Equivalence of Wilson actions

Transcription

1 Prog. Theor. Ex. Phys. 05, 03B0 7 ages DOI: 0.093/te/tv30 Equivalence of Wilson actions Physics Deartment, Kobe University, Kobe , Jaan hsonoda@kobe-u.ac.j Received June 6, 05; Revised August 5, 05; Acceted August 3, 05; Published October, We introduce the concet of equivalence among Wilson actions. Alying the concet to a real scalar theory on a Euclidean sace, we derive the exact renormalization grou transformation of K. G. Wilson, and give a simle roof of universality of the critical exonents at any fixed oint of the exact renormalization grou transformation. We also show how to reduce the original formalism of Wilson to the simlified formalism by J. Polchinski.... ubject Index B30, B3, B39. Introduction The urose of this aer is to introduce the concet of equivalence among Wilson actions. We consider a generic real scalar theory in D-dimensional Euclidean sace, and denote the Fourier comonent of the scalar field with momentum by φ. A Wilson action [φ is a real functional of φ. A momentum cutoff is incororated so that the exonentiated action e [φ can be integrated with no ultraviolet divergences [. An examle is given by [φ K φφ + I [φ, where the cutoff function K is a ositive function of that is at 0, and decreases toward 0 raidly for. The first term of the action can suress the modes with momenta higher than sufficiently that e [φ can be integrated over φ of all momenta. The second term consists of local interaction terms. In the continuum aroach adoted here, correlation functions are defined for φ of all momenta, even those above. A Wilson action is meant to describe low momentum energy hysics accurately, but not hysics at or above the cutoff scale. If two Wilson actions describe the same low energy hysics, we regard them as equivalent. It is the urose of this aer to rovide a concrete definition of equivalence using the continuum aroach. The aer is organized as follows. In ect. we introduce modified correlation functions, and then define equivalence of Wilson actions as the equality of the modified correlation functions. Basically, two Wilson actions are equivalent if their differences can be removed if we give them a massage at their resective cutoff scales. We derive two versions of exlicit formulas that relate two equivalent Wilson actions. The concet of equivalence is alied in the rest of the aer. In ect. 3 we derive the exact renormalization grou ERG transformation of Wilson [ by considering a articular tye of equivalence. We derive the ERG differential equation from our equivalence, which amounts to an integral solution to the differential equation. In ect. 4 we discuss The Authors 05. Published by Oxford University Press on behalf of the Physical ociety of Jaan. This is an Oen Access article distributed under the terms of the Creative Commons Attribution License htt://creativecommons.org/licenses/by/4.0/, which ermits unrestricted reuse, distribution, and reroduction in any medium, rovided the original work is roerly cited. Funded by COAP 3

2 PTEP 05, 03B0 the relation between the original formulation of ERG transformation by Wilson and the formulation by J. Polchinski [ which is more convenient for erturbation theory. In the revious literature only assing remarks have been given on this relation [3,4. Our short discussion of their relation is comlete and hoefully illuminating. ection 5 reares us for the discussion of universality in ect. 6. We generalize the definition of equivalence so that the exact renormalization grou transformation can have fixed oints. In ect. 6, we assume a fixed oint of the ERG transformation, and show that the critical exonents defined at the fixed oint are indeendent of the choice of cutoff functions. This is what we mean by universality. We conclude the aer with ect. 7. Throughout the aer we work in D-dimensional Euclidean momentum sace. We use the following abbreviated notation. Equivalence d D π D, δ πd δ D. Given a Wilson action [φ, we denote the correlation functions by φ φ n [dφ φ φ n e [φ. 3 We consider modifying the correlation functions for high momenta without touching them for small momenta. We define modified correlation functions by φ φ n K,k n i K i k ex δφδφ φ φ n 4 where K and k are non-negative functions of. We will call them cutoff functions. As 0, we must find K, k 0 5 so that the correlation functions are not modified at small momenta. In addition we constrain K by K 0. 6 In other words K is small for larger than the squared cutoff momentum of the Wilson action. The fluctuations of φ with larger than the cutoff are suressed, and we enhance their correlations by the large factor /K in 4. The exonential on the right-hand side of 4 amounts to mixing a free scalar with the roagator k/ to the original scalar field φ. incek0 0, the free scalar has no dynamics of its own. For examle, we may take K e 7 and k or e e where is the momentum cutoff of the Wilson action., 8 /7

