Electric Dipole Moments in the MSSM
Jae Sik Lee
National Center for Theoretical Sciences, Hsinchu, TAIWAN
NCTS HEP Journal Club, December 23, 2008
∗
based on JHEP 0810:049,2008, arXiv:0808.1819 [hep-ph] with J. Ellis and A. Pilaftsis
♠ Preliminary
• Electric Dipole Moments (EDMs): T violation ⇒ CP violation (under CPT )
HEDM = −d E · ŝ
|dTl| < 9 × 10−25 e cm ,
|dHg| < 2 × 10−28 e cm ,
|dn| < 3 × 10−26 e cm
B. C. Regan, E. D. Commins, C. J. Schmidt and D. DeMille, PRL88 (2002) 071805; M. V. Romalis, W. C. Griffith
and E. N. Fortson, PRL86 (2001) 2505; C. A. Baker et al., PRL97 (2006) 131801
Question: No large CP phases?
♠ Contents
♠ CP phases in the MSSM
♠ One-loop EDMs of leptons and quarks
♠ Higher-order contributions
♠ Observable EDMs
♠ EDM constraints
♠ Summary
♠ Introduction (1/3)
• CP phases in the MSSM:
– Φµ [1]: W ⊃ µ Ĥ2 · Ĥ1
fW
f + M1 B
eB
e + h.c.)
– Φi [3]: −Lsoft ⊃ 12 (M3 gege + M2 W
2
[5 × 3 → 0NFV ]:
– Arg MQe ,Le ,eu ,de,ee
i<j
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
– Arg (Au,d,e)i,j [3 × 9 → (3 × 3)NFV ]:
– Arg
e 2 − de∗R Ad QH
e 1 − ee∗R Ae LH
e 1 + h.c.)
−Lsoft ⊃ +(e
u∗R Au QH
m212
[1]: −Lsoft ⊃ −(m212H1H2 + h.c.)
♠ Introduction (2/3)
• Physical observables depend on : Arg(Mi µ (m212)∗) and Arg(Af µ (m212)∗) M. Dugan,
B. Grinstein and L. J. Hall, Nucl. Phys. B 255 (1985) 413; S. Dimopoulos and S. Thomas, Nucl. Phys. B 465 (1996)
23
Without flavour-mixing terms, we have the 12 physical CP phases
Arg(M1 µ) , Arg(M2 µ) , Arg(M3 µ) ;
Arg(Ae µ) , Arg(Aµ µ) , Arg(Aτ µ) ;
Arg(Ad µ) , Arg(As µ) , Arg(Ab µ) ;
Arg(Au µ) , Arg(Ac µ) , Arg(At µ)
∗ We will take the Arg(m212 ) = 0 convention throughout this talk
♠ Introduction (3/3)
• Our convention for EDMs and CEDMs:
L(C)EDM = −
dE
f /e
dC
q
=
i
i E µν ¯
a µν
a
df F f σµν γ5 f − dC
G
q̄
σ
γ
T
q,
µν
5
q
2
2
E
χ̃±
(df /e)
±
+
E
χ̃0
(df /e)
0
g̃
E
H
+ (dE
f /e) + (df /e)
χ̃
C χ̃
C g̃
C H
= (dC
)
+
(d
)
+
(d
)
+
(d
q
q
q
q)
where f = e, u, d, s and q = u, d
♠ One-loop EDMs (1/5)
• Generically, the χ-mediated one-loop f EDM is given by
See, for example, T. Ibrahim and
P. Nath,Rev. Mod. Phys. 80 (2008) 577,[arXiv:0705.2008 [hep-ph]]; S. Abel, S. Khalil and O. Lebedev,Nucl. Phys. B
606 (2001) 151, [arXiv:hep-ph/0103320]
LχχA
Lf˜′f˜′A
Lχf f˜′
=
−e Qχ (χ̄γµχ)Aµ
=
−ie Qf˜′ f˜′∗ ∂ µ f˜′Aµ
=
χf f˜′
gL ij
(χ̄iPLf ) f˜j′∗
χf f˜′
gR ij
(χ̄iPRf ) f˜j′∗ + h.c.
