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The static potential in quantum chromodynamics
Volume 69B, number 2
PttYSICS LETTERS
1 August 1977
THE STATIC POTENTIAL IN QUANTUM CHROMODYNAMICS Thomas APPELQUIST 1,2 and Michael DINE 1 J. WillardGibbs Laboratory, Yale University, New Haven, CT 06520, USA
and I.J. MUZINICH 3 Brookhaven National Laboratory, Upton, N.Y. 11973, USA Received 18 May 1977 The static potential between a heavy quark and antiquark in quantum chromodynamics is analyzed in the weak coupling expansion. At the one loop level, a physical explanation of asymptotic freedom is offered. In higher orders, the problem of separating an effectively instantaneous two body potential from iterations of the potential and from multiparticle (:~ 3) configurations is discussed. Scaling properties of the potential are studied and some phenomenological observations are made. In this letter, we report some preliminary results of a study of the heavy quark-antiquark interaction in quantum chromodynamics (QCD). The static limit is analyzed in the weak coupling expansion with several goals in mind. Intuition suggests that the static problem is simpler than the full dynamical one and it is important to explore its simplifying features. We do not attempt direct computation of binding energies but instead make some general observations about the structure of the static interaction. Foundations are laid for further studies of the static problem using the renormalization group and also non-perturbative methods 4:1 . Some speculative phenomenological conclusions can already be drawn and they are discussed. For simplicity, this paper will be restricted to a consideration of the static QO potential with all quark motion effects neglected. In a subsequent paper, we intend to extend our observations on the potential and consider the full bound state problem including spin I Research (Yale report no. COO-3075-172) supported in part by ERDA Grant No. EY-76-C-02-3075. 2 Alfred P. Sloan Foundation Fellow. 3 Research supported, in part, by Energy Research and Development Administration. ,l The limitations of a Feynman graphical analysis are well known. On the other hand, it is possible that the structure we axe observing may provide some insight into the nature of the non-relativistic quark-antiquark interaction at all distances. The analysis also serves as a starting point for the study of perturbation theory in high orders.
dependent forces. As a point of reference, it is worth commenting that this problem is completely trivial in quantum electrodynamics. There the static potential is e2/r and arises from the exchange of a single Coulomb photon. It is important to point out that the attempt to isolate a static potential is quite different from the computation of binding energies using the Bethe-Salpeter equation [1 ]. There one would include all contributions to each order in g2, be they corrections to the potential, or kinetic energies, spin dependent effects, etc. Apart from renormalization, the only scale is then the constituent mass M. Neglecting renormalization for the moment, the binding energy will be of the form f(g2)M and one is really only computing a dimensionless function f(g2). In the bound state, the typical moment transfer q is o f order g2M. The static potential, on the other hand, is isolated by treating q/M as an independent parameter. It then makes sense to speak of the large mass limit and the potential is obtained by neglecting the quark kinetic energy and working to zeroth order in q/M. The presence of the renormalization scale/~ complicates this comparison slightly but the essential point o f distinction remains. Whether or not a potential defined in this way is relevant to experiment is not clear. It must be assumed that the neglected relativistic correction terms stay small in the presence of higher order corrections. The M-+ oo limit is examined and the compariso n i,s made with the computation of the potential between fixed classical sources ~:2. 231
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The separation of potential from iterations of the potential is discussed in some detail. There are also multiparticle (/>3) contributions signaled by the presence of singularities in the static limit. The separation of an effectively instantaneous potential from these is discussed briefly and will be treated in more detail in a forthcoming paper. Preliminaries. In a theory with massless fields, the longest range component of the force (V ~ 1/r) is expected to correspond to single quantum exchange. If there are no self-coupled massless fields, then this is in fact true but it can be far from obvious. For example, in a theory with a massless neutral scalar field ¢ coupled to a massive fermion ~b the exchange of one meson gives the 1/~ 2 Coulomb potential in the static limit. Double exchange can take place either with the ¢ lines crossing (a contribution to the Bethe-Salpeter kernel) or with the ¢ lines uncrossed (an iteration of the lowest order kernel). The crossed box takes the following form in the static limit A
£
1 1 M -M crossed ~ J d4k k 2 + ie (k + q)2 + ie ~/k0 + ie M k 0 + ie
(1) where q~ = (0, ~) is the momentum transfer. The energy contour can be closed to avoid the fermion poles, giving a long range and infrared divergent contribution 1/~ 2 In ~2/t52 where ~5is an infrared cutoff. The box diagram in the static limit becomes Auncrossed f d 4k
(2) 1 1 M -M k 2 + ie (k +q-i2 ÷ ie M k 0 +ie - M k 0 +ie"
The energy integral will pick up contributions from the meson poles which precisely cancel the crossed box. In addition, one of the fermion poles must be picked up giving a k 0 = 0 term. This piece can be identified as the iteration of the potential (energy transfer = 0 in both lines) and in fact is singular in the static limit. The singularity is shielded by keeping the quark kinetic energy non zero and one then discovers the well known 1/Oquark singularities which begin to build up the Coulomb bound state. :#2 A covariant gauge computation of the potential between f'Lxed sources at the one loop level has been given by Susskind [2]. It is being extended to the two loop level bv W. Fischler and L. Susskind.
