The excitation spectrum of rotating strings with
masses at the ends
Jochen Zahn
Fakult¨at f¨ur Physik, Universit¨at Wien,
Boltzmanngasse 5, 1090 Wien, Austria.
jochen.zahn@univie.ac.at
October 30, 2018
Abstract
We compute the spectrum of excitations of the rotating Nambu–
Goto string with masses at the ends. We ﬁnd interesting quasi-massless
modes in the limit of slow rotation and comment on the nontrivial
relation between world-sheet and target space energy.
1 Introduction
The Nambu–Goto string is a well-established phenomenological model for a
vortex line connecting two quarks, cf. [1] and [2] for reviews on the situation
in mesons and on the lattice. In recent years, various aspects of this corre-
spondence could be tested in lattice simulations, for example the corrections
of order γ−1L−3 to the energy (where L is the length of the string and γ
its tension), or the spectrum of the excitations of the string, at order L−1.
For such studies, one usually considers the L¨uscher–Weisz string [3], i.e.,
a string with ﬁxed endpoints, corresponding to a string stretched between
two stationary D0 branes.1 However, at least for light quarks, this does
not seem to be a good approximation, as for reasonable values of the quark
mass, the string tension, and the distance, one expects the quarks to rotate
relativistically around each other. It is thus desirable to work with more
realistic conﬁgurations.
Here, we study the Nambu–Goto string with masses at the ends, and
compute the spectrum of excitations of the rigid solution, at ﬁrst order in
perturbation theory (corresponding to O(L−1)). We ﬁnd that, when moving
away from the static limit, the degeneracy of the frequency of planar and
1The exact energy for (excitations of) this conﬁguration was calculated by Arvis [4].
We refer to [5] for a recent critical reexamination of this result.
1
arXiv:1310.0253v3 [hep-th] 12 Dec 2013

scalar excitations (polarized in the plane of rotation or orthogonal to it) is
lifted. The degeneracy is restored in the relativistic limit, in which the ends
of the string move at the speed of light. Furthermore, we identify interesting
quasi-massless modes in the limit of slow rotation, which are not present in
the L¨uscher–Weisz string.
The ﬁrst treatment of the Nambu–Goto string with masses at the ends
seems to have been the work of Chodos and Thorn [6]. They discussed the
rigid rotating string solution, but did not study its excitations. Such a study
was performed by Hadasz [7], but for the case in which also a Gauß–Bonnet
term is present. This leads to too many boundary conditions.2 In particular,
the limit in which the coeﬃcient of the Gauß–Bonnet term vanishes does
not reduce to our case.
A non-relativistic approximation to the rotating string was studied by
Nesterenko [8]. We reproduce some of his results in the non-relativistic limit.
However, our results diﬀer in several aspects, in particular planar and scalar
polarizations are not distinguished in [8].
The most recent study of the excitation spectrum of the rotating string
with masses at the ends seems to have been the work of Baker and Steinke [9].
They use path integral techniques and obtain results for the spectrum of the
excitations, which they study further in the limits of massless endpoints and
one massless and one inﬁnitely heavy endpoint. They use these to obtain
Regge trajectories for the excited states. The main diﬀerences to our work
are the following: We use perturbations that are normal to the background
string, which simpliﬁes the calculation considerably. We discuss the evo-
lution of the spectrum over the whole range of parameters. In particular,
we ﬁnd interesting quasi-massless modes in the limit of a slowly rotating
string. Finally, we have a diﬀerent point of view regarding the signiﬁcance
of the spectrum, i.e., world-sheet energies, for the calculation of target space
energies.
The article is structured as follows: In the next section, we introduce
the setup and recall some basic results about the rotating rigid string. In
Section 3, we consider perturbations of the rigid string and in particular
compute their spectrum. For the limiting cases of a static and a massless
string, this can be done analytically, while in the intermediate regime we use
numerical methods. In Section 4 we argue that the relation of target space
and world-sheet energies is more complicated than assumed, for example,
in [9]. In particular, at the order usually considered, one already has to take
the interaction into account. We conclude with a summary and an outlook.
2Concretely, a complex potential is used, and four boundary conditions are imposed at
each end.
2

2 Setup
The Nambu–Goto action with boundary terms accounting for masses is given
by3
S = Sbulk + Sboundary = −γ
∫
Σ
√−gd2x − m
∫
∂Σ
√−hdx. (1)
Here Σ is the source space and g the determinant of the induced metric,
pulled back by the embedding X : Σ → Rd:
gµν = ∂µXaηab∂ν Xb.
Here η is the Minkowski metric on Rd, with signature (−, +, . . . , +). Like-
wise, h is the induced metric pulled back to ∂Σ.
On Σ, we choose coordinates (τ, σ) ∈ R × [−S, S]. We determine the
boundary conditions corresponding to the action (1), by the requirement
that the boundary terms vanish. This leads to
(
γg1ν ∂ν Xb√−g ± m∂0
( ∂0Xb
√−g00
))
|σ=±S = 0. (2)
Let us now consider the rigid rotating string solution, which we param-
eterize as
X = R(τ, cos τ sin σ, sin τ sin σ, 0), (3)
where 0 stands for the 0 of Rd−3. Here, we have to choose S < π/2. One
easily computes
gµν = R2 cos2 σηµν , √−ggµν = ηµν ,
so that the boundary conditions are satisﬁed, provided that
γR
m = tan S
cos S . (4)
Two limits are of particular interest: The relativistic limit γR/m → ∞, i.e.,
S → π/2, in which the velocity of the ends of the strings approaches unity.
And the static limit γR/m → 0, while keeping L = 2γR2/m ﬁxed, which
corresponds to the string connecting two stationary D0 branes, separated
by the distance L.
The energy of a string conﬁguration is given by [10]
E =
∫ S
−S
δSbulk
δ∂0X0(0, σ) dσ + δSboundary
δ∂0X0(0, −S) + δSboundary
δ∂0X0(0, S) . (5)
For the rotating string solution (3), we compute, using (4), the energy
E = 2γRS + 2γR 1
tan S ,
3For simplicity, we here assume equal masses at the two ends.
3

