Sum rules and 2-quark flux-tube structure

TL;DR

This paper investigates sum rules connecting the static quark potential to flux-tube field distributions, using them to derive beta-functions, check flux-tube models, and estimate quark self-energies, with comparisons to string models.

Contribution

It introduces new applications of sum rules for extracting beta-functions, validating flux-tube models, and estimating self-energies in the context of quark confinement.

Findings

Derived generalized beta-functions from sum rules.

Validated flux-tube profiles against hadronic string models.

Provided estimates for quark self-energies.

Abstract

Sum rules -- relating the static quark potential V(R) to the spatial distribution of the action and energy in the colour fields of flux-tubes -- are applied in three ways: 1) To extract generalised beta-functions: 2) As a consistency check for the use of excited gluon flux-tubes and as an estimate of the quark self-energies: 3) To extract approximate sum rules using a simplified form of V(R). Also the flux-tube profiles are compared with hadronic string and flux-tube models.

Equations

$$\frac{-1}{b}\left(V+R\frac{\partial V}{\partial R}\right)+S_0=-\sum ({\mathcal{E}}_L+2{\mathcal{E}}_T+2{\mathcal{B}}_T+{\mathcal{B}}_L)$$

$$\frac{1}{4\beta f}\left(V+R\frac{\partial V}{\partial R}\right)+E_0=\sum (-{\mathcal{E}}_L+{\mathcal{B}}_L)$$

$$\frac{1}{4\beta f}\left(V-R\frac{\partial V}{\partial R}\right)+E_0=\sum (-{\mathcal{E}}_T+{\mathcal{B}}_T).$$

$$\frac{\partial {\beta }_{ij}}{\partial \ln a_k}=Sifk=iorjand\frac{\partial {\beta }_{ij}}{\partial \ln a_k}=Uifk\ne iorj$$

$$b=2(S+U)=-0.3715(1+0.49\alpha +\dots ),2\beta f=U-S=2\beta (1-1.13\alpha +\dots ),$$

$$f=-\frac{\left[V(R_1)-V(R_2)\right]}{2\beta \left[(A+B{)}_{R_1}-(A+B{)}_{R_2}\right]}$$

$$b=\frac{2[V(R_1)-V(R_2)]{\left[1+\frac{A_{R_1}-A_{R_2}}{B_{R_1}-B_{R_2}}\right]}^{-1}}{C_{R_1}-C_{R_2}},$$

$$A_R=\sum ({\mathcal{E}}_T-{\mathcal{B}}_T{)}_R,B_R=\sum ({\mathcal{E}}_L-{\mathcal{B}}_L{)}_R$$

$$C_R=\sum ({\mathcal{E}}_L+2{\mathcal{E}}_T+2{\mathcal{B}}_T+{\mathcal{B}}_L{)}_R.$$

$$V(R{)}_{A1g}=0.562+0.0696R-0.255/R-0.045/R^2$$

$$V(R{)}_{Eu}-V(R{)}_{A1g}=\pi /R-4.24/R^2+3.983/R^4.$$

$$c+2b_{s}R=\sum 2(S+U)[{\mathcal{E}}_L+2{\mathcal{E}}_P+2{\mathcal{B}}_P+{\mathcal{B}}_L]$$

$$c+2b_{s}R=\sum 2(S-U)[{\mathcal{E}}_L-{\mathcal{B}}_L]\mathrm{and}c+\frac{2e}{R}=\sum 2(S-U)[{\mathcal{E}}_P-{\mathcal{B}}_P],$$

$$\sum _{x,y,z}[S^{\mathrm{Ex}}]=\sum _{x,y,z}[S^{\mathrm{GS}}]\mathrm{and}\sum _{x,y,z}[E_L^{\mathrm{Ex}}]=\sum _{x,y,z}[E_L^{\mathrm{GS}}]$$

$$\frac{\Delta e}{R}\approx \beta f\sum _{x,y,z}[E^{\mathrm{Ex}}-E^{\mathrm{GS}}],$$

Full text

SUM RULES AND 2-QUARK FLUX-TUBE STRUCTURE

A.M.GREEN, P.S.SPENCER

  

C.MICHAEL

Sum rules – relating the static quark potential $V(R)$ to the spatial distribution of the action and energy in the colour fields of flux-tubes – are applied in three ways:

1. To extract generalised $\beta$-functions:

2. As a consistency check for the use of excited gluon flux-tubes and as an estimate of the quark self-energies:

3. To extract approximate sum rules using a simplified form of $V(R)$.

Also the flux-tube profiles are compared with hadronic string and flux-tube models.

