Remarks on the determination of the Landau gauge OPE for the asymmetric three gluon vertex

Remarks on the determination of the Landau gauge OPE for the asymmetric three gluon vertex F. De Soto Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain J. Rodrı ´guez-Quintero Departamento de Fı ´sica Aplicada e Ingenierı ´a Ele ´ctrica, E.P.S. La Ra ´bida, Universidad de Huelva, 21819 Palos de la Fra., Spain 共Received 8 May 2001; published 24 October 2001兲 We compute a compact operator product expansion 共OPE兲formula describing power corrections to the perturbative expression for the asymmetric momentum subtraction–(MOM g -)renormalized running coupling constant up to the leading logarithm. By the use of a phenomenological hypothesis leading to the factorization of the condensates through a perturbative vacuum insertion, the only relevant condensate in the game is 具 A2 典 . The validity of the OPE formula is tested by searching for a good-quality coherent description of previous lattice evaluations of the MOM g -renormalized gluon propagator and running coupling. DOI: 10.1103/PhysRevD.64.114003 PACS number共s兲: 12.38.Aw, 11.15.Ha, 12.38.Cy, 12.38.Gc That the running coupling constant can be extracted from the three-gluon vertex in the Landau gauge was proposed several years ago in a seminal work 关1兴. This method appears closer to a physical interpretation than Schro ¨dinger functionals 关2兴but, and mainly for the same reason, more systematic effects should be included to produce a reliable prediction for the perturbative ␣ S. In particular, the first statistically meaningful attempts to follow that method missed the impact of power corrections 关3兴and failed to give an estimate for the coupling constant comparable to others in literature 共that of Ref. 关2兴, for instance兲. In parallel, these power corrections to the gluon propagator and to the running coupling constant have been greatly studied in recent years 关4,5兴. Although the first trials were rather inconclusive 关6兴, momentum power contributions have been manifestly put in evidence 关7,8兴for the Landau gauge three-gluon coupling constant renormalized in both symmetric 共MOM兲and asymmetric (MOM g ) momentum subtraction schemes. In Ref. 关7兴 the parameter ⌳MS is estimated from the matching of MOM g ␣ Slattice results to a perturbative three-loop formula corrected by an unavoidable 1/p2term. This naive ansatz used in Ref. 关7兴seems to eliminate most of the systematic deviation from the three-gluon vertex estimate of the perturbative ␣ Sand a precise prediction of ⌳MS emerges in full agreement with that of Ref. 关2兴. Unfortunately, the errors quoted in this work were clearly underestimated. In fact, the prediction of ⌳MS is so sensitive to a logarithmic dependence on the momentum scale of the coefficient of 1/p2that it appears to range over an interval of 40 MeV 关9兴. On the other hand, in Refs. 关8,9兴a description in terms of operator product expansion 共OPE兲for the power corrections to the MOM three-gluon ␣ Swas successfully tried consistently with an analogous description for the gluon propagator. The OPE approach provides through Shifman- Vainshtein-Zakharov 共SVZ兲factorization 关10兴a perturbative tool to obtain the leading logarithmic dependence of nontrivial Wilson coefficients. In particular, it can be applied to compute the coefficient of A2in the MOM case for the Landau gauge three-gluon coupling constant 关9兴. Obviously, the expectation value of A2in a nonperturbative vacuum is also estimated via this OPE approach. A