Supersymmetry and a new prediction for sfermion masses
Andrea RomaninoSISSA
Nardecchia, R, Ziegler arXiv:0909.3058 (JHEP)arXiv:0912.5482 (JHEP)
The Standard Model (SM) of particle physics
An extremely successful synthesis of particle physics
+ non renormalizable neutrinos mass operator: LLHH
in compact notations
i = 1,2,3: family index
ΨiiγµDµΨi −
14F a
µνF aµν gauge
LrenSM = +λijΨiΨjH + h.c. flavor
+|DµH|2 − V (H) symmetry breaking
few ‰
few %
indirect
Many reasons to go beyond the SM
Experimental “problems” of the SMGravityDark matterBaryon asymmetry
Experimental “hints” of physics beyond the SMNeutrino massesQuantum number unification
Theoretical puzzles of the SM<H> « MPl
Family replicationSmall Yukawa couplings, pattern of masses and mixingsGauge group, no anomaly, charge quantization, quantum numbers
Theoretical problems of the SMNaturalness/unitarity problemCosmological constant problemStrong CP problem
The unitarity/naturalness argument
Theorem (does not assume the existence of the Higgs in the SM)
Either physics becomes strongly interacting at TeV or
Physics is weakly interacting up or beyond the TeV scale in this case the Higgs h exists and m2h ≈ (m2h)0 + (115GeV)2 (QNP/0.5TeV)2
In the latter case, QNP » TeV needs delicate cancellations, so that
NP @ TeV cuts-off δm2h and the electroweak scale is “natural”, or
the electroweak scale is accidentally smaller than its radiative corrections, or the naturalness argument is not relevant at all
The LHC: status and plans as of 05.02.10
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ABC+DE BFB BG GH IB
AGJ+DE BF= B> GI JH
AJI+DE >F= BH IG HK
AIL+DE >FJ = C> GJ
ALH+DE >FL BJ GB JK
Steve Myers, summary of the Chamonix LHC performance workshop
The landscape of theory modelsand its consequences
Lack of signals, also indirect, from the TeV scale → proliferation of theory models
Higgsless: TC, ETC, walking-TC, EWSB in 5D or more, etc
TeV cutoff for δm2h:
Fundamental scale (large, TeV, susy, flat, warped, etc)
Higgs compositeness (plain, various Little, etc)
Supersymmetry breaking scale (MSSM, xMSSM, etc)
Fine-tuned models (SM, SpS, SuperSpS, etc)
The experiment provides an interesting perspective
the LEP&C heritage: EWPTs and the “little hierarchy” problem
quantum number and gauge coupling unification
the flavour problem
The little residual hierarchy
– <H> = 174 GeV
– MPl
– QNP
NP
SM
E
δm2h ∼
3GF√2π2
m2t Q
2NP
maximally B, L-violating NP: QNP > 1012 TeV
maximally Bi, Li, CP-violating NP: QNP > 103 TeV
B, L, Fl, CP conserving: QNP > λ 5 TeV
ν’s, p-decay, GUTs (4D, 5D)
why is TeV flavour violation
“small”?
EWPTs, conservative
= (115GeV)2(
QNP
0.5 TeV
)2
The little residual hierarchy
– <H> = 174 GeV
– MPl
– QNP
NP
SM
E
δm2h ∼
3GF√2π2
m2t Q
2NP = (115GeV)2
(QNP
0.5 TeV
)2
ci
Q2NP
! 1(5 TeV)2
No hint of physics at QNP:
LeffSM(E < QNP) = Lren
SM +∑
i
ci
Q2NP
Oi + . . .
QNP ! √ci · 5 TeV ≈
50 TeV composite e
5 TeV composite Ga, h
0.5 TeV 1-loop perturbative
[Barbieri Strumia, hep-ph/0007265]
MSSM
QNP ! √ci · 5 TeV ≈ 0.5 TeV
δm2h ∼
3GF√2π2
m2t Q
2NP log
M2Pl
Q2NP
SUSY
unificationneutrino massesbaryogenesis
?
