Resummation in QCD Fractional Analytic Perturbation Theory

We describe the generalization of Analytic Perturbation Theory (APT) for QCD observables, initiated by Radyushkin, Krasnikov, Pivovarov, Shirkov, and Solovtsov, to fractional powers of coupling — Fractional APT (FAPT). The basic aspects of FAPT are shortly summarized. We describe how to treat heavy-...

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Datum:	2010
1. Verfasser:	Bakulev, A.P.
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Schlagworte:	Поля та елементарні частинки
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spelling	irk-123456789-132812010-11-04T14:33:59Z Resummation in QCD Fractional Analytic Perturbation Theory Bakulev, A.P. Поля та елементарні частинки We describe the generalization of Analytic Perturbation Theory (APT) for QCD observables, initiated by Radyushkin, Krasnikov, Pivovarov, Shirkov, and Solovtsov, to fractional powers of coupling — Fractional APT (FAPT). The basic aspects of FAPT are shortly summarized. We describe how to treat heavy-quark thresholds in FAPT and then show how to resum perturbative series in both the one-loop APT and FAPT. As an application, we consider the FAPT description of the Higgs boson decay Hº → bb¯. The main conclusion is: To achieve an accuracy of the order of 1%, it is enough to consider up to the third correction. Представлено узагальнення аналiтичної теорiї збурень (АТЗ) для КХД-амплiтуд, iнiцiйованої роботами Джонса, Соловцова i Ширкова, на дробовi степенi ефективного заряду – дробово-аналiтична теорiя збурень (ДАТЗ). Обговорено проблему порогiв важких кваркiв в ДАТЗ, пiсля чого показано, як можна пiдсумувати весь пертурбативний ряд в однопетльовiй АТЗ i ДАТЗ. Як додаток розглянуто розрахунок ширини розпаду хiггсiвського бозона Hº → bb¯. 2010 Article Resummation in QCD Fractional Analytic Perturbation Theory / A.P. Bakulev // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 15-19. — Бібліогр.: 18 назв. — англ. 2071-0194 http://dspace.nbuv.gov.ua/handle/123456789/13281 en Відділення фізики і астрономії НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Поля та елементарні частинки Поля та елементарні частинки
spellingShingle	Поля та елементарні частинки Поля та елементарні частинки Bakulev, A.P. Resummation in QCD Fractional Analytic Perturbation Theory
description	We describe the generalization of Analytic Perturbation Theory (APT) for QCD observables, initiated by Radyushkin, Krasnikov, Pivovarov, Shirkov, and Solovtsov, to fractional powers of coupling — Fractional APT (FAPT). The basic aspects of FAPT are shortly summarized. We describe how to treat heavy-quark thresholds in FAPT and then show how to resum perturbative series in both the one-loop APT and FAPT. As an application, we consider the FAPT description of the Higgs boson decay Hº → bb¯. The main conclusion is: To achieve an accuracy of the order of 1%, it is enough to consider up to the third correction.
format	Article
author	Bakulev, A.P.
author_facet	Bakulev, A.P.
author_sort	Bakulev, A.P.
title	Resummation in QCD Fractional Analytic Perturbation Theory
title_short	Resummation in QCD Fractional Analytic Perturbation Theory
title_full	Resummation in QCD Fractional Analytic Perturbation Theory
title_fullStr	Resummation in QCD Fractional Analytic Perturbation Theory
title_full_unstemmed	Resummation in QCD Fractional Analytic Perturbation Theory
title_sort	resummation in qcd fractional analytic perturbation theory
publisher	Відділення фізики і астрономії НАН України
publishDate	2010
topic_facet	Поля та елементарні частинки
url	http://dspace.nbuv.gov.ua/handle/123456789/13281
citation_txt	Resummation in QCD Fractional Analytic Perturbation Theory / A.P. Bakulev // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 15-19. — Бібліогр.: 18 назв. — англ.
