Wilsonian effective action for SU(2) Yang-Mills theory with Cho-Faddeev-Niemi-Shabanov deco

更新时间：2023-06-09 00:15:01 阅读量：实用文档文档下载

说明：文章内容仅供预览，部分内容可能不全。下载后的文档，内容与下面显示的完全一致。下载之前请确认下面内容是否您想要的，是否完整无缺。

wilsonianism推荐度：
相关推荐

The Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2) Yang-Mills field is employed for the calculation of the corresponding Wilsonian effective action to one-loop order with covariant gauge fixing. The generation of a mass scale is observed, and the fl

a r

:h e

/01

A p

Wilsonian e?ective action for SU(2)Yang-Mills theory with Cho-Faddeev-Niemi-Shabanov decomposition Holger Gies ?Institut f¨u r theoretische Physik,Universit¨a t T¨u bingen,D-72076T¨u bingen,Germany and Theory Division,CERN,CH-1211Geneva,Switzerland E-mail:holger.gies@cern.ch February 1,2008Abstract The Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2)Yang-Mills ?eld is employed for the calculation of the corresponding Wilsonian e?ective action to one-loop order with covariant gauge ?xing.The generation of a mass scale is observed,and the ?ow of the marginal couplings is studied.Our results indicate that higher-derivative terms of the color-unit-vector n ?eld are necessary for the description of topologically stable knotlike solitons which have been conjectured to be the large-distance degrees of freedom.

1Introduction

The fact that quarks and gluons are not observed as asymptotic states in our world indicates that a description in terms of these ?elds is not the most appropriate language for discussing low-energy QCD.On the other hand,there seems to be little predictive virtue in describing the low-energy domain only by observable quantities,such as mesons and baryons.A purposive procedure can be the identi?cation of those (not necessarily observable)degrees of freedom of the system that allow for a “simple”description of the observable states.The required “simplicity”can be measured in terms of the simplicity of the action that governs those degrees of freedom.Clearly,a clever guess of such degrees of freedom is halfway to the solution of the theory;the remaining problem is to prove that these degrees of freedom truly arise from the fundamental theory by integrating out the high-energy modes.

For the pure Yang-Mills(YM)sector of QCD,such a guess has recently been made by Faddeev and Niemi[1]inspired by the work of Cho[2].For the gauge group SU(2),they decomposed the(implicitly gauge-?xed)gauge potential Aµinto an“abelian”component Cµ,a unit color vector n and a complex scalar?eld?;here,Cµis the local projection of Aµonto some direction in color space de?ned by the space-dependent n.Faddeev and Niemi conjectured that the important low-energy dynamics of SU(2)YM theory1is determined by the n?eld;its e?ective action of nonlinear sigma-model type,the Skyrme-Faddeev model,should then arise from integrating out the further degrees of freedom:Cµ,?,...:

ΓFN e?= d4x m2(?µn)2+1

1Di?erent generalizations of the gauge?eld decomposition for higher gauge groups can be found in[3], [4]and[9].

2In a very recent paper[7],Faddeev and Niemi generalized their decomposition in order to obtain a manifest duality between the here-considered“magnetic”and additional“electric”variables,involving an abelian scalar multiplet with two complex scalars.This electric sector will not be considered in the present work.

3A di?erent approach was put forward in[8],where the n?eld was identi?ed by constructing an unconstraint version of SU(2)Yang-Mills theory in a Hamiltonian context.

gauge we shall choose.We face this problem by?xing the gauge in such a way that Lorentz invariance and global color transformations remain as residual symmetries;these are the symmetries of the Skyrme-Faddeev model and must mandatorily be respected.

The Wilsonian e?ective action is characterized by the fact that it governs the dynamics of the low-energy modes below a certain cuto?k;it incorporates the interactions that are induced by high-energy?uctuations with momenta between k and the ultraviolet(UV) cuto?Λwhich have been integrated out.Following the Faddeev-Niemi conjecture,we only retain the n?eld as low-energy degree of freedom.Actually,we integrate over the high-energy modes in two di?erent ways:?rst,we integrate out the k<p<Λ?uctuations of all?elds except for the n?eld,which is left untouched(Sec.3).Secondly,we integrate all ?elds including the n?eld over the same momentum shell(Sec.4).In this way,we can study the e?ect of the n?eld?uctuations on the?ow of the mass scale and the couplings in detail.