1 Introduction

In ref. $\!{}^{{\bf?}}$ sum rules were derived to relate the sum over all spatial positions of colour fields (${\cal E,\ \cal B}$) to the static quark potential $V(R)$ and its derivative:

[TABLE]

[TABLE]

[TABLE]

Here $L(T)$ refers to longitudinal(transverse) with respect to the interquark separation axis. The ${\cal E}({\cal B})$ have the interpretation of gauge invariant averages of the fluctuation of squared strengths of the colour electric(magnetic)fields. The parameters $b$ and $f$ are related to the generalised $\beta$-functions

[TABLE]

by the expressions

[TABLE]

where the perturbative series for these quantities in terms of the bare lattice coupling $\alpha=g^{2}/4\pi=1/\pi\beta$ for $SU(2)$ colour fields are also given $\!{}^{{\bf?}}$. The $S_{0}$ and $E_{0}$ are the self energy terms. The general strategy for utilizing these sum rules is to insert known values or forms of $V(R)$ on the LHS and to measure on a lattice the ${\cal E,\ B}$ on the RHS. In this talk three such applications are discussed. In addition to these integrated colour field quantities, the individual field profiles are also studied.

2 Three applications of the above sum rules

The $V(R)$, ${\cal E}$ and ${\cal B}$ are all calculated in quenched SU(2) on the same $16^{3}\times 32$ lattice with $\beta=2.4$ using basis states with different degrees of fuzzing. The latter enables the excited gluon states with cubic symmetry $E_{u}$ and $A_{1g}^{\prime}$ to be studied.

1.1 $V(R)$ directly from the lattice to give Beta-functions

With $V(R)$ calculated on the same lattices as the ${\cal E,B}$ , combinations of the above sum rules are made at two values of $R$ to eliminate the ${\partial V\over\partial R}$ and also the self-energy terms giving:

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

This results in best estimates of $b=$ –0.35(2) and $f=$ 0.61(3). Therefore, $b$ is seen to be far from the perturbation estimate of –0.42 with $\alpha_{effective}=0.26$. A more detailed account can be found in ref. $\!{}^{{\bf?}}$

1.2 A parametrization of the lattice potential $V(R)$

Armed with estimates for $b$ and $f$, Eqs. 1-3 can be studied separately using the following interpolations of the lattice $V(R)$ for the ground ($A_{1g}$) and first excited ($E_{u}$) state:

[TABLE]

[TABLE]

For the ground state case it is found that all three sum rules are well satisfied with $S_{0}$=0 and $E_{0}$=0.1. However, the comparison is not so good for the $E_{u}$ state – indicating that this state is still somewhat contaminated for the values of Euclidean time $T\leq 4$, for which a signal could be measured. More details can be found in ref. $\!{}^{{\bf?}}$

1.3 A simple parametrization of $V(R)$

If the form $V(R)=c+b_{s}R+e/R$ is used in the sum rules, then they reduce to

[TABLE]

[TABLE]

where $e\approx-0.25(\pi)$ for the ground($E_{u}$) state. Since Eqs. 10 and 11a are independent of $e$, they reduce further to

[TABLE]

and the third becomes

[TABLE]

where $S^{i}(E^{i})$ refers to the action(energy) combinations of ${\cal E}({\cal B})$ in Eqs.  10, 11. For sufficiently large $R$, the replacement $\sum_{x,y,z}\rightarrow\frac{R}{a}\sum_{x,y}$ should be a reasonable approximation. In this case the 3-d sum rules in Eqs. 12,13 reduce to 2-d sum rules involving less Monte Carlo data and are useful when discussing flux-tube profiles in the next section. More details can be found in ref. $\!{}^{{\bf?}}$

3 Flux-tube profiles

So far only the sum rules for combinations of ${\cal E,\ B}$ have been discussed. In this section the non-integrated combinations of ${\cal E,B}$ are compared with two models. In Fig. 1a comparison is made for energy profiles at the centre of the line connecting the two quarks. There MC refers to the Monte Carlo result at $R=6$, BBZ to the dual superconducting model of ref. $\!{}^{{\bf?}}$ and IP to the string motivated model of ref $\!{}^{{\bf?}}$. In the latter, the string energy was tuned to fit the MC result on the central axis. The disagreement with the other two curves is not surprising, since the IP model includes a sizeable zero-point energy in its description of the gluon fields. Therefore, only differences between the profiles of the separate states should be compared. The similarity between the MC and BBZ results can then be interpreted as the two having similar self-energies . In Figs. 1b the action profiles for the ground and $E_{u},A_{1g}^{\prime}$ states are shown. Here, it is of interest to see that the $A_{1g}^{\prime}$ case has a dip at $r\approx 2a$ and is reminiscent of the node expected in excited s-wave states. The total action and energy profiles for the ground state are in agreement with refs. $\!{}^{{\bf?}}$  $\!{}^{{\bf?}}$.

The authors wish to acknowledge that these calculations were carried out at the Centre for Scientific Computing’s C94 in Helsinki and the RAL(UK) CRAY Y-MP and J90. This work is part of the EC Programme “Human Capital and Mobility” – project number ERB-CHRX-CT92-0051.