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Unitary in the ghost-free axial gauge
Nuclear Physics B108 (1976) 397-408 © North-Holland Publishing Company
UNITARITY IN THE GHOST-FREE A X I A L GAUGE W. KONETSCHNY and W. KUMMER Institut fur Theoretische Physik, Technische Universittlt Wien, Vienna, Austri~r
Received 30 January 1976 It is shown that gauge theories formulated in the ghost-free axial gauge n • A = 0 yield a unitary S-matrix.
1. Introduction The formalism of gauge theories in the so-called axial gauge n • A = 0 has been developed to a large extent [1,2]. At the expense of introducing a non-covariant vector n u this gauge avoids the complications stemming from the Faddeev-Popov ghosts. Renormalization and gauge as well as Lorentz invariance of the S-matrix have been demonstrated [1,2]. In addition, the latter must be unitary. As mentioned previously [1 ], it is not possible to reach a unitary gauge in a continuous way avoiding the introduction of Faddeev-Popov ghosts in each step of the argument. Thus unitarity has to be established in a different manner. The most direct way is the investigation of the imaginary part of the S-matrix, thereby checking the cancellation of the unphysical degrees of freedom explicitly. This was the approach already pursued in several pioneering papers on gauge theories [ 3 - 5 ] . In our opinion it is the most satisfactory one * and shall be followed here. The intrinsic simplicity of the axial gauge due to the absence of Faddeev-Popov ghosts is visible again. Just like in our previous work we have to worry instead about the problematic point n • k = 0, where k is the momentum vector of the Yang-Mills particle. This difficulty emphasizes the "singular" nature of the gauge. For internal lines it can be overcome by the principal value prescription. A slight subtlety remains for the definition of physical polarization vectors. In sect. 2 we first discuss the modification of the Cutkosky cutting rules due to the n . k denominators. Then we show unitarity for a pure Yang-Mills theory. In sect. 3 we do the same for the Abelian Higgs model. Finally this p r o o f is extended to the general non-Abelian case in sect. 4.
* One may, alternatively, invoke the equivalence of the generating functional to a Hamiltonian formalism; compare ref. [2]. 397
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398
2. Cutkosky rules and the pure Yang-Mills case In order to study the peculiarities characteristic of the axial gauge we start with a pure self-interacting * Yang-Mills theory. The propagator of the vector field reads ** [6]
ab
6ab
Ig
kunv + kvnu
~"~(~)- k~ -- ie -"~
;~. ~
kukvn27
+~j.
(1)
One of the necessary ingredients for unitarity are the Cutkosky rules [7]. Their validity is not evident here due to the presence of the n • k denominators. We tackle this question by going through the Veltman derivation [8] of the Cutkosky cutting rules once more. That derivation crucially depends on the decomposition of the propagator A~b(x) into positive and negative frequency parts. This decomposition can now be shown to hold true for A~b(x) if we remember that (n • k) -r, r = 1,2, was to be interpreted as a principal value [6]. Closing the contour of the k 0 integration in a typical integral appearing in Auv(x), say
l j d4keigxk2~ie
Iu=21
k'n+ie'
,]
+k'n-ie
(2)
above (below) the real axis for x 0 < 0 (x 0 > 0) we obtain contributions A~v(x ) from the "normal" pole at k 2 = +ie and from the "abnormal" ones at (k" n) = +-ie'. By differentiation of (2) with respect to n v another integral, relevant for the Fourier transform of the last term of (1) can be obtained. It is easy to verify that in fact for
A.v(X ) = i J d4k eikX Auv(k )
(3)
the decomposition becomes +
~.~(x) = O(xo) ~ . . ( x ) + O(-xo) ~;.(x) .
(4)
We note furthermore
,a~,~(x) -- ~L(-x), +
[Auv(X)]* =
Auv(X).
* The notorious infrared difficulties in the non-Abelian case may perhaps be curable in a manner similar to quantum electrodynamics [ 11 ]. ** Our metric is -goo =gx 1 = g 2 2 = g 3 3 = 1.
(sa) (5b)
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399
+
In addition, the derivation in ref. [8] requires Auv and Au-v to be contributions of positive and negative frequencies, respectively. This is not true for (2) due to abnormal poles, except for the special choice * n = 0, n o = 1 of n u. In this case (k" n) = k 0 = 0 at those poles, which does not spoil the energy flow argument of ref. [8]. Also the propagator (1) simplifies considerably:
(
ki_k]]
1
(6)
(i,] = space indices).
Ai0 = A00 = 0 ,
Thus all properties required for the derivation [8] of the cutting rules are seen to hold. In the Cutkosky rules [7,8] for any diagram in the axial gauge the contribution of a cut line is given by Aq-(x),
A~(x):27rfd4keikx[(6i]
i
k2 ]O(+-ko) 8 ( k 2 ) ¥ 2k 2 8'(k0) ] .
(7)
In order to show the disappearance of all terms in the unitarity sum involving the "abnormal" part with 8'(k0) in eq. (7) we recall the general off-shell Ward identity in the ghost-free axial gauge [1] in momentum space. In the compact notation where i stands for all field components and where the external leg a represents the auxiliary field Ca we have the relation for the Green functions:
(n " p)
Ga, i l . . . i n
(p, Pl "'" Pn) = ~$
X 6(p +Ps) + ~
S
Aas(p) G i l . . A ' s . . . i n (Pl ""t)s "'" Pn)
gills/Gil...j...i n (Pl .... P +Ps .... Pn)"
(8)
We amputate all propagators at the lines i 1 ... i n and go to the respective ("normal") mass -shells. The first term on the r.h.s, of (8) will not contribute, because there is no pole at p2 = _ m 2. From the second term a contribution may only emerge if (p + ps) 2 ~ - m 2 together w i t h p 2 ~ - m 2. This restricts the values of Pu t e a surface in Minkowski space. In any integration f d4p the contribution of such a surface is of measure zero and therefore vanishes. Since the auxiliary field C a only has the mixed propagators Ga, i = AT./(p" n) [1], the C line is amputated too:
Ga, il...i n (P, Pl "'" Pn)]amp = Ga, i(P) Ti, i,...i n (P, Pl "'" Pn) " Here Ti, ix...i n is a Green function with the lines i, i 1 ... i n amputated. This is exactly the quantity obtained by the cutting procedure. From the preceding argu* The n independence of the S-matrix [i ] allows us to pick out this special case.
