PROBING GLUON SATURATION THROUGH DI-HADRON AND TRI-HADRON - - PowerPoint PPT Presentation
PROBING GLUON SATURATION THROUGH DI-HADRON AND TRI-HADRON CORRELATIONS AT A FUTURE ELECTRON ION COLLIDER MARTIN HENTSCHINSKI martin.hentschinski@gmail.com IN COLLABORATION WITH A. AYALA, J. JALILIAN-MARIAN, M.E. TEJEDA YEOMANS, XV MEXICAN
MARTIN HENTSCHINSKI
martin.hentschinski@gmail.com
IN COLLABORATION WITH
XV MEXICAN WORKSHOP ON PARTICLES AND FIELDS
(02.-06. NOV. 2015)
parton distribution functions grow like powers for x→0 with x=Q2/2p･q ∈[0,1]
to find a quark, gluon with proton momentum fraction x in proton
→integral over x does not convergent at x=0 → invalidates probability interpretation at some x, new QCD dynamics must set in
k p X k' q
0.2 0.4 0.6 0.8 1
10
10
10
10 1
HERAPDF2.0 NLO uncertainties: experimental model parameterisation HERAPDF2.0AG NLO
x xf
2
= 10 GeV
2 f
µ
v
xu
v
xd 0.05) × xS ( 0.05) × xg (
H1 and ZEUS
HERA collider (92-07): Deep Inelastic Scattering (DIS) of
Photon virtuality Q2 = −q2
Open Questions
Geometric Scaling
Y = ln 1/x
non-perturbative region ln Q2 Q2
s(Y)
s a t u r a t i
r e g i
Λ2
QCD
αs < < 1 αs ~ 1 BK/JIMWLK DGLAP BFKL
I effective finite size 1/Q of
I at some x ⌧ 1, partons
I turning it around: system is
I grows with energy Qs ⇠ x−∆,
THEORY PREDICTIONS FOR HIGH & SATURATED GLUON DENSITIES
x =Q2/2p･q→0 limit corresponds to perturbative high energy limit 2p･q→∞ for fixed resolution Q2
high energy limit
“color dipole factor” (universal, resums ln1/x)
splits into color dipole (quark- antiquark pair) which interacts with Lorentz contracted target field
gluon densities multiple scatterings
γ∗
→
γ∗ x → 0: a single interaction with a strong & Lorentz contracted gluon field
φ Δ φ Δ φ Δ φ Δ
≡ ≡ A+,a(z−, z) = ↵a(z)(z−)
k p X k' q
σγ∗A
L,T (x, Q2) = 2
X
f
Z d2bd2r
1
Z dz
L,T (r, z; Q2)
N(x, r, b)
in Hentschinski (ICN-UNAM) The glue that binds us all November 3, 2015
4
PROPAGATORS IN THE PRESENCE OF A STRONG BACKGROUND FIELD
p q
= 2⇡(p− − q−)n
Z dd−2ze−iz·(p−q) · n ✓(p−)[V (z) − 1] − ✓(−p−)[V †(z) − 1]
q
= −2⇡(p− − q−)2p− Z dd−2ze−iz·(p−q) · n ✓(p−)[U(z) − 1] − ✓(−p−)[U †(z) − 1]
Z ∞
−∞
dx−A+,c(x−, z)tc U(z) ≡ U ab(z) ≡ P exp ig Z ∞
−∞
dx−A+,c(x−, z)T c
p q
= (2π)dδ(d)(p − q) ˜ S(0)
F (p) + ˜
S(0)
F (p)
p q
˜ S(0)
F (q)
p, µ q, ν
= (2π)dδ(d)(p − q) ˜ G(0)
µν (p) + ˜
G(0)
µα(p)
p q
˜ G(0)
αν (q)
˜ S(0)
F (p) =
ip + m p2 − m2 + i0 ˜ G(0)
µν (p) = idµν(p)
p2 + i0
dµν(p) = −gµν + n−
µ pν + pµn− ν
n− · p
interaction with the background field: strong background field resummed into path ordered exponentials (Wilson lines)
[Balitsky, Belitsky; NPB 629 (2002) 290], [Ayala, Jalilian-Marian, McLerran, Venugopalan, PRD 52 (1995) 2935-2943], …
use light-cone gauge, with k-=n-･k, (n-)2=0, n-~ target momentum
photon wave function and dipole density
JIMWLK or BK evolution equation in ln(1/x)
allows description of combined HERA data, but also (dilute!) DGLAP describes data
0.5 1 1.5
Data Theory
r
!
