Novel solutions of RG equations for α(s) and β(α) in the large N c limit

Transcription

1 Novel solutions of RG equations for α(s) and β(α) in the large N c limit Yu.A. Simonov Abstract General solution of RG equations in the framework of background perturbation theory is written down in the large N c limit. A simplified (model) approximation to the general solution is suggested which allows to write β(α) andα(β) to any loop order. The resulting α B (Q ) coincides asymptotically at large Q with standard (free) α s,saturates at small Q 0 and has poles at time-like Q in agreement with analytic properties of physical amplitudes in the large N c limit. 1

2 1 The running α s of Standard Perturbation Theory (SPT) has unphysical singularities (Landau ghost pole and cuts) in the Euclidean region; which contradict analiticity of physical amplitudes and are not observed in lattice calculations of α s (Q )[1]. As it was shown in [, 3] the Background Perturbation Theory (BPT) is free of these defects and the coupling constant α B of BPT displays an important property of IR freezing (saturation) with α B (Q =0) 0.5 [,3]. This behaviour of α B is well confirmed by experimental data on spin splitting of quarkonia levels [4, 5] and by lattice data [6]. Analytic properties of α B (Q )intheq plane were not studied heretofore, and is the topic of the present letter. To this end the large N c limit is exploited which ensures that any physical amplitude has only simple poles [7]. This requires that α B (Q ) should be a meromorphic function of Q. To find it explicitly we study below the general solution of RG equations and suggest a simple model approximation to it, yielding β(α) toallorders and explicit form of α B (Q ). Resulting α B (Q ) can be compared with α s (Q ) of SPT and coincides with the latter asymptotically at large Q. Separating background field B µ in the total gluonic field A µ = B µ + a µ and expanding in ga µ one gets the BPT series, where coefficients depend both on external momenta and the background field gb µ, which turns out to be RG invariant, if background gauge is chosen [8]. Correspondingly α B g satisfies the standard RG equations, where momenta and averaged 4 background characteristic tr <F µ1 ν 1 (B)F µ ν (B)...F µnνn (B) > enter together and will be denoted {Pi }. One starts with the standard definition dα B (µ) d ln µ = β(α B) (1) and represents β(α B ) in the form β(α) = β 0 α [1 β 1 β 0 ϕ ( 1 α )] ()

3 where ϕ(z) is an arbitrary function with some conditions to be imposed below, and prime means derivative. Solving (1) with the help of () one has α B = 4 β[ln µ + χ + β 1 ϕ( 1 β0 α B )] (3) Here χ is an arbitrary dimensionless function of {Pi }, while ϕ is partly fixed by few coefficients of expansion of β(α), β(α) = β 0 α β 1 4 α3 β 64 3 α4 β 3α 5 (4) 4 (4) (Note that β i are defined as in [9], for n f = 0,N c =3andMS scheme β 0 =11, β 1 =51, β = 857, β 3 = ) At this point we shall use the large N c limit and require that α B (Q )be a meromorphic function of Q. Consider e.g. the process e + e into hadrons, and the photon self-energy part Π(Q ), which has a pole expansion [3] Π(Q,α B )= n=0 C n (α B ) Q + Mn (α + subtractions (5) B) Expanding in α B and doing a partial summation (equivalent to the summation of leading logarithms) one obtains a running α B (Q ) which has poles at some generally speaking, shifted positions as compared to the poles of the lowest order Mn (0). The latter are proportional to n; M n (0) = m 0 n + M 0 ; n =0, 1,,..., m 0 =4σ, C n(0) = m 0 [3]. The lowest (partonic) approximation thus reduces to Π(Q, 0) ψ (Q +M0 ),whereψ(z) = Γ (z). m Γ(z) To specify the form of the function ϕ( 1 α B )andχ(q ), we shall use two requirements: i) the correspondence principle with the SPT, which tells that for Q κ, where κ is the scale of nonperturbative vacuum fields, our solution should coincide with that of SPT. ii) α B (Q ) should be meromorphic function of Q, and hence both χ and ϕ should be meromorphic functions of their arguments (Q and 1 α B respectively). 3