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ment we deduce the general on-shell Ward-identity (9)
(n . p) Ga, i,...i n (p, Pl "'" Pn)lamp = Aa Ti, i,...i n = 0 on-shell
in the ghost-free axial gauge. For the pure Y.M. field Aa(p) is just pu6ab for i = (U, 6). Eq. (8) suffices to show the compensation of all terms in the unitarity sum with one line cut at the "abnormal" pole: f d 4 k r / (1) (k)--2T.2 k i k i 6'(k 0) T~Z)*(k) = 0 . /¢
In this expression other (mass-shell) indices of external lines, internal indices and phase space integrations over (normal) mass-shells are not exhibited. As shown in appendix B similar arguments may be found for the general case with more "abnormal" lines cut. Thus all contributions from 6'(k0) in (7) vanish. The above result is not at all surprising because the principal value is real by definition and therefore does not contribute to the imaginary part involved in the unitarity equation. Alternatively speaking, the unitarity relation evaluates s-channel discontinuities. The corresponding thresholds are of course independent of n. In the Landau equations for an arbitrary Feynman graph determining the position of these thresholds, oq-(q2 + m 2) = O,
~j(n . q/) = O,
the ~j therefore have to be zero. In terms of Cutkosky's discontinuity formula [7] this implies that n • qi stays in the denominator and does not give rise to a delta function. The fact that the proof of unitarity can be based on eqs. (4,5,7) we consider to be the strongest technical argument for the principal value prescription. Next we have to construct the physical polarization vectors for the Yang-Mills particle and to work out the sum over polarizations. We have derived earlier [1 ] the conditions imposed on the physical polarizations etz , k- e = 0 ,
(10a)
n. e = 0 ,
(10b)
with k 2 = 0, since we are only interested in on-shell quantities. For time-like n 2 the situation is exactly as in quantum electrodynamics, where the freedom e u e u + 71ku allowed by the first equation is removed by the second equation, because n. k 4: 0. As is well known [9,10] there exist two transversal vectors eu(X), X = 1,2, and the sum over polarizations becomes 2 kukvn2 kun~ + kvnu eu(70 ev()t) = guy + - (11) k.= 1 (n- k) 2 n" k In the power-counting argument of ref. [6] we used a space-like n 2, although this
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was probably only necessary as a technical assumption. In this case (11) is expected to remain true as well, which can indeed be verified (cf. appendix A). However, this construction fails if n • k = 0, which is possible now. It turns out that only one transversal polarization can be found. We anticipate here that the same situation prevails for massive vector mesons; at n • k = 0 one always loses one polarization. In this context one statement made earlier [ 1] has to be straightened out: at n • k = 0 the polarization vector is singular in the sense above, but it does not have a pole. Rather the result of cutting a line does not coincide with the sum over physical polarizations. However, this mismatch occurs only in the integral over intermediate states (where it cannot be avoided by choosing the direction of n u suitably) on a set of measure zero. Another way around this difficulty is to take n 2 time-like as was actually' required in the proof of the cutting rule. Since the left-hand side of the unitafity relation is independent of n (being the imaginary part of an S-matrix element which is n-independent), so must be the right-hand side. The phase-space integrations involved in T T + go over time-like momentum vectors and hence n • k = 0 is impossible. Going back to the expression for Aub we notice that the square bracket in (1) becomes the "normal" term in eq. (7) in the system h~ = 0 and coincides exactly with what we obtained from summing over physical polarizations, eq. (11). Therefore unitarity holds. This almost trivial proof works for any pure Yang-Mills theory in the ghost-free gauge.
3. The Abelian Higgs model The Abelian Higgs model represents the simplest example of a gauge theory with spontaneous symmetry breaking. I t will turn out that it already contains the essential features of the non-Abelian case as far as unitarity is concerned. To the Lagrangian of the model 0 = - ¼ F u v F u v - (Dugp)*(Du(~) -/22~b*q~ + h(¢*q~) 2 , Fu~, = OuA,, - O~A u ,
(12)
D u = a u - igA u ,
we add the gauge breaking term [1] Z?g = ( n . A ) C symmetry breaking the (real) shift v in
1
¢ = (v + ~o + i × ) x / ~
In the case of spontaneous
(13)
provides the vector field A with the mass m = go. The propagators of the theory (we need only those of the A - X sector) are obtained by inverting the quadratic part of the Lagrangian ~ +.~g,
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A(AA)(k'~_ uu
I -
" "
(
-
guu
k2 +m2
kunv+kvnu n" k
+
kukvn2) (n" k) 2 '
(14a)
irn n 2 k u - n " kn u A(uAx)(k) - k 2 + m 2 (n" k) 2 '
A(xx)(k)
1 ( m2n2 k 2 + m 2 (n" k) 2
-
(14b)
) +
1
(14c)
"
A compact form of eqs. (14)is obtained from the observation that the only change from the pure Yang-Mills case is the replacement
a7
a; a = a7 +g
e} °)
(is)
Thus in the Abelian case (14)is nothing else but (1) with the 5-vector (k u, ira) instead of k u in the space of the vector particle and the Higgs-ghost X. The mixing of the latter with the vector field is also reflected in a more complicated condition for physical sources. In our previous work [ 1] we have derived A k ek = 0, which for the Abelian model becomes (g t~. e}0) -+ go = m): •
k. e =-ime(X),
ta
(16)
where e(X) is the source of the x-particle. In addition, n • e = 0 and k 2 = - m 2. The meaning of eq. (16) becomes clear if the combination etz, = eu - mi kue(X)
(17) t
is introduced. The corresponding field A # = A # - ~#y~/m diagonalizes the free part of the Lagrangian [4] (i.e. upon its introduction the X-field disappears) and has the propagator of a massive vector meson, A(A'A') (k~ 1 uv ,, k2 ~ +m
(g + k u k v 1 _uz' - ~ - ] "
(18)
It is nothing else but the vector field in the "unitary" gauge. Thus e' is the polarization of a heavy spin-one meson and satisfies k- e' -- 0 , ,
(19a)
in"
n. e -
--
k
m
e(X) .
