IL NUOVO CIhIE~TTO VoL. 56 A, N. 3 1 Aprile 1980

Electron-Muon Symmetry of Callan-Symanzik Function: Two-Lepton Case (*).

~[~. _5~CItAI~¥A a n d B . P . ~-IGAM

Department of Physics, Arizona State University - Tempe, Ariz. 85281

(ricevuto il 13 Febbraio 1980)

Summary. - - The symmetry of the (~ asymptotic part )> of the photon proper self-energy ~' 2 2 ~ 2 (m~lmo)J~ 2 - - k / m ~ , In in the two-lepton theory is examined. This leads to constraints among the sixth-order terms of ~ . These constraints agree with the explicit calculations of these terms done previously. An analysis of the mass dimensions of the photon renormalization constant Z3(A, m~, m~., m~/m~, ~) shows that only the sum of the electron and muon Callen-Symanzik functions fl,(~, mJmg) and fi~(~, too~rag) would vanish in finite QED.

1 . - I n t r o d u c t i o n .

I t has been conjectm~ed (1,2) t h a t t he first zero of the Ca l lan-Symunz ik (a)

f u n c t i o n a w a y f r o m the origin de te rmines t he f ine-s t ruc ture cons tun t ~. This

resu l t (4) is supposed to be i n d e p e n d e n t of the n u m b e r of leptons in t e rac t ing

wi th the e lec t romagne t i c field. There are a t least two wel l -es tabl ished leptons

which h a v e nonze ro mass , n a m e l y t he e lec t ron a n d the muon . I t is, therefore ,

necessa ry to expl ic i t ly ca lcula te t he Ca l l an -Symanz ik func t ion for t he two-

(*) To speed up publication, the authors of this paper have agree4 to not receive the proofs for correction. (1) K. W~LsoN: Phys. l~ev. D, 3, 1818 (1971). (2) S . L . AJ)L~: Phys. trey. D, 5, 3021 (1972). (a) C. CALLAN: Phys. l~ev. D, 2, 1541 (1970); K. SY~A~ZTK: Commun. Math. Phys., 18, 227 (1970). (4) M. P . FRY: Phys. l~ev. D, 16, 2271 (1977).

17 - I1 Nuovo Cimento A. 253

254 R. ACHARYA and B. P. NXGA~

lepton theory. This has been done in a previous paper (5). I t was found that the two-lepton Callan-Symanzik function fl,~(~, ?), where ?-----ln(m~/m~), is different from the one-lepton fl,(~) beyond the fourth order; to sixth order fi~7(a, ~) does not exhibit F-dependence, but is not a multiple of fl,(a). The explicit ~-dependence in floe(a, ?) is expected to appear for the first time in the eighth order.

In this paper we have examined the symmetry properties of the (~ asymp- totic part ~ of the photon proper self-energy ~ ( x , y, g, y) with respect to electron +-~ muon. This will impose constraints on the sixth-order terms of ~ . Some of these terms were calculated previously from the sixth-order terms in fl, a. In sect. 2, we have summarized the results of our previous paper, in order to make this paper self-contained. In sect. 3, we have examined the constraints imposed by the symmetry of ~ ( x , y, a, ~) under electron~--~ muon. In sect. 4, we have analysed the photon renormalization constant Za and ar- rived at the definition of the two-lepton Callan-Symanzik function fl,~(~, ~).

2. - S u m m a r y and exp lanat ion o f previous results .

a) One-lepton case. The Callan-Symanzik equation for the asymptotic part of the photon propagator ~ ( x , ~) is given by (')

)} o (1) [ ~m -F/Te(~) ~ - -~ - -1 {1-Focx~( ,m~, = ,

2 2 where x ~ - k /me, m. the electron mass, a the fine-structure constant and fle(~) is the electron Callan-Symanzik function. The power series expansions of fl~(:¢) and ~ 2 ~ a) are (s)

(3) o~Ti.(x,~) (al H-b l lnx ) °: H-(a~ H - b ~ l n x ) ( ~ ) ~ 7~

....

The relationship between fl~ and the coefficients a, b, c is established through the Callan-Symanzik equation (1). The coefficients a, b, c are obtained by

(5) l~. ACHARYA and B. P. NmAM: Natl. Phys. B, 141, 178 (1979); and Erratum (in press). (~) E. DE RArA~L and J. ROS~V,R: Ann. Phys. (2(. 17.), 82, 369 (1974); J. Rosl~v.l~: Ann. Phys. (2(. Y.), 44, 11 (1967).

