Even More Perturb at If Qcd

8/6/2019 Even More Perturb at If Qcd

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Even More Perturbative QCD ...

Lorenzo Magnea

Universita di Torino

I.N.F.N. Torino

[email protected]

XI Seminario Nazionale di Fisica Teorica

Abstract

These lectures describe the treatment of mass divergencesin perturbative QCD, concentrating on hadron productionin electron-positron annihilation. I perform an explicit anddetailed one-loop calculation, and use it to infer someresults and techniques valid to all orders in perturbationtheory. I introduce some of the tools necessary to provefactorization theorems, and show how they can be usedalso to resum certain classes of logarithmic contributionsto all orders in perturbation theory.

Ubi maior ... :G. Sterman: An introduction to quantum field theory.P. Nason: http://castore.mib.infn.it/~nason/misc/QCD...M.L. Mangano: http://home.cern.ch/~mlm/talks/cern98...G. Sterman: hep-ph/9606312.

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Outline

On mass divergences in perturbative QCD

Mass divergences, low energies e long distances.

Cancellation for physical observables, KLN theorem.

Factorizable and IR/C safe observables in PQCD.

An explicit example: Re+e

Definitions, cut diagrams, tree level. O(s): the calculation. Approximations and observations.

Other infrared finite quantities

StermanWeinberg jets.

Event shapes.

Methods for all-order calculations Singularities of Feynman diagrams and trapped surfaces. Landau equations and Coleman-Norton picture.

Infrared powercounting and finiteness of Re+e.

Factorization and resummation

Multi-scale problems and large logarithms.

From factorization to resummation.

Examples: the quark form factor; the thrust.

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Mass divergences: qualitative discussion

Fact: in quantum field theory, two kinds of divergences areassociated with the presence of massless particles.

Infrared (IR). Emission of particles with vanishing four

momentum (DB ); in gauge theories only; they arepresent also when matter particles are massive.

Collinear (C). Emission of particles moving parallel to the

emitter; they are present if all particles in the interaction

vertex are massless.

Example: a massless fermion emits a gauge boson in the finalstate.

p + k

k

p

M

igu(p)/(k)tai(p/ + k/)

(p + k)2 + iM ,

Singularities: 2p k = 2p0k0(1 cos pk) = 0 , k0 = 0 (IR); cos pk = 0 (C).

Note: p0 = 0 not a problem (singularities are always

integrable).

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Origin of mass singularities

In covariant perturbation theory ( p conserved in everyvertex; intermediate particles generally offshell): the

emitting fermion is onshell, so that it can propagate

indefinitely.

In time-ordered perturbation theory (all particles are on

shell, energy however is not generally conserved in the

interaction vertices): the IR/C emission vertex conserves

energy, so it can be placed at arbitrary distance from theprimary process.

Mass divergences originate from physical processes

happening at large distances.

Available terapies .

The sickness is serious. Because of mass divergences, the S

matrix cannot be constructed in the Fock space of quarksand gluons

Observe. Mass divergences are associated with the

existence of experimentally indistinguishable, energy

degenerate states. All physical detectors have finite

resolution in energy and angle.

KLN Theorem. Physically measurable quantities (such

as transition probabilities, cross sections, after summingcoherently over all physically indistinguishable states) are

finite, mass divergences cancel.

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KLN theorem.

Consider a theory defined by its hamiltonianH

, and let D(E

0)be the set of eigenstates of H characterized by energiesE0 E E0 + , with = 0. Let P(i j) be thetransition probability per unit volume and per unit time fromeigenstate i to eigenstate j. Then the quantity

P(E0, ) X

i,jD(E0)P(i j)

is finite in the massless limit to all orders in perturbation

theory

Note. In an asimptotically free field theory the limit m 0and the high-energy limit formally coincide. Masses acquire a

scale dependences such that m2(2) 0 as 2 . The situation in perturbative QCD

Long distance (d > 1fm), or low energy (E


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The strategy of perturbative QCD

All calculations are performed at partonic level, with an infraredregulator (e. g.: = 2 d/2 < 0), requiring the presence ofat least one hard scale Q2. One computes

part = part

Q2

2,s(

2),

(m2(2)

2,

)!.

IRsafe quantities are selected, having a finite limit when the

IR regulator removed ( 0, m2

(2

) 0).

part = part

Q2

2,s(

2), {0, 0}

!+ O

(m2

2

!p,

)!.

These partonic, inclusive quantities, admitting a perturbativeexpansion in powers of s(Q2) 1, are interpreted as

estimates of the corresponding hadronic quantities, validmodulo O `(QCD /Q)p corrections. With hadrons in the initial state, the goal is constructing

factorizable quantities, such that

part = f

m2

2F

! bpart

Q2

2,

2F2

!+ O

m2

2F

!p!.

