Preprint typeset in JHEP style - HYPER VERSION
Boundary action and profile of effective bosonic
strings
A. S. Bakry[1],M. A. Deliyergiyev[1,2],A. A. Galal[3],M. KhalilA Williams[4,5,6,3]
[1] Institute of Modern Physics, Chinese Academy of Sciences, Gansu 730000, China
[2] Institute of Physics, The Jan Kochanowski University in Kielce, 25-406, Poland
[3] Department of Physics, Al Azhar University, Cairo 11651, Egypt
[4] Department of Mathematics, Bergische Universitat Wuppertal, 42097 Germany
[5] Department of Physics, University of Ferrara, Ferrara 44121, Italy
[6] Computation-based science Research center, Cyprus Institute, Nicosia 2121, Cyprus.
*E-mail: [email protected]
Abstract: The mean-square width of the energy profile of bosonic string is calculated
considering two boundary terms in the effective action. The perturbative expansion of the
Lorentz-invariant boundary terms at the second and the fourth order in the effective action
is taken around the free Nambu-Goto string. The calculation are presented for open strings
with Dirichlet boundary condition on cylinder.
Keywords: Bosonic Strings, Boundary terms, Width profile, Effective actions.
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Contents
1. Introduction 1
2. Effective action of bosonic strings 3
3. The Lorentz invariant boundary terms 5
4. The boundary terms contribution to the energy width 7
5. Summary and Conclusion 13
A. Appendix: Functions and Identities 14
B. Appendix: Mean-square width W 2b2
18
B.1 The expectation value of W 21,b2 18
B.2 The expectation value of W 22,b2
23
C. Appendix(C): Mean-square width W 2b4
28
C.1 The expectation value of W 41,b4
28
C.2 The expectation value of W 22,b4
31
1. Introduction
The confinement of quarks is an essential property of quantum chromodynamics (QCD)
and strong interactions. Despite the tremendous research efforts to provide mechanisms
of quark confinement based dynamical gluonic degrees of freedom of the QCD, there is no
compiling analytical construction for the phenomenon of confinement starting from basic
principles.
The confinement property can be probed directly in the Monte-Carlo evaluations of
QCD path integrals. The computer simulations of the infinitely heavy quark-antiquark
(QQ) bound state revealed the linear rise property [1–4] of confining potential.
It is understood from this observation that the linear rise of the potential at long
separation between two static QQ is due to the gluonic field [5–14] which appears to be
condensed into a stringlike flux tube. The formation of stringlike flux tube in between
the color sources provides a prospective technique for the confinement of quarks. In this
model, the nonperturbave properties of the QCD flux-tubes are expected to conform with
that of an effective bosonic strings with fixed edges in the long string limit.
The string formation manifests in many strongly correlated systems [15–18] and can
be described after roughening transition by an effective string action. The effective string
– 1 –
action is a low energy effective field theory [19] which is obtained by integrating out the
degrees of freedom of Yang-Mills (YM) vacuum in the presence of two static quarks. This
string description supplies a tool to predict a set of infrared (IR) observables [68, 69, 69–77]
which can be confronted with the outcomes from the numerical lattice data.
The Luscher term is an essential [20, 21] prediction associated with the string’s quan-
tum fluctuations at zero temperature. This Casimir energy is a detectable subleading cor-
rection to the linearly rising QQ potential [22, 23] which is universal to any gauge group.
The Luscher term has been detected in the numerical simulations of either Wilson [24] or
Polyakov loop correlation function representing QQ static pair [25–32]. The Y-string [33–
35] binding the three quark [37, 38] in baryonic configurations produces a counterpart
Luscher-like term which have been detected in abelian lattice gauge system [35].
Not only the static potential but also the energy chart of the QCD vacuum in the
presence of the confined color sources [36] and its characteristic broadening is a funda-
mental source to probe the physics of the confinement from first principles. Many lattice
simulations of many gauge groups have unambiguously verified [26, 39–48] the predicted
logarithmic broadening [49] of the confining strings and the linear behavior [50] near the
confinement point and large distances.
Nevertheless, the analysis of string’s fine structure in the lattice numerical data for
the broadening profile revealed substantial deviations [51–53] from the free-string Nambu-
Goto (NG) model in the intermediate distance region at high temperatures. The excited
spectrum [47] and finite temperature string tension [54–56] obtained from the partition
function of the leading-order Gaussian formulation of NG string action show a similar
disagreement with the numerical data [26, 52, 56, 57] for distance scales less than 1.0 fm.
The higher-order corrections are terms of the asymptotic expansion in the inverse
powers of color source separation distance 1/R associated with the interaction terms of
growing dimensions. The next to leading correction terms of the NG action [58–60] have
been subjected to numerical investigations [42, 48, 61–67] to ascertain the relevance order
by order to the discrepancy from the numerical data in each gauge model. In fact, there is
no proof neither evidence that all orders [22, 23] of the power expansion are universal.
The effective bosonic string theory serves as a functional tool in many QCD processes
[68, 69, 69–77]. The precise study of the strings effects suggests considering other features
beyond the free NG action [78], in particular, possible stiffness properties [79, 80] and
boundary term corrections [81–84].
The Lorentz invariant boundary corrections [83] to the static QQ potential have shown
viable in both the Wilson and the Polyakov-loop correlators cases [57, 81, 82]. The bound-
ary corrections to the static quark potential provide an explanation to the deviations lately
discovered among predictions of the effective string and numerical outcomes [24].
The broadening profile of the energy field should receive similar corrections from the
Lorentz invariant boundary terms in the action. However, the contributions of the bound-
ary action to the width profile have neither been theoretically calculated nor confronted
with the numerical lattice data. These corrections are hoped to account to many features
of the fine structure of the profile of QCD flux-tube in IR region. In particular it could
account for the well known deviations at relatively short distances at low temperatures, or
– 2 –
larger distances for the excited spectrum of the flux-tubes [82, 86] and at high tempera-
tures [57, 85].
The goal of the present paper is to analytically estimate the mean-square width re-
sulting from the boundary terms in Luscher-Weisz (LW) effective string action. The calcu-
lations are laid out for open string with Dirichlet boundary condition on a cylinder. This
could be compatible with the energy fields set up by a static mesonic configurations. We
consider the perturbative expansion of two boundary terms at the order of fourth and six
derivative and evaluate the modification to the mean-square width around the free NG
string.
2. Effective action of bosonic strings
Long stringlike objects are not uncommon in field theory. The magnetic vortices in super-
conductors [90, 91], cosmic strings [93, 94] admits an effective string descriptions as well.
The property of linearly rising potential suggests that the YM vacuum admits the presence
of an object such as a quite thin long string, which is responsible for transmitting the
strongly interacting forces among the quarks. The conjecture is in consistency [87] with
the dual superconductive [28, 88–92] QCD vacuum and the formation of a vortex line dual
to the Abrikosov by the virtue of the dual Meissner effect.
The effective field theory description holds at distance scales larger than the scale
intrinsic thickness of the string [95, 96]. The classical long string solution entails, however,
the breaking of the (D − 2) transverse translational symmetry leading to the associated
transverse oscillations of massless Goldston modes [97, 98].
The Lagrangian of the low energy effective field theory [31] may be constructed from all
the terms respecting the imposed symmetries on the system [99]. To constrain the effective
action of the confining string Polchinski-Strominger (PS) [100] introduced a conformal
gauge on the worldsheet. Within this formalism, the action is constraint by requiring the
effective field theory to be ghost free which fixes the central charge to be D=26 [77, 100].
The conformal theory is manifestly Lorentz-invariant and the leading term of PS action
coincides with the 4-derivative term of the NG string [77]. However, it appears that this
technique is untraceable especially at higher orders.
However, Luscher, Symanzik and Weisz [101, 102] suggested that the effective action
of the string connecting a stable pair of QQ in D-dimensional confining the theory of
Yang-Mills can be constructed by the leading term of NG, free, action. In fact, the LW
effective action admits all terms which preserves Lorentz and transnational-invariance [103]
and is expressed in physical degrees of freedom. The effective action respects Lorentz
symmetry through nonlinear realization of this symmetry [105–107] since the worldsheet
gauge diffeomorphism is fixed to static/physical gauge. The nonlinearly-realized Lorentz
symmetry constrains the coefficients of the higher-order terms in the action which is found
to coincide with that of the NG action up to O(1/R3) in the long string expansion.
As mentioned above, an effective string action can be constructed from the derivative
expansion of collective string co-ordinates fulfilling Poincare and parity invariance. The
operators of the Goldstone fields in the action are perturbations around the classical so-
– 3 –
lution and necessarily are space-time derivatives to preserve the translation invariance. In
addition, the terms in the action that are proportional to the equation of motion or its
derivatives do not contribute in perturbation theory and can be absorbed by field redef-
inition. These fluctuations Xi (in D − 2 space-time dimensions) are massless and in the
static/physical gauge X1 = ζ1, X4 = ζ2 are restricted to transverse directions C. With this
prescription, the LW effective action [101, 102] up to 4-derivative term come into the form
SLW [X] = Scl +σ
2
∫d2ζ( ∂X
∂ζα· ∂X
∂ζα
)+ σ
∫d2ζ
[c2
( ∂X
∂ζα· ∂X
∂ζα
)2+ c3
( ∂X
∂ζα· ∂X
∂ζβ
)2]
+ γ
∫dζ2√gR+ α
∫dζ2√gK2 + ...+ Sb,
(2.1)
where we have considered the Euclidean signature. The vector Xµ(ζ0, ζ1) maps the area
C ⊂ R2 into R4, the geometrical invariants R and K define Ricii-scalar and the extrinsic
curvature [79, 80] of the corresponding configuration of world sheet, respectively. The term
Scl characterizes the classical term, on the quantum level the Weyl invariance of the action
is broken in four dimensions; however, the anomaly is known to vanish at large distances
[87].
Cylindrical boundary conditions explicitly break translation invariance of the classical
solution, this would entail the emergence of terms other than that in bulk of the effective
action. The boundary action Sb is a surface term located at the boundaries ζ1 = 0 and
ζ1 = R. This ought to signifies the interplay of the effective string with either the Polyakov
loops on the fixed ends of the string or Wilson’s loop which we will discuss in detail in the
next section.
The couplings c1, c2 in LW action Eq. (2.1) are the parameters of effective low-energy
theory. For the next-to-leading order terms in D dimension the open-closed duality [47]
imposes further constraint on these kinematically-dependent couplings to
(D − 2)c2 + c3 =
(D − 4
2σ
). (2.2)
Moreover, it has been shown [60] through a nonlinear Lorentz-transform in terms of the
string collective variables Xi [108] that the action is invariant under SO(1, D−1). By this
symmetry the couplings Eq.(2.2) of the four derivative term in the LW action Eq.(2.1) are
not arbitrary and are fixed in any dimension D by
c2 + c3 =−1
8σ. (2.3)
Condition (2.3) implies that all the terms with only first derivatives in the effective
string action Eq.(2.1) coincide with the corresponding one in Polyakov-Kleinert action in
the derivative expansion.
S = −∫d2ζ√−detgαβ
(σ +
1
α(Kαβ)2
), (2.4)
where g is the two-dimensional induced metric on the world sheet embedded in the back-
ground R4, α is the rigidity parameter.
– 4 –
3. The Lorentz invariant boundary terms
The Lorentz symmetry which is a basic feature of YM theory dictates constraints [59, 106,
107, 109] over the arbitrary coefficients appearing on the effective action expansion. The
string field operators of the action should be built such that these crucial symmetries are
preserved. In this paper we theoretically calculate the width profile of effective open strings
with boundary action made up of Lorentzian invariants.
First we naively set out the possible form of each order in the Lagrangian density and
then guided by requirement of fulfilling Lorentz symmetry the relevant terms are deduced
at each coupling. We consider the case of Dirichlet boundary conditions Xi = 0 at both
ends, pure ζ0-derivatives vanish on the boundary (∂n0X = 0). A Generic boundary action
is defined as
Sb =
∫dζ0 [L1 + L2 + L3 + L4 + ...] , (3.1)
with the Lagrangian density Li for each effective low-energy coupling bi. The worldsheet
coordinates have dimensions of length, a bulk term has the same order as a boundary term
with one less derivative. The operator has to have an odd number n of spatial derivatives
∂n1X so that it has non-vanishing value.
At the lowest order, the general allowed form is schematically proportional to ∂2X2
and the only possible term [102] in the Lagrangian is therefore
L1 = b1∂1X · ∂1X. (3.2)
Consistency with the open-closed string duality [47] implies a vanishing value of the
first boundary coupling b1 = 0, as we will discuss below the Lorentz invariance requirement
implies the vanishing of b1 as well.
