Squark and Gaugino Hadroproduction and Decays in Non-Minimal Flavour
  Violating Supersymmetry
Squark and Gaugino Hadroproduction and Decays in Non-Minimal
Flavour Violating Supersymmetry
Giuseppe Bozzi
Institut für Theoretische Physik, Universität Karlsruhe, Postfach 6980, D-76128 Karlsruhe, Germany
Benjamin Fuks, Björn Herrmann, and Michael Klasen∗
Laboratoire de Physique Subatomique et de Cosmologie,
Université Joseph Fourier/CNRS-IN2P3, 53 Avenue des Martyrs, F-38026 Grenoble, France
(Dated: October 29, 2018)
We present an extensive analysis of squark and gaugino hadroproduction and decays in non-
minimal flavour violating supersymmetry. We employ the so-called super-CKM basis to define the
possible misalignment of quark and squark rotations, and we use generalized (possibly complex)
charges to define the mutual couplings of (s)quarks and gauge bosons/gauginos. The cross sections
for all squark-(anti-)squark/gaugino pair and squark-gaugino associated production processes as
well as their decay widths are then given in compact analytic form. For four different constrained
supersymmetry breaking models with non-minimal flavour violation in the second/third generation
squark sector only, we establish the parameter space regions allowed/favoured by low-energy, elec-
troweak precision, and cosmological constraints and display the chirality and flavour decomposition
of all up- and down-type squark mass eigenstates. Finally, we compute numerically the dependence
of a representative sample of production cross sections at the LHC on the off-diagonal mass matrix
elements in the experimentally allowed/favoured ranges.
I. INTRODUCTION
KA-TP-07-2007
LPSC 07-023
SFB/CPP-07-09
Weak scale supersymmetry (SUSY) remains a both theoretically and phenomenologically attractive extension of
the Standard Model (SM) of particle physics [1, 2]. Apart from linking bosons with fermions and unifying internal and
external (space-time) symmetries, SUSY allows for a stabilization of the gap between the Planck and the electroweak
scale and for gauge coupling unification at high energies. It appears naturally in string theories, includes gravity, and
contains a stable lightest SUSY particle (LSP) as a dark matter candidate. Spin partners of the SM particles have not
yet been observed, and in order to remain a viable solution to the hierarchy problem, SUSY must be broken at low
energy via soft mass terms in the Lagrangian. As a consequence, the SUSY particles must be massive in comparison
to their SM counterparts, and the Tevatron and the LHC will perform a conclusive search covering a wide range of
masses up to the TeV scale.
If SUSY particles exist, they should also appear in virtual particle loops and affect low-energy and electroweak
precision observables. In particular, flavour-changing neutral currents (FCNC), which appear only at the one-loop
level even in the SM, put severe constraints on new physics contributions appearing at the same perturbative order.
Extended technicolour and many composite models have thus been ruled out, while the Minimal Supersymmetric
Standard Model (MSSM) has passed these crucial tests. This is largely due to the assumption of constrained Minimal
Flavour Violation (cMFV) [3, 4] or Minimal Flavour Violation (MFV) [5, 6, 7], where heavy SUSY particles may
appear in the loops, but flavour changes are either neglected or completely dictated by the structure of the Yukawa
couplings and thus the CKM-matrix [8, 9].
The squark mass matrices M2
, and M2
are usually expressed in the super-CKM flavour basis [10]. In MFV
SUSY scenarios, their flavour violating non-diagonal entries ∆ij , where i, j = L,R refer to the helicity of the (SM
partner of the) squark, stem from the trilinear Yukawa couplings of the fermion and Higgs supermultiplets and the
resulting different renormalizations of the quark and squark mass matrices, which induce additional flavour violation
at the weak scale through renormalization group running [11, 12, 13, 14], while in cMFV scenarios, these flavour
violating off-diagonal entries are simply neglected at both the SUSY-breaking and the weak scale.
When SUSY is embedded in larger structures such as grand unified theories (GUTs), new sources of flavour violation
can appear [15]. For example, local gauge symmetry allows for R-parity violating terms in the SUSY Lagrangian, but
these terms are today severely constrained by proton decay and collider searches. In non-minimal flavour violating
(NMFV) SUSY, additional sources of flavour violation are included in the mass matrices at the weak scale, and
∗klasen@lpsc.in2p3.fr
http://arxiv.org/abs/0704.1826v2
mailto:klasen@lpsc.in2p3.fr
their flavour violating off-diagonal terms cannot be simply deduced from the CKM matrix alone. NMFV is then
conveniently parameterized in the super-CKM basis by considering them as free parameters. The scaling of these
entries with the SUSY-breaking scale MSUSY implies a hierarchy ∆LL ≫ ∆LR,RL ≫ ∆RR [15].
Squark mixing is expected to be largest for the second and third generations due to the large Yukawa couplings
involved [16]. In addition, stringent experimental constraints for the first generation are imposed by precise mea-
surements of K0 − K̄0 mixing and first evidence of D0 − D̄0 mixing [17, 18, 19]. Furthermore, direct searches of
flavour violation depend on the possibility of flavour tagging, established experimentally only for heavy flavours. We
therefore consider here only flavour mixings of second- and third-generation squarks.
The direct search for SUSY particles constitutes a major physics goal of present (Tevatron) and future (LHC)
hadron colliders. SUSY particle hadroproduction and decay has therefore been studied in detail theoretically. Next-
to-leading order (NLO) SUSY-QCD calculations exist for the production of squarks and gluinos [20], sleptons [21],
and gauginos [22] as well as for their associated production [23]. The production of top [24] and bottom [25] squarks
with large helicity mixing has received particular attention. Recently, both QCD one-loop and electroweak tree-level
contributions have been calculated for non-diagonal, diagonal and mixed top and bottom squark pair production [26].
However, flavour violation has never been considered in the context of collider searches apart from the CKM-matrix
appearing in the electroweak stop-sbottom production channel [26].
It is the aim of this paper to investigate for the first time the possible effects of non-minimal flavour violation
at hadron colliders. To this end, we re-calculate all squark and gaugino production and decay helicity amplitudes,
keeping at the same time the CKM-matrix and the quark masses to account for non-diagonal charged-current gaugino
and Higgsino Yukawa interactions, and generalizing the two-dimensional helicity mixing matrices, often assumed to be
real, to generally complex six-dimensional helicity and generational mixing matrices. We keep the notation compact
by presenting all analytical expressions in terms of generalized couplings. In order to obtain numerical predictions
for hadron colliders, we have implemented all our results in a flexible computer program. In our phenomenological
analysis of NMFV squark and gaugino production, we concentrate on the LHC due to its larger centre-of-mass energy
and luminosity. We pay particular attention to the interesting interplay of parton density functions (PDFs), which
are dominated by light quarks, strong gluino contributions, which are generally larger than electroweak contributions
and need not be flavour-diagonal, and the appearance of third-generation squarks in the final state, which are easily
identified experimentally and generally lighter than first- and second-generation squarks.
After reviewing the MSSM with NMFV and setting up our notation in Sec. II, we define in Sec. III generalized
couplings of quarks, squarks, gauge bosons, and gauginos. We then use these couplings to present our analytical
calculations in concise form. In particular, we have computed partonic cross sections for NMFV squark-antisquark
and squark-squark pair production, squark and gaugino associated and gaugino pair production as well as NMFV
two-body decay widths of all squarks and gauginos. Section IV is devoted to a precise numerical analysis of the
experimentally allowed NMFV SUSY parameter space, an investigation of the corresponding helicity and flavour
decomposition of the up- and down-type squarks, and the definition of four collider-friendly benchmark points. These
points are then investigated in detail in Sec. V so as to determine the possible sensitivity of the LHC experiments on
the allowed NMFV parameter regions in the above-mentioned production channels. Our conclusions are presented in
Sec. VI.
II. NON-MINIMAL FLAVOUR VIOLATION IN THE MSSM
Within the SM, the only source of flavour violation arises through the rotation of the up-type (down-type) quark
interaction eigenstates u′L,R (d
L,R) to the basis of physical mass eigenstates uL,R (dL,R), such that
dL,R = V
L,R d
L,R and uL,R = V
L,R u
L,R. (1)
The four bi-unitary matrices V
L,R diagonalize the quark Yukawa matrices and render the charged-current interactions
proportional to the unitary CKM-matrix [8, 9]
V = V uL V
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
 . (2)
In the super-CKM basis [10], the squark interaction eigenstates undergo the same rotations at high energy scale
as their quark counterparts, so that their charged-current interactions are also proportional to the SM CKM-matrix.
However, different renormalizations of quarks and squarks introduce a mismatch of quark and squark field rotations
at low energies, so that the squark mass matrices effectively become non-diagonal [11, 12, 13, 14]. NMFV is then
conveniently parameterized by non-diagonal entries ∆
ij with i, j = L,R in the squared squark mass matrices
M2ũ =

∆ucLL ∆
LL muXu ∆
∆cuLL M
∆ctLL ∆
RL mcXc ∆
∆tuLL ∆
∆tuRL ∆
RL mtXt
muXu ∆
∆ucRR ∆
∆cuLR mcXc ∆
∆ctRR
∆tuLR ∆
LR mtXt ∆


∆dsLL ∆
LL mdXd ∆
∆sdLL M
∆sbLL ∆
RL msXs ∆
∆bdLL ∆
∆bdRL ∆
RL mbXb
mdXd ∆
∆dsRR ∆
∆sdLR msXs ∆
∆sbRR
∆bdLR ∆
LR mbXb ∆

, (4)
where the diagonal elements are given by
+m2q + cos 2βM
q − eqs2W ), (5)
+m2q + cos 2βM
W for up− type squarks, (6)
+m2q + cos 2βM
W for down− type squarks, (7)
while the well-known squark helicity mixing is generated by the elements
Xq = Aq − µ
cotβ for up− type squarks,
tanβ for down− type squarks. (8)
Here, mq, T
q , and eq denote the mass, weak isospin quantum number, and electric charge of the quark q. mZ is
the Z-boson mass, and sW (cW ) is the sine (cosine) of the electroweak mixing angle θW . The soft SUSY-breaking
mass terms are MQ̃q and MŨq,D̃q for the left- and right-handed squarks. Aq and µ are the trilinear coupling and
off-diagonal Higgs mass parameter, respectively, and tanβ = vu/vd is the ratio of vacuum expectation values of the
two Higgs doublets. The scaling of the flavour violating entries ∆
ij with the SUSY-breaking scale MSUSY implies a
hierarchy ∆
LL ≫ ∆
LR,RL ≫ ∆
RR among them [15]. They are usually normalized to the diagonal entries [18], so that
ij = λ
ij MĩqMj̃q′
. (9)
Note also that SU(2) gauge invariance relates the (numerically largest) ∆
LL of up- and down-type quarks through
the CKM-matrix, implying that a large difference between them is not allowed.
