Perturbative width of open rigid-strings A. S. Bakry,1, ∗ X. Chen,1 M. Deliyergiyev,1 A. Galal,1 and P.M. Zhang1 1
Institute of Modern Physics, Chinese Academy of Sciences, Gansu 730000, China (Dated: September 28, 2017)
We present perturbative calculation of the width of the energy profile of stiff strings up to two loops in D dimensions. The perturbative expansion of the extrinsic curvature term signifying the rigidity of the string in Polyakov-Kleinert action is taken around the free Nambu-Goto string. The mean-square width of the string field is derived for open strings with Dirichlet boundary condition.
arXiv:1709.09446v1 [hep-th] 27 Sep 2017
Keywords: QCD Strings, Polyakov-Kleinert String, Stiff Strings, Extrinsic Curvature, Geometrical Quantum Strings, Nambu-Goto String, String Width
An interesting generalization of the Nambu-Goto string has been proposed by Polyakov [1] and Kleinert [2] to stabilize the Nambu-Goto (NG) action. The surface representation of the Polyakov-Kleinert (PK) string depends on the area as well as the geometrical configuration of the embedded sheet in the space-time. This is a bosonic string with an extrinsic curvature as a next-order operator after NG action [1, 2]. Polyakov-Kleinert string theory provides a QCD model of smooth flux sheets over long distances same as the ordinary Nambu-Goto string, however, with sharply creased surfaces filtered out. The parameterised extrinsic-curvature term weighs favorably smooth string configurations over those that are sharply curved. The model is relevant to QCD main features with an ultraviolet (UV) freedom and infrared (IR) confinement properties [1, 2] and is consistent with glueballs [3, 4] formation, in addition to a possible tachyonic free spectrum [5] ¯ potential [6–8]. above some critical coupling [4] and a real (QQ) With these attractive features, the theory induced research effort to well-found the relevant properties such its perturbative stability in critical dimensions [9], the dynamical generation of the string tension [10] and the exact potential in the large dimension D limit [11, 12]. This is in conjunction with revealing the novel thermodynamic characteristics such as the deconfinement transition point [13], partition function [14], free energy and string tension at finite temperature [15–17]. Recently, rigid string effects have been reported in the numerical simulations of both the confining potential and width of the abelian U (1) compact gauge group [18, 19]. The rigidity effects are conjectured to be significant in deviations from the NG string in the quenched QCD [20] just before deconfinement together with possible manifestation in the IR region of SU (N ) gauge theories in 3D [21]. This calls to extend the discussion to identify the relevant properties of the energy-field distribution by virtue of the rigidity effects. The target of this paper is to analytically evaluate the mean-square width of the Dirichlet string on a cylinder which is very relevant to the energy fields in static mesonic configurations. In the following we consider the perturbative expansion of the extrinsic-curvature term and evaluate the width in the leading and next to leading order approximations expanded around the free (NG) string. I.
WIDTH OF STIFF STRING IN LEADING-ORDER APPROXIMATION
The action of the Polyakov-Kleinert bosonic string with the extrinsic curvature term reads Z Z σ √ √ ∂X ∂X S PK = · + αr d2 z g∆X.∆X . d2 ζ gg αβ 2 ∂ζα ∂ζβ
(1)
The paramters σ and αr refers to the rigidity factor and string tension, respectively. The vector X α (ζ1 , ζ2 ) maps the region C ⊂ R2 into RD with the Laplacian operator defined as √ √ (2) △ ≡ (1/ g)∂α ( gg αβ ∂β ) with gαβ =
∗
[email protected]
∂X ∂X · , ∂ζα ∂ζβ
(α, β = 1, 2, ...., D − 2),
g = det(gαβ ).
(3)
2 where gαβ is the intrinsic metric induced on the two-dimensional world sheet, parameterised by Xµ (ζ) embedded in the background RD . The free Nambu-Goto string action is given by NG Sℓo [X]
σ = σR LT + 2
Z
dζ0
Z
dζ
∂X ∂X · ∂ζα ∂ζα
.
(4)
Several of the physical IR properties of such a world sheet are in agreement with what is expected for QCD flux tube at low temperatures. The perturbative expansion of the extrinsic curvature term next to the Gaussian term NG S PK [X] ≈ Sℓo [X] + S Pert [X],
(5)
with S Pert [X] = αr
Z
√ d2 z g ∆X · ∆X,
(6)
= S 1 + S 2 + ...., The extrinsic curvature term S Pert [X] is introduced to circumvent some of the undesirable features of the Nambu-Goto action such as ghost, tachyons, and an imaginary static potential between quarks when the distance becomes smaller than some critical distance [1-8]. The mean-squared 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) =
R
DX(X(ζ, ζ 0 ) − X0 )2 exp(S PK [X]) R , DX exp(S PK [X])
(7)
In the following, we calculate the width of the rigid string Eq. (5) considering the first term in the perturbative expansion S Pert [X] ≈ S(1) of the Polyakov extrinsic curvature term Eq. (1) given by Z R " 2 2 2 2 # Z LT ∂ X ∂ X 0 dζ S(1) = αr dζ . (8) + 2 ∂ζ ∂ζ02 0 0 In the next section we evaluate the width at the next leading order in the extrinsic curvature term S Pert [X] ≈ S(1) + S(2) . Expanding around the free-string action Eq. (4) the squared width of the string [22] is 2 W 2 (ζ) = WNG (ζ) − hX 2 (ζ, ζ 0 )S Pert i0 + 2θh∂α X 2 (ζ, ζ0 )i0 + θ2 h∂α ∂α X 2 (ζ, ζ0 )i0 Z ′ θ2 dζ0 dζ dζ0′ dζ ′ h∂α ∂α X(ζ, ζ0 )∂α′ ∂α′ X(ζ, ζ0 )i0 . − βR
(9)
where the vacuum expectation value h...i0 is with respect to the free-string partition function and θ is an effective low energy parameter. The expansion Eq. (9) reads 2 2 W 2 (R, LT ) = WNG (R, LT ) + WExt (R, LT ),
(10)
2 with WExt (ζ) = X 2 (ζ, ζ0 )S Pert . ′ ′ Let us define the Green function G(ζ; ζ ′ ) = hX(ζ, ζ0 )X(ζ , ζ0 )i as the two point Gaussian propagator. The Gaussian correlator on a cylindrical sheet of surface area RLT is given by ′ ′ ∞ πn(ζ0 −ζ0 ) πn(ζ0 −ζ0 ) 1 πnζ ′ πnζ 1 X − n R R , +e sin q e sin G(ζ; ζ ) = πσ n=1 n (1 − q n ) R R ′
LT
(11)
where q = e−π R is the complementary nome. The Dirichlet boundary condition corresponding to fixed displacement vector at the ends ζ = 0 and ζ = R and periodic boundary condition in time ζ0 with period LT [22] are encoded in the above propagator.
3 After some algebraic manipulations and using the following auxiliary identities iπLT π π ′ (ζ − ζ0 − iζ − iζ ′ ) − , ξ2 = −i (ζ0 ′ − ζ0 + iζ − iζ ′ ) − 2R 0 2R 2R π ′ iπLT π ′ ξ3 = −i (ζ0 − ζ0 − iζ + iζ ′ ) − , ξ4 = −i (ζ0 − ζ0 + iζ + iζ ′ ) − 2R 2R 2R
ξ1 = −i
iπLT , 2R iπLT , 2R
the above Gaussian correlator Eq. (77) can be recast into the following closed form ϑ4 (ξ1 , q)ϑ4 (ξ4 , q) 1 , log G(ξi ) = 4πσ ϑ4 (ξ2 , q)ϑ4 (ξ3 , q)
(12)
where ϑ4 (ξ, q) is the Jacobi theta function. The expectation value of the free NG string can be derived from the 30
160 NGNLO-σ=0.024 NGNLO-σ=0.034 NGNLO-σ=0.044
140
25
120 100
W2 a-2
W2 a-2
20 15
5 0
60
NG-LO PK-α=0.10 PK-α=0.15 PK-α=0.20
10
80
40 20
0
2
4
6
8
Ra
10
-1
12
14
16
0
2
4
6
R a-1
8
10
12
FIG. 1: The mean-square width of the string W 2 at the middle of the string versus the string length R and string tension at finite temperature LT = 8 slices, units used form a typical lattice system [20]. (a) Compares the width of the free Nambu-Goto string Eq. (13) and the corresponding leading order width of Polyakov-Kleinert string for the depicted values of the rigidity parameter αr Eq. (19) and a string tension σa−2 = 0.045 (b) The width of the two loop pure Nambu-Goto string [22] for each corresponding value of the string tension.
above form using Jacobi triple product formula. The mean square width of the non-interacting string at the middle of the string is 2 (R/2) = lim G(ζ; ζ ′ ), WNG ǫ→0 √ ϑ3 (0, q) D−2 = + R0 . log √ 4πσ ϑ4 (0, q)
(13)
with R0 is the regularization scale. The above formula can be pinned down to an equivalent form to that derived in Refs. [22, 23] through the standard relations between the elliptic Jacobi and Dedkind η functions (See Appendix.(II) for detail). The next term in Eq (10) representing the leading order perturbation due to the instantaneous string in terms of the corresponding Green functions is
2 WExt =(D − 2)αr lim ′
ǫ →0 ǫ→0
Z
0
R
dζ ′
Z
0
LT
dζ0′ G(ζ; ζ ′ )∂α2 ∂α2 ′ G(ζ ′ ; ζ) + ∂α2 G(ζ; ζ ′ ) ∂α2 ′ G(ζ ′ ; ζ) .
(14)
The limit, ǫ, ǫ′ → 0, is such that the integral is ultimately regularized by the point-split method. 2 WExt =
∞ ∞ 2 −π(D − 2)αr X X (q m + 1) q −m (n (q n + 1)) . 4R2 σ 2 (q n − 1) n=1 m=1
(15)
4 With the use of zeta function regularization of the divergent sums in Eq. (15), a generic form for the contribution of the extrinsic curvature term to the mean-square width can be extracted as iLT π(D − 2)αr 2 . (16) E2 WExt = 24R2 σ 2 2R Using the standard modular transform of Eisenstein series −4R2 2R 12R LT = + . E2 i E2 i 2 2R LT LT πLT
(17)
The contribution of the rigidity term to the mean-square width of the string at the limit R ≫ LT can be read off from Eq. (19)
2 WExt
=
−(D − 2)αr 2σ 2 RLT
+ C(LT ).
(18)
with C(LT ) being only a function in the thermodynamic scale. However, Eq. (19) in limit LT → ∞ corresponding to zero temperature is given by 2 WExt =
π(D − 2)αr . 24R2 σ 2
(19)
The effects of rigidity to the mean-square width inversely decreases with the square of the length scale. Apart from the blow up at very short distance which appears as well in the next order of the perturbative expansion of pure NG string, we shall see in the next section that this behavior is moderated by considering higher order loops in the expansion of the extrinsic curvature term. The plot in Fig. 1 illustrates that the rigidity noticeable effects are mostly at intermediate and short string length. II.
