Diagram Reduction in Problem of Critical Dynamics of Ferromagnets: 4-Loop Approximation
L. Ts. Adzhemyan, E. V. Ivanova, M. V. Kompaniets, S. Ye. Vorobyeva
DDiagram Reduction in Problem of CriticalDynamics of Ferromagnets: 4-Loop Approximation
L. Ts. Adzhemyan , E. V. Ivanova , M. V. Kompaniets andS. Ye. Vorobyeva Saint-Petersburg State University, 7-9 Universitetskaya nab. Saint-Petersburg,Russian FederationE-mail: [email protected] , [email protected] , [email protected]@gmail.com , Abstract.
Within the framework of the renormalization group approach to the mod-els of critical dynamics, we propose a method for a considerable reduction of the numberof integrals needed to calculate the critical exponents. With this method we performa calculation of the critical exponent z of model A at 4-loop level, where our methodallows to reduce number of integrals from 66 to 17. The way of constructing theintegrand in Feynman representation of such diagrams is discussed. Integrals wereestimated numerically with Sector Decomposition technique. Keywords : renormalization group, ε -expansion, multi-loop diagrams, critical exponents a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec iagram Reduction in Critical Dynamics
1. Introduction
Model A of critical dynamics describes critical slowing down effect for systems withnon-conserved order parameter [1, 2, 3]. Usually this model is used as a theoreticalmodel for critical behavior of ferromagnets [4]. Recently the new classes of materialswere investigated and it was found that this model describes phase transitions inmultiferroics [5] and in the systems with ordering phase transitions [6] as well. Anadditional motivation to study model A is also the fact that it is the simplest model ofcritical dynamics and new technical methods can be tested on it.Despite the fact that renormalization group is one of the well acknowledgedtheoretical methods for investigation of continuous phase transitions, the applicationof this method to the problems of critical dynamics faces much greater difficulties incomparison with the problems of critical statics. The analytic results obtained here arelimited in the best case to the third order of perturbation theory [7], whereas in thestatic ϕ theory the six-loop result [8, 9, 10] is currently reached, and for the anomalousdimension of the field – seven-loop [11]. In problems of critical dynamics a noticeablelag occurs also in numerical calculations, in which the Sector Decomposition techniqueof the calculation of Feynman diagrams [12] proved to be very effective in critical staticsproblems (5 loops and partially 6 loops in the theory ϕ [10, 13]), while in the dynamicproblems this method has so far been used only in the two-loop approximation [14].The calculation of multi-loop diagrams in critical dynamics, taking into accounttheir complexity, requires considerable time, thus the problem of reducing the number ofcalculated diagrams arises. In this paper, a numerical calculation of the renormalizationgroup functions of the model A [1, 2] is performed in the fourth order of perturbationtheory. We present a method that allows to reduce significantly the number of Feynmandiagrams to be calculated (“reduction” of diagrams) by appropriate grouping of theoriginal diagrams of the theory. The rules for constructing these diagrams and theirintegrand in the Feynman representation directly from the graph are formulated.The subsequent