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296. Global aspects of the scalar meson puzzle

Posted by Renata Jora

at March 7, 2009

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This is a guest post by Renata Jora from INFN. Dmitry.

I would like to thank Dmitry for inviting me to write this post about our paper (A. Fariborz, R. Jora and J. Schechter) “Global aspects of the scalar meson puzzle”, arXiv:0902.2825.

I will start with a few historical facts. It was first noticed that the strong interaction is approximately blind at the interchange of the neutron to the proton which lead to the introduction of an SU(2) invariance. Then another approximate symmetry was discovered, the hypercharge Y=B+S where is the baryon number and is the strangeness. As one tries to incorporate both of this groups into a larger one obtains the invariance under an SU(3) . Note that all these groups act in the flavor space. This leads to the the “constituent quark model” (Gell Mann) which states that quarks have different quantum numbers and the meson are consequently made out of one quark and one antiquark, while the baryons have a three quark structure. All these combinations should be color neutral. If all quark masses are set to zero the actual invariance group is a larger one, the chiral $SU(3)_L\times SU(3)_R$ group. However since this symmetry is not realized in the spectrum it is then assumed that is spontaneously broken. According to the Goldstone theorem this would lead to 8 massless mesons. In reality there are very light particles like the pions but none of them is massless which means that this chiral symmetry should be also explicitly broken by the quark masses.

If one further analyzes the mass spectrum of the pseudoscalar ( P=- ), scalar ( P=+ ) and vector mesons, one finds that for example for the vector meson nonet:

$I=0: m[f_0(600)] \approx 500\,\,{\rm MeV}$
$I=1/2:\hskip .7cm m[\kappa] \approx 800 \,\,{\rm MeV}$
$I=0: m[f_0(980)] \approx 980 \,\,{\rm MeV}$
$I=1: m[a_0(980)] \approx 980 \,\,{\rm MeV}$

the masses increase with the strange quark content:

$\rho \sim \frac{u{\bar u}-d{\bar d}}{\sqrt{2}} \hspace{0.2cm} \rightarrow 770\, {\rm MeV}$
$\omega \sim \frac{u{\bar u}+d{\bar d}}{\sqrt{2}} \hspace{0.2cm} \rightarrow 780\, {\rm MeV}$
$K* \sim u{\bar s} \hspace{0.2cm} \rightarrow 880 \, {\rm MeV}$
$\phi \sim s {\bar s} \hspace{0.2cm} \rightarrow 1020\, {\rm MeV}$

The same mass ordering is approximately valid for the pseudoscalar mesons. However if one assumes for the light scalar mesons a diquark structure one gets:
$\sigma (600)\sim s{\bar s}\hspace{0.2cm} \rightarrow 500 \,{\rm MeV}$
$\kappa(800) \sim u{\bar s} \hspace{0.2cm} \rightarrow 800\, {\rm MeV}$
$a_0 (980) \sim \frac{u{\bar u}-d{\bar d}}{\sqrt{2}}\rightarrow 980 \, {\rm MeV}$
$f_0 (980) \sim \frac{u{\bar u}+d{\bar d}}{\sqrt{2}}\hspace{0.2cm} \rightarrow 980 \, {\rm MeV}$

It is evident that the previous mass ordering is not respected. This is the core of the scalar meson puzzle. It was observed long time ago that this problem might be solved by assuming that the light scalar mesons have some four quark content. Moreover there are more scalar states that one can fit into a multiplet (one can actually form two multiplets). Inspired by this, in the paper “Global aspects of the scalar meson puzzle” (arXiv:0902.2825) we consider a linear sigma model with global $SU(3)_L\times SU(3)_R$ invariance and with two nonets ( $M=S+i\Phi$ and $M'=S'+i\Phi'$ where , are the scalars and $\Phi$ , $\Phi'$ are the pseudoscalars) one with a diquark structure the other one with a four quark structure. The symmetry is broken explicitly by an SU(2) invariant term. The corresponding Lagrangian has the form

${\cal L}=-\frac{1}{2}{\rm Tr}\left(\partial_\mu{}M\partial_\mu{}M^\dagger\right)-\frac{1}{2}{\rm Tr}\left(\partial_\mu{}M^\prime{}\partial_\mu M^{\prime \dagger}\right)-V_0-V_{SB}$ ,

where $V_0(M,M^\prime)$ stands for a function made from $SU(3) _{\rm L} \times SU(3) _{\rm R}$ (but not necessarily $U(1) _{\rm A}$ ) invariants formed out of and $M^\prime$ .
Here

$V_0=c_2{\rm Tr} (MM^{\dagger})+c_4^a{\rm Tr} (MM^{\dagger}MM^{\dagger})+d_2{\rm Tr}(M'M'^{\dagger})+$
$+e^a_3(\epsilon_{ijk}\epsilon^{lmn}M^i_l{}M^j_m{}M'{}^k_n+h.c.)+$
$+c_3\left[ \gamma_1 {\rm ln} (\frac{{\rm det} M}{{\rm det}M^{\dagger}})+(1-\gamma_1){\rm ln}\frac{{\rm Tr}(MM'^\dagger)}{{\rm Tr}(M'M^\dagger)}\right]^2$ .

We chose the terms such that the number of quark lines at each vertex does not exceed 8. All the terms except the last one also possess the $U(1) _{\rm A}$ invariance. The symmetry breaking term which models the QCD mass term takes the form:

$V_{SB}=– 2\, {\rm Tr} (A\, S)$

where $A={\rm diag}(A_1,A_2,A_3)$ are proportional to the three light quark masses.

The model allows for two-quark condensates, $\alpha_a=\langle S_a^a \rangle$ as well as four-quark condensates $\beta_a=\langle {S'}_a^a \rangle$ . Here we assume isotopic spin symmetry so A _1=A _2 and

$\alpha_1=\alpha_2 \ne \alpha_3, \hskip 2cm\beta_1=\beta_2 \ne \beta_3$

We also need the “minimum” conditions,

$\left< \frac{\partial V_0}{\partial S}\right> + \left< \frac{\partial{}V_{SB}}{\partial{}S}\right>=0, \quad \quad \left< \frac{\partial V_0}{\partial S'}\right>=0.$

There are 12 parameters in the model describing the Lagrangian and the vacuum. Together with the four minimum conditions this reduces the number of necessary inputs to 8. These are:

$m[a_0(980)]=984.7 \pm 1.2\, {\rm MeV}$
$m[a_0(1450)]=1474 \pm 19\, {\rm MeV}$
$m[\pi(1300)]=1300 \pm 100\, {\rm MeV}$
$m_\pi=137 \, {\rm MeV}$
$F_\pi=131 \, {\rm MeV}$

Because $m[\pi(1300)]$ has such a large uncertainty, we examine predictions depending on the choice of this mass within its experimental range. The sixth input will be taken as the light “quark mass ratio” A_3/A_1 , which will be varied over an appropriate range.

Thus we are left with the problem of determining 9 masses and 16 four quark percentages. The main result of the paper is that while the lightest pseudoscalars have low four quark percentage thus being mainly quark-antiquark structures as expected the low lying scalars have large four quark percentage (larger than percent).

How can this resolve the scalar meson puzzle? Let us take a closer look at the I=1 scalars for example, the a’s. In the quark antiquark picture as we previously mentioned the a’s were composed of u and d quarks fact which could not explain their relative heaviness. However in our model the a’s correspond to an admixture of S_1^2 and $S_1^{‘2}$ . Let us assume the molecule picture for the four quark states, i.e.