(19b)
If the point n. k = 0 is excluded again, it follows immediately that the sum over polarizations is the familiar one,
3
kukv t
t
eu(X) ev(X) = guy + - ~.=1 m2
(20)
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The second equation (19b) does not impose a restriction, because e(X) is still free. Once the e' are known, it is straightforward (the details are relegated to appendix A) to evaluate the sum over polarizations for the unprimed sources. The result 3 X= 1
eu(X) ev(X ) =guy +
kskvn2
kunv + kvnu
(n-k) 2
n" k
'
(21)
is identical to the expression (11) in the massless case despite the fact that the sum now extends over three polarizations. However, the physical polarizations are those of eqs. (19), (20) and a cancellation of the unphysical components is therefore required. It is guaranteed again by the Ward identities as we shall demonstrate in the following. In the Abelian Higgs model Ga, i contains mixed propagators of the C field with A s and X, so that from (15) and (9) puTv + i m T = 0
(22)
x) is obtained. The abbreviations T v and T for Tu, il...in(P, Pl "'" Pn) and T}tl...in (P, Pl "'" Pn) respectively have been introduced. Eq. (22) may be interpreted as the statement 3uA s = 0 (with A s defined before) formulated in terms of on-shell amplitudes. We are now ready to prove unitarity. In any unitarity relation each intermediate A u line is accompanied by two mixed A - X lines and one X line (fig.l). There can be any number of additional intermediate lines which have not been drawn in fig. 1. Cutting the intermediate lines we obtain the expression t
T(1) uv
t
Puny + pvn~ + p" n ( p . n)2_]
+ T(1)(-im)
n2pv - n" pn v T(2) . + T(1)im n2pu - n. pn u T(2) , + T(1) ~ m2n 2 11 T (2)* (n-p)2 (n : p ) 2 L(n: p-)2 + (23)
Fig. 1. Intermediate lines coming together in any unitarity relation. The wavy line represents the gauge field, the dashed line the Higgs ghost.
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With the help of our Ward identity (22) and its complex conjugate, this reduces to +PuPvl r (2)*
(24)
Thus the Yang-Mills field and the unphysical Higgs ghost conspire to produce exactly the transverse degrees of freedom of a physical massive vector meson.
4. Unitarity in the non-Abelian case and conclusions The proof of unitarity outlined here immediately carries over to the non-Abelian case if we diagonalize the Lagrangian by help ofthe matrix U as described previously [ 1]. Then each massive vector field A~ gets its longitudinal polarization from the associated ghost Xa and in the new basis the on-shell Ward identity becomes p v T (a) + i m T (a) = 0 .
(25) t(l
The interpretation of this equation as the transversality of the appropriate A u is again as before. The propagators for each index a are the same as in eqs. (14) and the proof for the cancellation of the unphysical degrees of freedom goes through unchanged. Let us compare with the analogous proof in the context of Faddeev-Popov quantization [3]. Our Ward identity is the exact counterpart of 't Hooft's Ward identity if all Faddeev-Popov insertions are set equal to zero. This is to be expected for a "ghost-free" gauge. Besides this obvious simplification the proof proceeds along the very same lines. Again - as in the proofs of renormalization and gauge invariance the intuitive expectation is confirmed that following the main ideas of the general quantization scheme with Faddeev-Popov ghosts simply all complications from such ghosts may be ignored. In view of the main result of this paper, together with ref. [1] we believe that the following is a fair statement: non-Abelian gauge fields in the ghost-free axial gauges considered here represent legitimate formulations of such field theories, because all the basic requirements (renormalizability, gauge invariance, unitarity) are met. The absence of ghosts in these gauges constitutes their main advantage, whereas the Feynman rules seem rather cumbersome. It will depend on the particular problem at hand whether one decides to calculate in a conventional or in the axial gauge.
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405
Appendix A Physical polarization vectors In the massless case the polarization vector e u satisfies k" e = 0 ,
(A.1)
n- e = 0,
(A.2)
with k 2 = 0. F r o m (A.1) e has to be space-like, i.e. it is normalized to e 2 = +1. For n 2 spacelike one m a y choose n = (0,0, n2, n3) ,
(A.3)
k = (k0,0,0,k0).
(A.4)
Then from (A.1) and (A.2), e 0 = e 3,
e 2 +e 2 = 1,
n2e 2 + n3e 3 = 0 .
(A.5) (A.6)
If n 3 4= 0, i.e. n • k ve 0, two i n d e p e n d e n t transverse polarizations are e(1) = ( 0 , 1 , 0 , 0 ) ,
e(2) =
(A.7)
_n2 n2 ) ~3 ,0,1,-~3 "
(A.8)
To w o r k out the completeness sum the orthogonal basis e(1) and = (0,0,1,0) = e(2) + n---Tk n . g k,
=(0,0,0,1)-
n - (n. ~)~ , N/n 2 _ ( , . d)2
~: = (1,0,0,0) .-- k_ ~ (k "_n_~ )~
x/(k.
(A.9)
(A.10)
(A.11)
in
guy = eta(l) ev(1) + OuOv + huhv - kuTcv can be used. The sum over physical polarizations as given in the text (eq. (11)): follows i m m e d i a t e l y .
