Instability induced renormalization

7 January 1999

Physics Letters B 445 Ž1999. 351–356

Instability induced renormalization Jean Alexandre a

a,1

, Vincenzo Branchina

a,2

, Janos Polonyi

a,b,3

Laboratory of Theoretical Physics, Louis Pasteur UniÕersity, 3 rue de l’UniÕersite´ 67087 Strasbourg, Cedex, France b Department of Atomic Physics, L. EotÕos ¨ ¨ UniÕersity, Puskin u. 5-7 1088 Budapest, Hungary Received 21 September 1998; revised 1 November 1998 Editor: L. Alvarez-Gaume´

Abstract It is pointed out that models with condensates have nontrivial renormalization group flow on the tree level. The infinitesimal form of the tree level renormalization group equation is obtained and solved numerically for the f 4 model in the symmetry broken phase. We find an attractive infrared fixed point that eliminates the metastable region and reproduces the Maxwell construction. q 1999 Elsevier Science B.V. All rights reserved.

We have two systematic nonperturbative methods to handle multi-particle or quantum systems, the saddle point approximation and the renormalization group. Our goal in this letter is to combine these two apparently independent approximation methods in order to obtain a better understanding of the instabilities and first order phase transitions. We start with the path integral, 1

Z s w Df x ey " S w f x ,

H

Ž 1.

over the configurations f Ž x .. We assume the presence of UV and IR cutoffs thus the dimension of the domain of integration is large but finite.

1

E-mail: [email protected] E-mail: [email protected] 3 E-mail: [email protected] 2

The saddle point approximation to Ž1. coincides with its perturbative expansion when the saddle points are trivial, c s 0. In this case the elementary excitations, the quasiparticles, are characterized by the eigenfunctions fnŽ x . of the inverse propagator, Gy1 Ž x, y . s d 2 Sw f xrdf Ž x . df Ž y .< fs c . The perturbation expansion is applicable for fixed values of the cutoffs if the restoring force of the fluctuations to f Ž x . s c Ž x . s 0 is nonvanishing, i.e. the eigenvalues of the inverse propagator, l n , are positive. When the absolute minimum of the action is reached at f Ž x . s c Ž x . / 0 then we follow the strategy of the saddle point expansion and the elementary excitations are the fluctuations around c Ž x . / 0. The saddle point expansion is applicable as long as l n ) 0. The zero modes, the elementary excitations with l n s 0, correspond to continuous symmetries and are integrated over exactly. If the inverse propagator has too many small eigenvalues the saddle point expansion breaks down and strong fluctuations develop

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 4 9 1 - 9

J. Alexandre et al.r Physics Letters B 445 (1999) 351–356

352

around c as we remove one of the cutoffs. This possibility brings us to the renormalization group method which is supposed to deal with such problems w1x. The basic idea of the renormalization group is the subsequent integration in Ž1.,

ton equation w2x is obtained by eliminating the plane waves within the shell k y D k - < p < - k of the momentum space,

E Sk w f x Ek

" s

1

Dk 2

Trk , D k

d 2Sw f x dfdf

1 dSw f x y "

df

1

Z s d f 1 d f 2 PPP d f N ey " S N w f x ,

H H

H

Ž 2. P

where the field is expanded according to a suitable N Ž . chosen basis, f Ž x . s Ý ns 1 f nFn x , and N - ` plays the role of the UV cutoff. The effective theory for the first N’ modes is defined through the effective action SN X by the help of the blocking transformation for D N s N y N X number of modes,

ž

d 2Sw f x dfdf hqd

y 2

2

1

ey " S N w f x s d f N Xq1 PPP d f N ey " S N w f x . X

H

H

P

Em f P

dSw f x df

dSw f x dEm f

fP

dSw f x df

,

Ž 5.

Ž 3.

The elimination step is usually followed by a rescaling in order to restore the cutoff to its original value. This rescaling, together with its result, the anomalous dimension, is an important device to distinguish the trivial Žtree-level. and the dynamical Žloop-level. scale dependence. The fluctuations of the IR modes with n - N can be described within the effective theory given by the action SN w f x s Ý g j Ž N . s j f , Em f ,I f , . . . ,

/

2yhyd q

1

y1

Ž 4.