↔
+
dE
f
e
!χ
mχi
χf f˜′ ∗ χf f˜′
2
2
2
2
Q
=
ℑm[(g
)
g
]
A(m
Q
B(m
/m
/m
)
+
)
˜
′
χ
′
˜
χ
χ
R
ij
L
ij
2
f
i
i
fj
f˜j′
16π 2mf˜′
j
1
A(r) =
2(1−r)2
“
ln r
3 − r + 21−r
”
,
1
B(r) =
2(1−r)2
“
ln r
1 + r + 2r1−r
”
♠ One-loop EDMs (2/5)
• Explicitly, the chargino-mediated one-loop EDMs of charged leptons and quarks:
dE
l
e
dE
u
e
dE
d
e
!χ̃±
!χ̃±
!χ̃±
=
m ±
1 X χ̃i
2
2
χ̃± lν̃ ∗ χ̃± lν̃
]
Q
A(m
)
g
/m
ℑm[(g
−
±
ν̃l )
χ̃
Li
Ri
2
χ̃
16π2
m
i
ν̃
i
l
=
m
±
±
1 X χ̃±
2
2
2
2
i ℑm[(g χ̃ ud˜)∗ g χ̃ ud˜] [Q
+ A(m ± /md˜ ) + Qd˜B(m ± /md˜ )]
χ̃
L
ij
R
ij
2
2
χ̃
χ̃
j
j
16π
m˜
i
i
i,j
dj
=
m ±
1 X χ̃i
2
2
2
2
χ̃± dũ ∗ χ̃± dũ
ℑm[(gR ij ) gL ij ] [Qχ̃− A(m ± /mũ ) + Qũ B(m ± /mũ )]
2
2
j
j
χ̃
χ̃
16π
mũ
i
i
i,j
j
where Qχ̃± = ±1, Qũ = 2/3, Qd˜ = −1/3, and
χ̃± lν̃
χ̃± lν̃
=
−g(CR )i1 ,
χ̃± ud̃
=
−g(CL )i1 (U )1j + hd(CL)i2 (U )2j ,
χ̃± dũ
=
−g(CR )i1 (U )1j + hu (CR)i2 (U )2j ,
gL i
gL ij
gL ij
∗
gR i
d̃ ∗
ũ ∗
∗
d̃ ∗
ũ ∗
χ̃± ud˜
gR ij
χ̃± dũ
gR ij
= h∗l (CL)i2 ,
∗
∗
d̃ ∗
= hu (CR)i2 (U )1j ,
∗
ũ ∗
= hd(CL)i2 (U )1j
♠ One-loop EDMs (3/5)
• The neutralino-mediated one-loop EDMs of charged leptons and quarks:
0
1 0
χ̃
dE
@ fA
e
m
1 X χ̃0i
2
2
χ̃0 f f˜ ∗ χ̃0 f f˜
)
B(m
/m
]
Q
)
g
ℑm[(g
=
˜
0
L ij
R ij
f
2
f˜j
χ̃
16π2
m
i
i,j
f˜j
with f = l, u, d. The neutralino-fermion-sfermion couplings are
χ̃0 f f˜
gL ij
=
χ̃0 f f˜
gR ij
=
√
√
˜
˜
f
f
∗
∗
∗
f˜ ∗
(U f )∗1j − hf Niα
− 2 g T3 Ni2 (U )1j − 2 g tW (Qf − T3 )Ni1
(U f )∗2j ,
√
f˜ ∗
∗
f˜ ∗
2 g tW Qf Ni1 (U )2j − hf Niα (U )1j
where the Higgsino index α = 3 (f = l, d) or 4 (f = u)
• The gluino-mediated one-loop EDMs of quarks:
dE
q
e
g̃q q̃
gL j
! g̃
=
=
1 X |M3 |
g̃q q̃ ∗ g̃q q̃
) gL j ] Qq̃ B(|M3 |2 /m2q̃ )
ℑm[(g
R
j
2
2
j
3π
mq̃
j
j
gs −iΦ3 /2 q̃ ∗
−√ e
(U )1j ,
2
gs +iΦ3 /2 q̃ ∗
g̃q q̃
gR j = + √ e
(U )2j
2
♠ One-loop EDMs (4/5)
• The chargino-, neutralino-, and gluino-mediated one-loop CEDMs of quarks:
“
“
” ±
C χ̃
du
“
” ±
C χ̃
dd
”χ̃0
C
dq=u,d
C
(dq=u,d)
1
where C(r) ≡
6(1−r)2
g̃
=
m
gs X χ̃±
2
2
χ̃± ud˜ ∗ χ̃± ud˜
i
),
/m
]
B(m
)
g
ℑm[(g
±
L ij
R ij
2
d˜j
χ̃
16π2
m
i
i,j
d˜j
=
m ±
gs X χ̃i
χ̃± dũ ∗ χ̃± dũ
2
2
/m
]
B(m
)
g
ℑm[(g
±
ũj ) ,
L ij
R ij
2
χ̃
16π2
m
i
ũ
i,j
j
=
m
gs X χ̃0i
χ̃0 q q̃ ∗ χ̃0 q q̃
2
2
]
B(m
/m
),
)
g
ℑm[(g
0
q̃
L ij
R ij
2
χ̃
j
16π2
m
i
q̃
i,j
j
=
