Point transformations and renormalization in the unitary gauge

Point transformations and renormalization in the unitary gauge

Nuclear Physics B79 (1974) 503-525. North-Holland Publishing Company POINT TRANSFORMATIONS AND RENORMALIZATION 1N THE UNITARY GAUGE G.B. MAINLAND, L...

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Nuclear Physics B79 (1974) 503-525. North-Holland Publishing Company

POINT TRANSFORMATIONS AND RENORMALIZATION 1N THE UNITARY GAUGE G.B. MAINLAND, L. O'RAIFEARTAIGH and T.N. SHERRY Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, lreland Received 30 April 1974

Abstract: It is shown by means of a model that the renormalization and unitary gauges can be connected by a point transformation, and this fact is used to construct a formal proof of renormalization in the unitary gauge. The formal proof is then verified by demonstrating that for a fourth-order on-shell scattering process the S-matrix calculated directly in the unitary gauge is exactly equal to that calculated in the renormalization gauge. The calculation is refined to the point where it becomes purely graphical and this allows one to see by inspection how the cancellation of divergences occurs in the unitary gauge. The model considered here is Abelian, but it will be generalized to the non-Abelian case subsequently.

1. Introduction In spite of the fact that the renormalizability and unitarity of unified gauge theory has now been established [1, 2] beyond reasonable doubt, a direct proof of renormalizability in the unitary gauge, i.e. a proof which does not make use of the existence of other gauges, is still lacking, and with it is lacking an insight into the mechanism by which the cancellation of divergences in the unitary gauge takes place. In the present paper we attempt to throw some light on this mechanism by making a direct fourth-order calculation in the unitary gauge using the canonical formalism, and comparing the results with the same calculation in the renormalization gauge. The reason that we prefer to use the canonical formalism, rather than, say, a dispersion calculation [3], is that it allows us to insert renormalization counter-terms in the Lagrangian in a simple gauge invariant way, and to calculate from first principles not only the F e y n m a n graphs with structure, but also to calculate all the renormalization constants in each gauge [4]. The calculation of the renormalization constants actually requires also that we take into account the fact that the gaugetransformation is for second-quantized fields, and this we do by noting that the gauge transformation can be replaced by an appropriate point transformation [5 ]*, * Footnote see next page.

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G.B. Mainland et aL, Unitary gauge

for which the correction due to second quantization is easily calculated. In practice, what we shall show is that the difference between the S-matrices calculated in the unitary and renormalization gauges is zero. It turns out that to evaluate the contribution to this difference from various graphs, we do not have to actually calculate any Feynman integrals, but only to use the symmetry properties. Thus any form of regularization which preserves the symmetry properties of the Feynman graphs (e.g. dimensional [1] or Pauli-Villars [7] regularization)will suffice. Indeed one could even reverse the argument and say that the implementation of the formal equivalence o f the two gauges in terms of Feynman graphs restricts the regularization procedures to those which preserve symmetry! In any case, with symmetric regularization, the demonstration of the equivalence of the two gauges is eventually refined to the point where it becomes a purely graphical demonstration, and which allows one to see by inspection how the cancellation of divergences (in the unitary gauge) and ghosts (in the renormalization gauge) takes place. The abelian model which we shall consider will be obtained as a special case of the Salam-Weinberg model [8] in which the four vector fields Au, B v are reduced to a single vector field Vu, with axial vector coupling. It is well known that such a field Vv can be written in the Stueckelberg [9] formalism O, where/a is the meson mass. It actually turns out that on setting # = 0 in the Lagrangian, the model can be reduced to one in which Vv is replaced by O,v i.e. by a gradient model, and it is instructive to reduce to this simple model from time to time during the discussion. In fact, all the essential problems are present in the gradient case, and the additional complications coming from the transverse part o f Vv are easily handled once the gradient case is understood.

2. Reduction of Salam-Weinberg Lagrangian The unified gauge model which we shall consider is based on a reduction of the Salam-Weinberg Lagrangian [8]

~ w = - ~ F 2 - ~ F 2 + R ( i ~ + g'I~)R + E(i~ +-~g'l~ + s1g-

r)L

+ ~liau~P + (~ ~ Au" r - ~ g' Bu)~P12 - Ge(/, • ~PR + J~P+ , L ) + V(~+~P), (2.1)

* Point transformations in unified gauge theories have also been considered by 't Hooft and Veltman [6]. However, 't Hooft and Veltman are concerned primarily with the general structure of Feynman graphs, and in particular, with Faddeev-Popov loops, whereas we are concerned with the detailed derivation of Feynman graphs from the canonical formalism. Thus our work complements theirs~and verifies their results in detail for a particular model.

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505

where V(q~+~b)is a potential term for the ~-field and

F=Fv=B,v-Bv,

u, F : F ~ = A , v - A v ,

L =-~(1 +~'5)(~)'

R =-~(1 -~,5)e,

u-gAu×A

v,

q~=(¢~).

(2.2)

The reduction consists in eliminating the non-abelian part of the Au field and the coupling of the Abelian field to the conserved part of the current, according to the scheme 4~0 = v = 0 , f f = 3 g '

A_+ =0, A 3 = B .

(2.3)

We then obtain the reduced Lagrangian .67(V, ~9, 0) = - ¼ F 2 + t~i~~O- f ~ V % 7 5 ~O- g(~ 6q~l- i~'56q~2) + ½ liOv0~- 2fV~b/2 + V(l~bl2),

(2.4)

where, in terms of the original Weinberg parameters f:~2,

Ge : g ,

V : x / 2 B v, F 2 = F v F

.,

F v = V,v - V,u , ~O= e, ~b= ~bt + iq~2.

