Table Of ContentTUM-HEP-725/09
The SUSY CP Problem and the MFV Principle
Paride Paradisi and David M. Straub
Physik-Department, Technische Universita¨t Mu¨nchen, 85748 Garching, Germany
We address the SUSY CP problem in the framework of Minimal Flavor Violation (MFV), where
theSUSYflavorproblemfindsanaturalsolution. Bycontrast,theMFVprincipledoesnotsolvethe
SUSY CP problem as it allows for the presence of new flavor blind CP-violating phases. Then, we
generalize the MFV ansatz accounting for a natural solution of it. The phenomenological implica-
tionsofthegeneralizedMFVansatzareexploredforMFVscenariosdefinedbothattheelectroweak
(EW) and at the GUT scales.
I. INTRODUCTION EW scale, the bounds on the EDMs of the electron and
neutron are violated by orders of magnitude: this is the
0 Supersymmetric (SUSY) extensions of the Standard so-called SUSY CP problem.
1 Model (SM) are broadly considered as the most moti- Either an extra assumption or a mechanism account-
0 vated and promising New Physics (NP) theories beyond ing for a natural suppression of these CPV phases are
2 theSM.Thesolutionofthegaugehierarchyproblem,the desirable.
n gaugecouplingunificationandthepossibilityofhavinga In this work, we assume a flavor blindness for the soft
a natural cold dark matter candidate, constitute the most sector,i.e. universalityofthesoftmassesandproportion-
J
convincing arguments in favor of SUSY. ality of the trilinear terms to the Yukawas, when SUSY
5 On the other hand, a generic SUSY scenario provides is broken. In this limit, we also assume CP conservation
2
many (dangerous) new sources of flavor and CP viola- and we allow for the breaking of CP only through the
] tion,hence,largenon-standardeffectsinflavorprocesses MFV compatible terms breaking the flavor blindness.
h would be typically expected. That is, CP is preserved by the sector responsible for
p However, the SM has been very successfully tested by SUSY breaking, while it is broken in the flavor sector.
-
p low-energyflavorobservablesbothfromthekaonandBd The generalized MFV scenario naturally solves the
e sectors. SUSY CP problem while leading to specific and testable
h Inparticular,thetwoB factorieshaveestablishedthat predictions in low energy CP violating processes.
[
Bd flavor and CP violating processes are well described
2 by the SM up to an accuracy of the (10 20)% level [1].
−
v This immediately implies a tension between the solu- II. CP VIOLATION IN SUSY MFV SCENARIOS
1
tionofthehierarchyproblem,callingforaNPscalebelow
5
the TeV, and the explanation of the Flavor Physics data
5 The hypothesis of MFV states that the SM Yukawa
requiringamulti-TeVNPscaleifthenewflavor-violating
4 matrices are the only source of flavor breaking, even in
. couplings are generic.
6 NP theories beyond the SM [2, 3]. The MFV ansatz
Anelegantwaytosimultaneouslysolvetheaboveprob-
0 offers a natural way to avoid unobserved large effects in
lemsisprovidedbytheMinimalFlavorViolation(MFV)
9 flavor physics and it relies on the observation that, for
0 hypothesis [2, 3], where flavor and CP violation are as- vanishing Yukawa couplings, the SM enjoys an enhanced
: sumed to be entirely described by the CKM matrix even
v global symmetry
in theories beyond the SM.
i
X However, the MFV principle does not provide in it-
G = SU(3) SU(3) SU(3) SU(3) SU(3) . (1)
r self any restriction to the presence of new CP-violating f u× d× Q× e× L
a
phases, hence, the assumption that the CKM phase pro-
The SM Yukawa couplings are formally invariant un-
vides the only source for CP violation (CPV) even in
der G if the Yukawa matrices are promoted to spurions
NP theories satisfying the MFV principle seems to be f
transforming in a suitable way under G . NP models are
not general and thus a restrictive assumption [4, 5] (see f
thenoftheMFVtypeiftheyareformallyinvariantunder
also [6–9]).
G , whentreatingtheSMYukawacouplingsasspurions.
In this context, we analyze the most general SUSY f
In the MSSM with conserved R-parity, the most gen-
scenario, compatible with the MFV principle, allowing
eral expressions for the low-energy soft-breaking terms
for the presence of new CP violating sources.
compatible with the MFV principle and relevant for our
Ingeneral,aMFVMSSMsuffersfromthesameSUSY
analysis read [4]
CP problem as the ordinary MSSM. In fact, the symme-
try principle of the MFV does not forbid the presence of (cid:20)
the dangerous flavor blind CP violating sources such as m2 = m2 1+r Y†Y +r Y†Y +
Q Q 1 u u 2 d d
the µ parameter in the Higgs potential or the trilinear
(cid:21)
scalar couplings A . When such phases assume natural
I + (c Y†Y Y†Y +h.c.) , (2)
(1) values and if the SUSY scale is not far from the 1 d d u u
O
2
(cid:20) (cid:18)
m2 = m2 1+Y r +r Y†Y +r Y†Y The naturalness problem of so small CP-violating
D D d 3 4 u u 5 d d phases, provided a SUSY scale of the order of the EW
(cid:19) (cid:21) scale, is commonly referred to as the SUSY CP problem.
+ (c2Yd†YdYu†Yu+h.c.) Yd† , (3) Hence, either an extra assumption or a mechanism ac-
counting for such a strong suppression in a natural way
are desirable.
