- Email: [email protected]
Muon g-2 constraints on SUSY dark matter reviewed and predicted
New Astronomy Reviews 49 (2005) 125–131 www.elsevier.com/locate/newastrev
Muon g-2 constraints on SUSY dark matter reviewed and predicted Priscilla Cushman Physics Department, University of Minnesota, 116 Church St. SE, Minneapolis, MN 55455, USA Available online 22 February 2005
Abstract A summary of experimental results and theoretical calculations related to the muon anomalous magnetic moment. Assuming that a discrepancy with the Standard Model is due to contributions from supersymmetric particles provides a framework which can be used to constrain the mass of the dark matter particles, assumed to be the lightest neutral supersymmetric particles. A new g-2 run in the near future, coupled with expected theoretical advances, would result in a 5r discrepancy if the mean Dal remains the same. In this case, we would expect to see SUSY particles at the LHC and use the g-2 results to measure tan b. If, instead, the Standard Model is confirmed to this precision, gauginos must have masses higher than 500 GeV/c2 and simple SUSY dark matter models will be strained. 2005 Elsevier B.V. All rights reserved.
1. Historical summary Precision measurements are a complementary approach to investigating the highest energy, smallest scale frontier of particles and interactions. Over the last decade, E821, the Brookhaven g-2 experiment, has successfully mounted a precision challenge to the standard model. The 2.6r discrepancy announced three years ago sparked debate on the theoretical calculation and encouraged further work on reducing the uncertainty in the 1st-order hadronic contribution. One of the more startling developments, approximately 6 months after the E-mail address: [email protected].
announcement of the first precision result (1.3 ppm) was the revelation by the Marseilles group (Knecht and Nyffeler, 2002) that one of the contributions to the Standard Model theory, namely the hadronic light-by-light term, had been independently assigned the wrong sign by at least 2 separate groups. Kinoshita (Hayakawa and Kinoshita, 2001) and Bijnens et al. (2002) studied their previous work and found that they both had used an incorrect sign convention in a matrix evaluation in a widely-used computer program. This moved the theoretical value by 17 · 1010 (by more than its stated uncertainty) in the direction of the E821 result, thus reducing the discrepancy to 1.6r. The next g-2 run, with an improved
1387-6473/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2005.01.027
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P. Cushman / New Astronomy Reviews 49 (2005) 125–131
precision of 0.7 ppm, left the mean unchanged, but reduced the error bars, again indicating a 2.6r discrepancy with the Standard Model. Meanwhile, in order to reduce the uncertainty on the hadronic correction, the use of vector spectral functions from the study of hadronic taudecays in ALEPH was introduced by Alemany et al. (1998). Previously, the only handle on the hadronic vacuum polarization contribution at the low center of mass energies which are relevant for g-2 came from the dispersion relation: Z 1 a2 KðsÞ ahad RðsÞ; ¼ ds l s 3p2 4m2p where RðsÞ ¼
rðeþ e ! hadronsÞ : rðeþ e ! lþ l Þ
R(s) is determined from a compilation of experimental results dominated by the CMD-2 experiment at Novosibirsk. When the s-decay data was first combined with the e+e data in 1998, it halved the error bars. However, over the past five years the continued operation and analysis of CMD-2 has improved the e+e data to such an extent that the two methods were found to be in disagreement with each other. Thus, in order to quote a discrepancy with theory, it became necessary to distinguish which hadronic correction you were referring to. In 2003, the Novosibirsk collaboration completely reanalyzed their pp channel (Akhmetshin et al., 2003) following the discovery of a mistake in their normalization (the t-channel leptonic vacuum polarization contribution was missing in the Bhabha scattering cross section). Their correction increased their published hadronic cross sections by 2.5%, thus reducing, but not erasing the discrepancy between the two theoretical approaches (especially for energies above 0.85 GeV). New results from KLOE, BaBar and perhaps Belle can provide an independent method to distinguish between the two by using radiative decay to scan the center of mass energies in the region relevant to g-2 (so-called ‘‘radiative-return’’ method). A precise measurement of the pion form factor has been reported by KLOE (Alioso et al., 2003 and DiFalco et al., 2003). It confirms the Novosibirsk e+e result. Preliminary results on 4prong final states by BaBar (Davier, 2003) also
bolsters confidence in the e+e data. On the other hand, branching ratios from CLEO and OPAL continue to confirm the ALEPH data, thus indicating that the s-decay construction may be affected by a fundamental misunderstanding in how we apply CVC, isospin corrections, or the electroweak symmetry breaking. Ghozzi and Jegerlehner (2004) argue that by simply allowing the mass of the charged-q to differ from q0 is sufficient to account for this. Davier (2003) shows that even assuming this modification, a detailed comparison of the shape of the pion form factor reveals some discrepancy. This in itself may be an indication of new physics. Most are agreed, however, that in any comparison of al (exp) to al (theory), the direct result using e+e data is more reliable at this time. The final g-2 experimental result was announced in December 2003 and published as Bennett et al. (2004). This was a 0.7 ppm result with opposite polarity muons. It was consistent with the previous data sets, despite reversing the magnetic field in the storage ring, demonstrating that systematic errors are being correctly estimated. However, the mean value was slightly higher than the earlier value, serving more to emphasize than detract from any Standard Model discrepancy. In Fig. 1, the BNL g-2 results are shown together with those from the old CERN experiment. The line represents the Standard Model calculation using e+e
Fig. 1. An historical look at the sequence of g-2 results in improving precision. The line represents the Standard Model calculation as of early 2004, using e+e data to deduce the hadronic contribution. The dotted lines indicate the uncertainty in the SM calculation.