(2.5)

We next introduce spontaneous symmetry breaking in the usual way [8], by letting q~l ~ ~1 + m/g. In that case, setting 2mfig =/a, we have

~OB(V, ~,4~+m/g) : - ¼ F 2 +L022 V2 + t~(i~ - m ) f f - ft~ VTv75 ~k

(2.6)

g(~ ¢ ¢ 1 - i ~ ' 5 ~¢2) +-~ liOvd~-2fVv~12+ttV'~(Ov¢2 + 2/-V'vq~l)+ V(I~ + m/g[2). The Lagrangian (2.6) is the model which we shall consider. The main properties of this Lagrangian are (i) It is invariant with respect to the 'vacuum-breaking' gauge transformation

ff ~ e-if~/sA/uff,

¢ + rn/g -~ e -2ifA/u (¢ + m/g),

V,, -+ W + A,v"

(2.7)

(ii) For q~= 0 it reduces to the non-renormalizable interaction A?NR -

~ F 2 +½/a2V 2 + f(i~ - m ) ¢ -

f~tvTsff.

(2.8)

The reason that the Lagrangian (2.6) is renormalizable whereas the Lagrangian (2.8) is not, lies in the presence of the C-field and in particular the meson-meson interaction/aV • (0¢2 + 2fV¢l) which is induced by the spontaneous symmetry breaking and thus has no parameters which are independent of the parameters occuring in (2.8). We shall see very explicitly in our calculation how the induced meson-meson interaction cancels the divergences of (2.8).

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Note that the renormalization and unitary gauges of (2.6) can be defined formally as renormalization gauge: OuVu = 0 (on states), unitary gauge: ~b2 = 0.

3. R e n o r m a l i z a t i o n c o u n t e r - t e r m s

The renormalization counter-terms which we shall construct for .t2 B(V, if, q~+ m/g) are

8Z?(V, t),~b+m/g) =6m~ [l+(g/m)(491-i75~2) ] ~ +½(Skl-~k2)(g/2m)2 I~)+m/gl 4 + -~(--~6k 1 + ~6k2)lq) + m/gl 2 -~61a U2.

(3.1)

These counter-terms are manifestly invariant with respect to the vacuum-breaking gauge transformations (2.7), are no higher than quartic in the fields * in any gauge, and provide mass counter-terms 6m, 6kl, 6k 2 and 6/a for the fields ~b, 4~1, ~b2 and U respectively. (We find that, at least to the order to which we shall calculate, the field 0 requires no mass counter-term, and that the linear counter-term-~(-~-6k 1 +~-6k2) (2rn/g)~ 1 is absorbed by normal-ordering.) The explicit charge renormalization terms g(6m/m)~(¢l-i3'5~2) ~ which are not Separately invariant and therefore take a different form in each gauge, will turn out to be necessary for the equivalence of the renormalization and unitary gauge Lagrangians. It should perhaps be emphasized that the counter-term prescription (3.1) is not unique. For example, we could also choose a term of the form -(6/~/2/a~)li0x4~ - (2f4) +/QVxl 2 as a mass-renormalization term for the V-field. We make the choice (3.1) because it is the simplest choice which leads to renormalization in the unitary gauge. We finally combine the Lagrangian ~ B (2.6) and the counter-terms 6/2 in (3.1) into the more compact f o r m L =.6? B + 512 ,where

L(V, ~,4)+m/g) = - ~ F 2 + ~ [i~ - M - f~3' 5 - G(q~1 - i3,5~2) ] ~ +~ liOn,b- (2fq~+/a)Vxl 2 + lnl(g/2m)210+m/gl4 + ~214~+m/gl 2 -161~ U 2,

(3.2)

M = m - 8m, G --g(!- 6m/m), K1= [-~2+6kl - 6k2]' K2= [½~2--~ 6kl +~26k2](3"3) We have taken V(l~b+m/gl 2) in (2.6) to be of the conventional form

V(Ig)+m/gl2)=~K 2 (g/2m) 2 1~ + m/gl 4 +¼~2 I~b+ m/gl 2,

(3.4)

which gives the field 4~1 a mass ~ while ~b2 remains massless.

* We neglect quartic counter-terms other than those in (3.1) as they do not contribute to the processes which we shall consider.

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4. Stueckelberg formalism and gradient model

In the Stueckelberg formalism we can write V=U+Iow, kt

au UriC> = 0

(4.1)

and in that case the Lagrangian (3.2) takes the form

L(U~+I o,~, ~, O+m/g)=-~ F(U)2+~[i~ -M- f~75-f Tv750~- G(q)l-i75~2)]~ + 2t, i3~,4~- (2fq~+/l) (U~ + 1 0 , x ) 2 +-~K1(g/2rn) 2

(~+m/gl4+½K210+m/g2-½6, U2. (4.2)

The renormalization and unitary gauges can then be formally defined as renormalization gauge: 0 = 0, unitary gauge: q~2 = 0.