(cid:18)
AU = A Y 1+c Y†Y +c Y†Y +
U u 3 d d 4 u u
(cid:19) III. A GENERALIZED MFV ANSATZ AND THE
+ c Y†Y Y†Y +c Y†Y Y†Y , (4) SUSY CP PROBLEM
5 d d u u 6 u u d d
The SUSY CP problem is automatically solved in the
(cid:18) MFVframeworkofD’Ambrosioetal.[3],astheyassume
AD = A Y 1+c Y†Y +c Y†Y + the extreme situation where the SM Yukawa couplings
D d 7 u u 8 d d
are the only source of CPV.
(cid:19)
+ c Y†Y Y†Y +c Y†Y Y†Y , (5) However, the MFV symmetry principle allows for the
9 d d u u 10 u u d d presenceofnewCPVphases,inparticularofflavor blind
phases that represent the main source of the SUSY CP
wheremQ,mD,AU andAD setthemassscaleofthesoft problem.
terms,whileri andci areunknown,orderone,numerical Instead of following the approch of D’Ambrosio et
coefficients. al. [3], we first observe that the assumption of the flavor
Notice that, in the above expansions, the SM Yukawa blindness corresponds to setting all the r and c coeffi-
i i
couplings are not assumed to be the only source of CPV cients to zero.
as done instead in [3]. In particular, while all the ri Inthislimit,weassumeCPconservationandweallow
parametersmustbereal,asthesquarkmassmatricesare for the breaking of CP only through the terms breaking
hermitian, the ci parameters are generally complex [4]. the flavor blindness.
AsintheordinaryMSSM,flavorconservingCPviolat- In this way, A , A , the gaugino masses and the µ
U D
ingsourcessuchastheµparameterintheHiggspotential term turn out to be real at the scale where the MFV
or the trilinear scalar couplings AI are unavoidable also holds while the leading imaginary components of the A
in SUSY MFV frameworks, as they are not forbidden by terms, induced by the complex parameters c , have a cu-
i
the symmetry principle of the MFV [4]. bic scaling with the Yukawas.
Physics observables will then depend only on the Notice that, after the infinite sum of MFV-compatible
phasesofthecombinationsMiµ,AIµandA∗IMi [11]and terms for Eqs. (2)-(5) is taken into account, the genera-
it is always possible to choose a basis where only the µ tion of CP-violating phases for A and A is unavoid-
U D
and AI parameters remain complex 1. able [4, 5, 10]. However, we have checked that these
These CP violating phases generally lead to too large phases are at most of order y2 10−4, hence, safely
effects for the electron and neutron EDMs, which are in- neglegible. ∼ c ∼
duced already at the one loop level through the virtual IfwenowdealwithalowscaleMFVscenario, theone
exchange of gauginos and sfermions of the first genera- loop contributions to the electron and neutron EDMs,
tion. that depend on the first generation A terms, are propor-
In particular, the current experimental bounds on the tional to the cube of light fermion masses, hence safely
electron [13] and neutron [14] EDMs imply that under control even for order one CPV phases of the c
i
parameters. Asaresult,thegeneralizedMFVansatzap-
(cid:16) m (cid:17)2(cid:18)10(cid:19)
sinφ (cid:46) 10−3 SUSY , plied to a low scale SUSY MFV scenario can completely
µ
| | 300 GeV tβ cure the SUSY CP problem.
(cid:16) m (cid:17)2 The situaton can drastically change if we define a
sinφ (cid:46) 10−2 SUSY ,
| A| 300 GeV SUSY MFV scenario at the GUT scale.
Inthislastcase, RGEeffectsstemmingfromtrilinears
if we impose the bounds on φµ and φA separately. In ofthethirdgenerationwillunavoidablygeneratecomplex
Eq. (6), tβ = tanβ and a common SUSY mass mSUSY trilinears for light generations and a complex µ term at
has been assumed. thelowscale. Asaresult,theEDMswillreceivebothone
and two loop contributions and the SUSY CP problem
might reappear. However, as we will discuss in detail
later, if CPV arises only from terms breaking the flavor
1 To be precise, such a statement is valid as long as the gaugino blindness,itwillbestillpossibletoaccountfortheSUSY
masses are universal at some scale. Even in this last case, two
CP problem in natural ways.
loop effects driven by a complex stop trilinear At generate an
ButhownaturalistheassumptionfortheoriginofCP
imaginary component for Mi [12] that we systematically take
intoaccountinournumericalanalysis. breaking in the generalized MFV scenarios?
3
As an attempt to address this question, we make a H˜ H˜ g˜ g˜
d u
comparison between the generalized MFV scenario and µ M
3
d d d d
SUSY flavor models. iR iL iR iL
In fact, one could envisage the possibility that the pe- Vti mtAt Vt∗i d˜Ri (δRdR)3i(δLqL)i3 d˜Li
cMuFliaVrpflraivnocripslterumcitguhrte bofetthheesroefmt-nseacnttorofdiacntautenddebrylytinhge t˜L t˜R ˜bR ˜bL
flavor symmetry holding at some high energy scale.