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406
For the singular case n 3 = 0, i.e. n • k = 0, we must have n 2 4= 0 and hence e0 e = (e0,1,0,e0) = e(1) +~00
k,
where e 0 remains a free parameter. These infinitely many vectors e are all connected by a gauge transformation and there is only one genuine polarization. In the massive case k 2 = - m 2 the sum 3 3 3" =
eu(X) ev(X) =
'
~ eu(X ) x=l
,
ev(X) + i ~ (kue,v(X)e(X)(X) ~ - k= 1
3
~ +e'u(X)kve(X)(X)) kukv m 2 X=l
[e(X)(~,)]2,
is evaluated by using (20) and (19b). From these two equations we infer 3 [e(X)(?t)] 2 m2n2 1 , x---l (n. k) 2 3
,
im ( n~z +n'k ku) m2
e~(X) e(X)(X) = ~
.
(A.12)
(A.13)
(A.14)
Substituting (A. 13,14) into (A. 12) we get the result quoted in the text (eq. (21)).
Appendix B
Ward identities at "abnormalpoles" In the notation of sect. 2 a contribution to the unitarity sum from two lines with momenta k,q at the " a b n o r m a l " pole becomes with (7) (up to irrelevant factors, integrations and indices of particles at their normal mass shells)
f dkodq 0 T(1)(k,q) kik~i' q/ql' q2
8,(ko)8,(qo) T(,~!*(k,q).
By partial integration we obtain terms of the form
T(1) kiki, q/q], ( ~2 ) I¢2 q2 Ok-o~qo T('2!*
k2
q2
v ~ u - - )*
(B1) ko=qo=0
(B2) ko=qo =0
and similar ones with the derivatives interchanged
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407
In order to prove the vanishing of (B.1) and (B.2) we consider a Green function
Ga,Ai2...in where the index A now includes a possible C leg as well. As shown in ref. [ 1], eigenvectors eA of the self-energy matrix related to unphysical sources may contain a C field component (cf. [1], eqs. (5.3) and (5.4)). In fact, near (p" n) = 0 the self-energy matrix has the eigenvalue X ~ 0v- n)2/n 2 and (for the pure Yang-Mills field) the corresponding eigenvector (eO): (})
( p : n ) (~)+(p" n ) v ~ pNz/~
~
n2
(B.3) '
in a 5-dimensional space spanned by the orthonormal basis vectors (1_)
¢
pu =
(1) ,
(2) fu = Inu
=0,
=o,
(p: n ) p d p2 .J
(P' n)27-1/2
2)
pi 3
= 0,
(B.4)
=1. (0) (0)
The pole part of the "propagators" e A e B/X following from (B.3) and (B.4) for A = ~, B = u precisely coincides with the second and third term on the r.h.s, of (1) or with the second term in (6), if the special system n O = 1, n = 0 is chosen Furthermore, (e°),(e0)/X is nothing else but the usual mixed C - A propagator [1] p J(p" n), ~( 0) (u)0 whereas e 5 e 5 does not exhibit any singular behaviour at (p" n) -- P0 = 0. We now amputate this "propagator" from Ga,Ai2...in. Since there is no C - C propagator, B in (0) (0)
G a,Ai2...in -
eA eB ~ ( B - a m p ) )~ ~a, Bi2...i n
(B.5)
cannot be a C line. For B = vector particle (v,b) with momentum q we have (0) (0) ev e O
(p • n) Ga,(vb) i2...intam p ='Pu
X
Toaa),(ob)i2""in
+ pvf(p + q) Ti2...i n lab .
(B.6)
The second term in (B.6) arises because the C line (a) can join directly onto (v,b) yielding the mixed propagator Pv/(P" n). We shall be interested in the poles at the "mass shell" q0 = 0. For the left-hand side of (B.6) we have again [he Ward identity (8) at our disposal, if i 1 -+ (v,b). Inserting (B.6) into (8) the second term of (B.6) drops out. If we amputate all lines i 2 ... i n and go to their mass-shells, by means of the same argument as in eq. (9) all terms on the r.h.s, of the Ward identity vanish
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408
except s = 1. In our special system n o = 1, n = 0 we thus obtain near q0 = O:
qi q/ (qi + Pi)(qj + Pj) q2q2 Pu To~a),qb)i2...in (P'q)=gtabc (qo + P 0 ) 2(q + p ) 2 X Tqc)i2...in (p + q),
(B.7)
where at the index ( j e ) the abnormal propagator has been amputated. We observe that the 1.h.s. of (B.7) is proportional to qi whereas the r.h.s, has a factor (Pi + qi)" Hence both sides of (B.7) must vanish. For the poles of second order this yields
q/Pu Toaa)qb)lqo=O = 0 ,
(B.8)
and for the first order pole:
aq o (q/ Pu T(ua)qb))lqo =0 = O.
(B.9)
Eqs. (B.8) and (B.9) are the kind of relations required to show that the terms in (B.1) and (B.2) vanish. F r o m this analysis it should be clear how one must proceed in expressions like (B.1), (B.2) with more internal lines at abnormal poles and hence with more derivatives: one just needs the Ward identity (8) with more than one leg is amputated at an " a b n o r m a l " pole. We note finally that the transcription into relations for the case o f spontaneously broken symmetry (massive Yang-Mills fields) simply implies the replacement (15).
References [ 1 ] W. Konetschny and W. Kummer, Nucl. Phys. B100 (1975) 106 and references cited therein. [2] J. Frenkel, University of Sao Paulo preprint IFUSP/P-71 (December 1975). [3] G. 't Hooft, Nucl. Phys. B33 (1971) 173; B35 (1971) 167. [4] B.W. Lee, Phys. Rev. D5 (1972) 823. [5] B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3137. [6] W. Kummer, Acta Phys. Austr. 41 (1975) 315. [7] R.E. Cutkosky, J. Math. Phys. 1 (1960) 429. [8] M. Veltman, Physica 29 (1963) 186; G. 't Hooft and M. Veltman, Diagrammar (CERN 73-9). [9] J.D. Bjorken and S.D. Drell, Relativistic quantum fields (MacGraw-Hill, New York, 1965). [10] W. Kummer, Acta Phys. Austr. 14 (1961) 149. [11] York-Peng Yao, Michigan preprint UM HE 75-38; T. Appelquist, J. Carazzone, H. Kluberg-Stern and M. Roth, FERMILAB-PUB-76/16-THY.
