A Geometric Approach to CP Violation:
Applications to the MCPMFV SUSY Model
Jae Sik Lee
National Center for Theoretical Sciences, Hsinchu, TAIWAN
NCTS HEP Journal Club, November 9, 2010
∗
based on JHEP 1010:049,2010, arXiv:1006.3087 [hep-ph] and arXiv:1009.1151 [math.OC] with J. Ellis and A. Pilaftsis
♠ Preliminary
• The SUSY models such as the MSSM contain many possible sources of flavour
and CP violation in the soft SUSY-breaking sector:
30 ⊕ 33 ⊕ 46 = 109 !!!
– Gaugino mass terms: 3 ⊕ 3 = 6
1
fW
f + M1 B
eB
e + h.c.)
−Lsoft ⊃ (M3 gege + M2 W
2
– Trilinear a terms af ij ≡ hf ij · Af ij : 3 × (3 ⊕ 6 ⊕ 9) = 54
e 2 − de∗R ad QH
e 1 − ee∗R ae LH
e 1 + h.c.)
−Lsoft ⊃ (e
u∗R au QH
– Sfermion mass terms: 5 × (3 ⊕ 3 ⊕ 3) = 45
2
∗
e∗ M2 deR + ee∗ M2 eeR
e † M2 Q
e+L
e † M2 L
e+u
−Lsoft ⊃ Q
M
u
e
+
d
e
R
R
R
R
e
e
e
e
e
u
e
Q
L
d
– Higgs mass terms: 3 ⊕ 1 = 4
†
†
2
2
2
−Lsoft ⊃ MH
H
M
H
H
+
H
−
(m
2
1
H
12H1 H2 + h.c.)
2
u
d 1
♠ Preliminary
• Recently, we have suggested MCPMFV framework with the maximal set of
flavour-singlet mass scales: J. Ellis, JSL and A. Pilaftsis, Phys. Rev. D 76 (2007) 115011, [arXiv:0708.2079
[hep-ph]]
M1,2,3 ,
3⊕3
2
2
f 2Q,L,U,D,E = M
fQ,L,U,D,E
MH
,
M
13 , Au,d,e = Au,d,e 13
u,d
2
3⊕3
5
13 ⊕ 6 = 19 Parameters !
For related approaches, see,
M. Argyrou, A. B. Lahanas and V. C. Spanos, JHEP 0805 (2008) 026; [arXiv:0804.2613 [hep-ph]]
G Colangelo, E. Nikolidakis and C. Smith, Eur. Phys. J. C 59 (2009) 75; [arXiv:0807.0801 [hep-ph]]
W. Altmannshofer, A. J. Buras and P. Paradisi, Phys. Lett. B 669 (2008) 239; [arXiv:0808.0707 [hep-ph]]
L. Mercolli and C. Smith, Nucl. Phys. B 817 (2009) 1; [arXiv:0902.1949 [hep-ph]]
A. L. Kagan, G. Perez, T. Volansky and J. Zupan, Phys. Rev. D 80 (2009) 076002; [arXiv:0903.1794 [hep-ph]]
R. Zwicky and T. Fischbacher, Phys. Rev. D 80 (2009) 076009; [arXiv:0908.4182 [hep-ph]]
J. Ellis, R. N. Hodgkinson, JSL and A. Pilaftsis, JHEP 1002 (2010) 016; [arXiv:0911.3611 [hep-ph]]
♠ Preliminary
• Then, who ordered ”more” CP violation beyond the SM CKM phase?A.
D. Sakharov,
JETP Letters 5(1967)24
CP violation in the SM is too weak to explain the matter dominance of the Universe J. Cline,
arXiv:hep-ph/0609145
The matter-dominated Universe did!