j

where s j is a suitable chosen complete set of local functionals without keeping the UV modes n ) N present explicitly. Thus the IR modes decouple from the UV ones since the correlations generated by the latters are contained in the numerical value of the effective coupling constant g j Ž N .. This trivial observation gives rise a powerful approximation scheme by the truncation of Ž4. when any IR field configuration can be used to reconstruct the same overdetermined effective action. In the infinitesimal form of the renormalization group method, w2x, where D N < N the small parameter of the loop expansion in Ž3. is " D NrN, thus we expect that the saddle point approximation is applicable in Ž3.. The Wegner-Hough-

where the trace and the ‘‘P’’ operation is taken in the subspace of the eliminable modes. The second line generates the rescaling assuming the anomalous dimension h Žinput. in dimension d. The second order approximation is used for the action as the functional of the eliminable field variables which is applicable only if the saddle point amplitude is O Ž " .. Our interest in this work is the case where there is a saddle point during the elimination of the modes and we shall see that there is no guarantee in general that the saddle point remains O Ž " . during the evolution. Thus we need a more reliable evolution equation which is applicable without imposing the condition O Ž " . on the saddle point amplitude. Systems with instabilities display fluctuations with large amplitudes which can be taken into account by means of the saddle point approximation. In order to demonstrate the importance of the saddle point during the blocking on the renormalized trajectory we shall consider the f 4 model in the symmetry broken phase in a box with linear size L and with periodic boundary conditions. The bare action of this model is SL s Hd d x w 12 Ž Em f . 2 q UL Ž f .x where UL Ž f . s ym2r2 f 2 q g f 4r4! Ž L is the UV cutoff. and we search for the vacuum in the presence of the constraint LydHd d x ² f Ž x .: s F . The constrained vacuum displays spinodal instability or metastability when it is unstable against fluctuations with infinitesimal or finite amplitude, respectively. The spinodal

J. Alexandre et al.r Physics Letters B 445 (1999) 351–356

instability can be detected by local methods, mode by mode inspection. In fact, each fluctuation of the form

c k Ž x . s c˜ k e i km xm q c˜ kw eyi km xm s 2 r k cos Ž km xm q a k .

353

The saddle point of the blocking transformation can be written in the form c k Ž x . s ÝXc˜p e i p x where the prime denotes the summation for k y D k - < p < k. The blocked potential is given by Ld Uky D k w F x

Ž 6.

with infinitesimal amplitude, r k f 0, represents an independent unstable mode of the tree level theory when p 2 - m 2 y gF 2r2. The Legendre transform of the tree level theory suggests that the vacua with 2 m2rg - F 2 - 6 m2rg are metastable. The vacuum is stable when F 2 ) 6 m2rg. It is difficult to identify the possible instabilities of the true vacuum where the local, mode by mode analysis is unreliable. This is because the decay of the unstable vacuum undergoes large amplitude modifications either in the metastable or the spinodal instable region. We show now that the renormalization group can be used to deal with the modes of the unstable vacuum in a simple one by one manner. The blocking transformation we employ consists of the elimination of the modes with momentum k y D k - < p < k, where k is the current cutoff and D k is a momentum parameter smaller than any characteristic momentum scale of the system. The effective coupling constants can be defined in the leading order of the gradient expansion by Uk Ž f . s Ý j g j Ž k .rj!f j. Since any infrared field configuration should yield the same effective action we take the simplest choice, a homogeneous field f Ž x . s F . The loop expansion for Ž3. yields the general result SN X s Ý j " j SNŽ j.X . The perturbation expansion retains the term O Ž " j . with j ) 0 and the tree level, O Ž " 0 . contribution represents a non-perturbative piece what must be considered before the perturbative pieces are taken into account. Since the tree level expressions are different from the loop corrections the tree level renormalization, if exists, consists of essentially new and different expressions than the loop contributions which have been considered so far w3x. Our constraint generates non-trivial saddle points and we consider here the leading order, tree level renormalization only. In this approximation the naive scaling is correct Žh s 0, Z s 1., and we ignore the rescaling step of the renormalization group method for the sake of simplicity.

1 y

s y"ln

H w Dh x e

s min  c 4

Hd x ž

d

1 2

"

Sk

w Fq h x

Ž Em c .

2

q Uk Ž F q c .

/

qOŽ ".

Ž 7.

where the fluctuation h contains only the modes within the shell w k y D k,k x. It is worthwhile noting that Ž7. reduces to the usual local potential approximation of the Wegner-Houghton equation w4x if the saddle point vanishes. There might be several minima c in which case one should sum over them. We retain only the single plane waÕe saddle points, Ž6.. The motivation of this rather drastic approximation is the assumption that the saddle point Ž6. seems to be optimal from the point of view of the energy-entropy balance. To see this first note that when D k ™ 0 the saddle point is non-vanishing in the momentum space on the sphere with radius k only. As far as the saddle point on this sphere is concerned, the introduction of other additional plane waves would increase the kinetic energy. The phase a k s ayk is a zero mode for each plane waves, corresponding to the translation invariance. The more involved saddle points built by several plane waves have the same single translational zero mode. Thus the kinetic energy-entropy balance is optimized for the plane waves. The resulting tree-level blocking relation is Uky D k Ž F . s min  r 4 k 2r 2 q 12 s k 2r k2 q 12

1

Hy1du U Ž F q 2 r cos Ž p u . . k

1

Hy1du U ŽF q 2 r cos Ž p u . . . k

k

Ž 8.

