VNU Joum al o f Science, M athem atics - Physics 23 (2007) 22-27
Eliminating on the divergences of the photon self - energy diagram
in (2+1) dimensional quantum electrodynamics
Nguyen Suan Han1, Nguyen Nhu Xuan2’*
1 Department o f Physics, Coỉỉege o f Science, VNƯ 334 Nguyen Trai, Hanoi, Vietnam
2D epartm ent o f Physics, Le Qui Don Technical U niversity Received 15 May 2007
A b stract: The divergence of the photon self-energy diagram in spinor quantum electrodynamics in (2 + 1) dimensional space time- (Q E D s) is studied by the Pauli-Villars regularization and dimensional regularization. Results obtained by two diíTerent methods are coincided if the gauge invariant of theory is considered carefully step by step in these calculations.
1 . Introductỉon
It is vvell known that the gauge theories in ( 2 + 1) dimensional space time though super- renormalizable theory [1], showing up inconsistence already at one loop, arising from the regular- ization procedures adopted to evaluate ultraviolet divergent amplitudes such as the photon self-encrgy in QED3. In the Iatter, if we use dimensional regularization [2] the photon is induced a topological mass in contrast with the result obtained through the Pauli-Villars scheme [3Ị, where the photon re- mains massless when we let the auxiliary mass go to iníìnity. Other side this problem is important for constructing quantum íield theory with low dimensional modem.
This report is devoted to show up the inconsistencics not arising in QED3t if the gauge invariance of theory is considered careíìỉlly step by step in those calculations by above methods of regularization for the photon self-energy diagram. The paper is organized as íòllovvs. In the second section the photon self-energy is calculated by the dimensional regularization. In the third section this problein is done by the Pauli-Villars method. Finally, we draw our conclusions.
2. Dimensional regularization
In this section, we calculate the photon se!f-energy diagram in Q ED 3 given by (Fig. 1)
* Corresponding author. Tcl: 84-4-069515341 E-mail: [email protected]
22
N.s. Han, N.N. Xuan / VNU JournaI o f Science, Mathematics - Physics 23 (2007) 22-27 23
Figure 1. The photon self-energy diagram.
Following the Standard notation, this graph is corresponding to the íormula:
d3pTr
p2 - m 2 + ie ịp — k) 2 - m2 + ie In dimensional regularization scheme, we have to make the change :
d3p
p — k + m
(
1
)[ JLP [ J p L
J
(2 ? r)ỉJ
( 2 j t ) ĩ ' (2)vvhcre 6 = 3 - n ,/i is some arbitrary mass scale which is introduced to preserve dimensional of system.
Makc to shift p by p + jẮr, the expression (1) has the form :
ru(fc) =
ỉ c2ịấ— ie2 /i
c [ dnp
J
( 2 t t ) t■
J
( 2 >r)» 7o' í(p + ÌÂ-) + m (p - 5Â:) + m
7 / 1 (p + 2 * ) 2 “ m 2 + ^ (p ~ ~ m 2+ ie (3)
cix P(m)
ịm2 — p2 + (x2 — x)fc2]
vvith
P(m) = 2 ịm 2gfÀU -f 2 + (1 - 2x)pưkỊA + 2(x2 - x)kịẰkị/
— [p2 4- (1 - 2x)pfc + (x2 - x)k2] — im íM|/aẨ:a | .
(4)
In ihe expression (3), we have used Feynman integration parameter [5j.