♠ Preliminary
• Electric Dipole Moments (EDMs): T violation ⇒ CP violation (under CPT )
HEDM = −d E · ŝ
|dTl| < 9 × 10−25 e cm ,
|dHg | < 3.1 × 10−29 e cm ,
|dn| < 3 × 10−26 e cm
B. C. Regan, E. D. Commins, C. J. Schmidt and D. DeMille, PRL88 (2002) 071805; W. C. Griffith, M. D. Swallows,
T. H. Loftus, M. V. Romalis, B. R. Heckel and E. N. Fortson, PRL102 (2009) 101601; C. A. Baker et al., PRL97
(2006) 131801
Question: No large CP phases?
♠ Preliminary
• A scan method:
See, for example, V. D. Barger, T. Falk, T. Han, J. Jiang, T. Li and T. Plehn, Phys. Rev. D
64 (2001) 056007 [arXiv:hep-ph/0101106].
15 parameter MSSM
2
2
θµ/π
θµ/π
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
0
⋄ 15 parameter MSSM:
100
100
0
0
0
0.5
1
1.5
φ1/π
0
GeV
GeV
GeV
GeV
2
≤
≤
≤
≤
≤
≤
θµ, φ1,3 , θA d,e,u,t
µ
2M1 , M2 , M3
|A|
mẽR , mũR
tan β
≤
≤
≤
≤
≤
≤
2π
1 TeV
1 TeV
1 TeV
1 TeV
10
2
φ3/π
2
2
θµ/π
θµ/π
1.5
1.5
1
1
Open circles suffer from parameter tuning ∆X/X worse than
0.5
0.5
0
0
2
4
6
8
10
tan β
1%. Green dots correspond to configurations with a light Higgs
mh < 113 GeV.
10
-2
10
-1
tuning ∆X/X
1
♠ Preliminary
• A scan method:
See, for example, V. D. Barger, T. Falk, T. Han, J. Jiang, T. Li and T. Plehn, Phys. Rev. D
64 (2001) 056007 [arXiv:hep-ph/0101106].
23 parameter MSSM
2
2
θµ/π
θµ/π
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
0
⋄ 23 parameter MSSM:
0
0.5
1
1.5
2
2
θµ/π
θµ/π
1.5
1.5
1
1
0.5
0.5
0
2
4
6
8
10
tan β
2
φ3/π
0 GeV ≤
mq̃L ,mũR ,md˜
≤ 1 TeV
0 GeV ≤
mq̃ 3 ,mt̃
≤ 1 TeV
2 ≤
0
10
-2
10
-1
tuning ∆X/X
1
2π
1 TeV
1 TeV
1 TeV
1 TeV
θµ , θMi , θAi
µ
2M1 , M2 , M3
|Ad,e,u,t |
mℓ̃ ,mẽR
100
100
0
0
φ1/π
≤
≤
≤
≤
≤
≤
≤
≤
≤
≤
0
GeV
GeV
GeV
GeV
L
L
tan β
R
R
≤ 10
♠ Preliminary
300
250
CS
10
100
200
dEe
150
100
1
50
10 3
350
E
200
300
10
ΦA [ o ]
-1
0
0
100
200
300
0
Φ3 [ o ]
100
200
250
d
ΦA [ o ]
350
300
250
200
150
100
G
200
10
Cbd
10
300
| dn(QCD) / dEXP |
100
300
10 2
150
100
ρ=1
o
ΦA = 60
1
Φ3 [ o ]
0
C
d u,d
du,d
50
0
-1
0
10 3
100
ρ=1
10 2
d u,d
C
ΦA = 60
CS
300
0
Φ3 [ o ]
ΦA = 60
o
100
200
10
CP(’)
10
300
ΦA [ o ]
350
300
C
d u,d
250
200
10
150
E
du,d
100
1
o
C4f
1
0
200
ρ=3
10 2
10 3
dEe
50
50
-1
0
10 3
100
200
ρ=3
10 2
300
Φ3 [ o ]
ΦA = 60
o
C
d u,d
10
1
50
0
0
dTl
| dHg / dEXP |
150
10 2
350
| dHg / dEXP |
200
o
Φ3 [ o ]
250
ΦA = 60
Φ3 [ o ]
300
10 3
| dn(QCD) / dEXP |
350
J. R. Ellis, JSL and A. Pilaftsis, JHEP 0810 (2008) 049 [arXiv:0808.1819 [hep-ph]]
Φ3 [ o ]
Φ3 [ o ]
| dTl / dEXP |
• A scan method:
100
200
300
10
ΦA [ ]
o
-1
0
0
100
QCD
dn
200
300
Φ3 [ ]
o
0
100
200
300
10
ΦA [ ]
o
-1
0
100
dHg
200
300
Φ3 [ o ]
♠ Preliminary
• A scan method is like “shooting in the dark” ...
