axial couplings g2and g3to calculate various decay widths of heavy baryons. At leading order in the chiral expansion, the widths for the strong decays S ! T are
½S ! T ¼ c2f 1 magni-tude of the pion momentum in the S rest frame,
jpj¼
The mQ ¼ 1 expression for can be found, for example, in [52]. In Eq. (102), we included the term J=mQ to account for the first-order corrections for a finite heavy-quark mass. The parameters Jare related to the additional couplings in the order-1=mQ HHPT Lagrangian [62].
Terms suppressed by ðm=Þ2 and ðQCD=mQÞ2, which are omitted from (102), lead to small systematic uncertain-ties in .
To determine 1=2and 3=2, we performed fits of experi-mental data [63] for the widths of the þþc , 0c (J ¼ 1=2)
The fit parameters J are correlated with g3, and there-fore we also show the covariances in Eq. (105). The value of the sum g3þmJ
Qin Eq. (102) is plotted as a function of 1=mQ in Fig.16. For mQ¼12MJ=c, the values of g3þmJ
Q
FIG. 15 (color online). Experimental data for ½ðÞc !
c from Ref. [63], along with fits using Eq. (102), for J ¼ 1=2 (solid curve) and J ¼ 3=2 (dashed curve).
are determined dominantly by the experimental input used to fit J,
g3þ 1=2
1
2mJ=c¼ 1:059ð49Þ; g3þ 3=2
1
2mJ=c¼1:008ð46Þ:
(106) Using the masses of the cand cbaryons from Ref. [63], we obtain predictions for ½þc ! þc0; 0cþ and
½0c ! 0c0; þc as shown in Table XIII. There, we also show other predictions from the literature, as well as upper limits from experiments [71,72]. Our results for
½þc and ½0c are compatible with these limits.
We can also make predictions for the radiative decay
0c ! 0c, which is forbidden at tree level but can be mediated by loops because of flavor-SUð3Þ breaking.
Using HHPT, it has been shown that the branching fraction of this decay is related to the axial coupling g2 as follows [73]:
B ½0c ! 0c ¼ ð1:0 0:3Þ 103g22: (107) Combining this with our lattice QCD result for g2, Eq. (97), and our calculated strong-decay width ½0c !
0c0; þc ¼ 2:78ð29Þ MeV, we obtain
B ½0c ! 0c ¼ ð7 4Þ 104;
½0c ! 0c ¼ ð2:0 1:1Þ keV: (108) Next, we discuss the strong decays of bottom baryons. To calculate these widths, we evaluated (102) for mQ ¼12M. In this case the values of g3þmJ
Q are determined domi-nantly by the lattice result (97) for g3,
g3þ1=2
1
2m ¼ 0:822ð87Þ; g3þ3=2
1
2m¼ 0:805ð87Þ:
(109)
Our calculated widths ½ðÞb ! b as functions of the
ðÞb b mass difference are shown as the curves in Fig. 17. Using the experimental values of the baryon masses [13,63], we obtain the results for ½ðÞb !
b shown in TableXIII, in agreement with the widths measured by the CDF collaboration [13].
In our previous work [21] we predicted that the widths of the 0band bare less than 1.1 and 2.8 MeV, respectively.
Very recently, the CMS collaboration has observed the FIG. 16 (color online). Value and uncertainty of the quantity
ðg3þ J=mQÞ, which enters in the strong-decay width (102), as a function of the inverse heavy-quark mass m1Q , for J ¼ 1=2 (solid curve) and J ¼ 3=2 (dashed curve). At m1Q ¼ 0 the function is equal to g3, which is given by our lattice QCD result (97). The vertical lines indicate our choices for the inverse bottom and charm quark masses.
FIG. 17 (color online). Widths of the decays ðÞb ! b as functions of the ðÞb bmass difference. The curves (solid:
b, dashed: b) and shaded regions show our predictions and their uncertainties. The experimental data points are from CDF [13].
TABLE XIII. Results in MeV for the total strong-decay widths of charm and bottom baryons.