$M_k^{‘n}=\epsilon_{ijk}\epsilon^{lmn}M^{\dagger i}_l M^{\dagger j}_m$ .

Then simply $M_1^{‘2}$ and consequently $S_1^{‘2}$ will contain S_3^3 which is associated with $s{\bar s}$ . Thus a large four quark percentage in this case correspond with large $s{\bar s}$ content.

Journal club, Master Mason, Quantum field theory

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295. Weak lensing signal in Unified Dark Matter models

Posted by Stefano Camera

at March 6, 2009

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This is a guest post by Stefano Camera (INFN and U. of Torino) about the work he has done in collaboration with D. Bertacca, A. Diaferio, N. Bartolo and S. Matarrese. Dmitry.

A particular description of DE is suggested by the Unified Dark Matter (UDM) models. While most of the models of DE rely on the potential energy of a scalar field to lead to the late time acceleration of the Universe, it is possible to have a situation where the accelerated expansion arises out of modifications to the kinetic energy of the scalar field. The major advantage of these models is that there is only one non-standard fluid, which can mimic both DM and DE. Thus, one of the main issues of these UDM models is to see whether the single dark fluid is able to cluster and produce the cosmic structures we observe in the Universe today. In fact, the effective speed of sound can be significantly different from zero at late times; the corresponding Jeans’ length (or sound horizon), below which the dark fluid cannot cluster, can be so large that the gravitational potential first strongly oscillates and then decays, thus preventing structure formation. Here we choose to investigate the class of scalar field Lagrangians with a non-canonical kinetic term that allow to obtain UDM models with a small effective sound speed.

Recently, it has been shown that the scalar field in UDM models can cluster, but it remains to be explored whether UDM models provide a good fit to the various sets of available data. Weak lensing is a powerful tool, indeed gravitational lens effects are due to the deflection of light occuring when photons travel near matter, i.e. in the presence of a non-neglegible gravitational field. The cosmic convergence and shear encapsulate information about both the source emitting light and the structures that photons cross before arriving at the telescope. Hence, weak lensing allows to explore both the basis of the cosmological model and LSS of the Universe, in other words it brings information about the geometry and the dynamics. Therefore the study of the power spectrum of weak lensing can be a crucial test.

The metric of the Universe is described by the standard Friedmann-Lematre-Robertson-Walker (FLRW) length element for an isotropic and homogeneous space filled with a perfect fluid, that, with the addition of scalar perturbations and in Newtonian gauge, takes the form

$ds^2\equiv g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu=a^2(\tau)\left[-(1+2\mathbf\Phi) d\tau^2+(1+2\mathbf\Psi) d \ell^2\right]$

where $\mathrm{d} \tau=\mathrm{d} t/a$ is the conformal time, with the spatial metric

$\mathrm{d}\ell^2\equiv\delta_{ij}\mathrm{d} x^i\mathrm{d} x^j=\mathrm{d}\chi^2+\chi^2\mathrm{d}\Omega^2$

being $\delta_{ij}$ the Kronecker delta, and the radial comoving distance is

$\chi(z)=r_H\int_0^z\!\!\mathrm{d}\tilde z\,\frac{H_0}{H(\tilde z)}$

where $H=\dot a/a$ is the Hubble parameter and $r_H=c/H_0\simeq3\,h^{-1}\,\mathrm{Gpc}$ is the Hubble radius.

In LCDM, gravity is given by GR in a 4-dimensional Universe filled with a perfect fluid of photons, baryons, DM and DE (as cosmological constant). In linear theory of scalar perturbations the gravitational potential is given by

$\mathbf\Phi_k(a)=A_k\left(1-\frac{H(a)}{a}\int_0^a\!\!\frac{d\tilde a}{H(\tilde a)}\right)$

where the constant of integration is $A_k=\mathbf\Phi_k(0)T(k)$ ,

with T(k) the matter transfer function, that describes the evolution of perturbations through the epochs of horizon crossing and radiaton-matter transition, $\mathbf\Phi_k(0)$ the large-scale potential during the radiation dominated era.

UDM models use a scalar field $\varphi(t,\mathbf x)$ that mimics both DM and DE. This can be achieved thanks to a non-canonical kinetic term, i.e. letting the kinetic energy be a generic function of the derivatives of the scalar field. The Lagrangian density can be written as

$\mathcal L=\mathcal L_G+\mathcal L_\varphi=\frac{1}{16\pi G}R+\mathcal L_\varphi(\varphi,X)$

where $X=-\frac{1}{2}\nabla_\mu\varphi\nabla^\mu\varphi$ . If is time-like, $\mathcal L_\varphi$ describes a perfect fluid. By requiring that $\mathcal L_\varphi\equiv p=-\Lambda/(8\pi G)$ on cosmological scales the background is identical to the background of LCDM and we easily get

$\rho\left[a(t)\right]=\rho_\mathrm{DM}(a=1)a^{-3}+\frac{\Lambda}{8\pi G}=\rho_\mathrm{DM}+\rho_\Lambda$

where $\rho_\Lambda$ behaves like a cosmological constant “dark energy” component ( $\rho_\Lambda=\mathrm{const.}$ ) and $\rho_\mathrm{DM}$ behaves like a “dark matter” component ( $\rho_\mathrm{DM}\propto a^{-3}$ ).

The Newtonian potential is solution of the differential equation

$v^{\prime\prime}-{c_s}^2\nabla^2v-\frac{\theta^{\prime\prime}}{\theta}v=0$

where a prime denote a derivative with respect to the conformal time, $\theta$ is a function of the energy density and the pressure of the scalar field, and

$v=\frac{\mathbf\Phi}{\sqrt{\rho+p}}.$

Here, c_s is the “speed of sound” relative to the pressure and energy density fluctuations of the scalar field. It accounts for the presence of intrinsic entropy perturbations of the fluid. Recently, it has been proposed a technique to construct UDM models where the scalar field can have a sound speed small enough to allow structure formation and to avoid a strong integrated Sachs-Wolfe effect in the CMB anisotropies which tipically plague UDM models. The parametric form for the sound speed is

${c_s}^2(a)=\frac{{\Omega_\Lambda c_\infty}^2}{\Omega_\Lambda+(1-{c_\infty}^2)\Omega_ma^{-3}}$

where $c_\infty$ is the value of the sound speed at late times.

Gravitational potential Fourier components

On the Fig. above I present some gravitational potential Fourier’s components, normalized to unity at early times, at different scales and different values of $c_\infty$ for the models considered. By increasing the sound speed, the potential starts to decay earlier in time, oscillating then around zero. Moreover at small scales, if the sound speed is small enough UDM reproduces LCDM. This reflects the dependence of the gravitational potential on the effective Jeans’ length

${\lambda_J}^2(\tau)={c_s}^2|\theta/\theta^{\prime\prime}|$ .

