Exact Lorentz-violating all-loop ultraviolet divergences in scalar field theories
aa r X i v : . [ h e p - t h ] A ug Exact Lorentz-violating all-loop ultraviolet divergences in scalarfield theories
P. R. S. Carvalho ∗ Departamento de F´ısica, Universidade Federal do Piau´ı, 64049-550, Teresina, PI, Brazil
M. I. Sena-Junior † Escola Polit´ecnica de Pernambuco, Universidade dePernambuco, 50720-001, Recife, PE, Brazil andInstituto de F´ısica, Universidade Federal de Alagoas, 57072-900, Macei´o, AL, Brazil
Abstract
In this work we evaluate analytically the ultraviolet divergences of Lorentz-violating massiveO( N ) λφ scalar field theories, which are exact in the Lorentz-violating mechanism, firstly explicitlyat next-to-leading order and latter at any loop level through an induction procedure based on atheorem following from the exact approach, for computing the corresponding critical exponents.For attaining that goal, we employ three different and independent field-theoretic renormalizationgroup methods. The results found for the critical exponents show that they are identical in thethree distinct methods and equal to their Lorentz invariant counterparts. Furthermore, we showthat the results obtained here, based on the single concept of loop order of the referred terms ofthe corresponding β -function and anomalous dimensions, reduce to the ones obtained through theearlier non-exact approach based on a joint redefinition of the field and coupling constant of thetheory, in the appropriate limit. ∗ [email protected] † [email protected] . INTRODUCTION Lorentz symmetry is one of the most fundamental symmetries of nature and the possi-bility of its violation was theme of great investigation in the last years, usually as a finiteperturbative expansion at some Lorentz-violating (LV) parameters and loop number, bothin high energy [1–8] as well as in low energy [9–11] physics. In the latter realm, the criticalexponents were computed, at least, at first order in the Lorentz-violating (LV) parameters K µν and any loop level for LV scalar field theories [9–11]. For that, this evaluation waspossible by means of the application of a non-exact approach based on a joint redefinitionof the field and coupling constant of the theory. In this work, we present an exact approach,which naturally takes into account the effect of the LV parameters exactly and furthermorefor all loop orders. Moreover, we will show that the referred exact approach gives expressionsfor the β -function as well as for the corresponding fixed point and anomalous dimensions,besides critical exponents and that these expressions reduce to the ones obtained in theearlier non-exact approach in the appropriate limit.In this work, we compute analytically the critical exponents for massive O( N ) λφ scalarfield theories with Lorentz violation. This computation is exact in the LV mechanism. Forthat, we apply three distinct field-theoretic renormalization group methods and they involvethe same theory renormalized at different renormalization schemes. In this field-theoreticformulation, if the critical exponents present the same values when obtained through thethree methods, this means that they are universal quantities and we have the confirmationof the universality hypothesis. These universal quantities characterize the critical behaviorof distinct systems as a fluid and a ferromagnet. When the critical behavior of two or moredistinct systems is characterized by the same critical exponents, we say that they belong tothe same universality class. The universality class inspected here is the O( N ) one, whichencompasses the particular models: Ising ( N = 1), XY ( N = 2), Heisenberg ( N = 3), self-avoiding random walk ( N = 0) and spherical ( N → ∞ ) for short-range interactions [12].The critical exponents depend on the dimension d of the system, N and symmetry of some N -component order