3 PTEP 05, 03B0 In articular, for n and n 4,4 gives φ φ K,k K φ φ k δ +, 9 [ 4 φ φ 4 K,k k 3 φ φ 4 K i φ φ 3 δ i φ 3 φ 4 k k 3 3 δ t-, u-channels δ + + k δ +. 0 For small momenta, the modified correlation functions 4 reduce to the ordinary correlation functions 3. Now, we would like to introduce the concet of equivalence among Wilson actions. Let us regard two Wilson actions, as equivalent if, with an aroriate choice of K, and k,, their modified correlation functions become identical for any n and momenta: φ φ n K,k φ φ n K,k. ince the functions K,, k, kee the low energy hysics intact, and describe the same low energy hysics. In the following we solve to obtain an exlicit relation between the two actions. We first rewrite as k ex n i K i K i δφδφ ex k φ φ n δφδφ φ φ n ince functional integration by arts gives k ex φ φ n δφδφ [dφe [φ k ex δφδφ k [dφφ φ n ex δφδφ φ φ n. e [φ, 3 we obtain n [dφ φ i i n [dφ i k ex K i K i φ i δφδφ k ex e [φ δφδφ e [φ. 4 3/7

4 PTEP 05, 03B0 This imlies that k [ ex e [φ k ex δφδφ δφδφ e [φ subst 5 where the suffix subst denotes the substitution of K φ 6 K for φ on the right-hand side. Hence, we obtain an intermediate result e [φ k [ k ex ex δφδφ e [φ. δφδφ subst 7 We can rewrite this in two ways. First, noting that under the substitution 6, we must substitute K δ K δφ 8 for δ δφ, we obtain the first relation [ e [φ ex { k k } K K δφδφ [ K ex φ. 9 K Note that the two actions are the same for φ with small, since the function of in the curly bracket above is negligible for small. In this sense the two actions differ only by local terms. Alternatively, we rewrite 7 as e [φ [ ex Using the Gaussian formula ex A δφδφ [dφ ex { K δ k k } K δφδφ ex [φ e [φ subst. 0 φ φ φ φ + [ φ A roven in Aendix A, we obtain the second relation [dφ e [φ ex k K K k φ K K φ φ K K φ [ + φ. 4/7

5 PTEP 05, 03B0 We have thus obtained two exlicit formulas 9, relating two equivalent Wilson actions. The remaining sections give alications of these formulas. In Aendix B we give corresonding results for a Dirac fermion field. 3. Exact renormalization grou Let us aly the results of the revious section to derive the exact renormalization grou transformationofk.g.wilsonect.of[ We choose K K, k k 3 K K e t, k k e t so that the two sets of cutoff functions differ only by the choice of a momentum cutoff. We demand that the modified correlation functions 4 be indeendent of the momentum cutoff: φ φ n K,k φ φ n K,k. 4 Now, using the first formula 9, we obtain e [φ ex ex k et K k et K δφδφ [ K K etφ. 5 By denoting as and as e t, and taking t infinitesimal, we obtain the exact renormalization grou ERG differential equation [ e[φ K δ φ δφ + K k k e [φ, 6 δφδφ where we define K. 7 This amounts to.8 of Ref. [. For the articular choice 6 gets simlified to k K K, 8 [ e[φ K δ φ δφ + e [φ. 9 δφδφ This was introduced first by J. Polchinski [. 5/7