non-gauge-invariant condensate, that, being in the Landau gauge, is invariant under infinitesimal gauge transformations and thus connected in some way to the gauge invariant 具 Amin 2 典 defined in 关11兴,is then computed. The same approach for MOM g coupling constant requires use of T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲…⫽共c0兲 ␮ ␯ ab共p兲1⫹共c1兲 ␮ ␯ a⬘ ab ␮ ⬘ 共p兲:A ␮ ⬘ a⬘ 共0兲: ⫹共c2兲 ␮ ␯ a⬘b⬘ ab ␮ ⬘ ␯ ⬘ 共p兲:A ␮ ⬘ a⬘ 共0兲A ␯ ⬘ b⬘ 共0兲: ⫹共c3兲 ␮ ␯ a⬘b⬘c⬘ ab ␮ ⬘ ␯ ⬘ ␳ ⬘ 共p兲:A ␮ ⬘ a⬘ 共0兲A ␯ ⬘ b⬘ 共0兲 ⫻A ␳ ⬘ c⬘ 共0兲:⫹•••,共1兲 where the expansion to the three-gluon local operator is necessary because the three-point Green function in MOM g can be written as 具 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲… 典 NP ⬅ 具 0 兩 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲… 兩 g ␳ c 典 NP ⫽共c1兲 ␮ ␯ a⬘ ab ␮ ⬘ 具 0 兩 :A ␮ ⬘ a⬘ 共0兲: 兩 g ␳ c 典 NP ⫹共c3兲 ␮ ␯ a⬘b⬘c⬘ ab ␮ ⬘ ␯ ⬘ ␳ ⬘ 具 0 兩 :A ␮ ⬘ a⬘ 共0兲A ␯ ⬘ b⬘ 共0兲A ␳ ⬘ c⬘ 共0兲: 兩 g ␳ c 典 NP ⫹•••.共2兲 The index NP refers to the nonperturbative nature of the vacuum state in Eq. 共2兲, while T*refers to the standard time ordered product in momentum space. The following points should be noticed. 共i兲No other local operators with the same dimension of A2are written in Eq. 共1兲because, unlike the identity or A2itself, they do not generate a non-null vacuum expectation value.1共ii兲Operators other than the three-gluon 1No Lorentz invariant tensor with an odd number of indices can PHYSICAL REVIEW D, VOLUME 64, 114003 0556-2821/2001/64共11兲/114003共7兲/$20.00 ©2001 The American Physical Society64 114003-1 local one for the same dimension, such as ⳵ ␮ A ␯ b(0)A ␳ c(0),do not appear explicitly because they are phenomenologically supposed not to survive, as will be argued below. The identity and A2clearly do not contribute to the matrix element considered in Eq. 共2兲. Following standard SVZ techniques to obtain the peturbative expansion of OPE Wilson coefficients, we compute the appropriate matrix element of Eq. 共1兲’s left-hand side 共LHS兲 to the wanted order. It is obvious that taking the perturbative vacuum leads to 具 0 兩 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲… 兩 0 典 ⫽共c1兲 ␮ ␯ a⬘ ab ␮ ⬘G(2) g ␮ ⬘ ␳ a⬘c共0兲, 共3兲 where c1can be straightforwardly identified with the perturbative expansion for the Green function with an amputated soft-gluon leg, ⌫ ␮ ␯ a⬘ ab ␮ ⬘ 共p,⫺p,0兲⬅⫺2p ␮ ⬘g ␮ ␯ ⬜共p兲fa⬘ abGpert (3)共p2兲,共4兲 which should be proportional to its Landau gauge tree-level tensor. Beyond this purely perturbative first contribution, the situation is slightly more complicated. We consider now the matrix element on the LHS of Eq. 共2兲between two external soft gluons and the perturbative vacuum. Then, we obtain up to the considered order in perturbation theory 具 0 兩 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲… 兩 g␭ lg ␴ s 典 connected ⫽共c3兲 ␮ ␯ a⬘b⬘c⬘ ab ␮ ⬘ ␯ ⬘ ␳ ⬘ 共p兲 具 0 兩 :A ␮ ⬘ a⬘ 共0兲A ␯ ⬘ b⬘ 共0兲A ␳ ⬘ c⬘ 共0兲: 兩 g ␳ cg␭ lg ␴ s 典 . 共5兲 For the matrix element on the right hand side 共RHS兲we have 具 0 兩 :A ␮ ⬘ a⬘ 共0兲A ␯ ⬘ b⬘ 共0兲A ␳ ⬘ c⬘ 共0兲: 兩 g ␳ cg␭ lg ␴ s 典 ⫽G ␴␴ ⬘ (2) g ss⬘共0兲G␭␭⬘ (2) g ll⬘共0兲G ␳ ␶ (2) g ct共0兲 ⫻ 兵 P ␴ ⬘␭⬘ ␶ s⬘l⬘tg ␮ ⬘ ␴ ⬘g ␯ ‘ ␭⬘g ␳ ⬘ ␶ ␦ s⬘ a⬘ ␦ l⬘ b⬘ ␦ t c⬘ ⫹O共 ␣ 兲 其 ,共6兲 where Prefers to all the possible permutations of the couples ( ␴ ⬘s⬘), (␭⬘l⬘), and ( ␶ t). The Wilson coefficient c3may thus be computed at tree-level order as P ␴ ⬘␭⬘ ␶ s⬘l⬘t共c3兲 ␮ ␯ s⬘l⬘t ab ␴ ⬘␭⬘ ␶ ⫽ 具 A ˜ ␭ l共0兲A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲A ˜ ␴ s共0兲 典 G共2兲 g ␴␴ ⬘ ss⬘ 共0兲G共2兲 g ␭␭⬘ ll⬘ 共0兲G共2兲 g ␳ ␶ ct 共0兲 . 共7兲 The ratio in Eq. 