– <H> = 174 GeV
– MPl
– QNP
SM
E
δm2h ∼
3GF√2π2
m2t Q
2NP = (115GeV)2
(QNP
0.5 TeV
)2
Supersymmetry: fermions ↔ bosons
H2H1
e
ν1
u
d
μ
ν2
c
s
τ
ν3
t
b
W
Z
γ
g
H
e
ν1
u
d
μ
ν2
c
s
τ
ν3
t
b
W
Z
γ
g
H2H1
~ ~ ~
~~ ~
~ ~ ~
~~~
~ ~
~
~
~
~
typically, m ≳ 100 GeV(supersymmetry breaking)
How supersymmetry solves the hierarchy problem
Note that it is crucial that the coupling are exactly equal. Supersymmetry breaking, if it is not to spoil the solution of the hierarchy problem should maintain this equality. Supersymmetry breaking should only affect masses (which is what we need), dimensionful parameters (“soft” breaking)
0 50 100 150 200 250 300H
u [GeV]
0
20
40
60
Hd [G
eV
]
Figure 7.1: A contour map of the Higgs potential, for a typical case with tan β ≈ − cot α ≈ 10.The minimum of the potential is marked by +, and the contours are equally spaced equipotentials.Oscillations along the shallow direction, with H0
u/H0d ≈ 10, correspond to the mass eigenstate h0, while
the orthogonal steeper direction corresponds to the mass eigenstate H0.
∆(m2h0) =
h0
t
+h0
t
+ h0
t
Figure 7.2: Contributions to the MSSM lightest Higgs mass from top-quark and top-squark one-loopdiagrams. Incomplete cancellation, due to soft supersymmetry breaking, leads to a large positivecorrection to m2
h0 in the limit of heavy top squarks.
and is traditionally chosen to be negative; it follows that −π/2 < α < 0 (provided mA0 > mZ). TheFeynman rules for couplings of the mass eigenstate Higgs scalars to the Standard Model quarks andleptons and the electroweak vector bosons, as well as to the various sparticles, have been worked outin detail in ref. [165, 166].
The masses of A0, H0 and H± can in principle be arbitrarily large since they all grow with b/ sin(2β).In contrast, the mass of h0 is bounded above. From eq. (7.20), one finds at tree-level [167]:
mh0 < mZ | cos(2β)| (7.23)
This corresponds to a shallow direction in the scalar potential, along the direction (H0u−vu,H0
d −vd) ∝(cos α,− sin α). The existence of this shallow direction can be traced to the fact that the quartic Higgscouplings are given by the square of the electroweak gauge couplings, via the D-term. A contour mapof the potential, for a typical case with tan β ≈ − cot α ≈ 10, is shown in figure 7.1. If the tree-levelinequality (7.23) were robust, the lightest Higgs boson of the MSSM would have been discovered atLEP2. However, the tree-level formula for the squared mass of h0 is subject to quantum correctionsthat are relatively drastic. The largest such contributions typically come from top and stop loops, asshown‡ in fig. 7.2. In the simple limit of top squarks that have a small mixing in the gauge eigenstatebasis and with masses mt1
, mt2much greater than the top quark mass mt, one finds a large positive
one-loop radiative correction to eq. (7.20):
∆(m2h0) =
3
4π2cos2α y2
t m2t ln
(mt1
mt2/m2
t
). (7.24)
This shows that mh0 can exceed the LEP bounds.‡In general, one-loop 1-particle-reducible tadpole diagrams should also be included. However, they just cancel against
tree-level tadpoles, and so both can be omitted, if the VEVs vu and vd are taken at the minimum of the loop-correctedeffective potential (see previous footnote).
68
34π2
λ2t Q
2 − 34π2
λtQ2
λ2t = λt
MotivationsPhenomenological
Solves the naturalness (hierarchy) problem
Precisely predicts gauge coupling unification
Provides a natural DM candidate (needs RP)
...
Theoretical
Unification of fermions and bosons
Local supersymmetry = supergravity + crucial in string theory
Completes the list of possible symmetries of S (under hypotheses)
Powerful technical tool
Supersymmetry breakingThe supersymmetrization of the SM is straightforward, essentially unique, and does not introduce new parameters (it actually predicts one)
Breaking supersymmetry is non-obvious, the mechanism is unknown (spontaneous?), a model-independent effective description is useful
But about 100 new physical parameters
And large FCNC (and CPV) processes in most of the parameter space, (SUSY flavour problem)
One solution of SUSY flavour problem: m2ij = m20 δij + rad corr
−Lsoft = (m2q)ij q
†i qj + (m2
uc)ij(uci )
†ucj + (m2
dc)ij(dci )
†dcj + (m2
l )ij l†i lj
+ (m2ec)ij(ec
i )†ec
j + m2hu
h†uhu + m2
hdh†
dhd
+M3
2gAgA +
M2
2WaWa +
M1
2BB + h.c.
