- Email: [email protected]
Top and Bottom at Threshold: Recent Results
Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230 www.elsevierphysics.com
Top and Bottom at Threshold: Recent Results M. Steinhausera∗ a
Institut f¨ ur Theoretische Teilchenphysik, Universit¨ at Karlsruhe, D-76128 Karlsruhe
Recent results obtained for the systems of two heavy quarks are discussed. In particular, we consider phenomenological applications for the third-order results for the energy levels and wave function. Special emphasis is put on the resummation of potentially large logarithms in the velocity of the heavy quarks. Predictions for the mass of the ηb meson are dicussed which still has to be confirmed experimentally. Furthermore, the ratio of the photon mediated production or annihilation rates of spin triplet and spin singlet heavy quarkonium states are considered.
1. Introduction The study of properties of quarkonia is among the primary applications of Quantum Chromodynamics (QCD). The basic idea is to compare accurate measurements with precise calculations in order to determine fundamental parameters of the theory. In the case of bottomonium this strategy has already been followed since quite some time. One of the most important tasks of a future linear collider is the precise measurement of the cross section for the production of top quark pairs close to their production threshold. Next to the mass and the width of the top quark also the strong coupling and — in case the Higgs boson is not too heavy — also the top quark Yukawa coupling can be determined with a quite high accuracy. In order to match the expected experimental precision [1] it is important to compute higher-order quantum corrections to this process. In these proceedings we want to discuss recent results both for bottomonium and the system of two top quarks. In the next section we review the theoretical framework and present applications in the remaining parts of the paper. 2. Framework The correct framework necessary to perfrom bound state calculations is provided by an ∗ This work was supported by the “Impuls- und Vernetzungsfonds” of the Helmholtz Association, contract number VH-NG-008 and the SFB/TR 9.
0920-5632/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2006.03.024
effective theory constructed from QCD, nonrelativistic QCD (NRQCD). The pioneering work in this direction has been performed more than ten years ago [2,3]. An effective theory has been constructed where the heavy quark mass is integrated out. It consists of effective operators accompanied by coefficient functions incorporating the remnance of the heavy quark mass dependence. However, the computation of higher order corrections in this framework is still quite tedious. The main reason for this is that there are still too many degrees of freedom present which are not realized in bound state systems. A further step was taken end of the nineties [4, 5] where an effective theory, potential NRQCD (pNRQCD), has been constructed containing only so-called potential quarks and ultra-soft gluons as active degrees of freedom. As far as practical calculations are concerned it is particularly appealing that pNRQCD can also be constructed with the help the threshold expansion [6]. There are quite a number of higher order calculations which have been performed in the framework of pNRQCD and which certainly would have been significantly more difficult without. We will review them in Section 3. However, not only for the fixed-order results but also for the resummation of logarithms in the heavy quark velocity it is possible to use pNRQCD. In Refs. [7– 9] the framework has been set and in Refs. [10– 12] the complete NNLL corrections for the spindependent part of the Lagrangian has been de-
227
M. Steinhauser / Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230
the series for the ground state energy in the pole mass scheme. Similar results for the MS quark mass can be found in Ref. [17]. In Ref. [14] a universal relation has been obtained between the resonance energy in tt¯ threshold production in e+ e− annihilation or γγ collisions and the top quark pole mass
-1.8 NLO NNLO N3LO
-2
S=1
E t (GeV)
-2.2
p.t.
-2.4 -2.6 -2.8
Eres
-3 10
15
20
25
30
35
40
45
Figure 1. Ground state energy E1p.t. of the tt¯ bound state in the zero-width approximation as a function of the renormalization scale μ for S = 1. The short-dashed (blue), long-dashed (green) and solid (red) line corresponds to the NLO, NNLO and N3 LO approximations.
