Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Multiple layer structure of non-Abelian vortex

Minoru Etoa b *, Toshiaki Fujimoric, Takayuki Nagashimac, Muneto Nittad, Keisuke Ohashie, Norisuke Sakaif

a ¡NFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy b Department of Physics, University of Pisa Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy c Department of Physics, Tokyo Institute ofTechnology, Tokyo 152-8551, Japan d Department of Physics, Keio University, Hiyoshi, Yokohama, Kanagawa 223-8521, Japan e Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 0WA, UK f Department of Mathematics, Tokyo Woman's Christian University, Tokyo 167-8585, Japan

ARTICLE INFO

ABSTRACT

Article history: Bogomolny-Prasad-Sommerfield (BPS) vortices in U(N) gauge theories have two layers corresponding to

Received 14 April 2009 non-Abelian and Abelian fluxes, whose widths depend nontrivially on the ratio of U (1) and SU(N) gauge

Acceded 20 May 2009 couplings. We find numerically and analytically that the widths differ significantly from the Compton

"M" May 2009 lengths of lightest massive particles with the appropriate quantum number.

1 or' ' ve 'c © 2009 Elsevier B.V. All rights reserved.

1. Introduction

Many important properties of Abelian (ANO) vortex were found [1-5] since its discovery [6]. Recently vortices in U(N) gauge theories (called non-Abelian vortices) were found [7,8] and have attracted much attention [9] because they play an important role in a dual picture of quark confinement [8,10] and are a candidate of cosmic strings [11] (see [12] for review report). The moduli space of U(N) non-Abelian vortices was determined in [13] and study on interactions between non-BPS configurations started in [14]. Non-Abelian vortices in other gauge groups have been studied in [15].

Although there have been much progress and wide applications, internal structures and dependence on gauge coupling constants have not yet been studied for (color) magnetic flux tubes. It is particularly important to study physical widths of vortices qualitatively and quantitatively, although it is not easy because no analytic solutions are known. It may be tempting to speculate that the width is determined by the Compton lengths of lightest massive particles with the appropriate quantum number. Purpose of this Letter is to clarify intricate multiple layer structures of non-Abelian vortices by investigating numerically and analytically the equations of motion. Non-Abelian vortices have two distinct widths for SU(N) and U (1) fluxes. We clarify properties of these widths by making use of several approximations. It turns out that non-Abelian vor-

* Corresponding author at: INFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy.

E-mail addresses: minoru@df.unipi.it (M. Eto), fujimori@th.phys.titech.ac.jp (T. Fujimori), nagashi@th.phys.titech.ac.jp (T. Nagashima), nitta@phys-h.keio.ac.jp (M. Nitta), K.Ohashi@damtp.cam.ac.uk (K. Ohashi), sakai@lab.twcu.ac.jp (N. Sakai).

0370-2693/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.05.061

tices are very different from ANO vortices and have much richer internal structures.

2. Vortex equations and solutions

Let us consider a U (N) gauge theory with gauge fields Wp for SU(N) and wp for U(1) and N Higgs fields H (N-by-N matrix) in the fundamental representation. We consider the Lagrangian L = K — V which can be embedded into supersymmetric theory with eight supercharges

K = Tr

- Tg? (F »vf + H V» H t] - ^ ( fßvf,

V = g^Tr[(HHt)2] + ^ (Tr[HHt - c1N])

(1) (2)

where (X} stands for a traceless part of a square matrix X. Our notation is VpH = (dp + iWp + iWplN)H, Fpv = dpWv — dv Wp + i[Wp, Wv] and f/JiV = dpwv — dvwp. We have three couplings: SU(N) gauge coupling g, U (1) gauge coupling e and Fayet-Iliopoulos parameter c > 0.

The Higgs vacuum H = «Jc1N is unique and is in a color-flavor SU(N)C+F locking phase. Mass spectrum is classified according to representations of SU(N)C+F as mg = g«Jc for non-Abelian fields 0N = (W,< H >) and me = eV2Nc for Abelian fields 0A = (w, Tr(H — Vc1N)). The non-Hermitian part of H is eaten by the U(N) gauge fields. A special case of mg = me [7] has been mostly considered so far, which is equivalently

Y = 1 with y = ■

mg me '

Î7 = l-

<7 = 1

Fig. 1. h12 (solid, broken lines) and Beg. The left panels (me = 1) for logy = 0,1,2,3,4, 5 and to. The right panels (mg = 1) for logy = 0, -0.5, — 1,..., —3 and —to.

but we study general cases in this Letter.

