Mapping of Nambu-Goto String in d Space-time Dimension and QCD Potential

Received: 27/Sept/2019, Accepted: 15/Oct/2019, Online: 31/Oct/2019 AbstractThe bosonic string potential model proposed by Nambu and Goto contains Coulombic Lüscher term, which is a function of the space-time dimension ‘d’. In this work, we report some novel features of QCD-String mapping, if the colourcum-flavour dependent coefficient of the QCD coulomb potential and the Luscher coefficient in d space-time dimension are conjectured to be equivalent.


I. INTRODUCTION
Inter quark potential is mainly generated by gluon dynamics, under static approximation, with valence quark effect. At distances smaller than 1 Fermi, the quark-antiquark potential is coulombic, due to Asymptotic Freedom. At large distances the potential should be linear due to formation of a confining flux tube [1]. When these tubes are much longer than their thickness we can describe them by semi-classical Nambu strings [2]. This explains the existence of approximately linear Regge trajectories [3]. Empirical evidence for a string-like structure of hadrons comes from arranging mesons and baryons into approximately linear Regge trajectories.
Throughout its history, string theory has been intertwined with the theory of strong interactions. The AdS/CFT [4] correspondence succeeded in making precise connections between 4-dimensional gauge theories and higher dimensional string theory.
In recent years, there has been growing interest in the possible correlation between QCD and string theory through ACD/QCD approach [5]. Inspired by such approach, we have analyzed some novel features of QCD-String correspondence, in this paper.
The strength of the quark-gluon interaction is characterized by the coupling constant α s (Q 2 ) [9], defined by convention in a particular dimensional regularization scheme. The coulombic term of QCD potential contains this colour-cumflavour dependent coupling constant.
In this paper, The Nambu-Goto string potential [6] and standard QCD potential [7] are compared. The colour-cumflavour dependent coefficient of the QCD coulomb potential and the Lüscher term of Nambu-Goto potential in d spacetime dimension are conjectured to be equivalent, and the possible QCD-string correspondence is studied. The correspondence of QCD parameters like Q 2 , N c and N f with the space-time dimension 'd' of string model is also reviewed.
Chapter 2 contains brief review of formalism. In chapter 3 we report our conjectured correspondence and chapter 4 contains conclusion and comments.

2.1
Running coupling constant in QCD: In the framework of perturbative QCD (pQCD) [8], predictions for observables are expressed in terms of renormalized coupling [9] , a function of an renormalization scale . When is close to the scale of momentum transfer Q in a given process, is a measure of the effective strength of the strong interaction in the process.
This running coupling satisfies the Renormalisation Group Equation (RGE) [10]: (1) where, (2) This is the one-loop beta function coefficient and This is the two-loop beta function coefficient.
Here, T R is an invariant in QCD colour algebra [11] and is conventionally taken to be ½.
. is called the colour factor which is given by : This gives, From equation (1), the perturbative expansion of running coupling is obtained as [12]: (8) with defines the scale where the perturbatively-defined coupling would diverge, i.e., it is the non-perturbative scale of QCD [13].

QCD and String Potentials:
It has been found that the most consistent inter-quark potential in QCD has the form [6] : (9) This formula is basically an interpolation between the long range non-perturbative (br) and short range repulsive ( ) parts of the force between pairs of quarks. Here the coefficient of the coulombic term depends upon the colour ( N c ) and flavor (N f ) quantum numbers and also on momentum transfer Q 2 .
Again, from Wilson loop calculation, it is now established that the static quark-antiquark potential V(r) for bosonic strings, asymptotically rises linearly with r as [14]: → This is the linear quark confinement. The generally accepted leading order correction to this quark-antiquark linear potential has been calculated as [2]: Considering this first order correction, the Nambu-Goto string potential can be expressed as [7] : Here is the universal Lüscher term with [15]. It does not depend upon the gauge group -depending only on the space-time dimension d; it is universal in the sense that its value is the same for a large class of string action.
is the string tension, which characterizes the strength of the confining force, its value is 0.89 GeV/fm ( ~ 0.178 GeV 2 ). is a regularisation constant.

III. CONJECTURED CORRESPONDENCE
We compare the Nambu-Goto string potential in equation (11) with the standard QCD potential of equation (9). In standard QCD potential, the first term is perturbative and coulombic, while the similar term in equation (11) is having non-perturbative origin. The standard value of b [16] in equation (9) is 0.183 Gev 2 (~ 0.915 GeV/fm) to be compared with the corresponding value of [17] as 0.178 GeV 2 (~ 0.89 GeV/fm). The constant is a regularisation constant in Nambu-Goto string potential, whereas that scale factor 'c' in equation (9) is assumed to have numerical value of 1 GeV [18], to make it compatible with the usual masses of the mesons.
Further, we would like to study the correspondence between coefficient of coulombic term in QCD potential and the Lüscher coefficient in Nambu-Goto potential. The conjectured equivalence of Lüscher coefficient and the coefficient of coulombic term in QCD potential leads to : (12) with (13) At leading order (LO), the running coupling expansion is given by [12] , with (14) is given by equation (2). From equation (12) and ( 14) , we obtain correspondence at LO as: (15) The next-to-leading order (NLO) of running coupling expansion [12] is, with b 1 given by equation (3). Equations (12) and (16) give the correspondence at NLO as: Equations (15) and (17) are the conjectured correspondence between coulombic term of QCD potential and Lüscher term in Nambu-Goto potential. These two equations immediately show that while d=2 corresponds to as -the force field limit of QCD, corresponds to the non-perturbative scale. Thus, is the allowed limit for the space-time dimension, if the conjectured correspondence is taken seriously.
In It shows that strings of a definite dimension have correspondence with QCD at a definite Q 2 . Here, since dimension, by definition, is positive integer, such correspondence is possible only for discrete values of Q 2 explored in QCD. Further, it is evident from table-1 and also from figures-1 that beyond d=5, the NLO effect in becomes insignificant.

IV. CONCLUSION AND COMMENTS
In this paper, we have showed that, a conjectured mapping between Lüscher term of Nambu-Goto Strings and QCD standard potential yields relationship between space-time dimension d in string theory and virtuality Q 2 explored in QCD. Here, since dimension d, by definition, is positive integer, such mapping is limited to a specific set of virtuality. For a one to one mapping, the dimension of the strings needs to be continuous or fractionala notion, perhaps too novel to be taken seriously at least at the present manner. Feasibility of calculative potentiality of such a mapping and its possible relationship with celebrated ADS/QCD [5] correspondence needs further study.