CP violating ZZZ vertex
PASCOS
June 2014
Per Osland
University of Bergen
Work with:
Bohdan Grzadkowski, Odd Magne Øgreid
in progress…
Preamble
p 2 , µ 2 , a2
ggg vertex in QCD
p 1 , µ 1 , a1
p 3 , µ 3 , a3
SU(3)
Structure:
i g f a1 a2 a3 [gµ1 µ2 (p1
p2 )µ3 + gµ2 µ3 (p2
2
p3 )µ1 + gµ3 µ1 (p3
p 1 ) µ2 ]
gf
3
1 2 3
[gµ1 µ2 (p1
p2 )µ3 + gµ2 µ3 (p2
p3 )µ1 + gµ3 µ1 (p3
p 1 ) µ2 ]
µ⌫
F
Fa µ⌫
Origin of ggg coupling, aSU(3):
L 3 TrF µ⌫ Fµ⌫
F µ⌫ = @ ⌫ Aµ
Faµ⌫ = @ ⌫ Aµa
@ µ A⌫ ig[Aµ , A⌫ ]
@ µ A⌫a + gfabc Aµb A⌫c
F µ⌫ = @ ⌫ Aµ @ µ A⌫ ig[Aµ , A⌫ ]
structure µ
constant
µ⌫
⌫ µ
µ ⌫
⌫
F
=
@
A
@
A
+
gf
A
A
SU(2): Completely
analogous.!
abc b c
a
a
a
!
But structure constant is zero unless !
all 3 gauge fields are present. !
2
2
p
M
+
µ2 µ1but!
µ3 µ1 µ 2
Have
W
1
ZW-Z vertex,
µ3
µ 1 µ2 µ 3 ⇢
=
[if
(p
g
+
p
g
)
+
if
✏
(p2
4 1
5
1
2
No M
ZZZ
Z vertex (at tree level).
|f4 | <
⇠ 0.01
3
p3 )⇢ ] (0.
(0.
Back to QCD
p3
a3 µ3
p3
a3 µ3
At one loop (ggg):
p1
p2
a1 µ1
p3
a2 µ2
a3 µ3
p1
p2
a1 µ1 a2 µ2
p1
p2
a1 µ1
p3
a2 µ2
a3 µ3
p3
p1
p2
a1 µ1
a2 µ2
p3
p1
p2
a1 µ1
a3 µ3
Σ
flavours
p1
p2
a1 µ1
a2 µ2
4
a3 µ3
a2 µ2
and in general Γµ1 µ2 µ3 can be written as a sum of these tensors multiplied by scalar
functions (see Appendix A). This decomposition is useful for extracting the corresponding
scalar
functions structures:
from the result of a calculation. Although bosonic symmetry of the vertex
More
Lorentz
puts some conditions on the corresponding scalar functions, the explicit symmetry of the
expression is broken, because one of the momenta was substituted in terms of two others.
tree
level structure
To avoid this, one can use a more
symmetric
decomposition of the general three-gluon
vertex, proposed by Ball and Chiu [13]7 ,
Γµ1 µ2 µ3 (p1 , p2 , p3 ) = A(p21 , p22 ; p23 ) gµ1 µ2 (p1 − p2 )µ3 + B(p21 , p22 ; p23 ) gµ1 µ2 (p1 + p2 )µ3
−C(p21 , p22 ; p23 )
+ 13
5
#
(p1 p2 )gµ1 µ2 − p1 µ2 p2 µ1
S(p21 , p22 , p23 )
!
#
$
(p1 − p2 )µ3
p1 µ3 p2 µ1 p3 µ2 + p1 µ2 p2 µ3 p3 µ1
" !
$
5
"
2 2
+F (p21 , pthe
)gµ1 µLagrangian
p1 µ3 (p2to
p3 )the
− SU(3)
p2 µ3 (pgroup,
Although
1 p3 )
2 ; pstandard
3 ) (p1 p2QCD
2 − p1 µ2 p2 µcorresponds
1
our results are valid for an
!
"$
# gauge group.
!
"
arbitrary semi-simple
2 2 2
1
6 +H(p1 , p2 , p3 ) −gµ1 µ2 p1 µ (p2 p3 )−p2 µ (p1 p3 ) + a1 a2p
p p3 µ2considered,
−p1 µ2 p2 µ3 pbut
3 µ1 they do not
In fact, also completely symmetric
colour structures
d 3 a31 µmight
3 2 µ1 be
3
3
appear in the perturbative
of QCD three-point
the one-loop level.