blind,
time consuming,
no guiding principle, etc
Any analytic, exact, and more effective method?
⇒
A Geometric Approach
to CP violation
♠ Optimal Direction
• A linear approximation: We consider the case with
N CP-violating phases
In the N -dimensional CP-phase space, we define
N −D phase vector Φ = (Φ1, Φ2, · · ·, ΦN )
and then any CP-odd observable O and any EDM E can be expanded as
O = Φ · O + ··· ;
E = Φ · E + ···
Formally, we define
O ≡ ∇O ;
with ∇ ≡ (∂/∂φ1, ∂/∂φ2, · · · , ∂/∂φN )
E ≡ ∇E
♠ Optimal Direction
• [Simple 3D example] with 3 CP phases and 1 EDM constraints: EDM-free subspace
and Optimal direction in the linear approximation
EDM vector
An Observable vector
∗
Φ = E × (O × E)
An Optimal direction
CP-phase space
Subspace of EDM-free directions
♠ Optimal Direction
•
The higher-dimensional generalization with
N
The
CP phases and
n EDM constraints
N -dimensional vector of the optimal direction
∗
Φ
α
(1)
= ǫαβ1···βnγ1···γ
N −n−1
(n)
E β1 · · · E βn B γ1···γN −n−1
where (N -n-1)-dimensional B form is
B γ1···γN −n−1 = ǫγ1···γ
(1)
N −n−1
The maximum allowed value of O is:
(n)
O E · · · E βn
σβ1 ···βn σ β1
b∗ · O
Omax = (φ∗)max Φ
with the
b ∗ and φ∗ which may be practically determined by the validity of the small-phase
normalized optimal-direction vector Φ
approximation of the EDMs.
♠ Optimal Direction
• How good is the linear approximation?
The quadratic fit to the Thallium EDM that is used
to obtain the 6D vector E dTl ≡ ∇(dTl /dEXP
eα = 0◦ for the MCPMFV scenario:
Tl ) in an expansion around ϕ
˛
˛
f2 = (100 GeV)2 , |Au | =
f2 = M
f2 = M
f2 = M
f2 = M
˛M1,2,3 ˛ = 250 GeV , M 2 = M 2 = M
E
L
D
U
Q
H
Hu
d
0
-100
500
250
-250
-0.2
10
o
Φ2 [ ]
-10
0.2
0
0
10
Φ3 [ ]
o
EXP
0
dTl / dTl
EXP
0
-0.4
-10
dTl / dTl
0.01
0.2
0
10
o
Φ1 [ ]
EXP
0
0.4
0
-500
-10
dTl / dTl
EXP
100
dTl / dTl
EXP
dTl / dTl
dTl / dTl
EXP
|Ad | = |Ae | = 100 GeV , and tan β = 40.
