text: eΝ(cid:174)ΜΝ P eΝ(cid:174)ΜΝ P (cid:76) (cid:72) (cid:76) (cid:72) eΝ(cid:174)ΜΝ P eΝ(cid:174)ΜΝ P (cid:76) (cid:72) (cid:76) (cid:72) L(cid:61)2300 km, ΕΜΜ(cid:61)0.10, ΕΤΤ(cid:61)(cid:45)0.10 Η(cid:61)0.10, Ζ(cid:61) 0 L(cid:61)2300 km, ΕΜΜ(cid:61)(cid:45)0.10, ΕΤΤ(cid:61)0.10 Η(cid:61)(cid:45)0.10, Ζ(cid:61) 0 0.14 (cid:72) (cid:76)(cid:144)(cid:72) Θ23(cid:61) 35°,Ex(cid:76)act 0.14 (cid:72) (cid:76)(cid:144)(cid:72) Θ23(cid:61) 35°,Exa(cid:76)ct Θ23(cid:61) 35°,Thiswork Θ23(cid:61) 35°,Thiswork 0.12 Θ23(cid:61) 45°,Exact 0.12 Θ23(cid:61) 45°,Exact 0.10 Θ23(cid:61) 45°,Thiswork 0.10 Θ23(cid:61) 45°,Thiswork Θ23(cid:61) 55°,Exact Θ23(cid:61) 55°,Exact 0.08 Θ23(cid:61) 55°,Thiswork 0.08 Θ23(cid:61) 55°,Thiswork 0.06 0.06 0.04 0.04 0.02 0.02 0.00 0.00 2 4 6 8 2 4 6 8 E GeV E GeV (cid:72) (cid:76) (cid:72) (cid:76) L(cid:61)8770 km, ΕΜΜ(cid:61)0.10, ΕΤΤ(cid:61)(cid:45)0.10 Η(cid:61)0.10, Ζ(cid:61) 0 L(cid:61)8770 km, ΕΜΜ(cid:61)(cid:45)0.10, ΕΤΤ(cid:61)0.10 Η(cid:61)(cid:45)0.10, Ζ(cid:61) 0 0.6 (cid:72) (cid:76)(cid:144)(cid:72) (cid:76) 0.6 (cid:72) (cid:76)(cid:144)(cid:72) (cid:76) Θ23(cid:61) 35°,Exact Θ23(cid:61) 35°,Exact 0.5 Θ23(cid:61) 35°,Thiswork 0.5 Θ23(cid:61) 35°,Thiswork Θ23(cid:61) 45°,Exact Θ23(cid:61) 45°,Exact 0.4 Θ23(cid:61) 45°,Thiswork 0.4 Θ23(cid:61) 45°,Thiswork Θ23(cid:61) 55°,Exact Θ23(cid:61) 55°,Exact 0.3 Θ23(cid:61) 55°,Thiswork 0.3 Θ23(cid:61) 55°,Thiswork 0.2 0.2 0.1 0.1 0.0 0.0 2 4 6 8 10 12 2 4 6 8 10 12 E GeV E GeV (cid:72) (cid:76) (cid:72) (cid:76) Figure 10. Comparison of our analytical expressions (solid curves) to the exact numerical results ◦ (dashed curves) for various values of θ assuming δ = 0 and δm2 > 0. Upper (lower) panels are 23 31 for 2300 km (8770 km) baseline. (dashed green curves). The correspondence between our η and ζ and the NSI parameters ε , ε , and ε used in Refs. [69, 201] can be obtained via Eqs. (2.4), (2.7), and (2.9) ee µµ ττ → ≡ → → → − which suggest the changes : a aˆ a(1 + ζ), ε 0, ε η, and ε η. We can ee µµ ττ see from Fig. 9 that for the 2300 km baseline, the Asano-Minakata expressions give better matches compared to our results, while for the 8770 km baseline, our expressions agree better with the exact numerical results. The accuracy of our analytical approximations as compared to the exact numerical results for diﬀerent vacuum values of θ is demonstrated in Fig. 10. Here we consider the 23 ◦ ◦ minimum (35 ) and maximum (55 ) values of θ which are allowed in the 3σ range [194]. 23 We also present the results for the maximal mixing choice. All the plots in Fig. 10 have ◦ been generated assuming δ = 0 and normal mass hierarchy (δm2 > 0). We consider the 31 same choices of η and ζ as in Fig. 9 and results are given for 2300 km (upper panels) and 8770 km (lower panels) baselines. As is evident, our approximation provides satisfactory match with exact numerical results for diﬀerent values of θ . 23 Fig. 11 compares our approximate probability expressions (solid curves) against the exact numerical results (dashed curves) assuming four diﬀerent values of the CP-violating ◦ phase δ (0, 90, 180, and 270 degrees) at 2300 km. 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text: (cid:200) (cid:144) (cid:200) 2 ∆ Λ gol(cid:45) 13(cid:109) Ε Π Π 2 2 2 1 3Π 3Π 8 8 0 (cid:45)Λ(cid:45)' Π Π (cid:45)1 4 4 Λ(cid:43)' (cid:45)2 Π Θ12 Π 8 8 (cid:45)3 Θ12'(cid:61)Θ12(cid:43)(cid:106) (cid:45)4 0 0 (cid:45)4 (cid:45)3 (cid:45)2 (cid:45)1 0 1 2 (cid:45)4 (cid:45)3 (cid:45)2 (cid:45)1 0 1 2 Β Β (a) (b) (cid:48) − | | (cid:48) Figure 19. (a) The dependence of θ on β = log (a/ δm2 ). (b) The β-dependence of λ±. 