+ { calculation
cyclic permutations
of (p1 , µ1vertices
), (p2 , µat
(2.4)
2 ), (p3 , µ3 ) } .
7
Another general decomposition of the three-gluon vertex was considered in ref. [22].
Here, the A, C and F functions are symmetric in the first two arguments, the H function
is totally symmetric, the B function is antisymmetric in the first two arguments, while
the S function is antisymmetric with respect to interchange of any pair of arguments.
Note that the contribution containing the F and H functions is totally transverse, i.e. it
gives zero when contracted with any of p1 µ1 , p2 µ2 or p3 µ3 .
Now, before proceeding further, we introduce some notation. For a quantity X (e.g.
any of the scalar functions contributing to the propagators or the vertices), we shall denote
the zero-loop-order contribution as X (0) , and the one-loop-order contribution as X (1) . In
this paper, as a rule,
X (1) = X (1,ξ) + X (1,q) ,
(2.5)
all conserve CP
5
where X (1,ξ) denotes a contribution of gluon and ghost loops in a general covariant gauge
µ 1 µ2
1
2 µ3
µ2 µ3
i g f a1 a2 a3 [gµ1 µ2 (p1
2
3 µ1
µ3 µ 1
p2 )µ3 + gµ2 µ3 (p2
3
1 µ2
p3 )µ1 + gµ3 µ1 (p3
ZZZ
vertex
loop
µ⌫ L 3µ⌫
⌫ µ at µone
⌫
TrF
F
F =@ A
@ A
ig[Aµ , A⌫ ]
µ⌫
Fa Fa µ⌫
µ⌫
L 3 TrF µ⌫ Fµ⌫
General structure!
µ⌫
⌫ µ
F =@ A
a
p 1 ) µ2 ]
a
@
µ
⌫
Aa
+
µ ⌫
gfabc Ab Ac
Gaemers
& ⌫Gounaris
(1979); !µ ⌫
µ⌫
µ
µ ⌫
F = @Peccei,
A µ⌫ @Zeppenfeld
A⌫ µ ig[A
, A ] µ! ⌫
Hagiwara,
(1986);
µ ⌫
@ ⌫A µ @ A⌫ µ ig[A
,A ]
⌫
µ⌫
⌫ F µ⌫
µ = @ ⌫µAµF⌫ @=µ A
abc
F =Baur
@ A&a Rainwater
@ AaF µ⌫ (2000)
ig[A
,
A
⌫a +
µ gf
µ A
⌫]b Ac
=@ A
@ A + gf Aµ A⌫
a
a
a
abc
b
c
and ⌫Lorentz invariance:
µ ⌫Coupling vanishes if all are on-shell
µ⌫ Bose
⌫2 symmetry
µM 2 ⇥µ
p
µ2 µ1abc
1 a
Z
µ3 b µc3 µ1 µ2
µ 1 µ2 µ3 ⇢
1 µ2 µ 3
a
a
=For two Zs2 (Z2, Zif
g
+ p1 g
) + if5 ✏
(p2
4 (p1 on-shell:
VV
3) being
M
2
2
p
M2 Z 2
F
µ3
=@ A
=
1
µ1 µ2Z
µ3
V2V V
MZ
@ A + gf
A A
p1 µ2 MµZ1 µ3
µ2
µ µ ⇢ µ1 µ2 µ 3 ⇢
13
µµ23 gµµ
µ3ifµ1✏
µµ
1µ
2 1 2 3 (p
[if
(p
g
+
p
)
+
= 4 1 2 [if4 (p11 g
+ p1 g 5 ) + if5 ✏ 2 p(p
3 )2⇢ ] p3 )⇢ ]
MZ
µ1
Violates/
CPp1
µµ
µ2 µ1 µ 3
µ3 µ1 µ2µ1
1 1 µ2 µ 3 ⇢
/
p
/
p
1
[if4 (p1plusg “scalar”+terms
p1 g
(p2
1) + if5 ✏
f
,
f
SM: quark loop contributes to
5
6
f
,
f
5
6
|f4 | <
⇠ 0.01
LHC: |f4 | <
⇠ 0.01
2
M ,
,
,
|f4 | <
⇠6 0.01
<
|f4 | ⇠ 0.01
Preserves CP
p3 )⇢ ] (0.1)
(0.2)
p3 ) ⇢
ZZZ vertex at one loop
Z: odd under C
ZZ: even under C
Z2
ZZZ vertex will in general violate CP
q
Z1
q
SM contribution via fermion loops!
preserves CP
q
Gounaris, Layssac, Renard (2000):!