6 PTEP 05, 03B0 Alternatively, we can use the second formula which gives [dφ e e t [φ ex k et K k K et φ K K etφ φ K K etφ [ + φ. 30 This is a well known integral solution of the ERG differential equation 6. This is discussed in detail, for examle, in [5. Though mathematically equivalent, our starting oint 4 of this section is easier to understand than the differential equation 6 or its integral solution 30. Earlier in Ref. [6 it was observed that renormalized correlation functions of QED can be constructed out of correlation functions of its Wilson action: the modified correlation functions 4 coincide with renormalized correlation functions. Cutoff indeendent correlation functions have also been discussed by O. J. Rosten [7. 4. Polchinski vs. Wilson As a second alication, we consider K K, k k, K K, k k K K 3. Note that k follows Polchinski s convention 8, which is convenient for erturbative alications. Given a solution of the ERG differential equation 6 with K, k, we wish to construct an equivalent that solves 9 with K, k. As has been shown in the revious section, the modified correlation functions are indeendent of, if the Wilson action satisfies 6. Hence, if and are equivalent at a articular, theygive the same modified correlation functions at any. In the following let us choose, and demand and give the same modified correlation functions. Using 9 and denoting as and as, we obtain [ e [φ ex k k K K [ ex δφδφ [ K K φ. 3 A articularly simle result follows if we choose K satisfying K k k. 33 K This gives K K K K k With this choice, we obtain [ K + k [φ K φ, 35 so that the two equivalent actions are simly related by a linear change of field variables. 6/7

7 PTEP 05, 03B0 For the articular choice made in ect. of [, K e, k, 36 we obtain K e 5. ERG for fixed oints We now aly the results of ect. to show the universality of critical exonents at a fixed oint of the ERG transformation, by which we mean the indeendence of critical exonents on the choice of cutoff functions K, k. For the ERG transformation to have a fixed oint, we must change the transformation given in ect. 3 in two ways [: first by adoting a dimensionless notation, and second by introducing an anomalous dimension to the scalar field. We elaborate more on these oints in Aendix C. After these changes, the Wilson action t deends on t such that φ e t φ n e t K,k e t n D+ +γ φ t+t φ n K,k t 38 for the same cutoff functions K, k indeendent of t. This is the new form of equivalence between t and t+t : their modified correlation functions are the same u to a scale transformation. On the right-hand side, D+ gives the canonical mass dimension of the field φ since this is the Fourier transform, we obtain D D D+,andγ is the anomalous dimension, taken for simlicity as a t-indeendent constant. Let us solve 38 to obtain t+t in terms of t. Following the same line of arguments given in ect., we obtain e t+t [φ k ex δφδφ where subst stands for the substitution of e t D+ γ [ k ex for φ. ince this substitution imlies the substitution of δφδφ e t [φ subst 39 K K e t φet 40 e t D D+ +γ K e t K δ δφe t 4 for δ δφ, we obtain [ e t+t [φ ex k ke t K et γ [ K e t e t [φ δφδφ subst. 4 7/7

8 PTEP 05, 03B0 Taking t infinitesimal, we obtain the ERG differential equation [ t e t [φ K + D + φ δ γ φ + μ μ δφ e t [φ + dk k + K d γ k δφδφ e t [φ. 43 For Polchinski s choice, k K K, 44 this gets simlified to [ t e t [φ K + D + φ δ γ φ + μ μ δφ e t [φ + { γ K K } δφδφ e t [φ, 45 which is given in [5. For Wilson s choice, K e, k, gives [ t e t D [φ φ + φ δ μ μ δφ e t [φ + γ + δ φ δφ + e t [φ, 47 δφδφ which reroduces.7 of [ under the identification dρt dt γ. 48 Now, the anomalous dimension γ is chosen for the existence of a fixed oint action that satisfies [ K + D + φ δ γ φ + μ μ δφ e [φ + dk k + K d γ k δφδφ e [φ At the fixed oint, the correlation functions obey the scaling law: φ e t φ n e t K,k e tn D+ +γ φ φ n K,k. 50 Only for secific choices of γ,49 has an accetable solution. For examle, if we assume to be quadratic in φ, the solution becomes non-local unless γ 0,,,... We then obtain γ ZK φφ, 5 + k γ 8/7

