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

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

P. R. S. Carvalho Departamento de Física, Universidade Federal do Piauí, 64049-550, Teresina, PI, Brazil    M. I. Sena-Junior Escola Politécnica de Pernambuco, Universidade de Pernambuco, 50720-001, Recife, PE, Brazil Instituto de Física, Universidade Federal de Alagoas, 57072-900, Maceió, AL, Brazil
###### Abstract

In this work we evaluate analytically the ultraviolet divergences of Lorentz-violating massive O() 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 -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

Lorentz symmetry is one of the most fundamental symmetries of nature and the possibility of its violation was theme of great investigation in the last years, usually as a finite perturbative expansion at some Lorentz-violating (LV) parameters and loop number, both in high energy Kislat and Krawczynski (2015); Amelino-Camelia et al. (2015); Ribeiro et al. (2015); van Tilburg and van Veghel (2015); Anacleto et al. (2012); Ferrero and Altschul (2011); Carvalho (2013, 2014) as well as in low energy Vieira and Carvalho (2014); Carvalho (2016); Vieira and de Carvalho (2016) physics. In the latter realm, the critical exponents were computed, at least, at first order in the Lorentz-violating (LV) parameters and any loop level for LV scalar field theories Vieira and Carvalho (2014); Carvalho (2016); Vieira and de Carvalho (2016). For that, this evaluation was possible by means of the application of a non-exact approach based on a joint redefinition of 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 furthermore for all loop orders. Moreover, we will show that the referred exact approach gives expressions for the -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 the earlier non-exact approach in the appropriate limit.

In this work, we compute analytically the critical exponents for massive O() scalar field theories with Lorentz violation. This computation is exact in the LV mechanism. For that, we apply three distinct field-theoretic renormalization group methods and they involve the same theory renormalized at different renormalization schemes. In this field-theoretic formulation, if the critical exponents present the same values when obtained through the three methods, this means that they are universal quantities and we have the confirmation of the universality hypothesis. These universal quantities characterize the critical behavior of distinct systems as a fluid and a ferromagnet. When the critical behavior of two or more distinct systems is characterized by the same critical exponents, we say that they belong to the same universality class. The universality class inspected here is the O() one, which encompasses the particular models: Ising (), XY (), Heisenberg (), self-avoiding random walk () and spherical () for short-range interactions Pelissetto and Vicari (2002). The critical exponents depend on the dimension of the system, and symmetry of some -component order parameter (magnetization for magnetic systems), and if the interactions present are of short- or long-range type. Many works probing the dependence of the critical exponents on the obvious parameters as Moukouri and Eidelstein (2012); Nishiyama (2005) and Codello and D’Odorico (2013); Butti and Toldin (2005); Patrascioiu and Seiler (1996) were published. Just a few of them were published in the less one, that of symmetry of the order parameter Oppermann and Schmidt (2008); Trugenberger (2005). 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() self-interacting 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() scalar field theory whose bare Lagrangian density in Euclidean spacetime is given by Ferrero and Altschul (2011); Carvalho (2013, 2014)

 L=12(δμν+Kμν)∂μϕB∂νϕB+12m2Bϕ2B+λB4!ϕ4B. (1)

In Eq. above, the bare parameters , and are the bare field, mass and coupling constant, respectively. The responsible for the symmetry breaking mechanism are the constant symmetric LV coefficients . We can now expand the bare primitively PI 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 in Eq. 1 initially at tree-level for its three-loop counterpart Amit and Martín-Mayor (2005); Carvalho and Leite (2013) we can achieve the desired aim. Now making from now on we have

 Γ(2)B(P2+KμνPμPν,mB,λB)=\parbox34.143307pt\includegraphics[scale=1.0]fig9.eps−1−λ2B6(\parbox34.143307pt\includegraphics[scale=1.0]fig6.eps−\parbox34.143307pt\includegraphics[scale=1.0]fig6.eps∣∣∣P2+KμνPμPν=0)+ λ3B4(\parbox28.452756pt\includegraphics[scale=.9]fig7.eps−\parbox28.452756pt\includegraphics[scale=.9]fig7.eps∣∣∣P2+KμνPμPν=0), (4)
 Γ(4)B(Pi,mB,λB)=λB−λ2B2(\parbox28.452756pt\includegraphics[scale=1.0]fig10.eps+2perm.)+λ3B4(\parbox45.524409pt\includegraphics[scale=1.0]fig11.eps+2perm.)+ λ3B2(\parbox45.524409pt\includegraphics[scale=.9]fig21.eps+5perm.), (5)
 Γ(2,1)B(P1,P2,Q3,mB,λB)=1−λB2\parbox36.988583pt\includegraphics[scale=1.0]fig14.eps+λ2B4\parbox31.298031pt\includegraphics[scale=1.0]fig16.eps+λ2B2\parbox36.988583pt\includegraphics[scale=.9]fig17.eps (6)