parameter (magnetization for magnetic systems), and if the interactionspresent are of short- or long-range type. Many works probing the dependence of the criticalexponents on the obvious parameters as d [13, 14] and N [15–17] were published. Just a fewof them were published in the less one, that of symmetry of the order parameter [18, 19].2he aim of this work is to probe the exact effect of the LV mechanism on the values for thecritical exponents.This paper is organized as follows: In next three Sects., we compute analytically andexplicitly the next-to-leading loop order quantum corrections to the critical exponents forLV O( N ) self-interacting λφ scalar field taking into account the LV mechanism exactly,by applying three distinct field-theoretic renormalization group methods. In Sect. V wegeneralize the results for all loop levels. At the end, we present our conclusions. II. EXACT LORENTZ-VIOLATING NEXT-TO-LEADING ORDER CRITICALEXPONENTS IN THE CALLAN-SYMANZIK METHOD
We consider a massive LV O( N ) scalar field theory whose bare Lagrangian density inEuclidean spacetime is given by [6–8] L = 12 ( δ µν + K µν ) ∂ µ φ B ∂ ν φ B + 12 m B φ B + λ B φ B . (1)In Eq. above, the bare parameters φ B , m B and λ B are the bare field, mass and couplingconstant, respectively. The responsible for the symmetry breaking mechanism are the con-stant symmetric LV coefficients K µν . We can now expand the bare primitively 1PI vertexparts up to next-to-leading loop order to obtain the desired expansion. But up to this looporder, we have so many diagrams. This number of diagrams can be reduced. We see thatthe 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. 1initially at tree-level for its three-loop counterpart m B,three − loop [20, 21] we can achieve thedesired aim. Now making m B,three − loop → m B from now on we haveΓ (2) B ( P + K µν P µ P ν , m B , λ B ) = − − λ B − (cid:12)(cid:12)(cid:12)(cid:12) P + K µν P µ P ν =0 ! + λ B − (cid:12)(cid:12)(cid:12)(cid:12) P + K µν P µ P ν =0 ! , (4)3 (4) B ( P i , m B , λ B ) = λ B − λ B (cid:16) + 2 perm. (cid:17) + λ B (cid:16) + 2 perm. (cid:17) + λ B (cid:18) + 5 perm. (cid:19) , (5)Γ (2 , B ( P , P , Q , m B , λ B ) = 1 − λ B λ B λ B Q = − ( P + P ). We can now define the dimensional and the dimensionless renor-malized coupling constants λ and u as λ = um ǫ , where m , at the loop level considered, isused as an arbitrary momentum scale, thus we can consider the momenta as dimensionlessquantities. The same relation between the corresponding bare quantities λ B and u can bealso defined as λ B = u m ǫ . We renormalize these correlation functions multiplicativelyΓ ( n,l ) ( P i , Q j , u, m ) = Z n/ φ Z lφ Γ ( n,l ) B ( P i , Q j , λ B , m B ) (7)which satisfies the Callan-Symanzik equation (cid:18) m ∂∂m + β ∂∂u − nγ φ + lγ φ (cid:19) Γ ( n,l ) R ( P i , Q j , u, m ) = m (2 − γ φ )Γ ( n,l +1) R ( P i , Q j , , u, m )(8)where β ( u ) = m ∂u∂m = − ǫ (cid:18) ∂ ln u ∂u (cid:19) − , (9) γ φ ( u ) = β ( u ) ∂ ln Z φ ∂u , (10) γ φ ( u ) = − β ( u ) ∂ ln Z φ ∂u , (11)where we use the function γ φ ( u ) = − β ( u ) ∂ ln Z φ ∂u ≡ γ φ ( u ) − γ φ ( u ) (12)instead of γ φ ( u ), for convenience reasons, by fixing the external momenta through thenormalization conditions Γ (2) ( P + K µν P µ P ν = 0; m, u ) = m , (13) ∂ Γ (2) ( P + K µν P µ P ν ; m , u ) ∂ ( P + K µν P µ P ν ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P + K µν P µ P ν =0 = 1 , (14)4 (4) R ( P i = 0; m, u ) = u, (15)Γ (2 , R ( P i = 0 , Q j = 0 , m, u ) = 1 . (16)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 lastone which will be applied, the BPHZ one, where, for attaining the same task, we have tocompute around of fourteen diagrams. As the computation of the 1PI vertex parts leads tomomentum integration involving just their internal bubbles and not their external legs, allwhat matters in this evaluation are their internal