Exact Lorentz-violating all-loop ultraviolet divergences in scalar field theories

In this work we evaluate analytically the ultraviolet divergences of Lorentz-violating massive O($N$) $\lambda\phi^{4}$ scalar field theories, which are exact in the Lorentz-violating mechanism, firstly explicitly at next-to-leading order and latter at any loop level through an induction procedure based on a theorem following from the exact approach, for computing the corresponding critical exponents. For attaining that goal, we employ three different and independent field-theoretic renormalization group methods. The results found for the critical exponents show that they are identical in the three distinct methods and equal to their Lorentz invariant counterparts. Furthermore, we show that the results obtained here, based on the single concept of loop order of the referred terms of the corresponding $\beta$-function and anomalous dimensions, reduce to the ones obtained through the earlier non-exact approach based on a joint redefinition of the field and coupling constant of the theory, in the appropriate limit.


I. INTRODUCTION
The aim of this work is to probe the exact effect of the LV mechanism on the values for the critical exponents. This paper is organized as follows: In next three Sects., we compute analytically and explicitly the next-to-leading loop order quantum corrections to the critical exponents for LV O(N) self-interacting λφ 4 scalar field taking into account the LV mechanism exactly, by applying three distinct field-theoretic renormalization group methods. In Sect. V we generalize the results for all loop levels. At the end, we present our conclusions.

II. EXACT LORENTZ-VIOLATING NEXT-TO-LEADING ORDER CRITICAL EXPONENTS IN THE CALLAN-SYMANZIK METHOD
We consider a massive LV O(N) scalar field theory whose bare Lagrangian density in Euclidean spacetime is given by [6][7][8] In Eq. above, the bare parameters φ B , m B and λ B are the bare field, mass and coupling constant, respectively. The responsible for the symmetry breaking mechanism are the constant symmetric LV coefficients K µν . We can now expand the bare primitively 1PI vertex parts up to next-to-leading loop order to obtain the desired expansion. But up to this loop order, we have so many diagrams. This number of diagrams can be reduced. We see that the diagrams containing tadpole insertions (2) and the one which is independent of external momenta (3) can be eliminated. It is known that if we substitute the bare mass m B,tree−level in Eq. 1 initially at tree-level for its three-loop counterpart m B,three−loop [20,21] we can achieve the desired aim. Now making m B,three−loop → m B from now on we have Γ (2) B (P 2 + K µν P µ P ν , m B , λ B ) = −1 − λ 2 Γ (2,1) where Q = −(P 1 + P 2 ). We can now define the dimensional and the dimensionless renormalized coupling constants λ and u as λ = um ǫ , where m, at the loop level considered, is used as an arbitrary momentum scale, thus we can consider the momenta as dimensionless quantities. The same relation between the corresponding bare quantities λ B and u 0 can be also defined as λ B = u 0 m ǫ . We renormalize these correlation functions multiplicatively which satisfies the Callan-Symanzik equation where we use the function instead of γ φ 2 (u), for convenience reasons, by fixing the external momenta through the normalization conditions ∂Γ (2) (P 2 + K µν P µ P ν ; m 2 , u) ∂(P 2 + K µν P µ P ν ) Γ (4) It is well known that we can reduce more yet the number of diagrams to be evaluated [20] because some of them are not independent. This makes this method simpler than the last one which will be applied, the BPHZ one, where, for attaining the same task, we have to compute around of fourteen diagrams. where and in the Callan-Symanzik method, the two-loop diagram as well threeloop one given by , do not contribute to the subsequent computations, since we evaluate them at fixed vanishing external momenta. In the right-hand side (rhs) of Eq. 8, the referred 1PI vertex part is one of l + 1 composite field insertions and the one in the left-hand side (lhs) has l such insertions. As it is known, an extra composite field insertion is responsible for one additional power of the propagator in the corresponding 1PI vertex part. We can then work in the ultraviolet limit, i. e., in the limit where the external momenta P i /m → ∞. After taking this limit, the rhs can be neglected in comparison with the lhs, order by order in perturbation theory. This is, in essence, the content of the Weinberg's theorem [22]. So, the 1PI vertex parts satisfy the renormalization group equation, thus permitting us to apply the theory of scaling for these functions and evaluate the β-function and anomalous dimensions as well as the corresponding critical exponents.
The LV coefficients can now been considered exactly by noting that q 2 + K µν q µ q ν ≡ (δ µν + K µν )q µ q ν = q t (I + K)q, where q is a d-dimensional vector whose representation is a column matrix and q t is a row matrix and I and K are matrix representations of the identity and K µν , respectively. Thus making q ′ = √ I + K q, the LV mechanism is shown explicitly through two contributions. The first of them is displayed through the volume elements LV full or exact contribution Π = 1/ det(I + K) reduces to its perturbative counterpart [6][7][8][9][10][11]. The other LV modification of the theory is that involving the external momenta. It can be seen in the momentum-dependent d-dimensional This definition is convenient as to each loop integration we have a factor ofŜ 4 at four dimensions, thus avoiding the appearance of Euler-Mascheroni constants in the middle of calculations [20]. Now making q ′ → P ′ and q → P , P ′2 = P 2 + K µν P µ P ν . As it is known, from all diagrams displayed above, we need to compute only four of them [20]. They are shown in A.
When we absorbŜ in a redefinition of the coupling constant and use the Feynman diagrams for computing the β-function and anomalous dimensions by writing the Laurent expansion where the constant coefficients a 1 , · · · , c 2 depend on the Feynman diagrams, evaluated in appendix A, just mentioned [20], we obtain We observe that the expression for the β-function of Eq. (35) can be read off based on a single concept, that of loop order of the referred term of the corresponding function. As we can see, its first term does not originate from a loop integral and the exact approach demands that it has not to be accompanied of a LV full Π factor, although it is a term of first order in u. This term is fundamental for making possible expansions in quantum field theory and is essential in the renormalization group and ǫ-expansion techniques developed by Wilson, specially with applications in critical phenomena [23][24][25]  The trivial one leads to the mean field or Landau critical exponents and can be obtained mathematically by a factorization procedure resulting in the factorization of a single power of u in Eq. for the β-function. This procedure results in the nontrivial fixed point given by It can be written as u * = u * (0) /Π, where u * (0) is its Lorentz-invariant (LI) counterpart. Now the LV corrections to mean field or Landau approximation to the critical exponents are given though the application of definitions η ≡ γ φ (u * ) and ν −1 ≡ 2 − η − γ φ 2 (u * ). They can be applied to obtain, to next-to-leading order, the two respective critical exponents where η (0) and ν (0) are their corresponding Lorentz-invariant (LI) counterparts [23]. As there are six critical exponents and four scaling relations among them, there are only two independent ones. Thus the two ones above are enough for evaluating the four remaining ones. In next Sect. we will attain the same task but now in a distinct renormalization method and will compare the results.

