QCD Constituent Counting Rules for Neutral Vector Mesons

QCD constituent counting rules define the scaling behavior of exclusive hadronic scattering and electromagnetic scattering amplitudes at high momentum transfer in terms of the total number of fundamental constituents in the initial and final states participating in the hard subprocess. The scaling laws reflect the twist of the leading Fock state for each hadron and hence the leading operator that creates the composite state from the vacuum. Thus, the constituent counting scaling laws can be used to identify the twist of exotic hadronic candidates such as tetraquarks and pentaquarks. Effective field theories must consistently implement the scaling rules in order to be consistent with the fundamental theory. Here we examine how one can apply constituent counting rules for the exclusive production of one or two neutral vector mesons $V^0$ in $e^+ e^-$ annihilation, processes in which the $V^0$ can couple via intermediate photons. In case of a (narrow) real $V^0$, the photon virtuality is fixed to a precise value $s_1 = m_{V^0}^2$, in effect treating the $V^0$ as a single fundamental particle. Each real $V^0$ thus contributes to the constituent counting rules with $N_{V_0} = 1$. In effect, the leading operator underlying the $V^0$ has twist 1. Thus, in the specific physical case of single or double on-shell $V^0$ production via intermediate photons, the predicted scaling from counting rules coincides with Vector Meson Dominance (VMD), an effective theory that treats $V^0$ as an elementary field. However, the VMD prediction fails in the general case where the $V^0$ is not coupled through an elementary photon field, and then the leading-twist interpolating operator has twist $N_{V_0} = 2$. Analogous effects appear in $pp$ scattering processes.


I. INTRODUCTION
One of the distinctive consequences of the underlying conformal features of gauge theories such as QCD is counting rules for hard exclusive processes. In such processes, one can factorize the physical scattering amplitude as the convolution of a hard-scattering quark and gluon amplitude T H with the product of hadronic distribution amplitudes φ H (x, Q).
The resulting scaling for the differential cross section at large momentum transfer reads [1][2][3] dσ/dt ∼ 1/S N −2 , where S is a generic hard scale, and N = N i + N f is the total number of fundamental constituents participating in the hard subprocess. The number of constituents of each hadron entering the scattering amplitude coincides with the number of particles in its leading Fock state and thus with the twist of the leading operator that creates the composite state from the vacuum. For example, the scaling prediction for exclusive cross sections such as fixed-angle hadron-hadron scattering is [1][2][3]: where N = N A + N B + N C + N D is the total twist or number of elementary constituents.
When dealing with hadrons, one must take into account their quark content and use N M = 2 and N B = 3 for each meson and baryon, respectively. One also predicts logarithmic corrections from the behavior of the running couplings entering T H and the Efremov-Radyushkin-Brodsky-Lepage (ERBL) evolution of the distribution amplitude.
The constituent counting rules are completely rigorous when they are applied properly.
The leading-twist contribution to the power-law falloff of a cross section for any exclusive or semi-inclusive process depends upon the twist of the operators that couple the hadron to the hard subprocess. The twist τ of a hadron that couples to a hard-scattering subprocess is computed from the number N of its fundamental constituent fields interacting in the hardscattering subprocess (called active in Ref. [4]), plus L, the relative orbital angular momenta in the contributing hadronic Fock state. In contrast, the cross section for hadrons produced through a soft intermediate state-such as a neutral vector meson V 0 produced via its direct coupling to a photon of finite virtuality, or a hadron produced from jet fragmentation-does not have increased power-law falloff.
Note also that effective field theories developed to describe hard hadronic processes must consistently implement the counting rules in order to be consistent with the underlying fundamental theory of QCD. AdS/QCD, which allows the calculation of hadronic amplitudes using light-front holography [5], is good example of such an effective theory.
In Refs. [6,7], the present authors studied the application of the constituent counting rules for the production of tetraquarks, pentaquarks, and V 0 in the exclusive reactions of electroproduction and pp and e − e + annihilation. The purpose of the present paper is to further clarify our point regarding single and double on-shell V 0 production in e + e − annihilation (see Figs. 1 and 2), where each V 0 couples to the hard subprocess via a virtual photon. In effect, the leading operator underlying the V 0 has twist N V 0 = 1, a point not fully appreciated in Ref. [7]. In fact, the possibility that some of the constituents in a given process counted in the scaling rule might not be hard is the essence of the critique of [6] in Ref. [8]. Thus, in the specific physical case of single or double V 0 production via intermediate photons, the predicted scaling from counting rules coincides with Vector Meson Dominance (VMD) [9], an effective theory that treats the V 0 s as elementary fields. A modified form of the constituent counting rules therefore holds, and QCD can be approximated at these exceptional kinematic points by an effective field theory, the VMD model, which treats V 0 as an elementary field. However, as we shall show, VMD in general is not consistent with QCD and the constituent counting rules. The VMD prediction fails in the general case in which the V 0 is not coupled to the hard subprocess via an elementary photon field; in that case, the leading-twist interpolating operator has twist N V 0 = 2.