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Quantum electrodynamics in a covariant gauge behaves in much the same way. However, the gauge freedom of QED can sometimes be used to eliminate spurious long range forces at the outset. For the static problem, the natural choice is Coulomb gauge in which 1/~/2 behavior is found only in single Coulomb photon exchange. Graphs with crossed Coulomb lines vanish in the static limit and uncrossed multiple Coulomb exchanges are strictly iterations of the potential. The Coulomb gauge continues to be extremely useful in examining the static potential in QCD. The dynamics is now considerably more complicated but spurious contributions are still eliminated. The Yang-Mills theory in Coulomb gauge has been studied by many authors [3, 4] and we shall summarize only a few of its features. The gluon propagator has an instantaneous part and a transverse part: D ab= i 5 ab ~ gv ~2
~=v=O (3)
=i~ab(6ij kikj) 1 - - - ~ - f k2 + iE
la = i , v = ] .
A Fadeev-Popov ghost must be included but it couples only to the transverse gluon. An important Ward identity for the quark-Coulomb gluon IPI vertex can be proven [5]. Its main consequence is that the primitively divergent part of this vertex is equal to the primitively divergent part of the quark wave function renormalization. Infinite coupling constant renormalization comes only from Coulomb gluon propagator corrections. For our purposes, it is most convenient to define the mass by subtracting on-mass-shell, certainly a possibility to any finite order of perturbation theory. This definition of the mass will be implicit in some of the computations and Ward identity applications we shall discuss. This procedure cannot be completely correct of course since it gives a renormalized mass proportional to the bare mass and thus neglects any dynamical component. In the case of the light quarks, the bulk of their mass (a few hundred MeV) is believed to be dynamically generated and inherently nonperturbative. It is reasonable to assume that the dynamical part of a heavy quark mass is also no more than a few hundred MeV. It can then be neglected since it is on the order of or smaller than relativistic corrections. Our on-shell definition can be viewed as working in this approximation.
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can be seen by noting that the energy denominator Ikl + Ill in old fashioned perturbation theory is of order q if the loop is subtracted at II ~ q. Thus this process takes place over a typical time scale of order 1/q which is not instantaneous but which is small compared to the time scale for quark recoil (of order
k
0
(b)
(a)
M/q2).
Fig. 1. Contributions to the running coupling constant in Coulomb gauge. The labels 5, ~ and ~ describe respectively the coordinate space locations of the quark, the antiquark and an intermediate interaction point.
One Loop. We restrict
our attention to one flavor of heavy quark and neglect the light quarks altogether. The Ward identity insures that there is no ultraviolet divergent contribution from the sum of quark self energy and quark-Coulomb gluon vertex graphs. This is easy to verify explicitly. Infrared divergences are present in vertex and quark self energy diagrams involving transverse gluons and are associated with the possibility of production in perturbation theory. However they vanish in the static limit due to the quark velocity factor in the transverse gluon-quark coupling. The vertex and quark self energy contributions together can in fact be shown to give no contribution to coupling constant renormalization in the static limit. The crossed ladder with Coulomb and/or transverse gluons vanishes in the static limit as does QQ annihilation and vacuum polarization by massive QQ pairs [6]. The two diagrams shown in fig. 1 are the only important one loop contributions and they have already been computed in the literature [1, 4]. We discuss them here from a physical point of view to pave the way for the examination of more complicated contributions and to identify the physical origin of asymptotic freedom in the context of the static problem. Fig. la is vacuum polarization, a shielding effect which should reduce the renormalized coupling strength. Computation bears this out, giving in the instantaneous approximation (q0 = 0) the following correction to the Coulomb propagator ~ab i/t~2 : UP
_
i
SCA Inq/la]
I+.~4g2
1 August 1977
(4)
/a is the renormalization scale, q --- 141 and C A is the Casimir operator of the adjoint representation. Although vacuum polarization is not instantaneous, it appears as effectively instantaneous to the quarks. This
The Coulomb field self energy contribution fig. lb is, by contrast, completely instantaneous. The full propagator can be written in the form 6
_~i I2g 2 ~
q~q][" d3k ~i]- kikf/k2
1 (5)
Momentum integration and subtraction at q =/a gives the resulting contribution to the renormalized Coulomb propagator:
DSE ab(q) ='ab ~i [-1692 2-4--~ CA In q/la] .