P. Cushman / New Astronomy Reviews 49 (2005) 125–131
data (Davier et al., 2003). Assuming CPT holds, combining all our experiments, and properly accounting for correlated systematics, the final experimental value for the anomalous magnetic moment is now at al(exp) = 11659208(6) · 1010 or a precision of 0.5 ppm. The difference between experiment and theory now stands at Dal = (23.9 ± 9.9) · 1010, which is slightly smaller 2.4r discrepancy compared to 2.7r than that quoted in the E821 announcement paper (Bennett et al., 2004), due to further advances in theory. The number quoted above comes from comparing the world average g-2 result with a SM calculation compiled by Davier and Marciano (2004) which uses the hadronic correction from e+e data (Akhmetshin et al., 2003) and an enhanced light-by-light calculation (Melnikov and Vainshtein, 2003). The light-by-light calculation moves the SM closer to the g-2 result by 56 · 1011, but may be slightly over-estimated (Marciano, 2004). An improved value for the a4 QED term (Kinoshita and Nio, 2004) also raises the SM calculation, but only by 13.6 · 1011, which is within the uncertainties assigned by the hadronic correction. How Dal may change over the next several years is now in the hands of the theorists until such time as a new g-2 experiment can be mounted.
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bution that can most easily generate masses large enough to explain the discrepancy. A fairly generic result for tan b > 5 is an inverse quadratic dependence on the SUSY loop mass given by 2 SUSY 10 100 GeV tan b; al ffi 13 10 mSUSY where tan b is the ratio of vacuum expectation values of the Higgs doublet (Czarnecki and Marciano, 2001). This (mSUSY)2 dependence is responsible for shapes of the shaded regions in the plot of Fig. 3, provided by Goto et al. (1999) for three tan b regions, under the minimal supersymmetric extension of the Standard Model and the framework of SU(5) GUT models. The current Dal value (solid line) and its 1-r bounds (dotted lines) are plotted on top, with vertical arrows to show how mSUSY is limited by the g-2 experiment for a particular value of tan b (tan b = 10). When such constraints are translated into a 2-D plot of gaugino (m1/2) vs slepton (m0) mass in constrained minimal SUSY, they form the quarter circle shape of the g-2 preferred mass region.
2. Muon g-2 constraints on SUSY dark matter If supersymmetry is responsible for the nonstandard part of the g-2 anomaly, there exist new diagrams which can contribute to al, specifically two new one-loop diagrams: one with an internal loop of smuons and neutralinos and one with a loop of sneutrinos and charginos, see Fig. 2. For minimal supersymmetry (MSSM) parameter space in the limit of large tan b, it is the chargino contri-
Fig. 2. The supersymmetric diagrams that contribute to the muon anomalous magnetic moment at one-loop.
Fig. 3. The shaded regions are the allowed values for the supersymmetric contribution to al as a function of the lefthanded scalar muon mass for minimal supergravity (based on plots from Goto et al., 1999). Contraints from the Higgs boson search are already imposed. Three different tan b values (10, 20 and 40) are shown. The mean and 1-r bounds of Dal from the most recent experiment and SM calculation provide a straight line from which to determine mSUSY limits.
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Fig. 4. The (m1/2m0) planes for tan b = 10, l > 0. (Courtesy of Keith Olive). The region allowed by the older cosmological constraint (0.1 < XCDMh2 < 0.3) is the lightly shaded boomerang, while the new WMAP constraint (0.094 < XCDMh2 < 0.129) is the dark, thinner one. The region favored by g-2 at the 1 s level (dotted lines) and 2 s level (solid lines) define a shaded region that makes a quarter circle connecting m0to m1/2. This plot follows Ellis et al. (2003), but the g-2 allowed band is the most recent result: Dal = 239 (99) · 1011.
