13jv = 2.55 \u00b1 0.25 (1.25) 16.0^B + 31.9<\/Be + 2.74<^ep + 8.71(p13N = -4.65 \u00b1 9.5 (1.26) i p B = 0.51 \u00b10 .072 (1.27) Substituting \/ r \/ res \\ X . \\ \\ ( d N \/ d r ) e \/ \/ K d N \/ d r ) r e , cos a \/ \\ \\ a ipSin cpi 1 Figure 4.8: Three possible paths with the effective jump probability for each one. Note the cos(a) term which modifies the neutrino density gradient at the resonance crossing. The angle is defined by sin(a) = ZxiisM have when it leaves the sun. The jump probability at each resonance is again modified because the effective density gradient has an added factor of cos(ojj) as in the previous case. Defining Vai and Uai to be components of the C K M matrix in vacuum and in matter at the point of production respectively, and Pi and P 2 to be the jump probabilities at each of the two resonances modified by the terms cos(ai) and cos(o;2), the probability that a ve produced in the sun emerges as a ve along the six possible paths is given by: ee,(i) \u2022= Pelf | K l | 2 + \\Ue2\\2 | K 2 | 2 + Pesf \\Ve 3| ee,(ii) \\Uel\\2{(l - 2P2 + 2P22) | K i | 2 + 2P2 (1 - P2) I K 2 I 2 } + \\ue2\\2 {2P2 (1 - P 2) | K i | 2 + (1 - 2P2 + 2P22) |ye2|2} + (4.21) (4.22) Chapter 4. Numerical Methods 138 (0-(iii)-(iv)-(v> (vi; 'res,l V > \u2022 r , sun 1 Figure 4.9: Six possible paths which a neutrino can take when leaving the sun with two resonance radii. The density gradient at the two resonances is modified by factors of cos(a1) and cos(a2) just as it was in the two neutrino case. Pee,(iii) = |t\/e l|2 {(1 - ^2) |^el|2 + ^2 + \\ue2\\2 {P2\\vel\\2 + (1 - P2)\\ve2\\2} + \\ue3\\2\\V*\\ (4.23) ee,(iv) \\Uel\\2{[(l - P2)2 + P\\ ( l - 2PX + 2P2)] | K i | 2 + (4.24) 2P2 (1 - P2) ( l - Pi + Pi) | K 2 | 2 + 2P2Px (1 - Pj) |V e 3 | 2} + |t\/e2|2 {2P2 (1 - P2) ( l - Pi + P 2 ) iVeil2 + [P22 + (1 - P 2 ) 2 ( l - 2PX + 2P2)] I K 2 I 2 + 2(1 - P2)PX (1 - Px) |K 3 | 2 } + |C7e3|2 {2P2 (1 - Pi) P 2 IKi | 2 + 2P, (1 - Px) (1 - P2) | K 2 | 2 + ( l - 2 P 1 + 2P12)|V;3|2} Pee,W = l ^ e l | 2 { ( l - P 2 ) | K l | 2 + P2|V;2| 2}4 | t \/ e 2 | 2 {( l -2P 1 + 2 P 2 ) P 2 | K i | 2 + (4.25) Chapter 4. Numerical Methods 139 (l - 2PX + 2P2) (1 - P 2)\\V e 2\\ 2 + 2A (1 - Px) |K 3 | 2 } + \\Ue3\\2{2P1(l-P1)P2\\Vel\\2 + 2Pi (1 - Px) (1 - P2) | K 2 | 2 + (l - 2PX + 2P 2) |K 3 | 2 } , Pee,(vi) = |^el | 2 {( l-P 2 ) |V;i | 2 + JP2|V; 2 | 2}+ (4.26) \\Ue2\\2 {(1 - PX)P 2 | V e l | 2 + (l - P a)(l - P2) |V e 2 | 2 + P2 |K 3 | 2 } + l ^ l 2 {PiP2 IKil 2 + A ( i - P 2 ) |v;2|2 + (i - P 2) |v;3|2} After averaging out the phase of the neutrino wave functions across each resonance crossing and the point at which it leaves the sun, each term in the formulae above can be interpreted as the product of a series of probabilities at each step\u2014phase effects are not present. The neutrino point of production relative to the resonance radii determines the path across resonances in the sun, and it is a geometric problem to determine which of the six probabilities above to use to translate the neutrino out of the sun. 4.4.5 Seasonal a n d D a y \/ N i g h t Effects Seasonal and day\/night effects are included in the calculation of neutrino oscillations, as illustrated in figure 4.10. Neutrinos which arrive on earth at night have much longer paths through the earth's interior and may undergo matter enhanced oscillations. Two coordinate systems XYZ and X'Y'Z' are defined as shown, such that Z is perpendicular to the plane of the solar system, Z' is the axis of rotation of the earth and X = X'. The Y'Z' axes are rotated 9 = 23\u00b0 relative to the YZ axes. Given the azimuthal angle 7 of the detector with respect to Z' (90\u00b0 minus latitude), the angles a in the X'Y' plane and (3 in the XY plane define the time of day and time of year respectively fully parameterise a neutrino path from the sun to the earth. The position vector of the detector is rotated from the primed coordinates to the unprimed Chapter 4. Numerical Methods 140 Z = galactic north Figure 4.10: Coordinate systems used to determine the path length of the neutrino through the earth. The XY axes define the plane of the ecliptic, while the X'Y' define the equatorial plane. The angle 7 is the azimuthal angle of the detector about the axis of rotation of the earth. The angle a defines the degree of freedom in the earth's rotation and determines the time of day at which the neutrino arrives. The angle (3 defines the angle which the earth makes relative to the sun, and determines the time of year. Chapter 4. Numerical Methods 141 as follows: 'y \\rz J rdet 1 0 0 0 ce sg \\ 0 -se ce J s^ca V c 7 J ^det CQS1S(X + sec1 y \u2014ses^Sa + cec1 J (4.27) where sn and cv are defined to be the sine and cosine of the angle r\\. The path of a neutrino through the detector when the earth is at angle j3 in the XY plane is defined by the line cpx + spy = cprx + spry (4.28) z = rz Combining these equations with x2 + y2 + z2 = r 2 a r t h determines two solutions (x, y, z) which define the two points where the neutrino path crosses the earth's surface. Elimi-nating x and z using equations 4.28 yields the equation y2 - 2s0 (cprx + spry) y + (r2z - r2earthj + (c0rx + s0ry)2 = 0 (4.29)' The discriminant for this quadratic equation can be reduced to 4ac = Aci earth Tlet + (S0rx ~ CpVyf > 0 (4.30) which is positive definite and guarantees two real solutions, one short path and one long path. If the neutrino arrives in the daytime (the short path solution) the constant density Ne = 1.34mol\/cm3 is used in regeneration calculations corresponding to the electron density of the earth crust. For nighttime neutrinos, a density closer to the average through the earth is used, and regeneration is calculated with Ne = 2.1 mol\/cm3. 4.4.6 Stability of the Results One last question which must be addressed is the level of statistics required for the Monte Carlo method to generate accurate results\u2014how many events are required to accurately Chapter 4. Numerical Methods 142 Source Chlorine Gallium Reaction Rci [SNU] o~ci o~ci\/Rci RGa [SNU] 0~Ga 0~Gd\/RGa hep 4.27 x IO\"3 2.4 x 10\" - 5 0.55 % 8.33 x 10\"3 7.0 x 10-- 5 0.84 % 8B 6.87 8.9 x 10\" - 2 1.3% 15.9 0.13 0.81% pep 0.204 5.6 x 10\" - 9 0.0% 2.86 4.0 x 10\" - 8 0.0% 7Be 1.60 3.9 x 10\" - 3 0.25 % 37.7 8.2 x 10-- 2 0.22 % \u2022 pp 0.0 0.0 n\/a 71.4 0.43 0.61 % 1 3 AT 0.156 1.1 x 10\" - 3 0.71 % 3.85 1.7 x 10-- 2 0.44 % 150 0.371 2.8 x 10\" - 3 0.75 % 6.24 3.2 x 10\" - 2 0.52 % 4.47 x IO\"3 3.3 x .10\" - 5 0.74% 7.47 x IO\"2 3.7 x 10-- 4 0.49% Total 9.21 8.7 x.10\" - 2 0.95% 138.1 0.41 0.30 % Table 4.1: Mean standard model neutrino rates for chlorine and gallium detectors from ten different trials generating 10,000 events for each neutrino source with different pseu-do-random number generator seeds. calculate neutrino oscillation probabilities on earth? In addition, how accurately are the energy and radial distributions of the neutrino sources reproduced by a given number of events? The first of these questions can be answered by calculating the event rate for a given number of events using different initial random number seeds. Table 4.1 shows the mean and standard deviations in event rates for each of the eight neutrino sources on both chlorine and gallium targets based on 10, 000 neutrino events with ten different random number seeds. Almost all of the sources have standard deviations which are less than one percent of the mean, and the discrete energy neutrinos have much more accurately determined rates. The rates for the chlorine and gallium experiments have errors of 0.95 % and 0.30 % respectively. In calculating the MSW effect in a plot with typically a 41 x 41 grid, the average standard model (no MSW) event rate is calculated over all events and all eight neutrino sources (41 x 41 x 10,000 x 8 = 1.34 x 108 events). Then the MSW rate at the ij grid point rateijtMsw is multiplied by the ratio of the overall standard model result with the Chapter 4. Numerical Methods 143 standard model rate rijtsTM at that grid point. rateij~~ ACPPTfJ, and AcpPer are negative. T violation is defined with the probability difference ATPap = P ( i \/ Q -> vp) - P (up -> ua) = \u00b143 j (5.36) where the signs are identical to above. T violation extends easily in the presence of matter to ATP\u2122p = \u00b14Jmfm where Jm and fm are direct analogues of their counterparts in vacuum. CP violation does not translate simply, because the effect of matter on the neutrino oscillation probability is different from the antineutrino oscillation probability. 