Massive minimal subtraction scheme and “partial-” in anisotropic Lifshitz space(time)s
We introduce the “partial-” operation in a massive Euclidean scalar field theory describing anisotropic Lifshitz critical behavior. We then develop a minimal subtraction a la renormalization scheme. As an application we compute critical exponents diagrammatically using the orthogonal approximation at least up to two-loop order and show their equivalence with other renormalization techniques. We discuss possible applications of the method in other field-theoretic contexts.
pacs:64.60.ae; 64.60.Bd; 64.60.fd; 64.60.Kw
Introduction - All relativistic higher derivative field theories were relegated to a minor role for a long time because of the presence of ghosts associated to higher time derivatives. Recently, it was realized that keeping second order time derivatives and permitting higher order space derivatives would have the virtue of maintaining unitarity, although it breaks the Lorentz invariance of the theory A1 (). Models including gauge fields have also been built A2 (), culminating with the Horava’s proposal of a quantum gravity model where space and time scale anisotropicallyH (). It produced a tide of new developments: from modifications to general relativity SVW () to new paradigms in inflationary cosmology Wang (). From the field-theoretic perspective, a Wick rotation of the time coordinate permits us to make a direct comparison of these field theories (“anisotropy exponent” ) with critical systems pertaining to the anisotropic Lifshitz universality classes HLS () “living” in Lifshitz spaces CL1 ().
Competition is a mechanism which can induce anisotropy. For instance, anisotropic Lifshitz critical behaviors HLS () arise in a variety of real physical systems, from high- superconductors 3 (); 4 (); 5 () to magnetic materials Sh (); Be (). A modified Ising model on a -dimensional lattice describing them consists of first-neighbor ferromagnetic interactions competing with second-neighbor antiferromagnetic couplings along directions (the competition axes). The two inequivalent subspaces are the - and -dimensional subsets (). Perturbatively, performing the Feynman integrals associated to this -axial Lifshitz field theory exactly in an analytical manner is still a far-off task nowadays. A preliminary, crude approximation was developed to solve analytically higher-loop diagrams called “the dissipative approximation” which yielded the critical exponents beyond one-loop level for the first time using massless fields AL (). Another higher order perturbative semianalytical method employing massless fields was presented immediately afterwards SD (). While the dissipative approximation does not conserve momentum at higher order diagrams, the second alternative failed in its attempt to produce analytical answers to the exponents. From the non-perturbative analysis, the renormalization group () treatment of the first method could not give any information concerning exponents along the -dimensional competition subspace, whereas the latter obtains scaling laws with unclear meaning. Both treatments fail to produce scaling laws in the isotropic case (). Those problems were overcome through new arguments using massless fields , where two independent transformations result in independent scaling relations for each subspace and lead to a complete determination of critical properties of this system. Moreover, the development of the “orthogonal approximation” in Leite1 () represented the first solution in perturbation theory which allows the analytical determination of arbitrary loop order diagrams. It was shown to be entirely equivalent with the massive formulation CL1 (). From the renormalizability viewpoint, the minimal subtraction offers no difficulty in the massless theory. However, the massive minimal subtraction renormalization scheme for -axial anisotropic Lifshitz critical behaviors poses a formidable obstacle: the manipulation of overlapping divergences.
What version of “partial-” tHV () operation should be defined in handling those divergences showing up in higher-loop contributions, e. g., of the one-particle irreducible () Amit () two-point vertex part? In this Letter we propose a new “partial-” operation for these anisotropic spaces. As the noncompeting subspace is quadratic in derivatives, the time coordinate in the aforementioned quantum field theories can be identified with one coordinate of the subspace without competition, for example, in the limit and our construction here goes beyond the context of critical phenomena.
As an application, we build up a version of Bogoliubov-Parasyuk-Hepp-Zimmermann () method using minimal subtraction in order to compute critical exponents at least up to two-loop order. We employ the orthogonal approximation in the calculation of critical indices and find universal results in exact agreement with previous outcomes from massless Leite1 () and massive field settings CL1 () using normalization conditions. We conclude with a few comments on the utilization of the present method in other quantum field-theoretic models.
