Alien Calculus and non perturbative effects in Quantum Field Theory

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by  Marc P. Bellon

Alien Calculus and non perturbative effects in Quantum Field Theory
Marc P. Bellon
Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7589, LPTHE, 75005, Paris, France and
CNRS, UMR 7589, LPTHE, 75005, Paris, France
In many domains of physics, methods are needed to deal with non-perturbative aspects. I want
heretoargue thatagood approach istowork on theBorel transforms ofthequantitiesof interest,
the singularities of which give non-perturbative contributions. These singularities in many cases
can be largely determined by using the alien calculus developed by Jean E´calle. My main example
will be the two point function of a massless theory given as a solution of a renormalization group
equation.
7 I. INTRODUCTION formed Green functions. This presentation is based
1 on work done for the supersymmetric Wess–Zumino
0 model [3], which are presented in the contribution of
2 AtECTinTrento,mathematiciansandphysicistsmet
Pierre Clavier in this volume. Our aim here is to show
forthesecondtimeinSeptember2014toexchangeabout
n howgeneralthesemethodscanbeandgivehintsonwhat
theirvisiononSchwinger–Dysonequations. I,asamath-
a results could be obtained in the future.
J ematical physicist, did not really know where to sit, but
enjoyedthejuxtapositionofconcretproblemsandsophis-
9
ticated tools. I was challenged to think a bigger picture
II. RENORMALIZATION GROUP
] for the use of a tool I recently learned, alien calculus.
h
Hereis anexpandedandmaturedversionofwhatItried
t
- to explain to the participants. It has been widely recognized that the renormaliza-
p Alien calculus is a technique initiated by Jean E´calle tion group is a major feature of quantum field theory.
e
It corresponds to a breaking of the scale invariance in a
h to study functions whichhavenonconvergentTaylorex-
classicallyscaleinvarianttheoryandisattheheartofthe
[ pansions. By providing tools for the study of the singu-
application of quantum field theory to critical phenom-
larities of Borel transforms, it allows to compute or at
1
ena in statistical mechanics as well as the application
leastestimate the changethe Borelsum incurs when the
v
of QCD to the study of high energy hadronic interac-
4 integrationpathcrossesalineofsingularities. Thissheds
tions. However, these applications are in fact limited to
9 new lights for example on the Stokes phenomenon. This
2 allowsalsotogiveanewkindofexpansions,calledresur- ratherspecialcasesofthe renormalizationgroup. Insta-
2 gent expansions, with explicit non-perturbative terms tisticalmechanics,one considersfixedpoints wherescale
0 proportional to exp(−f/a), which have all their deriva- invariance is recovered, eventually with anomalous di-
. mensions, and for the applications to QCD and particle
1 tives at the origin vanishing. Singularities of the Borel
0 transform on the positive axis, once viewed as liabilities physicsingeneral,thescaleswherequalitativelydifferent
7 phenomena arise,e.g.,the mass scale ofhadrons,remain
introducing ambiguities in the summation process, be-
1 inaccessible. However, if one wants to use the two-point
come tools for a better understanding of the sum. The
v: basic references are the works of E´calle [1], but a gentler functions defined through the renormalization group in
i introduction can be found in [2]. somegraphstoself-consistentlycomputerenormalization
X
groupfunctions,onewouldneedtogetresultsfortheen-
Thepossibilityofcomputingnonperturbativetermsis
r tire kinematical space.
a of tremendous interest in theories like QCD, since they
Toseethenatureoftheproblem,letuswritetherenor-
are the only ones which can bridge physical and renor-
malization group for the ratio of a generic propagator
malizationmass scales. In quantum mechanics,non per-
to its free version G(a,L), depending on a coupling a
turbative contributions important for the computation
and the logarithm of the invariant p2 of the momentum
of tunneling rates can be deduced from the action of so-
referred to a scale µ2, L = log(p2/µ2). This equation
lutions where the barrier is crossed in imaginary time,
involvestheanomalousdimensionγ andthescaledepen-
but the equivalent computation for a classically scale in-
dence of the coupling β:
variant theory as QCD is not clear, since the action of
theclassicalinstantondependsonthecouplingconstant:
∂
which value for the gauge coupling to choose, when the G(a,L)=(γ(a)+β(a)a∂ )G(a,L). (1)
a
∂L
renormalization group makes it dependent on the scale?