The leading order corrections due to second boundary terms with the coupling b2appears at the four derivative term in the bulk. On the boundary the general term is of
the form ∂3X2, apart from a term proportional to the equation of motion there is naively
only one possible term
L2 = b2∂0∂1X · ∂0∂1X , (3.3)
The effective action preserves the transverse rotation symmetry SO(d−2), the complete
SO(1, d − 1) Lorentz invariance is spontaneously broken by the classical solution around
which we expand, but the expanded action still respects this symmetry non-linearly the
rotations in (i, j) plane generates
δijXi = εXj . (3.4)
In order to keep the gauge-fixing σ1 = X1 we must make a diffeomorphism
δ12ζ1 =εX2(ζ1),
δ12ζ0 =0,
(3.5)
that is, Eq.(3.4) generates rotation which should be followed by reparamterization
Eq. (3.5) to put the system again in the static gauge. The full nonlinear Lorentz transfor-
mation is then
δαiXj = −ε(δijζα +Xi∂αXj
), (3.6)
– 5 –
where ε is an infinitesimal parameter of boosts and rotations. For each boundary action
Sbi =
∫∂Σdζ0Li. (3.7)
The application of the infinitesimal transformation Eq.(3.6) and requiring the action Sbito vanish we obtain constraints on the values of the couplings; or realize higher-order
derivatives in the choice of the action of a given coupling.
The nonlinear transformation Eq. (3.6) of the Lagrangian densities Eq. (3.3) and
Eq. (3.11) generates higher-order terms at the same scaling [60, 107] which should still
acquire the same couplings. The scaling of a given term defines the order of the generated
terms when applying the transformation. That is, the transform Eq. (3.6) of ∂mXn gen-
erates terms with the same value of the difference m − n. The Lorentz transform of the
Lagrangian density Eq. (3.3) we have two basis terms (∂1∂0X)2 and ∂1∂0X · ∂1X
L2 = b2
∞∑k=0
[αk∂1∂0X · ∂1∂0X(∂1X · ∂X)k + βk+1(∂1∂0X · ∂1X)2(∂1X · ∂1X)k
]. (3.8)
The invariance of (3.3) leads to recursion relations when solved Ref.[60] give rise to
L2 = b2
(∂0∂1X · ∂0∂1X
1 + ∂1X · ∂1X− (∂0∂1X · ∂1X)2
(1 + ∂1X · ∂1X)2
). (3.9)
However, for the third order we find then
L3 = b3 (∂1X · ∂1X)2 . (3.10)
The variation of the boundary actions at first and third orders with infinitesimal nonlin-
ear Lorentz transform Eq.(3.6) of the Lagrangian densities Eq.(3.2) and Eq.(3.10) entails
vanishing value for b1 = 0 and b3 = 0.
The next order, L4, the general effective Lagrangian on the boundary is
L4 = b4∂20∂1X · ∂2
0∂1X. (3.11)
The first two terms derived in Ref.[60] are given by
L4 = b4
(∂2
0∂1X · ∂20∂1X
1 + ∂1X · ∂1X− (∂2
0∂1X · ∂1X)2 + 4(∂20∂1X · ∂0∂1X)(∂0∂1X)
(1 + ∂1X · ∂1X)2+ ...
). (3.12)
The action at the boundaries drive infinitesimal diffusion from generic source/sink into
the energy density along the QCD flux tube. This causes perturbation from the free NG
string behavior which is expected to evidently affect the static potential [24, 57, 81, 82]
and the profile near the color sources, short separation distances as well as high tempera-
tures [57, 85] and excited spectrum [53]. In the next section, we shall lay out the perturba-
tive expansion of the boundary action and estimate the subsequent augmentation/lessening
of mean-square width of the effective string in D dimension at any temperature.
– 6 –
4. The boundary terms contribution to the energy width
Let us consider the free NG action around which the perturbative expansion is taken
S`o[X] = Scl +σ
2
∫dζ1
∫dζ0( ∂X
∂ζα· ∂X
∂ζα
), (4.1)
The next-to-leading NG term combined with the expansion of the surface term on the
boundary
SPert[X] = Sn`o + Sb,
= (Sn`o + Sb2 + Sb4 + ..) + ....,(4.2)
defines the perturbation from the effective LW action Eq. (2.1) with higher-order geomet-
rical terms left out 1.
The mean-square width of the string is defined as the second moment of the field with
respect to the center of mass of the string X0 and is given by
W 2(ζ) =
∫DX(X(ζ1, ζ0)−X0)2e−S[X]∫
DXe−S[X], (4.3)
Expanding around the free-string action Eq. (4.1) the squared width of the string [110]
is given by
W 2(ζ1) = W 2`o(ζ
1)− 〈X2(ζ0, ζ1)SPert〉0 + 2γ〈∂αX2(ζ0, ζ1)〉0 + γ2〈∂2αX2(ζ0, ζ1)〉0
− γ2
LTR
∫dζ0 dζ1 dζ0′ dζ1′〈∂2
αX(ζ0, ζ1) · ∂2α′X(ζ0′ , ζ1′)〉0.
(4.4)
where the vacuum expectation value 〈...〉0 is with respect to the free-string partition func-
tion, γ is an effective low energy parameter and LT is the length of compactified time
direction for cylindrical boundary condition.
Substituting Eq. (4.2) in the low-energy parameter expansion Eq. (4.4) the mean-
square width
W 2(R,LT ) =⟨(X2(ζ1, ζ0)S`o)
⟩−⟨(X2(ζ1, ζ0)Sn`o + X2(ζ1, ζ0)Sb2 + X2(ζ1, ζ0)Sb4 + ...)
⟩,
=W 2`o(R,LT ) +W 2
n`o +W 2b (R,LT ) + ...
with the contribution of the boundary action
W 2b (R,LT ) = W 2
b2(R,LT ) +W 2b4(R,LT ),
and the mean square width of the free string
W 2`o(R,LT ) =
∫DX(X(ζ1, ζ0))2e−S`o[X]∫
DXe−S`o[X]. (4.5)
The Green-function defines the two-point free propagator
G(ζ0, ζ1; ζ0′ , ζ1′
)= 〈X(ζ0, ζ1) ·X(ζ0′ , ζ1′)〉, (4.6)
1The extrinsic curvature correction to the width profile has been worked out elsewhere in Ref. [111]
– 7 –
which is the solution of Laplace equation on a cylindrical sheet [110] of surface area RLTand is given in spectral form [110] by
G(ζ; ζ ′) =1
πσ
∞∑n=1
1
n (1− qn)sin
(πnζ1
R
)sin
(πnζ1′
R
)(qne
πn(ζ0−ζ0′)
R + e−πn(ζ0−ζ0
′)
R
),
(4.7)
where q = e−πLTR is the complementary nome. The Dirichlet boundary condition corre-
sponding to fixed displacement vector at the ends ζ1 = 0 and ζ1 = R and periodic boundary
condition in time ζ0 with period LT [110] are encoded in the above propagator.
The expectation value [39, 65, 112] of the mean-square width corresponds to the
Green-function correlator of the free bosonic string theory in two dimensions
W 2`o(ζ1, τ) =
D − 2
2πσ0log
(R
R0(ζ1)
)+D − 2
2πσ0log
∣∣∣∣ ϑ2(π ζ1/R; τ)
ϑ′1(0; τ)
∣∣∣∣ , (4.8)
where θ are Jacobi elliptic functions
θ1(ζ; τ) = 2∞∑n=0
(−1)nqn(n+1)+ 1
41 sin((2n+ 1) ζ),
θ2(ζ; τ) = 2
∞∑n=0
qn(n+1)+ 1
41 cos((2n+ 1)ζ), (4.9)
with q1 = eiπ2τ , and R2
0 is the UV cutoff which has been generalized to be dependent on
distances from the sources. The above formula Eq. (4.8) can be pinned down through the
standard relations between the elliptic Jacobi and Dedkind η functions to either of the
equivalent forms derived in Refs. [110, 111].
The NLO term from the low energy parameter expansion Eq.(4) to the width of NG
string have been worked out in detail in Ref.[110], the width due to the self-interaction is
modified by
W 2n`o =
(D − 2)π
12σ2R2
{τ(qd
dq− D − 2
12E2(τ)
)[E2(2τ)− E2(τ)]− D − 2
8πE2(τ)
}+
π
12σR2[E2(τ)− 4E2(2τ)]
(W 2lo −
D − 2
4πσ
).
(4.10)
In the following we evaluate the correction to the mean-square width of the free string
with Lorentz invariant boundary action at these two orders. We consider the expecta-
tion value of the quadratic field excitation up to the leading order term in the boundary
action (4.20).
The perturbative expansion of the boundary action Sb2 corresponding to the La-
grangian density Eq.(3.3) is given by
Sb2 = b2
∫∂Σdζ0 (∂0∂1X · ∂0∂1X− (∂0∂1X · ∂0∂1X) (∂1X · ∂1X) + ...) . (4.11)
The generic form of the Wick-contracted term representing the perturbative contribution
of the boundary term to the expectation value is
〈X2Sb2〉 = b2(〈(X ·X)∂20∂
21(X ·X)〉+ 〈∂0∂1(X ·X)∂0∂1(X ·X)〉). (4.12)
– 8 –
Upon point-splitting the expectation value in terms of Green propagators the mean-square
width becomes
W 2b2 = −(D − 2)b2
∫∂Σdζ0′ lim
ζ′→ζ
[∂0∂1′G(ζ; ζ ′)∂0′∂1G(ζ ′; ζ) +G(ζ; ζ ′)∂0∂1∂0′∂1′G(ζ ′; ζ)
].
(4.13)
the above two terms define the sum of the expectation values
W 2b2 = −
(W 2
1,b2 +W 22,b2
). (4.14)
Substituting the free Green propagator Eq.(4.7) in the above expectation value. We per-
form the integrals over ζ0 after evaluating the derivatives and rewriting the resulting sum-
mations in closed form.
The evaluation of the two correlators Eq.(4.14) involve cumbersome manipulations, we
present the detailed calculus in Appendix(B). The ζ-function regularization of the divergent
sums is performed on each expectation value appearing in the limits, ε, ε′ → 0. The mean-
square width W 2b2
turn out to be
W 2b2 =
−πb2(D − 2)
4R3σ2
(1
8− 1
24E2(2τ)
), (4.15)
with E2(τ) is Eisenstein series defined by Eq.(A.24). The second correlator W 22,b2
vanishes
after explicitly taking the limit ε → 0. However, assuming a finite value of ε introduces
terms related to the cutoff R20 = − log(ε) of Eq. (4.8). The next to leading terms in the ε
expansion are renormalization correction [110] to the coupling b2 given by
R2b2 = −πb2(D − 2)
24R3σ2µ1(R,LT , ε), (4.16)
where µ1(R,LT , ε) are given by Eq. (C.29) Appendix(C). The ultraviolet cutoff, ε, occurs
naturally in lattice gauge models due to the lattice discretization and is interpreted as a
fit parameter.
In Wilson loops [24, 60, 113] operators the boundary action survives over both the
spatial and temporal extends, the boundary action is given by
Sb2 = b2
∫∂Σt
dζ0 (∂0∂1X · ∂0∂1X) + b2
∫∂Σs
dζ1 (∂0∂1X · ∂0∂1X) , (4.17)
this surface term lives on the boundary ∂Σ which in the action Eq. (4.17) is conveniently
chosen as a rectangle-shaped Wilson’s loop circumfering the spatial-temporal area of R×L,
the curves ∂Σt and ∂Σs stands for temporal and spatial parts of the loop, respectively.
The direct calculation of the expectation value of the 〈Sb〉 entails contributions from
both the temporal and spatial parts to the static quark potential [24]. As a consequence
of the symmetry of the propagator Eq. (4.7), these two corrections give similar formu-
las [24]; however, with the role between the source separation, R, and temporal extent, L,
exchanged. Moreover, the contribution from the temporal path is formally equivalent [24]
to that of two Polyakov loop with the identification of the temporal height of the Wilson’s
loop L instead of the cylindrical time extend LT → L.
– 9 –
The generalization of the expectation value of the width due to two Polyakov loops
Eq.(4.19) to that due to the Wilson’s loop is accordingly evaluated as the expectation value
of
W2b2 = 〈X2Sb2〉∂Σt + 〈X2Sb2〉∂Σs , (4.18)
for the boundary action given by Eq.(4.17). The spatial part of this expectation value is
an integral of the diffused energy of the string along the spatial path the infinitesimal-
time interval of either the creation/annihilation of the static quark-antiquark pair. As the
mean-square width can be deduced to have the form
W2b2 =
−πb2(D − 2)
4σ2
[1
R3
(1
8− 1
24E2
(iL
R
))+
1
L3
(1
8− 1
24E2
(iR
L
))], (4.19)
where L is the temporal extend of Wilson loop.