12 (cid:15) 31 Then,   (cid:48)(cid:48) † (cid:48) H = V H V  0  0 (cid:48) − (cid:48)  λ− 0 aˆc c s (cid:48) − 1(cid:48) 2 13 13 = 0 λ aˆs c s , − (cid:48) − (cid:48) + − 12 13 13 aˆc c s aˆs c s aˆs2 + δm2 12 13 13 12 13 13 13 31 (A.7) where (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) c = cos θ¯ , s = sin θ¯ , θ¯ = θ + ϕ , (A.8) 12 12 12 12 12 12 (cid:112) and − ± − (cid:48) (δm2 aˆc2 ) (δm2 + aˆc2 )2 4aˆc2 s2 δm2 λ± = 21 13 21 13 13 12 21 . (A.9) 2 (cid:48) The angle θ¯12 = θ12 + ϕ can be calculated directly wi(cid:0)thout calculat(cid:1)ing ϕ via (cid:48) δm2 sin 2θ (cid:48) ≤ tan 2θ¯ = 21 12 , 0 < θ¯ θ . (A.10) 12 δm2 cos 2θ + aˆc2 12 12 21 12 13 (cid:48) (cid:48) The dependences of θ¯ and λ± on β are plotted in Fig. 19. 12 (cid:48) Note that in contrast to the neutrino case, θ¯ decreases monotonically from θ to 12 (cid:48) (cid:48) (cid:48) (cid:48) 12 zero as β is increased. The β-dependences of s = sin θ and c = cos θ are 12 12 12 12 – 47 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: (cid:200) (cid:144) (cid:200) 2 ∆ Λ gol(cid:45) 13(cid:109) Ε (cid:200) (cid:144) (cid:200) 2 ∆ Λ gol(cid:45) 13(cid:109) Ε 2 2 (cid:45)Λ(cid:45)'' (cid:45)Λ(cid:45)'' 1 1 0 0 Λ(cid:43)'' (cid:45)Λ(cid:43)'' (cid:45)1 (cid:45)1 (cid:45)2 (cid:45)2 (cid:45)3 Λ(cid:43)' (cid:45)3 Λ(cid:43)' (cid:45)4 (cid:45)4 (cid:45)4 (cid:45)3 (cid:45)2 (cid:45)1 0 1 2 (cid:45)4 (cid:45)3 (cid:45)2 (cid:45)1 0 1 2 Β Β (a) (b) (cid:48)(cid:48) Figure 22. The β-dependence of λ± for the (a) normal and (b) inverted mass hierarchies. (cid:48)(cid:48) → − λ− aˆ + δm2 s2 + δm2 s2 c2 , (A.17) 31 13 21 12 13 (cid:48)(cid:48) (cid:48)(cid:48) for both mass hierarchies. Note that λ− < 0 < λ for the δm2 > 0 case, while (cid:48)(cid:48) + (cid:48)(cid:48) 31 both λ± < 0 for the δm2 < 0 case. The β-dependences of λ± are shown in Fig. 22. 31 Order-of-magnitude-wise, we have (cid:48)(cid:48) −| | − (cid:48)(cid:48) | | λ− = δm2 O((cid:15) β) , λ = δm2 O(1) , if δm2 > 0 , (cid:48)(cid:48) −| 31| − +(cid:48)(cid:48) −| 31| − 31 (A.18) λ = δm2 O((cid:15) min(β,0)) , λ− = δm2 O((cid:15) max(β,0)) , if δm2 < 0 . + 31 31 31 (cid:38) In particular, in the range β 0, we have (cid:48)(cid:48) −| | − (cid:48)(cid:48) | | λ− = δm2 O((cid:15) β) , λ = δm2 O(1) , if δm2 > 0 , (cid:48)(cid:48) −| 31| +(cid:48)(cid:48) −| 31| − 31 (A.19) λ = δm2 O(1) , λ− = δm2 O((cid:15) β) , if δm2 < 0 . + 31 31 31 (cid:48) | | For the oﬀ-diagonal terms, we have aˆs = δm2 O((cid:15)2), c = O(1), s = O((cid:15)), and 12 31 13 13 s = O((cid:15)), c = O(1), if δm2 > 0 , φ φ 31 (A.20) s = O(1) , c = O((cid:15)) , if δm2 < 0 . φ φ 31 (cid:48)(cid:48)(cid:48)   Thus, looking at the elements of H in that range, we ﬁnd: 0    −  O((cid:15) β/1) O((cid:15)4/(cid:15)3) 0 (cid:48)(cid:48)(cid:48) | | H = δm2 O((cid:15)4/(cid:15)3) O((cid:15)2) O((cid:15)3/(cid:15)4) , (A.21) 0 31 − 0 O((cid:15)3/(cid:15)4) O(1/(cid:15) β) – 50 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) − = R (θ , 0)R (θ , δ)R (θ , 0) Q R (ϕ, 0) R (φ, 0) R (χ, δ) 23 23 13 13 12 12 3 12 13 23 ∗ U V W X − (cid:124) (cid:123)(cid:122) (cid:125) = R (θ , 0)Q R (θ , 0)R (θ , 0)R (ϕ, 0)R (φ, 0)R (χ, δ) 23 23 3 13 13 12 12 12 13 2−3 = R (θ , 0)Q R (θ , 0)R (θ + ϕ, 0)R (φ, 0)R (χ, δ) 23 23 3 13 13 12 12 13 23 (cid:48) = θ (cid:48) 12 − = R (θ , 0)Q R (θ , 0)R (θ , 0)R (φ, 0)R (χ, δ) , (A.41) 23 23 3 13 13 12 12 13 23 where in the last and penultimate lines we have combined the two 12-rotations into one. We now commute R (φ, 0)R (χ, δ) through the other mixing matrices to the 13 23 left as follows: (cid:48) – Step 1: Commutation of R (φ, 0) through R (θ , 0). 13 12 12 (cid:38) (cid:48) In the range β 0, the angle θ is approximately equal to zero, so we can 12   approximate     1 0 0 (cid:48) ≈ R (θ , 0) R (0, 0) = 0 1 0 . (A.42) 12 12 12 0 0 1 Note that R (0, 0)R (φ, 0) = R (φ, 0)R (0, 0) (A.43) 12 13 13 12 (cid:46) − for any φ. On the other hand, in the range β 1 the angle φ is negligibly   small, so we can approximate     1 0 0 ≈ R (φ, 0) R (0, 0) = 0 1 0 . (A.44) 13 13 0 0 1 Note that (cid:48) (cid:48) R (θ , 0)R (0, 0) = R (0, 0)R (θ , 0) (A.45) 12 12 13 13 12 12 (cid:48) for any θ . Therefore, for all β we have 12 (cid:48) ≈ (cid:48) R (θ , 0)R (φ, 0) R (φ, 0)R (θ , 0) , (A.46) 12 12 13 13 12 12 and – 58 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

