R-evolution: Improving perturbative QCD
Andre H. Hoang, Ambar Jain, Ignazio Scimemi, Iain W. Stewart
aa r X i v : . [ h e p - ph ] A ug MIT–CTP 4062MPP-2009-156arXiv:0908.3189 R -evolution: Improving perturbative QCD Andr´e H. Hoang, Ambar Jain, Ignazio Scimemi, and Iain W. Stewart Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6, 80805 M¨unchen, Germany Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Departamento de Fisica Teorica II, Universidad Complutense de Madrid, 28040 Madrid, Spain
Perturbative QCD results in the MS scheme can be dramatically improved by switching to ascheme that accounts for the dominant power law dependence on the factorization scale in theoperator product expansion. We introduce the “MSR scheme” which achieves this in a Lorentz andgauge invariant way. The MSR scheme has a very simple relation to MS, and can be easily used toreanalyze MS results. Results in MSR depend on a cutoff parameter R , in addition to the µ of MS. R variations can be used to independently estimate i) the size of power corrections, and ii) higherorder perturbative corrections (much like µ in MS). We give two examples at three-loop order, theratio of mass splittings in the B ∗ – B and D ∗ – D systems, and the Ellis-Jaffe sum rule as a functionof momentum transfer Q in deep inelastic scattering. Comparing to data, the perturbative MSRresults work well even for Q ∼ Introduction and Formalism
The operator product expansion (OPE) is an impor-tant tool for QCD. In hard scattering processes two im-portant scales are Q , a large moment transfer or mass,and Λ QCD , the scale of nonperturbative matrix elements.The Wilsonian OPE introduces a factorization scale Λ f ,where Λ QCD < Λ f < Q , and expands in Λ QCD /Q . Con-sider a dimensionless observable σ whose OPE is σ = C W ( Q, Λ f ) θ W (Λ f ) + C W ( Q, Λ f ) θ W (Λ f ) Q p + . . . . (1)The C W , are dimensionless Wilson coefficients contain-ing contributions from momenta k > Λ f with pertur-bative expansions in α s , and θ W , = hO , i W are non-perturbative matrix element with mass dimensions 0 and p , containing contributions from k < Λ f . If C W , ( Q, Λ f )are expanded they contain an infinite series of terms,(Λ f /Q ) n , modulo ln m (Λ f /Q ) terms, and this reflects thefact that C W , only include contributions from momenta k > Λ f . The Wilsonian OPE provides a clean separationof momentum scales, but can be technically challengingto implement. In particular, it is difficult to define Λ f andretain gauge symmetry and Lorentz invariance, and per-turbative computations beyond one-loop are atrocious.A popular alternative is the OPE with dimensionalregularization and the MS scheme, which preserves thesymmetries of QCD and provides powerful techniques formultiloop computations. In this case Eq. (1) becomes σ = ¯ C ( Q, µ )¯ θ ( µ ) + ¯ C ( Q, µ ) ¯ θ ( µ ) Q p + . . . , (2)where µ is the renormalization scale and bars are usedfor MS quantities. In MS the ¯ C i are simple series in α s ,¯ C i ( Q, µ ) = 1 + ∞ X n =1 b n (cid:0) µQ (cid:1) h α s ( µ )(4 π ) i n , (3)with coefficients b n ( µ/Q ) = P k =0 b nk ln k ( µ/Q ) contain-ing only ln µ/Q . We will always rescale σ and the