1 renormalization: state-of the-art report with a new...
TRANSCRIPT
1
Renormalization: state-of the-art report with a new concept and new methods
Jan HelmInstitute of Physics, LondonEmail: [email protected]
AbstractFirst, we present the state-of-the-art of renormalization with its three main methods: the dimensionalregularization due to ‘t Hooft-Veltmann, the Pauli-Villars cut-off method, and the lattice regularization.Second, we derive a new, mathematically rigorous formulation of the dimensional regularization, analyze thelimit behavior of the lattice regularization numerically, and present relations between the three methods. Theapproach is illustrated by numerical and analytic calculations for the three first-order loop corrections of thequantum electrodynamics (QED).
1 Definition of renormalizabilityRenormalization is a collection of techniques in quantum field theory, that are used to treat infinities arising incalculated quantities by altering values of quantities to compensate for effects of their self-interactions.Mathematically speaking, we get for an Feynman graph expression Γ(κi) a series in ε , where 0 is theregularization parameter:
...)()(
)(),( 11
0
i
iii C
CC , which has a residuum C-1 at 0,
instead of a “normal” complex value )(0 iC , if the expression is non-divergent.
Renormalization is the procedure of finding additional counter-terms in the corresponding lagrangian tocompensate for the residuum-term in (finitely many) divergent Feynman graphs of an interaction.
2 Physical dimension of wave functions and the lagrangianIn the following, we use the usual convention c 1 , i.e. all physical entities have dimension cmn or cm-n ,e.g. [length]=cm, [energy]=cm-1 . If we are working in a space of (geometric) dimension d=4 (normally, ofcourse in Minkowski space of dimension d=4)), then we have the following (physical) dimension
4][ cmL for a lagrangian1][ cm for a vector wave function, e.g. photon, or a Klein-Gordon scalar (lagrangian term 2 )
2/3][ cm , for a spinor, e.g. electron (lagrangian term i )
Now, it can be shown that for a renormalizable interaction, the corresponding term in the lagrangian has the
dimension 4cm , like the lagrangian itself, i.e. the coupling constant is dimensionless.For instance, in QED, the lagrangian is [1]
and the interaction term has the dimension 4cm and the coupling constant 4e is dimensionless(α is the fine-structure constant). For QCD
, where the field tensor is
is the covariant derivative , the coupling constant g is dimensionless, and
is the gluon field with the gluons aA and the the gauge generators )3(SUa
23. Renormalization criteria and methodsThere are four methods for dealing with divergences in the quantum field theory (QFT): power counting,regulation and counterterms [1].Power countingBy simply counting the powers of p in any Feynman graph, we can, for large p, tell whether the integraldiverges by calculating the degree of divergence of that graph:each boson propagator contributes p-2,each fermion propagator contributes p-1,each loop contributes a loop integration with p4,and each vertex with n derivatives contributes at most n powers of p.If the overall power of p; that is, the degree of divergence D, is 0 or positive, then the graph diverges.By simply counting the p-power at infinity, we have a necessary condition for non-renormalizability.Renormalizability criteria:-The degree of divergence D of any graph must be a function only of the number of external legs; that is, itmust remain constant if we add more internal loops.-The number of classes of divergent N-point graphs must be finite.