共7兲represents symbolically all the tree-level diagrams with five gluon legs where the three carrying zero momentum are cut 共see Fig. 1兲. Had we directly dealt with the three-gluon local operator :A ␮ ⬘ a⬘(0)A ␯ ⬘ b⬘(0)A ␳ ⬘ c⬘(0):, the evaluation of the higher order tensors involved would require a much more tedious calculation. On the other hand, vacuum insertion between one gluon field and the other two leads to 关 具 0 兩 :A ␮ ⬘ a⬘ 共0兲A ␯ ⬘ b⬘ 共0兲A ␳ ⬘ c⬘ 共0兲: 兩 g ␳ c 典 NP兴R, ␮ ⫽ 具 A2 典 R, ␮ G ˜ ␶ ␳ tc 共0, ␮ 2兲 4共Nc 2⫺1兲 T ␮ ⬘ ␯ ⬘ ␳ ⬘t a⬘b⬘c⬘ ␶ 共8兲 with T ␮ ⬘ ␯ ⬘ ␳ ⬘t a⬘b⬘c⬘ ␶ ⫽g ␮ ⬘ ␯ ⬘ ␦ a⬘b⬘g ␳ ⬘ ␶ ␦ t c⬘ ⫹g ␮ ⬘ ␳ ⬘ ␦ a⬘c⬘g ␶ ␯ ⬘ ␦ t b⬘ ⫹g ␳ ⬘ ␯ ⬘ ␦ c⬘b⬘g ␮ ⬘ ␶ ␦ t a⬘.共9兲 We may phenomenologically assume vacuum insertion to work for a certain renormalization momentum scale. Thus, the replacement :A ␮ ⬘ a⬘ 共0兲A ␯ ⬘ b⬘ 共0兲A ␳ ⬘ c⬘ 共0兲:→:A2A ␶ t共0兲: 4共Nc 2⫺1兲 T ␮ ⬘ ␯ ⬘ ␳ ⬘t a⬘b⬘c⬘ ␶ 共10兲 can be done for Eq. 共2兲, the other components of the tensor on the LHS not being required for our purposes. Furthermore, the local gluon field A ␶ t(0) is to be contracted with the zero-momentum gluon field defined for the vertex. The same vacuum insertion assumption leads us to argue that, for instance, 关 具 0 兩 : ⳵ ␮ ⬘A ␯ ⬘ b⬘ 共0兲A ␳ ⬘ c⬘ 共0兲: 兩 g ␳ c 典 NP兴R, ␮ → 具 ⳵ ␮ ⬘A ␯ ⬘ b⬘ 共0兲 典 R, ␮ G ˜ ␳ ⬘ ␳ c⬘c共0, ␮ 2兲⫽0. 共11兲 Thus, the only nonzero surviving condensate comes from the three-gluon local operator. It is easy to see that, using Eq. 共10兲, the relevant coefficient multiplying the local operator in Eq. 共1兲is 共c3兲 ␮ ␯ a⬘b⬘c⬘ ab ␮ ⬘ ␯ ⬘ ␳ ⬘ 共p兲T ␮ ⬘ ␯ ⬘ ␳ ⬘t a⬘b⬘c⬘ ␶ ⫽1 2 具 A ˜ ␭ l共0兲A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲A ˜ ␴ s共0兲 典 G共2兲 g ␴ ␮ ⬘ sa⬘ 共0兲G共2兲 g ␭ ␯ ⬘ lb⬘ 共0兲G共2兲 g ␳ ␶ ct 共0兲 g ␮ ⬘ ␯ ⬘ ␦ a⬘b⬘ 共12兲 be built without nonzero momenta. FIG. 1. Diagrams contributing to the treelevel Wilson coefficient in Eq. 共7兲Crosses mark the soft-gluon legs coming from the condensate. F. De SOTO AND J. RODRI ´GUEZ-QUINTERO PHYSICAL REVIEW D 64 114003 114003-2 where the RHS may be straightforwardly obtained from Eq. 共7兲. This last Eq. 共12兲gives the prescription in which Lorentz and color indices of external gluon legs are contracted in the diagrams contributing to the tree-level Wilson coefficient 共see Fig. 1兲. Thus, since Eq. 共4兲’s is the only Landau gauge tensor for the asymmetric three-gluon vertex 关3兴, we can write, for p2 ⫽⫺k2, k4 具 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲… 典 NP ⫽⫺2p ␶ g ␮ ␯ ⬜共p兲ft ab 冋 c1 冉 g,k2 ⌳2 冊 G共2兲 g ␶ ␳ tc 共0兲 ⫹c3 冉 g,k2 ⌳2 冊 具 0 兩 :A2A ␶ t共0兲: 兩 g ␳ c 典 NP 4共Nc 2⫺1兲 1 ⫺k2 册 .共13兲 We know the scalar coefficients c1,c3in Eq. 共13兲to be dimensionless by OPE power counting rules, and hence both depend only on the bare coupling gand the dimensionless ratio of momentum to regularization scale.2The term c1can be obtained from Eq. 共3兲and, at tree level, c3is to be computed by projecting over the Landau gauge tree-level tensor in Eq. 共4兲the result from Eq. 共12兲, c3,tree level 冉 g,k2 ⌳2 冊 ⫽共c3兲 ␮ ␯ a⬘b⬘c⬘ ab ␮ ⬘ ␯ ⬘ ␳ ⬘ 共p兲T ␮ ⬘ ␯ ⬘ ␳ ⬘t a⬘b⬘c⬘ ␶ ⫻1 ⫺6NC共NC⫺1兲k2p ␶ g⬜ ␮ ␯ 共p兲fabt ⫽3g.