On QCD strings beyond non-interacting model
A. S. Bakry, M. A. Deliyergiyev, A. A. Galal, M. Khalil William
AAIP/123-QED
On QCD strings beyond non-interacting model
A. S. Bakry, M. A. Deliyergiyev,
2, 1, ∗ A. A. Galal, and M. KhalilA Williams
3, 4, 5, 6 Institute of Modern Physics, Chinese Academy of Sciences, Gansu 730000, China Institute of Physics, The Jan Kochanowski University in Kielce, 25-406, Poland Department of Physics, Al Azhar University, Cairo 11651, Egypt Department of Mathematics, Bergische Universit¨at Wuppertal, 42097 Germany Department of Physics, University of Ferrara, Ferrara 44121, Italy Research and computing center, The Cyprus Institute, Nicosia 2121, Cyprus (Dated: January 12, 2020)We investigate the implications of Nambu-Goto (NG), Lscher-Weisz (LW) and Polyakov-Kleinert(PK) string actions for the Casimir energy of the QCD flux-tube at one and two loop order at finitetemperature. We perform our numerical study on the 4-dim pure SU(3) Yang-Mills lattice gaugetheory at finite temperature and coupling β = 6 .
0. The static quark-antiquark potential is calculatedusing link-integrated Polyakov loop correlators. At a high temperature-close to the critical point-We find that the rigidity and self-interactions effects of the QCD string to become detectable. Theremarkable feature of this model is that it retrieves a correct dependency of the renormalized stringtension on the temperature. Good fit to static potential data at source separations R ≥ . R = 0 . PACS numbers: 12.38.Gc, 12.38.Lg, 12.38.Aw
I. INTRODUCTION
The formulation of a string theory for hadrons hasbeen an attractive proposal since the phenomenologicalsuccess in explaining Venziano formula [1] even beforethe formulation of quantum chromodynamics (QCD).Despite the difficulties encountered in the quantizationscheme in a fundamental string theory, the proposalto describe the long-distance dynamics of strong inter-actions inside the hadrons by a low energy effectivestring [2, 3] has remained an alluring conjecture.String formation [4–15] is realized in many stronglycorrelated systems and is not an exclusive property ofthe QCD color tubes [16–20]. The normalization groupequations imply that the system flows towards a rough-ening phase where the transverse string oscillations tothe classical world sheet effects becomes measurable andcan be verified in numerical simulations of lattice gaugetheories (LGT).In the leading Gaussian approximation of the NG ac-tion, the quantum fluctuations of the string bring fortha universal quantum correction to the linearly rising po-tential well known as the L¨uscher term in the mesonicconfigurations. In the baryon a geometrical L¨uscher-liketerms [21, 22] ought to manifest.The width due to the quantum delocalizations of thestring grows logarithmically [23] as the two color sourcesare pulled apart. The character of logarithmic broaden-ing is expected for the baryonic junction [24] as well. Pre- ∗ [email protected] cise lattice measurements of the Q ¯ Q potential in SU (3)gauge model are in consistency with the L¨uscher sublead-ing term for color source separation commencing fromdistance R = 0 . /T c [25] where the ef-fects of the intrinsic thickness [26] of the flux tube dimin-ish. Many gauge models have unambiguously identifiedthe L¨uscher correction to the potential with unpreceden-tial accuracy [27–32].In addition, the string model predictions for the loga-rithmic broadening [23] of the mean-square width of thestring at very low temperatures has been observed inseveral lattice simulations corresponding to the differentgauge groups [27, 33–43]In the high temperature regime of QCD, the overlap ofthe string’s excited states spectrum would lead to a newquantum state with different characteristics. The freestring approximation implies a decrease in the slop of the Q ¯ Q potential or in other-words a temperature-dependenteffective string tension [44, 45]. The leading-order cor-rection for the mesonic potential turns into a logarithmicterm of the Dedekind η function which encompasses theL¨uscher term as a zero temperature T = 0 limiting term[3, 46, 47]. With respect to the string’s width the loga-rithmic broadening turns into a linear pattern before thedeconfinement is reached from below [48–51].Nevertheless, this non-interacting model of the bosonicstring derived on the basis of the leading order formula-tion of NG action poorly describes the numerical data inthe intermediate distances at high temperatures. For in-stance, substantial deviations [52–55] from the free stringbehavior have been found for the lattice data corre-sponding to temperatures very close to the deconfinement a r X i v : . [ h e p - l a t ] F e b point.The comparison with the lattice Monte-Carlo data sup-ports the validity of the leading-order approximation atsource separations larger than R > . Q ¯ Q potential and the color-tube width profile.The descripancies casts over source separation distancesat which the leading-order string model predictions arevalid [3] at zero temperature. In the baryon [56–58],taking into account the length of the Y-string betweenany two quarks, we found a similar behavior [59].The fact that the lattice data substantially deviatefrom the free string description at the intermediate dis-tances and high temperatures has induced many numer-ical experiments to verify the validity of higher-ordermodel-dependent corrections for the NG action [60, 61]before the string breaks [62].In the Nambu-Goto (NG) framework it seems thatthere is no reason to believe that all orders of the powerexpansion are relevant to the correct behavior of QCDstrings [63]. For example, a first-order term deviatingfrom the universal behavior has been determined unam-biguously in 3D percolation model [63], no numericalevidence indicating universal features of the correctionsbeyond the L¨uscher term have been encountered among Z (2), SU (2) and SU (3) confining gauge models [60]. Nu-merical simulations of different gauge models in differentdimensions may culminate in describing both the inter-mediate and long string behavior with different effectivestrings [60].However, the modeling of QCD flux-tubes beyond thefree string approximation may suggest considering otherpossiblities such as string’s resistance to bending. Thesestrings with rigid structure ought to exhibit smooth fluc-tuations [64, 65]. The idea that QCD strings may berigid appeared long ago and been extensively scrutinizedby Kleinert and German.The perturbative two loop potential at T = 0, theexact potential in the large dimension limit [66, 67], thedynamical generation of the string tension [68] have beenstudied, for example. The theory has other well-foundedthermodynamical characteristics such as the partitionfunction [69], free energy and string tension at finitetemperature [70–72] and the deconfinement transitionpoint [73].In this effective string theory only the smooth flux-sheets over long distance are favored and the sharplycreased surfaces are excluded. This implies peculiar ge-ometrical chracteristics that is being controlled via theextrinsic curvature or shape tensor of the surfaces.The string’s rigidity can intuitively understood in rel-evance to the vortex line picture of the string which indi-cates a repulsive [74] nature among the flux tubes. Thisinterpretation seems consisent with flux network withinbaryon [56–58] which indicates that the sharply-creasedflux sheets are energtically unfavorable. The strings ap-pears aligning itself, either via temperature change or thecolor sources location, such that the angles between thethree flux tubes are equally divided into 120 o [56–58]. Reviving interests appeared recently to address therigidity of the QCD flux-tube in the numerical simula-tions of the confining potential. In fact, both U (1) com-pact gauge group [29, 75] and SU ( N ) gauge theories in3 D [76] has been reported.The contribution of the boundary action to the openstring partition function come into play to recover thesymmetry breaking by the cylindrical boundary condi-tion. Two-variant formulas for the Lorentz-invariantboundary corrections to the static Q ¯ Q potential havebeen calculated in both the Wilson and the Polyakov-loop correlators cases [75, 77–79]These corrections are hoped to reflect some featuresof the fine structure of the profile of QCD flux-tube [76,80 ? –83] at relatively short distances/low temperatures,larger distances/high temperature [27] or the excitedspectrum of the flux-tubes [27, 84, 85]. Indeed, detectableeffects for the boundary corrections to the static quarkpotential have been shown viable on mitigating the de-viations from predictions of the effective string and nu-merical outcomes [75, 76, 86]. We report similar findingsin regard to the width profile of the QCD flux-tube nearthe critical point [].One goal of the present paper is to examine theLorentz-invariant boundary terms in Lscher-Weisz (LW)effective string action for open string with Dirichletboundary condition on a cylinder. The analytic estimatelaid out for the static potential resulting from two bound-ary terms at the order of fourth and six derivative couldbe compatible with the energy fields set up by a staticmesonic configurations.The pure SU (3) Yang-Mills theory in four dimensionsis the closest approximation to full QCD. Even though,we are lacking detailed understanding for the string be-havior at high temperature and short distance scale. Inthis region the deviations from the free string behavioroccurs on scales that is relevant to full QCD before thestring breaks [62]. The nature of the QCD strings atfinite temperatures can be very relevant to many por-trayals involving high energy phenomena [88, 89] suchas mesonic spectroscopy [90–92], glueballs [93, 94] andstring fireballs [95], for example. This calls for a dis-cussion of the validity of string effects beyond the freebosonic Nambu-Goto string which is the target of thisreport.The map of the paper is as follows: In section(II), wereview the most relevant string model to QCD and dis-cuss the lattice data corresponding to the Casimir energyversus different approximation schemes. In section(III),the numerical investigation is focused on the width profileof the energy density and the predictions of the Nambu-Goto (NG) and Polyakov-Kleinert (PK) strings. Con-cluding remarks and summary are provided in the lastsection. II. STRING ACTIONS AND CASMIR ENERGY
