Crossing the Gribov horizon: an unconventional study of geometric properties of gauge-configuration space in Landau gauge
aa r X i v : . [ h e p - l a t ] N ov Crossing the Gribov horizon:an unconventional study of geometric propertiesof gauge-configuration space in Landau gauge
Attilio Cucchieri ∗ † Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970 SãoCarlos, SP, BrazilE-mail: [email protected]
Tereza Mendes
Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970 SãoCarlos, SP, BrazilE-mail: [email protected]
We prove a lower bound for the smallest nonzero eigenvalue of the Landau-gauge Faddeev-Popovmatrix in Yang-Mills theories. The bound is written in terms of the smallest nonzero momentumon the lattice and of a parameter characterizing the geometry of the first Gribov region. Thisallows a simple and intuitive description of the infinite-volume limit in the ghost sector. In partic-ular, we show how nonperturbative effects may be quantified by the rate at which typical thermal-ized and gauge-fixed configurations approach the Gribov horizon. Our analytic results are verifiednumerically in the SU(2) case through an informal, free and easy, approach. This analysis pro-vides the first concrete explanation of why the so-called scaling solution of the Dyson-Schwingerequations is not observed in lattice studies. ∗ Speaker. † We acknowledge partial support from FAPESP ( grant ) and from CNPq. We would like to ac-knowledge computing time provided on the Blue Gene/P supercomputer supported by the Research Computing SupportGroup (Rice University) and Laboratório de Computação Científica Avançada (Universidade de São Paulo). c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ rossing the Gribov horizon
Attilio Cucchieri
1. Infinite-Volume Limit and the Boundary of the First Gribov Region W Since the original work by Gribov [1], innumerous numerical and analytic studies (see, forexample, the reviews [2, 3]) have focused on the infrared (IR) behavior of Yang-Mills Green’sfunctions in minimal-Landau gauge and on its connection to color confinement [4]. On the lattice,a detailed description of the IR sector of Yang-Mills theories requires an extrapolation to infinitevolume. In minimal-Landau gauge, this extrapolation should be governed by a (widely accepted)axiom stating that At very large volumes, the functional integration gets concentrated on the boundary ¶ W ofthe first Gribov region W [defined by transverse gauge configurations with all nonnegativeeigenvalues of the Faddeev-Popov (FP) matrix M ]. Thus, the functional integration should be strongly dominated at very large volumes by configura-tions belonging to a thin layer close to ¶ W , i.e. typical configurations should be characterized byvery small values for the smallest nonzero eigenvalue l of M . Indeed, numerical studies showthat l goes to zero as the lattice volume increases.In Ref. [5] we have introduced the following inequalities for the ghost propagator G ( p ) inmomentum space 1 N c − l (cid:229) b | e y ( b , p ) | ≤ G ( p ) ≤ l , (1.1)where e y ( b , p ) is the (Fourier-transformed) eigenvector of the FP matrix M corresponding tothe eigenvalue l and b = , , . . . , N c − N c − N c ) gauge group. Note that, on the lattice and for large lattice volumes, the smallestnonzero momentum p min is of the order of 1 / L , where L is the lattice size. Thus, if l behaves as L − − a in the infinite-volume limit, the inequality a > G ( p min ) ∼ / p + k min (with k > | e y ( b , p min ) | behaves as L − g at large L , the inequality a − g > k >
0. Therefore, in orderto describe the extrapolation to infinite volume in the ghost sector, we can re-formulate the aboveaxiom and say that
The key point seems to be the rate at which l goes to zero, which, in turn, should berelated to the rate at which a thermalized and gauge-fixed configuration approaches ¶ W . This is, however, just a qualitative statement. Indeed, in order to make the above axiom quantitative,we need to relate the eigenvalue l to the geometry of the Gribov region W . A first step in thisdirection was taken in Ref. [7]. There, we proved a lower bound for l [see Eq. (2.9) below] that This axiom is explained by considering the interplay among the volume of configuration space, the Boltzmannweight associated to the gauge configurations and the step function used to constrain the functional integration to theregion W . At the same time, one needs a good projection of the eigenvector y ( b , x ) on the plane waves corresponding to themomentum p min . , i.e. the exponent g should not be too large. rossing the Gribov horizon Attilio Cucchieri relates this eigenvalue to the distance of the gauge configuration A ∈ W from the boundary ¶ W . Asa consequence, we were able to provide the first concrete explanation of why the scaling solutionof the DSEs is not observed in lattice studies. The main results of Ref. [7] are presented below.