EXPONENTS IN THE UNCONVENTIONAL MINIMAL SUBTRACTION SCHEME
This method is characterized by its elegance as compared with the earlier one since the external momenta remain at arbitrary values along all the renormalization program. This implies that we do not have to compute any parametric integral because they cancel out in the final expressions for the β-function and anomalous dimensions. Then, now we have that where the poles are minimally eliminated, thus being absorbed in the renormalization constants for the field Z φ and composite field Z φ 2 , respectively. Now absorbingŜ in a redefinition of the coupling constant and using the Feynman diagrams computed in B, we find The renormalization program proceeds so elegantly that all the momentum-dependent integrals, namely the ones L(P 2 +K µν P µ P ν , m 2 B ), L 3 (P 2 +K µν P µ P ν , m 2 B ),ĩ(P 2 +K µν P µ P ν , m 2 B ) have disappeared. Now, the only LV dependence of the theory is that through the LV full Π factor. The cancelling of the integrals aforementioned are associated to the the renormalization of the field and composite field. In fact, technically, the renormalization of these parameters comes from the terms proportional to P 2 + K µν P µ P ν in the diagrams and . But, unfortunately, we are yet left with a residual divergence and it originates from the terms proportional to m 2 in the diagrams and . It is show below The reduction of the number of diagrams to be computed though the redefining of the initial bare mass at tree-level to its three-loop order counterpart produces this residual divergence.
We can overcome this problem then subtracting this pole minimally by redefining the twopoint function as This turn out to connect the Unconventional minimal subtraction scheme to the conventional one in the massless theory [20], once the terms proportional to m 2 vanish in the latter case.
The final check of this redefinition can be shown by showing that it satisfies the normalization condition used in Sect. IIΓ (2) (P 2 + K µν P µ P ν = 0, u, m) = Once again, for computing the LV loop quantum corrections to the critical exponents, we need to evaluate the nontrivial fixed point though the nontrivial solution for the equation β(u * ) = 0. It is given by This value for the nontrivial fixed point when used for evaluating the critical exponents, leads to the same ones of the earlier Sect.. On more time we confirm the universality hypothesis, that the critical exponents are universal quantities, thus being the same when obtained in different renormalization schemes. Now we proceed to compute the critical exponents in a third renormalization scheme.