Fig. 4 shows such a plot for a particular choice of tan b and l (the Higgs mixing parameter). The dotted lines represent the 1-r contours and the solid lines bounding the shaded region correspond to the 2-r contours on a g-2 discrepancy presumed to be saturated by the SUSY contribution. For this plot, Olive has used the value quoted above of Dal = (23.9 ± 9.9) · 1010. As tan b is increased the quarter circle stretches and moves to higher mass. Both the positive nature of the g-2 discrepancy and the b ! sc branching ratio constraint prefer positive l. The power of the g-2 measurement to constrain SUSY dark matter lies in the contrasting way in which it cuts across m0 m1/2 parameter space compared to the cosmologically preferred region, which is a hyperbolically thin dark line with co-annihilation strips extending to high m1/2 and m0. Connecting these two high-mass extensions is the central ‘‘focus point’’, considerably shrunk from the fatter (light-shaded) region by the WMAP data
(0.094 < XCDMh2 < 0.129). LEP data excludes low m1/2 regions. The requirement that the dark matter particle be neutral eliminates the lower right triangle where m~s < m0~v . The next generation of collider searches will take place at CERN when the Large Hadron Collider comes on line in summer of 2007. The narrower the g-2 band, the more tan b itself will be constrained if supersymmetric particles are discovered and their mass measured at a collider. Improvement in the g-2 constraint will depend on future advances in theory and whether or not a new g-2 experiment can be mounted at Brookhaven in the near future. Only 20% of the CMD-2 e+e data (center of mass energies from 0.3– 1.4 GeV) has been analyzed. Over the next several years one should expect the precision in the dispersion integral to improve from 0.6% to 0.1%. An upgrade to the VEPP-2000 collider will provide increased luminosity and an improved detector (SND) to add statistics to the Novosibirsk data sample. An intensity upgrade at the BEPS machine will increase the sample of e+e data at the intermediate s = 2–5 GeV range. This energy range contributes less to the g-2 hadronic correction, since the kernel K(s) is decreasing with s, but it does provide an important handle on potential systematic bias in the region where is overlaps CMD2 and previous experiments. BaBar, KLOE and Belle will weigh in on differential cross sections using radiative return for multiple pion states. The BaBar data will be especially interesting, since the data can be directly normalized in the same apparatus measuring e+e ! l+l. Within the decade, the error on the 1st order hadronic correction should be reduced to dal 35 · 1011 which is comparable to the uncertainty on the hadronic light-by-light contribution. Since hadronic lightby-light scattering is model-dependent, it is hard to predict whether a breakthrough will occur there anytime soon. Lattice gauge calculations may have some successes in the next few years.
3. Future g-2 experiments To take advantage of this inevitable improvement in the theory, the precision of the experimen-
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tal value should be improved by a factor of 2. Originally, we requested an additional run designed to accumulate another 4 billion decay electrons in 6 months. However, in the RHIC era with a reduced duty cycle, such a run would take 8 months ($10M) just for a 0.35 ppm result. A better choice would be to make minor improvements in the experiment over the next two years, and accumulate 70 billion decay electrons (or a statistics limit of 0.2 ppm). To do this, we will need to increase the number of stored muons, improve the electronics and data acquisition to handle the increased throughput, reduce our systematic uncertainties to 0.2 despite higher rate-dependent effects. The g-2 collaboration is actively pursuing the following plan to accomplish this: 1. Increase muon flux a. Increase number of bunches per AGS cycle from 12 to 24. b. Increase intensity per bunch (requires new target design: $0.2 M). c. Double number of quads in beamline ($1 M). 2. Increase stored muons and reduce coherent betatron oscillations a. Open-ended inflector($1 M). b. A second muon kicker($0.2 M). 3. Improve data throughput and reduce rate dependence($0.5–1 M) a. Further segmentation of calorimeter and additional front scintillators. b. Continuous digitization of WFD and commercial MTDCÕs. c. Modernize DAQ. 4. Reduce error on magnetic field measurement. ($1 M) a. The magnetic field is uniform to 1 ppm across the storage region, but only 50 ppm around the ring azimuthally. This is less important, since the muons essentially average over azimuth in their cycles. However, by re-machining the pole tips and re-shimming, we can reduce this to 5 ppm. Since the error in the NMR measurement is due to inhomogeneities in the field and azimuthal position of the measuring trolley, this will improve the precision of the magnetic field measurement by a factor of 10.
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b. Monitor kicker eddy currents using the Faraday effect c. Improving the absolute NMR calibration with. d. More fixed probes and better temperature control.
