Kaon-Soliton Binding and Pentaquark Skyrmion

1. Kaon-Soliton Binding and Pentaquark Skyrmion Mannque Rho (Saclay & Hanyang) 2004 Based on work with D.-P. Min (and B.-Y. Park)

2. Pentaquark story • Subsequent to earlier predictions in 1980’s, Diakonov et al (DPP) predicted in the SU(3) skyrme model a Q+ at mQ~ 1540 MeV with spin–parity ½+. • The LEPS “confirmed” the prediction: mQ=1540 ±10 MeV with width < 25 MeV. • Exotic since it must involve 5 quarks and flavor 10. • Prediction highly controversial since the skyrmion picture used is inconsistent with NC counting.

3. Pentaquark Q+ from skyrmions • Theorists, i.e., Tom Cohen and Igor Klebanov and others, show that the rigid rotor quantization of the SU(3) skyrmions employed by DPP is consistent with large NC argument of QCD for S<0 but inconsistent for S>0 as in the case of the Q+. • The same theorists show that the “bound-state model” or Callan-Klebanov model where the kaon is bound to an SU(2) soliton is consistent in NC for bothS<0 and S>0. • But in the standard Skyrme model with (pseudo)Goldstone fields only, there is no K+-soliton binding because of the Wess-Zumino term repulsion and there cannot be Q+ with a narrow width. • Their bet: There is no sharp resonance Q+ and hence what the experimentalists saw is not real!

4. Bound kaon-soliton picture • Callan-Klebanov (1985) model for baryons containing strange quarks: Kaon-skyrmion bound complex with kaon as “vibration.” Very successful for S=-1 hyperons • Applied to S=+1 systems, consistent with large NC counting. But no binding since the WZ term is repulsive unless … Hedgehog Soliton bound kaon SU(2) collective Coordinate Quatization N, S, X, D, S*, X*, W

5. Kaon-soliton binding mechansism • Kaons vibrate in the presence of SU(2) hedgehog (skyrmion) l- =l+ =WZ <0 Accounts for O(NC0) in the strangeness direction but no Casimir contributions. Note: The WZ is repulsive in the K+-N channel, so no binding

6. Vector mesons figure importantly Vector mesons change the situation dramatically!! There are many different ways of introducing vector degrees of freedom. But we know of only one way that we know can address the problem in consistent way… We favor “hidden local symmetry (HLS)” approach: Allows going up in scale while making contact with QCD.

7. Hidden local symmetry (HLS) • Given a set of low-lying degrees of freedom effective up to a scale Lbut what is known well is the interaction at very low excitations. Question: How to extend the low energy effective field theory so as to make it valid as one approches the scale L? • Example: At very low energy, hadronic interactions are governed by the interactions of the (pseudo-)Goldstone bosons, i.e., the triplet of pions but at higher energies the vector mesons r and w can intervene. Question: How to extend the chiral theory of pions to the scale where the vector mesons become relevant?

8. HLS : “Theory Space” Arkani-Hamed, Cohen and Georgi, PRL 86, 4757 (01) Illustration: closed “moose” diagram Consider vector fields A1 and A2on two sites 1 and 2 connected by a nonlinear sigma model link field U U=exp (ip/f) representing the gauge symmetry SU(n)1ХSU(n)2 U transforms linearly under the gauge symmetries U=g2-1 U g1

9. HLS : Theory Space (2) Gauge invariant Lagrangian: with Unitary gauge U=1. Then Let g1=0. The massless A1field decouples and we are left with The vector field A2is now massive with mass mA2=g2f.

10. HLS : Theory Space (3) The physics of gauge-invariant Lagrangian with Goldstone bosons is the same as the physics of gauge-noninvariant Lagrangian without Goldstone bosons. But gauge-invariant theory has advantages: Power (Advantage) of Gauge-Invariant Theory 1. With Goldstone bosons, one can locate where the EFT breaks down, i.e., where “new physics” shows up: It becomes strong coupling at ~4pmA/g ~4pf where f is the Goldstone decay constant. Without them, very complicated and awkward … 2. Can write higher-order terms in terms of the power of covariant derivatives. In unitary gauge, these correspond to This is cumbersome at best and in practice it is very difficult to do power counting systematically. This is the problem with massive vector meson theories, e.g. massive YM or tensor formalism.