Let us consider static vortex-string solutions along x3-axis. The BPS equations for the non-Abelian vortex are

V H = 0, -42 =

<HHt> /12 Tr(HHÎ - c1N)

with V = (V1 + iV2)/2. The tension of k-vortex is Tk = —cf d2x x Tr[f121N] = 2nkc. No analytic solutions have been known whereas a numerical solution was first found in [8]. For k = 1 vortex, we take H = S—1 Ho, W = —iS—1dS [2W = (W1 + W1IN) + i(W2 + w21N)] with diagonal matrices H0 = diag(rei0, 1,..., 1) and cSSt = N logr2)lN+(ftg + N—1logr2)T. Here, T = diag(1, — N—■, ..., — N11) and and ftg are real functions of the radius r > 0 of the polar coordinates (r,8) in x1; x2 plane. Then we get

1 ft g + Ne (e-fg + (N - 1)eN=r) = 1

Aft g N -1 , , , *g , „

+-e-fe(e-fg - eN-T) = 0,

ml N v ' ,

with A/(r) = dr(rdr/(r))/r. The boundary conditions are fte ,ftg ^ 0(r ^ to) and Nfte, n^y ftg ^ - log r2(r ^ 0). The fluxes and the Higgs fields are expressed by

/12 = -T Afte, F12 = -T AftgT,

H = ^c diag(hi, h2 ,...,h2)

with h1 = +ie and h2 = e— 1 (fe—n— ). The amount of the

Abelian flux is 1/N and the non-Abelian flux is (N — 1)/N of the ANO vortex.

We found the following theorems for Eqs. (5) and (6)

(a) fte,g > 0, drfte,g < 0 and Afte,g > 0,

(b) |h 1| < |h21, |h 1I < 1 and dr\h!\ > 0,

(c) dr|h2| § 0 and 1 § h2 § VN/(N + y2 — 1) for y § 1.

All these can be proved by using the following theorem for an analytic function f (r) satisfying f (r) < 0 ^ A f (r) < 0: If drf (0) < 0 and f (to) = 0, then f (r) > 0 for Vr e (0, to). In the case of y = 1, we get Nfte = nn1 ftg = ftANO and the above equations reduce to AftANO = m2(1 — e—ftANO) with boundary condition ftANO ^ 0(r ^ to) and ftANO ^ — log r2(r ^ 0).

Numerical solutions for N = 2 for a wide range of y (including y = 0, to) are shown in Fig. 1. Winding field h1 is not sensitive on y while unwound field h2 is. As mg being sent to to (y ^ to), the non-Abelian flux F12 becomes very sharp and finally gets to singular. Interestingly, the Abelian flux f12 is kept finite there. In a region y < 1 (me > mg), on the other hand, the Abelian flux is a bit smaller than the non-Abelian tube. Surprisingly, the fluxes remain finite even in me ^ to limit.

Table 1

Numerical data for k =

1 U (2) vortex.

Y Ce cg be bg aY

0 - 1.1363 0 0.75905 ■J2

0.25 - 1.1853 0.31719 0.73163 1.31688

0.5 - 1.3090 0.47907 0.68393 1.18361

0.75 2.196 1.4852 0.55921 0.64006 1.07932

1 1.70786 1.70786 0.60329 0.60329 1

1.5 1.4715 2.3031 0.64726 0.54697 0.88820

2 1.4037 3.15 0.66773 0.50604 0.81226

2.5 1.3746 4.32 0.67897 0.47469 0.75640

3 1.3594 6.0 0.68584 0.44969 0.71301

œ 1.3267 - 0.70653 0 0

3. Asymptotic width

Le (mg = 1) Le (mg = 1)

Le (me = 1)

Le (me = 1)

Let us investigate the vortex solution by expanding (5) and (6) in region r ^ max{m-1, m_where \fe|, \fg| ^ 1. We keep only the lowest-order term in fe while keeping terms up to next to leading order in fg in Eq. (5):

= 0, (A - mg) fg = 0.