−
gs X |M3 |
2
2
g̃q q̃ ∗ g̃q q̃
]
C(|M
|
/m
)
g
ℑm[(g
3
q̃j )
Lj
Rj
2
8π2
m
q̃
j
j
“
”
2r
ln
r
18
ln
r
10r − 26 + 1−r − 1−r
, with C(1) = 19/18
♠ One-loop EDMs (5/5)
• Complex Yukawa couplings; effects of Φ3 via resummed non-holomorphic threshold
corrections:
hu
hd
√
2mu
1
,
=
vsβ 1 + ∆u/tβ
√
2md
1
=
,
vcβ 1 + ∆d tβ
1
2mc
,
vsβ 1 + ∆c/tβ
√
2ms
1
=
vcβ 1 + ∆stβ
hc =
hs
√
where
∆u
∆d
2αs ∗ ∗
2
µ M3 I(MŨ2 , MQ̃
=
, |M3|2) ,
1
1
3π
2αs ∗ ∗
2
2
µ M3 I(MD̃
, MQ̃
, |M3|2) ,
=
1
1
3π
where I(x, y, z) ≡
2αs ∗ ∗
2
µ M3 I(MŨ2 , MQ̃
∆c =
, |M3|2) ,
2
2
3π
2αs ∗ ∗
2
2
µ M3 I(MD̃
, MQ̃
∆s =
, |M3|2)
2
2
3π
xy ln(x/y) + yz ln(y/z) + xz ln(z/x)
(x−y) (y−z) (x−z)
♠ Higher-order Contributions (1/4)
• Weinberg operator;S. Weinberg,Phys. Rev. Lett. 63 (1989) 2333; J. Dai, H. Dykstra, R. G. Leigh, S. Paban
and D. Dicus,Phys. Lett. B 237 (1990) 216 [Erratum-ibid. B 242 (1990) 547]; D. A. Dicus,Phys. Rev. D 41 (1990)
999
g̃
t̃, b̃
Hi
t, b
t, b
1
1
LWeinberg = − d G fabc Gaµρ Gb ρν Gcλσ ǫµνλσ = d G fabc Gaρµ G̃b µν Gcν ρ
6
3
where
d G = (d G)g̃ + (d G)H
♠ Higher-order Contributions (2/4)
• Higgs-mediated Four-fermion interactions;
f′
f′
Hi
f
L4f =
where
X
f
Cf f ′ (f¯f )(f¯′iγ5f ′)
f,f ′
(Cf f ′ )H = gf gf ′
S
P
g
X gH
¯
f f H f¯′ f ′
i
i
i
2
MH
i
P
S
LHf f = −gf Hi f¯ (gH
¯f + igH f¯f γ5 ) f
f
i
i
♠ Higher-order Contributions (3/4)
• Higgs-mediated Barr-Zee graphs;
D. Chang, W. Y. Keung and A. Pilaftsis,Phys. Rev. Lett. 82 (1999)
900 [Erratum-ibid. 83 (1999) 3972], [arXiv:hep-ph/9811202]; A. Pilaftsis,A. Pilaftsis, Nucl. Phys. B 644 (2002) 263,
[arXiv:hep-ph/0207277]; J. R. Ellis, JSL and A. Pilaftsis, Phys. Rev. D 72 (2005) 095006 [arXiv:hep-ph/0507046]
H
C H
(dE,C
)H ⇒ (dE
e ) , (du,d )
f
γ, (g)
τ̃ , t̃, b̃
Hi
f
γ, (g)
f
f
γ, (g)
τ̃ , t̃, b̃
Hi
f
γ, (g)
f
γ
γ, (g)
τ, t, b
Hi
f
f
f
γ, (g)
χ±
Hi
f
f
γ
f
f
♠ Higher-order Contributions (4/4)
• The gluon-gluon-Higgs contribution to CS , LCS = CS ēiγ5e N̄ N ;
e−
e−
e−
Hi
g
t, b
e−
e−
Hi
g
g
Hi
g
t̃, b̃
e−
t̃, b̃
g
g
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
8
<
9
gH q̃ ∗ q̃ =
2
P
P
i j j
2
x
q
S
S
− v12
gH
=
gH
j=1,2
q=t,b
q̄q
gg
3
i
i
m2
;
:
q̃j
♠ Observable EDMs (1/9)
• 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]
2
−19
dTl [e cm] = −585 · dE
[e
cm]