bubbles contents. Thus, without take intoaccount the O( N ) symmetry factors, we have that ∝ , ∝ , ∝ ∝ ( ) . Finally, the only diagrams to be evaluated are the , , , ones. Thuswe can write the 1PI vertex parts asΓ (2) B ( P + K µν P µ P ν , u , m B ) = ( P + K µν P µ P ν )(1 − B u + B u ) , (17)Γ (4) B ( P i , u , m B ) = m ǫB u [1 − A u + ( A (1)2 + A (2)2 ) u ] , (18)Γ (2 , B ( P , P , Q , u , m B ) = 1 − C u + ( C (1)2 + C (2)2 ) u , (19)where A = ( N + 8)6 SP (20) A (1)2 = ( N + 6 N + 20)36 SP , (21) A (2)2 = (5 N + 22)9 SP , (22) B = ( N + 2)18 ′ , (23) B = ( N + 2)( N + 8)108 ′ , (24) C = ( N + 2)6 SP , (25)5 (1)2 = ( N + 2) SP , (26) C (2)2 = ( N + 2)6 SP (27)and in the Callan-Symanzik method, the two-loop diagram (cid:12)(cid:12)(cid:12)(cid:12) P + K µν P µ P ν =0 as well three-loop one given by (cid:12)(cid:12)(cid:12)(cid:12) P + K µν P µ P ν =0 , 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 onein the left-hand side (lhs) has l such insertions. As it is known, an extra composite fieldinsertion is responsible for one additional power of the propagator in the corresponding 1PIvertex part. We can then work in the ultraviolet limit, i. e. , in the limit where the externalmomenta P i /m → ∞ . After taking this limit, the rhs can be neglected in comparisonwith the lhs, order by order in perturbation theory. This is, in essence, the content ofthe Weinberg’s theorem [22]. So, the 1PI vertex parts satisfy the renormalization groupequation, thus permitting us to apply the theory of scaling for these functions and evaluatethe β -function and anomalous dimensions as well as the corresponding critical exponents.The LV coefficients can now been considered exactly by noting that q + 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 columnmatrix 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 explicitlythrough two contributions. The first of them is displayed through the volume elementsof d -dimensional integrals d d q ′ = p det ( I + K ) d d q , thus d d q = d d q ′ / p det ( I + K ). ThisLV full or exact contribution Π = 1 / p det ( I + K ) reduces to its perturbative counterpartΠ ≃ Π (0) + Π (1) + Π (2) for small violations of Lorentz symmetry, where Π ( i ) is the LVcontribution of order i in K µν [6–11]. The other LV modification of the theory is thatinvolving the external momenta. It can be seen in the momentum-dependent d -dimensionalintegrals when evaluated in dimensional regularization in d = 4 − ǫ Z d d q (2 π ) d q + 2 P q + M ) α = ˆ S d
12 Γ( d/ α ) Γ( α − d/ M − P ) α − d/ , (28)where ˆ S d = S d / (2 π ) d = 2 / (4 π ) d/ Γ( d/ S d = 2 π d/ / Γ( d/
2) is the surface area of a unit d -dimensional sphere. Its finite value in four-dimensional spacetime is ˆ S = 2 / (4 π ) . This6efinition is convenient as to each loop integration we have a factor of ˆ S 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 ′ = P + K µν P µ P ν . As it is known, from alldiagrams displayed above, we need to compute only four of them [20]. They are shown in A.When we absorb ˆ S in a redefinition of the coupling constant and use the Feynman diagramsfor computing the β -function and anomalous dimensions by writing the Laurent expansion u = u ∞ X i =1 a i ( ǫ ) u i ! , (29) Z φ = 1 + ∞ X i =1 b i ( ǫ ) u i , (30) Z φ = 1 + ∞ X i =1 c i ( ǫ ) u i , (31) β ( u ) = − ǫu [1 − a u + 2( a − a ) u ] , (32) γ φ ( u ) = − ǫu [2 b u + (3 b − b a ) u ] , (33) γ φ ( u ) = ǫu [ c + (2 c − c − a c ) u ] , (34)where the constant coefficients a , · · · , c depend on the Feynman diagrams, evaluated inappendix A, just mentioned [20], we obtain β ( u ) = − ǫu + N + 86 (cid:18) − ǫ (cid:19) Π u − N + 1412 Π u , (35) γ φ ( u ) = N + 272 (cid:18) − ǫ + Iǫ (cid:19) Π u − ( N + 2)( N + 8)432 ( I + 1) Π u , (36) γ φ ( u ) = N + 26 (cid:18) − ǫ (cid:19) Π u − N + 212 Π u . (37)We observe that the expression for the β -function of Eq. (35) can be read off based on asingle concept, that of loop order of the referred term of the corresponding function. As wecan see, its first term does not originate from a loop integral and the exact approach demandsthat it has not to be accompanied of a LV full Π factor, although it is a term of first order7n u . This term is fundamental for making possible expansions in quantum field theory andis essential in the renormalization group and ǫ -expansion techniques developed by Wilson,specially with applications in critical phenomena [23–25] in d <
