Filling the gaps between model predictions and their prerequisites in electric dipole moments

0.44 MB·English

by  T. Fukuyama

March 15, 2016
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1 Filling the gaps between model predictions and their
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2 prerequisites in electric dipole moments.

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Takeshi Fukuyamaa, 1 and Koichiro Asahib, 2

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p- a Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka, 567-0047, Japan
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h b Department of Physics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro, Tokyo 152-8551,
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Japan

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0 Abstract
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1
0 We clarify the conditions or assumptions under which theoretical predictions of various
5 models beyond the standard model give mainly in electric dipole moments. The correct
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interpretation of those conditions seems to be indispensable to the refinements of model
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building as well as to the mutual reliance in experimental and theoritical communities. The
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connections of these analyses to the recent experimental results at the LHC and the other
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a places are also discussed.
PACS numbers: 11.30.Er, 12.60.-i,
1E-mail:[email protected]
2E-mail:[email protected]
1 Introduction
Parmanentelectricdipolemoments(EDMs)ofparticlesaresmokinggunsofnewphysicsbeyond
the standard model (BSM). Indeed, the standard model (SM) predicts EDMs of all particles far
beyondtheupperboundsofongoingandnear-futureexperiments,whereasmanymodelsbeyond
the SM (BSM models) suggest many orders of magnitudes larger than those of the SM. Some
of them seem to give even larger values than the existing upper bound of experiments. Another
peculiar property of EDM is that it appears in a variety of hierarchical stages of matter. For
instance, electron EDM appears also in paramagnetic atoms and molecules in enhanced forms.
This drives us to study, beyond particle physics, a wide range of fields, nuclear physics, atomic
physics, chemical physics, and solid state physics. These properties, a variety of models and a
variety of EDM appearances, motivated us to write the review of EDM which resumes these
diverse fields in a compact and self-complete form [1]. However, experimental developments are
beyond our expectation. Especially, the recent improvement of the upper bound on electron
EDM by ACME [2] and the discovery of 126 GeV Higgs particle and the negative searches for
SUSYparticles by theLarge Hadronic Collider (LHC)at CERNare especially impressive[3] [4].
Theygiverichpreciousinformationstomodelsandforcethemtobemodified. Therehavearisen
many discussions between the experimental and theoretial communities. In these situations it is
very useful if many scientists over the wide regions can easily see the list of predictions made by
various models. One of the most excellent and well known one may be Figure 1 by Pendlebury
andHinds[5]. Unfortunately,wehaveseenveryfrequentlythatsomanypeoplescitethesemodel
results withoutcarefully considering the conditions or assumptions underwhich their results are
obtained. These situations are not happy for the mutual understanding between experimental
andtheoretical communities, andmodelbuildingitself. Thisisthemotivation ofthisletter. The
main objects of this letter, therefore, are to clarify the above mentioned conditions, referring
the details of calculations to [1], and to add some comments on the connections to the recent
great experimental results.
2 EDMs in the SM
In the SM, EDMs come from Kobayashi-Maskawa CP violating phase [6], apart from possible
very small contribution from θ term in quark sector. We resume in Table I the predicted EDM
values of various particles in the framework of the SM and experimental results. The detailed
explanations of it are given in [1].
As a result of loop corrections, the effective Lagrangian is written as
iψ AijP +AijP σµνψ F
− i L L R R j µν
(cid:16) (cid:17)
i 1
= − (Aij +Aij)ψσµνψF + (Aij Aij)ψσµνγ5ψF
2 L R µν 2 R − L µν
~σ B~ 0 0 ~σ E~
ij ij ij ij
= (A +A )ψ · ψ+i(A A )ψ · ψ, (1)
L R 0 ~σ B~ ! R − L ~σ E~ 0 !
