The Cheshire Cat Principle from Holography ^{1}^{1}1To appear in Multifaceted Skyrmion, Eds. G. E. Brown and M. Rho, World Scientific
Holger Bech Nielsen and Ismail Zahed
Niels Bohr Institute, 17 Blegdamsvej, Copenhagen, Denmark
Department of Physics and Astronomy, SUNY StonyBrook, NY 11794
March 4, 2021
The Cheshire cat principle states that hadronic observables at low energy do not distinguish between hard (quark) or soft (meson) constituents. As a result, the delineation between hard/soft (bag radius) is like the Cheshire cat smile in Alice in wonderland. This principle reemerges from current holographic descriptions of chiral baryons whereby the smile appears in the holographic direction. We illustrate this point for the baryonic form factor.
1 Introduction
Back in the eighties, quark bag models were proposed as models for hadrons that capture the essentials of asymptotic freedom through weakly interacting quarks and gluons within a bag, and the tenets of nuclear physics through strongly interacting mesons at the boundary. The delineation or bag radius was considered as a fundamental and physically measurable scale that separates ultraviolet from infrared QCD [1].
The Cheshire cat principle [2] suggested that this delineation was unphysical in low energy physics, whereby fermion and color degrees of freedom could readily leak through the bag radius, making the latter immaterial. In a way, the bag radius was like the smile of the Cheshire cat in Alice in wonderland. The leakage of the fundamental charges was the result of quantum effects or anomalies [3].
In 1+1 dimensions exact bosonization shows that a fermion can translate to a boson and viceversa making the separation between a fermionic or quark and a bosonic or meson degree of freedom arbitrary. In 3+1 dimensions there is no known exact bozonization transcription, but in large the Skyrme model has shown that baryons can be decently described by topological mesons. The Skyrmion is the ultimate topological bag model with zero size bag radius [4], lending further credence to the Cheshire cat principle.
The Skyrme model was recently seen to emerge from holographic QCD once chiral symmetry is enforced in bulk [5]. In holography, the Skyrmion is dual to a flavor instanton in bulk at large and strong t’Hooft coupling [5, 6]. The chiral Skyrme field is just the holonomy of the instanton in the conformal direction. This construction shows how a flavor instanton with instanton number one in bulk, transmutes to a baryon with fermion number one at the boundary.
Of course, QCD is not yet in a true correspondence with a known string theory, as SYM happens to be according to Maldacena’s conjecture [7]. Perhaps, one way to achieve this is through the downup string approach advocated in [8]. Throughout, we will assume that the correspondence when established will result in a model perhaps like the one suggested in [5] for the light mesons and to which we refer to as holographic QCD.
With this in mind, holographic QCD provides a simple realization of the Cheshire cat principle at strong coupling. In section 2, we review briefly the holonomy construction for the Skyrmion in holography and illustrate the Cheshire cat principle. In section 3 we outline the holographic model. In section 4 we construct the baryonic current. In section 5 we derive the baryonic form factor. Many of the points presented in this review are borrowed from recent arguments in [9].
2 The Principle and Holography
In holographic QCD, a baryon is initially described as a flavor instanton in the holographic Zdirection. The latter is warped by gravity. For large , the warped instanton configuration is not known. However, at large the warped instanton configuration is forced to due to the high cost in gravitational energy. As a result, the instanton in leading order is just the ADHM configuration with an additional U(1) barynonic field, with gauge components [5]
(1) 
with all other gauge components zero. The size is . We refer to [5] (last reference) for more details on the relevance of this configuration for baryons. The ADHM configuration has maximal spherical symmetry and satisfies
(2) 
with a rigid SO(3) rotation, and is SU(2) analogue..
The holographic baryon is just the holonomy of (1) along the gravity bearing and conformal direction ,
(3) 
The corresponding Skyrmion in large and leading order in the strong coupling is with the profile
(4) 
In a way, the holonomy (3) is just the fermion propagator for an infinitly heavy flavored quark with the conformal direction playing the role of time. (3) is the bosonization of this conformal quark in 3+1 dimensions.