9 PTEP 05, 03B0 which gives φφq K,k Z γδ + q, 5 where Z is an arbitrary ositive constant. This is discussed in the aendix of [. 6. Universality of critical exonents We now discuss universality of critical exonents at an arbitrary fixed oint of the ERG transformation, reviewed in the revious section. Universality within the ERG formalism has been shown in Ref. [8; our discussion below has the merit of conciseness. In Aendix D we derive those results of [8 relevant to the resent aer. deends on K, k, but we know from ect. that for any choice of K, k there is an equivalent action that gives the same modified correlation functions. Equation 9 gives the equivalent action for K, k as [ e [φ ex k K K k δφδφ [ KK ex φ. 53 ince the integrand of the exonent vanishes at 0, and differ by local terms. ince φ φ n K,k φ φ n K,k, 54 the anomalous dimension γ is indeendent of the choice of K, k. Now, the anomalous dimension γ is not the only critical exonent defined at the fixed oint. The other exonents aear as scale dimensions of local comosite oerators [. A comosite oerator O y with momentum is a functional of φ satisfying Oy e t φ e t φ n e t K,k { } e t y+n D+ +γ Oy φ φ n K,k, 55 where the modified correlation functions are defined by Oy φ φ n K,k n i K i k O y ex δφδφ φ φ n, 56 and y is the scale dimension of O y. The scale dimension of O y x eix O y in coordinate sace is D y. For the equivalent fixed oint action with K, k, the corresonding comosite oerator has the same modified correlation functions: Oy φ φ n K,k K O y φ,k φ n. 57 This gives O y as O y e [φ ex k K K k δφδφ [ O y e [φ subst, 58 9/7

10 PTEP 05, 03B0 where subst imlies substitution of K K φ 59 into φ. The scale dimension y is thus indeendent of the choice of K, k. We conclude that all the critical exonents are indeendent of K, k. Before closing this section, we would like to discuss two issues related to the fixed oint action. 6.. Ambiguity of the fixed oint action Given K, k, and an aroriate choice of γ, the fixed oint solution of the ERG differential equation is still not unique. This is because normalization of the scalar field can be arbitrary. Given, we can construct Z satisfying φ φ n K,k Z Z n φ φ n K,k. 60 To obtain Z, we set K ZK and k k in 9. We then get e Z [φ k ex Z e [ Z φ. 6 δφδφ For examle, the Z-deendence of the Gaussian fixed oint γ 0isgivenby G,Z [φ ZK φφ. 6 + k Taking Z + ɛ, whereɛ is infinitesimal, we obtain where N [φ {φ δ δφ + k δ δφ +ɛ [φ [φ ɛ N [φ, 63 δ δφ + } δφδφ is a local comosite oerator satisfying N [φφ φ n K,k n φ φ n K,k. 65 Obviously, N [φ, called an equation-of-motion oerator in [5, has scale dimension Universal fixed oint action? We have shown that the modified correlation functions are universal u to normalization of the scalar field. We now ask if there is a universal Wilson action univ that gives the universal modified correlation functions as its unmodified correlation functions: This imlies 64 φ φ n univ φ φ n K,k. 66 e univ [φ ex k δφδφ e [φ where is the fixed oint action for K, k, and subst denotes substitution of subst, 67 K φ 68 0/7

11 PTEP 05, 03B0 for φ. The above result is obtained from 9 by setting K, k We exect that the right-hand side is indeendent of K, k, i.e., univ has no cutoff. But we know that the use of a cutoff is essential for Wilson actions, and there must be something wrong with univ. Let us first consider the examle of the Gaussian fixed oint given by G [φ K φφ k This gives the modified two-oint function φφq K,k G δ + q. 7 Equation 67 indeed gives an action free from a cutoff: G,univ [φ φφ. 7 For interacting theories, though, we exect 67 makes no sense. Let us look at this a little more closely. As K, k, we choose { K K t K e t, k k t k e t 73. In the limit t +, we obtain 69: We then define t so that lim t + K t, lim k t t + φ φ n K t,k t t φ φ n K,k. 75 t is related to by the ERG transformation of ect. 3. ince the momentum cutoff of is of order we are using the dimensionless convention, that of t is of order et. Hence, univ has an infinite momentum cutoff. We then exect the terms of univ to have divergent coefficients. Thus, there is no fixed oint action univ that gives the correlation functions without modification. 7. Concluding remarks We have introduced the concet of equivalence among Wilson actions. Our equivalence is hysically more transarent than the other formulations of the exact renormalization grou via differential equations or integral formulas. In articular we have alied our equivalence to obtain a simle roof of universality of critical exonents within the ERG formalism. Acknowledgements This work was artially suorted by the JP Grant-in-Aid # Preliminary results in this work were resented at ERG04 held in Lefkada, Greece. I would like to thank the organizers of ERG04 for giving me the oortunity. Funding Oen Access funding: COAP 3. /7