Neglccting the integrals that contain the odd terms of p in P{m) which will vanish under the symmetric integration in p. Then we have
n„„(*)=
=2ie fi2 ieVjf J
0/ dx / 7 (2 ‘ Xt t ) t
í 2x(l - x)kịlkí/ 2x(l - x)k2gịẨU
\(p2—a2)2 ( p2— a2)2
_ * y [ ' t e [ - £ > { - ¥ Jo J {2 n ) ĩ \ ( p2 - ị"2x(l x)(Ẳ-* yfỉi/ kfikư')
- [ (P2 - a2)2
(p2 — a2 ) 2 p2 — a2 (p2
im e ^ g k 01 Ị (p2 - a2 ) 2 J
^PịiPư Qnt/
{ / - < ? ) 2 ■ (p2 — a2) im e ^ a k 0 ị
(p2 - ã 2)2 1
(5)
24 N.s. Han, N.N. Xuan / VNU Journal o f Science, Mathematics - Physics 23 (2007) 22-27
To carry out separating Ufit/(k) into three terms n^ ( k ) = + and using the following formulae of the dimensional regularization :
1 *(_„)* r ( a - t ) 1
we obtain:
ỉ = í 1 *(~7r) r (Q — ■
°
J
( 2 t t ) t Cp2 - a » ) “ ( 2 j t ) ĩ T ( q )Ị í dnp PvP" = 9^(-o<2) ,
ụu J
( 2 t t ) * ( p 2 - a 2 ) “ ( * _ ! _ » ) ' » ’r ( 2 - ĩ )
= ( 1- ? ) r
( ' - ? ) ■ r (2) = r ó ) = 1,n ìfẮự(k) = 2ie 2^ dnp IPyPv_______g„u
^ ./o “ ./ (2*r)S [ ( p 2 - a2)2 (p2 - a 2) j
n2/i.-(fc) = 2i e V / 2x (l — x) dx [ pa -
Vo V (27T) J (p2 — o
e2^ {kpkv - k2gụv) r 1 1 ( 1 - ì )
" y.
‘ Ị ' * Ị À * - Jo J
( 2 t t ) ỹ ( p
f l J _ 1
Jo
2\2
n 3#i,(fc) = 2e2me^uaụ.tk
= e2metẦt/Qụ.tk
[m2 — x (l — x) ^ 2 ] 1 ^ 2 1
F x (p2 - a*)*
(
6
)(7) (8)
(9)
(10)
\/2 7o [m2 - i ( l — i)/c2]lí/2 (11)
From the expression (1 0 ), we are easy to see that ĨĨ2^ (0) = 0. So the íinal result, vve find : npi/(fc)/fej=o = [nj^i/(fc) + II2J1I>(A) + n 3^(fc)] |fcj=o ^ n íiỉ/(/c)/ca=o = n 3íit/(fc)Ậ;j=0 ^ 0- (12) The expression (1 2) talk to us that in the dimensional regularization method the photon have additional mass that is diíTerential from zero, even its momentum equal zero.
In the next section, we will study this problem by Pauli-Villars regularization method.
3. Pauli-VHIars regularization
Pauli-Villars regularization consists in replacing the singular Green’s functions of the massive free íìeld with the linear combination [4] :
A(x) -» regM&(m) = A(m) + y^CjA(M j). (13)
t
Here the Symbol Ac(m) stands for the Green’s function of the field o f mass m, and the Symbol A{Mi) are auxiliary quantities representing Green’s function of fictitious fíelds with mass Mi, while Ci are certain coefficients satisíying special conditions. These conditions are chosen so that the regularized function reg&(x\m) considered in the coníìguration representation tums out to be suíĩìciently regular in the vicinity o f the light cone, or (what is equivalent) such that the íiinction Ã(p\ m) in the momentum representation falls off sufficiently fast in the region of large |p|2.
On the base of Pauli-Villars regularization we calculate the poIarization tensor operator in QEDi.
For the vacuum polarization tensor vve find the folIowing expression :
( 2 t t ) 3 / 2 E *
J
\m ? + (p - ỉ f c ) 2 ì X \m ? + (p + ị k ) 2 1N.s. Han, N.N. Xuan ỉ VNU Journaỉ o f Science, Mathematics - Physics 23 (2007) 22-27 25
vvith c , = 1; Mo = m \ M i = t t i X ù Y ^ Í q C ì — 0; C ịM i = 0 ( í = 1 , 2 , . . . n / ) .
