Physics Letters B 687 (2010) 232–235

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Physics Letters B
www.elsevier.com/locate/physletb

Sommerfeld enhancement from unparticle exchange for dark matter annihilation
Chuan-Hung Chen a,b , C.S. Kim c,d,∗a

Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan
National Center for Theoretical Sciences, Hsinchu 300, Taiwan
Department of Physics & IPAP, Yonsei University, Seoul 120-479, Republic of Korea
d
IPMU, University of Tokyo, Kashiwa, Chiba 277-8568, Japanbca r t i c l e

i n f oArticle history:
Received 2 December 2009
Received in revised form 7 March 2010
Accepted 15 March 2010
Available online 20 March 2010
Editor: M. Cvetič

a b s t r a c t
We investigate the implication of unparticle exchange for the possible Sommerfeld enhancement in dark
matter annihilation process. Assuming the unparticle exchange during WIMP collision, we solve the
Schrödinger equation for the effective potential, and ﬁnd that the Sommerfeld enhancement factor is
dictated by the scale dimension of unparticle as 1/ v 3−2dU . Numerically the Sommerfeld enhancement
could be O (10–103 ).
© 2010 Elsevier B.V. All rights reserved.

Keywords:
Unparticle
Sommerfeld factor
Dark matter
WIMP

1. Introduction
It has been known that our universe is made of not only the
stuff of the standard model (SM) with the occupancy of mere 4%,
but also dark matter and dark energy with the abundance of 22%
and 74%, respectively [1]. Therefore, it must be one of the mostimportant issue to understand what the dark matter is and how to
explore it by various observations. Hopefully, through high energy
colliders such as the Large Hadron Collider (LHC) at CERN, we may
directly observe dark matter soon. In the mean time we may also
have the chance to probe dark matter indirectly by the study ofthe high energy cosmic-ray.
Recently, the collaborations of PAMELA [2], ATIC [3], FERMILAT [4], HESS [5], etc. have published quite astonished events incosmic-ray measurements, in which PAMELA observes the excess
in the positron ﬂux ratio over 10 GeV till PAMELA’s observational
limit of about 100 GeV, whereas others measure consistent anomalies in the electron + positron ﬂux in the 300–1000 GeV range.
Inspired by the new founds at the satellite, balloon and groundbased experiments, the excess could be readily ascribed to dark
matter annihilation, even though there still are the possibilities of*Corresponding author at: Department of Physics & IPAP, Yonsei University, Seoul
120-479, Republic of Korea.
E-mail addresses: physchen@mail.ncku.edu.tw (C.-H. Chen), cskim@yonsei.ac.kr(C.S. Kim).
0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.physletb.2010.03.054

existing new young pulsars [6]. Although dark matter decays could
be the origin of such anomaly, however, for making the lifetime aslong as O (1025 ) seconds, the extreme ﬁne-tuning on the couplingof interaction [7,8] cannot be avoided. For escaping the ﬁne-tuning
problem, hereafter, we will focus on the mechanism of dark matter annihilation only. In addition, the candidate of dark matter in
our following analysis is regarded as weakly interacting massive
particle (WIMP).
Although the WIMP annihilation could be the source for theexcess of cosmic-ray, however, due to low reacting rate in the annihilating process, an enhanced boost factor of a few orders of magnitude, e.g. Sommerfeld enhancement [9–16], has to appear during
the annihilation of WIMP. Therefore, a new force carrier in the dark
matter annihilation to dictate the enhancement is required. As an
example, an interesting mechanism for the Sommerfeld enhancement is arisen from the light boson exchange between dark matter[13,17], where the resulted interaction is Yukawa potential and theforce carrier has the signiﬁcant inﬂuence in the range of Compton wavelength, denoted by α M χ with α being the ﬁne structure
constant of the interaction and of order 10−2 .
In this Letter, we study another kind of new force that may
be alive in an invisible sector and dictated by scale invariant. As
known that an exact scale invariant stuff cannot have a deﬁnite
mass unless it is zero, therefore for distinguishing from the conventional particles, Georgi named the stuff as unparticle [18,19].
Interestingly, it is found that the unparticle with the scaling dimension dU behaves like a non-integral number dU of invisible

C.-H. Chen, C.S. Kim / Physics Letters B 687 (2010) 232–235

233

Fig. 1. Dark matter annihilation with unparticle-mediated Sommerfeld enhancement.