Hadron Ref. [52] Ref. [64] Ref. [61] Ref. [65,66] Ref. [67] Ref. [68] Ref. [69] This work Experiment
þb . . . 6.0 . . . 4.35 3.5 4.2(1.0) 9:7þ3:8þ1:22:81:1 [13]
b . . . 7.7 . . . 5.77 4.7 4.8(1.1) 4:9þ3:12:1 1:1 [13]
þb . . . 11.0 . . . 8.50 7.5 7.3(1.6) 11:5þ2:7þ1:02:21:5[13]
b . . . 13.2 . . . 10.44 9.2 7.8(1.8) 7:5þ2:2þ0:91:81:4 [13]
0b . . . 0.85 0.51(16) 2:1 1:7 [70]
þc 1.2–4.1 1.81 3.04(37) 3.18(10) 2.7(2) . . . 1.13 2.44(26) <3:1ðCL ¼ 90%Þ [71]
0c 1.2–4.0 1.88 3.12(33) 3.03(10) 2.8(2) . . . 1.08 2.78(29) <5:5ðCL ¼ 90%Þ [72]
0b , finding a width of 2:1 1:7 MeV [70].1 The mass difference to the b was measured to be
M0 the CDF measurement reported in Ref. [74], we have
M0
b M0
b ¼ 157:5 5:8 MeV: (111) Using the results (110) and (111), we can update our calculation of the 0b width and find
½0b ! bþ; 0b0 ¼ 0:51 0:16 MeV: (112) Given the observed mass difference (110), and assuming that M
The chiral dynamics of mesons and baryons containing a heavy quark is controlled at leading order by three axial couplings g1, g2, and g3. Knowledge of the values of these couplings is an essential ingredient for precision QCD calculations in flavor physics. In this paper, we have dis-cussed in detail the first complete lattice QCD determina-tion of g1, g2, and g3. We have extracted the axial couplings by fitting numerical data for matrix elements of the axial current using the quark-mass and volume dependence calculated in SUð4j2Þ heavy-hadron chiral perturbation theory. Our final results are
g1 ¼ 0:449 0:047stat 0:019syst; g2 ¼ 0:84 0:20stat 0:04syst; g3 ¼ 0:71 0:12stat 0:04syst:
(113)
The systematic uncertainties in (113) are very small, be-cause our analysis is based on data at low pion masses, with a large volume, and at two different lattice spacings. We have also carefully removed the excited-state contamina-tion in the matrix elements by extrapolating the ratios of correlation functions to infinite source-sink separation.
Previous lattice calculations of heavy-hadron axial cou-plings had only considered the mesonic coupling g1. The early calculations of g1 did not include dynamical quarks and hence are contaminated by uncontrolled systematic errors. The nf ¼ 2 calculations typically used large quark masses and the fits to the quark-mass dependence were performed either linearly in m2or with an incorrect coef-ficient of the chiral logarithm. Had the correct coefcoef-ficient been used, significantly lower values of g1 would have been obtained in these previous studies.
For the range of pion masses considered in our work (230 MeV& m& 350 MeV), the chiral expansion of the
axial-current matrix elements between heavy-light hadron states is found to be well-behaved. The next-to-leading-order contributions are small compared to the leading-next-to-leading-order contributions, and NNLO contributions are negligible. The rapid convergence of the chiral expansion is also a conse-quence of the smallness of the static-light axial couplings (113). It is interesting to compare the chiral dynamics of hadrons containing a heavy quark with that of light baryons.
Being particularly light, the interactions of virtual pions (and other pseudo-Goldstone bosons) produce significant contributions to many properties of baryons, and generically these effects scale quadratically with the strength with which a given baryon sources pions. This, in turn, is deter-mined by the relevant axial coupling, g1;2;3 in the case of heavy hadrons, and gA 1:26, jgNj 1:6 and g
1:9 in the case of light baryons [75,76]. From the numeri-cal values of these couplings, it is apparent that chiral dynamics is more perturbative for heavy-light hadrons than that for light baryons.
Our results for the heavy-light axial couplings, Eq. (113), are significantly smaller than the values one obtains in the nonrelativistic quark-model, g1¼ gudA , g2¼ 2gudA and g3 ¼ ffiffiffi
p2
gudA , where gudA ¼ 1 is the axial coupling of the single-quark transition u ! d. Even if gudA is set to 0.75, as needed to reproduce the experimental value of the nucleon axial charge, the corresponding quark-model values of g1;2;3 are still significantly smaller than the results (113) from first-principles lattice QCD.
We have used our results for g2and g3to calculate strong and radiative decay widths of charm and bottom baryons. For the strong decays, we have taken into account the order-1=mQ corrections, which we have constrained by combining experimental data for charmed baryon decay rates with our lattice determination of g3. We found that the 1=mQ corrections are significant (their effect on the amplitudes for ðÞQ ! Q decays is about 40% at mQ ¼ mc and about 13% at mQ ¼ mb). As a consequence, the coupling g3 cannot be reliably extracted from experimental data for charmed baryon decays alone, and our lattice calcu-lation in the static limit is crucial to calculate the widths of bottom baryons. Our results for the widths of the ðÞb baryons are in agreement with recent measurements at Fermilab.