EXPONENTS IN THE BPHZ METHOD
The BPHZ (Bogoliubov-Parasyuk-Hepp-Zimmermann) method [26][27][28] is the most general from all known renormalization methods. It does not include any trick for reducing the total number of diagrams to be evaluated. Thus we have to compute all diagrams in the original expansion for a given loop order. As opposed to the earlier ones, in the BPHZ method, we start from the renormalized theory where Initially, considering the bare theory at one-loop order, we absorb that divergence by adding terms to the initial Lagrangian density. Then, a finite Lagrangian density is found. For considering the bare theory at the next loop level, we apply the same procedure and so on, order by order in perturbation theory. Thus we absorb the divergences in the renormalization constants. We expand the renormalization constants as The c i φ , c i g and c i m coefficients are the i-th loop order renormalization constants for the field, renormalized coupling constant and composite field, respectively. They are given by where S is the symmetry factor for the corresponding diagram and so on when some N-component field is considered. By using the diagrams in C, we have that the β-function and anomalous dimensions are given by One more time, we have to compute the nontrivial solution of Eq. (65). This procedure yields the value Now by applying the relations η ≡ γ φ (u * ) and ν −1 ≡ 2 − γ m 2 (u * ), we obtain once again that the LV critical exponents are identical to their LV counterparts. Now we evaluate the LV critical exponents for any loop levels.

V. EXACT LORENTZ-VIOLATING ALL-LOOP ORDER CRITICAL EXPONENTS
For computing the critical exponents for all loop levels, we can employ any of the methods aforementioned since the critical exponents, as being universal quantities, must be the same if evaluated at any renormalization scheme. For that, we will employ the BPHZ method which is the most general one. Before that, we need to assert the following theorem Theorem. Consider a given Feynman diagram in momentum space of any loop order in a theory represented by the Lagrangian density of Eq. (1). Its evaluated expression in dimensional regularization in d = 4 − ǫ can be written as a general functional Π L F (u, P 2 + K µν P µ P ν , ǫ, µ) if its LI counterpart is given by F (u, P 2 , ǫ, µ, m), where L is the number of loops of the corresponding diagram.
Proof. A general Feynman diagram of loop level L is a multidimensional integral in L distinct and independent momentum integration variables q 1 , q 2 ,...,q L , each one with volume element given by d d q i (i = 1, 2, ..., L). As showed in last Section, the substitution q ′ = √ I + K q transforms each volume element as Then, the integration in L variables results in a LV overall factor of Π L . Now making q ′ → P ′ in the substitution above, where P ′ is the transformed external momentum, then P ′2 = P 2 + K µν P µ P ν . So a given Feynman diagram, evaluated in dimensional regularization in d = 4−ǫ, assumes the expression Π L F (u, P 2 +K µν P µ P ν , ǫ, µ), where F is associated to the corresponding diagram if the LI Feynman diagram counterpart evaluation results in F (u, P 2 , ǫ, µ).
Now using the result of the theorem above and the one in which all momentum-dependent integrals cancel out order by order in perturbation theory for all levels in the renormalization process [26][27][28], we see that the only LV dependence of β-function and anomalous dimensions is due to the LV full Π factor, which comes from the volume elements of the diagrams contributing with a Π L factor, where L is the number of loops of the corresponding graph.
Thus we can write the β-function and anomalous dimensions for all loop levels where β

VI. CONCLUSIONS
We have evaluated analytically the ultraviolet divergences of Lorentz-violating massive O(N) λφ 4 scalar field theories, which are exact in the Lorentz-violating mechanism, firstly explicitly at next-to-leading order and latter at any loop level through an induction procedure based on a theorem following from the exact approach, for computing the corresponding critical exponents. For that, we have employed three different and independent field-theoretic renormalization group methods. We have found equal critical exponents in the three methods and furthermore identical to their Lorentz invariant counterparts. We have also showed that the exact approach, which reduces to the non-exact one in its limited range of applicability, besides exact, is capable of furnishing the expressions for the all-loop LV radiative quantum corrections to the β-function and anomalous dimensions considering just a single concept, that of loop number of the corresponding terms of these functions.
Furthermore, the present exact approach, when applied to the referred theory, is the first one in literature for our knowledge. Thus it can inspire the exact solution of problems involving considering the exact effect of LV mechanisms in many physical phenomena ranging from high-(standard model extension for example) to low energy physics (corrections to scaling, finite-size scaling, amplitude ratios, critical exponents in geometries subjected to different boundary conditions, Lifshitz points etc [29][30][31][32].
Appendix A: Integrals of Callan-Symanzik method where the integral I [31-33] is a residual number and is a consequence of the symmetry point chosen.