An Exact, Finite, Gauge-Invariant, Non-Perturbative Model of QCD Renormalization
H. M. Fried, P. H. Tsang, Y. Gabellini, T. Grandou, Y.-M. Sheu
AAn Exact, Finite, Gauge-Invariant, Non-Perturbative Model ofQCD Renormalization
H. M. Fried, ∗ P. H. Tsang, † Y. Gabellini, ‡ T. Grandou, § and Y.-M. Sheu
1, 2, ¶ Physics Department, Brown University, Providence, RI 02912, USA Universit´e de Nice Sophia-Antipolis, Institut Non Lin ´ e aire de Nice,UMR 6618 CNRS, 06560 Valbonne, France (Dated: September 23, 2018) Abstract
A particular choice of renormalization, within the simplifications provided by the non-perturbative property of Effective Locality, leads to a completely finite, renormalized theory ofQCD, in which all correlation functions can, in principle, be defined and calculated. In this Modelof renormalization, only the Bundle chain-Graphs of the cluster expansion are non-zero. All Bundlegraphs connecting to closed quark loops of whatever complexity, and attached to a single quark line,provided no ’self-energy’ to that quark line, and hence no effective renormalization. However, theexchange of momentum between one quark line and another, involves only the cluster-expansion’schain graphs, and yields a set of contributions which can be summed and provide a finite color-charge renormalization that can be incorporated into all other QCD processes. An application toHigh Energy elastic pp scattering is now underway. ∗ [email protected] † Peter [email protected] ‡ [email protected] § [email protected] ¶ [email protected] (corresponding author) a r X i v : . [ h e p - t h ] M a r . INTRODUCTION AND REVIEW OF REFERENCES A recent, analytic formulation of non-perturbative, gauge-invariant, realistic QCD [1–6]used elementary functional techniques to sum all gluon exchanges between any pair of quarkand/or antiquark lines, including cubic and quartic gluon interactions, yielding a realisticquark-binding potential not approximated by static quarks, as well as (to our knowledge)the first example of nucleon-binding directly from QCD. The result is that individual gluonexchanges disappear from the theory, within their infinite sums each replaced by a
GluonBundle (GB). What remains to be calculated, for any QCD correlation functions, is thenthe attachment of such GBs to quarks and to all possible combinations of closed-quark-loops(CQLs), as well as the definition of a procedure of quark/hadron renormalization.It is wise at this point to remember (or at least paraphrase) Schwinger’s remark definingrenormalization as the change, or more properly a return, from the ’field picture’ back to the’particle picture’. In Abelian QED, for example, where the sum of all radiative correctionsdefines a dressed photon or lepton propagator, with its Z or Z factor multiplying themass-shell pole of that propagator, renormalization is simply defined as the division by, andeffective removal of those Z and Z factors. But in non-Abelian, gauge-invariant QCD,where the initial gluon fields and propagators disappear from the final analysis, what shallbe the physically correct prescription of renormalization? In this paper we formulate onesuch definition, in which simplicity and finiteness both play a major role. We make noclaim to the uniqueness of our definition of renormalization, but only that it is a very simpleand obvious way of performing both quark and coupling constant renormalization, in whichevery step of the program is finite.For ease of presentation and clarity, we here make two simplifying approximations, whichcan easily be corrected and extended, as desired. The CQL with but two GBs attached iswritten in terms of the well-known, un-renormalized, QED lepton loop, with color factorsappropriated to QCD appended; in this way, the intrinsic spin dependence of the quark andanti-quark which form the CQL have been included, while there remains to be calculatedan extra spin dependence peculiar to QCD. In addition, the transverse arguments x (cid:48) ofeach [ f · χ ( x (cid:48) − v ( t (cid:48) ))] − , representing the connection of each loop to its pair of GBs, willbe approximated by the ’averaged’ x µ coordinate appropriate to that loop. As shown inAppendix B of Ref. [2], this makes no change in the single-loop amplitude; and a similar2tatement can be obtained for the chain-loop calculation of this paper.In order to keep this paper one of finite size, we urge all interested readers to first famil-iarize themselves with the material of Refs. [1–6]. Ref. [2] explains our method of achievingmanifest gauge-invariance, and ends with a non-perturbative description of asymptotic free-dom. It also contains what we believe to be the first exact statement of Effective Locality(EL), in which the two endpoint space-time coordinates of every GB are shown to be thesame, which property provides tremendous computational simplicity, reducing a basic func-tional integral (FI) to a set of ordinary integrals [5].Ref. [2] employs the same functional techniques to display the small absurdities whichappear in the amplitudes of all non-perturbative QCD processes – for example, the exchangeof a GB between two quarks – because use of the standard QCD Lagrangian makes noprovision for the experimental fact that asymptotic quarks are always found in bound states;and hence that, in principle, their transverse coordinates can never be measured nor specifiedprecisely. To avoid such difficulty, a ’transverse imprecision’ integration over an unknown,transverse probability amplitude, (cid:90) d x ¯ ψ ( x ) γ µ λ a ψ ( x ) A aµ ( x ) → (cid:90) d x (cid:90) d x (cid:48)⊥ a ( x ⊥ − x (cid:48)⊥ ) ¯ ψ ( x (cid:48) ) γ µ λ a ψ ( x (cid:48) ) A aµ ( x ) (1)where x (cid:48) µ = ( x , x L , x (cid:48)⊥ ) with a ( x (cid:48)⊥ − x ⊥ ) real and symmetric under the interchange of x (cid:48)⊥ and x ⊥ , is introduced into the quark-gluon part of the Lagrangian, which has the immediateeffect of removing all such absurdities. The theory described by this extended Lagrangianis what we have called ’Realistic QCD’.In Ref. [2], a particularly simple choice of the corresponding transverse probability am-plitude ϕ ( (cid:126)b ) = (cid:82) d (cid:126)q e i(cid:126)q · (cid:126)b ( (cid:101) a ( q )) is inserted into the standard QCD Lagrangian, along with asimple and physically correct method of identifying that effective potential which generatesthe easily-calculated, non-perturbative eikonal function that would give the desired quarkscattering and/or binding; and our result is the potential of form V ( r ) ∼ µ ( µr ) ξ , where µ is the scale parameter for quark binding, on the order of the pion mass, and ξ is a small, real,positive parameter of order 1 /
10. By inspection, there are here two parameters essential forquark binding, µ and ξ , rather than just µ alone.Ref. [3] carries the analysis one step further, and provides what we believe to be the firstexample of Nuclear Physics binding directly from QCD, using our eikonal method to obtainthe effective potential which easily binds two nucleons – two triads of bound quarks – into a3odel deuteron. In eikonal and quenched approximations, Refs. [4, 5] use an exact RandomMatrix calculation to show that the amplitude corresponding to a single GB exchangedbetween a quark and anti-quark can be written in terms of Meijer G-functions (indeed, afinite sum of finite products of them!), in agreement with general theoretical arguments [7];and that both expected SU(3) Casimir invariants contribute to this amplitude, in contrastto lattice-gauge and other model calculations of this amplitude, which contain but one suchinvariant. The extension to more complicated processes will presumably involve productsand integrals over such Meijer G-functions, but this remains to be studied. Finally, Ref. [6]provides a Brief Review of the first three papers mentioned above. II. FORMULATION
We begin by considering the dressed quark propagator as the simplest example of a QCDcorrelation function, S (cid:48) c ( x − y ) = N (cid:90) d[ χ ] e i (cid:82) χ · det[ f · χ ] − e ˆ D A G c ( x, y | A ) e L [ A ] (cid:12)(cid:12)(cid:12) A → , (2)where ˆ D A = i (cid:82) δδA ( gf · χ ) − δδA . G c [ A ] represents a ’potential theory’ quark propagator in thepresence of a fictious ’classical’ field A aµ ( x ); and L [ A ] is the ’CQL functional’, representingthe sum of a single CQL with all possible (even) numbers of A-fields attached, L [ A ] = tr ln[1 + igγ ( λ · A ) G c [0]]. Each ( f · χ ) − factor is associated with the exchange of a specificGB; and the normalization constant N is the product of the normalization of the Halpernfunctional integral (FI), divided by the vacuum expectation value (VEV) of the QCD S-matrix.The derivation of Eq. (2), the relation of ( f · χ ) − to the GB exchanged between any twoquark and/or antiquark lines, and the overall gauge-invariant structure is fully and clearlydescribed in Refs. [1, 2]. The only point which may require special emphasis is the definitionof the measure of FI over the Halpern’s field χ aµν ( x ) [8], in the conventional sense of breakingup all of space-time into n small 4-volumes of size δ with the understanding that n → ∞ as δ →
