Electromagnetic mass shifts, equal-time commutators (ETC), and their relation are analyzed, using the Jost-Lehmann representation for the forward Compton amplitude. The divergences in 

δ

m

 are determined by Lorentz scalars associated with the ETC (when these exist). These results are illustrated by models with currents bilinear and linear in canonical fields. Existence of the ETC implies that the Jost-Lehmann representation is unsubtracted

Boulware, David G.

Deser, Stanley

Caltech Authors - Main

VOLUME 20, NUMBER 24 PHYSICAL REVIEW LETTERS 10 JUNs 1968
ELECTROMAGNETIC MASS SHIFTS, EQUAL-TIME COMMUTATORS, AND
JOST-LE HMANN REPRESENTATION
David G. Boulware*
Physics Department, University of Washington, Seattle, Washington
and
Stanley Desert
Physics Department, Brandeis University, %@ltham, Massachusetts
(H,eceived 24 April 1968)
Electromagnetic mass shifts, equal-time commutators (ETC), and their relation are
analyzed, using the Jost-Lehmann representation for the forvrard Compton amplitude,
The divergences in 6m are determined by Lorentz scal@rs associated with the ETC
(when these exist). These results are illustrated by models with currents bilinear and
linear in canonical fields. Existence of the ETC implies that the Jost-Lehmann repre-
sentation is unsubtracted.
%e analyze the general connection between asymptotic behavior of the forward Compton amplitude
K&y(k', v) and equal-time electromagnetic (em) current commutators (ETC). When the invariant am-
plitudes satisfy unsubtracted v relations and behave as k s or k, Bjorken' has shown that the limit
k, -~ (k fixed) determines the k'- ~ behavior. Since this limit also identifies the appropriate ETC,
the latter then determine the quadratic and logarithmic divergences in mass shifts implied by the as-
ymptotic k and k dependence. Thp precise 6m-ETC relation will be given first when the Bjopken
assumptions hold and will be applied to models of current interest. %e then establish and use the
Jost-Lehmann' representation for K&y to study the relation under more general circumstances, and
to find the necessary conditions for the ETC to exist at all.
The forward, spin-averaged, Compton amplitude is
II (k, v) —= i fd xe (P1T*[j (x)j (0)]IP),
p, 3L P
=(g k -k k )SR (k, v)-m [k p p pk(p k -+p k )+g (pk) ]3K (k, v),
pA px
where iT*(j ~j ) is the conserved covariant current correlation function with disconnected graphs re-
moved; v= -pk/m and p +m' =0, (p Ip') = (2m)'2p'6{p-p'). With the above form, the invariant functions
Ki are free of kinematic singularities. The O(n) mass-shift is determined by the trace%&I (gauge
invariance follows from the conservation condition):
Om'=-3ie'fd4k(2s) 4(k'-i~) 'lk'lK + &[k' —2(pk)']SR ). (2)
Internal indices are suppressed, mass splits being given by appropriate isospin combinations of the
Ki. If the 3Ri behave as fthm '+ f&k ', a Wick rotation is permitted, yielding for the leading divergenc-
es 5m-(Jdk )(f, + afs), Jk adk (j,+ —,'f, ), resPectively, These combinations are related to (but not
identical with) the matrix elements of Ci —=[j0(r),ji(0)]= C00ikak& (r) and Cgk =- [s0ji(r)-si j0(r), jk(o)]
=C00ik5 (r). The coefficients of 8kB(r) and 5(r) are, as anticipated by our notation, components of
fourth-rank tensors C»ya, C@&y~, as is seen from the kinematic tensor form of the k, -~ limit of
Oi~
~0.- [&. fl + m (&. p p -p,p )f2]k
and likewise for the C. Thus, effectively,
Terms of the f,"' form are clearly not of the kinematically required type, but can occur, for exam-
ple, in SU(2)S SU(2) algebra of fields Ter.ms of the form pi"pvp~po do not arise in simple models,
and have been omitted, as have also possible structures with vanishing (00ij) components because
VOLUME 20, NUMBER 24 PHYSICAL REVIEW LETTERS 10 JuNz 1968
these do not contribute to 6m divergences; the latter depend on the combination%, +2%, + ~%, '",
which may be shown to be the matrix element of the Lorentz scalar S=(36) 1[C&~~~-(25)C&~"~].
