Multiloop integrals in dimensional
regularization made simple
Johannes M. Henn
Institute for Advanced Study
based on
• PRL 110 (2013) [arXiv:1304.1806],
• JHEP 1307 (2013) 128 [arXiv:1306.2799] with A.V. and V. A. Smirnov
• arXiv: 1307.4083 with V. A. Smirnov
supported in part by the Department of Energy grant DE-SC0009988
and the IAS AMIAS fund
September 23, 2013
RADCOR 2013
Johannes M. Henn, IAS
The Abc of loop integrals
• Part 1: Method
- iterated integrals
- solving differential equations for loop integrals
• Part 2: Application and results
- massless 2->2 scattering
- Bhabha scattering
- vector boson production
Iterated integrals
frequently encounter in loop computations
• Functions we
Z
z
log z =
1
dt
t
Li2 (z) =
• More general integration kernels:
Z
z
0
dt1
t1
Z
- harmonic polylogarithms (HPLs), 2d HPLs, ...
- Goncharov polylogarithms:
Z z
dt
Ga1 ,...an (z) =
Ga2 ,...,an (t)
a1
0 t
- Chen
Z iterated integrals
!1 !2 . . . !n
C
C : [0, 1] ! M
t1
0
dt2
1 t2
[Remiddi,Vermaseren, 1999]
[Gehrmann, Remiddi, 2001]
[Goncharov; Brown]
[Chen, 1977]
(space of kinematical variables)
Alphabet: set of differential forms !i = d log ↵i
We will discuss Feynman integrals that evaluate to these functions
Using them will make the structure of the answer completely transparent
Uniform weight
• Number of iterations: weight
• Useful concept: uniform weight
e.g. Ga1 ,...an (z) has weight n
1
1
2
f1 = log x +
Li2 (1
x
1+x
• Pure functions: Q
x)
-linear combinations of uniform weight functions
f2 (x, y) = Li4 (x/y) + 3 log x Li3 (1
y)
Note: derivatives of pure functions have weight reduced by one!
• Generalization: assign weight 1 to ✏ in dimensional regularization
- discuss pure functions in this more general sense
- extremely strong constraint!
Example: one-loop box
=
1
(
s)1+✏ t✏2
+✏
3

2H
⇢
x = t/s
4 + ✏ [ 2 log x] + ✏
1,0,0 (x)
2
+ 2H0,0,0 (x)

4 2
⇡ +
3
7 2
⇡ H 1 (x) + ⇡ H0 (x)
6
2
34
⇣3
3
+ O(✏4 )
Properties of Feynman integrals
• family of integrals: consider arbitrary
integer powers of propagators
a3
a2
a4
a1
• integration-by-parts identities (IBP): linear relations [Chetyrkin, Tkachov, 1981]
[A.V. Smirnov, Petukhov, 2010]
• finite basis of integrals for a given family
e.g. f = {
,
,
• integrals satisfy differential equations
}
[Kotikov, 1991] [Gehrmann, Remiddi, 1999]
fa basis of integrals a = 1 , . . . N
xi kinematical variables, e.g. s = (p1 + p2 )2 , m2 , . . .
@i f (xj , ✏) = Ai (xj , ✏)f (xj , ✏)
(Ai )ab N ⇥ N matrices
Differential equations
• First-order system of DE
• Flat connection
[@i
@i f (xj , ✏) = Ai (xj , ✏)f (xj , ✏)
Ai , @ j
from integrability conditions
Aj ] = 0
(@i @j
@j @i )f = 0
• Gauge transformation (change of basis)
f !Bf
Ai ! B 1 Ai B
B
1
(@i B)
• Idea: choose integrals that are pure functions as basis
- this defines a specific gauge transformation
• Pure functions of uniform weight found systematically:
- using (generalized) unitarity cuts
- using explicit integral parametrizations
[JMH, 2013]
Solving the differential equations
• We find the simplified equations
• Integrability conditions
• pre-integration
@ i Aj
@i f (xj , ✏) = ✏ Ai (xj )f (xj , ✏)
@ j Ai = 0
[Ai , Aj ] = 0
d f (✏, xj ) = ✏ d Ã(xj )f (✏, xj )
where @i à = Ai
observation: matrix à contains only logarithms!