(4.3)

The main purpose of introducing the Stueckelberg formalism is to construct a point transformation which connects the unitary and renormalization gauges. However, it also has the following useful property: if, among the four parameters satisfying =/.tg, we regard f as the dependent one and eliminate it from (4.2), we can expand (4.2) in powers of/l, and the zero-order term will be

2mr

L(O,x, ~,~+g) = ~[i~-M- .~--~TvTsO 1 \2m ,]

iT5 ~b2)]

g

t

(4.4)

But this is just the Lagrangian (4.2) for the special case in which the field is a gradient field. Hence we shall call it the gradient model. Since all subsequent results hold for any value ofbt, they will hold in particular for # = f = 0 and hence for the gradient model. Accordingly, at any stage in our subsequent work we can simplify the results, in order to obtain a clearer insight or to check a calculation, by simply setting/a = f = 0 and obtaining the corresponding result for the gradient model.

5. Equivalence of gauges with respect to second-quantized point transformations We have seen in the last section that the renormalization gauge Lagrangian L R of our general model (3.2) is obtained by setting V~ = U~., that is to say

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Let us now apply to L R(V, ~, ~b+ m/g) the point transformation

~ = eiJ,.rso/ur7' d~= (O +g ) e2ifo/u mg'

(5.2)

where o is a hermitian field. We obtain

_ ¼F 2 + ~ [i, - M - f Tx~isO,x- f~J75- Go] ~ ~~ li~xo-(2f ° +#) (ux + l o'x)l 2

But using the Stueckdberg formalism Vv = Uv + (1//a)0~ we see by inspection that this expression is just

-¼ F 2 + fl [i~ -M-flw/5 - Go]~ +-~[i3xo-(2fo +/~) Vxl2 +-~1 ( g ) 2

m 12 _-~ ~i/~ U 2, Io +~]4 + -~K21° + g

(5.4)

i.e. is just the Lagrangian (3.2) in the unitary gauge (¢2 = 0) except that the names of the fields ¢1 and qJ have been changed to o and ~/respectively. Thus we have the point equivalence result L(Ux, ~,qS+

m/g)=L(U x + 1/.t0 , x, r/, o+ m/g),

(5.5)

where if, ¢ and r/, o, 0 are related by the (non-singular) point transformation (5.2). The above derivation has been at a classical level. We must now take into account the modifications necessary because the gauge function should be a quantized field. The required modification can be obtained quite simply from the point transformation by noting that, in the latter case, the second-quantization manifests itself in the fact that the fields in the point transformations cannot all have zero vacuum expectation value. One can then correct for second-quantization by modifying the point transformation so that they do. The most convenient modification is to change the point transformation to = eif~lsO/~rT, 4~= (o + 7n/g) e2ifO/t~- m/g, where ffz (cos 2fO~bt)= m. (5.6) The point equivalence result (5.5) then becomes modified to

L(Ux, q~,,+m/g)=L(ux+lo,x,rl, o+Fn,g)='~(ux+lo,x, rl, o).

(5.7,

Thus the modification due to second-quantization is equivalent to an anomalous mass-renormalization in the unitary gauge. We shall find in our direct calculation that this anomalous mass-renormalization, which is calculable from (5.6) is necessary for the equiv~ence of the renormalization and unitary gauge Lagrangians.

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509

6. Formal proof of renormalizability to all orders in unitary gauge We can now use the point-equivalence result (5.7) to give a formal proof of the renormalizability of the unitary gauge Lagrangian to all orders. For this purpose, we note that on account of the anomalous mass-renormalization, the sets of fields (~b, ~1, ~2) and (r/, o, 0) have the property that

=rl+O(~,o,O), ~1 =o+a(rT, o,O), ~2 =0 +O(rT,o,O),

(6.1)

where Q denotes terms of quadratic and higher order. Thus the sets (~b,~1, ¢2) and (r/, o, 0) are different interpolations of the same asymptotic fields. It follows from the equivalence theorem [5] for such fields that the S-matrices constructed with them will be identical, in other words that

S(L(U x, 4s, ok)) =S(L(U?,, rT, o, 0)).

(6.2)

Note that we have not written down any explicit Faddeev-Popov [10] terms in (6.2), although, as we shall see in sect. 7, we do have one such term. The reason is that, strictly speaking, the Faddeev-Popov terms occur in the derivation of covariant Feynman rules from ~ (using either a path-integral method or the canonical Hamiltonian formalism) rather than in .6? itself, and hence they are implicit in the definition of S. Combining (6.2) with (5.7) we have

SR =-S(L(U, rT, a, O)) = S(~L(U~ +I o,~ , ~, o) ) = Su .

(6.3)

But (6.3) together with the finiteness o f S R after charge renormalization, implies the finiteness o f S U after the same charge renormallization *, and hence implies the renormalizability of ~ e unitary gauge Lagrangian L. In this way the formal proof of renormalizatiility of L has been reduced to the formal proof of equivalence for point transformations. 7. Verification for fourth-order scattering. Construction of Hamiltonians We now wish to verify the formal result of sect. 6 by carrying out an explicit fourth-order calculation. For this jaurpose we first construct the interaction Hamiltonians corresponding to L R and L. Apart from the log-term in ~ U , we obtain by inspection ~ R ( U ' if, ~) =ft~7.),5 ~ +Gt~(~bl - iT5~2) ~ _ 2f2U 2 (~b2

+-g-~l) 2m

1 2-2 , /'g '~2 ~+m14 -- 4fUX¢laX~b2 - 8m ~ff --~g
(7.1)

--IK2 ~+m[ g 2 + -~ 8g 2 U 2, * The common charge renormalization r~ferred to here is, of course, over and above the explicit charge renormalization coming from the counter-terms in (3.1) and which is not the same in both gauges.