Factorization, proved at parton level, is transcribed at hadronlevel. Distribution functions f are measured, cross sectionsbpart are derived with a perturbative calculation.

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An explicit example: Re+e

The prototype of IRsafe cross sections is the total annihilationcross section for e+e hadrons.

tot(q2) =

1

2q2

XX

ZdX

1

4

Xspin

|M(k1 + k2 X)|2 ,

normalized dividing out the total muon-pair production crosssection

Re+e tot

e+e hadrons

tot (e+e

+

)

In d = 4 2, to leading order in ,

tot(q2) =

1

2q2L(k1, k2)H

(q2) ,

L(k1, k2) =e22

q4

`k1 k

2 + k

1 k

2 k1 k2g

,H

(q

2

) = e

2

2

q

2

fXX 0|J(0)|XX|J(0)|0(2)dd(q pX ) .Current conservation implies qH = qH = 0, so that

H(q2) =

qq q2g

H(q2) .

This leads to

gH(q

2) = (3

2) q

2H(q2

) ,

tot(q2) =

e22

2q41

3 2

gH(q2)

.

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Exercise: Re+e at treelevel

k

qtot(q

2) = e22

2q41

32 (g)

H = e22q2fZ

ddk

(2)d2+(k

2)+((kq)2)Tr`

k/(k/ q/)

.

In he center of mass frame (kq)2 = q2 2

pq2k0; use the two +

distributions to perform k0 and |k| integrals; the trace evaluatesto 4(1 )q2.Summing over quark colors and flavors one finds

H = 2(1 )e22NcX

f

q2f

q2

4

!122

(2)22.

The d-dimensional solid angle is given by the classic formula

d =2dd/2(d/2)

(d).

In d = 4 2 one finds then

H = 2 (2 )(2 2) q

2

42

q2

!NcX

f

q2f ,

whence the famous result, for

0,

tot =42

3q2NcX

f

q2f R(0)e+e = NcX

f

q2f .

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Radiative corrections

Enumerating graphs contributing to the one-loop correction is

easy in terms of cut diagrams

hH

i(1)= + c.t.

Summing over positions of the cuts one obtains real and virtual

diagrams in turn. We examine them separately.

Real emission

It is convenient to compute separately the transition probabilityand three-body space. Following the rules for cut diagrams onefinds

hHi(1,R) = Z ddpddk(2)2d3+(p2)+(k2)+((p + k q)2) hHi .The transition probability depends on a single polar angle. Letu cos pk in the center of mass frame: one finds

+((p + k q)2) =

p0

s 2s( p + k) + 2 pk(1 u)

,

where p = |p|, k = |k|. One can integrate over the energies of p

and k using the respective +. All angular integrations are trivial

except the one on u.

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Introduce dimensionless variables (quark and gluon energyfractions)

z =2k

s, x =

2ps

,

and define y = (1 u)/2. The result ishH

i(1,R)=

1

8

2212(2)52

s

2

12 Z10

dxx12Z1

0dzz12

Z1

0

dy [y(1

y)]

1

1 yzx

1 z1 yzhHi .

The transition probability can be computed from the Feynmanrules.

H = 2e22X

f

q2fg

2

2Tr(tat

a)

0@Trh(p/ + k/)p/

(p/ k/)p/i

(2p k)(2p k)

+Tr h(p/ + k/)p/(p/ + k/)p/i

(2p k)21A ,

and can be simplified using Tr(tata) = NcCF, Clifford algebraidentities such as

p/

= 2(1 ) p/ ,p/k/

= 4p k

2 p/k/ ,

p/k/q/ = 2q/k/p/ + 2 p/k/q/ ,

and with the identifications pq = sx/2, pk = sxyz/2, kq = sz/2.

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One can integrate out the quark energy fraction x, using theremaining . The expression simplifies considerably and one gets

hHi(1,R) = 2NcCFXf

q2fs(1 )

22

12(2)34 q22

2

q2 !2

Z10

dzdy

(1 )(1 z)2(1 yz)22z12(1 y)1 1

y1++

(1 z)12(1 yz)22z12 [y(1 y)]

(1 yz)2yz2(1 y)

!#.

One recognizes the announced singularities

Infrared: z12, gives a pole in when the gluon energy ztends to 0.

Collinear: y1 e (1 y)1, singular when y 0 (gluoncollinear to the quark), and y 1 (gluon collinear to theantiquark).

Note: mass singularities are regulated choosing < 0.The y and z integrations yield Euler B functions (typical of one-loop calculations in one-scale problems). Expanding around = 0one gets the final result for real emission, displaying the doubleinfraredcollinear pole,

hH

i(1,R)

= NcCF

Xfq2f

s

q2

42

q2

!2

1 (2 2)

2

2+

3

2 + 19

2+ O()

.