Clearly S and C00ik (or even C00ii) are quite different quantities and it is only S which is relevant to
5m'.
The operative ETC are entirely model dependent for em currents. It is therefore instructive to il-
lustrate these results with some explicit models: (1) j& bilinear in boson (spin-1 or -0) fields, (2)j P
bilinear in spin--, field only, and (3) jI" linear in a canonical field.
In (1), S has the form
S=:[2q q+r3B B -(6m ) 6 G ]:2 -1 f /ivp pv (4)
where p, B~, G~q are the charged scalar and vector fields and G~p the spin-1 field strength. The di-
agonal element will not, in general, vanish and there are quadratic divergences. For (2), even as-
suming that C00ik is a c-number (and this depends also on the charged fermion's other interactions),
there will in general be a logarithmic divergence from the operator S.
In (3), two classes of electromagnetic couplings are, in some cases, more convergent than the bilin-
ear models. In the first, the current is proportional to a canonical field, and its curl is the conjugate
momentum. Then both [j0,ji] and [&0j;-Sij0,jk] are c-numbers and there are no divergences. In one
version of this class the current is proportional to the p' field, so that either there is a Yang-Mills
coupling (analyzed below) or there must be additional terms in the current corresponding to the p~, in
which case these bilinear boson terms yield a quadratic divergence. In the second version the canoni-
cal field is a new entity, coupled to what is normally called the electromagnetic current; the mass
shift is then finite, at least to O(n). In the second class, a Yang-Mills coupling is introduced and
is no longer a c number. Here the structure-constant indices include parity, so that, in SU(2) 8 SU(2),
both vector and axial fields enter. The logarithmic divergence is then determined by
S=gm (pI V V +A A:Ip)
p
(6)
and, in particular, the (I=2) mass splitting of a massive, rather than Goldstone, pion is logarithmi-
cally divergent. '
To investigate the general behavior of %&p, we use the Jost-I ehmann representation. ' While the
latter was formulated for the causal commutator, the spectral conditions in this problem imply dis-
joint frequency ranges in the two orderings of the commutator, and so permit one to write it for the
unordered and T* products as well. The invariant functions then read, in the rest frame of pl",
311 . = fdsd'u 4'.(s, lu l)[s+(u-k)2-k 2 —ie]
2 0
where crossing and rotational invariance have been used to eliminate terms odd in k, and angular de-
pendence in 4z. The amplitude%&y, in an arbitrary frame, is then of the kinematical form exhibited
in Eq. (1). Subtractions may in general be required in the%i. However, in that case, we shall see
later that the corresponding ETC are not defined. %e may therefore deal with unsubtracted%i in dis-
cussing the 6m-ETC relation; then the leading divergence is quadratic, the coefficient of fdk' being
proportional to fdsd'u(4, + —,'4, ), which is also the expectation value (p IS Ip) of the ETC when expressed
in terms of the %i. Similarly, if the quadratic divergences vanish then the logarithmic divergence is
determined by fdsd u(-s)(4', + 2%2) which is again (p IR Ip), so the commutators —when they exist —still
determine the leading divergences.
The mass shifts can be calculated in terms of the Jost-Lehmann forms whether or not the ETC ex-
ist; however, the question of a relation between the two can only be meaningful if the latter exist as
distributions in r. The Jost-Lehmann representation may also be used to find the necessary existence
criteria. The conditions are obtained by considering the unordered products F&~(x-iq) —= (P ljI"(x—iq)
xj~(0) Ip) as analytic functions in the forward cone. Then the equal-time commutator is defined as the
1400
VOLUME 20, NUMBER 24 PHYSICAL REVIEW LETTERS 10 JUNE 1968
limit
lim [F (x-iq)-E (-x-ig)], =- (p 1 [j (r), j (0)]lp).