(A - m2e) fe + 2(N _ i}

The solution is given by the second modified Bessel function K0(r) and approximated as

fe —

n e-mer 2mer ,

e-2mgr

fg - ci

4(N-1)(1-4y2) mgr

with ceg being dimensionless constants, see Table 1. The asymptotic behavior of fe changes at y = 1/2 (upper for y > 1/2 and lower y ^ 1/2). Similar phenomenon was observed for the non-BPS ANO vortex [3,4]. The origin of fe (fg) is (non-)Abelian fields 0A (0N) with mass me (mg), and the y = 1/2 threshold can be interpreted as follows. The expansion of the Lagrangian with respect to small 0A,N contains the triple couplings For

me < 2mg, asymptotics for 0A N are given by K0(me,gr) as the two-dimensional Green's function. When me > 2mg, the particles 0A decay into two particles ^ through these couplings, and thus, 0A exhibits the asymptotic behavior e_2mgr below y = 1/2 like Eq. (9). On the contrary, even for y > 2, does not behave as e_2mer since there is no triple coupling condition for 0N.

Let us define asymptotic width of the vortex by an inverse of the decay constant in Eq. (9):

A due to the traceless

{2/me for y ^ 1 /2, 2/(2mg) for y< 1/2,

Lg = 2/mg.

Here the factor 2 is put in the numerator to match with another definition in Eq. (14). The asymptotic width of Abelian vortex is bigger than the non-Abelian one when y > 1 and vice versa for 1/2 < y < 1. For y = 1, the two widths are the same. The case Y < 1/2 indicates a significant modification, where the Abelian flux tube is supported by the non-Abelian flux tube. This answers the question why the Abelian vortex does not collapse even in the me ^ < limit. When y ^ 1, the thin non-Abelian flux hidden by fat Abelian flux cannot be correctly measured by Lg. We now turn to another definition of vortex width which reflect the size of the vortex core more faithfully.

4. Core widths

Let us consider a region near the vortex core. We expand fields

Lg(me i s =D:\ a\ 4 a\ a\ a\ A3 • •

Lg(™g = 1)

Lg{mg = 1) 1 Lg(me 1) ^ 2/3

-4-2 2 4

Fig. 2. The core and asymptotic widths vs. log y .

2 N - 1

fe N logbemer, fg ~-2 n logbgmgr. (11)

Dimensionless constants beg are related to h2(r = 0) by

aY = h2(0) = (be/Y bg)1/N. (12)

See Table 1. These are important since they are related to the maximum values of the magnetic fluxes at r = 0

Be = -f 1 -

m2 N - 1 2

Bg =--g-aY.

g 2 N Y

Widths of the magnetic fluxes can be estimated by using a step function 0(x) as F12 = Bg&(Lg _ r)T and f12 = Be&(Le _r) keeping amounts of the fluxes as \Be\ x nL2 = 2n/N and \Bg\ x nL?g = 2(N _ 1)n/N:

L e = ■

mjN - (N - 1)aY

We call Le and Lg as the core widths of the vortex. In the case of Y = 1, Le and Lg coincide because of aY=1 = h2 = 1. In Fig. 2, we show the core widths numerically in the case of N = 2, which are analytically reinforced as we will discuss. We again observe that the Abelian core does not collapse even when me ^ 1 (y ^ 1).

Mass dependence of the core widths coincides with one of the asymptotic widths Leg given in Eq. (10), except for Lg(y > 1), see Fig. 2. The asymptotic width Lg is independent of me whereas the core width Lg depends on me. This is because Le,g more faithfully reflects the multilayer structure in the large intermediate region of r for the strong coupling regime (y ^ 1), to which we now turn.

