K π¯νν decays in the general MSSMrosiek/lectures/kpivv.pdfK→ π¯νν decays in the general...

K → πνν decays in the general

MSSM

Janusz Rosiek

Warsaw, 22 Nov 04

Based on: Buras, Ewerth, Jager, J.R., hep-ph/0408142

1. Introduction: K+ → π+νν and KL → π0νν

decays as a tool for the “new physics” searches

2. Basic formulae and diagrams

3. Supersymmetric parameters. Mass insertion

approximation (MIA) vs. “exact” calculations

4. Experimental constraints

5. “Adaptive” Monte Carlo scanning algorithm

6. Allowed ranges for the Br(K+ → π+νν) and

Br(KL → π0νν) branching ratios

7. Conclusions

J. Rosiek, “K → πνν decays in the general MSSM”

1. Introduction

K+ → π+νν and KL → π0νν decays:

• Generated at the loop-level only - very sen-

sitive to virtual “new physics” effects

• Very clean theoretical calculation in the field

of meson processes, all strong corrections

well under control. Remains clean in exten-

sions of the SM - not true for other pro-

cesses.

Example: B0B0 mixing:

SM formulae:

∆Ms

∆Md=FBsFBd

BSMBs CSMs

BSMBdCSMd

Cancellation of some hadronic uncertainities in

ratio of the hadronic matrix FBdBSMBs

/FBsBSMBd

elements.

General (e.g. MSSM) formulae:

∆Ms

∆Md=FBsFBd

∑

iBiBsCis

∑

iBiBdCid

No cancellations, Bi’s weighted with different

Wilson coefficients.


Problem: very small branching ratios for both

Br(K → πνν) decays. Expected SM values:

Br(K+ → π+νν)SM = (7.8 ± 1.2) · 10−11

Br(KL → π0νν)SM = (3.0 ± 0.6) · 10−11

SM prediction error is expected to go down to

about 5% within next few years with new B-

factories measurements.

BR’s small, but nevertheless, accessible experi-

mentally now or in near future!

I. Br(K+ → π+νν) measurements:

Currently, AGS E787 and AGS E949 give (3

events observed in total):

Br(K+ → π+νν) = (14.713.0−8.9) · 10−11

Large errors, but central value 2 times larger

than SM prediction - in agreement with some

hints from anomalies in B → πK decay (Buras

et al.)

In future: AGS E949, NA48, JPARC: 50-100

new events expected within next 5 years.


II. Br(KL → π0νν) measurements:

Currently KTeV gives:

Br(KL → π0νν)SM < 5.9 · 10−7

4 orders of magnitude above SM expectation...

Stronger indirect constraints from the so-called

Grossman-Nir bound - discussed later.

In future: new experiments KEK, E391a, KO-

PIO at Brookhaven: improvement by 3 orders

of magnitude in the first stage, few hundreds

events in the second stage!

Very important source of information about “new

physics” in the next several years!


2. Basic formulae

The effective Hamiltonian for K+ → π+νν and

KL → π0νν decays in the general MSSM:

Heff =GF√

2

α

2π sin2 θw

[

H(c)eff + H(t)

eff

]

Internal charm part is fully dominated by the SM

contributions:

H(c)eff =

∑

l=e,µ,τ

V ∗csVcdX

lNL(sd)V−A(νlνl)V−A

Second term:

H(t)eff =

∑

l=e,µ,τ

V ∗tsVtd

[

XL(sd)V−A(νlνl)V−A

+ XR(sd)V+A(νlνl)V−A]

XL receives both the SM and SUSY contribu-

tions, XR only the SUSY ones (+ charged Higgs

exchanges).

In the SM XL is real, for mt(mt) = (168.1±4.1)

GeV

XSML ≡ XSM = 1.53 ± 0.04


Strong interactions not sensitive to the sign of

γ5 - hadronic matrix elements of XL and XR are

equal. Both analysed BR’s can be parameterized

in terms of one quantity

X = XL +XR = |X|eiθX

In the SM XR always negligible. In SUSY also

|XR| ≪ |XL|, apart from some non-interesting

points where XL is small due to cancellations.