RegularizationThere are three regularization techniques: cutoff regularization(Pauli-Villars), dimensional regularization(t’Hooft-Veltman), and lattice regularization.Manipulating divergent integrals is not well-defined , so we must cutoff the integration over d4p in some wayin order to make the integral finite and then proceed to a limit.The cutoff method does it by introducing an additional massive particle with large mass M (a ghost) ,integrating to the cutoff M, and then letting M→∞ .The dimensional method extends the integral to the fractional dimension d=4-ε, where it converges, and letsthen ε →0 .The lattice method calculates the integral over a finite equidistant lattice with lattice distance ε and the upperlimit M(ε) and lets ε →0 , M(ε) →∞ .The first two methods are analytical, i.e. one must be able to calculate the integral as a closed expression, thelattice method is numerical, i.e. it is applicable also for integrals, which cannot be calculated analytically.CountertermsThe method of counterterms, pioneered by Bogoliubov, Parasiuk, Hepp, and Zimmerman (BPHZ), consists ofadding new terms directly to the Lagrangian to subtract off the divergent graphs. The coefficients of thesecounterterms are chosen so that they precisely kill the divergent graphs.In a renormalizable theory, there are only a finite number of counterterms needed to render the theory finite toany order, these counterterms are proportional to terms in the original action. Adding the counterterms to theoriginal action gives us a renormalization of the masses and coupling constants in the action.Multiplicative renormalizationThe method of multiplicative renormalization, pioneered by Dyson and Ward for QED, means in essence tosum formally over an infinite series of Feynman graphs with a fixed number of external lines.The divergent sum is then absorbed into a redefinition of the coupling constants and masses in the theory. Sincethe bare masses and bare coupling constants are unmeasurable, we can assume they are divergent and that theycancel against the divergences of corresponding Feynman graphs, and hence the theory has absorbed alldivergences at that level.Using the formula for the geometric series
one develops the fermion propagator into a series in p2 and sums the integral formally:
...~~11
~1
111)(
2
2
2
2
2
2
2
2222
im
pi
m
p
mi
m
pmimppF
4 Green function, S-matrix and cross-sectionsWe start with the Green function (inverse operator) of the Schrödinger Hamiltonian
3
, where HI the interaction part of the Hamiltonianwe can then describe the evolution of the Green operator under the interaction HI
and for the wavefunction under n interaction steps
From this we can extract the transition probability matrix (S-matrix for short)
From this, we calculate the differential cross section for the transition under the interaction HI
as the number of transitions per unit time per volume divided by the incoming particle fluxV
vvJ 21
dσ= , where Nf is the density of final states
We introduce the transition probability2
fiM in the center-of-mass frame
k p
iffifi
kVE
PPVTMS2
12)(
4422
and with this we get the formula for {1,2}→{3,4,...,N} process for the cross-section
, where the volume V cancels out
fiM is the Feynman integral for the process {1,2}→{3,4,...,N}
)()...()2(
...)2(
1
0
4
4
0
41
4
nn
lk pipipdpd
PI
with integration over internal momenta, where )( npi
is the propagator of the internal process
5 Feynman rules
QED rules
4
general QFT rulesExternal lines radiate from vertices and receive the following factors:(a) For spin-1/2 fermion of momentum p and spin state s• in initial state: u(p, s) on the right,• in final state: ¯u(p, s) on the left;(b) For spin-1/2 antifermion of momentum p and spin state s• in initial state: ¯v(p, s) on the left,• in final state: v(p, s) on the right;(c) For spin-0 boson• in either initial or final state: 1;(d) For spin-1 boson of helicity λ (if massless boson, λ = ±1; if massive boson, λ = 0,±1) • in initial state: εμ(λ), • in final state: ε*μ(λ).
Internal lines
each internal loop: integrate over
closed fermion loop: factor -1closed loop n bosons: factor 1/n!
QCD vertices
5
electroweak quark-boson vertices
electroweak lepton-boson vertices
6 Divergent graphs of QED
degree of divergence: ,where Eψ=number of external electron legs, EA=number of external photon legsddiv(Γ)=D: Γ is divergent with 1/ε(D+1)
divergent diagrams: 6 one-loop
)( p = D=1 , one-loop electron self-energy, momentum p ( pp Dirac dagger)
6
)',( pp = D=0 , one-loop electron-photon vertex, momemtum p, q, p’
)(k = D=2 , photon vacuum polarization, momentum k
D=1 , 3-photon vertex correction one-loop, momentum k, k’, k’’
D=1 , 3-photon vertex correction inverse one-loop, momentum k, k’, k’’
D=0 , 4-photon vertex correction one-loop, momentum k, k’, k’’,k’’’
Actual divergence only Γ1 , Γ2 , Γ3
Γ1 : =O(1/ε) Γ2 : =O(1/ε) Γ3 : =O(1/ε) Γ4 = -Γ5 cancel because of Furry’s theorem: Γ(EA )= -Γ(-EA ) if EA =odd
Γ6 is convergent because of 0kk
Mathematically speaking, we get for a divergent graph Γ a series in ε , where 0 is the regularizationparameter:
...)( 11
0
CC
C , which has a residuum C-1 at 0.