4. Evolution of the experimental uncertainties The evolution of the g-2 experiment from engineering runs in 1997 to the improved BNL experiment described above is shown in Table 1. It can be seen that each run was effectively statistics limited. Another run at BNL in 2009 will represent the best one can hope to do with the current ring and detector geometry, as it is limited by systematics. Combined with the theory precision expected a few years from now, the error on Dal would then be at 4.7 · 1010. If the mean of Dal remains stable at its present value, this represents a 5r departure from the Standard Model. In order to explore how this translates into dark matter constraints, Olive Table 1 Causes and degree of systematic and statistical uncertainties in the g-2 experiment
The first three columns refer to completed experimental runs. Only the most recent g-2 runs have been included. Note that the experiments have all been statistically limited. The last two columns represent future experiments: a BNL experiment in the near future with only minor modifications to the existing beamline, storage ring and detectors, and a possible second generation experiment to be staged at JHF in Japan.
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Fig. 5. The (m1/2m0) planes for tan b = 10, l > 0 as in Fig. 4. However, here the g-2 allowed regions correspond to two possible future scenarios. (Courtesy of Keith Olive). Left plot: Mean discrepancy remains the same, but precision doubles. Dal = 239 (47) · 1011 Right plot: A standard model value with doubled precision. Dal = 0 (47) · 1011.
(following Ellis et al., 2003) produced Fig. 5 for the g-2 collaboration when we met with funding agencies in March 2004. Both plots include the factor of two reduction in error bars expected from a new run at BNL, combined with the improvement in the hadronic correction precision expected from analysis of e+e data already in hand. The plot on the left represents the case where the mean discrepancy remains stable at its present value. The plot on the right represents the case where the mean shifts down to the SM value. Due to the nature of the constraints, a reduction in the error bars which leaves the mean Dal intact will significantly narrow the band of allowed masses, while a shift in the mean to the Standard Model will widen the allowed region, but reject SUSY masses smaller than 500 GeV/c2. A SM result will also allow for negative l, while a non-SM result to 5r will definitively rule out the negative l option. The next generation g-2 experiment would need a factor of 100 more data to make it worthwhile. The timescale and improvement possible are included in the table above. Such concepts are being explored at the JPARC facility in Japan, (see, for example, Miller (2002)), where JHF provides a factor of 10 increase in intensity
(100 bunches/cycle every 0.7 ms) and the rest would have to come from an improved match between beam line and storage ring, etc. Another way to improve the experiment would be to increase the energy of the muons (and their dilated lifetime) so that more g-2 cycles can be measured for the same number of stored particles. This means abandoning electrostatic focusing, which can only be used at 3.1 GeV, that ‘‘magic’’ momentum where the radial electric field term cancels and the precession is unaffected. A new ring structure has been proposed by Farley (2003) which replaces electrostatic quads with edge focusing. Due to the large inhomogeneities in the field, the NMR probes must be replaced by proton calibration of the field.
5. Conclusion Although the BNL g-2 experiment has succeeded in its goal to improve the precision of a fundamental constant by a factor of 20 since the last CERN experiment 30 years ago, the theoretical landscape has shifted considerably. The utility of the experiment has always been evaluated by the
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model it seeks to confirm. Originally, the BNL experiment was designed to search for the Higgs and to confirm electroweak symmetry-breaking by measuring al (weak) to 20%. As the mass limits on the Higgs moved upward over the last decade of Tevatron and LEP runs, the contribution to al from diagrams containing the Higgs shrank below our sensitivity. The popularity of SUSY as an answer to the hierarchy problem and as a means to unify gauge couplings has renewed interest in al, especially since the enduring hint of a discrepancy points to such convenient SUSY masses. The experimental improvement has lead theorists to uncover a number of errors, improve calculations involving both hadronic vacuum polarizations, as well as higher order QED terms, spurred further experimental work on R(s), and lead to a re-examination of CVC and pion form factors. In the end, no matter what the fad of the moment, precision measurements of fundamental constants are an important contribution to physics, since they confront our preconceptions with reality and guide future discussions.