Sound horizon

Here I show $\lambda_J(a)$ , the sound horizon, for different values of $c_\infty$

From GR we know that light beam paths are curved by the presence of matter. In the weak lensing framework the deflection of light is small and, consequently, we can use Born’s approximation, where lensing effects are evaluated on the null-geodesic of the unperturbed (unlensed) photon. All weak lensing observables may be expressed in terms of the projected potential

$\phi(\hat{\mathbf n})=\int\!\!\mathrm{d}\chi\,\frac{W(\chi)}{\chi^2}\mathbf\Phi(\hat{\mathbf n},\chi)$

where

$W(\chi)=-2\chi\int_\chi^\infty\mathrm{d}\chi'\,\frac{\chi'-\chi}{\chi'}n(\chi')$

is the weight function of weak lensing, with $n\left[\chi(z)\right]$ representing the redshift distribution of sources, such that $\int\!\!\mathrm{d}\chi\,n(\chi)=1$ . The cosmic convergence is defined by

$\kappa(\hat{\mathbf n})=\frac{1}{2}\left(\phi_{,11}(\hat{\mathbf n})+\phi_{,22}(\hat{\mathbf n})\right)$

and its power spectrum is

$C^{\kappa\kappa}(l)=\frac{l^4}{4}\int_0^\infty\!\!\mathrm{d}\chi\,\frac{{W(\chi)}^2}{\chi^6}P^\mathbf\Phi\left(\frac{l}{\chi},\chi\right)$

where we introduced Limber’s approximation, in which the only Fourier modes that contribute to the integral are those with $l\gg k\chi$ , and $P^\mathbf\Phi(k,\chi)$ is the 3D power spectrum of the gravitational potential.

Now I will present the weak lensing power spectra for the CMB, background galaxy and high redshift proto-galaxy photons.

In UDM models, the background evolution of the Universe is the same as in LCDM, while the evolution of the gravitational potential and the growth of LSS suffer the non negligible sound speed, that increases with time. The discriminant is the effective Jeans’ length of the gravitational potential. The Newtonian potential in UDM models behave like in the LCDM model at scales much larger than the sound horizon, while at smaller scales it starts to decay and oscillate. Weak lensing observables, like cosmic convergence or shear, are an integral over the line-of-sight, hence they do not show directly these oscillations. However, high values of the multipole correspond to small scales, and thus cosmic convergence at high ’s must show the decay of the deflecting potential.

Our analysis is made according to the linear theory of perturbations, and what we calculate is not correct for any . The multipole is related to the scale by a direct proportionality $k=l/\chi(z)$ . We estimated the window of multipoles of validity of our approximations in the following way: the lower limit is $l\simeq10$ , due to Limber’s approximation, but the upper one is floating, because at higher multipoles non-linear effects become more important. Weak lensing power spectra are made by integrating over the line-of-sight, thus over the wide range of the angular comoving distance. When we deal with high multipoles, wave numbers $k>k_\mathrm{nl}\simeq0.2\,h\,\mathrm{Mpc}^{-1}$ will appear in the integration. It is known that in linear theory $P^\mathbf\Phi(k>k_\mathrm{nl})$ is underestimated, but for now there is no linear to non-linear mapping in perturbation theory for UDM models. To estimate the upper limit of validity of our results, we proceed as follow: we made the $\mathrm{d}\chi$ integration in the convergence power spectrum, according to weak lensing theory, for $\chi\geq0$ , using only the linear power spectrum for the gravitational potential, obtaining obiouvsly a lensing signal lower than the real one, in which non-linear contributions to small scales are present. Then, we made the integral imposing a lower cut-off, setting it at $\chi_\mathrm{nl}\equiv l/k_\mathrm{nl}$ . What we get in this way is a weak lensing power spectrum much more suppressed at large ’s than the one obtained in the former integration, because in that case the linear power spectrum receives no contributions from non-linearities, but with this cut-off we set at zero, by hand, the integrand in the $C^{\kappa\kappa}(l)$ when it involves wave numbers larger than $k_\mathrm{nl}$ . With these two quantities, $C^{\phi\phi}_{\chi_\mathrm{min}=0}(l)$ and $C^{\phi\phi}_{\chi_\mathrm{min}=\chi_\mathrm{nl}}$ , we fixed an arbitrary threshold

$\left|\frac{C^{\phi\phi}_{\chi_\mathrm{min}=0}(l)-C^{\phi\phi}_{\chi_\mathrm{min}=\chi_\mathrm{nl}}(l)}{C^{\phi\phi}_{\chi_\mathrm{min}=0}(l)}\right|<\varepsilon$

that enables us to estimate $l_\varepsilon$ , under which our results are reliable, because most of the signal comes from the linear regime, and over which our ignorance on non-linear effects is too high and the real power spectra could be substantially different from those we obtaine.

For CMB light, the source is the last scattering surface at $z_\mathrm{rec}\simeq10^3$ , and the distribution is

$n_\mathrm{CMB}\left[\chi(z)\right]=\delta_D(\chi(z)-\chi(z_\mathrm{rec})).$

Weak lensing power spectra of CMB

The upper panel shows the weak lensing power spectra of CMB light for LCDM and UDM. For UDM we present three curves, obtained for ${c_\infty}^2=10^{-6},10^{-5},10^{-4}$ . In the lower panel every curve of the upper panel is divided by the convergence power spectrum of LCDM. As we can see, for small values of the sound speed ( ${c_\infty}^2=10^{-6}$ ), we cannot distinguish the convergence of CMB photons in UDM models from the standard LCDM behaviour. By increasing $c_\infty$ , while at large scales the agreement is still good, at small enough scales $C^{\kappa\kappa}_\mathrm{UDM}(l)$ is clearly suppressed.

Dealing with galaxy photons, the sources are spread over different redshifts, and the distribution is

$n_g\left[\chi(z)\right]=\frac{\beta z^\alpha}{{z_0}^{\alpha+1}}\frac{e^{{-\left(\frac{z}{z_0}\right)}^\beta}}{\Gamma\left(\frac{\alpha+1}{\beta}\right)}\frac{\mathrm{d} z}{\mathrm{d}\chi},$

that peaks at redshift $z_p\equiv z_p(\alpha,\beta,z_0)$ , where $\alpha$ , $\beta$ and z_0 are free parameters.

To better understand our results, it is useful to look at the weight functions W_g(z) for background galaxies. I present |W_g(z)| for the different choices of the source redshift distributions and z_p we use in this work. The distribution of Kaiser (1999) has $\alpha=1$ and $\beta=4$ , that of Wittman et al. (2000) has $\alpha=2$ and $\beta=1$ , and we also use a Dirac’s delta. As explained before, we consider UDM models which are able to reproduce the same expansion history as in LCDM. Peculiar dynamics of the scalar field become important starting from considering cosmological perturbations. The height of the peak of |W_g(z)| determines the order of magnitude of the weak lensing signal.

Weight functions for background galaxies

In the figures below,

Kaiser

Wittman

the upper panels, we show the weak lensing power spectra $l(l+1)C^{\kappa\kappa}(l)/(2\pi)$ of background galaxy light for LCDM and UDM. As for the CMB case, for UDM we present three curves, obtained for different values of the sound speed. As in the CMB case, for small sound speeds and large angular scales ( $l\lesssim100$ ), we cannot distinguish the convergence of background galaxy photons in UDM models from the standard LCDM behaviour. However, the agreement disappears at large $c_\infty$ and ’s.

The redshift distribution of sources n(z) selects the peak redshift z_p of the source emitting light. The power spectrum $P^\mathbf\Phi(k,z)$ of the gravitational potential, through the Poisson’s equation, encodes the distribution of overdensities, thus the structures crossed by the photons. The weight function $W(\chi)$ is a filter that selects mostly signals emitted at $z\lesssim z_p$ , following the information of n(z) . Consequently, for different values of the speed of sound, the weak lensing signal in UDM models is sensitive to the choice of z_p . At the same time, c_s has to be very small to let the scalar field cluster in order to form the LSS we observe today (while in the past, at high enough redshift, the gravitational potential is similar to that predicted by LCDM). However, at lower z_p , sources emit light that feels strongly the decay and the oscillations of the Newtonian potential, because it is sensitive to the sound horizon $\lambda_J\equiv\lambda_J(z,c_\infty)$ , that increases with time, and to the presence of an effective $\Omega_\Lambda$ , that plays the role of DE. It is easy to see that for small z_p the differences between UDM models with different $c_\infty$ and between UDM and LCDM are very pronounced even at large angular scales, while for CMB and proto-galaxies (as I will show) the power spectra are less sensitive to the sound speed.

