A SEARCH FOR THE RARE DECAY B0 ➔ T+T- AT THE BABAR EXPERIMENT by CHRISTOPHER THOMAS POTTER A DISSERTATION Presented to the Department of Physics and the Graduate School of the University of Oregon in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 2005 ii ' 1A Search for the Ram Decay B0 -t ,+,- at the Babar Experiment," a dissertation prepared by Christopher Thomas Potter in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Department of Physics. This dissertation has been approved and accepted by: Dr. James rau , Chair of the Examining Committee Date Committee in charge: Accepted by: Dr. James Brau, Chair Dr. David Strom Dr. N.G. Deshpande Dr. Dietrich Belitz Dr. Brad Shelton Vice Provost and Dean of the Graduate School Ill © 2005 Christopher Thomas Potter An Abstract of the Dissertation of Christopher T homas Potter in the Department of P hysics for the degree of to be taken IV Doctor of Philosophy June 2005 Title: A SEARCH FOR THE RARE DECAY B0 ~ r+r- AT THE BABAR EXPERIMENT Approved: Dr. David Strom, Adviser The Standard Model of particle physics predicts that the branching ratio for the rare decay B0 ➔ ,,-+,,-- is 3.1 x 10- 8 , though untested models which could supersede it predict large enhancements. T his dissertation describes the search for this rare decay in 210.4 fb-L of B0B0 data collected at the Y(4S) resonance in the Babar detector at the Stanford Linear Accelerator Center. In the analysis, one neutral B meson is fu lly reconstructed in a hadronic mode and recoil events which are consistent with each tau decaying in a mode r ➔ rrv, pv, or lvv are selected. There is no evidence for signal. T he result is consistent with a downward fluctuation by 1.1 statistical standard deviations of the exp('rted Standard Model background. Taking the expected background, the number of observed events and the expected statistical and systematic errors into account yields 2.7 x 10- 3 as the upper limit for B0 ~ ,,..+,,.­ at the 90% confidence level. V ACKNOWLEDGEME TS I gratefully acknowledge the advice and instruction I received from my advisor, Professor David Strom, and from Professor Jim Brau, the principal investigator in the University of Oregon HEP research group. I further acknowledge all members of the Oregon HEP group for providing a very positive work environment. I also acknowledge the excellent training I received while working at the Bahar experiment with colleagues in the Leptonic B&c analysis working group. I gratefully acknowledge the Bahar Collaboration members and the Stanford Linear Accelerator Center staff who work on PEPII and the SLAC linac. Finally, I gratefully acknowledge the US Department of Energy Office of Science, without whose generous support this research would not have been possible. CURRICULUM VITA NAME OF AUTHOR: Christopher Thomas Potter PLACE OF BIRTH: Lima, Peru DATE OF BIRTH: January 23, 1969 GRADUATE A D UNDERGRADUATE SCHOOLS ATTENDED: University of Nebraska-Lincoln University of Oregon DEGREES AWARDED: Doctor of Philosophy in Physics, 2005, University of Oregon Master of Science in Mathematics, 1997, University of Oregon Bachelor of Arts in History, 1993, University of Nebraska-Lincoln Bachelor of Science in Physics and Mathematics, University of Nebraska-Lincoln, 1992 AREAS OF SPECIAL INTEREST: Experimental Particle Physics PROFESSIONAL EXPERIE CE: Graduate Teaching Fellow, University of Oregon Mathematics Depart­ ment, 1996 Graduate Teaching Fellow, University of Oregon Physics Department, 2000-2005 Vl TABLE OF CONTENTS Chapter 1. INTRODUCTION 1.1. New Physics in Dilepton Neutral Meson Decays 1.2. CP Violation and the Bahar Detector 1.3. B 0 ➔ 7+7- Background at Bahar 1.4. Outline of the Dissertation 2.1. Introduction .. 2.2. The Standard Model 2.3. The Two-Higgs-Doublet Model . 2.4. Supersymmetry 2.5. Leptoquarks . . . . . 3. THE BABAR DETECTOR 3.1. Introduction ..... 3.2. Charged Particle Tracking 3.3. Electromagnetic and Hadronic Calorimetry 3.4. Particle Ident ification . . . 3.5. Global Detector Operation 4. SIMULATION AT BABAR 4.1. Introduction . . . . . 4.2. Primary Event Simulation 4.3. Detector Physics Simulation 5.1. Data and Simulation Samples 5.2. Particle Candidate Selection . Vil Page 1 1 4 8 11 13 13 15 29 35 48 54 54 64 73 80 84 90 90 92 119 129 129 133 5.3. Tag B Selection . . . 5.4. Signal B Selection . . 5.5. The Neural Network 5.6. Data Control Sample 6. STATISTICAL ANALYSIS . 6.1. Analysis Chain Efficiencies 6.2. Limit Setting Procedure . 6.3. Systematic Errors 6.4. Upper Limit Optimization 7. RESULTS .... .. ...... . 7.1. Blinded Expectation and Unblinded Results 7.2. Implications for Theoretical Models 7.3. Conclusion . APPENDICES . . . . . . A. DISTRIBUTIO S BEFORE AND AFTER PRESELECTIO Vlll 139 145 157 164 168 169 174 176 182 186 186 189 192 195 195 B. mEs FITS FOR EXTRACTING EFFICIENCIES . . . . . . . . . . . . 210 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 lX LIST OF FIGURES Figure Page 2.1 Standard Model box (middle), penguin (top left and right) and self-energy (bottom left and right) processes responsible for s0 --+ 7+7-. . . . . . . 27 2.2 Dominant 2HDM processes responsible for s0 --+ r +7-. These are similar to the SM diagrams but with a qtW± (q = b, d) vertex replaced by a qtH± or qtG± vertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Branching ratio B2HDM(S0 --+ r+7-) plotted against mJ-1± (left) and against tan.B (right) in the large tan.B limit. The SM value is shown in bold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Dominant MSSM processes responsible for s 0 --+ r +r - . At left is the penguin with gluino exchange with flavor changing mass insertion. At right is the penguin with chargino exchange. . . . . . . . . . . . . . . . 46 2.5 Vector (left) and scalar (right) leptoquark processes which mediate S 0 --+ 7+7-. The quark and lepton chirality labels have been suppressed. . . . 51 2.6 Branching ratio BLQ(S0 --+ 7 +7-) plotted against mv0 (left) and against AtfR>.t~L (right). The SM value is shown in bold. . . . . 53 3.1 The linear accelerator and PEPII storage ring at SLAC. 55 3.2 Side view of the Bahar detector. 59 3.3 End view of the Babar detector. 60 3.4 Longitudinal section of the Silicon Vertex Tracker.. 65 3.5 Longitudinal section of the Drift Chamber. 69 3.6 Track parameter resolutions obtained from two halves of a cosmic ray event [l]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3. 7 Electromagnetic calorimeter in longitudinal section. 3.8 The flux return steel barrel (left) and endcaps (right). 3.9 The Detector for Internally Reflected Cherenkov radiation .. 3.10 Measurement performance for the photon energy in the EMC (a), Cherenkov X 73 78 80 angle and timing in the DIRC (b) and dE/dx in the DCH (c) [l] . 83 3.11 Schematic diagram of trigger, data acquisition, event reconstruction and event logging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1 The three stages of simulation at Babar. 91 4.2 Feynman diagrams for the two dominant classes of tau decay. Leptonic decays (-r ➔ e11e11-r and T ➔ µ11µ11-r) account for approximately 35.2% of all tau decays, while hadronic decays of the type depicted (T ➔ 1r11-r, T ➔ p11-r, -r ➔ a 111-r, T ➔ I<11-r and T ➔ 1(*11-r) account for approximately 57.2% of all tau decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3 Tau momentum in the B frame with EvtGen (solid line) and Tauola (error bars). Clockwise from top left are TT ➔ ee11eDe11-rD-r, TT ➔ µµ11µDµ11-rD-r, 'TT ➔ pp11-rD-r and 'TT ➔ 1r1r11-rD-r. For the p mode, the EvtGen results using TAUHADNU and TAUVECTOR U are superimposed. . . . . . . . . . 106 4.4 Cosine of the angle between tau daughters in the B frame with EvtGen (solid line) and Tauola (error bars). Clockwise from top left are T'T ➔ ee11eDe11-rD-r, TT ➔ µµ11µDµ11-rD-r, TT ➔ pp11-rD-r and TT ➔ 1f1fV-rD-r. For the p mode, the EvtGen results using TAUHADNU and TAUVECTOR U are superimposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5 Tau momentum in the B frame with EvtGen (solid line) and Tauola (error bars). From top to bottom are TT ➔ e1r11e11-rD-r, 'TT ➔ e p11e11-rD-r and T'T ➔ 1rp11-rD-r. Charge conjugate modes are horizontally opposed. Charge conjugate modes are horizontally opposed. For the p mode, the EvtGen results using TAUHAD U and TAUVECTORNU are superimposed. . . 107 4.6 Cosine of the angle between tau daughters in the B frame with Evt- Gen (solid line) and Tauola (error bars). From top to bottom are TT -t e1rvev.,ii,,., TT -t epvev.,ii., and TT -t 1rpv.,ii.,. Charge conjugate modes XI are horizontally opposed. For the p mode, the EvtGen results using TAUHADNU and TAUVECTORNU are superimposed. . . . . . . . . . . 107 4.7 The 7 + momentum plotted against the T - momentum in EvtGen with the TAUHADNU model for t he p mode. Clockwise from top left are TT -t eeveiiev.,D,,., TT -t µµvµiiµv.,D,,., TT -t ppv.,D., and TT -t 1r1rv.,D.,. . . 108 4.8 The 7 + momentum plotted against the T - momentum in Tauola. Clock­ wise from top left are TT -t eeveiiev.,D,,. , TT -t µ/Wµiiµv.,ii,,., TT -t ppv.,D., and ,T -t 1r1rv.,D.,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.9 Feynman diagrams for four classes of B decay. Leptonic decays are rare in the Standard Model and have not yet been observed. The internal and external hadronic diagrams add constructively for two-body decay modes with branching ratios of order 2%. The semileptonic decay branching ratio for the neutral B is approximately 24%. Most B decays are considerably more complex, with many gluons connecting internal and external quarks. 