1
0.5
0
-0.5
-0.01
-0.2
-10
0
10
ΦAu [ ]
o
-1
-10
0
10
ΦAd [ ]
o
-10
0
10
ΦAe [ ]
o
♠ Numerical Illustration
• We are considering J. Ellis, JSL, and A. Pilaftsis, JHEP 0810:049,2008, arXiv:0808.1819 [hep-ph] ; K. Cheung,
O. C. W. Kong, and JSL, JHEP 0906:020,2009, arXiv:0904.4352 [hep-ph]; J. Ellis, JSL and A. Pilaftsis, Phys. Rev. D
76 (2007) 115011, [arXiv:0708.2079 [hep-ph]]
– Thallium EDM
– Neutron EDM
– Mercury EDM
– Deuteron EDM
– muon EDM
– ACP(b → sγ)
These are all implemented in CPsuperH2.2
♠ Numerical Illustration
• Thallium EDM;I.B. Khriplovich and S.K. Lamoreaux, CP Violation Without Strangeness (Springer, New York,
1997); M. Pospelov and A. Ritz, Annals Phys. 318 (2005) 119,[arXiv:hep-ph/0504231]
−19
dTl [e cm] = −585 · dE
[e cm] · (CS TeV2) + · · ·
e [e cm] − 8.5 × 10
dE
e
CS
=
=
E χ̃±
E χ̃0
(de ) + (de ) +
(CS )4f + (CS )g
H
(dE
e)
MeV
MeV
where (CS )4f = Cde 29m
+ Cse κ×220
with κ ≡ hN |mss̄s|N i/220 MeV ≃
ms
d
0.50 ± 0.25 and
3
S
P
X
g
g
m
H
gg
H
e
i
i ēe
(CS )g = (0.1 GeV) 2
2
v i=1
MH
i
♠ Numerical Illustration
• Neutron EDM [QCD sum rule techniques (QCD)];M. Pospelov and A. Ritz, Phys. Rev. Lett. 83
(1999) 2526, [arXiv:hep-ph/9904483]; M. Pospelov and A. Ritz, Nucl. Phys. B 573 (2000) 177, [arXiv:hep-ph/9908508];
M. Pospelov and A. Ritz, Phys. Rev. D 63 (2001) 073015, [arXiv:hep-ph/0010037]; D. A. Demir, M. Pospelov and
A. Ritz, Phys. Rev. D 67 (2003) 015007,[arXiv:hep-ph/0208257]; D. A. Demir, O. Lebedev, K. A. Olive, M. Pospelov
and A. Ritz, Nucl. Phys. B 680 (2004) 339, [arXiv:hep-ph/0311314]; K. A. Olive, M. Pospelov, A. Ritz and Y. Santoso,
Phys. Rev. D 72 (2005) 075001, [arXiv:hep-ph/0506106]
C
G
dn = dn(dE
q , dq ) + dn (d ) + dn (Cbd ) + · · · ,
C
E
E
C
C
dn(dE
,
d
)
=
(1.4
±
0.6)
(d
−
0.25
d
)
+
(1.1
±
0.5)
e
(d
+
0.5
d
q
q
d
u
d
u )/gs ,
dn(d G) ∼ ± e (20 ± 10) MeV d G ,
Cdb
Cbd
+ 0.75
dn(Cbd) ∼ ± e 2.6 × 10−3 GeV2
mb
mb
where dG = dG(1 GeV) ≃ 8.5 dG(EW)
♠ Numerical Illustration
• Mercury EDM;D.
