text: adjusting those values in each trial. The minimization was again performed using the Minuit tool library. The search based on fully random initial values was unsuccessful, in that the K+ν¯ mode lifetime consistently fell in the 1031-32 year-range for the type-II solution and was typically ∼1×1033 years for the type-I case;8 at the same time however all ﬁve other modes in question were usually near or above their respective limits for those same arbitrary mixing values. Hence it was clear that, even with the K+ν¯ mode failure, that the ansatz was having the desired eﬀect to some extent. Also, note that this type-I solution for K+ν¯ was short of the limit by only about a factor of ﬁve. This is surprising since the type-I-based Yukawas reported in eq. (38) fell short of meeting the ansatz criteria. Given the diﬀering behaviors of the two solutions, I will report the remaining details in separate subsections once again. Proton Partial Lifetimes for Type II Seesaw To further explore the properties of the “default behavior” of the lifetime values in the model, I considered the case in which x ∼ 1 and all other x and y are set to 0 i i zero; one can see this case as deﬁning a baseline for the partial lifetimes, in that any x 0 terms in the Cs not suppressed by the Yukawa textures are necessarily large, and whereas problematic contributions from some other x with k (cid:54)= 0 may be suppressed simply by k setting x (cid:28) 1, the x contributions can be mitigated only through cancellation. k 0 The corresponding baseline lifetimes for the dominant modes in the type-II case are given in Table VI. One can see that the K+ν¯ mode decay width must be lowered by two orders of magnitude through cancellation of x terms by the others. Since it is 0 |C|2 that appears in the decay width expressions, the needed cancellation amounts to an − A O(10 1) tuning among the C factors. Furthermore, as it would be equally unnatural to see x (cid:28) 1 for all k (cid:54)= 0, one should expect O(1) cancellations to be present anyway; k therefore, the needed “tuning” is little more than a very ordinary restriction of parameter space. In order to elucidate the signiﬁcance of the improvement created by the Yukawa 8 The Minuit tool used, Migrad, works using a local gradient-based algorithm, so that in large parameter spaces, initial values are crucial in locating global minima. 36 ##LTLine##

text: adjusting

text: those

text: values

text: in

text: each

text: trial.

text: The

text: minimization

text: was

text: again

text: performed

text: using

text: the

text: Minuit

text: tool

text: library.

text: The

text: search

text: based

text: on

text: fully

text: random

text: initial

text: values

text: was

text: unsuccessful,

text: in

text: that

text: the

text: K+ν¯

text: mode

text: lifetime

text: consistently

text: fell

text: in

text: the

text: year-range

text: for

text: the

text: type-II

text: solution

text: and

text: was

text: typically

text: years

text: for

text: the

text: type-I

text: at

text: the

text: same

text: time

text: however

text: all

text: ﬁve

text: other

text: modes

text: in

text: question

text: were

text: usually

text: near

text: or

text: above

text: their

text: respective

text: limits

text: for

text: those

text: same

text: arbitrary

text: mixing

text: values.

text: Hence

text: it

text: was

text: clear

text: that,

text: even

text: with

text: the

text: K+ν¯

text: mode

text: failure,

text: that

text: the

text: ansatz

text: was

text: having

text: the

text: desired

text: eﬀect

text: to

text: some

text: extent.

text: Also,

text: note

text: that

text: this

text: type-I

text: solution

text: for

text: K+ν¯

text: was

text: short

text: of

text: the

text: limit

text: by

text: only

text: about

text: a

text: factor

text: of

text: ﬁve.

text: This

text: is

text: surprising

text: since

text: the

text: type-I-based

text: Yukawas

text: reported

text: in

text: eq.

text: fell

text: short

text: of

text: meeting

text: the

text: ansatz

text: criteria.

text: Given

text: the

text: diﬀering

text: behaviors

text: of

text: the

text: two

text: solutions,

text: I

text: will

text: report

text: the

text: remaining

text: details

text: in

text: separate

text: subsections

text: once

text: again.