2. Lower bound for l The (lattice) Landau gauge is usually imposed by minimizing the functional E [ U ; w ] = − Tr (cid:229) x , m w ( x ) U m ( x ) w † ( x + a e m ) (2.1)with respect to the lattice gauge transformations w ( x ) ∈ SU( N c ). This defines the first Gribovregion W ≡ { U : ¶ · A = M = − D · ¶ ≥ } , (2.2)where D bc ( x , y )[ A ] is the covariant derivative and M ( b , x ; c , y )[ A ] is the FP matrix. One can show[8] that all gauge orbits intersect W and that this region is characterized by the following threeproperties [9] (see also [3, 7]): ) the trivial vacuum A m = W ; ) the region W isconvex; ) the region W is bounded in every direction.From the definition of M [ A ] [see Eq. (2.2)] it is clear that the FP matrix has a trivial nulleigenvalue, corresponding to constant vectors. Then, if we indicate with l [ M [ A ] ] the smallestnonzero eigenvalue of M [ A ] , we can introduce the definition l [ M [ A ] ] = min c ( c , [ M [ A ] ] c ) , (2.3)where c are non-constant vectors such that ( c , c ) =
1. Also, by noticing that the operators D [ A ] , M [ A ] = − ¶ + K [ A ] and K [ A ] are linear in the gauge field A , one can write M [ r A ] = − ¶ + K [ r A ] = ( − r ) ( − ¶ ) + r M [ A ] . (2.4)At the same time, for A ∈ W , r ∈ [ , ] and using the first and second properties above, we have that r A ∈ W . This result applies in particular to configurations A ′ belonging to the boundary ¶ W of W .Thus, by using the definition (2.3), Eq. (2.4) in the case A ′ ∈ ¶ W and the concavity of the minimumfunction [10], i.e. min c ( c , [ M + M ] c ) ≥ min c ( c , M c ) + min c ( c , M c ) (where M and M aretwo generic square matrices), we obtain [7] l (cid:2) M [ r A ′ ] (cid:3) = l (cid:2) ( − r ) ( − ¶ ) + r M [ A ′ ] (cid:3) (2.5) = min c (cid:0) c , (cid:2) ( − r ) ( − ¶ ) + r M [ A ′ ] (cid:3) c (cid:1) (2.6) ≥ ( − r ) min c (cid:0) c , ( − ¶ ) c (cid:1) + r min c (cid:0) c , M [ A ′ ] c (cid:1) . (2.7) Here, we indicate with e m a unit vector in the positive m direction and with a the lattice spacing. For the gauge field A we consider the usual (unimproved) lattice definition (see for example Ref. [2]). This follows immediately if we write r A as the convex combination ( − r ) A + r A , with A = A = A . rossing the Gribov horizon Attilio Cucchieri
Since A ′ ∈ ¶ W , i.e. the smallest non-trivial eigenvalue of the FP matrix M [ A ′ ] is null, and since thesmallest non-trivial eigenvalue of (minus) the Laplacian − ¶ is p min , we find l (cid:2) M [ r A ′ ] (cid:3) ≥ ( − r ) p min . (2.8)Therefore, as the lattice size L goes to infinity, the eigenvalue l [ M [ r A ′ ] ] cannot go to zero fasterthan ( − r ) p min . In particular, since p min ∼ / L for large lattice size L , we have that l behavesas L − − a in the same limit, with a >
0, only if 1 − r goes to zero at least as fast as L − a . Let usstress that this result applies to any Gribov copy belonging to W .With r A ′ = A , the above inequality (2.8) may also be written as l [ M [ A ] ] ≥ [ − r ] p min . (2.9)Note that, in the Abelian case, one has M = − ¶ and l = p min , i.e. non-Abelian effects areincluded in the factor ( − r ) . At the same time, the quantity 1 − r ≤ A ∈ W from the boundary ¶ W (in such a way that A ′ = r − A ∈ ¶ W ). Thus, the new bound (2.9) suggests all non-perturbative features of a minimal-Landau-gauge configuration A ∈ W to be related to its normalized distance r from the “origin” A = − r from the boundary ¶ W ]. One should alsostress that the above inequality becomes an equality if and only if the eigenvectors correspondingto the smallest nonzero eigenvalues of M [ A ] and − ¶ coincide.As a consequence of the result (2.9), we can also find several new bounds. In particular, usingthe upper bound in Eq. (1.1) and the definition of the Gribov ghost form-factor s ( p ) , we have1 p min − s ( p min ) ≡ G ( p min ) ≤ [ − r ] p min (2.10)and therefore s ( p min ) ≤ r . This result is a stronger version of the so-called no-pole condition s ( p ) ≤ p > W . Similarly, for the horizon function H , defined in Eq. (3.10) of [12], one can prove that[7] HdV ( N c − ) ≡ h ≤ r . (2.11)
