On the width of collective excitations in chiral soliton models
aa r X i v : . [ h e p - ph ] A p r On the width of collective excitations in chiral soliton models
Herbert
Weigel ∗ ) Fachbereich Physik, Siegen University, D–57068 Siegen, Germany
In chiral soliton models for baryons the computation of hadronic decay widths of baryonresonances is a long standing problem. For the three flavor Skyrme model I present a solutionto this problem that satisfies large– N C consistency conditions. As an application I focus onthe hadronic decay of the Θ and Θ ∗ pentaquarks. §
1. Statement of the problem
Hadronic decays of baryon resonances are commonly described by a Yukawainteraction of the generic structure L int ∼ g ¯ ψ B ′ φ ψ B , (1.1)where B ′ is the resonance that decays into baryon B and meson φ and g is a couplingconstant. It is crucial that this interaction Lagrangian is linear in the meson field.If φ is a pseudoscalar meson this interaction yields the decay width Γ ( B ′ → Bφ ) ∝ g | ~p φ | , with ~p φ being the momentum of the outgoing meson.The situation is significantly different in soliton models that are based on actionfunctionals of only meson degrees of freedom, Γ = Γ [ Φ ]. These action functionalscontain classical (static) soliton solutions, Φ sol , that are identified as baryons. Theinteraction of these baryons with mesons is described by the (small) meson fluctua-tions about the soliton: Φ = Φ sol + φ . By pure definition we have δΓ [ Φ ] δΦ (cid:12)(cid:12)(cid:12) Φ = Φ sol = 0 . (1.2)Thus there is no term linear in φ to be associated with the Yukawa interaction,eq. (1.1). This puzzle has become famous as the Yukawa problem in soliton mod-els. However, this does not mean that soliton models cannot describe resonancewidths. On the contrary, these widths can be extracted from meson baryon scat-tering amplitudes, just as in experiment. In soliton models two–meson processesacquire contributions from the second order term Γ (2) = 12 φ · δ Γ [ Φ ] δ Φ (cid:12)(cid:12)(cid:12) Φ = Φ sol · φ . (1.3)This expansion simultaneously represents an expansion in N C , the number of colordegrees of freedom: Γ = O ( N C ) and Γ (2) = O ( N C ). Terms O ( φ ) vanish in the limit N C → ∞ . This implies that Γ (2) contains all large– N C information about hadronicdecays of resonances. We may reverse this statement to argue about any computation ∗ ) e-mail address: [email protected] typeset using PTP
TEX.cls h Ver.0.9 i H. Weigel of hadronic decay widths in soliton models: For N C → ∞ it must identically matchthe result obtained from Γ (2) . Unfortunately, the most prominent baryon resonance,the ∆ isobar, becomes degenerate with the nucleon as N C → ∞ . It is stable in thatlimit and its decay is not subject to the above described litmus–test. The situationis more interesting in soliton models for flavor SU (3). In the so–called rigid rotatorapproach (RRA), that generates baryon states as (flavor) rotational excitations of thesoliton, exotic resonances emerge that dwell in the anti–decuplet representation offlavor SU (3). The most discussed (and disputed) such state is the Θ + pentaquarkwith zero isospin and strangeness S = +1. In the limit N C → ∞ the (would–be)anti–decuplet states maintain a non–zero mass difference with respect to the nucleon.Therefore the properties of Θ + predicted from any model treatment must also be seenin the S –matrix for koan–nucleon scattering as computed from Γ (2) . This (seeminglyalternative) quantization of strangeness degrees of freedom is called the bound stateapproach (BSA) because in the S = − φ yield a bound state. Its occupation serves to describe the