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References Akhmetshin, R., et al., 2003. hep-ex/0308008. Alemany, R., Davier, M., Hocker, A., 1998. Eur. Phys. J. C2, 123. Alioso, A., et al., 2003. hep-ex/0312056. Bennett, G.W., et al., 2004. Phys. Rev. Lett. 92, 161802. Bijnens, J., Pallante, E., Prades, J., 2002. Nucl. Phys. B 626, 410. Czarnecki, A., Marciano, W.J., 2001. Phys. Rev. D 64, 013014. Davier, M., 2003. hep-ex/0312064. Davier, M., Eidelman, S., Hocker, A., Zhang, Z., 2003. Eur. Phys. J. C31, 503. Davier, M., Marciano, W.J., 2004. Ann. Rev. Nucl. Part. Sci. 54, in press. DiFalco, S., et al., 2003. Acta Phys.. Polon. B 34, 5207. Ellis, J., Olive, K., Santoso, Y., Spanos, V., 2003. Phys. Lett. B 565, 176. Farley, F.J.M., 2003, hep-ex/0307024. Ghozzi, S., Jegerlehner, F., 2004. Phys. Lett. B583, 222. Goto, T., Okada, Y., Shimizu, Y., 1999. hep-ph/9908499. Hayakawa, M., Kinoshita, T., 2001. hep-ph/0112102. Kinoshita, T., Nio, M., 2004. hep-ph/0402206. Knecht, M., Nyffeler, A., 2002. Phys. Rev. D 65, 073034. Marciano,W.J., 2004. Private communication. Melnikov, K., Vainshtein, A., 2003. hep-ph/0312226. Miller, J., 2002. A Future Muon (g-2) Experiment, NP02, Sept. 2002, Kyoto.
Muon g-2 constraints on SUSY dark matter reviewed and predicted Priscilla Cushman Physics Department, University of Minnesota, 116 Church St. SE, Minneapolis, MN 55455, USA Available online 22 February 2005
Abstract A summary of experimental results and theoretical calculations related to the muon anomalous magnetic moment. Assuming that a discrepancy with the Standard Model is due to contributions from supersymmetric particles provides a framework which can be used to constrain the mass of the dark matter particles, assumed to be the lightest neutral supersymmetric particles. A new g-2 run in the near future, coupled with expected theoretical advances, would result in a 5r discrepancy if the mean Dal remains the same. In this case, we would expect to see SUSY particles at the LHC and use the g-2 results to measure tan b. If, instead, the Standard Model is confirmed to this precision, gauginos must have masses higher than 500 GeV/c2 and simple SUSY dark matter models will be strained. 2005 Elsevier B.V. All rights reserved.
1. Historical summary Precision measurements are a complementary approach to investigating the highest energy, smallest scale frontier of particles and interactions. Over the last decade, E821, the Brookhaven g-2 experiment, has successfully mounted a precision challenge to the standard model. The 2.6r discrepancy announced three years ago sparked debate on the theoretical calculation and encouraged further work on reducing the uncertainty in the 1st-order hadronic contribution. One of the more startling developments, approximately 6 months after the E-mail address: [email protected].
announcement of the first precision result (1.3 ppm) was the revelation by the Marseilles group (Knecht and Nyffeler, 2002) that one of the contributions to the Standard Model theory, namely the hadronic light-by-light term, had been independently assigned the wrong sign by at least 2 separate groups. Kinoshita (Hayakawa and Kinoshita, 2001) and Bijnens et al. (2002) studied their previous work and found that they both had used an incorrect sign convention in a matrix evaluation in a widely-used computer program. This moved the theoretical value by 17 · 1010 (by more than its stated uncertainty) in the direction of the E821 result, thus reducing the discrepancy to 1.6r. The next g-2 run, with an improved
1387-6473/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2005.01.027
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P. Cushman / New Astronomy Reviews 49 (2005) 125–131
precision of 0.7 ppm, left the mean unchanged, but reduced the error bars, again indicating a 2.6r discrepancy with the Standard Model. Meanwhile, in order to reduce the uncertainty on the hadronic correction, the use of vector spectral functions from the study of hadronic taudecays in ALEPH was introduced by Alemany et al. (1998). Previously, the only handle on the hadronic vacuum polarization contribution at the low center of mass energies which are relevant for g-2 came from the dispersion relation: Z 1 a2 KðsÞ ahad RðsÞ; ¼ ds l s 3p2 4m2p where RðsÞ ¼
rðeþ e ! hadronsÞ : rðeþ e ! lþ l Þ
R(s) is determined from a compilation of experimental results dominated by the CMD-2 experiment at Novosibirsk. When the s-decay data was first combined with the e+e data in 1998, it halved the error bars. However, over the past five years the continued operation and analysis of CMD-2 has improved the e+e data to such an extent that the two methods were found to be in disagreement with each other. Thus, in order to quote a discrepancy with theory, it became necessary to distinguish which hadronic correction you were referring to. In 2003, the Novosibirsk collaboration completely reanalyzed their pp channel (Akhmetshin et al., 2003) following the discovery of a mistake in their normalization (the t-channel leptonic vacuum polarization contribution was missing in the Bhabha scattering cross section). Their correction increased their published hadronic cross sections by 2.5%, thus reducing, but not erasing the discrepancy between the two theoretical approaches (especially for energies above 0.85 GeV). New results from KLOE, BaBar and perhaps Belle can provide an independent method to distinguish between the two by using radiative decay to scan the center of mass energies in the region relevant to g-2 (so-called ‘‘radiative-return’’ method). A precise measurement of the pion form factor has been reported by KLOE (Alioso et al., 2003 and DiFalco et al., 2003). It confirms the Novosibirsk e+e result. Preliminary results on 4prong final states by BaBar (Davier, 2003) also