3. Simulating the Math
In order to verify the new bounds presented in the previous section, we started by consideringthe third property of the region W , i.e. the fact that W is bounded in every direction. To this end we“simulate” the mathematical proof of this property, i.e. given a thermalized gauge configuration A m ( x ) , we apply the scale transformations b A ( i ) m ( x ) = t i A m ( x ) such that: a ) t = b ) t i = dt i − , See, for example, [11] and references therein. In our simulations we used 70 thermalized configurations, for the SU(2) case at b = .
2, for lattice volumes V = , 24 , , and 50 configurations (at the same b value) for lattice volumes V = , , , , . With this rescaling we withdraw the unitarity of the link variables, thus losing the connection with the usual MonteCarlo simulations. In this sense, our approach is an informal, free and easy one. Nevertheless, it gives us useful insightsinto the properties of the Faddeev-Popov matrix M and of the first Gribov region W . Note also that the rescaled fieldstill respects the gauge condition. rossing the Gribov horizon Attilio Cucchieri N max ( n ) min ( n ) h n i R before / R after / Table 1:
The maximum, minimum and average number of steps n , necessary to “cross the Gribov horizon”along the direction A b m ( x ) , as a function of the lattice size N . We also show the ratio R [see Eq. (3.1)],divided by 1000, for the modified gauge fields b A ( n − ) m ( x ) = t n − A m ( x ) and b A ( n ) m ( x ) = t n A m ( x ) , i.e. for theconfigurations immediately before and after crossing ¶ W . c ) d = .
001 if l ≥ × − , d ) d = . l ∈ [ × − , × − ) and e ) d = . l < × − , with l evaluated at the step i −
1. Clearly, after n steps, the modified gaugefield b A ( n ) m ( x ) does not belong to the region W anymore, i.e. the eigenvalue l of M [ b A ( n ) ] is negative(while the eigenvalue l is still positive). Results for the number of steps n necessary to “cross theGribov horizon” along the direction A b m ( x ) are reported in Table 1. It is interesting to note that thevalue of n decreases as the lattice side N increases, i.e. configurations with larger physical volumeare (on average) closer to the boundary ¶ W .In the same table we also show the value of the ratio R = ( E ′′′ ) l E ′′′′ (3.1)for the rescaled gauge fields b A ( n − ) m ( x ) = t n − A m ( x ) and b A ( n ) m ( x ) = t n A m ( x ) , i.e. for the configu-rations immediately before and after crossing the first Gribov horizon ¶ W . The same ratio is alsoshown in Fig. 1 (left plot), as a function of the iteration step i , for a typical configuration and forthe lattice volume V = . For the same configuration we also show (see Fig. 1, right plot) thedependence of l , | E ′′′ | and of E ′′′′ on the iteration step i . One clearly sees that these quantitieshave a slow and continuous dependence on the factors t i . On the other hand, since l decreasesas t i increases, we find that the ratio R usually increases with t i and that R n − ≈ − R n , due to thechange in sign of l as the first Gribov horizon is crossed (see the fourth and the fifth columns in Here, E ′′′ and E ′′′′ are the third and the fourth derivatives of the minimizing functional, defined in Eq. (2.1),evaluated along the direction of the eigenvector y ( b , x ) corresponding to the eigenvalue l . As shown in Ref. [13], thisratio characterizes the