THE INTERACTION-TERM CONTRIBUTION TO THE WIDTH
The next term in the perturbation expansion S Pert [X] ≈ S 1 + S 2 of the extrinsic curvature of rigid string of Polyakov-Kleinert action Eq .(1) is given by
S(2) =
Z
0
LT
dζ0
Z
0
R
σ dζ 8
∂X ∂ζα
4
σ − 4
∂X ∂X · ∂ζα ∂ζβ
2
!2 αr ∂X ∂ 2 X ∂2X 2 X − · +2αr ∂ζα ∂ζβ 2 ∂ζα ∂ζβ2 2 ∂2X ∂ X ∂X ∂X . · · − αr ∂ζα ∂ζβ ∂ζα ∂ζβ ∂ζγ2
(20)
The contribution to the width of the stiff string due to second loop interactions corresponds to the expectation value with respect to the free NG string given by
2 WExt−nlo = X 2 S(2)
= X 2 (S(a) + S(b) + S(c) + S(d) + S(e) ) .
(21)
The first two interaction terms have the dimensional coupling σ with Feynman diagrams corresponding to discon-
5 nected loops [6] After Wick contraction [6], the first interaction term reads * 4 +
2 σ ∂X X S(a) 0 = X ·X 8 ∂ζα 0
2 ∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X σ 2 + X X X = (D − 2) 8 ∂ζα 0 ∂ζα 0 ∂ζα ∂ζα 0 ∂ζα ∂ζα 0 ∂ζα ∂ζα 0
2 ∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X + (2D − 2) X X + X 0 . ∂ζα 0 ∂ζβ 0 ∂ζα ∂ζβ 0 ∂ζα ∂ζβ 0 ∂ζα ∂ζβ 0
(22)
The above correlator upon point split reformulation becomes a sum over the following Green strings
2 X S(a) 0 = lim
ι→0 ǫ→0
Z
R
0
Z
LT
0
σ (D − 2)2 (∂α′ G (ζ; ζ ′ ) ∂α′ ∂α′′ G (ζ ′ ; ζ ′′ ) ∂α′′ G (ζ ′′ ; ζ) + G (ζ; ζ ′ )) 8
(23)
× ∂α′ ∂α′′ G (ζ ′ ; ζ ′′ ∂α′′ ∂α G (ζ ′′ ; ζ)) + (2D − 2)(∂α′ G (ζ; ζ ′ ) ∂α′ ∂β ′′ G (ζ ′ ; ζ ′′ ) ∂β ′′ G (ζ ′′ ; ζ) ′′ ′ ′′ ′ + G (ζ; ζ ) ∂α′ ∂β ′′ G (ζ ; ζ ) ∂α′′ ∂β G (ζ ; ζ)) dζdζ0 .
In the limit R >> LT corresponding to high temperature and long distance, the correlator’s value is given by the sum over Green string fields (1) and (2) of Eq. (86) evaluated in detail in Appendix. (IV) as
2 (D − 2)2 X S(a) 0 = − 1536R2σ 2
10π 3 3 2 2 τ + π τ + 4 E2 (τ ). 9
(24)
T where γ = 0.5772 is the Euler-Mascheroni constant and τ = − L 2R is the modular parameter of the cylindrical sheet. In the limit LT >> R corresponding to low temperature and small string length, the correlator assumes the form
2 −λ1 (D − 2)2 E2 (τ ), X S(a) 0 = 128R2 σ 2
with λ1 = ((π + 6 − log(8))/24. The second Wick-contracted interaction term * 2 +
2 ∂X ∂X −σ X ·X · X S(b) 0 = 4 ∂ζα ∂ζβ 0
2
2 −σ ∂X ∂X ∂X ∂X ∂X ∂X ∂X = (D − 2)(D − 1) X 0 + X X 0 4 ∂ζα ∂ζβ 0 ∂ζα ∂ζβ 0 ∂ζα ∂ζβ 0 ∂ζα 0
∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X X . + X2 0 + (D − 2) X ∂ζα 0 ∂ζα 0 ∂ζα ∂ζα 0 ∂ζα ∂ζα 0 ∂ζα ∂ζα 0
(25)
(26)
The sum of the corresponding strings of Green correlators is given by R
LT
−σ (D − 2)(D − 1) ∂α′ G (ζ; ζ ′ ) ∂α′ ∂β ′′ G (ζ ′ ; ζ ′′ ) ∂β ′′ G (ζ ′′ ; ζ) + G (ζ; ζ ′ ) ι→0 4 0 ǫ→0 0 ∂α′ ∂β ′′ G (ζ ′ ; ζ ′′ ) ∂α′′ ∂β G (ζ ′′ ; ζ) + (D − 2) ∂α′ G (ζ; ζ ′ ) ∂α′ ∂α′′ G (ζ ′ ; ζ ′′ ) ∂α′′ G (ζ ′′ ; ζ) + G (ζ; ζ ′ ) ∂α′ ∂α′′ G (ζ ′ ; ζ ′′ ) ∂α′′ ∂α G (ζ ′′ ; ζ) dζdζ0 .
2 X S(b) 0 = lim
Z
Z
(27)
6 Similar to the previous interaction term, the correlator is given by the sum of the triple products (1) and (2) of Green strings Eq. (86) yielding the expression
2 (D − 2) 10π 3 3 2 2 τ + π τ + 4 E2 (τ ). (28) X S(b) 0 = 768R2σ 2 9 which is valid for R >> LT and
2 λ1 (D − 2) X S(b) 0 = E2 (τ ), 64R2 σ 2
(29)
at the limit LT >> R. The next three interaction terms in Eq. (21) have dimensionless couplings αr and correspond to both connected and disconnected loop [6]
The contraction of the third interaction term reads *
2 + ∂X ∂X (X.X) (30) · ∂ζα ∂ζβ 0 2
2
∂X ∂ 2 X ∂X ∂X ∂X ∂ 2 X ∂ X ∂2X + X2 0 = 2αr 2(D − 2) X 0 ∂ζα ∂ζγ ζγ 0 ∂ζα ∂ζγ ζγ 0 ∂ζα ∂ζα 0 ∂ζγ ζγ ∂ζγ ζγ 0 2 2 2 2
2 ∂ X ∂X ∂X ∂ X ∂ X ∂X ∂X ∂ X 2 X 0 X X + (D − 2) + ∂ζα ∂ζγ ζγ 0 ∂ζα 0 ∂ζγ ζγ 0 ∂ζα ζα ∂ζγ 0 ∂ζβ ζβ ∂ζγ 0 2 2 ∂X ∂X ∂ X ∂ X + X X . ∂ζγ ∂ζγ 0 ∂ζα ζα 0 ∂ζβ ζβ 0
2 X S(c) 0 = 2αr
2
The corresponding sum of the point split Green strings read
2 X S(c) 0 = lim
ι→0 ǫ→0
Z
R
0
Z
LT
0
2αr (D − 2) ∂α′ ∂α′ G (ζ; ζ ′ ) ∂α′ ∂α′′ G (ζ ′ ; ζ ′′ ) ∂β ∂β G (ζ ′′ ; ζ) + G (ζ; ζ ′ ) 2
(31)
× ∂ ∂ ∂ G (ζ ; ζ ) ∂ ∂ ∂α G (ζ ; ζ) + 2(D − 2) ∂β G (ζ; ζ ′ ) ∂α′ ∂α′ ∂α′′ ∂α′′ G (ζ ′ ; ζ ′′ ) ′′ ′ ′ ′′ ′′ ′′ ′ ′ ′′ ′′ ′′ × ∂β G (ζ ; ζ) + G (ζ; ζ ) ∂α ∂α ∂β G (ζ ; ζ ) ∂α ∂α ∂β G (ζ ; ζ) dζdζ0 . α′
α′′
α′′
′
′′
α′′
α′′
′′
For which the only non-vanishing Green strings Eq. (86) is given by the product (7). The expectation value of the correlator for R >> LT is then
2 E4 (τ ) + 1 π 2 αr (D − 2) −π 3 3 2 2 (32) τ + π τ − 4 X S(c) 0 = 8R4 σ 3 9 240 The approximation of the correlator in the limit LT >> R is given by
2 π 2 λ2 (D − 2)αr E (τ ) + 1 , X S(c) 0 = 4 8R4 σ 3
with λ2 = (π − 2 + log(2))/240.
(33)
7 The fourth interaction term is * !2 +
2 αr ∂X ∂ 2 X X S(d) 0 = − X.X · 2 ∂ζα ∂ζβ2 0 2
αr ∂X ∂X ∂ 2 X ∂X ∂ X ∂2X (D − 2)(D − 1) X =− X + X2 0 2 ∂ζα 0 ∂ζα 0 ∂ζγ ζγ ∂ζγ ζγ 0 ∂ζα ∂ζγ ζγ 0 2 2
∂X ∂ 2 X ∂ X ∂X ∂X ∂X ∂X ∂ X 2 + (D − 2) X X + X 0 × ∂ζα ∂ζγ ζγ 0 ∂ζα ∂ζα 0 ∂ζβ ζβ 0 ∂ζγ 0 ∂ζα ∂ζβ ζβ 0 2 2 2 ∂X ∂ X ∂X ∂X ∂ X ∂ X × + X X . ∂ζα ∂ζγ ζγ 0 ∂ζα 0 ∂ζα 0 ∂ζβ ζβ ∂ζγ ζγ 0 The point split form of the above correlator in terms of Green functions reads Z R Z LT
2 αr X S(d) 0 = − (D − 2)(D − 1) ∂β G (ζ; ζ ′ ) ∂α′ ∂α′ ∂α′′ ∂α′′ G (ζ ′ ; ζ ′′ ) ∂β ′′ G (ζ ′′ ; ζ) lim 2 ǫ′ →0 0 0 ǫ→0 + G (ζ; ζ ′ ) ∂α′′ ∂α′′ ∂β G (ζ ′′ ; ζ ′ ) + (D − 2) ∂α′ ∂α′ G (ζ; ζ ′ ) ∂β ∂β G (ζ ′′ ; ζ) ∂α′ ∂α′′ G (ζ ′ ; ζ ′′ )
(34)
(35)
+ G (ζ; ζ ′ ) ∂β ′ ∂α′′ ∂α′′ G (ζ ′ ; ζ ′′ ) ∂α′′ ∂α′′ ∂β G (ζ ′′ ; ζ) + G (ζ; ζ ′ ) ∂β ′ ∂β ′′ ∂α′′ G (ζ ′ ; ζ ′′ ) × ∂α′′ ∂α′′ ∂α G (ζ ′′ ; ζ) + ∂α′ G (ζ; ζ ′ ) ∂β ∂β ∂α′ ∂α′ G (ζ ′ ; ζ ′′ ) ∂α′′ G (ζ ′′ ; ζ) dζdζ 0 .