“Vertex can only be generated by fermionic triangles” Incorrect!
7
Z3
ZZZ vertex at one loop, in 2HDM
non-zero contribution from bosonic sector
Allow for CP violation in Higgs potential
8
=
=
VVV
MZ2
µ⌫
V(
1,
2)
=
1 3
+ p1 g
1 2
) + if5 ✏
(p2
1 2 3
p3 ) ⇢ ]
2HDM
notation
=@ A
@ A
ig[A , A ] 1
⌫
2
1
(
2
1h
m11
†
1
1)
1
2
|f
|<
0.01
µ4 ⇠
⌫
µ
F
⌫ µ
n @ Aa
Faµ⌫1 =
†
2
+
p21
[if4 (p1 g
+
1
@2
µ
+ m22
2
2
(
†
2
A† ⌫a
2
+ m12
1†
2
⌫
µ ⌫
+ hgf2abc†Ab Ac io
V += 3 ( 1(
nh 2
2)
i
2
µ
1
2
+ h.c.
† † 2
+
1 )(
1 21 )2 ) +
2
4(
2i
(
†
1
†
† 2
)
22)( 22) 1 +
h
1
3
o
†
†
†
+ µ3 6µ
( µ
)
+
(
)
(
M+Z2 2 5( †1µ22)2µ+1 µh.c.
†
†
7
2 µ µ 1⇢ 2 ) + h.c. †
1 21
2 µ1
3
1
2) +
3
+
(
)(
4
2
1
5 (p31)⇢
[if
(p
g
+
p
g
)
+
if
✏
(p
1
2
2
4
5
1
1 1
2
2
MZ
⌘ Yab̄
No FCNC:
Allow CPV:
†
ā
b + Zab̄cd¯(
2
†
ā
b )(
†
c̄
|f4 | 2<
⇥
⇠ 0.01
Im J1 =6
=v20;Im
1d )n 2
m11 (
2
(1) ⇤
⇤
v̂ā Y7ab̄ Zbd¯ v̂d
=0
†
1
1 )+
h
2
m12 (
(relax this)
⇤
⇥ ⇤ ⇤
2
Im J2 =2 4 Im v̂b̄ v̂c̄ Ybē Ycf¯Zeāf d¯v̂a v̂d
M12v, ⇥5 , 6 , 7 complex
⇤
⇤ ⇤ (1) (1)
Im J3 = Im v̂b̄ v̂c̄ Z9 bē Zcf¯ Zeāf d¯v̂a v̂d
†
1
2 )+
2
h
i
nh
1
+
(
)(
)
+
(
)
+
h.c.
n
h
i
†
†
†
4
2
1
5
2
2
1 + 14 ( † 12 )(
2
1
†
2 † 2 1 ) + 22 5†( 1 2 ) + h.c.2
2
1
+
(
)
+
h.c.
+
(
)
+
(
5
2
6
1
7
n
h
i
m
(
)+
m
(
)+h.c.
+m
(
1
1
2
1
2
1
1
11
12
22
1
2
†
†
2
2
n
h
i +m2 (
2
m
(
)+
m
(
)+h.c.
1
† 1 1
2 11
2 12 † 1 2
222
1
2
† m11 ( 1 1 )+ m
† 12 ( 1 † 2 )+h.c. +m22 (
⌘ 0;Yab̄ 27ā =b0+ Zab̄cd¯( ā b )( c̄ d )
6 =
2
6 = 0;2
7 = 0
2HDM notation 2
m122
5,
0; m127
6 =
5,
m212
5,
=0
j
= ei⇠j
✓
◆!
!
p
,
'
=
!
i
i
1
(v
⌘2j1(v+i +
i'⌘+ji)/
+ i 2i )
j +p
i =
+ +
'j 'i +
††
22
†
†2
2
oi
) (
22) o.
o.