Our determination of the axial couplings can also im-prove the precision of future lattice QCD calculations of other heavy-hadron properties such as masses, decay stants, and form factors, because the axial couplings con-trol the dependence of these properties on the light-quark masses. Therefore, the calculation of the axial couplings from first principles also has an impact on searches for beyond-the-standard-model physics at the LHC and the planned SuperB experiment. Importantly, our results in-clude the baryonic couplings g2 and g3. Heavy baryons may offer additional opportunities for probing the structure of new physics as a consequence of the different spin quantum numbers.
1Without a spin identification, there is a small possibility that the state observed by CMS is the 00b instead. We do not consider this further.
ACKNOWLEDGMENTS
We would like to thank Hai-Yang Cheng, Kostas Orginos, Martin Savage, Brian Tiburzi, Andre´ Walker-Loud, and Matt Wingate for helpful discussions. We are indebted to the RBC and UKQCD collaborations for access to the gauge field configurations used in this work and to Robert Edwards and Balint Joo´ for the development of theCHROMAlibrary [82]
with which some of the calculations were performed. The work of W. D. is supported in part by Jefferson Science Associates, LLC, under U. S. Department of Energy Contract No. DE-AC05-06OR-23177 and by the Jeffress Memorial Trust, J-968. W. D. and S. M. were supported by DOE Outstanding Junior Investigator Award DE-SC000-1784 and DOE Grant No. DE-FG02-04ER41302. C. J. D. L. is supported by NSC Taiwan Grant No. 99-2112-M-009-004-MY3. We acknowledge the hospitality of Academia Sinica Taipei, National Chiao-Tung University, National Centre for Theoretical Sciences Taiwan, The College of William and Mary, and Thomas Jefferson National Accelerator Facility.
This research made use of computational resources provided by NERSC and the NSF Teragrid (NCSA, TACC and NICS).
APPENDIX A: PLOTS OF RAW DATA
FIG. 18 (color online). Summary of all data points for R1ðtÞ, R2ðtÞ and R3ðtÞ. At each value of t=a, results from up to five different values of nHYPare shown (from left to right: nHYP¼ 1, 2, 3, 5, 10; points offset horizontally for legibility; in some cases there are no results for R3for the lowest values of nHYP, because the statistical fluctuations were too large to calculate the square root of the double ratio). In physical units, the range of the horizontal axis in all plots is from t ¼ 0:336 fm to t ¼ 1:23 fm.
APPENDIX B: COMPARISON OF STANDARD RATIO METHOD AND SUMMATION METHOD
To extract the effective axial couplings from the ratios Riðt; t0Þ defined in Eqs. (63)–(65), we defined RiðtÞ to be the average of Riðt; t0Þ over a symmetric range of t0 values around t=2 in a region where there was no discernible t0 dependence, which essentially amounts to using
Riðt; t=2Þ: (B1)
An alternative approach for extracting geff is the summa-tion method [19,77–80]. In the following, we only consider the case of the simple ratios (63) and (64) for degenerate spectra. One defines the summed ratio SiðtÞ by summing Riðt; t0Þ over all values of t0,
SiðtÞ ¼ aXt
t0¼0
Riðt; t0Þ: (B2) For large t, one expects [77,78]
SiðtÞ ciþ ðgiÞefft; (B3) with some constant ci. Thus, the coupling ðgiÞeff can be extracted by taking the derivative [19,80],
Rsumi ðtÞ ¼ d
dtSiðtÞ; (B4)
which is approximated by a finite difference on the lattice.
Assuming that there is a nonvanishing off-diagonal matrix element of the axial current between the ground-state hadron and an excited state with an energy gap (for our data, contamination from off-diagonal matrix elements actually appears to be very small, as discussed in Secs. IVA and IV B), one expects that the systematic uncertainties of (B1) and (B4) due to this excited state are of order [80]
RiðtÞ ðgiÞeff ¼ Oðe12 itÞ;
Rsumi ðtÞ ðgiÞeff ¼ Oðte itÞ (B5)
[see Eq. (71) for the spectral decomposition of RiðtÞ]. Thus, the excited-state contamination in Rsummedi ðtÞ decays effec-tively with twice the energy gap relevant for ðgiÞeffðtÞ, but at the cost of an additional factor of t in front of the exponential, which may be important at intermediate val-ues of t.