numerical four-loop calculation was carried out using the SectorDecomposition method.The paper is organized as follows: in section 2 we recall renormalization procedurefor the model A with the use of dimensional regularization ( d = 4 − ε ) andminimal subtraction scheme (MS). In the next section we present the diagrammaticrepresentation for model A. In section 4 we describe the method of diagram reduction.In subsequent section we present the four loop results for the dynamical critical exponent z , which are followed by conclusion. In Appendix A we present one more example ofthe diagram reduction and in Appendix B we discuss the Feynman representation fordiagrams in models of critical dynamics. iagram Reduction in Critical Dynamics
2. Renormalization of the model
Non-renormalized action of model A of critical dynamics in the space with the dimension d = 4 − ε is determined by the set φ of two non-renormalized fields φ ≡ { ψ , ψ (cid:48) } andhas the form [3]: S ( φ ) = λ ψ (cid:48) ψ (cid:48) + ψ (cid:48) [ − ∂ t ψ + λ δS st /δψ ] == λ ψ (cid:48) ψ (cid:48) + ψ (cid:48) [ − ∂ t ψ + λ ( ∂ ψ − τ ψ − g ψ )] (1)with non-renormalized static action S st ( φ ) = − ( ∂ψ ) / − τ ψ / − g ψ . (2)Renormalized action S R obtained by multiplicative renormalization of parameters andfields can be represented as the sum S R = S B + ∆ S of basic action S B and counterterms∆ S [3]: S B = λψ (cid:48) ψ (cid:48) + ψ (cid:48) [ − ∂ t ψ + λ ( ∂ ψ − τ ψ − µ ε gψ )] , (3) S R = Z λψ (cid:48) ψ (cid:48) + ψ (cid:48) [ − Z ∂ t ψ + λ ( Z ∂ ψ − Z τ ψ − Z µ ε gψ )] , (4)where λ = λZ λ , τ = τ Z τ , g = gµ ε Z g , ψ = ψZ ψ , ψ (cid:48) = ψ (cid:48) Z ψ (cid:48) , (5) Z = Z λ Z ψ (cid:48) , Z = Z ψ (cid:48) Z ψ , Z = Z ψ (cid:48) Z λ Z ψ , (6) Z = Z ψ (cid:48) Z λ Z τ Z ψ , Z = Z ψ (cid:48) Z λ Z g Z ψ . It follows from the multiplicative renormalizability of the models (1), (2), that therenormalization constants Z ψ , Z τ , Z g in this model coincide with the static ones (i.e. ofthe model (2)) Z ψ = ( Z ψ ) st , Z τ = ( Z τ ) st , , Z g = ( Z g ) st (7)and the relation Z ψ (cid:48) Z λ = Z ψ fulfilled [3]. This means that the renormalization constants Z , Z , Z are purely static and Z = Z . (8)The only new renormalization constant is Z λ = Z − Z ψ = Z − Z ψ . (9)For our purposes, it is convenient to calculate it through the renormalization constant Z , which is determined from the diagrams of the 1-irreducible function Γ ψ (cid:48) ψ (cid:48) = (cid:104) ψ (cid:48) ψ (cid:48) (cid:105) − irr / (2 λ ) on the zero external frequency ω and the momentum p . For theexpansion of this function into a perturbation theory series, we will use the couplingconstant u = S d (2 π ) d g , where S d = π d/ Γ( d/ is the area of the d -dimensional unit sphere. Thisexpansion has the formΓ Rψ (cid:48) ψ (cid:48) | ω =0 ,p =0 = Z (1 + u Z g ( µ /τ ) ε Z − ετ A (2) + u Z g ( µ /τ ) ε/ Z − ε/ τ A (3) + ... ) . (10) iagram Reduction in Critical Dynamics ε . The renormalization constants Z g and Z τ in (10) are known from the statics, while Z at the 4-loop approximation in theMS scheme has the following form Z = 1 + z ε u + (cid:16) z ε + z ε (cid:17) u + (cid:16) z ε + z ε + z ε (cid:17) u + O ( u ) , (11)coefficients z nk can be found from the condition of the absence of poles in ε in thefunction Γ Rψ (cid:48) ψ (cid:48) | ω =0 ,p =0 , thus the main technical problem is to calculate the coefficients A (2) , A (3) , A (4) in (10).