where . We can now define the dimensional and the dimensionless renormalized coupling constants and as , where , 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 and can be also defined as . We renormalize these correlation functions multiplicatively

 Γ(n,l)(Pi,Qj,u,m)=Zn/2ϕZlϕ2Γ(n,l)B(Pi,Qj,λB,mB) (7)

which satisfies the Callan-Symanzik equation

 (m∂∂m+β∂∂u−12nγϕ+lγϕ2)Γ(n,l)R(Pi,Qj,u,m)=m2(2−γϕ)Γ(n,l+1)R(Pi,Qj,0,u,m) (8)

where

 β(u)=m∂u∂m=−ϵ(∂lnu0∂u)−1, (9)
 γϕ(u)=β(u)∂lnZϕ∂u, (10)
 γϕ2(u)=−β(u)∂lnZϕ2∂u, (11)

where we use the function

 ¯¯¯γϕ2(u)=−β(u)∂ln¯¯¯¯Zϕ2∂u≡γϕ2(u)−γϕ(u) (12)

instead of , for convenience reasons, by fixing the external momenta through the normalization conditions

 Γ(2)(P2+KμνPμPν=0;m,u)=m2, (13)
 ∂Γ(2)(P2+KμνPμPν;m2,u)∂(P2+KμνPμPν)∣∣∣P2+KμνPμPν=0=1, (14)
 Γ(4)R(Pi=0;m,u)=u, (15)
 Γ(2,1)R(Pi=0,Qj=0,m,u)=1. (16)

It is well known that we can reduce more yet the number of diagrams to be evaluated Amit and Martín-Mayor (2005) 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. As the computation of the PI vertex parts leads to momentum integration involving just their internal bubbles and not their external legs, all what matters in this evaluation are their internal bubbles contents. Thus, without take into account the O() symmetry factors, we have that , ,        ( ). Finally, the only diagrams to be evaluated are the , , , ones. Thus we can write the PI vertex parts as

 Γ(2)B(P2+KμνPμPν,u0,mB)=(P2+KμνPμPν)(1−B2u20+B3u30), (17)
 Γ(4)B(Pi,u0,mB)=mϵBu0[1−A1u0+(A(1)2+A(2)2)u20], (18)
 Γ(2,1)B(P1,P2,Q3,u0,mB)=1−C1u0+(C(1)2+C(2)2)u20, (19)

where

 A1=(N+8)6 \parbox28.452756pt\includegraphics[scale=1.0]fig10.epsSP (20)
 A(1)2=(N2+6N+20)36 \parbox45.524409pt\includegraphics[scale=1.0]fig11.epsSP, (21)
 A(2)2=(5N+22)9 \parbox39.833858pt\includegraphics[scale=1.0]fig21.epsSP, (22)
 B2=(N+2)18 \parbox34.143307pt\includegraphics[scale=1.0]fig6.eps′, (23)
 B3=(N+2)(N+8)108 \parbox34.143307pt\includegraphics[scale=1.0]fig7.eps′, (24)
 C1=(N+2)6\parbox39.833858pt\includegraphics[scale=1.0]fig14.epsSP, (25)
 C(1)2=(N+2)236\parbox34.143307pt\includegraphics[scale=1.0]fig16.epsSP, (26)
 C(2)2=(N+2)6\parbox36.988583pt\includegraphics[scale=1.0]fig17.epsSP (27)

