aa r X i v : . [ h e p - ph ] N ov Diquark Substructure in φ Photoproduction
Richard F. Lebed ∗ Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA (Dated: October, 2015)Observed enhancements in the forward and backward directions for φ meson photoproduction offnucleons are shown to be explainable by the production of a nonresonant recoiling ( su ) diquark,(¯ sud ) triquark pair. We show that the necessity of maintaining approximate collinearity of thequarks within these units constrains configurations with the minimum momentum transfer, andhence maximal amplitudes, to lie preferentially along the reaction axis. PACS numbers: 14.20.Pt, 12.39.Mk, 12.39.-xKeywords: exotic baryons; pentaquarks; diquarks
I. INTRODUCTION
LHCb has recently observed [1] two exotic states, P + c (4380) and P + c (4450), at high significance in the J/ψ p spectrum of Λ b → J/ψ K − p . In addition to extractingthe state masses and widths, the collaboration also mea-sured the phases of the production amplitude and foundthem to be compatible with genuine resonant behavior.Such properties strongly support the assertion that truepentaquark states have at last been revealed. Shouldthe P + c states be confirmed at another experiment, theywill join X , Y , and Z exotic mesons (widely believed tohave valence tetraquark structure) as the first hadronsobserved at high significance to lie outside of the text-book q ¯ q -meson, qqq -baryon paradigm.Of particular note is that all of the observed exoticstates contain c ¯ c (or b ¯ b ) pairs. The possibility of light-flavor ( u , d , s ) tetraquarks or pentaquarks has been dis-cussed for decades, but despite intense experimental ef-forts, no unambiguous signal of such a state has ever sur-vived scrutiny. Nor has a singly-heavy hadron ( D s , Λ b , etc. ) been identified as unambiguously exotic. One possi-ble explanation for this first appearance in doubly-heavychannels, as argued implicitly in Ref. [2] and explicitly inRefs. [3, 4], is that the exotic consists of two components,each of which contains a heavy quark and therefore isfairly compact, and which are separated from each otherin the sense of having a suppressed large wave functionoverlap. Such a configuration is much more difficult torealize with light quarks. While this scenario could beachieved by hadronic molecules (for instance, if the fa-mous X (3872) tetraquark candidate has a c ¯ cu ¯ u valence-quark structure organized into a D ¯ D ∗ molecule), inthis paper we use the innovation of Ref. [2] to assert theexistence of hadrons formed of separating colored compo-nents that are individually held together by the attractive ⊗ ⊃ ¯ color interaction, and collectively preventedfrom separating asymptotically far due to color confine-ment.If the existence of observable exotics is contingent upon ∗ Electronic address: [email protected] each of its components containing a heavy quark, thenone may hope that some vestige of the exotic behaviorpersists in analogous production mechanisms with va-lence s ¯ s rather than c ¯ c ( b ¯ b ) quark pairs. In Ref. [2], theobserved tetraquark states were argued to arise from arapidly separating diquark-antidiquark pair, ( cq ) ¯ (¯ c ¯ q ′ ) ( q in the following generically indicates u or d quarks)that subsequently hadronizes only through the large- r tails of meson wave functions stretching from the quarksin the diquarks to the antiquarks in the antidiquarks.This picture was extended in Ref. [4] to explain the pen-taquark states as an analogous diquark-anti triquark pair(this combination having been first proposed for lighter-quark pentaquarks in Ref. [5]), ( cu ) [¯ c ( ud ) ¯ ] , where the( ud ) diquark is inherited from the parent Λ b baryon, andthe triquark is seen to assemble from a further ¯ ⊗ ¯ ⊃ attraction. A short review of these papers appears inRef. [6]. If this mechanism produces such prominent ef-fects in the c ¯ c system as pentaquark resonances, then onemay hope to see at least a remnant of the mechanism inthe s ¯ s system, in the form of peculiar features appearingin the data.Shortly after the LHCb announcement, multiple the-oretical papers appeared, discussing various interpreta-tions of the P + c states and advocating for experimentsin which to study them. Among