Partial- and method - In massive field theories the mass sets the natural scale in the renormalization group approach. The anisotropic -axial Lifshitz behaviors require an augmented parameter space with two mass scales characterizing the two subspaces involved owing to the two independent scaling transformations. The bare and renormalized fields inherit this dependence on the masses. The bare Euclidean Lagrangian density of the scalar field with symmetry representing this critical behavior can be written as
The Lifshitz critical region is characterized by and we will use this value henceforth. The label specifies the subspace associated to the bare mass . The noncompeting -dimensional corresponds to whereas refers to the -dimensional subspace. Formally, the multiplicatively renormalized one-particle irreducible () vertex parts are the mathematical entities necessary to our discussion of the renormalization scheme. In the subspace , we set inside the vertex parts the external momenta along the competing subspace at zero, , and perform scale transformations involving only (and vice-versa when considering the subspace ; see below).
The method follows closely the conventions employed in a recent work CL2 () for critical systems without competition. (See also the excellent description of this scheme given in the book by Kleinert and Schulte-Frohlinde SFK ().) We perform the redefinitions , and , where is the expansion parameter and are the dimensionless renormalized coupling constants. Those redefinitions make the original bare Lagrangian to be extended with additional extra terms, the counterterms, which generate extra Feynman diagrams order by order in perturbation theory.
The redefinitions permit the bare Lagrangian density to be rewritten in terms of renormalized amounts, namely
whose coefficients are -dependent. The two subspaces in momentum space can be labeled collectively in the form , where and . If the external momenta is denoted by , the counterterms generate additional diagrams with Feyman rules in momentum space, namely
Note that the mass scales are arbitrary, are the renormalized masses and and are the renormalization functions. The quantities and are the counterterms which are added at each diagram in arbitrary loop level in order to cancel the singular contributions of the primitively divergent bare vertex parts. At the loop order desired, they can be expanded in powers of the dimensionless renormalized coupling constants as , , and .
Overlapping divergences can be handled by utilizing the “partial-” operation tHV () in quadratic field theories. In our case, the complication is that the free propagator in momentum space is given by . What saves us from that situation is the proposal of the following anisotropic version of the partial- operation:
And now let us compute the set of divergent integrals required in the evaluation of critical exponents at least up to two-loop order within this method. The diagrams required involve the one-particle irreducible () vertex functions and . We actually need the singular parts of these integrals. (Denote the extraction of the singular parts of an arbitrary singular integral by CL2 (); Comment (); BPH (); Z () although we can actually neglect the subscript, provided this does not cause any confusion to the reader). The graphs of up to three-loop level can be better understood if we divide them in two main categories: the tadpole ones and the diagrams which depend on the external momenta (together with their counterterms). For the purposes of computing up to three-loop level, we shall need only the one- and two-loop tadpole graphs associated, respectively, with the integrals and . They can be trivially computed using the partial- using standard formulas from dimensional regularization (see, e.g., Ref. Leite1 ()). The corresponding diagrams can be solved using the parameter as
The nontrivial higher-loop graphs of the two-point function involve explicitly the external momenta and present overlapping divergences. The two-loop integral
is related to the “sunset” diagram through . We benefit ourselves from the partial- operation by applying it twice, just as we do in conventional quadratic field theories, on the integrand along with the orthogonal approximation. Then, we separate the polynomials in mass and momentum (beside logarithms involving both). One encounters that the solution reads
The three-loop graph
can be determined by applying the partial- three times. After performing standard manipulations using the orthogonal approximation, at the end of the day we find the following solution:
Note that the first term proportional to will only contribute to at three-loop order, which is beyond our present concern here of determining up to two-loop level. (We can implement in the first term in the determination of up to three-loops and also in all diagrams presenting these polynomials in the mass, symbolically as ). This concludes the utilization of the partial -operation in the multiplicatively renormalized vertex parts.
We can expand the 2-point vertex part including the counterterms up to three-loop order in the form:
where mean all tadpole diagrams at three-loop order, which only contribute to and can be safely neglected. Since the counterterms pick out only the singular terms of the diagrams, we conclude that . Let us make a pause to discuss the simplest aspect of the four-point vertex part in order to achieve a better comprehension of the counterterm loop diagrams.
