F.V.Tkachov, talk at QFTHEP'99 on PT with unstable fields 1





On the structure of systematic perturbation
theory with unstable fields1

Contribution to the XIII Int. Workshop on Quantum Field Theory and High Energy Physics
(QFTHEP'99, 27 May  2 June 1999, Moscow)


F. V. Tkachov a

Institute for Nuclear Research of Russian Academy of Sciences
60th October Ave. 7a, Moscow 117312, Russia

a ftkachov@ms2.inr.ac.ru


Abstract. Discussed is the structure of non-trivial counterterms that occur in the systematic gauge-
invariant perturbation theory with unstable fields introduced in .



I would like to discuss the structure of the systematic develop highly efficient calculational algorithms geared
perturbation theory for models with unstable fundamental towards automated large-scale calculations5.
fields (sysPT) proposed in [1]. The discussion below is The sysPT implements the same logic: it is based on the
intended to complement [1] to which I will freely refer. same powerful method of AO6, and it yields the expansions
Some general warnings are in order. (now the expansion parameter is the coupling) in a maximally
The industry of PT calculations for LEP etc. has been simple form. The form is somewhat unconventional (VP-
dealing with unstable particles in a more or less successful distributions are involved) but is otherwise rather simple. This,
fashion for some time [2] (see also refs. in [1]) -- but it has one may hope, provides a starting point for developing more
also been encountering difficulties (most notably, a lack of efficient calculational procedures than the currently employed.
gauge invariance; see e.g. [3]). It is clear that to avoid groping Now on to the sysPT.
in the dark one should have a clear formulaic understanding of The key difficulty in the construction of PT with unstable
the nature of the difficulties being encountered. fields is that the formal expansion of amplitudes in powers of
the coupling yields singular expressions which, when squared
The purpose of sysPT is to clarify in a system atic m anner to form probabilities, result in expressions that are non-
the m athem atical nature of the weak-coupling lim it in integrable at the zero-coupling mass shell of unstable particles.
situations with unstable fields. But integrability is a key physical requirement (taking into
account, say, initial state radiation amounts to a convolution
Just how, exactly, such an understanding may lead to with suitably chosen kernels [2]).
improved calculations for LEP2 etc., I don't care at this point;
I'm just happy to have obtained the understanding.1 All conventional approaches to resolution of this difficulty
abandon the idea of a complete expansion in powers of the
Some physicists may be uncomfortable with the adjective coupling: self-energies are left in the denominators, and then
"mathematical" above. But we are dealing here with the one performs a more or less sophisticated massaging of the
structures encountered in higher-order corrections and relevant
arXiv: 21 Jan 2000 resulting expressions aimed at obtaining manageable gauge-
for precision calculations, and the second and higher decimal invariant theoretical predictions.
digits in theoretical predictions cannot be accounted for by
simple visualizations. So the kind of intuition needed to The drawback of such an approach is that it involves huge
understand the sysPT concerns the structure of the formalism amounts of hand work, and there is an opinion rather
and therefore requires a (non-standard) mathematical unambiguously expressed by some experts involved in the
expression. LEP1 calculations [2] that it is impossible with the old
techniques to accomplish anything similar to the complete one-
Also recall the industry of multiloop calculations that loop level LEP1 calculations for LEP2 (at LEP2 one deals with
emerged thanks to the clarification of the short-distance and O
mass expansions2 within the framework of the method of (104) one-loop diagrams [9]). It is clear that what one needs
asymptotic operation3. The calculational efficiency is due to the
fact that the method of AO yields expansions in a maximally are essentially unique [4].
simple form, i.e. the resulting expansions run in pure powers 5
and logarithms of the expansion parameter4. This allows to With short-distance and mass expansions, it is the celebrated integration-
by-parts algorithms [7].
6
1 The required extension of the Euclidean AO to arbitrary asymptotic
Some related materials can be found at regimes in Minkowski space was achieved in [8], thus yielding a much
http://www.inr.ac.ru/~ftkachov/projects/unstable/index.htm. sought solution of the general asymptotic expansion problem in
2 For a review see [4], [5]. perturbative QFT. The recipes of [8] are valid for both loop and phase-
3 For a review and references to the Euclidean variant of the theory of AO space integrals because the distribution-theoretic nature of AO makes it
see [4]. insensitive to whether the individual factors are ordinary propagators or
4 The so-called property of perfectness [6]; also note that such expansions phase-space -functions.