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Virtual contribution

Purely virtual contributions to the production amplitude are given

to all orders by the quark form factor.

p1, p2;

2,

=

p1

p2

The calculation greatly simplifies by taking into account thegeneral properties of the form factor.

For massles quarks, the form factor is expressed in terms of asingle scalar function, multiplying the Dirac structure of thethreee amplitude.

(p1, p2; 2, ) p1, p2|J(0)|0

= ieqf u(p1)v(p2) q22

,s(2), ! .As a consequence, the transition probability is proportional to

the treelevel result, with an overall factor given by 2 Re. The form factor is renormalization group invariant (it has

a vanishing anomalous dimension), as a consequence of theconservation of the electromagnetic current.

+ (,s)

s

Q2

2,s,

! = 0 .QCD does not violate the QED Ward identity. Z1 = Z.

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Reducible (1PR) graphs on each fermion line, including therespective counterterms, reconstruct the residue R of the

quark propagator. Since, according to the reduction formulas,

every external line must be multiplied times R1/2

, it isnecessary to include these graphs on one of the two lines only.

In Feynman gauge and in dimensional regularisation all 1PRgraphs with the loop on the external fermion line vanish

because they are expressed by scale-less integrals (p2i = 0).

Note! This is false in general in axial gauges ( n pi);furthermore it depends on a cancellation of IR and UV

effects ...At one loop these observations are summarized by the identity

0

One is left with only one graph to be computed, the vertexcorrection

(1)

p1, p2;

2,

=

which can be written as

(1) = eqfg22CF

Zddk

(2)du(p1)(p/1 k/)(p/2 + k/)v(p2)

k2(p1 k)2(p2 + k)2.

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The steps to compute this diagram are standard. Summarizing

Sistematically use the massshell conditions (Dirac equation),

u(p1)p/1 = p/2v(p2) = 0, then isolate integrals with differentpowers of k.

(1) = eqfg22CFu(p1)

h2q

2I0 + 2(p/1 p/2)I

Ii

v(p2) .

The tensor integrals

I0 =Z ddk

(2)d

1

k2(p1 k)2(p2 + k)2,

I =

Zddk

(2)d

k

k2(p1 k)2(p2 + k)2,

I =

Zddk

(2)d

kk

k2(p1 k)2(p2 + k)2,

can be computed with the usual Feynman parametrization. Note: only I0 can have IR divergences, and only I can

have UV divergences. I may have collinear poles ...

It may be useful to decompose tensor integrals a la PassarinoVeltman, in order to get directly the scalar form factor . Theresult is

I = p1 I1 +p2 I2 ,

I

= g

I3 +p1p

1 I4 +p

2p

2 I5 + (p

1p

2 + p

1p

2 ) I6 .

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In terms of the scalar integrals I1, . . . , I 6 one finds

(1) = g22CF

h4(1 )2I3 2q2 (I0 + I2 I1 + (1 )I6)

i.

The final result for the form factor is

(1)

= s4

CF

42

q2

!

2(1 )(1 + )(1 2)

2

2+

3

+ 8 + O()

.

When taking the real part one must use

(q2 + i) = (q2) ei .

Note! The sign of q2 is dictated by the Cutkosky rules.

Because of the double pole, the factor exp(i) must beexpanded to second order in . It generates numerically

important contributions.

Result

We observe that, as expected, the virtual contribution has thesame IR-C poles as real emission. Summing the two, the polescancel and one can take the limit 0, with the result

tot =42

3q2NcX

f

q2f

1 +

s

3

4CF + O(

2s)

,

For SU(3), where CF = 4/3, the (classical) result is

R(0)

e+e = NcX

f

q2f

1 +

s

+ O(2s)

.

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Soft approximation

The cancellation that was just exhibited is possible only because,in the IR and C limits the amplitude for the emission of a real

gluon becomes proportional to the Born amplitude, just as was the

case for the virtual diagram. This result can be made systematic

introducing the soft approximation.

Aaij =

p,ip,i

p, jp, j

k, ak, a

Aaij = gtaiju(p)

"/(k)(p/ + k/)

2p k (p/

+ k/)/(k)2p k

#v(p

) .

When the gluon is soft one can

neglect k in the numerator, and in the definition of p; commute p/ and p/ in order to impose the mass-shell condition,

p/v(p) = u(p)p/ = 0.

The result is

Aaij soft = gtaij "

p

p

k p

p

k#A

0 ,

where A0 = u(p)v(p) is the Born amplitude (whatever the

explicit form of the vertex ).

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Remarks

The soft amplitude is gaugeinvariant (it vanishes if k). Soft gluon emission has universal characters. Long-wavelength

gluons cannot analyze the short-distance properties of the

emitter (spin, internal structure), they only detect the global

color charge and the direction of motion These considerations

generalize to multiple emission.