The results for both calculations are similar:the mass scale m appearing in Eq.(1)is indeed generated by the renormalization group?ow,and the gauge coupling is asymptoti-cally free.As far as the simplicity of the conjectured e?ective action Eq.(1)is concerned, our results are a bit disappointing:as discussed in Sec.5,further marginal terms(not displayed in Eq.(1))are of the same order as the displayed one and therefore have to be included in Eq.(1).Keeping only those terms that involve single derivatives acting on n results in an action without stable solitons;nevertheless,stability is in fact ensured owing to the presence of higher-derivative terms.The disadvantage is that these terms spoil the desired simplicity of the low-energy e?ective theory.

Of course,our perturbative results represent only a?rst glance at the true infrared behavior of the system and are far from providing qualitatively con?rmed results,not to mention quantitative predictions.To be precise,the one-loop calculation investigates only the form of the renormalization group trajectories of the couplings in the vicinity of the perturbative Gaussian?xed point.Nevertheless,various extrapolations of the perturba-tive trajectories can elucidate the question as to whether the Faddeev-Niemi conjecture is realizable or not.

2Quantum Yang-Mills theory in Cho-Faddeev-Niemi-Shabanov variables

In decomposing the Yang-Mills gauge connection,we follow[2,9,10].Let Aµbe an SU(2) connection where the color degrees of freedom are represented in vector notation.We parametrize Aµas

Aµ=n Cµ+(?µn)×n+Wµ,(2) where the cross product is de?ned via the SU(2)structure constants.Cµis an“abelian”connection,whereas n denotes a unit vector in color space,n·n=1.Wµshall be orthogonal to n in color space,obeying Wµ·n=0,so that Cµ=n·Aµ.For a given n,Cµ

and Wµ,the connection Aµis uniquely determined by Eq.(2).In the opposite direction, there is still some arbitrariness:for a given Aµ,n can generally be chosen at will,but then Cµand Wµare?xed(e.g.,Wµ=n×Dµ(A)n,where Dµdenotes the covariant derivative). While the LHS of Eq.(2)describes3color×4Lorentz=12o?-shell and gauge-un?xed degrees of freedom,the RHS up to now allows for(Cµ:)4Lorentz+(n:)2color+(Wµ:)3color×4Lorentz?4n·Wµ=0=14degrees of freedom.Two degrees of freedom on the RHS remain to

be?xed.For example,by?xing n to point along a certain direction and imposing gauge conditions on Wµ,we arrive at the class of abelian gauges which are known to induce monopole degrees of freedom in Cµ.In order to avoid these topological defects,we let n vary in spacetime and impose a general condition on Cµ,n and Wµ,

χ(n,Cµ,Wµ)=0,withχ·n=0,(3) which?xes the redundant two degrees of freedom on the RHS of Eq.(2).Moreover, Eq.(3)also determines how n,Cµand Wµtransform under gauge transformations of Aµ: by demanding thatδχ(n,Cµ(A),Wµ(A))=0(andδ(χ·n)=0),the transformationδn of n is uniquely determined,from whichδCµandδWµare also obtainable.

The thus established one-to-one correspondence between Aµand its decomposition(2) allows us to rewrite the generating functional of YM theory in terms of a functional integral over the new?elds[9,10]:

Z= D n D C D Wδ(χ)?S?FP e?S YM?S gf.(4) Beyond the usual Faddeev-Popov determinant?FP,the YM action S YM and the gauge ?xing action S gf,we?nd one further determinant introduced by Shabanov,?S;this de-terminant accompanies theδfunctional which enforces the constraintχ=0,in complete analogy to the Faddeev-Popov procedure:

?S:=det δχ

Incidentally,the gauge transformation properties of Cµand Wµalso become very simple with the choice(6):Wµalso transforms homogeneously,andδCµ=n·?µ?.