Branching ratios:

Br(K+ → π+νν) = κ+

(

Im(λtX)

λ5

)2

+

(

Reλc

λPc +

Re(λtX)

λ5

)2

Br(KL → π0νν) = κL

(

Im(λtX)

λ5

)2

with λ = 0.224, λt = V ⋆tsVtd, λc = V ⋆csVcd,

κ+ = (4.84 ± 0.06) · 10−11

κL = (2.12 ± 0.03) · 10−10

and (charm contribution)

Pc = 0.39 ± 0.07


Grossman-Nir (GN) bound.

From isopin symmetry one can derive model in-

dependent bound:

Br(KL → π0νν) < 4.4 ·Br(K+ → π+νν)

Current experimental data imply

Br(K+ → π+νν) < 3.8 · 10−10 (90% C.L.)

which gives

Br(KL → π0νν) < 1.7 · 10−9 (90% C.L.)

Still two orders of magnitude below the upper

bound from the KTeV experiment!


MSSM contributions:

Cs d

U U

Z0

ν ν

Us d

C C

Z0

ν ν

Us d

C C

Lν ν

+ neutralino diagrams (smaller)

+ gluino diagrams (penguins only, small)

+ charged Higgs diagrams (small, no depen-

dence on flavour violation in sfermion sector)


3. Supersymmetric parameters

Even assuming conserved R-parity, very numer-

ous, unfortunately!

Flavour conserving: tanβ, µ, Higgs mass, chargi-

no, neutralino, gluino, squark and slepton masses,

diagonal trilinear mixing parameters.

Flavour/CP violating: µ/gaugino mass phases,

off diagonal sfermion mass matrices entries.

Too many parameters for reasonable analysis?

Do it step-by step!

First, useful definition of so-called mass inser-

tions. Start from sfermion mass matrices in so-

called super-KM basis:

M2U

=

(M2U)LL +m2

u (M2U)LR − cβ

sβµmu

(M2U)†LR − cβ

sβµ⋆mu (M2

U)RR +m2

u

+ D − terms

M2D

=

(M2D)LL +m2

d (M2D)LR − sβ

cβµmd

(M2D)†LR − sβ

cβµ⋆md (M2

D)RR +m2

d

+ D − terms

SU(2) relation: (M2U)LL = V (M2

D)LLV

†


Useful notation:

(M2U)LL =

(m2U1)LL (∆12

U )LL (∆13U )LL

(∆21U )LL (m2

U2)LL (∆23U )LL

(∆31U )LL (∆32

U )LL (m2U3)LL

(δIJU )LR =(∆IJ

U )LR

(mUI)LL (mUJ)RRTwo methods of calculations:

1) “Mass insertion approximation” (MIA): as-

sume δ’s to be small (not always justified exper-

imentally) and treat off-diagonal terms in the

sfermion mass matrices as interactions; expand

diagrams to the first or second order in δ’s.

Advantages - simple(r) and more transparent

analytical formulae; disadvantages - not neces-

sarily accurate enough and dangerous - easy to

miss some important contributions.

2) “Exact” or “mass eigenstates” calculation:

diagonalize exactly sfermion mass matrices, use

interaction vertices with explicit diagonalization

matrices.

Advantages - accurate for general sfermion mass

structure, automatically include all possible ex-

pansion terms; disadvantages - relation with ini-

tial mass parameters hidden in the mixing an-

gles, more suitable for numerical analysis.


Several papers on K → πνν decays in the MSSM:

Buras, Romanino, Silvestrini 98

Colangelo, Isidori 98

Nir, Worah 98

Buras, Colangelo, Isidori, Romanino, Silvestrini

99

...