2
=0.85 GeV
2
Q
0.5 1 1.5
r
!
2
=4.5 GeV
2
Q
0.5 1 1.5
r
!
2
=10.0 GeV
2
Q
5 30.5 1 1.5
r
!
2
=15.0 GeV
2
Q
−5
10
−4
10
−3
10
−2
10
0.5 1 1.5
r
!
2
=35 GeV
2
Q x
5 32
=2.0 GeV
2
Q
2
=8.5 GeV
2
Q
2
=12.0 GeV
2
Q
5 32
=28.0 GeV
2
Q
−4
10
−3
10
−2
10
2
=45 GeV
2
Q
x
σγ∗A
L,T (x, Q2) = 2
X
f
Z d2bd2r
1
Z dz
L,T (r, z; Q2)
N(x, r, b)
in Hentschinski (ICN-UNAM) The glue that binds us all November 3, 2015
γ∗
splitting recombination
6 [Albacete, Armesto, Milhano,Quiroga, Salgado,EPJ C71 (2011) 1705]
Saturation: high densities in the fast nucleus
Expect those effects to be even more enhanced in boosted nuclei:
Boost
Q2
s ∼ # gluons/unit transverse area ∼ A1/3
COLOR GLASS CONDENSATE (CGC)= BUZZWORD WHICH REFERS TO THE PHYSICS OF SATURATION AND IN PARTICULAR THE DEVELOPED THEORY
d-Au collisions at RHIC: depletion of away side peak in central collisions described by CGC many more studies at RHIC, LHC in pp, pA, AA collisions plethora of interesting phenomena, but also subject to large theory uncertainties due to uncontrolled re- scatterings→ no ultimate proof
7
instead of going to higher energies (expensive), possible to study large nuclei ….
THE ELECTRON ION COLLIDER PROJECT
A COLLIDER TO SEARCH FOR A DEFINITE ANSWER:
the world’s first eA collider: will allow to probe heavy nuclei at small x (using 16GeV electrons on 100GeV/u ions) Brookhaven National Laboratory: supplement RHIC with Electron Recovery Linac (eRHIC) Jefferson Lab: supplement CEBAF with hadron accelerator (MEIC)
2015: ENDORSED BY NUCLEAR SCIENCE ADVISORY COMMITTEE (NSAC) AS HIGHEST PRIORITY FOR NEW FACILITY CONSTRUCTION IN US NUCLEAR SCIENCE LONG RANGE PLAN
AN EIC OBSERVABLE TO SEARCH FOR SATURATION EFFECTS: DI-HADRON DE-CORRELATION IN DIS
αs < < 1 αs ∼ 1 ΛQCD
know how to do physics here
?
Qs
kT ~ 1/kT kT φ(x, kT
2)
collinear factorization (dilute pQCD): gluon kT peaked at kT=0 - expect dihadrons back-to-back Saturation (CGC): gluon kT peaked at saturation scale - expect de-correlated di-hadrons measure azimuthal angle of di- hadron final state
γ∗
, y=0.7
2
=1 GeV
2
Q
(rad) φ Δ
2 2.5 3 3.5 4 4.5
) φ Δ C(
0.05 0.1 0.15 0.2 0.25
ep eCa eAu 20 GeV on 100 GeV
(rad) φ Δ
2 2.5 3 3.5 4 4.5
) φ Δ C(
0.1 0.2 0.3 0.4
e+Au - no-sat eAu - sat
pT
trig > 2 GeV/c
1 GeV/c < pT
assoc < pT trig
0.2 < zh
trig, zh assoc < 0.4
1 < Q2 < 2 GeV2 0.6 < y < 0.8
20 GeV on 100 GeV
expect also (large?) collinear logs + scale setting uncertainties → higher order correction can lead to large effects
[rad] φ ∆
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
) φ ∆ C(
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
ep, No Sudakov eAu, No Sudakov ep, With Sudakov eAu, With Sudakov
10 GeV x 100 GeV
2
= 1 GeV
2
Q
[Zheng,Aschenauer, Lee, Xiao, PRD89 (2014)7, 074037]