rived. This will be discussed in Section 4. 3. N3 LO results within pNRQCD 3.1. Top quark mass For the top quark system the relation between the resonance energy, i.e. the position of the peak of the total cross section, and the quark mass is given by Eres = 2mt +E1p.t. +δ Γt Eres , where E1p.t. is the perturbative contribution to the ground state energy and δ Γt Eres = 100 ± 10 MeV is the effect of higher-order resonances and the finite width of the top quark. Non-perturbative effects are negligible. To evaluate the perturbative contribution we use the result of Ref. [13,14] (see also Refs. [15, 16]) for E1p.t. up to N3 LO. In Fig. 1 E1p.t. is plotted in NLO, NNLO and N3 LO approximation as a function of the renormalization scale of the strong coupling constant for spin S = 1. One can see, that the N3 LO result shows a much weaker dependence on μ than the NNLO one. Moreover at the scale μ ≈ 15 GeV, which is close to the physically motivated soft scale μs ≈ 30 GeV, the N3 LO correction vanishes and furthermore becomes independent of μ, i.e. the N3 LO curve shows a local minimum. This suggests the convergence of
mt − 174.3 GeV 174.3 GeV
±0.0009) × mt .
50
μ (GeV)
= (1.9833 + 0.007
(1)
The central value is computed for mt = 174.3 GeV. For the estimate of the theoretical uncertainty in Eq. (1) we assume a ±100% error in the Pad´e approximation of the still unknown three-loop static potential coefficient and vary the normalization scale in the stability interval which is roughly given by 0.4μs < μ < μs . Furthermore we use αs (MZ ) = 0.1185 ± 0.002 and take into account the ±10 MeV uncertainty in δ Γt Eres . Due to the very nice behaviour of the perturbative expansion for the ground state energy we do not expect large higher order corrections to our result. We want to mention that mt extracted from Eq. (1) leads to a fixed-order quark mass which, by definition, does not suffer from renormalon ambiguities. Furthermore, it can be converted with high accuracy to the MS quark mass. 3.2. Top quark threshold production In contrast to the bottom system the nonperturbative effects in the case of the top quark are negligible. However, due to the relatively large top quark width, Γt , its effect has to be taken into account properly [18] since the Coulomb-like resonances below threshold are smeared out. Actually, the cross section only shows a small bump which is essentially the remnant of the ground state pole. The higher poles and continuum, however, affect the position of the resonance peak and move it to higher energy. The value of the normalized cross section R = σ(e+ e− → tt¯)/σ(e+ e− → μ+ μ− ) at the resonance energy is dominated by the contribution from the would-be toponium ground state which in the leadingapproximation reads R1LO = 6πNc Q2t |ψ1C (0)|2 / m2t Γt , where Qt = 2/3. Numerically we find R1
≈ R1LO (1 − 0.243NLO + 0.435NNLO
228
2
∧
R1/R1
M. Steinhauser / Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
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50
60
70
80
90
100
μ (GeV)
ˆ 1 ≡ R1LO Figure 2. R1 normalized to R . αs →αs (μs ) as a function of μ at LO (dotted), NLO (dashed), NNLO (dotted-dashed) and N3 LO (full line). For the N3 LO result, the band reflects the errors due to αs (MZ ) = 0.118 ± 0.003.