7 Renormalization of QEDFor the 3 divergent graphs Γ1 , Γ2 , Γ3 we introduce 4 infinite expressions Z2, Z1, Z3 , δm and the 4 fundamentalparameters ψ, Aμ, e, m are renormalized, i.e. made finite, with unrenormalized (infinite) parametersψ0, Aμ0, e0, m0 .We develop )( p around mp
, where )(m infinite and 1)(' m and set
)('1
12
mZ
)(mm
7
)(~
)( 3 kZk
)',(~
)',( 1 ppZpp , where the ~expressions are the finite parts.
Now we make a transformation from renormalized to unrenormalized parameters
and insert them into the lagrangian, making it unrenormalized
are the counterterms
yields finite graphs 321
~,
~,
~ , that is, )(
~0 jj C
The procedure of finding the appropriate counterterms , i.e. the appropriate transformation to theunrenormalized parameters using the infinite terms Z2, Z1, Z3 , δm is the essence of renormalization.Ward-Takahashi identitiesFrom the simple identity in γ-matrices
one can prove that
ppp
p,
)(
(Ward-Takahashi identity)
and from this followsZ1=Z2 , i.e. the renormalization factors of electron-photon vertex )',( pp and electron self-energy )( p are
equal.
8 Types of regularizationThere are basically 3 types of regularization, i.e. forming a series in ε for Feynman graphs: Pauli-Villars cut-off regularization, t’Hooft-Veltman dimensional regularization and lattice regularization.Pauli-Villars cutoff regularizationThis was the first widely used regularization scheme.For the high-energy limit (ultraviolet divergence) we cutoff the integrals by assuming the existence of a virtualparticle (ghost) of large mass M. The propagator is then modified by:
The propagator now behaves as 1/p4 , which renders all graphs finite. Then we let M→∞ , making thetransition to the original operator. This method preserves the gauge invariance and the Ward-Takahashiidentities, enabling cancellation of terms due to gauge symmetry.As an example, the inner photon contribution
becomes
))(()2(2)(
222
02
22
0
0
4
402
iMpimp
Mmpdipi
M
8
If we set
m
Mlog
1
, we get a series in ε identical to the dimensional regularization.
For the low-energy limit (infrared divergence) we modify the photon propagator by a cut-off mass μ
222
1),(
1)(
kkP
kkP and let 0 , the resulting divergence is of the form
mlog
1in
analogy to the high-energy limit.Dimensional regularizationThis method is widely used now, as it is analytically simpler than the cutoff method. It also preserves the gaugeinvariance and the Ward-Takahashi identities, enabling cancellation of terms due to gauge symmetry.We calculate n-dimensional integrals in spherical coordinates {r,φ1,..., φn-1}n-dim spatial angle
)2/(
2 2/
n
nn
n-dim integral in spherical coordinates
121
2
0
1
0
3
22
2
1
0 0
11 ,...,,...sinsin
nn
nnRr
r
n rfdddrdr
Now, we will find the rigorous mathematical definition for xd n
First, cartesian xi in spherical coordinates φ n=3: {x1, x2, x3}= {sin[φ2]cos[φ1], sin[φ2]sin[φ1], cos[φ2]}ݎn=4: {x1, x2, x3, x4}= ,sin[32]sin[φ2]cos[φ1]}ݎ sin[32]sin[φ2]sin[φ1], sin[φ3]cos[φ2], cos[φ3]}With the function
)1(2
2/)1()1(2
0
)1(2
sin)2/2()2/3(2
)2(sin
sin
1
)3(
)4(),(
d
fomega
)1(2sin)0),,(lim(
fomega
we can formulate the integral xd n
for n=4-ε correctly (so that it provides an analytic expression, which gives
the correct value for n=3 and n=4 )
121
2
0
3
0
221
0 0
13 ,...,,sin),(
n
Rr
r
rfddfomegadrdr
Lattice regularizationIn the lattice method a four dimensional hypercube lattice in space-time is introduced, and the integral iscalculated as a sum over lattice points. This method is numerical and can be also used for non-perturbative (notwith Feynman diagram series) calculations, but it violates gauge invariance and the Ward-Takahashi identities,cancellation of terms because of gauge symmetry is only approximative, and an integral can become thereforenumerically divergent of a higher degree, e.g. the photon vacuum polarization )(k is originally
quadratically divergent with D=2, but becomes logarithmically divergent with D=0 because of termcancellation).