Finally,the choice of the gauge-?xing condition must also be viewed as being part of the de?nition of the decomposition.Not only does the functional form of?FP and S gf depend on this choice,but the discrimination of high-and low-momentum modes is also determined by the gauge?xing.In fact,this gauge dependence of the mode momenta usually is the main obstacle against setting up a Wilsonian renormalization group study.But in the present context,it belongs to the conjecture that the particular gauge that we shall choose singles out those low-momentum modes which?nally provide for a simple description of low-energy QCD;in a di?erent gauge,we would encounter di?erent low-momentum modes, but we also would not expect to?nd the same simple description.

In this work,we choose the covariant gauge condition?µAµ=0.This automatically ensures covariance of the resulting e?ective action and,moreover,allows for the residual symmetry of global gauge transformations,?=const.Together with the choice(6),this residual symmetry coincides with the desired global color symmetry of the Skyrme-Faddeev model(1).This means that the demand for color and Lorentz symmetry of the action(1) is satis?ed exactly by a covariant gauge and condition(6).

3One-loop e?ective action without n?uctuations Our aim is the construction of the one-loop Wilsonian e?ective action for the n?eld by integrating out the C and W?eld over a momentum shell between the UV cuto?Λand an infrared cuto?k<Λ.In general,this will induce nonlinear and nonlocal self-interactions of the n?eld;since we are looking for an action of the type(1),we represent these interactions in a derivative expansion and neglect higher derivative terms of order O(?2n?2n)(later, we shall question this approach).

Furthermore,we do not integrate out n?eld?uctuations in this section(see Sect.4)and disregard any induced C or W interactions below the infrared cuto?k.From a technical viewpoint,the one-loop approximation of the desired e?ective actionΓk[n]is obtained by a Gaussian integration of the quadratic C and W terms in Eq.(4),neglecting higher-order terms of the action:

e??Γk[n]=e?S cl[n] k D C D W?S[n]?FP[n]δ(χ)(7)

×e?12M CµνCν+Wµ1

4g2(?µn×?νn)2+

We treat the δfunctional in Eq.(7)in its Fourier representation,

δ(χ)→ D φe ?i φ·?µW µ+φ·C µn ×W µ+(φ·n )(?µn ·W µ),(9)

where the second term in the exponent,the triple vertex,can actually be neglected,because it leads only to nonlocal terms (ter)or terms of higher order in derivatives.Inserting Eq.(9)into Eq.(7),we end up with three functional integrals over C ,W and φ,which can successively be performed,leading to three determinants,

e ??Γk [n ]→e ?S cl [n ]?S [n ]?FP [n ] det M C ?1/2

det M W )?1

µνQ φν ?1/2,

(10)

where we have omitted several nonlocal terms that arise from the completion of the square in the exponent during the Gaussian integration.In Appendix B,we argue that these nonlocal terms are unimportant in the present Wilsonian investigation.Again,details about the various operators in Eq.(10)are given in App.A.

The determinants are functionals of n only and have to be evaluated over the space of test functions with momenta between k and Λ.The determinants depend also on the gauge parameter α.Only for the Landau gauge α=0is the gauge-?xing δfunctional implemented exactly;in fact,α=0appears to be a ?xed point of the renormalization group ?ow [11].But this in turn ensures that the choice of α=α(k )≡αk at the cuto?scale k →Λis to some extent arbitrary,since αk ?ows to zero anyway as k is lowered.This allows us to conveniently choose αk =Λ=1at the cuto?scale and evaluate the determinants with this parameter choice.

As mentioned above,we evaluate the determinants in a derivative expansion based on the assumption that the low-order derivatives of n represent the essential degrees of freedom in the low-energy domain.There are various techniques for the calculation at our disposal;it turns out that a direct momentum expansion of the operators is most e?cient.4We shall demonstrate this method by means of the third determinant of Eq.(10),the “C determinant”;the key observation is that derivatives acting on the space of test functions create momenta of the order of p with k <p <Λ,whereas derivatives of the n ?eld are assumed to obey |?n |?k in agreement with the Faddeev-Niemi conjecture.This suggests an expansion of the form

ln det M C 1/2=?1

2Tr ln(??2L )+ln L +

?n ·?n

2Tr ln(??2L )?1

??2+1

??2 2

+O ((?n )6),

where we suppressed Lorentz (L)indices.Here,we neglected higher-derivative terms of n ,e.g.,?2n ,which is in the spirit of the Faddeev-Niemi conjecture;of course,this has to be checked later on.Employing the integral formulas given in App.C,we ?nally obtain for the C determinant

ln det M C 1/2??