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text: β

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text: that

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text: φ.

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text: the

text: other

text: hand,

text: in

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text: range

text: β

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text: angle

text: φ

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text: negligibly

text: small,

text: so

text: we

text: can

text: approximate

text: R

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text: R

text: (θ

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text: any

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text: for

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text: R

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text: A

text: Theory

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text: Oscillations

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text: OPAL

text: Collaboration,

text: K.

text: Ackerstaﬀ

text: et

text: al.,

text: Search√for

text: anomalous

text: production

text: of

text: photonic

text: events

text: with

text: missing

text: energy

text: in

text: e+e

text: collisions

text: at

text: s

text: to

text: Eur.Phys.J.

text: OPALx

text: Collaboration,

text: G.

text: Abbiendi

text: √et

text: al.,

text: Search

text: for

text: anomalous

text: photonic

text: events

text: with

text: missing

text: energy

text: in

text: e+e

text: collisions

text: at

text: s

text: and

text: Eur.Phys.J.

text: OPAL

text: Colla√boration,

text: G.

text: Abbiendi

text: et

text: al.,

text: Photonic

text: events

text: with

text: missing

text: energy

text: in

text: e+e

text: collisions

text: at

text: s

text: Eur.Phys.J.

text: DELPHI

text: Co√llaboration,

text: J.

text: Abdallah

text: et

text: al.,

text: Photon

text: events

text: with

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text: energy

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text: collisions

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text: Collaboration,

text: A.

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text: Oscillations

text: in

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text: Nucl.Phys.Proc.Suppl.

text: IceCube

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text: The

text: Design

text: and

text: Performance

text: of

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text: DeepCore,

text: Astropart.Phys.

text: KamLAND

text: Collaboration,

text: S.

text: Abe

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text: Measurement

text: of

text: Neutrino

text: Oscillation

text: Parameters

text: with

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text: Collaboration,

text: B.

text: Aharmim

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text: al.,

text: An

text: Independent

text: Measurement

text: of

text: the

text: Total

text: Active

text: Solar

text: Neutrino

text: Flux

text: Using

text: an

text: Array

text: of

text: Proportional

text: Counters

text: at

text: the

text: Sudbury

text: Neutrino

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text: C.

text: Arpesella

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text: Neutrino

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text: Bonﬁni,

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text: of

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text: solar

text: neutrino

text: interaction

text: rate

text: in

text: Borexino,

text: Phys.Rev.Lett.

text: Z.

text: Berezhiani

text: and

text: A.

text: Rossi,

text: Limits

text: on

text: the

text: nonstandard

text: interactions

text: of

text: neutrinos

text: from

text: e+e

text: colliders,

text: Phys.Lett.

text: Z.

text: Berezhiani,

text: R.

text: Raghavan,

text: and

text: A.

text: Rossi,

text: Probing

text: nonstandard

text: couplings

text: of

text: neutrinos

text: at

text: the

text: Borexino

text: detector,

text: Nucl.Phys.

text: M.

text: Hirsch,

text: E.

text: Nardi,

text: and

text: D.

text: Restrepo,

text: Bounds

text: on

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text: vector

text: and

text: axial

text: vector

text: charge

text: radius,

text: Phys.Rev.

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text: C.

text: Pena-Garay,

text: N.

text: Rius,

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text: Santamaria,

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text: and

text: future

text: bounds

text: on

text: nonstandard

text: neutrino

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text: JHEP

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text: Maltoni,

text: Standard

text: and

text: nonstandard

text: physics

text: in

text: neutrino

text: oscillations,

text: Nucl.Phys.Proc.Suppl.

text: A.

text: Friedland,

text: C.

text: Lunardini,

text: and

text: C.

text: Pena-Garay,

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text: probes

text: of

text: neutrino

text: matter

text: interactions,

text: Phys.Lett.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: this group of terms has a diﬀerent product of hypercharges from that of (11), but it still has a single common product among the three terms, so we can again factor it out, leaving us with yet another vanishing contribution by the Fierz argument. Hence, the entire Bino dressing contribution to the K+ν¯ mode also vanishes under the universal mass assumption. Wino Dressing (cid:102) (cid:102) As the ﬂavor-diagonal restrictions of the gluino and Bino also apply to the W 0 but ± not to the W , they must be considered separately. That said, one additional restriction applicable in both cases is the ability to interact with only left-handed particles; thus (cid:102) there will be no contribution here from the R-type operators. Neutral Wino. As noted, dressing with the W 0 is also restricted to UDSN con- tributions to the K+ν¯ mode. The terms to be dressed are the same as those in the Bino case, given by expressions (9) and (12); however, in applying the dressing, we ﬁnd a kink in the previous argument: (cid:16) (cid:17) (cid:102) (cid:16) (cid:17) −W→0 κ (cid:15) α2 {T 3T 3(uaν )(dbsc) + T 3T 3(dbν )(uasc) + T 3T 3(scν )(uadb)} abc 4π d s l u s l u d l κ (cid:15) α = abc 2 {(uaν )(dbsc) − (dbν )(uasc) − (scν )(uadb)}, (14) l l l 4 4π (cid:16) (cid:17) (cid:102) (cid:16) (cid:17) −W→0 κ (cid:15) α2 {T 3T 3(dbsc)(uaν ) + T 3T 3(uasc)(dbν ) + T 3T 3(uadb)(scν )} abc 4π u ν l d ν l s ν l κ (cid:15) α = abc 2 {(dbsc)(uaν ) − (uasc)(dbν ) − (uadb)(scν )}; (15) l l l 4 4π the negative weak isospins carried by the down-type ﬁelds prevent us from using the Fierz identity argument. Thus it seems we have ﬁnally found a non-vanishing contribution to proton decay, albeit to only this one mode. There is something yet to be gained from the Fierz identity in this case: the same zero sum we have seen in the previous cases tells us that in each expression here, the sum of the two negative terms is equal to the ﬁrst term; furthermore, note that the ﬁnal 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: this