Identical considerations apply to gluon emission from gluons.

Soft cross section

Its easy to recover the singular part of the real emission crosssection. The transition probability can be computed summingover colors and polarizations (using

P

= g, which is

allowed in this case).

|Asoft|2

= g2CF |A0|

2 2p p

p k p k.

To get the cross section one integrates over phase space

softqqg = g

2CFqq

Zd3k

2|k|(2)32p p

p k p k.

In the center of mass frame (q = 0) and in the soft approximationthe quark and the antiquark are still back to back. One thenrecovers the structure of IR and C singularities,

softqqg = qqCF

s

Z11

d cos pk

Z0

d|k|

|k|

2

(1 cos pk)(1 + cos pk).

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Virtual diagrams

The soft approximation can be applied to virtual diagrams as well,

with some care.

When k pq2, , one can neglect k2 with respect topi k in denominators, as well as k in numerators (eikonal

approximation).

Note: the approximation is not uniformly valid, in some

cases it becomes necessary to deform integration contours,

or the approximation may break down.

Using light-cone coordinates one can set

p =

p+, 0, 0

, (p)

=

0, (p)

, 0

.

Note: For a generic four-vector, v =

v+, v, v

, v =

(v0 v3)/

2, v2 = 2v+v |v|2.

Consider for example the integral I0, containing the virtualdouble pole. In the eikonal approximation and in d = 4

I(eik)0 =

1

324q2

Z dk+dkd2k(k + i)(k+ + i)(2k+k |k|2 + i)

.

There are three integration regions giving rise to divergences.

One can parametrize them introducing a scaling variable ,

according to

k

qq

2 ,

,

IR ;

k q

q2 , k 2

qq2 , |k|

qq2 , COLL .

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Angular ordering

The soft approximation is of great practical relevance inperturbative QCD.

It displays the universal properties of soft color radiation (colortransparency, angular ordering)

It links perturbative and nonperturbative regimes(resummations, hadronization Monte Carlo)

The simplest example of angular ordering is the differenzial crosssection dqqg in a frame in which the decaying photon has a largemomentum q. In that frame the quark and the antiquark aretypically emitted forming a small angle (pp ), and one has

dsoftqqg = dqqCFs

d|k|

|k|d cos k

dk

2

1 cos pp(1 cos pk)(1 cos pk)

= dqqCFs

d|k|

|k|d cos k

dk

2

1

2 `Wq + Wqwhere

Wq =1 cos pp

(1 cos pk)(1 cos pk)+

1

(1 cos pk) 1

(1 cos pk).

while Wq is found exchanging p p.

The full angular distribution is positive definite, but singular foremissions parallel to either the quark or the antiquark. The partial

distributions Wi are not positive definite, but they enjoy special

properties.

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Properties of Wq e Wq

Wq is singular only when cos pk 1, while the opposite istrue for Wq.

The azimuthal average ofWq (with respect to the axis definedby p) vanishes ifpk > pp.

1

2

Z20

d Wq() =2

1 cos pk

pp pk

,

as can be proven using

cos pk = cos pk cos pp + sin pk sin pp cos .

An identical equation applies to Wq.

Azimuthal averages are positive definite and can be interpretedas probability distributions for the emission of soft gluons

independently of the quark and of the antiquark.

Thus: the distribution of soft radiation on average is given by a

sum of uncorrelated contributions from the quark and from theantiquark, each vanishing outside the cones built rotating the

direction of one of the fermions around that of the other one.

Comments

This property can be generalized to higher orders. Gluonswhich are radiated later are forced on average to the inside of

the cones defined by previously emitted gluons and quarks.

Perturbative evolution in the soft limit is local in phase space,so that color singlet parton clusters are likely to be formed

inside collimated particle beams (jets).

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StermanWeinberg jets

Momentum configurations responsible for singularities and neededfor their cancellation are infrared and collinear, as expected.

Thus it is not necessary to integrate real emission over the entire

phase space in order to obtain a finite result, it is sufficient

to consider sufficiently inclusive observables, such that gluon

emission be integrated over IR and C configurations.

Prototype: two-jet cross section

Eout <

s

Ein > (1 )

s

Definition: an event is a two-jet event iff

two opposite cones

with opening angle , such that all the energy, except at most afraction , flows into the cones.

At parton level:

All events are two-jet events at leading order.

At O(s) two-jet events are those in which the gluon is IR(and emitted in any direction), or C (with any energy). All

other events are three-jet events. Virtual contributions are two-jet events. Therefore the partonic

two-jet cross section is finite.

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More precisely:

At LO one finds simply (0)2j (, ) =

(0)tot = Nc

Pf q

2f

42

3q2.

