ISSN 1063-7788, Physics of Atomic Nuclei, 2009, Vol. 72, No. 2, pp. 371–376. c© Pleiades Publishing, Ltd., 2009.

ELEMENTARY PARTICLES AND FIELDSTheory

On the Effective Action of Center Vorticesin Continuum Yang–Mills Theory*

P. V. Buividovich**

Joint Institute of Power and Nuclear Research “Sosny”, Minsk, Belarus;Institute of Theoretical and Experimental Physics, Moscow, Russia

Received March 21, 2008

Abstract—The effective action of sufficiently smooth center vortices in continuum Yang–Mills theory isinvestigated taking into account some of their basic properties observed in lattice simulations. The obtainedaction is given by the action of an ideal vortex configuration, which is local on the vortex worldsheet,nonlocal quantum corrections due to virtual gluon exchanges, which can be reliably calculated usingperturbation theory, and the contribution of Abelian magnetic monopoles. Smooth center vortices appearto have negative rigidity. This implies that, in agreement with lattice data, in the ground state of the theorycenter vortices are not regular surfaces.

PACS numbers: 12.38.Aw, 05.40.FbDOI: 10.1134/S1063778809020264

Topologically nontrivial objects are often consid-ered as the effective infrared degrees of freedom ofgauge theories. For instance, if the gauge group ofthe theory has a nontrivial center, there can be ex-tended string-like topologically stable solitons whichcarry quantized magnetic flux—center vortices [1].Although the pictures of confinement based on thedominance of center vortices in the ground state ofthe theory have been discussed for quite a long time(see, e.g., [2, 3]), their role in the infrared dynamics ofquantum gauge fields was understood only recently,with the development of methods which allow to ex-tract configurations of center vortices from latticeconfigurations of gauge fields [4].

It turned out that all characteristic infrared prop-erties of the vacuum of Yang–Mills theory such as theconfinement [4] and the spontaneous breaking of chi-ral symmetry [5] disappear when center vortices areremoved from lattice configurations. This indicatesthat at low energies (typically, below ΛQCD) it shouldbe possible to reformulate pure Yang–Mills theoryas an effective theory of center vortices, in the sensethat all other degrees of freedom do not contributeto asymptotic expectation values. By definition, theeffective vortex action Weff[Σ], where Σ denotes theunion of all vortex worldsheets, is obtained by inte-grating over all degrees of freedom except the geom-

∗The text was submitted by the authors in English.**E-mail: buividovich@tut.by

etry of center vortices:

exp(−Weff[Σ]) (1)

=∫

DAµδ[Σ − Σ[Aµ]] exp(−SYM[Aµ]),

where Σ[Aµ] is the configuration of vortex world-sheets constructed from the gauge field Aµ(x) andSYM[Aµ] is the action of Yang–Mills theory.

Center vortices are singular field configurations,which should be regularized at the UV-cutoff scale [1].A properly regularized path integral over all gaugefields should contain all vortex configurations. Themost natural regularization is the lattice gauge the-ory, where each configuration of link variables canbe exactly mapped onto some configuration of centervortices and where integration over all link variablesreproduces therefore all vortex configurations with acorrect weight [4, 6]. In lattice gauge theory it is alsopossible to explicitly eliminate the δ function in (1)and rewrite the integral in (1) as an integral over allfield configurations which do not contain vortices.After that a fixed configuration of center vorticescan be introduced by explicitly modifying some linkvariables as gl → zlgl, where zl belongs to the groupcenter [1, 7, 8]. On the other hand, on the latticeit is rather difficult to analyze the effective vortexaction, since one should use rather complicated finitedifferences instead of conventional geometric objectson regular continuous surfaces. If one is interested inthe effective action of sufficiently smooth vortices, itis more convenient to work in the continuum theory.

371

372 BUIVIDOVICH

Unfortunately, it is not known how to define thecontinuum path integral in (1) nonperturbatively.

This paper is an attempt to combine the advan-tages of both approaches and to investigate the ef-fective action of center vortices in the long-waveapproximation of lattice gauge theory, where bothvortex worldsheets and gauge fields without vorticesare sufficiently smooth. It should be noted from thevery beginning that this approximation is not validfor real center vortices which were observed in latticesimulations and which tend to form creased, percolat-ing fractal structures with very large genus. However,smooth center vortices appear to be anti-rigid andthus indeed unstable with respect to surface creasing,which allows one to understand their observed prop-erties at least qualitatively.

