D-term Dynamical SUSY Breaking

Nobuhito Maru

(Keio University)

with H. Itoyama (Osaka City University)

arXiv: 1109.2276 [hep-ph]

7/26/2012 @YITP workshop on ”Field Theory & String Theory”�

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 @tree level is well known,

but no known explicit model of D-term DSB as far as we know�

Tree � 1-loop

Plan�

!  Introduction !  Basic Idea !  Veff @1-loop !  Gap equation & nontrivial solution !  induced by !  Comments on model building !  Summary�

‹F0›≠ 0 ‹D0›≠ 0

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( ) yµ = xµ + iθσ µθ

Antoniadis, Partrouche & Taylor (1996) Fujiwara, Itoyama & Sakaguchi (2005)�

N=2�N=1 partial breaking models naturally applicable�

LU N( ) = Im d 4θTrΦeV ∂F Φ( )∂Φ

+ d 2θ 12∂2F Φ( )∂Φa ∂Φb W aαWα

b∫∫⎡

⎣⎢

⎤

⎦⎥ + d 2θW Φ( ) + h.c.∫( )

W Φ( ) = Tr 2eΦ+m ∂F Φ( )∂Φ

⎡⎣⎢

⎤⎦⎥

Electric & Magnetic FI terms�

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

D=5 operator

yµ = xµ + iθσ µθ

− 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�

<D0>: U(1) part of D-term�

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 bilinears condensation�

The value of <D> will be determined

by the gap equation�

Veff@1-loop�

V1−loop =d 4k2π( )4

ln∫a∑

ma2λ +( )2 − k2 − iε( ) ma

2λ −( )2 − k2 − iε( )ma2 − k2 − iε( ) −k2 − iε( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= 132π 2 ma

4

a∑ A d( ) Δ2 + 1

8Δ4⎛

⎝⎜⎞⎠⎟ − λ

+( )4 logλ +( )2 − λ −( )4 logλ −( )2⎡⎣⎢

⎤⎦⎥

1-loop effective potential for D-term �

A d( ) = 34−γ + 1

2 − d 2d=4⎯ →⎯ ∞

ma ≡ gaa ∂a∂aW ,λ ±( ) ≡ 121± 1+ Δ2⎡⎣

⎤⎦,Δ ≡

τ 0aaD2ma

SUSY counterterm added�

Vc.t . = − Im Λ

2d 2θWα 0Wα

0∫ = − Im Λ2D0( )2

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 �

′Λres ≡ c + β + Λres +1

64π 2 =1

32π 2 A d( ),

β ≡g00 ∂a∂aW

2

ma4 τ 0aa

2

a∑

, Λres ≡ImΛ( ) ∂a∂aW

22

ma4 τ 0aa

22

a∑

1ma

4

a∑

∂2V1−loopD( )

∂Δ( )2Δ=0

= 2c

Gap equation &

Nontrivial solution�

Gap equation�

0 =∂V1−loop

D( )

∂Δ

= Δ c + 164π 2 +

′Λres

4Δ2⎡

⎣⎢

− 164π 2 1+ Δ2

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

λ ±( ) ≡ 121± 1+ Δ2( )

Trivial solution Δ=0 is NOT lifted unlike NJL�

Δ2 1:Δ2 − 4cΛres' − 5 64π 2( ) c < 0( )

Δ2 1:c + 164π 2 + Λres

' Δ2

⎛⎝⎜

⎞⎠⎟2

1

32π 2Δ2

⎛⎝⎜

⎞⎠⎟2

log Δ2

⎛⎝⎜

⎞⎠⎟2

Λres' > 0( )

Approximate form�

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 � 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)�

our vac.�

Long-lived for ma<<� �

m: mass of Φ, Λ: cutoff scale�

ΔV�(ma Λ)2�

Δφ�Λ�

F0 ≠ 0 induced by D0 ≠ 0

$ Fermion mass

δV = 0⇒ F0 2+ m0

g00 ∂0g00F0 + 1

2D0 2

+ 2 g00 V1−loop = 0

D0 ≠ 0 induces nonvanishing� F0

V = gab ∂aW ∂bW − 12gabD

aDb +V1−loop +Vc.t .

m0

g00 ∂0g00F0 = m0

∗

g00 ∂0g00F 0

Effective potenital up to 1-loop�

Stationary condition�

with � δV = 0we further obtain�

These determine the value of nonvanishing F-term�

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

Comments on Model Building�

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

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

Take the gauge group (G’=U(1):hidden sector)�′G ×GSM

D=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�

Scalar gluon: distinct from the other proposals�

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�

!  New mechanism of DDSB proposed

!  Shown a nontrivial solution of the gap eq. with nonzero <D> in a self-consistent H-F approx.

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

� Phenomenological Application briefly discussed