and in the Callan-Symanzik method, the two-loop diagram as well three-loop 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 PI vertex part is one of composite field insertions and the one in the left-hand side (lhs) has such insertions. As it is known, an extra composite field insertion is responsible for one additional power of the propagator in the corresponding PI vertex part. We can then work in the ultraviolet limit, i. e., in the limit where the external momenta . 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 Weinberg (1960). So, the PI vertex parts satisfy the renormalization group equation, thus permitting us to apply the theory of scaling for these functions and evaluate the -function and anomalous dimensions as well as the corresponding critical exponents. The LV coefficients can now been considered exactly by noting that = + , where is a -dimensional vector whose representation is a column matrix and is a row matrix and and are matrix representations of the identity and , respectively. Thus making , the LV mechanism is shown explicitly through two contributions. The first of them is displayed through the volume elements of -dimensional integrals , thus . This LV full or exact contribution reduces to its perturbative counterpart for small violations of Lorentz symmetry, where is the LV contribution of order in Ferrero and Altschul (2011); Carvalho (2013, 2014); Vieira and Carvalho (2014); Carvalho (2016); Vieira and de Carvalho (2016). The other LV modification of the theory is that involving the external momenta. It can be seen in the momentum-dependent -dimensional integrals when evaluated in dimensional regularization in

 ∫ddq(2π)d1(q2+2Pq+M2)α=^Sd12Γ(d/2)Γ(α)Γ(α−d/2)(M2−P2)α−d/2, (28)

where , and is the surface area of a unit -dimensional sphere. Its finite value in four-dimensional spacetime is . This definition is convenient as to each loop integration we have a factor of at four dimensions, thus avoiding the appearance of Euler-Mascheroni constants in the middle of calculations Amit and Martín-Mayor (2005). Now making and , . As it is known, from all diagrams displayed above, we need to compute only four of them Amit and Martín-Mayor (2005). They are shown in A. When we absorb in a redefinition of the coupling constant and use the Feynman diagrams for computing the -function and anomalous dimensions by writing the Laurent expansion

 u0=u(1+∞∑i=1ai(ϵ)ui), (29)
 Zϕ=1+∞∑i=1bi(ϵ)ui, (30)
 ¯¯¯¯Zϕ2=1+∞∑i=1ci(ϵ)ui, (31)
 β(u)=−ϵu[1−a1u+2(a21−a2)u2], (32)
 γϕ(u)=−ϵu[2b2u+(3b3−2b2a1)u2], (33)
 ¯¯¯γϕ2(u)=ϵu[c1+(2c2−c21−2a1c1)u], (34)

where the constant coefficients , , depend on the Feynman diagrams, evaluated in appendix A, just mentioned Amit and Martín-Mayor (2005), we obtain

 β(u)=−ϵu+N+86(1−12ϵ)Πu2−3N+1412Π2u3, (35)
 γϕ(u)=N+272(1−14ϵ+Iϵ)Π2u2−(N+2)(N+8)432(I+1)Π3u3, (36)
 ¯¯¯γϕ2(u)=N+26(1−12ϵ)Πu−N+212Π2u2. (37)

We observe that the expression for the -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 factor, although it is a term of first order in . This term is fundamental for making possible expansions in quantum field theory and is essential in the renormalization group and -expansion techniques developed by Wilson, specially with applications in critical phenomena Wilson and Kogut (1974); Wilson and Fisher (1972); Wilson (1972) in . Its second one-loop term is of second order in , but it has acquired only a linear power of . The last one, although being of third order in , must be of second order in , since it is of two-loop order. Similar arguments 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 accompanied of a power of the LV full factor as it is shown by the general theorem displayed in last Sect. This procedure is valid at all intermediate steps of the program. Another interesting point to be mentioned is that in this method, the -function and anomalous dimensions depend on the LV coefficients at its exact form only through the LV factor and on the symmetry point employed. We need to compute the nontrivial solution of the -function. 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 in Eq. for the -function. This procedure results in the nontrivial fixed point given by

 u∗=6ϵ(N+8)Π{1+ϵ[3(3N+14)(N+8)2+12]}. (38)

It can be written as , where 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 and . They can be applied to obtain, to next-to-leading order, the two respective critical exponents

 η=(N+2)ϵ22(N+8)2{1+ϵ[6(3N+14)(N+8)2−14]}≡η(0), (39)
 ν=12+(N+2)ϵ4(N+8)+(N+2)(N2+23N+60)ϵ28(N+8)3≡ν(0), (40)

where and are their corresponding Lorentz-invariant (LI) counterparts Wilson and Kogut (1974). 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.