G.B. Mainlandet al., Unitarygauge

510

~U(V, r/, o) =J~ [z')'5r? + G~71o - 2f2V2~o / 2 + 2too + fft2-m2 ]\ g J

+i54(O)ln(l+go)-(~n6m-~n+m)-{~2o2

(7.2)

--IKl(-~m)2 ( O + g ) 4 - - ~ K 2 ( O + g ) 2 + 1 6 / t / U 2 . The log-term in ~ U comes from the rules derived by Lee and Yang [1 1] to allow for the cancellation of surface-dependent terms against the difference between (T(O,uO,v)) and OuOv(T(O0)) in the calculation of the S-matrix. In other words, the function of the log-term is to allow us to use the so-called naive or covariant Feynman rules belonging to the Hamiltonians (7.1) and (7.2). Had we used a path-integral method the same term would have appeared as a Faddeev-Popov term. We note in passing that we obtain the Hamiltonians for the gradient model on setting/~ = f = 0 in (7.1) and (7.2); Expanding the Hamiltonians (7.1) and (7.2) up to the order required for fourthorder fermion scattering, and normal ordering them, we obtain ~ R ( U , ~,~b) ~ f ~ ~'y5 ~ + G: ~ :

+ 2-m g

~l-iGf~75~(~2-2tff:U2: ~b1 - 4fUx~l~h~ 2

+ ~ g t~26 "~1".~b2. 2" --aRm ~

- - ii f R k l

l

2 ,2 +_~6R/~U 2,

(7.3)

~ u ( U , r/,e, 0) -~f~ grTgr/+ G :~r/: o +{~TxYs~O,x-2#f :U2: o

-- 4fUx0,z o + ~-~

~--

- 6 u m ~ r ~- ½~ukl o2 + ½~U/a U2 , and if we change the names c,f the fields in ~ R for direct comparison with 9~u, ~q(R becomes ~R(U, ~, o, 0) ~ f~ ~Jq,5r/+ G :~r/: o - iG~75~0 - 2btf :U2" o -4fUx°0'x +-~-g : ° 3 ¢2 2m

+ ½ RUV2.

: +-~-g K2a2m :02: - 6 Rrnr/r/-- -~Rkl °2--~8Rk202

(7 .s)

Once again the gradient model can be extracted by setting ~t = f = 0. Here the mass-

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511

renormalization constants are given by 6Rrn = 6rn,

8 urn = 8 m - 2f2m (0 2),

(7.6)

/,/2

6Rk 1 = 6k 1+4f2(U 2) _ 2f2K 2 (3(o 2) + (02)), /22

2 2 8if2 4 a u k I --8kl+4f ( U ) + ~ - 6 (0) 6fzt¢ 2 ((02) +(o2)), /22

6R/2 = 6/2 -- 4f2((O 2) + (02)),

8US = 8/2 -- 4f2((o 2) + (02)),

6Rk2 = 6k2+4f2(U2)

6uk 2 = 0.

- 2f2K2 (3(02)+(o2)),

/22 The expressions (7.6) are obtained from the counter-terms (3.1), normal-ordering, and the anomalous mass renormalization (5.6). Note that the three parameters 8rn, 8k I and 6/2 are free, but the fourth parameter 8k 2 is not free since it must be chosen so as to eliminate te~ms of the form constant "o in the interaction Hamiltonians. From (7.1) and (7.2 i we see that it must be of the form 8 k 2 _- g~ [ g(~r/) - 2/2f(U2) +tic 2 (3(02) + (02)) ] /2

(7.7)

A first importa.nt step in the verification that the S-matrices derived from ~ R and 9f U are the same is the following: If we make a direct calculation of the conventional second-order self-masses for the Hamiltonians ~ R and 9~U, without including the mass-counter-terms, we find that 8urn--SRm

-

2f2m (02), /22

_ 8if 2 4f2K 2 (02), 8uk 1 -- 8 R k 1 ----~- 64(0) -- /22

6U/2 -- 8R~

(7.8)

=0,

and

6uk 2 = 0, 8Rk 2 -_~g2 - (~r/) + 4f2~2 /22 ((o2) -- (02))

(7.9)

Comparing (7.8) and (7.9) with (7.6) and (7.7) we see that (a) the self-masses for 0 have the correct values in each gauge and (b) if the constants 8m, 8k 1 and 8/2 are chosen so as to cancel the conventional self-masses in one gauge, then the self-masses in the other gauge will be cancelled automatically. In other words, the differen-

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ces in the normal-ordering in the two gauges (together with the anomalous fermion mass-renormalization) exactly cancel the differences in the conventional self-masses. This result is of great practical value for our subsequent calculations, since it means that we are justified in subtracting out the on-shell self-masses in the two gauges simultaneously. A simple graphical proof of (7.8) and (7.9) will be given in sect. 8.

8. Derivation of two basic graphical identities. The Feynman graphs corresponding to the Hamiltonians ~ U and ~ R are shown in fig. 1. U~

Ca) R&

(b)

_1_

'

n

(a)

~L

M

Q

U

I

(b) whero n = -2km(~er~> + <62>) 0

U

~

F i g . 1.

where we have normalized g/2m = 1. The last two graphs correspond only to the charge and wave-function renormalizations which are different in the two gauges. The charge renormalization is the explicit one induced by the renormalization counter. terms in (3.1) and the wavefunction renormalization comes from the normal-ordering of the term 0202 i n ~ g. Apart from these two renormalization graphs one sees that the only difference between the two gauges occurs in the first two graphs of fig. 1. It is worth noting that these first two graphs do not involve the U-field and hence the es, sential difference between the two gauges is already present in the gradient model U = 0. Indeed we shall see that once the gradient model h~ts been treated, the extension to the general abelian model is quite trivial. The calculation of higher order graphs derived from fig. 1 (or at least the difference between these graphs in the two gauges) can be reduced to a graphical calculation by means of two basic identities.~Fhe two basic graphical identities are:

i_

F-k

_L _

p

_

/.,

=

L

-iy

,,..

