Confining Strings in Supersymmetric Theories with Higgs Branches
FFTPI-MINN-14/40, UMN-TH-3411/14
Confining Strings in Supersymmetric Theorieswith Higgs Branches
M. Shifman, a Gianni Tallarita, b and Alexei Yung a,c a William I. Fine Theoretical Physics Institute, University of Minnesota,Minneapolis, MN 55455, USA b Centro de Estudios Cient´ıficos (CECs), Casilla 1469, Valdivia, Chile c Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300,Russia
Abstract
We study flux tubes (strings) on the Higgs branches in supersymmetricgauge theories. In generic vacua on the Higgs branches strings were shownto develop long-range “tails” associated with massless fields, a characteristicfeature of the Higgs branch (the only exception is the vacuum at the baseof the Higgs branch). A natural infrared regularization for the above tails isprovided by a finite string length L .We perform a numerical study of these strings in generic vacua. Wefocus on the simplest example of strings in N = 1 supersymmetric QEDwith the Fayet-Iliopoulos term. In particular, we examine the accuracy ofa logarithmic approximation (proposed earlier by Evlampiev and Yung) forthe tension of such string solutions. In the Evlampiev-Yung formula thedependence of tension on the string length is logarithmic and the dependenceon the geodesic length from the base of the Higgs branch is quadratic. Weobserve a remarkable agreement of our numerical results for the string tensionwith the Evlampiev-Yung analytic expression. a r X i v : . [ h e p - t h ] F e b Introduction
Supersymmetric gauge theories provide an excellent theoretical laboratoryfor understanding strongly coupled non-Abelian dynamics. In particular,the dual Meissner effect as a mechanism of confinement suggested in themid-1970s [1] was first analytically observed in 1994 in the framework of N = 2 supersymmetric theories [2, 3]. The main feature of this mechanismis formation of the Abrikosov-Nielsen-Olesen (ANO) [4] flux tubes (confin-ing strings). If in a given vacuum quarks condense then the conventionalmagnetic ANO strings are formed. They confine monopoles. If, instead,monopoles condense, the electric ANO strings are formed. They confinequarks [2, 3] (see also [5, 6] for reviews of scenarios with confined monopoles).Quite often supersymmetric gauge theories have Higgs branches. Theseare flat directions of the scalar potential on which charged scalar fields candevelop vacuum expectation values (VEVs) breaking the gauge symmetry. Inmany instances this breaking provides topological reasons behind formationof the ANO strings. The dynamical side of the problem of the confining stringformation in the theories with Higgs branches was addressed in [7, 8, 9]. Apriori it is not clear at all whether or not stable string solutions exist inthis class of theories. The point is that the theories with a Higgs branchrepresent a limiting case of type I superconductor, with vanishing Higgsmass. In particular, it was shown in [7] that infinitely long strings cannot beformed in this case due to infrared divergences.Later this problem was studied [8] in a more realistic confinement setup,namely, the string in question was assumed to have a large but finite length L .Finite length provides an infrared regularization implying [8] that finite-sizeANO strings still exist on the Higgs branches. They become logarithmically“thick” due to the presence of massless fields and give a confining potentialfor two heavy trial charges of the form V ( L ) ∼ L log L . (1)Note that V ( L ), instead of being linear in separation L is modified by log( L Λ)in the denominator.The potential between heavy trial charges provides us with an order pa-rameter marking distinct phases with different dynamical behaviors. Thus,we see that theories with the Higgs branches develop a novel confining phasewith logarithmically nonlinear potential (1).1ormation of strings in more generic theories with non-flat Higgs branchescurved by the presence of the Fayet-Iliopoulos (FI) term was considered laterin [9]. In this case string profile functions can be approximated by an almostBPS core built from massive fields and a long-range “tail” built from masslessfields. In this approximation the confining potential for the simplest case ofthe U (1) N = 1 supersymmetric gauge theory with one flavor of chargedmatter was shown to be V ( L ) ∼ L (cid:18) l log L (cid:19) , (2)where l is the length of the geodesic line on the Higgs branch between thegiven vacuum and the base point of the