Perturbative analysis of twisted volume reduced theories
Margarita Garcia Perez, Antonio Gonzalez-Arroyo, Masanori Okawa
aa r X i v : . [ h e p - l a t ] N ov Perturbative analysis of twisted volume reducedtheories
Margarita García Pérez ∗ Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, E-28048-Madrid,SpainE-mail: [email protected]
Antonio González-Arroyo
Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, C-15, UniversidadAutónoma de Madrid, E-28049-Madrid, SpainE-mail: [email protected]
Masanori Okawa
Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526,JapanE-mail: [email protected]
We discuss the perturbative expansion of SU(N) Yang-Mills theories defined on a d-dimensionaltorus of linear size l with twisted boundary conditions, generalizing previous results in the litera-ture. For a specific class of twist tensors depending on a single integer flux value k , we show thatperturbative results to all orders depend on the combination lN / d and a flux-dependent angle ˜ q .This implies a new kind of volume independence that holds at finite N and for fixed values of ˜ q .Our results also provide interesting information about the possible occurrence of tachyonic insta-bilities at one-loop order. We support the prescription that instabilities are avoided, if the largeN limit is taken keeping ˜ q > ˜ q c , and appropriately scaling the magnetic flux k with N . Numeri-cal results in 2+1 dimensions provide a test of how these ideas extend into the non-perturbativeregime. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ erturbative analysis of twisted volume reduced theories
Margarita García Pérez
1. Introduction
In a recent paper [1], focused on the study of 2+1 dimensional SU( N ) Yang-Mills theorydefined on a 2-d torus, we have analyzed the interplay between the rank of the group N and thefinite volume effects in the presence of a non-trivial magnetic flux. We have presented perturbativeand numerical evidence of a kind of volume independence in the theory, reflected in the combined Nl dependence of physical quantities, l being the size of the 2-dimensional spatial manifold. Here,we will generalize the perturbative results to the case of a four dimensional set-up with the kind oftwisted boundary conditions relevant for Twisted Eguchi-Kawai (TEK) reduction, as put forwardby two of the present authors [2, 3]. We will start by presenting the perturbative set-up for a Yang-Mills theory with twisted boundary conditions, and the interplay between finite volume and finite N effects in this context. We will continue by discussing possible caveats to the volume independenceconjecture, including the presence of tachyonic instabilities [4] and the appearance of symmetrybreaking in the TEK model for large values of N [5]. We will argue that the prescription to scale themagnetic twist with N , put forward in Refs. [2, 3] to prevent the latter, also succeeds in avoidingthe tachyonic behaviour. We will end by presenting some numerical results in 2+1 dimensions thatprovide a test on the realization of volume independence at a non-perturbative level.
2. Perturbative set-up
To set the stage, let us start with a brief and general introduction on the formulation of theSU(N) Yang-Mills theory with twisted boundary conditions. The notation and the discussion willfollow the review [6]. We start with a manifold composed of a d-dimensional torus, of lengths l m , times some non-compact extended directions. The latter will not be relevant for our purposesand will be neglected in the discussion below. Gauge fields on this base space satisfy periodicityconditions in the compact directions given by: A m ( x + l n ˆ e n ) = W n ( x ) A m ( x ) W † n ( x ) + i W n ( x ) ¶ m W † n ( x ) , (2.1)where the SU( N ) twist matrices W m ( x ) are subject to the consistency conditions: W m ( x + l n ˆ e n ) W n ( x ) = Z mn W n ( x + l m ˆ e m ) W m ( x ) , (2.2)with Z mn = exp { p in mn / N } , and n mn ∈ ZZ N . In what follows, we will focus on the case of constanttwist matrices W m ( x ) = G m . They are known under the name of twist-eaters. For the so-called irre-ducible twists, it can be shown that they are uniquely defined modulo global gauge transformations(similarity transformations) and multiplication by an element of ZZ N [6]. We will be consideringhere the case of even number of twisted compactified directions d . If we choose N = L d / , with L ∈ ZZ , the twist n mn = e mn k N / L , with e mn = Q ( n − m ) − Q ( m − n ) (where Q is the step function),is irreducible if k and L are co-prime. In what follows we will focus on this specific twist choice.The periodicity constraint: A m ( x + l n ˆ n ) = G n A m ( x ) G † n , (2.3)can be resolved by introducing a basis of the space of N × N matrices satisfying: G m ˆ G ( p ( c ) ) G † m = e il m p ( c ) m ˆ G ( p ( c ) ) . (2.4)2 erturbative analysis of twisted volume reduced theories Margarita García Pérez
