botli Pauli- Villnr iiK'thofl and dimensional regularization method are used to remove divergences and dctmniuo the electron Green's function for Q E D i i after m[r]
(1)VNU JOURNAL O F SCIENCE M a t h e m a t i c s Physics t XVIII n ° l - 2002
R E N O R M A L IZ A T IO N E L E C T R O N G R E E N ’S
F U N C T I O N IN B L O C H - N O R S IE C K M O D E L F O R Q E D ; { Nguyen Nhu Xuan
F a cu lty o f Physics C olleg e o f N a tu l ScU'ticcs V \TU H
Abstract: Tin (■!('< Il'ou GỉViỉís fimctton m Bloch-Novsicrk fo r QE1):\ has brt'11
Hit /(’(<1 ht/ Mi (HIS of tin fa Iirf HHHil intrqraiioii method Both methods of Pa uli-Vtllar and iliiifj nsifnml nyulttrizaft.tm an ttsf'd to thill tin dwerqe.nt integrals, which i/r/.sT uv also Ịnor<<l that Hu quantum Crcfỵis function fo r Q E p z is t.kc same as Green's function of f i n fit id hff'tmsc Q E D \\ is lli<‘ su p cr-rm u n iL a liz a tio v theory.
1 Introduction
In nrcnt years, studying asymptotic property of Green's function ill infrared zone lia.s received considerable attention because of its relation to asymptotic property of (livens functions in quantum chromodynaniics (QCD) with (Ịiiark confinement [1-5]
Fill I h rn n o iv theorvtic of low dimensional systems have a lot of interesting properties,
lor (‘XHinpIc SchwiiijLi.iT mechanism, fractional statistics, quantum Hall effect and high-Tc MiỊXTcondiict ivitv So study the electron Green's function, a fundamental quantity ill Q E D \\ is an indispensable1 and interesting problem The general formula of the Crocu s function is (l<kt(Tiiiii»(‘(l by functional integration method [4-:
n i „ J G ị x %y\A).Sị)(A).ỔA
C (J - ■"> ° ( "
where G ( r %j/\A ) is (hr electron Grom's function in the external classical electTimmtfiietic licld A „ S {)( A ) is the 5-matrix in van mm expectation of fermion field when the external (*!(•( tiomagiK'tic held A „ appears The quantum Green's function is obtained after other humIkhIs to remove divergences being used Calculating functional integrals is a difficult problem, so ill this paper, we only study Bloch-Norsieck model and could set A u (A ) =
This paper is organized as follows Ill see.2 we shall use the proper time method of Folk-Feynman to calculate the electron Green s function ill the external field ill t,(Tills of thí' functional integral After integrating this function following external fi(‘l(I the electron (iiv rn ’s function ill momentum representation is obtained In sec.3-4 botli Pauli- Villnr iiK'thofl and dimensional regularization method are used to remove divergences and dctmniuo the electron Green's function for Q E D i i after mass renormalization Finally, ill sec*.5 wo shall compare and discuss the results, which have been obtained
2 Representation of the electron Green’s function in the external field by functional integral
Th<‘ Blocli-Norsieck model ill QED is proposed to avoid inframl catastrophe ill the interaction of particles with light at low frequencies by approximation not based oil
(2)N gay en N h u XutOLU
perturbation theory In the zero order approximation this model is equivalent to replacing T lie* Dirac limtrices by r-number u (t, with V 2 = 11 It turns out that the Green s functiion
Go for a I’m* electron (to omit vacuum polarization) is purely ‘ retarded" Therefoiiv matrix elonionts corresponding to closed fermion loops will contain at lc.'ivst one fun<’ti«on G „ <‘<|iial to zero Physically, the absence of a second pole ill the Green s function mea ns
that t l i n v aIV no Hiitiparticles in this model, hence* any pairs can be created T h u s then*
arc no radiative corrections to the Green s function in this model This model is (lefiiK'd by th(’ following interaction Lagrangian
m >
4>
A\=
- ^ t!J( x )+ ỹ(.r) [«/’(/#,
- c A „ )-
m]*(ar).