work_keys_str_mv	AT bakulevap resummationinqcdfractionalanalyticperturbationtheory
first_indexed	2025-07-02T15:13:16Z
last_indexed	2025-07-02T15:13:16Z
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fulltext	RESUMMATION IN QCD FRACTIONAL ANALYTIC RESUMMATION IN QCD FRACTIONAL ANALYTIC PERTURBATION THEORY A.P. BAKULEV Bogoliubov Laboratory of Theoretical Physics (Joint Institute for Nuclear Research, Dubna 141980, Russia; e-mail: bakulev@ theor. jinr. ru ) PACS 11.15.Bt, 11.10.Hi, 12.38.Bx, 12.38.Cy c©2010 We describe the generalization of Analytic Perturbation Theory (APT) for QCD observables, initiated by Radyushkin, Krasnikov, Pivovarov, Shirkov, and Solovtsov, to fractional powers of coupling — Fractional APT (FAPT). The basic aspects of FAPT are shortly summarized. We describe how to treat heavy-quark thresholds in FAPT and then show how to resum perturbative series in both the one-loop APT and FAPT. As an application, we consider the FAPT description of the Higgs boson decay H0 → bb̄. The main conclusion is: To achieve an accuracy of the order of 1%, it is enough to consider up to the third correction. 1. APT and FAPT in QCD In the standard QCD Perturbation Theory (PT), we know that the Renormalization Group (RG) equa- tion das[L]/dL = −a2 s − . . . for the effective cou- pling αs(Q2) = as[L]/βf with L = ln(Q2/Λ2), βf = b0(Nf )/(4π) = (11 − 2Nf/3)/(4π)1. Then the one-loop solution generates the Landau pole singularity, as[L] = 1/L. Strictly speaking, the QCD Analytic Perturbation Theory (APT) was initiated by the paper by N. N. Bo- goliubov et al. [1], where the ghost-free effective cou- pling for QED has been constructed. Then in 1982, Radyushkin [2] and Krasnikov and Pivovarov [3] sug- gested, by using the same dispersion technique, the regu- lar (for s ≥ Λ2) QCD running coupling in a Minkowskian region, the famous 1 π arctan π L . After that in 1995, Jones and Solovtsov discovered the coupling which appears to be finite for all s and coincides with the Radyushkin one for s ≥ Λ2, namely A1[L] in Eq. (2b). Just in the same time, Beneke et al. [4, 5] within the renormalization- based approach and Shirkov and Solovtsov [6] within the same dispersion approach of [1] discovered the ghost-free coupling A1[L], Eq. (2a), in a Euclidean region. 1 We use notations f(Q2) and f [L] in order to specify the ar- guments we mean — squared momentum Q2 or its logarithm L = ln(Q2/Λ2), that is f [L] = f(Λ2 · eL) and Λ2 is usually referred to Nf = 3 region. But the Shirkov–Solovtsov approach, named APT, was more powerful: in the Euclidean domain, −q2 = Q2, L = lnQ2/Λ2, it generates the following set of images for the effective coupling and its n-th powers: {An[L]}n∈N; whereas, in the Minkowskian domain, q2 = s, Ls = ln s/Λ2, it generates another set, {An[Ls]}n∈N (see also in [7]). APT is based on the RG and causality, which guarantees the standard perturbative UV asymptotics and spectral properties. The power series ∑ m dma m s [L] is transformed into a non-power series ∑ m dmAm[L] in APT. By the analytization in APT for an observable f(Q2), we mean the “Källén–Lehmann” representation [ f(Q2) ] an = ∞∫ 0 ρf (σ) σ +Q2 − iε dσ (1) with ρf (σ) = 1 π Im [ f(−σ) ] . Then, in the one-loop ap- proximation, ρ1(σ) = 1/ √ L2 σ + π2 and A1[L] = ∞∫ 0 ρ1(σ) σ +Q2 dσ = 1 L − 1 eL − 1 , (2a) A1[Ls] = ∞∫ s ρ1(σ) σ dσ = 1 π arccos Ls√ π2 + L2 s , (2b) whereas the analytic images of higher powers (n ≥ 2, n ∈ N) are(An[L] An[Ls] ) = 1 (n− 1) ( − d dL )n−1(A1[L] A1[Ls] ) . (3) At first glance, the APT is a complete theory pro- viding tools to produce an analytic answer for any per- turbative series in QCD. But Karanikas and Stefanis [8] suggested the principle of analytization “as a whole” in the Q2 plane for hadronic observables calculated pertur- batively. More precisely, they proposed the analytiza- tion recipe for terms like ∫ 1 0 dx ∫ 1 0 dy αs ( Q2xy ) f(x)f(y), ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 15 A.P. BAKULEV which can be treated as an effective account for the log- arithmic terms in the next-to-leading-order approxima- tion of the perturbative QCD. This actually generalizes the analytic approach suggested in [9]. Indeed, in the standard QCD PT, one has also: (i) the factorization procedure in QCD that gives rise to the appearance of logarithmic factors of the type aνs [L]L; (ii) the RG evolution that generates evolution factors of the type B(Q2) = [ Z(Q2)/Z(µ2) ] B(µ2) which are re- duced in the one-loop approximation to Z(Q2) ∼ aνs [L] with ν = γ0/(2b0) being a fractional number. All that means that, in order to generalize APT in the “analytization as a whole” direction, one needs to con- struct analytic images of new functions aνs , a ν s L m, . . . . This task has been performed in the frames of the so- called FAPT suggested in [10, 11]. Now we briefly de- scribe this approach. In the one-loop approximation using recursive relation (3), we can obtain explicit expressions for Aν [L] and Aν [L]: Aν [L] = 1 Lν − F (e−L, 1− ν) Γ(ν) , (4a) Aν [L] = sin [ (ν − 1) arccos ( L√ π2+L2 )] π(ν − 1) (π2 + L2)(ν−1)/2 . (4b) Here, F (z, ν) is the reduced Lerch transcendental function which is an analytic function in ν. They have very interesting properties which were discussed exten- sively in our previous papers [10–13]. The construction of FAPT with a fixed number of quark flavors, Nf , is a two-step procedure: we start with the perturbative result [ as(Q2) ]ν , generate the spectral density ρν(σ) using Eq. (1), and then obtain analytic couplings Aν [L] and Aν [L] via Eqs. (2b). Here, Nf is fixed and factorized out. We can proceed in the same manner for Nf -dependent quantities [ αs(Q 2;Nf ) ]ν ⇒ ρ̄ν(σ;Nf ) = ρ̄ν [Lσ;Nf ] ≡ ρν(σ)/βνf ⇒ Āν [L;Nf ] and Āν [L;Nf ]; here, Nf is fixed, but not factorized out. The global version of FAPT [12] which takes heavy- quark thresholds into account is constructed along the same lines but starting from the global perturbative cou- pling [ α glob s (Q2) ]ν , being a continuous function of Q2 due to choosing different values of QCD scales Λf cor- responding to different values of Nf . Here, we illustrate the case of only one heavy-quark threshold at s = m2 4, corresponding