5.4.2 Review of the Literature Several authors have begun to look into CP violation using long baseline neutrino oscil-lation experiments in the last year. Discussions have focussed on where to look for CP violation and how to separate it from matter effects. Chapter 5. Numerical Calculations with Three Neutrinos 166 Arafune and Sato [152] studied the oscillatory term \/ when A m ^ ~~~~ 4 x l 0 0) - 5 g 4x10-< 4x10\" 4x10\" > 4 x l 0 \" ~~* 4 x l 0 -\u00a3 4 x 1 0 ' 4x10\" 0 0.25 0.5 0.75 10 0.25 0.5 0.75 10 0.25 0.5 0.75 1 sin 2 ( i? 1 2 ) sin 2 (i3 1 2 ) sin 2 ( i5 1 2 ) E= 1 : 0 - 4 . 2 GeV E= 1 . 4 - 3 . 0 GeV E= 1 . 6 - 2 . 4 GeV Figure 5.10: Study of CP violation by varying the region of the neutrino reference spec-trum used, with a baseline of 250 km. Contours of constant A ^ f P are shown for s<$ = 0.0, 0.5 and 1.0, expressed as percentages. The bottom row of figures shows the difference between the ss = 0.0 and ss = 1.0 plots in the column, scaled by a factor of 100. Moving from left to right, the neutrino energy spectrum is successively compressed. Chapter 5. Numerical Calculations with Three Neutrinos 182 proportion to energy. However, that larger matter effect will enhance some of the second order CP violating terms in the oscillation probability differences. Figure 5.11 illustrates what would be seen at MINOS using the reference spectrum scaled upwards by a factor of five and a baseline of 732 km. It shows AP^P using F[d], and both APj?p and APf?p with the F[e] parameter set. The size of the CP term in APj?Tp is somewhat larger than in AP^f for F[d] and the oscillation probability difference is smaller, so it would provide an easier signal to detect. Both P M T and PTfl are about 27%. As expected, the CKM parameter set F[e] produces a larger CP violating term because it has a smaller Am^. Although AP^P is larger in magnitude than AP^P, it induces a larger violation of CP and the relative contribution of that violation to the oscillation probability difference is about the same\u2014less than \\ %. This ratio is notably less than F[d] in the first column where AP^P was much smaller. However, with only about 20,000 charged current events per year, MINOS will not be able to measure an effect as small as CP violation. Next, consider an experiment with the standard reference spectrum and a detector 1500 km away. The longer baseline probes to lower values in the mass differences and increases the parameter Q, which in turn raises the magnitude of the first order CP violating term. Figure 5.12 shows AP^ep for three different cases beginning with the CKM parameters F[d]. The CP term is very small, and much smaller than the oscillation probability differences. However by tuning the neutrino energy, in this case scaling the reference spectrum downwards by a factor of 0.75, the spectrum peaks at 1.35 GeV. Then at the peak energy, Q, \u00ab 2.33, and 1 - C 2 n - ^ S 2 0 = 0. The result is that AP\u00b0ep drops by about 40 %, while the contribution of CP the violating term is increased by about 50 %. While he effect is not dramatic in this case, it would improve if a smaller region of the spectrum was used Chapter 5. Numerical Calculations with Three Neutrinos 183 4 x l ( T 7 1 \u2014 ' \u2014 1 \u2014 i \u2014 ' \u2014 i \u2014 L ^ j\u2014 I\u2014 i I i | i I i 1 i I i i , i , 1 0 0.25 0.5 0.75 10 0.25 0.5 0.75 10 0.25 0.5 0 