In the long-wave approximation one can definethe noncompact gauge field Aµ by the equation gl =exp(iaAµeµ

l ), where a ∼ Λ−1UV is the lattice spacing

which eventually tends to zero and eµl is the unit vec-

tor in the direction of the link l. Further analysis willbe restricted to the theory with SU(2) gauge group,since most numerical results on center vortices wereobtained for SU(2) lattice gauge theory [4–6, 9]. Interms of the field Aµ the transformation gl → zlgl isAµ → Aµ + a−1ηleµl, where ηl is the element of su(2)Lie algebra which has the same direction as Aµeµ

land which generates the elements of the group center:exp(iηl) = zl. Thus in the continuum limit a → 0 thetransformation gl → zlgl corresponds to the replace-

ment Aµ → Aµ + A(s)µ [Σ], where A

(s)µ [Σ] is a singular

gauge field which is determined by the geometry ofthe vortex worldsheet Σ [10]. Correspondingly, theeffective action (1) reads:

exp(−Weff[Σ]) (2)

=∫

DA(0)µ exp(−SYM[A(0)

µ + A(s)µ [Σ]]),

where∫DA

(0)µ denotes integration over all gauge

fields without center vortices and SYM[Aµ] is the ac-tion of Yang–Mills theory.

The observation that all nonperturbative phenom-ena disappear when center vortices are removed fromlattice configurations of gauge fields implies that thepath integral in (2) can be calculated with sufficientlygood precision in the few lowest orders of perturbationtheory (see [11] for a more detailed discussion). Thiscan be considered as the nonperturbative definition ofthe path integral over all gauge fields

∫DAµ:∫

DAµF [Aµ] exp(−SYM[Aµ]) (3)

=∫

DΣ∫

DA(0)µ F [A(0)

µ + A(s)µ [Σ]]

× exp(−SYM[A(0)µ + A(s)

µ [Σ]])

where the integral over A(0)µ should be calculated as a

perturbative expansion at fixed Σ. Finally, one shouldalso take into account the contribution of Abelianmonopoles. In lattice simulations monopole world-lines almost completely lie on vortex worldsheets [9].Here, it will be assumed that for sufficiently smalllattice spacings all Abelian monopoles belong to vor-tex worldsheets, so that center vortices are the onlyfield singularities outside of perturbative sector of thetheory. This assumption fits nicely in the definition (3)since, as will be shown further, in this case monopoleworldlines automatically belong to vortex worldsheets

and are specified by the perturbative field A(0)µ on

Σ. This does not mean that Abelian monopoles arecontained in the perturbative sector of the theory–instead, this means that the configuration of Abelianmonopoles is determined by the configuration of cen-ter vortices, which are essentially nonperturbative ob-jects.

The first step in the calculation of the effectiveaction (2) is to find the singular part of the gauge field

A(s)µ [Σ], which describes center vortex with world-

sheet Σ. A defining property of center vortices is thatthe Wilson loop W [C] in any representation with half-integer spin should change by −1 whenever the loopC is crossed by the vortex worldsheet Σ [1]. Using thenon-Abelian version of the Stokes theorem [12], it iseasy to show that the curvature tensor for the gauge

field Aµ = A(0)µ + A

(s)µ [Σ] should have the following

form:

Fµν(x) = F (0)µν (x) (4)

+ η(x)∫

Σ

d2σ√

gt̃µν(σ)δ(x − x(σ)).

In this expression F(0)µν (x) is the Yang–Mills cur-

vature of the gauge field A(0)µ , η(x) is some ele-

ment of su(2) Lie algebra which satisfies the equationexp(iη(x)) = −1; σa, a = 1, 2, are the coordinates onΣ; x = x(σ) is the embedding of Σ into the physicaltarget space, gab = ∂ax

µ∂bxµ is the induced metric

on Σ and g = det gab is its determinant. In order topreserve the transformation properties of the gauge

fields Aµ and A(0)µ and also to satisfy the equation

exp(iη(x)) = −1 in any gauge, η(x) should trans-form under the adjoint representation of SU(2) gaugegroup. The tensor t̃µν(σ) is the Hodge dual to thenormal two-form to the vortex worldsheet Σ:

t̃µν(σ) =12εµναβtαβ(σ), (5)

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 2 2009

ON THE EFFECTIVE ACTION 373

tαβ =1√

gεab∂ax

α∂bxβ.

