ScanGate document - VNU Journal of Science

(1)

VNU JOURNAL OF SCIENCE. Mathematics - Physics, t XVIII, n°l - 2002

IN F R A R E D S IN G U L A R IT IE S OF F E R M IO N G R E E N ’S F U N C T IO N AND T H E W IL S O N L O O P

Nguyen Suan Han

Faculty o f Physics, College o f N atural Sciences. V N U H

A b s t r 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 o f infrared a sy m p to tics o f the gauge- invariant spinor Green function. T he interest in this problem com es from th e hope the problem o f quark confinement, ill th e framework o f Q CD can be solved in this way [1- 1 . Usually the standard fcrmion propagator < ():7’vỊ/(.r)xI'(//);() > is stu d ied that, as it is well known is a gauge-dependent quantity. At the sam e tim e it is know n th at the iufrarrd behavior o f the com p lete ferm ion propagator essentially depend s oil the gauge choice l.r> T hus, for exam ple, in th e A belian case in the class o f covariant rv-gauges th e fcrmion propagator lias a branch point at p 2 = n r which only at a — = 3 loads to a pole singularity of ỉli<‘ propagator [-1]. That- is w hy one can conclude that a consistent, stu d y oi (lie lic'orii's should 1)0 (lone oil the basis of the gauge-independent quantities. Ill particular, instead o f the standard spinor propagator 0110 can consider a gauge-invariant.

G m’11 function f(jlet ion [()•

G...( x , y | C ) = - < 0 |T I t y ( x ) Pexp - i e

Ị

‘ dZ,t A „ ( z )

\Ằritlì t lu* n r n n a ớ n f / ì r t i m r iiv ü

ỹ( v) >\0

> . (

1

)

*{y).

(

²

)

111 contradistin ction w ith th e stand ard propagator tile G reen function (1) contains tile exponential w ith the path integral taken over the gauge field along an arbitrary path c that connects the points X and y. Our aim con sists in stu d y in g the infrared behavior o f 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 6 [ f ( A ) ] < l / { x ) r e x p i - i e

J

⁹dzlẳA „ { z )

Hero, till1 integration m easure over th e gauge field DA includes som e gauge condition

D A = D A .6 [ f ( A ) ] . (3)

the explicit, lb nil o f which clue to the gauge independence o f ( 1) is not essen tial. Performing in (2) the integration over th e 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

J dzt, A t,{z)\

■ 1 r (4)

T y p e s e t by ^VfS-lfejX

(2)

w h c r r S\i .1 is t h í ’ Enrli(l(*cUi a r t ion o f th í' free e l e c t r o m a g n e t i c fi(‘l<l .Su .1 — - ( I1 1 i f , . (./').

I ) fl (ìịt T i.e.. is invariant derivative, and G(.v.tj\A) if the Ciiern function <>l the (•[(‘(•Iron in nil cxtrn m l field A fl. which satisfied th e equation

hi,Of, m\ G( x , y) A) = 0(.i: - y).

Ill wlmt follows we shall represent, th e field ill accordance with as a sum o f slowly ;m<l rapidly varying com p on en ts A {{)) and A [ i \ For th e Green function o f I’m n io n ill all eh'ctroinagnol ic Held use tlir approxim ate formula [7i

C !(x ,y \A m + A ^ ) e x p { i e l % l s tûA ^ { z ) ) , (.-»)

that is valid in an infrared lim it. T h e integration in (5) is perform ed along th e piece of the straight line II th a t con nects tli(' points X and y:

n Zf, = .i’„ 4- s(fj - ./% . 0 < ./• < 1. ((>) W illi m rount o f formula (5) and th e approxim ate relation

«lot

.[)t, D, ă

-

nt\

(let [«>„(#„ 4

- i c . A „ ) - m \

1

I f r\ 1 I ,. r n I ckt. 1 -f „ /f^yy-4^ -w 1,

( lo t

[Ỵ/tỠỊi - m

^I d e t ^ ỡ , ! - m ị

'yỊid,ê - m

th at holds for th e infrared lim it, it is not difficult to show th at propagator (4) can he represented as a product, of two factors

G inft{x. y\ C)

=

J u v J i n .

