Holographic baryons from oblate instantons
Moshe Rozali
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Jared B. Stang
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Mark Van Raamsdonk
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Department of Physics and Astronomy, University of British Columbia
, 6224 Agricultural Road,
Vancouver, B.C.
, V6T 1Z1,
Canada
We investigate properties of baryons in a family of holographic field theories related to the Sakai-Sugimoto model of holographic QCD. Starting with the Nf = 2 SakaiSugimoto model, we truncate to a 5D Yang-Mills action for the gauge fields associated with the noncompact directions of the flavor D8-branes. We define a free parameter that controls the strength of this Yang-Mills term relative to the Chern-Simons term that couples the Abelian gauge field to the SU(2) instanton density. Moving away from = 0 should incorporate some of the effects of taking the Sakai-Sugimoto model away from large 't Hooft coupling . In this case, the baryon ground state corresponds to an oblate SU(2) instanton on the bulk flavor branes: the usual SO(4) symmetric instanton is deformed to spread more along the field theory directions than the radial direction. We numerically construct these anisotropic instanton solutions for various values of and calculate the mass and baryon charge profile of the corresponding baryons. Using the value = 2.55 that has been found to best fit the mesonic spectrum of QCD, we find a value for the baryon mass of 1.19 GeV, significantly more realistic than the value 1.60 GeV computed previously using an SO(4) symmetric ansatz for the instanton.
Contents
1 Introduction
2 Baryons as solitons in the Sakai-Sugimoto model
3 Numerical setup and boundary conditions
3.1 Gauge fixing
3.2 Ansatz and boundary conditions
3.3 Numerical procedure
4 Solutions
4.1 The mass-energy
4.2 The baryon charge
5 Conclusion
1 Introduction
Perhaps the most successful holographic model of QCD has been the Sakai-Sugimoto
model [1, 2], defined by the physics of Nf probe D8-branes in the background dual to
the decoupling limit of Nc D4-branes compactified on a circle with antiperiodic boundary
conditions for the fermions. This model reproduces many features of real QCD, including
chiral symmetry breaking, a deconfinement transition [3, 4], and a realistic meson spectrum.
The description of baryons in the Sakai-Sugimoto model involves solitonic
configurations of the Yang-Mills field on the D8-brane.1 In a simplified ansatz where the Yang-Mills
field is taken to depend only on the four non-compact spatial directions in the bulk,
configurations with baryon charge are precisely those configurations with non-zero instanton
number for this reduced 4D Yang-Mills field [1, 57]. This connection between baryon
charge and bulk instanton number stems from a Chern-Simons term s tr (F F ) in the
reduced D8-brane action. Here, tr (F F ) is the instanton density for the SU(2) part of
the Yang-Mills field, and s is the U(1) part of the Yang-Mills field, dual to the baryon
current operator in the field theory.
To date, the study of baryons in the Sakai-Sugimoto model has been somewhat
unsatisfactory, for several reasons: I) While the action for the gauge field is of Born-Infeld
type, only the leading Yang-Mills terms are typically used when studying the instantons.
II) For large t Hooft coupling where the model can be studied most reliably, the size of
the instanton in the bulk has been argued to be much smaller than the size of the compact
directions in the bulk. In this case, the assumption that the gauge field does not depend
1Mesons correspond to pertubative excitations of the D8-branes.
on the compact directions is questionable. III) Rather than solving the bulk equations
to determine the precise solitonic configuration of the Yang-Mills field, the form has been
taken to be that of a flat-space SO(4) symmetric instanton, with the size of the instanton
as the only free parameter.
The assumptions in I) and II) here amount to replacing the original top-down
SakaiSugimoto model with a phenomenological (bottom-up) holographic model that retains
many of the same successes as the Sakai-Sugimoto model. For the present paper, we
continue to make these assumptions, though we hope to relax them in future work in order
to better understand baryons in the fully-consistent top-down model. Our goal in the
present paper is to overcome the third deficiency, by setting up and solving numerically a set
of partial differential equations that determine the proper form of the soliton.2 Using these
solutions, we are able to calculate the mass and baryon charge distribution of the baryons
as a function of the model parameter (proportional to the inverse t Hooft coupling )
that controls the strength of the Chern-Simons term relative to the Yang-Mills term.
One motivation for our work is the work of [11], which points out that the flat-space
instanton approximation used previously does not give the correct large radius
asymptotic behavior (known from model-independent constraints) for the baryon form factors
(computed for example in [1214]). Via a perturbative expa (...truncated)