The spectrum of closed loops of fundamental flux in D = 3+1 SU(N) gauge theories
aa r X i v : . [ h e p - l a t ] D ec The spectrum of closed loops of fundamental flux in D = + SU ( N ) gauge theories. Andreas Athenodorou ∗ NIC, DESY,Platanenallee 6, 15738 Zeuthen, GermanyandRudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordE-mail: [email protected]
Barak Bringoltz
Physics DepartmentUniversity of Washington, Seattle, WA 98195-1560, USAE-mail: [email protected]
Michael Teper
Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordE-mail: [email protected]
We study the spectrum of closed flux tubes in four dimensional SU ( N ) gauge theories. We doso by calculating the energies of the low lying states with the variational technique (whose basisconsists of about ∼
700 operators). We study states of different values of angular momentum,transversal parity, longitudinal parity, and longitudinal momentum, and compare the results witheffective string theories (ESTs) such as the Nambu-Goto (NG) model. Most of our states agreevery well with the Nambu-Goto predictions and since most of our flux-tubes’ lengths are outsidethe radius of convergence of the ESTs, then for some states it is only the NG that predicts thespectrum well. This strongly suggests that the ESTs can be re-summed. Nonetheless, there area few states (all with negative parity and in the same representation of the lattice rotation group)that exhibit large deviations from the NG predictions; these deviations might provide clues to thenature of the effective string theory describing the large- N QCD string.
The XXVII International Symposium on Lattice Field TheoryJuly 25-31 2009Beijing, China ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ pectrum of closed fundamental flux tubes in D = +
1. Introduction
We have recently shown [1] that the closed flux tube spectrum in D = + SU ( N ) gaugetheories can be well-approximated by the Nambu-Goto (NG) free string in flat space-time. Inparticular, such agreement was observed for flux tube lengths that are comparable to the width ofthe flux. Thus, the flux-tube approximately behaves like a fundamental string even when, naively,it looks more like a fat blob than a thin string.It is interesting to know how the closed flux tube spectrum in D = + SU ( N ) gauge theoriesbehaves and how this compares to NG and other effective string theories. A pioneering attemptto calculate the closed flux tube spectrum in D = + SU ( N ) gauge theories has been reportedin [2], which investigated the spectrum for N = a ≈ . N = b = . a ≃ . .
338 ( a ≃ . N = b = .
630 ( a ≃ . SU ( ) and SU ( ) at a ≈ . N >
2. Setup of the lattice calculation
We define the SU ( N ) gauge theory on a four-dimensional Euclidean space-time lattice whichis compactified along all directions with L k × L ⊥ × L ⊥ × L T sites. The length of the flux tubeis equal to L k , while L ⊥ , L ⊥ and L T were chosen to be large enough so to avoid finite volumeeffects. To extract the flux tube spectrum we perform Monte-Carlo simulations using the standardWilson plaquette action, S = (cid:229) (cid:3) b (cid:2) − N ReTr ( U (cid:3) ) (cid:3) , with b = Ng ( a ) , and in order to keep the valueof the lattice spacing a approximately fixed for different values of N we keep the ’t Hooft coupling l ( a ) = Ng ( a ) approximately fixed, so that b (cid:181) N . The simulation algorithm we use combinesstandard heat-bath and over-relaxation steps in the ratio 1:4; these are implemented by updating SU ( ) subgroups using the Cabibbo-Marinari algorithm. To measure the spectrum of energies weuse the variational technique (e.g. see Ref. [4] and its references).The closed flux tube states in D = + C . It is a subgroup of O ( ) that corresponds to rotations by integer multiples of p / J mod4 = J mod4 = ± J mod4 =