γ,g γ,g
Supersymmetricmodelswithabelian[15,16]andnon-
abelian [17, 18] flavor symmetries have been extensively f f f
discussed in the literature. They are based on the
t˜
Frogatt-Nielsen[19]mechanismwheretheflavorsymme- A0 R γ,g
tries are spontaneously broken by (generally complex)
µ
vacuum expectation values of some “flavon” fields Φ and
the hierarchical patterns in the fermion mass matrices t˜ mtAt t˜
n L R
can then be explained by suppression factors ( Φ /M) ,
(cid:104) (cid:105)
where M is the scale of integrated out physics and the
power n depends on the horizontal group charges of the γ,g
fermion, Higgs and flavon fields.
Then, such flavor symmetries, while being at the ori- FIG.1: RelevantSUSYcontributionstothefermionEDMsin
gin of the pattern of fermion masses and mixings, relate, theEWscaleMFVscenarios. UpperLeft: dominantone-loop
at the same time, the flavor structure of fermion and contribution to the quark EDM generated by Im(µAt) (cid:54)= 0.
Upper Right: flavor effect contributions to the quark EDM
sfermion mass matrices.
mediated by the one-loop exchange of gluino/down-squarks.
However, the CP violating effects to the EDMs driven
Lower: two-loop Barr-Zee type diagram generating an EDM
bytheflavor blindphasesφA,φµ are,ingeneral,notcon- for quarks (f =q) and leptons (f =(cid:96)) when Im(µAt)(cid:54)=0.
strained at all by the flavor symmetry and an additional
assumption is required.
The usual assumption employed by SUSY flavor mod- Fig. 1. It reads
els is that CP is a symmetry of the theory that is spon-
taneously broken only in the flavor sector as a result of (cid:26)d (cid:27) α 5 m m2
di 2 di t Im(A µ) V 2t , (6)
the flavor symmetry breaking [16, 18]. e (cid:39)− 16π18 m4 m2 t | ti| β
χ˜ q˜ W
Hence, we believe that the assumption we made on
theoriginofCPVinMFVscenariosisreminiscentofthe leadingtod 3 10−29ecmformaximumCPVphases,
n
usual approach followed in SUSY flavor models. m 500∼Ge×V and t = 10, still far from dexp (cid:46)
Inthelightoftheseconsiderations, weproceednowto 10S−U2S6Yec∼m. The enhancemβent to d induced by ImnA is
di t
analyze the phenomenological implications of the gener- compensated by the strong suppressing factor V 2.
ti
| |
alized MFV ansatz for MFV scenarios defined both at A much more important effect is provided by the two
the EW scale and at the GUT scale. loopBarr-ZeetypediagramofFig.1, alsoinvolvingonly
Inparticular,wewanttoaddressthequestionwhether the third sfermion generation [20]. These diagrams will
(1) phases for the MFV coefficients c , which are the generate the electron EDM d as well as the EDMs and
i e
O
onlysourceofCPVinoursetup,arephenomenologically chromo-EDMsforquarks. Inparticular,itturnsoutthat
allowed. d and the Mercury EDM d (as induced by the down-
e Hg
quark chromo-EDM) are the most sensitive observables
to this scenario. However, the theoretical estimation of
d passes through some nuclear calculations that un-
Hg
IV. EW SCALE MFV SCENARIOS avoidably suffer from sizable uncertainties [11] hence, in
the following, we focus on the predictions for d , to be
e
The generalized MFV ansatz described in the previ- conservative.
ous section, where the Ai terms are assumed to be the The induced electron EDM de reads
only sources of CPV, implies a hierarchical structure for
(cid:18) (cid:19)2(cid:18) (cid:19)
ImA . In particular, it turns out that ImA ImA , d 500GeV t
ImAbi (cid:29) ImAs,d and ImAτ (cid:29) ImAµ,e (astI(cid:29)mAi scca,ule ecme (cid:39)10−27 mSUSY 1β0 sin(φµ+φA), (7)
with the cube of the fermion masses) and this leads to a
naturalsuppressionfortheoneloopSUSYcontributions where in Eq. (7) we have assumed m = m . Thus,
SUSY A
to the EDMs. if (1) phases are allowed, d can reach the current ex-
e
O
Still,apotentiallyrelevantoneloopeffectforthedown perimental bound for mSUSY 500GeV and tβ =10.
∼
quark EDMs, proportional to ImA , is induced by the So far, we have not considered the contributions to
t
stop exchange, as shown in the left-hand diagram of the EDMs stemming from flavor effects [21]. Indeed, the
4
MFV flavor structures of Eqs. (2)–(5) provide additional
one loop “flavored” effects to the hadronic EDMs.
The off-diagonal terms of Eqs. (2)–(5) can be conve-
niently parameterized by means of the so-called MI pa-
rameters [22] defined as usual as
δL =(m2) /m2 and δR =(m2 ) /m2 (8)
ij Q ij Q ij D ij D
(cid:112)
with m2 = (m2 ) (m2 ) and X =Q,L. Then, from
X X ii X jj
Eqs. (2)–(5), it follows that
δL (r +c y2)V (9)
3i (cid:39) 1 1 b ti
δR y y (r +c y2)V . (10)
3i (cid:39) b di 4 2 b ti
One of the most important “flavored” effects to the
hadronic EDMs arises from the gluino/squark contribu-
tion shown in Fig. 1, leading to
(cid:26) (cid:27)
ddi αs mg˜ 4 Im(cid:2)δLδLRδR(cid:3) , (11)
e (cid:39)4πm2135 i3 33 3i
g˜ q˜
where δLR = m (A µt )/m2. The apparent bottom FIG. 2: Predictions for the electron EDM in a MFV frame-
Yukawa3e3nhancebmenbt−ofEβq.(11q˜),bymeansof(δd ) work defined at the EW scale. The upper band corresponds
m , is not effective within a SUSY MFV scenariLoRas33th∼e to the scenario where AU = AUYu(1 + c4Yu†Yu) with
b c = i, while the lower band refers to the scenario where
necessary δR MI turns out to be always proportional to A4U = A Y (1+c Y†Y ) with c = i. In both cases, the
light quark Yukawas, see Eq. (10). In the most favorable black poiUntsuare exc3luddeddby the co3nstraints from B physics
situation, where we assume maximum CPV and yb processes.