ordinary hyperons, Λ , Σ , Σ ∗ , etc.. The above discussed litmus–test requires that the BSA and RRA giveidentical results for the Θ + properties as N C → ∞ . This did not seem to be trueand it was argued that the prediction of pentaquarks would be a mere artifact of theRRA. Here we will show that this is a premature conclusion and that pentaquarkstates do indeed emerge in both approaches. Furthermore the comparison betweenthe BSA and RRA provides an unambiguous computation of pentaquark widths: Itdiffers substantially from previous approaches that adopted transition operatorsfor Θ + → KN from the axial current.This presentation is based on ref. which the interested reader may want toconsult for further details. §
2. The model
For simplicity we consider the Skyrme model as a particular example for chiralsoliton models. However, we stress that our qualitative results do indeed generalizeto all chiral soliton models because these results solely reflect the treatment of themodel degrees of freedom.Chiral soliton models are functionals of the chiral field, U , the non–linear real-ization of the pseudoscalar mesons ∗ ) , φ a U ( ~x , t ) = exp (cid:20) if π φ a ( ~x , t ) λ a (cid:21) , (2.1)with λ a being the Gell–Mann matrices of SU (3). For a convenient presentation ofthe model we split the action into three pieces Γ = Γ SK + Γ W Z + Γ SB . (2.2) ∗ ) A remark on notation: In what follows we adopt the convention that repeated indices aresummed over in the range a, b, c, . . . = 1 , . . . , α, β, γ, . . . = 4 , . . . , i, j, k, . . . = 1 , , n the width of collective excitations Γ SK = Z d x tr (cid:26) f π h ∂ µ U ∂ µ U † i + 132 ǫ h [ U † ∂ µ U, U † ∂ ν U ] i(cid:27) . (2.3)Here f π = 93MeV is the pion decay constant and ǫ is the dimensionless Skyrmeparameter. In principle this is a free model parameter. The two–flavor version ofthe Skyrme model suggests to put ǫ = 4 .
25 from reproducing the ∆ –nucleon massdifference ∗ ) . The QCD anomaly is incorporated via the Wess–Zumino action Γ W Z = − iN C π Z d x ǫ µνρστ tr [ α µ α ν α ρ α σ α τ ] , (2.4)with α µ = U † ∂ µ U . Note that Γ W Z vanishes in the two–flavor version of the model.The flavor symmetry breaking terms are contained in Γ SB Γ SB = f π Z d x tr h M (cid:16) U + U † − (cid:17)i , M = diag (cid:0) m π , m π , m K − m π (cid:1) . (2.5)We do not include terms that distinguish between pion and kaon decay constantseven though they differ by about 20% empirically. This omission is a matter ofconvenience and leads to an underestimation of symmetry breaking effects whichapproximately can be accounted for by rescaling the kaon mass m K → m K f K /f π .The action, eq. (2.2) allows for a topologically non–trivial classical solution, thefamous hedgehog soliton Φ sol ∼ U ( ~x ) = exp h i~λ · ˆ xF ( r ) i , r = | ~x | (2.6)embedded in the isospin subspace of flavor SU (3). The chiral angle, F ( r ) solves theclassical equation of motion subject to the boundary condition F (0) − F ( ∞ ) = π ensuring unit winding (baryon) number. The soliton can be constructed as a functionof the dimensionless variable ǫf π r and is thus not subject to N C scaling.In the RRA baryon states are generated by canonically quantizing collectivecoordinates A ∈ SU (3) that describe the (spin) flavor orientation of the soliton, A ( t ) U ( ~x ) A † ( t ). The resultant eigenstates may be classified according to SU (3)multiplets; see ref. for a review. §