The conjecture Yang-Mills (YM) vacuum admits theformation of a very thin string-like object [2, 96, 97] hasoriginated in the context of the linear rise property of theconfining potential between color sources. The intuitionis in consistency [98] with the dual superconductivity [29,47, 99–103] property of the QCD vacuum. The colorfields are squeezed into a confining thin string dual tothe Abrikosov line by the virtue of the dual Meissnereffect.The formation of the string condensate spontaneouslybreakdown the transnational and rotational symmetriesof the YM-vacuum and entails the manifestation of (D-2)massless transverse Goldstone modes in addition to theirinteractions [104, 105].To establish an effective string description, a string ac-tion can be constructed from the derivative expansion ofcollective string co-ordinates satisfying Poincare and par-ity invariance. One particular form of this action is theL¨uscher and Weisz [2, 3], in the physical gauge [106, 107],which encompasses built-in surface/boundary terms toaccount for the interaction of an open string with bound-aries. The L¨uscher and Weisz [41] (LW) effective actionup to four-derivative term read S LW [ X ] = σ A + σ (cid:90) d ζ (cid:34) (cid:18) ∂ X ∂ζ α · ∂ X ∂ζ α (cid:19) + κ (cid:18) ∂ X ∂ζ α · ∂ X ∂ζ α (cid:19) + κ (cid:18) ∂ X ∂ζ α · ∂ X ∂ζ β (cid:19) (cid:35) + γ (cid:90) d ζ √ g R + α r (cid:90) d ζ √ g K + S b (1)with the physical gauge X = ζ , X = ζ which re-stricts the string fluctuations to transverse directions C .The vector X µ ( ζ , ζ ) maps the region C ⊂ R into R and couplings κ , κ are effective low-energy parameters.Invariance under party transform would keep only evennumber derivative terms.The open-closed duality [41] imposes constraint on thekinematically-dependent couplings κ , κ κ + κ = − σ . (2)which can be shown [78, 108] through a nonlinear realiza-tion of Lorentz transform is valid in any dimension d . Theabove condition Eq. (2) implies that all the terms withonly first derivatives in the effective string action Eq. (1)coincide with the corresponding one of Nambu-Goto ac-tion in the derivative expansion. The Nambu-Goto actionis the most simple form of string actions proportional toarea of the world-sheet S NG [ X ] = σ (cid:90) d ζ (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) (cid:18) ∂X∂ζ (cid:19) + (cid:18) ∂X∂ζ (cid:19) (cid:33) S NG (cid:96) o [ X ] = σ A + σ (cid:90) dζ (cid:18) ∂ X ∂ζ α · ∂ X ∂ζ α (cid:19) ,S NGn (cid:96) o [ X ] = σ (cid:90) dζ (cid:34)(cid:18) ∂ X ∂ζ α · ∂ X ∂ζ α (cid:19) + (cid:18) ∂ X ∂ζ α · ∂ X ∂ζ β (cid:19) (cid:35) , (3)where g is the two-dimensional induced metric on theworld sheet embedded in the background R . On thequantum level the Weyl invariance of the NG actionis broken in four dimensions; however, the anomaly isknown to vanish at large distances [98].The boundary term S b describes the interplay betweenthe effective string with the Polyakov loops [41] at thefixed ends of the string and is given by S b = (cid:90) ∂ Σ dζ (cid:34) b ∂ X ∂ζ · ∂ X ∂ζ + b ∂∂ X ∂ζ ∂ζ · ∂∂ X ∂ζ ζ + b (cid:18) ∂ X ∂ζ · ∂ X ∂ζ (cid:19) + b ∂ ∂ X ∂ζ ∂ζ · ∂ ∂ X ∂ζ ∂ζ (cid:35) . (4)where b i are the couplings of the boundary terms. Con-sistency with the open-closed string duality [41] whichimplies a vanishing value of the first boundary coupling b = 0, the leading-order correction due to second bound-ary terms with the coupling b appears at higher orderthan the four derivative term in the bulk.An interesting generalization of the Nambu-Gotostring [98, 109, 110] has been proposed by Polyakov [64]and Kleinert [111] to stabilize the NG action in thecontext of fluid membranes. The Polyakov-Kleinertstring is a free bosonic string with additional Poincare-invariant term proportional to the extrinsic curvatureof the surface as a next order operator after NG ac-tion [64, 111]. That is, the surface representation of thePolyakov-Kleinert (PK) string depends on the geomet-rical configuration of the embedded sheet in the space-time. The bosonic free string action is equiped with theextrinsic curvature as a next-order operator after NG ac-tion [64, 65].The model preserves the fundamental properties ofQCD of the ultraviolet (UV) freedom and infrared (IR)confinement properties [64, 65], and is consistent withthe formation of the glueballs [112, 113], and a real ( Q ¯ Q )potential [114–116] with a possible tachyonic free spec-trum [117] above some critical coupling [113].The action of the Polyakov-Kleinert (PK) string withthe extrinsic curvature term reads S PK [ X ] = S NG (cid:96) o [ X ] + S R [ X ] , (5)with S R defined as S R [ X ] = α r (cid:90) d ζ √ g K . (6)The extrinsic curvature K is defined as K = (cid:52) ( g ) ∂ α [ √ gg αβ ∂ β ] , (7)where (cid:52) is Laplace operator and M = σ α r is the rigidityparameter. The term satisfies the Poincare and the paritysymmetries and can also be considered [75] in the generalclass of (LW) actions (1).The perturbative expansion [114] of the rigidity termEq. (6) reads S R [ X ] = S R(cid:96) o [ X ] + S R n (cid:96) o [ X ] + ..., (8)has the leading term is given by S R(cid:96) o = α r (cid:90) L T dζ (cid:90) R dζ (cid:34)(cid:18) ∂ X ∂ζ (cid:19) + (cid:18) ∂ X ∂ζ (cid:19) (cid:35) (9)The rigidity parameter is tuned so as to weigh favor-ably the smooth surface configuration over the creasedworldsheets. In non-abelian gauge theories this ratio isexpected to remain constant in the continuum limit [75].In the following, prior to drawing a comparison be-tween the numerical Yang-mills lattice data and the var-ious models of the Casimir energy, we review the corre-sponding confining potential due to each string action inthe remaining of this section.The Casimir energy is extracted from the string parti-tion function as V ( R, T ) = − L T log( Z ( R, T )) . (10)The partition function of the NG model in the physicalgauge is a functional integrals over all the world sheetconfigurations swept by the string Z ( R, T ) = (cid:90) C [ D X ] exp( − S ( X )) . (11)For a periodic boundary condition along the time direc-tion such that X ( ζ = 0 , ζ ) = X ( ζ = L T , ζ ) , (12)with an extent equals to the inverse of the temperature L T = T and Dirichlet boundary condition at the sourcesposition given by X ( ζ , ζ = 0) = X ( ζ , ζ = R ) = 0 , (13)the eigenfunctions are given by φ mn = e πi (cid:16) mR + nLT (cid:17) , (14) and eigenvalues of −(cid:52) are given byΓ nm = (cid:18) πnL T (cid:19) + (cid:18) πmR (cid:19) . (15)The determinant of the Laplacian after ζ function regu-larization [118] readsDet ( −(cid:52) ) = ( q ∞ (cid:89) n =1 (1 − q n )) , (16)where q = e πτ and τ = L T R is the modular parameterof the cylinder. The path integral Eq. (11) and Eq. (10)yields the static potential for the leading order contribu-tion of the NG action S NG(cid:96)o . The partition function andthe static potential are respectively given by Z ( NG ) (cid:96)o = e − σRT − µ ( T ) [Det ( −(cid:52) )] − (d − , (17) V NG (cid:96) o ( R, T ) = σ R + ( d − T log η ( τ ) + µT, (18)where µ is a UV-cutoff and η is the Dedekind η functiondefined on the real axis as η ( τ ) = q ∞ (cid:89) n =1 (1 − q n ) . (19)The second term on the right hand side encompasses theL¨uscher term of the interquark potential. This term sig-nifies a universal quantum effect which is a character-istic of the CFT in the infrared free-string limit and isindependent of the interaction terms of the correspond-ing effective theory. One can extract the string tensiondependency on temperature from the slop of the linearterms in R . Considering the modular transform of theEq. (18) τ → /τ and taking the limit of long string, therenormalized string tension to leading order is given by σ ( T ) = σ − π ( d − T + O ( T ) . (20)Deitz and Filk [118] extracted the next to leadingorder term [119] of the Casimir potential from the explicitcalculation of the two-loop approximation using the ζ regularization scheme. The static potential of NG stringat second loop order Eq. (3) is given by V NGn (cid:96) o ( R, T ) = σ R + ( d − T log η ( τ ) − T log (cid:18) − TR ( d − π σ (cid:2) E ( τ ) + ( d − E ( τ ) (cid:3) (cid:19) + µT, (21)with E and E are the second and forth-order Eisensteinseries defined as E ( τ ) = 1 − ∞ (cid:88) n =1 n q n − q n , (22) E ( τ ) = 1 + 1240 ∞ (cid:88) n =1 n q n − q n , (23)respectively. With the modular transform τ → /τ ofEq. (21) and considering the limit of long string, thestring tension, which defines the slop of the leading linearterm of the potential, as a function of the temperaturereads σ ( T ) = σ − π ( d − T − π ( d − σ T + O ( T ) . (24)where σ denotes the string tension of the string at zerotemperature. The coefficient of the next higher-orderterms T can be induced [63, 109] from the expansionof NG action and leads to the exact NG string tensiongiven by σ ( T ) = σ (cid:115) − π ( d − T σ . (25)The boundary term S b in L¨uscher-Weisz action dueto the symmetry breaking by the cylindrical boundaryconditions