A. Demir, O. Lebedev, K. A. Olive, M. Pospelov and A. Ritz, Nucl. Phys. B 680 (2004)
339, [arXiv:hep-ph/0311314]; K. A. Olive, M. Pospelov, A. Ritz and Y. Santoso, Phys. Rev. D 72 (2005) 075001,
[arXiv:hep-ph/0506106]
C
−2 E
dHg = 7 × 10−3 e (dC
de
u − dd )/gs + 10
Csd
Cbd
2 0.5Cdd
−5
+ 3.3κ
+ (1 − 0.25κ)
− 1.4 × 10 e GeV
md
ms
mb
+ (3.5 × 10−3 GeV) e CS
"
−4
+ (4 × 10
GeV) e CP +
Z −N
A
CP′
Hg
#
where LCP = CP ēe N̄ iγ5N + CP′ ēe N̄ iγ5τ3N with
CP = (CP )
4f
≃ − 375 MeV
X
q=c,s,t,b
CP′
=
(CP′ )4f
Ceq
,
mq
Ced
− 181 MeV
≃ − 806 MeV
md
X
q=c,s,t,b
Ceq
mq
♠ Numerical Illustration
• Deuteron EDM;O. Lebedev, K. A. Olive, M. Pospelov and A. Ritz, Phys. Rev. D 70 (2004) 016003,[arXiv:hepph/0402023]
dD ≃ −
5+11
−3
C
+ (0.6 ± 0.3) e (dC
−
d
u
d )/gs
C
E
E
−(0.2 ± 0.1) e (dC
+
d
)/g
+
(0.5
±
0.3)(d
+
d
s
u
d
u
d)
0.5C
C
C
dd
sd
bd
+ 3.3κ
+ (1 − 0.25κ)
+(1 ± 0.2) × 10−2 e GeV2
md
ms
mb
± e (20 ± 10) MeV d G
The projective sensitivity:
Y. K. Semertzidis et al. [EDM Collaboration], AIP Conf. Proc. 698 (2004)
200, [arXiv:hep-ex/0308063]; Y. F. Orlov, W. M. Morse and Y. K. Semertzidis, Phys. Rev. Lett. 96 (2006) 214802,
[arXiv:hep-ex/0605022]; and also see http://www.bnl.gov/edm/deuteron proposal 080423 final.pdf
|dD | < 3 × 10−27 − 10−29 e cm
For our numerical study, we take 3 × 10−27 e cm as a representative expected value
♠ Numerical Illustration
• CP-violating QCD θ-term:
O. Lebedev, K. A. Olive, M. Pospelov and A. Ritz, Phys. Rev. D 70 (2004)
016003, [arXiv:hep-ph/0402023]; M. Pospelov and A. Ritz, Annals Phys. 318 (2005) 119, [arXiv:hep-ph/0504231]; J.
Ellis, JSL, and A. Pilaftsis, arXiv:1006.3087 [hep-ph]
αs
L=
θ̄ Gaµν G̃µν,a
8π
with
θ̄ = θQCD + Arg Det Mq
|hq̄qi|
−16
dn(θ̄) = (1 ± 0.5)
θ̄
×
2.5
×
10
e cm
(225 MeV)3
dHg(θ̄) ≃ +2.0 × 10−6 θ̄ e GeV−1 ≃ 3.9 × 10−20 θ̄ e cm
dD (θ̄) ≃ −e [(3.5 ± 1.4) + (1.4 ± 0.4)] × 10−3 θ̄ GeV−1 ≃ −9.7 × 10−17 θ̄ e cm
♠ Numerical Illustration
• [Summary] EDMs and Observables under consideration
dTl/dEXP
, dn/dEXP
, dHg /dEXP
Tl
n
Hg ,
, ACP(b → sγ)[%] ,
, dµ/dEXP
dD/dEXP
µ
D
where we choose the following normalization factors
−29
−26
EXP
−25
EXP
=
3.1
×
10
e cm ,
=
3
×
10
e
cm
,
d
=
9
×
10
e
cm
,
d
dEXP
Hg
n
Tl
= 1 × 10−24 e cm
= 3 × 10−27 e cm , dEXP
dEXP
µ
D
♠ Numerical Illustration
