l.a. ferreira, nobuyuki sawado and kouichi toda- static hopfions in the extended skyrme-faddeev...

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  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

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    P|h|m` Fazcmaep me |fb bv|belbl Pk}rdb$Chllbbs dalbi

    \fmp hr|m`ib fhp obbe la~eiahlbl crad MAZp`mbe`b% Zibhpb p`raii la~e |a pbb |fb cii |bv| hr|m`ib%

    GFBZ88&?::=-8?:6!?::=!88!8?

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

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    GFBZ8

    8&?::=-8

    ?:) Pha @hriap$PZ) OrhwmioLbzhr|dbe| ac Zf}pm`p) \ak}a Remsbrpm|} ac P`mbe`b)

    Ealh) @fmoh ?>6$628:) Ghzhe`Lbzhr|dbe| ac Dh|fbdh|m`hi Zf}pm`p) \a}hdh Zrbcb`|rhi Remsbrpm|})

    Krakh~h 286:) Mdmw) \a}hdh) =0=$:0=6) Ghzhe

    B$dhmi7 ihcJmcp`%pz%or) ph~hlaJzf%ealh%|p%h`%gz )

    kam`fmJ}kh~h%k}a|a$%h`%gz

    Hop|rh`|7 ^b `aep|r`| p|h|m` paim|ae pai|maep ~m|f eae$wbra Fazc |azaianm`hi `fhrnbp

    |a h |fbar} ~fm`f mp he bv|bepmae ac |fb Pk}rdb$Chllbbs dalbi o} |fb hllm|mae ac h

    cr|fbr xhr|m` |brd me lbrmsh|msbp% ^b pb he hvmhii} p}ddb|rm` heph|w ohpbl ae |aramlhi

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    imdm|men `hpbp% \fb pai|maep zrbpbe| he me|brbp|men obfhsmar elbr |fb `fhenbp ac h pzb`mhi

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    wbra hp |fh| `adomeh|mae hzzrah`fbp h zhr|m`ihr pzb`mhi shib% ^b `hi`ih|b |fb bxmshibe|

    ac |fb Shkibeka hel Khzm|hepkmm bebrn} oael car |fb |fbar} hel el |fh| m| shempfbp

    h| |fh| phdb pzb`mhi shib ac |fb `azimen `aep|he|p% Me hllm|mae) |fb dalbi zrbpbe|p he

    me|bnrhoib pb`|ar ~m|f he meem|b edobr ac ia`hi `aepbrsbl `rrbe|p ~fm`f hzzhrbe|i} hrb

    ea| rbih|bl |a p}ddb|rmbp ac |fb h`|mae% Me |fb me|brpb`|mae ac |fapb |~a pzb`mhi pb`|arp |fb

    |fbar} zappbppbp bvh`| sar|bv pai|maep &p|h|m` hel |mdb lbzbelbe|- ~fm`f ~brb `aep|r`|bl

    me h zrbsmap zhzbr o} aeb ac |fb h|farp% M| mp obimbsbl |fh| p`f dalbi lbp`rmobp padb

    hpzb`|p ac |fb ia~ bebrn} imdm| ac |fb zrb PR&?- ]hen$Dmiip |fbar}) hel ar rbpi|p dh}

    ob mdzar|he| me mlbe|mc}men mdzar|he| p|r`|rbp me |fh| p|raen `azimen rbnmdb%

    Kb}~arlp7 Paim|aep Daeazaibp hel Mep|he|aep) Me|bnrhoib Cmbil \fbarmbp) Me|bnrhoib

    Bxh|maep me Zf}pm`p) Me|bnrhoib Fmbrhr`fmbp

    HrVms bZrme|7 :=:6%03>?

    PMPPH ?::= lam78:%8:66!88?3$3>:6!?::=!88!8?

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

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    GFBZ8

    8&?::=-8

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    < \fb bxh|maep ac da|mae 6

    2 \fb obfhsmar ac |fb pai|maep me imdm|men `hpbp 8:

    2%8 \fb obfhsmar h| pzh|mhi meem|} 8:

    2%? \fb obfhsmar hrael |fb `mr`ib v?8 + v?? 4 r

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    2%0 \fb obfhsmar ac |fb pai|maep hrael |fb v0$hvmp 80

    3 \fb edbrm`hi hehi}pmp 8 8%03

    & 8%>8

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    Boarel 4 3< ? D

    yby

    b? 8 X0!-) hel |fh| dh} ob h ~h} ac elbrp|helmen ~f} B&8---

    hel pa m|p `ae|rmo|mae shempfbp hp b? 8% \fbrbcarb) ~b hrb ibc| ~m|f |fb xhlrh|m`|brd B? aei}) ~fm`f `heea| ob ohihe`bl o} |fb xhr|m` |brdp he}darb% Pme`b |fh| |brd

    p`hibp hp B? B?) hp vm

    vm

    ) |fb pai|mae |belp |a pfrmek% Hea|fbr ch`| mp |fh| |fbbxh|maep ac da|mae me |fb pb`|ar ac |fb |fbar} ~fbrb b? 4 8) hel ~fbrb &8%8:- mp ph|mpbl)

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    mdzraohoib |a fhzzbe me p`f h |fbar}% @aepbxbe|i}) aei} sar|bv pai|maep imkb |fb aebp

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    \fb pai|maep car |fb heph|w ce`|maep n &w) - hel &w) - ph|mpc}men |fb oaelhr}

    `aelm|maep &?%8:-) &?%88-) &

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

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    GFBZ8

    8&?::=-8

    ?