75 1 sin2(tf12) sin2(tf12) sin2(tf12) F[d] A P M T F[e] A P M T F[e] A P \u201e e Figure 5.11: Study of CP violation at an experiment with a baseline of 732 km using the reference neutrino spectrum modified by scaling the energy upwards by a factor of 5. Contours of constant A^f P and A^fP for CKM parameter sets F[d] and F[e] are shown for ss = 0.0, 0.5 and 1.0, expressed as percentages. The bottom row of figures shows the difference between the s$ = 0.0 and s$ = 1.0 plots in the column, scaled by a factor of 100. Chapter 5. Numerical Calculations with Three Neutrinos 184 0 0.25 0.5 0.75 10 0.25 0.5 0.75 10 0.25 0.5 0.75 1 sin 2(i3 1 2) F[d] A P M e sin 2(tf 1 2) F [dd] A P M e sin : ( A 2 ) F[e] AP Figure 5.12: Study of CP violation at an experiment with a baseline of 1500 km using the reference neutrino spectrum. Contours of constant Aj^fP are shown for s$ = 0.0, 0.5 and 1.0, expressed as percentages. The bottom row of figures shows the difference between the s$ = 0.0 and s$ = 1.0 plots in the column, scaled by a factor of 100. Column 1 uses the CKM parameter set F[d] and the reference neutrino spectrum, while column 2 scales the energy down by a factor of 0.75. In column 3, the C K M parameters defined by F[e] are combined with the reference spectrum. Chapter 5. Numerical Calculations with Three Neutrinos 185 to increase the region over which the matter term was cancelled. The third column shows contours of the oscillation probability difference for the C K M parameter set F[e]. Because the baseline is longer, this experiment can probe to lower mass differences. The oscillation probabilities in this case are 12 %, CP violation con-tributes an absolute magnitude of about 0.2 % which is about ten times as large as was found using a 250 km baseline. However, the number of events falls as the square of the baseline, so the advantage gained in a larger effect is more than lost to reduced statistics. Finally, figure 5.13 shows the oscillation probability difference spectrum using Ffe] for each of the three baselines discussed. In each plot, the upper curve shows AP^P with s<5 = 1 while the lower curve uses Sg = 0. Note that E\\ reaches the value 1 when the neutrino energy is 2.8 GeV, at which point matter begins to dominate the probability. It is clear that the best signal would be obtained in the low energy region of the 250 km experiment. Here, matter does not play a large part and Q is relatively small. By going out to 1500 km, fl becomes large and the oscillatory terms in AP^P wash out the CP violating signal. Thus, bigger is not necessarily better. 5.4.5 Combining Reactor and Long Baseline Experiments The last section highlighted the importance of separating matter effects from CP vio-lating effects using long baseline experiments. The reason that most authors calculate the difference AP^P = P M e \u2014 P^ is to cancel the leading order term in the oscillation probabilities. The e\\ terms also cancel because P^ g(^ i)<^ ) \u2014 Pfie (~\u00a3i, \u2014 S), which is a fortunate side-effect given that S\\ is probably larger than \u00a32 . A new approach emerges if the two terms are added together instead of subtracted from one another. In that case, both the first order matter terms as well as the first order CP violating terms proportional to sg cancel. This leaves the leading order, CP even (in 5) and second order terms. But that leading order term is identical to the one Chapter 5. Numerical Calculations with Three Neutrinos 186 Energy |GeV] Energy [GeV] Figure 5.13: Oscillation probability differences ACPPfie and ACPPjJ,T as a function of neutrino energy for baselines of 250, 732 and 1500 km, using the CKM parameter set F[e]. Black curves compare probabilities with ss = 1.0 to grey curves with s$ = 0.0. Chapter 5. Numerical Calculations with Three Neutrinos 187 in Pep, at a nuclear reactors (even if that's not what is measured). Although they have much lower energy, they have much shorter baselines and