(6)
The asymptotic freedom of the Coulomb field self energy can be seen from the positivity and ultraviolet divergence of the square bracketed correction in expression (5). It is a q dependent and cutoff dependent increase in the magnitude of the field strength over and above the bare Coulomb field. The renormalized Coulomb propagator (6) shows the attendant weakening of the effective coupling strength at short distances. The asymptotic freedom of the theory follows from the dominance of this contribution over the polarization of the vacuum (4), a computational result without an apparent simple explanation. Adding (4) and (6) gives the usual running coupling constant of the pure gauge sector of the theory. The physics of asymptotic freedom is most apparent in configuration space. Suppose that a quark source is located at the origin and an antiquark probe is at R. The Fourier transform of expression (5) is a positive addition to the 1Ix Coulomb potential. The dominant term in the transverse propagator is the 6ii, so we exhibit only that part for simplicity. One of the external Coulomb propagators (e.g. the one connecting to the quark source) is then cancelled and the addition to the 1Ix potential is 292C A f d3z 1 1 1 A -1 Z2 Z IX -- ~'l "
(7) 233
Volume 69B, number 2
I
PHYSICS LFTTERS
>1
t
k
i k
I
I I
II A
I
>
>
Vl+k+q I I >
-k -I
Fig. 2. A multiple exchange contribution to the static potential Integration gives an expression of the form (1 Ix)lnx A. This is a scale dependent increase ( f o r x >> I/A) of the attractive (repulsive) force between the quark and antiquark if they are in a color singlet (octet) state. The source of the increase is the attractive potential 1/z 2 which operates between the Coulomb field lines at the position of the source and at the point f. It arises from the instantaneous exchange of the octet transverse gluon between these points and is attractive because the two points of the Coulomb field are in a relative singlet state. The field lines connecting the source to the point ~ are thus drawn together, increasing the magnitude of the energy stored in the field. This mechanism of logarithmic flux collimation is clearly unique to Yang-Mills theories. Higher Orders. At two loops, the general structure of the static problem begins to emerge. The two loop corrections to single gluon exchange continue to build up the running coupling constant and in addition, multiple gluon exchange enters in the form of fig. 2. At three loops, contributions appear which are singular in the static limit and yet which are not iterations of the potential. We first consider fig. 2 and next the corrections to single Coulomb gluon exchange. In the static limit, the quark propagator (say the upper one in fig. 2) becomes i(l 0 + ie) -1 . A power counting analysis of fig. 2 in the static limit leads to the conclusion that it is convergent in both the ultra° violet and infrared. Therefore, on dimensional grounds, it behaves like 1/~ 2. This is something new dynamically, a Coulombic force associated with multiple massless quantum exchange. It is also new in the algebraic sense since in the exchange channel, it transforms like 8 × 8. With the quarks in the fundamental representation, this diagram contains both an octet and a new singlet exchange part. The total potential is no longer just a feature of the gluonic sector of the theory but it depends on the representation content of the sources. 234
1 August 1977
The space-time properties of this diagram are similar to vacuum polarization. The energy denominator of non-relativistic perturbation theory is simply the transverse gluon energy k when the quark kinetic energy is neglected. Convergence of the integral implies that k is of order q so that the time interval At between production and disappearance of the virtual transverse gluon is of order 1/q. During this time, quark recoil can be neglected since it takes place over a time scale of order M/q 2. We next consider the vertex and self energy corrections to single gluon exchange. To any order, the Ward Identity insures that ultraviolet divergent coupling constant renormalization comes only from corrections to the gluon propagator. The Ward identity is not exactly of the Abelian electrodynamic form. There are additional pieces involving the Fadeev-Popov ghost but these pieces can be shown to be ultraviolet finite to all orders even in the limit M ~ oo [5 ]. Thus the combination of the quark self energy and the quarkCoulomb vertex is ultraviolet finite in the infinite quark mass limit. In the static limit (neglecting the quark kinetic energy and holding q finite but small compared to M), there is one two-loop vertex and one two-loop quark self energy which are infrared divergent. These are formed by considering the emission and reabsorption of a Coulomb gluon by the quark and then inserting a transverse gluon pair loop in the Coulomb propagator. The divergence corresponds to the possibility of the transverse gluon pair becoming soft and real. Fortunately, there are two remaining two-loop graphs which naturally pair with these two and which eliminate this infrared divergence for a color singlet QQ. pair. They are shown in fig. 3. Part of fig. 3b is simply the iteration of the potential. However, an additional piece survives which comes from the gluon energy denominators and has a similar structure to fig. 3a. Each graph is infrared divergent and the sum of
k÷q;, s
s~'
(o)
~.,
~
I I
6
k
I
Ib)
Fig. 3. Some two loop diagrams with infrared singularities in the static limit.
Volume 69B, number 2
i
PHYSICS LETTERS
k
!
I
Fig. 4. A three loop graph with a singular static limit.