where the ﬁrst term is due to the string and the second one due to the
masses at the ends. By comparison with the expression for the static string,
we conclude that the eﬀective length of the string is given by
Leﬀ = 2RS.
Note that this is greater than the length of the string measured in the
laboratory frame, which is given by 2R sin S.
We note that the solution (3) breaks a couple of symmetries of the target
Minkowski space. All spatial translations are broken, whereas time transla-
tion survives if accompanied by a rotation in the 1 − 2 plane. Furthermore,
all boosts are broken, as well as the rotations in the planes 1 − 2, 1 − i, 2 − i,
for i ≥ 3. Below, we identify d − 2 corresponding zero modes, consistently
with the fact that locally we break d − 2 translational symmetries [11]. In
the static limit, we will ﬁnd quasi-massless modes for the remaining broken
symmetries, with the exception of the boosts.4
3 Perturbations
Now we consider perturbations of the rotating string solution (3), henceforth
denoted by ¯X. That is, we write
Xa = ¯Xa + ϕa,
and expand the action in ϕ. As we are perturbing around a classical solution,
the component of ﬁrst order in ϕ vanishes. The action at second order in ϕ
deﬁnes the free part S0 of the action. Due to the diﬀeomorphism invariance
of the action, the equations of motion derived from S0 are not hyperbolic.
The physical degrees of freedom of the bulk part of the action are the normal
perturbations [12], i.e., those fulﬁlling
∂µ ¯Xaηabϕb = 0.
The gauge condition we will adopt here is to set the longitudinal pertur-
bations to 0. This was apparently ﬁrst proposed in [13]. Obviously, this
gauge choice preserves the target space symmetries that are unbroken by
the solution (3).
In the following, we distinguish between the d − 3 scalar polarizations,
ϕa = f a
s ea, a ≥ 3, which are orthogonal to the plane of rotation, and one
planar polarization, which may be written as
ϕa = fpua, u = (tan σ, − sin τ / cos σ, cos τ / cos σ, 0). (6)
4This is not surprising, as these can not be seen as inﬁnitesimal for very late or early
times.
4

Due to the boundary condition, we have a supplementary physical degree of
freedom at the boundary, namely radial perturbations. We may write them
as
ϕa = frva, v = (0, cos τ, sin τ, 0), (7)
where fr lives on the boundary, i.e., at σ = ±S. This mode will enter the
boundary conditions for the planar polarization. In terms of these modes,
the free action S0 can be written as
S0 = γ
2
∫
Σ
( ˙f 2
p − f ′
p
2 − 2
cos2 σ f 2
p + ˙f 2
s − f ′
s
2)
d2x (8)
+ γ
2
1
tan S
∫
∂Σ
( ˙f 2
p + ˙f 2
r + ˙f 2
s + 1
cos2 σ f 2
p + (1 + 2 tan2 σ)f 2
r
+ 2
cos σ ( ˙fpfr − fp ˙fr)
)
dx.
3.1 Equations of motion
From the ﬁrst term of (8), we obtain the bulk equations of motion
− ¨fs + f ′′
s = 0, − ¨fp + f ′′
p − 2
cos2 σ fp = 0.
With the usual ansatz fs/p(τ, σ) = e−ikτ fs/p,k(σ), we have to solve the mode
equations
f ′′
s,k = −k2fs,k, f ′′
p,k − 2
cos2 σ fp,k = −k2fp,k.
As we deal with a problem that is invariant under the reﬂection σ → −σ,
the solutions will be either even or odd. For the scalar polarization, the
even/odd solutions are given by
f +
s,k = cos kσ, f −
s,k = sin kσ. (9)
The planar mode equation can be solved by choosing the coordinate x =
sin σ and the ansatz f (x) = (1 − x2)1/4g(x), leading to
(1 − x2)g′′(x) − 2xg′(x) − 9
4 (1 − x2)−1g(x) + (k2 − 1
4 )g(x) = 0,
which is the deﬁning equation for Legendre functions [14]. It follows that
the general even/odd solutions of the planar mode equation are given by5
f ±
p,k = cos 1
2 σ
[ √π
2 sin((k − 1
2 ) π
2 )
(
P 3
2
k− 1
2
(sin σ) ± P 3
2
k− 1
2
(− sin σ)
)
(10)
+ 1
√π cos((k − 1
2 ) π
2 )
(
Q 3
2
k− 1
2
(sin σ) ± Q 3
2
k− 1
2
(− sin σ)
)]
.
5Note that Q3/2
1/2 = 0 and P 3/2
1/2 is symmetric, so that f −
p,k as deﬁned in (10) degenerates
for k → 1. An antisymmetric solution for k = 1 is given by f −
p,1(σ) = σ
cos σ + sin σ.
Similarly, f +
p,k as deﬁned in (10) degenerates for k = 0. The symmetric solution for k = 0
may instead be written as f +
p,0(σ) = σ tan σ + 1.
5

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The excitation spectrum of rotating strings with masses at the ends

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