We present an algorithm to automatically derive Feynman rules for lattice perturbation theory in background field gauge. Vertices with an arbitrary number of both background and quantum legs can be derived automatically from both gluonic and fermionic actions. The algorithm is a generalisation of our earlier algorithm based on prior work by L\"uscher and Weisz. We also present techniques allowing for the parallelisation of the evaluation of the often rather complex lattice Feynman rules that should allow for efficient implementation on GPUs, but also give a significant speed-up when calculating the derivatives of Feynman diagrams with respect to external momenta.
Deep Dive into Improved automated lattice perturbation theory in background field gauge.
We present an algorithm to automatically derive Feynman rules for lattice perturbation theory in background field gauge. Vertices with an arbitrary number of both background and quantum legs can be derived automatically from both gluonic and fermionic actions. The algorithm is a generalisation of our earlier algorithm based on prior work by L"uscher and Weisz. We also present techniques allowing for the parallelisation of the evaluation of the often rather complex lattice Feynman rules that should allow for efficient implementation on GPUs, but also give a significant speed-up when calculating the derivatives of Feynman diagrams with respect to external momenta.
While the lattice offers a fully nonperturbative regulator for Quantum Field Theory, perturbative calculations still play an important role in renormalising, matching and improving operators, both those appearing in the action and those used to measure observables. For the highly improved actions now widely used, such as the Highly Improved Staggered Quark (HISQ) action [1] or moving NRQCD, [2] manual derivation and implementation of the Feynman rules would be highly impractical. Automated methods are therefore required. A general algorithm to derive the Feynman rules for arbitrary traced closed Wilson loops was derived by Lüscher and Weisz in [3]. We have generalised this algorithm to include fermionic fields [4] and have implemented it in the HiPPy/HPsrc packages [5].
HiPPy is an is an automated tool for generating Feynman rules from arbitrary lattice actions, which is written entirely in Python [6]. Starting from the perturbative expansion
of the link variables, the action is expanded as
with vertex functions
giving the Feynman rules.
The algorithm for achieving this expansion starts from the encoding of individual terms in the vertex function as so-called entities
each of which carries an amplitude f i . The crucial property of entities is the multiplication rule
where c c c = x x xy y y and α k is defined via Γ α i Γ α j = φ α i α j Γ α k . Entities differing by only translations are equivalent by momentum conservation. Additional structure (e.g. a non-trivial colour structure) can also be encoded by adding additional fields to the entity and amending the entity algebra accordingly.
A field is then defined as a double mapping
which encodes a generic Wilson line, and multiplication of field objects is defined accordingly in terms of entity multiplication by
where C rs = (r + s)!/(r!s!) is a combinatorial factor and φ is the phase from the multiplication of the spin matrices belonging to the entities E and E , as defined above. Addition of field objects is defined by the addition of the amplitudes belonging to the individual entities, with the amplitude in the sum of an entity present in only one of the summands being set to its amplitude in that summand.
The basic building block from which smeared links, operators and actions are constructed in an iterative fashion is the simple link encoded as
and predefined building blocks (e.g. smeared links, covariant derivatives and field strength tensors) constructed from this are provided as part of HiPPy.
The HPsrc package consists of a suite of Fortran 95 modules complementing HiPPy. While the output of HiPPy is in principle suitable to being converted directly into an analytic form, this is not usually necessary or even useful in lattice perturbation theory. We therefore have implemented routines that read in the HiPPy-generated Feynman rules at runtime and use them to construct the vertex functions and propagators for given momenta on the fly. This also offers the great advantage of being able to write Feynman diagrams in an action-blind way, so as to be able to recompute the same quantities for another action without needing to write new code. HPsrc also provides facilities for automated differentiation of Feynman diagrams, so that neither analytic manipulations nor inaccurate numerical derivatives are needed.
The background field technique has long been known as a valuable tool in field theory. In [7], Lüscher and Weisz showed that the theorem about dimensional regularisation stating that renormalisation of the effective action does not require additional counterterms beyond those needed for the renormalisation of the action holds also in the case of lattice gauge theory. This makes it possible to use the background field technique to perform calculations such as relating the bare lattice coupling to the MS coupling [8]. To determine the coefficient of the σ σ σ • • • B B B term in the (moving) NRQCD action [9], only the background field technique can guarantee the gauge invariance of higher-dimensional operators which will necessarily be generated in an effective theory such as (m)NRQCD. This makes it desirable to incorporate support for the background field method into the HiPPy/HPsrc packages.
We decompose our fields into a background field B µ and quantum fluctuations q µ by parametrising the basic gauge link as
We note that this does not affect the combinations of x x x, y y y, v v v i, j that are possible, and that hence the entity algebra remains unaffected. However, the gluon fields living at each lattice point v v v can now be either quantum or background, with the quantum fields always appearing to the left of the background fields coming from the same link, and thus the field objects need to keep track of nature of individual gluon fields. This leads to a new mapping structure for field objects given now by
with an order-r entity being mapped to an 2 r -tuple of amplitudes. With this structure, multiplication of fields now assign
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