figs. 3a and 3b is of the form 1
#2 [X l [xa] Q (ha q/8
+ const)
(8)
where <5 is the infrared cutoff. For a color singlet QC~ pair, fig. 3a pairs with the infrared divergent two-loop vertex and fig. 3b pairs with the infrared divergent two-loop quark self energy. All ~ dependence cancels. The combination of all the two-loop vertex and self energy corrections to single Coulomb exchange along with figs. 3a and 3b thus produces an effective potential for a color singlet QQ pair of the form (1/ #2)f(q/la, g2) in the limit M ~ oo. The function f can be computed and compared with covariant gauge computations of the two-loop fl function [7]. In addition there is the new two-loop "skeleton" shown in fig. 2. At the three loop level and beyond, there exist diagrams which are singular in the static limit and which are uncancelled by other contributions. An example of a static limit singularity is the three loop diagram of fig. 4. This is most easily seen by doing the energy integrals leading to a transverse gluon integral d3k/k and two energy denominator factors 1/k. If more Coulomb exchange rungs are added, the degree of divergence presumably increases. Effects of this type must be expected at some level in a quantum field theory and have been encountered often in quantum electrodynamics. The most familiar example is the Lamb shift in hydrogenic systems. There the famous Bethe logarithm arises from low frequency contributions being cut off on the order of the electron kinetic energy (Za)2m. Another example arises in the hyperfine interval computation in positronium [8]. Single transverse photon exchange accompanied by one or more Coulomb photon exchanges contributes at the ,v5 level. Each additional Coulomb rung adds a power of ot and a compensating energy denominator. The difference between QED and QCD is that in the former, this type of configuration mixing is very naturally identified as a relativistic correction. Transverse photons must couple to the fermions giving ve-
1 August 1977
locity factors in the static limit. In QCD, this ,~eed not happen and then there are no velocity factor:, which vanish in the static limit. Of course, in a systematic perturbative computation of the binding energy using the Bethe-Salpeter equation, the distinction becomes blurred since u is of order g2. Summary. The following picture has emerged. There exists a class of diagrams (a subset of the two particle irreducible kernel graphs) which are finite in the static limit. This class includes single Coulomb gluon exchange. The next "skeleton" in this class is fig. 2. It behaves like 1/# 2 and contains octet and singlet exchange. It too can be corrected with vertex and self energy insertions which will modify the I/#2 behavior. Other such skeletons can easily be drawn and providing they obey one important condition, they too will be finite in the static limit and behave like 1/#2. For a bare skeleton, a sufficient condition for the existence of the static limit is that the diagram be connected in the gluonic sector. The class of all such diagrams can be used to define a static potential which on quite general grounds, can be written in the momentum space form
(1/#2)K(q/la, M/V, gv).
(9)
K contains both octet and singlet exchange. Suppose, first of all, that QQ loops are omitted. It is then simplest to define the subtraction constants so that gv is explicitly M independent. It has been shown through the two loop level that K is ultraviolet finite in the limit M ~ ,,o and free of infrared divergences. This result has followed from the Ward identity and some power counting. It is true to all orders for single Coulomb exchange with vertex and self energy corrections. It is conjectured to be true to all orders for all the other dressed skeletons. Assuming it is, the momentum scale in K is set by q and IX. It follows that QQ loop insertions will be suppressed by O(q/M) or O(,u/M) except for their contribution to infinite coupling constant renormalization [6]. The upshot of this (assuming it can be established to all orders for all the skeletons) is that the potential K def'med in this way is independent of the constituent mass M. While it depends on the representation content of the sources, it is dynamically a property of the gluonic sector of the theory. If light quarks are included, the potential will depend on the glue plus light quark sector. If more than one heavy flavor is in235
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PHYSICS LETTERS
cluded, the potential will be universal for each o f the heavy flavors unless the momentum transfer q for the heaviest is large enough to polarize the lighter heavies in the vacuum. This will provide some justification for the use of a universal potential to describe the c~ and bl~ (M b ~ 4 - 5 GeV) systems [9]. Disconnected diagrams remain which are not identifiable as iterations o f the two body potential. The ones we have examined so far are divergent in the static limit and therefore describe long time physics -multiple particle configurations. If the potential defined in terms of the connected graphs is to correspond to the phenomenological QQ potential, then it must be shown that configuration mixing is a small effect. In a subsequent paper we shall discuss in more detail the separation of the two b o d y potential at three loops and beyond. Conversations with Leonard Susskind are gratefully acknowledged. One o f us (TA) would like to thank members of the Cornell University theory group for several helpful discussions. Many of our conclusions have been independently arrived at by F. Feinberg. We thank him for communicating his results to us before publication and for pointing out an error in an earlier version o f this manuscript.
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References [1 ] A. Duncan, Phys. Rev. D10 (1976) 2866. [2] L. Susskind, in: Coarse grained quantum chromodynamics, lectures given at the Les Houches Summer School on High energy physics, 1976. [3] E.S. Fradkin and I.V. Tyutin, Phys. Rev. D2 (1970) 2841 A. Ali and J. Bernstein, Phys. Rev. D12 (1975) 503; R.N. Mohapatra, Phys. Rev. D4 (1971) 378; D4 (1971) 1007; D.P. Sidhu, Phys. Rev. D6 (1972) 565; E. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1. [4] J. Frenkel and J.C. Taylor, University of Oxford Preprint, 21/76 (1976). [5 ] The details will be presented in a subsequent paper. They follow closely the corresponding proof in covariant gauges given by F. Feinberg and E. Eichten, Phys. Rev. D10 (1974) 3254. A closely related Ward identity is proven by Taylor and Frenkel in ref. [4]. [6] T. Appelquist and J. Carazzone, Phys. Rev. D11 (1975) 2856. [7] Covariant gauge computations have been carried out by W.E. Caswell, Phys. Rev. Lett. 33 (1974) 244; D.R.T. Jones, Nucl. Phys. D15 (1974) 5321; A.A. Belavin and A.A. Migdal, unpublished (1974). [8] E.E. Salpeter, Phys. Rev. 81 (1952) 328; T. Fulton and R. Karplus, Phys. Rev. 93 (1954) 1109; R.J. Eden, Proc. Roy. Soc. (London) 219 (1953) 516. [9] E. Eichten and K. Gottfried, Cornell University Preprint, CLNS-350, November, 1976.