particles [18]. Further implications of the unparticle to colliders
and low energy physics could be referred to Refs. [20–22]. In order to concentrate on the Sommerfeld effect, here we don’t study
the general effective interactions with unparticle, e.g. in our analysis, we have suppressed the interactions between unparticle andHiggs [23]. Although WIMP and unparticle both weakly couple to
the SM particles, however, there is no any reason to limit thatthe interactions between them should be weak. Therefore, when
WIMPs collide each other with small speed, we speculate that theSommerfeld enhancement could be arisen from the unparticle exchange during the collision, sketched in Fig. 1, where the χ and U
denotes the WIMP and unparticle, respectively. And φ represents
the (generic light) particle that might weakly decay to SM particles and the constraints on the couplings will be controlled bycurrent observed ﬂuxes of cosmic rays such as electrons, positrons,
antiproton, etc. Here, its appearance is responsible for the possible connection between dark and visible sectors, but not for theSommerfeld enhancement. Hence, we don’t further discuss the detailed couplings to the SM stuff and the related issue for thisdecay. Our motivation is to understand that if there exists unparticle in the invisible sector, what are its unique character on theSommerfeld factor and the differences from Coulomb and Yukawa
interactions?
The Letter is organized as follows: First, we derive the static unparticle potential, solve the associated radial Schrödinger equation
with suitable boundary condition, and ﬁnd the resultant formulafor s-wave Sommerfeld enhancement. Then, we do the numerical analysis on the resulted Sommerfeld factor and present it as afunction of involved parameters in two-dimensional contour plots.
Finally we give the conclusion.

Although the nonperturbative Sommerfeld effect can be calculated by the combined contributions of a set of ladder diagrams
shown as in Fig. 1, however in the non-relativistic limit, the effect could be equivalent to solving the Schrödinger equation with
an effective potential, which is arisen from the single particle exchange. In the considered mechanism, here the exchanged particle
is the unparticle. Following the scheme proposed in Ref. [18], theinteraction of WIMP to unparticle can be written as
d −1

ΛUU

χ̄ (c) χ OU

A dU =(1)

where we have assumed that the WIMP is a Dirac (or Majorana)fermion, λ is dimensionless parameter and ΛU denotes the living
scale of unparticle. For displaying the character of scale invariant
stuff, we concentrate only on the scalar unparticle. Please notethat when unparticle is realized in the framework of conformal
ﬁeld theories, the propagators for vector and tensor unparticles
should be modiﬁed appropriately to preserve the conformal symmetry [24]. To obtain the unparticle potential in non-relativistic
limit, we use the propagator of the scalar unparticle operator givenby [18,19]

A dU

1

2 sin dU π (−q2 )2−dU

Γ (dU + 1/2).
(2π )2dU Γ (dU − 1)Γ (2dU )16π 5/2

,

where
(2)

Combining Eqs. (1) and (2), the four-fermion effective interacting
term in momentum space could be expressed by



χ̄ χ

2

λ

(c )

d −1

A dU



1

2 sin dU π (−q2 )2−dU

ΛUUχ̄ (c) χ .

(3)

By Fourier transformation and with q0 = 0, the static unparticle
potential in WIMP interaction resulted by Eq. (3) is found byα

V (r ) = − trwith t = 2dU − 1,

α=

ξΓ



λ

(4)

2

,
d −1
ΛUU
Γ (dU + 1/2)Γ (dU − 1/2)ξΓ =.
Γ (2dU )2π 2dU(5)

Interestingly, we see that the power of unparticle potential is associated with the scaling dimension dU and not an integer.
Since the unparticle potential is independent of the polar andazimuth angles in spherical coordinate system, the relevant piece
for the Sommerfeld factor is the radial Schrödinger equation, read
by



d2
dr 2

2. Unparticle potential and its Sommerfeld factor

λe iqx 0| T OU (x)OU (0)|0 = i+ −( + 1)
r2

+

2μα
rt



+k

2

uk (r ) = 0,

(6)where we have used the nature unit with h̄ = c = 1, E = k2 /2μwith μ being the reduced mass of system and R k (r ) = uk (r )/r.
Once we solve the differential equation with the proper boundary condition, the Sommerfeld effect associated with each angularmomentum is obtained by [15] (2 + 1)!! ∂ R k (r ) 


|k| !
∂r

S = 

r =0

2

 .


(7)

Due to the power t of unparticle potential in r being not an integer, there is no hope to ﬁnd a general close form for the solution.
Nevertheless, since the required Sommerfeld factor is estimated at
r = 0, our strategy for ﬁnding the solution is to look for a good
approximation to extract the behavior of radial wave function atr ∼ 0.
Before discussing the solution to the differential equation, ﬁrst
we analyze the limit on t = 2dU − 1. To avoid the crossing of
the branching cut at 1/ sin dU π in Eq. (2), we require 1 < dU < 2(1 < t < 3). In order to further understand whether the upper limitof t can be bounded by the boundary condition when solving differential equation, we examine the case with t = 2, in which thepotential could be exactly solved at r → 0. Hence, by taking t = 2
and uk ∼ r σ and considering r → 0, from Eq. (6) we get234

C.-H. Chen, C.S. Kim / Physics Letters B 687 (2010) 232–235



σ=

1
2



1

+

+

2
− 2μα .