    : % : : 6

    : % : : =

    8 ? 0 < 2

    : % < = <

    : % < = 2

    : % < = 3

    : % < = >

    : % < = 6

    : % < = =

    : % 2 : :

    ?

    &

    b

    ?

    $

    8

    -

    B

    : % < = 6

    : % < = =

    : % 2 : :

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    Cmnrb 8% \fb zia|p ae |fb i%f%p% `arrbpzael |a |fb pai|mae ~fbrb &d) e- 4 &8) 8- hel pa Fazc

    `fhrnb bxhi |a 8) hel |fapb ae |fb r%f%p |a |fb pai|mae ~fbrb & d) e- 4 &8) ?- hel pa Fazc `fhrnb

    bxhi |a ?% Ae |fb rp| hel pb`ael ra~p ~b zia| |fb |a|hi bebrn} B &pbb &8%8>-- hel |fb zhrhdb|br

    h &pbb &8%83-- rbpzb`|msbi}) hnhmep| b?% Ae |fb |fmrl hel car|f ra~p ~b pfa~ |fb crh`|mae ac |fb

    |a|hi bebrn} `arrbpzaelmen |a |fb |~a xhr|m` |brdp) B&8-< hel B

    &?-< rbpzb`|msbi}) hp h ce`|mae ac

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    B&8-< +

    b? 8 B&?-< )hel pa |fb pd ac |fb shibp pfa~e ae |fb zia|p ae |fb |fmrl hel car|f ra~p nmsbp : %2%

    8>

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

    20/26

    GFBZ8

    8&?::=-8

    ?

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

    21/26

    GFBZ8

    8&?::=-8

    ?

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

    22/26

    GFBZ8

    8&?::=-8

    ?

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

    23/26

    GFBZ8

    8&?::=-8

    ?

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

    24/26

    GFBZ8

    8&?::=-8

    ?2$XP>: [PZM_BPT; [PZM_BPT;

    H embl bil |fbar} ac dbpaep hel ohr}aep) E`i% Zf}p% 08 &8=3?- 223 [PZM_BPT;

    G%K% Zbrrmen hel \%F%_% Pk}rdb) H dalbi embl bil bxh|mae) E`i% Zf}p% 08 &8=3?- 22:

    [PZM_BPT%

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    cbrradhneb|p) GB\Z Ib||% ?? &8=>2- ?- 26 [fbz$|f!=38:8=0T

    [PZM_BPT%

    [2T _%H% Oh||}b hel Z%D% P|`imb) \a ob ar kea| |a ob1) Zf}p% _bs% Ib||% 68 &8==6- =6

    [fbz$|f!=6:68?=T [PZM_BPT%

    [3T Z% P|`imb) Kea|p me |fb Pk}rdb$Chllbbs dalbi) Zra`% _a}% Pa`% Iael% H - 0::8

    [hrVms7:>:2%8T G% Fmb|hrme|h hel Z% Phia) Chllbbs$Fazc kea|p7 l}ehdm`p ac imekbl e$kea|p) Zf}p% Ib||% O

    :8 [PZM_BPT%

    [=T G% Fmb|hrme|h) G% Gh}kkh hel Z% Phia) L}ehdm`p ac sar|m`bp hel kea|p me Chllbbsp dalbi)

    ZaP&REBPZ?::?-:8>;

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    dalbi) Zf}p% Ib||% H 0?8 &?:: [hrVms7:=: &?::0- 8:2::= [fbz$|f!:0:0:=?T [PZM_BPT;

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    3= &?::

  • 8/3/2019 L.A. Ferreira, Nobuyuki Sawado and Kouichi Toda- Static Hopfi ons in the extended Skyrme-Faddeev model

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    GFBZ8

    8&?::=-8

    ?:3=T [PZM_BPT%

    [86T ]%D% @fa) H rbp|rm`|bl nhnb |fbar}) Zf}p% _bs% L ?8 &8=6:- 8:6: [PZM_BPT%

    [8=T ]%D% @fa) Bv|belbl nhnb |fbar} hel m|p dhpp pzb`|rd) Zf}p% _bs% L ?0 &8=68- ?T A% Hishrbw) I%H% Cbrrbmrh hel G% Phe`fbw Nmiibe) H eb~ hzzrah`f |a me|bnrhoib |fbarmbp mehe} lmdbepmae) E`i% Zf}p% O 2?= &8==6- 36= [fbz$|f!=>8:8T [PZM_BPT%

    [?6T A% Hishrbw) I%H% Cbrrbmrh hel G% Phe`fbw$Nmiibe) Me|bnrhoib |fbarmbp hel iaaz pzh`bp7

    celhdbe|hip) hzzim`h|maep hel eb~ lbsbiazdbe|p) Me|% G% Dal% Zf}p% H ?< &?::=- 86?2

    [hrVms7:=:8%832

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    GFBZ8

    8&?::=-8

    ?