Inclusive tau lepton hadronic decay in vector and axial-vector channels within dispersive approach to QCD
aa r X i v : . [ h e p - ph ] A ug Inclusive t lepton hadronic decay in vector and axial–vectorchannels within dispersive approach to QCD A.V. Nesterenko
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,Dubna, 141980, Russian FederationE-mail: [email protected]
Abstract.
The dispersive approach to QCD, which properly embodies the intrinsically nonperturbative constraints originatingin the kinematic restrictions on relevant physical processes and extends the applicability range of perturbation theory towardsthe infrared domain, is briefly overviewed. The study of OPAL (update 2012) and ALEPH (update 2014) experimental dataon inclusive t lepton hadronic decay in vector and axial–vector channels within dispersive approach is presented. Keywords: nonperturbative methods, low–energy QCD, dispersion relations, t lepton hadronic decay PACS:
The theoretical particle physics widely employs the methods based on dispersion relations. In particular, suchmethods have proved to be efficient in the extension of the range of applicability of chiral perturbation theory [1, 2],assessment of the hadronic light–by–light scattering [3], precise determination of parameters of resonances [4], andmany other issues.The dispersion relations render the kinematic restrictions on pertinent physical processes into the mathematical formand impose stringent nonperturbative constraints on relevant quantities, such as the hadronic vacuum polarizationfunction P ( q ) . These constraints have been properly embodied within dispersive approach to QCD [7, 8], whichprovides unified integral representations for P ( q ) , related function R ( s ) , which is identified with the so–called R –ratioof electron–positron annihilation into hadrons, and Adler function D ( Q ) : DP ( q , q ) = DP ( ) ( q , q ) + Z ¥ m r ( s ) ln (cid:18) s − q s − q m − q m − q (cid:19) d ss , (1) R ( s ) = R ( ) ( s ) + q ( s − m ) Z ¥ s r ( s ) d ss , (2) D ( Q ) = D ( ) ( Q ) + Q Q + m Z ¥ m r ( s ) s − m s + Q d ss . (3)In these equations m denotes the value of hadronic production threshold, r ( s ) is the spectral density r ( s ) = p i dd ln s lim e → + h p ( s − i e ) − p ( s + i e ) i = − d r ( s ) d ln s = p i lim e → + h d ( − s − i e ) − d ( − s + i e ) i , (4) DP ( q , q ) = P ( q ) − P ( q ) stands for the subtracted hadronic vacuum polarization function, whereas p ( q ) , r ( s ) ,and d ( Q ) denote the strong corrections to the functions P ( q ) , R ( s ) , and D ( Q ) , respectively. The derivation of inte-gral representations (1)–(3) employs only the kinematic restrictions on the relevant physical processes, the asymptoticultraviolet behavior of the hadronic vacuum polarization function, and requires neither additional approximations norphenomenological assumptions, see Refs. [7, 8].The common prefactor N c (cid:229) n f f = Q f is omitted throughout the paper, where N c = Q f standsfor the electric charge of f –th quark, and n f is the number of active flavors. In Eqs. (1)–(3) Q = − q > s = q > q ( x ) is the unit step–function [ q ( x ) = x ≥ Its preliminary formulation was discussed in Refs. [5, 6]. .0 0.5 1.0 1.5 2.012345 APT PTDPT Q , GeV(Q ) FIGURE 1.