evolution of dipole etc. densities & higher correlators know up to NLO instabilities get addressed photon wave function: only inclusive (on the level of correlation functions)
[Balitsky, Chirilli; PRD 88 (2013) 111501, PRD 77 (2008) 014019]; [Kovner,Lublinsky, Mulian; PRD 89 (2014) 6, 061704] [Iancu, Madrigal, Mueller, Soyez, Triantafyllopoulos; PLB 744 (2015) 293] [Balitsky, Chirilli; PRD 87 (2013) 1, 014013], [Beuf; PRD 85 (2012) 034039]
expect also (large?) collinear logs + scale setting uncertainties → higher order correction can lead to large effects
[rad] φ ∆
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
) φ ∆ C(
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
ep, No Sudakov eAu, No Sudakov ep, With Sudakov eAu, With Sudakov
10 GeV x 100 GeV
2
= 1 GeV
2
Q
[Zheng,Aschenauer, Lee, Xiao, PRD89 (2014)7, 074037]
expect more stringent tests of CGC through more complex final state
reduce uncertainties + possibly identify overlap region between collinear factorisation and saturation physics
→ 15! = 1307674368000 individual terms (not all non-zero though) necessary to achieve (potential) cancelations of diagrams BEFORE evaluation require automatization of calculation (= use of Computer algebra codes)
= + +
di-hadrons at LO: paper & pencil calculation e.g.[Gelis, Jalilian-Marian,PRD67, 074019 (2003) ] each line & each final state splits into two terms (free + interaction) → real NLO: 16 diagrams (amp. level) → virtual NLO: 32 diagrams (amp. level)
each vertex + four momentum integral at each internal internal line
line separating positive & negative light-cone time
possible (~vertical cuts)
∆(0)
F (x) =
Z ddp (2π)d i · e−ip·x p2 − m2 + i0 = Z dp+ (2π) Z dp−dd−2p (2π)d−1 e−ip−x++ip·x 2p− · i · e−ip+x− p+ − p2+m2−i0
2p−
= Z dp−dd−2p (2π)d−1 e−ipx 2p− ⇥ θ(p−)θ(x−) − θ(−p−)θ(−x−) ⇤
p+= p2+m2
2p−
p−
1
p−
2
x−
1
x−
2
x−
3
k−
p−
1
p−
2
p−
3
x−
1
x−
2
k−
Z ∞
−∞
dx−
i →
Z 0
−∞
dx−
i +
Z ∞ dx−
i 14
p−
1
p−
2
x−
1
x−
2
x−
3
k−
not altered through interaction
→ interaction must be always placed at a z-=0 cut of the diagram. Note: this applies equally to configuration and momentum space
p q
∝ δ(p− − q−)
≡ ≡ A+,a(z−, z) = ↵a(z)(z−)
forbidden configurations: cannot be accommodated by vertical (s- channel type) cut
15
consider complete configuration space propagator (free + interacting part)
Z SF (x, y) = Z ddp (2π)d ddq (2π)d e−ipx ˜ S(0)
F (p)(2π)dδ(d)(p − q) + ˜
S(0)
F (p)τF (p, q) ˜
S(0)
F (q)
propagator proportional to complete Wilson line V (fermion)
cut at light-cone time 0 no direct translation to momentum space adding free propagation & interaction→ mixing of different mom. space diagrams but strong constraints on the structure of the full result
16 z− = 0 x y z− = 0 x y
p−
1
p−
2
p−
3
x−
1
x−
2
k−
∝ V †(y)V (x)ta
p−
1
p−
2
p−
3
x−
1
x−
2
k−
∝ V †(y)tbV (x)U ba(z)
p−
1
p−
2
p−
3
x−
1
x−
2
k−
∝ taV †(y)V (x)
p−
1
p−
2
p−
3
x−
1
x−
2
k−
∝ V †(y)tbV (x)U ba(z).