the latter, three sep-arate collaborations [7–9] proposed using γN → P c → J/ψN ( ∗ ) photoproduction of nucleons N as sensitive testsof the internal structure of the P c states. In each case,the c ¯ c pair arises through the dissociation of the incomingphoton. By the reasoning of the previous paragraph, onemay therefore ask if any unusual features have arisen inthe analogous s ¯ s process of φ photoproduction, γp → φp .In fact, a detailed experimental study of φ photopro-duction was also published fairly recently by the CLASCollaboration at Jefferson Lab [10]. Data in the neutral( K L K S ) mode are presented in Ref. [11] The most inter-esting feature in the cross section data is a forward-angle“bump” structure at √ s ≈ . √ s . However, this struc-ture appears only at the most forward angles (in which φ lies in the same direction as γ in the center-of-momentum[c.m.] frame), and as argued in Ref. [12], such a unidi-rectional structure almost certainly does not indicate aresonance. In addition, the cross section data indicatea small but clear increase at the most backward angles(a factor of perhaps 50 smaller than the forward peak );while Ref. [10] mentions a possible origin for backwardenhancement in the u -channel exchange of a nucleon,they also comment that such a direct φ - p coupling wouldrequire a nucleon strangeness content or a violation ofOZI suppression surprisingly larger than expected.Here we argue that both features can be explainedin a very simple way: A fraction of φ photoproduc-tion events proceed through a constituent exchange be-tween the γ (dissociated into an s ¯ s state) and the nucleon[treated as a bound state of a ( ud ) diquark and a lightquark q ] (Fig. 1). The scattering products are a diquark-antitriquark pair, ( sq ) ¯ [¯ s ( ud ) ¯ ] , completely analogousto the construction proposed for the P + c resonances, butas experiment indicates, nonresonant in this case. That γ s ¯ s N ( ud ) ( ud ) qs ¯ s θ q FIG. 1: The constituent exchange for φ photoproduction viacreation of an ( sq ) ¯ diquark-[¯ s ( ud ) ¯ ] antitriquark pair. is, the anomalies correspond to “would-be” pentaquarks.The enhancement in the forward and backward direc-tions, with the forward direction favored, is explained bythe preference of the diquark and triquark to minimizemomentum transfer by aligning with the γp process axis.This paper is organized as follows: In Sec. II we discussthe forward-backward enhancement of hadronic cross sec-tions driven by constituent-exchange momentum trans-fers. A simple model based upon minimal gluon ex-changes is presented in Sec. III, and the calculation andresults are presented in Sec. IV. Section V summarizesand concludes. M. Dugger, private communication.
II. MOMENTUM TRANSFER ENHANCEMENT
One of the quintessential triumphs of quantum fieldtheory is its natural prediction of a Rutherford-type dif-ferential scattering cross section, in which the invariantamplitude M for the process p p → p ′ p ′ (using mo-menta to label the particles) scales as the inverse of the4-momentum transfer t = q = ( p ′ − p ) = ( p − p ′ ) .A dependence M ∝ / ( q − m ) arises from the virtualexchange of a quantum of mass m of the mediating inter-action. For example, in pp elastic scattering the quantum(at least at low energies) may be considered a single neu-tral virtual meson, such as π . At higher energies, the rel-evant degrees of freedom exchanged become quarks andgluons.Elastic scattering via mediators of negligible mass alsofeatures the characteristic Rutherford angular depen-dence 1 /q ∝ / sin ( θ/ θ is the c.m. scatteringangle. Inelastic 2 → ( θ/ M in the forward di-rection, and for precisely the reason that Rutherford un-derstood: Greater particle deflections demand strongerforces, requiring scattering events that, for given fixedinitial-particle energies, are comparatively rarer. If thescattered particles p ′ , p ′ cannot be unambiguously asso-ciated with the initial particles p , p , respectively—asoccurs either if p , p represent identical particles, or ifthe reaction is sufficiently inelastic that the associationis less than perfect—then one also expects an enhance-ment in M corresponding to the minimization of the4-momentum transfer u = ( p ′ − p ) = ( p − p ′ ) . u contains the factor cos ( θ/ M alsopeaks in the backward direction.A prominent and historically important example ofthis forward-backward peaking occurs in pn elastic scat-tering. Indeed, the strong backward peak, now under-stood to be due to the charge-exchange reaction p ↔ n inwhich charged mesons are traded between the nucleons,is one of the best pieces of evidence for strong interac-tions respecting fundamental isospin symmetry, partic-ularly the equivalence of p and n under strong interac-tions: In pn scattering the backward peak is almost ashigh as the forward peak, because p turns into n and viceversa but remains otherwise undeflected. It is interest-ing to note (as discussed, e.g. , in Ref. [13]) that simpleone-pion exchange, which one might naively expect to beresponsible for the full peak, is actually well known tolead to a zero differential cross section at 180 ◦ . At theconstituent level, a quark exchange u ↔ d occurs.For inelastic processes, the rule of thumb for estimatingthe relative height of the forward and backward peaks ap-pears to follow from the similarity of the initial and finalparticles. Taking p ( p ′ ) to be the lighter initial (final)particle, one expects the forward peak to be genericallyhigher. As an explicit example, consider K Λ photopro-duction; here, p = γ → s ¯ s , p ′ = K + , p = p , and p ′ = Λ. The recent CLAS data for this process [14] showsa backward peak in the differential cross section that isa factor of a few smaller than the forward peak (the ra-tio depending strongly upon √ s , reaching a maximumof about 1/2 around 2.1 GeV), supporting the generaldynamical picture described here. Indeed, a small mod-ification of Fig. 1 provides a simple quark-exchange pic-ture for K Λ photoproduction: Just exchange s ↔ ¯ s . Thepreference of forward ( θ →
0) scattering indicates the rel-ative dynamical preference for not diverting the (heavier)baryon line. Alternately, at the constituent level for K Λone sees that the lightest constituent q (and s ) suffers abackward scattering for θ →
0, while the heavier ( ud )diquark (and ¯ s ) remains undeflected. Nevertheless, oneexpects a smaller enhancement for θ → π as well, wherethe momentum transfers of q and s are minimized. III. A SIMPLE DIQUARK-TRIQUARK MODEL
Once the scattered particles in a 2 → q is no longer the only independent kinematical quantitycontributing to angular dependence. The dominant dia-gram for γN → φN is generally attributed to Pomeronexchange [15, 16], in which the γ dissociates into the s ¯ s pair. At the level of fundamental QCD, the correspond-ing Feynman diagrams include as their most simple rep-resentative Fig. 2, although the full Pomeron would in-clude many more gluons, particularly those cross-linkingthe exchanged gluon pair in this figure. The gluon ex- γ s ¯ s N ( ud ) ( ud ) q s ¯ s θ qφN Alternate diagrams are possible in which γ couples directly tothe N , and the s ¯ s pair is created from gluons emitted from thestruck N . At high energies, the latter diagram actually appearsto be dominant [17]. FIG. 2: One of many gluon-exchange Feynman diagrams con-tributing to the Pomeron exchange mechanism for γN → φN . changed between the final-state s ¯ s pair merely indicatesthat, in order for the four constituents to interact, sharemomentum, and be diverted to their final directions, aminimum of three gluons must be exchanged . In fact,the complete diagrams are even more complicated thanindicated because we have represented the nucleon N asa bound pair of a light quark q and a diquark ( ud ); in-deed, the gluons can couple separately to either of thequarks in ( ud ), the nucleon can contain both scalar andvector diquarks, and the quark q must be properly an-tisymmetrized with the identical quark in ( ud ) to sat-isfy Fermi statistics [18]. Nevertheless, Fig. 2 illustratesthe central point that the Pomeron carries the entiretyof the momentum transfer q in the t -channel, suggesting(as is observed) strong forward peaking of the differentialcross section, while a u -channel backward peak would re-quire a complicated intermediate state carrying not onlya Pomeron, but also nonzero baryon number and hiddenstrangeness.We propose an additional mechanism for φ photopro-duction, namely, the