+ AUij u
ci qjhu + AD
ij dci qjhd + AE
ij eci ljhd + m2
udhuhd + h.c.
(MSSM)
A wide class of models of supersymmetry breaking
Hidden sector
Observable sector
SUSY breaking MSSM?
Z chiral superfield<Z> = Fθ2
F » (MZ)2SM singlet
Q chiral superfield(MSSM)
∫d4θ
Z†Z Q†Q
M2
M
→ m2Q†Q, m2 =F 2
M2
Examples: gravity mediation, gauge mediation, gaugino mediation...
Tree level gauge mediation
Z†
Z
Q†
Q
heavy vector superfieldSM singlet
non-anomalousassumed to part of a GUT
↑
V
∫d4θ
Z†Z Q†Q
M2
“Dietrologia”Supersymmetry breaking masses (Z*ZQ*Q) are obtained at the tree level from spontaneous SUSY breaking in a renormalizable theory
Two arguments seem to prevent this possibility
1. what about the supertrace formula?
2. what about gaugino masses?
The supertrace formulaStr M2 ≡ ∑bosons m2 - ∑fermions m2 (weighted by # of dofs)
Ren. Kähler + tree level + Tr(Ta) = 0:
Holds separately for each set of conserved quantum numbers
Implication 1:
Holds in particular for the set of quantum numbers associated to the SM fermions “f”
MSSM: incompatible with (Str M2)f,MSSM = ∑sfermions m2 - ∑fermions m2 > 0
In our case there are extra heavy dofs with same quantum numbers: (Str M2)f,extra < 0, so that (Str M2)f,tot = (Str M2)f,MSSM + (Str M2)f,extra = 0 (automatically)
Implication 2:
Assume G = GSM; consider fields with quantum numbers of d-quarks
(if DY < 0, consider up sfermions)
Need extra U(1) with same sign on up and down quarks: our V
StrM2 = gDa Tr(Ta) = 0
m2lightest d-sfermion ≤ m2
d −13g′DY
Gaugino massesMg = 0 at the tree level
Mg arise at the 1-loop level (as in ordinary gauge mediation)
Expect hierarchy between sfermion and gaugino masses:
Sfermions might be out of the reach of LHC + significant FT (split susy)
Here:
mf ! M2/(α/4π) ∼ 100 M2 ! 10 TeV
[Arkani-Hamed Dimopoulos Giudice R]
mf ! (mod dep)/O (10)×M2/(α/4π) ! (mod dep)× 1 TeV
A concrete exampleG = SO(10) “minimal” GUT (V heavy SM singlet means rank ≥ 5)
V associated to the SU(5)-invariant generator “X”
gives
The (usual) embedding of a MSSM family in a single 16 does not work (whatever the sign of XZ)
SO(10) SU(5)
16 = 5 + 10 + 1X -3 1 5
-SO(10) SU(5)
10 = 5 + 5X 2 -2
-
Z†
Z
Q†
QVm2
Q ∝ XQXZ
The three MSSM families are embedded in 16i + 10i , i=1,2,3 (needs XZ > 0)
Let us only consider SO(10) reps with d < 120
SO(10) breaking to the SM needs 16 + 16 + 45 16 + 16 needed to reduce the rank 16 = 5 + 10 + 1 16 = 5 + 10 + 1 <1> = <1> = M ≈ MGUT (or larger)
SUSY breaking: sfermion masses need Z SM singlet with XZ > 0 only option: Z is the singlet of a 16’ gauge invariance: 16’ ≠ 16 16’ = 5’ + 10’ + Z 16’ = 5’ + 10’ + Z <Z> = F θ2 (<Z> = 0 for simplicity)
SO(10) SU(5)
16i = 5i + 10i + 1iX -3 1 5
-SO(10) SU(5)
10i = 5i + 5i
X 2 -2
-
must be made heavy
_ _- - -
__
_ _- - -
m2Q =
XQ
2XZ
F 2
M2
Then
In particular
all sfermion masses are positive
sfermion masses are flavour universal, thus solving the supersymmetric flavour problem
(at M)
m2q = m2
uc = m2ec = m2
10 =110
m2, m2l = m2
dc = m25 =
15
m2, m =F
M
m2q,uc,ec =
12m2
l,dc
Higgs doublets: only possible embeddings are into 16, 10 (hd), 16, 10 (hu)
Let cos2θu,d be the size of the components of hu,d in 10’s
Then
_
m2hu
=−2c2
u + 3s2u
5m2
−25m2 < m2
hu<
35m2
m2hd
=2c2
d − 3s2d
5m2
−35m2 < m2
hd<
25m2
Splitting the SO(10) multiplets
Automatic!