共14兲 If we renormalize following the MOM g prescription at momentum scale ␮ 2, and then apply the assumed vacuum insertion factorization, we can write k4关 具 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲… 典 兴R, ␮ 2 ⫽⫺2p ␶ g ␮ ␯ ⬜共p兲ft abG(2) g ␶ ␳ tc 共0, ␮ 2兲 冋 c1 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫹c3 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 具 A2 典 R, ␮ 2 4共Nc 2⫺1兲 1 ⫺k2 册 ,共15兲 where, after renormalization, the Wilson coefficients depend on the ratio of the momentum and the renormalization scale, and on ␣ ( ␮ )⫽gR 2( ␮ )/(4 ␲ ), i.e., the coupling constant consistently renormalized in MOM g . In the MOM g scheme, the renormalized Green functions take formally, at the renormalization scale, the same treelevel value but in terms of the renormalized coupling constant instead of the bare one. This defines ZMOM( ␮ ) ⫽ ␮ 2G(2)( ␮ 2) to renormalize appropriately the two-point Green function, where G(2) is the scalar factor of the bare two-point Green function defined in Ref. 关3兴. The threegluon Green function is then renormalized by dividing by 关ZMOM( ␮ )兴3/2. One gets gR共k2兲⫽k4GR (3)共k2, ␮ 2兲 GR (2)共0, ␮ 2兲关k2GR (2)共k2, ␮ 2兲兴⫺1/2.共16兲 The renormalized scalar factor for the three-gluon Green function with an amputated soft gluon, GR (3)/GR (2) , can be projected out from Eq. 共15兲, as is done for c3in Eq. 共14兲, while for GR (2)(k2, ␮ 2) we can write k2GR (2)共k2, ␮ 2兲⫽c0 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫹c2 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 具 A2 典 R, ␮ 2 4共Nc 2⫺1兲 1 ⫺k2, 共17兲 where the scalar coefficients c0and c2can be derived from those in Eq. 共1兲and computed similarly to c1and c3,as explained in Ref. 关9兴. Thus, after replacing in Eq. 共16兲we will finally get gR共k2兲⫽c1 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 冋 c0 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 册 ⫺1/2 ⫻ 冋 1⫹ 冉 c3„k2/ ␮ 2, ␣ 共 ␮ 兲… c1„k2/ ␮ 2, ␣ 共 ␮ 兲… ⫺1 2 c2共k2/ ␮ 2, ␣ 共 ␮ 兲兲 c0共k2/ ␮ 2, ␣ 共 ␮ 兲兲 冊 具 A2 典 R, ␮ 2 4共Nc 2⫺1兲 1 ⫺k2 册 .共18兲 The purpose is now to compute to leading logarithms the subleading Wilson coefficients in Eq. 共15兲, as was done in Ref. 关9兴. To this end, it will be useful to consider the following operator expansion: 2 9k2p ␶ g⬜ ␮ ␯ 共p兲fab t关T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲…兴R, ␮ 2 ⫽c3„k2/ ␮ 2, ␣ 共 ␮ 兲… ⫺k6关:A2A ␶ t共0兲:A ˜ ␳ c共0兲兴R, ␮ ⫹•••; 共19兲 where the ellipsis refers to terms with powers of 1/kother than 6. If the vacuum expectation in the nonperturbative vacuum is considered for the RHS of Eq. 共19兲, the result under the vacuum insertion hypothesis is known to be diagonal in color and Lorentz spaces. Then we contract on both sides of Eq. 共19兲with g ␳ ␶ ␦ ct and take the following matrix element 共see Ref. 关12兴兲: 2The dependence on the regularization momentum scale (a⫺1in lattice regularization or ⑀ ⫺1 ␮ in dimensional, for instance兲has been omitted up to now to simplify the notation. REMARKS ON THE DETERMINATION OF THE LANDAU . . . PHYSICAL REVIEW D 64 114003 114003-3 2 9k2p ␳ g⬜ ␮ ␯ 共p兲fabc 具 0 兩 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲… 兩 g ␴ sg␭ l 典 R, ␮ 2 ⫽c3„k2/ ␮ 2, ␣ 共 ␮ 兲… ⫺k6 具 0 兩 :A2Ac ␳ 共0兲: 兩 g ␳ cg ␴ sg␭ l 典 R, ␮ 2⫹•••, 共20兲 where we take external gluons in a perturbative vacuum and carrying soft momenta. From Eq. 共20兲we get ⫺2 9k4p ␳ g⬜ ␮ ␯ 共p兲fabc 具 0 兩 T*„A ˜ ␮ a共⫺p兲A ˜ ␯ b共p兲A ˜ ␳ c共0兲… 兩 g ␴ sg␭ l 典 具 0 兩 :A2Ac ␳ 共0兲: 兩 g ␳ cg ␴ sg␭ l 典 ⫽Z3共 ␮ 2兲ZA3 ⫺1共 ␮ 2兲c3 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫽Z3 ⫺1/2共 ␮ 2兲Z ¯ ⫺1c3 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ,共21兲 where, always in MOM g prescription, A ˜ R⫽Z3 ⫺1/2A ˜ and ZA3is defined such that 关:A2Ac ␳ 共0兲:A ˜ ␳ c共0兲兴R, ␮ 2 ⫽ZA3 ⫺1共 ␮ 2兲Z3 ⫺1/2共 ␮ 2兲:A2Ac ␳ 共0兲:A ˜ ␳ c共0兲,共22兲 while Z ¯ ⬅ZA3Z3 ⫺3/2 is a useful notation for the constant renormalizing the matrix element for the three-gluon local operator where the external soft gluons are explicitly cut. If we recover the divergent factor Z ˆ⬅ZA2Z3 ⫺1introduced in Ref. 