1. Technical University Munich
FCNC Processes
in Supersymmetry
Diego Guadagnoli
Technical University Munich
Outline
Introduction: impact of FCNC processes on generic SUSY corrections ⇒ New-Physics ‘flavour problem’
Approach 1: use exp. info to constrain the general MSSM
(general = with completely free soft terms)
Approach 2: implement a natural “near-flavour-conservation” mechanism
within the MSSM ⇒ MFV-MSSM
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

2. So, New Physics is already there
Exp. Facts about New Physics ☞
No hints (yet) of particle d.o.f. beyond the SM ones
from collider experiments, directly exploring high energy/short distance 
Ironically, many hints from astrophysics/cosmology,
i.e. from the other end in the range of distance scales studied in physics.
On one of them there is now large agreement:
– Dark Matter –
Many, very different, pieces of evidence for non-barionic, non-relativistic matter, which cannot be accounted for within the SM
E.g., from CMB:
 𝐷𝑀 ℎ 2 ≡  𝑀 −  𝐵  ℎ 2 = −
83 %
17 %
most of the ‘matter cake’ is a mystery within the SM alone
(and a perfect subject for the EC)
So, New Physics is already there
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

3. ‘Experimental’ scorecard of SUSY
SUSY (with R-parity conservation) has a clear DM candidate,
the Lightest SUSY Particle 
R-Parity is appealing in many respects, so it is not simply some ad hoc quantum number
it emerges naturally in the context of SO(10) GUTs
(and string theory)
it is the same mechanism stabilizing proton decay
If the three SM couplings at the MZ scale
are run upwards using the MSSM -functions,
they meet at ≈ 1016 GeV 
SUSY running
From: M. Peskin
SLAC-PUB-7479
That three lines cross at one point after
14 decades running in energy may be a coincidence. Maybe not.
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

4. SUSY breaking and FCNC ☑ ☑ ☑ ☑
We know very well how to construct an R-conserving,
SUSY- conserving MSSM. ☑
We know very poorly how to (softly) break it.
Hence, we have to parameterize this sector in the most general way. ☑
The understanding of this sector is as difficult as ambitious. It would shed light on the structure of the theory far beyond the EW scale. ☑
FCNC are, at the moment, the most powerful tool in this task. ☑
In fact, in the SM, FCNC are forbidden at tree level, so they
come with a loop suppression factor
(+ the hierarchical pattern required by the CKM/quark masses).
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

5. NON-diag.: Mass Insertion
General parameterization of FCNC in SUSY
In the CKM basis for quarks
the squark mass matrices
are still off-diagonal
in flavour (and ‘chirality’)
The “rotation” that makes quark masses diagonal
does not need to diagonalize squark masses as well
Then for squarks one chooses between:
MI IJ
𝑞 𝐼
𝑔 
𝑞  𝐽
 ZIJ
mass eigenstates ➊
Propagators diagonal ⇒ exact calculations
However, many parameters: MJ , ZIJ
MIJ
 IJ
𝑞 𝐼
𝑔 
𝑞  𝐽
flavour eigenstates ➋
Propagators perturbatively diagonalized
through “Mass Insertions”
Mass Matrices
Interactions
𝑀≡  𝑀 11  12 LL ⋯  21 LL 𝑀 22 ⋯ ⋮ ⋮ ⋱  ≃ 𝑀  𝟏  0  12 LL ⋯  21 LL 0 ⋯ ⋮ ⋮ ⋱ 
Mass
Insertion
Approx
SUSY source
of FCNC's
squark propagator
(flavor basis + MIA) ≃
diagonal + ×
NON-diag.: Mass Insertion 
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

6. Naïve assessment of SUSY effects
hm
  𝑀 SUSY  2
SUSY corrections ~
× f (SUSY mass ratios)
Example: ∆F = 2 case ☞
kl
Since mixing measurements check (within errors)
with the SM, one has roughly:
| SUSY corrections |
≤  exp 2   th 2
th > 10 % (!)
Sensitivity to O(%) deviations from SM demands steady improvement on the non-pert. error
bounds on 
(or rather, on /MSUSY)
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

7. ⇒ ⇒ Analysing FCNCs in SUSY
The SUSY (and NP in general) flavour problem
a) assuming MSUSY ~ O(300 GeV) [if we want it to stabilize the EW scale]
|| < 10-2 ÷ 10-3 ⇒
in absence of a symmetry principle (e.g. GIM-like mechanism)
such small numbers are ugly
Let us write the down quark mass matrix, in the basis with diagonal interaction terms (the equivalent of the MIA basis) [values in GeV]
𝑚  𝑑 MIA basis ≡  𝑚 𝑑 𝑚 𝑠 𝑚 𝑏  ⋅ 𝑉 CKM + =  6.8× 10 −3 −1.6× 10 −3 5.22.3𝑖 10 −5 2.3× 10 −2 9.7× 10 −2 −4.2× 10 −3 0.91.4𝑖× 10 −2 1.8× 10 − 
( does this matrix “look better” ? )
b) assuming || ~ O(1)
MSUSY ≫ O(TeV) ⇒
back to “Separation-of-Scales” Problems
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

8. Theoretical approaches to SUSY flavour effects
➊ Constrain the ‘general’ MSSM
(with completely free soft terms)
Take the  bounds “as they come”
(from measured FCNCs)
and study allowed effects on still-to-measure quantities (e.g. ACP [Bs  ], ...)
not easy without simplifying assumptions:
bulkiness of the parameter space
 Maybe FCNC effects in SUSY are small, because already
those in the SM are.
Naturalness of “near-flavour-conservation”
in SUSY: MFV-MSSM
Hall & Randall
D’Ambrosio et al.
i.e. low-energy flavour breaking
“building blocks” are the same in the SM and in the MSSM
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