The ADHM configuration in bulk acts as a pointlike Skyrmion on the boundary. The baryon emerges from a semiclassical organization of the quantum fluctuations around the pointlike source (3). To achieve this, we define
(5) 
The collective coordinates and the fluctuations in (5) form a redundant set. The redundancy is lifted by constraining the fluctuations to be orthogonal to the zero modes. This can be achieved either rigidly [10] or nonrigidly [11]. We choose the latter as it is causality friendly. For the collective isorotations the nonrigid constraint reads
(6) 
with the real generators of .
For and the nonrigid constraints are more natural to implement since these modes are only soft near the origin at large . The vector fluctuations at the origin linearize through the modes
(7) 
with . In the spinisospin 1 channel they are easily confused with near the origin as we show in Fig. 1.
Using the nonrigid constraint, the double counting is removed by removing the origin from the vector mode functions
(8) 
with which becomes the origin for large . In the nonrigid semiclassical framework, the baryon at small is described by a flat or uncurved instanton located at the origin of and rattling in the vicinity of . At large , the rattling instanton sources the vector meson fields described by a semiclassical expansion with nonrigid Dirac constraints. Changes in (the core boundary) are reabsorbed by a residual gauge transformation on the core instanton. This is a holographic realization of the Cheshire cat principle [2] where plays the role of the Cheshire cat smile.
3 The Holographic Model
To illustrate the Cheshire cat mechanism more quantitatively, we now summarize the holographic YangMillsChernSimons action in 5D curved background. This is the leading term in a expansion of the Dbrane BornInfeld (DBI) action on D8 [5],
(9)  
(10)  
(11) 
where are 4D indices and the fifth(internal) coordinate is dimensionless. There are three things which are inherited by the holographic dual gravity theory: and . is the KaluzaKlein scale and we will set as our unit. and are defined as
(12) 
is the 5D 1form gauge field and and are the components of the 2form field strength . is the ChernSimons 5form for the gauge field
(13) 
We note that is of order , while is of order . These terms are sufficient to carry a semiclassical expansion around the holonomy (3) with as we now illustrate it for the baryon current.
4 The Baryon Current
To extract the baryon current, we source the reduced action with a flavor field on the boundary in the presence of the vector fluctuations (). The effective action for the source to order reads
(14) 
The first line is the free action of the massive vector meson which is
(15) 
in Lorentz gauge. The second line is the direct coupling between the core instanton and the source as displayed in Fig.2a while the last line corresponds to the vector omega, omega’, … mediated couplings (VMD) as displayed in Fig.2b. These couplings are
(16) 
which are large and of order since . When is set to after the bookkeeping noted above, the coupling scales like , or in the large limit taken first.
The direct coupling drops by the sum rule
(17) 
following from closure in curved space
(18) 
in complete analogy with VMD for the pion in holography [5].
Baryonic VMD is exact in holography provided that an infinite tower of radial omega’s are included in the mediation of the current. To order the baryon current is
(19) 
This point is in agreement with the effective holographic approach described in [13]. The static baryon charge distribution is
(20) 
with
(21) 
The extra follows the normalization for the baryon number source.
5 Baryonic Form Factor
The static baryon form factor is a purely surface contribution from
(22)  
(23) 
with
(24) 
The boundary contribution at vanishes since for large . In the limit we pick the baryon charge
(25) 
due to the sum rule (17), with the limits understood sequentially.
The surface density follows from the U(1) bulk equation
(26) 
The baryon number density lodged in integrates to 1 since
(27) 
as the spatial flux vanishes on is zero for a sufficiently localized SU(2) instanton in .
The isoscalar charge radius, can be read from the terms of the form factor
(28) 
with . The first contribution is from the core and of order ,
(29) 
The second contribution is from the cloud and of order ,
(30) 
with the inverse vector meson propagator in bulk.
The results presented in this section were derived in [9] using the cheshire cat descriptive. They were independently arrived at in [12] using the strong coupling source quantization. They also support, the effective 5dimensional nucleon approach described in [13] using the heavy nucleon expansion.