The choice of η(x) is constrained by the Jacobiidentities for the curvature tensor (4). In order toregularize these identities for singular gauge fields,it is convenient to go back to the underlying latticegauge theory. Lattice version of the Jacobi identi-ties can be written as

∏p∈c gl(p)gpg

−1l(p) = 1, where

c, p, and l denote cubes, plaquettes, and links onthe lattice, gl are the link variables, gp =

∏l∈p gl;

and l(p) denotes the link which connects the pla-quette p with some fixed lattice site at one of theedges of the cube c. If gl = exp(iaAµeµ

l ), for suffi-ciently small a the plaquette variable gp for p in theµ, ν plane is gp = exp(ia2Fµν + O(a2)). Since Ja-

cobi identities hold for the curvature tensor F(0)µν , the

most straightforward way to satisfy these identitiesis to assume that on the vortex worldsheet Ση(x)has the same direction in su(2) Lie algebra as theprojection of the Hodge dual of the curvature ten-

sor F(0)µν on the vortex worldsheet. Indeed, in this

case exp(ia2F(0)µν + iηt̃µν) = exp(ia2F

(0)µν ) exp(iη) =

−1 · exp(ia2F(0)µν ), and the Jacobi identities only con-

strain the number of boundary plaquettes with Fµν =F

(0)µν + ηt̃µν to be even for any lattice cube. This

simply means that the worldsheets of center vorticesshould be closed surfaces. Taking into account that|η(x)| =

√Trη2(x) =

√2π = const, one can rewrite

the function η(x) for x ∈ Σ as follows:

η(x(σ)) =√

2πF(0)µν (x(σ))t̃µν(σ)√

Tr(F (0)µν (x(σ))t̃µν(σ))2

. (6)

In order to regularize the δ function in (4) and at thesame time to preserve the transformation properties ofFµν , it is necessary to extend the definition of η(x) toall points in the vicinity of Σ. One possible extensionvalid for sufficiently smooth vortex worldsheets is

η(x) =√

2πF(0)µν (x)t̃µν(σ(x))√

Tr(F (0)µν (x)t̃µν(σ(x)))2

, (7)

where σ(x) is a point on Σ which is closest to x.Before calculating the action of vortex configu-

rations with the curvature tensor (4), an importantremark should be made concerning the functionη(x(σ)) defined by (6). By gauge transformations one

can always rotate F(0)µν (x(σ))t̃µν(σ) in such a way

that it has a fixed direction in su(2) Lie algebra. Sincecenter vortices percolate and therefore fill almost allphysical space (in the sense that for any fixed point

x the average distance between x and the vortexworldsheet tends to zero in the continuum limit), onecan expect that numerically such gauge transforma-tion is close to the maximal Abelian projection used

to detect Abelian monopoles [9]. Thus F(0)µν t̃µν is in

fact only a scalar function on Σ, and if F(0)µν t̃µν is

not equal to zero everywhere on Σ, then accordingto (7) η(x(σ)) is just a constant. In general the

equation F(0)µν t̃µν = 0 is satisfied on some lines on Σ,

on which the definition (7) fails. In order to define η(x)everywhere, the expression (7) should be regularized,for example, as follows:

η(x) =√

2πF(0)µν (x)t̃µν(σ(x))√

Tr(F (0)µν (x)t̃µν(σ(x)))2 + ε2

, (8)

where ε → 0, so that η(x(σ)) = 0 if F(0)µν (x(σ)) ×

t̃µν(x(σ)) = 0. The lines with F(0)µν t̃µν = 0 separate

the regions on Σ with opposite orientations of Abelianmagnetic flux and thus are nothing but the worldlinesof Abelian magnetic monopoles [10]. Thus in ourapproximation the configuration of monopole world-lines on the vortex worldsheets is determined by the

curvature tensor of perturbative gauge field A(0)µ on Σ.