to the transitionNf = 3→ Nf = 4. Then we obtain the discontinuous spectral density ρglob n (σ) = θ (Lσ < L4) ρ̄n [Lσ; 3]+ +θ (L4 ≤ Lσ) ρ̄n [Lσ + λ4; 4] , (5) with Lσ ≡ ln ( σ/Λ2 3 ) , Lf ≡ ln ( m2 f/Λ 2 3 ) and λf ≡ ln ( Λ2 3/Λ 2 f ) for f = 4 which is expressed in terms of fixed-flavor spectral densities with 3 and 4 flavors, ρ̄n[L; 3] and ρ̄n[L + λ4; 4]. However, it generates the continuous Minkowskian coupling Aglob ν [L] = θ (L<L4) ( Āν [L; 3] + Δ43Āν ) + +θ (L4≤L) Āν [L+ λ4; 4] (6a) with Δ43Āν = Āν [L4+λ4; 4]−Āν [L4; 3] and the analytic Euclidean coupling Aglob ν [L] Aglob ν [L] = Āν [L+ λ4; 4]+ + L4∫ −∞ ρ̄ν [Lσ; 3]− ρ̄ν [Lσ + λ4; 4] 1 + eL−Lσ dLσ (6b) (for more details, see [12]). 2. Resummation in the One-Loop APT and FAPT We consider now the perturbative expansion of a typical physical quantity, like the Adler function and the ratio R, in the one-loop APT. Due to a limited space of our presentation, we provide all formulas only for quantities in the Minkowski region: R[L] = ∞∑ n=1 dn An[L] . (7) We suggest that there exists the generating function P (t) for coefficients d̃n = dn/d1: d̃n = ∞∫ 0 P (t) tn−1dt with ∞∫ 0 P (t) dt = 1 . (8) To shorten our formulae, we use the following notation for the integral ∫∞ 0 f(t)P (t)dt: 〈〈f(t)〉〉P (t). Then the coefficients dn = d1 〈〈tn−1〉〉P (t), and we have, as has been shown in [14], the exact result for the sum in (7) R[L] = d1 〈〈A1[L− t]〉〉P (t) . (9) The integral with respect to the variable t here has a rig- orous meaning ensured by the finiteness of the coupling 16 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 RESUMMATION IN QCD FRACTIONAL ANALYTIC A1[t] ≤ 1 and the fast fall-off of the generating function P (t). In our previous publications [12, 15], we constructed generalizations of these results, first, to the case of the global APT, when heavy-quark thresholds are taken into account. Then one starts with a series of the type (7), where An[L] are substituted by their global analogs Aglob n [L] (note that, due to different normalizations of global couplings, Aglob n [L] ' An[L]/βf , the coefficients dn should be also changed). Then Rglob[L] = d1θ(L<L4) 〈〈 Δ4Ā1[t] + Ā1 [ L− t β3 ; 3 ]〉〉 P (t) + +d1θ(L≥L4) 〈〈 Ā1 [ L+λ4 − t β4 ; 4 ]〉〉 P (t) , (10) where Δ4Āν [t] ≡ Āν [ L4+λ4−t/β4; 4 ] −Āν [ L3−t/β3; 3 ] . The second generalization has been obtained for the case of the global FAPT. Then the starting point is a series of the type ∑∞ n=0 dn Aglob n+ν [L], and the result of summation is a complete analog of Eq. (10) with substi- tutions P (t)⇒ Pν(t) = 1∫ 0 P ( t 1− x ) ν xν−1dx 1− x , (11) d0 ⇒ d0 Āν [L], Ā1[L− t]⇒ Ā1+ν [L− t], and Δ4Ā1[t]⇒ Δ4Ā1+ν [t]. All needed formulas have been also obtained in parallel for the Euclidean case. 3. Applications to Higgs Boson Decay Here, we analyze the Higgs boson decay to a b̄b pair. For its width, we have Γ(H→ bb̄) = GF 4 √ 2π MH R̃S(M2 H) (12) with R̃S(M2 H) ≡ m2 b(M 2 H)RS(M2 H), and RS(s) is the R- ratio for the scalar correlator (see [10, 16] for details). In the one-loop FAPT, this generates the following non- power expansion2: R̃S[L] = 3m̂2 (1)× × { Aglob ν0 [L] + dS1 ∑ n≥1 d̃S n πn Aglob n+ν0 [L] } , (13) 2 Appearance of denominators πn in association with the coeffi- cients d̃n is due to the dn normalization. where m̂2 (1) = 9.05 ± 0.09 GeV2 is the RG-invariant of the one-loop m2 b(µ 2) evolution m2 b(Q 2) = m̂2 (1) α ν0 s (Q2) with ν0 = 2γ0/b0(5) = 1.04, and γ0 is the quark-mass anomalous dimension. This value of m̂2 (1) has been ob- tained using the one-loop relation [17] between the pole b-quark mass of [18] and the mass mb(mb). For the generating function P (t), we take the Lipatov- like model of [15] with {c = 2.4, β = −0.52} d̃ S n = cn−1 Γ(n+ 1) + β Γ(n) 1 + β , (14a) PS(t) = (t/c) + β c (1 + β) e−t/c . (14b) It gives a very good prediction for d̃ S n with n = 2, 3, 4, calculated in the QCD PT [16]: 7.50, 61.1, and 625 in comparison with 7.42, 62.3, and 620. Then we ap- ply the FAPT resummation technique to estimate how good is FAPT in approximating the whole sum R̃S[L] in the range L ∈ [11.5, 13.7] which corresponds to the range MH ∈ [60, 180] GeV2 with ΛNf=3 QCD = 189 MeV and Aglob 1 (m2 Z) = 0.122. In this range, we have (L6 = ln(m2 t/Λ 2 3)) R̃S[L] 3 m̂2 (1) = Aglob ν0 [L] + d S 1 π 〈〈Ā1+ν0 [ L+λ5− t πβ5 ; 5 ] 〉〉P S ν0 + + d S 1 π 〈〈Δ6Ā1+ν0 [ t π ] 〉〉P S ν0 (15) with P S ν0(t) defined via Eqs. (14) and (11). Now we analyze the accuracy of the truncated FAPT expressions R̃S[L;N ] = 3m̂2 (1) [ Aglob ν0 [L] + d S 1 N∑ n=1 d̃ S n πn Aglob n+ν0 [L] ] (16) and compare them with the total sum R̃S[L] in Eq. (15) using the relative errors ΔN [L] = 1 − R̃S[L;N ]/R̃S[L]. In Fig. 1, we show these errors for N = 2, N = 3, and N = 4 in the analyzed range of L ∈ [11, 13.8]. We see that already R̃S[L; 2] gives accuracy of the order of 2.5%, whereas R̃S[L; 3] of the order of 1%. Looking at Fig. 1, we understand that, only in order to have the accuracy better than 0.5%, we need to take the 4-th correction into account. We verified also that the uncertainty due to P (t)-modelling is small . 0.6%, while the on-shell mass uncertainty is of the order of 2%. The overall uncertainty then is of the order of 3% (see Fig. 2). ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 17 A.P. BAKULEV 12 12.5 13 13.5 0.005 0.01 0.015 0.02 0.025 L ∆2[L] ∆3[L] ∆4[L] Fig. 1. Relative errors ΔN [L], N = 2, 3, and 4, of the truncated FAPT in comparison with the exact summation result, Eq. (15) 100 120 140 160 180 1.5 2.0 2.5 3.0 3.5 MH [GeV] Γ∞ H→b̄b [MeV] Fig. 2. Width ΓH→bb̄ as a function of the Higgs boson mass MH in the resummed FAPT. The width of the shaded strip is due to the overall uncertainties induced by the uncertainties of the resummation procedure and the pole mass error-bars 4. Conclusions In this report, we have described the resummation ap- proach in the global versions of the one-loop APT and FAPT and argued that it produces finite answers, pro- vided the generating function P (t) of perturbative coef- ficients dn is known. The main conclusion is: To achieve an accuracy of the order of 1% it is enough to consider up to the third correction—in complete agreement with Kataev–Kim [17]. The d4 coefficient is needed only to estimate the corresponding generating functions P (t). This work was supported in part by the Russian Foundation for Fundamental Research, grants No. ь 07- 02-91557 and 08-01-00686, the BRFBR–JINR Coop- eration Programme, contract No. F08D-001, and the Heisenberg–Landau Programme under grant 2009. 