The basis can be constructed in terms of the twist matrices as:ˆ G ( p ( c ) ) = √ N e i a ( p ( c ) ) G s · · · G s d − d − (2.5)with a ( p ( c ) ) an arbitrary phase factor, and: p ( c ) m = p L l m e mn k s n , (2.6)with s n integers defined modulo L . In the case of irreducible twists, one can show that there are N linearly independent such matrices which, excluding the identity, provide a basis of the SU ( N ) Liealgebra. The phase factors, a ( p ( c ) ) , can be chosen to satisfy the following commutation relations: [ ˆ G ( p ) , ˆ G ( q )] = i F ( p , q , − p − q )) ˆ G ( p + q ) , (2.7)with F ( p , q , − p − q ) = − r N sin (cid:18) q mn p m q n (cid:19) , (2.8)playing the role of the SU(N) structure constants in this particular basis. Here, q mn = L l m l n p × ˜ e mn ˜ q , (2.9)with ˜ e mn e ns = d ms , and the angle ˜ q = p ¯ k / L , where ¯ k is an integer satisfying k ¯ k = L ).We can now expand the gauge fields in this basis: A n ( x ) = √ V ′ (cid:229) p e ip · x ˆ A n ( p ) ˆ G ( p ) , (2.10)with momenta decomposed as p m = p ( s ) m + p ( c ) m , the sum of a colour-momentum part p ( c ) m , and aspatial-momentum part quantized in the usual way: p ( s ) m = p m m / l m , with m m ∈ ZZ . The primeimplies that the zero colour-momentum component is excluded from the sum. Neglecting thisissue, the momentum is quantized as if the theory lived in an effective torus with sizes l eff m = L l m .Note also that the Fourier coefficients ˆ A n ( p ) are just complex numbers, so that all the effect of thecolour is translated into the momentum dependence of the group structure constants.One can now easily generalize the Feynman rules derived for d = g √ V F ( p , q , − p − q ) = − s l V eff sin (cid:16) q mn p m q n (cid:17) , (2.11)where V eff ≡ (cid:213) m l eff m . This peculiar momentum dependent Feynman rules relate the twisted theorywith a non-commutative Yang-Mills theory with non-commutativity parameter q mn [7]-[9].3 erturbative analysis of twisted volume reduced theories Margarita García Pérez
3. Volume independence
Let us now analyze the dependence of the results on the rank of the gauge group N and thesizes of the torus l m . Recalling the definition of q mn in Eq. (2.9) and the momentum quantizationrule, it is clear that all the N and l m dependence enters only through the combination l eff m = Ll m ,and the angle ˜ q = p ¯ k / L . This implies that the perturbative expansion, at fixed ˜ q and l eff , dependsin an indistinguishable way on N and the torus size. We dub this phenomenon volume reduction or volume independence at finite N . A limiting case would be TEK reduction which applies to adiscretized version of the Yang-Mills theory in which l m = a (the lattice spacing). As a matter offact, TEK models have been used as a regularized version of non-commutative gauge theories withnon-commutativity parameter: q TEK mn = L a p × ˜ e mn ˜ q , (3.1)Beyond perturbation theory, there are, however, possible caveats to the volume independenceconjecture. As mentioned previously, several authors realized the presence of tachyonic insta-bilities in certain non-commutative theories [4]. These extend to ordinary theories with twistedboundary conditions and present a menace to the volume independence mechanism. The problemoccurs at one loop in perturbation theory. In Ref. [1] we saw how the problem does not arise in2+1 dimensions if one adopts the large N prescription given in Ref. [3]. The argument extends to4 dimensions as well, and goes as follows. The transverse part of the 2-point vertex function has anon-zero value at leading order, since twisted boundary conditions eliminate zero-momentum glu-ons. This contribution is proportional to | p | ∼ p / l . At one loop, the self-energy contributionis negative and proportional to l ˜ p m ˜ p n | ˜ p | (3.2)where ˜ p m = q mn p n . Instability arises if the second term is larger than the first. This occurs forlarge enough l where the calculation is unreliable. However, since the first term goes to zeroas l eff goes to infinity, one might wonder if instability could arise in that limit. The prescriptiongiven in Ref. [3] amounts to taking the l eff −→ ¥ limit with q mn given by Eq. (2.9) and keeping˜ q fixed. Plugging this expression in Eq. (3.2), one sees that the negative self-energy also goes tozero as l eff goes to infinity and the critical l remains finite, and of order ˜ q . Thus, as supported byour numerical results, no instability should arise for ˜ q > ˜ q c . The previous evidence for instabilityoccurred when taking a