• The EDMs and Observables under consideration are functions of
7 parameters :
Φ1 , Φ2 , Φ3 , ΦAu , ΦAd , ΦAe , θ̄
and then
∇α ≡
∂
∂
∂
∂
∂
∂
∂
,
,
,
,
,
,
∂Φ1 ∂Φ2 ∂Φ3 ∂ΦAu ∂ΦAd ∂ΦAe ∂ θb
The CP-violating phases Φ1,2,3 and ΦAu ,Ad,Ae are specified in degrees and we
normalize θ̄ in units of 10−10: θb ≡ θ̄ × 1010
♠ Numerical Illustration
• A scenario: We consider CP-violating variants of a typical CMSSM scenario with
|M1,2,3| = 250 GeV ,
2
2
2
2
fQ
fU2 = M
fD
fL2 = M
fE2 = (100 GeV)2 ,
MH
=
M
=
M
=
M
=
M
H
u
d
|Au| = |Ad| = |Ae| = 100 GeV ,
at the GUT scale, varying tan β (MSUSY )
We adopt the convention that Φµ = 0◦, and we vary independently the following
six MCPMFV phases at the GUT scale:
Φ1 , Φ2 , Φ3 , ΦAu , ΦAd , ΦAe
in addition to the QCD θ term: θ̄
This scenario becomes the SPS1a point when tan β = 10, Φ1,2,3 = 0◦ and ΦAu ,A ,Ae = 180◦
d
♠ Numerical Illustration
b∗
• Components of the vectors E ≡ ∇E , O ≡ ∇O ; Φ
∧
-3
10
1
10
10
10
-1
10
1
10
-3
20
40
tanβ
2
10
-1
-3
20
40
tanβ
10
10
10
| [ ACP optimal ]
α
| [ dµ optimal ]
-2
10
40
|Φ
∧*
|Φ
20
∧*
α
| [ dD optimal ]
α
-3
10
-3
20
tanβ
40
1
10
10
10
-1
-2
-3
20
tanβ
40
tanβ
∧
1
10
-2
10
40
10
1
-1
-2
-3
-4
20
40
10
10
-1
-2
∧
 : | Φ*2|
10
∧
 : | Φ*3|
-1
∧
− − − : | Φ*Au|
10
-2
∧
− − − : | Φ*Ad|
∧
− − − : | Φ*Ae|
10
-3
20
tanβ
d
 : | Φ*1|
1
∧
10
10
-2
20
-2
tanβ
| Oα | [ ACP ( % ) ]
EXP
10
2
-3
tanβ
| Oα | [ dµ / dµ ]
EXP
| Oα | [ dD / dD ]
tanβ
10
40
-2
10
| [ ∆Φ2,3 = θ = 0 ]
20
-1
-1
1
α
10
40
-1
∧
20
10
10
∧*
-3
-2
10
1
|Φ
10
-1
∧*
10
1
|Φ
1
10
| [ ∆Φ1,Ae = θ = 0 ]
10
-2
10
10 2
α
10
-1
10 2
∧*
1
− − − : ΦAd
|Φ
10
− − − : ΦAu
EXP
EXP
10 2
10
 : Φ3
| Eα | [ dHg / dHg ]
 : Φ2
| Eα | [ dn(QCD) / dn ]
EXP
| Eα | [ dTl / dTl ]
 : Φ1
−−−:θ
− − − : ΦAe
40
10
-3
20
tanβ
40
∧
−−− :|θ|
tanβ
dTl dn dHg
∗
b
Φ α = N εαβγδµνρ Eβ Eγ Eδ Bµνρ
dHg
with the 3-form Bµνρ = εµνρλστ ω Oλ EσTl Eτdn Eω
(1)
(2)
(3)
or Nµ Nν Nρ
for some reference directions
♠ Numerical Illustration
b∗ · O
• The products Φ
b∗ · O
Recall the relation O max = (φ∗)max Φ
∧
∧
-2
∼
φ =0
EXP
α
1
o
α
10
-3
10
10
-1
10
-2
20
40
tanβ
∼
φ =0
10
-4
-5
20
40
tanβ
o
α
10
∧*
∧
∧*
| Φ ⋅ O [ dD /
| Φ* ⋅ O [ dµ / dµ ] |
∼
φ = 0o
10
| Φ ⋅ O [ ACP ( % ) ] |
10
EXP
dD ]
|
 : dD dir. − − : dµ dir. ⋅ ⋅ ⋅ ⋅ : ACP dir.  : ∆Φ1,Ae=θ=0 dir.  : ∆Φ2,3=θ=0 dir.