PttYSICS LETTERS
1 August 1977
THE STATIC POTENTIAL IN QUANTUM CHROMODYNAMICS Thomas APPELQUIST 1,2 and Michael DINE 1 J. WillardGibbs Laboratory, Yale University, New Haven, CT 06520, USA
and I.J. MUZINICH 3 Brookhaven National Laboratory, Upton, N.Y. 11973, USA Received 18 May 1977 The static potential between a heavy quark and antiquark in quantum chromodynamics is analyzed in the weak coupling expansion. At the one loop level, a physical explanation of asymptotic freedom is offered. In higher orders, the problem of separating an effectively instantaneous two body potential from iterations of the potential and from multiparticle (:~ 3) configurations is discussed. Scaling properties of the potential are studied and some phenomenological observations are made. In this letter, we report some preliminary results of a study of the heavy quark-antiquark interaction in quantum chromodynamics (QCD). The static limit is analyzed in the weak coupling expansion with several goals in mind. Intuition suggests that the static problem is simpler than the full dynamical one and it is important to explore its simplifying features. We do not attempt direct computation of binding energies but instead make some general observations about the structure of the static interaction. Foundations are laid for further studies of the static problem using the renormalization group and also non-perturbative methods 4:1 . Some speculative phenomenological conclusions can already be drawn and they are discussed. For simplicity, this paper will be restricted to a consideration of the static QO potential with all quark motion effects neglected. In a subsequent paper, we intend to extend our observations on the potential and consider the full bound state problem including spin I Research (Yale report no. COO-3075-172) supported in part by ERDA Grant No. EY-76-C-02-3075. 2 Alfred P. Sloan Foundation Fellow. 3 Research supported, in part, by Energy Research and Development Administration. ,l The limitations of a Feynman graphical analysis are well known. On the other hand, it is possible that the structure we axe observing may provide some insight into the nature of the non-relativistic quark-antiquark interaction at all distances. The analysis also serves as a starting point for the study of perturbation theory in high orders.
dependent forces. As a point of reference, it is worth commenting that this problem is completely trivial in quantum electrodynamics. There the static potential is e2/r and arises from the exchange of a single Coulomb photon. It is important to point out that the attempt to isolate a static potential is quite different from the computation of binding energies using the Bethe-Salpeter equation [1 ]. There one would include all contributions to each order in g2, be they corrections to the potential, or kinetic energies, spin dependent effects, etc. Apart from renormalization, the only scale is then the constituent mass M. Neglecting renormalization for the moment, the binding energy will be of the form f(g2)M and one is really only computing a dimensionless function f(g2). In the bound state, the typical moment transfer q is o f order g2M. The static potential, on the other hand, is isolated by treating q/M as an independent parameter. It then makes sense to speak of the large mass limit and the potential is obtained by neglecting the quark kinetic energy and working to zeroth order in q/M. The presence of the renormalization scale/~ complicates this comparison slightly but the essential point o f distinction remains. Whether or not a potential defined in this way is relevant to experiment is not clear. It must be assumed that the neglected relativistic correction terms stay small in the presence of higher order corrections. The M-+ oo limit is examined and the compariso n i,s made with the computation of the potential between fixed classical sources ~:2. 231
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PHYSICS LETTERS
The separation of potential from iterations of the potential is discussed in some detail. There are also multiparticle (/>3) contributions signaled by the presence of singularities in the static limit. The separation of an effectively instantaneous potential from these is discussed briefly and will be treated in more detail in a forthcoming paper. Preliminaries. In a theory with massless fields, the longest range component of the force (V ~ 1/r) is expected to correspond to single quantum exchange. If there are no self-coupled massless fields, then this is in fact true but it can be far from obvious. For example, in a theory with a massless neutral scalar field ¢ coupled to a massive fermion ~b the exchange of one meson gives the 1/~ 2 Coulomb potential in the static limit. Double exchange can take place either with the ¢ lines crossing (a contribution to the Bethe-Salpeter kernel) or with the ¢ lines uncrossed (an iteration of the lowest order kernel). The crossed box takes the following form in the static limit A
£
1 1 M -M crossed ~ J d4k k 2 + ie (k + q)2 + ie ~/k0 + ie M k 0 + ie
(1) where q~ = (0, ~) is the momentum transfer. The energy contour can be closed to avoid the fermion poles, giving a long range and infrared divergent contribution 1/~ 2 In ~2/t52 where ~5is an infrared cutoff. The box diagram in the static limit becomes Auncrossed f d 4k
(2) 1 1 M -M k 2 + ie (k +q-i2 ÷ ie M k 0 +ie - M k 0 +ie"
The energy integral will pick up contributions from the meson poles which precisely cancel the crossed box. In addition, one of the fermion poles must be picked up giving a k 0 = 0 term. This piece can be identified as the iteration of the potential (energy transfer = 0 in both lines) and in fact is singular in the static limit. The singularity is shielded by keeping the quark kinetic energy non zero and one then discovers the well known 1/Oquark singularities which begin to build up the Coulomb bound state. :#2 A covariant gauge computation of the potential between f'Lxed sources at the one loop level has been given by Susskind [2]. It is being extended to the two loop level bv W. Fischler and L. Susskind.