· ·
Figure 1: Graph on the left:History of experimental upper limits on the electron and neutron
EDM. Right: Characterestic EDM predictions in various particle theory models. Predictions
for the electron EDM tend to lie lower than for the neutron and are hatched with double lines
cited from [5] by permision.
where notations follow those in [1]. For the EDM and magnetic dipole moment (MDM), we take
zero momentum of the photon. Then imaginary part of coefficients of the effective interaction
vanishes because of the optical theorem (imaginary part of the forward scattering amplitude is
given by the sum of possible cuts of intermediate states). We have an anomalous MDM a and
ψ
EDM d of particle ψ as
ψ
g 2 2m
a = − = ψ (Aii +Aii), (2)
ψ 2 −eQ ℜ R L
ψ
d = 2 (Aii Aii). (3)
ψ ℑ R− L
Here and express taking the real and imaginary parts, respectively.
ℜ ℑ
Table 1: The resume of EDM values or bounds from theoretical and experimental sides. The
second, third, and fourth columns are the number of loops in the first graph giving nonzero
value in the SM, EDM value in the SM, and experimental upper bounds, repectively. (*) The
SM bound of nucleons from Schiff moment of Hg is much weaker than those of 3-loop diagram,
d < 4.0 10 25, d < 3.8 10 24 e cm [8].
n − p −
× ×
particle loop number EDM value in SM/(e cm) experimental upper bound/(e cm)
quark 3 d 10 34
d −
≈
neutron 3 d 10 32 [7](*) 2.9 10 26 [9]
n − −
≈ ×
proton 3 d 10 32 (*) 7.9 10 25 [10]
p − −
≈ ×
deuteron d 1.5 10 31 [11]
D −
≈ ×
W boson 3 d 8 10 30
W −
≈ ×
lepton 4 d 8 10 41 d 8.7 10 29 [2]
e − e −
| |≈ × | |≤ ×
d = 1.9 10 19[12]
µ −
| | ×
For neutron EDM, we must take a contribution of chromo EDM d˜[13]
< qq >
d = (1 0.5)| | 0.55e(d˜ +0.5d˜ )+0.7(d 0.25d ) . (4)
n ± 225MeV3 d u d − u
h i
The experimental bound on neucleon may be obtained from the experimental value of 199Hg
EDM [10],
d 3.1 10 29 [e cm] (95% C.L.) (5)
Hg −
≤ ×
and theoretical estimations
d = (d˜ d˜ 0.012d˜) 3.2 10 2e (6)
Hg d u s −
− − − × ×
by the QCD sum rule [14] and
d = (d˜ d˜ 0.0051d˜) 8.7 10 3e (7)
Hg d u s −
− − − × ×
by the chiral Lagrangian method [15].
EDMs of light nuclear systems are calculated by combining effective ∆S = 1 four quark
interactions and π, η interaction [11].
For electron EDM there are many ongoing and near-future ambitious experiments using
polarized molecules. The advantage of diatomic molecule is due to strong internal electric field
and close rotation-vibration energy levels etc. [1, 16, 17]. So experimental d will severely
e
constrain BSM models in near-future.
3 EDMs in New Physics
In this section, we explain on the model predictions and thir prerequisites in the typical models,
two Higgs doublet model, left-right symmetric models, MSSM, and SUSY SO(10) GUT models.
3.1 Two Higgs Doublet Models (2HDMs)
As the simplest extension of the Higgs sector of the SM which has only one Higgs doublet φ ,
1
another Higgs doublet φ is introduced in the 2HDMs [18, 19, 20]. 3 There are several types of
2
the model depending on which doublet couples with which fermion:
type I (SM-like) : φ couples with all fermions
1
φ decouples from fermions
2
type II (MSSM-like) : φ couples with down-type quarks and charged leptons
1
φ couples with up-type quarks
2
type III (general) : both of Higgs doublets couple with all fermions
etc.
φ have vacuum expectation values (vevs) φ = v i = 1,2 and they are rotated by β =
i i i
h i
tan 1v /v as
− 2 1
G+ H+
Φ = , Φ = (8)
1 v+H1+iG0 2 H2+iA
! !