We choose the lattice equidistant with upper limit M=Mvr m/ε , and number of intervals ]1
[
n , where [] is
the next higher integer and Mvr is the step adaptation factor. The lattice step constant is then mMn
Md vr
independent of ε , and four-dimensional lattice-hypercube has4
nN cells with cell volume4
4
n
Mv
9
We approximate the integral by
N
i ikfam
MSkfkd1vr
vr4 )()M()M
,()( by the sum over the lattice
points ki , where a(Mvr) is the integral adaptation factor .The Taylor-series of f(k) around a lattice point ki gives then the error formula
dfkfkfkdMSerN
i i )()()()),((1
4 for M=const., where ξ is a point within the lattice range.
That means for the ε-series of Sε(ε ,Mvr m/ ε) that both terms are shifted by a constant:
C-1(Sε(ε ,Mvr m/ ε))= C-1( )(4 kfkd )+c-1
C0(Sε(ε ,Mvr m/ ε))= C0( )(4 kfkd )+c0 .
The term C-1(Sε) can be calculated numerically from
),()
2/,2/()(1
mMS
mMSSC vrvr , the constant
term of the integral C0( )(4 kfkd ) from )0),(),(())(( 14
0 SCMSLimkfkdC numerically for
small ε with M kept constant.
9 Divergent one-loops in QED9.1 Electron self-energyWe discuss here the first divergent QED graph
Γ1= )( p = D=1 , one-loop electron self-energy, momentum p
Dimensional regularization
expressed as kd d ,
)( p =
or mathematically correct
22
02
0
2
0
3
0
22
)1(2
1
0 0
14
32
0)(
)(sinsin
16)(
kmkp
mkpddd
kdkeip
then kd d transformed into integral 1
0
dx (Feynman trick) and with substitution q=k- p x
= =and with d=4-ε ,
=which yields the ε-series coefficients
C-1(Γ1)= )4(8
02
2
0 mpe
C0(Γ1)/e02 ( with pd= , p2=p2)
=
10
which is practically =const*m0 for =m0, p2= m02 :
2 4 6 8 10
0.2
0.4
0.6
0.8
The renormalization factor becomes
1
21
1
81
2
2
02
eZ , Z1=Z2 because of Ward-Takashi identity.
Pauli-Villars cut-offWe introduce the cutoff at M by the replacement
222
02
22
222
022
02
111
Mpmp
Mm
Mpmpmp
and get for the cross section
2222
02
22
004
42
0)()(
))()((
16)(
kMkpmkp
Mmmkpkdep
If we set
0
log1
m
M
, we get the same ε-series as for the dimensional regularization, i.e. the same
coefficients C-1(Γ1) and C0(Γ1) .Lattice regularization
We get for )( p for ε=π/60 nε=20, i.e.4
nN , Mvr =0.1 ,with a CPU-time per lattice point tε=0.0108s ,
the following values:a(Mvr)=214.4 the integral adaptation factor , ),( vrMk =0.913 the numerical ε-power (should be 1).
Physical effectsThe electron self-energy renormalization factor contributes to the density distribution of the electron
200 Z , e.g. the effective mass of the electron in matter is [11]
obsm , where Ecut-off is the limit energy of the interaction.
The measured energy values in matter are then
, where the correction factor
C=This leads to the famous Lamb-shift (difference in energy between two energy levels 2S1/2 and 2P1/2 of thehydrogen atom)
, where nj is transition energy between the two levels.