3. Diagrammatic representation after integration over internal timevariables
Propagators of the model (3) in the time-momentum ( t, k ) representation have the form: t t = (cid:104) ψ ( t ) ψ ( t ) (cid:105) = 1 E k exp − λE k | t − t | , t t = (cid:104) ψ ( t ) ψ (cid:48) ( t ) (cid:105) = θ ( t − t ) exp − λE k ( t − t ) , t t = (cid:104) ψ (cid:48) ( t ) ψ (cid:48) ( t ) (cid:105) = 0 , where E k ≡ k + τ . (12)The simple exponential dependence of these propagators on time makes it easy tointegrate diagrams in ( t, k ) representation over internal time variables and reducethe problem to integration over momenta (momentum representation). The result ofintegration over time can be expressed in a diagram language using the technique of“time versions” (see, for example, [3]). Let us remind this technique with the followingdiagram, considered at zero external frequency ω : t t t (cid:12)(cid:12)(cid:12) ω =0 . (13)Taking into account the θ –function in the propagator (cid:104) ψψ (cid:48) (cid:105) (12), the domain ofintegration in (13) can be represented as 3 contributions (time versions)( t > t > t ) + ( t > t > t ) + ( t > t > t ) . (14)Explicit integration over internal times for each time version (14) can be representedwith new diagrammatic technique: (cid:12)(cid:12)(cid:12) ω =0 = + + . (15) iagram Reduction in Critical Dynamics /E k i with each solid line ( (cid:104) ψψ (cid:105) ),factor 1 with each solid line with dash ( (cid:104) ψψ (cid:48) (cid:105) ) and factor 1 / (cid:80) i E k i with dotted line,the last sum is going over all “energies” (12) of the diagram lines, which are crossed bythe dotted line, where k i are the momenta of the corresponding lines.Than integrand in the momentum representation for the first diagram on the right-hand side (15) has the form:
123 45 ∼ E E E E · E + E + E ) · E + E + E ) . (16)Here and in the following, we denote E i ≡ E k i . The integrands of the remaining timeversions, shown in the figure (15), are constructed in a similar way.
4. Reduction of diagrams
A complicating circumstance in the problems of critical dynamics, in comparison withthe static case, is a significantly larger number of momentum integrals arising as aresult of integration over time (and a more complicated form of them). In a numberof papers [15, 16, 17, 18] the fact that, turning to certain sums of diagrams, one canappreciably simplify the integrands was used. We propose a systematic procedure forsuch a reduction of diagrams, that makes it possible to automate the calculations, whichis necessary for calculations in the higher orders of perturbation theory.The possibility of such a reduction is actually seen from the relation (7).The equality (7) of static and dynamic counterterms is a consequence of a moregeneral statement about the coincidence of 1-irreducible static functions (cid:104) ψψ (cid:105) − irr | st , (cid:104) ψψψψ (cid:105) − irr | st and dynamic functions (cid:104) ψ (cid:48) ψ (cid:105) − irr , (cid:104) ψ (cid:48) ψψψ (cid:105) − irr at zero frequency: (cid:104) ψ (cid:48) ψ (cid:105) − irr | ω =0 = (cid:104) ψψ (cid:105) − irr | st , (cid:104) ψ (cid:48) ψψψ (cid:105) − irr | ω =0 = (cid:104) ψψψψ (cid:105) − irr | st . (17)In the diagram language the equalities (17) mean that for these functions the sum of thedynamic diagrams is reduced to a simpler object – to the sum of the static diagrams.We will consider examples of the technical implementation of such a procedure, andthen apply similar techniques to simplify the function of interest to us (cid:104) ψ (cid:48) ψ (cid:48) (cid:105) − irr | ω =0 .Let us prove the equality:12 | ω =0 = st , (18)where and are symmetry coefficients of diagrams. Performing integration over time,we can write the left-hand side as12 | ω =0 = 12 · ∼