bolsters confidence in the e+e data. On the other hand, branching ratios from CLEO and OPAL continue to confirm the ALEPH data, thus indicating that the s-decay construction may be affected by a fundamental misunderstanding in how we apply CVC, isospin corrections, or the electroweak symmetry breaking. Ghozzi and Jegerlehner (2004) argue that by simply allowing the mass of the charged-q to differ from q0 is sufficient to account for this. Davier (2003) shows that even assuming this modification, a detailed comparison of the shape of the pion form factor reveals some discrepancy. This in itself may be an indication of new physics. Most are agreed, however, that in any comparison of al (exp) to al (theory), the direct result using e+e data is more reliable at this time. The final g-2 experimental result was announced in December 2003 and published as Bennett et al. (2004). This was a 0.7 ppm result with opposite polarity muons. It was consistent with the previous data sets, despite reversing the magnetic field in the storage ring, demonstrating that systematic errors are being correctly estimated. However, the mean value was slightly higher than the earlier value, serving more to emphasize than detract from any Standard Model discrepancy. In Fig. 1, the BNL g-2 results are shown together with those from the old CERN experiment. The line represents the Standard Model calculation using e+e
Fig. 1. An historical look at the sequence of g-2 results in improving precision. The line represents the Standard Model calculation as of early 2004, using e+e data to deduce the hadronic contribution. The dotted lines indicate the uncertainty in the SM calculation.
P. Cushman / New Astronomy Reviews 49 (2005) 125–131
data (Davier et al., 2003). Assuming CPT holds, combining all our experiments, and properly accounting for correlated systematics, the final experimental value for the anomalous magnetic moment is now at al(exp) = 11659208(6) · 1010 or a precision of 0.5 ppm. The difference between experiment and theory now stands at Dal = (23.9 ± 9.9) · 1010, which is slightly smaller 2.4r discrepancy compared to 2.7r than that quoted in the E821 announcement paper (Bennett et al., 2004), due to further advances in theory. The number quoted above comes from comparing the world average g-2 result with a SM calculation compiled by Davier and Marciano (2004) which uses the hadronic correction from e+e data (Akhmetshin et al., 2003) and an enhanced light-by-light calculation (Melnikov and Vainshtein, 2003). The light-by-light calculation moves the SM closer to the g-2 result by 56 · 1011, but may be slightly over-estimated (Marciano, 2004). An improved value for the a4 QED term (Kinoshita and Nio, 2004) also raises the SM calculation, but only by 13.6 · 1011, which is within the uncertainties assigned by the hadronic correction. How Dal may change over the next several years is now in the hands of the theorists until such time as a new g-2 experiment can be mounted.
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bution that can most easily generate masses large enough to explain the discrepancy. A fairly generic result for tan b > 5 is an inverse quadratic dependence on the SUSY loop mass given by 2 SUSY 10 100 GeV tan b; al ffi 13 10 mSUSY where tan b is the ratio of vacuum expectation values of the Higgs doublet (Czarnecki and Marciano, 2001). This (mSUSY)2 dependence is responsible for shapes of the shaded regions in the plot of Fig. 3, provided by Goto et al. (1999) for three tan b regions, under the minimal supersymmetric extension of the Standard Model and the framework of SU(5) GUT models. The current Dal value (solid line) and its 1-r bounds (dotted lines) are plotted on top, with vertical arrows to show how mSUSY is limited by the g-2 experiment for a particular value of tan b (tan b = 10). When such constraints are translated into a 2-D plot of gaugino (m1/2) vs slepton (m0) mass in constrained minimal SUSY, they form the quarter circle shape of the g-2 preferred mass region.
2. Muon g-2 constraints on SUSY dark matter If supersymmetry is responsible for the nonstandard part of the g-2 anomaly, there exist new diagrams which can contribute to al, specifically two new one-loop diagrams: one with an internal loop of smuons and neutralinos and one with a loop of sneutrinos and charginos, see Fig. 2. For minimal supersymmetry (MSSM) parameter space in the limit of large tan b, it is the chargino contri-
Fig. 2. The supersymmetric diagrams that contribute to the muon anomalous magnetic moment at one-loop.
Fig. 3. The shaded regions are the allowed values for the supersymmetric contribution to al as a function of the lefthanded scalar muon mass for minimal supergravity (based on plots from Goto et al., 1999). Contraints from the Higgs boson search are already imposed. Three different tan b values (10, 20 and 40) are shown. The mean and 1-r bounds of Dal from the most recent experiment and SM calculation provide a straight line from which to determine mSUSY limits.
