D-term Dynamical SUSY Breaking

Nobuhito Maru (Keio University)

with H. Itoyama (Osaka City University)

Int. J. Mod. Phys. A27 (2012) 1250159

[arXiv: 1109.2276 [hep-ph]]

12/6/2012 SCGT2012@Nagoya University

Plan

l  Introduction l  Basic Idea l  Gap equation l  Some Comments on Phenomenological Application l  Summary

Introduction

SUPERSYMMETRY

is one of the attractive scenarios

solving the hierarchy problem,

but it must be broken at low energy

Dynamical SUSY breaking(DSB) is most desirable to solve

the hierarchy problem

F-term DSB is induced by non-perturbative effects due to nonrenormalization theorem and well studied so far

D-term SUSY breaking is NOT affected by the nonrenormalization theorem

In principle, D-term DSB is possible, but no known explicit model as far as we know

Basic Idea

L = d 4θK Φa ,Φa ,V( ) + d 2θ Im 1

2τ ab Φa( )∫ W aαWα

b + d 2θW Φa( ) + h.c.∫⎡⎣ ⎤⎦∫

N=1 SUSY U(N) gauge theory with an adjoint chiral multiplet

Wαa = −iλα

a y( ) + δαβDa y( )− i

2σ µσ ν( )α

βFµνa y( )⎡

⎣⎢⎤⎦⎥θβ

Φa = φ a y( ) + 2θψ a y( ) +θθFa y( )

d 2θτ ab Φ( )W aαWαb∫ ⊃τ abc Φ( )ψ cλ aDb +τ abc Φ( )Fcλ aλ b

d 2θW Φ( )∫ ⊃ − 12∂a∂bW Φ( )ψ aψ b

τ abc ≡ ∂τ ab Φ( ) ∂φ c

Dirac mass term

Fermion masses Important

dim 5 operator

yµ = xµ + iθσ µθ

∂W(Φa)/∂Φa = 0 @tree level assumed

− 12

λ a ψ a( )0 − 2

4τ abcD

b

− 24

τ abcDb ∂ac∂cW

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

λ c

ψ c

⎛

⎝⎜⎞

⎠⎟+ h.c.

Fermion mass terms

MF ≡0 − 2

4τ 0aaD

0

− 24

τ 0aaD0 ∂ac∂aW

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Mixed Majorana-Dirac type masses (<F>=0 assumed)

Mass matrix

Only U(1) part of D-term VEV is assumed

m± =12

∂a∂aW 1± 1+ 2 D∂a∂aW

⎛

⎝⎜⎞

⎠⎟

2⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

D ≠ 0& ∂a∂aW ≠ 0if

Gaugino becomes massive by nonzero <D>

⇒ SUSY is broken

D ≡ − 24

τ 0aaD0

D0 = − 12 2

g00 τ 0cdψdλ c +τ 0cdψ

dλ c( )D-term equation of motion:

Dirac bilinear condensation

The value of <D> will be determined

by the gap equation

Gap equation

V1−loopD( ) = ma

4

a∑ c + 1

64π 2⎛⎝⎜

⎞⎠⎟ Δ

2 + ′ΛresΔ4

8⎧⎨⎩

− 132π 2 λ +( )4 logλ +( )2 + λ −( )4 logλ −( )2⎡⎣ ⎤⎦

⎫⎬⎭

1-loop effective potential for D-term

Tree level D-term pot. + 1-loop CW pot. + counter term (-Im Λ/2 ∫d2Θ WαWα)

ma ≡ ∂a∂aW , λ ±( ) ≡ 121± 1+ Δ2⎡⎣

⎤⎦, Δ ≡

τ 0aaD0

2ma

, ′Λres ≡ c + β + Λres +1

64π 2 ,

1ma

4

a∑

∂2V∂Δ( )2

Δ=0

= 2c, β ≡g00 ∂a∂aW

2

ma4 τ 0aa

2

a∑

, Λres ≡ImΛ( ) ∂a∂aW

22

ma4 τ 0aa

22

a∑

Gap equation

0 =∂V1−loop

D( )

∂Δ= Δ c + 1

64π 2 +′Λres

4Δ2⎡

⎣⎢

− 164π 2 1+ Δ2

λ +( )3 2 logλ +( )2 +1( )− λ −( )3 2 logλ −( )2 +1( ){ }⎤⎦⎥

5 10 15 20 25 30

!1.0

!0.5

0.5

1.0

Δ

c +1 64π2 = 1, ′Λres 8 = 0.001

Nontrivial solution!!