(2) \\’h vvcFtll/ = d t,A „ ( r ) - d uA f , ( x ) is the electromagnetic intensity tensorThr equation for th(' Green’s function in the external field will have the following form
{ w"[/ỡo + e.At, ( x ) ] - m } G { x , y \ A ) = - d ( x - y). ( 3)
Using the proper time method proposed by Feynman and Fock [4] we obtain tllio electron Green s function ill the external field in the following form
(7(.r.//Ll) = — X
Ị
dưI
(lnp e xp{ — //) w ( m - u p 4- i f ) + R ( r i ' ) } ( 4)(2ã)m Vo /
wlinv
/?(•<- V) = - f y ; I M -\u A (k ))e ' k' Ị ' (■5)
The quantum Green s function of electron in momentum representation is obtained by moans of expecting functional over external field
with:
G{ p) = i / e x p[—w ( m - up - i t ) + Ị{ ụ )\ (lV '
Jo
/(/') = ~ j f ỹ ị X Ị < P k (u >lD t,u (k)v.v ) Ị d v x d//2c‘(,‘A )|/ J and D nl,{ k ) is the five photon propagator, a is the gauge parameter
£>,,„(*•) = - £ ■
111 substituting (8) into (7) and we have to note that D ụ u ( k ) is the even fuuiction respect to k. After that we have
/ M - - ^ X I ’ m r < I „ f « ft « W L ,
(2 7ĩ);ỉ /() /() / [ k2 + ừ ( F + i e )
IP* f 1' f Í f fp r íìỉ(
- ~ 7 T? x / di/\ / dV2 / it k < ~r~0 -0 ~ a ) T 777
(2* r \/o ./ U - + íẽ v đ u ị L(fc2
1 t I I I I X I I • I 1 »
(.6 )
(•7)
(fH)
, / (hA‘ )i
ữt{uk)i/2 "Ị _{k2 + Ü Ÿ \
(í9)
1 111 Bltich-Norsifck model, the Dirac matrices is replaced by onumlnT u and list* in»ta’iofNN
1 i f n =
- I i f n - ,
(3)R e n o r m a l i z a t i o n e l e c t r o n G r e e n ' s f u n c t i o n i n b l o c h - n o r s i e c k m o d e l
T1'I<* alxnv-niiMit ionod integrals are divergent so ill this paper, we shall present the Pimli-Villar regularization method and the dimensional regularization Iiiothoil to pa.ss over I lh‘sf ‘ (livcTgrnees
3 J he Pauli-Villar method
111 tin* Pauli-Villar method, we introduce a new field, which has a large mass M 2( M - ỷ oc) and tlu* free photon propagator is replaced by: p j p -f- p —j/T (tlie auxiliary mas> A is used to move the infrared divergences, A —» 0) Thus
/ ( " )
- - ( * ? *
L
rf" 'I
d u ’ { F " ' M - 1 - ■ (1 ) with^
- / "
‘‘ [ ï r b + ï ỵ W
F {2)(i'->) =
ị
(l:ik i( r<Ắ.:)iẠỉ (1 1 )(A’2 + A2)2 (A-ịA/2)2
The integral (11 ) is turned into the form of Gauss's integral by using a representation
o f till-* in v e rs e ' o p e r a te ) !'
77
integrating over rf.s* and using formula [6]
- s / Y
^ f/.s.s'"'2 exp ^ -A 2.s - % 7T2r ( l + m)zm- 3/2A1/2- ”,e - Ax
F ^ ( „ 2) = (2n Ỳ ' ' J ị
(1 )
(13)
(14)
■i 1'2 V A
Substituting (14) into (1Ü) and integrating over d i'2 and d u1 W(‘ shall obtain the tthrtl result
/(;,) -
i l l —, i lJK ’ (2n ) :i 2 V
7T - f / >
+ i/r(Aỉ/ 0),
(15
( V 2 ' (2tt)3/2 V
with P(A/a0) being the Gamma incomplete When A —» it is asymptotic to logarithmic
funct ion
Replacing mass m by renonnalization mass
7T ( a 4*
-r(Ai/,0)
VV(* obtain the quantum Green's function in momentum representation G \ ( p ) = i / di/exp[—ii^(mi — up + ze)] =
./<) /71 ! — u p -f if • ( 16)
(4)N g u y en N h u X'uan
4 T h e d i m e n s i o n a l r e g u l a r i z a t i o n m e t h o d
Now we are going to use the dimensional regularization method 111' for the integrals in (9) One of the advantages of this method, in special cases, we don’t care about the at t rihution of fictit icms mass of photon though it is related to the infrarod divergences This problem must be noticed if we use the Pauli-Villar method By means of representation of the inverse operator (12) and Gauss integral in arbitrary //-dimension space, we find out