−0.268N3LO + . . .) ,
(2)
where the prime reminds that the N3 LO corrections are not complete but contains only the logarithmically enhanced and the β03 corrections which are expected to provide the dominant contributions (cf. Ref. [15]). Note that the thirdorder corrections proportional to β03 amount to approximately −7% of the LO approximation at the soft scale which is the same order of magnitude as the O(α3s ) linear logarithmic term. The available N3 LO terms improve the stability of the result with respect to the scale variation as can be seen in Fig. 2. The absence of a rapid growth of the coefficients along with the alternating-sign character of the series and the weak scale dependence suggest that the missing perturbative corrections are moderate and most likely are in the few-percent range. It is interesting to note that the perturbative contributions of different orders, which are relatively large when taken separately, cancel in the sum to give only a few percent variation of the leading order result. 4. Non-relativistic renormalization group 4.1. Prediction of M (ηb ) In order to determine M (ηb ) we exploit the relation M (ηb ) = M (Υ(1S)) − Ehfs where
M (Υ(1S)) = 9.46030(26) GeV and Ehfs is the hyperfine splitting. The latter can be determined from the spin-dependent part of the effective Lagrangian. The next-to-leading order (NLO) approximation of Ehfs is easily obtained from the N3 LO corrections to the energy level [14]. In order to improve the approximation one has to resum the logarithms in the velocity of the heavy quarks contained in the corresponding matching coefficient. This leads to Ehfs in next-to-leading logarithmic (NLL) accuracy. Its computation can be divided into three steps: First, the renormalization group (RG) equations have to be established within potential non-relativistic QCD (pNRQCD) [19]. One has to make sure to include the running of all relevant operators and to consider the soft, potential and ultra-soft regions. In a second step the RG equations have to be solved. In the case at hand this could be done analytically. As a result one obtains the matching coefficient of (2) the spin-dependent operator, DS 2 ,s , to the NLL accuracy. This expression is used in the third step to evaluate within perturbation theory the corrections to the energy level. These steps have been performed in Refs. [10,11]. A compact analytical expression for the HFS to NLL NLL, Ehfs , which is a function of αs (μ)/αs (mb ), is given in Eq. (1) of Ref. [10]. In Fig. 3, the HFS for the bottomonium ground state is plotted as a function of μ in the LO, NLO, LL, and NLL approximations. As can be seen, the LL curve shows a weaker scale dependence compared to the LO one. The scale dependence of the NLO and NLL expressions is further reduced, and, moreover, the NLL approximation remains stable up to smaller scales than the fixed-order calculation. At the scale μ ≈ 1.3 GeV the NLL correction vanishes. Furthermore, at μ ≈ 1.5 GeV, the result becomes independent of μ; i.e., the NLL curve shows a local maximum. This suggests a nice convergence of the logarithmic expansion despite the presence of the ultrasoft contribution with αs normalized at the rather low scale μ ¯2 /mb ∼ 0.8 GeV. It is also interesting to apply our formulae to the charm system where the experimental result for the hyperfine splitting is available. The result is shown in Fig. 4 along with the experimental
229
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Ehfs (MeV)
Ehfs (MeV)
M. Steinhauser / Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230
45
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35 30
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25 20
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15
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10 20
5 0
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
0
3.5
μ (GeV)
Figure 3. HFS of 1S bottomonium as a function of the renormalization scale μ in the LO (dotted line), NLO (dashed line), LL (dot-dashed line), and NLL (solid line) approximations.
value 117.7 ± 1.3 MeV. The local maximum of the NLL curve corresponds to Ehfs = 104 MeV and thus shows an impressive agreement with experiment. We should emphasize the crucial role of the resummation to bring the perturbative prediction closer to the experimental value. From these observations the following prediction of the mass of the as-yet undiscovered ηb meson has been obtained [10] M (ηb ) = 9421 ± 11 (th) +9 −8 (δαs ) MeV ,
(3)
where the errors due to the high-order perturbative corrections and the nonperturbative effects are added up in quadrature in “th”, whereas “δαs ” stands for the uncertainty in αs (MZ ) = 0.118 ± 0.003. If the experimental error in future measurements of M (ηb ) will not exceed a few MeV, the bottomonium HFS will become a competitive source of αs (MZ ) with an estimated accuracy of ±0.003, as can be seen from Fig. 3. 4.2. Spin dependence of heavy quarkonium production and annihilation In Ref. [12] a further step was undertaken and the ratio of the photon mediated production or annihilation rates of spin triplet and spin singlet heavy quarkonium states has been considered. In particular, we define Rq
=
σ(e+ e− → Q(n3 S1 )) σ(γγ → Q(n1 S0 ))
0.7
0.8
0.9
1
1.1
1.2
1.3
μ (GeV)
Figure 4. HFS of 1S charmonium as a function of the renormalization scale μ in the LO (dotted line), NLO (dashed line), LL (dot-dashed line), and NLL (solid line) approximations. The green band corresponds to the experimental result.