132π2ln Λ

32π2ln Λ

64π2

x (?µn )2+1k x ?µn ×?νn 2?1k x (?µn )4,ln(det

64π2

x (?µn )2?5k x ?µn ×?νn 2+35k x (?µn )4,ln(det ? Q φ128π2 x (?µn )2+49k x ?µn ×?νn 2?5k x (?µn )4.(13)The determinant ?S does not contribute,because it is independent of n in one-loop ap-proximation.Inserting these results into Eq.

(10)leads us to the desired Wilsonian e?ective action to one-loop order for the n ?eld in a derivative expansion:?Γk [n ]=1316π2 1?e 2t x (?µn )2+1g 2+716π2t x (?µn ×?νn )2

?1αg 2+516π2t x

(?µn )4,(14)where t =ln k/Λ∈]?∞,0]denotes the “renormalization group time”.We would like to stress once more that ?Γ

k [n ]does not contain the result of ?uctuations of the n ?eld itself;in other words,it represents (an approximation to)the “tree-level action”for the complete quantum theory of the n ?eld.

Indeed,the generation of a “kinetic”term ～(?µn )2growing under the ?ow of increasing k as conjectured by Faddeev and Niemi is observed.Moreover,it has the correct sign (+),implying that an “e?ective ?eld theory”interpretation seems possible.The second term which is proportional to the classical action reveals information about the renormalization of the Yang-Mills coupling:

g 2+7

16π2t ?

?βg 2:=?t ?g 2R =?716π2?g 4R .(15)7

The resulting?βfunction is a factor of44/7smaller than theβfunction of full Yang-Mills theory for SU(2).This is an expected result,since we did not integrate over all degrees of freedom of the gauge?eld;the n integration still remains.Nevertheless,the ?βfunction implies asymptotic freedom,which indicates that the decomposition of the

Yang-Mills?eld is not a pathologically absurd choice.It is interesting to observe that the C and W determinants contribute positively to?βg2,whereas the Faddeev-Popov and theφdeterminant contribute negatively;the latter,which arises from the W?xing,even dominates:?7/3=[6C?4FP+40W?49φ]/3.

The third term of Eq.(14)contains information about the renormalization of the gauge parameterαunder the?ow:

αg2+

16π2

t,??t?αR=

?αR ?g2R

Integrating out the hard modes A H results in two determinants in one-loop approximation,Γk[A S]=

?? µν A=A S,(20)

where Dµdenotes the covariant derivative and Fµνthe?eld strength tensor.The explicit representation of Eq.(20)in terms of the n?eld is again given in App.A,Eqs.(A.6) and(A.7).The determinants in Eq.(19)can be calculated in a derivative expansion in the same way as described in the preceding section.Since the computation of the term ～(?n)2is already very laborious,we do not calculate the marginal terms～(?µn×?νn)2 etc.directly,but take over the known one-loop results for the running coupling and the gauge parameter from[11].The?nal result for the Wilsonian one-loop e?ective action for the soft modes of the n?eld reads

Γk[n]=

Λ2

4 131

2 1312

131 3

by Faddeev and Niemi.This term is relevant in the renormalization group sense and perturbatively exhibits a quadratic dependence on the UV cuto?Λ.

Furthermore,we studied the renormalization group?ow of the marginal couplings of the n?eld self-interactions given by the Yang-Mills coupling and the gauge parameter. These terms are responsible for the stabilization of possible topological excitations of the n?eld,as suggested by the Skyrme-Faddeev model.In total,the di?erence between?Γk andΓk is only of quantitative nature:the inclusion of hard n?eld?uctuations increases the running of the marginal couplings and reduces the new mass scale;qualitative features such as stability of possible solitons remain untouched.