At NLO one finds only two- or three-jet events, so that

(1)2j

(, ) = (1)tot

(1)3j

(, ) ,

(1)3j is easily computed from the real emission matrix elementwith appropriate cuts on phase space. In d = 4

hH

i(1,R)

3j= 2NcCF

Xfq2f

s

q2

Z1

2dz

Z12(1z2/2)2

dy

z(1 y)

y(1 yz)2 +1 z

yz(1 y)

.

It is easy to compute the dominant contributions as , 0.Combining with the leptonic tensor one finds

(1)3j (, ) =

(0)totCF

s

"4 log() log(2) + 3 log() +2

3

7

4# .Observe:

The total cross section is dominated by two-jet events at largeq2 (asymptotic freedom for jets ...).

For increasing q2 the perturbative result remains reliable fornarrower cones, jets are more collimated.

It is possible to compute (and verify experimentally) theangular distribution of two-jet events d2j/d cos 1+cos2 ,as expected for spin 1/2 quarks.

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Eventshape variables

The mechanism underlying the cancellation of IR and C

divergences suggests a further generalization: study distributions

of observables constructed with final state momenta so as to

assign equal weights to events differing by IR or C emissions.

Given a final state with m partons, let Em(p1, . . . , pm) be theobservable. The corresponding distribution is defined by

d

de

=1

2q2Xm ZdLIPSm |Mm|2 (e Em(p1, . . . , pm)) ,and its moments (and in particular the average value) are

en =Zemax

emin

de en d

de.

Note: These are weighted cross sections.At order m1s one must sum contributions with m + 1 partons

in the final state, with one virtual m real partons, and so on.

(e)

Om+1s

= Zd(R)m+1 + Zd(1V)m + . . . .IR-C cancellation is preserved if the observable takes the same

value for those configurations differing only by the IR/C radiation.

lim

pj 0

Em+1(p1, . . . , pj , . . .) = Em(p1, . . . , pj

1, pj+1, . . .) ,

limp

k

pj

Em+1(p1, . . . , pj , . . . , pk, . . .) = Em(p1, . . . , pj +pk, . . .) .

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Examples of eventshape variablesThere is a variety of available IR/C safe event shapes.

Thrust

Tm = maxn

Pmi=1 |pi n|Pm

i=1 |pi|.

Clearly 0 < Tm 1, and Tm = 1 corresponds to two preciselycollimated back to back particle beams.

C parameter

Cm = 3 3

2

mXi,j=1

`pi pj2(pi q) pj q

.Also in this case 0 Cm 1; two-jet eventi have C = 0. The definitioncan be expressed in terms of the eigenvalues of the space part of the energy-

momentum tensor of the final state, C = 3(12 + 13 + 23).

Jet masses

(H)m =

1

q2 0@XpiHpi1A2

H is one of the two hemispheres identified by the thrust axis.

Observe Perturbative distributions are singular in the two-jet limit

(logarithms of the form ns log2n1 C), but expectation values

are finite.

Great phenomenological relevance (for example: determination

ofs, hadronization corrections). Jet algorithms and the related definitions of multi-jet events

can be seen as particular event-shapes.

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A comparison with QED

QED also has IR divergences, as well as collinear divergences inthe massless limit. There are similarities and differences.

In QED, consider for example the process e+e +. (e+e + diverges starting at O(3). Therefore Born is not a good approximation for . This

is not a problem. The true observable is tot() =

Pn (e+e + + n,), and for tot(), whichis finite, Born is a good approximation.

IR divergences in QED can be explicitly resummed

tot() = fin exp

"

log

2

q2

!f(m2, q2)

#.

so that lim0 tot() = 0.

Interpretation: it is not possibile to produce only +

;asimptotic states of QED are not isolated fermions.

In QCD, considering e+e qq, the situation is almostanalogous.

(e+e qq diverges starting at O(2s). Therefore Born is not a good approximation for = 0

(confinement). On the other hand it is a good

approximation for tot(e+e hadrons), which is finite. Interpretation: it is not possible to produce only qq; the

asimptotic states of QCD are not quarks and gluons.

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On singularities of Feynman diagrams

To study IR and C divergences to all orders one must characterizethe generic singularity structure of Feynman diagrams. Let us

begin with a simple example, I0.

Example: the scalar form factorIntroduce Feynman parameters y1, y2, y3. Then

I0 = 2 Z ddk(2)d Z1

0

3Yi=1

dyi(1

y1

y2

y3)y1k2 + y2(p k)2 + y3(p + k)2 + i.

Let D0 be the denominator. The only possible singularities of I0must lie on surfaces where D0 = 0. Furthermore

The integrand is a function of the complex variables k, yi,with an analytic structure determined by the i prescription.

The vanishing ofD0 on one integration contour is not sufficientto determine a singularity of the integral. The contour can be

deformed.