The next Lorentz invariant non-vanishing Lagrangian density is Eq.(3.11) correspond-
ing to the coupling b4, we consider the leading term in the expansion of boundary action
Sb4 = b4
∫∂Σdζ0 (∂0∂1X.∂0∂1X− (∂0∂1X.∂0∂1X) (∂1X.∂1X) + ...) . (4.20)
The Wick-contraction of the operator X2 with the leading term yields the expectation
value
〈X2Sb4〉 = b4〈(X ·X)∂20∂
21(X ·X)〉+ 〈∂0∂1(X ·X)∂0∂1(X ·X)〉. (4.21)
The expectation value Eq.(4.21) in terms of Green propagators after point-splitting
become
W 2b4 = −(D − 2)b4
∫∂Σdζ0′ lim
ζ′→ζ
(∂0∂1′G(ζ; ζ ′)∂0′∂1G(ζ ′; ζ) +G(ζ; ζ ′)∂0∂1∂0′∂1′G(ζ ′; ζ)
).
(4.22)
Similarly, substituting the free propagator Eq.(4.7) in the above expectation value and
integrating over ζ0, see Appendix(C) for details. The expectation value of the fourth-order
boundary term in the action
W 2b4 = −
(W 2
1,b4 +W 22,b4
), (4.23)
which turn up as,
W 2b4 =
−π3(D − 2)b432R5σ2
(E2(τ)− 5
4
)(11E2(τ/2)
36− 5E2(τ)
9− 55
12
), (4.24)
with the second correlator W 22,b2
vanishing in the limit ε→ 0. The ε expansion of the
correlator which only assuming finite value to UV cutoff ε put fourth to a renormalisation
of the coupling b2 given by
R2b4 = −π
3(D − 2)b4R5σ2
µ2(R,LT , ε) (4.25)
On the otherhand, the corresponding boundary corrections of the width of Wilson loop
at coupling b4 can be deduced following up the same line of reasoning leading to Eq.(4.19).
– 10 –
2 4 6 8 10 12]
1String length [Ra
0
5
10
15
20
25]2
a2
Str
ing w
idth
[w
=0.24
=1.0, b2
boundary: b
=0.34
=1.0, b2
boundary: b
=0.24
=1.0, b2
boundary: b
=0.34
=1.0, b2
boundary: b
=8.0, NG@LO1aT
L
(a)
2 4 6 8 10 12]
1String length [Ra
0
5
10
15
20
25]2
a2
Str
ing w
idth
[w
=0.24
=1.0, b2
boundary: b
=0.34
=1.0, b2
boundary: b
=0.24
=1.0, b2
boundary: b
=0.34
=1.0, b2
boundary: b
=8.0, NG@NLO1aT
L
(b)
Figure 1: The plot compares the mean-square width of the effective string bounded by two
Polyakov-loops at the depicted values of the boundary coupling b2 and high temperature LTa−1 = 8
in the middle plane. (a)Shows the contribution of the boundary action Eq.(4.15) to the width profile
W 2(R/2) = W 2NG(`o)
(R/2) +W 2b to the leading order NG string Eq.(4.8). (b)Similar to the plot in
(a); however, the boundary contribution to the mean-square width is at the next-to-leading order
in NG action Eq.(4.10) W 2(R/2) = W 2NG(`o)
(R/2) +W 2NG(n`o)
+W 2b .
The next-to-leading boundary correction for Wilson loop of rectangular area L × R is,
accordingly, given by
W2b4 =
π3(D − 2)b432σ2
[1
R5
(E2
(iL
2R
)− 5
4
)(11
36E2
(iL
4R
)− 5
9E2
(iL
2R
)− 55
12
)
+1
L5
(E2
(iR
2L
)− 5
4
)(11
36E2
(iR
4L
)− 5
9E2
(iR
2L
)− 55
12
)].
(4.26)
The leading non-vanishing boundary corrections to the flux-tube width Eq.(4.15) in-
dicate an inverse decrease with the third power of the length scale R. This suggests effects
that are more noticeable near the intermediate and short string length scale. In Fig.1
the mean-square width W 2, of the free NG string Eq.(4.8) and self-interacting NG string
Eq.(4.10) togather with the boundary corrections Eq.(4.15), are plotted at the middle plane
versus the string length such that
W 2(R/2) = W 2`o(R/2) +W 2
n`o +W 2b2 +W 2
b4 , (4.27)
On Fig.1 we compare modifications in the width profile which were received from the
depicted positive/negative values of the coupling parameter b2 and b4 in Eq.(4.15) and
Eq.(4.24).
The boundary term Sb2 in the effective string action may increase or decrease the
mean-square width of NG string depending on whether positive or negative values of the
coupling parameter b2 are considered. However, the resultant effects on the width is a
compromise between the value of the two couplings b2 and b4 and the corresponding signs.
– 11 –
As expected from the dimensional considerations the remarkable effects of the cor-
rections occur over the intermediate and short string length scale. The diffusion of the
interaction from the sources at the boundaries along the string is more stringent for rela-
tively short string length. For the values of b4 considered in Fig.1 the Sb4 boundary term
in the action appears to dominate and can in principle fine tune the width at shorter
distances.
2 4 6 8 10 12]
1String length [Ra
0
5
10
15
20
25
30]2
a2
Str
ing w
idth
[w
=0.24
=1.0, b2
boundary: b
=0.44
=1.0, b2
boundary: b
=0.24
=1.0, b2
boundary: b
=0.44
=1.0, b2
boundary: b
L×Wilson Loops R
=12, T=01aT
L
(a)
2 4 6 8 10 12]
1String length [Ra
0
5
10
15
20
25
30]2
a2
Str
ing w
idth
[w
=0.24
=1.0, b2=8: b1aTL
=0.24
=1.0, b2=8: b1aTL
=0.24
=1.0, b2=12: b1aTL
=0.24
=1.0, b2=12: b1aTL
=0.24
=1.0, b2=15: b1aTL
=0.24
=1.0, b2=15: b1aTL
=0.24
=1.0, b2=20: b1aTL
=0.24
=1.0, b2=20: b1aTL
Polyakov Loops
(b)
Figure 2: (a)Compares the profile of the effective string at next to leading order in NG action for
the depicted values of the couplings b2 and b4 (W 2 = W 2`o(R/2) +W2
b2+W2
b4) of Wilson loop of
area R × L with La−1 = 12 time slices at zero temperature T = 0. (b) The plot shows the profile
with the decrease of the temperatures to low values for two Ployakov loops. The dashed and solid
lines are for two different values of the coupling b2, and b4.
The comparison of the mean-square width of the free NG string with the corresponding
self-interacting string for a fixed value of the coupling b2 and b4 reveals subtle dilution of
the boundary action effects if considered with the associated self-interaction of NG string.
This suggests diffusion attenuated to some extend with the viscose medium driven by the
self-interacting string field.
In Fig.2(a) we plot the width of the effective string versus the source separation and
high temperature LTa−1 = 8 in the middle plane of the string. The plot compares the
width profile of the self-interacting string -Eq.(4.8) and Eq.(4.10)- setup by Wilson loop
W 2(R/2) = W 2`o(R/2) +W2
b2 +W2b4 , (4.28)
with W2b2
and W2b4
are given by Eq.(4.19) and Eq.(4.26), respectively. The first term in
Eq.(4.28) signifies the zero temperature logarithmic prodening of the effective string. A
similar plot in Fig.2(b) illustrates; however, the perturbative mean-square width with the
decrease of the temperature Eq.(4.28). The plot shows the convergence of the solutions in
the limit of infinite temporal LT →∞ extend.
The corrections provided by the boundary action to staticQQ potential seem to explain
to some extend the deviations appearing when constructing the static mesonic states with
Polyakov loop correlators [86]. The QQ potential is well described by the up to surprisingly
– 12 –
small distances as R = 0.3 fm using the boundary action at temperature near the end of
QCD Plateau T/Tc = 0.8 fm. At a higher temperature, the inclusion of the boundary
corrections up to the fourth order b4 together with string rigidity has been found [85] to
be viable in providing good fits for distances as small as R = 0.5 fm.
The lattice simulations of the energy-density profile is a more challenging quantity to
measure compared to the QQ potential. The exponentially decaying field corelators involve
the field strength tensor leading [35, 51–53] to a substantial numerical effort [57, 65, 85,
110] to attain a good enough precision.
The boundary corrections to the mean-square width of the flux-tube dictated by
Eq.(4.13) could be relevant to fine deviations such as those famed to occur on interme-
diate source separation scale. The detection of these subtleties is, of course, subject to
the continuous improvement in the resolution of the lattice data and computational power.
Nevertheless, one would expect an impact of these terms if to carry out the numerical
investigations with the Wilson loop operators [39].
5. Summary and Conclusion
In this work, the width of the energy profile of string bounded by two Polyakov loops is
derived as a function of the temperature in any dimension D. We have considered LW
string with two Lorentz invariant subleading boundary terms in the effective action.
The mean-square width have been estimated as a low energy perturbative expansion
around the free string action. The width is derived for open strings with Dirichlet boundary
condition on cylinder. We have implemented the technique of the ζ-function regularization
to the quadratic operators in the corresponding expectation values of the mean-square
width of the string.
The main findings in the present paper is the boundary correction to mean-square
width of the flux-tube constructed by two Polyakov loops which given by Eq.(4.13) at the
coupling order b2 of the Lagrangian density at the boundaries. The corrections induced by
the next Lorentz invariant term at coupling b4 have been estimated as well and are given
in Eq.(4.23). The expectation values of two Polyakov loops can be directly generalized
to Wilson’s loop, where the boundary action survives in both the spatial and temporal
extents, are given by Eq.(4.19) and Eq.(4.26) at coupling b2 and b4, respectively.
The fine structure of the mean-square width dictated by Eq.(4.13) and Eq.(4.23) in-
dicates effects at the third order in the inverse length scale 1/R3. This could be relevant
to QCD strings at relatively small quark separations but where the long effective string
theory still holds.
It would be interesting to investigate the width profile taking into account the boundary
action in the numerical simulation of Abelian and non-Abelian [28, 64, 81, 114–117] gauge
groups. In particular, the generalization of the calculation presented in this paper to
Wilson loops [24, 60, 113, 120] operators where more impact of the boundaries is expected.
The excited meson [118, 119] or baryonic configurations [37, 38, 121–125] are very relevant
system to discuss the boundary corrections.
– 13 –
Acknowledgments
This work has been supported by the Chinese Academy of Sciences President’s International
Fellowship Initiative grants No.2015PM062 and No.2016PM043, the Recruitment Program
of Foreign Experts, the Polish National Science Centre (NCN) grant 2016/23/B/ST2/00692,
NSFC grants (Nos. 11035006, 11175215, 11175220), the Hundred Talent Program of the
Chinese Academy of Sciences (Y101020BR0).
A. Appendix: Functions and Identities
Identity I The Jacobi elliptic ϑ1 function [126] of the first kind is defined as
ϑ1(z, q) = 2 4√q sin(z)
∞∏k=1
(1− q2k
)(1− 2q2k cos(2z) + q4k
), (A.1)
ϑ2(z, q) = 2 4√q cos(z)
∞∏k=1
(1− q2k
)(1 + 2q2k cos(2z) + q4k
), (A.2)
ϑ3(z, q) =∞∏k=1
(1− q2k
)(1 + 2q2k−1 cos(2z) + q4k−2
), (A.3)
ϑ4(z, q) =∞∏k=1
(1− q2k
)(1− 2q2k−1 cos(2z) + q4k−2
), (A.4)
(A.5)
with the nome defined as q = eiπτ ∧ =(τ) > 0.