Comparison of the hadronic vacuum polarization function ¯ P ( Q ) = DP ( , − Q ) with relevant lattice simulationdata [14], see Ref. [15] for the details. and q ( x ) = DP ( ) ( q , q ) = j − tan j tan j − j − tan j tan j , (5) R ( ) ( s ) = q ( s − m ) (cid:18) − m s (cid:19) / , (6) D ( ) ( Q ) = + x h − p + x − sinh − (cid:0) x / (cid:1)i , (7)where sin j = q / m , sin j = q / m , and x = Q / m , see papers [8, 9, 10] and references therein for the details.There is still no unambiguous method to restore the complete expression for the spectral density r ( s ) (4) (discussionof this issue can be found in, e.g., Refs. [9, 10, 11]). Nonetheless, the perturbative contribution to r ( s ) can becalculated by making use of the perturbative expression for either of the strong corrections to the functions on hand(see, e.g., Refs. [12, 13]): r pert ( s ) = p dd ln s Im lim e → + p pert ( s − i e ) = − d r pert ( s ) d ln s = p Im lim e → + d pert ( − s − i e ) . (8)In this paper the model [8] for the spectral density will be employed: r ( s ) = b ( s / L ) + p + L s , (9)where b = − n f / L denotes the QCD scale parameter. The first term on the right–hand side of Eq. (9) isthe one–loop perturbative contribution, whereas the second term represents intrinsically nonperturbative part of thespectral density, see paper [8] and references therein for the details.It is worthwhile to mention also that in the massless limit ( m =
0) for the case of perturbative spectral function[ r ( s ) = Im d pert ( − s − i + ) / p ] two equations (2) and (3) become identical to those of the analytic perturbationtheory (APT) [16] (see also Refs. [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]). However, it is essential to keepthe value of hadronic production threshold nonvanishing, since the massless limit loses some of the substantialnonperturbative constraints, which relevant dispersion relations impose on the functions on hand, see Refs. [7, 8,15, 29].The dispersively improved perturbation theory (DPT) [7, 8] extends the applicability range of perturbative approachtowards the infrared domain. In particular, the Adler function (3) conforms with relevant experimental prediction The studies of Adler function within other approaches can be found in Refs. [30, 31, 32, 33, 34, 35, 36, 37]. .0 0.5 1.0 1.5 2.0 2.5 3.00.40.81.21.6 A OPAL (update 2012)
QCDV , GeV B ALEPH (update 2014)
QCDV , GeV C OPAL (update 2012)
QCDA , GeV D ALEPH (update 2014)
QCDA , GeV
FIGURE 2.
Comparison of the perturbative expression D V/Apert (12) (solid curves) with relevant experimental data (horizontal shadedbands). Vertical dashed bands denote solutions for the QCD scale parameter L . The plots A, C and B, D correspond to experimentaldata [45] and [46], respectively. in the entire energy range [7, 29, 38] and the hadronic vacuum polarization function (1) agrees with pertinent latticesimulation data [15], see Fig. 1. Furthermore, the representations (1)–(3) conform with the results obtained in Ref. [39]as well as in Ref. [40]. Additionally, the respective hadronic contributions to the muon anomalous magnetic momentand to the shift of the electromagnetic fine structure constant at the scale of Z boson mass evaluated in the frameworkof DPT proved to be in a good agreement with recent estimations of these quantities [15]. All this testifies to theefficiency of dispersive approach [7, 8] in the studies of nonperturbative aspects of the strong interaction.The study of the inclusive t lepton hadronic decay represents a particular interest, since this process probes thelow–energy hadron dynamics. Specifically, the theoretical expression for the relevant experimentally measurablequantity reads R J = t , V/A = N c | V ud | S EW (cid:16) D V/AQCD + d ′ EW (cid:17) . (10)In this equation | V ud | = . ± . d ′ EW = . S EW = . ± . D V/AQCD = p Z M t m V/A (cid:18) − sM t (cid:19) (cid:18) + sM t (cid:19) Im P V/A ( s + i + ) dsM t (11)stands for the hadronic contribution, see Refs. [43, 44]. In Eq. (11) M t ≃ .