In fact,the question of stability turns out to be delicate:truncating our results for?Γk or Γk in Eqs.(14)or(21)at the level of the original Faddeev-Niemi proposal Eq.(1)(the?rst lines of Eqs.(14)and(21),respectively),we?nd an action that allows for stable knotlike solitons,since the coe?cients of both terms are positive(as long as we stay away from the Landau pole,which we consider as unphysical).Taking additionally the(?n)4term of?Γk orΓk into account,which is also marginal and does not contain second-order derivatives on n,stability is lost,since the coupling coe?cient is negative in Eqs.(14)and(21);for stable solitons,a strictly positive coe?cient would be required for this truncation,as was shown in[12].

Finally dropping the demand for?rst-order derivatives,we can include one further marginal term～?2n·?2n as given in Eq.(21)forΓk.With the aid of the identity

x(?2n×n)2= x[?2n·?2n?(?µn)4],(22)

we?nd that the second line of Eq.(21)represents a strictly positive contribution to the action which again stabilizes possible solitons.5

Of course,this game could be continued by including further destabilizing and stabiliz-ing higher-order terms again and again,but such terms are irrelevant in a renormalization group sense;that means their corresponding couplings are accompanied by inverse powers of the UV cuto?Λand are thereby expected to vanish in the limit of large cuto?.

To summarize,our perturbative renormalization group analysis suggests enlarging the Faddeev-Niemi proposal for the e?ective low-energy action of Yang-Mills theory by taking all marginal operators of a derivative expansion into account.The original proposal of Eq.(1)was inspired by a desired Hamiltonian interpretation of the action that demands the absence of third-or higher-order time derivatives.But,as demonstrated,the covariant renormalization group does not care about a Hamiltonian interpretation of the?nal result. In some sense,the desired“simplicity”of the?nal result is spoiled by the presence of higher-derivative terms;moreover,it remains questionable as to whether the importance of the ?2n·?2n term is still consistent with the derivative expansion of the action.Unfortunately, this cannot be checked within the present approach.

It should be stressed once again that the perturbative investigation performed here hardly su?ces to con?rm results about the infrared domain of Yang-Mills theories.On the

contrary,it is only a valid approximation in the vicinity of the Gaussian UV?xed point of

the theory.Nevertheless,our study might lend some intuition to possible nonperturbative scenarios:for example,let us assume that the Landau gaugeα=0indeed is an infrared

?xed point in covariant gauges.Then the stabilizing term～(?2n×n)2is enhanced in the

infrared,provided that the increase of the running coupling g obeysαg2→0for k→0; this would be realized,e.g.,if g approached an infrared?xed point.Such a scenario thus

supports the idea of topological knotlike solitons as important infrared degrees of freedom

of Yang-Mills theories.

Perhaps the main drawback of our study lies in the fact that the new mass scale is not

renormalization-group invariant;for example,we can read o?from Eq.(21)that

m2k=1

Λ≤0.(23)

The new mass scale m k is necessarily proportional toΛ,because there simply is no other

scale in our system.But contrary to the gauge coupling or the gauge parameter,which can be made independent ofΛby adjusting the bare parameters,theΛdependence of m k

persists,since there is no bare mass parameter to adjust.One may speculate that this problem is solved by“renormalization group improvement”of the kind

Λ2→Λ2e?3·16π2

q m,where q denotes the value of the coe?cient in front of the(?µn×?νn)2term[12,14].For couplings of order1,

we end up with soliton masses of the order of M～O(1)GeV;this is in accordance with

lattice results for glue ball masses:e.g.,M GB?1.5GeV for the lowest lying state in SU(2) [15].Of course,this rough and speculative estimate should not be viewed as a“serious

prediction”of our work.

With all these reservations in mind,the Faddeev-Niemi conjecture about possible low-energy degrees of freedom of Yang-Mills theories provides an interesting working hypothesis which deserves further exploration.

