Anti-Screening by Quarks and the Structure of the Inter-Quark Potential

The inter-quark potential is dominated by anti-screening effects which underly asymptotic freedom. We calculate the order g^6 anti-screening contribution from light fermions and demonstrate that these effects introduce a non-local divergence. These divergences are shown to make it impossible to define a coupling renormalisation scheme that renormalises this minimal, anti-screening potential. Hence the beta function cannot be divided into screening and anti-screening parts beyond lowest order. However, we then demonstrate that renormalisation can be carried out in terms of the anti-screening potential.

Asymptotic freedom is the paradigm effect of QCD. It has been shown in many approaches [1][2][3][4][5][6][7][8][9][10][11][12] that the leading order beta function can be divided into screening and anti-screening effects. In QCD with n f light fermions this decomposition reads: where the dominant term (the 4) corresponds to anti-screening and the smaller − 1 3 − 2n f 3 terms describe screening by glue and by matter. It is well known that the gluonic screening effects are due to physical (gauge invariant) glue. Anti-screening [9] is due to the contribution of glue which is needed to construct a gauge invariant definition of a coloured charge [13]. The static inter-quark potential [14,15] would seem to offer a direct way to study the screening and anti-screening effects in QCD. The potential can be calculated via Wilson loops. At order g 4 the momentum space bare potential,Ṽ 0 , in D = 4 − 2ǫ dimensions is, up to some finite terms, where ν is a dimensional scale parameter and, in SU(N), C F = (N 2 − 1)/(2N) and C A = N. This may be renormalised by the standard charge renormalisation where the bare coupling α 0 = ν −2ǫ Z α α where This yields to order α 2 The Wilson loop approach does not, however, display the screening/anti-screening decomposition of the potential. We will therefore study here the interaction between two gauge invariant descriptions of the colour charges. This has previously been seen [9,11] to show the decomposition of screening and anti-screening in 3 + 1 dimensions as well as in the potential in 2 + 1 dimensions [10] (where the beta function vanishes). The structure of this paper is as follows. We first review how a gauge invariant description of physical charges directly shows how anti-screening effects are produced. Then we calculate, for the first time, how at order g 6 anti-screening effects due to light fermions arise in the quark potential and that, due to non-local divergences, a minimal (or anti-screening) charge renormalisation approach breaks down here. Renormalising the potential directly is shown to consistently handle the non-local structures and the full result for the renormalised potential at this order is given.
It has previously been shown [16,17] that the correct gauge invariant description of static charges in the ground state is given by h −1 ψ where h −1 is a field dependent dressing that surrounds the matter field ψ. The dressing has a rich structure which in QED is as follows: Here the second exponential is the minimal dressing which ensures that the minimally dressed matter field, exp −ie ψ, is gauge invariant. The other factor, which we call the additional dressing, is itself gauge invariant. 4 The structure of the dressing is reflected in calculations of the potential between charges: anti-screening coming from the minimal dressing [9] while the effects of screening were shown to come from the additional gauge invariant dressing [11]. The different factors in the dressing also make their presence separately felt in the infra-red structure of the on-shell Green's functions of dressed fields: soft divergences are cancelled by the effects of the minimal dressing, while phase divergences are removed by the additional dressing.
The description (5) has been extended to QCD order by order in perturbation theory. With the inclusion of colour, the minimal dressing up to order g 3 is given by [13] h −1 (x) = exp (gχ(x)) + O(g 4 ) , with χ = χ a T a = (χ a 1 + gχ a 2 + g 2 χ a 3 )T a and where we have defined and To now calculate the potential between such charges, we take a quark and an antiquark, both dressed according to Eq. 6, average over colours and study the expectation value of the QCD Hamiltonian. The potential is given by the dependence of the energy on the quark separation, r := |y − y ′ |. The lowest order contribution from either charge is of order g and so, to calculate the potential at order g 4 between two charges, we only need to expand the two dressings up to order g 3 . The potential is therefore where the trace is over colour and we have used the fact that B a i commutes with the minimal dressing.