Identity II The series expansion of the derivative ofϑ′1ϑ1
(z, q)
∞∑k=−∞
csch(πkλ+ z)2 =ϑ′′1(iz, e−πλ
)ϑ1 (iz, e−πλ)
−ϑ′1(iz, e−πλ
)2ϑ1 (iz, e−πλ)
2 . (A.6)
Identity III The square of logarithmic derivative [127]
(ϑ′1(z, q)
ϑ1(z, q)
)2
=ϑ′′1(z, q)
ϑ1(z, q)−(ϑ′1(z, q)
ϑ1(z, q)
)′. (A.7)
Identity IV The logarithmic derivatives of ϑ1(z, q) can be expressed as a series sum of
hyperbolic functions as
∞∑k=−∞
coth(πkλ+ z) =ϑ′1(iz, e−πλ
)ϑ1 (iz, e−πλ)
. (A.8)
Identity V
– 14 –
Let ϕ = π(LT−4ζ0)4R and
(z1 = x+ iy; z2 = −x+ iy),
x =πζ0
R; y = −πζ
1
R,
(A.9)
the sum of the series
4
∞∑m=1
e−πLTm
2R
(sinh
(2mϕ− 2iπmζ1
R
))1− e−
πLTm
R
=ϑ′4ϑ4
(z1 +πτ
2), (A.10)
can be expressed in the form of Jacobi elliptic function ϑ4(τ) (Identity(XIII, XVIII)). In
addition, the folowing invariant subtraction follows from Identity(XIV)
ϑ′4ϑ4
(z2 +πτ
2)− ϑ′4
ϑ4(z1 +
πτ
2) =
ϑ′1ϑ1
(z2, q)−ϑ′1ϑ1
(z1, q), (A.11)
From both of Eq.(A.10) and Eq.(A.11), the following equivalence relation holds
ϑ′1ϑ1
(z2 +πτ
2)− ϑ
′1
ϑ1(z1 +
πτ
2) =
∞∑m=1
e−πLTm
2R
(sinh
(2mϕ− 2iπmζ1
R
)− sinh
(2mϕ− 2iπmζ1
R
))1− e−
πLTm
R
(A.12)
Identity VI The modular πτ increase to the argument of ϑ1 function has the prop-
erty [126]
ϑ1(πτ + z, q) = −e−2iz
qϑ1(z, q). (A.13)
For m ∈ Z
ϑ1(z +mπτ, q) = (−1)mq−mϑ1(z, q) exp(−i(π(m− 1)mτ + 2mz)). (A.14)
The modular increase of the logarithmic derivative can be deduced from properties of
ϑ1 function, for m = 1 we obtain
ϑ′1(z ± πτ, q)ϑ1(z ± πτ, q)
=ϑ′1(z, q)
ϑ1(z, q)± 2i, (A.15)
and m = −1,+1ϑ′′1(z ± πτ, q)ϑ1(z ± πτ, q)
=ϑ′′ (z, q)
ϑ1 (z, q)∓ 4
iϑ′1 (z, q)
ϑ1 (z, q)− 4. (A.16)
Identity VII The series sum
∞∑n=1
qn
(1− qn)2=
1
8
(1− θ′′2
θ2− θ′′3θ3
),
=−1
8+
1
6E2(2τ)− 1
24E2(τ),
(A.17)
has the above two quivalent functional form in terms of either elliptic ϑ2(0; q) and ϑ3(0; q)
or Eisenstein series E2(τ).
– 15 –
Identity VIII The logarithm of ϑ1(z, q) is given by
log(ϑ1(z, q)) = 4∞∑k=1
q2k sin2(kz)
k (1− q2k)+ log(ϑ′1(0, q)) + log(sin(z)). (A.18)
The first derivative of the logarithm of ϑ1(z, q)
ϑ′1(z, q)
ϑ1(z, q)= 4
∞∑k=1
q2k sin(2kz)
1− q2k+ cot(z). (A.19)
Identity IX The derivative of the Eq.(A.19) at z = 0 blow up due to the divergence of
csc2(z). Let x = e−iz and expanding csc(z)2 around x = 0
csc(x)2 = x2 + 2x4 + 3x6 + 4x8 + 5x10 + ....,
limx→1
csc(x)2 = (1 + 2 + 3 + 4 + ...) =
∞∑k=1
k,
= ζ(1),
= −1/12.
(A.20)
The divergent sum is regularized with ζ function.
Identity X The following identity (See Ref. [128]) gives series representation for the
quotient of elliptic ϑ function and its second derivative
ϑ′′1(z, q)
ϑ1(z, q)= 16
∞∑n=1
qn cos (2iπn (z))
(1− qn)2 + 8∞∑n=1
nqn
1− qn− 1. (A.21)
Identity XI The Eisenstein series is defined as
E2k(τ) = 1 + (−1)k4k
Bk
∞∑n=1
n2k−1qn
1− qn, (A.22)
where Bk are Bernoulli numbers which are given in terms of Riemann zeta functions as
ζ(2k) =Bk(2π)k
2k!, ζ(1− 2k) = (−1)k
Bk2k. (A.23)
for even and odd numbers respectively.
In particular E2 is given by
E2(τ) = 1− 24∞∑n=1
nqn
1− qn.
– 16 –
Identity XII The following products appear in Eq.(B.40) as a result of the algebraic
manipulations involved in the sums and are related to elliptic ϑ4 through the identities
given below
P1 =∞∏k=1
(e−
πkLR− 2πL
R+ 2πt
R − 1)(
1− e−πkLR
+πLR− 2πt
R
),
=∞∏k=1
(e−πkLR
+πLR− 2πt
R + e−πkLR− 2πL
R+ 2πt
R − e−2πkLR−πL
R − 1),
=∞∏k=1
(−1− e−πkLR−πL
2R + e−12( 2πkL
R+πL
R )(eπkLR
+ 2πLR− 2πt
R + eπkLR−πL
R+ 2πt
R
)/e−
12( 2πkL
R+πL
R ))
=∞∏k=1
(−1− e−πkLR−πL
2R + e12( 2πkL
R+πL
R ) cosh
(3πL
2R− πt
R
)),
=ϑ4
(i(3Lπ)
4R − i(πt)R , e−
Lπ2R
)G1
; with G1 =
∞∏k=1
(1− e−
πkLR
).
(A.24)
Identity XIII The modular increase of the argument of the elliptic ϑ4 is given by the
identity
ϑ4(z + im log(q), q) = (−1)mq−m2e2imzϑ4(z, q), (A.25)
∀m ∈ Z
Identity XIV The elliptic ϑ4(z, q) are related to ϑ1(z, q) functions through the relatio
ϑ1(z, q) = i 4√qe−izϑ4
(z +
πτ
2, q), (A.26)
Identity XV The following identity for logarithm of quotient involving ϑ1 of holds for
any α, β ∈ C
log
(ϑ1(α+ β, q)
ϑ1(α− β, q)
)= 4
∞∑k=1
q2k sin(2αk) sin(2βk)
k (1− q2k)+ log
(sin(α+ β)
sin(α− β)
). (A.27)
Identity XVI The logarithm of quotient of the following ϑ4 functions in series form is
given by
log
(ϑ4(z, q)
ϑ4(0, q)
)= 4
∞∑n=1
qn sin2(inz)
n (1− q2n). (A.28)
Identity XVII The Jonquiere [129] function Li(q) is defined as
Lis(z) =∑k=1
zk
ks, (A.29)
– 17 –
which has useful property [130] known as the Jonquiere inversion formula
Lis(z) + Lis(1
z) =−(2πi)s
s!Bs
(1
2− log(−z)
2πi
), (A.30)
z ∈ C and Bn are Bernoulli polynomials for which the generating function is given by
∞∑n=0
Bn(x)tn
n!=
text
et − 1. (A.31)
Identity XVIII The logarithmic derivative of ϑ′4(z, q) in series form is given by
ϑ′4(z, q)
ϑ4(z, q)= 4
∞∑n=1
qn sin(2nz)
1− q2n. (A.32)
B. Appendix: Mean-square width W 2b2
In this appendix we show in detail the evaluation of the correlator
〈X2Sb2〉 = b2〈(X ·X) ∂0∂1X · ∂0∂1X)〉, (B.1)
which gives the modifications of the mean-square width of open string by virtue of boundary
action Sb2 .
B.1 The expectation value of W 21,b2
The Wick contraction corresponding to
W 21,b2 = 〈(∂0∂1X.∂0∂1X)2〉, (B.2)
upon point splitting, the expectation value of this correlator is given by
W 21,b2 =− (D − 2)b2
∫∂Σdζ ′0 lim
ζ′→ζ
[∂0∂1G(ζ; ζ ′)∂0∂1G(ζ ′; ζ)
]. (B.3)
The correlator G(ζ ′, ζ ′0; ζ, ζ0) ≡ 〈X(ζ0, ζ1)X(ζ0′ , ζ1′)〉 on a cylinder of size RLT with
fixed boundary conditions at ζ1 = 0 and ζ1 = R and periodic boundary conditions in ζ0
with period LT is given by
G(ζ; ζ ′) =1
πσ
∞∑n=1
1
n (1− qn)sin
(πnζ
R
)sin
(πnζ ′
R
)(qne
πn(ζ0−ζ0′
R + e−πn(ζ0−ζ0
′
R
). (B.4)
The expectation value corresponding to the above correlator
W 21,b2 =− π2
R4σ2
∫∂Σdζ0 lim
ζ′→ζ
∑n,m
mnπ2
(−1 + qm)(−1 + qn)
{2e−
π(m+n)(ζ0−ζ0′)
R
(−1 + e
2πm(ζ0−ζ0′)
R qm)
× (−1 + e2πm(ζ0−ζ0
′)
R qn) cos
(πnζ1
R
)cos
(πmζ1′
R
)sin
(πmζ1
R
)sin
(πnζ1′
R
)}.
(B.5)
– 18 –
Making the substitutions
ζ1′ = ζ1 + ε; ζ0′ = LT − ζ0 − ε′; (B.6)
the expectation value now reads
W 21,b2 =− 2π2(D − 2)
R4σ2
∫∂Σdζ0 lim
ε′→0ε→0
∑m=1,n=1
mn
(−1 + qm)(−1 + qn)eπ(m+n)(LT−2ζ0−ε′)
R
×(− 1 + e
2πm(−LT+2ζ0+ε′)R qm
)(− 1 + e
2πn(−LT+2ζ0+ε′)R qn
)× cos
(πnζ1
R
)cos
(πm(ζ1 + ε)
R
)sin
(πmζ1
R
)sin
(πn(ζ1 + ε)
R
).
(B.7)
After evaluating the limits ε→ 0 and ε′ → 0
W 21,b2 =− π2b2(D − 2)
2R4σ2
∫∂Σdζ0
∞∑m=1
meπm(LT−2ζ0)
R
qm − 1sin
(2πmζ1
R
)(qme
2πm(2ζ0−LT )
R − 1
)
×∞∑n=1
neπn(LT−2ζ0)
R
qn − 1sin
(2πnζ1
R
)(qne
2πn(2ζ0−LT )
R − 1
).
(B.8)
Let us consider one of the above symmetric sums over m and n
S =∞∑m=1
meπm(LT−2ζ0)
R
qm − 1sin
(2πmζ1
R
)(qme
2πm(2ζ0−LT )
R − 1
). (B.9)
Expanding the denominator
S =
∞∑n=1
∞∑k=0
n
(− i
2e
(−πkLT n
R− 2πLT n
R+ 2πnζ0
R+ 2iπnζ1
R
)+i
2e
(−πkLT n
R+πLT n
R− 2πnζ0
R+ 2iπnζ1
R
)
+i
2e
(−πkLT n
R− 2πLT n
R+ 2πnζ0
R− 2iπnζ1
R
)− i
2e
(−πkLT n
R+πLT n
R− 2πnζ0
R− 2iπnζ1
R
)).
(B.10)
The sum over n of each term can be put in closed forms in terms of hyperbolic function
S =
∞∑k=0
i
8
(− csch
(−πLT (−k − 2)
2R− πζ0
R− iπζ1
R
)2
− csch
(−πLT (k − 1)
2R− πζ0
R− iπζ1
R
)2
+ csch
(−πLT (−k − 2)
2R− πζ0
R+iπζ1
R
)2
+ csch
(−πLT (k − 1)
2R− πζ0
R+iπζ1
R
)2).
(B.11)
The sum index can be redefined k to k − 1 in the second and fourth term of the above
equation such that∑∞
k=0 →∑∞
k=1, followed by sign inversion to k to −k such that∑∞
k=1 →
– 19 –
∑−∞k=−1 the sum reads
S =
i
8
(−∞∑k=0
csch
(−πLT (k − 1)
2R− πζ0
R− iπζ1
R
)2
−−∞∑k=−1
csch
(−πLT (k − 1)
2R− πζ0
R− iπζ1
R
)2
+
∞∑k=0
csch
(−πLT (k − 1)
2R− πζ0
R+iπζ1
R
)2
+
−∞∑k=−1
csch
(−πLT (k − 1)
2R− πζ0
R+iπζ1
R
)2).