232
1 August 1977
Quantum electrodynamics in a covariant gauge behaves in much the same way. However, the gauge freedom of QED can sometimes be used to eliminate spurious long range forces at the outset. For the static problem, the natural choice is Coulomb gauge in which 1/~/2 behavior is found only in single Coulomb photon exchange. Graphs with crossed Coulomb lines vanish in the static limit and uncrossed multiple Coulomb exchanges are strictly iterations of the potential. The Coulomb gauge continues to be extremely useful in examining the static potential in QCD. The dynamics is now considerably more complicated but spurious contributions are still eliminated. The Yang-Mills theory in Coulomb gauge has been studied by many authors [3, 4] and we shall summarize only a few of its features. The gluon propagator has an instantaneous part and a transverse part: D ab= i 5 ab ~ gv ~2
~=v=O (3)
=i~ab(6ij kikj) 1 - - - ~ - f k2 + iE
la = i , v = ] .
A Fadeev-Popov ghost must be included but it couples only to the transverse gluon. An important Ward identity for the quark-Coulomb gluon IPI vertex can be proven [5]. Its main consequence is that the primitively divergent part of this vertex is equal to the primitively divergent part of the quark wave function renormalization. Infinite coupling constant renormalization comes only from Coulomb gluon propagator corrections. For our purposes, it is most convenient to define the mass by subtracting on-mass-shell, certainly a possibility to any finite order of perturbation theory. This definition of the mass will be implicit in some of the computations and Ward identity applications we shall discuss. This procedure cannot be completely correct of course since it gives a renormalized mass proportional to the bare mass and thus neglects any dynamical component. In the case of the light quarks, the bulk of their mass (a few hundred MeV) is believed to be dynamically generated and inherently nonperturbative. It is reasonable to assume that the dynamical part of a heavy quark mass is also no more than a few hundred MeV. It can then be neglected since it is on the order of or smaller than relativistic corrections. Our on-shell definition can be viewed as working in this approximation.
Volume 69B, number 2
PHYSICS LETTERS
can be seen by noting that the energy denominator Ikl + Ill in old fashioned perturbation theory is of order q if the loop is subtracted at II ~ q. Thus this process takes place over a typical time scale of order 1/q which is not instantaneous but which is small compared to the time scale for quark recoil (of order
k
0
(b)
(a)
M/q2).
Fig. 1. Contributions to the running coupling constant in Coulomb gauge. The labels 5, ~ and ~ describe respectively the coordinate space locations of the quark, the antiquark and an intermediate interaction point.
One Loop. We restrict
our attention to one flavor of heavy quark and neglect the light quarks altogether. The Ward identity insures that there is no ultraviolet divergent contribution from the sum of quark self energy and quark-Coulomb gluon vertex graphs. This is easy to verify explicitly. Infrared divergences are present in vertex and quark self energy diagrams involving transverse gluons and are associated with the possibility of production in perturbation theory. However they vanish in the static limit due to the quark velocity factor in the transverse gluon-quark coupling. The vertex and quark self energy contributions together can in fact be shown to give no contribution to coupling constant renormalization in the static limit. The crossed ladder with Coulomb and/or transverse gluons vanishes in the static limit as does QQ annihilation and vacuum polarization by massive QQ pairs [6]. The two diagrams shown in fig. 1 are the only important one loop contributions and they have already been computed in the literature [1, 4]. We discuss them here from a physical point of view to pave the way for the examination of more complicated contributions and to identify the physical origin of asymptotic freedom in the context of the static problem. Fig. la is vacuum polarization, a shielding effect which should reduce the renormalized coupling strength. Computation bears this out, giving in the instantaneous approximation (q0 = 0) the following correction to the Coulomb propagator ~ab i/t~2 : UP
_
i
SCA Inq/la]
I+.~4g2
1 August 1977
(4)
/a is the renormalization scale, q --- 141 and C A is the Casimir operator of the adjoint representation. Although vacuum polarization is not instantaneous, it appears as effectively instantaneous to the quarks. This
The Coulomb field self energy contribution fig. lb is, by contrast, completely instantaneous. The full propagator can be written in the form 6
_~i I2g 2 ~
q~q][" d3k ~i]- kikf/k2
1 (5)
Momentum integration and subtraction at q =/a gives the resulting contribution to the renormalized Coulomb propagator:
DSE ab(q) ='ab ~i [-1692 2-4--~ CA In q/la] .
