^{3}.

We consider QCD at strong coupling with scalar quarks coupled to a chemical potential. Performing the link integrals we present a diagrammatic representation of the path integral weight. It is based on mesonic and baryonic building blocks, in close analogy to fermionic QCD. Likewise, the baryon loops are subject to a manifest conservation of the baryon number. The sign problem is expected to disappear in this representation and we do confirm this for three flavors, where a scalar baryon can be built and, thus, a dependence on the chemical potential occurs. For higher flavor number, we analyze examples for a potential sign problem in the baryon sector and conjecture that all weights are positive upon exploring the current conservation of each flavor.

Diagrammatic Monte Carlo methods are one of the most successful approaches to the sign problem of quantum field theories at nonzero chemical potential, e.g., see

Integrating out the quarks instead gives the determinant of the non-Hermitian Dirac operator and, therefore, a complex weight.

leads to certain building blocks along the lattice bonds. They are of mesonic and baryonic nature and (together with fermion saturation at every site) facilitate an intuitive diagrammatic representation of the QCD path integral.Nonetheless, positivity of the so-obtained weight is not guaranteed and terms of opposite sign indeed appear, even at zero chemical potential. This has hampered further use of this approach in simulations of realistic QCD even though worm algorithms have proven capable of simulating such constrained systems (for recent attempts see

One may view the problem of this approach as a fermionic sign problem. For staggered fermions there are clearly four sources of negative signs: (i) the relative sign between the hopping terms in the Dirac operator which is of first order in derivatives, (ii) antiperiodic boundary conditions in Euclidean time, (iii) the Grassmann nature of the quark fields in the path integral, (iv) the staggered signs. All of them are absent if quarks were Lorentz scalars, and one may expect scalar QCD (sQCD) to be free of the sign problem. Note, however, that the gauge links are complex and SU(3) group integrals are not necessarily positive

Gauge theories with scalar matter might be relevant beyond the Standard Model; here we compare sQCD to QCD at nonzero chemical potential. One of the main differences is the flavor-antisymmetric nature of the baryons of sQCD; as a consequence at least three scalar quark flavors are necessary to generate a dependence on the chemical potential (see Sec.

Scalar quarks are not subject to the Pauli exclusion principle and thus the building blocks of sQCD diagrams come with less constrained occupation numbers, e.g., baryon worldlines may intersect. This shall be of advantage for numerical simulations as well as for treating higher flavor numbers, which we conjecture to be free of the sign problem as well.

We treat sQCD with

At real

The diagrammatic representation of this system emerges after integrating out all gauge links. To do so for a particular link, we collect the two terms in which

For fermionic quarks

The integration over SU(3) group elements (with Haar measure) can be turned into a fivefold sum

We have further expanded the two terms

When interpreting

In

As is typical for bosonic systems, occupation numbers are unbounded from above and do not exclude each other. A configuration in this new representation can easily be determined by a list of all integers

Example of a diagram on a

As numbers, the mesons

Interestingly, imaginary

The objects potentially inducing a sign problem in the diagrammatic representation of sQCD are

For

For arbitrary

Note that this formula is of the type “determinant of a sum is a sum of determinants" (!), which holds for the outer product structure in which we are interested here. For fermions, this formula receives an additional sign due to reordering

So far we have not performed the matter field integrations. The

However, there is one important feature of the

Phase integrations typically cause U(1) current conservation in diagrammatic approaches to bosonic systems, for fermionic systems this role is played by the saturation of Grassmann integrals by equal numbers of

To derive an immediate consequence for the diagrams, consider the coarser constraints that occur after summing these constraints over all indices but the site,

According to Eq.

Since

Coming back to the sign problem at

One might argue that a negative sign occurs upon permuting the flavors under one of the determinants, but according to Eq.

In the diagrammatic representation this system can therefore be simulated, presumably with a hybrid approach for the updates: for unconstrained variables such as

We close by discussing the technicalities faced at more than three flavors, say at

It seems necessary to explore finer constraints than above. For instance, the summand

At first sight, mesons may change this positivity argument. A mesonic building block, say

Building up slightly more complicated configurations, consider a baryon loop connected to a line of unit mesonic

Two simple examples, where a closed baryon loop is connected to a single meson line at two sites

In our opinion, these examples point at the positivity of all diagrammatic weights even for

We have shown that, in the strong coupling limit, the sign problem in sQCD can, indeed, be solved for

For more than three flavors,

The case of

Then it would be interesting to analyze its phase diagram. We expect a phase transition at zero temperature, if

On the technical side, the ability to overcome the sign problem and to simulate sQCD could give benchmarks in a “QCD-like” theory for current approaches like reweighting, analytic continuation in

The authors are supported by the DFG (BR 2872/6-2 and BR 2872/7-1) and thank Jacques Bloch and Christof Gattringer for helpful discussions. F. B. is grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work.