−
8.5
×
10
[e
cm]
·
(C
TeV
) + ···
S
e
dE
e
CS
=
=
E χ̃0
E χ̃±
(de ) + (de ) +
(CS )4f + (CS )g
H
(dE
e)
MeV
MeV
where (CS )4f = Cde 29m
with κ ≡ hN |mss̄s|N i/220 MeV ≃
+ Cse κ×220
ms
d
0.50 ± 0.25
♠ Observable EDMs (2/9)
• Neutron EDM [Chiral Quark Model (CQM)];A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984)
189; R. Arnowitt, J. L. Lopez and D. V. Nanopoulos, Phys. Rev. D 42 (1990) 2423; R. Arnowitt, M. J. Duff and
K. S. Stelle, Phys. Rev. D 43 (1991) 3085; T. Ibrahim and P. Nath, Phys. Rev. D 57 (1998) 478 [Erratum-ibid. D 58
(1998 ERRAT,D60,079903.1999 ERRAT,D60,119901.1999) 019901] [arXiv:hep-ph/9708456]
dn =
dNDA
q=u,d
1
4 NDA
dd
− dNDA
,
u
3
3
= η
E (C)
E
dE
q
e C
G eΛ G
d + η
d ,
+ η
4π q
4π
dq=u,d = (dqE (C))χ̃
where η E ≃ 1.53 ,
Λ ≃ 1.19 GeV
C
±
0
+ (dqE (C))χ̃ + (dqE (C)) g̃ + (dqE (C)) H
η C ≃ η G ≃ 3.4 and the chiral symmetry breaking scale
♠ Observable EDMs (3/9)
• Neutron EDM [Parton Quark Model (PQM)];J. R. Ellis and R. A. Flores, Phys. Lett. B 377 (1996)
83, [arXiv:hep-ph/9602211]
E
PQM E
PQM E
dn = η E (∆PQM
d
+
∆
d
+
∆
ds ) ,
d
u
u
s
d
with
∆PQM
= 0.746 ,
d
∆PQM
= −0.508 ,
u
∆PQM
= −0.226
s
The isospin symmetry between the neutron n and the proton p implies that ∆d = (∆u )p = 4/3, ∆u = (∆d )p =
−1/3. Furthermore, in the relativistic Naive Quark Model (NQM), one has ∆s = (∆s )p = 0.
♠ Observable EDMs (4/9)
• 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
q , dq ) = (1.4 ± 0.6) (dd − 0.25 du ) + (1.1 ± 0.5) e (dd + 0.5 du )/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)
♠ Observable EDMs (5/9)
• 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 )
CP′
=
4f
(CP′ )4f
≃ − 375 MeV
X
q=c,s,t,b
Ceq
,
mq
Ced
− 181 MeV
≃ − 806 MeV
md
X
q=c,s,t,b
Ceq
mq
♠ Observable EDMs (6/9)
• More on Mercury EDM;
M. Pospelov and A. Ritz, Annals Phys. 318 (2005) 119,[arXiv:hep-ph/0504231]
(1)
−3
dHg = (1.8 × 10−3 GeV−1) e ḡπN N + 10−2dE
e + (3.5 × 10 GeV) e CS
"
#
Z −N
+ (4 × 10−4 GeV) e CP +
CP′ ,
A
Hg
(1)
ḡπN N
(1)
ḡπN N
C
(dC
|hq̄qi|
u − dd )/gs
=
× 10
,
−26
3
10 cm (225 MeV)
Csd
Cbd
0.5Cdd
+ 3.3κ
+ (1 − 0.25κ)
∼ −8 × 10−3 GeV3
md
ms
mb
2+4
−1
−12
(1)
where LπN N ⊃ ḡπN N N̄ N π 0. Note that the factors 1.8 × 10−3 GeV−1 and
8 × 10−3 GeV3 are known only up to 50 % accuracy.