In this case nj =1.041GHz, which was measured by W.E. Lamb and first calculated by Hans Bethe in 1947.
9.2 Electron-photon vertex
Γ2= )',( pp = D=0 , one-loop electron-photon vertex, momemtum p, q, p’
11Dimensional regularization
expressed as kd d ,
)',( pp =
or mathematically correct
22
022
02
002
0
3
0
22
)1(2
1
0 0
14
32
0)'()(
)()'(sinsin
16)',(
kmkpmkp
mkpmkpddd
kdkeipp
then kd d transformed into double integral x
dydx1
0
1
0
(double Feynman trick) and with substitution
k→k- p x-p’ y
)',( pp =
and with d=4-ε the result is up to O(ε)
2
1
12
''
3
1
22 2
0
22
m
ppppand therefore
2
1
12
''
3
1
228)',( 2
0
22
2
2
0
m
ppppepp
and the ε-series is
E
m
ppppepp
64
''
12
11
8)',( 2
0
22
2
2
0 , where γE=0.557 is the Euler-Mascheroni-number.
The renormalization factor becomes
1
21
1
81
2
2
01
eZ , Z1=Z2 because of Ward-Takashi identity.
Pauli-Villars cut-off
If we set
0
log1
m
M
, we get the same ε-series as for the dimensional regularization, i.e. the same
coefficients C-1(Γ2) and C0(Γ2) .Lattice regularization
We get for )',( pp for ε=π/60 nε=20, i.e.4
nN , Mvr =0.1 ,with a CPU-time per lattice point
tε=0.00371s , the following values:a(Mvr)=62.6 the integral adaptation factor , ),( vrMk =0.868 the numerical ε-power (should be 1).
Physical effectsThe electron-photon vertex correction contributes to the gyromagnetic factor of the electron:
0011614.1*22
12
S
Mg s
s , which is verified experimentally up to 10-10, making the magnetic
moment of the electron the most accurately verified prediction in the history of physics .
9.3 Photon vacuum polarization
Γ3= )(k = D=2 , photon vacuum polarization, momentum k
Dimensional regularization
expressed as kd d ,
12
)(k =
or mathematically correct
2
022
02
002
0
3
0
22
)1(2
1
0 0
14
32
0)(
))((sinsin
16)(
mkpmp
mkpmpTrddd
pdpeik
then kd d transformed into integral 1
0
dx (Feynman trick) and with substitution
p→q- k x ,
=and after cancellation of the first and the third term under the integral
)(k =-4
and with d=4-ε the result is up to O(ε)
)(k =
setting μ=m0 we get the ε-series:
C-1(Γ3)= )(6
2
2
2
0 kgkke
and with k2=k2
C0(Γ3)= )( 22
0 kgkke
C0(Γ3) in dependence on m0:
The renormalization factor becomes
1
3
21
1
61
2
2
03
eZ
Pauli-Villars cut-off
If we set
0
log1
m
M
, we get the same ε-series as for the dimensional regularization, i.e. the same
coefficients C-1(Γ3) and C0(Γ3) .Lattice regularizationWith the lattice regularization the cancellation of the first and third term under the integral above happens onlyin the limit, so we use the formula with cancellation, which is only logarithmically divergent:
13
2220
2
21
0
4
42
0))1((
))(1(2
24)(
xxkmq
kgkkxxqddxeik
We get for )(k for ε=π/80 nε=26, i.e.4
nN , Mvr =0.1 ,with a CPU-time per lattice point tε=0.00131s ,
the following values:a(Mvr)=204.4 the integral adaptation factor , ),( vrMk =0.99 the numerical ε-power (should be 1).
Physical effectsThe photon vacuum polarization correction contributes to the electrostatic potential of the electron [11]
, where the reduced Compton wavelength
This was confirmed experimentally at the TRISTAN experiment in Japan in 1997.
The effects of vacuum polarization become significant when the external field approaches:
10. Infrared divergence: vertex correction
A general interaction graph can have an infrared divergence (right above) [5].