Higgs branch.In this paper our task is to confirm the onset of the regime (2) for suffi-ciently large L . This will allow us to better understand the limits of applica-bility of the analytic consideration in [9]. To this end we numerically studythe string solution in N = 1 supersymmetric QED with the FI term. Wefind the string profile functions and calculate the string tension. In agree-ment with the analytic formula (2) we observe that our numeric solutionreproduces (with good accuracy) both features: the logarithmic dependenceof the “tail” tension on L and the quadratic dependence on l .The paper is organized as follows. In Sec. 2 we briefly review a basic con-struction of length- L flux tubes on curved Higgs branches in N = 1 SQED.Then we summarize main results concerning the analytic approximation [9]for their tension in terms of the distance from the base of the Higgs branchand L . In Sec. 3 we obtain the full numerical result for the profile functionsof the string solution following the general guidelines of [8]. We then put theanalytical tension formula (2) to test. Our numeric data establishes the onsetof the analytic approximation (2). In Sec. 4 we present some conclusions. N = 1 supersymmetric QED We begin by reviewing the construction of flux tubes on curved Higgs branchesin the Abelian gauge theories [8, 9]. The starting point is N = 1 SQED withthe action S QED = (cid:90) d x (cid:18) g F µν + | D µ q | + | D µ ˜ q | + V ( q, ˜ q ) (cid:19) (3)2here the covariant derivative is defined as D µ = ∂ µ − i A µ (4)and the complex scalar fields q and ˜ q have opposite charges under the U (1)gauge symmetry. We assume the charges for the scalar fields n e to be | n e | =1 /
2. The scalar potential is V ( q, ˜ q ) = g (cid:0) | q | − | ˜ q | − ξ (cid:1) . (5)It is obtained from the Fayet-Iliopoulus (FI) coupling for the U (1) vectorsuperfield with FI parameter ξ after its auxiliary field D is integrated out.This model has a rich vacuum structure dictated by the vacuum condition | (cid:104) q (cid:105) | − | (cid:104) ˜ q (cid:105) | − ξ = 0 , (6)which describes a Higgs branch of dimension two: two complex scalars subjectto one constraint after reduction of a gauge phase. As is clear from thecondition (6) in the vacuum the scalar fields develop vacuum expectationvalues thus completely breaking the U (1) gauge symmetry. Correspondinglythe photon acquires the mass m γ = 12 g v , (7)where v = | (cid:104) q (cid:105) | + | (cid:104) ˜ q (cid:105) | . (8)The scalar mass matrix has three zero eigenvalues corresponding to one“eaten” combination and two massless scalar components of chiral multipletsliving on the Higgs branch. In addition, the mass matrix has one non-zeroeigenvalue corresponding to a massive scalar field which is the superpartnerof the massive vector supermultiplet, with mass equal to the mass of thephoton m H = m γ .Consider now the low-energy effective action for the theory (3), see [9].To integrate out all massive fields in (3), namely, the photon and the heavyscalar, we use the following parametrization of the Higgs branch: q = (cid:112) ξ e i ( α + β ) cosh( ρ ) , (9)¯˜ q = (cid:112) ξ e i ( α − β ) sinh( ρ ) , (10)3here ρ ( x ), α ( x ) and β ( x ) are three real fields parametrizing q and ˜ q subjectto condition (6). Once the gauge field is massive at low energies we canneglect its kinetic term and eliminate A µ using the algebraic equation A µ = − i ¯ q∂ µ q − ∂ µ ¯ qq + ˜ q∂ µ ¯˜ q − ∂ µ ˜ q ¯˜ q | q | + | ˜ q | = 2 (cid:18) ∂ µ α + ∂ µ β cosh 2 ρ (cid:19) . (11)Substituting this into the action (3) we arrive at S eff = ξ (cid:90) d x cosh 2 ρ (cid:8) ( ∂ µ ρ ) + ( ∂ µ β ) tanh ρ (cid:9) . (12)This is the low energy-action in SQED, see (3), containing only massless fieldson the Higgs branch. The gauge phase α ( x ) is canceled out as expected.In the simplest case, at the base of the Higgs branch, the vacuum is (cid:104) ˜ q (cid:105) = 0 , (cid:104) q (cid:105) = (cid:112) ξ . (13)Far away from the base we can parametrize vacua on the Higgs branch asfollows: (cid:104) q (cid:105) = (cid:112) ξ e iβ cosh( ρ ) , (cid:104) ¯˜ q (cid:105) = (cid:112) ξ e − iβ sinh( ρ ) . (14)Here ρ = ρ ( ∞ ) is a real dimensionless parameter describing how far thegiven vacuum lies from the base of the Higgs branch at ρ = 0, while β isthe residual phase which cannot be gauged away. Each vacuum on the Higgsbranch is characterized by two parameters ρ and β . Consider first the vacuum (13) located on the base of the Higgs branch. Thisvacuum admits the standard Abrikosov-Nielsen-Olesen vortices of infinitelength [4] in which the phase of the scalar field q winds while its absolutevalue rapidly tends to its vacuum expectation value at spatial infinity. Thesestrings are BPS saturated, with the tension T BPS = 2 πnξ . (15)Here n the winding number of the solution. Below we consider elementarystrings with n = 1. 