(B.12)
Each two consecutive terms in the above equation form a complete sum from −∞ to
∞ that is
S =−iR8π
(∂U(ζ)
∂ζ0
). (B.13)
with
U =∞∑
k=−∞coth
(−πLTk
2R+ φ∗
)−
∞∑k=−∞
coth
(−πLTk
2R+ φ
)(B.14)
The integral Eq.(B.8) can now be brought into the closed form
W 21,b2 =
b2(D − 2)
512R2σ2
∫∂Σdζ0
(∂
∂ζ0U(ζ0)
)2
. (B.15)
The above equation Eq.(B.15) involves an integral over the square of the derivative
with respect to ζ0 which is nontrivial to evaluate directly. However, the square of the first
derivative is related to the second differentiation via,
2
(∂U(ζ0)
∂ζ0
)2
=∂
∂ζ0
∂U(ζ0)2
∂ζ0− 2U(ζ0)
∂
∂ζ0
∂U(ζ0)
∂ζ0, (B.16)
which reads after integrating both sides
∫ LT
0
(∂U(ζ0)
∂ζ0
)2
dζ0 =
[U(ζ0)
∂U(ζ0)
∂ζ0
]LT0
−∫ LT
0U(ζ0)
∂
∂ζ0
∂U(ζ0)
∂ζ0dζ0, (B.17)
The last term on lhs can be maniplulated further by integrating by parts∫ LT
0U(ζ0)
∂
∂ζ0
∂U(ζ0)
∂ζ0dζ0 =
[V (ζ0)
∂2U(ζ0)
∂ζ02
]LT0
−∫ LT
0V (ζ0)
∂
∂ζ0
∂2U(ζ0)
∂ζ02dζ0, (B.18)
with
V (ζ0) =
∫U(ζ0)dζ0. (B.19)
The function U(ζ0) can conveniently be represented (See Identity(V)) as the series
sum,
U(ζ0) =∞∑m=1
e−πLTm
2R
(sinh
(2mϕ− 2iπmζ1
R
)− sinh
(2mϕ+ 2iπmζ1
R
))1− e−
πLTm
R
, (B.20)
– 20 –
with ϕ = π(LT−4ζ0)4R . Differentiating the last term in the above integral with respect to ζ0,
∂3U(ζ0)
∂ζ03= S1 + S2
=8π2
R2
∞∑m=1
m3e−πLTm
2R
1− e−πLTm
R
×(
sinh
(πm(LT − 4ζ0)
2R− 2iπmζ1
R
)− sinh
(πm(LT − 4ζ0)
2R+
2iπmζ1
R
)),
(B.21)
Expanding the denumerator,
S1 + S2 =4π3
R3
∞∑k=0
∞∑m=1
m3e−(2k+1)(πLTm)
2R
×
(sinh
(πm(LT − 4ζ0)
2R− 2iπmζ1
R
)− sinh
(πm(LT − 4ζ0)
2R+
2iπmζ1
R
)).
(B.22)
The series sum with some algebraic manipulation of hyperbolic functions reads
S1 + S2 =4π3
R3
∞∑k=0
∞∑m=1
m3
(− sinh
(πkmLTR
+2πmζ0
R+
2iπmζ1
R
)− sinh
(πkmLTR
+πmLTR
− 2πmζ0
R+
2iπmζ1
R
)+ sinh
(πkmLTR
+2πmζ0
R− 2iπmζ1
R
)+ sinh
(πkmLTR
+πmLTR
− 2πmζ0
R− 2iπmζ1
R
)+ cosh
(πkmLTR
+2πmζ0
R+
2iπmζ1
R
)+ cosh
(πkmLTR
+πmLTR
− 2πmζ0
R+
2iπmζ1
R
)− cosh
(πkmLTR
+2πmζ0
R− 2iπmζ1
R
)− cosh
(πkmLTR
+πmLTR
− 2πmζ0
R− 2iπmζ1
R
)),
(B.23)
with the first series sum
S1 =4π3
R3
∞∑k=0
∞∑m=1
m3
2
(− sinh
(πkmLTR
+2πmζ0
R+
2iπmζ1
R
)− sinh
(πkmLTR
+πLTm
R− 2πmζ0
R+
2iπmζ1
R
)+ sinh
(πkmLTR
+2πmζ0
R− 2iπmζ1
R
)+ sinh
(πkmLTR
+πLTm
R− 2πmζ0
R− 2iπmζ1
R
)),
= 0,
(B.24)
– 21 –
which identically vanishes.
The second series sum is
S2 =4π3
R3
( ∞∑m=1
m3
( ∞∑k=0
cosh
(πkLTm
R+
2πmζ0
R+
2iπmζ1
R
)
−∞∑k=1
cosh
(πkLTm
R− 2πmζ0
R− 2iπmζ1
R
))
−∞∑m=1
m3
( ∞∑k=0
cosh
(πkLTm
R+
2πmζ0
R− 2iπmζ1
R
)
+
∞∑k=1
cosh
(πkLTm
R− 2πmζ0
R+
2iπmζ1
R
))).
(B.25)
This can be shown after redefining the running indices m, k to yield a trivial value
S2 =4π3
R3
∞∑m=−∞
m3
( ∞∑k=0
cosh
(πkLTm
R+
2πmζ0
R+
2iπmζ1
R
)+
−∞∑k=−1
cosh
(−πkLTm
R− 2πmζ0
R− 2iπmζ1
R
)
−∞∑k=0
cosh
(πkLTm
R+
2πmζ0
R− 2iπmζ1
R
)−−∞∑k=−1
cosh
(−πkLTm
R− 2πmζ0
R+
2iπmζ1
R
))
=4iπ3
R3
∞∑m,k=−∞
m3
(cosh
(πkLTm
R+
2πmζ0
R+
2iπmζ1
R
)− cosh
(πkLTm
R+
2πmζ0
R− 2iπmζ1
R
))= 0.
(B.26)
The integral of the quadratic derivative can be read off from Eq.(B.17), Eq.(B.18) and
Eq.(B.23) as ∫ LT
0
(∂U(ζ0)
∂ζ0
)2
dζ0 =
[V (ζ0)
∂2U(ζ0)
∂ζ02+ U(ζ0)
∂U(ζ0)
∂ζ0
]LT0
. (B.27)
The sums over the hyperbolic functions can be represented using Jacobi elliptic θ
function (with the help of Identity(IV)-Eq.(A.8)). The functions U(ζ0) and the cor-
responding derivatives upon that are expressed using the complex differential operator∂∂z = R
π
(∂∂ζ1− i ∂
∂ζ0
)as follows
∂
∂ζ0U(ζ) =
iR
π
∂
∂ζ0
(1
ϑ1
∂ϑ1
∂ζ0
(−iπζ0
R+ πτ +
πζ1
R, q
)− 1
ϑ1
∂ϑ1
∂ζ0
(−iπζ0
R+ πτ − πζ1
R, q
)
+1
ϑ1
∂ϑ1
∂ζ1
(−iπζ0
R+ πτ +
πζ1
R, q
)+
1
ϑ1
∂ϑ1
∂ζ1
(−iπζ0
R+ πτ − πζ1
R, q
)),
(B.28)
– 22 –
with q = e−πLT
2R .
Substituting Eq.(B.27) and Eq.(B.28) into Eq.(B.15), the correlator assumes the form
W 21,b2 =
b2(D − 2)π
512R3σ2
{limζ1→0
+ limζ1→R
}[R
π
∂U(ζ0)
∂ζ0
{1
ϑ1
∂ϑ1
∂ζ0
(−iπζ0
R+ πτ +
πζ1
R, q
)− 1
ϑ1
∂ϑ1
∂ζ0
(−iπζ0
R+ πτ − πζ1
R, q
))−(
1
ϑ1
∂ϑ1
∂ζ1
(−iπζ0
R+ πτ +
πζ1
R, q
)
+1
ϑ1
∂ϑ1
∂ζ1
(−iπζ0
R+ πτ − πζ1
R, q
)}+R2
π2
∂2U(ζ)
∂ζ02
{log
ϑ1
(−iπζ0R + πτ + πζ1
R , q)
ϑ1
(−iπζ0R + πτ − πζ1
R , q)
+ log
(ϑ1
(−iπζ0
R+ πτ +
πζ1
R, q
)ϑ1
(−iπζ0
R+ πτ − πζ1
R, q
))}]LT0
.
(B.29)
The integration limits are evaluated making use of Eq.(A.13), Eq.(A.14) and Eq.(A.16)
of Identity(V) for the modular increase/decrease πτ of the argument of the logarithmic
derivatives, the correlator presently is cast to
W 21,b2 =
b2(D − 2)π
512R3σ2
{limζ1→0
+ limζ1→R
}(8R2
π2
ϑ′1ϑ1
(πζ1
R, q
))′. (B.30)
However, the divergent part of the square of the logarithmic derivatives at the poles ζ1 =
(0, R) can be regularized with ζ-function (using Identities(VII-X)) such that
ϑ′
ϑ(0, q)′ =
(4∞∑n=1
2nq2n
1− q2n+ csc2(ε)
),
=8π2
R2
(1
24(1− E2(τ)) +
1
12
).
(B.31)
The expectation value finally turn out to be
W 21,b2 =
πb2(D − 2)
4R3σ2
(1
8− 1
24E2(τ)
). (B.32)
B.2 The expectation value of W 22,b2
The next possible Wick-contraction of (B.1) yields the correlator
W 22,b2 = 〈(X.X)∂2
0∂21(X.X)〉. (B.33)
The integral over Green function after point splitting of the correlator
W 22,b2 = −(D − 2)b2
∫∂Σdζ ′0 lim
ζ′→ζ
[G(ζ; ζ ′)∂2
0∂21G(ζ ′; ζ)
]. (B.34)
– 23 –
Substituting the Green function free propagator (B.4) into above (B.34)
W 22,b2 =
−π2b2(D − 2)
R4σ2
∫∂Σ
limζ′→ζ
dζ0∑m=1
∑n=1
m3
n (qm − 1) (qn − 1)
× cos
(πmζ1
R
)sin
(πnζ1
R
)sin
(πnζ1′
R
)cos
(πmζ1′
R
)
× e−π(m+n)
(ζ0−ζ0
′)R
(qme
2πm(ζ0−ζ0
′)R + 1
)(qne
2πn(ζ0−ζ0
′)R + 1
).
(B.35)
With the substitutions
ζ1′ = ζ1 + ε; ζ0′ = LT − ζ0 − ε′; (B.36)
the expectation value becomes
W 22,b2 =
−π2b2(D − 2)
R4σ2
∫∂Σ
limε′→0ε→0
dζ0
{−m3 cos
(πmζ1
R
)sin(πnζ1
R
)cos(πm(ζ1+ε)
R
)sin(πn(ζ1+ε)
R
)n (qm − 1) (qn − 1)
× eπ(m+n)(LT−2t−ε′)
R
(qme
2πm(−LT+2ζ0+ε′)R + 1
)(qne
2πn(−LT+2ζ0+ε′)R + 1
)}.
(B.37)
Taking the limit ε→ 0, ε′ → 0 the expectation value in compact form reads
W 22,b2 = −π
2b2(D − 2)
R4σ2
∫∂Σdζ0 (S1S2) , (B.38)
with
S1 =∞∑m=1
m3csch
(πLTm
2R
)cos2
(πmζ1
R
)cosh
(πm(3LT − 4ζ0)
2R
), (B.39)
and
S2 =∞∑n=1
1
ncsch
(πLTn
2R
)sin2
(πnζ1
R
)cosh
(πn(3LT − 4ζ0)
2R
). (B.40)
Let us consider each sum in the multiplication separately, plugging the series expansion
of the hyperbolic function
csch
(πLm
2R
)= −2
∞∑k=0
e(2k+1)(πLTm)
2R , (B.41)
into (B.39) for the first sum S1. For convenience we write each term in the sum
separately
S1 =
( ∞∑k=0
∞∑m=1
(S1,1(m, k) + S1,2(m, k))
), (B.42)
– 24 –
such that
S1,1(m, k) = m3
[− 1
2cosh
(−πkLTm
R+ nπα1
)− 1
2cosh
(πkLTm
R+ nπα2
)− 1
4cosh
(−πkLTm
R+ nπβ1
)− 1
4cosh
(πkLTm
R+ nπβ2
)− 1
4cosh
(−πkLTm
R+ nπβ∗1
)− 1
4cosh
(πkLTm
R+ nπβ∗2
)],
(B.43)
and
S1,2(m, k) = m3
[− 1
2sinh
(πkLTm
R+ nπα2
)+
1
2sinh
(−πkLTm
R+ nπα1
)+
1
4sinh
(πkLTm
R+ nπβ∗2
)+
1
4sinh
(−πkLTm
R+ nπβ∗1
)− 1
4sinh
(πkLTm
R+ nπβ2
)+
1
4sinh
(−πkLTm
R+ nπβ1
)],
(B.44)
with
α1 =LTR− ζ0
R,α2 =
2LTR− 2ζ0
R,
β1 =2LTR− 2ζ0
R− 2iζ1
R, β2 =
LTR− 2ζ0
R− 2iζ1
R.
(B.45)
The sum of terms in the above matrices S1,1(m, k) = 0 and S1,2(m, k) = 0 vanishes term
by term ∀(m, k) in the sum.
The regularization of the first series in the sum proceeds as follows
∞∑k=0
∞∑m=1
S1,1(m, k) =∞∑k=0
∞∑m=1
m3 cosh
(πm(2k + 1)LT
2R
)(cosh
(πm(3LT − 4ζ0 + 4iζ1)
2R
)
+ cosh
(πm(3LT − 4ζ0)
2R
)),
(B.46)
where the sum over the running k produces divergent sum over zero
∞∑k=0
∞∑m=1
S1,1(m, k) =
∞∑m=1
m3
(1
4csch
(πLm
2R
)(cosh
(πm(3LT − 4ζ0 + 4iζ1)
2R
)+ cosh
(πm(3LT − 4ζ0)
2R
))
− 1
4csch
(πLm
2R
))(cosh
(πm(3LT − 4ζ0 + 4iζ1)
2R
)+ cosh
(πm(3LT − 4ζ0)
2R
)),
(B.47)
– 25 –
that is,
S1(m3, k)] =
( ∞∑k=0
∞∑m=1
(S1,1(m, k) + S1,2(m, k))
),
=ζ(∞),
=1.
