Paper
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Radiation of Supersymmetric Particles from Aharonov-Bohm R-string
Authors
Yutaka Ookouchi,Takahiro Yonemoto
SOLOMON SACKEY, Ph.D., A.M.ASCE,Zosya Kanarskaya,Irina Davydova,Марина Батюшкина,Ekaterina Nikolaeva,Alexey Merinov,Wei Hu,Maria Dainotti,Krzysztof Jakubiak,Liliya Komalova,Il'nur Mirgaleev,Andrey Poliakov,Amir Sharipov,Tatyana Zamarina,Ольга Новікова,Daria Danilenko,Svetlana Volgina,Svetlana Tsygankova,Максим Корнієнко,Катерина Пономаренко,Наталья Колосова,Anton Kordonskiy,Konstantin Alekseev,Olesja Eliseeva,Rafaela Almeida Cordeiro,Tatiana Taranushenko,Maria Loginova,Alexander Pletzer,Novikov Yurii,Наталия Анатольевна Ладик,Fabrice Vandebrouck,Владимир Васильчиков,Rajmund Porada,Sergey Vasilyev,Inna Markova,Nina Melnikova,Kirk Jensen,Sarvenaz Sarabipour,Виктор Лукьянчиков,HALİL KARA,Елена Мошненко,Sergey Zelentsov,N/A ,Galyna Marchuk,John Lord,Nikita Misuna,Anzhela Ignatyuk,Kirill Kupavykh,Valentin Morenov,Manuela Kulaxizi,Hartmut Gimpel,Timothy Cohen,Aurelien Bigot,Anish Goyal,Sergey Lugovoy,Mahendra Ramachandran,Bharat Char,Garrett Poe,Jesse Thaler,Erdal Can Alkoclar,Benjamin Bellenie,Andy Van Brocklin,Joseph Fotsing,Vasilis Niarchos,Ana Maria Bernardo,Marie Connett,Thorsten Alexander Kern,Subhash Tummala,Jayanta Saha,Dagnachew Muluye Fetene,Norbert Keutgen,Fabio Anselmi,Vladimir Shats,narayan chandra nayak,Mika Tarkka,Марина Мазитова,Eleni Leontaridi,Ingo Jordan,Pavel Surynek,Steven Hollis,Suvendu Giri,Tatsuhiro Misumi,Gregory Carven,Leopoldo Pando Zayas,Elena Sokolova,Pavel Lazarev,Jin Matsumoto,Kailash Sahu,Bernhard Kliem,Liudmila Larionova,Ryan Trainor,Anna Lazareva,Sergio Martínez-González,Arantza Oyanguren,jonathan peter warner,Jose Queiruga,E.A. Galova,Serohin Vitalii,Jinmian Li,Markus von Kienlin,Sergei Obukhov,CARLOS JULIAO VARGAS,Li Li,Mikhail Alfimov,YUE ZHANG,Javier Tarrio,Eduardo Casali,Brian Balgley,Andrea Guerrieri,Albert Sesé,Oksana Molochkova,Дмитрий Плоткин,Mikhail Reshetnikov,Menachem Kojman,Shayakhmetov Salim,Maria Galas,Zhemal Rakhmanova,Alena Kharevich (Shalagina),Laura Pilozzi,Benoit Truax,Przemysław Mańkowski,Jean-Paul Roux,Soloveva Anna,Daria Fomina,Nina Dmitrieva,也铃 周,Charles Amory,Amar Vutha,Volodymyr Labay,Olha Lazorko,Olga Smirnova,Malcolm Clark,Máximo Pló Casasús,Alexandr Frid,Dimitri Gilis,Beatriz L. 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arXiv:1409.8384v3 [hep-th] 8 Feb 2015
KYUSHU-HET-143
Radiation of Supersymmetric Particles
from Aharonov-Bohm R-string
Yutaka Ookouchi1,2 and Takahiro Yonemoto2
1Faculty of Arts and Science, Kyushu University, Fukuoka 819-0395, Japan
2Department of Physics, Kyushu University, Fukuoka 810-8581, Japan
Abstract
We study radiation of supersymmetric particles from an Aharonov-Bohm string associated
with a discrete R-symmetry. Radiation of the lightest supersymmetric particle, when com-
bined with the observed dark matter density, imposes constraints on the string tension or the
freeze-out temperature of the particle. We also calculate the amplitude for Aharonov-Bohm
radiation of massive spin 3/2 particles.