The surviving Green strings Eq. (86) are the triple product (7) and (12) calculated in detail in Appendix(IV). The resultant interaction term in the limit R >> LT is
2 −π 2 D (D − 2)αr −π 3 3 E4 (τ ) + 1 2 2 . (36) X S(d) 0 = τ + π τ − 4 32R4 σ 3 9 240 In the limit R >> LT the correlator (Appendix(IV)) becomes
2 −π 2 λ2 (D)(D − 2)αr E4 (τ ) + 1 , X S(d) 0 = 64R4 σ 3
(37)
The last interaction term 2
2 ∂X ∂X ∂2X ∂ X X S(e) 0 = −αr X.X (38) · · ∂ζα ∂ζβ ∂ζα ∂ζβ ∂ζγ2 0 ∂2X ∂2X ∂X ∂ 2 X ∂X ∂ 2 X ∂X ∂X 2 X X + hX.Xi = −αr (D − 2) ∂ζα ∂ζα ∂ζβ ζβ ∂ζγ ζγ ∂ζα ∂ζα ζβ ∂ζβ ∂ζγ ζγ ∂X ∂X ∂2X ∂2X ∂2X ∂2X ∂X ∂X + X X ) + (2D − 2) X X ∂ζβ ∂ζβ ∂ζα ζβ ∂ζγ ζγ ∂ζα ∂ζβ ∂ζγ ζγ ∂ζα ζβ 2
2 ∂X ∂ 2 X ∂X ∂ X , + X ∂ζα ∂ζα ζβ ∂ζβ ∂ζγ ζγ for which the sum of Green strings of the fields is Z R Z LT
2 2 X S(e) 0 = −αr lim (D − 2) ∂α′ ∂α′′ G (ζ ′ ; ζ ′′ ) ∂α ∂α G (ζ; ζ ′ ) ∂β ∂β G (ζ ′′ ; ζ) + G (ζ; ζ ′ ) ′ ǫ →0 ǫ→0
0
(39)
0
∂α′′ ∂β ′′ ∂α G (ζ ′′ ; ζ) ∂α′ ∂α′ ∂β ′′ G (ζ ′ ; ζ ′′ ) + ∂β G (ζ; ζ ′ ) ∂α′ ∂α′ ∂α′′ ∂α′′ G (ζ ′ ; ζ ′′ ∂β ′′ G (ζ ′′ ; ζ)) + ∂β G (ζ; ζ ′ ) ′′ ′ ′′ ′′ ′′ ′′ ′ ′ ∂β ∂β ∂α ∂α G (ζ ; ζ ) ∂β G (ζ ; ζ) + 2(D − 2) ∂α ∂β ′ G (ζ; ζ ′ ) ∂α′′ ∂β ′′ G (ζ ′′ ; ζ) ∂α′ ∂α′ G (ζ ′ ; ζ ′′ ) ′ ′′ ′′ ′ ′′ ′′ ′ ′′ + G (ζ; ζ ) ∂α ∂α ∂β G (ζ ; ζ) ∂β ∂α ∂α G (ζ ; ζ ) dζdζ0 .
8 The non-vanishing Green string is given only by the triple product (12) of Eq. (86). At finite temperature R >> LT , the contribution of the above term to the width would read
2
X S(e)
0
−π 2 (D − 2)2 αr = 16R4 σ 3
−π 3 3 τ + π2 τ 2 − 4 9
E4 (τ ) + 1 . 240
(40)
In the other limit LT > R limit the contribution is
2 π 2 (D − 2)2 αr X S(e) 0 = −λ2 E4 (τ ) + 1 32R4 σ 3 70
30 28
PKLO-α=0.3 PKLO-α=0.4 PKLO-α=0.5 PKNLO-α=0.3 PKNLO-α=0.4 PKNLO-α=0.5
60 50
PKLO-α=0.1 PKLO-α=0.15 PKLO-α=0.2 PKNLO-α=0.1 PKNLO-α=0.15 PKNLO-α=0.2
26 24 22
40
W2 a-2
W2 a-2
(41)
30
20 18 16
20
14 12
10
10 0
2
4
6
Ra
-1
8
10
12
8
2
4
6
8
R a-1
10
12
FIG. 2: The mean-square width of the string W2 (z) versus the string length Ra−1 at finite temperature LT a−1 = 8 slices and string tension σa−2 = 0.044.(a)Compares the width of LO Eqs. (19), (13) and NLO of Polyakov-Kleinert string at finite temperature given by Eq. (9) and Eq. (42) Eq. (43) for the depicted values of the rigidity parameters. The left plot is the same as plot (a); however at temperature T = 0 corresponding to infinite length in time direction LT → ∞ Eq. (43).
The second loop interaction terms contribution to the width are given Eqs. (24), (28), (32), (36) and (40) at finite temperatures. This is, 2 WExt−nlo =
(D − 2)2 − 2(D − 2) 10π 3 3 π 2 (D − 2)(3D − 8)αr 2 2 τ + π τ + 4 E (τ ) + 2 1536R2σ 2 9 32R4 σ 3 −π 3 3 E4 (τ ) + 1 × . τ + π2 τ 2 − 4 9 240
(42)
At T = 0 or infinite time extend LT → ∞, Eisentein series terms can be readily evaluated E2k (τ ) → 1 in the the total mean square width corresponding to the sum of the correlators Eqs. (25), (29), (33), (37) and (41) such that 2 WExt−nlo
2
= 3π αr λ1
(D − 2)(D − 4) 64R4 σ 3
(E4 (τ ) + 1) + αr λ2
(D − 2)(D − 4) 128R2σ 2
E2 (τ ).
(43)
A plot comparing the width of the PK to the NG string at next to leading order is depicted in Fig. 2. The effect of including the next term in the expansion of the extrinsic curvature term on the extracted width of the energy profile is noticeable at the tail of the intermediate distances with significant improvements at zero temperature in particular. The effects of string stiffness are not expected to be dominant in non-abelian gauge models even in the continuum limit [18]. Indeed, a comparison of figure 1 for the profile of the leading order PK string with the next to leading order approximation, Figure 2, Eq. (42) and Eq. (43) for the depicted values of the rigidity parameter, indicates that the next to leading order formulas can serve as a good approximation on most application levels. In conclusion we have derived the width of the energy profile of Polyakov-Kleinert stiff strings as a function of the stiffness parameter in any dimension D at both low and high temperatures. Our main results are given by Eqs. (19)
9 at leading order level and Eqs. (42) and (43) for the next higher loops at finite and zero temperatures, respectively. This has been calculated by the perturbative expansion of the extrinsic curvature term around the free NG string action. The perturbative arguments regarding string’s rigidity have been embodied in the expansion of the encompassed NG string action with boundary terms in the identification of these effects in the numerical simulation of the confining potential [18–21]. It would be desirable in the future to extend the theoretical considerations to the closed string channel as well as other boundary conditions involving open strings such as the mixed Dirichlet/Neumann. This ought to be of direct relevance to the glueballs [24], mesonic spectrum, the potential and energy profile of knotted string configurations [25, 26] which is the target of our next reports. Acknowledgments
We thank F. Gliozi, M. Pepe, and U. Wiese for useful comments. This work has been funded by the Chinese Academy of Sciences President’s International Fellowship Initiative grants No.2015PM062 and No.2016PM043, the Recruitment Program of Foreign Experts, NSFC grants (Nos. 11035006, 11175215, 11175220) and the Hundred Talent Program of the Chinese Academy of Sciences (Y101020BR0).
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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III.
APPENDIX (I):FUNCTIONS AND IDENTITIES
• Dedekind η Function η(τ ) = e
2πiτ 24
∞ Y
k=1
with e−
πkLT R
T = e2πikτ and τ = − L 2R .
1 − e2πikτ ,
(44)
10
∞ X j=1
1 log 1 − q j = log q − 24 η(τ ) .
• Eisenstein series is defined as E2k (τ ) = 1 + (1)k
∞ 4k X n2k1 . Bk n=1 1 − q n
(45)
(46)
where Bk are Bernoulli numbers defined as ζ(2k) =
Bk (2)2k , 2k!
ζ(1 − 2k) =
Bk . 2k!
(47)
for even and odd numbers respectively. The modular transform τ →
1 τ
of Dedekind η function and the Eisenstein series E2k is given by −1 1 , η(τ ) = √ η τ iτ 1 1 −1 E2 (τ ) = 2 E2 − , τ τ 6iπτ
(48) (49)
respectively. • Identity(I): P The sum n
n(qn +1) qn −1
can be expressed in the form of Eisenstein series X n (q n + 1) n
qn − 1
=
X nq n n , − qn − 1 1 − qn n n X n X nq n = , − qn − 1 1 − qn n n X n 1 − (1 − E2 (τ )), = n−1 q 24 n X n (q n − q n + 1) 1 = − (1 − E2 (τ )), n q −1 24 n X 2 = − (1 − E2 (τ )) − n, 24 n
X
(50)
1 2 − (1 − E2 (τ )), 12 24 = (1/12)E2 (τ ). =
• Identity(II): The logarithmic divergent series ∞ ∞ X X q kǫ log 1 − q j − lim log 1 − q j = , ǫ→0 k j=1 j=0
∞ X
k=1
= =
∞ X
1 log 1 − q j − lim (ζ(1 − α) + ζ(α + 1)), α→0 2 j=1
∞ X j=1
log 1 − q j − γ.
(51)
11 where γ = 0.5772 Euler-Mascheroni constant. • Identity(III): The regularization of the series ∞ X ∞ ∞ X X 1 1 1 = − , m + n m + n n m=0 n=1 n=1 m=1 n=1 ∞ X ∞ X ∞ X
(52)
∞ X 1 = Φ(1, 1, n) − , n n=1 n=1
= Φ(1, 1, 1) −
∞ X 1 , n n=1
= ζ(1) − ζ(1) = 0,
yields a vanishing value, where Φ(1, 1, 1) is the Lerch transcendent which is a generalization of the Hurwitz zeta function and the Jonquire’s function and is given by Φ(z, s, a) =
∞ X
k=0
zk . (a + k)s
(53)
In the following listed are identities (IV) to (IX) which are used in the evaluation of the sums that appear in the triple product of Green strings in the limit R >> LT . • Identity(IV): ∞ X
m=1
bq −n b − 2πdn coth nq −n = − 2 2 4n m 8n b2 + d2
2πdn b
.
(54)
• Identity(V): ∞ X
∞ X n2 q −n q −n 1 + = (H−n + Hn − 2γ) q −n , 2 2 m (m − n ) m 2 m=1 m=1
(55)
where H is the harmonic series, the harmonic numbers are given by Hnr =
n X 1 r i i=1
with
Hn = Hn1 .
(56)
• Identity(VI): ∞ X 1 1 1 1 2 q−1 − , − πq −n coth(πn) = − π − log 2n 2 2 e2πj q − 1 q − 1 2 q j=0
∞ −n X q
n=1
=
(57)
π γ − . 2 2
• Identity(VII): ∞ X
n=1
γq −n =
γ . q−1
(58)
12 • Identity(V): ∞ X
Hn q −n
n=1
log 1 − 1q . =− 2 1 − q1
(59)
• Identity(VIII): lim
q→0
∞ X
Hn q −n = 0,
(60)
n=1
• Identity(IX): ∞ X
k=1
k3 qk − qk + 1 k3 = , 1 − qk 1 − qk =
(61)
∞ ∞ X X k3 qk + k3 , 1 − qk k=1
k=1
1 = (E4 (τ ) − 1) + ζ(−3). 240 The identities listed below are used in the evaluation of the sums of Green strings in the other limit LT >> R. • Identity(X) ∞ X ∞ X m(−1)m+n 1 m(−1)m+n n(−1)m+n = + , n(m + n) 2 n(m + n) m(m + n) m=1 n=1 (−1)m+n m2 + n2 = , 2mn(m + n) ∞ X ∞ X (−1)m+n+1 (−1)m+n , + = 2n 2m m=1 n=1 1 0 1 0 2 − 1 ζ(1) − 2 − 1 ζ(1), = 4 4 = 0.