4 The first requirement means that coefficients C n in the sum over poles should have finite limit C, i.e. one can represent this sum as the Euler function ψ(z) = Γ (z) plus a sum with fast decreasing coefficients, which we Γ(z) write symbolically as a finite sum. Thus the general solution satisfying i) and ii) can be written as ln µ + χ =ln m Λ + ψ(q + M 0 m )+ N 1 n=1 ) ϕ( 1 )=ψ( λ ( 1 + )+P α B α B α B b n Q + m n where is some constant and λ = 4 β 0. Note that our solution (6), (7) satisfies (1) and reproduces the first two coefficients in the expansion (4) when all b n =0andP 0, while our β,β 3 appear to be smaller; however the latter are scheme-dependent. It is easy to reproduce β, β 3 by choosing in (7) P 0. This choice of ϕ( 1 α B ) leads to a small numerical (around %) change in α B as compared to the minimal choice (8) for all Q 0. 3 In what follows we confine ourselves to the minimal model of (6), (7) with all b n,andp identically zero, which yields α B = 4 [ ln m β 0 Λ + + M ψ(q 0 )+ β 1 ψ( λ ] + ) (8) m β0 α B Eq.(8) defines α B as a meromorphic function of Q in the whole Q complex plane. Moreover, Eq.(8) defines α B to all orders in the loop expansion obtained by iteration of the last term in the denominator of (8) and will be compared below to the SPT loop expansion. Consider e.g. large Q, Q +M0 1. Then m using the asymptotic expansion for z ψ(z) =lnz 1 z B k, (9) k=1 kzk where B n are Bernoulli numbers, one obtains to the lowest order α (0) B = 4 (10) β 0 ln( Q +M0 ) Λ 4 (6) (7)

5 which explicitly shows the AF behaviour for large Q and absence of Landau pole (for M0 > Λ ). The forms () and (8) admit in general poles of α B (Q ) for Q 0 and poles of β(α B ) for α B 0, which would be unphysical. The latter poles are defined by equation 1 = β 1λ β 0 ψ ( λ + ), and are excluded from the α region α 0when > 0 =1.55. This latter condition also excludes singularities of α B (Q )(8),atQ > 0. At the same time β(α) has a point of condensing zeros on the negative side of α = 0, andβ(α) is monotonically decreasing when α, as it is shown in Table 1. Note that a pole of β(α) at α = α > 0 leads to a situation when real solutions α(q) exist only for α α,q>q. 4 Although α B (Q ) does not have poles for > 0 in the Euclidean region, Q > 0, it has an infinite number of poles for Q < 0. These poles are given by zeros of the denominator in (8), which can be written as Q s n = s (0) n δ n, s (0) n M n = M 0 + nm (11) and δ is then found from (8) to be [10] δ 1 n = ln M n Λ. (1) Hence poles of α B are shifted as compared with poles of ψ ( ) Q +M0 m,the latter describe the physical states of (double) hybrids - the background analog of gluon loops renormalizing α s in SPT. In the vicinity of the pole α B has the form [10] α B (s s n δ) = (4/β 0 ψ ( )) ψ (z n )(z z n ), z = M 0 s m (13) Hence perturbative series of BPT is a sum over poles of α B and nonperturbative poles of Π(Q, 0) in (5) (dual to partonic contribution), all poles being in the region s>m 0 δ 0 > 0 and this situation is in accord with the large N c analytic properties. Now one has to show that the physically averaged value of α B (s) corresponds to the AF behaviour, as it is seen e.g. 5