3. Large N C P –wave channel phase shifts with strangeness As motivated after eq. (1.3) we introduce fluctuations φ ∼ η α ( ~x , t ) U ( ~x , t ) = p U ( ~x ) exp (cid:20) if π λ α η α ( ~x , t ) (cid:21) p U ( ~x ) , (3.1)for the kaon fields. Expanding the action in powers of these fluctuations is anexpansion in η α /f π and thus a systematic series in 1 / √ N C . The term quadratic ∗ ) To ensure that the (perturbative) n –point functions scale as N − n/ C we substitute f π =93MeV p N C / ǫ = 4 . p /N C in the study of the N C dependence. H. Weigel in η α describes meson scattering off a potential generated by the classical soliton,eq. (2.6). The P –wave mode is characterized by a single radial function (cid:18) η + iη η + iη (cid:19) P ( ~x , t ) = Z ∞−∞ dω e iωt η ω ( r ) ˆ x · ~τ χ ( ω ) . (3.2)In future we will omit the subscript that indicates the Fourier frequency. Uponquantization the components of the two–component iso–spinor χ ( ω ) are elevated tocreation– and annihilation operators. It is straightforward to deduce the Schr¨odingertype equation h η ( r ) + ω [2 λ ( r ) − ωM K ( r )] η ( r ) = 0 with h = − d dr − r ddr + V eff ( r ) . (3.3)The radial functions arise from the chiral angle F ( r ) and may be readily takenfrom the literature. The equation of motion (3.3) is not invariant under particleconjugation ω ↔ − ω , and thus different for kaons ( ω >
0) and anti–kaons ( ω < ω = − ω Λ , that gives the mass difference between the Λ –hyperon and thenucleon in the large– N C limit. As this energy eigenvalue is negative it correspondsto a kaon, i.e. it carries strangeness S = −
1. In the symmetric case ( m K = m π ) thebound state is simply the zero mode of SU (3) flavor symmetry. The WZ–term movesthe potential bound state with S = +1 to the positive continuum and we expect aresonance structure in that channel. The corresponding phase shift is shown in theleft panel of figure 1. No clear resonance structure is visible; the phase shifts hardlyreach π/
2. The absence of such a resonance has previously lead to the prematurecriticism that there would not exist a bound pentaquark in the large– N C limit. §
4. Constraint fluctuations
To study the coupling between the fluctuations and the collective excitations wegeneralize eq. (3.1) to U ( ~x , t ) = A ( t ) p U ( ~x ) exp (cid:20) if π λ α e η α ( ~x , t ) (cid:21) p U ( ~x ) A † ( t ) . (4.1) pha s e s h i ft kaon momentum [GeV/c]BOUND STATE APPROACHm K = 495 MeV -1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 pha s e s h i ft kaon momentum [GeV/c]BOUND STATE APPROACHm K = 495 MeV resonancephase-shiftbackgroundphase shift Fig. 1. Large N C P –wave phase shifts with strangeness S = +1 as function of the kaon momentum.Left panel: unconstrained; right panel: constrained to be orthogonal to the collective rotation. n the width of collective excitations P –wave is subject to the modified integro–differential equation h e η ( r ) + ω [2 λ ( r ) − ωM K ( r )] e η ( r ) = − z ( r ) (cid:20)Z ∞ r ′ dr ′ z ( r ′ )2 λ ( r ′ ) e η ( r ′ ) (cid:21) × (cid:20) λ ( r ) − ( ω + ω ) M K ( r ) − ω (cid:18) X Θ ω Θ − ω + X Λ ω (cid:19) (2 λ ( r ) − ω M K ( r )) (cid:21) , (4.2)for the flavor symmetric case ∗ ) . The radial function e η ( r ) is defined according toeq. (3.2) and z ( r ) = √ π f π √ Θ K sin F ( r )2 is the collective mode wave–function nor-malized with respect to the moment of inertia for flavor rotations into strangenessdirection, Θ K = f π R d rM K ( r ) sin F ( r )2 = O ( N C ). The non–local terms without X Λ,Θ reflect the constraint R drr z ( r ) M K ( r ) e η ( r ) = 0 which avoids double countingof rotational modes in strangeness direction. The interesting coupling is containedin the interaction Hamiltonian H int = 2 √ πΘ K d iαβ D γα R β Z d r z ( r ) [2 λ ( r ) − ω M K ( r )] ˆ x i e ξ γ ( ~x , t ) , (4.3)where e ξ a = D ab e η b