2. Anadditional useful quantum number is the parity P ⊥ which is associated with reflections aroundthe axis ˆ ⊥ . Such parity flips the sign of J and so we can choose a basis in which states arecharacterised by their value of J (which can be of either sign), or by their value of | J | and P ⊥ .In our calculations we use the latter choice. While in the continuum states of nonzero J areparity degenerate, on the lattice this is exactly true only for the odd J sector. This means wecan denote our states by the 5 irreducible representations A , , E , B , of C whose J and P ⊥ as-signments are: { A : | J mod4 | = , P ⊥ = + } , { A : | J mod4 | = , P ⊥ = −} , { E : | J mod4 | = , P ⊥ = ±} , Since the global rotational symmetries of a flux-tube wound around a compact direction are those of a two-dimensional space. pectrum of closed fundamental flux tubes in D = + { B : | J mod4 | = , P ⊥ = + } , and { B : | J mod4 | = , P ⊥ = −} . All the representations of C are one-dimensional except for E which is two-dimensional.Two additional useful quantum numbers include the longitudinal momentum p || carried by theflux-tube along its axis (which is quantized in the form p || = p q / L || ; q ∈ Z ) and the parity P || withrespect to reflections across the string midpoint. Since P || and p || do not commute, we can use bothto simultaneously characterise a state only when q =
0. Also, since the energy does not depend onthe sign of q , we only focused on those with q ≥ J , P ⊥ , P || , and q . Thisis achieved by choosing a linear combination of Polyakov loops whose paths consist of varioustransverse deformations and various smearing and blocking levels (again see [4]). All the pathsused for the construction of the operators are presented in Table 1 and all together form a basis ofaround 700 operators. Let us show how to construct an operator with a certain value of J mod4 : beginwith the operator f a that has a deformation extending in angle a within the plane of transversedirections. We can construct an operator f ( J ) that belongs to a specific representation of C byusing the formula: f ( J ) = (cid:229) n = , , , e iJn p f n p . It is straight-forward to show that f ( ) belongs toeither A or A (depending on its value of P ⊥ ), that f ( ) belongs to E , and that f ( ) belongs to B or B . The projection onto certain values of P ⊥ and P || is demonstrated pictorially in Eq. (2.1) foran operator of J mod4 = f = Tr " j ik (2.1)If i = j = k = + f projects onto { A , P || = + } , if i = + , j = k = − { A , P || = + } , if i = − , j = + , k = − { A , P || = −} and finally, if i = j = − , k = +
1, it projects onto { A , P || = −} . Table 1:
All the transverse deformations used for the construction of the operators.
3. Theoretical expectations for the spectrum from effective string theories
Let us first think about the flux-tube as a string of length l = aL k winding around the torus.The classical configuration of the string spontaneously breaks translation symmetry and so weexpect a set of Nambu-Goldstone massless bosons to appear at low energies. These bosons reflectthe transverse fluctuations of the flux-tube around its classical configuration, and in D space-time3 pectrum of closed fundamental flux tubes in D = + dimensions there are D − The Nambu-Goto model describes relativistic strings [5]. The action of the model is propor-tional to the area of the world sheet swept by the string as it propagates in time. This model isself-consistent quantum mechanically only in D =
26 dimensions (see for example second refer-ence in [5]), but there are claims in the literature that, for any value of D , this model can serve asan effective low energy field theory for long strings [6]. From here on we refer to such low energytheories as Effective String Theories (ESTs). The spectrum of the Nambu-Goto model for D = E NG ( l ) = s ( s l ) + ps (cid:16) N + L + N − L + N + R + N − R − (cid:17) + (cid:18) p ql (cid:19) . (3.1)Here s is the string tension and N ± L and N ± R are the occupation numbers of the bosons that move tothe left and to the right (the ± superscript denotes the spin that they carry). This means that each ofthese occupation numbers is defined to count the energy units carried by the bosons: N = (cid:229) ¥ k = k n k (here n k is the number of bosons carrying