12 PTEP 05, 03B0 Aendix A. Gaussian formula In this aendix we rove the formula [ ex A δφδφ [dφ [ ex ex [[φ A φ φ + [ φ + φ. A Though the following roof requires the ositivity of A, we exect the formula to remain valid as long as both sides make sense. The left-hand side makes sense for any A, and the right-hand side makes sense even if A <0 for some as long as convergence of functional integration is rovided by the Wilson action. It is easy to understand this formula in terms of Feynman grahs. The right-hand side imlies the couling of a scalar field φ whose roagator is A. Contracting the airs of φ, we obtain the lefthand side. More formally, we can rove the equality by comaring the generating functionals of both sides for arbitrary source J. Let us first comute the generating functional of the left-hand side: e WL[J [ [ [dφex J φ ex A δφδφ ex [[φ. A Integrating this by arts, we obtain e WL[J [ [dφ ex[[φ ex [ [dφex [φ + [ ex J φ A δφδφ J φ + J AJ. A3 We next comute the generating functional of the right-hand side: e WR[J [ [dφex [dφ ex J φ [ We first shift φ by φ, and then shift φ by +φ to obtain A φ φ + [ φ + φ. A4 e WR[J [dφ [ dφ [ ex J φ + φ [ ex A φφ + [ φ [ [dφex φφ + J φ A [dφ [ ex J φ + [ φ. A5 /7

13 PTEP 05, 03B0 If A is ositive, we can erform the Gaussian integral over φ to obtain e W R[J [dφ [ ex J AJ + J φ + [ φ. A6 We thus obtain W L [J W R [J A7 for arbitrary J. This roves the Gaussian formula A. Finally, shifting φ by φ, we rewrite A as [ ex A δφδφ [dφ [ ex This is the form used in ect.. ex [[φ φ φ φ φ + [ φ. A8 A Aendix B. Equivalence of fermionic Wilson actions For a Dirac sinor field ψ and its comlex conjugate ψ, we define modified correlation functions by ψ ψ n ψq n ψq K,k ψ ψ n ex n i K i K q i δ k δ δψ / δ ψ ψq n ψq B so that ψ ψq K,k [ ψ ψq K k δ + q. B / Two Wilson actions, are equivalent if K, and k, exist so that ψ ψ n ψq n ψq K,k ψ ψ n ψq n ψq K,k. B3 The formula analogous to isgivenby e [ψ, ψ [dψ d ψ ex ψ K K ψ ψ K K ψ + [ ψ, ψ. / k K K k B4 The formula analogous to 9 is somewhat more comlicated to write down. Denoting A ab / ab k k K K, B5 3/7

14 PTEP 05, 03B0 we obtain [ e [ψ, ψ Tr ex n n0 ex n! [ δ δ δψ A K ex ψ, K ψ B6 δ ψ K K n δ δ A ai b i i δ ψ bn n δ ψ b,, n i [ K K ψ, K K ψ δ δψ a δ δψ an n, B7 where the sinor indices are summed over. The exonential imlies contraction of ψ ψq by Aδ + q. Aendix C. Derivation of Eq. 38 In thisaendixwe rovide more detailsbehindthenewform of equivalence 38. tarting from the original equivalence 4, we obtain 38 in two stes: first by rescaling dimensionful quantities, and second by introducing an anomalous dimension of the scalar field. C.. Rescaling We first rewrite the equivalence 4 by rescaling dimensionful quantities such as momenta and field variables. Note that K and K differ only by a rescaling of momentum K e t K. C Likewise, we have k e t k. We wish to rewrite in such a way that its cutoff functions become K, k. For this urose, we introduce a rescaled field variable C D+ t φ e φ e t C3 so that δ D δ e t δ φ δφ e t. C4 We then define [ φ [φ. C5 In other words, [φ is obtained from [φ by substituting e t D+ φ e t for φ. For examle, given [φ K φφ, C6 we obtain [φ K etd+ φ e t φ e t K e t φφ K φφ. C7 4/7

