Uncovering the Scaling Laws of Hard Exclusive Hadronic Processes in a Comprehensive Endpoint Model
UUncovering the Scaling Laws of Hard ExclusiveHadronic Processes in a Comprehensive EndpointModel
Sumeet Dagaonkar ∗ , Pankaj Jain † Department of Physics, Indian Institute of Technology, Kanpur 208016, India
John P. Ralston ‡ Department of Physics & Astronomy, University of Kansas,Lawrence, KS - 66045, USA
Abstract
We show that an endpoint overlap model can explain the scaling laws observed in exclu-sive hadronic reactions at large momentum transfer. The model assumes one of the valencequarks carries most of the hadron momentum. Hadron form factors and fixed angle scatteringare related directly to the quark wave function, which can be directly extracted from experi-mental data. A universal linear endpoint behavior explains the proton electromagnetic formfactor, proton-proton fixed angle scattering, and the t -dependence of proton-proton scatteringat large s >> t . Endpoint constituent counting rules relate the number of quarks in a hadronto the power-law behavior. All proton reactions surveyed are consistent with three quarks par-ticipating. The model is applicable at laboratory energies and does not need assumptions ofasymptotically-high energy regime. A rich phenomenology of lepton-hadron scattering andhadron-hadron scattering processes is found in remarkably simple relationships between di-verse processes. The experimental study of differential cross sections of hard exclusive hadronic reactions at highenergy reveals a remarkable pattern:
They are described by power laws [1–3]. A model explanationexists [4–7], yet it is not satisfactory [8] at the energies of experimental measurements. We aredriven to find a consistent explanation of experimental regularities by re-examining all the factsfrom a fresh point of view.“Hard” reactions are those which depend on a single large scale Q > GeV , or several largescales with a fixed ratio. It is remarkable that the proton electromagnetic form factor F ( Q ) agrees well with a decreasing power of Q for Q (cid:38) GeV [9]. For large momentum transfer, itis remarkable that pp → pp fixed-angle cross section d σ / dt agrees well with a decreasing powerof Q ∼ s [10], where √ s is the center of mass energy. There are many other examples. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] A p r . Dagaonkar et al. – Uncovering the Scaling Laws . . . the endpoint overlap model [11–13]. There is actually much to be learned andmuch that is new when the model is objectively explored. In their evaluation of the endpoint region, Brodsky and Mueller [14] wrote that “its contributiondepends sensitively on the hadronic wave functions”. The discussion discovered no actual faultin the endpoint contribution. Instead of finding a flaw, the section ends with a weak suggestionto assume validity of a short-distance perturbative model, as “at least plausible”, adding “in anycase there is currently no comprehensive alternative theory of these processes”.
We suggest that the lack of a comprehensive alternative theory came out of a historical failureof the endpoint contribution to be fully appreciated and developed.
In the concluding remarks we linkthis to the history of early development of perturbative QCD, which is an era long past.In reviewing the current status we noticed several facts: • The predictions of all models depend on the wave functions. For reasons we believe areobsolete, the opportunity to learn about wave functions using data was bypassed in thepromotion of short-distance ( SD ) models [4–7]. • Great emphasis was imposed early on asymptotic limits. The motivation was not to learnabout hadrons, but an attempt to make hadrons irrelevant for the goal of establishing QCD. • The asymptotic limits of QCD are now understood to be of negligible experimental rele-vance. The asymptotic limits of QCD predictions have also never actually been established.Instead limits of models have been established. None of the work assuming a model hasgone an inch beyond the boundaries of the model itself. In particular, it has not been shownanywhere that the pion form factor of the general theory known as QCD necessarily falls fasterthan 1/ Q , despite considerable effort to force such a conclusion. Careful reading is neededto verify this. For example, Farrar and Jackson [4] claim an asymptotic limit in openinglines, without actually supporting the claim: The contrary information about regions out-side the model is buried in the footnote labeled Ref. 13 in the paper. • Now that QCD is established, every integration regime, including those contradicting theassumptions of SD models, needs to be considered. Interest in the larger theory and hadronstructure has eclipsed the goal of exhibiting a model based solely on perturbation theory. • The main reason for early interest in perturbative models was power-law behavior. Inex-perience with more general models created a folk-lore that “soft” non-perturbative wavefunctions would lead to exponential dependence on a large Q scale. This is false. As wereview, power-law behavior is generic from the endpoint region. • Divisions in the field separated groups into two camps. Relying on perturbation theoryappears to be more theoretically ambitious, but it is actually less general than representingdynamics with wave functions. For one thing, an arbitrary order of perturbation theorycan be subsumed into equivalent wave functions, but not vice versa. • Calculations in perturbation theory use the Fock state basis of free field theory. Consider-able effort has been dedicated to making the endpoint region of perturbative calculations go . Dagaonkar et al. – Uncovering the Scaling Laws . . . perturbative self-consistency.None of that work is relevant to non-perturbative wave functions, which use a differentbasis of fully-interacting quanta. It is not logically self-consistent to extend the asymptoticpQCD-based suppression of the endpoint region to wave functions extracted from experi-mental data. • Despite years of study, very little is known about pions and protons. The proof comes fromthe dearth of definite information about pions and protons in terms of non-perturbativewave functions. Contrary to the bias of perturbative QCD, it is definitely possible and abso-lutely productive to use experiments to learn about non-perturbative wave functions.Recently Chang, Clo`et, Roberts, Schmidt, and Tandy [15, 16] have computed the pion formfactor with a method described as self-consistent for all space-like Q . The paper highlightsthe asymptotic SD model’s prediction being about 3 times smaller than the experimental formfactor. Ref. [15] states the asymptotic estimate is incapable of converging to a realistic value below Q ∼ GeV . This is typical of asymptotic SD estimates: The estimates require fabulouslyhigh momentum transfer to apply. We agree that the assumptions made in setting up SD modelsare contradicted by the application of the models to existing momentum transfers.The best test of the SD models comes from hadronic helicity conservation [17]. These tests aremuch less demanding than asymptotic limits, and apply to an expansion of leading power be-havior. The tests fail in almost every case experiments exist [18–34]. That is convincing evidencethat the observed experimental regularities are not explained by the SD model.The calculations of [15] are made in an overlap model emphasizing first-principles predic-tions of the pion wave