y 1, we find d 3 10−29ecm for m ≈=
t ≈ |{ di}g˜| (cid:39) × SUSY
500GeV and (1) parameters r , c .
i i
O | | | |
Still, the two loop contributions of Fig. 1 are largely twolooplevelwhileCPVeffectsinB-physicsobservables
dominant. The same conclusion holds for all the other arise already at one loop [7]; hence large effects in B-
flavoredeffectstothehadronicEDMs,hence,wewillnot physicscanbestillexpectedwhilebeingcompatiblewith
discuss them here. the EDM constraints. In particular, the phenomenology
Having discussed the dominant contributions to quark arising from the scenario discussed in this section is very
and lepton EDMs in the low-scale MFV setup, we pro- similar to that discussed in Ref. [7].
ceednowtoassessitsphenomenologicalviabilityinlight
of the experimental bounds on EDMs. As an illustrative
example, we choose a common SUSY mass m and V. GUT SCALE MFV SCENARIOS
SUSY
consider separately the two leading terms in the MFV
expansionofAU, assumingpurelyimaginarycoefficients Intheprevioussection,wehaveassumedthattheMFV
to be fair. expansion for the soft-breaking terms of Eqs. (2), (5)
Consequently,inFig.2,weshowthepredictionsforthe holds at the weak scale.
electronEDMde,asafunctionofacommonSUSYmass Incontrast,inthissection,weaddressthephenomeno-
mSUSY, arising within an EW scale MFV framework in logical implications for a MFV scenario defined at the
these two cases: high scale [4, 23]. In fact, even if we start with uni-
versal soft masses and proportional trilinear terms at
i) AU =A Y (1+c Y†Y ) with c =i,
U u 4 u u 4 the high-energy SUSY breaking scale M (correspond-
X
ii) AU =A Y (1+c Y†Y ) with c =i. ing to setting all the coefficients ri and ci to zero) RGE
U u 3 d d 3
effectsdonotpreservesuchauniversality. TheMFVco-
The most prominent feature of the two scenarios is their efficients r are RGE generated and their typical size is
i
dwihffielereinntcsacsaeliniig) dpero∼petr3βti.esMwoirtehovteβr:, tinhecapsreedii)ctdioen∼s ftoβr (M1/X4,πt)h2elneMffeXc2t/cma2SnUbSYe,sisgoniffiocrasnutffi. ciently large values of
d in the case ii) are suppressed compared to those of Moreover, as already discussed in Sec. II, it might be
e
case i) by a factor of y2/y2. Interestingly, Fig. 2 shows possible that the MFV flavor structure of the soft-sector
b t
that d is safely under control, but it can reach experi- can arise from an underlying flavor symmetry holding at
e
mentally visible levels, in both scenarios i) and ii) even some high energy scale.
for maximum CPV phases and a light SUSY spectrum. Inthisrespect,itseemsquitenaturaltodefineaMFV
We conclude noting that, within an EW scale MFV scenario at the high scale.
scenario, the EDMs receive the dominant effects at the As seen in Sec. IV, a remarkable virtue of a low-scale
5
MFV scenario is its natural solution to the SUSY CP 1
problem by means of hierarchical A terms.
However, generational hierarchies in the trilinear cou- (cid:76)At 0.1
plingsareaffectedbyRGeffectssincetheAtermsarenot (cid:72)m
0.01
I
protected by the non-renormalization theorem. There- (cid:76)(cid:144)
u
fore, even if these couplings are assumed to vanish at A 0.001
(cid:72)m
tehffeecGtsU.Tscale,theycanberegeneratedthroughrunning (cid:45)I 10(cid:45)4 5 tanΒ 40
Thisfactisparticularlyrelevantfortheimpactofcom-
10(cid:45)5
plex trilinears on quark or lepton EDMs. For example, 2 4 6 8 10 12 14 16
considertheRGequationfortheup-squarktrilinear; ne-
1
glecting Yukawa couplings of the two light generations
and U(1) gauge couplings, it reads 0.1
(cid:200)At
16π2ddtAu =6Atyt2−6g22M2− 332g32M3, (12) (cid:76)(cid:144)(cid:200)At 0.01
(cid:72)m 0.001
wsidheeroeftE=q.l(n1(2µ)/cµl0e)a.rlyThsheofiwrsstthtaetr,mevoenntihfethreighgta-uhgainndo I 10(cid:45)4 5 tanΒ 40
mass terms are real, Au can receive a sizable imaginary 10(cid:45)5
part if the stop trilinear A is complex, with potentially 2 4 6 8 10 12 14 16
t
dangerous impact on the one-loop contribution to the
log(cid:72)Μ(cid:144)GeV(cid:76)
neutron EDM.