different limit, in which ˜ q was decreasing to zero as 1 / L .Additional complications could arise from the fact that it is impossible to strictly keep ˜ q fixedas N changes. This is so because ˜ q is a rational number with coprime rational factors ¯ k and L .Smoothness of physical quantities on ˜ q is thus required for volume independence to hold. Our re-sults for the electric flux energies and the perturbative glueball spectrum in 2+1 dimensions exhibitsuch a smooth dependence, but the issue is difficult to settle in general terms and it has indeed beendiscussed profusely in the context of non-commutative gauge theories without conclusive results(see e.g. [10], [11]). Finally, effects arising from non-perturbative physics might not respect the l eff dependence. There are indications that this is so for certain twist choices in TEK models, wherereduction fails due to spontaneous symmetry breaking at large N [5]. It has been recently shown,however, that symmetry breaking can be avoided if (in addition to keeping ˜ q > ˜ q c ) the magneticflux is scaled with L , as L goes to infinity [2, 3]. 4 erturbative analysis of twisted volume reduced theories Margarita García Pérez
Most of these questions can only be addressed non-perturbatively. In what follows we willsummarize the results of a numerical analysis for the case of 2+1 dimensions [1]. The analysis for d =
4. Non-perturbative results in 2+1 dimensional SU(N) Yang-Mills theory
Our analysis will be focused on the study of the electric flux energies, E , extracted fromPolyakov loop correlators. It is easy to show that these operators carry electric flux e i = − e i j n j ¯ k ,determined by the gluon colour momentum ~ p = p ~ n / ( Nl ) . In perturbation theory at one-loop, acompact formula, exhibiting a smooth ˜ q dependence, has been derived in Ref. [1]: E l = | ~ n | x − x G (cid:16) ~ eN (cid:17) , (4.1)where G ( z ) = p Z ¥ dt √ t (cid:16) q ( , t ) − (cid:213) i = q ( z , it ) − t (cid:17) , (4.2)in terms of the Jacobi theta function: q ( z , it ) = (cid:229) n ∈ Z exp {− t p n + p inz } . We have introducedthe dimensionless parameter x = Nl l / p . Note that in 3 dimensions the coupling constant isdimensionful. Thus, all energy scales can be expressed in units of l and the resulting dimensionlessquantities should appear in perturbation theory as a power series in l l . Combining this informationwith volume independence, we conclude that the relevant scale parameter in perturbation theory isprecisely x (for a similar statement involving Nl L QCD in 4-d see [12] and [10]).In Ref. [1] we have presented evidence that x -scaling holds beyond perturbation theory forchoices of the twist that do not exhibit tachyonic instability. As an illustration, we present inthe left plot of Fig. 1 an analysis of electric-flux states with minimal momentum ~ p = ( p / Nl , ) .We display the x dependence of the combination x E / l , for ( N , ¯ k ) = ( , ) and ( , ) , corre-sponding to very close values of ˜ q . We stress the striking similarity of the results despite the verydifferent values of N . Other results, for varying N and ¯ k , confirm this conclusion, giving support tothe conjecture of a universal x -dependence for fixed ˜ q . The data show that the small- x behaviourfollows the perturbative formula, starting at the tree-level result x E / l = .
25. At large torussizes, we expect a linear growth of the energy that can be cast in a form that also exhibits x -scaling: E ( ~ e / N ) l = p x s ′ l f (cid:16) ~ eN (cid:17) , (4.3)where the string tension, for electric flux ~ e , has been parametrized as: s ~ e = N s ′ f ( ~ e / N ) . Higherorder string corrections to this formula can also be taken into account, including the contributionof Kalb-Ramond B -fields which play an important role in the twisted set-up [10]. Indeed, theobservation that the B -field contribution amounts precisely to the perturbative tree-level term inEq. (4.1) has guided us in the search for an x -dependent parameterization that fits very well the data.The reader is referred to Ref. [1] for further details and a full account of the results. Incidentally, letus mention that we have analyzed the dependence of the string tension on the electric flux, finding aclear preference for Sine scaling with f ( z ) = sin ( p z ) / p , over Casimir scaling with f ( z ) = z ( − z ) .Our results allow to analyze in detail the issue of tachyonic instabilities. The general perturba-tive argument presented in Sec. 3 can be refined using Eq. (4.1). As already discussed, the one-loop5 erturbative analysis of twisted volume reduced theories Margarita García Pérez -1 0 1 2 3 4 5 0 1 2 3 4 x e / l x= l Nl/(4 p )N=17, k=7, k_ =5N=7, k=3, k_=2 0 1 2 3 4 5 6 7 8x= l Nl/(4 p )N=17, k=1, k_ =1 0 1 2 3 4 5 6 7 8x= l Nl/(4 p )N=7, k=2, k_ = 3N=17, k=2, k_ = 8 Figure 1:
We display x E / l , as a function of x , for electric flux states with momentum: p = ( p / Nl , ) (Left, Center), and p = ( p / Nl , ) (Right). correction, being negative, could give rise to a tachyonic excitation. This occurs above a criticalcoupling: x c ( ~ e ) = | ~ n | / ( G ( ~ e / N )) . The quantity G ( ~ z ) diverges as 1 / | ~ z | for small | ~ z | . This wouldseem to unavoidably drive x c to zero as N goes to infinity. However, given the relation n i = − e i j ke j ,it suffices to scale k ∼ √ N to push x c into the non-perturbative domain. On the opposite side, if welook at minimal momentum | ~ n | =
1, the electric flux is given by ¯ k and the critical coupling occursat x c ( ~ e ) = p ¯ k / N ≡ p ˜ q . Thus, keeping ˜ q > ˜ q c the perturbative instability is avoided.One can still worry about the possible appearance of non-perturbative instability. An argumentbased on the effective string description has been used in Ref. [1] to indicate that this is avoidedif: | ~ n | | ~ e | ( N − | ~ e | ) / N > /
12. Still, a full proof should rely on numerical results. In our previouswork [1], we have performed simulations at several values of ˜ q and k / N . All the results with˜ q > p / k / N > /
17 showed no indication of instability. A representative sample is presentedin Fig. 1. In addition to the stable case already discussed, we display two cases in which k / N becomes small. They correspond to electric flux | ~ e | =
1, with k = ¯ k = k = k = ( N − ) / N = x regime though, the linearly rising potential overcomesthis behaviour and restores the standard x -dependence. In the second case, we display two values of k / N = / /
17. They are still large enough to prevent the appearance of instability. However,we observe a strong decrease in the energy of electric flux with decreasing k / N . Assuming thistrend continues, one expects an instability to set-in below a given value of this ratio.6 erturbative analysis of twisted volume reduced theories Margarita García Pérez
5. Conclusions
We have given a unified description of the perturbative expansion of SU(N) Yang-Mills theoryon an even dimensional torus traversed by Z(N) magnetic flux through each plane. We stress theemergence of an effective size parameter combining spatial and group degrees of freedom. Ourresults, valid at finite N , provide important information on large N reduced models and the twistedvolume reduction mechanism. In particular, they support the prescription given in Ref. [3] on howthe flux has to scale with the rank N. A few numerical results in 2+1 dimensions for the energiesof electric flux sectors support the applicability of these ideas at the non-perturbative level. A gooddescription is given of the evolution of these energies at all scales. Acknowledgments
M.G.P. and A.G-A acknowledge support from the grants FPA2012-31686 and FPA2012-31880,the MINECO Centro de Excelencia Severo Ochoa Program SEV-2012-0249, the Comunidad Autónomade Madrid HEPHACOS S2009/ESP-1473, and the EU PITN-GA-2009-238353 (STRONGnet).They participate in the Consolider-Ingenio 2010 CPAN (CSD2007-00042). M. O. is supportedby the Japanese MEXT grant No 23540310. We acknowledge the use of the IFT clusters.
References [1] M. García Pérez, A. González-Arroyo and M. Okawa, JHEP (2013) 003; PoS LATTICE (2012) 219.[2] A. González-Arroyo and M. Okawa, P hys. Lett. B120 (1983) 174;
Phys. Rev.
D27 (1983) 2397.[3] A. González-Arroyo and M. Okawa, JHEP (2010) 043; Phys. Lett. B (2013) 1524.[4] Z. Guralnik, R. C. Helling, K. Landsteiner and E. López, JHEP (2002) 025. W. Bietenholz, e.a.,JHEP (2006) 042; W. Bietenholz, F. Hofheinz and J. Nishimura, JHEP (2002) 009.[5] T. Ishikawa and M. Okawa, talk given at the
Annual Meeting of the Physical Society of Japan , March28-31, Sendai, Japan (2003); M. Teper and H. Vairinhos, Phys. Lett. B (2007) 359; T. Azeyanagi,M. Hanada, T. Hirata and T. Ishikawa, JHEP (2008) 025.[6] A. González-Arroyo, “Yang-Mills fields on the four-dimensional torus. Part 1.: Classical theory,” InPeñiscola 1997 57-91 [hep-th/9807108], and references therein.[7] A. González-Arroyo and C. P. Korthals Altes, Phys. Lett. B (1983) 396.[8] A. Connes, M. R. Douglas and A. S. Schwarz, JHEP (1998) 003.[9] J. Ambjorn, Y. M. Makeenko, J. Nishimura and R. J. Szabo, JHEP (2000) 023.[10] Z. Guralnik and J. Troost, JHEP (2001) 022.[11] L. Alvarez-Gaume and J. L. F. Barbon, Nucl. Phys. B (2002) 165.[12] M. Unsal and L. G. Yaffe, JHEP (2010) 030.[13] A. González-Arroyo and C. P. Korthals Altes, Nucl. Phys. B (1988) 433.[14] Z. Guralnik, JHEP (2000) 003.[15] A. Athenodorou, B. Bringoltz and M. Teper, JHEP (2011) 042.(2011) 042.