g-0
The requirement that the limit exist as a distribution in r imposes conditions on the Jost-Lehmann
spectral functions. These are most conveniently presented as conditions on the moments gi(n)(s)
-=fd uu ne, (s, u) which are well-defined by virtue of the compact support in u. Explicit calculation of
the small-x~ behavior of I I"~ allows one to show that the e ual-time commutator can exist, in all Lo-
rentz frames, only if the large-s behavior of i!Ii(n)(s) is (i(n (s)-Pi(n)(s)+A(i(n)(s), with Pi(n)(s) an
arbitrary polynomial in s and fo dsh(i(0)(s) (~, sn Ibgi(n)(s)-0 for n)0. If Pi(n)(s) is of higher
order than sn 1, the coefficients of the higher powers must satisfy sets of linear equations. These
conditions, in addition to guaranteeing the existence of the commutator, also guarantee that the com-
mutator transforms under Lorentz transformations as a tensor —of arbitrarily high rank. The coeffi-
cients of the s" 1 term and f, dsgi(0)(s) determine the commutator and, if there are contributions of
gi(n) to the commutator for n -K, the commutator transforms as a (2N)th-rank tensor, in addition to
the fourth-rank transformation properties dictated by the kinematical factors; The conditions also im-
ply that the Jost-Lehmann representation does not require subtractions, hence the existence of the
equal-time commutators implies that the Jost-Lehmann representation is unsubtracted.
The equal-time commutator can, in this case, still be identified as the coefficient of 1/k as k —~;
there is an additional polynomial in k, of arbitrary order, in%I" ~. However, the asymptotic behav-
ior of the Ki is, in general, no longer 1/km; there may be arbitrary terms7 of O(k2n) for n ) 0 and,
further, the terms of O(1/k') will include forms such as (1/k2)(v /k2)" which are responsible for the
higher rank tensor structure discussed above.
The mass shift can now be calculated in terms of the asymptotic behavior of the%&. If the ampli-
tudes are not of O(l/km), there are, in general, terms more than quadratically divergent even when
ETC exist. These divergences do not even have a well-defined coefficient-different methods of evalu-
ation give different weightings for the various terms of the same order. When the mass shift is no
more than quadratically divergent, the amplitude m2 is of the form
(n)
-2( 2/ 2)n
n 2
The contribution to the [j0,j"] commutator is the limit ko —~, i.e. ,
where the Ai(n) are linear combinations of the coefficients of (i(~)(s) for x~n. On the other hand, the
contribution to the mass shift is
=3e (4m) J dk (Q X ((P k) )+ —,'Q A ((P k) -2(P k) )},
where () denotes an angular average in four dimensions,
((P k) )=r(n+-,')[(n+1)!r(-'.)]
Comparison of these two formulas then yields the relation between 6m and the ETC in the most gener-
al case where the latter exist, and 5m is quadratic. The case n=0 reduces, of course, to the form
(f, + 2f, ) derived earlier. The logarithmic divergences become interesting if the above quadratic ones
are absent. In that case, a completely analogous treatment' can be carried through to link the param-
eters (p I [80ji-&ij0, jk]!p) to the logarithmic divergence coefficient.
%e have seen that the divergent parts of the mass shift constitute a probe of the small-distance be-
havior of a theory through the highly model-dependent equal-time commutators which determine it.
Conversely, knowledge of the latter in any model provides a statement of the modifications it will pro-
1401
VOLUME 20, NUMBER 24 PHYSICAL RKVIKW LKTTKRS 10 JUNz 1968
duce in radiative corrections. Detailed statements of the conditions under which the ETC exist have
been provided, as have precise statements of the relation between the latter and the divergences. A
more complete derivation will be given elsewhere.
One of us (D.G. B.) thanks Brandeis University for hospitality while part of this work was performed.
*Work supported by the U. S. Atomic Energy Commission under Contract No. RLO 1388B.