CONFIGURATION SPACE PREDICTS WHICH OPERATORS HAVE NON-ZERO COEFFICIENTS
momentum space: necessary coefficients from only 4 (instead of 16) diagrams
(cancelation of all other contributions verified by explicit calculations) 17
virtual corrections: similar result, necessary coefficients from 8 (instead of 32) diagrams
something slightly strange:
technical reason:
reduction procedure
evaluate momentum integrals trivially
absent intuitive picture: background field = t-channel gluons interacting with the target→ naturally provide a loop which is factorized & (partially) absorbed into the projectile in the high energy limit
for the rest of the talk: focus on real corrections/3 partons
a 1-loop and a 2-loop topology k1 and k2 are loop momenta new complication: exponentials/Fourier factors conventional: e.g. k1
+ integration by taking residues, then transverse integrals
particular for 2 loop case: complicated transverse integrals developed a new technique
★ complete exponential factors to 4 d ★ evaluate integral using “standard” momentum space techniques
p k q l −q − k −k1 l − k1 p k q l k2 k2 − k1 −k1 l − k1
20
start with integral which contains delta functions transverse exponential factors introduce relative coordinate r=x-y represent delta function by integral introduce dummy integral over r+ ➜ obtain 4 (d) dimensional integral next step: Schwinger-/α-parameters complete square in exponent, Wick rotation, Gauss integration, etc. reconstruct delta function to evaluate (some) integrals over α-parameters to facilitate these steps for 2, 3 loops (virtual!): “developed” Mathematica package ARepCGC; implements necessary text-book methods [V. Smirnov, Springer 2006]
I(p1, p2) = Z ddk1 i⇡d/2 1 [k2
1][(l − k1)2]eixt(·k1,t−p1,t)e−iyt·(k1,t+p2,t)(2⇡)2(p− 1 − k− 1 )(l− − k− 1 − p− 2 )
Z Z Z
− − ✓ i k2 − m2 + i0 ◆λ = 1 Γ() Z ∞ d↵ ↵λ−1eiα(k2−m2+i0)
Z − I(p1, p2) = 2⇡(l− − p−
1 − p− 2 )e−iyt·(p1,t+p2,t)
Z dr+ Z dr−(r+) Z ddk1 i⇡d/2 1 [k2
1][(l − k1)2]eir·k1
◆ Z
21
ξ, 𝞻1, 𝞻3max in terms of external momenta
◆ Z I(p1, p2) = 8⇡2(l− − p−
1 − p− 2 )e−ixt·p1,t
l− e−iyt·p2,tK0 ⇣p ↵(1 − ↵)Q2(x − y)2 ⌘ , ↵ = p−
1 /l−
Z d4k1 (2⇡)4 Z d4k3 (2⇡)4 (2⇡)3(k−
1 k− 3 p− 1 )(l− k− 1 p− 2 )(k− 3 p− 3 )
[k2
1][(l k1)2][(k1 k3)2][k2 3]
eixt·(k1,t−k3,t−p1,t)eiyt·(lt−k1,t−p2,t)eizt·(k3,t−p3,t) / e−ixt·p1,te−iyt·p2,te−izt·p3,t Z ⇢max
3
d⇢3 ⇢3 K0 "s ⇢1(1 ⇢1)Q2((x y)2 + ⇢3(x(1 ⇠) y + ⇠z)2) ⇢3 #
2-loop integral: evaluated into infinite sum over Bessel functions; numerics: keeping integral might be most stable tensor integrals from differentiation w.r.t. external coordinates inclusive: obtain (unexpected) endpoint contributions
22
Dirac trace to two general tensor integrals
[Vermaseren, math-ph/0010025]
usable (~23 pages)
simplification through reduction
(work in progress)