2

(8)

Thus, the corresponding solution for uk (r ) can be expressed byuk (r ) = r σ e ikr f (r ), where the function of f (r ) is controlled by

r

d2 f

df

dr

dr

+ (2σ + 2ikr )
2

r

d f
dr 2

+ 2( + 1) + 2ikr

(9)

df
dr



+ 2ik( + 1) +2μα

f = 0.

r t −1(10)
Although this equation cannot be solved for arbitrary value of t,
we ﬁnd that if we concentrate on small r, the terms associatedwith 2ikr and 2ik( + 1) can be safely neglected and the solution
to the simpliﬁed differential equation can be found by1
f t (r ) = d−(2 +1)/2(2−t ) (2 − t )(2 +1)/(2−t ) (2 +1)/2
r



×Γ

2 −t +3 

uk (r )
r2−t

J

2 +1 22−t,dr (2−t )/22−t= r e ikr f t (r ),

,

(11)

∞

dr  G r , r  U r  uk r 

(12)

0

with U (r ) = 2μ V (r ) and

G r, r =j (kr )kj (kr )+

∞dr n kr  j kr  U r  .

k2

(14)

j (kr )n (kr  )/kn (kr ) j (kr  )/kfor r < r  ,for r > r  ,Now we match both solutions at r ≈ 0 by solving differential equation and the integration with Green function. It has been
known that the Sommerfeld factor of Coulomb potential (i.e. t = 1)
for s-wave can be obtained exactly as

S0 =2πα / v
1 − exp(−2απ / v )(13)

where G (r , r  ) is the Green’s function of the differential equation.
After compared to the partial wave expansion given by exp(ikz) =

(15)

.

For the case of v
1, we have S 0 ≈ 2πα / v. Based on the known
result, we ﬁnd that in order to get correct approximation for
Coulomb potential, matching condition for Eqs. (11) and (14) has
to be adopted as

N 2 r +1 f (r ) =

∞

4
k

dr  n kr  j kr  U r  .

j (kr )
2
0

Using the asymptotic results f (r → 0) −→ 1 and j (kr → 0) −→2 !
+1 , the normalization constant is found by(2 +1)! (kr )N 2 = 8μα

2 !
k +t −2 X(2 + 1)!

(16)∞

with X = 0 dz j ( z)n ( z)/ zt . According to Eq. (7), the Sommerfeld factor for s-wave is obtained by S 0 = 8α | X 0 |μt −1 / v 2−t , where
we have used k = μ v. One can see that for t = 1, X 0 = −π /4 andS 0 = 2πα / v. Hence, the s-wave Sommerfeld factor induced by unparticle exchange is given by

S0 =where a suitable normalization for the wave function has to befurther dealt with.
Unlike conventional approach that the normalization of wave
function for scattering process is chosen at r → ∞, in order to
get a suitable normalization for the wave function at near origin,
we have to consider an alternative method which could provide
correct value of wave function at r = 0. We ﬁnd that the purpose
can be achieved by using integral form instead of directly solving
Eq. (6). In other words, the solution for uk (r ) could be expressedbyuk (r ) ∝ j (kr ) +

uk =



√

where J ν (x) = J (√
ν , x) is the Bessel function and the asymptotic
form for 0 < xν + 1 is J ν → (x/2)ν /Γ (ν + 1). Hence, the solution to radial Schrödinger equation in the region of small r canbe expressed by

R k (r ) =+ 1) j (kr ) P (cos θ)/kr, we ﬁnd that the solution for

r → 0 is0

+ 2ikσ f (r ) = 0.

Compare to the conﬂuent hypergeometric differential equation,xy  + (c − x) y  − ay = 0, in which the solution is conﬂuent hypergeometric function of the ﬁrst kind y (x) = 1 F 1 (a, c ; x), we seethat the solution to Eq. (9) can be found by taking a = σ , c = 2σ
and x = −2ikr. For = 0, from Eq. (8) it is easy to ﬁnd σ < 1.
Since 1 F 1 (σ , 2σ , −2ikr ) → 1 when r → 0, as a result, the wave
function R k (r ) is singular at origin. In other words, t = 2 leads
to an ill-deﬁned solution and should be taken as the new upper
limit. Therefore, we ﬁnd the new bound on the scale dimension of
unparticle as 1 < dU < 3/2 (1 < t < 2).
After obtaining the allowed power of unparticle potential, next
we are looking for a good approximation to get the proper solution
for near origin. As used before, if we regard uk (r ) = r +1 e ikr f (r )as a general solution to the radial Schrödinger equation, according
to Eq. (6), the function f (r ) should satisfy the differential equation2∞=0 i (2

2λ

π dU

2


ξΓ

μ
ΛU

2(dU −1)

| X0|
v 3−2dU.