are the fluctuations in the laboratory frame, that we actuallydetect in KN scattering. The collective coordinates are parameterized via the adjointrepresentation D ab ( A ) = tr (cid:2) λ a Aλ b A † (cid:3) and the SU (3) generators R a . Integratingout the collective degrees of freedom by means of standard perturbation theoryinduces the separable potential |h Θ | H int | ( KN ) I =0 i| ω Θ − ω + |h Λ | H int | ( KN ) I =0 i| ω Λ + ω . (4.4)These matrix elements concern the T –matrix elements in the laboratory frame. Sincethe laboratory and intrinsic T –matrix elements are identical for the Θ + channel, we may add the exchange potential, eq. (4.4) in the intrinsic frame. We define matrixelements of collective coordinate operators h Θ + | d αβ D + α R β | n i =: X Θ r N C
32 and h Λ | d αβ D − α R β | p i =: X Λ r N C , (4.5)to end up with eq. (4.2). The first factor in the coefficient ω = 2 (cid:18) √ Θ K q N C (cid:19) = N C Θ K arises in the equation of motion because the potential, eq. (4.4) is quadratic inthe fluctuations. The remaining (squared) factors stem from the definitions of X Θ,Λ and the constant of proportionality in H int . The X Θ,Λ must be computed with themethods provided in ref. but generalized to arbitrary (odd) N C . For N C → ∞ we have X Θ → X Λ →
0. From the orthogonality conditions of the equation ofmotion (3.3) we straightforwardly verify that e η ( r ) = η ( r ) − az ( r ) with a = Z ∞ drr z ( r ) M K ( r ) η ( r ) . (4.6) ∗ ) The more complicated case m K = m π is at length discussed in ref. H. Weigel solves eq. (4.2) for large N C . This is essential because, as z ( r ) is localized in space, η and e η have identical phase shifts! Hence the litmus–test discussed in the introduc-tion is indeed satisfied. The physics behind e η is best understood when introducingreduced wave–functions η ( r ) that solve eq. (4.2) modified with X Θ ≡ X Λ ≡ i. e. the collective excitations are decoupled. These wave–functions are still orthogonalto the collective modes and actually lead to the background phase shift shown infigure 1. Having obtained the η ( r ) we may again switch on the exchange contribu-tions, eq. (4.4). The additional separable potential is treated by standard R–matrixtechniques and augments the phase shift bytan ( δ R ( k )) = Γ ( ω k ) / ω Θ − ω k + ∆ ( ω k ) . (4.7)Here ω Θ = N C +34 Θ K is the RRA result for the excitation energy of Θ . This phase shiftexhibits the canonical resonance structure with the width and the pole shift Γ ( ω k ) = 2 kω X Θ (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r dr z ( r )2 λ ( r ) η ω k ( r ) (cid:12)(cid:12)(cid:12)(cid:12) , (4.8) ∆ ( ω k ) = 12 πω k P Z ∞ qdq (cid:20) Γ ( ω q ) ω k − ω q + Γ ( − ω q ) ω k + ω q (cid:21) , (4.9)respectively. We have numerically verified that in the large– N C limit with X Θ = 1,the phase shift from eq. (4.7) is identical to what is labeled resonance phase shift infigure 1, that we calculated as the difference between the total ( η ) and background ( η )phase shifts. For finite N C we have X Θ = 1 and X Λ = 0 so the R–matrix formalismbecomes two–dimensional ( Λ and Θ + exchange). Contrary to earlier criticisms thelarge N C pentaquark phase shift indeed resonates! §
5. Results