by the Polyakov lines. In Refs. [77, 108] thefirst and second nonvanishing Lorentz-Invariant bound-ary terms contribution to the potential have been calcu-lated. The modification to the potential received whenconsidering Dirichlet boundary condition are given by V B = V b + V b ,V b = b ( d − π L T R E ( q ) , (26)The contribution to the partition function coming fromthe action Z = (cid:90) DXe − S NG(cid:96)o − S b − S b (27)Expanding around the free action yields Z = Z (cid:18) (1 − (cid:104) S b (cid:105) − (cid:104) S b (cid:105) ) + 12 (cid:68) ( S b + S b + .... ) (cid:69)(cid:19) + ... (28)The Lorentzian-Invariance imply b = 0 , b = 0, thenext two non-vanishing Lorentzian-Invariant terms comeat order four and six derivative terms at coupling b (cid:104) S b (cid:105) = b (cid:90) ∂ Σ dζ (cid:104) ∂ ∂ X · ∂ ∂ X (cid:105) , (29)and coupling b (cid:104) S b (cid:105) = b (cid:90) ∂ Σ dζ (cid:10) ∂ ∂ X · ∂ ∂ X (cid:11) , (30)respectively.The spectral Green function corresponding to solutionof Laplace equation with Dirichelet boundary conditionson cylinder G ( ζ , ζ , ζ (cid:48) , ζ (cid:48) ) =2 π RL T (cid:88) m,n (cid:88) m (cid:48) ,n (cid:48) sin (cid:18) nπζ R (cid:19) sin (cid:18) n (cid:48) πζ (cid:48) R (cid:19) e πimLT ( ζ − ζ (cid:48) ) n R + m L T . (31)The correlator (cid:104) S b (cid:105) , which is the line integral overeach of the Polyakov loops (cid:104) S b (cid:105) = (cid:90) ∂ Σ ∂ ∂ ∂ ∂ (cid:48) ∂ (cid:48) ∂ (cid:48) G ( ζ , ζ , ζ (cid:48) , ζ (cid:48) ) , (32)becomes after substituting the spectral Green functionEq. (31), (cid:104) S b (cid:105) = 32 π R L T (cid:34) (cid:88) mn m n n R + m L T (cid:88) m (cid:48) n (cid:48) m (cid:48) n (cid:48) n (cid:48) R + m (cid:48) L T (cid:26) lim ζ → + lim ζ → R (cid:27) (cid:90) ∂ Σ dζ cos (cid:18) nπζ R (cid:19) cos (cid:18) n (cid:48) πζ R (cid:19) (cid:35) . (33)Making use of the ζ function regularization of the seriessum (cid:88) m,n n l m kn R + m L T = ( − k πR (cid:18) L T R (cid:19) k +1 ζ (1 − l − k ) E k + l ( q ) . (34)The correlator Eq. (33) yields the following correctionto the static potential from the boundary term at cou-pling b , V b = − b ( d − π L T R E ( τ ) . (35)The modular transforms of Eq.(26) and Eq.(35) do notyield a linear term proportional to R which changes theslop of the potential.The static potential for smooth open strings was eval-uated in [66, 71, 114, 120]. Employing ζ function regu-larization [69] the finite-temperature contribution is cal-culated in Ref. [70] without subtraction [72] to the firstloop. Here we show in more detail the calculation of thedeterminant of the Laplacian to unambiguously show theformulation of the static potential and link between dif-ferent forms used in the literature.The partition functions due to the leading order con-tribution from NG action and rigidity terms Z = Z ( NG ) (cid:96)o Z ( R ) (cid:96)o , (36)are given by Z = (cid:90) DX exp[ − S NG (cid:96) S R (cid:96) o ] , = (cid:90) D X exp − σ [1 + 12 X (1 − (cid:52) M )( −(cid:52) ) X ] . (37)With the use of the transformation X (cid:48) = (cid:52) M X , (38)the partition function can be decoupled into the leading-order NG partition function Eq. (16) multiplied by thecorresponding rigidity contribution which appears as theJacobian of the transformation Z ( NG ) (cid:96)o = e − σRT − µ ( T ) [Det ( −(cid:52) )] − (d − , (39)and Z ( R ) = [Det (cid:18) − (cid:52) M (cid:19) ] − (d − . (40)The eigenvalues of the trace of the triangle operators canbe calculated from the eigenvalue equations correspond-ing to periodic and Dirichlet boundary condition definedas ( −(cid:52) ) ψ nm = λ nm ψ nm . (41)The eigen functions and eigenvalues are given by ψ mn = e πi (cid:16) mR + nLT (cid:17) , (42)and λ nm = (2 πnT ) + (cid:18) πmR (cid:19) , (43)respectively. The traces then readln (cid:18) Det (cid:18) − (cid:52) M (cid:19)(cid:19) = − lim s → dds (cid:88) n,m (cid:18) π M (cid:20) m R + n L T + M π (cid:21)(cid:19) s , (44)where s is an auxilary parameter. In the following weillustrate the ζ function regularization scheme used forthe evaluation of the two summations appearing in theabove integral.The sum over m is the Epstein-Hurwitz ζ func-tion [121]. Using Sommerfeld-Watson transform [122] thesummation over n in the last expression can be presentedas a contor integral in the complex plane t as ln( Z R ) = ( d − (cid:32) lim s → dds (cid:88) n,m (cid:34) (cid:18) π M R (cid:19) − s (cid:32) (cid:73) c dt (cid:88) m (cid:18) e iπt i sin( πt ) + 12 (cid:19) (cid:18) m + t + M R π (cid:19) − s (cid:35)(cid:33) . (45)Solving the above integral with the contor (+ ∞ + i(cid:15) )to ( −∞ + i(cid:15) ) yields the two terms22 − d log( Z R ) = 4 ∞ (cid:88) m =0 log(1 − e π ( m + M R π ) ) − lim s → dds (cid:18) π M R (cid:19) − s sin( πs )cos( πs ) (2 τ ) − s Γ (1 − s )Γ(2 − s ) (cid:88) m m + M / π ) s − / . (46)The second sum over m is also the Epstein-Hurwitz ζ function. However, we proceed in the regularizationusing different integral representation [72] to bring thefinal closed form in terms of the standard mathematicalfunctions. Each term in this sum can be transformed intothe integral presentation corresponding to Euler-Gammafunction [123].For general s the Epstein-Hurwitz ζ function reads ζ ( s, M/ π ) = 1Γ( s ) (cid:90) ∞ t s − ∞ (cid:88) n =1 e − t ( n + M R ) dt, (47)then with the modular transform of t −→ /t of theJacobi ϑ , θ ( t ) = (cid:80) ∞ n =1 e − n t appearing in the above sum,the zeta function turns into the integral presentation ζ ( s, M/ π ) = − ( M R ) − s √ π Γ( s − )2Γ( s ) ( M R ) − s + + √ π Γ( s ) ∞ (cid:88) n =1 (cid:90) ∞ t s − exp (cid:18) − t M R π − π n t (cid:19) dt. (48)The integral in the last term can be presented interms of sum over the modified Bessel functions [124] ofthe second kind (cid:80) n n s − / K s − / (2 nM R ). EmployingEq. (47) in Eq. (48) in the partition function Z R Eq. (48)the resultant expression would read Z R = exp (cid:34) ( d − M π (cid:88) n =1 n − K (2 nM R ) (cid:35) × ∞ (cid:89) n =0 (cid:18) − e πτ ( n + M ) (cid:19) . (49)The potential corresponding to the total partion func-tion Z NG(cid:96)o Z R(cid:96)o is thus V R(cid:96)o ( R, T ) = σ R − ( d − T log ( η ( τ )) − λ ( T )+ ( d − M π ∞ (cid:88) n =1 n − K (2 nM R )]+ ( d − T ∞ (cid:88) n =0 (cid:32) log (cid:16) − e πτ √ n + M (cid:17) . (50)The massless limit corresponding to M = σ / α r → V R(cid:96)o ( R, T ) = σ R − d − T log ( η ( τ )) − ν ( T ) . (51)Let us define a string model for the quark-antiquarkpotential with the leading extrinsic curvature term inconjunction with two subsequent orders of the NG per-turbative expansion corresponding to the free and theleading self-interacting components Z = Z NG(cid:96)o Z NGn(cid:96)o Z R(cid:96)o . (52)The potential of the rigid string with the next to lead-ing order NG contribution is V R ( (cid:96)o ) n(cid:96)o ( R, T ) = V R ( (cid:96)o ) (cid:96)o ( R, T ) − T ln (cid:32) − ( d − π T σ o R (cid:2) E ( τ ) + ( d − E ( τ ) (cid:3) (cid:33) . (53)The string tension form this model can be calculatedfrom the power expansion of T in the large R limit andit turns out to be λ = 2 α r σ ren T π ,S ( α r , σ r , T ) = 1 λ ∞ (cid:88) n =1 ( (cid:112) λ + n − λ n − n ) ,σ ( T ) = σ r − ( d − σ r (cid:32)(cid:114) α r σ r T + π σ r T + α r π S ( α r , σ r , T ) (cid:33) . (54)In the limit of high and low temperatures the seriessum in Eq. (54) S takes the asymptotic forms S = − ζ (3)8 λ + ζ (5)16 λ − ζ (7)128 λ + ..... ; (55)and S = − π √ λ ∞ (cid:88) n =1 n K (cid:16) πn √ λ (cid:17) . (56)At short distances, renormalization corrections to thezero temperature string tension have to be taken intoaccount σ [71, 114]. The lattice spacing, however, nat-urally introduces cutoff-scale which affects the value ofthe returned fit parameters over large source separationdistances.The free NG string is known to reproduce [3] the sub-leading aspects of the QCD string over long distances R ≥ . R . The rigidityeffects may not be dominant but can not be neglectedwhen discussing static potential of spectrum of excitedstates [27].In the following, we numerically measure the two pointPolyakov-loop correlators and explore to what extenteach of the above string actions can be a sufficiently gooddescription for the potential between the two static colorsources. III. LATTICE Q ¯ Q POTENTIAL AND STRINGPHENOMENOLOGYA. Simulation setup
At fixed temperature T , the Polyakov loop correlatorsaddress the free energy of a system of two static colorcharges coupled to a heatbath [125]. Within the transfermatrix formalism [3] the two point Polyakov-loop corre-lators are the partition function of the string.The Monte-Carlo evaluation of the temperature depen-dent quark–antiquark potential at each R is calculatedthrough the expectation value of the Polyakov loop cor-relators P = (cid:90) d [ U ] P (0) P † ( R ) exp( − S w ) , = exp( − V ( R, T ) /T ) . (57) V Q ¯ Q ( R, T ) fm − e ( R ) n = R/a T / T c = . R in lattice units, β = 6 .
0, spatial volume N s =36 and N t = 8 time slices and temperature scale T /T c = 0 . with the Polyakov loop defined as P ( (cid:126)r i ) = 13 Tr (cid:34) N t (cid:89) n t =1 U µ =4 ( (cid:126)r i , n t ) (cid:35) , (58)Making use of the space-time symmetries of the torus,the above correlator is evaluated at each point of the lat-tice and then averaged. We perform simulations on largeenough lattice sizes to gain high statistics in a gauge-independent manner [126] in addition to reduce correla-tions across the boundaries. The two lattices employedin this investigation are of a typical spatial size of 3 . fm with a lattice spacing a = 0 . N t = 8, and N t = 10 slices at a cou-pling of value β = 6 .