(17)

3. Numerical analysis and discussions
We now analyze the formula for S 0 numerically. From Eq. (17),we see that the involved free parameters are λ, dU , ΛU and v.
Unlike the case of Coulomb or Yukawa potential, the velocitydependent factor appears by 1/ v 3−2dU with 1 < dU < 3/2. The
Sommerfeld factor is increasing when dU approaches to unity.
Consequently, if we focus on the maximum of S 0 , then we ﬁndthat S 0 is insensitive to the values of μ and ΛU , where they show
up by (μ/ΛU )2(dU −1) . Note that to explain the excess of cosmic
rays from both PAMELA and ATIC/Fermi-LAT, it has been known
that the mass of dark matter is of order of TeV. On the other
hand, in order to produce the unusual stuff at LHC, the interesting scale to form unparticle should be also at TeV scale. Therefore,
without lose of generality we set μ = ΛU in our numerical estimates.
In order to see the inﬂuence of remaining parameters, we willﬁx one parameter in turn when we make two-dimensional contours as a function of remaining two parameters. First, we analyze
the contour as a function of λ and dU when the speed of WIMP is
ﬁxed. Although dark stuff weakly couples to visible particle, however, the interaction between invisible particles may not be small.
Accordingly, we set the allowed range for λ be within one orderof magnitude, i.e. 1 < λ < 10. As a result, the contour for Sommerfeld factor S 0 induced by unparticle contributions as a function of
coupling λ and scale dimension dU with v = 10−3 is shown in
Fig. 2, where the numbers in the plot stand for the values of S 0 .
By the ﬁgure, we see that S 0 is increasing while dU is decreasing.

C.-H. Chen, C.S. Kim / Physics Letters B 687 (2010) 232–235

Fig. 2. Contour for Sommerfeld factor induced by unparticle-mediated as a functionof λ and dU with v = 10−3 . The numbers in the plot denote the values of S 0 .

235

investigated the impact of unparticle which is ruled by scale invariant.
We ﬁnd that the power of the static unparticle potential in r,
resulted from the exchange of scalar unparticle, is non-integral
number and it depends on the scale dimension of unparticle, expressed by 1/r 2dU −1 . By the boundary condition for wave function at r = 0, the upper bound of scale dimension is found by
dU |max < 3/2. By looking for the suitable boundary condition andmatching condition for the radial wave function at r → 0, we are
led to the Sommerfeld factor for s-wave collision, in which theresult could return to that of Coulomb potential. However, unlike
Coulomb or Yukawa potential, the speed dependence is related toscale dimension of unparticle and dictated by 1/ v 3−2dU . Although
the Sommerfeld factor is associated several parameters, however,
our results are only sensitive to the parameters λ, dU and v. According to our numerical calculations, we conclude that with theallowed range of free parameters, the Sommerfeld enhancement
induced by unparticle-mediated effects could be O (10–103 ) andthe factor could provide the necessity for enhancing the cross section of dark matter annihilation.
Acknowledgements
This work for C.H.C. is supported by the National Science Council of R.O.C. under Grant No: NSC-97-2112-M-006-001-MY3. The
work of C.S.K. is supported in part by Basic Science Research Program through the NRF of Korea funded by MOEST (2009-0088395),
in part by KOSEF through the Joint Research Program (F01-2009000-10031-0), and in part by WPI Initiative, MEXT, Japan.
References

Fig. 3. Legend is similar to Fig. 2, where dU = 1.1 and the variables for the contourare λ and v (in units of 10−3 ).

Furthermore, the value of dU for S 0 ∼ O (103 ) can be somewhat
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Secondly, we also study the behavior of S 0 in λ and v (in unitsof 10−3 ) by ﬁxing the value of dU . We present the resultant contour with dU = 1.1 in Fig. 3. Clearly, with the value of dU thatis close to unity, S 0 can easily reach O (103 ) even with λ ∼ 1 andv ∼ 10−3 . On the other hand, S 0 of O (103 ) can be preserved whenv and λ both are increasing simultaneously.
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