We also computed the convergence power spectra for proto-galaxy photons. The aim is to test UDM models at an intermediate redshift between background galaxies and the last scattering surface. We want to test a redshift distribution of sources more spread around the peak than that of background galaxies, but with no tail at small redshifts. We choose a peak redshift of z_p=15 and we use the source distribution with parameters $\alpha=10$ and $\beta=0.5$ .

The shape is consistent with what we find for CMB photons and background galaxies.

Lensing signal in linear theory

Here, we have shown the lensing signal in linear theory as produced in LCDM and UDM; we have considered a number of sources: CMB photons, proto-galaxies at very high redshifts and background galaxies, whith different values of the peak redshift of the distribution, and different shapes of the redshift distribution of sources. For sound speed lower than $c_\infty=10^{-3}$ , in the window of multipoles $l\gtrsim10$ (Limber’s approximation) and $l<l_\varepsilon$ (where our ignorance on non-linear effects due to small scales dynamics become relevant), the power spectra of the cosmic convergence in UDM and LCDM are not distinguishable. When the Jeans’ length $\lambda_J(a)$ increases, the Newtonian potential starts to decay earlier in time (for a fixed scale), or yet at greater scales (for a fixed epoch). This reflects on weak lensing by suppressing the convergence power spectra at high multipoles. We find that, for values of the sound speed between $c_\infty=10^{-3}$ and $c_\infty=10^{-2}$ , UDM models are still comparable with LCDM, while for higher values they are ruled out because of inhibition of structure formation. Moreover, we find that for low redshift sources the dependence of UDM weak lensing signal from the sound speed $c_\infty$ is much stronger.

Cosmological perturbations and large scale structure, Cosmology, Journal club, Master Postdoc, Quantum field theory

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294. The leptonic Higgs as a messenger of dark matter

Posted by Piyush Kumar

at March 5, 2009

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This is a guest post by Piyush Kumar from U. of California, Berkeley. Dmitry.

I would like to thank Dmitry for inviting me to post a guest blog on my recent paper “The leptonic Higgs as a messenger of dark matter” with Lawrence Hall and Hock-seng Goh on Dark Matter (DM).

The story begins with some data reported in the summer and fall of 2008. The PAMELA experiment reported an excess of positrons in the few GeV to 100 GeV range, providing further support to the earlier results of HEAT and AMS. In addition, results from the ATIC balloon experiment suggests an excess of electrons and positrons in the 300 GeV to 600 GeV range roughly consistent with earlier observations of PPB-BETS. Of course, these observations are still preliminary and need to be corroborated with more data. Moreover, even if confirmed, the data may have reasonable astrophysical explanations. While that remains a logical possibility, nonetheless, these observations have generated tremendous interest in the particle physics community, as they might provide the first non-gravitational evidence for Dark Matter (DM).

There are two interesting challenges faced by any model trying to explain the excesses in the above experiments. First, for annihilating DM these signals require that the annihilation cross-section for DM particles is typically two to three orders of magnitude larger than that expected from the thermal freezeout of WIMP DM. On the other hand, for decaying DM, the life-time of the DM particles must be extremely large, of ${\cal O}(10^{25-26})$ seconds. Second, the signals apparently require annihilations or decays dominantly into leptons rather than hadrons, since there is no reported excess in anti-proton cosmic rays.

In our paper, we have explored a general framework for DM based on our current understanding of particle interactions and taking into account the challenges mentioned above. In addition to the known physics of quarks and leptons interacting via $SU(3)\times{}SU(2)\times{}U(1)$ gauge interactions, we expect new physics to include both a Higgs sector, responsible for $SU(2)\times{}U(1)$ symmetry breaking, and a dark matter sector. The idea of a WIMP DM sector is particularly interesting as it ties in with general ideas about electroweak symmetry breaking. We are thus led to the three sectors in the figure below.

Three sectors and their interactions

The interactions between the higgs sector and the Standard Model sector are well known; the interactions of the WIMP DM sector with the other two sectors are more speculative on the other hand. However, if we think about it, the WIMP idea is that the mass scales of the dark matter and Higgs sectors are both related to the weak scale, and this strongly suggests some connection between these two sectors. In our work, we explore the possibility that any direct couplings between the WIMP sector and the visible sector are subdominant; thus the Higgs sector is seen to be the messenger that makes the WIMP visible to us, as shown in the Figure. The question now arises as to what constraints the cosmic ray data imposes on the Higgs sector. It turns out that the Higgs sector in the SM or in the MSSM does not give rise to a good fit to the data as in these cases the higgs decays predominantly to pairs of bosons, bottom quarks or top quarks (if heavy enough). This gives rise typically to too many anti-protons than actually observed by PAMELA. Also, the signal in $e^+{}+{}e^-$ is smooth, and does not give the peak shown by balloon experiments like ATIC. Thus, from the data we are quite generally led to a framework in which the Higgs couples dominantly to leptons, hence the title of our paper.

Since the Higgs couples via Yukawa couplings, the leptonic Higgs will dominantly decay to pairs of tau leptons which will subsequently cascade to electrons and positrons. A straightforward consequence of this is that the ATIC peak is broader and fitting the PAMELA data requires a larger DM mass than in models where the WIMP cascades directly to electrons or muons. Also, the signal for the positron fraction in PAMELA is expected to continue to energies larger than 100 GeV, and not to show a very sharp peak. Another very interesting feature of the framework results from the production of tau leptons. Since decays of taus give rise to an ${\cal O}(1)$ fraction of photons (mostly from $\pi^0$ s coming from tau decays) and neutrinos (from three-body tau decays), there is the possibility of energetic gamma rays and neutrinos in the energy range 100-1000 GeV. However, these signals depend on whether the taus originate from DM annihilations or decays.

The main focus of the paper was to emphasize the general nature of the Higgs sector through which the WIMP DM sector couples to the visible sector, and not to specialize to a particular DM sector. Therefore, both annhilation and decay modes of the DM particles were studied and simple models giving rise to both modes were provided. Also, depending on whether DM is a scalar or fermion, the leptonic cosmic ray signals could arise from a variety of channels: DM annihilation to $\tau^4$ , $\tau^2\nu^2$ or DM decay to $\tau^4$ , $\tau^2$ , $\tau^2\nu$ , $\tau\nu l$ via intermediate Higgs states. Good fits to both PAMELA and ATIC data could be typically obtained for $m_{DM}$ around 4 TeV. If the ATIC data is ignored, then $m_{DM}$ around a TeV can explain the PAMELA data. Regarding constraints from energetic gamma rays and neutrinos coming from the direction of the Galactic Center, it turns out that annihilations provide much more stringent bounds on the parameter space of models compared to decays. But at the same time, they also lead to better detection possibilities for future experiments. The primary reason for this difference is the fact that flux of gamma rays and neutrinos depends on $\rho_{DM}^2$ for annihilations in contrast to $\rho_{DM}$ for decays, and because $\rho_{DM}$ , the mass-density of DM particles in the Galactic Center, is expected to be quite large (although there are large theoretical uncertainties).