The contribution of Abelian monopoles to theeffective vortex action will be discussed later, forthe time being consider vortices without Abelianmonopoles or, more realistically, the regions on Σbounded by monopole worldlines. If the function η(x)is defined as (7), the action of the field configurationwith the curvature tensor (4) is

S =1

4g2s

∫d4xTr(FµνFµν) (9)

=1

4g2s

∫d4xTr

(F (0)

µν (x)

+ η(x)∫

Σ

d2σ√

gt̃µν(σ)δ(x − x(σ))

),

(F (0)

µν (x) + η(x)∫

Σ

d2σ√

gt̃µν(σ)δ(x − x(σ))

)

= S(0)YM +

π

2√

2g2s

∫

Σ

d2σ√

g

√Tr(t̃µνF

(0)µν )2 + Scv[Σ],

where S(0)YM =

14g2

s

∫d4xTr(F (0)

µν F(0)µν ) is the Yang–

Mills action for the gauge field A(0)µ , gs is the coupling

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 2 2009

374 BUIVIDOVICH

constant of Yang–Mills theory, and Scv[Σ] is the ac-tion of the ideal vortex configuration [10]:

Scv[Σ] =2π2

4g2s

∫

Σ

d2σ1

√g(σ1) (10)

×∫

Σ

d2σ2

√g(σ2)tµν(σ1)tµν(σ2)δ(x(σ1) − x(σ2)).

Substituting (9) into (2), one can express the effectivevortex action as:

Weff[Σ] = Scv[Σ] (11)

− ln

⟨exp

(− π

2√

2g2s

∫

Σ

d2σ√

gF (σ))⟩

0

,

where F (σ) =√

Tr(t̃µν(σ)F (0)µν (x(σ)))2 and 〈. . .〉0

denotes averaging over all gauge fields without centervortices.

Consider first the action of the ideal vortex con-figurations Scv[Σ]. In order to calculate the UV-divergent integral over σ1, σ2 in (11), the δ function inthe integrand should be regularized at the UV-cutoffscale. Instead of considering lattice gauge theoryagain, in the case of sufficiently smooth vortices itis more convenient to regularize this δ function as aGaussian distribution with dispersion Λ−1

UV:

δ(x) = limΛUV→∞

δΛ(x), (12)

δΛ(x) =Λ4

UV

(2π)2exp(−Λ2

UVx2/2).

In the limit ΛUV → ∞ one can change the integrationvariables in the last summand of (9) to σa = (σa

1 +σa

2)/2, ra = σa2 − σa

1 and assume that the range ofintegration over r is the whole real plane. For suffi-ciently smooth vortex worldsheets the double integralover σ1 and σ2 in (9) can be taken by expanding theintegrand in powers of r and taking into account onlythose terms that yield nonnegative powers of ΛUV.The integrand is proportional to Λ4

UV and each powerof r or integration over r gives a factor of order Λ−1

UV,thus only the terms up to second order in r shouldbe retained in the expansion of

√g(σ1,2)tµν(σ1,2) =√

g(σ ± r/2)tµν(σ ± r/2):∫

Σ

d2σ1

√g(σ1)

∫

Σ

d2σ2

√g(σ2) (13)

× tµν(σ1)tµν(σ2)δΛ(x(σ1) − x(σ2))

=∫

Σ

d2σ

∫d2r

√g(σ + r/2)

√g(σ − r/2)

× tµν(σ + r/2)tµν(σ − r/2)δΛ(x(σ + r/2)

− x(σ − r/2)) =∫

Σ

d2σ

∫d2r

(gtµνtµν

+√

gtµν∂2ab(

√gtµν)

rarb

4

− ∂a(√

gtµν)∂b(√

gtµν)rarb

4

)Λ4

UV

(2π)2

× exp(−

Λ2UV

2(x(σ + r/2) − x(σ − r/2))2

).

To the lowest order in ΛUV, the argument of the

exponent isΛ2

UV

2gabr

arb, and the integrals in the last

equation of (13) are just Gaussian. One more termwhich contributes to the integral at the order Λ0

UVis the first nonharmonic term in the expansion of(x(σ + r/2) − x(σ − r/2))2 in powers of r:

(x(σ + r/2) − x(σ − r/2))2 = gabrarb

+112

∂3abcx

µ∂dxµrarbrcrd + O(r6).

Indeed, if one expands

exp(−Λ2

UV

2(x(σ + r/2) − x(σ − r/2))2

)

= exp(−Λ2

UV

2gabr

arb

)

×(

1 − Λ2UV

24∂3

abcxµ∂dx

µrarbrcrd + O(Λ2UVr6)

),

the integral of the fourth-order term in r is of orderΛ0

UV, since it contains two additional powers of ΛUV.Thus to perform integration over r in (13), one needsthe following Gaussian integrals:∫

d2rΛ4

UV

(2π)2exp

(−Λ2

UV

2gabr

arb

)=

Λ2UV

2π√

g, (14)

∫d2r

Λ4UV

(2π)2rarb exp

(−

Λ2UV

2gabr

arb

)(15)

=gab

2π√

g,

∫d2r

Λ4UV

(2π)2rarbrcrd exp

(−Λ2

UV

2gabr

arb

)(16)

=gabgcd + gacgbd + gadgbc

2πΛ2UV

√g

.