A1squared = + qminus * ( DENn(k)*dot(p,k)*IntR1(nminus,nminus,nminus,1,1,1,p)* IntR1c(muc1,muc1,nminus,1,1,1,p) + DENn(k)*dot(p,k)*IntR1(nminus, nminus,nminus,1,1,1,p)*IntR1c(muc2,muc2,nminus,1,1,1,p) - DENn(k)* dot(p,k)*IntR1(nminus,mu2,nminus,1,1,1,p)*IntR1c(nminus,mu2,nminus,1, 1,1,p) - DENn(k)*dot(p,k)*IntR1(nminus,muc2,nminus,1,1,1,p)*IntR1c( nminus,muc2,nminus,1,1,1,p) - DENn(k)*dot(p,k)*IntR1(mu1,nminus, nminus,1,1,1,p)*IntR1c(mu1,nminus,nminus,1,1,1,p) + DENn(k)*dot(p,k)* IntR1(mu1,mu1,nminus,1,1,1,p)*IntR1c(nminus,nminus,nminus,1,1,1,p) + DENn(k)*dot(p,k)*IntR1(mu2,mu2,nminus,1,1,1,p)*IntR1c(nminus,nminus, nminus,1,1,1,p) - DENn(k)*dot(p,k)*IntR1(muc1,nminus,nminus,1,1,1,p)* IntR1c(muc1,nminus,nminus,1,1,1,p) - IntR1(nminus,nminus,p,1,1,1,p)* IntR1c(muc1,muc1,nminus,1,1,1,p) + IntR1(nminus,nminus,p,1,1,1,p)* IntR1c(muc2,muc2,nminus,1,1,1,p) + IntR1(nminus,mu2,p,1,1,1,p)* IntR1c(nminus,mu2,nminus,1,1,1,p) - IntR1(nminus,muc2,p,1,1,1,p)* IntR1c(nminus,muc2,nminus,1,1,1,p) + IntR1(mu1,p,mu1,1,1,1,p)*IntR1c( nminus,nminus,nminus,1,1,1,p) - IntR1(mu1,nminus,p,1,1,1,p)*IntR1c( mu1,nminus,nminus,1,1,1,p) - IntR1(mu1,nminus,mu1,1,1,1,p)*IntR1c(p, nminus,nminus,1,1,1,p) + IntR1(mu1,mu1,p,1,1,1,p)*IntR1c(nminus, nminus,nminus,1,1,1,p) - IntR1(mu2,mu2,p,1,1,1,p)*IntR1c(nminus, nminus,nminus,1,1,1,p) - IntR1(mu3,p,mu3,1,1,1,p)*IntR1c(nminus, nminus,nminus,1,1,1,p) + IntR1(mu3,nminus,mu3,1,1,1,p)*IntR1c(p, nminus,nminus,1,1,1,p) + IntR1(muc1,nminus,p,1,1,1,p)*IntR1c(muc1, nminus,nminus,1,1,1,p) ) + pminus*qminus * ( - DENn(k)*IntR1(k,nminus,nminus,1,1,1,p)*IntR1c( muc3,nminus,muc3,1,1,1,p) + DENn(k)*IntR1(k,nminus,mu3,1,1,1,p)* IntR1c(mu3,nminus,nminus,1,1,1,p) - DENn(k)*IntR1(k,mu3,mu3,1,1,1,p)* IntR1c(nminus,nminus,nminus,1,1,1,p) + DENn(k)*IntR1(k,muc3,nminus,1, 1,1,p)*IntR1c(nminus,nminus,muc3,1,1,1,p) + DENn(k)*IntR1(nminus,k, nminus,1,1,1,p)*IntR1c(nminus,muc3,muc3,1,1,1,p) - DENn(k)*IntR1( nminus,k,mu3,1,1,1,p)*IntR1c(nminus,mu3,nminus,1,1,1,p) + DENn(k)* IntR1(nminus,nminus,k,1,1,1,p)*IntR1c(muc1,muc1,nminus,1,1,1,p) - DENn(k)*IntR1(nminus,nminus,nminus,1,1,1,p)*IntR1c(k,muc3,muc3,1,1,1, p) + DENn(k)*IntR1(nminus,nminus,nminus,1,1,1,p)*IntR1c(muc2,muc2,k,1 ,1,1,p) + DENn(k)*IntR1(nminus,nminus,nminus,1,1,1,p)*IntR1c(muc3,k, muc3,1,1,1,p) + DENn(k)*IntR1(nminus,nminus,mu3,1,1,1,p)*IntR1c(k,mu3 ,nminus,1,1,1,p) - DENn(k)*IntR1(nminus,nminus,mu3,1,1,1,p)*IntR1c( mu3,k,nminus,1,1,1,p) - DENn(k)*IntR1(nminus,mu2,k,1,1,1,p)*IntR1c( nminus,mu2,nminus,1,1,1,p) + DENn(k)*IntR1(nminus,mu3,mu3,1,1,1,p)* IntR1c(nminus,k,nminus,1,1,1,p) - DENn(k)*IntR1(nminus,muc2,nminus,1, 1,1,p)*IntR1c(nminus,muc2,k,1,1,1,p) - DENn(k)*IntR1(nminus,muc3, nminus,1,1,1,p)*IntR1c(nminus,k,muc3,1,1,1,p) - DENn(k)*IntR1(mu1, nminus,nminus,1,1,1,p)*IntR1c(mu1,nminus,k,1,1,1,p) + DENn(k)*IntR1(
working on it
before, but never been exploited in a systematic way for this kind of calculation
advantage: benefit from standard techniques for higher orders in QCD (important: soft- and collinear singularities, ….)
EIC studies, but the developed techniques should also allow to evaluate NLO correction for saturation/CGC observables in e.g. pA at RHIC/LHC
to-leading order - requires a systematic approach
DANKESCHÖN!