(6)
The asymptotic freedom of the Coulomb field self energy can be seen from the positivity and ultraviolet divergence of the square bracketed correction in expression (5). It is a q dependent and cutoff dependent increase in the magnitude of the field strength over and above the bare Coulomb field. The renormalized Coulomb propagator (6) shows the attendant weakening of the effective coupling strength at short distances. The asymptotic freedom of the theory follows from the dominance of this contribution over the polarization of the vacuum (4), a computational result without an apparent simple explanation. Adding (4) and (6) gives the usual running coupling constant of the pure gauge sector of the theory. The physics of asymptotic freedom is most apparent in configuration space. Suppose that a quark source is located at the origin and an antiquark probe is at R. The Fourier transform of expression (5) is a positive addition to the 1Ix Coulomb potential. The dominant term in the transverse propagator is the 6ii, so we exhibit only that part for simplicity. One of the external Coulomb propagators (e.g. the one connecting to the quark source) is then cancelled and the addition to the 1Ix potential is 292C A f d3z 1 1 1 A -1 Z2 Z IX -- ~'l "
(7) 233
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>1
t
k
i k
I
I I
II A
I
>
>
Vl+k+q I I >
-k -I
Fig. 2. A multiple exchange contribution to the static potential Integration gives an expression of the form (1 Ix)lnx A. This is a scale dependent increase ( f o r x >> I/A) of the attractive (repulsive) force between the quark and antiquark if they are in a color singlet (octet) state. The source of the increase is the attractive potential 1/z 2 which operates between the Coulomb field lines at the position of the source and at the point f. It arises from the instantaneous exchange of the octet transverse gluon between these points and is attractive because the two points of the Coulomb field are in a relative singlet state. The field lines connecting the source to the point ~ are thus drawn together, increasing the magnitude of the energy stored in the field. This mechanism of logarithmic flux collimation is clearly unique to Yang-Mills theories. Higher Orders. At two loops, the general structure of the static problem begins to emerge. The two loop corrections to single gluon exchange continue to build up the running coupling constant and in addition, multiple gluon exchange enters in the form of fig. 2. At three loops, contributions appear which are singular in the static limit and yet which are not iterations of the potential. We first consider fig. 2 and next the corrections to single Coulomb gluon exchange. In the static limit, the quark propagator (say the upper one in fig. 2) becomes i(l 0 + ie) -1 . A power counting analysis of fig. 2 in the static limit leads to the conclusion that it is convergent in both the ultra° violet and infrared. Therefore, on dimensional grounds, it behaves like 1/~ 2. This is something new dynamically, a Coulombic force associated with multiple massless quantum exchange. It is also new in the algebraic sense since in the exchange channel, it transforms like 8 × 8. With the quarks in the fundamental representation, this diagram contains both an octet and a new singlet exchange part. The total potential is no longer just a feature of the gluonic sector of the theory but it depends on the representation content of the sources. 234
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The space-time properties of this diagram are similar to vacuum polarization. The energy denominator of non-relativistic perturbation theory is simply the transverse gluon energy k when the quark kinetic energy is neglected. Convergence of the integral implies that k is of order q so that the time interval At between production and disappearance of the virtual transverse gluon is of order 1/q. During this time, quark recoil can be neglected since it takes place over a time scale of order M/q 2. We next consider the vertex and self energy corrections to single gluon exchange. To any order, the Ward Identity insures that ultraviolet divergent coupling constant renormalization comes only from corrections to the gluon propagator. The Ward identity is not exactly of the Abelian electrodynamic form. There are additional pieces involving the Fadeev-Popov ghost but these pieces can be shown to be ultraviolet finite to all orders even in the limit M ~ oo [5 ]. Thus the combination of the quark self energy and the quarkCoulomb vertex is ultraviolet finite in the infinite quark mass limit. In the static limit (neglecting the quark kinetic energy and holding q finite but small compared to M), there is one two-loop vertex and one two-loop quark self energy which are infrared divergent. These are formed by considering the emission and reabsorption of a Coulomb gluon by the quark and then inserting a transverse gluon pair loop in the Coulomb propagator. The divergence corresponds to the possibility of the transverse gluon pair becoming soft and real. Fortunately, there are two remaining two-loop graphs which naturally pair with these two and which eliminate this infrared divergence for a color singlet QQ. pair. They are shown in fig. 3. Part of fig. 3b is simply the iteration of the potential. However, an additional piece survives which comes from the gluon energy denominators and has a similar structure to fig. 3a. Each graph is infrared divergent and the sum of
k÷q;, s
s~'
(o)
~.,
~
I I
6
k
I
Ib)
Fig. 3. Some two loop diagrams with infrared singularities in the static limit.
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k
!
I
Fig. 4. A three loop graph with a singular static limit.