=
Γ(Q(n3 S1 ) → e+ e− ) , Γ(Q(n1 S0 ) → γγ)
(4)
which can be written as Rq
=
(v,p)
c2s (ν) |ψnv (0)|2 + O(αs v 2 ) . 3Q2q |ψnp (0)|2
(5)
ψn ( r ) are the spin triplet (vector) and spin singlet (pseudoscalar) quarkonium wave functions of the principal quantum number n. The expression for |ψnv (0)|2 /|ψnp (0)|2 to NNLL can be found in Ref. [12]. cs is the ratio of the corresponding matching coefficients. With the help of the NLL (2) approximation of DS 2 ,s it was possible to obtain the NNLL corrections of cs and thus for the quantity Rq [12], i.e. for the first time for a physical quantity. It is interesting to consider the various approximations for Rq as a function of the renormalization scale, ν. One observes a very good convergence for the top quark system [12]. In the case of the bottom quark the result is shown in Fig. 5. A nice convergence of the logarithmic expansion is observed despite the presence of ultrasoft contributions with αs normalized at a rather low scale ν 2 /mb . At the same time, the perturbative corrections are important and reduce the leading order result by approximately 41%. The analog plot for the top quark system can be
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M. Steinhauser / Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230
3
0.75
2.5 0.7 0.65
ℜt
ℜb
2 1.5
0.6
1
0.55
0.5 0
1
1.5
2
2.5
3
ν (GeV)
3.5
4
4.5
5
Figure 5. Rb as the function of the renormalization scale ν in LO≡LL (dotted line), NLO (short-dashed line), NNLO (long-dashed line), NLL (dot-dashed line), and NNLL (solid line) approximation for the bottomonium ground state. The (yellow) band reflects the errors due to αs (MZ ) = 0.118 ± 0.003.
found in Fig. 6. As one can see the logarithmic expansion shows perfect convergence and the NNLL correction vanishes at the scale ν ≈ 13 GeV, which is close to the physically motivated scale of the inverse Bohr radius αs mt /2. More details can be found in Ref. [12]. REFERENCES 1. M. Martinez and R. Miquel, Eur. Phys. J. C 27 (2003) 49. 2. W. E. Caswell and G. P. Lepage, Phys. Lett. B 167 (1986) 437. 3. G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51 (1995) 1125 [Erratum-ibid. D 55 (1997) 5853] [arXiv:hep-ph/9407339]. 4. A. Pineda and J. Soto, Nucl. Phys. Proc. Suppl. 64 (1998) 428 [arXiv:hep-ph/9707481]. 5. N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B 566 (2000) 275 [arXiv:hep-ph/9907240]. 6. M. Beneke and V. A. Smirnov, Nucl. Phys. B 522 (1998) 321 [arXiv:hep-ph/9711391]. 7. M. E. Luke, A. V. Manohar and I. Z. Rothstein, Phys. Rev. D 61 (2000) 074025 [arXiv:hep-ph/9910209]. 8. A. Pineda, Phys. Rev. D 65 (2002) 074007
0.5
10
20
30
40
50
ν (GeV)
60
70
80
Figure 6. Same as Fig. 5 but for the top quark system.