_

/

_

\

(8.2)

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513

respectively, and they correspond term-by-term to the algebraic identities i/¢75 + 2im75 = (-i3'5)~--m) + (/b-/C-m)(~-i75),

(8.3)

-2k'k' - K2 = [ ( k - k ' ) 2 - K 2 ] - k 2 - k '2,

(8.4)

which relate the vertices of figs. la and lb to the propagators. From (8.3) (8.4) we see that if any of the lines in (8.1) and (8.2) are on the mass-shell, the corresponding graph on the right-hand side vanishes. Thus, for example, from (8.1) we have

-L-. but

J: I

__

', /

(8.6) (8.7)

and from (8.2) we have ,'-l'-"_

I"

=

-

-

(8.8)

but

(8.9) where all the external lines shown are assumed to be on the mass-shell. Note that eq. (8.7) is just the first-order equivalence theorem for pseudo-scalar pseudo-vector coupling. One sees from these examples that what happens essentially is that the difference between the vertices in the unitary and renormalization gauges "shortcircuits" the internal propagators. As a result, the identities (8.3) and (8.4) which are algebraically trivial, become quite powerful when translated into graphical language. As we shall see, the identities not only allow us to reduce the proof of equivalence of the two gauges to a purely graphical one, but they also exhibit explicitly the mechanism by which the cancellation of divergences (in the unitary gauge) and ghosts (in the renormalization gauge) comes about. In order to use the graphical identities (8.1) and (8.2) to best advantage it is convenient to introduce the following:

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514

Auxiliary vertices k" • (*)

k -2~.='~

-2t~

~.

K2

X2

Fig. 2. The general conventions will be those of Bjorken and Drell [12]. In particular the metric will be g00 --- -gii = 1 and there will be a factor (i) for fermion and spinzero boson propagators and ( - i ) for vector meson propagators and vertices. For convenience we normalize the constant g/2m to unity and drop an overall factor (27r) -4. It should be mentioned that some graphs acquire additional factors because of the number of ways of contracting the interaction Hamiltonian. Such factors will always appear in parentheses.

9. Graphical treatment of self-masses In this section we wish to establish the relations (7.8-7.9) for the self-masses in second order, using graphical methods. We treat the four particles U, T/, o, 0 in turn.

9.1. U-field. This is the simplest case since we see by inspection of fig. 1 that the second order self-mass of U is identically the same in both gauges.

9.2. rl-field. The only graphs of fig. 1 contributing to the r/self-mass and different in the two gauges are those of fig. 1a. Hence -iSum + i6Rm = ~

_

~

(on mass-shell)

(9.1)

To calculate the graphs in (9.1)we use the first basic identity (8.1)with the external lines in (9.1) on the mass-shell, and obtain ct

f ~d4k ~ = 0 (on mass shell)

=-2rn f d4k = 2irn(O2) (on mass shell)

(9.2)

(9.3)

Adding these two equations we have

6Urn - 5Rm =--2m(0 2)

(9.4)

as required. * There wiU be no ambiguity in the coupling -/~ because in practice we shall have/~ =/~' either identically or because the fermions are on mass-shell.

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515

9.3. a-field. The only graphs of fig. 1 contributing to the a self-mass and different in the two gauges are those of fig. 1b. Hence _ ~16ukll- + ½i6Rkl = " - ' I ~

.....

~__

(on mass-shell)

(9.5)

To calculate the graphs in (9.5) we use the second basic identity (8.2) with the external lines in (9.5) on the mass-shell and obtain -~ --~/,

.....

t~1---

.....

~

....

=

- ' ~ - = 4fd 4k = 464(0)

(on mass shell)

2--~-- = - 2 K 2 f d4k = 2iK2(02) k2

(on mass shell)

(9.6) (9.7)

Adding these equations we have

6 u k 1 - 6Rk 1 = 8 i 6 4 ( 0 ) - 4K2(02)

(9.8)

as required. As eq. (9.6) and (9.7) depend on symmetric regularization and hence are not completely straightforward, we shall give also the algebraic derivation in appendix A.

9.4. O-field. This is the most complicated case since we have to show that the selfmass vanishes in the unitary gauge, and because in the difference between the two gauges the graphs of both fig. la and fig. lb contribute. To show that the self-mass in the unitary gauge vanishes, we note that all the 0-couplings in the unitary gauge are derivative couplings and hence proportional to the external 0-momentum k. But then the self-mass must be proportional to k 2 and hence it must vanish on the mass-shell as required. We can use this result to help calculate the self-mass in the renormalization gauge by writing 2 + ~iSRk2 = ~igRk2 = --~tSuk 1. ~ [ , ~ ; i , , - '~:S,~,] + ~ [ " ~ Z ~ - " ~ : " ~ ] (on mass-shell). (9.9) To calculate the graphs in eq. (9.9) we use the basic identities (8.1) and (8.2). First, from the second basic identity (8.2) and symmetric regularization we have -= o

•,4$52~

(9.10) I

.... _

~

-_ _

~

-

~

b

=_~¢2 f 4 d k

+

.,k2_1¢

2

+K2 f d 4 k k2

= i• 2(o25 - it¢2(025.