For simplicity, but without loss of generality, we may choose both the electron mass and tvvo niass of auxiliary Helds to be positive, the coeíĩicients Ai ultimately go to infinity to recover the origina!
theory.
P(Mi)
" » ■ ( 2 ^ 7 d*[Mị — p2 + ( x2 — x )Ả :2]
P{Mị) = 2^M fgflv + 2 + (1 - 2x)pl/kịl + 2(x2 - x)khkư
— 9tiu\p2 + (1 - 2x ) p k + ( * 2 - x ) f c 2] - iA/iípucA:0 ! Neglecting the integrals that contain the odd terms of p in P(Mị) we get:
(15)
(16)
"/ r r1
X^C» J
d3pJ
dxx.n w = _
(2w)3/2
2 {A/,2^ ,, + 2pIẤpv + 2( i2 - I)k(lk„ - (x2- x) k2 - iM ịe ^a k 0 }
\M f - p2 + (x2 — x)A: 2 ] 2
The expression n£ý(fc) can be written in the form:
n" (k) = (g ^ - n^í*2) + imílluakan^(k2) + g ^ n ^ ik2).
Set á\ = M ị -f (X2 — x)fc2 = A/ t2 — x(l - x)k2y vve have:
n í V ) =4te2 1 « [ X(1 - x)dx /
^ 5
X ,+ Ỉ & Ị * *
n ỉ'(li! ) = 2Ịeí g c Ị
J ‘dt J + ỉ ị dxị-Ệ^—ĩL^
ry out these integrations in the momentum space, it is straightforward to arrive
(17)
(18)
(19) (20)
(21) lf vve carry out these integrations in the momentum space, it is straightforward to arrive: n£f (fc2) = 0
as cxpected by the gauge invariance.
Here comes the crucial point: we can’t blindly take only one auxiliary field with M = A771
as usual; this choice is missionary conditions £ ”ioCj = °iMi = 0 must be niatched. This is possible only íìxing À = 1 . Thus, the number of regulators must be at leat tvvo, otherwise we can’t gct the coeíTicients Aj becoming arbitrarily large. So, let us take : Ci = a - 1;C2 = - a ; c j = 0 vvhen j > 2.
Where the parameter a can assume any real value except zero and the unity, so that. condition (15) is satisíìed. For Ai,A2 -* oo and apply it to (18). To pay attention r ( l/2 ) = ự ĩ , we have
n/
whcre
n r ” 4“ 2 S c' r ‘tex(1 ~ 11 / õ ế 75 x
e2k2 A [ Co t Ci , c2
(2 7 T )-* /2 y o x ) [ ( 0 2 ) 1 / 2 + {a2 ) l / 2 + ( o | ) l / a _
(22)
áị = m2 —2 ( 1 — x )k z\ aị —A i m Ị — x ( l — x)k 2\ âị = A2T712 — x ( l — x)k2. (23)
26 N.s. Han, N.N. X uan / VNU Journal o f Science, Mathematìcs - Physics 23 (2007) 22-27
Thus , vvhen Ai,Ả2 -> oo :
[m2 — x(l — x)k2]1^2' (2 4 ) and consequently, rii(O) = 0.
From (20), we have :
n 2M(*2) = £ - --- — ---—
4mrr [m2 - x(l - x)*2] 1 / 2
(q - 1 )Mị___ ( a - l) A /2 (25)
[A/j2 — x (l - x)k2Ý^2 \M ị — z (l — x)k2Ỷ ^
Taking the limit Ai,A2 -» ± 0 0 (depending on couplings Ci and c2 having the same sign or diíĩerent sign X —>+ 0 0 or À —> -oo), for photon momentum k=0, yields:
• if A) —> +0 0; \ 2 -* 0 0: the couplings C1,C2 have the different sign, and a < 0 or a > 0, to
vvith s = sign (1 — - ) .