(62)
• Identity(XI) ∞ ∞ X X (−1)m 1 1−s = 2 − 1 . s s m m m=1 m=1
(63)
• Identity(XII)
Grandi’s series ∞ X
(−1)n =
n=1
1 . 2
(64)
• Identity(XIII) ∞ m(−1)m −n nj X
m=1
πLT R
j! (m2 − n2 )
j
π j (Φ(−1, 1, 1 − n) + Φ(−1, 1, n + 1))nj+1 = 2j!
LT R
j
.
(65)
13 • SOME GENERAL PROPERTIES OF ZETA-REGULARIZATION In the following, we quote some of basic properties of the zeta-regularized products [27] Let us consider the convergence index which is the least integer h such that the series for Z[h + 1] converges absolutely. 1-Partition Property. Let {ϕ1 }k and {ϕ2 }k be ζ- and let regularizable sequences {ϕ}k = {ϕ1 }k ∪ {ϕ2 }k disjoint union then Y
ϕk =
k
Y Y {ϕ1 }k {ϕ2 }k k
(66)
k
2-Splitting Property. If {ϕk } is zeta-regularizable and
Q
k
ak is convergent absolutely then Y Y Y ak ϕk = ak ϕk k
k
(67)
k
3-Conjugate Decomposition Property. If {ϕk } is a zeta-regularizable sequence with convergence index h < 2, then Y k
|ϕk − ϕ| 2 =
Y k
|ϕk ϕ|
Y k
|ϕk ϕ| ¯
(68)
4-Iteration Property. Suppose ϕ =QSmn is Q a double-indexed zeta-regularizable sequence, i.e. {ϕm }, ϕm = n ϕmn , m ϕm Y
ϕmn =
mn
A.
Y m
ϕm =
YY m
Q
mn
ϕmn is meaningful. Then for
ϕmn .
(69)
n
Appendix (II): The free string propagator and width on cylinder
The Green correlator can be expressed in terms of Jacobi theta functions ′ ′ ∞ πn(ζ0 −ζ0 ) πn(ζ0 −ζ0 ) πnζ ′ 1 πnζ 1 X n − R R sin q e sin +e , G(ζ; ζ ) = πσ n=1 n (1 − q n ) R R ′
LT
where q = e−π R is the complementary nome. Let us define the following auxiliary identities ′
κ1 = (ζ0 − ζ0 − iζ − iζ ′ ), κ2 = (ζ0 ′ − ζ0 + iζ − iζ ′ ), ′
κ3 = (ζ0 − ζ0 − iζ + iζ ′ ), ′
κ4 = (ζ0 − ζ0 + iζ + iζ ′ ).
(70)
14
∞ 1 X 1 4πσ n=1 n (1 − q n ) (πn)κ2 (πn)κ3 (πn)κ4 (πn)κ1 (πn)κ2 (πn)κ3 (πn)κ4 (πn)κ1 , × −q n e− R + q n e− R + q n e− R − q n e− R − e R + e R + e R − e R
G(κi ) =
(71)
Considering the general term
(πn)κ (πn)κ ∞ X q n e− R + e R , S(κi ) = n (1 − q n ) n=1
=
(πn)κ ∞ X q kn e− R
∞ X
n
n=1 k=1 ∞ X
=−
k=0
= − log = − log = log
πκ
log 1 − q k e R ∞ Y
k=0 ∞ Y
k=1 ∞ Y
k=1
k
1−q e
+
πκ R
1−q
!
(πn)κ ∞ X q kn e R
k=0 ∞ X
−
k=1
∞ Y
k=1
1 − q k−1 e k
(72)
πκ R
− log
n
!
,
πκ log 1 − q k e− R ,
k − πκ R
1−q e 1 − qk e ∞ Y
k=1
− πκ R
1−q
k
!
!
,
,
1−q
k−1
e
πκ R
k − πκ R
1−q e
!
.
Considering the sum of each separate S(κi ) the propagator becomes Q πκ πκ πκ ∞ Q∞ 1 − q k 1 − q k−1 e πκR1 k − R1 k − R4 k−1 R4 k 1 − q e 1 − q e 1 − q e 1 − q k=1 k=1 1 Q . log Q G(κi ) = πκ πκ πκ πκ ∞ ∞ 4πσ k e− R2 k e− R3 k ) 1 − q k−1 e R3 k ) 1 − q k−1 e R2 1 − q 1 − q (1 − q (1 − q k=1 k=1
(73)
In terms of Jacobi theta ϑi functions, the propagator reads (iπ)κ πL πL iπLT 4 − 2RT − 2RT T ϑ4 − 2R 1 − iπL ϑ4 − (iπ)κ 4R , e 2R − 4R , e 1 G(κi ) = log πLT πL 4πσ ϑ4 − (iπ)κ2 − iπLT , e− 2R ϑ4 − (iπ)κ3 − iπLT , e− 2R 2R
4R
2R
(74)
4R
using the transform
−iπ ξi = 2
κi LT + R 2R
.
(75)
The propagator can be recast into the following closed form " √ √ # θ4 ξ1 , q θ4 ξ4 , q 1 G(ξi ) = log √ √ . 4πσ θ4 ξ2 , q θ4 ξ3 , q IV.
(76)
APPENDIX (III): DETAILED CALCULATIONS OF THE LEADING MEAN-SQUARE WIDTH
The Gaussian correlator G(ζ ′ , ζ0′ ; ζ, ζ0 ) ≡ hX(ζ, ζ0 )X(ζ ′ , ζ0′ i on a cylinder of size RLT with fixed boundary conditions at ζ 1 = 0 and ζ 1 = R and periodic boundary conditions in ζ0 with period LT is given by
15
G(ζ; ζ ′ ) =
′ ′ ∞ πn(ζ0 −ζ0 ) πn(ζ0 −ζ0 ) 1 πnζ ′ πnζ 1 X n − R R sin q e sin , + e πσ n=1 n (1 − q n ) R R
(77)
By integrating the first part of Eq. (14) a vanishing series is obtained Z R Z LT ′ ′ I1 =(D − 2) lim dζ0 G(ζ; ζ ′ )∂α2 ∂α2 ′ G(ζ ′ ; ζ), dζ ′ ǫ,ǫ →0 0 0 ′ ′ 2πmǫ′ πn(ζ+ǫ) 3 − π(m−n)ǫ n − 2πnǫ R R qm + e R sin πm(ζ+ǫ) + 1 sin 2πn e q e R R = mR3 σ 2 (m2 − n2 ) (q m − 1) (q n − 1) × (m cos(πm) sin(πn) − n sin(πm) cos(πn)) = 0.
(78)
(79) (80)
The second part of Eq. (14)
I2 =(D − 2) Lim ′
ǫ,ǫ −→
Z
0
R
Z
LT
0
dζ0 dζ [∂ζ ∂ζ G(ζ, ζ0 ; ζ ′ , ζ0′ )∂ζ ′ ∂ζ ′ G(ζ, ζ0 ; ζ ′ , ζ0′ )
(81)
+ ∂ζ0 ∂ζ0 G(ζ, ζ0 ; ζ ′ , ζ0′ )∂ζ0′ ∂ζ0′ G(ζ, ζ0 ; ζ ′ , ζ0′ )] Z R Z LT X ∞ X ∞ 2πm(−LT +2ζ0 +ǫ′ ) 2πnǫ′ π 2 mn m n ′ R R +q e q +e =(D − 2) lim dζ ǫ,ǫ′ →0 0 R4 σ 2 (q m − 1) (q n − 1) 0 m=1 n=1 π(−LT m+ǫ′ (m+n)+2mζ0 ) πmζ πnζ πm(ζ + ǫ) πn(ζ + ǫ) R × sin sin sin sin × e− R R R R 2πnǫ′ Z LT ∞ X ∞ π 2 n2 − m2 (m(−2πn)) q n + e R X = (D − 2) lim dζ 0 ǫ′ →0 0 8R3 σ 2 (m − n)(m + n) (q m − 1) (q n − 1) m=1 n=1 π(−LT m+ǫ′ (m+n)+2mζ0 ) 2πm(−LT +2ζ0 +ǫ′ ) − m R R ×e e +q =
∞ ∞ X 2 X −π(D − 2) (q m + 1) q −m (n (q n + 1)) (q n − 1) (4R2 σ 2 ) m=1 n=1
Making use of Eisenstein series relations E2n = C2n
∞ X k 2n−1 q 2k k=1
1 − q 2k
+ 1,
(82)
with zeta function regularization tof he divergent series ∞ X m = ζ(−1), m=1
∞ X mq m 1 E2 (q) − ζ(−1). =− 2m 24 1−q 2 m=1
where q = e−
πLT 2R
and C2n =
−4n with B2n representing the corresponding Bernoulli numbers, the integral I2 becomes B2n −π(D − 2)αr iLT I2 = (83) E2 24R2 σ 2 2R
Using the modular transform −4R2 2R 12R LT = i + , E E2 i 2 2R L2T LT πLT
(84)