Approximatenumericalexpressionsaccountingforthe
FIG. 3: Running of the trilinear terms in MFV scenar-
low energy values of A (m ) and A (m ) as a function
u Z t Z ios defined at the GUT scale. Upper plot: predictions for
of the high energy input parameters, valid for low to in-
ImA /ImA as a function of the renormalization scale µ as-
u t
termediate tanβ, are suming the GUT scale boundary condition ImA (m ) = 0
u G
and A (m ) (cid:54)= 0. Lower plot: predictions for ImA /|A | as
Au(mZ)−≈(cid:0)A0u.0(m5yGt2y)b2−c30+.410y.1t21AyUt4c+4(cid:1)0A.0U3y−b2A2.D8m1/2(,13) as(ucafpulpenecbrttoioluinnnGedo)afratynhdecorAnendUoitri=monaclAizAaUti=oYncsY4cAa†UlYeYµuwaYsitsu†huYmcuinwg=ittthhiec(4lGotw=UeTri
3 U u d d 3
band). A =A =m was assumed for both plots.
U 0 1/2
A (m ) A (m ) 0.81y2A 0.09y2A
t Z +≈(cid:0)0t.04yG2y−2c +0.t10yU4c−(cid:1)A b D
t b 3 t 4 U see that, even if we start with purely imaginary A (m )
−(cid:0)0.03yt2yb2c7+0.01yb4c8(cid:1)AD−2.2m1/2(.14) at the GUT scale, such that ImAt(mG)/|At(mG)t| =G1,
RGE effects reduce the phase of A by more than one
t
where the Yukawa couplings are to be evaluated at the order of magnitude in case i) and up to four orders of
lowscaleandwehaveneglectedtermsofO(yi6). Eq.(13) magnitude in case ii) depending on the tanβ value.
shows that, irrespective of Au(mG), a sizable contribu- As a result, the attained values for Au(mZ) and
tion to Au(mZ) from a complex At(mG) is unavoidable. At(mZ) lie within an experimentally allowed level for
This is well illustrated in the upper plot of Fig. 3 largeregionsoftheparameterspaceevenfor (1)phases,
where we show the predictions for the ratio ImAu/ImAt ameliorating significantly the SUSY CP proOblem.
as a function of the renormalization scale µ assuming Another even more dangerous CP violating contribu-
the GUT scale boundary condition ImAu(mG) = 0 and tion driven by the RGE effects regards the µ term. To
ImAt(mG) = 0. Interestingly, the attained low energy see this point explicitly, let’s consider the one loop RGE
(cid:54)
values for ImAu and ImAt are very similar in spite of for the µ parameter
| | | |
their very different values at the GUT scale, as is con-
firmed by Eqs. (13), (14). dµ = µ (cid:0) 3g2+y2+3y2+3y2(cid:1) . (15)
At the same time, huge RGE effects driven by the dt 16π2 − 2 τ b t
SU(3) interactions strongly reduce the phase of A (m )
u Z
and A (m ) (see Eqs. (13), (14)), provided the gaugino As we can see, the phase of µ does not run and this is
t Z
masses are real, as we assume. still true at the two loop level. On the other hand, the
ThisisillustratedinthelowerplotofFig.3, wherewe RGE for the bilinear mass term B is
show the predictions for ImA /A as a function of the
renormalization scale µ assumtin|g tth|e GUT scale bound- ddBt = 8π12 (cid:0)−3g22M2+yτ2Aτ +3yb2Ab+3yt2At(cid:1) , (16)
aryconditioni)AU =c A Y Y†Y settingc =i(up-
4 U u u u 4
per line), ii) AU = c A Y Y†Y setting c = i (lower then, in contrast to the µ term, the phase of the B term
3 U u d d 3
band) and assuming A =A =m in both cases. We is affected, through RGE effects, by the phases of the A
U 0 1/2
6
terms. To have an idea of where we stand, it is useful to 0.1
provide a numerical solution to Eq. (16) as a function of
the high scale parameters
0.01
B B +(cid:0)0.15+0.60t˜2(cid:1)m
0 1/2
≈
0.41y2A 0.42y2A 0.30y2A
−− (cid:0)0.05ytt2yUb2c−3+0.11byt4Dc4−(cid:1)AU τ E ΜΠ(cid:76)(cid:144) 0.001
− (cid:0)0.12yt2yb2c7+0.05yb4c8(cid:1)AD. (17) (cid:72)rgB
wheret˜=tanβ/50,theYukawacouplingsaretobeeval- (cid:45)A 10(cid:45)4
uatedatthelowscaleandwehaveagainneglectedterms
of O(yi6). 10(cid:45)5 tanΒ
Recall that, since the overall phase of the µ and Bµ 5 40
termscanberemovedbyaPeccei-Quinntransformation,
only their overall phase is physical; moreover, the phase
and absolute value of the Bµ term at the low scale is 10(cid:45)6
2 4 6 8 10 12 14 16
dictated by the EWSB conditions: in fact, in the basis
wheretheHiggsVEVsarereal,theseconditionsrequirea log(cid:72)Μ(cid:144)GeV(cid:76)
real Bµ term at the leading order 2. Thus, the condition
thatthisrelativephasevanishesatthehighscaleimplies FIG. 4: Running of the phase φµ +φB in a MFV frame-
that the µ term must be complex. work defined at the GUT scale with respect to the renor-
malization scale assuming the GUT scale boundary condi-
This is shown in Fig. 4, where we consider the de-
tion AU =c A Y Y†Y with c =i, while the lower band
pendence of the phase of φ +φ on the renormaliza- 4 U u u u 4
µ B referstothescenariowhereAU =c A Y Y†Y withc =i.
tion scale assuming the GUT scale boundary condition 3 U u d d 3
B =A =A =m was assumed in both cases.