The diagonalization of the mass matrices M2ũ and M
requires the introduction of two additional 6 × 6 matrices
Ru and Rd with
diag (m2ũ1 , . . . ,m
) = RuM2ũ R
u† and diag (m2
, . . . ,m2
) = RdM2
Rd†. (10)
By convention, the masses are ordered according to mq̃1 < . . . < mq̃6 . The physical mass eigenstates are given by








. (11)
In the limit of vanishing off-diagonal parameters ∆
ij , the matrices R
q become flavour-diagonal, leaving only the
well-known helicity mixing already present in cMFV.
III. ANALYTICAL RESULTS FOR PRODUCTION CROSS SECTIONS AND DECAY WIDTHS
In this section, we introduce concise definitions of generalized strong and electroweak couplings in NMFV SUSY
and compute analytically the corresponding partonic cross sections for squark and gaugino production as well as their
decay widths. The cross sections of the production processes
aha(pa) bhb(pb) →
i (p1) q̃
j (p2),
χ̃±j (p1) q̃
i (p2),
i (p1) χ̃
j (p2)
are presented for definite helicities ha,b of the initial partons a, b = q, q̄, g and expressed in terms of the squark,
chargino, neutralino, and gluino masses mq̃k , mχ̃±
, mχ̃0
, and mg̃, the conventional Mandelstam variables,
s = (pa + pb)
2, t = (pa − p1)2, and u = (pa − p2)2, (13)
and the masses of the neutral and charged electroweak gauge bosons mZ and mW . Propagators appear as mass-
subtracted Mandelstam variables,
sw = s−m2W , sz = s−m2Z ,
= t−m2
, uχ̃0
= u−m2
tχ̃j = t−m2χ̃±
, uχ̃j = u−m2χ̃±
tg̃ = t−m2g̃ , ug̃ = u−m2g̃ ,
tq̃i = t−m2q̃i , uq̃i = u−m
Unpolarized cross sections, averaged over initial spins, can easily be derived from the expression
dσ̂ =
dσ̂1,1 + dσ̂1,−1 + dσ̂−1,1 + dσ̂−1,−1
, (15)
while single- and double-polarized cross sections, including the same average factor for initial spins, are given by
d∆σ̂L =
dσ̂1,1 + dσ̂1,−1 − dσ̂−1,1 − dσ̂−1,−1
and d∆σ̂LL =
dσ̂1,1 − dσ̂1,−1 − dσ̂−1,1 + dσ̂−1,−1
, (16)
so that the single- and double-spin asymmetries become
d∆σ̂L
and ALL =
d∆σ̂LL
. (17)
A. Generalized Strong and Electroweak Couplings in NMFV SUSY
Considering the strong interaction first, it is well known that the interaction of quarks, squarks, and gluinos, whose
coupling is normally just given by gs =
4παs, can in general lead to flavour violation in the left- and right-handed
sectors through non-diagonal entries in the matrices Rq,
Lq̃jqkg̃, Rq̃jqk g̃
jk,−R
j(k+3)
. (18)
Of course, the involved quark and squark both have to be up- or down-type, since the gluino is electrically neutral.
For the electroweak interaction, we define the square of the weak coupling constant g2W = e
2/ sin2 θW in terms
of the electromagnetic fine structure constant α = e2/(4π) and the squared sine of the electroweak mixing angle
xW = sin
2 θW = s
W = 1− cos2 θW = 1− c2W . Following the standard notation, the W± − χ̃0i − χ̃
j , Z − χ̃
i − χ̃
j , and
Z − χ̃0i − χ̃0j interaction vertices are proportional to [2]
OLij = −
j2 +Ni2V
j1 and O
N∗i3Uj2 +N
i2Uj1,
O′Lij = −Vi1V ∗j1 −
j2 + δijxW and O
ij = −U∗i1Uj1 −
U∗i2Uj2 + δijxW ,
O′′Lij = −
j4 and O
N∗i3Nj3 −
N∗i4Nj4. (19)
In NMFV, the coupling strengths of left- and right-handed (s)quarks to the electroweak gauge bosons are given by
{Lqq′Z , Rqq′Z} = (2T 3q − 2 eq xW )× δqq′ ,
{Lq̃iq̃jZ , Rq̃iq̃jZ} = (2T 3q̃ − 2 eq̃ xW )×
{Ruik Ru∗jk , Rui(3+k) Ru∗j(3+k)},
{Lqq′W , Rqq′W } = {
2 cW Vqq′ , 0},
{Lũid̃jW , Rũid̃jW } =
k,l=1
2 cW Vukdl R
jl , 0}, (20)
where the weak isospin quantum numbers are T 3
{q,q̃}
= ±1/2 for left-handed and T 3
{q,q̃}
= 0 for right-handed up- and
down-type (s)quarks, their fractional electromagnetic charges are denoted by e{q,q̃}, and Vkl are the elements of the
CKM-matrix defined in Eq. (2). To simplify the notation, we have introduced flavour indices in the latter, d1 = d,
d2 = s, d3 = b, u1 = u, u2 = c, and u3 = t.
The SUSY counterparts of these vertices correspond to the quark-squark-gaugino couplings,
Ld̃jdkχ̃0i
(eq − T 3q ) sW Ni1 + T 3q cW Ni2
Rd∗jk +
mdk cW Ni3R
j(k+3)
2mW cosβ
d̃jdkχ̃
= eq sW Ni1 R
j(k+3) −
mdk cW Ni3R
2mW cosβ
Lũjukχ̃0i =
(eq − T 3q ) sW Ni1 + T 3q cW Ni2
Ru∗jk +
muk cW Ni4 R
j(k+3)
2mW sinβ
−R∗ũjukχ̃0i = eq sW Ni1 R
j(k+3) −
muk cW Ni4 R
2mW sinβ
Ld̃julχ̃±i
Ui1 R
mdk Ui2R
j(k+3)√
2mW cosβ
Vuldk ,
d̃julχ̃
mul Vi2 V
Rdjk√
2mW sinβ
Lũjdlχ̃±i
V ∗i1 R
muk V
j(k+3)√
2mW sinβ
Vukdl ,
ũjdlχ̃
mdl U
Ru∗jk√
2mW cosβ
, (21)
where the matrices N , U and V relate to the gaugino/Higgsino mixing (see App. A). All other couplings vanish due
to (electromagnetic) charge conservation (e.g. Lũjulχ̃±i
). These general expressions can be simplified by neglecting the
Yukawa couplings except for the one of the top quark, whose mass is not small compared to mW . For the sake of
simplicity, we use the generic notation
C1abc, C2abc
= {Labc, Rabc} (22)
in the following.
B. Squark-Antisquark Pair Production
The production of charged squark-antisquark pairs
q(ha, pa) q̄
′(hb, pb) → ũi(p1) d̃∗j (p2), (23)
where i, j = 1, ..., 6 label up- and down-type squark mass eigenstates, ha,b helicities, and pa,b,1,2 four-momenta,
proceeds from an equally charged quark-antiquark initial state through the tree-level Feynman diagrams shown in
Fig. 1. The corresponding cross section can be written in a compact way as
FIG. 1: Tree-level Feynman diagrams for the production of charged squark-antisquark pairs in quark-antiquark collisions.
ha,hb
= (1− ha)(1 + hb)
k,l=1,...,4
N kl11
k=1,...,4
[NW ]k
[GW ]
tg̃ sw
+ (1− ha)(1 − hb)
k,l=1,...,4
N kl12
+ (1 + ha)(1 + hb)
k,l=1,...,4
N kl21
+ (1 + ha)(1 − hb)
k,l=1,...,4
N kl22
thanks to the form factors
W = π α
16 x2W (1− xW )2 s2
∣∣∣L∗qq′W Lũid̃jW
u t−m2ũi m
N klmn =
x2W (1− xW )2 s2
d̃jq′χ̃
Cm∗ũiqχ̃0k C
d̃jq′χ̃
Cmũiqχ̃0l
u t−m2ũi m
δmn +
(1− δmn)
Gmn =
2 π α2s
∣∣∣Cn∗
d̃jq′g̃
Cmũiqg̃
u t−m2ũi m
δmn +
m2g̃ s
(1− δmn)
[NW ]k = π α
6 x2W (1− xW )2 s2
L∗qq′W Lũid̃jW Lũiqχ̃0k
L∗q̃jq′χ̃0k
u t−m2ũi m
[GW ] = 4 π αs α
18 xW (1− xW ) s2
L∗ũiqg̃ Ld̃jq′g̃ L
qq′W Lũid̃jW
u t−m2ũi m
, (25)
which combine coupling constants and Dirac traces of the squared and interference diagrams. In cMFV, superpartners
of heavy flavours can only be produced through the purely left-handed s-channel W -exchange, since the t-channel
diagrams are suppressed by the small bottom and negligible top quark densities in the proton, and one recovers the
result in Ref. [26]. In NMFV, t-channel exchanges can, however, contribute to heavy-flavour final state production
from light-flavour initial states and even become dominant, due to the strong gluino coupling.
Neutral squark-antisquark pair production proceeds either from equally neutral quark-antiquark initial states
q(ha, pa) q̄
′(hb, pb) → q̃i(p1) q̃∗j (p2), (26)
through the five different gauge-boson/gaugino exchanges shown in Fig. 2 (top) or from gluon-gluon initial states
g(ha, pa) g(hb, pb) → q̃i(p1) q̃∗i (p2) (27)
through the purely strong couplings shown in Fig. 2 (bottom). The differential cross section for quark-antiquark
FIG. 2: Tree-level Feynman diagrams for the production of neutral squark-antisquark pairs in quark-antiquark (top) and
gluon-gluon collisions (bottom).
scattering
ha,hb
= (1− ha)(1 + hb)
[YZ]1
[G̃Y]1
tg̃ s
[G̃Z]1
tg̃ sz
[G̃G]1
tg̃ s
k,l=1,...,4
N kl11
k=1,...,4
[NY]k1
[NZ]k1
[NG]k1
+ (1 + ha)(1 − hb)
[YZ]2
[G̃Y]2
tg̃ s
[G̃Z]2
tg̃ sz
[G̃G]2
tg̃ s
k,l=1,...,4
N kl22
k=1,...,4
[NY]k2
[NZ]k2
[NG]k2
+ (1− ha)(1 − hb)
k,l=1,...,4
N kl12
+ (1 + ha)(1 + hb)
k,l=1,...,4
N kl21
involves many different form factors,
π α2 e2q e
q̃ δij δqq′
u t−m2q̃i m
16 s2 x2W (1 − xW )2
∣∣Lq̃iq̃jZ +Rq̃i q̃jZ
∣∣2 (Cmqq′Z
u t−m2q̃i m
G = 2 π α
s δij δqq′
u t−m2q̃i m
N klmn =
x2W (1 − xW )2 s2
Cm∗q̃iqχ̃0k C
q̃iqχ̃
Cnq̃jq′χ̃0kC
q̃jq′χ̃
u t−m2q̃i m
δmn +
(1− δmn)
G̃mn =
2 π α2s
∣∣∣Cmq̃iqg̃ C
q̃jq′g̃
u t−m2q̃i m
δmn +
m2g̃ s
(1− δmn)
[YZ]m =
π α2 eq eq̃ δij δqq′
2 s2 xW (1− xW )
Lq̃iq̃jZ +Rq̃i q̃jZ
Cmqq′Z
u t−m2q̃i m
[NY]km =
2 π α2 eq eq̃ δij δqq′
3 xW (1 − xW ) s2
Cmq̃iqχ̃0k C
q̃jq′χ̃
u t−m2q̃i m
FIG. 3: Tree-level Feynman diagrams for the production of one down-type squark (q̃i) and one up-type squark (q̃
j) in the
collision of an up-type quark (q) and a down-type quark (q′).