Aliencalculusgivesanalternativeapproachtodetermine It is easy to use such an equation to obtain the expan-
where the singularities of the Borel transform are and sion of G with respect to L to any finite degree, but
what are their properties, and it does not present such higher degree terms will involvehigher derivatives of the
ambiguities. functions γ and β, except in the case of a fixed point a
0
Iwillillustratethesegeneralconsiderationsbyastudy where β(a ) vanishes or in the extended supersymmetry
0
of the renormalization group equation for Borel trans- cases where the renormalization group functions reduce
2
to their lowerorder part. In more realistic theories, only but now, the β Ξ term gives an exponential as the or-
0
a few terms of the perturbative expansions of the renor- der 0 term. Using the renormalization condition that
malization group functions are known, we can presume G(a,0) = 1, or equivalently, Gˆ(ξ,0) = 0, we get for the
that these perturbative expansions are always divergent order 0 solution
and having the functional dependence of G on L seems
γˆ(ξ)
out of reach. Gˆ (ξ,L)= eβ0ξL−1 . (6)
0
β ξ
0 (cid:0) (cid:1)
III. BOREL PLANE FORMULATION Introducing this order zero solution in the convolution
products of the renormalization group equation (5) will
give successive approximations which will have in com-
A solutionis neverthelesspossible if we consider Borel
mon to be superpositions of those exponentials. The
transforms of all these functions. Indeed, it is desirable
proper parameterizationof this superpositionis however
toconsidertheBoreltransforms,whichareprobablywell
non trivial, since there are singular contributions from
definedasanalyticfunctions inthe vicinity ofthe origin,
theendpointsoftheinterval0andξ. Convolutionprod-
insteadofthepurelyformaloriginalpowerseries. Atthe
uctscanalsobedefinedbyintegratingoverapathdiffer-
level of power series, the Borel transform is defined by:
entfromthestraightlineandthisisimportanttoprevent
∞ ∞ spurioussingularitiestoappearwhenonedefinesanalytic
f
f(a)= fnan+1 −→ fˆ(ξ)= nξn. (2) extensions. It is therefore not clear how to obtain an ef-
n!
nX=0 nX=0 ficient representationfor Gˆ. Nevertheless, eβ0ξL is p2/µ2
tothepowerβ ξ. Thismeansthatlargerpowersofp2will
The Boreltransformationtransformspointwise products 0
appear in the representation of the propagator the fur-
offunctionsintoconvolutionproductsandequationslike
ther we go from the origin. Potentially, this means that
therenormalizationgroupequation(1)canbetranslated
all sorts of new divergences could appear when looking
to an equation for the Borel transforms. In its simplest
for the analytic extension of Gˆ which is necessary to go
version,the Boreltransforminvolvesashiftofdegree,so
that we isolate the constant part in G, so that Gˆ is in backtothepropagatorGintermsofthecouplinga. The
fact the Borel transform of G−1 viewed as a function nature of the divergences depends fundamentally on the
signofβ , making a cleardistinction betweenasymptot-
of the variable a and the degree counting operator a∂ 0
a
becomes ∂ ξ, which we write as ∇Ξ, with Ξ the opera- ically free theories and the other ones: in asymptotically
ξ
freetheorieswithβ negative,weobtainnegativepowers
tor of multiplication by the variable in the Borel plane. 0
Noting Gˆ(·,L) for the function evaluating to Gˆ(ξ,L) at ofp2 forthepositivevaluesofξ necessarytoevaluatethe
propagator for physical values of the coupling a and so
ξ, equation (1) becomes
we do not have new ultraviolet divergences,but infrared
∂ ones. The situation is reversedif β is positive.
Gˆ(·,L)=γˆ+γˆ⋆Gˆ(·,L)+βˆ⋆∇ΞGˆ(·,L) (3) 0
∂L
Atthispoint,onemaythinkthatnothingchanged,since
IV. PARAMETERIZATION OF THE
there still is a derivative acting on Gˆ. However the con- PROPAGATOR
volutionproductis defined throughanintegralonwhich
anintegrationby parttransformationcanbe performed.