• Solution in terms of Chen iterated integrals
P
f = k 0 ✏k f (k)
R
R
f (0) = const
f (1) = dà f (0)
f (2) = dà f (1)
makes weight properties manifest!
• In general: f = P e
here
C
✏
R
C
d Ã
g(✏)
is a contour in the kinematical space
boundary conditions at base point from physical information
Discussion
• simplified equations
• General solution
d f (✏, xj ) = ✏ d Ã(xj )f (✏, xj )
f =Pe
✏
R
C
d Ã
[JMH, 2013]
g(✏)
à contains (almost) all information about the solution in a compact way:
• expanding solution in ✏ is simple algebra
• contains all information about singularities, asymptotic behavior
• Ã and boundary info uniquely specify solution
• Ã specifies `alphabet’ of differential forms,
i.e. specific class of iterated integrals;
symbol of solution follows as corollary
[cf. Goncharov, Spradlin,Vergu,Volovich, 2010]
Chen iterated integrals are nice objects:
• monodromy invariance;
choice of contour related to integral identities
• underlying Hopf algebra (coproduct)
Part 2:
Applications and results
Example: massless 2 to 2 scattering
[Smirnov, 1999][Gehrmann, Remiddi, 1999]
• good choice of master integrals
[J.M.H., 2013]
4
[email protected]
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arXiv:1205.4991 [hep-ph].
[2] Z. Bern, L. J. Dixon, D. C.
Nucl. Phys., B425, 217 (19
∗
[3] R. Britto, F. Cachazo, B. F
[email protected]
Rev. Lett., 94, 181602 (200
[1] J. Blümlein, PoS, RADCOR2011, 048 (2011),
[4]
N. Arkani-Hamed, J. L. Bou
FIG.
1. Integral basis
corresponding to f1 , . . . f4 (first line)
arXiv:1205.4991
[hep-ph].
Huot,
and J. Trnka,
andZ.fBern,
line),
up Dunbar,
to overalland
factors.
dots
5 , . . . f8
[2]
L.(second
J. Dixon,
D. C.
D. A. Fat
Kosower,
arXiv:1008.2958 [hep-th].
indicate
propagators,
and the
dotted line an inverse
Nucl. doubled
Phys., B425,
217 (1994),
arXiv:hep-ph/9403226.
[5]
N. Arkani-Hamed,
J.
propagator.
incoming momenta
in a clock[3]
R. Britto,The
F. Cachazo,
B. Feng, are
andlabelled
E. Witten,
Phys.
0
0 0 0hazo,
0
0and 0 J. 0Trnka,
- 94,
2 0181602
0 p10
0 lower
0
0 0corner.
wise
order,
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in the
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Rev.
Lett.,
(2005),
hep-th/0501052.
0
0 J.
0 L.0Bourjaily,
0
0 F.0Cachazo,
0
0
0 0 0arXiv:1012.6032
0
0
0 [hep-th].
0
[4] N. Arkani-Hamed,
S. Caron[6]
J.
M.
Drummond
and J.
0
0J. 0
0 0JHEP,
0
0 0
0
0 0 0 0
0
0 0
Huot,
and
Trnka,
1101,
041 (2011),
(2011), arXiv:1008.2965 [he
0
0 [hep-th].
0 0 0
0
0 0
0
0 0 0 0
0
0 0
arXiv:1008.2958
[7] L. J. Dixon, J. M. Drummo
It would be3 interesting to find criteria for, or prove3 or
bä
0 0 0J. - 2L. 0 Bourjaily,
0 0
ä
[5]
N. aArkani-Hamed,
F. leads
- Cac-0 3 01201,
1 024
0 (2012),
0 0arXiv:111
2
disprove
the existence
of a matrix B of eq. (3) that
2
hazo,
and
J.
Trnka,
JHEP,
1206,
125
(2012),
1
1
0 0 that
- 2 beyond
0 0 the example
0
0 0 [8]0J. M.
0 Drummond,
2
0 0 J. M. H
to (4). We would
like0to stress
2
2 [hep-th].
arXiv:1012.6032
3
6 6 21104,
- 4 083
- 12(2011),
2 2arXiv:101
given
here,
we
- 3found
- 3 0this
12to apply
- 2JHEP,
0 to many
[6]
J. M.
Drummond
and0method
J. 4M. Henn,
1105,fur105
[9]
F.
Brown,
(2006),
arXiv:06
9
9
ther
cases
of
practical
interest.
