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: maksym.deliyergiyev@ujk.edu.pl

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)

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