)
2
2) .
j = 1, 2
(vi +i ⌘i + i i )
0 1
i =0
11
p
(vi + ⌘0
i1 i )
0H1 21
i ⌘+
1
HA
⌘1A
1 ⇥
⇤
2
@
@
(1)
0
1
0
1
⇤
H
⌘
=
R
,
2Im
2 A v̂
A
@
Im J1 = @
v̂
Y
Z
H
⌘
=
R
2
2bd¯ , d
a
b̄
ā
H
⌘
2
1
1
vHH3
⌘⌘3
@H A
3 = R @⌘ 3A ,
p
2
2
†
1
(1
()
2
(1
(1
(
(1
(1
(
2
H3 sin
2
2 2 2
2
3
+ ⌘cos
1diag(M
1
RM2 R⌘T3 =
=M
=
,
M
,
M
diag
1
2
3)
RM2 RT = M2diag = diag(M12 , M22 , M32 )
2 T
2
2
2
2
,
M
,
M
0RM R = Mdiag =1diag(M
0
1
2
3)
1
0
0
10
cos ↵2
0 sin ↵2
10
cos ↵1
CP violation in the 2HDM
• Lavoura and Silva, 1994
• Botella and Silva, 1995
• Gunion and Haber, 2005
• Ivanov, Nishi, Maniatis…, 2006-2007
11
⌘ Yab̄
†
ā
11 h
nh
†
† i
b ++ Zab̄cd(¯( † ā )b2 )(
d)
c̄
+
h.c.
+
6(
22 5 1 2
†
1
1)
+
7(
Gunion and Haber
expressed
invariants
1
⌘ Yab̄
†
ā
b
+
2
†
Zab̄cdvev
¯( ā
b )(
†
c̄
†
2
d)
⇥ ⇤
2
(1) ⇤
Im J1 =
Im
v̂
Y
Z
v̂
d
¯
a
b̄
ā
b
d
2
v
⇥ ⇤
2
(1) ⇤
2 Im J⇥1 ⇤= ⇤ v 2 Im v̂ā Yab̄ Zbd¯ v̂d⇤
Im J2 = 4 Im v̂b̄ v̂c̄ Ybē Y⇥cf¯Zeāf d¯v̂a v̂d
v Im J = 2 Im v̂ ⇤ v̂ ⇤ Y Y Z v̂ v̂ ⇤
⇥ ⇤ 2 ⇤ (1)4 (1) b̄ c̄ bē cf¯ ⇤eāf d¯ a d
Im J3 = Im v̂b̄ v̂c̄ Zbēv Z⇥ cf¯ Zeāf
¯v̂a v̂d
⇤
d
(1) (1)
Im J3 = Im
CP conserved iff
⇤ ⇤
v̂b̄ v̂c̄ Zbē
Zcf¯ Zeāf d¯v̂a v̂d
Im J1 = Im J2 = Im J3 = 0
12
What is the physical content?
Im J1 = Im J2 = Im J3 = 0
Can
Im J1 , Im J2 , Im J3
be rephrased in terms of
“physical” quantities?
1
13
“Tak”
14
“Ja”
Im J1 , Im J2 , Im J3
The physical content
Im J1 ,qIm
1 J2 , ImqJ
23
1
Im J1 = 5
v
e1
2
e1 M
q11
1
Im J1 = 5
v
q3
e2
e3
eq22M22 q3e3 M32
e1
e2
e3
e
e
e
e11M12 e2 M222 e3 M32 3
1
2
2
2
e
M
e
M
Im J2 = 9 e1 M
22 2 e3 3 3
1
e
e
1
v 1
42
4
2 4
2
e
M
e
M
e
M
Im J2 = 9 1e1 M11 e22M2 2e3 M33 3
v
e1 M14 e2 M24 e3 M34
e1e
e
e
2
e2
e3 3
1 1 1
Im J30Im=J30 =5 5 q1q1
q2q2
q3 q 3
v v q1 M22 q2 M 2 2q3 M 2 2
q1 M11 q2 M
q33 M3
2 2
Footnote:
Im J3 = Im J30 + terms / Im J1 , Im J2
15
v15 e1 2 feZ2(s 2= p2)evs3 s 2/M 2
q1 M1 q24M12 1q3 M13 Z
q2
q3
Im J30 = 5 q1
vCouplings:
2
2
2
2M
2
q
M
q
M
q
g
1
2
3
s
=
(M
+
M
)
,
(M
+
M
)
1
2
3
Z µ Hi Hj :
✏ijk1 ek (pZi p1 j )µ Z 2
2v cos ✓W g
antisymmetric µ
µ
Z Hi Hj :
✏ijk ei, kj,(p
k :i1, 2, 3pj )
+
H H Hµi :
iqi 2v cos
g ✓W
µ
Z
H
H
:
✏
e
(p
p
)
+
j
H H i Hji : 2v cos
iqi ✓W ijk k i
+
H H Hi :
iqi
ei = v1 Ri1 + v2 Ri2
3
ei = v1 Ri1 + v2 Ri2
ei = v1 Ri1 + v2 Ri2
q = ...