typically have similar values of -|. Moreover, the MeV energies of reactor neutrinos reduce the matter effect by three orders of magnitude compared to accelerators, so the matter term can be ignored. But because reactors measure the disappearance of Pe's, it is Pep + Pef = 1 \u2014 Pee which will have to be used to cancel the leading order term. This necessitates multiplying the reactor probability by a factor of s2 3 in the expression below. Thus, one can define: AA~RP,e = Pte + PA-2sl3[P\u00ab+px\\\u00a3i=o = 8e2QS2nS12C12S13cl3S23C23CS +4e2s23c23S^{2n2 (c23 - s23) + 4QS2n [s2 3c2 3 - (c23 - s23)2] + (1 - C2Q) [(4c23 - 1) (1 - 4s23) - 2tt2 (c23 - s23)] } (5.62) + 8 \u00a3 2 , f 2 2 C 2 3 { s 2 2 C 2 2 (4, - sj3) - 2S?2C12513523C23C (5 + (1 - C 2n) s12c12sl3sl3c23c$} +8cTie2^si2Ci25i3C 2 3 s 2 3 c 2 3 s < s (1 - 3s23) [s2 3sl3 (1 - C 2n) - flS2n] where the superscripts A and R refer to accelerator and reactor, respectively. It is assumed that the probabilities are measured at the same This type of subtraction, while analytically easy, would be experimentally difficult. It would have to be done over a limited range of energies because the accelerator beam is unlikely to exactly match the shape of the reactor spectrum. The accelerator energy would also have to be tuned so that the ratio ^ is the same for both experiments. Moreover, some systematic errors cancel out in the pure accelerator experiment because the baseline is identical for the and and this would not be the case here. Finally, reactors measure the disappearance of ve signal, so it is statistically more difficult to measure a small oscillation probability than in an accelerator appearance experiment. However if these challenges can be overcome, then the three different experimental Chapter 5. Numerical Calculations with Three Neutrinos 188 measurements (ie. PAe, P~ and P?--\\-PR) allow the cancellation of both the. leading order and O (s\\) terms. In addition, if s$ happens to be small, then the first order CP violating effect in the pure accelerator experiment will be negligible. Equation 5.62 measures c$ in the O (el) term, and this would be close to unify if s$ is small. Figure 5.14 plots contours of constant AA~RP\/J,e as a function of Am^i and si2 and the maximum CP violation, similar to the last section except now eg varies in successive rows. The accelerator is assumed to have a 250 km baseline while the reactor neutrinos are detected 500 m from where they are produced. The neutrinos are limited to a small region of their total spectrum for comparison in the plots, although one would expect that in any real experiment the data would all be kept binned by energy. In the first column, neutrinos in the 2.8 \u2014 3.0 GeV range at the accelerator are com-bined with 5.6 \u2014 6.0 MeV reactor neutrinos with the CKM parameter set F[c], which has a large A m ^ = 1.8 x 10 _ 2eV 2. Here, both ei and e2 are small and the second order terms contribute very little. As a result, AA~RPfie is dominated by the first order CP violating term. The centre column plots the oscillation probability difference using the scheme F[d] with 0.7 \u2014 0.9 GeV and 1.4 \u2014 1.6 MeV neutrinos. As might be expected the magnitude of the CP violation is larger because Am^ is lower. The signal is not as clean because matter terms have AA~RPne has increased, making it more difficult to unambiguously assign a measurable change in it to CP. However, CP violation is about 0.1 % for A m ^ = IO - 5 eV2. Finally the last column shows the F[e] CKM parameter set with neutrino energies of 4.8 \u2014 5.0 GeV and 9.6 \u2014 10.0 MeV. The second order matter terms are much more apparent, while the CP violating effect has, not increased in the lower regions of the Difference plot. It is, however, still in the range 0.1 %. Because the magnitude of CP violation for these combined experiments has increased Chapter 5. Numerical Calculations with Three Neutrinos 189 4 x l O - 4 0 0.25 0.5 