Running on the light front

SUPPLEMENTS Nuclear

Running Timothy

Physics

B (Proc.

Suppl.)

108 (2002) 264-266

www.elsevier.com/locate/npe

on the light front

S. Walhouta*

aEuropean Centre for Theoretical Studies in Nuclear Villa Tambosi, Strada delle Tabarelle, 286 I-38050 Villazzano (Trento) Italy

Physics

and Related

Areas

Using a particular choice of Similarity Renormalization framework developed by Dyson, the running coupling entering the light-front QCD Hamiltonian has been calculated to one-loop order. In Dyson’s scheme this is done by evaluating generalized Feynman diagrams that depend on an effective energy scale u, and the renormalization prescription may be tied to those of usual perturbation theory.

1. THE DYSON PICTURE In the early 1950’s, Dyson developed a scheme for calculating finite effective Hamiltonians in field theory [l]. He called this scheme the Intermediate Representation since it involves a partial transformation from the Interaction to the Heisenberg Pictures. Thus the unitary transformation Dyson developed puts those interactions corresponding to short time scales into the operators, thereby generating new effective operators, whereas interactions governed by long time scales must be solved for by finding those states that diagonalize the efffective operators. While predating development of the original renormalization group, Dyson’s work is evidently a precursor of the modern renormalization group; and even if his work went largely unnoticed, it certainly influenced Wilson [2]. Dyson’s transformation necessarily introduces an energy scale that separates long and short time-scale processes, and since it is the short time-scale processes that cause divergences, the transformed operators are in fact renormalized operators. Elsewhere [3], I have shown that Dyson’s transformation in fact meets all the requirements of the Similarity Renormalization scheme introduced much later by Glazek and Wilson [4] and Wegner [S]. Now the basic idea of the Similarity Renormalization scheme is to separate the solution of a bound state problem into two parts: *[email protected]

calculation of an effective Hamiltonian H, that suppresses interactions involving the exchange of particles with energy greater the scale a; diagonalization of the finite Hamiltonian H, with recourse to traditional methods of quantum mechanics (variational methods, basis function expansions, and so on). The suitability of this approach to QCD should be evident, for high-energy QCD processes can be calculated perturbatively; and while low-energy states are far removed from the perturbative regime, few-body constituent quark models work surprisingly well to describe low-energy strong interaction phenomena. The two step Similarity approach to QCD is then as follows: we deal with those degrees of freedom we know how to handle, those of high-energy, by putting them into H,, where 0 must be keep large enough to justify a perturbative calculation; and we hope that the resulting renormalized H, will be dominated at low-energies by few constituent states. 2. THE

LFQCD

HAMILTONIAN

Transforming to the Dyson Picture is not sufficient to guarantee that few-body states will dominate in theories with massless particles such as &CD, for many gluons can be exchanged between states of nearly the same energies, and such interactions will not be eliminated by a Similarity

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TS. Walhout/Nuclear Physics B (Proc. Suppl.) 108 (2002) 264-266

Transformation. Even building the vacuum becomes a hopeless infinite-body problem, at least from the perspective of equal-time quantization. If we choose to use light-front quantization, however, we can force a Constituent Quark Model (CQM) picture to emerge [6]. That gluons are massless still causes problems, but these are re-. fleeted in the complexity of the operator structure of the effective Hamiltonian, which we can hope to control through renormalization, the maintenance of symmetries, and - if necessary - phenomenology. The vacuum is trivial, and manyparticle interactions will be suppressed in the Similarity-Transformed Hamiltonian. Now Dyson showed that the various terms in the effective Hamiltonian H, can be calculated by means of generalized, off-energy-shell Feynman diagrams. Indeed, the Intermediate Representation Hamiltonian can be expressed as a timeordered exponential of the convolution integral over time of the usual Interaction Picture Hamiltonian (including counterterms) with a damping function that kills long time-scale processes. In momentum space, H, is evaluated by shifting the energy components of the momenta of internal lines in ordinary Feynman diagrams by attenuation factors originating from integral transforms of the damping functions. These factors are then integrated over to give the corresponding term in XT. Since the energy shifts in the generalized Feynman diagrams are proportional to (T, it is evident that the limit 0 + 0 gives the usual perturbative diagram and, moreover, divergences are independent of o. Thus we can evaluate the finite part of the generalized diagrams by subtracting out the divergent pieces in a way that reproduces whichever renormalization prescription we want in the limit cr + 0. The finite part will then differ from that of the usual perturbative expansion in terms proportional to g. 2.1. Masses For example, diagram gives

L(Pi,Pf) =

the generalized

s

quark self-energy

dr~drgyrl)F(r2)iur~

pi - py + ia(rl + r2)