more complicated
i
qi = . . .
qi = . . .
16
n
H1 = h
H2 = H
H3 = A
Recall CP-conserving 2HDM
Let
Then
H1 = h
H1 = h
H2 = H(ZHA) e1 ̸=H0 = H
(ZhA) e2 ̸= 02
H1 = h
H3 = A
Let
H1 = h
H2 = H
H3 = A
e3 =H
03
(ZhH)
=A
H2 = H
H3 = A
and
Then
+ −
(ZHA)
e1 ̸=
(hH
H0 ) q1 ̸= 0
(ZHA) e1 ̸=
0
+ −
Thene2 ̸=(ZhA)
(HH
) q2 ̸= 0
(ZhA)
0
e2 ̸= H
0 (ZHA)
+ −
(AH
(ZhH) e3 =(ZhH)
0
e = 0H ) q3 = 0
3
(hH + H − )
+
−
q1 ̸= 0
+
−17
(hH H )
e1 ̸= 0
(ZhA) (ZHA)
e2 ̸= 0 e1 =
̸
(ZhH) (ZhA)
e3 = 0 e2 =
̸
q1 ̸= 0
(ZhH)
e3 =
This result was actually published by Lavoura and Silva in 1994
[Im J2 ]Higgs Basis =
[Im J1 ]Higgs Basis =
[Im J30 ]Higgs Basis =
6
LS
v J1
3 LS
v J3
4 LS
v J2
and is reminiscent of the Jarlskog determinant
v
e1 M14 e2 M24 e3 M34
e1
e2
e3
1
q2
q3
= 5 q1
v
q1 M12 q2 M22 q3 M32
ZZZ vertex atImone
loop
J
30
Z2
Z2
Z2
g
Z Hi Hj :
✏ijk ek (pi
2v cos H
✓W
ej
ej
Hj
j
H + H Hi : Z1 iqeki
µ
Hj
Z1
ei
Z1
ek
Hk
Hi
ek
G0
Hi
ej
Z3
ei
Z
Hi
ei = v1 Ri1 + v2 Ri2
Z3
Z3
qi = . . .
Note sum over i, j, k all di↵erent.
All three neutral Higgs bosons involved.
19
ei
pj )
rent.
ns involved.
ei = v1 Ri1 + v2 Ri2
ZZZ vertex
at
one
loop
q = ...
s
f4Z (s1 = p21 ) v
qi = . . .
= (MZ + M1 )2 ,
1
Note sum over i, j, k all di↵erent.
Note sum over i, j, k all di↵erent.
All
three
neutral
Higgs
bosons
involved.
e1 eAll
e
2 3three neutral Higgs bosons involved.