1!

∂V1-loop(D)

∂!

E = D2/2 ≧ 0 in SUSY

⇒ Trivial solution Δ=0 is NOT lifted

⇒ Our SUSY breaking vac. is a local min.

Δ=0 1 2 3 4 5

5

10

15V(φ)

φ !≠ 0

Metastability of our false vacuum <D> = 0 tree vacuum is not lifted

⇒ check if our vacuum <D> 0 is sufficiently long-lived ≠

Decay rate of the false vacuum

∝ exp −

Δφ 4

ΔV⎡

⎣⎢⎢

⎤

⎦⎥⎥≈ exp − Λ2

ma2

⎡

⎣⎢

⎤

⎦⎥1

1 2 3 4 5

5

10

15V(φ)

φ

Coleman & De Luccia(1980) Δφ

ΔV our vac.

Long-lived for ma<< Λ

ma: mass of Φ, Λ: cutoff scale

Some Comments on Phenomenological Application

Following the model of Fox, Nelson & Weiner (2002),

consider a N=2 gauge sector & N=1 matter sector in MSSM ↑ Chirality, Asymptotic freedom of QCD

Take the gauge group (U(1)’:hidden gauge group) U 1( )′ ×GSM

Dim 5 gauge kinetic term provides Dirac gaugino mass term

d 2θτ abc Φ( )ΦSM

c ′W αaWαSMb∫ ⇒τ abc Φ( ) ′D a ψ SM

c λSMb

Gaugino masses are generated at tree level

Once gaugino masses are generated at tree level,

sfermion masses are generated by RGE effects

Msf2 ≈

Ci R( )α i

πM λi

2 log ma2

M λi2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Sfermion masses @1-loop

Flavor blind ⇒ No SUSY flavor & CP problems

(i = SU(3)C, SU(2)L, U(1)Y)

Figure 2: Loop contributions to scalar masses. The new contribution from the purely scalar loopcancels the logarithmic divergence resulting from a gaugino mass alone.

for the masses involving this spurion. The only possible such counterterm is proportional to

(2.3), and gives!

d4!!2!

2m4

D

!2Q†Q. (2.10)

Since we have four powers of mD, we have to introduce another scale to make this dimen-

sionfully consistent. Since the only other scale is the cuto" !, this operator is suppressed

by !2, and, in the limit that ! ! ", must vanish. Consequently, we conclude all radiative

corrections to the scalar soft masses are finite.

While a gaugino mass (including a Dirac mass) would ordinarily result in a logarith-

mic divergence, here this is cancelled by the new contribution from the scalar loop. The

contribution to the scalar soft mass squared is given by

4g2i Ci(")

!

d4k

(2#)41

k2#

1

k2 # m2i

+m2

i

k2(k2 # $2i )

, (2.11)

where mi is the mass of the gaugino of the gauge group i, and $2 is the SUSY breaking mass

squared of the real component of ai. If the term in (2.9) is absent, then $ = 2mi. As expected,

this integral is finite, yielding the result

m2 =Ci(r)%im2

i

#log

"

$2

m2i

#

. (2.12)

Note that as $ approaches mi from above, these one loop contributions will vanish! If A has

a Majorana mass of M , then this formula generalizes

m2 =Ci(r)%im2

i

#

$

log

"

M2 + $2

m2i

#

#M

2#log

%

2# + M

2# # M

&

'

, (2.13)

where #2 = M2/4 + m2i .

These contributions, arising from gauge interactions, are positive and flavor blind as in

gauge and gaugino mediation, but there are two other remarkable features of this result.

6

Fox, Nelson & Weiner, JHEP08 (2002) 035

Summary l  A new dynamical mechanism of DDSB proposed

l  Shown a nontrivial solution of the gap eq. with nonzero <D> in a self-consistent Hartree-Fock approx.