Only in two cases the singularity cannot be avoided by adeformation: either the contour is trapped between two poles,

or one has end-point singularities, a pole migrates to the edge

of an integration contour.

P1

P2

P

O

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Singularities of I0 dk integrals cannot have end-point singularities (I0 is UV

convergent). D0 however is quadratic in k, so that two

poles can trap the contour if they coalesce,

k D0 yi, k, p , p = 0 .

dyi integrals can only have end-point singularities (in yi =

0), since D0 is linear in yi. Alternatively, D0 can be

independent of yi on the surface D0 = 0, so that yibecomes useless for the deformation.

Landau equations for I0

A necessary condition for a singularity of I0 is that allintegration variables be trapped. This is expressed by theLandau equations

y1k y2(p k) + y3(p + k) = 0 and

yi = 0 or l2i = 0 ,

where l

i

is the momentum flowing through the line with

parameter yi.

Solutions of Landau equationsIts easy to find the expected solutions.

k = 0 ; y2/y1 = y3/y1 = 0 IR

k

= p

; y3 = 0 ; y1 = (1 )y2 Ck

=

p

; y2 = 0 ; y1 = (1

)y3 C

We recognize the expected IR and C singularities. Are there

other solutions of Landau equations?

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ColemanNorton physical picture

The search for solutions of Landau equations is simplifiedby the fact that they admit a simple physical representation.

Observe that

if line i in a loop is offshell it must be that yi = 0. Let xi yil

i , for each on-shell line li. Then

xi = x0i v

i ; v

i = 1,

li

l0

i ! .Interpretation: xi describes classical propagation of a

massless particle with momentum li. The Landau equations for I0 can now be written asX

i

(i) xi = 0 on shell

xi = 0 off shell .

Interpretation: the solutions of Landau equations are given byreduced diagrams, where

Offshell lines are contracted to points.

Onshell lines describe physically admissible processes for

the classical propagation of massless particles.

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General case

Feynman parametrization

Thanks to the identity

NYi=1

1

Daii

=

PNi=1 ai

QN

i=1 (ai)

Z10

NYi=1

dyiy

ai1i

1 PNi=1 yiPNi=1 yiDi

PNi=1 ai

,

an arbitrary Feynman diagram G(pr) can be written as

G(pi) =Y

linee

Z10

dyi

0@1 Xi

yi

1A Yloops

Zddkl

N`

yi, kl, pr

D

`yi, kl, pr

N

,

where the denominator D is a sum of propagatorsD`

yi, kl, pr

=X

linee

yi

l2i (p,k) m2i

+ i ,

and the momenta li are linear functions of pr and kl.

Landau equationsXi

ij xi

= 0 on shell , j/i j ,

xi = 0 off shell .

ColemanNorton pictureSolutions are again reduced diagrams (with offshell linescontracted to points), where all remaining loops can beinterpreted as classically permitted processes. For any loop,it must be possible to associate to all its vertices coordinatesx

k

, k = 1, . . . , M such that

x12 + . . . + x

M1

= 0 , xij x

i x

j .

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Example: the two-point function

Consider an arbitrary 1PI diagram for the two-point functionG(q2, m2), in a theory with only one species of particle with mass

m2.

qqG(q2, m2) =

Theorem: the only singularities of the diagram (and thus ofG(q2, m2)) are normal thresholds q2 = n2m2, n = 1, 2 . . ..

Proof:

Normal thresholds are solutions of Landau equations.

In fact, when q2 > 0 one can choose q =p

q2, 0

. The

ColemanNorton process is the creation of n > 1 particles at

rest, which do not move, and interact until they are absorbed,

for an arbitrarily long time. An example with n = 4 is

qq

No other reduced diagram satisfies ColemanNorton.

If one produced particle has non-vanishing momentum theother ones must compensate moving in the opposite direction.

Once separated they cannot meet again in free motion.

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IR/C power counting

The Landau equations are only necessary conditions for the

manifestation of IR/C divergences. Ifphase space has high enough

dimension singularities are suppressed (e.g.: IR divergences in 36).

One must develop power counting techniques, similar to those

employed in the UV, to determine the strength of the singularities.

Given a diagram, use the CN representation to identify a

trapped surfaceS

in the space{k

i, y

i}

. For every S, identify among the {ki } intrinsic coordinates(movement in S) and normal coordinates (distance from S).

n

i1

i2S

Example: for I0, kp, k+ is intrinsic, {k, k} are normal. Introduce a scaling variable determine the relative weight

(integration volume)/singularity. Set ni = ai ni and consider

0, ni finite.Example: for I0, kp, k 2p

q2, |k| pq2.

Construct the homogeneous integral for S, taking thedominant power of in every factor of the graph.