KYUSHU-HET-143
Radiation of Supersymmetric Particles
from Aharonov-Bohm R-string
Yutaka Ookouchi1,2 and Takahiro Yonemoto2
1Faculty of Arts and Science, Kyushu University, Fukuoka 819-0395, Japan
2Department of Physics, Kyushu University, Fukuoka 810-8581, Japan
Abstract
We study radiation of supersymmetric particles from an Aharonov-Bohm string associated
with a discrete R-symmetry. Radiation of the lightest supersymmetric particle, when com-
bined with the observed dark matter density, imposes constraints on the string tension or the
freeze-out temperature of the particle. We also calculate the amplitude for Aharonov-Bohm
radiation of massive spin 3/2 particles.
1 Introduction
A discrete symmetry is a useful tool to construct a realistic model in particle physics (for
example, see [1, 2] and references therein). In particular, R-symmetry is one of the impor-
tant ingredients in supersymmetric model building, since the lightest supersymmetric particle
stabilized by the symmetry offers a natural candidate for the dark matter.
Recently, Banks and Seiberg gave a new argument on discrete symmetries from a viewpoint
of a quantum theory of gravity [3]. They extended the so-called “no global symmetries
theorem” to include discrete symmetries. Accepting their arguments, one would be lead
to an interesting avenue for string phenomenology: A discrete symmetry, if not broken in
coupling to gravity, have to be gauged. Banks and Seiberg also showed the universal effective
Lagrangian of a discrete gauge theory by means of the BF coupling. In the effective theory, in
addition to a massive gauge field, there is a Kalb-Ramond 2-form field which naturally couples
to a string-like object, so-called Aharonov-Bohm (AB) string. Also, there is a particle, called
Aharonov-Bohm particle, coupling to the massive gauge field. This is the other ingredient of
the discrete gauge theory. As in the well-know Aharonov-Bohm effect for a solenoid [4], AB
strings and AB particles have quantum mechanical interactions. As was firstly pointed out in
[5], a moving AB string radiates AB particles by the interaction. Explicit calculation of the
AB radiation has been done quite recently [6, 7].
In [8], based on the remarkable progress of the AB radiation, one of the authors stud-
ied cosmological constraints arising from the Big Bang Nucleosynthesis and the diffuse γ-ray
background. Especially, in string theory such constraints are viable, and some of parameter
spaces are excluded in some compactification scenarios. In this paper, based on the study,
we would like to go a step further toward an application to supersymmetric (SUSY) model
building. One of the striking features of the SUSY model building is the existence of stabi-
lized supersymmetric particles. Throughout this paper, we simply assume that an AB string
associated with R-symmetry1 is formed at the early stage of the universe and that the lightest
supersymmetric particle carries a charge of the corresponding discrete symmetry.
The organization of this paper is as follows. In section 2, we briefly review the universal
effective Lagrangian of Zp gauge theory, and show a relationship between AB strings/particles
and the discrete gauge theory. Then, we summarize the results of calculations on the power of
the AB radiation shown in [6, 7, 8]. In section 3, we impose a cosmological constraint arising
from the observed dark matter density. Section 4 is devoted to conclusions and comments
1A cosmic string associated with R-symmetry has been studied in [9]. Rich physical aspects such as
instability of metastable vacua induced by the string and cosmological constraints coming from R-axion
radiated by the string have been discussed.
1
A discrete symmetry is a useful tool to construct a realistic model in particle physics (for
example, see [1, 2] and references therein). In particular, R-symmetry is one of the impor-
tant ingredients in supersymmetric model building, since the lightest supersymmetric particle
stabilized by the symmetry offers a natural candidate for the dark matter.
Recently, Banks and Seiberg gave a new argument on discrete symmetries from a viewpoint
of a quantum theory of gravity [3]. They extended the so-called “no global symmetries
theorem” to include discrete symmetries. Accepting their arguments, one would be lead
to an interesting avenue for string phenomenology: A discrete symmetry, if not broken in
coupling to gravity, have to be gauged. Banks and Seiberg also showed the universal effective
Lagrangian of a discrete gauge theory by means of the BF coupling. In the effective theory, in
addition to a massive gauge field, there is a Kalb-Ramond 2-form field which naturally couples
to a string-like object, so-called Aharonov-Bohm (AB) string. Also, there is a particle, called
Aharonov-Bohm particle, coupling to the massive gauge field. This is the other ingredient of
the discrete gauge theory. As in the well-know Aharonov-Bohm effect for a solenoid [4], AB
strings and AB particles have quantum mechanical interactions. As was firstly pointed out in
[5], a moving AB string radiates AB particles by the interaction. Explicit calculation of the
AB radiation has been done quite recently [6, 7].
In [8], based on the remarkable progress of the AB radiation, one of the authors stud-
ied cosmological constraints arising from the Big Bang Nucleosynthesis and the diffuse γ-ray
background. Especially, in string theory such constraints are viable, and some of parameter
spaces are excluded in some compactification scenarios. In this paper, based on the study,
we would like to go a step further toward an application to supersymmetric (SUSY) model
building. One of the striking features of the SUSY model building is the existence of stabi-
lized supersymmetric particles. Throughout this paper, we simply assume that an AB string
associated with R-symmetry1 is formed at the early stage of the universe and that the lightest
supersymmetric particle carries a charge of the corresponding discrete symmetry.