4. Its second one-loop termis of second order in u , but it has acquired only a linear power of Π . The last one, althoughbeing of third order in u , must be of second order in Π , since it is of two-loop order. Similararguments can be utilized to the others terms of the anomalous dimensions of Eqs. (36) and(37) as well. Thus, the exact approach permit us to see that each loop term is accompaniedof a power of the LV full Π factor as it is shown by the general theorem displayed in lastSect. This procedure is valid at all intermediate steps of the program. Another interestingpoint to be mentioned is that in this method, the β -function and anomalous dimensionsdepend on the LV coefficients at its exact form only through the LV Π factor and on thesymmetry point employed. We need to compute the nontrivial solution of the β -function.The trivial one leads to the mean field or Landau critical exponents and can be obtainedmathematically by a factorization procedure resulting in the factorization of a single powerof u in Eq. for the β -function. This procedure results in the nontrivial fixed point given by u ∗ = 6 ǫ ( N + 8) Π (cid:26) ǫ (cid:20) N + 14)( N + 8) + 12 (cid:21)(cid:27) . (38)It can be written as u ∗ = u ∗ (0) / Π , where u ∗ (0) is its Lorentz-invariant (LI) counterpart. Nowthe LV corrections to mean field or Landau approximation to the critical exponents are giventhough the application of definitions η ≡ γ φ ( u ∗ ) and ν − ≡ − η − γ φ ( u ∗ ). They can beapplied to obtain, to next-to-leading order, the two respective critical exponents η = ( N + 2) ǫ N + 8) (cid:26) ǫ (cid:20) N + 14)( N + 8) − (cid:21)(cid:27) ≡ η (0) , (39) ν = 12 + ( N + 2) ǫ N + 8) + ( N + 2)( N + 23 N + 60) ǫ N + 8) ≡ ν (0) , (40)where η (0) and ν (0) are their corresponding Lorentz-invariant (LI) counterparts [23]. Asthere are six critical exponents and four scaling relations among them, there are only twoindependent ones. Thus the two ones above are enough for evaluating the four remainingones. In next Sect. we will attain the same task but now in a distinct renormalizationmethod and will compare the results. 8 II. EXACT LORENTZ-VIOLATING NEXT-TO-LEADING ORDER CRITICALEXPONENTS IN THE UNCONVENTIONAL MINIMAL SUBTRACTION SCHEME
This method is characterized by its elegance as compared with the earlier one since theexternal momenta remain at arbitrary values along all the renormalization program. Thisimplies that we do not have to compute any parametric integral because they cancel out inthe final expressions for the β -function and anomalous dimensions. Then, now we have that A = ( N + 8)18 h + 2 perm. i (41) A (1)2 = ( N + 6 N + 20)108 h + 2 perm. i , (42) A (2)2 = (5 N + 22)54 (cid:20) + 5 perm. (cid:21) , (43) B = ( N + 2)18 (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) P =0 (cid:19) , (44) B = ( N + 2)( N + 8)108 (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) P =0 (cid:19) , (45) C = ( N + 2)6 , (46) C (1)2 = ( N + 2) , (47) C (2)2 = ( N + 2)6 , (48)where the poles are minimally eliminated, thus being absorbed in the renormalization con-stants for the field Z φ and composite field Z φ , respectively. Now absorbing ˆ S in a redefinitionof the coupling constant and using the Feynman diagrams computed in B, we find β ( u ) = − ǫu + N + 86 Π u − N + 1412 Π u , (49) γ φ ( u ) = N + 272 Π u − ( N + 2)( N + 8)1728 Π u , (50)9 φ ( u ) = N + 26 Π u − N + 212 Π u . (51)The renormalization program proceeds so elegantly that all the momentum-dependent inte-grals, namely the ones L ( P + K µν P µ P ν , m B ), L ( P + K µν P µ P ν , m B ), ˜ i ( P + K µν P µ P ν , m 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 renor-malization of the field and composite field. In fact, technically, the renormalization of theseparameters comes from the terms proportional to P + K µν P µ P ν in the diagrams and. But, unfortunately, we are yet left with a residual divergence and it originates fromthe terms proportional to m in the diagrams and . It is show belowΓ (2) ( P + K µν P µ P ν , u, m ) = P + K µν P µ P ν + m (cid:26) N + 2)24 ˜ I ( P + K µν P µ P ν , m B ) − ( N + 2)( N + 8)108 ǫ ˜ I ( P + K µν P µ P ν , m B ) u (cid:27) , (52)where ˜ I ( P + K µν P µ P ν , m B ) = Z dx Z dylny ddy (1 − y ) ln y (1 − y ) P + K µν P µ P ν m B + 1 − y + yx (1 − x ) − y + yx (1 − x ) . (53)The reduction of the number of diagrams to be computed though the redefining of the initialbare 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 two-point function as˜Γ (2) ( P + K µν P µ P ν , u, m ) =Γ (2) ( P + K µν P µ P ν , u, m ) + m (cid:26) ( N + 2)( N + 8)108 ǫ ˜ I ( P + K µν P µ P ν , m B ) u (cid:27) . (54)This turn out to connect the Unconventional minimal subtraction scheme to the conventionalone in the massless theory [20], once the terms proportional to m vanish in the latter case.The final check of this redefinition can be shown by showing that it satisfies the normalizationcondition used in Sect. II ˜Γ (2) ( P + K µν P µ P ν = 0 , u, m ) =Γ (2) ( P + K µν P µ P ν = 0 , u, m ) = m . (55)10nce again, for computing the LV loop quantum corrections to the critical exponents, weneed to evaluate the nontrivial fixed point though the nontrivial solution for the equation β ( u ∗ ) = 0. It is given by u ∗ = 6 ǫ ( N + 8)Π (cid:26) ǫ (cid:20) N + 14)( N + 8) (cid:21)(cid:27) . (56)This value for the nontrivial fixed point when used for evaluating the critical exponents, leadsto 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 indifferent renormalization schemes. Now we proceed to compute the critical exponents in athird renormalization scheme. IV. EXACT LORENTZ-VIOLATING NEXT-TO-LEADING ORDER CRITICALEXPONENTS IN THE BPHZ METHOD
The BPHZ (Bogoliubov-Parasyuk-Hepp-Zimmermann) method [26–28] is the most gen-eral from all known renormalization methods. It does not include any trick for reducingthe total number of diagrams to be evaluated. Thus we have to compute all diagrams inthe original expansion for a given loop order. As opposed to the earlier ones, in the BPHZmethod, we start from the renormalized theory L = 12 Z φ ( g µν + K µν ) ∂ µ φ∂ ν φ + µ ǫ u Z u φ + 12 tZ φ φ , (57)where φ = Z − / φ φ B , u = µ − ǫ Z φ Z u λ B , t = Z φ Z φ t B . (58)Initially, considering the bare theory at one-loop order, we absorb that divergence by addingterms to the initial Lagrangian density. Then, a finite Lagrangian density is found. Forconsidering 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 renormalizationconstants. We expand the renormalization constants as Z φ = 1 + ∞ X i =1 c iφ , (59)11 u = 1 + ∞ X i =1 c iu , (60) Z m = 1 + ∞ X i =1 c im . (61)The c iφ , c ig and c im coefficients are the i -th loop order renormalization constants for the field,renormalized coupling constant and composite field, respectively. They are given by Z φ ( u, ǫ − ) = 1 + 1 P + K µν P µ P ν " K (cid:16) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m =0 S +14 K (cid:16) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m =0 S + 13 K (cid:16) (cid:17) S , (62) Z u ( u, ǫ − ) = 1 + 1 µ ǫ u " K (cid:16) + 2 perm. (cid:17) S + 14 K (cid:16) + 2 perm. (cid:17) S +12 K (cid:16) + 5 perm. (cid:17) S + 12 K (cid:18) + 2 perm. (cid:19) S + K (cid:16) + 2 