UNITARITY IN THE GHOST-FREE A X I A L GAUGE W. KONETSCHNY and W. KUMMER Institut fur Theoretische Physik, Technische Universittlt Wien, Vienna, Austri~r
Received 30 January 1976 It is shown that gauge theories formulated in the ghost-free axial gauge n • A = 0 yield a unitary S-matrix.
1. Introduction The formalism of gauge theories in the so-called axial gauge n • A = 0 has been developed to a large extent [1,2]. At the expense of introducing a non-covariant vector n u this gauge avoids the complications stemming from the Faddeev-Popov ghosts. Renormalization and gauge as well as Lorentz invariance of the S-matrix have been demonstrated [1,2]. In addition, the latter must be unitary. As mentioned previously [1 ], it is not possible to reach a unitary gauge in a continuous way avoiding the introduction of Faddeev-Popov ghosts in each step of the argument. Thus unitarity has to be established in a different manner. The most direct way is the investigation of the imaginary part of the S-matrix, thereby checking the cancellation of the unphysical degrees of freedom explicitly. This was the approach already pursued in several pioneering papers on gauge theories [ 3 - 5 ] . In our opinion it is the most satisfactory one * and shall be followed here. The intrinsic simplicity of the axial gauge due to the absence of Faddeev-Popov ghosts is visible again. Just like in our previous work we have to worry instead about the problematic point n • k = 0, where k is the momentum vector of the Yang-Mills particle. This difficulty emphasizes the "singular" nature of the gauge. For internal lines it can be overcome by the principal value prescription. A slight subtlety remains for the definition of physical polarization vectors. In sect. 2 we first discuss the modification of the Cutkosky cutting rules due to the n . k denominators. Then we show unitarity for a pure Yang-Mills theory. In sect. 3 we do the same for the Abelian Higgs model. Finally this p r o o f is extended to the general non-Abelian case in sect. 4.
* One may, alternatively, invoke the equivalence of the generating functional to a Hamiltonian formalism; compare ref. [2]. 397
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398
2. Cutkosky rules and the pure Yang-Mills case In order to study the peculiarities characteristic of the axial gauge we start with a pure self-interacting * Yang-Mills theory. The propagator of the vector field reads ** [6]
ab
6ab
Ig
kunv + kvnu
~"~(~)- k~ -- ie -"~
;~. ~
kukvn27
+~j.
(1)
One of the necessary ingredients for unitarity are the Cutkosky rules [7]. Their validity is not evident here due to the presence of the n • k denominators. We tackle this question by going through the Veltman derivation [8] of the Cutkosky cutting rules once more. That derivation crucially depends on the decomposition of the propagator A~b(x) into positive and negative frequency parts. This decomposition can now be shown to hold true for A~b(x) if we remember that (n • k) -r, r = 1,2, was to be interpreted as a principal value [6]. Closing the contour of the k 0 integration in a typical integral appearing in Auv(x), say
l j d4keigxk2~ie
Iu=21
k'n+ie'
,]
+k'n-ie
(2)
above (below) the real axis for x 0 < 0 (x 0 > 0) we obtain contributions A~v(x ) from the "normal" pole at k 2 = +ie and from the "abnormal" ones at (k" n) = +-ie'. By differentiation of (2) with respect to n v another integral, relevant for the Fourier transform of the last term of (1) can be obtained. It is easy to verify that in fact for
A.v(X ) = i J d4k eikX Auv(k )
(3)
the decomposition becomes +
~.~(x) = O(xo) ~ . . ( x ) + O(-xo) ~;.(x) .
(4)
We note furthermore
,a~,~(x) -- ~L(-x), +
[Auv(X)]* =
Auv(X).
* The notorious infrared difficulties in the non-Abelian case may perhaps be curable in a manner similar to quantum electrodynamics [ 11 ]. ** Our metric is -goo =gx 1 = g 2 2 = g 3 3 = 1.
(sa) (5b)
W. Konetschny, W. Kummer / Unitarity
399
+
In addition, the derivation in ref. [8] requires Auv and Au-v to be contributions of positive and negative frequencies, respectively. This is not true for (2) due to abnormal poles, except for the special choice * n = 0, n o = 1 of n u. In this case (k" n) = k 0 = 0 at those poles, which does not spoil the energy flow argument of ref. [8]. Also the propagator (1) simplifies considerably:
(
ki_k]]
1
(6)
(i,] = space indices).
Ai0 = A00 = 0 ,
Thus all properties required for the derivation [8] of the cutting rules are seen to hold. In the Cutkosky rules [7,8] for any diagram in the axial gauge the contribution of a cut line is given by Aq-(x),
A~(x):27rfd4keikx[(6i]
i
k2 ]O(+-ko) 8 ( k 2 ) ¥ 2k 2 8'(k0) ] .
(7)
In order to show the disappearance of all terms in the unitarity sum involving the "abnormal" part with 8'(k0) in eq. (7) we recall the general off-shell Ward identity in the ghost-free axial gauge [1] in momentum space. In the compact notation where i stands for all field components and where the external leg a represents the auxiliary field Ca we have the relation for the Green functions:
(n " p)
Ga, i l . . . i n
(p, Pl "'" Pn) = ~$
X 6(p +Ps) + ~
S
Aas(p) G i l . . A ' s . . . i n (Pl ""t)s "'" Pn)
gills/Gil...j...i n (Pl .... P +Ps .... Pn)"
(8)
We amputate all propagators at the lines i 1 ... i n and go to the respective ("normal") mass -shells. The first term on the r.h.s, of (8) will not contribute, because there is no pole at p2 = _ m 2. From the second term a contribution may only emerge if (p + ps) 2 ~ - m 2 together w i t h p 2 ~ - m 2. This restricts the values of Pu t e a surface in Minkowski space. In any integration f d4p the contribution of such a surface is of measure zero and therefore vanishes. Since the auxiliary field C a only has the mixed propagators Ga, i = AT./(p" n) [1], the C line is amputated too:
Ga, il...i n (P, Pl "'" Pn)]amp = Ga, i(P) Ti, i,...i n (P, Pl "'" Pn) " Here Ti, ix...i n is a Green function with the lines i, i 1 ... i n amputated. This is exactly the quantity obtained by the cutting procedure. From the preceding argu* The n independence of the S-matrix [i ] allows us to pick out this special case.