function. They are similar in spirit to overlap models in the relativisticimpulse approximation [35–37]. These models are very successful in describing data at low Q .Our approach accepts the validity of these models as integral representations of form factors,while also extending the scope to other processes. We differ from most models by not attempt-ing to know wave functions in advance. Many studies have been restricted to estimating theendpoint region to order of magnitude. We will show that the end-point region of the wavefunctions is not only determined, but over-determined, by experimental regularities found inpower laws. By measuring the wave functions rather than predicting them we find the endpointoverlap model is a consistent comprehensive description. Rather unexpectedly, our approachextracting information from wave functions is quite consistent with the trend predicted by [15].Section 2 derives the endpoint overlap constituent counting rules for form factors. Section 3extends the rules to exclusive hadron-hadron reactions. These rules explain why scaling lawsshould be observed at the limited energies of laboratory experiments, and how scaling lawsare correlated with the number of quarks scattering. We cannot explain why this predictiveregularity of all exclusive reactions surveyed has been overlooked. Concluding remarks with briefhistorical commentary are given in Section 4. In this Section we derive the endpoint constituent counting rules , which predict the scaling powerof Q in terms of the number of constituents. Comparing experimental data to these rules findsconsistency with three (3) quarks scattered in every proton reaction we have surveyed. Thediscussion will be organized in increasing levels of detail, beginning with the simplest case of thepion electromagnetic form factor. . Dagaonkar et al. – Uncovering the Scaling Laws . . . F π ( Q ) : The Probability of a Slow Quark x’ P’x Px P x’ P’ ψ (k’ k’ ) ψ (k k ) Figure 1: Physical picture of the endpoint dominance model. Due to the change of direction of the fastmomentum, the transverse momentum scale of non-perturbative wave functions must overlapwith the range of x P / x (cid:48) P (cid:48) of spectator constituents. The coordinate system of variables k i and k (cid:48) i are defined in the text. The form factor is defined by < P (cid:48) | J µ em | P > = ( P + P (cid:48) ) µ F π ( Q ) , (1)where J µ em is the electromagnetic current operator. In a gauge-invariant local field theory the pho-ton interaction involves one (1) struck parton. The minimum number of constituents in the pionis two. The dynamical question of elastic scattering is how scattering one constituent can scat-ter the entire hadron. This is answered by the quantum mechanical overlap of wave functions.Figure 1 conveys the qualitative picture. The endpoint region is dominated by some transversehadronic scale “ Λ ” for which the slow parton obeys x (cid:46) Λ / Q , while the struck parton obeys1 − x (cid:46) Λ / Q . For the entire region the endpoint contribution is such that the transverse mo-mentum integrations do not contribute any power of Q to the form factor. This happens to bethe feature causing the short-distance model to be impossible to justify in this region.The form factor is F π ( Q ) = (cid:90) [ dk ][ dk (cid:48) ] ψ (cid:48) ∗ ( x (cid:48) i , (cid:126) k (cid:48) i ) T ψ ( x i , (cid:126) k i ) , [ dk ] = dx dx dk − dk − d k d k . Dagaonkar et al. – Uncovering the Scaling Laws . . . T contains the quark charges, a gamma ma-trix, and momentum-conserving delta functions. Wave functions with several independent spinstructures are possible, with discussion postponed to Section 2.1.1. We define the delta functionsto include factors representing momentum conservation, hence (cid:126) k + (cid:126) k = (cid:126) P , etc. in a frame wenow specify.Let P , P (cid:48) be the 4-momenta of the pions, with P (cid:48) = P + q , q = − Q . Choose a Lorentz framewith Cartesian labels ( E , p X , p Y , p Z ) where E is the energy. Thus q = ( Q , 0, 0 ) ; P = ( (cid:113) Q /2 + m π , − Q /2, 0, Q /2 ) ; P (cid:48) = ( (cid:113) Q /2 + m π , Q /2, 0, Q /2 ) . (2)Let k µ ( k (cid:48) µ ) be the momenta of the struck parton before (after) scattering. Due to the changeof direction of the fast momenta, the meaning of symbols “ (cid:126) k i ” must be adapted to be orthogonalto each hadron’s direction. We introduce a basis of transverse vectors adapted to the particularhadronic momenta : ˆ y =(
0, 0, 1, 0 ) = ˆ y (cid:48) ; (cid:126) P · ˆ y = (cid:126) P (cid:48) ˆ y (cid:48) = x =( −
1, 0, − ) , (cid:126) P · ˆ x = x (cid:48) =(
0, 1, 0, − ) , (cid:126) P (cid:48) · ˆ x (cid:48) = k i = x i P + k ix ˆ x + k iy ˆ y = ( x i (cid:113) Q /2 + m π , − x i Q /2, 0, x i Q /2 ) + ( − k ix , k iy , − k ix ) ; k (cid:48) i = x (cid:48) i P (cid:48) + k (cid:48) ix ˆ x (cid:48) + k (cid:48) iy ˆ y = ( x i (cid:113) Q /2 + m π , x i Q /2, 0, x i Q /2 ) + ( k (cid:48) ix , k (cid:48) iy , − k (cid:48) ix ) . (3)There are only three free parameters, and the quanta are not strictly constrained to the pertur-bative mass shell. By hypothesis, amplitudes are concentrated in the kinematic region shown.Integrating over a fourth (minus, or virtuality parameter) concentrated on the region is equiva-lent. Momentum conservation of the un-struck spectator is k = k (cid:48) . Momentum conservation ofthe struck quark k (cid:48) = k + q yields four constraints: k x = k (cid:48) x ≡ k x ; k y = k (cid:48) y ≡ k y ; x = x (cid:48) ; x (cid:48) Q /2 + k (cid:48) x = − x Q /2 + Q − k x (4)Solving gives k x = ( − x ) Q = x Q m π . . Dagaonkar et al. – Uncovering the Scaling Laws . . . PP’ k k’ q Figure 2: Endpoint kinematics in the pion case. Pion momenta are shown as dashed arrows, while quarkmomenta are solid arrows. Two isosceles triangles representing energy and momentum conserva-tion must close. The transverse and longitudinal momenta of one spectator covers the difference.By inspection, k x = xQ /2, where x is the momentum fraction of the slow quark. Evaluating F π gives F π ( Q ) = (cid:90) dx Φ π ( x , xQ /2 ) ; (5) Φ π ( x , k x ) = (cid:90) dk y ψ (cid:48) ∗ ( x , k x , k y ) ψ ( x , k x , k y ) . (6)A soft non-perturbative wave function means that Φ π ( x , k x ) is a rapidly falling function witha scale k x (cid:46) Λ , where Λ ∼
300 MeV. To see how power-law behavior emerges, consider anexponential function multiplied by a function of x : Φ π ( x , k x ) = e −| k x | / Λ φ ( x ) ; (7) F π ( Q ) = (cid:90) dx e − xQ /2 Λ φ ( x ) . (8)As Q → ∞ the exponential is driven toward the endpoint x (cid:46) Λ / Q .Integrals dominated by their endpoints have an asymptotic series expansion [38] developed The same expansion applies to a wide class of integrals with saddle points in x approaching the endpoint. Dagaonkar et al. – Uncovering the Scaling Laws . . . F π = (cid:90) dx − Λ Q (cid:18) ∂∂ x e − xQ /2 Λ (cid:19) φ ( x )= − Λ Q e − xQ /2 Λ φ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + Λ Q (cid:90) dx e − xQ /2 Λ ∂φ ( x ) ∂ x , = Λ Q φ ( ) + ...where x = x = − x . The last line dropped the exponentially small term from the upper limit.Using repeated integration by parts gives F π ( Q ) = Λ Q φ ( ) + (cid:18) Λ Q (cid:19) φ ( ) ( ) + (cid:18) Λ Q (cid:19) φ ( ) ( ) + ... (9)Here φ ( n ) = ∂ n φ / ∂ x ( n ) ( ) . It is generally expected that light cone wave functions have non-zero x -derivatives at the endpoints. Data from parton distributions supports this. The perception thatsoft wave function overlaps are incompatible with power-law dependence at large Q is in error.The result does not depend strongly on the exponential dependence or factored form of Eq.