Nevertheless a good parameterization of the propaga-
ξ tor is possible as a contour integral [3]:
f ⋆g(ξ)= f(η)g(ξ−η)dη
Z
0 dζ
Gˆ(ξ,L)= f(ξ,ζ)eζL (7)
so that IC ζ
ξ
f ⋆g′ =f′⋆g+f(0)g−g(0)f. (4) with C any contour enclosing 0 and β ξ. On a contour
ξ 0
Ifwe denote byβ andβ the firsttwocoefficientsofthe minimally including the endpoints, the jump of f along
0 1
β function, so that a cut from 0 to β0ξ gives a smooth integral, while the
singularities at the end points will contribute singular
β(a)=β a+β a2+··· , βˆ(ξ)=β +β ξ+··· , terms. The condition that Gˆ(ξ,0) is zero is also easily
0 1 0 1
obtainedinthisformalism,sincetheexponentialbecomes
therenormalizationgroupequationintheBorelplane(3)
1 for L=0 and the contour can be expanded to infinity.
becomes
It is therefore sufficient that f have limit 0 at infinity.
∂ The renormalization group equation for Gˆ (5) becomes
−β Ξ Gˆ(·,L)=γˆ+γˆ⋆Gˆ(·,L)+βˆ′⋆ΞGˆ(·,L). (5)
0 anequationonf,sinceonecanusethe samecontourfor
(cid:18)∂L (cid:19)
thecomputationofGˆ forallthenecessaryvaluesofηand
In the first form of the renormalization group equa- then, switching the orderof the contour integraland the
tion (3), treating the convolution products as pertur- other operations, one can write everything as a contour
bations gives a solution for Gˆ as a power series in L, integralonacommonpath,sinceeventheLindependent
3
term can be giventhe form of a similar contour integral, allows to extract the singular part of a function around
using that the point ξ and becomes a derivation with respect to
0
theconvolutionproductifonetakescertainweightedav-
dζ
1= eζL . erages of the analytic continuations along the different
ICξ ζ classes of paths. Applying this operator ∆ξ0, called an
alien derivative by E´calle, to a system of equations sat-
The definition of the contour integral includes a fac-
isfied by the propagator and the renormalization group
tor 1/(2πi) to simplify notations. ThHe annulation of
functionsthereforeproducesalinearsystemofequations
the integrand is certainly a sufficient condition for the
for the alien derivatives of these functions. For generic
renormalization group equation (5). One ends up with
values of ξ , the only possible solution for this system
0
the following equation for the functions f defined by
ζ of equations is zero. This means that ξ cannot be the
0
f (ξ)=f(ξ,ζ):
ζ position of a singularity of the solution unless ξ is the
0
(ζ−β Ξ)f =γˆ+γˆ⋆f +βˆ′⋆Ξf (8) sum of the positions of other singularities[7].
0 ζ ζ ζ
In the case of equation (8), the right hand side, which
Wecanexpandthisequationinpowersofζ−1 andobtain involves convolution products, will become less singu-
anexpansionwhichexactlycorrespondstotheexpansion lar than ∆ξ0fζ, so that a singularity is possible only if
ofGinpowersofLbyevaluatingthecontourintegralby ζ−β0ξ0 vanishes to allow the singularity degree of both
its residue at infinity. This however would defeat the sides to match. For generic values of ζ, such a singular-
purpose of this representation of the propagator which ity has no particular effects on the solution of the the-
was precisely to avoid an expansion in power of L. This ory, since one can always shift the integration contours
however shows that equation (8) is necessary, since as a to find an expression of the results which does not de-
function of ζ, it is holomorphic in a neighborhood of ∞ pend on this particular fζ. However the contributions
and has all of its Taylor coefficients 0. of the residues of the poles which must be crossed when
Exchanging again the order of integrations, the evalu- extendingthe integrationcontourwillbeproportionalto
ationofthe graphsappearingin Schwinger–Dysonequa- f−k for some integer k in the case of infrared poles. In
tionswiththemodifiedpropagatorsamounttofirsteval- this way, singularities will therefore also appear in the
uatethemwithsomepropagatorsmodifiedwiththepow- Borel transforms of the renormalization group functions
ersζ ofthe momenta,the Mellintransformofthe graph. and the propagator for ξ = −k/β0. If β0 is negative,
The perturbativesolutionofthe Schwinger–Dysonequa- thesesingularitieswillbeforpositiverealvaluesofξ and
tions only use the derivatives of this Mellin transform at will allow us to compute non-perturbative contributions
the origin, but analytic continuation of the Borel trans- to these quantities, proportional to e1/β0a and its pow-
form depends on its value at large. For small enough ζ, ers. Indeed, in Ecalle’s theory of summation, links exist
graphs will depend holomorphically on ζ: this is known between alien calculus and transseries, through the free-
sincealongtimebecausesuchmodificationsoftheprop- domin the definition intransmonomialswhenonly their
agatorsareattheheartofanalyticalregularization[4,5], values on a small sector are considered. The whole pro-
which had its limelight moment before being supplanted cedureofresummationhasbeendetailedinthe book[8],
by dimensional regularization. Indeed, holomorphy in but exploiting it would largely outstrip this elementary
thedomainwherethediagramisconvergentisquitenat- presentation.