If we modify the graph by a vertex correction with μ-modified photon propagator22
1),(
kkP , the
infrared divergence cancels out (left above).
We calculate below the μ-modified vertex correction ),',( ppc with the corresponding infrared divergence
and show, that it cancels out in the case of bremsstrahlung as the interaction graph above [1].
To show this, we will begin our calculation with the one-loop vertex correction:
),',( pp =
Our goal is to write this expression in the form:
sandwiched between u(p') and u(p), where F1 and F2 are the form factors that measure the deviation from the
simple , vertex. We will calculate explicit forms for these two form factors. We will find that F1 cancels
against the infrared divergence found in the bremsstrahlung calculation, giving us a finite result.
With and
14
we get
with
follows
where represents the Pauli-Villars contribution, where μ is replaced by Λ.
With this insertion, the integral converges for finite but large Λ. To perform this integration, we must do an
analytic continuation of the previous equation.
With
,
the result for F2 is
and
This gives us the correction to the magnetic moment of the electron
The result for F1 is:
15
compare
E
m
pppppp
64''
12
11
2)',(
2
0
22
with
4
2
0e ,
0
log1
m
M
, where M is the
high-energy cutoff
radiated energy
so
))'((log
2164
''
12
1log
2),',( 20
2
0
22
0
ppm
m
pppp
m
Mpp E
c
finally we get the correction factor
with
we recall that the bremsstrahlung amplitude was given by:
while the vertex correction graph yields for :
Adding these two amplitudes are added together, we find a finite, convergent result independent of μ2, as
desired. The cancellation of infrared divergences is thus shown for the bremsstrahlung. This can be shown for a
general interaction graph from above.
1611 The renormalization groupThe renormalization group equations are based on the plausible assumption that the physical theory cannotdepend on the mass μ which we use as the point of regularization. We have a multiplicative relation between
the vertex functions of the unrenormalized theory and the vertex functions of the renormalized theory
. But the unrenormalized vertex function is independent of the regularization point μ.
Let R represent some renormalization scheme. If Γ0 is an unrenormalized quantity and ΓR is the quantityrenormalized by R, then:
where Z(R) is the renormalization factor under R.For a different renormalization scheme R' we have
Then the relationship between these two renormalized quantities is given by:
Trivially, this satisfies a group multiplication law:Z(R", R')Z(R', R) = Z(R", R)where the identity element is given by:Z(R, R) = 1
The renormalization group equation
Let us make the following definitions (where we now take the limit as ε0):
we derive from the defining equationthe renormalization group equation
where g is the coupling constant, n is the order of g in β(g) , m is the mass, Zφ is thenormalization factor of the fermion wave function.We can solve this equation for β(g)and get for QED
β(g) , β(g)>0 , so the coupling constant increases with the energyFor QCD we obtain the expression
β(g) , where Cad =3 is the number of charges and 4Cf =6 is the number offermions in the interaction, so the coupling constant decreases with the energy, the theory is asymptoticallyfree.
17
References[1] M.Kaku, Quantum Field Theory, Oxford University Press 1993[2] J.D.Bjorken, S.D.Drell, Relativistic Quantum Mechanics, McGraw-Hill 1964[3] Quang Ho-Kim, Xuan-Yem Phan, Elementary particles and their interactions[4] Einan Gardi, Modern Quantum Field Theory, University of Edinburgh, Lecture 2015[5] Douglas Ross, Quantum Field Theory 3, University of Southampton, Lecture 2018[6] Joao Magueijo, Quantum Electrodynamics, Imperial College London, Lecture 2015[7] Andrey Grozin, Lectures on QED and QCD, [arxiv hep-ph/0508242], 2005[8] Robbert Rietkerk, One-loop amplitudes in perturbative quantum field theory, master thesis Utrechtuniversity, 2012[9] Yasumichi Aoki, Non-perturbative renormalization in lattice QCD, [arxiv hep-lat/1005.2339], 2010[10] Jim Branson, Quantum Physics, University of California San Diego, Lecture 2013[11] Wikipedia, Vacuum polarization, 2018