16 Eq (83) in the limit R >> LT is αr (D − 2) I2 = σ2 V.
π 1 − 2 − . 6LT 2RLT
(85)
APPENDIX (IV) DETAILED CALCULATIONS OF NLO COTRACTIONS
The listed below collects all the sixteen terms of Green strings involved in the second loop contraction Eq. (20)
1−lim lim ′
ǫ→0ǫ →0
2−lim lim ′
ǫ→0ǫ →0
3−lim lim ′
ǫ→0ǫ →0
4−lim lim ′
ǫ→0ǫ →0
5−lim lim ′
ǫ→0ǫ →0
6−lim lim ′
ǫ→0ǫ →0
7−lim lim ′
ǫ→0ǫ →0
8−lim lim ′
ǫ→0ǫ →0
9−lim lim ′
ǫ→0ǫ →0
10−lim lim ′
ǫ→0ǫ →0
11−lim lim ′
ǫ→0ǫ →0
12−lim lim ′
ǫ→0ǫ →0
13−lim lim ′
ǫ→0ǫ →0
14−lim lim ′
ǫ→0ǫ →0
15−lim lim ′
ǫ→0ǫ →0
16−lim lim ′
ǫ→0ǫ →0
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
R 0
0
R
Z
L
dζdζ0 [∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂α′′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 )],
R
Z
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α ∂α′′ [G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )], LT
0
0 R
Z
LT
R
Z
LT
dζdζ0 [∂α′ ∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂β G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [∂β G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂α′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂β ′′ ∂α′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α ∂α G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂β ∂α ∂α G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′′ ∂β ′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂β ′′ ∂β ′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂β ′ ∂β′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [∂β ∂α G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′ ∂α G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
LT
dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂β ′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′ ∂β ′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0 R
Z
0
dζdζ0 [∂α ∂α′ G ζ, ζ 0 ; ζ ′ , ζ0′ ∂α ∂β ′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂β ′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )], dζdζ0 [G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂β ′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
0
0
(86)
LT
0
0
0
Z
LT
dζdζ0 [∂β G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂β ′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )],
In the following we evaluate each string of the above inlisted Green functions Eq. (86) with the substitutions consistent [22] with the cylinderical boundary condition ζ ′ = ζ − R, ζ ′ = LT − ζ0 and ζ ′′ = ζ ′ + ǫ, ζ ′′ = ζ ′ + ǫ′
17 Green String (1) = lim ′
ǫ →0
Z
0
LT
π ((n−m)(LT −ζ0 +ǫ′ )−n(LT −ζ0 )+mζ0 ) m n π R e R4 σ 3 k(1 − q k ) (q m − 1) (q n − 1) ( 2πm(LT −2ζ0 +ǫ′ ) πk(LT −2ζ0 +ǫ′ ) πk(LT −2ζ0 +ǫ′ ) 2πnǫ′ R R R × q n e− R + 1 q m e +1 qk e + e−
(87)
R
h πkζ πk(ζ − R + ǫ) πmζ πn(ζ − R) sin sin cos cos ǫ→0 0 R R R R πn(ζ − R + ǫ) πm(ζ − R + ǫ) cos × cos R R ) πmζ πn(ζ − R) πm(ζ − R + ǫ) πn(ζ − R + ǫ) i + sin sin sin sin dζ dζ0 R R R R ∞ ∞ ∞ X 1 1 π X k qk + 1 X (88) =− 3 3 8R σ q k − 1 m=1 q m − 1 n=1 q n − 1 k=1 Z LT πm(ǫ′ −2ζ0 ) πm(ǫ′ −2ζ0 ) πn(2ζ0 ) πn(2ζ0 ) −n − m − 2n R R R R e lim × q e dζ0 , − q e − q e ǫ′ →0 0 ∞ ∞ ∞ X 1 1 π X k qk + 1 X =− 3 3 k m n 8R σ q − 1 m=1 q − 1 n=1 q − 1 k=1 −m q q 2m q m−n q −n q 2n q n−m q 2m+n q m+2n + + + − − − − . m+n m−n m−n m+n m−n m−n m+n m+n × lim
Z
Considering only the summation over n and m and noting the symmetry of the series expansion under the exchange of summation indices m → n −m ∞ ∞ X X 1 q q 2m q m−n q −n q 2n q n−m q 2m+n q m+2n 1 , (89) + + + − − − − S= q m − 1 n=1 q n − 1 m + n m − n m − n m + n m − n m − n m + n m+n m=1 ∞ ∞ X X 1 1 q −m q 2m q m−n q m+2n q 2m+n q −n q 2n q n−m = + + + − − + − − . q m − 1 n=1 q n − 1 m−n m+n m−n m−n m+n m+n m+n m−n m=1 Adding both sides of the above equation −m 1 1 1 q q 2m q m−n q −n q 2n q n−m q 2m+n q m+2n S= + + + − − − − 2 (−1 + q m ) (−1 + q n ) m + n m − n m − n m + n m − n m − n m + n m+n 1 q n−m q −m q 2m q m−n q m+2n q 2m+n q −n q 2n + m − , + + + − − + − (q − 1) (q n − 1) m−n m+n m−n m−n m+n m+n m+n m−n 2q −m−n (q m + q n ) (q m+n + 1) (m (q m + 1) (q n − 1) − n (q m − 1) (q n + 1)) = . (m − n)(m + n) (q m − 1) (q n − 1)
(90)
In case of R >> LT corresponding to finite temperature, we consider the approximation q = 1 + x and utilizing identity (III,IV) the sum becomes ∞ X ∞ X q −m−n (q m + q n ) (q m+n + 1) (m (q m + 1) (q n − 1) − n (q m − 1) (q n + 1)) S= , (m − n)(m + n) (q m − 1) (q n − 1) m=1 n=1
= =
∞ X ∞ X
m=1 n=1 ∞ X ∞ X m=1 n=1
=
−
(91)
(x(m + n) − 1)(x(m + n) + 2)2 , m+n
x m2 x2 + 2mnx2 + 4mx + n2 x2 + 4nx + 4 − x(x(m + n) + 4) −
2(−1)(−1)x3 3(−1)(−1)x2 3(−1)(−1)x2 + + , 12 × 12 2 × 12 2 × 12
4 m+n
,
18 , that is, S=
∞ X 1 1 m−1 n−1 q q n=1 m=1 Z LT πm(−2ζ0 ) πn(2ζ0 ) πm(−2ζ0 ) πn(2ζ0 ) e R q −n e− R − q 2n e R dζ0 × − q m e− R ∞ X
(92)
0
1 S= 72
πLT − R
2
πLT 18 − R
−4
The string of Green functions (1) is then lim lim ′
ǫ→0ǫ →0
=− =−
Z
R
0
Z
L
0 ∞ X
π 8R3 σ 3 k=1 1 E2 (τ ) 72
(93) ∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂α′′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) dζdζ0 ∞ Z LT ∞ X πm(ǫ′ −2ζ0 ) πm(ǫ′ −2ζ0 ) πn(2ζ0 ) k qk + 1 X 1 1 −n − πn(2t) m − 2n R R R R q e , −q e −q e dζ0 e q k − 1 m=1 q m − 1 n=1 q n − 1 0 2 πL − πL 18 − − 4 R R
12 (16R2 σ 3 )
In the limit LT → ∞ the sum S simplifies to S= =
∞ X ∞ X q −m + q −n , m+n m=1 n=1
(94)
∞ X ∞ X 2q −n , m+n m=1 n=1
∞ X ∞ X (m − n)q −n , m2 − n 2 m=1 n=1 ∞ ∞ X X mq −n nq −n , = − m2 − n 2 m2 − n 2 m=1 n=1 ∞ X ∞ X n2 q −n q −n nq −n = , + − m (m2 − n2 ) m m2 − n 2 m=1 n=1
=
∞ X q −n (πn cot(πn) − 1) 1 (H−n + Hn − 2γ) q −n − , 2 2n n=1 1 ∞ log 1 − X q 1 γ q −n (πn cot(πn) − 1) + (−γ − 2 + log(2)) − =− − , q−1 2 2n 2 1− 1 n=1
=
q
where we have employed identities from (IV-IX) in the above sum. The Green string is then
lim lim ′
ǫ→0ǫ →0
S=−
Z
0
R
Z
π 8R3 σ 3
L
∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂α′′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) dζdζ0 ∞ ∞ ∞ X X k qk + 1 X 1 1
0
k=1
qk − 1
q m − 1 n=1 q n − 1 Z LT πm(ǫ′ −2ζ0 ) πm(ǫ′ −2ζ0 ) πn(2ζ0 ) πn(2ζ0 ) −n − 2n m − R R R R q e × lim dζ0 . e − q e − q e ′
m=1
ǫ →0
(π − 2 + log(2))E2 (τ ) =− 24 (16R2 σ 3 )
0
(95)
19 Green String (2) Z
R
Z
LT
lim lim ′
ǫ→0ǫ →0
0
LT
Z
G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) 0 π ((n−m)(LT −ζ0 +ǫ′ )−n(LT −ζ0 )+mζ0 ) πmn exp R
(96)
= lim ǫ′ →0 0 kR4 σ 3 (1 − q k ) (q m − 1) (q n − 1) ′ 2πm(LT −2ζ0 +ǫ′ ) πk(LT −2ζ0 +ǫ′ ) πk(LT −2ζ0 +ǫ′ ) m n − 2πnǫ k − R R R R q e q e +1 q e +e +1 R
πk(−R + ζ + ǫ) πmζ πn(ζ − R) πm(−R + ζ + ǫ) πkζ sin (cos cos cos × lim sin ǫ→0 0 R R R R R πn(−R + ζ + ǫ) πmζ πn(ζ − R) πm(−R + ζ + ǫ) πn(−R + ζ + ǫ) × cos + sin sin sin sin dζ dζ0 R R R R R 2πnζ0 2πkζ 2πnζ πk(−2ζ0 ) −k − R 0 − R 0 2k − Z LT X ∞ X ∞ (−1)k (n(−1)n ) q n e R ∞ R m q e + e + q e X m (q + 1) π =− 3 3 dζ0 4R σ m=1 q m − 1 (k (q k − 1)) (q n − 1) 0 k=1 n=1 −k ∞ ∞ 1 X k(−1)k X (−1)n q q 2k q k−n q −n q 2n q n−k q 2k+n q k+2n π + + + − − − − =− 3 3 4R σ 2πR q k − 1 n=1 n (q n − 1) k + n k − n k − n k + n k − n k − n k+n k+n Z
k=1
Similar to the calculations in Green string (1) by making use of the symmetry of the summation over the indices m and n, this sum can be recast to the form −k ∞ ∞ 1 X X k(−1)k (−1)n q q 2k q k−n q −n q 2n q n−k q 2k+n q k+2n S= (97) + + + − − − − 2 (q k − 1) n (q n − 1) k + n k − n k − n k + n k − n k − n k+n k+n k=1 n=1 k(−1)k (−1)n q n−k q −k q 2k q k−n q k+2n q 2k+n q −n q 2n − + + + − − + − + k (q − 1) (n (q n − 1)) k−n k+n k−n k−n k+n k+n k+n k−n k+n k ∞ ∞ X k+n+1 −k−n k n k k X k(−1) q q +q q + 1 k q + 1 q − 1 − n q − 1 (q n + 1) = n(k − n)(k + n) (q k − 1) (q n − 1) n=1 k=1
In the case R >> LT with the use of identities (X-XII) the sum would read m(−1)m+n (x(m + n) − 1)(x(m + n) + 2)2 n(m + n) ∞ X ∞ 3 3 X 3m2 x2 (−1)m+n m x (−1)m+n 2 3 m+n 3 m+n 2 m+n + 2m x (−1) + + mnx (−1) + 3mx (−1) = n n m=1 n=1 2 2 3 2 − 1 x3 ζ(−2) + 20 − 1 24 − 1 x3 ζ(−3)ζ(1) = x3 22 − 1 ζ(−1) + 2 2 3 2 + 2 − 1 x ζ(−1) + 3 20 − 1 23 − 1 x2 ζ(−2)ζ(1) 2 1 = (x − 6)x2 16
S=
Z
R
Z
LT
(98)
(99) dζdζ 0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) 2πkζ0 2πnζ0 πk(−2ζ0 ) 2πnζ0 k n −k − 2k − Z ∞ ∞ ∞ R q n e R + e− R X m (q m + 1) LT X X (−1) (n(−1) ) q e R + q e π dζ0 =− 3 3 4R σ m=1 q m − 1 (k (q k − 1)) (q n − 1) 0 k=1 n=1 2 T T E2 (τ ) − πL − πL R −6 R =− 16 12 (8R2 σ 3 )
lim lim ′
ǫ→0ǫ →0
0
0
20 In case of LT → ∞ and using identity (XIII) the sum S would read S= =
∞ ∞ X m(−1)m X (−1)n q −m + q −n q m − 1 n=1 n (q n − 1) m + n m=1
(100)
∞ ∞ X X (m − n) (q −m + q −n ) 4n2 m2 b2 + d2 m=1 n=1
with b = 2 and d = 1 S=
∞ X
m(−1)m
m=1
=
1 2πR
∞ X
∞ X
n
(−1) +
2 n 4n b2 n=1
m2 d2
−
n mj
πLT R
j!
