arXiv:0904.1962v3
[hepph]
23 Jan 2010
Unparticle selfinteractions
Howard Georgi,
1
Yevgeny Kats,
2
Center for the Fundamental Laws of Nature
Jefferson Physical Laboratory
Harvard University
Cambridge, MA 02138
Abstract
We develop techniques for studying the effects of selfinteractions in the conformal
sector of an unparticle model. Their physics is encoded in the higher
n
point functions
of the conformal theory. We study inclusive processes and argue that the inclusive
production of unparticle stuff in standard model processes due to the unparticle self
interactions can be decomposed using the conformal partial wave expansion and its
generalizations into a sum over contributions from the production of various kinds
of unparticle stuff, corresponding to different primary conformal operators.
Such
processes typically involve the production of unparticle stuff associated with oper
ators other than those to which the standard model couples directly.
Thus just as
interactions between particles allow scattering processes to produce new particles in
the final state, so unparticle selfinteractions cause the production of various kinds
of unparticle stuff. We discuss both inclusive and exclusive methods for computing
these processes. The resulting picture, we believe, is a step towards understanding
what unparticle stuff “looks like” because it is quite analogous to way we describe
the production and scattering of ordinary particles in quantum field theory, with the
primary conformal operators playing the role of particles and the coefficients in the
conformal partial wave expansion (and its generalization to include more fields) play
ing the role of amplitudes. We exemplify our methods in the 2D toy model that we
discussed previously in which the BanksZaks theory is exactly solvable.
1
[email protected]
2
[email protected]
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Contents
1
Introduction
3
2
Unparticle selfinteractions
8
3
2D toy model of unparticle physics
12
3.1
Sommerfield model of a BanksZaks sector
. . . . . . . . . . . . . . . . . . .
12
3.2
Coupling to the “standard model”
. . . . . . . . . . . . . . . . . . . . . . . .
14
4
Correlation functions of the unparticle operator
15
4.1
2point function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.2
Higher
n
point functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.3
Momentum space and IR divergences
. . . . . . . . . . . . . . . . . . . . . .
20
5
Disappearance process:
φ
+
φ
→ U
22
6
Missing charge process:
φ
+
φ
→
φ
+
φ
+
U
24
6.1
Inclusive treatment: general
. . . . . . . . . . . . . . . . . . . . . . . . . . .
24
6.2
Results for small
a
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
6.3
Inclusive treatment: series expansion
. . . . . . . . . . . . . . . . . . . . . .
28
6.4
Interpretation in terms of exclusive processes
. . . . . . . . . . . . . . . . . .
30
6.5
Exclusive treatment: amputated 3point functions
. . . . . . . . . . . . . . .
33
7
Missing energy process:
φ
+
φ
→
φ
+
φ
+
U
35
7.1
Inclusive treatment: general
. . . . . . . . . . . . . . . . . . . . . . . . . . .
35
7.2
Inclusive treatment: series expansion
. . . . . . . . . . . . . . . . . . . . . .
37
7.3
Interpretation in terms of exclusive processes
. . . . . . . . . . . . . . . . . .
41
7.4
Exclusive treatment: amputated 3point functions
. . . . . . . . . . . . . . .
44
8
Comments about the massive bosons
46
8.1
Processes with a single massive boson
. . . . . . . . . . . . . . . . . . . . . .
46
8.2
Processes with multiple massive bosons
. . . . . . . . . . . . . . . . . . . . .
50
8.3
Massive boson decay
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
9
Conclusions
53
A Operators in the Sommerfield model
53
B
Basic properties of lightcone coordinates
56
C
Numerical treatment of IR divergences
56
D An alternative method for amputating the
j
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