ELECTRON-MUON SYMMETRY OF CALLAN-SYMANZIK FUNCTION : T W 0 - L E P T O N CASE 9 . ~

computa t ion of the second-, fourth- , and sixth-order vacuum polarizat ion

diagrams:

(4)

as = 5/9 , bl = - - 1/3 ,

a2 = 5/24 - - ~(3), b, --~ - - 1/4,

aa =- ? (unknown) , b3 ---- 47 /96- - (1/3)~(3),

cs = - - ( 1 / 2 ) b ~ b 2 -= - - 1/24,

where ~ ( 3 ) = ~ n -S is the Ricmann zeta-funct ion of argument 3. qt~l

are given as follows:

The fl~'s

(5)

2 /51~ = - - 2 b l = ~ ,

1

/~so ---- - - 2bs - - 2 b l a ~ = - - - - 121 144"

DE RAFAEL and R0SNER (6) obta ined the Baker-Johnson (?) result for the Gel l - ) iann-Low (s) funct ion ~(z) f rom the Callan-Symanzik funct ion fio~ eqs. (5) and (2), namely

} (6) ~o(z )=z ~ + ~ + ~(3)-- ~ + . . . ,

which satisfies the Gell-Mann-Low equation

(7)

where

~o(~,~)

In x = f dz J ~o(Z) '

(s) af(x, ~) = [1 + ~ 2 ( x , ~)]-~.

b) T w o - l e p t o n c a s e . In ref. (s), the asymptot ic pa r t of the photon prop- agator ~ ( x , y, ~) for the two-lepton (electron-muon) case was wri t ten in

(D M. BAKER and K. JORNSON: .Phys . R e v . , 183, 1292 (1969). (s) M. GELL-MANN and F. Low: .Phys . R e v . , 95, 1300 (1954).

256

analogy with eq. (3) in the foIlowing form:

:R. ACHAl~YA ~nd B. 1). NIGAI~

(9)

(lO)

where

(11)

~n~(x, y, g) = (al -k b~ In x) -~ -]- (a~ -~ bs In x) ~- ... 7g

. . . + ( a , + b ~ l n y ) - ~ ÷ ( a s + b s I n y ) ~ ÷ . . . - - 2"/:

=- ~z4(x,y,~ o~) = (K1 q- Ll lnX) ~o,- q- (Ks q- Zs in x) (~) = q-

q- (Ka q- Za In x q- Ma In s x) q- . . . ,

_ _ kSlm s x = - - k ~ / m 2 o , Y = i ~,

10 1 Kl(y) = 2al - - 4Y = -9- + 3 Y,

2 L1 = 2bi -- 3 '

5 Ks(y) = 2 a s - b2y - - 12

1 Zs = 2bs - - 2 '

Ka(y) = unknown,

1 - - - - 2 5 ( 3 ) + g 7,

7 = in ( ~ / m D ,

1 1 (12) Ma = -- ~ JB1~s -- 6"

I t is worth noting in eq. (6) t ha t the al, bl, as, bs terms are simply additive, since to fourth order there are no mixed electron-muon loops. The mixed loops appear for the first t ime in the sixth order and, therefore, aa, ba, ca terms are not additive for the electron and muon variables x and y, respectively. To the fourth order, since the electron and muon contributions to ~ (x, y , oc)

are additive, by comparing eqs. (3) and (10), it is easy to see tha t the two- lepton results are obtainable irom the one-lepton results by the replacement

(la) al, bl, a2, bs----~ K l ( ± y), L1, Ks ( - f - y ) , L s ,

where t~he q- (--) sign is for mixed loops with external electron (muon) loop and internal muon (electron) loop. Since the sixth-order two-loop contributions are obtainable in terms of the fourth-order contributions , eq. (13) suffices.

ELECTI%ON-I~IUON SY~IMETI~Y OF CALLAN-SYMANZIK FUNCTION: TWO-LEPTON CASE 257

The two-lepton sixth-order contribution fl~v is given as follows:

(1~) /~ = ~ , ~ + ~ , ~ . ,

= L,"aFflem + {fl~¢~)}(with a~, b,, a~, b~-+ .KI(y),-~1, K ~ ( 7 ) , - L 2 ) ] ~-

+ F~ va) + {fi~(*)}(with a~, b~, a2, bo. ---~ K~(-- 7), I~, K2(-- y), Is)] L V g

where superscripts (1) and (2) refer to one-loop and two-loop sixth-order dia- grams, /~°g(~)/Z¢~u)~ is the mixed loop contribution with external electron ~3~IL k~3ML /

(muon) and internal muon (electron) loop which are obtainable from the fourth- order result, eq. (13). l~rom eq. (8) of ~osner (6), eq. (3.35) of de Rafael and l~osner (o) and eq. (14), we obtain

2 3 (15) ? ; ~ = [ - ~ + {~ ( - - f fKl (7 )+ 3-~1)+ }] +

[ 1 {~( 23_K,(_,y)~.L13 ) ~'1 }] 233 72 + -- + -- + : (4=K~(-- ?) -k 8.L~ --

which is the two-lepton sixth-order result corresponding to the one-electron fi~ of eq. (5). The two-lepton second-order and fourth-order contributions are given by eq. (5):

(16) /3; ~ = 2/~;, fi:v = 2/~.