I knowldege of scattering enery + nucleon mass
+ measure scattered electron control kinematics k p X k' q Photon virtuality Q2 = −q2 Mass of system X W = (p + q)2 = M 2
N + 2p · q − Q2
Bjorken x = Q2 2p · q Resolution λ ∼ 1
Q
Inelasticity y = 2p · q 2p · k
27
proton structure functions = convolution of parton distribution functions (pdfs): = probability to find parton (quark, gluon) which carries the fraction x of the proton momentum (non- perturbative→ from fits to data) and coefficents C2q =1 + αs C2q
(1)+ …, C2g = αs C2g (1)+
… (calculated in perturbation theory) exact theory statement up to terms suppressed by 1/Q!
28
k=q,g
HERA@DESY (1992-2007): at the first time DIS on a proton at a Collider → access to small x region [large c.o.m. energy at fixed resolution Q] important HERA result: proton at small x dominated by gluons and seaquarks (qqbar pairs from gluon) powerlike rise of gluon distribution at small x BUT: rise cannot continue forever (probability distribution!)
0.2 0.4 0.6 0.8 1
10
10
10
10 1
HERAPDF2.0 NLO uncertainties: experimental model parameterisation HERAPDF2.0AG NLO
x xf
2
= 10 GeV
2 f
µ
v
xu
v
xd 0.05) × xS ( 0.05) × xg (
H1 and ZEUS
DONEC QUIS NUNC
10
10
10
10
10 1 10
10
10
10
10
10 1 10
x x Q2 (GeV2) Q2 (GeV2)
Q2
s,quark Model-I
b=0 Au, median b Ca, median b p, median b
Q2
s,quark, all b=0
Au, Model-II Ca, Model-II Ca, Model-I Au, Model-I xBJ × 300 ~ A
1/3
Au Au p Ca Ca
Saturation: high densities in the fast nucleus
Expect those effects to be even more enhanced in boosted nuclei:
Boost
Q2
s ∼ # gluons/unit transverse area ∼ A1/3
conventional pQCD (make use of know techniques) inclusion of finite masses (charm mass!) intuition: interaction at t=0 with Lorentz contracted target momentum space well explored complication, but doable lose intuitive picture at first -> large # of cancelations configuration space poorly explored very difficult many diagrams automatically zero
work in momentum space, but exploit relation to configuration space to set a large fraction of all diagrams to zero
Z d4q (2π)4 τ(p, q) ˜ ∆F (q)eiq·y
Z d4q (2π)4 d4q (2π)4 e−ip·x ˜ ∆F (p)τ(p, q) ˜ ∆F (q)eiq·y
Determine Fourier transform of “background field vertex” for propagator and final state Find light-cone time constraints and reason: conservation of light-cone momentum at vertex 𝞾 important consequence: interaction for each diagram only allowed along a certain time-slice =cut of diagrams
and x− > 0 > y− for p− < 0
y− > 0 > x− for p− > 0
p−
1
p−
2
p−
3
x−
1
x−
2
k−
p−
1
p−
2
p−
3
x−
1
x−
2
k−
example: 3 partons (real NLO): interaction term 𝞾 only allowed if the regarding line is “cut” examples of excluded configurations
32
33
p−
1
p−
2
x−
1
x−
2
x−
3
k− p−
1
p−
2
x−
1
x−
2
x−
3
k− p−
1
p−
2
x−
1
x−
2
x−
3
k− p−
1
p−
2
x−
1
x−
2
x−
3
k− p−
1
p−
2
x−
1
x−
2
x−
3
k− p−
1
p−
2
x−
1
x−
2
x−
3
k− p−
1
p−
2
x−
1
x−
2
x−
3
k−
applies also to virtual diagrams: organized into ‘cut’ configurations Note: different cuts can contain the same diagram
loop integraexpressls
Iµ1µ2µ3
1
(p1, p2) = Z ddk1 i⇡d/2 kµ1
1 (l − k1)µ2pµ3 1
[k2
1][(l − k1)2][p2 1]eixt·(k1,t−p1,t)eiyt·(−k1,t−p2,t)
(2⇡)2(p−
1 − k− 1 )(l− − k− 1 − p− 2 )
Iµ1µ2µ3µ4
R2
(p1, p2, p3) = = Z d4k1 (2⇡)4 Z d4k3 (2⇡)4 kµ1
1 (l − k1)µ2(k1 − k3)µ3kµ4 3
[k2
1 − m2][(l − k1)2 − m2][(k1 − k3)2 − m2][k2 3]
eixt·(k1,t−k3,t−p1,t)eiyt·(lt−k1,t−p2,t)eizt·(k3,t−p3,t) (2⇡)3(k−
1 − k− 3 − p− 1 )(l− − k− 1 − p− 2 )(k− 3 − p− 3 )