)Work supported by the U. S. Air Force, Office of Aerospace Research, under Office of Scientific Research
Grant No. 368-67.
J. D. Bjorken, Phys. Bev. 148, 1467 (1966).
2R. Jost and H. Lehmann, Nuovo Cimento 5, 1598 (1957).
It is possible, in particular models, for the (00) indices to be degenerate, with C"~~ -g" C~ . In general,
however, the C00i~'s can be components of arbitrarily high rank tensors, C00(p6[gp0. .., as we shall see later. In
the present context we restrict ourselves to foruth-rank structure.
T. D. Lee, S. Weinberg, and B. Zumino, Phys. Rev. Letters 18, 1029 (1967).
~T. D. Lee, to be published.
This result was established independently by the authors tS. Deser, in Lectures of the Summer Institute of the
Niels Bohr Institute, Copenhagen, Denmark, 1967 (to be published)]; M. Halpern and G. Segre, Phys. Rev. Let-
ters 19, 611, 1000(E) (1967); and G. C. Wick and B. Zumino, Phys. Letters 258, 479 (1967).
~These terms are also related to ETC, namely, to the higher-order derivatives of the 4 function which may oc-
cur in the ETC. Throughout, when referring to ETC, we mean the lowest-order derivative term of the commuta-
tor which is allowed to occur, e.g. , [j,j]-V6(r) and [80jt—Btj0,jt]-8(r).
The actual calculations in this ease require generalization of the conditions given in the quadratic case for the
asymptotic behavior of the gt(")(s). The relevant behavior of the SRt is k (-p2/k2)" in this case. It is amusing
that in some of our models, including algebra of fields (with Yang-Mills ETC), JRi has terms k (v2/k2) which
cancel parts of 5K2, exemplifying the possibility of nonkinematieal structures of the type we have discussed.
MISUSES OF THE FINITE-ENERGY SUM RULES*
Jeffrey E. Mandula and Richard C. Slansky
California Institute of Technology, Pasadena, California
(Received 22 April 1968)
It is shown that any model in which the scattering amplitude is given by finitely spaced
trajectories of direct-channel resonances does not yield Regge asymptotic behavior in
the direct channel. Several difficulties associated with the use of Regge asymptotic be-
havior at low values of the energy through the finite-energy sum rules are noted.
(I) The finite-energy sum rules (FESR),'~' or
generalized superconvergence relations, relate a
finite integral of the imaginary part of a scatter-
ing amplitude to the crossed-channel Regge poles
controlling the high-energy scattering. The
FESR are easily derived from the superconver-
gence of A (t, s)-Q;8;(t, s), the amplitude minus
the necessary number of leading Regge trajecto-
ries. The zeroth-moment sum rule for the scat-
tering of equal-mass scalar particles is
f ds' Im—A(t, s') =QP. (t)ss, , nt(t) + 17r l
Certain factors have been absorbed into Pt(t).
The absolute value of the difference of the two
sides of Eq. (1) goes to zero in the limit s- ~.
Note that for t ~0 the integral traverses part of
the s-channel physical region.
In practice the FESR are used in two distinct
ways. For very large s they are used to deter-
mine the parameters of a representation for
ImA(t, s) (other than the Regge representation),
which is assumed to be valid over the entire
range of integration. For values of s currently
accessible experimentally, they are used to ob-
tain information about the Regge parameters
from the low-energy data.
The most obvious and simple parametrization
of ImA(t, s') when s is large is a sum of direct-
channel poles. Such a model is particularly at-
tractive if the Regge trajectories rise indefinite-
ly, since one can then identify the Regge poles
with the direct-channel resonances, and the
1402


Electromagnetic Mass Shifts, Equal-Time Commutators, and Jost-Lehmann Representation

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Electromagnetic Mass Shifts, Equal-Time Commutators, and Jost-Lehmann Representation

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