Example: for I0, the homogeneous integral is the eikonal one.

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The degree of divergence is given by the power of associatedwith the homogeneous integral. If, for every line, l2i (p, k) m2i Aif(n), then

nS =Xi

ai Xi

Ai + nnum .

nS 0 signals a divergence, logarithmic when nS = 0.

Application: finiteness ofRe+e

The all-order finiteness oftot(e+e

hadrons) follows from its

relationship with the correlator of two elettromagnetic currents,

which in turn comes from unitarity,

pipi kjkj

2 ImP

C

C

generalizing T T = i(T T). Define then

(q) ie2Z

d4xeiqx0|TJ(x)J(0) |0=

qq q2g(q2) .

Unitarity gives2 Im

(q)

= e2

Xn

0|J(0)|nn|J(0)|0(2)44 (q pn) ,

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tot(e+e hadrons) = e

2

q2Imh(q2)

i.

It is then sufficient to prove the finiteness of(q2). This follows

from the Coleman-Norton representation.

In a frame in which q =p

q2, 0

one sees that there are no

allowed classical processes with non-vanishing momenta such

that the photon decays and then reconstitutes. Thus there

are no trapped surfaces with non-vanishing momenta. The only trapped surfaces are those with all particles having

vanishing momentum, and reduced diagram

H

S

These however are finite by power counting, as may be

expected (all lines in H are offshell).

To see it note that fermion lines at zero momentum are lesssingular than gluon lines. The worst case is then ifS containsonly gluons. In that case, in d dimensions, with LS loops andgS gluons in S,

nS = d LS 2gS = 2(1 )gS

which is positive in d > 2.

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Application: the quark form factor

Considering

(q

2

) one finds collinear divergences associated withobserved quarks. The most general reduced diagram is

J

H S

J

p1, p2;

2,

=

Further simplifications are possible

Gluons connecting S directly to H are suppressed (one moreoffshell propagator, plus a new soft propagator dominated by

the new soft loop).

Fermion lines connecting different subgraphs (except thenecessary q e q) are suppressed.

In an axial gauge, n A = 0, only the (anti)quark line connectsthe jet J to H. In fact the axial gauge gluon propagator is

Gax(k) =1

k2 + i

g +

nk + nk

n k n2 kk

(n k)2

.

Contracting with k cancels the denominator

k

Gax(k) =n

n k n2 k

(n k)2,

so that the degree of IR/C singularity is reduced.

These considerations suggest a factorization = J1J2SH.

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Diagrammatics of factorization

Identifying the leading integration regions in momentum space is

only the first step towards factorization. One must then

exploit simplifications in Feynman diagrams in the leadingregions (soft/collinear approximations, Ward identities).

organize all-order subtractions a tutti gli ordini so as toavoid double counting (different factor functions have operator

definitions, in general non local).

Microexample: again the one-loop form factor, collinear regionkp1, Feynman gauge.

Kinematics: p1 =

p+1 , 0, 0

, k =

k+, k, k

, con k+ {k, k}.

Grammer-Yennie approximation in the numerator

u(p1)(p/1 k/)(p/2 + k/)v(p2) u(p1)+(p/1 k/)(p/2 + k/)v(p2)

1k+

u(p1)+(p/1 k/)(p/2 + k/)k+v(p2)

1k u2

u(p1)+

(p/1 k/)(p/2 + k/) k/ v(p2) .

Ward identity: k/ = (p/2 + k/) p/2. One finds

(coll)

Zddku(p1) u/2 (p/1 k/) v(p2)

k2 (p1 k)2 k u2.

The gluon-antiquark coupling simplifies, recognizing only

charge and direction, it becomes eikonal.

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Eikonal lines

Graphically, one may introduce eikonal Feynman rules

= igutaij =iij

uk+iuj

j

i

i

, a

in terms of these rules the previous calculation reads

coll

More than one gluonSumming over all possible insertions of two or more collineargluons, one finds sistematic cancellations, due to Ward

identities. One uses then eikonal identities like1

k1 u

1

k2 u=

1

(k1 + k2) u

1

k1 u+

1

(k1 + k2) u

1

k2 u.

All collinear gluons couple to the same eikonal line.

Typical resultIn an axial gauge the form factor factorizes

q2

2!

= J1(p1 n)2

2n2!

J2(p2 n)2

2n2!S (ui n) H

q22!

.

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Multi-scale problems and large logarithms

For observables depending on a single hard scale all logarithms

are resummed using the renormalization group

q2

2,s(

2)

!=

1, s(q2

)

.

Most problems however have several hard scales. If for example

q21 q22 QCD, the reliability of perturbation theory is putin jeopardy by terms like ns log

p(q21/q22), with p n (single

logarithms) or p 2n (double logarithms). They all arise fromIR/C dynamics. Some examples. The Sudakov form factor.

q2

2

!= 1 s

4CF log

2

q2

2

!+ . . . .

