Search for dark photons from neutral meson decays in and Au collisions at =200 GeV
The standard model (SM) of particle physics is spectacularly successful, yet the measured value of the muon anomalous magnetic moment deviates from SM calculations by 3.6. Several theoretical models attribute this to the existence of a “dark photon,” an additional U(1) gauge boson, which is weakly coupled to ordinary photons. The PHENIX experiment at the Relativistic Heavy Ion Collider has searched for a dark photon, , in decays and obtained upper limits of on - mixing at 90% CL for the mass range MeV/. Combined with other experimental limits, the remaining region in the - mixing parameter space that can explain the deviation from its SM value is nearly completely excluded at the 90% confidence level, with only a small region of MeV/ remaining.
Introduction. The standard model (SM) of particle physics provides unprecedented numerical accuracy for quantities such as the anomalous magnetic moment of the electron , as well as predicting the existence of the vector bosons and and the recently discovered Higgs boson. Hence, measurements which lie outside SM predictions warrant special scrutiny. One such result is the measured value of for the muon Bennett et al. (2006), which deviates from SM calculations by 3.6 Olive et al. (2014). An intriguing explanation for this discrepancy has been proposed by adding a “dark” gauge boson Fayet (2007); Pospelov (2009); Endo et al. (2012); Davoudiasl et al. (2012). While the possibility of a hidden U(1) gauge sector had been considered shortly after the advent of the Standard Model Galison and Manohar (1984); Holdom (1986), it has recently gained more relevance, because it provides a simultaneous explanation of various beyond-the-standard-model phenomena in addition to . These include, for example, the discrepancy between the world’s data on proton charge radius Mohr et al. (2008) and that obtained by the Lamb shift in muonic hydrogen Pohl et al. (2010); Antognini et al. (2013), and the positron excess in cosmic rays observed by ATIC Chang et al. (2008), PAMELA Adriani et al. (2009) and AMS-II Aguilar et al. (2013) by providing a new mechanism for the decay of dark matter Arkani-Hamed et al. (2009); Tucker-Smith and Yavin (2011).
While a variety of mechanisms can be introduced to parameterize dark sector physics, a simple formulation postulates a “dark photon” of mass which mixes with QED photons via a “kinetic coupling” term in the Lagrangian Galison and Manohar (1984); Holdom (1986); Jaeckel (2012); Essig et al. ()
where parametrizes the mixing strength. Dark photons can then mix with QED photons through all processes that involve QED photons, with an effective strength . If the dark photon mass exceeds twice the electron mass, it can decay into an pair, and in the minimal version of the model, this is its dominant decay mode in the interval . To date, a wide range of searches Essig et al. () have excluded most of the parameter space that could explain the deviation of from its SM value. In this work, we report on new limits that exclude at the 90% confidence level essentially all of the remaining allowed parameter space, thereby rendering the dark photon an unlikely candidate to resolve the discrepancy of with the Standard Model.
Searching for . We search for possible decays of by examining the invariant mass of pairs in a large sample of Dalitz decays, for MeV/ in the dark photon parameter space, where the possibility of disentangling the anomaly by the dark photon survives at the 90% confidence level. The invariant yield of virtual photons from the Dalitz decays of is given by the Kroll-Wada equation Kroll and Wada (1955):
is the invariant yield of decays of , is the fine structure constant, and are masses for the electron, and , respectively. The deviation of the transition form factor from unity is 0.0157 even at MeV/ from the parameterization of with GeV Dzhelyadin et al. (1980). Therefore, the variation of is small enough in the mass range of interest to set in the calculation. The weak coupling of the dark photon to the QED photon implies that the natural width of the dark photon is very narrow, and as a result the expected line shape of the dark photon is set by the mass resolution, , of the detector
From the peak height ratio,
the dark photon mixing parameter can then be determined as:
Note that in this approach the efficiencies for detection of pairs from Dalitz decays and from dark photons cancel in the ratio .
The analysis presented here is based on a precise measurement of virtual photons from and Dalitz decays Adare et al. (2013a) across three PHENIX data sets at a collision energy of GeV with an integrated luminosity of 4.8 pb of collected in 2006, 82.3 nb of Au collected in 2008, and 6.0 pb of collected in 2009. Here, the Au statistics corresponds to nb = 32.4 pb of nucleon-nucleon collisions. All three data sets include an electron triggered sample, and the single electron trigger threshold for the Au run was higher than that for the runs. A hadron blind detector (HBD) Anderson et al. (2011), was installed in the experiment around the primary collision point prior to the 2009 data taking period. The additional material of the HBD resulted in a corresponding increase in the external photon conversion rate. The experiment was also operated with a reduced magnetic field integral during the period of HBD data taking. These effects substantially alter the shape of the 2009 mass spectrum below 35 MeV/ relative to the spectra from 2006 and 2008. Therefore, we restrict the 2009 analysis to the mass region above 40 MeV/ to avoid the edge effect at parameterization of the Dalitz contribution.