matrix elements ¯ θ i such that ¯ C i = 1 at tree level. In MS allpower law dependence on Q is manifest and unique ineach term of Eq. (2). Also simple renormalization groupequations in µ , like d ln ¯ C ( Q, µ ) /d ln µ = ¯ γ [ α s ( µ )], canbe used to sum large logs in Eq. (2) if Q ≫ Λ QCD . C Wi ( Q, Λ f ) and ¯ C j ( Q, µ ) are related to each other inperturbation theory, so Eqs. (1) and (2) are just the sameOPE in two different schemes. The renormalization scale µ in MS plays the role of Λ f . This is precisely true for log-arithmic contributions, ln µ ↔ ln Λ f , and here the Wilso-nian picture of scale separation in ¯ C i and ¯ θ i carries over.However, the same is not true for power law dependenceson Λ f . MS integrations are carried out over all momenta,so the ¯ C i actually contain some contributions from arbi-trary small momenta, and the ¯ θ i have contributions fromarbitrary large momenta. For the power law terms thereis no explicit scale separation in MS, and correspondinglyno powers of µ appear in Eq. (3). While this simplifieshigher order computations, it is known to lead to facto-rial growth in the perturbative coefficients. For ¯ C , onehas b n +1 ( µ/Q ) ≃ ( µ/Q ) p n ! [2 β /p ] n Z at large n [1], forconstant Z . In practice this sometimes leads to poor con-vergence already at one or two loop order in QCD. Thispoor behavior is canceled by corresponding instabilitiesin ¯ θ , and is referred to as an order- p infrared renor-malon in ¯ C canceling against an ultraviolet renormalonin ¯ θ [2, 3, 4]. The cancellation reflects the fact that theMS OPE does not strictly separate momentum scales.The OPE can be converted to a scheme that removesthis poor behavior, but still retains the simple compu-tational features of MS. Consider defining a new “ R -scheme” for C by subtracting a perturbative series C ( Q, R, µ ) = ¯ C ( Q, µ ) − δC ( Q, R, µ ) ,δC ( Q, R, µ ) = (cid:16) RQ (cid:17) p ∞ X n =1 d n (cid:0) µR (cid:1) h α s ( µ )(4 π ) i n , (4)with d n ( µ/R ) = P k =0 d nk ln k ( µ/R ). If for large n thecoefficients d n are chosen to have the same behavior as b n ,so d n +1 ( µ/Q ) ≃ ( µ/R ) p n ![2 β /p ] n Z , then the factorialgrowth in ¯ C ( Q, µ ) and δC ( Q, R, µ ) cancel, C ( Q, R, µ ) ∼ h µ p Q p − R p Q p µ p R p i X n n ! h β p i n Z . (5)Thus the R -scheme introduces power law dependenceon the cutoff, ( R/Q ) p , in C ( Q, R, µ ), which capturesthe dominant (Λ f /Q ) p behavior of the Wilsonian C W .In practice this improves the convergence in C evenat low orders in the α s series. The dominant effect ofthis change is compensated by a scheme change to ¯ θ ,¯ θ ( µ ) = θ ( R, µ ) − [ Q p δC ( Q, R, µ )]¯ θ ( µ ), and the new θ will exhibit improved stability. In the R -scheme theOPE becomes σ = C ( Q, R, µ )¯ θ ( µ ) + ¯ C ( Q, µ ) θ ( R, µ ) /Q p + ¯ C ′ ( Q, µ ) θ ′ ( R, µ ) /Q p + . . . , (6)where θ ′ = [ Q p δC ]¯ θ and ¯ C ′ = 1 − ¯ C ∼ α s . Both C and θ are free of order- p renormalons. The series in¯ C ′ θ ′ is Borel summable. In all examples below ¯ θ is alsorenormalon free. The above procedure may be repeatedfor higher renormalons and the higher power terms inthe OPE indicated by ellipses, to improve