[NZ]km =
6 x2W (1− xW )2 s2
Cmq̃iqχ̃0k C
q̃jq′χ̃
Lq̃iq̃jZ +Rq̃iq̃jZ
Cmqq′Z
u t−m2q̃i m
[NG]km =
8 π ααs δij δqq′
9 xW (1− xW ) s2
Cmq̃iqχ̃0k C
q̃jq′χ̃
u t−m2q̃i m
= −4 π α
s δij δqq′
27 s2
Cm∗q̃iqg̃ C
q̃jq′g̃
u t−m2q̃i m
[G̃Y]m =
8 π ααs eq eq̃ δij δqq′
Cm∗q̃iqg̃ C
q̃jq′g̃
u t−m2q̃i m
[G̃Z]m =
2 π ααs
9 xW (1− xW ) s2
Cm∗q̃iqg̃ C
q̃jq′g̃
Lq̃iq̃jZ +Rq̃i q̃jZ
Cmqq′Z
u t−m2q̃i m
, (29)
since only very few interferences (those between strong and electroweak channels of the same propagator type) are
eliminated due to colour conservation. On the other hand, the gluon-initiated cross section
ha,hb
128s2
1− 2 tq̃iuq̃i
(1− hahb)− 2
sm2q̃i
tq̃iuq̃i
(1− hahb)−
sm2q̃i
tq̃iuq̃i
involves only the strong coupling constant and is thus quite compact. In the case of cMFV, but diagonal or non-
diagonal squark helicity, our results agree with those in Ref. [26]. Diagonal production of identical squark-antisquark
mass eigenstates is, of course, dominated by the strong quark-antiquark and gluon-gluon channels. Their relative im-
portance depends on the partonic luminosity and thus on the type and energy of the hadron collider under considera-
tion. Non-diagonal production of squarks of different helicity or flavour involves only electroweak and gluino-mediated
quark-antiquark scattering, and the relative importance of these processes depends largely on the gluino mass.
C. Squark Pair Production
While squark-antisquark pairs are readily produced in pp̄ collisions, e.g. at the Tevatron, from valence quarks and
antiquarks, pp colliders have a larger quark-quark luminosity and will thus more easily lead to squark pair production.
The production of one down-type and one up-type squark
q(ha, pa) q
′(hb, pb) → d̃i(p1) ũj(p2), (31)
in the collision of an up-type quark q and a down-type quark q′ proceeds through the t-channel chargino or u-channel
neutralino and gluino exchanges shown in Fig. 3. The corresponding cross section
FIG. 4: Tree-level Feynman diagrams for the production of two up-type or down-type squarks.
ha,hb
= (1−ha)(1−hb)
k=1,2
l=1,2
Ckl11
tχ̃k tχ̃l
k=1,...,4
l=1,...,4
N kl11
k=1,2
l=1,...,4
[CN ]kl11
tχ̃k uχ̃0
k=1,2
[CG]k11
tχ̃k ug̃
+ (1+ha)(1+hb)
k=1,2
l=1,2
Ckl22
tχ̃k tχ̃l
k=1,...,4
l=1,...,4
N kl22
k=1,2
l=1,...,4
[CN ]kl22
tχ̃k uχ̃0l
k=1,2
[CG]k22
tχ̃k ug̃
+ (1−ha)(1+hb)
k=1,2
l=1,2
Ckl12
tχ̃k tχ̃l
k=1,...,4
l=1,...,4
N kl12
k=1,2
l=1,...,4
[CN ]kl12
tχ̃k uχ̃0
k=1,2
[CG]k12
tχ̃k ug̃
+ (1+ha)(1−hb)
k=1,2
l=1,2
Ckl21
tχ̃k tχ̃l
k=1,...,4
l=1,...,4
N kl21
k=1,2
l=1,...,4
[CN ]kl21
tχ̃k uχ̃0
k=1,2
[CG]k21
tχ̃k ug̃
involves the form factors
Cklmn =
4 x2W s
ũjq′χ̃
d̃iqχ̃
ũjq′χ̃
d̃iqχ̃
u t−m2
m2ũj
(1− δmn) +mχ̃±
s δmn
N klmn =
x2W (1− xW )2 s2
Cm∗ũjqχ̃0k C
d̃iq′χ̃
Cmũjqχ̃0l C
d̃iq′χ̃
u t−m2
m2ũj
(1− δmn) +mχ̃0
s δmn
Gmn =
2 π α2s
∣∣∣Cmũjqg̃ C
d̃iq′g̃
u t−m2
m2ũj
(1− δmn) +m2g̃ s δmn
[CN ]klmn =
3 x2W (1− xW ) s2
ũjq′χ̃
d̃iqχ̃
Cmũjqχ̃0l C
d̃iq′χ̃
u t−m2
m2ũj
(δmn − 1) +mχ̃±
s δmn
[CG]kmn =
4 π ααs
9 s2 xW
ũjq′χ̃
d̃iqχ̃
Cm∗ũjqg̃ C
d̃iq′g̃
u t−m2
m2ũj
(δmn − 1) +mχ̃±
mg̃ s δmn
, (33)
where the neutralino-gluino interference term is absent due to colour conservation. The cross section for the charge-
conjugate production of antisquarks from antiquarks can be obtained from the equations above by replacing ha,b →
−ha,b. Heavy-flavour final states are completely absent in cMFV due to the negligible top quark and small bottom
quark densities in the proton and can thus only be obtained in NMFV.
The Feynman diagrams for pair production of two up- or down-type squarks
q(ha, pa) q
′(hb, pb) → q̃i(p1) q̃j(p2) (34)
are shown in Fig. 4. In NMFV, neutralino and gluino exchanges can lead to identical squark flavours for different
quark initial states, so that both t- and u-channels contribute and may interfere. The cross section
ha,hb
= (1 − ha)(1− hb)
k=1,...,4
l=1,...,4
[NT ]kl11
[NU ]kl11
[NT U ]kl11
[GT ]11
[GU ]11
[GT U ]11
ug̃tg̃
k=1,...,4
[NGA]k11
[NGB]k11
1 + δij
+ (1 + ha)(1 + hb)
k=1,...,4
l=1,...,4
[NT ]kl22
[NU ]kl22
[NT U ]kl22
[GT ]22
[GU ]22
[GT U ]22
ug̃tg̃
k=1,...,4
[NGA]k22
[NGB]k22
1 + δij
+ (1 − ha)(1 + hb)
k=1,...,4
l=1,...,4
[NT ]kl12
[NU ]kl12
[NT U ]kl12
[GT ]12
[GU ]12
[GT U ]12
ug̃tg̃
k=1,...,4
[NGA]k12
[NGB]k12
1 + δij
+ (1 + ha)(1− hb)
k=1,...,4
l=1,...,4
[NT ]kl21
[NU ]kl21
[NT U ]kl21
[GT ]21
[GU ]21
[GT U ]21
ug̃tg̃
k=1,...,4
[NGA]k21
[NGB]k21
1 + δij
depends therefore on the form factors
[NT ]klmn =
x2W (1− xW )2 s2
Cn∗q̃jq′χ̃0k C
q̃iqχ̃
Cnq̃jq′χ̃0l C
q̃iqχ̃
u t−m2q̃i m
(1− δmn) +mχ̃0
s δmn
[NU ]klmn =
x2W (1− xW )2 s2
Cn∗q̃iq′χ̃0k C
q̃jqχ̃
Cnq̃iq′χ̃0l C
q̃jqχ̃
u t−m2q̃i m
(1− δmn) +mχ̃0
s δmn
[NT U ]klmn =
2 π α2
3 x2W (1− xW )2 s2
Cm∗q̃iqχ̃0k C
q̃jq′χ̃
Cnq̃iq′χ̃0l C
q̃jqχ̃
u t−m2q̃i m
(1− δmn) +mχ̃0
s δmn
[GT ]mn =
2 π α2s
∣∣∣Cnq̃jq′g̃ C
q̃iqg̃
u t−m2q̃i m
(1− δmn) +m2g̃ s δmn
[GU ]mn =
2 π α2s
∣∣∣Cmq̃iq′g̃ C
q̃jqg̃
u t−m2q̃i m
(1− δmn) +m2g̃ s δmn
[GT U ]mn =
−4 π α2s
27 s2
Cmq̃iqg̃ C
q̃jq′g̃
Cm∗q̃iq′g̃ C
q̃jqg̃
u t−m2q̃i m
(1− δmn) +m2g̃ s δmn
[NGA]kmn =
8 π ααs
9 s2 xW (1− xW )
Cn∗q̃jq′χ̃0k C
q̃iqχ̃
Cm∗q̃iq′g̃ C
q̃jqg̃
] [ (
u t−m2q̃i m
(1− δmn) +mχ̃0
mg̃ s δmn
[NGB]kmn =
8 π ααs
9 s2 xW (1− xW )
Cn∗q̃iq′χ̃0k C
q̃jqχ̃
Cn∗q̃jq′g̃ C
q̃iqg̃
] [ (
u t−m2q̃i m
(1− δmn) +mχ̃0
mg̃ s δmn
. (36)
Gluinos will dominate over neutralino exchanges due to their strong coupling, and the two will only interfere in the
mixed t- and u-channels due to colour conservation. At the LHC, up-type squark pair production should dominate
over mixed up-/down-type squark production and down-type squark pair production, since the proton contains two
valence up-quarks and only one valence down-quark. As before, the charge-conjugate production of antisquark pairs
FIG. 5: Tree-level Feynman diagrams for the associated production of squarks and gauginos.
is obtained by making the replacement ha,b → −ha,b. If we neglect electroweak contributions as well as squark flavour
and helicity mixing and sum over left- and right-handed squark states, our results agree with those of Ref. [27].