F.V.Tkachov, talk at QFTHEP'99 on PT with unstable fields 2





here is a very systematic, "mechanical" approach that would book-keeping has been lacking.
allow a high level of automation. (iv) Finally, there is nothing mysterious about the anomalous
The sysPT offers a new route to this goal. One observes that -functional terms: the simplest one has a clear physical
the difficulties of the conventional approaches reduce, from meaning, and the higher ones are simply corrections that form
mathematical viewpoint, to a non-commutativity of the a more or less regular pattern.
perturbative expansion in powers of the coupling and the For the squared propagator (the l.h.s. of 0.2) the expansion
operation of squaring of amplitudes to obtain observed problem reduces to a one-dimensional expansion problem
probabilities. Once this is realized, it is natural to attempt to which generalizes 0.1, and many felt that corrections to Eq. 0.2
obtain an expansion directly for probabilities. are bound to contain derivatives of the -function. The method
First one recalls the elementary formula (x is a real of AO (in particular, the secondary so-called "homogenization"
argument and is small): expansion [8]) make obtaining such expansions a mechanical
1 1 procedure.
= (x) + O 1
( ) . 0.1
2 2
x + The expansion 0.2 can be pushed to all orders in g : see [1]
0
for explicit expressions. For the purposes of illustration I show
This is easily verified e.g. by explicit integrations in infinite here only the O(1) term:
limits with rational functions that do not have zeros near x = 0 -
2 2 2 2 2 2 2
n VP c M - Q h + Z c M - Q h - Z 'cM - Q h , 0.4
such as bc + -
i xg . 1 2