RP-invariant superpotential interaction involving 16i 10i:
hij 16i 10j 16 → Mij 5i 5j when 16 → <16>
Mij = M hij (M = <116>) (hij may be related to light fermion masses)
(note also h’ij 16i 10j 16’ coupling 5i, 5j to supersymmetry breaking)
Reinforces the theoretical consistency of the framework
SO(10) SU(5)
16i = 5i + 10i + 1iX -3 1 5
-SO(10) SU(5)
10i = 5i + 5i
X 2 -2
-
must be made heavy
-
-
SM Yukawas
Down quark and charged lepton Yukawas:
10i 5j 5H (in SU(5) language) → hij 16i 10j 16H (where possibly 16H = 16)
Up quarks:
10i 10j 5H (in SU(5) language) → yij 16i 16j 10H
Note: down and up quarks described by two independent Yukawa matrices (room to explain their different structure despite the SO(10) constraints)
- -
Assume:
Minimal GUT implementation (SO(10))
Only SO(10) reps with d < 120
Pure embeddings of SM multiplets in 1 type of SO(10) reps (guarantees the solution of the SUSY flavour problem), or no matter mass terms
Then there are only two possible MSSM embeddings:
into 16i + 10i: needs XZ > 0, leads to
into 16i + 45i: needs XZ < 0, leads to
the top Yukawa needs embedding the Higgs in d = 210 or more
How general are the predictions?
m2q,uc,ec =
12m2
l,dc
m2q,uc,ec =
43m2
l,dc
SO(10) SU(5)
16i = 5i + 10i + 1iX -3 1 5
-SO(10) SU(5)
45i = 24i + 10i + 10i + 1iX 0 -4 4 0
_
A natural option is E6
The fundamental of E6 decomposes as 27i = 16i + 10i + 1i under SO(10)
yij 27i 27j 27 can in principle account for all Yukawas
Non minimal GUTs?
Gaugino massesVanish at the tree level
Arise at one-loop because of a built-in ordinary gauge mediation structure
Consider for example the 16i + 10i model
(W = hij 16i 10j 16 + h’ij 16i 10j 16’)
O(100) hierarchy → O(10): mt > O(10 TeV) → O(1 TeV) + model dep factor λ
(model dependent; the three messengers contribute at different scales; the enhancement also enhances two loop contributions to sfermion masses)
SO(10) SU(5)
16i = 5i + 10i + 1iX -3 1 5
SO(10) SU(5)
10i = 5i + 5i
X 2 -2
M2
mt
∣∣∣∣MGUT
=3√
10(4π)2
λ, λ =g2 Tr(h′h−1)
3
~
X
Z
5i 5j
5i 5j
h′ij
hijM
Vg Vg
Mg =α
4πTr(h′h−1) m, mt =
m√10
(m =
F
M
)
Sugra contributions to sfermion massesAdd to the tree level gauge mediated contribution and may induce FCNCs
Their size is less important than in loop gauge mediation (because no loop suppression here). As a consequence, a messenger scale as large as MGUT does not represent a potential problem for FCNCs
Assuming the gravity contribution to a generic entry of the sfermion mass matrix is (m2)sugra = (F/MPl)2 (MPl = 2.4 1018 GeV) we obtain
(m2)sugra < 2 10-3 (m2)stop iff M < 3 1016 GeV (guarantees FCNC effects from flavour-anarchical sugra contribution are under control)
A new solution to the μ-problemμ is a supersymmetric mass parameter that accounts for Higgsino masses
Its phenomenological window O(100 GeV) < μ < O(1 TeV) turns out to coincide with the window of supersymmetry breaking sfermion masses m: is it an accident or is there a connection between μ and m?