9. Constrain the general MSSM
Constrain
the general MSSM

10. Example: constraints from b  s FCNCs
y-axis: Im[(23)XY]
x-axis: Re[(23)XY]
Constraints
□ = ms
□ = b  s 
□ = b  s l+ l-
□ = all
LL only, tan =3 ⇩
[-0.15,+0.15] + i [-0.25,0.25]
RR only, tan =3 ⇩
[-0.4,+0.4] + i [-0.9,0.9]
LL=RR, tan =3 ⇩
[-0.05,+0.05] + i [-0.03,0.03]
LL=RR, tan =10 ⇩
[-0.03,+0.03] + i [-0.02,0.02]

11. Implication on the Bs – mixing phase
In the SM one has
Arg 𝑀 12 𝑆𝑀 ≡Arg 𝐵 𝑠 ∣ 𝐻 𝑒𝑓𝑓,𝑆𝑀 𝐵,𝑆=2 ∣ 𝐵 𝑠  =−2  2 ≃−0.04
 What is the allowed range for with the previous limits on the ’s ?
Arg 𝑀 12 𝑀𝑆𝑆𝑀
LR only, tan =3
no sizable deviations
from the SM
LL only, tan =3
~ 10 × SM value are allowed
RR only, tan =3
~ 100 × SM value are easy to get
(but RR is still mildly constrained...)
LL=RR, tan =3
~ 100 × SM value are again easy
(yet LL=RR is severely constrained!)
The CP asymmetry in Bs  
will provide a truly fantastic probe!
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

12. Minimal Flavour Violation
Minimal Flavour Violation
in the MSSM

13. Minimal Flavour Violation (MFV)
In the SM, FCNC are small, because of the GIM mechanism.
Can extensions of the SM incorporate a similar mechanism
of near-flavour-(and CP)-conservation?
Controversial issue on how to define MFV
‘pragmatic’ definition, Buras et al., ‘00:
in terms of allowed effective operators + explicit occurrence of the CKM ➊
EFT definition, D’Ambrosio et al., ‘02: in terms of the SM Yukawa couplings ➋
Def.  does not produce a consistent low-energy
limit for the MSSM, even at low tan 
Altmannshofer,
Buras, D.G., ‘07
MFV can then only be defined as a ‘symmetry requirement’ for such new sources .
In fact, in extensions of the SM one has (by def.)
new, a priori unrelated sources of flavour (and CP) violation.
The set of allowed operators and FV structures is an outcome of such requirement .
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

14. ☞ the SM Yukawa couplings are
MFV ‘principle’
☞ the SM Yukawa couplings are
the only structures responsible
for low-energy flavour and CP violation
every new source of flavour violation
must be expressed as function of
the SM Yukawa couplings
Example: soft mass term for ‘left-handed’ squarks
𝐿 soft =−  𝑚 𝑄 𝐼𝐽  2   𝑢  𝐿 𝐼  ∗ 𝑢  𝐿 𝐽   𝑑  𝐿 𝐼  ∗ 𝑑  𝐿 𝐽  ...
a priori new source
of flavour violation
FC effects are naturally small:
intuitively  = O(1)f(CKM)
𝑚 𝑄 2 𝑇 = 𝑚 2  𝑎 1 𝟏 𝑏 1 𝐾 + 𝑌 𝑢 2 𝐾 𝑏 2 𝑌 𝑑 2 𝑂 𝑌 𝑢 2 𝑌 𝑑 2 
MFV
expansion
squark
mass scale
expansion
coefficients
and
free parameters after
the MFV expansion
Strategy
 After expansions, mass scales are only a few. Then:
Dramatic increase
in the predictivity and
testability of the model
 Fix them to scenarios
 Extract just the expansion coefficients
(12 indep. parameters)
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

15. aligned with the SM value
∆F=2 example
for mass scales chosen as
Mg = 500 GeV (gluino mass)
M1,2 = (100, 500) GeV (U(1)SU(2) gaugino masses)
 = GeV (-parameter)
m = 200 GeV (squark scale)
Bs – Bs amplitude
Comments
Distributions of values, due to the extraction of the expansion parameters, are quite narrow ⇨
Corrections are naturally small ⇨
Corrections are dominantly positive. Signature of the
MFV-MSSM at low tan  ⇨
Due to MFV,
the mixing phase is
aligned with the SM value
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

16. Test of ‘constrained’ MFV (CMFV) within the MSSM
The case of Q1 -dominated MFV (so-called CMFV)
can be tested by looking at the ratio
RCMFV 
contrib. to operators other than Q1
contrib. to Q1
One can ‘define’ CMFV to hold when, e.g.
| RCMFV | < 0.05
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

17.   “Conclusions” In either case, to resolve such effects,
In the general MSSM, SUSY effects are typically
constrained to be small (exceptions: (Bs ), ...)
after imposing existing exp. input 
In the MFV-MSSM, SUSY effects are naturally small,
due to a ‘built-in’ GIM-like mechanism. 
In either case, to resolve such effects,
one needs a better control, O(few %),
of the effective operator matrix elements
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007

18. Analysis of the separate contributions
Note
When  is large, LR entries in the squark mass matrices become relevant, even for low tan .
Example with (GeV):
m = 300
Mg = 300
M1,2 = (100,500)
 = 1000 ☞
They manifest dominantly in gluino contributions,
which become competitive with chargino’s.
D.Guadagnoli, Kick-off Meeting EC, May 14, 2007