(B.48)
Similarly the series S2,
S2 =
( ∞∑k=0
∞∑n=1
(S2,1(n, k) + S2,2(n, k))
), (B.49)
corresponds to the sum of the two terms. The sum over the index n can be encapsulated
in logarithmic form, the above two series become
∑n
S2,1(n, k) =1
4log
(2− 2 cosh
(−πkLTR
+ α1
))+
1
4log
(2− 2 cosh
(πkLTR
+ α2
))− 1
8log
(2− 2 cosh
(−πkLTR
+ β1
))− 1
8log
(2− 2 cosh
(πkLTR
+ β2
))− 1
8log
(2− 2 cosh
(πkLTR− β1
))− 1
8log
(2− 2 cosh
(πkLTR
+ β∗2
))(B.50)
and
∑n
S2,2(n, k) = −1
4log(−e
−πkLTR
+α1
)− 1
4log(−e−
πkLTR−α2
)+
1
8log(−e
−πkLTR
+β1)
− 1
8log(−e
πkLTR
+β2)− 1
8log(−e
πkLTR−β1)
+1
8log(−e
−πkLTR−β∗2)
(B.51)
The exponential form of the product over the hyperbolic functions,
S2 =− 1
4log
(− 1
16
∞∏k=0
(−1 + e−
π((k−1)LT+2(ζ0−iζ1))R
)(−1 + e−
π(−kLT+LT−2(ζ0+iζ1))
R
)×(−1 + e−
π(−(k+2)LT+2(ζ0+iζ1))
R
)(−1 + e−
π((k+2)LT−2ζ0+2iζ1)
R
))+
∞∑k=0
π(2k + 1)LT4R
+1
2log
(1
4
∞∏k=0
(e−
πkLR− 2πL
R+ 2πt
R − 1)(
1− e−πkLR
+πLR− 2πt
R
))(B.52)
enables to find a closed form using elliptic ϑ4 defined by Eq.(A.4). Using Eq.(A.24) of
Identity(XII) the infinite product within each logarithmic term in the above Eq.(B.52) is
pinned down to
– 26 –
S2 =1
4log
ϑ4
(i(LT π)
4R − i(πζ0)2R , e−
LT π
2R
)2
ϑ4
(iπLT4R + πζ1
R −iπζ0
R , e−LT π
2R
)ϑ4
(iπLT4R −
iπζ0
R − πζ1
R , e−LT π
2R
) . (B.53)
Transforming ϑ4(z, q) into ϑ1(z, q) (Eq.(A.25),Eq.(A.26)-Identities(XIII-XIV)) in (B.53)
then we have
W 22,b2 = −π
2b2(D − 2)
4R4σ2
∫∂Σdζ0 log
ϑ1
(i(πζ0)
2R , e−LT π
2R
)2
ϑ1
(iπζ0
R − πζ1
R , e−LT π
2R
)ϑ1
(iπζ0
R + πζ1
R , e−LT π
2R
)
(B.54)
The integral over the second term can be manipulated further with the use of Eq.(A.27)-
Identity (XV)-for the logarithm of the quotient of the two elliptic ϑ1 functions
W 22,b2 = −π
2b2(D − 2)
4R4σ2
(∫ LT
0log
ϑ1
(iπζ0
R + πζ1
R , e−LT π
2R
)ϑ1
(iπζ0
2R , e−LT π
2R
)+ log
ϑ1
(iπζ0
R − πζ1
R , e−LT π
2R
)ϑ1
(iπζ0
2R , e−LT π
2R
) ,
=
∫ LT
0
1
4
log
(1
csch2(πtR
)sin2
(πxR
)+ 1
)− 8
∞∑k=1
e−k(πL)R cosh
(2πktR
)sin2
(πkxR
)k(
1− e−k(πL)R
) dζ0
).
(B.55)
Integrating over ζ0
W 22,b2 = −πb2(D − 2)
4R3σ2
∞∑k=1
e−k(πL)R sinh
(2πkLR
)sin2
(πkxR
)k2(
1− e−k(πL)R
) (B.56)
The integration of the last term in the above equation (B.56) yields an expression in terms
of Jonquiere functions
W 22,b2 = −πb2(D − 2)
24R3σ2
(3 Li2
(eπ(L+2ix)
R
)+ 3 Li2
(eπ(L−2ix)
R
)+ 3 Li2
(e−
2π(L+ix)R
)+ 3 Li2
(e−
π(L+2ix)R
)+ 3 Li2
(e−
2π(L−ix)R
)+ 3 Li2
(e−
π(L−2ix)R
)− 6 Li2
(eLπR
)− 6 Li2
(e−
LπR
)− 6 Li2
(e−
2LπR
)+ 3 Li2
(e
2iπxR
)+ 3 Li2
(e−
2iπxR
)− π2
)(B.57)
The Jonquiere functions Li2(z) in the above Eq.(B.57) are regularized with the Epstein-
Hurwitz zeta regularization which is omplemented in the so-called inversion formula
Eq.(A.30)-Identity(XVIII) so that
– 27 –
R2b2 = −πb2(D − 2)
24R3σ2µ1(R,L, ε) (B.58)
with
µ1(R,L, ε) = −(− π2 +
−(2πi)2
2!
4∑j=1
sjB2
(1
2− log(−zj)
2πi
)− 6
(1
8− 1
24E2(2τ)
)ε2.
− 6Li2
(e−
2LT π
R
)+ 3Li2
(e−
2π(LT+iε)
R
)+ 3Li2
(e−
2π(LT−iε)R
))(B.59)
and
z1 = ei(τ+ εR
) and z2 = ei(τ+2 εR
), sj = 3 for j = 1, 2, and , z3 = eiτ , z4 = e2iπεR , sj = 6.
for j = 3, 4.
(B.60)
The expectation value Eq.(B.57) vanishes in the limit ζ1 → ε unless for a finite cutoff ε
corresponding to the radius of small neighborhood of the Polyakov lines at the boundaries.
The UV cutoff is proportional to R0 = − log ε. The correlator at ζ1 = ε contributes to the
renormalization of the coupling b2.
C. Appendix(C): Mean-square width W 2b4
The next non-vanishing Lorentz invariant boundary term has a width expectation value
given by
W 2b4 = 〈X2Sb4〉
= b4〈(X ·X) ∂20∂1X · ∂2
0∂1X)〉.(C.1)
This expectation value has two Wick contractions that we work out separately in detail
in the following.
C.1 The expectation value of W 41,b4
The Wick contraction assumes the form
W 21,b4 = 〈∂2
0∂1(X ·X)∂20∂1(X ·X)〉 (C.2)
After point-splitting,
ζ1′ = ζ1 + ε,
ζ0′ = LT − ζ0 − ε′,(C.3)
the correlator in terms of Green function
W 21,b4 =
∫∂Σdζ0∂2
0∂1G(ζ0, ζ1; ζ0′ , ζ1′
)∂2
0′∂1′G(ζ0, ζ1; ζ0′ , ζ1′
), (C.4)
– 28 –
Substituting the free propagator Eq.(4.7) into Eq.(C.4) and taking the limits the ex-
pectation value would be
W 21,b4 = −π
4b4(D − 2)
4R6σ2
∫∂Σdζ0
∑m
1
(qm − 1)
(m2 sin
(2πmζ1
R
)(qme
πm(LT−2ζ0)
R
+ e−πm(LT−2ζ0)
R
))∑n
1
(qn − 1)
(n2 sin
(2πnζ1
R
)(qne−
πn(LT−2ζ0)
R + eπn(LT−2ζ0)
R
)),
(C.5)
This results in an integrand with two symmetric series
S =
∞∑m=1
m2 sin(
2πmζ1
R
)(qme
πm(LT−2ζ0)
R + e−πm(LT−2ζ0)
R
)qm − 1
,
=∞∑m=1
−m2csch
(πLTm
2R
)sin
(2πmζ1
R
)cosh
(πm(LT − 4ζ0)
2R
),
= −i∞∑m=1
m2e−πLTm
2R
(sinh
(2mϕ− 2iπmζ1
R
)− sinh
(2mϕ+ 2iπmζ1
R
))1− e−
πLTm
R
,
(C.6)
with ϕ = π(LT−4ζ0)4R . The sum can be represented as a derivative
S =−iR2
4π2
∂2
∂ζ0∂ζ0
∞∑m=1
e−πLTm
2R
(sinh
(2mϕ− 2iπmζ1
R
)− sinh
(2mϕ+ 2iπmζ1
R
))1− e−
πLTm
R
, (C.7)
Let us define the auxiliary function
U(ζ0) =∂
∂ζ0
∞∑m=1
e−πLTm
2R
(sinh
(2mϕ− 2iπmζ1
R
)− sinh
(2mϕ+ 2iπmζ1
R
))1− e−
πLTm
R
, (C.8)
The expectation value of the correlator Eq.(C.9) is obtained as an integral with respect
to ζ0 over the square of the sum S, using Eq.(C.8) this reads
W 21,b4 =
b4(D − 2)
64R2σ2
∫∂Σdζ0
(∂
∂ζ0U(ζ0, ζ1)
)2
. (C.9)
The above integrand may become cumbersome in evaluation since it involves two series
multiplication. In the following the square of the derivative of U(ζ0, ζ1) can be manipulated
with the help the relation
2
∫ LT
0
(∂U(ζ0)
∂ζ0
)2
dζ0 =
∫ LT
0
∂
∂ζ0
∂U(ζ0)2
∂ζ0dζ0 − 2
∫ LT
0U(ζ0)
∂
∂ζ0
∂U(ζ0)
∂ζ0(C.10)
– 29 –
Differentiating the last term in the above integral with respect to ζ0,
S3 =∂
∂ζ0
∂
∂ζ0U(ζ0, ζ1)
=4iπ2
R2
∞∑m=1
m3e−πLTm
2R
1− e−πLTm
R
×(
sinh
(πm(LT − 4ζ0)
2R− 2iπmζ1
R
)− sinh
(πm(LT − 4ζ0)
2R+
2iπmζ1
R
)),
(C.11)
Expanding the denumerator we found that,
S3 =4iπ2
R2
∞∑k=0
∞∑m=1
m3e−(2k+1)(πLTm)
2R
×
(sinh
(πm(LT − 4ζ0)
2R− 2iπmζ1
R
)− sinh
(πm(LT − 4ζ0)
2R+
2iπmζ1
R
)).
= 0.
(C.12)
since series sum Eq.(C.12) assumes the same form as Eq.(B.23) which yields a trivial
contribution.
The integral over the square derivative of U(ζ0) is now outright to evaluate in Eq.(C.10),
substituting the square derivative in Eq.(C.9) and using Eq.(A.26)-Identity(XIV) to con-
veniently express the series sums in terms of Jacobi function ϑ1, the correlator after inte-
gration becomes
W 21,b4 =
b4(D − 2)
1024R2σ2
{limζ1→0
+ limζ1→R
}[∂
∂ζ0
(∂
∂ζ0
ϑ′1ϑ1
(iφ, e−
LT π
2R
)− ∂
∂ζ0
ϑ′1ϑ1
(iφ∗, e−
LT π
2R
))2]LT
0
.
(C.13)
Differentiating with respect to ζ0
W 21,b4 =− π3(D − 2)b4
512R5σ2
{limζ1→0
+ limζ1→R
}[(ϑ′′1ϑ1
(iφ, q)− ϑ′1ϑ1
(iφ, q)2
)(ϑ′′′1ϑ1
(iφ, q)− 3ϑ′′1ϑ1
(iφ, q)
ϑ′1ϑ1
(iφ, q)− 2ϑ′1ϑ1
(iφ, q)3
)]φ2φ1
,
(C.14)
Making use of the properties of Jacobi θ function-Identities(VI-XI), and evaluating the
integration limits corresponding to φ1 = iπLT2R + iπζ1
R and φ2 = −iπLT2R + iπζ1
R . The correlator
assumes the form
W 21,b4 = −π
3(D − 2)b4512R5σ2
{limζ1→0
+ limζ1→R
}[− iϑ
′1
ϑ1
(πζ1
R, q
)′(−96i+ 48i
ϑ′1ϑ1
(πζ1
R, q
)2
− 24iϑ′′1ϑ1
(πζ1
R, q
))].
(C.15)
– 30 –
The above is the expectation value in the limit of infinitesmal ε at the end points
ζ1 → (0, R). The correlator diverges at both limits, the regularization of the infinities
proceeds by rewritting Eq.(C.15) accordingly with the series representations of logarithmic
derivative of elliptic θ functions, using Eq.(A.19) this would read
W 21,b4 =
−π3(D − 2)b4265R5σ2
[(8
∞∑k=1
kq2k cos(2k πζ
1
R )
1− q2k− csc2(
πζ1
R)
)(− 96− 24
(16
∞∑n=1
qn
(1− qn)2
+ 8
∞∑n=1
nqn
1− qn− 1
)+ 48
(8
∞∑k=1
kq2k cos(2k πζ
1
R )
1− q2k− csc2(
πζ1
R)
))].