In the absence of a connection, there would be no reason why μ should not be of the order of a much larger, susy-conserving mass scale, such as MGUT or MPl. Or, if μ is suppressed by a symmetry, there would be no reason why it should not be much smaller or vanish
Well known, appealing solutions of the μ-problem can be implemented
D=5: Giudice-Masiero can arise at loop-level; μ ≈ Mg ≈ 100 GeV, Bμ ≈ m ≈ TeV (because of μ/Bμ connection)
D=4: NMSSM S can have negative soft mass (unlike in ordinary gauge mediation) but should take care of quartic coupling
D=3: an intrinsic TGM solution
∫d4θ a
Z†
Mhuhd → µ = a
F
M
~~
∫d2θ λShuhd → µ = λ 〈S〉 ∼ λm
The D=3 solution of the μ-problem in TGM: μ and m arise from the same mass term in the superpotential
Reminder: we need
16 = 5 + 10 + 1 16 = 5 + 10 + 1 <1> = <1> = M ≈ MGUT
16’ = 5’ + 10’ + Z 16’ = 5’ + 10’ + Z <Z> = F θ2
The easiest way to get a susy-breaking <16’> is through W = m 16’ 16 [+ Y (16 16 - M2) + X 16’ 16] then F = m M (so that m = F/M)
Because of SO(10), m is also the mass term of the doublet components of 16’ 16, which can contain a Higgs component: 16’ = α’ hd + ..., 16 = α hu + ... so that μ = α α’ m = α α’ (F/M)
~
_ _- - -_ _-
_ _ _
_ _
_
LSP is the gravitino (in the regime in which sugra FCNC effects are under control), as in loop gauge mediation
Stable gravitino: a dilution mechanism is necessary not to overclose the universe, TR < 2 109 GeV
NLSP decay can spoil BBN
If the NLSP is a neutralino (typical case) a decay channel much faster than the Goldstino one is needed in order not to spoil BBN (e.g. a tiny amount of RP-violation; consistent with thermal leptogenesis and gravitino DM)
If the NLSP is a stau (the other possibility) BBN not a problem but the peculiar predictions of TGM are hidden by large loop gauge mediation contributions
(work in progress)
Cosmology
m3/2 =F√3MP
≈ 15 GeV(
m10
TeVM
2 · 1016 GeV
)
[Buchmuller, Covi, Hamaguchi, Ibarra, Yanagida,arXiv:hep-ph/0702184 (JHEP)]
An example of spectrum
Higgs: mh0 114
mH0 1543
mA 1543
mH± 1545
Gluinos: Mg 448
Neutralinos: mχ01
62
mχ02
124
mχ03
1414
mχ04
1415
Charginos: mχ±1
124
mχ±2
1416
Squarks: muL1092
muR1027
mdL1095
mdR1494
mt11007
mt21038
mb11069
mb21435
Sleptons: meL1420
meR1091
mτ1 992
mτ2 1387
mνe1418
mντ1382
h0
H0 A0
H±
N1
N2
N3 N4
C1
C2
g
dR
eL νe
dL uLeR
uR
b2
ντ τ2
t2b1
t1 τ1
100
500
1000
1500
GeV
Figure 2: An example of spectrum, corresponding to m = 3.2TeV, M1/2 = 150GeV, θd = π/6,tan β = 30 and sign(µ) = +, A = 0, η = 1. All the masses are in GeV, the first two familieshave an approximately equal mass.
8
ConclusionsTree level gauge mediation can be considered as the simplest way to communicate supersymmetry breaking: through the tree level renormalizable exchange of a gauge (GUT) messenger
This possibility is viable, despite the well known arguments associated to the supertrace formula
It offers new model building avenues (including μ-problem)
It solves the susy flavour problem
It leads to a description of supersymmetry breaking in terms of few parameters (as in the CMSSM), with peculiar relations among sfermion masses that can be tested at the LHC
Unification
103 106 109 1012 1015Q !GeV"
0
10
20
30
40
50
60
70
1#Α11#Α21#Α3 MSSM
SU(3) SU(2) U(1)
Li
eci
Qi
uci
dci
1 2 -1/2
1 1 1
3 2 1/6
3* 1 -2/3
3* 1 1/3
Y
SO(10)
16
103 106 109 1012 1015Q !GeV"
0
10
20
30
40
50
60
70
1#Α11#Α21#Α3 SM
+ MGUT prediction: ΛB < MGUT < MPl
back
Dark Matter
Ωχh2=688π5/2T 3
γ xf
99√
5g∗(H0/h)2M3Plσ
= 0.1pbσ
back
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