Constraints on Axion couplings from the CDEX-1 experiment at the China Jinping Underground Laboratory
We report the results of searches for solar axions and galactic dark matter axions or axion-like particles with CDEX-1 experiment at the China Jinping Underground Laboratory, using 335.6 kg-days of data from a p-type point-contact germanium detector. The data are compatible with the background model and no excess signals are observed. Limits of solar axions on the model independent coupling from Compton, bremsstrahlung, atomic-recombination and deexcitation channel and from Fe M1 transition at 90% confidence level are derived. Within the framework of the DFSZ and KSVZ models, our results exclude the axion mass heavier than 0.9 eV/c and 173 eV/c, respectively. The derived constraints for dark matter axions below 1 keV improves over the previous results.
pacs:95.35.+d, 98.70.Vc, 29.40.Wk
The China Dark Matter Experiment (CDEX) pursues direct searches of low mass weakly interacting massive particle (WIMP) and studies of double-beta decay in Ge Olive et al. (2014); Schumann (2015); Cushman et al. (2013); Barabash (2015a); *2015_Barabash_2; Bilenky and Giunti (2012); *2012_Elliott toward the goal of a ton-scale germanium detector array Kang et al. (2013) at the China Jinping Underground Laboratory (CJPL) Kang et al. (2010) . CJPL is located in the Jinping traffic tunnel, Sichuan province, China, with a vertical rock overburden of more than 2400 m, providing a measured muon flux of 61.7 y m Wu et al. (2013). A pilot measurement CDEX-0 with a germanium detector array with 20 g target mass, achieving the threshold of 177 eVee (“ee” represents electron equivalent energy), was reported Liu et al. (2014). CDEX-1 experiment adopted one single module of the p-type point-contact germanium (PCGe) detector with fiducial mass of 915 g Jiang et al. (2016). The phase-I of CDEX-1 measurement in the absence of anti-Compton detector and prior to surface suppression based on 14.6 kg-days of data was published with a threshold of 400 eVee Zhao et al. (2013). The phase-II measurements, featuring with an anti-Compton detector and bulk surface discrimination, based on the 53.9 kg-days Yue et al. (2014) and 335.6 kg-days of data Zhao et al. (2016) were reported. Both results strongly disfavors the allowed region implied by residual excess events from CoGeNT with an identical detector target.
Quantum chromodynamics (QCD), universally believed to be the best theory describing strong interactions, contains the term which could explicitly give a rise to a measurable CP-violation such as a large neutron electric dipole moment. The experimental bound is about ten orders of magnitude more stringent Baker et al. (2006) resulting in an unnaturally small upper limit () to the parameter. In order to solve this “strong CP problem”, Peccei and Quinn (PQ) postulated a new spontaneously broken symmetry that naturally and dynamically cancels CP violation in the strong interactions Peccei and Quinn (1977a); *1977_PQ. Weinberg Weinberg (1978) and Wilczek Wilczek (1978) later proposed that this new symmetry introduces a new pseudoscalar particle similar to neutral pions called axion. This original axion with a symmetry-breaking scale of the order of the electroweak scale has been excluded by experiments (see Kim (1978); *2010_Kim; Ringwald (2012) and refs therein) whereas “invisible” axion models such as non-hadronic model DFSZ (Dine-Fischler-Srednicki-Zhitnitskii) Dine et al. (1981); Zhitniskiy (1980) and hadronic model KSVZ (Kim-Shifman-Vainstein-Zakharov) Kim (1979); Shifman et al. (1980) arising from a higher symmetry-breaking energy scale are still allowed. In addition, axion-like-particles (ALPs) with the similar properties as the QCD axions also have the couplings to electrons (), photons () and nucleons (), though do not necessarily solve the strong CP problem.