ural since the integrand depends holomorphically on the Goingfromthis rapidsketchtoa fully consistenteval-
ζ’s. However,for larger values of ζ, new divergences can uationoftheseeffectswillnotbesimple,butweareconfi-
appear but the Mellin transformcan be shownto have a dent that an analytic approachto non-perturbativephe-
meromorphic extension with simple poles [6], in partic- nomena in quantum field theories can be obtained along
ular, infrared poles will appear for any negative integer theselinesandprovidenew insightsonsuchquestionsas
value for ζ. We therefore keep the independence on the the confinement of color and the generation of mass in
precise formofthe integrationcontour thatis important QCD, fulfilling the intuitions of Manfred Stingl [9]. The
for the validity of the representation(7), but when shift- singularitiesat−k/β shouldthereforetranslateinterms
0
ing the integration contour Cξ to accommodate a larger proportional to (Λ2QCD/p2)k which look like very bad
value of ξ, the crossing of these poles adds terms pro- news for the infrared behavior of the propagator. How-
portional to f−k which will contribute to the γˆ and βˆ ever, such singularities have contribution from ∆−k/β0
functions. but also from the effect of all products of alien deriva-
tives such that the sum of the arguments is −k/β , in
0
V. ALIEN CALCULUS AND THE particular the k-th power of ∆−1/β0. If this last term is
dominant, one should be able to see all these terms as
SINGULARITIES OF THE BOREL TRANSFORM
the expansion of a massive propagator, with a squared
mass proportional to Λ2 . To control the infrared be-
QCD
The alien calculus introduced by E´calle allows us to havior of our theory, we have gone to a representation
show that f has a singularity for ξ =ζ/β , with a com- withevenworseinfraredbehaviorbut whichcanbe seen
ζ 0
putableexponent. Indeed,thereisanoperator∆ which as the badly behaving expansion of a perfectly fine the-
ξ0
4
ory. However, the bad infrared behavior is unavoidable ishingquarkmasses. Thelinkwiththefundamentalscale
for the Borel transform which is a necessary step in the Λ2 wouldbehardertogetsinceitrequirescomparing
QCD
resolution of this conundrum since infrared divergences the actual terms in the perturbative coefficients of the
feed the singularities of the Borel transform which pro- β-function and the asymptotic behavior that can be de-
duce these non-perturbative terms which finally can be duced from the singularities of the Borel transform. As
interpreted as mass terms. in [10], some aspects of the computation of the possible
forms of the alien derivatives could be easier when using
Since the mass terms should all be linked to the same a formal multiplicative model of transseries, but the fi-
alien derivative of the β-function, ratio of masses of nal interpretation is better seen at the level of the Borel
hadrons should be computable in the chiral limit of van- transform where we deal with actual functions.
[1] J. Ecalle, Les fonctions r´esurgentes, Vol.1 (Pub. Math. [7] Convolutionproductshavegenericallysingularitiesatthe
Orsay,1981). positions of the singularities of each factors, but also at
[2] D. Sauzin, “Introduction to 1-summability and resur- thesumofthepositions,atleastinsomeofthesheetsof
gence,” (2014), arXiv:1405.0356v1 [math.DS]. the Riemann surface on which the convolution product
[3] M. P. Bellon and P. J. Clavier, is defined.
Lett.Math. Phys.105 (2015), 10.1007/s11005-015-0761-2, [8] J. Ecalle, Introduction aux fonctions analysables et
arXiv:1411.7190 [math-ph]. preuve constructive de laconjecture de Dulac (Hermann,
[4] E. R.Speer, J. Math. Phys.9, 1404 (1969). 1992).
[5] E. R.Speer, Commun. Math. Phys. 23, 23 (1971). [9] M. Stingl, “Field-theory amplitudes as resurgent func-
[6] M. Bellon and F. A. Schapos- tions,” (2002), arXiv:hep-ph/0207349.
nik, Lett. Math. Phys.103, 881 (2013), [10] M. P. Bellon and P. J. Clavier,
arXiv:1205.0022 [hep-th]. Lett. Math. Phys. 104, 1 (2014),
arXiv:1311.1160v2 [hep-th].