12 1 E E · E + E + E ) (19) iagram Reduction in Critical Dynamics E E · E + E + E ) → (20) → (cid:18) E E + 1 E E + 1 E E (cid:19) · E + E + E ) = 16 1 E E E , which coincides with the integrand of the right hand side of (18).In the diagram language, the symmetrization procedure can be written in the form:12 = 16 + + . (21)An analogous symmetrization for arbitrary diagrams can be written in the form of asymbolic equation: (22)Obviously, the following two equations are also valid (23)(24)Using these equalities, we consider a more complicated example of the sum of threediagrams J = t t t | ω =0 + 12 t t t | ω =0 + 12 t t t | ω =0 (25)Calculating this sum using time versions and performing symmetrization, we obtain: J = 12 (cid:32) (cid:33) + 12 (26)The first diagram in (25) has one time version and is divided into half the sum of the firsttwo diagrams in (26), the second diagram in (25) has two time versions correspondingto diagrams 3 and 4 in (26), and the last diagram has one time version corresponding iagram Reduction in Critical Dynamics J = 12 + 14 = 14 (27)To simplify the first brackets in (26) we used the equality (22), for the second one – (24)and for the last transition – (22). As a result, the sum of the dynamic diagrams (25)has been reduced to a single static one. This example shows how this technique reducesthe number of diagrams.As for diagrams of the one-irreducible function Γ ψ (cid:48) ψ (cid:48) , a complete reduction to staticdiagrams is not possible. However, the use of the relations (22)-(24) allows one even inthis case to reduce significantly the number of contributions and to simplify their form.Let us consider the sum of the last two diagrams in (15), which we rewrite as: J = + (28)Taking into account that the values of the diagrams do not depend on the numberingof the vertices, making symmetrization and using the relation (24), we obtain: J = 12 + + + = 12 (29)So one can see that even in this more complicated case reduction is possible.Now let us formulate a general recipe for the reduction of diagrams, illustrating itwith the example of the sum of the following two diagrams, containing in aggregate tentime versions: (30)The result of the reduction is the sum of the diagrams constructed according to thefollowing recipe. • Step I . Draw the diagrams of the static theory so that vertices with external legsare extreme left and right, while other (internal) vertices ordered in all possibleways so that nearest vertices are connected to each other.Possible order of vertices: (31)Forbidden one: (32) iagram Reduction in Critical Dynamics • Step II . On basis of the diagrams from step I, draw a set of diagrams with dashedlines (from one to (number of vertices -1) sections) starting from the left vertex:(33) • Step III . In all the lines coming from the left to the vertex located between the twosections, we arrange the strokes in all possible ways. If there are 2 similar lines, onwhich it is possible to arrange strokes, then we put only one, the remaining variantis taken into account by the symmetry coefficient. (34)Thus, the original sum of ten time versions has been reduced to the sum of three effectivediagrams (34). (See another example in the Appendix A)
5. Results
The result of the reduction of the diagrams of Γ ψ (cid:48) ψ (cid:48) up to the four-loop approximationis depicted on Figs. 1-2. The diagrams were calculated in the Feynman representation (a) (b) Figure 1: Diagrams of Γ ψ (cid:48) ψ (cid:48) after reduction: (a) two loops and (b) three loopsusing the Sector Decomposition method [12]. The required number of terms of their ε -expansion is given in the table 1, which also shows the corresponding symmetric factors S and additional weight factors f ( n ), which allow one to turn from the results for theone-component field with n = 1 to the results for an n -component O ( n )-symmetricmodel. The coefficients A ( m ) in the expansion (10) are determined from the data of thetable by the relation A ( m ) = (cid:88) j S ( m ) j f ( m ) j ( n ) D ( m ) j ( ε ) . (35)Here k = n + 89 , k = n + 6 n + 2027 , k = n + 23 , k = 5 n + 2227 . (36) iagram Reduction in Critical Dynamics A1B1C1D1D5 A2B2C2D2D6 A3B3C3D3D7 C4D4
Figure 2: Four loop diagrams of Γ ψ (cid:48) ψ (cid:48) after reduction grouped by graph topologyRenormalization constant Z g in (10) is known from the statics and with the requiredaccuracy is equal to Z g = 1 + u n ε + u (cid:18) (8 + n ) ε −