In figure 2 we show the resonance phase shift computed from eq. (4.7) for variousvalues of N C . First we observe that the resonance position quickly moves towardslarger energies as N C decreases. This is mainly due to the strong N C dependenceof ω Θ : For N C = 3 it is twice as large as in the limit N C → ∞ . The pole shift ∆ is actually quite small (some ten MeV) so that ω Θ is indeed a reliable estimate ofthe resonance energy. Second, the resonance becomes shaper as N C decreases. Tomajor parts this is caused by the reduction of X Θ .We now turn to more quantitative results for which we also include flavor sym-metry breaking effects. Then the resonance position changes to ω Θ = 12 "r ω + 3 Γ Θ K + ω + O (cid:18) N C (cid:19) . (5.1)where Γ = O ( N C ) is a functional of the soliton that is proportional to the mesonmass difference, m K − m π . The O (1 /N C ) piece is sizable for N C = 3 and we computeit in the scenario of ref. We then find ω Θ ≈ n the width of collective excitations r e s onan c e pha s e s h i ft kaon momentum [GeV/c]m K = m π N C =3N C =5N C =9N C =27N C → ∞ Fig. 2. The resonance phase shift as a function of N C for m K = m π . into account we expect the pentaquark to be about 600 . . . H sbint = (cid:0) m K − m π (cid:1) d iαβ D γα D β Z d r z ( r ) γ ( r ) e ξ γ ( ~x , t )ˆ x i , (5.2)The radial function γ ( r ) is again given in terms of the chiral angle. Second, the X Λ does not vanish as N C → ∞ and the R –matrix formalism is always two dimensional.Nevertheless, the large– N C solution is always of the form (4.6) and the BSA phaseshift is recovered. The width function turns to Γ ( ω k ) = 2 kω (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r dr z ( r ) (cid:20) X Θ λ ( r ) + Y Θ ω (cid:0) m K − m π (cid:1)(cid:21) η ω k ( r ) (cid:12)(cid:12)(cid:12)(cid:12) , (5.3)where X Θ and Y Θ = p N C / h Θ + | d αβ D + α D β | n i are to be computed in the RRAapproach with full inclusion of flavor symmetry breaking effects.This width function is shown (for N C = 3) in figure 3 for Θ and its isovectorpartner Θ ∗ . The latter merely requires the appropriate modification of the matrixelements in eq. (4.5). Most importantly, the k behavior of the width function, assuggested by the model, eq. (1.1) is reproduced only right above threshold, afterwardsit levels off. Second, and somewhat surprising, the width of the non–ground statepentaquark is smaller than that of the lowest lying pentaquark. Our particularmodel yields Γ Θ ≈ Γ Θ ∗ ≈ de c a y w i d t h [ M e V ] kaon momentum [GeV/c] Θ DECAY m K = m π m K = 495 MeV m K = 750 MeV 0 5 10 15 0 0.2 0.4 0.6 0.8 de c a y w i d t h [ M e V ] kaon momentum [GeV/c] Θ * DECAY m K = m π m K = 495 MeV m K = 750 MeV Fig. 3. Model prediction for the width, Γ ( ω ) of Θ + (left) and Θ ∗ + (right) for N C = 3 as functionof the momentum k = p ω − m K for three values of the kaon mass. Note the unequal scales. H. Weigel §
6. Conclusions
We have discussed the chiral soliton model approach to KN scattering in the S = +1 channel which contains the potential Θ + pentaquark, a state predicted as aflavor rotational excitation of the soliton. Though the exactly known large N C phaseshift suggests otherwise, the Θ emerges as a genuine resonance. Our central resultis the width function for Θ → KN . In the flavor symmetric case it contains only a single collective coordinate operator and is thus very different from estimates thatextract an effective Yukawa coupling from the axial current matrix element. Sinceour approach matches the exact large N C result, we must conclude that those axialcurrent scenarios are erroneous and that the cancellation among contributions tothis matrix element is an invalid argument for a small pentaquark width. Acknowledgments
This key note is based on a collaboration with H. Walliser, whose contributionis highly appreciated. I am very grateful to the organizers of this workshop for theinvitation to contribute to the proceedings despite I was unable to participate.
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