Several dark matter (DM) experiments aiming at direct detection of WIMPs have reported the axion searches results Ahmed et al. (2009); Bernabei et al. (2001); Aalseth et al. (2011); Alessandria et al. (2013); Armengaud et al. (2013); Abe et al. (2013); Aprile et al. (2014). These experiments mainly incorporate two detection mechanisms. The first is that axions from our sun have the couplings to photons () in detectors through the Primakoff effect, (Q stands for charged particles). These measurements utilize the Bragg diffraction effect in the crystal detectors Armengaud et al. (2013); Ahmed et al. (2009); Bernabei et al. (2001) in which the intense electric field would enhance the interaction cross-section. The constraints on from these experiments are typically much less sensitive than the helioscope experiment Arik et al. (2011) and microwave cavity experiment Rybka (2014). The second is that solar axions and dark matter ALPs have the couplings to electron () in detectors through the axioelectric effect:
which is similar to the photoelectric effect with the absorption of an axion instead of a photon Ahmed et al. (2009); Aalseth et al. (2011); Alessandria et al. (2013); Armengaud et al. (2013); Abe et al. (2013); Aprile et al. (2014).
We report the axion searches results from the CDEX-1 experiment based on an exposure of 335.6 kg-days of data which is the same data set as the ref. Zhao et al. (2016). We focus on the couplings from the solar axions and the galactic dark matter ALPs through axioelectric effect of Eq. 1. Studies on the coupling are not pursued since the Bragg diffraction methods are less sensitive and with larger systematic uncertainties.
Ii Axion searches with CDEX-1
ii.1 CDEX-1 setup and overview
CDEX-1 experiment adopted one single module of the p-type point-contact germanium (PCGe) detector at 994 g of mass Zhao et al. (2013); Yue et al. (2014); Zhao et al. (2016), featuring with a relative low threshold down to 475 eVee. A cylindrical NaI(Tl) crystal with well shaped cavity enclosing the cryostat of the PCGe, whose threshold was about 5 keVee, was served as the anti-Compton detector. Events in coincidence with the AC detector were discarded to get rid off -ray induced background.
The point-contact electrode after a pulsed-reset feedback preamplifier generated three identical energy-related signals. These three outputs were fed into the shaping amplifiers at 6 s (S), 12 s (S) shaping time, and a timing amplifier (T) respectively. The outputs from S provided the energy measurement and system trigger of the DAQ. Their dynamic ranges were limited to 12 keVee to achieve the maximal signal-to-noise ratio and maximal information for low-energy events. The output from T recording the raw fast pulse shape was employed to discriminate the bulk/surface events. Its energy dynamic range could be extended to 20 keVee, while it was slightly different from S below 2 keVee and had higher energy threshold. So in our following analysis, the spectrum below 12 keVee was from S and above 12 keVee was from T. Both calibrations with good linearity of less than 0.8% deviation were derived from the internal cosmogenic x-ray peaks and random trigger events Zhao et al. (2016).
ii.2 Axion sources
ii.2.1 Solar Axions
The Sun can be an abundant source of axions, which are generated by four production mechanisms that depend on Barth et al. (2013):
Compton-like scattering (C):
where is any charged particle, is electron, is ion and is its excited state.
where the unit is and axion energy is in keV. For RD solar axions, the flux also depend on and the tabulated spectrum in ref. Redondo (2013) is adopted.
The 14.4 keV monochromatic axions emitted in the M1 transition of Fe nuclei in the Sun
can be an additional important source of solar axions due to the large abundance of Fe among the heavy elements Haxton and Lee (1991); Moriyama (1995). Its flux is related to coupling and is given by Andriamonje et al. (2009); Armengaud et al. (2013).
where the unit is , and are the momenta of the outgoing axion and photon respectively. The effective nuclear coupling is model dependent, , where and are the model-dependent isoscalar and isovector axion-nucleon coupling constants, respectively Kaplan (1985); Srednicki (1985). Fig. 1 shows the evaluated fluxes of solar axions on Earth for the processes we are concerned.
ii.2.2 Galactic Dark matter ALPs
The non-thermal axions or ALPs, produced by the vacuum realignment mechanism and radiation from cosmic strings, are candidates to solve the dark matter problem in the universe. The total average flux independent on any axion coupling is given by
where is the dark matter halo density ( GeV/cm Green (2012)), is the axion mass, is the mean axion velocity distribution with respect to the Earth, is the ratio of the axion velocity to the speed of light.
ii.3 Experimental Signatures
We focus on the detection channel of axioelectric effect as illustrated in Eq.1, where the cross-section is given by
as described in Alessandria et al. (2013); Derevianko et al. (2010); Pospelov et al. (2008), where is the photoelectric cross section for Ge, is the fine structure constant, is the electron mass and is the ratio of the axion velocity to the speed of light.