00. The two lattices correspond totemperatures
T /T c = 0 . T /T c = 0 . SU (2) subgroup elements [130]. Eachupdate step/sweep consists of one heatbath and 5 micro-canonical reflections. The gauge configurations are ther-malized following 2000 sweeps. The measurements aretaken on 500 bins. Each bin consists of 4 measurementsseparated by 70 sweeps of updates.The correlator Eq.(60) is evaluated after averaging thetime links [131] in Eq.(60)¯ U t = (cid:82) dU U e − T r ( Q U † + U Q † ) (cid:82) dU e − T r ( Q U † + U Q † ) . (59)The temporal links are integrated out analytically byevaluating the equivalent contor integral of Eq.(59) asdetailed in Ref. [132].The lattice data of the ( Q ¯ Q ) potential are extracted V NG Fit Fit Parameters,
T /T c = 0 . σ µ χ V (cid:96) o [ R m , V n (cid:96) o [2,11] 0.036250(8) -0.35054(5) 627275.[3,11] 0.040406(9) -0.38757(7) 11136.[4,11] 0.04111(1) -0.3938(1) 1786.73[5,11] 0.0410(1) -0.393(1) 1694.91[6,11] 0.0407(2) -0.390(2) 1093.66[7,11] 0.04033(3) -0.3862(2) 525.181[8,11] 0.0398(3) -0.380(4) 168.203[9,11] 0.0392(6) -0.374(6) 21.832TABLE II. The χ values and the corresponding fit parame-ters returned from fits to the leading and the next-to-leadingorder (NLO) static potential of NG string Eqs. (18) and (21),respectively. from the two point Polyakov correlator V Q ¯ Q ( R ) = − T log (cid:104) P ( x ) P ( x + R ) (cid:105) (60) B. Temperature scale near critical point
T /T c = 0 . Thermal effects become dominant in the SU(3) Yang-Mills model [44, 127] under scrutiny if the temperature isscaled down close enough to the critical point
T /T c = 0 . Q ¯ Q po-tential Eq. (60) at this temperature are inlisted in Ta-ble. I.In the following, we consider examining three possibleansatz of the string potential and draw comparisons be-tween their possibly interesting combinations. The tar-get is to understand the relevance of each model at theselected source separation intervals for this temperaturescale.
1. Nambu-Goto string at leading and next-to-leading orders
The Q ¯ Q potential data are fitted to theoretical formu-las of the NG string potential at the leading and next-to-leading orders Eqs. (18) and Eq. (21), respectively. Weset the string tension σ a and the renormalization con-stant µ ( T ) as a free fitting parameters. The same value ofthe string tension taken as a fit parameter as in Ref. [44]and [45] is reproduced with the corresponding functionof the static potential [3, 46] and fit domain. ] -1 distance, [RaQQ po t en t i a l , [ V a ] QQ - - - - - - - - - - - - - Lattice data=0.044 s =0.041 s =0.039 s =6.0 b T/Tc=0.9 (R,T) lo V ] -1 distance, [RaQQ0 1 2 3 4 5 6 7 8 9 10 11 da t a - t heo r y - - - ] -1 distance, [RaQQ un c e r t a i n t y - - (a) ] -1 distance, [RaQQ po t en t i a l , [ V a ] QQ - - - - - - - - - - - - - Lattice data=0.044 s =0.041 s =0.039 s =6.0 b T/Tc=0.9 (R,T) nlo V ] -1 distance, [RaQQ0 1 2 3 4 5 6 7 8 9 10 11 da t a - t heo r y - - - ] -1 distance, [RaQQ un c e r t a i n t y - - (b) FIG. 1. The quark-antiquark Q ¯ Q potential measured at temperature T /T c = 0 .
9, the left and right plots correspond to thefits to LO and NLO Nambu-Goto string Eq.(18) and Eq. (21) for the depicted values of string tension σa , respectively. Thereturned fit parameters are inlisted in Table. III Table II enlists the returned value of the string tension σ a and χ for various source separations commenc-ing from R = 0 . , . , . . R = 1 . R = 0 . χ at σ a = 0 .
039 on R ∈ [0 . , . σ a = 0 .
037 on R ∈ [0 . , .
2] fm. However,the values of χ are outstandingly higher than the corre-sponding returned values considering the next-to-leadingapproximation Eq. (21).Higher-order terms in the free energy provide fine cor-rections for the value of the returned free-parameter σ a . This parameter is interpreted as the zero tem-perature string tension, that is, the value that should bereturned at zero or low temperature as in Ref. [133].To appreciate the role played by the string tension wedisclose the fit behavior of the lattice data at this temper-ature scale We systematically inspect the returned valuesof χ for an interval of selected values of the string tension σ a ∈ [0 . , . µ ( T ) for the corresponding σ a are inlisted inTable. II with plots in Fig. 1.Figure 3-(a) show the stability of the fits and a well-defined global Minimal in the ( σ, µ ) parameter space.The gradual descend of the string tension parameter from 0.045 to 0.041 reduces dramatically the values of χ (as inlisted in Table III) till a minimum is reachedat σ a = 0 .
041 for a fit interval from R = [0 . , .
2] fm.The plot of the static potential owing to the (LO) andthe (NLO) of the NG string at three selected values ofthe string tension σa is shown in Fig. 1.The two-dimensional version of Fig 3-(b) depicts howexcluding points at short distance, e.g, considering a fitinterval R ∈ [0 . , . χ with a shifted minimal at σ a = 0 .
039 as depicted inFig. 2.Larger residuals | T heory − data | at the (NLO) V n(cid:96)o appear to be stringent at shorter distances R < . σ a = 0 .
044 compared to theLO potential ansatz Eq. (21) of the NG string V (cid:96)o . Thissuggest that the string’s self-interactions are more rele-vant to the string configurations swept over long distancescales.The effective description based only on Nambu-Gotomodel does not accurately match the Q ¯ Q potential data.In spite of the fits beyond the free Gaussian approxima-tion, the inclusion of the NLO terms does not providean acceptable optimization for the potential data. Poorfits persist at both short distance and intermediate dis-tances. The fitted string potential, at its minimal sumof the residuals χ , produces deviation from the value ofthe measured value of the zero temperature string ten-sion [133].The fit to the Casimir energy of the self-interactingstring returns a value of the zero temperature string ten-0sion σ a = 0 .
039 which deviates at least by 11% ofthat measured at zero temperature σ a = 0 . T /T c V (cid:96)o V n(cid:96)o σ a χ σ χ I n t e r v a l R ∈ [ , ] χ returned from the fits for eachcorresponding value of the string tension, the table comparesboth values for fits to the leading order (LO) Eq.(18) andnext-to-leading order (NLO) Eq.(21). V b n(cid:96)o Fit Fit Parameters,
T /T c =0.9Interval σa µ b χ V n (cid:96) o + V b [ R m , R M ][2,11] 0.0411327(9) -0.39693(1) -0.1746(2) 2597.5[3,11] 0.04124(1) -0.39836(1) -0.202(2) 2428.1[4,11] 0.04094(2) -0.3888(4) 0.20(1) 1572.9[5,11] 0.0405(2) -0.35(1) 2.3(8) 852.6[6,11] 0.0401(3) -0.19(7) 11(4) 363.1[7,11] 0.03954(5) 0.47(4) 52(3) 99.4[8,11] 0.0389(2) 2.9(8) 205(50) 9.4[ R m , R M ][2,5] 0.04005(5) -0.3926(2) -0.1722(3) 373.7[3,6] 0.04232(5) -0.4062(4) -0.278(4) 0.049[3,7] 0.0423(4) -0.4057(3) -0.274(3) 2.2[3,9] 0.04189(2) -0.4032(2) -0.252(3) 205.2[4,9] 0.0417(3) -0.3994(4) -0.10(2) 120.5[5,10] 0.04102(3) -0.368(2) 1.46(9) 268.5TABLE IV. The χ values and the corresponding fit param-eters b µ returned from fits to the next to leading order(NLO) static potential with boundary terms V b n(cid:96)o given byEq. (61).
2. Boundary terms in L¨uscher-Weisz action
Effects such as the interaction of the string with thePloyakov lines at the boundaries may be relevant to thediscrepancies in the effective string description at theshort and intermediate string length. The contributionto the Casmir energy due to the two next-to-leading non-vanshing term S b in L¨uscher-Weisz action Eq. (1) does not not affect the slop of the linearly rising part of thepotential. However, the effects are received as inversepowers in R and are given by V b ,b n(cid:96)o of Eq (63).Since the leading nonvanshing boundary terms appearat the order of four derivative term Eq. (4) in L¨uscher-Weisz action Eq. 1, It may be more convenient to discussthe corresponding effects (26) in conjunction with theNLO form of the potential of the NG action Eqs. 21 withthe renormalization of the string tension included.For the purpose of the discussion of the numerical dataof the static meson potential, We define the following pos-sible combinations of LO and NLO Nambu-Goto staticpotential with boundary terms, V b n(cid:96)o = V n(cid:96)o + V b , (61) V b (cid:96)o,n(cid:96)o = V (cid:96)o,n(cid:96)o + V b , (62) V b ,b n(cid:96)o = V n(cid:96)o + V b + V b , (63)where subscripted V (cid:96)o,n(cid:96)o denotes either the LO NGstatic potential or the NLO Eq. (3). The Q ¯ Q potentialdata are fitted to the static potential with a two possi-bly interesting combinations of the boundary terms V b n(cid:96)o and V b ,b n(cid:96)o given by Eq. (61) and Eq. (63), respectively.The inspection of each boundary term allows for under-standing of the relevance of each boundary parameters tothe fit arrangement. The corresponding returned valuesof χ and fit parameters are enlisted in Table IV, VIIIand V, considering various fit intervals. The fit to thestatic potential V b n(cid:96)o of the model Eq. (61) returns valuesfor the parameter b which appear to vary dramaticallywith the considered range. The values of χ are still highwhen considering the entire fit-interval R ∈ [0 . , .
1] fm.Even though, none trivial improvements in the values of χ are retrieved as shown in Table. IV compared to thatobtained by merely considering the NG string potentialEqs. (21) (Table. III). The fits to the static potential withthe boundary term V b n(cid:96)o produce acceptable χ value forshorter fit intervals commencing from R ∈ [0 . , .
