of QCD Green’s functions within non-perturbative continuum approaches Alkofer:2000wg (); Fischer:2006ub () based on DSEs or FRGEs, for example. The precise definition of the underlying renormalisation scheme, the MiniMOM scheme, puts these approaches on a firmer ground, and should serve to resolve any previous misunderstandings. We have already stressed its utility in providing a route to a lattice determination of requiring only a calculation of two-point functions, which are relative easy to determine with high precision in current simulations. Here we show, as an added bonus, that the MiniMOM coupling may provide a useful alternative to the coupling in certain phenomenological applications. We do so by considering the expressions for the perturbative contribution to quantities relevant to a selection of phenomenological applications in the regime, expanded in terms of either the or the MiniMOM coupling. With , , , and from Eqs. (15), and
an observable, , whose expansion is
has an equivalent MiniMOM coupling expansion
In investigating the phenomenological utility of the MiniMOM coupling, we will consider the case, for which , and . We show that, in some of the considered cases, use of the MiniMOM coupling significantly improves the apparent convergence of the relevant perturbative series, while, in other cases, it does not. Whether or not it is useful to employ the MiniMOM coupling is thus something to be decided on a case to case basis.
4.1 The Adler function of the vector/axial vector current correlators
Our first example is the dimension contribution to the Adler function, , of the flavour vector (V) or axial vector (A) current scalar correlator, . At scales of phenomenological interest, is far and away the dominant term on the OPE side of finite energy sum rules (FESRs) which have been studied in the literature based on either electroproduction cross-sections or hadronic decay data. The -based FESRs are used in precision determinations of , the most recent of which are described in Refs. Beneke:2008ad (); Maltman:2008nf (); Narison:2009vy (). A combination of electroproduction- and -based FESRs has also been used to investigate the present discrepancy Davier:2003pw (); Davier:2007ua () between the electroproduction and version of the V spectral function Maltman:2005yk (), a discrepancy which prevents a clear decision as to whether or not the Standard Model (SM) prediction for is compatible with the current high-precision experimental result Bennett:2004pv ().
is known to 5-loops Baikov:2008jh () and, for , given in terms of , by
The equivalent expansion in terms of is
which displays significantly improved convergence at scales relevant to the phenomenological studies noted above (), even when one takes into account the increased size of as compared to . This improved convergence will also be manifest in FESR studies which employ “contour improved perturbation theory” (CIPT) in their evaluations of the relevant weighted integrals.555The CIPT prescription employs the truncated expansion of Eq. (25) point-by-point along the circle in the complex -plane. An alternate approach to evaluating the weighted contour integrals is to use the “fixed order perturbation theory” (FOPT) prescription, in which the series is expanded, and truncated, using the running coupling at the same fixed scale, e.g., , for all points on the circle. In the FOPT scheme, large logs are unavoidable over some portion of the contour. Nonetheless, recent arguments Beneke:2008ad (), based on a model constrained by known features of the large order behaviour of the perturbative series for , shows it is possible that the truncated FOPT form might provide a more reliable representation of the resummed series than would the CIPT form. This is potentially relevant here because the improvement in the convergence of the integrated FOPT series achieved through the use of the MiniMOM couplant is, for commonly used weights, far less compelling than that achieved in the CIPT case. While a study of the 5-loop FOPT approximation to a range of weighted integrals of the model for the resummed series in Ref. Beneke:2008ad () shows that a good representation of the corresponding data integrals is not possible, in contrast to the situation when the 5-loop CIPT evaluation is employed Maltman:2009ip (), the analogous study has not yet been performed for the full resummed model, and, as a result, the preference for the CIPT approach (where the improvement due to the re-ordering of the series using would be operative) is not yet conclusive.
4.2 The second derivative of the D=0 part of the scalar/ pseudoscalar correlator
As our second example, we consider the subtraction-constant-independent second derivatives, , of the part of the scalar (S) and pseudoscalar (PS) correlators, and , formed from the divergences of the flavour V or A currents. These quantities, which are equal for the S and PS cases, apart from the overall factors, are the dominant terms on the OPE side of S and PS FESRs and Borel sum rules (BSRs) which provide the most reliable sum rule determinations of and Maltman:2001gcMaltman:2001nx (); Jamin:2001zrJamin:2006tj (); Chetyrkin:2005kn (). Useful lower bounds on have also been obtained from the PS sum rules using a combination of the accurately known pole contribution and spectral positivity Becchi:1980vz (); Lellouch:1997hp (); Chetyrkin:2005kn (). A combination of FESRs and BSRs based on , in addition, provides a determination, not just of , but also of the decay constants of the first two excited resonances, and hence a useful, highly constrained model of the PS spectral function, a model which, combined with the S spectral function constructed in Refs. Jamin:2001zrJamin:2006tj (), allows the continuum spectral contributions to be subtracted from the experimental differential distribution in strange hadronic decays. This turns out to be a crucial input to the hadronic decay determination of Gamiz:2002nuGamiz:2004arGamiz:2007qs (); Maltman:2006stMaltman:2007icMaltman:2007pr (); Pich:2008ni (); Maltman:2008ib () since the OPE representation of the contributions is extremely badly behaved at all kinematically accessible scales, preventing one from employing FESRs based on the full experimental differential distribution Maltman:1998qzMaltman:2001sv ().
The expansion of in terms of is known to five loops Chetyrkin:2005kn () and, for , is given by
|where is the running quark mass in the scheme. Re-expressing the series in terms of yields|
which again displays significantly improved convergence. Such improved convergence is likely to allow a significant reduction in the errors on the determination of the light quark masses, and an improved version of the light quark mass bounds.