AU = c A Y Y†Y with c = i (upper line) and 0 U 0 1/2
4 U u u u 4
AU = c A Y Y†Y with c = i (lower band), and as-
3 U u d d 3
suming A =A =B =m for definiteness.
U 0 0 1/2
After discussing the RGE effects leading to important
As we can see, the phase φ + φ = 0 at the high
µ B
contributionstoquarkandleptonEDMsinthehigh-scale
scale as φ = 0 and φ = 0 singularly. However, at the
µ B
MFVsetup, weproceednowtoassessitsphenomenolog-
low scale, φ +φ = 0 as the phase φ is generated by
µ B B
(cid:54) ical viability.
RGE effects, in contrast to the phase φ that remains
µ
vanishing as it does not run. In Fig. 5, we show the predictions for the electron
In principle, one can now impose by hand a real µ EDM in a MFV framework defined at the GUT scale
term at the EWSB scale (and hence at all scales), but assuming the two cases i) and ii) for the trilinears AU
then the Bµ term will be complex at the high scale; this alreadydiscussedinthelow-scalescenario; moreover, we
is the approach that is commonly assumed e.g. in the set AU =A0 =m0 =m1/2 mSUSY.
≡
CMSSM. However, in our scenario, we assume that CP Aswecansee,thescenarioi)isruledoutforanytanβ
violation only arises from the soft flavor-breaking terms value up to a SUSY scale of order mSUSY (cid:38)1.5 TeV. On
of the MFV expansion, hence, the B parameter at the the contrary, the scenario ii) is still phenomenologically
high scale is assumed to be real. allowed even for mSUSY at the EW scale, provided tanβ
Thus, if we start with a SUSY MFV scenario at the is moderate to small.
GUT scale, where CP violating sources are confined to The above findings require some comments. In fact,
the third generation A terms, at the low scale we un- from a phenomenological perspective, it seems unlikely
avoidably generate complex trilinears for light genera- that the coefficient c of Eq. 4 can be an (1) complex
4
tions and a complex µ term via RGE effects 3. As a parameter, as it would lead to the problemOs met in the
result, the EDMs receive both one and two loop contri- above scenario i). Let’s try now to argue which could be
butions. However, in our MFV scenario defined at the theunderlyingtheoreticalmotivationleadingtoarealc
4
GUT scale, the dominant effects to the EDMs are by far making a comparison with the typical situations occur-
those induced by the one loop effects of Fig. 6, through ring in SUSY flavor models.
the phase of the µ term.
In these last cases, the flavor symmetries are sponta-
neously broken by the complex vacuum expectation val-
ues of some “flavon” fields Φ and the hierarchical pat-
terns for the Yukawa matrices are explained in terms of
2 Beyond the leading order, the Bµ term acquires a small imagi- suppressing factors ( Φ /M)n, as discussed in Sec. II.
narypartinthepresenceofCPviolationintheµorAterms,in (cid:104) (cid:105)
Clearly, given that the top Yukawa coupling is an order
order to compensate the CP-odd tadpole counterterms [24–26],
one parameter, it does not require any suppressing fac-
whileitsrealpartiscorrectedbyCP-eventadpoles.
3 ArelatedstudywithintheCMSSMcanbefoundinRef.[27] toranditisformallyofthezerothorderinthe( Φ /M)n
(cid:104) (cid:105)
7
couldenableustoreconstructtheunderlyingscenarioat
work.
VI. LEPTONIC DIPOLE MOMENTS: d , (g−2)
e µ
AND BR((cid:96) →(cid:96) γ)
i j
Inthefollowing,webrieflydiscussthecorrelationsaris-
ing among dipole transitions in the leptonic sector [29].
In particular, we consider the electric dipole moment of
the electron d , the anomalous magnetic moment of the
e
muon a = (g 2) /2 and the branching ratio of the
µ µ
−
lepton flavor violating (LFV) decay µ eγ as these ob-
→
servables are highly complementary in shedding light on
NP. In fact, while a and d are sensitive to the real
µ e
and imaginary flavor diagonal dipole amplitude, respec-
tively, BR((cid:96) (cid:96) γ) constrains the absolute value of
i j
→
off-diagonal dipole amplitudes.
Interestingly, mostrecentanalysesofthemuon(g 2)
pointtowardsa3σdiscrepancyinthe10−9range[30,−31]:
FIG. 5: Predictions for the electron EDM in a MFV frame-
∆a =aexp aSM (3 1) 10−9. Hence, the question
work defined at the GUT scale assuming the boundary con- µ µ − µ ≈ ± ×
ditions AU = A Y (1+c Y†Y ) with c = i (upper line) we intend to address now is which are the expected val-
U u 4 u u 4
and AU = AUYu(1+c3Yd†Yd) setting c3 = i (lower band). ues for de and BR((cid:96)i → (cid:96)jγ) if we interpret the above
A = A = m = m ≡ m was assumed in both sce- discrepancy in terms of NP effects, in particular coming
U 0 0 1/2 SUSY
narios. from SUSY.