0. This FI is NOT to be considered as a sum over all ”function space”, out of which onemay choose one or more convenient examples of χ aµν ( x ), anti-symmetric in µ, ν and carryinga color index a .This distinction becomes crucially important upon realizing that non-perturbative QCD4 IG. 1. Gluon Bundle (GB) self-energy (patterned moon shape) across a single quark line (solidline). is a theory which contains EL. What this signifies is that – in contradistinction to its per-turbation approximations – the sum of the gluon exchanges which define each GB takeplace locally – (cid:104) x | GB | y (cid:105) = [ gf · χ ( x )] − δ (4) ( x − y ) – so that the relevant Halpern FI can bereduced to an ordinary integration over small δ -volume in which the effective interactionoccurs. Integrations over all other δ -volume elements produce, with their normalizationfactors, multiple products of +1.In this simplest example of the dressed quark propagator, we shall employ a convenientform of Fradkin’s original representations [9] for G c [ A ] and L [ A ], both of which are Gaussianin A , and signify that the linkage operations of Eq. (2) can be carried through exactly; forclarity, these representations are reproduced in Appendix A. The immediate interest now isthe structure of the linkage operator upon the product e ˆ D A (cid:0) G c [ A ] e L [ A ] (cid:1) = ( e ˆ D A G c [ A ]) e ←→ D A ( e ˆ D A e L [ A ] ) , (3)where, with an obvious notation, ←→ D A = i (cid:82) ←− δδA ( gf · χ ) − −→ δδA , and each such ˆ D A operation hasthe effect of inserting a GB between the quark propagator G c [ A ] and a CQL functional L [ A ],as well as inserting a GB across a quark line, as represented by the ”self-energy” graph ofFig. 1, or across the interior of a loop, as in Fig. 2. In ordinary perturbation theory, where anindividual gluon would replace the GB above, such graphs are highly divergent. In realisticQCD, because of Effective Locality (EL), they vanish. For ease of presentation, we havemoved the proof of this statement to Appendix B, and here continue with the truly relevantpart of the formulation, which defines and uses the functional cluster expansion [10].Because of this simplification, ( e ˆ D A G c [ A ]) → G c [ A ] , (4)5 IG. 2. Closed-quark-loop (solid circular line) with an internal Gluon Bundle (patterned ovalshape). but the cross-linkage operation ( G c [ A ]) e ←→ D A ( e D A e L [ A ] ) (5)will link the quark propagator with every element of L [ A ], in a simple, but important”translational operation” way. If, for simplicity, we momentarily neglect the spin structureof G c [ A ], then all of its A -dependence will appear under its defining Fradkin represen-tation integral, in the factor exp (cid:2) − ig (cid:82) s d s (cid:48) Ω a ( s (cid:48) ) · u (cid:48) µ ( s (cid:48) ) · A aµ ( y (cid:48) − u ( s (cid:48) )) (cid:3) , where u µ ( s (cid:48) ) isthe functional variable whose integration defines the Fradkin representation of G c [ A ], andhence of the dressed quark propagator, while s is the proper-time variable associated withthe space-time properties of that propagator. What this means, in general, is that this s -dependence will be inserted into the defining structure of L [ A ], which will then exercise acertain measure of control over the subsequently needed (cid:82) ∞ d s e − ism over the s -dependencecontributed by G c [ A ] [11].We now turn to the functional cluster expansion, defined combinatorially and picturallyin Ref. [10], which takes the form e ˆ D A e L [ A ] = e (cid:80) ∞ (cid:96) =1 Q(cid:96)(cid:96) ! (6)where Q (cid:96) = e ˆ D A ( L [ A ]) (cid:96) | conn , and where the subscript ’conn’ means that only multiple loopsattached to each other by at least one GB are retained. For example, Q [ A ] = e ˆ D A L [ A ] ≡ L [ A ] (7)and Q = L [ A ] ( e ←→ D − L [ A ] (8)6 = + + + FIG. 3. Q term in the cluster expansion (solid circles as loops and patterned ovals as GBs). etc. But, as noted in Appendix B, every GB inserted across the same loop will always vanish,and therefore L [ A ] → L [ A ]. The multiplicative linkages of all the Q (cid:96) then correspond to allpossible GB insertions between different loops, with their ’self-energies’ missing.As an example, consider a pictorial representation of Q , taken from Ref. [10], and re-produced here as Fig. 3, where the patterned ovals represent all possible numbers of GBlinkages between the loops, and the integers on the left-side of each graph represent thestatistical weight of that arrangement of closed loops. The cross-linkages between exp { Q (cid:96) } and G c [ A ] will require two additional GBs linking the quark line with each of the diagramsof Q (cid:96) , in all possible ways.One immediate simplification is provided by the fact that the Fradkin representations forany loop will be non-zero for an even number of GB attachments to that loop. But thereare then still a huge number of possible linkages of Q (cid:96) to G c [ A ]; and the reduction to justa few such linkages, which can be easily summed, is the goal of the next sections. III. QUARK RENORMALIZATION
It will be most efficient at this point to temporally restrict attention to the class of ’chaingraphs’, such as the last RHS loop combination of Fig. 3. It has a statistical weight of12 = , and this numerator factor of 4! serves to cancel the in the sum of Eq. (6), leavingbehind a net multiplicative factor of . And such cancellation holds for every chain graph,for every value of (cid:96) .The simplest radiative corrections corresponding to the chain graphs attached to a singlequark line are pictured in Fig. 4. where there are two sorts of terms which enter the FI overthe closed loop, one due to the spin of virtual quarks, and the other due to the non-spin (cid:82) s d s (cid:48) u (cid:48) ( s (cid:48) ) Ω( s (cid:48) ) (cid:82) t d t (cid:48) v (cid:48) ( t (cid:48) ) (cid:98) Ω( t (cid:48) ) terms which multiply the attached GBs. For simplicity,7 ν α β FIG. 4. Simplest radiative corrections corresponding to the chain graph attached to a single quarkline. we here outline the calculation of the second contribution, and then, in words, simply statethe more obvious result of the spin dependence.Each of the two GBs of the loop of Fig. 4 is proportional to a 4-dimensional delta-functionof their end-point variables, v (cid:48) α ( t ) · δ (4) ( y (cid:48) − u ( s ) − x (cid:48) + v ( t )) · δ (4) ( x (cid:48)(cid:48) − v ( t ) − y (cid:48) + u ( s )) · v (cid:48) β ( t ) , (9)where v ( t (cid:48) ) is the space-time coordinate of the loop whose FI defines its Fradkin represen-tation, and, as always, z (cid:48) µ = ( z , z L , z (cid:48)⊥ ). At every intersection of a GB with a quark linethere will appear a transverse integration over the relevant probability amplitudes, in thiscase (cid:82) d x (cid:48)⊥ a ( x ⊥ − x (cid:48)⊥ ) · (cid:82) d x (cid:48)(cid:48)⊥ a ( x ⊥ − x (cid:48)(cid:48)⊥ ) · (cid:82) d y (cid:48) a ( y ⊥ − y (cid:48)⊥ ) · (cid:82) d y (cid:48)(cid:48) a ( y ⊥ − y (cid:48)(cid:48)⊥ ) and thegeneralization to higher numbers of loops forming the chain graphs is immediate.As a first step in the calculation, it will be useful to consider the product of the time-likeand longitudinal δ -functions of Eq. (9), δ (0 ,L ) ( y − u ( s ) − x + v ( t )) · δ (0 ,L ) ( x − v ( t ) − y + u ( s )),for upon integration over the corresponding loop coordinates (cid:82) d x (cid:82) d x L (required by theFradkin representation), one obtains the product δ (0) · δ ( L ) , or more compactly, δ (0 ,L ) ( u ( s ) − u ( s ) − v ( t ) + v ( t )) , (10)and we evaluate both the δ (0) and δ ( L ) of Eq. 10 by assuming that there are a set of points, t (cid:96) for the first and t m for the second, about which v ( t ) and v L ( t ) can be expanded – v ( t )about t (cid:96) , and v L ( t ) about t m – at which point the arguments of each δ -function vanishes.The result is, for the product of both functions, δ (0 ,L ) → (cid:88) (cid:96) δ ( t − t (cid:96) ) | v (cid:48) ( t (cid:96) ) | · (cid:88) m δ ( t − t m ) | v (cid:48) L ( t m ) | (cid:12)(cid:12)(cid:12)(cid:12) v ( t (cid:96) ) − v ( t m )= u ( s ) − u ( s ) v L ( t (cid:96) ) − v L ( t m )= u L ( s ) − u L ( s ) . (11)8ince v and v L , and u and u L , are completely arbitrary continuous functions of theirvariables, each possessing a first derivative – while the set of C [(0 , s ) → R ]-functions is ofmeasure zero in the Wiener-space relevant to the u , v -functions – the probability of findingpoints t (cid:96) and t m to fit the subsidiary conditions of Eq. (11) is arbitrary small. The onlyvalues of t (cid:96),m which can satisfy these conditions are t (cid:96),m = 0 or t , where we know that the pairof restrictions involving the difference of u ,L ( s ) and u ,L ( s ) are satisfied by construction.But now that t and t are either 0 or t , upon averaging all the transverse fluctuations,and integrating over the x ⊥ -dependence of the loop, the pair of transverse δ -functions ofEq. (9) lose all their v ⊥ ( t ) and v ⊥ ( t )-dependence, since v ⊥ (0) = v ⊥ ( t ) = 0. Immediately,the FI of the loop over its v -dependence will then vanish, (cid:90) d[ v ] v (cid:48) α ( t ) v (cid:48) β ( t ) e i (cid:82) v · (2 h ) − · v δ (4) ( v ( t )) = 0 , (12)and the contribution of this Bundle Graph to the quark’s ’ dressing ’ is zero. The sameresult appears for the purely spin contributions to the Fradkin representation of the loop,because these are given in terms of gradients of y-dependence, which dependence vanishesfrom the product of the δ -functions of Eq. (9), after the transverse fluctuations and the x ⊥ loop variables are integrated out.Higher-loop Chain Bundle Graphs, such as those pictured in Fig. 5, will also vanish. Theanalysis is simplest if one applies the above arguments to the ’end-loops’ – those carryinga GB attached to the quark line, as in Fig. 5 – which will always vanish, independentlyof the number of loops in the chain. In this way, one sees that the Chain Bundle Graphsdo not contribute to the quark ’self-energy’; they do not contribute to the ’dressing’ of thequark propagator. However, the Bundle Graphs will contribute to the processes involvingmomentum transfer between two quarks and/or antiquarks; and these subprocesses willprovide the basis for a finite color-charge renormalization. IV. GLUON BUNDLE RENORMALIZATION
We now give a definition of GB renormalization appropriate to the current situation inwhich individual gluon exchange has already been summed, and all gluons have effectivelydisappeared from the problem. This definition is most useful because it removes all of themany, non-chain Bundle Graphs in the cluster expansion of Section II.9
IG. 5. Higher-loop Chain Bundle Graphs across a single quark line.