14 + 3 n ε (cid:19) + O ( u ) . (37)Regarding the value of Z τ in (10), we need to make the following remark. The useof the static renormalization constants Z g and Z τ in (10) implies that in calculatingof the function Γ ψ (cid:48) ψ (cid:48) one takes into account all diagrams, including ones that containtadpole subgraphs. It is known that such diagrams can be ignored (which we did),if we do not take into account the tadpoles in counterterms as well. So, if tadpolesare not taken into account while calculating Γ ψ (cid:48) ψ (cid:48) , than to be consistent we shouldremove corresponding contributions from the renormalization constant Z τ . The resultingrenormalization constant ˜ Z τ with the necessary accuracy is given by the expression˜ Z τ = 1 − u (cid:18) n ε + 5(2 + n )144 ε (cid:19) + O ( u ) . (38)Note, that the problem of tadpoles is absent if the calculations of the renormalizationconstants are carried out in the “massless” theory with τ = 0, in which the tadpoles are iagram Reduction in Critical Dynamics № S ( m ) D ( m ) f ( m ) ( n ) m = 2 (2 loop) ε − ε ε / . − . . k m = 3 (3 loop) ε − ε − ε / . − . . k k / . − . . m = 4 (4 loop) ε − ε − ε − A1 1 / . − . . k k A2 1 / . − . . / . − . . / − . . k B2 1 /
12 0 . − . / − . . / . − . . k k C2 1 / . / . − . . − . . / . − . . k k D2 1 / . − . . / . / . − . . / . − . . / . − . / . − . ε -expansion of diagrams from Figs. 1-2defined by zeros. In this theory, instead of (10), the value Γ ψ (cid:48) ψ (cid:48) | ω =0 ,τ =0 is calculated forwhich the factor ( µ / ( τ Z τ )) ε/ on the right side of (10) is replaced by ( µ/p ) ε . However,with this approach, the integrands in the Feynman representation are slightly morecomplicated.Substituting the expressions (11), (37), (38) in (10), and calculating values of A ( m ) with (35), we can find the coefficients z nm from the requirement of the cancellationof the pole contributions in ε . According to renormalization theory, the coefficients z nm at the highest poles in ε ( m >
1) are expressed in a certain way in terms of thecoefficients z n of the first poles, which guarantees a cancellation of pole contributionsin (10). This fact can be used as an additional self-consistency check for the multi-looprenormalization group calculations. iagram Reduction in Critical Dynamics Z is associated with the RG-function γ γ ( u ) = β ( u ) ∂ u log Z . (39)The expression for the β -function is currently known with six-loop accuracy [9, 10].We do not need its explicit form, since the connection mentioned above between thecoefficients at the higher poles with the coefficient at the first pole makes it possible torepresent γ ( u ) in a simpler form γ ( u ) = − u∂ u ( z u + z u + z u + ... ) . (40)The dynamic critical exponent z is expressed in terms of the value γ ∗ ≡ γ ( u ∗ ) ofthe function γ ( u ) at the fixed point u ∗ and the Fisher exponent η by the relation [3] z = 2 + γ ∗ − η . (41)Substituting the values of z n into (40) and normalizing the result to the value of thetwo-loop contribution, we obtain: γ ∗ = k h u ∗ (cid:2) b k u ∗ + ( b k + b k + b k ) u ∗ (cid:3) + O ( u ∗ ) , (42)where h = 6 ln(4 / (cid:39) . , (43) b = − . , b = − . , (44) b = − . , b = 1 . . (45)The value u ∗ , determined by the condition β ( u ∗ ) = 0, with the required accuracy isgiven by the expression u ∗ = 6 n + 8 ε + 18(3 n + 14)( n + 8) ε + + 34( n + 8) (cid:16) − n + 110 n + 1760 n ++ 4544 − n + 8)(5 n + 22) ζ (3) (cid:17) ε + O ( ε ) . (46)The results of the dynamic exponent are usually presented in the form z = 2 + Rη . (47)The value of η can be written in a form similar to (40), with the required accuracy η = k u ∗ (cid:2) a k u ∗ + ( a k + a k + a k ) u ∗ (cid:3) + O ( u ∗ ) , (48)where a = − , a = − , a = − , a = 4532 . (49)From (47) and (41) taking into account (40), (48), (46) we get R = (6 ln (4 / − (cid:20) c ε + (cid:18) c + ( c + c n )( n + 8) (cid:19) ε + O ( ε ) (cid:21) , (50) iagram Reduction in Critical Dynamics c i are determined by the relations c = 23 hh − b − a ) ,c = 4 h h − (cid:18) a ( a − b ) + ( b − a ) (cid:19) , (51) c = 4 h h −