D. Associated Production of Squarks and Gauginos
The associated production of squarks and neutralinos or charginos
q(ha, pa) g(hb, pb) → χ̃j(p1) q̃i(p2) (37)
is a semi-weak process that originates from quark-gluon initial states and has both an s-channel quark and a t-channel
squark contribution. They involve both a quark-squark-gaugino vertex that can in general be flavour violating. The
corresponding Feynman diagrams can be seen in Fig. 5. The squark-gaugino cross section
ha,hb
π ααs
nχ̃ s2
−uχ̃j
(1− ha)(1 − hb)
∣∣Lq̃iqχ̃j
∣∣2 + (1 + ha)(1 + hb)
∣∣Rq̃iqχ̃j
t+m2q̃i
t2q̃i
(1 − ha)
∣∣Lq̃iqχ̃j
∣∣2 + (1 + ha)
∣∣Rq̃iqχ̃j
2 (u t−m2q̃i m
s tq̃i
(1 − ha)(1− hb)
∣∣Lq̃iqχ̃j
∣∣2 + (1 + ha)(1 + hb)
∣∣Rq̃iqχ̃j
tχ̃j (tχ̃j − uq̃i)
s tq̃i
(1 − ha)
∣∣Lq̃iqχ̃j
∣∣2 + (1 + ha)
∣∣Rq̃iqχ̃j
, (38)
where nχ̃ = 6xW (1− xW ) for neutralinos and nχ̃ = 12xW for charginos, is sufficiently compact to be written without
the definition of form factors. Note that the t-channel diagram involves the coupling of the gluon to scalars and
does thus not depend on its helicity hb. The cross section of the charge-conjugate process can be obtained by taking
ha → −ha. Third-generation squarks can only be produced in NMFV, preferably through a light (valence) quark in
the s-channel. For non-mixing squarks and gauginos, we agree again with the results of Ref. [27].
E. Gaugino Pair Production
Finally, we consider the purely electroweak production of gaugino pairs
q(ha, pa) q̄
′(hb, pb) → χ̃i(p1) χ̃j(p2) (39)
from quark-antiquark initial states, where flavour violation can occur via the quark-squark-gaugino vertices in the t-
and u-channels (see Fig. 6). However, if it were not for different parton density weights, summation over complete
squark multiplet exchanges would make these channels insensitive to the exchanged squark flavour. Furthermore there
are no final state squarks that could be experimentally tagged. The cross section can be expressed generically as
FIG. 6: Tree-level Feynman diagrams for the production of gaugino pairs.
ha,hb
(1− ha)(1 + hb)
|QuLL|
uχ̃iuχ̃j +
∣∣QtLL
∣∣2 tχ̃itχ̃j + 2Re[Q
LL]mχ̃imχ̃js
(1 + ha)(1− hb)
|QuRR|
uχ̃iuχ̃j +
∣∣QtRR
∣∣2 tχ̃itχ̃j + 2Re[Q
RR]mχ̃imχ̃js
(1 + ha)(1 + hb)
|QuRL|
uχ̃iuχ̃j +
∣∣QtRL
∣∣2 tχ̃itχ̃j +Re[Q
RL](ut−m2χ̃im
(1− ha)(1− hb)
|QuLR|
uχ̃iuχ̃j +
∣∣QtLR
∣∣2 tχ̃itχ̃j +Re[Q
LR](ut−m2χ̃im
, (40)
i.e. in terms of generalized charges. For χ̃−i χ̃
j -production, these charges are given by
Qu−+LL =
eqδijδqq′
Lqq′ZO
2 xW (1 − xW ) sz
Ld̃kq′χ̃±i
d̃kqχ̃
2 xW ud̃k
Qt−+LL =
eqδijδqq′
Lqq′ZO
2 xW (1 − xW ) sz
ũkq′χ̃
Lũkqχ̃±i
2 xW tũk
Qu−+RR =
eqδijδqq′
Rqq′ZO
2 xW (1 − xW ) sz
Rd̃kq′χ̃±i
d̃kqχ̃
2 xW ud̃k
Qt−+RR =
eqδijδqq′
Rqq′ZO
2 xW (1 − xW ) sz
ũkq′χ̃
Rũkqχ̃±i
2 xW tũk
Qu−+LR =
Rd̃kq′χ̃±i
d̃kqχ̃
2 xW ud̃k
Qt−+LR =
ũkq′χ̃
Lũkqχ̃±i
2 xW tũk
Qu−+RL =
Ld̃kq′χ̃±i
d̃kqχ̃
2 xW ud̃k
Qt−+RL =
ũkq′χ̃
Rũkqχ̃±i
2 xW tũk
. (41)
Note that there is no interference between t- and u-channel diagrams due to (electromagnetic) charge conservation.
The cross section for chargino-pair production in e+e−-collisions can be deduced by setting eq → el = −1, Lqq′Z →
LeeZ = (2T
l − 2 el xW ) and Rqq′Z → ReeZ = −2 el xW . Neglecting all Yukawa couplings, we can then reproduce the
calculations of Ref. [28].
The charges of the chargino-neutralino associated production are given by
Qu+0LL =
2 (1− xW )xW
OL∗ji L
qq′W√
ũkq′χ̃
ũkqχ̃
Qt+0LL =
2 (1− xW )xW
OR∗ji L
qq′W√
d̃kqχ̃
Ld̃kq′χ̃0j
Qu+0RR =
2 (1− xW )xW
ũkq′χ̃
ũkqχ̃
Qt+0RR =
2 (1− xW )xW
d̃kqχ̃
Rd̃kq′χ̃0j
Qu+0LR =
2 (1− xW )xW
ũkq′χ̃
ũkqχ̃
Qt+0LR =
2 (1− xW )xW
d̃kqχ̃
Rd̃kq′χ̃0j
Qu+0RL =
2 (1− xW )xW
ũkq′χ̃
ũkqχ̃
Qt+0RL =
2 (1− xW )xW
d̃kqχ̃
Ld̃kq′χ̃0j
. (42)
The charge-conjugate process is again obtained by making the replacement ha,b → −ha,b in Eq. (40). In the case of
non-mixing squarks with neglected Yukawa couplings, we agree with the results of Ref. [22], provided we correct a
sign in their Eq. (2) as described in Ref. [29].
Finally, the charges for the neutralino pair production are given by
Qu00LL =
xW (1− xW )
1 + δij
Lqq′ZO
LQ̃kq′χ̃0i
Q̃kqχ̃
Qt00LL =
xW (1− xW )
1 + δij
Lqq′ZO
Q̃kqχ̃
LQ̃kq′χ̃0j
Qu00RR =
xW (1− xW )
1 + δij
Rqq′ZO
RQ̃kq′χ̃0i
Q̃kqχ̃
Qt00RR =
xW (1− xW )
1 + δij
Rqq′ZO
Q̃kqχ̃
RQ̃kq′χ̃0j
Qu00LR =
xW (1− xW )
1 + δij
RQ̃kq′χ̃0i
Q̃kqχ̃
Qt00LR =
xW (1− xW )
1 + δij
Q̃kqχ̃
RQ̃kq′χ̃0j
Qu00RL =
xW (1− xW )
1 + δij
LQ̃kq′χ̃0i
Q̃kqχ̃
Qt00RL =
xW (1− xW )
1 + δij
Q̃kqχ̃
LQ̃kq′χ̃0j
, (43)
which agrees with the results of Ref. [30] in the case of non-mixing squarks.
FIG. 7: Tree-level Feynman diagrams for squark decays into gauginos and quarks (top) and into electroweak gauge bosons and
lighter squarks (bottom).
F. Squark Decays
We turn now from SUSY particle production to decay processes and show in Fig. 7 the possible decays of squarks
into gauginos and quarks (top) as well as into electroweak gauge bosons and lighter squarks (bottom). Both processes
can in general induce flavour violation. The decay widths of the former are given by
Γq̃i→χ̃0jqk =
2m3q̃i xW (1− xW )
m2q̃i −m
−m2qk
)( ∣∣∣Lq̃iqkχ̃0j
∣∣∣Rq̃iqkχ̃0j
− 4mχ̃0
mqk Re
Lq̃iqkχ̃0jR
q̃iqkχ̃
λ1/2(m2q̃i ,m
,m2qk), (44)
Γq̃i→χ̃±j q
4m3q̃i xW
m2q̃i −m
−m2q′
)( ∣∣∣Lq̃iq′kχ̃±j
∣∣∣Rq̃iq′kχ̃±j
− 4mχ̃±
Lq̃iq′kχ̃
λ1/2(m2q̃i ,m
,m2q′
), (45)
Γq̃i→g̃qk =
3m3q̃i xW
m2q̃i −m
g̃ −m2qk
|Lq̃iqkg̃|
+ |Rq̃iqk g̃|
− 4mg̃mqk Re
Lq̃iqkg̃R
q̃iqkg̃
× λ1/2(m2q̃i ,m
), (46)
while those of the latter are given by
Γq̃i→Zq̃k =
16m3q̃i m
Z xW (1− xW )
|Lq̃i q̃kZ +Rq̃i q̃kZ |
λ3/2(m2q̃i ,m
), (47)
Γq̃i→W±q̃′k =
16m3q̃i m
W xW (1− xW )
∣∣∣Lq̃iq̃′kW
λ3/2(m2q̃i ,m
). (48)
The usual Källen function is
λ(x, y, z) = x2 + y2 + z2 − 2(x y + y z + z x). (49)
In cMFV, our results agree with those of Ref. [31].
G. Gluino Decays
Heavy gluinos can decay strongly into squarks and quarks as shown in Fig. 8. The corresponding decay width
FIG. 8: Tree-level Feynman diagram for gluino decays into squarks and quarks.
FIG. 9: Tree-level Feynman diagrams for gaugino decays into squarks and quarks (left) and into lighter gauginos and electroweak
gauge bosons (centre and right).
Γg̃→q̃∗
8m3g̃
m2g̃ −m2q̃j +m
)( ∣∣Lq̃jqkg̃
∣∣2 +
∣∣Rq̃jqk g̃
+ 4mg̃mqk Re
Lq̃jqk g̃R
q̃jqk g̃
× λ1/2(m2g̃,m2q̃j ,m
) (50)
can in general also induce flavour violation. In cMFV, our result agrees again with the one of Ref. [31].
H. Gaugino Decays
Heavier gauginos can decay into squarks and quarks as shown in Fig. 9 (left) or into lighter gauginos and electroweak
gauge bosons (Fig. 9 centre and right). The analytical decay widths are
→q̃j q̄
−m2q̃j +m
)( ∣∣∣Lq̃jq′kχ̃±i
∣∣∣Rq̃jq′kχ̃±i
+ 4mχ̃±
Lq̃jq′kχ̃
λ1/2(m2
,m2q̃j ,m
) (51)
m2W xW
+m4χ̃0
− 2m4W +m2χ̃±
m2W +m
m2W − 2m2χ̃±
m2χ̃0
( ∣∣OLij
∣∣2 +
∣∣ORij
− 12mχ̃±
m2W mχ̃0
OLijO
λ1/2(m2
,m2χ̃0
,m2W ), (52)
m2Z xW (1− xW )
− 2m4Z +m2χ̃±
m2Z +m
m2Z − 2m2χ̃±
( ∣∣O′Lij
∣∣2 +
∣∣O′Rij
− 12mχ̃±
m2Z mχ̃±
O′Lij O
λ1/2(m2
,m2Z) (53)
for charginos and
→q̃j q̄k
xW (1− xW )
m2χ̃0
−m2q̃j +m
)( ∣∣∣Lq̃jqkχ̃0i
∣∣∣Rq̃jqkχ̃0i
+ 4mχ̃0
mqk Re
Lq̃jqkχ̃0iR
q̃jqkχ̃
λ1/2(m2χ̃0
,m2q̃j ,m
) (54)
m2W xW
m4χ̃0
− 2m4W +m2χ̃0
m2W +m
m2W − 2m2χ̃0
( ∣∣OLij
∣∣2 +
∣∣ORij
− 12mχ̃0
m2W mχ̃±
OLijO
λ1/2(m2χ̃0
,m2W ), (55)
m2Z xW (1− xW )
m4χ̃0
+m4χ̃0
− 2m4Z +m2χ̃0
m2Z +m
m2Z − 2m2χ̃0
m2χ̃0
( ∣∣O′′Lij
∣∣2 +
∣∣O′′Rij
− 12mχ̃0
m2Z mχ̃0j Re
O′′Lij O
λ1/2(m2χ̃0
,m2χ̃0
,m2Z) (56)
for neutralinos, respectively. Chargino decays into a slepton and a neutrino (lepton and sneutrino) can be deduced
from the previous equations by taking the proper limits, i.e. by removing colour factors and up-type masses in the
coupling definitions. Our results agree then with those of Ref. [32] in the limit of non-mixing sneutrinos. Note that
the same simplifications also permit a verification of our results for squark decays into a gaugino and a quark in Eqs.