(7). For example a Gaussian dependence of Φ ∼ e − k x / Λ → e − x Q / Λ uses ∂ / ∂ x integrationby parts. Pursuing that, the region x Q (cid:46) Λ produces just the same power expansion as theregion xQ (cid:46) Λ with different constants. It is also always possible to write Φ π ( x , k x ) = e − k x / Λ ˜ φ ( x , k x ) .This simply defines ˜ φ ( x , k x ) as a function with an exponential dependence removed, namely afunction that varies more slowly. Integration by parts proceeds as before, with ˜ φ ( n ) replacing φ ( n ) . The result is a series in powers of 1/ Q which is qualitatively unchanged so long as ˜ φ isslowly varying. Phenomenological quark models often violate current conservation. Projecting a γ µ vertex with g µν − q µ q ν / q hides the problem without actually curing it. The endpoint overlap model withmassless quarks satisfies electromagnetic gauge invariance automatically, passing the test < P (cid:48) | q µ J µ em | P > =
0. This is a very detailed topic, explaining why we postponed details of the Diracalgebra in Section 2.1.Let (cid:96) be the difference of quark momenta in one pion. The most general wave function for J P = − → f ermion , anti − f ermion has Dirac structure ψ = A γ / P + B γ [ / P / (cid:96) − / (cid:96) / P ] + C γ + D γ / (cid:96) .By permuting γ through the wave function, the terms with one gamma (“chirally even”) haveantiparallel helicity and are antisymmetric in spin quantum numbers, and vice versa for chirally-odd. The A , B terms are proportional to the large number P . These track the Fermion fieldswhich are the largest under a Lorentz boost, making them leading order in energy. Thus A isleading in power of Q , with zero orbital angular momentum while B is also leading in Q , whilerepresenting one unit along the momentum axis. (Power counting in SD models is different.Only the A wave function is large as (cid:96) →
0. ) The C , D terms are sub-leading on every basis. . Dagaonkar et al. – Uncovering the Scaling Laws . . . γ µ ) . Chiral selectionrules give zero unless even-even and odd-odd chirality wave functions are composed. Thereare six non-zero combinations denoted AA (cid:48) , BB (cid:48) , CC (cid:48) , DD (cid:48) , AD (cid:48) , BC (cid:48) , plus their complex con-jugates. Our demonstration of gauge invariance comes from contracting q µ with the vertex andcomputing the terms one by one.The final result is zero when evaluated using the kinematic conditions of Eq. 4. The resultis not obvious from manipulating perturbative Ward identities, and the full calculation is quiteextensive. The zero-result requires massless quark propagators, namely chirality conservation, aswe assume. Apparently a feature of the global chiral symmetry of the model protects the gaugesymmetry of the model. We would like to understand this better, while the issue goes beyondthe scope of this paper. Experimentally very little is known about the actual large Q dependence of F π ( Q ) . In thespace-like region of momentum transfer Q F π ( Q ) rises and appears to approach a plateau inthe region of Q ∼ f ew GeV [39–41]. That represents a serious problem for SD models, because Q is far too small for the asymptotic regime to set in. The attempt to interpret the flaw as a“bonus,” namely a confirmation of the model’s predictions despite contradicting the model’srequirements, does not stand up to careful examination. π πγ∗, ρ... H π πγ∗ = π π Figure 3: The short distance model (left panel) uses approximate pion wave functions developed from sum-ming a subset of diagrams evaluated at the pion pole and with pion quantum numbers. Thephysics of a t -channel ρ -meson resonance is absent. The more complete integration regions of theendpoint model (right side) can represent the resonance. It is a serious matter that existing data is well fit by vector meson dominance, associatedwith the pole at time-like Q of the ρ meson [42–44]. The importance of the ρ is a problem for SD models. The models [4, 45] are based on expanding the wave function of the pion aboutzero quark separation. That expansion contains no terms capable of representing the ρ meson’ssingularity: See Figure 3. The asymptotic contribution of the short distance model [4] is F π SD = . Dagaonkar et al. – Uncovering the Scaling Laws . . . const ./ ( Q log ( Q / Λ QCD ) . The normalization const = π F π /9 with F π =93 MeV is a test ofthe model. The normalization is too small compared to the experimental data. More ambitiouswork exploiting self-consistent Bethe-Salpeter wave functions is more convincing [46–48]. Yetnone of the debate about asymptotic limits matters for our approach, which is based on extractinginformation from what is observable at laboratory energies. We mentioned in the introduction that folklore about exponentially-dependent form factors is inerror. Analysis produces a sum rule for the endpoint-overlap model using Mellin transforms.Represent Φ ( k , x ) = π i (cid:90) i ∞ − i ∞ dN Φ N ( x ) k − N , where Φ N ( x ) = (cid:90) ∞ dk k N − Φ ( k , x ) .Positivity of Φ gives one constraint. The integral over N is done over a contour in a strip where Φ N is analytic. Then F π = π i (cid:90) i ∞ − i ∞ dN (cid:90) ∞ dx ( xQ /2 ) − N Φ N ( x )= π i (cid:90) i ∞ − i ∞ dN N Q − N Φ N , 1 − N .The right hand side is the Mellin transform of F π with respect to Q : F N = N Φ N , 1 − N ,where F N = (cid:90) ∞ dQ Q N − F π ( Q ) .The formula pinpoints what knowledge of F determines about Φ . The large Q dependenceis determined by singularities at N >
0. These come from any N k > k dependence, and any N x < x dependence. Thus: • If Φ falls exponentially with increasing k then Φ N ( x ) is analytic for N >
0. All large Q singularities are found from the x dependence. If F π ∼ Q , then Φ ( k ∼ x ) ∼ x ,modulo logarithmic factors. Term by term the x dependence determines the powers andterms in the asymptotic series in Q , while the k dependence contributes to the coefficients. • Suppose Φ has a power-law tail at large k . If Φ ∼ k − modulo logarithms, then Φ N ( x ) will have a pole at N =
2, producing Q − dependence. Term by term the k dependencedetermines the powers and terms in the asymptotic series in Q , while the x dependencecontributes to the coefficients, provided Φ is not as large as x • The combination of both x and k − can produce a double pole, which translates to possible Q − log ( Q ) dependence of F π .The last two options have been extensively explored by the short-distance models. By calculatingthe large k dependence in the first step, they select in advance one of the regions the analysis showsare possible in general. The perturbative calculations also agree with the general analysis by . Dagaonkar et al. – Uncovering the Scaling Laws . . . Q − / log ( Q ) behavior from wave functions with corresponding power law behavior.After many years and a considerable investment in manpower, the same calculations have beenfound not to apply at finite Q , and also to produce contributions that are too small to explainexperimental data [8]. That indicates the endpoint region as being dominant at large Q .That leaves the first option. Experimentally Q F π ( Q ) ∼ GeV , which leads to Φ − = GeV ,with the leading singularities determined by Φ ( x ) ∼ x . This contradicts the perturbative SD model wave function (sometimes inappropriately called the “asymptotic” wave function). Thereare no compelling arguments to calculate the x -dependence of a wave function with perturbationtheory. Ref. [15] fits a numerical calculation to the equivalent of Φ ( x ) ∼ x , which seems to bereasonably consistent. Summarizing this section, under the universal assumption that Φ ( x , k x ) falls rapidly to constrain x (cid:46) Q and φ ( x ) ∼ x A , the cost of overlapping to retain one spectator is a “slow quark proba-bility” factor of 1/ Q A + . F : The Probability of TwoSlow Quarks While our objective focuses on power-counting, we believe value is added by including consid-erable details in the calculation. Here we compute the proton electromagnetic form factor, F ,assuming end point domination. There have been several earlier calculations of this form factorin different models [49–59]. qP P'k k k k' k' k' Figure 4: The proton form factor