(9.1 1)

Then from the first basic identity (8.1) we obtain -

~

=

~

- O ,

(9.12)

516

G.B. Mainland et al., Unitary gauge

-

~-

~

6d4k

~

= am T r J ~--m = 4irn(~r~),

(9.13)

where (9.10-9.13) are all on mass shell. Adding these four equations we have 8 R k 2 = 4m(~r/) + 4~2((o 2) -- (02}),

(9.14)

as required. For verification and comparison the algebraic calculation corresponding to the four equations (9.10)-(9.13) is given in appendices B and C. All the mass relations (7.6) are now established, and as emphasized in sect. 7, this justifies our subtracting out the on-shell self-masses in both gauges simultaneously in our subsequent calculations.

10. Equality of the S-matrices in the two gauges In this and in sect. 11 we establish the main result of our paper, namely, the equality of the S-matrices for fourth-order fermion-fermion scattering in the two gauges. The Feynman graphs for this process fall into three groups for each of which separately we have equivalence, namely I scalar exchange in order ~t = 0 / . . . . . . ) (graalent moaet, U 0) II pseudo-scalar exchange in order g = 0J III all graphs containing U-propagators (order/a2). Indeed group III subdivides into two-particle exchange, one-particle scalar exchange, one-particle pseudo-scalar exchange, and U-exchange, for each of which we also have equality, but this group is so simple that we prefer to keep it together, and thus we shall prove equality for groups I, II and III in the following three sections respectively. Before embarking on these proofs it will be convenient to obtain first the difference between the fermion wave-function renormalizations in the two gauges, as this enters in all three groups, We give a graphical proof in this section and verify it by algebraic computation in appendix A. As in the case of the fermion self-mass the only contributory graphs which are different in the two gauges are those of fig. la. Using the first basic identity (8.1) we obtain _

t~,

=

~

+

_Q_ ~

~

=

¢'~ + ~'N

+ ~

(10.1)

where the second graph in the second expression vanishes by symmetric integration, and _

~'~

=

--Q-

4-

~

(10.2)

Adding these graphs we obtain (10.3)

G.B. Mainland et al., Unitary gauge

517

where we have symmetrized the last graph in (I0.1) with respect to the external line.(/b - rn) -1 which commutes with the loop. In (10.3) the last graph on the left is the on-shell self-mass subtraction and the last on the right is o f relative order (/b - m) 2 . It follows that the wave-function renormalization difference is given by the other two graphs on the right-hand side, which are of relative order (/b-m) i.e. that [13]

2a8:

+

+,Q

(,04)

t~bare = N/]-~ ~renonn' where A denotes the difference between the two gauges. The algebraic calculation in appendix D verifies this result, and a comparison of that calculation with the above graphical derivation demonstrates the economy o f the graphical method. We could, o f course, also calculate the wave-function renormalization difference for the meson fields, but for fourth-order fermion-fermion scattering the mesons are off mass-shell and hence it is more convenient to include the meson h/ave-function renormalization as part of the general calculation.

11. Equality of S-matrices I: scalar exchange (/l = O) The graphs for fourth-order fermion-fermion scattering via scalar exchange with ta = 0 which are different in the two gauges are shown in fig. 3.

÷

(=)

(2)

(b)

X-v .(2)

(j)

*

(,)

:

(,)

*

i

(~)

.(2)

÷=B

(a)

:

(f)

Fig. 3. where the dot beside the loop indicates that the on-shell self-mass is to be subtracted in both gauges, as justified in sect. 7 and 9, and the last graph comes from the fermion wave-function renormalization calculated in the last section. We first consider the two-particle exchange graphs. Using the first basic graphical identity, we have

518

G.B. Mainland et al., Unitary gauge

where the auxiliary vertices are defined in fig. 2. Hence,

It.X

-

I2. X

(11.3)

Similarly we have (11.4)

where the third graph in (11.5) vanishes by symmetric integration. Combining these results we obtain finally for the two-particle exchange

We see therefore that, as mentioned in sect. 8, the difference between unitary and renormalization graphs with a given number of vertices "short-circuits" them into auxiliary graphs with a smaller number of vertices (but with more particles entering at each vertex). What we now have to show is that the meson-meson interactions are such that they just cancel the auxiliary graphs. The meson-meson interactions are those of figs. 3c and 3d. First, by using the second basic identity (8.2) we have for fig. 3c (a,[A

" X-]

= ~ [K

- ~

-

~]+z[~

-

A

](11.7)

and a similar result for the inverted graph. For fig. 3d we could use the second basic identity (8.2) twice but it is simpler and more instructive to derive a graphical identity for it directly. This we do in appendix B and the result is (Z)

-

"

2~

- ?~ i+i. . . . tea ~ra~h~÷~

(11.8)

The first term in (1 1.7) and the last term in (11.8) clearly cancel the auxiliary graphs on the right-hand side of (11.6), as required. Then, however, the mesonmeson interactions (11.7) and (11.8) themselves have a remainder of the form ,'T--

..~ [

-2

"2-, :

,'7-+ Z

"

4- i n v e r t e d

graphs

(ll.9)

We have to show that this remainder (which takes the form of a vertex renormalization) is cancelled by the remaining graphs in fig. 3, namely fig. 3e and fig. 3f.