Remember that the measure of the Halpern integral is composed of a normalized productof n very small 4-volume elements, δ , which span the entire 4-volume, with the understand-ing of the subsequent limits, n → ∞ and δ →
0. The Gaussian weighting of the FI can bewritten as (cid:89) n N n (cid:90) d[ χ ] e i δ χ n det [ f · χ n ] − F [( f · χ n ) − ] , (13)where χ n = χ ( x n ), the subscript n labels the small space-time volume in which the variable χ aµν ( x n ) is defined; and that x n variable is to be integrated over all possible real values, from −∞ to + ∞ , independently of all other x m (cid:54) = n values. In Eq. (13), F contains the exponen-tiated dependence characteristic of the exchange of a GB, and N n is the normalization ofthat n -th integral, such that its value is unity when g → χ , defined by δ χ = ¯ χ , so that χ − = δ ¯ χ − . With this trivial change, the normalizations N n are now independent of δ , while the exponential interaction term now carries a multiplicative factor of δ , since( f · χ ) − → δ ( f · ¯ χ ) − . In pictorial terms, every GB now carries a δ factor, which may beimagined as a single δ factor appearing at each end of the GB.We now define ”GB renormalization” in the following way, effectively paraphrasingSchwinger’s comment that renormalization is what must be done in returning from the ’fieldpicture’ to the ’particle picture’, where in this case the particle is the quark. (It may alwaysbe asymptotically bound – as distinct from asymptotically free – but it is still the physical’particle’ in QCD.)When a GB connects to a quark, one which is or will eventually be bound, asymptotically,into a hadron, the δ at that end of the GB is to be replaced by a real, finite, non-zero δ q .But the δ at the other end of that GB, connecting a virtual quark loop – which is not aphysical particle – is to be maintained as a factor which is subsequently going to vanish.10he renormalization of that infinitesimal δ connected to the quark line can be viewed assimilar to the removal of the wave-function renormalization Z factors multiplying the ”freeparticle” part of a dressed propagator in conventional perturbation expansions; both the Z , whose inverse contains one or more UV divergences, and the δ , are redefined to obtainthe renormalized forms of each. And, as in the conventional theory, when one ”divides” bythat Z factor to obtain the renormalized propagator, one is ”dividing by zero”, to effectivelyreplace Z by 1.However, δ q has a dimension of length or time, and appropriate care must be taken inassigning it a numerical value. Since we expect the most significant contribution to anysuch high-energy scattering process to appear when the CM quark space-time indices areeither 4 or 3, corresponding to energy or to a related longitudinal momentum, we mightwell permit Quantum Mechanics (QM) to make the choice for us, replacing δ q by a factorproportional to E ≈ p . This introduces a reasonable energy dependence into the amplitude,which will act in such a way – as the Center of Mass (CM) energy increases – to decreasethe QM interference between separate chains linking the scattering quarks. Of course, thisassumption must be verified by comparison with extensive pp scattering data, at energiesranging from GeV to TeV; and this will be explored in a separate analysis.Now consider one loop of a Chain Bundle Graph whose Fradkin functional integral (cid:82) d[ v ]is evaluated. Each end of the two GBs which connects that loop contribute a factor of δ , sothat a net factor of δ → q passing through that loop, that integral has a well-defined, non-zerodependence on q , as well as a logarithmically-divergent UV factor which we shall call (cid:96) .Since δ is to vanish, and (cid:96) is to diverge, in their respective limits, and since they appearmultiplied together, we define the combination κ = δ (cid:96) as a real, finite, positive constant,whose value is to be determined subsequently.This definition is not unique; but it has the great advantage that Chain Bundle Graphswhich transfer momentum produce a finite contribution to their sum; and most importantly,all of the other loops of the cluster expansion vanish: a loop connected to four GBs isproportional to δ (cid:96) , while a loop connected to more than four GBs has no log divergence,and contributions of both groups vanish. By the above definition of GB renormalization,11 IG. 6. A single GB plus a single closed loop exchanged between two quarks contributions tonucleon-nucleon scattering. only the chain Bundle Graphs survive; and the essentially geometric sum of all of theircontributions is able to generate what might appropriately be termed a ’finite color-chargerenormalization’. This Model definition is quite appropriate to our nucleon-nucleon bindingpotential to form a model deuteron, as in Ref. [3], where the δ (cid:96) product is replaced by thefinite κ , combined with the coupling, and determined by the ground-state binding energy.Because such renormalized couplings will show a strong fall-off with increasing (cid:126)q ⊥ , thisproperty suggests an immediate application to the extensive experimental data of pp differ-ential cross-sections at high energies [12, 13]. Using the very approximate replacement of( f · χ ) factors by magnitudes R , neglecting all angular correlations between χ -projectionsin color space, one can easily evaluate an amplitude corresponding to the sum of a singleGB plus a single closed loop exchanged between two quarks, as in Fig. 6. One finds thequalitative result of Fig. 7 for the dσdt of two scattering nucleons, suppressing all dependenceon quark binding which produced those nucleons.The horizontal axis of Fig. 7 is given in units of nucleon mass m , while the scale ofthe vertical axis is arbitrary. The dashed line descending rapidly for small (cid:126)q ⊥ is due tothe single GB exchange, while the dotted line rising from the origin represents one-loopexchange. The | absolute value | of their sum is given by the solid line, and is of interestbecause it clearly shows the ”diffraction dip” in the region of 0 . m n , followed by the humpand subsequent descent at a higher (cid:126)q ⊥ value. When more loops are included, the dipshould very slightly exceed 0 . m n , while the decrease following the hump will be less rapid.Both features are exhibited by pp scattering data for (cid:126)q ⊥ values well past the Coulombinterference region. Work is currently underway to refine these calculations, and form aprecise representation of the shape of this curve, in particular, well past the hump; but12 n2 q d σ dt FIG. 7. The differential cross-section of two scattering nucleons. it is reassuring that the approximation evaluation used here shows a strong, qualitativeresemblance to the experimental data.