This paper is organized as follows: Section II provides general comments about e + e − → γ * γ and γ * γ * processes, both leptonic and hadronic. In Sec. III we examine the process e + e − → γ µ + µ − , for which the relevant cross sections have been explicitly computed, and for which the high-momentum scaling behavior is explicit, and infer the corresponding behavior for V 0 production. Section IV shows how the original constituent counting rules persist in inclusive e + e − processes involving vector (or scalar or tensor) meson production. In Sec. V we consider applications of these ideas in pp scattering processes, and in Sec. VI we conclude.

II. VECTOR-MESON PRODUCTION VIA INTERMEDIATE PHOTONS
The most straightforward scaling predicted by the counting rules is valid in most physical applications, e.g., in the pair-production of mesons, baryons, or tetraquarks in e + e − annihilation [6], and occurs whenever all constituents participate in the hard process, in which However, specific physical cases exist, e.g., single or double vector-boson V 0 production processes e + e − → γV 0 and e + e − → V 0 a V 0 b , in which each V 0 couples solely to an intermediate elementary photon field: γ → V 0 , or a weak gauge boson: Z 0 → V 0 , W ± → V ± . In such cases, the scale associated with the photon virtuality is fixed to a precise value s 1 = m 2 V 0 , where m V 0 is the vector meson mass. Therefore, one can treat the V 0 (with respect to the counting rules) as a single fundamental particle, and QCD reduces to the limit of the Vector Meson Dominance (VMD) model. In this specific case, the V 0 is approximated by an elementary field with N V 0 = 1 elementary constituents. Then one has N − 2 = 2 for both processes e + e − → γV 0 and e + e − → V 0 a V 0 b , which gives the differential cross-section scaling dσ/dt ∝ 1/s 2 , where s is the total c.m. energy of lepton pair, or 1/s|t| for forward scattering. This result follows from setting s 1 = m 2 V 0 in the γ → V 0 transition form factor G V (s 1 ) (calculated, e.g., using the soft-wall AdS/QCD approach) in Ref. [7], rather than introducing an O(|t|) scale in G V as advocated in that work. Independently, this scaling result can be shown explicitly by considering the related process e + e − → γ µ + µ − at high energy but small invariant mass for the µ + µ − pair (Sec. III), an exercise that is instructive in explicitly indicating where the various momentum scales appear. Let us stress again that the scaling of the differential cross section dσ/dt ∝ 1/s 2 in the particular processes of single or double vector-boson V 0 production does not violate the constituent counting rules, because the exclusive γ → V 0 transition necessarily implies soft QCD, leading one to approximate the V 0 (with respect to hard scales) as an elementary field. In other words, the presence of soft QCD vertices in hard processes leads to a decrease of the scaling power in the corresponding differential cross section by identifying each softly produced hadron composed of N a constituents with an elementary field: N a → 1.
Note that the production of a V 0 via a photon can be a subleading contribution to the matrix element of a hard process. An example of such process is V 0 production in the reaction e + e − → V 0 P 0 , where P 0 is a neutral pseudoscalar meson (e.g., π 0 , η, η ). In Ref. [8], VMD was the mechanism proposed for the γ → V 0 transition in such processes.
It is clear that this subprocess is O(α em )-suppressed in comparison with the leading QCD diagram discussed in Ref. [10] for direct production of a V 0 P 0 pair by a hard photon, γ * → V 0 P 0 . As was shown in Ref. [10], the matrix element for e + e − → V 0 P 0 contains a helicity-flip transition form factor F γ * V 0 P 0 (s), which encodes violation of hadron helicity selection rules and scales as 1/s 2 at large s. As result, the corresponding cross section scales as dσ/dt ∝ 1/s 5 , i.e., it has an additional 1/s falloff compared to helicity-favored modes of two-meson production (π + π − , K + K − , etc.) The mechanism for the e + e − → V 0 P 0 reaction considered in Ref. [8] gives dσ/dt ∝ 1/s 3 , but, as stressed above, is suppressed by a power of α em in comparison with the leading QCD diagram.
