Figure 4.4: b < f. Figure 4.5: f < F b < f. k 1 = F b, k 2 = F + b. ( ) < F b < f k 1 = f, k 2 = F + b. where:, (see Figure 4.6).

Transcription

1 Figure 4.4: b < f ii.) f < F b < f where: k = F b, k = F + b, (see Figure 4.5). Figure 4.5: f < F b < f. k = F b, k = F + b. ( ) < F b < f k = f, k = F + b iii.) f + b where:, (see Figure 4.6). 05

2 ( ) < F b < f k = f, k = F + b Figure 4.6: f + b.. Else the integral (4.6) equals to zero for:, which as it should be, is identical to the corresponding condition in.). This simplifies to: F > f + b. The above can be summarized to six cases where Ω F is non-zero and one where it is zero: ( ) = Ω F where: 0 3 ( ) π sgn b c f $ e i & % F 0 ; F > f + b c b + c b k ' ( +k ) ( ) * c sin b k k - ( ) +,. /; a), b), c), d),e) and f ) a) f < F b < f, ( b > f ); k = F b, k = f. b) f b < F b < f, ( b > f ); k = f, k = f. c) f < F + b < f, ( b > f ); k = f, k = F + b. d) f b < F b < f, ( b < f ); k = F b, k = f. e) f < F b < f, ( b < f ); k = F b, k = F + b. f) f + b, ( ); F b < ( b + f ), F b > f ( ) < F b < f b < f k = f, k = F + b ( ) (4.8) 06

3 The amplitude, ;, is thus given by: " ", ;, Ψ, Δ Ψ, ;,,, Ω, (4.9) where we have include the subscript,, to indicate the dependence of the respective terms on the neutrino mass spectrum. The derivative terms in (4.7) can be further evaluated to be more instructive if we employ the following velocities identities. From the energy conservation equation on the source side we have: +, +, + = Δ , +, +, + = Δ + + (4.0) where represents variation in responds to variation in Δ (Δ ) while keeping the momenta associated with state constant. From (4.0), we have the identity: =, (4.) and also, = 07 =,

4 (4.) with, (4.3) and, +, +, = +, +, =, (4.4) Similarly, keeping both the momenta of and the final state ( ) at the detection end constant, while enforcing the momentum conservation, = + : = = = +,,, (4.5) where (4.) is used in the last line. Using (4.), (4.) and (4.5), the terms in (4.7) involving the derivative, are given by: =, + "#. +,,, =,, (4.6) 08

5 For the sake of notational convenience, from here on, unless otherwise stated, all momenta and velocities are evaluated with = 0, with the conservation of energymomentum being enforced (cases in contrary, should be clear in their contexts). 4.. The Spatial Wave Packet: An Illustration With the identity, (4.), together with the linear approximation of the variable, the expression (3.4) for the neutrino spatial wave function in the last chapter can be solved. This will provide an intuitive picture of the situations in which the detection amplitude,, ;, is zero, and the required constraint on the space-time interval between emission and detection, for causality of the process to be preserved. At time, = + "#., this wave function is given by: + "#.,, + "#.,,, + "#. = Δ, ; Δ "#$ Δ "#. Δ Ψ, Δ "# Ψ, + Δ "#. + "#. Δ "#$ (4.7) The integral in the last line of (4.7) can then be solved using (4.):,, + "#. "#$, + "#. +, (4.8) 09

6 From (4.8) one could see that the spatial width of the wave function is: =, (4.9) Therefore, assuming that, <, (4.8) can be written as:,, + "#. "#$ + "#. + (4.30) This wave packet once emitted ( > ) will not spread with time. This feature is a consequence of the linear approximation on functions of Δ, which is true if the assumption (3.59) is true (spatial spread of ). The latter is conceivable if we assume that is produced through interactions that occur on a much shorter time scale than weak processes (through which decays and the neutrino is produced). Mathematically, the above statement can be justified as follows: the upper bound of is the lifetime of, as required by unitarity (see (3.54) and relevant discussions). If one repeats the same analysis performed on neutrino production to the production of, a formula similar to (4.9), but with the corresponding bound by the relevant time scales which characterize the interactions that lead to the production of is obtained. Using the same approximations, the normalizing factor,, can similarly be evaluated. From (3.4): 6 Ψ Δ "#$ Δ, + 8 Ψ =, + (4.3) 0