One of the most interesting aspects of this framework, at least in my opinion, is the fact that the cosmic-ray signals are necessarily correlated with LHC Higgs signals! Let me explain. Since the DM masses typically turn out to be a TeV or higher, it is hard to produce the DM particles directly at the LHC. However, the leptonic (and hadronic) Higgs states could be produced at the LHC (since they are ${\cal O}(100$ ) GeV) subsequently decaying to tau leptons. In order to provide concrete predictions, we studied a two-Higgs doublet model in which a (softly broken) discrete symmetry (parity) forces one of the Higgs to couple dominantly to leptons and the other to quarks. This gives rise to extremely interesting 4 $\tau$ signals at the LHC from Drell-Yan production (-channel exchange) of the leptonic Higgs and its pseudoscalar partner, followed by their decays to tau pairs. The same process also gives rise to production of charged higgs pairs $H^{\pm}$ which dominantly decay to $\tau^{\pm}\nu$ . This channel therefore provides a new search strategy for the Higgs which has not been well studied so far, and could provide a discovery channel for modest luminosities around 30 $fb^{-1}$ . In addition, we find that the $2\tau$ signal from single Higgs (both leptonic and hadronic) production by vector boson fusion followed by decay to tau pairs can be naturally enhanced compared to that for the Standard Model Higgs, and hence could also provide a discovery channel at modest luminosities. So, Higgs physics seems to be extremely promising.

Finally, I would like to comment on the general compatibility of DM models trying to explain cosmic ray signals, with the observed upper bound on the DM relic abundance. For annihilating DM, there is a mismatch in general in the cross-section required to explain the cosmic-ray signals and the cross-section required to obtain the correct relic abundance from a “standard” thermal freeze-out computation. Many models in the literature try to explain the required large annihilation cross-section for cosmic ray signals by a Sommerfeld enhancement which is operative at non-relativistic velocities ( $\beta{}\sim{}10^{-3}$ ) in the galactic halo, while still having a standard thermal relic abundance during thermal freeze-out. Although an interesting idea, this requires the presence of very light states ( $\lesssim$ GeV) whose mass scale has to be explained. In addition, these light particles have to survive existing constraints from low-energy particle physics experiments as well as astrophysics. Since there do not naturally exist any such light states within our framework, the mismatch between the two cross-sections has to be explained by a different mechanism. It turns out that the correct relic abundance can be obtained even for a larger cross-section compared to the thermal one, if one assumes the existence of late-decaying light ( $\sim{}m_{3/2}$ ) scalar fields (also known as “moduli”) because in this case the relic abundance is determined by the reheat temperature of the modulus rather than the freezeout temperature of DM. This gives rise to non-thermal production of Dark Matter. In fact, such light moduli fields automatically occur within “realistic” string theory compactifications, and hence non-thermal production of DM is quite natural. All this may sound a bit cryptic, but if you are interested, see some of my papers on this subject – “Non-thermal Dark Matter and the Moduli Problem in String Frameworks” and “Neutrino Masses, Baryon Asymmetry, Dark Matter and the Moduli Problem – A Complete Framework“. People always rave about the thermal WIMP ”miracle”, but there is always two-to-three orders of magnitude slop in the “miracle” in terms of the relic abundances. Indeed, the wino, one of the best motivated WIMPs, has a relic abundance about two orders of magnitude lower than the observed one for a weak scale mass ( $\sim{}200$ GeV). It seems to me that with some reasonable assumptions, it is possible to obtain the same parametrics for the relic abundance with a similar “slop” in the non-thermal case as well.

On the other hand, if the DM decay modes are relevant for cosmic-ray signals, then one has to explain the extremely long lifetime ( $\sim{}10^{26}$ s) required for the cosmic ray data. It turns out that if the lepton parity (for a leptonic Higgs) and dark parity (the parity which keeps the DM stable) are spontaneously broken by $\sim v_{EW}/M_{GUT}$ , then a lifetime of the correct magnitude can be naturally obtained. Note that signals for LHC Higgs physics are the same for both annihilation and decay modes since the parity breaking effects are extremely small.

Astrophysics, Cosmology, Journal club, Master Postdoc, Quantum field theory

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293. Power counting semi-classical inflation models and Higgs-Inflation

Posted by Michael Trott

at March 4, 2009

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This is a guest post by Michael Trott from Perimeter Institute. Dmitry.

Thanks Dmitry for offering me the chance to explain the results of a recent paper (arXiv:0902.4465) by Cliff Burgess, Hyun Min Lee and myself.

In this paper, we developed a power counting formalism to study the (often unspoken) limits of validity of various inflation models and then applied this tool to the examples of Higgs-Inflation and higher curvature inflation. To keep this post brief, I will simply set the stage, explain in some detail how the main result of the paper is obtained, and then describe the application to Higgs-Inflation, leaving all other details to the paper.

First the basic ideas. Inflation is now a well accepted paradigm in cosmology. During inflation, the universe underwent a period of accelerated expansion when a Lorentz invariant energy density dominated the equation of state. Inflation is a good theory, it efficiently explains away the flatness, isotropy, homogeneity, horizon and undesired relic problems of the early universe. The growth of quantum fluctuations during inflation seeds the large scale structure that we observe in the universe, and inflation leads qualitatively and quantitatively to the properties of the CMB that are observed by WMAP. One can be rather confident, given this set of facts, that inflation occurred in the early universe.

Cosmic inflation

However, one can also be relatively confident that we really have no singularly good idea as to what underlying physics lead to the inflationary epoch. Every theorist has their favourite model (interestingly, this is usually the particular model that a particular theorist came up with), and broadly speaking the inflationary energy density is generally the energy density of a scalar as it undergoes a classical “slow roll” due to a very flat potential. This scalar is coupled to gravity, and what we first studied was the general power counting of a Lagrangian of such scalars coupled to gravity.

When power counting, one determines the way that the scales of an effective Lagrangian will enter into a general amplitude. For the Lagrangian describing scalars $\theta^i$ coupled to gravity

$-\frac{ \L_\eff}{\sqrt{-g}}=v^4 V(\theta)+\frac{M_p^2}{2}g^{\mu\nu}\Bigl[W(\theta)R_{\mu\nu}+ G_{ij}(\theta)\partial_\mu\theta^i\partial_\nu\theta^j\Bigr]+A(\theta) (\partial\theta)^4$

$(1)\hspace{1cm}+B(\theta)R^2+C(\theta)R(\partial\theta)^2+\frac{E(\theta)}{M^2}(\partial\theta)^6+\frac{F(\theta)}{M^2} R^3+\cdots$

we have the following scales; the reduced Plank mass M_p , a potential scale , and the “cutoff scale” . This later scale is present as this is a nonrenormalizable effective Lagrangian that comes about by integrating out some new physics that comes in at the cutoff scale. Generally we don’t know what this scale is. What matters here is that any underlying theory that leads to an effective Lagrangian of this form, and hence an effective theory of inflation, can be studied consistently in this low energy theory. The nonrenormalizability does NOT mean that the theory is meaningless and not predictive, it simply means that the theory can only be used to predict things to a finite accuracy. If you haven’t met this thinking before, this post is not the right forum to discuss this, but many reviews of effective field theory exist in the literature.

From this Lagrangian one can derive that the following power counting formula for an amplitude with loops and external legs

$A_{N}(E)\simeq E^2M_p^2\left(\frac{1}{M_p}\right)^{N}\left(\frac{E}{4\pi M_p}\right)^{2L}\prod_{d_n=2}\Bigl(c_n\Bigr)^{V_n}$

$(2)\hspace{1cm}\prod_{d_n=0}\left[\lambda_n (\frac{v^4}{E^2 M_p^2})\right]^{V_n}\prod_{d_n\ge 4}\left[g_n\left(\frac{E}{M_p}\right)^2\left(\frac{E}{M}\right)^{d_n-4}\right]^{V_n}$

Here is the largest physical scale that appears in the propagators or vertices in the calculation and we have split up contribution to the amplitude from interactions with no derivatives d_n=0 from the terms with two derivatives d_n=2 and the terms with four or greater derivatives $d_n\ge4$ . This is an unassuming formula, but it is very useful and powerful when studying an inflation model!