Alternatively to taking the derivative as in Eq. (B4), one may fit SiðtÞ using the linear function (B3) with parameters ci and ðgiÞeff. In Fig. 19, we show numerical results for SiðtÞ, along with such fits. In Fig.20, we compare numeri-cal results for the standard ratio (B1), the derivative of the summed ratio (B4), and the results for ðgiÞeff from linear fits to SiðtÞ using Eq. (B3). For our data, the results from the summation method, especially for the derivative of the summed ratio, are seen to suffer from much larger statis-tical uncertainties than the standard ratio. This was also found in Ref. [80] and is not unexpected, because the relative statistical uncertainty in the difference of two similarly-sized observables (the discrete derivative used here) is much larger than the relative statistical uncertainty in the individual observables. Of course there are correla-tions which can improve the situation, and we did take these into account when calculating (B4), but because of the way that our lattice calculation was set up (data at successive values of t did not always have neighboring source locations), the correlations were not optimal.
It appears that the systematic errors of the results from the summation method at short t are similar in magnitude to the systematic errors of the results from the standard ratio method at the same t, but the deviations from ðgiÞeff have the opposite sign. This shows that valuable informa-tion about systematic errors can be obtained by comparing both methods. For the present data, our process of extrapolating the results from the standard ratio to infinite t is superior because of the much smaller statistical uncertainty.
Similarly to the work done in Ref. [80], we also studied models for the three-point and two-point functions with
FIG. 19 (color online). Fits to the summed ratios S1ðtÞ and S2ðtÞ, in the range t=a ¼ 8, 9, 10. The data are for a ¼ 0:112 fm, and a heavy-quark mass of mðvalÞu;d ¼ 0:04 (close to the physical strange-quark mass; the large mass was chosen here for the smaller statistical uncertainties) and nHYP¼ 3.
excited states. We found that at intermediate values of t, the systematic uncertainties of (B1) and (B4) were strongly dependent on the assumptions made in the model. For some models, the standard ratio showed an adantage while for others the summation method showed an advan-tage, so that again we were not able to draw definitive conclusions.
Further methods for the calculation of hadron-to-hadron matrix elements are based on the generalized eigenvalue problem [80] and the ‘‘generalized pencil-of-function’’
[81]. These techniques use matrices of correlation func-tions with multiple interpolating fields to reduce the excited-state contamination at finite t. Because we only have data from one interpolating field for each hadron, we cannot test these methods here.
[1] J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984).
[2] M. A. Shifman and M. B. Voloshin, Sov. J. Nucl. Phys. 47, 511 (1988).
[3] M. B. Wise,Phys. Rev. D 45, R2188 (1992).
[4] G. Burdman and J. F. Donoghue,Phys. Lett. B 280, 287 (1992).
[5] T.-M. Yan, H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y. C.
Lin, and H.-L. Yu,Phys. Rev. D 46, 1148 (1992);55, 5851 (E) (1997).
[6] P. L. Cho,Phys. Lett. B 285, 145 (1992).
[7] P. L. Cho, Nucl. Phys. B396, 183 (1993); B421, 683(E) (1994).
[8] H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin, T.-M.
Yan, and H.-L. Yu,Phys. Rev. D 49, 5857 (1994);55, 5851 (E) (1997).
[9] B. Grinstein, E. E. Jenkins, A. V. Manohar, M. J. Savage, and M. B. Wise, Nucl. Phys. B380, 369 (1992).
[10] J. L. Goity,Phys. Rev. D 46, 3929 (1992).
[11] C. G. Boyd and B. Grinstein, Nucl. Phys. B442, 205 (1995).
[12] A. S. Kronfeld and S. M. Ryan, Phys. Lett. B 543, 59 (2002).
[13] T. Aaltonen et al. (CDF Collaboration),Phys. Rev. D 85, 092011 (2012).
[14] G. M. de Divitiis, L. Del Debbio, M. Di Pierro, J. M.
Flynn, C. Michael, and J. Peisa (UKQCD Collaboration),J. High Energy Phys. 10 (1998) 010.
[15] A. Abada, D. Bec´irevic´, P. Boucaud, G. Herdoiza, J. P.
Leroy, A. Le Yaouanc, and O. Pene,J. High Energy Phys.
02 (2004) 016.
[16] S. Negishi, H. Matsufuru, and T. Onogi, Prog. Theor.