♠ Observable EDMs (7/9)
• 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
u − dd )/gs
C
E
E
−(0.2 ± 0.1) e (dC
+
d
)/g
+
(0.5
±
0.3)(d
+
d
s
u
d
u
d)
C
C
0.5C
sd
bd
dd
+ 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
♠ Observable EDMs (8/9)
• To summarize;
dTl = dTl(dE
e ) + dTl (CS )
C
G
dn (CQM) = dn(dE
u,d) + dn(du,d ) + dn (d )
E
E
dn (PQM) = dn(dE
u ) + dn (dd ) + dn (ds )
C
G
)
+
d
(d
) + dn(Cbd,db)
dn (QCD) = dn(dE
)
+
d
(d
n
n
u,d
u,d
(′)
C
dHg = dHg (dE
e ) + dHg (du,d) + dHg (C4f ) + dHg (CS ) + dHg (CP )
dD
C
G
= dD (dE
u,d) + dD (du,d) + dn(C4f ) + dn (d )
♠ Observable EDMs (9/9)
• The Thallium, neutron, Mercury and deuteron EDMs are implemented in an
updated CPsuperH2.0;
♠ EDM Constraints (CPX) (1/14)
• CPX scenario:
Fixed :
MQ̃3 = MŨ3 = MD̃3 = ML̃3 = MẼ3 = MSUSY ,
|µ| = 4 MSUSY , |At,b,τ | = 2 MSUSY , |M3| = 1 TeV
|M2| = 2|M1| = 100 GeV , MH ± = 300 GeV , MSUSY = 0.5 TeV
|Ae| = |Aτ |, |Au,c| = |At|, |Ad,s| = |Ab|
Φµ = ΦAτ = ΦAe = ΦAu = ΦAc = ΦAd = ΦAs = 0◦
Varying :
tan β ; ΦAt,b , Φ3 ; Φ1 , Φ2 , ρ
where the ρ parameter is defined as: MX̃1,2 = ρ MX̃3 with X = Q, U, D, L, E
♠ EDM Constraints (CPX) (2/14)
• LEP Limit in the CPX scenario :
CPX Scenario with ΦA = Φ3 = 90◦ for three values of mt (LEFT)
and with mt = 174.3 GeV for three values of ΦA = Φ3 (RIGHT) P. Bechtle, CPNSH Report, CERN-2006-009,
m t = 174.3 GeV
(a)
Excluded
by LEP
10
arg(A)=30
tanβ
tanβ
tanβ
m t = 169.3 GeV
Combined results with FeynHiggs
(b)
Excluded
by LEP
10
arg(A)=60
tanβ
hep-ph/0608079; ADLO, hep-ex/0602042
(b)
Excluded
by LEP
10
(c)
Excluded
by LEP
10
Theoretically
Inaccessible
Theoretically
Inaccessible
0
20
40
60
1
80 100 120 140
0
mH1 (GeV/c2)
Theoretically
Inaccessible
CPX
20
40
60
80 100 120 140
mH1 (GeV/c2)
1
Theoretically
Inaccessible
1
0
20
40
60
80 100 120 140
2
0
20
40
60
mH1 (GeV/c )
m t = 179.3 GeV
(c)
Excluded
by LEP
10
(c)
Excluded
by LEP
10
1
Theoretically
Inaccessible
CPX
0
20
40
60
80 100 120 140
mH1 (GeV/c2)
CL = 99.7 %
CL = 95.0 %
1
Theoretically
Inaccessible
CPX
0
20
40
80 100 120 140
2
mH1 (GeV/c )
arg(A)=90
tanβ
CPX
tanβ
1
60
80 100 120 140
2
mH1 (GeV/c )
♠ EDM Constraints (CPX) (3/14)
Φ3 [ o ]
| dTl / dEXP |
• Thallium EDM: CPX with Φ1 = Φ2 = 0◦ and ρ = 1;
|d/dEXP| = [< 1, 1 − 10, 10 − 100, > 100]
350
300
250
200
150
100
10 3
tanβ = 5
ΦA = 90
o
– dTl ∼ dTl(dE
e)
10 2
E H
– dE
e ∼ (de )
10
→ mild dependence on Φ3
1
50
0
100
200
10
300
ΦA [ o ]
Φ3 [ o ]
| dTl / dEXP |
0
350
300
250
200
150
100
-1
0
100
10 3
tanβ = 50
200
300
Φ3 [ o ]
ΦA = 90
o
10 2
dEe
10
CS
1
50
0
0
100