text: group

text: of

text: terms

text: has

text: a

text: diﬀerent

text: product

text: of

text: hypercharges

text: from

text: that

text: of

text: but

text: it

text: still

text: has

text: a

text: single

text: common

text: product

text: among

text: the

text: three

text: terms,

text: so

text: we

text: can

text: again

text: factor

text: it

text: out,

text: leaving

text: us

text: with

text: yet

text: another

text: vanishing

text: contribution

text: by

text: the

text: Fierz

text: argument.

text: Hence,

text: the

text: entire

text: Bino

text: dressing

text: contribution

text: to

text: the

text: K+ν¯

text: mode

text: also

text: vanishes

text: under

text: the

text: universal

text: mass

text: assumption.

text: Wino

label: section

text: Dressing

text: As

text: the

text: ﬂavor-diagonal

text: restrictions

text: of

text: the

text: gluino

text: and

text: Bino

text: also

text: apply

text: to

text: the

text: W

text: but

text: not

text: to

text: the

text: W

text: they

text: must

text: be

text: considered

text: separately.

text: That

text: said,

text: one

text: additional

text: restriction

text: applicable

text: in

text: both

text: cases

text: is

text: the

text: ability

text: to

text: interact

text: with

text: only

text: left-handed

text: particles;

text: thus

text: there

text: will

text: be

text: no

text: contribution

text: here

text: from

text: the

text: R-type

text: operators.

text: Neutral

text: Wino.

text: As

text: noted,

text: dressing

text: with

text: the

text: W

text: is

text: also

text: restricted

text: to

text: UDSN

text: con-

text: tributions

text: to

text: the

text: K+ν¯

text: mode.

text: The

text: terms

text: to

text: be

text: dressed

text: are

text: the

text: same

text: as

text: those

text: in

text: the

text: Bino

text: case,

text: given

text: by

text: expressions

text: and

text: however,

text: in

text: applying

text: the

text: dressing,

text: we

text: ﬁnd

text: a

text: kink

text: in

text: the

text: previous

text: argument:

text: −W→0

text: κ

text: α2

text: {T

text: )(dbsc)

text: T

text: )(uasc)

text: T

text: )(uadb)}

text: abc

text: 4π

text: d

text: s

text: l

text: u

text: s

text: l

text: u

text: d

text: l

text: κ

text: α

text: abc

text: {(uaν

text: )(dbsc)

text: (dbν

text: )(uasc)

text: (scν

text: )(uadb)},

text: l

text: l

text: l

text: 4π

text: −W→0

text: κ

text: α2

text: {T

text: T

text: T

text: abc

text: 4π

text: u

text: ν

text: l

text: d

text: ν

text: l

text: s

text: ν

text: l

text: κ

text: α

text: abc

text: {(dbsc)(uaν

text: (uasc)(dbν

text: (uadb)(scν

text: l

text: l

text: l

text: 4π

text: the

text: negative

text: weak

text: isospins

text: carried

text: by

text: the

text: down-type

text: ﬁelds

text: prevent

text: us

text: from

text: using

text: the

text: Fierz

text: identity

text: argument.

text: Thus

text: it

text: seems

text: we

text: have

text: ﬁnally

text: found

text: a

text: non-vanishing

text: contribution

text: to

text: proton

text: decay,

text: albeit

text: to

text: only

text: this

text: one

text: mode.

text: There

text: is

text: something

text: yet

text: to

text: be

text: gained

text: from

text: the

text: Fierz

text: identity

text: in

text: this

text: case:

text: the

text: same

text: zero

text: sum

text: we

text: have

text: seen

text: in

text: the

text: previous

text: cases

text: tells

text: us

text: that

text: in

text: each

text: expression

text: here,

text: the

text: sum

text: of

text: the

text: two

text: negative

text: terms

text: is

text: equal

text: to

text: the

text: ﬁrst

text: term;

text: furthermore,

text: note

text: that

text: the

text: ﬁnal

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: expressions in (14) and (15) are actually identical. Therefore, we can collect the above contributions into one expression: (cid:16) (cid:17) (cid:102) (cid:16) (cid:17) −W→0 2 × κ (cid:15)abc α2 (−2){(uasc)(dbν ) + (uadb)(scν )} l l 4 4π α = − κ (cid:15) 2 {(uasc)(dbν ) + (uadb)(scν )}. (16) abc l l 4π Including the factors from the triplet operator, we can write an operator for the entire neutral Wino contribu(cid:16)tion(cid:17)to K+ν¯: (cid:98) (cid:102) α − O = κ (cid:15) 2 MT 1CL {(uasc)(dbν ) + (uadb)(scν )}, (17) W0 abc 4π 1[12]l l l where the sign cancels with that from the UDDN term in eq. (6). The details of κ will be discussed in the next subsection. Note I could have instead written the above expression in terms of (dbsc)(uaν ) alone; I chose the two-operator version because the l up-up- and down-down-type pairings in the latter option are forbidden in Higgsino and charged Wino modes and so are not seen in the calculation otherwise. Charged Wino. The assumption of universal mass means that the sfermions are simultaneously ﬂavor and mass eigenstates; therefore, the would-be unitary rotation ma- trix for each is simply the identity, Uf˜ ∼ 1. As a result, the unitary matrix present in the fermion-sfermion-Wino couplings is not V , but rather the single unitary matrix ckm corresponding to the fermion quark rotation. Nonetheless, this rotation allows for the mixing of generations at the dressing vertices, and the limitations found on the neutral current dressings are not applicable. This is quite crucial since it allows for contributions from diagrams with any squark propagator not forbidden by the anti-symmetry of the CL operator. Proton decay modes involving neutral Kaons or pions, which have uu¯ or ijkl ¯ dd as external quarks, would be intractable without generation mixing. Such mixing will of course come at the expense of suppression from an oﬀ-diagonal element in the perti- − nent unitary matrix, which will typically be O(10 2-3); hence, one can begin to see an indication of why the K+ν¯ mode is so dominant in the full proton decay width. 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: expressions