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ment we deduce the general on-shell Ward-identity (9)
(n . p) Ga, i,...i n (p, Pl "'" Pn)lamp = Aa Ti, i,...i n = 0 on-shell
in the ghost-free axial gauge. For the pure Y.M. field Aa(p) is just pu6ab for i = (U, 6). Eq. (8) suffices to show the compensation of all terms in the unitarity sum with one line cut at the "abnormal" pole: f d 4 k r / (1) (k)--2T.2 k i k i 6'(k 0) T~Z)*(k) = 0 . /¢
In this expression other (mass-shell) indices of external lines, internal indices and phase space integrations over (normal) mass-shells are not exhibited. As shown in appendix B similar arguments may be found for the general case with more "abnormal" lines cut. Thus all contributions from 6'(k0) in (7) vanish. The above result is not at all surprising because the principal value is real by definition and therefore does not contribute to the imaginary part involved in the unitarity equation. Alternatively speaking, the unitarity relation evaluates s-channel discontinuities. The corresponding thresholds are of course independent of n. In the Landau equations for an arbitrary Feynman graph determining the position of these thresholds, oq-(q2 + m 2) = O,
~j(n . q/) = O,
the ~j therefore have to be zero. In terms of Cutkosky's discontinuity formula [7] this implies that n • qi stays in the denominator and does not give rise to a delta function. The fact that the proof of unitarity can be based on eqs. (4,5,7) we consider to be the strongest technical argument for the principal value prescription. Next we have to construct the physical polarization vectors for the Yang-Mills particle and to work out the sum over polarizations. We have derived earlier [1 ] the conditions imposed on the physical polarizations etz , k- e = 0 ,
(10a)
n. e = 0 ,
(10b)
with k 2 = 0, since we are only interested in on-shell quantities. For time-like n 2 the situation is exactly as in quantum electrodynamics, where the freedom e u e u + 71ku allowed by the first equation is removed by the second equation, because n. k 4: 0. As is well known [9,10] there exist two transversal vectors eu(X), X = 1,2, and the sum over polarizations becomes 2 kukvn2 kun~ + kvnu eu(70 ev()t) = guy + - (11) k.= 1 (n- k) 2 n" k In the power-counting argument of ref. [6] we used a space-like n 2, although this
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was probably only necessary as a technical assumption. In this case (11) is expected to remain true as well, which can indeed be verified (cf. appendix A). However, this construction fails if n • k = 0, which is possible now. It turns out that only one transversal polarization can be found. We anticipate here that the same situation prevails for massive vector mesons; at n • k = 0 one always loses one polarization. In this context one statement made earlier [ 1] has to be straightened out: at n • k = 0 the polarization vector is singular in the sense above, but it does not have a pole. Rather the result of cutting a line does not coincide with the sum over physical polarizations. However, this mismatch occurs only in the integral over intermediate states (where it cannot be avoided by choosing the direction of n u suitably) on a set of measure zero. Another way around this difficulty is to take n 2 time-like as was actually' required in the proof of the cutting rule. Since the left-hand side of the unitafity relation is independent of n (being the imaginary part of an S-matrix element which is n-independent), so must be the right-hand side. The phase-space integrations involved in T T + go over time-like momentum vectors and hence n • k = 0 is impossible. Going back to the expression for Aub we notice that the square bracket in (1) becomes the "normal" term in eq. (7) in the system h~ = 0 and coincides exactly with what we obtained from summing over physical polarizations, eq. (11). Therefore unitarity holds. This almost trivial proof works for any pure Yang-Mills theory in the ghost-free gauge.
3. The Abelian Higgs model The Abelian Higgs model represents the simplest example of a gauge theory with spontaneous symmetry breaking. I t will turn out that it already contains the essential features of the non-Abelian case as far as unitarity is concerned. To the Lagrangian of the model 0 = - ¼ F u v F u v - (Dugp)*(Du(~) -/22~b*q~ + h(¢*q~) 2 , Fu~, = OuA,, - O~A u ,
(12)
D u = a u - igA u ,
we add the gauge breaking term [1] Z?g = ( n . A ) C symmetry breaking the (real) shift v in
1
¢ = (v + ~o + i × ) x / ~
In the case of spontaneous
(13)
provides the vector field A with the mass m = go. The propagators of the theory (we need only those of the A - X sector) are obtained by inverting the quadratic part of the Lagrangian ~ +.~g,
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402
A(AA)(k'~_ uu
I -
" "
(
-
guu
k2 +m2
kunv+kvnu n" k
+
kukvn2) (n" k) 2 '
(14a)
irn n 2 k u - n " kn u A(uAx)(k) - k 2 + m 2 (n" k) 2 '
A(xx)(k)
1 ( m2n2 k 2 + m 2 (n" k) 2
-
(14b)
) +
1
(14c)
"
A compact form of eqs. (14)is obtained from the observation that the only change from the pure Yang-Mills case is the replacement
a7
a; a = a7 +g
e} °)
(is)
Thus in the Abelian case (14)is nothing else but (1) with the 5-vector (k u, ira) instead of k u in the space of the vector particle and the Higgs-ghost X. The mixing of the latter with the vector field is also reflected in a more complicated condition for physical sources. In our previous work [ 1] we have derived A k ek = 0, which for the Abelian model becomes (g t~. e}0) -+ go = m): •
k. e =-ime(X),
ta
(16)
where e(X) is the source of the x-particle. In addition, n • e = 0 and k 2 = - m 2. The meaning of eq. (16) becomes clear if the combination etz, = eu - mi kue(X)
(17) t
is introduced. The corresponding field A # = A # - ~#y~/m diagonalizes the free part of the Lagrangian [4] (i.e. upon its introduction the X-field disappears) and has the propagator of a massive vector meson, A(A'A') (k~ 1 uv ,, k2 ~ +m
(g + k u k v 1 _uz' - ~ - ] "
(18)
It is nothing else but the vector field in the "unitary" gauge. Thus e' is the polarization of a heavy spin-one meson and satisfies k- e' -- 0 , ,
(19a)
in"
n. e -
--
k
m
e(X) .