## Iii Exact Lorentz-violating next-to-leading order critical 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 -function and anomalous dimensions. Then, now we have that

 A1=(N+8)18[\parbox28.452756pt\includegraphics[scale=1.0]fig10.eps+2perm.] (41)
 A(1)2=(N2+6N+20)108[\parbox28.452756pt\includegraphics[scale=1.0]fig11.eps+2perm.], (42)
 A(2)2=(5N+22)54[\parbox28.452756pt\includegraphics[scale=1.0]fig21.eps+5perm.], (43)
 B2=(N+2)18(\parbox34.143307pt\includegraphics[scale=1.0]fig6.eps−\parbox34.143307pt\includegraphics[scale=1.0]fig6.eps∣∣∣P=0), (44)
 B3=(N+2)(N+8)108(\parbox34.143307pt\includegraphics[scale=1.0]fig7.eps−\parbox34.143307pt\includegraphics[scale=1.0]fig7.eps∣∣∣P=0), (45)
 C1=(N+2)6\parbox28.452756pt\includegraphics[scale=1.0]fig14.eps, (46)
 C(1)2=(N+2)236\parbox28.452756pt\includegraphics[scale=1.0]fig16.eps, (47)
 C(2)2=(N+2)6\parbox28.452756pt\includegraphics[scale=1.0]fig17.eps, (48)

where the poles are minimally eliminated, thus being absorbed in the renormalization constants for the field and composite field , respectively. Now absorbing in a redefinition of the coupling constant and using the Feynman diagrams computed in B, we find

 β(u)=−ϵu+N+86Πu2−3N+1412Π2u3, (49)
 γϕ(u)=N+272Π2u2−(N+2)(N+8)1728Π3u3, (50)
 ¯¯¯γϕ2(u)=N+26Πu−N+212Π2u2. (51)

The renormalization program proceeds so elegantly that all the momentum-dependent integrals, namely the ones , , have disappeared. Now, the only LV dependence of the theory is that through the LV full 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 in the diagrams and . But, unfortunately, we are yet left with a residual divergence and it originates from the terms proportional to in the diagrams and . It is show below

 Γ(2)(P2+KμνPμPν,u,m)=P2+KμνPμPν+ m2{1+(N+2)24~I(P2+KμνPμPν,m2B)−(N+2)(N+8)108ϵ~I(P2+KμνPμPν,m2B)u3}, (52)

where

 ~I(P2+KμνPμPν,m2B)= ∫10dx∫10dylnyddy⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(1−y)ln⎡⎢ ⎢ ⎢⎣y(1−y)P2+KμνPμPνm2B+1−y+yx(1−x)1−y+yx(1−x)⎤⎥ ⎥ ⎥⎦⎫⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪⎭. (53)

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 two-point function as

 ~Γ(2)(P2+KμνPμPν,u,m)= Γ(2)(P2+KμνPμPν,u,m)+m2{(N+2)(N+8)108ϵ~I(P2+KμνPμPν,m2B)u3}. (54)

This turn out to connect the Unconventional minimal subtraction scheme to the conventional one in the massless theory Amit and Martín-Mayor (2005), once the terms proportional to 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. II

 ~Γ(2)(P2+KμνPμPν=0,u,m)= Γ(2)(P2+KμνPμPν=0,u,m)=m2. (55)

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 . It is given by

 u∗=6ϵ(N+8)Π{1+ϵ[3(3N+14)(N+8)2]}. (56)

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.

## Iv Exact Lorentz-violating next-to-leading order critical exponents in the BPHZ method

The BPHZ (Bogoliubov-Parasyuk-Hepp-Zimmermann) method Bogoliubov and Parasyuk (1957); Hepp (1966); Zimmermann (1969) 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

 L=12Zϕ(gμν+Kμν)∂μϕ∂νϕ+μϵu4!Zuϕ4+12tZϕ2ϕ2,

where

 ϕ=Z−1/2ϕϕB,u=μ−ϵZ2ϕZuλB,t=ZϕZϕ2tB. (58)

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

 Zϕ=1+∞∑i=1ciϕ, (59)
 Zu=1+∞∑i=1ciu, (60)
 Zm2=1+∞∑i=1cim2. (61)

The , and coefficients are the -th loop order renormalization constants for the field, renormalized coupling constant and composite field, respectively. They are given by