2] fm.With the interchange of the term V b n(cid:96)o in place of V b n(cid:96)o in the string model Eq. (61), the fits are not surprisinglygood. However, the interesting observation is that thefits at shorter distances commencing from R = 0 . R = 0 . V b n(cid:96)o and V b ,b n(cid:96)o , areremarkably good with a returned σ a = 0 .
042 (Table V).Inspection of Tables IV and V indicates that fits tothe NLO form of (NG) string with boundary term V b n(cid:96)o produce very close value of σ a as the pure NG string V n(cid:96)o . The same observation holds for the fits to V b n(cid:96)o and V b ,b n(cid:96)o . Acceptable value of χ are returned σ a =0 . . , .
2] fm.Despite of the reductions in the minima of χ dof , thestring models with boundary terms have no significantlydifferent behavior with respect to the string tension pa-rameter. This is consistent with the modular transformswhere the inverse of the cyliner’s modular parameter doesnot produce terms linear in R . Therof, the boundary cor-1 ] -1 distance, [RaQQ - - - - - - - - - - - - - po t en t i a l , [ V a ] QQ Lattice data [5-11] ˛ fit range: R [6-11] ˛ fit range: R [7-11] ˛ fit range: R [8-11] ˛ fit range: R [9-11] ˛ fit range: R =6.0 b T/Tc=0.9 (R,T) lo NG: V -1 distance, [RaQQ - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y ] -1 distance, [RaQQ - - - - - - - - - - - - - po t en t i a l , [ V a ] QQ Lattice data [5-11] ˛ fit range: R [6-11] ˛ fit range: R [7-11] ˛ fit range: R [8-11] ˛ fit range: R [9-11] ˛ fit range: R =6.0 b T/Tc=0.9 (R,T) nlo
NG: V -1 distance, [RaQQ - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (a) (b) FIG. 2. The fits to the quark-antiquark Q ¯ Q potential data measured at temperature T /T c = 0 . V b , b n (cid:96) o Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) b (LU) b (LU) χ V n (cid:96) o + V b + V b [ Rm, RM ] [2,11] 0.04123(1) -0.3991(2) -0.236(5) 0.023(2) 2452.4[3,11] 0.04091(2) -0.332(2) 2.61(9) -1.84(6) 1511.9[4,11] 0.04053(2) 0.97(5) 58.1 (2.1) -44.7(1.6) 805.8[5,11] 0.0400(3) 27.7 (12.4) 1185.4 (520.8) -948.9 (417.7) 336.6[6,11] 0.0394(5) 553.3 (334.4) 23246.0 (14037.0) -18790.0 (11352.0) 89.2 [ Rm, RM ] [2,7] 0.0423(4) -0.4090(4) -0.408(7) 0.086(3) 4.8[3,8] 0.04198(4) -0.393(3) 0.2(1) -0.31(8) 19.4[4,9] 0.04142(4) 0.12(6) 22.3 (2.5) -17.2 (1.9) 41.3TABLE V. The χ values and the corresponding fit parameters b , b and µ returned from fits to the next-to-leading order(NLO) static potential with boundary terms V b ,b n(cid:96)o given by Eq. (63). rections to the static potential do not contribute to therenormalization of the string tension.
3. Rigidity terms in Polyakov-Kleinert action
Establishing a precise string description of the Q ¯ Q po-tential data and a correct thermodynamic behavior forthe string tension at high temperature have been a longwithstanding issues in many investigated gauge models.The consideration of the boundary terms solely does notprovide an optimal fits and other possible string prop-erties may be questioned in this context. The possible rigidity/stiffness/self-repulsion of QCD flux tube; or theresistance to sharp transverse-bending should manifestby the onset of the excited fluctuations at high tempera-tures.In order to unambiguously appreciate the changes onthe fits when the rigidity of the string is taken into ac-count, we discuss the modified static potential of rigidstring in conjunction with both the leading and the next-to-leading approximations to NG action V R(cid:96)o and V Rn(cid:96)o
Eqs. (64), separately.More variants of string models can be attained by in-2 (a) a σ χ (R,T): fit range: 5 11 lo NG:V (R,T): fit range: 5 11 nlo
NG:V (R,T): fit range: 9 11 lo NG:V (R,T): fit range: 9 11 nlo
NG:V (b)
FIG. 3. (a) The returned χ dof versus the string tension σ a and cutoff µ , from the fits of Q ¯ Q potential to leading orderapproximation of Nambu-Goto string Eq. (18) at T /T c = 0 . χ dof versus the string tension σ a ; how-ever, the fits are for the next-to-leading order approximationEq. (21). cluding other combinations of the rigid terms such as, V Rn(cid:96)o = V n(cid:96)o + V R , (64) V R,b n(cid:96)o = V n(cid:96)o + V R + V b , (65) V R,b n(cid:96)o = V n(cid:96)o + V R + V b , (66) V R,b ,b n(cid:96)o = V n(cid:96)o + V R + V b + V b , (67)the above compilations are particular choices of termsfrom the most general formalism L¨uscher-Weisz action.We proceed in the fit analysis of the Q ¯ Q static poten-tial data without fixing the value of the string tension.The rigidity factor, α r which weighs the extrinsic curva- V Rn(cid:96)o
Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) α χ R m (LU) V n (cid:96) o + V R [ Rm, [2,11] 0.0362498(8) -0.35054(5) 0(3.6) 627275.[3,11] 0.040(4) -0.39(1) 0(3.7) 11136[4,11] 0.04245(3) -0.3756(7) 0.059(2) 1148.11[5,11] 0.0435(2) -0.34(2) 0.16(5) 319.657[6,11] 0.0440(2) -0.30(4) 0.3(2) 79.2561[7,11] 0.0442(3) -0.2(1) 0.6(7) 11.592[8,11] 0.044(2) -0.2(4) 1(6) 1.80228 [ Rm, RM ] [2,5] 0.02677(6) -0.3095(3) 0.000291 460013.[3,6] 0.038(8) -0.37(2) 0.002391 5774.38[3,7] 0.039(5) -0.38(2) 0.002434 8395.19[4,10] 0.040(4) -0.38(1) 0.002473 10933.4[4,10] 0.04214(5) -0.3879(8) 0.033753 487.035[5,10] 0.04344(4) -0.360(2) 0.108(5) 119.563TABLE VI. The χ values and the corresponding fit param-eters rigidity α and cutoff µ returned from fits to the rigid-self-interacting string potential V Rn(cid:96)o given by Eq. (64). V R , b n (cid:96) o Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) b (LU) α r χ V n (cid:96) o + V R + V b [ Rm, RM ] [2,11] 4.169(4) -0.3929(5) 0.025(1) -0.199(2) 2487.2[3,11] 4.343(2) -0.359(1) 0.129(3) -0.533(6) 501.3[4,11] 4.395(2) -0.337(2) 0.244(9) -1.1(3) 161.6[5,11] 4.42(2) -0.32(4) 0.5(3) -3(2) 38.8[6,11] 4.42(8) -0.353(8) 1.0(2) -9(1) 4.7[7,11] 4.39(4) -0.48(4) 3(2.9) -25(7) 0.071 [ Rm, RM ] [2,6] 0.041(1) -0.39(1) 0.01(5) -0.18(7) 752.6[3,7] 0.0428(2) -0.400(3) 0.03(1) -0.31(3) 0.74[3,9] 0.04327(4) -0.385(1) 0.066(3) -0.40(1) 40.9[4,10] 0.04394(3) -0.355(2) 0.173(9) -0.97(4) 58.9[5,10] 0.04430(3) -0.343(4) 0.31(3) -2.6(2) 14.7TABLE VII. The χ values and the corresponding fit param-eters; rigidity α r , boundary parameter b and cutoff µ , re-turned from fits to the rigid-self-interacting string potential V R,b n(cid:96)o given by Eq. (65). ture tensor, and the ultraviolet cutoff µ are taken as afree fit parameters as well. Table. VI summarizes valuesof χ obtained from fits to V Rn(cid:96)o
Eq. (64). We remark thefollowing points:Figure 5 plots the returned χ values versus both thestring tension and rigidity. The plot indicates the qualityof the fit in the parameteric space ( α r , σ ), the oscillatorynature of χ dof when the rigidity is included and attain-ment of the global minima of χ at σ = 0 . χ in Ta-ble. II and Table VI reveals significant improvement inthe fit behavior with the rigidity term V Rn(cid:96)o of the string3 ] -1 distance, [RaQQ - - - - - - - - - - - - po t en t i a l , [ V a ] QQ Lattice data [4-11] ˛ fit range: R [5-11] ˛ fit range: R [6-11] ˛ fit range: R [7-11] ˛ fit range: R [8-11] ˛ fit range: R =6.0 b T/Tc=0.9 (R,T) b nlo V -1 distance, [RaQQ - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (a) ] -1 distance, [RaQQ - - - - - - - - - - - - po t en t i a l , [ V a ] QQ Lattice data [4-11] ˛ fit range: R [5-11] ˛ fit range: R [6-11] ˛ fit range: R [7-11] ˛ fit range: R =6.0 b T/Tc=0.9 (R,T) ,b b nlo V -1 distance, [RaQQ - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (b) FIG. 4. The quark-antiquark Q ¯ Q potential at temperature T /T c = 0 .
9, the lines correspond to the next to leading orderNambu-Goto string with two different boundary terms V b n(cid:96)o and V b n(cid:96)o Eqs. (61) and Eq. (63), respectively, at
T /T c = 0 . χ dof versus the string tension σ a and therigidity α from the fits of Q ¯ Q potential data to rigid stringansatz V Rn(cid:96)o
Eq. (64) at
T /T c = 0 . model Eq. (64) over the pure NG string potential givenby Eq. (18) and (21).The residuals are reduced on the fit interval R ∈ [0 . , .
1] fm, the returned χ dof indicate good values for R > . σa = 0 . R ∈ [0 . , .
1] fmand σa = 0 . R ∈ [0 . , .