The basic diagram for calculating proton electromagnetic form factors is shown in Fig. 4.The momenta P , P (cid:48) , Q are the same as in Eq.(2), and quark momenta use the same notation asEq. 3. Let Y be the proton wave function to three quarks, and let the electromagnetic vertex be . Dagaonkar et al. – Uncovering the Scaling Laws . . . Γ µ = − ie γ µ δ ( k + Q − k (cid:48) ) δ ( k − k (cid:48) ) δ ( k − k (cid:48) ) . The matrix element for the process is (cid:90) ∏ i d k i ( π ) d k (cid:48) i ( π ) ( Y (cid:48) ( k (cid:48) i ) × Γ µ × Y ( k i )) = − ieF ( Q )( N (cid:48) γ µ N ) + F ( Q )( N (cid:48) i σ µν q ν N ) , (10)where N , N (cid:48) are Dirac spinor functions. The momentum space wave functions with leadingpower of P is written as [60, 61] Y αβγ ( k i , P ) = f N √ N c { ( / PC ) αβ ( γ N ) γ V + ( / P γ C ) αβ N γ A + i ( σ µν P ν C ) αβ ( γ µ γ N ) γ T } . (11)Here α , β , γ are Dirac indices, V , A , T are scalar functions of the quark momenta( k i ), N c is thenumber of colors, C the charge conjugation operator, σ µν = i [ γ µ , γ ν ] , and f N is a normalization. PP’ k k’ q Figure 5: Endpoint kinematics in the proton case. Proton momenta are shown as dashed lines, quark mo-menta as solid lines. Isosceles triangles representing energy and momentum conservation closeas in Figure 2. The momenta of all spectator constituents sum to cover the differences P − k and P (cid:48) − k (cid:48) . Several combinations appearing in Y (cid:48) Γ µ Y give leading order contributions to F . Consider,for example, ( N (cid:48) γ γ ρ (cid:48) ) γ (cid:48) γ µγ (cid:48) γ ( γ ρ γ N ) γ = ( N (cid:48) γ µ N ) g ρ (cid:48) ρ + . . . (12)Collecting all coefficients proportional to this term written gives − ( C − / P (cid:48) ) αβ ( / PC ) αβ V (cid:48) V + ( C − γ / P (cid:48) ) αβ ( / P γ C ) αβ A (cid:48) A − g ab ( C − σ ν a P (cid:48) ν ) αβ ( σ b ν P ν C ) αβ T (cid:48) T (13)Inserting Eq. 13 in Eq. 10 gives F ( Q ) ∼ (cid:90) [ dxdk T ][ dx (cid:48) dk (cid:48) T ] Ψ tot δ ( k + q − k (cid:48) ) δ ( k − k (cid:48) ) δ ( P + q − P (cid:48) ) (14) . Dagaonkar et al. – Uncovering the Scaling Laws . . . Ψ tot = (cid:18) f N √ N c (cid:19) (cid:18) Q (cid:19) {− V (cid:48) V − A (cid:48) A + T (cid:48) T } and [ dxdk T ] = dx d (cid:126) k dx d (cid:126) k dx d (cid:126) k δ ( x + x + x − ) δ ( (cid:126) k + (cid:126) k + (cid:126) k ) (15)The delta functions lead to kinematics similar to Eq.(4). Momentum conservation requires k µ = k (cid:48) µ + q µ and k = k (cid:48) . One transverse momentum of the struck quark is unconstrained except bywave functions, while the other transverse momentum is constrained by the relation previouslyfound for the pion: − x Q /2 − k x + Q = x (cid:48) Q /2 + k (cid:48) x ; (16) k x = ( − x ) Q = ( x + x ) Q [ dxdk T ][ dx (cid:48) dk (cid:48) T ] → dx dx dk y dk y Q . (17)Integrating over the unconstrained variables gives (cid:90) dk y dk y Ψ tot = Φ P ( ˜ k x ( x ) , x , x ) .That leaves the integration depending on Q as F = (cid:90) dx dx Φ P ( k = ( − x ) Q /2, x , x ) .Once more consider an exponential ansatz V (cid:48) , A (cid:48) , T (cid:48) ∝ ( − x (cid:48) ) x (cid:48) ψ ( (cid:126) k (cid:48) T ) V , A , T ∝ ( − x ) x ψ ( (cid:126) k T ) (18)where ψ ( (cid:126) k T ) ∼ exp (cid:20) − (cid:16) (cid:126) k T (cid:17) / Λ (cid:21) . That leads to Φ P ( k , x , x ) ∼ e − k x / Λ φ ( x , x ) .Evaluated at k x = ( x + x ) Q /2, both x and x range over intervals of size Λ / Q . For wavefunctions that are uncorrelated products, ψ = ψ ( x ) ψ ( x ) , the probability of finding two slowquarks is precisely the product of two slow quark probabilities. For quark wave functions goinglike x A ( − x ) A we find Φ P ∼ x A x A + . . . ; F ∼ Q A .Experimental data finds that Q F ∼ constant for Q ≥ GeV . The data indicates that A ∼ . Dagaonkar et al. – Uncovering the Scaling Laws . . . x i near x i ∼
0. We emphasize that this result doesnot require extremely large Q (large logarithms of Q ). The estimates are based on comparisonwith the transverse size of the proton, Q >> Λ for Λ ∼ GeV . The approach based on short-distance perturbation theory typically asks how these relationsbehave when “soft gluons” are added.The notion of adding soft gluons is tied to a basis of Fock state wave functions used in pertur-bation theory. Since the interacting theory is not the free theory, Feynman diagrams represent theinteractions with gluons of all momenta. That does not represent our model: By construction, weare concerned with the full wave functions of the interacting theory. Thus the calculation usingthe fully interacting wave functions is self-consistent without adding gluons.The question of soft gluons is vital for the internal consistency of the SD model and its (con-ceptually different) estimate of the endpoint contribution. Assuming one desires a perturbativedescription, experience indicates that any finite number of soft gluon internal diagrams willnot revise the leading power of Q . Considerable effort has gone into arranging calculationsthat would be simultaneously compatible with the assumptions of short distance. That has ledto statements that Sudakov effects suppress the endpoint region. These statements refer to theshort-distance model, not ours. If it is true that the endpoint region of the SD model is negligible,it has no bearing on the endpoint region of all possible models expressed in different quantum-mechanical bases. None of it is our concern once the focus is on extracting non-perturbativeinformation from experiments.Nevertheless it is interesting to check that soft-gluons do not change the leading power be-havior. We feel that a specific calculation is more convincing than an estimate, and present onein the Appendix. F We have investigated the large Q dependence of the Pauli form factor F in the endpoint model.We find 1/ Q dependence occurs in more than one way, together with wave functions that go like x ( − x ) . This result is surprising and impinges directly on the issue of quark orbital angularmomentum [62] and “the shape of the proton” [63, 64]. Yet the calculations we have available arecomplicated, and too detailed to be appropriate to review here. We plan to present them in anfuture paper [65]. For the purposes of this survey, we can objectively report that it has not beenshown that experimental data [34] measuring F at large Q is in conflict with power-counting ofthe endpoint overlap model. The lack of previous work is itself remarkable because it has beenshown that F ∼ Q is incompatible with short distance models. We are now in a position to state the leading power endpoint constituent counting rules forform factors. Let there be n IN ( n OUT ) quarks in the