the behaviorof these terms as well. At the order at which we work, wewill consistently set ¯ C = 1 and drop θ ′ in the following.To setup an appropriate R-scheme it remains to de-fine the d n . In the renormalon literature such schemechanges are well known for masses. For OPE predictionsa “renormalon subtraction” (RS) scheme has been imple-mented in Ref. [5]. In RS an approximate result for theresidue of the leading Borel renormalon pole is used todefine the d n , which adds a source of uncertainty.For our analysis we define the “MSR” scheme for C by simply taking the coefficients of the subtraction to beexactly the MS coefficients. In general it is more con-venient to use ln ¯ C rather than ¯ C , since this simplifiesrenormalization group equations. Writing the series asln ¯ C ( Q, µ ) = ∞ X n =1 a n ( µ/Q ) h α s ( µ )(4 π ) i n , (7)with a n ( µ/Q ) = P k =0 a nk ln k µ/Q we define the MSRscheme by the seriesln C ( Q, R, µ ) ≡ ∞ X n =1 n a n (cid:0) µQ (cid:1) − R p Q p a n (cid:0) µR (cid:1)o α ns ( µ )(4 π ) n . (8)This definition still cancels the order- p renormalon forlarge n , as in Eq. (5). It yields the very simple relation C ( Q, R, µ ) = ¯ C ( Q, µ ) (cid:2) ¯ C ( R, µ ) (cid:3) − ( R/Q ) p , (9)which must be expanded order-by-order in α s ( µ ) to re-move the renormalon. Thus the coefficient C ( Q, R, µ )for the MSR scheme is obtained directly from the MSresult. Note C ( Q, Q, µ ) = 1 to all orders. The appropri-ate p is obtained from the MS OPE by p =dimension(¯ θ ) − dimension(¯ θ ). MSR preserves gauge invariance, Lorentzsymmetry, and the simplicity of MS.The appropriate values for R in Eqs. (4,6,9) are con-strained by power counting and the structure of largelogs in the OPE. The power counting ¯ θ ∼ Λ p QCD im-plies θ ∼ Λ p QCD , so for the matrix element we need R ∼ µ > ∼ Λ QCD (meaning a larger value where pertur-bation theory for the OPE still converges), which mini-mizes ln( µ/ Λ QCD ) and ln( µ/R ) terms in θ ( R, µ, Λ QCD ).On the other hand, C ( Q, R, µ ) has ln( µ/Q ) and ln( µ/R )terms, and for R ∼ Λ QCD no choice of µ avoids large logs.For R ∼ µ ∼ Q we can minimize the logs in C ( Q, R, µ ),but not in θ ( R, µ, Λ QCD ). When the OPE is carried outin MS this problem is dealt with using a µ -RGE to sumlarge logs between Q and Λ QCD . For MSR we must use R -evolution, an RGE in the R variable [6]. The appro-priate R-RGE is formulated with µ = R to ensure thereare no logs in the anomalous dimension. For C , R ddR ln C ( Q, R, R ) = ¯ γ [ α s ( R )] − (cid:16) RQ (cid:17) p γ [ α s ( R )] , (10)where ¯ γ [ α s ] = P ∞ n =0 ¯ γ n [ α s ( R ) / π ] n +1 and γ [ α s ] = P ∞ n =0 γ n [ α s ( R ) / π ] n +1 are the MS and R anomalous di-mensions. Here γ n − = pa n − P n − m =1 m a m β n − m − and we are using the MS β -function µd/dµα s ( µ ) = − α s ( µ ) / (2 π ) P ∞ n =0 β n [ α s ( µ ) / π ] n . The choice in Eq. (8)keeps Eq. (10) simple. In cases where ¯ γ is absent weexpect Eq. (10) to converge to lower scales due to the( R/Q ) p factor multiplying γ . For R > R the solutionof Eq. (10) is [ U µ = U µ ( R , R )] C ( Q, R , R ) = C ( Q, R , R ) U R ( Q, R , R ) U µ , (11)where U µ is a usual MS evolution factor