Calculations always in MIA scheme, various as-

sumptions and experimental bounds employed

or not employed. General result for the MSSM:

Br(K+ → π+νν) ≤ 1.7 · 10−10

Br(KL → π0νν) ≤ 1.2 · 10−10

θX ≡ arg(X) ∼ 0 (or small)

Mild enhancement over SM. Is this result re-

ally correct? We decided to check it with “mass

eigenstates” calculation and extensive numerical

analysis.


Parameter choice:

1) Chargino diagrams dominate (checked nu-

merically) - chargino and up-squark masses and

mixing parameters most important.

2) K → πνν is 2nd→1st generation transition -

in the 1st order depend on five (12) mass inser-

tions, δ12LL, δ

12DRR, δ12

URR, δ12DLR and δ12

ULR.

3) As stated in older papers, 2nd order transi-

tions depending on products of a (13) and (the

conjugate of) a (23) mass insertion are impor-

tant, actually dominate! 5 new (13) and 5 new

(23) mass insertion - 15 complex parameters!

Too many - check sensitivity to them. Only δ12LL,

δ13ULR and δ23

ULR really important.


“Minimal” set - 16 real parameters.

CKM phase −180◦ ≤ γ ≤ 180◦

CP-odd Higgs mass 150 ≤MA ≤ 400

SU(2) gaugino mass; M1 GUT-related to M2

50 ≤M2 ≤ 800

Gluino mass 195 ≤ mg ≤ 2000

Higgs mixing parameter −400 ≤ µ ≤ 400

Common flavour diagonal sleptonmass parameter

95 ≤ Msl ≤ 1000

Common mass parameter for thefirst 2 generations of squarks andbR

240 ≤ Msq ≤ 1000

Left stop and sbottom mass pa-rameter

50 ≤ MtL ≤ 1000

Right stop mass parameter 50 ≤MtR ≤ 1000

Flavour universal trilinear scalarmixing parameter (normalized tosfermion mass)

−1 ≤ A ≤ 1

Mass insertion δ12LL |δ12LL| ≤ 0.135

Mass insertion δ13ULR |δ13ULR| ≤ 1.65

Mass insertion δ23ULR |δ23ULR| ≤ 1.65


4. Experimental and theoretical bounds

Many parameters, but also many processes to

constrain them! Very nontrivial task, formulae

derived and collected over years and many pub-

lished papers.

Experimental bounds:

Quantity Measured value Exp. error

Lightest neutralino > 46.0 GeV

Second lightest neutralino > 62.4 GeV

Lightest chargino > 94.0 GeV

The two “sbottoms” > 89.0 GeV

The two “stops” > 95.7 GeV

all other squarks > 250.0 GeV

|εK| 2.280 · 10−3 0.013 · 10−3

∆MK (GeV) 3.489 · 10−15 0.008 · 10−15

∆Md (GeV) 3.31 · 10−13 0.04 · 10−13

∆Ms (GeV) > 9.5 · 10−12

Br(Bs → Xsγ) 3.28 · 10−4 +0.41−0.36 · 10−4

(sin 2β)ψKS0.736 0.049

All theoretical predictions calculated in mass eigen-

states approach, 1-loop SUSY plus best avail-

able QCD correction set.


To compare experimental and theoretical input,

we need to take into account both experimen-

tal and theoretical errors. For each low-energy

measurement Q we require:

|Qexp −Qth| ≤ 3∆Qexp + q|Qth|

or

(1 + q)|Qth| ≥ Qexp

where

q = 0.5

simulates both theoretical calculation and nu-

merical scan density errors.

Exception: q = 0 for (sin 2β)ψKSTheoretical

error here hard to estimate – conservative ap-

proach. As we checked, does not change final

results in a significant way.

Still ways for improvements and new bounds:

mh, ǫ′/ǫ...


Additional constraints: Charge and Color Break-

ing (CCB) and unbounded from below MSSM

scalar potential CCB/UFB bounds (Casas, Lleida,

Munoz).