11. HLS : Theory Space (3) Generalization When the EFT breaks down, need • Ultraviolet completion to “fundamental theory” that gives the Goldstones at low energies Higher dimension when N goes to infinity: Dimensional Deconstruction

12. HLS : simplest form Harada and Yamawaki, Phys. Rep. 381, 1-233 (03) • Consists of one site and two links with (L,R) boundaries. The relevant degrees of freedom are p’s, s’s and r’s. a1’s can be thought of as having been integrated out. • Theory sensible to a cutoff ~ 4p mr/g beyond which should be unltraviolet completed. In Harada-Yamawaki (HY), this is done by Wilsonian matching to QCD. Matching is chosen so that a1lies above the matching scale. • Parameters: gauge couplingg, a, pion decay constant fp

13. HLS : Matching to QCD Wilsonian Matching at the scale L • Formally integrate out the degrees of freedom above L where quarks and gluons live • The parameters of the bare EFT at L are to be given in terms of QCD variables at L • Matching is done with correlators. • QCD variables are given in terms of quark and gluon condensates and color gauge coupling etc. • Now the condensates depend on the background B such as temperature (T), density (n), number of flavors (NF) etc. Therefore the parameters of the bare Lagrangian depend on the background B.

14. HLS : RGE vector manifestation (VM) (1) At one loop • THE Fixed Point consistent with QCD: • If the quark condensate , then “vector manifestation” is realized:

15. HLS : RGEVM continued (2) Fixed point: Conclusion: In the chiral limit, as the critical point at which chiral symmetry changes from Goldstone to Wigner and vice-versa, the vector-meson (parametric as well as pole) mass must vanish in HLS theory. This result was confirmed by Harada and Yamawaki at the critical number of flavor NFc, by Harada, Kim and MR at the critical density nc and by Harada and Sasaki at the crtitcal temperature Tc. The HLS theory with VM: HLS/VM

16. Summary: VM Harada and Yamawaki Phys. Rept.381 (2003) 1 • (Wilsonian) matching to QCD at L and renormalization group analysis reveals the vector manifestation (VM) fixed point: • The VM fixed point is reached at the chiral phase transition (in the chiral limit) G.E. Brown, MR, 1991

17. HLS : two points where theory is known 1. “VM”: VM fixed point 2. “VAC”:T=n=0, nF < nC (matter-free vacuum) g≠0, a≠1, mr≠0 VM VAC VM Question: Where should the observer living between the VM and VAC fluctuate from, the point 1 or the point 2?

18. Meaning of a=1 <0|Vm|s(q)>=iqmFs , <0|Am|p(q)>=iqmFp • a≡Fs2/ Fp2=1 • “Mended Symmetry” ?

19. Evidence that a is near 1 in nature • Pion mass difference: Dmp2 =(mp+2 - mp02) • Nucleon EM form factor: GE/GM • Chiral doubling in heavy-light hadrons

20. HLS : Dmp2 =(mp+2 - mp02) from “VAC” • Standard current algebra treatment on the pion EM mass difference a la Das et al (1967) was anchored on Weinberg sum rules that invoke p, r, a1, leading to the cancellation of quadratic divergences between them • In HLS, the cancellation of the quadratic divergence takes place when a=1 which is the fixed point of a.

21. “Theory-space locality” • The vanishing of the quadratic divergence in Dmp2 is analogous to the “little Higgs” mechanism for resolving hierarchy problem in the Standard Model. The breaking of the theory-space locality by EM gives non-zero Dmp2. Harada, Tanabashi and Yamawaki 2003 r0 - photon coupling

22. Nucleon form factor for a ≈1 • Vector dominance (VD) corresponding toa≈2 observed in the pionic form factor is an “accident,” lying on an unstable RG trajectory. In non-zero temperature, a flows toward the fixed point a=1, i.e. the VD ismaximally broken(Harada and Yamawaki 03; Harada and Sasaki 04). • In baryonic environment, the VD is also maximally broken with a≈ 1. This was noticed a long ago (Iachello, Jackson and Lande 73, Brown, Rho and Weise 86) • JLab results can be interpreted with a=1; e.g., Iachello, Bijker, Wan 04. a≈1

23. Chiral doubling in heavy-light mesons • Consider hadrons made up of heavy quarks Q=c,b,t and light quarks q=u,d,s. Assume mQ=∞ and mq=0. Combine heavy-quark symmetry (HQS) and chiral symmetry (CS) to the leading order in mq and 1/mQ. • BaBar, CLEOII, Belle collaborations found/confirmed excited (C-anti-q) mesons denotedwith the splitting from the ground state of about 350 MeV which is much too small for standard quark models but of the same size as the constituent quark mass of the light quarks.

24. Confirmed by Belle

25. Chiral doubling: Concept • Consider specifically the D mesons (charm quark being the heavy quark) but arguments hold better for heavier quarks.