production of an ( sq ) color-¯ bounddiquark δ and an [¯ s ( ud )] color- bound antitriquark ¯ θ , asin Fig. 1. In this case, ( ud ) truly does refer to a diquarkcomponent in N , which at first blush can be either ofthe “good” (scalar-isoscalar) or “bad” (vector-isovector)variety, although data from charge and magnetic nucleonradii prefer the “good” component [18], once proper anti-symmetrization of the wave function between q and ( ud )is performed. In comparison, in the case of K Λ pho-toproduction described above, the Λ contains only the“good” ( ud ) diquark. The importance of including di-quark baryon substructure in AdS/QCD models to ob-tain Regge trajectories matching those of mesons (as isobserved), is emphasized in Refs. [19, 20].Since both δ and ¯ θ are colored objects, hadronizationof the pair is accomplished as in Refs. [2, 4], by meansof the hadron wave functions stretching across the spacebetween the colored bound states. One can obtain inthis way not only a φ = (¯ ss ), N = q ( ud ) final state,but also a contribution to the final state K = (¯ sq ), Λ = s ( ud ) expected to be smaller than the one discussed inthe previous section, due to the smaller strength of the ⊗ ⊃ ¯ attraction compared to that of ⊗ ¯ ⊃ .Our purpose is not to calculate the complete amplitudefor this entire process ( e.g. , the techniques of Ref. [21] areuseful for the large- θ region), but only to demonstratethat a natural physical mechanism exists to provide aninteraction producing an enhanced cross section in boththe forward and backward directions. We can thereforemake a number of simplifying assumptions. First, we ne- Note that the ( ud ) is treated as a single unit in the scattering. glect Fermi motion within the δ and ¯ θ , so that the con-stituents within each bound state move with the samevelocity (this assumption can be lifted by folding in ap-propriate distribution functions; e.g. , in the light-frontformalism, see Ref. [22]). Then it is clear from Fig. 1that forward ( θ →
0) scattering of ( sq ) also produces aforward-scattered φ , since in the c.m. ( sq ) and [¯ s ( ud )]have the same momentum magnitude, but the former islighter and therefore has a larger speed. In turn, s has alarger speed than ¯ s . Since m s = m ¯ s , the net momentumof φ = ( s ¯ s ) then points in the same direction as s , andhence, as ( sq ).This strategy has much in common with the “hardscattering approach” (HSA) developed in Refs. [23, 24],in that both take all constituents to move collinear withtheir parent hadrons, include the minimal necessary num-ber of gluon exchanges to accomplish the process, and re-quire folding in appropriate distribution amplitudes. TheHSA approach applied to φ photoproduction at high en-ergy scales (with genuinely hard gluons) has been studiedin Ref. [17], although using older data than here.Since the diagrams again have four constituents, theminimum number of gluon exchanges necessary for theprocess remains three. Unlike in Pomeron exchange,however, the δ -¯ θ production mechanism provides naturalalternatives that produce enhancements in the forwardand backward directions. Consider first Fig. 3; here, thescattering process is driven by the lightest constituent ( q )from either of the initial particles ( N ) exchanging a gluonwith the ¯ s constituent of the other initial particle ( γ ). As γ s ¯ s N ( ud ) ( ud ) qs ¯ s θ q FIG. 3: As in Fig. 1, but including a minimal gluon exchangethat naturally explains the enhancement of φ production inthe θ → just discussed, the δ -¯ θ configuration favored for forward φ production is the one in which ( sq ) is also producedin the forward direction, recoiling against [¯ s ( ud )]. Sincethe lightest constituent (here, q ) is the easiest to deflectthrough large angles, this particular exchange diagramprovides a natural mechanism for backward scattering of the q , as well as that of the ¯ s .Once the q and ¯ s are deflected through a large an-gle, the s and ( ud ) must each be deflected (through agenerically smaller angle) to become bound to into theirrespective ( sq ) and [¯ s ( ud )] combinations. This bindingis accomplished in our crude picture by the exchange ofa gluon between s and q , and between ¯ s and ( ud ), asdepicted in