It follows at leading order from (11) that Inserting the fundamental equal time commutator, , into this last equation gives at leading order We recognise the Coulombic inter-quark potential [19,20]. In general, and especially at higher orders, it is simpler to work in momentum space. Integral representations based upon the identity make the calculations much easier. In this way the contribution to the quark potential from the minimal dressing at O(g 4 ) has previously [9] been shown to be where d is the number of spatial dimensions (d = 3 − 2ǫ) and the tree level equal time gluon propagator in momentum space is given by The superscript T in (15) signifies projection upon the transverse components, k i A T i = 0. This shows the gauge invariance of (15) and it is straightforward, if tedious, to show that the longitudinal, gauge dependent A L i fields cancel in this result. At order g 4 this corresponds to inserting the free transverse projected, equal time propagator which yields the dominant part of the bare potential corresponding to anti-screening: This should be contrasted with the bare potential (2). The difference between the divergences in these two results is due to screening. The equations clearly show that gluons screen as well as anti-screen. The screening effect is due to transverse, gauge invariant glue from the additional dressing. The relative weighting of gluonic antiscreening to screening by glue is 12 to 1. (There is no anti-screening contribution from the matter fields at this order.) These effects have also been calculated [10] in 2 + 1 dimensions where it was seen that the relative weighting of anti-screening and screening in the potential is the same within 1%.
The Leading in n f Potential at Order g 6 At the next order in the coupling there are contributions from gluons and from light quarks. Here we will calculate the quark contribution, i.e., the n f dependent terms, to the minimal dressing. As is well known, quarks produce at next to leading order a screening of (electric and) colour charges and we have seen above that, at order g 4 , there are no contributions from quarks to the minimal anti-screening potential. However, we will now show that at next to next to leading order quarks also produce an anti-screening effect. This contribution is needed to ensure gauge invariance at higher orders. It occurs through the one loop, fermionic correction to the gluon propagator in (15).
In addition to (15) there are other contributions to the minimal potential at order g 6 . They arise by higher order expansions of the dressings and will involve Green's functions such as g 5 0|AAA|0 and g 6 0|AAAA|0 . These Green's functions will only depend on the n f light fermions through loops and it is easy to see that they will first introduce quark contributions beyond order g 6 in the coupling. We conclude that the first quark contribution to the anti-screening potential comes from (15) alone.
It should also be noted that although the QCD two point function 0|A T A T |0 in (15) is not generally gauge invariant at higher orders (see Appendix A of [13]), at one loop its n f dependent part is indeed gauge invariant. At order g 2 we have the well known n f dependent term from the one loop contribution to the gluon polarisation where we skip the obvious transverse projection tensor. This enters the one loop propagator via the contribution, iDiΠiD, which implies The one loop equal time propagator in momentum space, iD ij is now defined to be Projecting onto the transverse components (which are gauge invariant at this order in g) via δ il − p i p l /p 2 and δ jm − p j p m /p 2 gives Inserting (19) yields To calculate the n f dependent part of the potential, we now insert this into (15). It is helpful, though, to rewrite the resulting expression via where we have dropped terms that only contribute massless tadpoles in the subsequent integral and will hence vanish in dimensional regularisation. This leads to the order g 6 contribution to the potential (18) The divergent part of this contribution, in terms of the bare coupling α 0 , is: Note that these divergences are ultra-violet singularities as can be seen from power counting in (15) and (23). The leading singularity here is local, but the sub-leading divergences include the 1 ǫ log(k 2 ) term which is a non-local divergence. The immediate question is can renormalisation deal with this infinity?

