RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-5. Hadron Physics (Theory)

Holographic heavy-quark symmetry†
K. Hashimoto,∗1,∗2 N. Ogawa,∗1 and Y. Yamaguchi∗1Hadrons containing heavy quarks, i.e., charm andbottom quarks, have attracted much interest in hadronand nuclear physics. Accelerator experiments havefound rich spectra of heavy hadrons for conventional states, three-quark baryons and quark-antiquarkmesons, and exotic hadrons having complex structures such as multiquark states and hadronic composite states1) . Various structures of heavy hadrons aregenerated by the nature of internal quark potentials
and/or hadron-hadron interactions which result fromfundamental phenomena of quantum chromodynamics(QCD).
In heavy quark sectors, a new symmetry that has
not been appeared in the light-quark sector is expected
to be important. It is called heavy-quark spin symmetry2) , and it emerges because of the suppression of
the spin-dependent interaction between quarks, whichis inversely proportional to quark masses. This symmetry leads to a specific feature: mass degeneracy ofheavy hadrons having different spins. In the case ofmesons, we find small mass splittings between spin0 pseudo-scalar mesons and spin-1 vector mesons inexperimental results, e.g., the mass splitting of BB ∗(∼45 MeV) in the bottom-quark sector is much smaller
than that of KK ∗ (∼ 400 MeV) in the strange-quark
sector. The symmetry also affects decays and productions of heavy hadrons with different spins. Hence, theheavy-quark spin symmetry plays a significant role inheavy-hadron spectroscopy.
The spectra of heavy hadrons have been studied using various theoretical approaches, such as the quarkmodel and lattice QCD. In addition, we found thatthe gauge/gravity correspondence3) is a promising approach because it provides powerful methods to deal
with strongly coupled theories.
The gauge/gravity correspondence has been applied
to investigate hadron spectra by introducing dynamical quarks described by excitations on probe D-branes.
For example, N = 2 hypermultiplet flavors (= quarkmultiplets) added to N = 4 SU(Nc ) super Yang-Mills
theory are realized by introducing Nf probe D7-braneson the AdS5 × S5 background ( generated by Nc D3branes) on the gravity side4) . In this model, the quarks
are given as fundamental strings stretched between D3and D7-branes. The masses of pseudo-scalar and vector mesons are obtained as fluctuations of the scalarand vector fields on the flavor D-branes4) .
In the gauge theory holding supersymmetry, however, both pseudo-scalar and vector mesons are mem†

∗1
∗2

Condensed from the article in arXiv:1412.5590 [hep-th]
RIKEN Nishina Center
Department of Physics, Osaka University

bers of the same multiplet, and the masses are completely degenerate regardless of the value of the quark
mass. Even when the supersymmetry is broken byfinite temperature, Shark-Schwartz compactification,etc, the supersymmetry is recovered in the heavyquark limit (i.e., in the UV limit) in most of the
top-down models. Hence, the presence of the heavyquark spin symmetry has not been obvious in thegauge/gravity duality. In order to determine whether
the heavy-quark spin symmetry exists, we need to investigate meson mass degeneracies in theories that are
non-supersymmetric even in the UV region.
In this study, we propose a semi-bottom-up, deformed D3-D7 model. The background geometry is
deformed from the conventional AdS5 × S5 and obtained as 2dr2δ
2ds2str = r2α ηµν dxµ dxν + R2 r−2β+rdΩ,
5r2(1)eφ = g0 r−4γ ,(2)in the string frame, where R corresponds to the radius of AdS5 , dΩ25 represents the metric of S 5 , and φ isthe dilaton. Here r is the non-dimensional coordinate.
The background geometry is given in the UV leading
form and generally holds no supersymmetry or conformal symmetry. We introduce 4 deformation parameters (α, β, δ, γ) for the background, but they are constrained by several conditions so that the theory would
have physically reasonable properties. The standard
D3 background corresponds to (α, β, δ, γ) = (1, 0, 0, 0).
We investigate the spectra of the pseudo-scalar andvector quarkonia at the limit mQ → ∞, computed as
the fluctuations of the fields on the flavor D7-brane
placed on this background.
We observed a slight mass difference between
pseudo-scalar and vector quarkonia, which is at most 1.5%, even in the non-supersymmetric analysis. The
mass degeneracy is found in not only ground states but
also excited states.
References
1) N. Brambilla, S. Eidelman, B. K. Heltsley, R. Vogt,
G. T. Bodwin, E. Eichten, A. D. Frawley andA. B. Meyer et al.: Eur. Phys. J. C 71, 1534 (2011).
2) A. V. Manohar and M. B. Wise: Camb. Monogr. Part.
Phys. Nucl. Phys. Cosmol. 10, 1-191 (2000).
3) J. M. Maldacena: Int. J. Theor. Phys. 38, 1113 (1999)[Adv. Theor. Math. Phys. 2, 231 (1998)].
4) A. Karch and E. Katz: JHEP 0206, 043 (2002).
5) M. Kruczenski, D. Mateos, R. C. Myers, and D. J. Winters: JHEP 0307, 049 (2003).

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