It is obvious that, from (28), we saw: if 0 < Ct < 1 and s = - 1 the couplings C1,C2 havc the same sign 112(0) -ệ 0; in this case photon requires a topological mass, proportional to ri'2(0), coming from proper insertions o f the antisymmetry sector of the vacuum polarization tensor in the frec photon propagator. If vve assume that a is outside this range (0,1) and Ci and C2 have opposite signs and n 2(0) = 0.
We then conclude that this arbitrariness Q reAects in diíĩcrcnt values for the photon mass.
The nevv parameter s may be identiíĩed vvith the vvinding number o f homologically nontrivia! gauge transformations and also appears in lattice regularization [7].
Now vve face another problem: which value of a leads to the correct photon mass? A glance at equation (2 1) and vve rea!ize that ri2(Ar2) is ultraviolet finite by naive power counting. We were taught tliat a closed fermion loop must be regularized as a whole so to preserve gauge invariance. Hovvever having done that we have aíĩectcd a finite antisymmetric piece of the vacuum polarization tensor and, consequently, the photon mass. The same reasoning applies vvhen, using Pauli-Villars regularization, we calculate the anomalous magnetic mornent of the electron; again, if care is not taken, we may arrive at a wrong physical rcsult.
In order to get of this trouble we should pick out the value of Q that cancels the contribution Corning from the regulator íields. From expression (28), we easily find that this occurs for (cj = c2) because in this casc the signs o f the auxiliary masses are oppositc, in account of condition (15). From (28), we obtain ĨĨ2(0) = m , in agreement with the othcr approach already mentioncd. Wc sliould
n 2(0) = 0.
• if Ax —» +c»; A
2
—* -00
: the couplings C1
,C2
have the same sign, and 0 < a < 1(26)
(27) From the results (26) and (27), vve can be written them in the form
(28)
N.s. Han, N.N. Xuan / VNU Journal o f Science, M aihematics - Physics 23 (2007) 22-27 27
remember that Pauli Villars regularization violated party symmetry (2 + 1) dimensions. Nevertheless, for this particular choice a , this symmetry is restoređ as regulator mass get larger and larger.
a < 0, a > 1 0 < a < 1 photon mass Ci and C2 opposite sign n 2(0) = 0 equal zero C\ and C2 same sign n 2(0) / 0 unequal zero
d = c2; a = ị
Il2(0)
4. Conclusion
In d e p e n d in g on sig n o f th e c o u p lin g s C\ a n d C2, s a m e a n d o p p o s ite s ig n th e P a u li-V illa rs
regularization give a result n2(0) = - j ệ z r i (1 “ s)> where s = S2 0 7i ( l - £ ). Results obtained by regularization Pauli-Villars and dimensional methods are coincided if the gauge invariance o f theory is considered carefully step by step in these calculations. When Ci = C 2 \ a = 5, the expressions obtained
by the Pauli-Villars and the dimensional method have same results
112(
0) =
3/3in Q E D
3in
agreement with the other approaches for these problems [6).
Acknowledgments. This work was supported by Vietnam National Research Programme in National Sciences N406406.
R eíerences
[1] s. Dcsc, R. Jackiw, s. Templcton, Ann. oJ Phys. 140 (1982) 372.
[2] R. Dclbourggo, A.B. VVaites, Phys. Letỉ. B300 (1993) 241.
[3] B.M. Pimcntcl, A.T. Suzuki, J.L. Tomazclli, Int. J. Mod. Phys. A7 (1992) 5307.
[4] N.N. Bogoliubov, D.v. Shirkov, Introduction to the Theory o f Quantumzed Fìeldst John Wiley-Sons, Ncw York, 1984.
(5j J.M. Jauch, F. Rohrlich, The Theory o f Photons and Electrons, Addison-Wcslay Publishing Company, London, 1955.
[6 | B .M . P im cntcl, A .T . S uzuki, J.L . T o m a zc ỉli, Int. J. Mod. Phys. A 7 (1 9 9 2 ) 5307.
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