(44) and (45) when compared to their leptonic counterparts in Ref. [32].
IV. EXPERIMENTAL CONSTRAINTS, SCANS AND BENCHMARKS IN NMFV SUSY
In the absence of experimental evidence for supersymmetry, a large variety of data can be used to constrain the
MSSM parameter space. For example, sparticle mass limits can be obtained from searches of charginos (mχ̃±
GeV for heavier sneutrinos at LEP2), neutralinos (mχ̃0
≥ 59 GeV in minimal supergravity (mSUGRA) from the
combination of LEP2 results), gluinos (mg̃ ≥ 195 GeV from CDF), stops (mt̃1 ≥ 95 . . . 96 GeV for neutral- or
charged-current decays from the combination of LEP2 results), and other squarks (mq̃ ≥ 300 GeV for gluinos of equal
mass from CDF) at colliders [33]. Note that all of these limits have been obtained assuming minimal flavour violation.
For non-minimal flavour violation, rather strong constraints can be obtained from low-energy, electroweak precision,
and cosmological observables. These are discussed in the next subsection, followed by several scans for experimentally
allowed/favoured regions of the constrained MSSM parameter space and the definition of four NMFV benchmark
points/slopes. Finally, we exhibit the corresponding chirality and flavour decomposition of the various squark mass
eigenstates.
A. Low-Energy, Electroweak Precision, and Cosmological Constraints
In a rather complete analysis of FCNC constraints more than ten years ago [18], upper limits from the neutral
kaon sector (on ∆mK , ε, ε
′/ε), on B- (∆mB) and D-meson oscillations (∆mD), various rare decays (BR(b → sγ),
BR(µ → eγ), BR(τ → eγ), and BR(τ → µγ)), and electric dipole moments (dn and de) were used to impose
constraints on non-minimal flavour mixing in the squark and slepton sectors. The limit obtained for the absolute
value in the left-handed, down-type squark sector was rather weak (|λsbLL| < 4.4 . . .26 for varying gluino-to-squark
mass ratio), while the limits for the mixed/right-handed, imaginary or sleptonic parts were already several orders of
magnitude smaller. In the meantime, many of the experimental bounds have been improved or absolute values for the
observables have been determined, so that an updated analysis could be performed [34]. The results for the down-type
squark sector are cited in Tab. I. As can be seen and as has already been hinted at in the introduction, only mixing
between second- and third-generation squarks can be substantial, and this only in the left-left or right-right chiral
sectors, the latter being disfavoured by its scaling with the soft SUSY-breaking mass. Independent analyses focusing
TABLE I: The 95% probability bounds on |λ
ij | obtained in Ref. [34].
ij LL LR RL RR
12 1.4×10−2 9.0×10−5 9.0×10−5 9.0×10−3
13 9.0×10−2 1.7×10−2 1.7×10−2 7.0×10−2
23 1.6×10−1 4.5×10−3 6.0×10−3 2.2×10−1
on this particular sector, i.e. on BR(b → sγ), BR(b → sµµ), and ∆mBs , have been performed recently by two other
groups [35, 36] with very similar results.
In our own analysis, we take implicitly into account all of the previously mentioned constraints by restricting
ourselves to the case of only one real NMFV parameter, λ ≡ λsbLL = λctLL. Allowed regions for this parameter are then
obtained by imposing explicitly a number of low-energy, electroweak precision, and cosmological constraints. We start
by imposing the theoretically robust inclusive branching ratio
BR(b→ sγ) = (3.55± 0.26)× 10−4, (57)
obtained from the combined measurements of BaBar, Belle, and CLEO [37], at the 2σ-level on the two-loop QCD/one-
loop SUSY calculation [36, 38], which affects directly the allowed squark mixing between the second and third
generation.
A second important consequence of NMFV in the MSSM is the generation of large splittings between squark-mass
eigenvalues. The splitting within isospin doublets influences the Z- and W -boson self-energies at zero-momentum
ΣZ,W (0) in the electroweak ρ-parameter
∆ρ = ΣZ(0)/M
Z − ΣW (0)/M2W (58)
and consequently theW -boson massMW and the squared sine of the weak mixing angle sin
2 θW . The latest combined
fits of the Z-boson mass, width, pole asymmetry, W -boson and top-quark mass constrain new physics contributions
to T = −0.13± 0.11 [33] or
∆ρ = −αT = 0.00102± 0.00086, (59)
where we have used α(MZ) = 1/127.918. This value is then imposed at the 2σ-level on the one-loop NMFV and
two-loop cMFV SUSY calculation [39].
A third observable sensitive to SUSY loop-contributions is the anomalous magnetic moment aµ = (gµ − 2)/2 of the
muon, for which recent BNL data and the SM prediction disagree by [33]
∆aµ = (22± 10)× 10−10. (60)
In our calculation, we take into account the SM and MSSM contributions up to two loops [40, 41] and require them
to agree with the region above within two standard deviations.
For cosmological reasons, i.e. in order to have a suitable candidate for non-baryonic cold dark matter [42], we require
the lightest SUSY particle (LSP) to be stable, electrically neutral, and a colour singlet. The dark matter relic density
is then calculated using a modified version of DarkSUSY 4.1 [43], that takes into account the six-dimensional squark
helicity and flavour mixing, and constrained to the region
0.094 < ΩCDMh
2 < 0.136 (61)
at 95% (2σ) confidence level. This limit has recently been obtained from the three-year data of the WMAP satellite,
combined with the SDSS and SNLS survey and Baryon Acoustic Oscillation data and interpreted within an eleven-
parameter inflationary model [44], which is more general than the usual six-parameter “vanilla” concordance model
of cosmology. Note that this range is well compatible with the older, independently obtained range of 0.094 <
ΩCDMh
2 < 0.129 [45].
B. Scans of the Constrained NMFV MSSM Parameter Space
The above experimental limits are now imposed on the constrained MSSM (cMSSM), or minimal supergravity
(mSUGRA), model with five free parameters m0, m1/2, tanβ, A0, and sgn(µ) at the grand unification scale. Since
 [GeV]1/2m
500 1000 1500
=0λ<0, µ=10, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.03λ<0, µ=10, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.05λ<0, µ=10, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.1λ<0, µ=10, β=0, tan
FIG. 10: The (m0,m1/2)-planes for tan β = 10, A0 = 0 GeV, µ < 0, and λ = 0, 0.03, 0.05 and 0.1. We show WMAP (black)
favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in minimal (λ = 0)
and non-minimal (λ > 0) flavour violation.
our scans of the cMSSM parameter space in the common scalar mass m0 and the common fermion mass m1/2 depend
very little on the trilinear coupling A0, we set it to zero in the following. Furthermore, we fix a small (10), intermediate
(30), and large (50) value for the ratio of the Higgs vacuum expectation values tanβ. The impact of the sign of the
off-diagonal Higgs mass parameter µ is investigated for tanβ = 10 only, before we set it to µ > 0 for tanβ = 30 and
50 (see below).
With these boundary conditions at the grand unification scale, we solve the renormalization group equations numer-
ically to two-loop order using the computer program SPheno 2.2.3 [46] and compute the soft SUSY-breaking masses
at the electroweak scale with the complete one-loop formulas, supplemented by two-loop contributions in the case of
the neutral Higgs bosons and the µ-parameter. At this point we generalize the squark mass matrices as described
in Sec. II in order to account for flavour mixing in the left-chiral sector of the second- and third-generation squarks,
diagonalize these mass matrices, and compute the low-energy, electroweak precision, and cosmological observables
with the computer programs FeynHiggs 2.5.1 [47] and DarkSUSY 4.1 [43].
For the masses and widths of the electroweak gauge bosons and the mass of the top quark, we use the current
values of mZ = 91.1876 GeV, mW = 80.403 GeV, mt = 174.2 GeV, ΓZ = 2.4952 GeV, and ΓW = 2.141 GeV. The
CKM-matrix elements are computed using the parameterization
c12c13 s12c13 s13e
−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13
s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13
 , (62)
where sij = sin θij and cij = cos θij relate to the mixing of two specific generations i and j and δ is the SM CP -violating
complex phase. The numerical values are given by
s12 = 0.2243, s23 = 0.0413, s13 = 0.0037, and δ = 1.05. (63)
The squared sine of the electroweak mixing angle sin2 θW = 1 − m2W /m2Z and the electromagnetic fine structure
constant α =
W sin
2 θW /π are calculated in the improved Born approximation using the world average value
of GF = 1.16637 · 10−5 GeV−2 for Fermi’s coupling constant [33].
Typical scans of the cMSSM parameter space in m0 and m1/2 with a relatively small value of tanβ = 10 and A0 = 0
are shown in Figs. 10 and 11 for µ < 0 and µ > 0, respectively. All experimental limits described in Sec. IVA are
imposed at the 2σ-level. The b→ sγ excluded region depends strongly on flavour mixing, while the regions favoured
by gµ − 2 and the dark matter relic density are quite insensitive to variations of the λ-parameter. ∆ρ constrains the
parameter space only for heavy scalar masses m0 > 2000 GeV and heavy gaugino masses m1/2 > 1500 GeV, so that
the corresponding excluded regions are not shown here.
The dominant SUSY effects in the calculation of the anomalous magnetic moment of the muon come from induced
quantum loops of a gaugino and a slepton. Squarks contribute only at the two-loop level. This reduces the dependence
on flavour violation in the squark sector considerably. Furthermore, the region µ < 0 is disfavoured in all SUSY models,
since the one-loop SUSY contributions are approximatively given by [48]
aSUSY, 1−loopµ ≃ 13× 10−10
100 GeV
MSUSY
tanβ sgn(µ), (64)
if all SUSY particles (the relevant ones are the smuon, sneutralino, chargino, and neutralino) have a common mass
MSUSY. Negative values of µ would then increase, not decrease, the disagreement between the experimental measure-
ments and the theoretical SM value of aµ. Furthermore, the measured b → sγ branching ratio excludes virtually all
 [GeV]1/2m
500 1000 1500
=0λ>0, µ=10, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.03λ>0, µ=10, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.05λ>0, µ=10, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.1λ>0, µ=10, β=0, tan
FIG. 11: The (m0,m1/2)-planes for tanβ = 10, A0 = 0 GeV, µ > 0, and λ = 0, 0.03, 0.05 and 0.1. We show aµ (grey) and
WMAP (black) favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in
minimal (λ = 0) and non-minimal (λ > 0) flavour violation.