text: in

text: and

text: are

text: actually

text: identical.

text: Therefore,

text: we

text: can

text: collect

text: the

text: above

text: contributions

text: into

text: one

text: expression:

text: −W→0

text: κ

text: α2

text: (uadb)(scν

text: l

text: l

text: 4π

text: α

text: κ

text: {(uasc)(dbν

text: (uadb)(scν

text: abc

text: l

text: l

text: 4π

text: Including

text: the

text: factors

text: from

text: the

text: triplet

text: operator,

text: we

text: can

text: write

text: an

text: operator

text: for

text: the

text: entire

text: neutral

text: Wino

text: K+ν¯:

text: α

text: O

text: κ

text: MT

text: {(uasc)(dbν

text: (uadb)(scν

text: abc

text: 4π

text: l

text: l

text: where

text: the

text: sign

text: cancels

text: with

text: that

text: from

text: the

text: UDDN

text: term

text: in

text: eq.

text: The

text: details

text: of

text: κ

text: will

text: be

text: discussed

text: in

text: the

text: next

text: subsection.

text: Note

text: I

text: could

text: have

text: instead

text: written

text: the

text: above

text: expression

text: in

text: terms

text: of

text: (dbsc)(uaν

text: alone;

text: I

text: chose

text: the

text: two-operator

text: version

text: because

text: the

text: l

text: up-up-

text: and

text: down-down-type

text: pairings

text: in

text: the

text: latter

text: option

text: are

text: forbidden

text: in

text: Higgsino

text: and

text: charged

text: Wino

text: modes

text: and

text: so

text: are

text: not

text: seen

text: in

text: the

text: calculation

text: otherwise.

text: Charged

text: Wino.

text: The

text: assumption

text: of

text: universal

text: mass

text: means

text: that

text: the

text: sfermions

text: are

text: simultaneously

text: ﬂavor

text: and

text: mass

text: eigenstates;

text: therefore,

text: the

text: would-be

text: unitary

text: rotation

text: ma-

text: trix

text: for

text: each

text: is

text: simply

text: the

text: identity,

text: Uf˜

text: As

text: a

text: result,

text: the

text: unitary

text: matrix

text: present

text: in

text: the

text: fermion-sfermion-Wino

text: couplings

text: is

text: not

text: V

text: but

text: rather

text: the

text: single

text: unitary

text: matrix

text: ckm

text: corresponding

text: to

text: the

text: fermion

text: quark

text: rotation.

text: Nonetheless,

text: this

text: rotation

text: allows

text: for

text: the

text: mixing

text: of

text: generations

text: at

text: the

text: dressing

text: vertices,

text: and

text: the

text: limitations

text: found

text: on

text: the

text: neutral

text: current

text: dressings

text: are

text: not

text: applicable.

text: This

text: is

text: quite

text: crucial

text: since

text: it

text: allows

text: for

text: contributions

text: from

text: diagrams

text: with

text: any

text: squark

text: propagator

text: not

text: forbidden

text: by

text: the

text: anti-symmetry

text: of

text: the

text: CL

text: operator.

text: Proton

text: decay

text: modes

text: involving

text: neutral

text: Kaons

text: or

text: pions,

text: which

text: have

text: uu¯

text: or

text: ijkl

text: dd

text: as

text: external

text: quarks,

text: would

text: be

text: intractable

text: without

text: generation

text: mixing.

