291. Chiral symmetry breaking in soft wall AdS/QCD model

This is a guest post by Tom Kelley from the U. of Minnesota who works on AdS/QCD with Tony Gherghetta. Dmitry.

I would like to thank Dmitry for inviting me to talk a little about chiral symmetry breaking in the soft-wall AdS/QCD model, a topic in my recent work “Chiral Symmetry Breaking in Soft-Wall AdS/QCD” (arXiv:hep-ph/0902.1998). In this post, I’d like to paint, in broad strokes, the picture of how AdS/QCD incorporates chiral symmetry breaking.

At the end of the 20th century, several papers laid the groundwork for Anti-de Sitter Space/Conformal Field Theory (AdS/CFT) correspondence, relating type IIB string theory on to Super Yang-Mills (SYM) theory (arXiv:hep-th/9711200, arXiv:hep-th/9802109, arXiv:hep-th/9802150, arXiv:hep-th/9905104). This correspondence led to an effective dictionary relating strongly-coupled gauge field theories to higher-dimensional, weakly-coupled gravity theories. Such a correspondence removes the main difficulty of strongly-coupled systems, the breakdown of perturbative calculations. By relating strong-coupled systems to weakly-coupled analogs, perturbative  calculations are again valid and can be carried out with relative ease. Two methods are employed to formulate models using AdS/CFT correspondence: top-down and bottom-up. A top-down method uses a version of string theory to calculate an effective Lagrangian that, it is hoped, will contain certain characteristics of QCD. A bottom-up approach, commonly known as AdS/QCD, uses the basic tenets of QCD in d dimensions to formulate a dual gravity theory in AdS. Of course, constructing an AdS dual theory describing the richness of QCD presents a great challenge. Some may argue it isn’t possible with such simple assumptions in AdS/QCD. However, some aspects of this model come tantalizingly close to experimental results, I feel that an attempt is well worth it. In our work, we implemented a bottom-up approach evolving from the work of Randall, Erlich, Da Rold, and Karch (arXiv:hep-ph/9905221, arXiv:hep-ph/0501128, arXiv:hep-ph/0501218, arXiv:hep-ph/0602229).

Let’s begin with the question, what is chiral symmetry?

Chiral symmetry is a well-known property of massless fermions and is broken spontaneously and explicitly in QCD. The symmetry is easily seen from the Dirac Lagrangian,

.

Expressing the quark spinor in chiral components ,

we can rewrite the fermionic Lagrangian in Weyl representation as

.

For , right- and left-handed states can transform independently; in fact, for a particular number of quark flavors, , the Lagrangian contains a symmetry. But most of us want to describe massive particles, so for , right- and left-handed states mix and explicitly break the chiral symmetry. The QCD vacuum is responsible for the spontaneous breaking of chiral symmetry. The strong dynamics of QCD drive the composite field, , to obtain a nonzero value in the vacuum, a purely non-perturbative effect,

often called the chiral condensate. According to the Nambu-Goldstone theorem, for every spontaneously broken symmetry, a massless boson appears with quantum numbers corresponding to the broken symmetry. The pions are considered the Goldstone bosons of the spontaneous chiral breaking, only acquiring mass from the massive quarks. Much as the non-zero mass in our Lagrangian, the QCD vacuum mixes the chiral states, breaking the symmetry.

Spontaneous and explicit breaking of chiral symmetry affect meson mass spectra independently. The explicit breaking is put into the theory by hand and influences all quark states. Spontaneous breaking eliminates the degeneracy between the vector and the axial-vector mesons. An AdS/QCD model must incorporate two independent sources describing both of these breakings if there is any hope to connect it to reality.

How does AdS/QCD describe chiral symmetry breaking?

Chiral symmetry in QCD appears to revolve around the dimension-3 operator . According to the AdS/CFT dictionary, for every operator, , of dimension in the now broken CFT in d dimensions, a scalar field with mass

is introduced. The scalar field S obtains a -dependent vacuum expectation value whose asymptotic solution is of the form

,

where corresponds to the quark mass (hep-th/9905104). One can now can derive equations of motion from a 5-d action,

,

where is the soft-wall dilaton and the vector (axial-vector) gauge fields. The equations of motion become

,

where , the 5-d coupling constant, and n the resonance number. Most obviously, the equation for the vector meson is similar to that describing the axial-vector meson. The presence of the scalar VEV, , indeed breaks the chiral symmetry, since the term causes a mass splitting between the vector and the axial-vector mesons in this model.

It can also be shown from the equations of motion that we may choose to parametrize such that and in the IR region. The former behavior allows for the observed Regge behavior of , and the latter suggest no chiral restoration, i.e. vector and axial-vector mesons maintain a constant mass splitting even as . This parameterization has the added bonus of allowing the introduction of explicit AND spontaneous chiral symmetry breaking. Previously in the soft-wall model, the small -behavior of has always taken the form

where . As a result, taking the chiral limit eliminates all chiral symmetry breaking, explicit and spontaneous, leading to no pion. However, taking and solving for the dilaton, we can show that

,

where the quark mass and the chiral condensate can be set independently. Thus, the chiral limit does not destroy spontaneous symmetry breaking. Further, we can derive the Gell-Mann-Oakes-Renner relation in a similar manner as Erlich (hep-ph/0501128), showing that as , the pion mass vanishes.

Incorporating chiral symmetry in the soft-wall AdS/QCD model is non-trivial, but must be done if we are to draw accurate/valid analogies between it and reality, i.e. QCD. Of course, much work and possibly a new framework is needed to accurately describe QCD with gravity duals. What has been shown in (arXiv:hep-ph/0902.1998), though, that at the cost of modifying the dilaton in the UV and introducing another parameter, , we can represent explicit and spontaneous chiral symmetry breaking into the soft-wall model in a way not done before.