200
300
10
ΦA [ ]
o
-1
0
100
200
300
Φ3 [ o ]
– dTl(dE
e ) ∝ tan β
– dTl(CS ) ∝ tan2 β
♠ EDM Constraints (CPX) (4/14)
Φ3 [ o ]
| dn(CQM) / dEXP |
• Neutron EDM (CQM): CPX with Φ1 = Φ2 = 0◦ and ρ = 1;
|d/dEXP| = [< 1, 1 − 10, 10 − 100, > 100]
350
300
250
200
150
100
10 3
tanβ = 5
ΦA = 90
o
E
10 2
du,d
C
d u,d
10
G
d
1
50
0
200
300
ΦA [ o ]
350
300
250
200
150
100
| dn(CQM) / dEXP |
100
Φ3 [ o ]
0
10
-1
0
100
10 3
tanβ = 50
200
300
Φ3 [ o ]
ΦA = 90
o
10 2
0
100
200
300
10
1
10
ΦA [ ]
o
E,C g̃
– dE,C
∼
(d
) :
d
d
→ mild ΦA dependence
H
G
)
↑
and
d
↑
– subdominant (dE,C
d
as tan β ↑
– cancellation around Φ3 = 320◦
when tan β = 50
50
0
G
C
)
∼
d
(d
)
)
∼
d
(d
– dn(dE
n
n
u,d
u,d
-1
0
100
200
300
Φ3 [ o ]
♠ EDM Constraints (CPX) (5/14)
Φ3 [ o ]
| dn(PQM) / dEXP |
• Neutron EDM (PQM): CPX with Φ1 = Φ2 = 0◦ and ρ = 1;
|d/dEXP| = [< 1, 1 − 10, 10 − 100, > 100]
350
300
250
200
150
100
10 3
10 2
dEd
200
300
dEu
1
10
ΦA [ ]
350
300
250
200
150
100
tanβ = 5
-1
0
o
| dn(PQM) / dEXP |
100
Φ3 [ o ]
0
0
100
200
300
ΦA [ ]
o
200
300
Φ3 [ o ]
10 2
10
1
10
o
100
ΦA = 90
10 3
50
0
E g̃
– dE
with subdominant
s = (ds )
H
(dE
s)
10
50
0
– dn(dE
s ) dominates
dEs
tanβ = 50
-1
0
100
ΦA = 90
o
200
300
Φ3 [ o ]
♠ EDM Constraints (CPX) (6/14)
Φ3 [ o ]
| dn(QCD) / dEXP |
• Neutron EDM (QCD): CPX with Φ1 = Φ2 = 0◦ and ρ = 1;
|d/dEXP| = [< 1, 1 − 10, 10 − 100, > 100]
350
300
250
200
150
100
C
10 2
d u,d
E
du,d
G
d
tanβ = 5
50
200
300
ΦA [ ]
350
300
250
200
150
100
ΦA = 90
o
-1
0
o
| dn(QCD) / dEXP |
100
10
Φ3 [ o ]
0
100
200
10 3
300
Φ3 [ o ]
tanβ = 50
10 2
10
ΦA = 90
o
Cbd
1
50
0
0
100
200
300
10
ΦA [ ]
o
with
and
10
1
0
dominates
– dn(dC
d)
subdominant
dn(dE
d)
dn(dG)
10 3
-1
0
100
200
300
Φ3 [ o ]
CQM C
(dd )
(dC
– dQCD
n
d ) ∼ 3 dn
(dG)
dQCD
(dG) ∼ 0.5 dCQM
n
n
– cancellation around Φ3 = 320◦
when tan β = 50
♠ EDM Constraints (CPX) (7/14)
Φ3 [ o ]
| dHg / dEXP |
• Mercury EDM: CPX with Φ1 = Φ2 = 0◦ and ρ = 1;
|d/dEXP| = [< 1, 1 − 10, 10 − 100, > 100]
350
300
250
200
150
100
10 3
tanβ = 5
10 2
ΦA = 90
o
C
d u,d
10
1
dEe
50
0
100
200
300
10
ΦA [ ]
-1
0
o
Φ3 [ o ]
| dHg / dEXP |
0
350
300
250
200
150
100
200
10 3
300
Φ3 [ o ]
tanβ = 50
C
d u,d
10 2
ΦA = 90
o
10
dEe
CS
100
1
C4f
50
0
0
100
200
300
10
ΦA [ ]
o
CP(’)
-1
0
100
200
300
Φ3 [ o ]
dominates
– dHg (dC
d)
subdominant dHg (dE
e)
with
– For
large
tan β,
visible
contributions from C4f ≡
(′)
Cdd,sd,bd and CS,P
♠ EDM Constraints (CPX) (8/14)
10 3
ΦA = Φ3 = 90
o
tanβ=5
10 2
( Φ1 , Φ2 ) = (0 , 270 )
o
10
o
o
o
(0 , 0 )
| dTl / dEXP |
| dTl / dEXP |