I N ( OUT ) state hadron. For now choose n IN = n OUT = n . One constituent is scattered, requiring n − F ( n quarks ) = (cid:90) dx dx ... dx n − Φ P ( ˜ k = n − ∑ j = x j Q /2, n − ∑ j = x j ) . . Dagaonkar et al. – Uncovering the Scaling Laws . . . x j →
0, each extra constituent beyond the valenceconfiguration causes a suppression of the form factor by a power of 1/ Q . The leading powerdependence is F ( n quarks ) ∼ ( Q ) n − .These happen to be the same rules as the early “dimensional” counting rules [1], but for en-tirely different reasons. Hard propagator factors are not the explanation. The explanation lies inthe phase space to find quarks available to scatter. The general dominance of phase space overhard scattering is reminiscent of the independent scattering mechanism originally discovered byLandshoff [66, 67].We will be straightforward with what is new in the power law. It has been noticed again andagain that the endpoint contribution cropped up and competed with the short-distance model ofform factors. Many papers have dealt with the issue as a troublesome instability of short-distancedominance. Yet we are not aware of an explicit, positive statement of the predictive regularitybetween the number of scattered constituents and the observed power laws. We cannot explainwhy the universal potential of endpoint overlap models has not been not widely recognized. In this Section we find that the experimentally observed power laws of hadron-hadron exclusivereactions [5, 10, 68–80] are explained by the endpoint overlap model. Unlike the SD model, noapproximations of an asymptotic character are needed. The approximations assume only that Q >> Λ . We are not aware of a previous focused effort to study the contributions of the endpoint overlapmodel in 2 → SD and endpointmodels are expedited by a simple observation. The amplitude of almost all contributions scaleslike the combination of form factor amplitudes. This observation is quite old [81], and developedfor a different purpose, yet it is rather general. If there is another contribution with a qualitativelydifferent behavior, its momentum flow will go by a qualitatively different topology.In the high energy limit the differential cross section for 2 → M isgiven by d σ dt = const . MM ∗ s .The composition of two form factors F A F B with a 1/ Q exchange kernel scales like F A F B Q / Q .The numerator factor of Q accounts for the vector vertex factors not contained in F i .There is one significant difference between models, however. Multiple gluon exchanges inthe SD model have no strong selection rules from the color singlet nature of hadrons. Scatter-ing a single constituent in the endpoint overlap model requires at least two gluons in a singletcombination. It is well known that the box diagram of two gluon exchanges scales with justthe same power of Q as a single gluon exchange, times logarithmic factors that are exactlycomputable [82]. For our purposes multi-gluon exchanges are indistinguishable, and at mostlaboratory momentum transfers, probably necessary. . Dagaonkar et al. – Uncovering the Scaling Laws . . . fixed angle kinematics s ∼ t , the endpoint overlap model composing formfactors predicts: • For ππ → ππ , M ∼ Q ∼ s ; d σ dt ∼ s . • For π p → π p , M ∼ Q ∼ s ; d σ dt ∼ s .For pp → pp , M ∼ Q ∼ s ; d σ dt ∼ s . • For 2 → n and n valence constituents, M ∼ Q n + n + ∼ s ( n + n ) /2 + ; d σ dt ∼ s n + n + .Agreement with experiment [5, 10, 74–79, 83, 84] are explained by the endpoint overlap model.Unlike the SD model, no approximations of an asymptotic character are needed. The approxi-mations assume only that Q >> Λ , adds support to the valence state of the pion having twoconstituents, and the proton having three. As before, scattering constituents beyond the valencecomponents is suppressed by powers of 1/ s .The counting is different for the t dependence of amplitudes at fixed s >> GeV . In that case MM ∗ / s ∼ MM ∗ . With | t | << s the leading dependence replaces s → t , and multiplies theresults above by t . By far the most important example comes from pp → pp scattering, whichdisplays a stunning experimental dependence falling like t − [66, 85–87]: exactly the endpoint-overlap contribution. We mentioned that a qualitatively different momentum flow could change the counting. The in-dependent scattering model [66] is usually highlighted to explain the t − dependence. Landshoffhad earlier found the model by not making the same assumptions of the model of Brodsky andFarrar. The independent scattering ( IS ) model gets its power law partly from the phase space offast quarks with x ∼ t . In compar-ison the endpoint overlap contribution uses one (1) hard vector exchange, while obtaining thesame powers of 1/ t from the probability to find two quarks near the endpoint.The phenomenology of complex phases and spin dependence are very similar for the IS andendpoint models. When treated in Fock-basis perturbation theory both model have similar Su- . Dagaonkar et al. – Uncovering the Scaling Laws . . . P P' =P +ql l' P P' =P -qk k' k -rl +r Figure 6: Proton-proton elastic scattering with a minimal hard-scattering kernel. dakov factors. Such factors may well explain the oscillations seen in pp fixed angle scatteringand color transparency. Based on the results of Mueller [88], we conjecture that saddle pointinterpolation between the endpoint model and the SD model will be the dominant asymptoticamplitude. This is because the Sudakov suppression of one fast quark is less severe than thethree fast quarks of the IS model. As with the form factors, we believe that supporting calculations are at least as important asgeneral arguments.Consider proton-proton scattering of Fig.6, p ( P ) + p ( P ) → p ( P (cid:48) ) + p ( P (cid:48) ) in the limit s =( P + P ) ∼ t = ( P (cid:48) − P ) = q = − Q . In the center of mass frame the momenta of the twoincoming particles are P =( p , 0, 0, − p ) , P =( p , 0, 0, p ) .The amplitude for the scattering diagram is given by M ∝ (cid:18) − i g s (cid:19) (cid:90) d r ( π ) − i g µ ν ( q − r ) − i g µ ν r (cid:90) ∏ i d k i ( π ) d k (cid:48) i ( π ) Y α (cid:48) β (cid:48) γ (cid:48) ( k (cid:48) i , P (cid:48) ) (cid:20) γ µ / k − / r + m ( k − r ) − m + i (cid:101) γ µ (cid:21) γ (cid:48) γ δ α (cid:48) α δ β (cid:48) β Y αβγ ( k i , P ) (cid:90) ∏ j d l j ( π ) d l (cid:48) j ( π ) Y α (cid:48) β (cid:48) γ (cid:48) ( l (cid:48) i , P (cid:48) ) (cid:20) γ ν / l + / r + m ( l + r ) − m + i (cid:101) γ ν (cid:21) γ (cid:48) γ δ α (cid:48) α δ β (cid:48) β Y αβγ ( l i , P ) (19)As with the case of proton form factor, the interaction vertex will have delta functions enforc-ing the conservation of momentum in the quark interactions, which are implicit in the aboveexpression.Extract the integral over the free momentum r , given by (cid:90) d r ( π ) (cid:18) γ µ / k − / r + m ( k − r ) − m + i (cid:101) γ µ (cid:19) ( q − r ) r (cid:18) γ ν / l + / r + m ( l + r ) − m + i (cid:101) γ ν (cid:19) (20)Integration is performed using Feynman parametrization. Simplifying the denominator using . Dagaonkar et al. – Uncovering the Scaling Laws . . . ( a i ) leads to a denominator D given by, D = ( l − ∆ + i (cid:101) ) (21)where l = r − a k − a q + a l ; ∆ = [ a k + a q − a l ] − a k − a q − a l + a m − a m + i (cid:101) .We may neglect terms of the form k , l assuming