Combining eqs. (15) and (16), we obtain the two-lepton Callan-Symanzik func- tion fi~ up to sixth order

(17) /3o~ = ~ ~ \7# - ~ \?q + ... .

(18)

giving

(19)

(20)

The two-lepton ~p~ and I3(7) are obtained (5) from

1 fl~t~ : _ 2 ~ 1 K ~ ~ 2 ~ 2 K 1 ~ .~ V ~ - : - - 2 1 3 - - 2 I ~ K 2 ,

193 ~ = - - - - + - - ~ ( 3 ) ,

18

273 4 1 L3(?)-- 144 3~(3) -k~7"

The electron-muon Gell-Mann-Low function F~V(z, ?) is given by

32 z ~ ...]

258 ~. ACH~rA and m P. ~mA~

which satisfies the two-lepton (electron-muon) Gell-Mann-Low equation (5)

(22)

where

In x = ~o~(z, 7) '

a2(X, 7, 0¢) = [1 ÷ au~(X, 7, a)]- l .

We m a y point out t ha t the two-lepton Gell-Mann-Low equation, eq. (22),

appears only in the electron variable x, since the muon variable y h~s been scaled to variables x and 7 in eqs. (9) and (10).

The Gell-Mann-Low equation in terms of the variable 7 is as follows:

(23)

~(~,-v,~)

f In y = ~ ( z , - - 7)" ~e~'(1,-~,,~)

Equat ion (23) is equivalent to eq. (22). The Gell-Mann-Low equation which is manifest ly symmetric under x ¢-~ y (m, ~-+ m~) is obtained by adding eqs. (22) and (23).

3 . - S y m m e t r y p r o p e r t y of ~ f ( x , y, ~).

Equat ion (10) for ~ ( x , y, ~) should exhibit symmet ry under the exchange x ¢-+ y, since the vacuum polarization diagrams are symmetric with respect to e ~-+ ~ (too +-> m~-- y ¢-+-- 7). We, therefore, impose this condition on eq. (10) :

(24) ~ ( x , y, y, ~) = ~ ( y , x, - -7 , ~)"

Obviously, from eqs. (9) and (10), the a and a S terms exhibit the symmet ry x ~-~ y explicitly, viz.

(25) { K1(7) + Z1 In x = KI(-- 7) Jr Z~ in y = 2al + bl In x y ,

K2(7) -? Z2 In x ~-- K2(-- 7) + Z2 in y = 2a2 + b~ in x y .

From eq. (24), for order a3, we have (9)

(26) K~(7) + Ls(y) In x + M3 in ~ x = K3(-- Y) + L3(-- 7) In y ~- Ma ln* y .

(8) M 3 satisfies the renormalization group constraint M3=--½JDlJS,=---~ and is independent of 7.

ELECTRON-MUON SYMMETRY OF CALLAN-SYMANZIK FUNCTION: TW0-LEPTON CASE

The 1.h.s. of eq. (26) is

(27) Ka(?) ÷ yIa(?) + ?~Ma ÷ [La(r) ÷ 2?M3] in y ÷ Ms In ~ y.

Comparing eqs. (26) and (27), we obtain

(28a) /[3(7) - - g a ( - - y ) = - - ~J~a(Y) - - Y ~ M s ,

(28b) I3(y) - - Zs(-- y) -~ - - 2yMs.

From eq. (28b), the general form of .Sa(y) is

1 (29) Is(y) : a-- M a y ~ a ÷ ~ y,

259

which agrees with our (5) derivation of La(y), eq. (20). From eqs. (28a) and (20) we have

(30)

giving

(31)

[273_4 ] Ka(y)--Ks(--Y) = - - Y [:144 3~(3) ,

[273_3 ] K3(y) = c - - / 2 8 S ~ ~(3) y ÷ dy ~ ,

where c and d are undetermined. Thus the symmetry requirement alone, eq. (24), fixes uniquely the coefficient of the y-term in Ls(y), eq. (29). Also, the coefficient of the y-term in Ka(y) has been determined, eq. (31).

The actual rewriting of ~ ( x , y, ~), eq. (10) (using Ki(y), Zi(y) and Ms, eqs. (12) and (20), as derived by us (5)), into a manifestly symmetric form gives

(33)

-~ (2al + b, lnxy)~ + (2a~ ÷ b~lnxy)(~)~ +

12 . . . .

Applying the symmetry condition, eq. (24), to eq. (32) yields eqs. (30) and (31)9 making eq. (32) explicitly symmetric under x+--~y, ?¢---~--y.