 [GeV]1/2m
500 1000 1500
=0λ>0, µ=30, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.03λ>0, µ=30, β=0, tan 
 [GeV]1/2m
500 1000 1500
=0.05λ>0, µ=30, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.1λ>0, µ=30, β=0, tan
FIG. 12: The (m0,m1/2) planes for tan β = 30, A0 = 0 GeV, µ > 0, and λ = 0, 0.03, 0.05 and 0.1. We show aµ (grey) and
WMAP (black) favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in
minimal (λ = 0) and non-minimal (λ > 0) flavour violation.
of the region favoured by the dark matter relic density, except for very high scalar SUSY masses. We therefore do
not consider negative values of µ in the rest of this work. As stated above, we have also checked that the shape of the
different regions depends extremely weakly on the trilinear coupling A0.
In Figs. 12 and 13, we show the (m0,m1/2)-planes for larger tanβ, namely tanβ = 30 and tanβ = 50, and for
µ > 0. The regions which are favoured both by the anomalous magnetic moment of the muon and by the cold dark
matter relic density, and which are not excluded by the b→ sγ measurements, are stringently constrained and do not
allow for large flavour violation.
 [GeV]1/2m
500 1000 1500
=0λ>0, µ=50, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.03λ>0, µ=50, β=0, tan 
 [GeV]1/2m
500 1000 1500
=0.05λ>0, µ=50, β=0, tan
 [GeV]1/2m
500 1000 1500
=0.1λ>0, µ=50, β=0, tan
FIG. 13: The (m0,m1/2) planes for tan β = 50, A0 = 0 GeV, µ > 0, and λ = 0, 0.03, 0.05 and 0.1. We show aµ (grey) and
WMAP (black) favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in
minimal (λ = 0) and non-minimal (λ > 0) flavour violation.
TABLE II: Benchmark points allowing for flavour violation among the second and third generations for A0 = 0, µ > 0, and
three different values of tanβ. For comparison we also show the nearest pre-WMAP SPS [49, 50] and post-WMAP BDEGOP
[51] benchmark points and indicate the relevant cosmological regions.
m0 [GeV] m1/2 [GeV] A0 [GeV] tanβ sgn(µ) SPS BDEGOP Cosmol. Region
A 700 200 0 10 1 2 E’ Focus Point
B 100 400 0 10 1 3 C’ Co-Annihilation
C 230 590 0 30 1 1b I’ Co-Annihilation
D 600 700 0 50 1 4 L’ Bulk/Higgs-funnel
C. (c)MFV and NMFV Benchmark Points and Slopes
Restricting ourselves to non-negative values of µ, we now inspect the (m0,m1/2)-planes in Figs. 11-13 for cMSSM
scenarios that
• are allowed/favoured by low-energy, electroweak precision, and cosmological constraints,
• permit non-minimal flavour violation among left-chiral squarks of the second and third generation up to λ ≤ 0.1,
• and are at the same time collider-friendly, i.e. have relatively low values of m0 and m1/2.
Our choices are presented in Tab. II, together with the nearest pre-WMAP Snowmass Points (and Slopes, SPS) [49, 50]
and the nearest post-WMAP scenarios proposed in Ref. [51]. We also indicate the relevant cosmological region for
each point and attach a model line (slope) to it, given by
A : 180 GeV ≤ m1/2 ≤ 250 GeV , m0 = − 1936 GeV + 12.9m1/2,
B : 400 GeV ≤ m1/2 ≤ 900 GeV , m0 = 4.93 GeV + 0.229m1/2,
C : 500 GeV ≤ m1/2 ≤ 700 GeV , m0 = 54 GeV + 0.297m1/2,
D : 575 GeV ≤ m1/2 ≤ 725 GeV , m0 = 600 GeV. (65)
These slopes trace the allowed/favoured regions from lower to higher masses and can, of course, also be used in cMFV
scenarios with λ = 0. We have verified that in the case of MFV [7] the hierarchy ∆
LL ≫ ∆
LR,RL ≫ ∆
RR and the
equality of λsbLL = λ
LL are still reasonably well fulfilled numerically with the values of λ
LL ≈ λctLL ranging from zero
to 5× 10−3 . . . 1× 10−2 for our four typical benchmark points.
Starting with Fig. 11 and tanβ = 10, the bulk region of equally low scalar and fermion masses is all but excluded
by the b → sγ branching ratio. This leaves as a favoured region first the so-called focus point region of low fermion
massesm1/2, where the lightest neutralinos are relatively heavy, have a significant Higgsino component, and annihilate
dominantly into pairs of electroweak gauge bosons. Our benchmark point A lies in this region, albeit at smaller masses
than SPS 2 (m0 = 1450 GeV, m1/2 = 300 GeV) and BDEGOP E’ (m0 = 1530 GeV, m1/2 = 300 GeV), which lie
outside the region favoured by aµ (grey-shaded) and lead to collider-unfriendly squark and gaugino masses.
The second favoured region for small tanβ is the co-annihilation branch of low scalar masses m0, where the lighter
tau-slepton mass eigenstate is not much heavier than the lightest neutralino and the two have a considerable co-
annihilation cross section. This is where we have chosen our benchmark point B, which differs from the points SPS
3 (m0 = 90 GeV, m1/2 = 400 GeV) and BDEGOP C’ (m0 = 85 GeV, m1/2 = 400 GeV) only very little in the scalar
mass. This minor difference may be traced to the fact that we use DarkSUSY 4.1 [43] instead of the private dark
matter program SSARD of Ref. [51].
At the larger value of tanβ = 30 in Fig. 12, only the co-annihilation region survives the constraints coming from
b → sγ decays. Here we choose our point C, which has slightly higher masses than both SPS 1b (m0 = 200 GeV,
m1/2 = 400 GeV) and BDEGOP I’ (m0 = 175 GeV, m1/2 = 350 GeV), due to the ever more stringent constraints
from the above-mentioned rare B-decay.
For the very large value of tanβ = 50 in Fig. 13, the bulk region reappears at relatively heavy scalar and fermion
masses. Here, the couplings of the heavier scalar and pseudo-scalar Higgses H0 and A0 to bottom quarks and tau-
leptons and the charged-Higgs coupling to top-bottom pairs are significantly enhanced, resulting e.g. in increased
dark matter annihilation cross sections through s-channel Higgs-exchange into bottom-quark final states. So as tanβ
increases further, the so-called Higgs-funnel region eventually makes its appearance on the diagonal of large scalar and
fermion masses. We choose our point D in the concentrated (bulky) region favoured by cosmology and aµ at masses,
that are slightly higher than those of SPS 4 (m0 = 400 GeV, m1/2 = 300 GeV) and BDEGOP L’ (m0 = 300 GeV,
m1/2 = 450 GeV). We do so in order to escape again from the constraints of the b → sγ decay, which are stronger
today than they were a few years ago. In this scenario, squarks and gluinos are very heavy with masses above 1 TeV.
0 0.2 0.4 0.6 0.8 1
0.0001
0.001
0 0.2 0.4 0.6 0.8 1
γ s →b 
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Up-type squark masses
~ ≈ 4u
~ ≈ 3u
0 0.2 0.4 0.6 0.8 1
Down-type squark masses
 ≈ 4d
 ≈ 2d
FIG. 14: Dependence of the precision variables BR(b → sγ), ∆ρ, and the cold dark matter relic density ΩCDMh
2 (top) as well
as of the lightest SUSY particle, up- and down-type squark masses (bottom) on the NMFV parameter λ in our benchmark
scenario A. The experimentally allowed ranges (within 2σ) are indicated by horizontal dashed lines.
D. Dependence of Precision Observables and Squark-Mass Eigenvalues on Flavour Violation
Let us now turn to the dependence of the precision variables discussed in Sec. IVA on the flavour violating parameter
λ in the four benchmark scenarios defined in Sec. IVC. As already mentioned, we expect the leptonic observable aµ
to depend weakly (at two loops only) on the squark sector, and this is confirmed by our numerical analysis. We find
constant values of 6, 14, 16, and 13×10−10 for the benchmarks A, B, C, and D, all of which lie well within 2σ (the
latter three even within 1σ) of the experimentally favoured range (22± 10)× 10−10.
The electroweak precision observable ∆ρ is shown first in Figs. 14-17 for the four benchmark scenarios A, B, C, and
D. On our logarithmic scale, only the experimental upper bound of the 2σ-range is visible as a dashed line. While the
self-energy diagrams of the electroweak gauge bosons depend obviously strongly on the helicities, flavours, and mass
eigenvalues of the squarks in the loop, the SUSY masses in our scenarios are sufficiently small and the experimental
error is still sufficiently large to allow for relatively large values of λ ≤ 0.57, 0.52, 0.38, and 0.32 for the benchmark
points A, B, C, and D, respectively. As mentioned above, ∆ρ conversely constrains SUSY models in cMFV only for
masses above 2000 GeV for m0 and 1500 GeV for m1/2.
The next diagram in Figs. 14-17 shows the dependence of the most stringent low-energy constraint, coming from
the good agreement between the measured b → sγ branching ratio and the two-loop SM prediction, on the NMFV
parameter λ. The dashed lines of the 2σ-bands exhibit two allowed regions, one close to λ = 0 (vertical green line)
and a second one around λ ≃ 0.57, 0.75, 0.62, and 0.57, respectively. As is well-known, the latter are, however,
disfavoured by b→ sµ+µ− data constraining the sign of the b→ sγ amplitude to be the same as in the SM [52]. We
will therefore limit ourselves later to the regions λ ≤ 0.05 (points A, C, and D) and λ ≤ 0.1 (point B) in the vicinity
of (c)MFV (see also Tab. I).
The 95% confidence-level (or 2σ) region for the cold dark matter density was given in absolute values in Ref. [44]
and is shown as a dashed band in the upper right part of Figs. 14-17. However, only the lower bound (0.094) is of
relevance, as the relic density falls with increasing λ. This is not so pronounced in our model B as in our model A,
where squark masses are light and the lightest neutralino has a sizable Higgsino-component, so that squark exchanges
contribute significantly to the annihilation cross sections. For models C and D there is little sensitivity of ΩCDMh
(except at very large λ ≤ 1), as the squark masses are generally larger.
0 0.2 0.4 0.6 0.8 1
0.0001
0.001
0 0.2 0.4 0.6 0.8 1
γ s →b 
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Up-type squark masses
~, 4u
~, 3u
~, 2u
0 0.2 0.4 0.6 0.8 1
Down-type squark masses
 ≈ 4d
 ≈ 2d
FIG. 15: Same as Fig. 14 for our benchmark scenario B.