j
+
m mj
πLT R
j!
j
+
m nj
πLT R
j!
j
−
n nj
πLT R
j!
j
(101)
! πLT m m (−1) Φ(−1, 1, m + 1) e R − 1 − log(8)
m=1
2 − log(2) = 2πR
With the substitution of the above, the total Green string (2) becomes Z
R
Z
LT
(102) dζdζ0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) 2πkζ0 2πnζ0 πk(−2ζ0 ) 2πnζ0 k n −k − 2k − Z ∞ ∞ ∞ R q n e R + e− R X π m (q m + 1) LT X X (−1) (n(−1) ) q e R + q e =− 3 3 dζ0 4R σ m=1 q m − 1 (k (q k − 1)) (q n − 1) 0 n=1
lim lim ′
ǫ→0ǫ →0
0
0
k=1
π(2 − log(2))E2 (τ ) =− 12(2πR) (4R3 σ 3 ) (2 − log(2))E2 (τ ) =− 12 (8R2 σ 3 ) Green String (3) lim lim ′
ǫ→0ǫ →0
Z
0
R
Z
LT
0 R
dζdζ0 ∂α ∂α′ G ζ, ζ 0 ; ζ ′ , ζ0′ ∂α ∂β ′′ G (ζ ′′ , ζ ′′ ; ζ, ζ) ∂β ′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) LT
∞ X ∞ X ∞ X
′ πm m − 2πmǫ R q e − 1 ǫ→0ǫ →0 0 R4 σ 3 (q k − 1) (q m − 1) (q n − 1) 0 m=1 n=1 k=1 π ((k + m)(LT − ζ0 ) + ζ0 (k + n) + (n − m) (LT − ζ0 + ǫ′ )) exp − R 2πn(LT −ζ0 +ǫ′ ) 2πk(LT −ζ0 ) 2πkζ0 2πnζ0 k n R R R R q e q e −e +e πk(ζ − R) πm(x − R) πkζ sin sin sin R R R πm(−R + ζ + ǫ) πn(−R + ζ + ǫ) πnζ cos cos sin R R R 2πn L 2πk(LT −ζ0 ) ( T −ζ0 +ǫ′ ) 2πnζ0 2πkζ0 k n R R R R + e e +q e −q e πk(ζ − R) πm(ζ − R) πkζ cos cos sin R R R πnζ πm(−R + ζ + ǫ) πn(−R + ζ + ǫ) sin sin sin R R R
= lim lim ′
Z
Z
dζdζ0
(103)
21 The integrals over the spatial co-ordinates ζ yields a vanishing result I1 =
Z
0
=0
∞ X ∞ X ∞ R X
sin
πkζ R
m=1 n=1 k=1
πkζ R
sin
2πm(R − ζ) R
sin
πnζ R
πnζ R
sin
π(k − n)(R − ζ) R
dζ
(104)
and I2 =
Z
R
sin
0
=0
sin
2πm(R − ζ) R
sin
sin
π(k + n)(R − ζ) R
dζ
(105)
Green String (4) lim lim ′
ǫ→0ǫ →0
R
Z
0
Z
LT
0 R
dζdζ0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂β ′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) LT
(106)
∞ X ∞ X ∞ X
2πk(LT −2ζ0 ) 2πmǫ′ πmn m k R R q − e + q e ǫ→0ǫ →0 0 kR4 σ 3 (q k − 1) (q m − 1) (q n − 1) 0 m=1 n=1 k=1 π ((k + n)(LT − 2ζ0 ) + (m + n)ǫ′ ) exp − R πm(ζ − R) πnζ πm(−R + ζ + ǫ) πn(−R + ζ + ǫ) cos sin sin cos R R R R πnζ πm(−R + ζ + ǫ) πn(−R + ζ + ǫ) πm(ζ − R) cos cos sin + sin R R R R Z R Z LT ∞ X ∞ X ∞ 2 X π mn dζdζ0 = k (πkσ (1 − q )) (2R4 σ 2 (q n − 1)) 0 0 m=1 n=1 k=1 πn(ζ0 +2ζ0 ) πk(2ζ0 −LT ) 2πLT n πk(2ζ0 −LT ) 4πnζ0 R R R × qk e e− qn e R − e R + e− πk(ζ − R) 2πm(R − ζ) πn(R − 2ζ) πkζ sin sin sin sin R R R R
= lim lim ′
Z
Z
dζdζ0
The Green string has a zero value since the spatial integrals vanishes I=
Z
0
= 0.
R
dζ
∞ X ∞ X ∞ X
m=1 n=1 k=1
sin
πkζ R
sin
πk(ζ − R) R
sin
2πm(R − ζ) R
sin
πn(R − 2ζ) R
(107)
22 Green String (5) = lim lim ′
ǫ→0 ǫ →0
Z
R
LT
∞ X ∞ X ∞ X
π3 k
′′ −ζ ) πm(ζ0 0 m n R qm e n − 1) − 1) (q − 1) 0 m=1 n=1 k=1 ( ′ −ζ ′′ ) ′) ′ −ζ ′′ ) ′) πk(ζ0 πn(ζ0 −ζ0 πk(ζ0 πn(ζ0 −ζ0 0 0 R R R R qk e + e− + e− × qn e
Z
0
dζdζ0
R6 σ 3 (q k
πnζ +e sin sin R ′′ ′′ πm(ζ0 −ζ0 ) πm(ζ0 −ζ0 ) πkζ ′ πkζ ′′ πmζ ′′ πnζ ′ − m R R cos +e cos sin sin × q e R R R R ′ ′′ ′ ′ ′′ ′ πk(ζ0 −ζ0 ) πn(ζ0 −ζ0 ) πk(ζ0 −ζ0 ) πn(ζ0 −ζ0 ) R R R R × qn e q k −e − e− + e− ) πnζ πkζ ′ πnζ ′ πkζ ′′ πmζ ′′ πmζ sin sin sin sin sin × sin R R R R R R −
∞ X ∞ X ∞ X
=
m=1 n=1 k=1
′′ −ζ ) πm(ζ0 0 R
πmζ R
(108)
(q m
π3 k m n 8R5 σ 3 (q k − 1) (q m − 1) (q n − 1) ( πn(2ζ0 −LT ) πm(−2ζ0 ) πn2ζ0 πm(−2ζ0 ) R +e R + q 2n e R × q −n e− q k + 1 q m e− R − qk + 1
=0.
qm e
πm(−2ζ0 ) − R
+e
πm(−2ζ0 ) R
πn(2ζ0 −LT ) πn2ζ0 R q −n e− + q 2n e R
)
Green String (6) lim lim ′
ǫ→0ǫ →0
= lim lim ′
Z
ǫ→0 ǫ →0
R
LT
dζdζ0 ∂α′ ∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂β G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) dζdζ0
0
0
Z
Z
R
0
LT
= lim ′
ǫ →0
=
0
LT
dζdζ0
m=1 n=1 k=1
πmζ R
R6 σ 3 (q k
πnζ R
(q m
πkζ ′ πnζ ′ πkζ ′′ πmζ ′′ sin cos sin cos sin × sin R R R R ′ ′ ′′ ′ ′′ ′′ −ζ ′ ) ′ ′′ −ζ ′ ) πn(ζ0 −ζ0 ) πk(ζ0 −ζ0 ) πk(ζ0 −ζ0 ) πm(ζ0 πn(ζ0 −ζ0 ) πm(ζ0 0 0 n k − m − − R R R R R R + q e q −e −e q e +e +e ) πnζ πkζ ′ πnζ ′ πkζ ′′ πmζ ′′ πmζ sin sin sin sin sin × sin R R R R R R ∞ ∞ ∞ X X X π 3 (−1)k+n m2 n2 π (k(LT − 2ζ0 ) + n(LT + 2ζ0 ) + (m + n)ǫ′ )) exp − dζ0 8R5 σ 3 k(q k − 1) (q m − 1) (q n − 1) R m=1 n=1 k=1 2πk(LT −2ζ0 ) 2πmǫ′ 4πnζ0 2πn(LT −ǫ′ ) R R × e e R − qn e + qk e R − qm
∞ X ∞ X ∞ X
=0
π3 k
′′ −ζ ) πm(ζ0 0 m n R qm e n − 1) − 1) (q − 1) 0 m=1 n=1 k=1 ( ′) ′ −ζ ′′ ) ′′ −ζ ′ ) ′) ′ −ζ ′′ ) ′′ −ζ ′ ) πn(ζ0 −ζ0 πk(ζ0 πm(ζ0 πn(ζ0 −ζ0 πk(ζ0 πm(ζ0 0 0 0 0 n k m − − − R R R R R R × q e q e q e +e +e +e
Z
Z
∞ ∞ X ∞ X X
(109)
π 2 (−1)k+n nm2 (q n + 1) (q k+n − 1) k (q + 1)(q k − q n ) 16R4 σ 3 k(q k − 1) (q n − 1) q k+n
23 The Green string approaches zero owing to the zero value of the regularized divergent sum Green String (7)
lim lim ′
ǫ→0ǫ →0 ∞ X
Z
R
P∞
m=1
= ζ(2) = 0.