G.B. Mainland et aL, Unitary gauge

519

Using the first basic identity and the wave-function renormalization result of (10.4) we obtain for these graphs i =

----w-

Y

_~

-

---r---

--'-r

-~

.~

- ~',

~ +

~1

i =

~

:

"; *

: "',a~-

-~

-@ (11.10) ',

(11.ll)

respectively. One sees by inspection that the graphs (11.9), (11.10) and (11.11) exactly cancel as required.

12. Equality of S-matrices lh pseudo-scalar exchange (/1 = O) The graphs for fourth-order fermion-fermion scattering which are different in the two gauges in this case are

(.)

Z(.) +

(l)

(l)

(t~

(b)

(r)

"Y* t (o)

(a)

,. '

Ce)

.

Fig, 4.

and, as before, the dot indicates that the on-shell self-mass is to be subtracted out in both gauges as justified in sects. 7 and 9. We first consider the two-particle exchange. From the first basic graphical identity (8.1) we obtain

and similarly for the other three two-particle exchange graphs. Adding these four graphs we have

(12.2) where A denotes the difference between the graphs in the two gauges. Thus, as before, the difference between the vertices in the two gauges "short-circuits" the graphs

520

G.B. Mainland et al., Unitary gauge

into auxiliary graphs with fewer vertices (and more lines at each vertex). We now wish to show that, as before, the meson-meson vertex is just such as to cancel these auxiliary graphs. There are two such meson-meson interaction graphs, namely figs. 4b and 4f. Using the second basic graphical identity (8.2) we have for fig. 4f

(12.3)

and a similar result for the transposed and inverted graphs. For fig. 4b we could use the graphical identity twice, but it is simpler to proceed directly as shown in appendix B, and we obtain

(~') I~,- Q~'] m [~_~_,., .' -2~-Mr

( ~ ÷ inverted graphs 4"~'~',-~

(12,4)

The second graph in (12.3) and the last graphs in (12.3) and (12.4) clearly cancel the graphs on the right-hand side of (12.2) as required. However, (12.3) and (12.4) themselves then leave a remainder of the form

÷2

-2

-2

,,

+2

,,. "

* inverted graphs

, ,

(12.5)

~'P

which contributes to the vertex renormalization, and we have to show that this contribution is cancelled by the remaining graphs c, d, e, g, h, k in fig. 4. There is actually no contribution from graph c (appendix C) and for the others we obtain t

+z~B--~

= [-~.~_[-Z(~?] ÷ [ ~ •

- ~

~ ]

(12.6)

. inverted grsuhs

from the definition,and from the wave-function renormalization result(I 0.4) res-

pectively, ~

o

*.s

from the graphical definition of the self-mass to order ~u= 0, and finally, ~

,~s

- ,-~

.w,w

=

~

,s

+

• ~

.~,

(12.8)

(12.9)

521

G.B. Mainland et aL, Unitary gauge

from the first graphical identity (for a more complete derivation of (12.9) see appendix E). One sees by inspection that the graphs (12.5)-(12.9) exactly cancel, as required. The results of the last two sections establish the equality of the S-matrices in the two gauges to order/1 = O, i.e. for the gradient model.

13. Equality of S-matrices Ill: order/l 2, or U-field graphs The fourth-order fermion-fermion scattering graphs which include the U-field and are different in the two gauges are as shown in fig. 5:

Fig. 5. '

_Z

:

= ,\ • L +

(13.1)

Z+-v+Z

(13.2)

(13.3)

The calculations for this case are actually the simplest because the two-particle exchange, the a-exchange, the 0-exchange and the U-exchange graphs all vanish independently. We treat them in turn: 13.1. Two-particle exchange

Using the first basic graphical identity we have (13.4) where the term on the right vanishes because of the anti-commutation of 75 and the vector coupling 7u. Applying this result successively to the two-particle graphs we easily see that they vanish. 13. 2. Scalar exchange

Noting that the meson-meson vertex is the same in both gauges and applying the identity (13.4) above, we see that the two graphs in (13.1) cancel each other.

G.B. Mainlandet al., Unitarygauge

522

13.3. Pseudo-scalar exchange Using the first basic identity we easily see that the first three graphs vanish (firstorder equivalence theorem), while the fourth gives

;-L; BuI by the definition of the fermion on-shell self-mass in order/32 (i.e. self-mass due to the U-field) we see that the right-hand side of (13.5)just cancels the last graph in (13.3) as required.

13.4. Vector exchange The first two graphs vanish for the same reasons as the scalar-exchange graphs. The third graph gives

and by the result for the fermion wave-function renormalization (10.4) this exactly cancels the contribution from the last graph. Thus the equality of the S-matrices in order/2 2 (and hence for the full abelian model) is establishes. Appendices

The integration variable will always be k and we will omit a factor d4k/(2n) 4

Appendix A

.~.~ ~,

-- " - " ~ ' -

= f [(k 2 + k , 2 72)2 t<4] k2k '2

-r k2J by symmetry in k and k'. Hence using k' = r + k and symmetric integration we obtain =4fl -2r2f~2

+ (r4 t¢4)Yk2k'21 .

G.B. Mainlandet at, Unitarygauge

523

Expanding this expression in powers of 7"2-t~ 2 we have

f k-~k'2] ,

= [ 4 f l - - 2 k 2 < l J + ( r 2 - - K 2 ) [ - - 2 f 4 + 2kK22 f - ~ l J +k2k'2J (r2--K2)2 = [464(0) + 2iK2(0)2]

+ "r2--g:2 [--2 _ ~ r --~

+2

L

"] ('r2-~:2)2 + K4

-o-

O

The first bracket is the mass-renormalization term and the second bracket is the o-wave-function renormalization when o is taken to be on mass-shell.

