291. Chiral symmetry breaking in soft wall AdS/QCD model
Save This Post as PDFThis 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,
![S_{5}=-\int{}d^{5}x\sqrt{-g}{}e^{-\phi{}(z)}{\rm Tr}\left[|DS|^{2}+m_{S}^{2} |S|^{2}-\kappa{}|S|^{4}+\frac{1}{4g_{5}^{2}}(F_{V}^{2}+F_{A}^{2})\right]](http://www.nonequilibrium.net/latexrender/pictures/f9dca2b459fe57a3a9c420a3891b6cf2.gif "S_{5}=-\int{}d^{5}x\sqrt{-g}{}e^{-\phi{}(z)}{\rm Tr}\left[|DS|^{2}+m_{S}^{2} |S|^{2}-\kappa{}|S|^{4}+\frac{1}{4g_{5}^{2}}(F_{V}^{2}+F_{A}^{2})\right]")
,
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.
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Thank you for the nice post.
I really hope I could read it earlier, but I do have a couple of questions about your post and your paper.
1. In the 5-d action, it seems that the second term including the bulk field have a different sign with the one in hep-ph/0501128.
2. I don’t think you have included 3 flavors in the 5-d action and thus your fitting of scalar spectrum is quesionable. In the spectroscopy study, the isospin-1 scalar mesons (a_0, like rho and a_1) are more suitable than isospin-0 mesons f_0 as you used.
Thank you for your comment. I would have responded earlier but I have been away during the winter break. As for your questions,
, the described particle must take the corresponding quantum numbers, which are those of f_0.
1. Both 5-d actions are correct, the sign differences arise from our respective choices of metric signature (-++++) or (+—-).
2. You’re right that this is very much a light quark model which doesn’t necessitate heavier quarks, however, I do not believe that should affect the scalar fitting. If this is a problem I would suspect it would affect the scalar couplings to the vector/axial sectors.
Also, since the scalar field is dual to
Thank you for your response.
You are right, one must fit the isosinglet mesons f_0.
But since there is not heavier quark such as strange quark, in the fitting procedure one can not pick up the mesons made of ssbar. Right?
In the fitting procedure, the first two scalar mesons are f_0(550)(also refered as sigma) and f_0(980). Although their nature is still on the debate, they could be identified as the mixture of qqbar and ssbar and they have the same principle number (n). It is also similar for the other mesons. So in my opinion, the mesons which are composed of the strange quarks (many people use this kind of spectrum) are inappropriately used.
Thank you again.
I agree with you that the f_0 radial excitation spectrum is not fully understood yet, I chose the resonances after evaluating available literature, but still there are a lot of uncertainties. I see now the issue with ssbar; it may be a good think to look into incorporating the ssbar component.
I have learned a lot from your post and your explanation. Thank you.