and U R is the R -evolution. For p = 1 the complete solution for U R wasobtained in Ref. [6]. It is straightforward to generalizethis to any p . At N k +1 LL order the (real) result is U R ( Q, R , R ) = exp (cid:26)(cid:16) Λ ( k )QCD Q (cid:17) p k X j =0 S j ( − p ) j e iπp ˆ b × p ( p ˆ b ) (cid:2) Γ( − p ˆ b − j, pt ) − Γ( − p ˆ b − j, pt ) (cid:3)(cid:27) , (12)with Γ( c, t ) the incomplete gamma function and t , = − π/ ( β α s ( R , )). Λ (0)QCD = Re t , Λ (1)QCD = Re t ( − t ) ˆ b ,and Λ (2)QCD = Re t ( − t ) ˆ b e − ˆ b /t are evaluated at a large ref-erence R with t = − π/ ( β α s ( R )), and ˆ b = β / (2 β ),ˆ b = ( β − β β ) / (4 β ), and ˆ b = ( β − β β β + β β ) / (8 β ). Defining ˜ γ n = γ n / (2 β ) n +1 the coefficientsof U R needed for the first three orders of R -evolution are S = ˜ γ , S = ˜ γ − (ˆ b + p ˆ b )˜ γ , (13) S = ˜ γ − (ˆ b + p ˆ b )˜ γ + (cid:2) (1+ ˆ pb )ˆ b + p ( p ˆ b +ˆ b ) / (cid:3) ˜ γ . Eq. (6) becomes C ( Q, R , R ) U R ( Q, R , R ) U µ ( R , R ) × θ ( R ) + θ ( R , R ) /Q p , and this result sums logs be-tween R ∼ Q and R ∼ Λ QCD . This gives natural R scales for coefficients and matrix elements in the OPE. Heavy Meson Mass Splittings in MSR
The MS OPE for the mass-splitting of heavy mesons,∆ m H = m H ∗ − m H for H = B, D , is ∆ m H =¯ C G ( m Q , µ ) µ ( µ ) + P i ¯ C i ( m Q , µ ) 2 ρ i ( µ ) / (3 m Q ) + O (Λ /m Q ), where m Q = m b or m c . Here µ = −h B v | ¯ h v gσ µν G µν h v | B v i / ρ i for i = πG, A, LS, ¯Λ G are O (Λ ) matrix elements [7],with ρ G ( µ ) = (3 / µ ( µ ). At the order of ouranalysis tree level values for the ¯ C i suffice, so with¯Σ ρ ( µ ) = (2 / (cid:2) ρ πG ( µ ) + ρ A ( µ ) − ρ LS ( µ ) + ρ G ( µ ) (cid:3) ,∆ m H = ¯ C G ( m Q , µ ) µ ( µ ) + ¯Σ ρ ( µ ) /m Q + . . . . (14)Taking the ratio of mass splittings r = ∆ m B / ∆ m D gives r = ¯ C G ( m b , µ )¯ C G ( m c , µ ) + ¯Σ ρ ( µ ) µ G ( µ ) (cid:16) m b − m c (cid:17) + . . . . (15)The first term in this OPE gives a purely perturba-tive prediction for r . ¯ C G is known to suffer from an O (Λ QCD /m Q ) infrared renormalon ambiguity [7], witha corresponding ambiguity in ¯Σ ρ ( µ ). The three-loopcomputation of Ref. [8] yields, r = 1 − . | α s − . | α s − . | α s at fixed order with µ = m c , and r = (0 . LL + ( − . ∆NLL + ( − . ∆NNLL inRGE-improved perturbation theory, with no sign of con-vergence in either case. In MS these leading predictionsare unstable due to the p = 1 renormalon in ¯ C G .Lets examine the analogous result in the MSR scheme∆ m H = C G ( m Q , R, µ ) µ ( µ ) + Σ ρ ( R, µ ) m Q + . . . . (16)Since p = 1 the MSR definition in Eq. (9) gives C G ( m Q , R, µ ) ≡ ¯ C G ( m Q , µ )[ ¯ C G ( R, µ )] − R/m Q , (17)where ¯ C G ( m, µ ) is obtained from Ref. [8] and we expandin α s ( µ ). The OPE in MSR at a scale R > ∼ Λ QCD gives r = C G ( m b , R , R ) C G ( m c , R , R ) + Σ ρ ( R , R ) µ G ( R ) (cid:16) m b − m c (cid:17) . (18)Large logs in C G ( m Q , R , R ) can be summed with the R-RGE in Eqs. (11–13). For simplicity we integrate out the b and c -quarks simultaneously at a scale R ≃ √ m b m c ≫ R ≃ Λ QCD . This scale for R keeps ln( R /m b,c ) small.With R-evolution and U R from Eq. (12) we have r = C G ( m b , R , R ) U R ( m b , R , R ) C G ( m c , R , R ) U R ( m c , R , R ) (19)+ Σ ρ ( R , R ) µ G ( R ) (cid:16) m b − m c (cid:17) . This expression is independent of R and R . Order-by-order, varying R about √ m b m c yields an estimate ofhigher order perturbative uncertainties, much like vary-ing µ in MS. For R the dependence cancels between the r LL MSR LO MS and expt r (q=u,d) expt r (q=s) NLO MS NNLO MS LO MS N NLL