[arXiv:hep-ph/0109117]. 9. A. Pineda, Phys. Rev. D 66 (2002) 054022 [arXiv:hep-ph/0110216]. 10. B. A. Kniehl, A. A. Penin, A. Pineda, V. A. Smirnov and M. Steinhauser, Phys. Rev. Lett. 92 (2004) 242001 [arXiv:hepph/0312086]. 11. A. A. Penin, A. Pineda, V. A. Smirnov and M. Steinhauser, Phys. Lett. B 593 (2004) 124 [arXiv:hep-ph/0403080]. 12. A. A. Penin, A. Pineda, V. A. Smirnov and M. Steinhauser, Nucl. Phys. B 699 (2004) 183 [arXiv:hep-ph/0406175]. 13. B. A. Kniehl, A. A. Penin, V. A. Smirnov and M. Steinhauser, Nucl. Phys. B 635 (2002) 357 [arXiv:hep-ph/0203166]. 14. A. A. Penin and M. Steinhauser, Phys. Lett. B 538 (2002) 335 [arXiv:hep-ph/0204290]. 15. A. A. Penin, V. A. Smirnov and M. Steinhauser, Nucl. Phys. B 716 (2005) 303 [arXiv:hep-ph/0501042]. 16. M. Beneke, Y. Kiyo and K. Schuller, Nucl. Phys. B 714 (2005) 67 [arXiv:hepph/0501289]. 17. Y. Kiyo and Y. Sumino, Phys. Rev. D 67 (2003) 071501 [arXiv:hep-ph/0211299]. 18. V. S. Fadin and V. A. Khoze, Sov. J. Nucl. Phys. 48 (1988) 309 [Yad. Fiz. 48 (1988) 487]. 19. A. Pineda and J. Soto, Phys. Lett. B 495 (2000) 323 [arXiv:hep-ph/0007197].
Top and Bottom at Threshold: Recent Results M. Steinhausera∗ a
Institut f¨ ur Theoretische Teilchenphysik, Universit¨ at Karlsruhe, D-76128 Karlsruhe
Recent results obtained for the systems of two heavy quarks are discussed. In particular, we consider phenomenological applications for the third-order results for the energy levels and wave function. Special emphasis is put on the resummation of potentially large logarithms in the velocity of the heavy quarks. Predictions for the mass of the ηb meson are dicussed which still has to be confirmed experimentally. Furthermore, the ratio of the photon mediated production or annihilation rates of spin triplet and spin singlet heavy quarkonium states are considered.
1. Introduction The study of properties of quarkonia is among the primary applications of Quantum Chromodynamics (QCD). The basic idea is to compare accurate measurements with precise calculations in order to determine fundamental parameters of the theory. In the case of bottomonium this strategy has already been followed since quite some time. One of the most important tasks of a future linear collider is the precise measurement of the cross section for the production of top quark pairs close to their production threshold. Next to the mass and the width of the top quark also the strong coupling and — in case the Higgs boson is not too heavy — also the top quark Yukawa coupling can be determined with a quite high accuracy. In order to match the expected experimental precision [1] it is important to compute higher-order quantum corrections to this process. In these proceedings we want to discuss recent results both for bottomonium and the system of two top quarks. In the next section we review the theoretical framework and present applications in the remaining parts of the paper. 2. Framework The correct framework necessary to perfrom bound state calculations is provided by an ∗ This work was supported by the “Impuls- und Vernetzungsfonds” of the Helmholtz Association, contract number VH-NG-008 and the SFB/TR 9.
0920-5632/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2006.03.024
effective theory constructed from QCD, nonrelativistic QCD (NRQCD). The pioneering work in this direction has been performed more than ten years ago [2,3]. An effective theory has been constructed where the heavy quark mass is integrated out. It consists of effective operators accompanied by coefficient functions incorporating the remnance of the heavy quark mass dependence. However, the computation of higher order corrections in this framework is still quite tedious. The main reason for this is that there are still too many degrees of freedom present which are not realized in bound state systems. A further step was taken end of the nineties [4, 5] where an effective theory, potential NRQCD (pNRQCD), has been constructed containing only so-called potential quarks and ultra-soft gluons as active degrees of freedom. As far as practical calculations are concerned it is particularly appealing that pNRQCD can also be constructed with the help the threshold expansion [6]. There are quite a number of higher order calculations which have been performed in the framework of pNRQCD and which certainly would have been significantly more difficult without. We will review them in Section 3. However, not only for the fixed-order results but also for the resummation of logarithms in the heavy quark velocity it is possible to use pNRQCD. In Refs. [7– 9] the framework has been set and in Refs. [10– 12] the complete NNLL corrections for the spindependent part of the Lagrangian has been de-
227
M. Steinhauser / Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230
the series for the ground state energy in the pole mass scheme. Similar results for the MS quark mass can be found in Ref. [17]. In Ref. [14] a universal relation has been obtained between the resonance energy in tt¯ threshold production in e+ e− annihilation or γγ collisions and the top quark pole mass
-1.8 NLO NNLO N3LO
-2
S=1
E t (GeV)
-2.2
p.t.