1. N.N. Bogolyubov, A.A. Logunov, and D.V. Shirkov, So- viet Physics JETP 10, 574 (1960). 2. A.V. Radyushkin, JINR Rapid Commun. 78, 96 (1996); JINR Preprint, E2-82-159, Febr. 26, 1982; arXiv:[hep- ph]/9907228. 3. N.V. Krasnikov and A.A. Pivovarov, Phys. Lett. B 116, 168 (1982). 4. M. Beneke and V.M. Braun, Phys. Lett. B 348, 513 (1995). 5. P. Ball, M. Beneke, and V.M. Braun, Nucl. Phys. B 452, 563 (1995). 6. D.V. Shirkov and I.L. Solovtsov, JINR Rapid Commun. 2, [76] 5 (1996); Phys. Rev. Lett. 79, 1209 (1997); Theor. Math. Phys. 150, 132 (2007). 7. Y.A. Simonov, Phys. Atom. Nucl. 65, 135 (2002). 8. A.I. Karanikas and N.G. Stefanis, Phys. Lett. B 504, 225 (2001); 636, 330 (2006). 9. N.G. Stefanis, W. Schroers, and H.-C. Kim, Phys. Lett. B 449, 299 (1999); Eur. Phys. J. C 18, 137 (2000). 10. A.P. Bakulev, S.V. Mikhailov, and N.G. Stefanis, Phys. Rev. D 72, 074014, 119908(E) (2005); ibid. 75, 056005 (2007); 77, 079901(E) (2008). 11. A.P. Bakulev, A.I. Karanikas, and N.G. Stefanis, Phys. Rev. D 72, 074015 (2005). 12. A.P. Bakulev, Phys. Elem. Part. Nucl. 40, 715 (2009). 13. N.G. Stefanis, arXiv:0902.4805 [hep-ph]. 14. S.V. Mikhailov, JHEP 06, 009 (2007). 15. A.P. Bakulev and S.V. Mikhailov, in Proc. Int. Seminar on Contemp. Probl. of Part. Phys., dedicated to the mem- ory of I. L. Solovtsov, Dubna, Jan. 17–18, 2008, Eds. A. P. Bakulev et al. (JINR, Dubna, 2008), pp. 119–133 (arXiv:0803.3013). 16. P.A. Baikov, K.G. Chetyrkin, and J.H. Kühn, Phys. Rev. Lett. 96, 012003 (2006). 17. A.L. Kataev and V.T. Kim, in Proc. Int. Seminar on Contemp. Probl. of Part. Phys., dedicated to the mem- ory of I. L. Solovtsov, Dubna, Jan. 17–18, 2003, Eds. A.P. Bakulev et al. (JINR, Dubna, 2008), pp. 167–182 (arXiv:0804.3992); plenary talk presented by A.L.K. at XII Int. Work- shop on Advanced Computing and Analysis Techniques in Physics Research, November 3–7, 2008, Erice, Sicily, Italy (arXiv:0902.1442). 18. J.H. Kühn and M. Steinhauser, Nucl. Phys. B 619, 588 (2001). Received 05.10.09 18 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 RESUMMATION IN QCD FRACTIONAL ANALYTIC ПЕРЕПIДСУМОВУВАННЯ У ДРОБОВО-АНАЛIТИЧНIЙ КХД-ТЕОРIЇ ЗБУРЕНЬ О.П. Бакулев Р е з ю м е Представлено узагальнення аналiтичної теорiї збурень (АТЗ) для КХД-амплiтуд, iнiцiйованої роботами Джонса, Соловцова i Ширкова, на дробовi степенi ефективного заряду – дробово- аналiтична теорiя збурень (ДАТЗ). Обговорено проблему по- рогiв важких кваркiв в ДАТЗ, пiсля чого показано, як можна пiдсумувати весь пертурбативний ряд в однопетльовiй АТЗ i ДАТЗ. Як додаток розглянуто розрахунок ширини розпаду хiггсiвського бозона H → bb̄. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 19

Resummation in QCD Fractional Analytic Perturbation Theory

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