√2 √2
with v = v2+v2. Here H and A are CP-even and CP-odd, respectively. Fig. 1 contributes
1 2 i
to EDM via
p
1 sin2β Z
0n
H A = ℑ ,
h 1 i 2 q2 m2
n − n
X
1 cos2β Z Z˜
0n 0n
H A = ℑ −ℑ , (9)
h 2 i 2 q2 m2
n − n
X
where the summation is over all the mass eigenstates of neutral Higgs bosons, and the explicit
forms of Z and Z˜ are given in [21].
0n 0n
One-loop diagram (Fig. 2 (a)) contribution irrelevant to strong interaction is
e√2G tan2β
d1 loop = F (m3/m2)[ln(m2/m2)+3/2]( Z + Z˜ ) (10)
e,−µ (4π)2 0 0 ℑ 0 ℑ 0
in the limit of m2/m2 1. Here m is the mass of the lightest neutral Higgs boson and
0 ≪ 0
m = m , m [22].
e µ
However two-loop Barr-Zee diagram (Fig. 2 (b)) dominates over the one-loop diagram [23].
Dominant top loop diagram, for instance, gives
dtop = 16eme,µα√2GF [f(r )+g(r )] Z +[g(r ) f(r )] Z˜ (11)
e,µ − 3 (4π)3 { t t ℑ 0 t − t ℑ 0}
3We may consider the triplet Higgs model as a simpler model than 2HDM. However, it does not give any
predictable CP phaseand we donot discuss here.
(cid:13)
(cid:13)
t;b;(cid:28)(W)
A H1;H2 H1;H2
(cid:13);Z
A
(a) (b)
Figure 2: Diagrams contributing to EDMs of electron and muon in 2HDM.
with r = m2/m2. Here loop functions are given by
t t 0
r 1 1 2x(1 x) x(1 x)
f(r) = dx − − ln − , (12)
2 x(1 x) r r
Z0 − − (cid:20) (cid:21)
r 1 1 x(1 x)
g(r) = dx ln − (13)
2 x(1 x) r r
Z0 − − (cid:20) (cid:21)
and
emα√2G
F 1 10 25(m/(0.1GeV)) [e cm]. (14)
−
(4π)3 ≈ ×
The other loop contributions are given in [22] [24].
Condition;
♣
By comparing Eq.(11) with Fig.1, you can easily see that the estimated values in (14) and in [5]
are in units of Z . However, it is probable that Z 1. Indeed, the masses of neutral and
0 0
ℑ |ℑ |≪
Γ(Z bb) 0
charged Higgses and phases are tightly constrained from Rb ≡ Γ(Z h→adrons), Γ(b → sγ), B −B
→
mixing, ρ parameter etc., and we should take those constraints into account. Also Z satisfies
0
the sum rule [21]
Z˜ = Z = 0. (15)
n0 n0
n n
X X
Even if we assumed Z = 1, the estimated value amounts to be O(10 27) [e cm] for electron
0 −
|ℑ |
EDM. However, it is true that the recent experiment [2] gives severe constraints on the type II
model. It should beremarked that EDM is linear in its mass for two-loop and cubic for one-loop
diagrams. Here we have given the formulae for the type II model. It can be easily modified for
the type I and the other models [25] in which the numerical values of chromo EDMs in 2HDM
are also given under special assumptions of four unknown Higgs self coupling constants. One
of the most typical processes to check 2HDM is the 4σ excess of tauonic B decay from the SM,
B D( )τν . Unfortunately, however, the type II model (and probably the other types also)
∗ τ
→
may be excluded at least in their naive forms from the experiment, that is, mismatch of R(D)
and R(D ) in terms of tanβ/m [26]. Here
∗ H+
Br(B Dτ ν )
− τ
R(D) → (16)
≡ Br(B Dl ν )
− l
→
andl iseithereor µ. Alsoweneedmoreconstraints byapplyingthemodeltomanyphenomena.