Fig. 3. As promised, the minimal numberof gluon exchanges required to achieve the desired finalstates is precisely three. One therefore naively expectsthis diagram to peak in the forward direction, and in thenext section we see this expectation indeed to be realized.A natural exchange diagram producing a backwardpeak is depicted in Fig. 4. Here, the interaction gluonconnects the heaviest component [( ud )] to the s , and is γ s ¯ s N ( ud ) ( ud ) qs ¯ s θ q FIG. 4: As in Fig. 1, but including a minimal gluon exchangethat naturally explains the enhancement of φ production inthe θ → π direction, as described in the text. The angle θ exhibited here is identical to the one in Fig. 3, for the purposeof ease of comparison. expected to produce an enhancement when both of themare deflected in the backward direction, the θ → π limit ofFig. 4. In that case, in order to form the [¯ s ( ud )] and ( sq )combinations, the ¯ s and q deflect through a smaller an-gle, and binding is accomplished through gluon exchangesbetween ( ud ) and ¯ s , and between s and q , respectively.Again, a minimum of precisely three gluon exchanges isnecessary to obtain the desired final state. And sincebackward deflection of the heavier ( ud ) is not as easy asbackward deflection of the lighter q , one expects the sizeof the enhancement in the backward direction producedby Fig. 4 to be smaller than the enhancement in the for-ward direction produced by Fig. 3, which is verified inthe next section to be true.Before proceeding to a calculation, we hasten to em-phasize its extremely rudimentary nature. First, we areinvestigating a particular production mechanism ( δ -¯ θ )that has never been demonstrated unambiguously to oc-cur in any process. The ( sq ) and [¯ s ( ud )] are assumedto act as bound quasiparticles, allowing for analysis asa 2 → γ and N come unbound by gluonexchange first, and then the interaction occurs. Or, thebinding and unbinding gluons can stretch across the in-teraction . In real QCD, one expects all such gluons toappear—copiously—in a typical diagram.To establish a systematic calculation, we consider all12 diagrams that have the minimum three planar gluons,such that each of the constituents must couple at leastonce to a gluon. Figures 3 and 4 may be considered asrepresentatives of diagram classes [with amplitudes M (1) and M (2) , respectively], as defined by the placement ofthe central interaction gluon that connects initial and fi-nal states. The members of each class are defined bywhich two external states have their constituents con-nected by a gluon. In a third allowed class representedby Fig. 5 [amplitudes M (3) ], the constituents of each ofthree of the external states are connected by a gluon.The definition of the 12 diagrams is summarized in Ta-ble I; thus, for example, the amplitudes corresponding tothe literal diagrams of Figs. 3, 4, and 5 are M (1)1 , M (2)1 ,and M (3)1 , respectively. It is also simple to check thateach of these 12 diagrams also has exactly two internalconstituent propagators. Such gluons would be nonplanar, and give contributions formallysuppressed by O (1 /N c ) relative to diagrams without them, where N c = 3 is the number of QCD colors. γ s ¯ s N ( ud ) ( ud ) qs ¯ s θ q FIG. 5: As in Fig. 1, but including a minimal gluon exchangethat does not have a natural central interaction gluon con-necting initial and final states.TABLE I: Amplitudes based on the classes defined by Figs. 3,4, 5 [defined as M (1) , (2) , (3) , respectively]. Each amplitude isdefined by its class and the listed subset of initial and fi-nal states, each member of which exchanges a single bindinggluon. M (1)1 ( δ , ¯ θ ) M (2)1 ( δ , ¯ θ ) M (3)1 ( γ , δ , ¯ θ ) M (1)2 ( γ , N ) M (2)2 ( γ , N ) M (3)2 ( N , δ , ¯ θ ) M (1)3 ( δ , N ) M (2)3 ( δ , γ ) M (3)3 ( γ , N , δ ) M (1)4 ( γ , ¯ θ ) M (2)4 ( N , ¯ θ ) M (3)4 ( γ , N , ¯ θ ) IV. CALCULATION AND RESULTS