III. LESSONS FROM THE PROCESS e + e − → γ µ + µ − In order to verify or falsify the claim from Ref. [7] that the γ * V 0 transition form factor G V (q 2 ) scales as 1/ |t| for forward scattering in e + e − → γV 0 , one may study the related process e + e − → γ µ + µ − , which has been considered for decades [11,12] as a background to e + e − → µ + µ − , and more recently [13] in the initial-state radiation process, in which the real photon is hard but the µ + µ − pair is soft. Indeed, the result of Ref. [13] was used to estimate the yield of true muonium (µ + µ − ) atoms in the process e + e − → γ + (µ + µ − ) [14].
The process assumes the same topology as Fig. 3, with qq replaced by µ − µ + . To serve as an orientation, we exhibit the textbook result [15] of the Born-level cross section for the pair-annihilation process e + e − → γγ. One finds, neglecting masses, and in the forward direction (m 2 e |t| s), in agreement with the prediction of Eq. (1) with N = 4 and one forward (fermion) propagator.
The full Born-level cross section for e + e − → γ + (µ + µ − ), as can be seen in Eq. (14) of Ref. [12], possesses a second-order pole 1/(1 ± z) 2 in z ≡ cos θ + , where θ + is the µ + e − angle in the e + e − c.m. frame. The corresponding differential cross section dσ/dt therefore has separate terms scaling as 1/|t| 2 and as 1/|u| 2 . They clearly arise through the near-collinear kinematics in which µ + , µ − , and γ all lie close to the beam axis but have large relative momenta; in that case, both the fermion and photon propagators in Fig. 3 contribute the large momentum-transfer factors.
But now restrict to the kinematics of Ref. [13], in which the momentum transfer s 1 of the µ + µ − pair is small; in the exclusive qq case, s 1 = m 2 V 0 (labeled q 2 in Ref. [7]). The e + e − → γ + (µ + µ − ) forward differential cross section then reads where δ ≡ m 2 µ /s 1 , x ± ≡ E µ ± /( √ s/2) are the fractional µ ± energies, and m e → 0. The question then becomes how much the remaining integrals, those over dx − and ds 1 , influence the full high-momentum scaling of dσ/dt. One easily finds that where is the µ + µ − angle and θ is the e − γ angle in the c.m. frame. We are therefore interested in the hierarchy s 1 |t| s, or θ 1. Immediately one sees that small s 1 requires a small c.m. angle between the µ + µ − pair; however, at this stage no similar restriction requires the µ + and µ − to share their total energy √ s/2 equally, so that x − = 1 − x + can assume any value ∈ [0, 1].
We now turn to the exclusive hadronic case (µ − µ + → qq), in which Eq. One is therefore left to consider the s 1 integral. Strictly speaking, the allowed range of s 1 for a vector state V 0 of narrow width is vanishingly small, and |G V | 2 assumes the form of a decay constant F 2 V 0 (of dimension mass squared) times a delta function δ(s 1 − m 2 V 0 ): Requiring the virtual photon in Fig. 3 to produce only a single exclusive state V 0 of squared mass s 1 fixes the photon virtuality precisely to equal s 1 . However, the same result obtains if G V (s 1 ) is replaced by a properly normalized Breit-Wigner distribution representing a wide state such as ρ 0 . The form factor |G V | 2 must also decrease with s 1 in order to satisfy unitarity, but this decrease merely indicates that couplings to highly excited V 0 's must decrease with s 1 = m 2 V 0 in order to sum to a finite total. In the AdS/QCD calculation of Ref. [7], this dependence in terms of the AdS/QCD scale parameter κ would read κ 2 /s 1 .
The expected "large" momentum-scale suppression in G V in exclusive transitions due to constituent counting rules actually comes from s 1 .
Knowledge of the larger scale |t| by the V 0 is lost in the propagation of the photon. For the most naive form of the counting rules to hold, all propagators and fermion couplings must contribute large scales to the amplitude, and the virtual photon in this case contributes only 1/s 1 . One concludes that the forward cross section for a strictly two-body process e + e − → γV 0 in which V 0 contains two fundamental constituents [N = 5 in Eq. (1)] should scale as dσ/dt ∼ (α 3 /s|t|)|G V (s 1 )| 2 , with G V (s 1 ) ∼ 1/ √ s 1 .