After integration over r expression (13) reads:∫

Σ

d2σ√

g

8π(8Λ2

UV + g−1gab(tµν∂2abt

µν (17)

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 2 2009

ON THE EFFECTIVE ACTION 375

− ∂atµν∂bt

µν) − gabgcd∂3abcx

µ∂dxµ).

Now all the summands in (13) should be arrangedinto explicitly covariant combinations. The deriva-tives of the normal two-form tµν can be transformedas follows:

gabg−1(√

gtµν∂2ab

√gtµν (18)

− ∂a√

gtµν∂b√

gtµν) = gabtµν∂2abt

µν

− ∂atµν∂bt

µν − 2gab∂a ln√

g∂b ln√

g

+ 2gab 1√

g∂2

ab

√g = tµν∆tµν

−∇atµν∇atµν + 2gab∂aΓb,

where Γab;c are Cristoffel symbols constructed from the

metric gab, Γa = ∂a ln√

g = Γbb;a, ∇a is the covari-

ant derivative on the vortex worldsheet, ∆ = ∇a∇a

is the covariant two-dimensional Laplacian, and theidentities tµν∂tµν = 0 and ∂tµν∂tµν + tµν∂2tµν = 0which follow from tµνtµν = const were used. Now theterm gabgcd∂3

abcxµ∂dx

µ should be transformed. Thefirst step is to reduce the number of derivatives of xµ:

gabgcd∂3abcx

µ∂dxµ (19)

= gabgcd∂c(∂3abx

µ∂dxµ) − gabgcd∂2

abxµ∂2

cdxµ.

Now one can use the identity Γab;c = gad∂dx

µ∂2bcx

µ

which follows from gab = ∂axµ∂bx

µ to transform theexpression in brackets. The term gabgcd∂2

abxµ∂2

cdxµ

can be expressed in terms of ∆xµ∆xµ plus somenoncovariant terms:

gabgcd∂3abcx

µ∂dxµ (20)

= gab∂cΓca;b + ΓaΓa − ∆xµ∆xµ,

where Γa = gbcΓab;c. Combining (18) and (20), one

can rewrite (17) as∫

Σ

d2σ√

g

8π(8Λ2

UV + tµν∆tµν (21)

−∇atµν∇atµν + 2gab∂aΓb − gab∂cΓc

a;b

− ΓaΓa + ∆xµ∆xµ).

It turns out that the noncovariant terms in (21) timesthe square root of g can be grouped into minus twicethe scalar curvature of Σ plus the total derivativewhich do not contribute to the action, because Σ hasno boundary. Finally, the regularized action (10) canbe represented in the following covariant form:

Scv[Σ] =π

16g2s

∫

Σ

d2σ√

g(8Λ2UV − 3K − R), (22)

where K = ∇a∇bxµ∇a∇bxµ = 1

2∇atµν∇atµν andR = gbcRa

b;ac = ∆xµ∆xµ −∇a∇bxµ∇a∇bxµ are the

extrinsic and the internal curvatures of the vortexworldsheet. Note that in contrast to the result derivedin [10], the action (22) has an explicitly covariantform, because the contribution of the first anharmonicterm in the expansion of the exponent was taken intoaccount. As in [10, 13], smooth center vortices turnout to be anti-rigid and thus unstable with respect tosurface creasing. The meaning and the consequencesof such instability will be discussed somewhat later.

The second term in (11) is the quantum correctionto the vortex effective action due to virtual gluon ex-change between different points on the vortex world-sheet. It can be represented in a more convenient

form using the identity ln〈exp(x)〉 =∑+∞

k=1

1k!〈〈xk〉〉,

where 〈〈xk〉〉 denotes the kth-order connected cu-mulant of a random variable x: 〈〈x〉〉 = 〈x〉, 〈〈x2〉〉 =〈x2〉 − 〈x〉2, and so on. Thus the effective vortex ac-tion can be represented in the following form:

Weff[Σ] = Scv[Σ] ++∞∑k=1

(−1)k+1

k!πk

(2√

2g2s)k

(23)

×∫

Σ

. . .

∫

Σ

d2σ1 . . . d2σk〈〈F (σ1) . . . F (σk)〉〉.