109 5.1 At left, the reconstructed mass of the seed Dor D* used in reconstructing the tag B . At right, the b..E of the tag B. Only the peaking components of data (dots) , generic neutral B (solid) and the signal cocktail (dashed) are plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 At left, Runs 1-4 data (dots) and from top to bottom generic neutral B, BpBm, ccbar, uds and e+e- -> tau+tau- Monte carlo simula­ tion samples (solids) . The e+e- -> tau+tau- component is not large enough to be visible. At right, a four Argus fit to the mes data distribu- tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.3 At left, the very loose track multiplicity on the signal side. At right, the loose photon muliplicity on the signal side. Only the peaking components of data (dots), generic neutral B (solid) and the signal cocktail (dashed) are plotted. The generic neutral B histograms are normalized to the data histograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.4 Tag yield in the semiexclusive sample after preselection. Clockwise from top left are the generic neutral B, data, signal generic, and signal cocktail samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.5 Clockwise from top left , the net loose track charge on the signal side, loose kaon, default Ks and tight KL multiplicity on the signal side after all preceding requirements in the analysis chain have been imposed. Only the peaking components of data (dots), generic neut ral B (solid) and the signal cocktail (dashed) are plotted. The dark shaded solid (here and in subsequent plots) is the component of the neutral B containing a truth Xll verified KL· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.6 At top left, the remaining neutral energy after neutral pion reconstruction (blind). At top right, the ditau reconstruction mode (blind) . At bottom left , the mass of the rho candidate. At bottom right, the mass of the K s candidate in the extra K s control sample. Only the peaking components of data (dots), B0B0bar (solid) are plotted. The dark shaded histogram (here and in subsequent control sample plots) indicates events in which t he K s reconstruction is verified with Monte Carlo truth. . . . . . . . . 151 5.7 Candidate tau daughter pair momenta by mode in the signal (top) and background (bottom) cocktail Monte Carlo sample. All tag B and back­ ground rejection requirements are applied. For signal, the signal side mode reconstruction is required to be correct using Monte Carlo truth. 155 5.8 The selection for B -t 7+7- . At left, the mEs distribution in Monte Carlo simulation. At right, the neural network output in Monte Carlo simulation. Only t he peaking components of generic neutral B (solid) and the signal cocktail (dashed) are plotted. For the unblinded version of these plots, see Figure 7.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.9 Inputs to the neural network for the B -t 7+7- selection. Clockwise from top left are the signal multiplicity mode, cosine subtended by tau daughter candidate momenta, remaining neutral energy and magnitude of the tau daughter candidate momentum. Only the peaking components of data (dots), generic neutral B (solid) and the signal cocktail (dashed) are plotted. or the unblinded version of these plots, see Figure 7.3. . . . . . 156 5.10 The neural network topology. From left to right are input, hidden and output units. The activation level ai for each node i is displayed. . 158 5.11 The minimum RMSE plotted against number of training cycles. The cumulative miminum is 0.474054, obtained at cycle 75416. . . . . . 162 Xlll 5.12 Performance of the neural network in the cocktail (top) and generic (bot- tom) samples. All preceding requirements in the analysis chain are im­ posed. o background subtraction is performed. . . . . . . . . . . . . . . 163 5.13 The selection for the extra Ks control sample. At left, the mEs distri­ butions in data. At right, the neural network output in data and Monte Carlo samples. Only the peaking components of data (dots), B0B0bar (solid) and the signal cocktail ( dashed) are plotted. . . . . . . . . . . . 167 5.14 Inputs to the neural network for the extra Ks control sample. Clockwise from top left are the signal multiplicity mode, cosine subtended by tau daughter candidate momenta, remaining neutral energy and magnitude of the tau daughter candidate momentum. Only the peaking components of data (dots), generic neutral B (solid) and the signal cocktail (dashed) are plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.1 The expected 90% upper limit plotted against the error on the background expectation (in units of the nominal error, 37.2 events). Both the tag B yield error and the signal efficiency errors are varied. . . . . . . . . . . . 181 6.2 Contour plots of the expected 90% confidence level upper limit R;S plotted versus mode subset number m. Mode subsets m = 9, 27, 43 and 59 gener­ ate R~? < 4 x 10- 3 . The first contours are rejected as marking statistical outliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.3 Contour plots of the expected 90% confidence level upper limit R~? plotted versus cut values for lower bound for the neural network output (left) and upper bound for unassigned photon energy (right). The first contours are rejected as marking statistical outliers. . . . . . . . . . . . . . . . . . . 184 7.1 The distribution of the 90% confidence level upper limit in 105 Poisson trials with mean 281 assuming zero signal events. . . . . . . . . . . . . 187 7.2 The selection for B ➔ 7+7- . At left , the mes distribution in Monte Carlo simulation. At right, the neural network output in Monte Carlo simulation. Only the peaking components of generic neutral B (solid) and the signal cocktail ( dashed) are plotted. The mode selection has not been applied at right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.3 Inputs to the neural network for the B ➔ r+r- selection. Clockwise from top left are the signal multiplicity mode, cosine subtended by tau daughter candidate momenta, remaining neutral energy and magnitude of the tau daughter candidate momentum. Only the peaking components of data (dots), generic neutral B (solid) and the signal cocktail (dashed) are XlV plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.4 The regions of the tan /3 - mH+ plane which are excluded at the 90% confidence level by the upper limit measurements on B(B 0 -+ µ+ µ- ) and B(B0 -+ r +r-). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.5 The regions of the mLQ - )..ff )..f3Q plane which are excluded at the 90% confidence level by the upper limit measurements on B(B 0 -+ r+r-). . . 193 xv LIST OF APPENDIX FIGURES Figure Page A.1 Tag B 6.E before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail ( dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . 197 A.2 Tag B 6.E after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . 197 A.3 The cosine of the angle between the tag B and the rest of the event before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 A.4 The cosine of the angle between the tag B and the rest of the event after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . 198 A.5 The mass of the D or D* used for reconstructing the tag B before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 A.6 The mass of the D or D* used for reconstructing the tag B after preselec­ tion for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 A.7 The signal side energy in all GoodPhotonsLoose candidates before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 A.8 The signal side energy in all GoodPhotonsLoose candidates after pre­ selection for Runs 1-3 (left) and Run 4 (right). Only the peaking compo- XVl nents of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . 200 A.9 The signal side GoodPhotonsLoose multiplicity before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. 201 A.IO The signal side GoodPhotonsLoose multiplicity after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A.11 T he signal side energy in the piODefaul tMass candidates before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 A.12 The signal side energy in the piODefaul tMass candidates after pres­ election for Runs 1-3 (left) and Run 4 (right). Only the peaking compo- nents of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . 202 A.13 The signal side piODefaul tMass multiplicity before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. 203 A.14 The signal side pi ODefaul tMass multiplicity after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 A.15 The signal side GoodTracksVeryLoose multiplicity before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.16 The signal side GoodTracksVeryLoose multiplicity after preselection for Runs 1-3 (left) and Run 4 (right) . Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) xvii have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.17 The signal side GoodTracksLoose multiplicity before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. 205 A.18 The signal side GoodTracksLoose multiplicity after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data ( error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A.19 The total signal side charge in GoodTracksLoose before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail ( dashed) have been plotted. 206 A.20 The total signal side charge in GoodTracksLoose after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 A.21 The signal side eMicroTight multiplicity before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. 207 A.22 The signal side eMicroTight multiplicity after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted.207 A.23 The signal side muMicroTight multiplicity before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. 208 A.24 The signal side muMicroTight multiplicity after preselection for Runs 1-3 (left) and Run 4 (right) . Only thr peaking components of data (error bars) and generic neutral B (solid) and signal cocktail (dashed) have been plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.25 The signal side KMicroLoose multiplicity before (left) and after (right) preselection. Only the peaking components of data (error bars) and generic XVlll neutral B (solid) and signal cocktail (dashed) have been plotted. 209 A.26 The signal side KMicroLoose multiplicity after preselection for Runs 1-3 (left) and Run 4 (right). Only the peaking components of data (error bars) and generic neutral B (solid) and signal cocktail {dashed) have been plotted.209 B.l Cumulative Qcri = 0 B.2 Exclusive QcrL = 0 . B.3 Cumulative NcrL = 2 B.4 Exclusive NcrL = 2 B.5 Cumulative NK ML = 0 B.6 Exclusive NKML = 0 B.7 Cumulative NKsD = 0 B.8 Exclusive N 1 0.52 B.16 Exclusive NN > 0.52 .. 210 211 211 212 212 213 213 214 214 215 215 216 216 217 217 .. ... 218 XIX LIST OF TABLES Table Page 1.1 Branching ratio measurements of FC C- and helicity-suppressed neutral pseudoscalar meson decays to dilepton pairs from [2] unless otherwise indicated. Limits are 90% confidence limits. An x indicates that the decay is kinematically forbidden while a - indicates that no measurement has been reported. All three B0 measurements were made at Babar. 