momentum k ). The net longitudinal momentum carriedby the bosons is given by q = N + L + N − L − N + R − N − R , and the net angular momentum of a state isgiven by J = N + L + N + R − N − L − N − R . It is useful to make a connection to Regge theory by writing (cid:229) i = ± (cid:229) j = L , R N ij ≡ J + m and then interpreting the integer m as counting the daughter trajectories ofa certain angular momentum J . As usual, leading Regge trajectories of angular momentum J canbe degenerate in energy with daughter trajectories of lower spin states. Below we refer to statesthat have the same value of J + m as ‘being in the same NG level’.Since we think of the NG model as an EST, which may be justified only for long strings [6],we can expand Eq. (3.1) for l √ s ≫ q =
0, and follow theconvention in denoting J + m by n / E NG ( l ) = s l + p l (cid:18) n − D − (cid:19) − p s l (cid:18) n − D − (cid:19) + p s l (cid:18) n − D − (cid:19) + O ( l − ) . (3.2)We note in passing that because the NG model is only one possible candidate of an EST, then inthe language of effective field theories, it may differ from other candidate ESTs by the values ofcertain low energy constants (LECs). These LECs would make the most general EST spectrumdiffer from Eq. (3.2) by the coefficients in the 1 / l expansion. In the next subsection we discuss asystematically controlled approach to construct the most general EST of the QCD flux-tube. A systematic EST study that would describe the QCD flux-tube was pioneered by Lüscher,Symanzik, and Weisz in Ref. [7]. Such an EST approach produces predictions for the energy ofstates as an expansion in 1 / l . Terms in this expansion that are of O ( / l p ) are generated by ( p + ) -derivative terms in the EST action whose coefficients are a priori arbitrary LECs. Interestingly,these LECs were shown to obey strong constraints that reflect a non-linear realization of Lorentzsymmetry [8, 9, 10], and so to give parameter free predictions for certain terms in the 1 / l expansion.We review these predictions below. 4 pectrum of closed fundamental flux tubes in D = + First, since we focus on closed strings, p can only be odd (terms in the energy that come witheven powers of 1 / l appear only if there are boundary terms in the action of the EST and so donot exist for closed strings). The analysis with p = O ( / l ) ‘Lüscher term’ in the ground state energy, whose universal coefficient dependsonly on D . Since the 2 − derivative action is a free theory, then its spectrum of excited states isthat of ( D − ) massless bosons (the Lüscher term is the zero point energy of these bosons). Thepredictions of Ref. [7] were that to O ( / l ) the flux-tube energy is given by the first two terms inright hand-side of Eq. (3.2).The 4 − derivative terms were analysed in Ref. [8] and for D = O ( / l ) term that is identical to the third term in Eq. (3.2). Ref. [10] showed that this matchingbetween the O ( / l ) term in the NG prediction and in any general EST holds also for D = − derivative terms and showed that for D = D =
4, the coefficient of the O ( / l ) termmay differ from the one in Eq. (3.2). Nonetheless, the energy of n = D = O ( / l ) term is indeed given by Eq. (3.2), and that the O ( / l ) term in the average over the energies of states that are in the same NG level is identical tothe O ( / l ) in Eq. (3.2).A different approach to EST was proposed by Polchinski and Strominger in Ref. [11]. Tech-nically it uses a different gauge fixing of the embedding coordinates on the world-sheet (conformalgauge instead of the static gauge choice used in the ESTs following Ref. [7]). Here the constraintsobeyed by the LECs allows one to maintain the conformal symmetry of the world-sheet even out-side the critical dimension of D =
26. Ref. [11] showed that as a result of these constraints, the O ( / l ) term in the 1 / l expansion is the same as in the NG model — it is given by the Luscher termappearing in Eq. (3.2). Much more recently, Drummond [12] showed that the O ( / l ) term is alsoidentical to the one appearing in the 1 / l expansion of the NG model (Eq. (3.2)). Finally, the recentRefs.[13] claim that even higher order terms in the 1 / l expansion are identical to the correspondingterms in Eq. (3.2); these claims seem to contradict the results of Ref. [10] for D ≥