LT
Z
dζdζ0 ∂β G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂α′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )
0 0 3 3
(110)
′ π k πkǫ′ k − πkǫ R R + e q e R6 σ 3 (1 − q k ) k=1 ( Z LT ∞ X ∞ X πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) 1 1 m n − − R R R R q e × lim dζ q e + e + e 0 ǫ′ →0 0 (q m − 1) (q n − 1) m=1 n=1 Z R πnζ πk(ζ − R) πn(ζ − R) πk(ζ − R + ǫ) πn(ζ − R + ǫ) πmζ cos sin sin sin cos × lim dζsin ǫ→0 0 R R R R R R ′ ′ πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ ) πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ ) R R R R + qn e + qm e − e− − e−
=
R
πmζ πnζ πk(ζ − R) πn(ζ − R) × lim dζsin cos sin sin ǫ→0 0 R R R R ) πk(ζ − R + ǫ) πm(ζ − R + ǫ) × sin sin R R Z
=
∞ ∞ ∞ X 1 X k3 1 π3 X , 8R5 σ 3 m=1 q m − 1 n=1 q n − 1 1 − qk k=1 Z LT πm(ǫ′ −2ζ0 ) πm(ǫ′ −2ζ0 ) πn(2ζ0 ) πn(2ζ0 ) 2n m − −n − R R R R e − q e − q e dζ0 , lim × q e ′ 0
3
=
π 8R5 σ 3
∞ X
∞ X
ǫ →0
∞
1 X k3 1 q m − 1 n=1 q n − 1 1 − qk m=1 k=1 −m q q 2m q m−n q −n q 2n q n−m q 2m+n q m+2n . + + + − − − − m+n m−n m−n m+n m−n m−n m+n m+n
Let us separately evaluate the sum −m q q 2m q m−n q −n q 2n q n−m q 2m+n q m+2n 1 + + + − − − − S= 2 (q m − 1) (q n − 1) m + n m − n m − n m + n m − n m − n m + n m+n n−m −m 2m m−n m+2n 2m+n −n 1 q q q q q q q q 2n + − + + + − − + − 2 (q m − 1) (q n − 1) m−n m+n m−n m−n m+n m+n m+n m−n 2q −m−n (q m + q n ) (q m+n + 1) (m (q m + 1) (q n − 1) − n (q m − 1) (q n + 1)) = (m − n)(m + n) (q m − 1) (q n − 1)
(111)
In case of R >> LT finite temperature and utilizing identities (III), (IV) and (IX) the above sum becomes ∞ X ∞ X q −m−n (q m + q n ) (q m+n + 1) (m (q m + 1) (q n − 1) − n (q m − 1) (q n + 1)) S= (m − n)(m + n) (q m − 1) (q n − 1) m=1 n=1
=
∞ X ∞ X
m=1 n=1
= ∞ X
x m2 x2 + 2mnx2 + 4mx + n2 x2 + 4nx + 4 − x(x(m + n) + 4)
3(−1)(−1)x2 3(−1)(−1)x2 2(−1)(−1)x3 + + 12 × 12 2 × 12 2 × 12
Z LT ∞ X πn(2ζ0 ) πm(−2ζ0 ) πm(−2ζ0 ) πn(2ζ0 ) 1 1 −n − m − 2n R R R R q e dζ0 e − q e − q e q m − 1 n=1 q n − 1 0 m=1 2 1 πLT πLT = − 18 − −4 72 R R
S=
(112)
(113)
24 The total string of Green functions becomes lim lim ′
ǫ→0ǫ →0
Z
R
0
Z
LT
0 3
dζdζ0 ∂β G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂α′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) ,
(114)
∞ ∞ ∞ X X 1 1 π k3 X = 8R5 σ 3 1 − q k m=1 q m − 1 n=1 q n − 1 k=1 Z R πm(ǫ′ −2ζ0 ) πm(ǫ′ −2ζ0 ) πn(2ζ0 ) πn(2ζ0 ) − m R R −e × q −n e− R − q 2n e R dζ0 , q +e 0 2 πLT 1 2 T − πL 18 − − 4 π (E4 (τ ) + 1) 72 R R = 240 (16R4 σ 3 )
In case of LT → ∞ with the utilization of identities (IV-IX) ∞ X 1 1 S= m−1 n−1 q q n=1 m=1 ∞ X
= =
lim
ǫ′ →0
Z
LT
0
∞ X ∞ X q −m + q −n m+n m=1 n=1
q
−n −
e
πn(2ζ0 ) R
2n
−q e
πn(2ζ0 ) R
e
πm(ǫ′ −2ζ0 ) R
m −
−q e
πm(ǫ′ −2ζ0 ) R
dζ0
!
(115)
∞ X ∞ X 2q −n m+n m=1 n=1
∞ X 1 q −n (πn cot(πn) − 1) (H−n + Hn − 2γ) q −n − 2 2n n=1 ∞ log 1 − 1q X 1 q −n (πn cot(πn) − 1) γ + (−γ − 2 + log(2)) − − =− q−1 2 2n 2 1− 1 n=1
=
q
The total sum is then Z RZ lim lim ′ ǫ→0ǫ →0
0
=
LT 0 3
dζdζ0 ∂β G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂α′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )
∞ X k3 π 8R5 σ 3 1 − qk k=1
π 2 (π − 2 + log(2))(E4 (τ ) + 1) = (480) (16R4 σ 3 )
(116)
25 Green String (8)
lim lim ′
ǫ→0ǫ →0
Z
R 0
Z
LT
dζdζ0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂β ′′ ∂α′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )
0
R
LT
(117)
∞ X ∞ X ∞ X
π3 k2 m2 (−1)m+n+1 1 6 σ 3 (q k − 1) m − 1) n − 1) ǫ→0 ǫ →0 0 R (q n(q 0 m=1 n=1 k=1 π ((k + m)ǫ′ + (LT − 2ζ0 )(m + n)) × exp − R 2πn(LT −2ζ0 ) 2π ((k+m)ǫ′ +m(LT −2ζ0 )) 2πm(LT −2ζ0 +ǫ′ ) 2πkǫ′ k+m n m k R R R R × e q e +q +e q e +q πnζ π(k(ζ + ǫ) + mζ) πm(ζ + ǫ) πkζ sin2 cos sin × sin R R R R Z R Z LT ∞ ∞ ∞ 3 2 2 m+n+1 X XX π(LT −2ζ0 )(m+n) π k m (−1) 1 R dζdζ0 = e− 6 3 k m n R σ (q − 1) (q − 1) n(q − 1) 0 0 m=1 n=1 k=1 2πn(LT −2ζ0 ) 2πm(LT −2ζ0 ) π(k(ζ + ǫ) − mζ) k+m n R R q e +q + 1 cos × e R 2πm(LT −2ζ0 ) π(k(ζ + ǫ) + mζ) πkζ πnζ πm(ζ + ǫ) k m 2 2 2 R + q +q e cos sin sin sin R R R R Z LT ∞ X ∞ X ∞ 3 2 2 m+n X 2πm(L −2ζ ) 2πn(L −2ζ π(L −2ζ )(m+n) π k m (−1) 1 0 0) 0 T T T R R R qm e −1 e + qn dζ0 = e− 5 3 m n 8R σ (q − 1) n(q − 1) 0 m=1 n=1 k=1 πn(LT −2ζ0 ) 2πm(LT −2ζ0 ) πn(LT −2ζ0 ) 2 m ∞ (−1)n q n e− ∞ ∞ R R R qm e −1 X +e X π 3 X m (−1) =− k2 n − 1) 8R5 σ 3 m=1 (q m − 1) n(q n=1 Z
= lim lim ′
Z
dζdζ0
k=1
=0.
The integral vanishes by virtue of the regularization of the divergent sum Green String (9) lim lim ′
ǫ→0ǫ →0
= lim lim ′
Z
ǫ→0 ǫ →0
=
R
R 0
∞ X ∞ X ∞ X
k=1
k 2 = ζ(2) = 0.
LT
dζdζ0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂β ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′′ ∂β ′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )
0
0
Z
Z
P∞
LT
∞ X ∞ X ∞ X
m2 1 π4 k2 6 σ 2 (q k − 1) (q m − 1) n(q n − 1) R 0 m=1 n=1 k=1 πnζ πn(ζ − R) πmζ πk(ζ − R) × sin sin cos cos R R R R πm(ζ − R + ǫ) πk(ζ − R + ǫ) sin × sin R R πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πkǫ′ πkǫ′ R R R R × qn e qm e + e− + e− q k e− R + e R πk(ζ − R) πk(ζ − R + ǫ′ ) πm(ζ − R + ǫ′ ) πmζ sin sin sin + sin R R R R ′ ′ πm(LT −2ζ0 +ǫ′ ) πm(LT −2ζ0 +ǫ′ ) πkǫ πkǫ R R × qm e − e− e R − q k e− R
Z
dζdζ0
π3 k2 m2 1 5 3 k m n 8R σ (q − 1) (q − 1) n(q − 1) m=1 n=1 k=1 πkǫ′ ′ πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) m k − πkǫ − R R R R R R qn e q e −q e −e + e− × e
=0.
(118)
26 since the k-indexed summation Green String (10) R
Z
lim lim ′
ǫ→0ǫ →0
Z
R
Z
= lim lim ′
ǫ→0 ǫ →0
k 2 = 0 vanishes after regularization.
dζdζ0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂β ′′ ∂β ′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) LT
dζdζ0
∞ X ∞ X ∞ X
2πnζ0
∞ X ∞ X ∞ X
(119)
π4 k2 m2 R6 σ 2 (q k − 1) (q m − 1) 0 m=1 n=1 k=1 πmζ πk(ζ − R) πk(ζ − R) πmζ cos sin cos × sin R R R R πk(ζ − R + ǫ) πm(ζ − R + ǫ) × cos2 sin2 R R πkǫ′ πkǫ′ πkǫ′ πkǫ′ × e R − q k e− R e R + q k e− R πm(LT −2ζ0 +ǫ′ ) πm(LT −2ζ0 +ǫ′ ) πm(LT −2ζ0 +ǫ′ ) πm(LT −2ζ0 +ǫ′ ) R R R R qm e − e− + e− × qm e
Z
0
k=1
LT
0
0
P∞
2πnζ0
m2 (−1)m+n q −n e− R + q 2n e R π3 k2 = 8R5 σ 3 (q k − 1) (q m − 1) n(1 − q n ) m=1 n=1 k=1 πkǫ′ ′ πm(ǫ′ −2ζ0 ) πm(ǫ′ −2ζ0 ) k − πkǫ m − R R R R × e e −q e −q e
=0.
P∞
for the same reason, Green String (11) lim lim ′
ǫ→0ǫ →0
= lim lim ′
Z
ǫ→0 ǫ →0
= lim ′
ǫ →0
=0.