The PHENIX apparatus Adcox et al. (2003) was designed with only 0.39% of a radiation length () in front of the tracking detectors. It generates a small rate of conversions in the experimental aperture and provides excellent momentum resolution and electron identification. The HBD brought an additional material budget of for the 2009 run. The tracking system comprises drift wire and pad chambers with a momentum resolution of [GeV/]. Charged tracks with momenta above 0.2 GeV/ and pseudorapidity fall within the PHENIX acceptance. Electron identification requires hits in a Ring Imaging Čerenkov detector and energy-momentum matching in an electromagnetic calorimeter with an energy resolution of .
All combinations of electrons and positrons in an event are taken as pairs for the analysis. The contributions due to random combinations, correlated fake pairs from double Dalitz decays () and jet-induced correlations are evaluated using like-sign pairs. After scaling by the number of nucleon-nucleon collisions, the correlated backgrounds in and Au are very similar, indicating these background contributions are well understood. Pairs stemming from photon conversions in the material of the detector are removed by a cut on their characteristic angular orientation with respect to the magnetic field Adare et al. (2010). For the 2009 data, conversion pairs are rejected by a cut on the cluster size in the HBD, which depends on the pair opening angle Adare et al. (2013b), because the lower magnetic field of the 2009 run reduces the rejection power of the angular orientation cut. Conversions in the HBD readout plane were removed by an analysis technique of mass reconstruction assuming electrons come from the HBD readout plane Adare et al. (). In the 2009 dataset we consider pairs with an invariant mass above 40 MeV/, where the contribution of conversion pairs becomes negligible. Excluding these nonhadronic background pairs, we obtained 67k, 167k and 75k pairs for 2006 , 2008 Au, and 2009 , respectively in the mass range MeV/, where most pairs originate from Dalitz decays. Contributions to the electron pair spectrum are estimated by a GEANT3 based detector simulation using the measured invariant yields for hadrons as input. Effects such as the single electron trigger efficiency and inactive areas in the detector are taken into account. Figure 1 shows the raw spectra of pairs with the hadronic decay and background contributions for the 2006 , 2008 Au and 2009 data sets.
If the expected dark photon invariant mass distribution follows a normal distribution, then the standard deviation is equal to the detector mass resolution, as already described. This resolution is determined using a Monte Carlo procedure based on a GEANT3 description of the experimental apparatus. Spectra of dark photons with a flat distribution in transverse momentum for GeV/, covering the full azimuth, with rapidity , and with an initial vertex within 35 cm of the nominal vertex position are generated and forced to decay as . Dark photon masses from 20–90 MeV/ were investigated, with 20 million decays generated at each mass hypothesis. The reconstructed pairs were then weighted according to their pair to follow the experimental pair spectrum after background subtraction. The invariant mass resolution for the PHENIX detector in MeV/ is MeV/ with a 3% uncertainty. The calculated mass resolution is also confirmed with the data via a shape matching of the Dalitz peak around 5 MeV/.
To establish a limit on the dark photon yield, we first describe the shape of the background-subtracted spectrum with a physics motivated curve composed of the Kroll-Wada formula for virtual photon yield from both the and the multiplied by a 4-order Chebychev polynomial to allow for slight deviations due to various detector effects:
The ratio, , is fixed at 0.17, a value determined using a realistic “cocktail” of hadronic decays filtered through a model of the detector acceptance. The ratio is fixed at 0.03. The shapes of the mass spectra from and decays are indistinguishable for MeV/, and their combined yield relative to the , , is taken as the effective ratio for the analysis.
We divide the full mass ranges of MeV/ and MeV/ into lower and higher mass ranges after nonhadronic background subtraction, use Eq. 7 to describe each portion, and demand continuity of the model at the mass where the two ranges abut. A simultaneous fit to the three mass spectra, allowing each an independent normalization, results in a combined description of the Dalitz continuum. This procedure produces a lower reduced for the overall fit than using a single mass range for each dataset. The break point dividing the lower and upper mass ranges was allowed to vary, with 61 MeV/ giving the best reduced .
Figure 2 shows the best fit result to the Dalitz decay contribution in each dataset after subtraction of unphysical background pairs. The contribution of the fit procedure to the total uncertainty is explored by varying the break point above and below this preferred value until the reduced statistic rises by one and then taking the resulting 16% effect on the experimental sensitivity as the systematic uncertainty due to the procedure.