Renormalising the Minimal Potential
The minimal part of the inter-quark potential has been previously calculated at order g 4 in both four and three dimensions. The result in four dimensions (18) may be renormalised by using minimal charge renormalisation where we define the minimally renormalised coupling, α ′ , through This anti-screening renormalisation is defined so that is finite at this order in the minimal coupling: This is a very direct way to extract the minimal, anti-screening beta function which has also been observed in very different ways [4]. This minimal coupling clarifies the nature and importance of anti-screening in non-abelian gauge theories. It is, however, very simple to show that the 'anti-screening coupling' cannot be used at next order. We have seen that there is a non-local, n f dependent divergence in the minimal potential at order g 6 and this is the only n f dependence in the minimal potential (26) up to this order. There is a logarithm at order α 2 0 which might help produce nonlocal divergences at α 2 0 but it is not n f dependent and in the leading anti-screening charge renormalisation (27) there is no n f dependence either. Any n f dependence in Z min α ′ at order α ′ 3 would, of course, be local and not introduce any logs into the potential at order α ′ 3 . Thus nothing can cancel the non-local divergence at order α ′ 3 in this approach. We are forced to conclude that the anti-screening or minimal charge renormalisation of the minimal potential breaks down beyond leading order. It is, in other words, impossible to define a coupling renormalisation scheme that renormalises the minimal, anti-screening potential. We cannot, beyond lowest order, speak of screening and antiscreening structures in the beta function.
It is, however, not necessary to use the anti-screening coupling in the minimal potential. Instead one can use full coupling renormalisation (3) plus an additional multiplicative renormalisation of the minimal potential in (18): where we write The minimal potential is easily seen to be finite at this order if This corresponds toṼ Our interpretation of this additional factor, Z V , is that it is a renormalisation of the additional potential energy between excited, minimally dressed charges compared to the true ground state of the fully dressed system, i.e., with screening effects included.
We will now show that this second approach may still be used at the next order of perturbation theory, i.e., the minimal potential is indeed renormalised by the full coupling (3) and the potential renormalisation of (29) and (30). At order α 3 scheme dependence appears and we use the MS scheme. We require the standard two loop coupling renormalisation where We now define At order n f α 3 in the potential, we first consider the 1/ǫ 2 terms. Inserting all the above renormalisation constants and demanding the cancellation of 1/ǫ 2 terms leads to (Note that the n f independent term must be corrected by gluonic anti-screening effects which we neglect.) Inserting this into the potential and demanding the vanishing of the local 1/ǫ terms yields plus various n f independent terms from the purely gluonic contributions to the antiscreening potential.
Having now fixed the renormalisation constant, it is very satisfying to see that the non-local divergences in (26) are cancelled in this scheme. At order α 3 there are three such non-local terms: they are generated by n f dependent local divergences in the renormalisation constants multiplying the logarithm in the one loop potential (18). One is from the n f part of Z V : while there are two further n f dependent contributions from the coupling constant renormalisation (3) since α 0 occurs twice in (18). Each of these yields and adding all three of these terms together we see that the n f dependent, non-local divergences in V min at order α 3 indeed cancel. We stress that this cancellation is a stringent test of the method since there was no freedom in the calculation. We conclude that this renormalisation programme can be carried through. Our final result for the renormalised, anti-screening potential is whereṼ 2 min (k 2 ) = 4C A C F 3k 2 −7 + 6 ln(2) + 3 ln and the n f dependent terms +3 ln k 2 ν 2 −14 + 12 ln(2) + 3 ln where ν 2 = 4πν 2 e −γ E .

Conclusions
We have seen that the decomposition of the beta function into screening and antiscreening structures breaks down beyond one loop. This we saw by calculating the light quark contributions to the anti-screening potential: non-local divergences arose in fermion loops in the minimal potential at order g 6 which are not cancelled by an anti-screening beta function. This is due to anti-screening effects from light fermions which are necessary consequences of a gauge invariant construction of charges. However, we have seen that it is possible to renormalise this potential via full charge renormalisation plus a multiplicative renormalisation of the potential. This renormalisation provided a stringent test of the method. It is to be understood as a renormalisation of the additional energy due to the neglect of screening interactions in a minimally dressed construction of charges.
The results presented here suggest that a decomposition of the potential into a minimal, anti-screening part plus an additional screening structure is indeed possible. Further studies of this decomposition may help to clarify the structure of the forces between heavy quarks.