关9兴for the matrix element of the two-gluon local operator coming from proper vertex corrections, Z ¯ can be decomposed as Z ¯ 共 ␮ 2兲⬅Z ˆ共 ␮ 2兲Z ␬ 共 ␮ 2兲,共23兲 where Z ˆtakes the divergent part coming from the diagrams for the matrix element on the 共RHS兲of Eq. 共20兲which can be factorized 关diagrams 共a,b兲of Fig. 2兴as in Eq. 共8兲, and Z ␬ should be computed from those that cannot 关diagrams 共c,d兲 of Fig. 2兴. Then, taking the logarithmic derivatives with respect to ␮ on both sides of Eq. 共21兲, we get the following renormalization group differential equation: 再 ⫺2 ␥ ¯ 关 ␣ 共 ␮ 兲兴⫺ ␥ 关 ␣ 共 ␮ 兲兴⫹ ⳵ ⳵ ln ␮ ⫹ ␤ 关 ␣ 共 ␮ 兲兴 ⳵ ⳵ ␣ 冎 c3 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫽0, 共24兲 with the formal solution 共see Ref. 关12兴兲 c3 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫽c3„1, ␣ 共k兲… 冉 ␣ 共k兲 ␣ 共 ␮ 兲 冊 ⫺(2 ␥ ¯ 0⫹ ␥ 0)/2 ␤ 0, 共25兲 where ␥ ¯ 关 ␣ 共 ␮ 兲兴⫽d dln ␮ 2lnZ ¯ 共 ␮ 2兲⫽⫺ ␥ ¯ 0 ␣ 共 ␮ 兲 4 ␲ ⫹•••, ␥ 关 ␣ 共 ␮ 兲兴⫽d dln ␮ 2lnZ3共 ␮ 2兲 ⫽⫺ 冋 ␥ 0 ␣ 共 ␮ 兲 4 ␲ ⫹ ␥ 1 冉 ␣ 共 ␮ 兲 4 ␲ 冊 2 ⫹ ␥ 2 冉 ␣ 共 ␮ 兲 4 ␲ 冊 3 ⫹••• 册 , ␤ 关 ␣ 共 ␮ 兲兴⫽d dln ␮␣ 共 ␮ 兲 ⫽⫺ 冉 ␤ 0 2 ␲ ␣ 2共 ␮ 兲⫹ ␤ 1 4 ␲ 2 ␣ 3共 ␮ 兲 ⫹ ␤ 2 共4 ␲ 兲3 ␣ 4共 ␮ 兲⫹••• 冊 .共26兲 The boundary condition of Eq. 共24兲is given by our MOM g -like prescription for the renormalization of the condensate by ZA3: the condensate is renormalized such that the Wilson coefficient takes the tree-level form at the renormalization point. Thus the prefactor c3„1, ␣ (k)…has to be matched at tree level to Eq. 共14兲, and the only solution to the leading logarithm is FIG. 2. All the possible leg permutations from diagrams in the figure contribute to the renormalization constant Z ¯ defined in the text. 共a兲- and 共b兲-like diagrams, which do not break the assumed- to-work factorization hypothesis, are renormalized by Z ˆpreviously computed in 关9兴.共c兲and 共d兲factorization breaking diagrams give Z ␬ . The local operators are drawn as gray bullets; the two joined by a dashed line represent the ones contracted to give A2in Eqs. 共19兲– 共22兲through replacement in Eq. 共10兲 F. De SOTO AND J. RODRI ´GUEZ-QUINTERO PHYSICAL REVIEW D 64 114003 114003-4 c3„1, ␣ 共k兲…⫽3关gR共k兲兴3 冋 1⫹O 冉 1 log共k/⌳QCD兲 冊 册 .共27兲 Identifying the non-power-corrected term of Eq. 共18兲with the purely perturbative coupling constant, c1„k2/ ␮ 2, ␣ ( ␮ )…is known to be c1 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫽gR,pert共k2兲 冋 c0 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 册 1/2 .共28兲 Thus, we take from Ref. 关9兴the following leading order results: c0 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫽ 冉 ␣ 共k兲 ␣ 共 ␮ 兲 冊 ␥ 0/ ␤ 0 c2 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫽3gR 2共k2兲 冉 ␣ 共k兲 ␣ 共 ␮ 兲 冊 ⫺ ␥ ˆ0/ ␤ 0,共29兲 where, as usual, ␥ ˆ0is defined by d dln ␮ 2lnZ ˆ共 ␮ 2兲⫽⫺ ␥ ˆ0 ␣ 共 ␮ 兲 4 ␲ ⫹•••.共30兲 Equation 共18兲can then be applied to obtain ␣ MOM g ⫽gR 2/(4 ␲ ) which, after the appropriate Wick rotation, takes the following form in Euclidean space: ␣ MOM ˜ 共k2兲⫽ ␣ pert共k2兲 再 1⫹T共 ␮ 兲 k2 冋 ln 冉 k ⌳ 冊 册 ( ␥ ˆ0⫹ ␥ 0)/ ␤ 0⫺1 ⫻ 冉 2 冋 ln 冉 k ⌳ 冊 ln共 ␮ /⌳兲 册 ␬ 0/ ␤ 0 ⫺1 冊 冎 共31兲 with T共 ␮ 兲⫽6 ␲ 2 ␤ 0 具 A2 典 R, ␮ NC 2⫺1 冋 ln 冉 ␮ ⌳ 冊 册 ⫺( ␥ ˆ0⫹ ␥ 0)/ ␤ 0,共32兲 具 A2 典 R, ␮ being now the Euclidean condensate; and ␬ 0⫽ ␥ ¯ 0⫺ ␥ ˆ0,共33兲 as immediately follows from Eq. 共23兲if ␬ 0is defined from d dln ␮ 2lnZ ˆ ␬ 共 ␮ 2兲⫽⫺ ␬ 0 ␣ 共 ␮ 兲 4 ␲ ⫹•••.共34兲 The perturbative coefficients for the flavorless case, ␤ 0⫽11, ␤ 1⫽51, ␥ 0⫽13 2,共35兲 are universal, while for the coefficients in the MOM g scheme, ␤ 2was computed in 关3兴, and ␥ 1and ␥ 2in 关13兴: ␤ 2⯝4824, ␥ 1⫽29 8, ␥ 2⫽960. 