The first thing one notices is that for the theory to have a will defined loop expansion one must have

$(3)\hspace{1.5cm}\frac{E}{4\pi M_p} \ll1\hspace{1.5cm}g_n\left(\frac{E}{M_p}\right)^2\left(\frac{E}{M}\right)^{d_n-4}\ll 1$

It is common in the literature to study inflation models at tree level and not worry too much about all the loop corrections that one’s model necessarily leads too. It is dangerous at times to neglect constraints such as (3). Some quantum corrections that are simpler to include can (and are) incorporated in inflation models, especially when the help make the case for the model! However, a systematic study of loop corrections and all quantum effects can be boring and requires either a large number of post docs and students or one concise and powerful power counting formula.

Lets see how this can help one appreciate the strengths or limitations of a model by studying a particular example. Our interest in undertaking this work was sparked in part by the Higgs-Inflation model that was recently put forth by Bezrukov and Shaposhnikov (arXiv:0710.3755). This model is potentially very exciting as it economically uses the Higgs boson non-minimally coupled to gravity through the dimension four term $\xi{H^\dagger}{H}R$ to drive inflation. It is important to note that although this dimension four term is usually neglected when one studies the SM, it certainly exists as it is required when you renormalize the theory is curved space. The only argument is what is the coefficient. For the Higgs to be the scalar leading to inflation, requiring that the amplitude of primordial fluctuations agree with what is found by WMAP gives

$(4)\hspace{3cm}\xi \simeq 5 \times 10^4 \left(\frac{m_{H}}{\sqrt{2} v_{H}} \right) \gg 1.$

First of all this is very exciting. This model has a beautifully testable consequence in that once the Higgs is found and its mass determined, one can examine the CMB to see if it is consistent that the Higgs has lead to inflation as well as EW symmetry breaking. It is very remarkable that qualitatively the required mass lies within the stability and triviality bounds on the Higgs mass (due to entirely different physics) and is remarkably consistent with global fits to electroweak precision data. This did not have to be and is quite a conspiracy! However, one also notices that for a typical mass $m_{H}\sim{}120$ GeV that $\xi\sim{} 10^4$ . That is a troubling coupling to imagine putting in scattering diagrams. Does this theory really make sense? Perhaps it just violates unitarity for the energies of interest and all of this excitement is misplaced.

The power counting formula allows one to check this unitarity worry directly and efficiently. One can bound the cutoff scale of the theory by examining energetic graviton-higgs scattering (gh -> gh) or higgs-higgs scattering through graviton exchange in flat space. We use the power counting result to examine when these amplitudes saturate the bound as a function of the loop order . Here is the center of mass energy of the scattering and the relevant interactions come from the nonminimal coupling term, so d_n=2 . One easily finds that the most problematic amplitude for unitarity for (gh -> gh) scales as

$(5)\hspace{2.5cm}A_4^{\rm max}(E) \simeq \left(\frac{\xi E}{M_p} \right)^2\left( \frac{\xi E}{4 \pi \, M_p}\right)^{2L}$

Demanding that the cross section not saturate the unitarity bound $\sigma \propto 1/E^2$ gives an $\xi$ dependent upper bound on (that in turn gives an upper bound on ). This bound is

$(6)\hspace{3cm} E < E_{max} \simeq \frac{M_p}{\xi}$

Now as the Hubble scale in this theory is ${\rm H} \simeq \sqrt{\lambda_H}M_p/\xi$ this means that for this model to make sense as an effective theory, the cutoff scale lies in the uncomfortably small window

$(7)\hspace{3cm} 1\gg{\rm H}/M\gg\sqrt{\lambda_H}}$ .

The upper bound comes from the usual constraint of an adiabatic inflationary expansion. This is a small window of validity, although it is logically possible that such a cut off scale exists. This window gets even more uncomfortable if one has any other new physics coupled to the Higgs, as this scale is extremely unstable when the Higgs couples to any new physics particles.

Notice that this analysis is accomplished without the painful explicit calculation of partial wave scattering amplitudes of graviton Higgs or Higgs-Higgs scattering. If one actually calculates this explicitly, and some brave souls have calculated Higgs-Higgs scattering through graviton exchange (Han and Willenbrock hep-ph/0404182) before,the unitarity cut off constraint they find agrees with the scaling one obtains directly with the power counting directly and virtually instantaneously.

In summary, if one has a viable inflation model of some scalar(s) couples to gravity through some potential, and you want to check the validity of such a model through examining its quantum corrections and unitarity constraints, power counting is the right way to go about such a check if you don’t want to suffer too much. Eqn. (2) provides a powerful tool to check the quantum consequences of your inflationary theory. We have used this tool to examine the Higgs-Inflation theory and have found that the required cut off scale must lie in a rather small window for this to be a sensible low energy theory with underlying physics that is unitary. We hope we have provided a useful tool to those who might propose interesting and exciting semi-classical theories of inflation in the future to easily check some quantum consequences of their theory.

Update: Since our paper came out, a follow up paper by Barbon and Espinosa appeared (2 days later!) that agrees with some of our findings but makes the stronger claim that the cutoff scale rules out the higgs inflation scenario entirely (0903.0355). People sure are excited about Higgs inflation!

Cosmological perturbations and large scale structure, Cosmology, Journal club, Master Mason, Quantum field theory

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292. Universal properties of the U(1) current at deconfined quantum critical points

Posted by Flavio Nogueira

at March 3, 2009

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This is a guest post by Flavio Nogueira from the U. of Berlin. Dmitry.

Before I start talking my recent preprint [http://arxiv.org/abs/0902.0364], let me thank Dmitry for inviting me to write this contribution in his blog.

I cannot talk about my preprint without first talking a little bit about the paper of Pawel Kovtun and Adam Ritz [http://arxiv.org/abs/0806.0110; Phys. Rev. D 78, 066009 (2008)] entitled “Universal conductivity and central charges”. For theories having a gravitational dual, Kovtun and Ritz have derived interesting relations between the central charges and the universal amplitudes of certain thermodynamic quantities. Before talking about it more concretely, we have to recall some basic aspects of a conformal field theory (CFT).

Let us consider two important conserved physical quantities in a CFT, namely, the energy-momentum tensor $T_{\mu\nu}(x)$ and the U(1) current $J_\mu(x)$ . At zero temperature, a -dimensional CFT has correlation functions for these quantities given by

$\langle T_{\mu\nu}(x)T_{\alpha\beta}(0)\rangle=\frac{c}{S_dx^{2d}}\left(I_{\mu\alpha}I_{\nu\beta}+I_{\mu\beta}I_{\nu\alpha}-\frac{2}{d}\delta_{\mu\nu}\delta_{\alpha\beta}\right),$ (1)

$\langle{}J_\mu(x)J_\nu(0)\rangle=\frac{k}{S_dx^{2(d-1)}}I_{\mu\nu},$ (2)

where $I_{\mu\nu}=\delta_{\mu\nu}-2x_\mu x_\nu/x^2$ and $S_d=2\pi^{d/2}/\Gamma(d/2)$ is the surface of a unit sphere in dimensions. The numbers and are the so called central charges of the operators $T_{\mu\nu}(x)$ and $J_\mu(x)$ , respectively. Thus, we see that the above correlation functions of a CFT are simply determined by dimensional analysis and rotation invariance. This follows from the fact that a CFT is scale invariant. Interestingly, at finite temperature the pressure and the charge susceptibility of a CFT are also determined by dimensional analysis,

P=c'T^d , $\chi=k'T^{d-2}$ .

