Non-universal corrections to the tension ratios in softly broken N = 2 SU ( N ) gauge theory

. Calculation by Douglas and Shenker of the tension ratios for vortices of different N -alities in the softly broken N = 2 supersymmetric SU ( N ) Yang– Mills theory is carried to the second order in the adjoint multiplet mass m . Corrections to the ratios violating the well known sine formula are found, showing that it is not a universal quantity. Recently the tension ratios among the conﬁning vortices corresponding to sources of different N -alities in SU ( N ) gauge theories have been the subject of some attention, as a quantity characterizing quantitatively the conﬁning phase of these systems. After an interesting suggestion from MQCD that such ratios might have universal values

Recently the tension ratios among the confining vortices corresponding to sources of different N -alities in SU(N ) gauge theories have been the subject of some attention, as a quantity characterizing quantitatively the confining phase of these systems. After an interesting suggestion from MQCD that such ratios might have universal values [1], a more recent study in string theory based on supergravity duals [2], gave model-dependent results for two N = 1 SQCD-like theories. The result of direct measurement in the lattice (non-supersymmetric) SU(N ) gauge theories is consistent with equation (1) [3,4].
Derivation of formulae such as equation (1) in the standard, continuous SU(N ) gauge theories still defies us. The first field-theoretic result on this issue was obtained by Douglas and Shenker [5], in the N = 2 supersymmetric SU(N ) pure Yang-Mills theory, with supersymmetry softly broken to N = 1 by a small adjoint scalar multiplet mass m. They found equation (1) for the ratios of the tensions of Abelian (Abrikosov-Nielsen-Olesen (ANO)) vortices corresponding to different U (1) factors of the low-energy effective (magnetic) U (1) N −1 theory.

59.2
The nth colour component of the quark has charges with respect to the various electric U k (1) gauge groups. The source of the kth ANO string thus corresponds to the N -ality k multiquark state, |k = |q 1 q 2 , . . . , q k , allowing a reinterpretation of equation (1) as referring to the ratio of the tension for different N -ality confining strings [6]. However, physics of the softly broken N = 2 SU(N ) pure Yang-Mills theory is quite different from what is expected in QCD. Dynamical SU(N ) → U (1) N −1 breaking introduces multiple meson Regge trajectories with different slopes at low masses [6,7], a feature which is neither seen in nature nor expected in QCD †. For instance, another N -ality k state |k = |q 2 q 3 , . . . q k+1 acts as the source of the U k+1 (1) vortex and as the sink of the U 2 (1) vortex, which together bind |k and anti |k states with a tension different from T k . The Douglas-Shenker prediction is, so to speak, a good prediction for a wrong theory! Only in the limit of N = 1 does one expect to find one stable vortex for each N -ality, corresponding to the conserved Z N charges [6].
Within the softly broken N = 2 SU(N ) theory, the two regimes can be in principle smoothly interpolated by varying the adjoint mass m from zero to infinity, adjusting Λ appropriately. At small m one has a good local description of the low-energy effective dual, magnetic U (1) N −1 theory. The transition towards a large m regime involves both perturbative and nonperturbative effects. Perturbatively, there are higher corrections due to the N = 1 perturbation, m Tr Φ 2 . Non-perturbatively-in the dual theory-there are productions of massive gauge bosons of the broken SU(N )/U (1) N −1 generators, which mix different U (1) N −1 vortices and eventually lead to the unique stable vortex with a given N -ality. There seem to be no general reasons to believe that the tension ratios found in the small m limit are not renormalized in such processes.
Below we report the result on the first type of effects: perturbative corrections to the tension ratios equation (1) due to the next-to-lowest contributions in m. We shall find a small non-universal correction to the sine formula equation (1). Our point is, of course, not that such a result is of interest in itself as a physical prediction but that it gives a strong indication for the non-universality of this formula, even though it could be an approximately good one.
The problem of the next-to-lowest contributions in m has already been studied in SU(2) theory, by Vainshtein and Yung [7] and by Hou [12], although in that case there is only one U (1) factor so that the author's interest was different. When only up to the order A D term in the expansion is kept, the effective low-energy theory turns out to be an N = 2 SQED, A D being an N = 2 analogue of the Fayet-Iliopoulos term. As a result, the vortex remains BPS-saturated, and its tension is proportional to the monopole charge. When the A 2 D term is taken into account, the vortex ceases to be BPS-saturated: the correction to the vortex tension can be calculated perturbatively, giving rise to the result that the vacuum behaves as a type I superconductor. † In fact, the same problem is expected in any confining vacuum in which such a dynamical breaking takes place. t' Hooft's original suggestion for QCD ground state [8] is of this type.