(1)

x {c(pi + where F(F)

=

is2) f

is the

265

c(pi -

i0r2)j,

transform

s

Here we use the two-component effective theory obtained after substitution for the constrained quark and gluon degrees of freedom [7], and the Mandelstam-Leibbrandt

(q+, and dependence the factors enters in the of the through the p2 -+ + inp+I’. simple choice the damping that fits requirements of Similarity Transformais F(r) 6(l? + #(I’ 1). Using and setting initial and quarks on shell, we the contribution

=

+ &T(P),

with g, finite function does not at the quark mass. Likewise, is a u-dependent renormalization of gluon mass, independent of we prescribe the c 0 limit be. 2.2.

Confining Potential tree level, logarithmic confining potential in the effective Hamiltonian from energy gluon [8]. is a feature light-front Similarity-Transformed

degrees of

perturbatively,

vanish. Thus in QED, we should treat such terms as artifacts scheme. In &CD, however, we know that low-energy handled in a perturbative colored states are in fact suppressed.

T.S. Walhout/Nuclear

266

Physics B (Proc. Suppl.) 108 (2002) 264-266

take the logarithmic potential and the quark and gluon masses seriously, for they provide precisely the physics we would expect for a constituent quark picture. The challenge is to show that this simple picture survives both higher-order corrections from step (1) and detailed analysis of H,, in step (2) of the Similarity program. To proceed in both of these steps, we must determine how the Similarity scale cr is involved in the perturbative running of the LFQCD coupling. 3. THE RUNNING

COUPLING

In principle, we should allow all possible canonical terms in the interacting Hamiltonian, including entire functions of longitudinal momenta if we want to be assured of including all zero mode effects; and we should allow the corresponding couplings to vary independently. However, we know that QCD depends on a single scale, and thus on a single coupling; and so I will proceed under the assumption that all the couplings are a functions of a single coupling. Moreover, I assume for now that no terms beyond those of the usual canonical LFQCD Hamiltonian plus counterterms to remove divergences are needed. This means I am assuming that the operator structure arising from the elimination of constrained zero modes can entirely be accounted for through the renormalization of divergences. We take the fundamental coupling to be the Q + qg vertex. Details of the calculation will be presented elsewhere, but it should be clear enough from the example above how the calculation goes. The logarithmic divergence is independent of r, and using dimensional regularization we find the usual covariant dependence on an arbitrary mass scale p2. The complete vertex is:

with Sg, finite and independent

of p, and

in H, is multiplied

Q4 + (Q+d2) /J4 .

(7)

If we consider the usual rewriting of the ,u2dependence in terms of the QCD scale A, we have o(Q2)

cx

lnIQ4 +iF”)2

1) -’

Since we expect Q2 N A2 in solving for bound states in step (2), then we must keep Q+o >> A2 to justify the perturbative treatment of step (1). It remains to be seen if the simple constituent picture of the previous section is consistent with this constraint.

REFERENCES 1.

2.

3. 4. 5. 6.

(5) The coupling that appears a Similarity function

that vanishes as the energy difference w = p_ p!_ - q_ increases. As cr + 0, fc also vanishes unless w = 0, in which case it is equal to unity. The variable Q2 above represents the various mass scales involved at the qqg vertex. Now it is important to realize that although the divergences in the diagrams that contribute to this coupling are independent of u, the logarithmic scaling of the coupling picks up a a-dependence. Roughly, we can understand this as follows: in the generalized diagrams we find dependence on shifted momenta, so that Q2 + Q2 f iQ+a, and thus we should also shift

7.

by

8.

F.J. Dyson, Phys. Rev. 82, 428 (1951); 83, 608 (1951); 83, 1207 (1951); and Proc. Roy. Sot. London bf A207,395 (1951). K.G. Wilson, in Nobel Lectures in Physics, 1981-1990, ed. by T. Frangsmyr and G. Ekspong (World Scientific, Singapore, 1993). T.S. Walhout, Phys. Rev. D59 65009 (1998). St.D. Glazek and K.G. Wilson, Phys. Rev. D48, 5863 (1993), D49, 4214 (1994). K. Wegner, Ann. Physik 3, 77 (1994). K.G. Wilson, T.S. Walhout, A. Harindranath, W.M. Zhang, R.J. Perry and St.D. Glazek, Phys. Rev. D49, 6720 (1994). W.-M. Zhang and A. Harindranath, Phys. Rev. D48, 4881 (1993). R.J. Perry, in Hadron Physics 94, ed. by V.E. Herscovitz and C. Vasconcellos (World Scientific, Singapore, 1995);