perm. (cid:17) S + K (cid:16) + 2 perm. (cid:17) S , (63) Z m ( u, ǫ − ) = 1 + 1 m " K (cid:16) (cid:17) S + 14 K (cid:18) (cid:19) S + 12 K (cid:16) (cid:17) S +12 K (cid:16) (cid:17) S + 16 K (cid:16) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P + K µν P µ P ν =0 S , (64)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 -functionand anomalous dimensions are given by β ( u ) = − ǫu + N + 86 Π u − N + 1412 Π u , (65) γ φ ( u ) = N + 272 Π u − ( N + 2)( N + 8)1728 Π u , (66) γ m ( u ) = N + 26 Π u − N + 2)72 Π u . (67)12ne more time, we have to compute the nontrivial solution of Eq. (65). This procedureyields the value u ∗ = 6 ǫ ( N + 8) Π (cid:26) ǫ (cid:20) N + 14)( N + 8) (cid:21)(cid:27) . (68)Now by applying the relations η ≡ γ φ ( u ∗ ) and ν − ≡ − γ m ( u ∗ ), we obtain once again thatthe LV critical exponents are identical to their LV counterparts. Now we evaluate the LVcritical 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 methodsaforementioned since the critical exponents, as being universal quantities, must be the sameif evaluated at any renormalization scheme. For that, we will employ the BPHZ methodwhich 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 ina theory represented by the Lagrangian density of Eq. (1). Its evaluated expression indimensional regularization in d = 4 − ǫ can be written as a general functional Π L F ( u, P + K µν P µ P ν , ǫ, µ ) if its LI counterpart is given by F ( u, P , ǫ, µ, m ) , where L is the number ofloops of the corresponding diagram.Proof. A general Feynman diagram of loop level L is a multidimensional integral in L distinctand independent momentum integration variables q , q ,..., q L , each one with volume elementgiven by d d q i ( i = 1 , , ..., L ). As showed in last Section, the substitution q ′ = √ I + K q transforms each volume element as d d q ′ = p det ( I + K ) d d q . Thus d d q = d d q ′ / p det ( I + K ) ≡ Π d d q ′ , Π = 1 / p det ( I + K ). Then, the integration in L variables results in a LV overallfactor of Π L . Now making q ′ → P ′ in the substitution above, where P ′ is the transformedexternal momentum, then P ′ = P + K µν P µ P ν . So a given Feynman diagram, evaluated indimensional regularization in d = 4 − ǫ , assumes the expression Π L F ( u, P + K µν P µ P ν , ǫ, µ ),where F is associated to the corresponding diagram if the LI Feynman diagram counterpartevaluation results in F ( u, P , ǫ, µ ).Now using the result of the theorem above and the one in which all momentum-dependentintegrals cancel out order by order in perturbation theory for all levels in the renormalization13rocess[26–28], we see that the only LV dependence of β -function and anomalous dimensionsis due to the LV full Π factor, which comes from the volume elements of the diagramscontributing 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 β ( u ) = − ǫu + ∞ X n =2 β (0) n Π n − u n , (69) γ ( u ) = ∞ X n =2 γ (0) n Π n u n , (70) γ m ( u ) = ∞ X n =1 γ (0) m ,n Π n u n , (71)where β (0) n , γ (0) n and γ m ,n are the LI nth-loop corrections to the referred functions. Byapplying the same factorization process employed in the finite loop scenario for the any looprealm, we obtain that u ∗ = u ∗ (0) / Π where u ∗ (0) is the LI fixed point for all loop levels. Then,we can substitute this all-loop order fixed point in the β -function and anomalous dimensionsto obtain the LV critical exponents valid for any loop levels as being identical to their anyloop orders LI counterparts. VI. CONCLUSIONS