777 GeV [41] is the mass of t lepton,whereas m V/A denotes the total mass of the lightest allowed hadronic decay mode of t lepton in the correspondingchannel.It is worthwhile to mention that the perturbative description of the inclusive t lepton hadronic decay completelyleaves out the effects due to the nonvanishing hadronic production threshold. Moreover, the perturbative approach3 .00 0.25 0.50 0.750.40.81.21.6 A OPAL (update 2012)
QCDV , GeV B ALEPH (update 2014)
QCDV , GeV C OPAL (update 2012)
QCDA , GeV D ALEPH (update 2014)
QCDA , GeV
FIGURE 3.
Comparison of the expression D V/AQCD (14) (solid curves) with relevant experimental data (horizontal shaded bands).Vertical dashed bands denote solutions for the QCD scale parameter L . The plots A, C and B, D correspond to experimentaldata [45] and [46], respectively. suffers from its inherent difficulties, such as the infrared unphysical singularities. These facts eventually result inthe identity of the perturbative predictions for functions (11) in vector and axial–vector channels (i.e., D Vpert ≡ D Apert ),that contradicts experimental data. In particular, within perturbative approach the expression (11) acquires the form(in what follows the one–loop level with n f = D V/Apert = + b Z p l A ( q ) + q A ( q ) p ( l + q ) d q , (12)where l = ln (cid:0) M t / L (cid:1) , and A ( q ) = + ( q ) − ( q ) − cos ( q ) , A ( q ) = ( q ) − ( q ) − sin ( q ) , (13)see Refs. [8, 47]. Furthermore, the perturbative approach is incapable of describing the experimental data on theinclusive semileptonic branching ratio in axial–vector channel, see Fig. 2 and Table 1. It is worth noting also that forvector channel perturbative approach returns two equally justified solutions for the QCD scale parameter L , one ofwhich is commonly discarded, see paper [8] and references therein for the details.The inclusive t lepton hadronic decay was also studied within analytic perturbation theory and a number of itsmodifications [33, 48, 49]. However, these papers basically deal either with the total sum of vector and axial–vector terms (10) or with the vector term only. Additionally, APT disregards valuable effects due to nonvanishinghadronic production threshold and, similarly to perturbative approach, yields identical predictions for functions (11)in vector and axial–vector channels. For the vector channel APT returns a rather large value for the QCD scaleparameter ( L ≃
900 MeV). As for the axial–vector channel, the APT fails to describe the experimental data onthe inclusive t lepton hadronic decay, since for any value of L the APT expression for function (11) exceeds itsexperimental measurement, see also Ref. [9]. 4 ABLE 1.
Values of the QCD scale parameter L [MeV] obtained within perturbative and dispersive approaches from OPAL [45]and ALEPH [46] experimental data on inclusive t lepton hadronic decay (one–loop level, n f = (update 2012) (update 2014) (update 2012) (update 2014) Vector channel 445 + − + − ±
53 409 ± ±
61 419 ± The dispersive approach to QCD (contrary to perturbative and analytic approaches) properly accounts for the effectsdue to nonvanishing hadronic production threshold. The hadronic contribution (11) to the inclusive semileptonicbranching ratio within dispersive approach can eventually be represented as D V/AQCD = g (cid:18) c V/A (cid:19)p − c V/A − g (cid:18) c V/A (cid:19) ln (cid:18)q c − V/A + q c − V/A − (cid:19) + Z ¥ m V/A G (cid:16) s M t (cid:17) r ( s ) d ss , (14)where G ( x ) = g ( x ) q ( − x ) + g ( ) q ( x − ) − g ( c V/A ) , g ( x ) = x ( − x + x ) , c V/A = m V/A / M t , m V ≃ .
075 GeV , m A ≃ .
288 GeV , spectral density r ( s ) is specified in Eq. (9), and g ( x ) = + x − x + x , g ( x ) = x ( + x − x ) , (15)see papers [8, 9, 10, 47] and references therein. The comparison of Eq. (14) with OPAL (update 2012, Ref. [45]) andALEPH (update 2014, Ref. [46]) experimental data is presented in Fig. 3 and the respective values of the QCD scaleparameter L are given in Table 1. As one may infer from Fig. 3, the dispersive approach is capable of describingthe experimental data [45, 46] on inclusive t lepton hadronic decay in vector and axial–vector channels. The obtainedvalues of the QCD scale parameter L appear to be nearly identical in both channels, that testifies to the self–consistencyof the approach on hand.The author is grateful to D. Boito, R. Kaminski, B. Malaescu, E. Passemar, M. Passera, J. Portoles, and H. Wittigfor the stimulating discussions and useful comments. REFERENCES
1. F. Guerrero and A. Pich, Phys. Lett. B , 382 (1997); A. Pich and J. Portoles, Phys. Rev. D , 093005 (2001); D. GomezDumm and P. Roig, Eur. Phys. J. C , 2528 (2013); P. Roig, A. Guevara, and G.L. Castro, Phys. Rev. D , 073016 (2014).2. V. Bernard and E. Passemar, Phys. Lett. B , 95 (2008); V. Bernard, M. Oertel, E. Passemar, and J. Stern, Phys. Rev. D ,034034 (2009).3. G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP , 091 (2014); G. Colangelo, M. Hoferichter, A. Nyffeler,M. Passera, and P. Stoffer, Phys. Lett. B , 90 (2014); G. Colangelo, M. Hoferichter, B. Kubis, M. Procura, and P. Stoffer, ibid. , 6 (2014).4. R. Garcia–Martin, R. Kaminski, J.R. Pelaez, and J.R. Elvira, Phys. Rev. Lett. , 072001 (2011); R. Garcia–Martin,R. Kaminski, J.R. Pelaez, J.R. Elvira, and F.J. Yndurain, Phys. Rev. D , 074004 (2011).5. A.V. Nesterenko and J. Papavassiliou, Phys. Rev. D , 016009 (2005).6. A.V. Nesterenko and J. Papavassiliou, Int. J. Mod. Phys. A , 4622 (2005); Nucl. Phys. B (Proc. Suppl.) , 47 (2005); ,304 (2007).7. A.V. Nesterenko and J. Papavassiliou, J. Phys. G , 1025 (2006).8. A.V. Nesterenko, Phys. Rev. D , 056009 (2013).9. A.V. Nesterenko, Nucl. Phys. B (Proc. Suppl.) , 199 (2013); SLAC eConf C1106064, 23 (2011); arXiv:1110.3415 [hep-ph].10. A.V. Nesterenko, PoS (Confinement X), 350 (2013); arXiv:1401.0620 [hep-ph].11. A.V. Nesterenko, Phys. Rev. D , 094028 (2000); , 116009 (2001); Int. J. Mod. Phys. A , 5475 (2003); Nucl. Phys. B(Proc. Suppl.) , 59 (2004).12. A.V. Nesterenko and C. Simolo, Comput. Phys. Commun. , 1769 (2010); , 2303 (2011).13. A.P. Bakulev and V.L. Khandramai, Comput. Phys. Commun. , 183 (2013); C. Ayala and G. Cvetic, ibid. , 182 (2015);J. Phys. Conf. Ser. , 012064 (2015).14. M. Della Morte, B. Jaeger, A. Juttner, and H. Wittig, AIP Conf. Proc. , 337 (2011); PoS (LATTICE 2011), 161 (2011);PoS (LATTICE 2012), 175 (2012); JHEP , 055 (2012).15. A.V. Nesterenko, J. Phys. G , 085004 (2015).