Tree-level expressions in the form with “fudge

factor” f = 2:

|AIJu |2 ≤ f |Y Lu |2[

(m2Q)II + (m2

U)JJ +m2H2

+ |µ|2]

|AIJd |2 ≤ f |Y Ld |2[

(m2Q)II + (m2

D)JJ +m2H1

+ |µ|2]

|AIJl |2 ≤ f |Y Ll |2[

(m2L)II + (m2

E)JJ +m2H1

+ |µ|2]

|AIJu |2 ≤ f |Y Lu |2[

(m2Q)II + (m2

U)JJ + (m2L)KK

+ (m2E)KK

]

|AIJd |2 ≤ f |Y Ld |2[

(m2Q)II + (m2

D)JJ + (m2L)KK

]

|AIJl |2 ≤ f |Y Ll |2[

(m2L)II + (m2

E)JJ + (m2L)KK

]

where L = max(I, J), I 6= J, K 6= I, J.

Strict tree-level bounds correspond to f = 1.


5. Numerical analysis

1) Generate point in 16-parameter space

2) Check if it passes experimental and theoreti-

cal constraints (most of the CPU time)

3) If yes, calculate K+ → π+νν and KL → π0νν;

store the results for further analysis

Simple algorithm, but how to do it in efficient

way? Point 1) crucial!

“Grid” scanning - very inefficient! 1.6M points

gives following plot:

Scatter plot of X, Br(K+ → π+νν) and Br(KL → π0νν) distributions

for tanβ = 2 and uniform grid scan. Parameters varied in the ranges

0 ≤ γ ≤ 180◦, 200 ≤ Msq ≤ 500, 150 ≤ Msl ≤ 300, −400 ≤ µ ≤ 400,

200 ≤ M2 ≤ 600, 150 ≤ MA ≤ 300, −1 ≤ A ≤ 1, mass insertions

between 0.003 and 0.3, their phases set to 0◦, 45◦ or 90◦.


Random Monte-Carlo scan – much better, in-

creased effective sampling density!

Example: allowed range for f(x, y) = eay

1+x2for

0.2 ≤ x, y ≤ 1 and a = 1,1/15

Strong y dependence Weak y dependence

a = 1 a = 1/15

Grid scan Grid scan

Random scan Random scan


In general: N-dimensional space with M “unim-

portant” parameters - random scan gives ef-

fective scan density of “important” directions

nN/M , where n would be the grid scan density.

Problems: uniform random scan still not very ef-

ficient if interesting effects concentrate in small

“corners” of the parameter space.

Further improvement: “Adaptive” Monte Carlo

scanning algorithm.

Idea (Brein 04): use VEGAS procedure! Define:

f =

0 parameter set rejected∣

∣

∣X −XSM∣

∣

∣

nconstraints satisfied

with n = 3..8 and integrate function f over

parameter space, storing the points generated

by VEGAS routine during integration.

Result of the integration is irrelevant, but the

generated points are what we need! VEGAS works

iteratively, starts from uniform density and in

the following iteration it concentrates most of

the generated points in areas where function

f varies quickly and where it passes exp. bounds,

i.e. those most interesting for us.


Some dangers:

• “Probability density” of generated points mean-

ingless, just upper bounds important.

• VEGAS assume approximately factorizable

function f – works in practice, fortunately.

• How stable are results

–against the choice of exponent n?

–against the choice of RNG?

Tested in practice - variations affect calcu-

lation time, not the final distributions.

Very efficient procedure - eliminates semi-auto-

matically unimportant parameters, consumption

of CPU time grows only very slowly with adding

new ones, if calculated quantities are not very

sensitive to them.

One can in principle blindly start from very large

parameter set and only afterwards judge from

the generated distributions which are really im-

portant - no place for mistake!


6. Results

Sensitivity to input parameters

Distributions of X versus γ, MtR, M2 and µ for tan β = 2

in data set generated during VEGAS integration.


Box to penguin ratio:

Absolute value of the ratio of box to penguin contributions

as a function of slepton mass for tan β = 2. Only points

for which Br(K+ → π+νν) ≥ 1.5 · 10−10 are plotted.