1) (21( b − a ) − b − a ) + 18( b − a ) + 22( b − a )) ,c = 2 h h −
1) (9( b − a ) − b − a ) + 18( b − a ) + 10( b − a )) . The first two terms of the ε -expansion (50) do not depend on the number of componentsof the field n . The first of them was calculated in the work [19], the second – in thework [7], where the expression for b was obtained b = π / − F (1 / / −
34 + 138 ln 4 −
218 ln 3 (cid:39) − . , (52) F ( x ) = (cid:90) x ln tt − dt , (53)which, according to (49), (51) and (52) corresponds to c ∼ − . . (54)In the work [19] the value R was calculated in the leading order of the 1 /n expansionfor an arbitrary dimension d : R ∞ = 44 − d (cid:32) d Γ ( d/ − / Γ( d − (cid:82) / dx [ x (2 − x )] d/ − − (cid:33) . (55)The first terms in the expansion of this quantity with respect to ε = 4 − d have the form R ∞ = (6 ln (4 / − (cid:0) − . ε − . ε + O ( ε ) (cid:1) . (56)Taking into account that the first two terms of the ε -expansion (50) do not depend on n , they coincide with the corresponding contributions to (56), which is confirmed by theresults of [19], [7]. The expansion (56) also determines the coefficient c in the quadraticby ε contribution in (50): c = − . . (57)The values of the coefficients c i in (50) obtained in this paper are c = − . , c = − . ,c = 21 . , c = 4 . . (58)Results obtained are in full agreement with three loop calculations [7], as well aswith 1 /n -expansion [19], as for the four loop contribution, the coefficients c i were firstcalculated in [20] by a different method with much less accuracy: c = − . , c = − . , c = 21 . , c = 4 . , (59) iagram Reduction in Critical Dynamics ε -expansion directly for the dynamic index z for n = 1: z = 2 + 0 . ε + 0 . ε − . ε + O ( ε ) . (60)
6. Conclusion
In this paper we performed four loop calculation of the critical exponent z in theframework of ε -expansion and renormalization group. To perform this calculation wedeveloped a method of reduction of the diagrams in the models of critical dynamicswhich allows to significantly reduce a number of diagrams to be calculated. This methodcombined with Feynman representation and Sector Decomposition technique [12] allowsus to obtain high precision numbers for four-loop contribution to dynamic exponent.The necessity of high loop calculations for model A was pointed out in [21] whereBorel resummation of the results of the work [20] was performed. It was shown thatresults of resummation are very sensitive to particular realizations of the summation,which must be a consequence of the insufficient number of terms of the ε -expansion.As it was noted, the model A is the simplest model of critical dynamics, for morecomplicated models renormalization group calculations are limited at maximum by twoloop order. The lack of perturbative information in this models does not allow to makesolid theoretical predictions, moreover in some models (e.g. model E) it is not possibleto confidently distinguish concurrent asymptotic regimes.The method discussed in this paper allows to significantly reduce the totalcalculation time for such problems and opens the possibility to extend this calculationsto higher loops and more complicated models. For example, in model A at 5 loop levelnumber of diagrams is reduced from 1025 to 201 and with more simple integrands,which gives us a possibility to reach high accuracy of numerical calculations. While formore complicated theories like model E of critical dynamics [1, 14, 16, 22] (where RGcalculations are limited only by 2 loop order) our preliminary estimations show that thisfactor may be even greater than 5 and this gives us a hope that 3 and 4 loop calculationsin this models can be feasible. Acknowledgements iagram Reduction in Critical Dynamics Appendix A. Example of diagram reduction
As a second example of the diagram reduction, let us consider the sum of 28 timeversions of the diagrams of the Γ ψ (cid:48) ψ (cid:48) of the following type: (A.1) • Step I . (A.2) • Step II . (A.3) • Step III . (A.4)As a result, 28 versions of the diagrams (A.1) were reduced to 8 diagrams (A.4).