Phys. 117, 275 (2007).
[17] H. Ohki, H. Matsufuru, and T. Onogi,Phys. Rev. D 77, 094509 (2008).
[18] D. Bec´irevic´, B. Blossier, E. Chang, and B. Haas,Phys.
Lett. B 679, 231 (2009).
[19] J. Bulava, M. A. Donnellan, and R. Sommer (ALPHA Collaboration), Proc. Sci., LATTICE2010 (2010) 303.
[20] W. Detmold, C.-J. D. Lin, and S. Meinel,Phys. Rev. D 84, 094502 (2011).
[21] W. Detmold, C.-J. D. Lin, and S. Meinel,Phys. Rev. Lett.
108, 172003 (2012).
[22] E. Eichten and B. R. Hill,Phys. Lett. B 240, 193 (1990).
[23] M. Della Morte, S. Du¨rr, J. Heitger, H. Molke, J. Rolf, A.
Shindler, and R. Sommer (ALPHA Collaboration),Phys.
Lett. B 581, 93 (2004);612, 313(E) (2005).
FIG. 20 (color online). Comparison of the standard ratio method and the summation method for R1and R2at a ¼ 0:112 fm, for a heavy-quark mass of amðvalÞu;d ¼ 0:04 (close to the physical strange-quark mass; the large mass was chosen here for the smaller statistical uncertainties) and nHYP¼ 3. Shown is the dependence on the source-sink separation t. The derivative of the summed ratio, Eq. (B4), is approximated using Rsumi ¼ ½SiðtÞ þ Siðt þ aÞ=a and plotted with square symbols at t þ12a. The results for ðgiÞefffrom linear fits to SiðtÞ using Eq. (B3) are plotted with diamond symbols. We performed fits in the range t ¼ tmin. . . 10a, and the results are plotted at t ¼ tminþ14a. For the data from the standard ratio method (circles), the curves indicate the results of fits with the form RiðtÞ ¼ ðgiÞeff Aie itwith free parameters ðgiÞeff, Ai, and i see Sec.IV B.
[24] D. B. Kaplan,Phys. Lett. B 288, 342 (1992).
[25] Y. Shamir,Nucl. Phys. B406, 90 (1993).
[26] V. Furman and Y. Shamir,Nucl. Phys. B439, 54 (1995).
[27] Y. Aoki et al. (RBC/UKQCD Collaboration),Phys. Rev. D 83, 074508 (2011).
[28] H. Georgi,Nucl. Phys. B348, 293 (1991).
[29] T. Aaltonen et al. (CDF Collaboration),Phys. Rev. Lett.
99, 202001 (2007).
[30] H.-Y. Cheng,Phys. Lett. B 399, 281 (1997).
[31] W. Detmold, K. Orginos, and M. J. Savage,Phys. Rev. D 76, 114503 (2007).
[32] D. J. Antonio et al. (RBC/UKQCD Collaboration),Phys.
Rev. D 75, 114501 (2007).
[33] Y. Iwasaki, Report No. UTHEP-118, 1983.
[34] Y. Iwasaki and T. Yoshie,Phys. Lett. B 143, 449 (1984).
[35] Y. Aoki et al.,Phys. Rev. D 69, 074504 (2004).
[36] S. Meinel,Phys. Rev. D 82, 114502 (2010).
[37] A. Hasenfratz and F. Knechtli,Phys. Rev. D 64, 034504 (2001).
[38] G. P. Lepage,Nucl. Phys. B, Proc. Suppl. 26, 45 (1992).
[39] T. Burch, C. Hagen, C. B. Lang, M. Limmer, and A.
Scha¨fer,Phys. Rev. D 79, 014504 (2009).
[40] C. Michael, A. Shindler, and M. Wagner (ETM Collaboration),J. High Energy Phys. 08 (2010) 009.
[41] B. Yoon, Y. C. Jang, C. Jung, and W. Lee, arXiv:1101.2248.
[42] B. Sheikholeslami and R. Wohlert,Nucl. Phys. B259, 572 (1985).
[43] D. Becirevic, E. Chang, and A. L. Yaouanc, arXiv:1203.0167.
[44] S. Fajfer and J. F. Kamenik, Phys. Rev. D 74, 074023 (2006).
[45] P. Colangelo, F. De Fazio, and G. Nardulli,Phys. Lett. B 334, 175 (1994).
[46] D. Bec´irevic´ and A. L. Yaouanc,J. High Energy Phys. 03 (1999) 021.