Appendix B

k

k2 k,2_r 2 2_K4

:~"

f(

) k'2(k2 t~2)

[.(k2_k, _r2+K2 k2_K2 K,2_r2 J"

a

)(

)

k'2(k 2_ t~2)

=f(k2-k'2-r2+n2) fk2-k'2-r2+K 2 _ r 2 f k2-k'2-r2+~ 2 k,2

k 2_/¢2

k,2(k2_t¢2)

r2f~,2 + r2fk~ 2' K 2 r 2 f--'r2+2K__._~ 2

= f K2 f--2"r2+t¢ 2 d k-72-" ~

--

%,2(k2 g2)'

where we have used k' = r + k and symmetric integration, K4

t. k ~

1~2[' k2_~ 2 k 2 "'"

k,Z(k2_t¢2) J ~

g4 "'""

T4

,--,

= [--it~2(02) +iK2(O2)]+ r2 [ 3 ..X4.,,. -J t~4 L K2 Again the first bracket is the mass-renormalization term, and the second bracket is the 0-wave-function renormalization when 0 is taken to be on mass-shell.

Appendix C

P

,k ~.

P

-

~

=

d

1 r ~ % - ~ + e1_ m (~75-2m75)/¢-m

1 m]=O ' -

~

1 1 ----~r(*rs + 2m't's),+-q-c-~_m 2m't's~-m

= f t r ~ + 2m ~ _ m + 2rn '~ = 4m f

I¢-m,}

tr lcl_m= , , ~

Note that the only contribution of this loop is a mass renormalization.

J

524

G.B. Mainland et aL, Unitary gauge

Appendix D

, w

tb+~_m[~+'-m)')<5+')'5(c-m]} =DA[~+(/J")'5~+;_m'YS)(C-m)] =f-k2@75t6+,l_rn"lS)(c-m) =-S-1212m'YsIJ+--+-~---m_m"f5(c-m)

=f~2{/~"/5

+ (c-m~5~+-~-~y--rn_m3,S(c-m)+ (c-m , -

~

=

2

2

Adding the above two identities we obtain ~'5~+-~-2-m-m/¢Ts+2m~'5~+~-m 2m75 + 2m =

l~+¢_m "Y5+

0b-m)

+(c-m){-f-~[~s~2m~5 1

1

+1

1 3,5] (C-m). [] } -(c-m) [fk13'5 ~_,_ m

This equation is just the algebraic form of (10.3). Appendix E

~-

4,- = . ;

4,-4,

o J.4-

= 4~=~

GB. Ma&land et al., Unitary gauge

525

Adding these graphs we obtain the result stated in (12.9). The only novel point to notice here is that when the internal fermien line to the right in graph (El) is short-circuited, the central coupling (~-~)3'5 goes to the right of the loop, and since, after integration, the loop is a function o f f only, (~-~)75 can then be replaced by -2m75. This is why there is no derivative coupling at the two-meson vertex in graph (E2). The authors wish to thank Drs. R. Acharya, Z. Horv~th, J. Nilsson, B. Sredniawa and D. Wallace for discussions on particular points.

References [1] G. 't Hooft, Nucl. Phys. B35 (1971) 167; G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [2] B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3155; B.W, Lee, Proc. 16th Int. Conf. on high-energy physics, Batavia (NAL, Illinois, 1972); E. Abers and B.W. Lee, Phys. Reports 9 (1973) 1. [3] T. Appelquist et al., Phys. Rev. D6 (1972) 2998; D8 (1973) 1747; S. Borchardt and K. Mahanthappa, Nucl. Phys. B65 (1973) 445; S. Baran and'A. Barut, Nuovo Cimento Letters 8 (1973) 716. [4] D. Ross and J.C. Taylor, Nucl. Phys. B51 (1973) 125; D.A. Ross, Nucl. Phys. B51 (1973) 116; B59 (1973) 23. * [5] H. Borchers, Nuovo Cimento 15 (1960) 784; J. Chisholm, Nucl. Phys. 26 (1961) 469; S. Kamefuchi, L. O'Raifeartaigh and A. Salam, Nucl. Phys. 28 (1961) 529. [6] G. 't Hooft and M. Veltman, CERN report 7 3 - 9 , Lab I (1973). [7] W. Pauli and F. Villars, Rev. Mod. Phys. 21 (1949) 434. [8] S. Weinberg, Phys. Rev. Letters 19 (1967) 1264. [9] E. Stueckelberg, Helv. Phys. Acta 11 (1938) 299. [10] L. Faddeev and V. Popov, Phys. Letters 25B (1967); 29; A. Salam and J. Strathdee, Nuovo Cimento l l A (1972) 397. [11] T.D. Lee and C.N. Yang, Phys. Rev. 128 (1962) 885. [12] J.D. Bjorken and S. Drell, Relativistic quantum fields (McGraw-Hill, New York, 1965). [13] J.M. Jauch and F. Rohrlich, The theory of photons and electrons (Addison-Wesley, Cambridge, USA, 1955) ch. 10.

* The essential difference between our calculation and that of Ross (apart from the fact that ours is graphical) is that we insert our counter-terms in the Lagrangian, and hence derive all further relationships between the renormalization constants such as the relations (7.6), (7.8) and (10.4), whereas Ross uses some of these relations as input by demanding that they should hold as a consequence of gauge invariance.