-2.4 -2.6 -2.8
Eres
-3 10
15
20
25
30
35
40
45
Figure 1. Ground state energy E1p.t. of the tt¯ bound state in the zero-width approximation as a function of the renormalization scale μ for S = 1. The short-dashed (blue), long-dashed (green) and solid (red) line corresponds to the NLO, NNLO and N3 LO approximations.
rived. This will be discussed in Section 4. 3. N3 LO results within pNRQCD 3.1. Top quark mass For the top quark system the relation between the resonance energy, i.e. the position of the peak of the total cross section, and the quark mass is given by Eres = 2mt +E1p.t. +δ Γt Eres , where E1p.t. is the perturbative contribution to the ground state energy and δ Γt Eres = 100 ± 10 MeV is the effect of higher-order resonances and the finite width of the top quark. Non-perturbative effects are negligible. To evaluate the perturbative contribution we use the result of Ref. [13,14] (see also Refs. [15, 16]) for E1p.t. up to N3 LO. In Fig. 1 E1p.t. is plotted in NLO, NNLO and N3 LO approximation as a function of the renormalization scale of the strong coupling constant for spin S = 1. One can see, that the N3 LO result shows a much weaker dependence on μ than the NNLO one. Moreover at the scale μ ≈ 15 GeV, which is close to the physically motivated soft scale μs ≈ 30 GeV, the N3 LO correction vanishes and furthermore becomes independent of μ, i.e. the N3 LO curve shows a local minimum. This suggests the convergence of
mt − 174.3 GeV 174.3 GeV
±0.0009) × mt .
50
μ (GeV)
= (1.9833 + 0.007
(1)
The central value is computed for mt = 174.3 GeV. For the estimate of the theoretical uncertainty in Eq. (1) we assume a ±100% error in the Pad´e approximation of the still unknown three-loop static potential coefficient and vary the normalization scale in the stability interval which is roughly given by 0.4μs < μ < μs . Furthermore we use αs (MZ ) = 0.1185 ± 0.002 and take into account the ±10 MeV uncertainty in δ Γt Eres . Due to the very nice behaviour of the perturbative expansion for the ground state energy we do not expect large higher order corrections to our result. We want to mention that mt extracted from Eq. (1) leads to a fixed-order quark mass which, by definition, does not suffer from renormalon ambiguities. Furthermore, it can be converted with high accuracy to the MS quark mass. 3.2. Top quark threshold production In contrast to the bottom system the nonperturbative effects in the case of the top quark are negligible. However, due to the relatively large top quark width, Γt , its effect has to be taken into account properly [18] since the Coulomb-like resonances below threshold are smeared out. Actually, the cross section only shows a small bump which is essentially the remnant of the ground state pole. The higher poles and continuum, however, affect the position of the resonance peak and move it to higher energy. The value of the normalized cross section R = σ(e+ e− → tt¯)/σ(e+ e− → μ+ μ− ) at the resonance energy is dominated by the contribution from the would-be toponium ground state which in the leadingapproximation reads R1LO = 6πNc Q2t |ψ1C (0)|2 / m2t Γt , where Qt = 2/3. Numerically we find R1
≈ R1LO (1 − 0.243NLO + 0.435NNLO
228
2
∧
R1/R1
M. Steinhauser / Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
10
20
30
40
50
60
70
80
90
100
μ (GeV)
ˆ 1 ≡ R1LO Figure 2. R1 normalized to R . αs →αs (μs ) as a function of μ at LO (dotted), NLO (dashed), NNLO (dotted-dashed) and N3 LO (full line). For the N3 LO result, the band reflects the errors due to αs (MZ ) = 0.118 ± 0.003.
