1. The Electric Dipole Moment of the Neutron revisited
Schedar Marchetti
Euro-GDR 26/11/04
S. Marchetti - INFN Roma Tre

2. Motivations: 1
The electric dipole moment (edm) of particles is an important window to CP beyond SM in ΔF=0 sector
In the SM the edm’s arise from δCKM this contribution is much
smaller than current exp. limits (~ 6 order of magnitude)
|dN| < e · cm
In Supersymmetric models there are new sources of CP from the
complex phase of the soft SUSY breaking parameters and μ term
The presence of new CP is needed to have an efficient baryogenesis
Unconstrained MSSM ~ 40 phases
GUT allowed phases but only 2
mSUGRA are physical φA and φμ
Flavour Universality
Are this phases costrained using the current exp. costraints from neutron edm’s ?

3. Motivations: 2 Neutron edm SUSY problem
naive estimation dN ~ 2( ) sin φA,μ · e · cm
Two regions:
M > O(TeV) φA,μ ~ O(1) Hierarchy Problem
M < O(TeV) φA,μ ~ O(10 –2) Fine tuning? δCKM ~ O(1)?
To go beyond complete one loop analysis of neutron edm’s
scan of the parameter space just to see what happens varying
M in this regions
Alternative picture some cancellations mechanism beyond the
various contributions? 2
GeV M
Supersymmetric
scale

4. Electric Dipole Moment
The electric dipole moment of a classical distribution of charge is
defined as
d ≡ ∫ d3x x ρ(x)
The presence of an edm violates CP d = d
d → d , J → – J Under Time Reversal
d → – d , J → J Under Parity
d ≠ 0 violate T and P and so CP (CPT theorem)
In QFT the classical interaction Hint = – d • E corresponds to
Heff = – dE Ψ σµν γ5 Ψ Fµν ≡ C7 O (prove that dE = d)
spin J
|J| i 2

5. Electric Dipole Moment
If we consider strong interacting particle, such as atoms and hadrons
(e.g. mercury and neutron), other operators must take into account:
Heff = [ C7(μ) O7(μ) + C8(μ) O8(μ) + CG(μ) OG(μ) ]
O7 = – q σµν γ5 q Fµν Electric dipole
O8 = – gs q σµν γ5 ta q Gµν Chromoelectric dipole
OG = – gs fabc Gµρ Gbρν Gλσ εµνλσ Gluonic dipole
The Weinberg operator OG can be neglected at the leading order
only O7 and O8 enter in the game
EDM i 2 i a 2 1 3 a c 6
Euro-GDR 26/11/04
S. Marchetti - INFN Roma Tre

6. Electric Dipole Moment
Outline of the calculation Effective Theories
Projection of full theory on dipole operators
Matching
MSUSY
Ci = Ci(M)
Renormalization group equations μ
Evolution from matching to hadronic scale
governed by anomalous dimension matrix
Mixing
Operator Matrix Elements
Few informations
ΛQCD
e.g. naive dimensional analysis, ChPT, QCD Sum Rules d u 1
dN = [4C7(μ) – C7(μ)] 3
μ = 2 GeV

7. Electric Dipole Moment
One Loop diagrams
Gluino exchange
Chargino exchange
Neutralino exchange CP CP 2 mu ~
ΔLR u CP
Mass Insertion
Approximation 2 L
mu = ~ 2
ΔLR u * mu ~ R
Euro-GDR 26/11/04
S. Marchetti - INFN Roma Tre

8. CP Violation Sources beyond CKM
Quark-Squark sector
Super-CKM ~ ~ ~ ~ ′ ′ ′ ′
uL= Vu uL uR = Vu uR
uL= Vu uL uR = Vu uR L R L R ~ ~ ~ ~ ′ ′ ′ ′
dL = Vd dL dR = Vd dR
dL = Vd dL dR = Vd dR L R L R †
Quark mass
diagonal
Vu mu Vu = mu
diag R L
Gauge int. governed
by CKM ~ † † uL
L2 = – (uL uR) mu ~ ~ ~ 2 ~ uR
CP & F
CP & F CP 2 † † † 2
diag
diag
New source
of flavour and
CP violation
Vu mQ Vu + mu + Δu І ~
(Vu Au Vu - μ cot β) mu 2 L L L L L
mu = ~ † 2
diag 2
mu (Vu Au Vu – μ cot β)
diag
Vu mU Vu + mu + Δu І ~ L L R R R
CP & F
CP & F CP