/V2 r ( ? ? / - 1) R y ) = 8
ttII/ - n [2
+ (l
- a ) ( n -3)1
( u - i f ) I - n { - i f ) 4 - n
4 — n
(17) In order to obtain the Q E D , i we replace the respective number of space dimension into (7) then renormalize mass of the Green’s function (7) After replacing n = — // into (14) and taking the limit cts IỊ —» Ü we have
/(* ) = i ( 2 ( v — i e ) l n
•1 \Z2tt
Replacing (18) into (7) wo obtain
m
-V tc
,
4\/27r
(18)
G ( p ) = i d u exp I - i v
M aking m ass renorm alization
m — up + ừ
4\Z2tt
th ru
p ( \ , i v n i l = m -7=— I n -1
4v/27T V *
roc Ị
G ị ( p ) = i / civexp I- il / ( i n I — u p -+- /f)l =
-./() m - uup + — ie (19)
Evidently, with dimensional regularization method the expression of quantum Green s function ( 19) iis absolutely similar to the expression of quantum Green's function ( 16) which has been obtained by the Pauli-Villàr method
5 C onclusions
(5)I ỉ e ĩ ì O v m a l i z a t i ( ) Ị e l e c t r o n G r e e n ' s f u n c t i o n i n b l o c h - n o r s i e c k m o d e l
to tilt- Green’s function of free field In dimensional regularization f { v ) doesn’t depend on Miuiilarri paraiiH'tcT a but in Pauli-Villar method f ( u ) still depends on standard parame ter iHVtUisr W’V used auxiliary 1IIỈUSS A which causes theory not to have standard invariant
Moan while, in Q E D \ in zero order app roxim ation o f infrared ca ta stro p h e the electron
'Cỉrccii s (unci ion | l| would Ixx’oiiK’ the Green s function of free field
Acknowledgements: Tile author would like to thank all the mrinlxTs of t lie Department
of Theoretical Physics (or stim u la tin g discussions T h e financial su p p o rt of the N ational
Basis' Research Program ill National Sciences K T 4.1.1 s 4.13 is highly appreciated
6 R e fe re n ce s
1 H Page Is Phys Rev D15(1977) 2991
2 M.Baker Y s.Ball and F.Zachariascu N u rl P h y s3116(1981) 531
3 B A A rbuzov Phys Lett B (1 ) 497.
*4 N.N Bogoliubov and D.V.Shirkov In t ro d u c tio n to the T h e o ry o f quantized Field s. 3rd Edition, Now York, 1984
r> Nguyen Suan I Ian J o u r n a l C o m m u n ic a tio n s in T h eo retica l P h y sics. China, 2001 () F.Bloch and A.Nordsieck Phys Rev 52(1937) 54 :
7 D.Yennie S.Frautschi and Suura A n n Phys 13(1961) 379
S ,I.M Jauch and Rohrlich T h e o ry of ph oto n s and electrons. Addison Wesley Read ing, Mass 1955
9 Sviilzinski Determination of Green’s function in the Bloch-Nordsieck modoỉ by functional Integration, Pkys Rev., D 6(1954)
10 R.Castillans and R.Mruidclnunus Nud Pliys 1363(1973) 277
11 VV.J.Maciauo D im ensional regularization o f infrared divergen ces Nuclear physics
888(1975) 86-98
12 R.Jackiw N ucl P h y s13 (Proc.Suppl) 18a(1990) 107
13 H.O.Giroti M.Gomes and A.JdaSilva Preprint IFUSP/P-977 Nucl P h ys 1364(1978)
T A P CHÍ KHOA HỌC DHQGHN Toán - Lý t XVIII n ° l - 2002
TÁI C H U Ấ N H O Á HÀM G R E E N E L E C T R O N
TRONG MƠ HÌNH BLOCH-NORSIECK CHO QED.ị
N g u y ễ n N h X u â n
Khoa Lý D í l Khoa hục T ự nliiên Đ H Q G H N
Trong khn khổ phương pháp tích phân phiếm hàm Green electron trons mơ hình
Bloch-Norsieck (BN) cho Q E D ị đă nhận Đ ể loại bỏ tích phân phân kỳ
phương pháp Pauli-V illars điều chinh thứ n gu yên dà sứ dụ ng C ác phương pháp
này chứng m inh hàm Green fecm ion lượng tử QE D ị hoàn toàn giống với