1] fm isshifted above the value obtained from considering fits tomerely the static potential of the pure NG string. The fitto Eq. (64) results in a value of the string tension which,within the numerical uncertainities, is equivalent to that V R , b n (cid:96) o Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) α r b (LU) χ V n (cid:96) o + V R + V b [ Rm, RM ] [2,11] 0.0409(5) -0.390(2) 0.01(3) -0.067(7) 4460.22[3,11] 0.04327(2) -0.354(1) 0.109(2) -0.323(4) 558.3[4,11] 0.04393(2) -0.312(2) 0.23(8) -0.89(3) 154.9[5,11] 0.04422(2) -0.248(7) 0.45(3) -2.6(1) 36.02[6,11] 0.04415(9) -0.11(3) 1.1(2) -7.9(9) 4.44 [ Rm, RM ] [2,6] 0.0409(9) -0.390(4) 0.01(6) -0.06(1) 3835.6[3,8] 0.0427(1) -0.391(2) 0.032(5) -0.205(9) 13.5[4,10] 0.0439(4) -0.335(3) 0.162(8) -0.71(3) 56.3[5,10] 0.0443(3) -0.284(9) 0.30(3) -2.0(2) 14.0TABLE VIII. The χ values and the corresponding fit pa-rameters b (LU) and µ (LU) returned from fits to the nextto leading order (NLO) static potential with boundary terms V b n(cid:96)o given by Eq.(66). reproduced at zero temperature measurements [133].The consideration of the two-parameter rigid stringmodel ( µ, α r ) with the next-to-leading order NG poten-tial V Rn(cid:96)o of Eq. (64) results in a smaller χ compared tothe fit with the two parameter ( µ, α ) string models V b n(cid:96)o of formula Eq. (61). Nevertheless, the fit with bound-ary action models (Table. V)compares to the rigid stringwhen considering larger parametric space,i.e., the threeparameter ( µ, b , b ) model of V b ,b n(cid:96)o Eqs. (63).The plot in Fig. 6(a) is the fit of the static potential of4 ] -1 distance, [RaQQ - - - - - - - - - - - - po t en t i a l , [ V a ] QQ Lattice data[4,11] ˛ fit range R [5,11] ˛ fit range R [6,11] ˛ fit range R [7,11] ˛ fit range R =6.0 b T/Tc=0.9 (R,T) R nlo V -1 distance, [RaQQ - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (a) ] distance, [RaQQ − − − − − − − − − − − − po t en t i a l , [ V a ] QQ Lattice data [2 11] ∈ fit range: R [3 11] ∈ fit range: R [4 11] ∈ fit range: R [5 11] ∈ fit range: R =6.0 β T/Tc=0.9 (R,T) nlo R,b V ] distance, [RaQQ − − da t a t heo r y ] distance, [RaQQ − − − × un c e r t a i n t y (b) FIG. 6. (a)The quark-antiquark Q ¯ Q potential at T /T c = 0 .
9, the lines correspond to the fits (Table VI) to the static potentialof rigid string model V Rn(cid:96)o of Eq. (64). (b) Similar to (a), the lines correspond to the fits (Table VII) to the rigid string butwith the boundary term b given by the model V R,b n(cid:96)o of Eq. (65). V R , b , b n (cid:96) o Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) α r b (LU) b (LU) χ V n (cid:96) o + V R + V b + V b [ Rm, RM ] [2,11] 0.043519, 0.0000183 -0.36374, 0.000937 -0.140876, 0.0032 -0.903213, 0.013 0.222968, 0.00464167 437.185[3,11] 0.0439943, 0.0000204 -0.404129, 0.00240 0.258954, 0.0100 -4.24603, 0.188 2.25785, 0.120102 145.932[4,11] 0.0442363, 0.0000234 -1.36259, 0.0873 0.502746, 0.038 -47.2974, 3.9 34.353, 3.07522 34.16[5,11] 0.0441876, 0.0000808 -13.6639, 2.41 1.03577, 0.206 -568.27, 102.6 689.797, 82.3587 3.96 [ Rm, RM ] [2,8] 0.0432653, 0.0000787 -0.395082, 0.00194 0.0545925, 0.00554004 -0.569233, 0.0288 0.123925, 0.00867162 5.41[3,8] 0.0437036, 0.000155 -0.401344, 0.00342 -0.0855039, 0.0161917 -1.42324, 0.399 0.628589, 0.23702 0.9[3,9] 0.043877, 0.000071 -0.401914, 0.0030 0.120593, 0.0111493 -2.10003, 0.285 1.02458, 0.173838 7.4[4,9] 0.0442943, 0.000106759 -0.78807, 0.146409 0.198267, 0.0383622 -19.6914, 6.71211 14.3284, 5.07934 0.7[4,10] 0.0443398, 0.0000340 -1.08879, 0.109 0.330004, 0.0378015 -34.0812, 4.99 25.1703, 3.7973 12.5TABLE IX. The χ values and the corresponding fit parameters; rigidity α , boundary parameters ( b , b ) and string tensionand cutoff ( σ, µ ), returned from fits to the rigid-self-interacting string potential with two boundary terms V R,b ,b n(cid:96)o given byEq. (67). the rigid string for the fit intervals over the given in Ta-ble VI. A descending sequence of the values of rigidity pa-rameter α r are returned. The values of α decreases from α = 1 . α = 0 .
16 as one includes smaller distancesinto the fit range over intermediate distances [0 . ,
11] to[0 . ,
11] fm.Large uncertainties in the rigidity parameter α r arereturned from the fits with the decrease of minimal sourceseparation R m of the fit range R ∈ [ R m , R M ]. This is perhaps owing to higher-order terms in the perturbativeexpansion of the extrinsic curvature in the rigid stringaction.The renormalization of the string tension has been ex-plicitly given by German [71, 134] long ago. The misfor-tune that we are lacking an ansatz for the potential forthe two-loop static quark-antiquark [71, 114, 134] at fi-nite temperature scales and dimension. We evaluate thethermodynamic properties of this string gas at two-loop5 ] -1 distance, [RaQQ - - - - - - - - - - - - - ] - a po t en t i a l , [ R QQ Lattice data [2-8] ˛ fit range: R [3-9] ˛ fit range: R [2-11] ˛ fit range: R [3-11] ˛ fit range: R [5-11] ˛ fit range: R =6.0 b T/Tc=0.9 (R,T) ,b R,b nlo V ] -1 distance, [RaQQ - - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y FIG. 7. The quark-antiquark Q ¯ Q potential T /T c = 0 .
9. Thelines correspond to the fits to self-interacting NG potentialwith two boundary terms V b ,b n(cid:96)o given by Eq. (63),in additionthe rigidity of the string has been included in the fitting ansatz V R,b ,b n(cid:96)o Eq. (67). orders and examine the effects on the relatively short dis-tance physics in a separate report [135].Despite of the outstanding match between the staticpotential curve with α r = 0 . R ∈ [0 . , . R < . χ and string tension overlong distances. An optimization of both models is ex-pected, thereof, to provide a prospect for an extendeddistance scale of validity.The models given by Eq. (65), Eq. (66) and Eq. (67)define selected compilations grasping both aspects of therigidity and boundary effects. Tables. IX, VIII and VIIenlist the parameters and χ returned from the fits of thenumerical data of the static potential to these models,respectively.Inspection of the above mentioned tables reveals thatthe two models given by the interchanging the twoboundary terms V b and V b in Eq. (65), Eq. (66) areyielding almost comparable values of χ for each givenfit interval. However, the model encompassing both of the bound-ary terms Eq. (65) provide the best fit for the targeteddistance which is the intermediate distance scale [5,11], χ = 3 .
96. Interestingly, the fit over short distances in-terval in the second panel of Table. IX is as well indi-cating a good χ for most fit intervals. The plot of thestatic potential in Figs. 6(a), Fig. 6(b) and Fig. 7 respec-tively illustrates the subsequent diminish of the errors | Data − M odel | as the boundary terms are included torigid model.The panel of Fig. 8 congregates subfigures each showthe variation in the parametric subspace versus the belowbound R m of the fit interval [ R m , R M ] whereas the lastpoint has kept fixed at R M = 1 . σ a un-less otherwise the rigidity is considered is decreasing withthe increase of lower bound of the fit R M to achieve ac-ceptable reduction of χ . The remarkable feature is thatthe contrary of this behavior is observed when we stringmodels with rigidity are considered.The rigidity parameter α r appears to assume stablevalues α r ≤ . R m ∈ [0 . , .
6] fm. At R = 0 . b and b .The optimal χ is attained at R m = 0 . χ is almost dramatic for all other models R ≤ . χ indicating the long distancebehavior of effective bosonic strings are obtained fromdistances R m = 0 . α r , b , b )to R m = 0 . C. Temperature scale near the Plateau
T /T c = 0 . The analysis of the pure gluonic configuration near theend of QCD plateau is very interesting since a smallchange in the temperature could produce essential dif-ferent effects on the properties of the confining force. Itought to be instructive to extend the above reported anal-ysis to the lower temperature scale
T /T c = 0 .
1. The pure Nambu-Goto action
A large value of χ is returned for fits of color sourcesseparations commencing from R = 0 .
4. For separations6 -1 distance, [RaQQ - - LONLO NLO+b +b NLO+b m -1 distance, [RaQQ - - - rigid: NLO rigid: NLO+b +b rigid: NLO+b m -1 distance, [RaQQ s -1 distance, [RaQQ s -1 distance, [RaQQ - - - a -1 distance, [RaQQ - b -1 distance, [RaQQ - -
10 110 dof /N c -1 distance, [RaQQ - - - - - - b FIG. 8. Parameter chart of returned from the fit of each corresponding string model versus the fit rang lower bound R m ofthe interval R ∈ [ R m , R M ] whereas the upper [ R m ] bound is kept fixed at R M = 1 . χ corresponding to each model. V Q ¯ Q ( R, T ) fm − e ( R ) n = R/a T / T c = . R and temperature T /T c = 0 .