This happens because and $\chi$ are related to $T_{\mu\nu}(x)$ and $J_\mu(x)$ , respectively. For instance, $\chi=\langle Q^2\rangle/V$ , where is the conserved charge associated to the current $J_\mu$ and is the (infinite) volume.

For a two-dimensional CFT, we have the exact universal relations [I. Affleck, Phys. Rev. Lett. 56, 746 (1986); H. W. J. Bloete, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986)],

$\frac{c'}{c}=\frac{\pi}{6}$ , $\frac{k'}{k}=\frac{1}{2\pi}$ . (3)

The above result can be intuitively understood in the following way. To compute the correlation functions (1) and (2) at finite temperature, we need to consider the system in in a spacetime where the time direction is made periodic with period . In two dimensions this corresponds to map the plane into a cylinder and such a transformation is conformal. Thus, for a theory with conformal invariance we can expect that zero and finite temperature universal properties are related. In this case the central charges are determining the thermodynamics of the theory.

What Kovtun and Ritz have shown, is that for -dimensional theories with a gravitational dual the following relations should hold:

$\frac{c'}{c}=\frac{1}{4\pi^{d/2}}\left(\frac{4\pi}{d}\right)^d\frac{(d-1)\Gamma^3(d/2)}{d(d+1)\Gamma(d)}$ , (4)

$\frac{k'}{k}=\frac{1}{2\pi^{d/2}}\left(\frac{4\pi}{d}\right)^{d-2}\frac{\Gamma^3(d/2)}{\Gamma(d)}$ . (5)

For d=2 the above formulas reduce correctly to the result in (3). Note, however, that if by chance we find a CFT at d>2 fulffilling the above equations, this would not necessarily mean that the corresponding theory has a gravitational dual. However, this may well be the case, but a proof would certainly not be easy.

Now, note that many interesting theories are actually not conformal. However, there is a class of theories which are conformal at a precise value of the coupling constant. This special value of the coupling constant defines what is called a quantum critical point (QCP). In field theory language it corresponds to a non-trivial fixed point of the renormalization group (RG) $\beta$ function. Thus, the result given in Eqs. (4) and (5) should also be valid for a theory whose QCP has a gravitational dual.

A QCP separates different phases of a theory and governs a second-order phase transition. In condensed matter physics there are many examples of theories having a QCP. Particularly interesting are some theories defined in 2+1 dimensions. For example, there are phase transitions in magnetic Mott insulators which involve competing ordering states. A paradigmatic example is the quantum phase transition between a so called Neel state (i.e., an antiferromagnetically ordered state) and a crystal of singlet valence bonds, the so called valence-bond solid (VBS) state. Read and Sachdev [N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989); Phys. Rev. B 42, 4568 (1990)] have shown that the effective field theory governing this quantum phase transition is a gauge theory. The quantum critical point of such a system is governed by an Abelian Higgs model with two complex scalar fields, z and z_2 . These fields are the elementary constituents of the theory. They are called spinons. The spinons are the building blocks of the spin orientation field,

$\vec n=z_\alpha^*\vec \sigma_{\alpha\beta}z_\beta,$

where $\vec \sigma\equiv(\sigma_1,\sigma_2,\sigma_3)$ , $\sigma_i$ being a Pauli matrix, and the constraint $\vec n^2=1$ must be satisfied. We can write a Lagrangian describing the dynamics of the field $\vec n$ . From textbooks on solid state physics we learn that spin-waves in an antiferromagnet have a relativistic spectrum, i.e., $\omega\sim|\vec k|$ . Therefore, the simplest Lagrangian for an antiferromagnet reads

${\cal L}=\frac{1}{2g}(\partial_\mu\vec n)^2$ .

In terms of the spinon fields this becomes

${\cal L}=\frac{1}{g}|(\partial_\mu-iA_\mu)z_a|^2$ , (6)

with the additional constraint that |z_1|^2+|z_2|^2=1 , as required by $\vec n^2=1$ . This is the CP^1 model. It will be convenient to consider a version with complex fields, i.e., the $CP^{N-1}$ model, in which case the constraint becomes z_a^*z_a=1 , with the summation over running from to .

From the equations of motion we obtain

$A_\mu=\frac{i}{2}(z_a^*\partial_\mu z_a-z_a\partial_\mu z_a^*)$ .

Thus, at the classical level the gauge field $A_\mu$ emerges from the spinon fields. If we specialize to d=2+1 and consider the dual field $B_\mu=\epsilon_{\mu\nu\lambda}\partial_\nu A_\lambda$ , we can easily show that the flux of $B_\mu$ through a closed surface is quantized:

$\oint_{S}dS_\mu B_\mu=2\pi q$ ,

where is an integer. Here it is useful to note the differences between the above flux quantization and the flux quantization in a superconductor. In a superconductor it is the flux through an open surface that is quantized, such that Stokes theorem can be applied to show that the circulation of the vector potential is quantized. In our case it is the flux through a closed surface that is quantized. Thus, in the superconducting case flux quantization leads us to consider lines as topological defects, i.e., vortex lines. For the $CP^{N-1}$ model case, on the other and, the flux emerges from a point, a magnetic monopole in space time, the so called instanton.

The instantons play a crucial role in an antiferromagnetic Mott insulator, especially in the VBS phase. The reason for this is an important ingredient not yet mentioned here: the Berry phases. By quantizing a spin system with the path integral we always obtain a Berry phase in addition to the action describing the elementary excitations of the system. In an antiferromagnet this Berry phase is alternating and is actually responsible for the appearance of a non-trivial paramagnetic phase like the VBS one. As shown by Read and Sachdev, the Berry phases are not very important in the Neel phase, but are crucial in the VBS phase. Recently the important role of the interplay between the Berry phases and the intantons was discussed by Senthil and collaborators [T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science, 303, 1490 (2004), http://arxiv.org/abs/cond-mat/031132; T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B 70, 144407 (2004), http://arxiv.org/abs/cond-mat/0312617]. They realized that near the phase transition the instantons are actualy suppressed by the Berry phases, which leads to spinon deconfinement at the phase transition. Recent Monte Carlo simulations confirmed that this meachanism for instanton suppression works for the case of an easy-plane antiferromagnet [S. Kragset, E. Smorgrav, J. Hove, F. S. Nogueira, and A. Sudbo, Phys. Rev. Lett. 97, 247201 (2006), http://arxiv.org/abs/cond-mat/0609336], but in this case the phase transition was shown to be first-order [see also the simulations of A. B. Kuklov, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer, Ann. Phys. (N.Y.) 321, 1602 (2006); http://arxiv.org/abs/cond-mat/0602466], i.e., no QCP in this case. In cases where a QCP exist, which is the interesting situation to us, the QCP of the instanton-suppressed theory governs a completely new universality class featuring a large anomalous dimension of the magnetic order parameter. Note that in a Landau-Ginzburg-Wilson (LGW) type of approach the magnetization and the VBS order parameters would compete and never lead to a QCP. Moreover, for a case where the paramagnetic case has no broken symmetry (i.e., no VBS) and only the magnetic order parameter is available, there is a second-order phase transition with a very small anomalous dimension. The reason why the anomalous dimension is large in the case of the instanton-suppressed antiferromagnet is simple. As we have mentioned earlier, the spinons are the building blocks of the magnetic order parameter. So, in order to obtain the anomalous dimension of it we have to calculate the anomalous dimension of a composite operator, and this leads to a large anomalous dimension for $\vec n$ . This is impossible in the case of a LGW theory, since there $\vec n$ is the elementary field.