Results. The fitted background describes the yield of counts absent a dark photon signal. We employ the CL statistical approach Read (2002) to determine a limit on the number of dark photon candidates, which is in line with the current practice of setting limits for a hypothetical particle. This method has the effect of reducing the strength of the limit determination in the case of low (or no) signal strength, generally resulting in a conservative estimate of the CL. We step through the full mass range with a 1 MeV/ step repeatedly refitting the spectrum with the addition of a Gaussian of width equal to the mass resolution and centered at each mass hypothesis. This determines the observed yield as a function of , which may be greater or less than the experimental sensitivity at each mass, with a significance that is determined by the underlying probability distribution of the background, which is calculated by a likelihood ratio between the signal + background and background only hypotheses. The assumed background yield in any mass window will have uncertainties due to statistical fluctuations in the data used to determine the parameters describing the background by Eq. 7 and from systematic uncertainties in alternative background shapes. We evaluated the variation in the experimental sensitivity due to fluctuations in these uncertainties in addition to the uncertainty in the mass resolution. The observed value, the experimental sensitivity, and one- and two-standard deviation bands around the experimental sensitivity (shown as green and yellow bands) are all indicated on the plots for the different data sets as well as the combined result in Fig. 3.
The -value under the null hypothesis from the combined result is calculated considering only the statistical uncertainty and is always greater than 0.27 in the entire range MeV/. The minimum -value is consistent with the background only hypothesis if the look-elsewhere effect Gross and Vitells (2010) is taken into account. Therefore the limit on the number of dark photon candidate events can be translated directly into a limit on the dark photon coupling parameter using the peak-height ratio, Eq. 5. Figure 4 shows the limit determined by PHENIX along with the 90% confidence level (CL) limits from the WASA Adlarson et al. (2013), HADES Agakishiev et al. (2014), KLOE Babusci et al. (2013), A1(MAMI) Merkel et al. (2014) and BaBar Lees et al. (2014) experiments and the upper limit theoretically calculated from Davoudiasl et al. (2014). The bands indicate the range of parameters which would allow the dark photon to explain the anomalies with the 90% CL. The upward fluctuation apparent in the 2008 Au data compensates for a downward fluctuation of similar scale in the 2009 data, leading to the slightly modulated limit of the combined result. The PHENIX results cover the mass range MeV/, and over that range set a stricter limit than those of WASA, HADES or KLOE, and complement the A1(MAMI) results for their less sensitive region below 50 MeV/. The PHENIX limits exclude the values of the coupling favored by the anomaly above MeV/. Recently, BaBar reported stricter limits from a search of the reaction , excluding values of the preferred region for MeV/, and covering a mass range up to 10.2 GeV/. As a result, nearly all the available parameter space which would allow the dark photon to explain the results are ruled out at the 90% CL by independent experiments. Figure 5 shows the PHENIX limits in the dark photon parameter space with different confidence levels, focusing on the small remaining parameter space for MeV/. The entire parameter space to explain the anomaly by the dark photon can be excluded at the 85% CL by the PHENIX data alone. The level of the compatibility between our data and the coupling strength favored for the anomaly is 10% with a statistical test Maltoni and Schwetz (2003).
Conclusions. In summary, the PHENIX results set limits for the coupling of a dark photon to the QED photon over the mass range MeV/, improving upon the recent results of the KLOE, WASA, HADES, and A1 experiments. Combining with the BaBar results, the dark photon is ruled out at the 90% CL as an explanation for the anomaly for MeV/, leaving only a small remaining part of parameter space in the region MeV/. The probability that the theoretically predicted coupling strength required to explain the anomaly is compatible with the PHENIX results is only 10%. Future analyses by PHENIX would be able to provide even more stringent limits due to both increased data sets and improved detector technology that allow measurement of displaced vertices. As the coupling to the dark photon gets weaker, the distance traveled by the dark photon before decaying into grows longer Bjorken et al. (2009). The high statistics dataset taken after the recently commissioned PHENIX silicon vertex detector was installed in 2011 is being analyzed to look for such weakly coupled dark photons to provide limits even more restrictive than those reported here.
Acknowledgments. We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions. We also thank William Marciano and Hye-Sung Lee for useful discussions and theoretical calculations, and we thank the WASA, HADES and BaBar collaborations for useful interactions. We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, a sponsored research grant from Renaissance Technologies LLC, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Science, Education, and Sports (Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l’Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), OTKA NK 101 428 grant and the Ch. Simonyi Fund (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), Basic Science Research Program through NRF of the Ministry of Education (Korea), Physics Department, Lahore University of Management Sciences (Pakistan), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, and the US-Israel Binational Science Foundation.
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