• Thallium EDM: CPX with ΦA = Φ3 = 90◦ as functions of ρ;
10 3
tanβ=50
10 2
o
(90 , 0 )
o
o
(0 , 90 )
-1
10
| dEe | [ e cm ]
10
10
10
10
-24
tanβ=5
ΦA = Φ3 = 90
o
( Φ1 , Φ2 ) = (0 , 90 )
o
-25
o
±
χ
-26
0
10
10
10
H
-27
10
χ
0
-1
1
ρ
| dEe | [ e cm ]
1
10
– As ρ ↑; ’decrease’ → ’dip’ → ’flat’
1
o
10
-24
tanβ=50
-28
10
1
10
ρ
ρ
ΦA = Φ3 = 90
o
– ’decrease’: suppressed one-loop
contribution
-25
– ’dip’: cancellation between oneand two-loop contributions
-26
-27
( Φ1 , Φ2 ) = (0 , 90 )
o
10
– When (Φ1, Φ2) = (0◦, 0◦), dTl =
E
E H
dTl(dE
→ ρ
e ) with de = (de )
independence
10
1
10
ΦA = Φ3 = 90
o
-28
1
o
10
ρ
– ’flat’: two-loop (higher-order)
contribution dominates
♠ EDM Constraints (CPX) (9/14)
ΦA = Φ3 = 90
o
tanβ=5
10 2
10
(Φ1 , Φ2) = (0 , 90 )
o
1
o
10
o
o
(0 , 270 )
-1
1
10
ρ
ΦA = Φ3 = 90
o
10
2
o
o
(0 , 90 )
10
o
(0 , 270 )
o
1
10
tanβ=5
-1
1
2
o
(90 , 0 )
o o
(0 , 0 )
10
tanβ=5
10
o
ρ
ΦA = Φ3 = 90
o
10
1
10
-1
1
10
ρ
| dn(QCD) / dEXP | | dn(PQM) / dEXP | | dn(CQM) / dEXP |
| dn(QCD) / dEXP | | dn(PQM) / dEXP | | dn(CQM) / dEXP |
• Neutron EDM: CPX with ΦA = Φ3 = 90◦ as functions of ρ;
ΦA = Φ3 = 90
o
10 2
10
1
10
– PQM is most sensitive to Φ2
tanβ=50
-1
1
– no sensitive to Φ1
10
ρ
– QCD least sensitive to Φ2
10 2
– position of dips depends on
models/approaches
10
tanβ=50
o
ΦA = Φ3 = 90
1
10
-1
1
10
ρ
10
ρ
10 2
10
tanβ=50
o
ΦA = Φ3 = 90
1
10
-1
1
♠ EDM Constraints (CPX) (10/14)
tanβ=5
10 2
o
G
d
10
C
d u,d
E
du,d
1
10
-1
10
10 3
10
tanβ=5
ΦA = Φ3 = 90
o
( Φ1 , Φ2 ) = (0 , 90 )
o
10 2
o
10
1
o
– cancellation between dn(dC
d ) and
dn(dG)
– ’flat’: dG and (dE,C
)H
d
10
1
-1
1
ρ
3
tanβ=50
ΦA = Φ3 = 90
o
o
( Φ1 , Φ2 ) = (0 , 90 )
10 2
10
1
| dn(QCD) / dEXP |
ΦA = Φ3 = 90
o
o
( Φ1 , Φ2 ) = (0 , 90 )
| dn(CQM) / dEXP |
10 3
| dn(QCD) / dEXP |
| dn(CQM) / dEXP |
• Neutron EDM (CQM and QCD): CPX with ΦA = Φ3 = 90◦ as functions of ρ;
10
10
3
ρ
– tan β = 5: CQM dip around ρ =
2 and QCD dip around ρ = 3.6
( Φ1 , Φ2 ) = (0 , 90 )
o
o
– ±50% dn(dG) uncertainty: no
cancellation and/or dip position
δρ ∼ ±1
C
10 2
d u,d
E
du,d
G
d
10
tanβ=50
Cbd
1
ΦA = Φ3 = 90
o
10
-1
10
1
10
ρ
-1
1
10
ρ
– more significant dn(dE
d ) in CQM
→ more sensitive to Φ2
♠ EDM Constraints (CPX) (11/14)
ΦA = Φ3 = 90
o
tanβ=5
( Φ1 , Φ2 ) = (0 , 270 )
o
10 2
o
dEs
dEd
10
dEu
1
10
-1
| dEs | [ e cm ]
10
10
10 3
10
-22
o
~
ΦA = Φ3 = 90
o
g
-23
o
tanβ=5
±
χ
-24
ΦA = Φ3 = 90
±
1
tanβ=50
-1
1
10
10
10
-22
( Φ1 , Φ2 ) = (0 , 270 )
o
o
ΦA = Φ3 = 90
o
-23
10
10
10
χ
0
-26
-24
10
1
10
ρ
-25
tanβ=50
-26
1
10
– cancellation between the two
dominant one-loop EDMs
ρ
H
-25