i, j, k : 1,
2
3
v
Z
Proportional to Im J2
2
MZ e21 e2 e3 2
2f Z (p22) = 2 2↵2
2 2↵2
2
2 2
M
e1 e2 e3
, MZ ,4 M1i , M⇡jsin
,M
Z
3 Z ) + 2C001 (p
2 1 ,23MZ , MZ , MZ , Mj , Mk )Z
f4Z (pv1 ) =
(2✓W ) p1 M
3
2
2
3
X
⇥
p
M
v
⇡
sin
(2✓
)
W 21 2
2
2
2
2
2
2
Z
⇥
✏ijk C001 (p21 , MZ2 , MZ2 , Mi2 , M
,
M
)
+
C
(p
,
M
,
M
,
M
,
M
,
M
001
j
Z ⇥
1
Z
Z
Z
j
k)
X
2
2
2
2
2
2
2
2
2
,
M
,
M
)
C
(p
,
M
,
M
,
M
,
M
,
M
) 001 (p2 , M 2 , M 2 , M 2 , M 23, M
i,j,k
001 1
i
Z
k
Z
Z ⇥i
j✏ijk kC
Z2
i
j
⇤
2
2
2
2
2
2
2
2
21
2 Z 2
+ 2C001 (p1 , MZ , MZ , Mi , MZ , Mk ) C001 (p1 , MZ , MZ , Mi , Mj , Mk )
i,j,k
Mi2 , MZ2 , M
⇤
k)2
2
2
2
2
2
2
+ MZ C1 (p1 , MZ , MZ , Mi , MZ , Mk )
+ C001 (p21 , MZ2 , MZ2 , Mi2 , MZ2 , Mk2 ) C
⇤
2
2
2
2
2
2
2
Z2 , Z3 assumed on shell + MZ C1 (p1 , MZ , MZ , Mi , MZ , Mk )
i
non-symmetric expression
2
2
20
grals. The prime error in this respect is the confusion of the ext
The integrals with a tensor structure can be reduced to linear combinations of Lorentzthe momenta ki appearing
in the denominators, which are the
covariant tensors constructed from the
metric tensor gµν and a linearly independent set
(1.2)).
of the momenta [PaV79].
The choice of this basis is not unique.
Three-point functions
Consider
for example
followingbut
diagram:
LoopTools provides not
the tensor
integrals the
themselves,
the coefficients of these
Lorentz-covariant tensors. It works in a basis formed from gµν and the momenta ki ,
p1
which are the sums of the external momenta pi (see Eq. (1.2)) [De93]. In this basis the
tensor-coefficient functions are totally symmetric in their indices. For the integrals up
q + k1
to the four-point function the decomposition reads explicitly
q m1
Bµ = k1µ B1 ,
Bµν = gµν B00 + k1µ k1ν B11 ,
Tensor
coefficients:
m2
m3
q + k2
p2
2
Cµ = k1µ C1 + k2µ C2 =
∑ kiµCi ,
i=1
p3
2
Cµν =The
gµν Cthree-point
00 + ∑ kiµ kfunction
jν Cij ,
corresponding to this diagram can be
i,j=1
of2 the
! external momenta "as
Cµνρ =
∑
i=1
3
2
gµν kiρ + gνρ kiµ + gµρ kiν C00i +
!
kiµ kjν kℓρ Cijℓ ,
∑
2 2
2
2
C pi,j,ℓ=1
1 , p2 , (p1 + p2 )
21
ki as
Dµ =or
Di , of the momenta
∑inkiµterms
, m1 , m22 , m23
"
e1 e2 e3
e1 e32 e3 = O(0.01)
v3 =
O(0.01)
Numerical
example
v
M1 = 125 GeV,
M1 = 125 GeV,
M2 = (200, 250, 300, 350) GeV,
M2 = (200, 250, 300, 350) GeV,
M3 = 400 GeV
M3 = 400 GeV
Z
f4 (s1
2
= p1 )
s1 =
22
2
vs2 s1 /MZ
p1
ZZZ vertex at one loop
f Z4 [10-3]v 3/e 1e2e3
1
250
200
300
0
-1
4
6
8
10
12
23
14
16
18
s1 /
M2Z
20
e1 e2 e3
1
max
= p ' 0.1925
3
v
3 3
e1 e2 e3
= O(0.01)