Box diagrams assumed to be small and neglected

in older analyses!


Most interesting: X and branching ratios distri-

butions:

Distributions of X and Br(KL → π0νν), Br(K+ → π+νν)

for tan β = 2.


Few observations. Even after satisfying all other

constraints on parameters

|X| can be as large as 10 (SM value 1.53)

X phase, θX can be large (SM value 0) and

tends to be negative, which actually enhances

the branching ratios.

Br(K+ → π+νν) can be easily larger than cur-

rent experimental upper bound 3.8 · 10−10

Large Br(KL → π0νν)/(Br(K+ → π+νν) ratios

are possible, also when can (Br(K+ → π+νν) is

large.

Results stable also against the tanβ variations.

In general, large deviations from the SM are pos-

sible, in contradiction to the claims from older

papers.

Very interesting and promising process to look

for flavour violation in the MSSM! Some bounds

on the sfermion mass matrices come already

from current experimental measurement.


Example:

Parameter Example 1 Example 2

tanβ 2 20

MA 333 260

µ -375 -344

Mg 437 928

M2 181 750

Msq 308 608

MtL 138 215

MtR 279 338

Msl 105 884

A -0.289 -0.342

γ 64◦ 38◦

δ12LL (2.18 − 5.02i) · 10−5 (7.57 − 0.87i) · 10−4

δ13ULR (−1.52 + 0.75i) · 10−4 0.292 − 0.213i

δ23⋆ULR 0.001 − 0.604i 0.239 − 0.195i

|εK| 2.35 · 10−3 2.10 · 10−3

∆Md 3.15 · 10−13 2.55 · 10−13

∆Ms 1.03 · 10−11 1.19 · 10−11

Br(B → Xsγ) 3.88 · 10−4 3.93 · 10−4

Br(K+ → π+νν) 1.78 · 10−10 2.07 · 10−10

Br(KL → π0νν) 3.08 · 10−11 4.34 · 10−10

Examples of MSSM parameter points passing experimen-

tal constraints at the assumed accuracy and giving en-

hanced K+ → π+νν and KL → π0νν decay rates.


Generalization: start from (almost) completely

free low-energy parameterization of the MSSM.

Use as free parameters:

• the angle γ (real)

• CP-odd Higgs mass MA (real)

• U(1) gaugino mass M1 (complex)

• SU(2) gaugino mass M2 (complex)

• gluino mass mg (real)

• µ parameter (complex)

• diagonal left slepton mass m2L, common for all gen-

erations (real)

• diagonal right slepton mass m2R, common for all gen-

erations (real)

• 9 independent diagonal mass parameters in squarkmass matrices, 3 parameters for each of left, up-rightand down-right mass matrix (all real)

• common sfermion LR mixing parameter A (real)

• 3 independent LL mass insertions in squark mass ma-trices: δ12LL, δ

13LL, δ

32LL (all complex)

• 6 independent RR mass insertions in squark mass ma-trices: δ12DRR, δ

13DRR, δ

32DRR, δ

12URR, δ

13URR, δ

32URR (all com-

plex)

• 12 independent LR up- and down-squark mass inser-tions δ12DLR, δ

13DLR, δ

32DLR, δ

21DLR, δ

31DLR, δ

23DLR, δ

12ULR, δ

13ULR,

δ32ULR, δ21ULR, δ

31ULR, δ

23ULR (all complex)

63 free parameters, more than half of maximum

existing in R-parity conserving MSSM! Are the

results strongly affected?


No dramatic change - see figures!

Distributions of X, Br(KL → π0νν) and Br(K+ → π+νν)

for tan β = 2 in the 63-parameter scan.

CPU time increased only by about factor 2!


Byproduct of our analysis: new limits on allowed

size of mass insertions, i.e. on size of flavour

violation in sfermion sector.

Standard approach - use MIA expansion, pick up

the biggest term, compare with full experimental

result, find bound on given mass insertion.

Such procedure neglects interference with SM

contributions (which typically saturate the ex-

perimental measurement by itself!), interference

with other mass insertions, higher order terms in

expansion (often large and important etc.).