From Eq. 0.1 one obtains where Z i can be explicitly expressed in terms of the
2 2
1 2 coefficients of the Taylor expansion of (Q2 ) at Q = M .
2 2 2 2
M - Q - g (Q ) Eq. 0.4 is to be compared with the nave expression
obtained by squaring the tree-level unstable particle's
1
2
- 2 2
= g M
e - Q j + O 1
( ) . 0.2 propagator:
g0 2
Im ( M )
M 2 Q2 2
- -
c h . 0.5
Here M is the Lagrangian mass of the unstable particle (call it
2 2
X ), is its one-loop self-energy (Dyson-resummed into This has a non-integrable singularity at Q = M .
denominators), and g is the coupling responsible for the We see that the correct expression (Eq. 0.4) differs from the
instability. nave one (Eq. 0.5) in two respects:
Eq. 0.2 is nothing but a well-known relation in disguise: (i) In the correct expansion, the non-integrable singularity of
(q q X l l )  0.5 regulated by the VP prescription (see [1] for a definition).
1 2 1 2 (q q X ) Br( X l l )
1 2 1 2 . 0.3
This only affects the nave expression 0.5 exactly at Q2 M 2
= .
Indeed, the -function on the r.h.s. of 0.2 describes free This may seem unusual but such things are common in the
propagation of X in the final state. So Eqs. 0.2 and 0.3 only theory of distributions: an integrable distribution can be
describe the fact that the unstable particle becomes stable in obtained from a non-integrable singular function by modifying
the zero-coupling limit. This is a fundamental boundary the latter only exactly at the point of singularity.
condition for any systematic expansion of probabilities in
powers of the coupling. We see: (ii) On top of the VP prescription, there are -functional
counterterms. The presence of the first derivative of the -
(i) The exact expansion is bound to contain anomalous - function correlates with the fact that the singularity of 0.5 is
functional terms. The simplest such term (the one shown on linear by power counting.
the r.h.s. of 0.2) has a fundamental physical meaning. The described pattern (an intermediate regularization plus a
(ii) The -functional terms spoil the nave counting of powers fine-tuning by -functional "counterterms") is very general and
of the coupling. Indeed, the l.h.s. of 0.2 corresponds to tree + occurs systematically in the theory of AO. The first time this
one-loop approximation and one would normally expect the pattern occurred in the theory of UV renormalization by
r.h.s. to contain O (1) and O (g 2) terms. However, the Bogolyubov [10], and it is instructive to compare the structure
coefficient of -function contains an O(g -2) contribution from of 0.4 with the R-operation.
the one-loop of self-energy. Because only the imaginary part The logical structure of the reasoning that led to sysPT
are involved, this is equivalent to the total width of the decay essentially follows Bogolyubov's theory of UV renormalization
of X in the lowest order (the decay vertex is taken in the tree [10]. In both cases:
approximation, and the cut loop comes from phase space).
(i) The difficulty is traced back to an incorrect formal
(iii) It is clear that if one includes two-loop O (g 4) terms into manipulation which ignores the generalized nature of the
the self-energy ( g2 g2 loop + g4
1- 2-loop ) then it is safe mathematical objects involved. In the case of UV divergences,
to Taylor-expand in all occurrences of g except the one it is the formal multiplication of singular functions
corresponding to the one-loop self-energy. The two-loop (propagators in chronological products). In the case of unstable
contributions to self-energy contribute at the level O (1) in the particles, it is an unjustified expansion of amplitudes into a
r.h.s. Since only imaginary parts will contribute, this series in powers of the coupling prior to squaring.
corresponds to one-loop total width (one-loop decay vertices (ii) The final result is required to be a locally integrable
and another loop from phase space of decay products). This distribution (in the space of coordinates in the case of R-
anomaly of power counting for the coupling has long been felt operation, and in the aggregate space of loop and phase space
to occur (cf. the discussion in [3]) but a systematic method of momenta in the case of sysPT) and it is observed that to fix its


F.V.Tkachov, talk at QFTHEP'99 on PT with unstable fields 3





structure at the point of singularity, one needs a special The fat lines correspond to unstable propagators (forming
procedure. chains in the sense of [1]), there is a "photon" line, and the
(iii) The final answer has a characteristic form "nave blobs correspond to subgraphs which do not belong to the
expression + regularization + counterterms". The counterterms singular configuration. Note that the singular subgraphs need
are proportional to -functions (and their derivatives) localized not be connected subgraphs in the usual graph-theoretic sense.
exactly at the points of singularity, and the number of At this point to avoid confusion I would like to emphasize
derivatives of -functions is determined by a power counting. that the diagrammatic images that emerge here are entirely
There are also differences: determined by the underlying analytical structures, so there
need not be any direct correspondence with, say, singular
 UV singularities are localized on flat manifolds (linear subgraphs and any standard graph-theoretic notions invented a
subspaces) in the space of coordinates whereas the priori.
singularities due to unstable particles are localized on non- The following figure represents some more families of
linear manifolds (the mass shell Q2 M 2
= in the simplest case singular configurations; each diagram depicts only one member
of 0.5). of a family, with the total number of the diagrams shown as a
 The finite parts of the UV counterterms of the R-operation factor:
are arbitrary whereas the coefficients of the -functional  4  4
counterterms of sysPT are uniquely fixed: they are explicitly
expressed in terms of special integrals obtained via the so-
called consistency conditions of AO supplemented by the
homogenization procedure (a special secondary expansion
designed to yield an expansion of a purely power-and-log type).  4  2 (final+initial)