4s was mentioned, in this paper we are interested in the flux tube solu-tions at a generic point on the Higgs branch. Such solutions can be foundthrough the procedure of dividing the radial separation from the string centerinto two distinct spatial domains suggested in [9].First, one can safely assume that the photon field and the massive scalarfield will form a BPS core of a finite radius determined by their commonmass, namely, R c ∼ /g (cid:112) ξ . This implies that for r ≤ R c we can look for the solutions in which ˜ q ≈
0. Thisdomain is described by the standard BPS ANO string for which T = T BPS .Second, outside the above core, at r ≥ R c , the photon field vanishes.However, the massless fields are excited, and their dynamics is determinedby the low-energy action (12). This leads to a long-range logarithmic tail,contributing both, to the profile functions and the string tension [9].The above long-range logarithmic tails require an infrared (IR) regulariza-tion. This statement is equivalent to the well-known result that the infinite-length strings are not allowed on Higgs branches [7]. We will regularize oursolutions by considering strings of a finite length L .The finite length IR regularization is physically motivated because it cor-responds to considering the string in the confinement setup. Namely, weassume that finite length string is stretched between infinitely heavy trialmonopole and antimonopole at separation L . As we already mentioned theproblem with infinite string arises because at large r outside the string corescalar fields satisfy free equations of motion and therefore, have logarithmicbehavior in two dimensions. Now for the case of the finite length string scalarfields also have logarithmic tails for R c (cid:28) r (cid:28) L . However, as r becomes oforder of the string length L the problem becomes three-dimensional ratherthen two-dimensional, see [8] for details. In three dimensions solution of thefree equation of motion for the scalar field behaves as 1 / | x n | (rather thenlog r ), where x n , n = 1 , , /L plays the role of the IR regularization for the logarithmicbehavior of scalar fields at large r . In other words the finite length L alongthe string axis translates into the IR regularization in the plane orthogonalto the string axis.The total tension of the finite- L solutions will be given by T = T BPS + T tail , (16)5here T tail denotes the contribution to the tension from the long-range tail.It is given by T tail = ξ (cid:90) d x cosh 2 ρ (cid:104) ( ∂ i ρ ) + ( ∂ i β ) tanh ρ (cid:105) , (17)where we assume that the string is a static solution aligned along the x axis, so the string profile functions in (17) depend only on coordinates x i with i = 1 ,
2, if r (cid:28) L .Although the tail profile function were not found in [9] it was shown thatthe tail tension is determined by the universal formula depending on thelength l of the geodesic line from the given vacuum to the base of the Higgsbranch. In our model this length reduces to l = (cid:90) ρ (cid:112) cosh(2 ρ ) dρ , (18)where the upper limit is the position of the vacuum on the Higgs branch, see(14). The final result for strings of length L (in the limit L (cid:29) R c ) is T tail ≈ πξ log (cid:0) g √ ξL (cid:1) l , (19)see [9] for a detailed derivation. Hence, the expression for the total tension(16) is T πξ ≈ (cid:0) g √ ξL (cid:1) l . (20)Formation of such strings leads to confinement of monopoles with the con-fining potential (2). It is not strictly linear in L .Another IR regularization more suitable for numerical calculations is tolift the Higgs branch giving massless fields a small mass without breaking N = 1 supersymmetry. One particular way to do this is considered in [9].One can start from N = 2 QED and deform it with the mass term µ forthe neutral chiral multiplet. This term breaks N = 2 supersymmetry downto N = 1 and at large masses µ the deformed theory flows to N = 1 QED.Integrating out the massive neutral multiplet one obtains the scalar potential V ( q, ˜ q ) = g (cid:0) | q | − | ˜ q | − ξ (cid:1) + 14 µ ( | q | + | ˜ q | ) (cid:12)(cid:12)(cid:12) q ˜ q − η (cid:12)(cid:12)(cid:12) , (21)6here η is a new parameter which we take to be real, see [9, 10] for detailes.We will consider this potential as an IR regularization of the one in (5). TheHiggs branch is now lifted and we have an isolated vacuum with the vacuumvalue ρ given by sinh 2 ρ = ηξ (22)The light scalar fields ρ and β in the low-energy action (12) are no longermassless. They acquire the mass m L = v µ , (23)where v is the VEV given by (8). In terms of parameters of the potential(21) v can be expressed as v = ξ + η . (24)The relation between the two IR regularizations introduced above is m L ∼ L , (25)and the result (20) for the string tension reads T πξ ≈ l log (cid:0) g √ ξ/m L (cid:1) . (26)We use the latter IR regularization for the numerical calculations below.This regularization allows us to consider infinitely long string and look forsolutions for the string profile functions in ( x , x ) plane. In this section we will construct full numerical solutions describing stringsat a generic point on the Higgs branch and, with these solutions in hand,we can directly verify the validity of the Evlampiev-Yung analytic formula(26). Our numerical solver involves a second order central finite differenceprocedure with accuracy O (10 − ). From here on we set g = 1 .
7t is convenient to define dimensionless quantities as ρ = (cid:112) ξr , ˜ µ = µ ξ , ˜ η = ηξ . (27)Then the energy minimization equations, after using the ansatz A = A r = 0 , A θ = 2(1 − f ( ρ )) ,q = (cid:112) ξq ( ρ ) e iθ , ˜ q = (cid:112) ξ ˜ q ( ρ ) e − iθ , (28)reduce to q (cid:48)(cid:48) + q (cid:48) ρ = 1 ρ qf + 14 (cid:0) q − ˜ q − (cid:1) q + 14˜ µ (cid:18) q ˜ q − ˜ η (cid:19) (cid:20) q (cid:18) q ˜ q − ˜ η (cid:19) + ˜ q (cid:0) q + ˜ q (cid:1)(cid:21) , ˜ q (cid:48)(cid:48) + ˜ q (cid:48) ρ = 1 ρ ˜ qf − (cid:0) q − ˜ q − (cid:1) ˜ q + 14˜ µ (cid:18) q ˜ q − ˜ η (cid:19) (cid:20) ˜ q (cid:18) q ˜ q − ˜ η (cid:19) + q (cid:0) q + ˜ q (cid:1)(cid:21) ,f (cid:48)(cid:48) = 12 f (cid:0) q + ˜ q (cid:1) + f (cid:48) ρ , (29)where prime denotes differentiation with respect to ρ and θ is the polar anglein ( x , x ) plane.For large regularization parameter ˜ µ , far from the base of the Higgsbranch, where f = 0, the solution is basically determined by the Higgsconstraint q − ˜ q − . (30)Then, as is easily seen from Eqs. (29), the fields q and ˜ q obey the freeequations of motion, ( ρq (cid:48) ) (cid:48) = 0 , ( ρ ˜ q (cid:48) ) (cid:48) = 0 , (31)with the standard logarithmic solutions. Correspondingly, the tension of theflux tube will be dominated by this large logarithmic tail. Numerically the8 l (a) m L / √ ξ log( √ ξ/m L ) ˜ µ . ˜ µ . (b) Figure 1:
Numerical values of (a) l and (b) log( √ ξ/m L ) for characteristic param-eters used in the numerical solutions. We put g = 1. ˜ µ . and ˜ µ . show the valuesof ˜ µ at ρ = 0 . . m L used in the table. strategy is the following: the IR regularization is implemented as a massregularization on the scalar fields, as explained in section 2. Then, once wefix m L (making sure that m L << m γ ) we impose boundary conditions onthe fields at a fixed radial distance R >> /m L . In this scheme, in whichwe fix R we must ensure that ρ is sufficiently small so that the BPS coreapproximation holds. If ρ becomes too large then the ˜ q field will develop inthe core and spoil the theoretical approximation.We are interested in solutions of (29) with the following boundary condi-tions: q (0) = ˜ q (0) = 0 ,q ( R ) = cosh( ρ ) , ˜ q ( R ) = sinh( ρ ) ,f (0) = 1 , f ( R ) = 0 . (32)Figure 1 includes reference tables for the numerical values of the param-eters l and log( √ ξ/m L ) for characteristic values of ρ and m L used below.Solutions for the field profiles are shown in Figures 2 and 3 for varying valuesof m L and ρ . We fix √ ξR = 120. Some important expected features can beseen in these plots: there is a BPS core formed by the photon field and thefield q ; in this domain the field ˜ q almost vanishes; outside the BPS core thegauge field vanishes, and the massless scalar fields exhibit large logarithmictails.Figures 4 and 5 show the results of the numerical analysis of the tensionformula (20). As seen from the plots, at larger values of the parameters wefind that for some particular combinations the accuracy of our procedure is9ot enough to find a solution. These points are excluded from the plots. Theresults involve the difference between the numerical result for the tension andits BPS part coming from the core,∆ Tξ = T − T BPS πξ = 1log (cid:0) g √ ξ/m L (cid:1) l . (33)In particular, Fig. 4 shows a plot of (cid:112) ∆ T /ξ in which we fix m L and vary l .We observe a number of important features.First, the numerical and theoretical results coincide (within numerical ac-curacy) at ρ = 0. This is expected, of course, since at this point we are at thebase of the Higgs branch and the tension coincides with the BPS result. Aswe move along the Higgs branch by increasing ρ we see an increasing dis-agreement between the numerical solution and the analytic (approximate)theoretical prediction. Once again, this is expected as in this domain onepicks up large l effects. Second, we observe that the numerical solution for (cid:112) ∆ T /ξ is a linear function of l , in perfect agreement with the theoretical ex-pression. A slight deviation in the slope can be explained by the logarithmicaccuracy in the denominator in the theoretical prediction, the log (cid:0) √ ξ/m L (cid:1) term in the denominator can be shifted by a constant non logarithmic termof the order of unity. In fact, this conjecture is supported by the subsequentplots.Figure 5 shows similar plots in which we fix ρ in order to verify the loga-rithmic dependence on m L . Once again, by observing the linear dependencein the plot, we verify that the logarithm dependence up to small values of m L . Given that the theoretical approximation is for large values of logarithmin (26) it is not surprising to find a better agreement as m L decreases (andthus 1 / log( √ ξ/m L ) decreases). Quantitatively we find that at the small-est value of m L the agreement between the theoretical T their and numerical T number results, respectively is T numer − T theor T numer × ≈ , (34)(this value holds for both values of ρ investigated). The agreement is quitesatisfactory given the magnitude of the parameters used at this point (seeFig. 1). Adding a non-logarithmic term in the denominator of the theoreticalexpression and fitting it we could have dramatically improved the agreement(by two orders of magnitude!). 10ndeed, since Eq. (20) is an approximation with logarithmic accuracy wepropose a simple modification of this formula which we can test numerically.Let us replace (20) by T πξ ≈ (cid:0) √ ξ/m L (cid:1) − c l , (35)where c is a constant to be fitted numerically. We find that, for ρ = 0 . R = 120 T numer − T theor T c numer × ≈ . , (36)provided that c ≈ . . In other words, the value of the non-logarithmic constant in (35) turns outto be less than one, a complete success. For values of ρ greater than thoseused in the plots we find that one cannot ignore the effects of the ˜ q field inthe core. In this paper we analyzed magnetic flux tubes (strings) on the Higgs branchin supersymmetric QED. In generic vacua on the Higgs branches these stringswere previously shown to develop long-range tails due to massless fields ex-isting on the Higgs branch. A natural infrared regularization for the abovetails can be provided by a finite string length L . Numerically a small super-symmetry preserving mass regularization was used, the two being related by m L ∼ /L .We performed a detailed numerical analysis of flux tube solutions atgeneric points on the Higgs branch in N = 1 SQED. We found numericalsolutions for the field profile functions defining such strings that (i) containa BPS core and (ii) besides the core contain long logarithmic tails due tothe massless scalar fields characteristic to the Higgs branch. Using thesesolutions we then analyzed the Evlampiev-Yung analytic formula (26) (pre-senting the small- m L geodesic approximation for the tail contribution to thestring tension) comparing it to numerical results . We found good agreementfor the predicted functional dependence on m L and l .11 cknowledgments This work is supported in part by DOE grant de-sc0011842 and Fondecytgrant No. 3140122. G.T. would like to thank the William I. Fine Theoret-ical Physics Institute of the University of Minnesota for hospitality duringcompletion of this work. The work of A.Y. was supported by William I.Fine Theoretical Physics Institute of the University of Minnesota, by Rus-sian Foundation for Basic Research under Grant No. 13-02-00042a and byRussian State Grant for Scientific Schools RSGSS-657512010.2. The work ofA.Y. was supported by Russian Scientific Foundation under Grant No. 14-22-00281. The Centro de Estudios Cient´ıficos (CECS) is funded by the ChileanGovernment through the Centers of Excellence Base Financing Program ofConicyt.