 Zϕ(u,ϵ−1)=1+1P2+KμνPμPν[16K(\parbox34.143307pt\includegraphics[scale=1.0]fig6.eps)∣∣∣m2=0S\parbox28.452756pt\includegraphics[scale=0.5]fig6.eps+ 14K(\parbox34.143307pt\includegraphics[scale=1.0]fig7.eps)∣∣∣m2=0S\parbox17.071654pt\includegraphics[scale=0.5]fig7.eps+13K(\parbox34.143307pt\includegraphics[scale=1.0]fig26.eps)S\parbox17.071654pt\includegraphics[scale=0.5]fig26.eps], (62)
 Zu(u,ϵ−1)=1+1μϵu[12K(\parbox28.452756pt\includegraphics[scale=1.0]fig10.eps+2perm.)S\parbox28.452756pt\includegraphics[scale=0.5]fig10.eps+14K(\parbox48.369685pt\includegraphics[scale=1.0]fig11.eps+2perm.)S\parbox28.452756pt\includegraphics[scale=0.5]fig11.eps+ 12K(\parbox34.143307pt\includegraphics[scale=.8]fig21.eps+5perm.)S\parbox28.452756pt\includegraphics[scale=0.4]fig21.eps+12K(\parbox28.452756pt\includegraphics[scale=1.0]fig13.eps+2perm.)S\parbox28.452756pt\includegraphics[scale=0.5]fig13.eps+ K(\parbox28.452756pt\includegraphics[scale=1.0]fig25.eps+2perm.)S\parbox17.071654pt\includegraphics[scale=0.5]fig25.eps+K(\parbox28.452756pt\includegraphics[scale=1.0]fig24.eps+2perm.)S\parbox17.071654pt\includegraphics[scale=0.5]fig24.eps], (63)
 Zm2(u,ϵ−1)=1+1m2[12K(\parbox34.143307pt\includegraphics[scale=1.0]fig1.eps)S\parbox28.452756pt\includegraphics[scale=0.5]fig1.eps+14K(\parbox34.143307pt\includegraphics[scale=1.0]fig2.eps)S\parbox17.071654pt\includegraphics[scale=0.5]fig2.eps+12K(\parbox34.143307pt\includegraphics[scale=1.0]fig22.eps)S\parbox17.071654pt\includegraphics[scale=0.5]fig22.eps+ 12K(\parbox34.143307pt\includegraphics[scale=1.0]fig23.eps)S\parbox17.071654pt\includegraphics[scale=0.5]fig23.eps+16K(\parbox34.143307pt\includegraphics[scale=1.0]fig6.eps)∣∣∣P2+KμνPμPν=0S\parbox17.071654pt\includegraphics[scale=0.5]fig6.eps], (64)

where is the symmetry factor for the corresponding diagram and so on when some -component field is considered. By using the diagrams in C, we have that the -function and anomalous dimensions are given by

 β(u)=−ϵu+N+86Πu2−3N+1412Π2u3, (65)
 γϕ(u)=N+272Π2u2−(N+2)(N+8)1728Π3u3, (66)
 γm2(u)=N+26Πu−5(N+2)72Π2u2. (67)

One more time, we have to compute the nontrivial solution of Eq. (65). This procedure yields the value

 u∗=6ϵ(N+8)Π{1+ϵ[3(3N+14)(N+8)2]}. (68)

Now by applying the relations and , 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 can be written as a general functional if its LI counterpart is given by , where is the number of loops of the corresponding diagram.

###### Proof.

A general Feynman diagram of loop level is a multidimensional integral in distinct and independent momentum integration variables , ,…,, each one with volume element given by (). As showed in last Section, the substitution transforms each volume element as . Thus , . Then, the integration in variables results in a LV overall factor of . Now making in the substitution above, where is the transformed external momentum, then . So a given Feynman diagram, evaluated in dimensional regularization in , assumes the expression , where is associated to the corresponding diagram if the LI Feynman diagram counterpart evaluation results in . ∎

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 processBogoliubov and Parasyuk (1957); Hepp (1966); Zimmermann (1969), we see that the only LV dependence of -function and anomalous dimensions is due to the LV full factor, which comes from the volume elements of the diagrams contributing with a factor, where is the number of loops of the corresponding graph. Thus we can write the -function and anomalous dimensions for all loop levels

 β(u)=−ϵu+∞∑n=2β(0)nΠn−1un, (69)
 γ(u)=∞∑n=2γ(0)nΠnun, (70)
 γm2(u)=∞∑n=1γ(0)m