0 0.2 0.4 0.6 0.8 1
0.00001
0.0001
0.001
0 0.2 0.4 0.6 0.8 1
γ s →b 
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Up-type squark masses
~, 4u
~,  3u
0 0.2 0.4 0.6 0.8 1
Down-type squark masses
FIG. 16: Same as Fig. 14 for our benchmark scenario C.
0 0.2 0.4 0.6 0.8 1
0.00001
0.0001
0.001
0 0.2 0.4 0.6 0.8 1
γ s →b 
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Up-type squark masses
~ ,4u
~, 3u
0 0.2 0.4 0.6 0.8 1
Down-type squark masses
 ≈ 4d
FIG. 17: Same as Fig. 14 for our benchmark scenario D.
The rapid fall-off of the relic density for very large λ ≤ 1 can be understood by looking at the resulting lightest up-
and down-type squark mass eigenvalues in the lower left part of Figs. 14-17. For maximal flavour violation, the off-
diagonal squark mass matrix elements are of similar size as the diagonal ones, leading to one squark mass eigenvalue
that approaches and finally falls below the lightest neutralino (dark matter) mass. Light squark propagators and
co-annihilation processes thus lead to a rapidly falling dark matter relic density and finally to cosmologically excluded
NMFV SUSY models, since the LSP must be electrically neutral and a colour singlet.
An interesting phenomenon of level reordering between neighbouring states can be observed in the lower central
diagrams of Figs. 14-17 for the two lowest mass eigenvalues of up-type squarks. The squark mass eigenstates are, by
definition, labeled in ascending order with the mass eigenvalues, so that ũ1 represents the lightest, ũ2 the second-
lightest, and ũ6 the heaviest up-type squark. As λ and the off-diagonal entries in the mass matrix increase, the splitting
between the lightest and highest mass eigenvalues naturally increases, whereas the intermediate squark masses (of
ũ3,4,5) are practically degenerate and insensitive to λ. These remarks also hold for the down-type squark masses
shown in the lower right diagrams of Figs. 14-17. However, for up-type squarks it is first the second-lowest mass that
decreases up to intermediate values of λ = 0.2...0.5, whereas the lowest mass is constant, and only at this point the
second-lowest mass becomes constant and takes approximately the value of the until here lowest squark mass, whereas
the lowest squark mass starts to decrease further with λ. These “avoided crossings” are a common phenomenon for
Hermitian matrices and reminiscent of meta-stable systems in quantum mechanics. At the point where the two
levels should cross, the corresponding squark eigenstates mix and change character, as will be explained in the next
subsection. For scenario C (Fig. 16), the phenomenon occurs even a second time with an additional avoided crossing
between the states ũ2 and ũ3 at λ ≃ 0.05. For scenario B (Fig. 15), this takes place at λ ≃ 0.1, and there is even
another crossing at λ ≃ 0.02. For down-type squarks, the level-reordering phenomenon is not so pronounced.
E. Chirality and Flavour Decomposition of Squark Mass Eigenstates
In NMFV, squarks will not only exhibit the traditional mixing of left- and right-handed helicities of third-generation
flavour eigenstates, but will in addition exhibit generational mixing. As discussed before, we restrict ourselves here to
the simultaneous mixing of left-handed second- and third-generation up- and down-type squarks. For our benchmark
scenario A, the helicity and flavour decomposition of the six up-type (left) and down-type (right) squark mass eigen-
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
FIG. 18: Dependence of the chirality (L, R) and flavour (u, c, t; d, s, and b) content of up- (ũi) and down-type (d̃i) squark
mass eigenstates on the NMFV parameter λ ∈ [0; 1] for benchmark point A.
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
FIG. 19: Same as Fig. 18 for λ ∈ [0; 0.1].
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
FIG. 20: Same as Fig. 18 for benchmark point B.
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
FIG. 21: Same as Fig. 20 for λ ∈ [0; 0.1].
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
FIG. 22: Same as Fig. 18 for benchmark point C.
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
FIG. 23: Same as Fig. 22 for λ ∈ [0; 0.1].
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
FIG. 24: Same as Fig. 18 for benchmark point D.
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
FIG. 25: Same as Fig. 24 for λ ∈ [0; 0.1].
states is shown in Fig. 18 for the full range of the parameter λ ∈ [0; 1] and in Fig. 19 for the experimentally favoured
range in the vicinity of (c)MFV, λ ∈ [0; 0.1]. First-generation and right-handed second-generation squarks remain, of
course, helicity- and flavour-diagonal,
ũ3 = c̃R , d̃2 = s̃R,
ũ4 = ũR , d̃3 = d̃R,
ũ5 = ũL , d̃5 = d̃L, (66)
with the left-handed and first-generation squarks being slightly heavier due their weak isospin coupling (see Eqs. (5)-
(7)) and different renormalization-group running effects. Production of these states will benefit from t- and u-channel
contributions of first- and second-generation quarks with enhanced parton densities in the external hadrons, but they
will not be identified easily with heavy-flavour tagging and are of little interest for our study of flavour violation.
The lightest up-type squark ũ1 remains the traditional mixture of left- and right-handed stops over a large region of
λ ≤ 0.4, but it shows at this point an interesting flavour transition, which is in fact expected from the level reordering
phenomenon discussed in the lower central plot of Fig. 14. The transition happens, however, above the experimental
limit of λ ≤ 0.1. Below this limit, it is the states ũ2, ũ6, d̃1, and in particular d̃4 and d̃6 that show, in addition to
helicity mixing, the most interesting and smooth variation of second- and third-generation flavour content (see Fig.
19). Note that at very low λ ≃ 0.002 the states d̃L and s̃L rapidly switch levels. This numerically small change was
not visible on the linear scale in Fig. 14.
For the benchmark point B, whose helicity and flavour decomposition is shown in Fig. 20, level reordering occurs
at λ ≃ 0.4 for the intermediate-mass up-type squarks,
ũ3,4 = c̃R , d̃2 = s̃R,
ũ4,3 = ũR , d̃3 = d̃R,
ũ5 = ũL , d̃5 = d̃L (67)
whereas the ordering of down-type squarks is very similar to scenario A. Close inspection of Fig. 21 shows, however,
that also d̃R and s̃R switch levels at low values of λ ≃ 0.02. At λ ≃ 0.01, in addition s̃R and b̃L switch levels, and at
λ ≃ 0.002 it is the states ũL and c̃L. The lightest up-type squark is again nothing but a mix of left- and right-handed
stops up to λ ≤ 0.4. Phenomenologically smooth transitions below λ ≤ 0.1 involving taggable third-generation squarks
are observed for ũ4, ũ6, d̃1, and d̃6.
The helicity and flavour decomposition for our scenario D, shown in Fig. 24, is rather similar to the one in scenario
A, i.e.
ũ3 = c̃R , d̃3 = s̃R,
ũ4 = ũR , d̃4 = d̃R,
ũ5 = ũL , d̃5 = d̃L (68)
are exactly the same in the up-squark sector, and only the mixed down-type state d̃4 is now lighter and becomes d̃2.
The lightest up-type squark, ũ1, is again mostly a mix of left- and right-handed top squarks up to λ ≃ 0.4, where
the level reordering and generation mixing occurs (see lower central part of Fig. 17). At the experimentally favoured
lower values of λ ≤ 0.1, the states ũ2, ũ6, d̃1, d̃2, and d̃6 exhibit some smooth variations, shown in detail in Fig. 25,
albeit to a lesser extent than in scenario A. At very low λ ≃ 0.004, it is now the up-type squarks ũL and c̃L that
rapidly switch levels. This numerically small change was again not visible on a linear scale (see Fig. 17).
For our scenario C, shown in Fig. 22, the assignment of the intermediate states
ũ3 = c̃R , d̃3 = s̃R,
ũ4 = ũR , d̃4 = d̃R,
ũ5 = ũL , d̃5 = d̃L (69)
is the same as for scenario D above λ ≥ 0.1. Just below, ũR and c̃R as well as d̃R and s̃R rapidly switch levels, whereas
ũL and c̃L switch levels at very low λ ≃ 0.002. These changes were already visible upon close inspection of the lower
central and right plots in Fig. 16. On the other hand, the lightest squarks ũ1 and d̃1 only acquire significant flavour
admixtures at relatively large λ ≃ 0.2...0.4, whereas they are mostly superpositions of left- and right-handed stops
and sbottoms in the experimentally favourable range of λ ≤ 0.1 shown in Fig. 23. Here, the heaviest states ũ6 and d̃6
show already smooth admixtures of third-generation squarks as it was the case for the scenarios A and D discussed
above. The most interesting states are, however, ũ2, ũ4, d̃2, and d̃4, respectively, since they represent combinations
of up to four different helicity and flavour states and have a significant, taggable third-generation flavour content.
V. NUMERICAL PREDICTIONS FOR NMFV SUSY PARTICLE PRODUCTION AT THE LHC
In this section, we present numerical predictions for the production cross sections of squark-antisquark pairs, squark
pairs, the associated production of squarks and gauginos, and gaugino pairs in NMFV SUSY at the CERN LHC, i.e.
for pp-collisions at
S = 14 TeV centre-of-mass energy. Thanks to the QCD factorization theorem, total unpolarized
hadronic cross sections
4m2/S
∫ 1/2 ln τ
−1/2 ln τ
∫ tmax
dt fa/A(xa,M
a ) fb/B(xb,M
can be calculated by convolving the relevant partonic cross sections dσ̂/dt, computed in Sec. III, with universal parton
densities fa/A and fb/B of partons a, b in the hadrons A,B, which depend on the longitudinal momentum fractions
of the two partons xa,b =
τe±y and on the unphysical factorization scales Ma,b. For consistency with our leading
order (LO) QCD calculation in the collinear approximation, where all squared quark masses (except for the top-quark
mass)m2q ≪ s, we employ the LO set of the latest CTEQ6 global parton density fit [53], which includes nf = 5 “light”
(including the bottom) quark flavours and the gluon, but no top-quark density. Whenever it occurs, i.e. for gluon
initial states and gluon or gluino exchanges, the strong coupling constant αs(µR) is calculated with the corresponding
LO value of Λ
LO = 165 MeV. We identify the renormalization scale µR with the factorization scales Ma =Mb and
set the scales to the average mass of the final state SUSY particles i and j, m = (mi +mj)/2.
The numerical cross sections for charged squark-antisquark and squark-squark production, neutral up- and down-
type squark-antisquark and squark-squark pair production, associated production of squarks with charginos and
neutralinos, and gaugino pair production are shown in Fig. 26 for our benchmark scenario A, in Fig. 27 for scenario
B, in Fig. 28 for scenario C, and in Fig. 29 for scenario D. The magnitudes of the cross sections vary from the barely
visible level of 10−2 fb for weak production of heavy final states over the semi-strong production of average squarks
and gauginos and quark-gluon initial states to large cross sections of 102 to 103 fb for the strong production of diagonal
squark-(anti)squark pairs or weak production of very light gaugino pairs. Unfortunately, these processes, whose cross
sections are largest (top right, center left, and lower right parts of Figs. 26-29), are practically insensitive to the flavour
violation parameter λ, as the strong gauge interaction is insensitive to quark flavours and gaugino pair production
cross sections are summed over exchanged squark flavours.