4. Results q = , , channels In Figure 1 we present a comparison of the q = , , SU ( ) and a ≃ . SU ( ) and a ≃ . SU ( ) and a ≃ . q = { J = , P ⊥ = + , P || = + } . Indeed our data shows that the ground stateis in the A representation with q = P || = + . We use the measured energy of this state toextract the string tension by fitting it with the ansatz E gs ( l ) = q ( s l ) − ps + s C ( √ s l ) ; we find C ∼ O ( − ) and that for l √ s & q = J = P ⊥ degenerate, and so we measureonly the energy of the P ⊥ = + channel in the E representation. This measurement is presented inFig. 1 where we see that the NG prediction is in agreement with our data. Finally, NG predicts thatthe ground state for q = { J = , P ⊥ = + } ,two states with { J = , P ⊥ = ±} , and two states with { J = , P ⊥ = ±} . We find this to be consistentwith the ground states in the A , E and B , representation (again, as for q =
1, we measure only the5 pectrum of closed fundamental flux tubes in D = + P ⊥ = + states in the E representation). Comparing the three plots in Fig. 1 we see that the O ( a ) and O ( / N ) corrections of our data are small compared to our statistical errors. Thus, within ourlevel of accuracy, the agreement of our data with the NG model is largely insensitive to the latticespacing and 1 / N corrections. l √ s f E / √ s f NGwith n = n = n = { J = , P ⊥ =+ , P || =+ , q = } g.s { J = , q = } g.s { J = , P || =+ , q = } g.s { J = , q = } g.s { J = , P || =+ , q = } g.s { J = , P || =+ , q = } l √ s f E / √ s f NG with n = n = n = { J = , P ⊥ =+ , P || =+ , q = } g.s { J = , q = } g.s { J = , P || =+ , q = } g.s { J = , q = } g.s { J = , P || =+ , q = } g.s { J = , P || =+ , q = } l √ s f E / √ s f NGwith n = n = n = { J = , P ⊥ =+ , P || =+ , q = } g.s { J = , q = } g.s { J = , P || =+ , q = } g.s { J = , q = } g.s { J = , P || =+ , q = } g.s { J = , P || =+ , q = } SU ( ) , b = . SU ( ) , b = . SU ( ) , b = . Figure 1:
Energies of the lightest states that correspond to q = , , q = states We now turn to examine the first excited state in the q = { J = , P ⊥ = P || = + } , { J = , P ⊥ = P || = −} , { J = , P ⊥ = P || =+ } , and { J = , P ⊥ = − , P || = + } . On the lattice, this would imply that states from ( A , P || =+) , ( A , P || = − ) , ( B , P || = +) , and ( B , P || = +) are all degenerate (up to O ( a ) corrections).Instead, what we find is that while the states belonging to ( A , P || = +) , ( B , P || = +) , and ( B , P || = +) are all quite close to each other and to the NG model, the state in ( A , P || = − ) is anomalously different and shows substantial deviation from NG. It is tempting to expect thatthis state’s energy would eventually approach the NG prediction (and is perhaps reflecting a largecoefficient multiplying the term that controls this deviation from NG in the EST). Nonetheless,an equally likely possibility is that it does not converge to NG. In fact, observe that the energy ofthis state is higher than the ground state energy by approximately an equal amount throughout thedistance range that our simulations are able to explore. Therefore, at large enough l , it might crossthe NG prediction, as a massive state would. We do not know which possibility provides a betterexplanation for the strikingly ‘anomalous’ way that the energy of this state behaves. This behaviourdoes not change as we decrease the lattice spacing (central plot) or increase N (right-most plot).In Figure 2 we also provide a comparison of the four states with expansions of the Nambu-Goto square root order by order in terms of 1 / l up to O ( / l ) – see Eq. (3.2) (note that accordingto Ref. [10] it is only up to O ( / l ) that we can trust Eq. (3.2)). Excluding the anomalouslybehaving ground state in the { A , P || = −} channel, the other three states are