VNU JOURNAL OF SCIENCE Mathematics - Physics, t XVIII, n°l - 2002 IN FR A R E D SIN G U L A R IT IE S OF FE R M IO N G R E E N ’S F U N C T IO N AND T H E W ILSO N LO O P Nguyen Suan Han Faculty o f Physics, College o f N atural Sciences V N U H A b s tr a c t : Gauge-invariant, and path-dependent objects being infrared asymptotic* of a gauge-invariant ÿpinor Green fu n ctio n in the Q E D have studied It is proved that the infrared singularities o f fe rm io n Green's function can be factorized as the Wilson loop that contains the prim a ry Path and the straight'line contour and accu mulâtes all the dependence on the fo rm o f the path o f the in itial Green function Till! present report is devoted to the stu d y of infrared asym ptotics o f the gaugeinvariant spinor Green function T he interest in this problem com es from th e hope the problem of quark confinement, ill the framework o f QCD can be solved in this way [11 Usually the standard fcrmion propagator < ():7’vỊ/(.r)xI'(//);() > is stud ied that, as it is well known is a gauge-dependent quantity At the sam e tim e it is known that the iufrarrd behavior o f the com plete fermion propagator essentially depends oil the gauge choice l.r> Thus, for exam ple, in the A belian case in the class of covariant rv-gauges the fcrmion propagator lias a branch point at p = n r which only at a — = loads to a pole singularity of ỉli\ > ( 1) \Ằritlì t lu* r n n a n f / ì r ttile im Green r iiv ü 111 contradistinction w ith th e standard npropagator function (1) contains tile exponential with the path integral taken over the gauge field along an arbitrary path c that connects the points X and y Our aim consists in stud ying the infrared behavior of the path-dependent, gauge-invariant propagator (1) Let us use th e representation (1) in a form of the functional integral over the spinor and vector fields G (.c ,y \C ) = - I D * D * D A [ f ( A ) ] < l / { x ) r e x p i - i e J dzlẳA „ { z ) *{y) (2 ) Hero, till1 integration measure over th e gauge field DA includes som e gauge condition D A = D A [ f ( A ) ] (3) the explicit, lb nil of which clue to the gauge independence of ( 1) is not essential Performing in (2) the integration over the ferm ionic fields we get (w ith f ( A ) = d ^ A ^ ( x )) G(.r.y\C) = Ị e x p [ - S 0( A)\ G{x,y\ A) exp[-ic ■1 J dzt, A t,{z)\ r (4) T y p e s e t by ^VfS-lfejX In fra re d sin g u la rities o f f e m i i o n green9s fu n c t io n and w h c r r S\i is t h í ’ Enrli(l(*cUi a r t ion o f thí' free e l e c tr o m a g n e ti c fi(‘l (9) N g a y en Suan H an Thus, we have arrived to an interesting conclusion th at all th e properties of th e infrared behavior o f the gauge-invariant propagator (1) are accum ulated in the W ilson loop (9) Let us consider, as an exam ple, a particular choice of th e path c and the price of the straight line from the point X up to th e point y W ith such a choice o f the path th e propagator (1) takes the fon n y \ c = IT) = - < |r $ C r ) e x p ie / d a ( y - x ) uA ụ( x 4- a ( y - x ) ) Jo ỹ (y )\0 > (10) whore tlie integration in th e exp on en tia l is perform ed along th e piece o f the straight line c = 11, ,; o f form (6) It is clear that in th is case j) dztlA ,Ẽ{z ) = J d z,,A t,(z ) = 0, (11) SO Jin — T hus, add ition al infrared singularities (like a branch p oin t) th at appear due to th e interaction w ith th e "soft” p h oton s not appear here, and Fourier transform ation o f t ilt* propagator in the infrared lim it has a sim ple pole G ,IẼ,/(p |C = n ) =2 (l/jü -h m ) T his result exactly agrees w ith th a t o f th e calculation of the infrared asym ptotics of th e propagator (1) don e ill [()] T he author would like to thank Profs I.L Solovtsov, O.Yu Shevchenko for stim ulating discussions T he financial support of the N ational B asic Research Program ill Natural Sciences K T -410701 is highly appreciated R e fe r e n c e s II Page Is Phy$.Rev D (1 7 ), 2991 M Baker Y s.B a ll and F.Z achariasen Nv.cl.Phys B 1 ( 1981), 531 B A A rbuzov, P h y s.L e tL B (1 ) 497 N N B ogoliubov and D V Shirkov Introduction to the Theory of quantized Fields 3rd E dition, New York 1984 N guyen Suan Han Jo u rn a l C om m u nica tions in Theoretical P h ysics , China 37(2002) 167 A A Sissakian, [.L Solovtsov, O Yu Shevchenko Elcrn.Chast Atom Yad 2jf(1990), 664 V N P opov C o n tin u a l Integrals in Quantum F ie ld Theory and Statistical Physics M ,A tom izdat, 1976 Infrared singularities o f fcrrnion green's fu n c tio n and !) TAP CHI KHOA HỌC ĐHQGHN Toán - Lý t XVIII n°l - 2002 CÁC KỲ DỊ HỒNG NGOẠI CUA HÀM GREEN FECMION VÀ C H U T U Y E N W ILSO N Nguyễn Xuân Hàn K h o a Ly D U K lio a học T ự nhiên Đ H Q G H N Các đối tượng bất biến chuán phụ thuộc vào quỹ đạo tiệm cận hồng ncoại cùa hàm Green spinor bat hiên chuẩn diện động lực học lượng tử (QED) dược nghiên cứu Đã chứng minh lảng: kỳ dị hồng ngoại hàm Green fecmion có lh(i dược mai thừa hố chu tuyến Wilson, mà chứa quỷ đạo ban đáu đường Ihủnc viên quanh tập truns phụ thuộc vào dạne quỹ đạo cùa hàm Green ban đấu ... s tûA ^ { z ) ) , (.-») that is valid in an infrared lim it T he integration in (5) is performed along the piece of the straight line II th at connects tli(' points X and y: Zf, = i’„ 4- s(fj... ple, a particular choice of th e path c and the price of the straight line from the point X up to th e point y W ith such a choice o f the path th e propagator (1) takes the fon n y c = IT) =... we have arrived to an interesting conclusion th at all th e properties of th e infrared behavior o f the gauge-invariant propagator (1) are accum ulated in the W ilson loop (9) Let us consider,