The expected axion event rate at measurable energy is obtained by the convolution of the flux, the axioelectric cross section and the energy resolution of the detector:
where represents the different axion sources of fluxes . The detector energy resolution () is 90 eV at 10.37 keV Zhao et al. (2016).
where A is mass number for germanium and the units of and are kgday and barns/atom, respectively. We note that the sensitivity dependence on the coupling strength are different for different sources and detector channels. The event rate varies as , , for CBRD, Fe, DM, respectively. The difference in coupling dependence of ALP DM rates compared to those of solar axions is a consequence of the DM flux being fixed by cosmological data given a certain . The expected for various channels in CDEX-1 are depicted in Fig. 2.
Iii Data Analysis
iii.1 Candidate Event Selection
(i) Stability check, which discards the time periods of calibration or laboratory construction;
(ii) Physics versus electronic noise, which differentiates physical events from the electronic noise and spurious signals.
(iii) Anti-Compton selection, which removes the events in coincidence with the anti-Compton detector.
In particular, there exists an inactive layer of about 1 mm in thickness at the surface electrode. These surface events are rejected by pulse shape analysis using their characteristic slower rise-time Yue et al. (2014); Zhao et al. (2016). Procedures have been established to derive their signal-retaining and background-leakage efficiencies Li et al. (2013); *2014_LHB.
iii.2 Background description and background model simplification
In this work, we analyze the same data set as the ref.Zhao et al. (2016) with an exposure of 335.6 kg-days of data. The bulk spectrum () from 475 eV up to 20 keV after the data selection described above and after efficiency correction is shown in Fig. 3. The background consists of six distinct K-shell x-ray peaks from the cosmogenic nuclides and their corresponding L-shell x-rays (dotted red line), and a continuous component with a smooth, slightly increasing profile as the energy decreasing (solid blue line)Yue et al. (2014); Zhao et al. (2016).
The contribution of the ambient radioactivity at CJPL external to the shielding system was greatly suppressed to kg keV day at the energy range below 20 keV Zhao et al. (2016). The continuous background below 20 keV was expected to mainly originate from residual U, Th, K in the experimental hardware in the vicinity of the PCGe detector, radon gas penetrating inside shielding, and cosmogenic H inside the crystal. Quantitative studies of their relative contributions are our current research efforts and beyond the scope of this work.
However all the axion signals have the signatures which are significantly different from the continuous background especially in the local energy range. As shown in Fig. 2, for Fe solar axions and dark matter ALPs, compared with continuous background, these event signatures are monochromatic and Poisson distributions whose widths are determined by the energy resolution. As to the continuous CBRD solar axion, the event rate have the distinct signature that it has a saw-tooth-like profile between the local energy from 0.9 keV to 1.6 keV if we only consider the mass within 1 keV/c. As discussed in ref.Redondo (2013), the most recent and accurate calculation for solar axion flux is valid for light axions, hence we only consider the axion mass .
The accurate quantitative study of the continuous background is not essential for this axion sensitivity experiment. Therefore we interpret the background in a simplified way: the combination of K/L x-ray peaks and a continuous background. A constant background within a local energy range of interest is sufficient for this analysis. The formulation of the analysis algorithms and evaluation of systematic uncertainties are discussed in subsequent sections.
iii.3 Analysis method
The unbinned maximum likelihood method Ahmed et al. (2009); Aprile et al. (2014) is adopted to derive constraints in axion couplings from the measured spectrum. Every measured event is categorized as bulk or surface event, denoted by and , respectively, according to its rise time. The relationships between the measured spectrum (, ) and the efficiency corrected spectrum (, ) can be derived from the following coupled equations, which are illustrated in Ref. Li et al. (2013); *2014_LHB; Yue et al. (2014); Zhao et al. (2016):
refers to the combined efficiencies of physics versus electronic noise selection and Anti-Compton selection mentioned in Sec.III.A. The efficiencies of and , representing the bulk event retaining and surface background rejection, can translate (, ) to (, ).