10
10
-2
-3
-4
20
40
tanβ
♠ Numerical Illustration
• The maximum values of the observables along the optimal directions tan β = 10
b∗ · O
Again, recall the relation O max = (φ∗ )max Φ
∧
∧
o
α
6
4
o
α
0.3
0.2
4
3
1
0
0
0
-2
-0.1
-1
-4
-0.2
-6
-0.3
-8
-0.4
0
0
o
α
60
0.4
~
φ =0
o
0.2
40
0
0
0
-0.1
-0.02
-0.2
-0.04
-0.3
-0.06
-50
0
50
φ*
-0.4
-50
0
50
φ*
50
~
⋄ The maximum values of the observables
o
α
0.04
-20
-100
0
φ =0
0.06
0.02
-80
-50
0.08
0.1
-60
which are mainly constrained by dTl , dTl , dTl , dHg ,
and dTl , respectively.
φ*
20
-40
e
*
α
0.3
-4
50
φ
EXP
~
φ =0
80
-50
for each
` ∗ ´max
φ
∼
25 (dD -optimal) , 25 (dµ -optimal) , 50 (ACP -optimal) ,
40 (∆Φ1,A = θb = 0) , and 45 (∆Φ2,3 = θb = 0),
-3
*
dµ / dµ
100
-0.5
50
⋄ The maximum values of φ∗
EDM-free direction from the figure:
o
α
-2
ACP [ % ]
-50
∼
φ =0
2
0.1
φ
EXP
∼
φ =0
 : ∆Φ2,3=θ=0 dir.
2
-10
dD / dD
0.4
EXP
φ =0
8
0.5
dHg / dHg
∼
EXP
10
dn(QCD) / dn
dTl / dTl
EXP
 : dD dir. − − : dµ dir. ⋅⋅⋅⋅⋅ : ACP dir.  : ∆Φ1,Ae=θ=0 dir.
-0.08
-50
0
50
φ*
♠ Numerical Illustration
• The maximum values of the CP phases along the optimal directions tan β = 10
∧
20
o
Φ3 [ ]
10
o
Φ2 [ ]
40
o
Φ1 [ ]
 : dD dir. − − : dµ dir. ⋅⋅⋅⋅⋅ : ACP dir.  : ∆Φ1,Ae=θ=0 dir.
5
∧
 : ∆Φ2,3=θ=0 dir.
2
1
0
0
0
-20
-5
-1
In the top panels we see that Φ1 and Φ2` can´ be as large
max
as ∼ 20◦ and ∼ 5◦, respectively, for φ∗
∼ 25
along the dD - and dµ -optimal directions denoted by the
thick solid and dashed lines.
-2
-50
0
50
*
φ
-50
o
o
20
40
20
0
0
-20
-20
-40
-40
-40
50
*
φ
-50
0
50
*
φ
181
180
179
We also note in the middle and bottom panels that the
phases of Ad,u,e could be are large, in general, though
they are suppressed at the MSUSY scale.
-50
ΦAτ ( MSUSY )[ ]
190
0
50
*
φ
220
o
o
182
50
*
φ
20
0
0
0
40
-20
ΦAb ( MSUSY )[ ]
o
-10
50
*
φ
40
-50
ΦAt ( MSUSY )[ ]
0
ΦAe [ ]
-50
ΦAd [ ]
o
ΦAu [ ]
-40
185
180
175
200
Finally, we note (not shown) that θ̄ could be as large as
180
160
178
-50
0
50
*
φ
170
-50
0
50
*
φ
140
-50
0
50
*
φ
∼ 2 × 10−9 along the dD- and dµ-optimal directions
` ´max
with φ∗
∼ 25.
♠ Summary
• We are proposing a geometric method which provides an accurate parametric
determination of the optimal cancellation regions where any given physical
observable is maximized in the linear approximation
• Our geometric approach is exact, efficient and less computationally-intensive than
a naive scan of a multi-dimensional space
• This constitutes an analytic solution to the so-called linear programming problem
• You may want to apply this method to your problem if you are trying to achieve
the best outcome in a given requirements expressed in linear equations