● Our vacuum is metastable & can be made long-lived

● Phenomenological Application briefly discussed

Backup

δV = 0

⇒ F0 2+ m0

g00 ∂0g00F0 + 1

2D0 2

+ 2 g00 V1−loop = 0

D0 ≠ 0 ⇒ F0 ≠ 0

V = gab ∂aW ∂bW − 12gabD

aDb +V1−loop +Vc.t .

Effective potenital up to 1-loop

Stationary condition

This determines the value of nonvanishing <F>

Lmassholo( ) = − 1

2g0a,a F 0 ψ aψ a + i

4F0aa F0 λ aλ a

− 12

∂a∂aW ψ aψ a + 24

F0aa D0 ψ aλ a

Fermion masses

SU(N) part:

U(1) part:

NG fermion: admixture of and λ 0 ψ 0

Fermion masses are modified as follows

Once SUSY is broken, the next issue we should consider is

how to mediate its breaking to our world

SUSY SSM

SM gauge interaction loops

δ12d( )

LL≡mQ ,122

m2≤ 0.01

m100 GeV

⎛⎝⎜

⎞⎠⎟

SUSY breaking mediation mechanism reflects the pattern of sparticle spectrum,

which must satisfy severe constraints from experiments

ex. K0 − K 0 mixing

Gauge mediation is one of the attractive scenarios

Super-Higgs mechanism

e−1Lgravitino mass = −eK 2W †ψ µσ

µνψν +ψ µσµ − g

2Daλ

a + eK 2 i2DaW

†ψ a⎡⎣⎢

⎤⎦⎥+ h.c.

′ψ µ =ψ µ +i2e−K 2

W † σ µ − g2Daλ

a + eK 2 i2DaW

†ψ a⎡⎣⎢

⎤⎦⎥

e−1Lgravitino mass = −eK 2W † ′ψ µσ

µν ′ψν +i2eK 2

W † − g2Daλ

a + eK 2 i2DaW

†ψ a⎡⎣⎢

⎤⎦⎥

2

+ h.c.

If we couple our theory to supergravity, 　　　　　　　　　　　　　　　　　　　　　a super-Higgs mechanism works Gravitino mass

can be diagonalized by the following redefinition 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　of the gravitino

m3 2 = e

K 2 WMP

2 ≈0≈ V e K 2

DaW2+ g

2

2Da 2

3MP2

Gravitino mass

LU N( ) = Im d 4θTrΦeV ∂F Φ( )∂Φ∫ + d 2θ 1

2∂2F Φ( )∂Φa ∂Φb WαaWα

b∫⎡

⎣⎢

⎤

⎦⎥

+ d 2θTr 2eΦ+m ∂F Φ( )∂Φ

⎡⎣⎢

⎤⎦⎥+ h.c.∫

⎛⎝⎜

⎞⎠⎟

N = 2→N = 1 ModelFujiwara, Itoyama & Sakaguchi (2005, 2006)

“Electric” & “Magnetic” FI terms

F Φ( ) :"prepotential" holomorphic function of Φ

Take the following N=2 SYM

λ a ψ a( )→ ψ a − λ a( )

N=2 theory can be obtained from N=1 theory by imposing SU(2)R invariance

δλ a

ψ a

⎛

⎝⎜⎞

⎠⎟= 2 N gab

0 eδb0 +mF0b

− eδb0 +mF0b 0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

η1η2⎛⎝⎜

⎞⎠⎟

= 2 N gab 0e− e( )δb

0η1

⎛⎝⎜

⎞⎠⎟

SUSY transformation

0 = ∂V∂φ a = gbdFadeg

ec eδb0 +mF0b eδ c

0 +mF0c

=0

eδb

0 +mF0b = 0Vacuum condition chosen as (F=0) from

N=2 SUSY is partially broken to N=1

NG fermion: ψ

Λa±( ) = maλ

±( ), λ ±( ) ≡ 121± 1+ Δ2( )

ma ≡ 2Nm gaaF0aa , Δ2 ≡

D0( )24Nm2

In the present model, the gaugino masses are

Adjoint scalar mass

V(Aµ, λ), Φ(φ, ψ)

Mass

0

ma Φ(φ, ψ)

V(Aµ, λ)

N=2 → N=１