Z
0
k=1
R 0
Z
Z
LT
0
R
0
k 2 = 0 as Green strings (6), (8) and (9).
dζ0
∞ X ∞ X ∞ X
π4 k2 m2 (−1)m+n 8R6 σ 2 (q k − 1) (q m − 1) 0 m=1 n=1 k=1 2πk(2ζ + ǫ) πm(2ζ + ǫ) 2πnζ 1 + cos 1 − cos × 1 − cos R R R πk(2ζ + ǫ) πm(2ζ + ǫ) × sin sin R R πkǫ′ ′ ′ πm(ǫ −2ζ0 ) πm(ǫ′ −2ζ0 ) k − πkǫ m − R R R R × e e −q e −q e πkǫ′ ′ πm(ǫ′ −2ζ0 ) πm(ǫ′ −2ζ0 ) k − πkǫ m − R R R R e −q e +q e × −e
Z
LT
LT
dζdζ0 G ζ, ζ 0 ; ζ ′ , ζ 0′ ∂α ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂β ′′ ∂β ′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) dζdζ0
∞ X ∞ X ∞ X
m=1 n=1 k=1
m2 (−1)m+n 1 m − 1) (q − 1) πnσ(1 − q n ) πkǫ′ πm(ǫ′ −2ζ0 ) πn(2ζ0 ) πm(ǫ′ −2ζ0 ) πkǫ′ R R e R − q k e− R e + q 2n e R − q m e− π4 k2
8R6 σ 2 (q k
πn(2ζ0 ) × q −n e− R
(120)
27 Green String (12)
lim lim ′
ǫ→0ǫ →0
Z
R
Z
LT
dζdζ0 ∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂β ′ ∂β′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )
0 0 3 3
(121)
1 1 π k R6 σ 3 (1 − q k ) (1 − q m ) (1 − q n ) πmǫ′ πkǫ′ πn(2ζ0 −LT ) πn(2ζ0 −LT ) πkǫ′ πmǫ′ R R qn e e− R + q m e R + e− × e R + q k e− R Z R πk(ζ − R) πm(ζ − R) πnζ + sin sin sin R R R 0 πn(ζ − R) πk(ζ − R + ǫ) πm(ζ − R + ǫ) × cos cos sin dζ R R R π3 k3 1 1 + 6 3 R σ (1 − q k ) (1 − q m ) (1 − q n ) πn(2ζ0 −LT ) πkǫ′ πn(2ζ0 −LT ) πkǫ′ πmǫ′ πmǫ′ R R − qn e e− q m e R − e− R × e R + q k e− R Z R πk(ζ − R) πm(ζ − R) πnζ sin + sin sin R R R 0 πn(ζ − R) πk(ζ − R + ǫ) πm(ζ − R + ǫ) × sin sin sin dζ R R R ∞ Z LT ∞ ∞ X πm(−2ζ0 ) πn(2ζ0 ) πn(2ζ0 ) πm(−2ζ0 ) 1 1 π2 X k3 qk + 1 X m − 2n −n − R R R R − q e − q e q e e =− 32R4 σ 3 q k − 1 m=1 q m − 1 n=1 q n − 1 0 =−
k=1
Similar to Equation(23) one can calculate the contributions of the sums at finite temperature R >> LT as Z LT ∞ X πm(−2ζ0 ) πn(2ζ0 ) πm(−2ζ0 ) πn(2ζ0 ) 1 1 −n − m − 2n R R R R e q e − q e − q e q m − 1 n=1 q n − 1 0 m=1 ! 2 πLT πLT π2 − 18 − =− −4 72 × 32R4 σ 3 R R
S=
∞ X
(122)
The total sum is R
Z
LT
dζdζ0 ∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂β ′ ∂β′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) 2 πL 1 − πL 18 − − 4 π 2 (E4(τ ) + 1) 72 R R
lim lim ′
ǫ→0ǫ →0
=
Z
(123)
0
0
240 (16R4 σ 3 )
In the zero temperature limit LT → ∞ the string of Green functions takes the form lim lim ′
ǫ→0ǫ →0
=
Z
0
R
Z
0
LT
dζdζ0 ∂α′ G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂β ′ ∂β′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ )
π 2 (π − 2 + log(2))(E4(τ ) + 1) (480) (16R4 σ 3 )
(124)
28 Green string (13) lim lim ′
ǫ→0ǫ →0
= lim lim ′
Z
ǫ→0 ǫ →0
R
0
Z
0
Z
LT
0
R
LT
dζdζ0 ∂β ∂α G ζ, ζ 0 ; ζ ′ , ζ 0′ ∂β ∂α′′ G (ζ ′′ , ζ0′′ ; ζ, ζ0 ) ∂α′′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) , ∞ X ∞ X ∞ X
πnζ R 0 m=1 n=1 k=1 πk(ζ − R + ǫ) πm(ζ − R + ǫ) πk(ζ − R) cos cos × cos R R R πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) R R R R × qn e qm e − e− − e− −πkǫ′ −πkǫ′ πkǫ′ πkǫ′ × q k −e R −e R q k −e R −e R πmζ πmζ cos + sin R R πk(ζ − R) πk(ζ − R + ǫ) πm(ζ − R + ǫ) × sin sin sin R R R ′ ′ πm(LT −2ζ0 +ǫ ) πm(LT −2ζ0 +ǫ ) R R × qm e . − e−
Z
dζdζ0
sin
πn(ζ − R) R
(125)
cos
(126) (127)
Performing integrals over spatial co-ordinates
I1 =
Z
R
dζ
0
=0. I2 =
Z
0
= 0,
∞ X ∞ X ∞ X
πmζ πn(ζ − R) πnζ sin cos R R R m=1 n=1 k=1 πm(ζ − R + ǫ) πk(ζ − R + ǫ) sin × sin R R
∞ X ∞ X ∞ R X
m=1 n=1 k=1
sin
πk(ζ − R) R
sin
πkζ R
yields a zero value for the Green string.
cos
cos
πmζ R
sin
2πnζ R
sin
πk(ζ + ǫ) R
sin
πm(ζ + ǫ) R
(128)
dζ
29 Green String (14) lim lim ′
ǫ→0ǫ →0
Z
R
dζdζ0
0
0
R
∞ X ∞ X ∞ X
LT
Z
G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′ ∂α G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) ,
m=1 n=1 k=1 ∞ X ∞ X ∞ X
LT
m2 1 π4 k2 m ǫ→0 ǫ →0 0 − 1) (q − 1) πnσ(1 − qn ) 0 m=1 n=1 k=1 ( πn(ζ − R) πnζ πk(ζ − R) πmζ × sin sin sin cos R R R R πk(ζ − R + ǫ) πm(ζ − R + ǫ) × cos sin R R πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) m − − n R R R R q e +e +e × q e πkǫ′ πkǫ′ × q k e− R + e R πmζ πk(ζ − R + ǫ) πm(ζ − R + ǫ) πk(ζ − R) sin sin sin + sin R R R R ) ′ ′ πm(LT −2ζ0 +ǫ ) πm(LT −2ζ0 +ǫ ) πkǫ′ πkǫ′ R R × q k e− R − e R qm e − e−
= lim lim ′
= lim ′
ǫ →0
Z
Z
Z
LT
dζ0
0
dζdζ0
∞ X ∞ X
n=1 m=1
= 0.
−
π 3 m2 q −n e−
R6 σ 2 (q k
2πζ0 (m+n) R
qm e
4πmζ0 R
−1
q 3n e
4πnζ0 R
8nR5 σ 3 (q m − 1) (q n − 1)
+1
∞ X
k2
k=1
Green String 15 lim lim ′
ǫ→0ǫ →0
Z
0
R
Z
R
LT
dζdζ0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′ ∂α G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) ,
0 LT
∞ X ∞ X ∞ X
π4 k2 m2 1 6 2 k m ǫ→0 ǫ →0 0 R σ (q − 1) (q − 1) πnσ(1 − q n ) 0 m=1 n=1 k=1 ( πnζ ′ πmζ πkζ ′ πkζ ′′ πmζ ′′ πnζ sin sin cos sin cos × sin R R R R R R ′ ′ πn(ζ0 −ζ0 ) πn(ζ0 −ζ0 ) R R + e− × qn e ′′ −ζ ) ′ −ζ ′′ ) ′′ −ζ ) ′ −ζ ′′ ) πm(ζ0 πk(ζ0 πm(ζ0 πk(ζ0 0 0 0 0 R R R R × qm e qk e + e− + e− πkζ ′ πkζ ′′ πmζ ′′ πmζ cos cos cos + cos R R R R )
= lim lim ′
Z
Z
× qm e
dζdζ0
′′ −ζ ) πm(ζ0 0 R
− e−
−
′′ −ζ ) πm(ζ0 0 R
qk e
′ −ζ ′′ ) πk(ζ0 0 R
− e−
′ −ζ ′′ ) πk(ζ0 0 R
(129)
30 = lim lim ′
ǫ→0 ǫ →0
×
R
Z
0
(
× cos
Z
LT 0
q k e−
πkǫ′ R
πk(ζ − R) R
× cos
πk(ζ − R) R
R
lim lim ′
Z
ǫ→0ǫ →0
=
−
(130)
πmζ R
πn(ζ − R) R
πnζ πk(ζ − R + ǫ) πm(ζ − R + ǫ) cos cos R R R ′
πmζ R
πn(ζ − R) R
πnζ R
cos sin sin πm(LT −2ζ0 +ǫ ) πm(LT −2ζ0 +ǫ′ ) πkǫ′ R R +e R + e− qm e
πkǫ′ + q k e− R
Z
∞ X ∞ X ∞ X
πn(2ζ0 −LT ) πn(2ζ0 −LT ) m2 1 π3 k2 n − R R q e + e R6 σ 3 (q k − 1) (q m − 1) n(1 − q n ) m=1 n=1 k=1 πm(LT −2ζ0 +ǫ′ ) πm(LT −2ζ0 +ǫ′ ) πkǫ′ R R qm e −e R − e−
dζdζ0
sin
sin
sin
sin
πk(ζ − R + ǫ) R
cos
πm(ζ − R + ǫ) R
)
LT
0 0 ∞ X ∞ X ∞ X
k=1 m=1 n=1
dζdζ0 G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′ ∂α′′ ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′ ∂α G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) , lim ′
ǫ →0
Z
LT
0
= 0.
dζ0 k 2
(131)
πn(2ζ0 −LT ) πn(2ζ0 −LT ) π 3 m2 n − R R q e + e 8nR5 σ 3 (q m − 1) (1 − q n ) πm(LT −2ζ0 +ǫ′ ) πm(LT −2ζ0 +ǫ′ ) R R − e− × qm e
Green String 16 lim lim ′
ǫ→0ǫ →0
Z
R
0
Z
LT
0 R
dζdζ0 ∂β G (ζ, ζ0 ; ζ ′ , ζ0′ ) ∂α′′ G (ζ ′′ , ζ0′′ ; ζ ′ , ζ0′ ) ∂α′′ ∂α′′ ∂β ′ ∂α′ G (ζ ′ , ζ0′ ; ζ ′′ , ζ0′′ ) , LT
∞ X ∞ X ∞ X
(132)
1 1 π3 k3 6 3 k m n ǫ→0 ǫ →0 0 R σ (q − 1) (q − 1) (q − 1) 0 m=1 n=1 k=1 ( πk(ζ − R) πmζ πn(ζ − R) πnζ πk(ζ − R + ǫ) πm(ζ − R + ǫ) × cos sin sin sin sin cos R R R R R R πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πkǫ′ πkǫ′ R R R R × q k e− R − e R qn e qm e − e− − e− πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) πm(LT −2ζ0 +ǫ′ ) πn(2ζ0 −LT ) πkǫ′ πkǫ′ R R R R qm e + e− − e− qn e × q k e− R − e R ) πmζ πn(ζ − R) πnζ πk(ζ − R + ǫ) πm(ζ − R + ǫ) πk(ζ − R) sin sin cos sin sin . × cos R R R R R R
= lim lim ′
Z
Z
dζdζ0
−
Integrals over spatial co-ordinates Z R πk(ζ − R) πmζ πnζ I1 = lim cos sin sin ǫ→0 0 R R R πn(ζ − R) πk(−R + ζ + ǫ) πm(−R + ζ + ǫ) × sin sin cos dζ, R R R = 0,
(133)
31 and I2 = lim
ǫ→0
= 0.
Z
0
R
cos
πmζ πn(ζ − R) sin sin R R πk(−R + ζ + ǫ) πm(−R + ζ + ǫ) πnζ sin sin dζ, × cos R R R
πk(ζ − R) R
yield a vanishing value for the Green string.
(134)