V. SIMPLIFYING THE CHAIN-LOOP CONTRIBUTIONS
But before that stage can be realized, it is important to point out one property thatthe careful reader will observe, in this qualitative presentation, where the only quark spin-dependence retained comes from the quark forming the loop – which has the same form asthat of a QED closed fermion loop, with the addition of appropriate color factors – insteadof the more correct result that follows from including the complete spin dependence of thatloop, which can be inferred from the exact Fradkin representations of Appendix A. Thecomplete and relevant Halpern sub-integral over that portion of the ”interior” loops, thoselying between the ”end-point” or ”exterior” loops that connect to the scattering quarks, willalso have a role to play in this analysis.The point to be made here is that the detailed loop and Halpern integral computationsmust generate a result in which momentum transfer across each loop, and across the sum13f all loops, is such that the momentum transfer leaving one quark is received by the otherquark, an obvious necessity, but one which is hidden by the details of the computations.What shall be done here is to simplify matters, adopting a simplified form of the loop result,in which this property is guaranteed. Proper orders of magnitude of the q -dependence aremaintained in this simplification, which guarantees momentum-transfer conservation.Specifically, suppose that a momentum transfer q , moving left-to-right, enters an ’interior’loop which bears the overall, space-time matrix indices of α and β , corresponding to amomentum q entering the loops as q (I) α on its left-hand-side, and exiting as q (II) β on its right-hand-side. These are transverse momenta of two components, e.g. , q (I) α and q (II) β with α, β =1 ,
2; and if this momentum is going to be transferred across the loop, then the result of theexact Fradkin FI of the loop, together with the exact Halpern sub-integral over the verysmall space-time interval in which that integral is defined, must combine to produce theeffective statement that q (I)1 = q (II)1 , and that q (I)2 = q (II)2 . In other words, the indices α and β are not arbitrary, but, in effect, must be the same.The Fradkin FI of a corresponding QED loop has the form [11]( q α q β − δ αβ q ) Π( q ) , (14)where Π( q ) = (cid:90) ∞ d tt e − itm e π (cid:90) d y y (1 − y ) e − itq y (1 − y ) , (15)and contains an obvious logarithmic UV divergence, coming from the behavior of the inte-grand near its lower limit. This differs from the proper QCD loop integral which containscolor-factors, with indices a (cid:48) and b (cid:48) , associated with locations connected to GBs on eachside of the loop, (cid:82) t d t v (cid:48) α ( t ) Ω a (cid:48) ( t ) · (cid:82) t d t v (cid:48) β ( t ) Ω b (cid:48) ( t ), where the α , β , a (cid:48) , b (cid:48) indices arejoined to neighboring GBs. There is also another QCD quark-spin contribution, which istied to the Halpern integral in a moderately complicated way. It should also be mentionedthat in the product of any two neighboring loops, the factors of q α q β q γ q δ , appearing in theproduct of two of the neighboring brackets of Eq. (14), will give no contribution becausethe symmetric combination q β q γ will multiply the antisymmetric – in space-time and colorindices – factor ( f · χ ) − | βγ between those loops.The simplification noted above and now made is simply to retain only the − δ αβ q factorof Eq. (14), multiplied by a parameter λ which is to represent the result of the Fradkin andHalpern integrations, for as noted above their evaluations must produce such a δ αβ q factor.14 a' b' c' d' b μ α β γ ε ν FIG. 8. Quark-Quark interaction with chain graph of two loops.
The parameter λ will multiply the constant κ , and their product will enter into the definitionof the renormalized charge. Since we shall extract only the divergent part of each such loop– which provides a finite contribution in the limit as the width of the Halpern sub-integralvanishes – it turns out that the color indices a (cid:48) , b (cid:48) across each loop are also going to be thesame. The fact that both transverse and both color indices of interior loops are the samewill generate a significant simplification in the final result.Perhaps the simplest approach is to first consider the two-loop amplitude of Fig. 8, writingonly the factors needed for this evaluation, and to then state a sequence of operationsthat can be easily performed, along with their results. The amplitude for this process isproportional to the factors (cid:90) s d s u (cid:48) µ ( s ) Ω a ( s ) (cid:90) ¯ s d¯ s ¯ u (cid:48) ν (¯ s ) Ω b (¯ s ) · (cid:90) t d t v (cid:48) α ( t ) (cid:98) Ω a (cid:48) ( t ) (16) · (cid:90) t d t v (cid:48) β ( t ) (cid:98) Ω b (cid:48) ( t ) · (cid:90) ¯ t d¯ t ¯ v (cid:48) γ ˘Ω c (cid:48) (¯ t ) · (cid:90) ¯ t d¯ t v (cid:48) (cid:15) (¯ t ) ˘Ω d (cid:48) ( ¯ t ) · (cid:90) d y (cid:48)⊥ a ( y ⊥ − y (cid:48)⊥ ) · (cid:90) d ¯ y (cid:48)⊥ a (¯ y ⊥ − ¯ y (cid:48)⊥ ) · (cid:90) d x (cid:48)⊥ a ( x ⊥ − x (cid:48)⊥ ) · (cid:90) d x (cid:48)(cid:48)⊥ a ( x ⊥ − x (cid:48)(cid:48)⊥ ) · (cid:90) d ¯ x (cid:48)⊥ a (¯ x ⊥ − ¯ x (cid:48)⊥ ) · (cid:90) d ¯ x (cid:48)(cid:48)⊥ a (¯ x ⊥ − ¯ x (cid:48)(cid:48)⊥ ) · δ (4) ( y (cid:48) − u ( s ) − x (cid:48) + v ( t )) · δ (4) ( x (cid:48)(cid:48) − v ( t ) − ¯ x (cid:48) + ¯ v (¯ t )) · δ (4) (¯ x (cid:48)(cid:48) − ¯ v (¯ t ) − ¯ y (cid:48) + ¯ u (¯ s )) · [ f · χ ( y (cid:48) − u ( s ))] − | aa (cid:48) µα · [ f · χ (¯ x (cid:48) − ¯ v (¯ t ))] − | b (cid:48) c (cid:48) βγ · [ f · χ (¯ y (cid:48) − ¯ u (¯ s ))] − | d (cid:48) b(cid:15)ν , where the a ( z ⊥ − z (cid:48)⊥ ) represent the probability amplitudes of each quark to be found at aperpendicular distance z (cid:48)⊥ close to its ’average’, or ’Abelian’ value z ⊥ ; the square of the 2-DFourier transform of this quantity, ˜ ϕ ( q ) = [˜ a ( q )] , represents the probability of an individualGB event delivering a momentum transfer q . The three, 4-dimensional delta-functions ofEq. (16) are the statement of Effective Locality for each of the three GBs, and the primes ontheir arguments correspond to z (cid:48) µ = ( z , z L ; z (cid:48)⊥ ), where the subscripts 0 and L signify time-15ike and longitudinal components, respectively. The three ( f · χ ) − correspond to the threeGBs of this problem, while the x and ¯ x coordinates are the space-time coordinates of eachloop (which must be integrated over); and y and ¯ y represent the coordinates of each quark,with u ( s (cid:48) ) and u ( s (cid:48) ) their Fradkin functional variables. The forms written in this Section areappropriate to the simplest situation of a single GB chain exchanged between the scatteringquarks; the most general formulation of multiple GB chains exchanged between the quarksis noted in Section VII.Our simplified and justifiable prescriptions are as follows:(a) Suppress the primes in the arguments of each ( f · χ ) − ; the justification for this stepis given in Appendix B of Ref. [2].(b) Assume that [ f · χ (¯ x − ¯ v (¯ t ))] − is labeled only by its transverse arguments, an assump-tion made for convenience, which is consistent with the final results of this exercise.(c) Write an integral representation for each of the time-like and longitudinal δ -functionsof Eq. (16), thereby introducing the Fourier variables q , q L , p , p L , k , k L . Exactlyas shown in Section 3 of Ref. [3], assume the two quarks of Fig. 8 are scatteringat high energy, and adopt a simple, Eikonal Model description of that amplitude;this approximation removes the need for an integration over the Fradkin u - and ¯ u -dependence. It then follows that all of the Fourier variables q , q L , p , p L , k , k L vanish,so that only transverse q ⊥ , p ⊥ , k ⊥ dependence is relevant.(d) Write Fourier representations for the remaining three transverse delta-functions ofEq. (16), and calculate the integrals (cid:82) d y (cid:48)⊥ · (cid:82) d ¯ y (cid:48)⊥ · (cid:82) d x (cid:48)⊥ · (cid:82) d x (cid:48)(cid:48)⊥ · (cid:82) d ¯ x (cid:48)⊥ · (cid:82) d ¯ x (cid:48)(cid:48)⊥ to obtain factors of ˜ ϕ ( q ) · ˜ ϕ ( p ) · ˜ ϕ ( k ) where all previous z (cid:48)⊥ are effectively replaced by z ⊥ .(e) Calculate (cid:82) d x ⊥ · (cid:82) d ¯ x ⊥ and find that p ⊥ = k ⊥ = q ⊥ , so that there is but onetransverse integral, (cid:82) d q ⊥ ≡ (cid:82) d q , which remains.One final question remains: How is one to understand and represent [ f · χ (¯ x − ¯ v ( t ))] − ?The three transverse δ -functions multiplying the last line of Eq. (16) can be used to re-write16his term as (cid:20) f · χ (cid:18)