text: Proton

text: Partial

text: Lifetimes

text: for

text: Type

text: II

text: Seesaw

text: To

text: further

text: explore

text: the

text: properties

text: of

text: the

text: “default

text: behavior”

text: of

text: the

text: lifetime

text: values

text: in

text: the

text: model,

text: I

text: considered

text: the

text: case

text: in

text: which

text: x

text: and

text: all

text: other

text: x

text: and

text: y

text: are

text: set

text: to

text: i

text: i

text: zero;

text: one

text: can

text: see

text: this

text: case

text: as

text: deﬁning

text: a

text: baseline

text: for

text: the

text: partial

text: lifetimes,

text: in

text: that

text: any

text: x

text: terms

text: in

text: the

text: Cs

text: not

text: suppressed

text: by

text: the

text: Yukawa

text: textures

text: are

text: necessarily

text: large,

text: and

text: whereas

text: problematic

text: contributions

text: from

text: some

text: other

text: x

text: with

text: k

text: may

text: be

text: suppressed

text: simply

text: by

text: k

text: setting

text: x

text: the

text: x

text: contributions

text: can

text: be

text: mitigated

text: only

text: through

text: cancellation.

text: k

text: The

text: corresponding

text: baseline

text: lifetimes

text: for

text: the

text: dominant

text: modes

text: in

text: the

text: type-II

text: case

text: are

text: given

text: in

text: Table

text: VI.

text: One

text: can

text: see

text: that

text: the

text: K+ν¯

text: mode

text: decay

text: width

text: must

text: be

text: lowered

text: by

text: two

text: orders

text: of

text: magnitude

text: through

text: cancellation

text: of

text: x

text: terms

text: by

text: the

text: others.

text: Since

text: it

text: is

text: that

text: appears

text: in

text: the

text: decay

text: width

text: expressions,

text: the

text: needed

text: cancellation

text: amounts

text: to

text: an

text: A

text: tuning

text: among

text: the

text: C

text: factors.

text: Furthermore,

text: as

text: it

text: would

text: be

text: equally

text: unnatural

text: to

text: see

text: x

text: for

text: all

text: k

text: one

text: should

text: expect

text: cancellations

text: to

text: be

text: present

text: anyway;

text: k

text: therefore,

text: the

text: needed

text: “tuning”

text: is

text: little

text: more

text: than

text: a

text: very

text: ordinary

text: restriction

text: of

text: parameter

text: space.

text: In

text: order

text: to

text: elucidate

text: the

text: signiﬁcance

text: of

text: the

text: improvement

text: created

text: by

text: the

text: Yukawa

text: The

text: Minuit

text: tool

text: used,

text: Migrad,

text: works

text: using

text: a

text: local

text: gradient-based

text: algorithm,

text: so

text: that

text: in

text: large

text: parameter

text: spaces,

text: initial

text: values

text: are

text: crucial

text: in

text: locating

text: global

text: minima.

text: ##LTLine##


text:        −   0.972 0.2098 − 0.1044i −10 5 − 0.010i  Uu = −0.210 − 0.1043i 0.971 −0.00012 − 0.0414i −0.0043 − 0.001i −0.001 − 0.0423i 0.999          0.9998 0.00633 − 0.0095i 0.00765 − 0.01117i  Ud = −0.00708 − 0.0095i 0.9983 0.03386 − 0.04514i , (40) −0.00785 − 0.01054i −0.03401 − 0.04514i 0.9983 we see that the oﬀ-diagonal entries are the same size or smaller for the type-I case in every entry except Ud ; furthermore, several of the elements involving the third generation 12 are smaller by an order of magnitude. These diﬀerences may seem rather benign, but in fact each of these slightly suppressed values individually translates into a factor of 10 suppression in most of the dominant Cs, which all tend to involve third generation A elements. In some cases two or even three such suppressions may aﬀect a single C factor. The squaring of factors in the decay width then gives suppressions of generally 2-4 orders of magnitude in the lifetimes, which is precisely what one can see when comparing Tables VI and VII. Due to the more favorable circumstances, I was able to locate an allowed region of parameter space for type-I simply by running a large number of trials with the type-II parameter seeds. I repeated the process of expanding the range of x by again choosing 0 ﬁve seeds that gave every mode suﬃcient and τ(K+ν¯) roughly twice the experimental bound, and I again used those seeds to create scatter plots for a large number of trials. Figure 8 gives the relationships between the K+ν¯ mode and other representative modes and the distribution of τ(K+ν¯) as a function of x , and Figure 9 shows the relationships 0 between other more closely related modes. Note the bifurcation of the solution set in each plot; I have not yet been able to discover the cause of this behavior. Again I performed scans to determine a statistical upper bound for the value of τ(K+ν¯) in the model. I consistently found that τ > 1037 years was rare and did not see 42