As an illustrative case, if we consider the limit of a
degenerate SUSY spectrum, the SUSY contributions to
g˜ g˜ H˜u W˜ ∆a and d (as induced by flavor blind phases) read
diR M3 diL eR H˜d µ v M2 W˜ eL µ e
α 5 m2
2 µ
∆a t ,
d˜Ri d˜Li ν˜,e˜ µ (cid:39) 4π β 12(cid:18)m2SUSY (cid:19)
(δLdR)ii de α2 t 5 me sinθ , (18)
e (cid:39) 4π β 24 m2 µ
γ,g γ SUSY
leading to
FIG. 6: Dominant one loop contributions to the EDMs of
quarks(left-handdiagram)andleptons(right-handdiagram) (cid:18) (cid:19)
d ∆a θ
in a MFV framework defined at the GUT scale. | e| 10−27 µ | µ| . (19)
ecm ≈ × 3 10−9 10−3
×
The result of Eq. (19) immediately leads to the conclu-
expansion. Concerning the low energy phenomenology
sionthat, aslongasSUSYeffectsaccountforthe(g 2)
of the GUT MFV scenario discussed in this section, we −
anomaly, the prediction for d typically exceeds its ex-
want to stress that the EDMs, arising at one loop level, e
perimental bound d (cid:46) 10−27 unless θ (cid:46) 10−3. An
are the most promising observables and they generally e | µ|
explanation for such a strong suppression of θ can nat-
prevent any visible effect in CPV B-physics observables. µ
urally arise within the general GUT MFV framework, as
ThisisincontrastwiththeEWMFVscenariowherethe
discussed in the previous section. In fact, even assuming
EDMconstraintswerelessstringent(astheyariseatthe
maximum CP violation in the high-scale trilinears and
two loop level) and large B-physics signals, correlated
setting the unknown MFV coefficient c = 1, we have
with the predictions for the EDMs, were still allowed. 3
found that θ (cid:38)10−4, as shown in Fig. 4.
However, the above features of the EW and GUT sce- | µ|
Passing to BR((cid:96) (cid:96) γ) and assuming again a degen-
narios still cannot be considered as an unambiguous tool i → j
erate SUSY spectrum, it is straightforward to find [32]
to disentangle the two models. In fact, also in the GUT
MariFseVastcetnhaeriotw, toh-leoompailnevceoln(tsreibeuFtiiogn.s1t)oitnhethEeDcMonstceaxnt BR(µ eγ) 2 10−12(cid:34)∆aSµUSY (cid:35)2(cid:12)(cid:12)(cid:12) δµLe (cid:12)(cid:12)(cid:12)2. (20)
of hierarchical sfermions with light third and heavy first → ≈ × 3 10−9 (cid:12)10−4(cid:12)
×
generations [28]. Should this be the case, the low-energy
footprints of the EW and GUT MFV scenarios would where we have assumed that BR(µ eγ) is generated
→
turn out to be indistinguishable and only a synergy of onlybytheflavorstructuresamongleft-handedsleptons,
flavor data with the LHC data for the SUSY spectrum i.e. δL , as it happens in SUSY see-saw scenarios.
µe
8
The main messages from the above relations is that the MFV ansatz accounting for a natural solution of the
within a GUT MFV SUSY scenario, with generalized SUSY CP problem.
MFVansatz,anexplanationforthemuon(g 2)anomaly We have assumed flavor blindness, i.e. universality of
−
leads to predictions for d that are close to the cur- thesoftmassesandproportionalityofthetrilinearterms
e
rent experimental upper bound d (cid:46) 10−27e cm while to the Yukawas, when SUSY is broken.
e
BR(µ eγ)typicallylieswithintheexpectedMEGres- In this limit, we have assumed CP conservation al-
olution→s [33] for values of δL covering the predictions of lowing for the breaking of CP only through the MFV
µe
many SUSY see-saw scenarios. compatible terms breaking the flavor blindness.
That is, CP is preserved by the sector responsible for
SUSY breaking, while it is broken in the flavor sector.
VII. CONCLUSIONS Wehaveexploredthephenomenologicalimplicationsof
this generalized MFV ansatz for MFV scenarios defined
InthisworkwehaveaddressedtheSUSYCPproblem both at the electroweak and at the GUT scales, point-
in the framework of the MFV, where the SUSY flavor ing out the profound differences ofthe two scenariosand
problem finds a natural solution. By contrast, the MFV their peculiar and testable predictions in low energy CP
principle does not solve the SUSY CP problem as the violating processes.
MFV symmetry principle allows for the presence of new Acknowledgments: We thank A. J. Buras for useful
flavor blind CP-violating phases [4, 5] (see also [6–9]). discussionsandcommentsonthemanuscript. Thiswork
Hence,theMFVansatzhastobesupplementedeither has been supported in part by the Cluster of Excellence
by an extra assumption or by a mechanism accounting “Origin and Structure of the Universe” and by the Ger-
foranaturalsuppressionoftheflavor blindCPVphases. man Bundesministerium fu¨r Bildung und Forschung un-
Inthelightoftheseconsiderations,wehavegeneralized der contract 05HT6WOA.