共36兲 In a recent paper 关9兴we computed ␥ ˆ0from diagrams identical to these in Figs. 2共a兲and 2共b兲; ␬ 0, defined in Eq. 共33兲can be computed from diagrams 共c兲and 共d兲in Fig. 2 to give ␥ ˆ0⫽3NC 4, ␬ 0⫽⫺ 9NC 136 .共37兲 We proceeded to evaluate the diagrams for ␬ 0in total analogy with the calculation of those for ␥ ˆ0. For instance, we made the diagrams infrared safe by considering a momentum flow incoming to the local operator. The details of the procedure can be found in Ref. 关9兴. It should be noticed that ␬ 0Ⰶ ␥ ˆ0and that, in practice, ␬ 0/ ␤ 0⯝0 works as a good approximation to simplify Eq. 共31兲. In other words, the scheme given by vacuum insertion factorization reveals itself to be coherent: leading logarithm corrections violating the factorization induce a very small running with the renormalization scale for the factorized tree-level Wilson coefficient. Nevertheless, it is important to prove the assumption to work in order to obtain a good estimate of the Wilson coefficient for the asymmetric three-gluon Green function. To examine this question, we have performed the same combined fits shown in Ref. 关9兴for two- and three-point Green functions, at three loops for leading Wilson coefficients, to match lattice data taken from Refs. 关3,13兴. The Euclidean OPE formula for the two-point Green function is, from Ref. 关9兴, ZLatt MOM共k2,a兲⫽ZLatt MOM共 ␮ 2,a兲c0 冉 k2 ␮ 2, ␣ 共 ␮ 兲 冊 ⫻ 冋 1⫹R共 ␮ 兲 冉 lnk ⌳ 冊 ( ␥ 0⫹ ␥ ˆ0)/ ␤ 0⫺11 k2 册 , 共38兲 where ZLatt MOM共k2,a兲 ZLatt MOM共 ␮ 2,a兲 ⫽k2GR (2)共k2, ␮ 2兲⫹O共a2兲共39兲 and R共 ␮ 兲⫽6 ␲ 2 ␤ 0共Nc 2⫺1兲 冉 ln ␮ ⌳ 冊 ⫺ ␥ 0⫹ ␥ ˆ0/ ␤ 0 具 A2 典 R, ␮ .共40兲 The coefficient c0„k2/ ␮ 2, ␣ ( ␮ )…is taken now to be expanded at three loops in terms of the MOM g scheme ␣ (k); i.e., it will satisfy the same differential equation as ␥ ( ␣ ) in Eq. 共26兲 with a boundary condition that is apparent from Eq. 共29兲.In the following, all the scale-dependent quantities will be shown at ␮ ⫽10 GeV. Furthermore, we have checked that REMARKS ON THE DETERMINATION OF THE LANDAU . . . PHYSICAL REVIEW D 64 114003 114003-5 indeed neither the ratio of gluon condensate estimates nor ⌳MS depends on this last momentum scale. The quality of the fits as a function of the free exponent of ln(k/⌳)inEq.共31兲 has been explored and the results are shown in Fig. 3共a兲.We can conclude from these results that the approach given by vacuum insertion factorization provides a good estimate of this exponent. The results for two particular values of the exponent rin Fig. 3共a兲are of interest for the sake of comparison. The case r⫽⫺1 corresponds to the formula proposed in Ref. 关7兴to be matched to the lattice data: a perturbative three-loop formula ⫹a term c/p2,cbeing a constant. On the other hand, had we neglected the leading logarithm contributions ␥ ˆ0⫽ ␥ 0 ⫽0, the exponent would be r⫽1. It can be seen from Fig. 3 that both values of rgenerate rather less good fits to the lattice data. Then, Eqs. 共31兲,共32兲can be used to perform fits at two and three loops for the leading Wilson coefficients in order to estimate, from the asymmetric three-gluon Green function, the gluon condensate. The results of such fits, plotted in Fig. 3共b兲, are 兵 冑 具 A2 典 R, ␮ 其 alpha 兵 冑 具 A2 典 R, ␮ 其 prop ⫽3.65共4兲,⌳MS⫽283共15兲MeV, ␹ 2⫽1.95 共41兲 for the two-loop fit, and 兵 冑 具 A2 典 R, ␮ 其 alpha 兵 冑 具 A2 典 R, ␮ 其 prop ⫽1.7共3兲,⌳MS⫽260共18兲MeV, ␹ 2⫽1.18 共42兲 for the three-loop one. The impressive improvement from two to three loops suggests that the approach presented in this work permits a reasonable approximation to the Wilson coefficient. The ratio decreases to almost ␴ away from 1 and both estimates of ⌳MS and of the gluon condensate turn out to be close to the previous estimates obtained from the symmetric three-gluon Green function in Ref. 