6 in the e + e annihilation. To this end one one can choose two equivalent procedures: i) to introduce the width of the pole in α B (s), or ii) equivalently to consider the BPT series at the shifted value of s, s = s(1 + iγ) (see e.g.[11]). In what follows we shall exploit the second path. Considering Imα B above the real axis, at s(1 + iγ) with s large and γ fixed γ, allows to use asymptotics (10) (in one loop order) with the result 4( γ) Imα( s(1 + iγ)) (14) b 0 [ln M 0 s +( γ) Λ ] Another form of discontinuity results from the expansion of the Adler function D(Q ) = Q dπ(q ), from which the hadronic ratio R(s) canbe d ln Q obtained as [1]. R(s) = 1 s+iε D(σ) dσ i s iε σ = 1 D(σ) dσ (15) i C(s) σ Here C(s) is the circle of radius s around the origin, comprising all singularities of D(σ) in Euclidean region. Similarly to [1] one can define the operation Φ{ α B } transforming the known perturbative series for R(s); [ { } { (αb ) }] R(s) =N c e αb q 1+ Φ + d Φ (16) where q { (αb ) } k Φ = 1 i C(s) ( αb (σ) ) k dσ σ (17) We shall consider the situation, when one can use the logarithmic asymptotics (10) for α B in the Minkowskian region. This happens in the physical situation described above: when poles have imaginary parts (widths of resonances), or equivalently when one consider effective R(s) = 1 (Π(s(1 + i iγ)) Π(s(1 iγ)) with fixed γ (so that requirement arg(z = M 0 s ) <is m fulfilled). In this case one obtains, inserting in the integral (17) the asymptotic form (10) (see [10] for details) Φ } {ᾱb = 4 b 0 { 1 arctg ln s/λ + M 0 ln 3 s/λ +0 6 ( )} M 0 sln s/λ (18)

7 5 One can now compare the power ( loop expansion ) of α B in (8) in powers of α (0) B (10) with the SPT expression for α s [9, 13]. To this end we shall write it as follows: { α B = α (0) B 1 β [ 1 ln L β0l + 4β 1 (ln L 1 β0l 4 ) + b] } (19) where L ln( Q +M0 ) and for n Λ f =0, =1.5 b = β 0 β 1 ( 1 ) 1 4 = (0) This should be compared with the MS value of SPT, b MS =0.6 (n f =0). To bring our theoretical expressions for β, β 3 in agreement with the computed MS values (see [9] for the corresponding references), one should generalized the minimal model considered above by adding a term to ϕ( 1 ): α ϕ ( 1 α )=λψ ( λ α + )+ Cα 1+α η (1) Choosing C =1.53 and arbitrary real η, one obtains both β, β 3 in close agreement (within 1%) with MS values. In this case also b in (0) gets contribution from the new term and becomes b = b + β 0 β 1 C =0.63, which agrees with the SPT [9]. We study numerically both minimal model and the corrected one as explained above and present the results for β(α) intable1 and for α B (Q ) in Table. From Table 1 it is clear that nonperturbative β(α), namely β min (α), calculated as in (), (7) with P 0andβ ext (α) calculated as in (), (1), both decrease more moderately (as (-const α )) at α>1ascomparedto purely perturbative 3 loop expression (4). This might indicate that the series (4) is asymptotic and should be cut off after few first terms (note that β 3 is larger than previous three coefficients). For Q =(10GeV) and M 0 = 1 GeV the correction in α B (Q ) introduced by replacements (1) is around % and it decreases for Q growing. Defining Λ Λ B =Λ MS (since α B and α s practically coincide for Q > 40 (GeV) ) one can compare exact α B (Q ) obtained by solving (8) with α s (Q ) 7