Appendix B. Feynman presentation
The dependence on the integration momenta in the diagrams after the integration overtime has a structure that makes it possible to turn to the Feynman representation. Thisallows us to use the Sector Decomposition method [12], as in problems of critical statics[13]. As an example we will consider the second diagram from (34):
21 7 6543 (B.1)The numbers on the lines denote the integration momenta flowing from left to right.The integral that corresponds to diagram (B.1) looks as follows: J = (cid:90) d k ... (cid:90) d k δ ( k + k + k ) δ ( k + k + k ) δ ( k + k − k − k ) E k E k E k E k E k E k ( E k + E k + E k ) ( E k + E k + E k ) . (B.2) iagram Reduction in Critical Dynamics v i and using the Feynman formula, we obtain: J = (cid:90) (cid:89) dv i δ (cid:16)(cid:88) v i − (cid:17) F ( { v } ) , (B.3)where F ( { v } ) = (cid:90) d k ... (cid:90) d k δ ( k + k + k ) δ ( k + k + k ) δ ( k + k − k − k ) Q α , (B.4) Q = v E k + v E k + v E k + v E k + v E k + v E k ++ v ( E k + E k + E k ) + v ( E k + E k + E k ) , (B.5) α = 8 – the number of factors in the denominator of (B.2). Writing Q in the form Q = u E k + u E k + u E k + u E k + u E k + u E k + u E k , (B.6)where u = v + v , u = v , u = v + v , u = v + v ,u = v , u = v , u = v + v + v , (B.7)from (B.4), (B.5) we obtain F ( { v } ) = (cid:90) d k ...d k δ ( k + k + k ) δ ( k + k + k ) δ ( k + k − k − k )( (cid:80) j =1 u j E k j ) α . (B.8)Choosing in (B.8) as independent variables a certain set { k i , k i , k i , k i } , andperforming the integration with the help of δ -functions, we arrive at an expressionof the form F ( { v } ) = (cid:90) d k i (cid:90) d k i (cid:90) d k i (cid:90) d k i C + V i j ,i l k i j k i l ) α , C ≡ τ (cid:88) j =1 u j . (B.9)Calculating the integral of the power of the quadratic form, we obtain F ( { v } ) = π dL/ C d/ − α Γ( α − dL/ α ) (det V ) − d/ , (B.10)where L is the number of loops in the diagram, in the case under consideration L = 4.The value of the determinant det V in (B.10) does not depend on the choice ofthe variables of integration { k i , k i , k i , k i } and can be determined directly from thediagram view. By construction, det V is the sum of products of four factors u i . Forany set of independent variables of integration { k i , k i , k i , k i } the diagonal elementsof the matrix V are equal to u i , u i , u i , u i , their product contributes to det V withcoefficient one. The nondiagonal elements of matrix V do not contain the parameters u i , u i , u i , u i , consequently, det V does not contain the highest powers of u i . Obviously,we can not choose as independent variables some sets of { k i , k i , k i , k i } which formconservation laws. As a result such a products of u i will not appear in det V . iagram Reduction in Critical Dynamics { k , k , k } , { k , k , k } and { k , k , k , k } , which are defined by the δ -function arguments in (B.2), “conservationlaws” are also formed by sets { k , k , k } , { k , k , k , k } and { k , k , k , k } . Thus, outof 35 possible quadruples of products of the parameters u i
15 products do not contributeto det V , and det V for the diagram (B.1) is given by the expressiondet V = u u u u + u u u u + u u u u + u u u u + u u u u ++ u u u u + u u u u + u u u u + u u u u + u u u u ++ u u u u + u u u u + u u u u + u u u u + u u u u ++ u u u u + u u u u + u u u u + u u u u + u u u u (B.11)in which u i must be expressed in terms of v i according to (B.7). For the first and thirddiagrams of the formula (34), the expression (B.11) is preserved, only the connections(B.7) of the variables u i and v i will change accordingly. This can be easily found by theform of the diagram. References [1] Hohenberg P.C. ,Halperin B.I. 1977, “Theory of dynamic critical phenomena” Rev. Mod. Phys. “Critical dynamics: a field-theoretical approach” , J. Phys. A: Math.Gen. R207–R313[3] Vasil’ev A.N. 2004, “The Field Theoretic Renormalization Group in Critical Behavior Theory andStochastic Dynamics, Chapman and Hall” , Chapman& Hall/CRC Boca Raton[4] Marinelli M., Mercuri F., Belanger D.P. 1995, “ Specific heat, thermal diffusivity, and thermalconductivity of
F eF at the N´eel temperature” , Phys. Rev. B “Critical SlowingDown near the Multiferroic Phase Transition in M nW O ” , Phys. Rev. Lett. “Ordering fluctuation dynamics in