(C.16)
The ζ function of the divergent series in Eq. (C.16) (Identities(VIII)) yields a finite
value of the correlator which turns out to be
W 21,b4 =
π3(D − 2)b432R5σ2
(E2(τ)− 5
4
)(11E2(τ/2)
36− 5E2(τ)
9− 55
12
)(C.17)
C.2 The expectation value of W 22,b4
The second Wick contraction of (C.1) is
W 22,b4 = 〈(X ·X) ∂2
0∂1∂20∂1 (X ·X)〉 (C.18)
The point-split correlator in terms of Green functions reads
W 22,b4 =
∫∂Σdζ0G
(ζ0, ζ1; ζ0′ , ζ1′
)∂2
0∂1∂20′∂1′G
(ζ0, ζ1; ζ0′ , ζ1′
). (C.19)
Substituting the free propagator Eq.(4.7) into the above Eq.(C.19), the expectation value
is
W 22,b4 =
∫∂Σdζ0
(π4m5
nR6σ2 (qm − 1) (qn − 1)e−
π(m+n)(ζ0−ζ0
′)R
(qme
2πm(ζ0−ζ0
′)R + 1
)(qne
2πn(ζ0−ζ0
′)R + 1
)cos
(πmζ1′
R
)cos
(πmζ1
R
)sin
(πnζ
R
)sin
(πnζ1′
R
)).
(C.20)
the point-splitting implies the infinitesimal transform
ζ1′ = ζ1 + ε,
ζ0′ = LT − ζ0 − ε′.(C.21)
– 31 –
Substituting into Eq.(C.20) and considering the limits ε → 0 and ε′ → 0, the expectation
value becomes
W 22,b4 =
∫∂Σdζ0
(π4m5
nR6σ2 (qm − 1) (qn − 1)e−
2πm(LT−2ζ0)
R
(e
2πm(LT−2ζ0)
R + qm)
(qne
π(m−n)(LT−2ζ0)
R + eπ(m+n)(LT−2ζ0)
R
)cos2
(πmζ1
R
)sin2
(πnζ1
R
)).
(C.22)
Consider each sum over the separate indicies in the above integral, the first series
S1 =∞∑m=1
1
q2m − 1
(m5e−
2mπ(LT−2ζ0)
R emπ(LT−2ζ0)
R cos2
(mπζ1
R
)q2m + e
2πm(LT−2ζ0)
R
),
= m5csch
(πmLT
2R
)cos2
(πmζ1
R
)cosh
(πm(3LT − 4ζ0)
2R
),
(C.23)
have similar functional form to Eq.(B.48) with the replacement m5 → m3 in Eq.(B.43) and
Eq.(B.44). Similar ζ regularization procedures allows to reduce the series into the
S1 =∞∑k=0
∞∑m=1
(S1,1(m5, k) + S1,2(m5, k)),
=ζ(∞),
=1.
(C.24)
The second series
S2 =∞∑n=1
1
ncsch
(πnLT
2R
)sin2
(πnζ1
R
)cosh
(πn(3LT − 4ζ0)
2R
). (C.25)
assumes the same form as Eq.(B.40). Substituting Eq.(C.25) into Eq.(C.22) and integrating
over ζ0 yields a contribution to renormalisation R2b4
W 22,b4 = −(D − 2)π4LT b4
24R5σ2
(3 Li2
(eπ(LT+2iε)
R
)+ 3 Li2
(eπ(LT−2iε)
R
)+ 3 Li2
(e−
2π(LT+iε)
R
)+ 3 Li2
(e−
π(LT+2iε)
R
)+ 3 Li2
(e−
2π(LT−iε)R
)+ 3 Li2
(e−
π(LT−2iε)
R
)− 6 Li2
(eLT π
R
)− 6 Li2
(e−
LT π
R
)− 6 Li2
(e−
2LT π
R
)+ 3 Li2
(e
2iπεR
)+ 3 Li2
(e−
2iπεR
)− π2
)(C.26)
R2b4 = −πb2(D − 2)
R5σ2µ2(R,L, ε) (C.27)
– 32 –
with
µ2(R,L, ε) =−1
24
(− π2 +
−(2πi)2
2!
4∑j=1
sjB2
(1
2− log(−zj)
2πi
)
− 6Li2
(e−
2LT π
R
)+ 3Li2
(e−
2π(LT+iε)
R
)+ 3Li2
(e−
2π(LT−iε)R
))+
1
32
(E2(τ)− 5
4
)(11E2(τ/2)
36− 5E2(τ)
9− 55
12
)ε2
(C.28)
and
sj = 3, z1 = ei(τ+ εR
) and z2 = ei(τ+2 εR
) for j = 1, 2, and , sj = 6, z3 = eiτ , z4 = e2iπεR ,
for j = 3, 4.
(C.29)
References
[1] Creutz M (1980) Monte carlo study of quantized SU(2) gauge theory. Phys. Rev.
D21:2308–2315.
[2] Creutz M, Jacobs L, Rebbi C (1983) Monte carlo computations in lattice gauge the-
ories. Phys. Rept. 95:201.
[3] Creutz M (1981) Monte carlo study of renormalization in lattice gauge theory. Phys.
Rev. D23:1815.
[4] Creutz M (1980) Quark confinement. Madison H.E. Physics 296.
[5] M. Fukugita and T. Niuya (1983), Distribution of Chromoelectric Flux in SU(2)
Lattice Gauge Theory. Phys. Lett. 132B, 374 .
[6] J. W. Flower and S. W. Otto (1985), The Field Distribution in SU(3) Lattice Gauge
Theory. Phys. Lett. 160B, 128 .
[7] J. Wosiek and R. W. Haymaker (1987), On the Space Structure of Confining Strings.
Phys. Rev. D 36, 3297.
[8] R. Sommer (1987), Chromoflux Distribution in Lattice QCD. Nucl. Phys. B 291,
673.
[9] A. Di Giacomo, M. Maggiore and S. Olejnik (1990), Evidence for Flux Tubes From
Cooled QCD Configurations. Phys. Lett. B 236, 199.
[10] A. Di Giacomo, M. Maggiore and S. Olejnik (1990), Confinement and Chromoelectric
Flux Tubes in Lattice QCD. Nucl. Phys. B 347, 441.
[11] G. S. Bali, K. Schilling and C. Schlichter (1995), Observing long color flux tubes in
SU(2) lattice gauge theory. Phys. Rev. D 51, 5165 [hep-lat/9409005].
– 33 –
[12] R. W. Haymaker, V. Singh, Y. C. Peng and J. Wosiek (1996), Distribution of the
color fields around static quarks: Flux tube profiles. Phys. Rev. D 53, 389 [hep-
lat/9406021].
[13] P. Cea and L. Cosmai (1995), Dual superconductivity in the SU(2) pure gauge vacuum:
A Lattice study. Phys. Rev. D 52, 5152 [hep-lat/9504008].
[14] F. Okiharu and R. M. Woloshyn (2004), A Study of color field distributions in the
baryon. Nucl. Phys. Proc. Suppl. 129, 745 [hep-lat/0310007].
[15] Alford MG, Good G (2008) Flux tubes and the type-i/type-ii transition in a super-
conductor coupled to a superfluid. Phys. Rev. B 78(2):024510.
[16] Kasamatsu K, Tsubota M (2007) Quantized vortices in atomic Bose-Einstein con-
densates. ArXiv e-prints.
[17] Nielsen H, Olesen P (1973) Vortex-line models for dual strings. Nuclear Physics B
61:45 – 61.
[18] Lo AS, Wright EL (2005) Signatures of cosmic strings in the cosmic microwave back-
ground.
[19] Callan CG, Coleman S, Wess J, Zumino B (1969) Structure of phenomenological
lagrangians. ii. Phys. Rev. 177(5):2247–2250.
[20] Jaimungal S, Semenoff GW, Zarembo K (1999) Universality in effective strings. JETP
Lett. 69:509–515.
[21] Caselle M, Feo A, Panero M, Pellegrini R (2011) Universal signatures of the effective
string in finite temperature lattice gauge theories. JHEP 04:020.
[22] Alvarez O (1981) The Static Potential in String Models. Phys. Rev. D24:440.
[23] Arvis JF (1983) The Exact qq Potential in Nambu String Theory. Phys. Lett.
127B:106–108.
[24] Billo M, Caselle M, Pellegrini R (2012) New numerical results and novel effective
string predictions for Wilson loops. JHEP 01:104. [Erratum: JHEP04,097(2013)].
[25] Caselle M, Hasenbusch M, Panero M (2005) Comparing the Nambu-Goto string with
lgt results. JHEP. 03(hep-lat/0501027. DFTT-2005-04. DIAS-STP-2005-01. IFUP-
TH-2005-04):026. 25 p.
[26] Juge KJ, Kuti J, Morningstar C (2003) Fine structure of the QCD string spectrum.
Phys. Rev. Lett. 90:161601.
[27] Dass NH, Majumdar P (2008) Continuum limit of string formation in 3d su(2) lgt.
Phys. Lett. B 658(5):273 – 278.
– 34 –
[28] Caselle M, Panero M, Vadacchino D (2016) Width of the flux tube in compact U(1)
gauge theory in three dimensions. JHEP 02:180.
[29] Caselle M, Panero M, Provero P, Hasenbusch M (2003) String effects in Polyakov
loop correlators. Nucl. Phys. Proc. Suppl. 119:499–501.
[30] Pennanen P, Green AM, Michael C (1997) Flux-tube structure and beta-functions in
SU(2). Phys. Rev. D56:3903–3916.
[31] Brandt BB, Meineri M (2016) Effective string description of confining flux tubes. Int.
J. Mod. Phys. A31(22):1643001.
[32] Gliozzi F (1994) Quantum behavior of the flux tube: A Comparison between QFT
predictions and lattice gauge theory simulations.
[33] Jahn O, Forcrand PD (2004) Baryons and confining strings. Nucl. Phys. B - Proc.
Suppl. 129-130:700 – 702. Lattice 2003.
[34] M. Pfeuffer, G.S.Bali, M.Panero, Fluctuations of the baryonic flux-tube
junction from effective string theory , Phys. Rev. D79 (2009) 025022.
https://arxiv.org/abs/0810.1649.
[35] de Forcrand P, Jahn O (2005) The baryon static potential from lattice QCD. Nucl.
Phys. A755:475–480.
[36] N. Battelli and C. Bonati (2019), Color flux tubes in SU(3) Yang-Mills theory:
an investigation with the connected correlator Phys. Rev. D 99 no.11, 114501
[arXiv:1903.10463 [hep-lat]].
[37] Bakry AS, Chen X, Zhang PM (2016) Confining potential of Y-string on the lattice
at finite T. EPJ Web Conf. 126:05001.
[38] Bakry AS, Chen X, Zhang PM (2015) Y-stringlike behavior of a static baryon at
finite temperature. Phys. Rev. D91:114506.
[39] Caselle M, Gliozzi F, Magnea U, Vinti S (1996) Width of long colour flux tubes in
lattice gauge systems. Nucl. Phys. B460:397–412.
[40] Bonati C (2011) Finite temperature effective string corrections in (3+1)D SU(2)
lattice gauge theory. Phys.Lett. B703:376–378.
[41] Hasenbusch M, Marcu M, Pinn K (1994) High precision renormalization group study
of the roughening transition. Physica A: Statistical Mechanics and its Applications
208(1):124 – 161.
[42] Caselle M, Hasenbusch M, Panero M (2006) High precision Monte Carlo simulations
of interfaces in the three-dimensional ising model: A Comparison with the Nambu-
Goto effective string model. JHEP 03:084.
– 35 –
[43] Bringoltz B, Teper M (2008) Closed k-strings in SU(N) gauge theories : 2+1 dimen-
sions. Phys. Lett. B663:429–437.
[44] Athenodorou A, Bringoltz B, Teper M (2009) On the spectrum of closed k = 2 flux
tubes in D=2+1 SU(N) gauge theories. JHEP 05:019.
[45] Hari Dass ND, Majumdar P (2006) String-like behaviour of 4-D SU(3) Yang-Mills
flux tubes. JHEP 10:020.
[46] Giudice P, Gliozzi F, Lottini S (2007) Quantum broadening of k-strings in gauge
theories. JHEP 01:084.
[47] Luscher M, Weisz P (2004) String excitation energies in SU(N) gauge theories beyond
the free-string approximation. JHEP 07:014.
[48] Pepe M (2010) String effects in Yang-Mills theory. PoS LATTICE2010:017.
[49] Luscher M, Munster G, Weisz P (1981) How Thick Are Chromoelectric Flux Tubes?
Nucl. Phys. B180:1–12.