8, the latticeparameters β = 6 . N t = 10 time slices and spatial volume N s = 36 . The avaraging is taken for two Polyakov linesseparated by distance R Eq. (57). V NG Fit Fit Parameters,
T /T c = 0 . σa µ χ V (cid:96) o [ R m , R M ][2,11] 0.02520(8) -0.4000(2) 21875.7[3,11] 0.0392(1) -0.4586(5) 795.844[4,11] 0.0435(2) -0.482(1) 18.6047[5,11] 0.045(3) -0.49(2) 00[3,6] 0.0373(2) -0.4523(5) 399.716[3,7] 0.0381(1) -0.4553(5) 566.884[3,10] 0.0391(1) -0.4583(4) 779.503[4,10] 0.0435(2) -0.4816(9) 18.5844TABLE XI. The χ values and the corresponding fit parame-ters; string tension and cutoff ( σ, µ ), returned from fits to thefree string potential V (cid:96)o given by Eq. (18). distance R ≤ . ] -1 distance, [RaQQ - - - - - - - - - - - - - ] - a po t en t i a l , [ R QQ Lattice data [3-11] ˛ fit range: R [4-11] ˛ fit range: R [5-11] ˛ fit range: R [6-11] ˛ fit range: R =6.0 b T/Tc=0.8 (R,T) lo V ] -1 distance, [RaQQ - - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (a) ] -1 distance, [RaQQ - - - - - - - - - - - - - ] - a po t en t i a l , [ R QQ Lattice data [3-11] ˛ fit range: R [4-11] ˛ fit range: R [5-11] ˛ fit range: R [6-11] ˛ fit range: R =6.0 b T/Tc=0.8 (R,T) nlo V ] -1 distance, [RaQQ - - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (b) FIG. 9. The quark-antiquark Q ¯ Q potential data at temperature T /T c = 0 . R and is givenin Lattice units (a) The LO Nambu-Goto string model V (cid:96)o given by Eq. (18) for the depicted fit interval R ∈ [ R m , R M ] (b)The corresponding fits to the NLO Nambu-Goto string model V n(cid:96)o Eq. (21). V NG Fit Fit Parameters,
T /T c = 0 . σa µ χ V n (cid:96) o [ R m , R M ][2,11] 0.0240(2) -0.3208(8) 340026[3,11] 0.0325(2) -0.4084(6) 4609.75[4,11] 0.0422(2) -0.463(1) 119.701[5,11] 0.0449(3) -0.481(2) 2.60999TABLE XII. The χ values and the corresponding fit param-eters; string tension and cutoff ( σ, µ ), returned from fits tothe self-interacting string potential at next to leading order V n(cid:96)o given by Eq. (21). idealization of NG string. In Refs. [25, 26] the intrinsicthickness of the flux-tube has been discussed.As Table XI depicts, Excluding the point R = 0 . χ for boththe leading order and the next to leading approximationEqs.(18) and Eqs. (21), respectively. The returned valuesof the string tension parameter quickly reaches stabilityeven by the exclusion of further points at short distances R = 0 . R = 0 . σ a = 0 . σ a at T /T c = 0 . σ = 0 . Q ¯ Q po-tential match both the free leading-order NG string andthe NLO self-interacting form of Eq.(18)Eq.(21). Ap-proximately the same difference in the value of the stringtension is retrieved for fit domains involving short to large Q ¯ Q separation distances.Considering the fit of the same data of Q ¯ Q potentialto the two-loop expression of the NG string Eq. (21),the value of χ does not apprise mismatches for sourceseparation distances commencing from R > . R = 1 . χ with subtle changes in the free fit parameter σ a .The absence of the mismatch between Eq. (21) and thenumerical data at this temperature scale does not ruleout the validity at this temperature scale.This points out to the minor role of the higher ordermodes at the end of the QCD plateau T /T c = 0 .
8. Thepale out of the thermal effects together with a flat plateauregion at this temperature is present as well in the stringtension measurements [44] and the more recent Monte-Carlo measurements [133] which reproduces the samevalue of 0 .
044 of the zero temperature string tension.It is worth noting, on the otherhand, that the NLOterms alone esclates the fit on intervals
R.le. . ] -1 distance, [RaQQ - - - - - - - - - - - - - ] - a po t en t i a l , [ R QQ Lattice data fit range: [3-11] blo V fit range: [4-11] blo V fit range: [3-11] bnlo V fit range: [4-11] bnlo V =6.0 b T/Tc=0.8 (R,T) b nlo V , (R,T) b lo V ] -1 distance, [RaQQ - - - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (a) ] -1 distance, [RaQQ - - - - - - - - - - - - - ] - a po t en t i a l , [ R QQ Lattice data [2-7] ˛ fit range: R [3-8] ˛ fit range: R [2-11] ˛ fit range: R [3-11] ˛ fit range: R [4-11] ˛ fit range: R =6.0 b T/Tc=0.9 (R,T) ,b b nlo V ] -1 distance, [RaQQ - - - da t a - t heo r y ] -1 distance, [RaQQ - - un c e r t a i n t y (b) FIG. 10. The quark-antiquark Q ¯ Q potential data at temperature T /T c = 0 . R and is givenin Lattice units (a) Compares the fits from both LO and NLO Nambu-Goto string with boundary term V b , the fitted modelsare given by V b (cid:96)o Eq. (61) and Eq. (61) (b) Similar to (a); however, the fits are for both boundary terms V b ,b . The fittedmodels are given by V b ,b (cid:96)o Eq. (63) and V b ,b n(cid:96)o Eq. (63) V b , b n (cid:96) o Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) b (LU) b (LU) χ V b (cid:96) o (a) [2,5] 0.0400(2) -0.4689(9) -0.138(1) 0.0 50.9115[2,11] 0.0422(1) -0.4767(6) -0.148(1) 0.0 197.566[3,11] 0.0452(2) -0.489(1) -0.607(9) 0.0 5.91231 V b (cid:96) o (b) [2,5] 0.0376637, 0.000209043 -0.456079, 0.000749928 0.0 -0.0481111,0.000478584 107.329[3,7] 0.0444179, 0.000296234 -0.486774, 0.00141769 0.0 -0.213006, 0.00896163 1.93[2,11] 0.040609, 0.000129929 -0.466179, 0.000499672 0.0 -0.0536944,0.000366839 451.37[3,11] 0.0445357, 0.000226426 -0.487308, 0.00111589 0.0 -0.215779, 0.00766315 2.96961 V b , b (cid:96) o (c) [2,11] 0.0451733, 0.00025135 -0.496249, 0.00150301 -0.436979, 0.0205995 0.104738, 0.00747759 1.37428[3,11] 0.0450192, 0.000425963 -0.494044, 0.00514882 -0.327809, 0.24461 0.0244465, 0.17942 1.17367TABLE XIII. The χ values and the corresponding fit parameters; string tension and cutoff ( σ, µ ) together with the boundaryparameters ( b , b ) returned from fits three possible combinations of the boundary terms at LO level (a) V b (cid:96)o , (b) V b (cid:96)o and (c) V b ,b (cid:96)o given by Eq. (61),Eq. (62) and Eq. (63), respectively. ing an increase in the χ values by around 6 times larger.The enclosure of the fourth derivative term of the NLOterm in the NG action appears, thereof, neither to alterthe poor parameterisation behavior nor to indicate sig-nificant changes on the value of the string tension shownin Table. XII.
2. Boundary terms in L¨uscher-Weisz action
The values of χ returned from the fits to V b (cid:96)o and V b n(cid:96)o Eq. (61) are enlisted in Table XIV. The considera-tion of the boundary terms V b (cid:96)o or V b n(cid:96)o persist to provide9 V b , b n (cid:96) o Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) b (LU) b (LU) χ V b n (cid:96) o (a) [2,11] 0.0393(2) -0.4521(7) -0.3189(7) 0.0 1005.75[2,5] 0.0332(3) -0.427(1) -0.3077(8) 0.0 259.723[3,11] 0.0452(2) -0.489(1) -0.607(9) 0.0 5.91231 V b n (cid:96) o (b) [2,11] 0.0357542, 0.000149607 -0.427408, 0.000648593 0.0 -0.119262, 0.000259866 2447.1[2,5] 0.0262228, 0.000337028 -0.387406, 0.00151795 0.0 -0.115406, 0.000226769 688.624[3,6] 0.0429852, 0.00039019 -0.470346, 0.0019032 0.0 -0.424624, 0.00898908 6.57496[3,7] 0.0436562, 0.000301773 -0.473489, 0.00150711 0.0 -0.436586, 0.00787107 13.9662[3,11] 0.0442668, 0.000228102 -0.476414, 0.00117044 0.0 -0.448646, 0.00685283 23.7994 V b , b n (cid:96) o (c) [2,11] 0.0456057, 0.000249587 -0.495268, 0.0015125 -0.901874, 0.0186775 0.217754, 0.00695054 1.91735[3,11] 0.0459421, 0.000418268 -0.50015, 0.00509954 -1.14027, 0.238561 0.394536, 0.176489 0.912238TABLE XIV. The χ values and the corresponding fit parameters at NLO level in NG action; string tension and cutoff ( σ, µ )together with the boundary parameters ( b , b ) returned from the fits to three possible combinations of the boundary terms(a) V b n(cid:96)o , (b) V b n(cid:96)o and (c) V b ,b n(cid:96)o given by Eq. (61), Eq. (62) and Eq. (63), respectively. -0.6-0.4-0.2 0 0.2 0.4 2 4 6 8 10 12 V a R a -1 V R,b nlo (R,T/T c =0.8)Lattice Data α =0.21-R ε [3,11] α =0.21-R ε [4,11] α =0.21-R ε [5,11] (a) -0.6-0.4-0.2 0 0.2 0.4 2 4 6 8 10 12 V a R a -1 V R,b ,b nlo (R,T/T c =0.8)Lattice Data, β = 6.0 α =0.21-R ε [3,11] α =0.21-R ε [4,11] α =0.21-R ε [5,11] (b) FIG. 11. The quark-antiquark Q ¯ Q potential T /T c = 0 .
8, (a)Compares the rigid string models with the boundary parameter V R,b (cid:96)o given by Eq. (64) and Eq. (65) for the depicted fit interval R ∈ [ R m , R M ] (b) Similar to (a); however, the correspondingfits are considered for two boundary parameters ( b , b ) together with the leading order rigid string model V R,b ,b (cid:96)o and thecorresponding self-interacting model V R,b ,b n(cid:96)o Eq. (67). good χ values over short distance intervals R ∈ [0 . , . T /T c = 0 .