In
particular,
we
expect
3
3
1
4
18
1
1
3
3
1
4
18
1
1
(2011), arXiv:1008.2965
[hep-th].
[10] A. B. Goncharov, (2009), a
2
2
theL.method
to J.
apply
equally to massive
non-planar
[7]
J. Dixon,
M. Drummond,
and J. and
M. Henn,
JHEP, [11] A. B. Goncharov, M. S
integrals.
A more
detailed
discussion [hep-th].
and applications to
1201, 024
(2012),
arXiv:1111.1704
A. Volovich, Phys.Rev.L
arXiv:1006.5703 [hep-th].
[8]
J. M.
Drummond,
J. M. will
Henn,
and J.
Trnka, JHEP,
more
general
loop integrals
be given
elsewhere.
[12] F. Cachazo, (2008), arXiv:
1104,
(2011), arXiv:1010.3679
[hep-th].or processes
In
more083
complicated
multi-leg processes,
[13] N. Arkani-Hamed, J. L
[9] F. Brown, (2006), arXiv:0606419 [math.AG].
involving masses, the appropriate set of integral functions
A. B. Goncharov, A. P
[10] A. B. Goncharov, (2009), arXiv:0908.2238 [math.AG].
may
not
yet
be
known.
We
anticipate
that
our
method
arXiv:1212.5605 [hep-th].
[11] A. B. Goncharov, M. Spradlin, C. Vergu,
and
willA.be aVolovich,
convenient
way
of
solving
this
problem,
and
lead
[14]
T. Gehrmann, J. M. Henn,
Phys.Rev.Lett., 105, 151605 (2010),
to arXiv:1006.5703
investigations of [hep-th].
generalized functions appropriate for
101 (2012), arXiv:1112.4524
[15] K. Chetyrkin and F. Tkac
those
scattering(2008),
processes.
In this context
we also wish to
[12]
F. Cachazo,
arXiv:0803.1988
[hep-th].
(1981).
stress
the differential
can firstF.be trivially
[13]
N. that
Arkani-Hamed,
J. equations
L. Bourjaily,
Cachazo,
[16] C. Anastasiou and A. La
A. B.
Goncharov,
A. Postnikov,
al.,
(2012),
solved
in terms
of symbols,
and possibleet simplifications
(2004), arXiv:hep-ph/0404
arXiv:1212.5605
[hep-th].
identified,
before the
problem of finding a convenient inibid., 0810, 107 (2008)
[14]
T. Gehrmann,
J. M.isHenn,
and T. Huber, JHEP, 1203,
tegral
representation
addressed.
A.
Smirnov
and
V
101 (2012), arXiv:1112.4524 [hep-th].
• Knizhnik-Zamolodchikov equations

a
b
@x f = ✏
+
f
x 1+x
xIt would
= bet/s
interesting to find criteria for, or prove or
FIG. 1. Integral basis corresponding to f1 , . . . f4 (first line)
and f5 , . . . f8 (second line), up to overall factors. Fat dots
indicate doubled propagators, and the dotted line an inverse
propagator. The incoming momenta are labelled in a clockwise order, starting with p1 in the lower left corner.
disprove the existence of a matrix B of eq. (3) that leads
to (4). We would like to stress that beyond the example
given here, we found this method to apply to many further cases of practical interest. In particular, we expect
the method to apply equally to massive and non-planar
integrals. A more detailed discussion and applications to
more general loop integrals will be given elsewhere.
In more complicated multi-leg processes, or processes
involving masses, the appropriate set of integral functions
may not yet be known. We anticipate that our method
will be a convenient way of solving this problem, and lead
to investigations of generalized functions appropriate for
• (regular) singular points
s = 0,
t = 0,
u=
s
t=0
• boundary conditions: finiteness as
u!0
∗
Generalization to 3 loops
• all planar 2 to 2 three-loop master integrals
[J.M.H., A.V. Smirnov,V.A. Smirnov, 2013]
(26 master integrals) (41 master integrals)

a
b
@x f = ✏
+
f
x = t/s
x 1+x
• a, b constant N ⇥ N matrices, N = 26 or N = 41
k
• solution to arbitrary order ✏ in terms of HPLs of weight k
by simple algebra
• boundary conditions from finiteness at x = 1
• single-scale form factor integrals determined algebraically!