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Fig. 4. The (m1/2m0) planes for tan b = 10, l > 0. (Courtesy of Keith Olive). The region allowed by the older cosmological constraint (0.1 < XCDMh2 < 0.3) is the lightly shaded boomerang, while the new WMAP constraint (0.094 < XCDMh2 < 0.129) is the dark, thinner one. The region favored by g-2 at the 1 s level (dotted lines) and 2 s level (solid lines) define a shaded region that makes a quarter circle connecting m0to m1/2. This plot follows Ellis et al. (2003), but the g-2 allowed band is the most recent result: Dal = 239 (99) · 1011.
Fig. 4 shows such a plot for a particular choice of tan b and l (the Higgs mixing parameter). The dotted lines represent the 1-r contours and the solid lines bounding the shaded region correspond to the 2-r contours on a g-2 discrepancy presumed to be saturated by the SUSY contribution. For this plot, Olive has used the value quoted above of Dal = (23.9 ± 9.9) · 1010. As tan b is increased the quarter circle stretches and moves to higher mass. Both the positive nature of the g-2 discrepancy and the b ! sc branching ratio constraint prefer positive l. The power of the g-2 measurement to constrain SUSY dark matter lies in the contrasting way in which it cuts across m0 m1/2 parameter space compared to the cosmologically preferred region, which is a hyperbolically thin dark line with co-annihilation strips extending to high m1/2 and m0. Connecting these two high-mass extensions is the central ‘‘focus point’’, considerably shrunk from the fatter (light-shaded) region by the WMAP data
(0.094 < XCDMh2 < 0.129). LEP data excludes low m1/2 regions. The requirement that the dark matter particle be neutral eliminates the lower right triangle where m~s < m0~v . The next generation of collider searches will take place at CERN when the Large Hadron Collider comes on line in summer of 2007. The narrower the g-2 band, the more tan b itself will be constrained if supersymmetric particles are discovered and their mass measured at a collider. Improvement in the g-2 constraint will depend on future advances in theory and whether or not a new g-2 experiment can be mounted at Brookhaven in the near future. Only 20% of the CMD-2 e+e data (center of mass energies from 0.3– 1.4 GeV) has been analyzed. Over the next several years one should expect the precision in the dispersion integral to improve from 0.6% to 0.1%. An upgrade to the VEPP-2000 collider will provide increased luminosity and an improved detector (SND) to add statistics to the Novosibirsk data sample. An intensity upgrade at the BEPS machine will increase the sample of e+e data at the intermediate s = 2–5 GeV range. This energy range contributes less to the g-2 hadronic correction, since the kernel K(s) is decreasing with s, but it does provide an important handle on potential systematic bias in the region where is overlaps CMD2 and previous experiments. BaBar, KLOE and Belle will weigh in on differential cross sections using radiative return for multiple pion states. The BaBar data will be especially interesting, since the data can be directly normalized in the same apparatus measuring e+e ! l+l. Within the decade, the error on the 1st order hadronic correction should be reduced to dal 35 · 1011 which is comparable to the uncertainty on the hadronic light-by-light contribution. Since hadronic lightby-light scattering is model-dependent, it is hard to predict whether a breakthrough will occur there anytime soon. Lattice gauge calculations may have some successes in the next few years.
3. Future g-2 experiments To take advantage of this inevitable improvement in the theory, the precision of the experimen-
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tal value should be improved by a factor of 2. Originally, we requested an additional run designed to accumulate another 4 billion decay electrons in 6 months. However, in the RHIC era with a reduced duty cycle, such a run would take 8 months ($10M) just for a 0.35 ppm result. A better choice would be to make minor improvements in the experiment over the next two years, and accumulate 70 billion decay electrons (or a statistics limit of 0.2 ppm). To do this, we will need to increase the number of stored muons, improve the electronics and data acquisition to handle the increased throughput, reduce our systematic uncertainties to 0.2 despite higher rate-dependent effects. The g-2 collaboration is actively pursuing the following plan to accomplish this: 1. Increase muon flux a. Increase number of bunches per AGS cycle from 12 to 24. b. Increase intensity per bunch (requires new target design: $0.2 M). c. Double number of quads in beamline ($1 M). 2. Increase stored muons and reduce coherent betatron oscillations a. Open-ended inflector($1 M). b. A second muon kicker($0.2 M). 3. Improve data throughput and reduce rate dependence($0.5–1 M) a. Further segmentation of calorimeter and additional front scintillators. b. Continuous digitization of WFD and commercial MTDCÕs. c. Modernize DAQ. 4. Reduce error on magnetic field measurement. ($1 M) a. The magnetic field is uniform to 1 ppm across the storage region, but only 50 ppm around the ring azimuthally. This is less important, since the muons essentially average over azimuth in their cycles. However, by re-machining the pole tips and re-shimming, we can reduce this to 5 ppm. Since the error in the NMR measurement is due to inhomogeneities in the field and azimuthal position of the measuring trolley, this will improve the precision of the magnetic field measurement by a factor of 10.