(19b)
If the point n. k = 0 is excluded again, it follows immediately that the sum over polarizations is the familiar one,
3
kukv t
t
eu(X) ev(X) = guy + - ~.=1 m2
(20)
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403
The second equation (19b) does not impose a restriction, because e(X) is still free. Once the e' are known, it is straightforward (the details are relegated to appendix A) to evaluate the sum over polarizations for the unprimed sources. The result 3 X= 1
eu(X) ev(X ) =guy +
kskvn2
kunv + kvnu
(n-k) 2
n" k
'
(21)
is identical to the expression (11) in the massless case despite the fact that the sum now extends over three polarizations. However, the physical polarizations are those of eqs. (19), (20) and a cancellation of the unphysical components is therefore required. It is guaranteed again by the Ward identities as we shall demonstrate in the following. In the Abelian Higgs model Ga, i contains mixed propagators of the C field with A s and X, so that from (15) and (9) puTv + i m T = 0
(22)
x) is obtained. The abbreviations T v and T for Tu, il...in(P, Pl "'" Pn) and T}tl...in (P, Pl "'" Pn) respectively have been introduced. Eq. (22) may be interpreted as the statement 3uA s = 0 (with A s defined before) formulated in terms of on-shell amplitudes. We are now ready to prove unitarity. In any unitarity relation each intermediate A u line is accompanied by two mixed A - X lines and one X line (fig.l). There can be any number of additional intermediate lines which have not been drawn in fig. 1. Cutting the intermediate lines we obtain the expression t
T(1) uv
t
Puny + pvn~ + p" n ( p . n)2_]
+ T(1)(-im)
n2pv - n" pn v T(2) . + T(1)im n2pu - n. pn u T(2) , + T(1) ~ m2n 2 11 T (2)* (n-p)2 (n : p ) 2 L(n: p-)2 + (23)
Fig. 1. Intermediate lines coming together in any unitarity relation. The wavy line represents the gauge field, the dashed line the Higgs ghost.
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w. Konetschny, W. Kummer / Unitarity
With the help of our Ward identity (22) and its complex conjugate, this reduces to +PuPvl r (2)*
(24)
Thus the Yang-Mills field and the unphysical Higgs ghost conspire to produce exactly the transverse degrees of freedom of a physical massive vector meson.
4. Unitarity in the non-Abelian case and conclusions The proof of unitarity outlined here immediately carries over to the non-Abelian case if we diagonalize the Lagrangian by help ofthe matrix U as described previously [ 1]. Then each massive vector field A~ gets its longitudinal polarization from the associated ghost Xa and in the new basis the on-shell Ward identity becomes p v T (a) + i m T (a) = 0 .
(25) t(l
The interpretation of this equation as the transversality of the appropriate A u is again as before. The propagators for each index a are the same as in eqs. (14) and the proof for the cancellation of the unphysical degrees of freedom goes through unchanged. Let us compare with the analogous proof in the context of Faddeev-Popov quantization [3]. Our Ward identity is the exact counterpart of 't Hooft's Ward identity if all Faddeev-Popov insertions are set equal to zero. This is to be expected for a "ghost-free" gauge. Besides this obvious simplification the proof proceeds along the very same lines. Again - as in the proofs of renormalization and gauge invariance the intuitive expectation is confirmed that following the main ideas of the general quantization scheme with Faddeev-Popov ghosts simply all complications from such ghosts may be ignored. In view of the main result of this paper, together with ref. [1] we believe that the following is a fair statement: non-Abelian gauge fields in the ghost-free axial gauges considered here represent legitimate formulations of such field theories, because all the basic requirements (renormalizability, gauge invariance, unitarity) are met. The absence of ghosts in these gauges constitutes their main advantage, whereas the Feynman rules seem rather cumbersome. It will depend on the particular problem at hand whether one decides to calculate in a conventional or in the axial gauge.
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405
Appendix A Physical polarization vectors In the massless case the polarization vector e u satisfies k" e = 0 ,
(A.1)
n- e = 0,
(A.2)
with k 2 = 0. F r o m (A.1) e has to be space-like, i.e. it is normalized to e 2 = +1. For n 2 spacelike one m a y choose n = (0,0, n2, n3) ,
(A.3)
k = (k0,0,0,k0).
(A.4)
Then from (A.1) and (A.2), e 0 = e 3,
e 2 +e 2 = 1,
n2e 2 + n3e 3 = 0 .
(A.5) (A.6)
If n 3 4= 0, i.e. n • k ve 0, two i n d e p e n d e n t transverse polarizations are e(1) = ( 0 , 1 , 0 , 0 ) ,
e(2) =
(A.7)
_n2 n2 ) ~3 ,0,1,-~3 "
(A.8)
To w o r k out the completeness sum the orthogonal basis e(1) and = (0,0,1,0) = e(2) + n---Tk n . g k,
=(0,0,0,1)-
n - (n. ~)~ , N/n 2 _ ( , . d)2
~: = (1,0,0,0) .-- k_ ~ (k "_n_~ )~
x/(k.
(A.9)
(A.10)
(A.11)
in
guy = eta(l) ev(1) + OuOv + huhv - kuTcv can be used. The sum over physical polarizations as given in the text (eq. (11)): follows i m m e d i a t e l y .
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406
For the singular case n 3 = 0, i.e. n • k = 0, we must have n 2 4= 0 and hence e0 e = (e0,1,0,e0) = e(1) +~00
k,
where e 0 remains a free parameter. These infinitely many vectors e are all connected by a gauge transformation and there is only one genuine polarization. In the massive case k 2 = - m 2 the sum 3 3 3" =
eu(X) ev(X) =
'
~ eu(X ) x=l
,
ev(X) + i ~ (kue,v(X)e(X)(X) ~ - k= 1
3
~ +e'u(X)kve(X)(X)) kukv m 2 X=l
[e(X)(~,)]2,
is evaluated by using (20) and (19b). From these two equations we infer 3 [e(X)(?t)] 2 m2n2 1 , x---l (n. k) 2 3
,
im ( n~z +n'k ku) m2
e~(X) e(X)(X) = ~
.
