The Dirac Form Factor Predicts the Pauli Form Factor in the Endpoint Model

We compute the momentum-transfer dependence of the proton Pauli form factor $F_{2}$ in the endpoint overlap model. We find the model correctly reproduces the scaling of the ratio of $F_{2}$ with the Dirac Form factor $F_{1}$ observed at the Jefferson Laboratory. The calculation uses the leading-power, leading twist Dirac structure of the quark light-cone wave function, and the same endpoint dependence previously determined from the Dirac form factor $F_{1}$. There are no parameters and no adjustable functions in the endpoint model's prediction for $F_{2}$. The model's predicted ratio $F_{2}(Q^{2})/F_{1}(Q^{2})$ is quite insensitive to the endpoint wave function, which explains why the observed ratio scales like $1/Q$ down to rather low momentum transfers. The endpoint model appears to be the only comprehensive model consistent with all form factor information as well as reproducing fixed-angle proton-proton scattering at large momentum transfer. Any one of the processes is capable of predicting the others.


Introduction
The electromagnetic form factors know as F 1 and F 2 are an important probe of the internal structure of nucleons. A popular theoretical model assumes that at high momentum transfer these quantities can be factorized into a hard scattering contribution and a so-called distribution amplitude. The distribution amplitude has no information about the proton wave function except the parton momentum fraction Feynman-x dependence and some spin factors of a short-distance expansion. The focus of the short-distance (SD) model [1][2][3][4][5][6] is a perturbatively calculable hard scattering kernel. The model generates an order by order expansion in powers of the inverse momentum transfer-squared, 1/Q 2 . The expansion has often been claimed to be the unique prediction of QCD. However the task of comparing the model to the larger theory of QCD was never completed, and obviously cannot be explored within the SD model itself.
already have very small momenta. Their momenta are of the order of the QCD chiral symmetry breaking scale Λ. Then an equally small contribution from a quark mass is not a relatively small effect, and it cannot be neglected. Let us repeat that the attempt to banish small momentum regions from QCD never worked out. An unexpected consequence of small momenta appearing at leading power order is that mass effects can appear at the same order.
Another effect makes this even more interesting. Under a Lorentz transformation with rapidity y in the z direction, the big light cone + component transforms by e y and the small component like e −y . All previous calculations known to us at leading power order integrate quark wave functions over the small momentum in the first step. This appears to be much more safe than integrating over the transverse momentum components, which scale like 1 compared to e −y . Yet we have discovered a limit-interchange error occurs. Integrating away the small components is the first step of the SD model producing a visible factorization into separated hadronic parts. The assumption, actually a hope, that some factorization dominates is what demands that step. Yet that step instantly causes F 2 to scale no larger than 1/Q 6 . When the small momenta components are retained in the scattering process we find a contribution to F 2 scaling like 1/Q 5 . The integrals cannot be represented by effective, pre-integrated quantities that depend only on Feynman-x. This phenomenon contradicts the tenets of factorization. In the EP model, the leading power contribution to F 2 comes from an inseparable union of initial and final state proton states.
Finally all of this occurs with one simple wave function, which happens to be the most often cited, leading twist example. There is no particular reason to favor leading twist coming from a short distance expansion. There is every reason to use a wave function of leading power in the large momentum P . It is seldom noticed that the leading power, leading twist wave function has both chirally-even and chirally-odd components. A single wave function can both maintain the proton's chirality in F 1 , and flip the chirality in F 2 .
In section 2, we show that by respecting the necessary integration region, while using the endpoint dependence of the proton wave function obtained in [17], we obtain the experimentally observed scaling behavior for F 2 /F 1 . This is a remarkable prediction of the model: If attention had been given 30 years ago, it would have predicted F 2 in advance of the data. Reversing the argument, the observed scaling dependence of F 2 /F 1 predicts F 1 and pp scattering at high momentum transfer. None of these facts requires appealing to an unusually large logarithmic correction, or an unusually large dimensionful scale. As far as we know it is the first time that one model is actually consistent with the known data.
Quark orbital angular momentum is a topic of great interest. No orbital angular momentum (OAM ) enters the SD model, because a theoretical preference for factorization demands integrating over quark transverse momenta before the actual reaction has even been set up. Information about transverse size is lost by that step. When OAM is re-cast into a twist expansion [16] the sequence of operations dictated by the SD model produces a 1/Q 6 dependence for F 2 . References [18,19] showed that avoiding the SD assumptions and performing the transverse momentum integrations to compute F 2 led to power law dependence for F 2 intermediate between 1/Q 4 and 1/Q 6 . That is, the integration region assumed to dominate asymptotically was not the actually dominant region, whether or not an endpoint issue was considered. While the asymmetry of the endpoint integration regions produces a rather obvious role for OAM , that is not the focus of this paper. This paper is about using the same leading twist Dirac and endpoint structure found in the F 1 calculation to calculate F 2 . The calculation is relatively simple, and agrees remarkably with data. Even more remarkably, the ratio F 2 (Q 2 )/F 1 (Q 2 ) is quite insensitive to the endpoint wave function, explaining why the observed ratio goes like 1/Q down to rather small momentum transfer. Here we describe the calculation of F 2 through the quark mass contribution. One quark is struck by the virtual photon. The remaining quarks will be in a small momentum region, such that their incoming and outgoing momenta are entirely determined by their wave functions. No interactions are computed for those particles, because perturbative interactions would doublecount what is already included in the wave functions. We will use the same wave functions to compute F 2 as previously determined [17] from F 1 , found to be consistent with pp scattering.