The best-fit solution to ( and is evaluated by maximizing the likelihood function COWAN (1998):
where and are the numbers of bulk measurement events and surface measurement events respectively. and are the p.d.f.s (probability density function) of bulk measurement and surface measurement respectively, which are described by
According to previous discussion of simplified background model, the first background component: represents the normalized p.d.f. of local continuous background using a zeroth polynomial function. The fitting range is constrained to local by different kinds of axion sources. As to CBRD solar axion, the fitting range is limited to 0.9 keV to 1.6 keV as shown in Fig.4 (a); for Fe, that is limited to 13.0 keV to 16.0 keV as depicted in Fig.6; and to ALP DM axion, the range is constrained to 8 range. The other component: is the normalized p.d.f of the K/L shell x-rays peaks. is the normalized p.d.f. describing the axion events as shown in Fig. 2. represents the normalized p.d.f. of the efficiency corrected surface spectrum , derived from fitting by a smooth curve. The systematic uncertainties of the p.d.f. selection of is negligible by comparing bin-by-bin p.d.f from spectrum.
The relative contributions of each components in the bulk measurements are represented by , , and , while , , and are the their individual relative contributions to the surface measurement data. The measured bulk and surface components are related by
where * represents and . In addition, the efficiency-corrected component is given by
The goodness-of-fit of this maximum likelihood analysis is tested with binned the data, where ( represents the degrees of freedom) at the energy of 300 eV of DM ALPs.
iii.4 Systematic Uncertainties
The effects of systematic uncertainties have been evaluated for all of analyses from the three different axion sources. Systematic uncertainties may originate from bulk surface events selection, signal selection, fiducial mass as well as the background assumption.
According to the evaluation in the previous work Zhao et al. (2016), the contribution of the uncertainty of bulk surface event selection in the low energy range is dominated. This component has been taken into account in the likelihood function of Eq. 10, and introduced via the uncertainties of and in Eq. III.3. This contributes about 55%, 15% and well below 1% systematic uncertainties to the constraints on galactic dark matter axion below 1 keV/c, CBRD solar axion and Fe solar axion, respectively.
The uncertainties of the background assumption have been evaluated by the different background assumptions between the flat background, polynomial background and exponential background. The variation of the background assumptions gives the uncertainties of about 7%, 5% and 8% for the galactic dark matter axion below 1 keV/c, CBRD solar axion and Fe solar axion, respectively.
Compared to the uncertainties arisen from bulk surface events selection and background assumption, the contributions of signal selection, fiducial mass and energy resolution uncertainties of the detector to the systematic uncertainties are negligible.
iv.1 Solar axions
For the CBRD solar axion, the saw-tooth signature is within the energy range as the shadow displayed in Fig. 4 (a), since we only consider the mass below 1 keV/c. The local fitting range is limited to the shadow region 0.9 keV 1.6 keV and the fitting results of 90% C.L. at the mass of 0 keV is shown in Fig. 4 (b), as well as the spectrum and the background model described. The data are compatible with the background model and no excess signals are observed. The solid red line in Fig. 5 (a) shows our limit on at 90% C.L. which is restricted to the mass below 1 keV/c, together with the bounds from astrophysical bounds Gondolo and Raffelt (2009); Viaux et al. (2013); Raffelt (2008), other representative experiments including CBRD axion and Fe axion Armengaud et al. (2013); Aprile et al. (2014); Abe et al. (2013); Alessandria et al. (2013).
As illustrated in the inset of Fig. 5 (a), the improved energy threshold of CDEX-1 gives rise to a 90% C.L. limit of for , which is comparable to that of EDELWEISS experiment Armengaud et al. (2013) which also adopts germanium detectors.
As to a specific axion model, DFSZ or KSVZ Dine et al. (1981); Zhitniskiy (1980); Kim (1979); Shifman et al. (1980), the limit can be translated into the limit of axion mass . In the DFSZ model, on the assumption of model-dependent parameter cos, where is an arbitrary angle, CDEX-1 excludes axion masses above 0.9 eV/c. In the KSVZ model, on the assumption of model-dependent parameters , where is the ratio of the electromagnetic to color anomalies of the Peccei-Quinn symmetry Srednicki (1985), our result excludes axion masses above 265 eV/c.