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b. Monitor kicker eddy currents using the Faraday effect c. Improving the absolute NMR calibration with. d. More fixed probes and better temperature control.
4. Evolution of the experimental uncertainties The evolution of the g-2 experiment from engineering runs in 1997 to the improved BNL experiment described above is shown in Table 1. It can be seen that each run was effectively statistics limited. Another run at BNL in 2009 will represent the best one can hope to do with the current ring and detector geometry, as it is limited by systematics. Combined with the theory precision expected a few years from now, the error on Dal would then be at 4.7 · 1010. If the mean of Dal remains stable at its present value, this represents a 5r departure from the Standard Model. In order to explore how this translates into dark matter constraints, Olive Table 1 Causes and degree of systematic and statistical uncertainties in the g-2 experiment
The first three columns refer to completed experimental runs. Only the most recent g-2 runs have been included. Note that the experiments have all been statistically limited. The last two columns represent future experiments: a BNL experiment in the near future with only minor modifications to the existing beamline, storage ring and detectors, and a possible second generation experiment to be staged at JHF in Japan.
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Fig. 5. The (m1/2m0) planes for tan b = 10, l > 0 as in Fig. 4. However, here the g-2 allowed regions correspond to two possible future scenarios. (Courtesy of Keith Olive). Left plot: Mean discrepancy remains the same, but precision doubles. Dal = 239 (47) · 1011 Right plot: A standard model value with doubled precision. Dal = 0 (47) · 1011.
(following Ellis et al., 2003) produced Fig. 5 for the g-2 collaboration when we met with funding agencies in March 2004. Both plots include the factor of two reduction in error bars expected from a new run at BNL, combined with the improvement in the hadronic correction precision expected from analysis of e+e data already in hand. The plot on the left represents the case where the mean discrepancy remains stable at its present value. The plot on the right represents the case where the mean shifts down to the SM value. Due to the nature of the constraints, a reduction in the error bars which leaves the mean Dal intact will significantly narrow the band of allowed masses, while a shift in the mean to the Standard Model will widen the allowed region, but reject SUSY masses smaller than 500 GeV/c2. A SM result will also allow for negative l, while a non-SM result to 5r will definitively rule out the negative l option. The next generation g-2 experiment would need a factor of 100 more data to make it worthwhile. The timescale and improvement possible are included in the table above. Such concepts are being explored at the JPARC facility in Japan, (see, for example, Miller (2002)), where JHF provides a factor of 10 increase in intensity
(100 bunches/cycle every 0.7 ms) and the rest would have to come from an improved match between beam line and storage ring, etc. Another way to improve the experiment would be to increase the energy of the muons (and their dilated lifetime) so that more g-2 cycles can be measured for the same number of stored particles. This means abandoning electrostatic focusing, which can only be used at 3.1 GeV, that ‘‘magic’’ momentum where the radial electric field term cancels and the precession is unaffected. A new ring structure has been proposed by Farley (2003) which replaces electrostatic quads with edge focusing. Due to the large inhomogeneities in the field, the NMR probes must be replaced by proton calibration of the field.
5. Conclusion Although the BNL g-2 experiment has succeeded in its goal to improve the precision of a fundamental constant by a factor of 20 since the last CERN experiment 30 years ago, the theoretical landscape has shifted considerably. The utility of the experiment has always been evaluated by the
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model it seeks to confirm. Originally, the BNL experiment was designed to search for the Higgs and to confirm electroweak symmetry-breaking by measuring al (weak) to 20%. As the mass limits on the Higgs moved upward over the last decade of Tevatron and LEP runs, the contribution to al from diagrams containing the Higgs shrank below our sensitivity. The popularity of SUSY as an answer to the hierarchy problem and as a means to unify gauge couplings has renewed interest in al, especially since the enduring hint of a discrepancy points to such convenient SUSY masses. The experimental improvement has lead theorists to uncover a number of errors, improve calculations involving both hadronic vacuum polarizations, as well as higher order QED terms, spurred further experimental work on R(s), and lead to a re-examination of CVC and pion form factors. In the end, no matter what the fad of the moment, precision measurements of fundamental constants are an important contribution to physics, since they confront our preconceptions with reality and guide future discussions.
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