In contrast, Ref. [7] concluded that G V (s 1 ) ∼ 1/ |t|, the large scale |t| reemerging through a hard-gluon exchange needed to bind the otherwise noncollinear pair qq into the bound state V 0 . However, as noted above, the formation of a single photon of virtuality s 1 |t| Consider instead a process such as that illustrated in Fig. 3, except that the photon virtuality s 1 does not precisely equal m 2 V 0 , but rather assumes a value of O(|t|) (because the process is still one of forward scattering). The inclusive process e + e − → γqq has a much greater phase space than does the exclusive process e + e − → γV 0 , and its cross section scales in the forward direction as dσ/dt ∼ 1/s|t| 2 (as seen above for e − e + → γµ + µ − ). This inclusive rate does indeed include a portion of the exclusive channel e + e − → γV 0 , but only from the large-|t| tail of the V 0 line-shape. It also includes contributions from e + e − → γV 0 plus additional soft hadrons such that the total hadronic system has invariant mass-square of O(|t|), which can be misidentified as the exclusive channel e + e − → γV 0 if the soft hadrons escape detection.
In summary, the correct high-momentum forward-angle scaling for the genuine two-body exclusive e + e − → γV 0 cross section is dσ/dt ∼ 1/s|t|, rather than 1/s|t| 2 as given in Ref. [7].
However, tails of the original process and processes that can be misidentified as e + e − → γV 0 give contributions scaling as 1/s|t| 2 ; and since they have much greater available phase space, they may dominate the observed rate of e + e − → γV 0 even if |t| is rather larger than m 2 V 0 . Completely analogous comments hold for the process e + e − → V 0 a V 0 b .

IV. INCLUSIVE e + e − PROCESSES WITH VIRTUAL PHOTONS VS. VMD
When the V 0 s are off-shell, an extra power falloff in the large scale appears for each meson state. The forward-scattered virtual photons then carry O(|t|) momentum transfers, and the constituent counting rules read [7]: Crucially, the full QCD theory differs from an effective field theory developed from VMD in their applications to physical processes. In particular, VMD makes no distinction between on-shell and off-shell V 0 , which leads to an incorrect off-shell behavior of the γ * → V 0 transition. In Ref. [7] we explicitly showed that a constant value for this transition is ruled out by the nontrivial form factor G V (q 2 ), where q is the photon (or V 0 ) four-momentum.
In Ref. [7], we pointed out another property distinguishing pQCD from effective field theories treating V 0 as elementary fields, i.e., in its application to electromagnetic form factors of hadrons. Let us review this important point: In effective theories (like VMD [9], chiral perturbation theory [16], or the hidden-symmetry approach with vector mesons as dynamical gauge bosons [17]) that treat the V 0 as an elementary field, the form factor G V (q 2 ) is a constant. One obtains different scaling contributions of the relevant diagrams with elementary V 0 and with pQCD. For example, in the case of the pion electromagnetic form factor F π (Q 2 ), one may split the VMD result into the contact and the vector-meson exchange diagrams (Fig. 4). The contact diagram gives 1 (a constant contribution at large Q 2 = −q 2 ), whereas the vector-meson exchange diagram gives a (−1+m 2 V 0 /Q 2 ) contribution. Summing, one arrives at a m 2 V 0 /Q 2 scaling. In contrast, using pQCD counting rules, the contact diagram turns out to be of leading order (1/Q 2 ), whereas the V 0 -exchange diagram is subleading [(1/Q 2 ) 3/2 ] at large Q 2 . Of course, at fixed Q 2 = m 2 V 0 the contributions do not cleanly separate. Another crucial point is that, while the scaling of F π (Q 2 ) in VMD is formally m 2 V 0 /Q 2 due to the V 0 propagator, this factor has no connection with 1/Q 2 scaling in pQCD due to hard-gluon exchanges between constituent quarks in the pion. Therefore, the 1/Q 2 falloff of F π (Q 2 ) in VMD coincides with pQCD accidentally, and is not related to the physical nature of strong interactions at high scales. As a consequence, an effective field theory of VMD completely fails in the description of the electromagnetic form factors of baryons and multi-constituent hadronic systems (tetraquarks, pentaquarks, etc.) at high scales. Without pQCD it is impossible to produce the 1/Q 2(N −1) falloff of the electromagnetic form factors of hadrons containing N constituents. Note that a criticism of VMD in the description of data for photon-hadron interactions at high energies was also stressed in Ref. [18]. In particular, [18] argues that VMD is not a suitable framework for a description of deep-inelastic scattering over the full kinematical range. Additionally, in the generic case of exclusive on-shell hadron production, it is not possible to approximate the hadrons as elementary fields. In the recent paper Ref. [19], the exclusive production processes of scalar S = 0 ++ and tensor T = 2 ++ mesons through single-photon annihilation e + e − → γ * → γ S(T ) were analyzed. Here, the transition form factors of γ * → γS and γ * → γT are not constants, and scale as F γ * γS (s) ∼ 1/s and F γ * γT (s) ∼ 1/s 2 at large s, consistent with the scaling of the corresponding form factors at large values of virtual-photon squared Euclidean momentum Q 2 : F γ * γS (Q 2 ) ∼ 1/Q 2 [20] and F γ * γT (Q 2 ) ∼ 1/Q 4 [21]. The scaling of the form factors follows directly from using the differing twist counting for the S-and T -creating operators [10]. As a result, both differential cross sections scale as dσ dt (e + e − → γ + S(T )) ∼ 1/s 3 , in agreement with constituent-quark counting rules that treat real scalar and tensor mesons as qq systems with N S(T ) = 2 substituted into the counting formula (1) for dσ/dt. Consistent with Ref. [6], when scalar and tensor mesons are considered as tetraquark systems of two tightly bound color diquarks, the corresponding transition form factors and differential cross sections have the same falloffs as in the qq case.