8 (two loop) of SPT and observe a significant difference only at small Q, Q 10 GeV. One can see in Table that the difference of α (min) B and α s (loop) is 0% at 1 GeV and drops to 10% at 5 GeV, which implies that the use if α (min) B instead of α s in the Euclidean region would not produce much change in the region Q > 1GeV while it improves the situation for Q 1GeV,where α (min) B saturates in agreement with experiment and lattice data (see [1]-[6] and [14] for the discussion of this point). 6 We have used the large N c limit to simplify analysis of analytic properties of background coupling constant α B. This has enabled us to study α B (Q ) also for time-like Q and to suggest a simplified model solution of exact RG equations, which yields α B to all orders and prescribes the nonperturbative expression for β(α). The resulting solution for α B coincides with the standard α s at large Q and displays familiar AF properties. At the same time α B (Q ) stays finite everywhere in the Euclidean region, demonstrating phenomenon of IR saturation. For time-like Q the model α B (Q )inthelargen c limit acquires pole singularities, which have been treated above in the averaging procedure. It is worth nothing that our minimal model correctly reproduces both scheme-independent coefficients β 0,β 1. At the same time scheme-dependent coefficients β,β 3 etc. are defined by additional pole terms with arbitrary constants in them. Several improvements are possible with the suggested model solution. First, one can consider n f 0,N c finite, in which case poles would acquire width. This will not change much the averaged values of α B (s) calculated above, but will enable one to make local predictions of resonance behaviour. Secondly, our model is too simplified, since ψ( Q +M0 ) m appearing in (8), is responsible only for the equidistant spectrum of hybrids, (equivalent of gluon loops), but this spectrum should not be equidistant for lowest states, which yields power corrections, considered in [10]. The author is grateful to N.O.Agasian for helpful discussions; a partial financial support of RFFI grants, , and INTAS grants and are gratefully acknowledged. 8

9 α β min (α) β ext (α) β (3loop) MS Table 1: The QCD β-function β(α) for n f =0,N c = 3 obtained numerically to all orders in α for the minimal model β min (α) Eq.()withϕ = λ +, and α =1.5; λ =1.14, and for the extended model β ext with ϕ from (1). The bottom line refers to β(α) inms as given in Eq. (4). Q (GeV ) α (min) B α (ext) B α (loop) B α s (loop) Table : α B (Q ) computed numerically to all orders within minimal model, Eq.(8), with Λ = 0.37 GeV, = 1.5, m = M 0 = 1 GeV and in extended model, Eqs. (3), (1), with η =3, C =1.53 reproducing all known coefficients β n,n=0,...3 inms scheme for n f = 0. The last two lines refer to the two-loop approximation of α B and α s (in SPT) with the same parameters. References [1] S.P.Booth et al. Phys. Lett. B94 (199) 385; G.S.Bali and K.Schilling, Phys. Rev. D47(1993) 661; G.S.Bali, Phys. Lett. B460 (1999) 170. [] Yu.A.Simonov, Yad. Fiz. 58 (1995) 113, JETP Lett. 57 (1993) 513, hepph/ Yu.A.Simonov, in: Lecture Notes in Physics, v. 479, p. 139 (1996). [3] A.M.Badalian, Yu.A.Simonov, Phys. At. Nuclei 60 (1997) 630. [4] A.M.Badalian, V.L.Morgunov, Phys. Rev. D60 (1999) [5] A.M.Badalian and B.L.G.Bakker, Phys. Rev. D6 (000)

10 [6] A.M.Badalian and D.S.Kuzmenko, hep-ph/ [7] G. thooft, Nucl. Phys. B7 (1974) 461, E.Witten, Nucl. Phys. B156 (1979) 69. [8] L.F.Abbot, Nucl. Phys. 185 (1981) 189. [9] Review of Particle Physics, Europ. Phys. Journ C15 (000) 9 [10] Yu.A.Simomov, hep-ph/ ; Phys. At. Nucl. (in press) [11] E.C.Poggio, H.R.Quinn and S.Weinberg, Phys. Rev. D13 (1976) 1958 [1] A.Radyushkin, JINR preprint E (198) and hep-ph/99018; N.V.Krasnikov and A.A.Pivovarov, Phys. Lett. 116 B (198) 168; see also D.V.Shirkov hep-ph/00183 [13] F.J.Yndurain, The Theory of Quark and Gluon Interactions, 3d edition, Springer- Verlag, Berlin-Heidelberg, 1999 [14] A.C.Mattingly and P.M.Stevenson, Phys. Rev. D49 (1994) 437; Yu.L.Dokshitzer, hep-ph/ ; hep-ph/