We estimate the relative size of the diagrams in Table Iby establishing kinematics and definitions of momenta,and then by calculating the angular dependence origi-nating from the product of momentum transfer factorsforming the denominators of the three gluon propagatorsand two constituent propagators appearing in each dia-gram. Spin structures in the form of Dirac matrices orLorentz tensors are also ignored; indeed, the issue of howto parametrize the coupling of a gluon to the ( ud ) diquarkis avoided with this assumption. The precise recipe fortreating masses appearing in these factors is describedbelow. In short, we estimate the relative sizes and an-gular dependences of the 12 amplitudes M in Table I byusing just a few of the factors appearing in each of them.We begin with kinematical conventions. The externalcomposite particle masses, not only m γ = 0 and m p , butalso the diquark m ( sq ) and antitriquark m [¯ s ( ud )] masses,are assumed known, uniquely determining the c.m. en-ergy and momenta of the external particles in terms ofthe total c.m. energy √ s of the process, as in any 2 → √ s = E c . m . tot = q m N (2 E lab γ + m N ) ,E c . m .γ = s − m N √ s ,E c . m .N = s + m N √ s ,p c . m .γ = p c . m .N = E c . m .γ ,E c . m . ( sq ) = s − m s ( ud )] + m sq ) √ s ,E c . m . [¯ s ( ud )] = s − m sq ) + m s ( ud )] √ s , (1)and p c . m . ( sq ) = p c . m . [¯ s ( ud )] = rh s − (cid:0) m [¯ s ( ud )] + m ( sq ) (cid:1) i h s − (cid:0) m [¯ s ( ud )] − m ( sq ) (cid:1) i √ s . (2)Since the constituents of each composite particle areassumed to move with the same velocity, the fraction ofthe total momentum carried by each is assumed to be theratio of its mass to the total mass of the constituents. Inparticular, if we define r ≡ m s m s + m ¯ s ,r ≡ m q m q + m ( ud ) ,r ≡ m q m s + m q ,r ≡ m ¯ s m ¯ s + m ( ud ) , (3)and denote initial and final constituent momenta withprimes on the latter and not on the former, then p s = r p γ ,p ¯ s = (1 − r ) p γ ,p q = r p N ,p ( ud ) = (1 − r ) p N ,p ′ s = (1 − r ) p ( sq ) ,p ′ ¯ s = r p [¯ s ( ud )] ,p ′ q = r p ( sq ) ,p ′ ( ud ) = (1 − r ) p [¯ s ( ud )] . (4)This simple apportionment of momenta among con-stituents is completely analogous to the result in light-front quantum field theory [25]. Conservation of 4-momentum at each vertex then completely determinesthe momentum transfers in terms of the r i factors and scalar products between the external momenta, which inturn are completely determined by the external particlemasses and the scattering angle θ .Out of the 12 diagrams, only six distinct gluon mo-menta appear: q s ≡ p ′ s − p s ,q ¯ s ≡ p ′ ¯ s − p ¯ s q q ≡ p ′ q − p q ,q ( ud ) ≡ p ′ ( ud ) − p ( ud ) ,q ( sq ) ≡ p ( sq ) − p s − p q = p ¯ s + p ( ud ) − p [¯ s ( ud )] ,q s ¯ s ≡ p ′ s + p ′ ¯ s − p γ = p N − p ′ q − p ′ ( ud ) , (5)which satisfy between them three simple identities: q ( sq ) = q s + q q = − q ¯ s − q ( ud ) ,q s ¯ s = q s + q ¯ s = − q q − q ( ud ) , q s + q ¯ s + q q + q ( ud ) , (6)so that only three of them are linearly independent, ex-actly as one expects for momentum transfers betweenfour independent external momenta. Compared to thefull Feynman calculation with gluons in the amplitudes M , we treat the gluon propagator factors q j as the onlyrelevant ones for this analysis ( i.e. , we neglect polariza-tion tensor structures). However, using skeletal diagramssuch as Figs. 3, 4, 5 to model much more complicated di-agrams with extensive gluon and internal q ¯ q exchangesintroduces the potential for artificial kinematic singular-ities to arise when some of the internal lines go on massshell. Part of this problem is caused by the assumptionof zero transverse constituent momenta, but much of itsimply is due to the usual soft-gluon infrared singularitiesone expects in a perturbative treatment. Physically, anon-shell gluon corresponds to an infinite-range interac-tion between the constituents, which does not comportwith the finite range associated with confinement. Toaccount for this important dynamics, we provide the glu-ons with a finite mass scale m conf (with a specific valuechosen below), and introduce dimensionless gluon prop-agator factors Q j ≡ − m q j − m , (7)where the sign makes Q j positive in the usual case ofscattering four-momenta.Furthermore, eight distinct internal constituent mo-menta (in which the subscript refers to the particularconstituent line) appear: k s ≡ p ( sq ) − p q ,k ¯ s ≡ p [¯ s ( ud )] − p ( ud ) ,k q ≡ p ( sq ) − p s ,k ( ud ) ≡ p [¯ s ( ud )] − p ¯ s ,k ′ s ≡ p γ − p ′ ¯ s ,k ′ ¯ s ≡ p γ − p ′ s ,k ′ q ≡ p N − p ′ ( ud ) ,k ′ ( ud ) ≡ p N − p ′ q , (8)which are linearly independent except for the overall mo-mentum conservation constraint. In the full amplitudes M , the factors ¯ u j ( p f ) γ µ ( /k j − m j ) − γ ν u j ( p i ) appear forthe fermionic constituents. By treating the bosonizedpropagator denominator ( k j − m j ) as the only signifi-cant factor for this analysis, i.e. , by ignoring the effectof the spinor algebra except to use the Dirac equationof motion to eliminate /k j from the numerator, by usingthe conventional 2 m j normalization for spinors, and bynoting that scattering momentum transfers are negative,the relevant dimensionless factors for constituent lines inthis analysis are: K ( ′ ) j ≡ − m j k ( ′ ) 2 j − m j . (9)For convenience, we employ the same form for the(bosonic) diquark constituent ( ud ). From standard rela-tivistic kinematics using Eqs. (3)–(9), one finds that thepropagator factors Q s , Q ( ud ) , Q ( sq ) , Q s ¯ s , K ¯ s , K q , K ′ ¯ s ,and K ′ q are enhanced in the forward direction, while Q ¯ s , Q q , K s , K ( ud ) , K ′ s , and K ′ ( ud ) are enhanced in the back-ward direction. These results for Q s , Q ( ud ) , Q ¯ s , Q q wereanticipated in the last section.The choice of appropriate constituent masses also re-quires some care. Current quark masses would onlybe appropriate in a fully perturbative analysis in whichclasses of diagrams are resummed to avoid singularitiesassociated with particles going on shell. However, tra-ditional constituent masses are not entirely appropriatefor this calculation either, as they are typically obtainedfrom static processes, not a dynamical scattering such asphotoproduction. The choice of mass parameters, as seenbelow, lies somewhere in between these extremes.Even then, the appropriate mass to choose for a giveninternal constituent line suffers from ambiguity. Take,for example, the factor k q in Eq. (8), which refers to themomentum of a light q emerging from the breakup of thenucleon N en route to binding into a diquark ( sq ). Ac-cording to Eq. (4), should its mass be considered a frac-tion r of m N , or a fraction r of m ( sq ) ? Or indeed, sincethe propagator lies deep in the diagram, should one usethe confinement mass scale m conf previously introduced?We adopt the convention that the appropriate effectiveconstituent mass to appear in the propagators Eq. (9) is the maximum of the two suggested by the initial ( i ) andfinal ( f ) states into which it binds [according to Eqs. (3)and (4)] plus a contribution from the confinement scale m conf , to account for the expected off-shell behavior: m j, eff = max( m j,i , m j,f ) + m . (10)The model amplitudes of Table I in the simplified no-tation of Eqs. (7) and (9) read M (1)1 = Q s Q ( ud ) Q ( sq ) K ¯ s K q , M (1)2 = Q s Q ( ud ) Q s ¯ s K ′ ¯ s K ′ q , M (1)3 = Q s Q ( ud ) Q ¯ s K q K ′ q , M (1)4 = Q s Q ( ud ) Q q K ¯ s K ′ ¯ s , M (2)1 = Q ¯ s Q q Q ( sq ) K s K ( ud ) , M (2)2 = Q ¯ s Q q Q s ¯ s K ′ s K ′ ( ud ) , M (2)3 = Q ¯ s Q q Q ( ud ) K s K ′ s , M (2)4 = Q ¯ s Q q Q s K ( ud ) K ′ ( ud ) , M (3)1 = Q q Q ( ud ) Q ( sq ) K s K ¯ s , M (3)2 = Q s Q ¯ s Q ( sq ) K q K ( ud ) , M (3)3 = Q ¯ s Q ( ud ) Q s ¯ s K ′ s K ′ q , M (3)4 = Q s Q q Q s ¯ s K ′ ¯ s K ′ ( ud ) . (11)The chosen input masses, all in MeV, are listed in Ta-ble II. Again, guidance for this choice is suggested, butnot determined, by an interpolation between typical cur-rent and constituent masses used in the literature. Forexample, one can make an argument for ( ud ) diquarkmasses anywhere from ≃
30 MeV ( i.e. , not many timesmore than the sum of current quark masses) up to ≃