−0.268N3LO + . . .) ,
(2)
where the prime reminds that the N3 LO corrections are not complete but contains only the logarithmically enhanced and the β03 corrections which are expected to provide the dominant contributions (cf. Ref. [15]). Note that the thirdorder corrections proportional to β03 amount to approximately −7% of the LO approximation at the soft scale which is the same order of magnitude as the O(α3s ) linear logarithmic term. The available N3 LO terms improve the stability of the result with respect to the scale variation as can be seen in Fig. 2. The absence of a rapid growth of the coefficients along with the alternating-sign character of the series and the weak scale dependence suggest that the missing perturbative corrections are moderate and most likely are in the few-percent range. It is interesting to note that the perturbative contributions of different orders, which are relatively large when taken separately, cancel in the sum to give only a few percent variation of the leading order result. 4. Non-relativistic renormalization group 4.1. Prediction of M (ηb ) In order to determine M (ηb ) we exploit the relation M (ηb ) = M (Υ(1S)) − Ehfs where
M (Υ(1S)) = 9.46030(26) GeV and Ehfs is the hyperfine splitting. The latter can be determined from the spin-dependent part of the effective Lagrangian. The next-to-leading order (NLO) approximation of Ehfs is easily obtained from the N3 LO corrections to the energy level [14]. In order to improve the approximation one has to resum the logarithms in the velocity of the heavy quarks contained in the corresponding matching coefficient. This leads to Ehfs in next-to-leading logarithmic (NLL) accuracy. Its computation can be divided into three steps: First, the renormalization group (RG) equations have to be established within potential non-relativistic QCD (pNRQCD) [19]. One has to make sure to include the running of all relevant operators and to consider the soft, potential and ultra-soft regions. In a second step the RG equations have to be solved. In the case at hand this could be done analytically. As a result one obtains the matching coefficient of (2) the spin-dependent operator, DS 2 ,s , to the NLL accuracy. This expression is used in the third step to evaluate within perturbation theory the corrections to the energy level. These steps have been performed in Refs. [10,11]. A compact analytical expression for the HFS to NLL NLL, Ehfs , which is a function of αs (μ)/αs (mb ), is given in Eq. (1) of Ref. [10]. In Fig. 3, the HFS for the bottomonium ground state is plotted as a function of μ in the LO, NLO, LL, and NLL approximations. As can be seen, the LL curve shows a weaker scale dependence compared to the LO one. The scale dependence of the NLO and NLL expressions is further reduced, and, moreover, the NLL approximation remains stable up to smaller scales than the fixed-order calculation. At the scale μ ≈ 1.3 GeV the NLL correction vanishes. Furthermore, at μ ≈ 1.5 GeV, the result becomes independent of μ; i.e., the NLL curve shows a local maximum. This suggests a nice convergence of the logarithmic expansion despite the presence of the ultrasoft contribution with αs normalized at the rather low scale μ ¯2 /mb ∼ 0.8 GeV. It is also interesting to apply our formulae to the charm system where the experimental result for the hyperfine splitting is available. The result is shown in Fig. 4 along with the experimental
229
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50
Ehfs (MeV)
Ehfs (MeV)
M. Steinhauser / Nuclear Physics B (Proc. Suppl.) 157 (2006) 226–230
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15
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10 20
5 0
1.25
1.5
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2
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2.5
2.75
3
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0
3.5
μ (GeV)
Figure 3. HFS of 1S bottomonium as a function of the renormalization scale μ in the LO (dotted line), NLO (dashed line), LL (dot-dashed line), and NLL (solid line) approximations.