9. CP Violation Sources beyond CKM
Chargino sector
Neutralino sector ~ ~
Ψ+ = ( W− , h+ ) ~ ~ ~ ~
Ψ0 = ( B− , W0 , h0 , h0 ) u ~ ~ d u
Ψ− = ( W−, h− ) d 1
L = − [(Ψ0)T MN Ψ0 + h.c. ]
L = − [(Ψ−)T MC Ψ+ + h.c. ] 2 CP CP M1
−cβswmz
sβswmz M2
cβcwmz
−sβcwmz
−cβswmz
cβcwmz
− μ
−cβswmz
cβcwmz
− μ M2
√2 sβ mw
MN =
MC =
√2 cβ mw μ CP CP
diag
diag
MC = UT MC V
MN = ZT MN Z

10. Electric Dipole Moment
Explicit expression of the supersymmetric contributions
Gluino exchange 2 αs
Im(ΔLR ) q
Qq A( , ) mg 2 ~ mg ~ q q C8
Qq A → B
C7 = 3 2 2 ~ 4π mg ~ mq mq ~ L R u C8
C → 0, Qd → 1
Chargino exchange 2 2 2 mχ ~ mχ ~ u αe
Im(Γi ) u
[(Qu− Qd) C( ) + Qd D ( )] ∑ i i
C7 = 2 2 2 mχ ~
4π sw mq ~ mq ~
i=1 i L L q C8
Qq → 1
Neutralino exchange 2 2 2 4 mχ ~ mχ ~ mχ ~ αe Qq
[∑ Im(ζ j ) D( ) + Im(ζ LR) A( , )] i q ∑ q q i i
C7 = − 2 2 2 2
4π sw mχ ~ mq ~ mq ~ mq ~
i=1
j=L,R i j L R

11. An Exploratory Analysis - 1
Switch on a single phase: φμ
First Scenario mass parameters above TeV scale
Parameter range :
Mi = (1000,1500),
|Au| = |Ad| = (1000,1500),
mq = (1000,1500),
|μ| = (1000,1500),
(GeV)
φA = φA = (-π, π),
φμ = (-π, π) ~ u d
positive sign
negative sign
exp. limit

12. An Exploratory Analysis - 2
Second Scenario mass parameters under TeV scale
Parameter range :
Mi = (100,150),
|Au| = |Ad| = (100,150),
mq = (100,150),
|μ| = (100,150),
(GeV)
φA = φA = (-π, π),
φμ = (-π, π) ~ u d
positive sign
negative sign
exp. limit

13. An Exploratory Analysis - 3
Two independent phase φA φA φA φμ φμ φμ
neglecting terms suppressed by a factor
O( ) mq φμ 2
φA , φμ MW
Gluino contribution seems to depend mainly on φμ this might
depend strongly on the assumption φA = φA u d

14. Preliminary Conclusions
Two observations:
1 Hierarchies gluino > chargino >> neutralino
2 Gluino tend to be opposite in sign to chargino possibility of a
systematic negative interference between the two contributions ?
in the previous literature* the sensitivity to this cancellations was
limited to more or less limited areas, and requiring at least an O(TeV)
parameter fine tuning ?
* Ibrahim et al. hep-ph/
Pokorski et al. hep-ph/

15. Improved Analysis (work in progress)
This is only a quick look… to go more in details:
The previous choice of parameters is clearly inconsistent
universal soft breaking at the GUT scale and running the parameters
down to EW scale
Inclusion of the NLO contributions at least for the dominant gluino
exchange
resummation of the terms through RGE
Possible contribution from Chromoelectric and Weinberg operators
Hadronic matrix elements beyond naive estimation
Another edm could take into the game (e.g. electron, mercury)
crossing analysis αs μ
( ) n
αs ln 4π M
Frascati 26/11/04
S. Marchetti - INFN Roma Tre