As discussed above, since removing center vor-tices from gauge field configurations typically re-moves also all nonperturbative physics, in the re-gions on Σ bounded by monopole worldlines thecumulants of F [σ] can be reliably calculated usingperturbation theory. The first cumulant 〈F (σ)〉 yieldsjust a constant contribution to the vortex tension.The second cumulant 〈〈F (σ1)F (σ2)〉〉 gives a neg-ative contribution to the integral (23). By analogywith the calculation (13) and (22) one can expectthat this decreases the string tension and increasesvortex rigidity, although this should be checked byan explicit calculation. In any case, the results oflattice simulations show that the effect of gluonexchanges on vortex dynamics is rather small, at leastat presently available lattices, and much strongereffects are associated with Abelian monopoles lo-calized on vortex worldsheets [14]. Some qualita-tive features of the effective theory of vortices andAbelian monopoles on them can be inferred alreadyfrom (9), (11), and (23). On monopole worldlines,

where F(0)µν (x(σ))t̃µν (x(σ)) = 0, the function η(x)

goes to zero in a region with the width of order ofthe lattice spacing a in the direction perpendicularto the worldline. Correspondingly, the action densityon the vortex worldsheet (10) also goes from theUV-divergent value Λ2

UV to zero. The contributionof monopoles to the vortex action can be therefore

estimated as δS(mon)cv ∼ −Λ2

UVaL ∼ −ΛUVL, where

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 2 2009

376 BUIVIDOVICH

L is the total length of monopole worldlines. Thus themonopoles appear to have a tachyonic mass, whichagrees well with their percolation which was observedin lattice simulations [9]. Since the total density ofmonopoles scales in physical units [9], there shouldexist some other mechanism which stops monopolecondensation at scales of order of ΛQCD. The re-sults of lattice simulations suggest that monopoleworldlines are rigid [9], and it is also reasonable toexpect that the calculation similar to (13) and (22)would give a negative monopole mass and positiverigidity of monopole worldlines. In order to checkthis explicitly, the calculation similar to (13) and (22)should be repeated for one-dimensional objects—monopole worldlines. According to Eq. (6) and thediscussion which follows it, the information aboutmonopole worldlines is encoded in the fields A

(0)µ and

the configuration of vortex worldsheets Σ, thus thecorresponding Jacobian for the change of variables

from A(0)µ to the configurations of monopole currents

should also be calculated.To conclude, in this work some basic observations

which come from lattice studies of center vorticeswere applied to analyze the effective action of cen-ter vortices in continuum Yang–Mills theory withSU(2) gauge group. The effective vortex action wasfound to be a sum of the action of the ideal vortexconfigurations, i.e., the field configurations whichconsist of center vortices only, the contribution fromAbelian monopoles and the quantum corrections dueto virtual-gluon exchanges. As discussed in [11],center vortices seem to be responsible for mostnonperturbative phenomena, which allows one to cal-culate these quantum corrections reliably in the fewlowest orders of perturbation theory. An importantresult is that the effective vortex action contains theterms which favor nonregular, creased surfaces withvery large values of extrinsic and internal curvatures.Although such terms in the action can provide aqualitative explanation of the observed geometricproperties of vortices [15], in fact anti-rigidity givesno useful quantitative information, except for theindication that the string vacuum of center vorticesis not the vacuum of smooth strings, for which theexpressions (13) and (22) were derived. In this respectanti-rigidity is similar to the imaginary mass of Higgsparticle in the perturbative vacuum with 〈φ〉 = 0,which gives no information about the physical Higgsmass. Center vortices observed on the lattice are rigid,but are not smooth [14, 15]. A true string vacuumof center vortices should be therefore described us-ing completely different techniques, which take intoaccount the existence of percolating vortex clusterswith characteristic scale of geometric structures of

order of UV cutoff. Technically this means that inthe expansions like (13) and (22) one should counteach partial derivative as a power of ΛUV, whichmakes the very idea of such expansions meaningless.Unfortunately, up to now it is not known what are theadequate methods for addressing such problems inquantum field theory.

ACKNOWLEDGMENTS

This work was partly supported by Federal Pro-gram of the Russian Ministry of Industry, Scienceand Technology no. 40.052.1.1.1112 and by RussianFederal Agency for Nuclear Power.

The author is grateful to all members of ITEPlattice group for their kind hospitality, and to M.I. Po-likarpov, V.I. Zakharov, and E.T. Akhmedov for inter-esting and stimulating discussions.

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