the quarks are nearly light like. Terms ofthe form k · q , l · q , k · l are of the same order as Q , assuming t ∼ s . Thus the dominantcontribution in the ∆ goes like Q .Terms in the numerator of the form l µ k ν , l µ q ν , l µ l ν . . . vanish upon integration. The otherterms can be integrated using the standard substitution l µ l ν term ⇒ g µν ∆ ∝ g µν Q (22)In comparison, the other terms in the numerator scale like 1/ Q . To leading power we keep the l µ l ν term. The amplitude is given by − (cid:16) g s (cid:17) Q (cid:90) [ da i ] × (cid:90) ∏ i d k i ( π ) d k (cid:48) i ( π ) Y α (cid:48) β (cid:48) γ (cid:48) ( k (cid:48) i , P (cid:48) )[ γ µ γ µ γ µ ] γ (cid:48) γ δ α (cid:48) α δ β (cid:48) β Y αβγ ( k i , P ) (23) × (cid:90) ∏ j d l j ( π ) d l (cid:48) j ( π ) Y α (cid:48) β (cid:48) γ (cid:48) ( l (cid:48) i , P (cid:48) )[ γ µ γ µ γ µ ] γ (cid:48) γ δ α (cid:48) α δ β (cid:48) β Y αβγ ( l i , P ) .Here [ da i ] = ∏ i da i δ ( ∑ j a j − ) .The calculation can be simplified by a Lorentz transformation to a frame where the momenta ofthe protons becomes equivalent to the momenta of Eq. (2). For example k + k + k = P = ( p , 0, 0, − p ) Lorentz transform −−−−−−→ k L + k L + k L = P L = ( Q / √ − Q /2, 0, Q /2 ) k (cid:48) + k (cid:48) + k (cid:48) = P + q =( p − q , − q , − q , − p − q ) Lorentz transform −−−−−−→ k (cid:48) L + k (cid:48) L + k (cid:48) L = ( P − q ) L = ( Q / √ + Q /2, 0, Q /2 ) Such a transformation will allow the use of results of the proton form factor calculation.Substituting the wave function from Eq. (11), a single term from the wave function is suffi-cient to understand the behavior of this integral. We illustrate the term M V going like ( / PC ) αβ ( γ N ) γ V , . Dagaonkar et al. – Uncovering the Scaling Laws . . . M V =( − g s ) Q (cid:90) ∏ i [ da i ] (cid:90) [ dxdk T ][ dx (cid:48) dk (cid:48) T ]( / P C ) αβ ( C − ( / P − / Q )) αβ [ N P − Q γ µ γ µ γ µ N P ] V ( k i , P ) V (cid:48) ( k (cid:48) i , P − Q ) × (cid:90) [ dydl T ][ dy (cid:48) dl (cid:48) T ]( / P C ) αβ ( C − ( / P + / Q )) αβ [ N P + Q γ µ γ µ γ µ N P ] V ( l i , P ) V (cid:48) ( l (cid:48) i , P + Q ) (24)From the calculations following Eq.(14), the integrations become ( / P C ) αβ ( C − ( / P − / Q )) αβ (cid:90) [ dxdk T ][ dx (cid:48) dk (cid:48) T ] V ( k i , P ) V (cid:48) ( k (cid:48) i , P − Q ) ∝ ( Q ) ( / P C ) αβ ( C − ( / P + / Q )) αβ (cid:90) [ dydl T ][ dy (cid:48) dl (cid:48) T ] V ( l i , P ) V ( l (cid:48) i , P + Q ) ∝ ( Q ) That implies M ∝ − (cid:16) g s (cid:17) (cid:18) Q (cid:19) (cid:90) { da i } ( Q ) [ N P − Q γ µ γ µ γ µ N P ] ( Q ) [ N P + Q γ µ γ µ γ µ N P ] Calculate the cross section using d σ dt ∼ |M| s ∝ s (cid:18) ( Q ) (cid:19) (cid:18) Q (cid:19) Tr [( / P − / Q ) γ µ γ µ γ µ / P γ ν γ ν γ ν ] Tr [( / P + / Q ) γ µ γ µ γ µ / P γ ν γ ν γ ν ] The leading term from simplifying the trace goes like p . Using s ∼ t we find d σ dt ∝ s ( Q ) Q p ∝ s Many other terms give a similar s dependence. We have shown that the endpoint overlap model stands as a comprehensive theory of hadronicreactions at large momentum transfer. It explains the observed experimental regularities in allcases we have investigated. The history of endpoint dominance is curious, and possibly explainswhy the model failed to be completely developed.In 1970 Drell and Yan [12], and later West [13] (
DYW ) discussed a partonic model connectinghadronic form factors to deeply inelastic scattering. Using symbol η for the parton momentumfraction since called x , the central region 1 − η ≥ Λ / Q was found to predict a form factor fallingtoo fast to agree with data. From this region F ( Q ) ∼ g ( Q ) / Q , with g ( Q ) ∼ exp ( − Q / Λ ) is expected. In comparison, the endpoint region 1 − η ≤ Λ / Q was observed to predict F ( Q ) ∼ ( Q ) ( p + ) /2 , if the structure function ν W ∼ ( − η ) p . The value p = two constituent calculation. The value p = p ≥
2. While the two components of the toy model were . Dagaonkar et al. – Uncovering the Scaling Laws . . .
19a pion and a nucleon, the 1970 calculation in our view constitutes the prototype “constituentcounting” relation connecting the scaling power with the number of constituents. The paperappears approximately five years before the papers of Brodsky and Farrar [1] and Matveev etal. [2], which found counting rules on a different basis.Subsequently many workers noticed that a calculation of the endpoint contribution wouldbe revised by a power of 1/ Q for each additional constituent added to a given process. Manyworkers also concluded that the endpoint contribution might be dominating the calculation oftheir particular process. Yet the endpoint region never saw anything like the degree of develop-ment of the short-distance perturbative model. For reasons we cannot explain, we cannot finda reference strongly advocating for the endpoint region, and developing it as a “comprehensivetheoretical picture” that explains the observed power law dependence.There exists a possible explanation coming from the drive of the early era. That time wasconcerned with testing the Lagrangian of QCD, and without needing to know wave functions.Exclusive reactions were hardly a good testing ground. Imagine trying to test perturbative quan-tum electrodynamics in a world where the Hydrogen atom bound states had not been solved. Insuch a Universe, calculations actually depending on unknown wave functions would be “bad.”That is, they would be bad for establishing a Lagrangian by perturbation theory. Calculations forHydrogen-Hydrogen scattering not depending on unknown wave functions would be very dif-ficult to concoct. However, once any scheme self-consistent with perturbation theory was found,it would be “good” whether or not it was incomplete and “wrong.”We believe that the lure of a strictly perturbative procedure caused a false perception that“correct physics” could only depend on operator product expansion moments of wave functions.The operator product expansion is an ansatz of great power when it applies, while creating hugegaps when it does not, but this was not obvious right away. Once the attitude was adopted, theopportunity to use data to learn about wave functions was rejected. As a result, the opportunityto actively use data to learn about hadron structure remains a relatively unexplored field. It hastaken 30 years of more and more detailed calculations to find that the tiny integration region ofthe short-distance model can at most be relevant for extremely large Q which need momentumtransfers that are tens, hundreds, or thousands of times larger than laboratory scales to be rele-vant. For emphasis, we do not know of a single calculation that strongly supports the numericaldominance of the short-distance region, relentless advocacy notwithstanding. Indeed the firststep towards arranging for short distance dominance has been to banish the endpoint region asperturbatively inconsistent, which (we maintain) is a signal of a concept error .Through the entire period the endpoint contribution has never gone away. The time has cometo accept endpoint contribution, and explore it further. Acknowledgments:
Some of this work was initiated during a University of Washington-INT workshop. We thank G. A. Miller and Toly Radyushkin for discussions. We also thankLeonard Gamberg, Ron Gilman, Simonetta Liuti, Zein Eddine Meziani, Charles Perdrisat andOscar Rondon for insights. We also thank Dipankar Chakrabarti for collaborating at the initialstages of this work.