6 Conclusions
The holography model presented here provides a simple realization of the Cheshire principle, whereby a zero size Skyrmion emerges to order through a holonomy in 5 dimensions. The latter is a bosonized form of a heavy quark sitting still in the conformal direction viewed as time. The baryon has zero size.
To order , the core Skyrmion is dressed by an infinite tower of vector mesons which couple in the holographic direction a distance away from the core. The emergence of follows from a nonrigid semiclassical quantization constraint to prevent double counting. divides the holographic direction into a core dominated by an instanton and a cloud described by vector mesons.
Observables are independent provided that the curvature in both the core and the cloud is correctly accounted for. This is the Cheshire cat mechanism in holography with playing the role of the Cheshire cat smile. We have illustrated this point using the baryon form factor, where was taken to zero using the uncurved or flat ADHM instanton. The curved instanton is not known. Most of these observations carry to other baryonic observables [9, 12] and baryonic matter [14] (and references therein).
7 Acknowledgments
IZ thanks KeunYoung Kim for his collaboration on numerous aspects of holographic QCD. This work was supported in part by USDOE grants DEFG0288ER40388 and DEFG0397ER4014.
References

[1]
A. Chodos, R. Jaffe, K. Johnson and C. Thorn,
“Baryon Structure in the Bag Theory”,
Phys. Rev. D 10, 2599 (1974);
G.E. Brown and M. Rho, “The Little Bag,” Phys. Lett. B 82, 177 (1979).  [2] S. Nadkarni, H. B. Nielsen and I. Zahed, “Bosonization Relations As Bag Boundary Conditions,” Nucl. Phys. B 253, 308 (1985).

[3]
H.B. Nielsen, M. Rho, A. Wirzba and I. Zahed,
“Color Anomaly in Hybrid Bag Model,”
Phys. Lett. B 269, 389 (1991);
M. Rho, “The Cheshire Cat Hadrons Revisited,” Phys. Rep. 240, 1 (1994).  [4] I. Zahed and G.E. Brown, “The Skyrme Model”, Phys. Rep. 142, 1 (19986).

[5]
T. Sakai and S. Sugimoto,
“Low energy hadron physics in holographic QCD,”
Prog. Theor. Phys. 113, 843 (2005);
T. Sakai and S. Sugimoto, “More on a holographic dual of QCD,” Prog. Theor. Phys. 114, 1083 (2006);
H. Hata, T. Sakai, S. Sugimoto and S. Yamato, “Baryons from instantons in holographic QCD,” arXiv:hepth/0701280.  [6] D. Hong, M. Rho, H. Yee and P. Yi, “Chiral Dynamics of Baryons from String Theory” Phys. Rev. bf D76, 061901 (2007).
 [7] J. Maldacena, “The Large N limit of Superconformal Field Theories and Supergravity” Adv. Theor. Math. Phys. 2, 231 (1998).
 [8] C. Csaki and M. Reece, “Toward a Systematic Holographic QCD: A Braneless Approach”, arXiv:hepth/0608266.
 [9] K. Y. Kim and I. Zahed, “Electromagnetic Baryon Form Factors from Holographic QCD.” JHEP 0809, 007 (1998).
 [10] C. Adami and I. Zahed, “Soliton quantization in chiral models with vector mesons,” Phys. Lett. B 215 (1988) 387.
 [11] H. Verschelde and H. Verbeke, “Nonrigid quantization of the skyrmion,” Nucl. Phys. A 495 (1989) 523.
 [12] K. Hashimoto, T. Sakai and S. Sugimoto, “Holographic Baryons : Static Properties and Form Factors from Gauge/String Duality,” arXiv:0806.3122 [hepth].

[13]
D. K. Hong, M. Rho, H. U. Yee and P. Yi,
“Nucleon Form Factors and Hidden Symmetry in Holographic QCD,”
arXiv:0710.4615 [hepph];
M. Rho, “Baryons and Vector Dominance in Holographic Dual QCD,” arXiv:0805.3342 [hepph].  [14] K.Y. Kim, S.J. Sin and I. Zahed, “Dense Holographic QCD in the Wigner Seitz Approximation,” JHEP 0809, 001 (2008).