RENSSELAER POLYTECHNIC INSTITUTE
In what follows, I will take up each of these issues, explain why the Standard Model is deficient, and how theories of new physics are perhaps able to address them. In the process, we will see that string theory plays a prominent, encompassing explanation, which is one reason for its popularity. [Under construction...]
Gauge-gravity duality N=4 super-Yang-Mills is the unique four-dimensional renormalizable field theory with maximal supersymmetry, meaning that it has the maximum number of conserved supercharges. These are four Majorana spinor charges, hence N=4. In the late 90s it was discovered by Maldacena (and elaborated by others) that this theory with gauge group U(Nc) is dual to Type IIB string theory with Nc coincident D3 branes compactified on the space AdS(5) × S(5). Since the latter is a gravitational theory, and the former is a gauge theory, this is an example of a gauge-gravity duality. Subsequently, many other gauge-gravity duals have been discovered, all of which are supersymmetric.
| construction | dimensions | supersymmetry | string background | conformal/confining |
|---|---|---|---|---|
| Maldacena | 1+3 | N=4 | AdS(5) × S(5) | conformal |
| Klebanov-Witten | 1+3 | N=1 | AdS(5) × T(1,1) | conformal |
| Klebanov-Strassler | 1+3 | N=1 | resolved AdS(5) × T(1,1) | confining |
Background: Because of asymptotic freedom, there are classes of QCD-like theories that possess an infrared fixed point (IRFP). This occurs when at long distance the screening due to the fermions balances the attraction due to the gluons, in such a way that the running coupling no longer runs. Some evidence that this may occur can be seen from the two-loop beta function, where for a well-chosen fermion content, the one-loop and two-loop coefficients have opposite sign. Naturally the existence of the IRFP has to be verified nonpertubatively because typically the fixed point coupling obtained from the two-loop calculation is rather large. When there is an IRFP, the theory will have an enhanced symmetry in the infrared, with the Poincare group enlarged to the conformal group, yielding a conformal field theory (CFT). Such theories are of great theoretical interest. In particular, they may be dual to a gravitational theory because the conformal group is identical to the isometry group of an anti-deSitter geometry.
The challenge: We are involved in the study of a theory with an IRFP from first principles on the lattice. This is challenging for a number of reasons. First, the CFT is strongly interacting, and so a nonperturbative approach such as lattice gauge theory must be used in order to obtain meaningful results. Second, the underlying gauge theory has dynamics that span a wide range of scales, from the ultraviolet where the theory is defined to the infrared where the conformal behavior is to be observed. For this reason, it is difficult to directly simulate a system that incorporates these scales all at once, and a renormalization group approach is quite useful as a method to circumvent this problem. Third, the fermions in the CFT are massless because a mass is a relevant parameter which would drive the theory away from the IRFP. This makes such theories very expensive to study on a computer, because the matrix problem becomes ill-conditioned.
Goals: (1) Use Monte Carlo renormalization group to identify the infrared fixed point in SU(2) gauge theory with two Dirac flavors of adjoint representation fermions (Adj2). (2) Compute the anomalous mass dimension in this theory. (3) Show that we are in the basin of attraction of the Gaussian fixed point. What do these things mean? For this, we turn to the:
Method: We are using the two lattice matching method. This involves simulations on a "fine" lattice which are matched to simulations on a "coarse" lattice, using a number of "blocked" observables. Here blocking refers to an intelligent sort of averaging or coarse-graining. Bare couplings on the two lattices are adjusted until the observables give matching values. Evolution of the couplings under this procedure corresponds to renormalization group flow. For more details click here.
Fixed Point: In Adj2, it is now believed that an IRFP exists. This means that under renormalization group flow, the gauge coupling approaches a point in parameter space where it ceases to change. This would be indicated by a zero of the bare step scaling function (discrete beta function) So far all that we have found is that the bare step scaling function is small and could be zero once systematic uncertainties are taken into account. We are currently working to reduce these uncertainties by: (1) going to larger lattices where more blocking steps can be taken, (2) using O(a) improved actions (adding the clover term). Both of these would reduce scaling violations, which are the source of disagreement between different observables as to the matching of the bare couplings on the fine and coarse lattices.