15 PTEP 05, 03B0 We rewrite the left-hand side of 4 as φ φ n K D+,k nt e φ e t φ n e t K,k D+ n nt e K i k ex δφδφ i φ e t φ n e t. C8 [φ Using C andc4, we obtain k δφδφ e td Hence, using C, we obtain φ φ n K D+,k nt e n i k e t k δ φδ φ. δφ e t δφ e t K i e t k ex δ φδ φ C9 φ e t φ n e t. C0 [φ Using C5 and rewriting integration variables φ as φ, we obtain φ φ n K D+,k nt e φ e t φ n e t K,k. C Thus, by rescaling, has been converted to with the cutoff functions K, k. We can now write 4 as D+ nt e φ e t φ n e t K,k φ φ n K,k. C Relacing t by t, we obtain φ e t φ n e t K,k D+ t n e φ φ n K,k. C3 By writing as t and as t+t, we obtain 38 forγ 0. C.. Anomalous dimension Given a Wilson action [φ, we can construct an action Z [φ whose modified correlation functions differ only by normalization of the field: φ φ n K,k Z Z n φ φ n K,k. C4 To obtain Z, we set K ZK and k k in 9. We then get k ex Z [φ ex Z δφδφ ex We have used the same transformation for the fixed oint action in ect. 6.. [ φ Z. C5 5/7

16 PTEP 05, 03B0 Given, we construct such that φ φ n K,k e nγt φ φ n K,k, C6 where γ is an arbitrary constant. Then, C3 becomes φ e t φ n e t K,k e t n D+ +γ φ φ n K,k, C7 This gives 38, which defines the renormalization grou transformation with an anomalous dimension. Note that we have introduced an anomalous dimension γ by hand. A articular γ must be chosen for the new renormalization grou transformation to have a fixed oint. Aendix D. Relation to the results of Latorre and Morris In [8, Latorre and Morris have shown that the change of a cutoff function can be comensated by a change of field variables. We would like to exlain briefly how their result can be reroduced from the results of the resent aer. The relation between two equivalent actions with K, k and with K, k has been given by 9. Choosing { K K, k k, D K K + δk, k k + δk, where δk and δk are infinitesimal, we obtain from 9 [φ [φ e [φ δ [ θ e [φ, D δφ where δk θ K φ + δ δk kδk K δφ. D3 In deriving D, we have taken only the terms of first order in δk or δk, and we have ignored a field indeendent constant. The relation D gives the change of the action under an infinitesimal change of φ by θ. Uon the choice of the Polchinski convention k K K, D3 reduces to δk θ δ δφ + K φ, D4 which reroduces 3.5 of [8. In addition, Latorre and Morris have shown that the ERG transformation is also a change of variables. Our ERG differential equation 43 can be rewritten as where D + t t e t [φ δ δφ [ t e t [φ, γ + φ φ + μ K μ + dk k + K d γ k δt δφ. D5 D6 6/7

17 PTEP 05, 03B0 Thus, t t is the change of the action by an infinitesimal change of φ by t. Uon the choice k K K, the above reduces to D + t γ + φ φ + μ K μ + γ K K δt δφ, which reroduces.3 of [8 ifγ 0. References [ K. G. Wilson and J. B. Kogut, Phys. Re., [ J. Polchinski, Nucl. Phys. B 3, [3 T. R. Morris, Phys. Lett. B 39, [4 M. D Attanasio and T. R. Morris, Phys. Lett. B 409, [5 Y. Igarashi, K. Itoh, and, Prog. Theor. Phys. ul. 8, 00. [6, J. Phys. A 40, [7 O. J. Rosten, Phys. Re. 5, [8 J. I. Latorre and T. R. Morris, J. High Energy Phys. 00, D7 7/7