figs. 3a and 3b is of the form 1
#2 [X l [xa] Q (ha q/8
+ const)
(8)
where <5 is the infrared cutoff. For a color singlet QC~ pair, fig. 3a pairs with the infrared divergent two-loop vertex and fig. 3b pairs with the infrared divergent two-loop quark self energy. All ~ dependence cancels. The combination of all the two-loop vertex and self energy corrections to single Coulomb exchange along with figs. 3a and 3b thus produces an effective potential for a color singlet QQ pair of the form (1/ #2)f(q/la, g2) in the limit M ~ oo. The function f can be computed and compared with covariant gauge computations of the two-loop fl function [7]. In addition there is the new two-loop "skeleton" shown in fig. 2. At the three loop level and beyond, there exist diagrams which are singular in the static limit and which are uncancelled by other contributions. An example of a static limit singularity is the three loop diagram of fig. 4. This is most easily seen by doing the energy integrals leading to a transverse gluon integral d3k/k and two energy denominator factors 1/k. If more Coulomb exchange rungs are added, the degree of divergence presumably increases. Effects of this type must be expected at some level in a quantum field theory and have been encountered often in quantum electrodynamics. The most familiar example is the Lamb shift in hydrogenic systems. There the famous Bethe logarithm arises from low frequency contributions being cut off on the order of the electron kinetic energy (Za)2m. Another example arises in the hyperfine interval computation in positronium [8]. Single transverse photon exchange accompanied by one or more Coulomb photon exchanges contributes at the ,v5 level. Each additional Coulomb rung adds a power of ot and a compensating energy denominator. The difference between QED and QCD is that in the former, this type of configuration mixing is very naturally identified as a relativistic correction. Transverse photons must couple to the fermions giving ve-
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locity factors in the static limit. In QCD, this ,~eed not happen and then there are no velocity factor:, which vanish in the static limit. Of course, in a systematic perturbative computation of the binding energy using the Bethe-Salpeter equation, the distinction becomes blurred since u is of order g2. Summary. The following picture has emerged. There exists a class of diagrams (a subset of the two particle irreducible kernel graphs) which are finite in the static limit. This class includes single Coulomb gluon exchange. The next "skeleton" in this class is fig. 2. It behaves like 1/# 2 and contains octet and singlet exchange. It too can be corrected with vertex and self energy insertions which will modify the I/#2 behavior. Other such skeletons can easily be drawn and providing they obey one important condition, they too will be finite in the static limit and behave like 1/#2. For a bare skeleton, a sufficient condition for the existence of the static limit is that the diagram be connected in the gluonic sector. The class of all such diagrams can be used to define a static potential which on quite general grounds, can be written in the momentum space form
(1/#2)K(q/la, M/V, gv).
(9)
K contains both octet and singlet exchange. Suppose, first of all, that QQ loops are omitted. It is then simplest to define the subtraction constants so that gv is explicitly M independent. It has been shown through the two loop level that K is ultraviolet finite in the limit M ~ ,,o and free of infrared divergences. This result has followed from the Ward identity and some power counting. It is true to all orders for single Coulomb exchange with vertex and self energy corrections. It is conjectured to be true to all orders for all the other dressed skeletons. Assuming it is, the momentum scale in K is set by q and IX. It follows that QQ loop insertions will be suppressed by O(q/M) or O(,u/M) except for their contribution to infinite coupling constant renormalization [6]. The upshot of this (assuming it can be established to all orders for all the skeletons) is that the potential K def'med in this way is independent of the constituent mass M. While it depends on the representation content of the sources, it is dynamically a property of the gluonic sector of the theory. If light quarks are included, the potential will depend on the glue plus light quark sector. If more than one heavy flavor is in235
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cluded, the potential will be universal for each o f the heavy flavors unless the momentum transfer q for the heaviest is large enough to polarize the lighter heavies in the vacuum. This will provide some justification for the use of a universal potential to describe the c~ and bl~ (M b ~ 4 - 5 GeV) systems [9]. Disconnected diagrams remain which are not identifiable as iterations o f the two body potential. The ones we have examined so far are divergent in the static limit and therefore describe long time physics -multiple particle configurations. If the potential defined in terms of the connected graphs is to correspond to the phenomenological QQ potential, then it must be shown that configuration mixing is a small effect. In a subsequent paper we shall discuss in more detail the separation of the two b o d y potential at three loops and beyond. Conversations with Leonard Susskind are gratefully acknowledged. One o f us (TA) would like to thank members of the Cornell University theory group for several helpful discussions. Many of our conclusions have been independently arrived at by F. Feinberg. We thank him for communicating his results to us before publication and for pointing out an error in an earlier version o f this manuscript.
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References [1 ] A. Duncan, Phys. Rev. D10 (1976) 2866. [2] L. Susskind, in: Coarse grained quantum chromodynamics, lectures given at the Les Houches Summer School on High energy physics, 1976. [3] E.S. Fradkin and I.V. Tyutin, Phys. Rev. D2 (1970) 2841 A. Ali and J. Bernstein, Phys. Rev. D12 (1975) 503; R.N. Mohapatra, Phys. Rev. D4 (1971) 378; D4 (1971) 1007; D.P. Sidhu, Phys. Rev. D6 (1972) 565; E. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1. [4] J. Frenkel and J.C. Taylor, University of Oxford Preprint, 21/76 (1976). [5 ] The details will be presented in a subsequent paper. They follow closely the corresponding proof in covariant gauges given by F. Feinberg and E. Eichten, Phys. Rev. D10 (1974) 3254. A closely related Ward identity is proven by Taylor and Frenkel in ref. [4]. [6] T. Appelquist and J. Carazzone, Phys. Rev. D11 (1975) 2856. [7] Covariant gauge computations have been carried out by W.E. Caswell, Phys. Rev. Lett. 33 (1974) 244; D.R.T. Jones, Nucl. Phys. D15 (1974) 5321; A.A. Belavin and A.A. Migdal, unpublished (1974). [8] E.E. Salpeter, Phys. Rev. 81 (1952) 328; T. Fulton and R. Karplus, Phys. Rev. 93 (1954) 1109; R.J. Eden, Proc. Roy. Soc. (London) 219 (1953) 516. [9] E. Eichten and K. Gottfried, Cornell University Preprint, CLNS-350, November, 1976.