[50] Caselle M (2010) Flux tube delocalization at the deconfinement point. JHEP 08:063.
[51] Bakry AS, , et al. (2010,hep-lat/1004.0782) String effects and the distribution of the
glue in static mesons at finite temperature. Phys. Rev. D 82(9):094503.
[52] Bakry AS, Leinweber DB, Williams AG (2012) Bosonic stringlike behavior and the
Ultraviolet filtering of QCD. Phys.Rev. D85:034504.
[53] Bicudo P, Cardoso N, Cardoso M (2017) Pure gauge QCD flux tubes and their widths
at finite temperature.
[54] Pisarski RD, Alvarez O (1982) Strings at finite temperature and deconfinement. Phys.
Rev. D 26(12):3735–3737.
[55] Cardoso N, Bicudo P (2012) Lattice qcd computation of the su(3) string tension
critical curve. Phys. Rev. D 85(7):077501.
[56] Kaczmarek O, Karsch F, Laermann E, Lutgemeier M (2000) Heavy quark potentials
in quenched qcd at high temperature. Phys. Rev. D 62(3):034021.
[57] Bakry A, et al. (2017) Stiff self-interacting string near QCD deconfinement point.
[58] Dietz K, Filk T (1983) Renormalization of string functionals. Phys. Rev. D
27(12):2944–2955.
[59] Aharony O, Field M (2011) On the effective theory of long open strings. JHEP
01:065.
[60] Billo M, Caselle M, Gliozzi F, Meineri M, Pellegrini R (2012) The Lorentz-invariant
boundary action of the confining string and its universal contribution to the inter-
quark potential. JHEP 05:130.
– 36 –
[61] Caselle M, Pepe M, Rago A (2004) Static quark potential and effective string correc-
tions in the (2+1)-d SU(2) Yang-Mills theory. JHEP 10:005.
[62] Caselle M, Hasenbusch M, Panero M (2005) Comparing the Nambu-Goto string with
LGT results. JHEP 03:026.
[63] Caselle M, et al. (1994) Rough interfaces beyond the Gaussian approximation. Nucl.
Phys. B432:590–620.
[64] Caselle M, Hasenbusch M, Panero M (2004) Short distance behavior of the effective
string. JHEP 05:032.
[65] Gliozzi F, Pepe M, Wiese UJ (2010) The Width of the Confining String in Yang-Mills
Theory. Phys. Rev. Lett. 104:232001.
[66] Athenodorou A, Bringoltz B, Teper M (2011) Closed flux tubes and their string
description in D=3+1 SU(N) gauge theories. JHEP 02:030.
[67] Athenodorou A, Bringoltz B, Teper M (2011) Closed flux tubes and their string
description in D=2+1 SU(N) gauge theories. JHEP 05:042.
[68] Caselle M, Nada A, Panero M (2015) Hagedorn spectrum and thermodynamics of
SU(2) and SU(3) Yang-Mills theories. JHEP 07:143. [Erratum: JHEP11,016(2017)].
[69] Giddings SB (1989) Strings at the hagedorn temperature. Physics Letters B 226(1):55
– 61.
[70] Bali G, et al. (2013) The meson spectrum in large-N QCD. [PoSConfine-
mentX,278(2012)].
[71] Kalashnikova YuS (2002) Open questions of meson spectroscopy: Lattice, data, phe-
nomenology in Quarks. Proceedings, 12th International Seminar on High Energy
Physics, Quarks’2002, Novgorod, Russia, June 1-7, 2002.
[72] Grach IL, Narodetskii IM, Trusov MA, Veselov AI (2008) Heavy baryon spectroscopy
in the QCD string model in Particles and nuclei. Proceedings, 18th International
Conference, PANIC08, Eilat, Israel, November 9-14, 2008.
[73] Caselle M, Pellegrini R (2013) Finite-Temperature Behavior of Glueballs in Lattice
Gauge Theories. Phys. Rev. Lett. 111(13):132001.
[74] Johnson RW, Teper MJ (2002) String models of glueballs and the spectrum of SU(N)
gauge theories in (2+1)-dimensions. Phys. Rev. D66:036006.
[75] Kalaydzhyan T, Shuryak E (2014) Self-interacting QCD strings and string balls.
Phys. Rev. D90(2):025031.
[76] Makeenko Y (2011) Effective String Theory and QCD Scattering Amplitudes. Phys.
Rev. D83:026007.
– 37 –
[77] Dubovsky S, Flauger R, Gorbenko V (2012) Effective String Theory Revisited. JHEP
09:044.
[78] Ambjorn J, Makeenko Y, Sedrakyan A (2014) Effective QCD string beyond the
Nambu-Goto action. Phys. Rev. D89(10):106010.
[79] Polyakov A (1986) Fine structure of strings. Nuclear Physics B 268(2):406 – 412.
[80] Kleinert H (1986) The Membrane Properties of Condensing Strings. Phys. Lett.
B174:335–338.
[81] Caselle M, Panero M, Pellegrini R, Vadacchino D (2015) A different kind of string.
JHEP 01:105.
[82] Brandt BB (2017) Spectrum of the open QCD flux tube and its effective string
description I: 3d static potential in SU(N = 2,3). JHEP 07:008.
[83] Aharony O, Klinghoffer N (2010) Corrections to Nambu-Goto energy levels from the
effective string action. JHEP 12:058.
[84] S. Hellerman and I. Swanson (2017), Boundary Operators in Effective String Theory.
JHEP 1704 085. [arXiv:1609.01736 [hep-th]].
[85] Bakry AS, et al. (2018) Stiff self-interacting strings at high temperature QCD. EPJ
Web Conf. 175:12004.
[86] Brandt BB (2011) Probing boundary-corrections to Nambu-Goto open string energy
levels in 3d SU(2) gauge theory. JHEP 02:040.
[87] Olesen P (1985) Strings and QCD. Phys. Lett. B160:144–148.
[88] Mandelstam S (1976) Vortices and quark confinement in nonabelian gauge theories.
Phys. Rept. 23:245–249.
[89] Bali GS, Bornyakov V, Muller-Preussker M, Schilling K (1996) Dual superconductor
scenario of confinement: A Systematic study of Gribov copy effects. Phys. Rev.
D54:2863–2875.
[90] Di Giacomo A, Lucini B, Montesi L, Paffuti G (2000) Colour confinement and dual
superconductivity of the vacuum. I. Phys. Rev. D61:034503.
[91] Di Giacomo A, Lucini B, Montesi L, Paffuti G (2000) Colour confinement and dual
superconductivity of the vacuum. II. Phys. Rev. D61:034504.
[92] Carmona JM, D’Elia M, Di Giacomo A, Lucini B, Paffuti G (2001) Color confinement
and dual superconductivity of the vacuum. III. Phys. Rev. D64:114507.
[93] Bukenov AK, Pochinsky AV, Polikarpov MI, Polley L, Wiese UJ (1993) Cosmic
strings on the lattice. Nucl. Phys. Proc. Suppl. 30:723–726.
– 38 –
[94] Hindmarsh M, Rummukainen K, Tenkanen TVI, Weir DJ (2014) Improving cos-
mic string network simulations. Phys. Rev. D90(4):043539. [Erratum: Phys.
Rev.D94,no.8,089902(2016)].
[95] Vyas V (2010) Intrinsic Thickness of QCD Flux-Tubes.
[96] Caselle M, Grinza P (2012) On the intrinsic width of the chromoelectric flux tube in
finite temperature LGTs. JHEP 1211:174.
[97] Goddard P, Goldstone J, Rebbi C, Thorn C (1973) Quantum dynamics of a massless
relativistic string. Nuclear Physics B 56(1):109 – 135.
[98] Low I, Manohar AV (2002) Spontaneously broken space-time symmetries and Gold-
stone’s theorem. Phys. Rev. Lett. 88:101602.
[99] Cohn JD, Periwal V (1993) Lorentz invariance of effective strings. Nucl. Phys.
B395:119–128.
[100] Polchinski J, Strominger A (1991) Effective string theory. Phys. Rev. Lett.
67(13):1681–1684.
[101] Luscher M, Symanzik K, Weisz P (1980) Anomalies of the free loop wave equation
in the WKB approximation. Nucl. Phys. B173:365.
[102] Luscher M, Weisz P (2002) Quark confinement and the bosonic string. JHEP 07:049.
[103] Meyer HB (2006) Poincare invariance in effective string theories. JHEP 05:066.
[104] Creutz M (1980) Monte carlo study of quantized SU(2) gauge theory. Phys. Rev.
D21:2308–2315.
[105] Isham C, Salam A, Strathdee J (1971) Nonlinear realizations of space-time symme-
tries. scalar and tensor gravity. Annals of Physics 62(1):98 – 119.
[106] Gliozzi F (2011) Dirac-Born-Infeld action from spontaneous breakdown of Lorentz
symmetry in brane-world scenarios. Phys. Rev. D84:027702.
[107] Aharony O, Dodelson M (2012) Effective String Theory and Nonlinear Lorentz In-
variance. JHEP 02:008.
[108] Aharony O, Karzbrun E (2009) On the effective action of confining strings. JHEP
06:012.
[109] Meyer HB (2006) Poincar invariance in effective string theories. Journal of High
Energy Physics 2006(05):066.
[110] Gliozzi F, Pepe M, Wiese UJ (2010) The Width of the Color Flux Tube at 2-Loop
Order. JHEP 11:053.
– 39 –
[111] Bakry AS, Chen X, Deliyergiyev M, Galal A, Zhang PM (2017) Perturbative width
of open rigid-strings.
[112] Allais A, Caselle M (2009) On the linear increase of the flux tube thickness near the
deconfinement transition. JHEP 01:073.
[113] Caselle M, Zago M (2011) A new approach to the study of effective string corrections
in LGTs. Eur. Phys. J. C71:1658.
[114] Bakry AS, Leinweber DB, Williams AG (2011) On the ground state of Yang-Mills
theory. Annals Phys. 326:2165–2173.
[115] Bakry AS, Leinweber DB, Moran PJ, Sternbeck A, Williams AG (2010) String ef-
fects and the distribution of the glue in mesons at finite temperature. Phys. Rev.
D82:094503.
[116] Bakry AS, Leinweber DB, Williams AG (2015) Gluonic profile of the static baryon
at finite temperature. Phys. Rev. D91:094512.
[117] Bakry AS, Leinweber DB, Williams AG (2011) The thermal delocalization of the flux
tubes in mesons and baryons. AIP Conference Proceedings 1354(1):178–183.
[118] Bicudo P, Cardoso M, Cardoso N (2018) Colour fields of the quark-antiquark excited
flux tube. EPJ Web Conf. 175:14009.
[119] Majumdar P (2003) The string spectrum from large Wilson loops. Nucl. Phys.
B664:213–232.
[120] C. Bonati, S. Cal, M. D’Elia, M. Mesiti, F. Negro, A. Rucci and F. Sanfilippo
(2018)Effects of a strong magnetic field on the QCD flux tube. Phys. Rev. D 98
no.5, 054501 [arXiv:1807.01673 [hep-lat]].
[121] Alexandrou C, de Forcrand P, Jahn O (2003) The ground state of three quarks. Nucl.
Phys. Proc. Suppl. 119:667–669.
[122] Alexandrou C, De Forcrand P, Tsapalis A (2002) The static three-quark SU(3) and
four-quark SU(4) potentials. Phys. Rev. D65:054503.
[123] O. Borisenko, V. Chelnokov, E. Mendicelli and A. Papa (2019) Three-quark potentials
in an SU(3) effective Polyakov loop model. Nucl. Phys. B 940 214 [arXiv:1812.05384
[hep-lat]].
[124] Bissey F, , et al. (2007) Gluon flux-tube distribution and linear confinement in
baryons. Phys. Rev. D 76:114512. hep-lat/0606016.
[125] Koma Y, Koma M (2017) Precise determination of the three-quark potential in SU(3)
lattice gauge theory. Phys. Rev. D95(9):094513.
– 40 –
[126] Whittaker ET, Watson GN (1996) A Course of Modern Analysis, Cambridge Math-
ematical Library. (Cambridge University Press), 4 edition.
[127] Yaling M, Jiaolian Z (2015) A Logarithmic Derivative of Theta Function and Impli-
cation. Pure and Applied Mathematics Journal. Special Issue: Mathematical Aspects
of Engineering Disciplines. Vol. 4, No. 5-1:55–59.
[128] Shen L (1993) On the logarithmic derivative of a theta function and a fundamental
identity of ramanujan. Journal of Mathematical Analysis and Applications 177(1):299
– 307.
[129] Abramowitz M, Stegun IA (1964) Handbook of Mathematical Functions with For-
mulas, Graphs, and Mathematical Tables. (Dover, New York), ninth dover printing,
tenth gpo printing edition.
[130] Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG (1953) Higher Transcendental
Functions. (McGraw-Hill, New York) Vol. II.
– 41 –