2 1.2 The 90% confidence intervals for the proportions of background by type remaining in the B0B0bar generic simulation sample after all B0 ➔ r+r- requirements are imposed. . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Typical background modes to a B 0 ➔ r+r- analysis and their branching ratios. Kinematic background is rejected either by rejecting high momen- tum candidates or by using a neural network analysis. . . . . . . . . . . 11 2.1 The charge and spin of fermion (left) and boson (right) fields in the SM Lagrangian. The W, Z and A fields are generated by the electroweak group (U(l) x SU(2)) generators, and the eight fields Gk are generated by the strong interaction group SU(3) generators. . . . . . . . . . . . . 16 2.2 The interactions allowed in the Standard Model. 2.3 The minimal additional field content of the MSSM, excluding the Higgs sector. These are the superpartners of the leptons, quarks and gauge 25 vectors from Table 2.1 ( which are also included in the MSSM). 41 2.4 The MSSM Higgs sector. 42 2.5 The spin J, fermion number F = 3B + L and charge Q for leptoquarks allowed for renormalizable SM gauge group invariant interactions. . . . . 50 2.6 Leptoquark couplings which enter the Wilson coefficients for B 0 ➔ r+r-. 52 3.1 Cross sections, production rates (for £ = 3 x 1033 cm- 2 s- 1) and trigger efficiencies at the Y( 4S) resonance [l] [3] . . . . . . . . . . . . . . . . . . 58 3.2 Masses, lifetimes and detection method for particles typically detected in Babar. Longlived particles (above) are directly detected, while shortlived xx particles (below) are detected by their decay products. . . . . . . . . . . 64 3.3 Composition of the 13 Physics Accept sample at £ = 2.6 x 1033 cm-2 s- 1 [l ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.1 Particle properties used by EvtGen for a few select particles. Taken from the EvtGen evt.pdl file. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 The complete list of tau decay modes simulated by EvtGen together with their branching ratios and decay models. The branching ratios sum to unity. The parameters in the TAUHADNU decay model are /3, the p mass and width, the p' mass and width, and the a1 mass and width. The branching ratios in brackets () signify that the models for r -+ pv,,. and r-+ a1v,,. were changed from TAUVECTORNU to TAUHAD ru between simulation production cycles. Taken from the EvtGen DECAY.DEC file. 104 4.3 Above, a select list of semileptonic I3° decay modes simulated by EvtGen together with their branching ratios and decay models. The three param- eters in the HQET decay model are the form factor slope p~ and form factor ratios R1 and R2 , with values taken from CLEO measurements [4]. Below, a select list of hadronic f3° decay modes simulated by EvtGen to­ gether with their branching ratios and decay models. The parameters in the SVV _HELAMP decay models are the magnitude and phase of the am­ plitudes in the helicity state expansion of the D. Taken from the EvtGen DECAY.DEC file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4 The complete list of hadronic D* decay modes simulated by EvtGen to­ gether with their branching ratios and decay models. The branching ratios sum to unity. Taken from the EvtGen DECAY.DEC file. . . . . . . . . 117 4.5 Above, a select list of hadronic n + decay modes with one kaon in the final state simulated by EvtGen together with their branching ratios and decay models. Below, a select list of hadronic D0 decay modes ·with one kaon in the final state simulated by EvtGen together with their branching ratios and decay models. The branching ratios sum to < 77.525%. Taken from the EvtGen DECAY.DEC file. . . . . . . . . . . . . . . . . . . . . . . . 120 XXI 4.6 The Construct method for the BgsDchFullModel class in Bogus. This method defines the geometry for the DCH in GEA T4. The code has been simplified for ease of reading. . . . . . . . . . . . . . . . 127 5.1 Da ta samples used in this analysis. Runs 1-4 correspond to data taken from startup in 1999 to summer shutdown in 2004. The effective cross section is obtained from the B count to luminosity ratio. . . . . . . . . . 130 5.2 Signal and generic background Monte Carlo simulation samples from the BSemiExcl skim used in t his analysis. Cross sections are obtained from [3] . The Standard Model expectation B5 M (B0 -+ T + T - ) = 3.1 x 10-s is assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.3 Cuts used by the selectors employed in this analysis. For comparison, the cuts used by the selector with different criteria are also listed. . . . . . . 135 5.4 At top, the prescribed signal and background cocktail tag B decays B 0 -+ D(*)x. At bottom, the decay modes reconstructed. . . . . . . . . . . . . 140 5.5 Branching ratio, signal mode requirements, reconstruction efficiencies and sample composition with respect to all tau pair decay modes in the signal cocktail. The composition is the fraction of events reconstructed in a given mode which are truth verified to be that mode. All requirements are imposed except the neural network selection. . . . . . . . . . . . . . 152 5.6 The inputs for the neural network and their scale maps. The scale maps faciliatate training by mapping the inputs to the unit interval . . . . . . 157 5.7 Cumulative yields and efficiencies for the extra Ks control sample in data and Monte Carlo simulation. Here E res is the residual neutral energy EcPL - EpDM · The preselection requires four or fewer loose signal side photons and four very loose signal side tracks. . . . . . . . . . . . . . . 165 6.1 Cumulative yields and efficiencies (in % ) for the analysis chain in the Monte Carlo and data samples. The statistical error is added in quadra- ture with the fit uncertainty. For the unblinded version, see Table 7.3. 171 6.2 Exclusive yields and efficiencies (in %) for the analysis chain in the Monte Carlo and data samples. The statistical error is added in quadrature with the fit uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.3 Cumulative survival fraction Yi - Nd N0 and purities Pi (in %) in the data and background samples. The statistical error is added in quadrature with X,"'Cii the fit uncertainty. For the unblinded version, see Table 7.4 . . . . . . . 171 6.4 Cumulative yields and efficiencies for the analysis chain in the signal sam- ples. The statistical error is added in quadrature with the fit uncertainty. 172 6.5 Exclusive yields and efficiencies for the analysis chain in the signal samples. The statistical error is added in quadrature with the fit uncertainty. 172 6.6 Cumulative survival fraction Yi = Nd No and purities Pi in the signal sam- ples. The statistical error is added in quadrature with the fit uncertainty. 172 6.7 Relative systematic errors ol/f.. (in %) for the signal cocktail efficiency, signal generic efficiency and generic neutral B efficiency. Column three indicates the larger of the two signal systematic errors. . . . . . . . 177 7.1 The upper limit R~? for a 99.7% coverage range of outcomes nobs · The number of trial experiments for each result is 105 . The background ex­ pectation is obtained from the data preselection yield and the background efficiency from Table 6.1. The signal efficiency is obtained from Table 6.4. Systematic errors, which include simulation sample statistical error, are taken from Table 6.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.2 The signal efficiency, expected background and observed number of events by signal mode. The errors are statistical and fit error added in quadra- ture. The systematic errors are not included. . . . . . . . . . . . . . . . 190 7.3 Unblinded cumulative yields and efficiencies (in %) for the analysis chain in the Monte Carlo and data samples. Eres is the residual neutral energy EaPL - EpDM The statistical error is added in quadrature with the fit un­ certainty. The systematic errors are not included. Note in particular that the 3astat+ fit discrepancy after applying the residual energy requirement is well below lastat+sySl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.4 Unblinded survival fraction Yi = NdNo and purities Pi (in %) for the analysis chain in the Monte Carlo and data samples. E,-es is the residual neutral energy EaPL - EvD/11 The statistical error is added in quadrature with the fit uncertainty. The systemat ic errors are not included. 190 1 Chapter 1 INTRODUCTION 1.1. New Ph sics in Dile ton Neutral Meson Deca s The search for new physics in neutral pseudoscalar meson decays to a lepton pair has a hallowed history. The charm quark was discovered by direct production in 1974 [7, 8], but its possible existence had been inferred four years earlier through its effect on the neutral meson decay KL ➔ µ+ µ - [9]. Do undiscovered particles from new models of particle physics alter the expected rate for B0 ➔ 7+7-7 See Table 1.1 for the measured branching ratios for neutral meson decays to a lepton pair. These decays are examples of flavor changing neutral processes in which the final state flavor quantum number is different t han the initial state flavor, and they are mediated by flavor changing neutral currents (FC Cs) . These processes can proceed in the Standard Model (SM) only at higher order in perturbation theory and are therefore characterized by low branching ratios. Discovery has only been claimed for the decays I( L ➔ e+ e- and KL ➔ µ+ µ-. In all other kinematically allowed neutral pseudoscalar meson decays to a lepton pair, only upper limits on the branching ratios have been established. The Ks decays to a lepton pair violate GP conservation (see the following section) and are therefore even further suppressed. 