What are universal features of
gravitating Q-balls?
Takashi Tamaki(Nihon University)
with Nobuyuki Sakai
(Yamagata University)
I. Introduction
Q-balls:a kind of nontopological solitons consist
of scalar fields
Self-gravity may not be important?
Kaup, PR172, 1331 (1968)
m2 2
V ( ) 
 flat ×
2
with self-gravity ○ (mini-boson star)
2

m 2
    gravity-mediation type
Vgrav. ( ) 
 1  K ln    in the Affleck-Dine (AD)
2
 M   mechanism

2
K 0
flat
K 0
K 0
○
×
gravity
K 0
○
○ ``Q-balls”
surrounded by Q-matter
TT, N. Sakai, PRD83, 084046 (2011)
Here, gauge-mediation type
seek for universal features!!
2



4
Vgauge ( )  m ln 1  2 
 m 
TT, N. Sakai, PRD84, 044054 (2011)
II. Model
2

 
4
Vgauge ( )  m ln 1  2 
 m 
normalization


m
t  mt


m
r  mr
V
2
V  4  ln(1   )
m
III. flat case
dVgauge
2
 ''    '  2 
r
d
Particle motion in
V : Vgauge 
V
d V
(  0)  0
2
d
 : 2  
2

2
02
2
 ln(1   2 ) 
 2 2
V
2
2
 2 2
trivial
max(V )  0

02  2
r
2 0
V

Thick-wall solution

r
 2  2 Thin-wall solution

V

r
IV. gravitating case
2

dVgauge 

 2  ' A' 
2
 ''        ' A  2  


A
d 
r 

2

 2 : 2  2

Signature of
Particle motion in

2
depends on r. 
V
V depends on ``time”.
(a)
1.0003
  0.001
2
1.0002
A(r)
0.08
=0
1.0001
1
0.06
~
(r)
Thick-wall
(b)
0.1
0.9999
(r)
0.9998
0.04
0.9997
0.02
0.9996
=0.0005
0
0.9995
0
100
 : Gm 2  0.0005
200
~r
300
400
0
500
20
40
60
A(r)
140
10
=0
1
8
0.8
(r)
~
(r)
  1.65
2
120
(b)
12
1.2
100
(a)
1.4
Thin-wall
80
~r
6
0.6
4
=0.0005
0.4
2
0.2
0
2
4
~r
6
8
10
0
0
2
4
6
~r
8
10
Thick-wall
 2  0.001
 2  1.65
Thin-wall
5 10-5
4 10-7
infinity
~r=0
140
-7
3 10
4 10
-5
120
2 10-7
1 10-7
3 10
-5
100
0
-2 10-7
2 10-5
-V
-V
-1 10-7
0.005 0.01 0.015 0.02 0.025 0.03
80
60
40
1 10-5
20
infinity
~r=0
0
0
0.02
0.04
~
0.06
0.08
0
0.1
0
2
4
6
8
10
~
V changes ``quickly”.
For various
2?
12
strong gravity
C
1
 0
 0
Q
Q0
weak gravity
C
B
0.1
A
2
WHY ???
D
0.01
=0.004
0.001
=0
=0.0005
0.0001
0
200
400
600
Q
800
1000
1200
V. thick-wall case
weak gravity
take up to first order in h,f.
 2  1  h(r ) A2  1  f (r )
Let us evaluate 
0
:  (r  0)
V
 2   2 h(0)  0 2  0

Evaluate from
V  0
5
O
(

We used Maclaurin expansion and neglected
0 )
0  
For   0
For
 0
We should compare
h(0)
with
2

2 2



r  

t
i
2
Gt  Gi  
  8 r A  2  V 
 A 
 

 2  0, i.e.,  2  2 and weak gravity
2
  16 r 2 2
2

r
h
 
We assume
 (r )  0  1
and approximating
2
8
2 C
h(0)   0 2
3

for
r
C
C=const.

C
h ( )  h( )  0

substitute into 
2
  h(0)  0  0
2
2
3
0 
2
2
8 C  3
4
2
  C
2
  C
2
2
as in the flat case
0  
Q   r  0
2 3
2
1

2
0 

2
3
2C 2
Q   r  0
2 3
2
1

3
 0
3

1

V. conclusion
2

 
4
Gravitating Q-balls Vgauge ( )  m ln 1  2 
 m 
The thick-wall limit is completely different from the
flat case.
Q0
exists !!
From the analytic estimation, our results hold
for general potentials if a positive mass term
is a leading order.
3
0 
2
2
8 C  3
4
2
  C
2
  C
2
as in the flat case
0  
2
Q   r  0
2 3
2
1
2

3

1

0.001
0.0001
3
10-5
2C 2
0
0 

2
10-6
=10-4
10-7
Q   r  0
2 3
2
1

3
=10-5
=10-6
 0
10-8 -10
10
10-9
10-8
2
10-7
10-6
10-5
important quantities
Total energy
(Hamiltonian)
2
mE
mQ
E : 2 , Q :
M
M2
Basic equations
サイズの議論
M p2
mini-BS
1/mの半径GM~1/m 
BS  4
の相互作用項の存在が本質的  e.g.,
2 2 Λ大
2 2
U ()  m  (1 
U
m 2 M p2

GM~1/m_re
M
2
p
 )
1
mre
M
M

m
M p2 
M 3p
m
m2
m(neutron) 程度 M=M(太陽)
m