MSR
NNLL
MSR LL N MSR R (GeV) FIG. 1: Perturbative predictions at leading order in 1 /m Q forthe ratio r of the B - B ∗ and D - D ∗ mass splittings in the MSR-scheme (solid) versus MS (dashed). The R dependence of thesolid red curve provides an estimate for the power correction,independent of the comparison with the experimental data.Neither R nor µ variation is shown in the figure. first term in r and the Σ ρ power correction. In MSR theterm Σ ρ ( R , R ) is ∼ Λ and can be positive or neg-ative. One may expect that there is a value of R whereΣ ρ ( R , R ) vanishes. Thus keeping only the first termin Eq. (19) and varying R > ∼ Λ QCD yields an estimatefor the size of this power correction. This technique goesbeyond the dimensional analysis estimates used in MS.Fig. 1 gives perturbative predictions for r at differ-ent orders using the first terms in Eqs. (15,19) with m b = 4 . m c = 1 . α s ( √ m b m c ) = 0 . β -function. The solid lines are from theMSR scheme, plotted as functions of R . The dashedlines are the fixed order MS results with µ = √ m b m c .The MSR results exhibit a dramatic improvement inconvergence over MS for a wide range of R values.Varying R = √ m b m c / √ m b m c at N LL(MSR)gives ∆ r ≃ ± . µ variation in the same range for N LO(MS)where ∆ r ≃ ± . R dependent curve, whose dependence cancels againstΣ ρ ( R , R ), so the residual R dependence provides amethod to estimate the size of this power correction. Therange R = 0 . . R below m c andabove Λ QCD and yields r = 0 . ± (0 . Σ ρ ± (0 . pert . . (20)This estimate for the Σ ρ power correction in MSR is ingood agreement with experiment, r expt = 0 .
886 ( D ( ∗ ) u,d , B ( ∗ ) u,d ) and 0.854 ( D ( ∗ ) s , B ( ∗ ) s ). MSR achieves a conver-gent perturbative prediction for r at leading order in theOPE, and a 1 /m Q power correction of moderate size, ∼ . QCD (1 /m c − /m b ) ∼ . Ellis-Jaffe sum rule in MSR
In MS the Ellis-Jaffe sum rule [9] for the pro-ton in DIS with momentum transfer Q is M ( Q ) = (cid:2) ¯ C B ( Q, µ ) θ B + ¯ C ( Q, µ )ˆ a / (cid:3) + ¯ θ ( µ ) /Q . ¯ C B, are M (Q) MSR MS RS NLO MS NLL
MSR
NNLO MS LO MS N NNLL
MSR LL N MSR
NLONNLOLO N RSRSRS Q (GeV) FIG. 2: Perturbative results for the Ellis-Jaffe sum rule in theMSR, RS, and MS schemes, at leading order in 1 /Q . For allcurves the one parameter, ˆ a , is fixed by data at Q ≃ Q (GeV) M (Q) LL N MSR R variation R variation LO MS N m variation FIG. 3: Uncertainty estimates in the MSR scheme and MSscheme for the Ellis-Jaffe sum rule at leading order in 1 /Q . known at 3 loops [10]. The two leading order terms arewritten so that both coefficients and matrix elements areseparately µ -independent: θ B = g A /
12 + a /
36 is givenby the axial couplings g A = 1 . a = 0 .