5. Two strong coupling limits

Here we study two limits: (i) mg — to with me fixed, and (ii) me — to with mg fixed. In the former limit (y —to with me fixed), all the fields with mass mg become infinitely heavy and integrated out from the theory. As a result, the original U (N) gauge theory becomes the Abelian theory with one SU(N) singlet field B = det H. Note that the target space is C/ZN because U (1) charge of B is N. Eq. (6) is solved by fgTO = 0 while fe to is determined by Eq. (5)

Afe, to

= 1 - e fe,TO , fe,TO-^-T, logmebeTO (15)

r—0 N

where suffix to denotes y =to: fgTO = fg\y —^TO. The boundary condition tells that vorticity is fractional k = 1/N. This way the non-Abelian flux tube collapses and the U (N) non-Abelian vortex reduces to the 1 /N fractional Abelian vortex. This solution helps us to understand the non-Abelian vortex for y » 1 better. Since e-fe ^ (mebeTOr)2/N (r < 1/me) for y » 1, fg for r < 1/me is well approximated by a solution of the following

Afg N -1 2 , ,

g +--— (mgr) n(e-fg - eN-1

where the parameter mg = mg(beTOy ')N+1 has a mass dimension. Therefore fg has asymptotic behavior in the middle region

1/mg < r < 1/me

fg «cgKo

(mgr) N « 1,

and fg ^ - 2(Nn log(bgmgr) for r < 1/mg. Here bg, cg are determined numerically and independent of y, for instance, bg = 0.74672, cg = 0.63662 for N = 2. Comparing this with Eq. (11), we

~ 1 1 —1 i 1

find bg ^ bg[be,TOy-1] n+1 and aY ^ bgN [be,TOy-1] n+1 for y » 1.

In the second limit (y — 0 with mg fixed), all the fields with the mass me are integrated out. The model reduces to a CPn2-1 model with SU(N) isometry [in SU(N2 - 1)] gauged. Eq. (5) is

solved by efe-° = (e-fg-° + (N - 1)eN-1 )/N while fg,0 is determined by

2 (N - 1)(1 - e-N-f

Afg,0 = mg-

(N - 1) + e-N-T

where the suffix 0 denotes y — 0: fg,0 = fg\y—0. This is a new a-model lump with the non-Abelian flux accompanied with the internal orientation CPN-1. Again we can make use of this solution to understand the non-Abelian vortex for y < 1. Let us define a2 = Afg,0(0 = B1 \Y—0 which turns out to be finite

Afe,0(0)

a2 = (N - 1)/(1 + 4b2g\y—0). Since Afgfi(0) = m2g and Afe,0(0) = limY—0 m2gy-2(1 - (N - 1)a2Y/N), we find aY = a0(1 - Y2/(2a2) + ■ ■ ■), a0 = VN/(N - 1) for y << 1.

6. Summary and discussion

We have proposed two length scales for fluxes of non-Abelian vortices: asymptotic widths Leg in Eq. (10) and core widths Le,g in Eq. (14). By using the asymptotics of aY obtained above, the core width is summarized as

{Le, Lg

(Y < 1),

, 2£( — )N+ } (Y » 1), {meVN, mg (me ) } (/ » ),

where a and f$ depend only on N and are determined numerically, for instance a = 0.55010,^ = 0.97022 for N = 2. The core and asymptotic widths have the same mass dependence except for Lg and Lg for y » 1. Interestingly, the Abelian flux does not collapse even when me — to (y < 1). For y » 1, the thin non-Abelian flux is hidden by fat Abelian flux, so that the true width cannot be captured by the asymptotics at r » m-1. Instead we should use improved approximation given in Eq. (17) to measure the non-Abelian flux. Indeed, the decay constant in Eq. (17) is rh-1 whose mass dependence is the same as one of Lg for y » 1.

In the limit mg — 0, the original U (N) gauge theory reduces to U(1) gauge theory coupled to N2 Higgs fields. Eq. (19) tells us that the vortex is diluted and vanishes in this limit. This is consistent with the fact that there is no (smooth) vortex solution with a winding number 1/N in that U (1) theory. The minimal vortex in the U(1) theory corresponds to N vortices in the original theory. The dilution is expected to be avoided and all the fields with mass of the order of mg decouple, when N vortices are arranged as H = f (r)1N. In the limit me — 0, there is no BPS vortex solution since the U (1) gauge field is decoupled from the Higgs fields. Actually, according to Eq. (19), one can find both of the Abelian

and the non-Abelian fluxes are diluted again even in this limit due

to the factor y n+1 . Monopoles/instantons attached by vortices are known to exist [10]. The above observation implies that such configurations reduce to a monopole/instanton configurations in the SU(N) gauge theory in that limit, and strongly supports the correspondence between the moduli spaces of them.