text: Such

text: mixing

text: will

text: of

text: course

text: come

text: at

text: the

text: expense

text: of

text: suppression

text: from

text: an

text: oﬀ-diagonal

text: element

text: in

text: the

text: perti-

text: nent

text: unitary

text: matrix,

text: which

text: will

text: typically

text: be

text: hence,

text: one

text: can

text: begin

text: to

text: see

text: an

text: indication

text: of

text: why

text: the

text: K+ν¯

text: mode

text: is

text: so

text: dominant

text: in

text: the

text: full

text: proton

text: decay

text: width.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: proton of spin s. In principle β is not necessarily the same for cases where the quarks H have diﬀerent chiralities, but the values usually diﬀer only in sign, which is irrelevant when the entire factor is squared in the decay width expression. While lattice methods have advanced signiﬁcantly since the early years of SUSY GUT theory, there is still a substantial amount of uncertainty present in the calculation of both β and the matrix element factors; some groups have even obtained contradictory H results when applying the two methods in the same work [25]. Some more recent works (e.g. [26]) using more advanced statistics and larger lattices seem to be converging on trustworthy answers, but it is still normal to see results vary by factors of (1/2 - 5) for a single decay mode from one method to the next, where the values for the matrix elements themselves are O(10) × β . Thus I will simply take the admittedly favorable approach of H using (cid:104)M| (qq)q |p(s)(cid:105) ∼ β P u for all modes. H p It is not uncommon to see values as low as β = 0.003 used in other works calculating H proton decay [28], but while calculated values have indeed varied as much as (0.003 - 0.65) over the years [26], the value is now most commonly found in the range (0.006 - 0.03) [27], with a tendency to prefer β ∼ 0.015, as seen in [26]. Again, I will take a slightly H optimistic approach and use β = 0.008. H The p → M(cid:96)¯ Eﬀective Diagram and the Decay Width of the Proton. Ultimately it is a deceptively simple two-body decay that I am calculating, as shown in Figure 5. The corresponding decay width can be determined by the usual phase-space integral expression: (cid:90) (cid:90) (cid:88) 1 d3p d3p 1 Γ = (2π)4 δ4(p − p − p ) |M |2 (25) 2M (2π)3 2E (2π)3 2E p M (cid:96) 2 p M (cid:96) s where in this case (cid:88) (cid:0) (cid:1) (cid:88) (cid:102) 1 1 − (cid:48) |M |2 = β2 (A A )2 |O |2 + |O |2 |vT (p , s) C 1 u (p , s )|2. (26) 2 2 H L S W h˜ (cid:96) (cid:96) p p (cid:48) s s,s 25 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: proton

text: of

text: spin

text: s.

text: In

text: principle

text: β

text: is

text: not

text: necessarily

text: the

text: same

text: for

text: cases

text: where

text: the

text: quarks

text: H

text: have

text: diﬀerent

text: chiralities,

text: but

text: the

text: values

text: usually

text: diﬀer

text: only

text: in

text: sign,

text: which

text: is

text: irrelevant

text: when

text: the

text: entire

text: factor

text: is

text: squared

text: in

text: the

text: decay

text: width

text: expression.

text: While

text: lattice

text: methods

text: have

text: advanced

text: signiﬁcantly

text: since

text: the

text: early

text: years

text: of

text: SUSY

text: GUT

text: theory,

text: there

text: is

text: still

text: a

text: substantial

text: amount

text: of

text: uncertainty

text: present

text: in

text: the

text: calculation

text: of

text: both

text: β

text: and

text: the

text: matrix

text: element

text: factors;

text: some

text: groups

text: have

text: even

text: obtained

text: contradictory

text: H

text: results

text: when

text: applying

text: the

text: two

text: methods

text: in

text: the

text: same

text: work

text: Some

text: more

text: recent

text: works

text: (e.g.

text: using

text: more

text: advanced

text: statistics

text: and

text: larger

text: lattices

text: seem

text: to

text: be

text: converging

text: on

text: trustworthy

text: answers,

text: but

text: it

text: is

text: still

text: normal

text: to

text: see

text: results

text: vary

text: by

text: factors

text: of

text: for

text: a

text: single

text: decay

text: mode

text: from

text: one

text: method

text: to

text: the

text: next,

text: where

text: the

text: values

text: for

text: the

text: matrix

text: elements

text: themselves

text: are

text: β

text: Thus

text: I

text: will

text: simply

text: take

text: the

text: admittedly

text: favorable

text: approach

text: of

text: H

text: using

text: (qq)q

text: β

text: P

text: u

text: for

text: all

text: modes.

text: H

text: p

text: It

text: is

text: not

text: uncommon

text: to

text: see

text: values

text: as

text: low

text: as

text: β

text: used

text: in

text: other

text: works

text: calculating

text: H

text: proton

text: decay

text: but

text: while

text: calculated

text: values

text: have

text: indeed

text: varied

text: as

text: much

text: as

text: over

text: the

text: years

text: the

text: value

text: is

text: now

text: most

text: commonly

text: found

text: in

text: the

text: range

text: with

text: a

text: tendency

text: to

text: prefer

text: β

text: as

text: seen

text: in

text: Again,

text: I

text: will

text: take

text: a

text: slightly

text: H

text: optimistic

text: approach

text: and

text: use

text: β

text: H

text: The

text: p

text: Eﬀective

text: Diagram

text: and

text: the

text: Decay

text: Width

text: of

text: the

text: Proton.

text: Ultimately

text: it

text: is

text: a

text: deceptively

text: simple

text: two-body

text: decay

text: that

text: I

text: am

text: calculating,

text: as

text: shown

text: in

text: Figure

text: The

text: corresponding

text: decay

text: width

text: can

text: be

text: determined

text: by

text: the

text: usual

text: phase-space

text: integral

text: expression:

text: Γ

text: p

text: p

text: |M

text: p

text: M

text: p

text: M

text: s

text: where

text: in

text: this

text: case

text: |M

text: β2

text: (A

text: A

text: |O

text: |O

text: |vT

text: (p

text: s)