Soft gluon effects are an intrinsic difficulty of the SD models. When the effects of perturbationtheory produce large corrections they indicate that the first approximations were not dynamicallystable. We mentioned that soft gluon effects are not intrinsically present in the non-perturbative . Dagaonkar et al. – Uncovering the Scaling Laws . . . qP P'k k k k' k' k' p f2 p g2 p f1 p g1 Figure 7: A 2 gluon exchange contribution to the proton form factor quantum mechanical basis we use. Whether or not they are added they do not change the powerof Q . We demonstrate this explicitly here by considering a particular two gluon exchange dia-gram.Consider, as an example, the amplitude shown in Fig. 7. The hard scattering contributions ofsuch diagrams have been analyzed in [56]. We will extract the | Q | dependence of the amplitudein the endpoint region. The momenta of the virtual fermions and gluons are p g = k (cid:48) − k , p f = k (cid:48) + k (cid:48) − k , p g = P (cid:48) − P − k (cid:48) + k , p f = P − P (cid:48) + k (cid:48) . Considering only the endpointregion, it is understood that the momenta transferred to the spectator fermions is soft, hence thegluons and the spectator fermions both have low momenta. One of the terms in the amplitude is A = (cid:90) endpoint [ dxdk T ][ dx (cid:48) dk (cid:48) T ]( N (cid:48) γ ) γ (cid:48) ( C − / P (cid:48) ) α (cid:48) β (cid:48) Ψ (cid:48) ( k (cid:48) i ) (cid:34) γ µ ( / p f + m ) γ ρ p f − m (cid:35) γ (cid:48) γ (cid:34) γ λ ( / p f + m ) γ ρ ( p f − m ) (cid:35) α (cid:48) α × ( γ λ ) β (cid:48) β × p g × p g × ( / PC ) αβ ( γ N ) γ Ψ ( k i ) ∼ (cid:90) endpoint [ dxdk T ][ dx (cid:48) dk (cid:48) T ] (cid:34) N (cid:48) γ γ µ / p f + mp f − m γ ρ γ N (cid:35) (cid:34) Tr [( C − / P (cid:48) ) T ( γ λ ( / p f + m ) γ ρ )( / PC ) γ T λ ] p g p g ( p f − m ) (cid:35) × Ψ ( k i ) Ψ (cid:48) ( l i ) (25)The denominators have the form, ( p f − m ) ∝ ( − x (cid:48) ) x Q + ( (cid:126) k (cid:48) T + (cid:126) k T ) + m ( p f − m ) ∝ ( − x (cid:48) ) Q + (cid:126) k (cid:48) T + m (26)The denominator for the soft fermion, which has the momentum p f , has a ( − x (cid:48) ) x Q termwhich is suppressed in the endpoint region. The gluon denominators have similar behavior,and these terms do not give a Q dependence in the denominator. The Q dependence comesfrom the hard fermion ( p f ) denominator which is proportional to ( − x (cid:48) ) Q . The dominant Qdependence in the numerator is of the form Q × Q × ( − x (cid:48) ) Q × Q . Evaluating this term in theendpoint region using our wave function we obtain A ∼ (cid:90) endpoint [ dxdk T ][ dx (cid:48) dk (cid:48) T ][ N (cid:48) γ µ N ] ( − x (cid:48) ) Q ( − x (cid:48) ) Q x (cid:48) ( − x (cid:48) ) x ( − x ) ∝ [ N (cid:48) γ µ N ] × Q (27) . Dagaonkar et al. – Uncovering the Scaling Laws . . . References [1] S. J. Brodsky and G. R. Farrar, Phys. Rev. D 11, 1309 (1975).[2] V. A. Matveev, R.M. Muradian and A.N. Tavkhelidze, Lett. Nuovo Cim. , 719 (1973).[3] D. Sivers, S. J. Brodsky and R. Blankenbecler, Phys. Rep. 23, 1 (1976).[4] G. R. Farrar and D. R. Jackson, Phys. Rev. Lett.