12 [ x − v ( t ) + ¯ x − ¯ v (¯ t )] (cid:19)(cid:21) − (17) ⇒ (cid:20) f · χ (cid:18)
12 [ y − u ( s ) + v ( t ) − v ( t ) − ¯ v (¯ t ) + ¯ v (¯ t ) + ¯ y − ¯ u ( ¯ s )] (cid:19)(cid:21) − , and, as explained in Ref. [3], in the Center of Mass (CM) of the scattering quarks, withthe zero of time chosen as that time when both quarks’ longitudinal coordinates are zero,the Eikonal Model effectively replaces y − u ( s ) by y ⊥ , and ¯ y − ¯ u (¯ s ) by ¯ y ⊥ . This replacesEq. (17) by (cid:20) f · χ (cid:18)
12 [ y ⊥ + ¯ y ⊥ + ∆ v − ∆¯ v ] (cid:19)(cid:21) − , (18)where ∆ v = v ( t ) − v ( t ), and ∆¯ v = ¯ v (¯ t ) − ¯ v (¯ t ). And because the CM value of thetransverse vectors y ⊥ + ¯ y ⊥ = 0, Eq. (18) reduces to the simpler form (cid:20) f · χ (cid:18)
12 [∆ v − ∆¯ v ] (cid:19)(cid:21) − . (19)In contrast, the remaining transverse integral over d q ⊥ has as its integrand the factors e iq · [ y ⊥ − ¯ y ⊥ +∆ v +∆¯ v ] , (20)where y ⊥ − ¯ y ⊥ = (cid:126)b , the impact parameter. The ( f · χ ) − of Eq. (19) must now be included aspart of Fradkin’s v and ¯ v -integrals. For this, we write a Fourier representation of Eq. (19)as (cid:90) d K (2 π ) (cid:101) F ( K ) e i K [∆ v − ∆¯ v ] , (21)and immediately note that the UV divergent part of the Fradkin integrals over both loops, (cid:82) d[ v ] · (cid:82) d[¯ v ], is proportional to the product (cid:20) − λ δ αβ ( q + 12 K ) (cid:96) (cid:21) · (cid:20) − λ δ γ(cid:15) ( q − K ) (cid:96) (cid:21) , (cid:96) = ln(1 /m ) (22)using our initial approximation for the spin dependence of each loop. It should be notedthat the color indices of the two sides of the loop are forced to be identical in the divergentlimit of the loop.In the absence of its K -dependence, Eq. (22) is just given by the product of the twoloops’ q -factors; and that K -dependence appears in the form of a sum over products of17olynomial dependence on K components, multiplying the transform (cid:101) F . Let us now takethe inverse transform, writing (cid:101) F ( K ) = (cid:90) d B e − iK · B [ f · χ ( B )] − , (23)and noting that each K -component K α can be expressed as a derivative with respect to B α of the inverse transform, K α → i ∂∂B α . An integration-by-parts transforms this derivative,and all such derivatives arising from the polynomial K -dependence of Eq. (22), into oneor more derivatives operating upon [ f · χ ( B )] − . But now the (cid:82) d K can be immediatelyevaluated, yielding δ (2) ( B ), so that the result of all the K -dependence of Eq. (22) is a groupof derivatives taken at B = 0.To evaluate this result, remember that we have not yet allowed the small space-timeinterval of the ( f · χ ) − of this central GB to vanish, in conjunction with the loop UVdivergences becoming infinite. Upon what portion of this small transverse volume do these δδB α operate? Those derivatives cannot have any bearing upon differences of this smallvolume and neighboring volumes, because each such small volume is completely independentof its neighbors. These derivatives refer to possible transverse variations within the smallvolume of interest, centered about the point B = 0. But before taking its limit of zerovolume, we are free to define how that limit is to be taken; and the only natural definition,and surely the simplest, is to imagine that volume as ’flat’, without any curvature, so thateach and every such derivative within that volume vanishes, after which the limit B → q -factors, one from each loop, separated by the matrix quantity [ f · χ (0)] − | b (cid:48) c (cid:48) βγ which wenow replace by the simplified expression( − λq δ αβ )( − λq δ γ(cid:15) ) κ . (24)We leave it as an exercise for the interested reader to show that this result of the form ofEq. (24) will hold for every ’interior’ GB of the chain. For example, following exactly thesame procedures as for the two-loop amplitude above, the three-loop amplitude has four( f · χ ) − factors, and the central two may both be re-written as (cid:88) β,c (cid:48) [ f · χ (0)] − | b (cid:48) c (cid:48) αβ · [ f · χ (0)] − | c (cid:48) d (cid:48) βγ , (25)18r as [ f · χ (0)] − | b (cid:48) d (cid:48) αγ .In this way, the result for a chain with n ’interior’ GBs yields a term proportional to[ f · χ (0)] n | b (cid:48) d (cid:48) αγ which is inserted between the two ’exterior’ GBs, [ f · χ ( y ⊥ )] − | aa (cid:48) µα on the leftand [ f · χ (¯ y ⊥ )] − | d (cid:48) b(cid:15)ν on the right, multiplied by the remaining q -dependence, and integratedover all transverse q . With X = λ q κ g ˜ ϕ ( q ), all together one has, upon summing over allinterior loops which effectively form a geometric series, and including the amplitude withbut one loop,[ f · χ ( y ⊥ )] − (cid:12)(cid:12) aa (cid:48) µα (26) · gX ˜ ϕ (cid:2) iX [ f · χ (0)] − − X [ f · χ (0)] − − iX [ f · χ (0)] − + · · · (cid:3) a (cid:48) b (cid:48) αβ · [ f · χ (¯ y ⊥ )] − (cid:12)(cid:12) b (cid:48) bβν , or, suppressing matrix indices,[ f · χ ( y ⊥ )] − · gX ˜ ϕ (cid:20) iX [ f · χ (0)] − X [ f · χ (0)] − (cid:21) · [ f · χ (¯ y ⊥ )] − . (27)Eq. (27) can be replaced by gX ˜ ϕ [ f · χ ( y ⊥ )] − · [ f · χ (0)] (cid:20) iX [ f · χ (0)] − [ f · χ (0)] + X (cid:21) · [ f · χ (¯ y ⊥ )] − . (28)Since the α , β indices of χ aαβ (0) are transverse, all components of χ (0) can be chosen asreal; and since the f abc are also real, [ f · χ (0)] is positive, and the denominator of Eq. (28)is never zero. The χ (0) contribution to the amplitude is then proportional to g (cid:90) d q e iq · (cid:126)b · [ f · χ ( y ⊥ )] − · I ( q , g ) · [ f · χ (¯ y ⊥ )] − (29)or g R ( q ) = g I ( q , g ) q [ ˜ ϕ ( q )] λκ (30)with I ( q , g ) = N (cid:90) d χ (0) det[ f · χ (0)] − e i χ (0) [ f · χ (0)] [ f · χ (0)] + [ λκgq ˜ ϕ ] , (31)since the integral (cid:82) d n χ (0) over an odd function of [ f · χ (0)] − vanishes.While the integral of Eq. (31) may turn out to be complex, there is nothing really improperabout a complex quantity multiplying any matrix element. To put this into a conventionalform, where g R ( q ) is expected to be real, it may be possible to choose the product λκ sothat g R can be made real; but the reality of such a g R ( q ) is an intuitive nicety, rather thana QM-requirement. In general, g R is a matrix quantity, and the same remarks applies.19ne can see that there is no divergence in the integral of Eq. (29) for any value of q .The Fourier transform of this integral corresponds, in momentum space, to q dependence– following from this chain-graph form of renormalization – of an effective, or renormalized,charge dependence of the complete set of radiative corrections obtained by summation overall contributing graphs. This factor will re-appear in every process describing interactingquarks; and as such, it can be considered as the effective, or renormalized color-chargedependence of this Model renormalization, as the ’renormalized’ charge which appears inthe scattering of a pair of quarks and/or anti-quarks, at q -values somewhat different fromthose obtained from simple one-GB exchange. Integrals over this quantity are then finiteby virtue of the exponential cut-off appearing in ˜ ϕ ( q ), which is (slightly) less strong thanGaussian, reflecting the basic structure of confinement in this Model of Realistic QCD.This chain-graph Bundle structure will be repeated in all of the correlation functions, withcoordinates defined in terms of a basic CM frame; and while the integrations over coordinatecomponents may become somewhat complicated, and require numerical integration, they areall finite.Methods of Random Matrix theory [4, 5], requiring a certain measure of numerical com-putation, can be used to evaluate multiple-chain contributions to high-energy hadronic re-actions, in particular elastic pp scattering. Our intention in the next few paragraphs is todemonstrate something much simpler – the origin and appearance of the familiar ”diffractiondip” in the (momentum-transfer) region of m p – by adopting two intuitive, qualitative ap-proximations for the exchange of a single GB-loop chain between a pair of scattering quarks,each bound into a different proton, with the details of that binding suppressed.The first approximation is to represent the amplitude of a single chain by its first twoterms, as pictured in Fig. 6. The second approximation is to evaluate (cid:82) d n χ (0) by treating χ a as a vector in color space, with magnitude R = (cid:112)(cid:80) a ( χ a ) , greatly simplified by suppress-ing all of the normalized integrations over such angles, and retaining only the normalizedintegration over R .With an arbitrary normalization of the absolute value of that | amplitude | , its value asa function of q is represented in Fig. 7. The dashed curve of Fig. 7, largest at small q , isthe result of using only a single GB exchange while the curve defined by dots represents the | amplitude | of the single-loop exchange. The | amplitude | of the sum of both the GB andthe one-loop exchange is the solid curve of Fig. 7, and easily displays the expected diffraction20ip. When more loops are added to the total amplitude, the fall-off at larger q should bereduced, while the dip should be moved very slightly to the right of m p /