26. Viewed from “VAC” • The mass shift tags to the velocity, so the sign change in the mass shift for different parity states. • The constituent quark mass S≈mN/3≈310 MeV. • The heavier the heavy quark mass the better is the prediction. Nowak, Rho and Zahed 92 Bardeen and Hill 93 0- 0+ Goldberger-Treiman

27. HLS : two points where theory is known 1. “VM”: VM fixed point 2. “VAC”:T=n=0, nF < nC (matter-free vacuum) g≠0, a≠1, mr≠0 VM VAC VM Question: Where should the observer living between the VM and VAC fluctuate from, the point 1 or the point 2?

28. Viewed from “VM”

29. Near “VM” • Assume that “VM” is not too far from the real world. • At the VM, g=0, a=1, Mr=0. At this point, the HLS theory is completely known. Now fluctuating from the VM, write the Lagrangian for g≠0, Mr≠0 and in power of (a-1) due to chiral symmetry breaking. • To match to QCD, write the Lagrangian at the matching scale L.

30. Match to QCD • At L, match HLS to QCD using scalar density and pseudoscalar density correlators: GS, GP.

31. “Bare” mass splitting Compute Quantum Corrections by RGE

32. At the end of the day Harada, Sasaki, MR, Phys. Rev. D 04 • The r exchange is the only graph that contributes • To one-loop order a=1

33. Prediction • Solution of RGE • Parameters: Cquantum≈ 1.6

34. Prediction(Harada, Sasaki, MR. 04) ~ (1/3) mProton Numerically the same result starting from “vac” and from “VM” Up to the order considered a=1 works well!

35. Predicting a new structure for the pentaquark Byung-Yoon Park, Dong-Pil Min, MR, hep-ph/0405246

36. The result In HLS theory with a≈1, a bound K+-soliton can be formed with the quantum numbers of Q+ in consistency with NC counting. The crucial mechanism is the vector-meson contribution to “effective WZ” term that involves kaons without affecting the topological term.

37. Binding modified by gauge fields • Kaons vibrate in the presence of SU(2) hedgehog (skyrmion) • l- = WZ + v(a)- <0 • l+= WZ + v(a)+ • >0 for a<1.4, • <0 for a≥1.4. Vector mesons WZ (topological) Vector mesons (V)

38. Large NC limit • The HLS Lagrangian matched to QCD at L~ 1 GeV is a large NC Lagrangian, suitable for building skyrmions. • Harada and Yamawaki analyzed hadrons starting with this Lagrangian as a “bare” Lagrangian and determined a≈1.3, g≈3.69, fp≈145 MeV. • The K+-skyrmion system is bound with B.E.≈ 3 MeV

39. Varying “a” • The K-K* level repulsion suppresses the WZ, so bringing in attraction. • a<1.4 makes both K± be bound to the soliton • Not only the hyperons but also some pentaquark states can be kaon-skyrmion bound states. • For a<1.4 a bound pentaquark with the quantum numbers of Q+ can exist. • Nature favors a ~ 1. • But is this the Q+ “seen” in some experiments? Q+ L(1405) L(1116) Nature a < 1.4

40. What we have… • Nature favors a ~ 1. • For a<1.4, both K± are bound to the SU(2) soliton. • Fine for the hyperons with JP=1/2+but also for the odd parity S=-1 states with JP=1/2-and JP=3/2- states. • BUT the bound K+-skyrmion system (call it, T) when quantized is a bound object in the KN channel, i.e., MT < mN+ mK. So it cannot be the Q+ “seen” in some experiments as a resonance in the KN channel.

41. Fine-tuned “a” • When a is fine-tuned to a≈1.5, one can have a narrow-width resonance in the Q+ channel. But no justification for the fine-tuning. Q+ L(1405) L(1116) Nature a < 1.4

42. What it can be… • (a) If one can find a repulsive mechanism that makes MT > mN+ mK then T can be a CDD pole and produce a Feshbach-type resonance with tiny width and hence the Q+. The width can be tiny because of symmetry restoration (“mended symmetry”?) with a≈1. • (b) Otherwise the theory predicts a genuine bound state with the Q+ quantum numbers in the KN channel. This will be a new state, hitherto unobserved due to weak coupling, but may exist. But it will not be the Q+ believed to be seen in some experiments. • (c) If neither (a) nor (b), then the Q+ does not exist (Tom Cohen and Igor Klebanov are right?) and the experimentalists should get back and check their results again – which they are doing!! It boils down to: What does Nature choose for a ?

43. My conjecture for tiny width My guess is that in baryonic system, Nature chooses a≈ 1 Suppose that the bound K+-soliton for a≈1 becomes a CDD pole in the KN continuum. Then the theory-space locality is broken only by g≠0, in analogy to the EM mass difference of the pion and explains the weak coupling needed for understanding the tiny width. (Caveat: Disowned by BYP!)