D-term Dynamical SUSY Breakingqft.web/2012/slides/maru.pdf · Dynamical SUSY breaking(DSB) is most...
Transcript of D-term Dynamical SUSY Breakingqft.web/2012/slides/maru.pdf · Dynamical SUSY breaking(DSB) is most...
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