[J.M.H., A.V. Smirnov,V.A. Smirnov,
• similar pattern for non-planar integrals
in progress]
Bhabha scattering
• we will discuss the integral family
t
s
• kinematics
s
(1 x)2
=
2
m
x
p2i = m2
t
(1 y)2
=
2
m
y
• previous work (using Mellin-Barnes or DE)
[Smirnov, 2001] [Heinrich, Smirnov, 2004]
[Czakon, Gluza, Kajda, Riemann, 2004-2006]
[Bonciani, Ferroglia, Mastrolia, Remiddi, van der Bij, 2003, 2004]
many individual integrals computed, to some order in ✏
problems with coupled differential equations
One-loop warm-up
• differential equations
h
0
B
à = @
0
0
1
0
0
0
B
+@
0
B
+@
0
0
0
0
4
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
C
B
A log x + @
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
2
0
0
0
0
1
• alphabet:
d f = ✏ d à f
1
0
0
0
0
0
0
0
0
0
0
0
C
B
A log(1 + y) + @
1
0
C
B
A log(x + y) + @
0
0
2
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
[J.M.H.,V.A. Smirnov, 2013]
1
0
C
B
A log(1 + x) + @
0
0
0
2
0
0
0
0
0
2
0
0
0
0
2
0
0
0
0
1
1
1
C
A log(1
two-dimensional HPLs
0
1
0
2
0
0
0
0
0
4
0
0
0
0
0
C
A log y+
i
[Gehrmann, Remiddi, 2001]
• (regular) singular points of DE have physical meaning, e.g.:
x=1 $ s=0
x = 1 $ s = 4m2
0
0
0
0
0
y)+
C
A log(1 + xy)
{x , 1 ± x , y , 1 ± y , x + y , 1 + xy}
(natural generalization of)
0
0
0
2
0
1
x=0 $ s=1
x= y $ u=0
Two-loop case
• differential equations
d f = ✏ d à f
à =B1 log(x) + B2 log(1 + x) + B3 log(1
[J.M.H.,V.A. Smirnov, 2013]
x) + B4 log(y) + B5 log(1 + y)
+ B6 log(1
y) + B7 log(x + y) + B8 log(1 + xy)
✓
◆
1+Q
2
2
+ B9 log(x + y 4xy + x y + xy ) + B10 log
1 Q
✓
◆
✓
(1 + x) + (1 x)Q
(1 + y) + (1
+ B11 log
+ B12 log
(1 + x) (1 x)Q
(1 + y) (1
s
(x + y)(1 + xy)
Q=
x + y 4xy + x2 y + xy 2
Here
and Bi are constant 23 ⇥ 23 matrices
y)Q
y)Q
◆
• larger, 12-letter alphabet compared to one loop
• solution to any order in terms of Chen iterated integrals
• observation: except for one integral, up to weight 4, only the
one-loop alphabet is needed!
vector boson production
equal mass case: [Gehrmann, Tancredi, Weihs, 2013]
[see L.Tancredi’s talk]
general case: [J.M.H.,V.A. Smirnov, to appear]
• planar integral families
Q2
P2
• diff. equations
• example:
d f = ✏ d à f
parametrization (removes square roots):
P2
(x + 1)z
=
,
s
y+1
14
X
à =
Bi log(↵i )
t
= x,
s
• 14- letter alphabet:
i=1
x, 1 + x, y, 1 y, 1 + y, z, 1 z, y
y z + xy yz, y + xy xz xyz
1 + y z xz, x z xz + xy
Q2
(x + 1)y
=
s
(y + 1)z
z, y
• boundary constants from physical conditions
• other cases solved similarly
x, 1
xy
Conclusions and outlook
• criteria for finding optimal loop integral basis
- important concept: pure functions of uniform weight
- related questions in generalized unitarity approach
• new form of differential equations
- make properties of functions manifest
(analytic structure, discontinuities, monodromy invariance)
- trivial to solve in terms of Chen iterated integrals
- solution specified by constant matrices! (to all orders in ✏ )
- monodromy invariance, coproduct structure
helpful for rewriting answer (e.g. for faster numerical evaluation)
• further applications:
- a library for NNLO integrals relevant for phenomenology
- top quark amplitudes
- non-planar integrals
- phase space integrals
[von Manteuffel, Studerus 2011-2013;
JMH, von Manteuffel, Smirnov, to appear]