Now the expansion of the squared propagator 0.2 is far from
the whole story. The point is, the singularities due to
instabilities (i.e. the singularities of the expressions 0.5 and
0.4) may interact with the singularities of other factors in the 8  4
diagram such as photon, gluon, etc. propagators. This leads to
more complex singular configurations that also require addition
of the corresponding counterterms. The method of AO [4], [8]
offers a systematic way to determine such counterterms. 8
In the remaining part of this talk I would like to discuss the
structure of such counterterms.
0.7
The simplest example (a configuration with one photon line
attached by both ends to the unstable propagator) was
discussed in [1]. The observed features: the coefficient of the The thin solid lines correspond to stable "fermions" (the
corresponding -functional counterterm contains a logarithm of particle's spine is irrelevant because AO commutes with
the coupling; such logarithms cancel out in the sum over the multiplication by polynomials so the numerators may be
corresponding gauge-invariant subset of diagrams; this ignored).
cancellation was traced to a mechanism completely similar to Now I would like to present and discuss an explicit
that behind the well-known cancellations of IR singularities in expression for the counterterm for one of the configurations.
QED. The configuration is as follows:
In more complex cases the singular configurations of Q Q - k Q -k
propagators that require introduction of non-trivial counter- 1 1 1

terms are grouped into families, with the diagrams within a
family differing by the points on the chains of (unstable) k
propagators to which massless propagators are attached (cf.
gauge-invariant subsets of diagrams in QED). An example of Q Q + 0.8
2 Q k
such a family is as follows: 2 2

This corresponds to two chains of unstable propagators with
one massless exchange (imagine a W+W- pair exchanging a
photon; configurations with two massless exchanges may
correspond to gluons). For simplicity we consider the case
when the masses of the two unstable chains are equal.

The explicit expression for the product of propagators prior
to expansion in self-energies is as follows:

( ~ ~ 2
W W k
1
; ) (1
; ) (
- 2
; )
(
+ 2
; ) ( )
+ . 0.9

The notations are fixed by the correspondence with Fig. 0.8.
0.6 For instance, 2 2
= M - Q
1 1 , etc. Following [1], we denote
= g2 , and for the unstable propagator we write


F.V.Tkachov, talk at QFTHEP'99 on PT with unstable fields 4






b
; g =1 - h
( ) - i f
( ) , where h and f are the real where
5
and imaginary parts of the one-loop self-energy. We will also 1
A+ = -
be using the following notations: 2E 2 2 , 0.21
f M 2
0

h h(n)
b g f f (n
= = )
,
b g . 0.10 i L 4
NLO F 1 O
n n
= ~
0 =0 A M
f c
- = - - + 2 . 0.22
2 0 0 2 P
2Q M 2 f M HG I
KJ
N 0 QP
~
= + lnb
4 g + 2ln
c f Mh
0 . 0.11
The conclusions are as follows.

E = (Q + Q )2 , Q = E 2 - M 2
2 . (i) The number of non-trivial counterterms required for
1 2 0.12
writing out explicit expressions for probabilities within the
The expansion corresponding to the product 0.9 is as framework of sysPT is fairly large although finite -- O (10)
follows: rather than O (100) for a typical process with a single massless
b exchange.
Taylor, VPg + c
( )
( ) (k
0 1 2 )
(ii) Calculation of the coefficients of counterterms is quite
-
c 1 2

( )
( ) ( )
( )
( ) cumbersome but entirely straightforward and presents no
1 k
1 2 - c1 1 (k
2 )
difficulties.
- 
c
( )
( )
1 1 2 (k ) + O 1
 ( ). 0.13 (iii) There occur significant cancellations of logarithmic and
dilogarithmic contributions (closely related to cancellations of
The first term is the formal Taylor expansion with the products soft singularities in QED) -- and these can be observed prior
of unstable propagators regulated by VP prescriptions. to the cumbersome explicit evaluation of the integrals.
The explicit expressions for the coefficients c are as
follows. The most cumbersome expression is for the coefficient It may be useful to have a direct all-order verification of the
of the -function without derivatives: cancellations which ensure gauge invariance without relying on
the indirect arguments of [1] but this is not really needed in
c = cNLO + cNNLO
0 0 0 , 0.14 practical calculations.