We followed the tree level evolution of the potential by performing numerically the minimization in Ž8. at each blocking step. The most interesting aspect

354

J. Alexandre et al.r Physics Letters B 445 (1999) 351–356

of the result is that, starting from k s 'm 2 y D k, the saddle points c k are nonvanishing Žand consequently the potential receives a nontrivial renormal2 Ž . ization. for 0 - F 2 - Fvac k s 6Ž m2 y k 2 .rg. Thus the mode coupling provided by our successive elimination method extends the spinodal instability over the vacua which are seen as metastable by the tree level Legendre transformation. This can happen because once the tree level contributions are found at modes with p 2 - m2 for certain values of F 2 the saddle point contributions renormalize the potential in such a manner that spinodal instability occurs for other, larger values of F 2 . Other important results of the numerical analysis are in order: 1. the saddle point amplitude is a linear function of the field, 2 r k s yF q Fvac Ž k .; 2. we find the potential 1 3 2 Uk Ž F . s y k 2F 2 y Ž m2 y k 2 . 2 2g

Ž 9.

whenever the saddle points are nontrivial, i.e. in the unstable region, see Fig. 1, where the evolution of the potential with m2 s 0.1 and g s 0.2 is shown; 3. the blocking converges as D k ™ 0 despite the apparent absence of D k in Ž8.; 4. we checked that the previous results are uniÕersal with respect to the choice of the symmetry breaking bare potential.

Fig. 1. The potential Uk ŽF . for different values of k showing the RG evolution towards the Maxwell effective potential.

It is possible to understand the result Žiii. if we write Ž8. for two subsequent steps and take the difference: Uky 2 D k Ž F . s Uky D k Ž F . y D k 2 k r k2 q 2 k 2r k E k r k q 12

1

Hy1du E U ŽF q 2 r cos Ž p u . . k

k

1

Hy1du cos Ž p u . E

q

k

k

r k Ef Uk Ž F 2

q2 r k cos Ž p u . . q O Ž D k . ,

Ž 10 .

which shows explicitly that the correction to the potential due to one blocking step is proportional to D k. It is also worth remarking that the potential Ž9. is a solution of the Eq. Ž10., as can easily be verified. Few remarks are in order at this point. Ža. The disappearance of the metastable region is in agreement with the finding of Ref. w6x. The saddle points of the static problem reflect the spontaneous droplet formation dynamics w5x: Suppose that the stable vacuum bubble in a sphere of radius R and created in the false vacuum has the energy EŽ R . s 4p R 2s y 4p R 3D Er3 where s and D E stand for the surface tension and the Žfree.energy difference between the two vacua. The critical droplet size, R c s srD E, beyond which the droplets grow can be identified by the inverse of the highest momentum of the conden2 Ž . sate, R 2c s 6rg ŽFvac 0 y F 2 .. This relation establishes a connection between the static and the dynamical properties. Žb. The last term in the right hand side of Ž9. was added by hand to achieve a continuous matching of the partition function at the instability. As a result, Ž9. reproduces the Maxwell construction for the effective potential Veff ŽF . s Uks 0 ŽF . w7x, i.e. it remains unchanged Žat the tree 2 Ž . level. in the stable region, F 2 ) Fvac 0 , and turns 2 Ž . out to be constant and continuous for F 2 F Fvac 0. Žc. The potential Ž9. contains only one coupling constant, m2 Ž k . s Ef2 Uk Ž0. s yk 2 . The corresponding dimensionless coupling constant, m ˜ 2Žk. s 2Ž . 2 m k rk s y1, is trivially renormalization group invariant. In other words we recover the usual scaling laws in the stable region, while in the unstable region the potential is a fixed point given by Ž9.. Žd.