3
v
Qualitative features
Resonances, thresholds:
Z2
M1 = 125 GeV,
M2 = (200, 250, 300, 350) GeV, Hj
Z1
ej
M3 = 400 GeV
f4Z (s1 = p21 ) vs s1 /MZ2
at:
s1 = (MZ + M1 )2 ,
(MZ + M2 )2
24
Z
ei
Hk
ek
Z3
[Im
J
]
=
J
3v LS
1
Higgs
Basis
[Im J1 ]
=v J 3
Higgs Basis
3
6 LSLS
4v 4
LS1
v J2J
2
v 3 J3LS
[Im
J
=
2]]Higgs Basis==
[Im
J
]
v
J
[Im
J
30
30
Higgs
Basis
Higgs
Basis
Product of couplings:
[Im J1 ]Higgs Basis =
4
LS
[Im Je30
2 ]Higgs
2 Basis
2 = 2 v J2
+
e
+
e
21
22
32= v 2
e1 + e2 + e3 = v
e1 e22 e3 2 1 2
2
max e13 + e=2 +pe3 =
' v0.1925
1
v
3 3
max(e1 e2 e3 ) = p
e1 e2 e3
1 3 3
max
= p ' 0.1925
3
v
3 et al, 1205.6569:
For benchmarks studied
in3 Basso
e1 e2 e3
= O(0.01)
3
v
25
=
and
Then
SM@background
(ZhA)
A
ig[A
, A ]e 6= 0
and
and
andµ⌫
⌫ µ
and
F =@ A
µ
⌫
2
++
(hH
H )) q2q6=
0
1 6=
+H
(HH
0
+ Hµ ) ⌫
(hH
6=
⌫ µ
µ ⌫
(ZhH)
0 00
1 6=
andµ⌫
+ ) e3qq=
(hH
H
+
and
1
+ +H
abc
a
a
a (AH
10 0
(HH
=
6
2
0
(HH+HHb) ))cq3qq=
=
6
2
(HH
H
)
q
6= 0
+
2
++
and
+
+
(AH
(hH
20
30=
(hH
H+HH
) ) ))q1q6=
(AH
H
qq36=
=
00
1
V V V :! 0 Furry’s Theorem
(AH+ ++H+ ) q3 = 0
(HH
HH) )) q2q6=
6=06=300
(HH
H
2
(hH
q
1
V VV VV
AV:!
:⇠
Violates
P +
↵ 0Furry’s
V ✏:!
Furry’s
Theorem
V
0
Theorem
q0
+ +
2
(AH
H
)
q
=
(AH
H
)
q
=
0
(HH
H ) 3 µ3q2µ6=µ0 ⇢
VA
0 µ✏1Furry’s
Furry’s
Theorem
Theorem
V V:!
A
Violates
P
µ2✏:⇠
µ
ZV
↵
µ
µ
µ
3
3
1
2
1 2 3
VVVVAA
:⇠
Violates
P
↵
Z
1
+
4V AA
1 Theorem
1 ✏:!
(AH H )5 q3 = 0
V
V
A
:⇠
Violates
P
2
AAA
Violates
P
0
Furry’s
↵
V AA :! 0 Furry’s Theorem
F
p21
(ZhH) e e6=
3 =0 0
(ZHA)
1 0
(ZHA)
e
=
6
1
(ZhH)
e
(ZhH)
=
(ZhH) e33=
e30= 0
(ZhA) e2e6=
0
2 6=
(ZhA)
0
(ZHA)
e1 6= 0
+
(ZhH)
=01 6=
0 0
(hH
H
µ )e3⌫e=
3q
(ZhH)
=@ A
@ A + gf
(hHAHA)
q =
6 0
(HH H ) q =
6 0
(AH H ) q = 0
•M V
V V(p:!g0 Furry’s
[if
+ p Theorem
g
) + if ✏
M • VVV VV VA
:!
0✏0 Furry’s
Theorem
V
:!
Furry’s
Theorem
:⇠
Violates
AAA
:⇠
✏Furry’s
Violates
PP
V
AA
:!
0
Theorem
AAA
:⇠
Violates
VV V
✏✏:!
Violates
PPP
VA
AV:⇠
:⇠
Violates
V
V
0
Furry’s
Theorem
V
AA
:!
0
Furry’s
Theorem
• AAA
:⇠ ✏
Violates P
VV AA
:!
Theorem
AA
:!
Furry’s
Theorem
VV
A 0:⇠
Violates
P
AAA
:⇠
✏0 ✏Furry’s
Violates
P
/
p
AAA
:⇠
✏
Violates
PP
V
AA
:!
0
Furry’s
:⇠ ✏
ViolatesTheorem
• AAAAAA
:⇠ ✏
Violates P
Z
↵ ↵
↵↵
↵
↵
↵
↵
↵↵
µ1
1
↵
Contributions to
f 5 , f6
(p2
Z2
q
p3 ) ⇢ ]
q
Z3
Summary
• Invariants Im J , Im J , Im J
be expressed
in terms of physical couplings and masses
1
2
3 can
• All three neutral Higgses are “involved”
• Couplings lead to a calculable CP-violating
ZZZ coupling
• It will be a challenge to measure it