Coordinates
The basic diagram for proton electromagnetic form factor is given by Fig. 1. The initial and final proton 4-momenta are P and P , with q = P − P . Initial quark momenta k j (masses m j ) are unprimed, while final momenta use the same label with a prime. We let k 1 denote the struck quark, and k 2 , k 3 denote the spectators. Our coordinates are (energy, p x , p y , p z ). We use a Lorentz frame where the incoming and outgoing protons momenta are Here m P is the mass of the proton.
We introduce a basis for transverse momenta: y µ =(0, 0, 1, 0) = y ;P · y =P · y = 0; are the unit vectors along the direction of propagation of the incoming and outgoing protons respectively. The components of the quark 3-momenta are expressed as The four momenta of the quarks are then given by,

The Matrix element
With J µ the electromagnetic current operator and N standing for Dirac spinors, the matrix element for the interaction is parameterized by Let Ψ αβγ be the Bethe-Salpeter 3-quark wave function in the proton with spinor indices shown. Let symbol M µ stand for the quark-photon vertex, propagator factors, and momentum conservation factors, displayed in a moment. The model for the reaction is Here M µ is Note the delta functions δ 4 (k 2 −k 2 )δ 4 (k 3 −k 3 ) which explicitly enforce momentum conservation of spectator quarks.
The initial light cone coordinates are defined as Final state symbols have a prime. In literature, it is standard to use the co-ordinates κ − = k − p + ; κ + = k + /p + ; κ − = k − p + ; κ + = k + /p + which is just parameterizing the light cone co-ordinates with the momenta p + , p + .

Integration
It is generally assumed that wave functions Ψ (k i ), Ψ(k i ) of 4-momenta are peaked near the on-shell region. In that region, the actual wave function can be replaced by its integral over the small mo- The rest of the calculation cannot use the same approximation, because the delta-functions vary rapidly: Hence the process is indivisibly linked together by the integrations. The above expression for scattering kernel M µ has an important dependence on κ − 2 and κ − 3 , which cannot be overlooked. This is the point where our calculation begins to differ from previous ones.
The basic problem is that for the soft spectator quarks it is not reasonable to assume that their four momentum square, k 2 , is approximately zero. We expect k 2 to be of the order of Λ 2 . In a constituent quark model, these quarks are assumed to be approximately on-mass-shell with masses of the order of few hundred MeV for the up and down quarks. In general the behavior of the quark propagator is expected to be more complicated and one can model its form by solving a truncated Schwinger-Dyson equation [23][24][25]. For our purpose, it is adequate and self-consistent to assume that M µ is dominated by the on-shell region and the κ − , κ − dependence can be replaced by the on-shell expression, As explained above, we assume that the mass of the slow spectator quarks is of the order of a few hundred MeV. On the other hand, the struck quark is a perturbative object. Treating it consistently uses a mass of order of a few MeV. We ignore the tiny and power-suppressed helicity-flip contributions from the struck quark. Now, doing a change of variables gives the standard form with The delta functions of Eq.(9) and Eq.(6) lead to the following conditions, The light cone wave function Y of leading twist and leading power of large P is [26,27], Here V, A, T are scalar functions of the quark momenta, N is the proton spinor, N c the number of colors, C the charge conjugation operator, σ µν = i 2 [γ µ , γ ν ], and f N is a normalization. This wave function was previously used to compute F 1 , and is now being applied to compute F 2 .
It may come as a surprise that the same chirality structure creating F 1 can predict F 2 . Fig.  2 shows a cartoon of the chirality flow. Each term in the Y αβγ collection has been classified as chirally even or chirally odd depending on whether it conserves helicity (even, anti-commutes with γ 5 ) or flips helicity (odd, commutes with γ 5 ). Since momentum conservation is trivial it is not shown. The chirality flow of the V, A, T terms are shown at the top. A typical combination of diagrams flipping the final state proton chirality is shown at the bottom. This diagram needs one (1) internal flip of low momentum spectator quark chirality, which appears as the closed loop with a mass insertion indicated by "X." The cartoon shows how the Dirac algebra works without needing to do the algebra.
Returning to Eq. 9, inserting the wave function Eq. 10, and extracting the terms which lead to F 2 yields The 1/Q 2 factor after the integration measure comes from the Q dependence of δ(k 0 ). Evaluating the first two terms in the above expression and isolating the F 2 contribution gives The other terms are similar.