4
LF 1 O A c k n o wl ed g m en t s . The presented explicit results are
cNLO = - ~ A B
0 , 0.15
2
f 2
4 E NMHG I
KJ + QP based on the calculations performed in collaboration with
0 M.Nekrasov [11]. This work was supported in part by the
L h + i f O Russian Foundation for Basic Research (grant 99-02-18365).
cNNLO 2 2
M h h f f 1- 0 0 P cNLO
= - +
0 1 0 1 0
M E 2 0
4
N QP References
4
Lh + if F 1 h f
~ + h f O
0 0 1 0 0 1
+ M - AP , 0.16 [1] F.V. Tkachov, Perturbation theory with unstable
2 f 2 E M E
2 M 2 HG I
KJ - f P
0 N 0 Q fundamental fields, in: Proc. XXIII PNPI Winter School
(812 February, 1999, Gatchina), eds.: Ya.I. Azimov et
where al., PNPI, Gatchina, 1999 .
1 E Q 1 L F 2Q I F 2Q IO [2] D. Bardin and G. Passarino, Precision calculations in the
A = +
ln , B = MLi Li P . 0.17 Standard Model, Oxford Univ. Press, 1999;
Q E - Q Q M HG E + QKJ - -
HG E - QKJ
N QP G. Montagna, O. Nicrosini, and F. Piccinini, Riv. Nuovo
Both A and B are expressed via simple 1-dimensional Cim. (1998) .
integrals, and in fact the efforts invested into the obtaining of [3] S.Dittmaier, preprint CERN-TH/97-302 (1997) [hep-
expressions such as 0.17 are usually wasted because such .
logarithmic and dilogarithmic contributions tend to cancel in [4] F.V. Tkachov, Phys. Part. Nucl. 25 (1994) 649 [hep-
sums over entire families of singular configurations, and such .
cancellations can be observed already at the stage of 1- [5] M. Samuel and L. Surguladze, Rev. Mod. Phys. 68 (1996)
dimensional integrals. The reasons behind the cancellations are 259; J.H. Khn, in: Cracow Int. Symp. on Radiative
the same as discussed in [1]. Corrections, Cracow, Poland, 1996 .
[6] F.V.Tkachov, Phys. Lett. 124B (1983) 212.
The poles in are a result of the use of dimensional [7] F.V.Tkachov, Phys. Lett. 100B (1981) 65;
regularization (VP prescriptions alone are insufficient), and the F.V.Tkachov, Nucl. Instr. Meth. Phys. Res. A389 (1996)
cancellation of (di)logarithms follows the pattern of 309 .
cancellation of such poles. [8] F.V. Tkachov, Phys. Lett. B412 (1997) 350 [hep-
For the coefficients of -functions with derivatives one has: .

4
[9] G. Altarelli, T. Sjstrand and F. Zwirner (Eds.), Physics
NLO L F 1 ~ IO
c 1 =
ch + if hc + - i - +
1 0 0 0 2 ,
0.18 at LEP2, CERN, 1996.
f M 2 H
NM KQP
0 [10] N.N. Bogoliubov, Doklady USSR Acad. Sci. 82 (1952)

c 2 = ch + i f h cNLO 217; N.N. Bogoliubov and D.V. Shirkov, Introduction to
1 0 0 0 , 0.19
the Theory of Quantized Fields. 4th ed., NAUKA, 1984
    
c = Q
e + Q A
j + Q
e - Q A
j (in Russian).
1 1 2 + 1 2 - , 0.20
[11] M.L. Nekrasov and F.V. Tkachov, in preparation.