6. D.V. Shirkov and I.L. Solovtsov, Phys. Rev. Lett. , 1209 (1997); Theor. Math. Phys. , 132 (2007); K.A. Milton andI.L. Solovtsov, Phys. Rev. D , 5295 (1997); , 107701 (1999).17. G. Cvetic and C. Valenzuela, Braz. J. Phys. , 371 (2008); Phys. Rev. D , 114030 (2006); , 019902(E) (2011);A.P. Bakulev, Phys. Part. Nucl. , 715 (2009); N.G. Stefanis, ibid. , 494 (2013); G. Cvetic and A.V. Kotikov, J. Phys. G ,065005 (2012).18. G. Cvetic, A.Y. Illarionov, B.A. Kniehl, and A.V. Kotikov, Phys. Lett. B , 350 (2009); A.V. Kotikov,PoS (Baldin ISHEPP XXI), 033 (2013); A.V. Kotikov and B.G. Shaikhatdenov, Phys. Part. Nucl. , 543 (2013);Phys. Atom. Nucl. , 525 (2015).19. C. Contreras, G. Cvetic, O. Espinosa, and H.E. Martinez, Phys. Rev. D , 074005 (2010); C. Ayala, C. Contreras, andG. Cvetic, ibid. , 114043 (2012); G. Cvetic and C. Villavicencio, ibid. , 116001 (2012); C. Ayala and G. Cvetic, ibid. ,054008 (2013); P. Allendes, C. Ayala, and G. Cvetic, ibid. , 054016 (2014).20. M. Baldicchi and G.M. Prosperi, Phys. Rev. D , 074008 (2002); AIP Conf. Proc. , 152 (2005); M. Baldicchi,G.M. Prosperi, and C. Simolo, ibid. , 340 (2007).21. A.C. Aguilar, A.V. Nesterenko, and J. Papavassiliou, J. Phys. G , 997 (2005); Nucl. Phys. B (Proc. Suppl.) , 300 (2007).22. K.A. Milton, I.L. Solovtsov, and O.P. Solovtsova, Phys. Lett. B , 421 (1998); Phys. Rev. D , 016001 (1999);R.S. Pasechnik, D.V. Shirkov, and O.V. Teryaev, ibid. , 071902 (2008); R. Pasechnik, R. Enberg, and G. Ingelman, ibid. ,054036 (2010); R.S. Pasechnik, J. Soffer, and O.V. Teryaev, ibid. , 076007 (2010).23. K.A. Milton, I.L. Solovtsov, and O.P. Solovtsova, Phys. Lett. B , 104 (1997); K.A. Milton, I.L. Solovtsov, O.P. Solovtsova,and V.I. Yasnov, Eur. Phys. J. C , 495 (2000).24. G. Ganbold, Phys. Rev. D , 034034 (2009); , 094008 (2010); PoS (Confinement X), 065 (2013).25. N.G. Stefanis, Nucl. Phys. B (Proc. Suppl.) , 245 (2006); A.P. Bakulev, K. Passek–Kumericki, W. Schroers, andN.G. Stefanis, Phys. Rev. D , 033014 (2004); , 079906(E) (2004); A.P. Bakulev, A.V. Pimikov, and N.G. Stefanis, ibid. , 093010 (2009).26. G. Cvetic, R. Kogerler, and C. Valenzuela, Phys. Rev. D , 114004 (2010); J. Phys. G , 075001 (2010); G. Cvetic andR. Kogerler, Phys. Rev. D , 056005 (2011).27. A. Courtoy and S. Liuti, Phys. Lett. B , 320 (2013); A. Courtoy, arXiv:1405.6567 [hep-ph].28. O. Teryaev, Nucl. Phys. B (Proc. Suppl.) , 195 (2013); A.V. Sidorov and O.P. Solovtsova, Nonlin. Phenom. ComplexSyst. , 397 (2013); Mod. Phys. Lett. A , 1450194 (2014); PoS (Baldin ISHEPP XXII), 019 (2015).29. A.V. Nesterenko, SLAC eConf C0706044, 25 (2007).30. C.J. Maxwell, Phys. Lett. B , 382 (1997); D.M. Howe