It is interesting to study relation between non-BPS vortices and monopoles. It was found that monopoles do not collapse when the Higgs mass is very large [16].

Acknowledgements

M.E. and K.O. would like to thank S.B. Gudnason, N. Manton, D. Tong and W. Vinci for useful discussions. This work is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan Nos. 17540237, 18204024 (N.S.) and 20740141 (M.N.). The work is also supported by the Research Fellowships of the Japan Society for the Promotion of Science for Research Abroad (M.E. and K.O.) and for Young Scientists (T.F. and T.N.).

References

[1] H.J. de Vega, F.A. Schaposnik, Phys. Rev. D 14 (1976) 1100.

[2] C.H. Taubes, Commun. Math. Phys. 72 (1980) 277.

[3] B. Plohr, J. Math. Phys. 22 (1981) 2184.

[4] L. Perivolaropoulos, Phys. Rev. D 48 (1993) 5961.

[5] J.M. Speight, Phys. Rev. D 55 (1997) 3830.

[6] H.B. Nielsen, P. Olesen, Nucl. Phys. B 61 (1973 ) 45.

[7] A. Hanany, D. Tong, JHEP 0307 (2003) 037.

[8] R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi, A. Yung, Nucl. Phys. B 673 (2003)

[9] Y. Isozumi, M. Nitta, K. Ohashi, N. Sakai, Phys. Rev. D 71 (2005) 065018;

M. Eto, M. Nitta, N. Sakai, Nucl. Phys. B 701 (2004) 247;

A. Gorsky, M. Shifman, A. Yung, Phys. Rev. D 71 (2005) 045010;

M. Shifman, A. Yung, Phys. Rev. D 72 (2005) 085017;

M. Eto, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci, N. Yokoi, Phys.

Rev. D 74 (2006) 065021;

M. Edalati, D. Tong, JHEP 0705 (2007) 005;

D. Tong, JHEP 0709 (2007) 022;

M. Shifman, A. Yung, Phys. Rev. D 77 (2008) 125016;

J.M. Baptista, arXiv:0810.3220 [hep-th]. [10] D. Tong, Phys. Rev. D 69 (2004) 065003;

M. Shifman, A. Yung, Phys. Rev. D 70 (2004) 045004;

A. Hanany, D. Tong, JHEP 0404 (2004) 066;

M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, N. Sakai, Phys. Rev. D 72 (2005) 025011;

M. Eto, et al., Nucl. Phys. B 780 (2007) 161.

[11] K. Hashimoto, D. Tong, JCAP 0509 (2005) 004;

M. Eto, K. Hashimoto, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci, Phys. Rev. Lett. 98 (2007) 091602.

[12] D. Tong, arXiv:hep-th/0509216;

M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, N. Sakai, J. Phys. A 39 (2006) R315; M. Shifman, A. Yung, Rev. Mod. Phys. 79 (2007) 1139; D. Tong, arXiv:0809.5060 [hep-th]; K. Konishi, arXiv:0809.1370 [hep-th].

[13] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, N. Sakai, Phys. Rev. Lett. 96 (2006) 161601.

[14] R. Auzzi, M. Eto, W. Vinci, JHEP 0711 (2007) 090; R. Auzzi, M. Eto, W. Vinci, JHEP 0802 (2008) 100.

[15] L. Ferretti, S.B. Gudnason, K. Konishi, Nucl. Phys. B 789 (2008) 84;

M. Eto, T. Fujimori, S.B. Gudnason, K. Konishi, M. Nitta, K. Ohashi, W. Vinci, Phys. Lett. B 669 (2008) 98.

[16] E.B. Bogomolny, M.S. Marinov, Yad. Fiz. 23 (1976) 676.