关9兴共see Table I兲. The scheme in this work and that of Ref. 关9兴differ only by the kinematics of the renormalization point. Such a different renormalization for the Green functions implies a different renormalization of the gluon condensate. However, the discrepancy for estimates of 具 A2 典 Rin the two works is expected not to be important,3as indeed can be seen in Table I. A comparison of these estimates, mainly the ones obtained from the gluon propagator, strongly supports the claimed rather large contribution from the A2condensate 关7–9兴which might be connected to with the tachyonic gluon mass scale studied in Ref. 关11兴. A negative hint regarding previous results from the symmetric three-point Green function is nevertheless the higher central value of the ratio in Eq. 42 (1.2 in Ref. 关9兴兲.In principle two possible sources of discrepancies can be expected: either three loops is still insufficiently accurate for the estimate of the perturbative part in the MOM g renormalization scheme, or there is a deviation from the assumed vacuum insertion 共or factorization兲approximation. Both effects would have a direct impact on the bigger ratio we obtain for the asymmetric MOM g scheme. The very good agreement between the gluon condensates estimated from the gluon propagator previously discussed seems to point to factorization breaking as the major contributing factor. Still, the ratio in Eq. 共42兲is only about 2 ␴ from 1, which is in our opinion a rather encouraging result. The 具 A2 典 deduced from the propagator is in fair agreement with previous estimates, even though it is biased by the factorization hypothesis through the fit of ⌳MS , which combines the propagator and ␣ S. This good result of the propagator, as well as Fig. 3共a兲, 3To estimate the discrepancy for the nonperturbative estimates of 具 A2 典 Rwe need to compute beyond leading logarithm corrections, which is beyond the scope of this work. FIG. 3. 共a兲Quality of the fit as a function of the exponent r⫽1⫺( ␥ ˆ0⫹ ␥ 0)/ ␤ 0in Eq. 共31兲. The dot stands for the value of rcomputed in this paper. In 共b兲is shown the fit of our lattice data for ␣ MOM g at three loops with the calculated anomalous dimension. F. De SOTO AND J. RODRI ´GUEZ-QUINTERO PHYSICAL REVIEW D 64 114003 114003-6 suggest that our formula describing the power corrections to ␣ Sup to the leading logarithm yields a good approximation of the exact one. A two-sided goal is thus achieved. 共i兲The results of Ref. 关9兴turn out to be confirmed by the use of a slightly different renormalization scheme. 共ii兲Vacuum insertion factorization applied to condensates playing the role of the OPE for the asymmetric three-gluon Green function results in a compact prediction for its OPE power corrections. The coefficient of the power correction has been computed to the leading logarithm, and thus a most important source of systematic uncertainty for the estimate of ⌳MS in Ref. 关7兴is eliminated. The latter is a positive feature because the ␤ function is perturbatively known at four loops in the asymmetric MOM g 关14兴and lattice evaluations, on the other hand, turn out to be statistically more precise in this last renormalization scheme. A last consequence of this work and those from Refs. 