(A.12)
(A.13)
(A.14)
Substituting (A. 13,14) into (A. 12) we get the result quoted in the text (eq. (21)).
Appendix B
Ward identities at "abnormalpoles" In the notation of sect. 2 a contribution to the unitarity sum from two lines with momenta k,q at the " a b n o r m a l " pole becomes with (7) (up to irrelevant factors, integrations and indices of particles at their normal mass shells)
f dkodq 0 T(1)(k,q) kik~i' q/ql' q2
8,(ko)8,(qo) T(,~!*(k,q).
By partial integration we obtain terms of the form
T(1) kiki, q/q], ( ~2 ) I¢2 q2 Ok-o~qo T('2!*
k2
q2
v ~ u - - )*
(B1) ko=qo=0
(B2) ko=qo =0
and similar ones with the derivatives interchanged
W. Konetschny, W. Kummer / Unitarity
407
In order to prove the vanishing of (B.1) and (B.2) we consider a Green function
Ga,Ai2...in where the index A now includes a possible C leg as well. As shown in ref. [ 1], eigenvectors eA of the self-energy matrix related to unphysical sources may contain a C field component (cf. [1], eqs. (5.3) and (5.4)). In fact, near (p" n) = 0 the self-energy matrix has the eigenvalue X ~ 0v- n)2/n 2 and (for the pure Yang-Mills field) the corresponding eigenvector (eO): (})
( p : n ) (~)+(p" n ) v ~ pNz/~
~
n2
(B.3) '
in a 5-dimensional space spanned by the orthonormal basis vectors (1_)
¢
pu =
(1) ,
(2) fu = Inu
=0,
=o,
(p: n ) p d p2 .J
(P' n)27-1/2
2)
pi 3
= 0,
(B.4)
=1. (0) (0)
The pole part of the "propagators" e A e B/X following from (B.3) and (B.4) for A = ~, B = u precisely coincides with the second and third term on the r.h.s, of (1) or with the second term in (6), if the special system n O = 1, n = 0 is chosen Furthermore, (e°),(e0)/X is nothing else but the usual mixed C - A propagator [1] p J(p" n), ~( 0) (u)0 whereas e 5 e 5 does not exhibit any singular behaviour at (p" n) -- P0 = 0. We now amputate this "propagator" from Ga,Ai2...in. Since there is no C - C propagator, B in (0) (0)
G a,Ai2...in -
eA eB ~ ( B - a m p ) )~ ~a, Bi2...i n
(B.5)
cannot be a C line. For B = vector particle (v,b) with momentum q we have (0) (0) ev e O
(p • n) Ga,(vb) i2...intam p ='Pu
X
Toaa),(ob)i2""in
+ pvf(p + q) Ti2...i n lab .
(B.6)
The second term in (B.6) arises because the C line (a) can join directly onto (v,b) yielding the mixed propagator Pv/(P" n). We shall be interested in the poles at the "mass shell" q0 = 0. For the left-hand side of (B.6) we have again [he Ward identity (8) at our disposal, if i 1 -+ (v,b). Inserting (B.6) into (8) the second term of (B.6) drops out. If we amputate all lines i 2 ... i n and go to their mass-shells, by means of the same argument as in eq. (9) all terms on the r.h.s, of the Ward identity vanish
14:.Konetschny, W. Kummer / Unitarity
408
except s = 1. In our special system n o = 1, n = 0 we thus obtain near q0 = O:
qi q/ (qi + Pi)(qj + Pj) q2q2 Pu To~a),qb)i2...in (P'q)=gtabc (qo + P 0 ) 2(q + p ) 2 X Tqc)i2...in (p + q),
(B.7)
where at the index ( j e ) the abnormal propagator has been amputated. We observe that the 1.h.s. of (B.7) is proportional to qi whereas the r.h.s, has a factor (Pi + qi)" Hence both sides of (B.7) must vanish. For the poles of second order this yields
q/Pu Toaa)qb)lqo=O = 0 ,
(B.8)
and for the first order pole:
aq o (q/ Pu T(ua)qb))lqo =0 = O.
(B.9)
Eqs. (B.8) and (B.9) are the kind of relations required to show that the terms in (B.1) and (B.2) vanish. F r o m this analysis it should be clear how one must proceed in expressions like (B.1), (B.2) with more internal lines at abnormal poles and hence with more derivatives: one just needs the Ward identity (8) with more than one leg is amputated at an " a b n o r m a l " pole. We note finally that the transcription into relations for the case o f spontaneously broken symmetry (massive Yang-Mills fields) simply implies the replacement (15).
References [ 1 ] W. Konetschny and W. Kummer, Nucl. Phys. B100 (1975) 106 and references cited therein. [2] J. Frenkel, University of Sao Paulo preprint IFUSP/P-71 (December 1975). [3] G. 't Hooft, Nucl. Phys. B33 (1971) 173; B35 (1971) 167. [4] B.W. Lee, Phys. Rev. D5 (1972) 823. [5] B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3137. [6] W. Kummer, Acta Phys. Austr. 41 (1975) 315. [7] R.E. Cutkosky, J. Math. Phys. 1 (1960) 429. [8] M. Veltman, Physica 29 (1963) 186; G. 't Hooft and M. Veltman, Diagrammar (CERN 73-9). [9] J.D. Bjorken and S.D. Drell, Relativistic quantum fields (MacGraw-Hill, New York, 1965). [10] W. Kummer, Acta Phys. Austr. 14 (1961) 149. [11] York-Peng Yao, Michigan preprint UM HE 75-38; T. Appelquist, J. Carazzone, H. Kluberg-Stern and M. Roth, FERMILAB-PUB-76/16-THY.