3, 246 (1979).[5] S. J. Brodsky and G. P. Lepage, Phys. Rev. D 22, 2257 (1980).[6] A. V. Efremov and A. V. Radyushkin Theoretical and Mathematical Physics 42, 97 (1980).[7] A. V. Efremov and A. V. Radyushkin Phys. Lett. B94, 245 (1980).[8] N. Isgur and C. Llewelyn-Smith, Phys. Rev. Lett. 52, 1080, (1984)[9] L. Andivahis et al , Phys. Rev. D 50, 5491 (1994).[10] P. V. Landshoff and J. C. Polkinghorne, Phys. Lett. B44, 293 (1973).[11] R. P. Feynman, Phys. Rev. Lett. 23, 1415 (1969).[12] S. D. Drell and T.-M. Yan, Phys. Rev. Lett. 24, 181 (1970).[13] G. B. West, Phys. Rev. Lett. 24, 1206 (1970).[14] A. H. Mueller, ”Perturbative Quantum Chromodynamics”, World Scientific, Singapore (1989).[15] L. Chang, I. C. Clo´et, C. D. Roberts, S. M. Schmidt and P. C. Tandy, Phys. Rev. Lett. 111, 141802(2013).[16] L. Chang, I. C. Clo´et, J. J. Cobos-Martinez, C. D. Roberts, S. M. Schmidt and P. C. Tandy, Phys. Rev.Lett. 110, 132001 (2013).[17] S. J. Brodsky and G. P. Lepage, Phys. Rev. D 24, 2848 (1981).[18] G. Preparata and J. Soffer, Phys. Lett. B 86, 304 (1979).[19] S. J. Brodsky, C. C. Carlson and H. J. Lipkin, Phys. Rev. D 20, 2278 (1979).[20] T. Gousset and B. Pire, In *Cambridge 1995, Confinement physics* 111-143 [hep-ph/9511274]; T.Gousset, B. Pire and J. P. Ralston, “Hadron Helicity Violation in Exclusive Processes: QuantitativeCalculations in Leading Order QCD,”
Physical Review D ,
3, 1202 (1996).[21] J. R. O’Fallon et al. , Phys. Rev. Lett. 39, 733 (1977).[22] A. Lin et al. , Phys. Lett. B 74, 273 (1978).[23] S. L. Lin et al. , Phys. Rev. D 26, 550 (1982).[24] D. G. Aschman et al. , Nucl. Phys. B 125, 349 (1977).[25] K. Abe et al. , Phys. Lett. B 63, 239 (1976).[26] J. Antille et al. , Nucl. Phys. B 185, 1 (1981).[27] P. H. Hansen et al. , Phys. Rev. Lett. 50, 802 (1983).[28] D. G. Crabb et al. , Phys. Rev. Lett. 41, 1257 (1978).[29] E. A. Crosbie et al. , Phys. Rev. D 23, 600 (1981).[30] D. C. Peaslee et al. , Phys. Rev. Lett. 51, 2359 (1983).[31] F. Z. Khiari et al. , Phys. Rev. D 39, 45 (1989).[32] A. Wijesoriya et al , Phys. Rev. Lett. 86, 2975 (2001).[33] R. Gilman and F. Gross, J. Phys. G 28, R37 (2002).[34] M. K. Jones et al , Phys. Rev. Lett. 84, 1398 (2000).[35] E. P. Biernat, F. Gross, M. T. Pena and A. Stadler, Phys. Rev. D 89, 016005 (2014).[36] E. P. Biernat, F. Gross, M. T. Pena and A. Stadler, Phys. Rev. D 89, 016006 (2014). . Dagaonkar et al. – Uncovering the Scaling Laws . . . [37] F. Gross, G. Ramalho and M. T. Pena, Phys. Rev. D 85, 093005[38] Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturba-tion Theory (v. 1) [Hardcover] Carl M. Bender (Author), Steven A. Orszag (Author)[39] J. Volmer et al. , Phys. Rev. Lett. 86, 1713 (2001).[40] T. Horn et al. , Phys. Rev. Lett., 97, 192001 (2006).[41] G. Huber et al. , Phys. Rev. C 78, 045203 (2008).[42] G.J. Gounaris and J.J. Sakurai, Phys. Rev. Lett. 21, 244 (1968).[43] H. B. O’Connell, B. C. Pearce, A. W. Thomas and A. G. Williams, Prog. Part. Nucl. Phys. 39, 201(1997).[44] J. F. Donoghue and E. S. Na, Phys. Rev. D
6, 7073 (1997) [hep-ph/9611418].[45] H.-N. Li and G. Sterman, Nucl. Phys. B 381, 129 (1992).[46] P. Jain and H. Munczek, Phys. Rev. D 48, 5403 (1993).[47] R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001).[48] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994).[49] I.G. Aznauryan, S.V. Esaybegyan, N.L. Ter-Isaakyan, Phys. Lett. B 90, 151 (1980).[50] A. Duncan and A. H. Mueller, Phys. Rev. D 21, 1636 (1980)[51] V. A. Avdeenko, V. L. Chernyak and S. A. Korenbilt, Yad. Fiz. 33, 481 (1981) [Sov. J. Nucl. Phys. 33,252 (1981)].[52] A. V. Radyushkin, Acta Phys. Polonica, B15, 403 (1984); A. P. Bakulev and A. V. Radyushkin, Phys.Lett. B 271, 223 (1991).[53] C. R. Ji, A. F. Sill and R. M. Lombard-Nelsen, Phys. Rev. D 36, 165 (1987).[54] C. E. Carlson and F. Gross, Phys. Rev. D 36, 2060 (1987).[55] R. F. Wagenbrunn, S. Boffi, W. Klink, W. Plessas and M. Radici, Phys. Lett. B 511, 33 (2001)[56] H.-N. Li Phys. Rev. D 48, 4243 (1993).[57] R. Jakob and P. Kroll, Phys. Lett. B 315, 463 (1993).[58] J. Bolz, R. Jakob, P. Kroll, M. Bergmann, and N. G. Stefanis, Z. Phys. C 66, 267 (1995).[59] B. Kundu, H.-N. Li, J. Samuelsson and P. Jain, hep-ph/9806419, Euro. Phys. Journal C 8, Vol 4, 637(1999).[60] V. L. Chernyak and A. R. Zitnitsky, Yad. Fiz. 31, 1053 (1980) [Sov. J. Nucl. Phys. 31, 544 (1980)]; Phys.Rep. 112, 173 (1984); Nucl. Phys. B216, 373 (1983); Nucl. Phys. B246, 52 (1984).[61] V. M. Belyaev and B. L. Ioffe, Zh. Eksp. Teor. Phys. 83, 876 (1982) [Sov. Phys. JETP 56, 493 (1982)].[62] P. Jain and J. P. Ralston, Phys. Rev. D 69, 053008 (2004).[63] G. A. Miller, Phys. Rev. C 68 022201 (2003).[64] F. Gross and P. Agbakpe, Phys. Rev. C73 015203 (2006).[65] S. Dagaonkar, P. Jain and J. P. Ralston, in preparation[66] P. V. Landshoff, Phys. Rev. D 10, 1024 (1974).[67] A. Donnachie and P. V. Landshoff, Nucl. Phys. B 231, 189 (1984).[68] R. Fiore, L. L. Jenkovszky, V. K. Magas and F. Paccanoni, Phys. Rev. D 60, 116003 (1999).[69] M. Quiros, Z. Phys. C 11, 179 (1981).[70] C. K. Chen, Phys. Rev. D 18, 3297 (1978).[71] M. M. Islam, Lett. Nuovo Cim. 14, 627 (1975).[72] P. M. Fishbane and I. J. Muzinich, Phys. Rev. D8, 4015 (1973).[73] P. D. B. Collins, F. D. Gault and A. D. Martin, Nucl. Phys. B 83, 241 (1974).[74] Z. Asad et al. , Nucl. Phys. B 255, 273 (1985). . Dagaonkar et al. – Uncovering the Scaling Laws . . . [75] J. L. Stone, J. P. Chanowski, H. R. Gustafson, and M. J. Longo, Nucl. Phys. B 143, 1 (1978).[76] C. W. Akerlof et al. , Phys. Rev. Lett. 17, 1105 (1966).[77] J. V. Allaby et al. Phys. Lett. B 25, 156 (1967); B 27, 49 (1968); B28, 67 (1968); B34, 431 (1971).[78] R. A. Carrigan et al. , Phys. Rev. Lett. 24, 683 (1970).[79] G. Cocconi et al. , Phys. Rev. 138, B165 (1965).[80] B. Pire and J. P. Ralston, Phys. Lett. B
17 (1982) 233.[81] T. T. Chou and C. -N. Yang, Phys. Rev.
70 (1968) 1591.[82] S. Papadopoulos, A. P. Contogouris and J. P. Ralston, “Calculation of Box Graph with LightlikeParticles,”
Phys. Rev. , D
25, 2218(1982).[83] K. A. Jenkins, L. E. Price, R. Klem, R. J. Miller, P. Schreiner, H. Courant, Y. I. Makdisi and M. L. Mar-shak et al. , Phys. Rev. Lett.
0, 425 (1978).[84] J. L. Stone, J. P. Chanowski and M. J. Longo, Phys. Rev. Lett.3