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