The supression of the instantons by the Berry phases at the QCP can be consistently achieved by including a Maxwell term in the Lagrangian (6). We will also soften the $CP^{N-1}$ constraint. This softening does not affect the quantum critical properties of the theory near the QCP. Therefore, we are led to consider the Lagrangian,

${\cal L}=\frac{1}{4e_0^2}F_{\mu\nu}^2+\sum_{a=1}^{N}|(\partial_\mu-iA_\mu)z_\alpha|^2+r_0\sum_{a=1}^{N}|z_\alpha|^2+$
$+\frac{u_0}{2}\left(\sum_{a=1}^{N}|z_\alpha|^2\right)^2$ .

In terms of this theory, the Neel state corresponds to a phase where the spinons are condensed. This is the Higgs phase. In the VBS phase, away from the QCP, the instantons are more relevant. In this case, the spinons are confined and we have a non-zero VBS order parameter. The spinons are deconfined at the QCP. For this reason, the scenario described here is called deconfined quantum criticality. A schematic phase diagram is shown in the figure.

Neel-VBS phase transition

The conserved U(1) current of this theory is

$j_\mu=-\sum_{a=1}^N[ie_0(z_a^*\partial_\mu z_a-z_a\partial_\mu z_a^*)+2e_0^2A_\mu|z_a|^2]$ .

In my recent paper, I was mainly interested in the current correlation function of this theory and how close the universal aspects of this U(1) current to the predictions of the gauge-gravity duality are. The results I obtained are actually very simple. Let us discuss the main results of the paper.

As we know from textbooks, the renormalized gauge coupling is given simply by multiplying the bare coupling e_0^2 by the wavefunction renormalization of the gauge field Z_A , i.e., e^2=Z_Ae_0^2 . Recall that Z_A is directly calculated from the vacuum polarization $\Pi(p)$ . Indeed, the gauge field propagator is written in Euclidean space as

$\langle A_\mu(p)A_\nu(-p)\rangle=\frac{1}{p^2[1+\Pi(p)]}\left(\delta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}\right).$

Thus, we have,

$Z_A(p)=\frac{e_0^2}{1+\Pi(p)}$ .

The current correlation function in momentum space is

$\langle j_\mu(p)j_\nu(-p)\rangle=K(p)\left(\delta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}\right)$ ,

where the function K(p) is related to the vacuum polarization by

$K(p)=-\frac{1}{e_0^2}p^2\Pi(p)$ .

From the above equations we immediately see that

$Z_A(p)=\left[1-\frac{e_0^2}{(d-1)p^2}\langle j(p)\cdot j(-p)\rangle\right]^{-1}$ .

On the other hand, by Fourier transforming Eq. (2), we obtain

$\langle j(p)\cdot j(-p)\rangle=\frac{(d-2)\Gamma(1-d/2)\Gamma^2(d/2)}{(4\pi)^{d/2}\Gamma(d-1)}~k|p|^{d-2}$ ,

and therefore we can write an exact formula for the renormalized gauge coupling in terms of the central charge :

$\frac{1}{e^2(p)}=\frac{1}{e_0^2}-\frac{(d-2)\Gamma(1-d/2)\Gamma^2(d/2)}{(4\pi)^{d/2}\Gamma(d)}k|p|^{d-4}$ .

By defining the dimensionless gauge coupling as $f(p)=|p|^{d-4}e^2(p)$ , we can find the value of at the QCP by taking the limit $f_*=\lim_{|p|\to 0}f(p)$ . This QCP is the sameas the fixed point value of the RG $\beta$ function for . Note that we can determine the exact value of f_* in terms of , although we cannot determine the exact $\beta$ function. This is because e^2(p) is the renormalized gauge coupling already at the QCP, since we have calculated it using the quantum critical current correlation function. Therefore, we obtain the following interesting formula for :

$k=\frac{2^{d-1}\pi^{d/2}\Gamma(d)}{\Gamma(2-d/2)\Gamma^2(d/2)}\frac{1}{f_*}$ . (7)

There is a somewhat similar formula for a theory having a gravitational dual, except that in this case , which is the central charge for the non-gravitational theory living on the boundary, is related to the dimensionless coupling of the gravitational theory in the bulk. In fact, Freedman and collaborators [D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Nucl. Phys. B 546, 96 (1999);
http://arxiv.org/abs/hep-th/9804058] have derived the result:

$k=\frac{2\pi^{d/2}(d-2)\Gamma(d)}{\Gamma^3(d/2)}\frac{1}{\hat g_{d+1}^2}$ ,

where $\hat g_{d+1}^2$ is the dimensionless gauge coupling of the theory in (d+1) -dimensional anti-de Sitter spacetime. For d=2+1 both results for agree, except that f_* is not the coupling constant in the bulk. Indeed, we have k=32/f_* for the former, and k=32/g_4^2 for the latter. So, we see that both results look very similar.

Now, Kovtun and Ritz were able to derive also a formula for in terms of $\hat g_{d+1}^2$ by using gauge-gravity duality arguments. They have shown that

$k'=\frac{d-2}{\hat g_{d+1}^2}\left(\frac{4\pi}{d}\right)^{d-2}$ ,

Unfortunately, I was not able to derive a similar formula for for a deconfined QCP. Desirable would be a formula such that $k'=\alpha/f_*$ , where $\alpha$ is some constant. Then it would be of course very nice if in such a formula we get for d=2+1 the same value as predicted in the gauge-gravity duality. But such a comparison will have to wait because at the moment I still don’t know how to derive $\alpha$ . However, I was able to calculate the value of for a free theory featuring complex scalar fields. This actually corresponds to the same result as obtained from the interacting theory at the large . Since we are interested in comparing the ratio k'/k with the prediction (5), let us first consider in the large limit. After calculating f_* in large limit and using Eq. (7), that

$k=\frac{2N}{d-2}$ ,

which is also the value of for a free theory with complex scalar fields.

From the calculation of the charge susceptibility we obtain

$k'=N\pi^{(d-5)/2}(d-3)\Gamma\left(\frac{3-d}{2}\right)\zeta(3-d)$ .

The ratio k'/k at d=2 agrees with the exact result for that case, as we would expect. Interestingly, for d=3 the same value as for d=2 is obtained.

Note that the large result holds also in the absence of the Maxwell term and by strictly assuming the $CP^{N-1}$ constraint. It would be interesting to consider a finite result, like the case of interest N=2 , in which case the CP^1 model is equivalent to the non-linear $\sigma$ -model. There is actually a conjecture by Klebanov and Polyakov [Phys. Lett. B 550, 213 (2002); http://arxiv.org/abs/hep-th/0210114] where they argue that there should be a duality between the critical O(n) model at d=3 and a spin gauge theory defined in AdS ${}_4$ . For n=3 this is simply CP^1 . Unfortunately, it is not an easy task to consider finite theories within the framework of the gauge-gravity duality. While such an analysis would be of little interest for particle physics phenomenology, it would be a very important achievement in a condensed matter physics context as discussed here. I hope that with the time more condensed matter physicists will get interested on the AdS/CFT correspondence. At the moment most of the papers making condensed matter applications were written by string theorists and/or nuclear physicists, with only a few exceptions like Subir Sachdev, Markus Muller, and Lars Fritz. But I believe that the number of condensed matter physicists working on the subject is likely to increase considerably by the end of 2009.