关8,9兴: together they lead to the conclusion that the Green function methods, the three-gluon vertex in particular, provide us with a reliable and precise enough estimate for the running coupling constant and ⌳MS , once power corrections are properly taken in consideration. We warmly acknowledge Ph. Boucaud, A. Le Yaouanc, J. P. Leroy, J. Micheli, and O. Pe `ne for fruitful comments and stimulating discussions held at the LPT in Orsay. The authors are also indebted to J. A. Caballero and M. Lozano for carefully reading the manuscript. This work was supported in part by the European Community’s Human Potential Program under contract HPRN-CT-2000-00145, Hadrons Lattice QCD, the Spanish Fundacio ´nCa ´mara and Spanish DGICyT under contracts PB98-1111 and FPA2000-1592-C03-02. 关1兴B. Alles, D. Henty, H. Panagopoulos, C. Parrinello, C. Pittori, and D.G. Richards, Nucl. Phys. B502, 325 共1997兲. 关2兴S. Capitani, M. Guagnelli, M. Lu ¨scher, S. Sint, R. Sommer, P. Weisz, and H. Wittig, Nucl. Phys. B 共Proc. Suppl.兲63, 153 共1998兲; Nucl. Phys. B544, 669 共1999兲. 关3兴Ph. Boucaud, J.P. Leroy, J. Micheli, O. Pene, and C. Roiesnel, J. High Energy Phys. 10, 017 共1998兲;12, 004 共1998兲. 关4兴G Burgio, F. Di Renzo, G. Marchesini, and E. Onofri, Phys. Lett. B 422, 219 共1998兲. For reviews and classic references, see V.I. Zakharov, Nucl. Phys. B385, 452 共1992兲; A.H. Mueller, QCD 20 Years Later 共World Scientific, Singapore, 1993兲, Vol. 1; B. Lautrup, Phys. Lett. 69B, 109 共1977兲; G. Parisi, ibid. 76B,65共1977兲; Nucl. Phys. B150, 163 共1979兲; G. ’t Hooft, in The Whys of Subnuclear Physics, Erice 1977, edited by A. Zichichi 共Plenum, New York, 1977兲; M. Beneke and V.I. Zakharov, Phys. Lett. B 312, 340 共1993兲; M. Beneke, Nucl. Phys. B307, 154 共1993兲; A.H. Mueller, ibid. B250, 327 共1985兲; Phys. Lett. B 308, 355 共1993兲; G. Grunberg, ibid. 304, 183 共1993兲; 325, 441 共1994兲. 关5兴M. Lavelle and M. Oleszczuk, Mod. Phys. Lett. A 7, 3617 共1991兲; J. Ahlbach, M. Lavelle, M. Schaden, and A. Streibl, Phys. Lett. B 275, 124 共1992兲. 关6兴G. Burgio, F. Di Renzo, C. Parrinello, and C. Pittori, Nucl. Phys. B 共Proc. Suppl.兲73, 623 共1999兲;74, 388 共1999兲; hep-ph/9808258. 关7兴Ph. Boucaud et al., J. High Energy Phys. 04, 006 共2000兲. 关8兴Ph. Boucaud, A. Le Yaouanc, J.P. Leroy, J. Micheli, O. Pe `ne, and J. Rodriguez-Quintero, Phys. Lett. B 493, 315 共2000兲. 关9兴Ph. Boucaud, A. Le Yaouanc, J.P. Leroy, J. Micheli, O. Pe `ne, and J. Rodriguez-Quintero, Phys. Rev. D 63, 114003 共2001兲. 关10兴M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, Nucl. Phys. B147, 385 共1979兲;B147, 447 共1979兲;B147, 519 共1979兲; M.A. Shifman, A.I. Vainshtein, M.B. Voloshin, and V.I. Zakharov, Phys. Lett. 77B,80共1978兲; S. Weinberg, The Quantum Theory of Fields 共Cambridge University Press, Cambridge, England, 1996兲, Vol. 2. 关11兴F.V. Gubarev and V.I. Zakharov, Phys. Lett. B 501,28共2001兲; F.V. Gubarev, M.I. Polikarpov, and V.I. Zakharov, hep-ph/9908292; K.G. Chetyrkin, S. Narison, and V.I. Zakharov, Nucl. Phys. B550, 353 共1999兲. 关12兴P. Pascual and E. de Rafael, Z. Phys. C 12, 127 共1982兲. 关13兴D. Becirevic, Ph. Boucaud, J.P. Leroy, J. Micheli, O. Pene, J. Rodriguez-Quintero, and C. Roiesnel, Phys. Rev. D 60, 094509 共1999兲;61, 114508 共2000兲. 关14兴K.G. Chetyrkin and T. Seidensticker, Phys. Lett. B 495,74 共2000兲. TABLE I. Comparison between results obtained for the threeloop fit in the present work and in a previous one 关9兴. This work Symmetric three point ⌳MS 260共18兲MeV 233共28兲MeV 兵 冑 具 A2 典 R, ␮ 其 prop 1.39共14兲GeV 1.55共17兲GeV 兵 冑 具 A2 典 R, ␮ 其 alpha 2.3共6兲GeV 1.9共3兲GeV REMARKS ON THE DETERMINATION OF THE LANDAU . . . PHYSICAL REVIEW D 64 114003 114003-7