2. This approximateevaluation of what appears to be the largest contributions to a process such as elastic ppscattering suggests that a detailed fit, fixing the as yet open parameters g , λκ , δ q ( E ) will atthe very least be able to reproduce the essential features of the data. This project is nowunder detailed study. VI. PERTURBATIVE AND NON-PERTURBATIVE APPROXIMATIONS
Whether the chain-graph Model of QCD presented in the Previous Sections meets the es-sential criteria of experimental observation remains to be seen. It does permit a descriptionof interactions between its fundamental quarks, from binding to scattering, and althoughthe hadrons we measure are themselves bound states of quarks, the fall-off of measuredhadronic scattering amplitudes with increasing momentum transfer has its counterpart inthe ’renormalized’ coupling constant of the chain-graph Model. If that fall-off turns out tobe incorrect, then the Model must be discarded; but it does, at the very least, raise inter-esting questions about the structure, and comparison, of perturbative and non-perturbativetheories.Consider first QED, and in particular leptonic QED, in which photons are coupled toleptons which, not merely by definition but by experiment, seem to be fundamental, in thesense of having no sub-structure. The sequential calculations of their radiative correctionsacross the last half-century have always led to logarithmic UV divergences, whose sequentialrenormalization approximations – that is, expressing all results in terms of the measurable,or renormalized charges and leptons masses to the same order of approximation – have beenshown to agree with experiment to a remarkable degree of accuracy. Nevertheless, as manyauthors have attempted to understand Ref. [14], are there really divergences in QED, or isthe appearance of such terms tied to the method of approximation?That question has recently been partially answered in Ref. [11] by the application of amethod of functional summation – not just over a handful of Feynman graphs, but overinfinite numbers of interactions, in each member of an infinite class, each containing aninfinite number of Feynman graphs – which strongly suggests that charge renormalization inQED is indeed finite. That analysis has not yet been extended to leptonic mass renormal-21zation, nor to wave-function and vertex renormalization; but since the latter two quantitiesare equal and gauge-dependent, and always cancel in any physical measurement, one canaccept their presence as an artefact of calculation. Therefore, if gauge-independent chargerenormalization is finite, one can accept QED not as an approximate theory, containing astill-hidden sub-structure, but rather as a True Theory of Nature.Now consider QCD, which since its inception has always been defined in terms of aLorentz-covariant Lagrangian, similar to but more complicated than those of Abelian theo-ries. From the equations of motion of those theories, it is possible to define single-particle,asymptotic field operators whose quanta are described in terms of coordinates which can, inprinciple, be specified exactly, just as in QED. But we have known for several decades thatquarks and antiquarks are always asymptotically bound to each other, and therefore thattheir transverse momenta and/or position coordinates can only, in principle, be describedwith quantum-mechanical precision, rather than specified exactly.When the functional techniques referred to in the above Sections are applied to QCD,using a special rearrangement which guarantees manifest gauge invariance in Ref. [1], theresult of such a mismatch is the appearance of absurdities in non-perturbative amplitudesfor all processes, divergences multiplying otherwise reasonable and finite factors. Once thismost-inappropriate mismatch of the QM description is removed, as has been done phe-nomenologically in Ref. [2], all such absurdities vanish, and one can see the essential differ-ence of non-perturbative summations of a non-Abelian theory containing confinement, ascompared to summations over one which violates that basic quantum-mechanical principle.There is another difficulty with perturbative approximations in QCD, but one which couldnot have been known until summations over infinite numbers of gluons were functionallyobtained. This is tied to the fact that the coupling constant g appears in two differentplaces in the Lagrangian, once as the coupling of quarks to gluons, and again in that part ofthe interaction coupling of gluons to each other. For definiteness, we shall refer to the firstcoupling as g , and to the second as g ; they are, of course, to be set equal to each other,but it will be instructive to keep the distinction for a few more lines.The summation of all gluon exchanges between any two quark lines leads to amplitudesdepending upon factors of g g ; and at this stage one can see the difficulties which arise whenattempting perturbative expansions in g , that is, at this stage in g and g , for one cannotexpand a function of g about g = 0. Such an expansion of both g and g corresponds22o treating QCD as a mixture of QED and Yang-Mills, and all points in-between; andleads to irrelevant divergences and confusion. This could not have been foreseen until thesummations over all gluon exchanges were performed, but it illustrates the difficulties of apurely perturbative approach to QCD.Finally, it may be useful to consider perturbative approximations of radiative correctionsto particles which are themselves bound states of more fundamental objects, such as nu-cleons are of quarks, in mock-Abelian theories, such as QED applied directly to protons.The simplest self-energy Feynman graph of a proton emitting and re-absorbing a photon,illustrates the point: Why is there a UV divergence associated with this graph? The answeris simply because it has been tacitly assumed that the proton still exists, after emittingand then re-absorbing a virtual photon of sufficiently high energy, which produced the UVdivergence.Alternatively, consider the absorptive part of that amplitude, corresponding to the ab-sorption of a photon by the struck proton, and the emission of the final photon by theproton. If the initial photon energy is far less than the binding energy of that three-quarkproton state, then the Feynman graph is perfectly relevant, and the contribution of thecorresponding dispersion relation to the self-energy graph is finite. But when that photon’senergy is far greater than the 3-quark binding energy, it will split the proton apart into itsthree fundamental quarks, which, according to the chain-Model of this paper – will yield aperfectly finite amplitude, as will that of the re-combinations of each of those quarks intowhichever asymptotic states they may form themselves. Again, it is the tacit assumptionthat the proton is itself a fundamental particle which leads to the UV divergence; and theremoval of that and all other such UV divergences is simply to insert a cut-off statement,into the basic Lagrangian, carrying the information that the Lagrangian is only true wheninteracting photon energies are less than the proton’s bound-state energy. VII. SUMMARY