In this sence, recent indications of diphoton excess at 750 GeV found at the LHC [27, 28] might
be a good chance for it. This is because 2HDM has additional neutral Higgses, H , A χ.
0 0
≡
Unfortunately these contributions to χ γγ come from top quark loop and severly suppressed
→
by m m 750 GeV [29]. In order to realise the observed data, in most cases, heavy
t χ
≪ ≈
vectorlike fermions are introduced. This is the very drastic change of physics. We need more
definiteandbroadeventssupportingsuchfermions. Inthissencemoreprecisevaluesofresonance
at 750 GeV.
3.2 Left-Right (LR) Symmetric Models
There are many LR models. The smallest gauge group of LR symmetry is
SU(2) SU(2) U(1) (17)
L R B L
× × −
Then charge quantization [30][31],
1
Q = I +I + (B L), (18)
3L 3R
2 −
is realized. If we consider it as a remnant from SO(10), SO(10) SU(4) SU(2) SU(2)
c L R
→ × ×
(If we start from this group, it is called the Pati-Salam (PS) model.), it satisfies at a certain
energy scale v
PS
g = g (19)
L R
and PS model is unified at M as
GUT
M M M M
4 2L 2LR 1/2
= = = . (20)
α α α α
4 2L 2R GUT
Also mixing matrices of left-handed and right-handed fermions are the same. Of course, these
constraints are realized at v but break down as the energy goes down to the SM scale by
PS
renormalization effects. However, we do not assume such scheme here. Matters and Higgs are
assinged as
φ0 φ+
Φ 1 2 = (1,2,2,0), (21)
≡ φ−1 φ02 !
ui ui
Q = L = (3,2,1,1/3), Q = R = (3,1,2,1/3) (22)
L di ! R di !
L R
under SU(3) SU(2) SU(2) U(1) . Left and right-handed doublets of lepton, L
c L R B L L
× × × −
and L , having the quantum number (1,2,1, 1) and (1,1,2, 1), are also incorporated.
R
− −
Φ couples with Q Q , so (B L)(Φ) = 0. This and (18) indicate that the symmetry
L R −
breaking
κ 0
Φ = (23)
h i 0 κ′eiλ !
leads to U(1) U(1) and not to U(1) [32]. So we need additionally, for instance,
I3L+I3R × B−L Q
∆ = (3,1,2), ∆ = (1,3,2). (24)
L R
Then the mass matrix of charged L-R weak bosons becomes
1g2(κ2+κ2+2v2) g2κκ
2 ′ L ′ , (25)
g2κκ′ 12g2(κ2 +κ′2+2vR2) !
where v and v are vevs of ∆ and ∆ , respectively (v v ). The transformation angle ζ
L R L R R L
≫
from W to mass eigenstates W ,
L,R 1,2
W = W cosζ W sinζeiλ, W = W sinζe iλ+W cosζ, (26)
1 L R 2 L − R
−
is given by
2κκ M2
tan2ζ = ′ WL. (27)
v2 v2 ≈ M2
R− L WR
(cid:13)
WL WR
eL (cid:23)L NR eR
Figure 3: LR symmetric diagram giving rise to electron EDM.