7 Figure 4.7 and 4.8 depict the graphs of the neutrino wave function and the conditions under which it can be absorbed by the detector particle,. It is assumed that the range of the interaction is short and can be approximated by a contact interaction (which is the case for (3.7), the effective weak interaction). The spatial spread of the detector particle is also assumed to be small compared to the neutrino ( ), and as before, >,,,. Figure 4.7 shows the lower limit of + "#. at = + "#.. One could see that at = + "#. +, the gap between the neutrino and detector particle wave packets (, and ),,, will be just closed ( >,,, ) if is at position at = + "#.. Hence below this limit no wave functions overlap and therefore, no absorption (contact interaction assumption) is possible by = + "#. +. Figure 4.8 shows the upper bound of + "#. at time, = + "#.. As can be seen from the figure, since >,, beginning with an initial nonoverlapping borderline case, with the wave function of the neutrino just ahead of that of, the latter can never catch up with the former in the time interval. The results of this intuitive picture are summarized as the inequalities:, ; = 0; + "#. <,, + "#. > +, (4.3)

8 Figure 4.7: Shown is the lower limit of + "#. at = + "#., below which no overlap (hence no absorption) between, and is possible by the time = + "#. +. Figure 4.8: Shown is the upper limit of + "#. at = + "#., above which no wave functions overlap is possible within the next time interval. From the explicit calculation of, ; from (4.8) and (4.9):

9 , ; = 0; > + (4.33) > + > + " < + (4.34) applying (4.7) and (4.6) to (4.34): < +, + "#. +,,, <,, (4.35) since, >,,,, =,,, =,, (4.36) we get: + "#. <, which agrees with the first inequality of (4.3). Similarly, > +,, + "#. + > +,,,, (4.37) and we get: + "#. > +, 3

10 which agrees with the second of the inequalities (4.3). The above show that the intuitive wave packet picture of the neutrino emission and absorption process of Figure 4.7 and Figure 4.8 correspond correctly to the explicit formula (4.9). Since the relevant velocities,,, and, are all less than the speed of light by construction ( ), this picture naturally satisfy causality. For example, consider the propagation and spreading of the neutrino wave packet during < :,, "#$ = "#$,, +, +, (4.38) the wave front velocity, which should be the part of the wave traveling at the highest speed, is thus obtained by combining the speed at which the wave packet is spreading, with the speed of its center: "#$% =,, + + = (4.39) (where again, (4.36) is used) which automatically satisfies the light speed limit. Also, for >, both the neutrino wave packet and that of travel within the light speed limit and must overlap spatially for absorption to occur, leading to the preservation of causality of detection and emission. 4.. Conditions For Non-Zero Detection Amplitude. In this section, we recast the conditions for non-zero detection amplitude, ;, expressed in (4.8) in terms, and, into relations between the space-time parameters,,, "#. and the particle velocities,,, and,, 4

11 using (4.6). The corresponding amplitude parameters ( +,, see (4.8)) to various non-zero subcategories, ), ), ),), ) and ) are also expressed in terms of these parameters. Throughout all subsequent developments, we shall assumed (4.36) is true. From (4.8): ) (For, >, ) < <, < + "#. +,, <, +,, 0 < + "#. <, (4.40) This corresponds to: + = + =, + "#. (4.4) and, = + =, + "#. + +, (4.4) ) (For, >, ) < < 5

12 < + "#. < 0,,, < + "#. < 0 (4.43) This corresponds to: + = 0, = = (4.44) ) (For, >, ) <+ < 0 < + "#. +, <,, < + "#. <,, (4.45) This corresponds to: + = + =, + "#. +,, (4.46) and, = + + =, + "#. +, (4.47) ) (For, <, ) 6

13 < < < + "#. <,,, < + "#. <, (4.48) This corresponds to: + = + = + "#., (4.49) and, = + =, + "#. + +, (4.50) ) (For, <, ) 0 < + "#. <, (4.5) + = =, + "#. +,, (4.5) = =,, (4.53) ) (For, <, ) + < < 7