shape of the minimizing functional E , around the local minimum considered, when one appliesto E a fourth-order Taylor expansion (see in particular Figure 2 of the same reference). However, for a few configurations, we found [7] a very small value for the ratio R for all factors t i , i.e. also whenthe configuration b A ( i ) m ( x ) is very close to ¶ W . We interpret these configurations as possible candidates to belong to thecommon boundary ¶ W ∩ ¶ L . Here L is the fundamental modular region [12, 14], obtained by considering absoluteminima of the minimizing functional E [ U ; w ] defined in Eq. (2.1). rossing the Gribov horizon Attilio Cucchieri
Figure 1:
Left: plot of the ratio R [see Eq. (3.1)], as a function of the iteration step i , for a typicalconfiguration and lattice volume V = . Right: plot of l ( full circes ), | E ′′′ | ( full squares ) and E ′′′′ ( fulltriangles ) as a function of the iteration step i , for the same configuration considered in the left plot. Table 1 and the left plot in Fig. 1). At the same time one can check (see the right plot in Figure1) that the second smallest (non-trivial) eigenvalue l stays positive, i.e. the final configuration b A ( n ) m ( x ) = t n A m ( x ) belongs to the second Gribov region [1, 12].Once we have found a configuration b A ( n ) m / ∈ W , we can use the definition A ′ m ( x ) = b A ( n − ) m ( x ) + b A ( n ) m ( x ) = t n − + t n A m ( x ) ≡ e t A m ( x ) (3.2)as a candidate for a configuration belonging to the boundary ¶ W . This gives us an estimate forthe parameter r = / e t < A are very close to the first Gribovhorizon ¶ W , i.e. one usually finds r ≈ − r goes to zero reasonably fast. On the other hand, the inequality (2.9) is far from being saturated bythe lattice data (see Figure 2, right plot), and the situation seems to become worst in the infinite-volume limit. The same observation applies (see Fig. 2, right plot) to the lower bound in Eq.(1.1). Finally, for large lattice size L one finds G ( p min ) ∼ < / l (see again Figure 2, right plot).These results are all consistent with the fact that the eigenvector y is very different from the planewaves corresponding to p min , which clarifies why the ghost propagator G ( p ) is not enhanced inthe IR limit. Conversely, configurations producing an IR-enhanced ghost propagator should almostsaturate the new bound (2.9), i.e. their eigenvector y should have a large projection on at leastone of the plane waves corresponding to p min . Thus, in the scaling solution [6], nonperturbativeeffects, such as color confinement, should be driven by configurations whose FP matrix M is“dominated” by an eigenvector y very similar to the corresponding eigenvector of M = − ¶ ,i.e. to the eigenvector y of the free case! This would constitute a very odd situation indeed. Onthe contrary, the massive solution of the DSEs of gluon and ghost propagators [16] is consistentwith the more reasonable hypothesis that the eigenvector y is in general very different from a freewave. 6 rossing the Gribov horizon Attilio Cucchieri
Figure 2:
Left: plot of the (normalized) horizon function h ( empty circles ), of the Gribov ghost form-factor s ( p min ) ( full triangles ), of the quantity 1 − l / p min ( full squares ) and of their upper bound r ( full circles )as a function of the inverse lattice size 1 / N . Let us note that, in Ref. [15], it was proven that s ( ) = h to allorders in the gauge coupling. Right: plot of the inverse of the lower bound in Eq. (1.1) ( empty circles ), of1 / G ( p min ) ( full triangles ), of l ( full squares ) and of the quantity ( − r ) p min ( full circles ) as a functionof the inverse lattice size 1 / N . References [1] V. N. Gribov,
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