The organization of this paper is as follows. In section 2, we briefly review the universal
effective Lagrangian of Zp gauge theory, and show a relationship between AB strings/particles
and the discrete gauge theory. Then, we summarize the results of calculations on the power of
the AB radiation shown in [6, 7, 8]. In section 3, we impose a cosmological constraint arising
from the observed dark matter density. Section 4 is devoted to conclusions and comments
1A cosmic string associated with R-symmetry has been studied in [9]. Rich physical aspects such as
instability of metastable vacua induced by the string and cosmological constraints coming from R-axion
radiated by the string have been discussed.
1
on an application to string theories. In appendix A, we exhibit explicit calculations of AB
radiation of massive spin 3/2 particles. In appendix B, we briefly summarize the loop number
density for cosmic strings loosing the energy via particle and gravitational radiation.
2 Review of Aharonov-Bohm radiation
In this section, we first review the universal effective Lagrangian of Zp discrete gauge theory
and discuss Aharonov-Bohm (AB) particles/strings associated with the symmetry along the
lines of [3]. The effective Lagrangian is described by BF coupling (or St¨uckelberg coupling),
p
∫
4D
B2 ∧ dA , (2.1)
where A is the massive gauge field one-form and B2 is the Kalb-Ramond two-form field. The
gauge transformation for each field is
A → A + dλ, φ → φ + pλ,
B2 → B2 + dΛ, V → V + pΛ, (2.2)
where Λ is a one-form. The dual one-form gauge field V transforms non-linearly, indicating
the breaking of continuous U(1) symmetry. Also, φ is the dual field of the Kalb-Ramond field
B2.
Following the arguments shown in [3], let us review an interaction between AB strings and
AB particles in Zp gauge theory. An AB particle is a particle-like object coupling electrically
to the massive gauge field A. By exploiting a closed world-line or an infinite length of world-
line (we denote σ1), the AB particle can be written as a line operator,
Oparticle ∼ exp
(
i
∫
σ1
A
)
, (2.3)
where we assumed the minimum charge2. On the other hand, an AB string is a string-like
object coupling to B2 electrically. In the same way, an operator of the AB string can be
represented as a surface operator,
Ostring ∼ exp
(
i
∫
σ2
B2
)
, (2.4)
where σ2 is a closed surface or an infinitely large world-sheet. Note that p AB strings annihi-
late with the junction operator e−i ∫
L V : Clearly, this junction operator is not invariant under
2Hereafter, we assume that AB particles and AB strings carry the minimum charges in the fundamental
unit.
2
radiation of massive spin 3/2 particles. In appendix B, we briefly summarize the loop number
density for cosmic strings loosing the energy via particle and gravitational radiation.
2 Review of Aharonov-Bohm radiation
In this section, we first review the universal effective Lagrangian of Zp discrete gauge theory
and discuss Aharonov-Bohm (AB) particles/strings associated with the symmetry along the
lines of [3]. The effective Lagrangian is described by BF coupling (or St¨uckelberg coupling),
p
∫
4D
B2 ∧ dA , (2.1)
where A is the massive gauge field one-form and B2 is the Kalb-Ramond two-form field. The
gauge transformation for each field is
A → A + dλ, φ → φ + pλ,
B2 → B2 + dΛ, V → V + pΛ, (2.2)
where Λ is a one-form. The dual one-form gauge field V transforms non-linearly, indicating
the breaking of continuous U(1) symmetry. Also, φ is the dual field of the Kalb-Ramond field
B2.
Following the arguments shown in [3], let us review an interaction between AB strings and
AB particles in Zp gauge theory. An AB particle is a particle-like object coupling electrically
to the massive gauge field A. By exploiting a closed world-line or an infinite length of world-
line (we denote σ1), the AB particle can be written as a line operator,
Oparticle ∼ exp
(
i
∫
σ1
A
)
, (2.3)
where we assumed the minimum charge2. On the other hand, an AB string is a string-like
object coupling to B2 electrically. In the same way, an operator of the AB string can be
represented as a surface operator,
Ostring ∼ exp
(
i
∫
σ2
B2
)
, (2.4)
where σ2 is a closed surface or an infinitely large world-sheet. Note that p AB strings annihi-
late with the junction operator e−i ∫
L V : Clearly, this junction operator is not invariant under
2Hereafter, we assume that AB particles and AB strings carry the minimum charges in the fundamental
unit.