9. These are well-known deviations from thefree bosonic string over short distance even at zero tem-perature as well. This may suggest a role to the bound-ary terms for the deviation from the free non-interactingmodel over short distances
R < . V b n(cid:96)o , V b n(cid:96)o or V b ,b n(cid:96)o atNLO in NG string and V b (cid:96)o , V b (cid:96)o or V b ,b (cid:96)o at leading orderdoes not produce good χ for any fit interval involving R = 0 . R ∈ [0 . , .
1] fm is provided bythe ansatz V b ,b (cid:96)o which produces χ dof = 139 . / (9 − R = 0 . R tomagnify the long distances fits and the correspondingresiduals. The plot exhibits the large value of the residu-0 Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) α r χ V R (cid:96) o [ Rm, RM ] (a)[3,11] 0.0392(1) -0.4586(4) 0.000139251 795.844;2[4,11] 0.0436(2) -0.4816(9) 0.000190263 18.6047[5,11] 0.045 -0.5(2) 0.747119 V R n (cid:96) o (b)[3,11] 0.0326(1) -0.4085(7) 0.000334052 4609.75[4,11] 0.0422(2) -0.464(1) 0.000259934 23.94[5,11] 0.0449(1) -0.481(2) 0.00349991 2.61[6,11] 0.0463(6) -0.46(6) 0.138479 0.312231TABLE XV. The χ values and the corresponding fit param-eters at both LO and NLO level in NG action; string tensionand cutoff ( σ, µ ) together with the rigidity parameter α r re-turned from the fits to (a) V R(cid:96)o and (b) V Rn(cid:96)o given by Eq. (64),respectively. als and deviations over long distances when the point at R = 0 .
3. Rigidity terms in Polyakov-Kleinert action
Unlike the relative reduction in the square of the resid-uals χ at T /T c = 0 . T /T c = 0 .
8. The inclusure of the rigidity terms as wellas self-interactions at this temperature do not return anysigficant improvement in the fit behavior at short dis-tances at this temperature scale.Effects such as the string’s rigidity and self-interactionsseem to become noticeable at higher temperatures andenergies, the question should be posed here is whetherthese terms with the returned values at higher tempera-ture
T /T c = 0 . T /T c = 0 . χ at the inter-mediate distances 0 . ≤ R ≤ . T /T c = 0 . σ a = 0 . R ∈ [4 ,
11] and R ∈ [5 ,
11] with goodvalues χ , respectively. The fit parameters are given Estimate Standard Error R − . . b − . . b . . , (68) and Estimate Standard Error R − . . b − . . b . . . (69)Taking into account the additional degree of freedom en-dowed by b , the fits to Eq. (67) are returning χ = 3 . R ∈ [5 ,
11] the parameter values
Estimate Standard Error R . . b .
955 34 . b − .
381 41 . (70)the rigidity parameter in both cases is α r = 0 .
21 which isthe same measured at
T /T c = 0 . D. String tension
A discussion concerning the order of the phase tran-sition and the value of the temperature at the criticalpoint would be out of the scope of the present discus-sion. However, a correct string tension dependancy onthe temperature entails that σ ( T ) a at higher tempera-ture would fall into the same theoretical curve fixed bythe plateau value of σ a . This is equivalent to say thatthat all fits to the Q ¯ Q potential are returning the samevalue of σ a measured at zero temperature.In Fig 12 each theoretical curve is a plot correspond-ing to the respective order in the NG power expansion.Each of these curves is well defined by the plateau valueof σ a and endows the string tension dependency onthe temperature. Usually, this envolves a measurementextracted from the lattice data through the slop of thelinearly rising potential at low-enough temperature.The perturbative string tension up to the fourth powerin the temperature is layed out through Eq.(20) andEq.(24). The exact temperature-dependent NG stringtension is given by Eq. (25). A probable role of morehigher-order terms of the power expansion of NG stringaction may be discussed in the context of the string ten-sion dependency on the temperature.At the leading order NG string-tension the lattice datapoint at T /T c = 0 . σ (0 . T c ) a = 0 .
038 at thefixed parameter σ a = 0 . T /T c = 0 . σ (cid:96)o (0 . T c ) a = 0 . σ a = 0 . σ (0 . T c ) a = 0 . R ∈ [0 . , .
1] fm.These deviations at
T /T c = 0 . σ n(cid:96)o ( T ) a =0 . V R , b (cid:96) o Fit Interval Fit Parameters,
T/Tc = 0 . R ∈ I σ a µ (LU) α (LU) b b χ V R , b (cid:96) o [ Rm, RM ] V R , b n (cid:96) o χ values and the corresponding fit parameters returned from fits to the leading order (NLO) static potentialwith boundary terms Eq. (61) and Eq. (63). T L a s =0.036 s NG@LO: =0.044 s NG@LO: =0.044 s =0.8, c Lattice data: T/T =0.036 s =0.9, c Lattice data: T/T =0.044 s =0.0, c Lattice data: T/T (a) T L a s =0.039 s NG@NLO: =0.044 s NG@NLO: =0.039 s NG@NNLO: =0.044 s NG@NNLO: =0.044 s =0.8, c Lattice data: T/T =0.039 s =0.9, c Lattice data: T/T =0.044 s =0.0, c Lattice data: T/T (b)
FIG. 12. The temperature-dependence of the string tensionfor Nambu-Goto string action at LO, NLO and NNLO per-turbative expansion. The dashed lines correspond to σ a =0 .
044 and solid line corresponds to σ a = 0 . largly deviates from LO of NG string curve as shownin Fig. 12. Moreover, very small correction are receivedfrom the term proportional to the six power in the tem- σ a L T Rigid String, σ =0.045NG-NLO String, σ =0.045LGT, T=0, σ =0.045LGT Data FIG. 13. The solid line shows the temperature-dependence ofthe string tension for rigid string at NLO in NG perturbativeexpansionfor two values of the parameter α = 0 . α = 0 . σ a = 0 . perature. At the fourth and sixth order the string tensionis σ ( T ) a = and σ ( T ) a = 0 . σ a = 0 .
045 indicate that all or-ders in NG action do not provide the correct behav-ior of the temperature-dependent string tension in thepresent four-dimensional Yang-Mills model. In additionto that, the boundary corrections ( V b , V b ) Eq. (26) andEq. (35) do not contribute to the renormalization of thestring tension, since the modular transform of the poten-tial does not produce terms linearly proprtional to thestring length R .Nevertheless, the extrinsic curvature terms in PK ac-tion do redefine the temperature-dependence of the stringtension. The corrections to the NG string tension areuniquely determined by the value of the rigidity param-2eter.In Tables VI, the enlisted parameters of the rigid stringmodel at T /T c = 0 . σ a = 0 .
044 for fits over both intervals R ∈ [0 . , . R ∈ [0 . , .
1] fm. Similarly, in Table VII wherethe rigid string fit ansatz includes boundary correction V b Eq. (65) a stable value is returned over R ∈ [0 . , . R ∈ [0 . , .
1] fm.The string tension dependency on the temperature inrigid string model is given by the series expansion Eq. 54,with the asymptotic forms given by Eq. (56) and Eq. (55)at low and high temperatures, respectively.Figure. 13 compares the fourth power NG string ten-sion at next to leading order given by Eq.(24) and thecorresponding rigid string-tension given by Eq. (55) ver-sus the temperature for the returned α = 0 . α = 0 . σ a = 0 . T /T c = 0 . T /T c = 0 .
9. It would be desirableto include more lattice data [44] at lower and higher tem-perature to well-establish the flatness of QCD transitioncurve, which we report in the future.
IV. SUMMARY AND CONCLUDINGREMARKS
In this work, we compared the static quark-antiquarkpotential of effective bosonic string model of confinementbeyond free Nambu-Goto approximation to the Mont-Carlo lattice data [136]. The study mainly targets thecolor source separation R = 0 . R = 1 . SU (3)Yang-Mills lattice data in four dimensions. The theo-retical predictions laid down by the effects of boundaryterms in L¨uscher-Weisz (LW) action and extrinsic cur-vature in Polyakov-Kleinert (PK) action have been alsoconsidered.Both the LO and the NLO approximations of Nambu-Goto string show a good fit behavior for the data corre-sponding to the Q ¯ Q potential near the end of the QCDplateau region, namely, at T /T c = 0 .
8. The fit returnsalmost the same parameterization behavior with negligi-ble differences for the measured zero temperature stringtension σ a . The returned value of this fit paramter isin agreement with the measurements at zero tempera-ture [133].We detect signatures of two boundary terms of theLscher-Weisz (LW) string action. The (LW) string withboundary action is yielding a static potential which is ina good agreement with the static potential lattice data as well, however, for color source separation as short as R = 0 . T /T c = 0 . Q ¯ Q potential data for the fit region span the distances underscrutiny R ∈ [0 . , .
2] fm.Nevertheless, the fits show reduction of the residualsfor the next-to-leading order approximation of the NGstring on each corresponding fit interval. In both LO andNLO of NG string good χ is attained by the exclusionof the data points at short distances, i.e., R ∈ [0 . , . Q ¯ Q potential datawhich occur as a deviation from the standard value of thestring tension and the static potential data. The fit tothe Casimir energy of the self-interacting string returns avalue of the zero temperature string tension σ a = 0 . T /T c = 0 . σ a do not diminish.The fit of the static potential considering boundaryterms of LW action and contributions from the extrin-sic curvature of PK action show a significant improve-ment compared to that considering merely the ordinaryNambu-Goto string for the intermediate and asymptoticcolor source separation distances R ∈ [0 . , . χ and a zero temper-ature string tension σ a measured at T /T c = 0 . T = 0 [133], thus, indicating a correct temperature de-pendence of the string tension.The following enlists intervals over which the optimalvalue of χ is returned from the fit of the correspondingstring model: • The model encompassing one boundary term V b n(cid:96)o given by Eq. (61) return best fit on the interval R ∈ [0 . , .
1] fm. The rigid string model V Rn(cid:96)o ofEq. (64) retrieves best fit over interval R ∈ [0 . , . • Considering both rigidity and boundary term b the model V R,b n(cid:96)o of Eq. (65) extendsbest fit tothe interval R ∈ [0 . , .
1] fm. Eventually, thenext boundary-correction b of the model V R,b ,b n(cid:96)o of Eq. (67) reproduces best fit over the interval R ∈ [0 . , .
1] fm.3
ACKNOWLEDGMENTS
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