text: C

text: u

text: (p

text: s

text: H

text: L

text: S

text: W

text: h˜

text: p

text: p

text: s

text: s,s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: so that these coeﬃcients contain the Uf/2 or yf factors as well as the C of the ijkl A C (cid:18) (cid:19) (cid:18) (cid:19) operators in (20)-(22). Now I can easily translate an operator expression like (cid:102) (cid:102) (cid:102) (cid:102) iα 1 M O (K+ν¯) (cid:39) 2 W {C I + C IV } (30) W 4π MT m2 W W q˜ into a partial decay width statement(cid:18), (cid:19) (cid:16) (cid:17) (cid:102) (cid:102) 1 α 2 1 M 2 (cid:102) (cid:102) Γ (p → K+ν¯) (cid:39) 2 W β2 (A A )2 p | CI + CIV |2, (31) W 4π 4π MT2 m2 H L S W W q˜ without losing either information or readability. Note though there is still a “black- A A C box” nature to the C (it was there in the operators as well), in that without specifying the generation indices of the external d (cid:48) quarks, the sums in eqs. (30) and (31) could j,i just as easily apply to π+ν¯. Furthermore, there are at least several channels present in A C each operator that contribute to any one mode, which are determined uniquely by the generations of the internal sfermions in addition to those of the external quarks.7 If the reader wishes to examine the decay widths at the full level of detail, he or she should utilize these expressions along with the operators in eqs. (20)-(22) and the diagrams in Appendix . All remaining limitations aside, I can now present relatively compact and intelligible expressions for the Wino- and Higgsino-dressed partial decay widths of the proton for generic mode p → M(cid:96)¯: (cid:18) (cid:19) (cid:16) (cid:17) (cid:12) (cid:88) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:102) Γ(cid:102)(p → M(cid:96)¯) (cid:39) 1(cid:18) α2 (cid:19)2 1 (cid:18)MW (cid:19)2 β2 (A A )2 p (cid:12) (cid:88) CA(cid:102)(cid:12) 2 (32) W 4π 4π MT2 m2 H L S (cid:12) W(cid:12) q˜ (cid:12)A∈M(cid:96)¯ (cid:12) 1 1 2 1 µ 2 A 2 Γ (p → M(cid:96)¯) (cid:39) β2 (A A )2 p C . (33) h˜ 4π 16π2 MT2 m2 H L S h˜ q˜ A∈M(cid:96)¯ A 7 Indeed I could have deﬁned the coeﬃcients with six indices: C , thereby creating a means of ijklmn alleviating all degeneracy, but I don’t expect such information-dense objects to be so enlightening to readers, especially since for most modes, at least the Higgsino-dressed expression would devolve into an entire pageful of terms corresponding to the individual channels. 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: so

text: that

text: these

text: coeﬃcients

text: contain

text: the

text: or

text: yf

text: factors

text: as

text: well

text: as

text: the

text: C

text: of

text: the

text: ijkl

text: A

text: C

text: operators

text: in

text: Now

text: I

text: can

text: easily

text: translate

text: an

text: operator

text: expression

text: like

text: iα

text: M

text: O

text: (K+ν¯)

text: W

text: {C

text: I

text: C

text: IV

text: W

text: 4π

text: MT

text: W

text: W

text: q˜

text: into

text: a

text: partial

text: decay

text: width

text: α

text: M

text: Γ

text: (p

text: K+ν¯)

text: W

text: β2

text: (A

text: A

text: p

text: CI

text: CIV

text: W

text: 4π

text: 4π

text: H

text: L

text: S

text: W

text: W

text: q˜

text: without

text: losing

text: either

text: information

text: or

text: readability.

text: Note

text: though

text: there

text: is

text: still

text: a

text: “black-

text: A

text: A

text: C

text: box”

text: nature

text: to

text: the

text: C

text: (it

text: was

text: there

text: in

text: the

text: operators

text: as

text: well),

text: in

text: that

text: without

text: specifying

text: the

text: generation

text: indices

text: of

text: the

text: external

text: d

text: quarks,

text: the

text: sums

text: in

text: eqs.

text: and

text: could

text: j,i

text: just

text: as

text: easily

text: apply

text: to

text: π+ν¯.

text: Furthermore,

text: there

text: are

text: at

text: least

text: several

text: channels

text: present

text: in

text: A

text: C

text: each

text: operator

text: that

text: contribute

text: to

text: any

text: one

text: mode,

text: which

text: are

text: determined

text: uniquely

text: by

text: the

text: generations

text: of

text: the

text: internal

text: sfermions

text: in

text: addition

text: to

text: those

text: of

text: the

text: external

text: If

text: the

text: reader

text: wishes

text: to

text: examine

text: the

text: decay

text: widths

text: at

text: the

text: full

text: level

text: of

text: detail,

text: he

text: or

text: she

text: should

text: utilize

text: these

text: expressions

text: along

text: with

text: the

text: operators

text: in

text: eqs.

text: and

text: the

text: diagrams

text: in

text: Appendix

text: All

text: remaining

text: limitations

text: aside,

text: I

text: can

text: now

text: present

text: relatively

text: compact

text: and

text: intelligible

text: expressions

text: for

text: the

text: Wino-

text: and

text: Higgsino-dressed

text: partial

text: decay

text: widths

text: of

text: the

text: proton

text: for

text: generic

text: mode

text: p

text: α2

text: β2

text: (A

text: A

text: p

text: W

text: 4π

text: 4π

text: H

text: L

text: S

text: q˜

text: µ

text: A

text: Γ

text: (p

text: β2

text: (A

text: A

text: p

text: C

text: h˜

text: 4π

text: H

text: L

text: S

text: h˜

text: q˜

text: A

text: Indeed

text: I

text: could

text: have

text: deﬁned

text: the

text: coeﬃcients

text: with

text: six

text: indices:

text: C

text: thereby

text: creating

text: a

text: means

text: of

text: ijklmn

text: alleviating

text: all

text: degeneracy,

text: but

text: I

text: don’t

text: expect

text: such

text: information-dense

text: objects

text: to

text: be

text: so

text: enlightening

text: to

text: readers,

text: especially

text: since

text: for

text: most

text: modes,

text: at

text: least

text: the

text: Higgsino-dressed

text: expression

text: would

text: devolve

text: into

text: an

text: entire

text: pageful

text: of

text: terms

text: corresponding

text: to

text: the

text: individual

text: channels.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: a stable numerical ﬁt to all fermion mass and mixing parameters, including the neutrino sector, where values are predictions of the model, and (b) then ﬁnding adequately large lifetimes for the dominant modes of proton decay using the Yukawa couplings ﬁxed by the fermion ﬁt. I grounded the analysis in conservative assumptions, including large tan β, and a comprehensive calculation relying on as few approximations as necessary. The pertinent modes of proton decay I checked for suﬃcient partial lifetimes are p → K+ν¯, K0(cid:96)+, π+ν¯, and π0(cid:96)+, where (cid:96) = e, µ. I will present solutions for both type-I and II seesaw neutrino masses. The results not only give satisfactory predictions for the neutrino sector based on corresponding charged sector ﬁts, but also adequately predict suﬃciently long-lived pro- tons without relying on the usual large degree of cancellation. Furthermore, I ﬁnd that the ansatz is completely successful in satisfying the proton lifetime limits without any − need for tuning in the type-I seesaw scenario, while a modest O(10 1) cancellation is still needed in the type-II case to satisfy the partial lifetime limit of the often-problematic p → K+ν¯ mode. This combination of type-I and II results is precisely contrary to the tentative expectations of the authors in [18]; the discrepancy is due mainly to the unex- pected signiﬁcance of the eﬀect of rotation to mass basis on the results of the decay width calculations, combined with the numerical details of the rotation matrices arising from the charged sector mass and CKM ﬁt. The paper is organized as follows: in section , I give an overview of the SO(10) superpotential and the fermion mass matrices following from it, followed by the details of the Yukawa texture ansatz; in section , I expand further on the model speciﬁcs and examine general GUT proton-decay logistics in order to derive the needed partial decay widths; in section , I present the fermion sector results of the numerical ﬁtting to the measured masses and mixings; in section , I present the results of the calculation of the important partial lifetimes of the proton; and in section , I discuss the implications of the results and give my conclusions. II. DETAILS OF THE MODEL As mentioned in the introduction, the SO(10) model in question has 10-, 126-, and 4 ##LTLine##

text: a

text: stable

text: numerical

text: ﬁt

text: to

text: all

text: fermion

text: mass

text: and

text: mixing

text: parameters,

text: including

text: the

text: neutrino

text: sector,

text: where

text: values

text: are

text: predictions

text: of

text: the

text: model,

text: and

text: (b)

text: then

text: ﬁnding

text: adequately

text: large

text: lifetimes

text: for

text: the

text: dominant

text: modes

text: of

text: proton

text: decay

text: using

text: the

text: Yukawa

text: couplings

text: ﬁxed

text: by

text: the

text: fermion

text: ﬁt.

text: I

text: grounded

text: the

text: analysis

text: in

text: conservative

text: assumptions,

text: including

text: large

text: tan

text: β,

text: and

text: a

text: comprehensive

text: calculation

text: relying

text: on

text: as

text: few

text: approximations

text: as

text: necessary.

text: The

text: pertinent

text: modes

text: of

text: proton

text: decay

text: I

text: checked

text: for

text: suﬃcient

text: partial

text: lifetimes

text: are

text: p

text: K+ν¯,

text: π+ν¯,

text: and

text: where

text: e,

text: µ.

text: I

text: will

text: present

text: solutions

text: for

text: both

text: type-I

text: and

text: II

text: seesaw

text: neutrino

text: masses.

text: The

text: results

text: not

text: only

text: give

text: satisfactory

text: predictions

text: for

text: the

text: neutrino

text: sector

text: based

text: on

text: corresponding

text: charged

text: sector

text: ﬁts,

text: but

text: also

text: adequately

text: predict

text: suﬃciently

text: long-lived

text: pro-

text: tons

text: without

text: relying

text: on

text: the

text: usual

text: large

text: degree

text: of

text: cancellation.

text: Furthermore,

text: I

text: ﬁnd

text: that

text: the

text: ansatz

text: is

text: completely

text: successful

text: in

text: satisfying

text: the

text: proton

text: lifetime

text: limits

text: without

text: any

text: need

text: for

text: tuning

text: in

text: the

text: type-I

text: seesaw

text: scenario,

text: while

text: a

text: modest

text: cancellation

text: is

text: still

text: needed

text: in

text: the

text: type-II

text: case

text: to

text: satisfy

text: the

text: partial

text: lifetime

text: limit

text: of

text: the

text: often-problematic

text: p

text: K+ν¯

text: mode.

text: This

text: combination

text: of

text: type-I

text: and

text: II

text: results

text: is

text: precisely

text: contrary

text: to

text: the

text: tentative

text: expectations

text: of

text: the

text: authors

text: in

text: the

text: discrepancy

text: is

text: due

text: mainly

text: to

text: the

text: unex-

text: pected

text: signiﬁcance

text: of

text: the

text: eﬀect

text: of

text: rotation

text: to

text: mass

text: basis

text: on

text: the

text: results

text: of

text: the

text: decay

text: width

text: calculations,

text: combined

text: with

text: the

text: numerical

text: details

text: of

text: the

text: rotation

text: matrices

text: arising

text: from

text: the

text: charged

text: sector

text: mass

text: and

text: CKM

text: ﬁt.

text: The

text: paper

text: is

text: organized

text: as

text: follows:

text: in

text: section

text: I

text: give

text: an

text: overview

text: of

text: the

text: superpotential

text: and

text: the

text: fermion

text: mass

text: matrices

text: following

text: from

text: it,

text: followed

text: by

text: the

text: details

text: of

text: the

text: Yukawa

text: texture

text: ansatz;

text: in

text: section

text: I

text: expand

text: further

text: on

text: the

text: model

text: speciﬁcs

text: and

text: examine

text: general

text: GUT

text: proton-decay

text: logistics

text: in

text: order

text: to

text: derive

text: the

text: needed

text: partial

text: decay

text: widths;

text: in

text: section

text: I

text: present

text: the

text: fermion

text: sector

text: results

text: of

text: the

text: numerical

text: ﬁtting

text: to

text: the

text: measured

text: masses

text: and

text: mixings;

text: in

text: section

text: I

text: present

text: the

text: results

text: of

text: the

text: calculation

text: of

text: the

text: important

text: partial

text: lifetimes

text: of

text: the

text: proton;

text: and

text: in

text: section

text: I

text: discuss

text: the

text: implications

text: of

text: the

text: results

text: and

text: give

text: my

text: conclusions.

text: II.

text: DETAILS

text: OF

text: THE

text: MODEL

text: As

text: mentioned

text: in

text: the

text: introduction,

text: the

text: model

text: in

text: question

text: has

text: and

text: ##LTLine##