In a massless theory one may choose 2 = q2, and be left

with IR/C poles. With masses one finds duble logarithms

log2 q2/m2.

DIS. The two hard scales are Q2 = q2 and W2 = (p +q)2 = Q2(1 x)/x. There are single log(1/x), resummed bythe BFKL equation, and double log(1 x), resummed a laSudakov.

DrellYan. The process is qq +(q2), the two hard scalesare s = (p1 +p2)

2 and q2. Double log(1

q2/s) are resummed

a la Sudakov. The transverse momentum distribution in DrellYan. The two

scales are q2 and q2, giving double log(q2/q

2).

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Factorization and resummationThe deep connection between factorization and resummation, can

already be seen from the renormalization group.

G(n)0 (pi,, g0) =

nYi=1

Z1/2i (/,g()) G

(n)R

(pi, , g()) ,

dG(n)0

d= 0 d log G

(n)R

d log =

nXi=1

i (g()) .

Solving this equation resums in exponential form the logarithmic

dependence on .

The AltarelliParisi equation similarly leds to the resummation of(single) logarithms of F. In terms of Mellin moments

eF2

N,Q2

m2, s(Q

2)

!= eCN, Q2

2F

,s(Q2)

! efN, 2Fm2

,s(Q2)

!,

d eF2dF = 0 d log efd log F = Ns(Q2) .Double logarithms are more difficult. Ordinary renormalization

group is not sufficient. Gauge invariance plays a key role. Or: use

effective filed theory (SCET).

The quark form factorConsider the factorization

q2

2

!= J1

(p1 n1)

2

2n21

!J2

(p2 n2)

2

2n22

!S (ui ni) H

q2

2, ni

!.

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The form factor is gauge invariant, so that

log

p1 n1= 0 log J1

log(p1 n1)= log H

log(p1 n1) logSlog(u1 n1)

.

The two functions on the r.h.s. have different arguments. Then

log J

log q= KJ

s(

2),

+ GJ

q2

2,s(

2),

!.

The K function contains all singularities in (H is finite as 0).

The function G contains all q2

dependence.The whole form factor obeys an equation of identical form.Furthermore the form factor is renormalization group invariant,so that

dG

d log = dK

d log = K (s()) ,

with a finite anomalous dimension, independent of q2.

The equation for can be solved. Since is divergent, one needsto keep consistently the dependence on < 0 to all orders. s(2)becomes s(2, ) which implies (q2 = 0, < 0) = 0; then

Q2

2,s(

2),

!= exp

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The thrust distribution

The thrust distribution is singular as T 1, behaving asns log2n1(1 T)/(1 T). These logs can be resummed withsimilar methods.

As T 1 the distribution can be factorized in a manner similarto . The jets J now enter the final state, so they have anon-vanishing invariant mass m2J (1 T)q2 as T 1.

(N) 1

0 Z1

0 dT T

N d

dT =

J1 q2

N 2 ,

(p1 n)2

n22 ! J2 S H .In an axial gauge, leading logarithms are in the jet functions J,which satisfy

log J

log q= KJ

q2

N2,s(

2)

!+ GJ

q2

2,s(

2)

!.

One can then solve for J(N) in terms of J(1), as

J

q2

N 2

!=J

q2

2

!exp

2412

Zq2q2/N

d2

2

log

J

s(

2)

J

s(

2)35 .

Leading logarithms are determined by J

= K + . . .. Neglectingrunning coupling effects one easily finds J exp(s log2 N).Inverting the Mellin transform,

10

ddT

= 2CFs

log(1 T)1 T exp CFs log2(1 T) .

Note: d/dT 0 as T 1 (Sudakov suppression).

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The borders of the perturbative regime

Resummations test the limits of the perturbative theory. One getsin fact integrals of the form

fa(q2) =

Zq20

dk2

k2(k2)as(k

2) ,

One sees esplicitly the nonconvergence of perturbation theory athigh orders, consequence of the Landau pole. In fact

s(k2

) =s(q

2)

1 + 0s(q2) log `k2/q2 .Letting z log q2/k2, one observes that The perturbative expansion in powers ofs(q2) diverges.

fa(q2

) = (q2

)a

Xn=0

n0

n+1s (q

2)

Z0

dzeaz

zn

=X

n=1

cnns n! .

The integral is ambiguous, but the ambiguity is suppressed bypowers of q2. In fact

fa(q2) = (q2)as(q

2)

Z0

dzeaz

1 0s(q2)z.

The Landau pole on the integrazion contour induces anambiguity of the same parametric size as the residue

fa(q

2)

exp a0s(q2)

=2QCD

q2

!a.

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