59.3
Our aim here is to generalize the analysis of Vainshtein, Yung and Hou [7,12] to SU(N ) theory. In fact, the Douglas-Shenker result (equation (1)) in SU(N ) theory was obtained in the BPS approximation, by keeping only the linear terms in a Di in the expansion The coefficients U 0k were computed by Douglas and Shenker [5]. Our first task is then to compute the coefficients of the second term U 0mn . In principle it is a straightforward matter, as one must simply invert the Seiberg-Witten formula †: which is explicitly known to second order. The only trouble is that a Dm and a m (m = 1, 2, . . . , N − 1) are given simply in terms of N -dependent vacuum parameters which follow easily by using the constraint, δφ i which are explicitly given at the N confining vacua in [5], one then finds The explicit values of B mi are (see [5]): The definition of u(a Di ) is the following: Then the desired coefficients can be found by the following expression, computed at a Di = 0: The first part of equation (10) becomes: The evaluation of the second term in equation (10) is reported in the appendix. The result is the following: We now use this result to calculate the corrections to the tension ratios (1) found in the lowest order. The effective Lagrangian near one of the N confining N = 1 vacua is The coupling constant e 2 Di is formally vanishing, as 4π e 2 Dk 1 2π ln Λ sin(θ k ) a Dk N whereθ m ≡ πn N and a Dk = 0 at the minimum. Physically, the monopole loop integrals are in fact cut off by masses caused by the N = 1 perturbation. The monopole becomes massive when m = 0, and √ 2a Dk should be replaced by the physical monopole mass (mΛ sin(θ k )) 1/2 which acts as the infrared cutoff for the coupling constant evolution. This is equivalent to the prescription of taking a Dn = MM 1/2 n , which is used in [5]. One finds thus As U 0mn is found to be diagonal, the description of the ANO vortices [9,10] in terms of effective magnetic Abelian theory description continues to be valid for each U (1) factor. In the linear approximation U (A D ) = mΛ 2 + µA D , where µ ≡ |4mΛ sin πk N | for the kth U (1) theory, the theory can be (for the static configurations) effectively reduced to an N = 4 theory in 2 + 1 dimensions. In this way, Bogomolny's equations for the BPS vortex can be easily found from the condition that the vacuum is supersymmetric: (17) The solutions of these equations are similar to the one considered by Nielsen and Olesen: with boundary conditions f (0) = g(0) = 0, f (r → ∞) = 1, g(r → ∞) = +1/2). The tension turns out to be independent of the coupling constant: for the minimum vortex † (20) † The fact that the absolute value of m appears in equation (20), as it should, may not be obvious. This can be shown by an appropriate redefinition of the field variables, used in [11], which makes all equations real. The correction term in (23) is thus negative independently of the phase of m.

59.5
When the second-order term in U (A D ) = µA D + 1 2 ηA 2 D , η ≡ U kk , is taken into account, the vortex ceases to be BPS-saturated. The corrections to the vortex tension due to η can be taken into account by perturbation theory, following [12]. To first order, the equation for A Dk = A D is where unperturbed expressions from equation (18) can be used for M ,M. The vortex tension becomes simply where the second term represents the correction. By restoring the k dependence, we finally get for the tension of the kth vortex, where C = 2 √ 2π(0.68) = 6.04. The correction term has a negative sign, independently of the phase of the adjoint mass. Note that the relation T k = T N −k continues to hold. Equation (23) is valid for m Λ. We end with a few remarks. In the above consideration, we have taken into account exactly the m 2 corrections in the F term of the effective low-energy action. On the other hand, the corrections to the D terms are subtler. Indeed, based on the physical consideration, a D in the argument of the logarithm in the effective low-energy coupling constant was replaced by the monopole mass of O( √ mΛ). This amounts to m insertion to all orders in the loops. Such a re-summation is necessitated by the infrared divergences, just as in the case of chiral perturbation theory. This explains the non-analytic dependence on m as well as on 1/N [13].
Also, there are corrections due to non-diagonal elements in the coupling constant matrix τ ij , which mix the different U (1) factors [14], neglected in equation (14). These non-diagonal elements are suppressed by O( 1 log Λ/m ) relatively to the diagonal ones, apparently of the same order of suppression as the correction calculated above. However, these non-diagonal elements give rise to terms of the form of the effective potential [14] When this is used in the equations of motion, one finds that the correction to the tension due to the non-diagonal (Im τ ) −1 ij is actually one order higher, O 1 log 2 Λ/m , hence is negligible to the order considered.
We thus find a non-universal correction to the Douglas-Shenker formula, equation (1). In the process of transition towards fully non-Abelian superconductivity at large m, non-perturbative effects such as W boson production are probably essential. Nevertheless, the presence of a calculable deviation from the sine formula is qualitatively significant and shows that such a ratio is not a universal quantity.

Appendix. Computation of equation (12)
We use the following identity (found by partial derivation of the identity ∂a Dk ∂a Dm = δ km -the first part of equation (6)-with respect to a Dl ): The expression (12) now becomes: δφ j δφ i can be found by first considering d 2 λ dφ i dφ j and integrating this along the circles α m (see [5] for the conventions; the variable θ is defined by x = 2 cos[θ]): We perform the integrations taking the residues at the poles:θ m = πm N : The last equality involves rather cumbersome trigonometric expressions: we found equation (A.7) by using Mathematica up to N = 50.