For other tetraquark or two-hadron molecular configurations, the transition form factors In this section we discuss V 0 production in pp scattering processes. The hadronic angular momentum dependence of hard exclusive QCD processes is controlled by the Brodsky-Lepage helicity selection rules [10,22], which state that the total hadron helicity is conserved from the initial to the final state, up to higher-twist corrections appearing as inverse powers of the hard scale. This result was used for e + e − → V 0 P 0 in Sec. II. Taking here V 0 = ρ 0 , we consider three specific cases of semi-inclusive ρ 0 -meson production [4,23,24]: (1) the reaction pp → ρ 0 X, with the ρ 0 produced from jet fragmentation and X being any hadrons; (2) the reaction pp → ρ 0 D X, with a "direct" ρ 0 D produced at high p T in isolation from other hadrons on the trigger side (i.e., without any same-side particles); and (3) the reaction pp → γ * X → ρ 0 D X, where a single virtual photon produces a "direct" ρ 0 , which again is isolated on the trigger side.
Reaction (1) has normal conformal scaling (modulo log corrections). Consistent with Eq. (1), the differential cross section for semi-inclusive production of a single hadron ρ 0 with form factor F scales as where p T and x T = 2p T / √ s are the transverse momentum and its light-cone fraction, respectively.
In case (2) the ρ 0 D couples via a qq to the hard underlying hadron subprocess. The corresponding differential cross section then has an additional 1/p 2 T falloff in comparison with case (1) and scales as reflecting the corresponding twist-2 operator and the |qq Fock state of the ρ 0 . Note that reaction (2) is power-suppressed at high p T (being higher twist), but the ρ 0 in this case exhibits color transparency [25]: It is produced directly from the hard subprocess as a small-sized color-singlet state and can propagate through a nuclear medium with minimal interactions. In contrast to reaction (2), the process pp → γX with an isolated photon occurs at leading twist since the photon can couple directly to the hard process without additional power suppression.
Consideration of pp → γX leads to case (3), which again scales like dσ d 3 p/E ∼ 1/p 4 T as in Eq. (6), since in this case the ρ 0 couples softly via the twist-1 photon field to the hard subprocess without an additional power suppression. Reaction (3) exhibits the same type of behavior as discussed Secs. II and III for e + e − → γV 0 and e + e − → V 0 a V 0 b .

VI. CONCLUSIONS
Let us summarize the main results of this paper. We examined the application of QCD constituent counting rules to exclusive processes involving neutral vector mesons V 0 . In particular, we considered exclusive production of one or two V 0 via intermediate photons from e + e − annihilation, and in pp scattering. In case of a real V 0 , the photon virtuality s 1 can be fixed to a precise value m 2 V 0 , in effect treating the V 0 as a single fundamental particle. Therefore, each real V 0 contributes to the constituent counting rules with N V 0 = 1. Because the leading operator underlying the V 0 has twist 1, in the case of single or double on-shell However, the VMD prediction fails in the general case where the V 0 is not coupled solely through an elementary photon field, and in that case the leading-twist interpolating operator has N V 0 = 2. Furthermore, VMD fails in the case of off-shell coupling of the electromagnetic field with hadrons at large momentum scales because this approach, by construction, does not respect the constituent structure of hadrons and hard-gluon exchange at large scales.
As a result, the large-Q 2 scaling of electromagnetic form factors of hadrons with N ≥ 3 constituents in VMD is not consistent with that from pQCD. Only in the case of qq systems (conventional mesons) is the VMD prediction of 1/Q 2 scaling formally similar to that of pQCD because of the 1/Q 2 behavior of the V 0 propagator. One should also note the criticism of VMD in the description of data for photon-hadron interactions at high energies stressed before in Ref. [18].