(7)

Here the first. J L v (C /V -ultraviolet) is th e quantum Green function ob tain ed only with account o f th e rapidity varying field A \ 1 \ T he second factor .JIU ( / / ? - infrared) is obtained w ith account o f slow ly varying com pon en t i4,,0) only. It reflects th e interaction w ith the soft photon and thus contains th e infrared singularities. T h is factor has th e form

J / W _ I 0 4 W I _ i c ị d z M X » I f ( 8 )

I n fr a r e d s in g u la r itie s o f f e m i i o n green9s f u n c t i o n a n d ...

7

= I D A {i]).oxp I —ie (j) rÌ2,,/lỊ,0) j .

whore. f , dztlA \li} (z ) is the integral over th e closed path L = c + n o f th e form L = c + Ii

+■

Ï

It is not difficult, to see th a t expression (8) is n othing but the W ilson loop J Uỉ W ( L ) whore.

\ V ( L ) = < 0 |T e x p - i e j d z p A ^ i z ) |0 > . (9)

(3)

T hus, 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 accu m u lated in the W ilson loop (9).

Let us consider, as an ex a m p le, a particular choice o f th e p a th c and the price of th e straight line from th e point X up to th e point y. W ith such a choice o f th e path th e propagator (1) takes the fo n n .

8

N g a y en S u a n H a n

y \ c = IT) = - < 0 |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 in tegration in th e e x p o n e n tia l is perform ed along th e piece o f th e straight line c = 11, ,; o f form (6). It is clear th at in th is case

j) dztlA ,Ẽ{z ) = J d z,,A t,(z ) = 0, (11)

SO .Jin — 1. T h u s, ad d ition al infrared singu larities (like a branch p o in t) th at appear due to th e interaction w ith th e " soft” p h o to n s do not appear here, and Fourier transform ation o f t ilt* propagator in the infrared lim it has a sim p le pole G ,IẼ,/(p |C = n ) =2 ( l/j ü - h m ) . T h is result e x a ctly agrees w ith th a t o f th e calcu lation of the infrared a sy m p to tics o f th e propagator (1) d o n e ill [()].

T h e author w ould like to th ank Profs I.L. S olovtsov, O .Yu. Shevchenko for stim  ulating discussion s. T h e financial sup p ort o f th e N ational B asic R esearch Program ill N atural S ciences K T -410701 is highly appreciated.

R e f e r e n c e s

1. II. Page Is. Phy$.Rev. D 1 5 (1 9 7 7 ) , 2991.

2. M .Baker. Y .s .B a ll and F .Z achariasen. Nv.cl.Phys. B 1 1 6 ( 1981), 531.

3. B .A . A rbuzov, P h y s.L e tL. B 1 2 5 ( 1 8 3 ) . 497.

4. N .N . B ogoliu b ov and D .V .Sh irk ov. Introduction to the Theory of quantized Fields.

3rd E d ition , N ew York. 1984.

5. N guyen Suan Han. Jo u rn a l C om m u nica tions in Theoretical P h ysics, C hina. 3 7 (2 0 0 2 ) 167.

6. A .A . Sissak ian , [.L. S olo v tso v , O .Y u. Shevchenko. E lcrn.C hast. Atom .Yad. 2jf(1990), 664.

7. V .N . P op ov. C o n tin u a l Integrals in Quantum. F ie ld Theory and Statistical Physics.

M .,A to m izd a t, 1976.

(4)

Infrared sin g u la rities 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 IL SO N Nguyễn Xuân Hàn

K h o a Ly. D U K lio a học T ự nhiên. Đ H Q G H N