We have evaluated analytically the ultraviolet divergences of Lorentz-violating massiveO( N ) λφ scalar field theories, which are exact in the Lorentz-violating mechanism, firstlyexplicitly at next-to-leading order and latter at any loop level through an induction pro-cedure based on a theorem following from the exact approach, for computing the corre-sponding critical exponents. For that, we have employed three different and independentfield-theoretic renormalization group methods. We have found equal critical exponents inthe three methods and furthermore identical to their Lorentz invariant counterparts. Wehave also showed that the exact approach, which reduces to the non-exact one in its limitedrange of applicability, besides exact, is capable of furnishing the expressions for the all-loopLV radiative quantum corrections to the β -function and anomalous dimensions consideringjust 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 one14n literature for our knowledge. Thus it can inspire the exact solution of problems involvingconsidering the exact effect of LV mechanisms in many physical phenomena ranging fromhigh- (standard model extension for example) to low energy physics (corrections to scaling,finite-size scaling, amplitude ratios, critical exponents in geometries subjected to differentboundary conditions, Lifshitz points etc [29–32]. Appendix A: Integrals of Callan-Symanzik method SP = 1 ǫ (cid:18) − ǫ (cid:19) Π , (A1) ′ = − ǫ (cid:18) − ǫ + Iǫ (cid:19) Π , (A2) ′ = − ǫ (cid:18) − ǫ + 32 Iǫ (cid:19) Π , (A3) SP = 12 ǫ (cid:18) − ǫ (cid:19) Π , (A4)where the integral I [31–33] I = Z (cid:26) − x (1 − x ) + x (1 − x )[1 − x (1 − x )] (cid:27) (A5)is a residual number and is a consequence of the symmetry point chosen. Appendix B: Integrals of Unconventional minimal subtraction scheme = 1 ǫ (cid:20) − ǫ − ǫL ( P + K µν P µ P ν , m B ) (cid:21) Π , (B1)= (cid:26) − m B ǫ (cid:20) ǫ + (cid:18) π
12 + 1 (cid:19) ǫ (cid:21) − m B i ( P + K µν P µ P ν , m B ) − ( P + K µν P µ P ν )8 ǫ (cid:20) ǫ − ǫL ( P + K µν P µ P ν , m B ) (cid:21)(cid:27) Π , (B2)15 (cid:26) − m B ǫ (cid:20) ǫ + (cid:18) π
24 + 154 (cid:19) ǫ (cid:21) − m B ǫ ˜ i ( P + K µν P µ P ν , m B ) − P + K µν P µ P ν ǫ (cid:20) ǫ − ǫL ( P + K µν P µ P ν , m B ) (cid:21)(cid:27) Π , (B3)= 1 ǫ (cid:20) − ǫ − ǫL ( P + K µν P µ P ν , m B ) (cid:21) Π , (B4)where L ( P + K µν P µ P ν , m B ) = Z dx ln[ x (1 − x )( P + K µν P µ P ν ) + m B ] , (B5) L ( P + K µν P µ P ν , m B ) = Z dx (1 − x ) ln[ x (1 − x )( P + K µν P µ P ν ) + m B ] , (B6)˜ i ( P + K µν P µ P ν , m B ) = Z dx Z dy ln y ddy (cid:18) (1 − y ) ln (cid:26) y (1 − y ) P + (cid:20) − y + yx (1 − x ) (cid:21) m B (cid:27)(cid:19) , (B7) Appendix C: Integrals of BPHZ method (cid:16) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m =0 = − u ( P + K µν P µ P ν )8 ǫ (cid:20) ǫ − ǫ J ( P + K µν P µ P ν ) (cid:21) Π , (C1) (cid:12)(cid:12)(cid:12)(cid:12) m =0 = ( P + K µν P µ P ν ) u ǫ (cid:20) ǫ − ǫ J ( P + K µν P µ P ν ) (cid:21) Π , (C2)= − P + K µν P µ P ν ) u ǫ (cid:20) ǫ − ǫ J ( P + K µν P µ P ν ) (cid:21) Π , (C3)= µ ǫ u ǫ (cid:20) − ǫ − ǫJ ( P + K µν P µ P ν ) (cid:21) Π , (C4)= − µ ǫ u ǫ (cid:2) − ǫ − ǫJ ( P + K µν P µ P ν ) (cid:3) Π , (C5)= − µ ǫ u ǫ (cid:20) − ǫ − ǫJ ( P + K µν P µ P ν ) (cid:21) Π , (C6)16 µ ǫ u ǫ J ( P + K µν P µ P ν ) Π , (C7)= 3 µ ǫ u ǫ (cid:20) − ǫ − ǫJ ( P + K µν P µ P ν ) (cid:21) Π , (C8)= − µ ǫ u ǫ J ( P + K µν P µ P ν ) Π , (C9)= m u (4 π ) ǫ (cid:20) − ǫ ln (cid:18) m πµ (cid:19)(cid:21) Π , (C10)= − m u (4 π ) ǫ (cid:20) − ǫ − ǫ ln (cid:18) m πµ (cid:19)(cid:21) Π , (C11)= m g ǫ (cid:20) − ǫ − ǫ ln (cid:18) m πµ (cid:19)(cid:21) Π , (C12)= 3 m u ǫ (cid:20) − ǫ ln (cid:18) m πµ (cid:19)(cid:21) Π , (C13) (cid:16) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P + K µν P µ P ν =0 = − m g ǫ (cid:20) ǫ − ǫ ln (cid:18) m πµ (cid:19)(cid:21) Π , (C14)where J ( P + K µν P µ P ν ) = Z dx ln (cid:20) x (1 − x )( P + K µν P µ P ν ) + m µ (cid:21) , (C15) J ( P + K µν P µ P ν ) = Z Z dx dy (1 − y ) × ln ( y (1 − y )( P + K µν P µ P ν ) µ + (cid:20) − y + yx (1 − x ) (cid:21) m µ ) , (C16) J ( P + K µν P µ P ν ) = m µ Z dx (1 − x ) x (1 − x )( P + K µν P µ P ν ) µ + m µ . (C17)17 CKNOWLEDGMENTS
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