For Fe M1 transition 14.4 keV axion, Fig. 6 displays the spectrum at the energy range of 13 – 16 keV as well as the background model. There is no hint of a line at 14.4 keV and the expected signal at 90 % C.L. is shown as the blue line. The model independent limit of is shown in Fig. 5 (b) compared with the EDELWEISS limits Armengaud et al. (2013). The is model-dependent coupling. In KSVZ models, it dependents on the flavor-singlet axial-vector matrix element Adams et al. (1997); Altarelli et al. (1997). In DFSZ models, besides element it also dependents on the tan, which is the ratio of two Higgs vacuum expectation values of the model. The dashed red line in Fig. 5 (a) shows the 90% C.L. limit at the DFSZ model with and cos. In DFSZ and KSVZ models, using the parameters described above, the axion mass can be constrained to 9 eV/c and 173 eV/c, respectively. Combining the results from CBRD channel and Fe channel, our results exclude the axion mass heavier than 0.9 eV/c and 173 eV/c according to the DFSZ and KSVZ model respectively.
iv.2 Galactic ALPs
No statistically significant excess signals are observed, scanning the energy range between 0.475 – 20 keV with the same method as Fe solar axion. Fig. 7 shows one of the fitting results at the mass of 0.9 keV. The 90% C.L. limit on is displayed in Fig. 8. The peaks in the limit plots corresponds to the K/L x-ray peaks in the spectrum and the steps around 1.3 keV and 10 keV due to the atomic energy levels. Because of the monochromatic signal and the good energy resolution, the limit is sensitive to the fluctuations of individual bins . The CDEX-1 limits are more stringent than improve over from CoGeNT Aalseth et al. (2011) at axion mass less than 1 keV, due to improved detector threshold, energy resolution and residual background.
V Summary AND Prospects
The CDEX-1 hardware, interested axion sources, the details of data analysis procedures and results have been described. The limits of couplings of solar axions and galactic dark matter ALPs are derived with a data size of 335.6 kg-days. We demonstrated that the PCGe detector is a potential technique to axion searches due to the excellent energy resolution and low energy threshold, especially for the peak searches such as the DM ALPs and Fe solar axions. The constraints on DM ALPs below the mass of 1 keV/c improves over the previous results.
The CDEX-1 data spanning over 17 months allows the studies of annual modulation. In additional, research efforts on lowering the detector threshold, controlling of radiopurity as well as understanding of background are being pursued. Improved sensitivities of the studies of WIMP dark matter and axions can be foreseen.
Acknowledgements.This work was supported by the National Natural Science Foundation of China (Contracts No. 11505101, 11175099, 11275107, 11475117 and 11475099), and the National Basic Research Program of China (973 Program) (2010CB833006), and China Postdoctoral Science Foundation, and MOST 104-2112-M-001-038-MY3, and the Academia Sinica Principle Investigator Award 2011-2015 from Taiwan.
- Olive et al. (2014) K. A. Olive et al., Chin. Phys. C 38, 090001 (2014).
- Schumann (2015) M. Schumann, Eur. Phys. J. Web Conf. 96, 01027 (2015).
- Cushman et al. (2013) P. Cushman et al., arXiv:1310.8327 (2013).
- Barabash (2015a) A. S. Barabash, Phys. Procedia 74, 416 (2015a).
- Barabash (2015b) A. S. Barabash, AIP Conf. Proc. 1686, 020003 (2015b).
- Bilenky and Giunti (2012) S. M. Bilenky and C. Giunti, Mod. Phys. Lett. A 27, 1230015 (2012).
- Elliott (2012) S. R. Elliott, Mod. Phys. Lett. A 27, 1230009 (2012).
- Kang et al. (2013) K. J. Kang et al., Front. Phys. 8, 412 (2013).
- Kang et al. (2010) K. J. Kang et al., J. Phys. Conf. Ser. 203, 012028 (2010).
- Wu et al. (2013) Y. C. Wu et al., Chin. Phys. C 37, 086001 (2013).
- Liu et al. (2014) S. K. Liu et al., Phys. Rev. D 90, 032003 (2014).
- Jiang et al. (2016) H. Jiang et al., Chin. Phys. C 40, 096001 (2016).
- Zhao et al. (2013) W. Zhao et al., Phys. Rev. D 88, 052004 (2013).