600 MeV ( i.e. , two-thirds of a nucleon, or the mass of a σ meson). The most important inputs in obtaining a re- TABLE II: Input masses in MeV. √ s m γ m N m ( sq ) m [¯ s ( ud )] m ( ud ) m s = m ¯ s m q m conf sult resembling experimental data appear to be choosing m ( sq ) + m [¯ s ( ud )] not terribly far below the observed for-ward peak at √ s = 2 . m conf . The necessary magnitude of m conf ap-pears to arise largely due to the feature of this simplifiedmodel that the initial s and ¯ s quarks from the dissocia-tion of γ are lightlike, as seen from the first two of Eq. (4);presumably, introducing substantial transverse momentain a more realistic calculation produces the same effectas m conf . In any case, the model presented here is sosimple that the input masses used should be viewed onlyas a qualitative guide to obtaining results in accord withexperiment.For the given inputs, the amplitudes of class M (1) areall strongly peaked in the forward direction, while thoseof class M (2) are all strongly peaked in the backward di-rection but possess in addition a smaller forward peak.In particular, the amplitude M (1)1 [ M (2)1 ] correspondingto Fig. 3 (Fig. 4) shows preferential forward (backward)scattering, as was anticipated from general arguments.The diagrams of class M (3) also turn out to be forwardenhanced, but many times smaller than those in class M (1) . To illustrate the forward-backward peaking, wepresent in Fig. 6 the simple coherent sum M of the 12amplitudes. Obviously, the absence of numerous factors FIG. 6: Coherent sum M of the 12 amplitudes of Eq. (11)using the inputs in Table II, as a function of cos θ . in the full Feynman amplitudes—not least of which arerelative signs leading to destructive interferences—meansthat the plot of Fig. 6 should only be taken seriously in itscoarsest features, namely, a large peak in the forward di-rection and a small peak in the backward direction. Theparticular value for the forward-to-backward ratio is sim-ple to adjust using slightly different inputs. Furthermore,we note that the CLAS data presented in Ref. [10] doesnot extend to the most forward and backward directions;the limitation − . ≤ cos θ ≤ +0 .
92 is used in Fig. 6.The literal ratio M (cos θ = +0 . /M (cos θ = − .
8) inFig. 6 is about 30.Lastly, one may ask whether the forward enhance-ment behaves away from the particular chosen peak value √ s = 2 . M (1)1 as a function of √ s but otherwise use the mass in-puts of Table II. The amplitude, rather than its square,is relevant because its chief contribution to data wouldpresumably appear through interference with the domi-nant Pomeron amplitude. The full width at half max- FIG. 7: Behavior of the amplitude M (1)1 as a function of √ s (in GeV), with the other inputs as in Table II. imum appears to extend from just below 2.0 GeV toabout 2.4 GeV, which is very much like the experimentalresult presented in Refs. [10, 12]. In this model, the en-hancement is indeed due to a special correlated five-quarkconfiguration, but it is not a literal resonant pentaquarkstate. That is, the anomalies may be said to correspondto “would-be” pentaquarks. V. CONCLUSIONS
We have seen that interesting forward-backward en-hancements observed in recent data for the photopro-duction process γp → φp can easily be explained throughthe production of a ( su ) color-¯ bound diquark and an[¯ s ( ud )] color- bound antitriquark, using the preferentialcolor couplings ⊗ ⊃ ¯ . Such a configuration (with s → c ) has previously been advocated as an explana-tion of the exotic charmoniumlike states. Here, however,no claim is made that the enhancements are the resultof true resonances, but they are predicted to create s -dependent bumps in the data.The model relies only on the preference of a systemwith largely collinear constituents to minimize the con-stituent momentum transfers. Essentially no dynamicsis included; the minimal set of gluon exchanges usedfor each diagram does not incorporate color physics inany fundamental way. Likewise, the constituents are as-sumed at any point in the diagrams to carry only mo-menta parallel to the bound state in which they oc-cur. Nevertheless, using plausible values for constituentmasses, one obtains results in good qualitative, and evensemi-quantitative, agreement with experiment. Numer-ous possible improvements leading to stable and predic-tive results for this process should certainly be under-taken.The possibility that diquark/triquark structure can bediscerned in lighter quark systems has long been dis-cussed, but the experimental signals have always beentantalizingly vague. In the analogues to the types ofexperiments already performed or proposed for heavy-quark exotics, one can hope to find a clear indication forthese novel structures. Acknowledgments
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