– dn ∼ dn(dE
s)
E χ̃
g̃
+ (dE
– dE
s)
s ∼ (ds )
10
10
0
o
o
ρ
( Φ1 , Φ2 ) = (0 , 270 )
( Φ1 , Φ2 ) = (0 , 270 )
o
10 2
10
1
10
| dn(PQM) / dEXP |
10 3
| dEs | [ e cm ]
| dn(PQM) / dEXP |
• Neutron EDM (PQM): CPX with ΦA = Φ3 = 90◦ as functions of ρ;
ρ
±
χ̃
– large (dE
→ sensitive to Φ2
s)
♠ EDM Constraints (CPX) (12/14)
10 3
ΦA = Φ3 = 90
o
tanβ=5
10 2
10
| dHg / dEXP |
| dHg / dEXP |
• Mercury EDM: CPX with ΦA = Φ3 = 90◦ as functions of ρ;
– dominance of dHg(dC
d ) → no
sensitive to Φ1,2
10 3
10 2
10
1
1
tanβ=50
-1
10
| dCd | [ cm ]
1
10
10
10
10
10
10
10
-23
ΦA = Φ3 = 90
o
tanβ=5
Φ1 = Φ2 = 90
o
-24
10
0
H
-25
10
~
g
±
-26
χ
-27
χ
1
10
10
0
10
-28
10
1
10
ρ
o
-1
ρ
| dCd | [ cm ]
10
ΦA = Φ3 = 90
g̃
)
– no cancellation between (dC
d
C H
and (dd )
10
ρ
-23
-24
0
H
~
g
-25
±
-26
χ
tanβ=50
o
ΦA = Φ3 = 90
o
Φ1 = Φ2 = 90
-27
-28
1
χ
0
10
ρ
H
– ’flat’: (dC
d)
♠ EDM Constraints (CPX) (13/14)
( Φ1 , Φ2 ) = (0 , 90 )
o
o
10 2
10
-1
| dEe | [ e cm ]
10
10
10
10
10 2
10
10
1
10
o
10 3
( Φ1 , Φ2 ) = (0 , 90 )
o
10
-24
10
o
o
-25
±
χ
-26
1
-1
0
H
10
1
ρ
( Φ1 , Φ2 ) = (0 , 90 )
10
10
3
( Φ1 , Φ2 ) = (0 , 90 )
o
o
10 2
d
-28
10
1
10
ρ
10
10
10
10
E
1
0
1
10
G
C
χ
du,d
-1
10
10
1
10
-1
ρ
d u,d
-27
o
10 2
1
| dn / dEXP |
10
( Φ1 , Φ2 ) = (0 , 90 )
o
| dCd | [ cm ]
1
10 3
| dHg / dEXP |
10 3
| dn / dEXP |
| dTl / dEXP |
• Thallium, neutron(QCD), and Mercury EDMs: CPX with Φ1 = 0◦, Φ2 = 90◦, ΦA =
Φ3 = 90◦ as functions of ρ when tan β = 5;
ρ
-23
10
ρ
( Φ1 , Φ2 ) = (0 , 90 )
o
o
-24
0
H
-25
~
g
±
χ
-26
χ
0
-27
-28
1
10
ρ
– ρ∼4?
♠ EDM Constraints (CPX) (14/14)
10 4
ΦA = Φ3 = 90
o
10 3
10
2
– no sensitive to Φ1,2
10 4
10 3
10
10
10
1
1
10
-1
10
1
| dD / dEXP |
| dD / dEXP |
tanβ=5
10
tanβ=5
ΦA = Φ3 = 90
o
o
( Φ1 , Φ2 ) = (0 , 90 )
o
10 4
C
10 3
G
d
d u,d
10 2
E
10
10
ΦA = Φ3 = 90
o
-1
1
10
ρ
( Φ1 , Φ2 ) = (0 , 90 )
o
o
10 4
10 3
10 2
1
C4f
-1
10
1
tanβ=50
10
du,d
1
– Very sensitive even to the higherorder corrections
2
ρ
| dD / dEXP |
| dD / dEXP |
• Deuteron EDM: CPX with ΦA = Φ3 = 90◦ as functions of ρ;
10
ρ
tanβ=50
ΦA = Φ3 = 90
o
-1
1
10
ρ
– 300 times more sensitive if the
projective 10−29e cm achieved
♠ Summary
• We present the most comprehensive study of the Thallium, neutron, Mercury, and
deuteron EDMs
• We improve upon earlier calculations by including the resummation effects due to
CP-violating Higgs-boson mixing and to threshold corrections
• Large CP phases may be possible avoiding all existing EDM constraints
• The analytic expressions for the EDMs are implemented in an updated version of
the code CPsuperH2.0