value 117.7 ± 1.3 MeV. The local maximum of the NLL curve corresponds to Ehfs = 104 MeV and thus shows an impressive agreement with experiment. We should emphasize the crucial role of the resummation to bring the perturbative prediction closer to the experimental value. From these observations the following prediction of the mass of the as-yet undiscovered ηb meson has been obtained [10] M (ηb ) = 9421 ± 11 (th) +9 −8 (δαs ) MeV ,
(3)
where the errors due to the high-order perturbative corrections and the nonperturbative effects are added up in quadrature in “th”, whereas “δαs ” stands for the uncertainty in αs (MZ ) = 0.118 ± 0.003. If the experimental error in future measurements of M (ηb ) will not exceed a few MeV, the bottomonium HFS will become a competitive source of αs (MZ ) with an estimated accuracy of ±0.003, as can be seen from Fig. 3. 4.2. Spin dependence of heavy quarkonium production and annihilation In Ref. [12] a further step was undertaken and the ratio of the photon mediated production or annihilation rates of spin triplet and spin singlet heavy quarkonium states has been considered. In particular, we define Rq
=
σ(e+ e− → Q(n3 S1 )) σ(γγ → Q(n1 S0 ))
0.7
0.8
0.9
1
1.1
1.2
1.3
μ (GeV)
Figure 4. HFS of 1S charmonium as a function of the renormalization scale μ in the LO (dotted line), NLO (dashed line), LL (dot-dashed line), and NLL (solid line) approximations. The green band corresponds to the experimental result.
=
Γ(Q(n3 S1 ) → e+ e− ) , Γ(Q(n1 S0 ) → γγ)
(4)
which can be written as Rq
=
(v,p)
c2s (ν) |ψnv (0)|2 + O(αs v 2 ) . 3Q2q |ψnp (0)|2
(5)
ψn ( r ) are the spin triplet (vector) and spin singlet (pseudoscalar) quarkonium wave functions of the principal quantum number n. The expression for |ψnv (0)|2 /|ψnp (0)|2 to NNLL can be found in Ref. [12]. cs is the ratio of the corresponding matching coefficients. With the help of the NLL (2) approximation of DS 2 ,s it was possible to obtain the NNLL corrections of cs and thus for the quantity Rq [12], i.e. for the first time for a physical quantity. It is interesting to consider the various approximations for Rq as a function of the renormalization scale, ν. One observes a very good convergence for the top quark system [12]. In the case of the bottom quark the result is shown in Fig. 5. A nice convergence of the logarithmic expansion is observed despite the presence of ultrasoft contributions with αs normalized at a rather low scale ν 2 /mb . At the same time, the perturbative corrections are important and reduce the leading order result by approximately 41%. The analog plot for the top quark system can be
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3
0.75
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ℜb
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0.6
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1
1.5
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2.5
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ν (GeV)
3.5
4
4.5
5
Figure 5. Rb as the function of the renormalization scale ν in LO≡LL (dotted line), NLO (short-dashed line), NNLO (long-dashed line), NLL (dot-dashed line), and NNLL (solid line) approximation for the bottomonium ground state. The (yellow) band reflects the errors due to αs (MZ ) = 0.118 ± 0.003.
found in Fig. 6. As one can see the logarithmic expansion shows perfect convergence and the NNLL correction vanishes at the scale ν ≈ 13 GeV, which is close to the physically motivated scale of the inverse Bohr radius αs mt /2. More details can be found in Ref. [12]. REFERENCES 1. M. Martinez and R. Miquel, Eur. Phys. J. C 27 (2003) 49. 2. W. E. Caswell and G. P. Lepage, Phys. Lett. B 167 (1986) 437. 3. G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51 (1995) 1125 [Erratum-ibid. D 55 (1997) 5853] [arXiv:hep-ph/9407339]. 4. A. Pineda and J. Soto, Nucl. Phys. Proc. Suppl. 64 (1998) 428 [arXiv:hep-ph/9707481]. 5. N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B 566 (2000) 275 [arXiv:hep-ph/9907240]. 6. M. Beneke and V. A. Smirnov, Nucl. Phys. B 522 (1998) 321 [arXiv:hep-ph/9711391]. 7. M. E. Luke, A. V. Manohar and I. Z. Rothstein, Phys. Rev. D 61 (2000) 074025 [arXiv:hep-ph/9910209]. 8. A. Pineda, Phys. Rev. D 65 (2002) 074007
0.5
10
20
30
40
50
ν (GeV)
60
70
80
Figure 6. Same as Fig. 5 but for the top quark system.
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