- Yue et al. (2014) Q. Yue et al., Phys. Rev. D 90, 091701(R) (2014).
- Zhao et al. (2016) W. Zhao et al., Phys. Rev. D 93, 092003 (2016).
- Baker et al. (2006) C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006).
- Peccei and Quinn (1977a) R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977a).
- Peccei and Quinn (1977b) R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977b).
- Weinberg (1978) S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
- Wilczek (1978) F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
- Kim (1978) J. E. Kim, Phys. Rept. 150, 1 (1978).
- Kim (2010) J. E. Kim, Rev. Mod. Phys. 82, 557 (2010).
- Ringwald (2012) A. Ringwald, Phys. Dark Univ. 1, 116 (2012).
- Dine et al. (1981) M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 104, 199 (1981).
- Zhitniskiy (1980) A. R. Zhitniskiy, Yad. Fiz. 31, 497 (1980).
- Kim (1979) J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
- Shifman et al. (1980) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 166, 493 (1980).
- Ahmed et al. (2009) Z. Ahmed et al., Phys. Rev. Lett. 103, 141802 (2009).
- Bernabei et al. (2001) R. Bernabei et al., Phys. Lett. B 515, 6 (2001).
- Aalseth et al. (2011) C. E. Aalseth et al., Phys. Rev. Lett. 106, 131301 (2011).
- Alessandria et al. (2013) F. Alessandria et al., J. Cosmol. Astropart. Phys. 05, 007 (2013).
- Armengaud et al. (2013) E. Armengaud et al., J. Cosmol. Astropart. Phys. 11, 067 (2013).
- Abe et al. (2013) K. Abe et al., Phys. Lett. B 724, 46 (2013).
- Aprile et al. (2014) E. Aprile et al., Phys. Rev. D 90, 062009 (2014).
- Arik et al. (2011) M. Arik et al., Phys. Rev. Lett. 107, 261302 (2011).
- Rybka (2014) G. Rybka, Phys. Dark Univ. 4, 14 (2014).
- Barth et al. (2013) K. Barth et al., J. Cosmol. Astropart. Phys. 05, 010 (2013).
- Redondo (2013) J. Redondo, J. Cosmol. Astropart. Phys. 12, 008 (2013).
- Haxton and Lee (1991) W. C. Haxton and K. Y. Lee, Phys. Rev. Lett. 66, 2557 (1991).
- Moriyama (1995) S. Moriyama, Phys. Rev. Lett. 75, 3222 (1995).
- Andriamonje et al. (2009) S. Andriamonje et al., J. Cosmol. Astropart. Phys. 12, 002 (2009).
- Kaplan (1985) D. B. Kaplan, Nucl. Phys. B 260, 215 (1985).
- Srednicki (1985) M. Srednicki, Nucl. Phys. B 260, 689 (1985).
- Green (2012) A. M. Green, Mod. Phys. Lett. A 27, 1230004 (2012).
- Derevianko et al. (2010) A. Derevianko, V. A. Dzuba, V. V. Flambaum, and M. Pospelov, Phys. Rev. D 82, 065006 (2010).
- Pospelov et al. (2008) M. Pospelov, A. Ritz, and M. Voloshin, Phys. Rev. D 78, 115012 (2008).
- Li et al. (2013) H. B. Li et al., Phys. Rev. Lett. 110, 261301 (2013).
- Li et al. (2014) H. B. Li et al., Astropart. Phys. 56, 1 (2014).
- COWAN (1998) G. COWAN, Statistical Data Analysis (Oxford University Press, 1998).
- Gondolo and Raffelt (2009) P. Gondolo and G. G. Raffelt, Phys. Rev. D 79, 107301 (2009).
- Viaux et al. (2013) N. Viaux et al., Phys. Rev. Lett. 111, 231301 (2013).
- Raffelt (2008) G. G. Raffelt, Lect. Notes Phys. 741, 51 (2008).
- Adams et al. (1997) D. Adams et al., Phys. Rev. D 56, 5330 (1997).
- Altarelli et al. (1997) G. Altarelli, R. D. Ball, S. Forte, and G. Ridolfi, Nucl. Phys. B 496, 337 (1997).