2 I Meson I vv K s < 1.4 X 10- 7 < 3.2 X 10- 7 X - KL (9~:) X 10- 12 (7.27 ± 0.14) X 10- 9 X - vo < 6.2 X 10- 6 < 4.1 X 10- 6 X - Bo < 6.1 X 10- 8 [5] < 8.3 X 10- 8 [5] - < 2.2 X 10- 4 [6] Table 1.1: Branching ratio measurements of FCNC- and helicity-suppressed neu­ tral pseudoscalar meson decays to dilepton pairs from [2] unless otherwise indicated. Limits are 90% confidence limits. An x indicates that the decay is kinematically forbidden while a - indicates that no measurement has been reported. All three B 0 measurements were made at Babar. In addition to FC C suppression, neutral pseudoscalar decays to a lepton pair are also suppressed by kinematic constraints. A spin zero meson decay to spin 1/2 fermions must produce fermions in the same helicity eigenstate: either both are left­ handed or both are righthanded. Otherwise the decay violates angular momentum conservation. But all observed processes change flavor only through the electroweak process of W boson emission. In the relat ivistic limit of zero mass leptons, these processes create only lefthanded leptons l- and righthanded antileptons z+ so that the total angular momentum is one. This helicity suppression is total in the case of massless a lepton pair, but is not total in the case of massive leptons. The branching ratios for a given pseudoscalar are ordered by mass in ratios m;. : m; : m~ : m'f, . Despite FC C and helicity suppression, par ticle physics as it was known in 1970 predicted the occurrence of the decay KL ➔ µ+ µ - at a far higher rate than was observed experimentally. Undiscovered particles are generally discovered either by direct onshell production, typically at the high ~'nergy frontier of accelerator tech- 3 nology, or through their effects as offshell virtual particles in loop processes at low energies. The latter type of discovery requires precision measurements to detect the often subtle influence of virtual particles. Glashow, Iliopolous and Maiani posited [9] that the unexpectedly low rate for ](L ➔ µ+ µ - was due an undiscovered particle, the charm quark, which contributed to the ](L ➔ µ+ µ - amplitude and nearly exactly canceled the expected amplitude. This interference, they surmised, accounted for the low rate for KL ➔ µ+ µ - in what came to be known as the GIM mechanism. Four years later, in 1974, the charm quark was directly produced and observed in a bound state J / 'lj; (cc). For K 0 and D 0, the decay to a tau pair is prohibited by energy conservation. With the direct production of B mesons, a neutral meson decay to a tau pair first becomes kinematically possible. The helicity suppression in B 0 ➔ 7 + 7 - is milder than in any other lepton pair case, with the enhancement over B 0 ➔ e+c of order m;/m~ ~ 107 . But the greatest cause for interest in B 0 ➔ 7+7- is the potential for discovery of particles in physics beyond the SM. The Higgs mechanism in the SM employs a single SU(2) doublet which yields, after electroweak symmetry breaking, a single neutral scalar Higgs boson. In some models beyond the SM, at least two Higgs doublets are required. Their residue after symmetry breaking includes two charged scalars which could provide a new tree level 4 flavor changing interaction. Models beyond the SM can yield B 0 -+ r +r - branching ratio predictions at up to two orders of magnitude larger than in the SM. In the SM, the prediction is [3] BsM (Bo-+ r +r - ) = 3.1 x 10-s [ TB ] [ !B ] 2 [ l½dl ] 2 (1.1) l.55ps 180MeV 0.004 This is the same order of magnitude as the current upper bound on B(B0 -+ e+e-) and B(B0 -+ µ+ µ-). But electrons and muons are both longlived and directly observable. Unfortunately, the tau decay length (er) is only about 0.1 mm, making it difficult to detect directly, and a tau pair decays to final states involving at least two and up to four undetectable neutrinos. This makes tau pair reconstruction difficult as there are no kinematic constraints which force the observable parameters into isolated regions of phase space. The consequences for measuring B 0 -+ r +r - will be considered in the next two sections. 1.2. CP Violation and the Bahar Detector The Babar detector was not built to measure B 0 -+ r +r - . It was built to optimally measure the phenomenon of charge-parity conservation violation ( C ?­ violation) , which explains why matter rather than antimatter has come to dominate 5 the particle content of the universe. Had physical interactions respected GP sym­ metry, no mechanism could have produced the imbalance between the two forms of matter we observe today. The magnitude of the asymmetry provides clues about whether the SM alone can account for all of the asymmetry or if there exists undis­ covered physics which must account for part of it . In the SM, C ? -violation can occur in the electroweak interaction of quarks. The electroweak flavor eigenstates are actually mixed version of the physically realized mass eigenstates: q{'av = ¼iqfass where ¼j is the Cabibbo-Kobayashi-Maiani ( CKM) matrix. The CKM matrix is a complex unitary rotation matrix whose 18 free param­ eters are reduced to four by unitarity and arbitrary quark phases. T hese free param­ eters are not fixed by SM theory. In a standard parameterization [2], one parameter o is an imaginary phase which appears as ei6 in five of the elements ¼j - One can show that GP-conservation in the SM cannot occur unless all CKM elements are real and, in fact, the magnitude of the GP asymmetry depends on o. If o = 0, GP-violation cannot occur in the SM. Since vtv = 1, there are nine unitarity constraints relating the elements ¼j - One of them involves terms which are a product ¼d¼b: 0 (1.2) In the complex plane, this equation represents a so-called unitarity triangle. One of the interior angles of this triangle, /3 = arg( - V cd VcUVid ½b) is accessible experimen- 6 tally through observation of neutral B(bd) meson decays. /3 is a function of all four CKM parameters, but in particular if 8 vanishes so does {3 . Neutral B mesons oscillate between B and B states with a frequency 6.m8 - ms - miJ through an electroweak process. The oscillation occurs by double W boson emission from quark vertices, so the amplitude for the process depends on the CKM elements ¼d and ¼b- If two neutral Bs are produced simultaneously, as they are at Babar, and one is reconstructed in a C P eigenstate, then the other must also be a C P eigenstate. The decay rate f (6.t) depends on the proper time 6.t which has elapsed between the two decays. Due to neutral B oscillation, the decay rates are damped harmonic functions of tlt with amplitudes determined by the CKM elements. If one denotes the by f + the decay rate for B and by f _ the decay rate for B, then one can define the C P asymmetry (J + - f - ) / (J + + f - ) . This asymmetry can be shown to be [3] -rJ1 sin 2/3 sin(tlm8 tlt) (1.3) where 'T/J = +l for CP even modes and T/J = -1 for CP odd modes. The flavor of the B meson (B or B) can be determined experimentally by observation of its decay products. The amplitude of oscillation sin 2/3 can be measured if the proper time between B decays can be measured in many events. Since the required time resolution is 7 unobtainable through experiment in the center of mass Lorentz frame, 6.t must be measured indirectly by the displacement of the two vertices in a frame in which the entire event is boosted. The Babar detector and PEPII storage rings were built to produce a displacement of B decay vertices wit h a boost of /3, = 0.56. Then if the displacement 6.z can be sufficiently well resolved, 6.t = {316.z/c can be measured. The Babar Collaboration published its first result with 529 GP eigenstate events in 2001 [10] measuring sin 2(3 to be inconsistent with zero and thereby observing GP violation in B meson decays, but at a level consistent with the SM. In 2005, with 7730 GP eigenstate events, the preliminary unpublished result is sin 2/3 = 0.722 ± 0.040(stat) ± 0.023(syst) . The primary design goal of the Babar detector has been achieved in measuring sin 2(3, leaving many other phenomena of interest to measure. A good measurement for any signal phenomenon of interest requires that the signal be separated from the background by observable parameters. Since the decay products of B 0 ➔ 7+7- con­ tain between two and four neutrinos, which are undetectable in collider detectors, measuring it requires that all decay products from background B decays are carefully accounted for in each event. The design goal of the Babar detector had two nega­ tive consequences for measuring B 0 ➔ 7 + 7 - with respect to careful accounting of particles. First, the large boost required to separate B decay vertices has the effect of boost­ ing many B decay products out the uninstruinented forward end of the detector, where 8 the beam pipe and steering magnets must be placed. Second, while Babar was built to detect KL mesons from B ➔ J / 'I/; KL decays, these K is typically have high momen­ tum and are therefore more likely to interact at detectable levels with sensitive areas in the detector. Babar was not built to detect low momentum KLs- Therefore many B decays which lose particles through the forward end of the detector or produce low momentum KLs may mimic B 0 ➔ 7 + 7 - decays with multiple neutrinos. The consequences for measuring B0 ➔ 7 + 7 - will be explored more fully in the following section. 1.3. B 0 ➔ 7+7- B ackground at B ahar Before elaborating a full analysis of B 0 ➔ 7 + 7 - , we first investigated the nature of the background to this mode in the Babar detector. We studied the simulation truth versions of decays in the generic neutral B simulation sample which passed a simplified B0 ➔ 7+7- analysis after detector simulation. The rough analysis first reconstructed a neutral B from a hadronic decay and then searched in the recoil for the electron, muon and pion multiplicities required for a ditau decay. The background sources are categorized as events with KL which have not been detected but are within the detector acceptance, events with extra particles which have gone outside the detector acceptance, events which may be classed as kinematic 9 Category 90% Confidence Interval Unidentified K 1., 0.54 ± 0.14 Acceptance 0.30 ± 0.13 Kinematic 0.09 ± 0.08 Unidentified K ±, Ks, n° 0.06 ± 0.07 Table 1.2: The 90% confidence intervals for the proportions of background by type remaining in the B0B0bar generic simulation sample after all B 0 ➔ 7+7- re­ quirements are imposed. background and events with an undetected I.33) and the coupling of the d to the T (>.31)- 2.2. The Standard Model Outline of the Standard Model The Standard Model (SM) of the electroweak and strong interactions in particle physics is defined by the Lagrangian £ sM- The guiding principle in constructing £ sM is local gauge symmetry: it is invariant under an arbitrary local gauge transformation of the underlying symmetry group U(l ) x SU(2) x SU(3) of the SM [13] [14]. This principle ensures that the interactions contained in the SM are renormalizable, an indispensable property of any viable theory because it allows the perturbative calcu­ lation of observable quantities. Most SM predictions for observables have been verified by experimental observation, and no experimental result has decisively contradicted the SM [15]. 