Some of the subleading, non-diagonal cross sections show, however, sharp transitions, in particular down-type
squark-antisquark production at the benchmark point B (centre-left part of Fig. 27), but also the other squark-
antisquark and squark-squark production processes. At λ = 0.02, the cross sections for d̃1d̃
6 and d̃3d̃
6 switch places.
Since the concerned sstrange and sbottom mass differences are rather small, this is mainly due to the different strange
and bottom quark densities in the proton. The cross section is mainly due to the exchange of strongly coupled gluinos
despite their larger mass. At λ = 0.035 the cross sections for d̃3d̃
6 and d̃1d̃
3 increase sharply, since d̃3 = d̃R can then
be produced from down-type valence quarks. The cross section of the latter process increases with the strange squark
content of d̃1.
At the benchmark point C (Fig. 28), sharp transitions occur between the ũ2/ũ4 and d̃2/d̃4 states, which are pure
charm/strange squarks below/above λ = 0.035, for all types of charged and neutral squark-antisquark and squark-
squark production and also squark-gaugino associated production. As a side-remark we note that an interesting
perspective might be the exploitation of these t-channel contributions to second- and third-generation squark pro-
duction for the determination of heavy-quark densities in the proton. This requires, of course, efficient experimental
techniques for heavy-flavour tagging.
Smooth transitions and semi-strong cross sections of about 1 fb are observed for the associated production of third-
generation squarks with charginos (lower left diagrams) and neutralinos (lower centre diagrams) and in particular
for the scenarios A and B. For benchmark point A (Fig. 26), the cross section for d̃4 production decreases with its
strange squark content, while the bottom squark content increases at the same time. For benchmark point B (Fig.
27), the same (opposite) happens for d̃6 (d̃1), while the cross sections for ũ6 increase/decrease with its charm/top
squark content. Even in minimal flavour violation, the associated production of stops and charginos is a particularly
useful channel for SUSY particle spectroscopy, as can be seen from the fact that cross sections vary over several orders
of magnitude among our four benchmark points (see also Ref. [54]).
An illustrative summary of flavour violating hadroproduction cross section contributions for third-generation squarks
and/or gauginos is presented in Tab. III, together with the competing flavour-diagonal contributions.
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
* + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
* + c.c.
 →p p 
 ≈* 4d
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
 →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ →p p 
∼ ≈ 1u
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
10000
 + c.c.
χ∼ →p p 
FIG. 26: Cross sections for charged squark-antisquark (top left) and squark-squark (top centre) production, neutral up-type (top
right) and down-type (centre left) squark-antisquark and squark-squark pair (centre and centre right) production, associated
production of squarks with charginos (bottom left) and neutralinos (bottom centre), and gaugino pair production (bottom
right) at the LHC in our benchmark scenario A.
VI. CONCLUSIONS
In conclusion, we have performed an extensive analysis of squark and gaugino hadroproduction and decays in non-
minimal flavour violating supersymmetry. Within the super-CKM basis, we have taken into account the possible
misalignment of quark and squark rotations and computed all squared helicity amplitudes for the production and the
decay widths of squarks and gauginos in compact analytic form, verifying that our results agree with the literature in
the case of non-mixing squarks whenever possible. Flavour violating effects have also been included in our analysis
of dark matter (co-)annihilation processes. We have then analyzed the NMFV SUSY parameter space for regions
allowed by low-energy, electroweak precision, and cosmological data and defined four new post-WMAP benchmark
points and slopes equally valid in minimal and non-minimal flavour violating SUSY. We found that left-chiral mixing
of second- and third-generation squarks is slightly stronger constrained than previously believed, mostly due to smaller
experimental errors on the b → sγ branching ratio and the cold dark matter relic density. For our four benchmark
points, we have presented the dependence of squark mass eigenvalues and the flavour and helicity decomposition of
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
 + c.c.
u~ →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
 + c.c.
u~ →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
* + c.c.
u~ →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
* + c.c.
 →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
 + c.c.
u~ →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
 + c.c.
 →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
 + c.c.
χ∼ →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
 + c.c.
χ∼ →p p 
λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1
 + c.c.
χ∼ →p p 
FIG. 27: Same as Fig. 26 for our benchmark scenario B.
TABLE III: Dominant s-, t-, and u-channel contributions to the flavour violating hadroproduction of third-generation squarks
and/or gauginos and the competing dominant flavour-diagonal contributions.
Exchange s t u
Final State
t̃b̃∗ W NMFV-g̃ -
b̃s̃∗ NMFV-Z NMFV-g̃ -
t̃c̃∗ NMFV-Z NMFV-g̃ -
t̃b̃ - - NMFV-g̃
b̃b̃ - g̃ g̃
t̃t̃ - NMFV-g̃ NMFV-g̃
χ̃0b̃ b b̃ -
χ̃±b̃ NMFV-c NMFV-b̃ -
χ̃0 t̃ NMFV-c NMFV-t̃ -
t̃ b t̃ -
χ̃χ̃ γ, Z,W q̃ q̃
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
* + c.c.
u~ iu
~ →p p 
u~ ≈*
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
* + c.c.
 →p p 
 ≈* 2d
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
 →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ iχ
∼ →p p 
FIG. 28: Same as Fig. 26 for our benchmark scenario C.
the squark mass eigenstates on the flavour violating parameter λ. We have computed numerically all production
cross sections for the LHC and discussed in detail their dependence on flavour violation. A full experimental study
including heavy-flavour tagging efficiencies, detector resolutions, and background processes would, of course, be very
interesting in order to establish the experimental significance of NMFV. While the implementation of our analytical
results in a general-purpose Monte Carlo generator should now be straight-forward, such a detailed experimental
study represents a research project of its own [55] and is beyond the scope of the work presented here.
Acknowledgments
A large part of this work has been performed in the context of the CERN 2006/2007 workshop on “Flavour in the
Era of the LHC”. The authors also acknowledge interesting discussions with J. Debove, A. Djouadi, W. Porod, J.M.
Richard, and P. Skands. This work was supported by two Ph.D. fellowships of the French ministry for education and
research.
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
* + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
* + c.c.
 →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
u~ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
 →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ →p p 
λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05
 + c.c.
χ∼ →p p 
FIG. 29: Same as Fig. 26 for our benchmark scenario D.
APPENDIX A: GAUGINO AND HIGGSINO MIXING
The soft SUSY-breaking terms in the minimally supersymmetric Lagrangian include a term
L ⊃ −1
(ψ0)T Y ψ0 + h.c., (A1)
which is bilinear in the (2-component) fermionic partners
ψ0j = (−iB̃,−iW̃ 3, H̃01 , H̃02 )T (A2)
of the neutral electroweak gauge and Higgs bosons and proportional to the, generally complex and symmetric, neu-
tralino mass matrix
M1 0 −mZ sW cβ mZ sW sβ
0 M2 mZ cW cβ −mZ cW sβ
−mZ sW cβ mZ cW cβ 0 −µ
mZ sW sβ −mZ cW sβ −µ 0
 . (A3)
Here, M1, M2, and µ are the SUSY-breaking bino, wino, and off-diagonal higgsino mass parameters with tanβ =
sβ/cβ = vu/vd being the ratio of the vacuum expectation values vu,d of the two Higgs doublets, while mZ is the SM Z-
boson mass and sW (cW ) is the sine (co-sine) of the electroweak mixing angle θW . After electroweak gauge-symmetry
breaking and diagonalization of the mass matrix Y , one obtains the neutralino mass eigenstates
χ0i = Nij ψ
j , i, j = 1, . . . , 4, (A4)
where N is a unitary matrix satisfying the relation
N∗ Y N−1 = diag (mχ̃0
,mχ̃0
,mχ̃0
,mχ̃0
). (A5)
In 4-component notation, the Majorana-fermionic neutralino mass eigenstates can be written as
χ̃0i =
, i = 1, . . . , 4. (A6)
Their mass eigenvalues mχ̃0
can, e.g., be found in analytic form in [56] and can be chosen to be real and non-negative.
The chargino mass term in the SUSY Lagrangian
L ⊃ −1
(ψ+ψ−)
+ h.c. (A7)
is bilinear in the (2-component) fermionic partners
ψ±j = (−iW̃
±, H̃±2,1)
T (A8)
of the charged electroweak gauge and Higgs bosons and proportional to the, generally complex, chargino mass matrix
M2 mW
2 cβ µ
, (A9)
where mW is the mass of the SM W -boson. Its diagonalization leads to the chargino mass eigenstates
χ+i = Vij ψ
χ−i = Uij ψ
, i, j = 1, 2, (A10)
where the matrices U and V satisfy the relation
U∗X V −1 = diag (mχ̃±
,mχ̃±
). (A11)
In 4-component notation, the Dirac-fermionic chargino mass eigenstates can be written as
χ̃±i =
. (A12)
The mass eigenvalues can be chosen to be real and non-negative and are given by [2]
M22 + µ
2 + 2m2W ∓
(M22 − µ2)2 + 4m4W c22β + 4m2W (M22 + µ2 + 2M2 µ s2β)
]1/2}
, (A13)
while the matrices
U = O− and V =
O+ , if detX ≥ 0
σ3 O+ , if detX < 0
with O± =
cos θ± sin θ±
− sin θ± cos θ±
(A14)
are determined by the mixing angles θ± with 0 ≤ θ± ≤ π/2 and
tan 2θ+ =
2mW (M2 sinβ + µ cosβ)
M22 − µ2 + 2m2W cos 2β
and tan 2θ− =
2mW (M2 cosβ + µ sinβ)
M22 − µ2 − 2m2W cos 2β
. (A15)
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http://arxiv.org/abs/hep-ph/0703200
http://arxiv.org/abs/hep-ph/0703204
http://arxiv.org/abs/hep-ph/0702144
http://arxiv.org/abs/hep-ph/0512315
http://arxiv.org/abs/hep-ex/0603003
	Introduction
	Non-Minimal Flavour Violation in the MSSM
	Analytical Results for Production Cross Sections and Decay Widths
	Generalized Strong and Electroweak Couplings in NMFV SUSY
	Squark-Antisquark Pair Production
	Squark Pair Production
	Associated Production of Squarks and Gauginos
	Gaugino Pair Production
	Squark Decays
	Gluino Decays
	Gaugino Decays
	Experimental Constraints, Scans and Benchmarks in NMFV SUSY
	Low-Energy, Electroweak Precision, and Cosmological Constraints
	Scans of the Constrained NMFV MSSM Parameter Space
	(c)MFV and NMFV Benchmark Points and Slopes
	Dependence of Precision Observables and Squark-Mass Eigenvalues on Flavour Violation
	Chirality and Flavour Decomposition of Squark Mass Eigenstates
	Numerical Predictions for NMFV SUSY Particle Production at the LHC
	Conclusions
	Acknowledgments
	Gaugino and Higgsino Mixing
	References