572 forthe nucleon and hyperon, while ˆ a is a Q independentMS matrix element. ¯ θ denotes all 1 /Q power correc-tions with their Wilson coefficients at tree level. The MScoefficients are affected by a p = 2 renormalon [11], whichis removed in the MSR scheme. Eq. (9) gives [ i = B, C i ( Q, R, R ) ≡ ¯ C i ( Q, R )[ ¯ C i ( R, R )] − R /Q . (21) With R -evolution the MSR OPE prediction is M ( Q ) = (cid:2) C B ( Q, R , R ) U BR ( Q, R , R ) θ B (22)+ C ( Q, R , R ) U R ( Q, R , R )ˆ a / (cid:3) + θ ( R , R ) /Q , where U B, R are given by Eq. (12) with p = 2 and the cor-responding ( a n ) B, determine the appropriate ( γ n ) B, .Figures 2,3 show perturbative predictions for the Ellis-Jaffe sum rule at leading power in 1 /Q , compared withproton data from Ref. [12]. We use α s (4 GeV) = 0 . β with 4 flavors. In Fig. 2, we show order-by-order results for the MS scheme at µ = Q , and for theresummed MSR scheme with R = Q and R = 0 . a = 0 .
141 so that MS and MSR agree with thedata for Q ≃ Q , but turns away at Q < ∼ Q values. The NLL MSR result already has theright curvature, and the NNLL and N LL curves furtherimprove the agreement. We also include predictions inthe RS scheme with subtraction scale ν f = 1 . µ in the range µ min ( Q ) < µ < Q .For Q > . µ min = Q/
2, while for
Q < . µ min = 1 . Q/ (1 . Q/ (1 GeV)). The red solid line is theMSR prediction, the red band is the perturbative uncer-tainty from varying R in the same range as was done for µ in MS, and the green band estimates the 1 /Q powercorrection by varying R = 0 . . − .
01 GeV < ∼ θ ( R , R ) < ∼ .
01 GeV in MSR, which isa much smaller power correction than the ∼ . es-timate obtained from naive dimensional analysis in MS.This work was supported by the EU network, MRTN-CT-2006-035482 (Flavianet), Spanish Ministry of Edu-cation, FPA2008-00592, the Office of Nuclear Physics ofthe U.S. Department of Energy, DE-FG02-94ER40818,the Alexander von Humboldt foundation, and the Max-Planck-Institut f¨ur Physik guest progam. [1] M. Beneke, Phys. Rept. , 1 (1999).[2] A. H. Mueller, Nucl. Phys. B250 , 327 (1985).[3] M. Luke et al., Phys. Rev.
D51 , 4924 (1995).[4] L. Maiani et al., Nucl. Phys.
B368 , 281 (1992).[5] F. Campanario and A. Pineda, Phys. Rev.
D72 , 056008(2005); A. Pineda, JHEP , 022 (2001).[6] A. H. Hoang, A. Jain, I. Scimemi, and I. W. Stewart,Phys. Rev. Lett. , 151602 (2008), 0803.4214.[7] A.Grozin and M. Neubert, Nucl. Phys. B508 , 311 (1997). [8] A. G. Grozin et al., Nucl. Phys.
B789 , 277 (2008).[9] J. R. Ellis and R. L. Jaffe, Phys. Rev. D9 , 1444 (1974).[10] S. A. Larin, T. van Ritbergen, and J. A. M. Vermaseren,Phys. Lett. B404 , 153 (1997).[11] D. J. Broadhurst and A. L. Kataev, Phys. Lett.
B315 ,179 (1993).[12] M. Osipenko et al., Phys. Rev.