The endpoint wave function and F 2
The leading power wave functions of Ref. [17] were determined in the endpoint region: The exponential dependence on the transverse momentum is a generic form that restricts the range of x 1 ∈ (1 − Λ Q , 1) and x 2 ∈ (0, Λ Q ). The dot products are In terms of the light cone variables, this gives

The ratio of form factors
In our estimate of the form factor F 2 we used the wave function given in Eq. 12, whose x dependence was determined by fitting the Dirac form factor, F 1 . However it is easy to see that the ratio F 2 /F 1 is independent of the precise form of the wave function within the end point model. Consider a rather arbitrary wave function This leads to the Dirac form factor [17], Similarly the form factor F 2 becomes, Taking the ratio gives Thus the ratio of form factors in the endpoint model is independent of the precise form of the wave function. The JLAB data [9] shows QF 2 /F 1 ∼ constant starting from Q 2 as low as 2 GeV 2 . At such low values F 1 differs significantly from its high-Q 2 scaling behavior, which is observed to set in for Q 2 > 5 GeV 2 [28]. In the low Q 2 regime a more complicated wave function is needed to fit the data. However Eq. 17 follows quite generally since the dependence on the wave function cancels out while taking the ratio.

Soft gluon exchange
It can be verified that addition of low momentum gluons in the interaction will not change the scaling behavior of the Pauli Form factor F 2 . Consider the simple case of 2 gluon exchange illustrated in Fig.[3].
The matrix element for this diagram is Evaluating the traces and extracting the co-efficient of N iσ µν q ν N we find Keeping only leading power term for the limit Q Λ, dropping transverse momentum integrals of order the hadronic scale and substituting V, T from Eq.12 gives Each integral dx over an interval of length Λ/Q contributes a power of 1/Q. The integration of 1 − x over 1 − Λ/Q < x < 1 contributes a power of 1/Q. It follows that Thus the gluon exchanges do not change the leading power behavior.

Conclusions
As mentioned in the Introduction, if the EP model had been given adequate attention 30 years ago, a fit to the known 1/Q 4 dependence of F 1 would have then predicted F 2 /F 1 ∼ 1/Q at large Q, just as eventually observed. The calculation was never done, despite the model's visibility after initial development by Drell, Yan, Feynman, and others. [29][30][31]. Between then and now came a period attempting to dispense with hadron structure in form factors, and replacing protons with perturbation theory, which revealed very little about hadron structure. We find that one simple pattern of an endpoint wave function, previously determined in Ref. [17] and going like 1 − x, explains many independent experiments. The endpoint region produces the original and earliest quark-counting model [30]. For each spectator integration dx restricted to x Λ/Q an integral goes like Λ/Q. For each hard struck quark with 1−Λ/Q x ≤ 1 an integral goes like Λ/Q. Thus three quarks leads to F 1 ∼ 1/Q 4 . The leading twist Dirac structure, which has no room for orbital angular momentum, still allows a reversal of the proton's chirality characterizing F 2 , and F 2 ∼ 1/Q 5 . These are not asymptotic limits, but generic results of power-counting that apply in the region Q >> Λ, namely Q GeV.
The fact that QF 2 (Q 2 )/F 1 (Q 2 ) is nearly constant with Q down to rather low values of momentum transfer is now understood. At small Q the details of the endpoint wave function enter the calculation, and replacing dx ∼ Λ/Q is not accurate. It is possible to fit that dependence from data for F 1 rather trivially. However the integrations for F 2 are so nearly like those for F 1 that the details of the wave function cancel out in the ratio F 2 /F 1 . The rule that F 2 /F 1 ∼ 1/Q for Q >>GeV naturally extends itself into the region of Q ∼ few GeV. When future experiments probe higher momentum transfers we are confident that QF 2 (Q 2 )/F 1 (Q 2 ) will remain constant, regardless of what might occur with the numerator and denominator.