J. Alexandre et al.r Physics Letters B 445 (1999) 351–356

Similar tree level potential has been found in the N-component f 4 model with smooth cutoff w8x where the would be unstable and the stable regime join and the naive metastable region is wiped out by the Goldstone modes. While the possibility of having a homogeneous c k2 Ž x . is an essential part of the argument, our result shows the more general origin of the potential. Furthermore note that the keeping of a single plane wave mode as a saddle point is not justified when a smooth cutoff is used. The analytical structure of the loop corrections in the vicinity of the instability indicates the same potential as well, w9x. The availability of such a diverse derivations and the microscopic dynamics-independent form suggests a more fundamental origin of this potential. We believe that underlying reason of Ž9. is the Maxwell construction and the actual form can easiest be identified by means of the renormalization group method where the dynamics of each mode can be dealt with individually. In fact, one can show that the differentiability of the renormalization group flow, i.e. the existence of the beta functions, and the nonvanishing of the saddle points as D k ™ 0 requires the form Ž9.. To see this consider the finite difference Uky D k Ž F . y Uk Ž F .

Dk s

k 2 q EF2 Uk Ž F . 2 LdD k q

EF3 Uk Ž F . 6 LdD k

Hdx c

Hdx c

3 k

2 k

Ž x.

Ž x . q PPP qO Ž " . ,

at O Ž " 0 .. In fact, let us insert the product of field variables in the integrand of Ž2. and follow the successive elimination of the modes in the path integral. The resulting tree level expression for the 2 n-point function on the background field F is GFŽ0. Ž p 1 , . . . , p 2 n . 2n

H

D w kˆ x D w a x Ł c˜ k jŽ pj .

H

s

where we expanded Ž7. in the saddle point, c k Ž x .. The convergence of the left hand side requires either the smallness of the saddle point, c k s O Ž D k 1r2 ., or k 2 q EF2 Uk ŽF . s O Ž D k . and EFn Uk ŽF . s O Ž D k .. Since the saddle point is nonvanishing the form Ž9. follows. Note the essential differences between the tree and the loop level renormalization: The former is nonanalytic in g and lacks the usual logarithms of the latter. Že. The result Ži. can be obtained by assuming Ž9. in the unstable region and requiring continuity of EF Uk ŽF . in F . The saddle point approximation includes the softest fluctuations, the zero modes, and one can obtain nontrivial contributions to the correlation functions

js1

H D w kˆ x H D w a x n

sÝ

d Ž 2p .

Ł d Ž pP Ž2 j. q pP Ž2 jy1. . V

P js1

d

d d k ŽF .

r p2j

Ž 12 . where c˜ k Ž p . s Hd d eyi p xc k Ž x ., V d stands for the solid angle, we integrate over the zero modes characterized by the unit vector kˆ Ž k . corresponding for each value of the cutoff, k, and the phase a Ž p .. The sum in the second equation is over the permutations P of the field variables and k 2 ŽF . s m2 y gF 2r6. Whenever we have an odd number of fields or 2 Ž . F 2 G Fvac 0 or one of the pj is such that pj2 ) m 2 2 y gF r6, G Ž0. s 0. The integration over the zero modes is reminiscent of the integration over the possible rearrangements of the domain walls in the mixed phase and restores the translation invariance of the correlations. The two point function GFŽ0. Ž p 1 , p 2 . s 2

Ž 11 .

355

d Ž 2p .

d

Vd k d ŽF .

r p21 d Ž p 1 q p 2 .

shows that the Fourier transform of c˜p can be interpreted as the domain wall structure for a given choice of the zero modes. The method put forward in this letter, the use of the saddle point approximation for the blocking transformation, can be used to handle some of the systems where large amplitude, inhomogeneous fluctuations are present. We studied here an Euclidean system describing the equilibrium situation. Whenever the time dependence far from equilibrium has nontrivial semiclassical limit this method can be extended to include the real time dependence, w10x. Another natural continuation of this work is the inclusion of the loop corrections in addition to the tree level pieces. The correlation functions computed

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J. Alexandre et al.r Physics Letters B 445 (1999) 351–356

in such a manner include the fluctuations in a systematic manner which is an improvement compared to Ref. w11x where the master equation was used to describe the most probable values of the correlation functions. Acknowledgements We thank Daniel Boyanowski for a useful discussion. References w1x K. Wilson, J. Kogut, Phys. Rep. C 12 Ž1974. 75; K. Wilson, Rev. Mod. Phys. 47 Ž1975. 773. w2x F.J. Wegner, A. Houghton, Phys. Rev. A 8 Ž1973. 40. w3x J. Polchinski, Nucl. Phys. B 231 Ž1984. 269; A. Hasenfratz, P. Hasenfratz, Nucl. Phys. B 295 Ž1988. 1; C. Wetterich, Nucl. Phys. B 352 Ž1991. 529.

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