Acknowledgment

The author wishes to thank W.Dittrich for helpful conversations and for carefully read-ing the manuscript.Furthermore,the author pro?ted from discussions with T.Tok, ngfeld and A.Sch¨a fke.This work was supported in part by the Deutsche Forschungs-gemeinschaft under DFG GI328/1-1.

Appendix

A Di?erential operators,tensors,currents,etc.

This appendix represents a collection of di?erential operators and other tensorial quantities which are required in the main text.

The Faddeev-Popov determinant?FP in Eq.(7)and(10)for covariant gauges involves the operator(in one-loop approximation)

??µDµ(A) C=0=W=??2c+(?2n?n?n??2n)+(?µn?n?n??µn)?µ,(A.1) so that?FP=det ??µDµ(A) C=0=W .

The objects occurring in the exponent of Eq.(7)are de?ned as follows:

M Cµν:=??2δµν+?µ?ν?1α?µn·?νn

M Wµν:=??2δµνc+?µ?νc?1

α ?µn?ν+?νn?µ+?µ?νn

K Cµ:=?ν(n·?νn×?µn)+

?µ(n×?2n).(A.2)

The determinants in Eq.(10)employ several composites of these operators.Since we?rst perform the C integration,the resulting determinant involves only M C,whereas the W determinant also receives contributions from the mixing term Q C,

The last determinant in Eq.(10)arises

from

the

integration and

receives contributions from the relevant parts of the exponent of Eq.(9),which we denote by

Q φµ:=i ??µc +?µn ?n ,

(A.5)so that δ(χ)→ D φexp(? W µ·Q φµφ)to one-loop order.Employing a notation similar

to Eq.(A.4),the di?erential operator accompanying the term ～φφin the exponent ?nally

reads Q φµ(

??2+?n ·?n µν

(n ·?κn ×?κνn ).(B.8)

Within the calculation of the determinants,we expanded the inverse operator assuming that ?n ·?n ???2.This was justi?ed,since the derivative operator acts on the test func-tion space with momenta p between k and Λ,which are large compared to the conjectured slow variation of the n ?eld.

In the present case,the situation is di?erent,because the derivative term ??2acts only on the n ?eld and its derivatives to the right (there is no test function to act on).In other words,the nonlocal terms are only numbers,not operators.The derivatives can thus be approximated by the (inverse)scale of variation of the n ?eld or its derivatives which is much smaller than k or Λ.This implies that the nonlocal terms do not depend on k or Λ,so that they cannot contribute to the ?ow of the couplings.

For example,a reasonable lowest-order approximation of the RHS of Eq.

(B.8)

given by its local limit,K C (M C )?1K C =(n ·?λn ×?λµn )

1(2π)4p λp κp µp ν

31k δλκδµν+δλµδκν+δλνδκµ .(C.10)

From this formula,we can also deduce upon index contraction that

[k,Λ]d 4p p 6=1k , [k,Λ]d 4p p 4

=1k .(C.11)

The last integral is,of course,standard and can be used to prove Eq.(C.10)in addition to symmetry arguments.The same philosophy applies to the second type of integrals:

[k,Λ]

d 4p p 4=1(2π)4116π

2(Λ2?k 2).(C.12)References

[1]L.Faddeev and A.J.Niemi,Phys.Rev.Lett.82,1624(1999)[hep-th/9807069].

[2]Y.M.Cho,Phys.Rev.D21,1080(1980);Phys.Rev.D23,2415(1981).

[3]V.Periwal,hep-th/9808127.

[4]L.Faddeev and A.J.Niemi,Phys.Lett.B449,214(1999)[hep-th/9812090].

[5]ngmann and A.J.Niemi,Phys.Lett.B463,252(1999)[hep-th/9905147].

[6]Y.M.Cho,H.Lee and D.G.Pak,hep-th/9905215.

[7]L.Faddeev and A.J.Niemi,hep-th/0101078.

[8]A.M.Khvedelidze and H.P.Pavel,Phys.Rev.D59,105017(1999)[hep-th/9808102].

[9]S.V.Shabanov,Phys.Lett.B458,322(1999)[hep-th/9903223].