Lepton EDM comes mainly from the diagram (Fig. 3) and is given by [33]
eG m2
d = F I ( R ,0)sin2ζ (m ) (28)
e 8√2π2 1 M2 ℑ D
WL
m2
= 2.1 10 24I ( R ,0)sin2ζ( (m )/1MeV) [e cm]. (29)
× − 1 M2 ℑ D
WL
Here m (m )is the mass of heavy right-handed neutrinoN (Dirac neutrino), andthe mass of
R D R
light left-handed neutrino ν is given by m m2 /m for the type I seesaw. The loop function
L ν ≈ D R
I is
1
2 11 1 3r2lnr
I (r,0) 1 r+ r2 . (30)
1 ≈ (1 r)2 − 4 4 − 2(1 r)
− (cid:18) − (cid:19)
Inserting the observed upper limit of
M2
sin2ζ < 2.74 10 3 or ζ WL (31)
| | × − ≈ M2
WR
with
M > 2.15 103 GeV 95%C.L., (32)
WR ×
we obtain
2
(m ) m
de < 2.8 10−27 |ℑ D | e cm for R 1. (33)
| | × MeV M ≫
(cid:18) WL(cid:19)
Here we have considered a TeV order seesaw. It is more probable that m is of order of top
D
quark masss but in this case v amounts to O(1013) GeV and d is much less than the value
R e
of (33). The LR model also predicts d [34] utilizing the CP parameters of the neutral kaon
n
system,
4sinλ+1.4sin(λ δ) 0.1sin(λ+δ)
dn (10−21[e cm])η+ η00 − − (34)
≈ | −− | (sinλ+sin(λ+δ)) γ
| |
via
η η tanζ(sinλ+sin(λ+δ))γ . (35)
+ 00
| −− | ≈ | |
Here quark charge current eigenstates (d0, s0) are transformed to mass eigenstatres (d, s),
generalizing to invoke complex vacuum expectation value,
d = d0 cosθ s0 sinθ eiδL,R, s = d0sinθ e iδL,R +s0cosθ (36)
L,R L,R L,R− L,R L,R L,R L,R − L,R
and δ = δ δ . γ is an O(1) numerical factor coming from strong interaction.
L R
−
Condition;
♣
The expression (34) has simple forms in the following two cases:
(i) δ = 0
d (2.7/ γ ) (10 21e cm)η η (37)
n − + 00
| |≈ || | × | − − |
and
(ii) λ = 0
d (1.5/ γ ) (10 21e cm)η η (38)
n − + 00
| |≈ || | × | − − |
Due to PDG2012 [12]
η = (2.222 0.010) 10 3, η = (2.233 0.010) 10 3. (39)
00 − + −
| | ± × | −| ± ×
Using (35) and (39), we obtain
ζ 10 5/γ 10 6 (40)
− −
| |≥ | | ≥
This is consistent with (31). As for the recent diphoton excess at 750 GeV, the situation is same
as the case of 2HDM.
3.3 Minimal Supersymmetric Standard Model (MSSM)
MSSM has many CP phases and EDM appears in one-loop level. Note that A and A in (1)
L R
must include a fermion mass (or sfermion mass in the loop) because the effective interaction
ψσµνψ changes the chirality which can be done by the mass term in the soft SUSY breaking
Lagrangian [35],
1
= M g˜g˜+M W˜ W˜ +M B˜B˜ +c.c.
soft 3 2 1
L −2
(cid:16) (cid:17)
u˜A Q˜H d˜A Q˜H ˜eA L˜H +c.c. (41)
u u d d e d
− − −
(cid:16) (cid:17)
− Q˜†m2QQ˜ −L˜†m2LL˜ −u˜†m2uu˜†−d˜m2dd˜†−˜em2e˜e†
m2 H H m2 H H (bH H +c.c.).
− Hu u∗ u− Hd d∗ d− u d
Here tilde marks the SUSY partner. So even if we do not incorporate new CP phase, we
have 19 parameters (3 gaugino masses + tanβ + µ + b + 10 sfermion masses + 3 trilinear
terms) in addition to the SM+neutrinoparameters (27 for Dirac ν or 29 for Majorana ν), called
phenomenologicalMSSM(pMSSM).UniversalSUSY-breakingisaverystrongassumpotionthat
itrequiresnotonlyflavour-blindnessbutalsouniversality over quarksandleptonsatGUTscale,
m2 = m2 = m2 = m2 = m2 = m21 , (42)
Q u d L e 0 3
m = m = m , (43)
Hu Hd 0
M M M M
3 2 1 1/2
= = = , (44)
g2 g2 g2 g2
3 2 1 u
A = A Y , A = A Y , A = A Y . (45)
u 0 u d 0 d e 0 e