16 I Lepton I Q I J II Quark I Q J II Boson I Q I J I e -1 1/2 u 2/3 1/2 -.k (k = 1, ... , 8) are the generators for SU(2) and SU(3) respectively, then [Ti , Ti] = il:,ijkTk and [>-.i, >-.i] = ifijk>-.k define the Lie Algebra for SU(2) and SU(3). The f and f are the structure constants for SU(2) and SU(3) respectively. Then any element of the SM and u3 = exp(-if3k(xµ)>-.k) are the elements of U(l), SU(2) and SU(3) respectively. The functions 0(xµ), aj(xµ) and f3k(xµ) are explicitly space and time dependent, as demanded by local gauge symmetry. The field content of the SM is summarized in Table 2 .1 1. The only scalar field in the SM, the Higgs field , is vital to the theory since, by virtue of its coupling to other fields it generates the mass of the particles associated with the fields. The fermionic fields are the leptons and quarks. The vector fields are generated by requiring gauge 1The notation in this section follows closely that of [13]. 17 symmetry: one distinct vector field is required for each generator of the underlying symmetry group. For the U(l) x SU(2) electroweak transformations, t he Higgs field and the fermionic fields which undergo electroweak interactions are arranged into SU(2) doublets or singlets. The Higgs field is a complex doublet <.T? (2.5) The lefthanded lepton fields are arranged in doublets, while the righthanded fields are arranged in singlets. Similarly, the quarks are arranged into lefthanded doublets and righthanded singlets. The SU(3) strong transformations act on quark SU(3) color triplets. Explicitly, (2.6) (2.7) ( q, q, q, r (q = u, d, c, s, t, b) (2.8) where r, g, b denote red, green and blue color charge. Since leptons do not carry color charge, they do not participate in the strong interaction and are therefore not subject to SU(3) transformations. 18 Local gauge invariance means that when the fields are transformed by a local gauge transformation, the Lagrangian remains invariant . This requires the introduction of gauge derivatives which incorporate the vector boson fields in such a way as to counteract the effect of gauge transforming the fields. Also introduced in the gauge derivatives are the gauge group U(l) x SU(2) x SU(3) coupling constants 91 , 92,93, which are undetermined in the SM and must be measured by experiment. The SM Lagrangian contains terms for each field: r r r r electroweak r strong .t..,4> + .t..,vectors + .t..,leptons + .t..,quarks + .1..,quark s and the term due to the scalar field is (2.9) (2.10) where m is the Higgs boson mass and ¢0 is t he vacuum expectation value of the Higgs field. Dµ, is the U(l) x SU(2) gauge derivative described below. The remaining terms in the SM Lagrangian will be developed below. After transforming the Higgs field in U(l ) space, ➔ ' = u 1 , the U(l ) sym- metry is obtained by requiring the existence of a vector gauge field Bµ, and the use 19 of the U(l ) gauge derivative. The transformations which preserve the Lagrangian are ➔ ' - U1

➔ ' = u2 requires the existence of three further vector gauge fields W,! , vV; and W; , combined together as the SU(2) operator Wµ = W;-ri (i = 1, 2, 3) where the -ri are the SU(2) generators. Then ➔ ' u2 (2.14) vvµ ➔ w; t 2 t (2.15) - u2Wµu2 + -Ti(8,,.u2)u2 92 8 ➔ n su(2) 1 . (2.16) µ µ - 8µ + 2ig2TWµ where 92 is the SU(2) coupling and T is the SU(2) charge (also called weak isospin charge). Weak isospin T is 1/2 for weak isospin doublets and O for weak isospin singlets. The third component of T, denoted T3, is + 1/2 for neutrino and up type quark fields and - 1/2 for lepton and down type quark fields. \t\Teak hypercharge is related to weak isospin charge T3 and electric charge Q by Y = 2(Q - T3 ) . Finally, rotating the quark triplets in SU(3) space requires the existence of eight vector gauge fields G~, ... , G!, one for each generator of SU(3) . The SU(3) operator 20 Gµ = G~>.i, where the >.i (i = 1, ... , 8) are the SU(3) generators. The simultaneous gauge transformation required to preserve the Lagrangian is ➔ ' - u3 (2.17) Gµ ➔ G~ - t 2 • ) t U3GµU3 + -i(8µu3 U3 93 (2.18) 8 ➔ Dsu(3) µ µ - 8µ + ~i93Gµ (2.19) where 93 is the SU(3) charge. The introduction of the vector bosons Bµ,, Wµ and Gµ requires that there be a dynamical term in the Lagrangian £dynamical (2.20) since these are the dynamical terms in the Lagrangian for vector bosons. The field strength tensors are defined by Bµ,v OµBv - OvBµ, (2.21) WJLV Oµ,VVv - OvWµ + ~i92[Wµ , Wv] (2.22) Gµv 1. OµGv - BvGµ, + 21,g3[Gµ, Gv]- (2.23) The commutator term in Bµv is zero since U(l) is Abelian. Since SU(2) and SU(3) are non-Abelian the commutators [Wµ , ltl 'v] and [Gµ, Gv] are nonzero. As a consequence, 21 there are no self-couplings of the B µ field but there are cubic and quartic self-couplings of the Wµ and Gµ fields. The vector mass terms are generated by SU(2) symmetry breaking, discussed below. Together the dynamical term and the mass terms for the vector fields comprise the vector boson Lagrangian £ vectors . According to the SM, nature chose to physically realize the fields mixed versions of Bµ and Wµ: the fields Aµ , Zµ, w: and w;. These fields are related by ( Aµ ) ( sin 0w co~ 0w ) ( vV; ) Zµ cos 0w - sin 0w Bµ (2.24) ( w + ) ( 1 i ) ( w 1 ) ~ i - 1 ~ (2.25) The mixing angle 0w is one of the undetermined parameters of the SM which has been measured experimentally. ·when these definitions are used to solve for the fields B µ and Wµ in terms of the physical fields, the terms B µvBµv and Wµv W µv in £ dynamical yield the interactions of the physical fields: the cubic interactions Aw+w- and zw+vv- and t he quartic interactions WWZZ, WW AA, and WvVZA. The term GµvGµv yields the cubic GGG and quartic GGGG gluonic self-interactions. 22 The Bµ , Wµ and Gµ vector boson mass terms are explicitly excluded from the Lagrangian term [,dynamical · In the SM they are generated when the SU(2) symmetry is broken when the Higgs field chooses a particular orientation in SU(2) space. This electroweak space symmet ry breaking takes the form (2.26) where ¢0 is t he vacuum expectation for contains W and Z mass terms as well as hvv+w - and hZZ direct couplings which are proportional to the masses. The A mass term is zero (as required for photons), and there is no tree level hAA coupling. Likewise, the G mass term is zero (rendering gluons massless) and there is no tree level hGG coupling. These properties of [, is the term oµhol-'h - m 2 h2, the Klein-Gordon Lagrangian for a scalar field h with mass m, and the cubic and quartic self-interactions hhh and hhhh of the Higgs scalar. 23 The fermions obtain mass by coupling to the Higgs field 4>. The electroweak Lagrangian for leptons Lteptons gives mass to e, µ , T {but not their neutrinos) with the terms m1(llln + tktL) for l = e, µ , r. Together with the dynamical term, this yields {2.27) where the symbols CJµ = (1, a') and aµ = (1, -o) are defined by the Pauli matrices CJ1 , CJ2, CJ3 • ote that the SU(2) gauge derivative does not act on righthandecl leptons, which are not arranged in SU(2) doublets as the lefthanded leptons are. This corre­ sponds to the experimental observation that righthancled lepton fields do not couple to the W field. After SU(2) symmetry breaking, and using the U(l) x SU(2) gauge derivative, £leptons contains Dirac Lagrangians for all leptons with mass terms deter­ mined by the constants Ct as well as the direct couplings httLn + CC, WllvlL + CC, ZltfL, Zlkln, AllzL and Alkln. The Higgs couplings are proportional to the mass terms Ct, The lepton masses (or, equivalently, their couplings to the Higgs scalar) are undetermined in the SM. The electroweak Lagrangian for quarks is similar to that for the leptons, but two factors somewhat complicate it. Whereas neutrinos are massless in Lteptons, up type quarks ui (u, c, t) must be given mass by coupling to . Furthermore, while 24 generational mixing of leptons is disallowed in .Cteptons, it is required in .c:tit,c'':'eak in order to conform to experimental observations. The electroweak quark Lagrangian is [, electroweak quarks (2.29) T he introduction of the matrices au and Gd (which operate in flavor space) account for the mixing of flavor eigenstates in mass eigenstates, and the use of the antisym­ metric tensor€ (Eu = E22 = 0 and E12 = -€21 = 1) allows up type quarks to couple to and thereby obtain mass. After symmetry breaking, .c:ti:r,_~weak contains the Dirac Lagrangian for quarks (with mass terms) as well as the interactions hqlqR + CC, be diagonalized with unitary matrices au= U1NruuR and Gd= DlMdDR where the mass matrices M are diagonal. Then the quark interactions with the W gauge boson can be written Wu!L Vii dj L where V = UiDl is the Cabibbo-Kobayashi-Maskawa ( CKM) matrix. The strength of each interaction is determined by the weak hyper­ charge Y, weak isospin T and the CKM elements Vii . 