[1] E. Barberio et al. [Heavy Flavor Averaging Group], 3741; A. Pomarol and D. Tommasini, Nucl. Phys. B466
arXiv:0808.1297 [hep-ex]. (1996)3;L.J.HallandH.Murayama,Phys.Rev.Lett.75
[2] R.S.ChivukulaandH.Georgi,Phys.Lett.B188(1987) (1995) 3985; C.D. Carone et al.,Phys. Rev. D54 (1996)
99;L.J.HallandL.Randall,Phys.Rev.Lett.65(1990) 2328; R. Barbieri et al, Phys. Lett. B 377, 76 (1996);
2939; A. J. Buras et al., Phys. Lett. B500 (2001) 161. Nucl. Phys. B 493, 3 (1997); Phys. Lett. B 401, 47
[3] G. D’Ambrosio et al., Nucl. Phys. B645 (2002) 155. (1997).
[4] G.Colangelo,E.NikolidakisandC.Smith,Eur.Phys.J. [18] G.G.Ross,L.Velasco-SevillaandO.Vives,Nucl.Phys.
C 59, 75 (2009). B 692, 50 (2004).
[5] L. Mercolli and C. Smith, Nucl. Phys. B 817, 1 (2009). [19] C.D.FrogattandH.B.Nielsen,Nucl.Phys.B147(1979)
[6] J. R. Ellis, J. S. Lee and A. Pilaftsis, Phys. Rev. D 76, 277.
115011 (2007). [20] D.Chang,W.Y.KeungandA.Pilaftsis,Phys.Rev.Lett.
[7] W. Altmannshofer, A. J. Buras and P. Paradisi, Phys. 82, 900 (1999) [Erratum-ibid. 83, 3972 (1999)].
Lett. B 669, 239 (2008). [21] J. Hisano, M. Nagai and P. Paradisi, Phys. Lett. B
[8] A. L. Kagan et al., arXiv:0903.1794 [hep-ph]. 642, 510 (2006); Phys. Rev. D 78, 075019 (2008);
[9] T. Feldmann, M. Jung and T. Mannel, arXiv:0906.1523 arXiv:0812.4283 [hep-ph].
[hep-ph]. [22] L.J.Hall,V.A.KosteleckyandS.Raby, Nucl.Phys.B
[10] We thank C. Smith for drawing this point to our atten- 267 (1986) 415; F. Gabbiani et al., Nucl. Phys. B 477,
tion. 321(1996);M.Ciuchinietal.,Nucl.Phys.B783(2007)
[11] For a review of EDMs please see, M. Pospelov and 112.
A.Ritz,AnnalsPhys.318,119(2005)andthereinrefer- [23] P. Paradisi, M. Ratz, R. Schieren and C. Simonetto,
ences;J.R.Ellis,J.S.LeeandA.Pilaftsis,JHEP0810, Phys. Lett. B 668 (2008) 202.
049 (2008). [24] A. Pilaftsis, Phys. Rev. D 58 (1998) 096010.
[12] K. A. Olive et al., Phys. Rev. D 72, 075001 (2005). [25] A. Pilaftsis and C. E. M. Wagner, Nucl. Phys. B 553
[13] B. C. Regan et al., Phys. Rev. Lett. 88, 071805 (2002). (1999) 3.
[14] C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006). [26] D. A. Demir, Phys. Rev. D 60 (1999) 055006.
[15] M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B398 [27] R.GaristoandJ.D.Wells,Phys.Rev.D55(1997)1611.
(1993) 319; Nucl. Phys. B420 (1994) 468; L.E. Ibanez [28] A. G. Cohen et al., Phys. Lett. B388, 588 (1996).
andG.G.Ross,Phys.Lett.B332(1994)100;P.Binetruy [29] Foradetailedanalysisofthissubject,pleaseseeJ.Hisano
and P. Ramond, Phys. Lett. B350 (1995) 49; V. Jain et al., arXiv:0904.2080 [hep-ph].
and R. Shrock, Phys. Lett. B352 (1995) 83; E. Dudas, [30] M.Passera,J.Phys.G31(2005)R75;Nucl.Phys.Proc.
S.PokorskiandC.A.Savoy,Phys.Lett.B356(1995)45; Suppl. 155 (2006) 365; M. Davier, Nucl. Phys. Proc.
P. Binetruy, S. Lavignac and P. Ramond, Nucl. Phys. Suppl. 169, 288 (2007); K. Hagiwara et al., Phys. Lett.
B477 (1996) 353. B 649, 173 (2007).
[16] Y. Nir and R. Rattazzi, Phys. Lett. B 382, 363 (1996). [31] M. Passera, W. J. Marciano and A. Sirlin, Phys. Rev. D
[17] D.B. Kaplan and M. Schmaltz, Phys. Rev. D49 (1994) 78, 013009 (2008).
9
[32] J. Hisano and K. Tobe, Phys. Lett. B 510, 197 (2001); ley, Italy (March 1-7, 2009). Presentatation is placed on
G. Isidori et al., Phys. Rev. D 75 (2007) 115019. http://agenda.infn.it/conferenceDisplay.py?confId=930.
[33] Talk given by Marco Grassi, Les Rencontres de
Physique de la Vallee D’Aoste, La Thuile, Aosta Val-