We can plot distribution of e.g. |X| vs. mass

insertions, and see how large MI are allowed on

the x-axis - this gives us a multi-process based

bound on them.

Detailed analysis - next paper, first impression -

bounds much more strict then cited previously

in the literature.


MI type tan β = 2 tan β = 10 tan β = 20

Re δ12DLL 0.01 0.01 0.01

Im δ12DLL 0.01 0.01 7.5 · 10−3

Re δ13DLL 0.1 0.2 0.15

Im δ13DLL 0.1 0.2 0.15

Re δ23DLL 0.35 0.3 0.3

Im δ23DLL 0.35 0.28 0.25

Re δ12DRR 0.03 0.02 0.01

Im δ12DRR 0.03 0.02 0.01

Re δ13DRR 0.3 0.2 0.1

Im δ13DRR 0.35 0.25 0.1

Re δ23DRR 0.9 0.7 0.5

Im δ23DRR 0.9 0.9 0.5

Re δ12URR 0.35 0.35 0.37

Im δ12URR 0.35 0.35 0.37

Re δ13URR 0.35 0.35 0.37

Im δ13URR 0.35 0.35 0.37

Re δ23URR 1 1 1

Im δ23URR 1 1 1

Bounds on mass insertions in the LL and RR squark mass

matrices. Limits on the entries with interchanged indices

are identical due to hermicity property of those 3 matrices.


MI type tan β = 2 tan β = 10 tan β = 20

Re δ12DLR 5 · 10−5 7 · 10−5 1 · 10−4

Im δ12DLR 5 · 10−5 6 · 10−5 1 · 10−4

Re δ21DLR 8 · 10−5 1.2 · 10−4 9 · 10−5

Im δ21DLR 7 · 10−5 1.2 · 10−4 9 · 10−5

Re δ13DLR 0.01 5 · 10−3 7 · 10−3

Im δ13DLR 0.01 5 · 10−3 5 · 10−3

Re δ31DLR 7 · 10−3 0.01 0.02

Im δ31DLR 7 · 10−3 0.01 0.015

Re δ23DLR 0.05 0.05 0.05

Im δ23DLR 0.01 0.02 0.02

Re δ32DLR 8 · 10−3 6 · 10−3 5 · 10−3

Im δ32DLR 8 · 10−3 7 · 10−3 4 · 10−3

Re δ12ULR 1.5 · 10−3 1.8 · 10−3 1.4 · 10−3

Im δ12ULR 1.2 · 10−3 1.5 · 10−3 1.4 · 10−3

Re δ21ULR 2.5 · 10−3 2.5 · 10−3 2.5 · 10−3

Im δ21ULR 2.5 · 10−3 2.5 · 10−3 2.5 · 10−3

Re δ13ULR 0.45 0.5 0.5

Im δ13ULR 0.45 0.5 0.5

Re δ31ULR 1 − −Im δ31ULR 1 − −Re δ23ULR 0.4 0.5 0.55

Im δ23ULR 0.4 0.5 0.55

Re δ32ULR 0.8 0.6 1

Im δ32ULR 0.6 0.5 1

Bounds on mass insertions in the LR squark matrices.


7. Conclusions

Physical:

• within the general MSSM large departures

from the SM expectations for K → πνν are

possible while satisfying all existing constraints.

• Br(K+ → π+νν) and Br(KL → π0νν) can be

both as large as few times 10−10 (SM expec-

tation is below 10−10) with Br(KL → π0νν)

often larger than Br(K+ → π+νν) and close

to its model independent upper bound.

• interesting predictions to compare with, com-

ing soon, new accurate experimental results!

Technical:

• efficient numerical exploration of the multi-

dimensional general MSSM parameter space

is possible with the use of appropriate tech-

niques

• such an analysis is reasonably predictive when

also sufficient number of appropriate exper-

imental and theoretical bounds is included

(although it may require substantial amount

of work!)


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