25 Leptons Quarks I Bosons (Cubic) I Bosons (Quartic) I hll l R + cc hql qn + CC hWW wwzz zztzL + z tktn Zql qL + Zqkqn hZZ WWAA AtlzL + Atktn Aql qL + Aqk qn AWW WWZA Wtlv1L + CC Wu!LdiL + CC zww wwww GijQ!iQj L GGG GGGG hhh hhhh Table 2.2: The interactions allowed in the Standard Model. Finally, the strong interaction Lagrangian involves only t he SU(3) vector bosons G~. The Lagrangian is t t qr qr qr qr .c,strong - E Qg (o-µi Dµ) Qg -mq Qg Qg (2.30) quarks q=v.,d,s ,c,b,t Qb Qb Qb Qb where t he quark masses mq are generated by coupling to the Higgs field above . .c,s;:;:rnfs gives the qqG couplings and the Dirac equation for quarks. Properties of this and the dynamical Lagrangian term G µvGµv have had success in explaining the strong interaction properties of confinement and asymptotic freedom. See Table 2.2 for a complete list of interactions allowed by the SM Lagrangian. B 0 ➔ 7+7- in the Standard Model Since the SM is a renormalizable field theory, the amplitude for B 0 -+ 7 + 7 - may be calculated perturbatively. If an interaction is not explicitly prohibi ted by kinematic or conservation considerations, it is expected to occur in t he SM. V-/e can 26 therefore proceed from first order in perturbation theory to higher orders to discover the dominant process which mediates B0 -+ 7 + 7 - . Three observations from Table 2.2 are most helpful in determining the process by which B0 -+ 7+7- proceeds in the SM. First, the only interactions involving quarks or leptons are cubic interactions. Second, there is no cubic interaction qq'X where X is neutral. An interaction qq' X would give rise to a tree level flavor-changing neutral current (FC C), which the SM does not admit. This forces the initial state quarks to interact at distinct interaction vertices. Third, there is no direct interaction involving both a quark and a lepton. The following argument applies to the general case of a neutral meson decaying to a lepton pair. The process cannot occur at first order in perturbation theory because there is no quartic interaction in the SM involving either quark or lepton fields. Nor can it occur at second order: there is no cubic interaction qq' X where X is neutral ( for an s-channel process), and there is no cubic interaction involving a quark and lepton field (for a t - or u - channel process). At third order, the final state leptons must meet at the same interaction vertex. But since the only allowed quark interactions are cubic, this forces the lepton vertex to be a quartic interaction. But this cannot be in the SM since there are no quartic lepton interactions. The process can proceed with four interaction vertices. Since the only tree level flavor changing interaction in the SM (Wu!LdiL) is charged and the meson is neutral, there must be at least two such interactions ( one mediated by w+ and the other 27 d' b e e e e d' b w w d' ________ b w Vt w z e z e e Figure 2.1: Standard Model box (middle), penguin (top left and right) and self-energy (bottom left and right) processes responsible for B0 ➔ T+T- . by w -), one each at the initial state quark vertices. Then either the Ws or the undetermined quarks meet at the two remaining free vertices. If the final state leptons share a common interaction vertex, then either the undetermined quarks may proceed to the last free vertex (in which case the W s connect the init ial state quark vertices) or the W s may proceed to t he last free vertex (in which case the undetermined quarks connect the initial state quark vertices). In either case, the qij or w+w- vertex must connect to the final state lepton pair vertex with a h, A or Z line. The resulting diagrams are called penguin diagrams. See F igure 2.1 (top left and right) for t he penguin diagrams which mediate B0 ➔ T+T- . Note also that if a penguin diagram with l,1/ emission from the d quark line reabsorbs the W , such a diagram is called a self-energy diagram. See Figure 2.1 (bottom left and right) for the diagrams. Self-energy diagrams contributing to B 0 ➔ 28 7+7- are also allowed at fourth order in the SM. If the final state leptons do not share the same vertex, t hen the undetermined quarks must connect the initial state quark vertices and the W s proceed to the lepton vertices (since there is no SM interaction involving both a quark and a lepton). The only interaction available for the lepton vertices is W tlv,L, in which case the resultant neutrino connects the final state lepton vertices. Such a diagram is called a box diagram. See Figure 2.1 (middle) for t he box diagram which mediates B 0 ➔ 7+7-. The SM amplitude for a neutral meson decay to a lepton pair was calculated in [16] for KL ➔ µ+ µ - . In [17] the amplit ude was calculated for Bq ➔ 7+7- ( q = d, s), and in [18] QCD corrections at first order in a 5 were calculated. The QCD corrections account for single gluon emission and reabsorption by quark lines in both box and penguin diagrams, and for B 0 ➔ t+ L- were found at most to be 13% effects by the authors of [18]. A more recent estimate for B 0 ➔ 7 + 7 - in [19] found the correction to be a 3% effect. In the SM, the penguin diagrams contribute to Gp and Cs, but these are sup­ pressed relat ive to CA by a factor of (mb/mw)2 ~ 2.6 x 10- 3 [12]. The box diagram dominates through its contribution to CA. Box top exchange diagrams dominate the contribut ion from box diagrams wit h up and charm exchange. From [12], CSM _ A - I * I aem(fiw) (( / )2) ½ d Ytb v'8 . 2 Y fit fiw 87f Slll 0w 29 (2.31) where Y (x) = Y0(x) + t Y1(x) includes the order as correction term. The terms CjM :::::: CjM :::::: 0 relative to C{M. From [19], the numerical expression for Y is 2 fit fit [ ( ) ] 1.55 Y((fit/fiw) ) = 0.997 166GeV (2.32) Collecting together these elements and inserting them into the expression for the branching ratio in Equat ion 2.4, from [3] 3l x io- s [ 78 ] [ f s ] 2 [~ ] 2 • l.55ps 180M eV 0.004 (2.33) 2.3. T he Two-Higgs-Doublet Model Outline of the Two-Higgs-Doublet Model (2HDM ) \ i\Thile no experimental observation has successfully challenged the SM as a model of fundamental particle interactions, there is still a key component which has not yet been observed: the Higgs scalar h . And while the simplest assumption of a single SU(2) doublet Higgs field is sufficient to generate mass and thereby explain SU(2) 30 symmetry breaking, it is not necessary. Nature may have chosen a more elaborate scheme. In the Two-Higgs-Doublet Model (2HDM), the non-Higgs sector field content is identical to the field content of the SM. But in the Higgs sector, two SU(2) doublets are employed for SU(2) symmetry breaking and mass generation rather than one as in the SM [20]. If only one of the doublets couples to fermions and the other is decoupled, the 2HDM is a 2HDM type 1. If one doublet couples to up type fermions and the other couples to down type fermions , the 2HDM is a 2HDM type 2. The Higgs sector of the 2HDM type 2 employs the doublets (2.34) (2.35) to generate mass. Hu generates mass for the up type quarks and up type leptons. Hd generates mass for the down type quarks and down type leptons. Hereafter 2HDM will refer to 2HDM type 2. 31 In the 2HDM, as in the SM, particles obtain mass when SU(2) symmetry is broken. In the 2HDM, the Higgs fields Hu and Hd assume particular orientations in SU(2) space: H. ➔ ( :. ) + ~ ( :n Hd ➔ ( : ) + ~ ( :; ) (2.36) (2.37) where hu and hd are excitations above the vacuum expectations Vu. and vd- Since there is no a priori connection between the vacuum expectation values Vu and vd, their relative magnitudes are encoded in a free parameter of the 2HDM, tan /3 = vu/vd. There is no experimentally preferred value for tan ,B, but in grand unified theories wit h SO(10) as the unification gauge group it is found that the third generation Yukawa couplings unify at the unification scale when tan,B is large (of order mtfmb) [?]. T he fields Hu and Hd couple to the SM fermions in the natural way (see Equa­ t ion 2.55). But in the 2HDM, the quark Yukawa coupling matrix is not necessarily diagonalizable in the same basis as the quark mass matrix. This is the case for large tan ,B [21 ]. This generates off-diagonal entries in the Yukawa matrix, which introduces flavor changing interaction vertices qq' H±. The physically realized particles of the 2HDM Higgs sector are mixed versions of the scalar excitations h;;, h~, h~ and h"i. Goldstone bosons c ± and G0 are absorbed 32 by gauge vectors. The physical fields are ( :: ) ( ::: : ~;:: )( :; ) {2.38) ( G 0 ) _ J2 ( sin .B - cos .B ) Im ( h~ ) A0 cos .B sin .B h~ {2.39) ( h 0 ) = J2 ( sin a - ~os a ) R e ( h~ ) H 0 cosa sma h~ {2.40) where the mixing angle a is another free parameter in the 2HDM. Since the Goldstone bosons are unphysical and are only required in Feynman- t'Hooft gauge, this leaves the charged Higgs H + and H -, the scalar Higgs h0 and H 0 and the pseudoscalar A0 . The 2HDM provides a mechanism for the enhancement of B0 ➔ 7+7- over the SM rate, but not at less than fourth order in perturbation t heory. The introduction of new 2HDM flavor changing couplings qq' H ± and qq'G± allow for additional processes similar to those in t he SM. Thus any w ± boson in the SM box and penguin diagrams may be replaced by a charged Higgs boson H ± or a charged Goldstone boson Q±. In addition, t here are three neutral Higgs scalars A0 , H 0 and h0 in t he 2HDM which may replace a zo or photon line. The amplit ude M '1ft~~~r- was first calculated in (17] and later in [19]. The authors of [19] found t he results in [17] to be gauge dependent 33 d' b d' t b ► ~ .. I I \IV ,✓'H+ w I H + I I : ho, H o,Ao I I I Vt I e e e e d' t b d' t b • ~ ► • ► ► ~ • • ► ' • ' , / ' , ' -,, I c+ ' , H + H +, c+ I f ', ho Ho Ao I : ho,Ho,Ao t t I I I I I e .. • .. e e .. • .. e Figure 2.2: Dominant 2HDM processes responsible for B0 ➔ T+T - . These are similar to the SM diagrams but with a qtW± (q = b, d) vertex replaced by a qtH± or qtG± vertex. and therefore necessarily incorrect. The dominant 2HDM diagrams which mediate B0 ➔ T+T- are shown in Figure 2.2. T hey are variations of t he box and penguin diagrams in the SM but with con­ tributions from the 2HDM Higgs sector. The first is the SM box diagram but with a single qt Hf± ( q = b, d) vertex replaced by a qtH± or qtG±, the second is the SM penguin but with a single qtW± vertex replaced by a qtH± or qtG± and the third is the SM self-energy diagram with a single qtW± vertex replaced by a qtH± or qtG± so that the d quark line changes into a b line by emission and reabsorption of a H ± or G±. Diagrams in which both btW± and dtvV± vertices are replaced by qtH± or qtG± vertices are suppressed by (mr/mw )2 ~ 4.9 x 10- 4 [22]. According to [19], there are no new contributions to the axial vector Wilson constant CA in the 2HDM. But the 34 ~ -$~----------~ µ \. ojo◄ lif 10-$ Figure 2.3: Branching ratio B2HDM (B0 ➔ r+r-) plotted against mH± (left) and against tan ,B (right) in the large tan ,B limit. The SM value is shown in bold. constants Gp and Cs are modified due to the 2HDM box, penguin and self-energy processes so that, for large tan ,B, cfIDM = C{M and C 2HDM _ C2HDM _ m,,. 2 ,8 logr s - s - -22 tan -­ mw r-l where r = m't-±/mz(mt)- The 2HDM branching ratio is then [19] (2.41) -8 [ 7 B l [ f B l 2 [ !Vtd l l 2 ( 2 \1 2•4 x lO l.54ps 210Nl eV 0.008 x a +~-,2) a b 4 2 log r 2 5.4 x 10- tan ,B 1 _ r - Y((mtfmw) ) (2.43) logr 4.0 x 10- 4 tan2 ,B-­ l - r (2.44) where Y(xt) is defined in Equation 2.32. See Figure 2.3 for a plot of the branching 35 ratio versus the charged Higgs mass m H ± and versus tan ,B. The SM and 2HDM amplitudes interfere destructively, but for large tan ,B the 2HDM enhancement domi­ nates. 