2
the gauge transformation (2.2). To compensate the non-invariance, one can add the world-
sheet operator as exp[−i ∫
L V + ip ∫
C B2], where ∂C = L. This imply that p world-sheets
corresponding to AB strings annihilate at the boundary ∂C with the junction operator, in-
dicating p periodicity of Zp theory. To see the topological interaction between the AB string
and the AB particle, let us put the AB string with the minimum charge in the space-time.
The action becomes
p
∫
4D
B2 ∧ dA +
∫
σ2
B2. (2.5)
Consider the holonomy picked up by the AB particle circling around the AB string,
hol(c) ≡ exp
(∫
c
A
)
= exp
(∫
S
F
)
= exp
( 2πi
p
)
≡ exp(iφ), (2.6)
where ∂S = c. We used the equation of motion for B2 in the third equality. We refer to the
total magnetic flux in the AB string as φ. When p > 1, by means of the topological term in
the Lagrangian, a non-trivial gauge potential is generated around the AB string, which gives
rise to the Aharonov-Bohm effect. The interaction between the AB particle and the massive
gauge field can be simply understood as ∫
4D A ∧ ∗4J .3
Now we are ready to review the total power of radiated particles from Aharonov-Bohm
strings studied initially in [8]. Basically, exploiting the analysis of [6, 7], one can evaluate the
radiation power since in the present situation, radiated particles are massive but much lighter
than the scale of the string tension. According to the results shown in [6, 7], the dominant
radiation of massive particle comes from cusps (or kinks) on loops. Hence, we simple apply
the formulae in [6, 7] for a cuspy loop to the current analysis. Also, as in the previous work
[8], since we are interested in order estimation of cosmological constraints arising from the
AB radiation, we will not carefully treat order one coefficients of the formulae.
From the equation of motion for the Kalb-Ramond field B2, we obtain the gauge potential
around the AB string,
p ∂ν Aµ = ˜Jµν . (2.7)
Here, we ignored the kinetic term which is irrelevant in our assumption of the string tension.
In the wire-approximation, the dual of the string current is written as
˜Jµν = ǫµναβ
∫
dτ dσ( ˙XαX′β − ˙Xβ X′α)δ(4)(x − X(σ, τ )). (2.8)
σ, τ are the world-sheet coordinates of the string. In momentum space, the solution of (2.7)
is written as follows:
Aµ = 1
pǫµναβ
kν
(kλkλ)Jαβ . (2.9)
3Here, ∗4 is the Hodge dual in four dimensional space-time.
3
sheet operator as exp[−i ∫
L V + ip ∫
C B2], where ∂C = L. This imply that p world-sheets
corresponding to AB strings annihilate at the boundary ∂C with the junction operator, in-
dicating p periodicity of Zp theory. To see the topological interaction between the AB string
and the AB particle, let us put the AB string with the minimum charge in the space-time.
The action becomes
p
∫
4D
B2 ∧ dA +
∫
σ2
B2. (2.5)
Consider the holonomy picked up by the AB particle circling around the AB string,
hol(c) ≡ exp
(∫
c
A
)
= exp
(∫
S
F
)
= exp
( 2πi
p
)
≡ exp(iφ), (2.6)
where ∂S = c. We used the equation of motion for B2 in the third equality. We refer to the
total magnetic flux in the AB string as φ. When p > 1, by means of the topological term in
the Lagrangian, a non-trivial gauge potential is generated around the AB string, which gives
rise to the Aharonov-Bohm effect. The interaction between the AB particle and the massive
gauge field can be simply understood as ∫
4D A ∧ ∗4J .3
Now we are ready to review the total power of radiated particles from Aharonov-Bohm
strings studied initially in [8]. Basically, exploiting the analysis of [6, 7], one can evaluate the
radiation power since in the present situation, radiated particles are massive but much lighter
than the scale of the string tension. According to the results shown in [6, 7], the dominant
radiation of massive particle comes from cusps (or kinks) on loops. Hence, we simple apply
the formulae in [6, 7] for a cuspy loop to the current analysis. Also, as in the previous work
[8], since we are interested in order estimation of cosmological constraints arising from the
AB radiation, we will not carefully treat order one coefficients of the formulae.
From the equation of motion for the Kalb-Ramond field B2, we obtain the gauge potential
around the AB string,
p ∂ν Aµ = ˜Jµν . (2.7)
Here, we ignored the kinetic term which is irrelevant in our assumption of the string tension.
In the wire-approximation, the dual of the string current is written as
˜Jµν = ǫµναβ
∫
dτ dσ( ˙XαX′β − ˙Xβ X′α)δ(4)(x − X(σ, τ )). (2.8)
σ, τ are the world-sheet coordinates of the string. In momentum space, the solution of (2.7)
is written as follows:
Aµ = 1
pǫµναβ
kν
(kλkλ)Jαβ . (2.9)
3Here, ∗4 is the Hodge dual in four dimensional space-time.
3
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