Anomalous Mass Dimension: This quantity characterizes how the running mass behaves with respect to the renormalization group. It also dictates the quantum mass dimension of the scalar fermion bilinear. According to a number of methods, the anomalous mass dimension is about 0.4. However, Monte Carlo renormalization group has been giving us confusing results: about 0 if we assume that we are near the fixed point where the couplings on the two lattices should be equal, and somewhere between -0.6 and 0.6 if we take into account our uncertainties in the bare step scaling function. We are currently working to include fermionic observables in the matching, in addition to reducing the scaling violations as mentioned above. We are hopeful that these improvements will lead to more definitive answers.
Publications:
The MCRG project has its own webpage. Click here for more details
Background: The Wess-Zumino model is the simplest interacting four-dimensional supersymmetric theory. It consists of a Majorana fermion and a complex scalar field, interacting through a Yukawa coupling. Formulating theories with scalars on the lattice is especially challenging, since the lattice regulator explicitly breaks supersymmetry and hence there is no symmetry to protect the scalar mass from quantum corrections. Another problem is that the form of the Yukawa coupling is dictated by U(1) R-symmetry, which is a chiral symmetry; such a symmetry requires fermions that satisfy the Ginsparg-Wilson relation if it is to be realized explicitly on the lattice (as opposed to emerging as an accidental symmetry in the infrared due to fine-tuning). Furthermore, supersymmetry requires that the bosonic couplings be related to the fermionic ones in a very specific way, and if supersymmetry is violated by the regulator, this will not be the case in the low-energy effective theory. Because of the interest in supersymmetric theories with scalars, such as super-QCD (used in phenomenology) and N=4 super-Yang-Mills (used in gauge-gravity duality), it makes sense to first try out methods on the relatively simple case of the Wess-Zumino model.
Goal: Using a formulation based on Ginsparg-Wilson fermions, conduct the necessary fine-tuning to achieve the supersymmetric continuum limit.
Method: The Ginsparg-Wilson chiral symmetry reduces the number of counterterms that must be adjusted nonperturbatively, thus reducing the dimensionality of the parameter space that must be searched. This is a significant savings. We are measuring the four-divergence of the supercurrent as a probe of supersymmetry violation. We are also measuring the effective masses of bosons and fermions, as these must be equal when the fine-tuning is successful.
Simulations: We have written graphics processing unit (GPU) code based on Nvidia's CUDA (a C interface) to perform our simulations. The Wess-Zumino model is ideally suited to this computing platform, because the memory requirements are not that great. That is, we are able to fit the problem onto a single GPU. We currently have four GPUs for these calculations.
Publications:
Definition: In theories with approximate scale invariance, where the coupling "walks" instead of running over some range of scales, some workers have argued that there will be a pseudo-Nambu-Goldstone boson associated with the spontaneous breakdown of this symmetry. This light scalar particle is called the dilaton. To learn more about our studies of this scenario, click here for more details.
In a project supported by the National Science Foundation, we are developing computer code to study scalar resonances in QCD-like theories, harvesting the significant computing power of GPUs. For further details, click here.
We have written code that integrates QUDA and CPS for clover fermions. For this, one should use a recent version of CPS and replace the file src/util/lattice/f_clover/f_clover.C with this file f_clover.C. The configure script needs some options, which are illustrated here do_configure.
Goals:
Facilities: The Computational Center for Nanotechnology Innovations (CCNI). We are currently exploiting some of the 16 BlueGene/L racks that are available to us at this facility, which was built as a partnership between Rensselaer, IBM and New York State.
Performance: Each rack provides 5.6 trillion floating point operations per second (TFlops), and we use software built on a modification of the Columbia Physics System. It has an approximate 10% sustained utilization.
Other Project Members: Richard Brower (Boston U.), Simon Catterall (Syracuse U.), George Fleming (Yale U.), Pavlos Vranas (Lawrence Livermore Natl. Lab.)
Publications:
We are exploring how the physics of flavor is predicted based on the geometry of the compact space, which is a seven-dimensional manifold with G(2) holonomy. For further details, click here.
"Hadrons" that don't fit neatly into the naive quark model. See for instance the recent Science feature.
It is important to get up out of the chair, take a walk, and think about physics!