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Generalized WZW-term fro m holographic QCD. Shigeki Sugimoto (YITP, Kyoto Univ.). Holographic QCD @ Nordita , July 26, 2019. Introduction. 1. Consider 4 dim SU(Nc) QCD with Nf flavors. Chiral symmetry for massless QCD. Chiral anomaly. Under the infinitesimal gauge transformation.

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## Generalized WZW-term fro m holographic QCD

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**Generalized WZW-termfromholographic QCD**Shigeki Sugimoto (YITP, Kyoto Univ.) Holographic QCD @ Nordita, July 26, 2019**Introduction**1 Consider 4 dim SU(Nc) QCD with Nf flavors Chiral symmetry for massless QCD Chiral anomaly Under the infinitesimal gauge transformation the partition function of QCD transforms as**’t Hooft argued that the low energy effective theory**should have the same anomaly. Low energy theory of QCD is given by the Nambu-Goldstone (NG) modes (pion) associated with the chiral symmetry breaking. Spontaneous chiral symmetry breaking NG modes Or,equivalently,**WZW term**reproduces the chiral anomaly**Original derivation of the WZW term**Witten: Nuclear Physics B223 (1983) 422-432 Now, there are more systematic ways to derive this. Holographic QCD provides one of them.**Today**I’d like to explain how to derive the WZW term based on holographic QCD for the cases with general SSB: unbroken subgroup global sym Sorry for the technical talk! You don’t have to use holographic QCD, if you don’t like it. We only use it as a useful guide. If you are interested in the couplings including the vector mesons, holographic QCD tells us how to get them. Holographic SO(Nc) and USp(Nc) QCD are known. For these theories, G=SU(Nf) and H=SO(Nf), USp(Nf) [Imoto-Sakai-S.S. 09]**Plan**• Introduction • CS-term in holo QCD 1 2 Generalized WZW-term from holo QCD 3 Conclusion 4**2**CS-term in holo QCD In this section, we consider the top-down holographic QCD based on a D4/D8 brane system in string theory. But, the following discussion also applies to many other models. [Sakai-S.S. 04,05] Meson effective theory turns out to be 5 dim U(Nf) YM-CS theory in a curved spacetime number of flavors 5th coordinate CS 5-form CS-term**CS-term**This term is not gauge invariant, if the manifold has boundaries: Reproduces the chiral anomaly in QCD with the standard identification external U(Nf)L x U(Nf)R gauge fields boundary values of the 5 dim gauge field**Problems of the naïve CS-term**We often want to consider the gauge configurations with e.g. Baryon In such cases, the naïve CS-term doesn’t work. In fact, Hata-Murata (’07) pointed out that an important constraint to get the correct baryon spectrumforNf=3: doesn’t show up correctly, if one uses the naïve CS-term. This constraint can be obtained in Skyrme model with WZW term.**[Lau-S.S. 16]**Modified CS-term Suppose the boundary values of the gauge field and the external gauge field are identified up to a gauge transformation: (previously written as ) where Then the CS-term we should use is • Invariant under the gauge transformation that keeps fixed. • Reproduces the naïve CS-term when are topologically equiv. • Reproduces the correct chiral anomaly. • Reproduces the constraint:**Generalized WZW-term from holo QCD**3 Consider the cases with general SSB: We assume that the low energy physics is given entirely by the Nambu-Goldstone modes. (No other light degrees of freedom that contribute to the chiral anomaly.) Notation: : Lie algebras of G and H : orthogonal complement of decomposition of the gauge field, w.r.t. gauge transfomation**Setup:**Consider a 5 dim gauge field A of gauge group G (UV boundary) (IR boundary) Boundary conditions (IR) (UV) G is broken to H**Action**C is chosen to reproduce the anomaly of the UV theory. Comment: One can check that the boundary conditions are consistent with the variational principle if H is an anomaly free group satisfying This condition is also needed to make sure that the action is invariant under the gauge transformation at z=0.**The Nambu-Goldstone mode is**Under the gauge transformation transforms as where Hidden local H symmetry global G symmetry For gauge, we have the NG mode appears in the UV boundary condition as**Derivation of the WZW term**Let us work in the gauge A mode expansion of the 5 dim gauge field consistent with the boundary condition is given by are interpreted as massive vector (axial-vector) mesons.**In principle, we can obtain the WZW term as follows:**1. Insert the mode expansion into the action and integrate over z. → a 4 dim meson action is obtained. 2. Integrate out the massive fields . →The action for the NG mode with the WZW term is obtained. A much better way is to choose s.t. the mixing terms between vanish. Then, we can simply set . It turns out that this is achieved by**Inserting**into and integrating over z, we finally obtain where**Example**Suppose Consider introducing an (imaginary) chemical potential associated with this U(1) by replacing chemical potential Then, one can easily show independent of**Conclusion**4 I explained how to derive the WZW term based on holographic QCD for the cases with general SSB: We should use the modified CS-term if the boundary value of the gauge field is not directly identified with the external field.

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