These next paragraphs are not intended to be a restatement of previously mentioneditems above, but rather a final insertion of a few points previously not emphasized.It may have been overlooked, but the final form of L [ A ] – after its A-dependence hasbeen translated so as to incorporate the u (cid:48) and ¯ u (cid:48) variables of the two quark lines – is in23n exponential, along with all the ( f · χ ) − of all GBs. (The Halpern integrals over thethree different ( f · χ ) − are non exponentiated.) The expansion of that exponential in itspowers of u (cid:48) and ¯ u (cid:48) corresponds to the interacting quarks exchanging more than one set ofGB chain graphs – but chains whose loop substructures can never interact with each other,since such loop interactions via new GBs would vanish under this Model renormalization.Whether more than one complete GB chain should be considered would depend upon howmuch time is allowed for any such reaction, how fast the quarks are moving, etc. It wouldallow the different chains to interfere with each other, in a QM way but as entities, and notallow their loop-sub-structures to interact with each other.The complete loop-exchange functional structure here has the form (cid:90) d n χ ( y ⊥ ) det[ f · χ ( y ⊥ )] − e i χ ( y ⊥ ) (32) · (cid:90) d n χ ( y ⊥ ) det[ f · χ ( y ⊥ )] − e i χ ( y ⊥ ) · (cid:90) d n χ (0) det[ f · χ (0)] − e i χ (0) · exp (cid:20)(cid:90) s d s u (cid:48) µ ( s )Ω a ( s ) ( f · χ ( y ⊥ )) − (cid:12)(cid:12) aa (cid:48) αµ (cid:21) δδA a (cid:48) α ( y − u ( s )) · exp (cid:20)(cid:90) ¯ s d¯ s ¯ u (cid:48) ν (¯ s )Ω b ( ¯ s ) ( f · χ (¯ y ⊥ )) − (cid:12)(cid:12) b (cid:48) bβν (cid:21) δδA bβ (¯ y − ¯ u (¯ s )) · exp (cid:110) L [ A ] + L [ A ] ( e ←→D − L [ A ] + · · · (cid:111) . Each of the L [ A ] entering into Eq. (32) is then to have an A -dependence, which itself entersin the form of an exponential, shifted by the translation operators of the u and ¯ u quantities,such that after the individual loop-functional integrations are performed, the result will bethe exponential factor exp (cid:20)(cid:90) d q e i(cid:126)q · (cid:126)b g X ˜ ϕ (cid:18) iX [ f · χ (0)] − X [ f · χ (0)] − (cid:19)(cid:21) , (33)so that the entire multiple GB chain contribution to the scattering – in which none of thesub-elements of any chain can interact with those of another chain – takes the formexp (cid:26)(cid:90) s d s u (cid:48) µ ( s ) Ω a ( s ) [ f · χ ( y ⊥ )] − (cid:12)(cid:12) aa (cid:48) µα (34) · (cid:90) d q e i(cid:126)q (cid:126)b g X (cid:101) ϕ (cid:18) iX ( f · χ (0)) − X ( f · χ (0)) − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a (cid:48) b (cid:48) αβ · (cid:90) ¯ s d¯ s ¯ u (cid:48) µ (¯ s ) Ω b (¯ s ) [ f · χ (¯ y ⊥ )] − (cid:12)(cid:12) b (cid:48) bβν (cid:27) u (cid:48) and ¯ u (cid:48) of Eq. (34) corresponds to the exchange of asingle GB chain, as in Section IV.The simplest interpretation, consistent with fast-moving quarks described by an eikonalrepresentation, is obtained by expanding the u (cid:48) and ¯ u (cid:48) , retaining only linear u (cid:48) and ¯ u (cid:48) dependence – and therefore their own ( f · χ ) − variables, and thereby bringing down theentire ( f · χ ) − dependence, as written in this paper – so that only one GB chain is exchanged.That analysis is sufficient to produce ( g R ) as a function of q , which quantity may becomplex and matrix-valued.Two things are important, and quite attractive, about this Model QCD renormalization:1. The fact that everything comes out finite, as each loop’s UV divergence is absorbed bythe vanishing ( δ ) of the Halpern FI; and because the final integral over d [ χ (0)] shouldbe perfectly finite, even with the ( f · χ (0)) − terms all up (as in the first paragraphabove) in the exponential. It can be easily estimated by suppressing all ”angular” colorand tranverse coordinate dependence, and simply integrating over ”the magnitude”of R of ( f · χ (0)) − ; and it produces, as expected, a strong dependence on g X (cid:101) ϕ ,especially for large q , so that one finds a strong fall-off with increasing q for theeffective, or ’renormalized’ charge, just as expected and needed. This can easily beseen by using Random Matrix methods, as in Ref. [4, 5].2. Just as one may now have confidence in QED as being a ”fundamental and true Theoryof Nature”, because its charge renormalization is almost surely finite [11], and there isno need to hunt for any ’underlying’ Theory which could magically produce that effect,so may QCD be tentatively called a ”fundamental and true Theory of Nature” [15],at least in this simplest renormalization Model. Whether or not the q dependencederived for quark-quark interactions, when incorporated into hadron scattering andproduction processes, turns out to be that required by experiment, is the crucial point.If so, then this model will have the right to be assumed correct and proper. If not,the experience gained with this functional approach will suggest that somewhat morecomplicated calculations must be done, difficult but certainly possible, before QCDcan be placed in the same, high category as QED.25 CKNOWLEDGMENTS
This publication was made possible through the partial support of a Grant from theJulian Schwinger Foundation. The opinions expressed in this publication are those of theauthors and do not necessarily reflect the views of the Julian Schwinger Foundation. Weespecially wish to thank Mario Gattobigio, of Universit´e de Nice Sophia-Antipolis, for hismany kind and informative conversations relevant to the Nuclear Physics aspects of ourwork. It is also a pleasure to thank Mark Restollan, of the American University of Paris,for his kind assistance in arranging sites for our collaborative research when in Paris.
Appendix A: Fradkin’s Representations for Green’s Function and Closed-Fermion-Loop Functional
The exact functional representations of these two functionals of A ( x ) are perhaps the mostuseful tools in all of QFT, for they allow that A -dependence of these functionals to be ex-tracted from inside ordered exponentials; and because they, themselves, are Gaussian in theirdependence upon A ( x ), they permit the functional operations of the Schwinger/Symanzikgenerating functional (Gaussian functional integration, or functional linkage operation) tobe performed exactly. This corresponds to an explicit sum over all Feynman graphs rele-vant to the process under consideration, with the results expressed in terms of functionalintegrals over the Fradkin variables; and in the present QCD case, because of EL, thosenon-perturbative results can be extracted and related to physical measurements.The causal quark Green’s function (which is essentially the most customary Feynmanone) can be written as [9, 10] G c [ A ] = [ m + iγ · Π][ m + ( γ · Π) ] − = [ m + iγ · Π] · i (cid:90) ∞ ds e − ism e is ( γ · Π) , (A1)where Π = i [ ∂ µ − igA aµ τ a ] and ( γ · Π) = Π + igσ µν F aµν τ a with σ µν = [ γ µ , γ ν ]. FollowingFradkin’s method [9, 10] and replacing Π µ with i δδv µ , one obtains G c ( x, y | A ) (A2)= i (cid:90) ∞ ds e − ism · e i (cid:82) s ds (cid:48) δ δv µ ( s (cid:48) ) · (cid:20) m − γ µ δδv µ ( s ) (cid:21) δ ( x − y + (cid:90) s ds (cid:48) v ( s (cid:48) )) × (cid:32) exp (cid:40) − ig (cid:90) s ds (cid:48) (cid:34) v µ ( s (cid:48) ) A aµ ( y − (cid:90) s (cid:48) v ) τ a + iσ µν F aµν ( y − (cid:90) s (cid:48) v ) τ a (cid:35)(cid:41)(cid:33) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v µ → . (cid:90) d[ u ] δ [ u ( s (cid:48) ) − (cid:90) s (cid:48) ds (cid:48)(cid:48) v ( s (cid:48)(cid:48) )] , (A3)and replace the delta-functional δ [ u ( s (cid:48) ) − (cid:82) s (cid:48) ds (cid:48)(cid:48) v ( s (cid:48)(cid:48) )] with a functional integral over Ω,and then the Green’s function becomes [16] G c ( x, y | A ) (A4)= i (cid:90) ∞ ds e − ism e − Tr ln (2 h ) (cid:90) d [ u ] e i (cid:82) s ds (cid:48) [ u (cid:48) ( s (cid:48) )] δ (4) ( x − y + u ( s )) × (cid:2) m + igγ µ A aµ ( y − u ( s )) τ a (cid:3) (cid:16) e − ig (cid:82) s ds (cid:48) u (cid:48) µ ( s (cid:48) ) A aµ ( y − u ( s (cid:48) )) τ a + g (cid:82) s ds (cid:48) σ µν F aµν ( y − u ( s (cid:48) )) τ a (cid:17) + , where h ( s , s ) = (cid:82) s ds (cid:48) Θ( s − s (cid:48) )Θ( s − s (cid:48) ). To remove the A -dependence out of the lin-ear (mass) term, one can replace igA aµ ( y − u ( s )) τ a with − δδu (cid:48) µ ( s ) operating on the orderedexponential so that G c ( x, y | A ) (A5)= i (cid:90) ∞ ds e − ism e − Tr ln (2 h ) (cid:90) d [ u ] e i (cid:82) s ds (cid:48) [ u (cid:48) ( s (cid:48) )] δ (4) ( x − y + u ( s )) × (cid:20) m − γ µ δδu (cid:48) µ ( s ) (cid:21) (cid:16) e − ig (cid:82) s ds (cid:48) u (cid:48) µ ( s (cid:48) ) A aµ ( y − u ( s (cid:48) )) τ a + g (cid:82) s ds (cid:48) σ µν F aµν ( y − u ( s (cid:48) )) τ a (cid:17) + . To extract the A -dependence out of the ordered exponential, one may use the followingidentities, 1 = (cid:90) d [ α ] δ (cid:2) α a ( s (cid:48) ) + gu (cid:48) µ ( s (cid:48) ) A aµ ( y − u ( s (cid:48) )) (cid:3) , (A6)1 = (cid:90) d [ Ξ ] δ (cid:2) Ξ aµν ( s (cid:48) ) − g F aµν ( y − u ( s (cid:48) )) (cid:3) , and the ordered exponential becomes (cid:16) e − ig (cid:82) s ds (cid:48) u (cid:48) µ ( s (cid:48) ) A aµ ( y − u ( s (cid:48) )) τ a + g (cid:82) s ds (cid:48) σ µν F aµν ( y − u ( s (cid:48) )) τ a (cid:17) + (A7)= N Ω N Φ (cid:90) d [ α ] (cid:90) d [ Ξ ] (cid:90) d [Ω] (cid:90) d [ Φ ] (cid:16) e i (cid:82) s ds (cid:48) [ α a ( s (cid:48) ) − iσ µν Ξ aµν ( s (cid:48) ) ] τ a (cid:17) + × e − i (cid:82) ds (cid:48) Ω a ( s (cid:48) ) α a ( s (cid:48) ) − i (cid:82) ds (cid:48) Φ aµν ( s (cid:48) ) Ξ aµν ( s (cid:48) ) × e − ig (cid:82) ds (cid:48) u (cid:48) µ ( s (cid:48) ) Ω a ( s (cid:48) ) A aµ ( y − u ( s (cid:48) ))+ ig (cid:82) ds (cid:48) Φ aµν ( s (cid:48) ) F aµν ( y − u ( s (cid:48) )) , where N Ω and N Φ are constants that normalize the functional representations of the delta-functionals. All A -dependence is removed from the ordered exponential and the resultingform of the Green’s function is exact (it entails no approximation). Alternatively, extracting27he A -dependence out of the ordered exponential can also be achieved by using the functionaltranslation operator, and one writes (cid:16) e + g (cid:82) s ds (cid:48) [ σ µν F aµν ( y − u ( s (cid:48) )) τ a ] (cid:17) + (A8)= e g (cid:82) s ds (cid:48) F aµν ( y − u ( s (cid:48) )) δδ Ξ aµν ( s (cid:48) ) · (cid:16) e (cid:82) s ds (cid:48) [ σ µν Ξ aµν ( s (cid:48) ) τ a ] (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) Ξ → . For the closed-fermion-loop functional L [ A ], one can write [10] L [ A ] = − (cid:90) ∞ dss e − ism (cid:110) Tr (cid:104) e − is ( γ · Π) (cid:105) − { g = 0 } (cid:111) , (A9)where the trace Tr sums over all degrees of freedom, space-time coordinates, spin and color.The Fradkin representation proceeds along the same steps as in the case of G c [ A ], and theclosed-fermion-loop functional reads L [ A ] = − (cid:90) ∞ dss e − ism e − Tr ln (2 h ) (A10) × (cid:90) d [ v ] δ (4) ( v ( s )) e i (cid:82) s ds (cid:48) [ v (cid:48) ( s (cid:48) )] × (cid:90) d x tr (cid:16) e − ig (cid:82) s ds (cid:48) v (cid:48) µ ( s (cid:48) ) A aµ ( x − v ( s (cid:48) )) τ a + g (cid:82) s ds (cid:48) σ µν F aµν ( x − v ( s (cid:48) )) τ a (cid:17) + − { g = 0 } , where the trace tr sums over color and spinor indices. Also, Fradkin’s variables have beendenoted by v ( s (cid:48) ), instead of u ( s (cid:48) ), in order to distinguish them from those appearing in theGreen’s function G c [ A ]. One finds L [ A ] = − (cid:90) ∞ dss e − ism e − Tr ln (2 h ) (A11) ×N Ω N Φ (cid:90) d x (cid:90) d[ α ] (cid:90) d[Ω] (cid:90) d[ Ξ ] (cid:90) d[ Φ ] × (cid:90) d [ v ] δ (4) ( v ( s )) e i (cid:82) s ds (cid:48) [ v (cid:48) ( s (cid:48) )] × e − i (cid:82) ds (cid:48) Ω a ( s (cid:48) ) α a ( s (cid:48) ) − i (cid:82) ds (cid:48) Φ aµν ( s (cid:48) ) Ξ aµν ( s (cid:48) ) · tr (cid:16) e i (cid:82) s ds (cid:48) [ α a ( s (cid:48) ) − iσ µν Ξ aµν ( s (cid:48) ) ] τ a (cid:17) + × e − ig (cid:82) s ds (cid:48) v (cid:48) µ ( s (cid:48) ) Ω a ( s (cid:48) ) A aµ ( x − v ( s (cid:48) )) − ig (cid:82) d z ( ∂ ν Φ aνµ ( z ) ) A aµ ( z ) × e + ig (cid:82) ds (cid:48) f abc Φ aµν ( s (cid:48) ) A bµ ( x − v ( s (cid:48) )) A cν ( x − v ( s (cid:48) )) − { g = 0 } , where the same properties as those of G c [ A ] can be read off readily.28 ppendix B: Vanishing ’Bundle Self-Energy’ Diagrams of Fig. 1 and 2. For simplicity and clarity, we first consider the non-spin dependence of the Fradkin rep-resentation of G c [ A ], and then discuss the spin terms separately. Because of the EffectiveLocality (EL), the 4-dimensional delta-function multiplying the ( f · χ ) − factor of the GBof Fig. 1 is given by δ (4) ( u ( s ) − u ( s )). This suggests but does not necessarily require that s = s ; but that condition is obtained by considering the time-like and longitudinal inte-grals separately, δ ( u ( s ) − u ( s )) and δ ( u L ( s ) − u L ( s )). Suppose now that there are aset of points s (cid:96) for which the argument of the time-like δ (0) vanishes, and a set of points s m for which the argument of the longitudinal δ ( L ) -function vanishes, δ (0) = (cid:88) (cid:96) δ ( s − s (cid:96) ) | u (cid:48) ( s (cid:96) ) | (cid:12)(cid:12)(cid:12)(cid:12) u ( s (cid:96) )= u ( s ) , (B1) δ ( L ) = (cid:88) m δ ( s s − s m ) | u (cid:48) L ( s m ) | (cid:12)(cid:12)(cid:12)(cid:12) u L ( s )= u L ( s m ) . Their product is then given by (cid:88) (cid:96),m δ ( s − s (cid:96) ) δ ( s − s m ) | u (cid:48) ( s (cid:96) ) u (cid:48) L ( s m ) | (cid:12)(cid:12)(cid:12)(cid:12) u ( s (cid:96) )= u ( s m ) u L ( s (cid:96) )= u L ( s m ) , (B2)and it is the subsidiary conditions which are most relevant. Since u and u L are continuousbut completely independent functions, the probability of finding sets of points s (cid:96) and s m at which u takes on the same value, and at which u L simultaneously has the same value,would appear to be less than (cid:15) , and (cid:15) →
0. However, there are two s -values for whichthis is possible, where initial conditions specify that u µ (0) = 0, and that u µ ( s ) = − z µ .Therefore, only s = s = 0, or else s = s = s . Then, for either case, s = s , and thecoefficients u (cid:48) µ ( s ) and u (cid:48) ν ( s ) are symmetric in µ and ν , and are multiplying ( f · χ ) − | µν which is antisymmetric in those indices; and the result is zero.The spin dependence for this particular process will also vanish, but for two differentreasons. Those terms coming from the linear A -dependence of the G c [ A ] representation willhave gradient terms differentiating the y -dependence of the δ -functions representing EL , butthat y -dependence trivially cancels for this ’self-energy’ process, and hence those terms givea zero result. The antisymmetric spin dependence coming from quadratic A -terms findsitself multiplying a different set of u (cid:48) µ ( s ) and u (cid:48) ν ( s ) coefficients; and then the analysis ofthe previous paragraph again rules out any non-zero contribution.29he vanishing of the Bundle Diagram of Fig. 2 may be inferred from that of Fig. 1, byimagining the two ends of the quark line of Fig. 1 to be wrapped around and form a closedloop; and then, without performing the loop integrations, the result is zero. Or, one mayfollow the argument used in the text following Eq. (11) for chain-graph loops but applied tothis single loop containing an internal GB; and again the result is zero. [1] H. M. Fried, Y. Gabellini, T. Grandou, and Y.-M. Sheu, Eur. Phys. J. C (2010) 395-411.[2] H. M. Fried, T. Grandou, and Y.-M. Sheu, Ann. Phys. (2012) 2666-2690.[3] H. M. Fried, Y. Gabellini, T. Grandou, and Y.-M. Sheu,
Ann. Phys. (2013) 107-122.[4] T. Grandou,
Eur. Phys. Lett. (2014) 11001; arXiv:1402.7273 [hep-th].[5] H. M. Fried, T. Grandou, and Y.-M. Sheu,
Ann. Phys. (2014) 78-96.[6] H. M. Fried,
Modern Phys. Letts.
A 28 (2013) 1230045.[7] D. D. Ferrante, G. S. Guralnik, Z. Guralnik, and C. Pehlevan, arXiv:1301.4233 [hep-th].[8] M. B. Halpern,
Phys. Rev. D (1977) 1798; ibib , (1977) 3515.[9] E. S. Fradkin, Dokl. Akad. Nauk SSSR (1954) 47; and E. S. Fradkin, Nucl. Phys. (1966)588-624.[10] For both the standard combinatoric and differential analysis, see H. M. Fried, Basics of Func-tional Methods and Eikonal Models , Editions Fronti´eres, Gif-sur-Yvette Cedex, France (1990).[11] H. M. Fried and Y. Gabellini,
Ann. Phys. (2012) 1645-1667.[12] The TOTEM Collaboration et al ., Eur. Phys. Lett. (2011) 41001.[13] The TOTEM Collaboration et al ., Eur. Phys. Lett. (2013) 21002.[14] M. Baker and K. Johnson,
Phys. Rev. (1969) 1292.[15] F. Wilczek,
QCD and Natural Philosophy , Proceedings of International Conference on The-oretical Physics TH-2002,
Annales Henri Poincar´e (2003) S211-S228, D. Iagolnitzer,V. Rivasseau, and J. Zinn-Justin, Editors, Paris July 22-27, 2002; arXiv:physics/0212025[physics.ed-ph][16] Y.-M. Sheu, Finite-Temperature Quantum Electrodynamics: General Theory and Bloch-Nordsieck Estimates of Fermion Damping in a Hot Medium , PhD Thesis, Brown University,May 2008., PhD Thesis, Brown University,May 2008.