260 n. ACHARYA and B: P. I~IGAIK

4 . - E q u a t i o n f o r Z3, t w o - . l e p t o n c a s e .

The general form of the photon renormalization constant Z3 for the two- lepton ease with masses m e and m~ is

(33 ) Z 3 = Z~(A, me, my. , me/ma) ,

where A is the cut-off parameter. Since Z3 has zero mass dimension, the scal- ing (lo) of A, me, m v into ~A, umo, xm~, respectively, will not change it. Thus

(34) Z3(~A , xme, xm~, xme/~cm~) -~ na Za(A, me, Orbit , me~my) ,

where the mass dimension

(35) d(Z3) = O .

From eqs. (34) and (35), we have

(~) ~ ~ ( ~ A , ~ o , ~ , ~ . / ~ ) = ~ ~ ( A , ~., ~ , ~ . / ~ ) : 0

Using the chain rule of differentiation

(37) d 8(~A) ~ 8(nine) ~ 8(zm~) d~ -- 8 ~ ~(ZA) + 8~¢ ~(zme) 4 ~ 8(~m~) '

we obtain from eq. (36)

(38) g ~ Za(zA, zme, zmv~ me~my. ) =

= ~A + ~mo ~(~m.--~ + ~m~ Z~(~A, ~mo, em~., meier) = 6,

which can be rewrit ten into

(39) A ~ + me ~ -]- m• Za(A, me, mt~, ----0.

(lo) K..WILso~: Phys..Rev., 179, 1499 (1969):

ELECTRON-IKUON SYMIKETI%Y OF CALLAN-SYMANZIK FUNCTION: TWO-LEPTON CASE 261

I f Z~ =/= 0, then (11)

(~o)

where

(4~) /~o = mo ~=:- l n Z s ,

(42) /3v. : m~ - - - in Z~.

F r o m eq. (40) we notice t ha t in the two-lepton case -- A(~/~A) In Z3 is the sum of the electron and muon Cal lan-Symanzik functions. I f Z3 is in- dependent of A, as in finite (s) QED, then

(43) ~o~(~, ~)=~o(~, ~) +fl~(~, r ) = 0 ,

which is the eigenvalue equat ion relat ing a and ~. I t is wor th not ing tha t , in

the two-lepton case, finite QED does not require fi~ and fl~ to vanish indepen-

dently. I f i t is possible to go beyond finite QED and make fi~(~, y) and fi~(~, y) vanish independent ly , then one could de termine both ~ and 7 = In (mv/m).2 2

5 . - C o n c l u s i o n .

The s y m m e t r y requi rements on aT~°(x, y, a, y) under electron +-+ muon

enables us to obta in constra ints on the s ixth-order te rms Ks(y), L3(y) and Ms. These agree wi th our (~) explicit calculation of ~n~o. We also find tha t , in finite QED wi th two leptons, the sum of the electron and muou Cal lan-Symanzik

funct ions fie(% 7) and fi~(a, y) vanishes, thus establishing a relat ionship between ~nd ?. The independent determina t ion of a (and y) would require t h a t we

invoke t h a t fio(~, y) and D~(~, y) vanish separa te ly ; this would involve going beyond finite QED.

* * *

One of the authors (BPN) would like to acknowledge fruitful discussions wi th Profs. S. ~N. BlSWAS and S. ]~o¥ CHOWDI~I.

(11) As an example, a general form Z 3 may be

lnZ~ ~ , y , - - , = I n 7~-[- ~- A2 ~ A2 A2

Then it is easy to verify that

8 8 Z - - A --SA In Z 3 = m~ ~mo In ~ + m v ~mmv In Z 3.

This will also hold good for any polynomial form on the right-hand side.

262 1~. ACHARYA ~nd B. P. NIGA:~

• R I A S S U N T O (*)

Si esamina la s immet r ia della ~ par te as intot iea ~> dell 'au~oenergia propr ia del fo tone co 2 2 ~z~ (--I~ line, --l~2 /m~, ~, In ~ 2 (m~/m~)) nella teor ia a due leptoni. Questa po r t a a v incol i

f ra i t e rmin i del sesto ordine di ~z~. Questi v incol i sono in accordo con i ealcoli esplieit i di quest i t e rmin i ef fe t tuat i in precedenza. Un 'ana l i s i delle dimensioni di massa della cos tan te di r inormal izzazione del fo tone Zz(A, m~, m~, mdm~, ~) mos t ra che solo la so~n~na delle funzioni di Callan e Symanz ik de l l ' e le t t rone e del muone /3~(~, mdmg 3 e fl~(~, mdm~) si annul lerebbe nel la QED finita.

(*) Tmduzione a eura della t~edazione.

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(*) 1-1epeeeOeuo pe3amlue~.