[47] Z. Guralnik, M. E. Luke, and A. V. Manohar,Nucl. Phys.
B390, 474 (1993).
[48] P. Colangelo, G. Nardulli, A. Deandrea, N. Di Bartolomeo, R. Gatto, and F. Feruglio, Phys. Lett. B 339, 151 (1994).
[49] V. M. Belyaev, V. M. Braun, A. Khodjamirian, and R.
Ru¨ckl,Phys. Rev. D 51, 6177 (1995).
[50] H. G. Dosch and S. Narison,Phys. Lett. B 368, 163 (1996).
[51] P. Colangelo and F. De Fazio, Eur. Phys. J. C 4, 503
[55] M. C. Arnesen, B. Grinstein, I. Z. Rothstein, and I. W.
Stewart,Phys. Rev. Lett. 95, 071802 (2005).
[56] X.-Q. Li, F. Su, and Y.-D. Yang,Phys. Rev. D 83, 054019 (2011).
[57] E. E. Jenkins,Phys. Lett. B 315, 431 (1993).
[58] J. F. Amundson, C. G. Boyd, E. E. Jenkins, M. Luke, A. V.
Manohar, J. L. Rosner, M. J. Savage, and M. B. Wise, Phys. Lett. B 296, 415 (1992).
[59] I. W. Stewart,Nucl. Phys. B529, 62 (1998).
[60] A. Anastassov et al. (CLEO Collaboration),Phys. Rev. D 65, 032003 (2002).
[61] M. A. Ivanov, J. G. Ko¨rner, V. E. Lyubovitskij, and A. G.
Rusetsky,Phys. Rev. D 60, 094002 (1999).
[62] H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin, T.-M.
Yan, and H.-L. Yu,Phys. Rev. D 49, 2490 (1994).
[63] K. Nakamura et al. (Particle Data Group),J. Phys. G 37, 075021 (2010).
[64] S. Tawfiq, P. J. O’Donnell, and J. G. Ko¨rner,Phys. Rev. D 58, 054010 (1998).
[65] C. Albertus, E. Hernandez, J. Nieves, and J. M. Verde-Velasco,Phys. Rev. D 72, 094022 (2005).
[66] E. Hernandez and J. Nieves, Phys. Rev. D 84, 057902 (2011).
[67] H.-Y. Cheng and C.-K. Chua,Phys. Rev. D 75, 014006 (2007).
[68] C.-W. Hwang,Eur. Phys. J. C 50, 793 (2007).
[69] C. Chen, X.-L. Chen, X. Liu, W.-Z. Deng, and S.-L. Zhu, Phys. Rev. D 75, 094017 (2007).
[70] S. Chatrchyan et al. (CMS Collaboration), arXiv:1204.5955.
[71] L. Gibbons et al. (CLEO Collaboration),Phys. Rev. Lett.
77, 810 (1996).
[72] P. Avery et al. (CLEO Collaboration),Phys. Rev. Lett. 75, 4364 (1995).
[73] M. Lu, M. J. Savage, and J. Walden,Phys. Lett. B 369, 337 (1996).
[74] T. Aaltonen et al. (CDF Collaboration),Phys. Rev. Lett.
107, 102001 (2011).
[75] E. E. Jenkins and A. V. Manohar, Proceedings of the Workshop on Effective Field Theories of the Standard Model, Dobogo´ko˝, Hungary, 1991, edited by U.-G.
Meissner (World Scientific, Singapore, 1992) (Report No. UCSD/PTH 91-30).
[76] E. E. Jenkins and A. V. Manohar,Phys. Lett. B 255, 558 (1991).
[77] L. Maiani, G. Martinelli, M. L. Paciello, and B. Taglienti, Nucl. Phys. B293, 420 (1987).
[78] S. Gu¨sken, U. Low, K. H. Mutter, R. Sommer, A. Patel, and K. Schilling,Phys. Lett. B 227, 266 (1989).
[79] S. Capitani, B. Knippschild, M. Della Morte, and H.
Wittig, Proc. Sci., LATTICE2010 (2010) 147.
[80] J. Bulava, M. Donnellan, and R. Sommer,J. High Energy Phys. 01 (2012) 140.
[81] C. Aubin and K. Orginos, AIP Conf. Proc. 1374, 621 (2011).
[82] R. G. Edwards and B. Joo´ (SciDAC Collaboration, LHPC Collaboration, and UKQCD Collaboration), Nucl. Phys.
B, Proc. Suppl. 140, 832 (2005).