References [1] Y. Nambu, Phys. Rev. D , 4262 (1974);G. ’t Hooft, Gauge theories with unified weak, electromagnetic and strong in-teractions, in Proc. of the E.P.S. Int. Conf. on High Energy Physics, Palermo,23-28 June, 1975 ed. A. Zichichi (Editrice Compositori, Bologna, 1976); Nucl.Phys. B , 455 (1981); S. Mandelstam, Phys. Rept. , 245 (1976).[2] N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensa-tion, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl.Phys.
B426 , 19 (1994), (E)
B430 , 485 (1994) [hep-th/9407087].[3] N. Seiberg and E. Witten,
Monopoles, duality and chiral symmetry break-ing in N = 2 supersymmetric QCD, Nucl. Phys.
B431 , 484 (1994) [hep-th/9408099].[4] A. Abrikosov, Sov. Phys. JETP Solitons andParticles , Eds. C. Rebbi and G. Soliani (World Scientific, Singapore, 1984),p. 356]; H. Nielsen and P. Olesen, Nucl. Phys.
B61
45 (1973) [Reprintedin
Solitons and Particles , Eds. C. Rebbi and G. Soliani (World Scientific,Singapore, 1984), p. 365].[5] M. Shifman and A. Yung, Rev. Mod. Phys. , 1139 (2007), [arXiv:hep-th/0703267]; an expanded version in Supersymmetric Solitons, (CambridgeUniversity Press, 2009).
6] M. Shifman and A. Yung,
Lessons from Supersymmetry: Instead-of-Confinement Mechanism,
Int. J. Mod. Phys.
A29 , 1430064 (2014)arxiv:1410:2900 [hep-th].[7] A. A. Penin, V. A. Rubakov, P. G. Tinyakov and S. V. Troitsky,
What becomesof vortices in theories with flat directions,
Phys. Lett. B , 13 (1996) [hep-ph/9609257].[8] A. Yung,
Vortices on the Higgs Branch of the Seiberg-Witten Theory,
Nucl.Phys. B , 191 (1999) [hep-th/9906243].[9] K. Evlampiev and A. Yung,
Flux Tubes on Higgs Branches in SUSY GaugeTheories,
Nucl. Phys. B , 120 (2003) [hep-th/0303047].[10] A. I. Vainshtein and A. Yung,
Type I superconductivity upon monopolecondensation in Seiberg-Witten theory,
Nucl. Phys. B (2001) 3 [hep-th/0012250]. ρ (a)
20 40 60 80 100 120 ρ (b) Figure 2:
Numerical solutions for field profile functions varying m L , the curvelabels in (a) also apply to plot (b). In (a) we use ρ = 0 . ρ = 0 .
3. The values of m L used are reported in Figure 1. ρ Figure 3:
Field profiles varying ρ at m L = 0 . ρ = 0 . , . , . , . .2 0.3 0.4 0.5 0.6 0.7 0.8 l0.10.20.30.4 Δ T Figure 4:
Difference between numerical and theoretical tensions ∆
T /ξ for varying l . Solid line corresponds to numerical result, dashed line to theoretical. The valuesof ρ used are 0.1 to 0.7 in steps of 0.1. .26 0.28 0.30 0.32 1Log m γ m L Δ T / ξ (a) m γ m L Δ T / ξ (b) Figure 5:
Difference between numerical and theoretical tensions ∆ T for varying m L . Solid line corresponds to numerical result, dashed line to theoretical. (a) ρ = 0 .
2. (b) ρ = 0 .
3. The values of m L used are reported in Figure 1, plot(a) excludes the point in which our numerical procedure could not determine asolution.used are reported in Figure 1, plot(a) excludes the point in which our numerical procedure could not determine asolution.