and C.J. Maxwell, Phys. Rev. D , 014002 (2004); P.M. Brooksand C.J. Maxwell, ibid. , 065012 (2006).31. S. Peris and E. de Rafael, Nucl. Phys. B , 325 (1997).32. S. Eidelman, F. Jegerlehner, A.L. Kataev, and O. Veretin, Phys. Lett. B , 369 (1999); A.L. Kataev, arXiv:hep-ph/9906534;Phys. Lett. B , 350 (2008); JETP Lett. , 789 (2011).33. K.A. Milton, I.L. Solovtsov, and O.P. Solovtsova, Phys. Rev. D , 016005 (2001); Mod. Phys. Lett. A , 1355 (2006).34. G. Cvetic and T. Lee, Phys. Rev. D , 014030 (2001); G. Cvetic, C. Dib, T. Lee, and I. Schmidt, ibid. , 093016 (2001);G. Cvetic, ibid. , 036003 (2014); G. Cvetic and C. Valenzuela, J. Phys. G , L27 (2006); G. Cvetic, C. Valenzuela, andI. Schmidt, Nucl. Phys. B (Proc. Suppl.) , 308 (2007).35. M. Beneke and M. Jamin, JHEP , 044 (2008).36. I. Caprini and J. Fischer, Eur. Phys. J. C , 35 (2009); Phys. Rev. D , 054019 (2011).37. G. Abbas, B. Ananthanarayan, and I. Caprini, Phys. Rev. D , 094018 (2012); G. Abbas, B. Ananthanarayan, I. Caprini, andJ. Fischer, ibid. , 014008 (2013).38. A.V. Nesterenko, Nucl. Phys. B (Proc. Suppl.) , 207 (2009).39. M. Baldicchi, A.V. Nesterenko, G.M. Prosperi, D.V. Shirkov, and C. Simolo, Phys. Rev. Lett. , 242001 (2007); M. Baldicchi,A.V. Nesterenko, G.M. Prosperi, and C. Simolo, Phys. Rev. D , 034013 (2008).40. B. Blossier et al. , Phys. Rev. D , 034503 (2012); , 014507 (2014); Nucl. Phys. B (Proc. Suppl.) , 217 (2013).41. J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D , 010001 (2012).42. W.J. Marciano and A. Sirlin, Phys. Rev. Lett. , 22 (1986); , 1815 (1988); E. Braaten and C.S. Li, Phys. Rev. D , 3888(1990).43. E. Braaten, S. Narison, and A. Pich, Nucl. Phys. B , 581 (1992); A.A. Pivovarov, Z. Phys. C , 461 (1992); F. Le Diberderand A. Pich, Phys. Lett. B , 147 (1992); , 165 (1992).44. M. Davier, A. Hocker, and Z. Zhang, Rev. Mod. Phys. , 1043 (2006); A. Pich, Prog. Part. Nucl. Phys. , 41 (2014).45. K. Ackerstaff et al. [OPAL Collaboration], Eur. Phys. J. C , 571 (1999); D. Boito, et al. , Phys. Rev. D , 113006 (2011); ,093015 (2012); , 034003 (2015); Nucl. Part. Phys. Proc. , 134 (2015).46. S. Schael et al. [ALEPH Collaboration], Phys. Rept. , 191 (2005); M. Davier, A. Hocker, B. Malaescu, C. Yuan, andZ. Zhang, Eur. Phys. J. C , 2803 (2014).47. A.V. Nesterenko, Nucl. Part. Phys. Proc. , 177 (2015).48. K.A. Milton, I.L. Solovtsov, and O.P. Solovtsova, Phys. Lett. B , 104 (1997); K.A. Milton, I.L. Solovtsov, O.P. Solovtsova,and V.I. Yasnov, Eur. Phys. J. C , 495 (2000).49. G. Cvetic and C. Valenzuela, Phys. Rev. D , 114030 (2006); , 019902(E) (2011); C. Ayala, C. Contreras, and G. Cvetic, ibid. , 114043 (2012)., 114043 (2012).