label: equation

text: Uu

text: Ud

text: we

label: paragraph

text: see

text: that

text: the

text: oﬀ-diagonal

text: entries

text: are

text: the

text: same

text: size

text: or

text: smaller

text: for

text: the

text: type-I

text: case

text: in

text: every

text: entry

text: except

text: Ud

text: furthermore,

text: several

text: of

text: the

text: elements

text: involving

text: the

text: third

text: generation

text: are

text: smaller

text: by

text: an

text: order

text: of

text: magnitude.

text: These

text: diﬀerences

text: may

text: seem

text: rather

text: benign,

text: but

text: in

text: fact

text: each

text: of

text: these

text: slightly

text: suppressed

text: values

text: individually

text: translates

text: into

text: a

text: factor

text: of

text: suppression

text: in

text: most

text: of

text: the

text: dominant

text: Cs,

text: which

text: all

text: tend

text: to

text: involve

text: third

text: generation

text: A

text: elements.

text: In

text: some

text: cases

text: two

text: or

text: even

text: three

text: such

text: suppressions

text: may

text: aﬀect

text: a

text: single

text: C

text: factor.

text: The

text: squaring

text: of

text: factors

text: in

text: the

text: decay

text: width

text: then

text: gives

text: suppressions

text: of

text: generally

text: orders

text: of

text: magnitude

text: in

text: the

text: lifetimes,

text: which

text: is

text: precisely

text: what

text: one

text: can

text: see

text: when

text: comparing

text: Tables

text: VI

text: and

text: VII.

text: Due

text: to

text: the

text: more

text: favorable

text: circumstances,

text: I

text: was

text: able

text: to

text: locate

text: an

text: allowed

text: region

text: of

text: parameter

text: space

text: for

text: type-I

text: simply

text: by

text: running

text: a

text: large

text: number

text: of

text: trials

text: with

text: the

text: type-II

text: parameter

text: seeds.

text: I

text: repeated

text: the

text: process

text: of

text: expanding

text: the

text: range

text: of

text: x

text: by

text: again

text: choosing

text: ﬁve

text: seeds

text: that

text: gave

text: every

text: mode

text: suﬃcient

text: and

text: τ(K+ν¯)

text: roughly

text: twice

text: the

text: experimental

text: bound,

text: and

text: I

text: again

text: used

text: those

text: seeds

text: to

text: create

text: scatter

text: plots

text: for

text: a

text: large

text: number

text: of

text: trials.

text: Figure

text: gives

text: the

text: relationships

text: between

text: the

text: K+ν¯

text: mode

text: and

text: other

text: representative

text: modes

text: and

text: the

text: distribution

text: of

text: τ(K+ν¯)

text: as

text: a

text: function

text: of

text: x

text: and

text: Figure

text: shows

text: the

text: relationships

text: between

text: other

text: more

text: closely

text: related

text: modes.

text: Note

text: the

text: bifurcation

text: of

text: the

text: solution

text: set

text: in

text: each

text: plot;

text: I

text: have

text: not

text: yet

text: been

text: able

text: to

text: discover

text: the

text: cause

text: of

text: this

text: behavior.

text: Again

text: I

text: performed

text: scans

text: to

text: determine

text: a

text: statistical

text: upper

text: bound

text: for

text: the

text: value

text: of

text: τ(K+ν¯)

text: in

text: the

text: model.

text: I

text: consistently

text: found

text: that

text: τ

text: years

text: was

text: rare

text: and

text: did

text: not

text: see