2.4. Supersymmetry Outline of Supersymmetry While the SM remains experimentally unchallenged as a model for fundamental particle interactions, there is a strong motivation to revise it on theoretical grounds. While experiment has determined that the vacuum expectation value for mh must be of order 100 GeV, theoretical calculations show that first order corrections to m1 are of order Mi1 = (2.4 x 1018 GeV) 2 , the energy scale at which quantum gravity must alter the SM if it is to be a complete theory. Keeping the Higgs mass at the electroweak scale despite an unavoidable Planck scale correction is known as the hierarchy problem. Supersymmetry theory [23] introduces a new symmetry Q and develops much in the spirit of the gauge symmetry of the SM. The supersymmetry Q 36 acts on bosonic particle to create fermionic superpartner, and acts on fermionic par­ ticles to create bosonic superpartners. The commutation relations 2 [Qt,Qt] =[Q,Q] - 0 [Pµ, Qt] = [Pµ, Q] - 0 (2.45) (2.46) (2.47) define the supersymmetry algebra [23]. Since pµ commutes with Q, the superparticle mass is equal to the par ticle mass if the supersymmetry is unbroken. But the sym­ metry is slightly broken, causing the difference in particle and super partner masses. Thus in supersymmetry, particles acquire mass after SU(2) symmetry breaking, and their superpartners acquire different masses from their partners after supersymmetry breaking. The huge SM correction to m~ comes from the fermionic quark loop in the Higgs propagator. This diagram arises from the hqkqL interaction in Table 2.2. Unless this diagram is miraculously cancelled by higher order corrections, the experimentally preferred value for mh is excluded. Supersymmetry solves the hierarchy problem since, for each fermionic loop correction to mt there is a scalar loop correction with the opposite sign which exactly cancels it . If the supersymmetry is broken, the cancellat ion is not exact. But if it is only mildly broken, the Higgs mass remains at 2T he notation in this section follows [23). Throughout, represents a scalar, 1/J a chiral fermion, >. a gaugino fermion and F a vector. 37 the electroweak scale. The cancellat ion only occurs if the superpartner of the fermion quark is a scalar. From { Q, Qt} = pµ one can show [23] that the number of degrees of freedom in a particle must equal the number of degrees of freedom in the superpartner. From this requirement and that of renormalizability, it can be shown that there are only two ways of grouping particles with superpartners: chiral or gauge supermultiplets. Chiral supermultiplets group Weyl fermions with complex scalars. Gauge supermultiplets group massless vectors with Weyl fermion~. Only chiral supermultiplets allow different gauge transformation rules for lefthanded and righthanded fermions, so quarks and leptons must be grouped together with their squark and slepton scalar superpartners in a chiral supermultiplet . T he gauge vectors must be grouped together with their fermionic gaugino superpartners in a gauge supermultiplet. Supersymmetry posits that the gauge group generators commute with Q, so that the superpartner gauge quantum numbers (electric charge, weak hypercharge, weak isospin) are the same as those of the partner. Whereas a single Higgs doublet was sufficient in the SM to generate mass, super­ symmetry requires at least two. This is due to a divergence (the triangle anomaly) [23] which is uncancelled in supersymmetry with one Higgs doublet but cancelled with two. Minimally, supersymmetry employs two Higgs doublets (as in the 2HDM) to generate mass for the chiral fie lds. Hu generate mass for the up type (s)quarks and up type (s)leptons. H d generate mass for the down type (s)quarks and down type (s) leptons. 38 The general supersymmetric Lagrangian Css allows for undiscovered fermions and bosons and their superpartners. But beyond the superpartners of the fields already discovered and the Higgs sector, it does not require them. The Lagrangian is (2.48) The chiral supermultiplet Lagrangian contains kinetic terms for all scalars (Higgs, sleptons and squarks) and chiral fermions (higgsinos, leptons and quarks), a scalar­ fermion interaction term and a term for an auxiliary field F: (2.49) All scalar fields are indexed by i and all chiral fermions are indexed by j. The auxilliary fields are necessary to maintain the required balance between fermionic and bosonic degrees of freedom in the supermultiplet when the field particles are offshell. The interaction term is determined by the superpotential W 1 .. . - 2W 1 J1Pi1Pj + W 1 *Wi + cc w 1 .. . 'k 2 MtJ as we as eir equiva en S 1Or Sep OnS and squarks. It also contains the supersymmetric analog of .C~ , which involves the doublets Hu and Hd· The gauge supermultiplet Lagrangian contains kinetic terms for the gauge vectors (B, W, G) as well as gaugino fermions (B, W , G ) . It also contains an auxiliary field D for the gauge fields: L gauge (2.52) where i indexes gauge fields and j indexes gaugino fields. The gauge fields and gauginos are massless before SU(2) symmetry breaking. The D field is required again in order to maintain the equality betwen the fermionic and bosonic degrees of freedom. .C9auge contains the term .Cdynamical from the SM Lagrangian. Using the equations of motion for the auxiliary fields, they can be replaced by Fi = Wt and Dj = - gj(*TJ) where TJ is the jth generator and gj is the jth coupling of the gauge group. The superpotential defines all Yukawa interactions in the MSSM. The gauge and gaugino interactions are determined by the action of gauge covariant derivatives on 40 the fields as in the SM. The only remaining term in Lss contains all gauge invariant and renormalizable interactions between scalars, chiral fermions and gauginos which are not generated by the covariant derivatives Dµ: (2.53) £ 55 is the most general renormalizable Lagrangian which is invariant under a supersymmetric transformation ¢ -+ Q, 'I/; -+ Q'lj;, ,\ -+ Q,\, F -+ QF [23]. It remains general until the field content {¢,'I/;, ,\, F } and superpotential W are specified. The M inimal Supersymmetric Model (MSSM) Any supersymmetry model must be phenomenologically viable. This means that it must include all fields which are already known to exist, it must specify a super­ potential which excludes interactions in Lss which violate conservation laws already observed to hold true, and it must specify the terms in £ 55 which break the symmetry and render sparticle masses at values above the current experimental reach. At a minimum, the supersymmetry field content must include all SM fields (less the Higgs doublet ) and the SM interactions. It must also include all supersymmetric partners of the SM fields as well as the two Higgs doublets Hu and Hd- See Tables 2.3 a nd 2.4 for the minimal addit ional content in the MSSM. The scalar content ¢ of the MSSM includes squark fields, slepton fields and Higgs fields Hu and Hd. The chiral fermion c-ontent 'I/; includes the SM quark fields, SM 41 I Slepton I Q I J II Squark I Q I J II Gaugino I Q J e -1 0 u 2/3 0 ite 0 0 ci -1/3 0 A(,} 0 1/2 ii -1 0 c 2/3 0 z 0 1/2 vµ 0 0 s -1/3 0 w+ +1 1/2 :;- -1 0 t 2/3 0 w- -1 1/2 1/7 0 0 b -1/3 0 Gk(g) 0 1/2 Table 2.3: The minimal additional field content of the MSSM, excluding the Higgs sector. These are the superpartners of the leptons, quarks and gauge vectors from Table 2.1 (which are also included in the MSSM). lepton fields and the higgsino fields flu and Hd- The gaugino fermion content >. includes the superpartners of the SM gauge fields .B, vv+, vv- , vV0 and G. The gaugino vector content includes the SM gauge fields B , w+, w - w 0 and G. The superpotential for the MSSM is (2.54) (2.55) Here yJ, yJ and y} are Yukawa couplings. Note in particular the flavor changing Yukawa interactions involving the scalars Hu and Hd, which are not present in the SM. While there are many interactions in the MSSM, and t herefore many poten­ t ially free parameters, large subgroups of interactions occur wi th the same coupling 42 I Higgs I Q I J II Higgsino I Q J H+ u +1 0 H+ u +1 1/ 2 Ho u 0 0 fIO u 0 1/2 Ho 0 0 ~o 0 1/2 d Hd H-d -1 0 iI- d -1 1/ 2 Table 2.4: The MSSM Higgs sector. strength. For example, consider three flavor changing (s)quark interactions mediated b h d H • . dd-t H+ - dd-t H- + d ddt H- + -y a c arge u or its superpartner. Yj j L u UjR, Yj jL u Uj n , an Yj jL u UjR· All three interactions occur with the same coupling yf. In all observed processes, lepton number L and baryon number B are conserved. The MSSM explicitly excludes interactions which violate L or B conservation by postulat ing a new conservation law. It first defines R-parity Pn = (-1)3(B-L)+2S where S is spin. If this multiplicative quantum number is conserved, as the MSSM stipulates, then B and L are also conserved. Since particles have Pn = + l and spar ticles have Pn = -1, Pn conservation has some interesting phenomenological consequences: the lightest supersymmetric partner (the LSP ) is stable against decay, sparticles are produced in even numbers at colliders colliding Pn = + 1 particles, and sparticle decays contain an odd number of spart icles in t he final state. T he MSSM adopts the 2HDM Higgs sector in its entirety. At t ree-level, the masses of h0 , H 0 , A0 , and H ± are fixed by MSSM parameters. Radiative corrections impose an upper limit of 130 GeV on mho given reasonable assumptions about the stop masses [23]. As this upper limit is approached, the remaining Higgs sector masses 43 become large and the MSSM couplings approach those of the SM in what is called the decoupling limit. In principle, any sparticles in Tables 2.3 and 2.4 which share the same spin, electric charge and color charge may mix to produce mass eigenstates. The neutral gauginos B and W may mix with the neutral higgsinos il2 and il~ to form any of four neutralinos N1, N2, N3 or N4 . The charged gauginos w+ and vv- may mix with charged higgsinos ilt and Hi to form any of four charginos Ct, C1, Ct or C2. The gluino cannot mix because it is the only neutral fermion which carries color charge. The squarks and sleptons are not expected to mix except in the third generation, where the left­ and righthanded fields mix to produce mass eigenstates t1, i2, b1 , b2, f 1 and f2. B 0 -+ r+r- in the MSSM The amplitude for B 0 -+ µ+ µ - in t he MSSM was first calculated in [21], again in [22] and again in [24]. It is easily translated to B0 -+ r+r- . The MSSM incorporates the 2HDM and therefore inherits the 2HDM mechanism for enhancing