We momentarily focus on the four-point vertex function at one-loop. Its diagrammatic expansion is
with and . This diagram can be calculated and yields
where . Finiteness of the last vertex part at one-loop implies that .
Coming back to the two-point vertex part, the counterterm diagram can be understood as the four-point one-loop diagram computed at zero external momentum, with the upper coupling constant replaced by . Expanding in powers of the contribution of the counterterm is given by:
The next counterterm diagram is the product of a one-loop tadpole with a one-loop four-point insertion where the loop has shrunken to zero, picking out the coupling constant at . Similar expansions in as performed in the previous counterterm diagram lead us to
Combining the two-loop contribution eliminates all the terms proportional to at . The divergences are cancelled provided and . The three-loop counterterm diagram is the “sunset” with one of the couplings replaced by the coupling constant counterterm at one-loop. The combination of the diagrams at this loop order in the simplified form displayed eliminates the and we can read off the value .
We are left with the computation of . The two-loop contribution of the four-point vertex part is given by
The first diagram is the one-loop contribution to the four-point function to the square. The nontrivial two-loop diagram of the four-point function
can be expanded in as
The singular part of the third diagram is canceled by the (counterterm) fourth graph. The last counterterm diagram can be easily constructed from our previous discussion. Summing up all diagrams, the contributions do vanish in the singular terms. The four-point function becomes finite if .
The Wilson functions are defined in terms of the dimensionless coupling constants by , and . The fixed point is defined by which implies . Through the identifications and we obtain the exponents
Performing the identifications (), () we retrieve the expressions for these exponents already found in refs. CL1 (); Leite1 (). Utilizing the scaling laws derived in Leite1 () we obtain all other exponents which are identical to those determined before.
Conclusions The proposed -partial operation for -axial anisotropic Lifshitz scalar field theory does circumvent the problem of overlapping divergences in higher-loop Feynman integrals as explicitly demonstrated herein in the computation of the two-point vertex part. On the other hand, the present massive minimal subtraction in the computation of critical exponents closes the circle and proves the complete mathematical consistency of the orthogonal approximation with a great deal of information using quite different renormalization schemes and in agreement with the universality hypothesis. We wish to expand the panorama of minimal subtraction within this massive formulation in conjunction with the orthogonal approximation by developing the appropriate version of the unconventional approach first introduced in Ref. CL2 () for ordinary critical systems.
Besides, the perturbative treatment of anisotropic quantum field theories in Lifshitz spacetimes can be greatly benefited from the method just developed. It might prove interesting to see how the orthogonal approximation can address the higher-loop computations of observables in those sort of field-theoretic models. For example, at spacetime dimension the Hoava-Lifshitz gravity for a careful choice of parameters has a classical Weyl invariance for . For corresponding to four-dimensional spacetime, a scalar field coupled with this gravity system in a Weyl-invariant way was shown to possess an anomaly computed in position space Theisen (). With the development just obtained, we could compute certain flat space -point correlators in momentum space at arbitrary loop order in the determination of the anomaly and make a comparison with the previous result. This is certainly a missing part in a better understanding of fields propagating in such backgrounds. Our betting is that the generation of new effects could broaden up our present knowledge of the subject, sheding light on these issues just as the pioneer works on the description of phase transitions did on unveiling the perturbative structure of ordinary (quadratic) quantum field theories.
Finally, we can adapt the aforementioned technique to tackle the partial- operations in generic anisotropic competing systems of the Lifshitz type Leite2 () where arbitrary even momentum powers are present in the free propagator. This problem is connected with field theories in anisotropic spacetimes with (even) arbitrary anisotropy exponent . We believe it can be formulated similarly as discussed in the present Letter. An extended version of the present work will be presented elsewhere.
Acknowledgements EVS would like to thank CAPES (Brazilian agency) for financial support. PRSC acknowledges partial support from FAPEPI (State of Piauí Foundation) and CNPq (Brazilian agency) grant number CCN-052/2010. MML was partially supported by CAPES and FACEPE (State of Pernambuco Foundation).
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