0.75 10 0.25 0.5 0.75 10 0.25 0.5 0.75 1 sin 2(i? 1 2) sin 2(i9 1 2) sin 2(i5 1 2) F[c] L \/ E = 86.2 F[d] L \/ E = 312.5 F[e] L \/ E = 51.0 Figure 5.14: Study of CP violation by combining long baseline accelerator and reactor experiments with constant L\/E. Contours of constant AA~RP are shown for cs = 0.0, 0.5 and 1.0, expressed as percentages. The bottom row of figures shows the difference between the c\u00ab$ = 0.0 and cs = 1.0 plots in the column, scaled by a factor of 100. Each column uses a different CKM parameter set and ratio L\/E as labelled. Chapter 5. Numerical Calculations with Three Neutrinos 190 compared to the purely accelerator case, fewer events would be needed to measure a CP signal. Approximately 106 events would be required. However, scaling the baselines at the Chooz and Palo Verde reactor experiments to 500 m would produce one the order of 100 events per day, or about 30,000 per year. Thus, the detectors would have to be scaled up by about an order of magnitude to approach the event rates required to contribute to a measurement of CP violation. Using reactors to subtract off the leading order terms can also be applied to the dimensions of the MINOS experiment, but with less success. With the higher neutrino energies, \u00a3i is relatively larger than e2. Matter effects do not dominate AA~RPfie at MINOS in CKM parameter sets F[b] and F[c] but they are large enough to prevent a clean CP violating signal from being seen. F[d] has larger A m ^ and the e2 terms in equation 5.62 are beginning to approach the size of the e\\ terms, so that matter effects again compete with CP violation. In Ffe], e\\ is close to unity so cancelling the first order matter term offers no significant advantage and the approximation breaks down. However the numerical results indicate that CP still does not contribute enough of a signal to be measured. 5.4.6 C P V i o l a t i o n a n d A t m o s p h e r i c N e u t r i n o s At particle accelerators, experimentalists decide whether to produce a beam of neutrinos or a beam of antineutrinos by focussing a beam of either positively or negatively charged mesons. They first produce neutrinos and measure P^, then antineutrinos to measure Pfxe- If the two oscillation probabilities differ, it must either be due to a CP violating effect or the presence of matter (which is not CP invariant). In this case, the choice to study either matter or antimatter is explicitly made by the experimentor at the neutrino source. Chapter 5. Numerical Calculations with Three Neutrinos 191 The situation with atmospheric neutrinos is quite different. There, the \"source\" con-sists of four different neutrinos: identical and fluxes, combined with ve and ve fluxes of equal magnitude, but which differ from the v^. However, because the detection processes are dominated by matter they see the four neutrino types differently. In par-ticular, the cross section for detecting a ve is different from that of a i>e, and the rate of production of pT from impinging on nucleons is different from the rate of p+ produced by *V So, whereas accelerator based experiments can measure CP violation by alternately producing neutrino and antineutrino beams at the source and looking for differences in the oscillation rates, atmospheric neutrino experiments can allow the natural detection processes differentiate the two identical fluxes. It.turns out, as will be explicitly calcu-lated, that the experiments need not be able to differentiate p~ events from p,+. CP violation modifies the ratio of e-type to \/u-type events. First, define the four neutrino fluxes and ratio A as follows: $ M = $ ^ = $ ^ (5.63) * c = $ \u201e . = $ P e = A $ \u201e (5.64) Next, make the following definitions about the detection cross sections of the neutrinos and the ratios between them: : \u2022 1 = S ^ P ^ o ^ + SvP-wO-Vv + ^ V e P e ^ + ^ D e P m a ^ = $ M < 7 [ P w + ' A M P ^ + A P e \/ 1 + AA M P g p] Taking the ratio of these two, $^ *