B 0 -+ r+r- over the SM rate. In addition, t he MSSM introduces two further mechanisms: the flavor changing supersymmetrized qq' iI±, qq'G± and qq'vV± couplings, and the flavor changing mass insertion qq' due to off-diagonal entries in the squark mass matrix. either mechanism occurs at less than fourth order in perturbation t heory or in processes with Feynman diagrams topologically distinct from those of the SM. 44 The MSSM couplings qq' fI±, qq'G± and qq'W± allow for the decay of a quark into a squark of a different flavor with the emission of a chargino with higgsino, goldstino or wino content. Thus any qq'W vertex in the SM box and penguin diagrams could be replaced by a qq'C1 or a qq'C2 vertex. Moreover, if the MSSM squark mass matrix is not diagonalizable in the same basis as the quark mass matrix, off-diagonal entries in the squark mass matrix entries induce flavor changing mass insertions ijq' [21]. A neutral Higgs penguin diagram with gluino exchange between initial quark lines (with ddg and bbg vertices) and a mass insertion bd also contributes to B 0 ➔ t+z- in the MSSM. See Figure 2.4 for Feynman diagrams of B 0 ➔ 7+7- through chargino and gluino exchange. The authors of [22] found that the box diagram with chargino or gluino exchange does not appreciably alter the Wilson coefficient CA over the SM value. Ior do the Z penguin diagrams with chargino or gluino exchange alter Gp or CA over the 2HDM values. But for large tan,B, the neutral Higgs (h0 ,H0 ,A0 ) penguin diagrams with chargino exchange or gluino exchange with mass insertion enhance Gp and Cs by a factor of tan3 /3. For simplicity, we consider only the case when t he gluino exchange diagram is negligible. In the limit of diagonal squark matrix, the gluino exchange penguin con­ tributes nothing to t he process B 0 ➔ 7+7- and the chargino exchange penguin dominates. 45 The Wilson coefficients for chargino exchange are found to be C1;fSSM = C{M and [22] 2 C MSSM cMSSM c2HDM m.,. mt 3 /3 - C ( X ) p = s = p + -2 2 -2 tan Atmel 0 me,, mi1 , mt2 2.56 mwmA where the three point function C0 is 1 [ x x y yl C0(x,y,z) = -- --log- - -- log- x-y x-z z y-z z (2.57) and At = At+ µcot/3. Here At is the top Yukawa coupling. The corresponding branching ratio is enhanced by a factor of tan6 /3 in the MSSM, and is calculated to be [22] _ 2 4 x 10-s [ 78 ] [ f s ] 2 [~] 2 • l.54ps 21OMeV 0.008 x (c2 + d2 ) (2.58) (2.59) The gluino exchange penguin amplitude is found also to increase as tan3 /3 [22]. But this amplitude may interfere either constructively or destructively with the chargino exchange penguin. For some regions of parameter space, the interference is destructive ( see the following section). d d ,,, c , -'t. t ',r' I I h/H/A f 46 b Figure 2.4: Dominant MSSM processes responsible for B0 ➔ 7+7- . At left is the penguin with gluino exchange with flavor changing mass insertion. At right is the penguin with chargino exchange. B 0 ➔ 7+7- in Constrained MSSMs In order to account for the mass differences between particles and their superpart­ ners, the terms in .lss which break supersymmetry at the electroweak scale must be specified. In the MSSM, these are the gaugino, higgsino, slepton and squark mass terms as well as t he squark-Higgs and slepton-Higgs couplings. The mechanism of supersymmetry breaking in the MSSM is still unknown, but is thought to be due to radiative corrections to the sparticle masses due to particles in a hidden sector which do not interact with the SM particles. One can specify t he mechanism of supersym­ metry breaking and thereby further constrain the MSSM. Many of the masses and couplings in the MSSM can be seen to originate from common values at the unification scale. vVhen the mechanism of supersymmetry breaking is specified, a specific set of unification scale masses and couplings can then be evolved down to the electroweak scale using the renormalization group equations. 47 In gravity-mediated supersymmetry breaking, the gravitino (the superpart ner of the graviton) mediates the radiative corrections. This framework is called minimal Su pergravi ty ( mS U G RA). In gauge-mediated su persymmetry breaking, the messenger particles with SM gauge group quantum numbers mediates the radiative corrections, and the framework here is called minimal Gauge Mediated Supersymmetry Breaking (mGMSB). Neither mSUGRA nor mGMSB is fully satisfactory, with the Particle Data Group review referring to mSUGRA as "too simplistic" and mGMSB as "not a fully realized model" [25]. onetheless, the amplitude for B 0 ➔ µ+ µ- has been calculated in mSUGRA and mGMSB, first in [26], [27] and [28] and later in [29]. The mechanism for enhancing the rate for B 0 ➔ r+r- in mSUGRA and mGMSB is the same as it is in the MSSM (chargino and gluino exchange) and the enhancement is also proportional to tan6 (3. T hese models simply constrain the electroweak scale masses and couplings to orig­ inate in common unification scale values which diverge after renormalization group evolution downward in energy to t he electroweak scale. In mSUGRA, only five parameters are free: the scalar mass m0 at the unification scale, the gaugino mass m1; 2 at the unification scale, the scalar cubic coupling Ao at the unification scale, tan,B and the sign of the superpotential parameter sgn(µ) . The authors of [29] found that, after scanning this parameter space, only a very small portion of the m0 - m 1; 2 plane is excluded by assuming tan f3 > 35 and B0 ➔ r+r- > 10- 6 . T hey furt her found, however, that the penguin diagrams with gluino 48 exchange and chargino exchange may interfere destructively and therefore decrease the enhancement below the tan6 /3 expectation. Even so, in the parameter space covered the branching ratio was found never to fall below half the SM expectation. In mGMSB, the authors of [29] found that the enhancement is much lower than in mSUGRA. Here the free parameters are the sparticle mass scale A, the messenger mass scale NI, the number of SU(5) vector representations in the messenger sector, tan /3 and sgn(µ). In scans of this parameter space, after evolving the parameters down to the electroweak scale, the branching ration for B 0 ➔ r+r- was found never to exceed 1 o-7 . 2.5. Leptoguarks Models Containing Leptoquarks The symmetry between quark and lepton generations suggests that there may be a hidden connection between quarks and leptons. There is no compelling theoretical reason in the SM why there should be the same number of quark generations as lepton generations, but the SM relies on the equal and opposite contributions from leptons and quarks to the hypercharge anomaly cancellation [30]. Leptoquarks codify this connection in a physically realized object. 49 The 2HDM and MSSM are only two of many possible extensions of the SM. Other models include: • Grand Unified Theories (GUTs). GUTs seek to unify t he electroweak, strong and gravitational interactions based on the groups SU(5), SO(10) or SU(15). • Composite Models. These model~ identify substructure in the SM. Quarks and leptons are made of preons. • Technicolor Theories. Technicolor introduces a new non-Abelian gauge interac­ tion and requires any scalars to be bound states of fermions and anti-fermions. All of these models incorporate leptoquarks [31], which carry both lepton number L and baryon number B and are SU(3) color triplets. The SM does not. In the context of GUTs, leptoquarks were first introduced in an SU( 4) model in which lepton number is considered to be a fourth strong interaction color charge, extending the SU(3) color symmetry [32] . Pati-Salam leptoquarks are then simply the gauge boson which mediate the SU( 4) strong interaction. The SU( 4) symmetry is broken down to SU(3) symmetry, giving mass to the leptoquarks but leaving the gluons massless. Technicolor theories require all scalars to be bound states of a fermion and an anti-fermion, and a bound state lij or [q would be a natural candidate for a leptoquark. Compositie theories build leptons and quarks from preons, which therefore must carry lepton number and baryon number. Bound states of two preons 50 I Leptoquark I J I F I Q II Leptoquark I J I F I Q So 0 -2 1/3 Vo 1 0 2/ 3 So 0 -2 4/3 Vo 1 0 5/3 ~1/2 0 0 5/3,2/3 ~ 1/2 1 -2 4/3,1/3 S1;2 0 0 2/3,-1/3 Vi;2 1 -2 1/3,-2/3 81 0 -2 4/3,1/ 3,-2/3 Vi 1 0 5/3,2/3,-1/3 Table 2.5: The spin J , fermion number F = 3B + L and charge Q for leptoquarks allowed for renormalizable SM gauge group invariant interactions. could then carry both lepton and baryon number and would also be natural candidates for leptoquarks. Assuming that the leptoquark interactions are renormalizable and invariant under the SM gauge group transformation, the number of leptoquark SU(2) multiplets is fixed at ten: the scalars So , So, S 1;2, S1;2, S1 and the vectors Vo , Vo, Vi;2, V1;2, Vi [30]. The subscripts indicate the weak isospin quantum number of the leptoquark. Leptoquarks are either scalars or vectors and carry fermion number F = - 2 or F = 0 (F = L + 3B). Different models incorporate different subsets: superstring E 6 includes only S0 while the SU(15) GUT includes all ten leptoquarks. See Table 2.5 for the leptoquarks and t heir quantum numbers. The Lagrangian which includes all interactions of the leptoquarks with leptons and quarks is given by LLQ = LL L >.t~lf1qJ (2.62) i,j m,n LQ where >.f1~ is t he coupling of the lepton from generation i to the quark of generation j with chirality n to the leptoquark LQ. The lepton chirality m is determined by d - - - - - - - - - - - >.;;' LQ = Vo/V,;2 d >-!l ----- - ----- >-tf LQ = S,12/So b 51 Figure 2.5: Vector (left) and scalar (right) leptoquark processes which mediate B 0 -+ 7+7-. The quark and lepton chirality labels have been suppressed. angular momentum conservation and JLQ· The F = - 2 leptoquarks couple lq and the F = 0 leptoquarks couple lq. Phenomenological viability requires that some constraints must be placed on some leptoquark couplings and masses. A given interaction involving a given leptoquark can be suppressed by i) requiring separate B and L conservation, ii) requiring a sufficiently small coupling A, iii) requiring the leptoquark mass to be sufficiently large or iv) requiring the leptoquark to couple only to one generation. For example, FCNC constraints from the upper limit on B(I(0 ➔ µe) are satisfied if the leptoquark is not allowed to couple to both th