obviously betterdescribed by NG than by any other EST. This reflects a simple fact: nearly all our data is beyondthe radius of convergence of the 1 / l expansion (which can be estimated from the EST series to be ( l √ s ) converge ≃ . + pectrum of closed fundamental flux tubes in D = + N.N.L.O NGwith n = n = n = n = { J = , P ⊥ = − , P k =+ } g.s { J = , P ⊥ =+ , P k =+ } g.s { J = , P ⊥ = − , P k = −} f.e.s { J = , P ⊥ =+ , P k =+ } l √ s f E / √ s f N.N.L.O NGwith n = n = n = n = { J = , P ⊥ = − , P k =+ } g.s { J = , P ⊥ =+ , P k =+ } g.s { J = , P ⊥ = − , P k = −} f.e.s { J = , P ⊥ =+ , P k =+ } l √ s f E / √ s f N.N.L.O NGwith n = n = n = n = { J = , P ⊥ = − , P k =+ } g.s { J = , P ⊥ =+ , P k =+ } g.s { J = , P ⊥ = − , P k = −} f.e.s { J = , P ⊥ =+ , P k =+ } l √ s f E / √ s f SU ( ) , b = . SU ( ) , b = . SU ( ) , b = . Figure 2:
Energies of the four states with q = n = q = states We now discuss further results for states with q = n = /
2) should be ten-fold degenerate. These states are the ground states of { J = , P ⊥ = + } , { J = , P ⊥ = −} , { J = , P ⊥ = ±} , { J = , P ⊥ = + } , { J = , P ⊥ = −} and thefirst excited states of { J = , P ⊥ = ±} . On the lattice these states fall into two degenerate pairsof states in E , and four more states that belong to A , and B , . The parity degeneracy in the E representation allows us to calculate only the P ⊥ = + states and we therefore expect six degeneratestates in the NG model.Unfortunately, our basis of operators was insufficient to successfully isolate all these six states,and we were able to extract only four of them. We found that the energy of the q = A and B representation, and second lowest energy states in the E representation,agree fairly well with the energy of the n = / q = A representation (which is naively associated with the J = P ⊥ = − channel) appearsanomalous: it has a large deviation from the NG curve and does not show any sign of convergence.This is true also on our finer lattice (central plot) and for SU ( ) (right-most plot). NG with n = / n = / { B , q = }{ E , q = }{ A , q = }{ A , q = } l √ s f E / √ s f NG with n = / n = / { B , q = }{ E , q = }{ A , q = }{ A , q = } l √ s f E / √ s f NG with n = / n = / { B , q = }{ E , q = }{ A , q = }{ A , q = } l √ s f E / √ s f SU ( ) , b = . SU ( ) , b = . SU ( ) , b = . Figure 3:
Energies of the lightest five distinguishable states with q = pectrum of closed fundamental flux tubes in D = +
5. Summary
We calculated energies of 13 states in the spectrum of closed strings in 3 + SU ( N ) gauge theories. We study the gauge groups SU ( ) (with lattice spacing a ≈ . , . SU ( ) (with a ≈ . . . l . . + A representation of the lattice group (whichnaively corresponds to zero angular momentum). We see these deviations also for states that carryboth zero and one unit of longitudinal momentum.There are many avenues one could take on the lattice to make progress towards establishingwhat is the effective string theory of the QCD flux-tube. These future studies may include the searchfor massive excitations like breathing modes, attempting to accurately test the current theoreticalpredictions (see Section 3) within their radius of convergence, studying the open string spectrum,etc. We also look forward to theoretical progress that would allow one to understand the results wepresented in this proceedings. For example, how can the flux-tube behave like a NG string belowthe radius of convergence of the effective string theory expansion? and what makes the states inthe A representation have large deviations from the NG model? Acknowledgements
BB thanks O. Aharony for multiple useful discussions and the Weizmann institute for its kindhospitality. The computations were carried out on EPSRC and Oxford funded computers in OxfordTheoretical Physics. AA acknowledges the support of the Leventis Foundation and the LATTICE2009 organizers for supporting him financially. BB was supported by the U.S. Department ofEnergy under Grant No. DE-FG02-96ER40956.
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